LMFDB - Embedded newform 8046.2.a.p.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8046,2,Mod(1,8046)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8046, 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("8046.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8046 = 2 \cdot 3^{3} \cdot 149 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8046.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:	$64.2476334663$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 5 x^{11} - 23 x^{10} + 142 x^{9} + 104 x^{8} - 1302 x^{7} + 607 x^{6} + 4323 x^{5} - 4461 x^{4} + \cdots - 553 $ x^12 - 5x^11 - 23x^10 + 142x^9 + 104x^8 - 1302x^7 + 607x^6 + 4323x^5 - 4461x^4 - 3333x^3 + 4805x^2 - 224*x - 553
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.302744$ of defining polynomial
Character	$\chi$	$=$	8046.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -0.302744 q^{5} -1.13869 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} -0.302744 q^{5} -1.13869 q^{7} +1.00000 q^{8} -0.302744 q^{10} -4.19192 q^{11} +2.26439 q^{13} -1.13869 q^{14} +1.00000 q^{16} +0.509122 q^{17} -0.322285 q^{19} -0.302744 q^{20} -4.19192 q^{22} +2.53341 q^{23} -4.90835 q^{25} +2.26439 q^{26} -1.13869 q^{28} +1.17124 q^{29} -3.06906 q^{31} +1.00000 q^{32} +0.509122 q^{34} +0.344732 q^{35} +7.76152 q^{37} -0.322285 q^{38} -0.302744 q^{40} -0.564009 q^{41} -1.22686 q^{43} -4.19192 q^{44} +2.53341 q^{46} +2.74601 q^{47} -5.70338 q^{49} -4.90835 q^{50} +2.26439 q^{52} +13.3891 q^{53} +1.26908 q^{55} -1.13869 q^{56} +1.17124 q^{58} +7.37465 q^{59} -10.1082 q^{61} -3.06906 q^{62} +1.00000 q^{64} -0.685528 q^{65} +13.0727 q^{67} +0.509122 q^{68} +0.344732 q^{70} +10.9236 q^{71} +5.03088 q^{73} +7.76152 q^{74} -0.322285 q^{76} +4.77332 q^{77} +9.20954 q^{79} -0.302744 q^{80} -0.564009 q^{82} +8.66971 q^{83} -0.154133 q^{85} -1.22686 q^{86} -4.19192 q^{88} +6.47316 q^{89} -2.57844 q^{91} +2.53341 q^{92} +2.74601 q^{94} +0.0975698 q^{95} +8.10932 q^{97} -5.70338 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{2} + 12 q^{4} + 5 q^{5} + 6 q^{7} + 12 q^{8}+O(q^{10}) $ 12 * q + 12 * q^2 + 12 * q^4 + 5 * q^5 + 6 * q^7 + 12 * q^8	$ 12 q + 12 q^{2} + 12 q^{4} + 5 q^{5} + 6 q^{7} + 12 q^{8} + 5 q^{10} + 6 q^{11} + 3 q^{13} + 6 q^{14} + 12 q^{16} + 6 q^{17} + 8 q^{19} + 5 q^{20} + 6 q^{22} + 11 q^{23} + 11 q^{25} + 3 q^{26} + 6 q^{28} + 29 q^{29} + 2 q^{31} + 12 q^{32} + 6 q^{34} + 4 q^{35} + 5 q^{37} + 8 q^{38} + 5 q^{40} + 22 q^{41} + 9 q^{43} + 6 q^{44} + 11 q^{46} + 15 q^{47} + 14 q^{49} + 11 q^{50} + 3 q^{52} + 12 q^{53} + 13 q^{55} + 6 q^{56} + 29 q^{58} + 34 q^{59} - 4 q^{61} + 2 q^{62} + 12 q^{64} + 12 q^{65} + q^{67} + 6 q^{68} + 4 q^{70} + 21 q^{71} - 2 q^{73} + 5 q^{74} + 8 q^{76} + 34 q^{77} + 9 q^{79} + 5 q^{80} + 22 q^{82} + 10 q^{83} + 5 q^{85} + 9 q^{86} + 6 q^{88} - 2 q^{89} + 17 q^{91} + 11 q^{92} + 15 q^{94} + 69 q^{95} - 13 q^{97} + 14 q^{98}+O(q^{100}) $ 12 * q + 12 * q^2 + 12 * q^4 + 5 * q^5 + 6 * q^7 + 12 * q^8 + 5 * q^10 + 6 * q^11 + 3 * q^13 + 6 * q^14 + 12 * q^16 + 6 * q^17 + 8 * q^19 + 5 * q^20 + 6 * q^22 + 11 * q^23 + 11 * q^25 + 3 * q^26 + 6 * q^28 + 29 * q^29 + 2 * q^31 + 12 * q^32 + 6 * q^34 + 4 * q^35 + 5 * q^37 + 8 * q^38 + 5 * q^40 + 22 * q^41 + 9 * q^43 + 6 * q^44 + 11 * q^46 + 15 * q^47 + 14 * q^49 + 11 * q^50 + 3 * q^52 + 12 * q^53 + 13 * q^55 + 6 * q^56 + 29 * q^58 + 34 * q^59 - 4 * q^61 + 2 * q^62 + 12 * q^64 + 12 * q^65 + q^67 + 6 * q^68 + 4 * q^70 + 21 * q^71 - 2 * q^73 + 5 * q^74 + 8 * q^76 + 34 * q^77 + 9 * q^79 + 5 * q^80 + 22 * q^82 + 10 * q^83 + 5 * q^85 + 9 * q^86 + 6 * q^88 - 2 * q^89 + 17 * q^91 + 11 * q^92 + 15 * q^94 + 69 * q^95 - 13 * q^97 + 14 * q^98

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
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−0.302744	−0.135391	−0.0676955	−	0.997706i	$-0.521565\pi$
$5$	−0.302744	−0.135391	−0.0676955	+	0.997706i	$0.521565\pi$
$6$	0	0
$7$	−1.13869	−0.430386	−0.215193	−	0.976572i	$-0.569038\pi$
$7$	−1.13869	−0.430386	−0.215193	+	0.976572i	$0.569038\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−0.302744	−0.0957359
$11$	−4.19192	−1.26391	−0.631956	−	0.775004i	$-0.717748\pi$
$11$	−4.19192	−1.26391	−0.631956	+	0.775004i	$0.717748\pi$
$12$	0	0
$13$	2.26439	0.628027	0.314014	−	0.949418i	$-0.398326\pi$
$13$	2.26439	0.628027	0.314014	+	0.949418i	$0.398326\pi$
$14$	−1.13869	−0.304329
$15$	0	0
$16$	1.00000	0.250000
$17$	0.509122	0.123480	0.0617401	−	0.998092i	$-0.480335\pi$
$17$	0.509122	0.123480	0.0617401	+	0.998092i	$0.480335\pi$
$18$	0	0
$19$	−0.322285	−0.0739373	−0.0369686	−	0.999316i	$-0.511770\pi$
$19$	−0.322285	−0.0739373	−0.0369686	+	0.999316i	$0.511770\pi$
$20$	−0.302744	−0.0676955
$21$	0	0
$22$	−4.19192	−0.893721
$23$	2.53341	0.528254	0.264127	−	0.964488i	$-0.414916\pi$
$23$	2.53341	0.528254	0.264127	+	0.964488i	$0.414916\pi$
$24$	0	0
$25$	−4.90835	−0.981669
$26$	2.26439	0.444083
$27$	0	0
$28$	−1.13869	−0.215193
$29$	1.17124	0.217494	0.108747	−	0.994069i	$-0.465316\pi$
$29$	1.17124	0.217494	0.108747	+	0.994069i	$0.465316\pi$
$30$	0	0
$31$	−3.06906	−0.551219	−0.275610	−	0.961270i	$-0.588880\pi$
$31$	−3.06906	−0.551219	−0.275610	+	0.961270i	$0.588880\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	0.509122	0.0873136
$35$	0.344732	0.0582704
$36$	0	0
$37$	7.76152	1.27599	0.637993	−	0.770042i	$-0.279765\pi$
