LMFDB - Embedded newform 8021.2.a.c.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8021,2,Mod(1,8021)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8021, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("8021.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8021 = 13 \cdot 617 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8021.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:	$64.0480074613$
Analytic rank:	$0$
Dimension:	$169$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Character	$\chi$	$=$	8021.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.61542 q^{2} +2.30971 q^{3} +4.84044 q^{4} -1.40185 q^{5} -6.04088 q^{6} +1.96303 q^{7} -7.42897 q^{8} +2.33477 q^{9} +O(q^{10})$	\(q-2.61542 q^{2} +2.30971 q^{3} +4.84044 q^{4} -1.40185 q^{5} -6.04088 q^{6} +1.96303 q^{7} -7.42897 q^{8} +2.33477 q^{9} +3.66644 q^{10} +1.00105 q^{11} +11.1800 q^{12} -1.00000 q^{13} -5.13417 q^{14} -3.23788 q^{15} +9.74901 q^{16} -5.21836 q^{17} -6.10642 q^{18} +1.16632 q^{19} -6.78560 q^{20} +4.53405 q^{21} -2.61816 q^{22} +4.51097 q^{23} -17.1588 q^{24} -3.03480 q^{25} +2.61542 q^{26} -1.53648 q^{27} +9.50196 q^{28} -1.53104 q^{29} +8.46843 q^{30} +1.74435 q^{31} -10.6399 q^{32} +2.31213 q^{33} +13.6482 q^{34} -2.75189 q^{35} +11.3013 q^{36} +9.52085 q^{37} -3.05042 q^{38} -2.30971 q^{39} +10.4143 q^{40} -7.60783 q^{41} -11.8585 q^{42} -4.02619 q^{43} +4.84551 q^{44} -3.27301 q^{45} -11.7981 q^{46} -0.823555 q^{47} +22.5174 q^{48} -3.14649 q^{49} +7.93730 q^{50} -12.0529 q^{51} -4.84044 q^{52} +7.68690 q^{53} +4.01855 q^{54} -1.40332 q^{55} -14.5833 q^{56} +2.69386 q^{57} +4.00431 q^{58} +10.3962 q^{59} -15.6728 q^{60} -1.89300 q^{61} -4.56221 q^{62} +4.58324 q^{63} +8.32974 q^{64} +1.40185 q^{65} -6.04721 q^{66} +9.65531 q^{67} -25.2592 q^{68} +10.4190 q^{69} +7.19736 q^{70} +6.03530 q^{71} -17.3450 q^{72} +4.94587 q^{73} -24.9011 q^{74} -7.00952 q^{75} +5.64550 q^{76} +1.96509 q^{77} +6.04088 q^{78} -8.42714 q^{79} -13.6667 q^{80} -10.5532 q^{81} +19.8977 q^{82} -9.65474 q^{83} +21.9468 q^{84} +7.31538 q^{85} +10.5302 q^{86} -3.53625 q^{87} -7.43675 q^{88} +4.45736 q^{89} +8.56032 q^{90} -1.96303 q^{91} +21.8351 q^{92} +4.02894 q^{93} +2.15394 q^{94} -1.63501 q^{95} -24.5750 q^{96} +10.3170 q^{97} +8.22942 q^{98} +2.33722 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 169 q + 9 q^{2} + 9 q^{3} + 199 q^{4} + 12 q^{5} + 22 q^{6} + 36 q^{7} + 30 q^{8} + 198 q^{9}+O(q^{10}) $ 169 * q + 9 * q^2 + 9 * q^3 + 199 * q^4 + 12 * q^5 + 22 * q^6 + 36 * q^7 + 30 * q^8 + 198 * q^9	\( 169 q + 9 q^{2} + 9 q^{3} + 199 q^{4} + 12 q^{5} + 22 q^{6} + 36 q^{7} + 30 q^{8} + 198 q^{9} - 3 q^{10} + 59 q^{11} + 11 q^{12} - 169 q^{13} + 30 q^{14} + 50 q^{15} + 267 q^{16} + q^{17} + 53 q^{18} + 107 q^{19} + 48 q^{20} + 36 q^{21} + 14 q^{22} + 12 q^{23} + 78 q^{24} + 217 q^{25} - 9 q^{26} + 39 q^{27} + 99 q^{28} + 30 q^{29} - q^{30} + 106 q^{31} + 74 q^{32} + 16 q^{33} + 56 q^{34} + 46 q^{35} + 271 q^{36} + 73 q^{37} + 2 q^{38} - 9 q^{39} - 16 q^{40} + 52 q^{41} - 2 q^{42} + 64 q^{43} + 124 q^{44} + 84 q^{45} + 105 q^{46} + 55 q^{47} + 26 q^{48} + 257 q^{49} + 60 q^{50} + 117 q^{51} - 199 q^{52} + 7 q^{53} + 78 q^{54} - 4 q^{55} + 63 q^{56} + 51 q^{57} + 84 q^{58} + 98 q^{59} + 94 q^{60} + 32 q^{61} - 25 q^{62} + 128 q^{63} + 380 q^{64} - 12 q^{65} + 16 q^{66} + 170 q^{67} - 10 q^{68} + 55 q^{69} + 70 q^{70} + 124 q^{71} + 173 q^{72} + 81 q^{73} + 54 q^{74} + 120 q^{75} + 212 q^{76} + 20 q^{77} - 22 q^{78} + 92 q^{79} + 66 q^{80} + 265 q^{81} + 21 q^{82} + 62 q^{83} + 98 q^{84} + 139 q^{85} + 51 q^{86} - 33 q^{87} + 31 q^{88} + 58 q^{89} + 16 q^{90} - 36 q^{91} + 40 q^{92} + 37 q^{93} + 55 q^{94} + 23 q^{95} + 164 q^{96} + 78 q^{97} + 69 q^{98} + 307 q^{99}+O(q^{100}) \) 169 * q + 9 * q^2 + 9 * q^3 + 199 * q^4 + 12 * q^5 + 22 * q^6 + 36 * q^7 + 30 * q^8 + 198 * q^9 - 3 * q^10 + 59 * q^11 + 11 * q^12 - 169 * q^13 + 30 * q^14 + 50 * q^15 + 267 * q^16 + q^17 + 53 * q^18 + 107 * q^19 + 48 * q^20 + 36 * q^21 + 14 * q^22 + 12 * q^23 + 78 * q^24 + 217 * q^25 - 9 * q^26 + 39 * q^27 + 99 * q^28 + 30 * q^29 - q^30 + 106 * q^31 + 74 * q^32 + 16 * q^33 + 56 * q^34 + 46 * q^35 + 271 * q^36 + 73 * q^37 + 2 * q^38 - 9 * q^39 - 16 * q^40 + 52 * q^41 - 2 * q^42 + 64 * q^43 + 124 * q^44 + 84 * q^45 + 105 * q^46 + 55 * q^47 + 26 * q^48 + 257 * q^49 + 60 * q^50 + 117 * q^51 - 199 * q^52 + 7 * q^53 + 78 * q^54 - 4 * q^55 + 63 * q^56 + 51 * q^57 + 84 * q^58 + 98 * q^59 + 94 * q^60 + 32 * q^61 - 25 * q^62 + 128 * q^63 + 380 * q^64 - 12 * q^65 + 16 * q^66 + 170 * q^67 - 10 * q^68 + 55 * q^69 + 70 * q^70 + 124 * q^71 + 173 * q^72 + 81 * q^73 + 54 * q^74 + 120 * q^75 + 212 * q^76 + 20 * q^77 - 22 * q^78 + 92 * q^79 + 66 * q^80 + 265 * q^81 + 21 * q^82 + 62 * q^83 + 98 * q^84 + 139 * q^85 + 51 * q^86 - 33 * q^87 + 31 * q^88 + 58 * q^89 + 16 * q^90 - 36 * q^91 + 40 * q^92 + 37 * q^93 + 55 * q^94 + 23 * q^95 + 164 * q^96 + 78 * q^97 + 69 * q^98 + 307 * 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 $

$2$	−2.61542	−1.84938	−0.924692	−	0.380716i	$-0.875678\pi$
$2$	−2.61542	−1.84938	−0.924692	+	0.380716i	$0.875678\pi$
$3$	2.30971	1.33351	0.666757	−	0.745275i	$-0.267682\pi$
$3$	2.30971	1.33351	0.666757	+	0.745275i	$0.267682\pi$
$4$	4.84044	2.42022
$5$	−1.40185	−0.626928	−0.313464	−	0.949600i	$-0.601490\pi$
$5$	−1.40185	−0.626928	−0.313464	+	0.949600i	$0.601490\pi$
$6$	−6.04088	−2.46618
$7$	1.96303	0.741957	0.370979	−	0.928641i	$-0.379022\pi$
$7$	1.96303	0.741957	0.370979	+	0.928641i	$0.379022\pi$
$8$	−7.42897	−2.62654
$9$	2.33477	0.778258
$10$	3.66644	1.15943
$11$	1.00105	0.301827	0.150914	−	0.988547i	$-0.451778\pi$
$11$	1.00105	0.301827	0.150914	+	0.988547i	$0.451778\pi$
$12$	11.1800	3.22740
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−5.13417	−1.37216
$15$	−3.23788	−0.836017
$16$	9.74901	2.43725
$17$	−5.21836	−1.26564	−0.632819	−	0.774300i	$-0.718102\pi$
$17$	−5.21836	−1.26564	−0.632819	+	0.774300i	$0.718102\pi$
$18$	−6.10642	−1.43930
$19$	1.16632	0.267572	0.133786	−	0.991010i	$-0.457287\pi$
$19$	1.16632	0.267572	0.133786	+	0.991010i	$0.457287\pi$
$20$	−6.78560	−1.51731
$21$	4.53405	0.989410
$22$	−2.61816	−0.558194
