LMFDB - Embedded newform 4030.2.a.c.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4030,2,Mod(1,4030)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4030, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4030.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4030 = 2 \cdot 5 \cdot 13 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4030.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:	$32.1797120146$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.3081125.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 8x^{4} + 6x^{3} + 16x^{2} - 10x - 5 $ x^6 - x^5 - 8x^4 + 6x^3 + 16x^2 - 10x - 5
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.44295$ of defining polynomial
Character	$\chi$	$=$	4030.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.13283 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.13283 q^{6} -1.36085 q^{7} -1.00000 q^{8} -1.71670 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.13283 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.13283 q^{6} -1.36085 q^{7} -1.00000 q^{8} -1.71670 q^{9} +1.00000 q^{10} -4.03486 q^{11} -1.13283 q^{12} +1.00000 q^{13} +1.36085 q^{14} +1.13283 q^{15} +1.00000 q^{16} -0.934050 q^{17} +1.71670 q^{18} +4.43798 q^{19} -1.00000 q^{20} +1.54161 q^{21} +4.03486 q^{22} +0.0668778 q^{23} +1.13283 q^{24} +1.00000 q^{25} -1.00000 q^{26} +5.34321 q^{27} -1.36085 q^{28} +3.64335 q^{29} -1.13283 q^{30} +1.00000 q^{31} -1.00000 q^{32} +4.57080 q^{33} +0.934050 q^{34} +1.36085 q^{35} -1.71670 q^{36} +3.52570 q^{37} -4.43798 q^{38} -1.13283 q^{39} +1.00000 q^{40} +6.16029 q^{41} -1.54161 q^{42} +10.9140 q^{43} -4.03486 q^{44} +1.71670 q^{45} -0.0668778 q^{46} +4.68512 q^{47} -1.13283 q^{48} -5.14808 q^{49} -1.00000 q^{50} +1.05812 q^{51} +1.00000 q^{52} -2.16268 q^{53} -5.34321 q^{54} +4.03486 q^{55} +1.36085 q^{56} -5.02746 q^{57} -3.64335 q^{58} -5.66877 q^{59} +1.13283 q^{60} -7.38270 q^{61} -1.00000 q^{62} +2.33618 q^{63} +1.00000 q^{64} -1.00000 q^{65} -4.57080 q^{66} -7.18954 q^{67} -0.934050 q^{68} -0.0757610 q^{69} -1.36085 q^{70} -5.63632 q^{71} +1.71670 q^{72} +10.3728 q^{73} -3.52570 q^{74} -1.13283 q^{75} +4.43798 q^{76} +5.49086 q^{77} +1.13283 q^{78} -2.73521 q^{79} -1.00000 q^{80} -0.902834 q^{81} -6.16029 q^{82} +6.85404 q^{83} +1.54161 q^{84} +0.934050 q^{85} -10.9140 q^{86} -4.12729 q^{87} +4.03486 q^{88} -3.44321 q^{89} -1.71670 q^{90} -1.36085 q^{91} +0.0668778 q^{92} -1.13283 q^{93} -4.68512 q^{94} -4.43798 q^{95} +1.13283 q^{96} -0.160657 q^{97} +5.14808 q^{98} +6.92665 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{2} - q^{3} + 6 q^{4} - 6 q^{5} + q^{6} + 4 q^{7} - 6 q^{8} - q^{9}+O(q^{10}) $ 6 * q - 6 * q^2 - q^3 + 6 * q^4 - 6 * q^5 + q^6 + 4 * q^7 - 6 * q^8 - q^9	\( 6 q - 6 q^{2} - q^{3} + 6 q^{4} - 6 q^{5} + q^{6} + 4 q^{7} - 6 q^{8} - q^{9} + 6 q^{10} - 6 q^{11} - q^{12} + 6 q^{13} - 4 q^{14} + q^{15} + 6 q^{16} - 4 q^{17} + q^{18} + 3 q^{19} - 6 q^{20} - q^{21} + 6 q^{22} - 7 q^{23} + q^{24} + 6 q^{25} - 6 q^{26} - q^{27} + 4 q^{28} - 14 q^{29} - q^{30} + 6 q^{31} - 6 q^{32} - 2 q^{33} + 4 q^{34} - 4 q^{35} - q^{36} + 14 q^{37} - 3 q^{38} - q^{39} + 6 q^{40} - 12 q^{41} + q^{42} + 3 q^{43} - 6 q^{44} + q^{45} + 7 q^{46} - 2 q^{47} - q^{48} - 6 q^{49} - 6 q^{50} - 7 q^{51} + 6 q^{52} - 4 q^{53} + q^{54} + 6 q^{55} - 4 q^{56} + 13 q^{57} + 14 q^{58} - 17 q^{59} + q^{60} - 5 q^{61} - 6 q^{62} - q^{63} + 6 q^{64} - 6 q^{65} + 2 q^{66} + 8 q^{67} - 4 q^{68} - 8 q^{69} + 4 q^{70} - 16 q^{71} + q^{72} + 11 q^{73} - 14 q^{74} - q^{75} + 3 q^{76} - 15 q^{77} + q^{78} - 2 q^{79} - 6 q^{80} - 26 q^{81} + 12 q^{82} + 4 q^{83} - q^{84} + 4 q^{85} - 3 q^{86} + 7 q^{87} + 6 q^{88} - 8 q^{89} - q^{90} + 4 q^{91} - 7 q^{92} - q^{93} + 2 q^{94} - 3 q^{95} + q^{96} + 5 q^{97} + 6 q^{98} + 15 q^{99}+O(q^{100}) \) 6 * q - 6 * q^2 - q^3 + 6 * q^4 - 6 * q^5 + q^6 + 4 * q^7 - 6 * q^8 - q^9 + 6 * q^10 - 6 * q^11 - q^12 + 6 * q^13 - 4 * q^14 + q^15 + 6 * q^16 - 4 * q^17 + q^18 + 3 * q^19 - 6 * q^20 - q^21 + 6 * q^22 - 7 * q^23 + q^24 + 6 * q^25 - 6 * q^26 - q^27 + 4 * q^28 - 14 * q^29 - q^30 + 6 * q^31 - 6 * q^32 - 2 * q^33 + 4 * q^34 - 4 * q^35 - q^36 + 14 * q^37 - 3 * q^38 - q^39 + 6 * q^40 - 12 * q^41 + q^42 + 3 * q^43 - 6 * q^44 + q^45 + 7 * q^46 - 2 * q^47 - q^48 - 6 * q^49 - 6 * q^50 - 7 * q^51 + 6 * q^52 - 4 * q^53 + q^54 + 6 * q^55 - 4 * q^56 + 13 * q^57 + 14 * q^58 - 17 * q^59 + q^60 - 5 * q^61 - 6 * q^62 - q^63 + 6 * q^64 - 6 * q^65 + 2 * q^66 + 8 * q^67 - 4 * q^68 - 8 * q^69 + 4 * q^70 - 16 * q^71 + q^72 + 11 * q^73 - 14 * q^74 - q^75 + 3 * q^76 - 15 * q^77 + q^78 - 2 * q^79 - 6 * q^80 - 26 * q^81 + 12 * q^82 + 4 * q^83 - q^84 + 4 * q^85 - 3 * q^86 + 7 * q^87 + 6 * q^88 - 8 * q^89 - q^90 + 4 * q^91 - 7 * q^92 - q^93 + 2 * q^94 - 3 * q^95 + q^96 + 5 * q^97 + 6 * q^98 + 15 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.13283	−0.654038	−0.327019	−	0.945018i	$-0.606044\pi$
$3$	−1.13283	−0.654038	−0.327019	+	0.945018i	$0.606044\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	1.13283	0.462475
$7$	−1.36085	−0.514354	−0.257177	−	0.966364i	$-0.582792\pi$
$7$	−1.36085	−0.514354	−0.257177	+	0.966364i	$0.582792\pi$
$8$	−1.00000	−0.353553
$9$	−1.71670	−0.572234
$10$	1.00000	0.316228
$11$	−4.03486	−1.21656	−0.608278	−	0.793724i	$-0.708140\pi$
$11$	−4.03486	−1.21656	−0.608278	+	0.793724i	$0.708140\pi$
$12$	−1.13283	−0.327019
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	1.36085	0.363703
$15$	1.13283	0.292495
$16$	1.00000	0.250000
$17$	−0.934050	−0.226540	−0.113270	−	0.993564i	$-0.536133\pi$
$17$	−0.934050	−0.226540	−0.113270	+	0.993564i	$0.536133\pi$
$18$	1.71670	0.404630
$19$	4.43798	1.01814	0.509071	−	0.860725i	$-0.329989\pi$
$19$	4.43798	1.01814	0.509071	+	0.860725i	$0.329989\pi$
$20$	−1.00000	−0.223607
$21$	1.54161	0.336407
$22$	4.03486	0.860235
$23$	0.0668778	0.0139450	0.00697249	−	0.999976i	$-0.497781\pi$
$23$	0.0668778	0.0139450	0.00697249	+	0.999976i	$0.497781\pi$
$24$	1.13283	0.231238
$25$	1.00000	0.200000
$26$	−1.00000	−0.196116
$27$	5.34321	1.02830
$28$	−1.36085	−0.257177
$29$	3.64335	0.676553	0.338277	−	0.941047i	$-0.390156\pi$
$29$	3.64335	0.676553	0.338277	+	0.941047i	$0.390156\pi$
$30$	−1.13283	−0.206825
