LMFDB - Embedded newform 7938.2.a.ca.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7938,2,Mod(1,7938)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7938, 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("7938.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7938 = 2 \cdot 3^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7938.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:	$63.3852491245$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.321.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 1 $ x^3 - x^2 - 4*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 126)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.239123$ of defining polynomial
Character	$\chi$	$=$	7938.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +3.18194 q^{5} +1.00000 q^{8} +3.18194 q^{10} -3.18194 q^{11} -5.70370 q^{13} +1.00000 q^{16} -1.52175 q^{17} +1.28263 q^{19} +3.18194 q^{20} -3.18194 q^{22} -2.23912 q^{23} +5.12476 q^{25} -5.70370 q^{26} -7.08126 q^{29} -9.42107 q^{31} +1.00000 q^{32} -1.52175 q^{34} -1.00000 q^{37} +1.28263 q^{38} +3.18194 q^{40} -5.60301 q^{41} -6.82846 q^{43} -3.18194 q^{44} -2.23912 q^{46} +5.82846 q^{47} +5.12476 q^{50} -5.70370 q^{52} +2.05718 q^{53} -10.1248 q^{55} -7.08126 q^{58} +1.12476 q^{59} +3.12476 q^{61} -9.42107 q^{62} +1.00000 q^{64} -18.1488 q^{65} +10.9669 q^{67} -1.52175 q^{68} -8.69002 q^{71} +4.96690 q^{73} -1.00000 q^{74} +1.28263 q^{76} -4.13844 q^{79} +3.18194 q^{80} -5.60301 q^{82} -8.06758 q^{83} -4.84213 q^{85} -6.82846 q^{86} -3.18194 q^{88} +0.225450 q^{89} -2.23912 q^{92} +5.82846 q^{94} +4.08126 q^{95} -14.8421 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{4} + q^{5} + 3 q^{8} + q^{10} - q^{11} - 8 q^{13} + 3 q^{16} - 4 q^{17} + 3 q^{19} + q^{20} - q^{22} - 7 q^{23} - 2 q^{25} - 8 q^{26} - 5 q^{29} - 20 q^{31} + 3 q^{32} - 4 q^{34}+ \cdots - 28 q^{97}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^4 + q^5 + 3 * q^8 + q^10 - q^11 - 8 * q^13 + 3 * q^16 - 4 * q^17 + 3 * q^19 + q^20 - q^22 - 7 * q^23 - 2 * q^25 - 8 * q^26 - 5 * q^29 - 20 * q^31 + 3 * q^32 - 4 * q^34 - 3 * q^37 + 3 * q^38 + q^40 + 6 * q^43 - q^44 - 7 * q^46 - 9 * q^47 - 2 * q^50 - 8 * q^52 + 15 * q^53 - 13 * q^55 - 5 * q^58 - 14 * q^59 - 8 * q^61 - 20 * q^62 + 3 * q^64 - 12 * q^65 - q^67 - 4 * q^68 - 7 * q^71 - 19 * q^73 - 3 * q^74 + 3 * q^76 - 5 * q^79 + q^80 + 2 * q^83 + 2 * q^85 + 6 * q^86 - q^88 - 9 * q^89 - 7 * q^92 - 9 * q^94 - 4 * q^95 - 28 * q^97

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
$5$	3.18194	1.42301	0.711504	−	0.702682i	$-0.248014\pi$
$5$	3.18194	1.42301	0.711504	+	0.702682i	$0.248014\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	0	0
$10$	3.18194	1.00622
$11$	−3.18194	−0.959392	−0.479696	−	0.877435i	$-0.659253\pi$
$11$	−3.18194	−0.959392	−0.479696	+	0.877435i	$0.659253\pi$
$12$	0	0
$13$	−5.70370	−1.58192	−0.790960	−	0.611867i	$-0.790419\pi$
$13$	−5.70370	−1.58192	−0.790960	+	0.611867i	$0.790419\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	−1.52175	−0.369079	−0.184540	−	0.982825i	$-0.559079\pi$
$17$	−1.52175	−0.369079	−0.184540	+	0.982825i	$0.559079\pi$
$18$	0	0
$19$	1.28263	0.294256	0.147128	−	0.989117i	$-0.452997\pi$
$19$	1.28263	0.294256	0.147128	+	0.989117i	$0.452997\pi$
$20$	3.18194	0.711504
$21$	0	0
$22$	−3.18194	−0.678393
$23$	−2.23912	−0.466889	−0.233445	−	0.972370i	$-0.575000\pi$
$23$	−2.23912	−0.466889	−0.233445	+	0.972370i	$0.575000\pi$
$24$	0	0
$25$	5.12476	1.02495
$26$	−5.70370	−1.11859
$27$	0	0
$28$	0	0
$29$	−7.08126	−1.31496	−0.657478	−	0.753474i	$-0.728377\pi$
$29$	−7.08126	−1.31496	−0.657478	+	0.753474i	$0.728377\pi$
$30$	0	0
$31$	−9.42107	−1.69207	−0.846037	−	0.533125i	$-0.821018\pi$
$31$	−9.42107	−1.69207	−0.846037	+	0.533125i	$0.821018\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−1.52175	−0.260979
$35$	0	0
$36$	0	0
$37$	−1.00000	−0.164399	−0.0821995	−	0.996616i	$-0.526194\pi$
$37$	−1.00000	−0.164399	−0.0821995	+	0.996616i	$0.526194\pi$
$38$	1.28263	0.208070
$39$	0	0
$40$	3.18194	0.503109
$41$	−5.60301	−0.875043	−0.437522	−	0.899208i	$-0.644144\pi$
$41$	−5.60301	−0.875043	−0.437522	+	0.899208i	$0.644144\pi$
$42$	0	0
$43$	−6.82846	−1.04133	−0.520665	−	0.853761i	$-0.674316\pi$
$43$	−6.82846	−1.04133	−0.520665	+	0.853761i	$0.674316\pi$
$44$	−3.18194	−0.479696
$45$	0	0
$46$	−2.23912	−0.330141
$47$	5.82846	0.850168	0.425084	−	0.905154i	$-0.360245\pi$
$47$	5.82846	0.850168	0.425084	+	0.905154i	$0.360245\pi$
$48$	0	0
$49$	0	0
$50$	5.12476	0.724751
$51$	0	0
$52$	−5.70370	−0.790960
$53$	2.05718	0.282575	0.141288	−	0.989969i	$-0.454876\pi$
$53$	2.05718	0.282575	0.141288	+	0.989969i	$0.454876\pi$
$54$	0	0
$55$	−10.1248	−1.36522
$56$	0	0
$57$	0	0
$58$	−7.08126	−0.929815
$59$	1.12476	0.146432	0.0732159	−	0.997316i	$-0.476674\pi$
$59$	1.12476	0.146432	0.0732159	+	0.997316i	$0.476674\pi$
$60$	0	0
$61$	3.12476	0.400085	0.200042	−	0.979787i	$-0.435892\pi$
$61$	3.12476	0.400085	0.200042	+	0.979787i	$0.435892\pi$
$62$	−9.42107	−1.19648
$63$	0	0
$64$	1.00000	0.125000
$65$	−18.1488	−2.25109
$66$	0	0
$67$	10.9669	1.33982	0.669910	−	0.742442i	$-0.266333\pi$
$67$	10.9669	1.33982	0.669910	+	0.742442i	$0.266333\pi$
