LMFDB - Embedded newform 7800.2.a.bg.1.2

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:	$1$
Dimension:	$3$
Coefficient field:	3.3.788.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 7x - 3 $ x^3 - x^2 - 7*x - 3
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			1.2
Root			$-0.476452$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +0.476452 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +0.476452 q^{7} +1.00000 q^{9} -3.34364 q^{11} +1.00000 q^{13} +1.00000 q^{17} -6.77299 q^{19} -0.476452 q^{21} -4.77299 q^{23} -1.00000 q^{27} +7.59308 q^{29} +7.93672 q^{31} +3.34364 q^{33} +7.82009 q^{37} -1.00000 q^{39} +9.46027 q^{41} +1.82009 q^{43} -9.06953 q^{47} -6.77299 q^{49} -1.00000 q^{51} +9.50736 q^{53} +6.77299 q^{57} +3.16373 q^{59} -9.59308 q^{61} +0.476452 q^{63} -1.34364 q^{67} +4.77299 q^{69} -4.86719 q^{71} -14.4132 q^{73} -1.59308 q^{77} -2.17991 q^{79} +1.00000 q^{81} +9.16373 q^{83} -7.59308 q^{87} -10.5931 q^{89} +0.476452 q^{91} -7.93672 q^{93} +7.04710 q^{97} -3.34364 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} - q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 - q^7 + 3 * q^9	$ 3 q - 3 q^{3} - q^{7} + 3 q^{9} - 3 q^{11} + 3 q^{13} + 3 q^{17} - 6 q^{19} + q^{21} - 3 q^{27} - q^{29} - 7 q^{31} + 3 q^{33} + 14 q^{37} - 3 q^{39} - 4 q^{43} - q^{47} - 6 q^{49} - 3 q^{51} + 5 q^{53} + 6 q^{57} - 7 q^{59} - 5 q^{61} - q^{63} + 3 q^{67} - 10 q^{71} - 10 q^{73} + 19 q^{77} - 16 q^{79} + 3 q^{81} + 11 q^{83} + q^{87} - 8 q^{89} - q^{91} + 7 q^{93} + 26 q^{97} - 3 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 - q^7 + 3 * q^9 - 3 * q^11 + 3 * q^13 + 3 * q^17 - 6 * q^19 + q^21 - 3 * q^27 - q^29 - 7 * q^31 + 3 * q^33 + 14 * q^37 - 3 * q^39 - 4 * q^43 - q^47 - 6 * q^49 - 3 * q^51 + 5 * q^53 + 6 * q^57 - 7 * q^59 - 5 * q^61 - q^63 + 3 * q^67 - 10 * q^71 - 10 * q^73 + 19 * q^77 - 16 * q^79 + 3 * q^81 + 11 * q^83 + q^87 - 8 * q^89 - q^91 + 7 * q^93 + 26 * q^97 - 3 * 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
$7$	0.476452	0.180082	0.0900410	−	0.995938i	$-0.471300\pi$
$7$	0.476452	0.180082	0.0900410	+	0.995938i	$0.471300\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.34364	−1.00814	−0.504072	−	0.863661i	$-0.668165\pi$
$11$	−3.34364	−1.00814	−0.504072	+	0.863661i	$0.668165\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.00000	0.242536	0.121268	−	0.992620i	$-0.461304\pi$
$17$	1.00000	0.242536	0.121268	+	0.992620i	$0.461304\pi$
$18$	0	0
$19$	−6.77299	−1.55383	−0.776916	−	0.629605i	$-0.783217\pi$
$19$	−6.77299	−1.55383	−0.776916	+	0.629605i	$0.783217\pi$
$20$	0	0
$21$	−0.476452	−0.103970
$22$	0	0
$23$	−4.77299	−0.995238	−0.497619	−	0.867396i	$-0.665792\pi$
$23$	−4.77299	−0.995238	−0.497619	+	0.867396i	$0.665792\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	7.59308	1.41000	0.705000	−	0.709207i	$-0.250947\pi$
$29$	7.59308	1.41000	0.705000	+	0.709207i	$0.250947\pi$
$30$	0	0
$31$	7.93672	1.42548	0.712739	−	0.701430i	$-0.247455\pi$
$31$	7.93672	1.42548	0.712739	+	0.701430i	$0.247455\pi$
$32$	0	0
$33$	3.34364	0.582053
$34$	0	0
$35$	0	0
$36$	0	0
$37$	7.82009	1.28561	0.642807	−	0.766028i	$-0.277770\pi$
$37$	7.82009	1.28561	0.642807	+	0.766028i	$0.277770\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	9.46027	1.47745	0.738723	−	0.674009i	$-0.235429\pi$
$41$	9.46027	1.47745	0.738723	+	0.674009i	$0.235429\pi$
$42$	0	0
$43$	1.82009	0.277561	0.138781	−	0.990323i	$-0.455682\pi$
$43$	1.82009	0.277561	0.138781	+	0.990323i	$0.455682\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−9.06953	−1.32293	−0.661464	−	0.749977i	$-0.730064\pi$
$47$	−9.06953	−1.32293	−0.661464	+	0.749977i	$0.730064\pi$
$48$	0	0
$49$	−6.77299	−0.967570
$50$	0	0
$51$	−1.00000	−0.140028
$52$	0	0
$53$	9.50736	1.30594	0.652968	−	0.757385i	$-0.273523\pi$
$53$	9.50736	1.30594	0.652968	+	0.757385i	$0.273523\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	6.77299	0.897105
$58$	0	0
$59$	3.16373	0.411882	0.205941	−	0.978564i	$-0.433974\pi$
$59$	3.16373	0.411882	0.205941	+	0.978564i	$0.433974\pi$
$60$	0	0
$61$	−9.59308	−1.22827	−0.614134	−	0.789202i	$-0.710495\pi$
$61$	−9.59308	−1.22827	−0.614134	+	0.789202i	$0.710495\pi$
$62$	0	0
$63$	0.476452	0.0600273
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−1.34364	−0.164151	−0.0820757	−	0.996626i	$-0.526155\pi$
$67$	−1.34364	−0.164151	−0.0820757	+	0.996626i	$0.526155\pi$
$68$	0	0
$69$	4.77299	0.574601
$70$	0	0
$71$	−4.86719	−0.577629	−0.288814	−	0.957385i	$-0.593261\pi$
$71$	−4.86719	−0.577629	−0.288814	+	0.957385i	$0.593261\pi$
$72$	0	0
