LMFDB - Embedded newform 106.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [106,2,Mod(1,106)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(106, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("106.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 106 = 2 \cdot 53 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	106.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:	$0.846414261426$
Analytic rank:	$0$
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	$\chi$	$=$	106.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.00000 q^{6} -2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.00000 q^{6} -2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +5.00000 q^{11} +2.00000 q^{12} -4.00000 q^{13} +2.00000 q^{14} +2.00000 q^{15} +1.00000 q^{16} +3.00000 q^{17} -1.00000 q^{18} -4.00000 q^{19} +1.00000 q^{20} -4.00000 q^{21} -5.00000 q^{22} -3.00000 q^{23} -2.00000 q^{24} -4.00000 q^{25} +4.00000 q^{26} -4.00000 q^{27} -2.00000 q^{28} -6.00000 q^{29} -2.00000 q^{30} +7.00000 q^{31} -1.00000 q^{32} +10.0000 q^{33} -3.00000 q^{34} -2.00000 q^{35} +1.00000 q^{36} -6.00000 q^{37} +4.00000 q^{38} -8.00000 q^{39} -1.00000 q^{40} +2.00000 q^{41} +4.00000 q^{42} +7.00000 q^{43} +5.00000 q^{44} +1.00000 q^{45} +3.00000 q^{46} +4.00000 q^{47} +2.00000 q^{48} -3.00000 q^{49} +4.00000 q^{50} +6.00000 q^{51} -4.00000 q^{52} +1.00000 q^{53} +4.00000 q^{54} +5.00000 q^{55} +2.00000 q^{56} -8.00000 q^{57} +6.00000 q^{58} +7.00000 q^{59} +2.00000 q^{60} +2.00000 q^{61} -7.00000 q^{62} -2.00000 q^{63} +1.00000 q^{64} -4.00000 q^{65} -10.0000 q^{66} +16.0000 q^{67} +3.00000 q^{68} -6.00000 q^{69} +2.00000 q^{70} +12.0000 q^{71} -1.00000 q^{72} -12.0000 q^{73} +6.00000 q^{74} -8.00000 q^{75} -4.00000 q^{76} -10.0000 q^{77} +8.00000 q^{78} -7.00000 q^{79} +1.00000 q^{80} -11.0000 q^{81} -2.00000 q^{82} -14.0000 q^{83} -4.00000 q^{84} +3.00000 q^{85} -7.00000 q^{86} -12.0000 q^{87} -5.00000 q^{88} +17.0000 q^{89} -1.00000 q^{90} +8.00000 q^{91} -3.00000 q^{92} +14.0000 q^{93} -4.00000 q^{94} -4.00000 q^{95} -2.00000 q^{96} +3.00000 q^{97} +3.00000 q^{98} +5.00000 q^{99} +O(q^{100})\)

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	2.00000	1.15470	0.577350	−	0.816497i	$-0.304087\pi$
$3$	2.00000	1.15470	0.577350	+	0.816497i	$0.304087\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	−2.00000	−0.816497
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	5.00000	1.50756	0.753778	−	0.657129i	$-0.228229\pi$
$11$	5.00000	1.50756	0.753778	+	0.657129i	$0.228229\pi$
$12$	2.00000	0.577350
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	2.00000	0.534522
$15$	2.00000	0.516398
$16$	1.00000	0.250000
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	−1.00000	−0.235702
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	1.00000	0.223607
$21$	−4.00000	−0.872872
$22$	−5.00000	−1.06600
$23$	−3.00000	−0.625543	−0.312772	−	0.949828i	$-0.601257\pi$
$23$	−3.00000	−0.625543	−0.312772	+	0.949828i	$0.601257\pi$
$24$	−2.00000	−0.408248
$25$	−4.00000	−0.800000
$26$	4.00000	0.784465
$27$	−4.00000	−0.769800
$28$	−2.00000	−0.377964
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	−2.00000	−0.365148
$31$	7.00000	1.25724	0.628619	−	0.777714i	$-0.283621\pi$
$31$	7.00000	1.25724	0.628619	+	0.777714i	$0.283621\pi$
$32$	−1.00000	−0.176777
$33$	10.0000	1.74078
$34$	−3.00000	−0.514496
$35$	−2.00000	−0.338062
$36$	1.00000	0.166667
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	4.00000	0.648886
$39$	−8.00000	−1.28103
$40$	−1.00000	−0.158114
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	4.00000	0.617213
$43$	7.00000	1.06749	0.533745	−	0.845645i	$-0.320784\pi$
$43$	7.00000	1.06749	0.533745	+	0.845645i	$0.320784\pi$
$44$	5.00000	0.753778
$45$	1.00000	0.149071
$46$	3.00000	0.442326
$47$	4.00000	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$48$	2.00000	0.288675
$49$	−3.00000	−0.428571
$50$	4.00000	0.565685
$51$	6.00000	0.840168
$52$	−4.00000	−0.554700
$53$	1.00000	0.137361
$53$	1.00000	0.137361
$54$	4.00000	0.544331
$55$	5.00000	0.674200
$56$	2.00000	0.267261
$57$	−8.00000	−1.05963
$58$	6.00000	0.787839
$59$	7.00000	0.911322	0.455661	−	0.890153i	$-0.349403\pi$
$59$	7.00000	0.911322	0.455661	+	0.890153i	$0.349403\pi$
$60$	2.00000	0.258199
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	−7.00000	−0.889001
$63$	−2.00000	−0.251976
$64$	1.00000	0.125000
$65$	−4.00000	−0.496139
$66$	−10.0000	−1.23091
$67$	16.0000	1.95471	0.977356	−	0.211604i	$-0.0678686\pi$
$67$	16.0000	1.95471	0.977356	+	0.211604i	$0.0678686\pi$
$68$	3.00000	0.363803
$69$	−6.00000	−0.722315
$70$	2.00000	0.239046
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	−1.00000	−0.117851
$73$	−12.0000	−1.40449	−0.702247	−	0.711934i	$-0.747820\pi$
$73$	−12.0000	−1.40449	−0.702247	+	0.711934i	$0.747820\pi$
$74$	6.00000	0.697486
$75$	−8.00000	−0.923760
$76$	−4.00000	−0.458831
$77$	−10.0000	−1.13961
$78$	8.00000	0.905822
$79$	−7.00000	−0.787562	−0.393781	−	0.919204i	$-0.628833\pi$
$79$	−7.00000	−0.787562	−0.393781	+	0.919204i	$0.628833\pi$
$80$	1.00000	0.111803
$81$	−11.0000	−1.22222
$82$	−2.00000	−0.220863
$83$	−14.0000	−1.53670	−0.768350	−	0.640030i	$-0.778922\pi$
