LMFDB - Embedded newform 7800.2.a.bn.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7800,2,Mod(1,7800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("7800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7800.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:	$62.2833135766$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.568.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x - 2 $ x^3 - x^2 - 6*x - 2
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			1.3
Root			$3.12489$ of defining polynomial
Character	$\chi$	$=$	7800.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +2.12489 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +2.12489 q^{7} +1.00000 q^{9} +3.15516 q^{11} -1.00000 q^{13} -0.515138 q^{17} +6.73463 q^{19} +2.12489 q^{21} +4.24977 q^{23} +1.00000 q^{27} +0.0302761 q^{29} +7.15516 q^{31} +3.15516 q^{33} -5.76491 q^{37} -1.00000 q^{39} -3.76491 q^{41} +3.28005 q^{43} +4.60975 q^{47} -2.48486 q^{49} -0.515138 q^{51} -0.280047 q^{53} +6.73463 q^{57} +11.3444 q^{59} -5.01468 q^{61} +2.12489 q^{63} -15.1093 q^{67} +4.24977 q^{69} +2.23509 q^{71} +2.96972 q^{73} +6.70436 q^{77} -8.98440 q^{79} +1.00000 q^{81} -5.40493 q^{83} +0.0302761 q^{87} -15.5298 q^{89} -2.12489 q^{91} +7.15516 q^{93} -0.909172 q^{97} +3.15516 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} - 2 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 - 2 * q^7 + 3 * q^9	$ 3 q + 3 q^{3} - 2 q^{7} + 3 q^{9} + 2 q^{11} - 3 q^{13} - 2 q^{17} + 3 q^{19} - 2 q^{21} - 4 q^{23} + 3 q^{27} + q^{29} + 14 q^{31} + 2 q^{33} - q^{37} - 3 q^{39} + 5 q^{41} - 6 q^{43} + 5 q^{47} - 7 q^{49} - 2 q^{51} + 15 q^{53} + 3 q^{57} + 8 q^{59} + 18 q^{61} - 2 q^{63} - 3 q^{67} - 4 q^{69} + 23 q^{71} + 8 q^{73} + 2 q^{77} + 7 q^{79} + 3 q^{81} + 8 q^{83} + q^{87} - 14 q^{89} + 2 q^{91} + 14 q^{93} + 2 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 - 2 * q^7 + 3 * q^9 + 2 * q^11 - 3 * q^13 - 2 * q^17 + 3 * q^19 - 2 * q^21 - 4 * q^23 + 3 * q^27 + q^29 + 14 * q^31 + 2 * q^33 - q^37 - 3 * q^39 + 5 * q^41 - 6 * q^43 + 5 * q^47 - 7 * q^49 - 2 * q^51 + 15 * q^53 + 3 * q^57 + 8 * q^59 + 18 * q^61 - 2 * q^63 - 3 * q^67 - 4 * q^69 + 23 * q^71 + 8 * q^73 + 2 * q^77 + 7 * q^79 + 3 * q^81 + 8 * q^83 + q^87 - 14 * q^89 + 2 * q^91 + 14 * q^93 + 2 * 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 $

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.12489	0.803131	0.401566	−	0.915830i	$-0.368466\pi$
$7$	2.12489	0.803131	0.401566	+	0.915830i	$0.368466\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.15516	0.951317	0.475658	−	0.879630i	$-0.342210\pi$
$11$	3.15516	0.951317	0.475658	+	0.879630i	$0.342210\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−0.515138	−0.124939	−0.0624697	−	0.998047i	$-0.519898\pi$
$17$	−0.515138	−0.124939	−0.0624697	+	0.998047i	$0.519898\pi$
$18$	0	0
$19$	6.73463	1.54503	0.772515	−	0.634996i	$-0.218998\pi$
$19$	6.73463	1.54503	0.772515	+	0.634996i	$0.218998\pi$
$20$	0	0
$21$	2.12489	0.463688
$22$	0	0
$23$	4.24977	0.886138	0.443069	−	0.896487i	$-0.353890\pi$
$23$	4.24977	0.886138	0.443069	+	0.896487i	$0.353890\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	0.0302761	0.00562213	0.00281106	−	0.999996i	$-0.499105\pi$
$29$	0.0302761	0.00562213	0.00281106	+	0.999996i	$0.499105\pi$
$30$	0	0
$31$	7.15516	1.28510	0.642552	−	0.766242i	$-0.277875\pi$
$31$	7.15516	1.28510	0.642552	+	0.766242i	$0.277875\pi$
$32$	0	0
$33$	3.15516	0.549243
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−5.76491	−0.947745	−0.473873	−	0.880593i	$-0.657144\pi$
$37$	−5.76491	−0.947745	−0.473873	+	0.880593i	$0.657144\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−3.76491	−0.587980	−0.293990	−	0.955808i	$-0.594983\pi$
$41$	−3.76491	−0.587980	−0.293990	+	0.955808i	$0.594983\pi$
$42$	0	0
$43$	3.28005	0.500202	0.250101	−	0.968220i	$-0.419536\pi$
$43$	3.28005	0.500202	0.250101	+	0.968220i	$0.419536\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.60975	0.672401	0.336200	−	0.941790i	$-0.390858\pi$
$47$	4.60975	0.672401	0.336200	+	0.941790i	$0.390858\pi$
$48$	0	0
$49$	−2.48486	−0.354980
$50$	0	0
$51$	−0.515138	−0.0721338
$52$	0	0
$53$	−0.280047	−0.0384674	−0.0192337	−	0.999815i	$-0.506123\pi$
$53$	−0.280047	−0.0384674	−0.0192337	+	0.999815i	$0.506123\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	6.73463	0.892024
$58$	0	0
$59$	11.3444	1.47691	0.738456	−	0.674301i	$-0.235555\pi$
$59$	11.3444	1.47691	0.738456	+	0.674301i	$0.235555\pi$
$60$	0	0
$61$	−5.01468	−0.642064	−0.321032	−	0.947068i	$-0.604030\pi$
$61$	−5.01468	−0.642064	−0.321032	+	0.947068i	$0.604030\pi$
$62$	0	0
$63$	2.12489	0.267710