$37$	7.76152	1.27599	0.637993	+	0.770042i	$0.279765\pi$
$38$	−0.322285	−0.0522816
$39$	0	0
$40$	−0.302744	−0.0478680
$41$	−0.564009	−0.0880835	−0.0440417	−	0.999030i	$-0.514023\pi$
$41$	−0.564009	−0.0880835	−0.0440417	+	0.999030i	$0.514023\pi$
$42$	0	0
$43$	−1.22686	−0.187095	−0.0935473	−	0.995615i	$-0.529821\pi$
$43$	−1.22686	−0.187095	−0.0935473	+	0.995615i	$0.529821\pi$
$44$	−4.19192	−0.631956
$45$	0	0
$46$	2.53341	0.373532
$47$	2.74601	0.400547	0.200274	−	0.979740i	$-0.435817\pi$
$47$	2.74601	0.400547	0.200274	+	0.979740i	$0.435817\pi$
$48$	0	0
$49$	−5.70338	−0.814768
$50$	−4.90835	−0.694145
$51$	0	0
$52$	2.26439	0.314014
$53$	13.3891	1.83913	0.919567	−	0.392933i	$-0.128540\pi$
$53$	13.3891	1.83913	0.919567	+	0.392933i	$0.128540\pi$
$54$	0	0
$55$	1.26908	0.171123
$56$	−1.13869	−0.152164
$57$	0	0
$58$	1.17124	0.153791
$59$	7.37465	0.960098	0.480049	−	0.877242i	$-0.340619\pi$
$59$	7.37465	0.960098	0.480049	+	0.877242i	$0.340619\pi$
$60$	0	0
$61$	−10.1082	−1.29422	−0.647112	−	0.762395i	$-0.724023\pi$
$61$	−10.1082	−1.29422	−0.647112	+	0.762395i	$0.724023\pi$
$62$	−3.06906	−0.389771
$63$	0	0
$64$	1.00000	0.125000
$65$	−0.685528	−0.0850293
$66$	0	0
$67$	13.0727	1.59709	0.798543	−	0.601938i	$-0.205605\pi$
$67$	13.0727	1.59709	0.798543	+	0.601938i	$0.205605\pi$
$68$	0.509122	0.0617401
$69$	0	0
$70$	0.344732	0.0412034
$71$	10.9236	1.29639	0.648196	−	0.761474i	$-0.275524\pi$
$71$	10.9236	1.29639	0.648196	+	0.761474i	$0.275524\pi$
$72$	0	0
$73$	5.03088	0.588820	0.294410	−	0.955679i	$-0.404877\pi$
$73$	5.03088	0.588820	0.294410	+	0.955679i	$0.404877\pi$
$74$	7.76152	0.902258
$75$	0	0
$76$	−0.322285	−0.0369686
$77$	4.77332	0.543970
$78$	0	0
$79$	9.20954	1.03615	0.518077	−	0.855334i	$-0.326648\pi$
$79$	9.20954	1.03615	0.518077	+	0.855334i	$0.326648\pi$
$80$	−0.302744	−0.0338478
$81$	0	0
$82$	−0.564009	−0.0622844
$83$	8.66971	0.951625	0.475812	−	0.879547i	$-0.342154\pi$
$83$	8.66971	0.951625	0.475812	+	0.879547i	$0.342154\pi$
$84$	0	0
$85$	−0.154133	−0.0167181
$86$	−1.22686	−0.132296
$87$	0	0
$88$	−4.19192	−0.446861
$89$	6.47316	0.686153	0.343077	−	0.939307i	$-0.388531\pi$
$89$	6.47316	0.686153	0.343077	+	0.939307i	$0.388531\pi$
$90$	0	0
$91$	−2.57844	−0.270294
$92$	2.53341	0.264127
$93$	0	0
$94$	2.74601	0.283230
$95$	0.0975698	0.0100104
$96$	0	0
$97$	8.10932	0.823377	0.411688	−	0.911325i	$-0.364939\pi$
$97$	8.10932	0.823377	0.411688	+	0.911325i	$0.364939\pi$
$98$	−5.70338	−0.576128
$99$	0	0
$100$	−4.90835	−0.490835
$101$	−17.7845	−1.76963	−0.884814	−	0.465944i	$-0.845715\pi$
$101$	−17.7845	−1.76963	−0.884814	+	0.465944i	$0.845715\pi$
$102$	0	0
$103$	−17.3242	−1.70700	−0.853501	−	0.521091i	$-0.825525\pi$
$103$	−17.3242	−1.70700	−0.853501	+	0.521091i	$0.825525\pi$
$104$	2.26439	0.222041
$105$	0	0
$106$	13.3891	1.30046
$107$	−3.54307	−0.342522	−0.171261	−	0.985226i	$-0.554784\pi$
$107$	−3.54307	−0.342522	−0.171261	+	0.985226i	$0.554784\pi$
$108$	0	0
$109$	−10.4860	−1.00438	−0.502189	−	0.864758i	$-0.667472\pi$
$109$	−10.4860	−1.00438	−0.502189	+	0.864758i	$0.667472\pi$
$110$	1.26908	0.121002
$111$	0	0
$112$	−1.13869	−0.107596
$113$	−1.22465	−0.115205	−0.0576027	−	0.998340i	$-0.518346\pi$
$113$	−1.22465	−0.115205	−0.0576027	+	0.998340i	$0.518346\pi$
$114$	0	0
$115$	−0.766975	−0.0715208
$116$	1.17124	0.108747
$117$	0	0
$118$	7.37465	0.678892
$119$	−0.579734	−0.0531441
$120$	0	0
$121$	6.57223	0.597475
$122$	−10.1082	−0.915154
$123$	0	0
$124$	−3.06906	−0.275610
$125$	2.99969	0.268300
$126$	0	0
$127$	1.82281	0.161748	0.0808740	−	0.996724i	$-0.474229\pi$
$127$	1.82281	0.161748	0.0808740	+	0.996724i	$0.474229\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−0.685528	−0.0601248
$131$	13.0355	1.13892	0.569459	−	0.822019i	$-0.307153\pi$
$131$	13.0355	1.13892	0.569459	+	0.822019i	$0.307153\pi$
$132$	0	0
$133$	0.366984	0.0318216
$134$	13.0727	1.12931
$135$	0	0
$136$	0.509122	0.0436568
$137$	13.7648	1.17600	0.588001	−	0.808860i	$-0.299915\pi$
$137$	13.7648	1.17600	0.588001	+	0.808860i	$0.299915\pi$
$138$	0	0
$139$	−4.12288	−0.349698	−0.174849	−	0.984595i	$-0.555944\pi$
$139$	−4.12288	−0.349698	−0.174849	+	0.984595i	$0.555944\pi$
$140$	0.344732	0.0291352
$141$	0	0
$142$	10.9236	0.916687
$143$	−9.49213	−0.793772
$144$	0	0
$145$	−0.354585	−0.0294467
$146$	5.03088	0.416358
$147$	0	0
$148$	7.76152	0.637993
$149$	−1.00000	−0.0819232
$149$	−1.00000	−0.0819232
$150$	0	0
$151$	23.2099	1.88879	0.944397	−	0.328806i	$-0.106646\pi$
$151$	23.2099	1.88879	0.944397	+	0.328806i	$0.106646\pi$
$152$	−0.322285	−0.0261408
$153$	0	0
$154$	4.77332	0.384645
$155$	0.929138	0.0746302
$156$	0	0
$157$	1.38471	0.110512	0.0552560	−	0.998472i	$-0.482403\pi$
$157$	1.38471	0.110512	0.0552560	+	0.998472i	$0.482403\pi$
$158$	9.20954	0.732672
$159$	0	0
$160$	−0.302744	−0.0239340
$161$	−2.88478	−0.227353
$162$	0	0
$163$	9.83981	0.770713	0.385357	−	0.922768i	$-0.374078\pi$
$163$	9.83981	0.770713	0.385357	+	0.922768i	$0.374078\pi$
$164$	−0.564009	−0.0440417
$165$	0	0
$166$	8.66971	0.672900
$167$	7.15760	0.553872	0.276936	−	0.960888i	$-0.410681\pi$
$167$	7.15760	0.553872	0.276936	+	0.960888i	$0.410681\pi$
$168$	0	0
$169$	−7.87256	−0.605581
$170$	−0.154133	−0.0118215
$171$	0	0
$172$	−1.22686	−0.0935473
$173$	−16.7109	−1.27051	−0.635254	−	0.772303i	$-0.719104\pi$
$173$	−16.7109	−1.27051	−0.635254	+	0.772303i	$0.719104\pi$
