LMFDB - Embedded newform 82.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [82,2,Mod(1,82)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(82, 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("82.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 82 = 2 \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	82.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:	$0.654773296574$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	82.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} +2.00000 q^{6} -4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} +2.00000 q^{6} -4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} -2.00000 q^{11} -2.00000 q^{12} +4.00000 q^{13} +4.00000 q^{14} +4.00000 q^{15} +1.00000 q^{16} -2.00000 q^{17} -1.00000 q^{18} +6.00000 q^{19} -2.00000 q^{20} +8.00000 q^{21} +2.00000 q^{22} -8.00000 q^{23} +2.00000 q^{24} -1.00000 q^{25} -4.00000 q^{26} +4.00000 q^{27} -4.00000 q^{28} -4.00000 q^{30} -8.00000 q^{31} -1.00000 q^{32} +4.00000 q^{33} +2.00000 q^{34} +8.00000 q^{35} +1.00000 q^{36} +2.00000 q^{37} -6.00000 q^{38} -8.00000 q^{39} +2.00000 q^{40} -1.00000 q^{41} -8.00000 q^{42} -12.0000 q^{43} -2.00000 q^{44} -2.00000 q^{45} +8.00000 q^{46} +4.00000 q^{47} -2.00000 q^{48} +9.00000 q^{49} +1.00000 q^{50} +4.00000 q^{51} +4.00000 q^{52} -4.00000 q^{53} -4.00000 q^{54} +4.00000 q^{55} +4.00000 q^{56} -12.0000 q^{57} +8.00000 q^{59} +4.00000 q^{60} -14.0000 q^{61} +8.00000 q^{62} -4.00000 q^{63} +1.00000 q^{64} -8.00000 q^{65} -4.00000 q^{66} -2.00000 q^{67} -2.00000 q^{68} +16.0000 q^{69} -8.00000 q^{70} +8.00000 q^{71} -1.00000 q^{72} +10.0000 q^{73} -2.00000 q^{74} +2.00000 q^{75} +6.00000 q^{76} +8.00000 q^{77} +8.00000 q^{78} +4.00000 q^{79} -2.00000 q^{80} -11.0000 q^{81} +1.00000 q^{82} +12.0000 q^{83} +8.00000 q^{84} +4.00000 q^{85} +12.0000 q^{86} +2.00000 q^{88} -14.0000 q^{89} +2.00000 q^{90} -16.0000 q^{91} -8.00000 q^{92} +16.0000 q^{93} -4.00000 q^{94} -12.0000 q^{95} +2.00000 q^{96} +6.00000 q^{97} -9.00000 q^{98} -2.00000 q^{99} +O(q^{100})\)

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	$-0.695913\pi$
$3$	−2.00000	−1.15470	−0.577350	+	0.816497i	$0.695913\pi$
$4$	1.00000	0.500000
$5$	−2.00000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	2.00000	0.816497
$7$	−4.00000	−1.51186	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000	−1.51186	−0.755929	+	0.654654i	$0.772814\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	2.00000	0.632456
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	−2.00000	−0.577350
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	4.00000	1.06904
$15$	4.00000	1.03280
$16$	1.00000	0.250000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	−1.00000	−0.235702
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	−2.00000	−0.447214
$21$	8.00000	1.74574
$22$	2.00000	0.426401
$23$	−8.00000	−1.66812	−0.834058	−	0.551677i	$-0.813988\pi$
$23$	−8.00000	−1.66812	−0.834058	+	0.551677i	$0.813988\pi$
$24$	2.00000	0.408248
$25$	−1.00000	−0.200000
$26$	−4.00000	−0.784465
$27$	4.00000	0.769800
$28$	−4.00000	−0.755929
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	−4.00000	−0.730297
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	−1.00000	−0.176777
$33$	4.00000	0.696311
$34$	2.00000	0.342997
$35$	8.00000	1.35225
$36$	1.00000	0.166667
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	−6.00000	−0.973329
$39$	−8.00000	−1.28103
$40$	2.00000	0.316228
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	−8.00000	−1.23443
$43$	−12.0000	−1.82998	−0.914991	−	0.403473i	$-0.867803\pi$
$43$	−12.0000	−1.82998	−0.914991	+	0.403473i	$0.867803\pi$
$44$	−2.00000	−0.301511
$45$	−2.00000	−0.298142
$46$	8.00000	1.17954
$47$	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
$49$	9.00000	1.28571
$50$	1.00000	0.141421
$51$	4.00000	0.560112
$52$	4.00000	0.554700
$53$	−4.00000	−0.549442	−0.274721	−	0.961524i	$-0.588586\pi$
$53$	−4.00000	−0.549442	−0.274721	+	0.961524i	$0.588586\pi$
$54$	−4.00000	−0.544331
$55$	4.00000	0.539360
$56$	4.00000	0.534522
$57$	−12.0000	−1.58944
$58$	0	0
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	4.00000	0.516398
$61$	−14.0000	−1.79252	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−14.0000	−1.79252	−0.896258	+	0.443533i	$0.853725\pi$
$62$	8.00000	1.01600
$63$	−4.00000	−0.503953
$64$	1.00000	0.125000
$65$	−8.00000	−0.992278
$66$	−4.00000	−0.492366
$67$	−2.00000	−0.244339	−0.122169	−	0.992509i	$-0.538985\pi$
$67$	−2.00000	−0.244339	−0.122169	+	0.992509i	$0.538985\pi$
$68$	−2.00000	−0.242536
$69$	16.0000	1.92617
$70$	−8.00000	−0.956183
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	−1.00000	−0.117851
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	−2.00000	−0.232495
$75$	2.00000	0.230940
$76$	6.00000	0.688247
$77$	8.00000	0.911685
$78$	8.00000	0.905822
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	−2.00000	−0.223607
$81$	−11.0000	−1.22222