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	−1.00000	−0.176777
$33$	4.57080	0.795675
$34$	0.934050	0.160188
$35$	1.36085	0.230026
$36$	−1.71670	−0.286117
$37$	3.52570	0.579621	0.289811	−	0.957084i	$-0.406408\pi$
$37$	3.52570	0.579621	0.289811	+	0.957084i	$0.406408\pi$
$38$	−4.43798	−0.719935
$39$	−1.13283	−0.181398
$40$	1.00000	0.158114
$41$	6.16029	0.962076	0.481038	−	0.876700i	$-0.340260\pi$
$41$	6.16029	0.962076	0.481038	+	0.876700i	$0.340260\pi$
$42$	−1.54161	−0.237876
$43$	10.9140	1.66437	0.832186	−	0.554497i	$-0.187089\pi$
$43$	10.9140	1.66437	0.832186	+	0.554497i	$0.187089\pi$
$44$	−4.03486	−0.608278
$45$	1.71670	0.255911
$46$	−0.0668778	−0.00986059
$47$	4.68512	0.683395	0.341697	−	0.939810i	$-0.388998\pi$
$47$	4.68512	0.683395	0.341697	+	0.939810i	$0.388998\pi$
$48$	−1.13283	−0.163510
$49$	−5.14808	−0.735440
$50$	−1.00000	−0.141421
$51$	1.05812	0.148166
$52$	1.00000	0.138675
$53$	−2.16268	−0.297067	−0.148534	−	0.988907i	$-0.547455\pi$
$53$	−2.16268	−0.297067	−0.148534	+	0.988907i	$0.547455\pi$
$54$	−5.34321	−0.727119
$55$	4.03486	0.544061
$56$	1.36085	0.181852
$57$	−5.02746	−0.665904
$58$	−3.64335	−0.478396
$59$	−5.66877	−0.738011	−0.369005	−	0.929427i	$-0.620302\pi$
$59$	−5.66877	−0.738011	−0.369005	+	0.929427i	$0.620302\pi$
$60$	1.13283	0.146247
$61$	−7.38270	−0.945258	−0.472629	−	0.881261i	$-0.656695\pi$
$61$	−7.38270	−0.945258	−0.472629	+	0.881261i	$0.656695\pi$
$62$	−1.00000	−0.127000
$63$	2.33618	0.294331
$64$	1.00000	0.125000
$65$	−1.00000	−0.124035
$66$	−4.57080	−0.562627
$67$	−7.18954	−0.878342	−0.439171	−	0.898404i	$-0.644728\pi$
$67$	−7.18954	−0.878342	−0.439171	+	0.898404i	$0.644728\pi$
$68$	−0.934050	−0.113270
$69$	−0.0757610	−0.00912056
$70$	−1.36085	−0.162653
$71$	−5.63632	−0.668908	−0.334454	−	0.942412i	$-0.608552\pi$
$71$	−5.63632	−0.668908	−0.334454	+	0.942412i	$0.608552\pi$
$72$	1.71670	0.202315
$73$	10.3728	1.21405	0.607025	−	0.794683i	$-0.292363\pi$
$73$	10.3728	1.21405	0.607025	+	0.794683i	$0.292363\pi$
$74$	−3.52570	−0.409854
$75$	−1.13283	−0.130808
$76$	4.43798	0.509071
$77$	5.49086	0.625741
$78$	1.13283	0.128267
$79$	−2.73521	−0.307736	−0.153868	−	0.988091i	$-0.549173\pi$
$79$	−2.73521	−0.307736	−0.153868	+	0.988091i	$0.549173\pi$
$80$	−1.00000	−0.111803
$81$	−0.902834	−0.100315
$82$	−6.16029	−0.680290
$83$	6.85404	0.752329	0.376165	−	0.926553i	$-0.377243\pi$
$83$	6.85404	0.752329	0.376165	+	0.926553i	$0.377243\pi$
$84$	1.54161	0.168204
$85$	0.934050	0.101312
$86$	−10.9140	−1.17689
$87$	−4.12729	−0.442492
$88$	4.03486	0.430118
$89$	−3.44321	−0.364980	−0.182490	−	0.983208i	$-0.558416\pi$
$89$	−3.44321	−0.364980	−0.182490	+	0.983208i	$0.558416\pi$
$90$	−1.71670	−0.180956
$91$	−1.36085	−0.142656
$92$	0.0668778	0.00697249
$93$	−1.13283	−0.117469
$94$	−4.68512	−0.483233
$95$	−4.43798	−0.455327
$96$	1.13283	0.115619
$97$	−0.160657	−0.0163123	−0.00815615	−	0.999967i	$-0.502596\pi$
$97$	−0.160657	−0.0163123	−0.00815615	+	0.999967i	$0.502596\pi$
$98$	5.14808	0.520034
$99$	6.92665	0.696155
$100$	1.00000	0.100000
$101$	−17.8570	−1.77684	−0.888420	−	0.459032i	$-0.848196\pi$
$101$	−17.8570	−1.77684	−0.888420	+	0.459032i	$0.848196\pi$
$102$	−1.05812	−0.104769
$103$	−0.858733	−0.0846134	−0.0423067	−	0.999105i	$-0.513471\pi$
$103$	−0.858733	−0.0846134	−0.0423067	+	0.999105i	$0.513471\pi$
$104$	−1.00000	−0.0980581
$105$	−1.54161	−0.150446
$106$	2.16268	0.210058
$107$	−6.79606	−0.657000	−0.328500	−	0.944504i	$-0.606543\pi$
$107$	−6.79606	−0.657000	−0.328500	+	0.944504i	$0.606543\pi$
$108$	5.34321	0.514151
$109$	10.2029	0.977264	0.488632	−	0.872490i	$-0.337496\pi$
$109$	10.2029	0.977264	0.488632	+	0.872490i	$0.337496\pi$
$110$	−4.03486	−0.384709
$111$	−3.99401	−0.379095
$112$	−1.36085	−0.128589
$113$	−19.5945	−1.84330	−0.921648	−	0.388026i	$-0.873157\pi$
$113$	−19.5945	−1.84330	−0.921648	+	0.388026i	$0.873157\pi$
$114$	5.02746	0.470865
$115$	−0.0668778	−0.00623639
$116$	3.64335	0.338277
$117$	−1.71670	−0.158709
$118$	5.66877	0.521853
$119$	1.27111	0.116522
$120$	−1.13283	−0.103413
$121$	5.28010	0.480010
$122$	7.38270	0.668398
$123$	−6.97855	−0.629234
$124$	1.00000	0.0898027
$125$	−1.00000	−0.0894427
$126$	−2.33618	−0.208123
$127$	19.8002	1.75698	0.878490	−	0.477761i	$-0.158552\pi$
$127$	19.8002	1.75698	0.878490	+	0.477761i	$0.158552\pi$
$128$	−1.00000	−0.0883883
$129$	−12.3637	−1.08856
$130$	1.00000	0.0877058
$131$	−6.47048	−0.565329	−0.282664	−	0.959219i	$-0.591218\pi$
$131$	−6.47048	−0.565329	−0.282664	+	0.959219i	$0.591218\pi$
$132$	4.57080	0.397837
$133$	−6.03943	−0.523685
$134$	7.18954	0.621081
$135$	−5.34321	−0.459870
$136$	0.934050	0.0800941
$137$	2.94609	0.251701	0.125851	−	0.992049i	$-0.459834\pi$
$137$	2.94609	0.251701	0.125851	+	0.992049i	$0.459834\pi$
$138$	0.0757610	0.00644921
$139$	−14.3501	−1.21716	−0.608581	−	0.793491i	$-0.708261\pi$
$139$	−14.3501	−1.21716	−0.608581	+	0.793491i	$0.708261\pi$
$140$	1.36085	0.115013
$141$	−5.30743	−0.446966
$142$	5.63632	0.472989
$143$	−4.03486	−0.337412
$144$	−1.71670	−0.143058
$145$	−3.64335	−0.302564
$146$	−10.3728	−0.858463
$147$	5.83189	0.481006
$148$	3.52570	0.289811
$149$	0.0405321	0.00332052	0.00166026	−	0.999999i	$-0.499472\pi$
$149$	0.0405321	0.00332052	0.00166026	+	0.999999i	$0.499472\pi$
$150$	1.13283	0.0924950
$151$	−5.19280	−0.422584	−0.211292	−	0.977423i	$-0.567767\pi$
$151$	−5.19280	−0.422584	−0.211292	+	0.977423i	$0.567767\pi$
$152$	−4.43798	−0.359967
$153$	1.60348	0.129634
$154$	−5.49086	−0.442466
$155$	−1.00000	−0.0803219
$156$	−1.13283	−0.0906988
$157$	−10.6477	−0.849778	−0.424889	−	0.905245i	$-0.639687\pi$
$157$	−10.6477	−0.849778	−0.424889	+	0.905245i	$0.639687\pi$
$158$	2.73521	0.217602
$159$	2.44995	0.194293
$160$	1.00000	0.0790569
$161$	−0.0910109	−0.00717266
$162$	0.902834	0.0709333
$163$	−12.6497	−0.990798	−0.495399	−	0.868666i	$-0.664978\pi$
$163$	−12.6497	−0.990798	−0.495399	+	0.868666i	$0.664978\pi$
$164$	6.16029	0.481038
$165$	−4.57080	−0.355837
$166$	−6.85404	−0.531977
$167$	15.7124	1.21586	0.607929	−	0.793991i	$-0.292000\pi$
$167$	15.7124	1.21586	0.607929	+	0.793991i	$0.292000\pi$
$168$	−1.54161	−0.118938
$169$	1.00000	0.0769231
$170$	−0.934050	−0.0716384
$171$	−7.61868	−0.582615
$172$	10.9140	0.832186