$83$	−14.0000	−1.53670	−0.768350	+	0.640030i	$0.778922\pi$
$84$	−4.00000	−0.436436
$85$	3.00000	0.325396
$86$	−7.00000	−0.754829
$87$	−12.0000	−1.28654
$88$	−5.00000	−0.533002
$89$	17.0000	1.80200	0.900998	−	0.433823i	$-0.142836\pi$
$89$	17.0000	1.80200	0.900998	+	0.433823i	$0.142836\pi$
$90$	−1.00000	−0.105409
$91$	8.00000	0.838628
$92$	−3.00000	−0.312772
$93$	14.0000	1.45173
$94$	−4.00000	−0.412568
$95$	−4.00000	−0.410391
$96$	−2.00000	−0.204124
$97$	3.00000	0.304604	0.152302	−	0.988334i	$-0.451331\pi$
$97$	3.00000	0.304604	0.152302	+	0.988334i	$0.451331\pi$
$98$	3.00000	0.303046
$99$	5.00000	0.502519
$100$	−4.00000	−0.400000
$101$	−3.00000	−0.298511	−0.149256	−	0.988799i	$-0.547688\pi$
$101$	−3.00000	−0.298511	−0.149256	+	0.988799i	$0.547688\pi$
$102$	−6.00000	−0.594089
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	4.00000	0.392232
$105$	−4.00000	−0.390360
$106$	−1.00000	−0.0971286
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	−4.00000	−0.384900
$109$	19.0000	1.81987	0.909935	−	0.414751i	$-0.136131\pi$
$109$	19.0000	1.81987	0.909935	+	0.414751i	$0.136131\pi$
$110$	−5.00000	−0.476731
$111$	−12.0000	−1.13899
$112$	−2.00000	−0.188982
$113$	11.0000	1.03479	0.517396	−	0.855746i	$-0.326901\pi$
$113$	11.0000	1.03479	0.517396	+	0.855746i	$0.326901\pi$
$114$	8.00000	0.749269
$115$	−3.00000	−0.279751
$116$	−6.00000	−0.557086
$117$	−4.00000	−0.369800
$118$	−7.00000	−0.644402
$119$	−6.00000	−0.550019
$120$	−2.00000	−0.182574
$121$	14.0000	1.27273
$122$	−2.00000	−0.181071
$123$	4.00000	0.360668
$124$	7.00000	0.628619
$125$	−9.00000	−0.804984
$126$	2.00000	0.178174
$127$	−13.0000	−1.15356	−0.576782	−	0.816898i	$-0.695692\pi$
$127$	−13.0000	−1.15356	−0.576782	+	0.816898i	$0.695692\pi$
$128$	−1.00000	−0.0883883
$129$	14.0000	1.23263
$130$	4.00000	0.350823
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	10.0000	0.870388
$133$	8.00000	0.693688
$134$	−16.0000	−1.38219
$135$	−4.00000	−0.344265
$136$	−3.00000	−0.257248
$137$	−20.0000	−1.70872	−0.854358	−	0.519685i	$-0.826049\pi$
$137$	−20.0000	−1.70872	−0.854358	+	0.519685i	$0.826049\pi$
$138$	6.00000	0.510754
$139$	16.0000	1.35710	0.678551	−	0.734553i	$-0.262608\pi$
$139$	16.0000	1.35710	0.678551	+	0.734553i	$0.262608\pi$
$140$	−2.00000	−0.169031
$141$	8.00000	0.673722
$142$	−12.0000	−1.00702
$143$	−20.0000	−1.67248
$144$	1.00000	0.0833333
$145$	−6.00000	−0.498273
$146$	12.0000	0.993127
$147$	−6.00000	−0.494872
$148$	−6.00000	−0.493197
$149$	14.0000	1.14692	0.573462	−	0.819232i	$-0.305600\pi$
$149$	14.0000	1.14692	0.573462	+	0.819232i	$0.305600\pi$
$150$	8.00000	0.653197
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	4.00000	0.324443
$153$	3.00000	0.242536
$154$	10.0000	0.805823
$155$	7.00000	0.562254
$156$	−8.00000	−0.640513
$157$	−19.0000	−1.51637	−0.758183	−	0.652042i	$-0.773912\pi$
$157$	−19.0000	−1.51637	−0.758183	+	0.652042i	$0.773912\pi$
$158$	7.00000	0.556890
$159$	2.00000	0.158610
$160$	−1.00000	−0.0790569
$161$	6.00000	0.472866
$162$	11.0000	0.864242
$163$	1.00000	0.0783260	0.0391630	−	0.999233i	$-0.487531\pi$
$163$	1.00000	0.0783260	0.0391630	+	0.999233i	$0.487531\pi$
$164$	2.00000	0.156174
$165$	10.0000	0.778499
$166$	14.0000	1.08661
$167$	3.00000	0.232147	0.116073	−	0.993241i	$-0.462969\pi$
$167$	3.00000	0.232147	0.116073	+	0.993241i	$0.462969\pi$
$168$	4.00000	0.308607
$169$	3.00000	0.230769
$170$	−3.00000	−0.230089
$171$	−4.00000	−0.305888
$172$	7.00000	0.533745
$173$	1.00000	0.0760286	0.0380143	−	0.999277i	$-0.487897\pi$
$173$	1.00000	0.0760286	0.0380143	+	0.999277i	$0.487897\pi$
$174$	12.0000	0.909718
$175$	8.00000	0.604743
$176$	5.00000	0.376889
$177$	14.0000	1.05230
$178$	−17.0000	−1.27420
$179$	−22.0000	−1.64436	−0.822179	−	0.569230i	$-0.807242\pi$
$179$	−22.0000	−1.64436	−0.822179	+	0.569230i	$0.807242\pi$
$180$	1.00000	0.0745356
$181$	−6.00000	−0.445976	−0.222988	−	0.974821i	$-0.571581\pi$
$181$	−6.00000	−0.445976	−0.222988	+	0.974821i	$0.571581\pi$
$182$	−8.00000	−0.592999
$183$	4.00000	0.295689
$184$	3.00000	0.221163
$185$	−6.00000	−0.441129
$186$	−14.0000	−1.02653
$187$	15.0000	1.09691
$188$	4.00000	0.291730
$189$	8.00000	0.581914
$190$	4.00000	0.290191
$191$	−21.0000	−1.51951	−0.759753	−	0.650211i	$-0.774680\pi$
$191$	−21.0000	−1.51951	−0.759753	+	0.650211i	$0.774680\pi$
$192$	2.00000	0.144338
$193$	−8.00000	−0.575853	−0.287926	−	0.957653i	$-0.592966\pi$
$193$	−8.00000	−0.575853	−0.287926	+	0.957653i	$0.592966\pi$
$194$	−3.00000	−0.215387
$195$	−8.00000	−0.572892
$196$	−3.00000	−0.214286
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	−5.00000	−0.355335
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	4.00000	0.282843
$201$	32.0000	2.25711
$202$	3.00000	0.211079
$203$	12.0000	0.842235
$204$	6.00000	0.420084
$205$	2.00000	0.139686
$206$	−8.00000	−0.557386
$207$	−3.00000	−0.208514
$208$	−4.00000	−0.277350
$209$	−20.0000	−1.38343
$210$	4.00000	0.276026
$211$	−5.00000	−0.344214	−0.172107	−	0.985078i	$-0.555058\pi$
$211$	−5.00000	−0.344214	−0.172107	+	0.985078i	$0.555058\pi$
$212$	1.00000	0.0686803
$213$	24.0000	1.64445
$214$	0	0
$215$	7.00000	0.477396
$216$	4.00000	0.272166
$217$	−14.0000	−0.950382
$218$	−19.0000	−1.28684