$174$	0	0
$175$	5.58910	0.422497
$176$	−4.19192	−0.315978
$177$	0	0
$178$	6.47316	0.485184
$179$	6.22326	0.465148	0.232574	−	0.972579i	$-0.425285\pi$
$179$	6.22326	0.465148	0.232574	+	0.972579i	$0.425285\pi$
$180$	0	0
$181$	22.9159	1.70332	0.851662	−	0.524092i	$-0.175595\pi$
$181$	22.9159	1.70332	0.851662	+	0.524092i	$0.175595\pi$
$182$	−2.57844	−0.191127
$183$	0	0
$184$	2.53341	0.186766
$185$	−2.34975	−0.172757
$186$	0	0
$187$	−2.13420	−0.156068
$188$	2.74601	0.200274
$189$	0	0
$190$	0.0975698	0.00707846
$191$	0.0115679	0.000837026 0	0.000418513	−	1.00000i	$-0.499867\pi$
$191$	0.0115679	0.000837026 0	0.000418513		1.00000i	$0.499867\pi$
$192$	0	0
$193$	1.07058	0.0770621	0.0385310	−	0.999257i	$-0.487732\pi$
$193$	1.07058	0.0770621	0.0385310	+	0.999257i	$0.487732\pi$
$194$	8.10932	0.582215
$195$	0	0
$196$	−5.70338	−0.407384
$197$	18.7000	1.33232	0.666160	−	0.745809i	$-0.267937\pi$
$197$	18.7000	1.33232	0.666160	+	0.745809i	$0.267937\pi$
$198$	0	0
$199$	26.9630	1.91136	0.955680	−	0.294409i	$-0.0951227\pi$
$199$	26.9630	1.91136	0.955680	+	0.294409i	$0.0951227\pi$
$200$	−4.90835	−0.347072
$201$	0	0
$202$	−17.7845	−1.25132
$203$	−1.33368	−0.0936063
$204$	0	0
$205$	0.170750	0.0119257
$206$	−17.3242	−1.20703
$207$	0	0
$208$	2.26439	0.157007
$209$	1.35100	0.0934503
$210$	0	0
$211$	−8.89198	−0.612149	−0.306074	−	0.952008i	$-0.599016\pi$
$211$	−8.89198	−0.612149	−0.306074	+	0.952008i	$0.599016\pi$
$212$	13.3891	0.919567
$213$	0	0
$214$	−3.54307	−0.242199
$215$	0.371424	0.0253309
$216$	0	0
$217$	3.49472	0.237237
$218$	−10.4860	−0.710203
$219$	0	0
$220$	1.26908	0.0855613
$221$	1.15285	0.0775489
$222$	0	0
$223$	3.71985	0.249099	0.124550	−	0.992213i	$-0.460251\pi$
$223$	3.71985	0.249099	0.124550	+	0.992213i	$0.460251\pi$
$224$	−1.13869	−0.0760822
$225$	0	0
$226$	−1.22465	−0.0814625
$227$	14.7757	0.980697	0.490349	−	0.871526i	$-0.336869\pi$
$227$	14.7757	0.980697	0.490349	+	0.871526i	$0.336869\pi$
$228$	0	0
$229$	−29.5499	−1.95271	−0.976355	−	0.216175i	$-0.930642\pi$
$229$	−29.5499	−1.95271	−0.976355	+	0.216175i	$0.930642\pi$
$230$	−0.766975	−0.0505729
$231$	0	0
$232$	1.17124	0.0768957
$233$	−1.24854	−0.0817944	−0.0408972	−	0.999163i	$-0.513022\pi$
$233$	−1.24854	−0.0817944	−0.0408972	+	0.999163i	$0.513022\pi$
$234$	0	0
$235$	−0.831339	−0.0542305
$236$	7.37465	0.480049
$237$	0	0
$238$	−0.579734	−0.0375786
$239$	−9.32136	−0.602948	−0.301474	−	0.953474i	$-0.597479\pi$
$239$	−9.32136	−0.602948	−0.301474	+	0.953474i	$0.597479\pi$
$240$	0	0
$241$	−9.06545	−0.583957	−0.291979	−	0.956425i	$-0.594314\pi$
$241$	−9.06545	−0.583957	−0.291979	+	0.956425i	$0.594314\pi$
$242$	6.57223	0.422479
$243$	0	0
$244$	−10.1082	−0.647112
$245$	1.72666	0.110312
$246$	0	0
$247$	−0.729778	−0.0464347
$248$	−3.06906	−0.194886
$249$	0	0
$250$	2.99969	0.189717
$251$	6.14955	0.388156	0.194078	−	0.980986i	$-0.437828\pi$
$251$	6.14955	0.388156	0.194078	+	0.980986i	$0.437828\pi$
$252$	0	0
$253$	−10.6199	−0.667666
$254$	1.82281	0.114373
$255$	0	0
$256$	1.00000	0.0625000
$257$	3.94539	0.246107	0.123053	−	0.992400i	$-0.460731\pi$
$257$	3.94539	0.246107	0.123053	+	0.992400i	$0.460731\pi$
$258$	0	0
$259$	−8.83799	−0.549166
$260$	−0.685528	−0.0425147
$261$	0	0
$262$	13.0355	0.805337
$263$	26.6881	1.64566	0.822830	−	0.568288i	$-0.192394\pi$
$263$	26.6881	1.64566	0.822830	+	0.568288i	$0.192394\pi$
$264$	0	0
$265$	−4.05346	−0.249002
$266$	0.366984	0.0225012
$267$	0	0
$268$	13.0727	0.798543
$269$	−20.4579	−1.24734	−0.623671	−	0.781687i	$-0.714359\pi$
$269$	−20.4579	−1.24734	−0.623671	+	0.781687i	$0.714359\pi$
$270$	0	0
$271$	7.14871	0.434253	0.217127	−	0.976143i	$-0.430332\pi$
$271$	7.14871	0.434253	0.217127	+	0.976143i	$0.430332\pi$
$272$	0.509122	0.0308700
$273$	0	0
$274$	13.7648	0.831559
$275$	20.5754	1.24074
$276$	0	0
$277$	−16.5076	−0.991847	−0.495924	−	0.868366i	$-0.665170\pi$
$277$	−16.5076	−0.991847	−0.495924	+	0.868366i	$0.665170\pi$
$278$	−4.12288	−0.247274
$279$	0	0
$280$	0.344732	0.0206017
$281$	6.49409	0.387405	0.193702	−	0.981060i	$-0.437950\pi$
$281$	6.49409	0.387405	0.193702	+	0.981060i	$0.437950\pi$
$282$	0	0
$283$	19.8510	1.18002	0.590011	−	0.807395i	$-0.299124\pi$
$283$	19.8510	1.18002	0.590011	+	0.807395i	$0.299124\pi$
$284$	10.9236	0.648196
$285$	0	0
$286$	−9.49213	−0.561282
$287$	0.642234	0.0379099
$288$	0	0
$289$	−16.7408	−0.984753
$290$	−0.354585	−0.0208220
$291$	0	0
$292$	5.03088	0.294410
$293$	29.4732	1.72185	0.860923	−	0.508736i	$-0.169887\pi$
$293$	29.4732	1.72185	0.860923	+	0.508736i	$0.169887\pi$
$294$	0	0
$295$	−2.23263	−0.129989
$296$	7.76152	0.451129
$297$	0	0
$298$	−1.00000	−0.0579284
$299$	5.73663	0.331758
$300$	0	0
$301$	1.39702	0.0805228
$302$	23.2099	1.33558
$303$	0	0
$304$	−0.322285	−0.0184843
$305$	3.06020	0.175226
$306$	0	0
$307$	14.0890	0.804103	0.402051	−	0.915617i	$-0.368297\pi$
$307$	14.0890	0.804103	0.402051	+	0.915617i	$0.368297\pi$
$308$	4.77332	0.271985
$309$	0	0
$310$	0.929138	0.0527715
$311$	23.9808	1.35982	0.679912	−	0.733294i	$-0.262018\pi$
$311$	23.9808	1.35982	0.679912	+	0.733294i	$0.262018\pi$
$312$	0	0
$313$	4.56586	0.258078	0.129039	−	0.991640i	$-0.458811\pi$
$313$	4.56586	0.258078	0.129039	+	0.991640i	$0.458811\pi$
$314$	1.38471	0.0781438
$315$	0	0
$316$	9.20954	0.518077
$317$	−14.8773	−0.835593	−0.417797	−	0.908541i	$-0.637198\pi$
$317$	−14.8773	−0.835593	−0.417797	+	0.908541i	$0.637198\pi$
