LMFDB - Embedded newform 54.6.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [54,6,Mod(1,54)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(54, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("54.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 54 = 2 \cdot 3^{3} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	54.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:	$8.66072626990$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	54.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-4.00000 q^{2} +16.0000 q^{4} -24.0000 q^{5} +77.0000 q^{7} -64.0000 q^{8} +O(q^{10})$

\(q-4.00000 q^{2} +16.0000 q^{4} -24.0000 q^{5} +77.0000 q^{7} -64.0000 q^{8} +96.0000 q^{10} -408.000 q^{11} +89.0000 q^{13} -308.000 q^{14} +256.000 q^{16} -2088.00 q^{17} -2617.00 q^{19} -384.000 q^{20} +1632.00 q^{22} -1752.00 q^{23} -2549.00 q^{25} -356.000 q^{26} +1232.00 q^{28} +7296.00 q^{29} +2348.00 q^{31} -1024.00 q^{32} +8352.00 q^{34} -1848.00 q^{35} -4993.00 q^{37} +10468.0 q^{38} +1536.00 q^{40} +6528.00 q^{41} -6232.00 q^{43} -6528.00 q^{44} +7008.00 q^{46} +29832.0 q^{47} -10878.0 q^{49} +10196.0 q^{50} +1424.00 q^{52} -22608.0 q^{53} +9792.00 q^{55} -4928.00 q^{56} -29184.0 q^{58} -19608.0 q^{59} -22045.0 q^{61} -9392.00 q^{62} +4096.00 q^{64} -2136.00 q^{65} +48131.0 q^{67} -33408.0 q^{68} +7392.00 q^{70} -51120.0 q^{71} +30737.0 q^{73} +19972.0 q^{74} -41872.0 q^{76} -31416.0 q^{77} +38219.0 q^{79} -6144.00 q^{80} -26112.0 q^{82} -8112.00 q^{83} +50112.0 q^{85} +24928.0 q^{86} +26112.0 q^{88} +44280.0 q^{89} +6853.00 q^{91} -28032.0 q^{92} -119328. q^{94} +62808.0 q^{95} -136651. q^{97} +43512.0 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−4.00000	−0.707107
$2$	−4.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	16.0000	0.500000
$5$	−24.0000	−0.429325	−0.214663	−	0.976688i	$-0.568865\pi$
$5$	−24.0000	−0.429325	−0.214663	+	0.976688i	$0.568865\pi$
$6$	0	0
$7$	77.0000	0.593944	0.296972	−	0.954886i	$-0.404023\pi$
$7$	77.0000	0.593944	0.296972	+	0.954886i	$0.404023\pi$
$8$	−64.0000	−0.353553
$9$	0	0
$10$	96.0000	0.303579
$11$	−408.000	−1.01667	−0.508333	−	0.861160i	$-0.669738\pi$
$11$	−408.000	−1.01667	−0.508333	+	0.861160i	$0.669738\pi$
$12$	0	0
$13$	89.0000	0.146060	0.0730301	−	0.997330i	$-0.476733\pi$
$13$	89.0000	0.146060	0.0730301	+	0.997330i	$0.476733\pi$
$14$	−308.000	−0.419982
$15$	0	0
$16$	256.000	0.250000
$17$	−2088.00	−1.75230	−0.876149	−	0.482040i	$-0.839896\pi$
$17$	−2088.00	−1.75230	−0.876149	+	0.482040i	$0.839896\pi$
$18$	0	0
$19$	−2617.00	−1.66311	−0.831553	−	0.555446i	$-0.812548\pi$
$19$	−2617.00	−1.66311	−0.831553	+	0.555446i	$0.812548\pi$
$20$	−384.000	−0.214663
$21$	0	0
$22$	1632.00	0.718892
$23$	−1752.00	−0.690581	−0.345290	−	0.938496i	$-0.612220\pi$
$23$	−1752.00	−0.690581	−0.345290	+	0.938496i	$0.612220\pi$
$24$	0	0
$25$	−2549.00	−0.815680
$26$	−356.000	−0.103280
$27$	0	0
$28$	1232.00	0.296972
$29$	7296.00	1.61098	0.805489	−	0.592610i	$-0.201903\pi$
$29$	7296.00	1.61098	0.805489	+	0.592610i	$0.201903\pi$
$30$	0	0
$31$	2348.00	0.438828	0.219414	−	0.975632i	$-0.429586\pi$
$31$	2348.00	0.438828	0.219414	+	0.975632i	$0.429586\pi$
$32$	−1024.00	−0.176777
$33$	0	0
$34$	8352.00	1.23906
$35$	−1848.00	−0.254995
$36$	0	0
$37$	−4993.00	−0.599594	−0.299797	−	0.954003i	$-0.596919\pi$
$37$	−4993.00	−0.599594	−0.299797	+	0.954003i	$0.596919\pi$
$38$	10468.0	1.17599
$39$	0	0
$40$	1536.00	0.151789
$41$	6528.00	0.606486	0.303243	−	0.952913i	$-0.401931\pi$
$41$	6528.00	0.606486	0.303243	+	0.952913i	$0.401931\pi$
$42$	0	0
$43$	−6232.00	−0.513992	−0.256996	−	0.966412i	$-0.582733\pi$
$43$	−6232.00	−0.513992	−0.256996	+	0.966412i	$0.582733\pi$
$44$	−6528.00	−0.508333
$45$	0	0
$46$	7008.00	0.488314
$47$	29832.0	1.96987	0.984935	−	0.172923i	$-0.0553212\pi$
$47$	29832.0	1.96987	0.984935	+	0.172923i	$0.0553212\pi$
$48$	0	0
$49$	−10878.0	−0.647230
$50$	10196.0	0.576773
$51$	0	0
$52$	1424.00	0.0730301
$53$	−22608.0	−1.10553	−0.552767	−	0.833336i	$-0.686428\pi$
$53$	−22608.0	−1.10553	−0.552767	+	0.833336i	$0.686428\pi$
$54$	0	0
$55$	9792.00	0.436480
$56$	−4928.00	−0.209991
$57$	0	0
$58$	−29184.0	−1.13913
$59$	−19608.0	−0.733336	−0.366668	−	0.930352i	$-0.619502\pi$
$59$	−19608.0	−0.733336	−0.366668	+	0.930352i	$0.619502\pi$
$60$	0	0
$61$	−22045.0	−0.758552	−0.379276	−	0.925284i	$-0.623827\pi$
$61$	−22045.0	−0.758552	−0.379276	+	0.925284i	$0.623827\pi$
$62$	−9392.00	−0.310298
$63$	0	0
$64$	4096.00	0.125000
$65$	−2136.00	−0.0627073
$66$	0	0
$67$	48131.0	1.30990	0.654950	−	0.755673i	$-0.272690\pi$
$67$	48131.0	1.30990	0.654950	+	0.755673i	$0.272690\pi$
$68$	−33408.0	−0.876149
$69$	0	0
$70$	7392.00	0.180309
$71$	−51120.0	−1.20350	−0.601748	−	0.798686i	$-0.705529\pi$
$71$	−51120.0	−1.20350	−0.601748	+	0.798686i	$0.705529\pi$
$72$	0	0
$73$	30737.0	0.675079	0.337539	−	0.941311i	$-0.390405\pi$