$219$	−24.0000	−1.62177
$220$	5.00000	0.337100
$221$	−12.0000	−0.807207
$222$	12.0000	0.805387
$223$	−4.00000	−0.267860	−0.133930	−	0.990991i	$-0.542760\pi$
$223$	−4.00000	−0.267860	−0.133930	+	0.990991i	$0.542760\pi$
$224$	2.00000	0.133631
$225$	−4.00000	−0.266667
$226$	−11.0000	−0.731709
$227$	−4.00000	−0.265489	−0.132745	−	0.991150i	$-0.542379\pi$
$227$	−4.00000	−0.265489	−0.132745	+	0.991150i	$0.542379\pi$
$228$	−8.00000	−0.529813
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	3.00000	0.197814
$231$	−20.0000	−1.31590
$232$	6.00000	0.393919
$233$	−12.0000	−0.786146	−0.393073	−	0.919507i	$-0.628588\pi$
$233$	−12.0000	−0.786146	−0.393073	+	0.919507i	$0.628588\pi$
$234$	4.00000	0.261488
$235$	4.00000	0.260931
$236$	7.00000	0.455661
$237$	−14.0000	−0.909398
$238$	6.00000	0.388922
$239$	9.00000	0.582162	0.291081	−	0.956698i	$-0.405985\pi$
$239$	9.00000	0.582162	0.291081	+	0.956698i	$0.405985\pi$
$240$	2.00000	0.129099
$241$	23.0000	1.48156	0.740780	−	0.671748i	$-0.234456\pi$
$241$	23.0000	1.48156	0.740780	+	0.671748i	$0.234456\pi$
$242$	−14.0000	−0.899954
$243$	−10.0000	−0.641500
$244$	2.00000	0.128037
$245$	−3.00000	−0.191663
$246$	−4.00000	−0.255031
$247$	16.0000	1.01806
$248$	−7.00000	−0.444500
$249$	−28.0000	−1.77443
$250$	9.00000	0.569210
$251$	−2.00000	−0.126239	−0.0631194	−	0.998006i	$-0.520105\pi$
$251$	−2.00000	−0.126239	−0.0631194	+	0.998006i	$0.520105\pi$
$252$	−2.00000	−0.125988
$253$	−15.0000	−0.943042
$254$	13.0000	0.815693
$255$	6.00000	0.375735
$256$	1.00000	0.0625000
$257$	2.00000	0.124757	0.0623783	−	0.998053i	$-0.480131\pi$
$257$	2.00000	0.124757	0.0623783	+	0.998053i	$0.480131\pi$
$258$	−14.0000	−0.871602
$259$	12.0000	0.745644
$260$	−4.00000	−0.248069
$261$	−6.00000	−0.371391
$262$	−12.0000	−0.741362
$263$	11.0000	0.678289	0.339145	−	0.940734i	$-0.389862\pi$
$263$	11.0000	0.678289	0.339145	+	0.940734i	$0.389862\pi$
$264$	−10.0000	−0.615457
$265$	1.00000	0.0614295
$266$	−8.00000	−0.490511
$267$	34.0000	2.08077
$268$	16.0000	0.977356
$269$	20.0000	1.21942	0.609711	−	0.792624i	$-0.291286\pi$
$269$	20.0000	1.21942	0.609711	+	0.792624i	$0.291286\pi$
$270$	4.00000	0.243432
$271$	−32.0000	−1.94386	−0.971931	−	0.235267i	$-0.924404\pi$
$271$	−32.0000	−1.94386	−0.971931	+	0.235267i	$0.924404\pi$
$272$	3.00000	0.181902
$273$	16.0000	0.968364
$274$	20.0000	1.20824
$275$	−20.0000	−1.20605
$276$	−6.00000	−0.361158
$277$	19.0000	1.14160	0.570800	−	0.821089i	$-0.306633\pi$
$277$	19.0000	1.14160	0.570800	+	0.821089i	$0.306633\pi$
$278$	−16.0000	−0.959616
$279$	7.00000	0.419079
$280$	2.00000	0.119523
$281$	−11.0000	−0.656205	−0.328102	−	0.944642i	$-0.606409\pi$
$281$	−11.0000	−0.656205	−0.328102	+	0.944642i	$0.606409\pi$
$282$	−8.00000	−0.476393
$283$	18.0000	1.06999	0.534994	−	0.844856i	$-0.320314\pi$
$283$	18.0000	1.06999	0.534994	+	0.844856i	$0.320314\pi$
$284$	12.0000	0.712069
$285$	−8.00000	−0.473879
$286$	20.0000	1.18262
$287$	−4.00000	−0.236113
$288$	−1.00000	−0.0589256
$289$	−8.00000	−0.470588
$290$	6.00000	0.352332
$291$	6.00000	0.351726
$292$	−12.0000	−0.702247
$293$	16.0000	0.934730	0.467365	−	0.884064i	$-0.345203\pi$
$293$	16.0000	0.934730	0.467365	+	0.884064i	$0.345203\pi$
$294$	6.00000	0.349927
$295$	7.00000	0.407556
$296$	6.00000	0.348743
$297$	−20.0000	−1.16052
$298$	−14.0000	−0.810998
$299$	12.0000	0.693978
$300$	−8.00000	−0.461880
$301$	−14.0000	−0.806947
$302$	16.0000	0.920697
$303$	−6.00000	−0.344691
$304$	−4.00000	−0.229416
$305$	2.00000	0.114520
$306$	−3.00000	−0.171499
$307$	17.0000	0.970241	0.485121	−	0.874447i	$-0.338776\pi$
$307$	17.0000	0.970241	0.485121	+	0.874447i	$0.338776\pi$
$308$	−10.0000	−0.569803
$309$	16.0000	0.910208
$310$	−7.00000	−0.397573
$311$	16.0000	0.907277	0.453638	−	0.891186i	$-0.350126\pi$
$311$	16.0000	0.907277	0.453638	+	0.891186i	$0.350126\pi$
$312$	8.00000	0.452911
$313$	−10.0000	−0.565233	−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000	−0.565233	−0.282617	+	0.959233i	$0.591202\pi$
$314$	19.0000	1.07223
$315$	−2.00000	−0.112687
$316$	−7.00000	−0.393781
$317$	32.0000	1.79730	0.898650	−	0.438667i	$-0.144549\pi$
$317$	32.0000	1.79730	0.898650	+	0.438667i	$0.144549\pi$
$318$	−2.00000	−0.112154
$319$	−30.0000	−1.67968
$320$	1.00000	0.0559017
$321$	0	0
$322$	−6.00000	−0.334367
$323$	−12.0000	−0.667698
$324$	−11.0000	−0.611111
$325$	16.0000	0.887520
$326$	−1.00000	−0.0553849
$327$	38.0000	2.10140
$328$	−2.00000	−0.110432
$329$	−8.00000	−0.441054
$330$	−10.0000	−0.550482
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	−14.0000	−0.768350
$333$	−6.00000	−0.328798
$334$	−3.00000	−0.164153
$335$	16.0000	0.874173
$336$	−4.00000	−0.218218
$337$	−14.0000	−0.762629	−0.381314	−	0.924445i	$-0.624528\pi$
$337$	−14.0000	−0.762629	−0.381314	+	0.924445i	$0.624528\pi$
$338$	−3.00000	−0.163178
$339$	22.0000	1.19488
$340$	3.00000	0.162698
$341$	35.0000	1.89536
$342$	4.00000	0.216295
$343$	20.0000	1.07990
$344$	−7.00000	−0.377415
$345$	−6.00000	−0.323029
$346$	−1.00000	−0.0537603
$347$	−29.0000	−1.55680	−0.778401	−	0.627768i	$-0.783969\pi$
$347$	−29.0000	−1.55680	−0.778401	+	0.627768i	$0.783969\pi$
$348$	−12.0000	−0.643268
$349$	1.00000	0.0535288	0.0267644	−	0.999642i	$-0.491480\pi$
