LMFDB - Embedded newform 2240.2.a.b.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:	$1$
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)

$f(q)$

$=$

$q-3.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} +6.00000 q^{9} +O(q^{10})$

$q-3.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} +6.00000 q^{9} -3.00000 q^{11} -1.00000 q^{13} +3.00000 q^{15} -1.00000 q^{17} +4.00000 q^{19} -3.00000 q^{21} -4.00000 q^{23} +1.00000 q^{25} -9.00000 q^{27} +9.00000 q^{29} -6.00000 q^{31} +9.00000 q^{33} -1.00000 q^{35} +8.00000 q^{37} +3.00000 q^{39} +6.00000 q^{41} -8.00000 q^{43} -6.00000 q^{45} +7.00000 q^{47} +1.00000 q^{49} +3.00000 q^{51} +8.00000 q^{53} +3.00000 q^{55} -12.0000 q^{57} +4.00000 q^{59} -10.0000 q^{61} +6.00000 q^{63} +1.00000 q^{65} -8.00000 q^{67} +12.0000 q^{69} +12.0000 q^{71} -14.0000 q^{73} -3.00000 q^{75} -3.00000 q^{77} +5.00000 q^{79} +9.00000 q^{81} -16.0000 q^{83} +1.00000 q^{85} -27.0000 q^{87} -14.0000 q^{89} -1.00000 q^{91} +18.0000 q^{93} -4.00000 q^{95} -17.0000 q^{97} -18.0000 q^{99} +O(q^{100})$

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
$3$	−3.00000	−1.73205	−0.866025	−	0.500000i	$-0.833333\pi$
$3$	−3.00000	−1.73205	−0.866025	+	0.500000i	$0.833333\pi$
$4$	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
$9$	6.00000	2.00000
$10$	0	0
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	0	0
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	0	0
$15$	3.00000	0.774597
$16$	0	0
$17$	−1.00000	−0.242536	−0.121268	−	0.992620i	$-0.538696\pi$
$17$	−1.00000	−0.242536	−0.121268	+	0.992620i	$0.538696\pi$
$18$	0	0
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	−3.00000	−0.654654
$22$	0	0
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−9.00000	−1.73205
$28$	0	0
$29$	9.00000	1.67126	0.835629	−	0.549294i	$-0.185103\pi$
$29$	9.00000	1.67126	0.835629	+	0.549294i	$0.185103\pi$
$30$	0	0
$31$	−6.00000	−1.07763	−0.538816	−	0.842424i	$-0.681128\pi$
$31$	−6.00000	−1.07763	−0.538816	+	0.842424i	$0.681128\pi$
$32$	0	0
$33$	9.00000	1.56670
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	8.00000	1.31519	0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000	1.31519	0.657596	+	0.753371i	$0.271573\pi$
$38$	0	0
$39$	3.00000	0.480384
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	0	0
$45$	−6.00000	−0.894427
$46$	0	0
$47$	7.00000	1.02105	0.510527	−	0.859861i	$-0.329450\pi$
$47$	7.00000	1.02105	0.510527	+	0.859861i	$0.329450\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	3.00000	0.420084
$52$	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
$55$	3.00000	0.404520
$56$	0	0
$57$	−12.0000	−1.58944
$58$	0	0
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	0	0
$63$	6.00000	0.755929
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	0	0
$69$	12.0000	1.44463
$70$	0	0
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	0	0
$73$	−14.0000	−1.63858	−0.819288	−	0.573382i	$-0.805631\pi$
$73$	−14.0000	−1.63858	−0.819288	+	0.573382i	$0.805631\pi$
$74$	0	0
$75$	−3.00000	−0.346410
$76$	0	0
$77$	−3.00000	−0.341882
$78$	0	0
$79$	5.00000	0.562544	0.281272	−	0.959628i	$-0.409244\pi$
$79$	5.00000	0.562544	0.281272	+	0.959628i	$0.409244\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	−16.0000	−1.75623	−0.878114	−	0.478451i	$-0.841198\pi$
$83$	−16.0000	−1.75623	−0.878114	+	0.478451i	$0.841198\pi$
$84$	0	0
$85$	1.00000	0.108465
$86$	0	0
$87$	−27.0000	−2.89470
$88$	0	0
$89$	−14.0000	−1.48400	−0.741999	−	0.670402i	$-0.766122\pi$
$89$	−14.0000	−1.48400	−0.741999	+	0.670402i	$0.766122\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	18.0000	1.86651
$94$	0	0
$95$	−4.00000	−0.410391
$96$	0	0
$97$	−17.0000	−1.72609	−0.863044	−	0.505128i	$-0.831445\pi$
$97$	−17.0000	−1.72609	−0.863044	+	0.505128i	$0.831445\pi$
$98$	0	0
$99$	−18.0000	−1.80907
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	−1.00000	−0.0985329	−0.0492665	−	0.998786i	$-0.515688\pi$
$103$	−1.00000	−0.0985329	−0.0492665	+	0.998786i	$0.515688\pi$
$104$	0	0
