LMFDB - Embedded newform 3192.2.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3192,2,Mod(1,3192)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3192, 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("3192.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3192 = 2^{3} \cdot 3 \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3192.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:	$25.4882483252$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	3192.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{3} -4.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$

$q+1.00000 q^{3} -4.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +2.00000 q^{11} -4.00000 q^{13} -4.00000 q^{15} +4.00000 q^{17} +1.00000 q^{19} +1.00000 q^{21} -6.00000 q^{23} +11.0000 q^{25} +1.00000 q^{27} -2.00000 q^{29} +4.00000 q^{31} +2.00000 q^{33} -4.00000 q^{35} +2.00000 q^{37} -4.00000 q^{39} -2.00000 q^{41} -8.00000 q^{43} -4.00000 q^{45} +4.00000 q^{47} +1.00000 q^{49} +4.00000 q^{51} -10.0000 q^{53} -8.00000 q^{55} +1.00000 q^{57} -4.00000 q^{59} +6.00000 q^{61} +1.00000 q^{63} +16.0000 q^{65} -10.0000 q^{67} -6.00000 q^{69} -16.0000 q^{71} -6.00000 q^{73} +11.0000 q^{75} +2.00000 q^{77} -10.0000 q^{79} +1.00000 q^{81} -12.0000 q^{83} -16.0000 q^{85} -2.00000 q^{87} +6.00000 q^{89} -4.00000 q^{91} +4.00000 q^{93} -4.00000 q^{95} +16.0000 q^{97} +2.00000 q^{99} +O(q^{100})$

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
$5$	−4.00000	−1.78885	−0.894427	−	0.447214i	$-0.852416\pi$
$5$	−4.00000	−1.78885	−0.894427	+	0.447214i	$0.852416\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	0	0
$15$	−4.00000	−1.03280
$16$	0	0
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	0	0
$25$	11.0000	2.20000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	0	0
$33$	2.00000	0.348155
$34$	0	0
$35$	−4.00000	−0.676123
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	0	0
$39$	−4.00000	−0.640513
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	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
$45$	−4.00000	−0.596285
$46$	0	0
$47$	4.00000	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	4.00000	0.560112
$52$	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
$55$	−8.00000	−1.07872
$56$	0	0
$57$	1.00000	0.132453
$58$	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	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	16.0000	1.98456
$66$	0	0
$67$	−10.0000	−1.22169	−0.610847	−	0.791748i	$-0.709171\pi$
$67$	−10.0000	−1.22169	−0.610847	+	0.791748i	$0.709171\pi$
$68$	0	0
$69$	−6.00000	−0.722315
$70$	0	0
$71$	−16.0000	−1.89885	−0.949425	−	0.313993i	$-0.898333\pi$
$71$	−16.0000	−1.89885	−0.949425	+	0.313993i	$0.898333\pi$
$72$	0	0
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	0	0
$75$	11.0000	1.27017
$76$	0	0
$77$	2.00000	0.227921
$78$	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
$83$	−12.0000	−1.31717	−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000	−1.31717	−0.658586	+	0.752506i	$0.728845\pi$
$84$	0	0
$85$	−16.0000	−1.73544
$86$	0	0
$87$	−2.00000	−0.214423
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	−4.00000	−0.419314
$92$	0	0
$93$	4.00000	0.414781
$94$	0	0
$95$	−4.00000	−0.410391
$96$	0	0
$97$	16.0000	1.62455	0.812277	−	0.583272i	$-0.198228\pi$
$97$	16.0000	1.62455	0.812277	+	0.583272i	$0.198228\pi$
$98$	0	0
$99$	2.00000	0.201008
$100$	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
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	0	0
$105$	−4.00000	−0.390360
$106$	0	0
$107$	8.00000	0.773389	0.386695	−	0.922208i	$-0.373617\pi$
$107$	8.00000	0.773389	0.386695	+	0.922208i	$0.373617\pi$
$108$	0	0
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	0	0
$113$	−18.0000	−1.69330	−0.846649	−	0.532152i	$-0.821383\pi$
$113$	−18.0000	−1.69330	−0.846649	+	0.532152i	$0.821383\pi$