$23$	4.51097	0.940603	0.470301	−	0.882506i	$-0.344145\pi$
$23$	4.51097	0.940603	0.470301	+	0.882506i	$0.344145\pi$
$24$	−17.1588	−3.50252
$25$	−3.03480	−0.606961
$26$	2.61542	0.512927
$27$	−1.53648	−0.295696
$28$	9.50196	1.79570
$29$	−1.53104	−0.284306	−0.142153	−	0.989845i	$-0.545403\pi$
$29$	−1.53104	−0.284306	−0.142153	+	0.989845i	$0.545403\pi$
$30$	8.46843	1.54612
$31$	1.74435	0.313294	0.156647	−	0.987655i	$-0.449932\pi$
$31$	1.74435	0.313294	0.156647	+	0.987655i	$0.449932\pi$
$32$	−10.6399	−1.88088
$33$	2.31213	0.402491
$34$	13.6482	2.34065
$35$	−2.75189	−0.465154
$36$	11.3013	1.88356
$37$	9.52085	1.56522	0.782609	−	0.622514i	$-0.213889\pi$
$37$	9.52085	1.56522	0.782609	+	0.622514i	$0.213889\pi$
$38$	−3.05042	−0.494843
$39$	−2.30971	−0.369850
$40$	10.4143	1.64665
$41$	−7.60783	−1.18814	−0.594072	−	0.804412i	$-0.702480\pi$
$41$	−7.60783	−1.18814	−0.594072	+	0.804412i	$0.702480\pi$
$42$	−11.8585	−1.82980
$43$	−4.02619	−0.613988	−0.306994	−	0.951711i	$-0.599323\pi$
$43$	−4.02619	−0.613988	−0.306994	+	0.951711i	$0.599323\pi$
$44$	4.84551	0.730489
$45$	−3.27301	−0.487912
$46$	−11.7981	−1.73954
$47$	−0.823555	−0.120128	−0.0600639	−	0.998195i	$-0.519130\pi$
$47$	−0.823555	−0.120128	−0.0600639	+	0.998195i	$0.519130\pi$
$48$	22.5174	3.25011
$49$	−3.14649	−0.449499
$50$	7.93730	1.12250
$51$	−12.0529	−1.68775
$52$	−4.84044	−0.671249
$53$	7.68690	1.05588	0.527938	−	0.849283i	$-0.322965\pi$
$53$	7.68690	1.05588	0.527938	+	0.849283i	$0.322965\pi$
$54$	4.01855	0.546856
$55$	−1.40332	−0.189224
$56$	−14.5833	−1.94878
$57$	2.69386	0.356811
$58$	4.00431	0.525792
$59$	10.3962	1.35347	0.676733	−	0.736228i	$-0.263395\pi$
$59$	10.3962	1.35347	0.676733	+	0.736228i	$0.263395\pi$
$60$	−15.6728	−2.02335
$61$	−1.89300	−0.242373	−0.121187	−	0.992630i	$-0.538670\pi$
$61$	−1.89300	−0.242373	−0.121187	+	0.992630i	$0.538670\pi$
$62$	−4.56221	−0.579401
$63$	4.58324	0.577434
$64$	8.32974	1.04122
$65$	1.40185	0.173879
$66$	−6.04721	−0.744360
$67$	9.65531	1.17958	0.589792	−	0.807555i	$-0.299210\pi$
$67$	9.65531	1.17958	0.589792	+	0.807555i	$0.299210\pi$
$68$	−25.2592	−3.06313
$69$	10.4190	1.25431
$70$	7.19736	0.860249
$71$	6.03530	0.716259	0.358129	−	0.933672i	$-0.383415\pi$
$71$	6.03530	0.716259	0.358129	+	0.933672i	$0.383415\pi$
$72$	−17.3450	−2.04412
$73$	4.94587	0.578870	0.289435	−	0.957198i	$-0.406533\pi$
$73$	4.94587	0.578870	0.289435	+	0.957198i	$0.406533\pi$
$74$	−24.9011	−2.89469
$75$	−7.00952	−0.809390
$76$	5.64550	0.647583
$77$	1.96509	0.223943
$78$	6.04088	0.683995
$79$	−8.42714	−0.948127	−0.474064	−	0.880491i	$-0.657213\pi$
$79$	−8.42714	−0.948127	−0.474064	+	0.880491i	$0.657213\pi$
$80$	−13.6667	−1.52798
$81$	−10.5532	−1.17257
$82$	19.8977	2.19733
$83$	−9.65474	−1.05975	−0.529873	−	0.848077i	$-0.677760\pi$
$83$	−9.65474	−1.05975	−0.529873	+	0.848077i	$0.677760\pi$
$84$	21.9468	2.39459
$85$	7.31538	0.793465
$86$	10.5302	1.13550
$87$	−3.53625	−0.379126
$88$	−7.43675	−0.792760
$89$	4.45736	0.472479	0.236240	−	0.971695i	$-0.424085\pi$
$89$	4.45736	0.472479	0.236240	+	0.971695i	$0.424085\pi$
$90$	8.56032	0.902337
$91$	−1.96303	−0.205782
$92$	21.8351	2.27647
$93$	4.02894	0.417782
$94$	2.15394	0.222162
$95$	−1.63501	−0.167748
$96$	−24.5750	−2.50818
$97$	10.3170	1.04753	0.523764	−	0.851864i	$-0.324527\pi$
$97$	10.3170	1.04753	0.523764	+	0.851864i	$0.324527\pi$
$98$	8.22942	0.831297
$99$	2.33722	0.234899
$100$	−14.6898	−1.46898
$101$	−6.93166	−0.689726	−0.344863	−	0.938653i	$-0.612075\pi$
$101$	−6.93166	−0.689726	−0.344863	+	0.938653i	$0.612075\pi$
$102$	31.5235	3.12129
$103$	19.9266	1.96343	0.981713	−	0.190365i	$-0.0609672\pi$
$103$	19.9266	1.96343	0.981713	+	0.190365i	$0.0609672\pi$
$104$	7.42897	0.728470
$105$	−6.35607	−0.620289
$106$	−20.1045	−1.95272
$107$	12.2832	1.18746	0.593729	−	0.804665i	$-0.297655\pi$
$107$	12.2832	1.18746	0.593729	+	0.804665i	$0.297655\pi$
$108$	−7.43726	−0.715650
$109$	19.8783	1.90399	0.951996	−	0.306111i	$-0.0990280\pi$
$109$	19.8783	1.90399	0.951996	+	0.306111i	$0.0990280\pi$
$110$	3.67029	0.349948
$111$	21.9904	2.08724
$112$	19.1376	1.80834
$113$	−0.543866	−0.0511626	−0.0255813	−	0.999673i	$-0.508144\pi$
$113$	−0.543866	−0.0511626	−0.0255813	+	0.999673i	$0.508144\pi$
$114$	−7.04559	−0.659880
$115$	−6.32373	−0.589691
$116$	−7.41090	−0.688084
$117$	−2.33477	−0.215850
$118$	−27.1904	−2.50308
$119$	−10.2438	−0.939050
$120$	24.0541	2.19583
$121$	−9.99790	−0.908900
$122$	4.95099	0.448242
$123$	−17.5719	−1.58441
$124$	8.44341	0.758241
$125$	11.2636	1.00745
$126$	−11.9871	−1.06790
$127$	15.5102	1.37631	0.688155	−	0.725563i	$-0.258421\pi$
$127$	15.5102	1.37631	0.688155	+	0.725563i	$0.258421\pi$
$128$	−0.506069	−0.0447306
$129$	−9.29934	−0.818761
$130$	−3.66644	−0.321568
$131$	0.654623	0.0571947	0.0285973	−	0.999591i	$-0.490896\pi$
$131$	0.654623	0.0571947	0.0285973	+	0.999591i	$0.490896\pi$
$132$	11.1917	0.974117
$133$	2.28952	0.198527
$134$	−25.2527	−2.18150
$135$	2.15392	0.185380
$136$	38.7670	3.32424
$137$	−12.2981	−1.05070	−0.525348	−	0.850887i	$-0.676065\pi$
$137$	−12.2981	−1.05070	−0.525348	+	0.850887i	$0.676065\pi$
$138$	−27.2502	−2.31969
$139$	18.5379	1.57236	0.786182	−	0.617995i	$-0.212055\pi$
$139$	18.5379	1.57236	0.786182	+	0.617995i	$0.212055\pi$
$140$	−13.3204	−1.12578
$141$	−1.90217	−0.160192
$142$	−15.7849	−1.32464
$143$	−1.00105	−0.0837118
$144$	22.7617	1.89681
$145$	2.14629	0.178240
$146$	−12.9355	−1.07055
$147$	−7.26750	−0.599413
$148$	46.0851	3.78817
$149$	−12.0580	−0.987829	−0.493915	−	0.869510i	$-0.664434\pi$
$149$	−12.0580	−0.987829	−0.493915	+	0.869510i	$0.664434\pi$
$150$	18.3329	1.49687
$151$	2.71320	0.220797	0.110399	−	0.993887i	$-0.464787\pi$
$151$	2.71320	0.220797	0.110399	+	0.993887i	$0.464787\pi$
$152$	−8.66454	−0.702787
$153$	−12.1837	−0.984993
$154$	−5.13955	−0.414157
$155$	−2.44532	−0.196413
$156$	−11.1800	−0.895119
$157$	−13.8715	−1.10706	−0.553531	−	0.832828i	$-0.686720\pi$
$157$	−13.8715	−1.10706	−0.553531	+	0.832828i	$0.686720\pi$
$158$	22.0405	1.75345
$159$	17.7545	1.40803
$160$	14.9155	1.17918
$161$	8.85519	0.697887
$162$	27.6010	2.16854
$163$	12.2254	0.957568	0.478784	−	0.877933i	$-0.341078\pi$
$163$	12.2254	0.957568	0.478784	+	0.877933i	$0.341078\pi$
$164$	−36.8253	−2.87557
$165$	−3.24127	−0.252333