$73$	30737.0	0.675079	0.337539	+	0.941311i	$0.390405\pi$
$74$	19972.0	0.423977
$75$	0	0
$76$	−41872.0	−0.831553
$77$	−31416.0	−0.603843
$78$	0	0
$79$	38219.0	0.688988	0.344494	−	0.938789i	$-0.388051\pi$
$79$	38219.0	0.688988	0.344494	+	0.938789i	$0.388051\pi$
$80$	−6144.00	−0.107331
$81$	0	0
$82$	−26112.0	−0.428850
$83$	−8112.00	−0.129251	−0.0646253	−	0.997910i	$-0.520585\pi$
$83$	−8112.00	−0.129251	−0.0646253	+	0.997910i	$0.520585\pi$
$84$	0	0
$85$	50112.0	0.752306
$86$	24928.0	0.363447
$87$	0	0
$88$	26112.0	0.359446
$89$	44280.0	0.592560	0.296280	−	0.955101i	$-0.404254\pi$
$89$	44280.0	0.592560	0.296280	+	0.955101i	$0.404254\pi$
$90$	0	0
$91$	6853.00	0.0867516
$92$	−28032.0	−0.345290
$93$	0	0
$94$	−119328.	−1.39291
$95$	62808.0	0.714013
$96$	0	0
$97$	−136651.	−1.47463	−0.737316	−	0.675548i	$-0.763907\pi$
$97$	−136651.	−1.47463	−0.737316	+	0.675548i	$0.763907\pi$
$98$	43512.0	0.457661
$99$	0	0
$100$	−40784.0	−0.407840
$101$	−23808.0	−0.232231	−0.116115	−	0.993236i	$-0.537044\pi$
$101$	−23808.0	−0.232231	−0.116115	+	0.993236i	$0.537044\pi$
$102$	0	0
$103$	173969.	1.61577	0.807884	−	0.589342i	$-0.200613\pi$
$103$	173969.	1.61577	0.807884	+	0.589342i	$0.200613\pi$
$104$	−5696.00	−0.0516400
$105$	0	0
$106$	90432.0	0.781731
$107$	188856.	1.59467	0.797336	−	0.603536i	$-0.206242\pi$
$107$	188856.	1.59467	0.797336	+	0.603536i	$0.206242\pi$
$108$	0	0
$109$	−208654.	−1.68213	−0.841067	−	0.540931i	$-0.818072\pi$
$109$	−208654.	−1.68213	−0.841067	+	0.540931i	$0.818072\pi$
$110$	−39168.0	−0.308638
$111$	0	0
$112$	19712.0	0.148486
$113$	155640.	1.14663	0.573317	−	0.819333i	$-0.305656\pi$
$113$	155640.	1.14663	0.573317	+	0.819333i	$0.305656\pi$
$114$	0	0
$115$	42048.0	0.296484
$116$	116736.	0.805489
$117$	0	0
$118$	78432.0	0.518547
$119$	−160776.	−1.04077
$120$	0	0
$121$	5413.00	0.0336105
$122$	88180.0	0.536377
$123$	0	0
$124$	37568.0	0.219414
$125$	136176.	0.779517
$126$	0	0
$127$	111332.	0.612507	0.306253	−	0.951950i	$-0.400925\pi$
$127$	111332.	0.612507	0.306253	+	0.951950i	$0.400925\pi$
$128$	−16384.0	−0.0883883
$129$	0	0
$130$	8544.00	0.0443407
$131$	−133152.	−0.677906	−0.338953	−	0.940803i	$-0.610073\pi$
$131$	−133152.	−0.677906	−0.338953	+	0.940803i	$0.610073\pi$
$132$	0	0
$133$	−201509.	−0.987792
$134$	−192524.	−0.926239
$135$	0	0
$136$	133632.	0.619531
$137$	−285816.	−1.30102	−0.650512	−	0.759496i	$-0.725446\pi$
$137$	−285816.	−1.30102	−0.650512	+	0.759496i	$0.725446\pi$
$138$	0	0
$139$	53009.0	0.232709	0.116354	−	0.993208i	$-0.462879\pi$
$139$	53009.0	0.232709	0.116354	+	0.993208i	$0.462879\pi$
$140$	−29568.0	−0.127498
$141$	0	0
$142$	204480.	0.851001
$143$	−36312.0	−0.148494
$144$	0	0
$145$	−175104.	−0.691634
$146$	−122948.	−0.477353
$147$	0	0
$148$	−79888.0	−0.299797
$149$	−210480.	−0.776685	−0.388343	−	0.921515i	$-0.626952\pi$
$149$	−210480.	−0.776685	−0.388343	+	0.921515i	$0.626952\pi$
$150$	0	0
$151$	−30721.0	−0.109646	−0.0548230	−	0.998496i	$-0.517459\pi$
$151$	−30721.0	−0.109646	−0.0548230	+	0.998496i	$0.517459\pi$
$152$	167488.	0.587996
$153$	0	0
$154$	125664.	0.426982
$155$	−56352.0	−0.188400
$156$	0	0
$157$	−453538.	−1.46847	−0.734234	−	0.678896i	$-0.762459\pi$
$157$	−453538.	−1.46847	−0.734234	+	0.678896i	$0.762459\pi$
$158$	−152876.	−0.487188
$159$	0	0
$160$	24576.0	0.0758947
$161$	−134904.	−0.410166
$162$	0	0
$163$	175241.	0.516615	0.258307	−	0.966063i	$-0.416835\pi$
$163$	175241.	0.516615	0.258307	+	0.966063i	$0.416835\pi$
$164$	104448.	0.303243
$165$	0	0
$166$	32448.0	0.0913940
$167$	56712.0	0.157356	0.0786781	−	0.996900i	$-0.474930\pi$
$167$	56712.0	0.157356	0.0786781	+	0.996900i	$0.474930\pi$
$168$	0	0
$169$	−363372.	−0.978666
$170$	−200448.	−0.531961
$171$	0	0
$172$	−99712.0	−0.256996
$173$	249216.	0.633083	0.316542	−	0.948579i	$-0.397478\pi$
$173$	249216.	0.633083	0.316542	+	0.948579i	$0.397478\pi$
$174$	0	0
$175$	−196273.	−0.484468
$176$	−104448.	−0.254167
$177$	0	0
$178$	−177120.	−0.419003
$179$	−177552.	−0.414184	−0.207092	−	0.978322i	$-0.566400\pi$
$179$	−177552.	−0.414184	−0.207092	+	0.978322i	$0.566400\pi$
$180$	0	0
$181$	453053.	1.02790	0.513952	−	0.857819i	$-0.328181\pi$
$181$	453053.	1.02790	0.513952	+	0.857819i	$0.328181\pi$
$182$	−27412.0	−0.0613426
$183$	0	0
$184$	112128.	0.244157
$185$	119832.	0.257421
$186$	0	0
$187$	851904.	1.78150
$188$	477312.	0.984935
$189$	0	0
$190$	−251232.	−0.504883
$191$	588408.	1.16707	0.583533	−	0.812090i	$-0.301670\pi$
$191$	588408.	1.16707	0.583533	+	0.812090i	$0.301670\pi$
$192$	0	0
$193$	586439.	1.13326	0.566630	−	0.823972i	$-0.308247\pi$
$193$	586439.	1.13326	0.566630	+	0.823972i	$0.308247\pi$
$194$	546604.	1.04272
$195$	0	0
$196$	−174048.	−0.323615
$197$	−374328.	−0.687206	−0.343603	−	0.939115i	$-0.611647\pi$
$197$	−374328.	−0.687206	−0.343603	+	0.939115i	$0.611647\pi$
$198$	0	0
$199$	−303415.	−0.543131	−0.271565	−	0.962420i	$-0.587541\pi$
$199$	−303415.	−0.543131	−0.271565	+	0.962420i	$0.587541\pi$
