LMFDB - Embedded newform 2240.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2240,2,Mod(1,2240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2240, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2240.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2240 = 2^{6} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2240.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:	$17.8864900528$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1120)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2240.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{3} -1.00000 q^{5} -1.00000 q^{7} -2.00000 q^{9} +O(q^{10})$

$q-1.00000 q^{3} -1.00000 q^{5} -1.00000 q^{7} -2.00000 q^{9} -5.00000 q^{11} -5.00000 q^{13} +1.00000 q^{15} -5.00000 q^{17} +1.00000 q^{21} +8.00000 q^{23} +1.00000 q^{25} +5.00000 q^{27} +1.00000 q^{29} +2.00000 q^{31} +5.00000 q^{33} +1.00000 q^{35} -4.00000 q^{37} +5.00000 q^{39} +2.00000 q^{41} -4.00000 q^{43} +2.00000 q^{45} +13.0000 q^{47} +1.00000 q^{49} +5.00000 q^{51} +8.00000 q^{53} +5.00000 q^{55} -4.00000 q^{59} +2.00000 q^{61} +2.00000 q^{63} +5.00000 q^{65} +8.00000 q^{67} -8.00000 q^{69} -12.0000 q^{71} -6.00000 q^{73} -1.00000 q^{75} +5.00000 q^{77} +11.0000 q^{79} +1.00000 q^{81} -8.00000 q^{83} +5.00000 q^{85} -1.00000 q^{87} -18.0000 q^{89} +5.00000 q^{91} -2.00000 q^{93} -5.00000 q^{97} +10.0000 q^{99} +O(q^{100})$

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	−0.288675	−	0.957427i	$-0.593215\pi$
$3$	−1.00000	−0.577350	−0.288675	+	0.957427i	$0.593215\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−2.00000	−0.666667
$10$	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
$13$	−5.00000	−1.38675	−0.693375	−	0.720577i	$-0.743877\pi$
$13$	−5.00000	−1.38675	−0.693375	+	0.720577i	$0.743877\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−5.00000	−1.21268	−0.606339	−	0.795206i	$-0.707363\pi$
$17$	−5.00000	−1.21268	−0.606339	+	0.795206i	$0.707363\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	8.00000	1.66812	0.834058	−	0.551677i	$-0.186012\pi$
$23$	8.00000	1.66812	0.834058	+	0.551677i	$0.186012\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.00000	0.962250
$28$	0	0
$29$	1.00000	0.185695	0.0928477	−	0.995680i	$-0.470403\pi$
$29$	1.00000	0.185695	0.0928477	+	0.995680i	$0.470403\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	0	0
$33$	5.00000	0.870388
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000	−0.657596	−0.328798	+	0.944400i	$0.606644\pi$
$38$	0	0
$39$	5.00000	0.800641
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	0	0
$45$	2.00000	0.298142
$46$	0	0
$47$	13.0000	1.89624	0.948122	−	0.317905i	$-0.102979\pi$
$47$	13.0000	1.89624	0.948122	+	0.317905i	$0.102979\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	5.00000	0.700140
$52$	0	0
$53$	8.00000	1.09888	0.549442	−	0.835532i	$-0.314840\pi$
$53$	8.00000	1.09888	0.549442	+	0.835532i	$0.314840\pi$
$54$	0	0
$55$	5.00000	0.674200
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	0	0
$63$	2.00000	0.251976
$64$	0	0
$65$	5.00000	0.620174
$66$	0	0
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	0	0
$69$	−8.00000	−0.963087
$70$	0	0
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	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
$75$	−1.00000	−0.115470
$76$	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	$-0.644687\pi$
$83$	−8.00000	−0.878114	−0.439057	+	0.898459i	$0.644687\pi$
$84$	0	0
$85$	5.00000	0.542326
$86$	0	0
$87$	−1.00000	−0.107211
$88$	0	0
$89$	−18.0000	−1.90800	−0.953998	−	0.299813i	$-0.903076\pi$
$89$	−18.0000	−1.90800	−0.953998	+	0.299813i	$0.903076\pi$
$90$	0	0
$91$	5.00000	0.524142
$92$	0	0
$93$	−2.00000	−0.207390
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−5.00000	−0.507673	−0.253837	−	0.967247i	$-0.581693\pi$
$97$	−5.00000	−0.507673	−0.253837	+	0.967247i	$0.581693\pi$
$98$	0	0
$99$	10.0000	1.00504
$100$	0	0
$101$	−12.0000	−1.19404	−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000	−1.19404	−0.597022	+	0.802225i	$0.703650\pi$
$102$	0	0
$103$	5.00000	0.492665	0.246332	−	0.969185i	$-0.420775\pi$
