LMFDB - Embedded newform 5712.2.a.w.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5712,2,Mod(1,5712)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5712, 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("5712.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5712 = 2^{4} \cdot 3 \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5712.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:	$45.6105496346$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 357)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	5712.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

$q+1.00000 q^{3} +1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} -3.00000 q^{11} +3.00000 q^{13} +1.00000 q^{15} +1.00000 q^{17} -3.00000 q^{19} +1.00000 q^{21} -7.00000 q^{23} -4.00000 q^{25} +1.00000 q^{27} -6.00000 q^{29} -10.0000 q^{31} -3.00000 q^{33} +1.00000 q^{35} +4.00000 q^{37} +3.00000 q^{39} -9.00000 q^{41} -9.00000 q^{43} +1.00000 q^{45} -6.00000 q^{47} +1.00000 q^{49} +1.00000 q^{51} -10.0000 q^{53} -3.00000 q^{55} -3.00000 q^{57} +2.00000 q^{59} +1.00000 q^{63} +3.00000 q^{65} +12.0000 q^{67} -7.00000 q^{69} +12.0000 q^{71} +6.00000 q^{73} -4.00000 q^{75} -3.00000 q^{77} -10.0000 q^{79} +1.00000 q^{81} -10.0000 q^{83} +1.00000 q^{85} -6.00000 q^{87} -4.00000 q^{89} +3.00000 q^{91} -10.0000 q^{93} -3.00000 q^{95} +8.00000 q^{97} -3.00000 q^{99} +O(q^{100})$

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
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	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
$13$	3.00000	0.832050	0.416025	−	0.909353i	$-0.363423\pi$
$13$	3.00000	0.832050	0.416025	+	0.909353i	$0.363423\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−3.00000	−0.688247	−0.344124	−	0.938924i	$-0.611824\pi$
$19$	−3.00000	−0.688247	−0.344124	+	0.938924i	$0.611824\pi$
$20$	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
$25$	−4.00000	−0.800000
$26$	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
$33$	−3.00000	−0.522233
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	4.00000	0.657596	0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000	0.657596	0.328798	+	0.944400i	$0.393356\pi$
$38$	0	0
$39$	3.00000	0.480384
$40$	0	0
$41$	−9.00000	−1.40556	−0.702782	−	0.711405i	$-0.748059\pi$
$41$	−9.00000	−1.40556	−0.702782	+	0.711405i	$0.748059\pi$
$42$	0	0
$43$	−9.00000	−1.37249	−0.686244	−	0.727372i	$-0.740742\pi$
$43$	−9.00000	−1.37249	−0.686244	+	0.727372i	$0.740742\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	1.00000	0.140028
$52$	0	0
$53$	−10.0000	−1.37361	−0.686803	−	0.726844i	$-0.740986\pi$
$53$	−10.0000	−1.37361	−0.686803	+	0.726844i	$0.740986\pi$
$54$	0	0
$55$	−3.00000	−0.404520
$56$	0	0
$57$	−3.00000	−0.397360
$58$	0	0
$59$	2.00000	0.260378	0.130189	−	0.991489i	$-0.458442\pi$
$59$	2.00000	0.260378	0.130189	+	0.991489i	$0.458442\pi$
$60$	0	0
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	3.00000	0.372104
$66$	0	0
$67$	12.0000	1.46603	0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000	1.46603	0.733017	+	0.680211i	$0.238112\pi$
$68$	0	0
$69$	−7.00000	−0.842701
$70$	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
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	0	0
$75$	−4.00000	−0.461880
$76$	0	0
$77$	−3.00000	−0.341882
$78$	0	0
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−10.0000	−1.09764	−0.548821	−	0.835940i	$-0.684923\pi$
$83$	−10.0000	−1.09764	−0.548821	+	0.835940i	$0.684923\pi$
$84$	0	0
$85$	1.00000	0.108465
$86$	0	0
$87$	−6.00000	−0.643268
$88$	0	0
$89$	−4.00000	−0.423999	−0.212000	−	0.977270i	$-0.567998\pi$
$89$	−4.00000	−0.423999	−0.212000	+	0.977270i	$0.567998\pi$
$90$	0	0
$91$	3.00000	0.314485
$92$	0	0
$93$	−10.0000	−1.03695
$94$	0	0
$95$	−3.00000	−0.307794
$96$	0	0
$97$	8.00000	0.812277	0.406138	−	0.913812i	$-0.366875\pi$
$97$	8.00000	0.812277	0.406138	+	0.913812i	$0.366875\pi$
$98$	0	0
$99$	−3.00000	−0.301511
$100$	0	0
$101$	12.0000	1.19404	0.597022	−	0.802225i	$-0.296350\pi$
$101$	12.0000	1.19404	0.597022	+	0.802225i	$0.296350\pi$