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−15.1093	−1.84589	−0.922947	−	0.384928i	$-0.874226\pi$
$67$	−15.1093	−1.84589	−0.922947	+	0.384928i	$0.874226\pi$
$68$	0	0
$69$	4.24977	0.511612
$70$	0	0
$71$	2.23509	0.265257	0.132628	−	0.991166i	$-0.457658\pi$
$71$	2.23509	0.265257	0.132628	+	0.991166i	$0.457658\pi$
$72$	0	0
$73$	2.96972	0.347580	0.173790	−	0.984783i	$-0.444399\pi$
$73$	2.96972	0.347580	0.173790	+	0.984783i	$0.444399\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	6.70436	0.764032
$78$	0	0
$79$	−8.98440	−1.01082	−0.505412	−	0.862878i	$-0.668660\pi$
$79$	−8.98440	−1.01082	−0.505412	+	0.862878i	$0.668660\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−5.40493	−0.593268	−0.296634	−	0.954991i	$-0.595864\pi$
$83$	−5.40493	−0.593268	−0.296634	+	0.954991i	$0.595864\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0.0302761	0.00324594
$88$	0	0
$89$	−15.5298	−1.64616	−0.823079	−	0.567927i	$-0.807745\pi$
$89$	−15.5298	−1.64616	−0.823079	+	0.567927i	$0.807745\pi$
$90$	0	0
$91$	−2.12489	−0.222749
$92$	0	0
$93$	7.15516	0.741956
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−0.909172	−0.0923124	−0.0461562	−	0.998934i	$-0.514697\pi$
$97$	−0.909172	−0.0923124	−0.0461562	+	0.998934i	$0.514697\pi$
$98$	0	0
$99$	3.15516	0.317106
$100$	0	0
$101$	16.0450	1.59653	0.798266	−	0.602305i	$-0.205751\pi$
$101$	16.0450	1.59653	0.798266	+	0.602305i	$0.205751\pi$
$102$	0	0
$103$	10.5601	1.04052	0.520258	−	0.854009i	$-0.325836\pi$
$103$	10.5601	1.04052	0.520258	+	0.854009i	$0.325836\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	11.0450	1.06776	0.533878	−	0.845561i	$-0.320734\pi$
$107$	11.0450	1.06776	0.533878	+	0.845561i	$0.320734\pi$
$108$	0	0
$109$	0.734633	0.0703651	0.0351825	−	0.999381i	$-0.488799\pi$
$109$	0.734633	0.0703651	0.0351825	+	0.999381i	$0.488799\pi$
$110$	0	0
$111$	−5.76491	−0.547181
$112$	0	0
$113$	7.93945	0.746880	0.373440	−	0.927654i	$-0.378178\pi$
$113$	7.93945	0.746880	0.373440	+	0.927654i	$0.378178\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	−1.09461	−0.100343
$120$	0	0
$121$	−1.04496	−0.0949960
$122$	0	0
$123$	−3.76491	−0.339470
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−3.51514	−0.311918	−0.155959	−	0.987764i	$-0.549847\pi$
$127$	−3.51514	−0.311918	−0.155959	+	0.987764i	$0.549847\pi$
$128$	0	0
$129$	3.28005	0.288792
$130$	0	0
$131$	7.76491	0.678423	0.339212	−	0.940710i	$-0.389840\pi$
$131$	7.76491	0.678423	0.339212	+	0.940710i	$0.389840\pi$
$132$	0	0
$133$	14.3103	1.24086
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−18.2039	−1.55526	−0.777632	−	0.628720i	$-0.783579\pi$
$137$	−18.2039	−1.55526	−0.777632	+	0.628720i	$0.783579\pi$
$138$	0	0
$139$	−12.7493	−1.08138	−0.540691	−	0.841221i	$-0.681837\pi$
$139$	−12.7493	−1.08138	−0.540691	+	0.841221i	$0.681837\pi$
$140$	0	0
$141$	4.60975	0.388211
$142$	0	0
$143$	−3.15516	−0.263848
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−2.48486	−0.204948
$148$	0	0
$149$	−9.52982	−0.780713	−0.390357	−	0.920664i	$-0.627648\pi$
$149$	−9.52982	−0.780713	−0.390357	+	0.920664i	$0.627648\pi$
$150$	0	0
$151$	18.6244	1.51563	0.757817	−	0.652467i	$-0.226266\pi$
$151$	18.6244	1.51563	0.757817	+	0.652467i	$0.226266\pi$
$152$	0	0
$153$	−0.515138	−0.0416464
$154$	0	0
$155$	0	0
$156$	0	0
$157$	11.6741	0.931693	0.465847	−	0.884866i	$-0.345750\pi$
$157$	11.6741	0.931693	0.465847	+	0.884866i	$0.345750\pi$
$158$	0	0
$159$	−0.280047	−0.0222092
$160$	0	0
$161$	9.03028	0.711685
$162$	0	0
$163$	−1.03028	−0.0806975	−0.0403487	−	0.999186i	$-0.512847\pi$
$163$	−1.03028	−0.0806975	−0.0403487	+	0.999186i	$0.512847\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−10.7346	−0.830671	−0.415335	−	0.909668i	$-0.636336\pi$
$167$	−10.7346	−0.830671	−0.415335	+	0.909668i	$0.636336\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	6.73463	0.515010
$172$	0	0
$173$	−3.74931	−0.285055	−0.142527	−	0.989791i	$-0.545523\pi$
$173$	−3.74931	−0.285055	−0.142527	+	0.989791i	$0.545523\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	11.3444	0.852696
$178$	0	0
$179$	10.4390	0.780247	0.390123	−	0.920763i	$-0.372432\pi$
$179$	10.4390	0.780247	0.390123	+	0.920763i	$0.372432\pi$
$180$	0	0
$181$	5.17454	0.384620	0.192310	−	0.981334i	$-0.438402\pi$
$181$	5.17454	0.384620	0.192310	+	0.981334i	$0.438402\pi$
$182$	0	0
$183$	−5.01468	−0.370696
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−1.62534	−0.118857
$188$	0	0
$189$	2.12489	0.154563
$190$	0	0
$191$	7.15894	0.518003	0.259001	−	0.965877i	$-0.416607\pi$
$191$	7.15894	0.518003	0.259001	+	0.965877i	$0.416607\pi$
$192$	0	0
$193$	−14.5601	−1.04806	−0.524029	−	0.851700i	$-0.675572\pi$
