LMFDB - Embedded newform 1682.2.a.a.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:	yes
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)

$f(q)$

$=$

$q-1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} +2.00000 q^{6} -5.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} +2.00000 q^{6} -5.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} -2.00000 q^{12} +2.00000 q^{13} +5.00000 q^{14} +4.00000 q^{15} +1.00000 q^{16} +7.00000 q^{17} -1.00000 q^{18} +6.00000 q^{19} -2.00000 q^{20} +10.0000 q^{21} +1.00000 q^{23} +2.00000 q^{24} -1.00000 q^{25} -2.00000 q^{26} +4.00000 q^{27} -5.00000 q^{28} -4.00000 q^{30} +3.00000 q^{31} -1.00000 q^{32} -7.00000 q^{34} +10.0000 q^{35} +1.00000 q^{36} -2.00000 q^{37} -6.00000 q^{38} -4.00000 q^{39} +2.00000 q^{40} -2.00000 q^{41} -10.0000 q^{42} -10.0000 q^{43} -2.00000 q^{45} -1.00000 q^{46} -7.00000 q^{47} -2.00000 q^{48} +18.0000 q^{49} +1.00000 q^{50} -14.0000 q^{51} +2.00000 q^{52} -2.00000 q^{53} -4.00000 q^{54} +5.00000 q^{56} -12.0000 q^{57} -12.0000 q^{59} +4.00000 q^{60} +8.00000 q^{61} -3.00000 q^{62} -5.00000 q^{63} +1.00000 q^{64} -4.00000 q^{65} -6.00000 q^{67} +7.00000 q^{68} -2.00000 q^{69} -10.0000 q^{70} +11.0000 q^{71} -1.00000 q^{72} -7.00000 q^{73} +2.00000 q^{74} +2.00000 q^{75} +6.00000 q^{76} +4.00000 q^{78} -3.00000 q^{79} -2.00000 q^{80} -11.0000 q^{81} +2.00000 q^{82} +4.00000 q^{83} +10.0000 q^{84} -14.0000 q^{85} +10.0000 q^{86} +13.0000 q^{89} +2.00000 q^{90} -10.0000 q^{91} +1.00000 q^{92} -6.00000 q^{93} +7.00000 q^{94} -12.0000 q^{95} +2.00000 q^{96} +11.0000 q^{97} -18.0000 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−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
$7$	−5.00000	−1.88982	−0.944911	−	0.327327i	$-0.893852\pi$
$7$	−5.00000	−1.88982	−0.944911	+	0.327327i	$0.893852\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	2.00000	0.632456
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	−2.00000	−0.577350
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	5.00000	1.33631
$15$	4.00000	1.03280
$16$	1.00000	0.250000
$17$	7.00000	1.69775	0.848875	−	0.528594i	$-0.177281\pi$
$17$	7.00000	1.69775	0.848875	+	0.528594i	$0.177281\pi$
$18$	−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
$21$	10.0000	2.18218
$22$	0	0
$23$	1.00000	0.208514	0.104257	−	0.994550i	$-0.466753\pi$
$23$	1.00000	0.208514	0.104257	+	0.994550i	$0.466753\pi$
$24$	2.00000	0.408248
$25$	−1.00000	−0.200000
$26$	−2.00000	−0.392232
$27$	4.00000	0.769800
$28$	−5.00000	−0.944911
$29$	0	0
$29$	0	0
$30$	−4.00000	−0.730297
$31$	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
$33$	0	0
$34$	−7.00000	−1.20049
$35$	10.0000	1.69031
$36$	1.00000	0.166667
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	−6.00000	−0.973329
$39$	−4.00000	−0.640513
$40$	2.00000	0.316228
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	−10.0000	−1.54303
$43$	−10.0000	−1.52499	−0.762493	−	0.646997i	$-0.776025\pi$
$43$	−10.0000	−1.52499	−0.762493	+	0.646997i	$0.776025\pi$
$44$	0	0
$45$	−2.00000	−0.298142
$46$	−1.00000	−0.147442
$47$	−7.00000	−1.02105	−0.510527	−	0.859861i	$-0.670550\pi$
$47$	−7.00000	−1.02105	−0.510527	+	0.859861i	$0.670550\pi$
$48$	−2.00000	−0.288675
$49$	18.0000	2.57143
$50$	1.00000	0.141421
$51$	−14.0000	−1.96039
$52$	2.00000	0.277350
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	−4.00000	−0.544331
$55$	0	0
$56$	5.00000	0.668153
$57$	−12.0000	−1.58944
$58$	0	0
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	4.00000	0.516398
$61$	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
$63$	−5.00000	−0.629941
$64$	1.00000	0.125000
$65$	−4.00000	−0.496139
$66$	0	0
$67$	−6.00000	−0.733017	−0.366508	−	0.930415i	$-0.619447\pi$
$67$	−6.00000	−0.733017	−0.366508	+	0.930415i	$0.619447\pi$
$68$	7.00000	0.848875
$69$	−2.00000	−0.240772
$70$	−10.0000	−1.19523
$71$	11.0000	1.30546	0.652730	−	0.757591i	$-0.273624\pi$
$71$	11.0000	1.30546	0.652730	+	0.757591i	$0.273624\pi$
$72$	−1.00000	−0.117851
$73$	−7.00000	−0.819288	−0.409644	−	0.912245i	$-0.634347\pi$
$73$	−7.00000	−0.819288	−0.409644	+	0.912245i	$0.634347\pi$
$74$	2.00000	0.232495
$75$	2.00000	0.230940
$76$	6.00000	0.688247
$77$	0	0
$78$	4.00000	0.452911
$79$	−3.00000	−0.337526	−0.168763	−	0.985657i	$-0.553977\pi$
$79$	−3.00000	−0.337526	−0.168763	+	0.985657i	$0.553977\pi$
$80$	−2.00000	−0.223607
$81$	−11.0000	−1.22222
$82$	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	$0.429548\pi$
$84$	10.0000	1.09109
$85$	−14.0000	−1.51851
$86$	10.0000	1.07833
$87$	0	0
$88$	0	0
$89$	13.0000	1.37800	0.688999	−	0.724763i	$-0.258051\pi$
$89$	13.0000	1.37800	0.688999	+	0.724763i	$0.258051\pi$
$90$	2.00000	0.210819
$91$	−10.0000	−1.04828
$92$	1.00000	0.104257
$93$	−6.00000	−0.622171
$94$	7.00000	0.721995
$95$	−12.0000	−1.23117