$318$	0	0
$319$	−4.90975	−0.274893
$320$	−0.302744	−0.0169239
$321$	0	0
$322$	−2.88478	−0.160763
$323$	−0.164082	−0.00912979
$324$	0	0
$325$	−11.1144	−0.616515
$326$	9.83981	0.544977
$327$	0	0
$328$	−0.564009	−0.0311422
$329$	−3.12687	−0.172390
$330$	0	0
$331$	−28.4548	−1.56402	−0.782009	−	0.623267i	$-0.785805\pi$
$331$	−28.4548	−1.56402	−0.782009	+	0.623267i	$0.785805\pi$
$332$	8.66971	0.475812
$333$	0	0
$334$	7.15760	0.391646
$335$	−3.95768	−0.216231
$336$	0	0
$337$	22.7248	1.23790	0.618948	−	0.785432i	$-0.287559\pi$
$337$	22.7248	1.23790	0.618948	+	0.785432i	$0.287559\pi$
$338$	−7.87256	−0.428211
$339$	0	0
$340$	−0.154133	−0.00835905
$341$	12.8653	0.696693
$342$	0	0
$343$	14.4653	0.781050
$344$	−1.22686	−0.0661479
$345$	0	0
$346$	−16.7109	−0.898384
$347$	6.13192	0.329179	0.164589	−	0.986362i	$-0.447370\pi$
$347$	6.13192	0.329179	0.164589	+	0.986362i	$0.447370\pi$
$348$	0	0
$349$	−32.7601	−1.75361	−0.876804	−	0.480848i	$-0.840329\pi$
$349$	−32.7601	−1.75361	−0.876804	+	0.480848i	$0.840329\pi$
$350$	5.58910	0.298750
$351$	0	0
$352$	−4.19192	−0.223430
$353$	28.7010	1.52760	0.763801	−	0.645452i	$-0.223331\pi$
$353$	28.7010	1.52760	0.763801	+	0.645452i	$0.223331\pi$
$354$	0	0
$355$	−3.30705	−0.175520
$356$	6.47316	0.343077
$357$	0	0
$358$	6.22326	0.328909
$359$	−18.9227	−0.998700	−0.499350	−	0.866400i	$-0.666428\pi$
$359$	−18.9227	−0.998700	−0.499350	+	0.866400i	$0.666428\pi$
$360$	0	0
$361$	−18.8961	−0.994533
$362$	22.9159	1.20443
$363$	0	0
$364$	−2.57844	−0.135147
$365$	−1.52307	−0.0797209
$366$	0	0
$367$	−27.3441	−1.42735	−0.713675	−	0.700477i	$-0.752971\pi$
$367$	−27.3441	−1.42735	−0.713675	+	0.700477i	$0.752971\pi$
$368$	2.53341	0.132063
$369$	0	0
$370$	−2.34975	−0.122158
$371$	−15.2461	−0.791537
$372$	0	0
$373$	24.6569	1.27668	0.638342	−	0.769753i	$-0.279621\pi$
$373$	24.6569	1.27668	0.638342	+	0.769753i	$0.279621\pi$
$374$	−2.13420	−0.110357
$375$	0	0
$376$	2.74601	0.141615
$377$	2.65214	0.136592
$378$	0	0
$379$	−9.51003	−0.488497	−0.244249	−	0.969713i	$-0.578541\pi$
$379$	−9.51003	−0.488497	−0.244249	+	0.969713i	$0.578541\pi$
$380$	0.0975698	0.00500522
$381$	0	0
$382$	0.0115679	0.000591867 0
$383$	−22.4974	−1.14956	−0.574781	−	0.818307i	$-0.694913\pi$
$383$	−22.4974	−1.14956	−0.574781	+	0.818307i	$0.694913\pi$
$384$	0	0
$385$	−1.44509	−0.0736487
$386$	1.07058	0.0544911
$387$	0	0
$388$	8.10932	0.411688
$389$	4.68766	0.237674	0.118837	−	0.992914i	$-0.462083\pi$
$389$	4.68766	0.237674	0.118837	+	0.992914i	$0.462083\pi$
$390$	0	0
$391$	1.28982	0.0652288
$392$	−5.70338	−0.288064
$393$	0	0
$394$	18.7000	0.942092
$395$	−2.78813	−0.140286
$396$	0	0
$397$	9.81638	0.492670	0.246335	−	0.969185i	$-0.420774\pi$
$397$	9.81638	0.492670	0.246335	+	0.969185i	$0.420774\pi$
$398$	26.9630	1.35153
$399$	0	0
$400$	−4.90835	−0.245417
$401$	18.3259	0.915153	0.457576	−	0.889170i	$-0.348718\pi$
$401$	18.3259	0.915153	0.457576	+	0.889170i	$0.348718\pi$
$402$	0	0
$403$	−6.94953	−0.346181
$404$	−17.7845	−0.884814
$405$	0	0
$406$	−1.33368	−0.0661896
$407$	−32.5357	−1.61273
$408$	0	0
$409$	−12.2531	−0.605878	−0.302939	−	0.953010i	$-0.597968\pi$
$409$	−12.2531	−0.605878	−0.302939	+	0.953010i	$0.597968\pi$
$410$	0.170750	0.00843275
$411$	0	0
$412$	−17.3242	−0.853501
$413$	−8.39747	−0.413213
$414$	0	0
$415$	−2.62470	−0.128841
$416$	2.26439	0.111021
$417$	0	0
$418$	1.35100	0.0660793
$419$	−11.2802	−0.551074	−0.275537	−	0.961290i	$-0.588856\pi$
$419$	−11.2802	−0.551074	−0.275537	+	0.961290i	$0.588856\pi$
$420$	0	0
$421$	−18.0498	−0.879695	−0.439847	−	0.898073i	$-0.644967\pi$
$421$	−18.0498	−0.879695	−0.439847	+	0.898073i	$0.644967\pi$
$422$	−8.89198	−0.432855
$423$	0	0
$424$	13.3891	0.650232
$425$	−2.49894	−0.121217
$426$	0	0
$427$	11.5102	0.557015
$428$	−3.54307	−0.171261
$429$	0	0
$430$	0.371424	0.0179117
$431$	12.4956	0.601891	0.300945	−	0.953641i	$-0.402698\pi$
$431$	12.4956	0.601891	0.300945	+	0.953641i	$0.402698\pi$
$432$	0	0
$433$	27.9729	1.34429	0.672146	−	0.740419i	$-0.265373\pi$
$433$	27.9729	1.34429	0.672146	+	0.740419i	$0.265373\pi$
$434$	3.49472	0.167752
$435$	0	0
$436$	−10.4860	−0.502189
$437$	−0.816482	−0.0390576
$438$	0	0
$439$	30.6628	1.46346	0.731728	−	0.681597i	$-0.238714\pi$
$439$	30.6628	1.46346	0.731728	+	0.681597i	$0.238714\pi$
$440$	1.26908	0.0605009
$441$	0	0
$442$	1.15285	0.0548354
$443$	34.2986	1.62958	0.814788	−	0.579758i	$-0.196853\pi$
$443$	34.2986	1.62958	0.814788	+	0.579758i	$0.196853\pi$
$444$	0	0
$445$	−1.95971	−0.0928990
$446$	3.71985	0.176140
$447$	0	0
$448$	−1.13869	−0.0537982
$449$	−29.6723	−1.40032	−0.700162	−	0.713984i	$-0.746889\pi$
$449$	−29.6723	−1.40032	−0.700162	+	0.713984i	$0.746889\pi$
$450$	0	0
$451$	2.36428	0.111330
$452$	−1.22465	−0.0576027
$453$	0	0
$454$	14.7757	0.693458
$455$	0.780607	0.0365954
$456$	0	0
$457$	15.8126	0.739680	0.369840	−	0.929095i	$-0.379413\pi$
$457$	15.8126	0.739680	0.369840	+	0.929095i	$0.379413\pi$
$458$	−29.5499	−1.38077
$459$	0	0
$460$	−0.766975	−0.0357604
$461$	−32.3273	−1.50563	−0.752817	−	0.658230i	$-0.771305\pi$
$461$	−32.3273	−1.50563	−0.752817	+	0.658230i	$0.771305\pi$
$462$	0	0
$463$	−26.5600	−1.23435	−0.617173	−	0.786828i	$-0.711722\pi$
$463$	−26.5600	−1.23435	−0.617173	+	0.786828i	$0.711722\pi$
$464$	1.17124	0.0543735
$465$	0	0
$466$	−1.24854	−0.0578374
$467$	−22.5118	−1.04172	−0.520861	−	0.853642i	$-0.674389\pi$
