LMFDB - Embedded newform 21.8.a.c.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [21,8,Mod(1,21)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(21, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 8, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("21.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 21 = 3 \cdot 7 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	21.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$6.56008553517$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{67}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 67 $ x^2 - 67
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$8.18535$ of defining polynomial
Character	$\chi$	$=$	21.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+22.3707 q^{2} -27.0000 q^{3} +372.448 q^{4} +118.966 q^{5} -604.009 q^{6} -343.000 q^{7} +5468.48 q^{8} +729.000 q^{9} +O(q^{10})$	\(q+22.3707 q^{2} -27.0000 q^{3} +372.448 q^{4} +118.966 q^{5} -604.009 q^{6} -343.000 q^{7} +5468.48 q^{8} +729.000 q^{9} +2661.35 q^{10} -3521.80 q^{11} -10056.1 q^{12} -5256.76 q^{13} -7673.15 q^{14} -3212.07 q^{15} +74660.5 q^{16} -37023.1 q^{17} +16308.2 q^{18} +5165.26 q^{19} +44308.6 q^{20} +9261.00 q^{21} -78785.1 q^{22} +34550.9 q^{23} -147649. q^{24} -63972.2 q^{25} -117598. q^{26} -19683.0 q^{27} -127750. q^{28} +6382.04 q^{29} -71856.3 q^{30} +155842. q^{31} +970241. q^{32} +95088.5 q^{33} -828233. q^{34} -40805.2 q^{35} +271515. q^{36} +18789.3 q^{37} +115550. q^{38} +141933. q^{39} +650562. q^{40} +261595. q^{41} +207175. q^{42} -134629. q^{43} -1.31169e6 q^{44} +86726.0 q^{45} +772928. q^{46} +1.08722e6 q^{47} -2.01583e6 q^{48} +117649. q^{49} -1.43110e6 q^{50} +999624. q^{51} -1.95787e6 q^{52} +404711. q^{53} -440323. q^{54} -418973. q^{55} -1.87569e6 q^{56} -139462. q^{57} +142771. q^{58} +2.34681e6 q^{59} -1.19633e6 q^{60} -306843. q^{61} +3.48630e6 q^{62} -250047. q^{63} +1.21484e7 q^{64} -625374. q^{65} +2.12720e6 q^{66} -2.72369e6 q^{67} -1.37892e7 q^{68} -932874. q^{69} -912841. q^{70} -1.21376e6 q^{71} +3.98653e6 q^{72} -4.24836e6 q^{73} +420331. q^{74} +1.72725e6 q^{75} +1.92379e6 q^{76} +1.20798e6 q^{77} +3.17513e6 q^{78} -4.67709e6 q^{79} +8.88203e6 q^{80} +531441. q^{81} +5.85206e6 q^{82} +1.16593e6 q^{83} +3.44925e6 q^{84} -4.40448e6 q^{85} -3.01174e6 q^{86} -172315. q^{87} -1.92589e7 q^{88} +2.59661e6 q^{89} +1.94012e6 q^{90} +1.80307e6 q^{91} +1.28684e7 q^{92} -4.20774e6 q^{93} +2.43219e7 q^{94} +614488. q^{95} -2.61965e7 q^{96} -1.06794e7 q^{97} +2.63189e6 q^{98} -2.56739e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 12 q^{2} - 54 q^{3} + 352 q^{4} - 24 q^{5} - 324 q^{6} - 686 q^{7} + 7008 q^{8} + 1458 q^{9}+O(q^{10}) $ 2 * q + 12 * q^2 - 54 * q^3 + 352 * q^4 - 24 * q^5 - 324 * q^6 - 686 * q^7 + 7008 * q^8 + 1458 * q^9	\( 2 q + 12 q^{2} - 54 q^{3} + 352 q^{4} - 24 q^{5} - 324 q^{6} - 686 q^{7} + 7008 q^{8} + 1458 q^{9} + 4144 q^{10} + 2124 q^{11} - 9504 q^{12} - 1084 q^{13} - 4116 q^{14} + 648 q^{15} + 61312 q^{16} - 29256 q^{17} + 8748 q^{18} - 25816 q^{19} + 47232 q^{20} + 18522 q^{21} - 137336 q^{22} + 68316 q^{23} - 189216 q^{24} - 121658 q^{25} - 160872 q^{26} - 39366 q^{27} - 120736 q^{28} + 211308 q^{29} - 111888 q^{30} + 435840 q^{31} + 911616 q^{32} - 57348 q^{33} - 908784 q^{34} + 8232 q^{35} + 256608 q^{36} - 28428 q^{37} + 436848 q^{38} + 29268 q^{39} + 430464 q^{40} + 749760 q^{41} + 111132 q^{42} + 397096 q^{43} - 1427136 q^{44} - 17496 q^{45} + 422760 q^{46} + 840168 q^{47} - 1655424 q^{48} + 235298 q^{49} - 832860 q^{50} + 789912 q^{51} - 2043200 q^{52} - 246684 q^{53} - 236196 q^{54} - 1226128 q^{55} - 2403744 q^{56} + 697032 q^{57} - 1982456 q^{58} + 2199504 q^{59} - 1275264 q^{60} - 1951108 q^{61} + 582528 q^{62} - 500094 q^{63} + 14465024 q^{64} - 1221936 q^{65} + 3708072 q^{66} + 1532048 q^{67} - 13948032 q^{68} - 1844532 q^{69} - 1421392 q^{70} + 2024004 q^{71} + 5108832 q^{72} - 1709028 q^{73} + 910008 q^{74} + 3284766 q^{75} + 2557312 q^{76} - 728532 q^{77} + 4343544 q^{78} + 1048168 q^{79} + 10790400 q^{80} + 1062882 q^{81} + 789440 q^{82} - 4894296 q^{83} + 3259872 q^{84} - 5514912 q^{85} - 8526096 q^{86} - 5705316 q^{87} - 10567104 q^{88} - 60864 q^{89} + 3020976 q^{90} + 371812 q^{91} + 12177984 q^{92} - 11767680 q^{93} + 26884080 q^{94} + 5043744 q^{95} - 24613632 q^{96} - 26046852 q^{97} + 1411788 q^{98} + 1548396 q^{99}+O(q^{100}) \) 2 * q + 12 * q^2 - 54 * q^3 + 352 * q^4 - 24 * q^5 - 324 * q^6 - 686 * q^7 + 7008 * q^8 + 1458 * q^9 + 4144 * q^10 + 2124 * q^11 - 9504 * q^12 - 1084 * q^13 - 4116 * q^14 + 648 * q^15 + 61312 * q^16 - 29256 * q^17 + 8748 * q^18 - 25816 * q^19 + 47232 * q^20 + 18522 * q^21 - 137336 * q^22 + 68316 * q^23 - 189216 * q^24 - 121658 * q^25 - 160872 * q^26 - 39366 * q^27 - 120736 * q^28 + 211308 * q^29 - 111888 * q^30 + 435840 * q^31 + 911616 * q^32 - 57348 * q^33 - 908784 * q^34 + 8232 * q^35 + 256608 * q^36 - 28428 * q^37 + 436848 * q^38 + 29268 * q^39 + 430464 * q^40 + 749760 * q^41 + 111132 * q^42 + 397096 * q^43 - 1427136 * q^44 - 17496 * q^45 + 422760 * q^46 + 840168 * q^47 - 1655424 * q^48 + 235298 * q^49 - 832860 * q^50 + 789912 * q^51 - 2043200 * q^52 - 246684 * q^53 - 236196 * q^54 - 1226128 * q^55 - 2403744 * q^56 + 697032 * q^57 - 1982456 * q^58 + 2199504 * q^59 - 1275264 * q^60 - 1951108 * q^61 + 582528 * q^62 - 500094 * q^63 + 14465024 * q^64 - 1221936 * q^65 + 3708072 * q^66 + 1532048 * q^67 - 13948032 * q^68 - 1844532 * q^69 - 1421392 * q^70 + 2024004 * q^71 + 5108832 * q^72 - 1709028 * q^73 + 910008 * q^74 + 3284766 * q^75 + 2557312 * q^76 - 728532 * q^77 + 4343544 * q^78 + 1048168 * q^79 + 10790400 * q^80 + 1062882 * q^81 + 789440 * q^82 - 4894296 * q^83 + 3259872 * q^84 - 5514912 * q^85 - 8526096 * q^86 - 5705316 * q^87 - 10567104 * q^88 - 60864 * q^89 + 3020976 * q^90 + 371812 * q^91 + 12177984 * q^92 - 11767680 * q^93 + 26884080 * q^94 + 5043744 * q^95 - 24613632 * q^96 - 26046852 * q^97 + 1411788 * q^98 + 1548396 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	22.3707	1.97731	0.988655	−	0.150205i	$-0.0479934\pi$
$2$	22.3707	1.97731	0.988655	+	0.150205i	$0.0479934\pi$
$3$	−27.0000	−0.577350
$3$	−27.0000	−0.577350
$4$	372.448	2.90975
$5$	118.966	0.425624	0.212812	−	0.977093i	$-0.431738\pi$
$5$	118.966	0.425624	0.212812	+	0.977093i	$0.431738\pi$
$6$	−604.009	−1.14160
$7$	−343.000	−0.377964
$7$	−343.000	−0.377964
$8$	5468.48	3.77617
$9$	729.000	0.333333
$10$	2661.35	0.841591
$11$	−3521.80	−0.797793	−0.398896	−	0.916996i	$-0.630607\pi$
$11$	−3521.80	−0.797793	−0.398896	+	0.916996i	$0.630607\pi$
$12$	−10056.1	−1.67995
$13$	−5256.76	−0.663616	−0.331808	−	0.943347i	$-0.607659\pi$
$13$	−5256.76	−0.663616	−0.331808	+	0.943347i	$0.607659\pi$
$14$	−7673.15	−0.747353
$15$	−3212.07	−0.245734
$16$	74660.5	4.55691
