LMFDB - Embedded newform 816.4.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [816,4,Mod(1,816)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(816, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("816.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 816 = 2^{4} \cdot 3 \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	816.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:	$48.1455585647$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 102)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	816.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-3.00000 q^{3} -5.00000 q^{5} +32.0000 q^{7} +9.00000 q^{9} -27.0000 q^{11} -69.0000 q^{13} +15.0000 q^{15} -17.0000 q^{17} +83.0000 q^{19} -96.0000 q^{21} +117.000 q^{23} -100.000 q^{25} -27.0000 q^{27} +94.0000 q^{29} -198.000 q^{31} +81.0000 q^{33} -160.000 q^{35} -244.000 q^{37} +207.000 q^{39} +169.000 q^{41} -227.000 q^{43} -45.0000 q^{45} +382.000 q^{47} +681.000 q^{49} +51.0000 q^{51} +686.000 q^{53} +135.000 q^{55} -249.000 q^{57} -450.000 q^{59} -700.000 q^{61} +288.000 q^{63} +345.000 q^{65} -540.000 q^{67} -351.000 q^{69} +276.000 q^{71} -298.000 q^{73} +300.000 q^{75} -864.000 q^{77} +182.000 q^{79} +81.0000 q^{81} -282.000 q^{83} +85.0000 q^{85} -282.000 q^{87} -1468.00 q^{89} -2208.00 q^{91} +594.000 q^{93} -415.000 q^{95} -1140.00 q^{97} -243.000 q^{99} +O(q^{100})$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	−5.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−5.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	0	0
$7$	32.0000	1.72784	0.863919	−	0.503631i	$-0.168003\pi$
$7$	32.0000	1.72784	0.863919	+	0.503631i	$0.168003\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−27.0000	−0.740073	−0.370037	−	0.929017i	$-0.620655\pi$
$11$	−27.0000	−0.740073	−0.370037	+	0.929017i	$0.620655\pi$
$12$	0	0
$13$	−69.0000	−1.47209	−0.736044	−	0.676933i	$-0.763309\pi$
$13$	−69.0000	−1.47209	−0.736044	+	0.676933i	$0.763309\pi$
$14$	0	0
$15$	15.0000	0.258199
$16$	0	0
$17$	−17.0000	−0.242536
$17$	−17.0000	−0.242536
$18$	0	0
$19$	83.0000	1.00218	0.501092	−	0.865394i	$-0.332932\pi$
$19$	83.0000	1.00218	0.501092	+	0.865394i	$0.332932\pi$
$20$	0	0
$21$	−96.0000	−0.997567
$22$	0	0
$23$	117.000	1.06070	0.530352	−	0.847778i	$-0.322060\pi$
$23$	117.000	1.06070	0.530352	+	0.847778i	$0.322060\pi$
$24$	0	0
$25$	−100.000	−0.800000
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	94.0000	0.601909	0.300955	−	0.953638i	$-0.402695\pi$
$29$	94.0000	0.601909	0.300955	+	0.953638i	$0.402695\pi$
$30$	0	0
$31$	−198.000	−1.14716	−0.573578	−	0.819151i	$-0.694445\pi$
$31$	−198.000	−1.14716	−0.573578	+	0.819151i	$0.694445\pi$
$32$	0	0
$33$	81.0000	0.427282
$34$	0	0
$35$	−160.000	−0.772712
$36$	0	0
$37$	−244.000	−1.08414	−0.542072	−	0.840332i	$-0.682360\pi$
$37$	−244.000	−1.08414	−0.542072	+	0.840332i	$0.682360\pi$
$38$	0	0
$39$	207.000	0.849911
$40$	0	0
$41$	169.000	0.643741	0.321870	−	0.946784i	$-0.395688\pi$
$41$	169.000	0.643741	0.321870	+	0.946784i	$0.395688\pi$
$42$	0	0
$43$	−227.000	−0.805051	−0.402525	−	0.915409i	$-0.631867\pi$
$43$	−227.000	−0.805051	−0.402525	+	0.915409i	$0.631867\pi$
$44$	0	0
$45$	−45.0000	−0.149071
$46$	0	0
$47$	382.000	1.18554	0.592770	−	0.805371i	$-0.298034\pi$
$47$	382.000	1.18554	0.592770	+	0.805371i	$0.298034\pi$
$48$	0	0
$49$	681.000	1.98542
$50$	0	0
$51$	51.0000	0.140028
$52$	0	0
$53$	686.000	1.77791	0.888956	−	0.457992i	$-0.151431\pi$
$53$	686.000	1.77791	0.888956	+	0.457992i	$0.151431\pi$
$54$	0	0
$55$	135.000	0.330971
$56$	0	0
$57$	−249.000	−0.578612
$58$	0	0
$59$	−450.000	−0.992966	−0.496483	−	0.868046i	$-0.665376\pi$
$59$	−450.000	−0.992966	−0.496483	+	0.868046i	$0.665376\pi$
$60$	0	0
$61$	−700.000	−1.46928	−0.734638	−	0.678459i	$-0.762648\pi$
$61$	−700.000	−1.46928	−0.734638	+	0.678459i	$0.762648\pi$
$62$	0	0
$63$	288.000	0.575946
$64$	0	0
$65$	345.000	0.658338
$66$	0	0
$67$	−540.000	−0.984649	−0.492325	−	0.870412i	$-0.663853\pi$
$67$	−540.000	−0.984649	−0.492325	+	0.870412i	$0.663853\pi$
$68$	0	0
$69$	−351.000	−0.612398
$70$	0	0
$71$	276.000	0.461340	0.230670	−	0.973032i	$-0.425908\pi$
$71$	276.000	0.461340	0.230670	+	0.973032i	$0.425908\pi$
$72$	0	0
$73$	−298.000	−0.477784	−0.238892	−	0.971046i	$-0.576784\pi$
$73$	−298.000	−0.477784	−0.238892	+	0.971046i	$0.576784\pi$
$74$	0	0
$75$	300.000	0.461880
$76$	0	0
$77$	−864.000	−1.27873
