LMFDB - Embedded newform 5712.2.a.v.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 714)
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)

$f(q)$

$=$

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

$q+1.00000 q^{3} +1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} -3.00000 q^{11} -3.00000 q^{13} +1.00000 q^{15} +1.00000 q^{17} -6.00000 q^{19} +1.00000 q^{21} +2.00000 q^{23} -4.00000 q^{25} +1.00000 q^{27} +6.00000 q^{29} -4.00000 q^{31} -3.00000 q^{33} +1.00000 q^{35} -11.0000 q^{37} -3.00000 q^{39} +12.0000 q^{41} -3.00000 q^{43} +1.00000 q^{45} -12.0000 q^{47} +1.00000 q^{49} +1.00000 q^{51} +5.00000 q^{53} -3.00000 q^{55} -6.00000 q^{57} -4.00000 q^{59} -6.00000 q^{61} +1.00000 q^{63} -3.00000 q^{65} -9.00000 q^{67} +2.00000 q^{69} -12.0000 q^{71} +15.0000 q^{73} -4.00000 q^{75} -3.00000 q^{77} +11.0000 q^{79} +1.00000 q^{81} -7.00000 q^{83} +1.00000 q^{85} +6.00000 q^{87} -13.0000 q^{89} -3.00000 q^{91} -4.00000 q^{93} -6.00000 q^{95} +11.0000 q^{97} -3.00000 q^{99} +O(q^{100})$

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	$-0.636577\pi$
$13$	−3.00000	−0.832050	−0.416025	+	0.909353i	$0.636577\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	2.00000	0.417029	0.208514	−	0.978019i	$-0.433137\pi$
$23$	2.00000	0.417029	0.208514	+	0.978019i	$0.433137\pi$
$24$	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	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	0	0
$33$	−3.00000	−0.522233
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	−11.0000	−1.80839	−0.904194	−	0.427121i	$-0.859528\pi$
$37$	−11.0000	−1.80839	−0.904194	+	0.427121i	$0.859528\pi$
$38$	0	0
$39$	−3.00000	−0.480384
$40$	0	0
$41$	12.0000	1.87409	0.937043	−	0.349215i	$-0.113552\pi$
$41$	12.0000	1.87409	0.937043	+	0.349215i	$0.113552\pi$
$42$	0	0
$43$	−3.00000	−0.457496	−0.228748	−	0.973486i	$-0.573463\pi$
$43$	−3.00000	−0.457496	−0.228748	+	0.973486i	$0.573463\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	−12.0000	−1.75038	−0.875190	−	0.483779i	$-0.839264\pi$
$47$	−12.0000	−1.75038	−0.875190	+	0.483779i	$0.839264\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	1.00000	0.140028
$52$	0	0
$53$	5.00000	0.686803	0.343401	−	0.939189i	$-0.388421\pi$
$53$	5.00000	0.686803	0.343401	+	0.939189i	$0.388421\pi$
$54$	0	0
$55$	−3.00000	−0.404520
$56$	0	0
$57$	−6.00000	−0.794719
$58$	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
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	−3.00000	−0.372104
$66$	0	0
$67$	−9.00000	−1.09952	−0.549762	−	0.835321i	$-0.685282\pi$
$67$	−9.00000	−1.09952	−0.549762	+	0.835321i	$0.685282\pi$
$68$	0	0
$69$	2.00000	0.240772
$70$	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
$73$	15.0000	1.75562	0.877809	−	0.479012i	$-0.159005\pi$
$73$	15.0000	1.75562	0.877809	+	0.479012i	$0.159005\pi$
$74$	0	0
$75$	−4.00000	−0.461880
$76$	0	0
$77$	−3.00000	−0.341882
$78$	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
$83$	−7.00000	−0.768350	−0.384175	−	0.923260i	$-0.625514\pi$
$83$	−7.00000	−0.768350	−0.384175	+	0.923260i	$0.625514\pi$
$84$	0	0
$85$	1.00000	0.108465
$86$	0	0
$87$	6.00000	0.643268
$88$	0	0
$89$	−13.0000	−1.37800	−0.688999	−	0.724763i	$-0.741949\pi$
$89$	−13.0000	−1.37800	−0.688999	+	0.724763i	$0.741949\pi$
$90$	0	0
$91$	−3.00000	−0.314485
$92$	0	0
$93$	−4.00000	−0.414781
$94$	0	0
$95$	−6.00000	−0.615587
$96$	0	0
$97$	11.0000	1.11688	0.558440	−	0.829545i	$-0.311400\pi$
$97$	11.0000	1.11688	0.558440	+	0.829545i	$0.311400\pi$
$98$	0	0
$99$	−3.00000	−0.301511
$100$	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
$103$	−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
$107$	−16.0000	−1.54678	−0.773389	−	0.633932i	$-0.781440\pi$
$107$	−16.0000	−1.54678	−0.773389	+	0.633932i	$0.781440\pi$
$108$	0	0
$109$	6.00000	0.574696	0.287348	−	0.957826i	$-0.407226\pi$
$109$	6.00000	0.574696	0.287348	+	0.957826i	$0.407226\pi$