$96$	2.00000	0.204124
$97$	11.0000	1.11688	0.558440	−	0.829545i	$-0.311400\pi$
$97$	11.0000	1.11688	0.558440	+	0.829545i	$0.311400\pi$
$98$	−18.0000	−1.81827
$99$	0	0
$100$	−1.00000	−0.100000
$101$	8.00000	0.796030	0.398015	−	0.917379i	$-0.369699\pi$
$101$	8.00000	0.796030	0.398015	+	0.917379i	$0.369699\pi$
$102$	14.0000	1.38621
$103$	−7.00000	−0.689730	−0.344865	−	0.938652i	$-0.612075\pi$
$103$	−7.00000	−0.689730	−0.344865	+	0.938652i	$0.612075\pi$
$104$	−2.00000	−0.196116
$105$	−20.0000	−1.95180
$106$	2.00000	0.194257
$107$	4.00000	0.386695	0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000	0.386695	0.193347	+	0.981130i	$0.438066\pi$
$108$	4.00000	0.384900
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	4.00000	0.379663
$112$	−5.00000	−0.472456
$113$	9.00000	0.846649	0.423324	−	0.905978i	$-0.360863\pi$
$113$	9.00000	0.846649	0.423324	+	0.905978i	$0.360863\pi$
$114$	12.0000	1.12390
$115$	−2.00000	−0.186501
$116$	0	0
$117$	2.00000	0.184900
$118$	12.0000	1.10469
$119$	−35.0000	−3.20844
$120$	−4.00000	−0.365148
$121$	−11.0000	−1.00000
$122$	−8.00000	−0.724286
$123$	4.00000	0.360668
$124$	3.00000	0.269408
$125$	12.0000	1.07331
$126$	5.00000	0.445435
$127$	−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
$129$	20.0000	1.76090
$130$	4.00000	0.350823
$131$	−6.00000	−0.524222	−0.262111	−	0.965038i	$-0.584419\pi$
$131$	−6.00000	−0.524222	−0.262111	+	0.965038i	$0.584419\pi$
$132$	0	0
$133$	−30.0000	−2.60133
$134$	6.00000	0.518321
$135$	−8.00000	−0.688530
$136$	−7.00000	−0.600245
$137$	3.00000	0.256307	0.128154	−	0.991754i	$-0.459095\pi$
$137$	3.00000	0.256307	0.128154	+	0.991754i	$0.459095\pi$
$138$	2.00000	0.170251
$139$	−2.00000	−0.169638	−0.0848189	−	0.996396i	$-0.527031\pi$
$139$	−2.00000	−0.169638	−0.0848189	+	0.996396i	$0.527031\pi$
$140$	10.0000	0.845154
$141$	14.0000	1.17901
$142$	−11.0000	−0.923099
$143$	0	0
$144$	1.00000	0.0833333
$145$	0	0
$146$	7.00000	0.579324
$147$	−36.0000	−2.96923
$148$	−2.00000	−0.164399
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	−2.00000	−0.163299
$151$	−7.00000	−0.569652	−0.284826	−	0.958579i	$-0.591936\pi$
$151$	−7.00000	−0.569652	−0.284826	+	0.958579i	$0.591936\pi$
$152$	−6.00000	−0.486664
$153$	7.00000	0.565916
$154$	0	0
$155$	−6.00000	−0.481932
$156$	−4.00000	−0.320256
$157$	−18.0000	−1.43656	−0.718278	−	0.695756i	$-0.755069\pi$
$157$	−18.0000	−1.43656	−0.718278	+	0.695756i	$0.755069\pi$
$158$	3.00000	0.238667
$159$	4.00000	0.317221
$160$	2.00000	0.158114
$161$	−5.00000	−0.394055
$162$	11.0000	0.864242
$163$	6.00000	0.469956	0.234978	−	0.972001i	$-0.424498\pi$
$163$	6.00000	0.469956	0.234978	+	0.972001i	$0.424498\pi$
$164$	−2.00000	−0.156174
$165$	0	0
$166$	−4.00000	−0.310460
$167$	−20.0000	−1.54765	−0.773823	−	0.633402i	$-0.781658\pi$
$167$	−20.0000	−1.54765	−0.773823	+	0.633402i	$0.781658\pi$
$168$	−10.0000	−0.771517
$169$	−9.00000	−0.692308
$170$	14.0000	1.07375
$171$	6.00000	0.458831
$172$	−10.0000	−0.762493
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	0	0
$175$	5.00000	0.377964
$176$	0	0
$177$	24.0000	1.80395
$178$	−13.0000	−0.974391
$179$	14.0000	1.04641	0.523205	−	0.852207i	$-0.324736\pi$
$179$	14.0000	1.04641	0.523205	+	0.852207i	$0.324736\pi$
$180$	−2.00000	−0.149071
$181$	−8.00000	−0.594635	−0.297318	−	0.954779i	$-0.596092\pi$
$181$	−8.00000	−0.594635	−0.297318	+	0.954779i	$0.596092\pi$
$182$	10.0000	0.741249
$183$	−16.0000	−1.18275
$184$	−1.00000	−0.0737210
$185$	4.00000	0.294086
$186$	6.00000	0.439941
$187$	0	0
$188$	−7.00000	−0.510527
$189$	−20.0000	−1.45479
$190$	12.0000	0.870572
$191$	5.00000	0.361787	0.180894	−	0.983503i	$-0.442101\pi$
$191$	5.00000	0.361787	0.180894	+	0.983503i	$0.442101\pi$
$192$	−2.00000	−0.144338
$193$	−2.00000	−0.143963	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000	−0.143963	−0.0719816	+	0.997406i	$0.522932\pi$
$194$	−11.0000	−0.789754
$195$	8.00000	0.572892
$196$	18.0000	1.28571
$197$	12.0000	0.854965	0.427482	−	0.904024i	$-0.359401\pi$
$197$	12.0000	0.854965	0.427482	+	0.904024i	$0.359401\pi$
$198$	0	0
$199$	5.00000	0.354441	0.177220	−	0.984171i	$-0.443289\pi$
$199$	5.00000	0.354441	0.177220	+	0.984171i	$0.443289\pi$
$200$	1.00000	0.0707107
$201$	12.0000	0.846415
$202$	−8.00000	−0.562878
$203$	0	0
$204$	−14.0000	−0.980196
$205$	4.00000	0.279372
$206$	7.00000	0.487713
$207$	1.00000	0.0695048
$208$	2.00000	0.138675
$209$	0	0
$210$	20.0000	1.38013
$211$	−6.00000	−0.413057	−0.206529	−	0.978441i	$-0.566217\pi$
$211$	−6.00000	−0.413057	−0.206529	+	0.978441i	$0.566217\pi$
$212$	−2.00000	−0.137361
$213$	−22.0000	−1.50742
$214$	−4.00000	−0.273434
$215$	20.0000	1.36399
$216$	−4.00000	−0.272166
$217$	−15.0000	−1.01827
$218$	−2.00000	−0.135457
$219$	14.0000	0.946032
$220$	0	0
$221$	14.0000	0.941742
$222$	−4.00000	−0.268462
$223$	−17.0000	−1.13840	−0.569202	−	0.822198i	$-0.692748\pi$