$17$	−37023.1	−1.82769	−0.913844	−	0.406066i	$-0.866900\pi$
$17$	−37023.1	−1.82769	−0.913844	+	0.406066i	$0.866900\pi$
$18$	16308.2	0.659103
$19$	5165.26	0.172764	0.0863822	−	0.996262i	$-0.472469\pi$
$19$	5165.26	0.172764	0.0863822	+	0.996262i	$0.472469\pi$
$20$	44308.6	1.23846
$21$	9261.00	0.218218
$22$	−78785.1	−1.57748
$23$	34550.9	0.592123	0.296061	−	0.955169i	$-0.404327\pi$
$23$	34550.9	0.592123	0.296061	+	0.955169i	$0.404327\pi$
$24$	−147649.	−2.18018
$25$	−63972.2	−0.818844
$26$	−117598.	−1.31217
$27$	−19683.0	−0.192450
$28$	−127750.	−1.09978
$29$	6382.04	0.0485922	0.0242961	−	0.999705i	$-0.492266\pi$
$29$	6382.04	0.0485922	0.0242961	+	0.999705i	$0.492266\pi$
$30$	−71856.3	−0.485893
$31$	155842.	0.939549	0.469774	−	0.882786i	$-0.344335\pi$
$31$	155842.	0.939549	0.469774	+	0.882786i	$0.344335\pi$
$32$	970241.	5.23425
$33$	95088.5	0.460606
$34$	−828233.	−3.61390
$35$	−40805.2	−0.160871
$36$	271515.	0.969918
$37$	18789.3	0.0609825	0.0304913	−	0.999535i	$-0.490293\pi$
$37$	18789.3	0.0609825	0.0304913	+	0.999535i	$0.490293\pi$
$38$	115550.	0.341609
$39$	141933.	0.383139
$40$	650562.	1.60723
$41$	261595.	0.592769	0.296384	−	0.955069i	$-0.404219\pi$
$41$	261595.	0.592769	0.296384	+	0.955069i	$0.404219\pi$
$42$	207175.	0.431484
$43$	−134629.	−0.258225	−0.129112	−	0.991630i	$-0.541213\pi$
$43$	−134629.	−0.258225	−0.129112	+	0.991630i	$0.541213\pi$
$44$	−1.31169e6	−2.32138
$45$	86726.0	0.141875
$46$	772928.	1.17081
$47$	1.08722e6	1.52748	0.763741	−	0.645523i	$-0.223360\pi$
$47$	1.08722e6	1.52748	0.763741	+	0.645523i	$0.223360\pi$
$48$	−2.01583e6	−2.63093
$49$	117649.	0.142857
$50$	−1.43110e6	−1.61911
$51$	999624.	1.05522
$52$	−1.95787e6	−1.93096
$53$	404711.	0.373405	0.186702	−	0.982417i	$-0.440220\pi$
$53$	404711.	0.373405	0.186702	+	0.982417i	$0.440220\pi$
$54$	−440323.	−0.380533
$55$	−418973.	−0.339560
$56$	−1.87569e6	−1.42726
$57$	−139462.	−0.0997456
$58$	142771.	0.0960818
$59$	2.34681e6	1.48763	0.743816	−	0.668385i	$-0.233014\pi$
$59$	2.34681e6	1.48763	0.743816	+	0.668385i	$0.233014\pi$
$60$	−1.19633e6	−0.715027
$61$	−306843.	−0.173086	−0.0865431	−	0.996248i	$-0.527582\pi$
$61$	−306843.	−0.173086	−0.0865431	+	0.996248i	$0.527582\pi$
$62$	3.48630e6	1.85778
$63$	−250047.	−0.125988
$64$	1.21484e7	5.79283
$65$	−625374.	−0.282451
$66$	2.12720e6	0.910760
$67$	−2.72369e6	−1.10636	−0.553179	−	0.833063i	$-0.686585\pi$
$67$	−2.72369e6	−1.10636	−0.553179	+	0.833063i	$0.686585\pi$
$68$	−1.37892e7	−5.31812
$69$	−932874.	−0.341862
$70$	−912841.	−0.318092
$71$	−1.21376e6	−0.402465	−0.201233	−	0.979543i	$-0.564495\pi$
$71$	−1.21376e6	−0.402465	−0.201233	+	0.979543i	$0.564495\pi$
$72$	3.98653e6	1.25872
$73$	−4.24836e6	−1.27818	−0.639089	−	0.769133i	$-0.720688\pi$
$73$	−4.24836e6	−1.27818	−0.639089	+	0.769133i	$0.720688\pi$
$74$	420331.	0.120581
$75$	1.72725e6	0.472760
$76$	1.92379e6	0.502702
$77$	1.20798e6	0.301537
$78$	3.17513e6	0.757584
$79$	−4.67709e6	−1.06729	−0.533643	−	0.845710i	$-0.679177\pi$
$79$	−4.67709e6	−1.06729	−0.533643	+	0.845710i	$0.679177\pi$
$80$	8.88203e6	1.93953
$81$	531441.	0.111111
$82$	5.85206e6	1.17209
$83$	1.16593e6	0.223821	0.111910	−	0.993718i	$-0.464303\pi$
$83$	1.16593e6	0.223821	0.111910	+	0.993718i	$0.464303\pi$
$84$	3.44925e6	0.634960
$85$	−4.40448e6	−0.777908
$86$	−3.01174e6	−0.510590
$87$	−172315.	−0.0280547
$88$	−1.92589e7	−3.01260
$89$	2.59661e6	0.390428	0.195214	−	0.980761i	$-0.437460\pi$
$89$	2.59661e6	0.390428	0.195214	+	0.980761i	$0.437460\pi$
$90$	1.94012e6	0.280530
$91$	1.80307e6	0.250823
$92$	1.28684e7	1.72293
$93$	−4.20774e6	−0.542449
$94$	2.43219e7	3.02031
$95$	614488.	0.0735328
$96$	−2.61965e7	−3.02200
$97$	−1.06794e7	−1.18808	−0.594040	−	0.804435i	$-0.702468\pi$
$97$	−1.06794e7	−1.18808	−0.594040	+	0.804435i	$0.702468\pi$
$98$	2.63189e6	0.282473
$99$	−2.56739e6	−0.265931
$100$	−2.38263e7	−2.38263
$101$	−246324.	−0.0237893	−0.0118946	−	0.999929i	$-0.503786\pi$
$101$	−246324.	−0.0237893	−0.0118946	+	0.999929i	$0.503786\pi$
$102$	2.23623e7	2.08649
$103$	1.10660e7	0.997841	0.498920	−	0.866648i	$-0.333730\pi$
$103$	1.10660e7	0.997841	0.498920	+	0.866648i	$0.333730\pi$
$104$	−2.87465e7	−2.50593
$105$	1.10174e6	0.0928789
$106$	9.05368e6	0.738337
$107$	−2.58805e6	−0.204234	−0.102117	−	0.994772i	$-0.532562\pi$
$107$	−2.58805e6	−0.204234	−0.102117	+	0.994772i	$0.532562\pi$
$108$	−7.33090e6	−0.559982
$109$	1.35463e7	1.00191	0.500955	−	0.865473i	$-0.332982\pi$
$109$	1.35463e7	1.00191	0.500955	+	0.865473i	$0.332982\pi$
$110$	−9.37272e6	−0.671415
$111$	−507312.	−0.0352083
$112$	−2.56085e7	−1.72235
$113$	5.00074e6	0.326032	0.163016	−	0.986623i	$-0.447878\pi$
$113$	5.00074e6	0.326032	0.163016	+	0.986623i	$0.447878\pi$
$114$	−3.11986e6	−0.197228
$115$	4.11037e6	0.252022
$116$	2.37698e6	0.141391
$117$	−3.83218e6	−0.221205
$118$	5.24997e7	2.94151
$119$	1.26989e7	0.690801
$120$	−1.75652e7	−0.927936
$121$	−7.08411e6	−0.363527
$122$	−6.86430e6	−0.342245
$123$	−7.06306e6	−0.342235
$124$	5.80432e7	2.73386
$125$	−1.69047e7	−0.774144
$126$	−5.59373e6	−0.249118
$127$	1.40147e7	0.607116	0.303558	−	0.952813i	$-0.401825\pi$
$127$	1.40147e7	0.607116	0.303558	+	0.952813i	$0.401825\pi$
$128$	1.47578e8	6.21996
$129$	3.63497e6	0.149086
$130$	−1.39901e7	−0.558493
$131$	1.85461e7	0.720780	0.360390	−	0.932802i	$-0.382644\pi$
$131$	1.85461e7	0.720780	0.360390	+	0.932802i	$0.382644\pi$
$132$	3.54156e7	1.34025
$133$	−1.77168e6	−0.0652988
$134$	−6.09308e7	−2.18761
$135$	−2.34160e6	−0.0819115
$136$	−2.02460e8	−6.90167
$137$	1.88670e7	0.626877	0.313438	−	0.949609i	$-0.398519\pi$
$137$	1.88670e7	0.626877	0.313438	+	0.949609i	$0.398519\pi$
$138$	−2.08691e7	−0.675968
$139$	−2.19684e7	−0.693819	−0.346910	−	0.937899i	$-0.612769\pi$
$139$	−2.19684e7	−0.693819	−0.346910	+	0.937899i	$0.612769\pi$
$140$	−1.51978e7	−0.468095
$141$	−2.93550e7	−0.881892
$142$	−2.71526e7	−0.795798
$143$	1.85133e7	0.529428
$144$	5.44275e7	1.51897
$145$	759244.	0.0206820
$146$	−9.50388e7	−2.52735
$147$	−3.17652e6	−0.0824786
$148$	6.99806e6	0.177444
$149$	−5.67614e7	−1.40573	−0.702864	−	0.711325i	$-0.748096\pi$
$149$	−5.67614e7	−1.40573	−0.702864	+	0.711325i	$0.748096\pi$
$150$	3.86398e7	0.934792
$151$	5.28273e7	1.24864	0.624322	−	0.781167i	$-0.285375\pi$
$151$	5.28273e7	1.24864	0.624322	+	0.781167i	$0.285375\pi$
$152$	2.82461e7	0.652389
$153$	−2.69899e7	−0.609229
$154$	2.70233e7	0.596233
$155$	1.85399e7	0.399895
$156$	5.28626e7	1.11484
$157$	7.51444e7	1.54970	0.774850	−	0.632145i	$-0.217825\pi$
$157$	7.51444e7	1.54970	0.774850	+	0.632145i	$0.217825\pi$
$158$	−1.04630e8	−2.11035
$159$	−1.09272e7	−0.215585
$160$	1.15425e8	2.22783
$161$	−1.18510e7	−0.223801
$162$	1.18887e7	0.219701
$163$	−6.33322e7	−1.14543	−0.572714	−	0.819755i	$-0.694110\pi$
$163$	−6.33322e7	−1.14543	−0.572714	+	0.819755i	$0.694110\pi$
$164$	9.74306e7	1.72481
$165$	1.13123e7	0.196045
$166$	2.60827e7	0.442563