$166$	25.2512	1.95988
$167$	−17.0040	−1.31581	−0.657903	−	0.753103i	$-0.728556\pi$
$167$	−17.0040	−1.31581	−0.657903	+	0.753103i	$0.728556\pi$
$168$	−33.6833	−2.59872
$169$	1.00000	0.0769231
$170$	−19.1328	−1.46742
$171$	2.72309	0.208240
$172$	−19.4885	−1.48599
$173$	−10.4005	−0.790739	−0.395370	−	0.918522i	$-0.629383\pi$
$173$	−10.4005	−0.790739	−0.395370	+	0.918522i	$0.629383\pi$
$174$	9.24881	0.701150
$175$	−5.95743	−0.450339
$176$	9.75922	0.735629
$177$	24.0122	1.80487
$178$	−11.6579	−0.873796
$179$	−17.2636	−1.29034	−0.645171	−	0.764039i	$-0.723214\pi$
$179$	−17.2636	−1.29034	−0.645171	+	0.764039i	$0.723214\pi$
$180$	−15.8428	−1.18086
$181$	23.1972	1.72424	0.862118	−	0.506707i	$-0.169137\pi$
$181$	23.1972	1.72424	0.862118	+	0.506707i	$0.169137\pi$
$182$	5.13417	0.380570
$183$	−4.37228	−0.323208
$184$	−33.5119	−2.47053
$185$	−13.3468	−0.981279
$186$	−10.5374	−0.772639
$187$	−5.22383	−0.382004
$188$	−3.98637	−0.290736
$189$	−3.01617	−0.219394
$190$	4.27624	0.310231
$191$	9.23640	0.668323	0.334161	−	0.942516i	$-0.391547\pi$
$191$	9.23640	0.668323	0.334161	+	0.942516i	$0.391547\pi$
$192$	19.2393	1.38848
$193$	−10.9863	−0.790809	−0.395404	−	0.918507i	$-0.629396\pi$
$193$	−10.9863	−0.790809	−0.395404	+	0.918507i	$0.629396\pi$
$194$	−26.9832	−1.93728
$195$	3.23788	0.231870
$196$	−15.2304	−1.08789
$197$	−15.2536	−1.08677	−0.543386	−	0.839483i	$-0.682858\pi$
$197$	−15.2536	−1.08677	−0.543386	+	0.839483i	$0.682858\pi$
$198$	−6.11282	−0.434419
$199$	21.1243	1.49746	0.748731	−	0.662873i	$-0.230663\pi$
$199$	21.1243	1.49746	0.748731	+	0.662873i	$0.230663\pi$
$200$	22.5455	1.59420
$201$	22.3010	1.57299
$202$	18.1292	1.27557
$203$	−3.00548	−0.210943
$204$	−58.3414	−4.08472
$205$	10.6651	0.744881
$206$	−52.1165	−3.63113
$207$	10.5321	0.732031
$208$	−9.74901	−0.675972
$209$	1.16754	0.0807605
$210$	16.6238	1.14715
$211$	9.45446	0.650872	0.325436	−	0.945564i	$-0.394489\pi$
$211$	9.45446	0.650872	0.325436	+	0.945564i	$0.394489\pi$
$212$	37.2080	2.55546
$213$	13.9398	0.955141
$214$	−32.1257	−2.19606
$215$	5.64413	0.384926
$216$	11.4145	0.776657
$217$	3.42421	0.232451
$218$	−51.9901	−3.52121
$219$	11.4235	0.771931
$220$	−6.79271	−0.457964
$221$	5.21836	0.351025
$222$	−57.5143	−3.86011
$223$	24.8789	1.66601	0.833006	−	0.553264i	$-0.186618\pi$
$223$	24.8789	1.66601	0.833006	+	0.553264i	$0.186618\pi$
$224$	−20.8864	−1.39553
$225$	−7.08558	−0.472372
$226$	1.42244	0.0946193
$227$	−9.50598	−0.630934	−0.315467	−	0.948937i	$-0.602161\pi$
$227$	−9.50598	−0.630934	−0.315467	+	0.948937i	$0.602161\pi$
$228$	13.0395	0.863561
$229$	23.2184	1.53431	0.767157	−	0.641460i	$-0.221671\pi$
$229$	23.2184	1.53431	0.767157	+	0.641460i	$0.221671\pi$
$230$	16.5392	1.09056
$231$	4.53880	0.298631
$232$	11.3740	0.746741
$233$	26.1076	1.71036	0.855182	−	0.518328i	$-0.173445\pi$
$233$	26.1076	1.71036	0.855182	+	0.518328i	$0.173445\pi$
$234$	6.10642	0.399189
$235$	1.15450	0.0753115
$236$	50.3221	3.27569
$237$	−19.4643	−1.26434
$238$	26.7919	1.73666
$239$	−1.08901	−0.0704422	−0.0352211	−	0.999380i	$-0.511214\pi$
$239$	−1.08901	−0.0704422	−0.0352211	+	0.999380i	$0.511214\pi$
$240$	−31.5661	−2.03759
$241$	21.4674	1.38284	0.691419	−	0.722454i	$-0.256986\pi$
$241$	21.4674	1.38284	0.691419	+	0.722454i	$0.256986\pi$
$242$	26.1488	1.68091
$243$	−19.7653	−1.26795
$244$	−9.16295	−0.586598
$245$	4.41093	0.281804
$246$	45.9580	2.93017
$247$	−1.16632	−0.0742111
$248$	−12.9587	−0.822878
$249$	−22.2997	−1.41319
$250$	−29.4592	−1.86316
$251$	18.6261	1.17567	0.587835	−	0.808981i	$-0.299980\pi$
$251$	18.6261	1.17567	0.587835	+	0.808981i	$0.299980\pi$
$252$	22.1849	1.39752
$253$	4.51570	0.283899
$254$	−40.5658	−2.54533
$255$	16.8964	1.05810
$256$	−15.3359	−0.958493
$257$	−4.29611	−0.267984	−0.133992	−	0.990982i	$-0.542780\pi$
$257$	−4.29611	−0.267984	−0.133992	+	0.990982i	$0.542780\pi$
$258$	24.3217	1.51420
$259$	18.6898	1.16132
$260$	6.78560	0.420825
$261$	−3.57462	−0.221264
$262$	−1.71212	−0.105775
$263$	−0.0823898	−0.00508037	−0.00254019	−	0.999997i	$-0.500809\pi$
$263$	−0.0823898	−0.00508037	−0.00254019	+	0.999997i	$0.500809\pi$
$264$	−17.1768	−1.05716
$265$	−10.7759	−0.661959
$266$	−5.98808	−0.367153
$267$	10.2952	0.630058
$268$	46.7360	2.85485
$269$	6.87557	0.419211	0.209605	−	0.977786i	$-0.432782\pi$
$269$	6.87557	0.419211	0.209605	+	0.977786i	$0.432782\pi$
$270$	−5.63343	−0.342839
$271$	9.49688	0.576894	0.288447	−	0.957496i	$-0.406861\pi$
$271$	9.49688	0.576894	0.288447	+	0.957496i	$0.406861\pi$
$272$	−50.8738	−3.08468
$273$	−4.53405	−0.274413
$274$	32.1647	1.94314
$275$	−3.03798	−0.183197
$276$	50.4328	3.03570
$277$	−7.42871	−0.446348	−0.223174	−	0.974779i	$-0.571642\pi$
$277$	−7.42871	−0.446348	−0.223174	+	0.974779i	$0.571642\pi$
$278$	−48.4845	−2.90791
$279$	4.07265	0.243823
$280$	20.4437	1.22174
$281$	25.4852	1.52032	0.760159	−	0.649737i	$-0.225121\pi$
$281$	25.4852	1.52032	0.760159	+	0.649737i	$0.225121\pi$
$282$	4.97499	0.296257
$283$	−14.8814	−0.884606	−0.442303	−	0.896866i	$-0.645838\pi$
$283$	−14.8814	−0.884606	−0.442303	+	0.896866i	$0.645838\pi$
$284$	29.2136	1.73351
$285$	−3.77640	−0.223695
$286$	2.61816	0.154815
$287$	−14.9344	−0.881552
$288$	−24.8417	−1.46381
$289$	10.2313	0.601840
$290$	−5.61346	−0.329634
$291$	23.8292	1.39689
$292$	23.9402	1.40099
$293$	26.0621	1.52256	0.761282	−	0.648421i	$-0.224570\pi$
$293$	26.0621	1.52256	0.761282	+	0.648421i	$0.224570\pi$
$294$	19.0076	1.10855
$295$	−14.5739	−0.848526
$296$	−70.7300	−4.11110
$297$	−1.53809	−0.0892491
$298$	31.5368	1.82688
$299$	−4.51097	−0.260876
$300$	−33.9292	−1.95890
$301$	−7.90355	−0.455553
$302$	−7.09617	−0.408339
$303$	−16.0101	−0.919759
$304$	11.3705	0.652140
$305$	2.65371	0.151951
$306$	31.8655	1.82163
$307$	−6.52736	−0.372536	−0.186268	−	0.982499i	$-0.559639\pi$
$307$	−6.52736	−0.372536	−0.186268	+	0.982499i	$0.559639\pi$
$308$	9.51191	0.541992
$309$	46.0247	2.61826
$310$	6.39555	0.363243
$311$	−22.7640	−1.29083	−0.645415	−	0.763832i	$-0.723316\pi$
$311$	−22.7640	−1.29083	−0.645415	+	0.763832i	$0.723316\pi$
$312$	17.1588	0.971424
$313$	−23.0390	−1.30224	−0.651120	−	0.758975i	$-0.725701\pi$
$313$	−23.0390	−1.30224	−0.651120	+	0.758975i	$0.725701\pi$
$314$	36.2797	2.04738
$315$	−6.42504	−0.362010