$349$	1.00000	0.0535288	0.0267644	+	0.999642i	$0.491480\pi$
$350$	−8.00000	−0.427618
$351$	16.0000	0.854017
$352$	−5.00000	−0.266501
$353$	4.00000	0.212899	0.106449	−	0.994318i	$-0.466052\pi$
$353$	4.00000	0.212899	0.106449	+	0.994318i	$0.466052\pi$
$354$	−14.0000	−0.744092
$355$	12.0000	0.636894
$356$	17.0000	0.900998
$357$	−12.0000	−0.635107
$358$	22.0000	1.16274
$359$	3.00000	0.158334	0.0791670	−	0.996861i	$-0.474774\pi$
$359$	3.00000	0.158334	0.0791670	+	0.996861i	$0.474774\pi$
$360$	−1.00000	−0.0527046
$361$	−3.00000	−0.157895
$362$	6.00000	0.315353
$363$	28.0000	1.46962
$364$	8.00000	0.419314
$365$	−12.0000	−0.628109
$366$	−4.00000	−0.209083
$367$	−4.00000	−0.208798	−0.104399	−	0.994535i	$-0.533292\pi$
$367$	−4.00000	−0.208798	−0.104399	+	0.994535i	$0.533292\pi$
$368$	−3.00000	−0.156386
$369$	2.00000	0.104116
$370$	6.00000	0.311925
$371$	−2.00000	−0.103835
$372$	14.0000	0.725866
$373$	15.0000	0.776671	0.388335	−	0.921518i	$-0.373050\pi$
$373$	15.0000	0.776671	0.388335	+	0.921518i	$0.373050\pi$
$374$	−15.0000	−0.775632
$375$	−18.0000	−0.929516
$376$	−4.00000	−0.206284
$377$	24.0000	1.23606
$378$	−8.00000	−0.411476
$379$	−16.0000	−0.821865	−0.410932	−	0.911666i	$-0.634797\pi$
$379$	−16.0000	−0.821865	−0.410932	+	0.911666i	$0.634797\pi$
$380$	−4.00000	−0.205196
$381$	−26.0000	−1.33202
$382$	21.0000	1.07445
$383$	−9.00000	−0.459879	−0.229939	−	0.973205i	$-0.573853\pi$
$383$	−9.00000	−0.459879	−0.229939	+	0.973205i	$0.573853\pi$
$384$	−2.00000	−0.102062
$385$	−10.0000	−0.509647
$386$	8.00000	0.407189
$387$	7.00000	0.355830
$388$	3.00000	0.152302
$389$	−13.0000	−0.659126	−0.329563	−	0.944134i	$-0.606901\pi$
$389$	−13.0000	−0.659126	−0.329563	+	0.944134i	$0.606901\pi$
$390$	8.00000	0.405096
$391$	−9.00000	−0.455150
$392$	3.00000	0.151523
$393$	24.0000	1.21064
$394$	−6.00000	−0.302276
$395$	−7.00000	−0.352208
$396$	5.00000	0.251259
$397$	−27.0000	−1.35509	−0.677546	−	0.735481i	$-0.736956\pi$
$397$	−27.0000	−1.35509	−0.677546	+	0.735481i	$0.736956\pi$
$398$	4.00000	0.200502
$399$	16.0000	0.801002
$400$	−4.00000	−0.200000
$401$	6.00000	0.299626	0.149813	−	0.988714i	$-0.452133\pi$
$401$	6.00000	0.299626	0.149813	+	0.988714i	$0.452133\pi$
$402$	−32.0000	−1.59601
$403$	−28.0000	−1.39478
$404$	−3.00000	−0.149256
$405$	−11.0000	−0.546594
$406$	−12.0000	−0.595550
$407$	−30.0000	−1.48704
$408$	−6.00000	−0.297044
$409$	−11.0000	−0.543915	−0.271957	−	0.962309i	$-0.587671\pi$
$409$	−11.0000	−0.543915	−0.271957	+	0.962309i	$0.587671\pi$
$410$	−2.00000	−0.0987730
$411$	−40.0000	−1.97305
$412$	8.00000	0.394132
$413$	−14.0000	−0.688895
$414$	3.00000	0.147442
$415$	−14.0000	−0.687233
$416$	4.00000	0.196116
$417$	32.0000	1.56705
$418$	20.0000	0.978232
$419$	−26.0000	−1.27018	−0.635092	−	0.772437i	$-0.719038\pi$
$419$	−26.0000	−1.27018	−0.635092	+	0.772437i	$0.719038\pi$
$420$	−4.00000	−0.195180
$421$	−13.0000	−0.633581	−0.316791	−	0.948495i	$-0.602605\pi$
$421$	−13.0000	−0.633581	−0.316791	+	0.948495i	$0.602605\pi$
$422$	5.00000	0.243396
$423$	4.00000	0.194487
$424$	−1.00000	−0.0485643
$425$	−12.0000	−0.582086
$426$	−24.0000	−1.16280
$427$	−4.00000	−0.193574
$428$	0	0
$429$	−40.0000	−1.93122
$430$	−7.00000	−0.337570
$431$	8.00000	0.385346	0.192673	−	0.981263i	$-0.438284\pi$
$431$	8.00000	0.385346	0.192673	+	0.981263i	$0.438284\pi$
$432$	−4.00000	−0.192450
$433$	−14.0000	−0.672797	−0.336399	−	0.941720i	$-0.609209\pi$
$433$	−14.0000	−0.672797	−0.336399	+	0.941720i	$0.609209\pi$
$434$	14.0000	0.672022
$435$	−12.0000	−0.575356
$436$	19.0000	0.909935
$437$	12.0000	0.574038
$438$	24.0000	1.14676
$439$	22.0000	1.05000	0.525001	−	0.851101i	$-0.324065\pi$
$439$	22.0000	1.05000	0.525001	+	0.851101i	$0.324065\pi$
$440$	−5.00000	−0.238366
$441$	−3.00000	−0.142857
$442$	12.0000	0.570782
$443$	4.00000	0.190046	0.0950229	−	0.995475i	$-0.469708\pi$
$443$	4.00000	0.190046	0.0950229	+	0.995475i	$0.469708\pi$
$444$	−12.0000	−0.569495
$445$	17.0000	0.805877
$446$	4.00000	0.189405
$447$	28.0000	1.32435
$448$	−2.00000	−0.0944911
$449$	−14.0000	−0.660701	−0.330350	−	0.943858i	$-0.607167\pi$
$449$	−14.0000	−0.660701	−0.330350	+	0.943858i	$0.607167\pi$
$450$	4.00000	0.188562
$451$	10.0000	0.470882
$452$	11.0000	0.517396
$453$	−32.0000	−1.50349
$454$	4.00000	0.187729
$455$	8.00000	0.375046
$456$	8.00000	0.374634
$457$	−26.0000	−1.21623	−0.608114	−	0.793849i	$-0.708074\pi$
$457$	−26.0000	−1.21623	−0.608114	+	0.793849i	$0.708074\pi$
$458$	14.0000	0.654177
$459$	−12.0000	−0.560112
$460$	−3.00000	−0.139876
$461$	38.0000	1.76984	0.884918	−	0.465746i	$-0.154214\pi$
$461$	38.0000	1.76984	0.884918	+	0.465746i	$0.154214\pi$
$462$	20.0000	0.930484
$463$	19.0000	0.883005	0.441502	−	0.897260i	$-0.354446\pi$
$463$	19.0000	0.883005	0.441502	+	0.897260i	$0.354446\pi$
$464$	−6.00000	−0.278543
$465$	14.0000	0.649234
$466$	12.0000	0.555889
$467$	36.0000	1.66588	0.832941	−	0.553362i	$-0.186655\pi$
$467$	36.0000	1.66588	0.832941	+	0.553362i	$0.186655\pi$
$468$	−4.00000	−0.184900
$469$	−32.0000	−1.47762
$470$	−4.00000	−0.184506
$471$	−38.0000	−1.75095
$472$	−7.00000	−0.322201
$473$	35.0000	1.60930
$474$	14.0000	0.643041
$475$	16.0000	0.734130
$476$	−6.00000	−0.275010
$477$	1.00000	0.0457869