$78$	0	0
$79$	182.000	0.259197	0.129599	−	0.991567i	$-0.458631\pi$
$79$	182.000	0.259197	0.129599	+	0.991567i	$0.458631\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−282.000	−0.372934	−0.186467	−	0.982461i	$-0.559704\pi$
$83$	−282.000	−0.372934	−0.186467	+	0.982461i	$0.559704\pi$
$84$	0	0
$85$	85.0000	0.108465
$86$	0	0
$87$	−282.000	−0.347512
$88$	0	0
$89$	−1468.00	−1.74840	−0.874200	−	0.485565i	$-0.838614\pi$
$89$	−1468.00	−1.74840	−0.874200	+	0.485565i	$0.838614\pi$
$90$	0	0
$91$	−2208.00	−2.54353
$92$	0	0
$93$	594.000	0.662311
$94$	0	0
$95$	−415.000	−0.448191
$96$	0	0
$97$	−1140.00	−1.19329	−0.596647	−	0.802504i	$-0.703501\pi$
$97$	−1140.00	−1.19329	−0.596647	+	0.802504i	$0.703501\pi$
$98$	0	0
$99$	−243.000	−0.246691
$100$	0	0
$101$	−1400.00	−1.37926	−0.689630	−	0.724162i	$-0.742227\pi$
$101$	−1400.00	−1.37926	−0.689630	+	0.724162i	$0.742227\pi$
$102$	0	0
$103$	−1939.00	−1.85491	−0.927453	−	0.373939i	$-0.878007\pi$
$103$	−1939.00	−1.85491	−0.927453	+	0.373939i	$0.878007\pi$
$104$	0	0
$105$	480.000	0.446126
$106$	0	0
$107$	−39.0000	−0.0352362	−0.0176181	−	0.999845i	$-0.505608\pi$
$107$	−39.0000	−0.0352362	−0.0176181	+	0.999845i	$0.505608\pi$
$108$	0	0
$109$	−428.000	−0.376101	−0.188050	−	0.982159i	$-0.560217\pi$
$109$	−428.000	−0.376101	−0.188050	+	0.982159i	$0.560217\pi$
$110$	0	0
$111$	732.000	0.625931
$112$	0	0
$113$	715.000	0.595235	0.297617	−	0.954685i	$-0.403808\pi$
$113$	715.000	0.595235	0.297617	+	0.954685i	$0.403808\pi$
$114$	0	0
$115$	−585.000	−0.474361
$116$	0	0
$117$	−621.000	−0.490696
$118$	0	0
$119$	−544.000	−0.419062
$120$	0	0
$121$	−602.000	−0.452292
$122$	0	0
$123$	−507.000	−0.371664
$124$	0	0
$125$	1125.00	0.804984
$126$	0	0
$127$	−797.000	−0.556869	−0.278434	−	0.960455i	$-0.589815\pi$
$127$	−797.000	−0.556869	−0.278434	+	0.960455i	$0.589815\pi$
$128$	0	0
$129$	681.000	0.464796
$130$	0	0
$131$	−373.000	−0.248772	−0.124386	−	0.992234i	$-0.539696\pi$
$131$	−373.000	−0.248772	−0.124386	+	0.992234i	$0.539696\pi$
$132$	0	0
$133$	2656.00	1.73161
$134$	0	0
$135$	135.000	0.0860663
$136$	0	0
$137$	−2706.00	−1.68751	−0.843756	−	0.536727i	$-0.819661\pi$
$137$	−2706.00	−1.68751	−0.843756	+	0.536727i	$0.819661\pi$
$138$	0	0
$139$	2086.00	1.27289	0.636447	−	0.771321i	$-0.280403\pi$
$139$	2086.00	1.27289	0.636447	+	0.771321i	$0.280403\pi$
$140$	0	0
$141$	−1146.00	−0.684472
$142$	0	0
$143$	1863.00	1.08945
$144$	0	0
$145$	−470.000	−0.269182
$146$	0	0
$147$	−2043.00	−1.14628
$148$	0	0
$149$	−1446.00	−0.795040	−0.397520	−	0.917594i	$-0.630129\pi$
$149$	−1446.00	−0.795040	−0.397520	+	0.917594i	$0.630129\pi$
$150$	0	0
$151$	744.000	0.400966	0.200483	−	0.979697i	$-0.435749\pi$
$151$	744.000	0.400966	0.200483	+	0.979697i	$0.435749\pi$
$152$	0	0
$153$	−153.000	−0.0808452
$154$	0	0
$155$	990.000	0.513024
$156$	0	0
$157$	207.000	0.105225	0.0526127	−	0.998615i	$-0.483245\pi$
$157$	207.000	0.105225	0.0526127	+	0.998615i	$0.483245\pi$
$158$	0	0
$159$	−2058.00	−1.02648
$160$	0	0
$161$	3744.00	1.83272
$162$	0	0
$163$	1746.00	0.839002	0.419501	−	0.907755i	$-0.362205\pi$
$163$	1746.00	0.839002	0.419501	+	0.907755i	$0.362205\pi$
$164$	0	0
$165$	−405.000	−0.191086
$166$	0	0
$167$	1153.00	0.534262	0.267131	−	0.963660i	$-0.413924\pi$
$167$	1153.00	0.534262	0.267131	+	0.963660i	$0.413924\pi$
$168$	0	0
$169$	2564.00	1.16705
$170$	0	0
$171$	747.000	0.334062
$172$	0	0
$173$	−3521.00	−1.54738	−0.773690	−	0.633565i	$-0.781591\pi$
$173$	−3521.00	−1.54738	−0.773690	+	0.633565i	$0.781591\pi$
$174$	0	0
$175$	−3200.00	−1.38227
$176$	0	0
$177$	1350.00	0.573289
$178$	0	0
$179$	−2646.00	−1.10487	−0.552434	−	0.833557i	$-0.686301\pi$
$179$	−2646.00	−1.10487	−0.552434	+	0.833557i	$0.686301\pi$
$180$	0	0
$181$	−1454.00	−0.597099	−0.298550	−	0.954394i	$-0.596503\pi$
$181$	−1454.00	−0.597099	−0.298550	+	0.954394i	$0.596503\pi$
$182$	0	0
$183$	2100.00	0.848287
$184$	0	0
$185$	1220.00	0.484844
$186$	0	0
$187$	459.000	0.179494
$188$	0	0
$189$	−864.000	−0.332522
$190$	0	0
$191$	−3730.00	−1.41305	−0.706527	−	0.707686i	$-0.749739\pi$
$191$	−3730.00	−1.41305	−0.706527	+	0.707686i	$0.749739\pi$
$192$	0	0
$193$	−2478.00	−0.924199	−0.462099	−	0.886828i	$-0.652904\pi$
$193$	−2478.00	−0.924199	−0.462099	+	0.886828i	$0.652904\pi$
$194$	0	0
$195$	−1035.00	−0.380092
$196$	0	0
$197$	−2141.00	−0.774314	−0.387157	−	0.922014i	$-0.626543\pi$
$197$	−2141.00	−0.774314	−0.387157	+	0.922014i	$0.626543\pi$
