LMFDB - Embedded newform 1682.2.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1682,2,Mod(1,1682)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1682, 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("1682.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1682 = 2 \cdot 29^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1682.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:	$13.4308376200$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 58)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1682.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -2.00000 q^{7} -1.00000 q^{8} -2.00000 q^{9} -1.00000 q^{10} +3.00000 q^{11} +1.00000 q^{12} -1.00000 q^{13} +2.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} -8.00000 q^{17} +2.00000 q^{18} +1.00000 q^{20} -2.00000 q^{21} -3.00000 q^{22} +4.00000 q^{23} -1.00000 q^{24} -4.00000 q^{25} +1.00000 q^{26} -5.00000 q^{27} -2.00000 q^{28} -1.00000 q^{30} +3.00000 q^{31} -1.00000 q^{32} +3.00000 q^{33} +8.00000 q^{34} -2.00000 q^{35} -2.00000 q^{36} -8.00000 q^{37} -1.00000 q^{39} -1.00000 q^{40} -2.00000 q^{41} +2.00000 q^{42} +11.0000 q^{43} +3.00000 q^{44} -2.00000 q^{45} -4.00000 q^{46} -13.0000 q^{47} +1.00000 q^{48} -3.00000 q^{49} +4.00000 q^{50} -8.00000 q^{51} -1.00000 q^{52} -11.0000 q^{53} +5.00000 q^{54} +3.00000 q^{55} +2.00000 q^{56} +1.00000 q^{60} +8.00000 q^{61} -3.00000 q^{62} +4.00000 q^{63} +1.00000 q^{64} -1.00000 q^{65} -3.00000 q^{66} -12.0000 q^{67} -8.00000 q^{68} +4.00000 q^{69} +2.00000 q^{70} +2.00000 q^{71} +2.00000 q^{72} -4.00000 q^{73} +8.00000 q^{74} -4.00000 q^{75} -6.00000 q^{77} +1.00000 q^{78} -15.0000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +2.00000 q^{82} +4.00000 q^{83} -2.00000 q^{84} -8.00000 q^{85} -11.0000 q^{86} -3.00000 q^{88} +10.0000 q^{89} +2.00000 q^{90} +2.00000 q^{91} +4.00000 q^{92} +3.00000 q^{93} +13.0000 q^{94} -1.00000 q^{96} +2.00000 q^{97} +3.00000 q^{98} -6.00000 q^{99} +O(q^{100})\)

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	0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	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	$0.428217\pi$
$6$	−1.00000	−0.408248
$7$	−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
$9$	−2.00000	−0.666667
$10$	−1.00000	−0.316228
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	1.00000	0.288675
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	2.00000	0.534522
$15$	1.00000	0.258199
$16$	1.00000	0.250000
$17$	−8.00000	−1.94029	−0.970143	−	0.242536i	$-0.922021\pi$
$17$	−8.00000	−1.94029	−0.970143	+	0.242536i	$0.922021\pi$
$18$	2.00000	0.471405
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	1.00000	0.223607
$21$	−2.00000	−0.436436
$22$	−3.00000	−0.639602
$23$	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
$26$	1.00000	0.196116
$27$	−5.00000	−0.962250
$28$	−2.00000	−0.377964
$29$	0	0
$29$	0	0
$30$	−1.00000	−0.182574
$31$	3.00000	0.538816	0.269408	−	0.963026i	$-0.413172\pi$
$31$	3.00000	0.538816	0.269408	+	0.963026i	$0.413172\pi$
$32$	−1.00000	−0.176777
$33$	3.00000	0.522233
$34$	8.00000	1.37199
$35$	−2.00000	−0.338062
$36$	−2.00000	−0.333333
$37$	−8.00000	−1.31519	−0.657596	−	0.753371i	$-0.728427\pi$
$37$	−8.00000	−1.31519	−0.657596	+	0.753371i	$0.728427\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	−1.00000	−0.158114
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	2.00000	0.308607
$43$	11.0000	1.67748	0.838742	−	0.544529i	$-0.183292\pi$
$43$	11.0000	1.67748	0.838742	+	0.544529i	$0.183292\pi$
$44$	3.00000	0.452267
$45$	−2.00000	−0.298142
$46$	−4.00000	−0.589768
$47$	−13.0000	−1.89624	−0.948122	−	0.317905i	$-0.897021\pi$
$47$	−13.0000	−1.89624	−0.948122	+	0.317905i	$0.897021\pi$
$48$	1.00000	0.144338
$49$	−3.00000	−0.428571
$50$	4.00000	0.565685
$51$	−8.00000	−1.12022
$52$	−1.00000	−0.138675
$53$	−11.0000	−1.51097	−0.755483	−	0.655168i	$-0.772598\pi$
$53$	−11.0000	−1.51097	−0.755483	+	0.655168i	$0.772598\pi$
$54$	5.00000	0.680414
$55$	3.00000	0.404520
$56$	2.00000	0.267261
$57$	0	0
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	1.00000	0.129099
$61$	8.00000	1.02430	0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000	1.02430	0.512148	+	0.858898i	$0.328850\pi$
$62$	−3.00000	−0.381000
$63$	4.00000	0.503953
$64$	1.00000	0.125000
$65$	−1.00000	−0.124035
$66$	−3.00000	−0.369274
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	−8.00000	−0.970143
$69$	4.00000	0.481543
$70$	2.00000	0.239046
$71$	2.00000	0.237356	0.118678	−	0.992933i	$-0.462134\pi$
$71$	2.00000	0.237356	0.118678	+	0.992933i	$0.462134\pi$
$72$	2.00000	0.235702
$73$	−4.00000	−0.468165	−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	8.00000	0.929981
$75$	−4.00000	−0.461880
$76$	0	0
$77$	−6.00000	−0.683763
$78$	1.00000	0.113228
$79$	−15.0000	−1.68763	−0.843816	−	0.536633i	$-0.819696\pi$
$79$	−15.0000	−1.68763	−0.843816	+	0.536633i	$0.819696\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	2.00000	0.220863