$478$	−9.00000	−0.411650
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	−2.00000	−0.0912871
$481$	24.0000	1.09431
$482$	−23.0000	−1.04762
$483$	12.0000	0.546019
$484$	14.0000	0.636364
$485$	3.00000	0.136223
$486$	10.0000	0.453609
$487$	−4.00000	−0.181257	−0.0906287	−	0.995885i	$-0.528888\pi$
$487$	−4.00000	−0.181257	−0.0906287	+	0.995885i	$0.528888\pi$
$488$	−2.00000	−0.0905357
$489$	2.00000	0.0904431
$490$	3.00000	0.135526
$491$	−2.00000	−0.0902587	−0.0451294	−	0.998981i	$-0.514370\pi$
$491$	−2.00000	−0.0902587	−0.0451294	+	0.998981i	$0.514370\pi$
$492$	4.00000	0.180334
$493$	−18.0000	−0.810679
$494$	−16.0000	−0.719874
$495$	5.00000	0.224733
$496$	7.00000	0.314309
$497$	−24.0000	−1.07655
$498$	28.0000	1.25471
$499$	16.0000	0.716258	0.358129	−	0.933672i	$-0.383415\pi$
$499$	16.0000	0.716258	0.358129	+	0.933672i	$0.383415\pi$
$500$	−9.00000	−0.402492
$501$	6.00000	0.268060
$502$	2.00000	0.0892644
$503$	−8.00000	−0.356702	−0.178351	−	0.983967i	$-0.557076\pi$
$503$	−8.00000	−0.356702	−0.178351	+	0.983967i	$0.557076\pi$
$504$	2.00000	0.0890871
$505$	−3.00000	−0.133498
$506$	15.0000	0.666831
$507$	6.00000	0.266469
$508$	−13.0000	−0.576782
$509$	10.0000	0.443242	0.221621	−	0.975133i	$-0.428865\pi$
$509$	10.0000	0.443242	0.221621	+	0.975133i	$0.428865\pi$
$510$	−6.00000	−0.265684
$511$	24.0000	1.06170
$512$	−1.00000	−0.0441942
$513$	16.0000	0.706417
$514$	−2.00000	−0.0882162
$515$	8.00000	0.352522
$516$	14.0000	0.616316
$517$	20.0000	0.879599
$518$	−12.0000	−0.527250
$519$	2.00000	0.0877903
$520$	4.00000	0.175412
$521$	13.0000	0.569540	0.284770	−	0.958596i	$-0.408083\pi$
$521$	13.0000	0.569540	0.284770	+	0.958596i	$0.408083\pi$
$522$	6.00000	0.262613
$523$	−7.00000	−0.306089	−0.153044	−	0.988219i	$-0.548908\pi$
$523$	−7.00000	−0.306089	−0.153044	+	0.988219i	$0.548908\pi$
$524$	12.0000	0.524222
$525$	16.0000	0.698297
$526$	−11.0000	−0.479623
$527$	21.0000	0.914774
$528$	10.0000	0.435194
$529$	−14.0000	−0.608696
$530$	−1.00000	−0.0434372
$531$	7.00000	0.303774
$532$	8.00000	0.346844
$533$	−8.00000	−0.346518
$534$	−34.0000	−1.47132
$535$	0	0
$536$	−16.0000	−0.691095
$537$	−44.0000	−1.89874
$538$	−20.0000	−0.862261
$539$	−15.0000	−0.646096
$540$	−4.00000	−0.172133
$541$	−38.0000	−1.63375	−0.816874	−	0.576816i	$-0.804295\pi$
$541$	−38.0000	−1.63375	−0.816874	+	0.576816i	$0.804295\pi$
$542$	32.0000	1.37452
$543$	−12.0000	−0.514969
$544$	−3.00000	−0.128624
$545$	19.0000	0.813871
$546$	−16.0000	−0.684737
$547$	0	0		−	1.00000i	$-0.5\pi$
$547$	0	0			1.00000i	$0.5\pi$
$548$	−20.0000	−0.854358
$549$	2.00000	0.0853579
$550$	20.0000	0.852803
$551$	24.0000	1.02243
$552$	6.00000	0.255377
$553$	14.0000	0.595341
$554$	−19.0000	−0.807233
$555$	−12.0000	−0.509372
$556$	16.0000	0.678551
$557$	3.00000	0.127114	0.0635570	−	0.997978i	$-0.479756\pi$
$557$	3.00000	0.127114	0.0635570	+	0.997978i	$0.479756\pi$
$558$	−7.00000	−0.296334
$559$	−28.0000	−1.18427
$560$	−2.00000	−0.0845154
$561$	30.0000	1.26660
$562$	11.0000	0.464007
$563$	16.0000	0.674320	0.337160	−	0.941447i	$-0.390534\pi$
$563$	16.0000	0.674320	0.337160	+	0.941447i	$0.390534\pi$
$564$	8.00000	0.336861
$565$	11.0000	0.462773
$566$	−18.0000	−0.756596
$567$	22.0000	0.923913
$568$	−12.0000	−0.503509
$569$	−24.0000	−1.00613	−0.503066	−	0.864248i	$-0.667795\pi$
$569$	−24.0000	−1.00613	−0.503066	+	0.864248i	$0.667795\pi$
$570$	8.00000	0.335083
$571$	−26.0000	−1.08807	−0.544033	−	0.839064i	$-0.683103\pi$
$571$	−26.0000	−1.08807	−0.544033	+	0.839064i	$0.683103\pi$
$572$	−20.0000	−0.836242
$573$	−42.0000	−1.75458
$574$	4.00000	0.166957
$575$	12.0000	0.500435
$576$	1.00000	0.0416667
$577$	−3.00000	−0.124892	−0.0624458	−	0.998048i	$-0.519890\pi$
$577$	−3.00000	−0.124892	−0.0624458	+	0.998048i	$0.519890\pi$
$578$	8.00000	0.332756
$579$	−16.0000	−0.664937
$580$	−6.00000	−0.249136
$581$	28.0000	1.16164
$582$	−6.00000	−0.248708
$583$	5.00000	0.207079
$584$	12.0000	0.496564
$585$	−4.00000	−0.165380
$586$	−16.0000	−0.660954
$587$	29.0000	1.19696	0.598479	−	0.801138i	$-0.295772\pi$
$587$	29.0000	1.19696	0.598479	+	0.801138i	$0.295772\pi$
$588$	−6.00000	−0.247436
$589$	−28.0000	−1.15372
$590$	−7.00000	−0.288185
$591$	12.0000	0.493614
$592$	−6.00000	−0.246598
$593$	−6.00000	−0.246390	−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000	−0.246390	−0.123195	+	0.992382i	$0.539314\pi$
$594$	20.0000	0.820610
$595$	−6.00000	−0.245976
$596$	14.0000	0.573462
$597$	−8.00000	−0.327418
$598$	−12.0000	−0.490716
$599$	−10.0000	−0.408589	−0.204294	−	0.978909i	$-0.565490\pi$
$599$	−10.0000	−0.408589	−0.204294	+	0.978909i	$0.565490\pi$
$600$	8.00000	0.326599
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	14.0000	0.570597
$603$	16.0000	0.651570
$604$	−16.0000	−0.651031
$605$	14.0000	0.569181
$606$	6.00000	0.243733
$607$	42.0000	1.70473	0.852364	−	0.522949i	$-0.175168\pi$
$607$	42.0000	1.70473	0.852364	+	0.522949i	$0.175168\pi$
$608$	4.00000	0.162221
$609$	24.0000	0.972529
$610$	−2.00000	−0.0809776
$611$	−16.0000	−0.647291
$612$	3.00000	0.121268
$613$	39.0000	1.57520	0.787598	−	0.616190i	$-0.211325\pi$
$613$	39.0000	1.57520	0.787598	+	0.616190i	$0.211325\pi$
$614$	−17.0000	−0.686064
$615$	4.00000	0.161296
$616$	10.0000	0.402911