$167$	8.08576e6	0.134342	0.0671712	−	0.997741i	$-0.478603\pi$
$167$	8.08576e6	0.134342	0.0671712	+	0.997741i	$0.478603\pi$
$168$	5.06436e7	0.824029
$169$	−3.51150e7	−0.559614
$170$	−9.85313e7	−1.53817
$171$	3.76547e6	0.0575882
$172$	−5.01422e7	−0.751370
$173$	4.62544e7	0.679190	0.339595	−	0.940572i	$-0.389710\pi$
$173$	4.62544e7	0.679190	0.339595	+	0.940572i	$0.389710\pi$
$174$	−3.85481e6	−0.0554729
$175$	2.19425e7	0.309494
$176$	−2.62939e8	−3.63547
$177$	−6.33638e7	−0.858884
$178$	5.80879e7	0.771998
$179$	1.24705e8	1.62517	0.812586	−	0.582841i	$-0.198059\pi$
$179$	1.24705e8	1.62517	0.812586	+	0.582841i	$0.198059\pi$
$180$	3.23009e7	0.412821
$181$	−7.77417e7	−0.974494	−0.487247	−	0.873264i	$-0.661999\pi$
$181$	−7.77417e7	−0.974494	−0.487247	+	0.873264i	$0.661999\pi$
$182$	4.03359e7	0.495955
$183$	8.28477e6	0.0999314
$184$	1.88941e8	2.23596
$185$	2.23529e6	0.0259557
$186$	−9.41302e7	−1.07259
$187$	1.30388e8	1.45812
$188$	4.04935e8	4.44460
$189$	6.75127e6	0.0727393
$190$	1.37465e7	0.145397
$191$	1.88867e7	0.196128	0.0980639	−	0.995180i	$-0.468735\pi$
$191$	1.88867e7	0.196128	0.0980639	+	0.995180i	$0.468735\pi$
$192$	−3.28008e8	−3.34449
$193$	−1.49723e8	−1.49913	−0.749564	−	0.661932i	$-0.769737\pi$
$193$	−1.49723e8	−1.49913	−0.749564	+	0.661932i	$0.769737\pi$
$194$	−2.38906e8	−2.34920
$195$	1.68851e7	0.163073
$196$	4.38182e7	0.415679
$197$	−6.84902e7	−0.638259	−0.319129	−	0.947711i	$-0.603390\pi$
$197$	−6.84902e7	−0.638259	−0.319129	+	0.947711i	$0.603390\pi$
$198$	−5.74343e7	−0.525828
$199$	1.04209e8	0.937390	0.468695	−	0.883360i	$-0.344724\pi$
$199$	1.04209e8	0.937390	0.468695	+	0.883360i	$0.344724\pi$
$200$	−3.49831e8	−3.09210
$201$	7.35395e7	0.638756
$202$	−5.51043e6	−0.0470387
$203$	−2.18904e6	−0.0183661
$204$	3.72309e8	3.07042
$205$	3.11208e7	0.252297
$206$	2.47555e8	1.97304
$207$	2.51876e7	0.197374
$208$	−3.92472e8	−3.02404
$209$	−1.81910e7	−0.137830
$210$	2.46467e7	0.183650
$211$	2.55635e7	0.187341	0.0936703	−	0.995603i	$-0.470140\pi$
$211$	2.55635e7	0.187341	0.0936703	+	0.995603i	$0.470140\pi$
$212$	1.50734e8	1.08652
$213$	3.27715e7	0.232363
$214$	−5.78964e7	−0.403835
$215$	−1.60162e7	−0.109907
$216$	−1.07636e8	−0.726725
$217$	−5.34539e7	−0.355116
$218$	3.03041e8	1.98109
$219$	1.14706e8	0.737956
$220$	−1.56046e8	−0.988036
$221$	1.94622e8	1.21288
$222$	−1.13489e7	−0.0696177
$223$	−2.25176e8	−1.35974	−0.679869	−	0.733333i	$-0.737964\pi$
$223$	−2.25176e8	−1.35974	−0.679869	+	0.733333i	$0.737964\pi$
$224$	−3.32793e8	−1.97836
$225$	−4.66357e7	−0.272948
$226$	1.11870e8	0.644666
$227$	−2.20854e8	−1.25318	−0.626591	−	0.779348i	$-0.715550\pi$
$227$	−2.20854e8	−1.25318	−0.626591	+	0.779348i	$0.715550\pi$
$228$	−5.19424e7	−0.290235
$229$	−2.35878e8	−1.29796	−0.648982	−	0.760804i	$-0.724805\pi$
$229$	−2.35878e8	−1.29796	−0.648982	+	0.760804i	$0.724805\pi$
$230$	9.19519e7	0.498326
$231$	−3.26154e7	−0.174093
$232$	3.49001e7	0.183493
$233$	−3.34403e8	−1.73191	−0.865953	−	0.500125i	$-0.833287\pi$
$233$	−3.34403e8	−1.73191	−0.865953	+	0.500125i	$0.833287\pi$
$234$	−8.57286e7	−0.437391
$235$	1.29342e8	0.650134
$236$	8.74065e8	4.32864
$237$	1.26281e8	0.616197
$238$	2.84084e8	1.36593
$239$	−1.48232e8	−0.702345	−0.351172	−	0.936311i	$-0.614217\pi$
$239$	−1.48232e8	−0.702345	−0.351172	+	0.936311i	$0.614217\pi$
$240$	−2.39815e8	−1.11979
$241$	−4.10445e8	−1.88884	−0.944421	−	0.328738i	$-0.893377\pi$
$241$	−4.10445e8	−1.88884	−0.944421	+	0.328738i	$0.893377\pi$
$242$	−1.58477e8	−0.718805
$243$	−1.43489e7	−0.0641500
$244$	−1.14283e8	−0.503638
$245$	1.39962e7	0.0608035
$246$	−1.58006e8	−0.676705
$247$	−2.71525e7	−0.114649
$248$	8.52221e8	3.54790
$249$	−3.14802e7	−0.129223
$250$	−3.78170e8	−1.53072
$251$	8.73950e7	0.348842	0.174421	−	0.984671i	$-0.444195\pi$
$251$	8.73950e7	0.348842	0.174421	+	0.984671i	$0.444195\pi$
$252$	−9.31296e7	−0.366595
$253$	−1.21681e8	−0.472391
$254$	3.13519e8	1.20046
$255$	1.18921e8	0.449126
$256$	1.74643e9	6.50596
$257$	9.99571e7	0.367323	0.183661	−	0.982990i	$-0.441205\pi$
$257$	9.99571e7	0.367323	0.183661	+	0.982990i	$0.441205\pi$
$258$	8.13169e7	0.294789
$259$	−6.44474e6	−0.0230492
$260$	−2.32920e8	−0.821863
$261$	4.65251e6	0.0161974
$262$	4.14889e8	1.42521
$263$	−3.69452e7	−0.125231	−0.0626156	−	0.998038i	$-0.519944\pi$
$263$	−3.69452e7	−0.125231	−0.0626156	+	0.998038i	$0.519944\pi$
$264$	5.19990e8	1.73933
$265$	4.81468e7	0.158930
$266$	−3.96338e7	−0.129116
$267$	−7.01084e7	−0.225414
$268$	−1.01443e9	−3.21923
$269$	−1.07622e8	−0.337106	−0.168553	−	0.985693i	$-0.553909\pi$
$269$	−1.07622e8	−0.337106	−0.168553	+	0.985693i	$0.553909\pi$
$270$	−5.23833e7	−0.161964
$271$	3.47931e8	1.06194	0.530971	−	0.847390i	$-0.321827\pi$
$271$	3.47931e8	1.06194	0.530971	+	0.847390i	$0.321827\pi$
$272$	−2.76416e9	−8.32861
$273$	−4.86829e7	−0.144813
$274$	4.22069e8	1.23953
$275$	2.25297e8	0.653268
$276$	−3.47448e8	−0.994735
$277$	4.33727e8	1.22613	0.613066	−	0.790031i	$-0.289936\pi$
$277$	4.33727e8	1.22613	0.613066	+	0.790031i	$0.289936\pi$
$278$	−4.91448e8	−1.37190
$279$	1.13609e8	0.313183
$280$	−2.23143e8	−0.607477
$281$	1.95616e8	0.525935	0.262967	−	0.964805i	$-0.415299\pi$
$281$	1.95616e8	0.525935	0.262967	+	0.964805i	$0.415299\pi$
$282$	−6.56693e8	−1.74377
$283$	−2.02622e8	−0.531416	−0.265708	−	0.964054i	$-0.585606\pi$
$283$	−2.02622e8	−0.531416	−0.265708	+	0.964054i	$0.585606\pi$
$284$	−4.52063e8	−1.17107
$285$	−1.65912e7	−0.0424542
$286$	4.14155e8	1.04684
$287$	−8.97270e7	−0.224046
$288$	7.07306e8	1.74475
$289$	9.60373e8	2.34044
$290$	1.69848e7	0.0408948
$291$	2.88344e8	0.685939
$292$	−1.58229e9	−3.71918
$293$	2.99557e8	0.695734	0.347867	−	0.937544i	$-0.386906\pi$
$293$	2.99557e8	0.695734	0.347867	+	0.937544i	$0.386906\pi$
$294$	−7.10611e7	−0.163086
$295$	2.79189e8	0.633172
$296$	1.02749e8	0.230281
$297$	6.93195e7	0.153535
$298$	−1.26979e9	−2.77956
$299$	−1.81626e8	−0.392942
$300$	6.43311e8	1.37561
$301$	4.61776e7	0.0975998
$302$	1.18178e9	2.46896
$303$	6.65073e6	0.0137347
$304$	3.85641e8	0.787273
$305$	−3.65038e7	−0.0736697
$306$	−6.03782e8	−1.20463
$307$	−3.74127e8	−0.737964	−0.368982	−	0.929437i	$-0.620294\pi$
$307$	−3.74127e8	−0.737964	−0.368982	+	0.929437i	$0.620294\pi$
$308$	4.49909e8	0.877399
$309$	−2.98783e8	−0.576104
$310$	4.14750e8	0.790716
$311$	−9.65713e8	−1.82048	−0.910241	−	0.414078i	$-0.864104\pi$
$311$	−9.65713e8	−1.82048	−0.910241	+	0.414078i	$0.864104\pi$
$312$	7.76156e8	1.44680
$313$	4.56220e8	0.840948	0.420474	−	0.907305i	$-0.361864\pi$
$313$	4.56220e8	0.840948	0.420474	+	0.907305i	$0.361864\pi$
$314$	1.68103e9	3.06424
$315$	−2.97470e7	−0.0536236
$316$	−1.74197e9	−3.10554
$317$	9.27114e8	1.63465	0.817327	−	0.576174i	$-0.195455\pi$
$317$	9.27114e8	1.63465	0.817327	+	0.576174i	$0.195455\pi$
$318$	−2.44449e8	−0.426279
$319$	−2.24763e7	−0.0387665
$320$	1.44525e9	2.46557
$321$	6.98773e7	0.117915
