LMFDB - Embedded newform 342.6.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [342,6,Mod(1,342)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(342, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("342.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 342 = 2 \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	342.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:	$54.8512663760$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 38)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	342.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+4.00000 q^{2} +16.0000 q^{4} -31.0000 q^{5} -27.0000 q^{7} +64.0000 q^{8} +O(q^{10})$

\(q+4.00000 q^{2} +16.0000 q^{4} -31.0000 q^{5} -27.0000 q^{7} +64.0000 q^{8} -124.000 q^{10} +323.000 q^{11} -676.000 q^{13} -108.000 q^{14} +256.000 q^{16} +1107.00 q^{17} -361.000 q^{19} -496.000 q^{20} +1292.00 q^{22} -1384.00 q^{23} -2164.00 q^{25} -2704.00 q^{26} -432.000 q^{28} -2870.00 q^{29} +1372.00 q^{31} +1024.00 q^{32} +4428.00 q^{34} +837.000 q^{35} -7982.00 q^{37} -1444.00 q^{38} -1984.00 q^{40} -1202.00 q^{41} -9911.00 q^{43} +5168.00 q^{44} -5536.00 q^{46} -3463.00 q^{47} -16078.0 q^{49} -8656.00 q^{50} -10816.0 q^{52} -17764.0 q^{53} -10013.0 q^{55} -1728.00 q^{56} -11480.0 q^{58} -27270.0 q^{59} +20867.0 q^{61} +5488.00 q^{62} +4096.00 q^{64} +20956.0 q^{65} +15228.0 q^{67} +17712.0 q^{68} +3348.00 q^{70} -40642.0 q^{71} -66711.0 q^{73} -31928.0 q^{74} -5776.00 q^{76} -8721.00 q^{77} +68960.0 q^{79} -7936.00 q^{80} -4808.00 q^{82} +12396.0 q^{83} -34317.0 q^{85} -39644.0 q^{86} +20672.0 q^{88} -41220.0 q^{89} +18252.0 q^{91} -22144.0 q^{92} -13852.0 q^{94} +11191.0 q^{95} -113432. q^{97} -64312.0 q^{98} +O(q^{100})\)

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
$5$	−31.0000	−0.554545	−0.277272	−	0.960791i	$-0.589430\pi$
$5$	−31.0000	−0.554545	−0.277272	+	0.960791i	$0.589430\pi$
$6$	0	0
$7$	−27.0000	−0.208266	−0.104133	−	0.994563i	$-0.533207\pi$
$7$	−27.0000	−0.208266	−0.104133	+	0.994563i	$0.533207\pi$
$8$	64.0000	0.353553
$9$	0	0
$10$	−124.000	−0.392122
$11$	323.000	0.804861	0.402430	−	0.915451i	$-0.368166\pi$
$11$	323.000	0.804861	0.402430	+	0.915451i	$0.368166\pi$
$12$	0	0
$13$	−676.000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−676.000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	−108.000	−0.147266
$15$	0	0
$16$	256.000	0.250000
$17$	1107.00	0.929021	0.464510	−	0.885568i	$-0.346230\pi$
$17$	1107.00	0.929021	0.464510	+	0.885568i	$0.346230\pi$
$18$	0	0
$19$	−361.000	−0.229416
$19$	−361.000	−0.229416
$20$	−496.000	−0.277272
$21$	0	0
$22$	1292.00	0.569123
$23$	−1384.00	−0.545527	−0.272764	−	0.962081i	$-0.587938\pi$
$23$	−1384.00	−0.545527	−0.272764	+	0.962081i	$0.587938\pi$
$24$	0	0
$25$	−2164.00	−0.692480
$26$	−2704.00	−0.784465
$27$	0	0
$28$	−432.000	−0.104133
$29$	−2870.00	−0.633705	−0.316852	−	0.948475i	$-0.602626\pi$
$29$	−2870.00	−0.633705	−0.316852	+	0.948475i	$0.602626\pi$
$30$	0	0
$31$	1372.00	0.256419	0.128209	−	0.991747i	$-0.459077\pi$
$31$	1372.00	0.256419	0.128209	+	0.991747i	$0.459077\pi$
$32$	1024.00	0.176777
$33$	0	0
$34$	4428.00	0.656917
$35$	837.000	0.115493
$36$	0	0
$37$	−7982.00	−0.958534	−0.479267	−	0.877669i	$-0.659097\pi$
$37$	−7982.00	−0.958534	−0.479267	+	0.877669i	$0.659097\pi$
$38$	−1444.00	−0.162221
$39$	0	0
$40$	−1984.00	−0.196061
$41$	−1202.00	−0.111672	−0.0558361	−	0.998440i	$-0.517782\pi$
$41$	−1202.00	−0.111672	−0.0558361	+	0.998440i	$0.517782\pi$
$42$	0	0
$43$	−9911.00	−0.817422	−0.408711	−	0.912664i	$-0.634022\pi$
$43$	−9911.00	−0.817422	−0.408711	+	0.912664i	$0.634022\pi$
$44$	5168.00	0.402430
$45$	0	0
$46$	−5536.00	−0.385746
$47$	−3463.00	−0.228669	−0.114335	−	0.993442i	$-0.536474\pi$
$47$	−3463.00	−0.228669	−0.114335	+	0.993442i	$0.536474\pi$
$48$	0	0
$49$	−16078.0	−0.956625
$50$	−8656.00	−0.489657
$51$	0	0
$52$	−10816.0	−0.554700
$53$	−17764.0	−0.868663	−0.434331	−	0.900753i	$-0.643015\pi$
$53$	−17764.0	−0.868663	−0.434331	+	0.900753i	$0.643015\pi$
$54$	0	0
$55$	−10013.0	−0.446331
$56$	−1728.00	−0.0736332
$57$	0	0
$58$	−11480.0	−0.448097
$59$	−27270.0	−1.01989	−0.509947	−	0.860206i	$-0.670335\pi$
$59$	−27270.0	−1.01989	−0.509947	+	0.860206i	$0.670335\pi$
$60$	0	0
$61$	20867.0	0.718018	0.359009	−	0.933334i	$-0.383115\pi$
$61$	20867.0	0.718018	0.359009	+	0.933334i	$0.383115\pi$
$62$	5488.00	0.181315
$63$	0	0
$64$	4096.00	0.125000
$65$	20956.0	0.615212
$66$	0	0
$67$	15228.0	0.414434	0.207217	−	0.978295i	$-0.433559\pi$
$67$	15228.0	0.414434	0.207217	+	0.978295i	$0.433559\pi$
$68$	17712.0	0.464510
$69$	0	0
$70$	3348.00	0.0816658
$71$	−40642.0	−0.956818	−0.478409	−	0.878137i	$-0.658786\pi$
$71$	−40642.0	−0.956818	−0.478409	+	0.878137i	$0.658786\pi$
$72$	0	0
$73$	−66711.0	−1.46518	−0.732589	−	0.680671i	$-0.761688\pi$
$73$	−66711.0	−1.46518	−0.732589	+	0.680671i	$0.761688\pi$