$83$	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
$86$	−11.0000	−1.18616
$87$	0	0
$88$	−3.00000	−0.319801
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	2.00000	0.210819
$91$	2.00000	0.209657
$92$	4.00000	0.417029
$93$	3.00000	0.311086
$94$	13.0000	1.34085
$95$	0	0
$96$	−1.00000	−0.102062
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	3.00000	0.303046
$99$	−6.00000	−0.603023
$100$	−4.00000	−0.400000
$101$	8.00000	0.796030	0.398015	−	0.917379i	$-0.369699\pi$
$101$	8.00000	0.796030	0.398015	+	0.917379i	$0.369699\pi$
$102$	8.00000	0.792118
$103$	14.0000	1.37946	0.689730	−	0.724066i	$-0.257729\pi$
$103$	14.0000	1.37946	0.689730	+	0.724066i	$0.257729\pi$
$104$	1.00000	0.0980581
$105$	−2.00000	−0.195180
$106$	11.0000	1.06841
$107$	−2.00000	−0.193347	−0.0966736	−	0.995316i	$-0.530820\pi$
$107$	−2.00000	−0.193347	−0.0966736	+	0.995316i	$0.530820\pi$
$108$	−5.00000	−0.481125
$109$	5.00000	0.478913	0.239457	−	0.970907i	$-0.423031\pi$
$109$	5.00000	0.478913	0.239457	+	0.970907i	$0.423031\pi$
$110$	−3.00000	−0.286039
$111$	−8.00000	−0.759326
$112$	−2.00000	−0.188982
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	4.00000	0.373002
$116$	0	0
$117$	2.00000	0.184900
$118$	0	0
$119$	16.0000	1.46672
$120$	−1.00000	−0.0912871
$121$	−2.00000	−0.181818
$122$	−8.00000	−0.724286
$123$	−2.00000	−0.180334
$124$	3.00000	0.269408
$125$	−9.00000	−0.804984
$126$	−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
$129$	11.0000	0.968496
$130$	1.00000	0.0877058
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	3.00000	0.261116
$133$	0	0
$134$	12.0000	1.03664
$135$	−5.00000	−0.430331
$136$	8.00000	0.685994
$137$	12.0000	1.02523	0.512615	−	0.858619i	$-0.328677\pi$
$137$	12.0000	1.02523	0.512615	+	0.858619i	$0.328677\pi$
$138$	−4.00000	−0.340503
$139$	−20.0000	−1.69638	−0.848189	−	0.529694i	$-0.822307\pi$
$139$	−20.0000	−1.69638	−0.848189	+	0.529694i	$0.822307\pi$
$140$	−2.00000	−0.169031
$141$	−13.0000	−1.09480
$142$	−2.00000	−0.167836
$143$	−3.00000	−0.250873
$144$	−2.00000	−0.166667
$145$	0	0
$146$	4.00000	0.331042
$147$	−3.00000	−0.247436
$148$	−8.00000	−0.657596
$149$	15.0000	1.22885	0.614424	−	0.788976i	$-0.289388\pi$
$149$	15.0000	1.22885	0.614424	+	0.788976i	$0.289388\pi$
$150$	4.00000	0.326599
$151$	2.00000	0.162758	0.0813788	−	0.996683i	$-0.474068\pi$
$151$	2.00000	0.162758	0.0813788	+	0.996683i	$0.474068\pi$
$152$	0	0
$153$	16.0000	1.29352
$154$	6.00000	0.483494
$155$	3.00000	0.240966
$156$	−1.00000	−0.0800641
$157$	−18.0000	−1.43656	−0.718278	−	0.695756i	$-0.755069\pi$
$157$	−18.0000	−1.43656	−0.718278	+	0.695756i	$0.755069\pi$
$158$	15.0000	1.19334
$159$	−11.0000	−0.872357
$160$	−1.00000	−0.0790569
$161$	−8.00000	−0.630488
$162$	−1.00000	−0.0785674
$163$	−9.00000	−0.704934	−0.352467	−	0.935824i	$-0.614657\pi$
$163$	−9.00000	−0.704934	−0.352467	+	0.935824i	$0.614657\pi$
$164$	−2.00000	−0.156174
$165$	3.00000	0.233550
$166$	−4.00000	−0.310460
$167$	−2.00000	−0.154765	−0.0773823	−	0.997001i	$-0.524656\pi$
$167$	−2.00000	−0.154765	−0.0773823	+	0.997001i	$0.524656\pi$
$168$	2.00000	0.154303
$169$	−12.0000	−0.923077
$170$	8.00000	0.613572
$171$	0	0
$172$	11.0000	0.838742
$173$	−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
$175$	8.00000	0.604743
$176$	3.00000	0.226134
$177$	0	0
$178$	−10.0000	−0.749532
$179$	−10.0000	−0.747435	−0.373718	−	0.927543i	$-0.621917\pi$
$179$	−10.0000	−0.747435	−0.373718	+	0.927543i	$0.621917\pi$
$180$	−2.00000	−0.149071
$181$	7.00000	0.520306	0.260153	−	0.965567i	$-0.416227\pi$
$181$	7.00000	0.520306	0.260153	+	0.965567i	$0.416227\pi$
$182$	−2.00000	−0.148250
$183$	8.00000	0.591377
$184$	−4.00000	−0.294884
$185$	−8.00000	−0.588172
$186$	−3.00000	−0.219971
$187$	−24.0000	−1.75505
$188$	−13.0000	−0.948122
$189$	10.0000	0.727393
$190$	0	0
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	1.00000	0.0721688
$193$	−14.0000	−1.00774	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000	−1.00774	−0.503871	+	0.863779i	$0.668091\pi$
$194$	−2.00000	−0.143592
$195$	−1.00000	−0.0716115
$196$	−3.00000	−0.214286
$197$	18.0000	1.28245	0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000	1.28245	0.641223	+	0.767354i	$0.278427\pi$
$198$	6.00000	0.426401
$199$	−10.0000	−0.708881	−0.354441	−	0.935079i	$-0.615329\pi$
$199$	−10.0000	−0.708881	−0.354441	+	0.935079i	$0.615329\pi$
$200$	4.00000	0.282843
$201$	−12.0000	−0.846415
$202$	−8.00000	−0.562878
$203$	0	0
$204$	−8.00000	−0.560112
$205$	−2.00000	−0.139686
$206$	−14.0000	−0.975426
$207$	−8.00000	−0.556038
$208$	−1.00000	−0.0693375
$209$	0	0
$210$	2.00000	0.138013
$211$	3.00000	0.206529	0.103264	−	0.994654i	$-0.467071\pi$
$211$	3.00000	0.206529	0.103264	+	0.994654i	$0.467071\pi$
$212$	−11.0000	−0.755483
$213$	2.00000	0.137038
$214$	2.00000	0.136717
$215$	11.0000	0.750194
$216$	5.00000	0.340207
$217$	−6.00000	−0.407307
$218$	−5.00000	−0.338643
$219$	−4.00000	−0.270295
$220$	3.00000	0.202260
$221$	8.00000	0.538138
$222$	8.00000	0.536925