$617$	−36.0000	−1.44931	−0.724653	−	0.689114i	$-0.758000\pi$
$617$	−36.0000	−1.44931	−0.724653	+	0.689114i	$0.758000\pi$
$618$	−16.0000	−0.643614
$619$	3.00000	0.120580	0.0602901	−	0.998181i	$-0.480797\pi$
$619$	3.00000	0.120580	0.0602901	+	0.998181i	$0.480797\pi$
$620$	7.00000	0.281127
$621$	12.0000	0.481543
$622$	−16.0000	−0.641542
$623$	−34.0000	−1.36218
$624$	−8.00000	−0.320256
$625$	11.0000	0.440000
$626$	10.0000	0.399680
$627$	−40.0000	−1.59745
$628$	−19.0000	−0.758183
$629$	−18.0000	−0.717707
$630$	2.00000	0.0796819
$631$	−8.00000	−0.318475	−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000	−0.318475	−0.159237	+	0.987240i	$0.550904\pi$
$632$	7.00000	0.278445
$633$	−10.0000	−0.397464
$634$	−32.0000	−1.27088
$635$	−13.0000	−0.515889
$636$	2.00000	0.0793052
$637$	12.0000	0.475457
$638$	30.0000	1.18771
$639$	12.0000	0.474713
$640$	−1.00000	−0.0395285
$641$	38.0000	1.50091	0.750455	−	0.660922i	$-0.229834\pi$
$641$	38.0000	1.50091	0.750455	+	0.660922i	$0.229834\pi$
$642$	0	0
$643$	35.0000	1.38027	0.690133	−	0.723683i	$-0.257552\pi$
$643$	35.0000	1.38027	0.690133	+	0.723683i	$0.257552\pi$
$644$	6.00000	0.236433
$645$	14.0000	0.551249
$646$	12.0000	0.472134
$647$	46.0000	1.80845	0.904223	−	0.427060i	$-0.140451\pi$
$647$	46.0000	1.80845	0.904223	+	0.427060i	$0.140451\pi$
$648$	11.0000	0.432121
$649$	35.0000	1.37387
$650$	−16.0000	−0.627572
$651$	−28.0000	−1.09741
$652$	1.00000	0.0391630
$653$	−8.00000	−0.313064	−0.156532	−	0.987673i	$-0.550031\pi$
$653$	−8.00000	−0.313064	−0.156532	+	0.987673i	$0.550031\pi$
$654$	−38.0000	−1.48592
$655$	12.0000	0.468879
$656$	2.00000	0.0780869
$657$	−12.0000	−0.468165
$658$	8.00000	0.311872
$659$	−10.0000	−0.389545	−0.194772	−	0.980848i	$-0.562397\pi$
$659$	−10.0000	−0.389545	−0.194772	+	0.980848i	$0.562397\pi$
$660$	10.0000	0.389249
$661$	−40.0000	−1.55582	−0.777910	−	0.628376i	$-0.783720\pi$
$661$	−40.0000	−1.55582	−0.777910	+	0.628376i	$0.783720\pi$
$662$	4.00000	0.155464
$663$	−24.0000	−0.932083
$664$	14.0000	0.543305
$665$	8.00000	0.310227
$666$	6.00000	0.232495
$667$	18.0000	0.696963
$668$	3.00000	0.116073
$669$	−8.00000	−0.309298
$670$	−16.0000	−0.618134
$671$	10.0000	0.386046
$672$	4.00000	0.154303
$673$	−19.0000	−0.732396	−0.366198	−	0.930537i	$-0.619341\pi$
$673$	−19.0000	−0.732396	−0.366198	+	0.930537i	$0.619341\pi$
$674$	14.0000	0.539260
$675$	16.0000	0.615840
$676$	3.00000	0.115385
$677$	−26.0000	−0.999261	−0.499631	−	0.866239i	$-0.666531\pi$
$677$	−26.0000	−0.999261	−0.499631	+	0.866239i	$0.666531\pi$
$678$	−22.0000	−0.844905
$679$	−6.00000	−0.230259
$680$	−3.00000	−0.115045
$681$	−8.00000	−0.306561
$682$	−35.0000	−1.34022
$683$	−33.0000	−1.26271	−0.631355	−	0.775494i	$-0.717501\pi$
$683$	−33.0000	−1.26271	−0.631355	+	0.775494i	$0.717501\pi$
$684$	−4.00000	−0.152944
$685$	−20.0000	−0.764161
$686$	−20.0000	−0.763604
$687$	−28.0000	−1.06827
$688$	7.00000	0.266872
$689$	−4.00000	−0.152388
$690$	6.00000	0.228416
$691$	−40.0000	−1.52167	−0.760836	−	0.648944i	$-0.775211\pi$
$691$	−40.0000	−1.52167	−0.760836	+	0.648944i	$0.775211\pi$
$692$	1.00000	0.0380143
$693$	−10.0000	−0.379869
$694$	29.0000	1.10082
$695$	16.0000	0.606915
$696$	12.0000	0.454859
$697$	6.00000	0.227266
$698$	−1.00000	−0.0378506
$699$	−24.0000	−0.907763
$700$	8.00000	0.302372
$701$	−46.0000	−1.73740	−0.868698	−	0.495342i	$-0.835043\pi$
$701$	−46.0000	−1.73740	−0.868698	+	0.495342i	$0.835043\pi$
$702$	−16.0000	−0.603881
$703$	24.0000	0.905177
$704$	5.00000	0.188445
$705$	8.00000	0.301297
$706$	−4.00000	−0.150542
$707$	6.00000	0.225653
$708$	14.0000	0.526152
$709$	23.0000	0.863783	0.431892	−	0.901926i	$-0.357846\pi$
$709$	23.0000	0.863783	0.431892	+	0.901926i	$0.357846\pi$
$710$	−12.0000	−0.450352
$711$	−7.00000	−0.262521
$712$	−17.0000	−0.637102
$713$	−21.0000	−0.786456
$714$	12.0000	0.449089
$715$	−20.0000	−0.747958
$716$	−22.0000	−0.822179
$717$	18.0000	0.672222
$718$	−3.00000	−0.111959
$719$	28.0000	1.04422	0.522112	−	0.852877i	$-0.325144\pi$
$719$	28.0000	1.04422	0.522112	+	0.852877i	$0.325144\pi$
$720$	1.00000	0.0372678
$721$	−16.0000	−0.595871
$722$	3.00000	0.111648
$723$	46.0000	1.71076
$724$	−6.00000	−0.222988
$725$	24.0000	0.891338
$726$	−28.0000	−1.03918
$727$	−24.0000	−0.890111	−0.445055	−	0.895503i	$-0.646816\pi$
$727$	−24.0000	−0.890111	−0.445055	+	0.895503i	$0.646816\pi$
$728$	−8.00000	−0.296500
$729$	13.0000	0.481481
$730$	12.0000	0.444140
$731$	21.0000	0.776713
$732$	4.00000	0.147844
$733$	52.0000	1.92066	0.960332	−	0.278859i	$-0.0899564\pi$
$733$	52.0000	1.92066	0.960332	+	0.278859i	$0.0899564\pi$
$734$	4.00000	0.147643
$735$	−6.00000	−0.221313
$736$	3.00000	0.110581
$737$	80.0000	2.94684
$738$	−2.00000	−0.0736210
$739$	0	0		−	1.00000i	$-0.5\pi$
$739$	0	0			1.00000i	$0.5\pi$
$740$	−6.00000	−0.220564
$741$	32.0000	1.17555
$742$	2.00000	0.0734223
$743$	−6.00000	−0.220119	−0.110059	−	0.993925i	$-0.535104\pi$
$743$	−6.00000	−0.220119	−0.110059	+	0.993925i	$0.535104\pi$
$744$	−14.0000	−0.513265
$745$	14.0000	0.512920
$746$	−15.0000	−0.549189
$747$	−14.0000	−0.512233
$748$	15.0000	0.548454
$749$	0	0
$750$	18.0000	0.657267
$751$	−2.00000	−0.0729810	−0.0364905	−	0.999334i	$-0.511618\pi$