$105$	3.00000	0.292770
$106$	0	0
$107$	10.0000	0.966736	0.483368	−	0.875417i	$-0.339413\pi$
$107$	10.0000	0.966736	0.483368	+	0.875417i	$0.339413\pi$
$108$	0	0
$109$	−5.00000	−0.478913	−0.239457	−	0.970907i	$-0.576969\pi$
$109$	−5.00000	−0.478913	−0.239457	+	0.970907i	$0.576969\pi$
$110$	0	0
$111$	−24.0000	−2.27798
$112$	0	0
$113$	16.0000	1.50515	0.752577	−	0.658505i	$-0.228811\pi$
$113$	16.0000	1.50515	0.752577	+	0.658505i	$0.228811\pi$
$114$	0	0
$115$	4.00000	0.373002
$116$	0	0
$117$	−6.00000	−0.554700
$118$	0	0
$119$	−1.00000	−0.0916698
$120$	0	0
$121$	−2.00000	−0.181818
$122$	0	0
$123$	−18.0000	−1.62301
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−20.0000	−1.77471	−0.887357	−	0.461084i	$-0.847461\pi$
$127$	−20.0000	−1.77471	−0.887357	+	0.461084i	$0.847461\pi$
$128$	0	0
$129$	24.0000	2.11308
$130$	0	0
$131$	8.00000	0.698963	0.349482	−	0.936943i	$-0.386358\pi$
$131$	8.00000	0.698963	0.349482	+	0.936943i	$0.386358\pi$
$132$	0	0
$133$	4.00000	0.346844
$134$	0	0
$135$	9.00000	0.774597
$136$	0	0
$137$	−14.0000	−1.19610	−0.598050	−	0.801459i	$-0.704058\pi$
$137$	−14.0000	−1.19610	−0.598050	+	0.801459i	$0.704058\pi$
$138$	0	0
$139$	−10.0000	−0.848189	−0.424094	−	0.905618i	$-0.639408\pi$
$139$	−10.0000	−0.848189	−0.424094	+	0.905618i	$0.639408\pi$
$140$	0	0
$141$	−21.0000	−1.76852
$142$	0	0
$143$	3.00000	0.250873
$144$	0	0
$145$	−9.00000	−0.747409
$146$	0	0
$147$	−3.00000	−0.247436
$148$	0	0
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	5.00000	0.406894	0.203447	−	0.979086i	$-0.434786\pi$
$151$	5.00000	0.406894	0.203447	+	0.979086i	$0.434786\pi$
$152$	0	0
$153$	−6.00000	−0.485071
$154$	0	0
$155$	6.00000	0.481932
$156$	0	0
$157$	−14.0000	−1.11732	−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−14.0000	−1.11732	−0.558661	+	0.829396i	$0.688685\pi$
$158$	0	0
$159$	−24.0000	−1.90332
$160$	0	0
$161$	−4.00000	−0.315244
$162$	0	0
$163$	−2.00000	−0.156652	−0.0783260	−	0.996928i	$-0.524958\pi$
$163$	−2.00000	−0.156652	−0.0783260	+	0.996928i	$0.524958\pi$
$164$	0	0
$165$	−9.00000	−0.700649
$166$	0	0
$167$	7.00000	0.541676	0.270838	−	0.962625i	$-0.412699\pi$
$167$	7.00000	0.541676	0.270838	+	0.962625i	$0.412699\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	24.0000	1.83533
$172$	0	0
$173$	−19.0000	−1.44454	−0.722272	−	0.691609i	$-0.756902\pi$
$173$	−19.0000	−1.44454	−0.722272	+	0.691609i	$0.756902\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	−12.0000	−0.901975
$178$	0	0
$179$	−24.0000	−1.79384	−0.896922	−	0.442189i	$-0.854202\pi$
$179$	−24.0000	−1.79384	−0.896922	+	0.442189i	$0.854202\pi$
$180$	0	0
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	0	0
$183$	30.0000	2.21766
$184$	0	0
$185$	−8.00000	−0.588172
$186$	0	0
$187$	3.00000	0.219382
$188$	0	0
$189$	−9.00000	−0.654654
$190$	0	0
$191$	−9.00000	−0.651217	−0.325609	−	0.945505i	$-0.605569\pi$
$191$	−9.00000	−0.651217	−0.325609	+	0.945505i	$0.605569\pi$
$192$	0	0
$193$	18.0000	1.29567	0.647834	−	0.761781i	$-0.275675\pi$
$193$	18.0000	1.29567	0.647834	+	0.761781i	$0.275675\pi$
$194$	0	0
$195$	−3.00000	−0.214834
$196$	0	0
$197$	−18.0000	−1.28245	−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000	−1.28245	−0.641223	+	0.767354i	$0.721573\pi$
$198$	0	0
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	0	0
$201$	24.0000	1.69283
$202$	0	0
$203$	9.00000	0.631676
$204$	0	0
$205$	−6.00000	−0.419058
$206$	0	0
$207$	−24.0000	−1.66812
$208$	0	0
$209$	−12.0000	−0.830057
$210$	0	0
$211$	19.0000	1.30801	0.654007	−	0.756489i	$-0.273087\pi$
$211$	19.0000	1.30801	0.654007	+	0.756489i	$0.273087\pi$
$212$	0	0
$213$	−36.0000	−2.46668
$214$	0	0
$215$	8.00000	0.545595
$216$	0	0
$217$	−6.00000	−0.407307
$218$	0	0
$219$	42.0000	2.83810
$220$	0	0
$221$	1.00000	0.0672673
$222$	0	0
$223$	−17.0000	−1.13840	−0.569202	−	0.822198i	$-0.692748\pi$
$223$	−17.0000	−1.13840	−0.569202	+	0.822198i	$0.692748\pi$
$224$	0	0
$225$	6.00000	0.400000
$226$	0	0
$227$	−3.00000	−0.199117	−0.0995585	−	0.995032i	$-0.531743\pi$