$467$	−22.5118	−1.04172	−0.520861	+	0.853642i	$0.674389\pi$
$468$	0	0
$469$	−14.8858	−0.687363
$470$	−0.831339	−0.0383468
$471$	0	0
$472$	7.37465	0.339446
$473$	5.14291	0.236471
$474$	0	0
$475$	1.58189	0.0725820
$476$	−0.579734	−0.0265720
$477$	0	0
$478$	−9.32136	−0.426349
$479$	16.7309	0.764453	0.382227	−	0.924069i	$-0.375157\pi$
$479$	16.7309	0.764453	0.382227	+	0.924069i	$0.375157\pi$
$480$	0	0
$481$	17.5751	0.801354
$482$	−9.06545	−0.412920
$483$	0	0
$484$	6.57223	0.298738
$485$	−2.45504	−0.111478
$486$	0	0
$487$	32.8751	1.48971	0.744856	−	0.667225i	$-0.232518\pi$
$487$	32.8751	1.48971	0.744856	+	0.667225i	$0.232518\pi$
$488$	−10.1082	−0.457577
$489$	0	0
$490$	1.72666	0.0780026
$491$	40.0650	1.80811	0.904053	−	0.427420i	$-0.140577\pi$
$491$	40.0650	1.80811	0.904053	+	0.427420i	$0.140577\pi$
$492$	0	0
$493$	0.596304	0.0268562
$494$	−0.729778	−0.0328343
$495$	0	0
$496$	−3.06906	−0.137805
$497$	−12.4386	−0.557948
$498$	0	0
$499$	−10.5335	−0.471544	−0.235772	−	0.971808i	$-0.575762\pi$
$499$	−10.5335	−0.471544	−0.235772	+	0.971808i	$0.575762\pi$
$500$	2.99969	0.134150
$501$	0	0
$502$	6.14955	0.274468
$503$	−31.6695	−1.41207	−0.706036	−	0.708176i	$-0.749518\pi$
$503$	−31.6695	−1.41207	−0.706036	+	0.708176i	$0.749518\pi$
$504$	0	0
$505$	5.38416	0.239592
$506$	−10.6199	−0.472111
$507$	0	0
$508$	1.82281	0.0808740
$509$	8.08562	0.358389	0.179194	−	0.983814i	$-0.442651\pi$
$509$	8.08562	0.358389	0.179194	+	0.983814i	$0.442651\pi$
$510$	0	0
$511$	−5.72863	−0.253420
$512$	1.00000	0.0441942
$513$	0	0
$514$	3.94539	0.174024
$515$	5.24478	0.231113
$516$	0	0
$517$	−11.5111	−0.506257
$518$	−8.83799	−0.388319
$519$	0	0
$520$	−0.685528	−0.0300624
$521$	−9.43477	−0.413345	−0.206672	−	0.978410i	$-0.566263\pi$
$521$	−9.43477	−0.413345	−0.206672	+	0.978410i	$0.566263\pi$
$522$	0	0
$523$	−5.13209	−0.224411	−0.112205	−	0.993685i	$-0.535791\pi$
$523$	−5.13209	−0.224411	−0.112205	+	0.993685i	$0.535791\pi$
$524$	13.0355	0.569459
$525$	0	0
$526$	26.6881	1.16366
$527$	−1.56252	−0.0680646
$528$	0	0
$529$	−16.5818	−0.720948
$530$	−4.05346	−0.176071
$531$	0	0
$532$	0.366984	0.0159108
$533$	−1.27713	−0.0553188
$534$	0	0
$535$	1.07264	0.0463744
$536$	13.0727	0.564655
$537$	0	0
$538$	−20.4579	−0.882004
$539$	23.9081	1.02980
$540$	0	0
$541$	−23.7350	−1.02045	−0.510223	−	0.860042i	$-0.670437\pi$
$541$	−23.7350	−1.02045	−0.510223	+	0.860042i	$0.670437\pi$
$542$	7.14871	0.307063
$543$	0	0
$544$	0.509122	0.0218284
$545$	3.17458	0.135984
$546$	0	0
$547$	4.93778	0.211124	0.105562	−	0.994413i	$-0.466336\pi$
$547$	4.93778	0.211124	0.105562	+	0.994413i	$0.466336\pi$
$548$	13.7648	0.588001
$549$	0	0
$550$	20.5754	0.877339
$551$	−0.377473	−0.0160809
$552$	0	0
$553$	−10.4868	−0.445946
$554$	−16.5076	−0.701342
$555$	0	0
$556$	−4.12288	−0.174849
$557$	−35.3569	−1.49812	−0.749060	−	0.662502i	$-0.769495\pi$
$557$	−35.3569	−1.49812	−0.749060	+	0.662502i	$0.769495\pi$
$558$	0	0
$559$	−2.77809	−0.117500
$560$	0.344732	0.0145676
$561$	0	0
$562$	6.49409	0.273936
$563$	18.7209	0.788993	0.394496	−	0.918898i	$-0.370919\pi$
$563$	18.7209	0.788993	0.394496	+	0.918898i	$0.370919\pi$
$564$	0	0
$565$	0.370755	0.0155978
$566$	19.8510	0.834401
$567$	0	0
$568$	10.9236	0.458343
$569$	24.2461	1.01645	0.508225	−	0.861224i	$-0.330302\pi$
$569$	24.2461	1.01645	0.508225	+	0.861224i	$0.330302\pi$
$570$	0	0
$571$	−40.8599	−1.70993	−0.854967	−	0.518682i	$-0.826423\pi$
$571$	−40.8599	−1.70993	−0.854967	+	0.518682i	$0.826423\pi$
$572$	−9.49213	−0.396886
$573$	0	0
$574$	0.642234	0.0268063
$575$	−12.4349	−0.518570
$576$	0	0
$577$	0.182177	0.00758414	0.00379207	−	0.999993i	$-0.498793\pi$
$577$	0.182177	0.00758414	0.00379207	+	0.999993i	$0.498793\pi$
$578$	−16.7408	−0.696325
$579$	0	0
$580$	−0.354585	−0.0147234
$581$	−9.87215	−0.409566
$582$	0	0
$583$	−56.1261	−2.32451
$584$	5.03088	0.208179
$585$	0	0
$586$	29.4732	1.21753
$587$	−12.0851	−0.498804	−0.249402	−	0.968400i	$-0.580234\pi$
$587$	−12.0851	−0.498804	−0.249402	+	0.968400i	$0.580234\pi$
$588$	0	0
$589$	0.989113	0.0407557
$590$	−2.23263	−0.0919159
$591$	0	0
$592$	7.76152	0.318996
$593$	41.4789	1.70333	0.851667	−	0.524083i	$-0.175592\pi$
$593$	41.4789	1.70333	0.851667	+	0.524083i	$0.175592\pi$
$594$	0	0
$595$	0.175511	0.00719524
$596$	−1.00000	−0.0409616
$597$	0	0
$598$	5.73663	0.234588
$599$	0.922883	0.0377080	0.0188540	−	0.999822i	$-0.493998\pi$
$599$	0.922883	0.0377080	0.0188540	+	0.999822i	$0.493998\pi$
$600$	0	0
$601$	−6.29019	−0.256582	−0.128291	−	0.991737i	$-0.540949\pi$
$601$	−6.29019	−0.256582	−0.128291	+	0.991737i	$0.540949\pi$
$602$	1.39702	0.0569382
$603$	0	0
$604$	23.2099	0.944397
$605$	−1.98970	−0.0808928
$606$	0	0
$607$	9.58728	0.389136	0.194568	−	0.980889i	$-0.437670\pi$
$607$	9.58728	0.389136	0.194568	+	0.980889i	$0.437670\pi$
$608$	−0.322285	−0.0130704
$609$	0	0
$610$	3.06020	0.123904
$611$	6.21804	0.251555
$612$	0	0
$613$	−3.05404	−0.123351	−0.0616757	−	0.998096i	$-0.519644\pi$
$613$	−3.05404	−0.123351	−0.0616757	+	0.998096i	$0.519644\pi$
$614$	14.0890	0.568587
$615$	0	0
$616$	4.77332	0.192323
$617$	−15.9413	−0.641773	−0.320886	−	0.947118i	$-0.603981\pi$
$617$	−15.9413	−0.641773	−0.320886	+	0.947118i	$0.603981\pi$
$618$	0	0
$619$	−3.29285	−0.132351	−0.0661755	−	0.997808i	$-0.521080\pi$
$619$	−3.29285	−0.132351	−0.0661755	+	0.997808i	$0.521080\pi$
$620$	0.929138	0.0373151
$621$	0	0