$193$	−14.5601	−1.04806	−0.524029	+	0.851700i	$0.675572\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	14.5601	1.03736	0.518682	−	0.854967i	$-0.326423\pi$
$197$	14.5601	1.03736	0.518682	+	0.854967i	$0.326423\pi$
$198$	0	0
$199$	17.1055	1.21258	0.606289	−	0.795245i	$-0.292658\pi$
$199$	17.1055	1.21258	0.606289	+	0.795245i	$0.292658\pi$
$200$	0	0
$201$	−15.1093	−1.06573
$202$	0	0
$203$	0.0643332	0.00451531
$204$	0	0
$205$	0	0
$206$	0	0
$207$	4.24977	0.295379
$208$	0	0
$209$	21.2489	1.46981
$210$	0	0
$211$	14.1892	0.976826	0.488413	−	0.872613i	$-0.337576\pi$
$211$	14.1892	0.976826	0.488413	+	0.872613i	$0.337576\pi$
$212$	0	0
$213$	2.23509	0.153146
$214$	0	0
$215$	0	0
$216$	0	0
$217$	15.2039	1.03211
$218$	0	0
$219$	2.96972	0.200675
$220$	0	0
$221$	0.515138	0.0346519
$222$	0	0
$223$	4.06055	0.271915	0.135957	−	0.990715i	$-0.456589\pi$
$223$	4.06055	0.271915	0.135957	+	0.990715i	$0.456589\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−14.6850	−0.974676	−0.487338	−	0.873213i	$-0.662032\pi$
$227$	−14.6850	−0.974676	−0.487338	+	0.873213i	$0.662032\pi$
$228$	0	0
$229$	−24.3250	−1.60744	−0.803721	−	0.595007i	$-0.797149\pi$
$229$	−24.3250	−1.60744	−0.803721	+	0.595007i	$0.797149\pi$
$230$	0	0
$231$	6.70436	0.441114
$232$	0	0
$233$	13.6509	0.894302	0.447151	−	0.894459i	$-0.352439\pi$
$233$	13.6509	0.894302	0.447151	+	0.894459i	$0.352439\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−8.98440	−0.583600
$238$	0	0
$239$	15.0341	0.972472	0.486236	−	0.873827i	$-0.338369\pi$
$239$	15.0341	0.972472	0.486236	+	0.873827i	$0.338369\pi$
$240$	0	0
$241$	14.5601	0.937898	0.468949	−	0.883225i	$-0.344633\pi$
$241$	14.5601	0.937898	0.468949	+	0.883225i	$0.344633\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−6.73463	−0.428514
$248$	0	0
$249$	−5.40493	−0.342524
$250$	0	0
$251$	−9.20482	−0.581003	−0.290501	−	0.956875i	$-0.593822\pi$
$251$	−9.20482	−0.581003	−0.290501	+	0.956875i	$0.593822\pi$
$252$	0	0
$253$	13.4087	0.842999
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−13.7952	−0.860520	−0.430260	−	0.902705i	$-0.641578\pi$
$257$	−13.7952	−0.860520	−0.430260	+	0.902705i	$0.641578\pi$
$258$	0	0
$259$	−12.2498	−0.761164
$260$	0	0
$261$	0.0302761	0.00187404
$262$	0	0
$263$	3.03028	0.186855	0.0934274	−	0.995626i	$-0.470218\pi$
$263$	3.03028	0.186855	0.0934274	+	0.995626i	$0.470218\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−15.5298	−0.950409
$268$	0	0
$269$	−0.469266	−0.0286116	−0.0143058	−	0.999898i	$-0.504554\pi$
$269$	−0.469266	−0.0286116	−0.0143058	+	0.999898i	$0.504554\pi$
$270$	0	0
$271$	−2.87420	−0.174595	−0.0872975	−	0.996182i	$-0.527823\pi$
$271$	−2.87420	−0.174595	−0.0872975	+	0.996182i	$0.527823\pi$
$272$	0	0
$273$	−2.12489	−0.128604
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−1.87890	−0.112892	−0.0564459	−	0.998406i	$-0.517977\pi$
$277$	−1.87890	−0.112892	−0.0564459	+	0.998406i	$0.517977\pi$
$278$	0	0
$279$	7.15516	0.428368
$280$	0	0
$281$	23.2342	1.38603	0.693017	−	0.720921i	$-0.256281\pi$
$281$	23.2342	1.38603	0.693017	+	0.720921i	$0.256281\pi$
$282$	0	0
$283$	−26.8099	−1.59368	−0.796841	−	0.604190i	$-0.793497\pi$
$283$	−26.8099	−1.59368	−0.796841	+	0.604190i	$0.793497\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−8.00000	−0.472225
$288$	0	0
$289$	−16.7346	−0.984390
$290$	0	0
$291$	−0.909172	−0.0532966
$292$	0	0
$293$	10.4702	0.611675	0.305837	−	0.952084i	$-0.401064\pi$
$293$	10.4702	0.611675	0.305837	+	0.952084i	$0.401064\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	3.15516	0.183081
$298$	0	0
$299$	−4.24977	−0.245771
$300$	0	0
$301$	6.96972	0.401728
$302$	0	0
$303$	16.0450	0.921759
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−26.3856	−1.50590	−0.752952	−	0.658076i	$-0.771371\pi$
$307$	−26.3856	−1.50590	−0.752952	+	0.658076i	$0.771371\pi$
$308$	0	0
$309$	10.5601	0.600743
$310$	0	0
$311$	−24.2791	−1.37674	−0.688372	−	0.725358i	$-0.741674\pi$
$311$	−24.2791	−1.37674	−0.688372	+	0.725358i	$0.741674\pi$
$312$	0	0
$313$	−14.2800	−0.807156	−0.403578	−	0.914945i	$-0.632234\pi$
$313$	−14.2800	−0.807156	−0.403578	+	0.914945i	$0.632234\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−21.5298	−1.20924	−0.604618	−	0.796516i	$-0.706674\pi$
$317$	−21.5298	−1.20924	−0.604618	+	0.796516i	$0.706674\pi$
$318$	0	0
$319$	0.0955260	0.00534843
$320$	0	0
$321$	11.0450	0.616469
$322$	0	0
$323$	−3.46927	−0.193035
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0.734633	0.0406253
$328$	0	0
$329$	9.79518	0.540026
$330$	0	0
$331$	31.5904	1.73636	0.868182	−	0.496246i	$-0.165289\pi$
$331$	31.5904	1.73636	0.868182	+	0.496246i	$0.165289\pi$
$332$	0	0