$74$	−31928.0	−0.677786
$75$	0	0
$76$	−5776.00	−0.114708
$77$	−8721.00	−0.167625
$78$	0	0
$79$	68960.0	1.24317	0.621584	−	0.783348i	$-0.286490\pi$
$79$	68960.0	1.24317	0.621584	+	0.783348i	$0.286490\pi$
$80$	−7936.00	−0.138636
$81$	0	0
$82$	−4808.00	−0.0789641
$83$	12396.0	0.197509	0.0987544	−	0.995112i	$-0.468514\pi$
$83$	12396.0	0.197509	0.0987544	+	0.995112i	$0.468514\pi$
$84$	0	0
$85$	−34317.0	−0.515184
$86$	−39644.0	−0.578005
$87$	0	0
$88$	20672.0	0.284561
$89$	−41220.0	−0.551611	−0.275806	−	0.961213i	$-0.588945\pi$
$89$	−41220.0	−0.551611	−0.275806	+	0.961213i	$0.588945\pi$
$90$	0	0
$91$	18252.0	0.231051
$92$	−22144.0	−0.272764
$93$	0	0
$94$	−13852.0	−0.161694
$95$	11191.0	0.127221
$96$	0	0
$97$	−113432.	−1.22407	−0.612035	−	0.790831i	$-0.709649\pi$
$97$	−113432.	−1.22407	−0.612035	+	0.790831i	$0.709649\pi$
$98$	−64312.0	−0.676436
$99$	0	0
$100$	−34624.0	−0.346240
$101$	129098.	1.25926	0.629631	−	0.776894i	$-0.283206\pi$
$101$	129098.	1.25926	0.629631	+	0.776894i	$0.283206\pi$
$102$	0	0
$103$	−116166.	−1.07891	−0.539456	−	0.842014i	$-0.681370\pi$
$103$	−116166.	−1.07891	−0.539456	+	0.842014i	$0.681370\pi$
$104$	−43264.0	−0.392232
$105$	0	0
$106$	−71056.0	−0.614237
$107$	172862.	1.45962	0.729810	−	0.683650i	$-0.239608\pi$
$107$	172862.	1.45962	0.729810	+	0.683650i	$0.239608\pi$
$108$	0	0
$109$	−169560.	−1.36696	−0.683482	−	0.729967i	$-0.739535\pi$
$109$	−169560.	−1.36696	−0.683482	+	0.729967i	$0.739535\pi$
$110$	−40052.0	−0.315604
$111$	0	0
$112$	−6912.00	−0.0520665
$113$	24866.0	0.183193	0.0915967	−	0.995796i	$-0.470803\pi$
$113$	24866.0	0.183193	0.0915967	+	0.995796i	$0.470803\pi$
$114$	0	0
$115$	42904.0	0.302519
$116$	−45920.0	−0.316852
$117$	0	0
$118$	−109080.	−0.721174
$119$	−29889.0	−0.193484
$120$	0	0
$121$	−56722.0	−0.352199
$122$	83468.0	0.507716
$123$	0	0
$124$	21952.0	0.128209
$125$	163959.	0.938556
$126$	0	0
$127$	223758.	1.23103	0.615516	−	0.788124i	$-0.288948\pi$
$127$	223758.	1.23103	0.615516	+	0.788124i	$0.288948\pi$
$128$	16384.0	0.0883883
$129$	0	0
$130$	83824.0	0.435021
$131$	−283647.	−1.44411	−0.722054	−	0.691836i	$-0.756802\pi$
$131$	−283647.	−1.44411	−0.722054	+	0.691836i	$0.756802\pi$
$132$	0	0
$133$	9747.00	0.0477795
$134$	60912.0	0.293049
$135$	0	0
$136$	70848.0	0.328458
$137$	−223253.	−1.01624	−0.508120	−	0.861287i	$-0.669659\pi$
$137$	−223253.	−1.01624	−0.508120	+	0.861287i	$0.669659\pi$
$138$	0	0
$139$	−168315.	−0.738900	−0.369450	−	0.929251i	$-0.620454\pi$
$139$	−168315.	−0.738900	−0.369450	+	0.929251i	$0.620454\pi$
$140$	13392.0	0.0577465
$141$	0	0
$142$	−162568.	−0.676572
$143$	−218348.	−0.892913
$144$	0	0
$145$	88970.0	0.351418
$146$	−266844.	−1.03604
$147$	0	0
$148$	−127712.	−0.479267
$149$	107775.	0.397697	0.198849	−	0.980030i	$-0.436280\pi$
$149$	107775.	0.397697	0.198849	+	0.980030i	$0.436280\pi$
$150$	0	0
$151$	389522.	1.39024	0.695120	−	0.718894i	$-0.255351\pi$
$151$	389522.	1.39024	0.695120	+	0.718894i	$0.255351\pi$
$152$	−23104.0	−0.0811107
$153$	0	0
$154$	−34884.0	−0.118529
$155$	−42532.0	−0.142196
$156$	0	0
$157$	−108242.	−0.350467	−0.175233	−	0.984527i	$-0.556068\pi$
$157$	−108242.	−0.350467	−0.175233	+	0.984527i	$0.556068\pi$
$158$	275840.	0.879052
$159$	0	0
$160$	−31744.0	−0.0980306
$161$	37368.0	0.113615
$162$	0	0
$163$	−527476.	−1.55501	−0.777506	−	0.628876i	$-0.783515\pi$
$163$	−527476.	−1.55501	−0.777506	+	0.628876i	$0.783515\pi$
$164$	−19232.0	−0.0558361
$165$	0	0
$166$	49584.0	0.139660
$167$	574782.	1.59482	0.797411	−	0.603437i	$-0.206203\pi$
$167$	574782.	1.59482	0.797411	+	0.603437i	$0.206203\pi$
$168$	0	0
$169$	85683.0	0.230769
$170$	−137268.	−0.364290
$171$	0	0
$172$	−158576.	−0.408711
$173$	80586.0	0.204712	0.102356	−	0.994748i	$-0.467362\pi$
$173$	80586.0	0.204712	0.102356	+	0.994748i	$0.467362\pi$
$174$	0	0
$175$	58428.0	0.144220
$176$	82688.0	0.201215
$177$	0	0
$178$	−164880.	−0.390048
$179$	557430.	1.30034	0.650171	−	0.759788i	$-0.274697\pi$
$179$	557430.	1.30034	0.650171	+	0.759788i	$0.274697\pi$
$180$	0	0
$181$	767962.	1.74238	0.871191	−	0.490945i	$-0.163348\pi$
$181$	767962.	1.74238	0.871191	+	0.490945i	$0.163348\pi$
$182$	73008.0	0.163377
$183$	0	0
$184$	−88576.0	−0.192873
$185$	247442.	0.531550
$186$	0	0
$187$	357561.	0.747732
$188$	−55408.0	−0.114335
$189$	0	0
$190$	44764.0	0.0899591
$191$	−542657.	−1.07632	−0.538161	−	0.842842i	$-0.680881\pi$
$191$	−542657.	−1.07632	−0.538161	+	0.842842i	$0.680881\pi$
$192$	0	0
$193$	362004.	0.699552	0.349776	−	0.936833i	$-0.386258\pi$
$193$	362004.	0.699552	0.349776	+	0.936833i	$0.386258\pi$
$194$	−453728.	−0.865548
$195$	0	0
$196$	−257248.	−0.478313
$197$	831662.	1.52680	0.763398	−	0.645928i	$-0.223529\pi$
$197$	831662.	1.52680	0.763398	+	0.645928i	$0.223529\pi$
$198$	0	0
$199$	−778975.	−1.39441	−0.697206	−	0.716871i	$-0.745574\pi$
$199$	−778975.	−1.39441	−0.697206	+	0.716871i	$0.745574\pi$
