LMFDB - Embedded newform 1104.6.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1104,6,Mod(1,1104)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1104, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("1104.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 1104 = 2^{4} \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1104.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:	$177.063737074$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 552)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1104.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+9.00000 q^{3} -82.0000 q^{5} +64.0000 q^{7} +81.0000 q^{9} +O(q^{10})$

$q+9.00000 q^{3} -82.0000 q^{5} +64.0000 q^{7} +81.0000 q^{9} +412.000 q^{11} +622.000 q^{13} -738.000 q^{15} +466.000 q^{17} -204.000 q^{19} +576.000 q^{21} -529.000 q^{23} +3599.00 q^{25} +729.000 q^{27} +1030.00 q^{29} -5688.00 q^{31} +3708.00 q^{33} -5248.00 q^{35} +7862.00 q^{37} +5598.00 q^{39} +4026.00 q^{41} -6548.00 q^{43} -6642.00 q^{45} +22016.0 q^{47} -12711.0 q^{49} +4194.00 q^{51} +4974.00 q^{53} -33784.0 q^{55} -1836.00 q^{57} -35972.0 q^{59} +17870.0 q^{61} +5184.00 q^{63} -51004.0 q^{65} +22708.0 q^{67} -4761.00 q^{69} +59288.0 q^{71} -22038.0 q^{73} +32391.0 q^{75} +26368.0 q^{77} -85656.0 q^{79} +6561.00 q^{81} -59100.0 q^{83} -38212.0 q^{85} +9270.00 q^{87} -113174. q^{89} +39808.0 q^{91} -51192.0 q^{93} +16728.0 q^{95} -75038.0 q^{97} +33372.0 q^{99} +O(q^{100})$

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
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	0	0
$5$	−82.0000	−1.46686	−0.733430	−	0.679765i	$-0.762082\pi$
$5$	−82.0000	−1.46686	−0.733430	+	0.679765i	$0.762082\pi$
$6$	0	0
$7$	64.0000	0.493668	0.246834	−	0.969058i	$-0.420610\pi$
$7$	64.0000	0.493668	0.246834	+	0.969058i	$0.420610\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	412.000	1.02663	0.513317	−	0.858199i	$-0.328417\pi$
$11$	412.000	1.02663	0.513317	+	0.858199i	$0.328417\pi$
$12$	0	0
$13$	622.000	1.02078	0.510390	−	0.859943i	$-0.329501\pi$
$13$	622.000	1.02078	0.510390	+	0.859943i	$0.329501\pi$
$14$	0	0
$15$	−738.000	−0.846892
$16$	0	0
$17$	466.000	0.391078	0.195539	−	0.980696i	$-0.437354\pi$
$17$	466.000	0.391078	0.195539	+	0.980696i	$0.437354\pi$
$18$	0	0
$19$	−204.000	−0.129642	−0.0648211	−	0.997897i	$-0.520648\pi$
$19$	−204.000	−0.129642	−0.0648211	+	0.997897i	$0.520648\pi$
$20$	0	0
$21$	576.000	0.285019
$22$	0	0
$23$	−529.000	−0.208514
$23$	−529.000	−0.208514
$24$	0	0
$25$	3599.00	1.15168
$26$	0	0
$27$	729.000	0.192450
$28$	0	0
$29$	1030.00	0.227427	0.113714	−	0.993514i	$-0.463725\pi$
$29$	1030.00	0.227427	0.113714	+	0.993514i	$0.463725\pi$
$30$	0	0
$31$	−5688.00	−1.06305	−0.531527	−	0.847041i	$-0.678382\pi$
$31$	−5688.00	−1.06305	−0.531527	+	0.847041i	$0.678382\pi$
$32$	0	0
$33$	3708.00	0.592727
$34$	0	0
$35$	−5248.00	−0.724142
$36$	0	0
$37$	7862.00	0.944123	0.472062	−	0.881566i	$-0.343510\pi$
$37$	7862.00	0.944123	0.472062	+	0.881566i	$0.343510\pi$
$38$	0	0
$39$	5598.00	0.589347
$40$	0	0
$41$	4026.00	0.374037	0.187018	−	0.982356i	$-0.440118\pi$
$41$	4026.00	0.374037	0.187018	+	0.982356i	$0.440118\pi$
$42$	0	0
$43$	−6548.00	−0.540054	−0.270027	−	0.962853i	$-0.587033\pi$
$43$	−6548.00	−0.540054	−0.270027	+	0.962853i	$0.587033\pi$
$44$	0	0
$45$	−6642.00	−0.488954
$46$	0	0
$47$	22016.0	1.45376	0.726882	−	0.686763i	$-0.240969\pi$
$47$	22016.0	1.45376	0.726882	+	0.686763i	$0.240969\pi$
$48$	0	0
$49$	−12711.0	−0.756292
$50$	0	0
$51$	4194.00	0.225789
$52$	0	0
$53$	4974.00	0.243229	0.121615	−	0.992577i	$-0.461193\pi$
$53$	4974.00	0.243229	0.121615	+	0.992577i	$0.461193\pi$
$54$	0	0
$55$	−33784.0	−1.50593
$56$	0	0
$57$	−1836.00	−0.0748489
$58$	0	0
$59$	−35972.0	−1.34535	−0.672674	−	0.739939i	$-0.734854\pi$
$59$	−35972.0	−1.34535	−0.672674	+	0.739939i	$0.734854\pi$
$60$	0	0
$61$	17870.0	0.614894	0.307447	−	0.951565i	$-0.400525\pi$
$61$	17870.0	0.614894	0.307447	+	0.951565i	$0.400525\pi$
$62$	0	0
$63$	5184.00	0.164556
$64$	0	0
$65$	−51004.0	−1.49734
$66$	0	0
$67$	22708.0	0.618005	0.309002	−	0.951061i	$-0.400005\pi$
$67$	22708.0	0.618005	0.309002	+	0.951061i	$0.400005\pi$
$68$	0	0
$69$	−4761.00	−0.120386
$70$	0	0
$71$	59288.0	1.39579	0.697896	−	0.716199i	$-0.254120\pi$
$71$	59288.0	1.39579	0.697896	+	0.716199i	$0.254120\pi$
$72$	0	0
$73$	−22038.0	−0.484022	−0.242011	−	0.970274i	$-0.577807\pi$
$73$	−22038.0	−0.484022	−0.242011	+	0.970274i	$0.577807\pi$
$74$	0	0
$75$	32391.0	0.664923
$76$	0	0
