LMFDB - Embedded newform 4026.2.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4026,2,Mod(1,4026)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4026, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4026.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4026 = 2 \cdot 3 \cdot 11 \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4026.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:	$32.1477718538$
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	$\chi$	$=$	4026.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -1.00000 q^{11} -1.00000 q^{12} -3.00000 q^{13} +2.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} +8.00000 q^{17} +1.00000 q^{18} +4.00000 q^{19} -1.00000 q^{20} -2.00000 q^{21} -1.00000 q^{22} +4.00000 q^{23} -1.00000 q^{24} -4.00000 q^{25} -3.00000 q^{26} -1.00000 q^{27} +2.00000 q^{28} -9.00000 q^{29} +1.00000 q^{30} -3.00000 q^{31} +1.00000 q^{32} +1.00000 q^{33} +8.00000 q^{34} -2.00000 q^{35} +1.00000 q^{36} +6.00000 q^{37} +4.00000 q^{38} +3.00000 q^{39} -1.00000 q^{40} +3.00000 q^{41} -2.00000 q^{42} +4.00000 q^{43} -1.00000 q^{44} -1.00000 q^{45} +4.00000 q^{46} -1.00000 q^{48} -3.00000 q^{49} -4.00000 q^{50} -8.00000 q^{51} -3.00000 q^{52} +8.00000 q^{53} -1.00000 q^{54} +1.00000 q^{55} +2.00000 q^{56} -4.00000 q^{57} -9.00000 q^{58} -3.00000 q^{59} +1.00000 q^{60} -1.00000 q^{61} -3.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} +3.00000 q^{65} +1.00000 q^{66} +4.00000 q^{67} +8.00000 q^{68} -4.00000 q^{69} -2.00000 q^{70} +6.00000 q^{71} +1.00000 q^{72} +12.0000 q^{73} +6.00000 q^{74} +4.00000 q^{75} +4.00000 q^{76} -2.00000 q^{77} +3.00000 q^{78} -4.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +3.00000 q^{82} +4.00000 q^{83} -2.00000 q^{84} -8.00000 q^{85} +4.00000 q^{86} +9.00000 q^{87} -1.00000 q^{88} -17.0000 q^{89} -1.00000 q^{90} -6.00000 q^{91} +4.00000 q^{92} +3.00000 q^{93} -4.00000 q^{95} -1.00000 q^{96} +13.0000 q^{97} -3.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	−1.00000	−0.408248
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−1.00000	−0.288675
$13$	−3.00000	−0.832050	−0.416025	−	0.909353i	$-0.636577\pi$
$13$	−3.00000	−0.832050	−0.416025	+	0.909353i	$0.636577\pi$
$14$	2.00000	0.534522
$15$	1.00000	0.258199
$16$	1.00000	0.250000
$17$	8.00000	1.94029	0.970143	−	0.242536i	$-0.0779791\pi$
$17$	8.00000	1.94029	0.970143	+	0.242536i	$0.0779791\pi$
$18$	1.00000	0.235702
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	−1.00000	−0.223607
$21$	−2.00000	−0.436436
$22$	−1.00000	−0.213201
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	−1.00000	−0.204124
$25$	−4.00000	−0.800000
$26$	−3.00000	−0.588348
$27$	−1.00000	−0.192450
$28$	2.00000	0.377964
$29$	−9.00000	−1.67126	−0.835629	−	0.549294i	$-0.814897\pi$
$29$	−9.00000	−1.67126	−0.835629	+	0.549294i	$0.814897\pi$
$30$	1.00000	0.182574
$31$	−3.00000	−0.538816	−0.269408	−	0.963026i	$-0.586828\pi$
$31$	−3.00000	−0.538816	−0.269408	+	0.963026i	$0.586828\pi$
$32$	1.00000	0.176777
$33$	1.00000	0.174078
$34$	8.00000	1.37199
$35$	−2.00000	−0.338062
$36$	1.00000	0.166667
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	4.00000	0.648886
$39$	3.00000	0.480384
$40$	−1.00000	−0.158114
$41$	3.00000	0.468521	0.234261	−	0.972174i	$-0.424733\pi$
$41$	3.00000	0.468521	0.234261	+	0.972174i	$0.424733\pi$
$42$	−2.00000	−0.308607
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	−1.00000	−0.150756
$45$	−1.00000	−0.149071
$46$	4.00000	0.589768
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	−1.00000	−0.144338
$49$	−3.00000	−0.428571
$50$	−4.00000	−0.565685
$51$	−8.00000	−1.12022
$52$	−3.00000	−0.416025
$53$	8.00000	1.09888	0.549442	−	0.835532i	$-0.314840\pi$
$53$	8.00000	1.09888	0.549442	+	0.835532i	$0.314840\pi$
$54$	−1.00000	−0.136083
$55$	1.00000	0.134840
$56$	2.00000	0.267261
$57$	−4.00000	−0.529813
$58$	−9.00000	−1.18176
$59$	−3.00000	−0.390567	−0.195283	−	0.980747i	$-0.562563\pi$
$59$	−3.00000	−0.390567	−0.195283	+	0.980747i	$0.562563\pi$
$60$	1.00000	0.129099
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	−3.00000	−0.381000
$63$	2.00000	0.251976
$64$	1.00000	0.125000
$65$	3.00000	0.372104
$66$	1.00000	0.123091
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	8.00000	0.970143
$69$	−4.00000	−0.481543
$70$	−2.00000	−0.239046
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	1.00000	0.117851
$73$	12.0000	1.40449	0.702247	−	0.711934i	$-0.252180\pi$
$73$	12.0000	1.40449	0.702247	+	0.711934i	$0.252180\pi$
$74$	6.00000	0.697486
$75$	4.00000	0.461880
$76$	4.00000	0.458831
$77$	−2.00000	−0.227921
$78$	3.00000	0.339683
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	3.00000	0.331295
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	−2.00000	−0.218218
$85$	−8.00000	−0.867722
$86$	4.00000	0.431331
$87$	9.00000	0.964901
$88$	−1.00000	−0.106600
$89$	−17.0000	−1.80200	−0.900998	−	0.433823i	$-0.857164\pi$
$89$	−17.0000	−1.80200	−0.900998	+	0.433823i	$0.857164\pi$
$90$	−1.00000	−0.105409
$91$	−6.00000	−0.628971
$92$	4.00000	0.417029
