LMFDB - Embedded newform 4014.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4014,2,Mod(1,4014)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4014, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4014.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4014 = 2 \cdot 3^{2} \cdot 223 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4014.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:	$32.0519513713$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 446)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	4014.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{8} +O(q^{10})$

$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{8} -1.00000 q^{11} -2.00000 q^{13} +1.00000 q^{16} -1.00000 q^{17} -4.00000 q^{19} -1.00000 q^{22} -1.00000 q^{23} -5.00000 q^{25} -2.00000 q^{26} +3.00000 q^{29} -10.0000 q^{31} +1.00000 q^{32} -1.00000 q^{34} -3.00000 q^{37} -4.00000 q^{38} +5.00000 q^{41} -6.00000 q^{43} -1.00000 q^{44} -1.00000 q^{46} -6.00000 q^{47} -7.00000 q^{49} -5.00000 q^{50} -2.00000 q^{52} +9.00000 q^{53} +3.00000 q^{58} +1.00000 q^{59} +4.00000 q^{61} -10.0000 q^{62} +1.00000 q^{64} +9.00000 q^{67} -1.00000 q^{68} -4.00000 q^{71} -5.00000 q^{73} -3.00000 q^{74} -4.00000 q^{76} +5.00000 q^{82} -14.0000 q^{83} -6.00000 q^{86} -1.00000 q^{88} +5.00000 q^{89} -1.00000 q^{92} -6.00000 q^{94} +2.00000 q^{97} -7.00000 q^{98} +O(q^{100})$

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	0	0
$11$	−1.00000	−0.301511	−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000	−0.301511	−0.150756	+	0.988571i	$0.548171\pi$
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	−1.00000	−0.242536	−0.121268	−	0.992620i	$-0.538696\pi$
$17$	−1.00000	−0.242536	−0.121268	+	0.992620i	$0.538696\pi$
$18$	0	0
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	0	0
$22$	−1.00000	−0.213201
$23$	−1.00000	−0.208514	−0.104257	−	0.994550i	$-0.533247\pi$
$23$	−1.00000	−0.208514	−0.104257	+	0.994550i	$0.533247\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	−2.00000	−0.392232
$27$	0	0
$28$	0	0
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	0	0
$31$	−10.0000	−1.79605	−0.898027	−	0.439941i	$-0.854999\pi$
$31$	−10.0000	−1.79605	−0.898027	+	0.439941i	$0.854999\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−1.00000	−0.171499
$35$	0	0
$36$	0	0
$37$	−3.00000	−0.493197	−0.246598	−	0.969118i	$-0.579313\pi$
$37$	−3.00000	−0.493197	−0.246598	+	0.969118i	$0.579313\pi$
$38$	−4.00000	−0.648886
$39$	0	0
$40$	0	0
$41$	5.00000	0.780869	0.390434	−	0.920631i	$-0.372325\pi$
$41$	5.00000	0.780869	0.390434	+	0.920631i	$0.372325\pi$
$42$	0	0
$43$	−6.00000	−0.914991	−0.457496	−	0.889212i	$-0.651253\pi$
$43$	−6.00000	−0.914991	−0.457496	+	0.889212i	$0.651253\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	−1.00000	−0.147442
$47$	−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
$49$	−7.00000	−1.00000
$50$	−5.00000	−0.707107
$51$	0	0
$52$	−2.00000	−0.277350
$53$	9.00000	1.23625	0.618123	−	0.786082i	$-0.287894\pi$
$53$	9.00000	1.23625	0.618123	+	0.786082i	$0.287894\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	3.00000	0.393919
$59$	1.00000	0.130189	0.0650945	−	0.997879i	$-0.479265\pi$
$59$	1.00000	0.130189	0.0650945	+	0.997879i	$0.479265\pi$
$60$	0	0
$61$	4.00000	0.512148	0.256074	−	0.966657i	$-0.417571\pi$
$61$	4.00000	0.512148	0.256074	+	0.966657i	$0.417571\pi$
$62$	−10.0000	−1.27000
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	9.00000	1.09952	0.549762	−	0.835321i	$-0.314718\pi$
$67$	9.00000	1.09952	0.549762	+	0.835321i	$0.314718\pi$
$68$	−1.00000	−0.121268
$69$	0	0
$70$	0	0
$71$	−4.00000	−0.474713	−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000	−0.474713	−0.237356	+	0.971423i	$0.576281\pi$
$72$	0	0
$73$	−5.00000	−0.585206	−0.292603	−	0.956234i	$-0.594521\pi$
$73$	−5.00000	−0.585206	−0.292603	+	0.956234i	$0.594521\pi$
$74$	−3.00000	−0.348743
$75$	0	0
$76$	−4.00000	−0.458831
$77$	0	0
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	0	0
$82$	5.00000	0.552158
$83$	−14.0000	−1.53670	−0.768350	−	0.640030i	$-0.778922\pi$
$83$	−14.0000	−1.53670	−0.768350	+	0.640030i	$0.778922\pi$