$227$	−3.00000	−0.199117	−0.0995585	+	0.995032i	$0.531743\pi$
$228$	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
$231$	9.00000	0.592157
$232$	0	0
$233$	−24.0000	−1.57229	−0.786146	−	0.618041i	$-0.787927\pi$
$233$	−24.0000	−1.57229	−0.786146	+	0.618041i	$0.787927\pi$
$234$	0	0
$235$	−7.00000	−0.456630
$236$	0	0
$237$	−15.0000	−0.974355
$238$	0	0
$239$	21.0000	1.35838	0.679189	−	0.733964i	$-0.262332\pi$
$239$	21.0000	1.35838	0.679189	+	0.733964i	$0.262332\pi$
$240$	0	0
$241$	4.00000	0.257663	0.128831	−	0.991667i	$-0.458877\pi$
$241$	4.00000	0.257663	0.128831	+	0.991667i	$0.458877\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	−4.00000	−0.254514
$248$	0	0
$249$	48.0000	3.04188
$250$	0	0
$251$	−6.00000	−0.378717	−0.189358	−	0.981908i	$-0.560641\pi$
$251$	−6.00000	−0.378717	−0.189358	+	0.981908i	$0.560641\pi$
$252$	0	0
$253$	12.0000	0.754434
$254$	0	0
$255$	−3.00000	−0.187867
$256$	0	0
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	0	0
$259$	8.00000	0.497096
$260$	0	0
$261$	54.0000	3.34252
$262$	0	0
$263$	18.0000	1.10993	0.554964	−	0.831875i	$-0.312732\pi$
$263$	18.0000	1.10993	0.554964	+	0.831875i	$0.312732\pi$
$264$	0	0
$265$	−8.00000	−0.491436
$266$	0	0
$267$	42.0000	2.57036
$268$	0	0
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	0	0
$271$	2.00000	0.121491	0.0607457	−	0.998153i	$-0.480652\pi$
$271$	2.00000	0.121491	0.0607457	+	0.998153i	$0.480652\pi$
$272$	0	0
$273$	3.00000	0.181568
$274$	0	0
$275$	−3.00000	−0.180907
$276$	0	0
$277$	2.00000	0.120168	0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000	0.120168	0.0600842	+	0.998193i	$0.480863\pi$
$278$	0	0
$279$	−36.0000	−2.15526
$280$	0	0
$281$	−17.0000	−1.01413	−0.507067	−	0.861906i	$-0.669271\pi$
$281$	−17.0000	−1.01413	−0.507067	+	0.861906i	$0.669271\pi$
$282$	0	0
$283$	11.0000	0.653882	0.326941	−	0.945045i	$-0.393982\pi$
$283$	11.0000	0.653882	0.326941	+	0.945045i	$0.393982\pi$
$284$	0	0
$285$	12.0000	0.710819
$286$	0	0
$287$	6.00000	0.354169
$288$	0	0
$289$	−16.0000	−0.941176
$290$	0	0
$291$	51.0000	2.98967
$292$	0	0
$293$	9.00000	0.525786	0.262893	−	0.964825i	$-0.415323\pi$
$293$	9.00000	0.525786	0.262893	+	0.964825i	$0.415323\pi$
$294$	0	0
$295$	−4.00000	−0.232889
$296$	0	0
$297$	27.0000	1.56670
$298$	0	0
$299$	4.00000	0.231326
$300$	0	0
$301$	−8.00000	−0.461112
$302$	0	0
$303$	0	0
$304$	0	0
$305$	10.0000	0.572598
$306$	0	0
$307$	−33.0000	−1.88341	−0.941705	−	0.336440i	$-0.890777\pi$
$307$	−33.0000	−1.88341	−0.941705	+	0.336440i	$0.890777\pi$
$308$	0	0
$309$	3.00000	0.170664
$310$	0	0
$311$	4.00000	0.226819	0.113410	−	0.993548i	$-0.463823\pi$
$311$	4.00000	0.226819	0.113410	+	0.993548i	$0.463823\pi$
$312$	0	0
$313$	21.0000	1.18699	0.593495	−	0.804838i	$-0.297748\pi$
$313$	21.0000	1.18699	0.593495	+	0.804838i	$0.297748\pi$
$314$	0	0
$315$	−6.00000	−0.338062
$316$	0	0
$317$	−10.0000	−0.561656	−0.280828	−	0.959758i	$-0.590609\pi$
$317$	−10.0000	−0.561656	−0.280828	+	0.959758i	$0.590609\pi$
$318$	0	0
$319$	−27.0000	−1.51171
$320$	0	0
$321$	−30.0000	−1.67444
$322$	0	0
$323$	−4.00000	−0.222566
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	0	0
$327$	15.0000	0.829502
$328$	0	0
$329$	7.00000	0.385922
$330$	0	0
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	0	0
$333$	48.0000	2.63038
$334$	0	0
$335$	8.00000	0.437087
$336$	0	0
$337$	−14.0000	−0.762629	−0.381314	−	0.924445i	$-0.624528\pi$
$337$	−14.0000	−0.762629	−0.381314	+	0.924445i	$0.624528\pi$
$338$	0	0
$339$	−48.0000	−2.60700
$340$	0	0
$341$	18.0000	0.974755
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−12.0000	−0.646058
$346$	0	0
$347$	−4.00000	−0.214731	−0.107366	−	0.994220i	$-0.534242\pi$
$347$	−4.00000	−0.214731	−0.107366	+	0.994220i	$0.534242\pi$
$348$	0	0
$349$	0	0		−	1.00000i	$-0.5\pi$
$349$	0	0			1.00000i	$0.5\pi$
$350$	0	0
$351$	9.00000	0.480384
$352$	0	0
$353$	3.00000	0.159674	0.0798369	−	0.996808i	$-0.474560\pi$
$353$	3.00000	0.159674	0.0798369	+	0.996808i	$0.474560\pi$
$354$	0	0
$355$	−12.0000	−0.636894
$356$	0	0
$357$	3.00000	0.158777