$68$	−1.52175	−0.184540
$69$	0	0
$70$	0	0
$71$	−8.69002	−1.03132	−0.515658	−	0.856794i	$-0.672452\pi$
$71$	−8.69002	−1.03132	−0.515658	+	0.856794i	$0.672452\pi$
$72$	0	0
$73$	4.96690	0.581331	0.290666	−	0.956825i	$-0.406123\pi$
$73$	4.96690	0.581331	0.290666	+	0.956825i	$0.406123\pi$
$74$	−1.00000	−0.116248
$75$	0	0
$76$	1.28263	0.147128
$77$	0	0
$78$	0	0
$79$	−4.13844	−0.465610	−0.232805	−	0.972523i	$-0.574790\pi$
$79$	−4.13844	−0.465610	−0.232805	+	0.972523i	$0.574790\pi$
$80$	3.18194	0.355752
$81$	0	0
$82$	−5.60301	−0.618749
$83$	−8.06758	−0.885532	−0.442766	−	0.896637i	$-0.646003\pi$
$83$	−8.06758	−0.885532	−0.442766	+	0.896637i	$0.646003\pi$
$84$	0	0
$85$	−4.84213	−0.525203
$86$	−6.82846	−0.736332
$87$	0	0
$88$	−3.18194	−0.339196
$89$	0.225450	0.0238977	0.0119488	−	0.999929i	$-0.496196\pi$
$89$	0.225450	0.0238977	0.0119488	+	0.999929i	$0.496196\pi$
$90$	0	0
$91$	0	0
$92$	−2.23912	−0.233445
$93$	0	0
$94$	5.82846	0.601160
$95$	4.08126	0.418728
$96$	0	0
$97$	−14.8421	−1.50699	−0.753495	−	0.657453i	$-0.771634\pi$
$97$	−14.8421	−1.50699	−0.753495	+	0.657453i	$0.771634\pi$
$98$	0	0
$99$	0	0
$100$	5.12476	0.512476
$101$	−18.5893	−1.84971	−0.924854	−	0.380322i	$-0.875813\pi$
$101$	−18.5893	−1.84971	−0.924854	+	0.380322i	$0.875813\pi$
$102$	0	0
$103$	−0.282630	−0.0278484	−0.0139242	−	0.999903i	$-0.504432\pi$
$103$	−0.282630	−0.0278484	−0.0139242	+	0.999903i	$0.504432\pi$
$104$	−5.70370	−0.559293
$105$	0	0
$106$	2.05718	0.199811
$107$	11.3776	1.09991	0.549955	−	0.835194i	$-0.314645\pi$
$107$	11.3776	1.09991	0.549955	+	0.835194i	$0.314645\pi$
$108$	0	0
$109$	4.42107	0.423461	0.211731	−	0.977328i	$-0.432090\pi$
$109$	4.42107	0.423461	0.211731	+	0.977328i	$0.432090\pi$
$110$	−10.1248	−0.965358
$111$	0	0
$112$	0	0
$113$	−3.21505	−0.302446	−0.151223	−	0.988500i	$-0.548321\pi$
$113$	−3.21505	−0.302446	−0.151223	+	0.988500i	$0.548321\pi$
$114$	0	0
$115$	−7.12476	−0.664388
$116$	−7.08126	−0.657478
$117$	0	0
$118$	1.12476	0.103543
$119$	0	0
$120$	0	0
$121$	−0.875237	−0.0795670
$122$	3.12476	0.282903
$123$	0	0
$124$	−9.42107	−0.846037
$125$	0.396990	0.0355079
$126$	0	0
$127$	20.1053	1.78406	0.892030	−	0.451976i	$-0.149281\pi$
$127$	20.1053	1.78406	0.892030	+	0.451976i	$0.149281\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−18.1488	−1.59176
$131$	−6.36389	−0.556015	−0.278008	−	0.960579i	$-0.589674\pi$
$131$	−6.36389	−0.556015	−0.278008	+	0.960579i	$0.589674\pi$
$132$	0	0
$133$	0	0
$134$	10.9669	0.947396
$135$	0	0
$136$	−1.52175	−0.130489
$137$	−2.74145	−0.234218	−0.117109	−	0.993119i	$-0.537363\pi$
$137$	−2.74145	−0.234218	−0.117109	+	0.993119i	$0.537363\pi$
$138$	0	0
$139$	7.96690	0.675743	0.337872	−	0.941192i	$-0.390293\pi$
$139$	7.96690	0.675743	0.337872	+	0.941192i	$0.390293\pi$
$140$	0	0
$141$	0	0
$142$	−8.69002	−0.729251
$143$	18.1488	1.51768
$144$	0	0
$145$	−22.5322	−1.87119
$146$	4.96690	0.411063
$147$	0	0
$148$	−1.00000	−0.0821995
$149$	23.2599	1.90553	0.952764	−	0.303712i	$-0.0982261\pi$
$149$	23.2599	1.90553	0.952764	+	0.303712i	$0.0982261\pi$
$150$	0	0
$151$	−8.12476	−0.661184	−0.330592	−	0.943774i	$-0.607248\pi$
$151$	−8.12476	−0.661184	−0.330592	+	0.943774i	$0.607248\pi$
$152$	1.28263	0.104035
$153$	0	0
$154$	0	0
$155$	−29.9773	−2.40783
$156$	0	0
$157$	−11.2632	−0.898901	−0.449451	−	0.893305i	$-0.648380\pi$
$157$	−11.2632	−0.898901	−0.449451	+	0.893305i	$0.648380\pi$
$158$	−4.13844	−0.329236
$159$	0	0
$160$	3.18194	0.251555
$161$	0	0
$162$	0	0
$163$	3.98057	0.311782	0.155891	−	0.987774i	$-0.450175\pi$
$163$	3.98057	0.311782	0.155891	+	0.987774i	$0.450175\pi$
$164$	−5.60301	−0.437522
$165$	0	0
$166$	−8.06758	−0.626166
$167$	5.23912	0.405416	0.202708	−	0.979239i	$-0.435026\pi$
$167$	5.23912	0.405416	0.202708	+	0.979239i	$0.435026\pi$
$168$	0	0
$169$	19.5322	1.50247
$170$	−4.84213	−0.371375
$171$	0	0
$172$	−6.82846	−0.520665
$173$	−2.55159	−0.193994	−0.0969968	−	0.995285i	$-0.530924\pi$
$173$	−2.55159	−0.193994	−0.0969968	+	0.995285i	$0.530924\pi$
$174$	0	0
$175$	0	0
$176$	−3.18194	−0.239848
$177$	0	0
$178$	0.225450	0.0168982
$179$	7.03775	0.526026	0.263013	−	0.964792i	$-0.415284\pi$
$179$	7.03775	0.526026	0.263013	+	0.964792i	$0.415284\pi$
$180$	0	0
$181$	−12.9669	−0.963822	−0.481911	−	0.876220i	$-0.660057\pi$
$181$	−12.9669	−0.963822	−0.481911	+	0.876220i	$0.660057\pi$
$182$	0	0
$183$	0	0
$184$	−2.23912	−0.165070
$185$	−3.18194	−0.233941
$186$	0	0
$187$	4.84213	0.354092
$188$	5.82846	0.425084
$189$	0	0
$190$	4.08126	0.296085
$191$	−1.98057	−0.143309	−0.0716545	−	0.997430i	$-0.522828\pi$
$191$	−1.98057	−0.143309	−0.0716545	+	0.997430i	$0.522828\pi$
$192$	0	0
$193$	−4.54583	−0.327216	−0.163608	−	0.986525i	$-0.552313\pi$
$193$	−4.54583	−0.327216	−0.163608	+	0.986525i	$0.552313\pi$
$194$	−14.8421	−1.06560
$195$	0	0
$196$	0	0
$197$	21.8148	1.55424	0.777120	−	0.629353i	$-0.216680\pi$
$197$	21.8148	1.55424	0.777120	+	0.629353i	$0.216680\pi$
$198$	0	0
$199$	−12.2826	−0.870693	−0.435346	−	0.900263i	$-0.643374\pi$
$199$	−12.2826	−0.870693	−0.435346	+	0.900263i	$0.643374\pi$
$200$	5.12476	0.362375
$201$	0	0
$202$	−18.5893	−1.30794
$203$	0	0
$204$	0	0
$205$	−17.8285	−1.24519
$206$	−0.282630	−0.0196918