$200$	−138496.	−0.244829
$201$	0	0
$202$	516392.	0.890433
$203$	77490.0	0.131979
$204$	0	0
$205$	37262.0	0.0619272
$206$	−464664.	−0.762906
$207$	0	0
$208$	−173056.	−0.277350
$209$	−116603.	−0.184648
$210$	0	0
$211$	−298158.	−0.461042	−0.230521	−	0.973067i	$-0.574043\pi$
$211$	−298158.	−0.461042	−0.230521	+	0.973067i	$0.574043\pi$
$212$	−284224.	−0.434331
$213$	0	0
$214$	691448.	1.03211
$215$	307241.	0.453297
$216$	0	0
$217$	−37044.0	−0.0534034
$218$	−678240.	−0.966590
$219$	0	0
$220$	−160208.	−0.223166
$221$	−748332.	−1.03066
$222$	0	0
$223$	−308566.	−0.415514	−0.207757	−	0.978180i	$-0.566616\pi$
$223$	−308566.	−0.415514	−0.207757	+	0.978180i	$0.566616\pi$
$224$	−27648.0	−0.0368166
$225$	0	0
$226$	99464.0	0.129537
$227$	−77108.0	−0.0993196	−0.0496598	−	0.998766i	$-0.515814\pi$
$227$	−77108.0	−0.0993196	−0.0496598	+	0.998766i	$0.515814\pi$
$228$	0	0
$229$	626465.	0.789420	0.394710	−	0.918806i	$-0.370845\pi$
$229$	626465.	0.789420	0.394710	+	0.918806i	$0.370845\pi$
$230$	171616.	0.213913
$231$	0	0
$232$	−183680.	−0.224048
$233$	−1.26058e6	−1.52118	−0.760589	−	0.649233i	$-0.775090\pi$
$233$	−1.26058e6	−1.52118	−0.760589	+	0.649233i	$0.775090\pi$
$234$	0	0
$235$	107353.	0.126807
$236$	−436320.	−0.509947
$237$	0	0
$238$	−119556.	−0.136814
$239$	1.51052e6	1.71053	0.855264	−	0.518192i	$-0.173395\pi$
$239$	1.51052e6	1.71053	0.855264	+	0.518192i	$0.173395\pi$
$240$	0	0
$241$	1.26542e6	1.40344	0.701718	−	0.712455i	$-0.252417\pi$
$241$	1.26542e6	1.40344	0.701718	+	0.712455i	$0.252417\pi$
$242$	−226888.	−0.249042
$243$	0	0
$244$	333872.	0.359009
$245$	498418.	0.530492
$246$	0	0
$247$	244036.	0.254514
$248$	87808.0	0.0906577
$249$	0	0
$250$	655836.	0.663659
$251$	394973.	0.395716	0.197858	−	0.980231i	$-0.436602\pi$
$251$	394973.	0.395716	0.197858	+	0.980231i	$0.436602\pi$
$252$	0	0
$253$	−447032.	−0.439074
$254$	895032.	0.870471
$255$	0	0
$256$	65536.0	0.0625000
$257$	−1.86645e6	−1.76272	−0.881360	−	0.472446i	$-0.843371\pi$
$257$	−1.86645e6	−1.76272	−0.881360	+	0.472446i	$0.843371\pi$
$258$	0	0
$259$	215514.	0.199630
$260$	335296.	0.307606
$261$	0	0
$262$	−1.13459e6	−1.02114
$263$	919581.	0.819786	0.409893	−	0.912134i	$-0.365566\pi$
$263$	919581.	0.819786	0.409893	+	0.912134i	$0.365566\pi$
$264$	0	0
$265$	550684.	0.481712
$266$	38988.0	0.0337852
$267$	0	0
$268$	243648.	0.207217
$269$	1.19688e6	1.00849	0.504243	−	0.863562i	$-0.331772\pi$
$269$	1.19688e6	1.00849	0.504243	+	0.863562i	$0.331772\pi$
$270$	0	0
$271$	2.19767e6	1.81777	0.908886	−	0.417044i	$-0.136934\pi$
$271$	2.19767e6	1.81777	0.908886	+	0.417044i	$0.136934\pi$
$272$	283392.	0.232255
$273$	0	0
$274$	−893012.	−0.718590
$275$	−698972.	−0.557350
$276$	0	0
$277$	−858767.	−0.672475	−0.336237	−	0.941777i	$-0.609154\pi$
$277$	−858767.	−0.672475	−0.336237	+	0.941777i	$0.609154\pi$
$278$	−673260.	−0.522481
$279$	0	0
$280$	53568.0	0.0408329
$281$	566698.	0.428140	0.214070	−	0.976818i	$-0.431328\pi$
$281$	566698.	0.428140	0.214070	+	0.976818i	$0.431328\pi$
$282$	0	0
$283$	335829.	0.249260	0.124630	−	0.992203i	$-0.460226\pi$
$283$	335829.	0.249260	0.124630	+	0.992203i	$0.460226\pi$
$284$	−650272.	−0.478409
$285$	0	0
$286$	−873392.	−0.631385
$287$	32454.0	0.0232575
$288$	0	0
$289$	−194408.	−0.136921
$290$	355880.	0.248490
$291$	0	0
$292$	−1.06738e6	−0.732589
$293$	2.27104e6	1.54545	0.772725	−	0.634741i	$-0.218893\pi$
$293$	2.27104e6	1.54545	0.772725	+	0.634741i	$0.218893\pi$
$294$	0	0
$295$	845370.	0.565577
$296$	−510848.	−0.338893
$297$	0	0
$298$	431100.	0.281214
$299$	935584.	0.605208
$300$	0	0
$301$	267597.	0.170241
$302$	1.55809e6	0.983048
$303$	0	0
$304$	−92416.0	−0.0573539
$305$	−646877.	−0.398173
$306$	0	0
$307$	−339132.	−0.205363	−0.102682	−	0.994714i	$-0.532742\pi$
$307$	−339132.	−0.205363	−0.102682	+	0.994714i	$0.532742\pi$
$308$	−139536.	−0.0838126
$309$	0	0
$310$	−170128.	−0.100548
$311$	1.14571e6	0.671699	0.335850	−	0.941916i	$-0.390977\pi$
$311$	1.14571e6	0.671699	0.335850	+	0.941916i	$0.390977\pi$
$312$	0	0
$313$	2.31661e6	1.33657	0.668287	−	0.743904i	$-0.267028\pi$
$313$	2.31661e6	1.33657	0.668287	+	0.743904i	$0.267028\pi$
$314$	−432968.	−0.247817
$315$	0	0
$316$	1.10336e6	0.621584
$317$	374562.	0.209351	0.104676	−	0.994506i	$-0.466620\pi$
$317$	374562.	0.209351	0.104676	+	0.994506i	$0.466620\pi$
$318$	0	0
$319$	−927010.	−0.510044
$320$	−126976.	−0.0693181
$321$	0	0
$322$	149472.	0.0803378
$323$	−399627.	−0.213132
$324$	0	0
$325$	1.46286e6	0.768238
$326$	−2.10990e6	−1.09956
$327$	0	0
$328$	−76928.0	−0.0394821
$329$	93501.0	0.0476241
$330$	0	0
$331$	1.72749e6	0.866655	0.433327	−	0.901237i	$-0.357339\pi$
$331$	1.72749e6	0.866655	0.433327	+	0.901237i	$0.357339\pi$
$332$	198336.	0.0987544
$333$	0	0
$334$	2.29913e6	1.12771
$335$	−472068.	−0.229822
$336$	0	0
$337$	−2.61987e6	−1.25662	−0.628312	−	0.777961i	$-0.716254\pi$