$114$	0	0
$115$	24.0000	2.23801
$116$	0	0
$117$	−4.00000	−0.369800
$118$	0	0
$119$	4.00000	0.366679
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	−2.00000	−0.180334
$124$	0	0
$125$	−24.0000	−2.14663
$126$	0	0
$127$	2.00000	0.177471	0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000	0.177471	0.0887357	+	0.996055i	$0.471717\pi$
$128$	0	0
$129$	−8.00000	−0.704361
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	−4.00000	−0.344265
$136$	0	0
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	0	0
$139$	20.0000	1.69638	0.848189	−	0.529694i	$-0.177693\pi$
$139$	20.0000	1.69638	0.848189	+	0.529694i	$0.177693\pi$
$140$	0	0
$141$	4.00000	0.336861
$142$	0	0
$143$	−8.00000	−0.668994
$144$	0	0
$145$	8.00000	0.664364
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	−18.0000	−1.47462	−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000	−1.47462	−0.737309	+	0.675556i	$0.763904\pi$
$150$	0	0
$151$	−2.00000	−0.162758	−0.0813788	−	0.996683i	$-0.525932\pi$
$151$	−2.00000	−0.162758	−0.0813788	+	0.996683i	$0.525932\pi$
$152$	0	0
$153$	4.00000	0.323381
$154$	0	0
$155$	−16.0000	−1.28515
$156$	0	0
$157$	−22.0000	−1.75579	−0.877896	−	0.478852i	$-0.841053\pi$
$157$	−22.0000	−1.75579	−0.877896	+	0.478852i	$0.841053\pi$
$158$	0	0
$159$	−10.0000	−0.793052
$160$	0	0
$161$	−6.00000	−0.472866
$162$	0	0
$163$	−24.0000	−1.87983	−0.939913	−	0.341415i	$-0.889094\pi$
$163$	−24.0000	−1.87983	−0.939913	+	0.341415i	$0.889094\pi$
$164$	0	0
$165$	−8.00000	−0.622799
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	1.00000	0.0764719
$172$	0	0
$173$	−22.0000	−1.67263	−0.836315	−	0.548250i	$-0.815294\pi$
$173$	−22.0000	−1.67263	−0.836315	+	0.548250i	$0.815294\pi$
$174$	0	0
$175$	11.0000	0.831522
$176$	0	0
$177$	−4.00000	−0.300658
$178$	0	0
$179$	8.00000	0.597948	0.298974	−	0.954261i	$-0.403356\pi$
$179$	8.00000	0.597948	0.298974	+	0.954261i	$0.403356\pi$
$180$	0	0
$181$	16.0000	1.18927	0.594635	−	0.803996i	$-0.297296\pi$
$181$	16.0000	1.18927	0.594635	+	0.803996i	$0.297296\pi$
$182$	0	0
$183$	6.00000	0.443533
$184$	0	0
$185$	−8.00000	−0.588172
$186$	0	0
$187$	8.00000	0.585018
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	−6.00000	−0.434145	−0.217072	−	0.976156i	$-0.569651\pi$
$191$	−6.00000	−0.434145	−0.217072	+	0.976156i	$0.569651\pi$
$192$	0	0
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	0	0
$195$	16.0000	1.14578
$196$	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
$201$	−10.0000	−0.705346
$202$	0	0
$203$	−2.00000	−0.140372
$204$	0	0
$205$	8.00000	0.558744
$206$	0	0
$207$	−6.00000	−0.417029
$208$	0	0
$209$	2.00000	0.138343
$210$	0	0
$211$	−2.00000	−0.137686	−0.0688428	−	0.997628i	$-0.521931\pi$
$211$	−2.00000	−0.137686	−0.0688428	+	0.997628i	$0.521931\pi$
$212$	0	0
$213$	−16.0000	−1.09630
$214$	0	0
$215$	32.0000	2.18238
$216$	0	0
$217$	4.00000	0.271538
$218$	0	0
$219$	−6.00000	−0.405442
$220$	0	0
$221$	−16.0000	−1.07628
$222$	0	0
$223$	16.0000	1.07144	0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000	1.07144	0.535720	+	0.844396i	$0.320040\pi$
$224$	0	0
$225$	11.0000	0.733333
$226$	0	0
$227$	4.00000	0.265489	0.132745	−	0.991150i	$-0.457621\pi$
$227$	4.00000	0.265489	0.132745	+	0.991150i	$0.457621\pi$
$228$	0	0
$229$	22.0000	1.45380	0.726900	−	0.686743i	$-0.240960\pi$
$229$	22.0000	1.45380	0.726900	+	0.686743i	$0.240960\pi$
$230$	0	0
$231$	2.00000	0.131590
$232$	0	0
$233$	−18.0000	−1.17922	−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000	−1.17922	−0.589610	+	0.807688i	$0.700718\pi$
$234$	0	0
$235$	−16.0000	−1.04372
$236$	0	0
$237$	−10.0000	−0.649570
$238$	0	0
$239$	−6.00000	−0.388108	−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000	−0.388108	−0.194054	+	0.980991i	$0.562164\pi$
$240$	0	0
$241$	−4.00000	−0.257663	−0.128831	−	0.991667i	$-0.541123\pi$
