LMFDB - Embedded newform 7800.2.a.g.1.1

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:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1560)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
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)

$f(q)$

$=$

$q-1.00000 q^{3} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$

$q-1.00000 q^{3} +1.00000 q^{7} +1.00000 q^{9} -5.00000 q^{11} -1.00000 q^{13} -3.00000 q^{17} -6.00000 q^{19} -1.00000 q^{21} -7.00000 q^{23} -1.00000 q^{27} -6.00000 q^{29} -10.0000 q^{31} +5.00000 q^{33} +11.0000 q^{37} +1.00000 q^{39} +5.00000 q^{41} +4.00000 q^{43} +2.00000 q^{47} -6.00000 q^{49} +3.00000 q^{51} +5.00000 q^{53} +6.00000 q^{57} +1.00000 q^{61} +1.00000 q^{63} +4.00000 q^{67} +7.00000 q^{69} -5.00000 q^{71} -6.00000 q^{73} -5.00000 q^{77} +11.0000 q^{79} +1.00000 q^{81} +8.00000 q^{83} +6.00000 q^{87} +7.00000 q^{89} -1.00000 q^{91} +10.0000 q^{93} +13.0000 q^{97} -5.00000 q^{99} +O(q^{100})$

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
$7$	1.00000	0.377964	0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.00000	−1.50756	−0.753778	−	0.657129i	$-0.771771\pi$
$11$	−5.00000	−1.50756	−0.753778	+	0.657129i	$0.771771\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	0	0
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−7.00000	−1.45960	−0.729800	−	0.683660i	$-0.760387\pi$
$23$	−7.00000	−1.45960	−0.729800	+	0.683660i	$0.760387\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	−10.0000	−1.79605	−0.898027	−	0.439941i	$-0.854999\pi$
$31$	−10.0000	−1.79605	−0.898027	+	0.439941i	$0.854999\pi$
$32$	0	0
$33$	5.00000	0.870388
$34$	0	0
$35$	0	0
$36$	0	0
$37$	11.0000	1.80839	0.904194	−	0.427121i	$-0.140472\pi$
$37$	11.0000	1.80839	0.904194	+	0.427121i	$0.140472\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	5.00000	0.780869	0.390434	−	0.920631i	$-0.372325\pi$
$41$	5.00000	0.780869	0.390434	+	0.920631i	$0.372325\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2.00000	0.291730	0.145865	−	0.989305i	$-0.453403\pi$
$47$	2.00000	0.291730	0.145865	+	0.989305i	$0.453403\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	0	0
$51$	3.00000	0.420084
$52$	0	0
$53$	5.00000	0.686803	0.343401	−	0.939189i	$-0.388421\pi$
$53$	5.00000	0.686803	0.343401	+	0.939189i	$0.388421\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	6.00000	0.794719
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	1.00000	0.128037	0.0640184	−	0.997949i	$-0.479608\pi$
$61$	1.00000	0.128037	0.0640184	+	0.997949i	$0.479608\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	0	0
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	0	0
$69$	7.00000	0.842701
$70$	0	0
$71$	−5.00000	−0.593391	−0.296695	−	0.954972i	$-0.595885\pi$
$71$	−5.00000	−0.593391	−0.296695	+	0.954972i	$0.595885\pi$
$72$	0	0
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−5.00000	−0.569803
$78$	0	0
$79$	11.0000	1.23760	0.618798	−	0.785550i	$-0.287620\pi$
$79$	11.0000	1.23760	0.618798	+	0.785550i	$0.287620\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	8.00000	0.878114	0.439057	−	0.898459i	$-0.355313\pi$
$83$	8.00000	0.878114	0.439057	+	0.898459i	$0.355313\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	6.00000	0.643268
$88$	0	0
$89$	7.00000	0.741999	0.370999	−	0.928633i	$-0.379015\pi$
$89$	7.00000	0.741999	0.370999	+	0.928633i	$0.379015\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	10.0000	1.03695
$94$	0	0
$95$	0	0
$96$	0	0
$97$	13.0000	1.31995	0.659975	−	0.751288i	$-0.270567\pi$
$97$	13.0000	1.31995	0.659975	+	0.751288i	$0.270567\pi$
$98$	0	0
$99$	−5.00000	−0.502519
$100$	0	0
$101$	−4.00000	−0.398015	−0.199007	−	0.979998i	$-0.563772\pi$
$101$	−4.00000	−0.398015	−0.199007	+	0.979998i	$0.563772\pi$
$102$	0	0
$103$	16.0000	1.57653	0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000	1.57653	0.788263	+	0.615338i	$0.210980\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.00000	0.0966736	0.0483368	−	0.998831i	$-0.484608\pi$