$93$	3.00000	0.311086
$94$	0	0
$95$	−4.00000	−0.410391
$96$	−1.00000	−0.102062
$97$	13.0000	1.31995	0.659975	−	0.751288i	$-0.270567\pi$
$97$	13.0000	1.31995	0.659975	+	0.751288i	$0.270567\pi$
$98$	−3.00000	−0.303046
$99$	−1.00000	−0.100504
$100$	−4.00000	−0.400000
$101$	−1.00000	−0.0995037	−0.0497519	−	0.998762i	$-0.515843\pi$
$101$	−1.00000	−0.0995037	−0.0497519	+	0.998762i	$0.515843\pi$
$102$	−8.00000	−0.792118
$103$	12.0000	1.18240	0.591198	−	0.806527i	$-0.298655\pi$
$103$	12.0000	1.18240	0.591198	+	0.806527i	$0.298655\pi$
$104$	−3.00000	−0.294174
$105$	2.00000	0.195180
$106$	8.00000	0.777029
$107$	13.0000	1.25676	0.628379	−	0.777908i	$-0.283719\pi$
$107$	13.0000	1.25676	0.628379	+	0.777908i	$0.283719\pi$
$108$	−1.00000	−0.0962250
$109$	3.00000	0.287348	0.143674	−	0.989625i	$-0.454108\pi$
$109$	3.00000	0.287348	0.143674	+	0.989625i	$0.454108\pi$
$110$	1.00000	0.0953463
$111$	−6.00000	−0.569495
$112$	2.00000	0.188982
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	−4.00000	−0.374634
$115$	−4.00000	−0.373002
$116$	−9.00000	−0.835629
$117$	−3.00000	−0.277350
$118$	−3.00000	−0.276172
$119$	16.0000	1.46672
$120$	1.00000	0.0912871
$121$	1.00000	0.0909091
$122$	−1.00000	−0.0905357
$123$	−3.00000	−0.270501
$124$	−3.00000	−0.269408
$125$	9.00000	0.804984
$126$	2.00000	0.178174
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	1.00000	0.0883883
$129$	−4.00000	−0.352180
$130$	3.00000	0.263117
$131$	1.00000	0.0873704	0.0436852	−	0.999045i	$-0.486090\pi$
$131$	1.00000	0.0873704	0.0436852	+	0.999045i	$0.486090\pi$
$132$	1.00000	0.0870388
$133$	8.00000	0.693688
$134$	4.00000	0.345547
$135$	1.00000	0.0860663
$136$	8.00000	0.685994
$137$	14.0000	1.19610	0.598050	−	0.801459i	$-0.295942\pi$
$137$	14.0000	1.19610	0.598050	+	0.801459i	$0.295942\pi$
$138$	−4.00000	−0.340503
$139$	−19.0000	−1.61156	−0.805779	−	0.592216i	$-0.798253\pi$
$139$	−19.0000	−1.61156	−0.805779	+	0.592216i	$0.798253\pi$
$140$	−2.00000	−0.169031
$141$	0	0
$142$	6.00000	0.503509
$143$	3.00000	0.250873
$144$	1.00000	0.0833333
$145$	9.00000	0.747409
$146$	12.0000	0.993127
$147$	3.00000	0.247436
$148$	6.00000	0.493197
$149$	20.0000	1.63846	0.819232	−	0.573462i	$-0.194400\pi$
$149$	20.0000	1.63846	0.819232	+	0.573462i	$0.194400\pi$
$150$	4.00000	0.326599
$151$	4.00000	0.325515	0.162758	−	0.986666i	$-0.447961\pi$
$151$	4.00000	0.325515	0.162758	+	0.986666i	$0.447961\pi$
$152$	4.00000	0.324443
$153$	8.00000	0.646762
$154$	−2.00000	−0.161165
$155$	3.00000	0.240966
$156$	3.00000	0.240192
$157$	−7.00000	−0.558661	−0.279330	−	0.960195i	$-0.590112\pi$
$157$	−7.00000	−0.558661	−0.279330	+	0.960195i	$0.590112\pi$
$158$	−4.00000	−0.318223
$159$	−8.00000	−0.634441
$160$	−1.00000	−0.0790569
$161$	8.00000	0.630488
$162$	1.00000	0.0785674
$163$	−5.00000	−0.391630	−0.195815	−	0.980641i	$-0.562735\pi$
$163$	−5.00000	−0.391630	−0.195815	+	0.980641i	$0.562735\pi$
$164$	3.00000	0.234261
$165$	−1.00000	−0.0778499
$166$	4.00000	0.310460
$167$	24.0000	1.85718	0.928588	−	0.371113i	$-0.121024\pi$
$167$	24.0000	1.85718	0.928588	+	0.371113i	$0.121024\pi$
$168$	−2.00000	−0.154303
$169$	−4.00000	−0.307692
$170$	−8.00000	−0.613572
$171$	4.00000	0.305888
$172$	4.00000	0.304997
$173$	11.0000	0.836315	0.418157	−	0.908375i	$-0.362676\pi$
$173$	11.0000	0.836315	0.418157	+	0.908375i	$0.362676\pi$
$174$	9.00000	0.682288
$175$	−8.00000	−0.604743
$176$	−1.00000	−0.0753778
$177$	3.00000	0.225494
$178$	−17.0000	−1.27420
$179$	14.0000	1.04641	0.523205	−	0.852207i	$-0.324736\pi$
$179$	14.0000	1.04641	0.523205	+	0.852207i	$0.324736\pi$
$180$	−1.00000	−0.0745356
$181$	1.00000	0.0743294	0.0371647	−	0.999309i	$-0.488167\pi$
$181$	1.00000	0.0743294	0.0371647	+	0.999309i	$0.488167\pi$
$182$	−6.00000	−0.444750
$183$	1.00000	0.0739221
$184$	4.00000	0.294884
$185$	−6.00000	−0.441129
$186$	3.00000	0.219971
$187$	−8.00000	−0.585018
$188$	0	0
$189$	−2.00000	−0.145479
$190$	−4.00000	−0.290191
$191$	18.0000	1.30243	0.651217	−	0.758891i	$-0.274259\pi$
$191$	18.0000	1.30243	0.651217	+	0.758891i	$0.274259\pi$
$192$	−1.00000	−0.0721688
$193$	−17.0000	−1.22369	−0.611843	−	0.790979i	$-0.709572\pi$
$193$	−17.0000	−1.22369	−0.611843	+	0.790979i	$0.709572\pi$
$194$	13.0000	0.933346
$195$	−3.00000	−0.214834
$196$	−3.00000	−0.214286
$197$	−14.0000	−0.997459	−0.498729	−	0.866758i	$-0.666200\pi$
$197$	−14.0000	−0.997459	−0.498729	+	0.866758i	$0.666200\pi$
$198$	−1.00000	−0.0710669
$199$	14.0000	0.992434	0.496217	−	0.868199i	$-0.334722\pi$
$199$	14.0000	0.992434	0.496217	+	0.868199i	$0.334722\pi$
$200$	−4.00000	−0.282843
$201$	−4.00000	−0.282138
$202$	−1.00000	−0.0703598
$203$	−18.0000	−1.26335
$204$	−8.00000	−0.560112
$205$	−3.00000	−0.209529
$206$	12.0000	0.836080
$207$	4.00000	0.278019
$208$	−3.00000	−0.208013
$209$	−4.00000	−0.276686
$210$	2.00000	0.138013
$211$	−25.0000	−1.72107	−0.860535	−	0.509390i	$-0.829871\pi$
$211$	−25.0000	−1.72107	−0.860535	+	0.509390i	$0.829871\pi$
$212$	8.00000	0.549442
$213$	−6.00000	−0.411113
$214$	13.0000	0.888662
$215$	−4.00000	−0.272798
$216$	−1.00000	−0.0680414