$103$	5.00000	0.492665	0.246332	+	0.969185i	$0.420775\pi$
$104$	0	0
$105$	−1.00000	−0.0975900
$106$	0	0
$107$	−2.00000	−0.193347	−0.0966736	−	0.995316i	$-0.530820\pi$
$107$	−2.00000	−0.193347	−0.0966736	+	0.995316i	$0.530820\pi$
$108$	0	0
$109$	11.0000	1.05361	0.526804	−	0.849987i	$-0.323390\pi$
$109$	11.0000	1.05361	0.526804	+	0.849987i	$0.323390\pi$
$110$	0	0
$111$	4.00000	0.379663
$112$	0	0
$113$	−12.0000	−1.12887	−0.564433	−	0.825479i	$-0.690905\pi$
$113$	−12.0000	−1.12887	−0.564433	+	0.825479i	$0.690905\pi$
$114$	0	0
$115$	−8.00000	−0.746004
$116$	0	0
$117$	10.0000	0.924500
$118$	0	0
$119$	5.00000	0.458349
$120$	0	0
$121$	14.0000	1.27273
$122$	0	0
$123$	−2.00000	−0.180334
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	20.0000	1.77471	0.887357	−	0.461084i	$-0.152539\pi$
$127$	20.0000	1.77471	0.887357	+	0.461084i	$0.152539\pi$
$128$	0	0
$129$	4.00000	0.352180
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−5.00000	−0.430331
$136$	0	0
$137$	22.0000	1.87959	0.939793	−	0.341743i	$-0.111017\pi$
$137$	22.0000	1.87959	0.939793	+	0.341743i	$0.111017\pi$
$138$	0	0
$139$	10.0000	0.848189	0.424094	−	0.905618i	$-0.360592\pi$
$139$	10.0000	0.848189	0.424094	+	0.905618i	$0.360592\pi$
$140$	0	0
$141$	−13.0000	−1.09480
$142$	0	0
$143$	25.0000	2.09061
$144$	0	0
$145$	−1.00000	−0.0830455
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	−18.0000	−1.47462	−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000	−1.47462	−0.737309	+	0.675556i	$0.763904\pi$
$150$	0	0
$151$	−13.0000	−1.05792	−0.528962	−	0.848645i	$-0.677419\pi$
$151$	−13.0000	−1.05792	−0.528962	+	0.848645i	$0.677419\pi$
$152$	0	0
$153$	10.0000	0.808452
$154$	0	0
$155$	−2.00000	−0.160644
$156$	0	0
$157$	18.0000	1.43656	0.718278	−	0.695756i	$-0.244931\pi$
$157$	18.0000	1.43656	0.718278	+	0.695756i	$0.244931\pi$
$158$	0	0
$159$	−8.00000	−0.634441
$160$	0	0
$161$	−8.00000	−0.630488
$162$	0	0
$163$	2.00000	0.156652	0.0783260	−	0.996928i	$-0.475042\pi$
$163$	2.00000	0.156652	0.0783260	+	0.996928i	$0.475042\pi$
$164$	0	0
$165$	−5.00000	−0.389249
$166$	0	0
$167$	−19.0000	−1.47026	−0.735132	−	0.677924i	$-0.762880\pi$
$167$	−19.0000	−1.47026	−0.735132	+	0.677924i	$0.762880\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	0	0
$171$	0	0
$172$	0	0
$173$	9.00000	0.684257	0.342129	−	0.939653i	$-0.388852\pi$
$173$	9.00000	0.684257	0.342129	+	0.939653i	$0.388852\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	4.00000	0.300658
$178$	0	0
$179$	24.0000	1.79384	0.896922	−	0.442189i	$-0.145798\pi$
$179$	24.0000	1.79384	0.896922	+	0.442189i	$0.145798\pi$
$180$	0	0
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	0	0
$183$	−2.00000	−0.147844
$184$	0	0
$185$	4.00000	0.294086
$186$	0	0
$187$	25.0000	1.82818
$188$	0	0
$189$	−5.00000	−0.363696
$190$	0	0
$191$	−15.0000	−1.08536	−0.542681	−	0.839939i	$-0.682591\pi$
$191$	−15.0000	−1.08536	−0.542681	+	0.839939i	$0.682591\pi$
$192$	0	0
$193$	−10.0000	−0.719816	−0.359908	−	0.932988i	$-0.617192\pi$
$193$	−10.0000	−0.719816	−0.359908	+	0.932988i	$0.617192\pi$
$194$	0	0
$195$	−5.00000	−0.358057
$196$	0	0
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	0	0
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	0	0
$201$	−8.00000	−0.564276
$202$	0	0
$203$	−1.00000	−0.0701862
$204$	0	0
$205$	−2.00000	−0.139686
$206$	0	0
$207$	−16.0000	−1.11208
$208$	0	0
$209$	0	0
$210$	0	0
$211$	13.0000	0.894957	0.447478	−	0.894295i	$-0.352322\pi$
$211$	13.0000	0.894957	0.447478	+	0.894295i	$0.352322\pi$
$212$	0	0
$213$	12.0000	0.822226
$214$	0	0
$215$	4.00000	0.272798
$216$	0	0
$217$	−2.00000	−0.135769
$218$	0	0
$219$	6.00000	0.405442
$220$	0	0
$221$	25.0000	1.68168
$222$	0	0
$223$	−3.00000	−0.200895	−0.100447	−	0.994942i	$-0.532027\pi$
$223$	−3.00000	−0.200895	−0.100447	+	0.994942i	$0.532027\pi$
$224$	0	0
$225$	−2.00000	−0.133333
$226$	0	0
$227$	7.00000	0.464606	0.232303	−	0.972643i	$-0.425374\pi$
$227$	7.00000	0.464606	0.232303	+	0.972643i	$0.425374\pi$
$228$	0	0
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	0	0
$231$	−5.00000	−0.328976
$232$	0	0
$233$	24.0000	1.57229	0.786146	−	0.618041i	$-0.212073\pi$