$107$	1.00000	0.0966736	0.0483368	+	0.998831i	$0.484608\pi$
$108$	0	0
$109$	20.0000	1.91565	0.957826	−	0.287348i	$-0.0927736\pi$
$109$	20.0000	1.91565	0.957826	+	0.287348i	$0.0927736\pi$
$110$	0	0
$111$	−11.0000	−1.04407
$112$	0	0
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	−3.00000	−0.275010
$120$	0	0
$121$	14.0000	1.27273
$122$	0	0
$123$	−5.00000	−0.450835
$124$	0	0
$125$	0	0
$126$	0	0
$127$	14.0000	1.24230	0.621150	−	0.783692i	$-0.286666\pi$
$127$	14.0000	1.24230	0.621150	+	0.783692i	$0.286666\pi$
$128$	0	0
$129$	−4.00000	−0.352180
$130$	0	0
$131$	6.00000	0.524222	0.262111	−	0.965038i	$-0.415581\pi$
$131$	6.00000	0.524222	0.262111	+	0.965038i	$0.415581\pi$
$132$	0	0
$133$	−6.00000	−0.520266
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−22.0000	−1.87959	−0.939793	−	0.341743i	$-0.888983\pi$
$137$	−22.0000	−1.87959	−0.939793	+	0.341743i	$0.888983\pi$
$138$	0	0
$139$	13.0000	1.10265	0.551323	−	0.834292i	$-0.314123\pi$
$139$	13.0000	1.10265	0.551323	+	0.834292i	$0.314123\pi$
$140$	0	0
$141$	−2.00000	−0.168430
$142$	0	0
$143$	5.00000	0.418121
$144$	0	0
$145$	0	0
$146$	0	0
$147$	6.00000	0.494872
$148$	0	0
$149$	−1.00000	−0.0819232	−0.0409616	−	0.999161i	$-0.513042\pi$
$149$	−1.00000	−0.0819232	−0.0409616	+	0.999161i	$0.513042\pi$
$150$	0	0
$151$	−20.0000	−1.62758	−0.813788	−	0.581161i	$-0.802599\pi$
$151$	−20.0000	−1.62758	−0.813788	+	0.581161i	$0.802599\pi$
$152$	0	0
$153$	−3.00000	−0.242536
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	0	0
$159$	−5.00000	−0.396526
$160$	0	0
$161$	−7.00000	−0.551677
$162$	0	0
$163$	−11.0000	−0.861586	−0.430793	−	0.902451i	$-0.641766\pi$
$163$	−11.0000	−0.861586	−0.430793	+	0.902451i	$0.641766\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−6.00000	−0.458831
$172$	0	0
$173$	22.0000	1.67263	0.836315	−	0.548250i	$-0.184706\pi$
$173$	22.0000	1.67263	0.836315	+	0.548250i	$0.184706\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	18.0000	1.34538	0.672692	−	0.739923i	$-0.265138\pi$
$179$	18.0000	1.34538	0.672692	+	0.739923i	$0.265138\pi$
$180$	0	0
$181$	−11.0000	−0.817624	−0.408812	−	0.912619i	$-0.634057\pi$
$181$	−11.0000	−0.817624	−0.408812	+	0.912619i	$0.634057\pi$
$182$	0	0
$183$	−1.00000	−0.0739221
$184$	0	0
$185$	0	0
$186$	0	0
$187$	15.0000	1.09691
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	0	0
$193$	−9.00000	−0.647834	−0.323917	−	0.946085i	$-0.605000\pi$
$193$	−9.00000	−0.647834	−0.323917	+	0.946085i	$0.605000\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	0	0
$199$	−20.0000	−1.41776	−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000	−1.41776	−0.708881	+	0.705328i	$0.750800\pi$
$200$	0	0
$201$	−4.00000	−0.282138
$202$	0	0
$203$	−6.00000	−0.421117
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−7.00000	−0.486534
$208$	0	0
$209$	30.0000	2.07514
$210$	0	0
$211$	−28.0000	−1.92760	−0.963800	−	0.266627i	$-0.914091\pi$
$211$	−28.0000	−1.92760	−0.963800	+	0.266627i	$0.914091\pi$
$212$	0	0
$213$	5.00000	0.342594
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−10.0000	−0.678844
$218$	0	0
$219$	6.00000	0.405442
$220$	0	0
$221$	3.00000	0.201802
$222$	0	0
$223$	8.00000	0.535720	0.267860	−	0.963458i	$-0.413684\pi$
$223$	8.00000	0.535720	0.267860	+	0.963458i	$0.413684\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−18.0000	−1.19470	−0.597351	−	0.801980i	$-0.703780\pi$
$227$	−18.0000	−1.19470	−0.597351	+	0.801980i	$0.703780\pi$
$228$	0	0
$229$	26.0000	1.71813	0.859064	−	0.511868i	$-0.171046\pi$
$229$	26.0000	1.71813	0.859064	+	0.511868i	$0.171046\pi$
$230$	0	0
$231$	5.00000	0.328976
$232$	0	0
$233$	15.0000	0.982683	0.491341	−	0.870967i	$-0.336507\pi$
$233$	15.0000	0.982683	0.491341	+	0.870967i	$0.336507\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−11.0000	−0.714527
$238$	0	0
$239$	11.0000	0.711531	0.355765	−	0.934575i	$-0.384220\pi$
$239$	11.0000	0.711531	0.355765	+	0.934575i	$0.384220\pi$
$240$	0	0