$173$	−6.86527	−0.521957	−0.260978	−	0.965345i	$-0.584045\pi$
$173$	−6.86527	−0.521957	−0.260978	+	0.965345i	$0.584045\pi$
$174$	4.12729	0.312889
$175$	−1.36085	−0.102871
$176$	−4.03486	−0.304139
$177$	6.42174	0.482688
$178$	3.44321	0.258080
$179$	−5.15978	−0.385660	−0.192830	−	0.981232i	$-0.561767\pi$
$179$	−5.15978	−0.385660	−0.192830	+	0.981232i	$0.561767\pi$
$180$	1.71670	0.127955
$181$	7.84176	0.582874	0.291437	−	0.956590i	$-0.405867\pi$
$181$	7.84176	0.582874	0.291437	+	0.956590i	$0.405867\pi$
$182$	1.36085	0.100873
$183$	8.36333	0.618235
$184$	−0.0668778	−0.00493030
$185$	−3.52570	−0.259214
$186$	1.13283	0.0830630
$187$	3.76876	0.275599
$188$	4.68512	0.341697
$189$	−7.27133	−0.528911
$190$	4.43798	0.321965
$191$	21.1349	1.52927	0.764635	−	0.644464i	$-0.222919\pi$
$191$	21.1349	1.52927	0.764635	+	0.644464i	$0.222919\pi$
$192$	−1.13283	−0.0817548
$193$	8.05818	0.580041	0.290020	−	0.957020i	$-0.406338\pi$
$193$	8.05818	0.580041	0.290020	+	0.957020i	$0.406338\pi$
$194$	0.160657	0.0115345
$195$	1.13283	0.0811235
$196$	−5.14808	−0.367720
$197$	−14.3984	−1.02584	−0.512921	−	0.858436i	$-0.671437\pi$
$197$	−14.3984	−1.02584	−0.512921	+	0.858436i	$0.671437\pi$
$198$	−6.92665	−0.492256
$199$	17.7013	1.25481	0.627404	−	0.778694i	$-0.284117\pi$
$199$	17.7013	1.25481	0.627404	+	0.778694i	$0.284117\pi$
$200$	−1.00000	−0.0707107
$201$	8.14451	0.574469
$202$	17.8570	1.25641
$203$	−4.95807	−0.347988
$204$	1.05812	0.0740831
$205$	−6.16029	−0.430253
$206$	0.858733	0.0598307
$207$	−0.114809	−0.00797979
$208$	1.00000	0.0693375
$209$	−17.9066	−1.23863
$210$	1.54161	0.106381
$211$	27.9009	1.92078	0.960389	−	0.278664i	$-0.0898915\pi$
$211$	27.9009	1.92078	0.960389	+	0.278664i	$0.0898915\pi$
$212$	−2.16268	−0.148534
$213$	6.38498	0.437492
$214$	6.79606	0.464569
$215$	−10.9140	−0.744330
$216$	−5.34321	−0.363559
$217$	−1.36085	−0.0923808
$218$	−10.2029	−0.691030
$219$	−11.7506	−0.794035
$220$	4.03486	0.272030
$221$	−0.934050	−0.0628310
$222$	3.99401	0.268060
$223$	−0.983455	−0.0658570	−0.0329285	−	0.999458i	$-0.510483\pi$
$223$	−0.983455	−0.0658570	−0.0329285	+	0.999458i	$0.510483\pi$
$224$	1.36085	0.0909259
$225$	−1.71670	−0.114447
$226$	19.5945	1.30341
$227$	21.8640	1.45116	0.725582	−	0.688135i	$-0.241571\pi$
$227$	21.8640	1.45116	0.725582	+	0.688135i	$0.241571\pi$
$228$	−5.02746	−0.332952
$229$	−10.7272	−0.708874	−0.354437	−	0.935080i	$-0.615328\pi$
$229$	−10.7272	−0.708874	−0.354437	+	0.935080i	$0.615328\pi$
$230$	0.0668778	0.00440979
$231$	−6.22019	−0.409259
$232$	−3.64335	−0.239198
$233$	3.97892	0.260668	0.130334	−	0.991470i	$-0.458395\pi$
$233$	3.97892	0.260668	0.130334	+	0.991470i	$0.458395\pi$
$234$	1.71670	0.112224
$235$	−4.68512	−0.305623
$236$	−5.66877	−0.369005
$237$	3.09853	0.201271
$238$	−1.27111	−0.0823935
$239$	−4.63269	−0.299664	−0.149832	−	0.988712i	$-0.547873\pi$
$239$	−4.63269	−0.299664	−0.149832	+	0.988712i	$0.547873\pi$
$240$	1.13283	0.0731237
$241$	−8.73533	−0.562692	−0.281346	−	0.959606i	$-0.590781\pi$
$241$	−8.73533	−0.562692	−0.281346	+	0.959606i	$0.590781\pi$
$242$	−5.28010	−0.339418
$243$	−15.0069	−0.962692
$244$	−7.38270	−0.472629
$245$	5.14808	0.328899
$246$	6.97855	0.444936
$247$	4.43798	0.282382
$248$	−1.00000	−0.0635001
$249$	−7.76445	−0.492052
$250$	1.00000	0.0632456
$251$	−14.6179	−0.922676	−0.461338	−	0.887224i	$-0.652631\pi$
$251$	−14.6179	−0.922676	−0.461338	+	0.887224i	$0.652631\pi$
$252$	2.33618	0.147165
$253$	−0.269843	−0.0169649
$254$	−19.8002	−1.24237
$255$	−1.05812	−0.0662619
$256$	1.00000	0.0625000
$257$	0.399855	0.0249423	0.0124711	−	0.999922i	$-0.496030\pi$
$257$	0.399855	0.0249423	0.0124711	+	0.999922i	$0.496030\pi$
$258$	12.3637	0.769730
$259$	−4.79796	−0.298131
$260$	−1.00000	−0.0620174
$261$	−6.25455	−0.387147
$262$	6.47048	0.399748
$263$	−16.2600	−1.00264	−0.501319	−	0.865263i	$-0.667152\pi$
$263$	−16.2600	−1.00264	−0.501319	+	0.865263i	$0.667152\pi$
$264$	−4.57080	−0.281313
$265$	2.16268	0.132853
$266$	6.03943	0.370301
$267$	3.90057	0.238711
$268$	−7.18954	−0.439171
$269$	−18.0559	−1.10089	−0.550443	−	0.834872i	$-0.685541\pi$
$269$	−18.0559	−1.10089	−0.550443	+	0.834872i	$0.685541\pi$
$270$	5.34321	0.325177
$271$	−25.2448	−1.53352	−0.766758	−	0.641937i	$-0.778131\pi$
$271$	−25.2448	−1.53352	−0.766758	+	0.641937i	$0.778131\pi$
$272$	−0.934050	−0.0566351
$273$	1.54161	0.0933026
$274$	−2.94609	−0.177980
$275$	−4.03486	−0.243311
$276$	−0.0757610	−0.00456028
$277$	10.3352	0.620983	0.310491	−	0.950576i	$-0.399506\pi$
$277$	10.3352	0.620983	0.310491	+	0.950576i	$0.399506\pi$
$278$	14.3501	0.860664
$279$	−1.71670	−0.102776
$280$	−1.36085	−0.0813266
$281$	3.02665	0.180555	0.0902775	−	0.995917i	$-0.471225\pi$
$281$	3.02665	0.180555	0.0902775	+	0.995917i	$0.471225\pi$
$282$	5.30743	0.316053
$283$	5.59690	0.332701	0.166351	−	0.986067i	$-0.446802\pi$
$283$	5.59690	0.332701	0.166351	+	0.986067i	$0.446802\pi$
$284$	−5.63632	−0.334454
$285$	5.02746	0.297801
$286$	4.03486	0.238586
$287$	−8.38325	−0.494848
$288$	1.71670	0.101158
$289$	−16.1276	−0.948679
$290$	3.64335	0.213945
$291$	0.181997	0.0106689
$292$	10.3728	0.607025
$293$	0.721285	0.0421379	0.0210690	−	0.999778i	$-0.493293\pi$
$293$	0.721285	0.0421379	0.0210690	+	0.999778i	$0.493293\pi$
$294$	−5.83189	−0.340122
$295$	5.66877	0.330049
$296$	−3.52570	−0.204927
$297$	−21.5591	−1.25099
$298$	−0.0405321	−0.00234796
$299$	0.0668778	0.00386764
$300$	−1.13283	−0.0654038
$301$	−14.8524	−0.856077
$302$	5.19280	0.298812
$303$	20.2289	1.16212
$304$	4.43798	0.254535
$305$	7.38270	0.422732
$306$	−1.60348	−0.0916651
$307$	−0.0583655	−0.00333110	−0.00166555	−	0.999999i	$-0.500530\pi$
$307$	−0.0583655	−0.00333110	−0.00166555	+	0.999999i	$0.500530\pi$
$308$	5.49086	0.312871
$309$	0.972796	0.0553404
$310$	1.00000	0.0567962
$311$	−29.5903	−1.67791	−0.838955	−	0.544201i	$-0.816833\pi$
$311$	−29.5903	−1.67791	−0.838955	+	0.544201i	$0.816833\pi$
$312$	1.13283	0.0641337
$313$	−23.6364	−1.33601	−0.668004	−	0.744157i	$-0.732851\pi$
$313$	−23.6364	−1.33601	−0.668004	+	0.744157i	$0.732851\pi$
$314$	10.6477	0.600884
$315$	−2.33618	−0.131629
$316$	−2.73521	−0.153868
$317$	4.07304	0.228765	0.114382	−	0.993437i	$-0.463511\pi$
$317$	4.07304	0.228765	0.114382	+	0.993437i	$0.463511\pi$
$318$	−2.44995	−0.137386