$102$	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
$105$	1.00000	0.0975900
$106$	0	0
$107$	17.0000	1.64345	0.821726	−	0.569883i	$-0.193011\pi$
$107$	17.0000	1.64345	0.821726	+	0.569883i	$0.193011\pi$
$108$	0	0
$109$	0	0		−	1.00000i	$-0.5\pi$
$109$	0	0			1.00000i	$0.5\pi$
$110$	0	0
$111$	4.00000	0.379663
$112$	0	0
$113$	9.00000	0.846649	0.423324	−	0.905978i	$-0.360863\pi$
$113$	9.00000	0.846649	0.423324	+	0.905978i	$0.360863\pi$
$114$	0	0
$115$	−7.00000	−0.652753
$116$	0	0
$117$	3.00000	0.277350
$118$	0	0
$119$	1.00000	0.0916698
$120$	0	0
$121$	−2.00000	−0.181818
$122$	0	0
$123$	−9.00000	−0.811503
$124$	0	0
$125$	−9.00000	−0.804984
$126$	0	0
$127$	13.0000	1.15356	0.576782	−	0.816898i	$-0.304308\pi$
$127$	13.0000	1.15356	0.576782	+	0.816898i	$0.304308\pi$
$128$	0	0
$129$	−9.00000	−0.792406
$130$	0	0
$131$	−17.0000	−1.48530	−0.742648	−	0.669681i	$-0.766431\pi$
$131$	−17.0000	−1.48530	−0.742648	+	0.669681i	$0.766431\pi$
$132$	0	0
$133$	−3.00000	−0.260133
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	2.00000	0.170872	0.0854358	−	0.996344i	$-0.472772\pi$
$137$	2.00000	0.170872	0.0854358	+	0.996344i	$0.472772\pi$
$138$	0	0
$139$	−2.00000	−0.169638	−0.0848189	−	0.996396i	$-0.527031\pi$
$139$	−2.00000	−0.169638	−0.0848189	+	0.996396i	$0.527031\pi$
$140$	0	0
$141$	−6.00000	−0.505291
$142$	0	0
$143$	−9.00000	−0.752618
$144$	0	0
$145$	−6.00000	−0.498273
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	0	0
$153$	1.00000	0.0808452
$154$	0	0
$155$	−10.0000	−0.803219
$156$	0	0
$157$	15.0000	1.19713	0.598565	−	0.801074i	$-0.295738\pi$
$157$	15.0000	1.19713	0.598565	+	0.801074i	$0.295738\pi$
$158$	0	0
$159$	−10.0000	−0.793052
$160$	0	0
$161$	−7.00000	−0.551677
$162$	0	0
$163$	−14.0000	−1.09656	−0.548282	−	0.836293i	$-0.684718\pi$
$163$	−14.0000	−1.09656	−0.548282	+	0.836293i	$0.684718\pi$
$164$	0	0
$165$	−3.00000	−0.233550
$166$	0	0
$167$	−7.00000	−0.541676	−0.270838	−	0.962625i	$-0.587301\pi$
$167$	−7.00000	−0.541676	−0.270838	+	0.962625i	$0.587301\pi$
$168$	0	0
$169$	−4.00000	−0.307692
$170$	0	0
$171$	−3.00000	−0.229416
$172$	0	0
$173$	−11.0000	−0.836315	−0.418157	−	0.908375i	$-0.637324\pi$
$173$	−11.0000	−0.836315	−0.418157	+	0.908375i	$0.637324\pi$
$174$	0	0
$175$	−4.00000	−0.302372
$176$	0	0
$177$	2.00000	0.150329
$178$	0	0
$179$	14.0000	1.04641	0.523205	−	0.852207i	$-0.324736\pi$
$179$	14.0000	1.04641	0.523205	+	0.852207i	$0.324736\pi$
$180$	0	0
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	4.00000	0.294086
$186$	0	0
$187$	−3.00000	−0.219382
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	2.00000	0.144715	0.0723575	−	0.997379i	$-0.476948\pi$
$191$	2.00000	0.144715	0.0723575	+	0.997379i	$0.476948\pi$
$192$	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
$197$	−3.00000	−0.213741	−0.106871	−	0.994273i	$-0.534083\pi$
$197$	−3.00000	−0.213741	−0.106871	+	0.994273i	$0.534083\pi$
$198$	0	0
$199$	12.0000	0.850657	0.425329	−	0.905039i	$-0.360158\pi$
$199$	12.0000	0.850657	0.425329	+	0.905039i	$0.360158\pi$
$200$	0	0
$201$	12.0000	0.846415
$202$	0	0
$203$	−6.00000	−0.421117
$204$	0	0
$205$	−9.00000	−0.628587
$206$	0	0
$207$	−7.00000	−0.486534
$208$	0	0
$209$	9.00000	0.622543
$210$	0	0
$211$	−26.0000	−1.78991	−0.894957	−	0.446153i	$-0.852794\pi$
$211$	−26.0000	−1.78991	−0.894957	+	0.446153i	$0.852794\pi$
$212$	0	0
$213$	12.0000	0.822226
$214$	0	0
$215$	−9.00000	−0.613795
$216$	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
$223$	5.00000	0.334825	0.167412	−	0.985887i	$-0.446459\pi$
$223$	5.00000	0.334825	0.167412	+	0.985887i	$0.446459\pi$
$224$	0	0
$225$	−4.00000	−0.266667
$226$	0	0
$227$	−17.0000	−1.12833	−0.564165	−	0.825662i	$-0.690802\pi$
$227$	−17.0000	−1.12833	−0.564165	+	0.825662i	$0.690802\pi$
$228$	0	0
$229$	−26.0000	−1.71813	−0.859064	−	0.511868i	$-0.828954\pi$
$229$	−26.0000	−1.71813	−0.859064	+	0.511868i	$0.828954\pi$
$230$	0	0
$231$	−3.00000	−0.197386
$232$	0	0
$233$	3.00000	0.196537	0.0982683	−	0.995160i	$-0.468670\pi$
$233$	3.00000	0.196537	0.0982683	+	0.995160i	$0.468670\pi$