$358$	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
$361$	−3.00000	−0.157895
$362$	0	0
$363$	6.00000	0.314918
$364$	0	0
$365$	14.0000	0.732793
$366$	0	0
$367$	3.00000	0.156599	0.0782994	−	0.996930i	$-0.475051\pi$
$367$	3.00000	0.156599	0.0782994	+	0.996930i	$0.475051\pi$
$368$	0	0
$369$	36.0000	1.87409
$370$	0	0
$371$	8.00000	0.415339
$372$	0	0
$373$	−14.0000	−0.724893	−0.362446	−	0.932005i	$-0.618058\pi$
$373$	−14.0000	−0.724893	−0.362446	+	0.932005i	$0.618058\pi$
$374$	0	0
$375$	3.00000	0.154919
$376$	0	0
$377$	−9.00000	−0.463524
$378$	0	0
$379$	−8.00000	−0.410932	−0.205466	−	0.978664i	$-0.565871\pi$
$379$	−8.00000	−0.410932	−0.205466	+	0.978664i	$0.565871\pi$
$380$	0	0
$381$	60.0000	3.07389
$382$	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
$385$	3.00000	0.152894
$386$	0	0
$387$	−48.0000	−2.43998
$388$	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
$391$	4.00000	0.202289
$392$	0	0
$393$	−24.0000	−1.21064
$394$	0	0
$395$	−5.00000	−0.251577
$396$	0	0
$397$	−7.00000	−0.351320	−0.175660	−	0.984451i	$-0.556206\pi$
$397$	−7.00000	−0.351320	−0.175660	+	0.984451i	$0.556206\pi$
$398$	0	0
$399$	−12.0000	−0.600751
$400$	0	0
$401$	−27.0000	−1.34832	−0.674158	−	0.738587i	$-0.735493\pi$
$401$	−27.0000	−1.34832	−0.674158	+	0.738587i	$0.735493\pi$
$402$	0	0
$403$	6.00000	0.298881
$404$	0	0
$405$	−9.00000	−0.447214
$406$	0	0
$407$	−24.0000	−1.18964
$408$	0	0
$409$	−2.00000	−0.0988936	−0.0494468	−	0.998777i	$-0.515746\pi$
$409$	−2.00000	−0.0988936	−0.0494468	+	0.998777i	$0.515746\pi$
$410$	0	0
$411$	42.0000	2.07171
$412$	0	0
$413$	4.00000	0.196827
$414$	0	0
$415$	16.0000	0.785409
$416$	0	0
$417$	30.0000	1.46911
$418$	0	0
$419$	24.0000	1.17248	0.586238	−	0.810139i	$-0.300608\pi$
$419$	24.0000	1.17248	0.586238	+	0.810139i	$0.300608\pi$
$420$	0	0
$421$	−1.00000	−0.0487370	−0.0243685	−	0.999703i	$-0.507758\pi$
$421$	−1.00000	−0.0487370	−0.0243685	+	0.999703i	$0.507758\pi$
$422$	0	0
$423$	42.0000	2.04211
$424$	0	0
$425$	−1.00000	−0.0485071
$426$	0	0
$427$	−10.0000	−0.483934
$428$	0	0
$429$	−9.00000	−0.434524
$430$	0	0
$431$	−5.00000	−0.240842	−0.120421	−	0.992723i	$-0.538424\pi$
$431$	−5.00000	−0.240842	−0.120421	+	0.992723i	$0.538424\pi$
$432$	0	0
$433$	18.0000	0.865025	0.432512	−	0.901628i	$-0.357627\pi$
$433$	18.0000	0.865025	0.432512	+	0.901628i	$0.357627\pi$
$434$	0	0
$435$	27.0000	1.29455
$436$	0	0
$437$	−16.0000	−0.765384
$438$	0	0
$439$	−26.0000	−1.24091	−0.620456	−	0.784241i	$-0.713053\pi$
$439$	−26.0000	−1.24091	−0.620456	+	0.784241i	$0.713053\pi$
$440$	0	0
$441$	6.00000	0.285714
$442$	0	0
$443$	−40.0000	−1.90046	−0.950229	−	0.311553i	$-0.899151\pi$
$443$	−40.0000	−1.90046	−0.950229	+	0.311553i	$0.899151\pi$
$444$	0	0
$445$	14.0000	0.663664
$446$	0	0
$447$	−18.0000	−0.851371
$448$	0	0
$449$	−17.0000	−0.802280	−0.401140	−	0.916017i	$-0.631386\pi$
$449$	−17.0000	−0.802280	−0.401140	+	0.916017i	$0.631386\pi$
$450$	0	0
$451$	−18.0000	−0.847587
$452$	0	0
$453$	−15.0000	−0.704761
$454$	0	0
$455$	1.00000	0.0468807
$456$	0	0
$457$	22.0000	1.02912	0.514558	−	0.857455i	$-0.327956\pi$
$457$	22.0000	1.02912	0.514558	+	0.857455i	$0.327956\pi$
$458$	0	0
$459$	9.00000	0.420084
$460$	0	0
$461$	−40.0000	−1.86299	−0.931493	−	0.363760i	$-0.881493\pi$
$461$	−40.0000	−1.86299	−0.931493	+	0.363760i	$0.881493\pi$
$462$	0	0
$463$	−14.0000	−0.650635	−0.325318	−	0.945605i	$-0.605471\pi$
$463$	−14.0000	−0.650635	−0.325318	+	0.945605i	$0.605471\pi$
$464$	0	0
$465$	−18.0000	−0.834730
$466$	0	0
$467$	−5.00000	−0.231372	−0.115686	−	0.993286i	$-0.536907\pi$
$467$	−5.00000	−0.231372	−0.115686	+	0.993286i	$0.536907\pi$
$468$	0	0
$469$	−8.00000	−0.369406
$470$	0	0
$471$	42.0000	1.93526
$472$	0	0
$473$	24.0000	1.10352
$474$	0	0
$475$	4.00000	0.183533
$476$	0	0
$477$	48.0000	2.19777
$478$	0	0
$479$	40.0000	1.82765	0.913823	−	0.406112i	$-0.133116\pi$
$479$	40.0000	1.82765	0.913823	+	0.406112i	$0.133116\pi$
$480$	0	0
$481$	−8.00000	−0.364769
$482$	0	0
$483$	12.0000	0.546019
$484$	0	0