$241$	22.0000	1.41714	0.708572	−	0.705638i	$-0.249340\pi$
$241$	22.0000	1.41714	0.708572	+	0.705638i	$0.249340\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	6.00000	0.381771
$248$	0	0
$249$	−8.00000	−0.506979
$250$	0	0
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	0	0
$253$	35.0000	2.20043
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	0	0
$259$	11.0000	0.683507
$260$	0	0
$261$	−6.00000	−0.371391
$262$	0	0
$263$	−16.0000	−0.986602	−0.493301	−	0.869859i	$-0.664210\pi$
$263$	−16.0000	−0.986602	−0.493301	+	0.869859i	$0.664210\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−7.00000	−0.428393
$268$	0	0
$269$	4.00000	0.243884	0.121942	−	0.992537i	$-0.461088\pi$
$269$	4.00000	0.243884	0.121942	+	0.992537i	$0.461088\pi$
$270$	0	0
$271$	−2.00000	−0.121491	−0.0607457	−	0.998153i	$-0.519348\pi$
$271$	−2.00000	−0.121491	−0.0607457	+	0.998153i	$0.519348\pi$
$272$	0	0
$273$	1.00000	0.0605228
$274$	0	0
$275$	0	0
$276$	0	0
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	0	0
$279$	−10.0000	−0.598684
$280$	0	0
$281$	2.00000	0.119310	0.0596550	−	0.998219i	$-0.481000\pi$
$281$	2.00000	0.119310	0.0596550	+	0.998219i	$0.481000\pi$
$282$	0	0
$283$	16.0000	0.951101	0.475551	−	0.879688i	$-0.342249\pi$
$283$	16.0000	0.951101	0.475551	+	0.879688i	$0.342249\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	5.00000	0.295141
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	−13.0000	−0.762073
$292$	0	0
$293$	−20.0000	−1.16841	−0.584206	−	0.811605i	$-0.698594\pi$
$293$	−20.0000	−1.16841	−0.584206	+	0.811605i	$0.698594\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	5.00000	0.290129
$298$	0	0
$299$	7.00000	0.404820
$300$	0	0
$301$	4.00000	0.230556
$302$	0	0
$303$	4.00000	0.229794
$304$	0	0
$305$	0	0
$306$	0	0
$307$	21.0000	1.19853	0.599267	−	0.800549i	$-0.295459\pi$
$307$	21.0000	1.19853	0.599267	+	0.800549i	$0.295459\pi$
$308$	0	0
$309$	−16.0000	−0.910208
$310$	0	0
$311$	−24.0000	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	0	0
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	28.0000	1.57264	0.786318	−	0.617822i	$-0.211985\pi$
$317$	28.0000	1.57264	0.786318	+	0.617822i	$0.211985\pi$
$318$	0	0
$319$	30.0000	1.67968
$320$	0	0
$321$	−1.00000	−0.0558146
$322$	0	0
$323$	18.0000	1.00155
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−20.0000	−1.10600
$328$	0	0
$329$	2.00000	0.110264
$330$	0	0
$331$	28.0000	1.53902	0.769510	−	0.638635i	$-0.220501\pi$
$331$	28.0000	1.53902	0.769510	+	0.638635i	$0.220501\pi$
$332$	0	0
$333$	11.0000	0.602796
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−20.0000	−1.08947	−0.544735	−	0.838608i	$-0.683370\pi$
$337$	−20.0000	−1.08947	−0.544735	+	0.838608i	$0.683370\pi$
$338$	0	0
$339$	6.00000	0.325875
$340$	0	0
$341$	50.0000	2.70765
$342$	0	0
$343$	−13.0000	−0.701934
$344$	0	0
$345$	0	0
$346$	0	0
$347$	21.0000	1.12734	0.563670	−	0.826000i	$-0.309389\pi$
$347$	21.0000	1.12734	0.563670	+	0.826000i	$0.309389\pi$
$348$	0	0
$349$	−16.0000	−0.856460	−0.428230	−	0.903670i	$-0.640863\pi$
$349$	−16.0000	−0.856460	−0.428230	+	0.903670i	$0.640863\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	−2.00000	−0.106449	−0.0532246	−	0.998583i	$-0.516950\pi$
$353$	−2.00000	−0.106449	−0.0532246	+	0.998583i	$0.516950\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	3.00000	0.158777
$358$	0	0
$359$	−32.0000	−1.68890	−0.844448	−	0.535638i	$-0.820071\pi$
$359$	−32.0000	−1.68890	−0.844448	+	0.535638i	$0.820071\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	0	0
$363$	−14.0000	−0.734809
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−28.0000	−1.46159	−0.730794	−	0.682598i	$-0.760850\pi$
$367$	−28.0000	−1.46159	−0.730794	+	0.682598i	$0.760850\pi$
$368$	0	0
$369$	5.00000	0.260290
$370$	0	0
$371$	5.00000	0.259587
$372$	0	0
$373$	−32.0000	−1.65690	−0.828449	−	0.560065i	$-0.810776\pi$
$373$	−32.0000	−1.65690	−0.828449	+	0.560065i	$0.810776\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	6.00000	0.309016
$378$	0	0