$316$	−40.7911	−2.29468
$317$	−2.34136	−0.131504	−0.0657520	−	0.997836i	$-0.520945\pi$
$317$	−2.34136	−0.131504	−0.0657520	+	0.997836i	$0.520945\pi$
$318$	−46.4356	−2.60398
$319$	−1.53264	−0.0858114
$320$	−11.6771	−0.652769
$321$	28.3706	1.58349
$322$	−23.1601	−1.29066
$323$	−6.08627	−0.338649
$324$	−51.0819	−2.83789
$325$	3.03480	0.168341
$326$	−31.9746	−1.77091
$327$	45.9131	2.53900
$328$	56.5183	3.12070
$329$	−1.61667	−0.0891297
$330$	8.47731	0.466660
$331$	−21.8049	−1.19850	−0.599252	−	0.800561i	$-0.704535\pi$
$331$	−21.8049	−1.19850	−0.599252	+	0.800561i	$0.704535\pi$
$332$	−46.7332	−2.56482
$333$	22.2290	1.21814
$334$	44.4726	2.43343
$335$	−13.5353	−0.739515
$336$	44.2025	2.41144
$337$	15.4060	0.839217	0.419608	−	0.907705i	$-0.362167\pi$
$337$	15.4060	0.839217	0.419608	+	0.907705i	$0.362167\pi$
$338$	−2.61542	−0.142260
$339$	−1.25617	−0.0682260
$340$	35.4097	1.92036
$341$	1.74617	0.0945606
$342$	−7.12204	−0.385116
$343$	−19.9179	−1.07547
$344$	29.9104	1.61266
$345$	−14.6060	−0.786360
$346$	27.2018	1.46238
$347$	28.1501	1.51118	0.755588	−	0.655048i	$-0.227351\pi$
$347$	28.1501	1.51118	0.755588	+	0.655048i	$0.227351\pi$
$348$	−17.1170	−0.917570
$349$	27.5176	1.47299	0.736493	−	0.676446i	$-0.236481\pi$
$349$	27.5176	1.47299	0.736493	+	0.676446i	$0.236481\pi$
$350$	15.5812	0.832850
$351$	1.53648	0.0820114
$352$	−10.6510	−0.567701
$353$	31.3003	1.66595	0.832974	−	0.553312i	$-0.186636\pi$
$353$	31.3003	1.66595	0.832974	+	0.553312i	$0.186636\pi$
$354$	−62.8020	−3.33789
$355$	−8.46062	−0.449043
$356$	21.5756	1.14351
$357$	−23.6603	−1.25224
$358$	45.1516	2.38634
$359$	35.0335	1.84900	0.924498	−	0.381186i	$-0.124484\pi$
$359$	35.0335	1.84900	0.924498	+	0.381186i	$0.124484\pi$
$360$	24.3151	1.28152
$361$	−17.6397	−0.928405
$362$	−60.6706	−3.18878
$363$	−23.0923	−1.21203
$364$	−9.50196	−0.498038
$365$	−6.93339	−0.362910
$366$	11.4354	0.597736
$367$	−1.92115	−0.100283	−0.0501416	−	0.998742i	$-0.515967\pi$
$367$	−1.92115	−0.100283	−0.0501416	+	0.998742i	$0.515967\pi$
$368$	43.9775	2.29249
$369$	−17.7626	−0.924682
$370$	34.9077	1.81476
$371$	15.0896	0.783415
$372$	19.5019	1.01112
$373$	−23.5902	−1.22145	−0.610726	−	0.791842i	$-0.709122\pi$
$373$	−23.5902	−1.22145	−0.610726	+	0.791842i	$0.709122\pi$
$374$	13.6625	0.706472
$375$	26.0157	1.34345
$376$	6.11816	0.315520
$377$	1.53104	0.0788524
$378$	7.88856	0.405744
$379$	−31.6948	−1.62805	−0.814027	−	0.580828i	$-0.802729\pi$
$379$	−31.6948	−1.62805	−0.814027	+	0.580828i	$0.802729\pi$
$380$	−7.91417	−0.405988
$381$	35.8242	1.83533
$382$	−24.1571	−1.23599
$383$	0.711648	0.0363635	0.0181818	−	0.999835i	$-0.494212\pi$
$383$	0.711648	0.0363635	0.0181818	+	0.999835i	$0.494212\pi$
$384$	−1.16887	−0.0596488
$385$	−2.75477	−0.140396
$386$	28.7337	1.46251
$387$	−9.40024	−0.477841
$388$	49.9386	2.53525
$389$	−12.6019	−0.638943	−0.319471	−	0.947596i	$-0.603505\pi$
$389$	−12.6019	−0.638943	−0.319471	+	0.947596i	$0.603505\pi$
$390$	−8.46843	−0.428816
$391$	−23.5399	−1.19046
$392$	23.3752	1.18063
$393$	1.51199	0.0762699
$394$	39.8946	2.00986
$395$	11.8136	0.594408
$396$	11.3132	0.568509
$397$	−3.23202	−0.162211	−0.0811053	−	0.996706i	$-0.525845\pi$
$397$	−3.23202	−0.162211	−0.0811053	+	0.996706i	$0.525845\pi$
$398$	−55.2490	−2.76938
$399$	5.28814	0.264738
$400$	−29.5863	−1.47932
$401$	18.7439	0.936023	0.468012	−	0.883722i	$-0.344971\pi$
$401$	18.7439	0.936023	0.468012	+	0.883722i	$0.344971\pi$
$402$	−58.3266	−2.90906
$403$	−1.74435	−0.0868921
$404$	−33.5523	−1.66929
$405$	14.7940	0.735119
$406$	7.86060	0.390115
$407$	9.53082	0.472425
$408$	89.5407	4.43292
$409$	31.4410	1.55466	0.777328	−	0.629095i	$-0.216575\pi$
$409$	31.4410	1.55466	0.777328	+	0.629095i	$0.216575\pi$
$410$	−27.8937	−1.37757
$411$	−28.4051	−1.40112
$412$	96.4536	4.75193
$413$	20.4080	1.00421
$414$	−27.5459	−1.35381
$415$	13.5345	0.664385
$416$	10.6399	0.521662
$417$	42.8172	2.09677
$418$	−3.05361	−0.149357
$419$	−7.96627	−0.389178	−0.194589	−	0.980885i	$-0.562337\pi$
$419$	−7.96627	−0.389178	−0.194589	+	0.980885i	$0.562337\pi$
$420$	−30.7662	−1.50124
$421$	12.3375	0.601293	0.300646	−	0.953736i	$-0.402798\pi$
$421$	12.3375	0.601293	0.300646	+	0.953736i	$0.402798\pi$
$422$	−24.7274	−1.20371
$423$	−1.92281	−0.0934904
$424$	−57.1057	−2.77330
$425$	15.8367	0.768193
$426$	−36.4585	−1.76642
$427$	−3.71602	−0.179831
$428$	59.4559	2.87391
$429$	−2.31213	−0.111631
$430$	−14.7618	−0.711877
$431$	−4.68920	−0.225871	−0.112935	−	0.993602i	$-0.536025\pi$
$431$	−4.68920	−0.225871	−0.112935	+	0.993602i	$0.536025\pi$
$432$	−14.9792	−0.720686
$433$	−28.8385	−1.38589	−0.692946	−	0.720990i	$-0.743688\pi$
$433$	−28.8385	−1.38589	−0.692946	+	0.720990i	$0.743688\pi$
$434$	−8.95577	−0.429891
$435$	4.95732	0.237685
$436$	96.2196	4.60808
$437$	5.26123	0.251679
$438$	−29.8774	−1.42760
$439$	26.4461	1.26220	0.631102	−	0.775700i	$-0.282603\pi$
$439$	26.4461	1.26220	0.631102	+	0.775700i	$0.282603\pi$
$440$	10.4252	0.497004
$441$	−7.34635	−0.349826
$442$	−13.6482	−0.649180
$443$	−25.7745	−1.22458	−0.612292	−	0.790632i	$-0.709752\pi$
$443$	−25.7745	−1.22458	−0.612292	+	0.790632i	$0.709752\pi$
$444$	106.443	5.05158
$445$	−6.24857	−0.296211
$446$	−65.0688	−3.08110
$447$	−27.8505	−1.31728
$448$	16.3516	0.772539
$449$	24.4923	1.15586	0.577932	−	0.816085i	$-0.303860\pi$
$449$	24.4923	1.15586	0.577932	+	0.816085i	$0.303860\pi$
$450$	18.5318	0.873597
$451$	−7.61580	−0.358614
$452$	−2.63255	−0.123825
$453$	6.26672	0.294436
$454$	24.8622	1.16684
$455$	2.75189	0.129011
$456$	−20.0126	−0.937176
$457$	−26.8559	−1.25627	−0.628134	−	0.778106i	$-0.716181\pi$
$457$	−26.8559	−1.25627	−0.628134	+	0.778106i	$0.716181\pi$
$458$	−60.7259	−2.83753
$459$	8.01792	0.374244
$460$	−30.6096	−1.42718
$461$	−26.1036	−1.21577	−0.607883	−	0.794027i	$-0.707981\pi$
$461$	−26.1036	−1.21577	−0.607883	+	0.794027i	$0.707981\pi$
$462$	−11.8709	−0.552283
$463$	−0.387612	−0.0180139	−0.00900693	−	0.999959i	$-0.502867\pi$
$463$	−0.387612	−0.0180139	−0.00900693	+	0.999959i	$0.502867\pi$
$464$	−14.9261	−0.692926
$465$	−5.64799	−0.261919
$466$	−68.2824	−3.16312
$467$	4.81368	0.222750	0.111375	−	0.993778i	$-0.464474\pi$
$467$	4.81368	0.222750	0.111375	+	0.993778i	$0.464474\pi$
$468$	−11.3013	−0.522405