$200$	163136.	0.288386
$201$	0	0
$202$	95232.0	0.164212
$203$	561792.	0.956831
$204$	0	0
$205$	−156672.	−0.260379
$206$	−695876.	−1.14252
$207$	0	0
$208$	22784.0	0.0365150
$209$	1.06774e6	1.69082
$210$	0	0
$211$	229445.	0.354791	0.177395	−	0.984140i	$-0.443233\pi$
$211$	229445.	0.354791	0.177395	+	0.984140i	$0.443233\pi$
$212$	−361728.	−0.552767
$213$	0	0
$214$	−755424.	−1.12760
$215$	149568.	0.220670
$216$	0	0
$217$	180796.	0.260639
$218$	834616.	1.18945
$219$	0	0
$220$	156672.	0.218240
$221$	−185832.	−0.255941
$222$	0	0
$223$	1.06668e6	1.43638	0.718192	−	0.695845i	$-0.244970\pi$
$223$	1.06668e6	1.43638	0.718192	+	0.695845i	$0.244970\pi$
$224$	−78848.0	−0.104995
$225$	0	0
$226$	−622560.	−0.810793
$227$	−267888.	−0.345055	−0.172528	−	0.985005i	$-0.555193\pi$
$227$	−267888.	−0.345055	−0.172528	+	0.985005i	$0.555193\pi$
$228$	0	0
$229$	−1.01921e6	−1.28432	−0.642160	−	0.766571i	$-0.721962\pi$
$229$	−1.01921e6	−1.28432	−0.642160	+	0.766571i	$0.721962\pi$
$230$	−168192.	−0.209646
$231$	0	0
$232$	−466944.	−0.569567
$233$	−187488.	−0.226247	−0.113124	−	0.993581i	$-0.536086\pi$
$233$	−187488.	−0.226247	−0.113124	+	0.993581i	$0.536086\pi$
$234$	0	0
$235$	−715968.	−0.845715
$236$	−313728.	−0.366668
$237$	0	0
$238$	643104.	0.735934
$239$	−1.04107e6	−1.17892	−0.589462	−	0.807796i	$-0.700660\pi$
$239$	−1.04107e6	−1.17892	−0.589462	+	0.807796i	$0.700660\pi$
$240$	0	0
$241$	743585.	0.824685	0.412342	−	0.911029i	$-0.364711\pi$
$241$	743585.	0.824685	0.412342	+	0.911029i	$0.364711\pi$
$242$	−21652.0	−0.0237662
$243$	0	0
$244$	−352720.	−0.379276
$245$	261072.	0.277872
$246$	0	0
$247$	−232913.	−0.242913
$248$	−150272.	−0.155149
$249$	0	0
$250$	−544704.	−0.551202
$251$	−1.66608e6	−1.66921	−0.834606	−	0.550847i	$-0.814305\pi$
$251$	−1.66608e6	−1.66921	−0.834606	+	0.550847i	$0.814305\pi$
$252$	0	0
$253$	714816.	0.702090
$254$	−445328.	−0.433108
$255$	0	0
$256$	65536.0	0.0625000
$257$	−1.20120e6	−1.13444	−0.567221	−	0.823565i	$-0.691982\pi$
$257$	−1.20120e6	−1.13444	−0.567221	+	0.823565i	$0.691982\pi$
$258$	0	0
$259$	−384461.	−0.356125
$260$	−34176.0	−0.0313536
$261$	0	0
$262$	532608.	0.479352
$263$	163248.	0.145532	0.0727660	−	0.997349i	$-0.476817\pi$
$263$	163248.	0.145532	0.0727660	+	0.997349i	$0.476817\pi$
$264$	0	0
$265$	542592.	0.474634
$266$	806036.	0.698474
$267$	0	0
$268$	770096.	0.654950
$269$	970776.	0.817972	0.408986	−	0.912541i	$-0.365883\pi$
$269$	970776.	0.817972	0.408986	+	0.912541i	$0.365883\pi$
$270$	0	0
$271$	−828601.	−0.685365	−0.342683	−	0.939451i	$-0.611336\pi$
$271$	−828601.	−0.685365	−0.342683	+	0.939451i	$0.611336\pi$
$272$	−534528.	−0.438075
$273$	0	0
$274$	1.14326e6	0.919962
$275$	1.03999e6	0.829274
$276$	0	0
$277$	−616342.	−0.482639	−0.241319	−	0.970446i	$-0.577580\pi$
$277$	−616342.	−0.482639	−0.241319	+	0.970446i	$0.577580\pi$
$278$	−212036.	−0.164550
$279$	0	0
$280$	118272.	0.0901544
$281$	−2.61638e6	−1.97668	−0.988338	−	0.152273i	$-0.951341\pi$
$281$	−2.61638e6	−1.97668	−0.988338	+	0.152273i	$0.951341\pi$
$282$	0	0
$283$	−2.05034e6	−1.52180	−0.760902	−	0.648866i	$-0.775243\pi$
$283$	−2.05034e6	−1.52180	−0.760902	+	0.648866i	$0.775243\pi$
$284$	−817920.	−0.601748
$285$	0	0
$286$	145248.	0.105001
$287$	502656.	0.360219
$288$	0	0
$289$	2.93989e6	2.07055
$290$	700416.	0.489059
$291$	0	0
$292$	491792.	0.337539
$293$	1.64460e6	1.11916	0.559579	−	0.828777i	$-0.310963\pi$
$293$	1.64460e6	1.11916	0.559579	+	0.828777i	$0.310963\pi$
$294$	0	0
$295$	470592.	0.314840
$296$	319552.	0.211988
$297$	0	0
$298$	841920.	0.549200
$299$	−155928.	−0.100866
$300$	0	0
$301$	−479864.	−0.305283
$302$	122884.	0.0775315
$303$	0	0
$304$	−669952.	−0.415776
$305$	529080.	0.325665
$306$	0	0
$307$	−2.17154e6	−1.31499	−0.657493	−	0.753461i	$-0.728383\pi$
$307$	−2.17154e6	−1.31499	−0.657493	+	0.753461i	$0.728383\pi$
$308$	−502656.	−0.301922
$309$	0	0
$310$	225408.	0.133219
$311$	168312.	0.0986766	0.0493383	−	0.998782i	$-0.484289\pi$
$311$	168312.	0.0986766	0.0493383	+	0.998782i	$0.484289\pi$
$312$	0	0
$313$	−1.38371e6	−0.798333	−0.399166	−	0.916879i	$-0.630700\pi$
$313$	−1.38371e6	−0.798333	−0.399166	+	0.916879i	$0.630700\pi$
$314$	1.81415e6	1.03836
$315$	0	0
$316$	611504.	0.344494
$317$	−485616.	−0.271422	−0.135711	−	0.990748i	$-0.543332\pi$
$317$	−485616.	−0.271422	−0.135711	+	0.990748i	$0.543332\pi$
$318$	0	0
$319$	−2.97677e6	−1.63783
$320$	−98304.0	−0.0536656
$321$	0	0
$322$	539616.	0.290031
$323$	5.46430e6	2.91426
$324$	0	0
$325$	−226861.	−0.119138
$326$	−700964.	−0.365302
$327$	0	0
$328$	−417792.	−0.214425
$329$	2.29706e6	1.16999
$330$	0	0
$331$	1.14210e6	0.572975	0.286488	−	0.958084i	$-0.407512\pi$
$331$	1.14210e6	0.572975	0.286488	+	0.958084i	$0.407512\pi$
$332$	−129792.	−0.0646253
$333$	0	0
$334$	−226848.	−0.111268
$335$	−1.15514e6	−0.562373
$336$	0	0
$337$	2.38059e6	1.14185	0.570926	−	0.821002i	$-0.306584\pi$
