LMFDB - Embedded newform 768.4.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,4,Mod(1,768)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(768, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("768.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 768 = 2^{8} \cdot 3 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	768.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$45.3134668844$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 384)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	768.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-3.00000 q^{3} -8.00000 q^{5} +12.0000 q^{7} +9.00000 q^{9} +O(q^{10})$

$q-3.00000 q^{3} -8.00000 q^{5} +12.0000 q^{7} +9.00000 q^{9} +12.0000 q^{11} +20.0000 q^{13} +24.0000 q^{15} +62.0000 q^{17} -108.000 q^{19} -36.0000 q^{21} -72.0000 q^{23} -61.0000 q^{25} -27.0000 q^{27} -128.000 q^{29} +204.000 q^{31} -36.0000 q^{33} -96.0000 q^{35} -228.000 q^{37} -60.0000 q^{39} +22.0000 q^{41} +204.000 q^{43} -72.0000 q^{45} +600.000 q^{47} -199.000 q^{49} -186.000 q^{51} +256.000 q^{53} -96.0000 q^{55} +324.000 q^{57} +828.000 q^{59} -84.0000 q^{61} +108.000 q^{63} -160.000 q^{65} -348.000 q^{67} +216.000 q^{69} +456.000 q^{71} -822.000 q^{73} +183.000 q^{75} +144.000 q^{77} +1356.00 q^{79} +81.0000 q^{81} -108.000 q^{83} -496.000 q^{85} +384.000 q^{87} +938.000 q^{89} +240.000 q^{91} -612.000 q^{93} +864.000 q^{95} +1278.00 q^{97} +108.000 q^{99} +O(q^{100})$

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
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	−8.00000	−0.715542	−0.357771	−	0.933809i	$-0.616463\pi$
$5$	−8.00000	−0.715542	−0.357771	+	0.933809i	$0.616463\pi$
$6$	0	0
$7$	12.0000	0.647939	0.323970	−	0.946068i	$-0.394982\pi$
$7$	12.0000	0.647939	0.323970	+	0.946068i	$0.394982\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	12.0000	0.328921	0.164461	−	0.986384i	$-0.447412\pi$
$11$	12.0000	0.328921	0.164461	+	0.986384i	$0.447412\pi$
$12$	0	0
$13$	20.0000	0.426692	0.213346	−	0.976977i	$-0.431564\pi$
$13$	20.0000	0.426692	0.213346	+	0.976977i	$0.431564\pi$
$14$	0	0
$15$	24.0000	0.413118
$16$	0	0
$17$	62.0000	0.884542	0.442271	−	0.896882i	$-0.354173\pi$
$17$	62.0000	0.884542	0.442271	+	0.896882i	$0.354173\pi$
$18$	0	0
$19$	−108.000	−1.30405	−0.652024	−	0.758199i	$-0.726080\pi$
$19$	−108.000	−1.30405	−0.652024	+	0.758199i	$0.726080\pi$
$20$	0	0
$21$	−36.0000	−0.374088
$22$	0	0
$23$	−72.0000	−0.652741	−0.326370	−	0.945242i	$-0.605826\pi$
$23$	−72.0000	−0.652741	−0.326370	+	0.945242i	$0.605826\pi$
$24$	0	0
$25$	−61.0000	−0.488000
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	−128.000	−0.819621	−0.409810	−	0.912171i	$-0.634405\pi$
$29$	−128.000	−0.819621	−0.409810	+	0.912171i	$0.634405\pi$
$30$	0	0
$31$	204.000	1.18192	0.590959	−	0.806701i	$-0.298749\pi$
$31$	204.000	1.18192	0.590959	+	0.806701i	$0.298749\pi$
$32$	0	0
$33$	−36.0000	−0.189903
$34$	0	0
$35$	−96.0000	−0.463627
$36$	0	0
$37$	−228.000	−1.01305	−0.506527	−	0.862224i	$-0.669071\pi$
$37$	−228.000	−1.01305	−0.506527	+	0.862224i	$0.669071\pi$
$38$	0	0
$39$	−60.0000	−0.246351
$40$	0	0
$41$	22.0000	0.0838006	0.0419003	−	0.999122i	$-0.486659\pi$
$41$	22.0000	0.0838006	0.0419003	+	0.999122i	$0.486659\pi$
$42$	0	0
$43$	204.000	0.723482	0.361741	−	0.932279i	$-0.382183\pi$
$43$	204.000	0.723482	0.361741	+	0.932279i	$0.382183\pi$
$44$	0	0
$45$	−72.0000	−0.238514
$46$	0	0
$47$	600.000	1.86211	0.931053	−	0.364884i	$-0.118891\pi$
$47$	600.000	1.86211	0.931053	+	0.364884i	$0.118891\pi$
$48$	0	0
$49$	−199.000	−0.580175
$50$	0	0
$51$	−186.000	−0.510690
$52$	0	0
$53$	256.000	0.663477	0.331739	−	0.943371i	$-0.392365\pi$
$53$	256.000	0.663477	0.331739	+	0.943371i	$0.392365\pi$
$54$	0	0
$55$	−96.0000	−0.235357
$56$	0	0
$57$	324.000	0.752892
$58$	0	0
$59$	828.000	1.82706	0.913529	−	0.406774i	$-0.133346\pi$
$59$	828.000	1.82706	0.913529	+	0.406774i	$0.133346\pi$
$60$	0	0
$61$	−84.0000	−0.176313	−0.0881565	−	0.996107i	$-0.528098\pi$
$61$	−84.0000	−0.176313	−0.0881565	+	0.996107i	$0.528098\pi$
$62$	0	0
$63$	108.000	0.215980
$64$	0	0
$65$	−160.000	−0.305316
$66$	0	0
$67$	−348.000	−0.634552	−0.317276	−	0.948333i	$-0.602768\pi$
$67$	−348.000	−0.634552	−0.317276	+	0.948333i	$0.602768\pi$
$68$	0	0
$69$	216.000	0.376860
$70$	0	0
$71$	456.000	0.762215	0.381107	−	0.924531i	$-0.375543\pi$
$71$	456.000	0.762215	0.381107	+	0.924531i	$0.375543\pi$
$72$	0	0
$73$	−822.000	−1.31792	−0.658958	−	0.752180i	$-0.729002\pi$