$110$	0	0
$111$	−11.0000	−1.04407
$112$	0	0
$113$	−15.0000	−1.41108	−0.705541	−	0.708669i	$-0.749296\pi$
$113$	−15.0000	−1.41108	−0.705541	+	0.708669i	$0.749296\pi$
$114$	0	0
$115$	2.00000	0.186501
$116$	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
$123$	12.0000	1.08200
$124$	0	0
$125$	−9.00000	−0.804984
$126$	0	0
$127$	−2.00000	−0.177471	−0.0887357	−	0.996055i	$-0.528283\pi$
$127$	−2.00000	−0.177471	−0.0887357	+	0.996055i	$0.528283\pi$
$128$	0	0
$129$	−3.00000	−0.264135
$130$	0	0
$131$	10.0000	0.873704	0.436852	−	0.899533i	$-0.356093\pi$
$131$	10.0000	0.873704	0.436852	+	0.899533i	$0.356093\pi$
$132$	0	0
$133$	−6.00000	−0.520266
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	8.00000	0.683486	0.341743	−	0.939793i	$-0.388983\pi$
$137$	8.00000	0.683486	0.341743	+	0.939793i	$0.388983\pi$
$138$	0	0
$139$	1.00000	0.0848189	0.0424094	−	0.999100i	$-0.486497\pi$
$139$	1.00000	0.0848189	0.0424094	+	0.999100i	$0.486497\pi$
$140$	0	0
$141$	−12.0000	−1.01058
$142$	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
$149$	−15.0000	−1.22885	−0.614424	−	0.788976i	$-0.710612\pi$
$149$	−15.0000	−1.22885	−0.614424	+	0.788976i	$0.710612\pi$
$150$	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
$155$	−4.00000	−0.321288
$156$	0	0
$157$	−18.0000	−1.43656	−0.718278	−	0.695756i	$-0.755069\pi$
$157$	−18.0000	−1.43656	−0.718278	+	0.695756i	$0.755069\pi$
$158$	0	0
$159$	5.00000	0.396526
$160$	0	0
$161$	2.00000	0.157622
$162$	0	0
$163$	22.0000	1.72317	0.861586	−	0.507611i	$-0.169471\pi$
$163$	22.0000	1.72317	0.861586	+	0.507611i	$0.169471\pi$
$164$	0	0
$165$	−3.00000	−0.233550
$166$	0	0
$167$	5.00000	0.386912	0.193456	−	0.981109i	$-0.438030\pi$
$167$	5.00000	0.386912	0.193456	+	0.981109i	$0.438030\pi$
$168$	0	0
$169$	−4.00000	−0.307692
$170$	0	0
$171$	−6.00000	−0.458831
$172$	0	0
$173$	10.0000	0.760286	0.380143	−	0.924928i	$-0.375875\pi$
$173$	10.0000	0.760286	0.380143	+	0.924928i	$0.375875\pi$
$174$	0	0
$175$	−4.00000	−0.302372
$176$	0	0
$177$	−4.00000	−0.300658
$178$	0	0
$179$	−10.0000	−0.747435	−0.373718	−	0.927543i	$-0.621917\pi$
$179$	−10.0000	−0.747435	−0.373718	+	0.927543i	$0.621917\pi$
$180$	0	0
$181$	20.0000	1.48659	0.743294	−	0.668965i	$-0.233262\pi$
$181$	20.0000	1.48659	0.743294	+	0.668965i	$0.233262\pi$
$182$	0	0
$183$	−6.00000	−0.443533
$184$	0	0
$185$	−11.0000	−0.808736
$186$	0	0
$187$	−3.00000	−0.219382
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	11.0000	0.795932	0.397966	−	0.917400i	$-0.369716\pi$
$191$	11.0000	0.795932	0.397966	+	0.917400i	$0.369716\pi$
$192$	0	0
$193$	−24.0000	−1.72756	−0.863779	−	0.503871i	$-0.831909\pi$
$193$	−24.0000	−1.72756	−0.863779	+	0.503871i	$0.831909\pi$
$194$	0	0
$195$	−3.00000	−0.214834
$196$	0	0
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	−9.00000	−0.634811
$202$	0	0
$203$	6.00000	0.421117
$204$	0	0
$205$	12.0000	0.838116
$206$	0	0
$207$	2.00000	0.139010
$208$	0	0
$209$	18.0000	1.24509
$210$	0	0
$211$	−14.0000	−0.963800	−0.481900	−	0.876226i	$-0.660053\pi$
$211$	−14.0000	−0.963800	−0.481900	+	0.876226i	$0.660053\pi$
$212$	0	0
$213$	−12.0000	−0.822226
$214$	0	0
$215$	−3.00000	−0.204598
$216$	0	0
$217$	−4.00000	−0.271538
$218$	0	0
$219$	15.0000	1.01361
$220$	0	0
$221$	−3.00000	−0.201802
$222$	0	0
$223$	−16.0000	−1.07144	−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000	−1.07144	−0.535720	+	0.844396i	$0.679960\pi$
$224$	0	0
$225$	−4.00000	−0.266667
$226$	0	0
$227$	−8.00000	−0.530979	−0.265489	−	0.964114i	$-0.585534\pi$
$227$	−8.00000	−0.530979	−0.265489	+	0.964114i	$0.585534\pi$
$228$	0	0
$229$	19.0000	1.25556	0.627778	−	0.778393i	$-0.283965\pi$
$229$	19.0000	1.25556	0.627778	+	0.778393i	$0.283965\pi$
$230$	0	0
$231$	−3.00000	−0.197386
$232$	0	0
$233$	−3.00000	−0.196537	−0.0982683	−	0.995160i	$-0.531330\pi$
$233$	−3.00000	−0.196537	−0.0982683	+	0.995160i	$0.531330\pi$