$82$	1.00000	0.110432
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	8.00000	0.872872
$85$	4.00000	0.433861
$86$	12.0000	1.29399
$87$	0	0
$88$	2.00000	0.213201
$89$	−14.0000	−1.48400	−0.741999	−	0.670402i	$-0.766122\pi$
$89$	−14.0000	−1.48400	−0.741999	+	0.670402i	$0.766122\pi$
$90$	2.00000	0.210819
$91$	−16.0000	−1.67726
$92$	−8.00000	−0.834058
$93$	16.0000	1.65912
$94$	−4.00000	−0.412568
$95$	−12.0000	−1.23117
$96$	2.00000	0.204124
$97$	6.00000	0.609208	0.304604	−	0.952479i	$-0.401476\pi$
$97$	6.00000	0.609208	0.304604	+	0.952479i	$0.401476\pi$
$98$	−9.00000	−0.909137
$99$	−2.00000	−0.201008
$100$	−1.00000	−0.100000
$101$	−12.0000	−1.19404	−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000	−1.19404	−0.597022	+	0.802225i	$0.703650\pi$
$102$	−4.00000	−0.396059
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	−4.00000	−0.392232
$105$	−16.0000	−1.56144
$106$	4.00000	0.388514
$107$	−4.00000	−0.386695	−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000	−0.386695	−0.193347	+	0.981130i	$0.561934\pi$
$108$	4.00000	0.384900
$109$	−12.0000	−1.14939	−0.574696	−	0.818367i	$-0.694880\pi$
$109$	−12.0000	−1.14939	−0.574696	+	0.818367i	$0.694880\pi$
$110$	−4.00000	−0.381385
$111$	−4.00000	−0.379663
$112$	−4.00000	−0.377964
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	12.0000	1.12390
$115$	16.0000	1.49201
$116$	0	0
$117$	4.00000	0.369800
$118$	−8.00000	−0.736460
$119$	8.00000	0.733359
$120$	−4.00000	−0.365148
$121$	−7.00000	−0.636364
$122$	14.0000	1.26750
$123$	2.00000	0.180334
$124$	−8.00000	−0.718421
$125$	12.0000	1.07331
$126$	4.00000	0.356348
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	−1.00000	−0.0883883
$129$	24.0000	2.11308
$130$	8.00000	0.701646
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	4.00000	0.348155
$133$	−24.0000	−2.08106
$134$	2.00000	0.172774
$135$	−8.00000	−0.688530
$136$	2.00000	0.171499
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	−16.0000	−1.36201
$139$	8.00000	0.678551	0.339276	−	0.940687i	$-0.389818\pi$
$139$	8.00000	0.678551	0.339276	+	0.940687i	$0.389818\pi$
$140$	8.00000	0.676123
$141$	−8.00000	−0.673722
$142$	−8.00000	−0.671345
$143$	−8.00000	−0.668994
$144$	1.00000	0.0833333
$145$	0	0
$146$	−10.0000	−0.827606
$147$	−18.0000	−1.48461
$148$	2.00000	0.164399
$149$	12.0000	0.983078	0.491539	−	0.870855i	$-0.336434\pi$
$149$	12.0000	0.983078	0.491539	+	0.870855i	$0.336434\pi$
$150$	−2.00000	−0.163299
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	−6.00000	−0.486664
$153$	−2.00000	−0.161690
$154$	−8.00000	−0.644658
$155$	16.0000	1.28515
$156$	−8.00000	−0.640513
$157$	4.00000	0.319235	0.159617	−	0.987179i	$-0.448974\pi$
$157$	4.00000	0.319235	0.159617	+	0.987179i	$0.448974\pi$
$158$	−4.00000	−0.318223
$159$	8.00000	0.634441
$160$	2.00000	0.158114
$161$	32.0000	2.52195
$162$	11.0000	0.864242
$163$	12.0000	0.939913	0.469956	−	0.882690i	$-0.344270\pi$
$163$	12.0000	0.939913	0.469956	+	0.882690i	$0.344270\pi$
$164$	−1.00000	−0.0780869
$165$	−8.00000	−0.622799
$166$	−12.0000	−0.931381
$167$	8.00000	0.619059	0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000	0.619059	0.309529	+	0.950890i	$0.399829\pi$
$168$	−8.00000	−0.617213
$169$	3.00000	0.230769
$170$	−4.00000	−0.306786
$171$	6.00000	0.458831
$172$	−12.0000	−0.914991
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	0	0
$175$	4.00000	0.302372
$176$	−2.00000	−0.150756
$177$	−16.0000	−1.20263
$178$	14.0000	1.04934
$179$	18.0000	1.34538	0.672692	−	0.739923i	$-0.265138\pi$
$179$	18.0000	1.34538	0.672692	+	0.739923i	$0.265138\pi$
$180$	−2.00000	−0.149071
$181$	−16.0000	−1.18927	−0.594635	−	0.803996i	$-0.702704\pi$
$181$	−16.0000	−1.18927	−0.594635	+	0.803996i	$0.702704\pi$
$182$	16.0000	1.18600
$183$	28.0000	2.06982
$184$	8.00000	0.589768
$185$	−4.00000	−0.294086
$186$	−16.0000	−1.17318
$187$	4.00000	0.292509
$188$	4.00000	0.291730
$189$	−16.0000	−1.16383
$190$	12.0000	0.870572
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	−2.00000	−0.144338
$193$	6.00000	0.431889	0.215945	−	0.976406i	$-0.430717\pi$
$193$	6.00000	0.431889	0.215945	+	0.976406i	$0.430717\pi$
$194$	−6.00000	−0.430775
$195$	16.0000	1.14578
$196$	9.00000	0.642857
$197$	−22.0000	−1.56744	−0.783718	−	0.621117i	$-0.786679\pi$
$197$	−22.0000	−1.56744	−0.783718	+	0.621117i	$0.786679\pi$
$198$	2.00000	0.142134
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	1.00000	0.0707107
$201$	4.00000	0.282138
$202$	12.0000	0.844317
$203$	0	0
$204$	4.00000	0.280056
$205$	2.00000	0.139686
$206$	0	0
$207$	−8.00000	−0.556038
$208$	4.00000	0.277350
$209$	−12.0000	−0.830057
$210$	16.0000	1.10410
$211$	14.0000	0.963800	0.481900	−	0.876226i	$-0.339947\pi$
$211$	14.0000	0.963800	0.481900	+	0.876226i	$0.339947\pi$
$212$	−4.00000	−0.274721
$213$	−16.0000	−1.09630
$214$	4.00000	0.273434
$215$	24.0000	1.63679
$216$	−4.00000	−0.272166
$217$	32.0000	2.17230
$218$	12.0000	0.812743
$219$	−20.0000	−1.35147
$220$	4.00000	0.269680
$221$	−8.00000	−0.538138