$207$	0	0
$208$	−5.70370	−0.395480
$209$	−4.08126	−0.282306
$210$	0	0
$211$	16.6569	1.14671	0.573355	−	0.819307i	$-0.305642\pi$
$211$	16.6569	1.14671	0.573355	+	0.819307i	$0.305642\pi$
$212$	2.05718	0.141288
$213$	0	0
$214$	11.3776	0.777754
$215$	−21.7278	−1.48182
$216$	0	0
$217$	0	0
$218$	4.42107	0.299432
$219$	0	0
$220$	−10.1248	−0.682611
$221$	8.67962	0.583854
$222$	0	0
$223$	10.6569	0.713640	0.356820	−	0.934173i	$-0.383861\pi$
$223$	10.6569	0.713640	0.356820	+	0.934173i	$0.383861\pi$
$224$	0	0
$225$	0	0
$226$	−3.21505	−0.213862
$227$	14.5081	0.962935	0.481468	−	0.876464i	$-0.340104\pi$
$227$	14.5081	0.962935	0.481468	+	0.876464i	$0.340104\pi$
$228$	0	0
$229$	10.2495	0.677308	0.338654	−	0.940911i	$-0.390028\pi$
$229$	10.2495	0.677308	0.338654	+	0.940911i	$0.390028\pi$
$230$	−7.12476	−0.469793
$231$	0	0
$232$	−7.08126	−0.464907
$233$	1.08126	0.0708355	0.0354177	−	0.999373i	$-0.488724\pi$
$233$	1.08126	0.0708355	0.0354177	+	0.999373i	$0.488724\pi$
$234$	0	0
$235$	18.5458	1.20980
$236$	1.12476	0.0732159
$237$	0	0
$238$	0	0
$239$	−12.3204	−0.796939	−0.398470	−	0.917182i	$-0.630459\pi$
$239$	−12.3204	−0.796939	−0.398470	+	0.917182i	$0.630459\pi$
$240$	0	0
$241$	−13.0000	−0.837404	−0.418702	−	0.908124i	$-0.637515\pi$
$241$	−13.0000	−0.837404	−0.418702	+	0.908124i	$0.637515\pi$
$242$	−0.875237	−0.0562623
$243$	0	0
$244$	3.12476	0.200042
$245$	0	0
$246$	0	0
$247$	−7.31573	−0.465489
$248$	−9.42107	−0.598238
$249$	0	0
$250$	0.396990	0.0251079
$251$	−5.11109	−0.322609	−0.161305	−	0.986905i	$-0.551570\pi$
$251$	−5.11109	−0.322609	−0.161305	+	0.986905i	$0.551570\pi$
$252$	0	0
$253$	7.12476	0.447930
$254$	20.1053	1.26152
$255$	0	0
$256$	1.00000	0.0625000
$257$	−7.66019	−0.477830	−0.238915	−	0.971041i	$-0.576792\pi$
$257$	−7.66019	−0.477830	−0.238915	+	0.971041i	$0.576792\pi$
$258$	0	0
$259$	0	0
$260$	−18.1488	−1.12554
$261$	0	0
$262$	−6.36389	−0.393162
$263$	3.09493	0.190842	0.0954208	−	0.995437i	$-0.469580\pi$
$263$	3.09493	0.190842	0.0954208	+	0.995437i	$0.469580\pi$
$264$	0	0
$265$	6.54583	0.402107
$266$	0	0
$267$	0	0
$268$	10.9669	0.669910
$269$	−26.8903	−1.63953	−0.819765	−	0.572700i	$-0.805896\pi$
$269$	−26.8903	−1.63953	−0.819765	+	0.572700i	$0.805896\pi$
$270$	0	0
$271$	22.2164	1.34955	0.674776	−	0.738023i	$-0.264240\pi$
$271$	22.2164	1.34955	0.674776	+	0.738023i	$0.264240\pi$
$272$	−1.52175	−0.0922699
$273$	0	0
$274$	−2.74145	−0.165617
$275$	−16.3067	−0.983331
$276$	0	0
$277$	−14.6375	−0.879482	−0.439741	−	0.898125i	$-0.644930\pi$
$277$	−14.6375	−0.879482	−0.439741	+	0.898125i	$0.644930\pi$
$278$	7.96690	0.477823
$279$	0	0
$280$	0	0
$281$	23.3984	1.39583	0.697915	−	0.716181i	$-0.254111\pi$
$281$	23.3984	1.39583	0.697915	+	0.716181i	$0.254111\pi$
$282$	0	0
$283$	−26.1248	−1.55296	−0.776478	−	0.630144i	$-0.782996\pi$
$283$	−26.1248	−1.55296	−0.776478	+	0.630144i	$0.782996\pi$
$284$	−8.69002	−0.515658
$285$	0	0
$286$	18.1488	1.07316
$287$	0	0
$288$	0	0
$289$	−14.6843	−0.863780
$290$	−22.5322	−1.32313
$291$	0	0
$292$	4.96690	0.290666
$293$	25.8629	1.51093	0.755465	−	0.655190i	$-0.227411\pi$
$293$	25.8629	1.51093	0.755465	+	0.655190i	$0.227411\pi$
$294$	0	0
$295$	3.57893	0.208374
$296$	−1.00000	−0.0581238
$297$	0	0
$298$	23.2599	1.34741
$299$	12.7713	0.738582
$300$	0	0
$301$	0	0
$302$	−8.12476	−0.467528
$303$	0	0
$304$	1.28263	0.0735639
$305$	9.94282	0.569324
$306$	0	0
$307$	3.53216	0.201591	0.100795	−	0.994907i	$-0.467861\pi$
$307$	3.53216	0.201591	0.100795	+	0.994907i	$0.467861\pi$
$308$	0	0
$309$	0	0
$310$	−29.9773	−1.70260
$311$	−1.70370	−0.0966078	−0.0483039	−	0.998833i	$-0.515382\pi$
$311$	−1.70370	−0.0966078	−0.0483039	+	0.998833i	$0.515382\pi$
$312$	0	0
$313$	−2.84213	−0.160647	−0.0803234	−	0.996769i	$-0.525595\pi$
$313$	−2.84213	−0.160647	−0.0803234	+	0.996769i	$0.525595\pi$
$314$	−11.2632	−0.635619
$315$	0	0
$316$	−4.13844	−0.232805
$317$	24.9201	1.39965	0.699827	−	0.714313i	$-0.253261\pi$
$317$	24.9201	1.39965	0.699827	+	0.714313i	$0.253261\pi$
$318$	0	0
$319$	22.5322	1.26156
$320$	3.18194	0.177876
$321$	0	0
$322$	0	0
$323$	−1.95185	−0.108604
$324$	0	0
$325$	−29.2301	−1.62139
$326$	3.98057	0.220463
$327$	0	0
$328$	−5.60301	−0.309374
$329$	0	0
$330$	0	0
$331$	−7.17154	−0.394183	−0.197092	−	0.980385i	$-0.563150\pi$
$331$	−7.17154	−0.394183	−0.197092	+	0.980385i	$0.563150\pi$
$332$	−8.06758	−0.442766
$333$	0	0
$334$	5.23912	0.286672
$335$	34.8960	1.90657
$336$	0	0
$337$	21.8421	1.18982	0.594908	−	0.803793i	$-0.297188\pi$
$337$	21.8421	1.18982	0.594908	+	0.803793i	$0.297188\pi$
$338$	19.5322	1.06241
$339$	0	0
$340$	−4.84213	−0.262602
$341$	29.9773	1.62336
$342$	0	0
$343$	0	0
$344$	−6.82846	−0.368166
$345$	0	0
$346$	−2.55159	−0.137174
$347$	2.11109	0.113329	0.0566646	−	0.998393i	$-0.481953\pi$
$347$	2.11109	0.113329	0.0566646	+	0.998393i	$0.481953\pi$
$348$	0	0
$349$	−36.2164	−1.93862	−0.969310	−	0.245840i	$-0.920936\pi$
$349$	−36.2164	−1.93862	−0.969310	+	0.245840i	$0.920936\pi$
$350$	0	0
$351$	0	0
$352$	−3.18194	−0.169598
$353$	10.4887	0.558255	0.279127	−	0.960254i	$-0.409955\pi$
$353$	10.4887	0.558255	0.279127	+	0.960254i	$0.409955\pi$
$354$	0	0
$355$	−27.6512	−1.46757
$356$	0.225450	0.0119488
$357$	0	0