$223$	−26.0000	−1.74109	−0.870544	−	0.492090i	$-0.836233\pi$
$223$	−26.0000	−1.74109	−0.870544	+	0.492090i	$0.836233\pi$
$224$	2.00000	0.133631
$225$	8.00000	0.533333
$226$	−6.00000	−0.399114
$227$	18.0000	1.19470	0.597351	−	0.801980i	$-0.296220\pi$
$227$	18.0000	1.19470	0.597351	+	0.801980i	$0.296220\pi$
$228$	0	0
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	−4.00000	−0.263752
$231$	−6.00000	−0.394771
$232$	0	0
$233$	−1.00000	−0.0655122	−0.0327561	−	0.999463i	$-0.510428\pi$
$233$	−1.00000	−0.0655122	−0.0327561	+	0.999463i	$0.510428\pi$
$234$	−2.00000	−0.130744
$235$	−13.0000	−0.848026
$236$	0	0
$237$	−15.0000	−0.974355
$238$	−16.0000	−1.03713
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	1.00000	0.0645497
$241$	17.0000	1.09507	0.547533	−	0.836784i	$-0.315567\pi$
$241$	17.0000	1.09507	0.547533	+	0.836784i	$0.315567\pi$
$242$	2.00000	0.128565
$243$	16.0000	1.02640
$244$	8.00000	0.512148
$245$	−3.00000	−0.191663
$246$	2.00000	0.127515
$247$	0	0
$248$	−3.00000	−0.190500
$249$	4.00000	0.253490
$250$	9.00000	0.569210
$251$	−27.0000	−1.70422	−0.852112	−	0.523359i	$-0.824679\pi$
$251$	−27.0000	−1.70422	−0.852112	+	0.523359i	$0.824679\pi$
$252$	4.00000	0.251976
$253$	12.0000	0.754434
$254$	8.00000	0.501965
$255$	−8.00000	−0.500979
$256$	1.00000	0.0625000
$257$	13.0000	0.810918	0.405459	−	0.914113i	$-0.367112\pi$
$257$	13.0000	0.810918	0.405459	+	0.914113i	$0.367112\pi$
$258$	−11.0000	−0.684830
$259$	16.0000	0.994192
$260$	−1.00000	−0.0620174
$261$	0	0
$262$	12.0000	0.741362
$263$	−9.00000	−0.554964	−0.277482	−	0.960731i	$-0.589500\pi$
$263$	−9.00000	−0.554964	−0.277482	+	0.960731i	$0.589500\pi$
$264$	−3.00000	−0.184637
$265$	−11.0000	−0.675725
$266$	0	0
$267$	10.0000	0.611990
$268$	−12.0000	−0.733017
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	5.00000	0.304290
$271$	13.0000	0.789694	0.394847	−	0.918747i	$-0.370798\pi$
$271$	13.0000	0.789694	0.394847	+	0.918747i	$0.370798\pi$
$272$	−8.00000	−0.485071
$273$	2.00000	0.121046
$274$	−12.0000	−0.724947
$275$	−12.0000	−0.723627
$276$	4.00000	0.240772
$277$	−2.00000	−0.120168	−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000	−0.120168	−0.0600842	+	0.998193i	$0.519137\pi$
$278$	20.0000	1.19952
$279$	−6.00000	−0.359211
$280$	2.00000	0.119523
$281$	27.0000	1.61068	0.805342	−	0.592810i	$-0.201981\pi$
$281$	27.0000	1.61068	0.805342	+	0.592810i	$0.201981\pi$
$282$	13.0000	0.774139
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	2.00000	0.118678
$285$	0	0
$286$	3.00000	0.177394
$287$	4.00000	0.236113
$288$	2.00000	0.117851
$289$	47.0000	2.76471
$290$	0	0
$291$	2.00000	0.117242
$292$	−4.00000	−0.234082
$293$	−14.0000	−0.817889	−0.408944	−	0.912559i	$-0.634103\pi$
$293$	−14.0000	−0.817889	−0.408944	+	0.912559i	$0.634103\pi$
$294$	3.00000	0.174964
$295$	0	0
$296$	8.00000	0.464991
$297$	−15.0000	−0.870388
$298$	−15.0000	−0.868927
$299$	−4.00000	−0.231326
$300$	−4.00000	−0.230940
$301$	−22.0000	−1.26806
$302$	−2.00000	−0.115087
$303$	8.00000	0.459588
$304$	0	0
$305$	8.00000	0.458079
$306$	−16.0000	−0.914659
$307$	7.00000	0.399511	0.199756	−	0.979846i	$-0.435985\pi$
$307$	7.00000	0.399511	0.199756	+	0.979846i	$0.435985\pi$
$308$	−6.00000	−0.341882
$309$	14.0000	0.796432
$310$	−3.00000	−0.170389
$311$	8.00000	0.453638	0.226819	−	0.973937i	$-0.427167\pi$
$311$	8.00000	0.453638	0.226819	+	0.973937i	$0.427167\pi$
$312$	1.00000	0.0566139
$313$	9.00000	0.508710	0.254355	−	0.967111i	$-0.418137\pi$
$313$	9.00000	0.508710	0.254355	+	0.967111i	$0.418137\pi$
$314$	18.0000	1.01580
$315$	4.00000	0.225374
$316$	−15.0000	−0.843816
$317$	12.0000	0.673987	0.336994	−	0.941507i	$-0.390590\pi$
$317$	12.0000	0.673987	0.336994	+	0.941507i	$0.390590\pi$
$318$	11.0000	0.616849
$319$	0	0
$320$	1.00000	0.0559017
$321$	−2.00000	−0.111629
$322$	8.00000	0.445823
$323$	0	0
$324$	1.00000	0.0555556
$325$	4.00000	0.221880
$326$	9.00000	0.498464
$327$	5.00000	0.276501
$328$	2.00000	0.110432
$329$	26.0000	1.43343
$330$	−3.00000	−0.165145
$331$	23.0000	1.26419	0.632097	−	0.774889i	$-0.282194\pi$
$331$	23.0000	1.26419	0.632097	+	0.774889i	$0.282194\pi$
$332$	4.00000	0.219529
$333$	16.0000	0.876795
$334$	2.00000	0.109435
$335$	−12.0000	−0.655630
$336$	−2.00000	−0.109109
$337$	32.0000	1.74315	0.871576	−	0.490261i	$-0.163099\pi$
$337$	32.0000	1.74315	0.871576	+	0.490261i	$0.163099\pi$
$338$	12.0000	0.652714
$339$	6.00000	0.325875
$340$	−8.00000	−0.433861
$341$	9.00000	0.487377
$342$	0	0
$343$	20.0000	1.07990
$344$	−11.0000	−0.593080
$345$	4.00000	0.215353
$346$	6.00000	0.322562
$347$	−2.00000	−0.107366	−0.0536828	−	0.998558i	$-0.517096\pi$
$347$	−2.00000	−0.107366	−0.0536828	+	0.998558i	$0.517096\pi$
$348$	0	0
$349$	−15.0000	−0.802932	−0.401466	−	0.915874i	$-0.631499\pi$
$349$	−15.0000	−0.802932	−0.401466	+	0.915874i	$0.631499\pi$
$350$	−8.00000	−0.427618
$351$	5.00000	0.266880
$352$	−3.00000	−0.159901
$353$	−26.0000	−1.38384	−0.691920	−	0.721974i	$-0.743235\pi$
$353$	−26.0000	−1.38384	−0.691920	+	0.721974i	$0.743235\pi$