$333$	−5.76491	−0.315915
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−7.42431	−0.404428	−0.202214	−	0.979341i	$-0.564814\pi$
$337$	−7.42431	−0.404428	−0.202214	+	0.979341i	$0.564814\pi$
$338$	0	0
$339$	7.93945	0.431212
$340$	0	0
$341$	22.5757	1.22254
$342$	0	0
$343$	−20.1542	−1.08823
$344$	0	0
$345$	0	0
$346$	0	0
$347$	13.9541	0.749097	0.374548	−	0.927207i	$-0.377798\pi$
$347$	13.9541	0.749097	0.374548	+	0.927207i	$0.377798\pi$
$348$	0	0
$349$	−4.06055	−0.217356	−0.108678	−	0.994077i	$-0.534662\pi$
$349$	−4.06055	−0.217356	−0.108678	+	0.994077i	$0.534662\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	24.3856	1.29791	0.648956	−	0.760826i	$-0.275206\pi$
$353$	24.3856	1.29791	0.648956	+	0.760826i	$0.275206\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−1.09461	−0.0579329
$358$	0	0
$359$	−3.76869	−0.198904	−0.0994519	−	0.995042i	$-0.531709\pi$
$359$	−3.76869	−0.198904	−0.0994519	+	0.995042i	$0.531709\pi$
$360$	0	0
$361$	26.3553	1.38712
$362$	0	0
$363$	−1.04496	−0.0548460
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−19.2654	−1.00564	−0.502822	−	0.864390i	$-0.667705\pi$
$367$	−19.2654	−1.00564	−0.502822	+	0.864390i	$0.667705\pi$
$368$	0	0
$369$	−3.76491	−0.195993
$370$	0	0
$371$	−0.595068	−0.0308944
$372$	0	0
$373$	14.3553	0.743288	0.371644	−	0.928375i	$-0.378794\pi$
$373$	14.3553	0.743288	0.371644	+	0.928375i	$0.378794\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−0.0302761	−0.00155930
$378$	0	0
$379$	33.1845	1.70457	0.852287	−	0.523074i	$-0.175215\pi$
$379$	33.1845	1.70457	0.852287	+	0.523074i	$0.175215\pi$
$380$	0	0
$381$	−3.51514	−0.180086
$382$	0	0
$383$	30.8851	1.57815	0.789077	−	0.614294i	$-0.210559\pi$
$383$	30.8851	1.57815	0.789077	+	0.614294i	$0.210559\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	3.28005	0.166734
$388$	0	0
$389$	0.295643	0.0149897	0.00749486	−	0.999972i	$-0.497614\pi$
$389$	0.295643	0.0149897	0.00749486	+	0.999972i	$0.497614\pi$
$390$	0	0
$391$	−2.18922	−0.110714
$392$	0	0
$393$	7.76491	0.391688
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−16.3250	−0.819328	−0.409664	−	0.912236i	$-0.634354\pi$
$397$	−16.3250	−0.819328	−0.409664	+	0.912236i	$0.634354\pi$
$398$	0	0
$399$	14.3103	0.716412
$400$	0	0
$401$	28.4390	1.42018	0.710088	−	0.704113i	$-0.248655\pi$
$401$	28.4390	1.42018	0.710088	+	0.704113i	$0.248655\pi$
$402$	0	0
$403$	−7.15516	−0.356424
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−18.1892	−0.901606
$408$	0	0
$409$	−16.3784	−0.809862	−0.404931	−	0.914347i	$-0.632704\pi$
$409$	−16.3784	−0.809862	−0.404931	+	0.914347i	$0.632704\pi$
$410$	0	0
$411$	−18.2039	−0.897932
$412$	0	0
$413$	24.1055	1.18615
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−12.7493	−0.624337
$418$	0	0
$419$	31.4158	1.53476	0.767382	−	0.641190i	$-0.221559\pi$
$419$	31.4158	1.53476	0.767382	+	0.641190i	$0.221559\pi$
$420$	0	0
$421$	23.5298	1.14677	0.573387	−	0.819285i	$-0.305629\pi$
$421$	23.5298	1.14677	0.573387	+	0.819285i	$0.305629\pi$
$422$	0	0
$423$	4.60975	0.224134
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−10.6556	−0.515662
$428$	0	0
$429$	−3.15516	−0.152333
$430$	0	0
$431$	−19.3553	−0.932311	−0.466155	−	0.884703i	$-0.654361\pi$
$431$	−19.3553	−0.932311	−0.466155	+	0.884703i	$0.654361\pi$
$432$	0	0
$433$	−19.7044	−0.946931	−0.473465	−	0.880812i	$-0.656997\pi$
$433$	−19.7044	−0.946931	−0.473465	+	0.880812i	$0.656997\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	28.6206	1.36911
$438$	0	0
$439$	8.98440	0.428802	0.214401	−	0.976746i	$-0.431220\pi$
$439$	8.98440	0.428802	0.214401	+	0.976746i	$0.431220\pi$
$440$	0	0
$441$	−2.48486	−0.118327
$442$	0	0
$443$	2.92385	0.138916	0.0694582	−	0.997585i	$-0.477873\pi$
$443$	2.92385	0.138916	0.0694582	+	0.997585i	$0.477873\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−9.52982	−0.450745
$448$	0	0
$449$	−17.3553	−0.819046	−0.409523	−	0.912300i	$-0.634305\pi$
$449$	−17.3553	−0.819046	−0.409523	+	0.912300i	$0.634305\pi$
$450$	0	0
$451$	−11.8789	−0.559355
$452$	0	0
$453$	18.6244	0.875052
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−7.05964	−0.330236	−0.165118	−	0.986274i	$-0.552800\pi$
$457$	−7.05964	−0.330236	−0.165118	+	0.986274i	$0.552800\pi$
$458$	0	0
$459$	−0.515138	−0.0240446
$460$	0	0
$461$	−1.52982	−0.0712507	−0.0356254	−	0.999365i	$-0.511342\pi$
$461$	−1.52982	−0.0712507	−0.0356254	+	0.999365i	$0.511342\pi$
$462$	0	0
$463$	27.3132	1.26935	0.634676	−	0.772779i	$-0.281134\pi$
$463$	27.3132	1.26935	0.634676	+	0.772779i	$0.281134\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−6.29564	−0.291328	−0.145664	−	0.989334i	$-0.546532\pi$