$485$	17.0000	0.771930
$486$	0	0
$487$	−18.0000	−0.815658	−0.407829	−	0.913058i	$-0.633714\pi$
$487$	−18.0000	−0.815658	−0.407829	+	0.913058i	$0.633714\pi$
$488$	0	0
$489$	6.00000	0.271329
$490$	0	0
$491$	−5.00000	−0.225647	−0.112823	−	0.993615i	$-0.535989\pi$
$491$	−5.00000	−0.225647	−0.112823	+	0.993615i	$0.535989\pi$
$492$	0	0
$493$	−9.00000	−0.405340
$494$	0	0
$495$	18.0000	0.809040
$496$	0	0
$497$	12.0000	0.538274
$498$	0	0
$499$	37.0000	1.65635	0.828174	−	0.560471i	$-0.189380\pi$
$499$	37.0000	1.65635	0.828174	+	0.560471i	$0.189380\pi$
$500$	0	0
$501$	−21.0000	−0.938211
$502$	0	0
$503$	9.00000	0.401290	0.200645	−	0.979664i	$-0.435696\pi$
$503$	9.00000	0.401290	0.200645	+	0.979664i	$0.435696\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	36.0000	1.59882
$508$	0	0
$509$	−18.0000	−0.797836	−0.398918	−	0.916987i	$-0.630614\pi$
$509$	−18.0000	−0.797836	−0.398918	+	0.916987i	$0.630614\pi$
$510$	0	0
$511$	−14.0000	−0.619324
$512$	0	0
$513$	−36.0000	−1.58944
$514$	0	0
$515$	1.00000	0.0440653
$516$	0	0
$517$	−21.0000	−0.923579
$518$	0	0
$519$	57.0000	2.50202
$520$	0	0
$521$	16.0000	0.700973	0.350486	−	0.936568i	$-0.386016\pi$
$521$	16.0000	0.700973	0.350486	+	0.936568i	$0.386016\pi$
$522$	0	0
$523$	20.0000	0.874539	0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000	0.874539	0.437269	+	0.899331i	$0.355946\pi$
$524$	0	0
$525$	−3.00000	−0.130931
$526$	0	0
$527$	6.00000	0.261364
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	24.0000	1.04151
$532$	0	0
$533$	−6.00000	−0.259889
$534$	0	0
$535$	−10.0000	−0.432338
$536$	0	0
$537$	72.0000	3.10703
$538$	0	0
$539$	−3.00000	−0.129219
$540$	0	0
$541$	1.00000	0.0429934	0.0214967	−	0.999769i	$-0.493157\pi$
$541$	1.00000	0.0429934	0.0214967	+	0.999769i	$0.493157\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	5.00000	0.214176
$546$	0	0
$547$	42.0000	1.79579	0.897895	−	0.440209i	$-0.145096\pi$
$547$	42.0000	1.79579	0.897895	+	0.440209i	$0.145096\pi$
$548$	0	0
$549$	−60.0000	−2.56074
$550$	0	0
$551$	36.0000	1.53365
$552$	0	0
$553$	5.00000	0.212622
$554$	0	0
$555$	24.0000	1.01874
$556$	0	0
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	0	0
$559$	8.00000	0.338364
$560$	0	0
$561$	−9.00000	−0.379980
$562$	0	0
$563$	−36.0000	−1.51722	−0.758610	−	0.651546i	$-0.774121\pi$
$563$	−36.0000	−1.51722	−0.758610	+	0.651546i	$0.774121\pi$
$564$	0	0
$565$	−16.0000	−0.673125
$566$	0	0
$567$	9.00000	0.377964
$568$	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
$571$	−28.0000	−1.17176	−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000	−1.17176	−0.585882	+	0.810397i	$0.699252\pi$
$572$	0	0
$573$	27.0000	1.12794
$574$	0	0
$575$	−4.00000	−0.166812
$576$	0	0
$577$	33.0000	1.37381	0.686904	−	0.726748i	$-0.258969\pi$
$577$	33.0000	1.37381	0.686904	+	0.726748i	$0.258969\pi$
$578$	0	0
$579$	−54.0000	−2.24416
$580$	0	0
$581$	−16.0000	−0.663792
$582$	0	0
$583$	−24.0000	−0.993978
$584$	0	0
$585$	6.00000	0.248069
$586$	0	0
$587$	−28.0000	−1.15568	−0.577842	−	0.816149i	$-0.696105\pi$
$587$	−28.0000	−1.15568	−0.577842	+	0.816149i	$0.696105\pi$
$588$	0	0
$589$	−24.0000	−0.988903
$590$	0	0
$591$	54.0000	2.22126
$592$	0	0
$593$	1.00000	0.0410651	0.0205325	−	0.999789i	$-0.493464\pi$
$593$	1.00000	0.0410651	0.0205325	+	0.999789i	$0.493464\pi$
$594$	0	0
$595$	1.00000	0.0409960
$596$	0	0
$597$	48.0000	1.96451
$598$	0	0
$599$	35.0000	1.43006	0.715031	−	0.699093i	$-0.246413\pi$
$599$	35.0000	1.43006	0.715031	+	0.699093i	$0.246413\pi$
$600$	0	0
$601$	−14.0000	−0.571072	−0.285536	−	0.958368i	$-0.592172\pi$
$601$	−14.0000	−0.571072	−0.285536	+	0.958368i	$0.592172\pi$
$602$	0	0
$603$	−48.0000	−1.95471
$604$	0	0
$605$	2.00000	0.0813116
$606$	0	0
$607$	41.0000	1.66414	0.832069	−	0.554672i	$-0.187156\pi$
$607$	41.0000	1.66414	0.832069	+	0.554672i	$0.187156\pi$
$608$	0	0
$609$	−27.0000	−1.09410
$610$	0	0
$611$	−7.00000	−0.283190
$612$	0	0
$613$	−34.0000	−1.37325	−0.686624	−	0.727013i	$-0.740908\pi$
$613$	−34.0000	−1.37325	−0.686624	+	0.727013i	$0.740908\pi$
$614$	0	0