$469$	18.9537	0.875201
$470$	−3.01952	−0.139280
$471$	−32.0391	−1.47628
$472$	−77.2328	−3.55493
$473$	−4.03041	−0.185318
$474$	50.9073	2.33825
$475$	−3.53955	−0.162406
$476$	−49.5847	−2.27271
$477$	17.9472	0.821744
$478$	2.84822	0.130275
$479$	−3.25933	−0.148923	−0.0744613	−	0.997224i	$-0.523724\pi$
$479$	−3.25933	−0.148923	−0.0744613	+	0.997224i	$0.523724\pi$
$480$	34.4506	1.57245
$481$	−9.52085	−0.434113
$482$	−56.1464	−2.55740
$483$	20.4530	0.930642
$484$	−48.3943	−2.19974
$485$	−14.4629	−0.656725
$486$	51.6947	2.34492
$487$	−34.3504	−1.55656	−0.778282	−	0.627916i	$-0.783908\pi$
$487$	−34.3504	−1.55656	−0.778282	+	0.627916i	$0.783908\pi$
$488$	14.0630	0.636603
$489$	28.2372	1.27693
$490$	−11.5364	−0.521163
$491$	15.7045	0.708732	0.354366	−	0.935107i	$-0.384697\pi$
$491$	15.7045	0.708732	0.354366	+	0.935107i	$0.384697\pi$
$492$	−85.0558	−3.83461
$493$	7.98950	0.359829
$494$	3.05042	0.137245
$495$	−3.27644	−0.147265
$496$	17.0057	0.763576
$497$	11.8475	0.531434
$498$	58.3231	2.61352
$499$	−13.1107	−0.586915	−0.293457	−	0.955972i	$-0.594806\pi$
$499$	−13.1107	−0.586915	−0.293457	+	0.955972i	$0.594806\pi$
$500$	54.5210	2.43825
$501$	−39.2743	−1.75465
$502$	−48.7152	−2.17427
$503$	33.8941	1.51126	0.755632	−	0.654997i	$-0.227330\pi$
$503$	33.8941	1.51126	0.755632	+	0.654997i	$0.227330\pi$
$504$	−34.0487	−1.51665
$505$	9.71718	0.432409
$506$	−11.8105	−0.525039
$507$	2.30971	0.102578
$508$	75.0764	3.33098
$509$	13.3634	0.592322	0.296161	−	0.955138i	$-0.404294\pi$
$509$	13.3634	0.592322	0.296161	+	0.955138i	$0.404294\pi$
$510$	−44.1913	−1.95683
$511$	9.70892	0.429497
$512$	41.1220	1.81735
$513$	−1.79203	−0.0791200
$514$	11.2361	0.495605
$515$	−27.9342	−1.23093
$516$	−45.0129	−1.98158
$517$	−0.824417	−0.0362578
$518$	−48.8816	−2.14774
$519$	−24.0223	−1.05446
$520$	−10.4143	−0.456699
$521$	−10.6959	−0.468594	−0.234297	−	0.972165i	$-0.575279\pi$
$521$	−10.6959	−0.468594	−0.234297	+	0.972165i	$0.575279\pi$
$522$	9.34916	0.409201
$523$	−2.98975	−0.130733	−0.0653663	−	0.997861i	$-0.520822\pi$
$523$	−2.98975	−0.130733	−0.0653663	+	0.997861i	$0.520822\pi$
$524$	3.16867	0.138424
$525$	−13.7599	−0.600533
$526$	0.215484	0.00939556
$527$	−9.10263	−0.396517
$528$	22.5410	0.980971
$529$	−2.65114	−0.115267
$530$	28.1836	1.22422
$531$	24.2727	1.05335
$532$	11.0823	0.480479
$533$	7.60783	0.329532
$534$	−26.9264	−1.16522
$535$	−17.2192	−0.744451
$536$	−71.7290	−3.09822
$537$	−39.8739	−1.72069
$538$	−17.9825	−0.775282
$539$	−3.14979	−0.135671
$540$	10.4260	0.448662
$541$	−23.9273	−1.02871	−0.514357	−	0.857576i	$-0.671969\pi$
$541$	−23.9273	−1.02871	−0.514357	+	0.857576i	$0.671969\pi$
$542$	−24.8384	−1.06690
$543$	53.5790	2.29929
$544$	55.5226	2.38051
$545$	−27.8664	−1.19367
$546$	11.8585	0.507495
$547$	−26.4288	−1.13002	−0.565008	−	0.825086i	$-0.691127\pi$
$547$	−26.4288	−1.13002	−0.565008	+	0.825086i	$0.691127\pi$
$548$	−59.5282	−2.54292
$549$	−4.41972	−0.188629
$550$	7.94561	0.338802
$551$	−1.78568	−0.0760724
$552$	−77.4028	−3.29448
$553$	−16.5428	−0.703470
$554$	19.4292	0.825469
$555$	−30.8274	−1.30855
$556$	89.7317	3.80547
$557$	−15.9153	−0.674352	−0.337176	−	0.941442i	$-0.609472\pi$
$557$	−15.9153	−0.674352	−0.337176	+	0.941442i	$0.609472\pi$
$558$	−10.6517	−0.450923
$559$	4.02619	0.170290
$560$	−26.8282	−1.13370
$561$	−12.0655	−0.509407
$562$	−66.6546	−2.81165
$563$	22.1164	0.932093	0.466047	−	0.884760i	$-0.345678\pi$
$563$	22.1164	0.932093	0.466047	+	0.884760i	$0.345678\pi$
$564$	−9.20737	−0.387700
$565$	0.762421	0.0320753
$566$	38.9211	1.63598
$567$	−20.7162	−0.869999
$568$	−44.8361	−1.88128
$569$	24.4010	1.02294	0.511471	−	0.859301i	$-0.329101\pi$
$569$	24.4010	1.02294	0.511471	+	0.859301i	$0.329101\pi$
$570$	9.87689	0.413698
$571$	−17.7091	−0.741103	−0.370552	−	0.928812i	$-0.620831\pi$
$571$	−17.7091	−0.741103	−0.370552	+	0.928812i	$0.620831\pi$
$572$	−4.84551	−0.202601
$573$	21.3334	0.891217
$574$	39.0599	1.63033
$575$	−13.6899	−0.570909
$576$	19.4480	0.810335
$577$	25.7465	1.07184	0.535919	−	0.844269i	$-0.319965\pi$
$577$	25.7465	1.07184	0.535919	+	0.844269i	$0.319965\pi$
$578$	−26.7591	−1.11303
$579$	−25.3751	−1.05455
$580$	10.3890	0.431380
$581$	−18.9526	−0.786286
$582$	−62.3235	−2.58339
$583$	7.69495	0.318692
$584$	−36.7427	−1.52042
$585$	3.27301	0.135322
$586$	−68.1634	−2.81580
$587$	−9.14305	−0.377374	−0.188687	−	0.982037i	$-0.560423\pi$
$587$	−9.14305	−0.377374	−0.188687	+	0.982037i	$0.560423\pi$
$588$	−35.1779	−1.45071
$589$	2.03446	0.0838286
$590$	38.1170	1.56925
$591$	−35.2314	−1.44923
$592$	92.8188	3.81483
$593$	−33.5590	−1.37810	−0.689051	−	0.724712i	$-0.741973\pi$
$593$	−33.5590	−1.37810	−0.689051	+	0.724712i	$0.741973\pi$
$594$	4.02276	0.165056
$595$	14.3604	0.588717
$596$	−58.3660	−2.39077
$597$	48.7911	1.99689
$598$	11.7981	0.482460
$599$	−13.6925	−0.559459	−0.279730	−	0.960079i	$-0.590245\pi$
$599$	−13.6925	−0.559459	−0.279730	+	0.960079i	$0.590245\pi$
$600$	52.0735	2.12589
$601$	−11.7517	−0.479363	−0.239682	−	0.970852i	$-0.577043\pi$
$601$	−11.7517	−0.479363	−0.239682	+	0.970852i	$0.577043\pi$
$602$	20.6711	0.842492
$603$	22.5430	0.918020
$604$	13.1331	0.534378
$605$	14.0156	0.569815
$606$	41.8733	1.70099
$607$	23.6603	0.960342	0.480171	−	0.877175i	$-0.340575\pi$
$607$	23.6603	0.960342	0.480171	+	0.877175i	$0.340575\pi$
$608$	−12.4095	−0.503270
$609$	−6.94179	−0.281296
$610$	−6.94057	−0.281015
$611$	0.823555	0.0333175
$612$	−58.9745	−2.38390
$613$	−20.0410	−0.809448	−0.404724	−	0.914439i	$-0.632632\pi$
$613$	−20.0410	−0.809448	−0.404724	+	0.914439i	$0.632632\pi$
$614$	17.0718	0.688962
$615$	24.6333	0.993309
$616$	−14.5986	−0.588194
$617$	1.00000	0.0402585
$617$	1.00000	0.0402585
$618$	−120.374	−4.84216
$619$	−48.3637	−1.94390	−0.971950	−	0.235187i	$-0.924430\pi$
$619$	−48.3637	−1.94390	−0.971950	+	0.235187i	$0.924430\pi$
$620$	−11.8364	−0.475363
$621$	−6.93103	−0.278133
$622$	59.5376	2.38724
$623$	8.74996	0.350560
$624$	−22.5174	−0.901418
$625$	−0.615950	−0.0246380
$626$	60.2567	2.40834
$627$	2.69668	0.107695
$628$	−67.1440	−2.67934
$629$	−49.6832	−1.98100
$630$	16.8042	0.669495
$631$	−2.56697	−0.102189	−0.0510947	−	0.998694i	$-0.516271\pi$
$631$	−2.56697	−0.102189	−0.0510947	+	0.998694i	$0.516271\pi$
$632$	62.6049	2.49029