$241$	−4.00000	−0.257663	−0.128831	+	0.991667i	$0.541123\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−4.00000	−0.255551
$246$	0	0
$247$	−4.00000	−0.254514
$248$	0	0
$249$	−12.0000	−0.760469
$250$	0	0
$251$	−8.00000	−0.504956	−0.252478	−	0.967603i	$-0.581245\pi$
$251$	−8.00000	−0.504956	−0.252478	+	0.967603i	$0.581245\pi$
$252$	0	0
$253$	−12.0000	−0.754434
$254$	0	0
$255$	−16.0000	−1.00196
$256$	0	0
$257$	10.0000	0.623783	0.311891	−	0.950118i	$-0.399037\pi$
$257$	10.0000	0.623783	0.311891	+	0.950118i	$0.399037\pi$
$258$	0	0
$259$	2.00000	0.124274
$260$	0	0
$261$	−2.00000	−0.123797
$262$	0	0
$263$	30.0000	1.84988	0.924940	−	0.380114i	$-0.124115\pi$
$263$	30.0000	1.84988	0.924940	+	0.380114i	$0.124115\pi$
$264$	0	0
$265$	40.0000	2.45718
$266$	0	0
$267$	6.00000	0.367194
$268$	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
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	0	0
$273$	−4.00000	−0.242091
$274$	0	0
$275$	22.0000	1.32665
$276$	0	0
$277$	22.0000	1.32185	0.660926	−	0.750451i	$-0.270164\pi$
$277$	22.0000	1.32185	0.660926	+	0.750451i	$0.270164\pi$
$278$	0	0
$279$	4.00000	0.239474
$280$	0	0
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	0	0
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	0	0
$285$	−4.00000	−0.236940
$286$	0	0
$287$	−2.00000	−0.118056
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	16.0000	0.937937
$292$	0	0
$293$	10.0000	0.584206	0.292103	−	0.956387i	$-0.405645\pi$
$293$	10.0000	0.584206	0.292103	+	0.956387i	$0.405645\pi$
$294$	0	0
$295$	16.0000	0.931556
$296$	0	0
$297$	2.00000	0.116052
$298$	0	0
$299$	24.0000	1.38796
$300$	0	0
$301$	−8.00000	−0.461112
$302$	0	0
$303$	−12.0000	−0.689382
$304$	0	0
$305$	−24.0000	−1.37424
$306$	0	0
$307$	−32.0000	−1.82634	−0.913168	−	0.407583i	$-0.866372\pi$
$307$	−32.0000	−1.82634	−0.913168	+	0.407583i	$0.866372\pi$
$308$	0	0
$309$	4.00000	0.227552
$310$	0	0
$311$	32.0000	1.81455	0.907277	−	0.420534i	$-0.138157\pi$
$311$	32.0000	1.81455	0.907277	+	0.420534i	$0.138157\pi$
$312$	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
$315$	−4.00000	−0.225374
$316$	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
$319$	−4.00000	−0.223957
$320$	0	0
$321$	8.00000	0.446516
$322$	0	0
$323$	4.00000	0.222566
$324$	0	0
$325$	−44.0000	−2.44068
$326$	0	0
$327$	10.0000	0.553001
$328$	0	0
$329$	4.00000	0.220527
$330$	0	0
$331$	10.0000	0.549650	0.274825	−	0.961494i	$-0.411380\pi$
$331$	10.0000	0.549650	0.274825	+	0.961494i	$0.411380\pi$
$332$	0	0
$333$	2.00000	0.109599
$334$	0	0
$335$	40.0000	2.18543
$336$	0	0
$337$	−18.0000	−0.980522	−0.490261	−	0.871576i	$-0.663099\pi$
$337$	−18.0000	−0.980522	−0.490261	+	0.871576i	$0.663099\pi$
$338$	0	0
$339$	−18.0000	−0.977626
$340$	0	0
$341$	8.00000	0.433224
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	24.0000	1.29212
$346$	0	0
$347$	−10.0000	−0.536828	−0.268414	−	0.963304i	$-0.586500\pi$
$347$	−10.0000	−0.536828	−0.268414	+	0.963304i	$0.586500\pi$
$348$	0	0
$349$	10.0000	0.535288	0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000	0.535288	0.267644	+	0.963518i	$0.413755\pi$
$350$	0	0
$351$	−4.00000	−0.213504
$352$	0	0
$353$	12.0000	0.638696	0.319348	−	0.947638i	$-0.396536\pi$
$353$	12.0000	0.638696	0.319348	+	0.947638i	$0.396536\pi$
$354$	0	0
$355$	64.0000	3.39677
$356$	0	0
$357$	4.00000	0.211702
$358$	0	0
$359$	22.0000	1.16112	0.580558	−	0.814219i	$-0.302835\pi$
$359$	22.0000	1.16112	0.580558	+	0.814219i	$0.302835\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−7.00000	−0.367405
$364$	0	0
$365$	24.0000	1.25622
$366$	0	0
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	0	0
$369$	−2.00000	−0.104116
$370$	0	0
$371$	−10.0000	−0.519174
$372$	0	0
$373$	22.0000	1.13912	0.569558	−	0.821951i	$-0.307114\pi$
$373$	22.0000	1.13912	0.569558	+	0.821951i	$0.307114\pi$
$374$	0	0