$234$	0	0
$235$	−12.0000	−0.782794
$236$	0	0
$237$	11.0000	0.714527
$238$	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
$241$	14.0000	0.901819	0.450910	−	0.892570i	$-0.351100\pi$
$241$	14.0000	0.901819	0.450910	+	0.892570i	$0.351100\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	18.0000	1.14531
$248$	0	0
$249$	−7.00000	−0.443607
$250$	0	0
$251$	−9.00000	−0.568075	−0.284037	−	0.958813i	$-0.591674\pi$
$251$	−9.00000	−0.568075	−0.284037	+	0.958813i	$0.591674\pi$
$252$	0	0
$253$	−6.00000	−0.377217
$254$	0	0
$255$	1.00000	0.0626224
$256$	0	0
$257$	−27.0000	−1.68421	−0.842107	−	0.539311i	$-0.818685\pi$
$257$	−27.0000	−1.68421	−0.842107	+	0.539311i	$0.818685\pi$
$258$	0	0
$259$	−11.0000	−0.683507
$260$	0	0
$261$	6.00000	0.371391
$262$	0	0
$263$	19.0000	1.17159	0.585795	−	0.810459i	$-0.300782\pi$
$263$	19.0000	1.17159	0.585795	+	0.810459i	$0.300782\pi$
$264$	0	0
$265$	5.00000	0.307148
$266$	0	0
$267$	−13.0000	−0.795587
$268$	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
$271$	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	$-0.318138\pi$
$277$	18.0000	1.08152	0.540758	+	0.841178i	$0.318138\pi$
$278$	0	0
$279$	−4.00000	−0.239474
$280$	0	0
$281$	−22.0000	−1.31241	−0.656205	−	0.754583i	$-0.727839\pi$
$281$	−22.0000	−1.31241	−0.656205	+	0.754583i	$0.727839\pi$
$282$	0	0
$283$	13.0000	0.772770	0.386385	−	0.922338i	$-0.373724\pi$
$283$	13.0000	0.772770	0.386385	+	0.922338i	$0.373724\pi$
$284$	0	0
$285$	−6.00000	−0.355409
$286$	0	0
$287$	12.0000	0.708338
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	11.0000	0.644831
$292$	0	0
$293$	−8.00000	−0.467365	−0.233682	−	0.972313i	$-0.575078\pi$
$293$	−8.00000	−0.467365	−0.233682	+	0.972313i	$0.575078\pi$
$294$	0	0
$295$	−4.00000	−0.232889
$296$	0	0
$297$	−3.00000	−0.174078
$298$	0	0
$299$	−6.00000	−0.346989
$300$	0	0
$301$	−3.00000	−0.172917
$302$	0	0
$303$	−12.0000	−0.689382
$304$	0	0
$305$	−6.00000	−0.343559
$306$	0	0
$307$	8.00000	0.456584	0.228292	−	0.973593i	$-0.426686\pi$
$307$	8.00000	0.456584	0.228292	+	0.973593i	$0.426686\pi$
$308$	0	0
$309$	−1.00000	−0.0568880
$310$	0	0
$311$	−3.00000	−0.170114	−0.0850572	−	0.996376i	$-0.527107\pi$
$311$	−3.00000	−0.170114	−0.0850572	+	0.996376i	$0.527107\pi$
$312$	0	0
$313$	−22.0000	−1.24351	−0.621757	−	0.783210i	$-0.713581\pi$
$313$	−22.0000	−1.24351	−0.621757	+	0.783210i	$0.713581\pi$
$314$	0	0
$315$	1.00000	0.0563436
$316$	0	0
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	0	0
$319$	−18.0000	−1.00781
$320$	0	0
$321$	−16.0000	−0.893033
$322$	0	0
$323$	−6.00000	−0.333849
$324$	0	0
$325$	12.0000	0.665640
$326$	0	0
$327$	6.00000	0.331801
$328$	0	0
$329$	−12.0000	−0.661581
$330$	0	0
$331$	3.00000	0.164895	0.0824475	−	0.996595i	$-0.473726\pi$
$331$	3.00000	0.164895	0.0824475	+	0.996595i	$0.473726\pi$
$332$	0	0
$333$	−11.0000	−0.602796
$334$	0	0
$335$	−9.00000	−0.491723
$336$	0	0
$337$	14.0000	0.762629	0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000	0.762629	0.381314	+	0.924445i	$0.375472\pi$
$338$	0	0
$339$	−15.0000	−0.814688
$340$	0	0
$341$	12.0000	0.649836
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	2.00000	0.107676
$346$	0	0
$347$	−9.00000	−0.483145	−0.241573	−	0.970383i	$-0.577663\pi$
$347$	−9.00000	−0.483145	−0.241573	+	0.970383i	$0.577663\pi$
$348$	0	0
$349$	−1.00000	−0.0535288	−0.0267644	−	0.999642i	$-0.508520\pi$
$349$	−1.00000	−0.0535288	−0.0267644	+	0.999642i	$0.508520\pi$
$350$	0	0
$351$	−3.00000	−0.160128
$352$	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
$357$	1.00000	0.0529256
$358$	0	0
$359$	−16.0000	−0.844448	−0.422224	−	0.906492i	$-0.638750\pi$
$359$	−16.0000	−0.844448	−0.422224	+	0.906492i	$0.638750\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	0	0
$363$	−2.00000	−0.104973
$364$	0	0
$365$	15.0000	0.785136
$366$	0	0
$367$	−20.0000	−1.04399	−0.521996	−	0.852948i	$-0.674812\pi$