$223$	−17.0000	−1.13840	−0.569202	+	0.822198i	$0.692748\pi$
$224$	5.00000	0.334077
$225$	−1.00000	−0.0666667
$226$	−9.00000	−0.598671
$227$	−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
$229$	−16.0000	−1.05731	−0.528655	−	0.848837i	$-0.677303\pi$
$229$	−16.0000	−1.05731	−0.528655	+	0.848837i	$0.677303\pi$
$230$	2.00000	0.131876
$231$	0	0
$232$	0	0
$233$	23.0000	1.50678	0.753390	−	0.657574i	$-0.228417\pi$
$233$	23.0000	1.50678	0.753390	+	0.657574i	$0.228417\pi$
$234$	−2.00000	−0.130744
$235$	14.0000	0.913259
$236$	−12.0000	−0.781133
$237$	6.00000	0.389742
$238$	35.0000	2.26871
$239$	−9.00000	−0.582162	−0.291081	−	0.956698i	$-0.594015\pi$
$239$	−9.00000	−0.582162	−0.291081	+	0.956698i	$0.594015\pi$
$240$	4.00000	0.258199
$241$	−19.0000	−1.22390	−0.611949	−	0.790897i	$-0.709614\pi$
$241$	−19.0000	−1.22390	−0.611949	+	0.790897i	$0.709614\pi$
$242$	11.0000	0.707107
$243$	10.0000	0.641500
$244$	8.00000	0.512148
$245$	−36.0000	−2.29996
$246$	−4.00000	−0.255031
$247$	12.0000	0.763542
$248$	−3.00000	−0.190500
$249$	−8.00000	−0.506979
$250$	−12.0000	−0.758947
$251$	6.00000	0.378717	0.189358	−	0.981908i	$-0.439359\pi$
$251$	6.00000	0.378717	0.189358	+	0.981908i	$0.439359\pi$
$252$	−5.00000	−0.314970
$253$	0	0
$254$	8.00000	0.501965
$255$	28.0000	1.75343
$256$	1.00000	0.0625000
$257$	−17.0000	−1.06043	−0.530215	−	0.847863i	$-0.677889\pi$
$257$	−17.0000	−1.06043	−0.530215	+	0.847863i	$0.677889\pi$
$258$	−20.0000	−1.24515
$259$	10.0000	0.621370
$260$	−4.00000	−0.248069
$261$	0	0
$262$	6.00000	0.370681
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	4.00000	0.245718
$266$	30.0000	1.83942
$267$	−26.0000	−1.59117
$268$	−6.00000	−0.366508
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	8.00000	0.486864
$271$	−11.0000	−0.668202	−0.334101	−	0.942537i	$-0.608433\pi$
$271$	−11.0000	−0.668202	−0.334101	+	0.942537i	$0.608433\pi$
$272$	7.00000	0.424437
$273$	20.0000	1.21046
$274$	−3.00000	−0.181237
$275$	0	0
$276$	−2.00000	−0.120386
$277$	−20.0000	−1.20168	−0.600842	−	0.799368i	$-0.705168\pi$
$277$	−20.0000	−1.20168	−0.600842	+	0.799368i	$0.705168\pi$
$278$	2.00000	0.119952
$279$	3.00000	0.179605
$280$	−10.0000	−0.597614
$281$	3.00000	0.178965	0.0894825	−	0.995988i	$-0.471479\pi$
$281$	3.00000	0.178965	0.0894825	+	0.995988i	$0.471479\pi$
$282$	−14.0000	−0.833688
$283$	22.0000	1.30776	0.653882	−	0.756596i	$-0.273139\pi$
$283$	22.0000	1.30776	0.653882	+	0.756596i	$0.273139\pi$
$284$	11.0000	0.652730
$285$	24.0000	1.42164
$286$	0	0
$287$	10.0000	0.590281
$288$	−1.00000	−0.0589256
$289$	32.0000	1.88235
$290$	0	0
$291$	−22.0000	−1.28966
$292$	−7.00000	−0.409644
$293$	−2.00000	−0.116841	−0.0584206	−	0.998292i	$-0.518606\pi$
$293$	−2.00000	−0.116841	−0.0584206	+	0.998292i	$0.518606\pi$
$294$	36.0000	2.09956
$295$	24.0000	1.39733
$296$	2.00000	0.116248
$297$	0	0
$298$	−6.00000	−0.347571
$299$	2.00000	0.115663
$300$	2.00000	0.115470
$301$	50.0000	2.88195
$302$	7.00000	0.402805
$303$	−16.0000	−0.919176
$304$	6.00000	0.344124
$305$	−16.0000	−0.916157
$306$	−7.00000	−0.400163
$307$	22.0000	1.25561	0.627803	−	0.778372i	$-0.283954\pi$
$307$	22.0000	1.25561	0.627803	+	0.778372i	$0.283954\pi$
$308$	0	0
$309$	14.0000	0.796432
$310$	6.00000	0.340777
$311$	−16.0000	−0.907277	−0.453638	−	0.891186i	$-0.649874\pi$
$311$	−16.0000	−0.907277	−0.453638	+	0.891186i	$0.649874\pi$
$312$	4.00000	0.226455
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	18.0000	1.01580
$315$	10.0000	0.563436
$316$	−3.00000	−0.168763
$317$	−24.0000	−1.34797	−0.673987	−	0.738743i	$-0.735420\pi$
$317$	−24.0000	−1.34797	−0.673987	+	0.738743i	$0.735420\pi$
$318$	−4.00000	−0.224309
$319$	0	0
$320$	−2.00000	−0.111803
$321$	−8.00000	−0.446516
$322$	5.00000	0.278639
$323$	42.0000	2.33694
$324$	−11.0000	−0.611111
$325$	−2.00000	−0.110940
$326$	−6.00000	−0.332309
$327$	−4.00000	−0.221201
$328$	2.00000	0.110432
$329$	35.0000	1.92961
$330$	0	0
$331$	−10.0000	−0.549650	−0.274825	−	0.961494i	$-0.588620\pi$
$331$	−10.0000	−0.549650	−0.274825	+	0.961494i	$0.588620\pi$
$332$	4.00000	0.219529
$333$	−2.00000	−0.109599
$334$	20.0000	1.09435
$335$	12.0000	0.655630
$336$	10.0000	0.545545
$337$	5.00000	0.272367	0.136184	−	0.990684i	$-0.456516\pi$
$337$	5.00000	0.272367	0.136184	+	0.990684i	$0.456516\pi$
$338$	9.00000	0.489535
$339$	−18.0000	−0.977626
$340$	−14.0000	−0.759257
$341$	0	0
$342$	−6.00000	−0.324443
$343$	−55.0000	−2.96972
$344$	10.0000	0.539164
$345$	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
$349$	24.0000	1.28469	0.642345	−	0.766415i	$-0.277962\pi$
$349$	24.0000	1.28469	0.642345	+	0.766415i	$0.277962\pi$
$350$	−5.00000	−0.267261
$351$	8.00000	0.427008
$352$	0	0
$353$	−29.0000	−1.54351	−0.771757	−	0.635917i	$-0.780622\pi$
$353$	−29.0000	−1.54351	−0.771757	+	0.635917i	$0.780622\pi$