$354$	0	0
$355$	2.00000	0.106149
$356$	10.0000	0.529999
$357$	16.0000	0.846810
$358$	10.0000	0.528516
$359$	25.0000	1.31945	0.659725	−	0.751507i	$-0.270673\pi$
$359$	25.0000	1.31945	0.659725	+	0.751507i	$0.270673\pi$
$360$	2.00000	0.105409
$361$	−19.0000	−1.00000
$362$	−7.00000	−0.367912
$363$	−2.00000	−0.104973
$364$	2.00000	0.104828
$365$	−4.00000	−0.209370
$366$	−8.00000	−0.418167
$367$	32.0000	1.67039	0.835193	−	0.549957i	$-0.185356\pi$
$367$	32.0000	1.67039	0.835193	+	0.549957i	$0.185356\pi$
$368$	4.00000	0.208514
$369$	4.00000	0.208232
$370$	8.00000	0.415900
$371$	22.0000	1.14218
$372$	3.00000	0.155543
$373$	−21.0000	−1.08734	−0.543669	−	0.839299i	$-0.682965\pi$
$373$	−21.0000	−1.08734	−0.543669	+	0.839299i	$0.682965\pi$
$374$	24.0000	1.24101
$375$	−9.00000	−0.464758
$376$	13.0000	0.670424
$377$	0	0
$378$	−10.0000	−0.514344
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	0	0
$381$	−8.00000	−0.409852
$382$	−8.00000	−0.409316
$383$	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
$385$	−6.00000	−0.305788
$386$	14.0000	0.712581
$387$	−22.0000	−1.11832
$388$	2.00000	0.101535
$389$	0	0		−	1.00000i	$-0.5\pi$
$389$	0	0			1.00000i	$0.5\pi$
$390$	1.00000	0.0506370
$391$	−32.0000	−1.61831
$392$	3.00000	0.151523
$393$	−12.0000	−0.605320
$394$	−18.0000	−0.906827
$395$	−15.0000	−0.754732
$396$	−6.00000	−0.301511
$397$	−17.0000	−0.853206	−0.426603	−	0.904439i	$-0.640290\pi$
$397$	−17.0000	−0.853206	−0.426603	+	0.904439i	$0.640290\pi$
$398$	10.0000	0.501255
$399$	0	0
$400$	−4.00000	−0.200000
$401$	27.0000	1.34832	0.674158	−	0.738587i	$-0.264507\pi$
$401$	27.0000	1.34832	0.674158	+	0.738587i	$0.264507\pi$
$402$	12.0000	0.598506
$403$	−3.00000	−0.149441
$404$	8.00000	0.398015
$405$	1.00000	0.0496904
$406$	0	0
$407$	−24.0000	−1.18964
$408$	8.00000	0.396059
$409$	−30.0000	−1.48340	−0.741702	−	0.670729i	$-0.765981\pi$
$409$	−30.0000	−1.48340	−0.741702	+	0.670729i	$0.765981\pi$
$410$	2.00000	0.0987730
$411$	12.0000	0.591916
$412$	14.0000	0.689730
$413$	0	0
$414$	8.00000	0.393179
$415$	4.00000	0.196352
$416$	1.00000	0.0490290
$417$	−20.0000	−0.979404
$418$	0	0
$419$	−10.0000	−0.488532	−0.244266	−	0.969708i	$-0.578547\pi$
$419$	−10.0000	−0.488532	−0.244266	+	0.969708i	$0.578547\pi$
$420$	−2.00000	−0.0975900
$421$	−32.0000	−1.55958	−0.779792	−	0.626038i	$-0.784675\pi$
$421$	−32.0000	−1.55958	−0.779792	+	0.626038i	$0.784675\pi$
$422$	−3.00000	−0.146038
$423$	26.0000	1.26416
$424$	11.0000	0.534207
$425$	32.0000	1.55223
$426$	−2.00000	−0.0969003
$427$	−16.0000	−0.774294
$428$	−2.00000	−0.0966736
$429$	−3.00000	−0.144841
$430$	−11.0000	−0.530467
$431$	32.0000	1.54139	0.770693	−	0.637207i	$-0.219910\pi$
$431$	32.0000	1.54139	0.770693	+	0.637207i	$0.219910\pi$
$432$	−5.00000	−0.240563
$433$	16.0000	0.768911	0.384455	−	0.923144i	$-0.374389\pi$
$433$	16.0000	0.768911	0.384455	+	0.923144i	$0.374389\pi$
$434$	6.00000	0.288009
$435$	0	0
$436$	5.00000	0.239457
$437$	0	0
$438$	4.00000	0.191127
$439$	20.0000	0.954548	0.477274	−	0.878755i	$-0.341625\pi$
$439$	20.0000	0.954548	0.477274	+	0.878755i	$0.341625\pi$
$440$	−3.00000	−0.143019
$441$	6.00000	0.285714
$442$	−8.00000	−0.380521
$443$	−4.00000	−0.190046	−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000	−0.190046	−0.0950229	+	0.995475i	$0.530292\pi$
$444$	−8.00000	−0.379663
$445$	10.0000	0.474045
$446$	26.0000	1.23114
$447$	15.0000	0.709476
$448$	−2.00000	−0.0944911
$449$	10.0000	0.471929	0.235965	−	0.971762i	$-0.424175\pi$
$449$	10.0000	0.471929	0.235965	+	0.971762i	$0.424175\pi$
$450$	−8.00000	−0.377124
$451$	−6.00000	−0.282529
$452$	6.00000	0.282216
$453$	2.00000	0.0939682
$454$	−18.0000	−0.844782
$455$	2.00000	0.0937614
$456$	0	0
$457$	−2.00000	−0.0935561	−0.0467780	−	0.998905i	$-0.514895\pi$
$457$	−2.00000	−0.0935561	−0.0467780	+	0.998905i	$0.514895\pi$
$458$	10.0000	0.467269
$459$	40.0000	1.86704
$460$	4.00000	0.186501
$461$	−2.00000	−0.0931493	−0.0465746	−	0.998915i	$-0.514831\pi$
$461$	−2.00000	−0.0931493	−0.0465746	+	0.998915i	$0.514831\pi$
$462$	6.00000	0.279145
$463$	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
$465$	3.00000	0.139122
$466$	1.00000	0.0463241
$467$	27.0000	1.24941	0.624705	−	0.780860i	$-0.285219\pi$
$467$	27.0000	1.24941	0.624705	+	0.780860i	$0.285219\pi$
$468$	2.00000	0.0924500
$469$	24.0000	1.10822
$470$	13.0000	0.599645
$471$	−18.0000	−0.829396
$472$	0	0
$473$	33.0000	1.51734
$474$	15.0000	0.688973
$475$	0	0
$476$	16.0000	0.733359
$477$	22.0000	1.00731
$478$	0	0
$479$	5.00000	0.228456	0.114228	−	0.993455i	$-0.463561\pi$
$479$	5.00000	0.228456	0.114228	+	0.993455i	$0.463561\pi$
$480$	−1.00000	−0.0456435
$481$	8.00000	0.364769
$482$	−17.0000	−0.774329
$483$	−8.00000	−0.364013
$484$	−2.00000	−0.0909091
$485$	2.00000	0.0908153
$486$	−16.0000	−0.725775
$487$	−22.0000	−0.996915	−0.498458	−	0.866914i	$-0.666100\pi$
$487$	−22.0000	−0.996915	−0.498458	+	0.866914i	$0.666100\pi$
$488$	−8.00000	−0.362143
$489$	−9.00000	−0.406994
$490$	3.00000	0.135526