$367$	−20.0000	−1.04399	−0.521996	+	0.852948i	$0.674812\pi$
$368$	0	0
$369$	12.0000	0.624695
$370$	0	0
$371$	5.00000	0.259587
$372$	0	0
$373$	−20.0000	−1.03556	−0.517780	−	0.855514i	$-0.673242\pi$
$373$	−20.0000	−1.03556	−0.517780	+	0.855514i	$0.673242\pi$
$374$	0	0
$375$	−9.00000	−0.464758
$376$	0	0
$377$	−18.0000	−0.927047
$378$	0	0
$379$	−22.0000	−1.13006	−0.565032	−	0.825069i	$-0.691136\pi$
$379$	−22.0000	−1.13006	−0.565032	+	0.825069i	$0.691136\pi$
$380$	0	0
$381$	−2.00000	−0.102463
$382$	0	0
$383$	−20.0000	−1.02195	−0.510976	−	0.859595i	$-0.670716\pi$
$383$	−20.0000	−1.02195	−0.510976	+	0.859595i	$0.670716\pi$
$384$	0	0
$385$	−3.00000	−0.152894
$386$	0	0
$387$	−3.00000	−0.152499
$388$	0	0
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	2.00000	0.101144
$392$	0	0
$393$	10.0000	0.504433
$394$	0	0
$395$	11.0000	0.553470
$396$	0	0
$397$	12.0000	0.602263	0.301131	−	0.953583i	$-0.402636\pi$
$397$	12.0000	0.602263	0.301131	+	0.953583i	$0.402636\pi$
$398$	0	0
$399$	−6.00000	−0.300376
$400$	0	0
$401$	14.0000	0.699127	0.349563	−	0.936913i	$-0.386330\pi$
$401$	14.0000	0.699127	0.349563	+	0.936913i	$0.386330\pi$
$402$	0	0
$403$	12.0000	0.597763
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	33.0000	1.63575
$408$	0	0
$409$	−14.0000	−0.692255	−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000	−0.692255	−0.346128	+	0.938187i	$0.612504\pi$
$410$	0	0
$411$	8.00000	0.394611
$412$	0	0
$413$	−4.00000	−0.196827
$414$	0	0
$415$	−7.00000	−0.343616
$416$	0	0
$417$	1.00000	0.0489702
$418$	0	0
$419$	6.00000	0.293119	0.146560	−	0.989202i	$-0.453180\pi$
$419$	6.00000	0.293119	0.146560	+	0.989202i	$0.453180\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	0	0
$423$	−12.0000	−0.583460
$424$	0	0
$425$	−4.00000	−0.194029
$426$	0	0
$427$	−6.00000	−0.290360
$428$	0	0
$429$	9.00000	0.434524
$430$	0	0
$431$	4.00000	0.192673	0.0963366	−	0.995349i	$-0.469287\pi$
$431$	4.00000	0.192673	0.0963366	+	0.995349i	$0.469287\pi$
$432$	0	0
$433$	32.0000	1.53782	0.768911	−	0.639356i	$-0.220799\pi$
$433$	32.0000	1.53782	0.768911	+	0.639356i	$0.220799\pi$
$434$	0	0
$435$	6.00000	0.287678
$436$	0	0
$437$	−12.0000	−0.574038
$438$	0	0
$439$	32.0000	1.52728	0.763638	−	0.645644i	$-0.223411\pi$
$439$	32.0000	1.52728	0.763638	+	0.645644i	$0.223411\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−4.00000	−0.190046	−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000	−0.190046	−0.0950229	+	0.995475i	$0.530292\pi$
$444$	0	0
$445$	−13.0000	−0.616259
$446$	0	0
$447$	−15.0000	−0.709476
$448$	0	0
$449$	1.00000	0.0471929	0.0235965	−	0.999722i	$-0.492488\pi$
$449$	1.00000	0.0471929	0.0235965	+	0.999722i	$0.492488\pi$
$450$	0	0
$451$	−36.0000	−1.69517
$452$	0	0
$453$	16.0000	0.751746
$454$	0	0
$455$	−3.00000	−0.140642
$456$	0	0
$457$	−17.0000	−0.795226	−0.397613	−	0.917553i	$-0.630161\pi$
$457$	−17.0000	−0.795226	−0.397613	+	0.917553i	$0.630161\pi$
$458$	0	0
$459$	1.00000	0.0466760
$460$	0	0
$461$	−30.0000	−1.39724	−0.698620	−	0.715493i	$-0.746202\pi$
$461$	−30.0000	−1.39724	−0.698620	+	0.715493i	$0.746202\pi$
$462$	0	0
$463$	16.0000	0.743583	0.371792	−	0.928316i	$-0.378744\pi$
$463$	16.0000	0.743583	0.371792	+	0.928316i	$0.378744\pi$
$464$	0	0
$465$	−4.00000	−0.185496
$466$	0	0
$467$	−32.0000	−1.48078	−0.740392	−	0.672176i	$-0.765360\pi$
$467$	−32.0000	−1.48078	−0.740392	+	0.672176i	$0.765360\pi$
$468$	0	0
$469$	−9.00000	−0.415581
$470$	0	0
$471$	−18.0000	−0.829396
$472$	0	0
$473$	9.00000	0.413820
$474$	0	0
$475$	24.0000	1.10120
$476$	0	0
$477$	5.00000	0.228934
$478$	0	0
$479$	11.0000	0.502603	0.251301	−	0.967909i	$-0.419141\pi$
$479$	11.0000	0.502603	0.251301	+	0.967909i	$0.419141\pi$
$480$	0	0
$481$	33.0000	1.50467
$482$	0	0
$483$	2.00000	0.0910032
$484$	0	0
$485$	11.0000	0.499484
$486$	0	0
$487$	13.0000	0.589086	0.294543	−	0.955638i	$-0.404833\pi$
$487$	13.0000	0.589086	0.294543	+	0.955638i	$0.404833\pi$