$337$	2.38059e6	1.14185	0.570926	+	0.821002i	$0.306584\pi$
$338$	1.45349e6	0.692022
$339$	0	0
$340$	801792.	0.376153
$341$	−957984.	−0.446141
$342$	0	0
$343$	−2.13174e6	−0.978363
$344$	398848.	0.181724
$345$	0	0
$346$	−996864.	−0.447657
$347$	−3.15787e6	−1.40790	−0.703948	−	0.710251i	$-0.748581\pi$
$347$	−3.15787e6	−1.40790	−0.703948	+	0.710251i	$0.748581\pi$
$348$	0	0
$349$	−1.54337e6	−0.678276	−0.339138	−	0.940737i	$-0.610135\pi$
$349$	−1.54337e6	−0.678276	−0.339138	+	0.940737i	$0.610135\pi$
$350$	785092.	0.342571
$351$	0	0
$352$	417792.	0.179723
$353$	2.07701e6	0.887159	0.443579	−	0.896235i	$-0.353708\pi$
$353$	2.07701e6	0.887159	0.443579	+	0.896235i	$0.353708\pi$
$354$	0	0
$355$	1.22688e6	0.516691
$356$	708480.	0.296280
$357$	0	0
$358$	710208.	0.292872
$359$	−2.46362e6	−1.00888	−0.504439	−	0.863448i	$-0.668301\pi$
$359$	−2.46362e6	−1.00888	−0.504439	+	0.863448i	$0.668301\pi$
$360$	0	0
$361$	4.37259e6	1.76592
$362$	−1.81221e6	−0.726838
$363$	0	0
$364$	109648.	0.0433758
$365$	−737688.	−0.289828
$366$	0	0
$367$	813785.	0.315388	0.157694	−	0.987488i	$-0.449594\pi$
$367$	813785.	0.315388	0.157694	+	0.987488i	$0.449594\pi$
$368$	−448512.	−0.172645
$369$	0	0
$370$	−479328.	−0.182024
$371$	−1.74082e6	−0.656626
$372$	0	0
$373$	3.20436e6	1.19253	0.596265	−	0.802788i	$-0.296651\pi$
$373$	3.20436e6	1.19253	0.596265	+	0.802788i	$0.296651\pi$
$374$	−3.40762e6	−1.25971
$375$	0	0
$376$	−1.90925e6	−0.696454
$377$	649344.	0.235300
$378$	0	0
$379$	−1.94680e6	−0.696182	−0.348091	−	0.937461i	$-0.613170\pi$
$379$	−1.94680e6	−0.696182	−0.348091	+	0.937461i	$0.613170\pi$
$380$	1.00493e6	0.357006
$381$	0	0
$382$	−2.35363e6	−0.825240
$383$	3.12269e6	1.08776	0.543878	−	0.839164i	$-0.316955\pi$
$383$	3.12269e6	1.08776	0.543878	+	0.839164i	$0.316955\pi$
$384$	0	0
$385$	753984.	0.259245
$386$	−2.34576e6	−0.801336
$387$	0	0
$388$	−2.18642e6	−0.737316
$389$	−4.56809e6	−1.53060	−0.765298	−	0.643676i	$-0.777408\pi$
$389$	−4.56809e6	−1.53060	−0.765298	+	0.643676i	$0.777408\pi$
$390$	0	0
$391$	3.65818e6	1.21010
$392$	696192.	0.228830
$393$	0	0
$394$	1.49731e6	0.485928
$395$	−917256.	−0.295800
$396$	0	0
$397$	−3.27733e6	−1.04362	−0.521812	−	0.853061i	$-0.674744\pi$
$397$	−3.27733e6	−1.04362	−0.521812	+	0.853061i	$0.674744\pi$
$398$	1.21366e6	0.384051
$399$	0	0
$400$	−652544.	−0.203920
$401$	−5.44027e6	−1.68951	−0.844753	−	0.535157i	$-0.820253\pi$
$401$	−5.44027e6	−1.68951	−0.844753	+	0.535157i	$0.820253\pi$
$402$	0	0
$403$	208972.	0.0640952
$404$	−380928.	−0.116115
$405$	0	0
$406$	−2.24717e6	−0.676582
$407$	2.03714e6	0.609587
$408$	0	0
$409$	4.42722e6	1.30865	0.654325	−	0.756214i	$-0.272953\pi$
$409$	4.42722e6	1.30865	0.654325	+	0.756214i	$0.272953\pi$
$410$	626688.	0.184116
$411$	0	0
$412$	2.78350e6	0.807884
$413$	−1.50982e6	−0.435561
$414$	0	0
$415$	194688.	0.0554905
$416$	−91136.0	−0.0258200
$417$	0	0
$418$	−4.27094e6	−1.19559
$419$	−1.59583e6	−0.444071	−0.222035	−	0.975039i	$-0.571270\pi$
$419$	−1.59583e6	−0.444071	−0.222035	+	0.975039i	$0.571270\pi$
$420$	0	0
$421$	−6.04537e6	−1.66233	−0.831165	−	0.556025i	$-0.812326\pi$
$421$	−6.04537e6	−1.66233	−0.831165	+	0.556025i	$0.812326\pi$
$422$	−917780.	−0.250875
$423$	0	0
$424$	1.44691e6	0.390866
$425$	5.32231e6	1.42932
$426$	0	0
$427$	−1.69746e6	−0.450538
$428$	3.02170e6	0.797336
$429$	0	0
$430$	−598272.	−0.156037
$431$	−3.92220e6	−1.01704	−0.508518	−	0.861051i	$-0.669807\pi$
$431$	−3.92220e6	−1.01704	−0.508518	+	0.861051i	$0.669807\pi$
$432$	0	0
$433$	4.74163e6	1.21537	0.607685	−	0.794178i	$-0.292098\pi$
$433$	4.74163e6	1.21537	0.607685	+	0.794178i	$0.292098\pi$
$434$	−723184.	−0.184300
$435$	0	0
$436$	−3.33846e6	−0.841067
$437$	4.58498e6	1.14851
$438$	0	0
$439$	425588.	0.105397	0.0526985	−	0.998610i	$-0.483218\pi$
$439$	425588.	0.105397	0.0526985	+	0.998610i	$0.483218\pi$
$440$	−626688.	−0.154319
$441$	0	0
$442$	743328.	0.180978
$443$	2.60232e6	0.630016	0.315008	−	0.949089i	$-0.397993\pi$
$443$	2.60232e6	0.630016	0.315008	+	0.949089i	$0.397993\pi$
$444$	0	0
$445$	−1.06272e6	−0.254401
$446$	−4.26670e6	−1.01568
$447$	0	0
$448$	315392.	0.0742430
$449$	−167256.	−0.0391531	−0.0195765	−	0.999808i	$-0.506232\pi$
$449$	−167256.	−0.0391531	−0.0195765	+	0.999808i	$0.506232\pi$
$450$	0	0
$451$	−2.66342e6	−0.616594
$452$	2.49024e6	0.573317
$453$	0	0
$454$	1.07155e6	0.243991
$455$	−164472.	−0.0372446
$456$	0	0
$457$	1.18146e6	0.264624	0.132312	−	0.991208i	$-0.457760\pi$
$457$	1.18146e6	0.264624	0.132312	+	0.991208i	$0.457760\pi$
$458$	4.07682e6	0.908151
$459$	0	0
$460$	672768.	0.148242
$461$	5.61660e6	1.23090	0.615448	−	0.788178i	$-0.288975\pi$
$461$	5.61660e6	1.23090	0.615448	+	0.788178i	$0.288975\pi$
$462$	0	0
$463$	−1.12567e6	−0.244038	−0.122019	−	0.992528i	$-0.538937\pi$
$463$	−1.12567e6	−0.244038	−0.122019	+	0.992528i	$0.538937\pi$
$464$	1.86778e6	0.402745
$465$	0	0
$466$	749952.	0.159981
$467$	−4.74732e6	−1.00729	−0.503647	−	0.863910i	$-0.668009\pi$