$491$	33.0000	1.48927	0.744635	−	0.667472i	$-0.232624\pi$
$491$	33.0000	1.48927	0.744635	+	0.667472i	$0.232624\pi$
$492$	−2.00000	−0.0901670
$493$	0	0
$494$	0	0
$495$	−6.00000	−0.269680
$496$	3.00000	0.134704
$497$	−4.00000	−0.179425
$498$	−4.00000	−0.179244
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	−9.00000	−0.402492
$501$	−2.00000	−0.0893534
$502$	27.0000	1.20507
$503$	−19.0000	−0.847168	−0.423584	−	0.905857i	$-0.639228\pi$
$503$	−19.0000	−0.847168	−0.423584	+	0.905857i	$0.639228\pi$
$504$	−4.00000	−0.178174
$505$	8.00000	0.355995
$506$	−12.0000	−0.533465
$507$	−12.0000	−0.532939
$508$	−8.00000	−0.354943
$509$	−15.0000	−0.664863	−0.332432	−	0.943127i	$-0.607869\pi$
$509$	−15.0000	−0.664863	−0.332432	+	0.943127i	$0.607869\pi$
$510$	8.00000	0.354246
$511$	8.00000	0.353899
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−13.0000	−0.573405
$515$	14.0000	0.616914
$516$	11.0000	0.484248
$517$	−39.0000	−1.71522
$518$	−16.0000	−0.703000
$519$	−6.00000	−0.263371
$520$	1.00000	0.0438529
$521$	−13.0000	−0.569540	−0.284770	−	0.958596i	$-0.591917\pi$
$521$	−13.0000	−0.569540	−0.284770	+	0.958596i	$0.591917\pi$
$522$	0	0
$523$	24.0000	1.04945	0.524723	−	0.851273i	$-0.324169\pi$
$523$	24.0000	1.04945	0.524723	+	0.851273i	$0.324169\pi$
$524$	−12.0000	−0.524222
$525$	8.00000	0.349149
$526$	9.00000	0.392419
$527$	−24.0000	−1.04546
$528$	3.00000	0.130558
$529$	−7.00000	−0.304348
$530$	11.0000	0.477809
$531$	0	0
$532$	0	0
$533$	2.00000	0.0866296
$534$	−10.0000	−0.432742
$535$	−2.00000	−0.0864675
$536$	12.0000	0.518321
$537$	−10.0000	−0.431532
$538$	0	0
$539$	−9.00000	−0.387657
$540$	−5.00000	−0.215166
$541$	8.00000	0.343947	0.171973	−	0.985102i	$-0.444986\pi$
$541$	8.00000	0.343947	0.171973	+	0.985102i	$0.444986\pi$
$542$	−13.0000	−0.558398
$543$	7.00000	0.300399
$544$	8.00000	0.342997
$545$	5.00000	0.214176
$546$	−2.00000	−0.0855921
$547$	38.0000	1.62476	0.812381	−	0.583127i	$-0.198171\pi$
$547$	38.0000	1.62476	0.812381	+	0.583127i	$0.198171\pi$
$548$	12.0000	0.512615
$549$	−16.0000	−0.682863
$550$	12.0000	0.511682
$551$	0	0
$552$	−4.00000	−0.170251
$553$	30.0000	1.27573
$554$	2.00000	0.0849719
$555$	−8.00000	−0.339581
$556$	−20.0000	−0.848189
$557$	−2.00000	−0.0847427	−0.0423714	−	0.999102i	$-0.513491\pi$
$557$	−2.00000	−0.0847427	−0.0423714	+	0.999102i	$0.513491\pi$
$558$	6.00000	0.254000
$559$	−11.0000	−0.465250
$560$	−2.00000	−0.0845154
$561$	−24.0000	−1.01328
$562$	−27.0000	−1.13893
$563$	11.0000	0.463595	0.231797	−	0.972764i	$-0.425539\pi$
$563$	11.0000	0.463595	0.231797	+	0.972764i	$0.425539\pi$
$564$	−13.0000	−0.547399
$565$	6.00000	0.252422
$566$	−4.00000	−0.168133
$567$	−2.00000	−0.0839921
$568$	−2.00000	−0.0839181
$569$	30.0000	1.25767	0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000	1.25767	0.628833	+	0.777541i	$0.283533\pi$
$570$	0	0
$571$	−28.0000	−1.17176	−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000	−1.17176	−0.585882	+	0.810397i	$0.699252\pi$
$572$	−3.00000	−0.125436
$573$	8.00000	0.334205
$574$	−4.00000	−0.166957
$575$	−16.0000	−0.667246
$576$	−2.00000	−0.0833333
$577$	−8.00000	−0.333044	−0.166522	−	0.986038i	$-0.553254\pi$
$577$	−8.00000	−0.333044	−0.166522	+	0.986038i	$0.553254\pi$
$578$	−47.0000	−1.95494
$579$	−14.0000	−0.581820
$580$	0	0
$581$	−8.00000	−0.331896
$582$	−2.00000	−0.0829027
$583$	−33.0000	−1.36672
$584$	4.00000	0.165521
$585$	2.00000	0.0826898
$586$	14.0000	0.578335
$587$	28.0000	1.15568	0.577842	−	0.816149i	$-0.303895\pi$
$587$	28.0000	1.15568	0.577842	+	0.816149i	$0.303895\pi$
$588$	−3.00000	−0.123718
$589$	0	0
$590$	0	0
$591$	18.0000	0.740421
$592$	−8.00000	−0.328798
$593$	39.0000	1.60154	0.800769	−	0.598973i	$-0.204424\pi$
$593$	39.0000	1.60154	0.800769	+	0.598973i	$0.204424\pi$
$594$	15.0000	0.615457
$595$	16.0000	0.655936
$596$	15.0000	0.614424
$597$	−10.0000	−0.409273
$598$	4.00000	0.163572
$599$	5.00000	0.204294	0.102147	−	0.994769i	$-0.467429\pi$
$599$	5.00000	0.204294	0.102147	+	0.994769i	$0.467429\pi$
$600$	4.00000	0.163299
$601$	−2.00000	−0.0815817	−0.0407909	−	0.999168i	$-0.512988\pi$
$601$	−2.00000	−0.0815817	−0.0407909	+	0.999168i	$0.512988\pi$
$602$	22.0000	0.896653
$603$	24.0000	0.977356
$604$	2.00000	0.0813788
$605$	−2.00000	−0.0813116
$606$	−8.00000	−0.324978
$607$	−3.00000	−0.121766	−0.0608831	−	0.998145i	$-0.519392\pi$
$607$	−3.00000	−0.121766	−0.0608831	+	0.998145i	$0.519392\pi$
$608$	0	0
$609$	0	0
$610$	−8.00000	−0.323911
$611$	13.0000	0.525924
$612$	16.0000	0.646762
$613$	−31.0000	−1.25208	−0.626039	−	0.779792i	$-0.715325\pi$
$613$	−31.0000	−1.25208	−0.626039	+	0.779792i	$0.715325\pi$
$614$	−7.00000	−0.282497
$615$	−2.00000	−0.0806478
$616$	6.00000	0.241747
$617$	12.0000	0.483102	0.241551	−	0.970388i	$-0.422344\pi$
$617$	12.0000	0.483102	0.241551	+	0.970388i	$0.422344\pi$
$618$	−14.0000	−0.563163
$619$	35.0000	1.40677	0.703384	−	0.710810i	$-0.251671\pi$
$619$	35.0000	1.40677	0.703384	+	0.710810i	$0.251671\pi$
$620$	3.00000	0.120483
$621$	−20.0000	−0.802572
