LMFDB - Embedded newform 1666.2.a.h.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1666,2,Mod(1,1666)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1666, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1666.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1666 = 2 \cdot 7^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1666.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:	$13.3030769767$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 238)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1666.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} -3.00000 q^{6} -1.00000 q^{8} +6.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} -3.00000 q^{6} -1.00000 q^{8} +6.00000 q^{9} +2.00000 q^{10} -5.00000 q^{11} +3.00000 q^{12} -3.00000 q^{13} -6.00000 q^{15} +1.00000 q^{16} -1.00000 q^{17} -6.00000 q^{18} -2.00000 q^{19} -2.00000 q^{20} +5.00000 q^{22} -8.00000 q^{23} -3.00000 q^{24} -1.00000 q^{25} +3.00000 q^{26} +9.00000 q^{27} -6.00000 q^{29} +6.00000 q^{30} -4.00000 q^{31} -1.00000 q^{32} -15.0000 q^{33} +1.00000 q^{34} +6.00000 q^{36} +8.00000 q^{37} +2.00000 q^{38} -9.00000 q^{39} +2.00000 q^{40} +6.00000 q^{41} +4.00000 q^{43} -5.00000 q^{44} -12.0000 q^{45} +8.00000 q^{46} -10.0000 q^{47} +3.00000 q^{48} +1.00000 q^{50} -3.00000 q^{51} -3.00000 q^{52} +9.00000 q^{53} -9.00000 q^{54} +10.0000 q^{55} -6.00000 q^{57} +6.00000 q^{58} +4.00000 q^{59} -6.00000 q^{60} +4.00000 q^{61} +4.00000 q^{62} +1.00000 q^{64} +6.00000 q^{65} +15.0000 q^{66} -10.0000 q^{67} -1.00000 q^{68} -24.0000 q^{69} -5.00000 q^{71} -6.00000 q^{72} +2.00000 q^{73} -8.00000 q^{74} -3.00000 q^{75} -2.00000 q^{76} +9.00000 q^{78} -1.00000 q^{79} -2.00000 q^{80} +9.00000 q^{81} -6.00000 q^{82} -12.0000 q^{83} +2.00000 q^{85} -4.00000 q^{86} -18.0000 q^{87} +5.00000 q^{88} +9.00000 q^{89} +12.0000 q^{90} -8.00000 q^{92} -12.0000 q^{93} +10.0000 q^{94} +4.00000 q^{95} -3.00000 q^{96} +4.00000 q^{97} -30.0000 q^{99} +O(q^{100})\)

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
$3$	3.00000	1.73205	0.866025	−	0.500000i	$-0.166667\pi$
$3$	3.00000	1.73205	0.866025	+	0.500000i	$0.166667\pi$
$4$	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	$0.647584\pi$
$6$	−3.00000	−1.22474
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	6.00000	2.00000
$10$	2.00000	0.632456
$11$	−5.00000	−1.50756	−0.753778	−	0.657129i	$-0.771771\pi$
$11$	−5.00000	−1.50756	−0.753778	+	0.657129i	$0.771771\pi$
$12$	3.00000	0.866025
$13$	−3.00000	−0.832050	−0.416025	−	0.909353i	$-0.636577\pi$
$13$	−3.00000	−0.832050	−0.416025	+	0.909353i	$0.636577\pi$
$14$	0	0
$15$	−6.00000	−1.54919
$16$	1.00000	0.250000
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	−6.00000	−1.41421
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	−2.00000	−0.447214
$21$	0	0
$22$	5.00000	1.06600
$23$	−8.00000	−1.66812	−0.834058	−	0.551677i	$-0.813988\pi$
$23$	−8.00000	−1.66812	−0.834058	+	0.551677i	$0.813988\pi$
$24$	−3.00000	−0.612372
$25$	−1.00000	−0.200000
$26$	3.00000	0.588348
$27$	9.00000	1.73205
$28$	0	0
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	6.00000	1.09545
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	−1.00000	−0.176777
$33$	−15.0000	−2.61116
$34$	1.00000	0.171499
$35$	0	0
$36$	6.00000	1.00000
$37$	8.00000	1.31519	0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000	1.31519	0.657596	+	0.753371i	$0.271573\pi$
$38$	2.00000	0.324443
$39$	−9.00000	−1.44115
$40$	2.00000	0.316228
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	−5.00000	−0.753778
$45$	−12.0000	−1.78885
$46$	8.00000	1.17954
$47$	−10.0000	−1.45865	−0.729325	−	0.684167i	$-0.760166\pi$
$47$	−10.0000	−1.45865	−0.729325	+	0.684167i	$0.760166\pi$
$48$	3.00000	0.433013
$49$	0	0
$50$	1.00000	0.141421
$51$	−3.00000	−0.420084
$52$	−3.00000	−0.416025
$53$	9.00000	1.23625	0.618123	−	0.786082i	$-0.287894\pi$
$53$	9.00000	1.23625	0.618123	+	0.786082i	$0.287894\pi$
$54$	−9.00000	−1.22474
$55$	10.0000	1.34840
$56$	0	0
$57$	−6.00000	−0.794719
$58$	6.00000	0.787839
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	−6.00000	−0.774597
$61$	4.00000	0.512148	0.256074	−	0.966657i	$-0.417571\pi$
$61$	4.00000	0.512148	0.256074	+	0.966657i	$0.417571\pi$
$62$	4.00000	0.508001
$63$	0	0
$64$	1.00000	0.125000