$233$	24.0000	1.57229	0.786146	+	0.618041i	$0.212073\pi$
$234$	0	0
$235$	−13.0000	−0.848026
$236$	0	0
$237$	−11.0000	−0.714527
$238$	0	0
$239$	3.00000	0.194054	0.0970269	−	0.995282i	$-0.469067\pi$
$239$	3.00000	0.194054	0.0970269	+	0.995282i	$0.469067\pi$
$240$	0	0
$241$	−16.0000	−1.03065	−0.515325	−	0.856995i	$-0.672329\pi$
$241$	−16.0000	−1.03065	−0.515325	+	0.856995i	$0.672329\pi$
$242$	0	0
$243$	−16.0000	−1.02640
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	0	0
$248$	0	0
$249$	8.00000	0.506979
$250$	0	0
$251$	−26.0000	−1.64111	−0.820553	−	0.571571i	$-0.806334\pi$
$251$	−26.0000	−1.64111	−0.820553	+	0.571571i	$0.806334\pi$
$252$	0	0
$253$	−40.0000	−2.51478
$254$	0	0
$255$	−5.00000	−0.313112
$256$	0	0
$257$	14.0000	0.873296	0.436648	−	0.899632i	$-0.356166\pi$
$257$	14.0000	0.873296	0.436648	+	0.899632i	$0.356166\pi$
$258$	0	0
$259$	4.00000	0.248548
$260$	0	0
$261$	−2.00000	−0.123797
$262$	0	0
$263$	14.0000	0.863277	0.431638	−	0.902047i	$-0.357936\pi$
$263$	14.0000	0.863277	0.431638	+	0.902047i	$0.357936\pi$
$264$	0	0
$265$	−8.00000	−0.491436
$266$	0	0
$267$	18.0000	1.10158
$268$	0	0
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	0	0
$271$	−22.0000	−1.33640	−0.668202	−	0.743980i	$-0.732936\pi$
$271$	−22.0000	−1.33640	−0.668202	+	0.743980i	$0.732936\pi$
$272$	0	0
$273$	−5.00000	−0.302614
$274$	0	0
$275$	−5.00000	−0.301511
$276$	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
$279$	−4.00000	−0.239474
$280$	0	0
$281$	−9.00000	−0.536895	−0.268447	−	0.963294i	$-0.586511\pi$
$281$	−9.00000	−0.536895	−0.268447	+	0.963294i	$0.586511\pi$
$282$	0	0
$283$	17.0000	1.01055	0.505273	−	0.862960i	$-0.331392\pi$
$283$	17.0000	1.01055	0.505273	+	0.862960i	$0.331392\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2.00000	−0.118056
$288$	0	0
$289$	8.00000	0.470588
$290$	0	0
$291$	5.00000	0.293105
$292$	0	0
$293$	−11.0000	−0.642627	−0.321313	−	0.946973i	$-0.604124\pi$
$293$	−11.0000	−0.642627	−0.321313	+	0.946973i	$0.604124\pi$
$294$	0	0
$295$	4.00000	0.232889
$296$	0	0
$297$	−25.0000	−1.45065
$298$	0	0
$299$	−40.0000	−2.31326
$300$	0	0
$301$	4.00000	0.230556
$302$	0	0
$303$	12.0000	0.689382
$304$	0	0
$305$	−2.00000	−0.114520
$306$	0	0
$307$	29.0000	1.65512	0.827559	−	0.561379i	$-0.189729\pi$
$307$	29.0000	1.65512	0.827559	+	0.561379i	$0.189729\pi$
$308$	0	0
$309$	−5.00000	−0.284440
$310$	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
$313$	1.00000	0.0565233	0.0282617	−	0.999601i	$-0.491003\pi$
$313$	1.00000	0.0565233	0.0282617	+	0.999601i	$0.491003\pi$
$314$	0	0
$315$	−2.00000	−0.112687
$316$	0	0
$317$	−2.00000	−0.112331	−0.0561656	−	0.998421i	$-0.517887\pi$
$317$	−2.00000	−0.112331	−0.0561656	+	0.998421i	$0.517887\pi$
$318$	0	0
$319$	−5.00000	−0.279946
$320$	0	0
$321$	2.00000	0.111629
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−5.00000	−0.277350
$326$	0	0
$327$	−11.0000	−0.608301
$328$	0	0
$329$	−13.0000	−0.716713
$330$	0	0
$331$	12.0000	0.659580	0.329790	−	0.944054i	$-0.393022\pi$
$331$	12.0000	0.659580	0.329790	+	0.944054i	$0.393022\pi$
$332$	0	0
$333$	8.00000	0.438397
$334$	0	0
$335$	−8.00000	−0.437087
$336$	0	0
$337$	18.0000	0.980522	0.490261	−	0.871576i	$-0.336901\pi$
$337$	18.0000	0.980522	0.490261	+	0.871576i	$0.336901\pi$
$338$	0	0
$339$	12.0000	0.651751
$340$	0	0
$341$	−10.0000	−0.541530
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	8.00000	0.430706
$346$	0	0
$347$	−24.0000	−1.28839	−0.644194	−	0.764862i	$-0.722807\pi$
$347$	−24.0000	−1.28839	−0.644194	+	0.764862i	$0.722807\pi$
$348$	0	0
$349$	−20.0000	−1.07058	−0.535288	−	0.844670i	$-0.679797\pi$
$349$	−20.0000	−1.07058	−0.535288	+	0.844670i	$0.679797\pi$
$350$	0	0
$351$	−25.0000	−1.33440
$352$	0	0
$353$	−9.00000	−0.479022	−0.239511	−	0.970894i	$-0.576987\pi$
$353$	−9.00000	−0.479022	−0.239511	+	0.970894i	$0.576987\pi$
$354$	0	0
$355$	12.0000	0.636894
$356$	0	0
$357$	−5.00000	−0.264628
$358$	0	0
$359$	8.00000	0.422224	0.211112	−	0.977462i	$-0.432292\pi$
$359$	8.00000	0.422224	0.211112	+	0.977462i	$0.432292\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	−14.0000	−0.734809
$364$	0	0