$622$	−8.00000	−0.320771
$623$	−20.0000	−0.801283
$624$	−1.00000	−0.0400320
$625$	11.0000	0.440000
$626$	−9.00000	−0.359712
$627$	0	0
$628$	−18.0000	−0.718278
$629$	64.0000	2.55185
$630$	−4.00000	−0.159364
$631$	−38.0000	−1.51276	−0.756378	−	0.654135i	$-0.773033\pi$
$631$	−38.0000	−1.51276	−0.756378	+	0.654135i	$0.773033\pi$
$632$	15.0000	0.596668
$633$	3.00000	0.119239
$634$	−12.0000	−0.476581
$635$	−8.00000	−0.317470
$636$	−11.0000	−0.436178
$637$	3.00000	0.118864
$638$	0	0
$639$	−4.00000	−0.158238
$640$	−1.00000	−0.0395285
$641$	8.00000	0.315981	0.157991	−	0.987441i	$-0.449498\pi$
$641$	8.00000	0.315981	0.157991	+	0.987441i	$0.449498\pi$
$642$	2.00000	0.0789337
$643$	34.0000	1.34083	0.670415	−	0.741987i	$-0.266116\pi$
$643$	34.0000	1.34083	0.670415	+	0.741987i	$0.266116\pi$
$644$	−8.00000	−0.315244
$645$	11.0000	0.433125
$646$	0	0
$647$	−32.0000	−1.25805	−0.629025	−	0.777385i	$-0.716546\pi$
$647$	−32.0000	−1.25805	−0.629025	+	0.777385i	$0.716546\pi$
$648$	−1.00000	−0.0392837
$649$	0	0
$650$	−4.00000	−0.156893
$651$	−6.00000	−0.235159
$652$	−9.00000	−0.352467
$653$	26.0000	1.01746	0.508729	−	0.860927i	$-0.330115\pi$
$653$	26.0000	1.01746	0.508729	+	0.860927i	$0.330115\pi$
$654$	−5.00000	−0.195515
$655$	−12.0000	−0.468879
$656$	−2.00000	−0.0780869
$657$	8.00000	0.312110
$658$	−26.0000	−1.01359
$659$	−15.0000	−0.584317	−0.292159	−	0.956370i	$-0.594373\pi$
$659$	−15.0000	−0.584317	−0.292159	+	0.956370i	$0.594373\pi$
$660$	3.00000	0.116775
$661$	−18.0000	−0.700119	−0.350059	−	0.936727i	$-0.613839\pi$
$661$	−18.0000	−0.700119	−0.350059	+	0.936727i	$0.613839\pi$
$662$	−23.0000	−0.893920
$663$	8.00000	0.310694
$664$	−4.00000	−0.155230
$665$	0	0
$666$	−16.0000	−0.619987
$667$	0	0
$668$	−2.00000	−0.0773823
$669$	−26.0000	−1.00522
$670$	12.0000	0.463600
$671$	24.0000	0.926510
$672$	2.00000	0.0771517
$673$	9.00000	0.346925	0.173462	−	0.984841i	$-0.444505\pi$
$673$	9.00000	0.346925	0.173462	+	0.984841i	$0.444505\pi$
$674$	−32.0000	−1.23259
$675$	20.0000	0.769800
$676$	−12.0000	−0.461538
$677$	−38.0000	−1.46046	−0.730229	−	0.683202i	$-0.760587\pi$
$677$	−38.0000	−1.46046	−0.730229	+	0.683202i	$0.760587\pi$
$678$	−6.00000	−0.230429
$679$	−4.00000	−0.153506
$680$	8.00000	0.306786
$681$	18.0000	0.689761
$682$	−9.00000	−0.344628
$683$	24.0000	0.918334	0.459167	−	0.888350i	$-0.348148\pi$
$683$	24.0000	0.918334	0.459167	+	0.888350i	$0.348148\pi$
$684$	0	0
$685$	12.0000	0.458496
$686$	−20.0000	−0.763604
$687$	−10.0000	−0.381524
$688$	11.0000	0.419371
$689$	11.0000	0.419067
$690$	−4.00000	−0.152277
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	−6.00000	−0.228086
$693$	12.0000	0.455842
$694$	2.00000	0.0759190
$695$	−20.0000	−0.758643
$696$	0	0
$697$	16.0000	0.606043
$698$	15.0000	0.567758
$699$	−1.00000	−0.0378235
$700$	8.00000	0.302372
$701$	27.0000	1.01978	0.509888	−	0.860241i	$-0.329687\pi$
$701$	27.0000	1.01978	0.509888	+	0.860241i	$0.329687\pi$
$702$	−5.00000	−0.188713
$703$	0	0
$704$	3.00000	0.113067
$705$	−13.0000	−0.489608
$706$	26.0000	0.978523
$707$	−16.0000	−0.601742
$708$	0	0
$709$	15.0000	0.563337	0.281668	−	0.959512i	$-0.409112\pi$
$709$	15.0000	0.563337	0.281668	+	0.959512i	$0.409112\pi$
$710$	−2.00000	−0.0750587
$711$	30.0000	1.12509
$712$	−10.0000	−0.374766
$713$	12.0000	0.449404
$714$	−16.0000	−0.598785
$715$	−3.00000	−0.112194
$716$	−10.0000	−0.373718
$717$	0	0
$718$	−25.0000	−0.932992
$719$	−50.0000	−1.86469	−0.932343	−	0.361576i	$-0.882239\pi$
$719$	−50.0000	−1.86469	−0.932343	+	0.361576i	$0.882239\pi$
$720$	−2.00000	−0.0745356
$721$	−28.0000	−1.04277
$722$	19.0000	0.707107
$723$	17.0000	0.632237
$724$	7.00000	0.260153
$725$	0	0
$726$	2.00000	0.0742270
$727$	−8.00000	−0.296704	−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000	−0.296704	−0.148352	+	0.988935i	$0.547397\pi$
$728$	−2.00000	−0.0741249
$729$	13.0000	0.481481
$730$	4.00000	0.148047
$731$	−88.0000	−3.25480
$732$	8.00000	0.295689
$733$	−24.0000	−0.886460	−0.443230	−	0.896408i	$-0.646168\pi$
$733$	−24.0000	−0.886460	−0.443230	+	0.896408i	$0.646168\pi$
$734$	−32.0000	−1.18114
$735$	−3.00000	−0.110657
$736$	−4.00000	−0.147442
$737$	−36.0000	−1.32608
$738$	−4.00000	−0.147242
$739$	5.00000	0.183928	0.0919640	−	0.995762i	$-0.470686\pi$
$739$	5.00000	0.183928	0.0919640	+	0.995762i	$0.470686\pi$
$740$	−8.00000	−0.294086
$741$	0	0
$742$	−22.0000	−0.807645
$743$	−44.0000	−1.61420	−0.807102	−	0.590412i	$-0.798965\pi$
$743$	−44.0000	−1.61420	−0.807102	+	0.590412i	$0.798965\pi$
$744$	−3.00000	−0.109985
$745$	15.0000	0.549557
$746$	21.0000	0.768865
$747$	−8.00000	−0.292705
$748$	−24.0000	−0.877527
$749$	4.00000	0.146157
$750$	9.00000	0.328634
$751$	−32.0000	−1.16770	−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000	−1.16770	−0.583848	+	0.811863i	$0.698454\pi$
$752$	−13.0000	−0.474061
$753$	−27.0000	−0.983935
$754$	0	0
$755$	2.00000	0.0727875
$756$	10.0000	0.363696
$757$	−8.00000	−0.290765	−0.145382	−	0.989376i	$-0.546441\pi$
$757$	−8.00000	−0.290765	−0.145382	+	0.989376i	$0.546441\pi$