$65$	6.00000	0.744208
$66$	15.0000	1.84637
$67$	−10.0000	−1.22169	−0.610847	−	0.791748i	$-0.709171\pi$
$67$	−10.0000	−1.22169	−0.610847	+	0.791748i	$0.709171\pi$
$68$	−1.00000	−0.121268
$69$	−24.0000	−2.88926
$70$	0	0
$71$	−5.00000	−0.593391	−0.296695	−	0.954972i	$-0.595885\pi$
$71$	−5.00000	−0.593391	−0.296695	+	0.954972i	$0.595885\pi$
$72$	−6.00000	−0.707107
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	−8.00000	−0.929981
$75$	−3.00000	−0.346410
$76$	−2.00000	−0.229416
$77$	0	0
$78$	9.00000	1.01905
$79$	−1.00000	−0.112509	−0.0562544	−	0.998416i	$-0.517916\pi$
$79$	−1.00000	−0.112509	−0.0562544	+	0.998416i	$0.517916\pi$
$80$	−2.00000	−0.223607
$81$	9.00000	1.00000
$82$	−6.00000	−0.662589
$83$	−12.0000	−1.31717	−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000	−1.31717	−0.658586	+	0.752506i	$0.728845\pi$
$84$	0	0
$85$	2.00000	0.216930
$86$	−4.00000	−0.431331
$87$	−18.0000	−1.92980
$88$	5.00000	0.533002
$89$	9.00000	0.953998	0.476999	−	0.878904i	$-0.341725\pi$
$89$	9.00000	0.953998	0.476999	+	0.878904i	$0.341725\pi$
$90$	12.0000	1.26491
$91$	0	0
$92$	−8.00000	−0.834058
$93$	−12.0000	−1.24434
$94$	10.0000	1.03142
$95$	4.00000	0.410391
$96$	−3.00000	−0.306186
$97$	4.00000	0.406138	0.203069	−	0.979164i	$-0.434908\pi$
$97$	4.00000	0.406138	0.203069	+	0.979164i	$0.434908\pi$
$98$	0	0
$99$	−30.0000	−3.01511
$100$	−1.00000	−0.100000
$101$	−2.00000	−0.199007	−0.0995037	−	0.995037i	$-0.531726\pi$
$101$	−2.00000	−0.199007	−0.0995037	+	0.995037i	$0.531726\pi$
$102$	3.00000	0.297044
$103$	14.0000	1.37946	0.689730	−	0.724066i	$-0.257729\pi$
$103$	14.0000	1.37946	0.689730	+	0.724066i	$0.257729\pi$
$104$	3.00000	0.294174
$105$	0	0
$106$	−9.00000	−0.874157
$107$	−9.00000	−0.870063	−0.435031	−	0.900415i	$-0.643263\pi$
$107$	−9.00000	−0.870063	−0.435031	+	0.900415i	$0.643263\pi$
$108$	9.00000	0.866025
$109$	−16.0000	−1.53252	−0.766261	−	0.642529i	$-0.777885\pi$
$109$	−16.0000	−1.53252	−0.766261	+	0.642529i	$0.777885\pi$
$110$	−10.0000	−0.953463
$111$	24.0000	2.27798
$112$	0	0
$113$	16.0000	1.50515	0.752577	−	0.658505i	$-0.228811\pi$
$113$	16.0000	1.50515	0.752577	+	0.658505i	$0.228811\pi$
$114$	6.00000	0.561951
$115$	16.0000	1.49201
$116$	−6.00000	−0.557086
$117$	−18.0000	−1.66410
$118$	−4.00000	−0.368230
$119$	0	0
$120$	6.00000	0.547723
$121$	14.0000	1.27273
$122$	−4.00000	−0.362143
$123$	18.0000	1.62301
$124$	−4.00000	−0.359211
$125$	12.0000	1.07331
$126$	0	0
$127$	−6.00000	−0.532414	−0.266207	−	0.963916i	$-0.585770\pi$
$127$	−6.00000	−0.532414	−0.266207	+	0.963916i	$0.585770\pi$
$128$	−1.00000	−0.0883883
$129$	12.0000	1.05654
$130$	−6.00000	−0.526235
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	−15.0000	−1.30558
$133$	0	0
$134$	10.0000	0.863868
$135$	−18.0000	−1.54919
$136$	1.00000	0.0857493
$137$	−19.0000	−1.62328	−0.811640	−	0.584158i	$-0.801425\pi$
$137$	−19.0000	−1.62328	−0.811640	+	0.584158i	$0.801425\pi$
$138$	24.0000	2.04302
$139$	9.00000	0.763370	0.381685	−	0.924292i	$-0.375344\pi$
$139$	9.00000	0.763370	0.381685	+	0.924292i	$0.375344\pi$
$140$	0	0
$141$	−30.0000	−2.52646
$142$	5.00000	0.419591
$143$	15.0000	1.25436
$144$	6.00000	0.500000
$145$	12.0000	0.996546
$146$	−2.00000	−0.165521
$147$	0	0
$148$	8.00000	0.657596
$149$	−17.0000	−1.39269	−0.696347	−	0.717705i	$-0.745193\pi$
$149$	−17.0000	−1.39269	−0.696347	+	0.717705i	$0.745193\pi$
$150$	3.00000	0.244949
$151$	2.00000	0.162758	0.0813788	−	0.996683i	$-0.474068\pi$
$151$	2.00000	0.162758	0.0813788	+	0.996683i	$0.474068\pi$
$152$	2.00000	0.162221
$153$	−6.00000	−0.485071
$154$	0	0
$155$	8.00000	0.642575
$156$	−9.00000	−0.720577
$157$	11.0000	0.877896	0.438948	−	0.898513i	$-0.355351\pi$
$157$	11.0000	0.877896	0.438948	+	0.898513i	$0.355351\pi$
$158$	1.00000	0.0795557
$159$	27.0000	2.14124
$160$	2.00000	0.158114
$161$	0	0
$162$	−9.00000	−0.707107
$163$	12.0000	0.939913	0.469956	−	0.882690i	$-0.344270\pi$
$163$	12.0000	0.939913	0.469956	+	0.882690i	$0.344270\pi$
$164$	6.00000	0.468521
$165$	30.0000	2.33550
$166$	12.0000	0.931381
$167$	5.00000	0.386912	0.193456	−	0.981109i	$-0.438030\pi$
$167$	5.00000	0.386912	0.193456	+	0.981109i	$0.438030\pi$
$168$	0	0
$169$	−4.00000	−0.307692
$170$	−2.00000	−0.153393
$171$	−12.0000	−0.917663
$172$	4.00000	0.304997
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	18.0000	1.36458