$354$	−24.0000	−1.27559
$355$	−22.0000	−1.16764
$356$	13.0000	0.688999
$357$	70.0000	3.70479
$358$	−14.0000	−0.739923
$359$	−11.0000	−0.580558	−0.290279	−	0.956942i	$-0.593748\pi$
$359$	−11.0000	−0.580558	−0.290279	+	0.956942i	$0.593748\pi$
$360$	2.00000	0.105409
$361$	17.0000	0.894737
$362$	8.00000	0.420471
$363$	22.0000	1.15470
$364$	−10.0000	−0.524142
$365$	14.0000	0.732793
$366$	16.0000	0.836333
$367$	−7.00000	−0.365397	−0.182699	−	0.983169i	$-0.558483\pi$
$367$	−7.00000	−0.365397	−0.182699	+	0.983169i	$0.558483\pi$
$368$	1.00000	0.0521286
$369$	−2.00000	−0.104116
$370$	−4.00000	−0.207950
$371$	10.0000	0.519174
$372$	−6.00000	−0.311086
$373$	−18.0000	−0.932005	−0.466002	−	0.884783i	$-0.654306\pi$
$373$	−18.0000	−0.932005	−0.466002	+	0.884783i	$0.654306\pi$
$374$	0	0
$375$	−24.0000	−1.23935
$376$	7.00000	0.360997
$377$	0	0
$378$	20.0000	1.02869
$379$	4.00000	0.205466	0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000	0.205466	0.102733	+	0.994709i	$0.467241\pi$
$380$	−12.0000	−0.615587
$381$	16.0000	0.819705
$382$	−5.00000	−0.255822
$383$	17.0000	0.868659	0.434330	−	0.900754i	$-0.356985\pi$
$383$	17.0000	0.868659	0.434330	+	0.900754i	$0.356985\pi$
$384$	2.00000	0.102062
$385$	0	0
$386$	2.00000	0.101797
$387$	−10.0000	−0.508329
$388$	11.0000	0.558440
$389$	36.0000	1.82527	0.912636	−	0.408773i	$-0.134043\pi$
$389$	36.0000	1.82527	0.912636	+	0.408773i	$0.134043\pi$
$390$	−8.00000	−0.405096
$391$	7.00000	0.354005
$392$	−18.0000	−0.909137
$393$	12.0000	0.605320
$394$	−12.0000	−0.604551
$395$	6.00000	0.301893
$396$	0	0
$397$	−38.0000	−1.90717	−0.953583	−	0.301131i	$-0.902636\pi$
$397$	−38.0000	−1.90717	−0.953583	+	0.301131i	$0.902636\pi$
$398$	−5.00000	−0.250627
$399$	60.0000	3.00376
$400$	−1.00000	−0.0500000
$401$	21.0000	1.04869	0.524345	−	0.851506i	$-0.324310\pi$
$401$	21.0000	1.04869	0.524345	+	0.851506i	$0.324310\pi$
$402$	−12.0000	−0.598506
$403$	6.00000	0.298881
$404$	8.00000	0.398015
$405$	22.0000	1.09319
$406$	0	0
$407$	0	0
$408$	14.0000	0.693103
$409$	−6.00000	−0.296681	−0.148340	−	0.988936i	$-0.547393\pi$
$409$	−6.00000	−0.296681	−0.148340	+	0.988936i	$0.547393\pi$
$410$	−4.00000	−0.197546
$411$	−6.00000	−0.295958
$412$	−7.00000	−0.344865
$413$	60.0000	2.95241
$414$	−1.00000	−0.0491473
$415$	−8.00000	−0.392705
$416$	−2.00000	−0.0980581
$417$	4.00000	0.195881
$418$	0	0
$419$	20.0000	0.977064	0.488532	−	0.872546i	$-0.337533\pi$
$419$	20.0000	0.977064	0.488532	+	0.872546i	$0.337533\pi$
$420$	−20.0000	−0.975900
$421$	16.0000	0.779792	0.389896	−	0.920859i	$-0.372511\pi$
$421$	16.0000	0.779792	0.389896	+	0.920859i	$0.372511\pi$
$422$	6.00000	0.292075
$423$	−7.00000	−0.340352
$424$	2.00000	0.0971286
$425$	−7.00000	−0.339550
$426$	22.0000	1.06590
$427$	−40.0000	−1.93574
$428$	4.00000	0.193347
$429$	0	0
$430$	−20.0000	−0.964486
$431$	−25.0000	−1.20421	−0.602104	−	0.798418i	$-0.705671\pi$
$431$	−25.0000	−1.20421	−0.602104	+	0.798418i	$0.705671\pi$
$432$	4.00000	0.192450
$433$	19.0000	0.913082	0.456541	−	0.889702i	$-0.349088\pi$
$433$	19.0000	0.913082	0.456541	+	0.889702i	$0.349088\pi$
$434$	15.0000	0.720023
$435$	0	0
$436$	2.00000	0.0957826
$437$	6.00000	0.287019
$438$	−14.0000	−0.668946
$439$	−7.00000	−0.334092	−0.167046	−	0.985949i	$-0.553423\pi$
$439$	−7.00000	−0.334092	−0.167046	+	0.985949i	$0.553423\pi$
$440$	0	0
$441$	18.0000	0.857143
$442$	−14.0000	−0.665912
$443$	20.0000	0.950229	0.475114	−	0.879924i	$-0.342407\pi$
$443$	20.0000	0.950229	0.475114	+	0.879924i	$0.342407\pi$
$444$	4.00000	0.189832
$445$	−26.0000	−1.23252
$446$	17.0000	0.804973
$447$	−12.0000	−0.567581
$448$	−5.00000	−0.236228
$449$	34.0000	1.60456	0.802280	−	0.596948i	$-0.203620\pi$
$449$	34.0000	1.60456	0.802280	+	0.596948i	$0.203620\pi$
$450$	1.00000	0.0471405
$451$	0	0
$452$	9.00000	0.423324
$453$	14.0000	0.657777
$454$	18.0000	0.844782
$455$	20.0000	0.937614
$456$	12.0000	0.561951
$457$	−38.0000	−1.77757	−0.888783	−	0.458329i	$-0.848448\pi$
$457$	−38.0000	−1.77757	−0.888783	+	0.458329i	$0.848448\pi$
$458$	16.0000	0.747631
$459$	28.0000	1.30693
$460$	−2.00000	−0.0932505
$461$	−38.0000	−1.76984	−0.884918	−	0.465746i	$-0.845786\pi$
$461$	−38.0000	−1.76984	−0.884918	+	0.465746i	$0.845786\pi$
$462$	0	0
$463$	−41.0000	−1.90543	−0.952716	−	0.303863i	$-0.901724\pi$
$463$	−41.0000	−1.90543	−0.952716	+	0.303863i	$0.901724\pi$
$464$	0	0
$465$	12.0000	0.556487
$466$	−23.0000	−1.06545
$467$	6.00000	0.277647	0.138823	−	0.990317i	$-0.455668\pi$
$467$	6.00000	0.277647	0.138823	+	0.990317i	$0.455668\pi$
$468$	2.00000	0.0924500
$469$	30.0000	1.38527
$470$	−14.0000	−0.645772
$471$	36.0000	1.65879
$472$	12.0000	0.552345
$473$	0	0
$474$	−6.00000	−0.275589
$475$	−6.00000	−0.275299
$476$	−35.0000	−1.60422
$477$	−2.00000	−0.0915737
$478$	9.00000	0.411650
$479$	−25.0000	−1.14228	−0.571140	−	0.820853i	$-0.693499\pi$