$319$	−14.7004	−0.823065
$320$	−1.00000	−0.0559017
$321$	7.69877	0.429703
$322$	0.0910109	0.00507184
$323$	−4.14529	−0.230650
$324$	−0.902834	−0.0501574
$325$	1.00000	0.0554700
$326$	12.6497	0.700600
$327$	−11.5582	−0.639168
$328$	−6.16029	−0.340145
$329$	−6.37576	−0.351507
$330$	4.57080	0.251614
$331$	−31.6892	−1.74180	−0.870899	−	0.491462i	$-0.836463\pi$
$331$	−31.6892	−1.74180	−0.870899	+	0.491462i	$0.836463\pi$
$332$	6.85404	0.376165
$333$	−6.05257	−0.331679
$334$	−15.7124	−0.859742
$335$	7.18954	0.392806
$336$	1.54161	0.0841019
$337$	−8.65145	−0.471275	−0.235637	−	0.971841i	$-0.575718\pi$
$337$	−8.65145	−0.471275	−0.235637	+	0.971841i	$0.575718\pi$
$338$	−1.00000	−0.0543928
$339$	22.1972	1.20559
$340$	0.934050	0.0506560
$341$	−4.03486	−0.218500
$342$	7.61868	0.411971
$343$	16.5318	0.892631
$344$	−10.9140	−0.588444
$345$	0.0757610	0.00407884
$346$	6.86527	0.369079
$347$	26.2366	1.40845	0.704227	−	0.709975i	$-0.251294\pi$
$347$	26.2366	1.40845	0.704227	+	0.709975i	$0.251294\pi$
$348$	−4.12729	−0.221246
$349$	1.44446	0.0773200	0.0386600	−	0.999252i	$-0.487691\pi$
$349$	1.44446	0.0773200	0.0386600	+	0.999252i	$0.487691\pi$
$350$	1.36085	0.0727407
$351$	5.34321	0.285199
$352$	4.03486	0.215059
$353$	−5.33197	−0.283792	−0.141896	−	0.989882i	$-0.545320\pi$
$353$	−5.33197	−0.283792	−0.141896	+	0.989882i	$0.545320\pi$
$354$	−6.42174	−0.341312
$355$	5.63632	0.299145
$356$	−3.44321	−0.182490
$357$	−1.43994	−0.0762099
$358$	5.15978	0.272703
$359$	28.7092	1.51521	0.757607	−	0.652711i	$-0.226368\pi$
$359$	28.7092	1.51521	0.757607	+	0.652711i	$0.226368\pi$
$360$	−1.71670	−0.0904781
$361$	0.695624	0.0366118
$362$	−7.84176	−0.412154
$363$	−5.98145	−0.313945
$364$	−1.36085	−0.0713281
$365$	−10.3728	−0.542939
$366$	−8.36333	−0.437158
$367$	−24.4435	−1.27594	−0.637970	−	0.770061i	$-0.720226\pi$
$367$	−24.4435	−1.27594	−0.637970	+	0.770061i	$0.720226\pi$
$368$	0.0668778	0.00348625
$369$	−10.5754	−0.550532
$370$	3.52570	0.183292
$371$	2.94309	0.152798
$372$	−1.13283	−0.0587344
$373$	−27.1639	−1.40650	−0.703248	−	0.710945i	$-0.748268\pi$
$373$	−27.1639	−1.40650	−0.703248	+	0.710945i	$0.748268\pi$
$374$	−3.76876	−0.194878
$375$	1.13283	0.0584990
$376$	−4.68512	−0.241616
$377$	3.64335	0.187642
$378$	7.27133	0.373997
$379$	4.93432	0.253459	0.126730	−	0.991937i	$-0.459552\pi$
$379$	4.93432	0.253459	0.126730	+	0.991937i	$0.459552\pi$
$380$	−4.43798	−0.227663
$381$	−22.4302	−1.14913
$382$	−21.1349	−1.08136
$383$	11.6951	0.597591	0.298795	−	0.954317i	$-0.403415\pi$
$383$	11.6951	0.597591	0.298795	+	0.954317i	$0.403415\pi$
$384$	1.13283	0.0578094
$385$	−5.49086	−0.279840
$386$	−8.05818	−0.410151
$387$	−18.7361	−0.952409
$388$	−0.160657	−0.00815615
$389$	−15.2773	−0.774591	−0.387295	−	0.921956i	$-0.626591\pi$
$389$	−15.2773	−0.774591	−0.387295	+	0.921956i	$0.626591\pi$
$390$	−1.13283	−0.0573630
$391$	−0.0624672	−0.00315910
$392$	5.14808	0.260017
$393$	7.32994	0.369747
$394$	14.3984	0.725380
$395$	2.73521	0.137624
$396$	6.92665	0.348077
$397$	24.0178	1.20542	0.602710	−	0.797960i	$-0.294088\pi$
$397$	24.0178	1.20542	0.602710	+	0.797960i	$0.294088\pi$
$398$	−17.7013	−0.887284
$399$	6.84164	0.342510
$400$	1.00000	0.0500000
$401$	5.35932	0.267632	0.133816	−	0.991006i	$-0.457277\pi$
$401$	5.35932	0.267632	0.133816	+	0.991006i	$0.457277\pi$
$402$	−8.14451	−0.406211
$403$	1.00000	0.0498135
$404$	−17.8570	−0.888420
$405$	0.902834	0.0448622
$406$	4.95807	0.246065
$407$	−14.2257	−0.705142
$408$	−1.05812	−0.0523846
$409$	−11.3612	−0.561777	−0.280889	−	0.959740i	$-0.590629\pi$
$409$	−11.3612	−0.561777	−0.280889	+	0.959740i	$0.590629\pi$
$410$	6.16029	0.304235
$411$	−3.33741	−0.164622
$412$	−0.858733	−0.0423067
$413$	7.71437	0.379599
$414$	0.114809	0.00564256
$415$	−6.85404	−0.336452
$416$	−1.00000	−0.0490290
$417$	16.2562	0.796071
$418$	17.9066	0.875841
$419$	−11.2438	−0.549295	−0.274647	−	0.961545i	$-0.588561\pi$
$419$	−11.2438	−0.549295	−0.274647	+	0.961545i	$0.588561\pi$
$420$	−1.54161	−0.0752230
$421$	25.1546	1.22596	0.612981	−	0.790097i	$-0.289970\pi$
$421$	25.1546	1.22596	0.612981	+	0.790097i	$0.289970\pi$
$422$	−27.9009	−1.35819
$423$	−8.04295	−0.391061
$424$	2.16268	0.105029
$425$	−0.934050	−0.0453081
$426$	−6.38498	−0.309353
$427$	10.0468	0.486198
$428$	−6.79606	−0.328500
$429$	4.57080	0.220680
$430$	10.9140	0.526320
$431$	−28.5286	−1.37417	−0.687086	−	0.726576i	$-0.741110\pi$
$431$	−28.5286	−1.37417	−0.687086	+	0.726576i	$0.741110\pi$
$432$	5.34321	0.257075
$433$	−13.7268	−0.659670	−0.329835	−	0.944039i	$-0.606993\pi$
$433$	−13.7268	−0.659670	−0.329835	+	0.944039i	$0.606993\pi$
$434$	1.36085	0.0653231
$435$	4.12729	0.197888
$436$	10.2029	0.488632
$437$	0.296802	0.0141980
$438$	11.7506	0.561468
$439$	−35.9822	−1.71734	−0.858668	−	0.512532i	$-0.828708\pi$
$439$	−35.9822	−1.71734	−0.858668	+	0.512532i	$0.828708\pi$
$440$	−4.03486	−0.192354
$441$	8.83771	0.420843
$442$	0.934050	0.0444282
$443$	24.4222	1.16033	0.580167	−	0.814497i	$-0.302987\pi$
$443$	24.4222	1.16033	0.580167	+	0.814497i	$0.302987\pi$
$444$	−3.99401	−0.189547
$445$	3.44321	0.163224
$446$	0.983455	0.0465679
$447$	−0.0459159	−0.00217175
$448$	−1.36085	−0.0642943
$449$	−39.0842	−1.84450	−0.922248	−	0.386599i	$-0.873650\pi$
$449$	−39.0842	−1.84450	−0.922248	+	0.386599i	$0.873650\pi$
$450$	1.71670	0.0809261
$451$	−24.8559	−1.17042
$452$	−19.5945	−0.921648
$453$	5.88254	0.276386
$454$	−21.8640	−1.02613
$455$	1.36085	0.0637978
$456$	5.02746	0.235432
$457$	31.2170	1.46027	0.730136	−	0.683302i	$-0.239457\pi$
$457$	31.2170	1.46027	0.730136	+	0.683302i	$0.239457\pi$
$458$	10.7272	0.501250
$459$	−4.99083	−0.232952
$460$	−0.0668778	−0.00311819
$461$	−3.30841	−0.154088	−0.0770441	−	0.997028i	$-0.524548\pi$
$461$	−3.30841	−0.154088	−0.0770441	+	0.997028i	$0.524548\pi$
$462$	6.22019	0.289390
$463$	0.804307	0.0373793	0.0186897	−	0.999825i	$-0.494051\pi$
$463$	0.804307	0.0373793	0.0186897	+	0.999825i	$0.494051\pi$
$464$	3.64335	0.169138
$465$	1.13283	0.0525336
$466$	−3.97892	−0.184320
$467$	4.96392	0.229703	0.114851	−	0.993383i	$-0.463361\pi$
$467$	4.96392	0.229703	0.114851	+	0.993383i	$0.463361\pi$
$468$	−1.71670	−0.0793545
$469$	9.78391	0.451779
$470$	4.68512	0.216108
$471$	12.0620	0.555788
$472$	5.66877	0.260926