$84$	0	0
$85$	0	0
$86$	−6.00000	−0.646997
$87$	0	0
$88$	−1.00000	−0.106600
$89$	5.00000	0.529999	0.264999	−	0.964249i	$-0.414628\pi$
$89$	5.00000	0.529999	0.264999	+	0.964249i	$0.414628\pi$
$90$	0	0
$91$	0	0
$92$	−1.00000	−0.104257
$93$	0	0
$94$	−6.00000	−0.618853
$95$	0	0
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	−7.00000	−0.707107
$99$	0	0
$100$	−5.00000	−0.500000
$101$	−2.00000	−0.199007	−0.0995037	−	0.995037i	$-0.531726\pi$
$101$	−2.00000	−0.199007	−0.0995037	+	0.995037i	$0.531726\pi$
$102$	0	0
$103$	13.0000	1.28093	0.640464	−	0.767988i	$-0.278742\pi$
$103$	13.0000	1.28093	0.640464	+	0.767988i	$0.278742\pi$
$104$	−2.00000	−0.196116
$105$	0	0
$106$	9.00000	0.874157
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	0	0
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	0	0
$116$	3.00000	0.278543
$117$	0	0
$118$	1.00000	0.0920575
$119$	0	0
$120$	0	0
$121$	−10.0000	−0.909091
$122$	4.00000	0.362143
$123$	0	0
$124$	−10.0000	−0.898027
$125$	0	0
$126$	0	0
$127$	−2.00000	−0.177471	−0.0887357	−	0.996055i	$-0.528283\pi$
$127$	−2.00000	−0.177471	−0.0887357	+	0.996055i	$0.528283\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	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
$133$	0	0
$134$	9.00000	0.777482
$135$	0	0
$136$	−1.00000	−0.0857493
$137$	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
$139$	22.0000	1.86602	0.933008	−	0.359856i	$-0.117174\pi$
$139$	22.0000	1.86602	0.933008	+	0.359856i	$0.117174\pi$
$140$	0	0
$141$	0	0
$142$	−4.00000	−0.335673
$143$	2.00000	0.167248
$144$	0	0
$145$	0	0
$146$	−5.00000	−0.413803
$147$	0	0
$148$	−3.00000	−0.246598
$149$	−22.0000	−1.80231	−0.901155	−	0.433497i	$-0.857280\pi$
$149$	−22.0000	−1.80231	−0.901155	+	0.433497i	$0.857280\pi$
$150$	0	0
$151$	−9.00000	−0.732410	−0.366205	−	0.930534i	$-0.619343\pi$
$151$	−9.00000	−0.732410	−0.366205	+	0.930534i	$0.619343\pi$
$152$	−4.00000	−0.324443
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−8.00000	−0.626608	−0.313304	−	0.949653i	$-0.601436\pi$
$163$	−8.00000	−0.626608	−0.313304	+	0.949653i	$0.601436\pi$
$164$	5.00000	0.390434
$165$	0	0
$166$	−14.0000	−1.08661
$167$	1.00000	0.0773823	0.0386912	−	0.999251i	$-0.487681\pi$
$167$	1.00000	0.0773823	0.0386912	+	0.999251i	$0.487681\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	0	0
$172$	−6.00000	−0.457496
$173$	16.0000	1.21646	0.608229	−	0.793762i	$-0.291880\pi$
$173$	16.0000	1.21646	0.608229	+	0.793762i	$0.291880\pi$
$174$	0	0
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	5.00000	0.374766
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	0	0
$181$	9.00000	0.668965	0.334482	−	0.942402i	$-0.391439\pi$
$181$	9.00000	0.668965	0.334482	+	0.942402i	$0.391439\pi$
$182$	0	0
$183$	0	0
$184$	−1.00000	−0.0737210
$185$	0	0
$186$	0	0
$187$	1.00000	0.0731272
$188$	−6.00000	−0.437595
$189$	0	0
$190$	0	0
$191$	3.00000	0.217072	0.108536	−	0.994092i	$-0.465384\pi$
$191$	3.00000	0.217072	0.108536	+	0.994092i	$0.465384\pi$
$192$	0	0
$193$	−4.00000	−0.287926	−0.143963	−	0.989583i	$-0.545985\pi$
$193$	−4.00000	−0.287926	−0.143963	+	0.989583i	$0.545985\pi$
$194$	2.00000	0.143592
$195$	0	0
$196$	−7.00000	−0.500000
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	0	0
$199$	−14.0000	−0.992434	−0.496217	−	0.868199i	$-0.665278\pi$
$199$	−14.0000	−0.992434	−0.496217	+	0.868199i	$0.665278\pi$
$200$	−5.00000	−0.353553
$201$	0	0
$202$	−2.00000	−0.140720
$203$	0	0
$204$	0	0
$205$	0	0
$206$	13.0000	0.905753
$207$	0	0
$208$	−2.00000	−0.138675
$209$	4.00000	0.276686
$210$	0	0
$211$	−18.0000	−1.23917	−0.619586	−	0.784929i	$-0.712699\pi$
$211$	−18.0000	−1.23917	−0.619586	+	0.784929i	$0.712699\pi$
$212$	9.00000	0.618123
$213$	0	0
$214$	−12.0000	−0.820303
$215$	0	0
$216$	0	0
$217$	0	0
$218$	−10.0000	−0.677285
$219$	0	0
$220$	0	0
$221$	2.00000	0.134535
$222$	0	0
$223$	−1.00000	−0.0669650
$223$	−1.00000	−0.0669650
$224$	0	0
$225$	0	0
$226$	−6.00000	−0.399114
$227$	8.00000	0.530979	0.265489	−	0.964114i	$-0.414466\pi$
$227$	8.00000	0.530979	0.265489	+	0.964114i	$0.414466\pi$
$228$	0	0
$229$	8.00000	0.528655	0.264327	−	0.964433i	$-0.414850\pi$