$358$	7.03775	0.371957
$359$	32.4419	1.71222	0.856108	−	0.516796i	$-0.172876\pi$
$359$	32.4419	1.71222	0.856108	+	0.516796i	$0.172876\pi$
$360$	0	0
$361$	−17.3549	−0.913414
$362$	−12.9669	−0.681525
$363$	0	0
$364$	0	0
$365$	15.8044	0.827239
$366$	0	0
$367$	−18.1111	−0.945391	−0.472696	−	0.881226i	$-0.656719\pi$
$367$	−18.1111	−0.945391	−0.472696	+	0.881226i	$0.656719\pi$
$368$	−2.23912	−0.116722
$369$	0	0
$370$	−3.18194	−0.165421
$371$	0	0
$372$	0	0
$373$	−11.6706	−0.604280	−0.302140	−	0.953263i	$-0.597701\pi$
$373$	−11.6706	−0.604280	−0.302140	+	0.953263i	$0.597701\pi$
$374$	4.84213	0.250381
$375$	0	0
$376$	5.82846	0.300580
$377$	40.3893	2.08016
$378$	0	0
$379$	14.2690	0.732947	0.366474	−	0.930428i	$-0.380565\pi$
$379$	14.2690	0.732947	0.366474	+	0.930428i	$0.380565\pi$
$380$	4.08126	0.209364
$381$	0	0
$382$	−1.98057	−0.101335
$383$	1.64979	0.0843001	0.0421501	−	0.999111i	$-0.486579\pi$
$383$	1.64979	0.0843001	0.0421501	+	0.999111i	$0.486579\pi$
$384$	0	0
$385$	0	0
$386$	−4.54583	−0.231377
$387$	0	0
$388$	−14.8421	−0.753495
$389$	32.0676	1.62589	0.812946	−	0.582340i	$-0.197863\pi$
$389$	32.0676	1.62589	0.812946	+	0.582340i	$0.197863\pi$
$390$	0	0
$391$	3.40739	0.172319
$392$	0	0
$393$	0	0
$394$	21.8148	1.09901
$395$	−13.1683	−0.662568
$396$	0	0
$397$	37.9338	1.90384	0.951921	−	0.306343i	$-0.0991054\pi$
$397$	37.9338	1.90384	0.951921	+	0.306343i	$0.0991054\pi$
$398$	−12.2826	−0.615673
$399$	0	0
$400$	5.12476	0.256238
$401$	−10.6192	−0.530296	−0.265148	−	0.964208i	$-0.585421\pi$
$401$	−10.6192	−0.530296	−0.265148	+	0.964208i	$0.585421\pi$
$402$	0	0
$403$	53.7349	2.67673
$404$	−18.5893	−0.924854
$405$	0	0
$406$	0	0
$407$	3.18194	0.157723
$408$	0	0
$409$	5.54583	0.274224	0.137112	−	0.990556i	$-0.456218\pi$
$409$	5.54583	0.274224	0.137112	+	0.990556i	$0.456218\pi$
$410$	−17.8285	−0.880485
$411$	0	0
$412$	−0.282630	−0.0139242
$413$	0	0
$414$	0	0
$415$	−25.6706	−1.26012
$416$	−5.70370	−0.279647
$417$	0	0
$418$	−4.08126	−0.199621
$419$	5.54910	0.271091	0.135546	−	0.990771i	$-0.456721\pi$
$419$	5.54910	0.271091	0.135546	+	0.990771i	$0.456721\pi$
$420$	0	0
$421$	6.84213	0.333465	0.166733	−	0.986002i	$-0.446678\pi$
$421$	6.84213	0.333465	0.166733	+	0.986002i	$0.446678\pi$
$422$	16.6569	0.810846
$423$	0	0
$424$	2.05718	0.0999055
$425$	−7.79863	−0.378289
$426$	0	0
$427$	0	0
$428$	11.3776	0.549955
$429$	0	0
$430$	−21.7278	−1.04781
$431$	33.1078	1.59475	0.797374	−	0.603486i	$-0.206222\pi$
$431$	33.1078	1.59475	0.797374	+	0.603486i	$0.206222\pi$
$432$	0	0
$433$	−12.1111	−0.582022	−0.291011	−	0.956720i	$-0.593992\pi$
$433$	−12.1111	−0.582022	−0.291011	+	0.956720i	$0.593992\pi$
$434$	0	0
$435$	0	0
$436$	4.42107	0.211731
$437$	−2.87197	−0.137385
$438$	0	0
$439$	−8.83422	−0.421634	−0.210817	−	0.977526i	$-0.567612\pi$
$439$	−8.83422	−0.421634	−0.210817	+	0.977526i	$0.567612\pi$
$440$	−10.1248	−0.482679
$441$	0	0
$442$	8.67962	0.412847
$443$	−17.5185	−0.832328	−0.416164	−	0.909290i	$-0.636626\pi$
$443$	−17.5185	−0.832328	−0.416164	+	0.909290i	$0.636626\pi$
$444$	0	0
$445$	0.717370	0.0340066
$446$	10.6569	0.504620
$447$	0	0
$448$	0	0
$449$	−31.2301	−1.47384	−0.736920	−	0.675980i	$-0.763720\pi$
$449$	−31.2301	−1.47384	−0.736920	+	0.675980i	$0.763720\pi$
$450$	0	0
$451$	17.8285	0.839509
$452$	−3.21505	−0.151223
$453$	0	0
$454$	14.5081	0.680898
$455$	0	0
$456$	0	0
$457$	−32.1248	−1.50273	−0.751367	−	0.659885i	$-0.770605\pi$
$457$	−32.1248	−1.50273	−0.751367	+	0.659885i	$0.770605\pi$
$458$	10.2495	0.478929
$459$	0	0
$460$	−7.12476	−0.332194
$461$	2.46457	0.114787	0.0573933	−	0.998352i	$-0.481721\pi$
$461$	2.46457	0.114787	0.0573933	+	0.998352i	$0.481721\pi$
$462$	0	0
$463$	−30.3469	−1.41034	−0.705171	−	0.709037i	$-0.749130\pi$
$463$	−30.3469	−1.41034	−0.705171	+	0.709037i	$0.749130\pi$
$464$	−7.08126	−0.328739
$465$	0	0
$466$	1.08126	0.0500882
$467$	−15.9636	−0.738709	−0.369354	−	0.929289i	$-0.620421\pi$
$467$	−15.9636	−0.738709	−0.369354	+	0.929289i	$0.620421\pi$
$468$	0	0
$469$	0	0
$470$	18.5458	0.855455
$471$	0	0
$472$	1.12476	0.0517714
$473$	21.7278	0.999044
$474$	0	0
$475$	6.57318	0.301598
$476$	0	0
$477$	0	0
$478$	−12.3204	−0.563521
$479$	23.1729	1.05880	0.529399	−	0.848373i	$-0.322418\pi$
$479$	23.1729	1.05880	0.529399	+	0.848373i	$0.322418\pi$
$480$	0	0
$481$	5.70370	0.260066
$482$	−13.0000	−0.592134
$483$	0	0
$484$	−0.875237	−0.0397835
$485$	−47.2268	−2.14446
$486$	0	0
$487$	−3.41315	−0.154665	−0.0773323	−	0.997005i	$-0.524640\pi$
$487$	−3.41315	−0.154665	−0.0773323	+	0.997005i	$0.524640\pi$
$488$	3.12476	0.141451
$489$	0	0
$490$	0	0
$491$	−19.1683	−0.865052	−0.432526	−	0.901621i	$-0.642378\pi$
$491$	−19.1683	−0.865052	−0.432526	+	0.901621i	$0.642378\pi$
$492$	0	0
$493$	10.7759	0.485323
$494$	−7.31573	−0.329150
$495$	0	0
$496$	−9.42107	−0.423018
$497$	0	0
$498$	0	0
$499$	41.1696	1.84301	0.921503	−	0.388371i	$-0.126962\pi$
$499$	41.1696	1.84301	0.921503	+	0.388371i	$0.126962\pi$
$500$	0.396990	0.0177539
$501$	0	0
$502$	−5.11109	−0.228119
$503$	26.4542	1.17953	0.589767	−	0.807574i	$-0.299220\pi$
$503$	26.4542	1.17953	0.589767	+	0.807574i	$0.299220\pi$
$504$	0	0
$505$	−59.1502	−2.63215
$506$	7.12476	0.316734
$507$	0	0