$473$	−44.0365	−2.02480
$474$	−3.09853	−0.142320
$475$	4.43798	0.203628
$476$	1.27111	0.0582610
$477$	3.71268	0.169992
$478$	4.63269	0.211894
$479$	−1.51705	−0.0693158	−0.0346579	−	0.999399i	$-0.511034\pi$
$479$	−1.51705	−0.0693158	−0.0346579	+	0.999399i	$0.511034\pi$
$480$	−1.13283	−0.0517063
$481$	3.52570	0.160758
$482$	8.73533	0.397883
$483$	0.103100	0.00469120
$484$	5.28010	0.240005
$485$	0.160657	0.00729508
$486$	15.0069	0.680726
$487$	11.6995	0.530155	0.265077	−	0.964227i	$-0.414603\pi$
$487$	11.6995	0.530155	0.265077	+	0.964227i	$0.414603\pi$
$488$	7.38270	0.334199
$489$	14.3299	0.648020
$490$	−5.14808	−0.232566
$491$	3.71782	0.167783	0.0838914	−	0.996475i	$-0.473265\pi$
$491$	3.71782	0.167783	0.0838914	+	0.996475i	$0.473265\pi$
$492$	−6.97855	−0.314617
$493$	−3.40307	−0.153267
$494$	−4.43798	−0.199674
$495$	−6.92665	−0.311330
$496$	1.00000	0.0449013
$497$	7.67021	0.344056
$498$	7.76445	0.347933
$499$	21.8335	0.977401	0.488700	−	0.872452i	$-0.337471\pi$
$499$	21.8335	0.977401	0.488700	+	0.872452i	$0.337471\pi$
$500$	−1.00000	−0.0447214
$501$	−17.7994	−0.795218
$502$	14.6179	0.652431
$503$	−16.7867	−0.748482	−0.374241	−	0.927332i	$-0.622097\pi$
$503$	−16.7867	−0.748482	−0.374241	+	0.927332i	$0.622097\pi$
$504$	−2.33618	−0.104062
$505$	17.8570	0.794627
$506$	0.269843	0.0119960
$507$	−1.13283	−0.0503107
$508$	19.8002	0.878490
$509$	−26.4036	−1.17032	−0.585159	−	0.810919i	$-0.698968\pi$
$509$	−26.4036	−1.17032	−0.585159	+	0.810919i	$0.698968\pi$
$510$	1.05812	0.0468542
$511$	−14.1159	−0.624452
$512$	−1.00000	−0.0441942
$513$	23.7130	1.04696
$514$	−0.399855	−0.0176368
$515$	0.858733	0.0378403
$516$	−12.3637	−0.544281
$517$	−18.9038	−0.831388
$518$	4.79796	0.210810
$519$	7.77717	0.341380
$520$	1.00000	0.0438529
$521$	−6.26931	−0.274664	−0.137332	−	0.990525i	$-0.543853\pi$
$521$	−6.26931	−0.274664	−0.137332	+	0.990525i	$0.543853\pi$
$522$	6.25455	0.273754
$523$	−17.3817	−0.760050	−0.380025	−	0.924976i	$-0.624085\pi$
$523$	−17.3817	−0.760050	−0.380025	+	0.924976i	$0.624085\pi$
$524$	−6.47048	−0.282664
$525$	1.54161	0.0672815
$526$	16.2600	0.708972
$527$	−0.934050	−0.0406879
$528$	4.57080	0.198919
$529$	−22.9955	−0.999806
$530$	−2.16268	−0.0939409
$531$	9.73158	0.422315
$532$	−6.03943	−0.261843
$533$	6.16029	0.266832
$534$	−3.90057	−0.168794
$535$	6.79606	0.293819
$536$	7.18954	0.310541
$537$	5.84514	0.252237
$538$	18.0559	0.778445
$539$	20.7718	0.894704
$540$	−5.34321	−0.229935
$541$	−29.8201	−1.28206	−0.641032	−	0.767514i	$-0.721493\pi$
$541$	−29.8201	−1.28206	−0.641032	+	0.767514i	$0.721493\pi$
$542$	25.2448	1.08436
$543$	−8.88337	−0.381222
$544$	0.934050	0.0400471
$545$	−10.2029	−0.437046
$546$	−1.54161	−0.0659749
$547$	38.7832	1.65825	0.829124	−	0.559065i	$-0.188840\pi$
$547$	38.7832	1.65825	0.829124	+	0.559065i	$0.188840\pi$
$548$	2.94609	0.125851
$549$	12.6739	0.540909
$550$	4.03486	0.172047
$551$	16.1691	0.688827
$552$	0.0757610	0.00322460
$553$	3.72223	0.158285
$554$	−10.3352	−0.439101
$555$	3.99401	0.169536
$556$	−14.3501	−0.608581
$557$	0.225345	0.00954817	0.00477409	−	0.999989i	$-0.498480\pi$
$557$	0.225345	0.00954817	0.00477409	+	0.999989i	$0.498480\pi$
$558$	1.71670	0.0726738
$559$	10.9140	0.461614
$560$	1.36085	0.0575066
$561$	−4.26936	−0.180252
$562$	−3.02665	−0.127672
$563$	2.53847	0.106984	0.0534919	−	0.998568i	$-0.482965\pi$
$563$	2.53847	0.106984	0.0534919	+	0.998568i	$0.482965\pi$
$564$	−5.30743	−0.223483
$565$	19.5945	0.824347
$566$	−5.59690	−0.235255
$567$	1.22862	0.0515974
$568$	5.63632	0.236495
$569$	−35.4332	−1.48544	−0.742718	−	0.669605i	$-0.766464\pi$
$569$	−35.4332	−1.48544	−0.742718	+	0.669605i	$0.766464\pi$
$570$	−5.02746	−0.210577
$571$	−5.76473	−0.241246	−0.120623	−	0.992698i	$-0.538489\pi$
$571$	−5.76473	−0.241246	−0.120623	+	0.992698i	$0.538489\pi$
$572$	−4.03486	−0.168706
$573$	−23.9422	−1.00020
$574$	8.38325	0.349910
$575$	0.0668778	0.00278900
$576$	−1.71670	−0.0715292
$577$	−5.91165	−0.246105	−0.123053	−	0.992400i	$-0.539268\pi$
$577$	−5.91165	−0.246105	−0.123053	+	0.992400i	$0.539268\pi$
$578$	16.1276	0.670818
$579$	−9.12853	−0.379369
$580$	−3.64335	−0.151282
$581$	−9.32735	−0.386964
$582$	−0.181997	−0.00754403
$583$	8.72613	0.361399
$584$	−10.3728	−0.429231
$585$	1.71670	0.0709769
$586$	−0.721285	−0.0297960
$587$	19.0058	0.784454	0.392227	−	0.919868i	$-0.371705\pi$
$587$	19.0058	0.784454	0.392227	+	0.919868i	$0.371705\pi$
$588$	5.83189	0.240503
$589$	4.43798	0.182864
$590$	−5.66877	−0.233380
$591$	16.3109	0.670941
$592$	3.52570	0.144905
$593$	5.41330	0.222298	0.111149	−	0.993804i	$-0.464547\pi$
$593$	5.41330	0.222298	0.111149	+	0.993804i	$0.464547\pi$
$594$	21.5591	0.884581
$595$	−1.27111	−0.0521102
$596$	0.0405321	0.00166026
$597$	−20.0525	−0.820693
$598$	−0.0668778	−0.00273484
$599$	−3.17042	−0.129540	−0.0647700	−	0.997900i	$-0.520631\pi$
$599$	−3.17042	−0.129540	−0.0647700	+	0.997900i	$0.520631\pi$
$600$	1.13283	0.0462475
$601$	22.6476	0.923816	0.461908	−	0.886928i	$-0.347165\pi$
$601$	22.6476	0.923816	0.461908	+	0.886928i	$0.347165\pi$
$602$	14.8524	0.605338
$603$	12.3423	0.502617
$604$	−5.19280	−0.211292
$605$	−5.28010	−0.214667
$606$	−20.2289	−0.821744
$607$	3.00024	0.121776	0.0608880	−	0.998145i	$-0.480607\pi$
$607$	3.00024	0.121776	0.0608880	+	0.998145i	$0.480607\pi$
$608$	−4.43798	−0.179984
$609$	5.61664	0.227598
$610$	−7.38270	−0.298917
$611$	4.68512	0.189540
$612$	1.60348	0.0648170
$613$	9.81594	0.396462	0.198231	−	0.980155i	$-0.436480\pi$
$613$	9.81594	0.396462	0.198231	+	0.980155i	$0.436480\pi$
$614$	0.0583655	0.00235544
$615$	6.97855	0.281402
$616$	−5.49086	−0.221233
$617$	−29.2079	−1.17586	−0.587932	−	0.808910i	$-0.700058\pi$
$617$	−29.2079	−1.17586	−0.587932	+	0.808910i	$0.700058\pi$
$618$	−0.972796	−0.0391316
$619$	45.8421	1.84255	0.921275	−	0.388911i	$-0.127149\pi$
$619$	45.8421	1.84255	0.921275	+	0.388911i	$0.127149\pi$
$620$	−1.00000	−0.0401610
$621$	0.357342	0.0143396
$622$	29.5903	1.18646
$623$	4.68571	0.187729
$624$	−1.13283	−0.0453494
$625$	1.00000	0.0400000
$626$	23.6364	0.944701
$627$	20.2851	0.810109
$628$	−10.6477	−0.424889
$629$	−3.29318	−0.131308
$630$	2.33618	0.0930756
$631$	−33.1043	−1.31786	−0.658931	−	0.752204i	$-0.728991\pi$
$631$	−33.1043	−1.31786	−0.658931	+	0.752204i	$0.728991\pi$