$229$	8.00000	0.528655	0.264327	+	0.964433i	$0.414850\pi$
$230$	0	0
$231$	0	0
$232$	3.00000	0.196960
$233$	−4.00000	−0.262049	−0.131024	−	0.991379i	$-0.541827\pi$
$233$	−4.00000	−0.262049	−0.131024	+	0.991379i	$0.541827\pi$
$234$	0	0
$235$	0	0
$236$	1.00000	0.0650945
$237$	0	0
$238$	0	0
$239$	6.00000	0.388108	0.194054	−	0.980991i	$-0.437836\pi$
$239$	6.00000	0.388108	0.194054	+	0.980991i	$0.437836\pi$
$240$	0	0
$241$	13.0000	0.837404	0.418702	−	0.908124i	$-0.362485\pi$
$241$	13.0000	0.837404	0.418702	+	0.908124i	$0.362485\pi$
$242$	−10.0000	−0.642824
$243$	0	0
$244$	4.00000	0.256074
$245$	0	0
$246$	0	0
$247$	8.00000	0.509028
$248$	−10.0000	−0.635001
$249$	0	0
$250$	0	0
$251$	−10.0000	−0.631194	−0.315597	−	0.948893i	$-0.602205\pi$
$251$	−10.0000	−0.631194	−0.315597	+	0.948893i	$0.602205\pi$
$252$	0	0
$253$	1.00000	0.0628695
$254$	−2.00000	−0.125491
$255$	0	0
$256$	1.00000	0.0625000
$257$	9.00000	0.561405	0.280702	−	0.959795i	$-0.409433\pi$
$257$	9.00000	0.561405	0.280702	+	0.959795i	$0.409433\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	10.0000	0.617802
$263$	−9.00000	−0.554964	−0.277482	−	0.960731i	$-0.589500\pi$
$263$	−9.00000	−0.554964	−0.277482	+	0.960731i	$0.589500\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	9.00000	0.549762
$269$	16.0000	0.975537	0.487769	−	0.872973i	$-0.337811\pi$
$269$	16.0000	0.975537	0.487769	+	0.872973i	$0.337811\pi$
$270$	0	0
$271$	5.00000	0.303728	0.151864	−	0.988401i	$-0.451472\pi$
$271$	5.00000	0.303728	0.151864	+	0.988401i	$0.451472\pi$
$272$	−1.00000	−0.0606339
$273$	0	0
$274$	2.00000	0.120824
$275$	5.00000	0.301511
$276$	0	0
$277$	−8.00000	−0.480673	−0.240337	−	0.970690i	$-0.577258\pi$
$277$	−8.00000	−0.480673	−0.240337	+	0.970690i	$0.577258\pi$
$278$	22.0000	1.31947
$279$	0	0
$280$	0	0
$281$	13.0000	0.775515	0.387757	−	0.921761i	$-0.373250\pi$
$281$	13.0000	0.775515	0.387757	+	0.921761i	$0.373250\pi$
$282$	0	0
$283$	−6.00000	−0.356663	−0.178331	−	0.983970i	$-0.557070\pi$
$283$	−6.00000	−0.356663	−0.178331	+	0.983970i	$0.557070\pi$
$284$	−4.00000	−0.237356
$285$	0	0
$286$	2.00000	0.118262
$287$	0	0
$288$	0	0
$289$	−16.0000	−0.941176
$290$	0	0
$291$	0	0
$292$	−5.00000	−0.292603
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	0	0
$295$	0	0
$296$	−3.00000	−0.174371
$297$	0	0
$298$	−22.0000	−1.27443
$299$	2.00000	0.115663
$300$	0	0
$301$	0	0
$302$	−9.00000	−0.517892
$303$	0	0
$304$	−4.00000	−0.229416
$305$	0	0
$306$	0	0
$307$	19.0000	1.08439	0.542194	−	0.840254i	$-0.317594\pi$
$307$	19.0000	1.08439	0.542194	+	0.840254i	$0.317594\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−21.0000	−1.19080	−0.595400	−	0.803429i	$-0.703007\pi$
$311$	−21.0000	−1.19080	−0.595400	+	0.803429i	$0.703007\pi$
$312$	0	0
$313$	−26.0000	−1.46961	−0.734803	−	0.678280i	$-0.762726\pi$
$313$	−26.0000	−1.46961	−0.734803	+	0.678280i	$0.762726\pi$
$314$	2.00000	0.112867
$315$	0	0
$316$	0	0
$317$	−21.0000	−1.17948	−0.589739	−	0.807594i	$-0.700769\pi$
$317$	−21.0000	−1.17948	−0.589739	+	0.807594i	$0.700769\pi$
$318$	0	0
$319$	−3.00000	−0.167968
$320$	0	0
$321$	0	0
$322$	0	0
$323$	4.00000	0.222566
$324$	0	0
$325$	10.0000	0.554700
$326$	−8.00000	−0.443079
$327$	0	0
$328$	5.00000	0.276079
$329$	0	0
$330$	0	0
$331$	−7.00000	−0.384755	−0.192377	−	0.981321i	$-0.561620\pi$
$331$	−7.00000	−0.384755	−0.192377	+	0.981321i	$0.561620\pi$
$332$	−14.0000	−0.768350
$333$	0	0
$334$	1.00000	0.0547176
$335$	0	0
$336$	0	0
$337$	14.0000	0.762629	0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000	0.762629	0.381314	+	0.924445i	$0.375472\pi$
$338$	−9.00000	−0.489535
$339$	0	0
$340$	0	0
$341$	10.0000	0.541530
$342$	0	0
$343$	0	0
$344$	−6.00000	−0.323498
$345$	0	0
$346$	16.0000	0.860165
$347$	−16.0000	−0.858925	−0.429463	−	0.903085i	$-0.641297\pi$
$347$	−16.0000	−0.858925	−0.429463	+	0.903085i	$0.641297\pi$
$348$	0	0
$349$	15.0000	0.802932	0.401466	−	0.915874i	$-0.368501\pi$
$349$	15.0000	0.802932	0.401466	+	0.915874i	$0.368501\pi$
$350$	0	0
$351$	0	0
$352$	−1.00000	−0.0533002
$353$	14.0000	0.745145	0.372572	−	0.928003i	$-0.378476\pi$
$353$	14.0000	0.745145	0.372572	+	0.928003i	$0.378476\pi$
$354$	0	0
$355$	0	0
$356$	5.00000	0.264999