$222$	4.00000	0.268462
$223$	−24.0000	−1.60716	−0.803579	−	0.595198i	$-0.797074\pi$
$223$	−24.0000	−1.60716	−0.803579	+	0.595198i	$0.797074\pi$
$224$	4.00000	0.267261
$225$	−1.00000	−0.0666667
$226$	−2.00000	−0.133038
$227$	−18.0000	−1.19470	−0.597351	−	0.801980i	$-0.703780\pi$
$227$	−18.0000	−1.19470	−0.597351	+	0.801980i	$0.703780\pi$
$228$	−12.0000	−0.794719
$229$	−8.00000	−0.528655	−0.264327	−	0.964433i	$-0.585150\pi$
$229$	−8.00000	−0.528655	−0.264327	+	0.964433i	$0.585150\pi$
$230$	−16.0000	−1.05501
$231$	−16.0000	−1.05272
$232$	0	0
$233$	10.0000	0.655122	0.327561	−	0.944830i	$-0.393773\pi$
$233$	10.0000	0.655122	0.327561	+	0.944830i	$0.393773\pi$
$234$	−4.00000	−0.261488
$235$	−8.00000	−0.521862
$236$	8.00000	0.520756
$237$	−8.00000	−0.519656
$238$	−8.00000	−0.518563
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	4.00000	0.258199
$241$	−2.00000	−0.128831	−0.0644157	−	0.997923i	$-0.520518\pi$
$241$	−2.00000	−0.128831	−0.0644157	+	0.997923i	$0.520518\pi$
$242$	7.00000	0.449977
$243$	10.0000	0.641500
$244$	−14.0000	−0.896258
$245$	−18.0000	−1.14998
$246$	−2.00000	−0.127515
$247$	24.0000	1.52708
$248$	8.00000	0.508001
$249$	−24.0000	−1.52094
$250$	−12.0000	−0.758947
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	−4.00000	−0.251976
$253$	16.0000	1.00591
$254$	8.00000	0.501965
$255$	−8.00000	−0.500979
$256$	1.00000	0.0625000
$257$	−22.0000	−1.37232	−0.686161	−	0.727450i	$-0.740706\pi$
$257$	−22.0000	−1.37232	−0.686161	+	0.727450i	$0.740706\pi$
$258$	−24.0000	−1.49417
$259$	−8.00000	−0.497096
$260$	−8.00000	−0.496139
$261$	0	0
$262$	0	0
$263$	−28.0000	−1.72655	−0.863277	−	0.504730i	$-0.831592\pi$
$263$	−28.0000	−1.72655	−0.863277	+	0.504730i	$0.831592\pi$
$264$	−4.00000	−0.246183
$265$	8.00000	0.491436
$266$	24.0000	1.47153
$267$	28.0000	1.71357
$268$	−2.00000	−0.122169
$269$	30.0000	1.82913	0.914566	−	0.404436i	$-0.132532\pi$
$269$	30.0000	1.82913	0.914566	+	0.404436i	$0.132532\pi$
$270$	8.00000	0.486864
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	−2.00000	−0.121268
$273$	32.0000	1.93673
$274$	−6.00000	−0.362473
$275$	2.00000	0.120605
$276$	16.0000	0.963087
$277$	18.0000	1.08152	0.540758	−	0.841178i	$-0.318138\pi$
$277$	18.0000	1.08152	0.540758	+	0.841178i	$0.318138\pi$
$278$	−8.00000	−0.479808
$279$	−8.00000	−0.478947
$280$	−8.00000	−0.478091
$281$	−14.0000	−0.835170	−0.417585	−	0.908638i	$-0.637123\pi$
$281$	−14.0000	−0.835170	−0.417585	+	0.908638i	$0.637123\pi$
$282$	8.00000	0.476393
$283$	−16.0000	−0.951101	−0.475551	−	0.879688i	$-0.657751\pi$
$283$	−16.0000	−0.951101	−0.475551	+	0.879688i	$0.657751\pi$
$284$	8.00000	0.474713
$285$	24.0000	1.42164
$286$	8.00000	0.473050
$287$	4.00000	0.236113
$288$	−1.00000	−0.0589256
$289$	−13.0000	−0.764706
$290$	0	0
$291$	−12.0000	−0.703452
$292$	10.0000	0.585206
$293$	8.00000	0.467365	0.233682	−	0.972313i	$-0.424922\pi$
$293$	8.00000	0.467365	0.233682	+	0.972313i	$0.424922\pi$
$294$	18.0000	1.04978
$295$	−16.0000	−0.931556
$296$	−2.00000	−0.116248
$297$	−8.00000	−0.464207
$298$	−12.0000	−0.695141
$299$	−32.0000	−1.85061
$300$	2.00000	0.115470
$301$	48.0000	2.76667
$302$	8.00000	0.460348
$303$	24.0000	1.37876
$304$	6.00000	0.344124
$305$	28.0000	1.60328
$306$	2.00000	0.114332
$307$	−8.00000	−0.456584	−0.228292	−	0.973593i	$-0.573314\pi$
$307$	−8.00000	−0.456584	−0.228292	+	0.973593i	$0.573314\pi$
$308$	8.00000	0.455842
$309$	0	0
$310$	−16.0000	−0.908739
$311$	8.00000	0.453638	0.226819	−	0.973937i	$-0.427167\pi$
$311$	8.00000	0.453638	0.226819	+	0.973937i	$0.427167\pi$
$312$	8.00000	0.452911
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	−4.00000	−0.225733
$315$	8.00000	0.450749
$316$	4.00000	0.225018
$317$	4.00000	0.224662	0.112331	−	0.993671i	$-0.464168\pi$
$317$	4.00000	0.224662	0.112331	+	0.993671i	$0.464168\pi$
$318$	−8.00000	−0.448618
$319$	0	0
$320$	−2.00000	−0.111803
$321$	8.00000	0.446516
$322$	−32.0000	−1.78329
$323$	−12.0000	−0.667698
$324$	−11.0000	−0.611111
$325$	−4.00000	−0.221880
$326$	−12.0000	−0.664619
$327$	24.0000	1.32720
$328$	1.00000	0.0552158
$329$	−16.0000	−0.882109
$330$	8.00000	0.440386
$331$	18.0000	0.989369	0.494685	−	0.869072i	$-0.335284\pi$
$331$	18.0000	0.989369	0.494685	+	0.869072i	$0.335284\pi$
$332$	12.0000	0.658586
$333$	2.00000	0.109599
$334$	−8.00000	−0.437741
$335$	4.00000	0.218543
$336$	8.00000	0.436436
$337$	6.00000	0.326841	0.163420	−	0.986557i	$-0.447747\pi$
$337$	6.00000	0.326841	0.163420	+	0.986557i	$0.447747\pi$
$338$	−3.00000	−0.163178
$339$	−4.00000	−0.217250
$340$	4.00000	0.216930
$341$	16.0000	0.866449
$342$	−6.00000	−0.324443
$343$	−8.00000	−0.431959
$344$	12.0000	0.646997
$345$	−32.0000	−1.72282
$346$	6.00000	0.322562
$347$	−10.0000	−0.536828	−0.268414	−	0.963304i	$-0.586500\pi$
$347$	−10.0000	−0.536828	−0.268414	+	0.963304i	$0.586500\pi$
$348$	0	0
$349$	−10.0000	−0.535288	−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000	−0.535288	−0.267644	+	0.963518i	$0.586245\pi$
$350$	−4.00000	−0.213809