$234$	0	0
$235$	−6.00000	−0.391397
$236$	0	0
$237$	−10.0000	−0.649570
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	−28.0000	−1.80364	−0.901819	−	0.432113i	$-0.857768\pi$
$241$	−28.0000	−1.80364	−0.901819	+	0.432113i	$0.857768\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	−9.00000	−0.572656
$248$	0	0
$249$	−10.0000	−0.633724
$250$	0	0
$251$	24.0000	1.51487	0.757433	−	0.652913i	$-0.226453\pi$
$251$	24.0000	1.51487	0.757433	+	0.652913i	$0.226453\pi$
$252$	0	0
$253$	21.0000	1.32026
$254$	0	0
$255$	1.00000	0.0626224
$256$	0	0
$257$	−12.0000	−0.748539	−0.374270	−	0.927320i	$-0.622107\pi$
$257$	−12.0000	−0.748539	−0.374270	+	0.927320i	$0.622107\pi$
$258$	0	0
$259$	4.00000	0.248548
$260$	0	0
$261$	−6.00000	−0.371391
$262$	0	0
$263$	16.0000	0.986602	0.493301	−	0.869859i	$-0.335790\pi$
$263$	16.0000	0.986602	0.493301	+	0.869859i	$0.335790\pi$
$264$	0	0
$265$	−10.0000	−0.614295
$266$	0	0
$267$	−4.00000	−0.244796
$268$	0	0
$269$	−21.0000	−1.28039	−0.640196	−	0.768211i	$-0.721147\pi$
$269$	−21.0000	−1.28039	−0.640196	+	0.768211i	$0.721147\pi$
$270$	0	0
$271$	9.00000	0.546711	0.273356	−	0.961913i	$-0.411866\pi$
$271$	9.00000	0.546711	0.273356	+	0.961913i	$0.411866\pi$
$272$	0	0
$273$	3.00000	0.181568
$274$	0	0
$275$	12.0000	0.723627
$276$	0	0
$277$	−18.0000	−1.08152	−0.540758	−	0.841178i	$-0.681862\pi$
$277$	−18.0000	−1.08152	−0.540758	+	0.841178i	$0.681862\pi$
$278$	0	0
$279$	−10.0000	−0.598684
$280$	0	0
$281$	−16.0000	−0.954480	−0.477240	−	0.878773i	$-0.658363\pi$
$281$	−16.0000	−0.954480	−0.477240	+	0.878773i	$0.658363\pi$
$282$	0	0
$283$	22.0000	1.30776	0.653882	−	0.756596i	$-0.273139\pi$
$283$	22.0000	1.30776	0.653882	+	0.756596i	$0.273139\pi$
$284$	0	0
$285$	−3.00000	−0.177705
$286$	0	0
$287$	−9.00000	−0.531253
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	8.00000	0.468968
$292$	0	0
$293$	4.00000	0.233682	0.116841	−	0.993151i	$-0.462723\pi$
$293$	4.00000	0.233682	0.116841	+	0.993151i	$0.462723\pi$
$294$	0	0
$295$	2.00000	0.116445
$296$	0	0
$297$	−3.00000	−0.174078
$298$	0	0
$299$	−21.0000	−1.21446
$300$	0	0
$301$	−9.00000	−0.518751
$302$	0	0
$303$	12.0000	0.689382
$304$	0	0
$305$	0	0
$306$	0	0
$307$	20.0000	1.14146	0.570730	−	0.821138i	$-0.306660\pi$
$307$	20.0000	1.14146	0.570730	+	0.821138i	$0.306660\pi$
$308$	0	0
$309$	−1.00000	−0.0568880
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	20.0000	1.13047	0.565233	−	0.824931i	$-0.308786\pi$
$313$	20.0000	1.13047	0.565233	+	0.824931i	$0.308786\pi$
$314$	0	0
$315$	1.00000	0.0563436
$316$	0	0
$317$	30.0000	1.68497	0.842484	−	0.538721i	$-0.181092\pi$
$317$	30.0000	1.68497	0.842484	+	0.538721i	$0.181092\pi$
$318$	0	0
$319$	18.0000	1.00781
$320$	0	0
$321$	17.0000	0.948847
$322$	0	0
$323$	−3.00000	−0.166924
$324$	0	0
$325$	−12.0000	−0.665640
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−6.00000	−0.330791
$330$	0	0
$331$	−3.00000	−0.164895	−0.0824475	−	0.996595i	$-0.526274\pi$
$331$	−3.00000	−0.164895	−0.0824475	+	0.996595i	$0.526274\pi$
$332$	0	0
$333$	4.00000	0.219199
$334$	0	0
$335$	12.0000	0.655630
$336$	0	0
$337$	2.00000	0.108947	0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000	0.108947	0.0544735	+	0.998515i	$0.482652\pi$
$338$	0	0
$339$	9.00000	0.488813
$340$	0	0
$341$	30.0000	1.62459
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−7.00000	−0.376867
$346$	0	0
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	0	0
$349$	17.0000	0.909989	0.454995	−	0.890494i	$-0.349641\pi$
$349$	17.0000	0.909989	0.454995	+	0.890494i	$0.349641\pi$
$350$	0	0
$351$	3.00000	0.160128
$352$	0	0
$353$	−18.0000	−0.958043	−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000	−0.958043	−0.479022	+	0.877803i	$0.659008\pi$
$354$	0	0
$355$	12.0000	0.636894
$356$	0	0
$357$	1.00000	0.0529256
$358$	0	0
$359$	32.0000	1.68890	0.844448	−	0.535638i	$-0.179929\pi$
$359$	32.0000	1.68890	0.844448	+	0.535638i	$0.179929\pi$
$360$	0	0
$361$	−10.0000	−0.526316
$362$	0	0
$363$	−2.00000	−0.104973
$364$	0	0
$365$	6.00000	0.314054
$366$	0	0
$367$	4.00000	0.208798	0.104399	−	0.994535i	$-0.466708\pi$