$479$	−25.0000	−1.14228	−0.571140	+	0.820853i	$0.693499\pi$
$480$	−4.00000	−0.182574
$481$	−4.00000	−0.182384
$482$	19.0000	0.865426
$483$	10.0000	0.455016
$484$	−11.0000	−0.500000
$485$	−22.0000	−0.998969
$486$	−10.0000	−0.453609
$487$	−40.0000	−1.81257	−0.906287	−	0.422664i	$-0.861095\pi$
$487$	−40.0000	−1.81257	−0.906287	+	0.422664i	$0.861095\pi$
$488$	−8.00000	−0.362143
$489$	−12.0000	−0.542659
$490$	36.0000	1.62631
$491$	−30.0000	−1.35388	−0.676941	−	0.736038i	$-0.736695\pi$
$491$	−30.0000	−1.35388	−0.676941	+	0.736038i	$0.736695\pi$
$492$	4.00000	0.180334
$493$	0	0
$494$	−12.0000	−0.539906
$495$	0	0
$496$	3.00000	0.134704
$497$	−55.0000	−2.46709
$498$	8.00000	0.358489
$499$	−8.00000	−0.358129	−0.179065	−	0.983837i	$-0.557307\pi$
$499$	−8.00000	−0.358129	−0.179065	+	0.983837i	$0.557307\pi$
$500$	12.0000	0.536656
$501$	40.0000	1.78707
$502$	−6.00000	−0.267793
$503$	−40.0000	−1.78351	−0.891756	−	0.452517i	$-0.850526\pi$
$503$	−40.0000	−1.78351	−0.891756	+	0.452517i	$0.850526\pi$
$504$	5.00000	0.222718
$505$	−16.0000	−0.711991
$506$	0	0
$507$	18.0000	0.799408
$508$	−8.00000	−0.354943
$509$	−6.00000	−0.265945	−0.132973	−	0.991120i	$-0.542452\pi$
$509$	−6.00000	−0.265945	−0.132973	+	0.991120i	$0.542452\pi$
$510$	−28.0000	−1.23986
$511$	35.0000	1.54831
$512$	−1.00000	−0.0441942
$513$	24.0000	1.05963
$514$	17.0000	0.749838
$515$	14.0000	0.616914
$516$	20.0000	0.880451
$517$	0	0
$518$	−10.0000	−0.439375
$519$	−12.0000	−0.526742
$520$	4.00000	0.175412
$521$	−19.0000	−0.832405	−0.416203	−	0.909272i	$-0.636639\pi$
$521$	−19.0000	−0.832405	−0.416203	+	0.909272i	$0.636639\pi$
$522$	0	0
$523$	−24.0000	−1.04945	−0.524723	−	0.851273i	$-0.675831\pi$
$523$	−24.0000	−1.04945	−0.524723	+	0.851273i	$0.675831\pi$
$524$	−6.00000	−0.262111
$525$	−10.0000	−0.436436
$526$	0	0
$527$	21.0000	0.914774
$528$	0	0
$529$	−22.0000	−0.956522
$530$	−4.00000	−0.173749
$531$	−12.0000	−0.520756
$532$	−30.0000	−1.30066
$533$	−4.00000	−0.173259
$534$	26.0000	1.12513
$535$	−8.00000	−0.345870
$536$	6.00000	0.259161
$537$	−28.0000	−1.20829
$538$	6.00000	0.258678
$539$	0	0
$540$	−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	$0.568964\pi$
$542$	11.0000	0.472490
$543$	16.0000	0.686626
$544$	−7.00000	−0.300123
$545$	−4.00000	−0.171341
$546$	−20.0000	−0.855921
$547$	26.0000	1.11168	0.555840	−	0.831289i	$-0.312397\pi$
$547$	26.0000	1.11168	0.555840	+	0.831289i	$0.312397\pi$
$548$	3.00000	0.128154
$549$	8.00000	0.341432
$550$	0	0
$551$	0	0
$552$	2.00000	0.0851257
$553$	15.0000	0.637865
$554$	20.0000	0.849719
$555$	−8.00000	−0.339581
$556$	−2.00000	−0.0848189
$557$	16.0000	0.677942	0.338971	−	0.940797i	$-0.389921\pi$
$557$	16.0000	0.677942	0.338971	+	0.940797i	$0.389921\pi$
$558$	−3.00000	−0.127000
$559$	−20.0000	−0.845910
$560$	10.0000	0.422577
$561$	0	0
$562$	−3.00000	−0.126547
$563$	14.0000	0.590030	0.295015	−	0.955493i	$-0.404675\pi$
$563$	14.0000	0.590030	0.295015	+	0.955493i	$0.404675\pi$
$564$	14.0000	0.589506
$565$	−18.0000	−0.757266
$566$	−22.0000	−0.924729
$567$	55.0000	2.30978
$568$	−11.0000	−0.461550
$569$	−33.0000	−1.38343	−0.691716	−	0.722170i	$-0.743145\pi$
$569$	−33.0000	−1.38343	−0.691716	+	0.722170i	$0.743145\pi$
$570$	−24.0000	−1.00525
$571$	2.00000	0.0836974	0.0418487	−	0.999124i	$-0.486675\pi$
$571$	2.00000	0.0836974	0.0418487	+	0.999124i	$0.486675\pi$
$572$	0	0
$573$	−10.0000	−0.417756
$574$	−10.0000	−0.417392
$575$	−1.00000	−0.0417029
$576$	1.00000	0.0416667
$577$	25.0000	1.04076	0.520382	−	0.853934i	$-0.325790\pi$
$577$	25.0000	1.04076	0.520382	+	0.853934i	$0.325790\pi$
$578$	−32.0000	−1.33102
$579$	4.00000	0.166234
$580$	0	0
$581$	−20.0000	−0.829740
$582$	22.0000	0.911929
$583$	0	0
$584$	7.00000	0.289662
$585$	−4.00000	−0.165380
$586$	2.00000	0.0826192
$587$	28.0000	1.15568	0.577842	−	0.816149i	$-0.303895\pi$
$587$	28.0000	1.15568	0.577842	+	0.816149i	$0.303895\pi$
$588$	−36.0000	−1.48461
$589$	18.0000	0.741677
$590$	−24.0000	−0.988064
$591$	−24.0000	−0.987228
$592$	−2.00000	−0.0821995
$593$	−42.0000	−1.72473	−0.862367	−	0.506284i	$-0.831019\pi$
$593$	−42.0000	−1.72473	−0.862367	+	0.506284i	$0.831019\pi$
$594$	0	0
$595$	70.0000	2.86972
$596$	6.00000	0.245770
$597$	−10.0000	−0.409273
$598$	−2.00000	−0.0817861
$599$	−16.0000	−0.653742	−0.326871	−	0.945069i	$-0.605994\pi$
$599$	−16.0000	−0.653742	−0.326871	+	0.945069i	$0.605994\pi$
$600$	−2.00000	−0.0816497
$601$	−23.0000	−0.938190	−0.469095	−	0.883148i	$-0.655420\pi$
$601$	−23.0000	−0.938190	−0.469095	+	0.883148i	$0.655420\pi$
$602$	−50.0000	−2.03785
$603$	−6.00000	−0.244339
$604$	−7.00000	−0.284826
$605$	22.0000	0.894427
$606$	16.0000	0.649956
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	−6.00000	−0.243332
$609$	0	0
$610$	16.0000	0.647821
$611$	−14.0000	−0.566379
$612$	7.00000	0.282958