$351$	16.0000	0.854017
$352$	2.00000	0.106600
$353$	14.0000	0.745145	0.372572	−	0.928003i	$-0.378476\pi$
$353$	14.0000	0.745145	0.372572	+	0.928003i	$0.378476\pi$
$354$	16.0000	0.850390
$355$	−16.0000	−0.849192
$356$	−14.0000	−0.741999
$357$	−16.0000	−0.846810
$358$	−18.0000	−0.951330
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	2.00000	0.105409
$361$	17.0000	0.894737
$362$	16.0000	0.840941
$363$	14.0000	0.734809
$364$	−16.0000	−0.838628
$365$	−20.0000	−1.04685
$366$	−28.0000	−1.46358
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	−8.00000	−0.417029
$369$	−1.00000	−0.0520579
$370$	4.00000	0.207950
$371$	16.0000	0.830679
$372$	16.0000	0.829561
$373$	14.0000	0.724893	0.362446	−	0.932005i	$-0.381942\pi$
$373$	14.0000	0.724893	0.362446	+	0.932005i	$0.381942\pi$
$374$	−4.00000	−0.206835
$375$	−24.0000	−1.23935
$376$	−4.00000	−0.206284
$377$	0	0
$378$	16.0000	0.822951
$379$	−8.00000	−0.410932	−0.205466	−	0.978664i	$-0.565871\pi$
$379$	−8.00000	−0.410932	−0.205466	+	0.978664i	$0.565871\pi$
$380$	−12.0000	−0.615587
$381$	16.0000	0.819705
$382$	0	0
$383$	8.00000	0.408781	0.204390	−	0.978889i	$-0.434479\pi$
$383$	8.00000	0.408781	0.204390	+	0.978889i	$0.434479\pi$
$384$	2.00000	0.102062
$385$	−16.0000	−0.815436
$386$	−6.00000	−0.305392
$387$	−12.0000	−0.609994
$388$	6.00000	0.304604
$389$	18.0000	0.912636	0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000	0.912636	0.456318	+	0.889817i	$0.349168\pi$
$390$	−16.0000	−0.810191
$391$	16.0000	0.809155
$392$	−9.00000	−0.454569
$393$	0	0
$394$	22.0000	1.10834
$395$	−8.00000	−0.402524
$396$	−2.00000	−0.100504
$397$	8.00000	0.401508	0.200754	−	0.979642i	$-0.435661\pi$
$397$	8.00000	0.401508	0.200754	+	0.979642i	$0.435661\pi$
$398$	4.00000	0.200502
$399$	48.0000	2.40301
$400$	−1.00000	−0.0500000
$401$	2.00000	0.0998752	0.0499376	−	0.998752i	$-0.484098\pi$
$401$	2.00000	0.0998752	0.0499376	+	0.998752i	$0.484098\pi$
$402$	−4.00000	−0.199502
$403$	−32.0000	−1.59403
$404$	−12.0000	−0.597022
$405$	22.0000	1.09319
$406$	0	0
$407$	−4.00000	−0.198273
$408$	−4.00000	−0.198030
$409$	−26.0000	−1.28562	−0.642809	−	0.766027i	$-0.722231\pi$
$409$	−26.0000	−1.28562	−0.642809	+	0.766027i	$0.722231\pi$
$410$	−2.00000	−0.0987730
$411$	−12.0000	−0.591916
$412$	0	0
$413$	−32.0000	−1.57462
$414$	8.00000	0.393179
$415$	−24.0000	−1.17811
$416$	−4.00000	−0.196116
$417$	−16.0000	−0.783523
$418$	12.0000	0.586939
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	−16.0000	−0.780720
$421$	0	0		−	1.00000i	$-0.5\pi$
$421$	0	0			1.00000i	$0.5\pi$
$422$	−14.0000	−0.681509
$423$	4.00000	0.194487
$424$	4.00000	0.194257
$425$	2.00000	0.0970143
$426$	16.0000	0.775203
$427$	56.0000	2.71003
$428$	−4.00000	−0.193347
$429$	16.0000	0.772487
$430$	−24.0000	−1.15738
$431$	−24.0000	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	4.00000	0.192450
$433$	14.0000	0.672797	0.336399	−	0.941720i	$-0.390791\pi$
$433$	14.0000	0.672797	0.336399	+	0.941720i	$0.390791\pi$
$434$	−32.0000	−1.53605
$435$	0	0
$436$	−12.0000	−0.574696
$437$	−48.0000	−2.29615
$438$	20.0000	0.955637
$439$	8.00000	0.381819	0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000	0.381819	0.190910	+	0.981608i	$0.438856\pi$
$440$	−4.00000	−0.190693
$441$	9.00000	0.428571
$442$	8.00000	0.380521
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	−4.00000	−0.189832
$445$	28.0000	1.32733
$446$	24.0000	1.13643
$447$	−24.0000	−1.13516
$448$	−4.00000	−0.188982
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	1.00000	0.0471405
$451$	2.00000	0.0941763
$452$	2.00000	0.0940721
$453$	16.0000	0.751746
$454$	18.0000	0.844782
$455$	32.0000	1.50018
$456$	12.0000	0.561951
$457$	−34.0000	−1.59045	−0.795226	−	0.606313i	$-0.792648\pi$
$457$	−34.0000	−1.59045	−0.795226	+	0.606313i	$0.792648\pi$
$458$	8.00000	0.373815
$459$	−8.00000	−0.373408
$460$	16.0000	0.746004
$461$	2.00000	0.0931493	0.0465746	−	0.998915i	$-0.485169\pi$
$461$	2.00000	0.0931493	0.0465746	+	0.998915i	$0.485169\pi$
$462$	16.0000	0.744387
$463$	4.00000	0.185896	0.0929479	−	0.995671i	$-0.470371\pi$
$463$	4.00000	0.185896	0.0929479	+	0.995671i	$0.470371\pi$
$464$	0	0
$465$	−32.0000	−1.48396
$466$	−10.0000	−0.463241
$467$	8.00000	0.370196	0.185098	−	0.982720i	$-0.440740\pi$
$467$	8.00000	0.370196	0.185098	+	0.982720i	$0.440740\pi$
$468$	4.00000	0.184900
$469$	8.00000	0.369406
$470$	8.00000	0.369012
$471$	−8.00000	−0.368621
$472$	−8.00000	−0.368230
$473$	24.0000	1.10352
$474$	8.00000	0.367452
$475$	−6.00000	−0.275299
$476$	8.00000	0.366679
$477$	−4.00000	−0.183147
$478$	24.0000	1.09773
$479$	36.0000	1.64488	0.822441	−	0.568850i	$-0.192612\pi$
$479$	36.0000	1.64488	0.822441	+	0.568850i	$0.192612\pi$
$480$	−4.00000	−0.182574
$481$	8.00000	0.364769
$482$	2.00000	0.0910975
$483$	−64.0000	−2.91210
$484$	−7.00000	−0.318182
$485$	−12.0000	−0.544892
$486$	−10.0000	−0.453609
$487$	8.00000	0.362515	0.181257	−	0.983436i	$-0.441983\pi$
$487$	8.00000	0.362515	0.181257	+	0.983436i	$0.441983\pi$