$337$	−2.61987e6	−1.25662	−0.628312	+	0.777961i	$0.716254\pi$
$338$	342732.	0.163178
$339$	0	0
$340$	−549072.	−0.257592
$341$	443156.	0.206381
$342$	0	0
$343$	887895.	0.407499
$344$	−634304.	−0.289002
$345$	0	0
$346$	322344.	0.144754
$347$	986437.	0.439790	0.219895	−	0.975524i	$-0.429428\pi$
$347$	986437.	0.439790	0.219895	+	0.975524i	$0.429428\pi$
$348$	0	0
$349$	1.02008e6	0.448304	0.224152	−	0.974554i	$-0.428039\pi$
$349$	1.02008e6	0.448304	0.224152	+	0.974554i	$0.428039\pi$
$350$	233712.	0.101979
$351$	0	0
$352$	330752.	0.142281
$353$	226806.	0.0968764	0.0484382	−	0.998826i	$-0.484576\pi$
$353$	226806.	0.0968764	0.0484382	+	0.998826i	$0.484576\pi$
$354$	0	0
$355$	1.25990e6	0.530598
$356$	−659520.	−0.275806
$357$	0	0
$358$	2.22972e6	0.919481
$359$	−1.57300e6	−0.644160	−0.322080	−	0.946712i	$-0.604382\pi$
$359$	−1.57300e6	−0.644160	−0.322080	+	0.946712i	$0.604382\pi$
$360$	0	0
$361$	130321.	0.0526316
$362$	3.07185e6	1.23205
$363$	0	0
$364$	292032.	0.115525
$365$	2.06804e6	0.812507
$366$	0	0
$367$	−2.30163e6	−0.892012	−0.446006	−	0.895030i	$-0.647154\pi$
$367$	−2.30163e6	−0.892012	−0.446006	+	0.895030i	$0.647154\pi$
$368$	−354304.	−0.136382
$369$	0	0
$370$	989768.	0.375863
$371$	479628.	0.180913
$372$	0	0
$373$	−4.76396e6	−1.77295	−0.886473	−	0.462780i	$-0.846852\pi$
$373$	−4.76396e6	−1.77295	−0.886473	+	0.462780i	$0.846852\pi$
$374$	1.43024e6	0.528727
$375$	0	0
$376$	−221632.	−0.0808468
$377$	1.94012e6	0.703032
$378$	0	0
$379$	−4.57179e6	−1.63489	−0.817444	−	0.576007i	$-0.804610\pi$
$379$	−4.57179e6	−1.63489	−0.817444	+	0.576007i	$0.804610\pi$
$380$	179056.	0.0636107
$381$	0	0
$382$	−2.17063e6	−0.761074
$383$	582676.	0.202969	0.101485	−	0.994837i	$-0.467641\pi$
$383$	582676.	0.202969	0.101485	+	0.994837i	$0.467641\pi$
$384$	0	0
$385$	270351.	0.0929557
$386$	1.44802e6	0.494658
$387$	0	0
$388$	−1.81491e6	−0.612035
$389$	2.73808e6	0.917430	0.458715	−	0.888583i	$-0.348310\pi$
$389$	2.73808e6	0.917430	0.458715	+	0.888583i	$0.348310\pi$
$390$	0	0
$391$	−1.53209e6	−0.506806
$392$	−1.02899e6	−0.338218
$393$	0	0
$394$	3.32665e6	1.07961
$395$	−2.13776e6	−0.689392
$396$	0	0
$397$	−477067.	−0.151916	−0.0759579	−	0.997111i	$-0.524201\pi$
$397$	−477067.	−0.151916	−0.0759579	+	0.997111i	$0.524201\pi$
$398$	−3.11590e6	−0.985998
$399$	0	0
$400$	−553984.	−0.173120
$401$	−4.70877e6	−1.46233	−0.731167	−	0.682198i	$-0.761024\pi$
$401$	−4.70877e6	−1.46233	−0.731167	+	0.682198i	$0.761024\pi$
$402$	0	0
$403$	−927472.	−0.284471
$404$	2.06557e6	0.629631
$405$	0	0
$406$	309960.	0.0933234
$407$	−2.57819e6	−0.771486
$408$	0	0
$409$	−2.91422e6	−0.861418	−0.430709	−	0.902491i	$-0.641737\pi$
$409$	−2.91422e6	−0.861418	−0.430709	+	0.902491i	$0.641737\pi$
$410$	149048.	0.0437891
$411$	0	0
$412$	−1.85866e6	−0.539456
$413$	736290.	0.212409
$414$	0	0
$415$	−384276.	−0.109527
$416$	−692224.	−0.196116
$417$	0	0
$418$	−466412.	−0.130566
$419$	−5.84430e6	−1.62629	−0.813144	−	0.582063i	$-0.802246\pi$
$419$	−5.84430e6	−1.62629	−0.813144	+	0.582063i	$0.802246\pi$
$420$	0	0
$421$	−3.94173e6	−1.08388	−0.541940	−	0.840417i	$-0.682310\pi$
$421$	−3.94173e6	−1.08388	−0.541940	+	0.840417i	$0.682310\pi$
$422$	−1.19263e6	−0.326006
$423$	0	0
$424$	−1.13690e6	−0.307119
$425$	−2.39555e6	−0.643328
$426$	0	0
$427$	−563409.	−0.149539
$428$	2.76579e6	0.729810
$429$	0	0
$430$	1.22896e6	0.320530
$431$	−6.32167e6	−1.63923	−0.819613	−	0.572918i	$-0.805811\pi$
$431$	−6.32167e6	−1.63923	−0.819613	+	0.572918i	$0.805811\pi$
$432$	0	0
$433$	5.49775e6	1.40918	0.704589	−	0.709616i	$-0.251132\pi$
$433$	5.49775e6	1.40918	0.704589	+	0.709616i	$0.251132\pi$
$434$	−148176.	−0.0377619
$435$	0	0
$436$	−2.71296e6	−0.683482
$437$	499624.	0.125153
$438$	0	0
$439$	−1.32559e6	−0.328283	−0.164141	−	0.986437i	$-0.552485\pi$
$439$	−1.32559e6	−0.328283	−0.164141	+	0.986437i	$0.552485\pi$
$440$	−640832.	−0.157802
$441$	0	0
$442$	−2.99333e6	−0.728784
$443$	−7.66602e6	−1.85593	−0.927963	−	0.372673i	$-0.878441\pi$
$443$	−7.66602e6	−1.85593	−0.927963	+	0.372673i	$0.878441\pi$
$444$	0	0
$445$	1.27782e6	0.305893
$446$	−1.23426e6	−0.293813
$447$	0	0
$448$	−110592.	−0.0260333
$449$	501280.	0.117345	0.0586725	−	0.998277i	$-0.481313\pi$
$449$	501280.	0.117345	0.0586725	+	0.998277i	$0.481313\pi$
$450$	0	0
$451$	−388246.	−0.0898805
$452$	397856.	0.0915967
$453$	0	0
$454$	−308432.	−0.0702295
$455$	−565812.	−0.128128
$456$	0	0
$457$	1.86680e6	0.418127	0.209063	−	0.977902i	$-0.432958\pi$
$457$	1.86680e6	0.418127	0.209063	+	0.977902i	$0.432958\pi$
$458$	2.50586e6	0.558204
$459$	0	0
$460$	686464.	0.151260
$461$	5.60264e6	1.22784	0.613918	−	0.789370i	$-0.289592\pi$
$461$	5.60264e6	1.22784	0.613918	+	0.789370i	$0.289592\pi$
$462$	0	0
$463$	2.41406e6	0.523354	0.261677	−	0.965156i	$-0.415725\pi$
$463$	2.41406e6	0.523354	0.261677	+	0.965156i	$0.415725\pi$
$464$	−734720.	−0.158426
$465$	0	0
$466$	−5.04232e6	−1.07564