$175$	0	0
$176$	−5.00000	−0.376889
$177$	12.0000	0.901975
$178$	−9.00000	−0.674579
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	−12.0000	−0.894427
$181$	−12.0000	−0.891953	−0.445976	−	0.895045i	$-0.647144\pi$
$181$	−12.0000	−0.891953	−0.445976	+	0.895045i	$0.647144\pi$
$182$	0	0
$183$	12.0000	0.887066
$184$	8.00000	0.589768
$185$	−16.0000	−1.17634
$186$	12.0000	0.879883
$187$	5.00000	0.365636
$188$	−10.0000	−0.729325
$189$	0	0
$190$	−4.00000	−0.290191
$191$	−24.0000	−1.73658	−0.868290	−	0.496058i	$-0.834780\pi$
$191$	−24.0000	−1.73658	−0.868290	+	0.496058i	$0.834780\pi$
$192$	3.00000	0.216506
$193$	−16.0000	−1.15171	−0.575853	−	0.817554i	$-0.695330\pi$
$193$	−16.0000	−1.15171	−0.575853	+	0.817554i	$0.695330\pi$
$194$	−4.00000	−0.287183
$195$	18.0000	1.28901
$196$	0	0
$197$	−2.00000	−0.142494	−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000	−0.142494	−0.0712470	+	0.997459i	$0.522698\pi$
$198$	30.0000	2.13201
$199$	−15.0000	−1.06332	−0.531661	−	0.846957i	$-0.678432\pi$
$199$	−15.0000	−1.06332	−0.531661	+	0.846957i	$0.678432\pi$
$200$	1.00000	0.0707107
$201$	−30.0000	−2.11604
$202$	2.00000	0.140720
$203$	0	0
$204$	−3.00000	−0.210042
$205$	−12.0000	−0.838116
$206$	−14.0000	−0.975426
$207$	−48.0000	−3.33623
$208$	−3.00000	−0.208013
$209$	10.0000	0.691714
$210$	0	0
$211$	20.0000	1.37686	0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000	1.37686	0.688428	+	0.725304i	$0.258301\pi$
$212$	9.00000	0.618123
$213$	−15.0000	−1.02778
$214$	9.00000	0.615227
$215$	−8.00000	−0.545595
$216$	−9.00000	−0.612372
$217$	0	0
$218$	16.0000	1.08366
$219$	6.00000	0.405442
$220$	10.0000	0.674200
$221$	3.00000	0.201802
$222$	−24.0000	−1.61077
$223$	−12.0000	−0.803579	−0.401790	−	0.915732i	$-0.631612\pi$
$223$	−12.0000	−0.803579	−0.401790	+	0.915732i	$0.631612\pi$
$224$	0	0
$225$	−6.00000	−0.400000
$226$	−16.0000	−1.06430
$227$	3.00000	0.199117	0.0995585	−	0.995032i	$-0.468257\pi$
$227$	3.00000	0.199117	0.0995585	+	0.995032i	$0.468257\pi$
$228$	−6.00000	−0.397360
$229$	2.00000	0.132164	0.0660819	−	0.997814i	$-0.478950\pi$
$229$	2.00000	0.132164	0.0660819	+	0.997814i	$0.478950\pi$
$230$	−16.0000	−1.05501
$231$	0	0
$232$	6.00000	0.393919
$233$	−8.00000	−0.524097	−0.262049	−	0.965055i	$-0.584398\pi$
$233$	−8.00000	−0.524097	−0.262049	+	0.965055i	$0.584398\pi$
$234$	18.0000	1.17670
$235$	20.0000	1.30466
$236$	4.00000	0.260378
$237$	−3.00000	−0.194871
$238$	0	0
$239$	−4.00000	−0.258738	−0.129369	−	0.991596i	$-0.541295\pi$
$239$	−4.00000	−0.258738	−0.129369	+	0.991596i	$0.541295\pi$
$240$	−6.00000	−0.387298
$241$	18.0000	1.15948	0.579741	−	0.814801i	$-0.303154\pi$
$241$	18.0000	1.15948	0.579741	+	0.814801i	$0.303154\pi$
$242$	−14.0000	−0.899954
$243$	0	0
$244$	4.00000	0.256074
$245$	0	0
$246$	−18.0000	−1.14764
$247$	6.00000	0.381771
$248$	4.00000	0.254000
$249$	−36.0000	−2.28141
$250$	−12.0000	−0.758947
$251$	2.00000	0.126239	0.0631194	−	0.998006i	$-0.479895\pi$
$251$	2.00000	0.126239	0.0631194	+	0.998006i	$0.479895\pi$
$252$	0	0
$253$	40.0000	2.51478
$254$	6.00000	0.376473
$255$	6.00000	0.375735
$256$	1.00000	0.0625000
$257$	23.0000	1.43470	0.717350	−	0.696713i	$-0.245355\pi$
$257$	23.0000	1.43470	0.717350	+	0.696713i	$0.245355\pi$
$258$	−12.0000	−0.747087
$259$	0	0
$260$	6.00000	0.372104
$261$	−36.0000	−2.22834
$262$	−12.0000	−0.741362
$263$	−18.0000	−1.10993	−0.554964	−	0.831875i	$-0.687268\pi$
$263$	−18.0000	−1.10993	−0.554964	+	0.831875i	$0.687268\pi$
$264$	15.0000	0.923186
$265$	−18.0000	−1.10573
$266$	0	0
$267$	27.0000	1.65237
$268$	−10.0000	−0.610847
$269$	−14.0000	−0.853595	−0.426798	−	0.904347i	$-0.640358\pi$
$269$	−14.0000	−0.853595	−0.426798	+	0.904347i	$0.640358\pi$
$270$	18.0000	1.09545
$271$	−4.00000	−0.242983	−0.121491	−	0.992592i	$-0.538768\pi$
$271$	−4.00000	−0.242983	−0.121491	+	0.992592i	$0.538768\pi$
$272$	−1.00000	−0.0606339
$273$	0	0
$274$	19.0000	1.14783
$275$	5.00000	0.301511
$276$	−24.0000	−1.44463
$277$	−4.00000	−0.240337	−0.120168	−	0.992754i	$-0.538343\pi$
$277$	−4.00000	−0.240337	−0.120168	+	0.992754i	$0.538343\pi$
$278$	−9.00000	−0.539784
$279$	−24.0000	−1.43684
$280$	0	0
$281$	−27.0000	−1.61068	−0.805342	−	0.592810i	$-0.798019\pi$
$281$	−27.0000	−1.61068	−0.805342	+	0.592810i	$0.798019\pi$
$282$	30.0000	1.78647