$367$	4.00000	0.208798	0.104399	+	0.994535i	$0.466708\pi$
$368$	0	0
$369$	−9.00000	−0.468521
$370$	0	0
$371$	−10.0000	−0.519174
$372$	0	0
$373$	10.0000	0.517780	0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000	0.517780	0.258890	+	0.965907i	$0.416643\pi$
$374$	0	0
$375$	−9.00000	−0.464758
$376$	0	0
$377$	−18.0000	−0.927047
$378$	0	0
$379$	−4.00000	−0.205466	−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000	−0.205466	−0.102733	+	0.994709i	$0.532759\pi$
$380$	0	0
$381$	13.0000	0.666010
$382$	0	0
$383$	16.0000	0.817562	0.408781	−	0.912633i	$-0.365954\pi$
$383$	16.0000	0.817562	0.408781	+	0.912633i	$0.365954\pi$
$384$	0	0
$385$	−3.00000	−0.152894
$386$	0	0
$387$	−9.00000	−0.457496
$388$	0	0
$389$	−24.0000	−1.21685	−0.608424	−	0.793612i	$-0.708198\pi$
$389$	−24.0000	−1.21685	−0.608424	+	0.793612i	$0.708198\pi$
$390$	0	0
$391$	−7.00000	−0.354005
$392$	0	0
$393$	−17.0000	−0.857537
$394$	0	0
$395$	−10.0000	−0.503155
$396$	0	0
$397$	0	0		−	1.00000i	$-0.5\pi$
$397$	0	0			1.00000i	$0.5\pi$
$398$	0	0
$399$	−3.00000	−0.150188
$400$	0	0
$401$	−25.0000	−1.24844	−0.624220	−	0.781248i	$-0.714583\pi$
$401$	−25.0000	−1.24844	−0.624220	+	0.781248i	$0.714583\pi$
$402$	0	0
$403$	−30.0000	−1.49441
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	−12.0000	−0.594818
$408$	0	0
$409$	−23.0000	−1.13728	−0.568638	−	0.822588i	$-0.692530\pi$
$409$	−23.0000	−1.13728	−0.568638	+	0.822588i	$0.692530\pi$
$410$	0	0
$411$	2.00000	0.0986527
$412$	0	0
$413$	2.00000	0.0984136
$414$	0	0
$415$	−10.0000	−0.490881
$416$	0	0
$417$	−2.00000	−0.0979404
$418$	0	0
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	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
$423$	−6.00000	−0.291730
$424$	0	0
$425$	−4.00000	−0.194029
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−9.00000	−0.434524
$430$	0	0
$431$	−8.00000	−0.385346	−0.192673	−	0.981263i	$-0.561716\pi$
$431$	−8.00000	−0.385346	−0.192673	+	0.981263i	$0.561716\pi$
$432$	0	0
$433$	11.0000	0.528626	0.264313	−	0.964437i	$-0.414855\pi$
$433$	11.0000	0.528626	0.264313	+	0.964437i	$0.414855\pi$
$434$	0	0
$435$	−6.00000	−0.287678
$436$	0	0
$437$	21.0000	1.00457
$438$	0	0
$439$	−16.0000	−0.763638	−0.381819	−	0.924237i	$-0.624702\pi$
$439$	−16.0000	−0.763638	−0.381819	+	0.924237i	$0.624702\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	14.0000	0.665160	0.332580	−	0.943075i	$-0.392081\pi$
$443$	14.0000	0.665160	0.332580	+	0.943075i	$0.392081\pi$
$444$	0	0
$445$	−4.00000	−0.189618
$446$	0	0
$447$	−6.00000	−0.283790
$448$	0	0
$449$	−26.0000	−1.22702	−0.613508	−	0.789689i	$-0.710242\pi$
$449$	−26.0000	−1.22702	−0.613508	+	0.789689i	$0.710242\pi$
$450$	0	0
$451$	27.0000	1.27138
$452$	0	0
$453$	16.0000	0.751746
$454$	0	0
$455$	3.00000	0.140642
$456$	0	0
$457$	13.0000	0.608114	0.304057	−	0.952654i	$-0.401659\pi$
$457$	13.0000	0.608114	0.304057	+	0.952654i	$0.401659\pi$
$458$	0	0
$459$	1.00000	0.0466760
$460$	0	0
$461$	6.00000	0.279448	0.139724	−	0.990190i	$-0.455378\pi$
$461$	6.00000	0.279448	0.139724	+	0.990190i	$0.455378\pi$
$462$	0	0
$463$	−8.00000	−0.371792	−0.185896	−	0.982569i	$-0.559519\pi$
$463$	−8.00000	−0.371792	−0.185896	+	0.982569i	$0.559519\pi$
$464$	0	0
$465$	−10.0000	−0.463739
$466$	0	0
$467$	−38.0000	−1.75843	−0.879215	−	0.476425i	$-0.841932\pi$
$467$	−38.0000	−1.75843	−0.879215	+	0.476425i	$0.841932\pi$
$468$	0	0
$469$	12.0000	0.554109
$470$	0	0
$471$	15.0000	0.691164
$472$	0	0
$473$	27.0000	1.24146
$474$	0	0
$475$	12.0000	0.550598
$476$	0	0
$477$	−10.0000	−0.457869
$478$	0	0
$479$	41.0000	1.87334	0.936669	−	0.350216i	$-0.113892\pi$
$479$	41.0000	1.87334	0.936669	+	0.350216i	$0.113892\pi$
$480$	0	0
$481$	12.0000	0.547153
$482$	0	0
$483$	−7.00000	−0.318511
$484$	0	0
$485$	8.00000	0.363261
$486$	0	0
$487$	−2.00000	−0.0906287	−0.0453143	−	0.998973i	$-0.514429\pi$
$487$	−2.00000	−0.0906287	−0.0453143	+	0.998973i	$0.514429\pi$
$488$	0	0
$489$	−14.0000	−0.633102
$490$	0	0
$491$	−34.0000	−1.53440	−0.767199	−	0.641409i	$-0.778350\pi$
$491$	−34.0000	−1.53440	−0.767199	+	0.641409i	$0.778350\pi$