$322$	−2.65114e8	−0.442525
$323$	−1.91234e8	−0.315759
$324$	1.97934e8	0.323306
$325$	3.36287e8	0.543398
$326$	−1.41679e9	−2.26487
$327$	−3.65751e8	−0.578453
$328$	1.43053e9	2.23840
$329$	−3.72917e8	−0.577334
$330$	2.53063e8	0.387642
$331$	9.78966e8	1.48378	0.741890	−	0.670522i	$-0.233930\pi$
$331$	9.78966e8	1.48378	0.741890	+	0.670522i	$0.233930\pi$
$332$	4.34250e8	0.651263
$333$	1.36974e7	0.0203275
$334$	1.80884e8	0.265637
$335$	−3.24025e8	−0.470893
$336$	6.91430e8	0.994400
$337$	4.88207e7	0.0694863	0.0347431	−	0.999396i	$-0.488939\pi$
$337$	4.88207e7	0.0694863	0.0347431	+	0.999396i	$0.488939\pi$
$338$	−7.85546e8	−1.10653
$339$	−1.35020e8	−0.188235
$340$	−1.64044e9	−2.26352
$341$	−5.48845e8	−0.749565
$342$	8.42363e7	0.113870
$343$	−4.03536e7	−0.0539949
$344$	−7.36214e8	−0.975101
$345$	−1.10980e8	−0.145505
$346$	1.03474e9	1.34297
$347$	−1.24915e8	−0.160495	−0.0802474	−	0.996775i	$-0.525571\pi$
$347$	−1.24915e8	−0.160495	−0.0802474	+	0.996775i	$0.525571\pi$
$348$	−6.41785e7	−0.0816323
$349$	−7.80485e8	−0.982823	−0.491411	−	0.870928i	$-0.663519\pi$
$349$	−7.80485e8	−0.982823	−0.491411	+	0.870928i	$0.663519\pi$
$350$	4.90868e8	0.611965
$351$	1.03469e8	0.127713
$352$	−3.41699e9	−4.17585
$353$	6.57963e8	0.796141	0.398070	−	0.917355i	$-0.369680\pi$
$353$	6.57963e8	0.796141	0.398070	+	0.917355i	$0.369680\pi$
$354$	−1.41749e9	−1.69828
$355$	−1.44396e8	−0.171299
$356$	9.67103e8	1.13605
$357$	−3.42871e8	−0.398834
$358$	2.78975e9	3.21347
$359$	−1.17200e9	−1.33689	−0.668446	−	0.743761i	$-0.733040\pi$
$359$	−1.17200e9	−1.33689	−0.668446	+	0.743761i	$0.733040\pi$
$360$	4.74260e8	0.535744
$361$	−8.67192e8	−0.970152
$362$	−1.73914e9	−1.92688
$363$	1.91271e8	0.209882
$364$	6.71551e8	0.729834
$365$	−5.05409e8	−0.544023
$366$	1.85336e8	0.197595
$367$	−1.53053e9	−1.61625	−0.808127	−	0.589008i	$-0.799519\pi$
$367$	−1.53053e9	−1.61625	−0.808127	+	0.589008i	$0.799519\pi$
$368$	2.57959e9	2.69825
$369$	1.90703e8	0.197590
$370$	5.00049e7	0.0513224
$371$	−1.38816e8	−0.141134
$372$	−1.56717e9	−1.57839
$373$	1.08425e9	1.08181	0.540903	−	0.841085i	$-0.318083\pi$
$373$	1.08425e9	1.08181	0.540903	+	0.841085i	$0.318083\pi$
$374$	2.91687e9	2.88315
$375$	4.56426e8	0.446952
$376$	5.94546e9	5.76804
$377$	−3.35489e7	−0.0322466
$378$	1.51031e8	0.143828
$379$	1.59392e9	1.50394	0.751968	−	0.659199i	$-0.229105\pi$
$379$	1.59392e9	1.50394	0.751968	+	0.659199i	$0.229105\pi$
$380$	2.28865e8	0.213962
$381$	−3.78398e8	−0.350518
$382$	4.22509e8	0.387805
$383$	−7.14242e8	−0.649605	−0.324803	−	0.945782i	$-0.605298\pi$
$383$	−7.14242e8	−0.649605	−0.324803	+	0.945782i	$0.605298\pi$
$384$	−3.98461e9	−3.59110
$385$	1.43708e8	0.128342
$386$	−3.34941e9	−2.96424
$387$	−9.81442e7	−0.0860749
$388$	−3.97753e9	−3.45702
$389$	1.27967e9	1.10223	0.551116	−	0.834429i	$-0.314202\pi$
$389$	1.27967e9	1.10223	0.551116	+	0.834429i	$0.314202\pi$
$390$	3.77732e8	0.322446
$391$	−1.27918e9	−1.08222
$392$	6.43362e8	0.539453
$393$	−5.00744e8	−0.416143
$394$	−1.53217e9	−1.26203
$395$	−5.56413e8	−0.454263
$396$	−9.56221e8	−0.773793
$397$	3.71121e7	0.0297679	0.0148840	−	0.999889i	$-0.495262\pi$
$397$	3.71121e7	0.0297679	0.0148840	+	0.999889i	$0.495262\pi$
$398$	2.33123e9	1.85351
$399$	4.78355e7	0.0377003
$400$	−4.77619e9	−3.73140
$401$	1.16802e9	0.904574	0.452287	−	0.891872i	$-0.350608\pi$
$401$	1.16802e9	0.904574	0.452287	+	0.891872i	$0.350608\pi$
$402$	1.64513e9	1.26302
$403$	−8.19226e8	−0.623499
$404$	−9.17428e7	−0.0692209
$405$	6.32232e7	0.0472916
$406$	−4.89704e7	−0.0363155
$407$	−6.61723e7	−0.0486514
$408$	5.46643e9	3.98468
$409$	−1.71549e8	−0.123982	−0.0619909	−	0.998077i	$-0.519745\pi$
$409$	−1.71549e8	−0.123982	−0.0619909	+	0.998077i	$0.519745\pi$
$410$	6.96194e8	0.498869
$411$	−5.09410e8	−0.361927
$412$	4.12152e9	2.90347
$413$	−8.04955e8	−0.562272
$414$	5.63464e8	0.390270
$415$	1.38706e8	0.0952635
$416$	−5.10033e9	−3.47353
$417$	5.93146e8	0.400577
$418$	−4.06945e8	−0.272533
$419$	1.61069e9	1.06970	0.534852	−	0.844946i	$-0.320367\pi$
$419$	1.61069e9	1.06970	0.534852	+	0.844946i	$0.320367\pi$
$420$	4.10342e8	0.270255
$421$	2.37944e8	0.155413	0.0777064	−	0.996976i	$-0.475240\pi$
$421$	2.37944e8	0.155413	0.0777064	+	0.996976i	$0.475240\pi$
$422$	5.71874e8	0.370430
$423$	7.92586e8	0.509161
$424$	2.21316e9	1.41004
$425$	2.36845e9	1.49659
$426$	7.33121e8	0.459454
$427$	1.05247e8	0.0654204
$428$	−9.63914e8	−0.594272
$429$	−4.99858e8	−0.305665
$430$	−3.58293e8	−0.217320
$431$	7.32984e8	0.440985	0.220493	−	0.975389i	$-0.429233\pi$
$431$	7.32984e8	0.440985	0.220493	+	0.975389i	$0.429233\pi$
$432$	−1.46954e9	−0.876978
$433$	−3.01359e9	−1.78392	−0.891962	−	0.452111i	$-0.850671\pi$
$433$	−3.01359e9	−1.78392	−0.891962	+	0.452111i	$0.850671\pi$
$434$	−1.19580e9	−0.702175
$435$	−2.04996e7	−0.0119408
$436$	5.04531e9	2.91531
$437$	1.78464e8	0.102298
$438$	2.56605e9	1.45917
$439$	9.23499e8	0.520967	0.260484	−	0.965478i	$-0.416118\pi$
$439$	9.23499e8	0.520967	0.260484	+	0.965478i	$0.416118\pi$
$440$	−2.29115e9	−1.28224
$441$	8.57661e7	0.0476190
$442$	4.35383e9	2.39824
$443$	−1.24106e9	−0.678236	−0.339118	−	0.940744i	$-0.610129\pi$
$443$	−1.24106e9	−0.678236	−0.339118	+	0.940744i	$0.610129\pi$
$444$	−1.88948e8	−0.102447
$445$	3.08907e8	0.166176
$446$	−5.03735e9	−2.68862
$447$	1.53256e9	0.811597
$448$	−4.16691e9	−2.18948
$449$	2.34362e9	1.22187	0.610936	−	0.791680i	$-0.290793\pi$
$449$	2.34362e9	1.22187	0.610936	+	0.791680i	$0.290793\pi$
$450$	−1.04327e9	−0.539703
$451$	−9.21284e8	−0.472907
$452$	1.86252e9	0.948672
$453$	−1.42634e9	−0.720905
$454$	−4.94065e9	−2.47793
$455$	2.14503e8	0.106756
$456$	−7.62646e8	−0.376657
$457$	−1.41919e9	−0.695560	−0.347780	−	0.937576i	$-0.613064\pi$
$457$	−1.41919e9	−0.695560	−0.347780	+	0.937576i	$0.613064\pi$
$458$	−5.27675e9	−2.56648
$459$	7.28726e8	0.351739
$460$	1.53090e9	0.733322
$461$	3.22509e9	1.53317	0.766583	−	0.642145i	$-0.221955\pi$
$461$	3.22509e9	1.53317	0.766583	+	0.642145i	$0.221955\pi$
$462$	−7.29629e8	−0.344235
$463$	−2.60402e9	−1.21930	−0.609651	−	0.792670i	$-0.708690\pi$
$463$	−2.60402e9	−1.21930	−0.609651	+	0.792670i	$0.708690\pi$
$464$	4.76486e8	0.221430
$465$	−5.00577e8	−0.230879
$466$	−7.48083e9	−3.42452
$467$	6.16965e8	0.280318	0.140159	−	0.990129i	$-0.455239\pi$
$467$	6.16965e8	0.280318	0.140159	+	0.990129i	$0.455239\pi$
$468$	−1.42729e9	−0.643653
$469$	9.34224e8	0.418164
$470$	2.89348e9	1.28552
$471$	−2.02890e9	−0.894720
$472$	1.28335e10	5.61756
$473$	4.74135e8	0.206010
$474$	2.82500e9	1.21841
$475$	−3.30433e8	−0.141467
$476$	4.72970e9	2.01006
$477$	2.95035e8	0.124468
$478$	−3.31606e9	−1.38875
$479$	2.90828e9	1.20910	0.604550	−	0.796567i	$-0.293353\pi$
$479$	2.90828e9	1.20910	0.604550	+	0.796567i	$0.293353\pi$
$480$	−3.11648e9	−1.28624
$481$	−9.87711e7	−0.0404690
$482$	−9.18195e9	−3.73483
$483$	3.19976e8	0.129212
$484$	−2.63847e9	−1.05777