$217$	−6.00000	−0.407307
$218$	3.00000	0.203186
$219$	−12.0000	−0.810885
$220$	1.00000	0.0674200
$221$	−24.0000	−1.61441
$222$	−6.00000	−0.402694
$223$	−12.0000	−0.803579	−0.401790	−	0.915732i	$-0.631612\pi$
$223$	−12.0000	−0.803579	−0.401790	+	0.915732i	$0.631612\pi$
$224$	2.00000	0.133631
$225$	−4.00000	−0.266667
$226$	2.00000	0.133038
$227$	24.0000	1.59294	0.796468	−	0.604681i	$-0.206699\pi$
$227$	24.0000	1.59294	0.796468	+	0.604681i	$0.206699\pi$
$228$	−4.00000	−0.264906
$229$	−4.00000	−0.264327	−0.132164	−	0.991228i	$-0.542192\pi$
$229$	−4.00000	−0.264327	−0.132164	+	0.991228i	$0.542192\pi$
$230$	−4.00000	−0.263752
$231$	2.00000	0.131590
$232$	−9.00000	−0.590879
$233$	−20.0000	−1.31024	−0.655122	−	0.755523i	$-0.727383\pi$
$233$	−20.0000	−1.31024	−0.655122	+	0.755523i	$0.727383\pi$
$234$	−3.00000	−0.196116
$235$	0	0
$236$	−3.00000	−0.195283
$237$	4.00000	0.259828
$238$	16.0000	1.03713
$239$	−16.0000	−1.03495	−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000	−1.03495	−0.517477	+	0.855697i	$0.673129\pi$
$240$	1.00000	0.0645497
$241$	−8.00000	−0.515325	−0.257663	−	0.966235i	$-0.582952\pi$
$241$	−8.00000	−0.515325	−0.257663	+	0.966235i	$0.582952\pi$
$242$	1.00000	0.0642824
$243$	−1.00000	−0.0641500
$244$	−1.00000	−0.0640184
$245$	3.00000	0.191663
$246$	−3.00000	−0.191273
$247$	−12.0000	−0.763542
$248$	−3.00000	−0.190500
$249$	−4.00000	−0.253490
$250$	9.00000	0.569210
$251$	−19.0000	−1.19927	−0.599635	−	0.800274i	$-0.704687\pi$
$251$	−19.0000	−1.19927	−0.599635	+	0.800274i	$0.704687\pi$
$252$	2.00000	0.125988
$253$	−4.00000	−0.251478
$254$	0	0
$255$	8.00000	0.500979
$256$	1.00000	0.0625000
$257$	26.0000	1.62184	0.810918	−	0.585160i	$-0.198968\pi$
$257$	26.0000	1.62184	0.810918	+	0.585160i	$0.198968\pi$
$258$	−4.00000	−0.249029
$259$	12.0000	0.745644
$260$	3.00000	0.186052
$261$	−9.00000	−0.557086
$262$	1.00000	0.0617802
$263$	−6.00000	−0.369976	−0.184988	−	0.982741i	$-0.559225\pi$
$263$	−6.00000	−0.369976	−0.184988	+	0.982741i	$0.559225\pi$
$264$	1.00000	0.0615457
$265$	−8.00000	−0.491436
$266$	8.00000	0.490511
$267$	17.0000	1.04038
$268$	4.00000	0.244339
$269$	23.0000	1.40233	0.701167	−	0.712997i	$-0.252663\pi$
$269$	23.0000	1.40233	0.701167	+	0.712997i	$0.252663\pi$
$270$	1.00000	0.0608581
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	8.00000	0.485071
$273$	6.00000	0.363137
$274$	14.0000	0.845771
$275$	4.00000	0.241209
$276$	−4.00000	−0.240772
$277$	26.0000	1.56219	0.781094	−	0.624413i	$-0.214662\pi$
$277$	26.0000	1.56219	0.781094	+	0.624413i	$0.214662\pi$
$278$	−19.0000	−1.13954
$279$	−3.00000	−0.179605
$280$	−2.00000	−0.119523
$281$	20.0000	1.19310	0.596550	−	0.802576i	$-0.296538\pi$
$281$	20.0000	1.19310	0.596550	+	0.802576i	$0.296538\pi$
$282$	0	0
$283$	−32.0000	−1.90220	−0.951101	−	0.308879i	$-0.900046\pi$
$283$	−32.0000	−1.90220	−0.951101	+	0.308879i	$0.900046\pi$
$284$	6.00000	0.356034
$285$	4.00000	0.236940
$286$	3.00000	0.177394
$287$	6.00000	0.354169
$288$	1.00000	0.0589256
$289$	47.0000	2.76471
$290$	9.00000	0.528498
$291$	−13.0000	−0.762073
$292$	12.0000	0.702247
$293$	−8.00000	−0.467365	−0.233682	−	0.972313i	$-0.575078\pi$
$293$	−8.00000	−0.467365	−0.233682	+	0.972313i	$0.575078\pi$
$294$	3.00000	0.174964
$295$	3.00000	0.174667
$296$	6.00000	0.348743
$297$	1.00000	0.0580259
$298$	20.0000	1.15857
$299$	−12.0000	−0.693978
$300$	4.00000	0.230940
$301$	8.00000	0.461112
$302$	4.00000	0.230174
$303$	1.00000	0.0574485
$304$	4.00000	0.229416
$305$	1.00000	0.0572598
$306$	8.00000	0.457330
$307$	−11.0000	−0.627803	−0.313902	−	0.949456i	$-0.601636\pi$
$307$	−11.0000	−0.627803	−0.313902	+	0.949456i	$0.601636\pi$
$308$	−2.00000	−0.113961
$309$	−12.0000	−0.682656
$310$	3.00000	0.170389
$311$	16.0000	0.907277	0.453638	−	0.891186i	$-0.350126\pi$
$311$	16.0000	0.907277	0.453638	+	0.891186i	$0.350126\pi$
$312$	3.00000	0.169842
$313$	−4.00000	−0.226093	−0.113047	−	0.993590i	$-0.536061\pi$
$313$	−4.00000	−0.226093	−0.113047	+	0.993590i	$0.536061\pi$
$314$	−7.00000	−0.395033
$315$	−2.00000	−0.112687
$316$	−4.00000	−0.225018
$317$	3.00000	0.168497	0.0842484	−	0.996445i	$-0.473151\pi$
$317$	3.00000	0.168497	0.0842484	+	0.996445i	$0.473151\pi$
$318$	−8.00000	−0.448618
$319$	9.00000	0.503903
$320$	−1.00000	−0.0559017
$321$	−13.0000	−0.725589
$322$	8.00000	0.445823
$323$	32.0000	1.78053
$324$	1.00000	0.0555556
$325$	12.0000	0.665640
$326$	−5.00000	−0.276924
$327$	−3.00000	−0.165900
$328$	3.00000	0.165647
$329$	0	0
$330$	−1.00000	−0.0550482
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	4.00000	0.219529
$333$	6.00000	0.328798
$334$	24.0000	1.31322
$335$	−4.00000	−0.218543
$336$	−2.00000	−0.109109
$337$	15.0000	0.817102	0.408551	−	0.912735i	$-0.366034\pi$
$337$	15.0000	0.817102	0.408551	+	0.912735i	$0.366034\pi$
$338$	−4.00000	−0.217571
$339$	−2.00000	−0.108625
$340$	−8.00000	−0.433861
$341$	3.00000	0.162459
$342$	4.00000	0.216295
$343$	−20.0000	−1.07990
$344$	4.00000	0.215666
$345$	4.00000	0.215353
$346$	11.0000	0.591364
$347$	−25.0000	−1.34207	−0.671035	−	0.741426i	$-0.734150\pi$