$357$	0	0
$358$	4.00000	0.211407
$359$	36.0000	1.90001	0.950004	−	0.312239i	$-0.101079\pi$
$359$	36.0000	1.90001	0.950004	+	0.312239i	$0.101079\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	9.00000	0.473029
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−10.0000	−0.521996	−0.260998	−	0.965339i	$-0.584052\pi$
$367$	−10.0000	−0.521996	−0.260998	+	0.965339i	$0.584052\pi$
$368$	−1.00000	−0.0521286
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	16.0000	0.828449	0.414224	−	0.910175i	$-0.364053\pi$
$373$	16.0000	0.828449	0.414224	+	0.910175i	$0.364053\pi$
$374$	1.00000	0.0517088
$375$	0	0
$376$	−6.00000	−0.309426
$377$	−6.00000	−0.309016
$378$	0	0
$379$	4.00000	0.205466	0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000	0.205466	0.102733	+	0.994709i	$0.467241\pi$
$380$	0	0
$381$	0	0
$382$	3.00000	0.153493
$383$	−1.00000	−0.0510976	−0.0255488	−	0.999674i	$-0.508133\pi$
$383$	−1.00000	−0.0510976	−0.0255488	+	0.999674i	$0.508133\pi$
$384$	0	0
$385$	0	0
$386$	−4.00000	−0.203595
$387$	0	0
$388$	2.00000	0.101535
$389$	−37.0000	−1.87597	−0.937987	−	0.346670i	$-0.887312\pi$
$389$	−37.0000	−1.87597	−0.937987	+	0.346670i	$0.887312\pi$
$390$	0	0
$391$	1.00000	0.0505722
$392$	−7.00000	−0.353553
$393$	0	0
$394$	−6.00000	−0.302276
$395$	0	0
$396$	0	0
$397$	8.00000	0.401508	0.200754	−	0.979642i	$-0.435661\pi$
$397$	8.00000	0.401508	0.200754	+	0.979642i	$0.435661\pi$
$398$	−14.0000	−0.701757
$399$	0	0
$400$	−5.00000	−0.250000
$401$	26.0000	1.29838	0.649189	−	0.760627i	$-0.275108\pi$
$401$	26.0000	1.29838	0.649189	+	0.760627i	$0.275108\pi$
$402$	0	0
$403$	20.0000	0.996271
$404$	−2.00000	−0.0995037
$405$	0	0
$406$	0	0
$407$	3.00000	0.148704
$408$	0	0
$409$	6.00000	0.296681	0.148340	−	0.988936i	$-0.452607\pi$
$409$	6.00000	0.296681	0.148340	+	0.988936i	$0.452607\pi$
$410$	0	0
$411$	0	0
$412$	13.0000	0.640464
$413$	0	0
$414$	0	0
$415$	0	0
$416$	−2.00000	−0.0980581
$417$	0	0
$418$	4.00000	0.195646
$419$	−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
$421$	10.0000	0.487370	0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000	0.487370	0.243685	+	0.969854i	$0.421644\pi$
$422$	−18.0000	−0.876226
$423$	0	0
$424$	9.00000	0.437079
$425$	5.00000	0.242536
$426$	0	0
$427$	0	0
$428$	−12.0000	−0.580042
$429$	0	0
$430$	0	0
$431$	15.0000	0.722525	0.361262	−	0.932464i	$-0.382346\pi$
$431$	15.0000	0.722525	0.361262	+	0.932464i	$0.382346\pi$
$432$	0	0
$433$	14.0000	0.672797	0.336399	−	0.941720i	$-0.390791\pi$
$433$	14.0000	0.672797	0.336399	+	0.941720i	$0.390791\pi$
$434$	0	0
$435$	0	0
$436$	−10.0000	−0.478913
$437$	4.00000	0.191346
$438$	0	0
$439$	−15.0000	−0.715911	−0.357955	−	0.933739i	$-0.616526\pi$
$439$	−15.0000	−0.715911	−0.357955	+	0.933739i	$0.616526\pi$
$440$	0	0
$441$	0	0
$442$	2.00000	0.0951303
$443$	24.0000	1.14027	0.570137	−	0.821549i	$-0.306890\pi$
$443$	24.0000	1.14027	0.570137	+	0.821549i	$0.306890\pi$
$444$	0	0
$445$	0	0
$446$	−1.00000	−0.0473514
$447$	0	0
$448$	0	0
$449$	30.0000	1.41579	0.707894	−	0.706319i	$-0.249646\pi$
$449$	30.0000	1.41579	0.707894	+	0.706319i	$0.249646\pi$
$450$	0	0
$451$	−5.00000	−0.235441
$452$	−6.00000	−0.282216
$453$	0	0
$454$	8.00000	0.375459
$455$	0	0
$456$	0	0
$457$	36.0000	1.68401	0.842004	−	0.539471i	$-0.181376\pi$
$457$	36.0000	1.68401	0.842004	+	0.539471i	$0.181376\pi$
$458$	8.00000	0.373815
$459$	0	0
$460$	0	0
$461$	−17.0000	−0.791769	−0.395884	−	0.918300i	$-0.629562\pi$
$461$	−17.0000	−0.791769	−0.395884	+	0.918300i	$0.629562\pi$
$462$	0	0
$463$	−8.00000	−0.371792	−0.185896	−	0.982569i	$-0.559519\pi$
$463$	−8.00000	−0.371792	−0.185896	+	0.982569i	$0.559519\pi$
$464$	3.00000	0.139272
$465$	0	0
$466$	−4.00000	−0.185296
$467$	−9.00000	−0.416470	−0.208235	−	0.978079i	$-0.566772\pi$
$467$	−9.00000	−0.416470	−0.208235	+	0.978079i	$0.566772\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	1.00000	0.0460287
$473$	6.00000	0.275880
$474$	0	0
$475$	20.0000	0.917663
$476$	0	0
$477$	0	0
$478$	6.00000	0.274434
$479$	−6.00000	−0.274147	−0.137073	−	0.990561i	$-0.543770\pi$
$479$	−6.00000	−0.274147	−0.137073	+	0.990561i	$0.543770\pi$
$480$	0	0
$481$	6.00000	0.273576