$77$	26368.0	0.506816
$78$	0	0
$79$	−85656.0	−1.54415	−0.772076	−	0.635530i	$-0.780782\pi$
$79$	−85656.0	−1.54415	−0.772076	+	0.635530i	$0.780782\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	−59100.0	−0.941656	−0.470828	−	0.882225i	$-0.656045\pi$
$83$	−59100.0	−0.941656	−0.470828	+	0.882225i	$0.656045\pi$
$84$	0	0
$85$	−38212.0	−0.573657
$86$	0	0
$87$	9270.00	0.131305
$88$	0	0
$89$	−113174.	−1.51451	−0.757254	−	0.653120i	$-0.773460\pi$
$89$	−113174.	−1.51451	−0.757254	+	0.653120i	$0.773460\pi$
$90$	0	0
$91$	39808.0	0.503926
$92$	0	0
$93$	−51192.0	−0.613755
$94$	0	0
$95$	16728.0	0.190167
$96$	0	0
$97$	−75038.0	−0.809752	−0.404876	−	0.914372i	$-0.632685\pi$
$97$	−75038.0	−0.809752	−0.404876	+	0.914372i	$0.632685\pi$
$98$	0	0
$99$	33372.0	0.342211
$100$	0	0
$101$	−52114.0	−0.508336	−0.254168	−	0.967160i	$-0.581802\pi$
$101$	−52114.0	−0.508336	−0.254168	+	0.967160i	$0.581802\pi$
$102$	0	0
$103$	69104.0	0.641815	0.320908	−	0.947110i	$-0.396012\pi$
$103$	69104.0	0.641815	0.320908	+	0.947110i	$0.396012\pi$
$104$	0	0
$105$	−47232.0	−0.418084
$106$	0	0
$107$	116364.	0.982560	0.491280	−	0.871002i	$-0.336529\pi$
$107$	116364.	0.982560	0.491280	+	0.871002i	$0.336529\pi$
$108$	0	0
$109$	92430.0	0.745155	0.372578	−	0.928001i	$-0.378474\pi$
$109$	92430.0	0.745155	0.372578	+	0.928001i	$0.378474\pi$
$110$	0	0
$111$	70758.0	0.545090
$112$	0	0
$113$	111138.	0.818779	0.409389	−	0.912360i	$-0.365742\pi$
$113$	111138.	0.818779	0.409389	+	0.912360i	$0.365742\pi$
$114$	0	0
$115$	43378.0	0.305862
$116$	0	0
$117$	50382.0	0.340260
$118$	0	0
$119$	29824.0	0.193063
$120$	0	0
$121$	8693.00	0.0539767
$122$	0	0
$123$	36234.0	0.215950
$124$	0	0
$125$	−38868.0	−0.222493
$126$	0	0
$127$	156472.	0.860850	0.430425	−	0.902626i	$-0.358364\pi$
$127$	156472.	0.860850	0.430425	+	0.902626i	$0.358364\pi$
$128$	0	0
$129$	−58932.0	−0.311801
$130$	0	0
$131$	284772.	1.44984	0.724918	−	0.688835i	$-0.241877\pi$
$131$	284772.	1.44984	0.724918	+	0.688835i	$0.241877\pi$
$132$	0	0
$133$	−13056.0	−0.0640002
$134$	0	0
$135$	−59778.0	−0.282297
$136$	0	0
$137$	421450.	1.91842	0.959212	−	0.282687i	$-0.0912259\pi$
$137$	421450.	1.91842	0.959212	+	0.282687i	$0.0912259\pi$
$138$	0	0
$139$	−234836.	−1.03093	−0.515463	−	0.856912i	$-0.672380\pi$
$139$	−234836.	−1.03093	−0.515463	+	0.856912i	$0.672380\pi$
$140$	0	0
$141$	198144.	0.839331
$142$	0	0
$143$	256264.	1.04797
$144$	0	0
$145$	−84460.0	−0.333604
$146$	0	0
$147$	−114399.	−0.436645
$148$	0	0
$149$	−22914.0	−0.0845542	−0.0422771	−	0.999106i	$-0.513461\pi$
$149$	−22914.0	−0.0845542	−0.0422771	+	0.999106i	$0.513461\pi$
$150$	0	0
$151$	152704.	0.545014	0.272507	−	0.962154i	$-0.412147\pi$
$151$	152704.	0.545014	0.272507	+	0.962154i	$0.412147\pi$
$152$	0	0
$153$	37746.0	0.130359
$154$	0	0
$155$	466416.	1.55935
$156$	0	0
$157$	217230.	0.703349	0.351674	−	0.936122i	$-0.385612\pi$
$157$	217230.	0.703349	0.351674	+	0.936122i	$0.385612\pi$
$158$	0	0
$159$	44766.0	0.140429
$160$	0	0
$161$	−33856.0	−0.102937
$162$	0	0
$163$	−161836.	−0.477096	−0.238548	−	0.971131i	$-0.576671\pi$
$163$	−161836.	−0.477096	−0.238548	+	0.971131i	$0.576671\pi$
$164$	0	0
$165$	−304056.	−0.869448
$166$	0	0
$167$	470184.	1.30460	0.652299	−	0.757962i	$-0.273805\pi$
$167$	470184.	1.30460	0.652299	+	0.757962i	$0.273805\pi$
$168$	0	0
$169$	15591.0	0.0419911
$170$	0	0
$171$	−16524.0	−0.0432140
$172$	0	0
$173$	20134.0	0.0511464	0.0255732	−	0.999673i	$-0.491859\pi$
$173$	20134.0	0.0511464	0.0255732	+	0.999673i	$0.491859\pi$
$174$	0	0
$175$	230336.	0.568547
$176$	0	0
$177$	−323748.	−0.776737
$178$	0	0
$179$	497780.	1.16119	0.580597	−	0.814191i	$-0.302819\pi$
$179$	497780.	1.16119	0.580597	+	0.814191i	$0.302819\pi$
$180$	0	0
$181$	−253002.	−0.574021	−0.287010	−	0.957927i	$-0.592661\pi$
$181$	−253002.	−0.574021	−0.287010	+	0.957927i	$0.592661\pi$
$182$	0	0
$183$	160830.	0.355009
$184$	0	0
$185$	−644684.	−1.38490
$186$	0	0
$187$	191992.	0.401494
$188$	0	0
$189$	46656.0	0.0950064
$190$	0	0
$191$	425184.	0.843322	0.421661	−	0.906754i	$-0.361447\pi$
$191$	425184.	0.843322	0.421661	+	0.906754i	$0.361447\pi$
$192$	0	0
$193$	443266.	0.856586	0.428293	−	0.903640i	$-0.359115\pi$
$193$	443266.	0.856586	0.428293	+	0.903640i	$0.359115\pi$
$194$	0	0
$195$	−459036.	−0.864490
$196$	0	0
$197$	95502.0	0.175326	0.0876631	−	0.996150i	$-0.472060\pi$
$197$	95502.0	0.175326	0.0876631	+	0.996150i	$0.472060\pi$
$198$	0	0
$199$	290864.	0.520664	0.260332	−	0.965519i	$-0.416168\pi$
$199$	290864.	0.520664	0.260332	+	0.965519i	$0.416168\pi$