$492$	0	0
$493$	−6.00000	−0.270226
$494$	0	0
$495$	−3.00000	−0.134840
$496$	0	0
$497$	12.0000	0.538274
$498$	0	0
$499$	−26.0000	−1.16392	−0.581960	−	0.813217i	$-0.697714\pi$
$499$	−26.0000	−1.16392	−0.581960	+	0.813217i	$0.697714\pi$
$500$	0	0
$501$	−7.00000	−0.312737
$502$	0	0
$503$	−21.0000	−0.936344	−0.468172	−	0.883637i	$-0.655087\pi$
$503$	−21.0000	−0.936344	−0.468172	+	0.883637i	$0.655087\pi$
$504$	0	0
$505$	12.0000	0.533993
$506$	0	0
$507$	−4.00000	−0.177646
$508$	0	0
$509$	20.0000	0.886484	0.443242	−	0.896402i	$-0.353828\pi$
$509$	20.0000	0.886484	0.443242	+	0.896402i	$0.353828\pi$
$510$	0	0
$511$	6.00000	0.265424
$512$	0	0
$513$	−3.00000	−0.132453
$514$	0	0
$515$	−1.00000	−0.0440653
$516$	0	0
$517$	18.0000	0.791639
$518$	0	0
$519$	−11.0000	−0.482846
$520$	0	0
$521$	−17.0000	−0.744784	−0.372392	−	0.928076i	$-0.621462\pi$
$521$	−17.0000	−0.744784	−0.372392	+	0.928076i	$0.621462\pi$
$522$	0	0
$523$	−20.0000	−0.874539	−0.437269	−	0.899331i	$-0.644054\pi$
$523$	−20.0000	−0.874539	−0.437269	+	0.899331i	$0.644054\pi$
$524$	0	0
$525$	−4.00000	−0.174574
$526$	0	0
$527$	−10.0000	−0.435607
$528$	0	0
$529$	26.0000	1.13043
$530$	0	0
$531$	2.00000	0.0867926
$532$	0	0
$533$	−27.0000	−1.16950
$534$	0	0
$535$	17.0000	0.734974
$536$	0	0
$537$	14.0000	0.604145
$538$	0	0
$539$	−3.00000	−0.129219
$540$	0	0
$541$	−4.00000	−0.171973	−0.0859867	−	0.996296i	$-0.527404\pi$
$541$	−4.00000	−0.171973	−0.0859867	+	0.996296i	$0.527404\pi$
$542$	0	0
$543$	14.0000	0.600798
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−16.0000	−0.684111	−0.342055	−	0.939680i	$-0.611123\pi$
$547$	−16.0000	−0.684111	−0.342055	+	0.939680i	$0.611123\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	18.0000	0.766826
$552$	0	0
$553$	−10.0000	−0.425243
$554$	0	0
$555$	4.00000	0.169791
$556$	0	0
$557$	18.0000	0.762684	0.381342	−	0.924434i	$-0.375462\pi$
$557$	18.0000	0.762684	0.381342	+	0.924434i	$0.375462\pi$
$558$	0	0
$559$	−27.0000	−1.14198
$560$	0	0
$561$	−3.00000	−0.126660
$562$	0	0
$563$	−18.0000	−0.758610	−0.379305	−	0.925272i	$-0.623837\pi$
$563$	−18.0000	−0.758610	−0.379305	+	0.925272i	$0.623837\pi$
$564$	0	0
$565$	9.00000	0.378633
$566$	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
$571$	−32.0000	−1.33916	−0.669579	−	0.742741i	$-0.733526\pi$
$571$	−32.0000	−1.33916	−0.669579	+	0.742741i	$0.733526\pi$
$572$	0	0
$573$	2.00000	0.0835512
$574$	0	0
$575$	28.0000	1.16768
$576$	0	0
$577$	21.0000	0.874241	0.437121	−	0.899403i	$-0.355998\pi$
$577$	21.0000	0.874241	0.437121	+	0.899403i	$0.355998\pi$
$578$	0	0
$579$	18.0000	0.748054
$580$	0	0
$581$	−10.0000	−0.414870
$582$	0	0
$583$	30.0000	1.24247
$584$	0	0
$585$	3.00000	0.124035
$586$	0	0
$587$	−40.0000	−1.65098	−0.825488	−	0.564419i	$-0.809100\pi$
$587$	−40.0000	−1.65098	−0.825488	+	0.564419i	$0.809100\pi$
$588$	0	0
$589$	30.0000	1.23613
$590$	0	0
$591$	−3.00000	−0.123404
$592$	0	0
$593$	−46.0000	−1.88899	−0.944497	−	0.328521i	$-0.893450\pi$
$593$	−46.0000	−1.88899	−0.944497	+	0.328521i	$0.893450\pi$
$594$	0	0
$595$	1.00000	0.0409960
$596$	0	0
$597$	12.0000	0.491127
$598$	0	0
$599$	−30.0000	−1.22577	−0.612883	−	0.790173i	$-0.709990\pi$
$599$	−30.0000	−1.22577	−0.612883	+	0.790173i	$0.709990\pi$
$600$	0	0
$601$	14.0000	0.571072	0.285536	−	0.958368i	$-0.407828\pi$
$601$	14.0000	0.571072	0.285536	+	0.958368i	$0.407828\pi$
$602$	0	0
$603$	12.0000	0.488678
$604$	0	0
$605$	−2.00000	−0.0813116
$606$	0	0
$607$	30.0000	1.21766	0.608831	−	0.793300i	$-0.291639\pi$
$607$	30.0000	1.21766	0.608831	+	0.793300i	$0.291639\pi$
$608$	0	0
$609$	−6.00000	−0.243132
$610$	0	0
$611$	−18.0000	−0.728202
$612$	0	0
$613$	11.0000	0.444286	0.222143	−	0.975014i	$-0.428695\pi$
$613$	11.0000	0.444286	0.222143	+	0.975014i	$0.428695\pi$
$614$	0	0
$615$	−9.00000	−0.362915
$616$	0	0
$617$	22.0000	0.885687	0.442843	−	0.896599i	$-0.353970\pi$
$617$	22.0000	0.885687	0.442843	+	0.896599i	$0.353970\pi$
$618$	0	0
$619$	−10.0000	−0.401934	−0.200967	−	0.979598i	$-0.564408\pi$
$619$	−10.0000	−0.401934	−0.200967	+	0.979598i	$0.564408\pi$