$467$	−8.43680e6	−1.79013	−0.895067	−	0.445931i	$-0.852873\pi$
$467$	−8.43680e6	−1.79013	−0.895067	+	0.445931i	$0.852873\pi$
$468$	0	0
$469$	−411156.	−0.0863127
$470$	429412.	0.0896664
$471$	0	0
$472$	−1.74528e6	−0.360587
$473$	−3.20125e6	−0.657911
$474$	0	0
$475$	781204.	0.158866
$476$	−478224.	−0.0967418
$477$	0	0
$478$	6.04206e6	1.20953
$479$	−342260.	−0.0681581	−0.0340790	−	0.999419i	$-0.510850\pi$
$479$	−342260.	−0.0681581	−0.0340790	+	0.999419i	$0.510850\pi$
$480$	0	0
$481$	5.39583e6	1.06340
$482$	5.06169e6	0.992379
$483$	0	0
$484$	−907552.	−0.176099
$485$	3.51639e6	0.678802
$486$	0	0
$487$	−5.00700e6	−0.956655	−0.478328	−	0.878182i	$-0.658757\pi$
$487$	−5.00700e6	−0.956655	−0.478328	+	0.878182i	$0.658757\pi$
$488$	1.33549e6	0.253858
$489$	0	0
$490$	1.99367e6	0.375114
$491$	5.91163e6	1.10663	0.553316	−	0.832971i	$-0.313362\pi$
$491$	5.91163e6	1.10663	0.553316	+	0.832971i	$0.313362\pi$
$492$	0	0
$493$	−3.17709e6	−0.588725
$494$	976144.	0.179969
$495$	0	0
$496$	351232.	0.0641047
$497$	1.09733e6	0.199273
$498$	0	0
$499$	−7.81152e6	−1.40438	−0.702190	−	0.711990i	$-0.747794\pi$
$499$	−7.81152e6	−1.40438	−0.702190	+	0.711990i	$0.747794\pi$
$500$	2.62334e6	0.469278
$501$	0	0
$502$	1.57989e6	0.279813
$503$	−3.08564e6	−0.543783	−0.271892	−	0.962328i	$-0.587649\pi$
$503$	−3.08564e6	−0.543783	−0.271892	+	0.962328i	$0.587649\pi$
$504$	0	0
$505$	−4.00204e6	−0.698317
$506$	−1.78813e6	−0.310472
$507$	0	0
$508$	3.58013e6	0.615516
$509$	−2.21256e6	−0.378530	−0.189265	−	0.981926i	$-0.560611\pi$
$509$	−2.21256e6	−0.378530	−0.189265	+	0.981926i	$0.560611\pi$
$510$	0	0
$511$	1.80120e6	0.305147
$512$	262144.	0.0441942
$513$	0	0
$514$	−7.46579e6	−1.24643
$515$	3.60115e6	0.598305
$516$	0	0
$517$	−1.11855e6	−0.184047
$518$	862056.	0.141160
$519$	0	0
$520$	1.34118e6	0.217510
$521$	−3.50975e6	−0.566477	−0.283238	−	0.959050i	$-0.591409\pi$
$521$	−3.50975e6	−0.566477	−0.283238	+	0.959050i	$0.591409\pi$
$522$	0	0
$523$	−6.15833e6	−0.984484	−0.492242	−	0.870459i	$-0.663822\pi$
$523$	−6.15833e6	−0.984484	−0.492242	+	0.870459i	$0.663822\pi$
$524$	−4.53835e6	−0.722054
$525$	0	0
$526$	3.67832e6	0.579676
$527$	1.51880e6	0.238218
$528$	0	0
$529$	−4.52089e6	−0.702400
$530$	2.20274e6	0.340622
$531$	0	0
$532$	155952.	0.0238898
$533$	812552.	0.123889
$534$	0	0
$535$	−5.35872e6	−0.809425
$536$	974592.	0.146525
$537$	0	0
$538$	4.78752e6	0.713107
$539$	−5.19319e6	−0.769950
$540$	0	0
$541$	410267.	0.0602661	0.0301331	−	0.999546i	$-0.490407\pi$
$541$	410267.	0.0602661	0.0301331	+	0.999546i	$0.490407\pi$
$542$	8.79069e6	1.28536
$543$	0	0
$544$	1.13357e6	0.164229
$545$	5.25636e6	0.758043
$546$	0	0
$547$	2.67459e6	0.382198	0.191099	−	0.981571i	$-0.438795\pi$
$547$	2.67459e6	0.382198	0.191099	+	0.981571i	$0.438795\pi$
$548$	−3.57205e6	−0.508120
$549$	0	0
$550$	−2.79589e6	−0.394106
$551$	1.03607e6	0.145382
$552$	0	0
$553$	−1.86192e6	−0.258910
$554$	−3.43507e6	−0.475511
$555$	0	0
$556$	−2.69304e6	−0.369450
$557$	−1.44745e7	−1.97681	−0.988407	−	0.151825i	$-0.951485\pi$
$557$	−1.44745e7	−1.97681	−0.988407	+	0.151825i	$0.951485\pi$
$558$	0	0
$559$	6.69984e6	0.906848
$560$	214272.	0.0288732
$561$	0	0
$562$	2.26679e6	0.302741
$563$	9.92053e6	1.31906	0.659529	−	0.751679i	$-0.270756\pi$
$563$	9.92053e6	1.31906	0.659529	+	0.751679i	$0.270756\pi$
$564$	0	0
$565$	−770846.	−0.101589
$566$	1.34332e6	0.176253
$567$	0	0
$568$	−2.60109e6	−0.338286
$569$	−626360.	−0.0811042	−0.0405521	−	0.999177i	$-0.512912\pi$
$569$	−626360.	−0.0811042	−0.0405521	+	0.999177i	$0.512912\pi$
$570$	0	0
$571$	−8.51787e6	−1.09330	−0.546652	−	0.837360i	$-0.684098\pi$
$571$	−8.51787e6	−1.09330	−0.546652	+	0.837360i	$0.684098\pi$
$572$	−3.49357e6	−0.446456
$573$	0	0
$574$	129816.	0.0164456
$575$	2.99498e6	0.377767
$576$	0	0
$577$	1.48465e7	1.85645	0.928227	−	0.372015i	$-0.121333\pi$
$577$	1.48465e7	1.85645	0.928227	+	0.372015i	$0.121333\pi$
$578$	−777632.	−0.0968176
$579$	0	0
$580$	1.42352e6	0.175709
$581$	−334692.	−0.0411344
$582$	0	0
$583$	−5.73777e6	−0.699152
$584$	−4.26950e6	−0.518019
$585$	0	0
$586$	9.08414e6	1.09280
$587$	−6.53076e6	−0.782292	−0.391146	−	0.920329i	$-0.627921\pi$
$587$	−6.53076e6	−0.782292	−0.391146	+	0.920329i	$0.627921\pi$
$588$	0	0
$589$	−495292.	−0.0588265
$590$	3.38148e6	0.399923
$591$	0	0
$592$	−2.04339e6	−0.239633
$593$	−595334.	−0.0695223	−0.0347611	−	0.999396i	$-0.511067\pi$
$593$	−595334.	−0.0695223	−0.0347611	+	0.999396i	$0.511067\pi$
$594$	0	0
$595$	926559.	0.107295
$596$	1.72440e6	0.198849
$597$	0	0
$598$	3.74234e6	0.427947
$599$	973620.	0.110872	0.0554361	−	0.998462i	$-0.482345\pi$
$599$	973620.	0.110872	0.0554361	+	0.998462i	$0.482345\pi$
$600$	0	0
$601$	7.20788e6	0.813995	0.406997	−	0.913429i	$-0.366576\pi$
$601$	7.20788e6	0.813995	0.406997	+	0.913429i	$0.366576\pi$
$602$	1.07039e6	0.120379
$603$	0	0
$604$	6.23235e6	0.695120
$605$	1.75838e6	0.195310
$606$	0	0