$200$	0	0
$201$	204372.	0.356805
$202$	0	0
$203$	65920.0	0.112273
$204$	0	0
$205$	−330132.	−0.548660
$206$	0	0
$207$	−42849.0	−0.0695048
$208$	0	0
$209$	−84048.0	−0.133095
$210$	0	0
$211$	519700.	0.803612	0.401806	−	0.915725i	$-0.368383\pi$
$211$	519700.	0.803612	0.401806	+	0.915725i	$0.368383\pi$
$212$	0	0
$213$	533592.	0.805861
$214$	0	0
$215$	536936.	0.792185
$216$	0	0
$217$	−364032.	−0.524796
$218$	0	0
$219$	−198342.	−0.279450
$220$	0	0
$221$	289852.	0.399205
$222$	0	0
$223$	994456.	1.33913	0.669566	−	0.742753i	$-0.266480\pi$
$223$	994456.	1.33913	0.669566	+	0.742753i	$0.266480\pi$
$224$	0	0
$225$	291519.	0.383893
$226$	0	0
$227$	523892.	0.674803	0.337402	−	0.941361i	$-0.390452\pi$
$227$	523892.	0.674803	0.337402	+	0.941361i	$0.390452\pi$
$228$	0	0
$229$	−328250.	−0.413634	−0.206817	−	0.978380i	$-0.566310\pi$
$229$	−328250.	−0.413634	−0.206817	+	0.978380i	$0.566310\pi$
$230$	0	0
$231$	237312.	0.292610
$232$	0	0
$233$	500778.	0.604304	0.302152	−	0.953260i	$-0.402295\pi$
$233$	500778.	0.604304	0.302152	+	0.953260i	$0.402295\pi$
$234$	0	0
$235$	−1.80531e6	−2.13247
$236$	0	0
$237$	−770904.	−0.891517
$238$	0	0
$239$	642256.	0.727300	0.363650	−	0.931536i	$-0.381530\pi$
$239$	642256.	0.727300	0.363650	+	0.931536i	$0.381530\pi$
$240$	0	0
$241$	−533710.	−0.591920	−0.295960	−	0.955200i	$-0.595639\pi$
$241$	−533710.	−0.591920	−0.295960	+	0.955200i	$0.595639\pi$
$242$	0	0
$243$	59049.0	0.0641500
$244$	0	0
$245$	1.04230e6	1.10937
$246$	0	0
$247$	−126888.	−0.132336
$248$	0	0
$249$	−531900.	−0.543665
$250$	0	0
$251$	−1.76898e6	−1.77231	−0.886153	−	0.463393i	$-0.846632\pi$
$251$	−1.76898e6	−1.77231	−0.886153	+	0.463393i	$0.846632\pi$
$252$	0	0
$253$	−217948.	−0.214068
$254$	0	0
$255$	−343908.	−0.331201
$256$	0	0
$257$	1.45005e6	1.36946	0.684731	−	0.728796i	$-0.259920\pi$
$257$	1.45005e6	1.36946	0.684731	+	0.728796i	$0.259920\pi$
$258$	0	0
$259$	503168.	0.466083
$260$	0	0
$261$	83430.0	0.0758090
$262$	0	0
$263$	1.14442e6	1.02023	0.510114	−	0.860107i	$-0.329603\pi$
$263$	1.14442e6	1.02023	0.510114	+	0.860107i	$0.329603\pi$
$264$	0	0
$265$	−407868.	−0.356784
$266$	0	0
$267$	−1.01857e6	−0.874402
$268$	0	0
$269$	−991050.	−0.835055	−0.417527	−	0.908664i	$-0.637103\pi$
$269$	−991050.	−0.835055	−0.417527	+	0.908664i	$0.637103\pi$
$270$	0	0
$271$	381608.	0.315642	0.157821	−	0.987468i	$-0.449553\pi$
$271$	381608.	0.315642	0.157821	+	0.987468i	$0.449553\pi$
$272$	0	0
$273$	358272.	0.290942
$274$	0	0
$275$	1.48279e6	1.18235
$276$	0	0
$277$	−266698.	−0.208843	−0.104422	−	0.994533i	$-0.533299\pi$
$277$	−266698.	−0.208843	−0.104422	+	0.994533i	$0.533299\pi$
$278$	0	0
$279$	−460728.	−0.354351
$280$	0	0
$281$	1.92398e6	1.45356	0.726782	−	0.686868i	$-0.241015\pi$
$281$	1.92398e6	1.45356	0.726782	+	0.686868i	$0.241015\pi$
$282$	0	0
$283$	−1.60927e6	−1.19443	−0.597217	−	0.802080i	$-0.703727\pi$
$283$	−1.60927e6	−1.19443	−0.597217	+	0.802080i	$0.703727\pi$
$284$	0	0
$285$	150552.	0.109793
$286$	0	0
$287$	257664.	0.184650
$288$	0	0
$289$	−1.20270e6	−0.847058
$290$	0	0
$291$	−675342.	−0.467510
$292$	0	0
$293$	−18274.0	−0.0124355	−0.00621777	−	0.999981i	$-0.501979\pi$
$293$	−18274.0	−0.0124355	−0.00621777	+	0.999981i	$0.501979\pi$
$294$	0	0
$295$	2.94970e6	1.97344
$296$	0	0
$297$	300348.	0.197576
$298$	0	0
$299$	−329038.	−0.212847
$300$	0	0
$301$	−419072.	−0.266608
$302$	0	0
$303$	−469026.	−0.293488
$304$	0	0
$305$	−1.46534e6	−0.901963
$306$	0	0
$307$	1.07756e6	0.652520	0.326260	−	0.945280i	$-0.394211\pi$
$307$	1.07756e6	0.652520	0.326260	+	0.945280i	$0.394211\pi$
$308$	0	0
$309$	621936.	0.370552
$310$	0	0
$311$	−1.63745e6	−0.959990	−0.479995	−	0.877271i	$-0.659361\pi$
$311$	−1.63745e6	−0.959990	−0.479995	+	0.877271i	$0.659361\pi$
$312$	0	0
$313$	3.15535e6	1.82049	0.910243	−	0.414075i	$-0.135895\pi$
$313$	3.15535e6	1.82049	0.910243	+	0.414075i	$0.135895\pi$
$314$	0	0
$315$	−425088.	−0.241381
$316$	0	0
$317$	−2.34585e6	−1.31115	−0.655575	−	0.755130i	$-0.727573\pi$
$317$	−2.34585e6	−1.31115	−0.655575	+	0.755130i	$0.727573\pi$
$318$	0	0
$319$	424360.	0.233484
$320$	0	0
$321$	1.04728e6	0.567281
$322$	0	0
$323$	−95064.0	−0.0507002
$324$	0	0
$325$	2.23858e6	1.17561
$326$	0	0
$327$	831870.	0.430216
$328$	0	0
$329$	1.40902e6	0.717676
$330$	0	0
$331$	−1.11260e6	−0.558171	−0.279086	−	0.960266i	$-0.590031\pi$
$331$	−1.11260e6	−0.558171	−0.279086	+	0.960266i	$0.590031\pi$
$332$	0	0
$333$	636822.	0.314708
$334$	0	0
$335$	−1.86206e6	−0.906527
$336$	0	0