$73$	−822.000	−1.31792	−0.658958	+	0.752180i	$0.729002\pi$
$74$	0	0
$75$	183.000	0.281747
$76$	0	0
$77$	144.000	0.213121
$78$	0	0
$79$	1356.00	1.93116	0.965582	−	0.260100i	$-0.0837554\pi$
$79$	1356.00	1.93116	0.965582	+	0.260100i	$0.0837554\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−108.000	−0.142826	−0.0714129	−	0.997447i	$-0.522751\pi$
$83$	−108.000	−0.142826	−0.0714129	+	0.997447i	$0.522751\pi$
$84$	0	0
$85$	−496.000	−0.632927
$86$	0	0
$87$	384.000	0.473208
$88$	0	0
$89$	938.000	1.11717	0.558583	−	0.829449i	$-0.311345\pi$
$89$	938.000	1.11717	0.558583	+	0.829449i	$0.311345\pi$
$90$	0	0
$91$	240.000	0.276471
$92$	0	0
$93$	−612.000	−0.682381
$94$	0	0
$95$	864.000	0.933100
$96$	0	0
$97$	1278.00	1.33774	0.668872	−	0.743377i	$-0.266777\pi$
$97$	1278.00	1.33774	0.668872	+	0.743377i	$0.266777\pi$
$98$	0	0
$99$	108.000	0.109640
$100$	0	0
$101$	608.000	0.598993	0.299496	−	0.954097i	$-0.403181\pi$
$101$	608.000	0.598993	0.299496	+	0.954097i	$0.403181\pi$
$102$	0	0
$103$	948.000	0.906886	0.453443	−	0.891285i	$-0.350196\pi$
$103$	948.000	0.906886	0.453443	+	0.891285i	$0.350196\pi$
$104$	0	0
$105$	288.000	0.267675
$106$	0	0
$107$	−1044.00	−0.943246	−0.471623	−	0.881800i	$-0.656332\pi$
$107$	−1044.00	−0.943246	−0.471623	+	0.881800i	$0.656332\pi$
$108$	0	0
$109$	1780.00	1.56416	0.782078	−	0.623180i	$-0.214160\pi$
$109$	1780.00	1.56416	0.782078	+	0.623180i	$0.214160\pi$
$110$	0	0
$111$	684.000	0.584887
$112$	0	0
$113$	−622.000	−0.517813	−0.258906	−	0.965902i	$-0.583362\pi$
$113$	−622.000	−0.517813	−0.258906	+	0.965902i	$0.583362\pi$
$114$	0	0
$115$	576.000	0.467063
$116$	0	0
$117$	180.000	0.142231
$118$	0	0
$119$	744.000	0.573129
$120$	0	0
$121$	−1187.00	−0.891811
$122$	0	0
$123$	−66.0000	−0.0483823
$124$	0	0
$125$	1488.00	1.06473
$126$	0	0
$127$	−204.000	−0.142536	−0.0712680	−	0.997457i	$-0.522705\pi$
$127$	−204.000	−0.142536	−0.0712680	+	0.997457i	$0.522705\pi$
$128$	0	0
$129$	−612.000	−0.417702
$130$	0	0
$131$	−348.000	−0.232098	−0.116049	−	0.993243i	$-0.537023\pi$
$131$	−348.000	−0.232098	−0.116049	+	0.993243i	$0.537023\pi$
$132$	0	0
$133$	−1296.00	−0.844943
$134$	0	0
$135$	216.000	0.137706
$136$	0	0
$137$	−106.000	−0.0661036	−0.0330518	−	0.999454i	$-0.510523\pi$
$137$	−106.000	−0.0661036	−0.0330518	+	0.999454i	$0.510523\pi$
$138$	0	0
$139$	−1188.00	−0.724927	−0.362463	−	0.931998i	$-0.618064\pi$
$139$	−1188.00	−0.724927	−0.362463	+	0.931998i	$0.618064\pi$
$140$	0	0
$141$	−1800.00	−1.07509
$142$	0	0
$143$	240.000	0.140348
$144$	0	0
$145$	1024.00	0.586473
$146$	0	0
$147$	597.000	0.334964
$148$	0	0
$149$	2648.00	1.45592	0.727962	−	0.685618i	$-0.240468\pi$
$149$	2648.00	1.45592	0.727962	+	0.685618i	$0.240468\pi$
$150$	0	0
$151$	3420.00	1.84315	0.921575	−	0.388200i	$-0.126903\pi$
$151$	3420.00	1.84315	0.921575	+	0.388200i	$0.126903\pi$
$152$	0	0
$153$	558.000	0.294847
$154$	0	0
$155$	−1632.00	−0.845712
$156$	0	0
$157$	60.0000	0.0305001	0.0152501	−	0.999884i	$-0.495146\pi$
$157$	60.0000	0.0305001	0.0152501	+	0.999884i	$0.495146\pi$
$158$	0	0
$159$	−768.000	−0.383059
$160$	0	0
$161$	−864.000	−0.422936
$162$	0	0
$163$	228.000	0.109560	0.0547802	−	0.998498i	$-0.482554\pi$
$163$	228.000	0.109560	0.0547802	+	0.998498i	$0.482554\pi$
$164$	0	0
$165$	288.000	0.135883
$166$	0	0
$167$	4128.00	1.91278	0.956390	−	0.292093i	$-0.0943517\pi$
$167$	4128.00	1.91278	0.956390	+	0.292093i	$0.0943517\pi$
$168$	0	0
$169$	−1797.00	−0.817934
$170$	0	0
$171$	−972.000	−0.434682
$172$	0	0
$173$	−1352.00	−0.594166	−0.297083	−	0.954852i	$-0.596014\pi$
$173$	−1352.00	−0.594166	−0.297083	+	0.954852i	$0.596014\pi$
$174$	0	0
$175$	−732.000	−0.316194
$176$	0	0
$177$	−2484.00	−1.05485
$178$	0	0
$179$	1716.00	0.716536	0.358268	−	0.933619i	$-0.383367\pi$
$179$	1716.00	0.716536	0.358268	+	0.933619i	$0.383367\pi$
$180$	0	0
$181$	−3692.00	−1.51616	−0.758078	−	0.652164i	$-0.773861\pi$
$181$	−3692.00	−1.51616	−0.758078	+	0.652164i	$0.773861\pi$
$182$	0	0
$183$	252.000	0.101794
$184$	0	0
$185$	1824.00	0.724882
$186$	0	0
$187$	744.000	0.290945
$188$	0	0
$189$	−324.000	−0.124696
$190$	0	0
$191$	−48.0000	−0.0181841	−0.00909204	−	0.999959i	$-0.502894\pi$
$191$	−48.0000	−0.0181841	−0.00909204	+	0.999959i	$0.502894\pi$
$192$	0	0
$193$	2414.00	0.900329	0.450165	−	0.892946i	$-0.351365\pi$
$193$	2414.00	0.900329	0.450165	+	0.892946i	$0.351365\pi$
$194$	0	0
$195$	480.000	0.176274
$196$	0	0
$197$	4304.00	1.55659	0.778293	−	0.627902i	$-0.216086\pi$
$197$	4304.00	1.55659	0.778293	+	0.627902i	$0.216086\pi$
$198$	0	0