$365$	6.00000	0.314054
$366$	0	0
$367$	17.0000	0.887393	0.443696	−	0.896177i	$-0.353667\pi$
$367$	17.0000	0.887393	0.443696	+	0.896177i	$0.353667\pi$
$368$	0	0
$369$	−4.00000	−0.208232
$370$	0	0
$371$	−8.00000	−0.415339
$372$	0	0
$373$	6.00000	0.310668	0.155334	−	0.987862i	$-0.450355\pi$
$373$	6.00000	0.310668	0.155334	+	0.987862i	$0.450355\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	−5.00000	−0.257513
$378$	0	0
$379$	8.00000	0.410932	0.205466	−	0.978664i	$-0.434129\pi$
$379$	8.00000	0.410932	0.205466	+	0.978664i	$0.434129\pi$
$380$	0	0
$381$	−20.0000	−1.02463
$382$	0	0
$383$	20.0000	1.02195	0.510976	−	0.859595i	$-0.329284\pi$
$383$	20.0000	1.02195	0.510976	+	0.859595i	$0.329284\pi$
$384$	0	0
$385$	−5.00000	−0.254824
$386$	0	0
$387$	8.00000	0.406663
$388$	0	0
$389$	15.0000	0.760530	0.380265	−	0.924878i	$-0.375833\pi$
$389$	15.0000	0.760530	0.380265	+	0.924878i	$0.375833\pi$
$390$	0	0
$391$	−40.0000	−2.02289
$392$	0	0
$393$	−12.0000	−0.605320
$394$	0	0
$395$	−11.0000	−0.553470
$396$	0	0
$397$	13.0000	0.652451	0.326226	−	0.945292i	$-0.394223\pi$
$397$	13.0000	0.652451	0.326226	+	0.945292i	$0.394223\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	37.0000	1.84769	0.923846	−	0.382765i	$-0.125028\pi$
$401$	37.0000	1.84769	0.923846	+	0.382765i	$0.125028\pi$
$402$	0	0
$403$	−10.0000	−0.498135
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	20.0000	0.991363
$408$	0	0
$409$	−26.0000	−1.28562	−0.642809	−	0.766027i	$-0.722231\pi$
$409$	−26.0000	−1.28562	−0.642809	+	0.766027i	$0.722231\pi$
$410$	0	0
$411$	−22.0000	−1.08518
$412$	0	0
$413$	4.00000	0.196827
$414$	0	0
$415$	8.00000	0.392705
$416$	0	0
$417$	−10.0000	−0.489702
$418$	0	0
$419$	−24.0000	−1.17248	−0.586238	−	0.810139i	$-0.699392\pi$
$419$	−24.0000	−1.17248	−0.586238	+	0.810139i	$0.699392\pi$
$420$	0	0
$421$	39.0000	1.90074	0.950372	−	0.311116i	$-0.100703\pi$
$421$	39.0000	1.90074	0.950372	+	0.311116i	$0.100703\pi$
$422$	0	0
$423$	−26.0000	−1.26416
$424$	0	0
$425$	−5.00000	−0.242536
$426$	0	0
$427$	−2.00000	−0.0967868
$428$	0	0
$429$	−25.0000	−1.20701
$430$	0	0
$431$	5.00000	0.240842	0.120421	−	0.992723i	$-0.461576\pi$
$431$	5.00000	0.240842	0.120421	+	0.992723i	$0.461576\pi$
$432$	0	0
$433$	−14.0000	−0.672797	−0.336399	−	0.941720i	$-0.609209\pi$
$433$	−14.0000	−0.672797	−0.336399	+	0.941720i	$0.609209\pi$
$434$	0	0
$435$	1.00000	0.0479463
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−6.00000	−0.286364	−0.143182	−	0.989696i	$-0.545733\pi$
$439$	−6.00000	−0.286364	−0.143182	+	0.989696i	$0.545733\pi$
$440$	0	0
$441$	−2.00000	−0.0952381
$442$	0	0
$443$	−36.0000	−1.71041	−0.855206	−	0.518289i	$-0.826569\pi$
$443$	−36.0000	−1.71041	−0.855206	+	0.518289i	$0.826569\pi$
$444$	0	0
$445$	18.0000	0.853282
$446$	0	0
$447$	18.0000	0.851371
$448$	0	0
$449$	−9.00000	−0.424736	−0.212368	−	0.977190i	$-0.568118\pi$
$449$	−9.00000	−0.424736	−0.212368	+	0.977190i	$0.568118\pi$
$450$	0	0
$451$	−10.0000	−0.470882
$452$	0	0
$453$	13.0000	0.610793
$454$	0	0
$455$	−5.00000	−0.234404
$456$	0	0
$457$	34.0000	1.59045	0.795226	−	0.606313i	$-0.207352\pi$
$457$	34.0000	1.59045	0.795226	+	0.606313i	$0.207352\pi$
$458$	0	0
$459$	−25.0000	−1.16690
$460$	0	0
$461$	8.00000	0.372597	0.186299	−	0.982493i	$-0.440351\pi$
$461$	8.00000	0.372597	0.186299	+	0.982493i	$0.440351\pi$
$462$	0	0
$463$	18.0000	0.836531	0.418265	−	0.908325i	$-0.362638\pi$
$463$	18.0000	0.836531	0.418265	+	0.908325i	$0.362638\pi$
$464$	0	0
$465$	2.00000	0.0927478
$466$	0	0
$467$	33.0000	1.52706	0.763529	−	0.645774i	$-0.223465\pi$
$467$	33.0000	1.52706	0.763529	+	0.645774i	$0.223465\pi$
$468$	0	0
$469$	−8.00000	−0.369406
$470$	0	0
$471$	−18.0000	−0.829396
$472$	0	0
$473$	20.0000	0.919601
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−16.0000	−0.732590
$478$	0	0
$479$	−12.0000	−0.548294	−0.274147	−	0.961688i	$-0.588395\pi$
$479$	−12.0000	−0.548294	−0.274147	+	0.961688i	$0.588395\pi$
$480$	0	0
$481$	20.0000	0.911922
$482$	0	0
$483$	8.00000	0.364013
$484$	0	0
$485$	5.00000	0.227038
$486$	0	0
$487$	18.0000	0.815658	0.407829	−	0.913058i	$-0.366286\pi$