$620$	0	0
$621$	−7.00000	−0.280900
$622$	0	0
$623$	−4.00000	−0.160257
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	9.00000	0.359425
$628$	0	0
$629$	4.00000	0.159490
$630$	0	0
$631$	13.0000	0.517522	0.258761	−	0.965941i	$-0.416686\pi$
$631$	13.0000	0.517522	0.258761	+	0.965941i	$0.416686\pi$
$632$	0	0
$633$	−26.0000	−1.03341
$634$	0	0
$635$	13.0000	0.515889
$636$	0	0
$637$	3.00000	0.118864
$638$	0	0
$639$	12.0000	0.474713
$640$	0	0
$641$	−15.0000	−0.592464	−0.296232	−	0.955116i	$-0.595730\pi$
$641$	−15.0000	−0.592464	−0.296232	+	0.955116i	$0.595730\pi$
$642$	0	0
$643$	24.0000	0.946468	0.473234	−	0.880937i	$-0.343087\pi$
$643$	24.0000	0.946468	0.473234	+	0.880937i	$0.343087\pi$
$644$	0	0
$645$	−9.00000	−0.354375
$646$	0	0
$647$	18.0000	0.707653	0.353827	−	0.935311i	$-0.384880\pi$
$647$	18.0000	0.707653	0.353827	+	0.935311i	$0.384880\pi$
$648$	0	0
$649$	−6.00000	−0.235521
$650$	0	0
$651$	−10.0000	−0.391931
$652$	0	0
$653$	−27.0000	−1.05659	−0.528296	−	0.849060i	$-0.677169\pi$
$653$	−27.0000	−1.05659	−0.528296	+	0.849060i	$0.677169\pi$
$654$	0	0
$655$	−17.0000	−0.664245
$656$	0	0
$657$	6.00000	0.234082
$658$	0	0
$659$	−6.00000	−0.233727	−0.116863	−	0.993148i	$-0.537284\pi$
$659$	−6.00000	−0.233727	−0.116863	+	0.993148i	$0.537284\pi$
$660$	0	0
$661$	−11.0000	−0.427850	−0.213925	−	0.976850i	$-0.568625\pi$
$661$	−11.0000	−0.427850	−0.213925	+	0.976850i	$0.568625\pi$
$662$	0	0
$663$	3.00000	0.116510
$664$	0	0
$665$	−3.00000	−0.116335
$666$	0	0
$667$	42.0000	1.62625
$668$	0	0
$669$	5.00000	0.193311
$670$	0	0
$671$	0	0
$672$	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
$675$	−4.00000	−0.153960
$676$	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
$679$	8.00000	0.307012
$680$	0	0
$681$	−17.0000	−0.651441
$682$	0	0
$683$	27.0000	1.03313	0.516563	−	0.856249i	$-0.327211\pi$
$683$	27.0000	1.03313	0.516563	+	0.856249i	$0.327211\pi$
$684$	0	0
$685$	2.00000	0.0764161
$686$	0	0
$687$	−26.0000	−0.991962
$688$	0	0
$689$	−30.0000	−1.14291
$690$	0	0
$691$	0	0		−	1.00000i	$-0.5\pi$
$691$	0	0			1.00000i	$0.5\pi$
$692$	0	0
$693$	−3.00000	−0.113961
$694$	0	0
$695$	−2.00000	−0.0758643
$696$	0	0
$697$	−9.00000	−0.340899
$698$	0	0
$699$	3.00000	0.113470
$700$	0	0
$701$	−16.0000	−0.604312	−0.302156	−	0.953259i	$-0.597706\pi$
$701$	−16.0000	−0.604312	−0.302156	+	0.953259i	$0.597706\pi$
$702$	0	0
$703$	−12.0000	−0.452589
$704$	0	0
$705$	−6.00000	−0.225973
$706$	0	0
$707$	12.0000	0.451306
$708$	0	0
$709$	−22.0000	−0.826227	−0.413114	−	0.910679i	$-0.635559\pi$
$709$	−22.0000	−0.826227	−0.413114	+	0.910679i	$0.635559\pi$
$710$	0	0
$711$	−10.0000	−0.375029
$712$	0	0
$713$	70.0000	2.62152
$714$	0	0
$715$	−9.00000	−0.336581
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−25.0000	−0.932343	−0.466171	−	0.884694i	$-0.654367\pi$
$719$	−25.0000	−0.932343	−0.466171	+	0.884694i	$0.654367\pi$
$720$	0	0
$721$	−1.00000	−0.0372419
$722$	0	0
$723$	−28.0000	−1.04133
$724$	0	0
$725$	24.0000	0.891338
$726$	0	0
$727$	32.0000	1.18681	0.593407	−	0.804902i	$-0.297782\pi$
$727$	32.0000	1.18681	0.593407	+	0.804902i	$0.297782\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−9.00000	−0.332877
$732$	0	0
$733$	−14.0000	−0.517102	−0.258551	−	0.965998i	$-0.583245\pi$
$733$	−14.0000	−0.517102	−0.258551	+	0.965998i	$0.583245\pi$
$734$	0	0
$735$	1.00000	0.0368856
$736$	0	0
$737$	−36.0000	−1.32608
$738$	0	0
$739$	25.0000	0.919640	0.459820	−	0.888012i	$-0.347914\pi$
$739$	25.0000	0.919640	0.459820	+	0.888012i	$0.347914\pi$
$740$	0	0
$741$	−9.00000	−0.330623
$742$	0	0
$743$	16.0000	0.586983	0.293492	−	0.955962i	$-0.405183\pi$
$743$	16.0000	0.586983	0.293492	+	0.955962i	$0.405183\pi$
$744$	0	0
$745$	−6.00000	−0.219823
$746$	0	0
$747$	−10.0000	−0.365881
$748$	0	0
$749$	17.0000	0.621166
$750$	0	0
$751$	−10.0000	−0.364905	−0.182453	−	0.983215i	$-0.558404\pi$
$751$	−10.0000	−0.364905	−0.182453	+	0.983215i	$0.558404\pi$
$752$	0	0
$753$	24.0000	0.874609
$754$	0	0
$755$	16.0000	0.582300
$756$	0	0
$757$	21.0000	0.763258	0.381629	−	0.924316i	$-0.375363\pi$