$485$	−1.27048e9	−0.505676
$486$	−3.20995e8	−0.126844
$487$	1.46877e8	0.0576240	0.0288120	−	0.999585i	$-0.490828\pi$
$487$	1.46877e8	0.0576240	0.0288120	+	0.999585i	$0.490828\pi$
$488$	−1.67797e9	−0.653604
$489$	1.70997e9	0.661313
$490$	3.13105e8	0.120227
$491$	−3.73895e9	−1.42549	−0.712745	−	0.701423i	$-0.752548\pi$
$491$	−3.73895e9	−1.42549	−0.712745	+	0.701423i	$0.752548\pi$
$492$	−2.63062e9	−0.995821
$493$	−2.36283e8	−0.0888114
$494$	−6.07422e8	−0.226697
$495$	−3.05431e8	−0.113187
$496$	1.16353e10	4.28144
$497$	4.16319e8	0.152118
$498$	−7.04234e8	−0.255514
$499$	2.15882e9	0.777793	0.388896	−	0.921282i	$-0.372856\pi$
$499$	2.15882e9	0.777793	0.388896	+	0.921282i	$0.372856\pi$
$500$	−6.29612e9	−2.25257
$501$	−2.18316e8	−0.0775626
$502$	1.95509e9	0.689768
$503$	−4.39095e9	−1.53840	−0.769202	−	0.639006i	$-0.779346\pi$
$503$	−4.39095e9	−1.53840	−0.769202	+	0.639006i	$0.779346\pi$
$504$	−1.36738e9	−0.475753
$505$	−2.93040e7	−0.0101253
$506$	−2.72210e9	−0.934064
$507$	9.48104e8	0.323093
$508$	5.21976e9	1.76656
$509$	3.89569e9	1.30940	0.654699	−	0.755889i	$-0.272795\pi$
$509$	3.89569e9	1.30940	0.654699	+	0.755889i	$0.272795\pi$
$510$	2.66035e9	0.888060
$511$	1.45719e9	0.483106
$512$	2.01789e10	6.64434
$513$	−1.01668e8	−0.0332485
$514$	2.23611e9	0.726311
$515$	1.31648e9	0.424705
$516$	1.35384e9	0.433804
$517$	−3.82898e9	−1.21861
$518$	−1.44173e8	−0.0455755
$519$	−1.24887e9	−0.392131
$520$	−3.41985e9	−1.06658
$521$	3.49790e9	1.08362	0.541808	−	0.840503i	$-0.317740\pi$
$521$	3.49790e9	1.08362	0.541808	+	0.840503i	$0.317740\pi$
$522$	1.04080e8	0.0320273
$523$	1.69050e9	0.516725	0.258362	−	0.966048i	$-0.416817\pi$
$523$	1.69050e9	0.516725	0.258362	+	0.966048i	$0.416817\pi$
$524$	6.90746e9	2.09729
$525$	−5.92446e8	−0.178686
$526$	−8.26490e8	−0.247621
$527$	−5.76977e9	−1.71720
$528$	7.09935e9	2.09894
$529$	−2.21106e9	−0.649390
$530$	1.07708e9	0.314254
$531$	1.71082e9	0.495877
$532$	−6.59861e8	−0.190004
$533$	−1.37514e9	−0.393371
$534$	−1.56837e9	−0.445713
$535$	−3.07889e8	−0.0869272
$536$	−1.48944e10	−4.17780
$537$	−3.36704e9	−0.938294
$538$	−2.40757e9	−0.666563
$539$	−4.14336e8	−0.113970
$540$	−8.72126e8	−0.238342
$541$	−1.34175e9	−0.364318	−0.182159	−	0.983269i	$-0.558309\pi$
$541$	−1.34175e9	−0.364318	−0.182159	+	0.983269i	$0.558309\pi$
$542$	7.78347e9	2.09979
$543$	2.09903e9	0.562624
$544$	−3.59214e10	−9.56658
$545$	1.61155e9	0.426438
$546$	−1.08907e9	−0.286340
$547$	6.96823e9	1.82040	0.910200	−	0.414169i	$-0.135928\pi$
$547$	6.96823e9	1.82040	0.910200	+	0.414169i	$0.135928\pi$
$548$	7.02700e9	1.82406
$549$	−2.23689e8	−0.0576954
$550$	5.04005e9	1.29171
$551$	3.29649e7	0.00839501
$552$	−5.10141e9	−1.29093
$553$	1.60424e9	0.403396
$554$	9.70278e9	2.42444
$555$	−6.03527e7	−0.0149855
$556$	−8.18209e9	−2.01884
$557$	−5.01746e9	−1.23024	−0.615121	−	0.788433i	$-0.710893\pi$
$557$	−5.01746e9	−1.23024	−0.615121	+	0.788433i	$0.710893\pi$
$558$	2.54151e9	0.619260
$559$	7.07711e8	0.171362
$560$	−3.04654e9	−0.733075
$561$	−3.52047e9	−0.841843
$562$	4.37606e9	1.03994
$563$	2.11565e9	0.499648	0.249824	−	0.968291i	$-0.419627\pi$
$563$	2.11565e9	0.499648	0.249824	+	0.968291i	$0.419627\pi$
$564$	−1.09332e10	−2.56609
$565$	5.94917e8	0.138767
$566$	−4.53280e9	−1.05077
$567$	−1.82284e8	−0.0419961
$568$	−6.63742e9	−1.51978
$569$	1.64764e9	0.374947	0.187474	−	0.982270i	$-0.439970\pi$
$569$	1.64764e9	0.374947	0.187474	+	0.982270i	$0.439970\pi$
$570$	−3.71157e8	−0.0839451
$571$	5.07137e9	1.13999	0.569993	−	0.821650i	$-0.306946\pi$
$571$	5.07137e9	1.13999	0.569993	+	0.821650i	$0.306946\pi$
$572$	6.89523e9	1.54050
$573$	−5.09941e8	−0.113234
$574$	−2.00726e9	−0.443008
$575$	−2.21030e9	−0.484856
$576$	8.85621e9	1.93094
$577$	4.71321e9	1.02141	0.510707	−	0.859755i	$-0.329384\pi$
$577$	4.71321e9	1.02141	0.510707	+	0.859755i	$0.329384\pi$
$578$	2.14842e10	4.62778
$579$	4.04253e9	0.865522
$580$	2.82779e8	0.0601796
$581$	−3.99915e8	−0.0845962
$582$	6.45046e9	1.35631
$583$	−1.42531e9	−0.297900
$584$	−2.32321e10	−4.82662
$585$	−4.55898e8	−0.0941504
$586$	6.70131e9	1.37568
$587$	6.75924e9	1.37932	0.689659	−	0.724134i	$-0.257760\pi$
$587$	6.75924e9	1.37932	0.689659	+	0.724134i	$0.257760\pi$
$588$	−1.18309e9	−0.239992
$589$	8.04966e8	0.162321
$590$	6.24566e9	1.25198
$591$	1.84924e9	0.368499
$592$	1.40282e9	0.277892
$593$	−5.51671e9	−1.08640	−0.543199	−	0.839604i	$-0.682787\pi$
$593$	−5.51671e9	−1.08640	−0.543199	+	0.839604i	$0.682787\pi$
$594$	1.55073e9	0.303587
$595$	1.51074e9	0.294022
$596$	−2.11407e10	−4.09032
$597$	−2.81365e9	−0.541202
$598$	−4.06310e9	−0.776968
$599$	−4.81408e9	−0.915208	−0.457604	−	0.889156i	$-0.651292\pi$
$599$	−4.81408e9	−0.915208	−0.457604	+	0.889156i	$0.651292\pi$
$600$	9.44543e9	1.78522
$601$	−3.48220e8	−0.0654324	−0.0327162	−	0.999465i	$-0.510416\pi$
$601$	−3.48220e8	−0.0654324	−0.0327162	+	0.999465i	$0.510416\pi$
$602$	1.03303e9	0.192985
$603$	−1.98557e9	−0.368786
$604$	1.96754e10	3.63325
$605$	−8.42766e8	−0.154726
$606$	1.48782e8	0.0271578
$607$	8.43348e8	0.153055	0.0765273	−	0.997067i	$-0.475617\pi$
$607$	8.43348e8	0.153055	0.0765273	+	0.997067i	$0.475617\pi$
$608$	5.01155e9	0.904293
$609$	5.91041e7	0.0106037
$610$	−8.16616e8	−0.145668
$611$	−5.71527e9	−1.01366
$612$	−1.00523e10	−1.77271
$613$	−9.13491e9	−1.60174	−0.800871	−	0.598837i	$-0.795630\pi$
$613$	−9.13491e9	−1.60174	−0.800871	+	0.598837i	$0.795630\pi$
$614$	−8.36949e9	−1.45918
$615$	−8.40261e8	−0.145664
$616$	6.60580e9	1.13866
$617$	−4.80233e9	−0.823103	−0.411552	−	0.911386i	$-0.635013\pi$
$617$	−4.80233e9	−0.823103	−0.411552	+	0.911386i	$0.635013\pi$
$618$	−6.68398e9	−1.13914
$619$	7.91703e9	1.34167	0.670834	−	0.741608i	$-0.265936\pi$
$619$	7.91703e9	1.34167	0.670834	+	0.741608i	$0.265936\pi$
$620$	6.90515e9	1.16360
$621$	−6.80065e8	−0.113954
$622$	−2.16037e10	−3.59966
$623$	−8.90636e8	−0.147568
$624$	1.05968e10	1.74593
$625$	2.98675e9	0.489349
$626$	1.02060e10	1.66281
$627$	4.91157e8	0.0795763
$628$	2.79874e10	4.50925
$629$	−6.95640e8	−0.111457
$630$	−6.65461e8	−0.106031
$631$	−2.99670e8	−0.0474832	−0.0237416	−	0.999718i	$-0.507558\pi$
$631$	−2.99670e8	−0.0474832	−0.0237416	+	0.999718i	$0.507558\pi$
$632$	−2.55766e10	−4.03026
$633$	−6.90215e8	−0.108161
$634$	2.07402e10	3.23222
$635$	1.66727e9	0.258403
$636$	−4.06982e9	−0.627300
$637$	−6.18453e8	−0.0948022
$638$	−5.02810e8	−0.0766534
$639$	−8.84830e8	−0.134155
$640$	1.75567e10	2.64737
$641$	−5.58996e8	−0.0838312	−0.0419156	−	0.999121i	$-0.513346\pi$
$641$	−5.58996e8	−0.0838312	−0.0419156	+	0.999121i	$0.513346\pi$
$642$	1.56320e9	0.233154
$643$	−6.60776e9	−0.980203	−0.490102	−	0.871665i	$-0.663040\pi$
$643$	−6.60776e9	−0.980203	−0.490102	+	0.871665i	$0.663040\pi$
$644$	−4.41387e9	−0.651207
$645$	4.32437e8	0.0634547
$646$	−4.27804e9	−0.624354
$647$	−6.38752e9	−0.927187	−0.463593	−	0.886048i	$-0.653440\pi$