$73$	−14.4132	−1.68693	−0.843467	−	0.537181i	$-0.819489\pi$
$73$	−14.4132	−1.68693	−0.843467	+	0.537181i	$0.819489\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.59308	−0.181549
$78$	0	0
$79$	−2.17991	−0.245259	−0.122630	−	0.992453i	$-0.539133\pi$
$79$	−2.17991	−0.245259	−0.122630	+	0.992453i	$0.539133\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	9.16373	1.00585	0.502925	−	0.864330i	$-0.332257\pi$
$83$	9.16373	1.00585	0.502925	+	0.864330i	$0.332257\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−7.59308	−0.814064
$88$	0	0
$89$	−10.5931	−1.12286	−0.561432	−	0.827523i	$-0.689749\pi$
$89$	−10.5931	−1.12286	−0.561432	+	0.827523i	$0.689749\pi$
$90$	0	0
$91$	0.476452	0.0499457
$92$	0	0
$93$	−7.93672	−0.823000
$94$	0	0
$95$	0	0
$96$	0	0
$97$	7.04710	0.715524	0.357762	−	0.933813i	$-0.383540\pi$
$97$	7.04710	0.715524	0.357762	+	0.933813i	$0.383540\pi$
$98$	0	0
$99$	−3.34364	−0.336048
$100$	0	0
$101$	−18.3661	−1.82749	−0.913746	−	0.406285i	$-0.866824\pi$
$101$	−18.3661	−1.82749	−0.913746	+	0.406285i	$0.866824\pi$
$102$	0	0
$103$	−13.5545	−1.33556	−0.667780	−	0.744358i	$-0.732755\pi$
$103$	−13.5545	−1.33556	−0.667780	+	0.744358i	$0.732755\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	16.2333	1.56933	0.784664	−	0.619921i	$-0.212835\pi$
$107$	16.2333	1.56933	0.784664	+	0.619921i	$0.212835\pi$
$108$	0	0
$109$	−16.4132	−1.57210	−0.786048	−	0.618165i	$-0.787876\pi$
$109$	−16.4132	−1.57210	−0.786048	+	0.618165i	$0.787876\pi$
$110$	0	0
$111$	−7.82009	−0.742250
$112$	0	0
$113$	8.59308	0.808369	0.404185	−	0.914677i	$-0.367555\pi$
$113$	8.59308	0.808369	0.404185	+	0.914677i	$0.367555\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	0.476452	0.0436763
$120$	0	0
$121$	0.179911	0.0163555
$122$	0	0
$123$	−9.46027	−0.853004
$124$	0	0
$125$	0	0
$126$	0	0
$127$	2.86719	0.254422	0.127211	−	0.991876i	$-0.459398\pi$
$127$	2.86719	0.254422	0.127211	+	0.991876i	$0.459398\pi$
$128$	0	0
$129$	−1.82009	−0.160250
$130$	0	0
$131$	−16.1475	−1.41082	−0.705409	−	0.708801i	$-0.749237\pi$
$131$	−16.1475	−1.41082	−0.705409	+	0.708801i	$0.749237\pi$
$132$	0	0
$133$	−3.22701	−0.279817
$134$	0	0
$135$	0	0
$136$	0	0
$137$	22.2333	1.89952	0.949758	−	0.312986i	$-0.101329\pi$
$137$	22.2333	1.89952	0.949758	+	0.312986i	$0.101329\pi$
$138$	0	0
$139$	9.28036	0.787150	0.393575	−	0.919293i	$-0.371238\pi$
$139$	9.28036	0.787150	0.393575	+	0.919293i	$0.371238\pi$
$140$	0	0
$141$	9.06953	0.763793
$142$	0	0
$143$	−3.34364	−0.279609
$144$	0	0
$145$	0	0
$146$	0	0
$147$	6.77299	0.558627
$148$	0	0
$149$	−22.1391	−1.81370	−0.906852	−	0.421450i	$-0.861521\pi$
$149$	−22.1391	−1.81370	−0.906852	+	0.421450i	$0.861521\pi$
$150$	0	0
$151$	−16.0224	−1.30389	−0.651944	−	0.758267i	$-0.726046\pi$
$151$	−16.0224	−1.30389	−0.651944	+	0.758267i	$0.726046\pi$
$152$	0	0
$153$	1.00000	0.0808452
$154$	0	0
$155$	0	0
$156$	0	0
$157$	6.22701	0.496969	0.248485	−	0.968636i	$-0.420067\pi$
$157$	6.22701	0.496969	0.248485	+	0.968636i	$0.420067\pi$
$158$	0	0
$159$	−9.50736	−0.753983
$160$	0	0
$161$	−2.27410	−0.179224
$162$	0	0
$163$	17.6935	1.38586	0.692932	−	0.721003i	$-0.256319\pi$
$163$	17.6935	1.38586	0.692932	+	0.721003i	$0.256319\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0.179911	0.0139219	0.00696095	−	0.999976i	$-0.497784\pi$
$167$	0.179911	0.0139219	0.00696095	+	0.999976i	$0.497784\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−6.77299	−0.517944
$172$	0	0
$173$	−12.4603	−0.947337	−0.473668	−	0.880703i	$-0.657070\pi$
$173$	−12.4603	−0.947337	−0.473668	+	0.880703i	$0.657070\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−3.16373	−0.237800
$178$	0	0
$179$	−5.55446	−0.415160	−0.207580	−	0.978218i	$-0.566559\pi$
$179$	−5.55446	−0.415160	−0.207580	+	0.978218i	$0.566559\pi$
$180$	0	0
$181$	24.0063	1.78437	0.892185	−	0.451669i	$-0.149171\pi$
$181$	24.0063	1.78437	0.892185	+	0.451669i	$0.149171\pi$
$182$	0	0
$183$	9.59308	0.709141
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−3.34364	−0.244511
$188$	0	0
$189$	−0.476452	−0.0346568
$190$	0	0
$191$	3.18617	0.230543	0.115271	−	0.993334i	$-0.463226\pi$
$191$	3.18617	0.230543	0.115271	+	0.993334i	$0.463226\pi$
$192$	0	0
$193$	−14.8672	−1.07016	−0.535082	−	0.844800i	$-0.679719\pi$
$193$	−14.8672	−1.07016	−0.535082	+	0.844800i	$0.679719\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	23.7259	1.69040	0.845200	−	0.534450i	$-0.179481\pi$
$197$	23.7259	1.69040	0.845200	+	0.534450i	$0.179481\pi$
$198$	0	0
$199$	−9.64018	−0.683374	−0.341687	−	0.939814i	$-0.610998\pi$
$199$	−9.64018	−0.683374	−0.341687	+	0.939814i	$0.610998\pi$