$615$	18.0000	0.725830
$616$	0	0
$617$	18.0000	0.724653	0.362326	−	0.932051i	$-0.381983\pi$
$617$	18.0000	0.724653	0.362326	+	0.932051i	$0.381983\pi$
$618$	0	0
$619$	−36.0000	−1.44696	−0.723481	−	0.690344i	$-0.757459\pi$
$619$	−36.0000	−1.44696	−0.723481	+	0.690344i	$0.757459\pi$
$620$	0	0
$621$	36.0000	1.44463
$622$	0	0
$623$	−14.0000	−0.560898
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	36.0000	1.43770
$628$	0	0
$629$	−8.00000	−0.318981
$630$	0	0
$631$	−37.0000	−1.47295	−0.736473	−	0.676467i	$-0.763510\pi$
$631$	−37.0000	−1.47295	−0.736473	+	0.676467i	$0.763510\pi$
$632$	0	0
$633$	−57.0000	−2.26555
$634$	0	0
$635$	20.0000	0.793676
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	72.0000	2.84828
$640$	0	0
$641$	6.00000	0.236986	0.118493	−	0.992955i	$-0.462194\pi$
$641$	6.00000	0.236986	0.118493	+	0.992955i	$0.462194\pi$
$642$	0	0
$643$	−19.0000	−0.749287	−0.374643	−	0.927169i	$-0.622235\pi$
$643$	−19.0000	−0.749287	−0.374643	+	0.927169i	$0.622235\pi$
$644$	0	0
$645$	−24.0000	−0.944999
$646$	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
$649$	−12.0000	−0.471041
$650$	0	0
$651$	18.0000	0.705476
$652$	0	0
$653$	−24.0000	−0.939193	−0.469596	−	0.882881i	$-0.655601\pi$
$653$	−24.0000	−0.939193	−0.469596	+	0.882881i	$0.655601\pi$
$654$	0	0
$655$	−8.00000	−0.312586
$656$	0	0
$657$	−84.0000	−3.27715
$658$	0	0
$659$	9.00000	0.350590	0.175295	−	0.984516i	$-0.443912\pi$
$659$	9.00000	0.350590	0.175295	+	0.984516i	$0.443912\pi$
$660$	0	0
$661$	4.00000	0.155582	0.0777910	−	0.996970i	$-0.475213\pi$
$661$	4.00000	0.155582	0.0777910	+	0.996970i	$0.475213\pi$
$662$	0	0
$663$	−3.00000	−0.116510
$664$	0	0
$665$	−4.00000	−0.155113
$666$	0	0
$667$	−36.0000	−1.39393
$668$	0	0
$669$	51.0000	1.97177
$670$	0	0
$671$	30.0000	1.15814
$672$	0	0
$673$	10.0000	0.385472	0.192736	−	0.981251i	$-0.438264\pi$
$673$	10.0000	0.385472	0.192736	+	0.981251i	$0.438264\pi$
$674$	0	0
$675$	−9.00000	−0.346410
$676$	0	0
$677$	19.0000	0.730229	0.365115	−	0.930963i	$-0.381030\pi$
$677$	19.0000	0.730229	0.365115	+	0.930963i	$0.381030\pi$
$678$	0	0
$679$	−17.0000	−0.652400
$680$	0	0
$681$	9.00000	0.344881
$682$	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
$685$	14.0000	0.534913
$686$	0	0
$687$	−78.0000	−2.97589
$688$	0	0
$689$	−8.00000	−0.304776
$690$	0	0
$691$	−8.00000	−0.304334	−0.152167	−	0.988355i	$-0.548625\pi$
$691$	−8.00000	−0.304334	−0.152167	+	0.988355i	$0.548625\pi$
$692$	0	0
$693$	−18.0000	−0.683763
$694$	0	0
$695$	10.0000	0.379322
$696$	0	0
$697$	−6.00000	−0.227266
$698$	0	0
$699$	72.0000	2.72329
$700$	0	0
$701$	−47.0000	−1.77517	−0.887583	−	0.460648i	$-0.847617\pi$
$701$	−47.0000	−1.77517	−0.887583	+	0.460648i	$0.847617\pi$
$702$	0	0
$703$	32.0000	1.20690
$704$	0	0
$705$	21.0000	0.790906
$706$	0	0
$707$	0	0
$708$	0	0
$709$	1.00000	0.0375558	0.0187779	−	0.999824i	$-0.494022\pi$
$709$	1.00000	0.0375558	0.0187779	+	0.999824i	$0.494022\pi$
$710$	0	0
$711$	30.0000	1.12509
$712$	0	0
$713$	24.0000	0.898807
$714$	0	0
$715$	−3.00000	−0.112194
$716$	0	0
$717$	−63.0000	−2.35278
$718$	0	0
$719$	−36.0000	−1.34257	−0.671287	−	0.741198i	$-0.734258\pi$
$719$	−36.0000	−1.34257	−0.671287	+	0.741198i	$0.734258\pi$
$720$	0	0
$721$	−1.00000	−0.0372419
$722$	0	0
$723$	−12.0000	−0.446285
$724$	0	0
$725$	9.00000	0.334252
$726$	0	0
$727$	16.0000	0.593407	0.296704	−	0.954970i	$-0.404113\pi$
$727$	16.0000	0.593407	0.296704	+	0.954970i	$0.404113\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	8.00000	0.295891
$732$	0	0
$733$	35.0000	1.29275	0.646377	−	0.763018i	$-0.276283\pi$
$733$	35.0000	1.29275	0.646377	+	0.763018i	$0.276283\pi$
$734$	0	0
$735$	3.00000	0.110657
$736$	0	0
$737$	24.0000	0.884051
$738$	0	0
$739$	−19.0000	−0.698926	−0.349463	−	0.936950i	$-0.613636\pi$
$739$	−19.0000	−0.698926	−0.349463	+	0.936950i	$0.613636\pi$
$740$	0	0
$741$	12.0000	0.440831
$742$	0	0
$743$	4.00000	0.146746	0.0733729	−	0.997305i	$-0.476624\pi$
$743$	4.00000	0.146746	0.0733729	+	0.997305i	$0.476624\pi$
$744$	0	0