$487$	18.0000	0.815658	0.407829	+	0.913058i	$0.366286\pi$
$488$	0	0
$489$	−2.00000	−0.0904431
$490$	0	0
$491$	−35.0000	−1.57953	−0.789764	−	0.613411i	$-0.789797\pi$
$491$	−35.0000	−1.57953	−0.789764	+	0.613411i	$0.789797\pi$
$492$	0	0
$493$	−5.00000	−0.225189
$494$	0	0
$495$	−10.0000	−0.449467
$496$	0	0
$497$	12.0000	0.538274
$498$	0	0
$499$	35.0000	1.56682	0.783408	−	0.621508i	$-0.213480\pi$
$499$	35.0000	1.56682	0.783408	+	0.621508i	$0.213480\pi$
$500$	0	0
$501$	19.0000	0.848857
$502$	0	0
$503$	11.0000	0.490466	0.245233	−	0.969464i	$-0.421136\pi$
$503$	11.0000	0.490466	0.245233	+	0.969464i	$0.421136\pi$
$504$	0	0
$505$	12.0000	0.533993
$506$	0	0
$507$	−12.0000	−0.532939
$508$	0	0
$509$	−2.00000	−0.0886484	−0.0443242	−	0.999017i	$-0.514113\pi$
$509$	−2.00000	−0.0886484	−0.0443242	+	0.999017i	$0.514113\pi$
$510$	0	0
$511$	6.00000	0.265424
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−5.00000	−0.220326
$516$	0	0
$517$	−65.0000	−2.85870
$518$	0	0
$519$	−9.00000	−0.395056
$520$	0	0
$521$	−4.00000	−0.175243	−0.0876216	−	0.996154i	$-0.527927\pi$
$521$	−4.00000	−0.175243	−0.0876216	+	0.996154i	$0.527927\pi$
$522$	0	0
$523$	−4.00000	−0.174908	−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000	−0.174908	−0.0874539	+	0.996169i	$0.527873\pi$
$524$	0	0
$525$	1.00000	0.0436436
$526$	0	0
$527$	−10.0000	−0.435607
$528$	0	0
$529$	41.0000	1.78261
$530$	0	0
$531$	8.00000	0.347170
$532$	0	0
$533$	−10.0000	−0.433148
$534$	0	0
$535$	2.00000	0.0864675
$536$	0	0
$537$	−24.0000	−1.03568
$538$	0	0
$539$	−5.00000	−0.215365
$540$	0	0
$541$	−39.0000	−1.67674	−0.838370	−	0.545101i	$-0.816491\pi$
$541$	−39.0000	−1.67674	−0.838370	+	0.545101i	$0.816491\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−11.0000	−0.471188
$546$	0	0
$547$	−22.0000	−0.940652	−0.470326	−	0.882493i	$-0.655864\pi$
$547$	−22.0000	−0.940652	−0.470326	+	0.882493i	$0.655864\pi$
$548$	0	0
$549$	−4.00000	−0.170716
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−11.0000	−0.467768
$554$	0	0
$555$	−4.00000	−0.169791
$556$	0	0
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	0	0
$559$	20.0000	0.845910
$560$	0	0
$561$	−25.0000	−1.05550
$562$	0	0
$563$	−28.0000	−1.18006	−0.590030	−	0.807382i	$-0.700884\pi$
$563$	−28.0000	−1.18006	−0.590030	+	0.807382i	$0.700884\pi$
$564$	0	0
$565$	12.0000	0.504844
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	4.00000	0.167395	0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000	0.167395	0.0836974	+	0.996491i	$0.473327\pi$
$572$	0	0
$573$	15.0000	0.626634
$574$	0	0
$575$	8.00000	0.333623
$576$	0	0
$577$	−35.0000	−1.45707	−0.728535	−	0.685009i	$-0.759798\pi$
$577$	−35.0000	−1.45707	−0.728535	+	0.685009i	$0.759798\pi$
$578$	0	0
$579$	10.0000	0.415586
$580$	0	0
$581$	8.00000	0.331896
$582$	0	0
$583$	−40.0000	−1.65663
$584$	0	0
$585$	−10.0000	−0.413449
$586$	0	0
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	6.00000	0.246807
$592$	0	0
$593$	−3.00000	−0.123195	−0.0615976	−	0.998101i	$-0.519620\pi$
$593$	−3.00000	−0.123195	−0.0615976	+	0.998101i	$0.519620\pi$
$594$	0	0
$595$	−5.00000	−0.204980
$596$	0	0
$597$	8.00000	0.327418
$598$	0	0
$599$	−3.00000	−0.122577	−0.0612883	−	0.998120i	$-0.519521\pi$
$599$	−3.00000	−0.122577	−0.0612883	+	0.998120i	$0.519521\pi$
$600$	0	0
$601$	2.00000	0.0815817	0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000	0.0815817	0.0407909	+	0.999168i	$0.487012\pi$
$602$	0	0
$603$	−16.0000	−0.651570
$604$	0	0
$605$	−14.0000	−0.569181
$606$	0	0
$607$	27.0000	1.09590	0.547948	−	0.836512i	$-0.315409\pi$
$607$	27.0000	1.09590	0.547948	+	0.836512i	$0.315409\pi$
$608$	0	0
$609$	1.00000	0.0405220
$610$	0	0
$611$	−65.0000	−2.62962
$612$	0	0
$613$	−2.00000	−0.0807792	−0.0403896	−	0.999184i	$-0.512860\pi$
$613$	−2.00000	−0.0807792	−0.0403896	+	0.999184i	$0.512860\pi$
$614$	0	0
$615$	2.00000	0.0806478
$616$	0	0
$617$	10.0000	0.402585	0.201292	−	0.979531i	$-0.435486\pi$
$617$	10.0000	0.402585	0.201292	+	0.979531i	$0.435486\pi$
$618$	0	0
$619$	16.0000	0.643094	0.321547	−	0.946894i	$-0.395797\pi$
$619$	16.0000	0.643094	0.321547	+	0.946894i	$0.395797\pi$
$620$	0	0
$621$	40.0000	1.60514