$199$	−204.000	−0.0726692	−0.0363346	−	0.999340i	$-0.511568\pi$
$199$	−204.000	−0.0726692	−0.0363346	+	0.999340i	$0.511568\pi$
$200$	0	0
$201$	1044.00	0.366359
$202$	0	0
$203$	−1536.00	−0.531064
$204$	0	0
$205$	−176.000	−0.0599628
$206$	0	0
$207$	−648.000	−0.217580
$208$	0	0
$209$	−1296.00	−0.428929
$210$	0	0
$211$	4020.00	1.31160	0.655801	−	0.754933i	$-0.272331\pi$
$211$	4020.00	1.31160	0.655801	+	0.754933i	$0.272331\pi$
$212$	0	0
$213$	−1368.00	−0.440065
$214$	0	0
$215$	−1632.00	−0.517681
$216$	0	0
$217$	2448.00	0.765811
$218$	0	0
$219$	2466.00	0.760899
$220$	0	0
$221$	1240.00	0.377427
$222$	0	0
$223$	516.000	0.154950	0.0774751	−	0.996994i	$-0.475314\pi$
$223$	516.000	0.154950	0.0774751	+	0.996994i	$0.475314\pi$
$224$	0	0
$225$	−549.000	−0.162667
$226$	0	0
$227$	1428.00	0.417532	0.208766	−	0.977966i	$-0.433055\pi$
$227$	1428.00	0.417532	0.208766	+	0.977966i	$0.433055\pi$
$228$	0	0
$229$	−6028.00	−1.73948	−0.869741	−	0.493508i	$-0.835714\pi$
$229$	−6028.00	−1.73948	−0.869741	+	0.493508i	$0.835714\pi$
$230$	0	0
$231$	−432.000	−0.123046
$232$	0	0
$233$	−2630.00	−0.739472	−0.369736	−	0.929137i	$-0.620552\pi$
$233$	−2630.00	−0.739472	−0.369736	+	0.929137i	$0.620552\pi$
$234$	0	0
$235$	−4800.00	−1.33241
$236$	0	0
$237$	−4068.00	−1.11496
$238$	0	0
$239$	−4416.00	−1.19518	−0.597588	−	0.801803i	$-0.703874\pi$
$239$	−4416.00	−1.19518	−0.597588	+	0.801803i	$0.703874\pi$
$240$	0	0
$241$	4830.00	1.29099	0.645493	−	0.763766i	$-0.276652\pi$
$241$	4830.00	1.29099	0.645493	+	0.763766i	$0.276652\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	1592.00	0.415139
$246$	0	0
$247$	−2160.00	−0.556427
$248$	0	0
$249$	324.000	0.0824605
$250$	0	0
$251$	5532.00	1.39114	0.695571	−	0.718457i	$-0.255151\pi$
$251$	5532.00	1.39114	0.695571	+	0.718457i	$0.255151\pi$
$252$	0	0
$253$	−864.000	−0.214700
$254$	0	0
$255$	1488.00	0.365420
$256$	0	0
$257$	−254.000	−0.0616501	−0.0308251	−	0.999525i	$-0.509813\pi$
$257$	−254.000	−0.0616501	−0.0308251	+	0.999525i	$0.509813\pi$
$258$	0	0
$259$	−2736.00	−0.656397
$260$	0	0
$261$	−1152.00	−0.273207
$262$	0	0
$263$	−4272.00	−1.00161	−0.500804	−	0.865561i	$-0.666962\pi$
$263$	−4272.00	−1.00161	−0.500804	+	0.865561i	$0.666962\pi$
$264$	0	0
$265$	−2048.00	−0.474746
$266$	0	0
$267$	−2814.00	−0.644996
$268$	0	0
$269$	4544.00	1.02994	0.514968	−	0.857210i	$-0.327804\pi$
$269$	4544.00	1.02994	0.514968	+	0.857210i	$0.327804\pi$
$270$	0	0
$271$	−2076.00	−0.465343	−0.232672	−	0.972555i	$-0.574747\pi$
$271$	−2076.00	−0.465343	−0.232672	+	0.972555i	$0.574747\pi$
$272$	0	0
$273$	−720.000	−0.159620
$274$	0	0
$275$	−732.000	−0.160514
$276$	0	0
$277$	484.000	0.104985	0.0524923	−	0.998621i	$-0.483283\pi$
$277$	484.000	0.104985	0.0524923	+	0.998621i	$0.483283\pi$
$278$	0	0
$279$	1836.00	0.393973
$280$	0	0
$281$	−406.000	−0.0861919	−0.0430960	−	0.999071i	$-0.513722\pi$
$281$	−406.000	−0.0861919	−0.0430960	+	0.999071i	$0.513722\pi$
$282$	0	0
$283$	8172.00	1.71652	0.858260	−	0.513216i	$-0.171546\pi$
$283$	8172.00	1.71652	0.858260	+	0.513216i	$0.171546\pi$
$284$	0	0
$285$	−2592.00	−0.538726
$286$	0	0
$287$	264.000	0.0542977
$288$	0	0
$289$	−1069.00	−0.217586
$290$	0	0
$291$	−3834.00	−0.772347
$292$	0	0
$293$	4960.00	0.988963	0.494482	−	0.869188i	$-0.335358\pi$
$293$	4960.00	0.988963	0.494482	+	0.869188i	$0.335358\pi$
$294$	0	0
$295$	−6624.00	−1.30734
$296$	0	0
$297$	−324.000	−0.0633010
$298$	0	0
$299$	−1440.00	−0.278520
$300$	0	0
$301$	2448.00	0.468772
$302$	0	0
$303$	−1824.00	−0.345829
$304$	0	0
$305$	672.000	0.126159
$306$	0	0
$307$	−6684.00	−1.24259	−0.621296	−	0.783576i	$-0.713394\pi$
$307$	−6684.00	−1.24259	−0.621296	+	0.783576i	$0.713394\pi$
$308$	0	0
$309$	−2844.00	−0.523591
$310$	0	0
$311$	−4992.00	−0.910194	−0.455097	−	0.890442i	$-0.650395\pi$
$311$	−4992.00	−0.910194	−0.455097	+	0.890442i	$0.650395\pi$
$312$	0	0
$313$	−5402.00	−0.975524	−0.487762	−	0.872977i	$-0.662187\pi$
$313$	−5402.00	−0.975524	−0.487762	+	0.872977i	$0.662187\pi$
$314$	0	0
$315$	−864.000	−0.154542
$316$	0	0
$317$	7856.00	1.39191	0.695957	−	0.718083i	$-0.254980\pi$
$317$	7856.00	1.39191	0.695957	+	0.718083i	$0.254980\pi$
$318$	0	0
$319$	−1536.00	−0.269591
$320$	0	0
$321$	3132.00	0.544583
$322$	0	0
$323$	−6696.00	−1.15348
$324$	0	0
$325$	−1220.00	−0.208226
$326$	0	0
$327$	−5340.00	−0.903066
$328$	0	0
$329$	7200.00	1.20653
$330$	0	0
$331$	−3732.00	−0.619726	−0.309863	−	0.950781i	$-0.600283\pi$
$331$	−3732.00	−0.619726	−0.309863	+	0.950781i	$0.600283\pi$
$332$	0	0
$333$	−2052.00	−0.337684
$334$	0	0
$335$	2784.00	0.454048