$375$	−24.0000	−1.23935
$376$	0	0
$377$	8.00000	0.412021
$378$	0	0
$379$	−14.0000	−0.719132	−0.359566	−	0.933120i	$-0.617075\pi$
$379$	−14.0000	−0.719132	−0.359566	+	0.933120i	$0.617075\pi$
$380$	0	0
$381$	2.00000	0.102463
$382$	0	0
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	−8.00000	−0.407718
$386$	0	0
$387$	−8.00000	−0.406663
$388$	0	0
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	−24.0000	−1.21373
$392$	0	0
$393$	0	0
$394$	0	0
$395$	40.0000	2.01262
$396$	0	0
$397$	−30.0000	−1.50566	−0.752828	−	0.658217i	$-0.771311\pi$
$397$	−30.0000	−1.50566	−0.752828	+	0.658217i	$0.771311\pi$
$398$	0	0
$399$	1.00000	0.0500626
$400$	0	0
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	0	0
$403$	−16.0000	−0.797017
$404$	0	0
$405$	−4.00000	−0.198762
$406$	0	0
$407$	4.00000	0.198273
$408$	0	0
$409$	−24.0000	−1.18672	−0.593362	−	0.804936i	$-0.702200\pi$
$409$	−24.0000	−1.18672	−0.593362	+	0.804936i	$0.702200\pi$
$410$	0	0
$411$	−2.00000	−0.0986527
$412$	0	0
$413$	−4.00000	−0.196827
$414$	0	0
$415$	48.0000	2.35623
$416$	0	0
$417$	20.0000	0.979404
$418$	0	0
$419$	−24.0000	−1.17248	−0.586238	−	0.810139i	$-0.699392\pi$
$419$	−24.0000	−1.17248	−0.586238	+	0.810139i	$0.699392\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	0	0
$423$	4.00000	0.194487
$424$	0	0
$425$	44.0000	2.13431
$426$	0	0
$427$	6.00000	0.290360
$428$	0	0
$429$	−8.00000	−0.386244
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	16.0000	0.768911	0.384455	−	0.923144i	$-0.374389\pi$
$433$	16.0000	0.768911	0.384455	+	0.923144i	$0.374389\pi$
$434$	0	0
$435$	8.00000	0.383571
$436$	0	0
$437$	−6.00000	−0.287019
$438$	0	0
$439$	28.0000	1.33637	0.668184	−	0.743996i	$-0.267072\pi$
$439$	28.0000	1.33637	0.668184	+	0.743996i	$0.267072\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	18.0000	0.855206	0.427603	−	0.903967i	$-0.359358\pi$
$443$	18.0000	0.855206	0.427603	+	0.903967i	$0.359358\pi$
$444$	0	0
$445$	−24.0000	−1.13771
$446$	0	0
$447$	−18.0000	−0.851371
$448$	0	0
$449$	−10.0000	−0.471929	−0.235965	−	0.971762i	$-0.575825\pi$
$449$	−10.0000	−0.471929	−0.235965	+	0.971762i	$0.575825\pi$
$450$	0	0
$451$	−4.00000	−0.188353
$452$	0	0
$453$	−2.00000	−0.0939682
$454$	0	0
$455$	16.0000	0.750092
$456$	0	0
$457$	6.00000	0.280668	0.140334	−	0.990104i	$-0.455182\pi$
$457$	6.00000	0.280668	0.140334	+	0.990104i	$0.455182\pi$
$458$	0	0
$459$	4.00000	0.186704
$460$	0	0
$461$	−12.0000	−0.558896	−0.279448	−	0.960161i	$-0.590151\pi$
$461$	−12.0000	−0.558896	−0.279448	+	0.960161i	$0.590151\pi$
$462$	0	0
$463$	8.00000	0.371792	0.185896	−	0.982569i	$-0.440481\pi$
$463$	8.00000	0.371792	0.185896	+	0.982569i	$0.440481\pi$
$464$	0	0
$465$	−16.0000	−0.741982
$466$	0	0
$467$	12.0000	0.555294	0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000	0.555294	0.277647	+	0.960683i	$0.410445\pi$
$468$	0	0
$469$	−10.0000	−0.461757
$470$	0	0
$471$	−22.0000	−1.01371
$472$	0	0
$473$	−16.0000	−0.735681
$474$	0	0
$475$	11.0000	0.504715
$476$	0	0
$477$	−10.0000	−0.457869
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	−8.00000	−0.364769
$482$	0	0
$483$	−6.00000	−0.273009
$484$	0	0
$485$	−64.0000	−2.90609
$486$	0	0
$487$	10.0000	0.453143	0.226572	−	0.973995i	$-0.427248\pi$
$487$	10.0000	0.453143	0.226572	+	0.973995i	$0.427248\pi$
$488$	0	0
$489$	−24.0000	−1.08532
$490$	0	0
$491$	30.0000	1.35388	0.676941	−	0.736038i	$-0.263305\pi$
$491$	30.0000	1.35388	0.676941	+	0.736038i	$0.263305\pi$
$492$	0	0
$493$	−8.00000	−0.360302
$494$	0	0
$495$	−8.00000	−0.359573
$496$	0	0
$497$	−16.0000	−0.717698
$498$	0	0
$499$	0	0		−	1.00000i	$-0.5\pi$
$499$	0	0			1.00000i	$0.5\pi$
$500$	0	0
$501$	0	0
$502$	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
$505$	48.0000	2.13597
$506$	0	0
$507$	3.00000	0.133235
$508$	0	0
$509$	−30.0000	−1.32973	−0.664863	−	0.746965i	$-0.731510\pi$