$633$	21.8371	0.867946
$634$	6.12365	0.243201
$635$	−21.7431	−0.862848
$636$	85.9398	3.40773
$637$	3.14649	0.124669
$638$	4.00850	0.158698
$639$	14.0911	0.557434
$640$	0.709435	0.0280429
$641$	8.18530	0.323300	0.161650	−	0.986848i	$-0.448318\pi$
$641$	8.18530	0.323300	0.161650	+	0.986848i	$0.448318\pi$
$642$	−74.2010	−2.92848
$643$	−31.7918	−1.25375	−0.626874	−	0.779121i	$-0.715666\pi$
$643$	−31.7918	−1.25375	−0.626874	+	0.779121i	$0.715666\pi$
$644$	42.8631	1.68904
$645$	13.0363	0.513305
$646$	15.9182	0.626292
$647$	−27.1426	−1.06709	−0.533543	−	0.845773i	$-0.679140\pi$
$647$	−27.1426	−1.06709	−0.533543	+	0.845773i	$0.679140\pi$
$648$	78.3990	3.07980
$649$	10.4071	0.408513
$650$	−7.93730	−0.311326
$651$	7.90895	0.309976
$652$	59.1764	2.31753
$653$	−38.6136	−1.51107	−0.755533	−	0.655111i	$-0.772622\pi$
$653$	−38.6136	−1.51107	−0.755533	+	0.655111i	$0.772622\pi$
$654$	−120.082	−4.69558
$655$	−0.917686	−0.0358570
$656$	−74.1688	−2.89581
$657$	11.5475	0.450510
$658$	4.22827	0.164835
$659$	−31.7127	−1.23535	−0.617675	−	0.786433i	$-0.711925\pi$
$659$	−31.7127	−1.23535	−0.617675	+	0.786433i	$0.711925\pi$
$660$	−15.6892	−0.610701
$661$	−28.6927	−1.11602	−0.558009	−	0.829835i	$-0.688435\pi$
$661$	−28.6927	−1.11602	−0.558009	+	0.829835i	$0.688435\pi$
$662$	57.0289	2.21649
$663$	12.0529	0.468096
$664$	71.7248	2.78346
$665$	−3.20958	−0.124462
$666$	−58.1383	−2.25281
$667$	−6.90646	−0.267419
$668$	−82.3067	−3.18454
$669$	57.4630	2.22165
$670$	35.4007	1.36765
$671$	−1.89498	−0.0731549
$672$	−48.2416	−1.86096
$673$	28.2792	1.09008	0.545042	−	0.838409i	$-0.316514\pi$
$673$	28.2792	1.09008	0.545042	+	0.838409i	$0.316514\pi$
$674$	−40.2931	−1.55203
$675$	4.66292	0.179476
$676$	4.84044	0.186171
$677$	8.89393	0.341821	0.170911	−	0.985287i	$-0.445329\pi$
$677$	8.89393	0.341821	0.170911	+	0.985287i	$0.445329\pi$
$678$	3.28543	0.126176
$679$	20.2525	0.777221
$680$	−54.3457	−2.08406
$681$	−21.9561	−0.841359
$682$	−4.56699	−0.174879
$683$	26.5559	1.01613	0.508066	−	0.861318i	$-0.330361\pi$
$683$	26.5559	1.01613	0.508066	+	0.861318i	$0.330361\pi$
$684$	13.1810	0.503987
$685$	17.2401	0.658712
$686$	52.0938	1.98895
$687$	53.6278	2.04603
$688$	−39.2513	−1.49644
$689$	−7.68690	−0.292847
$690$	38.2009	1.45428
$691$	−8.15259	−0.310139	−0.155070	−	0.987904i	$-0.549560\pi$
$691$	−8.15259	−0.310139	−0.155070	+	0.987904i	$0.549560\pi$
$692$	−50.3433	−1.91376
$693$	4.58804	0.174285
$694$	−73.6244	−2.79474
$695$	−25.9874	−0.985760
$696$	26.2707	0.995789
$697$	39.7004	1.50376
$698$	−71.9703	−2.72412
$699$	60.3010	2.28079
$700$	−28.8366	−1.08992
$701$	−8.80177	−0.332438	−0.166219	−	0.986089i	$-0.553156\pi$
$701$	−8.80177	−0.332438	−0.166219	+	0.986089i	$0.553156\pi$
$702$	−4.01855	−0.151671
$703$	11.1043	0.418808
$704$	8.33846	0.314268
$705$	2.66657	0.100429
$706$	−81.8636	−3.08098
$707$	−13.6071	−0.511747
$708$	116.230	4.36817
$709$	39.2220	1.47301	0.736506	−	0.676431i	$-0.236474\pi$
$709$	39.2220	1.47301	0.736506	+	0.676431i	$0.236474\pi$
$710$	22.1281	0.830453
$711$	−19.6755	−0.737887
$712$	−33.1136	−1.24098
$713$	7.86870	0.294685
$714$	61.8817	2.31586
$715$	1.40332	0.0524813
$716$	−83.5634	−3.12291
$717$	−2.51530	−0.0939356
$718$	−91.6274	−3.41951
$719$	−13.6336	−0.508448	−0.254224	−	0.967145i	$-0.581820\pi$
$719$	−13.6336	−0.508448	−0.254224	+	0.967145i	$0.581820\pi$
$720$	−31.9086	−1.18916
$721$	39.1166	1.45678
$722$	46.1353	1.71698
$723$	49.5835	1.84403
$724$	112.285	4.17304
$725$	4.64640	0.172563
$726$	60.3961	2.24151
$727$	40.8477	1.51496	0.757478	−	0.652861i	$-0.226431\pi$
$727$	40.8477	1.51496	0.757478	+	0.652861i	$0.226431\pi$
$728$	14.5833	0.540494
$729$	−13.9927	−0.518249
$730$	18.1338	0.671161
$731$	21.0101	0.777087
$732$	−21.1638	−0.782236
$733$	48.2321	1.78150	0.890748	−	0.454498i	$-0.150181\pi$
$733$	48.2321	1.78150	0.890748	+	0.454498i	$0.150181\pi$
$734$	5.02462	0.185462
$735$	10.1880	0.375789
$736$	−47.9961	−1.76916
$737$	9.66542	0.356030
$738$	46.4566	1.71009
$739$	24.1195	0.887251	0.443626	−	0.896212i	$-0.353692\pi$
$739$	24.1195	0.887251	0.443626	+	0.896212i	$0.353692\pi$
$740$	−64.6046	−2.37491
$741$	−2.69386	−0.0989615
$742$	−39.4658	−1.44884
$743$	−24.2779	−0.890668	−0.445334	−	0.895364i	$-0.646915\pi$
$743$	−24.2779	−0.890668	−0.445334	+	0.895364i	$0.646915\pi$
$744$	−29.9309	−1.09732
$745$	16.9036	0.619298
$746$	61.6983	2.25894
$747$	−22.5416	−0.824755
$748$	−25.2856	−0.924535
$749$	24.1123	0.881043
$750$	−68.0422	−2.48455
$751$	46.3631	1.69181	0.845907	−	0.533331i	$-0.179060\pi$
$751$	46.3631	1.69181	0.845907	+	0.533331i	$0.179060\pi$
$752$	−8.02884	−0.292782
$753$	43.0210	1.56777
$754$	−4.00431	−0.145828
$755$	−3.80352	−0.138424
$756$	−14.5996	−0.530982
$757$	48.0795	1.74748	0.873740	−	0.486394i	$-0.161688\pi$
$757$	48.0795	1.74748	0.873740	+	0.486394i	$0.161688\pi$
$758$	82.8954	3.01090
$759$	10.4300	0.378584
$760$	12.1464	0.440597
$761$	−30.6173	−1.10988	−0.554938	−	0.831892i	$-0.687258\pi$
$761$	−30.6173	−1.10988	−0.554938	+	0.831892i	$0.687258\pi$
$762$	−93.6954	−3.39423
$763$	39.0217	1.41268
$764$	44.7083	1.61749
$765$	17.0798	0.617520
$766$	−1.86126	−0.0672501
$767$	−10.3962	−0.375384
$768$	−35.4215	−1.27816
$769$	−25.0953	−0.904959	−0.452479	−	0.891775i	$-0.649460\pi$
$769$	−25.0953	−0.904959	−0.452479	+	0.891775i	$0.649460\pi$
$770$	7.20490	0.259647
$771$	−9.92277	−0.357360
$772$	−53.1784	−1.91393
$773$	24.3609	0.876201	0.438100	−	0.898926i	$-0.355651\pi$
$773$	24.3609	0.876201	0.438100	+	0.898926i	$0.355651\pi$
$774$	24.5856	0.883711
$775$	−5.29375	−0.190157
$776$	−76.6443	−2.75137
$777$	43.1680	1.54864
$778$	32.9594	1.18165
$779$	−8.87315	−0.317914
$780$	15.6728	0.561176
$781$	6.04163	0.216186
$782$	61.5668	2.20162
$783$	2.35241	0.0840683
$784$	−30.6752	−1.09554
$785$	19.4458	0.694049
$786$	−3.95450	−0.141052
$787$	−37.8969	−1.35088	−0.675439	−	0.737416i	$-0.736046\pi$
$787$	−37.8969	−1.35088	−0.675439	+	0.737416i	$0.736046\pi$
$788$	−73.8341	−2.63023
$789$	−0.190297	−0.00677475
$790$	−30.8976	−1.09929
$791$	−1.06763	−0.0379605
$792$	−17.3631	−0.616972
$793$	1.89300	0.0672223
$794$	8.45311	0.299990
$795$	−24.8893	−0.882731
$796$	102.251	3.62419
$797$	7.15616	0.253484	0.126742	−	0.991936i	$-0.459548\pi$