$647$	−6.38752e9	−0.927187	−0.463593	+	0.886048i	$0.653440\pi$
$648$	2.90618e9	0.419575
$649$	−8.26498e9	−1.18682
$650$	7.52297e9	1.07447
$651$	1.44326e9	0.205026
$652$	−2.35880e10	−3.33291
$653$	−5.31297e9	−0.746691	−0.373346	−	0.927692i	$-0.621789\pi$
$653$	−5.31297e9	−0.746691	−0.373346	+	0.927692i	$0.621789\pi$
$654$	−8.18211e9	−1.14378
$655$	2.20635e9	0.306782
$656$	1.95308e10	2.70120
$657$	−3.09705e9	−0.426059
$658$	−8.34243e9	−1.14157
$659$	1.31533e10	1.79035	0.895173	−	0.445719i	$-0.147052\pi$
$659$	1.31533e10	1.79035	0.895173	+	0.445719i	$0.147052\pi$
$660$	4.21324e9	0.570443
$661$	−7.31153e9	−0.984698	−0.492349	−	0.870398i	$-0.663862\pi$
$661$	−7.31153e9	−0.984698	−0.492349	+	0.870398i	$0.663862\pi$
$662$	2.19002e10	2.93389
$663$	−5.25479e9	−0.700258
$664$	6.37588e9	0.845185
$665$	−2.10770e8	−0.0277928
$666$	3.06421e8	0.0401938
$667$	2.20505e8	0.0287726
$668$	3.01153e9	0.390903
$669$	6.07976e9	0.785046
$670$	−7.24867e9	−0.931101
$671$	1.08064e9	0.138087
$672$	8.98540e9	1.14221
$673$	3.62533e9	0.458453	0.229227	−	0.973373i	$-0.426380\pi$
$673$	3.62533e9	0.458453	0.229227	+	0.973373i	$0.426380\pi$
$674$	1.09215e9	0.137396
$675$	1.25916e9	0.157587
$676$	−1.30785e10	−1.62834
$677$	7.55267e9	0.935492	0.467746	−	0.883863i	$-0.345066\pi$
$677$	7.55267e9	0.935492	0.467746	+	0.883863i	$0.345066\pi$
$678$	−3.02049e9	−0.372198
$679$	3.66304e9	0.449052
$680$	−2.40858e10	−2.93752
$681$	5.96305e9	0.723525
$682$	−1.22780e10	−1.48212
$683$	−1.10499e10	−1.32704	−0.663522	−	0.748157i	$-0.730939\pi$
$683$	−1.10499e10	−1.32704	−0.663522	+	0.748157i	$0.730939\pi$
$684$	1.40244e9	0.167567
$685$	2.24453e9	0.266814
$686$	−9.02739e8	−0.106765
$687$	6.36870e9	0.749380
$688$	−1.00514e10	−1.17671
$689$	−2.12747e9	−0.247797
$690$	−2.48270e9	−0.287708
$691$	1.91692e9	0.221020	0.110510	−	0.993875i	$-0.464752\pi$
$691$	1.91692e9	0.221020	0.110510	+	0.993875i	$0.464752\pi$
$692$	1.72274e10	1.97628
$693$	8.80615e8	0.100512
$694$	−2.79443e9	−0.317348
$695$	−2.61348e9	−0.295306
$696$	−9.42303e8	−0.105940
$697$	−9.68505e9	−1.08340
$698$	−1.74600e10	−1.94335
$699$	9.02888e9	0.999917
$700$	8.17243e9	0.900551
$701$	8.88383e9	0.974064	0.487032	−	0.873384i	$-0.338080\pi$
$701$	8.88383e9	0.974064	0.487032	+	0.873384i	$0.338080\pi$
$702$	2.31467e9	0.252528
$703$	9.70518e7	0.0105356
$704$	−4.27843e10	−4.62147
$705$	−3.49224e9	−0.375355
$706$	1.47191e10	1.57422
$707$	8.44890e7	0.00899150
$708$	−2.35997e10	−2.49914
$709$	1.32525e10	1.39649	0.698244	−	0.715859i	$-0.253965\pi$
$709$	1.32525e10	1.39649	0.698244	+	0.715859i	$0.253965\pi$
$710$	−3.23023e9	−0.338711
$711$	−3.40960e9	−0.355762
$712$	1.41995e10	1.47433
$713$	5.38449e9	0.556328
$714$	−7.67027e9	−0.788618
$715$	2.20244e9	0.225337
$716$	4.64463e10	4.72885
$717$	4.00227e9	0.405499
$718$	−2.62184e10	−2.64345
$719$	−3.38868e9	−0.340000	−0.170000	−	0.985444i	$-0.554377\pi$
$719$	−3.38868e9	−0.340000	−0.170000	+	0.985444i	$0.554377\pi$
$720$	6.47500e9	0.646511
$721$	−3.79565e9	−0.377148
$722$	−1.93997e10	−1.91829
$723$	1.10820e10	1.09052
$724$	−2.89548e10	−2.83554
$725$	−4.08273e8	−0.0397894
$726$	4.27887e9	0.415003
$727$	−1.18748e9	−0.114619	−0.0573096	−	0.998356i	$-0.518252\pi$
$727$	−1.18748e9	−0.114619	−0.0573096	+	0.998356i	$0.518252\pi$
$728$	9.86006e9	0.947152
$729$	3.87420e8	0.0370370
$730$	−1.13063e10	−1.07570
$731$	4.98437e9	0.471954
$732$	3.08565e9	0.290776
$733$	−1.18089e10	−1.10750	−0.553752	−	0.832681i	$-0.686805\pi$
$733$	−1.18089e10	−1.10750	−0.553752	+	0.832681i	$0.686805\pi$
$734$	−3.42390e10	−3.19584
$735$	−3.77897e8	−0.0351049
$736$	3.35227e10	3.09932
$737$	9.59227e9	0.882643
$738$	4.26615e9	0.390696
$739$	7.76067e9	0.707365	0.353683	−	0.935366i	$-0.384929\pi$
$739$	7.76067e9	0.707365	0.353683	+	0.935366i	$0.384929\pi$
$740$	8.32529e8	0.0755246
$741$	7.33119e8	0.0661928
$742$	−3.10541e9	−0.279065
$743$	8.01232e9	0.716634	0.358317	−	0.933600i	$-0.383351\pi$
$743$	8.01232e9	0.716634	0.358317	+	0.933600i	$0.383351\pi$
$744$	−2.30100e10	−2.04838
$745$	−6.75266e9	−0.598312
$746$	2.42555e10	2.13907
$747$	8.49965e8	0.0746069
$748$	4.85628e10	4.24276
$749$	8.87700e8	0.0771934
$750$	1.02106e10	0.883763
$751$	−7.92531e8	−0.0682774	−0.0341387	−	0.999417i	$-0.510869\pi$
$751$	−7.92531e8	−0.0682774	−0.0341387	+	0.999417i	$0.510869\pi$
$752$	8.11726e10	6.96060
$753$	−2.35966e9	−0.201404
$754$	−7.50512e8	−0.0637614
$755$	6.28463e9	0.531453
$756$	2.51450e9	0.211653
$757$	1.47532e10	1.23609	0.618047	−	0.786141i	$-0.287924\pi$
$757$	1.47532e10	1.23609	0.618047	+	0.786141i	$0.287924\pi$
$758$	3.56571e10	2.97375
$759$	3.28539e9	0.272735
$760$	3.36032e9	0.277673
$761$	−3.42569e9	−0.281775	−0.140887	−	0.990026i	$-0.544996\pi$
$761$	−3.42569e9	−0.281775	−0.140887	+	0.990026i	$0.544996\pi$
$762$	−8.46502e9	−0.693084
$763$	−4.64639e9	−0.378687
$764$	7.03432e9	0.570684
$765$	−3.21087e9	−0.259303
$766$	−1.59781e10	−1.28447
$767$	−1.23366e10	−0.987216
$768$	−4.71536e10	−3.75622
$769$	1.89295e10	1.50106	0.750530	−	0.660837i	$-0.229798\pi$
$769$	1.89295e10	1.50106	0.750530	+	0.660837i	$0.229798\pi$
$770$	3.21484e9	0.253771
$771$	−2.69884e9	−0.212074
$772$	−5.57642e10	−4.36209
$773$	−2.17372e10	−1.69268	−0.846340	−	0.532642i	$-0.821199\pi$
$773$	−2.17372e10	−1.69268	−0.846340	+	0.532642i	$0.821199\pi$
$774$	−2.19556e9	−0.170197
$775$	−9.96957e9	−0.769344
$776$	−5.84002e10	−4.48640
$777$	1.74008e8	0.0133075
$778$	2.86270e10	2.17945
$779$	1.35120e9	0.102409
$780$	6.28883e9	0.474503
$781$	4.27461e9	0.321084
$782$	−2.86162e10	−2.13988
$783$	−1.25618e8	−0.00935157
$784$	8.78373e9	0.650988
$785$	8.93960e9	0.659590
$786$	−1.12020e10	−0.822843
$787$	1.69867e10	1.24221	0.621107	−	0.783726i	$-0.286683\pi$
$787$	1.69867e10	1.24221	0.621107	+	0.783726i	$0.286683\pi$
$788$	−2.55091e10	−1.85718
$789$	9.97520e8	0.0723023
$790$	−1.24473e10	−0.898218
$791$	−1.71525e9	−0.123228
$792$	−1.40397e10	−1.00420
$793$	1.61300e9	0.114863
$794$	8.30224e8	0.0588604
$795$	−1.29996e9	−0.0917584
$796$	3.88126e10	2.72757
$797$	1.21624e10	0.850972	0.425486	−	0.904965i	$-0.360103\pi$
$797$	1.21624e10	0.850972	0.425486	+	0.904965i	$0.360103\pi$
$798$	1.07011e9	0.0745452
$799$	−4.02524e10	−2.79176
$800$	−6.20684e10	−4.28604
$801$	1.89293e9	0.130143
$802$	2.61294e10	1.78862
$803$	1.49619e10	1.01972
$804$	2.73897e10	1.85862
$805$	−1.40986e9	−0.0952554
$806$	−1.83267e10	−1.23285
$807$	2.90578e9	0.194628
$808$	−1.34702e9	−0.0898324
$809$	−4.64855e8	−0.0308672	−0.0154336	−	0.999881i	$-0.504913\pi$
$809$	−4.64855e8	−0.0308672	−0.0154336	+	0.999881i	$0.504913\pi$
$810$	1.41435e9	0.0935101
$811$	−8.27001e9	−0.544419	−0.272209	−	0.962238i	$-0.587754\pi$
$811$	−8.27001e9	−0.544419	−0.272209	+	0.962238i	$0.587754\pi$
$812$	−8.15305e8	−0.0534409
$813$	−9.39415e9	−0.613113
$814$	−1.48032e9	−0.0961989
$815$	−7.53436e9	−0.487522
$816$	7.46324e10	4.80853