$467$	−4.74732e6	−1.00729	−0.503647	+	0.863910i	$0.668009\pi$
$468$	0	0
$469$	3.70609e6	0.778007
$470$	2.86387e6	0.598011
$471$	0	0
$472$	1.25491e6	0.259274
$473$	2.54266e6	0.522558
$474$	0	0
$475$	6.67073e6	1.35656
$476$	−2.57242e6	−0.520384
$477$	0	0
$478$	4.16429e6	0.833626
$479$	9.78072e6	1.94774	0.973872	−	0.227096i	$-0.0729231\pi$
$479$	9.78072e6	1.94774	0.973872	+	0.227096i	$0.0729231\pi$
$480$	0	0
$481$	−444377.	−0.0875768
$482$	−2.97434e6	−0.583140
$483$	0	0
$484$	86608.0	0.0168052
$485$	3.27962e6	0.633096
$486$	0	0
$487$	−2.34782e6	−0.448582	−0.224291	−	0.974522i	$-0.572007\pi$
$487$	−2.34782e6	−0.448582	−0.224291	+	0.974522i	$0.572007\pi$
$488$	1.41088e6	0.268189
$489$	0	0
$490$	−1.04429e6	−0.196485
$491$	4.79762e6	0.898095	0.449048	−	0.893508i	$-0.351763\pi$
$491$	4.79762e6	0.898095	0.449048	+	0.893508i	$0.351763\pi$
$492$	0	0
$493$	−1.52340e7	−2.82292
$494$	931652.	0.171766
$495$	0	0
$496$	601088.	0.109707
$497$	−3.93624e6	−0.714810
$498$	0	0
$499$	7.03722e6	1.26517	0.632586	−	0.774490i	$-0.281994\pi$
$499$	7.03722e6	1.26517	0.632586	+	0.774490i	$0.281994\pi$
$500$	2.17882e6	0.389758
$501$	0	0
$502$	6.66432e6	1.18031
$503$	3.98858e6	0.702908	0.351454	−	0.936205i	$-0.385687\pi$
$503$	3.98858e6	0.702908	0.351454	+	0.936205i	$0.385687\pi$
$504$	0	0
$505$	571392.	0.0997024
$506$	−2.85926e6	−0.496453
$507$	0	0
$508$	1.78131e6	0.306253
$509$	−1.07277e7	−1.83533	−0.917664	−	0.397358i	$-0.869927\pi$
$509$	−1.07277e7	−1.83533	−0.917664	+	0.397358i	$0.869927\pi$
$510$	0	0
$511$	2.36675e6	0.400959
$512$	−262144.	−0.0441942
$513$	0	0
$514$	4.80480e6	0.802172
$515$	−4.17526e6	−0.693689
$516$	0	0
$517$	−1.21715e7	−2.00270
$518$	1.53784e6	0.251819
$519$	0	0
$520$	136704.	0.0221704
$521$	−5.86332e6	−0.946345	−0.473172	−	0.880970i	$-0.656891\pi$
$521$	−5.86332e6	−0.946345	−0.473172	+	0.880970i	$0.656891\pi$
$522$	0	0
$523$	−2.31968e6	−0.370830	−0.185415	−	0.982660i	$-0.559363\pi$
$523$	−2.31968e6	−0.370830	−0.185415	+	0.982660i	$0.559363\pi$
$524$	−2.13043e6	−0.338953
$525$	0	0
$526$	−652992.	−0.102907
$527$	−4.90262e6	−0.768957
$528$	0	0
$529$	−3.36684e6	−0.523098
$530$	−2.17037e6	−0.335617
$531$	0	0
$532$	−3.22414e6	−0.493896
$533$	580992.	0.0885834
$534$	0	0
$535$	−4.53254e6	−0.684633
$536$	−3.08038e6	−0.463119
$537$	0	0
$538$	−3.88310e6	−0.578393
$539$	4.43822e6	0.658017
$540$	0	0
$541$	−3.26629e6	−0.479801	−0.239901	−	0.970797i	$-0.577115\pi$
$541$	−3.26629e6	−0.479801	−0.239901	+	0.970797i	$0.577115\pi$
$542$	3.31440e6	0.484627
$543$	0	0
$544$	2.13811e6	0.309766
$545$	5.00770e6	0.722182
$546$	0	0
$547$	4.31680e6	0.616870	0.308435	−	0.951245i	$-0.400195\pi$
$547$	4.31680e6	0.616870	0.308435	+	0.951245i	$0.400195\pi$
$548$	−4.57306e6	−0.650512
$549$	0	0
$550$	−4.15997e6	−0.586386
$551$	−1.90936e7	−2.67923
$552$	0	0
$553$	2.94286e6	0.409220
$554$	2.46537e6	0.341277
$555$	0	0
$556$	848144.	0.116354
$557$	−5.78052e6	−0.789458	−0.394729	−	0.918798i	$-0.629161\pi$
$557$	−5.78052e6	−0.789458	−0.394729	+	0.918798i	$0.629161\pi$
$558$	0	0
$559$	−554648.	−0.0750737
$560$	−473088.	−0.0637488
$561$	0	0
$562$	1.04655e7	1.39772
$563$	4.95725e6	0.659128	0.329564	−	0.944133i	$-0.393098\pi$
$563$	4.95725e6	0.659128	0.329564	+	0.944133i	$0.393098\pi$
$564$	0	0
$565$	−3.73536e6	−0.492279
$566$	8.20134e6	1.07608
$567$	0	0
$568$	3.27168e6	0.425500
$569$	−3.33502e6	−0.431834	−0.215917	−	0.976412i	$-0.569274\pi$
$569$	−3.33502e6	−0.431834	−0.215917	+	0.976412i	$0.569274\pi$
$570$	0	0
$571$	5.56927e6	0.714839	0.357419	−	0.933944i	$-0.383657\pi$
$571$	5.56927e6	0.714839	0.357419	+	0.933944i	$0.383657\pi$
$572$	−580992.	−0.0742472
$573$	0	0
$574$	−2.01062e6	−0.254713
$575$	4.46585e6	0.563293
$576$	0	0
$577$	−226861.	−0.0283675	−0.0141837	−	0.999899i	$-0.504515\pi$
$577$	−226861.	−0.0283675	−0.0141837	+	0.999899i	$0.504515\pi$
$578$	−1.17595e7	−1.46410
$579$	0	0
$580$	−2.80166e6	−0.345817
$581$	−624624.	−0.0767677
$582$	0	0
$583$	9.22406e6	1.12396
$584$	−1.96717e6	−0.238676
$585$	0	0
$586$	−6.57840e6	−0.791364
$587$	−1.95550e6	−0.234240	−0.117120	−	0.993118i	$-0.537366\pi$
$587$	−1.95550e6	−0.234240	−0.117120	+	0.993118i	$0.537366\pi$
$588$	0	0
$589$	−6.14472e6	−0.729816
$590$	−1.88237e6	−0.222625
$591$	0	0
$592$	−1.27821e6	−0.149898
$593$	9.83938e6	1.14903	0.574514	−	0.818495i	$-0.305191\pi$
$593$	9.83938e6	1.14903	0.574514	+	0.818495i	$0.305191\pi$
$594$	0	0
$595$	3.85862e6	0.446828
$596$	−3.36768e6	−0.388343
$597$	0	0
$598$	623712.	0.0713233
$599$	−4.45862e6	−0.507731	−0.253866	−	0.967240i	$-0.581702\pi$
$599$	−4.45862e6	−0.507731	−0.253866	+	0.967240i	$0.581702\pi$
$600$	0	0
$601$	1.20944e7	1.36583	0.682917	−	0.730496i	$-0.260711\pi$
$601$	1.20944e7	1.36583	0.682917	+	0.730496i	$0.260711\pi$
$602$	1.91946e6	0.215867
$603$	0	0
$604$	−491536.	−0.0548230
$605$	−129912.	−0.0144298
$606$	0	0
$607$	3.54029e6	0.390003	0.195001	−	0.980803i	$-0.437529\pi$