$379$	14.0000	0.719132	0.359566	−	0.933120i	$-0.382925\pi$
$379$	14.0000	0.719132	0.359566	+	0.933120i	$0.382925\pi$
$380$	0	0
$381$	−14.0000	−0.717242
$382$	0	0
$383$	−18.0000	−0.919757	−0.459879	−	0.887982i	$-0.652107\pi$
$383$	−18.0000	−0.919757	−0.459879	+	0.887982i	$0.652107\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	4.00000	0.203331
$388$	0	0
$389$	4.00000	0.202808	0.101404	−	0.994845i	$-0.467667\pi$
$389$	4.00000	0.202808	0.101404	+	0.994845i	$0.467667\pi$
$390$	0	0
$391$	21.0000	1.06202
$392$	0	0
$393$	−6.00000	−0.302660
$394$	0	0
$395$	0	0
$396$	0	0
$397$	3.00000	0.150566	0.0752828	−	0.997162i	$-0.476014\pi$
$397$	3.00000	0.150566	0.0752828	+	0.997162i	$0.476014\pi$
$398$	0	0
$399$	6.00000	0.300376
$400$	0	0
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	0	0
$403$	10.0000	0.498135
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−55.0000	−2.72625
$408$	0	0
$409$	22.0000	1.08783	0.543915	−	0.839140i	$-0.316941\pi$
$409$	22.0000	1.08783	0.543915	+	0.839140i	$0.316941\pi$
$410$	0	0
$411$	22.0000	1.08518
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−13.0000	−0.636613
$418$	0	0
$419$	34.0000	1.66101	0.830504	−	0.557012i	$-0.188052\pi$
$419$	34.0000	1.66101	0.830504	+	0.557012i	$0.188052\pi$
$420$	0	0
$421$	16.0000	0.779792	0.389896	−	0.920859i	$-0.372511\pi$
$421$	16.0000	0.779792	0.389896	+	0.920859i	$0.372511\pi$
$422$	0	0
$423$	2.00000	0.0972433
$424$	0	0
$425$	0	0
$426$	0	0
$427$	1.00000	0.0483934
$428$	0	0
$429$	−5.00000	−0.241402
$430$	0	0
$431$	8.00000	0.385346	0.192673	−	0.981263i	$-0.438284\pi$
$431$	8.00000	0.385346	0.192673	+	0.981263i	$0.438284\pi$
$432$	0	0
$433$	20.0000	0.961139	0.480569	−	0.876957i	$-0.340430\pi$
$433$	20.0000	0.961139	0.480569	+	0.876957i	$0.340430\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	42.0000	2.00913
$438$	0	0
$439$	−7.00000	−0.334092	−0.167046	−	0.985949i	$-0.553423\pi$
$439$	−7.00000	−0.334092	−0.167046	+	0.985949i	$0.553423\pi$
$440$	0	0
$441$	−6.00000	−0.285714
$442$	0	0
$443$	−7.00000	−0.332580	−0.166290	−	0.986077i	$-0.553179\pi$
$443$	−7.00000	−0.332580	−0.166290	+	0.986077i	$0.553179\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	1.00000	0.0472984
$448$	0	0
$449$	21.0000	0.991051	0.495526	−	0.868593i	$-0.334975\pi$
$449$	21.0000	0.991051	0.495526	+	0.868593i	$0.334975\pi$
$450$	0	0
$451$	−25.0000	−1.17720
$452$	0	0
$453$	20.0000	0.939682
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−7.00000	−0.327446	−0.163723	−	0.986506i	$-0.552350\pi$
$457$	−7.00000	−0.327446	−0.163723	+	0.986506i	$0.552350\pi$
$458$	0	0
$459$	3.00000	0.140028
$460$	0	0
$461$	−3.00000	−0.139724	−0.0698620	−	0.997557i	$-0.522256\pi$
$461$	−3.00000	−0.139724	−0.0698620	+	0.997557i	$0.522256\pi$
$462$	0	0
$463$	1.00000	0.0464739	0.0232370	−	0.999730i	$-0.492603\pi$
$463$	1.00000	0.0464739	0.0232370	+	0.999730i	$0.492603\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−15.0000	−0.694117	−0.347059	−	0.937843i	$-0.612820\pi$
$467$	−15.0000	−0.694117	−0.347059	+	0.937843i	$0.612820\pi$
$468$	0	0
$469$	4.00000	0.184703
$470$	0	0
$471$	10.0000	0.460776
$472$	0	0
$473$	−20.0000	−0.919601
$474$	0	0
$475$	0	0
$476$	0	0
$477$	5.00000	0.228934
$478$	0	0
$479$	1.00000	0.0456912	0.0228456	−	0.999739i	$-0.492727\pi$
$479$	1.00000	0.0456912	0.0228456	+	0.999739i	$0.492727\pi$
$480$	0	0
$481$	−11.0000	−0.501557
$482$	0	0
$483$	7.00000	0.318511
$484$	0	0
$485$	0	0
$486$	0	0
$487$	11.0000	0.498458	0.249229	−	0.968445i	$-0.419823\pi$
$487$	11.0000	0.498458	0.249229	+	0.968445i	$0.419823\pi$
$488$	0	0
$489$	11.0000	0.497437
$490$	0	0
$491$	−40.0000	−1.80517	−0.902587	−	0.430507i	$-0.858335\pi$
$491$	−40.0000	−1.80517	−0.902587	+	0.430507i	$0.858335\pi$
$492$	0	0
$493$	18.0000	0.810679
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−5.00000	−0.224281
$498$	0	0
$499$	−10.0000	−0.447661	−0.223831	−	0.974628i	$-0.571856\pi$
$499$	−10.0000	−0.447661	−0.223831	+	0.974628i	$0.571856\pi$