$622$	23.9808	0.961541
$623$	−7.37094	−0.295311
$624$	0	0
$625$	23.6336	0.945344
$626$	4.56586	0.182488
$627$	0	0
$628$	1.38471	0.0552560
$629$	3.95156	0.157559
$630$	0	0
$631$	−16.8515	−0.670848	−0.335424	−	0.942067i	$-0.608880\pi$
$631$	−16.8515	−0.670848	−0.335424	+	0.942067i	$0.608880\pi$
$632$	9.20954	0.366336
$633$	0	0
$634$	−14.8773	−0.590854
$635$	−0.551844	−0.0218992
$636$	0	0
$637$	−12.9146	−0.511697
$638$	−4.90975	−0.194379
$639$	0	0
$640$	−0.302744	−0.0119670
$641$	1.57957	0.0623892	0.0311946	−	0.999513i	$-0.490069\pi$
$641$	1.57957	0.0623892	0.0311946	+	0.999513i	$0.490069\pi$
$642$	0	0
$643$	5.13247	0.202405	0.101202	−	0.994866i	$-0.467731\pi$
$643$	5.13247	0.202405	0.101202	+	0.994866i	$0.467731\pi$
$644$	−2.88478	−0.113676
$645$	0	0
$646$	−0.164082	−0.00645573
$647$	−38.4970	−1.51348	−0.756738	−	0.653719i	$-0.773208\pi$
$647$	−38.4970	−1.51348	−0.756738	+	0.653719i	$0.773208\pi$
$648$	0	0
$649$	−30.9140	−1.21348
$650$	−11.1144	−0.435942
$651$	0	0
$652$	9.83981	0.385357
$653$	3.27021	0.127973	0.0639866	−	0.997951i	$-0.479618\pi$
$653$	3.27021	0.127973	0.0639866	+	0.997951i	$0.479618\pi$
$654$	0	0
$655$	−3.94642	−0.154199
$656$	−0.564009	−0.0220209
$657$	0	0
$658$	−3.12687	−0.121898
$659$	−29.6700	−1.15578	−0.577889	−	0.816116i	$-0.696123\pi$
$659$	−29.6700	−1.15578	−0.577889	+	0.816116i	$0.696123\pi$
$660$	0	0
$661$	−39.9496	−1.55386	−0.776929	−	0.629588i	$-0.783224\pi$
$661$	−39.9496	−1.55386	−0.776929	+	0.629588i	$0.783224\pi$
$662$	−28.4548	−1.10593
$663$	0	0
$664$	8.66971	0.336450
$665$	−0.111102	−0.00430836
$666$	0	0
$667$	2.96724	0.114892
$668$	7.15760	0.276936
$669$	0	0
$670$	−3.95768	−0.152898
$671$	42.3728	1.63579
$672$	0	0
$673$	3.24537	0.125100	0.0625499	−	0.998042i	$-0.480077\pi$
$673$	3.24537	0.125100	0.0625499	+	0.998042i	$0.480077\pi$
$674$	22.7248	0.875325
$675$	0	0
$676$	−7.87256	−0.302791
$677$	47.6753	1.83231	0.916156	−	0.400822i	$-0.131275\pi$
$677$	47.6753	1.83231	0.916156	+	0.400822i	$0.131275\pi$
$678$	0	0
$679$	−9.23403	−0.354370
$680$	−0.154133	−0.00591074
$681$	0	0
$682$	12.8653	0.492637
$683$	21.9642	0.840435	0.420218	−	0.907423i	$-0.361954\pi$
$683$	21.9642	0.840435	0.420218	+	0.907423i	$0.361954\pi$
$684$	0	0
$685$	−4.16719	−0.159220
$686$	14.4653	0.552286
$687$	0	0
$688$	−1.22686	−0.0467736
$689$	30.3181	1.15503
$690$	0	0
$691$	15.4653	0.588326	0.294163	−	0.955755i	$-0.404959\pi$
$691$	15.4653	0.588326	0.294163	+	0.955755i	$0.404959\pi$
$692$	−16.7109	−0.635254
$693$	0	0
$694$	6.13192	0.232764
$695$	1.24818	0.0473460
$696$	0	0
$697$	−0.287149	−0.0108766
$698$	−32.7601	−1.23999
$699$	0	0
$700$	5.58910	0.211248
$701$	38.6560	1.46002	0.730008	−	0.683438i	$-0.239516\pi$
$701$	38.6560	1.46002	0.730008	+	0.683438i	$0.239516\pi$
$702$	0	0
$703$	−2.50142	−0.0943429
$704$	−4.19192	−0.157989
$705$	0	0
$706$	28.7010	1.08018
$707$	20.2512	0.761623
$708$	0	0
$709$	1.82518	0.0685462	0.0342731	−	0.999413i	$-0.489088\pi$
$709$	1.82518	0.0685462	0.0342731	+	0.999413i	$0.489088\pi$
$710$	−3.30705	−0.124111
$711$	0	0
$712$	6.47316	0.242592
$713$	−7.77520	−0.291184
$714$	0	0
$715$	2.87368	0.107470
$716$	6.22326	0.232574
$717$	0	0
$718$	−18.9227	−0.706188
$719$	−2.06396	−0.0769728	−0.0384864	−	0.999259i	$-0.512254\pi$
$719$	−2.06396	−0.0769728	−0.0384864	+	0.999259i	$0.512254\pi$
$720$	0	0
$721$	19.7269	0.734670
$722$	−18.8961	−0.703241
$723$	0	0
$724$	22.9159	0.851662
$725$	−5.74885	−0.213507
$726$	0	0
$727$	−33.1941	−1.23110	−0.615551	−	0.788097i	$-0.711066\pi$
$727$	−33.1941	−1.23110	−0.615551	+	0.788097i	$0.711066\pi$
$728$	−2.57844	−0.0955634
$729$	0	0
$730$	−1.52307	−0.0563712
$731$	−0.624621	−0.0231025
$732$	0	0
$733$	36.1364	1.33473	0.667363	−	0.744732i	$-0.267423\pi$
$733$	36.1364	1.33473	0.667363	+	0.744732i	$0.267423\pi$
$734$	−27.3441	−1.00929
$735$	0	0
$736$	2.53341	0.0933829
$737$	−54.7998	−2.01858
$738$	0	0
$739$	13.8334	0.508870	0.254435	−	0.967090i	$-0.418111\pi$
$739$	13.8334	0.508870	0.254435	+	0.967090i	$0.418111\pi$
$740$	−2.34975	−0.0863785
$741$	0	0
$742$	−15.2461	−0.559701
$743$	49.9995	1.83430	0.917151	−	0.398539i	$-0.130483\pi$
$743$	49.9995	1.83430	0.917151	+	0.398539i	$0.130483\pi$
$744$	0	0
$745$	0.302744	0.0110917
$746$	24.6569	0.902752
$747$	0	0
$748$	−2.13420	−0.0780340
$749$	4.03448	0.147417
$750$	0	0
$751$	−9.26448	−0.338066	−0.169033	−	0.985610i	$-0.554064\pi$
$751$	−9.26448	−0.338066	−0.169033	+	0.985610i	$0.554064\pi$
$752$	2.74601	0.100137
$753$	0	0
$754$	2.65214	0.0965852
$755$	−7.02665	−0.255726
$756$	0	0
$757$	−38.2808	−1.39134	−0.695670	−	0.718362i	$-0.744892\pi$
$757$	−38.2808	−1.39134	−0.695670	+	0.718362i	$0.744892\pi$
$758$	−9.51003	−0.345420
$759$	0	0
$760$	0.0975698	0.00353923
$761$	12.2426	0.443796	0.221898	−	0.975070i	$-0.428775\pi$
$761$	12.2426	0.443796	0.221898	+	0.975070i	$0.428775\pi$
$762$	0	0
$763$	11.9404	0.432270
$764$	0.0115679	0.000418513 0
$765$	0	0
$766$	−22.4974	−0.812863
$767$	16.6991	0.602968
$768$	0	0
$769$	38.2355	1.37881	0.689404	−	0.724377i	$-0.257872\pi$
$769$	38.2355	1.37881	0.689404	+	0.724377i	$0.257872\pi$
$770$	−1.44509	−0.0520775
$771$	0	0
$772$	1.07058	0.0385310
$773$	7.48878	0.269353	0.134676	−	0.990890i	$-0.457001\pi$
$773$	7.48878	0.269353	0.134676	+	0.990890i	$0.457001\pi$
$774$	0	0
$775$	15.0640	0.541115
$776$	8.10932	0.291108
$777$	0	0
$778$	4.68766	0.168061