$467$	−6.29564	−0.291328	−0.145664	+	0.989334i	$0.546532\pi$
$468$	0	0
$469$	−32.1055	−1.48249
$470$	0	0
$471$	11.6741	0.537913
$472$	0	0
$473$	10.3491	0.475851
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−0.280047	−0.0128225
$478$	0	0
$479$	26.9806	1.23278	0.616388	−	0.787443i	$-0.288595\pi$
$479$	26.9806	1.23278	0.616388	+	0.787443i	$0.288595\pi$
$480$	0	0
$481$	5.76491	0.262857
$482$	0	0
$483$	9.03028	0.410892
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−11.8751	−0.538113	−0.269056	−	0.963124i	$-0.586712\pi$
$487$	−11.8751	−0.538113	−0.269056	+	0.963124i	$0.586712\pi$
$488$	0	0
$489$	−1.03028	−0.0465907
$490$	0	0
$491$	−41.8695	−1.88954	−0.944772	−	0.327728i	$-0.893717\pi$
$491$	−41.8695	−1.88954	−0.944772	+	0.327728i	$0.893717\pi$
$492$	0	0
$493$	−0.0155964	−0.000702425 0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	4.74931	0.213036
$498$	0	0
$499$	30.5786	1.36888	0.684442	−	0.729067i	$-0.260046\pi$
$499$	30.5786	1.36888	0.684442	+	0.729067i	$0.260046\pi$
$500$	0	0
$501$	−10.7346	−0.479588
$502$	0	0
$503$	32.9991	1.47136	0.735678	−	0.677331i	$-0.236864\pi$
$503$	32.9991	1.47136	0.735678	+	0.677331i	$0.236864\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	41.6197	1.84476	0.922381	−	0.386281i	$-0.126241\pi$
$509$	41.6197	1.84476	0.922381	+	0.386281i	$0.126241\pi$
$510$	0	0
$511$	6.31032	0.279152
$512$	0	0
$513$	6.73463	0.297341
$514$	0	0
$515$	0	0
$516$	0	0
$517$	14.5445	0.639666
$518$	0	0
$519$	−3.74931	−0.164577
$520$	0	0
$521$	6.47018	0.283464	0.141732	−	0.989905i	$-0.454733\pi$
$521$	6.47018	0.283464	0.141732	+	0.989905i	$0.454733\pi$
$522$	0	0
$523$	−37.8401	−1.65463	−0.827317	−	0.561735i	$-0.810134\pi$
$523$	−37.8401	−1.65463	−0.827317	+	0.561735i	$0.810134\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−3.68590	−0.160560
$528$	0	0
$529$	−4.93945	−0.214759
$530$	0	0
$531$	11.3444	0.492304
$532$	0	0
$533$	3.76491	0.163076
$534$	0	0
$535$	0	0
$536$	0	0
$537$	10.4390	0.450476
$538$	0	0
$539$	−7.84014	−0.337699
$540$	0	0
$541$	6.37844	0.274230	0.137115	−	0.990555i	$-0.456217\pi$
$541$	6.37844	0.274230	0.137115	+	0.990555i	$0.456217\pi$
$542$	0	0
$543$	5.17454	0.222061
$544$	0	0
$545$	0	0
$546$	0	0
$547$	10.0899	0.431413	0.215707	−	0.976458i	$-0.430794\pi$
$547$	10.0899	0.431413	0.215707	+	0.976458i	$0.430794\pi$
$548$	0	0
$549$	−5.01468	−0.214021
$550$	0	0
$551$	0.203898	0.00868636
$552$	0	0
$553$	−19.0908	−0.811825
$554$	0	0
$555$	0	0
$556$	0	0
$557$	4.96972	0.210574	0.105287	−	0.994442i	$-0.466424\pi$
$557$	4.96972	0.210574	0.105287	+	0.994442i	$0.466424\pi$
$558$	0	0
$559$	−3.28005	−0.138731
$560$	0	0
$561$	−1.62534	−0.0686221
$562$	0	0
$563$	−14.8027	−0.623861	−0.311931	−	0.950105i	$-0.600976\pi$
$563$	−14.8027	−0.623861	−0.311931	+	0.950105i	$0.600976\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	2.12489	0.0892368
$568$	0	0
$569$	−3.85574	−0.161641	−0.0808205	−	0.996729i	$-0.525754\pi$
$569$	−3.85574	−0.161641	−0.0808205	+	0.996729i	$0.525754\pi$
$570$	0	0
$571$	24.6282	1.03066	0.515329	−	0.856992i	$-0.327670\pi$
$571$	24.6282	1.03066	0.515329	+	0.856992i	$0.327670\pi$
$572$	0	0
$573$	7.15894	0.299069
$574$	0	0
$575$	0	0
$576$	0	0
$577$	6.02936	0.251006	0.125503	−	0.992093i	$-0.459946\pi$
$577$	6.02936	0.251006	0.125503	+	0.992093i	$0.459946\pi$
$578$	0	0
$579$	−14.5601	−0.605097
$580$	0	0
$581$	−11.4849	−0.476472
$582$	0	0
$583$	−0.883593	−0.0365947
$584$	0	0
$585$	0	0
$586$	0	0
$587$	14.0861	0.581397	0.290698	−	0.956815i	$-0.406112\pi$
$587$	14.0861	0.581397	0.290698	+	0.956815i	$0.406112\pi$
$588$	0	0
$589$	48.1874	1.98553
$590$	0	0
$591$	14.5601	0.598922
$592$	0	0
$593$	−28.2333	−1.15940	−0.579700	−	0.814830i	$-0.696830\pi$
$593$	−28.2333	−1.15940	−0.579700	+	0.814830i	$0.696830\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	17.1055	0.700082
$598$	0	0
$599$	−23.0596	−0.942191	−0.471096	−	0.882082i	$-0.656141\pi$
$599$	−23.0596	−0.942191	−0.471096	+	0.882082i	$0.656141\pi$
$600$	0	0
$601$	14.1901	0.578828	0.289414	−	0.957204i	$-0.406540\pi$
$601$	14.1901	0.578828	0.289414	+	0.957204i	$0.406540\pi$
$602$	0	0
$603$	−15.1093	−0.615298
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−43.9229	−1.78278	−0.891388	−	0.453240i	$-0.850268\pi$
$607$	−43.9229	−1.78278	−0.891388	+	0.453240i	$0.850268\pi$
$608$	0	0
$609$	0.0643332	0.00260691
$610$	0	0
$611$	−4.60975	−0.186490
$612$	0	0
$613$	1.62156	0.0654943	0.0327472	−	0.999464i	$-0.489574\pi$
$613$	1.62156	0.0654943	0.0327472	+	0.999464i	$0.489574\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−12.3250	−0.496186	−0.248093	−	0.968736i	$-0.579804\pi$
$617$	−12.3250	−0.496186	−0.248093	+	0.968736i	$0.579804\pi$
$618$	0	0