$337$	−1.15944e6	−0.556125	−0.278063	−	0.960563i	$-0.589692\pi$
$337$	−1.15944e6	−0.556125	−0.278063	+	0.960563i	$0.589692\pi$
$338$	0	0
$339$	1.00024e6	0.472722
$340$	0	0
$341$	−2.34346e6	−1.09137
$342$	0	0
$343$	−1.88915e6	−0.867025
$344$	0	0
$345$	390402.	0.176589
$346$	0	0
$347$	7724.00	0.00344365	0.00172182	−	0.999999i	$-0.499452\pi$
$347$	7724.00	0.00344365	0.00172182	+	0.999999i	$0.499452\pi$
$348$	0	0
$349$	1.99435e6	0.876472	0.438236	−	0.898860i	$-0.355604\pi$
$349$	1.99435e6	0.876472	0.438236	+	0.898860i	$0.355604\pi$
$350$	0	0
$351$	453438.	0.196449
$352$	0	0
$353$	−614622.	−0.262525	−0.131263	−	0.991348i	$-0.541903\pi$
$353$	−614622.	−0.262525	−0.131263	+	0.991348i	$0.541903\pi$
$354$	0	0
$355$	−4.86162e6	−2.04743
$356$	0	0
$357$	268416.	0.111465
$358$	0	0
$359$	1.56982e6	0.642854	0.321427	−	0.946934i	$-0.395837\pi$
$359$	1.56982e6	0.642854	0.321427	+	0.946934i	$0.395837\pi$
$360$	0	0
$361$	−2.43448e6	−0.983193
$362$	0	0
$363$	78237.0	0.0311635
$364$	0	0
$365$	1.80712e6	0.709993
$366$	0	0
$367$	−1.96092e6	−0.759967	−0.379983	−	0.924993i	$-0.624070\pi$
$367$	−1.96092e6	−0.759967	−0.379983	+	0.924993i	$0.624070\pi$
$368$	0	0
$369$	326106.	0.124679
$370$	0	0
$371$	318336.	0.120075
$372$	0	0
$373$	4.19076e6	1.55963	0.779813	−	0.626012i	$-0.215314\pi$
$373$	4.19076e6	1.55963	0.779813	+	0.626012i	$0.215314\pi$
$374$	0	0
$375$	−349812.	−0.128457
$376$	0	0
$377$	640660.	0.232153
$378$	0	0
$379$	566236.	0.202488	0.101244	−	0.994862i	$-0.467718\pi$
$379$	566236.	0.202488	0.101244	+	0.994862i	$0.467718\pi$
$380$	0	0
$381$	1.40825e6	0.497012
$382$	0	0
$383$	2.50262e6	0.871763	0.435882	−	0.900004i	$-0.356437\pi$
$383$	2.50262e6	0.871763	0.435882	+	0.900004i	$0.356437\pi$
$384$	0	0
$385$	−2.16218e6	−0.743429
$386$	0	0
$387$	−530388.	−0.180018
$388$	0	0
$389$	2.25294e6	0.754877	0.377438	−	0.926035i	$-0.376805\pi$
$389$	2.25294e6	0.754877	0.377438	+	0.926035i	$0.376805\pi$
$390$	0	0
$391$	−246514.	−0.0815454
$392$	0	0
$393$	2.56295e6	0.837064
$394$	0	0
$395$	7.02379e6	2.26506
$396$	0	0
$397$	3.49513e6	1.11298	0.556490	−	0.830854i	$-0.312148\pi$
$397$	3.49513e6	1.11298	0.556490	+	0.830854i	$0.312148\pi$
$398$	0	0
$399$	−117504.	−0.0369505
$400$	0	0
$401$	−1.22811e6	−0.381396	−0.190698	−	0.981649i	$-0.561075\pi$
$401$	−1.22811e6	−0.381396	−0.190698	+	0.981649i	$0.561075\pi$
$402$	0	0
$403$	−3.53794e6	−1.08514
$404$	0	0
$405$	−538002.	−0.162985
$406$	0	0
$407$	3.23914e6	0.969269
$408$	0	0
$409$	1.13379e6	0.335137	0.167569	−	0.985860i	$-0.446408\pi$
$409$	1.13379e6	0.335137	0.167569	+	0.985860i	$0.446408\pi$
$410$	0	0
$411$	3.79305e6	1.10760
$412$	0	0
$413$	−2.30221e6	−0.664155
$414$	0	0
$415$	4.84620e6	1.38128
$416$	0	0
$417$	−2.11352e6	−0.595206
$418$	0	0
$419$	3.81525e6	1.06167	0.530833	−	0.847476i	$-0.321879\pi$
$419$	3.81525e6	1.06167	0.530833	+	0.847476i	$0.321879\pi$
$420$	0	0
$421$	6.40509e6	1.76125	0.880624	−	0.473817i	$-0.157124\pi$
$421$	6.40509e6	1.76125	0.880624	+	0.473817i	$0.157124\pi$
$422$	0	0
$423$	1.78330e6	0.484588
$424$	0	0
$425$	1.67713e6	0.450397
$426$	0	0
$427$	1.14368e6	0.303553
$428$	0	0
$429$	2.30638e6	0.605044
$430$	0	0
$431$	1.98662e6	0.515137	0.257568	−	0.966260i	$-0.417079\pi$
$431$	1.98662e6	0.515137	0.257568	+	0.966260i	$0.417079\pi$
$432$	0	0
$433$	1.02181e6	0.261909	0.130954	−	0.991388i	$-0.458196\pi$
$433$	1.02181e6	0.261909	0.130954	+	0.991388i	$0.458196\pi$
$434$	0	0
$435$	−760140.	−0.192606
$436$	0	0
$437$	107916.	0.0270323
$438$	0	0
$439$	−234928.	−0.0581800	−0.0290900	−	0.999577i	$-0.509261\pi$
$439$	−234928.	−0.0581800	−0.0290900	+	0.999577i	$0.509261\pi$
$440$	0	0
$441$	−1.02959e6	−0.252097
$442$	0	0
$443$	−3.13954e6	−0.760075	−0.380038	−	0.924971i	$-0.624089\pi$
$443$	−3.13954e6	−0.760075	−0.380038	+	0.924971i	$0.624089\pi$
$444$	0	0
$445$	9.28027e6	2.22157
$446$	0	0
$447$	−206226.	−0.0488174
$448$	0	0
$449$	−2.98430e6	−0.698597	−0.349299	−	0.937011i	$-0.613580\pi$
$449$	−2.98430e6	−0.698597	−0.349299	+	0.937011i	$0.613580\pi$
$450$	0	0
$451$	1.65871e6	0.383999
$452$	0	0
$453$	1.37434e6	0.314664
$454$	0	0
$455$	−3.26426e6	−0.739189
$456$	0	0
$457$	2.83614e6	0.635239	0.317619	−	0.948218i	$-0.397117\pi$
$457$	2.83614e6	0.635239	0.317619	+	0.948218i	$0.397117\pi$
$458$	0	0
$459$	339714.	0.0752630
$460$	0	0
$461$	−2.82111e6	−0.618256	−0.309128	−	0.951020i	$-0.600037\pi$
$461$	−2.82111e6	−0.618256	−0.309128	+	0.951020i	$0.600037\pi$
$462$	0	0
$463$	2.50177e6	0.542369	0.271184	−	0.962527i	$-0.412585\pi$
$463$	2.50177e6	0.542369	0.271184	+	0.962527i	$0.412585\pi$