$488$	14.0000	0.633750
$489$	−24.0000	−1.08532
$490$	18.0000	0.813157
$491$	−8.00000	−0.361035	−0.180517	−	0.983572i	$-0.557777\pi$
$491$	−8.00000	−0.361035	−0.180517	+	0.983572i	$0.557777\pi$
$492$	2.00000	0.0901670
$493$	0	0
$494$	−24.0000	−1.07981
$495$	4.00000	0.179787
$496$	−8.00000	−0.359211
$497$	−32.0000	−1.43540
$498$	24.0000	1.07547
$499$	26.0000	1.16392	0.581960	−	0.813217i	$-0.302286\pi$
$499$	26.0000	1.16392	0.581960	+	0.813217i	$0.302286\pi$
$500$	12.0000	0.536656
$501$	−16.0000	−0.714827
$502$	0	0
$503$	36.0000	1.60516	0.802580	−	0.596544i	$-0.203460\pi$
$503$	36.0000	1.60516	0.802580	+	0.596544i	$0.203460\pi$
$504$	4.00000	0.178174
$505$	24.0000	1.06799
$506$	−16.0000	−0.711287
$507$	−6.00000	−0.266469
$508$	−8.00000	−0.354943
$509$	32.0000	1.41838	0.709188	−	0.705020i	$-0.249062\pi$
$509$	32.0000	1.41838	0.709188	+	0.705020i	$0.249062\pi$
$510$	8.00000	0.354246
$511$	−40.0000	−1.76950
$512$	−1.00000	−0.0441942
$513$	24.0000	1.05963
$514$	22.0000	0.970378
$515$	0	0
$516$	24.0000	1.05654
$517$	−8.00000	−0.351840
$518$	8.00000	0.351500
$519$	12.0000	0.526742
$520$	8.00000	0.350823
$521$	−10.0000	−0.438108	−0.219054	−	0.975713i	$-0.570297\pi$
$521$	−10.0000	−0.438108	−0.219054	+	0.975713i	$0.570297\pi$
$522$	0	0
$523$	−4.00000	−0.174908	−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000	−0.174908	−0.0874539	+	0.996169i	$0.527873\pi$
$524$	0	0
$525$	−8.00000	−0.349149
$526$	28.0000	1.22086
$527$	16.0000	0.696971
$528$	4.00000	0.174078
$529$	41.0000	1.78261
$530$	−8.00000	−0.347498
$531$	8.00000	0.347170
$532$	−24.0000	−1.04053
$533$	−4.00000	−0.173259
$534$	−28.0000	−1.21168
$535$	8.00000	0.345870
$536$	2.00000	0.0863868
$537$	−36.0000	−1.55351
$538$	−30.0000	−1.29339
$539$	−18.0000	−0.775315
$540$	−8.00000	−0.344265
$541$	−10.0000	−0.429934	−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000	−0.429934	−0.214967	+	0.976621i	$0.568964\pi$
$542$	16.0000	0.687259
$543$	32.0000	1.37325
$544$	2.00000	0.0857493
$545$	24.0000	1.02805
$546$	−32.0000	−1.36947
$547$	−6.00000	−0.256541	−0.128271	−	0.991739i	$-0.540943\pi$
$547$	−6.00000	−0.256541	−0.128271	+	0.991739i	$0.540943\pi$
$548$	6.00000	0.256307
$549$	−14.0000	−0.597505
$550$	−2.00000	−0.0852803
$551$	0	0
$552$	−16.0000	−0.681005
$553$	−16.0000	−0.680389
$554$	−18.0000	−0.764747
$555$	8.00000	0.339581
$556$	8.00000	0.339276
$557$	24.0000	1.01691	0.508456	−	0.861088i	$-0.330216\pi$
$557$	24.0000	1.01691	0.508456	+	0.861088i	$0.330216\pi$
$558$	8.00000	0.338667
$559$	−48.0000	−2.03018
$560$	8.00000	0.338062
$561$	−8.00000	−0.337760
$562$	14.0000	0.590554
$563$	2.00000	0.0842900	0.0421450	−	0.999112i	$-0.486581\pi$
$563$	2.00000	0.0842900	0.0421450	+	0.999112i	$0.486581\pi$
$564$	−8.00000	−0.336861
$565$	−4.00000	−0.168281
$566$	16.0000	0.672530
$567$	44.0000	1.84783
$568$	−8.00000	−0.335673
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	−24.0000	−1.00525
$571$	−6.00000	−0.251092	−0.125546	−	0.992088i	$-0.540068\pi$
$571$	−6.00000	−0.251092	−0.125546	+	0.992088i	$0.540068\pi$
$572$	−8.00000	−0.334497
$573$	0	0
$574$	−4.00000	−0.166957
$575$	8.00000	0.333623
$576$	1.00000	0.0416667
$577$	34.0000	1.41544	0.707719	−	0.706494i	$-0.249724\pi$
$577$	34.0000	1.41544	0.707719	+	0.706494i	$0.249724\pi$
$578$	13.0000	0.540729
$579$	−12.0000	−0.498703
$580$	0	0
$581$	−48.0000	−1.99138
$582$	12.0000	0.497416
$583$	8.00000	0.331326
$584$	−10.0000	−0.413803
$585$	−8.00000	−0.330759
$586$	−8.00000	−0.330477
$587$	−10.0000	−0.412744	−0.206372	−	0.978474i	$-0.566166\pi$
$587$	−10.0000	−0.412744	−0.206372	+	0.978474i	$0.566166\pi$
$588$	−18.0000	−0.742307
$589$	−48.0000	−1.97781
$590$	16.0000	0.658710
$591$	44.0000	1.80992
$592$	2.00000	0.0821995
$593$	−6.00000	−0.246390	−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000	−0.246390	−0.123195	+	0.992382i	$0.539314\pi$
$594$	8.00000	0.328244
$595$	−16.0000	−0.655936
$596$	12.0000	0.491539
$597$	8.00000	0.327418
$598$	32.0000	1.30858
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	−2.00000	−0.0816497
$601$	34.0000	1.38689	0.693444	−	0.720510i	$-0.256092\pi$
$601$	34.0000	1.38689	0.693444	+	0.720510i	$0.256092\pi$
$602$	−48.0000	−1.95633
$603$	−2.00000	−0.0814463
$604$	−8.00000	−0.325515
$605$	14.0000	0.569181
$606$	−24.0000	−0.974933
$607$	16.0000	0.649420	0.324710	−	0.945814i	$-0.394733\pi$
$607$	16.0000	0.649420	0.324710	+	0.945814i	$0.394733\pi$
$608$	−6.00000	−0.243332
$609$	0	0
$610$	−28.0000	−1.13369
$611$	16.0000	0.647291
$612$	−2.00000	−0.0808452
$613$	22.0000	0.888572	0.444286	−	0.895885i	$-0.353457\pi$
$613$	22.0000	0.888572	0.444286	+	0.895885i	$0.353457\pi$
$614$	8.00000	0.322854
$615$	−4.00000	−0.161296
$616$	−8.00000	−0.322329
$617$	−38.0000	−1.52982	−0.764911	−	0.644136i	$-0.777217\pi$
$617$	−38.0000	−1.52982	−0.764911	+	0.644136i	$0.777217\pi$
$618$	0	0
$619$	48.0000	1.92928	0.964641	−	0.263566i	$-0.0848986\pi$
$619$	48.0000	1.92928	0.964641	+	0.263566i	$0.0848986\pi$
$620$	16.0000	0.642575