$283$	1.00000	0.0594438	0.0297219	−	0.999558i	$-0.490538\pi$
$283$	1.00000	0.0594438	0.0297219	+	0.999558i	$0.490538\pi$
$284$	−5.00000	−0.296695
$285$	12.0000	0.710819
$286$	−15.0000	−0.886969
$287$	0	0
$288$	−6.00000	−0.353553
$289$	1.00000	0.0588235
$290$	−12.0000	−0.704664
$291$	12.0000	0.703452
$292$	2.00000	0.117041
$293$	−29.0000	−1.69420	−0.847099	−	0.531435i	$-0.821653\pi$
$293$	−29.0000	−1.69420	−0.847099	+	0.531435i	$0.821653\pi$
$294$	0	0
$295$	−8.00000	−0.465778
$296$	−8.00000	−0.464991
$297$	−45.0000	−2.61116
$298$	17.0000	0.984784
$299$	24.0000	1.38796
$300$	−3.00000	−0.173205
$301$	0	0
$302$	−2.00000	−0.115087
$303$	−6.00000	−0.344691
$304$	−2.00000	−0.114708
$305$	−8.00000	−0.458079
$306$	6.00000	0.342997
$307$	−26.0000	−1.48390	−0.741949	−	0.670456i	$-0.766098\pi$
$307$	−26.0000	−1.48390	−0.741949	+	0.670456i	$0.766098\pi$
$308$	0	0
$309$	42.0000	2.38930
$310$	−8.00000	−0.454369
$311$	15.0000	0.850572	0.425286	−	0.905059i	$-0.360174\pi$
$311$	15.0000	0.850572	0.425286	+	0.905059i	$0.360174\pi$
$312$	9.00000	0.509525
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	−11.0000	−0.620766
$315$	0	0
$316$	−1.00000	−0.0562544
$317$	12.0000	0.673987	0.336994	−	0.941507i	$-0.390590\pi$
$317$	12.0000	0.673987	0.336994	+	0.941507i	$0.390590\pi$
$318$	−27.0000	−1.51408
$319$	30.0000	1.67968
$320$	−2.00000	−0.111803
$321$	−27.0000	−1.50699
$322$	0	0
$323$	2.00000	0.111283
$324$	9.00000	0.500000
$325$	3.00000	0.166410
$326$	−12.0000	−0.664619
$327$	−48.0000	−2.65441
$328$	−6.00000	−0.331295
$329$	0	0
$330$	−30.0000	−1.65145
$331$	−8.00000	−0.439720	−0.219860	−	0.975531i	$-0.570560\pi$
$331$	−8.00000	−0.439720	−0.219860	+	0.975531i	$0.570560\pi$
$332$	−12.0000	−0.658586
$333$	48.0000	2.63038
$334$	−5.00000	−0.273588
$335$	20.0000	1.09272
$336$	0	0
$337$	−8.00000	−0.435788	−0.217894	−	0.975972i	$-0.569919\pi$
$337$	−8.00000	−0.435788	−0.217894	+	0.975972i	$0.569919\pi$
$338$	4.00000	0.217571
$339$	48.0000	2.60700
$340$	2.00000	0.108465
$341$	20.0000	1.08306
$342$	12.0000	0.648886
$343$	0	0
$344$	−4.00000	−0.215666
$345$	48.0000	2.58423
$346$	0	0
$347$	20.0000	1.07366	0.536828	−	0.843692i	$-0.319622\pi$
$347$	20.0000	1.07366	0.536828	+	0.843692i	$0.319622\pi$
$348$	−18.0000	−0.964901
$349$	−18.0000	−0.963518	−0.481759	−	0.876304i	$-0.660002\pi$
$349$	−18.0000	−0.963518	−0.481759	+	0.876304i	$0.660002\pi$
$350$	0	0
$351$	−27.0000	−1.44115
$352$	5.00000	0.266501
$353$	−1.00000	−0.0532246	−0.0266123	−	0.999646i	$-0.508472\pi$
$353$	−1.00000	−0.0532246	−0.0266123	+	0.999646i	$0.508472\pi$
$354$	−12.0000	−0.637793
$355$	10.0000	0.530745
$356$	9.00000	0.476999
$357$	0	0
$358$	−4.00000	−0.211407
$359$	20.0000	1.05556	0.527780	−	0.849381i	$-0.323025\pi$
$359$	20.0000	1.05556	0.527780	+	0.849381i	$0.323025\pi$
$360$	12.0000	0.632456
$361$	−15.0000	−0.789474
$362$	12.0000	0.630706
$363$	42.0000	2.20443
$364$	0	0
$365$	−4.00000	−0.209370
$366$	−12.0000	−0.627250
$367$	15.0000	0.782994	0.391497	−	0.920179i	$-0.371957\pi$
$367$	15.0000	0.782994	0.391497	+	0.920179i	$0.371957\pi$
$368$	−8.00000	−0.417029
$369$	36.0000	1.87409
$370$	16.0000	0.831800
$371$	0	0
$372$	−12.0000	−0.622171
$373$	−11.0000	−0.569558	−0.284779	−	0.958593i	$-0.591920\pi$
$373$	−11.0000	−0.569558	−0.284779	+	0.958593i	$0.591920\pi$
$374$	−5.00000	−0.258544
$375$	36.0000	1.85903
$376$	10.0000	0.515711
$377$	18.0000	0.927047
$378$	0	0
$379$	−5.00000	−0.256833	−0.128416	−	0.991720i	$-0.540989\pi$
$379$	−5.00000	−0.256833	−0.128416	+	0.991720i	$0.540989\pi$
$380$	4.00000	0.205196
$381$	−18.0000	−0.922168
$382$	24.0000	1.22795
$383$	−20.0000	−1.02195	−0.510976	−	0.859595i	$-0.670716\pi$
$383$	−20.0000	−1.02195	−0.510976	+	0.859595i	$0.670716\pi$
$384$	−3.00000	−0.153093
$385$	0	0
$386$	16.0000	0.814379
$387$	24.0000	1.21999
$388$	4.00000	0.203069
$389$	15.0000	0.760530	0.380265	−	0.924878i	$-0.375833\pi$
$389$	15.0000	0.760530	0.380265	+	0.924878i	$0.375833\pi$
$390$	−18.0000	−0.911465
$391$	8.00000	0.404577
$392$	0	0
$393$	36.0000	1.81596
$394$	2.00000	0.100759
$395$	2.00000	0.100631
$396$	−30.0000	−1.50756
$397$	32.0000	1.60603	0.803017	−	0.595956i	$-0.203227\pi$
$397$	32.0000	1.60603	0.803017	+	0.595956i	$0.203227\pi$
$398$	15.0000	0.751882
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	16.0000	0.799002	0.399501	−	0.916733i	$-0.369183\pi$