$464$	0	0
$465$	4.19774e6	0.900292
$466$	0	0
$467$	3.37399e6	0.715898	0.357949	−	0.933741i	$-0.383476\pi$
$467$	3.37399e6	0.715898	0.357949	+	0.933741i	$0.383476\pi$
$468$	0	0
$469$	1.45331e6	0.305089
$470$	0	0
$471$	1.95507e6	0.406079
$472$	0	0
$473$	−2.69778e6	−0.554438
$474$	0	0
$475$	−734196.	−0.149306
$476$	0	0
$477$	402894.	0.0810765
$478$	0	0
$479$	2.99102e6	0.595636	0.297818	−	0.954623i	$-0.403741\pi$
$479$	2.99102e6	0.595636	0.297818	+	0.954623i	$0.403741\pi$
$480$	0	0
$481$	4.89016e6	0.963742
$482$	0	0
$483$	−304704.	−0.0594306
$484$	0	0
$485$	6.15312e6	1.18779
$486$	0	0
$487$	−2.38898e6	−0.456446	−0.228223	−	0.973609i	$-0.573292\pi$
$487$	−2.38898e6	−0.456446	−0.228223	+	0.973609i	$0.573292\pi$
$488$	0	0
$489$	−1.45652e6	−0.275452
$490$	0	0
$491$	2.04552e6	0.382912	0.191456	−	0.981501i	$-0.438679\pi$
$491$	2.04552e6	0.382912	0.191456	+	0.981501i	$0.438679\pi$
$492$	0	0
$493$	479980.	0.0889418
$494$	0	0
$495$	−2.73650e6	−0.501976
$496$	0	0
$497$	3.79443e6	0.689058
$498$	0	0
$499$	−486860.	−0.0875292	−0.0437646	−	0.999042i	$-0.513935\pi$
$499$	−486860.	−0.0875292	−0.0437646	+	0.999042i	$0.513935\pi$
$500$	0	0
$501$	4.23166e6	0.753210
$502$	0	0
$503$	−3.60946e6	−0.636096	−0.318048	−	0.948075i	$-0.603027\pi$
$503$	−3.60946e6	−0.636096	−0.318048	+	0.948075i	$0.603027\pi$
$504$	0	0
$505$	4.27335e6	0.745658
$506$	0	0
$507$	140319.	0.0242436
$508$	0	0
$509$	−9.10137e6	−1.55708	−0.778542	−	0.627592i	$-0.784041\pi$
$509$	−9.10137e6	−1.55708	−0.778542	+	0.627592i	$0.784041\pi$
$510$	0	0
$511$	−1.41043e6	−0.238946
$512$	0	0
$513$	−148716.	−0.0249496
$514$	0	0
$515$	−5.66653e6	−0.941454
$516$	0	0
$517$	9.07059e6	1.49248
$518$	0	0
$519$	181206.	0.0295294
$520$	0	0
$521$	2.42750e6	0.391800	0.195900	−	0.980624i	$-0.437237\pi$
$521$	2.42750e6	0.391800	0.195900	+	0.980624i	$0.437237\pi$
$522$	0	0
$523$	−4.84472e6	−0.774488	−0.387244	−	0.921977i	$-0.626573\pi$
$523$	−4.84472e6	−0.774488	−0.387244	+	0.921977i	$0.626573\pi$
$524$	0	0
$525$	2.07302e6	0.328251
$526$	0	0
$527$	−2.65061e6	−0.415737
$528$	0	0
$529$	279841.	0.0434783
$530$	0	0
$531$	−2.91373e6	−0.448449
$532$	0	0
$533$	2.50417e6	0.381809
$534$	0	0
$535$	−9.54185e6	−1.44128
$536$	0	0
$537$	4.48002e6	0.670416
$538$	0	0
$539$	−5.23693e6	−0.776435
$540$	0	0
$541$	1.35670e6	0.199293	0.0996463	−	0.995023i	$-0.468229\pi$
$541$	1.35670e6	0.199293	0.0996463	+	0.995023i	$0.468229\pi$
$542$	0	0
$543$	−2.27702e6	−0.331411
$544$	0	0
$545$	−7.57926e6	−1.09304
$546$	0	0
$547$	9.35672e6	1.33707	0.668537	−	0.743679i	$-0.266921\pi$
$547$	9.35672e6	1.33707	0.668537	+	0.743679i	$0.266921\pi$
$548$	0	0
$549$	1.44747e6	0.204965
$550$	0	0
$551$	−210120.	−0.0294841
$552$	0	0
$553$	−5.48198e6	−0.762298
$554$	0	0
$555$	−5.80216e6	−0.799571
$556$	0	0
$557$	−483498.	−0.0660323	−0.0330162	−	0.999455i	$-0.510511\pi$
$557$	−483498.	−0.0660323	−0.0330162	+	0.999455i	$0.510511\pi$
$558$	0	0
$559$	−4.07286e6	−0.551277
$560$	0	0
$561$	1.72793e6	0.231803
$562$	0	0
$563$	26916.0	0.00357882	0.00178941	−	0.999998i	$-0.499430\pi$
$563$	26916.0	0.00357882	0.00178941	+	0.999998i	$0.499430\pi$
$564$	0	0
$565$	−9.11332e6	−1.20103
$566$	0	0
$567$	419904.	0.0548520
$568$	0	0
$569$	3.43231e6	0.444433	0.222217	−	0.974997i	$-0.428671\pi$
$569$	3.43231e6	0.444433	0.222217	+	0.974997i	$0.428671\pi$
$570$	0	0
$571$	−9.89055e6	−1.26949	−0.634746	−	0.772721i	$-0.718895\pi$
$571$	−9.89055e6	−1.26949	−0.634746	+	0.772721i	$0.718895\pi$
$572$	0	0
$573$	3.82666e6	0.486892
$574$	0	0
$575$	−1.90387e6	−0.240142
$576$	0	0
$577$	−8.31833e6	−1.04015	−0.520076	−	0.854120i	$-0.674096\pi$
$577$	−8.31833e6	−1.04015	−0.520076	+	0.854120i	$0.674096\pi$
$578$	0	0
$579$	3.98939e6	0.494550
$580$	0	0
$581$	−3.78240e6	−0.464865
$582$	0	0
$583$	2.04929e6	0.249708
$584$	0	0
$585$	−4.13132e6	−0.499114
$586$	0	0
$587$	1.13469e7	1.35919	0.679597	−	0.733585i	$-0.262155\pi$
$587$	1.13469e7	1.35919	0.679597	+	0.733585i	$0.262155\pi$
$588$	0	0
$589$	1.16035e6	0.137817
$590$	0	0
$591$	859518.	0.101225
$592$	0	0
$593$	−9.18069e6	−1.07211	−0.536054	−	0.844184i	$-0.680086\pi$
$593$	−9.18069e6	−1.07211	−0.536054	+	0.844184i	$0.680086\pi$
$594$	0	0
$595$	−2.44557e6	−0.283196
$596$	0	0
$597$	2.61778e6	0.300605
$598$	0	0
$599$	5.72775e6	0.652255	0.326127	−	0.945326i	$-0.394256\pi$
$599$	5.72775e6	0.652255	0.326127	+	0.945326i	$0.394256\pi$
$600$	0	0
$601$	−3.63309e6	−0.410290	−0.205145	−	0.978732i	$-0.565767\pi$
$601$	−3.63309e6	−0.410290	−0.205145	+	0.978732i	$0.565767\pi$