$488$	0	0
$489$	22.0000	0.994874
$490$	0	0
$491$	14.0000	0.631811	0.315906	−	0.948791i	$-0.397692\pi$
$491$	14.0000	0.631811	0.315906	+	0.948791i	$0.397692\pi$
$492$	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
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	0	0
$501$	5.00000	0.223384
$502$	0	0
$503$	−24.0000	−1.07011	−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000	−1.07011	−0.535054	+	0.844818i	$0.679709\pi$
$504$	0	0
$505$	−12.0000	−0.533993
$506$	0	0
$507$	−4.00000	−0.177646
$508$	0	0
$509$	8.00000	0.354594	0.177297	−	0.984157i	$-0.443265\pi$
$509$	8.00000	0.354594	0.177297	+	0.984157i	$0.443265\pi$
$510$	0	0
$511$	15.0000	0.663561
$512$	0	0
$513$	−6.00000	−0.264906
$514$	0	0
$515$	−1.00000	−0.0440653
$516$	0	0
$517$	36.0000	1.58328
$518$	0	0
$519$	10.0000	0.438951
$520$	0	0
$521$	−20.0000	−0.876216	−0.438108	−	0.898922i	$-0.644351\pi$
$521$	−20.0000	−0.876216	−0.438108	+	0.898922i	$0.644351\pi$
$522$	0	0
$523$	4.00000	0.174908	0.0874539	−	0.996169i	$-0.472127\pi$
$523$	4.00000	0.174908	0.0874539	+	0.996169i	$0.472127\pi$
$524$	0	0
$525$	−4.00000	−0.174574
$526$	0	0
$527$	−4.00000	−0.174243
$528$	0	0
$529$	−19.0000	−0.826087
$530$	0	0
$531$	−4.00000	−0.173585
$532$	0	0
$533$	−36.0000	−1.55933
$534$	0	0
$535$	−16.0000	−0.691740
$536$	0	0
$537$	−10.0000	−0.431532
$538$	0	0
$539$	−3.00000	−0.129219
$540$	0	0
$541$	−31.0000	−1.33279	−0.666397	−	0.745597i	$-0.732164\pi$
$541$	−31.0000	−1.33279	−0.666397	+	0.745597i	$0.732164\pi$
$542$	0	0
$543$	20.0000	0.858282
$544$	0	0
$545$	6.00000	0.257012
$546$	0	0
$547$	38.0000	1.62476	0.812381	−	0.583127i	$-0.198171\pi$
$547$	38.0000	1.62476	0.812381	+	0.583127i	$0.198171\pi$
$548$	0	0
$549$	−6.00000	−0.256074
$550$	0	0
$551$	−36.0000	−1.53365
$552$	0	0
$553$	11.0000	0.467768
$554$	0	0
$555$	−11.0000	−0.466924
$556$	0	0
$557$	42.0000	1.77960	0.889799	−	0.456354i	$-0.150845\pi$
$557$	42.0000	1.77960	0.889799	+	0.456354i	$0.150845\pi$
$558$	0	0
$559$	9.00000	0.380659
$560$	0	0
$561$	−3.00000	−0.126660
$562$	0	0
$563$	21.0000	0.885044	0.442522	−	0.896758i	$-0.354084\pi$
$563$	21.0000	0.885044	0.442522	+	0.896758i	$0.354084\pi$
$564$	0	0
$565$	−15.0000	−0.631055
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	−42.0000	−1.76073	−0.880366	−	0.474295i	$-0.842703\pi$
$569$	−42.0000	−1.76073	−0.880366	+	0.474295i	$0.842703\pi$
$570$	0	0
$571$	16.0000	0.669579	0.334790	−	0.942293i	$-0.391335\pi$
$571$	16.0000	0.669579	0.334790	+	0.942293i	$0.391335\pi$
$572$	0	0
$573$	11.0000	0.459532
$574$	0	0
$575$	−8.00000	−0.333623
$576$	0	0
$577$	30.0000	1.24892	0.624458	−	0.781058i	$-0.285320\pi$
$577$	30.0000	1.24892	0.624458	+	0.781058i	$0.285320\pi$
$578$	0	0
$579$	−24.0000	−0.997406
$580$	0	0
$581$	−7.00000	−0.290409
$582$	0	0
$583$	−15.0000	−0.621237
$584$	0	0
$585$	−3.00000	−0.124035
$586$	0	0
$587$	17.0000	0.701665	0.350833	−	0.936438i	$-0.385899\pi$
$587$	17.0000	0.701665	0.350833	+	0.936438i	$0.385899\pi$
$588$	0	0
$589$	24.0000	0.988903
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−34.0000	−1.39621	−0.698106	−	0.715994i	$-0.745974\pi$
$593$	−34.0000	−1.39621	−0.698106	+	0.715994i	$0.745974\pi$
$594$	0	0
$595$	1.00000	0.0409960
$596$	0	0
$597$	0	0
$598$	0	0
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	0	0
$601$	−46.0000	−1.87638	−0.938190	−	0.346122i	$-0.887498\pi$
$601$	−46.0000	−1.87638	−0.938190	+	0.346122i	$0.887498\pi$
$602$	0	0
$603$	−9.00000	−0.366508
$604$	0	0
$605$	−2.00000	−0.0813116
$606$	0	0
$607$	−12.0000	−0.487065	−0.243532	−	0.969893i	$-0.578306\pi$
$607$	−12.0000	−0.487065	−0.243532	+	0.969893i	$0.578306\pi$
$608$	0	0
$609$	6.00000	0.243132
$610$	0	0
$611$	36.0000	1.45640
$612$	0	0
$613$	−28.0000	−1.13091	−0.565455	−	0.824779i	$-0.691299\pi$
$613$	−28.0000	−1.13091	−0.565455	+	0.824779i	$0.691299\pi$
$614$	0	0
$615$	12.0000	0.483887
$616$	0	0