$622$	0	0
$623$	18.0000	0.721155
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	20.0000	0.797452
$630$	0	0
$631$	29.0000	1.15447	0.577236	−	0.816577i	$-0.304131\pi$
$631$	29.0000	1.15447	0.577236	+	0.816577i	$0.304131\pi$
$632$	0	0
$633$	−13.0000	−0.516704
$634$	0	0
$635$	−20.0000	−0.793676
$636$	0	0
$637$	−5.00000	−0.198107
$638$	0	0
$639$	24.0000	0.949425
$640$	0	0
$641$	38.0000	1.50091	0.750455	−	0.660922i	$-0.229834\pi$
$641$	38.0000	1.50091	0.750455	+	0.660922i	$0.229834\pi$
$642$	0	0
$643$	−17.0000	−0.670415	−0.335207	−	0.942144i	$-0.608806\pi$
$643$	−17.0000	−0.670415	−0.335207	+	0.942144i	$0.608806\pi$
$644$	0	0
$645$	−4.00000	−0.157500
$646$	0	0
$647$	−12.0000	−0.471769	−0.235884	−	0.971781i	$-0.575799\pi$
$647$	−12.0000	−0.471769	−0.235884	+	0.971781i	$0.575799\pi$
$648$	0	0
$649$	20.0000	0.785069
$650$	0	0
$651$	2.00000	0.0783862
$652$	0	0
$653$	−28.0000	−1.09572	−0.547862	−	0.836569i	$-0.684558\pi$
$653$	−28.0000	−1.09572	−0.547862	+	0.836569i	$0.684558\pi$
$654$	0	0
$655$	−12.0000	−0.468879
$656$	0	0
$657$	12.0000	0.468165
$658$	0	0
$659$	−9.00000	−0.350590	−0.175295	−	0.984516i	$-0.556088\pi$
$659$	−9.00000	−0.350590	−0.175295	+	0.984516i	$0.556088\pi$
$660$	0	0
$661$	−36.0000	−1.40024	−0.700119	−	0.714026i	$-0.746870\pi$
$661$	−36.0000	−1.40024	−0.700119	+	0.714026i	$0.746870\pi$
$662$	0	0
$663$	−25.0000	−0.970920
$664$	0	0
$665$	0	0
$666$	0	0
$667$	8.00000	0.309761
$668$	0	0
$669$	3.00000	0.115987
$670$	0	0
$671$	−10.0000	−0.386046
$672$	0	0
$673$	30.0000	1.15642	0.578208	−	0.815890i	$-0.303752\pi$
$673$	30.0000	1.15642	0.578208	+	0.815890i	$0.303752\pi$
$674$	0	0
$675$	5.00000	0.192450
$676$	0	0
$677$	39.0000	1.49889	0.749446	−	0.662066i	$-0.230320\pi$
$677$	39.0000	1.49889	0.749446	+	0.662066i	$0.230320\pi$
$678$	0	0
$679$	5.00000	0.191882
$680$	0	0
$681$	−7.00000	−0.268241
$682$	0	0
$683$	30.0000	1.14792	0.573959	−	0.818884i	$-0.305407\pi$
$683$	30.0000	1.14792	0.573959	+	0.818884i	$0.305407\pi$
$684$	0	0
$685$	−22.0000	−0.840577
$686$	0	0
$687$	−14.0000	−0.534133
$688$	0	0
$689$	−40.0000	−1.52388
$690$	0	0
$691$	−32.0000	−1.21734	−0.608669	−	0.793424i	$-0.708296\pi$
$691$	−32.0000	−1.21734	−0.608669	+	0.793424i	$0.708296\pi$
$692$	0	0
$693$	−10.0000	−0.379869
$694$	0	0
$695$	−10.0000	−0.379322
$696$	0	0
$697$	−10.0000	−0.378777
$698$	0	0
$699$	−24.0000	−0.907763
$700$	0	0
$701$	9.00000	0.339925	0.169963	−	0.985451i	$-0.445635\pi$
$701$	9.00000	0.339925	0.169963	+	0.985451i	$0.445635\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	13.0000	0.489608
$706$	0	0
$707$	12.0000	0.451306
$708$	0	0
$709$	49.0000	1.84023	0.920117	−	0.391644i	$-0.128094\pi$
$709$	49.0000	1.84023	0.920117	+	0.391644i	$0.128094\pi$
$710$	0	0
$711$	−22.0000	−0.825064
$712$	0	0
$713$	16.0000	0.599205
$714$	0	0
$715$	−25.0000	−0.934947
$716$	0	0
$717$	−3.00000	−0.112037
$718$	0	0
$719$	24.0000	0.895049	0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000	0.895049	0.447524	+	0.894272i	$0.352306\pi$
$720$	0	0
$721$	−5.00000	−0.186210
$722$	0	0
$723$	16.0000	0.595046
$724$	0	0
$725$	1.00000	0.0371391
$726$	0	0
$727$	40.0000	1.48352	0.741759	−	0.670667i	$-0.233992\pi$
$727$	40.0000	1.48352	0.741759	+	0.670667i	$0.233992\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	20.0000	0.739727
$732$	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
$735$	1.00000	0.0368856
$736$	0	0
$737$	−40.0000	−1.47342
$738$	0	0
$739$	11.0000	0.404642	0.202321	−	0.979319i	$-0.435152\pi$
$739$	11.0000	0.404642	0.202321	+	0.979319i	$0.435152\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−44.0000	−1.61420	−0.807102	−	0.590412i	$-0.798965\pi$
$743$	−44.0000	−1.61420	−0.807102	+	0.590412i	$0.798965\pi$
$744$	0	0
$745$	18.0000	0.659469
$746$	0	0
$747$	16.0000	0.585409
$748$	0	0
$749$	2.00000	0.0730784
$750$	0	0
$751$	15.0000	0.547358	0.273679	−	0.961821i	$-0.411759\pi$
$751$	15.0000	0.547358	0.273679	+	0.961821i	$0.411759\pi$
$752$	0	0
$753$	26.0000	0.947493
$754$	0	0
$755$	13.0000	0.473118
$756$	0	0
$757$	−4.00000	−0.145382	−0.0726912	−	0.997354i	$-0.523159\pi$
$757$	−4.00000	−0.145382	−0.0726912	+	0.997354i	$0.523159\pi$