$758$	20.0000	0.726433
$759$	12.0000	0.435572
$760$	0	0
$761$	42.0000	1.52250	0.761249	−	0.648459i	$-0.224586\pi$
$761$	42.0000	1.52250	0.761249	+	0.648459i	$0.224586\pi$
$762$	8.00000	0.289809
$763$	−10.0000	−0.362024
$764$	8.00000	0.289430
$765$	16.0000	0.578481
$766$	−14.0000	−0.505841
$767$	0	0
$768$	1.00000	0.0360844
$769$	20.0000	0.721218	0.360609	−	0.932717i	$-0.382569\pi$
$769$	20.0000	0.721218	0.360609	+	0.932717i	$0.382569\pi$
$770$	6.00000	0.216225
$771$	13.0000	0.468184
$772$	−14.0000	−0.503871
$773$	−14.0000	−0.503545	−0.251773	−	0.967786i	$-0.581013\pi$
$773$	−14.0000	−0.503545	−0.251773	+	0.967786i	$0.581013\pi$
$774$	22.0000	0.790774
$775$	−12.0000	−0.431053
$776$	−2.00000	−0.0717958
$777$	16.0000	0.573997
$778$	0	0
$779$	0	0
$780$	−1.00000	−0.0358057
$781$	6.00000	0.214697
$782$	32.0000	1.14432
$783$	0	0
$784$	−3.00000	−0.107143
$785$	−18.0000	−0.642448
$786$	12.0000	0.428026
$787$	−22.0000	−0.784215	−0.392108	−	0.919919i	$-0.628254\pi$
$787$	−22.0000	−0.784215	−0.392108	+	0.919919i	$0.628254\pi$
$788$	18.0000	0.641223
$789$	−9.00000	−0.320408
$790$	15.0000	0.533676
$791$	−12.0000	−0.426671
$792$	6.00000	0.213201
$793$	−8.00000	−0.284088
$794$	17.0000	0.603307
$795$	−11.0000	−0.390130
$796$	−10.0000	−0.354441
$797$	32.0000	1.13350	0.566749	−	0.823890i	$-0.308201\pi$
$797$	32.0000	1.13350	0.566749	+	0.823890i	$0.308201\pi$
$798$	0	0
$799$	104.000	3.67926
$800$	4.00000	0.141421
$801$	−20.0000	−0.706665
$802$	−27.0000	−0.953403
$803$	−12.0000	−0.423471
$804$	−12.0000	−0.423207
$805$	−8.00000	−0.281963
$806$	3.00000	0.105670
$807$	0	0
$808$	−8.00000	−0.281439
$809$	0	0		−	1.00000i	$-0.5\pi$
$809$	0	0			1.00000i	$0.5\pi$
$810$	−1.00000	−0.0351364
$811$	−18.0000	−0.632065	−0.316033	−	0.948748i	$-0.602351\pi$
$811$	−18.0000	−0.632065	−0.316033	+	0.948748i	$0.602351\pi$
$812$	0	0
$813$	13.0000	0.455930
$814$	24.0000	0.841200
$815$	−9.00000	−0.315256
$816$	−8.00000	−0.280056
$817$	0	0
$818$	30.0000	1.04893
$819$	−4.00000	−0.139771
$820$	−2.00000	−0.0698430
$821$	−33.0000	−1.15171	−0.575854	−	0.817553i	$-0.695330\pi$
$821$	−33.0000	−1.15171	−0.575854	+	0.817553i	$0.695330\pi$
$822$	−12.0000	−0.418548
$823$	16.0000	0.557725	0.278862	−	0.960331i	$-0.410043\pi$
$823$	16.0000	0.557725	0.278862	+	0.960331i	$0.410043\pi$
$824$	−14.0000	−0.487713
$825$	−12.0000	−0.417786
$826$	0	0
$827$	−13.0000	−0.452054	−0.226027	−	0.974121i	$-0.572574\pi$
$827$	−13.0000	−0.452054	−0.226027	+	0.974121i	$0.572574\pi$
$828$	−8.00000	−0.278019
$829$	−40.0000	−1.38926	−0.694629	−	0.719368i	$-0.744431\pi$
$829$	−40.0000	−1.38926	−0.694629	+	0.719368i	$0.744431\pi$
$830$	−4.00000	−0.138842
$831$	−2.00000	−0.0693792
$832$	−1.00000	−0.0346688
$833$	24.0000	0.831551
$834$	20.0000	0.692543
$835$	−2.00000	−0.0692129
$836$	0	0
$837$	−15.0000	−0.518476
$838$	10.0000	0.345444
$839$	−45.0000	−1.55357	−0.776786	−	0.629764i	$-0.783151\pi$
$839$	−45.0000	−1.55357	−0.776786	+	0.629764i	$0.783151\pi$
$840$	2.00000	0.0690066
$841$	0	0
$842$	32.0000	1.10279
$843$	27.0000	0.929929
$844$	3.00000	0.103264
$845$	−12.0000	−0.412813
$846$	−26.0000	−0.893898
$847$	4.00000	0.137442
$848$	−11.0000	−0.377742
$849$	4.00000	0.137280
$850$	−32.0000	−1.09759
$851$	−32.0000	−1.09695
$852$	2.00000	0.0685189
$853$	−14.0000	−0.479351	−0.239675	−	0.970853i	$-0.577041\pi$
$853$	−14.0000	−0.479351	−0.239675	+	0.970853i	$0.577041\pi$
$854$	16.0000	0.547509
$855$	0	0
$856$	2.00000	0.0683586
$857$	−27.0000	−0.922302	−0.461151	−	0.887322i	$-0.652563\pi$
$857$	−27.0000	−0.922302	−0.461151	+	0.887322i	$0.652563\pi$
$858$	3.00000	0.102418
$859$	25.0000	0.852989	0.426494	−	0.904490i	$-0.359748\pi$
$859$	25.0000	0.852989	0.426494	+	0.904490i	$0.359748\pi$
$860$	11.0000	0.375097
$861$	4.00000	0.136320
$862$	−32.0000	−1.08992
$863$	−46.0000	−1.56586	−0.782929	−	0.622111i	$-0.786275\pi$
$863$	−46.0000	−1.56586	−0.782929	+	0.622111i	$0.786275\pi$
$864$	5.00000	0.170103
$865$	−6.00000	−0.204006
$866$	−16.0000	−0.543702
$867$	47.0000	1.59620
$868$	−6.00000	−0.203653
$869$	−45.0000	−1.52652
$870$	0	0
$871$	12.0000	0.406604
$872$	−5.00000	−0.169321
$873$	−4.00000	−0.135379
$874$	0	0
$875$	18.0000	0.608511
$876$	−4.00000	−0.135147
$877$	13.0000	0.438979	0.219489	−	0.975615i	$-0.429561\pi$
$877$	13.0000	0.438979	0.219489	+	0.975615i	$0.429561\pi$
$878$	−20.0000	−0.674967
$879$	−14.0000	−0.472208
$880$	3.00000	0.101130
$881$	−42.0000	−1.41502	−0.707508	−	0.706705i	$-0.750181\pi$
$881$	−42.0000	−1.41502	−0.707508	+	0.706705i	$0.750181\pi$
$882$	−6.00000	−0.202031
$883$	−26.0000	−0.874970	−0.437485	−	0.899226i	$-0.644131\pi$
$883$	−26.0000	−0.874970	−0.437485	+	0.899226i	$0.644131\pi$
$884$	8.00000	0.269069
$885$	0	0
$886$	4.00000	0.134383
$887$	−33.0000	−1.10803	−0.554016	−	0.832506i	$-0.686905\pi$
$887$	−33.0000	−1.10803	−0.554016	+	0.832506i	$0.686905\pi$
$888$	8.00000	0.268462
$889$	16.0000	0.536623
$890$	−10.0000	−0.335201
$891$	3.00000	0.100504
$892$	−26.0000	−0.870544
$893$	0	0