$619$	−5.15138	−0.207051	−0.103526	−	0.994627i	$-0.533012\pi$
$619$	−5.15138	−0.207051	−0.103526	+	0.994627i	$0.533012\pi$
$620$	0	0
$621$	4.24977	0.170537
$622$	0	0
$623$	−32.9991	−1.32208
$624$	0	0
$625$	0	0
$626$	0	0
$627$	21.2489	0.848597
$628$	0	0
$629$	2.96972	0.118411
$630$	0	0
$631$	1.97064	0.0784500	0.0392250	−	0.999230i	$-0.487511\pi$
$631$	1.97064	0.0784500	0.0392250	+	0.999230i	$0.487511\pi$
$632$	0	0
$633$	14.1892	0.563971
$634$	0	0
$635$	0	0
$636$	0	0
$637$	2.48486	0.0984538
$638$	0	0
$639$	2.23509	0.0884188
$640$	0	0
$641$	4.33348	0.171162	0.0855811	−	0.996331i	$-0.472725\pi$
$641$	4.33348	0.171162	0.0855811	+	0.996331i	$0.472725\pi$
$642$	0	0
$643$	−6.85574	−0.270364	−0.135182	−	0.990821i	$-0.543162\pi$
$643$	−6.85574	−0.270364	−0.135182	+	0.990821i	$0.543162\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−5.93945	−0.233504	−0.116752	−	0.993161i	$-0.537248\pi$
$647$	−5.93945	−0.233504	−0.116752	+	0.993161i	$0.537248\pi$
$648$	0	0
$649$	35.7934	1.40501
$650$	0	0
$651$	15.2039	0.595888
$652$	0	0
$653$	9.61353	0.376206	0.188103	−	0.982149i	$-0.439766\pi$
$653$	9.61353	0.376206	0.188103	+	0.982149i	$0.439766\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	2.96972	0.115860
$658$	0	0
$659$	0.455503	0.0177439	0.00887193	−	0.999961i	$-0.497176\pi$
$659$	0.455503	0.0177439	0.00887193	+	0.999961i	$0.497176\pi$
$660$	0	0
$661$	−29.2635	−1.13822	−0.569110	−	0.822262i	$-0.692712\pi$
$661$	−29.2635	−1.13822	−0.569110	+	0.822262i	$0.692712\pi$
$662$	0	0
$663$	0.515138	0.0200063
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0.128666	0.00498199
$668$	0	0
$669$	4.06055	0.156990
$670$	0	0
$671$	−15.8221	−0.610806
$672$	0	0
$673$	−24.7796	−0.955183	−0.477591	−	0.878582i	$-0.658490\pi$
$673$	−24.7796	−0.955183	−0.477591	+	0.878582i	$0.658490\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	7.93945	0.305138	0.152569	−	0.988293i	$-0.451245\pi$
$677$	7.93945	0.305138	0.152569	+	0.988293i	$0.451245\pi$
$678$	0	0
$679$	−1.93189	−0.0741390
$680$	0	0
$681$	−14.6850	−0.562730
$682$	0	0
$683$	7.53360	0.288265	0.144133	−	0.989558i	$-0.453961\pi$
$683$	7.53360	0.288265	0.144133	+	0.989558i	$0.453961\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−24.3250	−0.928057
$688$	0	0
$689$	0.280047	0.0106689
$690$	0	0
$691$	−51.4490	−1.95721	−0.978606	−	0.205745i	$-0.934038\pi$
$691$	−51.4490	−1.95721	−0.978606	+	0.205745i	$0.934038\pi$
$692$	0	0
$693$	6.70436	0.254677
$694$	0	0
$695$	0	0
$696$	0	0
$697$	1.93945	0.0734618
$698$	0	0
$699$	13.6509	0.516325
$700$	0	0
$701$	−5.35620	−0.202301	−0.101150	−	0.994871i	$-0.532252\pi$
$701$	−5.35620	−0.202301	−0.101150	+	0.994871i	$0.532252\pi$
$702$	0	0
$703$	−38.8245	−1.46430
$704$	0	0
$705$	0	0
$706$	0	0
$707$	34.0937	1.28223
$708$	0	0
$709$	0.969724	0.0364187	0.0182094	−	0.999834i	$-0.494203\pi$
$709$	0.969724	0.0364187	0.0182094	+	0.999834i	$0.494203\pi$
$710$	0	0
$711$	−8.98440	−0.336941
$712$	0	0
$713$	30.4078	1.13878
$714$	0	0
$715$	0	0
$716$	0	0
$717$	15.0341	0.561457
$718$	0	0
$719$	23.5592	0.878609	0.439305	−	0.898338i	$-0.355225\pi$
$719$	23.5592	0.878609	0.439305	+	0.898338i	$0.355225\pi$
$720$	0	0
$721$	22.4390	0.835672
$722$	0	0
$723$	14.5601	0.541496
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−22.3179	−0.827725	−0.413862	−	0.910340i	$-0.635820\pi$
$727$	−22.3179	−0.827725	−0.413862	+	0.910340i	$0.635820\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−1.68968	−0.0624950
$732$	0	0
$733$	46.7034	1.72503	0.862515	−	0.506031i	$-0.168888\pi$
$733$	46.7034	1.72503	0.862515	+	0.506031i	$0.168888\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−47.6722	−1.75603
$738$	0	0
$739$	−29.2985	−1.07776	−0.538882	−	0.842382i	$-0.681153\pi$
$739$	−29.2985	−1.07776	−0.538882	+	0.842382i	$0.681153\pi$
$740$	0	0
$741$	−6.73463	−0.247403
$742$	0	0
$743$	51.0175	1.87165	0.935826	−	0.352462i	$-0.114655\pi$
$743$	51.0175	1.87165	0.935826	+	0.352462i	$0.114655\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−5.40493	−0.197756
$748$	0	0
$749$	23.4693	0.857548
$750$	0	0
$751$	4.36376	0.159236	0.0796179	−	0.996825i	$-0.474630\pi$
$751$	4.36376	0.159236	0.0796179	+	0.996825i	$0.474630\pi$
$752$	0	0
$753$	−9.20482	−0.335442
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−42.6344	−1.54957	−0.774787	−	0.632222i	$-0.782143\pi$
$757$	−42.6344	−1.54957	−0.774787	+	0.632222i	$0.782143\pi$
$758$	0	0
$759$	13.4087	0.486705
$760$	0	0
$761$	−3.73372	−0.135347	−0.0676736	−	0.997708i	$-0.521558\pi$
$761$	−3.73372	−0.135347	−0.0676736	+	0.997708i	$0.521558\pi$
$762$	0	0
$763$	1.56101	0.0565124
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−11.3444	−0.409622