$401$	16.0000	0.799002	0.399501	+	0.916733i	$0.369183\pi$
$402$	30.0000	1.49626
$403$	12.0000	0.597763
$404$	−2.00000	−0.0995037
$405$	−18.0000	−0.894427
$406$	0	0
$407$	−40.0000	−1.98273
$408$	3.00000	0.148522
$409$	23.0000	1.13728	0.568638	−	0.822588i	$-0.307470\pi$
$409$	23.0000	1.13728	0.568638	+	0.822588i	$0.307470\pi$
$410$	12.0000	0.592638
$411$	−57.0000	−2.81160
$412$	14.0000	0.689730
$413$	0	0
$414$	48.0000	2.35907
$415$	24.0000	1.17811
$416$	3.00000	0.147087
$417$	27.0000	1.32220
$418$	−10.0000	−0.489116
$419$	39.0000	1.90527	0.952637	−	0.304109i	$-0.0983586\pi$
$419$	39.0000	1.90527	0.952637	+	0.304109i	$0.0983586\pi$
$420$	0	0
$421$	30.0000	1.46211	0.731055	−	0.682318i	$-0.239028\pi$
$421$	30.0000	1.46211	0.731055	+	0.682318i	$0.239028\pi$
$422$	−20.0000	−0.973585
$423$	−60.0000	−2.91730
$424$	−9.00000	−0.437079
$425$	1.00000	0.0485071
$426$	15.0000	0.726752
$427$	0	0
$428$	−9.00000	−0.435031
$429$	45.0000	2.17262
$430$	8.00000	0.385794
$431$	5.00000	0.240842	0.120421	−	0.992723i	$-0.461576\pi$
$431$	5.00000	0.240842	0.120421	+	0.992723i	$0.461576\pi$
$432$	9.00000	0.433013
$433$	30.0000	1.44171	0.720854	−	0.693087i	$-0.243750\pi$
$433$	30.0000	1.44171	0.720854	+	0.693087i	$0.243750\pi$
$434$	0	0
$435$	36.0000	1.72607
$436$	−16.0000	−0.766261
$437$	16.0000	0.765384
$438$	−6.00000	−0.286691
$439$	7.00000	0.334092	0.167046	−	0.985949i	$-0.446577\pi$
$439$	7.00000	0.334092	0.167046	+	0.985949i	$0.446577\pi$
$440$	−10.0000	−0.476731
$441$	0	0
$442$	−3.00000	−0.142695
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	24.0000	1.13899
$445$	−18.0000	−0.853282
$446$	12.0000	0.568216
$447$	−51.0000	−2.41222
$448$	0	0
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	6.00000	0.282843
$451$	−30.0000	−1.41264
$452$	16.0000	0.752577
$453$	6.00000	0.281905
$454$	−3.00000	−0.140797
$455$	0	0
$456$	6.00000	0.280976
$457$	−10.0000	−0.467780	−0.233890	−	0.972263i	$-0.575146\pi$
$457$	−10.0000	−0.467780	−0.233890	+	0.972263i	$0.575146\pi$
$458$	−2.00000	−0.0934539
$459$	−9.00000	−0.420084
$460$	16.0000	0.746004
$461$	−11.0000	−0.512321	−0.256161	−	0.966634i	$-0.582458\pi$
$461$	−11.0000	−0.512321	−0.256161	+	0.966634i	$0.582458\pi$
$462$	0	0
$463$	−34.0000	−1.58011	−0.790057	−	0.613033i	$-0.789949\pi$
$463$	−34.0000	−1.58011	−0.790057	+	0.613033i	$0.789949\pi$
$464$	−6.00000	−0.278543
$465$	24.0000	1.11297
$466$	8.00000	0.370593
$467$	−6.00000	−0.277647	−0.138823	−	0.990317i	$-0.544332\pi$
$467$	−6.00000	−0.277647	−0.138823	+	0.990317i	$0.544332\pi$
$468$	−18.0000	−0.832050
$469$	0	0
$470$	−20.0000	−0.922531
$471$	33.0000	1.52056
$472$	−4.00000	−0.184115
$473$	−20.0000	−0.919601
$474$	3.00000	0.137795
$475$	2.00000	0.0917663
$476$	0	0
$477$	54.0000	2.47249
$478$	4.00000	0.182956
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	6.00000	0.273861
$481$	−24.0000	−1.09431
$482$	−18.0000	−0.819878
$483$	0	0
$484$	14.0000	0.636364
$485$	−8.00000	−0.363261
$486$	0	0
$487$	11.0000	0.498458	0.249229	−	0.968445i	$-0.419823\pi$
$487$	11.0000	0.498458	0.249229	+	0.968445i	$0.419823\pi$
$488$	−4.00000	−0.181071
$489$	36.0000	1.62798
$490$	0	0
$491$	−38.0000	−1.71492	−0.857458	−	0.514554i	$-0.827958\pi$
$491$	−38.0000	−1.71492	−0.857458	+	0.514554i	$0.827958\pi$
$492$	18.0000	0.811503
$493$	6.00000	0.270226
$494$	−6.00000	−0.269953
$495$	60.0000	2.69680
$496$	−4.00000	−0.179605
$497$	0	0
$498$	36.0000	1.61320
$499$	−11.0000	−0.492428	−0.246214	−	0.969216i	$-0.579187\pi$
$499$	−11.0000	−0.492428	−0.246214	+	0.969216i	$0.579187\pi$
$500$	12.0000	0.536656
$501$	15.0000	0.670151
$502$	−2.00000	−0.0892644
$503$	9.00000	0.401290	0.200645	−	0.979664i	$-0.435696\pi$
$503$	9.00000	0.401290	0.200645	+	0.979664i	$0.435696\pi$
$504$	0	0
$505$	4.00000	0.177998
$506$	−40.0000	−1.77822
$507$	−12.0000	−0.532939
$508$	−6.00000	−0.266207
$509$	26.0000	1.15243	0.576215	−	0.817298i	$-0.304529\pi$
$509$	26.0000	1.15243	0.576215	+	0.817298i	$0.304529\pi$
$510$	−6.00000	−0.265684
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	−18.0000	−0.794719
$514$	−23.0000	−1.01449
$515$	−28.0000	−1.23383
$516$	12.0000	0.528271
$517$	50.0000	2.19900
$518$	0	0
$519$	0	0
$520$	−6.00000	−0.263117
$521$	−30.0000	−1.31432	−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000	−1.31432	−0.657162	+	0.753749i	$0.728243\pi$