$198$	0	0
$199$	580.000	0.206609	0.103304	−	0.994650i	$-0.467058\pi$
$199$	580.000	0.206609	0.103304	+	0.994650i	$0.467058\pi$
$200$	0	0
$201$	1620.00	0.568488
$202$	0	0
$203$	3008.00	1.04000
$204$	0	0
$205$	−845.000	−0.287890
$206$	0	0
$207$	1053.00	0.353568
$208$	0	0
$209$	−2241.00	−0.741690
$210$	0	0
$211$	3826.00	1.24831	0.624153	−	0.781302i	$-0.285444\pi$
$211$	3826.00	1.24831	0.624153	+	0.781302i	$0.285444\pi$
$212$	0	0
$213$	−828.000	−0.266355
$214$	0	0
$215$	1135.00	0.360030
$216$	0	0
$217$	−6336.00	−1.98210
$218$	0	0
$219$	894.000	0.275849
$220$	0	0
$221$	1173.00	0.357034
$222$	0	0
$223$	1759.00	0.528212	0.264106	−	0.964494i	$-0.414923\pi$
$223$	1759.00	0.528212	0.264106	+	0.964494i	$0.414923\pi$
$224$	0	0
$225$	−900.000	−0.266667
$226$	0	0
$227$	−341.000	−0.0997047	−0.0498523	−	0.998757i	$-0.515875\pi$
$227$	−341.000	−0.0997047	−0.0498523	+	0.998757i	$0.515875\pi$
$228$	0	0
$229$	3958.00	1.14215	0.571074	−	0.820898i	$-0.306527\pi$
$229$	3958.00	1.14215	0.571074	+	0.820898i	$0.306527\pi$
$230$	0	0
$231$	2592.00	0.738273
$232$	0	0
$233$	3009.00	0.846035	0.423017	−	0.906122i	$-0.360971\pi$
$233$	3009.00	0.846035	0.423017	+	0.906122i	$0.360971\pi$
$234$	0	0
$235$	−1910.00	−0.530190
$236$	0	0
$237$	−546.000	−0.149648
$238$	0	0
$239$	−3464.00	−0.937521	−0.468761	−	0.883325i	$-0.655299\pi$
$239$	−3464.00	−0.937521	−0.468761	+	0.883325i	$0.655299\pi$
$240$	0	0
$241$	156.000	0.0416964	0.0208482	−	0.999783i	$-0.493363\pi$
$241$	156.000	0.0416964	0.0208482	+	0.999783i	$0.493363\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	−3405.00	−0.887908
$246$	0	0
$247$	−5727.00	−1.47530
$248$	0	0
$249$	846.000	0.215314
$250$	0	0
$251$	4464.00	1.12257	0.561285	−	0.827622i	$-0.310307\pi$
$251$	4464.00	1.12257	0.561285	+	0.827622i	$0.310307\pi$
$252$	0	0
$253$	−3159.00	−0.784999
$254$	0	0
$255$	−255.000	−0.0626224
$256$	0	0
$257$	5544.00	1.34562	0.672812	−	0.739814i	$-0.265086\pi$
$257$	5544.00	1.34562	0.672812	+	0.739814i	$0.265086\pi$
$258$	0	0
$259$	−7808.00	−1.87323
$260$	0	0
$261$	846.000	0.200636
$262$	0	0
$263$	−612.000	−0.143489	−0.0717444	−	0.997423i	$-0.522857\pi$
$263$	−612.000	−0.143489	−0.0717444	+	0.997423i	$0.522857\pi$
$264$	0	0
$265$	−3430.00	−0.795107
$266$	0	0
$267$	4404.00	1.00944
$268$	0	0
$269$	5601.00	1.26951	0.634757	−	0.772712i	$-0.281100\pi$
$269$	5601.00	1.26951	0.634757	+	0.772712i	$0.281100\pi$
$270$	0	0
$271$	5851.00	1.31152	0.655762	−	0.754968i	$-0.272348\pi$
$271$	5851.00	1.31152	0.655762	+	0.754968i	$0.272348\pi$
$272$	0	0
$273$	6624.00	1.46851
$274$	0	0
$275$	2700.00	0.592059
$276$	0	0
$277$	−5138.00	−1.11449	−0.557243	−	0.830350i	$-0.688141\pi$
$277$	−5138.00	−1.11449	−0.557243	+	0.830350i	$0.688141\pi$
$278$	0	0
$279$	−1782.00	−0.382385
$280$	0	0
$281$	7608.00	1.61514	0.807572	−	0.589770i	$-0.200781\pi$
$281$	7608.00	1.61514	0.807572	+	0.589770i	$0.200781\pi$
$282$	0	0
$283$	−8474.00	−1.77995	−0.889977	−	0.456005i	$-0.849280\pi$
$283$	−8474.00	−1.77995	−0.889977	+	0.456005i	$0.849280\pi$
$284$	0	0
$285$	1245.00	0.258763
$286$	0	0
$287$	5408.00	1.11228
$288$	0	0
$289$	289.000	0.0588235
$290$	0	0
$291$	3420.00	0.688948
$292$	0	0
$293$	7672.00	1.52970	0.764852	−	0.644207i	$-0.222812\pi$
$293$	7672.00	1.52970	0.764852	+	0.644207i	$0.222812\pi$
$294$	0	0
$295$	2250.00	0.444068
$296$	0	0
$297$	729.000	0.142427
$298$	0	0
$299$	−8073.00	−1.56145
$300$	0	0
$301$	−7264.00	−1.39100
$302$	0	0
$303$	4200.00	0.796316
$304$	0	0
$305$	3500.00	0.657080
$306$	0	0
$307$	3396.00	0.631335	0.315668	−	0.948870i	$-0.397772\pi$
$307$	3396.00	0.631335	0.315668	+	0.948870i	$0.397772\pi$
$308$	0	0
$309$	5817.00	1.07093
$310$	0	0
$311$	−6480.00	−1.18150	−0.590751	−	0.806854i	$-0.701168\pi$
$311$	−6480.00	−1.18150	−0.590751	+	0.806854i	$0.701168\pi$
$312$	0	0
$313$	−4940.00	−0.892094	−0.446047	−	0.895010i	$-0.647169\pi$
$313$	−4940.00	−0.892094	−0.446047	+	0.895010i	$0.647169\pi$
$314$	0	0
$315$	−1440.00	−0.257571
$316$	0	0
$317$	−3998.00	−0.708360	−0.354180	−	0.935177i	$-0.615240\pi$
$317$	−3998.00	−0.708360	−0.354180	+	0.935177i	$0.615240\pi$
$318$	0	0
$319$	−2538.00	−0.445457
$320$	0	0
$321$	117.000	0.0203436
$322$	0	0
$323$	−1411.00	−0.243065
$324$	0	0
$325$	6900.00	1.17767
$326$	0	0
$327$	1284.00	0.217142
$328$	0	0
$329$	12224.0	2.04842
$330$	0	0
$331$	−2105.00	−0.349551	−0.174775	−	0.984608i	$-0.555920\pi$