$509$	−30.0000	−1.32973	−0.664863	+	0.746965i	$0.731510\pi$
$510$	0	0
$511$	−6.00000	−0.265424
$512$	0	0
$513$	1.00000	0.0441511
$514$	0	0
$515$	−16.0000	−0.705044
$516$	0	0
$517$	8.00000	0.351840
$518$	0	0
$519$	−22.0000	−0.965693
$520$	0	0
$521$	6.00000	0.262865	0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000	0.262865	0.131432	+	0.991325i	$0.458042\pi$
$522$	0	0
$523$	−16.0000	−0.699631	−0.349816	−	0.936819i	$-0.613756\pi$
$523$	−16.0000	−0.699631	−0.349816	+	0.936819i	$0.613756\pi$
$524$	0	0
$525$	11.0000	0.480079
$526$	0	0
$527$	16.0000	0.696971
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	−4.00000	−0.173585
$532$	0	0
$533$	8.00000	0.346518
$534$	0	0
$535$	−32.0000	−1.38348
$536$	0	0
$537$	8.00000	0.345225
$538$	0	0
$539$	2.00000	0.0861461
$540$	0	0
$541$	−46.0000	−1.97769	−0.988847	−	0.148933i	$-0.952416\pi$
$541$	−46.0000	−1.97769	−0.988847	+	0.148933i	$0.952416\pi$
$542$	0	0
$543$	16.0000	0.686626
$544$	0	0
$545$	−40.0000	−1.71341
$546$	0	0
$547$	−18.0000	−0.769624	−0.384812	−	0.922995i	$-0.625734\pi$
$547$	−18.0000	−0.769624	−0.384812	+	0.922995i	$0.625734\pi$
$548$	0	0
$549$	6.00000	0.256074
$550$	0	0
$551$	−2.00000	−0.0852029
$552$	0	0
$553$	−10.0000	−0.425243
$554$	0	0
$555$	−8.00000	−0.339581
$556$	0	0
$557$	−18.0000	−0.762684	−0.381342	−	0.924434i	$-0.624538\pi$
$557$	−18.0000	−0.762684	−0.381342	+	0.924434i	$0.624538\pi$
$558$	0	0
$559$	32.0000	1.35346
$560$	0	0
$561$	8.00000	0.337760
$562$	0	0
$563$	36.0000	1.51722	0.758610	−	0.651546i	$-0.225879\pi$
$563$	36.0000	1.51722	0.758610	+	0.651546i	$0.225879\pi$
$564$	0	0
$565$	72.0000	3.02906
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	26.0000	1.08998	0.544988	−	0.838444i	$-0.316534\pi$
$569$	26.0000	1.08998	0.544988	+	0.838444i	$0.316534\pi$
$570$	0	0
$571$	20.0000	0.836974	0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000	0.836974	0.418487	+	0.908223i	$0.362561\pi$
$572$	0	0
$573$	−6.00000	−0.250654
$574$	0	0
$575$	−66.0000	−2.75239
$576$	0	0
$577$	−30.0000	−1.24892	−0.624458	−	0.781058i	$-0.714680\pi$
$577$	−30.0000	−1.24892	−0.624458	+	0.781058i	$0.714680\pi$
$578$	0	0
$579$	2.00000	0.0831172
$580$	0	0
$581$	−12.0000	−0.497844
$582$	0	0
$583$	−20.0000	−0.828315
$584$	0	0
$585$	16.0000	0.661519
$586$	0	0
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	0	0
$589$	4.00000	0.164817
$590$	0	0
$591$	−18.0000	−0.740421
$592$	0	0
$593$	24.0000	0.985562	0.492781	−	0.870153i	$-0.335980\pi$
$593$	24.0000	0.985562	0.492781	+	0.870153i	$0.335980\pi$
$594$	0	0
$595$	−16.0000	−0.655936
$596$	0	0
$597$	−16.0000	−0.654836
$598$	0	0
$599$	−40.0000	−1.63436	−0.817178	−	0.576386i	$-0.804463\pi$
$599$	−40.0000	−1.63436	−0.817178	+	0.576386i	$0.804463\pi$
$600$	0	0
$601$	24.0000	0.978980	0.489490	−	0.872009i	$-0.337183\pi$
$601$	24.0000	0.978980	0.489490	+	0.872009i	$0.337183\pi$
$602$	0	0
$603$	−10.0000	−0.407231
$604$	0	0
$605$	28.0000	1.13836
$606$	0	0
$607$	32.0000	1.29884	0.649420	−	0.760430i	$-0.275012\pi$
$607$	32.0000	1.29884	0.649420	+	0.760430i	$0.275012\pi$
$608$	0	0
$609$	−2.00000	−0.0810441
$610$	0	0
$611$	−16.0000	−0.647291
$612$	0	0
$613$	14.0000	0.565455	0.282727	−	0.959200i	$-0.408761\pi$
$613$	14.0000	0.565455	0.282727	+	0.959200i	$0.408761\pi$
$614$	0	0
$615$	8.00000	0.322591
$616$	0	0
$617$	14.0000	0.563619	0.281809	−	0.959470i	$-0.409065\pi$
$617$	14.0000	0.563619	0.281809	+	0.959470i	$0.409065\pi$
$618$	0	0
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	0	0
$621$	−6.00000	−0.240772
$622$	0	0
$623$	6.00000	0.240385
$624$	0	0
$625$	41.0000	1.64000
$626$	0	0
$627$	2.00000	0.0798723
$628$	0	0
$629$	8.00000	0.318981
$630$	0	0
$631$	−20.0000	−0.796187	−0.398094	−	0.917345i	$-0.630328\pi$
$631$	−20.0000	−0.796187	−0.398094	+	0.917345i	$0.630328\pi$
$632$	0	0
$633$	−2.00000	−0.0794929
$634$	0	0
$635$	−8.00000	−0.317470