$758$	0	0
$759$	40.0000	1.45191
$760$	0	0
$761$	12.0000	0.435000	0.217500	−	0.976060i	$-0.430210\pi$
$761$	12.0000	0.435000	0.217500	+	0.976060i	$0.430210\pi$
$762$	0	0
$763$	−11.0000	−0.398227
$764$	0	0
$765$	−10.0000	−0.361551
$766$	0	0
$767$	20.0000	0.722158
$768$	0	0
$769$	18.0000	0.649097	0.324548	−	0.945869i	$-0.394788\pi$
$769$	18.0000	0.649097	0.324548	+	0.945869i	$0.394788\pi$
$770$	0	0
$771$	−14.0000	−0.504198
$772$	0	0
$773$	15.0000	0.539513	0.269756	−	0.962929i	$-0.413057\pi$
$773$	15.0000	0.539513	0.269756	+	0.962929i	$0.413057\pi$
$774$	0	0
$775$	2.00000	0.0718421
$776$	0	0
$777$	−4.00000	−0.143499
$778$	0	0
$779$	0	0
$780$	0	0
$781$	60.0000	2.14697
$782$	0	0
$783$	5.00000	0.178685
$784$	0	0
$785$	−18.0000	−0.642448
$786$	0	0
$787$	−17.0000	−0.605985	−0.302992	−	0.952993i	$-0.597986\pi$
$787$	−17.0000	−0.605985	−0.302992	+	0.952993i	$0.597986\pi$
$788$	0	0
$789$	−14.0000	−0.498413
$790$	0	0
$791$	12.0000	0.426671
$792$	0	0
$793$	−10.0000	−0.355110
$794$	0	0
$795$	8.00000	0.283731
$796$	0	0
$797$	−23.0000	−0.814702	−0.407351	−	0.913272i	$-0.633547\pi$
$797$	−23.0000	−0.814702	−0.407351	+	0.913272i	$0.633547\pi$
$798$	0	0
$799$	−65.0000	−2.29953
$800$	0	0
$801$	36.0000	1.27200
$802$	0	0
$803$	30.0000	1.05868
$804$	0	0
$805$	8.00000	0.281963
$806$	0	0
$807$	−18.0000	−0.633630
$808$	0	0
$809$	49.0000	1.72275	0.861374	−	0.507971i	$-0.169604\pi$
$809$	49.0000	1.72275	0.861374	+	0.507971i	$0.169604\pi$
$810$	0	0
$811$	2.00000	0.0702295	0.0351147	−	0.999383i	$-0.488820\pi$
$811$	2.00000	0.0702295	0.0351147	+	0.999383i	$0.488820\pi$
$812$	0	0
$813$	22.0000	0.771574
$814$	0	0
$815$	−2.00000	−0.0700569
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−10.0000	−0.349428
$820$	0	0
$821$	−25.0000	−0.872506	−0.436253	−	0.899824i	$-0.643695\pi$
$821$	−25.0000	−0.872506	−0.436253	+	0.899824i	$0.643695\pi$
$822$	0	0
$823$	2.00000	0.0697156	0.0348578	−	0.999392i	$-0.488902\pi$
$823$	2.00000	0.0697156	0.0348578	+	0.999392i	$0.488902\pi$
$824$	0	0
$825$	5.00000	0.174078
$826$	0	0
$827$	−18.0000	−0.625921	−0.312961	−	0.949766i	$-0.601321\pi$
$827$	−18.0000	−0.625921	−0.312961	+	0.949766i	$0.601321\pi$
$828$	0	0
$829$	30.0000	1.04194	0.520972	−	0.853574i	$-0.325570\pi$
$829$	30.0000	1.04194	0.520972	+	0.853574i	$0.325570\pi$
$830$	0	0
$831$	−10.0000	−0.346896
$832$	0	0
$833$	−5.00000	−0.173240
$834$	0	0
$835$	19.0000	0.657522
$836$	0	0
$837$	10.0000	0.345651
$838$	0	0
$839$	32.0000	1.10476	0.552381	−	0.833592i	$-0.313719\pi$
$839$	32.0000	1.10476	0.552381	+	0.833592i	$0.313719\pi$
$840$	0	0
$841$	−28.0000	−0.965517
$842$	0	0
$843$	9.00000	0.309976
$844$	0	0
$845$	−12.0000	−0.412813
$846$	0	0
$847$	−14.0000	−0.481046
$848$	0	0
$849$	−17.0000	−0.583438
$850$	0	0
$851$	−32.0000	−1.09695
$852$	0	0
$853$	6.00000	0.205436	0.102718	−	0.994711i	$-0.467246\pi$
$853$	6.00000	0.205436	0.102718	+	0.994711i	$0.467246\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−54.0000	−1.84460	−0.922302	−	0.386469i	$-0.873695\pi$
$857$	−54.0000	−1.84460	−0.922302	+	0.386469i	$0.873695\pi$
$858$	0	0
$859$	44.0000	1.50126	0.750630	−	0.660722i	$-0.229750\pi$
$859$	44.0000	1.50126	0.750630	+	0.660722i	$0.229750\pi$
$860$	0	0
$861$	2.00000	0.0681598
$862$	0	0
$863$	6.00000	0.204242	0.102121	−	0.994772i	$-0.467437\pi$
$863$	6.00000	0.204242	0.102121	+	0.994772i	$0.467437\pi$
$864$	0	0
$865$	−9.00000	−0.306009
$866$	0	0
$867$	−8.00000	−0.271694
$868$	0	0
$869$	−55.0000	−1.86575
$870$	0	0
$871$	−40.0000	−1.35535
$872$	0	0
$873$	10.0000	0.338449
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	44.0000	1.48577	0.742887	−	0.669417i	$-0.233456\pi$
$877$	44.0000	1.48577	0.742887	+	0.669417i	$0.233456\pi$
$878$	0	0
$879$	11.0000	0.371021
$880$	0	0
$881$	−56.0000	−1.88669	−0.943344	−	0.331816i	$-0.892339\pi$
$881$	−56.0000	−1.88669	−0.943344	+	0.331816i	$0.892339\pi$
$882$	0	0
$883$	34.0000	1.14419	0.572096	−	0.820187i	$-0.306131\pi$
$883$	34.0000	1.14419	0.572096	+	0.820187i	$0.306131\pi$
$884$	0	0
$885$	−4.00000	−0.134459
$886$	0	0
$887$	−8.00000	−0.268614	−0.134307	−	0.990940i	$-0.542881\pi$