$331$	−2105.00	−0.349551	−0.174775	+	0.984608i	$0.555920\pi$
$332$	0	0
$333$	−2196.00	−0.361382
$334$	0	0
$335$	2700.00	0.440349
$336$	0	0
$337$	7818.00	1.26372	0.631860	−	0.775083i	$-0.282292\pi$
$337$	7818.00	1.26372	0.631860	+	0.775083i	$0.282292\pi$
$338$	0	0
$339$	−2145.00	−0.343659
$340$	0	0
$341$	5346.00	0.848980
$342$	0	0
$343$	10816.0	1.70265
$344$	0	0
$345$	1755.00	0.273873
$346$	0	0
$347$	1388.00	0.214731	0.107366	−	0.994220i	$-0.465758\pi$
$347$	1388.00	0.214731	0.107366	+	0.994220i	$0.465758\pi$
$348$	0	0
$349$	−9591.00	−1.47104	−0.735522	−	0.677501i	$-0.763063\pi$
$349$	−9591.00	−1.47104	−0.735522	+	0.677501i	$0.763063\pi$
$350$	0	0
$351$	1863.00	0.283304
$352$	0	0
$353$	−1442.00	−0.217422	−0.108711	−	0.994073i	$-0.534672\pi$
$353$	−1442.00	−0.217422	−0.108711	+	0.994073i	$0.534672\pi$
$354$	0	0
$355$	−1380.00	−0.206318
$356$	0	0
$357$	1632.00	0.241946
$358$	0	0
$359$	−4104.00	−0.603345	−0.301672	−	0.953412i	$-0.597545\pi$
$359$	−4104.00	−0.603345	−0.301672	+	0.953412i	$0.597545\pi$
$360$	0	0
$361$	30.0000	0.00437382
$362$	0	0
$363$	1806.00	0.261131
$364$	0	0
$365$	1490.00	0.213672
$366$	0	0
$367$	−3904.00	−0.555278	−0.277639	−	0.960685i	$-0.589552\pi$
$367$	−3904.00	−0.555278	−0.277639	+	0.960685i	$0.589552\pi$
$368$	0	0
$369$	1521.00	0.214580
$370$	0	0
$371$	21952.0	3.07194
$372$	0	0
$373$	4882.00	0.677695	0.338848	−	0.940841i	$-0.389963\pi$
$373$	4882.00	0.677695	0.338848	+	0.940841i	$0.389963\pi$
$374$	0	0
$375$	−3375.00	−0.464758
$376$	0	0
$377$	−6486.00	−0.886064
$378$	0	0
$379$	10324.0	1.39923	0.699615	−	0.714520i	$-0.253355\pi$
$379$	10324.0	1.39923	0.699615	+	0.714520i	$0.253355\pi$
$380$	0	0
$381$	2391.00	0.321508
$382$	0	0
$383$	10480.0	1.39818	0.699090	−	0.715034i	$-0.253589\pi$
$383$	10480.0	1.39818	0.699090	+	0.715034i	$0.253589\pi$
$384$	0	0
$385$	4320.00	0.571864
$386$	0	0
$387$	−2043.00	−0.268350
$388$	0	0
$389$	−9724.00	−1.26742	−0.633710	−	0.773571i	$-0.718469\pi$
$389$	−9724.00	−1.26742	−0.633710	+	0.773571i	$0.718469\pi$
$390$	0	0
$391$	−1989.00	−0.257258
$392$	0	0
$393$	1119.00	0.143629
$394$	0	0
$395$	−910.000	−0.115917
$396$	0	0
$397$	−14632.0	−1.84977	−0.924885	−	0.380246i	$-0.875839\pi$
$397$	−14632.0	−1.84977	−0.924885	+	0.380246i	$0.875839\pi$
$398$	0	0
$399$	−7968.00	−0.999747
$400$	0	0
$401$	−8907.00	−1.10921	−0.554606	−	0.832113i	$-0.687131\pi$
$401$	−8907.00	−1.10921	−0.554606	+	0.832113i	$0.687131\pi$
$402$	0	0
$403$	13662.0	1.68872
$404$	0	0
$405$	−405.000	−0.0496904
$406$	0	0
$407$	6588.00	0.802347
$408$	0	0
$409$	−4859.00	−0.587438	−0.293719	−	0.955892i	$-0.594893\pi$
$409$	−4859.00	−0.587438	−0.293719	+	0.955892i	$0.594893\pi$
$410$	0	0
$411$	8118.00	0.974286
$412$	0	0
$413$	−14400.0	−1.71568
$414$	0	0
$415$	1410.00	0.166781
$416$	0	0
$417$	−6258.00	−0.734905
$418$	0	0
$419$	444.000	0.0517681	0.0258840	−	0.999665i	$-0.491760\pi$
$419$	444.000	0.0517681	0.0258840	+	0.999665i	$0.491760\pi$
$420$	0	0
$421$	14731.0	1.70533	0.852666	−	0.522456i	$-0.174984\pi$
$421$	14731.0	1.70533	0.852666	+	0.522456i	$0.174984\pi$
$422$	0	0
$423$	3438.00	0.395180
$424$	0	0
$425$	1700.00	0.194029
$426$	0	0
$427$	−22400.0	−2.53867
$428$	0	0
$429$	−5589.00	−0.628996
$430$	0	0
$431$	9968.00	1.11402	0.557009	−	0.830507i	$-0.311949\pi$
$431$	9968.00	1.11402	0.557009	+	0.830507i	$0.311949\pi$
$432$	0	0
$433$	1167.00	0.129521	0.0647603	−	0.997901i	$-0.479372\pi$
$433$	1167.00	0.129521	0.0647603	+	0.997901i	$0.479372\pi$
$434$	0	0
$435$	1410.00	0.155412
$436$	0	0
$437$	9711.00	1.06302
$438$	0	0
$439$	−16532.0	−1.79733	−0.898667	−	0.438632i	$-0.855463\pi$
$439$	−16532.0	−1.79733	−0.898667	+	0.438632i	$0.855463\pi$
$440$	0	0
$441$	6129.00	0.661808
$442$	0	0
$443$	−10398.0	−1.11518	−0.557589	−	0.830117i	$-0.688273\pi$
$443$	−10398.0	−1.11518	−0.557589	+	0.830117i	$0.688273\pi$
$444$	0	0
$445$	7340.00	0.781909
$446$	0	0
$447$	4338.00	0.459016
$448$	0	0
$449$	4922.00	0.517335	0.258668	−	0.965966i	$-0.416717\pi$
$449$	4922.00	0.517335	0.258668	+	0.965966i	$0.416717\pi$
$450$	0	0
$451$	−4563.00	−0.476415
$452$	0	0
$453$	−2232.00	−0.231498
$454$	0	0
$455$	11040.0	1.13750
$456$	0	0
$457$	4573.00	0.468087	0.234044	−	0.972226i	$-0.424804\pi$
$457$	4573.00	0.468087	0.234044	+	0.972226i	$0.424804\pi$
$458$	0	0
$459$	459.000	0.0466760
$460$	0	0
$461$	−13586.0	−1.37259	−0.686294	−	0.727324i	$-0.740764\pi$
$461$	−13586.0	−1.37259	−0.686294	+	0.727324i	$0.740764\pi$