$347$	−25.0000	−1.34207	−0.671035	+	0.741426i	$0.734150\pi$
$348$	9.00000	0.482451
$349$	36.0000	1.92704	0.963518	−	0.267644i	$-0.0862451\pi$
$349$	36.0000	1.92704	0.963518	+	0.267644i	$0.0862451\pi$
$350$	−8.00000	−0.427618
$351$	3.00000	0.160128
$352$	−1.00000	−0.0533002
$353$	6.00000	0.319348	0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000	0.319348	0.159674	+	0.987170i	$0.448956\pi$
$354$	3.00000	0.159448
$355$	−6.00000	−0.318447
$356$	−17.0000	−0.900998
$357$	−16.0000	−0.846810
$358$	14.0000	0.739923
$359$	−33.0000	−1.74167	−0.870837	−	0.491572i	$-0.836422\pi$
$359$	−33.0000	−1.74167	−0.870837	+	0.491572i	$0.836422\pi$
$360$	−1.00000	−0.0527046
$361$	−3.00000	−0.157895
$362$	1.00000	0.0525588
$363$	−1.00000	−0.0524864
$364$	−6.00000	−0.314485
$365$	−12.0000	−0.628109
$366$	1.00000	0.0522708
$367$	−22.0000	−1.14839	−0.574195	−	0.818718i	$-0.694685\pi$
$367$	−22.0000	−1.14839	−0.574195	+	0.818718i	$0.694685\pi$
$368$	4.00000	0.208514
$369$	3.00000	0.156174
$370$	−6.00000	−0.311925
$371$	16.0000	0.830679
$372$	3.00000	0.155543
$373$	−20.0000	−1.03556	−0.517780	−	0.855514i	$-0.673242\pi$
$373$	−20.0000	−1.03556	−0.517780	+	0.855514i	$0.673242\pi$
$374$	−8.00000	−0.413670
$375$	−9.00000	−0.464758
$376$	0	0
$377$	27.0000	1.39057
$378$	−2.00000	−0.102869
$379$	−13.0000	−0.667765	−0.333883	−	0.942615i	$-0.608359\pi$
$379$	−13.0000	−0.667765	−0.333883	+	0.942615i	$0.608359\pi$
$380$	−4.00000	−0.205196
$381$	0	0
$382$	18.0000	0.920960
$383$	14.0000	0.715367	0.357683	−	0.933843i	$-0.383567\pi$
$383$	14.0000	0.715367	0.357683	+	0.933843i	$0.383567\pi$
$384$	−1.00000	−0.0510310
$385$	2.00000	0.101929
$386$	−17.0000	−0.865277
$387$	4.00000	0.203331
$388$	13.0000	0.659975
$389$	36.0000	1.82527	0.912636	−	0.408773i	$-0.134043\pi$
$389$	36.0000	1.82527	0.912636	+	0.408773i	$0.134043\pi$
$390$	−3.00000	−0.151911
$391$	32.0000	1.61831
$392$	−3.00000	−0.151523
$393$	−1.00000	−0.0504433
$394$	−14.0000	−0.705310
$395$	4.00000	0.201262
$396$	−1.00000	−0.0502519
$397$	−29.0000	−1.45547	−0.727734	−	0.685859i	$-0.759427\pi$
$397$	−29.0000	−1.45547	−0.727734	+	0.685859i	$0.759427\pi$
$398$	14.0000	0.701757
$399$	−8.00000	−0.400501
$400$	−4.00000	−0.200000
$401$	25.0000	1.24844	0.624220	−	0.781248i	$-0.285417\pi$
$401$	25.0000	1.24844	0.624220	+	0.781248i	$0.285417\pi$
$402$	−4.00000	−0.199502
$403$	9.00000	0.448322
$404$	−1.00000	−0.0497519
$405$	−1.00000	−0.0496904
$406$	−18.0000	−0.893325
$407$	−6.00000	−0.297409
$408$	−8.00000	−0.396059
$409$	31.0000	1.53285	0.766426	−	0.642333i	$-0.222033\pi$
$409$	31.0000	1.53285	0.766426	+	0.642333i	$0.222033\pi$
$410$	−3.00000	−0.148159
$411$	−14.0000	−0.690569
$412$	12.0000	0.591198
$413$	−6.00000	−0.295241
$414$	4.00000	0.196589
$415$	−4.00000	−0.196352
$416$	−3.00000	−0.147087
$417$	19.0000	0.930434
$418$	−4.00000	−0.195646
$419$	−36.0000	−1.75872	−0.879358	−	0.476162i	$-0.842028\pi$
$419$	−36.0000	−1.75872	−0.879358	+	0.476162i	$0.842028\pi$
$420$	2.00000	0.0975900
$421$	−31.0000	−1.51085	−0.755424	−	0.655237i	$-0.772569\pi$
$421$	−31.0000	−1.51085	−0.755424	+	0.655237i	$0.772569\pi$
$422$	−25.0000	−1.21698
$423$	0	0
$424$	8.00000	0.388514
$425$	−32.0000	−1.55223
$426$	−6.00000	−0.290701
$427$	−2.00000	−0.0967868
$428$	13.0000	0.628379
$429$	−3.00000	−0.144841
$430$	−4.00000	−0.192897
$431$	18.0000	0.867029	0.433515	−	0.901146i	$-0.357273\pi$
$431$	18.0000	0.867029	0.433515	+	0.901146i	$0.357273\pi$
$432$	−1.00000	−0.0481125
$433$	−4.00000	−0.192228	−0.0961139	−	0.995370i	$-0.530641\pi$
$433$	−4.00000	−0.192228	−0.0961139	+	0.995370i	$0.530641\pi$
$434$	−6.00000	−0.288009
$435$	−9.00000	−0.431517
$436$	3.00000	0.143674
$437$	16.0000	0.765384
$438$	−12.0000	−0.573382
$439$	13.0000	0.620456	0.310228	−	0.950662i	$-0.399595\pi$
$439$	13.0000	0.620456	0.310228	+	0.950662i	$0.399595\pi$
$440$	1.00000	0.0476731
$441$	−3.00000	−0.142857
$442$	−24.0000	−1.14156
$443$	−26.0000	−1.23530	−0.617649	−	0.786454i	$-0.711915\pi$
$443$	−26.0000	−1.23530	−0.617649	+	0.786454i	$0.711915\pi$
$444$	−6.00000	−0.284747
$445$	17.0000	0.805877
$446$	−12.0000	−0.568216
$447$	−20.0000	−0.945968
$448$	2.00000	0.0944911
$449$	−40.0000	−1.88772	−0.943858	−	0.330350i	$-0.892833\pi$
$449$	−40.0000	−1.88772	−0.943858	+	0.330350i	$0.892833\pi$
$450$	−4.00000	−0.188562
$451$	−3.00000	−0.141264
$452$	2.00000	0.0940721
$453$	−4.00000	−0.187936
$454$	24.0000	1.12638
$455$	6.00000	0.281284
$456$	−4.00000	−0.187317
$457$	−14.0000	−0.654892	−0.327446	−	0.944870i	$-0.606188\pi$
$457$	−14.0000	−0.654892	−0.327446	+	0.944870i	$0.606188\pi$
$458$	−4.00000	−0.186908
$459$	−8.00000	−0.373408
$460$	−4.00000	−0.186501
$461$	−22.0000	−1.02464	−0.512321	−	0.858794i	$-0.671214\pi$
$461$	−22.0000	−1.02464	−0.512321	+	0.858794i	$0.671214\pi$
$462$	2.00000	0.0930484
$463$	6.00000	0.278844	0.139422	−	0.990233i	$-0.455476\pi$
$463$	6.00000	0.278844	0.139422	+	0.990233i	$0.455476\pi$
$464$	−9.00000	−0.417815
$465$	−3.00000	−0.139122
$466$	−20.0000	−0.926482
$467$	33.0000	1.52706	0.763529	−	0.645774i	$-0.223465\pi$