$607$	3.54029e6	0.390003	0.195001	+	0.980803i	$0.437529\pi$
$608$	2.67981e6	0.293998
$609$	0	0
$610$	−2.11632e6	−0.230280
$611$	2.65505e6	0.287720
$612$	0	0
$613$	−1.30110e7	−1.39849	−0.699243	−	0.714884i	$-0.746480\pi$
$613$	−1.30110e7	−1.39849	−0.699243	+	0.714884i	$0.746480\pi$
$614$	8.68614e6	0.929835
$615$	0	0
$616$	2.01062e6	0.213491
$617$	3.36842e6	0.356216	0.178108	−	0.984011i	$-0.443002\pi$
$617$	3.36842e6	0.356216	0.178108	+	0.984011i	$0.443002\pi$
$618$	0	0
$619$	4.16277e6	0.436673	0.218336	−	0.975874i	$-0.429937\pi$
$619$	4.16277e6	0.436673	0.218336	+	0.975874i	$0.429937\pi$
$620$	−901632.	−0.0941998
$621$	0	0
$622$	−673248.	−0.0697749
$623$	3.40956e6	0.351948
$624$	0	0
$625$	4.69740e6	0.481014
$626$	5.53484e6	0.564506
$627$	0	0
$628$	−7.25661e6	−0.734234
$629$	1.04254e7	1.05067
$630$	0	0
$631$	1.65343e7	1.65315	0.826577	−	0.562823i	$-0.190285\pi$
$631$	1.65343e7	1.65315	0.826577	+	0.562823i	$0.190285\pi$
$632$	−2.44602e6	−0.243594
$633$	0	0
$634$	1.94246e6	0.191924
$635$	−2.67197e6	−0.262964
$636$	0	0
$637$	−968142.	−0.0945345
$638$	1.19071e7	1.15812
$639$	0	0
$640$	393216.	0.0379473
$641$	−8.10379e6	−0.779010	−0.389505	−	0.921024i	$-0.627354\pi$
$641$	−8.10379e6	−0.779010	−0.389505	+	0.921024i	$0.627354\pi$
$642$	0	0
$643$	−6.59706e6	−0.629250	−0.314625	−	0.949216i	$-0.601879\pi$
$643$	−6.59706e6	−0.629250	−0.314625	+	0.949216i	$0.601879\pi$
$644$	−2.15846e6	−0.205083
$645$	0	0
$646$	−2.18572e7	−2.06069
$647$	1.06116e7	0.996603	0.498301	−	0.867004i	$-0.333957\pi$
$647$	1.06116e7	0.996603	0.498301	+	0.867004i	$0.333957\pi$
$648$	0	0
$649$	8.00006e6	0.745558
$650$	907444.	0.0842435
$651$	0	0
$652$	2.80386e6	0.258307
$653$	1.14480e7	1.05063	0.525313	−	0.850909i	$-0.323948\pi$
$653$	1.14480e7	1.05063	0.525313	+	0.850909i	$0.323948\pi$
$654$	0	0
$655$	3.19565e6	0.291042
$656$	1.67117e6	0.151621
$657$	0	0
$658$	−9.18826e6	−0.827310
$659$	1.03409e7	0.927564	0.463782	−	0.885949i	$-0.346492\pi$
$659$	1.03409e7	0.927564	0.463782	+	0.885949i	$0.346492\pi$
$660$	0	0
$661$	−3.21066e6	−0.285819	−0.142909	−	0.989736i	$-0.545646\pi$
$661$	−3.21066e6	−0.285819	−0.142909	+	0.989736i	$0.545646\pi$
$662$	−4.56842e6	−0.405155
$663$	0	0
$664$	519168.	0.0456970
$665$	4.83622e6	0.424084
$666$	0	0
$667$	−1.27826e7	−1.11251
$668$	907392.	0.0786781
$669$	0	0
$670$	4.62058e6	0.397657
$671$	8.99436e6	0.771195
$672$	0	0
$673$	7.24765e6	0.616822	0.308411	−	0.951253i	$-0.400203\pi$
$673$	7.24765e6	0.616822	0.308411	+	0.951253i	$0.400203\pi$
$674$	−9.52236e6	−0.807411
$675$	0	0
$676$	−5.81395e6	−0.489333
$677$	−2.86630e6	−0.240353	−0.120176	−	0.992753i	$-0.538346\pi$
$677$	−2.86630e6	−0.240353	−0.120176	+	0.992753i	$0.538346\pi$
$678$	0	0
$679$	−1.05221e7	−0.875849
$680$	−3.20717e6	−0.265980
$681$	0	0
$682$	3.83194e6	0.315469
$683$	−2.12852e7	−1.74593	−0.872964	−	0.487785i	$-0.837805\pi$
$683$	−2.12852e7	−1.74593	−0.872964	+	0.487785i	$0.837805\pi$
$684$	0	0
$685$	6.85958e6	0.558562
$686$	8.52698e6	0.691807
$687$	0	0
$688$	−1.59539e6	−0.128498
$689$	−2.01211e6	−0.161475
$690$	0	0
$691$	208040.	0.0165749	0.00828747	−	0.999966i	$-0.497362\pi$
$691$	208040.	0.0165749	0.00828747	+	0.999966i	$0.497362\pi$
$692$	3.98746e6	0.316542
$693$	0	0
$694$	1.26315e7	0.995533
$695$	−1.27222e6	−0.0999077
$696$	0	0
$697$	−1.36305e7	−1.06274
$698$	6.17348e6	0.479613
$699$	0	0
$700$	−3.14037e6	−0.242234
$701$	−952488.	−0.0732090	−0.0366045	−	0.999330i	$-0.511654\pi$
$701$	−952488.	−0.0732090	−0.0366045	+	0.999330i	$0.511654\pi$
$702$	0	0
$703$	1.30667e7	0.997188
$704$	−1.67117e6	−0.127083
$705$	0	0
$706$	−8.30803e6	−0.627316
$707$	−1.83322e6	−0.137932
$708$	0	0
$709$	−8.96432e6	−0.669733	−0.334867	−	0.942266i	$-0.608691\pi$
$709$	−8.96432e6	−0.669733	−0.334867	+	0.942266i	$0.608691\pi$
$710$	−4.90752e6	−0.365356
$711$	0	0
$712$	−2.83392e6	−0.209502
$713$	−4.11370e6	−0.303046
$714$	0	0
$715$	871488.	0.0637524
$716$	−2.84083e6	−0.207092
$717$	0	0
$718$	9.85450e6	0.713384
$719$	−1.33824e7	−0.965407	−0.482703	−	0.875784i	$-0.660345\pi$
$719$	−1.33824e7	−0.965407	−0.482703	+	0.875784i	$0.660345\pi$
$720$	0	0
$721$	1.33956e7	0.959676
$722$	−1.74904e7	−1.24869
$723$	0	0
$724$	7.24885e6	0.513952
$725$	−1.85975e7	−1.31404
$726$	0	0
$727$	6.78844e6	0.476359	0.238179	−	0.971221i	$-0.423449\pi$
$727$	6.78844e6	0.476359	0.238179	+	0.971221i	$0.423449\pi$
$728$	−438592.	−0.0306713
$729$	0	0
$730$	2.95075e6	0.204939
$731$	1.30124e7	0.900668
$732$	0	0
$733$	5.21059e6	0.358201	0.179101	−	0.983831i	$-0.442681\pi$
$733$	5.21059e6	0.358201	0.179101	+	0.983831i	$0.442681\pi$
$734$	−3.25514e6	−0.223013
$735$	0	0
$736$	1.79405e6	0.122079
$737$	−1.96374e7	−1.33173
$738$	0	0
$739$	6.17470e6	0.415915	0.207958	−	0.978138i	$-0.433318\pi$
$739$	6.17470e6	0.415915	0.207958	+	0.978138i	$0.433318\pi$
$740$	1.91731e6	0.128710
$741$	0	0
$742$	6.96326e6	0.464305