$797$	7.15616	0.253484	0.126742	+	0.991936i	$0.459548\pi$
$798$	−13.8307	−0.489603
$799$	4.29760	0.152038
$800$	32.2899	1.14162
$801$	10.4069	0.367711
$802$	−49.0231	−1.73107
$803$	4.95105	0.174719
$804$	107.947	3.80699
$805$	−12.4137	−0.437525
$806$	4.56221	0.160697
$807$	15.8806	0.559023
$808$	51.4951	1.81159
$809$	22.6466	0.796214	0.398107	−	0.917339i	$-0.369667\pi$
$809$	22.6466	0.796214	0.398107	+	0.917339i	$0.369667\pi$
$810$	−38.6926	−1.35952
$811$	41.5697	1.45971	0.729855	−	0.683602i	$-0.239587\pi$
$811$	41.5697	1.45971	0.729855	+	0.683602i	$0.239587\pi$
$812$	−14.5478	−0.510529
$813$	21.9351	0.769296
$814$	−24.9271	−0.873696
$815$	−17.1382	−0.600327
$816$	−117.504	−4.11346
$817$	−4.69582	−0.164286
$818$	−82.2315	−2.87516
$819$	−4.58324	−0.160151
$820$	51.6237	1.80278
$821$	8.18126	0.285528	0.142764	−	0.989757i	$-0.454401\pi$
$821$	8.18126	0.285528	0.142764	+	0.989757i	$0.454401\pi$
$822$	74.2913	2.59121
$823$	41.8941	1.46034	0.730168	−	0.683268i	$-0.239442\pi$
$823$	41.8941	1.46034	0.730168	+	0.683268i	$0.239442\pi$
$824$	−148.034	−5.15701
$825$	−7.01687	−0.244296
$826$	−53.3757	−1.85718
$827$	44.7827	1.55725	0.778623	−	0.627492i	$-0.215918\pi$
$827$	44.7827	1.55725	0.778623	+	0.627492i	$0.215918\pi$
$828$	50.9800	1.77168
$829$	6.61798	0.229852	0.114926	−	0.993374i	$-0.463337\pi$
$829$	6.61798	0.229852	0.114926	+	0.993374i	$0.463337\pi$
$830$	−35.3986	−1.22870
$831$	−17.1582	−0.595211
$832$	−8.32974	−0.288782
$833$	16.4195	0.568903
$834$	−111.985	−3.87773
$835$	23.8371	0.824916
$836$	5.65141	0.195458
$837$	−2.68016	−0.0926398
$838$	20.8352	0.719739
$839$	15.5649	0.537362	0.268681	−	0.963229i	$-0.413412\pi$
$839$	15.5649	0.537362	0.268681	+	0.963229i	$0.413412\pi$
$840$	47.2191	1.62921
$841$	−26.6559	−0.919170
$842$	−32.2678	−1.11202
$843$	58.8635	2.02737
$844$	45.7638	1.57525
$845$	−1.40185	−0.0482253
$846$	5.02897	0.172900
$847$	−19.6262	−0.674365
$848$	74.9396	2.57344
$849$	−34.3717	−1.17963
$850$	−41.4197	−1.42068
$851$	42.9483	1.47225
$852$	67.4749	2.31165
$853$	−23.8013	−0.814939	−0.407470	−	0.913219i	$-0.633589\pi$
$853$	−23.8013	−0.814939	−0.407470	+	0.913219i	$0.633589\pi$
$854$	9.71897	0.332576
$855$	−3.81738	−0.130552
$856$	−91.2511	−3.11890
$857$	33.8190	1.15523	0.577617	−	0.816308i	$-0.303983\pi$
$857$	33.8190	1.15523	0.577617	+	0.816308i	$0.303983\pi$
$858$	6.04721	0.206448
$859$	−4.84130	−0.165183	−0.0825914	−	0.996583i	$-0.526320\pi$
$859$	−4.84130	−0.165183	−0.0825914	+	0.996583i	$0.526320\pi$
$860$	27.3201	0.931608
$861$	−34.4943	−1.17556
$862$	12.2643	0.417722
$863$	−50.8264	−1.73015	−0.865076	−	0.501641i	$-0.832730\pi$
$863$	−50.8264	−1.73015	−0.865076	+	0.501641i	$0.832730\pi$
$864$	16.3480	0.556169
$865$	14.5801	0.495737
$866$	75.4250	2.56305
$867$	23.6313	0.802562
$868$	16.5747	0.562582
$869$	−8.43597	−0.286171
$870$	−12.9655	−0.439571
$871$	−9.65531	−0.327158
$872$	−147.675	−5.00090
$873$	24.0877	0.815247
$874$	−13.7603	−0.465451
$875$	22.1109	0.747485
$876$	55.2950	1.86825
$877$	1.51534	0.0511692	0.0255846	−	0.999673i	$-0.491855\pi$
$877$	1.51534	0.0511692	0.0255846	+	0.999673i	$0.491855\pi$
$878$	−69.1678	−2.33430
$879$	60.1959	2.03036
$880$	−13.6810	−0.461187
$881$	53.8869	1.81549	0.907747	−	0.419517i	$-0.137801\pi$
$881$	53.8869	1.81549	0.907747	+	0.419517i	$0.137801\pi$
$882$	19.2138	0.646963
$883$	−44.8627	−1.50975	−0.754875	−	0.655868i	$-0.772303\pi$
$883$	−44.8627	−1.50975	−0.754875	+	0.655868i	$0.772303\pi$
$884$	25.2592	0.849558
$885$	−33.6616	−1.13152
$886$	67.4112	2.26472
$887$	5.00172	0.167941	0.0839706	−	0.996468i	$-0.473240\pi$
$887$	5.00172	0.167941	0.0839706	+	0.996468i	$0.473240\pi$
$888$	−163.366	−5.48221
$889$	30.4471	1.02116
$890$	16.3427	0.547808
$891$	−10.5642	−0.353914
$892$	120.425	4.03212
$893$	−0.960527	−0.0321428
$894$	72.8409	2.43616
$895$	24.2010	0.808952
$896$	−0.993430	−0.0331882
$897$	−10.4190	−0.347882
$898$	−64.0579	−2.13764
$899$	−2.67066	−0.0890715
$900$	−34.2973	−1.14324
$901$	−40.1130	−1.33636
$902$	19.9185	0.663215
$903$	−18.2549	−0.607486
$904$	4.04036	0.134380
$905$	−32.5192	−1.08097
$906$	−16.3901	−0.544525
$907$	43.7024	1.45112	0.725558	−	0.688161i	$-0.241582\pi$
$907$	43.7024	1.45112	0.725558	+	0.688161i	$0.241582\pi$
$908$	−46.0131	−1.52700
$909$	−16.1839	−0.536785
$910$	−7.19736	−0.238590
$911$	−3.67929	−0.121900	−0.0609502	−	0.998141i	$-0.519413\pi$
$911$	−3.67929	−0.121900	−0.0609502	+	0.998141i	$0.519413\pi$
$912$	26.2625	0.869638
$913$	−9.66486	−0.319860
$914$	70.2396	2.32332
$915$	6.12930	0.202628
$916$	112.387	3.71338
$917$	1.28505	0.0424360
$918$	−20.9703	−0.692122
$919$	20.3683	0.671887	0.335944	−	0.941882i	$-0.390945\pi$
$919$	20.3683	0.671887	0.335944	+	0.941882i	$0.390945\pi$
$920$	46.9787	1.54884
$921$	−15.0763	−0.496782
$922$	68.2719	2.24842
$923$	−6.03530	−0.198654
$924$	21.9698	0.722753
$925$	−28.8939	−0.950026
$926$	1.01377	0.0333146
$927$	46.5241	1.52805
$928$	16.2900	0.534746
$929$	48.6776	1.59706	0.798531	−	0.601954i	$-0.205611\pi$
$929$	48.6776	1.59706	0.798531	+	0.601954i	$0.205611\pi$
$930$	14.7719	0.484389
$931$	−3.66981	−0.120273
$932$	126.372	4.13946
$933$	−52.5784	−1.72134
$934$	−12.5898	−0.411951
$935$	7.32305	0.239489
$936$	17.3450	0.566937
$937$	−22.6422	−0.739687	−0.369844	−	0.929094i	$-0.620589\pi$
$937$	−22.6422	−0.739687	−0.369844	+	0.929094i	$0.620589\pi$
$938$	−49.5720	−1.61858
$939$	−53.2134	−1.73656
$940$	5.58831	0.182271
$941$	18.8079	0.613120	0.306560	−	0.951851i	$-0.400822\pi$
$941$	18.8079	0.613120	0.306560	+	0.951851i	$0.400822\pi$
$942$	83.7958	2.73021
$943$	−34.3187	−1.11757
$944$	101.352	3.29874
$945$	4.22823	0.137544
$946$	10.5412	0.342725
$947$	−26.9706	−0.876428	−0.438214	−	0.898871i	$-0.644389\pi$
$947$	−26.9706	−0.876428	−0.438214	+	0.898871i	$0.644389\pi$
$948$	−94.2157	−3.05998
$949$	−4.94587	−0.160550
$950$	9.25742	0.300350
$951$	−5.40787	−0.175362
$952$	76.1010	2.46645
$953$	−6.84168	−0.221624	−0.110812	−	0.993841i	$-0.535345\pi$
$953$	−6.84168	−0.221624	−0.110812	+	0.993841i	$0.535345\pi$
$954$	−46.9394	−1.51972
$955$	−12.9481	−0.418990
$956$	−5.27129	−0.170486
$957$	−3.53996	−0.114431
$958$	8.52454	0.275415
$959$	−24.1416	−0.779572
$960$	−26.9707	−0.870476
$961$	−27.9573	−0.901847
$962$	24.9011	0.802842