$500$	0	0
$501$	12.0000	0.536120
$502$	0	0
$503$	20.0000	0.891756	0.445878	−	0.895094i	$-0.352892\pi$
$503$	20.0000	0.891756	0.445878	+	0.895094i	$0.352892\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	−21.0000	−0.930809	−0.465404	−	0.885098i	$-0.654091\pi$
$509$	−21.0000	−0.930809	−0.465404	+	0.885098i	$0.654091\pi$
$510$	0	0
$511$	−6.00000	−0.265424
$512$	0	0
$513$	6.00000	0.264906
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−10.0000	−0.439799
$518$	0	0
$519$	−22.0000	−0.965693
$520$	0	0
$521$	6.00000	0.262865	0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000	0.262865	0.131432	+	0.991325i	$0.458042\pi$
$522$	0	0
$523$	−16.0000	−0.699631	−0.349816	−	0.936819i	$-0.613756\pi$
$523$	−16.0000	−0.699631	−0.349816	+	0.936819i	$0.613756\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	30.0000	1.30682
$528$	0	0
$529$	26.0000	1.13043
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−5.00000	−0.216574
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−18.0000	−0.776757
$538$	0	0
$539$	30.0000	1.29219
$540$	0	0
$541$	−30.0000	−1.28980	−0.644900	−	0.764267i	$-0.723101\pi$
$541$	−30.0000	−1.28980	−0.644900	+	0.764267i	$0.723101\pi$
$542$	0	0
$543$	11.0000	0.472055
$544$	0	0
$545$	0	0
$546$	0	0
$547$	36.0000	1.53925	0.769624	−	0.638497i	$-0.220443\pi$
$547$	36.0000	1.53925	0.769624	+	0.638497i	$0.220443\pi$
$548$	0	0
$549$	1.00000	0.0426790
$550$	0	0
$551$	36.0000	1.53365
$552$	0	0
$553$	11.0000	0.467768
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−26.0000	−1.10166	−0.550828	−	0.834619i	$-0.685688\pi$
$557$	−26.0000	−1.10166	−0.550828	+	0.834619i	$0.685688\pi$
$558$	0	0
$559$	−4.00000	−0.169182
$560$	0	0
$561$	−15.0000	−0.633300
$562$	0	0
$563$	−11.0000	−0.463595	−0.231797	−	0.972764i	$-0.574461\pi$
$563$	−11.0000	−0.463595	−0.231797	+	0.972764i	$0.574461\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	−24.0000	−1.00613	−0.503066	−	0.864248i	$-0.667795\pi$
$569$	−24.0000	−1.00613	−0.503066	+	0.864248i	$0.667795\pi$
$570$	0	0
$571$	−5.00000	−0.209243	−0.104622	−	0.994512i	$-0.533363\pi$
$571$	−5.00000	−0.209243	−0.104622	+	0.994512i	$0.533363\pi$
$572$	0	0
$573$	12.0000	0.501307
$574$	0	0
$575$	0	0
$576$	0	0
$577$	41.0000	1.70685	0.853426	−	0.521214i	$-0.174521\pi$
$577$	41.0000	1.70685	0.853426	+	0.521214i	$0.174521\pi$
$578$	0	0
$579$	9.00000	0.374027
$580$	0	0
$581$	8.00000	0.331896
$582$	0	0
$583$	−25.0000	−1.03539
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−6.00000	−0.247647	−0.123823	−	0.992304i	$-0.539516\pi$
$587$	−6.00000	−0.247647	−0.123823	+	0.992304i	$0.539516\pi$
$588$	0	0
$589$	60.0000	2.47226
$590$	0	0
$591$	0	0
$592$	0	0
$593$	48.0000	1.97112	0.985562	−	0.169316i	$-0.0541557\pi$
$593$	48.0000	1.97112	0.985562	+	0.169316i	$0.0541557\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	20.0000	0.818546
$598$	0	0
$599$	48.0000	1.96123	0.980613	−	0.195952i	$-0.0627798\pi$
$599$	48.0000	1.96123	0.980613	+	0.195952i	$0.0627798\pi$
$600$	0	0
$601$	3.00000	0.122373	0.0611863	−	0.998126i	$-0.480512\pi$
$601$	3.00000	0.122373	0.0611863	+	0.998126i	$0.480512\pi$
$602$	0	0
$603$	4.00000	0.162893
$604$	0	0
$605$	0	0
$606$	0	0
$607$	12.0000	0.487065	0.243532	−	0.969893i	$-0.421694\pi$
$607$	12.0000	0.487065	0.243532	+	0.969893i	$0.421694\pi$
$608$	0	0
$609$	6.00000	0.243132
$610$	0	0
$611$	−2.00000	−0.0809113
$612$	0	0
$613$	29.0000	1.17130	0.585649	−	0.810564i	$-0.300840\pi$
$613$	29.0000	1.17130	0.585649	+	0.810564i	$0.300840\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−34.0000	−1.36879	−0.684394	−	0.729112i	$-0.739933\pi$
$617$	−34.0000	−1.36879	−0.684394	+	0.729112i	$0.739933\pi$
$618$	0	0
$619$	42.0000	1.68812	0.844061	−	0.536247i	$-0.180158\pi$
$619$	42.0000	1.68812	0.844061	+	0.536247i	$0.180158\pi$
$620$	0	0
$621$	7.00000	0.280900
$622$	0	0
$623$	7.00000	0.280449
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−30.0000	−1.19808
$628$	0	0
$629$	−33.0000	−1.31580
$630$	0	0