$508$	20.1053	0.892030
$509$	−12.7713	−0.566077	−0.283039	−	0.959109i	$-0.591342\pi$
$509$	−12.7713	−0.566077	−0.283039	+	0.959109i	$0.591342\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	0.0441942
$513$	0	0
$514$	−7.66019	−0.337876
$515$	−0.899313	−0.0396285
$516$	0	0
$517$	−18.5458	−0.815645
$518$	0	0
$519$	0	0
$520$	−18.1488	−0.795879
$521$	−6.81230	−0.298452	−0.149226	−	0.988803i	$-0.547678\pi$
$521$	−6.81230	−0.298452	−0.149226	+	0.988803i	$0.547678\pi$
$522$	0	0
$523$	−29.5070	−1.29025	−0.645125	−	0.764077i	$-0.723195\pi$
$523$	−29.5070	−1.29025	−0.645125	+	0.764077i	$0.723195\pi$
$524$	−6.36389	−0.278008
$525$	0	0
$526$	3.09493	0.134945
$527$	14.3365	0.624510
$528$	0	0
$529$	−17.9863	−0.782014
$530$	6.54583	0.284333
$531$	0	0
$532$	0	0
$533$	31.9579	1.38425
$534$	0	0
$535$	36.2028	1.56518
$536$	10.9669	0.473698
$537$	0	0
$538$	−26.8903	−1.15932
$539$	0	0
$540$	0	0
$541$	−29.4016	−1.26408	−0.632038	−	0.774938i	$-0.717781\pi$
$541$	−29.4016	−1.26408	−0.632038	+	0.774938i	$0.717781\pi$
$542$	22.2164	0.954277
$543$	0	0
$544$	−1.52175	−0.0652446
$545$	14.0676	0.602589
$546$	0	0
$547$	−35.2301	−1.50633	−0.753165	−	0.657832i	$-0.771474\pi$
$547$	−35.2301	−1.50633	−0.753165	+	0.657832i	$0.771474\pi$
$548$	−2.74145	−0.117109
$549$	0	0
$550$	−16.3067	−0.695320
$551$	−9.08263	−0.386933
$552$	0	0
$553$	0	0
$554$	−14.6375	−0.621887
$555$	0	0
$556$	7.96690	0.337872
$557$	−6.73818	−0.285506	−0.142753	−	0.989758i	$-0.545595\pi$
$557$	−6.73818	−0.285506	−0.142753	+	0.989758i	$0.545595\pi$
$558$	0	0
$559$	38.9475	1.64730
$560$	0	0
$561$	0	0
$562$	23.3984	0.987001
$563$	1.45993	0.0615286	0.0307643	−	0.999527i	$-0.490206\pi$
$563$	1.45993	0.0615286	0.0307643	+	0.999527i	$0.490206\pi$
$564$	0	0
$565$	−10.2301	−0.430383
$566$	−26.1248	−1.09811
$567$	0	0
$568$	−8.69002	−0.364625
$569$	−19.5653	−0.820218	−0.410109	−	0.912036i	$-0.634509\pi$
$569$	−19.5653	−0.820218	−0.410109	+	0.912036i	$0.634509\pi$
$570$	0	0
$571$	−21.9259	−0.917569	−0.458785	−	0.888547i	$-0.651715\pi$
$571$	−21.9259	−0.917569	−0.458785	+	0.888547i	$0.651715\pi$
$572$	18.1488	0.758841
$573$	0	0
$574$	0	0
$575$	−11.4750	−0.478540
$576$	0	0
$577$	−24.7310	−1.02957	−0.514783	−	0.857320i	$-0.672128\pi$
$577$	−24.7310	−1.02957	−0.514783	+	0.857320i	$0.672128\pi$
$578$	−14.6843	−0.610785
$579$	0	0
$580$	−22.5322	−0.935597
$581$	0	0
$582$	0	0
$583$	−6.54583	−0.271101
$584$	4.96690	0.205532
$585$	0	0
$586$	25.8629	1.06839
$587$	−36.1592	−1.49245	−0.746226	−	0.665693i	$-0.768136\pi$
$587$	−36.1592	−1.49245	−0.746226	+	0.665693i	$0.768136\pi$
$588$	0	0
$589$	−12.0837	−0.497902
$590$	3.57893	0.147342
$591$	0	0
$592$	−1.00000	−0.0410997
$593$	−15.1078	−0.620404	−0.310202	−	0.950671i	$-0.600397\pi$
$593$	−15.1078	−0.620404	−0.310202	+	0.950671i	$0.600397\pi$
$594$	0	0
$595$	0	0
$596$	23.2599	0.952764
$597$	0	0
$598$	12.7713	0.522256
$599$	5.45417	0.222851	0.111426	−	0.993773i	$-0.464458\pi$
$599$	5.45417	0.222851	0.111426	+	0.993773i	$0.464458\pi$
$600$	0	0
$601$	6.73680	0.274800	0.137400	−	0.990516i	$-0.456125\pi$
$601$	6.73680	0.274800	0.137400	+	0.990516i	$0.456125\pi$
$602$	0	0
$603$	0	0
$604$	−8.12476	−0.330592
$605$	−2.78495	−0.113224
$606$	0	0
$607$	6.67059	0.270751	0.135376	−	0.990794i	$-0.456776\pi$
$607$	6.67059	0.270751	0.135376	+	0.990794i	$0.456776\pi$
$608$	1.28263	0.0520175
$609$	0	0
$610$	9.94282	0.402573
$611$	−33.2438	−1.34490
$612$	0	0
$613$	−1.30998	−0.0529094	−0.0264547	−	0.999650i	$-0.508422\pi$
$613$	−1.30998	−0.0529094	−0.0264547	+	0.999650i	$0.508422\pi$
$614$	3.53216	0.142546
$615$	0	0
$616$	0	0
$617$	34.4966	1.38878	0.694390	−	0.719599i	$-0.255674\pi$
$617$	34.4966	1.38878	0.694390	+	0.719599i	$0.255674\pi$
$618$	0	0
$619$	−16.4484	−0.661118	−0.330559	−	0.943785i	$-0.607237\pi$
$619$	−16.4484	−0.661118	−0.330559	+	0.943785i	$0.607237\pi$
$620$	−29.9773	−1.20392
$621$	0	0
$622$	−1.70370	−0.0683120
$623$	0	0
$624$	0	0
$625$	−24.3606	−0.974425
$626$	−2.84213	−0.113594
$627$	0	0
$628$	−11.2632	−0.449451
$629$	1.52175	0.0606763
$630$	0	0
$631$	−30.0118	−1.19475	−0.597375	−	0.801962i	$-0.703790\pi$
$631$	−30.0118	−1.19475	−0.597375	+	0.801962i	$0.703790\pi$
$632$	−4.13844	−0.164618
$633$	0	0
$634$	24.9201	0.989704
$635$	63.9740	2.53873
$636$	0	0
$637$	0	0
$638$	22.5322	0.892057
$639$	0	0
$640$	3.18194	0.125777
$641$	−27.8993	−1.10196	−0.550978	−	0.834520i	$-0.685745\pi$
$641$	−27.8993	−1.10196	−0.550978	+	0.834520i	$0.685745\pi$
$642$	0	0
$643$	−28.5048	−1.12412	−0.562060	−	0.827096i	$-0.689991\pi$
$643$	−28.5048	−1.12412	−0.562060	+	0.827096i	$0.689991\pi$
$644$	0	0
$645$	0	0
$646$	−1.95185	−0.0767944
$647$	16.7141	0.657099	0.328550	−	0.944487i	$-0.393440\pi$
$647$	16.7141	0.657099	0.328550	+	0.944487i	$0.393440\pi$
$648$	0	0
$649$	−3.57893	−0.140485
$650$	−29.2301	−1.14650
$651$	0	0
$652$	3.98057	0.155891
$653$	−38.1650	−1.49351	−0.746756	−	0.665098i	$-0.768390\pi$
$653$	−38.1650	−1.49351	−0.746756	+	0.665098i	$0.768390\pi$
$654$	0	0
$655$	−20.2495	−0.791214
$656$	−5.60301	−0.218761
$657$	0	0
$658$	0	0
$659$	8.74145	0.340518	0.170259	−	0.985399i	$-0.445539\pi$
$659$	8.74145	0.340518	0.170259	+	0.985399i	$0.445539\pi$
$660$	0	0