$745$	−6.00000	−0.219823
$746$	0	0
$747$	−96.0000	−3.51246
$748$	0	0
$749$	10.0000	0.365392
$750$	0	0
$751$	−47.0000	−1.71505	−0.857527	−	0.514439i	$-0.828000\pi$
$751$	−47.0000	−1.71505	−0.857527	+	0.514439i	$0.828000\pi$
$752$	0	0
$753$	18.0000	0.655956
$754$	0	0
$755$	−5.00000	−0.181969
$756$	0	0
$757$	20.0000	0.726912	0.363456	−	0.931611i	$-0.381597\pi$
$757$	20.0000	0.726912	0.363456	+	0.931611i	$0.381597\pi$
$758$	0	0
$759$	−36.0000	−1.30672
$760$	0	0
$761$	−8.00000	−0.290000	−0.145000	−	0.989432i	$-0.546318\pi$
$761$	−8.00000	−0.290000	−0.145000	+	0.989432i	$0.546318\pi$
$762$	0	0
$763$	−5.00000	−0.181012
$764$	0	0
$765$	6.00000	0.216930
$766$	0	0
$767$	−4.00000	−0.144432
$768$	0	0
$769$	−30.0000	−1.08183	−0.540914	−	0.841078i	$-0.681921\pi$
$769$	−30.0000	−1.08183	−0.540914	+	0.841078i	$0.681921\pi$
$770$	0	0
$771$	−18.0000	−0.648254
$772$	0	0
$773$	−45.0000	−1.61854	−0.809269	−	0.587439i	$-0.800136\pi$
$773$	−45.0000	−1.61854	−0.809269	+	0.587439i	$0.800136\pi$
$774$	0	0
$775$	−6.00000	−0.215526
$776$	0	0
$777$	−24.0000	−0.860995
$778$	0	0
$779$	24.0000	0.859889
$780$	0	0
$781$	−36.0000	−1.28818
$782$	0	0
$783$	−81.0000	−2.89470
$784$	0	0
$785$	14.0000	0.499681
$786$	0	0
$787$	−35.0000	−1.24762	−0.623808	−	0.781578i	$-0.714415\pi$
$787$	−35.0000	−1.24762	−0.623808	+	0.781578i	$0.714415\pi$
$788$	0	0
$789$	−54.0000	−1.92245
$790$	0	0
$791$	16.0000	0.568895
$792$	0	0
$793$	10.0000	0.355110
$794$	0	0
$795$	24.0000	0.851192
$796$	0	0
$797$	29.0000	1.02723	0.513616	−	0.858020i	$-0.328305\pi$
$797$	29.0000	1.02723	0.513616	+	0.858020i	$0.328305\pi$
$798$	0	0
$799$	−7.00000	−0.247642
$800$	0	0
$801$	−84.0000	−2.96799
$802$	0	0
$803$	42.0000	1.48215
$804$	0	0
$805$	4.00000	0.140981
$806$	0	0
$807$	18.0000	0.633630
$808$	0	0
$809$	−47.0000	−1.65243	−0.826216	−	0.563353i	$-0.809511\pi$
$809$	−47.0000	−1.65243	−0.826216	+	0.563353i	$0.809511\pi$
$810$	0	0
$811$	22.0000	0.772524	0.386262	−	0.922389i	$-0.373766\pi$
$811$	22.0000	0.772524	0.386262	+	0.922389i	$0.373766\pi$
$812$	0	0
$813$	−6.00000	−0.210429
$814$	0	0
$815$	2.00000	0.0700569
$816$	0	0
$817$	−32.0000	−1.11954
$818$	0	0
$819$	−6.00000	−0.209657
$820$	0	0
$821$	15.0000	0.523504	0.261752	−	0.965135i	$-0.415700\pi$
$821$	15.0000	0.523504	0.261752	+	0.965135i	$0.415700\pi$
$822$	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
$825$	9.00000	0.313340
$826$	0	0
$827$	18.0000	0.625921	0.312961	−	0.949766i	$-0.398679\pi$
$827$	18.0000	0.625921	0.312961	+	0.949766i	$0.398679\pi$
$828$	0	0
$829$	26.0000	0.903017	0.451509	−	0.892267i	$-0.350886\pi$
$829$	26.0000	0.903017	0.451509	+	0.892267i	$0.350886\pi$
$830$	0	0
$831$	−6.00000	−0.208138
$832$	0	0
$833$	−1.00000	−0.0346479
$834$	0	0
$835$	−7.00000	−0.242245
$836$	0	0
$837$	54.0000	1.86651
$838$	0	0
$839$	28.0000	0.966667	0.483334	−	0.875436i	$-0.339426\pi$
$839$	28.0000	0.966667	0.483334	+	0.875436i	$0.339426\pi$
$840$	0	0
$841$	52.0000	1.79310
$842$	0	0
$843$	51.0000	1.75653
$844$	0	0
$845$	12.0000	0.412813
$846$	0	0
$847$	−2.00000	−0.0687208
$848$	0	0
$849$	−33.0000	−1.13256
$850$	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
$855$	−24.0000	−0.820783
$856$	0	0
$857$	−22.0000	−0.751506	−0.375753	−	0.926720i	$-0.622616\pi$
$857$	−22.0000	−0.751506	−0.375753	+	0.926720i	$0.622616\pi$
$858$	0	0
$859$	36.0000	1.22830	0.614152	−	0.789188i	$-0.289498\pi$
$859$	36.0000	1.22830	0.614152	+	0.789188i	$0.289498\pi$
$860$	0	0
$861$	−18.0000	−0.613438
$862$	0	0
$863$	−34.0000	−1.15737	−0.578687	−	0.815550i	$-0.696435\pi$
$863$	−34.0000	−1.15737	−0.578687	+	0.815550i	$0.696435\pi$
$864$	0	0
$865$	19.0000	0.646019
$866$	0	0
$867$	48.0000	1.63017
$868$	0	0
$869$	−15.0000	−0.508840
$870$	0	0
$871$	8.00000	0.271070
$872$	0	0
$873$	−102.000	−3.45218
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	16.0000	0.540282	0.270141	−	0.962821i	$-0.412930\pi$
$877$	16.0000	0.540282	0.270141	+	0.962821i	$0.412930\pi$
$878$	0	0
$879$	−27.0000	−0.910687
$880$	0	0
$881$	24.0000	0.808581	0.404290	−	0.914631i	$-0.367519\pi$