$467$	33.0000	1.52706	0.763529	+	0.645774i	$0.223465\pi$
$468$	−3.00000	−0.138675
$469$	8.00000	0.369406
$470$	0	0
$471$	7.00000	0.322543
$472$	−3.00000	−0.138086
$473$	−4.00000	−0.183920
$474$	4.00000	0.183726
$475$	−16.0000	−0.734130
$476$	16.0000	0.733359
$477$	8.00000	0.366295
$478$	−16.0000	−0.731823
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	1.00000	0.0456435
$481$	−18.0000	−0.820729
$482$	−8.00000	−0.364390
$483$	−8.00000	−0.364013
$484$	1.00000	0.0454545
$485$	−13.0000	−0.590300
$486$	−1.00000	−0.0453609
$487$	20.0000	0.906287	0.453143	−	0.891438i	$-0.350303\pi$
$487$	20.0000	0.906287	0.453143	+	0.891438i	$0.350303\pi$
$488$	−1.00000	−0.0452679
$489$	5.00000	0.226108
$490$	3.00000	0.135526
$491$	−28.0000	−1.26362	−0.631811	−	0.775122i	$-0.717688\pi$
$491$	−28.0000	−1.26362	−0.631811	+	0.775122i	$0.717688\pi$
$492$	−3.00000	−0.135250
$493$	−72.0000	−3.24272
$494$	−12.0000	−0.539906
$495$	1.00000	0.0449467
$496$	−3.00000	−0.134704
$497$	12.0000	0.538274
$498$	−4.00000	−0.179244
$499$	−22.0000	−0.984855	−0.492428	−	0.870353i	$-0.663890\pi$
$499$	−22.0000	−0.984855	−0.492428	+	0.870353i	$0.663890\pi$
$500$	9.00000	0.402492
$501$	−24.0000	−1.07224
$502$	−19.0000	−0.848012
$503$	38.0000	1.69434	0.847168	−	0.531325i	$-0.178306\pi$
$503$	38.0000	1.69434	0.847168	+	0.531325i	$0.178306\pi$
$504$	2.00000	0.0890871
$505$	1.00000	0.0444994
$506$	−4.00000	−0.177822
$507$	4.00000	0.177646
$508$	0	0
$509$	−4.00000	−0.177297	−0.0886484	−	0.996063i	$-0.528255\pi$
$509$	−4.00000	−0.177297	−0.0886484	+	0.996063i	$0.528255\pi$
$510$	8.00000	0.354246
$511$	24.0000	1.06170
$512$	1.00000	0.0441942
$513$	−4.00000	−0.176604
$514$	26.0000	1.14681
$515$	−12.0000	−0.528783
$516$	−4.00000	−0.176090
$517$	0	0
$518$	12.0000	0.527250
$519$	−11.0000	−0.482846
$520$	3.00000	0.131559
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	−9.00000	−0.393919
$523$	−13.0000	−0.568450	−0.284225	−	0.958758i	$-0.591736\pi$
$523$	−13.0000	−0.568450	−0.284225	+	0.958758i	$0.591736\pi$
$524$	1.00000	0.0436852
$525$	8.00000	0.349149
$526$	−6.00000	−0.261612
$527$	−24.0000	−1.04546
$528$	1.00000	0.0435194
$529$	−7.00000	−0.304348
$530$	−8.00000	−0.347498
$531$	−3.00000	−0.130189
$532$	8.00000	0.346844
$533$	−9.00000	−0.389833
$534$	17.0000	0.735662
$535$	−13.0000	−0.562039
$536$	4.00000	0.172774
$537$	−14.0000	−0.604145
$538$	23.0000	0.991600
$539$	3.00000	0.129219
$540$	1.00000	0.0430331
$541$	16.0000	0.687894	0.343947	−	0.938989i	$-0.388236\pi$
$541$	16.0000	0.687894	0.343947	+	0.938989i	$0.388236\pi$
$542$	20.0000	0.859074
$543$	−1.00000	−0.0429141
$544$	8.00000	0.342997
$545$	−3.00000	−0.128506
$546$	6.00000	0.256776
$547$	11.0000	0.470326	0.235163	−	0.971956i	$-0.424438\pi$
$547$	11.0000	0.470326	0.235163	+	0.971956i	$0.424438\pi$
$548$	14.0000	0.598050
$549$	−1.00000	−0.0426790
$550$	4.00000	0.170561
$551$	−36.0000	−1.53365
$552$	−4.00000	−0.170251
$553$	−8.00000	−0.340195
$554$	26.0000	1.10463
$555$	6.00000	0.254686
$556$	−19.0000	−0.805779
$557$	27.0000	1.14403	0.572013	−	0.820244i	$-0.306163\pi$
$557$	27.0000	1.14403	0.572013	+	0.820244i	$0.306163\pi$
$558$	−3.00000	−0.127000
$559$	−12.0000	−0.507546
$560$	−2.00000	−0.0845154
$561$	8.00000	0.337760
$562$	20.0000	0.843649
$563$	−32.0000	−1.34864	−0.674320	−	0.738440i	$-0.735563\pi$
$563$	−32.0000	−1.34864	−0.674320	+	0.738440i	$0.735563\pi$
$564$	0	0
$565$	−2.00000	−0.0841406
$566$	−32.0000	−1.34506
$567$	2.00000	0.0839921
$568$	6.00000	0.251754
$569$	−35.0000	−1.46728	−0.733638	−	0.679540i	$-0.762179\pi$
$569$	−35.0000	−1.46728	−0.733638	+	0.679540i	$0.762179\pi$
$570$	4.00000	0.167542
$571$	−44.0000	−1.84134	−0.920671	−	0.390339i	$-0.872358\pi$
$571$	−44.0000	−1.84134	−0.920671	+	0.390339i	$0.872358\pi$
$572$	3.00000	0.125436
$573$	−18.0000	−0.751961
$574$	6.00000	0.250435
$575$	−16.0000	−0.667246
$576$	1.00000	0.0416667
$577$	8.00000	0.333044	0.166522	−	0.986038i	$-0.446746\pi$
$577$	8.00000	0.333044	0.166522	+	0.986038i	$0.446746\pi$
$578$	47.0000	1.95494
$579$	17.0000	0.706496
$580$	9.00000	0.373705
$581$	8.00000	0.331896
$582$	−13.0000	−0.538867
$583$	−8.00000	−0.331326
$584$	12.0000	0.496564
$585$	3.00000	0.124035
$586$	−8.00000	−0.330477
$587$	39.0000	1.60970	0.804851	−	0.593477i	$-0.202245\pi$
$587$	39.0000	1.60970	0.804851	+	0.593477i	$0.202245\pi$
$588$	3.00000	0.123718
$589$	−12.0000	−0.494451
$590$	3.00000	0.123508
$591$	14.0000	0.575883
$592$	6.00000	0.246598
$593$	6.00000	0.246390	0.123195	−	0.992382i	$-0.460686\pi$
$593$	6.00000	0.246390	0.123195	+	0.992382i	$0.460686\pi$
$594$	1.00000	0.0410305
$595$	−16.0000	−0.655936
$596$	20.0000	0.819232
$597$	−14.0000	−0.572982
$598$	−12.0000	−0.490716
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	4.00000	0.163299
$601$	−14.0000	−0.571072	−0.285536	−	0.958368i	$-0.592172\pi$
$601$	−14.0000	−0.571072	−0.285536	+	0.958368i	$0.592172\pi$
$602$	8.00000	0.326056
$603$	4.00000	0.162893
$604$	4.00000	0.162758
$605$	−1.00000	−0.0406558
$606$	1.00000	0.0406222