$522$	36.0000	1.57568
$523$	16.0000	0.699631	0.349816	−	0.936819i	$-0.386244\pi$
$523$	16.0000	0.699631	0.349816	+	0.936819i	$0.386244\pi$
$524$	12.0000	0.524222
$525$	0	0
$526$	18.0000	0.784837
$527$	4.00000	0.174243
$528$	−15.0000	−0.652791
$529$	41.0000	1.78261
$530$	18.0000	0.781870
$531$	24.0000	1.04151
$532$	0	0
$533$	−18.0000	−0.779667
$534$	−27.0000	−1.16840
$535$	18.0000	0.778208
$536$	10.0000	0.431934
$537$	12.0000	0.517838
$538$	14.0000	0.603583
$539$	0	0
$540$	−18.0000	−0.774597
$541$	−8.00000	−0.343947	−0.171973	−	0.985102i	$-0.555014\pi$
$541$	−8.00000	−0.343947	−0.171973	+	0.985102i	$0.555014\pi$
$542$	4.00000	0.171815
$543$	−36.0000	−1.54491
$544$	1.00000	0.0428746
$545$	32.0000	1.37073
$546$	0	0
$547$	5.00000	0.213785	0.106892	−	0.994271i	$-0.465910\pi$
$547$	5.00000	0.213785	0.106892	+	0.994271i	$0.465910\pi$
$548$	−19.0000	−0.811640
$549$	24.0000	1.02430
$550$	−5.00000	−0.213201
$551$	12.0000	0.511217
$552$	24.0000	1.02151
$553$	0	0
$554$	4.00000	0.169944
$555$	−48.0000	−2.03749
$556$	9.00000	0.381685
$557$	21.0000	0.889799	0.444899	−	0.895581i	$-0.353239\pi$
$557$	21.0000	0.889799	0.444899	+	0.895581i	$0.353239\pi$
$558$	24.0000	1.01600
$559$	−12.0000	−0.507546
$560$	0	0
$561$	15.0000	0.633300
$562$	27.0000	1.13893
$563$	16.0000	0.674320	0.337160	−	0.941447i	$-0.390534\pi$
$563$	16.0000	0.674320	0.337160	+	0.941447i	$0.390534\pi$
$564$	−30.0000	−1.26323
$565$	−32.0000	−1.34625
$566$	−1.00000	−0.0420331
$567$	0	0
$568$	5.00000	0.209795
$569$	−3.00000	−0.125767	−0.0628833	−	0.998021i	$-0.520030\pi$
$569$	−3.00000	−0.125767	−0.0628833	+	0.998021i	$0.520030\pi$
$570$	−12.0000	−0.502625
$571$	−40.0000	−1.67395	−0.836974	−	0.547243i	$-0.815677\pi$
$571$	−40.0000	−1.67395	−0.836974	+	0.547243i	$0.815677\pi$
$572$	15.0000	0.627182
$573$	−72.0000	−3.00784
$574$	0	0
$575$	8.00000	0.333623
$576$	6.00000	0.250000
$577$	−43.0000	−1.79011	−0.895057	−	0.445952i	$-0.852865\pi$
$577$	−43.0000	−1.79011	−0.895057	+	0.445952i	$0.852865\pi$
$578$	−1.00000	−0.0415945
$579$	−48.0000	−1.99481
$580$	12.0000	0.498273
$581$	0	0
$582$	−12.0000	−0.497416
$583$	−45.0000	−1.86371
$584$	−2.00000	−0.0827606
$585$	36.0000	1.48842
$586$	29.0000	1.19798
$587$	2.00000	0.0825488	0.0412744	−	0.999148i	$-0.486858\pi$
$587$	2.00000	0.0825488	0.0412744	+	0.999148i	$0.486858\pi$
$588$	0	0
$589$	8.00000	0.329634
$590$	8.00000	0.329355
$591$	−6.00000	−0.246807
$592$	8.00000	0.328798
$593$	9.00000	0.369586	0.184793	−	0.982777i	$-0.440839\pi$
$593$	9.00000	0.369586	0.184793	+	0.982777i	$0.440839\pi$
$594$	45.0000	1.84637
$595$	0	0
$596$	−17.0000	−0.696347
$597$	−45.0000	−1.84173
$598$	−24.0000	−0.981433
$599$	−36.0000	−1.47092	−0.735460	−	0.677568i	$-0.763034\pi$
$599$	−36.0000	−1.47092	−0.735460	+	0.677568i	$0.763034\pi$
$600$	3.00000	0.122474
$601$	−46.0000	−1.87638	−0.938190	−	0.346122i	$-0.887498\pi$
$601$	−46.0000	−1.87638	−0.938190	+	0.346122i	$0.887498\pi$
$602$	0	0
$603$	−60.0000	−2.44339
$604$	2.00000	0.0813788
$605$	−28.0000	−1.13836
$606$	6.00000	0.243733
$607$	−43.0000	−1.74532	−0.872658	−	0.488332i	$-0.837606\pi$
$607$	−43.0000	−1.74532	−0.872658	+	0.488332i	$0.837606\pi$
$608$	2.00000	0.0811107
$609$	0	0
$610$	8.00000	0.323911
$611$	30.0000	1.21367
$612$	−6.00000	−0.242536
$613$	35.0000	1.41364	0.706818	−	0.707395i	$-0.250130\pi$
$613$	35.0000	1.41364	0.706818	+	0.707395i	$0.250130\pi$
$614$	26.0000	1.04927
$615$	−36.0000	−1.45166
$616$	0	0
$617$	−14.0000	−0.563619	−0.281809	−	0.959470i	$-0.590935\pi$
$617$	−14.0000	−0.563619	−0.281809	+	0.959470i	$0.590935\pi$
$618$	−42.0000	−1.68949
$619$	25.0000	1.00483	0.502417	−	0.864625i	$-0.332444\pi$
$619$	25.0000	1.00483	0.502417	+	0.864625i	$0.332444\pi$
$620$	8.00000	0.321288
$621$	−72.0000	−2.88926
$622$	−15.0000	−0.601445
$623$	0	0
$624$	−9.00000	−0.360288
$625$	−19.0000	−0.760000
$626$	−6.00000	−0.239808
$627$	30.0000	1.19808
$628$	11.0000	0.438948
$629$	−8.00000	−0.318981
$630$	0	0
$631$	32.0000	1.27390	0.636950	−	0.770905i	$-0.280196\pi$
$631$	32.0000	1.27390	0.636950	+	0.770905i	$0.280196\pi$
$632$	1.00000	0.0397779
$633$	60.0000	2.38479
$634$	−12.0000	−0.476581
$635$	12.0000	0.476205
$636$	27.0000	1.07062
$637$	0	0
$638$	−30.0000	−1.18771
$639$	−30.0000	−1.18678
$640$	2.00000	0.0790569