$743$	309288.	0.0205537	0.0102769	−	0.999947i	$-0.496729\pi$
$743$	309288.	0.0205537	0.0102769	+	0.999947i	$0.496729\pi$
$744$	0	0
$745$	5.05152e6	0.333451
$746$	−1.28174e7	−0.843246
$747$	0	0
$748$	1.36305e7	0.890752
$749$	1.45419e7	0.947146
$750$	0	0
$751$	−9.42649e6	−0.609888	−0.304944	−	0.952370i	$-0.598638\pi$
$751$	−9.42649e6	−0.609888	−0.304944	+	0.952370i	$0.598638\pi$
$752$	7.63699e6	0.492468
$753$	0	0
$754$	−2.59738e6	−0.166382
$755$	737304.	0.0470738
$756$	0	0
$757$	−3.03790e7	−1.92679	−0.963393	−	0.268095i	$-0.913606\pi$
$757$	−3.03790e7	−1.92679	−0.963393	+	0.268095i	$0.913606\pi$
$758$	7.78719e6	0.492275
$759$	0	0
$760$	−4.01971e6	−0.252442
$761$	−1.28934e7	−0.807057	−0.403528	−	0.914967i	$-0.632216\pi$
$761$	−1.28934e7	−0.807057	−0.403528	+	0.914967i	$0.632216\pi$
$762$	0	0
$763$	−1.60664e7	−0.999093
$764$	9.41453e6	0.583533
$765$	0	0
$766$	−1.24908e7	−0.769160
$767$	−1.74511e6	−0.107111
$768$	0	0
$769$	2.42712e6	0.148005	0.0740023	−	0.997258i	$-0.476423\pi$
$769$	2.42712e6	0.148005	0.0740023	+	0.997258i	$0.476423\pi$
$770$	−3.01594e6	−0.183314
$771$	0	0
$772$	9.38302e6	0.566630
$773$	−1.14496e7	−0.689193	−0.344597	−	0.938751i	$-0.611984\pi$
$773$	−1.14496e7	−0.689193	−0.344597	+	0.938751i	$0.611984\pi$
$774$	0	0
$775$	−5.98505e6	−0.357943
$776$	8.74566e6	0.521361
$777$	0	0
$778$	1.82724e7	1.08229
$779$	−1.70838e7	−1.00865
$780$	0	0
$781$	2.08570e7	1.22355
$782$	−1.46327e7	−0.855673
$783$	0	0
$784$	−2.78477e6	−0.161808
$785$	1.08849e7	0.630450
$786$	0	0
$787$	3.22920e6	0.185848	0.0929240	−	0.995673i	$-0.470379\pi$
$787$	3.22920e6	0.185848	0.0929240	+	0.995673i	$0.470379\pi$
$788$	−5.98925e6	−0.343603
$789$	0	0
$790$	3.66902e6	0.209162
$791$	1.19843e7	0.681037
$792$	0	0
$793$	−1.96200e6	−0.110794
$794$	1.31093e7	0.737953
$795$	0	0
$796$	−4.85464e6	−0.271565
$797$	2.14248e7	1.19473	0.597367	−	0.801968i	$-0.296214\pi$
$797$	2.14248e7	1.19473	0.597367	+	0.801968i	$0.296214\pi$
$798$	0	0
$799$	−6.22892e7	−3.45180
$800$	2.61018e6	0.144193
$801$	0	0
$802$	2.17611e7	1.19466
$803$	−1.25407e7	−0.686330
$804$	0	0
$805$	3.23770e6	0.176095
$806$	−835888.	−0.0453221
$807$	0	0
$808$	1.52371e6	0.0821059
$809$	−1.55614e7	−0.835942	−0.417971	−	0.908460i	$-0.637259\pi$
$809$	−1.55614e7	−0.835942	−0.417971	+	0.908460i	$0.637259\pi$
$810$	0	0
$811$	−1.91972e7	−1.02491	−0.512454	−	0.858715i	$-0.671263\pi$
$811$	−1.91972e7	−1.02491	−0.512454	+	0.858715i	$0.671263\pi$
$812$	8.98867e6	0.478416
$813$	0	0
$814$	−8.14858e6	−0.431043
$815$	−4.20578e6	−0.221796
$816$	0	0
$817$	1.63091e7	0.854823
$818$	−1.77089e7	−0.925355
$819$	0	0
$820$	−2.50675e6	−0.130190
$821$	2.36303e7	1.22352	0.611760	−	0.791044i	$-0.290462\pi$
$821$	2.36303e7	1.22352	0.611760	+	0.791044i	$0.290462\pi$
$822$	0	0
$823$	−1.67796e6	−0.0863540	−0.0431770	−	0.999067i	$-0.513748\pi$
$823$	−1.67796e6	−0.0863540	−0.0431770	+	0.999067i	$0.513748\pi$
$824$	−1.11340e7	−0.571260
$825$	0	0
$826$	6.03926e6	0.307988
$827$	1.39053e7	0.706995	0.353497	−	0.935436i	$-0.384992\pi$
$827$	1.39053e7	0.706995	0.353497	+	0.935436i	$0.384992\pi$
$828$	0	0
$829$	−1.33464e7	−0.674493	−0.337247	−	0.941416i	$-0.609496\pi$
$829$	−1.33464e7	−0.674493	−0.337247	+	0.941416i	$0.609496\pi$
$830$	−778752.	−0.0392377
$831$	0	0
$832$	364544.	0.0182575
$833$	2.27133e7	1.13414
$834$	0	0
$835$	−1.36109e6	−0.0675569
$836$	1.70838e7	0.845412
$837$	0	0
$838$	6.38333e6	0.314005
$839$	2.95225e7	1.44793	0.723967	−	0.689834i	$-0.242317\pi$
$839$	2.95225e7	1.44793	0.723967	+	0.689834i	$0.242317\pi$
$840$	0	0
$841$	3.27205e7	1.59525
$842$	2.41815e7	1.17545
$843$	0	0
$844$	3.67112e6	0.177395
$845$	8.72093e6	0.420166
$846$	0	0
$847$	416801.	0.0199627
$848$	−5.78765e6	−0.276384
$849$	0	0
$850$	−2.12892e7	−1.01068
$851$	8.74774e6	0.414068
$852$	0	0
$853$	1.23083e7	0.579195	0.289597	−	0.957149i	$-0.406479\pi$
$853$	1.23083e7	0.579195	0.289597	+	0.957149i	$0.406479\pi$
$854$	6.78986e6	0.318578
$855$	0	0
$856$	−1.20868e7	−0.563802
$857$	1.65890e7	0.771559	0.385780	−	0.922591i	$-0.373933\pi$
$857$	1.65890e7	0.771559	0.385780	+	0.922591i	$0.373933\pi$
$858$	0	0
$859$	1.31286e7	0.607066	0.303533	−	0.952821i	$-0.401834\pi$
$859$	1.31286e7	0.607066	0.303533	+	0.952821i	$0.401834\pi$
$860$	2.39309e6	0.110335
$861$	0	0
$862$	1.56888e7	0.719153
$863$	−2.09802e7	−0.958918	−0.479459	−	0.877564i	$-0.659167\pi$
$863$	−2.09802e7	−0.958918	−0.479459	+	0.877564i	$0.659167\pi$
$864$	0	0
$865$	−5.98118e6	−0.271798
$866$	−1.89665e7	−0.859396
$867$	0	0
$868$	2.89274e6	0.130320
$869$	−1.55934e7	−0.700471
$870$	0	0
$871$	4.28366e6	0.191324
$872$	1.33539e7	0.594724
$873$	0	0
$874$	−1.83399e7	−0.812118
$875$	1.04856e7	0.462990
$876$	0	0
$877$	4.73882e6	0.208052	0.104026	−	0.994575i	$-0.466828\pi$
$877$	4.73882e6	0.208052	0.104026	+	0.994575i	$0.466828\pi$
$878$	−1.70235e6	−0.0745269
$879$	0	0
$880$	2.50675e6	0.109120
$881$	3.09880e7	1.34510	0.672549	−	0.740053i	$-0.265199\pi$