$336$	0	0
$337$	−5598.00	−0.904874	−0.452437	−	0.891796i	$-0.649445\pi$
$337$	−5598.00	−0.904874	−0.452437	+	0.891796i	$0.649445\pi$
$338$	0	0
$339$	1866.00	0.298959
$340$	0	0
$341$	2448.00	0.388758
$342$	0	0
$343$	−6504.00	−1.02386
$344$	0	0
$345$	−1728.00	−0.269659
$346$	0	0
$347$	8220.00	1.27168	0.635840	−	0.771821i	$-0.280654\pi$
$347$	8220.00	1.27168	0.635840	+	0.771821i	$0.280654\pi$
$348$	0	0
$349$	−11844.0	−1.81660	−0.908302	−	0.418315i	$-0.862621\pi$
$349$	−11844.0	−1.81660	−0.908302	+	0.418315i	$0.862621\pi$
$350$	0	0
$351$	−540.000	−0.0821170
$352$	0	0
$353$	−7006.00	−1.05635	−0.528175	−	0.849135i	$-0.677124\pi$
$353$	−7006.00	−1.05635	−0.528175	+	0.849135i	$0.677124\pi$
$354$	0	0
$355$	−3648.00	−0.545396
$356$	0	0
$357$	−2232.00	−0.330896
$358$	0	0
$359$	−7512.00	−1.10437	−0.552184	−	0.833722i	$-0.686205\pi$
$359$	−7512.00	−1.10437	−0.552184	+	0.833722i	$0.686205\pi$
$360$	0	0
$361$	4805.00	0.700539
$362$	0	0
$363$	3561.00	0.514887
$364$	0	0
$365$	6576.00	0.943023
$366$	0	0
$367$	−5076.00	−0.721976	−0.360988	−	0.932571i	$-0.617560\pi$
$367$	−5076.00	−0.721976	−0.360988	+	0.932571i	$0.617560\pi$
$368$	0	0
$369$	198.000	0.0279335
$370$	0	0
$371$	3072.00	0.429893
$372$	0	0
$373$	4860.00	0.674641	0.337321	−	0.941390i	$-0.390479\pi$
$373$	4860.00	0.674641	0.337321	+	0.941390i	$0.390479\pi$
$374$	0	0
$375$	−4464.00	−0.614720
$376$	0	0
$377$	−2560.00	−0.349726
$378$	0	0
$379$	5964.00	0.808312	0.404156	−	0.914690i	$-0.367565\pi$
$379$	5964.00	0.808312	0.404156	+	0.914690i	$0.367565\pi$
$380$	0	0
$381$	612.000	0.0822932
$382$	0	0
$383$	432.000	0.0576349	0.0288175	−	0.999585i	$-0.490826\pi$
$383$	432.000	0.0576349	0.0288175	+	0.999585i	$0.490826\pi$
$384$	0	0
$385$	−1152.00	−0.152497
$386$	0	0
$387$	1836.00	0.241161
$388$	0	0
$389$	8888.00	1.15846	0.579228	−	0.815165i	$-0.303354\pi$
$389$	8888.00	1.15846	0.579228	+	0.815165i	$0.303354\pi$
$390$	0	0
$391$	−4464.00	−0.577376
$392$	0	0
$393$	1044.00	0.134002
$394$	0	0
$395$	−10848.0	−1.38183
$396$	0	0
$397$	−2676.00	−0.338299	−0.169149	−	0.985590i	$-0.554102\pi$
$397$	−2676.00	−0.338299	−0.169149	+	0.985590i	$0.554102\pi$
$398$	0	0
$399$	3888.00	0.487828
$400$	0	0
$401$	13790.0	1.71731	0.858653	−	0.512557i	$-0.171302\pi$
$401$	13790.0	1.71731	0.858653	+	0.512557i	$0.171302\pi$
$402$	0	0
$403$	4080.00	0.504316
$404$	0	0
$405$	−648.000	−0.0795046
$406$	0	0
$407$	−2736.00	−0.333215
$408$	0	0
$409$	1974.00	0.238650	0.119325	−	0.992855i	$-0.461927\pi$
$409$	1974.00	0.238650	0.119325	+	0.992855i	$0.461927\pi$
$410$	0	0
$411$	318.000	0.0381649
$412$	0	0
$413$	9936.00	1.18382
$414$	0	0
$415$	864.000	0.102198
$416$	0	0
$417$	3564.00	0.418537
$418$	0	0
$419$	−4764.00	−0.555457	−0.277729	−	0.960660i	$-0.589582\pi$
$419$	−4764.00	−0.555457	−0.277729	+	0.960660i	$0.589582\pi$
$420$	0	0
$421$	−92.0000	−0.0106504	−0.00532518	−	0.999986i	$-0.501695\pi$
$421$	−92.0000	−0.0106504	−0.00532518	+	0.999986i	$0.501695\pi$
$422$	0	0
$423$	5400.00	0.620702
$424$	0	0
$425$	−3782.00	−0.431656
$426$	0	0
$427$	−1008.00	−0.114240
$428$	0	0
$429$	−720.000	−0.0810301
$430$	0	0
$431$	−10488.0	−1.17213	−0.586066	−	0.810263i	$-0.699324\pi$
$431$	−10488.0	−1.17213	−0.586066	+	0.810263i	$0.699324\pi$
$432$	0	0
$433$	−13138.0	−1.45813	−0.729067	−	0.684442i	$-0.760046\pi$
$433$	−13138.0	−1.45813	−0.729067	+	0.684442i	$0.760046\pi$
$434$	0	0
$435$	−3072.00	−0.338600
$436$	0	0
$437$	7776.00	0.851205
$438$	0	0
$439$	−3612.00	−0.392691	−0.196346	−	0.980535i	$-0.562907\pi$
$439$	−3612.00	−0.392691	−0.196346	+	0.980535i	$0.562907\pi$
$440$	0	0
$441$	−1791.00	−0.193392
$442$	0	0
$443$	12972.0	1.39124	0.695619	−	0.718411i	$-0.255130\pi$
$443$	12972.0	1.39124	0.695619	+	0.718411i	$0.255130\pi$
$444$	0	0
$445$	−7504.00	−0.799379
$446$	0	0
$447$	−7944.00	−0.840578
$448$	0	0
$449$	5998.00	0.630430	0.315215	−	0.949020i	$-0.397923\pi$
$449$	5998.00	0.630430	0.315215	+	0.949020i	$0.397923\pi$
$450$	0	0
$451$	264.000	0.0275638
$452$	0	0
$453$	−10260.0	−1.06414
$454$	0	0
$455$	−1920.00	−0.197826
$456$	0	0
$457$	8934.00	0.914475	0.457237	−	0.889345i	$-0.348839\pi$
$457$	8934.00	0.914475	0.457237	+	0.889345i	$0.348839\pi$
$458$	0	0
$459$	−1674.00	−0.170230
$460$	0	0
$461$	7448.00	0.752468	0.376234	−	0.926525i	$-0.377219\pi$
$461$	7448.00	0.752468	0.376234	+	0.926525i	$0.377219\pi$
$462$	0	0
$463$	4356.00	0.437236	0.218618	−	0.975810i	$-0.429845\pi$
$463$	4356.00	0.437236	0.218618	+	0.975810i	$0.429845\pi$
$464$	0	0
$465$	4896.00	0.488272
$466$	0	0