$462$	0	0
$463$	4120.00	0.413548	0.206774	−	0.978389i	$-0.433704\pi$
$463$	4120.00	0.413548	0.206774	+	0.978389i	$0.433704\pi$
$464$	0	0
$465$	−2970.00	−0.296195
$466$	0	0
$467$	174.000	0.0172415	0.00862073	−	0.999963i	$-0.497256\pi$
$467$	174.000	0.0172415	0.00862073	+	0.999963i	$0.497256\pi$
$468$	0	0
$469$	−17280.0	−1.70131
$470$	0	0
$471$	−621.000	−0.0607520
$472$	0	0
$473$	6129.00	0.595796
$474$	0	0
$475$	−8300.00	−0.801748
$476$	0	0
$477$	6174.00	0.592637
$478$	0	0
$479$	−18311.0	−1.74666	−0.873331	−	0.487128i	$-0.838045\pi$
$479$	−18311.0	−1.74666	−0.873331	+	0.487128i	$0.838045\pi$
$480$	0	0
$481$	16836.0	1.59596
$482$	0	0
$483$	−11232.0	−1.05812
$484$	0	0
$485$	5700.00	0.533657
$486$	0	0
$487$	−6766.00	−0.629562	−0.314781	−	0.949164i	$-0.601931\pi$
$487$	−6766.00	−0.629562	−0.314781	+	0.949164i	$0.601931\pi$
$488$	0	0
$489$	−5238.00	−0.484398
$490$	0	0
$491$	13846.0	1.27263	0.636315	−	0.771429i	$-0.280458\pi$
$491$	13846.0	1.27263	0.636315	+	0.771429i	$0.280458\pi$
$492$	0	0
$493$	−1598.00	−0.145984
$494$	0	0
$495$	1215.00	0.110324
$496$	0	0
$497$	8832.00	0.797121
$498$	0	0
$499$	5854.00	0.525172	0.262586	−	0.964909i	$-0.415425\pi$
$499$	5854.00	0.525172	0.262586	+	0.964909i	$0.415425\pi$
$500$	0	0
$501$	−3459.00	−0.308457
$502$	0	0
$503$	883.000	0.0782724	0.0391362	−	0.999234i	$-0.487539\pi$
$503$	883.000	0.0782724	0.0391362	+	0.999234i	$0.487539\pi$
$504$	0	0
$505$	7000.00	0.616824
$506$	0	0
$507$	−7692.00	−0.673794
$508$	0	0
$509$	6748.00	0.587622	0.293811	−	0.955863i	$-0.405076\pi$
$509$	6748.00	0.587622	0.293811	+	0.955863i	$0.405076\pi$
$510$	0	0
$511$	−9536.00	−0.825534
$512$	0	0
$513$	−2241.00	−0.192871
$514$	0	0
$515$	9695.00	0.829539
$516$	0	0
$517$	−10314.0	−0.877387
$518$	0	0
$519$	10563.0	0.893380
$520$	0	0
$521$	−9263.00	−0.778924	−0.389462	−	0.921043i	$-0.627339\pi$
$521$	−9263.00	−0.778924	−0.389462	+	0.921043i	$0.627339\pi$
$522$	0	0
$523$	−3868.00	−0.323395	−0.161698	−	0.986840i	$-0.551697\pi$
$523$	−3868.00	−0.323395	−0.161698	+	0.986840i	$0.551697\pi$
$524$	0	0
$525$	9600.00	0.798054
$526$	0	0
$527$	3366.00	0.278226
$528$	0	0
$529$	1522.00	0.125092
$530$	0	0
$531$	−4050.00	−0.330989
$532$	0	0
$533$	−11661.0	−0.947643
$534$	0	0
$535$	195.000	0.0157581
$536$	0	0
$537$	7938.00	0.637896
$538$	0	0
$539$	−18387.0	−1.46936
$540$	0	0
$541$	−6840.00	−0.543576	−0.271788	−	0.962357i	$-0.587615\pi$
$541$	−6840.00	−0.543576	−0.271788	+	0.962357i	$0.587615\pi$
$542$	0	0
$543$	4362.00	0.344735
$544$	0	0
$545$	2140.00	0.168197
$546$	0	0
$547$	5564.00	0.434917	0.217458	−	0.976070i	$-0.430223\pi$
$547$	5564.00	0.434917	0.217458	+	0.976070i	$0.430223\pi$
$548$	0	0
$549$	−6300.00	−0.489759
$550$	0	0
$551$	7802.00	0.603224
$552$	0	0
$553$	5824.00	0.447851
$554$	0	0
$555$	−3660.00	−0.279925
$556$	0	0
$557$	1618.00	0.123082	0.0615412	−	0.998105i	$-0.480398\pi$
$557$	1618.00	0.123082	0.0615412	+	0.998105i	$0.480398\pi$
$558$	0	0
$559$	15663.0	1.18511
$560$	0	0
$561$	−1377.00	−0.103631
$562$	0	0
$563$	21050.0	1.57576	0.787879	−	0.615830i	$-0.211179\pi$
$563$	21050.0	1.57576	0.787879	+	0.615830i	$0.211179\pi$
$564$	0	0
$565$	−3575.00	−0.266197
$566$	0	0
$567$	2592.00	0.191982
$568$	0	0
$569$	26484.0	1.95126	0.975630	−	0.219422i	$-0.0704171\pi$
$569$	26484.0	1.95126	0.975630	+	0.219422i	$0.0704171\pi$
$570$	0	0
$571$	10996.0	0.805899	0.402949	−	0.915222i	$-0.367985\pi$
$571$	10996.0	0.805899	0.402949	+	0.915222i	$0.367985\pi$
$572$	0	0
$573$	11190.0	0.815827
$574$	0	0
$575$	−11700.0	−0.848563
$576$	0	0
$577$	16089.0	1.16082	0.580411	−	0.814324i	$-0.302892\pi$
$577$	16089.0	1.16082	0.580411	+	0.814324i	$0.302892\pi$
$578$	0	0
$579$	7434.00	0.533586
$580$	0	0
$581$	−9024.00	−0.644369
$582$	0	0
$583$	−18522.0	−1.31579
$584$	0	0
$585$	3105.00	0.219446
$586$	0	0
$587$	−5700.00	−0.400791	−0.200395	−	0.979715i	$-0.564223\pi$
$587$	−5700.00	−0.400791	−0.200395	+	0.979715i	$0.564223\pi$
$588$	0	0
$589$	−16434.0	−1.14966
$590$	0	0
$591$	6423.00	0.447051
$592$	0	0
$593$	3474.00	0.240573	0.120287	−	0.992739i	$-0.461619\pi$
$593$	3474.00	0.240573	0.120287	+	0.992739i	$0.461619\pi$
$594$	0	0
$595$	2720.00	0.187410
$596$	0	0
$597$	−1740.00	−0.119286
$598$	0	0
$599$	12710.0	0.866972	0.433486	−	0.901160i	$-0.357283\pi$
$599$	12710.0	0.866972	0.433486	+	0.901160i	$0.357283\pi$
$600$	0	0
$601$	8806.00	0.597678	0.298839	−	0.954304i	$-0.403401\pi$