$200$	0	0
$201$	1.34364	0.0947729
$202$	0	0
$203$	3.61774	0.253916
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−4.77299	−0.331746
$208$	0	0
$209$	22.6464	1.56649
$210$	0	0
$211$	−10.8672	−0.748128	−0.374064	−	0.927403i	$-0.622036\pi$
$211$	−10.8672	−0.748128	−0.374064	+	0.927403i	$0.622036\pi$
$212$	0	0
$213$	4.86719	0.333494
$214$	0	0
$215$	0	0
$216$	0	0
$217$	3.78147	0.256703
$218$	0	0
$219$	14.4132	0.973952
$220$	0	0
$221$	1.00000	0.0672673
$222$	0	0
$223$	−22.2417	−1.48942	−0.744708	−	0.667390i	$-0.767411\pi$
$223$	−22.2417	−1.48942	−0.744708	+	0.667390i	$0.767411\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	23.4827	1.55860	0.779301	−	0.626650i	$-0.215574\pi$
$227$	23.4827	1.55860	0.779301	+	0.626650i	$0.215574\pi$
$228$	0	0
$229$	28.3723	1.87490	0.937448	−	0.348125i	$-0.113181\pi$
$229$	28.3723	1.87490	0.937448	+	0.348125i	$0.113181\pi$
$230$	0	0
$231$	1.59308	0.104817
$232$	0	0
$233$	−16.5931	−1.08705	−0.543524	−	0.839393i	$-0.682911\pi$
$233$	−16.5931	−1.08705	−0.543524	+	0.839393i	$0.682911\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	2.17991	0.141600
$238$	0	0
$239$	12.0224	0.777667	0.388833	−	0.921308i	$-0.372878\pi$
$239$	12.0224	0.777667	0.388833	+	0.921308i	$0.372878\pi$
$240$	0	0
$241$	−23.6935	−1.52623	−0.763117	−	0.646260i	$-0.776332\pi$
$241$	−23.6935	−1.52623	−0.763117	+	0.646260i	$0.776332\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−6.77299	−0.430955
$248$	0	0
$249$	−9.16373	−0.580728
$250$	0	0
$251$	18.7406	1.18290	0.591449	−	0.806342i	$-0.298556\pi$
$251$	18.7406	1.18290	0.591449	+	0.806342i	$0.298556\pi$
$252$	0	0
$253$	15.9592	1.00334
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−9.24173	−0.576484	−0.288242	−	0.957558i	$-0.593071\pi$
$257$	−9.24173	−0.576484	−0.288242	+	0.957558i	$0.593071\pi$
$258$	0	0
$259$	3.72590	0.231516
$260$	0	0
$261$	7.59308	0.470000
$262$	0	0
$263$	−17.9143	−1.10464	−0.552321	−	0.833632i	$-0.686258\pi$
$263$	−17.9143	−1.10464	−0.552321	+	0.833632i	$0.686258\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	10.5931	0.648286
$268$	0	0
$269$	−6.28036	−0.382920	−0.191460	−	0.981500i	$-0.561322\pi$
$269$	−6.28036	−0.382920	−0.191460	+	0.981500i	$0.561322\pi$
$270$	0	0
$271$	−18.2881	−1.11092	−0.555461	−	0.831543i	$-0.687458\pi$
$271$	−18.2881	−1.11092	−0.555461	+	0.831543i	$0.687458\pi$
$272$	0	0
$273$	−0.476452	−0.0288362
$274$	0	0
$275$	0	0
$276$	0	0
$277$	12.7645	0.766946	0.383473	−	0.923552i	$-0.374728\pi$
$277$	12.7645	0.766946	0.383473	+	0.923552i	$0.374728\pi$
$278$	0	0
$279$	7.93672	0.475159
$280$	0	0
$281$	−26.8263	−1.60033	−0.800163	−	0.599783i	$-0.795254\pi$
$281$	−26.8263	−1.60033	−0.800163	+	0.599783i	$0.795254\pi$
$282$	0	0
$283$	−21.4518	−1.27518	−0.637588	−	0.770377i	$-0.720068\pi$
$283$	−21.4518	−1.27518	−0.637588	+	0.770377i	$0.720068\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4.50736	0.266061
$288$	0	0
$289$	−16.0000	−0.941176
$290$	0	0
$291$	−7.04710	−0.413108
$292$	0	0
$293$	−11.3745	−0.664508	−0.332254	−	0.943190i	$-0.607809\pi$
$293$	−11.3745	−0.664508	−0.332254	+	0.943190i	$0.607809\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	3.34364	0.194018
$298$	0	0
$299$	−4.77299	−0.276029
$300$	0	0
$301$	0.867185	0.0499837
$302$	0	0
$303$	18.3661	1.05510
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−14.0857	−0.803914	−0.401957	−	0.915658i	$-0.631670\pi$
$307$	−14.0857	−0.803914	−0.401957	+	0.915658i	$0.631670\pi$
$308$	0	0
$309$	13.5545	0.771086
$310$	0	0
$311$	−17.4518	−0.989601	−0.494800	−	0.869007i	$-0.664759\pi$
$311$	−17.4518	−0.989601	−0.494800	+	0.869007i	$0.664759\pi$
$312$	0	0
$313$	−6.39844	−0.361661	−0.180831	−	0.983514i	$-0.557879\pi$
$313$	−6.39844	−0.361661	−0.180831	+	0.983514i	$0.557879\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	5.58683	0.313788	0.156894	−	0.987615i	$-0.449852\pi$
$317$	5.58683	0.313788	0.156894	+	0.987615i	$0.449852\pi$
$318$	0	0
$319$	−25.3885	−1.42148
$320$	0	0
$321$	−16.2333	−0.906052
$322$	0	0
$323$	−6.77299	−0.376859
$324$	0	0
$325$	0	0
$326$	0	0
$327$	16.4132	0.907650
$328$	0	0
$329$	−4.32120	−0.238235
$330$	0	0
$331$	10.4132	0.572360	0.286180	−	0.958176i	$-0.407615\pi$
$331$	10.4132	0.572360	0.286180	+	0.958176i	$0.407615\pi$
$332$	0	0
$333$	7.82009	0.428538
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−16.3661	−0.891517	−0.445758	−	0.895153i	$-0.647066\pi$
$337$	−16.3661	−0.891517	−0.445758	+	0.895153i	$0.647066\pi$
$338$	0	0
$339$	−8.59308	−0.466712
$340$	0	0
$341$	−26.5375	−1.43709
$342$	0	0
$343$	−6.56217	−0.354324