$636$	0	0
$637$	−4.00000	−0.158486
$638$	0	0
$639$	−16.0000	−0.632950
$640$	0	0
$641$	−34.0000	−1.34292	−0.671460	−	0.741041i	$-0.734332\pi$
$641$	−34.0000	−1.34292	−0.671460	+	0.741041i	$0.734332\pi$
$642$	0	0
$643$	−4.00000	−0.157745	−0.0788723	−	0.996885i	$-0.525132\pi$
$643$	−4.00000	−0.157745	−0.0788723	+	0.996885i	$0.525132\pi$
$644$	0	0
$645$	32.0000	1.26000
$646$	0	0
$647$	−32.0000	−1.25805	−0.629025	−	0.777385i	$-0.716546\pi$
$647$	−32.0000	−1.25805	−0.629025	+	0.777385i	$0.716546\pi$
$648$	0	0
$649$	−8.00000	−0.314027
$650$	0	0
$651$	4.00000	0.156772
$652$	0	0
$653$	42.0000	1.64359	0.821794	−	0.569785i	$-0.192974\pi$
$653$	42.0000	1.64359	0.821794	+	0.569785i	$0.192974\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−6.00000	−0.234082
$658$	0	0
$659$	40.0000	1.55818	0.779089	−	0.626913i	$-0.215682\pi$
$659$	40.0000	1.55818	0.779089	+	0.626913i	$0.215682\pi$
$660$	0	0
$661$	−16.0000	−0.622328	−0.311164	−	0.950356i	$-0.600719\pi$
$661$	−16.0000	−0.622328	−0.311164	+	0.950356i	$0.600719\pi$
$662$	0	0
$663$	−16.0000	−0.621389
$664$	0	0
$665$	−4.00000	−0.155113
$666$	0	0
$667$	12.0000	0.464642
$668$	0	0
$669$	16.0000	0.618596
$670$	0	0
$671$	12.0000	0.463255
$672$	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
$675$	11.0000	0.423390
$676$	0	0
$677$	−26.0000	−0.999261	−0.499631	−	0.866239i	$-0.666531\pi$
$677$	−26.0000	−0.999261	−0.499631	+	0.866239i	$0.666531\pi$
$678$	0	0
$679$	16.0000	0.614024
$680$	0	0
$681$	4.00000	0.153280
$682$	0	0
$683$	36.0000	1.37750	0.688751	−	0.724998i	$-0.258159\pi$
$683$	36.0000	1.37750	0.688751	+	0.724998i	$0.258159\pi$
$684$	0	0
$685$	8.00000	0.305664
$686$	0	0
$687$	22.0000	0.839352
$688$	0	0
$689$	40.0000	1.52388
$690$	0	0
$691$	4.00000	0.152167	0.0760836	−	0.997101i	$-0.475758\pi$
$691$	4.00000	0.152167	0.0760836	+	0.997101i	$0.475758\pi$
$692$	0	0
$693$	2.00000	0.0759737
$694$	0	0
$695$	−80.0000	−3.03457
$696$	0	0
$697$	−8.00000	−0.303022
$698$	0	0
$699$	−18.0000	−0.680823
$700$	0	0
$701$	34.0000	1.28416	0.642081	−	0.766637i	$-0.278071\pi$
$701$	34.0000	1.28416	0.642081	+	0.766637i	$0.278071\pi$
$702$	0	0
$703$	2.00000	0.0754314
$704$	0	0
$705$	−16.0000	−0.602595
$706$	0	0
$707$	−12.0000	−0.451306
$708$	0	0
$709$	10.0000	0.375558	0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000	0.375558	0.187779	+	0.982211i	$0.439871\pi$
$710$	0	0
$711$	−10.0000	−0.375029
$712$	0	0
$713$	−24.0000	−0.898807
$714$	0	0
$715$	32.0000	1.19673
$716$	0	0
$717$	−6.00000	−0.224074
$718$	0	0
$719$	48.0000	1.79010	0.895049	−	0.445968i	$-0.147140\pi$
$719$	48.0000	1.79010	0.895049	+	0.445968i	$0.147140\pi$
$720$	0	0
$721$	4.00000	0.148968
$722$	0	0
$723$	−4.00000	−0.148762
$724$	0	0
$725$	−22.0000	−0.817059
$726$	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
$729$	1.00000	0.0370370
$730$	0	0
$731$	−32.0000	−1.18356
$732$	0	0
$733$	−46.0000	−1.69905	−0.849524	−	0.527549i	$-0.823111\pi$
$733$	−46.0000	−1.69905	−0.849524	+	0.527549i	$0.823111\pi$
$734$	0	0
$735$	−4.00000	−0.147542
$736$	0	0
$737$	−20.0000	−0.736709
$738$	0	0
$739$	44.0000	1.61857	0.809283	−	0.587419i	$-0.199856\pi$
$739$	44.0000	1.61857	0.809283	+	0.587419i	$0.199856\pi$
$740$	0	0
$741$	−4.00000	−0.146944
$742$	0	0
$743$	−36.0000	−1.32071	−0.660356	−	0.750953i	$-0.729595\pi$
$743$	−36.0000	−1.32071	−0.660356	+	0.750953i	$0.729595\pi$
$744$	0	0
$745$	72.0000	2.63788
$746$	0	0
$747$	−12.0000	−0.439057
$748$	0	0
$749$	8.00000	0.292314
$750$	0	0
$751$	26.0000	0.948753	0.474377	−	0.880322i	$-0.342673\pi$
$751$	26.0000	0.948753	0.474377	+	0.880322i	$0.342673\pi$
$752$	0	0
$753$	−8.00000	−0.291536
$754$	0	0
$755$	8.00000	0.291150
$756$	0	0
$757$	30.0000	1.09037	0.545184	−	0.838316i	$-0.316460\pi$
$757$	30.0000	1.09037	0.545184	+	0.838316i	$0.316460\pi$
$758$	0	0
$759$	−12.0000	−0.435572
$760$	0	0