$607$	3.40957e6	0.375602	0.187801	−	0.982207i	$-0.439864\pi$
$607$	3.40957e6	0.375602	0.187801	+	0.982207i	$0.439864\pi$
$608$	−369664.	−0.0405554
$609$	0	0
$610$	−2.58751e6	−0.281551
$611$	2.34099e6	0.253686
$612$	0	0
$613$	4.24037e6	0.455777	0.227889	−	0.973687i	$-0.426818\pi$
$613$	4.24037e6	0.455777	0.227889	+	0.973687i	$0.426818\pi$
$614$	−1.35653e6	−0.145214
$615$	0	0
$616$	−558144.	−0.0592645
$617$	5.12649e6	0.542134	0.271067	−	0.962560i	$-0.412623\pi$
$617$	5.12649e6	0.542134	0.271067	+	0.962560i	$0.412623\pi$
$618$	0	0
$619$	1.86695e7	1.95842	0.979208	−	0.202857i	$-0.0650226\pi$
$619$	1.86695e7	1.95842	0.979208	+	0.202857i	$0.0650226\pi$
$620$	−680512.	−0.0710979
$621$	0	0
$622$	4.58285e6	0.474963
$623$	1.11294e6	0.114882
$624$	0	0
$625$	1.67977e6	0.172009
$626$	9.26646e6	0.945100
$627$	0	0
$628$	−1.73187e6	−0.175233
$629$	−8.83607e6	−0.890498
$630$	0	0
$631$	−1.66982e7	−1.66954	−0.834770	−	0.550599i	$-0.814399\pi$
$631$	−1.66982e7	−1.66954	−0.834770	+	0.550599i	$0.814399\pi$
$632$	4.41344e6	0.439526
$633$	0	0
$634$	1.49825e6	0.148034
$635$	−6.93650e6	−0.682662
$636$	0	0
$637$	1.08687e7	1.06128
$638$	−3.70804e6	−0.360656
$639$	0	0
$640$	−507904.	−0.0490153
$641$	1.15752e7	1.11272	0.556359	−	0.830942i	$-0.312198\pi$
$641$	1.15752e7	1.11272	0.556359	+	0.830942i	$0.312198\pi$
$642$	0	0
$643$	−7.26373e6	−0.692839	−0.346419	−	0.938080i	$-0.612603\pi$
$643$	−7.26373e6	−0.692839	−0.346419	+	0.938080i	$0.612603\pi$
$644$	597888.	0.0568074
$645$	0	0
$646$	−1.59851e6	−0.150707
$647$	−1.39869e7	−1.31359	−0.656797	−	0.754067i	$-0.728089\pi$
$647$	−1.39869e7	−1.31359	−0.656797	+	0.754067i	$0.728089\pi$
$648$	0	0
$649$	−8.80821e6	−0.820873
$650$	5.85146e6	0.543226
$651$	0	0
$652$	−8.43962e6	−0.777506
$653$	1.07611e7	0.987579	0.493789	−	0.869582i	$-0.335611\pi$
$653$	1.07611e7	0.987579	0.493789	+	0.869582i	$0.335611\pi$
$654$	0	0
$655$	8.79306e6	0.800823
$656$	−307712.	−0.0279180
$657$	0	0
$658$	374004.	0.0336753
$659$	1.78558e7	1.60164	0.800822	−	0.598902i	$-0.204396\pi$
$659$	1.78558e7	1.60164	0.800822	+	0.598902i	$0.204396\pi$
$660$	0	0
$661$	−349328.	−0.0310978	−0.0155489	−	0.999879i	$-0.504950\pi$
$661$	−349328.	−0.0310978	−0.0155489	+	0.999879i	$0.504950\pi$
$662$	6.90997e6	0.612817
$663$	0	0
$664$	793344.	0.0698299
$665$	−302157.	−0.0264959
$666$	0	0
$667$	3.97208e6	0.345703
$668$	9.19651e6	0.797411
$669$	0	0
$670$	−1.88827e6	−0.162509
$671$	6.74004e6	0.577905
$672$	0	0
$673$	−4.51115e6	−0.383928	−0.191964	−	0.981402i	$-0.561486\pi$
$673$	−4.51115e6	−0.383928	−0.191964	+	0.981402i	$0.561486\pi$
$674$	−1.04795e7	−0.888567
$675$	0	0
$676$	1.37093e6	0.115385
$677$	−3.44692e6	−0.289041	−0.144520	−	0.989502i	$-0.546164\pi$
$677$	−3.44692e6	−0.289041	−0.144520	+	0.989502i	$0.546164\pi$
$678$	0	0
$679$	3.06266e6	0.254932
$680$	−2.19629e6	−0.182145
$681$	0	0
$682$	1.77262e6	0.145934
$683$	984196.	0.0807290	0.0403645	−	0.999185i	$-0.487148\pi$
$683$	984196.	0.0807290	0.0403645	+	0.999185i	$0.487148\pi$
$684$	0	0
$685$	6.92084e6	0.563550
$686$	3.55158e6	0.288145
$687$	0	0
$688$	−2.53722e6	−0.204356
$689$	1.20085e7	0.963695
$690$	0	0
$691$	9.92073e6	0.790403	0.395201	−	0.918595i	$-0.370675\pi$
$691$	9.92073e6	0.790403	0.395201	+	0.918595i	$0.370675\pi$
$692$	1.28938e6	0.102356
$693$	0	0
$694$	3.94575e6	0.310979
$695$	5.21776e6	0.409753
$696$	0	0
$697$	−1.33061e6	−0.103746
$698$	4.08034e6	0.316999
$699$	0	0
$700$	934848.	0.0721101
$701$	6.20014e6	0.476548	0.238274	−	0.971198i	$-0.423418\pi$
$701$	6.20014e6	0.476548	0.238274	+	0.971198i	$0.423418\pi$
$702$	0	0
$703$	2.88150e6	0.219903
$704$	1.32301e6	0.100608
$705$	0	0
$706$	907224.	0.0685019
$707$	−3.48565e6	−0.262262
$708$	0	0
$709$	3.72177e6	0.278057	0.139029	−	0.990288i	$-0.455602\pi$
$709$	3.72177e6	0.278057	0.139029	+	0.990288i	$0.455602\pi$
$710$	5.03961e6	0.375190
$711$	0	0
$712$	−2.63808e6	−0.195024
$713$	−1.89885e6	−0.139883
$714$	0	0
$715$	6.76879e6	0.495160
$716$	8.91888e6	0.650171
$717$	0	0
$718$	−6.29202e6	−0.455490
$719$	−7.99873e6	−0.577030	−0.288515	−	0.957475i	$-0.593162\pi$
$719$	−7.99873e6	−0.577030	−0.288515	+	0.957475i	$0.593162\pi$
$720$	0	0
$721$	3.13648e6	0.224701
$722$	521284.	0.0372161
$723$	0	0
$724$	1.22874e7	0.871191
$725$	6.21068e6	0.438828
$726$	0	0
$727$	1.60855e7	1.12875	0.564376	−	0.825518i	$-0.309117\pi$
$727$	1.60855e7	1.12875	0.564376	+	0.825518i	$0.309117\pi$
$728$	1.16813e6	0.0816887
$729$	0	0
$730$	8.27216e6	0.574529
$731$	−1.09715e7	−0.759402
$732$	0	0
$733$	1.03402e7	0.710832	0.355416	−	0.934708i	$-0.384339\pi$
$733$	1.03402e7	0.710832	0.355416	+	0.934708i	$0.384339\pi$
$734$	−9.20653e6	−0.630748
$735$	0	0
$736$	−1.41722e6	−0.0964365
$737$	4.91864e6	0.333562
$738$	0	0
$739$	−3.50460e6	−0.236062	−0.118031	−	0.993010i	$-0.537658\pi$
$739$	−3.50460e6	−0.236062	−0.118031	+	0.993010i	$0.537658\pi$
$740$	3.95907e6	0.265775