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−36.3723	−1.95257	−0.976285	−	0.216491i	$-0.930539\pi$
$347$	−36.3723	−1.95257	−0.976285	+	0.216491i	$0.930539\pi$
$348$	0	0
$349$	−17.7259	−0.948846	−0.474423	−	0.880297i	$-0.657343\pi$
$349$	−17.7259	−0.948846	−0.474423	+	0.880297i	$0.657343\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	19.5375	1.03988	0.519938	−	0.854204i	$-0.325955\pi$
$353$	19.5375	1.03988	0.519938	+	0.854204i	$0.325955\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−0.476452	−0.0252165
$358$	0	0
$359$	−5.92825	−0.312881	−0.156440	−	0.987687i	$-0.550002\pi$
$359$	−5.92825	−0.312881	−0.156440	+	0.987687i	$0.550002\pi$
$360$	0	0
$361$	26.8734	1.41439
$362$	0	0
$363$	−0.179911	−0.00944286
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−27.2804	−1.42402	−0.712012	−	0.702168i	$-0.752216\pi$
$367$	−27.2804	−1.42402	−0.712012	+	0.702168i	$0.752216\pi$
$368$	0	0
$369$	9.46027	0.492482
$370$	0	0
$371$	4.52980	0.235176
$372$	0	0
$373$	−27.2866	−1.41285	−0.706424	−	0.707789i	$-0.749693\pi$
$373$	−27.2866	−1.41285	−0.706424	+	0.707789i	$0.749693\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	7.59308	0.391064
$378$	0	0
$379$	−33.4503	−1.71823	−0.859114	−	0.511784i	$-0.828985\pi$
$379$	−33.4503	−1.71823	−0.859114	+	0.511784i	$0.828985\pi$
$380$	0	0
$381$	−2.86719	−0.146890
$382$	0	0
$383$	2.60156	0.132933	0.0664666	−	0.997789i	$-0.478827\pi$
$383$	2.60156	0.132933	0.0664666	+	0.997789i	$0.478827\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1.82009	0.0925203
$388$	0	0
$389$	−0.781467	−0.0396219	−0.0198110	−	0.999804i	$-0.506306\pi$
$389$	−0.781467	−0.0396219	−0.0198110	+	0.999804i	$0.506306\pi$
$390$	0	0
$391$	−4.77299	−0.241381
$392$	0	0
$393$	16.1475	0.814536
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−6.49889	−0.326170	−0.163085	−	0.986612i	$-0.552144\pi$
$397$	−6.49889	−0.326170	−0.163085	+	0.986612i	$0.552144\pi$
$398$	0	0
$399$	3.22701	0.161552
$400$	0	0
$401$	21.0063	1.04900	0.524501	−	0.851410i	$-0.324252\pi$
$401$	21.0063	1.04900	0.524501	+	0.851410i	$0.324252\pi$
$402$	0	0
$403$	7.93672	0.395356
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−26.1475	−1.29609
$408$	0	0
$409$	−8.08572	−0.399813	−0.199907	−	0.979815i	$-0.564064\pi$
$409$	−8.08572	−0.399813	−0.199907	+	0.979815i	$0.564064\pi$
$410$	0	0
$411$	−22.2333	−1.09669
$412$	0	0
$413$	1.50736	0.0741725
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−9.28036	−0.454461
$418$	0	0
$419$	1.91428	0.0935188	0.0467594	−	0.998906i	$-0.485111\pi$
$419$	1.91428	0.0935188	0.0467594	+	0.998906i	$0.485111\pi$
$420$	0	0
$421$	−27.1089	−1.32121	−0.660604	−	0.750735i	$-0.729700\pi$
$421$	−27.1089	−1.32121	−0.660604	+	0.750735i	$0.729700\pi$
$422$	0	0
$423$	−9.06953	−0.440976
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−4.57064	−0.221189
$428$	0	0
$429$	3.34364	0.161432
$430$	0	0
$431$	−13.0386	−0.628048	−0.314024	−	0.949415i	$-0.601677\pi$
$431$	−13.0386	−0.628048	−0.314024	+	0.949415i	$0.601677\pi$
$432$	0	0
$433$	−26.3275	−1.26522	−0.632608	−	0.774472i	$-0.718016\pi$
$433$	−26.3275	−1.26522	−0.632608	+	0.774472i	$0.718016\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	32.3275	1.54643
$438$	0	0
$439$	7.69353	0.367192	0.183596	−	0.983002i	$-0.441226\pi$
$439$	7.69353	0.367192	0.183596	+	0.983002i	$0.441226\pi$
$440$	0	0
$441$	−6.77299	−0.322523
$442$	0	0
$443$	23.7259	1.12725	0.563626	−	0.826030i	$-0.309406\pi$
$443$	23.7259	1.12725	0.563626	+	0.826030i	$0.309406\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	22.1391	1.04714
$448$	0	0
$449$	9.73437	0.459393	0.229697	−	0.973262i	$-0.426227\pi$
$449$	9.73437	0.459393	0.229697	+	0.973262i	$0.426227\pi$
$450$	0	0
$451$	−31.6317	−1.48948
$452$	0	0
$453$	16.0224	0.752800
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−5.64865	−0.264233	−0.132116	−	0.991234i	$-0.542177\pi$
$457$	−5.64865	−0.264233	−0.132116	+	0.991234i	$0.542177\pi$
$458$	0	0
$459$	−1.00000	−0.0466760
$460$	0	0
$461$	−12.4540	−0.580041	−0.290021	−	0.957020i	$-0.593662\pi$
$461$	−12.4540	−0.580041	−0.290021	+	0.957020i	$0.593662\pi$
$462$	0	0
$463$	14.2642	0.662912	0.331456	−	0.943471i	$-0.392460\pi$
$463$	14.2642	0.662912	0.331456	+	0.943471i	$0.392460\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−18.7815	−0.869103	−0.434551	−	0.900647i	$-0.643093\pi$
$467$	−18.7815	−0.869103	−0.434551	+	0.900647i	$0.643093\pi$
$468$	0	0
$469$	−0.640179	−0.0295607
$470$	0	0
$471$	−6.22701	−0.286925
$472$	0	0
$473$	−6.08572	−0.279822
$474$	0	0
$475$	0	0
$476$	0	0
$477$	9.50736	0.435312
$478$	0	0