$631$	−20.0000	−0.796187	−0.398094	−	0.917345i	$-0.630328\pi$
$631$	−20.0000	−0.796187	−0.398094	+	0.917345i	$0.630328\pi$
$632$	0	0
$633$	28.0000	1.11290
$634$	0	0
$635$	0	0
$636$	0	0
$637$	6.00000	0.237729
$638$	0	0
$639$	−5.00000	−0.197797
$640$	0	0
$641$	8.00000	0.315981	0.157991	−	0.987441i	$-0.449498\pi$
$641$	8.00000	0.315981	0.157991	+	0.987441i	$0.449498\pi$
$642$	0	0
$643$	1.00000	0.0394362	0.0197181	−	0.999806i	$-0.493723\pi$
$643$	1.00000	0.0394362	0.0197181	+	0.999806i	$0.493723\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	27.0000	1.06148	0.530740	−	0.847535i	$-0.321914\pi$
$647$	27.0000	1.06148	0.530740	+	0.847535i	$0.321914\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	10.0000	0.391931
$652$	0	0
$653$	42.0000	1.64359	0.821794	−	0.569785i	$-0.192974\pi$
$653$	42.0000	1.64359	0.821794	+	0.569785i	$0.192974\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−6.00000	−0.234082
$658$	0	0
$659$	−16.0000	−0.623272	−0.311636	−	0.950202i	$-0.600877\pi$
$659$	−16.0000	−0.623272	−0.311636	+	0.950202i	$0.600877\pi$
$660$	0	0
$661$	−32.0000	−1.24466	−0.622328	−	0.782757i	$-0.713813\pi$
$661$	−32.0000	−1.24466	−0.622328	+	0.782757i	$0.713813\pi$
$662$	0	0
$663$	−3.00000	−0.116510
$664$	0	0
$665$	0	0
$666$	0	0
$667$	42.0000	1.62625
$668$	0	0
$669$	−8.00000	−0.309298
$670$	0	0
$671$	−5.00000	−0.193023
$672$	0	0
$673$	22.0000	0.848038	0.424019	−	0.905653i	$-0.360619\pi$
$673$	22.0000	0.848038	0.424019	+	0.905653i	$0.360619\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	21.0000	0.807096	0.403548	−	0.914959i	$-0.367777\pi$
$677$	21.0000	0.807096	0.403548	+	0.914959i	$0.367777\pi$
$678$	0	0
$679$	13.0000	0.498894
$680$	0	0
$681$	18.0000	0.689761
$682$	0	0
$683$	−16.0000	−0.612223	−0.306111	−	0.951996i	$-0.599028\pi$
$683$	−16.0000	−0.612223	−0.306111	+	0.951996i	$0.599028\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−26.0000	−0.991962
$688$	0	0
$689$	−5.00000	−0.190485
$690$	0	0
$691$	−26.0000	−0.989087	−0.494543	−	0.869153i	$-0.664665\pi$
$691$	−26.0000	−0.989087	−0.494543	+	0.869153i	$0.664665\pi$
$692$	0	0
$693$	−5.00000	−0.189934
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−15.0000	−0.568166
$698$	0	0
$699$	−15.0000	−0.567352
$700$	0	0
$701$	−28.0000	−1.05755	−0.528773	−	0.848763i	$-0.677348\pi$
$701$	−28.0000	−1.05755	−0.528773	+	0.848763i	$0.677348\pi$
$702$	0	0
$703$	−66.0000	−2.48924
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−4.00000	−0.150435
$708$	0	0
$709$	52.0000	1.95290	0.976450	−	0.215742i	$-0.0692169\pi$
$709$	52.0000	1.95290	0.976450	+	0.215742i	$0.0692169\pi$
$710$	0	0
$711$	11.0000	0.412532
$712$	0	0
$713$	70.0000	2.62152
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−11.0000	−0.410803
$718$	0	0
$719$	−12.0000	−0.447524	−0.223762	−	0.974644i	$-0.571834\pi$
$719$	−12.0000	−0.447524	−0.223762	+	0.974644i	$0.571834\pi$
$720$	0	0
$721$	16.0000	0.595871
$722$	0	0
$723$	−22.0000	−0.818189
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−26.0000	−0.964287	−0.482143	−	0.876092i	$-0.660142\pi$
$727$	−26.0000	−0.964287	−0.482143	+	0.876092i	$0.660142\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−12.0000	−0.443836
$732$	0	0
$733$	31.0000	1.14501	0.572506	−	0.819901i	$-0.305971\pi$
$733$	31.0000	1.14501	0.572506	+	0.819901i	$0.305971\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−20.0000	−0.736709
$738$	0	0
$739$	−46.0000	−1.69214	−0.846069	−	0.533074i	$-0.821037\pi$
$739$	−46.0000	−1.69214	−0.846069	+	0.533074i	$0.821037\pi$
$740$	0	0
$741$	−6.00000	−0.220416
$742$	0	0
$743$	30.0000	1.10059	0.550297	−	0.834969i	$-0.314515\pi$
$743$	30.0000	1.10059	0.550297	+	0.834969i	$0.314515\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	8.00000	0.292705
$748$	0	0
$749$	1.00000	0.0365392
$750$	0	0
$751$	−3.00000	−0.109472	−0.0547358	−	0.998501i	$-0.517432\pi$
$751$	−3.00000	−0.109472	−0.0547358	+	0.998501i	$0.517432\pi$
$752$	0	0
$753$	−12.0000	−0.437304
$754$	0	0
$755$	0	0
$756$	0	0