$641$	34.0000	1.34292	0.671460	−	0.741041i	$-0.265668\pi$
$641$	34.0000	1.34292	0.671460	+	0.741041i	$0.265668\pi$
$642$	27.0000	1.06561
$643$	21.0000	0.828159	0.414080	−	0.910241i	$-0.364104\pi$
$643$	21.0000	0.828159	0.414080	+	0.910241i	$0.364104\pi$
$644$	0	0
$645$	−24.0000	−0.944999
$646$	−2.00000	−0.0786889
$647$	4.00000	0.157256	0.0786281	−	0.996904i	$-0.474946\pi$
$647$	4.00000	0.157256	0.0786281	+	0.996904i	$0.474946\pi$
$648$	−9.00000	−0.353553
$649$	−20.0000	−0.785069
$650$	−3.00000	−0.117670
$651$	0	0
$652$	12.0000	0.469956
$653$	−8.00000	−0.313064	−0.156532	−	0.987673i	$-0.550031\pi$
$653$	−8.00000	−0.313064	−0.156532	+	0.987673i	$0.550031\pi$
$654$	48.0000	1.87695
$655$	−24.0000	−0.937758
$656$	6.00000	0.234261
$657$	12.0000	0.468165
$658$	0	0
$659$	−16.0000	−0.623272	−0.311636	−	0.950202i	$-0.600877\pi$
$659$	−16.0000	−0.623272	−0.311636	+	0.950202i	$0.600877\pi$
$660$	30.0000	1.16775
$661$	−18.0000	−0.700119	−0.350059	−	0.936727i	$-0.613839\pi$
$661$	−18.0000	−0.700119	−0.350059	+	0.936727i	$0.613839\pi$
$662$	8.00000	0.310929
$663$	9.00000	0.349531
$664$	12.0000	0.465690
$665$	0	0
$666$	−48.0000	−1.85996
$667$	48.0000	1.85857
$668$	5.00000	0.193456
$669$	−36.0000	−1.39184
$670$	−20.0000	−0.772667
$671$	−20.0000	−0.772091
$672$	0	0
$673$	10.0000	0.385472	0.192736	−	0.981251i	$-0.438264\pi$
$673$	10.0000	0.385472	0.192736	+	0.981251i	$0.438264\pi$
$674$	8.00000	0.308148
$675$	−9.00000	−0.346410
$676$	−4.00000	−0.153846
$677$	−42.0000	−1.61419	−0.807096	−	0.590421i	$-0.798962\pi$
$677$	−42.0000	−1.61419	−0.807096	+	0.590421i	$0.798962\pi$
$678$	−48.0000	−1.84343
$679$	0	0
$680$	−2.00000	−0.0766965
$681$	9.00000	0.344881
$682$	−20.0000	−0.765840
$683$	21.0000	0.803543	0.401771	−	0.915740i	$-0.368395\pi$
$683$	21.0000	0.803543	0.401771	+	0.915740i	$0.368395\pi$
$684$	−12.0000	−0.458831
$685$	38.0000	1.45191
$686$	0	0
$687$	6.00000	0.228914
$688$	4.00000	0.152499
$689$	−27.0000	−1.02862
$690$	−48.0000	−1.82733
$691$	4.00000	0.152167	0.0760836	−	0.997101i	$-0.475758\pi$
$691$	4.00000	0.152167	0.0760836	+	0.997101i	$0.475758\pi$
$692$	0	0
$693$	0	0
$694$	−20.0000	−0.759190
$695$	−18.0000	−0.682779
$696$	18.0000	0.682288
$697$	−6.00000	−0.227266
$698$	18.0000	0.681310
$699$	−24.0000	−0.907763
$700$	0	0
$701$	33.0000	1.24639	0.623196	−	0.782065i	$-0.285834\pi$
$701$	33.0000	1.24639	0.623196	+	0.782065i	$0.285834\pi$
$702$	27.0000	1.01905
$703$	−16.0000	−0.603451
$704$	−5.00000	−0.188445
$705$	60.0000	2.25973
$706$	1.00000	0.0376355
$707$	0	0
$708$	12.0000	0.450988
$709$	22.0000	0.826227	0.413114	−	0.910679i	$-0.364441\pi$
$709$	22.0000	0.826227	0.413114	+	0.910679i	$0.364441\pi$
$710$	−10.0000	−0.375293
$711$	−6.00000	−0.225018
$712$	−9.00000	−0.337289
$713$	32.0000	1.19841
$714$	0	0
$715$	−30.0000	−1.12194
$716$	4.00000	0.149487
$717$	−12.0000	−0.448148
$718$	−20.0000	−0.746393
$719$	−39.0000	−1.45445	−0.727227	−	0.686397i	$-0.759191\pi$
$719$	−39.0000	−1.45445	−0.727227	+	0.686397i	$0.759191\pi$
$720$	−12.0000	−0.447214
$721$	0	0
$722$	15.0000	0.558242
$723$	54.0000	2.00828
$724$	−12.0000	−0.445976
$725$	6.00000	0.222834
$726$	−42.0000	−1.55877
$727$	−32.0000	−1.18681	−0.593407	−	0.804902i	$-0.702218\pi$
$727$	−32.0000	−1.18681	−0.593407	+	0.804902i	$0.702218\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	4.00000	0.148047
$731$	−4.00000	−0.147945
$732$	12.0000	0.443533
$733$	17.0000	0.627909	0.313955	−	0.949438i	$-0.398346\pi$
$733$	17.0000	0.627909	0.313955	+	0.949438i	$0.398346\pi$
$734$	−15.0000	−0.553660
$735$	0	0
$736$	8.00000	0.294884
$737$	50.0000	1.84177
$738$	−36.0000	−1.32518
$739$	−16.0000	−0.588570	−0.294285	−	0.955718i	$-0.595081\pi$
$739$	−16.0000	−0.588570	−0.294285	+	0.955718i	$0.595081\pi$
$740$	−16.0000	−0.588172
$741$	18.0000	0.661247
$742$	0	0
$743$	−15.0000	−0.550297	−0.275148	−	0.961402i	$-0.588727\pi$
$743$	−15.0000	−0.550297	−0.275148	+	0.961402i	$0.588727\pi$
$744$	12.0000	0.439941
$745$	34.0000	1.24566
$746$	11.0000	0.402739
$747$	−72.0000	−2.63434
$748$	5.00000	0.182818
$749$	0	0
$750$	−36.0000	−1.31453
$751$	23.0000	0.839282	0.419641	−	0.907690i	$-0.362156\pi$
$751$	23.0000	0.839282	0.419641	+	0.907690i	$0.362156\pi$
$752$	−10.0000	−0.364662
$753$	6.00000	0.218652
$754$	−18.0000	−0.655521
$755$	−4.00000	−0.145575