$479$	−12.7954	−0.584638	−0.292319	−	0.956321i	$-0.594427\pi$
$479$	−12.7954	−0.584638	−0.292319	+	0.956321i	$0.594427\pi$
$480$	0	0
$481$	7.82009	0.356565
$482$	0	0
$483$	2.27410	0.103475
$484$	0	0
$485$	0	0
$486$	0	0
$487$	15.4378	0.699555	0.349777	−	0.936833i	$-0.386257\pi$
$487$	15.4378	0.699555	0.349777	+	0.936833i	$0.386257\pi$
$488$	0	0
$489$	−17.6935	−0.800129
$490$	0	0
$491$	−11.3745	−0.513326	−0.256663	−	0.966501i	$-0.582623\pi$
$491$	−11.3745	−0.513326	−0.256663	+	0.966501i	$0.582623\pi$
$492$	0	0
$493$	7.59308	0.341975
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−2.31898	−0.104020
$498$	0	0
$499$	43.7075	1.95662	0.978308	−	0.207155i	$-0.0664205\pi$
$499$	43.7075	1.95662	0.978308	+	0.207155i	$0.0664205\pi$
$500$	0	0
$501$	−0.179911	−0.00803781
$502$	0	0
$503$	10.5846	0.471944	0.235972	−	0.971760i	$-0.424173\pi$
$503$	10.5846	0.471944	0.235972	+	0.971760i	$0.424173\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	4.05335	0.179662	0.0898308	−	0.995957i	$-0.471367\pi$
$509$	4.05335	0.179662	0.0898308	+	0.995957i	$0.471367\pi$
$510$	0	0
$511$	−6.86719	−0.303786
$512$	0	0
$513$	6.77299	0.299035
$514$	0	0
$515$	0	0
$516$	0	0
$517$	30.3252	1.33370
$518$	0	0
$519$	12.4603	0.546945
$520$	0	0
$521$	19.1862	0.840561	0.420281	−	0.907394i	$-0.361932\pi$
$521$	19.1862	0.840561	0.420281	+	0.907394i	$0.361932\pi$
$522$	0	0
$523$	−40.2333	−1.75928	−0.879639	−	0.475642i	$-0.842216\pi$
$523$	−40.2333	−1.75928	−0.879639	+	0.475642i	$0.842216\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	7.93672	0.345729
$528$	0	0
$529$	−0.218533	−0.00950146
$530$	0	0
$531$	3.16373	0.137294
$532$	0	0
$533$	9.46027	0.409770
$534$	0	0
$535$	0	0
$536$	0	0
$537$	5.55446	0.239693
$538$	0	0
$539$	22.6464	0.975451
$540$	0	0
$541$	43.7708	1.88185	0.940926	−	0.338611i	$-0.109957\pi$
$541$	43.7708	1.88185	0.940926	+	0.338611i	$0.109957\pi$
$542$	0	0
$543$	−24.0063	−1.03021
$544$	0	0
$545$	0	0
$546$	0	0
$547$	1.72590	0.0737940	0.0368970	−	0.999319i	$-0.488253\pi$
$547$	1.72590	0.0737940	0.0368970	+	0.999319i	$0.488253\pi$
$548$	0	0
$549$	−9.59308	−0.409423
$550$	0	0
$551$	−51.4279	−2.19090
$552$	0	0
$553$	−1.03862	−0.0441667
$554$	0	0
$555$	0	0
$556$	0	0
$557$	16.6016	0.703430	0.351715	−	0.936107i	$-0.385599\pi$
$557$	16.6016	0.703430	0.351715	+	0.936107i	$0.385599\pi$
$558$	0	0
$559$	1.82009	0.0769816
$560$	0	0
$561$	3.34364	0.141168
$562$	0	0
$563$	−27.1453	−1.14404	−0.572020	−	0.820240i	$-0.693840\pi$
$563$	−27.1453	−1.14404	−0.572020	+	0.820240i	$0.693840\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0.476452	0.0200091
$568$	0	0
$569$	−24.0232	−1.00710	−0.503552	−	0.863965i	$-0.667974\pi$
$569$	−24.0232	−1.00710	−0.503552	+	0.863965i	$0.667974\pi$
$570$	0	0
$571$	37.9592	1.58854	0.794271	−	0.607564i	$-0.207853\pi$
$571$	37.9592	1.58854	0.794271	+	0.607564i	$0.207853\pi$
$572$	0	0
$573$	−3.18617	−0.133104
$574$	0	0
$575$	0	0
$576$	0	0
$577$	39.7020	1.65282	0.826408	−	0.563072i	$-0.190381\pi$
$577$	39.7020	1.65282	0.826408	+	0.563072i	$0.190381\pi$
$578$	0	0
$579$	14.8672	0.617859
$580$	0	0
$581$	4.36608	0.181135
$582$	0	0
$583$	−31.7892	−1.31657
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−32.6070	−1.34584	−0.672918	−	0.739717i	$-0.734960\pi$
$587$	−32.6070	−1.34584	−0.672918	+	0.739717i	$0.734960\pi$
$588$	0	0
$589$	−53.7554	−2.21495
$590$	0	0
$591$	−23.7259	−0.975953
$592$	0	0
$593$	−0.454013	−0.0186441	−0.00932204	−	0.999957i	$-0.502967\pi$
$593$	−0.454013	−0.0186441	−0.00932204	+	0.999957i	$0.502967\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	9.64018	0.394546
$598$	0	0
$599$	−8.31898	−0.339904	−0.169952	−	0.985452i	$-0.554361\pi$
$599$	−8.31898	−0.339904	−0.169952	+	0.985452i	$0.554361\pi$
$600$	0	0
$601$	36.3337	1.48208	0.741041	−	0.671459i	$-0.234332\pi$
$601$	36.3337	1.48208	0.741041	+	0.671459i	$0.234332\pi$
$602$	0	0
$603$	−1.34364	−0.0547171
$604$	0	0
$605$	0	0
$606$	0	0
$607$	34.9654	1.41920	0.709601	−	0.704604i	$-0.248875\pi$
$607$	34.9654	1.41920	0.709601	+	0.704604i	$0.248875\pi$
$608$	0	0
$609$	−3.61774	−0.146598
$610$	0	0
$611$	−9.06953	−0.366914
$612$	0	0
$613$	15.3745	0.620972	0.310486	−	0.950578i	$-0.399508\pi$
$613$	15.3745	0.620972	0.310486	+	0.950578i	$0.399508\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−11.8734	−0.478007	−0.239003	−	0.971019i	$-0.576821\pi$
$617$	−11.8734	−0.478007	−0.239003	+	0.971019i	$0.576821\pi$
$618$	0	0
$619$	13.1946	0.530337	0.265169	−	0.964202i	$-0.414572\pi$
$619$	13.1946	0.530337	0.265169	+	0.964202i	$0.414572\pi$
$620$	0	0
$621$	4.77299	0.191534
$622$	0	0