$613$	−16.0000	−0.646234	−0.323117	−	0.946359i	$-0.604731\pi$
$613$	−16.0000	−0.646234	−0.323117	+	0.946359i	$0.604731\pi$
$614$	−22.0000	−0.887848
$615$	−8.00000	−0.322591
$616$	0	0
$617$	42.0000	1.69086	0.845428	−	0.534089i	$-0.179345\pi$
$617$	42.0000	1.69086	0.845428	+	0.534089i	$0.179345\pi$
$618$	−14.0000	−0.563163
$619$	26.0000	1.04503	0.522514	−	0.852631i	$-0.324994\pi$
$619$	26.0000	1.04503	0.522514	+	0.852631i	$0.324994\pi$
$620$	−6.00000	−0.240966
$621$	4.00000	0.160514
$622$	16.0000	0.641542
$623$	−65.0000	−2.60417
$624$	−4.00000	−0.160128
$625$	−19.0000	−0.760000
$626$	6.00000	0.239808
$627$	0	0
$628$	−18.0000	−0.718278
$629$	−14.0000	−0.558217
$630$	−10.0000	−0.398410
$631$	25.0000	0.995234	0.497617	−	0.867397i	$-0.334208\pi$
$631$	25.0000	0.995234	0.497617	+	0.867397i	$0.334208\pi$
$632$	3.00000	0.119334
$633$	12.0000	0.476957
$634$	24.0000	0.953162
$635$	16.0000	0.634941
$636$	4.00000	0.158610
$637$	36.0000	1.42637
$638$	0	0
$639$	11.0000	0.435153
$640$	2.00000	0.0790569
$641$	−10.0000	−0.394976	−0.197488	−	0.980305i	$-0.563278\pi$
$641$	−10.0000	−0.394976	−0.197488	+	0.980305i	$0.563278\pi$
$642$	8.00000	0.315735
$643$	−14.0000	−0.552106	−0.276053	−	0.961142i	$-0.589027\pi$
$643$	−14.0000	−0.552106	−0.276053	+	0.961142i	$0.589027\pi$
$644$	−5.00000	−0.197028
$645$	−40.0000	−1.57500
$646$	−42.0000	−1.65247
$647$	−5.00000	−0.196570	−0.0982851	−	0.995158i	$-0.531336\pi$
$647$	−5.00000	−0.196570	−0.0982851	+	0.995158i	$0.531336\pi$
$648$	11.0000	0.432121
$649$	0	0
$650$	2.00000	0.0784465
$651$	30.0000	1.17579
$652$	6.00000	0.234978
$653$	20.0000	0.782660	0.391330	−	0.920250i	$-0.372015\pi$
$653$	20.0000	0.782660	0.391330	+	0.920250i	$0.372015\pi$
$654$	4.00000	0.156412
$655$	12.0000	0.468879
$656$	−2.00000	−0.0780869
$657$	−7.00000	−0.273096
$658$	−35.0000	−1.36444
$659$	24.0000	0.934907	0.467454	−	0.884018i	$-0.345171\pi$
$659$	24.0000	0.934907	0.467454	+	0.884018i	$0.345171\pi$
$660$	0	0
$661$	−42.0000	−1.63361	−0.816805	−	0.576913i	$-0.804257\pi$
$661$	−42.0000	−1.63361	−0.816805	+	0.576913i	$0.804257\pi$
$662$	10.0000	0.388661
$663$	−28.0000	−1.08743
$664$	−4.00000	−0.155230
$665$	60.0000	2.32670
$666$	2.00000	0.0774984
$667$	0	0
$668$	−20.0000	−0.773823
$669$	34.0000	1.31452
$670$	−12.0000	−0.463600
$671$	0	0
$672$	−10.0000	−0.385758
$673$	33.0000	1.27206	0.636028	−	0.771666i	$-0.280576\pi$
$673$	33.0000	1.27206	0.636028	+	0.771666i	$0.280576\pi$
$674$	−5.00000	−0.192593
$675$	−4.00000	−0.153960
$676$	−9.00000	−0.346154
$677$	46.0000	1.76792	0.883962	−	0.467559i	$-0.154866\pi$
$677$	46.0000	1.76792	0.883962	+	0.467559i	$0.154866\pi$
$678$	18.0000	0.691286
$679$	−55.0000	−2.11071
$680$	14.0000	0.536875
$681$	36.0000	1.37952
$682$	0	0
$683$	−30.0000	−1.14792	−0.573959	−	0.818884i	$-0.694593\pi$
$683$	−30.0000	−1.14792	−0.573959	+	0.818884i	$0.694593\pi$
$684$	6.00000	0.229416
$685$	−6.00000	−0.229248
$686$	55.0000	2.09991
$687$	32.0000	1.22088
$688$	−10.0000	−0.381246
$689$	−4.00000	−0.152388
$690$	−4.00000	−0.152277
$691$	50.0000	1.90209	0.951045	−	0.309053i	$-0.100012\pi$
$691$	50.0000	1.90209	0.951045	+	0.309053i	$0.100012\pi$
$692$	6.00000	0.228086
$693$	0	0
$694$	2.00000	0.0759190
$695$	4.00000	0.151729
$696$	0	0
$697$	−14.0000	−0.530288
$698$	−24.0000	−0.908413
$699$	−46.0000	−1.73988
$700$	5.00000	0.188982
$701$	30.0000	1.13308	0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000	1.13308	0.566542	+	0.824033i	$0.308281\pi$
$702$	−8.00000	−0.301941
$703$	−12.0000	−0.452589
$704$	0	0
$705$	−28.0000	−1.05454
$706$	29.0000	1.09143
$707$	−40.0000	−1.50435
$708$	24.0000	0.901975
$709$	0	0		−	1.00000i	$-0.5\pi$
$709$	0	0			1.00000i	$0.5\pi$
$710$	22.0000	0.825645
$711$	−3.00000	−0.112509
$712$	−13.0000	−0.487196
$713$	3.00000	0.112351
$714$	−70.0000	−2.61968
$715$	0	0
$716$	14.0000	0.523205
$717$	18.0000	0.672222
$718$	11.0000	0.410516
$719$	52.0000	1.93927	0.969636	−	0.244551i	$-0.0786406\pi$
$719$	52.0000	1.93927	0.969636	+	0.244551i	$0.0786406\pi$
$720$	−2.00000	−0.0745356
$721$	35.0000	1.30347
$722$	−17.0000	−0.632674
$723$	38.0000	1.41324
$724$	−8.00000	−0.297318
$725$	0	0
$726$	−22.0000	−0.816497
$727$	−8.00000	−0.296704	−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000	−0.296704	−0.148352	+	0.988935i	$0.547397\pi$
$728$	10.0000	0.370625
$729$	13.0000	0.481481
$730$	−14.0000	−0.518163
$731$	−70.0000	−2.58904
$732$	−16.0000	−0.591377
$733$	6.00000	0.221615	0.110808	−	0.993842i	$-0.464656\pi$
$733$	6.00000	0.221615	0.110808	+	0.993842i	$0.464656\pi$
$734$	7.00000	0.258375
$735$	72.0000	2.65576
$736$	−1.00000	−0.0368605
$737$	0	0
$738$	2.00000	0.0736210
$739$	14.0000	0.514998	0.257499	−	0.966279i	$-0.417102\pi$
$739$	14.0000	0.514998	0.257499	+	0.966279i	$0.417102\pi$
$740$	4.00000	0.147043
$741$	−24.0000	−0.881662
$742$	−10.0000	−0.367112