$779$	0.181772	0.00651265
$780$	0	0
$781$	−45.7908	−1.63853
$782$	1.28982	0.0461237
$783$	0	0
$784$	−5.70338	−0.203692
$785$	−0.419213	−0.0149623
$786$	0	0
$787$	28.1374	1.00299	0.501496	−	0.865160i	$-0.332783\pi$
$787$	28.1374	1.00299	0.501496	+	0.865160i	$0.332783\pi$
$788$	18.7000	0.666160
$789$	0	0
$790$	−2.78813	−0.0991972
$791$	1.39450	0.0495828
$792$	0	0
$793$	−22.8889	−0.812808
$794$	9.81638	0.348370
$795$	0	0
$796$	26.9630	0.955680
$797$	23.3172	0.825938	0.412969	−	0.910745i	$-0.364492\pi$
$797$	23.3172	0.825938	0.412969	+	0.910745i	$0.364492\pi$
$798$	0	0
$799$	1.39806	0.0494596
$800$	−4.90835	−0.173536
$801$	0	0
$802$	18.3259	0.647111
$803$	−21.0891	−0.744217
$804$	0	0
$805$	0.873350	0.0307815
$806$	−6.94953	−0.244787
$807$	0	0
$808$	−17.7845	−0.625658
$809$	−8.90199	−0.312977	−0.156489	−	0.987680i	$-0.550017\pi$
$809$	−8.90199	−0.312977	−0.156489	+	0.987680i	$0.550017\pi$
$810$	0	0
$811$	−9.89807	−0.347568	−0.173784	−	0.984784i	$-0.555600\pi$
$811$	−9.89807	−0.347568	−0.173784	+	0.984784i	$0.555600\pi$
$812$	−1.33368	−0.0468031
$813$	0	0
$814$	−32.5357	−1.14038
$815$	−2.97894	−0.104348
$816$	0	0
$817$	0.395399	0.0138333
$818$	−12.2531	−0.428420
$819$	0	0
$820$	0.170750	0.00596286
$821$	17.0013	0.593349	0.296675	−	0.954979i	$-0.404122\pi$
$821$	17.0013	0.593349	0.296675	+	0.954979i	$0.404122\pi$
$822$	0	0
$823$	28.8059	1.00411	0.502055	−	0.864836i	$-0.332578\pi$
$823$	28.8059	1.00411	0.502055	+	0.864836i	$0.332578\pi$
$824$	−17.3242	−0.603516
$825$	0	0
$826$	−8.39747	−0.292185
$827$	−0.181352	−0.00630624	−0.00315312	−	0.999995i	$-0.501004\pi$
$827$	−0.181352	−0.00630624	−0.00315312	+	0.999995i	$0.501004\pi$
$828$	0	0
$829$	−30.3181	−1.05299	−0.526496	−	0.850177i	$-0.676495\pi$
$829$	−30.3181	−1.05299	−0.526496	+	0.850177i	$0.676495\pi$
$830$	−2.62470	−0.0911047
$831$	0	0
$832$	2.26439	0.0785034
$833$	−2.90371	−0.100608
$834$	0	0
$835$	−2.16692	−0.0749893
$836$	1.35100	0.0467251
$837$	0	0
$838$	−11.2802	−0.389668
$839$	39.9824	1.38035	0.690173	−	0.723644i	$-0.257534\pi$
$839$	39.9824	1.38035	0.690173	+	0.723644i	$0.257534\pi$
$840$	0	0
$841$	−27.6282	−0.952696
$842$	−18.0498	−0.622038
$843$	0	0
$844$	−8.89198	−0.306074
$845$	2.38337	0.0819903
$846$	0	0
$847$	−7.48376	−0.257145
$848$	13.3891	0.459784
$849$	0	0
$850$	−2.49894	−0.0857131
$851$	19.6631	0.674044
$852$	0	0
$853$	−44.9051	−1.53752	−0.768760	−	0.639537i	$-0.779126\pi$
$853$	−44.9051	−1.53752	−0.768760	+	0.639537i	$0.779126\pi$
$854$	11.5102	0.393869
$855$	0	0
$856$	−3.54307	−0.121100
$857$	−16.5059	−0.563832	−0.281916	−	0.959439i	$-0.590970\pi$
$857$	−16.5059	−0.563832	−0.281916	+	0.959439i	$0.590970\pi$
$858$	0	0
$859$	−22.9828	−0.784162	−0.392081	−	0.919931i	$-0.628245\pi$
$859$	−22.9828	−0.784162	−0.392081	+	0.919931i	$0.628245\pi$
$860$	0.371424	0.0126655
$861$	0	0
$862$	12.4956	0.425601
$863$	7.21445	0.245583	0.122791	−	0.992433i	$-0.460815\pi$
$863$	7.21445	0.245583	0.122791	+	0.992433i	$0.460815\pi$
$864$	0	0
$865$	5.05912	0.172015
$866$	27.9729	0.950558
$867$	0	0
$868$	3.49472	0.118619
$869$	−38.6057	−1.30961
$870$	0	0
$871$	29.6016	1.00301
$872$	−10.4860	−0.355102
$873$	0	0
$874$	−0.816482	−0.0276179
$875$	−3.41573	−0.115473
$876$	0	0
$877$	−14.9344	−0.504298	−0.252149	−	0.967688i	$-0.581137\pi$
$877$	−14.9344	−0.504298	−0.252149	+	0.967688i	$0.581137\pi$
$878$	30.6628	1.03482
$879$	0	0
$880$	1.26908	0.0427806
$881$	−43.8896	−1.47868	−0.739339	−	0.673333i	$-0.764862\pi$
$881$	−43.8896	−1.47868	−0.739339	+	0.673333i	$0.764862\pi$
$882$	0	0
$883$	51.3850	1.72924	0.864621	−	0.502425i	$-0.167559\pi$
$883$	51.3850	1.72924	0.864621	+	0.502425i	$0.167559\pi$
$884$	1.15285	0.0387745
$885$	0	0
$886$	34.2986	1.15228
$887$	−17.4041	−0.584372	−0.292186	−	0.956362i	$-0.594383\pi$
$887$	−17.4041	−0.584372	−0.292186	+	0.956362i	$0.594383\pi$
$888$	0	0
$889$	−2.07562	−0.0696141
$890$	−1.95971	−0.0656895
$891$	0	0
$892$	3.71985	0.124550
$893$	−0.885000	−0.0296154
$894$	0	0
$895$	−1.88405	−0.0629769
$896$	−1.13869	−0.0380411
$897$	0	0
$898$	−29.6723	−0.990179
$899$	−3.59461	−0.119887
$900$	0	0
$901$	6.81668	0.227097
$902$	2.36428	0.0787221
$903$	0	0
$904$	−1.22465	−0.0407313
$905$	−6.93764	−0.230615
$906$	0	0
$907$	18.0235	0.598460	0.299230	−	0.954181i	$-0.403270\pi$
$907$	18.0235	0.598460	0.299230	+	0.954181i	$0.403270\pi$
$908$	14.7757	0.490349
$909$	0	0
$910$	0.780607	0.0258769
$911$	−55.1171	−1.82611	−0.913055	−	0.407836i	$-0.866284\pi$
$911$	−55.1171	−1.82611	−0.913055	+	0.407836i	$0.866284\pi$
$912$	0	0
$913$	−36.3428	−1.20277
$914$	15.8126	0.523033
$915$	0	0
$916$	−29.5499	−0.976355
$917$	−14.8435	−0.490175
$918$	0	0
$919$	16.1561	0.532940	0.266470	−	0.963843i	$-0.414143\pi$
$919$	16.1561	0.532940	0.266470	+	0.963843i	$0.414143\pi$
$920$	−0.766975	−0.0252864
$921$	0	0
$922$	−32.3273	−1.06464
$923$	24.7352	0.814169
$924$	0	0
$925$	−38.0962	−1.25260
$926$	−26.5600	−0.872814
$927$	0	0
$928$	1.17124	0.0384478
$929$	33.1722	1.08834	0.544172	−	0.838974i	$-0.316844\pi$
$929$	33.1722	1.08834	0.544172	+	0.838974i	$0.316844\pi$
$930$	0	0
$931$	1.83811	0.0602417
$932$	−1.24854	−0.0408972
$933$	0	0
$934$	−22.5118	−0.736609
$935$	0.646115	0.0211302
$936$	0	0
$937$	−53.7529	−1.75603	−0.878015	−	0.478634i	$-0.841132\pi$
$937$	−53.7529	−1.75603	−0.878015	+	0.478634i	$0.841132\pi$
$938$	−14.8858	−0.486039
$939$	0	0
$940$	−0.831339	−0.0271153