$607$	1.00000	0.0405887	0.0202944	−	0.999794i	$-0.493540\pi$
$607$	1.00000	0.0405887	0.0202944	+	0.999794i	$0.493540\pi$
$608$	4.00000	0.162221
$609$	18.0000	0.729397
$610$	1.00000	0.0404888
$611$	0	0
$612$	8.00000	0.323381
$613$	−33.0000	−1.33286	−0.666429	−	0.745569i	$-0.732178\pi$
$613$	−33.0000	−1.33286	−0.666429	+	0.745569i	$0.732178\pi$
$614$	−11.0000	−0.443924
$615$	3.00000	0.120972
$616$	−2.00000	−0.0805823
$617$	−19.0000	−0.764911	−0.382456	−	0.923974i	$-0.624922\pi$
$617$	−19.0000	−0.764911	−0.382456	+	0.923974i	$0.624922\pi$
$618$	−12.0000	−0.482711
$619$	−31.0000	−1.24600	−0.622998	−	0.782224i	$-0.714085\pi$
$619$	−31.0000	−1.24600	−0.622998	+	0.782224i	$0.714085\pi$
$620$	3.00000	0.120483
$621$	−4.00000	−0.160514
$622$	16.0000	0.641542
$623$	−34.0000	−1.36218
$624$	3.00000	0.120096
$625$	11.0000	0.440000
$626$	−4.00000	−0.159872
$627$	4.00000	0.159745
$628$	−7.00000	−0.279330
$629$	48.0000	1.91389
$630$	−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	$0.550904\pi$
$632$	−4.00000	−0.159111
$633$	25.0000	0.993661
$634$	3.00000	0.119145
$635$	0	0
$636$	−8.00000	−0.317221
$637$	9.00000	0.356593
$638$	9.00000	0.356313
$639$	6.00000	0.237356
$640$	−1.00000	−0.0395285
$641$	−6.00000	−0.236986	−0.118493	−	0.992955i	$-0.537806\pi$
$641$	−6.00000	−0.236986	−0.118493	+	0.992955i	$0.537806\pi$
$642$	−13.0000	−0.513069
$643$	−34.0000	−1.34083	−0.670415	−	0.741987i	$-0.733884\pi$
$643$	−34.0000	−1.34083	−0.670415	+	0.741987i	$0.733884\pi$
$644$	8.00000	0.315244
$645$	4.00000	0.157500
$646$	32.0000	1.25902
$647$	−10.0000	−0.393141	−0.196570	−	0.980490i	$-0.562980\pi$
$647$	−10.0000	−0.393141	−0.196570	+	0.980490i	$0.562980\pi$
$648$	1.00000	0.0392837
$649$	3.00000	0.117760
$650$	12.0000	0.470679
$651$	6.00000	0.235159
$652$	−5.00000	−0.195815
$653$	−12.0000	−0.469596	−0.234798	−	0.972044i	$-0.575443\pi$
$653$	−12.0000	−0.469596	−0.234798	+	0.972044i	$0.575443\pi$
$654$	−3.00000	−0.117309
$655$	−1.00000	−0.0390732
$656$	3.00000	0.117130
$657$	12.0000	0.468165
$658$	0	0
$659$	−39.0000	−1.51922	−0.759612	−	0.650376i	$-0.774611\pi$
$659$	−39.0000	−1.51922	−0.759612	+	0.650376i	$0.774611\pi$
$660$	−1.00000	−0.0389249
$661$	−21.0000	−0.816805	−0.408403	−	0.912802i	$-0.633914\pi$
$661$	−21.0000	−0.816805	−0.408403	+	0.912802i	$0.633914\pi$
$662$	−4.00000	−0.155464
$663$	24.0000	0.932083
$664$	4.00000	0.155230
$665$	−8.00000	−0.310227
$666$	6.00000	0.232495
$667$	−36.0000	−1.39393
$668$	24.0000	0.928588
$669$	12.0000	0.463947
$670$	−4.00000	−0.154533
$671$	1.00000	0.0386046
$672$	−2.00000	−0.0771517
$673$	11.0000	0.424019	0.212009	−	0.977268i	$-0.431999\pi$
$673$	11.0000	0.424019	0.212009	+	0.977268i	$0.431999\pi$
$674$	15.0000	0.577778
$675$	4.00000	0.153960
$676$	−4.00000	−0.153846
$677$	−50.0000	−1.92166	−0.960828	−	0.277145i	$-0.910612\pi$
$677$	−50.0000	−1.92166	−0.960828	+	0.277145i	$0.910612\pi$
$678$	−2.00000	−0.0768095
$679$	26.0000	0.997788
$680$	−8.00000	−0.306786
$681$	−24.0000	−0.919682
$682$	3.00000	0.114876
$683$	−30.0000	−1.14792	−0.573959	−	0.818884i	$-0.694593\pi$
$683$	−30.0000	−1.14792	−0.573959	+	0.818884i	$0.694593\pi$
$684$	4.00000	0.152944
$685$	−14.0000	−0.534913
$686$	−20.0000	−0.763604
$687$	4.00000	0.152610
$688$	4.00000	0.152499
$689$	−24.0000	−0.914327
$690$	4.00000	0.152277
$691$	33.0000	1.25538	0.627690	−	0.778464i	$-0.284001\pi$
$691$	33.0000	1.25538	0.627690	+	0.778464i	$0.284001\pi$
$692$	11.0000	0.418157
$693$	−2.00000	−0.0759737
$694$	−25.0000	−0.948987
$695$	19.0000	0.720711
$696$	9.00000	0.341144
$697$	24.0000	0.909065
$698$	36.0000	1.36262
$699$	20.0000	0.756469
$700$	−8.00000	−0.302372
$701$	15.0000	0.566542	0.283271	−	0.959040i	$-0.408580\pi$
$701$	15.0000	0.566542	0.283271	+	0.959040i	$0.408580\pi$
$702$	3.00000	0.113228
$703$	24.0000	0.905177
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	6.00000	0.225813
$707$	−2.00000	−0.0752177
$708$	3.00000	0.112747
$709$	21.0000	0.788672	0.394336	−	0.918966i	$-0.370975\pi$
$709$	21.0000	0.788672	0.394336	+	0.918966i	$0.370975\pi$
$710$	−6.00000	−0.225176
$711$	−4.00000	−0.150012
$712$	−17.0000	−0.637102
$713$	−12.0000	−0.449404
$714$	−16.0000	−0.598785
$715$	−3.00000	−0.112194
$716$	14.0000	0.523205
$717$	16.0000	0.597531
$718$	−33.0000	−1.23155
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	−1.00000	−0.0372678
$721$	24.0000	0.893807
$722$	−3.00000	−0.111648
$723$	8.00000	0.297523
$724$	1.00000	0.0371647
$725$	36.0000	1.33701
$726$	−1.00000	−0.0371135
$727$	−12.0000	−0.445055	−0.222528	−	0.974926i	$-0.571431\pi$
$727$	−12.0000	−0.445055	−0.222528	+	0.974926i	$0.571431\pi$
$728$	−6.00000	−0.222375
$729$	1.00000	0.0370370
$730$	−12.0000	−0.444140
$731$	32.0000	1.18356
$732$	1.00000	0.0369611
$733$	14.0000	0.517102	0.258551	−	0.965998i	$-0.416755\pi$
$733$	14.0000	0.517102	0.258551	+	0.965998i	$0.416755\pi$
$734$	−22.0000	−0.812035
$735$	−3.00000	−0.110657
$736$	4.00000	0.147442
$737$	−4.00000	−0.147342
$738$	3.00000	0.110432
$739$	−31.0000	−1.14035	−0.570177	−	0.821522i	$-0.693125\pi$
$739$	−31.0000	−1.14035	−0.570177	+	0.821522i	$0.693125\pi$