$757$	8.00000	0.290765	0.145382	−	0.989376i	$-0.453559\pi$
$757$	8.00000	0.290765	0.145382	+	0.989376i	$0.453559\pi$
$758$	0	0
$759$	−35.0000	−1.27042
$760$	0	0
$761$	30.0000	1.08750	0.543750	−	0.839248i	$-0.317004\pi$
$761$	30.0000	1.08750	0.543750	+	0.839248i	$0.317004\pi$
$762$	0	0
$763$	20.0000	0.724049
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−20.0000	−0.721218	−0.360609	−	0.932717i	$-0.617431\pi$
$769$	−20.0000	−0.721218	−0.360609	+	0.932717i	$0.617431\pi$
$770$	0	0
$771$	6.00000	0.216085
$772$	0	0
$773$	8.00000	0.287740	0.143870	−	0.989597i	$-0.454045\pi$
$773$	8.00000	0.287740	0.143870	+	0.989597i	$0.454045\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−11.0000	−0.394623
$778$	0	0
$779$	−30.0000	−1.07486
$780$	0	0
$781$	25.0000	0.894570
$782$	0	0
$783$	6.00000	0.214423
$784$	0	0
$785$	0	0
$786$	0	0
$787$	28.0000	0.998092	0.499046	−	0.866575i	$-0.333684\pi$
$787$	28.0000	0.998092	0.499046	+	0.866575i	$0.333684\pi$
$788$	0	0
$789$	16.0000	0.569615
$790$	0	0
$791$	−6.00000	−0.213335
$792$	0	0
$793$	−1.00000	−0.0355110
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−9.00000	−0.318796	−0.159398	−	0.987214i	$-0.550955\pi$
$797$	−9.00000	−0.318796	−0.159398	+	0.987214i	$0.550955\pi$
$798$	0	0
$799$	−6.00000	−0.212265
$800$	0	0
$801$	7.00000	0.247333
$802$	0	0
$803$	30.0000	1.05868
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−4.00000	−0.140807
$808$	0	0
$809$	−46.0000	−1.61727	−0.808637	−	0.588308i	$-0.799794\pi$
$809$	−46.0000	−1.61727	−0.808637	+	0.588308i	$0.799794\pi$
$810$	0	0
$811$	−12.0000	−0.421377	−0.210688	−	0.977553i	$-0.567571\pi$
$811$	−12.0000	−0.421377	−0.210688	+	0.977553i	$0.567571\pi$
$812$	0	0
$813$	2.00000	0.0701431
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−24.0000	−0.839654
$818$	0	0
$819$	−1.00000	−0.0349428
$820$	0	0
$821$	−15.0000	−0.523504	−0.261752	−	0.965135i	$-0.584300\pi$
$821$	−15.0000	−0.523504	−0.261752	+	0.965135i	$0.584300\pi$
$822$	0	0
$823$	−56.0000	−1.95204	−0.976019	−	0.217687i	$-0.930149\pi$
$823$	−56.0000	−1.95204	−0.976019	+	0.217687i	$0.930149\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−10.0000	−0.347734	−0.173867	−	0.984769i	$-0.555626\pi$
$827$	−10.0000	−0.347734	−0.173867	+	0.984769i	$0.555626\pi$
$828$	0	0
$829$	−2.00000	−0.0694629	−0.0347314	−	0.999397i	$-0.511058\pi$
$829$	−2.00000	−0.0694629	−0.0347314	+	0.999397i	$0.511058\pi$
$830$	0	0
$831$	−10.0000	−0.346896
$832$	0	0
$833$	18.0000	0.623663
$834$	0	0
$835$	0	0
$836$	0	0
$837$	10.0000	0.345651
$838$	0	0
$839$	35.0000	1.20833	0.604167	−	0.796858i	$-0.293506\pi$
$839$	35.0000	1.20833	0.604167	+	0.796858i	$0.293506\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	−2.00000	−0.0688837
$844$	0	0
$845$	0	0
$846$	0	0
$847$	14.0000	0.481046
$848$	0	0
$849$	−16.0000	−0.549119
$850$	0	0
$851$	−77.0000	−2.63953
$852$	0	0
$853$	5.00000	0.171197	0.0855984	−	0.996330i	$-0.472720\pi$
$853$	5.00000	0.171197	0.0855984	+	0.996330i	$0.472720\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−33.0000	−1.12726	−0.563629	−	0.826028i	$-0.690595\pi$
$857$	−33.0000	−1.12726	−0.563629	+	0.826028i	$0.690595\pi$
$858$	0	0
$859$	1.00000	0.0341196	0.0170598	−	0.999854i	$-0.494569\pi$
$859$	1.00000	0.0341196	0.0170598	+	0.999854i	$0.494569\pi$
$860$	0	0
$861$	−5.00000	−0.170400
$862$	0	0
$863$	14.0000	0.476566	0.238283	−	0.971196i	$-0.423415\pi$
$863$	14.0000	0.476566	0.238283	+	0.971196i	$0.423415\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	8.00000	0.271694
$868$	0	0
$869$	−55.0000	−1.86575
$870$	0	0
$871$	−4.00000	−0.135535
$872$	0	0
$873$	13.0000	0.439983
$874$	0	0
$875$	0	0
$876$	0	0
$877$	50.0000	1.68838	0.844190	−	0.536044i	$-0.180082\pi$
$877$	50.0000	1.68838	0.844190	+	0.536044i	$0.180082\pi$
$878$	0	0
$879$	20.0000	0.674583
$880$	0	0
$881$	−54.0000	−1.81931	−0.909653	−	0.415369i	$-0.863653\pi$
$881$	−54.0000	−1.81931	−0.909653	+	0.415369i	$0.863653\pi$
$882$	0	0
$883$	−8.00000	−0.269221	−0.134611	−	0.990899i	$-0.542978\pi$