$661$	−20.0837	−0.781167	−0.390584	−	0.920567i	$-0.627727\pi$
$661$	−20.0837	−0.781167	−0.390584	+	0.920567i	$0.627727\pi$
$662$	−7.17154	−0.278730
$663$	0	0
$664$	−8.06758	−0.313083
$665$	0	0
$666$	0	0
$667$	15.8558	0.613939
$668$	5.23912	0.202708
$669$	0	0
$670$	34.8960	1.34815
$671$	−9.94282	−0.383838
$672$	0	0
$673$	34.0528	1.31264	0.656319	−	0.754483i	$-0.272112\pi$
$673$	34.0528	1.31264	0.656319	+	0.754483i	$0.272112\pi$
$674$	21.8421	0.841328
$675$	0	0
$676$	19.5322	0.751237
$677$	0.717370	0.0275708	0.0137854	−	0.999905i	$-0.495612\pi$
$677$	0.717370	0.0275708	0.0137854	+	0.999905i	$0.495612\pi$
$678$	0	0
$679$	0	0
$680$	−4.84213	−0.185687
$681$	0	0
$682$	29.9773	1.14789
$683$	−21.0539	−0.805605	−0.402803	−	0.915287i	$-0.631964\pi$
$683$	−21.0539	−0.805605	−0.402803	+	0.915287i	$0.631964\pi$
$684$	0	0
$685$	−8.72313	−0.333294
$686$	0	0
$687$	0	0
$688$	−6.82846	−0.260333
$689$	−11.7335	−0.447012
$690$	0	0
$691$	5.84789	0.222464	0.111232	−	0.993794i	$-0.464520\pi$
$691$	5.84789	0.222464	0.111232	+	0.993794i	$0.464520\pi$
$692$	−2.55159	−0.0969968
$693$	0	0
$694$	2.11109	0.0801359
$695$	25.3502	0.961588
$696$	0	0
$697$	8.52640	0.322960
$698$	−36.2164	−1.37081
$699$	0	0
$700$	0	0
$701$	−10.2711	−0.387935	−0.193967	−	0.981008i	$-0.562136\pi$
$701$	−10.2711	−0.387935	−0.193967	+	0.981008i	$0.562136\pi$
$702$	0	0
$703$	−1.28263	−0.0483753
$704$	−3.18194	−0.119924
$705$	0	0
$706$	10.4887	0.394746
$707$	0	0
$708$	0	0
$709$	43.4854	1.63313	0.816564	−	0.577255i	$-0.195876\pi$
$709$	43.4854	1.63313	0.816564	+	0.577255i	$0.195876\pi$
$710$	−27.6512	−1.03773
$711$	0	0
$712$	0.225450	0.00844910
$713$	21.0949	0.790011
$714$	0	0
$715$	57.7486	2.15967
$716$	7.03775	0.263013
$717$	0	0
$718$	32.4419	1.21072
$719$	50.8824	1.89759	0.948796	−	0.315889i	$-0.102303\pi$
$719$	50.8824	1.89759	0.948796	+	0.315889i	$0.102303\pi$
$720$	0	0
$721$	0	0
$722$	−17.3549	−0.645881
$723$	0	0
$724$	−12.9669	−0.481911
$725$	−36.2898	−1.34777
$726$	0	0
$727$	−12.1442	−0.450403	−0.225202	−	0.974312i	$-0.572304\pi$
$727$	−12.1442	−0.450403	−0.225202	+	0.974312i	$0.572304\pi$
$728$	0	0
$729$	0	0
$730$	15.8044	0.584946
$731$	10.3912	0.384334
$732$	0	0
$733$	−46.1696	−1.70531	−0.852657	−	0.522470i	$-0.825011\pi$
$733$	−46.1696	−1.70531	−0.852657	+	0.522470i	$0.825011\pi$
$734$	−18.1111	−0.668493
$735$	0	0
$736$	−2.23912	−0.0825352
$737$	−34.8960	−1.28541
$738$	0	0
$739$	4.99208	0.183637	0.0918184	−	0.995776i	$-0.470732\pi$
$739$	4.99208	0.183637	0.0918184	+	0.995776i	$0.470732\pi$
$740$	−3.18194	−0.116971
$741$	0	0
$742$	0	0
$743$	−31.4120	−1.15240	−0.576198	−	0.817310i	$-0.695464\pi$
$743$	−31.4120	−1.15240	−0.576198	+	0.817310i	$0.695464\pi$
$744$	0	0
$745$	74.0118	2.71158
$746$	−11.6706	−0.427291
$747$	0	0
$748$	4.84213	0.177046
$749$	0	0
$750$	0	0
$751$	3.29630	0.120284	0.0601419	−	0.998190i	$-0.480845\pi$
$751$	3.29630	0.120284	0.0601419	+	0.998190i	$0.480845\pi$
$752$	5.82846	0.212542
$753$	0	0
$754$	40.3893	1.47089
$755$	−25.8525	−0.940870
$756$	0	0
$757$	−10.1384	−0.368488	−0.184244	−	0.982881i	$-0.558984\pi$
$757$	−10.1384	−0.368488	−0.184244	+	0.982881i	$0.558984\pi$
$758$	14.2690	0.518272
$759$	0	0
$760$	4.08126	0.148043
$761$	−14.0676	−0.509950	−0.254975	−	0.966948i	$-0.582067\pi$
$761$	−14.0676	−0.509950	−0.254975	+	0.966948i	$0.582067\pi$
$762$	0	0
$763$	0	0
$764$	−1.98057	−0.0716545
$765$	0	0
$766$	1.64979	0.0596092
$767$	−6.41531	−0.231643
$768$	0	0
$769$	−22.6922	−0.818301	−0.409151	−	0.912467i	$-0.634175\pi$
$769$	−22.6922	−0.818301	−0.409151	+	0.912467i	$0.634175\pi$
$770$	0	0
$771$	0	0
$772$	−4.54583	−0.163608
$773$	0.655544	0.0235783	0.0117891	−	0.999931i	$-0.496247\pi$
$773$	0.655544	0.0235783	0.0117891	+	0.999931i	$0.496247\pi$
$774$	0	0
$775$	−48.2807	−1.73430
$776$	−14.8421	−0.532802
$777$	0	0
$778$	32.0676	1.14968
$779$	−7.18659	−0.257486
$780$	0	0
$781$	27.6512	0.989436
$782$	3.40739	0.121848
$783$	0	0
$784$	0	0
$785$	−35.8389	−1.27914
$786$	0	0
$787$	0.540073	0.0192515	0.00962576	−	0.999954i	$-0.496936\pi$
$787$	0.540073	0.0192515	0.00962576	+	0.999954i	$0.496936\pi$
$788$	21.8148	0.777120
$789$	0	0
$790$	−13.1683	−0.468506
$791$	0	0
$792$	0	0
$793$	−17.8227	−0.632903
$794$	37.9338	1.34622
$795$	0	0
$796$	−12.2826	−0.435346
$797$	−25.1100	−0.889441	−0.444721	−	0.895669i	$-0.646697\pi$
$797$	−25.1100	−0.889441	−0.444721	+	0.895669i	$0.646697\pi$
$798$	0	0
$799$	−8.86948	−0.313780
$800$	5.12476	0.181188
$801$	0	0
$802$	−10.6192	−0.374976
$803$	−15.8044	−0.557725
$804$	0	0
$805$	0	0
$806$	53.7349	1.89273
$807$	0	0
$808$	−18.5893	−0.653971
$809$	−29.1729	−1.02567	−0.512833	−	0.858489i	$-0.671404\pi$
$809$	−29.1729	−1.02567	−0.512833	+	0.858489i	$0.671404\pi$
$810$	0	0
$811$	−15.4290	−0.541785	−0.270892	−	0.962610i	$-0.587319\pi$
$811$	−15.4290	−0.541785	−0.270892	+	0.962610i	$0.587319\pi$
$812$	0	0
$813$	0	0
$814$	3.18194	0.111527
$815$	12.6659	0.443669
$816$	0	0
$817$	−8.75839	−0.306417
$818$	5.54583	0.193905
$819$	0	0
$820$	−17.8285	−0.622597
$821$	−8.48727	−0.296208	−0.148104	−	0.988972i	$-0.547317\pi$
$821$	−8.48727	−0.296208	−0.148104	+	0.988972i	$0.547317\pi$
$822$	0	0
$823$	29.0974	1.01427	0.507136	−	0.861866i	$-0.330704\pi$