$602$	0	0
$603$	1.83935e6	0.206002
$604$	0	0
$605$	−712826.	−0.0791763
$606$	0	0
$607$	1.50751e7	1.66069	0.830343	−	0.557253i	$-0.188145\pi$
$607$	1.50751e7	1.66069	0.830343	+	0.557253i	$0.188145\pi$
$608$	0	0
$609$	593280.	0.0648211
$610$	0	0
$611$	1.36940e7	1.48397
$612$	0	0
$613$	−1.32798e7	−1.42739	−0.713694	−	0.700458i	$-0.752979\pi$
$613$	−1.32798e7	−1.42739	−0.713694	+	0.700458i	$0.752979\pi$
$614$	0	0
$615$	−2.97119e6	−0.316769
$616$	0	0
$617$	−9.67978e6	−1.02365	−0.511826	−	0.859089i	$-0.671031\pi$
$617$	−9.67978e6	−1.02365	−0.511826	+	0.859089i	$0.671031\pi$
$618$	0	0
$619$	−8.44042e6	−0.885396	−0.442698	−	0.896671i	$-0.645979\pi$
$619$	−8.44042e6	−0.885396	−0.442698	+	0.896671i	$0.645979\pi$
$620$	0	0
$621$	−385641.	−0.0401286
$622$	0	0
$623$	−7.24314e6	−0.747664
$624$	0	0
$625$	−8.05970e6	−0.825313
$626$	0	0
$627$	−756432.	−0.0768424
$628$	0	0
$629$	3.66369e6	0.369226
$630$	0	0
$631$	−4.29554e6	−0.429481	−0.214741	−	0.976671i	$-0.568891\pi$
$631$	−4.29554e6	−0.429481	−0.214741	+	0.976671i	$0.568891\pi$
$632$	0	0
$633$	4.67730e6	0.463966
$634$	0	0
$635$	−1.28307e7	−1.26275
$636$	0	0
$637$	−7.90624e6	−0.772008
$638$	0	0
$639$	4.80233e6	0.465264
$640$	0	0
$641$	4.36288e6	0.419400	0.209700	−	0.977766i	$-0.432751\pi$
$641$	4.36288e6	0.419400	0.209700	+	0.977766i	$0.432751\pi$
$642$	0	0
$643$	1.22816e7	1.17146	0.585732	−	0.810505i	$-0.300807\pi$
$643$	1.22816e7	1.17146	0.585732	+	0.810505i	$0.300807\pi$
$644$	0	0
$645$	4.83242e6	0.457368
$646$	0	0
$647$	1.17133e7	1.10007	0.550034	−	0.835142i	$-0.314615\pi$
$647$	1.17133e7	1.10007	0.550034	+	0.835142i	$0.314615\pi$
$648$	0	0
$649$	−1.48205e7	−1.38118
$650$	0	0
$651$	−3.27629e6	−0.302991
$652$	0	0
$653$	−1.92255e7	−1.76439	−0.882197	−	0.470881i	$-0.843936\pi$
$653$	−1.92255e7	−1.76439	−0.882197	+	0.470881i	$0.843936\pi$
$654$	0	0
$655$	−2.33513e7	−2.12671
$656$	0	0
$657$	−1.78508e6	−0.161341
$658$	0	0
$659$	1.11372e7	0.998991	0.499495	−	0.866317i	$-0.333519\pi$
$659$	1.11372e7	0.998991	0.499495	+	0.866317i	$0.333519\pi$
$660$	0	0
$661$	−2.07766e6	−0.184957	−0.0924784	−	0.995715i	$-0.529479\pi$
$661$	−2.07766e6	−0.184957	−0.0924784	+	0.995715i	$0.529479\pi$
$662$	0	0
$663$	2.60867e6	0.230481
$664$	0	0
$665$	1.07059e6	0.0938793
$666$	0	0
$667$	−544870.	−0.0474218
$668$	0	0
$669$	8.95010e6	0.773148
$670$	0	0
$671$	7.36244e6	0.631270
$672$	0	0
$673$	−1.18233e7	−1.00624	−0.503120	−	0.864216i	$-0.667815\pi$
$673$	−1.18233e7	−1.00624	−0.503120	+	0.864216i	$0.667815\pi$
$674$	0	0
$675$	2.62367e6	0.221641
$676$	0	0
$677$	194414.	0.0163026	0.00815128	−	0.999967i	$-0.497405\pi$
$677$	194414.	0.0163026	0.00815128	+	0.999967i	$0.497405\pi$
$678$	0	0
$679$	−4.80243e6	−0.399748
$680$	0	0
$681$	4.71503e6	0.389598
$682$	0	0
$683$	1.15464e7	0.947100	0.473550	−	0.880767i	$-0.342972\pi$
$683$	1.15464e7	0.947100	0.473550	+	0.880767i	$0.342972\pi$
$684$	0	0
$685$	−3.45589e7	−2.81406
$686$	0	0
$687$	−2.95425e6	−0.238812
$688$	0	0
$689$	3.09383e6	0.248284
$690$	0	0
$691$	−4.43088e6	−0.353016	−0.176508	−	0.984299i	$-0.556480\pi$
$691$	−4.43088e6	−0.353016	−0.176508	+	0.984299i	$0.556480\pi$
$692$	0	0
$693$	2.13581e6	0.168939
$694$	0	0
$695$	1.92566e7	1.51223
$696$	0	0
$697$	1.87612e6	0.146278
$698$	0	0
$699$	4.50700e6	0.348895
$700$	0	0
$701$	−1.27278e7	−0.978270	−0.489135	−	0.872208i	$-0.662688\pi$
$701$	−1.27278e7	−0.978270	−0.489135	+	0.872208i	$0.662688\pi$
$702$	0	0
$703$	−1.60385e6	−0.122398
$704$	0	0
$705$	−1.62478e7	−1.23118
$706$	0	0
$707$	−3.33530e6	−0.250949
$708$	0	0
$709$	−1.15842e7	−0.865463	−0.432732	−	0.901523i	$-0.642450\pi$
$709$	−1.15842e7	−0.865463	−0.432732	+	0.901523i	$0.642450\pi$
$710$	0	0
$711$	−6.93814e6	−0.514717
$712$	0	0
$713$	3.00895e6	0.221662
$714$	0	0
$715$	−2.10136e7	−1.53722
$716$	0	0
$717$	5.78030e6	0.419907
$718$	0	0
$719$	−7.56016e6	−0.545392	−0.272696	−	0.962100i	$-0.587915\pi$
$719$	−7.56016e6	−0.545392	−0.272696	+	0.962100i	$0.587915\pi$
$720$	0	0
$721$	4.42266e6	0.316844
$722$	0	0
$723$	−4.80339e6	−0.341745
$724$	0	0
$725$	3.70697e6	0.261923
$726$	0	0
$727$	1.64277e7	1.15276	0.576381	−	0.817181i	$-0.304464\pi$
$727$	1.64277e7	1.15276	0.576381	+	0.817181i	$0.304464\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	−3.05137e6	−0.211204
$732$	0	0
$733$	−8.91447e6	−0.612823	−0.306412	−	0.951899i	$-0.599128\pi$
$733$	−8.91447e6	−0.612823	−0.306412	+	0.951899i	$0.599128\pi$
$734$	0	0
$735$	9.38072e6	0.640498
$736$	0	0
$737$	9.35570e6	0.634465
$738$	0	0