$887$	−8.00000	−0.268614	−0.134307	+	0.990940i	$0.542881\pi$
$888$	0	0
$889$	−20.0000	−0.670778
$890$	0	0
$891$	−5.00000	−0.167506
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−24.0000	−0.802232
$896$	0	0
$897$	40.0000	1.33556
$898$	0	0
$899$	2.00000	0.0667037
$900$	0	0
$901$	−40.0000	−1.33259
$902$	0	0
$903$	−4.00000	−0.133112
$904$	0	0
$905$	0	0
$906$	0	0
$907$	16.0000	0.531271	0.265636	−	0.964073i	$-0.414418\pi$
$907$	16.0000	0.531271	0.265636	+	0.964073i	$0.414418\pi$
$908$	0	0
$909$	24.0000	0.796030
$910$	0	0
$911$	16.0000	0.530104	0.265052	−	0.964234i	$-0.414611\pi$
$911$	16.0000	0.530104	0.265052	+	0.964234i	$0.414611\pi$
$912$	0	0
$913$	40.0000	1.32381
$914$	0	0
$915$	2.00000	0.0661180
$916$	0	0
$917$	−12.0000	−0.396275
$918$	0	0
$919$	47.0000	1.55039	0.775193	−	0.631724i	$-0.217652\pi$
$919$	47.0000	1.55039	0.775193	+	0.631724i	$0.217652\pi$
$920$	0	0
$921$	−29.0000	−0.955582
$922$	0	0
$923$	60.0000	1.97492
$924$	0	0
$925$	−4.00000	−0.131519
$926$	0	0
$927$	−10.0000	−0.328443
$928$	0	0
$929$	12.0000	0.393707	0.196854	−	0.980433i	$-0.436928\pi$
$929$	12.0000	0.393707	0.196854	+	0.980433i	$0.436928\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	24.0000	0.785725
$934$	0	0
$935$	−25.0000	−0.817587
$936$	0	0
$937$	−25.0000	−0.816714	−0.408357	−	0.912822i	$-0.633898\pi$
$937$	−25.0000	−0.816714	−0.408357	+	0.912822i	$0.633898\pi$
$938$	0	0
$939$	−1.00000	−0.0326338
$940$	0	0
$941$	−38.0000	−1.23876	−0.619382	−	0.785090i	$-0.712617\pi$
$941$	−38.0000	−1.23876	−0.619382	+	0.785090i	$0.712617\pi$
$942$	0	0
$943$	16.0000	0.521032
$944$	0	0
$945$	5.00000	0.162650
$946$	0	0
$947$	34.0000	1.10485	0.552426	−	0.833562i	$-0.313702\pi$
$947$	34.0000	1.10485	0.552426	+	0.833562i	$0.313702\pi$
$948$	0	0
$949$	30.0000	0.973841
$950$	0	0
$951$	2.00000	0.0648544
$952$	0	0
$953$	−34.0000	−1.10137	−0.550684	−	0.834714i	$-0.685633\pi$
$953$	−34.0000	−1.10137	−0.550684	+	0.834714i	$0.685633\pi$
$954$	0	0
$955$	15.0000	0.485389
$956$	0	0
$957$	5.00000	0.161627
$958$	0	0
$959$	−22.0000	−0.710417
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	4.00000	0.128898
$964$	0	0
$965$	10.0000	0.321911
$966$	0	0
$967$	22.0000	0.707472	0.353736	−	0.935345i	$-0.384911\pi$
$967$	22.0000	0.707472	0.353736	+	0.935345i	$0.384911\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	18.0000	0.577647	0.288824	−	0.957382i	$-0.406736\pi$
$971$	18.0000	0.577647	0.288824	+	0.957382i	$0.406736\pi$
$972$	0	0
$973$	−10.0000	−0.320585
$974$	0	0
$975$	5.00000	0.160128
$976$	0	0
$977$	32.0000	1.02377	0.511885	−	0.859054i	$-0.328947\pi$
$977$	32.0000	1.02377	0.511885	+	0.859054i	$0.328947\pi$
$978$	0	0
$979$	90.0000	2.87641
$980$	0	0
$981$	−22.0000	−0.702406
$982$	0	0
$983$	15.0000	0.478426	0.239213	−	0.970967i	$-0.423111\pi$
$983$	15.0000	0.478426	0.239213	+	0.970967i	$0.423111\pi$
$984$	0	0
$985$	6.00000	0.191176
$986$	0	0
$987$	13.0000	0.413795
$988$	0	0
$989$	−32.0000	−1.01754
$990$	0	0
$991$	8.00000	0.254128	0.127064	−	0.991894i	$-0.459445\pi$
$991$	8.00000	0.254128	0.127064	+	0.991894i	$0.459445\pi$
$992$	0	0
$993$	−12.0000	−0.380808
$994$	0	0
$995$	8.00000	0.253617
$996$	0	0
$997$	−11.0000	−0.348373	−0.174187	−	0.984713i	$-0.555730\pi$
$997$	−11.0000	−0.348373	−0.174187	+	0.984713i	$0.555730\pi$
$998$	0	0
$999$	−20.0000	−0.632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.2.a.e.1.1		1
4.3	odd	2		2240.2.a.t.1.1		1
8.3	odd	2		1120.2.a.e.1.1	✓	1
8.5	even	2		1120.2.a.n.1.1	yes	1
40.19	odd	2		5600.2.a.n.1.1		1
40.29	even	2		5600.2.a.i.1.1		1
56.13	odd	2		7840.2.a.h.1.1		1
56.27	even	2		7840.2.a.p.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1120.2.a.e.1.1	✓	1	8.3	odd	2
1120.2.a.n.1.1	yes	1	8.5	even	2
2240.2.a.e.1.1		1	1.1	even	1	trivial
2240.2.a.t.1.1		1	4.3	odd	2
5600.2.a.i.1.1		1	40.29	even	2
5600.2.a.n.1.1		1	40.19	odd	2
7840.2.a.h.1.1		1	56.13	odd	2
7840.2.a.p.1.1		1	56.27	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:	\( N \)	\(=\)	\( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2240.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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