$601$	8806.00	0.597678	0.298839	+	0.954304i	$0.403401\pi$
$602$	0	0
$603$	−4860.00	−0.328216
$604$	0	0
$605$	3010.00	0.202271
$606$	0	0
$607$	−6022.00	−0.402678	−0.201339	−	0.979522i	$-0.564529\pi$
$607$	−6022.00	−0.402678	−0.201339	+	0.979522i	$0.564529\pi$
$608$	0	0
$609$	−9024.00	−0.600445
$610$	0	0
$611$	−26358.0	−1.74522
$612$	0	0
$613$	−2337.00	−0.153981	−0.0769907	−	0.997032i	$-0.524531\pi$
$613$	−2337.00	−0.153981	−0.0769907	+	0.997032i	$0.524531\pi$
$614$	0	0
$615$	2535.00	0.166213
$616$	0	0
$617$	23082.0	1.50607	0.753036	−	0.657979i	$-0.228589\pi$
$617$	23082.0	1.50607	0.753036	+	0.657979i	$0.228589\pi$
$618$	0	0
$619$	20510.0	1.33177	0.665886	−	0.746054i	$-0.268054\pi$
$619$	20510.0	1.33177	0.665886	+	0.746054i	$0.268054\pi$
$620$	0	0
$621$	−3159.00	−0.204133
$622$	0	0
$623$	−46976.0	−3.02095
$624$	0	0
$625$	6875.00	0.440000
$626$	0	0
$627$	6723.00	0.428215
$628$	0	0
$629$	4148.00	0.262944
$630$	0	0
$631$	−29453.0	−1.85817	−0.929085	−	0.369866i	$-0.879404\pi$
$631$	−29453.0	−1.85817	−0.929085	+	0.369866i	$0.879404\pi$
$632$	0	0
$633$	−11478.0	−0.720710
$634$	0	0
$635$	3985.00	0.249039
$636$	0	0
$637$	−46989.0	−2.92272
$638$	0	0
$639$	2484.00	0.153780
$640$	0	0
$641$	−4045.00	−0.249248	−0.124624	−	0.992204i	$-0.539772\pi$
$641$	−4045.00	−0.249248	−0.124624	+	0.992204i	$0.539772\pi$
$642$	0	0
$643$	30188.0	1.85148	0.925738	−	0.378167i	$-0.123445\pi$
$643$	30188.0	1.85148	0.925738	+	0.378167i	$0.123445\pi$
$644$	0	0
$645$	−3405.00	−0.207863
$646$	0	0
$647$	28246.0	1.71633	0.858164	−	0.513375i	$-0.171605\pi$
$647$	28246.0	1.71633	0.858164	+	0.513375i	$0.171605\pi$
$648$	0	0
$649$	12150.0	0.734868
$650$	0	0
$651$	19008.0	1.14437
$652$	0	0
$653$	−1301.00	−0.0779664	−0.0389832	−	0.999240i	$-0.512412\pi$
$653$	−1301.00	−0.0779664	−0.0389832	+	0.999240i	$0.512412\pi$
$654$	0	0
$655$	1865.00	0.111254
$656$	0	0
$657$	−2682.00	−0.159261
$658$	0	0
$659$	6450.00	0.381269	0.190635	−	0.981661i	$-0.438945\pi$
$659$	6450.00	0.381269	0.190635	+	0.981661i	$0.438945\pi$
$660$	0	0
$661$	12445.0	0.732306	0.366153	−	0.930555i	$-0.380675\pi$
$661$	12445.0	0.732306	0.366153	+	0.930555i	$0.380675\pi$
$662$	0	0
$663$	−3519.00	−0.206134
$664$	0	0
$665$	−13280.0	−0.774400
$666$	0	0
$667$	10998.0	0.638447
$668$	0	0
$669$	−5277.00	−0.304964
$670$	0	0
$671$	18900.0	1.08737
$672$	0	0
$673$	5566.00	0.318802	0.159401	−	0.987214i	$-0.449044\pi$
$673$	5566.00	0.318802	0.159401	+	0.987214i	$0.449044\pi$
$674$	0	0
$675$	2700.00	0.153960
$676$	0	0
$677$	−33539.0	−1.90400	−0.952000	−	0.306097i	$-0.900977\pi$
$677$	−33539.0	−1.90400	−0.952000	+	0.306097i	$0.900977\pi$
$678$	0	0
$679$	−36480.0	−2.06182
$680$	0	0
$681$	1023.00	0.0575645
$682$	0	0
$683$	−29517.0	−1.65364	−0.826820	−	0.562466i	$-0.809853\pi$
$683$	−29517.0	−1.65364	−0.826820	+	0.562466i	$0.809853\pi$
$684$	0	0
$685$	13530.0	0.754678
$686$	0	0
$687$	−11874.0	−0.659420
$688$	0	0
$689$	−47334.0	−2.61724
$690$	0	0
$691$	−1684.00	−0.0927097	−0.0463548	−	0.998925i	$-0.514760\pi$
$691$	−1684.00	−0.0927097	−0.0463548	+	0.998925i	$0.514760\pi$
$692$	0	0
$693$	−7776.00	−0.426242
$694$	0	0
$695$	−10430.0	−0.569255
$696$	0	0
$697$	−2873.00	−0.156130
$698$	0	0
$699$	−9027.00	−0.488459
$700$	0	0
$701$	34152.0	1.84009	0.920045	−	0.391812i	$-0.128152\pi$
$701$	34152.0	1.84009	0.920045	+	0.391812i	$0.128152\pi$
$702$	0	0
$703$	−20252.0	−1.08651
$704$	0	0
$705$	5730.00	0.306105
$706$	0	0
$707$	−44800.0	−2.38314
$708$	0	0
$709$	7994.00	0.423443	0.211721	−	0.977330i	$-0.432093\pi$
$709$	7994.00	0.423443	0.211721	+	0.977330i	$0.432093\pi$
$710$	0	0
$711$	1638.00	0.0863992
$712$	0	0
$713$	−23166.0	−1.21679
$714$	0	0
$715$	−9315.00	−0.487219
$716$	0	0
$717$	10392.0	0.541278
$718$	0	0
$719$	−22713.0	−1.17810	−0.589049	−	0.808098i	$-0.700497\pi$
$719$	−22713.0	−1.17810	−0.589049	+	0.808098i	$0.700497\pi$
$720$	0	0
$721$	−62048.0	−3.20498
$722$	0	0
$723$	−468.000	−0.0240735
$724$	0	0
$725$	−9400.00	−0.481527
$726$	0	0
$727$	−21712.0	−1.10764	−0.553819	−	0.832637i	$-0.686830\pi$
$727$	−21712.0	−1.10764	−0.553819	+	0.832637i	$0.686830\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	3859.00	0.195253
$732$	0	0
$733$	−5430.00	−0.273617	−0.136809	−	0.990597i	$-0.543685\pi$
$733$	−5430.00	−0.273617	−0.136809	+	0.990597i	$0.543685\pi$
$734$	0	0
$735$	10215.0	0.512634
$736$	0	0
$737$	14580.0	0.728713
$738$	0	0
$739$	30723.0	1.52932	0.764658	−	0.644437i	$-0.222908\pi$