$623$	−5.04710	−0.202208
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−22.6464	−0.904411
$628$	0	0
$629$	7.82009	0.311807
$630$	0	0
$631$	−2.63392	−0.104855	−0.0524274	−	0.998625i	$-0.516696\pi$
$631$	−2.63392	−0.104855	−0.0524274	+	0.998625i	$0.516696\pi$
$632$	0	0
$633$	10.8672	0.431932
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−6.77299	−0.268356
$638$	0	0
$639$	−4.86719	−0.192543
$640$	0	0
$641$	−18.4687	−0.729471	−0.364736	−	0.931111i	$-0.618841\pi$
$641$	−18.4687	−0.729471	−0.364736	+	0.931111i	$0.618841\pi$
$642$	0	0
$643$	−36.5074	−1.43971	−0.719855	−	0.694125i	$-0.755792\pi$
$643$	−36.5074	−1.43971	−0.719855	+	0.694125i	$0.755792\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−1.40692	−0.0553116	−0.0276558	−	0.999618i	$-0.508804\pi$
$647$	−1.40692	−0.0553116	−0.0276558	+	0.999618i	$0.508804\pi$
$648$	0	0
$649$	−10.5784	−0.415237
$650$	0	0
$651$	−3.78147	−0.148207
$652$	0	0
$653$	−38.7469	−1.51628	−0.758141	−	0.652090i	$-0.773892\pi$
$653$	−38.7469	−1.51628	−0.758141	+	0.652090i	$0.773892\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−14.4132	−0.562311
$658$	0	0
$659$	28.0857	1.09406	0.547032	−	0.837112i	$-0.315758\pi$
$659$	28.0857	1.09406	0.547032	+	0.837112i	$0.315758\pi$
$660$	0	0
$661$	−45.2071	−1.75835	−0.879177	−	0.476495i	$-0.841907\pi$
$661$	−45.2071	−1.75835	−0.879177	+	0.476495i	$0.841907\pi$
$662$	0	0
$663$	−1.00000	−0.0388368
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−36.2417	−1.40329
$668$	0	0
$669$	22.2417	0.859915
$670$	0	0
$671$	32.0758	1.23827
$672$	0	0
$673$	−21.2741	−0.820056	−0.410028	−	0.912073i	$-0.634481\pi$
$673$	−21.2741	−0.820056	−0.410028	+	0.912073i	$0.634481\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	19.5460	0.751213	0.375607	−	0.926779i	$-0.377434\pi$
$677$	19.5460	0.751213	0.375607	+	0.926779i	$0.377434\pi$
$678$	0	0
$679$	3.35760	0.128853
$680$	0	0
$681$	−23.4827	−0.899859
$682$	0	0
$683$	7.79765	0.298369	0.149184	−	0.988809i	$-0.452335\pi$
$683$	7.79765	0.298369	0.149184	+	0.988809i	$0.452335\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−28.3723	−1.08247
$688$	0	0
$689$	9.50736	0.362202
$690$	0	0
$691$	−20.7267	−0.788479	−0.394240	−	0.919008i	$-0.628992\pi$
$691$	−20.7267	−0.788479	−0.394240	+	0.919008i	$0.628992\pi$
$692$	0	0
$693$	−1.59308	−0.0605162
$694$	0	0
$695$	0	0
$696$	0	0
$697$	9.46027	0.358333
$698$	0	0
$699$	16.5931	0.627608
$700$	0	0
$701$	−32.7792	−1.23806	−0.619028	−	0.785369i	$-0.712473\pi$
$701$	−32.7792	−1.23806	−0.619028	+	0.785369i	$0.712473\pi$
$702$	0	0
$703$	−52.9654	−1.99763
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−8.75055	−0.329098
$708$	0	0
$709$	4.08572	0.153442	0.0767212	−	0.997053i	$-0.475555\pi$
$709$	4.08572	0.153442	0.0767212	+	0.997053i	$0.475555\pi$
$710$	0	0
$711$	−2.17991	−0.0817530
$712$	0	0
$713$	−37.8819	−1.41869
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−12.0224	−0.448986
$718$	0	0
$719$	30.0857	1.12201	0.561004	−	0.827813i	$-0.310415\pi$
$719$	30.0857	1.12201	0.561004	+	0.827813i	$0.310415\pi$
$720$	0	0
$721$	−6.45805	−0.240510
$722$	0	0
$723$	23.6935	0.881172
$724$	0	0
$725$	0	0
$726$	0	0
$727$	18.9121	0.701410	0.350705	−	0.936486i	$-0.385942\pi$
$727$	18.9121	0.701410	0.350705	+	0.936486i	$0.385942\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	1.82009	0.0673184
$732$	0	0
$733$	30.3893	1.12245	0.561227	−	0.827662i	$-0.310330\pi$
$733$	30.3893	1.12245	0.561227	+	0.827662i	$0.310330\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4.49264	0.165488
$738$	0	0
$739$	−5.60927	−0.206340	−0.103170	−	0.994664i	$-0.532899\pi$
$739$	−5.60927	−0.206340	−0.103170	+	0.994664i	$0.532899\pi$
$740$	0	0
$741$	6.77299	0.248812
$742$	0	0
$743$	9.51507	0.349074	0.174537	−	0.984651i	$-0.444157\pi$
$743$	9.51507	0.349074	0.174537	+	0.984651i	$0.444157\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	9.16373	0.335283
$748$	0	0
$749$	7.73437	0.282608
$750$	0	0
$751$	−31.5221	−1.15026	−0.575129	−	0.818063i	$-0.695048\pi$
$751$	−31.5221	−1.15026	−0.575129	+	0.818063i	$0.695048\pi$
$752$	0	0
$753$	−18.7406	−0.682946
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−25.5607	−0.929020	−0.464510	−	0.885568i	$-0.653770\pi$
$757$	−25.5607	−0.929020	−0.464510	+	0.885568i	$0.653770\pi$
$758$	0	0
$759$	−15.9592	−0.579281
$760$	0	0
$761$	10.4132	0.377477	0.188739	−	0.982027i	$-0.439560\pi$
$761$	10.4132	0.377477	0.188739	+	0.982027i	$0.439560\pi$
$762$	0	0
$763$	−7.82009	−0.283106
$764$	0	0
$765$	0	0
$766$	0	0
$767$	3.16373	0.114236
$768$	0	0
$769$	−2.95290	−0.106484	−0.0532422	−	0.998582i	$-0.516956\pi$