$740$	−6.00000	−0.220564
$741$	12.0000	0.440831
$742$	16.0000	0.587378
$743$	−5.00000	−0.183432	−0.0917161	−	0.995785i	$-0.529235\pi$
$743$	−5.00000	−0.183432	−0.0917161	+	0.995785i	$0.529235\pi$
$744$	3.00000	0.109985
$745$	−20.0000	−0.732743
$746$	−20.0000	−0.732252
$747$	4.00000	0.146352
$748$	−8.00000	−0.292509
$749$	26.0000	0.950019
$750$	−9.00000	−0.328634
$751$	2.00000	0.0729810	0.0364905	−	0.999334i	$-0.488382\pi$
$751$	2.00000	0.0729810	0.0364905	+	0.999334i	$0.488382\pi$
$752$	0	0
$753$	19.0000	0.692398
$754$	27.0000	0.983282
$755$	−4.00000	−0.145575
$756$	−2.00000	−0.0727393
$757$	−52.0000	−1.88997	−0.944986	−	0.327111i	$-0.893925\pi$
$757$	−52.0000	−1.88997	−0.944986	+	0.327111i	$0.893925\pi$
$758$	−13.0000	−0.472181
$759$	4.00000	0.145191
$760$	−4.00000	−0.145095
$761$	−12.0000	−0.435000	−0.217500	−	0.976060i	$-0.569790\pi$
$761$	−12.0000	−0.435000	−0.217500	+	0.976060i	$0.569790\pi$
$762$	0	0
$763$	6.00000	0.217215
$764$	18.0000	0.651217
$765$	−8.00000	−0.289241
$766$	14.0000	0.505841
$767$	9.00000	0.324971
$768$	−1.00000	−0.0360844
$769$	13.0000	0.468792	0.234396	−	0.972141i	$-0.424689\pi$
$769$	13.0000	0.468792	0.234396	+	0.972141i	$0.424689\pi$
$770$	2.00000	0.0720750
$771$	−26.0000	−0.936367
$772$	−17.0000	−0.611843
$773$	21.0000	0.755318	0.377659	−	0.925945i	$-0.376729\pi$
$773$	21.0000	0.755318	0.377659	+	0.925945i	$0.376729\pi$
$774$	4.00000	0.143777
$775$	12.0000	0.431053
$776$	13.0000	0.466673
$777$	−12.0000	−0.430498
$778$	36.0000	1.29066
$779$	12.0000	0.429945
$780$	−3.00000	−0.107417
$781$	−6.00000	−0.214697
$782$	32.0000	1.14432
$783$	9.00000	0.321634
$784$	−3.00000	−0.107143
$785$	7.00000	0.249841
$786$	−1.00000	−0.0356688
$787$	−35.0000	−1.24762	−0.623808	−	0.781578i	$-0.714415\pi$
$787$	−35.0000	−1.24762	−0.623808	+	0.781578i	$0.714415\pi$
$788$	−14.0000	−0.498729
$789$	6.00000	0.213606
$790$	4.00000	0.142314
$791$	4.00000	0.142224
$792$	−1.00000	−0.0355335
$793$	3.00000	0.106533
$794$	−29.0000	−1.02917
$795$	8.00000	0.283731
$796$	14.0000	0.496217
$797$	42.0000	1.48772	0.743858	−	0.668338i	$-0.232994\pi$
$797$	42.0000	1.48772	0.743858	+	0.668338i	$0.232994\pi$
$798$	−8.00000	−0.283197
$799$	0	0
$800$	−4.00000	−0.141421
$801$	−17.0000	−0.600665
$802$	25.0000	0.882781
$803$	−12.0000	−0.423471
$804$	−4.00000	−0.141069
$805$	−8.00000	−0.281963
$806$	9.00000	0.317011
$807$	−23.0000	−0.809638
$808$	−1.00000	−0.0351799
$809$	25.0000	0.878953	0.439477	−	0.898254i	$-0.355164\pi$
$809$	25.0000	0.878953	0.439477	+	0.898254i	$0.355164\pi$
$810$	−1.00000	−0.0351364
$811$	44.0000	1.54505	0.772524	−	0.634985i	$-0.218994\pi$
$811$	44.0000	1.54505	0.772524	+	0.634985i	$0.218994\pi$
$812$	−18.0000	−0.631676
$813$	−20.0000	−0.701431
$814$	−6.00000	−0.210300
$815$	5.00000	0.175142
$816$	−8.00000	−0.280056
$817$	16.0000	0.559769
$818$	31.0000	1.08389
$819$	−6.00000	−0.209657
$820$	−3.00000	−0.104765
$821$	−38.0000	−1.32621	−0.663105	−	0.748527i	$-0.730762\pi$
$821$	−38.0000	−1.32621	−0.663105	+	0.748527i	$0.730762\pi$
$822$	−14.0000	−0.488306
$823$	11.0000	0.383436	0.191718	−	0.981450i	$-0.438594\pi$
$823$	11.0000	0.383436	0.191718	+	0.981450i	$0.438594\pi$
$824$	12.0000	0.418040
$825$	−4.00000	−0.139262
$826$	−6.00000	−0.208767
$827$	−4.00000	−0.139094	−0.0695468	−	0.997579i	$-0.522155\pi$
$827$	−4.00000	−0.139094	−0.0695468	+	0.997579i	$0.522155\pi$
$828$	4.00000	0.139010
$829$	−20.0000	−0.694629	−0.347314	−	0.937749i	$-0.612906\pi$
$829$	−20.0000	−0.694629	−0.347314	+	0.937749i	$0.612906\pi$
$830$	−4.00000	−0.138842
$831$	−26.0000	−0.901930
$832$	−3.00000	−0.104006
$833$	−24.0000	−0.831551
$834$	19.0000	0.657916
$835$	−24.0000	−0.830554
$836$	−4.00000	−0.138343
$837$	3.00000	0.103695
$838$	−36.0000	−1.24360
$839$	9.00000	0.310715	0.155357	−	0.987858i	$-0.450347\pi$
$839$	9.00000	0.310715	0.155357	+	0.987858i	$0.450347\pi$
$840$	2.00000	0.0690066
$841$	52.0000	1.79310
$842$	−31.0000	−1.06833
$843$	−20.0000	−0.688837
$844$	−25.0000	−0.860535
$845$	4.00000	0.137604
$846$	0	0
$847$	2.00000	0.0687208
$848$	8.00000	0.274721
$849$	32.0000	1.09824
$850$	−32.0000	−1.09759
$851$	24.0000	0.822709
$852$	−6.00000	−0.205557
$853$	3.00000	0.102718	0.0513590	−	0.998680i	$-0.483645\pi$
$853$	3.00000	0.102718	0.0513590	+	0.998680i	$0.483645\pi$
$854$	−2.00000	−0.0684386
$855$	−4.00000	−0.136797
$856$	13.0000	0.444331
$857$	17.0000	0.580709	0.290354	−	0.956919i	$-0.406227\pi$
$857$	17.0000	0.580709	0.290354	+	0.956919i	$0.406227\pi$
$858$	−3.00000	−0.102418
$859$	31.0000	1.05771	0.528853	−	0.848713i	$-0.322622\pi$
$859$	31.0000	1.05771	0.528853	+	0.848713i	$0.322622\pi$
$860$	−4.00000	−0.136399
$861$	−6.00000	−0.204479
$862$	18.0000	0.613082
$863$	17.0000	0.578687	0.289343	−	0.957225i	$-0.406563\pi$
$863$	17.0000	0.578687	0.289343	+	0.957225i	$0.406563\pi$
$864$	−1.00000	−0.0340207
$865$	−11.0000	−0.374011
$866$	−4.00000	−0.135926
$867$	−47.0000	−1.59620
$868$	−6.00000	−0.203653
$869$	4.00000	0.135691
$870$	−9.00000	−0.305129
$871$	−12.0000	−0.406604
$872$	3.00000	0.101593