$467$	8580.00	0.850182	0.425091	−	0.905151i	$-0.360242\pi$
$467$	8580.00	0.850182	0.425091	+	0.905151i	$0.360242\pi$
$468$	0	0
$469$	−4176.00	−0.411151
$470$	0	0
$471$	−180.000	−0.0176093
$472$	0	0
$473$	2448.00	0.237969
$474$	0	0
$475$	6588.00	0.636375
$476$	0	0
$477$	2304.00	0.221159
$478$	0	0
$479$	−12648.0	−1.20648	−0.603238	−	0.797561i	$-0.706123\pi$
$479$	−12648.0	−1.20648	−0.603238	+	0.797561i	$0.706123\pi$
$480$	0	0
$481$	−4560.00	−0.432262
$482$	0	0
$483$	2592.00	0.244182
$484$	0	0
$485$	−10224.0	−0.957212
$486$	0	0
$487$	15036.0	1.39907	0.699534	−	0.714599i	$-0.253391\pi$
$487$	15036.0	1.39907	0.699534	+	0.714599i	$0.253391\pi$
$488$	0	0
$489$	−684.000	−0.0632547
$490$	0	0
$491$	−15684.0	−1.44157	−0.720783	−	0.693161i	$-0.756218\pi$
$491$	−15684.0	−1.44157	−0.720783	+	0.693161i	$0.756218\pi$
$492$	0	0
$493$	−7936.00	−0.724989
$494$	0	0
$495$	−864.000	−0.0784523
$496$	0	0
$497$	5472.00	0.493869
$498$	0	0
$499$	16308.0	1.46302	0.731509	−	0.681831i	$-0.238816\pi$
$499$	16308.0	1.46302	0.731509	+	0.681831i	$0.238816\pi$
$500$	0	0
$501$	−12384.0	−1.10434
$502$	0	0
$503$	10344.0	0.916931	0.458465	−	0.888712i	$-0.348399\pi$
$503$	10344.0	0.916931	0.458465	+	0.888712i	$0.348399\pi$
$504$	0	0
$505$	−4864.00	−0.428604
$506$	0	0
$507$	5391.00	0.472234
$508$	0	0
$509$	−5648.00	−0.491833	−0.245917	−	0.969291i	$-0.579089\pi$
$509$	−5648.00	−0.491833	−0.245917	+	0.969291i	$0.579089\pi$
$510$	0	0
$511$	−9864.00	−0.853929
$512$	0	0
$513$	2916.00	0.250964
$514$	0	0
$515$	−7584.00	−0.648915
$516$	0	0
$517$	7200.00	0.612487
$518$	0	0
$519$	4056.00	0.343042
$520$	0	0
$521$	−19498.0	−1.63958	−0.819792	−	0.572662i	$-0.805911\pi$
$521$	−19498.0	−1.63958	−0.819792	+	0.572662i	$0.805911\pi$
$522$	0	0
$523$	−22596.0	−1.88920	−0.944602	−	0.328217i	$-0.893552\pi$
$523$	−22596.0	−1.88920	−0.944602	+	0.328217i	$0.893552\pi$
$524$	0	0
$525$	2196.00	0.182555
$526$	0	0
$527$	12648.0	1.04546
$528$	0	0
$529$	−6983.00	−0.573929
$530$	0	0
$531$	7452.00	0.609019
$532$	0	0
$533$	440.000	0.0357571
$534$	0	0
$535$	8352.00	0.674932
$536$	0	0
$537$	−5148.00	−0.413692
$538$	0	0
$539$	−2388.00	−0.190832
$540$	0	0
$541$	−812.000	−0.0645298	−0.0322649	−	0.999479i	$-0.510272\pi$
$541$	−812.000	−0.0645298	−0.0322649	+	0.999479i	$0.510272\pi$
$542$	0	0
$543$	11076.0	0.875353
$544$	0	0
$545$	−14240.0	−1.11922
$546$	0	0
$547$	6132.00	0.479315	0.239658	−	0.970857i	$-0.422965\pi$
$547$	6132.00	0.479315	0.239658	+	0.970857i	$0.422965\pi$
$548$	0	0
$549$	−756.000	−0.0587710
$550$	0	0
$551$	13824.0	1.06882
$552$	0	0
$553$	16272.0	1.25128
$554$	0	0
$555$	−5472.00	−0.418511
$556$	0	0
$557$	2792.00	0.212389	0.106195	−	0.994345i	$-0.466133\pi$
$557$	2792.00	0.212389	0.106195	+	0.994345i	$0.466133\pi$
$558$	0	0
$559$	4080.00	0.308704
$560$	0	0
$561$	−2232.00	−0.167977
$562$	0	0
$563$	6468.00	0.484181	0.242090	−	0.970254i	$-0.422167\pi$
$563$	6468.00	0.484181	0.242090	+	0.970254i	$0.422167\pi$
$564$	0	0
$565$	4976.00	0.370517
$566$	0	0
$567$	972.000	0.0719932
$568$	0	0
$569$	−10522.0	−0.775229	−0.387614	−	0.921822i	$-0.626701\pi$
$569$	−10522.0	−0.775229	−0.387614	+	0.921822i	$0.626701\pi$
$570$	0	0
$571$	1068.00	0.0782739	0.0391370	−	0.999234i	$-0.487539\pi$
$571$	1068.00	0.0782739	0.0391370	+	0.999234i	$0.487539\pi$
$572$	0	0
$573$	144.000	0.0104986
$574$	0	0
$575$	4392.00	0.318537
$576$	0	0
$577$	3602.00	0.259884	0.129942	−	0.991522i	$-0.458521\pi$
$577$	3602.00	0.259884	0.129942	+	0.991522i	$0.458521\pi$
$578$	0	0
$579$	−7242.00	−0.519805
$580$	0	0
$581$	−1296.00	−0.0925424
$582$	0	0
$583$	3072.00	0.218232
$584$	0	0
$585$	−1440.00	−0.101772
$586$	0	0
$587$	−17940.0	−1.26144	−0.630718	−	0.776012i	$-0.717240\pi$
$587$	−17940.0	−1.26144	−0.630718	+	0.776012i	$0.717240\pi$
$588$	0	0
$589$	−22032.0	−1.54128
$590$	0	0
$591$	−12912.0	−0.898695
$592$	0	0
$593$	18034.0	1.24885	0.624425	−	0.781085i	$-0.285334\pi$
$593$	18034.0	1.24885	0.624425	+	0.781085i	$0.285334\pi$
$594$	0	0
$595$	−5952.00	−0.410098
$596$	0	0
$597$	612.000	0.0419556
$598$	0	0
$599$	12264.0	0.836550	0.418275	−	0.908320i	$-0.362635\pi$
$599$	12264.0	0.836550	0.418275	+	0.908320i	$0.362635\pi$
$600$	0	0
$601$	12634.0	0.857490	0.428745	−	0.903426i	$-0.358956\pi$
$601$	12634.0	0.857490	0.428745	+	0.903426i	$0.358956\pi$
$602$	0	0
$603$	−3132.00	−0.211517
$604$	0	0
$605$	9496.00	0.638128
$606$	0	0
$607$	−2796.00	−0.186962	−0.0934812	−	0.995621i	$-0.529799\pi$
$607$	−2796.00	−0.186962	−0.0934812	+	0.995621i	$0.529799\pi$
$608$	0	0
$609$	4608.00	0.306610
$610$	0	0