$741$	0	0
$742$	1.91851e6	0.127925
$743$	−1.34768e7	−0.895600	−0.447800	−	0.894134i	$-0.647792\pi$
$743$	−1.34768e7	−0.895600	−0.447800	+	0.894134i	$0.647792\pi$
$744$	0	0
$745$	−3.34102e6	−0.220541
$746$	−1.90558e7	−1.25366
$747$	0	0
$748$	5.72098e6	0.373866
$749$	−4.66727e6	−0.303990
$750$	0	0
$751$	126272.	0.00816972	0.00408486	−	0.999992i	$-0.498700\pi$
$751$	126272.	0.00816972	0.00408486	+	0.999992i	$0.498700\pi$
$752$	−886528.	−0.0571673
$753$	0	0
$754$	7.76048e6	0.497119
$755$	−1.20752e7	−0.770950
$756$	0	0
$757$	1.30450e7	0.827379	0.413689	−	0.910418i	$-0.364240\pi$
$757$	1.30450e7	0.827379	0.413689	+	0.910418i	$0.364240\pi$
$758$	−1.82872e7	−1.15604
$759$	0	0
$760$	716224.	0.0449795
$761$	1.12451e7	0.703888	0.351944	−	0.936021i	$-0.385521\pi$
$761$	1.12451e7	0.703888	0.351944	+	0.936021i	$0.385521\pi$
$762$	0	0
$763$	4.57812e6	0.284692
$764$	−8.68251e6	−0.538161
$765$	0	0
$766$	2.33070e6	0.143521
$767$	1.84345e7	1.13147
$768$	0	0
$769$	−9.74766e6	−0.594408	−0.297204	−	0.954814i	$-0.596054\pi$
$769$	−9.74766e6	−0.594408	−0.297204	+	0.954814i	$0.596054\pi$
$770$	1.08140e6	0.0657296
$771$	0	0
$772$	5.79206e6	0.349776
$773$	−1.43099e6	−0.0861369	−0.0430684	−	0.999072i	$-0.513713\pi$
$773$	−1.43099e6	−0.0861369	−0.0430684	+	0.999072i	$0.513713\pi$
$774$	0	0
$775$	−2.96901e6	−0.177565
$776$	−7.25965e6	−0.432774
$777$	0	0
$778$	1.09523e7	0.648721
$779$	433922.	0.0256193
$780$	0	0
$781$	−1.31274e7	−0.770105
$782$	−6.12835e6	−0.358366
$783$	0	0
$784$	−4.11597e6	−0.239156
$785$	3.35550e6	0.194349
$786$	0	0
$787$	−1.04363e7	−0.600631	−0.300315	−	0.953840i	$-0.597092\pi$
$787$	−1.04363e7	−0.600631	−0.300315	+	0.953840i	$0.597092\pi$
$788$	1.33066e7	0.763398
$789$	0	0
$790$	−8.55104e6	−0.487474
$791$	−671382.	−0.0381530
$792$	0	0
$793$	−1.41061e7	−0.796570
$794$	−1.90827e6	−0.107421
$795$	0	0
$796$	−1.24636e7	−0.697206
$797$	1.61641e7	0.901377	0.450689	−	0.892681i	$-0.351178\pi$
$797$	1.61641e7	0.901377	0.450689	+	0.892681i	$0.351178\pi$
$798$	0	0
$799$	−3.83354e6	−0.212438
$800$	−2.21594e6	−0.122414
$801$	0	0
$802$	−1.88351e7	−1.03403
$803$	−2.15477e7	−1.17926
$804$	0	0
$805$	−1.15841e6	−0.0630045
$806$	−3.70989e6	−0.201151
$807$	0	0
$808$	8.26227e6	0.445216
$809$	2.33677e7	1.25529	0.627646	−	0.778499i	$-0.284019\pi$
$809$	2.33677e7	1.25529	0.627646	+	0.778499i	$0.284019\pi$
$810$	0	0
$811$	1.84173e7	0.983273	0.491636	−	0.870801i	$-0.336399\pi$
$811$	1.84173e7	0.983273	0.491636	+	0.870801i	$0.336399\pi$
$812$	1.23984e6	0.0659896
$813$	0	0
$814$	−1.03127e7	−0.545523
$815$	1.63518e7	0.862324
$816$	0	0
$817$	3.57787e6	0.187529
$818$	−1.16569e7	−0.609115
$819$	0	0
$820$	596192.	0.0309636
$821$	1.54337e7	0.799119	0.399560	−	0.916707i	$-0.369163\pi$
$821$	1.54337e7	0.799119	0.399560	+	0.916707i	$0.369163\pi$
$822$	0	0
$823$	−2.80106e7	−1.44153	−0.720764	−	0.693181i	$-0.756209\pi$
$823$	−2.80106e7	−1.44153	−0.720764	+	0.693181i	$0.756209\pi$
$824$	−7.43462e6	−0.381453
$825$	0	0
$826$	2.94516e6	0.150196
$827$	−2.05570e7	−1.04519	−0.522596	−	0.852581i	$-0.675036\pi$
$827$	−2.05570e7	−1.04519	−0.522596	+	0.852581i	$0.675036\pi$
$828$	0	0
$829$	1.50528e6	0.0760731	0.0380365	−	0.999276i	$-0.487890\pi$
$829$	1.50528e6	0.0760731	0.0380365	+	0.999276i	$0.487890\pi$
$830$	−1.53710e6	−0.0774476
$831$	0	0
$832$	−2.76890e6	−0.138675
$833$	−1.77983e7	−0.888724
$834$	0	0
$835$	−1.78182e7	−0.884400
$836$	−1.86565e6	−0.0923239
$837$	0	0
$838$	−2.33772e7	−1.14996
$839$	−2.84888e7	−1.39723	−0.698617	−	0.715496i	$-0.746201\pi$
$839$	−2.84888e7	−1.39723	−0.698617	+	0.715496i	$0.746201\pi$
$840$	0	0
$841$	−1.22742e7	−0.598418
$842$	−1.57669e7	−0.766419
$843$	0	0
$844$	−4.77053e6	−0.230521
$845$	−2.65617e6	−0.127972
$846$	0	0
$847$	1.53149e6	0.0733511
$848$	−4.54758e6	−0.217166
$849$	0	0
$850$	−9.58219e6	−0.454902
$851$	1.10471e7	0.522906
$852$	0	0
$853$	1.82110e7	0.856959	0.428480	−	0.903551i	$-0.359049\pi$
$853$	1.82110e7	0.856959	0.428480	+	0.903551i	$0.359049\pi$
$854$	−2.25364e6	−0.105740
$855$	0	0
$856$	1.10632e7	0.516054
$857$	2.82204e7	1.31253	0.656267	−	0.754528i	$-0.272134\pi$
$857$	2.82204e7	1.31253	0.656267	+	0.754528i	$0.272134\pi$
$858$	0	0
$859$	2.66847e7	1.23390	0.616948	−	0.787004i	$-0.288369\pi$
$859$	2.66847e7	1.23390	0.616948	+	0.787004i	$0.288369\pi$
$860$	4.91586e6	0.226649
$861$	0	0
$862$	−2.52867e7	−1.15911
$863$	−2.98497e7	−1.36431	−0.682155	−	0.731208i	$-0.738957\pi$
$863$	−2.98497e7	−1.36431	−0.682155	+	0.731208i	$0.738957\pi$
$864$	0	0
$865$	−2.49817e6	−0.113522
$866$	2.19910e7	0.996439
$867$	0	0
$868$	−592704.	−0.0267017
$869$	2.22741e7	1.00058
$870$	0	0
$871$	−1.02941e7	−0.459774
$872$	−1.08518e7	−0.483295
$873$	0	0
$874$	1.99850e6	0.0884962
$875$	−4.42689e6	−0.195469
$876$	0	0
$877$	3.23649e7	1.42094	0.710469	−	0.703728i	$-0.248483\pi$
$877$	3.23649e7	1.42094	0.710469	+	0.703728i	$0.248483\pi$