$873$	13.0000	0.439983
$874$	16.0000	0.541208
$875$	18.0000	0.608511
$876$	−12.0000	−0.405442
$877$	−36.0000	−1.21563	−0.607817	−	0.794077i	$-0.707955\pi$
$877$	−36.0000	−1.21563	−0.607817	+	0.794077i	$0.707955\pi$
$878$	13.0000	0.438729
$879$	8.00000	0.269833
$880$	1.00000	0.0337100
$881$	52.0000	1.75192	0.875962	−	0.482380i	$-0.160227\pi$
$881$	52.0000	1.75192	0.875962	+	0.482380i	$0.160227\pi$
$882$	−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	$0.478560\pi$
$884$	−24.0000	−0.807207
$885$	−3.00000	−0.100844
$886$	−26.0000	−0.873487
$887$	−24.0000	−0.805841	−0.402921	−	0.915235i	$-0.632005\pi$
$887$	−24.0000	−0.805841	−0.402921	+	0.915235i	$0.632005\pi$
$888$	−6.00000	−0.201347
$889$	0	0
$890$	17.0000	0.569841
$891$	−1.00000	−0.0335013
$892$	−12.0000	−0.401790
$893$	0	0
$894$	−20.0000	−0.668900
$895$	−14.0000	−0.467968
$896$	2.00000	0.0668153
$897$	12.0000	0.400668
$898$	−40.0000	−1.33482
$899$	27.0000	0.900500
$900$	−4.00000	−0.133333
$901$	64.0000	2.13215
$902$	−3.00000	−0.0998891
$903$	−8.00000	−0.266223
$904$	2.00000	0.0665190
$905$	−1.00000	−0.0332411
$906$	−4.00000	−0.132891
$907$	16.0000	0.531271	0.265636	−	0.964073i	$-0.414418\pi$
$907$	16.0000	0.531271	0.265636	+	0.964073i	$0.414418\pi$
$908$	24.0000	0.796468
$909$	−1.00000	−0.0331679
$910$	6.00000	0.198898
$911$	15.0000	0.496972	0.248486	−	0.968635i	$-0.420067\pi$
$911$	15.0000	0.496972	0.248486	+	0.968635i	$0.420067\pi$
$912$	−4.00000	−0.132453
$913$	−4.00000	−0.132381
$914$	−14.0000	−0.463079
$915$	−1.00000	−0.0330590
$916$	−4.00000	−0.132164
$917$	2.00000	0.0660458
$918$	−8.00000	−0.264039
$919$	−49.0000	−1.61636	−0.808180	−	0.588935i	$-0.799547\pi$
$919$	−49.0000	−1.61636	−0.808180	+	0.588935i	$0.799547\pi$
$920$	−4.00000	−0.131876
$921$	11.0000	0.362462
$922$	−22.0000	−0.724531
$923$	−18.0000	−0.592477
$924$	2.00000	0.0657952
$925$	−24.0000	−0.789115
$926$	6.00000	0.197172
$927$	12.0000	0.394132
$928$	−9.00000	−0.295439
$929$	0	0		−	1.00000i	$-0.5\pi$
$929$	0	0			1.00000i	$0.5\pi$
$930$	−3.00000	−0.0983739
$931$	−12.0000	−0.393284
$932$	−20.0000	−0.655122
$933$	−16.0000	−0.523816
$934$	33.0000	1.07979
$935$	8.00000	0.261628
$936$	−3.00000	−0.0980581
$937$	−26.0000	−0.849383	−0.424691	−	0.905338i	$-0.639617\pi$
$937$	−26.0000	−0.849383	−0.424691	+	0.905338i	$0.639617\pi$
$938$	8.00000	0.261209
$939$	4.00000	0.130535
$940$	0	0
$941$	27.0000	0.880175	0.440087	−	0.897955i	$-0.354947\pi$
$941$	27.0000	0.880175	0.440087	+	0.897955i	$0.354947\pi$
$942$	7.00000	0.228072
$943$	12.0000	0.390774
$944$	−3.00000	−0.0976417
$945$	2.00000	0.0650600
$946$	−4.00000	−0.130051
$947$	−48.0000	−1.55979	−0.779895	−	0.625910i	$-0.784728\pi$
$947$	−48.0000	−1.55979	−0.779895	+	0.625910i	$0.784728\pi$
$948$	4.00000	0.129914
$949$	−36.0000	−1.16861
$950$	−16.0000	−0.519109
$951$	−3.00000	−0.0972817
$952$	16.0000	0.518563
$953$	−42.0000	−1.36051	−0.680257	−	0.732974i	$-0.738132\pi$
$953$	−42.0000	−1.36051	−0.680257	+	0.732974i	$0.738132\pi$
$954$	8.00000	0.259010
$955$	−18.0000	−0.582466
$956$	−16.0000	−0.517477
$957$	−9.00000	−0.290929
$958$	0	0
$959$	28.0000	0.904167
$960$	1.00000	0.0322749
$961$	−22.0000	−0.709677
$962$	−18.0000	−0.580343
$963$	13.0000	0.418919
$964$	−8.00000	−0.257663
$965$	17.0000	0.547249
$966$	−8.00000	−0.257396
$967$	−23.0000	−0.739630	−0.369815	−	0.929105i	$-0.620579\pi$
$967$	−23.0000	−0.739630	−0.369815	+	0.929105i	$0.620579\pi$
$968$	1.00000	0.0321412
$969$	−32.0000	−1.02799
$970$	−13.0000	−0.417405
$971$	−36.0000	−1.15529	−0.577647	−	0.816286i	$-0.696029\pi$
$971$	−36.0000	−1.15529	−0.577647	+	0.816286i	$0.696029\pi$
$972$	−1.00000	−0.0320750
$973$	−38.0000	−1.21822
$974$	20.0000	0.640841
$975$	−12.0000	−0.384308
$976$	−1.00000	−0.0320092
$977$	44.0000	1.40768	0.703842	−	0.710356i	$-0.251466\pi$
$977$	44.0000	1.40768	0.703842	+	0.710356i	$0.251466\pi$
$978$	5.00000	0.159882
$979$	17.0000	0.543322
$980$	3.00000	0.0958315
$981$	3.00000	0.0957826
$982$	−28.0000	−0.893516
$983$	6.00000	0.191370	0.0956851	−	0.995412i	$-0.469496\pi$
$983$	6.00000	0.191370	0.0956851	+	0.995412i	$0.469496\pi$
$984$	−3.00000	−0.0956365
$985$	14.0000	0.446077
$986$	−72.0000	−2.29295
$987$	0	0
$988$	−12.0000	−0.381771
$989$	16.0000	0.508770
$990$	1.00000	0.0317821
$991$	−10.0000	−0.317660	−0.158830	−	0.987306i	$-0.550772\pi$
$991$	−10.0000	−0.317660	−0.158830	+	0.987306i	$0.550772\pi$
$992$	−3.00000	−0.0952501
$993$	4.00000	0.126936
$994$	12.0000	0.380617
$995$	−14.0000	−0.443830
$996$	−4.00000	−0.126745
$997$	40.0000	1.26681	0.633406	−	0.773819i	$-0.281656\pi$
$997$	40.0000	1.26681	0.633406	+	0.773819i	$0.281656\pi$
$998$	−22.0000	−0.696398
$999$	−6.00000	−0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4026.2.a.f.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4026.2.a.f.1.1	✓	1	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4026 = 2 \cdot 3 \cdot 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4026.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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