$739$	−5.68060e6	−0.382634	−0.191317	−	0.981528i	$-0.561276\pi$
$739$	−5.68060e6	−0.382634	−0.191317	+	0.981528i	$0.561276\pi$
$740$	0	0
$741$	−1.14199e6	−0.0764043
$742$	0	0
$743$	2.20087e7	1.46259	0.731296	−	0.682060i	$-0.238916\pi$
$743$	2.20087e7	1.46259	0.731296	+	0.682060i	$0.238916\pi$
$744$	0	0
$745$	1.87895e6	0.124029
$746$	0	0
$747$	−4.78710e6	−0.313885
$748$	0	0
$749$	7.44730e6	0.485058
$750$	0	0
$751$	1.06320e7	0.687882	0.343941	−	0.938991i	$-0.388238\pi$
$751$	1.06320e7	0.687882	0.343941	+	0.938991i	$0.388238\pi$
$752$	0	0
$753$	−1.59208e7	−1.02324
$754$	0	0
$755$	−1.25217e7	−0.799460
$756$	0	0
$757$	−2.53289e7	−1.60648	−0.803242	−	0.595653i	$-0.796893\pi$
$757$	−2.53289e7	−1.60648	−0.803242	+	0.595653i	$0.796893\pi$
$758$	0	0
$759$	−1.96153e6	−0.123592
$760$	0	0
$761$	−1.51699e7	−0.949558	−0.474779	−	0.880105i	$-0.657472\pi$
$761$	−1.51699e7	−0.949558	−0.474779	+	0.880105i	$0.657472\pi$
$762$	0	0
$763$	5.91552e6	0.367859
$764$	0	0
$765$	−3.09517e6	−0.191219
$766$	0	0
$767$	−2.23746e7	−1.37330
$768$	0	0
$769$	1.33459e6	0.0813829	0.0406915	−	0.999172i	$-0.487044\pi$
$769$	1.33459e6	0.0813829	0.0406915	+	0.999172i	$0.487044\pi$
$770$	0	0
$771$	1.30504e7	0.790660
$772$	0	0
$773$	1.45025e7	0.872957	0.436479	−	0.899715i	$-0.356225\pi$
$773$	1.45025e7	0.872957	0.436479	+	0.899715i	$0.356225\pi$
$774$	0	0
$775$	−2.04711e7	−1.22430
$776$	0	0
$777$	4.52851e6	0.269093
$778$	0	0
$779$	−821304.	−0.0484909
$780$	0	0
$781$	2.44267e7	1.43297
$782$	0	0
$783$	750870.	0.0437684
$784$	0	0
$785$	−1.78129e7	−1.03171
$786$	0	0
$787$	2.05487e7	1.18263	0.591313	−	0.806442i	$-0.298610\pi$
$787$	2.05487e7	1.18263	0.591313	+	0.806442i	$0.298610\pi$
$788$	0	0
$789$	1.02998e7	0.589029
$790$	0	0
$791$	7.11283e6	0.404205
$792$	0	0
$793$	1.11151e7	0.627671
$794$	0	0
$795$	−3.67081e6	−0.205989
$796$	0	0
$797$	−2.90155e7	−1.61802	−0.809010	−	0.587795i	$-0.799996\pi$
$797$	−2.90155e7	−1.61802	−0.809010	+	0.587795i	$0.799996\pi$
$798$	0	0
$799$	1.02595e7	0.568535
$800$	0	0
$801$	−9.16709e6	−0.504836
$802$	0	0
$803$	−9.07966e6	−0.496913
$804$	0	0
$805$	2.77619e6	0.150994
$806$	0	0
$807$	−8.91945e6	−0.482119
$808$	0	0
$809$	2.91739e7	1.56720	0.783599	−	0.621267i	$-0.213382\pi$
$809$	2.91739e7	1.56720	0.783599	+	0.621267i	$0.213382\pi$
$810$	0	0
$811$	−2.92976e7	−1.56415	−0.782077	−	0.623182i	$-0.785840\pi$
$811$	−2.92976e7	−1.56415	−0.782077	+	0.623182i	$0.785840\pi$
$812$	0	0
$813$	3.43447e6	0.182236
$814$	0	0
$815$	1.32706e7	0.699834
$816$	0	0
$817$	1.33579e6	0.0700138
$818$	0	0
$819$	3.22445e6	0.167975
$820$	0	0
$821$	−7.24240e6	−0.374994	−0.187497	−	0.982265i	$-0.560038\pi$
$821$	−7.24240e6	−0.374994	−0.187497	+	0.982265i	$0.560038\pi$
$822$	0	0
$823$	−186400.	−0.00959282	−0.00479641	−	0.999988i	$-0.501527\pi$
$823$	−186400.	−0.00959282	−0.00479641	+	0.999988i	$0.501527\pi$
$824$	0	0
$825$	1.33451e7	0.682632
$826$	0	0
$827$	1.13311e7	0.576115	0.288058	−	0.957613i	$-0.406990\pi$
$827$	1.13311e7	0.576115	0.288058	+	0.957613i	$0.406990\pi$
$828$	0	0
$829$	3.18438e6	0.160931	0.0804653	−	0.996757i	$-0.474359\pi$
$829$	3.18438e6	0.160931	0.0804653	+	0.996757i	$0.474359\pi$
$830$	0	0
$831$	−2.40028e6	−0.120576
$832$	0	0
$833$	−5.92333e6	−0.295769
$834$	0	0
$835$	−3.85551e7	−1.91366
$836$	0	0
$837$	−4.14655e6	−0.204585
$838$	0	0
$839$	−1.59730e7	−0.783398	−0.391699	−	0.920093i	$-0.628113\pi$
$839$	−1.59730e7	−0.783398	−0.391699	+	0.920093i	$0.628113\pi$
$840$	0	0
$841$	−1.94502e7	−0.948277
$842$	0	0
$843$	1.73158e7	0.839216
$844$	0	0
$845$	−1.27846e6	−0.0615951
$846$	0	0
$847$	556352.	0.0266466
$848$	0	0
$849$	−1.44834e7	−0.689607
$850$	0	0
$851$	−4.15900e6	−0.196863
$852$	0	0
$853$	1.41773e7	0.667147	0.333573	−	0.942724i	$-0.391745\pi$
$853$	1.41773e7	0.667147	0.333573	+	0.942724i	$0.391745\pi$
$854$	0	0
$855$	1.35497e6	0.0633890
$856$	0	0
$857$	3.83069e7	1.78166	0.890829	−	0.454338i	$-0.150124\pi$
$857$	3.83069e7	1.78166	0.890829	+	0.454338i	$0.150124\pi$
$858$	0	0
$859$	−7.77458e6	−0.359496	−0.179748	−	0.983713i	$-0.557528\pi$
$859$	−7.77458e6	−0.359496	−0.179748	+	0.983713i	$0.557528\pi$
$860$	0	0
$861$	2.31898e6	0.106608
$862$	0	0
$863$	2.94086e6	0.134415	0.0672075	−	0.997739i	$-0.478591\pi$
$863$	2.94086e6	0.134415	0.0672075	+	0.997739i	$0.478591\pi$
$864$	0	0
$865$	−1.65099e6	−0.0750246
$866$	0	0
$867$	−1.08243e7	−0.489049
$868$	0	0
$869$	−3.52903e7	−1.58528
$870$	0	0
$871$	1.41244e7	0.630847
$872$	0	0
$873$	−6.07808e6	−0.269917
$874$	0	0
$875$	−2.48755e6	−0.109838
$876$	0	0