$823$	29.0974	1.01427	0.507136	+	0.861866i	$0.330704\pi$
$824$	−0.282630	−0.00984589
$825$	0	0
$826$	0	0
$827$	25.9396	0.902007	0.451003	−	0.892522i	$-0.351066\pi$
$827$	25.9396	0.902007	0.451003	+	0.892522i	$0.351066\pi$
$828$	0	0
$829$	−6.21642	−0.215905	−0.107953	−	0.994156i	$-0.534429\pi$
$829$	−6.21642	−0.215905	−0.107953	+	0.994156i	$0.534429\pi$
$830$	−25.6706	−0.891039
$831$	0	0
$832$	−5.70370	−0.197740
$833$	0	0
$834$	0	0
$835$	16.6706	0.576910
$836$	−4.08126	−0.141153
$837$	0	0
$838$	5.54910	0.191690
$839$	42.5893	1.47035	0.735174	−	0.677879i	$-0.237101\pi$
$839$	42.5893	1.47035	0.735174	+	0.677879i	$0.237101\pi$
$840$	0	0
$841$	21.1442	0.729110
$842$	6.84213	0.235795
$843$	0	0
$844$	16.6569	0.573355
$845$	62.1502	2.13803
$846$	0	0
$847$	0	0
$848$	2.05718	0.0706438
$849$	0	0
$850$	−7.79863	−0.267491
$851$	2.23912	0.0767562
$852$	0	0
$853$	21.3937	0.732507	0.366254	−	0.930515i	$-0.380640\pi$
$853$	21.3937	0.732507	0.366254	+	0.930515i	$0.380640\pi$
$854$	0	0
$855$	0	0
$856$	11.3776	0.388877
$857$	36.8435	1.25855	0.629275	−	0.777183i	$-0.283352\pi$
$857$	36.8435	1.25855	0.629275	+	0.777183i	$0.283352\pi$
$858$	0	0
$859$	−17.6375	−0.601783	−0.300892	−	0.953658i	$-0.597284\pi$
$859$	−17.6375	−0.601783	−0.300892	+	0.953658i	$0.597284\pi$
$860$	−21.7278	−0.740911
$861$	0	0
$862$	33.1078	1.12766
$863$	−0.760877	−0.0259005	−0.0129503	−	0.999916i	$-0.504122\pi$
$863$	−0.760877	−0.0259005	−0.0129503	+	0.999916i	$0.504122\pi$
$864$	0	0
$865$	−8.11901	−0.276054
$866$	−12.1111	−0.411552
$867$	0	0
$868$	0	0
$869$	13.1683	0.446703
$870$	0	0
$871$	−62.5519	−2.11949
$872$	4.42107	0.149716
$873$	0	0
$874$	−2.87197	−0.0971457
$875$	0	0
$876$	0	0
$877$	−41.4991	−1.40132	−0.700662	−	0.713494i	$-0.747112\pi$
$877$	−41.4991	−1.40132	−0.700662	+	0.713494i	$0.747112\pi$
$878$	−8.83422	−0.298140
$879$	0	0
$880$	−10.1248	−0.341306
$881$	−8.35486	−0.281482	−0.140741	−	0.990046i	$-0.544949\pi$
$881$	−8.35486	−0.281482	−0.140741	+	0.990046i	$0.544949\pi$
$882$	0	0
$883$	35.6181	1.19864	0.599322	−	0.800508i	$-0.295437\pi$
$883$	35.6181	1.19864	0.599322	+	0.800508i	$0.295437\pi$
$884$	8.67962	0.291927
$885$	0	0
$886$	−17.5185	−0.588545
$887$	37.1100	1.24603	0.623016	−	0.782209i	$-0.285907\pi$
$887$	37.1100	1.24603	0.623016	+	0.782209i	$0.285907\pi$
$888$	0	0
$889$	0	0
$890$	0.717370	0.0240463
$891$	0	0
$892$	10.6569	0.356820
$893$	7.47576	0.250167
$894$	0	0
$895$	22.3937	0.748540
$896$	0	0
$897$	0	0
$898$	−31.2301	−1.04216
$899$	66.7130	2.22500
$900$	0	0
$901$	−3.13052	−0.104293
$902$	17.8285	0.593623
$903$	0	0
$904$	−3.21505	−0.106931
$905$	−41.2599	−1.37153
$906$	0	0
$907$	−48.1502	−1.59880	−0.799401	−	0.600798i	$-0.794850\pi$
$907$	−48.1502	−1.59880	−0.799401	+	0.600798i	$0.794850\pi$
$908$	14.5081	0.481468
$909$	0	0
$910$	0	0
$911$	34.8856	1.15581	0.577906	−	0.816103i	$-0.303870\pi$
$911$	34.8856	1.15581	0.577906	+	0.816103i	$0.303870\pi$
$912$	0	0
$913$	25.6706	0.849573
$914$	−32.1248	−1.06259
$915$	0	0
$916$	10.2495	0.338654
$917$	0	0
$918$	0	0
$919$	51.7349	1.70658	0.853289	−	0.521439i	$-0.174605\pi$
$919$	51.7349	1.70658	0.853289	+	0.521439i	$0.174605\pi$
$920$	−7.12476	−0.234896
$921$	0	0
$922$	2.46457	0.0811664
$923$	49.5653	1.63146
$924$	0	0
$925$	−5.12476	−0.168501
$926$	−30.3469	−0.997262
$927$	0	0
$928$	−7.08126	−0.232454
$929$	50.8285	1.66763	0.833814	−	0.552046i	$-0.186153\pi$
$929$	50.8285	1.66763	0.833814	+	0.552046i	$0.186153\pi$
$930$	0	0
$931$	0	0
$932$	1.08126	0.0354177
$933$	0	0
$934$	−15.9636	−0.522346
$935$	15.4074	0.503876
$936$	0	0
$937$	2.54583	0.0831686	0.0415843	−	0.999135i	$-0.486759\pi$
$937$	2.54583	0.0831686	0.0415843	+	0.999135i	$0.486759\pi$
$938$	0	0
$939$	0	0
$940$	18.5458	0.604898
$941$	−1.15787	−0.0377454	−0.0188727	−	0.999822i	$-0.506008\pi$
$941$	−1.15787	−0.0377454	−0.0188727	+	0.999822i	$0.506008\pi$
$942$	0	0
$943$	12.5458	0.408548
$944$	1.12476	0.0366079
$945$	0	0
$946$	21.7278	0.706431
$947$	−9.81479	−0.318938	−0.159469	−	0.987203i	$-0.550978\pi$
$947$	−9.81479	−0.318938	−0.159469	+	0.987203i	$0.550978\pi$
$948$	0	0
$949$	−28.3297	−0.919620
$950$	6.57318	0.213262
$951$	0	0
$952$	0	0
$953$	−6.53791	−0.211784	−0.105892	−	0.994378i	$-0.533770\pi$
$953$	−6.53791	−0.211784	−0.105892	+	0.994378i	$0.533770\pi$
$954$	0	0
$955$	−6.30206	−0.203930
$956$	−12.3204	−0.398470
$957$	0	0
$958$	23.1729	0.748683
$959$	0	0
$960$	0	0
$961$	57.7565	1.86311
$962$	5.70370	0.183895
$963$	0	0
$964$	−13.0000	−0.418702
$965$	−14.4646	−0.465631
$966$	0	0
$967$	−28.8889	−0.929005	−0.464502	−	0.885572i	$-0.653767\pi$
$967$	−28.8889	−0.929005	−0.464502	+	0.885572i	$0.653767\pi$
$968$	−0.875237	−0.0281312
$969$	0	0
$970$	−47.2268	−1.51636
$971$	5.33654	0.171258	0.0856289	−	0.996327i	$-0.472710\pi$
$971$	5.33654	0.171258	0.0856289	+	0.996327i	$0.472710\pi$
$972$	0	0
$973$	0	0
$974$	−3.41315	−0.109364
$975$	0	0
$976$	3.12476	0.100021
$977$	48.0722	1.53797	0.768983	−	0.639269i	$-0.220763\pi$
$977$	48.0722	1.53797	0.768983	+	0.639269i	$0.220763\pi$
$978$	0	0
$979$	−0.717370	−0.0229272
$980$	0	0
$981$	0	0
$982$	−19.1683	−0.611684
$983$	−29.4627	−0.939714	−0.469857	−	0.882743i	$-0.655694\pi$