$632$	2.73521	0.108801
$633$	−31.6069	−1.25626
$634$	−4.07304	−0.161761
$635$	−19.8002	−0.785745
$636$	2.44995	0.0971467
$637$	−5.14808	−0.203974
$638$	14.7004	0.581995
$639$	9.67588	0.382772
$640$	1.00000	0.0395285
$641$	28.8735	1.14044	0.570218	−	0.821494i	$-0.306859\pi$
$641$	28.8735	1.14044	0.570218	+	0.821494i	$0.306859\pi$
$642$	−7.69877	−0.303846
$643$	−11.7193	−0.462164	−0.231082	−	0.972934i	$-0.574227\pi$
$643$	−11.7193	−0.462164	−0.231082	+	0.972934i	$0.574227\pi$
$644$	−0.0910109	−0.00358633
$645$	12.3637	0.486820
$646$	4.14529	0.163094
$647$	23.2127	0.912586	0.456293	−	0.889830i	$-0.349177\pi$
$647$	23.2127	0.912586	0.456293	+	0.889830i	$0.349177\pi$
$648$	0.902834	0.0354667
$649$	22.8727	0.897832
$650$	−1.00000	−0.0392232
$651$	1.54161	0.0604206
$652$	−12.6497	−0.495399
$653$	−20.5442	−0.803958	−0.401979	−	0.915649i	$-0.631677\pi$
$653$	−20.5442	−0.803958	−0.401979	+	0.915649i	$0.631677\pi$
$654$	11.5582	0.451960
$655$	6.47048	0.252823
$656$	6.16029	0.240519
$657$	−17.8071	−0.694720
$658$	6.37576	0.248553
$659$	20.8290	0.811382	0.405691	−	0.914010i	$-0.367031\pi$
$659$	20.8290	0.811382	0.405691	+	0.914010i	$0.367031\pi$
$660$	−4.57080	−0.177918
$661$	30.1505	1.17272	0.586360	−	0.810051i	$-0.300561\pi$
$661$	30.1505	1.17272	0.586360	+	0.810051i	$0.300561\pi$
$662$	31.6892	1.23164
$663$	1.05812	0.0410939
$664$	−6.85404	−0.265989
$665$	6.03943	0.234199
$666$	6.05257	0.234532
$667$	0.243659	0.00943453
$668$	15.7124	0.607929
$669$	1.11408	0.0430730
$670$	−7.18954	−0.277756
$671$	29.7882	1.14996
$672$	−1.54161	−0.0594690
$673$	5.94686	0.229234	0.114617	−	0.993410i	$-0.463436\pi$
$673$	5.94686	0.229234	0.114617	+	0.993410i	$0.463436\pi$
$674$	8.65145	0.333242
$675$	5.34321	0.205660
$676$	1.00000	0.0384615
$677$	34.1649	1.31306	0.656531	−	0.754299i	$-0.272023\pi$
$677$	34.1649	1.31306	0.656531	+	0.754299i	$0.272023\pi$
$678$	−22.1972	−0.852479
$679$	0.218631	0.00839030
$680$	−0.934050	−0.0358192
$681$	−24.7681	−0.949117
$682$	4.03486	0.154503
$683$	−47.5310	−1.81872	−0.909362	−	0.416006i	$-0.863430\pi$
$683$	−47.5310	−1.81872	−0.909362	+	0.416006i	$0.863430\pi$
$684$	−7.61868	−0.291307
$685$	−2.94609	−0.112564
$686$	−16.5318	−0.631185
$687$	12.1521	0.463631
$688$	10.9140	0.416093
$689$	−2.16268	−0.0823917
$690$	−0.0757610	−0.00288417
$691$	−24.3215	−0.925234	−0.462617	−	0.886558i	$-0.653090\pi$
$691$	−24.3215	−0.925234	−0.462617	+	0.886558i	$0.653090\pi$
$692$	−6.86527	−0.260978
$693$	−9.42616	−0.358070
$694$	−26.2366	−0.995927
$695$	14.3501	0.544332
$696$	4.12729	0.156445
$697$	−5.75402	−0.217949
$698$	−1.44446	−0.0546735
$699$	−4.50743	−0.170487
$700$	−1.36085	−0.0514354
$701$	−43.3251	−1.63637	−0.818184	−	0.574957i	$-0.805019\pi$
$701$	−43.3251	−1.63637	−0.818184	+	0.574957i	$0.805019\pi$
$702$	−5.34321	−0.201666
$703$	15.6470	0.590136
$704$	−4.03486	−0.152070
$705$	5.30743	0.199889
$706$	5.33197	0.200671
$707$	24.3008	0.913925
$708$	6.42174	0.241344
$709$	17.5146	0.657773	0.328886	−	0.944369i	$-0.393327\pi$
$709$	17.5146	0.657773	0.328886	+	0.944369i	$0.393327\pi$
$710$	−5.63632	−0.211527
$711$	4.69555	0.176097
$712$	3.44321	0.129040
$713$	0.0668778	0.00250459
$714$	1.43994	0.0538885
$715$	4.03486	0.150895
$716$	−5.15978	−0.192830
$717$	5.24804	0.195991
$718$	−28.7092	−1.07142
$719$	−14.9777	−0.558575	−0.279287	−	0.960208i	$-0.590098\pi$
$719$	−14.9777	−0.558575	−0.279287	+	0.960208i	$0.590098\pi$
$720$	1.71670	0.0639777
$721$	1.16861	0.0435213
$722$	−0.695624	−0.0258885
$723$	9.89562	0.368022
$724$	7.84176	0.291437
$725$	3.64335	0.135311
$726$	5.98145	0.221992
$727$	31.1405	1.15494	0.577469	−	0.816413i	$-0.304041\pi$
$727$	31.1405	1.15494	0.577469	+	0.816413i	$0.304041\pi$
$728$	1.36085	0.0504366
$729$	19.7087	0.729952
$730$	10.3728	0.383916
$731$	−10.1942	−0.377047
$732$	8.36333	0.309118
$733$	0.368493	0.0136106	0.00680529	−	0.999977i	$-0.497834\pi$
$733$	0.368493	0.0136106	0.00680529	+	0.999977i	$0.497834\pi$
$734$	24.4435	0.902226
$735$	−5.83189	−0.215112
$736$	−0.0668778	−0.00246515
$737$	29.0088	1.06855
$738$	10.5754	0.389285
$739$	−0.812617	−0.0298926	−0.0149463	−	0.999888i	$-0.504758\pi$
$739$	−0.812617	−0.0298926	−0.0149463	+	0.999888i	$0.504758\pi$
$740$	−3.52570	−0.129607
$741$	−5.02746	−0.184688
$742$	−2.94309	−0.108044
$743$	−32.5883	−1.19555	−0.597775	−	0.801664i	$-0.703948\pi$
$743$	−32.5883	−1.19555	−0.597775	+	0.801664i	$0.703948\pi$
$744$	1.13283	0.0415315
$745$	−0.0405321	−0.00148498
$746$	27.1639	0.994542
$747$	−11.7663	−0.430508
$748$	3.76876	0.137800
$749$	9.24844	0.337931
$750$	−1.13283	−0.0413650
$751$	−18.6194	−0.679432	−0.339716	−	0.940528i	$-0.610331\pi$
$751$	−18.6194	−0.679432	−0.339716	+	0.940528i	$0.610331\pi$
$752$	4.68512	0.170849
$753$	16.5596	0.603466
$754$	−3.64335	−0.132683
$755$	5.19280	0.188985
$756$	−7.27133	−0.264456
$757$	−46.8808	−1.70391	−0.851956	−	0.523613i	$-0.824584\pi$
$757$	−46.8808	−1.70391	−0.851956	+	0.523613i	$0.824584\pi$
$758$	−4.93432	−0.179223
$759$	0.305685	0.0110957
$760$	4.43798	0.160982
$761$	28.1307	1.01974	0.509869	−	0.860252i	$-0.329694\pi$
$761$	28.1307	1.01974	0.509869	+	0.860252i	$0.329694\pi$
$762$	22.4302	0.812559
$763$	−13.8847	−0.502660
$764$	21.1349	0.764635
$765$	−1.60348	−0.0579741
$766$	−11.6951	−0.422560
$767$	−5.66877	−0.204687
$768$	−1.13283	−0.0408774
$769$	−36.6682	−1.32229	−0.661145	−	0.750258i	$-0.729929\pi$
$769$	−36.6682	−1.32229	−0.661145	+	0.750258i	$0.729929\pi$
$770$	5.49086	0.197877
$771$	−0.452967	−0.0163132
$772$	8.05818	0.290020
$773$	−48.2191	−1.73432	−0.867160	−	0.498030i	$-0.834057\pi$
$773$	−48.2191	−1.73432	−0.867160	+	0.498030i	$0.834057\pi$
$774$	18.7361	0.673455
$775$	1.00000	0.0359211
$776$	0.160657	0.00576727
$777$	5.43526	0.194989
$778$	15.2773	0.547718
$779$	27.3392	0.979529
$780$	1.13283	0.0405617
$781$	22.7418	0.813764
$782$	0.0624672	0.00223382
$783$	19.4672	0.695701
$784$	−5.14808	−0.183860
$785$	10.6477	0.380032
$786$	−7.32994	−0.261450
$787$	−9.43385	−0.336280	−0.168140	−	0.985763i	$-0.553776\pi$
$787$	−9.43385	−0.336280	−0.168140	+	0.985763i	$0.553776\pi$
$788$	−14.3984	−0.512921
$789$	18.4198	0.655764
$790$	−2.73521	−0.0973146
$791$	26.6653	0.948107
$792$	−6.92665	−0.246128
$793$	−7.38270	−0.262167