$877$	−3.03135e7	−1.33088	−0.665438	−	0.746453i	$-0.731755\pi$
$877$	−3.03135e7	−1.33088	−0.665438	+	0.746453i	$0.731755\pi$
$878$	0	0
$879$	−164466.	−0.00717966
$880$	0	0
$881$	2.47840e7	1.07580	0.537900	−	0.843009i	$-0.319218\pi$
$881$	2.47840e7	1.07580	0.537900	+	0.843009i	$0.319218\pi$
$882$	0	0
$883$	−9.88604e6	−0.426698	−0.213349	−	0.976976i	$-0.568437\pi$
$883$	−9.88604e6	−0.426698	−0.213349	+	0.976976i	$0.568437\pi$
$884$	0	0
$885$	2.65473e7	1.13936
$886$	0	0
$887$	1.57903e7	0.673878	0.336939	−	0.941526i	$-0.390608\pi$
$887$	1.57903e7	0.673878	0.336939	+	0.941526i	$0.390608\pi$
$888$	0	0
$889$	1.00142e7	0.424974
$890$	0	0
$891$	2.70313e6	0.114070
$892$	0	0
$893$	−4.49126e6	−0.188469
$894$	0	0
$895$	−4.08180e7	−1.70331
$896$	0	0
$897$	−2.96134e6	−0.122887
$898$	0	0
$899$	−5.85864e6	−0.241767
$900$	0	0
$901$	2.31788e6	0.0951217
$902$	0	0
$903$	−3.77165e6	−0.153926
$904$	0	0
$905$	2.07462e7	0.842008
$906$	0	0
$907$	1.75470e7	0.708245	0.354123	−	0.935199i	$-0.384780\pi$
$907$	1.75470e7	0.708245	0.354123	+	0.935199i	$0.384780\pi$
$908$	0	0
$909$	−4.22123e6	−0.169445
$910$	0	0
$911$	2.62301e7	1.04714	0.523570	−	0.851983i	$-0.324600\pi$
$911$	2.62301e7	1.04714	0.523570	+	0.851983i	$0.324600\pi$
$912$	0	0
$913$	−2.43492e7	−0.966736
$914$	0	0
$915$	−1.31881e7	−0.520749
$916$	0	0
$917$	1.82254e7	0.715738
$918$	0	0
$919$	−1.35236e7	−0.528208	−0.264104	−	0.964494i	$-0.585076\pi$
$919$	−1.35236e7	−0.528208	−0.264104	+	0.964494i	$0.585076\pi$
$920$	0	0
$921$	9.69800e6	0.376733
$922$	0	0
$923$	3.68771e7	1.42480
$924$	0	0
$925$	2.82953e7	1.08733
$926$	0	0
$927$	5.59742e6	0.213938
$928$	0	0
$929$	−15406.0	−0.000585667 0	−0.000292833	−	1.00000i	$-0.500093\pi$
$929$	−15406.0	−0.000585667 0	−0.000292833		1.00000i	$0.500093\pi$
$930$	0	0
$931$	2.59304e6	0.0980473
$932$	0	0
$933$	−1.47370e7	−0.554250
$934$	0	0
$935$	−1.57433e7	−0.588936
$936$	0	0
$937$	1.41659e6	0.0527101	0.0263551	−	0.999653i	$-0.491610\pi$
$937$	1.41659e6	0.0527101	0.0263551	+	0.999653i	$0.491610\pi$
$938$	0	0
$939$	2.83982e7	1.05106
$940$	0	0
$941$	−1.08997e7	−0.401273	−0.200637	−	0.979666i	$-0.564301\pi$
$941$	−1.08997e7	−0.401273	−0.200637	+	0.979666i	$0.564301\pi$
$942$	0	0
$943$	−2.12975e6	−0.0779920
$944$	0	0
$945$	−3.82579e6	−0.139361
$946$	0	0
$947$	−2.53470e7	−0.918440	−0.459220	−	0.888323i	$-0.651871\pi$
$947$	−2.53470e7	−0.918440	−0.459220	+	0.888323i	$0.651871\pi$
$948$	0	0
$949$	−1.37076e7	−0.494080
$950$	0	0
$951$	−2.11126e7	−0.756992
$952$	0	0
$953$	−5.28501e7	−1.88501	−0.942504	−	0.334195i	$-0.891536\pi$
$953$	−5.28501e7	−1.88501	−0.942504	+	0.334195i	$0.891536\pi$
$954$	0	0
$955$	−3.48651e7	−1.23704
$956$	0	0
$957$	3.81924e6	0.134802
$958$	0	0
$959$	2.69728e7	0.947064
$960$	0	0
$961$	3.72419e6	0.130084
$962$	0	0
$963$	9.42548e6	0.327520
$964$	0	0
$965$	−3.63478e7	−1.25649
$966$	0	0
$967$	8.77261e6	0.301691	0.150846	−	0.988557i	$-0.451800\pi$
$967$	8.77261e6	0.301691	0.150846	+	0.988557i	$0.451800\pi$
$968$	0	0
$969$	−855576.	−0.0292718
$970$	0	0
$971$	2.53929e7	0.864300	0.432150	−	0.901802i	$-0.357755\pi$
$971$	2.53929e7	0.864300	0.432150	+	0.901802i	$0.357755\pi$
$972$	0	0
$973$	−1.50295e7	−0.508935
$974$	0	0
$975$	2.01472e7	0.678740
$976$	0	0
$977$	4.23118e7	1.41816	0.709080	−	0.705128i	$-0.249110\pi$
$977$	4.23118e7	1.41816	0.709080	+	0.705128i	$0.249110\pi$
$978$	0	0
$979$	−4.66277e7	−1.55485
$980$	0	0
$981$	7.48683e6	0.248385
$982$	0	0
$983$	8.26108e6	0.272680	0.136340	−	0.990662i	$-0.456466\pi$
$983$	8.26108e6	0.272680	0.136340	+	0.990662i	$0.456466\pi$
$984$	0	0
$985$	−7.83116e6	−0.257179
$986$	0	0
$987$	1.26812e7	0.414351
$988$	0	0
$989$	3.46389e6	0.112609
$990$	0	0
$991$	−8.90652e6	−0.288087	−0.144044	−	0.989571i	$-0.546011\pi$
$991$	−8.90652e6	−0.288087	−0.144044	+	0.989571i	$0.546011\pi$
$992$	0	0
$993$	−1.00134e7	−0.322260
$994$	0	0
$995$	−2.38508e7	−0.763741
$996$	0	0
$997$	−2.05454e7	−0.654603	−0.327301	−	0.944920i	$-0.606139\pi$
$997$	−2.05454e7	−0.654603	−0.327301	+	0.944920i	$0.606139\pi$
$998$	0	0
$999$	5.73140e6	0.181697

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1104.6.a.b.1.1		1
4.3	odd	2		552.6.a.a.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
552.6.a.a.1.1	✓	1	4.3	odd	2
1104.6.a.b.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:	\( N \)	\(=\)	\( 1104 = 2^{4} \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1104.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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