$482$	13.0000	0.592134
$483$	0	0
$484$	−10.0000	−0.454545
$485$	0	0
$486$	0	0
$487$	−4.00000	−0.181257	−0.0906287	−	0.995885i	$-0.528888\pi$
$487$	−4.00000	−0.181257	−0.0906287	+	0.995885i	$0.528888\pi$
$488$	4.00000	0.181071
$489$	0	0
$490$	0	0
$491$	−15.0000	−0.676941	−0.338470	−	0.940977i	$-0.609909\pi$
$491$	−15.0000	−0.676941	−0.338470	+	0.940977i	$0.609909\pi$
$492$	0	0
$493$	−3.00000	−0.135113
$494$	8.00000	0.359937
$495$	0	0
$496$	−10.0000	−0.449013
$497$	0	0
$498$	0	0
$499$	−8.00000	−0.358129	−0.179065	−	0.983837i	$-0.557307\pi$
$499$	−8.00000	−0.358129	−0.179065	+	0.983837i	$0.557307\pi$
$500$	0	0
$501$	0	0
$502$	−10.0000	−0.446322
$503$	−9.00000	−0.401290	−0.200645	−	0.979664i	$-0.564304\pi$
$503$	−9.00000	−0.401290	−0.200645	+	0.979664i	$0.564304\pi$
$504$	0	0
$505$	0	0
$506$	1.00000	0.0444554
$507$	0	0
$508$	−2.00000	−0.0887357
$509$	18.0000	0.797836	0.398918	−	0.916987i	$-0.369386\pi$
$509$	18.0000	0.797836	0.398918	+	0.916987i	$0.369386\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	0.0441942
$513$	0	0
$514$	9.00000	0.396973
$515$	0	0
$516$	0	0
$517$	6.00000	0.263880
$518$	0	0
$519$	0	0
$520$	0	0
$521$	18.0000	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$522$	0	0
$523$	−40.0000	−1.74908	−0.874539	−	0.484955i	$-0.838836\pi$
$523$	−40.0000	−1.74908	−0.874539	+	0.484955i	$0.838836\pi$
$524$	10.0000	0.436852
$525$	0	0
$526$	−9.00000	−0.392419
$527$	10.0000	0.435607
$528$	0	0
$529$	−22.0000	−0.956522
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−10.0000	−0.433148
$534$	0	0
$535$	0	0
$536$	9.00000	0.388741
$537$	0	0
$538$	16.0000	0.689809
$539$	7.00000	0.301511
$540$	0	0
$541$	−10.0000	−0.429934	−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000	−0.429934	−0.214967	+	0.976621i	$0.568964\pi$
$542$	5.00000	0.214768
$543$	0	0
$544$	−1.00000	−0.0428746
$545$	0	0
$546$	0	0
$547$	−2.00000	−0.0855138	−0.0427569	−	0.999086i	$-0.513614\pi$
$547$	−2.00000	−0.0855138	−0.0427569	+	0.999086i	$0.513614\pi$
$548$	2.00000	0.0854358
$549$	0	0
$550$	5.00000	0.213201
$551$	−12.0000	−0.511217
$552$	0	0
$553$	0	0
$554$	−8.00000	−0.339887
$555$	0	0
$556$	22.0000	0.933008
$557$	28.0000	1.18640	0.593199	−	0.805056i	$-0.297865\pi$
$557$	28.0000	1.18640	0.593199	+	0.805056i	$0.297865\pi$
$558$	0	0
$559$	12.0000	0.507546
$560$	0	0
$561$	0	0
$562$	13.0000	0.548372
$563$	20.0000	0.842900	0.421450	−	0.906852i	$-0.361521\pi$
$563$	20.0000	0.842900	0.421450	+	0.906852i	$0.361521\pi$
$564$	0	0
$565$	0	0
$566$	−6.00000	−0.252199
$567$	0	0
$568$	−4.00000	−0.167836
$569$	22.0000	0.922288	0.461144	−	0.887325i	$-0.347439\pi$
$569$	22.0000	0.922288	0.461144	+	0.887325i	$0.347439\pi$
$570$	0	0
$571$	−37.0000	−1.54840	−0.774201	−	0.632940i	$-0.781848\pi$
$571$	−37.0000	−1.54840	−0.774201	+	0.632940i	$0.781848\pi$
$572$	2.00000	0.0836242
$573$	0	0
$574$	0	0
$575$	5.00000	0.208514
$576$	0	0
$577$	−7.00000	−0.291414	−0.145707	−	0.989328i	$-0.546546\pi$
$577$	−7.00000	−0.291414	−0.145707	+	0.989328i	$0.546546\pi$
$578$	−16.0000	−0.665512
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−9.00000	−0.372742
$584$	−5.00000	−0.206901
$585$	0	0
$586$	0	0
$587$	−1.00000	−0.0412744	−0.0206372	−	0.999787i	$-0.506569\pi$
$587$	−1.00000	−0.0412744	−0.0206372	+	0.999787i	$0.506569\pi$
$588$	0	0
$589$	40.0000	1.64817
$590$	0	0
$591$	0	0
$592$	−3.00000	−0.123299
$593$	44.0000	1.80686	0.903432	−	0.428732i	$-0.141040\pi$
$593$	44.0000	1.80686	0.903432	+	0.428732i	$0.141040\pi$
$594$	0	0
$595$	0	0
$596$	−22.0000	−0.901155
$597$	0	0
$598$	2.00000	0.0817861
$599$	40.0000	1.63436	0.817178	−	0.576386i	$-0.195537\pi$
$599$	40.0000	1.63436	0.817178	+	0.576386i	$0.195537\pi$
$600$	0	0
$601$	−22.0000	−0.897399	−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000	−0.897399	−0.448699	+	0.893683i	$0.648113\pi$
$602$	0	0
$603$	0	0
$604$	−9.00000	−0.366205
$605$	0	0
$606$	0	0
$607$	33.0000	1.33943	0.669714	−	0.742619i	$-0.266417\pi$
$607$	33.0000	1.33943	0.669714	+	0.742619i	$0.266417\pi$
$608$	−4.00000	−0.162221
$609$	0	0
$610$	0	0
$611$	12.0000	0.485468
$612$	0	0
$613$	−32.0000	−1.29247	−0.646234	−	0.763139i	$-0.723657\pi$