$761$	24.0000	0.869999	0.435000	−	0.900431i	$-0.356748\pi$
$761$	24.0000	0.869999	0.435000	+	0.900431i	$0.356748\pi$
$762$	0	0
$763$	10.0000	0.362024
$764$	0	0
$765$	−16.0000	−0.578481
$766$	0	0
$767$	16.0000	0.577727
$768$	0	0
$769$	−22.0000	−0.793340	−0.396670	−	0.917961i	$-0.629834\pi$
$769$	−22.0000	−0.793340	−0.396670	+	0.917961i	$0.629834\pi$
$770$	0	0
$771$	10.0000	0.360141
$772$	0	0
$773$	−38.0000	−1.36677	−0.683383	−	0.730061i	$-0.739492\pi$
$773$	−38.0000	−1.36677	−0.683383	+	0.730061i	$0.739492\pi$
$774$	0	0
$775$	44.0000	1.58053
$776$	0	0
$777$	2.00000	0.0717496
$778$	0	0
$779$	−2.00000	−0.0716574
$780$	0	0
$781$	−32.0000	−1.14505
$782$	0	0
$783$	−2.00000	−0.0714742
$784$	0	0
$785$	88.0000	3.14085
$786$	0	0
$787$	−48.0000	−1.71102	−0.855508	−	0.517790i	$-0.826755\pi$
$787$	−48.0000	−1.71102	−0.855508	+	0.517790i	$0.826755\pi$
$788$	0	0
$789$	30.0000	1.06803
$790$	0	0
$791$	−18.0000	−0.640006
$792$	0	0
$793$	−24.0000	−0.852265
$794$	0	0
$795$	40.0000	1.41865
$796$	0	0
$797$	42.0000	1.48772	0.743858	−	0.668338i	$-0.232994\pi$
$797$	42.0000	1.48772	0.743858	+	0.668338i	$0.232994\pi$
$798$	0	0
$799$	16.0000	0.566039
$800$	0	0
$801$	6.00000	0.212000
$802$	0	0
$803$	−12.0000	−0.423471
$804$	0	0
$805$	24.0000	0.845889
$806$	0	0
$807$	−6.00000	−0.211210
$808$	0	0
$809$	34.0000	1.19538	0.597688	−	0.801729i	$-0.296086\pi$
$809$	34.0000	1.19538	0.597688	+	0.801729i	$0.296086\pi$
$810$	0	0
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	0	0
$813$	8.00000	0.280572
$814$	0	0
$815$	96.0000	3.36273
$816$	0	0
$817$	−8.00000	−0.279885
$818$	0	0
$819$	−4.00000	−0.139771
$820$	0	0
$821$	6.00000	0.209401	0.104701	−	0.994504i	$-0.466612\pi$
$821$	6.00000	0.209401	0.104701	+	0.994504i	$0.466612\pi$
$822$	0	0
$823$	−16.0000	−0.557725	−0.278862	−	0.960331i	$-0.589957\pi$
$823$	−16.0000	−0.557725	−0.278862	+	0.960331i	$0.589957\pi$
$824$	0	0
$825$	22.0000	0.765942
$826$	0	0
$827$	0	0		−	1.00000i	$-0.5\pi$
$827$	0	0			1.00000i	$0.5\pi$
$828$	0	0
$829$	36.0000	1.25033	0.625166	−	0.780492i	$-0.285031\pi$
$829$	36.0000	1.25033	0.625166	+	0.780492i	$0.285031\pi$
$830$	0	0
$831$	22.0000	0.763172
$832$	0	0
$833$	4.00000	0.138592
$834$	0	0
$835$	0	0
$836$	0	0
$837$	4.00000	0.138260
$838$	0	0
$839$	−16.0000	−0.552381	−0.276191	−	0.961103i	$-0.589072\pi$
$839$	−16.0000	−0.552381	−0.276191	+	0.961103i	$0.589072\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	6.00000	0.206651
$844$	0	0
$845$	−12.0000	−0.412813
$846$	0	0
$847$	−7.00000	−0.240523
$848$	0	0
$849$	−4.00000	−0.137280
$850$	0	0
$851$	−12.0000	−0.411355
$852$	0	0
$853$	−26.0000	−0.890223	−0.445112	−	0.895475i	$-0.646836\pi$
$853$	−26.0000	−0.890223	−0.445112	+	0.895475i	$0.646836\pi$
$854$	0	0
$855$	−4.00000	−0.136797
$856$	0	0
$857$	−18.0000	−0.614868	−0.307434	−	0.951569i	$-0.599470\pi$
$857$	−18.0000	−0.614868	−0.307434	+	0.951569i	$0.599470\pi$
$858$	0	0
$859$	−36.0000	−1.22830	−0.614152	−	0.789188i	$-0.710502\pi$
$859$	−36.0000	−1.22830	−0.614152	+	0.789188i	$0.710502\pi$
$860$	0	0
$861$	−2.00000	−0.0681598
$862$	0	0
$863$	4.00000	0.136162	0.0680808	−	0.997680i	$-0.478312\pi$
$863$	4.00000	0.136162	0.0680808	+	0.997680i	$0.478312\pi$
$864$	0	0
$865$	88.0000	2.99209
$866$	0	0
$867$	−1.00000	−0.0339618
$868$	0	0
$869$	−20.0000	−0.678454
$870$	0	0
$871$	40.0000	1.35535
$872$	0	0
$873$	16.0000	0.541518
$874$	0	0
$875$	−24.0000	−0.811348
$876$	0	0
$877$	18.0000	0.607817	0.303908	−	0.952701i	$-0.401708\pi$
$877$	18.0000	0.607817	0.303908	+	0.952701i	$0.401708\pi$
$878$	0	0
$879$	10.0000	0.337292
$880$	0	0
$881$	−44.0000	−1.48240	−0.741199	−	0.671286i	$-0.765742\pi$
$881$	−44.0000	−1.48240	−0.741199	+	0.671286i	$0.765742\pi$
$882$	0	0
$883$	40.0000	1.34611	0.673054	−	0.739594i	$-0.264982\pi$
$883$	40.0000	1.34611	0.673054	+	0.739594i	$0.264982\pi$
$884$	0	0
$885$	16.0000	0.537834
$886$	0	0