$611$	12000.0	0.794547
$612$	0	0
$613$	13788.0	0.908470	0.454235	−	0.890882i	$-0.349913\pi$
$613$	13788.0	0.908470	0.454235	+	0.890882i	$0.349913\pi$
$614$	0	0
$615$	528.000	0.0346195
$616$	0	0
$617$	18074.0	1.17931	0.589653	−	0.807657i	$-0.299264\pi$
$617$	18074.0	1.17931	0.589653	+	0.807657i	$0.299264\pi$
$618$	0	0
$619$	−5940.00	−0.385701	−0.192850	−	0.981228i	$-0.561773\pi$
$619$	−5940.00	−0.385701	−0.192850	+	0.981228i	$0.561773\pi$
$620$	0	0
$621$	1944.00	0.125620
$622$	0	0
$623$	11256.0	0.723856
$624$	0	0
$625$	−4279.00	−0.273856
$626$	0	0
$627$	3888.00	0.247642
$628$	0	0
$629$	−14136.0	−0.896088
$630$	0	0
$631$	8700.00	0.548877	0.274439	−	0.961605i	$-0.411508\pi$
$631$	8700.00	0.548877	0.274439	+	0.961605i	$0.411508\pi$
$632$	0	0
$633$	−12060.0	−0.757254
$634$	0	0
$635$	1632.00	0.101990
$636$	0	0
$637$	−3980.00	−0.247556
$638$	0	0
$639$	4104.00	0.254072
$640$	0	0
$641$	−16306.0	−1.00476	−0.502378	−	0.864648i	$-0.667541\pi$
$641$	−16306.0	−1.00476	−0.502378	+	0.864648i	$0.667541\pi$
$642$	0	0
$643$	−22668.0	−1.39026	−0.695131	−	0.718883i	$-0.744654\pi$
$643$	−22668.0	−1.39026	−0.695131	+	0.718883i	$0.744654\pi$
$644$	0	0
$645$	4896.00	0.298883
$646$	0	0
$647$	11928.0	0.724788	0.362394	−	0.932025i	$-0.381959\pi$
$647$	11928.0	0.724788	0.362394	+	0.932025i	$0.381959\pi$
$648$	0	0
$649$	9936.00	0.600959
$650$	0	0
$651$	−7344.00	−0.442141
$652$	0	0
$653$	−2552.00	−0.152936	−0.0764682	−	0.997072i	$-0.524364\pi$
$653$	−2552.00	−0.152936	−0.0764682	+	0.997072i	$0.524364\pi$
$654$	0	0
$655$	2784.00	0.166076
$656$	0	0
$657$	−7398.00	−0.439305
$658$	0	0
$659$	2196.00	0.129809	0.0649044	−	0.997891i	$-0.479326\pi$
$659$	2196.00	0.129809	0.0649044	+	0.997891i	$0.479326\pi$
$660$	0	0
$661$	−4260.00	−0.250673	−0.125336	−	0.992114i	$-0.540001\pi$
$661$	−4260.00	−0.250673	−0.125336	+	0.992114i	$0.540001\pi$
$662$	0	0
$663$	−3720.00	−0.217908
$664$	0	0
$665$	10368.0	0.604592
$666$	0	0
$667$	9216.00	0.535000
$668$	0	0
$669$	−1548.00	−0.0894606
$670$	0	0
$671$	−1008.00	−0.0579932
$672$	0	0
$673$	−2018.00	−0.115584	−0.0577921	−	0.998329i	$-0.518406\pi$
$673$	−2018.00	−0.115584	−0.0577921	+	0.998329i	$0.518406\pi$
$674$	0	0
$675$	1647.00	0.0939156
$676$	0	0
$677$	9256.00	0.525461	0.262730	−	0.964869i	$-0.415377\pi$
$677$	9256.00	0.525461	0.262730	+	0.964869i	$0.415377\pi$
$678$	0	0
$679$	15336.0	0.866777
$680$	0	0
$681$	−4284.00	−0.241062
$682$	0	0
$683$	29244.0	1.63835	0.819173	−	0.573546i	$-0.194433\pi$
$683$	29244.0	1.63835	0.819173	+	0.573546i	$0.194433\pi$
$684$	0	0
$685$	848.000	0.0472999
$686$	0	0
$687$	18084.0	1.00429
$688$	0	0
$689$	5120.00	0.283101
$690$	0	0
$691$	3684.00	0.202816	0.101408	−	0.994845i	$-0.467665\pi$
$691$	3684.00	0.202816	0.101408	+	0.994845i	$0.467665\pi$
$692$	0	0
$693$	1296.00	0.0710404
$694$	0	0
$695$	9504.00	0.518715
$696$	0	0
$697$	1364.00	0.0741251
$698$	0	0
$699$	7890.00	0.426934
$700$	0	0
$701$	13456.0	0.725002	0.362501	−	0.931983i	$-0.381923\pi$
$701$	13456.0	0.725002	0.362501	+	0.931983i	$0.381923\pi$
$702$	0	0
$703$	24624.0	1.32107
$704$	0	0
$705$	14400.0	0.769270
$706$	0	0
$707$	7296.00	0.388111
$708$	0	0
$709$	−6460.00	−0.342187	−0.171093	−	0.985255i	$-0.554730\pi$
$709$	−6460.00	−0.342187	−0.171093	+	0.985255i	$0.554730\pi$
$710$	0	0
$711$	12204.0	0.643721
$712$	0	0
$713$	−14688.0	−0.771487
$714$	0	0
$715$	−1920.00	−0.100425
$716$	0	0
$717$	13248.0	0.690036
$718$	0	0
$719$	17160.0	0.890070	0.445035	−	0.895513i	$-0.353191\pi$
$719$	17160.0	0.890070	0.445035	+	0.895513i	$0.353191\pi$
$720$	0	0
$721$	11376.0	0.587607
$722$	0	0
$723$	−14490.0	−0.745351
$724$	0	0
$725$	7808.00	0.399975
$726$	0	0
$727$	11820.0	0.602998	0.301499	−	0.953466i	$-0.402513\pi$
$727$	11820.0	0.602998	0.301499	+	0.953466i	$0.402513\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	12648.0	0.639950
$732$	0	0
$733$	23924.0	1.20553	0.602765	−	0.797919i	$-0.294066\pi$
$733$	23924.0	1.20553	0.602765	+	0.797919i	$0.294066\pi$
$734$	0	0
$735$	−4776.00	−0.239681
$736$	0	0
$737$	−4176.00	−0.208718
$738$	0	0
$739$	11796.0	0.587176	0.293588	−	0.955932i	$-0.405151\pi$
$739$	11796.0	0.587176	0.293588	+	0.955932i	$0.405151\pi$
$740$	0	0
$741$	6480.00	0.321253
$742$	0	0
$743$	−27024.0	−1.33434	−0.667170	−	0.744906i	$-0.732494\pi$
$743$	−27024.0	−1.33434	−0.667170	+	0.744906i	$0.732494\pi$
$744$	0	0
$745$	−21184.0	−1.04177
$746$	0	0
$747$	−972.000	−0.0476086
$748$	0	0
$749$	−12528.0	−0.611166
$750$	0	0
$751$	17340.0	0.842537	0.421269	−	0.906936i	$-0.361585\pi$