$768$	0	0
$769$	−20.3784	−0.734865	−0.367433	−	0.930050i	$-0.619763\pi$
$769$	−20.3784	−0.734865	−0.367433	+	0.930050i	$0.619763\pi$
$770$	0	0
$771$	−13.7952	−0.496821
$772$	0	0
$773$	−14.5289	−0.522568	−0.261284	−	0.965262i	$-0.584146\pi$
$773$	−14.5289	−0.522568	−0.261284	+	0.965262i	$0.584146\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−12.2498	−0.439458
$778$	0	0
$779$	−25.3553	−0.908447
$780$	0	0
$781$	7.05207	0.252343
$782$	0	0
$783$	0.0302761	0.00108198
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−5.00470	−0.178398	−0.0891991	−	0.996014i	$-0.528431\pi$
$787$	−5.00470	−0.178398	−0.0891991	+	0.996014i	$0.528431\pi$
$788$	0	0
$789$	3.03028	0.107881
$790$	0	0
$791$	16.8704	0.599843
$792$	0	0
$793$	5.01468	0.178076
$794$	0	0
$795$	0	0
$796$	0	0
$797$	48.3846	1.71387	0.856936	−	0.515423i	$-0.172365\pi$
$797$	48.3846	1.71387	0.856936	+	0.515423i	$0.172365\pi$
$798$	0	0
$799$	−2.37466	−0.0840093
$800$	0	0
$801$	−15.5298	−0.548719
$802$	0	0
$803$	9.36996	0.330659
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−0.469266	−0.0165189
$808$	0	0
$809$	9.18074	0.322778	0.161389	−	0.986891i	$-0.448403\pi$
$809$	9.18074	0.322778	0.161389	+	0.986891i	$0.448403\pi$
$810$	0	0
$811$	30.2224	1.06125	0.530625	−	0.847607i	$-0.321957\pi$
$811$	30.2224	1.06125	0.530625	+	0.847607i	$0.321957\pi$
$812$	0	0
$813$	−2.87420	−0.100803
$814$	0	0
$815$	0	0
$816$	0	0
$817$	22.0899	0.772828
$818$	0	0
$819$	−2.12489	−0.0742495
$820$	0	0
$821$	31.6197	1.10354	0.551768	−	0.833998i	$-0.313953\pi$
$821$	31.6197	1.10354	0.551768	+	0.833998i	$0.313953\pi$
$822$	0	0
$823$	−31.4158	−1.09509	−0.547544	−	0.836777i	$-0.684437\pi$
$823$	−31.4158	−1.09509	−0.547544	+	0.836777i	$0.684437\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−11.2838	−0.392377	−0.196189	−	0.980566i	$-0.562856\pi$
$827$	−11.2838	−0.392377	−0.196189	+	0.980566i	$0.562856\pi$
$828$	0	0
$829$	−4.83302	−0.167858	−0.0839289	−	0.996472i	$-0.526747\pi$
$829$	−4.83302	−0.167858	−0.0839289	+	0.996472i	$0.526747\pi$
$830$	0	0
$831$	−1.87890	−0.0651782
$832$	0	0
$833$	1.28005	0.0443510
$834$	0	0
$835$	0	0
$836$	0	0
$837$	7.15516	0.247319
$838$	0	0
$839$	−15.4693	−0.534058	−0.267029	−	0.963688i	$-0.586042\pi$
$839$	−15.4693	−0.534058	−0.267029	+	0.963688i	$0.586042\pi$
$840$	0	0
$841$	−28.9991	−0.999968
$842$	0	0
$843$	23.2342	0.800227
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−2.22041	−0.0762942
$848$	0	0
$849$	−26.8099	−0.920112
$850$	0	0
$851$	−24.4995	−0.839833
$852$	0	0
$853$	−41.2635	−1.41284	−0.706418	−	0.707795i	$-0.749690\pi$
$853$	−41.2635	−1.41284	−0.706418	+	0.707795i	$0.749690\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	9.87890	0.337457	0.168728	−	0.985663i	$-0.446034\pi$
$857$	9.87890	0.337457	0.168728	+	0.985663i	$0.446034\pi$
$858$	0	0
$859$	−20.7200	−0.706956	−0.353478	−	0.935443i	$-0.615001\pi$
$859$	−20.7200	−0.706956	−0.353478	+	0.935443i	$0.615001\pi$
$860$	0	0
$861$	−8.00000	−0.272639
$862$	0	0
$863$	−1.91915	−0.0653288	−0.0326644	−	0.999466i	$-0.510399\pi$
$863$	−1.91915	−0.0653288	−0.0326644	+	0.999466i	$0.510399\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−16.7346	−0.568338
$868$	0	0
$869$	−28.3472	−0.961614
$870$	0	0
$871$	15.1093	0.511959
$872$	0	0
$873$	−0.909172	−0.0307708
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−18.3250	−0.618791	−0.309396	−	0.950933i	$-0.600127\pi$
$877$	−18.3250	−0.618791	−0.309396	+	0.950933i	$0.600127\pi$
$878$	0	0
$879$	10.4702	0.353150
$880$	0	0
$881$	12.7649	0.430061	0.215030	−	0.976607i	$-0.431015\pi$
$881$	12.7649	0.430061	0.215030	+	0.976607i	$0.431015\pi$
$882$	0	0
$883$	−11.8789	−0.399757	−0.199878	−	0.979821i	$-0.564055\pi$
$883$	−11.8789	−0.399757	−0.199878	+	0.979821i	$0.564055\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−40.7493	−1.36823	−0.684114	−	0.729375i	$-0.739811\pi$
$887$	−40.7493	−1.36823	−0.684114	+	0.729375i	$0.739811\pi$
$888$	0	0
$889$	−7.46927	−0.250511
$890$	0	0
$891$	3.15516	0.105702
$892$	0	0
$893$	31.0450	1.03888
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−4.24977	−0.141896
$898$	0	0
$899$	0.216630	0.00722503
$900$	0	0
$901$	0.144263	0.00480609
$902$	0	0
$903$	6.96972	0.231938
$904$	0	0
$905$	0	0
$906$	0	0
$907$	32.5289	1.08010	0.540052	−	0.841632i	$-0.318404\pi$
$907$	32.5289	1.08010	0.540052	+	0.841632i	$0.318404\pi$
$908$	0	0
$909$	16.0450	0.532178
$910$	0	0
$911$	−22.2810	−0.738201	−0.369101	−	0.929389i	$-0.620334\pi$
$911$	−22.2810	−0.738201	−0.369101	+	0.929389i	$0.620334\pi$
$912$	0	0
$913$	−17.0534	−0.564386
$914$	0	0
$915$	0	0
$916$	0	0
$917$	16.4995	0.544863
$918$	0	0