$617$	13.0000	0.523360	0.261680	−	0.965155i	$-0.415723\pi$
$617$	13.0000	0.523360	0.261680	+	0.965155i	$0.415723\pi$
$618$	0	0
$619$	11.0000	0.442127	0.221064	−	0.975259i	$-0.429047\pi$
$619$	11.0000	0.442127	0.221064	+	0.975259i	$0.429047\pi$
$620$	0	0
$621$	2.00000	0.0802572
$622$	0	0
$623$	−13.0000	−0.520834
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	18.0000	0.718851
$628$	0	0
$629$	−11.0000	−0.438599
$630$	0	0
$631$	−32.0000	−1.27390	−0.636950	−	0.770905i	$-0.719804\pi$
$631$	−32.0000	−1.27390	−0.636950	+	0.770905i	$0.719804\pi$
$632$	0	0
$633$	−14.0000	−0.556450
$634$	0	0
$635$	−2.00000	−0.0793676
$636$	0	0
$637$	−3.00000	−0.118864
$638$	0	0
$639$	−12.0000	−0.474713
$640$	0	0
$641$	33.0000	1.30342	0.651711	−	0.758468i	$-0.274052\pi$
$641$	33.0000	1.30342	0.651711	+	0.758468i	$0.274052\pi$
$642$	0	0
$643$	−3.00000	−0.118308	−0.0591542	−	0.998249i	$-0.518840\pi$
$643$	−3.00000	−0.118308	−0.0591542	+	0.998249i	$0.518840\pi$
$644$	0	0
$645$	−3.00000	−0.118125
$646$	0	0
$647$	12.0000	0.471769	0.235884	−	0.971781i	$-0.424201\pi$
$647$	12.0000	0.471769	0.235884	+	0.971781i	$0.424201\pi$
$648$	0	0
$649$	12.0000	0.471041
$650$	0	0
$651$	−4.00000	−0.156772
$652$	0	0
$653$	−18.0000	−0.704394	−0.352197	−	0.935926i	$-0.614565\pi$
$653$	−18.0000	−0.704394	−0.352197	+	0.935926i	$0.614565\pi$
$654$	0	0
$655$	10.0000	0.390732
$656$	0	0
$657$	15.0000	0.585206
$658$	0	0
$659$	−48.0000	−1.86981	−0.934907	−	0.354892i	$-0.884518\pi$
$659$	−48.0000	−1.86981	−0.934907	+	0.354892i	$0.884518\pi$
$660$	0	0
$661$	10.0000	0.388955	0.194477	−	0.980907i	$-0.437699\pi$
$661$	10.0000	0.388955	0.194477	+	0.980907i	$0.437699\pi$
$662$	0	0
$663$	−3.00000	−0.116510
$664$	0	0
$665$	−6.00000	−0.232670
$666$	0	0
$667$	12.0000	0.464642
$668$	0	0
$669$	−16.0000	−0.618596
$670$	0	0
$671$	18.0000	0.694882
$672$	0	0
$673$	48.0000	1.85026	0.925132	−	0.379646i	$-0.123954\pi$
$673$	48.0000	1.85026	0.925132	+	0.379646i	$0.123954\pi$
$674$	0	0
$675$	−4.00000	−0.153960
$676$	0	0
$677$	33.0000	1.26829	0.634147	−	0.773213i	$-0.281352\pi$
$677$	33.0000	1.26829	0.634147	+	0.773213i	$0.281352\pi$
$678$	0	0
$679$	11.0000	0.422141
$680$	0	0
$681$	−8.00000	−0.306561
$682$	0	0
$683$	0	0		−	1.00000i	$-0.5\pi$
$683$	0	0			1.00000i	$0.5\pi$
$684$	0	0
$685$	8.00000	0.305664
$686$	0	0
$687$	19.0000	0.724895
$688$	0	0
$689$	−15.0000	−0.571454
$690$	0	0
$691$	−21.0000	−0.798878	−0.399439	−	0.916760i	$-0.630795\pi$
$691$	−21.0000	−0.798878	−0.399439	+	0.916760i	$0.630795\pi$
$692$	0	0
$693$	−3.00000	−0.113961
$694$	0	0
$695$	1.00000	0.0379322
$696$	0	0
$697$	12.0000	0.454532
$698$	0	0
$699$	−3.00000	−0.113470
$700$	0	0
$701$	−1.00000	−0.0377695	−0.0188847	−	0.999822i	$-0.506012\pi$
$701$	−1.00000	−0.0377695	−0.0188847	+	0.999822i	$0.506012\pi$
$702$	0	0
$703$	66.0000	2.48924
$704$	0	0
$705$	−12.0000	−0.451946
$706$	0	0
$707$	−12.0000	−0.451306
$708$	0	0
$709$	−19.0000	−0.713560	−0.356780	−	0.934188i	$-0.616125\pi$
$709$	−19.0000	−0.713560	−0.356780	+	0.934188i	$0.616125\pi$
$710$	0	0
$711$	11.0000	0.412532
$712$	0	0
$713$	−8.00000	−0.299602
$714$	0	0
$715$	9.00000	0.336581
$716$	0	0
$717$	21.0000	0.784259
$718$	0	0
$719$	−16.0000	−0.596699	−0.298350	−	0.954457i	$-0.596436\pi$
$719$	−16.0000	−0.596699	−0.298350	+	0.954457i	$0.596436\pi$
$720$	0	0
$721$	−1.00000	−0.0372419
$722$	0	0
$723$	14.0000	0.520666
$724$	0	0
$725$	−24.0000	−0.891338
$726$	0	0
$727$	47.0000	1.74313	0.871567	−	0.490277i	$-0.163104\pi$
$727$	47.0000	1.74313	0.871567	+	0.490277i	$0.163104\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−3.00000	−0.110959
$732$	0	0
$733$	−38.0000	−1.40356	−0.701781	−	0.712393i	$-0.747612\pi$
$733$	−38.0000	−1.40356	−0.701781	+	0.712393i	$0.747612\pi$
$734$	0	0
$735$	1.00000	0.0368856
$736$	0	0
$737$	27.0000	0.994558
$738$	0	0
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	0	0
$741$	18.0000	0.661247