$739$	30723.0	1.52932	0.764658	+	0.644437i	$0.222908\pi$
$740$	0	0
$741$	17181.0	0.851768
$742$	0	0
$743$	−28096.0	−1.38727	−0.693635	−	0.720326i	$-0.743992\pi$
$743$	−28096.0	−1.38727	−0.693635	+	0.720326i	$0.743992\pi$
$744$	0	0
$745$	7230.00	0.355553
$746$	0	0
$747$	−2538.00	−0.124311
$748$	0	0
$749$	−1248.00	−0.0608824
$750$	0	0
$751$	5458.00	0.265200	0.132600	−	0.991170i	$-0.457667\pi$
$751$	5458.00	0.265200	0.132600	+	0.991170i	$0.457667\pi$
$752$	0	0
$753$	−13392.0	−0.648116
$754$	0	0
$755$	−3720.00	−0.179317
$756$	0	0
$757$	−24319.0	−1.16762	−0.583810	−	0.811890i	$-0.698439\pi$
$757$	−24319.0	−1.16762	−0.583810	+	0.811890i	$0.698439\pi$
$758$	0	0
$759$	9477.00	0.453219
$760$	0	0
$761$	2126.00	0.101271	0.0506356	−	0.998717i	$-0.483875\pi$
$761$	2126.00	0.101271	0.0506356	+	0.998717i	$0.483875\pi$
$762$	0	0
$763$	−13696.0	−0.649841
$764$	0	0
$765$	765.000	0.0361551
$766$	0	0
$767$	31050.0	1.46173
$768$	0	0
$769$	−2503.00	−0.117374	−0.0586869	−	0.998276i	$-0.518691\pi$
$769$	−2503.00	−0.117374	−0.0586869	+	0.998276i	$0.518691\pi$
$770$	0	0
$771$	−16632.0	−0.776896
$772$	0	0
$773$	36780.0	1.71136	0.855682	−	0.517502i	$-0.173138\pi$
$773$	36780.0	1.71136	0.855682	+	0.517502i	$0.173138\pi$
$774$	0	0
$775$	19800.0	0.917725
$776$	0	0
$777$	23424.0	1.08151
$778$	0	0
$779$	14027.0	0.645147
$780$	0	0
$781$	−7452.00	−0.341426
$782$	0	0
$783$	−2538.00	−0.115837
$784$	0	0
$785$	−1035.00	−0.0470583
$786$	0	0
$787$	−7356.00	−0.333181	−0.166590	−	0.986026i	$-0.553276\pi$
$787$	−7356.00	−0.333181	−0.166590	+	0.986026i	$0.553276\pi$
$788$	0	0
$789$	1836.00	0.0828433
$790$	0	0
$791$	22880.0	1.02847
$792$	0	0
$793$	48300.0	2.16290
$794$	0	0
$795$	10290.0	0.459055
$796$	0	0
$797$	−19672.0	−0.874301	−0.437151	−	0.899388i	$-0.644012\pi$
$797$	−19672.0	−0.874301	−0.437151	+	0.899388i	$0.644012\pi$
$798$	0	0
$799$	−6494.00	−0.287536
$800$	0	0
$801$	−13212.0	−0.582800
$802$	0	0
$803$	8046.00	0.353595
$804$	0	0
$805$	−18720.0	−0.819619
$806$	0	0
$807$	−16803.0	−0.732954
$808$	0	0
$809$	18713.0	0.813244	0.406622	−	0.913597i	$-0.366707\pi$
$809$	18713.0	0.813244	0.406622	+	0.913597i	$0.366707\pi$
$810$	0	0
$811$	−16062.0	−0.695454	−0.347727	−	0.937596i	$-0.613046\pi$
$811$	−16062.0	−0.695454	−0.347727	+	0.937596i	$0.613046\pi$
$812$	0	0
$813$	−17553.0	−0.757209
$814$	0	0
$815$	−8730.00	−0.375213
$816$	0	0
$817$	−18841.0	−0.806809
$818$	0	0
$819$	−19872.0	−0.847844
$820$	0	0
$821$	−19653.0	−0.835438	−0.417719	−	0.908576i	$-0.637170\pi$
$821$	−19653.0	−0.835438	−0.417719	+	0.908576i	$0.637170\pi$
$822$	0	0
$823$	3276.00	0.138754	0.0693768	−	0.997591i	$-0.477899\pi$
$823$	3276.00	0.138754	0.0693768	+	0.997591i	$0.477899\pi$
$824$	0	0
$825$	−8100.00	−0.341825
$826$	0	0
$827$	11491.0	0.483170	0.241585	−	0.970380i	$-0.422333\pi$
$827$	11491.0	0.483170	0.241585	+	0.970380i	$0.422333\pi$
$828$	0	0
$829$	3606.00	0.151075	0.0755377	−	0.997143i	$-0.475933\pi$
$829$	3606.00	0.151075	0.0755377	+	0.997143i	$0.475933\pi$
$830$	0	0
$831$	15414.0	0.643449
$832$	0	0
$833$	−11577.0	−0.481536
$834$	0	0
$835$	−5765.00	−0.238929
$836$	0	0
$837$	5346.00	0.220770
$838$	0	0
$839$	36027.0	1.48247	0.741234	−	0.671247i	$-0.234241\pi$
$839$	36027.0	1.48247	0.741234	+	0.671247i	$0.234241\pi$
$840$	0	0
$841$	−15553.0	−0.637706
$842$	0	0
$843$	−22824.0	−0.932503
$844$	0	0
$845$	−12820.0	−0.521919
$846$	0	0
$847$	−19264.0	−0.781486
$848$	0	0
$849$	25422.0	1.02766
$850$	0	0
$851$	−28548.0	−1.14996
$852$	0	0
$853$	−31078.0	−1.24747	−0.623734	−	0.781637i	$-0.714385\pi$
$853$	−31078.0	−1.24747	−0.623734	+	0.781637i	$0.714385\pi$
$854$	0	0
$855$	−3735.00	−0.149397
$856$	0	0
$857$	−22106.0	−0.881128	−0.440564	−	0.897721i	$-0.645221\pi$
$857$	−22106.0	−0.881128	−0.440564	+	0.897721i	$0.645221\pi$
$858$	0	0
$859$	−23924.0	−0.950263	−0.475132	−	0.879915i	$-0.657600\pi$
$859$	−23924.0	−0.950263	−0.475132	+	0.879915i	$0.657600\pi$
$860$	0	0
$861$	−16224.0	−0.642175
$862$	0	0
$863$	−31860.0	−1.25669	−0.628347	−	0.777933i	$-0.716268\pi$
$863$	−31860.0	−1.25669	−0.628347	+	0.777933i	$0.716268\pi$
$864$	0	0
$865$	17605.0	0.692009
$866$	0	0
$867$	−867.000	−0.0339618
$868$	0	0
$869$	−4914.00	−0.191825
$870$	0	0
$871$	37260.0	1.44949
$872$	0	0
$873$	−10260.0	−0.397764
$874$	0	0
$875$	36000.0	1.39088
$876$	0	0
$877$	34122.0	1.31382	0.656909	−	0.753970i	$-0.271864\pi$
$877$	34122.0	1.31382	0.656909	+	0.753970i	$0.271864\pi$