$769$	−2.95290	−0.106484	−0.0532422	+	0.998582i	$0.516956\pi$
$770$	0	0
$771$	9.24173	0.332833
$772$	0	0
$773$	11.0471	0.397336	0.198668	−	0.980067i	$-0.436338\pi$
$773$	11.0471	0.397336	0.198668	+	0.980067i	$0.436338\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−3.72590	−0.133666
$778$	0	0
$779$	−64.0743	−2.29570
$780$	0	0
$781$	16.2741	0.582333
$782$	0	0
$783$	−7.59308	−0.271355
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−15.1553	−0.540226	−0.270113	−	0.962829i	$-0.587061\pi$
$787$	−15.1553	−0.540226	−0.270113	+	0.962829i	$0.587061\pi$
$788$	0	0
$789$	17.9143	0.637765
$790$	0	0
$791$	4.09419	0.145573
$792$	0	0
$793$	−9.59308	−0.340660
$794$	0	0
$795$	0	0
$796$	0	0
$797$	38.7962	1.37423	0.687116	−	0.726548i	$-0.258876\pi$
$797$	38.7962	1.37423	0.687116	+	0.726548i	$0.258876\pi$
$798$	0	0
$799$	−9.06953	−0.320857
$800$	0	0
$801$	−10.5931	−0.374288
$802$	0	0
$803$	48.1924	1.70067
$804$	0	0
$805$	0	0
$806$	0	0
$807$	6.28036	0.221079
$808$	0	0
$809$	−50.1391	−1.76280	−0.881398	−	0.472375i	$-0.843397\pi$
$809$	−50.1391	−1.76280	−0.881398	+	0.472375i	$0.843397\pi$
$810$	0	0
$811$	−17.5684	−0.616911	−0.308455	−	0.951239i	$-0.599812\pi$
$811$	−17.5684	−0.616911	−0.308455	+	0.951239i	$0.599812\pi$
$812$	0	0
$813$	18.2881	0.641391
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−12.3275	−0.431283
$818$	0	0
$819$	0.476452	0.0166486
$820$	0	0
$821$	1.19464	0.0416932	0.0208466	−	0.999783i	$-0.493364\pi$
$821$	1.19464	0.0416932	0.0208466	+	0.999783i	$0.493364\pi$
$822$	0	0
$823$	44.2458	1.54231	0.771155	−	0.636647i	$-0.219679\pi$
$823$	44.2458	1.54231	0.771155	+	0.636647i	$0.219679\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−6.84475	−0.238015	−0.119008	−	0.992893i	$-0.537971\pi$
$827$	−6.84475	−0.238015	−0.119008	+	0.992893i	$0.537971\pi$
$828$	0	0
$829$	1.04488	0.0362901	0.0181451	−	0.999835i	$-0.494224\pi$
$829$	1.04488	0.0362901	0.0181451	+	0.999835i	$0.494224\pi$
$830$	0	0
$831$	−12.7645	−0.442796
$832$	0	0
$833$	−6.77299	−0.234670
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−7.93672	−0.274333
$838$	0	0
$839$	38.2417	1.32025	0.660126	−	0.751155i	$-0.270503\pi$
$839$	38.2417	1.32025	0.660126	+	0.751155i	$0.270503\pi$
$840$	0	0
$841$	28.6549	0.988100
$842$	0	0
$843$	26.8263	0.923948
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0.0857188	0.00294533
$848$	0	0
$849$	21.4518	0.736224
$850$	0	0
$851$	−37.3252	−1.27949
$852$	0	0
$853$	44.9978	1.54069	0.770347	−	0.637624i	$-0.220083\pi$
$853$	44.9978	1.54069	0.770347	+	0.637624i	$0.220083\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	12.3723	0.422631	0.211315	−	0.977418i	$-0.432225\pi$
$857$	12.3723	0.422631	0.211315	+	0.977418i	$0.432225\pi$
$858$	0	0
$859$	−22.4750	−0.766837	−0.383418	−	0.923575i	$-0.625253\pi$
$859$	−22.4750	−0.766837	−0.383418	+	0.923575i	$0.625253\pi$
$860$	0	0
$861$	−4.50736	−0.153611
$862$	0	0
$863$	12.1251	0.412743	0.206372	−	0.978474i	$-0.433834\pi$
$863$	12.1251	0.412743	0.206372	+	0.978474i	$0.433834\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	16.0000	0.543388
$868$	0	0
$869$	7.28883	0.247257
$870$	0	0
$871$	−1.34364	−0.0455274
$872$	0	0
$873$	7.04710	0.238508
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−26.0449	−0.879473	−0.439737	−	0.898127i	$-0.644928\pi$
$877$	−26.0449	−0.879473	−0.439737	+	0.898127i	$0.644928\pi$
$878$	0	0
$879$	11.3745	0.383654
$880$	0	0
$881$	−13.4132	−0.451901	−0.225951	−	0.974139i	$-0.572549\pi$
$881$	−13.4132	−0.451901	−0.225951	+	0.974139i	$0.572549\pi$
$882$	0	0
$883$	−28.5158	−0.959634	−0.479817	−	0.877369i	$-0.659297\pi$
$883$	−28.5158	−0.959634	−0.479817	+	0.877369i	$0.659297\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	30.6464	1.02901	0.514503	−	0.857488i	$-0.327976\pi$
$887$	30.6464	1.02901	0.514503	+	0.857488i	$0.327976\pi$
$888$	0	0
$889$	1.36608	0.0458167
$890$	0	0
$891$	−3.34364	−0.112016
$892$	0	0
$893$	61.4279	2.05561
$894$	0	0
$895$	0	0
$896$	0	0
$897$	4.77299	0.159366
$898$	0	0
$899$	60.2642	2.00992
$900$	0	0
$901$	9.50736	0.316736
$902$	0	0
$903$	−0.867185	−0.0288581
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−36.9978	−1.22849	−0.614246	−	0.789115i	$-0.710540\pi$
$907$	−36.9978	−1.22849	−0.614246	+	0.789115i	$0.710540\pi$
$908$	0	0
$909$	−18.3661	−0.609164
$910$	0	0
$911$	35.3337	1.17066	0.585329	−	0.810796i	$-0.300965\pi$
$911$	35.3337	1.17066	0.585329	+	0.810796i	$0.300965\pi$
$912$	0	0
$913$	−30.6402	−1.01404
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−7.69353	−0.254063
$918$	0	0