$742$	0	0
$743$	46.0000	1.68758	0.843788	−	0.536676i	$-0.180320\pi$
$743$	46.0000	1.68758	0.843788	+	0.536676i	$0.180320\pi$
$744$	0	0
$745$	−15.0000	−0.549557
$746$	0	0
$747$	−7.00000	−0.256117
$748$	0	0
$749$	−16.0000	−0.584627
$750$	0	0
$751$	20.0000	0.729810	0.364905	−	0.931045i	$-0.381101\pi$
$751$	20.0000	0.729810	0.364905	+	0.931045i	$0.381101\pi$
$752$	0	0
$753$	−9.00000	−0.327978
$754$	0	0
$755$	16.0000	0.582300
$756$	0	0
$757$	−12.0000	−0.436147	−0.218074	−	0.975932i	$-0.569977\pi$
$757$	−12.0000	−0.436147	−0.218074	+	0.975932i	$0.569977\pi$
$758$	0	0
$759$	−6.00000	−0.217786
$760$	0	0
$761$	41.0000	1.48625	0.743124	−	0.669153i	$-0.233343\pi$
$761$	41.0000	1.48625	0.743124	+	0.669153i	$0.233343\pi$
$762$	0	0
$763$	6.00000	0.217215
$764$	0	0
$765$	1.00000	0.0361551
$766$	0	0
$767$	12.0000	0.433295
$768$	0	0
$769$	2.00000	0.0721218	0.0360609	−	0.999350i	$-0.488519\pi$
$769$	2.00000	0.0721218	0.0360609	+	0.999350i	$0.488519\pi$
$770$	0	0
$771$	−27.0000	−0.972381
$772$	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
$775$	16.0000	0.574737
$776$	0	0
$777$	−11.0000	−0.394623
$778$	0	0
$779$	−72.0000	−2.57967
$780$	0	0
$781$	36.0000	1.28818
$782$	0	0
$783$	6.00000	0.214423
$784$	0	0
$785$	−18.0000	−0.642448
$786$	0	0
$787$	−53.0000	−1.88925	−0.944623	−	0.328158i	$-0.893572\pi$
$787$	−53.0000	−1.88925	−0.944623	+	0.328158i	$0.893572\pi$
$788$	0	0
$789$	19.0000	0.676418
$790$	0	0
$791$	−15.0000	−0.533339
$792$	0	0
$793$	18.0000	0.639199
$794$	0	0
$795$	5.00000	0.177332
$796$	0	0
$797$	6.00000	0.212531	0.106265	−	0.994338i	$-0.466111\pi$
$797$	6.00000	0.212531	0.106265	+	0.994338i	$0.466111\pi$
$798$	0	0
$799$	−12.0000	−0.424529
$800$	0	0
$801$	−13.0000	−0.459332
$802$	0	0
$803$	−45.0000	−1.58802
$804$	0	0
$805$	2.00000	0.0704907
$806$	0	0
$807$	18.0000	0.633630
$808$	0	0
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	0	0
$811$	21.0000	0.737410	0.368705	−	0.929547i	$-0.379801\pi$
$811$	21.0000	0.737410	0.368705	+	0.929547i	$0.379801\pi$
$812$	0	0
$813$	9.00000	0.315644
$814$	0	0
$815$	22.0000	0.770626
$816$	0	0
$817$	18.0000	0.629740
$818$	0	0
$819$	−3.00000	−0.104828
$820$	0	0
$821$	50.0000	1.74501	0.872506	−	0.488603i	$-0.162493\pi$
$821$	50.0000	1.74501	0.872506	+	0.488603i	$0.162493\pi$
$822$	0	0
$823$	43.0000	1.49889	0.749443	−	0.662069i	$-0.230321\pi$
$823$	43.0000	1.49889	0.749443	+	0.662069i	$0.230321\pi$
$824$	0	0
$825$	12.0000	0.417786
$826$	0	0
$827$	−7.00000	−0.243414	−0.121707	−	0.992566i	$-0.538837\pi$
$827$	−7.00000	−0.243414	−0.121707	+	0.992566i	$0.538837\pi$
$828$	0	0
$829$	37.0000	1.28506	0.642532	−	0.766259i	$-0.277884\pi$
$829$	37.0000	1.28506	0.642532	+	0.766259i	$0.277884\pi$
$830$	0	0
$831$	18.0000	0.624413
$832$	0	0
$833$	1.00000	0.0346479
$834$	0	0
$835$	5.00000	0.173032
$836$	0	0
$837$	−4.00000	−0.138260
$838$	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
$843$	−22.0000	−0.757720
$844$	0	0
$845$	−4.00000	−0.137604
$846$	0	0
$847$	−2.00000	−0.0687208
$848$	0	0
$849$	13.0000	0.446159
$850$	0	0
$851$	−22.0000	−0.754150
$852$	0	0
$853$	−34.0000	−1.16414	−0.582069	−	0.813139i	$-0.697757\pi$
$853$	−34.0000	−1.16414	−0.582069	+	0.813139i	$0.697757\pi$
$854$	0	0
$855$	−6.00000	−0.205196
$856$	0	0
$857$	12.0000	0.409912	0.204956	−	0.978771i	$-0.434295\pi$
$857$	12.0000	0.409912	0.204956	+	0.978771i	$0.434295\pi$
$858$	0	0
$859$	38.0000	1.29654	0.648272	−	0.761409i	$-0.275492\pi$
$859$	38.0000	1.29654	0.648272	+	0.761409i	$0.275492\pi$
$860$	0	0
$861$	12.0000	0.408959
$862$	0	0
$863$	7.00000	0.238283	0.119141	−	0.992877i	$-0.461986\pi$
$863$	7.00000	0.238283	0.119141	+	0.992877i	$0.461986\pi$
$864$	0	0
$865$	10.0000	0.340010
$866$	0	0
$867$	1.00000	0.0339618
$868$	0	0
$869$	−33.0000	−1.11945
$870$	0	0
$871$	27.0000	0.914860
$872$	0	0
$873$	11.0000	0.372294
$874$	0	0
$875$	−9.00000	−0.304256
$876$	0	0