$621$	−32.0000	−1.28412
$622$	−8.00000	−0.320771
$623$	56.0000	2.24359
$624$	−8.00000	−0.320256
$625$	−19.0000	−0.760000
$626$	−14.0000	−0.559553
$627$	24.0000	0.958468
$628$	4.00000	0.159617
$629$	−4.00000	−0.159490
$630$	−8.00000	−0.318728
$631$	−32.0000	−1.27390	−0.636950	−	0.770905i	$-0.719804\pi$
$631$	−32.0000	−1.27390	−0.636950	+	0.770905i	$0.719804\pi$
$632$	−4.00000	−0.159111
$633$	−28.0000	−1.11290
$634$	−4.00000	−0.158860
$635$	16.0000	0.634941
$636$	8.00000	0.317221
$637$	36.0000	1.42637
$638$	0	0
$639$	8.00000	0.316475
$640$	2.00000	0.0790569
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	−8.00000	−0.315735
$643$	−38.0000	−1.49857	−0.749287	−	0.662246i	$-0.769604\pi$
$643$	−38.0000	−1.49857	−0.749287	+	0.662246i	$0.769604\pi$
$644$	32.0000	1.26098
$645$	−48.0000	−1.89000
$646$	12.0000	0.472134
$647$	−32.0000	−1.25805	−0.629025	−	0.777385i	$-0.716546\pi$
$647$	−32.0000	−1.25805	−0.629025	+	0.777385i	$0.716546\pi$
$648$	11.0000	0.432121
$649$	−16.0000	−0.628055
$650$	4.00000	0.156893
$651$	−64.0000	−2.50836
$652$	12.0000	0.469956
$653$	−24.0000	−0.939193	−0.469596	−	0.882881i	$-0.655601\pi$
$653$	−24.0000	−0.939193	−0.469596	+	0.882881i	$0.655601\pi$
$654$	−24.0000	−0.938474
$655$	0	0
$656$	−1.00000	−0.0390434
$657$	10.0000	0.390137
$658$	16.0000	0.623745
$659$	−30.0000	−1.16863	−0.584317	−	0.811525i	$-0.698638\pi$
$659$	−30.0000	−1.16863	−0.584317	+	0.811525i	$0.698638\pi$
$660$	−8.00000	−0.311400
$661$	14.0000	0.544537	0.272268	−	0.962221i	$-0.412226\pi$
$661$	14.0000	0.544537	0.272268	+	0.962221i	$0.412226\pi$
$662$	−18.0000	−0.699590
$663$	16.0000	0.621389
$664$	−12.0000	−0.465690
$665$	48.0000	1.86136
$666$	−2.00000	−0.0774984
$667$	0	0
$668$	8.00000	0.309529
$669$	48.0000	1.85579
$670$	−4.00000	−0.154533
$671$	28.0000	1.08093
$672$	−8.00000	−0.308607
$673$	−50.0000	−1.92736	−0.963679	−	0.267063i	$-0.913947\pi$
$673$	−50.0000	−1.92736	−0.963679	+	0.267063i	$0.913947\pi$
$674$	−6.00000	−0.231111
$675$	−4.00000	−0.153960
$676$	3.00000	0.115385
$677$	−18.0000	−0.691796	−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000	−0.691796	−0.345898	+	0.938272i	$0.612426\pi$
$678$	4.00000	0.153619
$679$	−24.0000	−0.921035
$680$	−4.00000	−0.153393
$681$	36.0000	1.37952
$682$	−16.0000	−0.612672
$683$	−38.0000	−1.45403	−0.727015	−	0.686622i	$-0.759093\pi$
$683$	−38.0000	−1.45403	−0.727015	+	0.686622i	$0.759093\pi$
$684$	6.00000	0.229416
$685$	−12.0000	−0.458496
$686$	8.00000	0.305441
$687$	16.0000	0.610438
$688$	−12.0000	−0.457496
$689$	−16.0000	−0.609551
$690$	32.0000	1.21822
$691$	−14.0000	−0.532585	−0.266293	−	0.963892i	$-0.585799\pi$
$691$	−14.0000	−0.532585	−0.266293	+	0.963892i	$0.585799\pi$
$692$	−6.00000	−0.228086
$693$	8.00000	0.303895
$694$	10.0000	0.379595
$695$	−16.0000	−0.606915
$696$	0	0
$697$	2.00000	0.0757554
$698$	10.0000	0.378506
$699$	−20.0000	−0.756469
$700$	4.00000	0.151186
$701$	2.00000	0.0755390	0.0377695	−	0.999286i	$-0.487975\pi$
$701$	2.00000	0.0755390	0.0377695	+	0.999286i	$0.487975\pi$
$702$	−16.0000	−0.603881
$703$	12.0000	0.452589
$704$	−2.00000	−0.0753778
$705$	16.0000	0.602595
$706$	−14.0000	−0.526897
$707$	48.0000	1.80523
$708$	−16.0000	−0.601317
$709$	16.0000	0.600893	0.300446	−	0.953799i	$-0.402864\pi$
$709$	16.0000	0.600893	0.300446	+	0.953799i	$0.402864\pi$
$710$	16.0000	0.600469
$711$	4.00000	0.150012
$712$	14.0000	0.524672
$713$	64.0000	2.39682
$714$	16.0000	0.598785
$715$	16.0000	0.598366
$716$	18.0000	0.672692
$717$	48.0000	1.79259
$718$	24.0000	0.895672
$719$	−16.0000	−0.596699	−0.298350	−	0.954457i	$-0.596436\pi$
$719$	−16.0000	−0.596699	−0.298350	+	0.954457i	$0.596436\pi$
$720$	−2.00000	−0.0745356
$721$	0	0
$722$	−17.0000	−0.632674
$723$	4.00000	0.148762
$724$	−16.0000	−0.594635
$725$	0	0
$726$	−14.0000	−0.519589
$727$	−8.00000	−0.296704	−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000	−0.296704	−0.148352	+	0.988935i	$0.547397\pi$
$728$	16.0000	0.592999
$729$	13.0000	0.481481
$730$	20.0000	0.740233
$731$	24.0000	0.887672
$732$	28.0000	1.03491
$733$	−22.0000	−0.812589	−0.406294	−	0.913742i	$-0.633179\pi$
$733$	−22.0000	−0.812589	−0.406294	+	0.913742i	$0.633179\pi$
$734$	−8.00000	−0.295285
$735$	36.0000	1.32788
$736$	8.00000	0.294884
$737$	4.00000	0.147342
$738$	1.00000	0.0368105
$739$	52.0000	1.91285	0.956425	−	0.291977i	$-0.0943129\pi$
$739$	52.0000	1.91285	0.956425	+	0.291977i	$0.0943129\pi$
$740$	−4.00000	−0.147043
$741$	−48.0000	−1.76332
$742$	−16.0000	−0.587378
$743$	−32.0000	−1.17397	−0.586983	−	0.809599i	$-0.699684\pi$
$743$	−32.0000	−1.17397	−0.586983	+	0.809599i	$0.699684\pi$
$744$	−16.0000	−0.586588
$745$	−24.0000	−0.879292
$746$	−14.0000	−0.512576
$747$	12.0000	0.439057
$748$	4.00000	0.146254
$749$	16.0000	0.584627
$750$	24.0000	0.876356
$751$	−8.00000	−0.291924	−0.145962	−	0.989290i	$-0.546628\pi$
$751$	−8.00000	−0.291924	−0.145962	+	0.989290i	$0.546628\pi$
$752$	4.00000	0.145865
$753$	0	0
$754$	0	0
$755$	16.0000	0.582300