$941$	55.7724	1.81813	0.909063	−	0.416658i	$-0.136799\pi$
$941$	55.7724	1.81813	0.909063	+	0.416658i	$0.136799\pi$
$942$	0	0
$943$	−1.42887	−0.0465304
$944$	7.37465	0.240024
$945$	0	0
$946$	5.14291	0.167210
$947$	8.50233	0.276289	0.138144	−	0.990412i	$-0.455886\pi$
$947$	8.50233	0.276289	0.138144	+	0.990412i	$0.455886\pi$
$948$	0	0
$949$	11.3918	0.369795
$950$	1.58189	0.0513232
$951$	0	0
$952$	−0.579734	−0.0187893
$953$	−4.14998	−0.134431	−0.0672155	−	0.997738i	$-0.521412\pi$
$953$	−4.14998	−0.134431	−0.0672155	+	0.997738i	$0.521412\pi$
$954$	0	0
$955$	−0.00350212	−0.000113326 0
$956$	−9.32136	−0.301474
$957$	0	0
$958$	16.7309	0.540550
$959$	−15.6738	−0.506135
$960$	0	0
$961$	−21.5809	−0.696157
$962$	17.5751	0.566643
$963$	0	0
$964$	−9.06545	−0.291979
$965$	−0.324111	−0.0104335
$966$	0	0
$967$	−50.1961	−1.61420	−0.807100	−	0.590415i	$-0.798964\pi$
$967$	−50.1961	−1.61420	−0.807100	+	0.590415i	$0.798964\pi$
$968$	6.57223	0.211239
$969$	0	0
$970$	−2.45504	−0.0788267
$971$	34.5794	1.10971	0.554853	−	0.831949i	$-0.312775\pi$
$971$	34.5794	1.10971	0.554853	+	0.831949i	$0.312775\pi$
$972$	0	0
$973$	4.69470	0.150505
$974$	32.8751	1.05339
$975$	0	0
$976$	−10.1082	−0.323556
$977$	17.4656	0.558774	0.279387	−	0.960179i	$-0.409869\pi$
$977$	17.4656	0.558774	0.279387	+	0.960179i	$0.409869\pi$
$978$	0	0
$979$	−27.1350	−0.867238
$980$	1.72666	0.0551562
$981$	0	0
$982$	40.0650	1.27852
$983$	−48.8778	−1.55896	−0.779481	−	0.626426i	$-0.784517\pi$
$983$	−48.8778	−1.55896	−0.779481	+	0.626426i	$0.784517\pi$
$984$	0	0
$985$	−5.66130	−0.180384
$986$	0.596304	0.0189902
$987$	0	0
$988$	−0.729778	−0.0232173
$989$	−3.10815	−0.0988333
$990$	0	0
$991$	41.0412	1.30372	0.651859	−	0.758340i	$-0.273989\pi$
$991$	41.0412	1.30372	0.651859	+	0.758340i	$0.273989\pi$
$992$	−3.06906	−0.0974428
$993$	0	0
$994$	−12.4386	−0.394529
$995$	−8.16289	−0.258781
$996$	0	0
$997$	52.2965	1.65625	0.828124	−	0.560546i	$-0.189409\pi$
$997$	52.2965	1.65625	0.828124	+	0.560546i	$0.189409\pi$
$998$	−10.5335	−0.333432
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.p.1.5	yes	12
3.2	odd	2		8046.2.a.i.1.8	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.i.1.8	✓	12	3.2	odd	2
8046.2.a.p.1.5	yes	12	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:	\( N \)	\(=\)	\( 8046 = 2 \cdot 3^{3} \cdot 149 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8046.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -0.302744 q^{5} -1.13869 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -0.302744 q^{5} -1.13869 q^{7} +1.00000 q^{8} -0.302744 q^{10} -4.19192 q^{11} +2.26439 q^{13} -1.13869 q^{14} +1.00000 q^{16} +0.509122 q^{17} -0.322285 q^{19} -0.302744 q^{20} -4.19192 q^{22} +2.53341 q^{23} -4.90835 q^{25} +2.26439 q^{26} -1.13869 q^{28} +1.17124 q^{29} -3.06906 q^{31} +1.00000 q^{32} +0.509122 q^{34} +0.344732 q^{35} +7.76152 q^{37} -0.322285 q^{38} -0.302744 q^{40} -0.564009 q^{41} -1.22686 q^{43} -4.19192 q^{44} +2.53341 q^{46} +2.74601 q^{47} -5.70338 q^{49} -4.90835 q^{50} +2.26439 q^{52} +13.3891 q^{53} +1.26908 q^{55} -1.13869 q^{56} +1.17124 q^{58} +7.37465 q^{59} -10.1082 q^{61} -3.06906 q^{62} +1.00000 q^{64} -0.685528 q^{65} +13.0727 q^{67} +0.509122 q^{68} +0.344732 q^{70} +10.9236 q^{71} +5.03088 q^{73} +7.76152 q^{74} -0.322285 q^{76} +4.77332 q^{77} +9.20954 q^{79} -0.302744 q^{80} -0.564009 q^{82} +8.66971 q^{83} -0.154133 q^{85} -1.22686 q^{86} -4.19192 q^{88} +6.47316 q^{89} -2.57844 q^{91} +2.53341 q^{92} +2.74601 q^{94} +0.0975698 q^{95} +8.10932 q^{97} -5.70338 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{2} + 12 q^{4} + 5 q^{5} + 6 q^{7} + 12 q^{8}+O(q^{10}) \) 12 * q + 12 * q^2 + 12 * q^4 + 5 * q^5 + 6 * q^7 + 12 * q^8	\( 12 q + 12 q^{2} + 12 q^{4} + 5 q^{5} + 6 q^{7} + 12 q^{8} + 5 q^{10} + 6 q^{11} + 3 q^{13} + 6 q^{14} + 12 q^{16} + 6 q^{17} + 8 q^{19} + 5 q^{20} + 6 q^{22} + 11 q^{23} + 11 q^{25} + 3 q^{26} + 6 q^{28} + 29 q^{29} + 2 q^{31} + 12 q^{32} + 6 q^{34} + 4 q^{35} + 5 q^{37} + 8 q^{38} + 5 q^{40} + 22 q^{41} + 9 q^{43} + 6 q^{44} + 11 q^{46} + 15 q^{47} + 14 q^{49} + 11 q^{50} + 3 q^{52} + 12 q^{53} + 13 q^{55} + 6 q^{56} + 29 q^{58} + 34 q^{59} - 4 q^{61} + 2 q^{62} + 12 q^{64} + 12 q^{65} + q^{67} + 6 q^{68} + 4 q^{70} + 21 q^{71} - 2 q^{73} + 5 q^{74} + 8 q^{76} + 34 q^{77} + 9 q^{79} + 5 q^{80} + 22 q^{82} + 10 q^{83} + 5 q^{85} + 9 q^{86} + 6 q^{88} - 2 q^{89} + 17 q^{91} + 11 q^{92} + 15 q^{94} + 69 q^{95} - 13 q^{97} + 14 q^{98}+O(q^{100}) \) 12 * q + 12 * q^2 + 12 * q^4 + 5 * q^5 + 6 * q^7 + 12 * q^8 + 5 * q^10 + 6 * q^11 + 3 * q^13 + 6 * q^14 + 12 * q^16 + 6 * q^17 + 8 * q^19 + 5 * q^20 + 6 * q^22 + 11 * q^23 + 11 * q^25 + 3 * q^26 + 6 * q^28 + 29 * q^29 + 2 * q^31 + 12 * q^32 + 6 * q^34 + 4 * q^35 + 5 * q^37 + 8 * q^38 + 5 * q^40 + 22 * q^41 + 9 * q^43 + 6 * q^44 + 11 * q^46 + 15 * q^47 + 14 * q^49 + 11 * q^50 + 3 * q^52 + 12 * q^53 + 13 * q^55 + 6 * q^56 + 29 * q^58 + 34 * q^59 - 4 * q^61 + 2 * q^62 + 12 * q^64 + 12 * q^65 + q^67 + 6 * q^68 + 4 * q^70 + 21 * q^71 - 2 * q^73 + 5 * q^74 + 8 * q^76 + 34 * q^77 + 9 * q^79 + 5 * q^80 + 22 * q^82 + 10 * q^83 + 5 * q^85 + 9 * q^86 + 6 * q^88 - 2 * q^89 + 17 * q^91 + 11 * q^92 + 15 * q^94 + 69 * q^95 - 13 * q^97 + 14 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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