$817$	−6.95392e8	−0.0446121
$818$	−3.83768e9	−0.245150
$819$	1.31444e9	0.0836077
$820$	1.15909e10	0.734122
$821$	1.90387e10	1.20070	0.600352	−	0.799736i	$-0.295027\pi$
$821$	1.90387e10	1.20070	0.600352	+	0.799736i	$0.295027\pi$
$822$	−1.13959e10	−0.715642
$823$	−2.71902e10	−1.70025	−0.850126	−	0.526579i	$-0.823474\pi$
$823$	−2.71902e10	−1.70025	−0.850126	+	0.526579i	$0.823474\pi$
$824$	6.05144e10	3.76802
$825$	−6.08302e9	−0.377164
$826$	−1.80074e10	−1.11179
$827$	−1.62749e10	−1.00057	−0.500286	−	0.865860i	$-0.666772\pi$
$827$	−1.62749e10	−1.00057	−0.500286	+	0.865860i	$0.666772\pi$
$828$	9.38108e9	0.574311
$829$	−6.34273e9	−0.386665	−0.193333	−	0.981133i	$-0.561930\pi$
$829$	−6.34273e9	−0.386665	−0.193333	+	0.981133i	$0.561930\pi$
$830$	3.10295e9	0.188365
$831$	−1.17106e10	−0.707908
$832$	−6.38615e10	−3.84421
$833$	−4.35573e9	−0.261098
$834$	1.32691e10	0.792064
$835$	9.61928e8	0.0571794
$836$	−6.77521e9	−0.401052
$837$	−3.06744e9	−0.180816
$838$	3.60323e10	2.11513
$839$	−2.76715e10	−1.61758	−0.808791	−	0.588096i	$-0.799878\pi$
$839$	−2.76715e10	−1.61758	−0.808791	+	0.588096i	$0.799878\pi$
$840$	6.02485e9	0.350727
$841$	−1.72091e10	−0.997639
$842$	5.32297e9	0.307299
$843$	−5.28163e9	−0.303648
$844$	9.52109e9	0.545115
$845$	−4.17747e9	−0.238185
$846$	1.77307e10	1.00677
$847$	2.42985e9	0.137400
$848$	3.02159e10	1.70157
$849$	5.47080e9	0.306813
$850$	5.29839e10	2.95922
$851$	6.49189e8	0.0361092
$852$	1.22057e10	0.676120
$853$	2.62262e10	1.44682	0.723409	−	0.690420i	$-0.242574\pi$
$853$	2.62262e10	1.44682	0.723409	+	0.690420i	$0.242574\pi$
$854$	2.35446e9	0.129356
$855$	4.47962e8	0.0245109
$856$	−1.41527e10	−0.771225
$857$	−1.11062e10	−0.602743	−0.301372	−	0.953507i	$-0.597444\pi$
$857$	−1.11062e10	−0.602743	−0.301372	+	0.953507i	$0.597444\pi$
$858$	−1.11822e10	−0.604395
$859$	−1.02974e10	−0.554307	−0.277153	−	0.960826i	$-0.589391\pi$
$859$	−1.02974e10	−0.554307	−0.277153	+	0.960826i	$0.589391\pi$
$860$	−5.96520e9	−0.319802
$861$	2.42263e9	0.129353
$862$	1.63974e10	0.871965
$863$	−1.09948e10	−0.582302	−0.291151	−	0.956677i	$-0.594038\pi$
$863$	−1.09948e10	−0.582302	−0.291151	+	0.956677i	$0.594038\pi$
$864$	−1.90973e10	−1.00733
$865$	5.50268e9	0.289080
$866$	−6.74161e10	−3.52737
$867$	−2.59301e10	−1.35125
$868$	−1.99088e10	−1.03330
$869$	1.64717e10	0.851472
$870$	−4.58590e8	−0.0236106
$871$	1.43178e10	0.734196
$872$	7.40779e10	3.78339
$873$	−7.78528e9	−0.396027
$874$	3.99237e9	0.202274
$875$	5.79831e9	0.292599
$876$	4.27219e10	2.14727
$877$	6.82900e9	0.341868	0.170934	−	0.985282i	$-0.445322\pi$
$877$	6.82900e9	0.341868	0.170934	+	0.985282i	$0.445322\pi$
$878$	2.06593e10	1.03011
$879$	−8.08804e9	−0.401682
$880$	−3.12807e10	−1.54735
$881$	7.79828e9	0.384223	0.192111	−	0.981373i	$-0.438466\pi$
$881$	7.79828e9	0.384223	0.192111	+	0.981373i	$0.438466\pi$
$882$	1.91865e9	0.0941576
$883$	1.04123e10	0.508960	0.254480	−	0.967078i	$-0.418096\pi$
$883$	1.04123e10	0.508960	0.254480	+	0.967078i	$0.418096\pi$
$884$	7.24866e10	3.52919
$885$	−7.53811e9	−0.365562
$886$	−2.77635e10	−1.34108
$887$	1.72418e10	0.829563	0.414781	−	0.909921i	$-0.363858\pi$
$887$	1.72418e10	0.829563	0.414781	+	0.909921i	$0.363858\pi$
$888$	−2.77423e9	−0.132953
$889$	−4.80705e9	−0.229468
$890$	6.91047e9	0.328581
$891$	−1.87163e9	−0.0886436
$892$	−8.38665e10	−3.95651
$893$	5.61579e9	0.263895
$894$	3.42844e10	1.60478
$895$	1.48356e10	0.691713
$896$	−5.06194e10	−2.35092
$897$	4.90390e9	0.226865
$898$	5.24285e10	2.41602
$899$	9.94592e8	0.0456548
$900$	−1.73694e10	−0.794211
$901$	−1.49837e10	−0.682467
$902$	−2.06098e10	−0.935083
$903$	−1.24680e9	−0.0563492
$904$	2.73465e10	1.23115
$905$	−9.24860e9	−0.414769
$906$	−3.19081e10	−1.42545
$907$	3.74992e10	1.66877	0.834385	−	0.551183i	$-0.185823\pi$
$907$	3.74992e10	1.66877	0.834385	+	0.551183i	$0.185823\pi$
$908$	−8.22566e10	−3.64645
$909$	−1.79570e8	−0.00792976
$910$	4.79859e9	0.211091
$911$	3.32750e9	0.145816	0.0729078	−	0.997339i	$-0.476772\pi$
$911$	3.32750e9	0.145816	0.0729078	+	0.997339i	$0.476772\pi$
$912$	−1.04123e10	−0.454532
$913$	−4.10618e9	−0.178562
$914$	−3.17483e10	−1.37534
$915$	9.85603e8	0.0425332
$916$	−8.78523e10	−3.77675
$917$	−6.36131e9	−0.272429
$918$	1.63021e10	0.695496
$919$	−9.77356e9	−0.415383	−0.207691	−	0.978194i	$-0.566595\pi$
$919$	−9.77356e9	−0.415383	−0.207691	+	0.978194i	$0.566595\pi$
$920$	2.24775e10	0.951679
$921$	1.01014e10	0.426064
$922$	7.21476e10	3.03154
$923$	6.38044e9	0.267082
$924$	−1.21475e10	−0.506567
$925$	−1.20200e9	−0.0499352
$926$	−5.82538e10	−2.41094
$927$	8.06713e9	0.332614
$928$	6.19212e9	0.254344
$929$	3.48231e10	1.42499	0.712496	−	0.701676i	$-0.247565\pi$
$929$	3.48231e10	1.42499	0.712496	+	0.701676i	$0.247565\pi$
$930$	−1.11983e10	−0.456520
$931$	6.07688e8	0.0246806
$932$	−1.24548e11	−5.03942
$933$	2.60742e10	1.05106
$934$	1.38019e10	0.554276
$935$	1.55117e10	0.620609
$936$	−2.09562e10	−0.835310
$937$	3.40725e10	1.35306	0.676528	−	0.736417i	$-0.263484\pi$
$937$	3.40725e10	1.35306	0.676528	+	0.736417i	$0.263484\pi$
$938$	2.08993e10	0.826839
$939$	−1.23179e10	−0.485521
$940$	4.81733e10	1.89173
$941$	3.17499e9	0.124216	0.0621082	−	0.998069i	$-0.480218\pi$
$941$	3.17499e9	0.124216	0.0621082	+	0.998069i	$0.480218\pi$
$942$	−4.53879e10	−1.76914
$943$	9.03833e9	0.350992
$944$	1.75214e11	6.77901
$945$	8.03169e8	0.0309596
$946$	1.06067e10	0.407345
$947$	−2.26834e10	−0.867927	−0.433964	−	0.900930i	$-0.642885\pi$
$947$	−2.26834e10	−0.867927	−0.433964	+	0.900930i	$0.642885\pi$
$948$	4.70333e10	1.79298
$949$	2.23326e10	0.848219
$950$	−7.39202e9	−0.279724
$951$	−2.50321e10	−0.943768
$952$	6.94439e10	2.60858
$953$	1.87055e10	0.700075	0.350037	−	0.936736i	$-0.386169\pi$
$953$	1.87055e10	0.700075	0.350037	+	0.936736i	$0.386169\pi$
$954$	6.60013e9	0.246112
$955$	2.24687e9	0.0834768
$956$	−5.52089e10	−2.04365
$957$	6.06859e8	0.0223819
$958$	6.50603e10	2.39076
$959$	−6.47140e9	−0.236937
$960$	−3.90217e10	−1.42350
$961$	−3.22580e9	−0.117248
$962$	−2.20958e9	−0.0800197
$963$	−1.88669e9	−0.0680781
$964$	−1.52870e11	−5.49607
$965$	−1.78119e10	−0.638065
$966$	7.15809e9	0.255492
$967$	−3.29948e10	−1.17342	−0.586709	−	0.809798i	$-0.699577\pi$
$967$	−3.29948e10	−1.17342	−0.586709	+	0.809798i	$0.699577\pi$
$968$	−3.87394e10	−1.37274
$969$	5.16332e9	0.182304
$970$	−2.84216e10	−0.999878
$971$	5.53455e10	1.94006	0.970029	−	0.242989i	$-0.0781279\pi$
$971$	5.53455e10	1.94006	0.970029	+	0.242989i	$0.0781279\pi$
$972$	−5.34423e9	−0.186661
$973$	7.53516e9	0.262239
$974$	3.28575e9	0.113940
$975$	−9.07974e9	−0.313731
$976$	−2.29091e10	−0.788739
$977$	−3.47635e10	−1.19259	−0.596297	−	0.802764i	$-0.703362\pi$
$977$	−3.47635e10	−1.19259	−0.596297	+	0.802764i	$0.703362\pi$
$978$	3.82532e10	1.30762
$979$	−9.14473e9	−0.311481
$980$	5.21286e9	0.176923
$981$	9.87528e9	0.333970
$982$	−8.36429e10	−2.81864