$881$	3.09880e7	1.34510	0.672549	+	0.740053i	$0.265199\pi$
$882$	0	0
$883$	−1.94710e6	−0.0840402	−0.0420201	−	0.999117i	$-0.513379\pi$
$883$	−1.94710e6	−0.0840402	−0.0420201	+	0.999117i	$0.513379\pi$
$884$	−2.97331e6	−0.127970
$885$	0	0
$886$	−1.04093e7	−0.445488
$887$	1.85254e7	0.790601	0.395301	−	0.918552i	$-0.370640\pi$
$887$	1.85254e7	0.790601	0.395301	+	0.918552i	$0.370640\pi$
$888$	0	0
$889$	8.57256e6	0.363795
$890$	4.25088e6	0.179889
$891$	0	0
$892$	1.70668e7	0.718192
$893$	−7.80703e7	−3.27610
$894$	0	0
$895$	4.26125e6	0.177819
$896$	−1.26157e6	−0.0524977
$897$	0	0
$898$	669024.	0.0276854
$899$	1.71310e7	0.706942
$900$	0	0
$901$	4.72055e7	1.93723
$902$	1.06537e7	0.435997
$903$	0	0
$904$	−9.96096e6	−0.405397
$905$	−1.08733e7	−0.441305
$906$	0	0
$907$	−3.61233e7	−1.45804	−0.729019	−	0.684493i	$-0.760023\pi$
$907$	−3.61233e7	−1.45804	−0.729019	+	0.684493i	$0.760023\pi$
$908$	−4.28621e6	−0.172528
$909$	0	0
$910$	657888.	0.0263359
$911$	−3.03119e7	−1.21009	−0.605044	−	0.796192i	$-0.706845\pi$
$911$	−3.03119e7	−1.21009	−0.605044	+	0.796192i	$0.706845\pi$
$912$	0	0
$913$	3.30970e6	0.131405
$914$	−4.72585e6	−0.187117
$915$	0	0
$916$	−1.63073e7	−0.642160
$917$	−1.02527e7	−0.402638
$918$	0	0
$919$	1.82688e7	0.713545	0.356772	−	0.934191i	$-0.383877\pi$
$919$	1.82688e7	0.713545	0.356772	+	0.934191i	$0.383877\pi$
$920$	−2.69107e6	−0.104823
$921$	0	0
$922$	−2.24664e7	−0.870375
$923$	−4.54968e6	−0.175783
$924$	0	0
$925$	1.27272e7	0.489077
$926$	4.50267e6	0.172561
$927$	0	0
$928$	−7.47110e6	−0.284784
$929$	9.19896e6	0.349703	0.174852	−	0.984595i	$-0.444055\pi$
$929$	9.19896e6	0.349703	0.174852	+	0.984595i	$0.444055\pi$
$930$	0	0
$931$	2.84677e7	1.07641
$932$	−2.99981e6	−0.113124
$933$	0	0
$934$	1.89893e7	0.712265
$935$	−2.04457e7	−0.764844
$936$	0	0
$937$	−6.01912e6	−0.223967	−0.111983	−	0.993710i	$-0.535720\pi$
$937$	−6.01912e6	−0.223967	−0.111983	+	0.993710i	$0.535720\pi$
$938$	−1.48243e7	−0.550134
$939$	0	0
$940$	−1.14555e7	−0.422857
$941$	3.23615e7	1.19139	0.595696	−	0.803210i	$-0.296876\pi$
$941$	3.23615e7	1.19139	0.595696	+	0.803210i	$0.296876\pi$
$942$	0	0
$943$	−1.14371e7	−0.418827
$944$	−5.01965e6	−0.183334
$945$	0	0
$946$	−1.01706e7	−0.369505
$947$	−3.36309e7	−1.21861	−0.609303	−	0.792937i	$-0.708551\pi$
$947$	−3.36309e7	−1.21861	−0.609303	+	0.792937i	$0.708551\pi$
$948$	0	0
$949$	2.73559e6	0.0986021
$950$	−2.66829e7	−0.959234
$951$	0	0
$952$	1.02897e7	0.367967
$953$	2.17482e7	0.775697	0.387848	−	0.921723i	$-0.373218\pi$
$953$	2.17482e7	0.775697	0.387848	+	0.921723i	$0.373218\pi$
$954$	0	0
$955$	−1.41218e7	−0.501050
$956$	−1.66572e7	−0.589462
$957$	0	0
$958$	−3.91229e7	−1.37726
$959$	−2.20078e7	−0.772735
$960$	0	0
$961$	−2.31160e7	−0.807430
$962$	1.77751e6	0.0619261
$963$	0	0
$964$	1.18974e7	0.412342
$965$	−1.40745e7	−0.486537
$966$	0	0
$967$	−3.28153e7	−1.12852	−0.564262	−	0.825596i	$-0.690839\pi$
$967$	−3.28153e7	−1.12852	−0.564262	+	0.825596i	$0.690839\pi$
$968$	−346432.	−0.0118831
$969$	0	0
$970$	−1.31185e7	−0.447667
$971$	1.98137e6	0.0674399	0.0337200	−	0.999431i	$-0.489265\pi$
$971$	1.98137e6	0.0674399	0.0337200	+	0.999431i	$0.489265\pi$
$972$	0	0
$973$	4.08169e6	0.138216
$974$	9.39128e6	0.317196
$975$	0	0
$976$	−5.64352e6	−0.189638
$977$	−3.85128e7	−1.29083	−0.645414	−	0.763833i	$-0.723315\pi$
$977$	−3.85128e7	−1.29083	−0.645414	+	0.763833i	$0.723315\pi$
$978$	0	0
$979$	−1.80662e7	−0.602436
$980$	4.17715e6	0.138936
$981$	0	0
$982$	−1.91905e7	−0.635049
$983$	−3.15858e7	−1.04258	−0.521288	−	0.853381i	$-0.674548\pi$
$983$	−3.15858e7	−1.04258	−0.521288	+	0.853381i	$0.674548\pi$
$984$	0	0
$985$	8.98387e6	0.295035
$986$	6.09362e7	1.99610
$987$	0	0
$988$	−3.72661e6	−0.121457
$989$	1.09185e7	0.354953
$990$	0	0
$991$	5.92538e7	1.91660	0.958302	−	0.285758i	$-0.0922453\pi$
$991$	5.92538e7	1.91660	0.958302	+	0.285758i	$0.0922453\pi$
$992$	−2.40435e6	−0.0775745
$993$	0	0
$994$	1.57450e7	0.505447
$995$	7.28196e6	0.233180
$996$	0	0
$997$	4.67726e7	1.49023	0.745115	−	0.666936i	$-0.232394\pi$
$997$	4.67726e7	1.49023	0.745115	+	0.666936i	$0.232394\pi$
$998$	−2.81489e7	−0.894612
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	54.6.a.b.1.1	✓	1
3.2	odd	2		54.6.a.e.1.1	yes	1
4.3	odd	2		432.6.a.d.1.1		1
9.2	odd	6		162.6.c.c.109.1		2
9.4	even	3		162.6.c.j.55.1		2
9.5	odd	6		162.6.c.c.55.1		2
9.7	even	3		162.6.c.j.109.1		2
12.11	even	2		432.6.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
54.6.a.b.1.1	✓	1	1.1	even	1	trivial
54.6.a.e.1.1	yes	1	3.2	odd	2
162.6.c.c.55.1		2	9.5	odd	6
162.6.c.c.109.1		2	9.2	odd	6
162.6.c.j.55.1		2	9.4	even	3
162.6.c.j.109.1		2	9.7	even	3
432.6.a.d.1.1		1	4.3	odd	2
432.6.a.g.1.1		1	12.11	even	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:	\( N \)	\(=\)	\( 54 = 2 \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	54.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more