$613$	−32.0000	−1.29247	−0.646234	+	0.763139i	$0.723657\pi$
$614$	19.0000	0.766778
$615$	0	0
$616$	0	0
$617$	29.0000	1.16750	0.583748	−	0.811935i	$-0.301586\pi$
$617$	29.0000	1.16750	0.583748	+	0.811935i	$0.301586\pi$
$618$	0	0
$619$	21.0000	0.844061	0.422031	−	0.906582i	$-0.361317\pi$
$619$	21.0000	0.844061	0.422031	+	0.906582i	$0.361317\pi$
$620$	0	0
$621$	0	0
$622$	−21.0000	−0.842023
$623$	0	0
$624$	0	0
$625$	25.0000	1.00000
$626$	−26.0000	−1.03917
$627$	0	0
$628$	2.00000	0.0798087
$629$	3.00000	0.119618
$630$	0	0
$631$	31.0000	1.23409	0.617045	−	0.786928i	$-0.288330\pi$
$631$	31.0000	1.23409	0.617045	+	0.786928i	$0.288330\pi$
$632$	0	0
$633$	0	0
$634$	−21.0000	−0.834017
$635$	0	0
$636$	0	0
$637$	14.0000	0.554700
$638$	−3.00000	−0.118771
$639$	0	0
$640$	0	0
$641$	2.00000	0.0789953	0.0394976	−	0.999220i	$-0.487424\pi$
$641$	2.00000	0.0789953	0.0394976	+	0.999220i	$0.487424\pi$
$642$	0	0
$643$	−16.0000	−0.630978	−0.315489	−	0.948929i	$-0.602169\pi$
$643$	−16.0000	−0.630978	−0.315489	+	0.948929i	$0.602169\pi$
$644$	0	0
$645$	0	0
$646$	4.00000	0.157378
$647$	−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
$649$	−1.00000	−0.0392534
$650$	10.0000	0.392232
$651$	0	0
$652$	−8.00000	−0.313304
$653$	14.0000	0.547862	0.273931	−	0.961749i	$-0.411676\pi$
$653$	14.0000	0.547862	0.273931	+	0.961749i	$0.411676\pi$
$654$	0	0
$655$	0	0
$656$	5.00000	0.195217
$657$	0	0
$658$	0	0
$659$	−12.0000	−0.467454	−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000	−0.467454	−0.233727	+	0.972302i	$0.575092\pi$
$660$	0	0
$661$	24.0000	0.933492	0.466746	−	0.884391i	$-0.345426\pi$
$661$	24.0000	0.933492	0.466746	+	0.884391i	$0.345426\pi$
$662$	−7.00000	−0.272063
$663$	0	0
$664$	−14.0000	−0.543305
$665$	0	0
$666$	0	0
$667$	−3.00000	−0.116160
$668$	1.00000	0.0386912
$669$	0	0
$670$	0	0
$671$	−4.00000	−0.154418
$672$	0	0
$673$	−1.00000	−0.0385472	−0.0192736	−	0.999814i	$-0.506135\pi$
$673$	−1.00000	−0.0385472	−0.0192736	+	0.999814i	$0.506135\pi$
$674$	14.0000	0.539260
$675$	0	0
$676$	−9.00000	−0.346154
$677$	−50.0000	−1.92166	−0.960828	−	0.277145i	$-0.910612\pi$
$677$	−50.0000	−1.92166	−0.960828	+	0.277145i	$0.910612\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	10.0000	0.382920
$683$	28.0000	1.07139	0.535695	−	0.844411i	$-0.320050\pi$
$683$	28.0000	1.07139	0.535695	+	0.844411i	$0.320050\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	−6.00000	−0.228748
$689$	−18.0000	−0.685745
$690$	0	0
$691$	−20.0000	−0.760836	−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000	−0.760836	−0.380418	+	0.924815i	$0.624220\pi$
$692$	16.0000	0.608229
$693$	0	0
$694$	−16.0000	−0.607352
$695$	0	0
$696$	0	0
$697$	−5.00000	−0.189389
$698$	15.0000	0.567758
$699$	0	0
$700$	0	0
$701$	−15.0000	−0.566542	−0.283271	−	0.959040i	$-0.591420\pi$
$701$	−15.0000	−0.566542	−0.283271	+	0.959040i	$0.591420\pi$
$702$	0	0
$703$	12.0000	0.452589
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	14.0000	0.526897
$707$	0	0
$708$	0	0
$709$	28.0000	1.05156	0.525781	−	0.850620i	$-0.323773\pi$
$709$	28.0000	1.05156	0.525781	+	0.850620i	$0.323773\pi$
$710$	0	0
$711$	0	0
$712$	5.00000	0.187383
$713$	10.0000	0.374503
$714$	0	0
$715$	0	0
$716$	4.00000	0.149487
$717$	0	0
$718$	36.0000	1.34351
$719$	16.0000	0.596699	0.298350	−	0.954457i	$-0.403564\pi$
$719$	16.0000	0.596699	0.298350	+	0.954457i	$0.403564\pi$
$720$	0	0
$721$	0	0
$722$	−3.00000	−0.111648
$723$	0	0
$724$	9.00000	0.334482
$725$	−15.0000	−0.557086
$726$	0	0
$727$	−16.0000	−0.593407	−0.296704	−	0.954970i	$-0.595887\pi$
$727$	−16.0000	−0.593407	−0.296704	+	0.954970i	$0.595887\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	6.00000	0.221918
$732$	0	0
$733$	13.0000	0.480166	0.240083	−	0.970752i	$-0.422825\pi$
$733$	13.0000	0.480166	0.240083	+	0.970752i	$0.422825\pi$
$734$	−10.0000	−0.369107
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	−9.00000	−0.331519
$738$	0	0
$739$	−52.0000	−1.91285	−0.956425	−	0.291977i	$-0.905687\pi$
$739$	−52.0000	−1.91285	−0.956425	+	0.291977i	$0.905687\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−6.00000	−0.220119	−0.110059	−	0.993925i	$-0.535104\pi$