$983$	8.82839e9	0.296445	0.148222	−	0.988954i	$-0.452645\pi$
$983$	8.82839e9	0.296445	0.148222	+	0.988954i	$0.452645\pi$
$984$	−3.86242e10	−1.29234
$985$	−8.14798e9	−0.271658
$986$	−5.28582e9	−0.175608
$987$	1.00688e10	0.333324
$988$	−1.01129e10	−0.333601
$989$	−4.65154e9	−0.152901
$990$	−6.83271e9	−0.223805
$991$	5.32608e10	1.73840	0.869200	−	0.494461i	$-0.164634\pi$
$991$	5.32608e10	1.73840	0.869200	+	0.494461i	$0.164634\pi$
$992$	1.51205e11	4.91784
$993$	−2.64321e10	−0.856661
$994$	9.31336e9	0.300783
$995$	1.23973e10	0.398976
$996$	−1.17247e10	−0.376007
$997$	1.69055e10	0.540249	0.270125	−	0.962825i	$-0.412935\pi$
$997$	1.69055e10	0.540249	0.270125	+	0.962825i	$0.412935\pi$
$998$	4.82943e10	1.53794
$999$	−3.69831e8	−0.0117361

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	21.8.a.c.1.2	✓	2
3.2	odd	2		63.8.a.c.1.1		2
4.3	odd	2		336.8.a.p.1.2		2
5.4	even	2		525.8.a.d.1.1		2
7.2	even	3		147.8.e.f.67.1		4
7.3	odd	6		147.8.e.e.79.1		4
7.4	even	3		147.8.e.f.79.1		4
7.5	odd	6		147.8.e.e.67.1		4
7.6	odd	2		147.8.a.d.1.2		2
21.20	even	2		441.8.a.h.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.8.a.c.1.2	✓	2	1.1	even	1	trivial
63.8.a.c.1.1		2	3.2	odd	2
147.8.a.d.1.2		2	7.6	odd	2
147.8.e.e.67.1		4	7.5	odd	6
147.8.e.e.79.1		4	7.3	odd	6
147.8.e.f.67.1		4	7.2	even	3
147.8.e.f.79.1		4	7.4	even	3
336.8.a.p.1.2		2	4.3	odd	2
441.8.a.h.1.1		2	21.20	even	2
525.8.a.d.1.1		2	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 21 = 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	21.a (trivial)

\(f(q)\)	\(=\)	\(q+22.3707 q^{2} -27.0000 q^{3} +372.448 q^{4} +118.966 q^{5} -604.009 q^{6} -343.000 q^{7} +5468.48 q^{8} +729.000 q^{9} +O(q^{10})\)	\(q+22.3707 q^{2} -27.0000 q^{3} +372.448 q^{4} +118.966 q^{5} -604.009 q^{6} -343.000 q^{7} +5468.48 q^{8} +729.000 q^{9} +2661.35 q^{10} -3521.80 q^{11} -10056.1 q^{12} -5256.76 q^{13} -7673.15 q^{14} -3212.07 q^{15} +74660.5 q^{16} -37023.1 q^{17} +16308.2 q^{18} +5165.26 q^{19} +44308.6 q^{20} +9261.00 q^{21} -78785.1 q^{22} +34550.9 q^{23} -147649. q^{24} -63972.2 q^{25} -117598. q^{26} -19683.0 q^{27} -127750. q^{28} +6382.04 q^{29} -71856.3 q^{30} +155842. q^{31} +970241. q^{32} +95088.5 q^{33} -828233. q^{34} -40805.2 q^{35} +271515. q^{36} +18789.3 q^{37} +115550. q^{38} +141933. q^{39} +650562. q^{40} +261595. q^{41} +207175. q^{42} -134629. q^{43} -1.31169e6 q^{44} +86726.0 q^{45} +772928. q^{46} +1.08722e6 q^{47} -2.01583e6 q^{48} +117649. q^{49} -1.43110e6 q^{50} +999624. q^{51} -1.95787e6 q^{52} +404711. q^{53} -440323. q^{54} -418973. q^{55} -1.87569e6 q^{56} -139462. q^{57} +142771. q^{58} +2.34681e6 q^{59} -1.19633e6 q^{60} -306843. q^{61} +3.48630e6 q^{62} -250047. q^{63} +1.21484e7 q^{64} -625374. q^{65} +2.12720e6 q^{66} -2.72369e6 q^{67} -1.37892e7 q^{68} -932874. q^{69} -912841. q^{70} -1.21376e6 q^{71} +3.98653e6 q^{72} -4.24836e6 q^{73} +420331. q^{74} +1.72725e6 q^{75} +1.92379e6 q^{76} +1.20798e6 q^{77} +3.17513e6 q^{78} -4.67709e6 q^{79} +8.88203e6 q^{80} +531441. q^{81} +5.85206e6 q^{82} +1.16593e6 q^{83} +3.44925e6 q^{84} -4.40448e6 q^{85} -3.01174e6 q^{86} -172315. q^{87} -1.92589e7 q^{88} +2.59661e6 q^{89} +1.94012e6 q^{90} +1.80307e6 q^{91} +1.28684e7 q^{92} -4.20774e6 q^{93} +2.43219e7 q^{94} +614488. q^{95} -2.61965e7 q^{96} -1.06794e7 q^{97} +2.63189e6 q^{98} -2.56739e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 12 q^{2} - 54 q^{3} + 352 q^{4} - 24 q^{5} - 324 q^{6} - 686 q^{7} + 7008 q^{8} + 1458 q^{9}+O(q^{10}) \) 2 * q + 12 * q^2 - 54 * q^3 + 352 * q^4 - 24 * q^5 - 324 * q^6 - 686 * q^7 + 7008 * q^8 + 1458 * q^9	\( 2 q + 12 q^{2} - 54 q^{3} + 352 q^{4} - 24 q^{5} - 324 q^{6} - 686 q^{7} + 7008 q^{8} + 1458 q^{9} + 4144 q^{10} + 2124 q^{11} - 9504 q^{12} - 1084 q^{13} - 4116 q^{14} + 648 q^{15} + 61312 q^{16} - 29256 q^{17} + 8748 q^{18} - 25816 q^{19} + 47232 q^{20} + 18522 q^{21} - 137336 q^{22} + 68316 q^{23} - 189216 q^{24} - 121658 q^{25} - 160872 q^{26} - 39366 q^{27} - 120736 q^{28} + 211308 q^{29} - 111888 q^{30} + 435840 q^{31} + 911616 q^{32} - 57348 q^{33} - 908784 q^{34} + 8232 q^{35} + 256608 q^{36} - 28428 q^{37} + 436848 q^{38} + 29268 q^{39} + 430464 q^{40} + 749760 q^{41} + 111132 q^{42} + 397096 q^{43} - 1427136 q^{44} - 17496 q^{45} + 422760 q^{46} + 840168 q^{47} - 1655424 q^{48} + 235298 q^{49} - 832860 q^{50} + 789912 q^{51} - 2043200 q^{52} - 246684 q^{53} - 236196 q^{54} - 1226128 q^{55} - 2403744 q^{56} + 697032 q^{57} - 1982456 q^{58} + 2199504 q^{59} - 1275264 q^{60} - 1951108 q^{61} + 582528 q^{62} - 500094 q^{63} + 14465024 q^{64} - 1221936 q^{65} + 3708072 q^{66} + 1532048 q^{67} - 13948032 q^{68} - 1844532 q^{69} - 1421392 q^{70} + 2024004 q^{71} + 5108832 q^{72} - 1709028 q^{73} + 910008 q^{74} + 3284766 q^{75} + 2557312 q^{76} - 728532 q^{77} + 4343544 q^{78} + 1048168 q^{79} + 10790400 q^{80} + 1062882 q^{81} + 789440 q^{82} - 4894296 q^{83} + 3259872 q^{84} - 5514912 q^{85} - 8526096 q^{86} - 5705316 q^{87} - 10567104 q^{88} - 60864 q^{89} + 3020976 q^{90} + 371812 q^{91} + 12177984 q^{92} - 11767680 q^{93} + 26884080 q^{94} + 5043744 q^{95} - 24613632 q^{96} - 26046852 q^{97} + 1411788 q^{98} + 1548396 q^{99}+O(q^{100}) \) 2 * q + 12 * q^2 - 54 * q^3 + 352 * q^4 - 24 * q^5 - 324 * q^6 - 686 * q^7 + 7008 * q^8 + 1458 * q^9 + 4144 * q^10 + 2124 * q^11 - 9504 * q^12 - 1084 * q^13 - 4116 * q^14 + 648 * q^15 + 61312 * q^16 - 29256 * q^17 + 8748 * q^18 - 25816 * q^19 + 47232 * q^20 + 18522 * q^21 - 137336 * q^22 + 68316 * q^23 - 189216 * q^24 - 121658 * q^25 - 160872 * q^26 - 39366 * q^27 - 120736 * q^28 + 211308 * q^29 - 111888 * q^30 + 435840 * q^31 + 911616 * q^32 - 57348 * q^33 - 908784 * q^34 + 8232 * q^35 + 256608 * q^36 - 28428 * q^37 + 436848 * q^38 + 29268 * q^39 + 430464 * q^40 + 749760 * q^41 + 111132 * q^42 + 397096 * q^43 - 1427136 * q^44 - 17496 * q^45 + 422760 * q^46 + 840168 * q^47 - 1655424 * q^48 + 235298 * q^49 - 832860 * q^50 + 789912 * q^51 - 2043200 * q^52 - 246684 * q^53 - 236196 * q^54 - 1226128 * q^55 - 2403744 * q^56 + 697032 * q^57 - 1982456 * q^58 + 2199504 * q^59 - 1275264 * q^60 - 1951108 * q^61 + 582528 * q^62 - 500094 * q^63 + 14465024 * q^64 - 1221936 * q^65 + 3708072 * q^66 + 1532048 * q^67 - 13948032 * q^68 - 1844532 * q^69 - 1421392 * q^70 + 2024004 * q^71 + 5108832 * q^72 - 1709028 * q^73 + 910008 * q^74 + 3284766 * q^75 + 2557312 * q^76 - 728532 * q^77 + 4343544 * q^78 + 1048168 * q^79 + 10790400 * q^80 + 1062882 * q^81 + 789440 * q^82 - 4894296 * q^83 + 3259872 * q^84 - 5514912 * q^85 - 8526096 * q^86 - 5705316 * q^87 - 10567104 * q^88 - 60864 * q^89 + 3020976 * q^90 + 371812 * q^91 + 12177984 * q^92 - 11767680 * q^93 + 26884080 * q^94 + 5043744 * q^95 - 24613632 * q^96 - 26046852 * q^97 + 1411788 * q^98 + 1548396 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more