$757$	21.0000	0.763258	0.381629	+	0.924316i	$0.375363\pi$
$758$	0	0
$759$	21.0000	0.762252
$760$	0	0
$761$	14.0000	0.507500	0.253750	−	0.967270i	$-0.418336\pi$
$761$	14.0000	0.507500	0.253750	+	0.967270i	$0.418336\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	1.00000	0.0361551
$766$	0	0
$767$	6.00000	0.216647
$768$	0	0
$769$	29.0000	1.04577	0.522883	−	0.852404i	$-0.324856\pi$
$769$	29.0000	1.04577	0.522883	+	0.852404i	$0.324856\pi$
$770$	0	0
$771$	−12.0000	−0.432169
$772$	0	0
$773$	4.00000	0.143870	0.0719350	−	0.997409i	$-0.477083\pi$
$773$	4.00000	0.143870	0.0719350	+	0.997409i	$0.477083\pi$
$774$	0	0
$775$	40.0000	1.43684
$776$	0	0
$777$	4.00000	0.143499
$778$	0	0
$779$	27.0000	0.967375
$780$	0	0
$781$	−36.0000	−1.28818
$782$	0	0
$783$	−6.00000	−0.214423
$784$	0	0
$785$	15.0000	0.535373
$786$	0	0
$787$	−44.0000	−1.56843	−0.784215	−	0.620489i	$-0.786934\pi$
$787$	−44.0000	−1.56843	−0.784215	+	0.620489i	$0.786934\pi$
$788$	0	0
$789$	16.0000	0.569615
$790$	0	0
$791$	9.00000	0.320003
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−10.0000	−0.354663
$796$	0	0
$797$	−48.0000	−1.70025	−0.850124	−	0.526583i	$-0.823473\pi$
$797$	−48.0000	−1.70025	−0.850124	+	0.526583i	$0.823473\pi$
$798$	0	0
$799$	−6.00000	−0.212265
$800$	0	0
$801$	−4.00000	−0.141333
$802$	0	0
$803$	−18.0000	−0.635206
$804$	0	0
$805$	−7.00000	−0.246718
$806$	0	0
$807$	−21.0000	−0.739235
$808$	0	0
$809$	27.0000	0.949269	0.474635	−	0.880183i	$-0.342580\pi$
$809$	27.0000	0.949269	0.474635	+	0.880183i	$0.342580\pi$
$810$	0	0
$811$	54.0000	1.89620	0.948098	−	0.317978i	$-0.103004\pi$
$811$	54.0000	1.89620	0.948098	+	0.317978i	$0.103004\pi$
$812$	0	0
$813$	9.00000	0.315644
$814$	0	0
$815$	−14.0000	−0.490399
$816$	0	0
$817$	27.0000	0.944610
$818$	0	0
$819$	3.00000	0.104828
$820$	0	0
$821$	−43.0000	−1.50071	−0.750355	−	0.661035i	$-0.770118\pi$
$821$	−43.0000	−1.50071	−0.750355	+	0.661035i	$0.770118\pi$
$822$	0	0
$823$	52.0000	1.81261	0.906303	−	0.422628i	$-0.138892\pi$
$823$	52.0000	1.81261	0.906303	+	0.422628i	$0.138892\pi$
$824$	0	0
$825$	12.0000	0.417786
$826$	0	0
$827$	−13.0000	−0.452054	−0.226027	−	0.974121i	$-0.572574\pi$
$827$	−13.0000	−0.452054	−0.226027	+	0.974121i	$0.572574\pi$
$828$	0	0
$829$	−26.0000	−0.903017	−0.451509	−	0.892267i	$-0.649114\pi$
$829$	−26.0000	−0.903017	−0.451509	+	0.892267i	$0.649114\pi$
$830$	0	0
$831$	−18.0000	−0.624413
$832$	0	0
$833$	1.00000	0.0346479
$834$	0	0
$835$	−7.00000	−0.242245
$836$	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
$843$	−16.0000	−0.551069
$844$	0	0
$845$	−4.00000	−0.137604
$846$	0	0
$847$	−2.00000	−0.0687208
$848$	0	0
$849$	22.0000	0.755038
$850$	0	0
$851$	−28.0000	−0.959828
$852$	0	0
$853$	14.0000	0.479351	0.239675	−	0.970853i	$-0.422959\pi$
$853$	14.0000	0.479351	0.239675	+	0.970853i	$0.422959\pi$
$854$	0	0
$855$	−3.00000	−0.102598
$856$	0	0
$857$	18.0000	0.614868	0.307434	−	0.951569i	$-0.400530\pi$
$857$	18.0000	0.614868	0.307434	+	0.951569i	$0.400530\pi$
$858$	0	0
$859$	−4.00000	−0.136478	−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000	−0.136478	−0.0682391	+	0.997669i	$0.521738\pi$
$860$	0	0
$861$	−9.00000	−0.306719
$862$	0	0
$863$	−56.0000	−1.90626	−0.953131	−	0.302558i	$-0.902160\pi$
$863$	−56.0000	−1.90626	−0.953131	+	0.302558i	$0.902160\pi$
$864$	0	0
$865$	−11.0000	−0.374011
$866$	0	0
$867$	1.00000	0.0339618
$868$	0	0
$869$	30.0000	1.01768
$870$	0	0
$871$	36.0000	1.21981
$872$	0	0
$873$	8.00000	0.270759
$874$	0	0
$875$	−9.00000	−0.304256
$876$	0	0
$877$	−22.0000	−0.742887	−0.371444	−	0.928456i	$-0.621137\pi$
$877$	−22.0000	−0.742887	−0.371444	+	0.928456i	$0.621137\pi$
$878$	0	0
$879$	4.00000	0.134917
$880$	0	0
$881$	6.00000	0.202145	0.101073	−	0.994879i	$-0.467773\pi$
$881$	6.00000	0.202145	0.101073	+	0.994879i	$0.467773\pi$
$882$	0	0
$883$	−11.0000	−0.370179	−0.185090	−	0.982722i	$-0.559258\pi$
$883$	−11.0000	−0.370179	−0.185090	+	0.982722i	$0.559258\pi$
$884$	0	0
$885$	2.00000	0.0672293
$886$	0	0
$887$	−13.0000	−0.436497	−0.218249	−	0.975893i	$-0.570034\pi$