$751$	17340.0	0.842537	0.421269	+	0.906936i	$0.361585\pi$
$752$	0	0
$753$	−16596.0	−0.803176
$754$	0	0
$755$	−27360.0	−1.31885
$756$	0	0
$757$	27236.0	1.30767	0.653837	−	0.756635i	$-0.273158\pi$
$757$	27236.0	1.30767	0.653837	+	0.756635i	$0.273158\pi$
$758$	0	0
$759$	2592.00	0.123957
$760$	0	0
$761$	14758.0	0.702992	0.351496	−	0.936189i	$-0.385673\pi$
$761$	14758.0	0.702992	0.351496	+	0.936189i	$0.385673\pi$
$762$	0	0
$763$	21360.0	1.01348
$764$	0	0
$765$	−4464.00	−0.210976
$766$	0	0
$767$	16560.0	0.779592
$768$	0	0
$769$	25774.0	1.20863	0.604314	−	0.796747i	$-0.293447\pi$
$769$	25774.0	1.20863	0.604314	+	0.796747i	$0.293447\pi$
$770$	0	0
$771$	762.000	0.0355937
$772$	0	0
$773$	−37424.0	−1.74133	−0.870665	−	0.491877i	$-0.836311\pi$
$773$	−37424.0	−1.74133	−0.870665	+	0.491877i	$0.836311\pi$
$774$	0	0
$775$	−12444.0	−0.576776
$776$	0	0
$777$	8208.00	0.378971
$778$	0	0
$779$	−2376.00	−0.109280
$780$	0	0
$781$	5472.00	0.250709
$782$	0	0
$783$	3456.00	0.157736
$784$	0	0
$785$	−480.000	−0.0218241
$786$	0	0
$787$	12804.0	0.579941	0.289970	−	0.957036i	$-0.406355\pi$
$787$	12804.0	0.579941	0.289970	+	0.957036i	$0.406355\pi$
$788$	0	0
$789$	12816.0	0.578278
$790$	0	0
$791$	−7464.00	−0.335511
$792$	0	0
$793$	−1680.00	−0.0752315
$794$	0	0
$795$	6144.00	0.274095
$796$	0	0
$797$	−32024.0	−1.42327	−0.711636	−	0.702548i	$-0.752046\pi$
$797$	−32024.0	−1.42327	−0.711636	+	0.702548i	$0.752046\pi$
$798$	0	0
$799$	37200.0	1.64711
$800$	0	0
$801$	8442.00	0.372389
$802$	0	0
$803$	−9864.00	−0.433491
$804$	0	0
$805$	6912.00	0.302629
$806$	0	0
$807$	−13632.0	−0.594633
$808$	0	0
$809$	−38090.0	−1.65534	−0.827672	−	0.561212i	$-0.810335\pi$
$809$	−38090.0	−1.65534	−0.827672	+	0.561212i	$0.810335\pi$
$810$	0	0
$811$	10428.0	0.451512	0.225756	−	0.974184i	$-0.427515\pi$
$811$	10428.0	0.451512	0.225756	+	0.974184i	$0.427515\pi$
$812$	0	0
$813$	6228.00	0.268666
$814$	0	0
$815$	−1824.00	−0.0783950
$816$	0	0
$817$	−22032.0	−0.943454
$818$	0	0
$819$	2160.00	0.0921569
$820$	0	0
$821$	−7984.00	−0.339395	−0.169698	−	0.985496i	$-0.554279\pi$
$821$	−7984.00	−0.339395	−0.169698	+	0.985496i	$0.554279\pi$
$822$	0	0
$823$	−28788.0	−1.21930	−0.609652	−	0.792669i	$-0.708691\pi$
$823$	−28788.0	−1.21930	−0.609652	+	0.792669i	$0.708691\pi$
$824$	0	0
$825$	2196.00	0.0926726
$826$	0	0
$827$	−468.000	−0.0196783	−0.00983915	−	0.999952i	$-0.503132\pi$
$827$	−468.000	−0.0196783	−0.00983915	+	0.999952i	$0.503132\pi$
$828$	0	0
$829$	28852.0	1.20877	0.604386	−	0.796692i	$-0.293419\pi$
$829$	28852.0	1.20877	0.604386	+	0.796692i	$0.293419\pi$
$830$	0	0
$831$	−1452.00	−0.0606129
$832$	0	0
$833$	−12338.0	−0.513189
$834$	0	0
$835$	−33024.0	−1.36867
$836$	0	0
$837$	−5508.00	−0.227460
$838$	0	0
$839$	1944.00	0.0799932	0.0399966	−	0.999200i	$-0.487265\pi$
$839$	1944.00	0.0799932	0.0399966	+	0.999200i	$0.487265\pi$
$840$	0	0
$841$	−8005.00	−0.328222
$842$	0	0
$843$	1218.00	0.0497629
$844$	0	0
$845$	14376.0	0.585266
$846$	0	0
$847$	−14244.0	−0.577839
$848$	0	0
$849$	−24516.0	−0.991033
$850$	0	0
$851$	16416.0	0.661261
$852$	0	0
$853$	−37044.0	−1.48694	−0.743472	−	0.668768i	$-0.766822\pi$
$853$	−37044.0	−1.48694	−0.743472	+	0.668768i	$0.766822\pi$
$854$	0	0
$855$	7776.00	0.311033
$856$	0	0
$857$	15046.0	0.599722	0.299861	−	0.953983i	$-0.403060\pi$
$857$	15046.0	0.599722	0.299861	+	0.953983i	$0.403060\pi$
$858$	0	0
$859$	−12180.0	−0.483791	−0.241895	−	0.970302i	$-0.577769\pi$
$859$	−12180.0	−0.483791	−0.241895	+	0.970302i	$0.577769\pi$
$860$	0	0
$861$	−792.000	−0.0313488
$862$	0	0
$863$	28752.0	1.13410	0.567051	−	0.823683i	$-0.308084\pi$
$863$	28752.0	1.13410	0.567051	+	0.823683i	$0.308084\pi$
$864$	0	0
$865$	10816.0	0.425150
$866$	0	0
$867$	3207.00	0.125623
$868$	0	0
$869$	16272.0	0.635201
$870$	0	0
$871$	−6960.00	−0.270758
$872$	0	0
$873$	11502.0	0.445915
$874$	0	0
$875$	17856.0	0.689878
$876$	0	0
$877$	31884.0	1.22765	0.613823	−	0.789443i	$-0.289631\pi$
$877$	31884.0	1.22765	0.613823	+	0.789443i	$0.289631\pi$
$878$	0	0
$879$	−14880.0	−0.570978
$880$	0	0
$881$	30802.0	1.17792	0.588959	−	0.808163i	$-0.299538\pi$
$881$	30802.0	1.17792	0.588959	+	0.808163i	$0.299538\pi$
$882$	0	0
$883$	−32460.0	−1.23711	−0.618554	−	0.785742i	$-0.712281\pi$
$883$	−32460.0	−1.23711	−0.618554	+	0.785742i	$0.712281\pi$
$884$	0	0
$885$	19872.0	0.754791
$886$	0	0
$887$	14832.0	0.561454	0.280727	−	0.959788i	$-0.409424\pi$
$887$	14832.0	0.561454	0.280727	+	0.959788i	$0.409424\pi$
$888$	0	0
$889$	−2448.00	−0.0923547
$890$	0	0
$891$	972.000	0.0365468
$892$	0	0