$919$	26.9239	0.888136	0.444068	−	0.895993i	$-0.353535\pi$
$919$	26.9239	0.888136	0.444068	+	0.895993i	$0.353535\pi$
$920$	0	0
$921$	−26.3856	−0.869434
$922$	0	0
$923$	−2.23509	−0.0735689
$924$	0	0
$925$	0	0
$926$	0	0
$927$	10.5601	0.346839
$928$	0	0
$929$	−36.4755	−1.19672	−0.598361	−	0.801227i	$-0.704181\pi$
$929$	−36.4755	−1.19672	−0.598361	+	0.801227i	$0.704181\pi$
$930$	0	0
$931$	−16.7346	−0.548455
$932$	0	0
$933$	−24.2791	−0.794863
$934$	0	0
$935$	0	0
$936$	0	0
$937$	7.26445	0.237319	0.118660	−	0.992935i	$-0.462140\pi$
$937$	7.26445	0.237319	0.118660	+	0.992935i	$0.462140\pi$
$938$	0	0
$939$	−14.2800	−0.466012
$940$	0	0
$941$	−8.08991	−0.263724	−0.131862	−	0.991268i	$-0.542096\pi$
$941$	−8.08991	−0.263724	−0.131862	+	0.991268i	$0.542096\pi$
$942$	0	0
$943$	−16.0000	−0.521032
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−48.9036	−1.58915	−0.794576	−	0.607165i	$-0.792307\pi$
$947$	−48.9036	−1.58915	−0.794576	+	0.607165i	$0.792307\pi$
$948$	0	0
$949$	−2.96972	−0.0964013
$950$	0	0
$951$	−21.5298	−0.698152
$952$	0	0
$953$	6.58325	0.213252	0.106626	−	0.994299i	$-0.465995\pi$
$953$	6.58325	0.213252	0.106626	+	0.994299i	$0.465995\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0.0955260	0.00308792
$958$	0	0
$959$	−38.6812	−1.24908
$960$	0	0
$961$	20.1963	0.651495
$962$	0	0
$963$	11.0450	0.355919
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−9.58659	−0.308284	−0.154142	−	0.988049i	$-0.549261\pi$
$967$	−9.58659	−0.308284	−0.154142	+	0.988049i	$0.549261\pi$
$968$	0	0
$969$	−3.46927	−0.111449
$970$	0	0
$971$	−36.1358	−1.15965	−0.579826	−	0.814740i	$-0.696880\pi$
$971$	−36.1358	−1.15965	−0.579826	+	0.814740i	$0.696880\pi$
$972$	0	0
$973$	−27.0908	−0.868492
$974$	0	0
$975$	0	0
$976$	0	0
$977$	27.0303	0.864775	0.432388	−	0.901688i	$-0.357671\pi$
$977$	27.0303	0.864775	0.432388	+	0.901688i	$0.357671\pi$
$978$	0	0
$979$	−48.9991	−1.56602
$980$	0	0
$981$	0.734633	0.0234550
$982$	0	0
$983$	−49.9338	−1.59264	−0.796321	−	0.604874i	$-0.793223\pi$
$983$	−49.9338	−1.59264	−0.796321	+	0.604874i	$0.793223\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	9.79518	0.311784
$988$	0	0
$989$	13.9394	0.443249
$990$	0	0
$991$	20.4849	0.650723	0.325362	−	0.945590i	$-0.394514\pi$
$991$	20.4849	0.650723	0.325362	+	0.945590i	$0.394514\pi$
$992$	0	0
$993$	31.5904	1.00249
$994$	0	0
$995$	0	0
$996$	0	0
$997$	40.9541	1.29703	0.648515	−	0.761202i	$-0.275390\pi$
$997$	40.9541	1.29703	0.648515	+	0.761202i	$0.275390\pi$
$998$	0	0
$999$	−5.76491	−0.182394

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7800.2.a.bn.1.3	yes	3
5.4	even	2		7800.2.a.bm.1.1	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7800.2.a.bm.1.1	✓	3	5.4	even	2
7800.2.a.bn.1.3	yes	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:	\( N \)	\(=\)	\( 7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7800.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +2.12489 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +2.12489 q^{7} +1.00000 q^{9} +3.15516 q^{11} -1.00000 q^{13} -0.515138 q^{17} +6.73463 q^{19} +2.12489 q^{21} +4.24977 q^{23} +1.00000 q^{27} +0.0302761 q^{29} +7.15516 q^{31} +3.15516 q^{33} -5.76491 q^{37} -1.00000 q^{39} -3.76491 q^{41} +3.28005 q^{43} +4.60975 q^{47} -2.48486 q^{49} -0.515138 q^{51} -0.280047 q^{53} +6.73463 q^{57} +11.3444 q^{59} -5.01468 q^{61} +2.12489 q^{63} -15.1093 q^{67} +4.24977 q^{69} +2.23509 q^{71} +2.96972 q^{73} +6.70436 q^{77} -8.98440 q^{79} +1.00000 q^{81} -5.40493 q^{83} +0.0302761 q^{87} -15.5298 q^{89} -2.12489 q^{91} +7.15516 q^{93} -0.909172 q^{97} +3.15516 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} - 2 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 - 2 * q^7 + 3 * q^9	\( 3 q + 3 q^{3} - 2 q^{7} + 3 q^{9} + 2 q^{11} - 3 q^{13} - 2 q^{17} + 3 q^{19} - 2 q^{21} - 4 q^{23} + 3 q^{27} + q^{29} + 14 q^{31} + 2 q^{33} - q^{37} - 3 q^{39} + 5 q^{41} - 6 q^{43} + 5 q^{47} - 7 q^{49} - 2 q^{51} + 15 q^{53} + 3 q^{57} + 8 q^{59} + 18 q^{61} - 2 q^{63} - 3 q^{67} - 4 q^{69} + 23 q^{71} + 8 q^{73} + 2 q^{77} + 7 q^{79} + 3 q^{81} + 8 q^{83} + q^{87} - 14 q^{89} + 2 q^{91} + 14 q^{93} + 2 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 - 2 * q^7 + 3 * q^9 + 2 * q^11 - 3 * q^13 - 2 * q^17 + 3 * q^19 - 2 * q^21 - 4 * q^23 + 3 * q^27 + q^29 + 14 * q^31 + 2 * q^33 - q^37 - 3 * q^39 + 5 * q^41 - 6 * q^43 + 5 * q^47 - 7 * q^49 - 2 * q^51 + 15 * q^53 + 3 * q^57 + 8 * q^59 + 18 * q^61 - 2 * q^63 - 3 * q^67 - 4 * q^69 + 23 * q^71 + 8 * q^73 + 2 * q^77 + 7 * q^79 + 3 * q^81 + 8 * q^83 + q^87 - 14 * q^89 + 2 * q^91 + 14 * q^93 + 2 * q^99

Introduction

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