$919$	−35.6851	−1.17714	−0.588571	−	0.808446i	$-0.700309\pi$
$919$	−35.6851	−1.17714	−0.588571	+	0.808446i	$0.700309\pi$
$920$	0	0
$921$	14.0857	0.464140
$922$	0	0
$923$	−4.86719	−0.160205
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−13.5545	−0.445187
$928$	0	0
$929$	−48.2009	−1.58142	−0.790710	−	0.612191i	$-0.790288\pi$
$929$	−48.2009	−1.58142	−0.790710	+	0.612191i	$0.790288\pi$
$930$	0	0
$931$	45.8734	1.50344
$932$	0	0
$933$	17.4518	0.571346
$934$	0	0
$935$	0	0
$936$	0	0
$937$	36.1538	1.18109	0.590547	−	0.807004i	$-0.298912\pi$
$937$	36.1538	1.18109	0.590547	+	0.807004i	$0.298912\pi$
$938$	0	0
$939$	6.39844	0.208805
$940$	0	0
$941$	−6.66629	−0.217315	−0.108657	−	0.994079i	$-0.534655\pi$
$941$	−6.66629	−0.217315	−0.108657	+	0.994079i	$0.534655\pi$
$942$	0	0
$943$	−45.1538	−1.47041
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−15.3760	−0.499653	−0.249827	−	0.968291i	$-0.580374\pi$
$947$	−15.3760	−0.499653	−0.249827	+	0.968291i	$0.580374\pi$
$948$	0	0
$949$	−14.4132	−0.467871
$950$	0	0
$951$	−5.58683	−0.181165
$952$	0	0
$953$	7.10266	0.230078	0.115039	−	0.993361i	$-0.463301\pi$
$953$	7.10266	0.230078	0.115039	+	0.993361i	$0.463301\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	25.3885	0.820694
$958$	0	0
$959$	10.5931	0.342068
$960$	0	0
$961$	31.9915	1.03198
$962$	0	0
$963$	16.2333	0.523110
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−28.8573	−0.927987	−0.463993	−	0.885839i	$-0.653584\pi$
$967$	−28.8573	−0.927987	−0.463993	+	0.885839i	$0.653584\pi$
$968$	0	0
$969$	6.77299	0.217580
$970$	0	0
$971$	−25.5909	−0.821250	−0.410625	−	0.911804i	$-0.634689\pi$
$971$	−25.5909	−0.821250	−0.410625	+	0.911804i	$0.634689\pi$
$972$	0	0
$973$	4.42165	0.141751
$974$	0	0
$975$	0	0
$976$	0	0
$977$	31.6402	1.01226	0.506129	−	0.862457i	$-0.331076\pi$
$977$	31.6402	1.01226	0.506129	+	0.862457i	$0.331076\pi$
$978$	0	0
$979$	35.4194	1.13201
$980$	0	0
$981$	−16.4132	−0.524032
$982$	0	0
$983$	−5.12289	−0.163395	−0.0816973	−	0.996657i	$-0.526034\pi$
$983$	−5.12289	−0.163395	−0.0816973	+	0.996657i	$0.526034\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	4.32120	0.137545
$988$	0	0
$989$	−8.68727	−0.276239
$990$	0	0
$991$	−26.3359	−0.836588	−0.418294	−	0.908312i	$-0.637372\pi$
$991$	−26.3359	−0.836588	−0.418294	+	0.908312i	$0.637372\pi$
$992$	0	0
$993$	−10.4132	−0.330452
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−42.0766	−1.33258	−0.666289	−	0.745694i	$-0.732118\pi$
$997$	−42.0766	−1.33258	−0.666289	+	0.745694i	$0.732118\pi$
$998$	0	0
$999$	−7.82009	−0.247417

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7800.2.a.bg.1.2	✓	3
5.4	even	2		7800.2.a.br.1.2	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7800.2.a.bg.1.2	✓	3	1.1	even	1	trivial
7800.2.a.br.1.2	yes	3	5.4	even	2

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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +0.476452 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +0.476452 q^{7} +1.00000 q^{9} -3.34364 q^{11} +1.00000 q^{13} +1.00000 q^{17} -6.77299 q^{19} -0.476452 q^{21} -4.77299 q^{23} -1.00000 q^{27} +7.59308 q^{29} +7.93672 q^{31} +3.34364 q^{33} +7.82009 q^{37} -1.00000 q^{39} +9.46027 q^{41} +1.82009 q^{43} -9.06953 q^{47} -6.77299 q^{49} -1.00000 q^{51} +9.50736 q^{53} +6.77299 q^{57} +3.16373 q^{59} -9.59308 q^{61} +0.476452 q^{63} -1.34364 q^{67} +4.77299 q^{69} -4.86719 q^{71} -14.4132 q^{73} -1.59308 q^{77} -2.17991 q^{79} +1.00000 q^{81} +9.16373 q^{83} -7.59308 q^{87} -10.5931 q^{89} +0.476452 q^{91} -7.93672 q^{93} +7.04710 q^{97} -3.34364 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} - q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 - q^7 + 3 * q^9	\( 3 q - 3 q^{3} - q^{7} + 3 q^{9} - 3 q^{11} + 3 q^{13} + 3 q^{17} - 6 q^{19} + q^{21} - 3 q^{27} - q^{29} - 7 q^{31} + 3 q^{33} + 14 q^{37} - 3 q^{39} - 4 q^{43} - q^{47} - 6 q^{49} - 3 q^{51} + 5 q^{53} + 6 q^{57} - 7 q^{59} - 5 q^{61} - q^{63} + 3 q^{67} - 10 q^{71} - 10 q^{73} + 19 q^{77} - 16 q^{79} + 3 q^{81} + 11 q^{83} + q^{87} - 8 q^{89} - q^{91} + 7 q^{93} + 26 q^{97} - 3 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 - q^7 + 3 * q^9 - 3 * q^11 + 3 * q^13 + 3 * q^17 - 6 * q^19 + q^21 - 3 * q^27 - q^29 - 7 * q^31 + 3 * q^33 + 14 * q^37 - 3 * q^39 - 4 * q^43 - q^47 - 6 * q^49 - 3 * q^51 + 5 * q^53 + 6 * q^57 - 7 * q^59 - 5 * q^61 - q^63 + 3 * q^67 - 10 * q^71 - 10 * q^73 + 19 * q^77 - 16 * q^79 + 3 * q^81 + 11 * q^83 + q^87 - 8 * q^89 - q^91 + 7 * q^93 + 26 * q^97 - 3 * q^99

Introduction

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