$794$	−24.0178	−0.852360
$795$	−2.44995	−0.0868907
$796$	17.7013	0.627404
$797$	20.8382	0.738128	0.369064	−	0.929404i	$-0.379678\pi$
$797$	20.8382	0.738128	0.369064	+	0.929404i	$0.379678\pi$
$798$	−6.84164	−0.242191
$799$	−4.37613	−0.154816
$800$	−1.00000	−0.0353553
$801$	5.91097	0.208854
$802$	−5.35932	−0.189244
$803$	−41.8530	−1.47696
$804$	8.14451	0.287235
$805$	0.0910109	0.00320771
$806$	−1.00000	−0.0352235
$807$	20.4542	0.720022
$808$	17.8570	0.628207
$809$	−1.46653	−0.0515604	−0.0257802	−	0.999668i	$-0.508207\pi$
$809$	−1.46653	−0.0515604	−0.0257802	+	0.999668i	$0.508207\pi$
$810$	−0.902834	−0.0317224
$811$	34.1155	1.19796	0.598978	−	0.800766i	$-0.295574\pi$
$811$	34.1155	1.19796	0.598978	+	0.800766i	$0.295574\pi$
$812$	−4.95807	−0.173994
$813$	28.5981	1.00298
$814$	14.2257	0.498611
$815$	12.6497	0.443098
$816$	1.05812	0.0370415
$817$	48.4361	1.69457
$818$	11.3612	0.397237
$819$	2.33618	0.0816327
$820$	−6.16029	−0.215127
$821$	28.0352	0.978434	0.489217	−	0.872162i	$-0.337283\pi$
$821$	28.0352	0.978434	0.489217	+	0.872162i	$0.337283\pi$
$822$	3.33741	0.116406
$823$	24.5316	0.855116	0.427558	−	0.903988i	$-0.359374\pi$
$823$	24.5316	0.855116	0.427558	+	0.903988i	$0.359374\pi$
$824$	0.858733	0.0299154
$825$	4.57080	0.159135
$826$	−7.71437	−0.268417
$827$	7.24646	0.251984	0.125992	−	0.992031i	$-0.459789\pi$
$827$	7.24646	0.251984	0.125992	+	0.992031i	$0.459789\pi$
$828$	−0.114809	−0.00398990
$829$	−4.85025	−0.168456	−0.0842281	−	0.996446i	$-0.526842\pi$
$829$	−4.85025	−0.168456	−0.0842281	+	0.996446i	$0.526842\pi$
$830$	6.85404	0.237907
$831$	−11.7080	−0.406147
$832$	1.00000	0.0346688
$833$	4.80856	0.166607
$834$	−16.2562	−0.562907
$835$	−15.7124	−0.543748
$836$	−17.9066	−0.619313
$837$	5.34321	0.184688
$838$	11.2438	0.388410
$839$	−11.7700	−0.406347	−0.203173	−	0.979143i	$-0.565126\pi$
$839$	−11.7700	−0.406347	−0.203173	+	0.979143i	$0.565126\pi$
$840$	1.54161	0.0531907
$841$	−15.7260	−0.542275
$842$	−25.1546	−0.866886
$843$	−3.42868	−0.118090
$844$	27.9009	0.960389
$845$	−1.00000	−0.0344010
$846$	8.04295	0.276522
$847$	−7.18545	−0.246895
$848$	−2.16268	−0.0742668
$849$	−6.34033	−0.217600
$850$	0.934050	0.0320376
$851$	0.235791	0.00808281
$852$	6.38498	0.218746
$853$	28.9918	0.992662	0.496331	−	0.868133i	$-0.334680\pi$
$853$	28.9918	0.992662	0.496331	+	0.868133i	$0.334680\pi$
$854$	−10.0468	−0.343794
$855$	7.61868	0.260553
$856$	6.79606	0.232285
$857$	−5.55008	−0.189587	−0.0947936	−	0.995497i	$-0.530219\pi$
$857$	−5.55008	−0.189587	−0.0947936	+	0.995497i	$0.530219\pi$
$858$	−4.57080	−0.156045
$859$	49.5732	1.69142	0.845708	−	0.533646i	$-0.179179\pi$
$859$	49.5732	1.69142	0.845708	+	0.533646i	$0.179179\pi$
$860$	−10.9140	−0.372165
$861$	9.49678	0.323649
$862$	28.5286	0.971686
$863$	−38.8519	−1.32253	−0.661267	−	0.750150i	$-0.729981\pi$
$863$	−38.8519	−1.32253	−0.661267	+	0.750150i	$0.729981\pi$
$864$	−5.34321	−0.181780
$865$	6.86527	0.233426
$866$	13.7268	0.466457
$867$	18.2697	0.620473
$868$	−1.36085	−0.0461904
$869$	11.0362	0.374378
$870$	−4.12729	−0.139928
$871$	−7.18954	−0.243608
$872$	−10.2029	−0.345515
$873$	0.275801	0.00933444
$874$	−0.296802	−0.0100395
$875$	1.36085	0.0460052
$876$	−11.7506	−0.397018
$877$	12.0366	0.406448	0.203224	−	0.979132i	$-0.434858\pi$
$877$	12.0366	0.406448	0.203224	+	0.979132i	$0.434858\pi$
$878$	35.9822	1.21434
$879$	−0.817091	−0.0275598
$880$	4.03486	0.136015
$881$	−3.08756	−0.104023	−0.0520113	−	0.998646i	$-0.516563\pi$
$881$	−3.08756	−0.104023	−0.0520113	+	0.998646i	$0.516563\pi$
$882$	−8.83771	−0.297581
$883$	−1.00591	−0.0338516	−0.0169258	−	0.999857i	$-0.505388\pi$
$883$	−1.00591	−0.0338516	−0.0169258	+	0.999857i	$0.505388\pi$
$884$	−0.934050	−0.0314155
$885$	−6.42174	−0.215864
$886$	−24.4222	−0.820481
$887$	−54.7200	−1.83732	−0.918658	−	0.395053i	$-0.870726\pi$
$887$	−54.7200	−1.83732	−0.918658	+	0.395053i	$0.870726\pi$
$888$	3.99401	0.134030
$889$	−26.9451	−0.903710
$890$	−3.44321	−0.115417
$891$	3.64281	0.122039
$892$	−0.983455	−0.0329285
$893$	20.7924	0.695792
$894$	0.0459159	0.00153566
$895$	5.15978	0.172472
$896$	1.36085	0.0454629
$897$	−0.0757610	−0.00252959
$898$	39.0842	1.30426
$899$	3.64335	0.121513
$900$	−1.71670	−0.0572234
$901$	2.02005	0.0672978
$902$	24.8559	0.827611
$903$	16.8252	0.559907
$904$	19.5945	0.651704
$905$	−7.84176	−0.260669
$906$	−5.88254	−0.195434
$907$	−8.53521	−0.283407	−0.141704	−	0.989909i	$-0.545258\pi$
$907$	−8.53521	−0.283407	−0.141704	+	0.989909i	$0.545258\pi$
$908$	21.8640	0.725582
$909$	30.6552	1.01677
$910$	−1.36085	−0.0451119
$911$	−47.0026	−1.55727	−0.778633	−	0.627480i	$-0.784087\pi$
$911$	−47.0026	−1.55727	−0.778633	+	0.627480i	$0.784087\pi$
$912$	−5.02746	−0.166476
$913$	−27.6551	−0.915251
$914$	−31.2170	−1.03257
$915$	−8.36333	−0.276483
$916$	−10.7272	−0.354437
$917$	8.80538	0.290779
$918$	4.99083	0.164722
$919$	22.6941	0.748611	0.374305	−	0.927305i	$-0.377881\pi$
$919$	22.6941	0.748611	0.374305	+	0.927305i	$0.377881\pi$
$920$	0.0668778	0.00220490
$921$	0.0661181	0.00217867
$922$	3.30841	0.108957
$923$	−5.63632	−0.185522
$924$	−6.22019	−0.204629
$925$	3.52570	0.115924
$926$	−0.804307	−0.0264312
$927$	1.47419	0.0484187
$928$	−3.64335	−0.119599
$929$	−4.05586	−0.133068	−0.0665342	−	0.997784i	$-0.521194\pi$
$929$	−4.05586	−0.133068	−0.0665342	+	0.997784i	$0.521194\pi$
$930$	−1.13283	−0.0371469
$931$	−22.8470	−0.748782
$932$	3.97892	0.130334
$933$	33.5207	1.09742
$934$	−4.96392	−0.162424
$935$	−3.76876	−0.123252
$936$	1.71670	0.0561121
$937$	1.30339	0.0425798	0.0212899	−	0.999773i	$-0.493223\pi$
$937$	1.30339	0.0425798	0.0212899	+	0.999773i	$0.493223\pi$
$938$	−9.78391	−0.319456
$939$	26.7760	0.873801
$940$	−4.68512	−0.152812
$941$	−2.51793	−0.0820822	−0.0410411	−	0.999157i	$-0.513067\pi$
$941$	−2.51793	−0.0820822	−0.0410411	+	0.999157i	$0.513067\pi$
$942$	−12.0620	−0.393001
$943$	0.411987	0.0134161
$944$	−5.66877	−0.184503
$945$	7.27133	0.236536
$946$	44.0365	1.43175
$947$	1.31435	0.0427106	0.0213553	−	0.999772i	$-0.493202\pi$
$947$	1.31435	0.0427106	0.0213553	+	0.999772i	$0.493202\pi$
$948$	3.09853	0.100635
$949$	10.3728	0.336717
$950$	−4.43798	−0.143987
$951$	−4.61405	−0.149621
$952$	−1.27111	−0.0411968
$953$	−19.7007	−0.638170	−0.319085	−	0.947726i	$-0.603375\pi$