$877$	−1.00000	−0.0337676	−0.0168838	−	0.999857i	$-0.505375\pi$
$877$	−1.00000	−0.0337676	−0.0168838	+	0.999857i	$0.505375\pi$
$878$	0	0
$879$	−8.00000	−0.269833
$880$	0	0
$881$	42.0000	1.41502	0.707508	−	0.706705i	$-0.249819\pi$
$881$	42.0000	1.41502	0.707508	+	0.706705i	$0.249819\pi$
$882$	0	0
$883$	−20.0000	−0.673054	−0.336527	−	0.941674i	$-0.609252\pi$
$883$	−20.0000	−0.673054	−0.336527	+	0.941674i	$0.609252\pi$
$884$	0	0
$885$	−4.00000	−0.134459
$886$	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
$889$	−2.00000	−0.0670778
$890$	0	0
$891$	−3.00000	−0.100504
$892$	0	0
$893$	72.0000	2.40939
$894$	0	0
$895$	−10.0000	−0.334263
$896$	0	0
$897$	−6.00000	−0.200334
$898$	0	0
$899$	−24.0000	−0.800445
$900$	0	0
$901$	5.00000	0.166574
$902$	0	0
$903$	−3.00000	−0.0998337
$904$	0	0
$905$	20.0000	0.664822
$906$	0	0
$907$	−36.0000	−1.19536	−0.597680	−	0.801735i	$-0.703911\pi$
$907$	−36.0000	−1.19536	−0.597680	+	0.801735i	$0.703911\pi$
$908$	0	0
$909$	−12.0000	−0.398015
$910$	0	0
$911$	−40.0000	−1.32526	−0.662630	−	0.748947i	$-0.730560\pi$
$911$	−40.0000	−1.32526	−0.662630	+	0.748947i	$0.730560\pi$
$912$	0	0
$913$	21.0000	0.694999
$914$	0	0
$915$	−6.00000	−0.198354
$916$	0	0
$917$	10.0000	0.330229
$918$	0	0
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	0	0
$921$	8.00000	0.263609
$922$	0	0
$923$	36.0000	1.18495
$924$	0	0
$925$	44.0000	1.44671
$926$	0	0
$927$	−1.00000	−0.0328443
$928$	0	0
$929$	0	0		−	1.00000i	$-0.5\pi$
$929$	0	0			1.00000i	$0.5\pi$
$930$	0	0
$931$	−6.00000	−0.196642
$932$	0	0
$933$	−3.00000	−0.0982156
$934$	0	0
$935$	−3.00000	−0.0981105
$936$	0	0
$937$	8.00000	0.261349	0.130674	−	0.991425i	$-0.458286\pi$
$937$	8.00000	0.261349	0.130674	+	0.991425i	$0.458286\pi$
$938$	0	0
$939$	−22.0000	−0.717943
$940$	0	0
$941$	−21.0000	−0.684580	−0.342290	−	0.939594i	$-0.611203\pi$
$941$	−21.0000	−0.684580	−0.342290	+	0.939594i	$0.611203\pi$
$942$	0	0
$943$	24.0000	0.781548
$944$	0	0
$945$	1.00000	0.0325300
$946$	0	0
$947$	−28.0000	−0.909878	−0.454939	−	0.890523i	$-0.650339\pi$
$947$	−28.0000	−0.909878	−0.454939	+	0.890523i	$0.650339\pi$
$948$	0	0
$949$	−45.0000	−1.46076
$950$	0	0
$951$	18.0000	0.583690
$952$	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
$955$	11.0000	0.355952
$956$	0	0
$957$	−18.0000	−0.581857
$958$	0	0
$959$	8.00000	0.258333
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	−16.0000	−0.515593
$964$	0	0
$965$	−24.0000	−0.772587
$966$	0	0
$967$	−16.0000	−0.514525	−0.257263	−	0.966342i	$-0.582821\pi$
$967$	−16.0000	−0.514525	−0.257263	+	0.966342i	$0.582821\pi$
$968$	0	0
$969$	−6.00000	−0.192748
$970$	0	0
$971$	−3.00000	−0.0962746	−0.0481373	−	0.998841i	$-0.515328\pi$
$971$	−3.00000	−0.0962746	−0.0481373	+	0.998841i	$0.515328\pi$
$972$	0	0
$973$	1.00000	0.0320585
$974$	0	0
$975$	12.0000	0.384308
$976$	0	0
$977$	0	0		−	1.00000i	$-0.5\pi$
$977$	0	0			1.00000i	$0.5\pi$
$978$	0	0
$979$	39.0000	1.24645
$980$	0	0
$981$	6.00000	0.191565
$982$	0	0
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−12.0000	−0.381964
$988$	0	0
$989$	−6.00000	−0.190789
$990$	0	0
$991$	−13.0000	−0.412959	−0.206479	−	0.978451i	$-0.566201\pi$
$991$	−13.0000	−0.412959	−0.206479	+	0.978451i	$0.566201\pi$
$992$	0	0
$993$	3.00000	0.0952021
$994$	0	0
$995$	0	0
$996$	0	0
$997$	60.0000	1.90022	0.950110	−	0.311916i	$-0.100971\pi$
$997$	60.0000	1.90022	0.950110	+	0.311916i	$0.100971\pi$
$998$	0	0
$999$	−11.0000	−0.348025

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5712.2.a.v.1.1		1
4.3	odd	2		714.2.a.b.1.1	✓	1
12.11	even	2		2142.2.a.o.1.1		1
28.27	even	2		4998.2.a.q.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
714.2.a.b.1.1	✓	1	4.3	odd	2
2142.2.a.o.1.1		1	12.11	even	2
4998.2.a.q.1.1		1	28.27	even	2
5712.2.a.v.1.1		1	1.1	even	1	trivial

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

Downloads

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