$883$	−8.00000	−0.269221	−0.134611	+	0.990899i	$0.542978\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	31.0000	1.04088	0.520439	−	0.853899i	$-0.325768\pi$
$887$	31.0000	1.04088	0.520439	+	0.853899i	$0.325768\pi$
$888$	0	0
$889$	14.0000	0.469545
$890$	0	0
$891$	−5.00000	−0.167506
$892$	0	0
$893$	−12.0000	−0.401565
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−7.00000	−0.233723
$898$	0	0
$899$	60.0000	2.00111
$900$	0	0
$901$	−15.0000	−0.499722
$902$	0	0
$903$	−4.00000	−0.133112
$904$	0	0
$905$	0	0
$906$	0	0
$907$	2.00000	0.0664089	0.0332045	−	0.999449i	$-0.489429\pi$
$907$	2.00000	0.0664089	0.0332045	+	0.999449i	$0.489429\pi$
$908$	0	0
$909$	−4.00000	−0.132672
$910$	0	0
$911$	28.0000	0.927681	0.463841	−	0.885919i	$-0.346471\pi$
$911$	28.0000	0.927681	0.463841	+	0.885919i	$0.346471\pi$
$912$	0	0
$913$	−40.0000	−1.32381
$914$	0	0
$915$	0	0
$916$	0	0
$917$	6.00000	0.198137
$918$	0	0
$919$	−5.00000	−0.164935	−0.0824674	−	0.996594i	$-0.526280\pi$
$919$	−5.00000	−0.164935	−0.0824674	+	0.996594i	$0.526280\pi$
$920$	0	0
$921$	−21.0000	−0.691974
$922$	0	0
$923$	5.00000	0.164577
$924$	0	0
$925$	0	0
$926$	0	0
$927$	16.0000	0.525509
$928$	0	0
$929$	−41.0000	−1.34517	−0.672583	−	0.740022i	$-0.734815\pi$
$929$	−41.0000	−1.34517	−0.672583	+	0.740022i	$0.734815\pi$
$930$	0	0
$931$	36.0000	1.17985
$932$	0	0
$933$	24.0000	0.785725
$934$	0	0
$935$	0	0
$936$	0	0
$937$	14.0000	0.457360	0.228680	−	0.973502i	$-0.426559\pi$
$937$	14.0000	0.457360	0.228680	+	0.973502i	$0.426559\pi$
$938$	0	0
$939$	−14.0000	−0.456873
$940$	0	0
$941$	31.0000	1.01057	0.505286	−	0.862952i	$-0.331387\pi$
$941$	31.0000	1.01057	0.505286	+	0.862952i	$0.331387\pi$
$942$	0	0
$943$	−35.0000	−1.13976
$944$	0	0
$945$	0	0
$946$	0	0
$947$	24.0000	0.779895	0.389948	−	0.920837i	$-0.372493\pi$
$947$	24.0000	0.779895	0.389948	+	0.920837i	$0.372493\pi$
$948$	0	0
$949$	6.00000	0.194768
$950$	0	0
$951$	−28.0000	−0.907962
$952$	0	0
$953$	−27.0000	−0.874616	−0.437308	−	0.899312i	$-0.644068\pi$
$953$	−27.0000	−0.874616	−0.437308	+	0.899312i	$0.644068\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−30.0000	−0.969762
$958$	0	0
$959$	−22.0000	−0.710417
$960$	0	0
$961$	69.0000	2.22581
$962$	0	0
$963$	1.00000	0.0322245
$964$	0	0
$965$	0	0
$966$	0	0
$967$	40.0000	1.28631	0.643157	−	0.765735i	$-0.277624\pi$
$967$	40.0000	1.28631	0.643157	+	0.765735i	$0.277624\pi$
$968$	0	0
$969$	−18.0000	−0.578243
$970$	0	0
$971$	−36.0000	−1.15529	−0.577647	−	0.816286i	$-0.696029\pi$
$971$	−36.0000	−1.15529	−0.577647	+	0.816286i	$0.696029\pi$
$972$	0	0
$973$	13.0000	0.416761
$974$	0	0
$975$	0	0
$976$	0	0
$977$	60.0000	1.91957	0.959785	−	0.280736i	$-0.0905785\pi$
$977$	60.0000	1.91957	0.959785	+	0.280736i	$0.0905785\pi$
$978$	0	0
$979$	−35.0000	−1.11860
$980$	0	0
$981$	20.0000	0.638551
$982$	0	0
$983$	20.0000	0.637901	0.318950	−	0.947771i	$-0.396670\pi$
$983$	20.0000	0.637901	0.318950	+	0.947771i	$0.396670\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−2.00000	−0.0636607
$988$	0	0
$989$	−28.0000	−0.890348
$990$	0	0
$991$	41.0000	1.30241	0.651204	−	0.758903i	$-0.274264\pi$
$991$	41.0000	1.30241	0.651204	+	0.758903i	$0.274264\pi$
$992$	0	0
$993$	−28.0000	−0.888553
$994$	0	0
$995$	0	0
$996$	0	0
$997$	20.0000	0.633406	0.316703	−	0.948525i	$-0.397424\pi$
$997$	20.0000	0.633406	0.316703	+	0.948525i	$0.397424\pi$
$998$	0	0
$999$	−11.0000	−0.348025

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7800.2.a.g.1.1		1
5.4	even	2		1560.2.a.h.1.1	✓	1
15.14	odd	2		4680.2.a.o.1.1		1
20.19	odd	2		3120.2.a.f.1.1		1
60.59	even	2		9360.2.a.br.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1560.2.a.h.1.1	✓	1	5.4	even	2
3120.2.a.f.1.1		1	20.19	odd	2
4680.2.a.o.1.1		1	15.14	odd	2
7800.2.a.g.1.1		1	1.1	even	1	trivial
9360.2.a.br.1.1		1	60.59	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)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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