$756$	−16.0000	−0.581914
$757$	−24.0000	−0.872295	−0.436147	−	0.899875i	$-0.643657\pi$
$757$	−24.0000	−0.872295	−0.436147	+	0.899875i	$0.643657\pi$
$758$	8.00000	0.290573
$759$	−32.0000	−1.16153
$760$	12.0000	0.435286
$761$	50.0000	1.81250	0.906249	−	0.422744i	$-0.138933\pi$
$761$	50.0000	1.81250	0.906249	+	0.422744i	$0.138933\pi$
$762$	−16.0000	−0.579619
$763$	48.0000	1.73772
$764$	0	0
$765$	4.00000	0.144620
$766$	−8.00000	−0.289052
$767$	32.0000	1.15545
$768$	−2.00000	−0.0721688
$769$	−14.0000	−0.504853	−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000	−0.504853	−0.252426	+	0.967616i	$0.581229\pi$
$770$	16.0000	0.576600
$771$	44.0000	1.58462
$772$	6.00000	0.215945
$773$	24.0000	0.863220	0.431610	−	0.902060i	$-0.357946\pi$
$773$	24.0000	0.863220	0.431610	+	0.902060i	$0.357946\pi$
$774$	12.0000	0.431331
$775$	8.00000	0.287368
$776$	−6.00000	−0.215387
$777$	16.0000	0.573997
$778$	−18.0000	−0.645331
$779$	−6.00000	−0.214972
$780$	16.0000	0.572892
$781$	−16.0000	−0.572525
$782$	−16.0000	−0.572159
$783$	0	0
$784$	9.00000	0.321429
$785$	−8.00000	−0.285532
$786$	0	0
$787$	−48.0000	−1.71102	−0.855508	−	0.517790i	$-0.826755\pi$
$787$	−48.0000	−1.71102	−0.855508	+	0.517790i	$0.826755\pi$
$788$	−22.0000	−0.783718
$789$	56.0000	1.99365
$790$	8.00000	0.284627
$791$	−8.00000	−0.284447
$792$	2.00000	0.0710669
$793$	−56.0000	−1.98862
$794$	−8.00000	−0.283909
$795$	−16.0000	−0.567462
$796$	−4.00000	−0.141776
$797$	−10.0000	−0.354218	−0.177109	−	0.984191i	$-0.556675\pi$
$797$	−10.0000	−0.354218	−0.177109	+	0.984191i	$0.556675\pi$
$798$	−48.0000	−1.69918
$799$	−8.00000	−0.283020
$800$	1.00000	0.0353553
$801$	−14.0000	−0.494666
$802$	−2.00000	−0.0706225
$803$	−20.0000	−0.705785
$804$	4.00000	0.141069
$805$	−64.0000	−2.25570
$806$	32.0000	1.12715
$807$	−60.0000	−2.11210
$808$	12.0000	0.422159
$809$	−26.0000	−0.914111	−0.457056	−	0.889438i	$-0.651096\pi$
$809$	−26.0000	−0.914111	−0.457056	+	0.889438i	$0.651096\pi$
$810$	−22.0000	−0.773001
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	0	0
$813$	32.0000	1.12229
$814$	4.00000	0.140200
$815$	−24.0000	−0.840683
$816$	4.00000	0.140028
$817$	−72.0000	−2.51896
$818$	26.0000	0.909069
$819$	−16.0000	−0.559085
$820$	2.00000	0.0698430
$821$	18.0000	0.628204	0.314102	−	0.949389i	$-0.398297\pi$
$821$	18.0000	0.628204	0.314102	+	0.949389i	$0.398297\pi$
$822$	12.0000	0.418548
$823$	−16.0000	−0.557725	−0.278862	−	0.960331i	$-0.589957\pi$
$823$	−16.0000	−0.557725	−0.278862	+	0.960331i	$0.589957\pi$
$824$	0	0
$825$	−4.00000	−0.139262
$826$	32.0000	1.11342
$827$	−14.0000	−0.486828	−0.243414	−	0.969923i	$-0.578267\pi$
$827$	−14.0000	−0.486828	−0.243414	+	0.969923i	$0.578267\pi$
$828$	−8.00000	−0.278019
$829$	−18.0000	−0.625166	−0.312583	−	0.949890i	$-0.601194\pi$
$829$	−18.0000	−0.625166	−0.312583	+	0.949890i	$0.601194\pi$
$830$	24.0000	0.833052
$831$	−36.0000	−1.24883
$832$	4.00000	0.138675
$833$	−18.0000	−0.623663
$834$	16.0000	0.554035
$835$	−16.0000	−0.553703
$836$	−12.0000	−0.415029
$837$	−32.0000	−1.10608
$838$	12.0000	0.414533
$839$	12.0000	0.414286	0.207143	−	0.978311i	$-0.433583\pi$
$839$	12.0000	0.414286	0.207143	+	0.978311i	$0.433583\pi$
$840$	16.0000	0.552052
$841$	−29.0000	−1.00000
$842$	0	0
$843$	28.0000	0.964371
$844$	14.0000	0.481900
$845$	−6.00000	−0.206406
$846$	−4.00000	−0.137523
$847$	28.0000	0.962091
$848$	−4.00000	−0.137361
$849$	32.0000	1.09824
$850$	−2.00000	−0.0685994
$851$	−16.0000	−0.548473
$852$	−16.0000	−0.548151
$853$	−14.0000	−0.479351	−0.239675	−	0.970853i	$-0.577041\pi$
$853$	−14.0000	−0.479351	−0.239675	+	0.970853i	$0.577041\pi$
$854$	−56.0000	−1.91628
$855$	−12.0000	−0.410391
$856$	4.00000	0.136717
$857$	−42.0000	−1.43469	−0.717346	−	0.696717i	$-0.754643\pi$
$857$	−42.0000	−1.43469	−0.717346	+	0.696717i	$0.754643\pi$
$858$	−16.0000	−0.546231
$859$	0	0		−	1.00000i	$-0.5\pi$
$859$	0	0			1.00000i	$0.5\pi$
$860$	24.0000	0.818393
$861$	−8.00000	−0.272639
$862$	24.0000	0.817443
$863$	56.0000	1.90626	0.953131	−	0.302558i	$-0.0978405\pi$
$863$	56.0000	1.90626	0.953131	+	0.302558i	$0.0978405\pi$
$864$	−4.00000	−0.136083
$865$	12.0000	0.408012
$866$	−14.0000	−0.475739
$867$	26.0000	0.883006
$868$	32.0000	1.08615
$869$	−8.00000	−0.271381
$870$	0	0
$871$	−8.00000	−0.271070
$872$	12.0000	0.406371
$873$	6.00000	0.203069
$874$	48.0000	1.62362
$875$	−48.0000	−1.62270
$876$	−20.0000	−0.675737
$877$	22.0000	0.742887	0.371444	−	0.928456i	$-0.378863\pi$
$877$	22.0000	0.742887	0.371444	+	0.928456i	$0.378863\pi$
$878$	−8.00000	−0.269987
$879$	−16.0000	−0.539667
$880$	4.00000	0.134840
$881$	30.0000	1.01073	0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000	1.01073	0.505363	+	0.862907i	$0.331359\pi$
$882$	−9.00000	−0.303046
$883$	−22.0000	−0.740359	−0.370179	−	0.928960i	$-0.620704\pi$
$883$	−22.0000	−0.740359	−0.370179	+	0.928960i	$0.620704\pi$
$884$	−8.00000	−0.269069
$885$	32.0000	1.07567
$886$	0	0
$887$	52.0000	1.74599	0.872995	−	0.487730i	$-0.162175\pi$