$894$	−15.0000	−0.501675
$895$	−10.0000	−0.334263
$896$	2.00000	0.0668153
$897$	−4.00000	−0.133556
$898$	−10.0000	−0.333704
$899$	0	0
$900$	8.00000	0.266667
$901$	88.0000	2.93171
$902$	6.00000	0.199778
$903$	−22.0000	−0.732114
$904$	−6.00000	−0.199557
$905$	7.00000	0.232688
$906$	−2.00000	−0.0664455
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	18.0000	0.597351
$909$	−16.0000	−0.530687
$910$	−2.00000	−0.0662994
$911$	13.0000	0.430709	0.215355	−	0.976536i	$-0.430909\pi$
$911$	13.0000	0.430709	0.215355	+	0.976536i	$0.430909\pi$
$912$	0	0
$913$	12.0000	0.397142
$914$	2.00000	0.0661541
$915$	8.00000	0.264472
$916$	−10.0000	−0.330409
$917$	24.0000	0.792550
$918$	−40.0000	−1.32020
$919$	30.0000	0.989609	0.494804	−	0.869004i	$-0.335240\pi$
$919$	30.0000	0.989609	0.494804	+	0.869004i	$0.335240\pi$
$920$	−4.00000	−0.131876
$921$	7.00000	0.230658
$922$	2.00000	0.0658665
$923$	−2.00000	−0.0658308
$924$	−6.00000	−0.197386
$925$	32.0000	1.05215
$926$	−4.00000	−0.131448
$927$	−28.0000	−0.919641
$928$	0	0
$929$	−30.0000	−0.984268	−0.492134	−	0.870519i	$-0.663783\pi$
$929$	−30.0000	−0.984268	−0.492134	+	0.870519i	$0.663783\pi$
$930$	−3.00000	−0.0983739
$931$	0	0
$932$	−1.00000	−0.0327561
$933$	8.00000	0.261908
$934$	−27.0000	−0.883467
$935$	−24.0000	−0.784884
$936$	−2.00000	−0.0653720
$937$	−2.00000	−0.0653372	−0.0326686	−	0.999466i	$-0.510401\pi$
$937$	−2.00000	−0.0653372	−0.0326686	+	0.999466i	$0.510401\pi$
$938$	−24.0000	−0.783628
$939$	9.00000	0.293704
$940$	−13.0000	−0.424013
$941$	37.0000	1.20617	0.603083	−	0.797679i	$-0.293939\pi$
$941$	37.0000	1.20617	0.603083	+	0.797679i	$0.293939\pi$
$942$	18.0000	0.586472
$943$	−8.00000	−0.260516
$944$	0	0
$945$	10.0000	0.325300
$946$	−33.0000	−1.07292
$947$	−33.0000	−1.07236	−0.536178	−	0.844105i	$-0.680132\pi$
$947$	−33.0000	−1.07236	−0.536178	+	0.844105i	$0.680132\pi$
$948$	−15.0000	−0.487177
$949$	4.00000	0.129845
$950$	0	0
$951$	12.0000	0.389127
$952$	−16.0000	−0.518563
$953$	−1.00000	−0.0323932	−0.0161966	−	0.999869i	$-0.505156\pi$
$953$	−1.00000	−0.0323932	−0.0161966	+	0.999869i	$0.505156\pi$
$954$	−22.0000	−0.712276
$955$	8.00000	0.258874
$956$	0	0
$957$	0	0
$958$	−5.00000	−0.161543
$959$	−24.0000	−0.775000
$960$	1.00000	0.0322749
$961$	−22.0000	−0.709677
$962$	−8.00000	−0.257930
$963$	4.00000	0.128898
$964$	17.0000	0.547533
$965$	−14.0000	−0.450676
$966$	8.00000	0.257396
$967$	−13.0000	−0.418052	−0.209026	−	0.977910i	$-0.567029\pi$
$967$	−13.0000	−0.418052	−0.209026	+	0.977910i	$0.567029\pi$
$968$	2.00000	0.0642824
$969$	0	0
$970$	−2.00000	−0.0642161
$971$	28.0000	0.898563	0.449281	−	0.893390i	$-0.351680\pi$
$971$	28.0000	0.898563	0.449281	+	0.893390i	$0.351680\pi$
$972$	16.0000	0.513200
$973$	40.0000	1.28234
$974$	22.0000	0.704925
$975$	4.00000	0.128103
$976$	8.00000	0.256074
$977$	13.0000	0.415907	0.207953	−	0.978139i	$-0.433320\pi$
$977$	13.0000	0.415907	0.207953	+	0.978139i	$0.433320\pi$
$978$	9.00000	0.287788
$979$	30.0000	0.958804
$980$	−3.00000	−0.0958315
$981$	−10.0000	−0.319275
$982$	−33.0000	−1.05307
$983$	−49.0000	−1.56286	−0.781429	−	0.623995i	$-0.785509\pi$
$983$	−49.0000	−1.56286	−0.781429	+	0.623995i	$0.785509\pi$
$984$	2.00000	0.0637577
$985$	18.0000	0.573528
$986$	0	0
$987$	26.0000	0.827589
$988$	0	0
$989$	44.0000	1.39912
$990$	6.00000	0.190693
$991$	22.0000	0.698853	0.349427	−	0.936964i	$-0.386376\pi$
$991$	22.0000	0.698853	0.349427	+	0.936964i	$0.386376\pi$
$992$	−3.00000	−0.0952501
$993$	23.0000	0.729883
$994$	4.00000	0.126872
$995$	−10.0000	−0.317021
$996$	4.00000	0.126745
$997$	−8.00000	−0.253363	−0.126681	−	0.991943i	$-0.540433\pi$
$997$	−8.00000	−0.253363	−0.126681	+	0.991943i	$0.540433\pi$
$998$	20.0000	0.633089
$999$	40.0000	1.26554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1682.2.a.d.1.1		1
29.12	odd	4		1682.2.b.a.1681.1		2
29.17	odd	4		1682.2.b.a.1681.2		2
29.28	even	2		58.2.a.b.1.1	✓	1
87.86	odd	2		522.2.a.b.1.1		1
116.115	odd	2		464.2.a.e.1.1		1
145.28	odd	4		1450.2.b.b.349.1		2
145.57	odd	4		1450.2.b.b.349.2		2
145.144	even	2		1450.2.a.c.1.1		1
203.202	odd	2		2842.2.a.e.1.1		1
232.115	odd	2		1856.2.a.f.1.1		1
232.173	even	2		1856.2.a.k.1.1		1
319.318	odd	2		7018.2.a.a.1.1		1
348.347	even	2		4176.2.a.n.1.1		1
377.376	even	2		9802.2.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
58.2.a.b.1.1	✓	1	29.28	even	2
464.2.a.e.1.1		1	116.115	odd	2
522.2.a.b.1.1		1	87.86	odd	2
1450.2.a.c.1.1		1	145.144	even	2
1450.2.b.b.349.1		2	145.28	odd	4
1450.2.b.b.349.2		2	145.57	odd	4
1682.2.a.d.1.1		1	1.1	even	1	trivial
1682.2.b.a.1681.1		2	29.12	odd	4
1682.2.b.a.1681.2		2	29.17	odd	4
1856.2.a.f.1.1		1	232.115	odd	2
1856.2.a.k.1.1		1	232.173	even	2
2842.2.a.e.1.1		1	203.202	odd	2
4176.2.a.n.1.1		1	348.347	even	2
7018.2.a.a.1.1		1	319.318	odd	2
9802.2.a.a.1.1		1	377.376	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:	\( N \)	\(=\)	\( 1682 = 2 \cdot 29^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1682.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more