$887$	−32.0000	−1.07445	−0.537227	−	0.843437i	$-0.680528\pi$
$887$	−32.0000	−1.07445	−0.537227	+	0.843437i	$0.680528\pi$
$888$	0	0
$889$	2.00000	0.0670778
$890$	0	0
$891$	2.00000	0.0670025
$892$	0	0
$893$	4.00000	0.133855
$894$	0	0
$895$	−32.0000	−1.06964
$896$	0	0
$897$	24.0000	0.801337
$898$	0	0
$899$	−8.00000	−0.266815
$900$	0	0
$901$	−40.0000	−1.33259
$902$	0	0
$903$	−8.00000	−0.266223
$904$	0	0
$905$	−64.0000	−2.12743
$906$	0	0
$907$	−30.0000	−0.996134	−0.498067	−	0.867139i	$-0.665957\pi$
$907$	−30.0000	−0.996134	−0.498067	+	0.867139i	$0.665957\pi$
$908$	0	0
$909$	−12.0000	−0.398015
$910$	0	0
$911$	−56.0000	−1.85536	−0.927681	−	0.373373i	$-0.878201\pi$
$911$	−56.0000	−1.85536	−0.927681	+	0.373373i	$0.878201\pi$
$912$	0	0
$913$	−24.0000	−0.794284
$914$	0	0
$915$	−24.0000	−0.793416
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−48.0000	−1.58337	−0.791687	−	0.610927i	$-0.790797\pi$
$919$	−48.0000	−1.58337	−0.791687	+	0.610927i	$0.790797\pi$
$920$	0	0
$921$	−32.0000	−1.05444
$922$	0	0
$923$	64.0000	2.10659
$924$	0	0
$925$	22.0000	0.723356
$926$	0	0
$927$	4.00000	0.131377
$928$	0	0
$929$	20.0000	0.656179	0.328089	−	0.944647i	$-0.393595\pi$
$929$	20.0000	0.656179	0.328089	+	0.944647i	$0.393595\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	32.0000	1.04763
$934$	0	0
$935$	−32.0000	−1.04651
$936$	0	0
$937$	22.0000	0.718709	0.359354	−	0.933201i	$-0.382997\pi$
$937$	22.0000	0.718709	0.359354	+	0.933201i	$0.382997\pi$
$938$	0	0
$939$	−22.0000	−0.717943
$940$	0	0
$941$	−22.0000	−0.717180	−0.358590	−	0.933495i	$-0.616742\pi$
$941$	−22.0000	−0.717180	−0.358590	+	0.933495i	$0.616742\pi$
$942$	0	0
$943$	12.0000	0.390774
$944$	0	0
$945$	−4.00000	−0.130120
$946$	0	0
$947$	−42.0000	−1.36482	−0.682408	−	0.730971i	$-0.739067\pi$
$947$	−42.0000	−1.36482	−0.682408	+	0.730971i	$0.739067\pi$
$948$	0	0
$949$	24.0000	0.779073
$950$	0	0
$951$	18.0000	0.583690
$952$	0	0
$953$	−2.00000	−0.0647864	−0.0323932	−	0.999475i	$-0.510313\pi$
$953$	−2.00000	−0.0647864	−0.0323932	+	0.999475i	$0.510313\pi$
$954$	0	0
$955$	24.0000	0.776622
$956$	0	0
$957$	−4.00000	−0.129302
$958$	0	0
$959$	−2.00000	−0.0645834
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	8.00000	0.257796
$964$	0	0
$965$	−8.00000	−0.257529
$966$	0	0
$967$	−28.0000	−0.900419	−0.450210	−	0.892923i	$-0.648651\pi$
$967$	−28.0000	−0.900419	−0.450210	+	0.892923i	$0.648651\pi$
$968$	0	0
$969$	4.00000	0.128499
$970$	0	0
$971$	36.0000	1.15529	0.577647	−	0.816286i	$-0.303971\pi$
$971$	36.0000	1.15529	0.577647	+	0.816286i	$0.303971\pi$
$972$	0	0
$973$	20.0000	0.641171
$974$	0	0
$975$	−44.0000	−1.40913
$976$	0	0
$977$	42.0000	1.34370	0.671850	−	0.740688i	$-0.265500\pi$
$977$	42.0000	1.34370	0.671850	+	0.740688i	$0.265500\pi$
$978$	0	0
$979$	12.0000	0.383522
$980$	0	0
$981$	10.0000	0.319275
$982$	0	0
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	72.0000	2.29411
$986$	0	0
$987$	4.00000	0.127321
$988$	0	0
$989$	48.0000	1.52631
$990$	0	0
$991$	22.0000	0.698853	0.349427	−	0.936964i	$-0.386376\pi$
$991$	22.0000	0.698853	0.349427	+	0.936964i	$0.386376\pi$
$992$	0	0
$993$	10.0000	0.317340
$994$	0	0
$995$	64.0000	2.02894
$996$	0	0
$997$	34.0000	1.07679	0.538395	−	0.842692i	$-0.319031\pi$
$997$	34.0000	1.07679	0.538395	+	0.842692i	$0.319031\pi$
$998$	0	0
$999$	2.00000	0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3192.2.a.l.1.1	✓	1
3.2	odd	2		9576.2.a.bb.1.1		1
4.3	odd	2		6384.2.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3192.2.a.l.1.1	✓	1	1.1	even	1	trivial
6384.2.a.a.1.1		1	4.3	odd	2
9576.2.a.bb.1.1		1	3.2	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:	\( N \)	\(=\)	\( 3192 = 2^{3} \cdot 3 \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3192.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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