$743$	−32.0000	−1.17397	−0.586983	−	0.809599i	$-0.699684\pi$
$743$	−32.0000	−1.17397	−0.586983	+	0.809599i	$0.699684\pi$
$744$	6.00000	0.219971
$745$	−12.0000	−0.439646
$746$	18.0000	0.659027
$747$	4.00000	0.146352
$748$	0	0
$749$	−20.0000	−0.730784
$750$	24.0000	0.876356
$751$	−23.0000	−0.839282	−0.419641	−	0.907690i	$-0.637844\pi$
$751$	−23.0000	−0.839282	−0.419641	+	0.907690i	$0.637844\pi$
$752$	−7.00000	−0.255264
$753$	−12.0000	−0.437304
$754$	0	0
$755$	14.0000	0.509512
$756$	−20.0000	−0.727393
$757$	34.0000	1.23575	0.617876	−	0.786276i	$-0.287994\pi$
$757$	34.0000	1.23575	0.617876	+	0.786276i	$0.287994\pi$
$758$	−4.00000	−0.145287
$759$	0	0
$760$	12.0000	0.435286
$761$	−15.0000	−0.543750	−0.271875	−	0.962333i	$-0.587644\pi$
$761$	−15.0000	−0.543750	−0.271875	+	0.962333i	$0.587644\pi$
$762$	−16.0000	−0.579619
$763$	−10.0000	−0.362024
$764$	5.00000	0.180894
$765$	−14.0000	−0.506171
$766$	−17.0000	−0.614235
$767$	−24.0000	−0.866590
$768$	−2.00000	−0.0721688
$769$	38.0000	1.37032	0.685158	−	0.728395i	$-0.259733\pi$
$769$	38.0000	1.37032	0.685158	+	0.728395i	$0.259733\pi$
$770$	0	0
$771$	34.0000	1.22448
$772$	−2.00000	−0.0719816
$773$	−44.0000	−1.58257	−0.791285	−	0.611448i	$-0.790588\pi$
$773$	−44.0000	−1.58257	−0.791285	+	0.611448i	$0.790588\pi$
$774$	10.0000	0.359443
$775$	−3.00000	−0.107763
$776$	−11.0000	−0.394877
$777$	−20.0000	−0.717496
$778$	−36.0000	−1.29066
$779$	−12.0000	−0.429945
$780$	8.00000	0.286446
$781$	0	0
$782$	−7.00000	−0.250319
$783$	0	0
$784$	18.0000	0.642857
$785$	36.0000	1.28490
$786$	−12.0000	−0.428026
$787$	32.0000	1.14068	0.570338	−	0.821410i	$-0.306812\pi$
$787$	32.0000	1.14068	0.570338	+	0.821410i	$0.306812\pi$
$788$	12.0000	0.427482
$789$	0	0
$790$	−6.00000	−0.213470
$791$	−45.0000	−1.60002
$792$	0	0
$793$	16.0000	0.568177
$794$	38.0000	1.34857
$795$	−8.00000	−0.283731
$796$	5.00000	0.177220
$797$	−28.0000	−0.991811	−0.495905	−	0.868377i	$-0.665164\pi$
$797$	−28.0000	−0.991811	−0.495905	+	0.868377i	$0.665164\pi$
$798$	−60.0000	−2.12398
$799$	−49.0000	−1.73350
$800$	1.00000	0.0353553
$801$	13.0000	0.459332
$802$	−21.0000	−0.741536
$803$	0	0
$804$	12.0000	0.423207
$805$	10.0000	0.352454
$806$	−6.00000	−0.211341
$807$	12.0000	0.422420
$808$	−8.00000	−0.281439
$809$	6.00000	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	−22.0000	−0.773001
$811$	−30.0000	−1.05344	−0.526721	−	0.850038i	$-0.676579\pi$
$811$	−30.0000	−1.05344	−0.526721	+	0.850038i	$0.676579\pi$
$812$	0	0
$813$	22.0000	0.771574
$814$	0	0
$815$	−12.0000	−0.420342
$816$	−14.0000	−0.490098
$817$	−60.0000	−2.09913
$818$	6.00000	0.209785
$819$	−10.0000	−0.349428
$820$	4.00000	0.139686
$821$	−48.0000	−1.67521	−0.837606	−	0.546275i	$-0.816045\pi$
$821$	−48.0000	−1.67521	−0.837606	+	0.546275i	$0.816045\pi$
$822$	6.00000	0.209274
$823$	−56.0000	−1.95204	−0.976019	−	0.217687i	$-0.930149\pi$
$823$	−56.0000	−1.95204	−0.976019	+	0.217687i	$0.930149\pi$
$824$	7.00000	0.243857
$825$	0	0
$826$	−60.0000	−2.08767
$827$	−16.0000	−0.556375	−0.278187	−	0.960527i	$-0.589734\pi$
$827$	−16.0000	−0.556375	−0.278187	+	0.960527i	$0.589734\pi$
$828$	1.00000	0.0347524
$829$	8.00000	0.277851	0.138926	−	0.990303i	$-0.455635\pi$
$829$	8.00000	0.277851	0.138926	+	0.990303i	$0.455635\pi$
$830$	8.00000	0.277684
$831$	40.0000	1.38758
$832$	2.00000	0.0693375
$833$	126.000	4.36564
$834$	−4.00000	−0.138509
$835$	40.0000	1.38426
$836$	0	0
$837$	12.0000	0.414781
$838$	−20.0000	−0.690889
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	20.0000	0.690066
$841$	0	0
$842$	−16.0000	−0.551396
$843$	−6.00000	−0.206651
$844$	−6.00000	−0.206529
$845$	18.0000	0.619219
$846$	7.00000	0.240665
$847$	55.0000	1.88982
$848$	−2.00000	−0.0686803
$849$	−44.0000	−1.51008
$850$	7.00000	0.240098
$851$	−2.00000	−0.0685591
$852$	−22.0000	−0.753708
$853$	10.0000	0.342393	0.171197	−	0.985237i	$-0.445237\pi$
$853$	10.0000	0.342393	0.171197	+	0.985237i	$0.445237\pi$
$854$	40.0000	1.36877
$855$	−12.0000	−0.410391
$856$	−4.00000	−0.136717
$857$	45.0000	1.53717	0.768585	−	0.639747i	$-0.220961\pi$
$857$	45.0000	1.53717	0.768585	+	0.639747i	$0.220961\pi$
$858$	0	0
$859$	−14.0000	−0.477674	−0.238837	−	0.971060i	$-0.576766\pi$
$859$	−14.0000	−0.477674	−0.238837	+	0.971060i	$0.576766\pi$
$860$	20.0000	0.681994
$861$	−20.0000	−0.681598
$862$	25.0000	0.851503
$863$	−31.0000	−1.05525	−0.527626	−	0.849477i	$-0.676918\pi$
$863$	−31.0000	−1.05525	−0.527626	+	0.849477i	$0.676918\pi$
$864$	−4.00000	−0.136083
$865$	−12.0000	−0.408012
$866$	−19.0000	−0.645646
$867$	−64.0000	−2.17355
$868$	−15.0000	−0.509133
$869$	0	0
$870$	0	0
$871$	−12.0000	−0.406604
$872$	−2.00000	−0.0677285
$873$	11.0000	0.372294
$874$	−6.00000	−0.202953
$875$	−60.0000	−2.02837
$876$	14.0000	0.473016
$877$	−2.00000	−0.0675352	−0.0337676	−	0.999430i	$-0.510751\pi$