$751$	−2.00000	−0.0729810	−0.0364905	+	0.999334i	$0.511618\pi$
$752$	4.00000	0.145865
$753$	−4.00000	−0.145768
$754$	−24.0000	−0.874028
$755$	−16.0000	−0.582300
$756$	8.00000	0.290957
$757$	−40.0000	−1.45382	−0.726912	−	0.686730i	$-0.759045\pi$
$757$	−40.0000	−1.45382	−0.726912	+	0.686730i	$0.759045\pi$
$758$	16.0000	0.581146
$759$	−30.0000	−1.08893
$760$	4.00000	0.145095
$761$	18.0000	0.652499	0.326250	−	0.945284i	$-0.394215\pi$
$761$	18.0000	0.652499	0.326250	+	0.945284i	$0.394215\pi$
$762$	26.0000	0.941881
$763$	−38.0000	−1.37569
$764$	−21.0000	−0.759753
$765$	3.00000	0.108465
$766$	9.00000	0.325183
$767$	−28.0000	−1.01102
$768$	2.00000	0.0721688
$769$	26.0000	0.937584	0.468792	−	0.883309i	$-0.344689\pi$
$769$	26.0000	0.937584	0.468792	+	0.883309i	$0.344689\pi$
$770$	10.0000	0.360375
$771$	4.00000	0.144056
$772$	−8.00000	−0.287926
$773$	21.0000	0.755318	0.377659	−	0.925945i	$-0.376729\pi$
$773$	21.0000	0.755318	0.377659	+	0.925945i	$0.376729\pi$
$774$	−7.00000	−0.251610
$775$	−28.0000	−1.00579
$776$	−3.00000	−0.107694
$777$	24.0000	0.860995
$778$	13.0000	0.466073
$779$	−8.00000	−0.286630
$780$	−8.00000	−0.286446
$781$	60.0000	2.14697
$782$	9.00000	0.321839
$783$	24.0000	0.857690
$784$	−3.00000	−0.107143
$785$	−19.0000	−0.678139
$786$	−24.0000	−0.856052
$787$	−2.00000	−0.0712923	−0.0356462	−	0.999364i	$-0.511349\pi$
$787$	−2.00000	−0.0712923	−0.0356462	+	0.999364i	$0.511349\pi$
$788$	6.00000	0.213741
$789$	22.0000	0.783221
$790$	7.00000	0.249049
$791$	−22.0000	−0.782230
$792$	−5.00000	−0.177667
$793$	−8.00000	−0.284088
$794$	27.0000	0.958194
$795$	2.00000	0.0709327
$796$	−4.00000	−0.141776
$797$	−17.0000	−0.602171	−0.301085	−	0.953597i	$-0.597349\pi$
$797$	−17.0000	−0.602171	−0.301085	+	0.953597i	$0.597349\pi$
$798$	−16.0000	−0.566394
$799$	12.0000	0.424529
$800$	4.00000	0.141421
$801$	17.0000	0.600665
$802$	−6.00000	−0.211867
$803$	−60.0000	−2.11735
$804$	32.0000	1.12855
$805$	6.00000	0.211472
$806$	28.0000	0.986258
$807$	40.0000	1.40807
$808$	3.00000	0.105540
$809$	22.0000	0.773479	0.386739	−	0.922189i	$-0.373601\pi$
$809$	22.0000	0.773479	0.386739	+	0.922189i	$0.373601\pi$
$810$	11.0000	0.386501
$811$	15.0000	0.526721	0.263361	−	0.964697i	$-0.415169\pi$
$811$	15.0000	0.526721	0.263361	+	0.964697i	$0.415169\pi$
$812$	12.0000	0.421117
$813$	−64.0000	−2.24458
$814$	30.0000	1.05150
$815$	1.00000	0.0350285
$816$	6.00000	0.210042
$817$	−28.0000	−0.979596
$818$	11.0000	0.384606
$819$	8.00000	0.279543
$820$	2.00000	0.0698430
$821$	25.0000	0.872506	0.436253	−	0.899824i	$-0.356305\pi$
$821$	25.0000	0.872506	0.436253	+	0.899824i	$0.356305\pi$
$822$	40.0000	1.39516
$823$	14.0000	0.488009	0.244005	−	0.969774i	$-0.421539\pi$
$823$	14.0000	0.488009	0.244005	+	0.969774i	$0.421539\pi$
$824$	−8.00000	−0.278693
$825$	−40.0000	−1.39262
$826$	14.0000	0.487122
$827$	28.0000	0.973655	0.486828	−	0.873498i	$-0.338154\pi$
$827$	28.0000	0.973655	0.486828	+	0.873498i	$0.338154\pi$
$828$	−3.00000	−0.104257
$829$	37.0000	1.28506	0.642532	−	0.766259i	$-0.277884\pi$
$829$	37.0000	1.28506	0.642532	+	0.766259i	$0.277884\pi$
$830$	14.0000	0.485947
$831$	38.0000	1.31821
$832$	−4.00000	−0.138675
$833$	−9.00000	−0.311832
$834$	−32.0000	−1.10807
$835$	3.00000	0.103819
$836$	−20.0000	−0.691714
$837$	−28.0000	−0.967822
$838$	26.0000	0.898155
$839$	−12.0000	−0.414286	−0.207143	−	0.978311i	$-0.566417\pi$
$839$	−12.0000	−0.414286	−0.207143	+	0.978311i	$0.566417\pi$
$840$	4.00000	0.138013
$841$	7.00000	0.241379
$842$	13.0000	0.448010
$843$	−22.0000	−0.757720
$844$	−5.00000	−0.172107
$845$	3.00000	0.103203
$846$	−4.00000	−0.137523
$847$	−28.0000	−0.962091
$848$	1.00000	0.0343401
$849$	36.0000	1.23552
$850$	12.0000	0.411597
$851$	18.0000	0.617032
$852$	24.0000	0.822226
$853$	−14.0000	−0.479351	−0.239675	−	0.970853i	$-0.577041\pi$
$853$	−14.0000	−0.479351	−0.239675	+	0.970853i	$0.577041\pi$
$854$	4.00000	0.136877
$855$	−4.00000	−0.136797
$856$	0	0
$857$	−33.0000	−1.12726	−0.563629	−	0.826028i	$-0.690595\pi$
$857$	−33.0000	−1.12726	−0.563629	+	0.826028i	$0.690595\pi$
$858$	40.0000	1.36558
$859$	−29.0000	−0.989467	−0.494734	−	0.869045i	$-0.664734\pi$
$859$	−29.0000	−0.989467	−0.494734	+	0.869045i	$0.664734\pi$
$860$	7.00000	0.238698
$861$	−8.00000	−0.272639
$862$	−8.00000	−0.272481
$863$	−12.0000	−0.408485	−0.204242	−	0.978920i	$-0.565473\pi$
$863$	−12.0000	−0.408485	−0.204242	+	0.978920i	$0.565473\pi$
$864$	4.00000	0.136083
$865$	1.00000	0.0340010
$866$	14.0000	0.475739
$867$	−16.0000	−0.543388
$868$	−14.0000	−0.475191
$869$	−35.0000	−1.18729
$870$	12.0000	0.406838
$871$	−64.0000	−2.16856
$872$	−19.0000	−0.643421
$873$	3.00000	0.101535
$874$	−12.0000	−0.405906
$875$	18.0000	0.608511
$876$	−24.0000	−0.810885
$877$	50.0000	1.68838	0.844190	−	0.536044i	$-0.180082\pi$
$877$	50.0000	1.68838	0.844190	+	0.536044i	$0.180082\pi$
$878$	−22.0000	−0.742464
$879$	32.0000	1.07933
$880$	5.00000	0.168550
$881$	−8.00000	−0.269527	−0.134763	−	0.990878i	$-0.543027\pi$
$881$	−8.00000	−0.269527	−0.134763	+	0.990878i	$0.543027\pi$
$882$	3.00000	0.101015
$883$	4.00000	0.134611	0.0673054	−	0.997732i	$-0.478560\pi$
$883$	4.00000	0.134611	0.0673054	+	0.997732i	$0.478560\pi$
$884$	−12.0000	−0.403604
$885$	14.0000	0.470605