$743$	−6.00000	−0.220119	−0.110059	+	0.993925i	$0.535104\pi$
$744$	0	0
$745$	0	0
$746$	16.0000	0.585802
$747$	0	0
$748$	1.00000	0.0365636
$749$	0	0
$750$	0	0
$751$	14.0000	0.510867	0.255434	−	0.966827i	$-0.417782\pi$
$751$	14.0000	0.510867	0.255434	+	0.966827i	$0.417782\pi$
$752$	−6.00000	−0.218797
$753$	0	0
$754$	−6.00000	−0.218507
$755$	0	0
$756$	0	0
$757$	−34.0000	−1.23575	−0.617876	−	0.786276i	$-0.712006\pi$
$757$	−34.0000	−1.23575	−0.617876	+	0.786276i	$0.712006\pi$
$758$	4.00000	0.145287
$759$	0	0
$760$	0	0
$761$	−24.0000	−0.869999	−0.435000	−	0.900431i	$-0.643252\pi$
$761$	−24.0000	−0.869999	−0.435000	+	0.900431i	$0.643252\pi$
$762$	0	0
$763$	0	0
$764$	3.00000	0.108536
$765$	0	0
$766$	−1.00000	−0.0361315
$767$	−2.00000	−0.0722158
$768$	0	0
$769$	35.0000	1.26213	0.631066	−	0.775729i	$-0.282618\pi$
$769$	35.0000	1.26213	0.631066	+	0.775729i	$0.282618\pi$
$770$	0	0
$771$	0	0
$772$	−4.00000	−0.143963
$773$	−16.0000	−0.575480	−0.287740	−	0.957709i	$-0.592904\pi$
$773$	−16.0000	−0.575480	−0.287740	+	0.957709i	$0.592904\pi$
$774$	0	0
$775$	50.0000	1.79605
$776$	2.00000	0.0717958
$777$	0	0
$778$	−37.0000	−1.32651
$779$	−20.0000	−0.716574
$780$	0	0
$781$	4.00000	0.143131
$782$	1.00000	0.0357599
$783$	0	0
$784$	−7.00000	−0.250000
$785$	0	0
$786$	0	0
$787$	3.00000	0.106938	0.0534692	−	0.998569i	$-0.482972\pi$
$787$	3.00000	0.106938	0.0534692	+	0.998569i	$0.482972\pi$
$788$	−6.00000	−0.213741
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−8.00000	−0.284088
$794$	8.00000	0.283909
$795$	0	0
$796$	−14.0000	−0.496217
$797$	−11.0000	−0.389640	−0.194820	−	0.980839i	$-0.562412\pi$
$797$	−11.0000	−0.389640	−0.194820	+	0.980839i	$0.562412\pi$
$798$	0	0
$799$	6.00000	0.212265
$800$	−5.00000	−0.176777
$801$	0	0
$802$	26.0000	0.918092
$803$	5.00000	0.176446
$804$	0	0
$805$	0	0
$806$	20.0000	0.704470
$807$	0	0
$808$	−2.00000	−0.0703598
$809$	36.0000	1.26569	0.632846	−	0.774277i	$-0.281886\pi$
$809$	36.0000	1.26569	0.632846	+	0.774277i	$0.281886\pi$
$810$	0	0
$811$	4.00000	0.140459	0.0702295	−	0.997531i	$-0.477627\pi$
$811$	4.00000	0.140459	0.0702295	+	0.997531i	$0.477627\pi$
$812$	0	0
$813$	0	0
$814$	3.00000	0.105150
$815$	0	0
$816$	0	0
$817$	24.0000	0.839654
$818$	6.00000	0.209785
$819$	0	0
$820$	0	0
$821$	−21.0000	−0.732905	−0.366453	−	0.930437i	$-0.619428\pi$
$821$	−21.0000	−0.732905	−0.366453	+	0.930437i	$0.619428\pi$
$822$	0	0
$823$	4.00000	0.139431	0.0697156	−	0.997567i	$-0.477791\pi$
$823$	4.00000	0.139431	0.0697156	+	0.997567i	$0.477791\pi$
$824$	13.0000	0.452876
$825$	0	0
$826$	0	0
$827$	9.00000	0.312961	0.156480	−	0.987681i	$-0.449985\pi$
$827$	9.00000	0.312961	0.156480	+	0.987681i	$0.449985\pi$
$828$	0	0
$829$	−36.0000	−1.25033	−0.625166	−	0.780492i	$-0.714969\pi$
$829$	−36.0000	−1.25033	−0.625166	+	0.780492i	$0.714969\pi$
$830$	0	0
$831$	0	0
$832$	−2.00000	−0.0693375
$833$	7.00000	0.242536
$834$	0	0
$835$	0	0
$836$	4.00000	0.138343
$837$	0	0
$838$	−24.0000	−0.829066
$839$	−23.0000	−0.794048	−0.397024	−	0.917808i	$-0.629957\pi$
$839$	−23.0000	−0.794048	−0.397024	+	0.917808i	$0.629957\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	10.0000	0.344623
$843$	0	0
$844$	−18.0000	−0.619586
$845$	0	0
$846$	0	0
$847$	0	0
$848$	9.00000	0.309061
$849$	0	0
$850$	5.00000	0.171499
$851$	3.00000	0.102839
$852$	0	0
$853$	28.0000	0.958702	0.479351	−	0.877623i	$-0.340872\pi$
$853$	28.0000	0.958702	0.479351	+	0.877623i	$0.340872\pi$
$854$	0	0
$855$	0	0
$856$	−12.0000	−0.410152
$857$	14.0000	0.478231	0.239115	−	0.970991i	$-0.423143\pi$
$857$	14.0000	0.478231	0.239115	+	0.970991i	$0.423143\pi$
$858$	0	0
$859$	36.0000	1.22830	0.614152	−	0.789188i	$-0.289498\pi$
$859$	36.0000	1.22830	0.614152	+	0.789188i	$0.289498\pi$
$860$	0	0
$861$	0	0
$862$	15.0000	0.510902
$863$	21.0000	0.714848	0.357424	−	0.933942i	$-0.383655\pi$
$863$	21.0000	0.714848	0.357424	+	0.933942i	$0.383655\pi$
$864$	0	0
$865$	0	0
$866$	14.0000	0.475739
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−18.0000	−0.609907
$872$	−10.0000	−0.338643
$873$	0	0
$874$	4.00000	0.135302
$875$	0	0
$876$	0	0
$877$	52.0000	1.75592	0.877958	−	0.478738i	$-0.158906\pi$