$953$	−19.7007	−0.638170	−0.319085	+	0.947726i	$0.603375\pi$
$954$	−3.71268	−0.120202
$955$	−21.1349	−0.683910
$956$	−4.63269	−0.149832
$957$	16.6530	0.538316
$958$	1.51705	0.0490137
$959$	−4.00919	−0.129464
$960$	1.13283	0.0365619
$961$	1.00000	0.0322581
$962$	−3.52570	−0.113673
$963$	11.6668	0.375957
$964$	−8.73533	−0.281346
$965$	−8.05818	−0.259402
$966$	−0.103100	−0.00331718
$967$	34.1590	1.09848	0.549240	−	0.835664i	$-0.314917\pi$
$967$	34.1590	1.09848	0.549240	+	0.835664i	$0.314917\pi$
$968$	−5.28010	−0.169709
$969$	4.69590	0.150854
$970$	−0.160657	−0.00515840
$971$	33.9896	1.09078	0.545389	−	0.838183i	$-0.316382\pi$
$971$	33.9896	1.09078	0.545389	+	0.838183i	$0.316382\pi$
$972$	−15.0069	−0.481346
$973$	19.5284	0.626053
$974$	−11.6995	−0.374876
$975$	−1.13283	−0.0362795
$976$	−7.38270	−0.236315
$977$	−37.5091	−1.20002	−0.600011	−	0.799992i	$-0.704837\pi$
$977$	−37.5091	−1.20002	−0.600011	+	0.799992i	$0.704837\pi$
$978$	−14.3299	−0.458219
$979$	13.8929	0.444019
$980$	5.14808	0.164449
$981$	−17.5154	−0.559223
$982$	−3.71782	−0.118640
$983$	−40.1073	−1.27922	−0.639612	−	0.768698i	$-0.720905\pi$
$983$	−40.1073	−1.27922	−0.639612	+	0.768698i	$0.720905\pi$
$984$	6.97855	0.222468
$985$	14.3984	0.458771
$986$	3.40307	0.108376
$987$	7.22264	0.229899
$988$	4.43798	0.141191
$989$	0.729905	0.0232096
$990$	6.92665	0.220143
$991$	9.03379	0.286968	0.143484	−	0.989653i	$-0.454169\pi$
$991$	9.03379	0.286968	0.143484	+	0.989653i	$0.454169\pi$
$992$	−1.00000	−0.0317500
$993$	35.8985	1.13920
$994$	−7.67021	−0.243284
$995$	−17.7013	−0.561168
$996$	−7.76445	−0.246026
$997$	28.0510	0.888385	0.444193	−	0.895931i	$-0.353491\pi$
$997$	28.0510	0.888385	0.444193	+	0.895931i	$0.353491\pi$
$998$	−21.8335	−0.691127
$999$	18.8385	0.596025

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4030.2.a.c.1.3	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4030.2.a.c.1.3	✓	6	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:	\( N \)	\(=\)	\( 4030 = 2 \cdot 5 \cdot 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4030.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.13283 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.13283 q^{6} -1.36085 q^{7} -1.00000 q^{8} -1.71670 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.13283 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.13283 q^{6} -1.36085 q^{7} -1.00000 q^{8} -1.71670 q^{9} +1.00000 q^{10} -4.03486 q^{11} -1.13283 q^{12} +1.00000 q^{13} +1.36085 q^{14} +1.13283 q^{15} +1.00000 q^{16} -0.934050 q^{17} +1.71670 q^{18} +4.43798 q^{19} -1.00000 q^{20} +1.54161 q^{21} +4.03486 q^{22} +0.0668778 q^{23} +1.13283 q^{24} +1.00000 q^{25} -1.00000 q^{26} +5.34321 q^{27} -1.36085 q^{28} +3.64335 q^{29} -1.13283 q^{30} +1.00000 q^{31} -1.00000 q^{32} +4.57080 q^{33} +0.934050 q^{34} +1.36085 q^{35} -1.71670 q^{36} +3.52570 q^{37} -4.43798 q^{38} -1.13283 q^{39} +1.00000 q^{40} +6.16029 q^{41} -1.54161 q^{42} +10.9140 q^{43} -4.03486 q^{44} +1.71670 q^{45} -0.0668778 q^{46} +4.68512 q^{47} -1.13283 q^{48} -5.14808 q^{49} -1.00000 q^{50} +1.05812 q^{51} +1.00000 q^{52} -2.16268 q^{53} -5.34321 q^{54} +4.03486 q^{55} +1.36085 q^{56} -5.02746 q^{57} -3.64335 q^{58} -5.66877 q^{59} +1.13283 q^{60} -7.38270 q^{61} -1.00000 q^{62} +2.33618 q^{63} +1.00000 q^{64} -1.00000 q^{65} -4.57080 q^{66} -7.18954 q^{67} -0.934050 q^{68} -0.0757610 q^{69} -1.36085 q^{70} -5.63632 q^{71} +1.71670 q^{72} +10.3728 q^{73} -3.52570 q^{74} -1.13283 q^{75} +4.43798 q^{76} +5.49086 q^{77} +1.13283 q^{78} -2.73521 q^{79} -1.00000 q^{80} -0.902834 q^{81} -6.16029 q^{82} +6.85404 q^{83} +1.54161 q^{84} +0.934050 q^{85} -10.9140 q^{86} -4.12729 q^{87} +4.03486 q^{88} -3.44321 q^{89} -1.71670 q^{90} -1.36085 q^{91} +0.0668778 q^{92} -1.13283 q^{93} -4.68512 q^{94} -4.43798 q^{95} +1.13283 q^{96} -0.160657 q^{97} +5.14808 q^{98} +6.92665 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{2} - q^{3} + 6 q^{4} - 6 q^{5} + q^{6} + 4 q^{7} - 6 q^{8} - q^{9}+O(q^{10}) \) 6 * q - 6 * q^2 - q^3 + 6 * q^4 - 6 * q^5 + q^6 + 4 * q^7 - 6 * q^8 - q^9	\( 6 q - 6 q^{2} - q^{3} + 6 q^{4} - 6 q^{5} + q^{6} + 4 q^{7} - 6 q^{8} - q^{9} + 6 q^{10} - 6 q^{11} - q^{12} + 6 q^{13} - 4 q^{14} + q^{15} + 6 q^{16} - 4 q^{17} + q^{18} + 3 q^{19} - 6 q^{20} - q^{21} + 6 q^{22} - 7 q^{23} + q^{24} + 6 q^{25} - 6 q^{26} - q^{27} + 4 q^{28} - 14 q^{29} - q^{30} + 6 q^{31} - 6 q^{32} - 2 q^{33} + 4 q^{34} - 4 q^{35} - q^{36} + 14 q^{37} - 3 q^{38} - q^{39} + 6 q^{40} - 12 q^{41} + q^{42} + 3 q^{43} - 6 q^{44} + q^{45} + 7 q^{46} - 2 q^{47} - q^{48} - 6 q^{49} - 6 q^{50} - 7 q^{51} + 6 q^{52} - 4 q^{53} + q^{54} + 6 q^{55} - 4 q^{56} + 13 q^{57} + 14 q^{58} - 17 q^{59} + q^{60} - 5 q^{61} - 6 q^{62} - q^{63} + 6 q^{64} - 6 q^{65} + 2 q^{66} + 8 q^{67} - 4 q^{68} - 8 q^{69} + 4 q^{70} - 16 q^{71} + q^{72} + 11 q^{73} - 14 q^{74} - q^{75} + 3 q^{76} - 15 q^{77} + q^{78} - 2 q^{79} - 6 q^{80} - 26 q^{81} + 12 q^{82} + 4 q^{83} - q^{84} + 4 q^{85} - 3 q^{86} + 7 q^{87} + 6 q^{88} - 8 q^{89} - q^{90} + 4 q^{91} - 7 q^{92} - q^{93} + 2 q^{94} - 3 q^{95} + q^{96} + 5 q^{97} + 6 q^{98} + 15 q^{99}+O(q^{100}) \) 6 * q - 6 * q^2 - q^3 + 6 * q^4 - 6 * q^5 + q^6 + 4 * q^7 - 6 * q^8 - q^9 + 6 * q^10 - 6 * q^11 - q^12 + 6 * q^13 - 4 * q^14 + q^15 + 6 * q^16 - 4 * q^17 + q^18 + 3 * q^19 - 6 * q^20 - q^21 + 6 * q^22 - 7 * q^23 + q^24 + 6 * q^25 - 6 * q^26 - q^27 + 4 * q^28 - 14 * q^29 - q^30 + 6 * q^31 - 6 * q^32 - 2 * q^33 + 4 * q^34 - 4 * q^35 - q^36 + 14 * q^37 - 3 * q^38 - q^39 + 6 * q^40 - 12 * q^41 + q^42 + 3 * q^43 - 6 * q^44 + q^45 + 7 * q^46 - 2 * q^47 - q^48 - 6 * q^49 - 6 * q^50 - 7 * q^51 + 6 * q^52 - 4 * q^53 + q^54 + 6 * q^55 - 4 * q^56 + 13 * q^57 + 14 * q^58 - 17 * q^59 + q^60 - 5 * q^61 - 6 * q^62 - q^63 + 6 * q^64 - 6 * q^65 + 2 * q^66 + 8 * q^67 - 4 * q^68 - 8 * q^69 + 4 * q^70 - 16 * q^71 + q^72 + 11 * q^73 - 14 * q^74 - q^75 + 3 * q^76 - 15 * q^77 + q^78 - 2 * q^79 - 6 * q^80 - 26 * q^81 + 12 * q^82 + 4 * q^83 - q^84 + 4 * q^85 - 3 * q^86 + 7 * q^87 + 6 * q^88 - 8 * q^89 - q^90 + 4 * q^91 - 7 * q^92 - q^93 + 2 * q^94 - 3 * q^95 + q^96 + 5 * q^97 + 6 * q^98 + 15 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more