$878$	−5.30236e6	−0.232131
$879$	0	0
$880$	−2.56333e6	−0.111583
$881$	3.03559e7	1.31766	0.658830	−	0.752291i	$-0.271051\pi$
$881$	3.03559e7	1.31766	0.658830	+	0.752291i	$0.271051\pi$
$882$	0	0
$883$	−1.11500e7	−0.481252	−0.240626	−	0.970618i	$-0.577353\pi$
$883$	−1.11500e7	−0.481252	−0.240626	+	0.970618i	$0.577353\pi$
$884$	−1.19733e7	−0.515328
$885$	0	0
$886$	−3.06641e7	−1.31234
$887$	−1.35844e6	−0.0579737	−0.0289868	−	0.999580i	$-0.509228\pi$
$887$	−1.35844e6	−0.0579737	−0.0289868	+	0.999580i	$0.509228\pi$
$888$	0	0
$889$	−6.04147e6	−0.256382
$890$	5.11128e6	0.216299
$891$	0	0
$892$	−4.93706e6	−0.207757
$893$	1.25014e6	0.0524603
$894$	0	0
$895$	−1.72803e7	−0.721098
$896$	−442368.	−0.0184083
$897$	0	0
$898$	2.00512e6	0.0829754
$899$	−3.93764e6	−0.162494
$900$	0	0
$901$	−1.96647e7	−0.807005
$902$	−1.55298e6	−0.0635551
$903$	0	0
$904$	1.59142e6	0.0647686
$905$	−2.38068e7	−0.966229
$906$	0	0
$907$	3.05036e7	1.23121	0.615607	−	0.788053i	$-0.288911\pi$
$907$	3.05036e7	1.23121	0.615607	+	0.788053i	$0.288911\pi$
$908$	−1.23373e6	−0.0496598
$909$	0	0
$910$	−2.26325e6	−0.0906001
$911$	−1.85013e7	−0.738597	−0.369298	−	0.929311i	$-0.620402\pi$
$911$	−1.85013e7	−0.738597	−0.369298	+	0.929311i	$0.620402\pi$
$912$	0	0
$913$	4.00391e6	0.158967
$914$	7.46721e6	0.295660
$915$	0	0
$916$	1.00234e7	0.394710
$917$	7.65847e6	0.300759
$918$	0	0
$919$	−3.62534e7	−1.41599	−0.707995	−	0.706217i	$-0.750400\pi$
$919$	−3.62534e7	−1.41599	−0.707995	+	0.706217i	$0.750400\pi$
$920$	2.74586e6	0.106957
$921$	0	0
$922$	2.24106e7	0.868212
$923$	2.74740e7	1.06149
$924$	0	0
$925$	1.72730e7	0.663765
$926$	9.65624e6	0.370067
$927$	0	0
$928$	−2.93888e6	−0.112024
$929$	−2.82890e7	−1.07542	−0.537710	−	0.843130i	$-0.680710\pi$
$929$	−2.82890e7	−1.07542	−0.537710	+	0.843130i	$0.680710\pi$
$930$	0	0
$931$	5.80416e6	0.219465
$932$	−2.01693e7	−0.760589
$933$	0	0
$934$	−3.37472e7	−1.26582
$935$	−1.10844e7	−0.414651
$936$	0	0
$937$	−3.99532e7	−1.48663	−0.743314	−	0.668943i	$-0.766747\pi$
$937$	−3.99532e7	−1.48663	−0.743314	+	0.668943i	$0.766747\pi$
$938$	−1.64462e6	−0.0610323
$939$	0	0
$940$	1.71765e6	0.0634037
$941$	4.07406e6	0.149987	0.0749934	−	0.997184i	$-0.476106\pi$
$941$	4.07406e6	0.149987	0.0749934	+	0.997184i	$0.476106\pi$
$942$	0	0
$943$	1.66357e6	0.0609202
$944$	−6.98112e6	−0.254974
$945$	0	0
$946$	−1.28050e7	−0.465213
$947$	−7.55423e6	−0.273725	−0.136863	−	0.990590i	$-0.543702\pi$
$947$	−7.55423e6	−0.273725	−0.136863	+	0.990590i	$0.543702\pi$
$948$	0	0
$949$	4.50966e7	1.62547
$950$	3.12482e6	0.112335
$951$	0	0
$952$	−1.91290e6	−0.0684068
$953$	−4.37822e7	−1.56158	−0.780792	−	0.624791i	$-0.785184\pi$
$953$	−4.37822e7	−1.56158	−0.780792	+	0.624791i	$0.785184\pi$
$954$	0	0
$955$	1.68224e7	0.596869
$956$	2.41682e7	0.855264
$957$	0	0
$958$	−1.36904e6	−0.0481950
$959$	6.02783e6	0.211648
$960$	0	0
$961$	−2.67468e7	−0.934249
$962$	2.15833e7	0.751936
$963$	0	0
$964$	2.02468e7	0.701718
$965$	−1.12221e7	−0.387933
$966$	0	0
$967$	−2.22403e6	−0.0764847	−0.0382424	−	0.999268i	$-0.512176\pi$
$967$	−2.22403e6	−0.0764847	−0.0382424	+	0.999268i	$0.512176\pi$
$968$	−3.63021e6	−0.124521
$969$	0	0
$970$	1.40656e7	0.479985
$971$	−4.29548e7	−1.46205	−0.731027	−	0.682349i	$-0.760959\pi$
$971$	−4.29548e7	−1.46205	−0.731027	+	0.682349i	$0.760959\pi$
$972$	0	0
$973$	4.54450e6	0.153888
$974$	−2.00280e7	−0.676457
$975$	0	0
$976$	5.34195e6	0.179505
$977$	4.20864e7	1.41061	0.705303	−	0.708906i	$-0.250811\pi$
$977$	4.20864e7	1.41061	0.705303	+	0.708906i	$0.250811\pi$
$978$	0	0
$979$	−1.33141e7	−0.443970
$980$	7.97469e6	0.265246
$981$	0	0
$982$	2.36465e7	0.782507
$983$	1.33583e7	0.440928	0.220464	−	0.975395i	$-0.429243\pi$
$983$	1.33583e7	0.440928	0.220464	+	0.975395i	$0.429243\pi$
$984$	0	0
$985$	−2.57815e7	−0.846677
$986$	−1.27084e7	−0.416291
$987$	0	0
$988$	3.90458e6	0.127257
$989$	1.37168e7	0.445926
$990$	0	0
$991$	4.08428e7	1.32109	0.660543	−	0.750788i	$-0.270326\pi$
$991$	4.08428e7	1.32109	0.660543	+	0.750788i	$0.270326\pi$
$992$	1.40493e6	0.0453289
$993$	0	0
$994$	4.38934e6	0.140907
$995$	2.41482e7	0.773264
$996$	0	0
$997$	−1.80068e7	−0.573719	−0.286860	−	0.957973i	$-0.592611\pi$
$997$	−1.80068e7	−0.573719	−0.286860	+	0.957973i	$0.592611\pi$
$998$	−3.12461e7	−0.993046
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	342.6.a.e.1.1		1
3.2	odd	2		38.6.a.a.1.1	✓	1
12.11	even	2		304.6.a.c.1.1		1
15.14	odd	2		950.6.a.b.1.1		1
57.56	even	2		722.6.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.6.a.a.1.1	✓	1	3.2	odd	2
304.6.a.c.1.1		1	12.11	even	2
342.6.a.e.1.1		1	1.1	even	1	trivial
722.6.a.b.1.1		1	57.56	even	2
950.6.a.b.1.1		1	15.14	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:	\( N \)	\(=\)	\( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	342.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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