$878$	0	0
$879$	−23016.0	−0.883175
$880$	0	0
$881$	−48062.0	−1.83797	−0.918984	−	0.394295i	$-0.870989\pi$
$881$	−48062.0	−1.83797	−0.918984	+	0.394295i	$0.870989\pi$
$882$	0	0
$883$	−6185.00	−0.235721	−0.117861	−	0.993030i	$-0.537604\pi$
$883$	−6185.00	−0.235721	−0.117861	+	0.993030i	$0.537604\pi$
$884$	0	0
$885$	−6750.00	−0.256383
$886$	0	0
$887$	41835.0	1.58363	0.791816	−	0.610760i	$-0.209136\pi$
$887$	41835.0	1.58363	0.791816	+	0.610760i	$0.209136\pi$
$888$	0	0
$889$	−25504.0	−0.962179
$890$	0	0
$891$	−2187.00	−0.0822304
$892$	0	0
$893$	31706.0	1.18813
$894$	0	0
$895$	13230.0	0.494112
$896$	0	0
$897$	24219.0	0.901504
$898$	0	0
$899$	−18612.0	−0.690484
$900$	0	0
$901$	−11662.0	−0.431207
$902$	0	0
$903$	21792.0	0.803092
$904$	0	0
$905$	7270.00	0.267031
$906$	0	0
$907$	12174.0	0.445679	0.222840	−	0.974855i	$-0.428467\pi$
$907$	12174.0	0.445679	0.222840	+	0.974855i	$0.428467\pi$
$908$	0	0
$909$	−12600.0	−0.459753
$910$	0	0
$911$	24089.0	0.876075	0.438037	−	0.898957i	$-0.355674\pi$
$911$	24089.0	0.876075	0.438037	+	0.898957i	$0.355674\pi$
$912$	0	0
$913$	7614.00	0.275998
$914$	0	0
$915$	−10500.0	−0.379365
$916$	0	0
$917$	−11936.0	−0.429838
$918$	0	0
$919$	55163.0	1.98004	0.990021	−	0.140917i	$-0.0450049\pi$
$919$	55163.0	1.98004	0.990021	+	0.140917i	$0.0450049\pi$
$920$	0	0
$921$	−10188.0	−0.364502
$922$	0	0
$923$	−19044.0	−0.679134
$924$	0	0
$925$	24400.0	0.867316
$926$	0	0
$927$	−17451.0	−0.618302
$928$	0	0
$929$	19555.0	0.690612	0.345306	−	0.938490i	$-0.387775\pi$
$929$	19555.0	0.690612	0.345306	+	0.938490i	$0.387775\pi$
$930$	0	0
$931$	56523.0	1.98976
$932$	0	0
$933$	19440.0	0.682140
$934$	0	0
$935$	−2295.00	−0.0802722
$936$	0	0
$937$	10066.0	0.350952	0.175476	−	0.984484i	$-0.443854\pi$
$937$	10066.0	0.350952	0.175476	+	0.984484i	$0.443854\pi$
$938$	0	0
$939$	14820.0	0.515051
$940$	0	0
$941$	−10998.0	−0.381004	−0.190502	−	0.981687i	$-0.561012\pi$
$941$	−10998.0	−0.381004	−0.190502	+	0.981687i	$0.561012\pi$
$942$	0	0
$943$	19773.0	0.682818
$944$	0	0
$945$	4320.00	0.148709
$946$	0	0
$947$	2268.00	0.0778248	0.0389124	−	0.999243i	$-0.487611\pi$
$947$	2268.00	0.0778248	0.0389124	+	0.999243i	$0.487611\pi$
$948$	0	0
$949$	20562.0	0.703341
$950$	0	0
$951$	11994.0	0.408972
$952$	0	0
$953$	23876.0	0.811563	0.405781	−	0.913970i	$-0.366999\pi$
$953$	23876.0	0.811563	0.405781	+	0.913970i	$0.366999\pi$
$954$	0	0
$955$	18650.0	0.631937
$956$	0	0
$957$	7614.00	0.257185
$958$	0	0
$959$	−86592.0	−2.91575
$960$	0	0
$961$	9413.00	0.315968
$962$	0	0
$963$	−351.000	−0.0117454
$964$	0	0
$965$	12390.0	0.413314
$966$	0	0
$967$	12113.0	0.402821	0.201410	−	0.979507i	$-0.435447\pi$
$967$	12113.0	0.402821	0.201410	+	0.979507i	$0.435447\pi$
$968$	0	0
$969$	4233.00	0.140334
$970$	0	0
$971$	1190.00	0.0393295	0.0196647	−	0.999807i	$-0.493740\pi$
$971$	1190.00	0.0393295	0.0196647	+	0.999807i	$0.493740\pi$
$972$	0	0
$973$	66752.0	2.19935
$974$	0	0
$975$	−20700.0	−0.679929
$976$	0	0
$977$	10474.0	0.342982	0.171491	−	0.985186i	$-0.445142\pi$
$977$	10474.0	0.342982	0.171491	+	0.985186i	$0.445142\pi$
$978$	0	0
$979$	39636.0	1.29394
$980$	0	0
$981$	−3852.00	−0.125367
$982$	0	0
$983$	48259.0	1.56584	0.782921	−	0.622121i	$-0.213729\pi$
$983$	48259.0	1.56584	0.782921	+	0.622121i	$0.213729\pi$
$984$	0	0
$985$	10705.0	0.346284
$986$	0	0
$987$	−36672.0	−1.18266
$988$	0	0
$989$	−26559.0	−0.853920
$990$	0	0
$991$	−37256.0	−1.19422	−0.597112	−	0.802158i	$-0.703685\pi$
$991$	−37256.0	−1.19422	−0.597112	+	0.802158i	$0.703685\pi$
$992$	0	0
$993$	6315.00	0.201813
$994$	0	0
$995$	−2900.00	−0.0923982
$996$	0	0
$997$	17830.0	0.566381	0.283190	−	0.959064i	$-0.408607\pi$
$997$	17830.0	0.566381	0.283190	+	0.959064i	$0.408607\pi$
$998$	0	0
$999$	6588.00	0.208644

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	816.4.a.c.1.1		1
3.2	odd	2		2448.4.a.j.1.1		1
4.3	odd	2		102.4.a.b.1.1	✓	1
12.11	even	2		306.4.a.g.1.1		1
68.67	odd	2		1734.4.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
102.4.a.b.1.1	✓	1	4.3	odd	2
306.4.a.g.1.1		1	12.11	even	2
816.4.a.c.1.1		1	1.1	even	1	trivial
1734.4.a.a.1.1		1	68.67	odd	2
2448.4.a.j.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:	\( N \)	\(=\)	\( 816 = 2^{4} \cdot 3 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	816.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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