$881$	24.0000	0.808581	0.404290	+	0.914631i	$0.367519\pi$
$882$	0	0
$883$	10.0000	0.336527	0.168263	−	0.985742i	$-0.446184\pi$
$883$	10.0000	0.336527	0.168263	+	0.985742i	$0.446184\pi$
$884$	0	0
$885$	12.0000	0.403376
$886$	0	0
$887$	8.00000	0.268614	0.134307	−	0.990940i	$-0.457119\pi$
$887$	8.00000	0.268614	0.134307	+	0.990940i	$0.457119\pi$
$888$	0	0
$889$	−20.0000	−0.670778
$890$	0	0
$891$	−27.0000	−0.904534
$892$	0	0
$893$	28.0000	0.936984
$894$	0	0
$895$	24.0000	0.802232
$896$	0	0
$897$	−12.0000	−0.400668
$898$	0	0
$899$	−54.0000	−1.80100
$900$	0	0
$901$	−8.00000	−0.266519
$902$	0	0
$903$	24.0000	0.798670
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−4.00000	−0.132818	−0.0664089	−	0.997792i	$-0.521154\pi$
$907$	−4.00000	−0.132818	−0.0664089	+	0.997792i	$0.521154\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	8.00000	0.265052	0.132526	−	0.991180i	$-0.457691\pi$
$911$	8.00000	0.265052	0.132526	+	0.991180i	$0.457691\pi$
$912$	0	0
$913$	48.0000	1.58857
$914$	0	0
$915$	−30.0000	−0.991769
$916$	0	0
$917$	8.00000	0.264183
$918$	0	0
$919$	−39.0000	−1.28649	−0.643246	−	0.765660i	$-0.722413\pi$
$919$	−39.0000	−1.28649	−0.643246	+	0.765660i	$0.722413\pi$
$920$	0	0
$921$	99.0000	3.26216
$922$	0	0
$923$	−12.0000	−0.394985
$924$	0	0
$925$	8.00000	0.263038
$926$	0	0
$927$	−6.00000	−0.197066
$928$	0	0
$929$	36.0000	1.18112	0.590561	−	0.806993i	$-0.298907\pi$
$929$	36.0000	1.18112	0.590561	+	0.806993i	$0.298907\pi$
$930$	0	0
$931$	4.00000	0.131095
$932$	0	0
$933$	−12.0000	−0.392862
$934$	0	0
$935$	−3.00000	−0.0981105
$936$	0	0
$937$	3.00000	0.0980057	0.0490029	−	0.998799i	$-0.484396\pi$
$937$	3.00000	0.0980057	0.0490029	+	0.998799i	$0.484396\pi$
$938$	0	0
$939$	−63.0000	−2.05593
$940$	0	0
$941$	−34.0000	−1.10837	−0.554184	−	0.832394i	$-0.686970\pi$
$941$	−34.0000	−1.10837	−0.554184	+	0.832394i	$0.686970\pi$
$942$	0	0
$943$	−24.0000	−0.781548
$944$	0	0
$945$	9.00000	0.292770
$946$	0	0
$947$	−22.0000	−0.714904	−0.357452	−	0.933932i	$-0.616354\pi$
$947$	−22.0000	−0.714904	−0.357452	+	0.933932i	$0.616354\pi$
$948$	0	0
$949$	14.0000	0.454459
$950$	0	0
$951$	30.0000	0.972817
$952$	0	0
$953$	−6.00000	−0.194359	−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000	−0.194359	−0.0971795	+	0.995267i	$0.530982\pi$
$954$	0	0
$955$	9.00000	0.291233
$956$	0	0
$957$	81.0000	2.61836
$958$	0	0
$959$	−14.0000	−0.452084
$960$	0	0
$961$	5.00000	0.161290
$962$	0	0
$963$	60.0000	1.93347
$964$	0	0
$965$	−18.0000	−0.579441
$966$	0	0
$967$	50.0000	1.60789	0.803946	−	0.594703i	$-0.202730\pi$
$967$	50.0000	1.60789	0.803946	+	0.594703i	$0.202730\pi$
$968$	0	0
$969$	12.0000	0.385496
$970$	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
$975$	3.00000	0.0960769
$976$	0	0
$977$	−36.0000	−1.15174	−0.575871	−	0.817541i	$-0.695337\pi$
$977$	−36.0000	−1.15174	−0.575871	+	0.817541i	$0.695337\pi$
$978$	0	0
$979$	42.0000	1.34233
$980$	0	0
$981$	−30.0000	−0.957826
$982$	0	0
$983$	−11.0000	−0.350846	−0.175423	−	0.984493i	$-0.556129\pi$
$983$	−11.0000	−0.350846	−0.175423	+	0.984493i	$0.556129\pi$
$984$	0	0
$985$	18.0000	0.573528
$986$	0	0
$987$	−21.0000	−0.668437
$988$	0	0
$989$	32.0000	1.01754
$990$	0	0
$991$	24.0000	0.762385	0.381193	−	0.924496i	$-0.375513\pi$
$991$	24.0000	0.762385	0.381193	+	0.924496i	$0.375513\pi$
$992$	0	0
$993$	−60.0000	−1.90404
$994$	0	0
$995$	16.0000	0.507234
$996$	0	0
$997$	25.0000	0.791758	0.395879	−	0.918303i	$-0.370440\pi$
$997$	25.0000	0.791758	0.395879	+	0.918303i	$0.370440\pi$
$998$	0	0
$999$	−72.0000	−2.27798

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1120.2.a.b.1.1	✓	1	8.3	odd	2
1120.2.a.p.1.1	yes	1	8.5	even	2
2240.2.a.b.1.1		1	1.1	even	1	trivial
2240.2.a.y.1.1		1	4.3	odd	2
5600.2.a.b.1.1		1	40.29	even	2
5600.2.a.u.1.1		1	40.19	odd	2
7840.2.a.a.1.1		1	56.13	odd	2
7840.2.a.y.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)

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L-functions

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