$877$	52.0000	1.75592	0.877958	+	0.478738i	$0.158906\pi$
$878$	−15.0000	−0.506225
$879$	0	0
$880$	0	0
$881$	39.0000	1.31394	0.656972	−	0.753915i	$-0.271837\pi$
$881$	39.0000	1.31394	0.656972	+	0.753915i	$0.271837\pi$
$882$	0	0
$883$	−36.0000	−1.21150	−0.605748	−	0.795656i	$-0.707126\pi$
$883$	−36.0000	−1.21150	−0.605748	+	0.795656i	$0.707126\pi$
$884$	2.00000	0.0672673
$885$	0	0
$886$	24.0000	0.806296
$887$	6.00000	0.201460	0.100730	−	0.994914i	$-0.467882\pi$
$887$	6.00000	0.201460	0.100730	+	0.994914i	$0.467882\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	−1.00000	−0.0334825
$893$	24.0000	0.803129
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	30.0000	1.00111
$899$	−30.0000	−1.00056
$900$	0	0
$901$	−9.00000	−0.299833
$902$	−5.00000	−0.166482
$903$	0	0
$904$	−6.00000	−0.199557
$905$	0	0
$906$	0	0
$907$	−16.0000	−0.531271	−0.265636	−	0.964073i	$-0.585582\pi$
$907$	−16.0000	−0.531271	−0.265636	+	0.964073i	$0.585582\pi$
$908$	8.00000	0.265489
$909$	0	0
$910$	0	0
$911$	8.00000	0.265052	0.132526	−	0.991180i	$-0.457691\pi$
$911$	8.00000	0.265052	0.132526	+	0.991180i	$0.457691\pi$
$912$	0	0
$913$	14.0000	0.463332
$914$	36.0000	1.19077
$915$	0	0
$916$	8.00000	0.264327
$917$	0	0
$918$	0	0
$919$	48.0000	1.58337	0.791687	−	0.610927i	$-0.209203\pi$
$919$	48.0000	1.58337	0.791687	+	0.610927i	$0.209203\pi$
$920$	0	0
$921$	0	0
$922$	−17.0000	−0.559865
$923$	8.00000	0.263323
$924$	0	0
$925$	15.0000	0.493197
$926$	−8.00000	−0.262896
$927$	0	0
$928$	3.00000	0.0984798
$929$	−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
$931$	28.0000	0.917663
$932$	−4.00000	−0.131024
$933$	0	0
$934$	−9.00000	−0.294489
$935$	0	0
$936$	0	0
$937$	54.0000	1.76410	0.882052	−	0.471153i	$-0.156162\pi$
$937$	54.0000	1.76410	0.882052	+	0.471153i	$0.156162\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−59.0000	−1.92335	−0.961673	−	0.274201i	$-0.911587\pi$
$941$	−59.0000	−1.92335	−0.961673	+	0.274201i	$0.911587\pi$
$942$	0	0
$943$	−5.00000	−0.162822
$944$	1.00000	0.0325472
$945$	0	0
$946$	6.00000	0.195077
$947$	−6.00000	−0.194974	−0.0974869	−	0.995237i	$-0.531080\pi$
$947$	−6.00000	−0.194974	−0.0974869	+	0.995237i	$0.531080\pi$
$948$	0	0
$949$	10.0000	0.324614
$950$	20.0000	0.648886
$951$	0	0
$952$	0	0
$953$	−36.0000	−1.16615	−0.583077	−	0.812417i	$-0.698151\pi$
$953$	−36.0000	−1.16615	−0.583077	+	0.812417i	$0.698151\pi$
$954$	0	0
$955$	0	0
$956$	6.00000	0.194054
$957$	0	0
$958$	−6.00000	−0.193851
$959$	0	0
$960$	0	0
$961$	69.0000	2.22581
$962$	6.00000	0.193448
$963$	0	0
$964$	13.0000	0.418702
$965$	0	0
$966$	0	0
$967$	−20.0000	−0.643157	−0.321578	−	0.946883i	$-0.604213\pi$
$967$	−20.0000	−0.643157	−0.321578	+	0.946883i	$0.604213\pi$
$968$	−10.0000	−0.321412
$969$	0	0
$970$	0	0
$971$	−13.0000	−0.417190	−0.208595	−	0.978002i	$-0.566889\pi$
$971$	−13.0000	−0.417190	−0.208595	+	0.978002i	$0.566889\pi$
$972$	0	0
$973$	0	0
$974$	−4.00000	−0.128168
$975$	0	0
$976$	4.00000	0.128037
$977$	8.00000	0.255943	0.127971	−	0.991778i	$-0.459153\pi$
$977$	8.00000	0.255943	0.127971	+	0.991778i	$0.459153\pi$
$978$	0	0
$979$	−5.00000	−0.159801
$980$	0	0
$981$	0	0
$982$	−15.0000	−0.478669
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	0	0
$986$	−3.00000	−0.0955395
$987$	0	0
$988$	8.00000	0.254514
$989$	6.00000	0.190789
$990$	0	0
$991$	−56.0000	−1.77890	−0.889449	−	0.457034i	$-0.848912\pi$
$991$	−56.0000	−1.77890	−0.889449	+	0.457034i	$0.848912\pi$
$992$	−10.0000	−0.317500
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	10.0000	0.316703	0.158352	−	0.987383i	$-0.449382\pi$
$997$	10.0000	0.316703	0.158352	+	0.987383i	$0.449382\pi$
$998$	−8.00000	−0.253236
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4014.2.a.e.1.1		1
3.2	odd	2		446.2.a.b.1.1	✓	1
12.11	even	2		3568.2.a.f.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
446.2.a.b.1.1	✓	1	3.2	odd	2
3568.2.a.f.1.1		1	12.11	even	2
4014.2.a.e.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:	\( N \)	\(=\)	\( 4014 = 2 \cdot 3^{2} \cdot 223 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4014.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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