$963$	28.6784	0.924148
$964$	103.912	3.34677
$965$	15.4011	0.495780
$966$	−53.4932	−1.72111
$967$	−6.92513	−0.222697	−0.111349	−	0.993781i	$-0.535517\pi$
$967$	−6.92513	−0.222697	−0.111349	+	0.993781i	$0.535517\pi$
$968$	74.2741	2.38726
$969$	−14.0575	−0.451593
$970$	37.8265	1.21454
$971$	48.0472	1.54191	0.770954	−	0.636891i	$-0.219780\pi$
$971$	48.0472	1.54191	0.770954	+	0.636891i	$0.219780\pi$
$972$	−95.6729	−3.06871
$973$	36.3905	1.16663
$974$	89.8407	2.87868
$975$	7.00952	0.224484
$976$	−18.4548	−0.590725
$977$	54.3492	1.73878	0.869392	−	0.494123i	$-0.164511\pi$
$977$	54.3492	1.73878	0.869392	+	0.494123i	$0.164511\pi$
$978$	−73.8522	−2.36153
$979$	4.46203	0.142607
$980$	21.3508	0.682028
$981$	46.4112	1.48180
$982$	−41.0738	−1.31072
$983$	13.1534	0.419528	0.209764	−	0.977752i	$-0.432730\pi$
$983$	13.1534	0.419528	0.209764	+	0.977752i	$0.432730\pi$
$984$	130.541	4.16150
$985$	21.3833	0.681329
$986$	−20.8959	−0.665462
$987$	−3.73404	−0.118856
$988$	−5.64550	−0.179607
$989$	−18.1620	−0.577519
$990$	8.56928	0.272350
$991$	47.5110	1.50924	0.754618	−	0.656165i	$-0.227822\pi$
$991$	47.5110	1.50924	0.754618	+	0.656165i	$0.227822\pi$
$992$	−18.5596	−0.589268
$993$	−50.3630	−1.59822
$994$	−30.9863	−0.982825
$995$	−29.6132	−0.938802
$996$	−107.940	−3.42022
$997$	−35.7714	−1.13289	−0.566446	−	0.824099i	$-0.691682\pi$
$997$	−35.7714	−1.13289	−0.566446	+	0.824099i	$0.691682\pi$
$998$	34.2900	1.08543
$999$	−14.6286	−0.462829

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8021.2.a.c.1.8	✓	169

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8021.2.a.c.1.8	✓	169	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8021 = 13 \cdot 617 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8021.a (trivial)

\(f(q)\)	\(=\)	\(q-2.61542 q^{2} +2.30971 q^{3} +4.84044 q^{4} -1.40185 q^{5} -6.04088 q^{6} +1.96303 q^{7} -7.42897 q^{8} +2.33477 q^{9} +O(q^{10})\)	\(q-2.61542 q^{2} +2.30971 q^{3} +4.84044 q^{4} -1.40185 q^{5} -6.04088 q^{6} +1.96303 q^{7} -7.42897 q^{8} +2.33477 q^{9} +3.66644 q^{10} +1.00105 q^{11} +11.1800 q^{12} -1.00000 q^{13} -5.13417 q^{14} -3.23788 q^{15} +9.74901 q^{16} -5.21836 q^{17} -6.10642 q^{18} +1.16632 q^{19} -6.78560 q^{20} +4.53405 q^{21} -2.61816 q^{22} +4.51097 q^{23} -17.1588 q^{24} -3.03480 q^{25} +2.61542 q^{26} -1.53648 q^{27} +9.50196 q^{28} -1.53104 q^{29} +8.46843 q^{30} +1.74435 q^{31} -10.6399 q^{32} +2.31213 q^{33} +13.6482 q^{34} -2.75189 q^{35} +11.3013 q^{36} +9.52085 q^{37} -3.05042 q^{38} -2.30971 q^{39} +10.4143 q^{40} -7.60783 q^{41} -11.8585 q^{42} -4.02619 q^{43} +4.84551 q^{44} -3.27301 q^{45} -11.7981 q^{46} -0.823555 q^{47} +22.5174 q^{48} -3.14649 q^{49} +7.93730 q^{50} -12.0529 q^{51} -4.84044 q^{52} +7.68690 q^{53} +4.01855 q^{54} -1.40332 q^{55} -14.5833 q^{56} +2.69386 q^{57} +4.00431 q^{58} +10.3962 q^{59} -15.6728 q^{60} -1.89300 q^{61} -4.56221 q^{62} +4.58324 q^{63} +8.32974 q^{64} +1.40185 q^{65} -6.04721 q^{66} +9.65531 q^{67} -25.2592 q^{68} +10.4190 q^{69} +7.19736 q^{70} +6.03530 q^{71} -17.3450 q^{72} +4.94587 q^{73} -24.9011 q^{74} -7.00952 q^{75} +5.64550 q^{76} +1.96509 q^{77} +6.04088 q^{78} -8.42714 q^{79} -13.6667 q^{80} -10.5532 q^{81} +19.8977 q^{82} -9.65474 q^{83} +21.9468 q^{84} +7.31538 q^{85} +10.5302 q^{86} -3.53625 q^{87} -7.43675 q^{88} +4.45736 q^{89} +8.56032 q^{90} -1.96303 q^{91} +21.8351 q^{92} +4.02894 q^{93} +2.15394 q^{94} -1.63501 q^{95} -24.5750 q^{96} +10.3170 q^{97} +8.22942 q^{98} +2.33722 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 169 q + 9 q^{2} + 9 q^{3} + 199 q^{4} + 12 q^{5} + 22 q^{6} + 36 q^{7} + 30 q^{8} + 198 q^{9}+O(q^{10}) \) 169 * q + 9 * q^2 + 9 * q^3 + 199 * q^4 + 12 * q^5 + 22 * q^6 + 36 * q^7 + 30 * q^8 + 198 * q^9	\( 169 q + 9 q^{2} + 9 q^{3} + 199 q^{4} + 12 q^{5} + 22 q^{6} + 36 q^{7} + 30 q^{8} + 198 q^{9} - 3 q^{10} + 59 q^{11} + 11 q^{12} - 169 q^{13} + 30 q^{14} + 50 q^{15} + 267 q^{16} + q^{17} + 53 q^{18} + 107 q^{19} + 48 q^{20} + 36 q^{21} + 14 q^{22} + 12 q^{23} + 78 q^{24} + 217 q^{25} - 9 q^{26} + 39 q^{27} + 99 q^{28} + 30 q^{29} - q^{30} + 106 q^{31} + 74 q^{32} + 16 q^{33} + 56 q^{34} + 46 q^{35} + 271 q^{36} + 73 q^{37} + 2 q^{38} - 9 q^{39} - 16 q^{40} + 52 q^{41} - 2 q^{42} + 64 q^{43} + 124 q^{44} + 84 q^{45} + 105 q^{46} + 55 q^{47} + 26 q^{48} + 257 q^{49} + 60 q^{50} + 117 q^{51} - 199 q^{52} + 7 q^{53} + 78 q^{54} - 4 q^{55} + 63 q^{56} + 51 q^{57} + 84 q^{58} + 98 q^{59} + 94 q^{60} + 32 q^{61} - 25 q^{62} + 128 q^{63} + 380 q^{64} - 12 q^{65} + 16 q^{66} + 170 q^{67} - 10 q^{68} + 55 q^{69} + 70 q^{70} + 124 q^{71} + 173 q^{72} + 81 q^{73} + 54 q^{74} + 120 q^{75} + 212 q^{76} + 20 q^{77} - 22 q^{78} + 92 q^{79} + 66 q^{80} + 265 q^{81} + 21 q^{82} + 62 q^{83} + 98 q^{84} + 139 q^{85} + 51 q^{86} - 33 q^{87} + 31 q^{88} + 58 q^{89} + 16 q^{90} - 36 q^{91} + 40 q^{92} + 37 q^{93} + 55 q^{94} + 23 q^{95} + 164 q^{96} + 78 q^{97} + 69 q^{98} + 307 q^{99}+O(q^{100}) \) 169 * q + 9 * q^2 + 9 * q^3 + 199 * q^4 + 12 * q^5 + 22 * q^6 + 36 * q^7 + 30 * q^8 + 198 * q^9 - 3 * q^10 + 59 * q^11 + 11 * q^12 - 169 * q^13 + 30 * q^14 + 50 * q^15 + 267 * q^16 + q^17 + 53 * q^18 + 107 * q^19 + 48 * q^20 + 36 * q^21 + 14 * q^22 + 12 * q^23 + 78 * q^24 + 217 * q^25 - 9 * q^26 + 39 * q^27 + 99 * q^28 + 30 * q^29 - q^30 + 106 * q^31 + 74 * q^32 + 16 * q^33 + 56 * q^34 + 46 * q^35 + 271 * q^36 + 73 * q^37 + 2 * q^38 - 9 * q^39 - 16 * q^40 + 52 * q^41 - 2 * q^42 + 64 * q^43 + 124 * q^44 + 84 * q^45 + 105 * q^46 + 55 * q^47 + 26 * q^48 + 257 * q^49 + 60 * q^50 + 117 * q^51 - 199 * q^52 + 7 * q^53 + 78 * q^54 - 4 * q^55 + 63 * q^56 + 51 * q^57 + 84 * q^58 + 98 * q^59 + 94 * q^60 + 32 * q^61 - 25 * q^62 + 128 * q^63 + 380 * q^64 - 12 * q^65 + 16 * q^66 + 170 * q^67 - 10 * q^68 + 55 * q^69 + 70 * q^70 + 124 * q^71 + 173 * q^72 + 81 * q^73 + 54 * q^74 + 120 * q^75 + 212 * q^76 + 20 * q^77 - 22 * q^78 + 92 * q^79 + 66 * q^80 + 265 * q^81 + 21 * q^82 + 62 * q^83 + 98 * q^84 + 139 * q^85 + 51 * q^86 - 33 * q^87 + 31 * q^88 + 58 * q^89 + 16 * q^90 - 36 * q^91 + 40 * q^92 + 37 * q^93 + 55 * q^94 + 23 * q^95 + 164 * q^96 + 78 * q^97 + 69 * q^98 + 307 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more