$886$	−4.00000	−0.134383
$887$	13.0000	0.436497	0.218249	−	0.975893i	$-0.429966\pi$
$887$	13.0000	0.436497	0.218249	+	0.975893i	$0.429966\pi$
$888$	12.0000	0.402694
$889$	26.0000	0.872012
$890$	−17.0000	−0.569841
$891$	−55.0000	−1.84257
$892$	−4.00000	−0.133930
$893$	−16.0000	−0.535420
$894$	−28.0000	−0.936460
$895$	−22.0000	−0.735379
$896$	2.00000	0.0668153
$897$	24.0000	0.801337
$898$	14.0000	0.467186
$899$	−42.0000	−1.40078
$900$	−4.00000	−0.133333
$901$	3.00000	0.0999445
$902$	−10.0000	−0.332964
$903$	−28.0000	−0.931782
$904$	−11.0000	−0.365855
$905$	−6.00000	−0.199447
$906$	32.0000	1.06313
$907$	29.0000	0.962929	0.481465	−	0.876466i	$-0.340105\pi$
$907$	29.0000	0.962929	0.481465	+	0.876466i	$0.340105\pi$
$908$	−4.00000	−0.132745
$909$	−3.00000	−0.0995037
$910$	−8.00000	−0.265197
$911$	−40.0000	−1.32526	−0.662630	−	0.748947i	$-0.730560\pi$
$911$	−40.0000	−1.32526	−0.662630	+	0.748947i	$0.730560\pi$
$912$	−8.00000	−0.264906
$913$	−70.0000	−2.31666
$914$	26.0000	0.860004
$915$	4.00000	0.132236
$916$	−14.0000	−0.462573
$917$	−24.0000	−0.792550
$918$	12.0000	0.396059
$919$	39.0000	1.28649	0.643246	−	0.765660i	$-0.277587\pi$
$919$	39.0000	1.28649	0.643246	+	0.765660i	$0.277587\pi$
$920$	3.00000	0.0989071
$921$	34.0000	1.12034
$922$	−38.0000	−1.25146
$923$	−48.0000	−1.57994
$924$	−20.0000	−0.657952
$925$	24.0000	0.789115
$926$	−19.0000	−0.624379
$927$	8.00000	0.262754
$928$	6.00000	0.196960
$929$	−43.0000	−1.41078	−0.705392	−	0.708817i	$-0.749229\pi$
$929$	−43.0000	−1.41078	−0.705392	+	0.708817i	$0.749229\pi$
$930$	−14.0000	−0.459078
$931$	12.0000	0.393284
$932$	−12.0000	−0.393073
$933$	32.0000	1.04763
$934$	−36.0000	−1.17796
$935$	15.0000	0.490552
$936$	4.00000	0.130744
$937$	53.0000	1.73143	0.865717	−	0.500533i	$-0.166863\pi$
$937$	53.0000	1.73143	0.865717	+	0.500533i	$0.166863\pi$
$938$	32.0000	1.04484
$939$	−20.0000	−0.652675
$940$	4.00000	0.130466
$941$	−30.0000	−0.977972	−0.488986	−	0.872292i	$-0.662633\pi$
$941$	−30.0000	−0.977972	−0.488986	+	0.872292i	$0.662633\pi$
$942$	38.0000	1.23811
$943$	−6.00000	−0.195387
$944$	7.00000	0.227831
$945$	8.00000	0.260240
$946$	−35.0000	−1.13795
$947$	−48.0000	−1.55979	−0.779895	−	0.625910i	$-0.784728\pi$
$947$	−48.0000	−1.55979	−0.779895	+	0.625910i	$0.784728\pi$
$948$	−14.0000	−0.454699
$949$	48.0000	1.55815
$950$	−16.0000	−0.519109
$951$	64.0000	2.07534
$952$	6.00000	0.194461
$953$	−42.0000	−1.36051	−0.680257	−	0.732974i	$-0.738132\pi$
$953$	−42.0000	−1.36051	−0.680257	+	0.732974i	$0.738132\pi$
$954$	−1.00000	−0.0323762
$955$	−21.0000	−0.679544
$956$	9.00000	0.291081
$957$	−60.0000	−1.93952
$958$	0	0
$959$	40.0000	1.29167
$960$	2.00000	0.0645497
$961$	18.0000	0.580645
$962$	−24.0000	−0.773791
$963$	0	0
$964$	23.0000	0.740780
$965$	−8.00000	−0.257529
$966$	−12.0000	−0.386094
$967$	−12.0000	−0.385894	−0.192947	−	0.981209i	$-0.561805\pi$
$967$	−12.0000	−0.385894	−0.192947	+	0.981209i	$0.561805\pi$
$968$	−14.0000	−0.449977
$969$	−24.0000	−0.770991
$970$	−3.00000	−0.0963242
$971$	39.0000	1.25157	0.625785	−	0.779996i	$-0.284779\pi$
$971$	39.0000	1.25157	0.625785	+	0.779996i	$0.284779\pi$
$972$	−10.0000	−0.320750
$973$	−32.0000	−1.02587
$974$	4.00000	0.128168
$975$	32.0000	1.02482
$976$	2.00000	0.0640184
$977$	−36.0000	−1.15174	−0.575871	−	0.817541i	$-0.695337\pi$
$977$	−36.0000	−1.15174	−0.575871	+	0.817541i	$0.695337\pi$
$978$	−2.00000	−0.0639529
$979$	85.0000	2.71661
$980$	−3.00000	−0.0958315
$981$	19.0000	0.606623
$982$	2.00000	0.0638226
$983$	−18.0000	−0.574111	−0.287055	−	0.957914i	$-0.592676\pi$
$983$	−18.0000	−0.574111	−0.287055	+	0.957914i	$0.592676\pi$
$984$	−4.00000	−0.127515
$985$	6.00000	0.191176
$986$	18.0000	0.573237
$987$	−16.0000	−0.509286
$988$	16.0000	0.509028
$989$	−21.0000	−0.667761
$990$	−5.00000	−0.158910
$991$	−8.00000	−0.254128	−0.127064	−	0.991894i	$-0.540555\pi$
$991$	−8.00000	−0.254128	−0.127064	+	0.991894i	$0.540555\pi$
$992$	−7.00000	−0.222250
$993$	−8.00000	−0.253872
$994$	24.0000	0.761234
$995$	−4.00000	−0.126809
$996$	−28.0000	−0.887214
$997$	50.0000	1.58352	0.791758	−	0.610835i	$-0.209166\pi$
$997$	50.0000	1.58352	0.791758	+	0.610835i	$0.209166\pi$
$998$	−16.0000	−0.506471
$999$	24.0000	0.759326

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	106.2.a.b.1.1	✓	1
3.2	odd	2		954.2.a.i.1.1		1
4.3	odd	2		848.2.a.a.1.1		1
5.2	odd	4		2650.2.b.b.849.1		2
5.3	odd	4		2650.2.b.b.849.2		2
5.4	even	2		2650.2.a.f.1.1		1
7.6	odd	2		5194.2.a.c.1.1		1
8.3	odd	2		3392.2.a.r.1.1		1
8.5	even	2		3392.2.a.e.1.1		1
12.11	even	2		7632.2.a.g.1.1		1
53.52	even	2		5618.2.a.f.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
106.2.a.b.1.1	✓	1	1.1	even	1	trivial
848.2.a.a.1.1		1	4.3	odd	2
954.2.a.i.1.1		1	3.2	odd	2
2650.2.a.f.1.1		1	5.4	even	2
2650.2.b.b.849.1		2	5.2	odd	4
2650.2.b.b.849.2		2	5.3	odd	4
3392.2.a.e.1.1		1	8.5	even	2
3392.2.a.r.1.1		1	8.3	odd	2
5194.2.a.c.1.1		1	7.6	odd	2
5618.2.a.f.1.1		1	53.52	even	2
7632.2.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:	\( N \)	\(=\)	\( 106 = 2 \cdot 53 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	106.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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