$893$	−64800.0	−2.42827
$894$	0	0
$895$	−13728.0	−0.512711
$896$	0	0
$897$	4320.00	0.160803
$898$	0	0
$899$	−26112.0	−0.968725
$900$	0	0
$901$	15872.0	0.586873
$902$	0	0
$903$	−7344.00	−0.270646
$904$	0	0
$905$	29536.0	1.08487
$906$	0	0
$907$	−6900.00	−0.252603	−0.126301	−	0.991992i	$-0.540311\pi$
$907$	−6900.00	−0.252603	−0.126301	+	0.991992i	$0.540311\pi$
$908$	0	0
$909$	5472.00	0.199664
$910$	0	0
$911$	−32832.0	−1.19404	−0.597021	−	0.802225i	$-0.703649\pi$
$911$	−32832.0	−1.19404	−0.597021	+	0.802225i	$0.703649\pi$
$912$	0	0
$913$	−1296.00	−0.0469785
$914$	0	0
$915$	−2016.00	−0.0728381
$916$	0	0
$917$	−4176.00	−0.150386
$918$	0	0
$919$	−8340.00	−0.299359	−0.149680	−	0.988735i	$-0.547824\pi$
$919$	−8340.00	−0.299359	−0.149680	+	0.988735i	$0.547824\pi$
$920$	0	0
$921$	20052.0	0.717411
$922$	0	0
$923$	9120.00	0.325231
$924$	0	0
$925$	13908.0	0.494370
$926$	0	0
$927$	8532.00	0.302295
$928$	0	0
$929$	−39826.0	−1.40651	−0.703255	−	0.710937i	$-0.748271\pi$
$929$	−39826.0	−1.40651	−0.703255	+	0.710937i	$0.748271\pi$
$930$	0	0
$931$	21492.0	0.756576
$932$	0	0
$933$	14976.0	0.525501
$934$	0	0
$935$	−5952.00	−0.208183
$936$	0	0
$937$	−28550.0	−0.995398	−0.497699	−	0.867350i	$-0.665822\pi$
$937$	−28550.0	−0.995398	−0.497699	+	0.867350i	$0.665822\pi$
$938$	0	0
$939$	16206.0	0.563219
$940$	0	0
$941$	−50632.0	−1.75404	−0.877022	−	0.480450i	$-0.840473\pi$
$941$	−50632.0	−1.75404	−0.877022	+	0.480450i	$0.840473\pi$
$942$	0	0
$943$	−1584.00	−0.0547000
$944$	0	0
$945$	2592.00	0.0892251
$946$	0	0
$947$	−18204.0	−0.624657	−0.312329	−	0.949974i	$-0.601109\pi$
$947$	−18204.0	−0.624657	−0.312329	+	0.949974i	$0.601109\pi$
$948$	0	0
$949$	−16440.0	−0.562345
$950$	0	0
$951$	−23568.0	−0.803622
$952$	0	0
$953$	4934.00	0.167710	0.0838552	−	0.996478i	$-0.473277\pi$
$953$	4934.00	0.167710	0.0838552	+	0.996478i	$0.473277\pi$
$954$	0	0
$955$	384.000	0.0130115
$956$	0	0
$957$	4608.00	0.155648
$958$	0	0
$959$	−1272.00	−0.0428311
$960$	0	0
$961$	11825.0	0.396932
$962$	0	0
$963$	−9396.00	−0.314415
$964$	0	0
$965$	−19312.0	−0.644223
$966$	0	0
$967$	13284.0	0.441763	0.220881	−	0.975301i	$-0.429107\pi$
$967$	13284.0	0.441763	0.220881	+	0.975301i	$0.429107\pi$
$968$	0	0
$969$	20088.0	0.665964
$970$	0	0
$971$	−50820.0	−1.67960	−0.839800	−	0.542896i	$-0.817328\pi$
$971$	−50820.0	−1.67960	−0.839800	+	0.542896i	$0.817328\pi$
$972$	0	0
$973$	−14256.0	−0.469709
$974$	0	0
$975$	3660.00	0.120219
$976$	0	0
$977$	11038.0	0.361450	0.180725	−	0.983534i	$-0.442156\pi$
$977$	11038.0	0.361450	0.180725	+	0.983534i	$0.442156\pi$
$978$	0	0
$979$	11256.0	0.367460
$980$	0	0
$981$	16020.0	0.521386
$982$	0	0
$983$	−44112.0	−1.43129	−0.715643	−	0.698466i	$-0.753866\pi$
$983$	−44112.0	−1.43129	−0.715643	+	0.698466i	$0.753866\pi$
$984$	0	0
$985$	−34432.0	−1.11380
$986$	0	0
$987$	−21600.0	−0.696591
$988$	0	0
$989$	−14688.0	−0.472246
$990$	0	0
$991$	−56196.0	−1.80134	−0.900668	−	0.434507i	$-0.856923\pi$
$991$	−56196.0	−1.80134	−0.900668	+	0.434507i	$0.856923\pi$
$992$	0	0
$993$	11196.0	0.357799
$994$	0	0
$995$	1632.00	0.0519979
$996$	0	0
$997$	−45588.0	−1.44813	−0.724065	−	0.689731i	$-0.757729\pi$
$997$	−45588.0	−1.44813	−0.724065	+	0.689731i	$0.757729\pi$
$998$	0	0
$999$	6156.00	0.194962

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.4.a.a.1.1		1
3.2	odd	2		2304.4.a.l.1.1		1
4.3	odd	2		768.4.a.c.1.1		1
8.3	odd	2		768.4.a.b.1.1		1
8.5	even	2		768.4.a.d.1.1		1
12.11	even	2		2304.4.a.k.1.1		1
16.3	odd	4		384.4.d.b.193.2	yes	2
16.5	even	4		384.4.d.a.193.2	yes	2
16.11	odd	4		384.4.d.b.193.1	yes	2
16.13	even	4		384.4.d.a.193.1	✓	2
24.5	odd	2		2304.4.a.f.1.1		1
24.11	even	2		2304.4.a.e.1.1		1
48.5	odd	4		1152.4.d.b.577.2		2
48.11	even	4		1152.4.d.g.577.2		2
48.29	odd	4		1152.4.d.b.577.1		2
48.35	even	4		1152.4.d.g.577.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.4.d.a.193.1	✓	2	16.13	even	4
384.4.d.a.193.2	yes	2	16.5	even	4
384.4.d.b.193.1	yes	2	16.11	odd	4
384.4.d.b.193.2	yes	2	16.3	odd	4
768.4.a.a.1.1		1	1.1	even	1	trivial
768.4.a.b.1.1		1	8.3	odd	2
768.4.a.c.1.1		1	4.3	odd	2
768.4.a.d.1.1		1	8.5	even	2
1152.4.d.b.577.1		2	48.29	odd	4
1152.4.d.b.577.2		2	48.5	odd	4
1152.4.d.g.577.1		2	48.35	even	4
1152.4.d.g.577.2		2	48.11	even	4
2304.4.a.e.1.1		1	24.11	even	2
2304.4.a.f.1.1		1	24.5	odd	2
2304.4.a.k.1.1		1	12.11	even	2
2304.4.a.l.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:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	768.a (trivial)

Introduction

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