$983$	−29.4627	−0.939714	−0.469857	+	0.882743i	$0.655694\pi$
$984$	0	0
$985$	69.4134	2.21170
$986$	10.7759	0.343175
$987$	0	0
$988$	−7.31573	−0.232744
$989$	15.2898	0.486186
$990$	0	0
$991$	−30.8285	−0.979298	−0.489649	−	0.871920i	$-0.662875\pi$
$991$	−30.8285	−0.979298	−0.489649	+	0.871920i	$0.662875\pi$
$992$	−9.42107	−0.299119
$993$	0	0
$994$	0	0
$995$	−39.0826	−1.23900
$996$	0	0
$997$	5.54583	0.175638	0.0878191	−	0.996136i	$-0.472010\pi$
$997$	5.54583	0.175638	0.0878191	+	0.996136i	$0.472010\pi$
$998$	41.1696	1.30320
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7938.2.a.ca.1.3		3
3.2	odd	2		7938.2.a.bv.1.1		3
7.2	even	3		1134.2.g.l.487.1		6
7.4	even	3		1134.2.g.l.163.1		6
7.6	odd	2		7938.2.a.bz.1.1		3
9.2	odd	6		882.2.f.n.589.2		6
9.4	even	3		2646.2.f.l.883.1		6
9.5	odd	6		882.2.f.n.295.2		6
9.7	even	3		2646.2.f.l.1765.1		6
21.2	odd	6		1134.2.g.m.487.3		6
21.11	odd	6		1134.2.g.m.163.3		6
21.20	even	2		7938.2.a.bw.1.3		3
63.2	odd	6		126.2.h.d.67.1	yes	6
63.4	even	3		378.2.h.c.289.3		6
63.5	even	6		882.2.e.o.655.1		6
63.11	odd	6		126.2.e.c.121.3	yes	6
63.13	odd	6		2646.2.f.m.883.3		6
63.16	even	3		378.2.h.c.361.3		6
63.20	even	6		882.2.f.o.589.2		6
63.23	odd	6		126.2.e.c.25.3	✓	6
63.25	even	3		378.2.e.d.37.1		6
63.31	odd	6		2646.2.h.o.667.1		6
63.32	odd	6		126.2.h.d.79.1	yes	6
63.34	odd	6		2646.2.f.m.1765.3		6
63.38	even	6		882.2.e.o.373.1		6
63.40	odd	6		2646.2.e.p.2125.3		6
63.41	even	6		882.2.f.o.295.2		6
63.47	even	6		882.2.h.p.67.3		6
63.52	odd	6		2646.2.e.p.1549.3		6
63.58	even	3		378.2.e.d.235.1		6
63.59	even	6		882.2.h.p.79.3		6
63.61	odd	6		2646.2.h.o.361.1		6
252.11	even	6		1008.2.q.g.625.1		6
252.23	even	6		1008.2.q.g.529.1		6
252.67	odd	6		3024.2.t.h.289.3		6
252.79	odd	6		3024.2.t.h.1873.3		6
252.95	even	6		1008.2.t.h.961.3		6
252.151	odd	6		3024.2.q.g.2305.1		6
252.191	even	6		1008.2.t.h.193.3		6
252.247	odd	6		3024.2.q.g.2881.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.2.e.c.25.3	✓	6	63.23	odd	6
126.2.e.c.121.3	yes	6	63.11	odd	6
126.2.h.d.67.1	yes	6	63.2	odd	6
126.2.h.d.79.1	yes	6	63.32	odd	6
378.2.e.d.37.1		6	63.25	even	3
378.2.e.d.235.1		6	63.58	even	3
378.2.h.c.289.3		6	63.4	even	3
378.2.h.c.361.3		6	63.16	even	3
882.2.e.o.373.1		6	63.38	even	6
882.2.e.o.655.1		6	63.5	even	6
882.2.f.n.295.2		6	9.5	odd	6
882.2.f.n.589.2		6	9.2	odd	6
882.2.f.o.295.2		6	63.41	even	6
882.2.f.o.589.2		6	63.20	even	6
882.2.h.p.67.3		6	63.47	even	6
882.2.h.p.79.3		6	63.59	even	6
1008.2.q.g.529.1		6	252.23	even	6
1008.2.q.g.625.1		6	252.11	even	6
1008.2.t.h.193.3		6	252.191	even	6
1008.2.t.h.961.3		6	252.95	even	6
1134.2.g.l.163.1		6	7.4	even	3
1134.2.g.l.487.1		6	7.2	even	3
1134.2.g.m.163.3		6	21.11	odd	6
1134.2.g.m.487.3		6	21.2	odd	6
2646.2.e.p.1549.3		6	63.52	odd	6
2646.2.e.p.2125.3		6	63.40	odd	6
2646.2.f.l.883.1		6	9.4	even	3
2646.2.f.l.1765.1		6	9.7	even	3
2646.2.f.m.883.3		6	63.13	odd	6
2646.2.f.m.1765.3		6	63.34	odd	6
2646.2.h.o.361.1		6	63.61	odd	6
2646.2.h.o.667.1		6	63.31	odd	6
3024.2.q.g.2305.1		6	252.151	odd	6
3024.2.q.g.2881.1		6	252.247	odd	6
3024.2.t.h.289.3		6	252.67	odd	6
3024.2.t.h.1873.3		6	252.79	odd	6
7938.2.a.bv.1.1		3	3.2	odd	2
7938.2.a.bw.1.3		3	21.20	even	2
7938.2.a.bz.1.1		3	7.6	odd	2
7938.2.a.ca.1.3		3	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:	\( N \)	\(=\)	\( 7938 = 2 \cdot 3^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7938.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +3.18194 q^{5} +1.00000 q^{8} +3.18194 q^{10} -3.18194 q^{11} -5.70370 q^{13} +1.00000 q^{16} -1.52175 q^{17} +1.28263 q^{19} +3.18194 q^{20} -3.18194 q^{22} -2.23912 q^{23} +5.12476 q^{25} -5.70370 q^{26} -7.08126 q^{29} -9.42107 q^{31} +1.00000 q^{32} -1.52175 q^{34} -1.00000 q^{37} +1.28263 q^{38} +3.18194 q^{40} -5.60301 q^{41} -6.82846 q^{43} -3.18194 q^{44} -2.23912 q^{46} +5.82846 q^{47} +5.12476 q^{50} -5.70370 q^{52} +2.05718 q^{53} -10.1248 q^{55} -7.08126 q^{58} +1.12476 q^{59} +3.12476 q^{61} -9.42107 q^{62} +1.00000 q^{64} -18.1488 q^{65} +10.9669 q^{67} -1.52175 q^{68} -8.69002 q^{71} +4.96690 q^{73} -1.00000 q^{74} +1.28263 q^{76} -4.13844 q^{79} +3.18194 q^{80} -5.60301 q^{82} -8.06758 q^{83} -4.84213 q^{85} -6.82846 q^{86} -3.18194 q^{88} +0.225450 q^{89} -2.23912 q^{92} +5.82846 q^{94} +4.08126 q^{95} -14.8421 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} + q^{5} + 3 q^{8} + q^{10} - q^{11} - 8 q^{13} + 3 q^{16} - 4 q^{17} + 3 q^{19} + q^{20} - q^{22} - 7 q^{23} - 2 q^{25} - 8 q^{26} - 5 q^{29} - 20 q^{31} + 3 q^{32} - 4 q^{34}+ \cdots - 28 q^{97}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^4 + q^5 + 3 * q^8 + q^10 - q^11 - 8 * q^13 + 3 * q^16 - 4 * q^17 + 3 * q^19 + q^20 - q^22 - 7 * q^23 - 2 * q^25 - 8 * q^26 - 5 * q^29 - 20 * q^31 + 3 * q^32 - 4 * q^34 - 3 * q^37 + 3 * q^38 + q^40 + 6 * q^43 - q^44 - 7 * q^46 - 9 * q^47 - 2 * q^50 - 8 * q^52 + 15 * q^53 - 13 * q^55 - 5 * q^58 - 14 * q^59 - 8 * q^61 - 20 * q^62 + 3 * q^64 - 12 * q^65 - q^67 - 4 * q^68 - 7 * q^71 - 19 * q^73 - 3 * q^74 + 3 * q^76 - 5 * q^79 + q^80 + 2 * q^83 + 2 * q^85 + 6 * q^86 - q^88 - 9 * q^89 - 7 * q^92 - 9 * q^94 - 4 * q^95 - 28 * q^97

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