$887$	52.0000	1.74599	0.872995	+	0.487730i	$0.162175\pi$
$888$	4.00000	0.134231
$889$	32.0000	1.07325
$890$	−28.0000	−0.938562
$891$	22.0000	0.737028
$892$	−24.0000	−0.803579
$893$	24.0000	0.803129
$894$	24.0000	0.802680
$895$	−36.0000	−1.20335
$896$	4.00000	0.133631
$897$	64.0000	2.13690
$898$	6.00000	0.200223
$899$	0	0
$900$	−1.00000	−0.0333333
$901$	8.00000	0.266519
$902$	−2.00000	−0.0665927
$903$	−96.0000	−3.19468
$904$	−2.00000	−0.0665190
$905$	32.0000	1.06372
$906$	−16.0000	−0.531564
$907$	48.0000	1.59381	0.796907	−	0.604102i	$-0.206468\pi$
$907$	48.0000	1.59381	0.796907	+	0.604102i	$0.206468\pi$
$908$	−18.0000	−0.597351
$909$	−12.0000	−0.398015
$910$	−32.0000	−1.06079
$911$	40.0000	1.32526	0.662630	−	0.748947i	$-0.269440\pi$
$911$	40.0000	1.32526	0.662630	+	0.748947i	$0.269440\pi$
$912$	−12.0000	−0.397360
$913$	−24.0000	−0.794284
$914$	34.0000	1.12462
$915$	−56.0000	−1.85130
$916$	−8.00000	−0.264327
$917$	0	0
$918$	8.00000	0.264039
$919$	40.0000	1.31948	0.659739	−	0.751495i	$-0.270667\pi$
$919$	40.0000	1.31948	0.659739	+	0.751495i	$0.270667\pi$
$920$	−16.0000	−0.527504
$921$	16.0000	0.527218
$922$	−2.00000	−0.0658665
$923$	32.0000	1.05329
$924$	−16.0000	−0.526361
$925$	−2.00000	−0.0657596
$926$	−4.00000	−0.131448
$927$	0	0
$928$	0	0
$929$	14.0000	0.459325	0.229663	−	0.973270i	$-0.426238\pi$
$929$	14.0000	0.459325	0.229663	+	0.973270i	$0.426238\pi$
$930$	32.0000	1.04932
$931$	54.0000	1.76978
$932$	10.0000	0.327561
$933$	−16.0000	−0.523816
$934$	−8.00000	−0.261768
$935$	−8.00000	−0.261628
$936$	−4.00000	−0.130744
$937$	−46.0000	−1.50275	−0.751377	−	0.659873i	$-0.770610\pi$
$937$	−46.0000	−1.50275	−0.751377	+	0.659873i	$0.770610\pi$
$938$	−8.00000	−0.261209
$939$	−28.0000	−0.913745
$940$	−8.00000	−0.260931
$941$	−46.0000	−1.49956	−0.749779	−	0.661689i	$-0.769840\pi$
$941$	−46.0000	−1.49956	−0.749779	+	0.661689i	$0.769840\pi$
$942$	8.00000	0.260654
$943$	8.00000	0.260516
$944$	8.00000	0.260378
$945$	32.0000	1.04096
$946$	−24.0000	−0.780307
$947$	−24.0000	−0.779895	−0.389948	−	0.920837i	$-0.627507\pi$
$947$	−24.0000	−0.779895	−0.389948	+	0.920837i	$0.627507\pi$
$948$	−8.00000	−0.259828
$949$	40.0000	1.29845
$950$	6.00000	0.194666
$951$	−8.00000	−0.259418
$952$	−8.00000	−0.259281
$953$	−58.0000	−1.87880	−0.939402	−	0.342817i	$-0.888619\pi$
$953$	−58.0000	−1.87880	−0.939402	+	0.342817i	$0.888619\pi$
$954$	4.00000	0.129505
$955$	0	0
$956$	−24.0000	−0.776215
$957$	0	0
$958$	−36.0000	−1.16311
$959$	−24.0000	−0.775000
$960$	4.00000	0.129099
$961$	33.0000	1.06452
$962$	−8.00000	−0.257930
$963$	−4.00000	−0.128898
$964$	−2.00000	−0.0644157
$965$	−12.0000	−0.386294
$966$	64.0000	2.05917
$967$	44.0000	1.41494	0.707472	−	0.706741i	$-0.249835\pi$
$967$	44.0000	1.41494	0.707472	+	0.706741i	$0.249835\pi$
$968$	7.00000	0.224989
$969$	24.0000	0.770991
$970$	12.0000	0.385297
$971$	−18.0000	−0.577647	−0.288824	−	0.957382i	$-0.593264\pi$
$971$	−18.0000	−0.577647	−0.288824	+	0.957382i	$0.593264\pi$
$972$	10.0000	0.320750
$973$	−32.0000	−1.02587
$974$	−8.00000	−0.256337
$975$	8.00000	0.256205
$976$	−14.0000	−0.448129
$977$	30.0000	0.959785	0.479893	−	0.877327i	$-0.340676\pi$
$977$	30.0000	0.959785	0.479893	+	0.877327i	$0.340676\pi$
$978$	24.0000	0.767435
$979$	28.0000	0.894884
$980$	−18.0000	−0.574989
$981$	−12.0000	−0.383131
$982$	8.00000	0.255290
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	−2.00000	−0.0637577
$985$	44.0000	1.40196
$986$	0	0
$987$	32.0000	1.01857
$988$	24.0000	0.763542
$989$	96.0000	3.05262
$990$	−4.00000	−0.127128
$991$	32.0000	1.01651	0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000	1.01651	0.508257	+	0.861206i	$0.330290\pi$
$992$	8.00000	0.254000
$993$	−36.0000	−1.14243
$994$	32.0000	1.01498
$995$	8.00000	0.253617
$996$	−24.0000	−0.760469
$997$	28.0000	0.886769	0.443384	−	0.896332i	$-0.353778\pi$
$997$	28.0000	0.886769	0.443384	+	0.896332i	$0.353778\pi$
$998$	−26.0000	−0.823016
$999$	8.00000	0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	82.2.a.a.1.1	✓	1
3.2	odd	2		738.2.a.h.1.1		1
4.3	odd	2		656.2.a.c.1.1		1
5.2	odd	4		2050.2.c.e.1149.1		2
5.3	odd	4		2050.2.c.e.1149.2		2
5.4	even	2		2050.2.a.g.1.1		1
7.6	odd	2		4018.2.a.j.1.1		1
8.3	odd	2		2624.2.a.c.1.1		1
8.5	even	2		2624.2.a.g.1.1		1
11.10	odd	2		9922.2.a.c.1.1		1
12.11	even	2		5904.2.a.s.1.1		1
41.40	even	2		3362.2.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
82.2.a.a.1.1	✓	1	1.1	even	1	trivial
656.2.a.c.1.1		1	4.3	odd	2
738.2.a.h.1.1		1	3.2	odd	2
2050.2.a.g.1.1		1	5.4	even	2
2050.2.c.e.1149.1		2	5.2	odd	4
2050.2.c.e.1149.2		2	5.3	odd	4
2624.2.a.c.1.1		1	8.3	odd	2
2624.2.a.g.1.1		1	8.5	even	2
3362.2.a.b.1.1		1	41.40	even	2
4018.2.a.j.1.1		1	7.6	odd	2
5904.2.a.s.1.1		1	12.11	even	2
9922.2.a.c.1.1		1	11.10	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:	\( N \)	\(=\)	\( 82 = 2 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	82.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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