$887$	−13.0000	−0.436497	−0.218249	+	0.975893i	$0.570034\pi$
$888$	0	0
$889$	13.0000	0.436006
$890$	0	0
$891$	−3.00000	−0.100504
$892$	0	0
$893$	18.0000	0.602347
$894$	0	0
$895$	14.0000	0.467968
$896$	0	0
$897$	−21.0000	−0.701170
$898$	0	0
$899$	60.0000	2.00111
$900$	0	0
$901$	−10.0000	−0.333148
$902$	0	0
$903$	−9.00000	−0.299501
$904$	0	0
$905$	14.0000	0.465376
$906$	0	0
$907$	30.0000	0.996134	0.498067	−	0.867139i	$-0.334043\pi$
$907$	30.0000	0.996134	0.498067	+	0.867139i	$0.334043\pi$
$908$	0	0
$909$	12.0000	0.398015
$910$	0	0
$911$	−19.0000	−0.629498	−0.314749	−	0.949175i	$-0.601920\pi$
$911$	−19.0000	−0.629498	−0.314749	+	0.949175i	$0.601920\pi$
$912$	0	0
$913$	30.0000	0.992855
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−17.0000	−0.561389
$918$	0	0
$919$	37.0000	1.22052	0.610259	−	0.792202i	$-0.291065\pi$
$919$	37.0000	1.22052	0.610259	+	0.792202i	$0.291065\pi$
$920$	0	0
$921$	20.0000	0.659022
$922$	0	0
$923$	36.0000	1.18495
$924$	0	0
$925$	−16.0000	−0.526077
$926$	0	0
$927$	−1.00000	−0.0328443
$928$	0	0
$929$	−51.0000	−1.67326	−0.836628	−	0.547772i	$-0.815476\pi$
$929$	−51.0000	−1.67326	−0.836628	+	0.547772i	$0.815476\pi$
$930$	0	0
$931$	−3.00000	−0.0983210
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−3.00000	−0.0981105
$936$	0	0
$937$	−22.0000	−0.718709	−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000	−0.718709	−0.359354	+	0.933201i	$0.617003\pi$
$938$	0	0
$939$	20.0000	0.652675
$940$	0	0
$941$	−18.0000	−0.586783	−0.293392	−	0.955992i	$-0.594784\pi$
$941$	−18.0000	−0.586783	−0.293392	+	0.955992i	$0.594784\pi$
$942$	0	0
$943$	63.0000	2.05156
$944$	0	0
$945$	1.00000	0.0325300
$946$	0	0
$947$	−52.0000	−1.68977	−0.844886	−	0.534946i	$-0.820332\pi$
$947$	−52.0000	−1.68977	−0.844886	+	0.534946i	$0.820332\pi$
$948$	0	0
$949$	18.0000	0.584305
$950$	0	0
$951$	30.0000	0.972817
$952$	0	0
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	0	0
$955$	2.00000	0.0647185
$956$	0	0
$957$	18.0000	0.581857
$958$	0	0
$959$	2.00000	0.0645834
$960$	0	0
$961$	69.0000	2.22581
$962$	0	0
$963$	17.0000	0.547817
$964$	0	0
$965$	18.0000	0.579441
$966$	0	0
$967$	23.0000	0.739630	0.369815	−	0.929105i	$-0.379421\pi$
$967$	23.0000	0.739630	0.369815	+	0.929105i	$0.379421\pi$
$968$	0	0
$969$	−3.00000	−0.0963739
$970$	0	0
$971$	−54.0000	−1.73294	−0.866471	−	0.499227i	$-0.833617\pi$
$971$	−54.0000	−1.73294	−0.866471	+	0.499227i	$0.833617\pi$
$972$	0	0
$973$	−2.00000	−0.0641171
$974$	0	0
$975$	−12.0000	−0.384308
$976$	0	0
$977$	−18.0000	−0.575871	−0.287936	−	0.957650i	$-0.592969\pi$
$977$	−18.0000	−0.575871	−0.287936	+	0.957650i	$0.592969\pi$
$978$	0	0
$979$	12.0000	0.383522
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−21.0000	−0.669796	−0.334898	−	0.942254i	$-0.608702\pi$
$983$	−21.0000	−0.669796	−0.334898	+	0.942254i	$0.608702\pi$
$984$	0	0
$985$	−3.00000	−0.0955879
$986$	0	0
$987$	−6.00000	−0.190982
$988$	0	0
$989$	63.0000	2.00328
$990$	0	0
$991$	−52.0000	−1.65183	−0.825917	−	0.563791i	$-0.809342\pi$
$991$	−52.0000	−1.65183	−0.825917	+	0.563791i	$0.809342\pi$
$992$	0	0
$993$	−3.00000	−0.0952021
$994$	0	0
$995$	12.0000	0.380426
$996$	0	0
$997$	−18.0000	−0.570066	−0.285033	−	0.958518i	$-0.592005\pi$
$997$	−18.0000	−0.570066	−0.285033	+	0.958518i	$0.592005\pi$
$998$	0	0
$999$	4.00000	0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5712.2.a.w.1.1		1
4.3	odd	2		357.2.a.b.1.1	✓	1
12.11	even	2		1071.2.a.b.1.1		1
20.19	odd	2		8925.2.a.s.1.1		1
28.27	even	2		2499.2.a.h.1.1		1
68.67	odd	2		6069.2.a.d.1.1		1
84.83	odd	2		7497.2.a.h.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
357.2.a.b.1.1	✓	1	4.3	odd	2
1071.2.a.b.1.1		1	12.11	even	2
2499.2.a.h.1.1		1	28.27	even	2
5712.2.a.w.1.1		1	1.1	even	1	trivial
6069.2.a.d.1.1		1	68.67	odd	2
7497.2.a.h.1.1		1	84.83	odd	2
8925.2.a.s.1.1		1	20.19	odd	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:	\( N \)	\(=\)	\( 5712 = 2^{4} \cdot 3 \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5712.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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