$877$	−2.00000	−0.0675352	−0.0337676	+	0.999430i	$0.510751\pi$
$878$	7.00000	0.236239
$879$	4.00000	0.134917
$880$	0	0
$881$	−30.0000	−1.01073	−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000	−1.01073	−0.505363	+	0.862907i	$0.668641\pi$
$882$	−18.0000	−0.606092
$883$	−8.00000	−0.269221	−0.134611	−	0.990899i	$-0.542978\pi$
$883$	−8.00000	−0.269221	−0.134611	+	0.990899i	$0.542978\pi$
$884$	14.0000	0.470871
$885$	−48.0000	−1.61350
$886$	−20.0000	−0.671913
$887$	−39.0000	−1.30949	−0.654746	−	0.755849i	$-0.727224\pi$
$887$	−39.0000	−1.30949	−0.654746	+	0.755849i	$0.727224\pi$
$888$	−4.00000	−0.134231
$889$	40.0000	1.34156
$890$	26.0000	0.871522
$891$	0	0
$892$	−17.0000	−0.569202
$893$	−42.0000	−1.40548
$894$	12.0000	0.401340
$895$	−28.0000	−0.935937
$896$	5.00000	0.167038
$897$	−4.00000	−0.133556
$898$	−34.0000	−1.13459
$899$	0	0
$900$	−1.00000	−0.0333333
$901$	−14.0000	−0.466408
$902$	0	0
$903$	−100.000	−3.32779
$904$	−9.00000	−0.299336
$905$	16.0000	0.531858
$906$	−14.0000	−0.465119
$907$	32.0000	1.06254	0.531271	−	0.847202i	$-0.321714\pi$
$907$	32.0000	1.06254	0.531271	+	0.847202i	$0.321714\pi$
$908$	−18.0000	−0.597351
$909$	8.00000	0.265343
$910$	−20.0000	−0.662994
$911$	13.0000	0.430709	0.215355	−	0.976536i	$-0.430909\pi$
$911$	13.0000	0.430709	0.215355	+	0.976536i	$0.430909\pi$
$912$	−12.0000	−0.397360
$913$	0	0
$914$	38.0000	1.25693
$915$	32.0000	1.05789
$916$	−16.0000	−0.528655
$917$	30.0000	0.990687
$918$	−28.0000	−0.924138
$919$	0	0		−	1.00000i	$-0.5\pi$
$919$	0	0			1.00000i	$0.5\pi$
$920$	2.00000	0.0659380
$921$	−44.0000	−1.44985
$922$	38.0000	1.25146
$923$	22.0000	0.724139
$924$	0	0
$925$	2.00000	0.0657596
$926$	41.0000	1.34734
$927$	−7.00000	−0.229910
$928$	0	0
$929$	−3.00000	−0.0984268	−0.0492134	−	0.998788i	$-0.515671\pi$
$929$	−3.00000	−0.0984268	−0.0492134	+	0.998788i	$0.515671\pi$
$930$	−12.0000	−0.393496
$931$	108.000	3.53956
$932$	23.0000	0.753390
$933$	32.0000	1.04763
$934$	−6.00000	−0.196326
$935$	0	0
$936$	−2.00000	−0.0653720
$937$	−29.0000	−0.947389	−0.473694	−	0.880689i	$-0.657080\pi$
$937$	−29.0000	−0.947389	−0.473694	+	0.880689i	$0.657080\pi$
$938$	−30.0000	−0.979535
$939$	12.0000	0.391605
$940$	14.0000	0.456630
$941$	−32.0000	−1.04317	−0.521585	−	0.853199i	$-0.674659\pi$
$941$	−32.0000	−1.04317	−0.521585	+	0.853199i	$0.674659\pi$
$942$	−36.0000	−1.17294
$943$	−2.00000	−0.0651290
$944$	−12.0000	−0.390567
$945$	40.0000	1.30120
$946$	0	0
$947$	36.0000	1.16984	0.584921	−	0.811090i	$-0.301125\pi$
$947$	36.0000	1.16984	0.584921	+	0.811090i	$0.301125\pi$
$948$	6.00000	0.194871
$949$	−14.0000	−0.454459
$950$	6.00000	0.194666
$951$	48.0000	1.55651
$952$	35.0000	1.13436
$953$	26.0000	0.842223	0.421111	−	0.907009i	$-0.361640\pi$
$953$	26.0000	0.842223	0.421111	+	0.907009i	$0.361640\pi$
$954$	2.00000	0.0647524
$955$	−10.0000	−0.323592
$956$	−9.00000	−0.291081
$957$	0	0
$958$	25.0000	0.807713
$959$	−15.0000	−0.484375
$960$	4.00000	0.129099
$961$	−22.0000	−0.709677
$962$	4.00000	0.128965
$963$	4.00000	0.128898
$964$	−19.0000	−0.611949
$965$	4.00000	0.128765
$966$	−10.0000	−0.321745
$967$	41.0000	1.31847	0.659236	−	0.751936i	$-0.270880\pi$
$967$	41.0000	1.31847	0.659236	+	0.751936i	$0.270880\pi$
$968$	11.0000	0.353553
$969$	−84.0000	−2.69847
$970$	22.0000	0.706377
$971$	−38.0000	−1.21948	−0.609739	−	0.792602i	$-0.708726\pi$
$971$	−38.0000	−1.21948	−0.609739	+	0.792602i	$0.708726\pi$
$972$	10.0000	0.320750
$973$	10.0000	0.320585
$974$	40.0000	1.28168
$975$	4.00000	0.128103
$976$	8.00000	0.256074
$977$	−2.00000	−0.0639857	−0.0319928	−	0.999488i	$-0.510185\pi$
$977$	−2.00000	−0.0639857	−0.0319928	+	0.999488i	$0.510185\pi$
$978$	12.0000	0.383718
$979$	0	0
$980$	−36.0000	−1.14998
$981$	2.00000	0.0638551
$982$	30.0000	0.957338
$983$	23.0000	0.733586	0.366793	−	0.930303i	$-0.380456\pi$
$983$	23.0000	0.733586	0.366793	+	0.930303i	$0.380456\pi$
$984$	−4.00000	−0.127515
$985$	−24.0000	−0.764704
$986$	0	0
$987$	−70.0000	−2.22812
$988$	12.0000	0.381771
$989$	−10.0000	−0.317982
$990$	0	0
$991$	−32.0000	−1.01651	−0.508257	−	0.861206i	$-0.669710\pi$
$991$	−32.0000	−1.01651	−0.508257	+	0.861206i	$0.669710\pi$
$992$	−3.00000	−0.0952501
$993$	20.0000	0.634681
$994$	55.0000	1.74449
$995$	−10.0000	−0.317021
$996$	−8.00000	−0.253490
$997$	40.0000	1.26681	0.633406	−	0.773819i	$-0.281656\pi$
$997$	40.0000	1.26681	0.633406	+	0.773819i	$0.281656\pi$
$998$	8.00000	0.253236
$999$	−8.00000	−0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1682.2.a.a.1.1	✓	1
29.12	odd	4		1682.2.b.d.1681.1		2
29.17	odd	4		1682.2.b.d.1681.2		2
29.28	even	2		1682.2.a.h.1.1	yes	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1682.2.a.a.1.1	✓	1	1.1	even	1	trivial
1682.2.a.h.1.1	yes	1	29.28	even	2
1682.2.b.d.1681.1		2	29.12	odd	4
1682.2.b.d.1681.2		2	29.17	odd	4

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