$756$	0	0
$757$	23.0000	0.835949	0.417975	−	0.908459i	$-0.362740\pi$
$757$	23.0000	0.835949	0.417975	+	0.908459i	$0.362740\pi$
$758$	5.00000	0.181608
$759$	120.000	4.35572
$760$	−4.00000	−0.145095
$761$	45.0000	1.63125	0.815624	−	0.578582i	$-0.196394\pi$
$761$	45.0000	1.63125	0.815624	+	0.578582i	$0.196394\pi$
$762$	18.0000	0.652071
$763$	0	0
$764$	−24.0000	−0.868290
$765$	12.0000	0.433861
$766$	20.0000	0.722629
$767$	−12.0000	−0.433295
$768$	3.00000	0.108253
$769$	−23.0000	−0.829401	−0.414701	−	0.909958i	$-0.636114\pi$
$769$	−23.0000	−0.829401	−0.414701	+	0.909958i	$0.636114\pi$
$770$	0	0
$771$	69.0000	2.48497
$772$	−16.0000	−0.575853
$773$	3.00000	0.107903	0.0539513	−	0.998544i	$-0.482818\pi$
$773$	3.00000	0.107903	0.0539513	+	0.998544i	$0.482818\pi$
$774$	−24.0000	−0.862662
$775$	4.00000	0.143684
$776$	−4.00000	−0.143592
$777$	0	0
$778$	−15.0000	−0.537776
$779$	−12.0000	−0.429945
$780$	18.0000	0.644503
$781$	25.0000	0.894570
$782$	−8.00000	−0.286079
$783$	−54.0000	−1.92980
$784$	0	0
$785$	−22.0000	−0.785214
$786$	−36.0000	−1.28408
$787$	−52.0000	−1.85360	−0.926800	−	0.375555i	$-0.877452\pi$
$787$	−52.0000	−1.85360	−0.926800	+	0.375555i	$0.877452\pi$
$788$	−2.00000	−0.0712470
$789$	−54.0000	−1.92245
$790$	−2.00000	−0.0711568
$791$	0	0
$792$	30.0000	1.06600
$793$	−12.0000	−0.426132
$794$	−32.0000	−1.13564
$795$	−54.0000	−1.91518
$796$	−15.0000	−0.531661
$797$	17.0000	0.602171	0.301085	−	0.953597i	$-0.402651\pi$
$797$	17.0000	0.602171	0.301085	+	0.953597i	$0.402651\pi$
$798$	0	0
$799$	10.0000	0.353775
$800$	1.00000	0.0353553
$801$	54.0000	1.90800
$802$	−16.0000	−0.564980
$803$	−10.0000	−0.352892
$804$	−30.0000	−1.05802
$805$	0	0
$806$	−12.0000	−0.422682
$807$	−42.0000	−1.47847
$808$	2.00000	0.0703598
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	18.0000	0.632456
$811$	−37.0000	−1.29925	−0.649623	−	0.760257i	$-0.725073\pi$
$811$	−37.0000	−1.29925	−0.649623	+	0.760257i	$0.725073\pi$
$812$	0	0
$813$	−12.0000	−0.420858
$814$	40.0000	1.40200
$815$	−24.0000	−0.840683
$816$	−3.00000	−0.105021
$817$	−8.00000	−0.279885
$818$	−23.0000	−0.804176
$819$	0	0
$820$	−12.0000	−0.419058
$821$	0	0		−	1.00000i	$-0.5\pi$
$821$	0	0			1.00000i	$0.5\pi$
$822$	57.0000	1.98810
$823$	29.0000	1.01088	0.505438	−	0.862863i	$-0.331331\pi$
$823$	29.0000	1.01088	0.505438	+	0.862863i	$0.331331\pi$
$824$	−14.0000	−0.487713
$825$	15.0000	0.522233
$826$	0	0
$827$	3.00000	0.104320	0.0521601	−	0.998639i	$-0.483389\pi$
$827$	3.00000	0.104320	0.0521601	+	0.998639i	$0.483389\pi$
$828$	−48.0000	−1.66812
$829$	−5.00000	−0.173657	−0.0868286	−	0.996223i	$-0.527673\pi$
$829$	−5.00000	−0.173657	−0.0868286	+	0.996223i	$0.527673\pi$
$830$	−24.0000	−0.833052
$831$	−12.0000	−0.416275
$832$	−3.00000	−0.104006
$833$	0	0
$834$	−27.0000	−0.934934
$835$	−10.0000	−0.346064
$836$	10.0000	0.345857
$837$	−36.0000	−1.24434
$838$	−39.0000	−1.34723
$839$	36.0000	1.24286	0.621429	−	0.783470i	$-0.286552\pi$
$839$	36.0000	1.24286	0.621429	+	0.783470i	$0.286552\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	−30.0000	−1.03387
$843$	−81.0000	−2.78979
$844$	20.0000	0.688428
$845$	8.00000	0.275208
$846$	60.0000	2.06284
$847$	0	0
$848$	9.00000	0.309061
$849$	3.00000	0.102960
$850$	−1.00000	−0.0342997
$851$	−64.0000	−2.19389
$852$	−15.0000	−0.513892
$853$	−10.0000	−0.342393	−0.171197	−	0.985237i	$-0.554763\pi$
$853$	−10.0000	−0.342393	−0.171197	+	0.985237i	$0.554763\pi$
$854$	0	0
$855$	24.0000	0.820783
$856$	9.00000	0.307614
$857$	28.0000	0.956462	0.478231	−	0.878234i	$-0.341278\pi$
$857$	28.0000	0.956462	0.478231	+	0.878234i	$0.341278\pi$
$858$	−45.0000	−1.53627
$859$	34.0000	1.16007	0.580033	−	0.814593i	$-0.303040\pi$
$859$	34.0000	1.16007	0.580033	+	0.814593i	$0.303040\pi$
$860$	−8.00000	−0.272798
$861$	0	0
$862$	−5.00000	−0.170301
$863$	−16.0000	−0.544646	−0.272323	−	0.962206i	$-0.587792\pi$
$863$	−16.0000	−0.544646	−0.272323	+	0.962206i	$0.587792\pi$
$864$	−9.00000	−0.306186
$865$	0	0
$866$	−30.0000	−1.01944
$867$	3.00000	0.101885
$868$	0	0
$869$	5.00000	0.169613
$870$	−36.0000	−1.22051
$871$	30.0000	1.01651
$872$	16.0000	0.541828
$873$	24.0000	0.812277
$874$	−16.0000	−0.541208
$875$	0	0
$876$	6.00000	0.202721
$877$	−6.00000	−0.202606	−0.101303	−	0.994856i	$-0.532301\pi$
$877$	−6.00000	−0.202606	−0.101303	+	0.994856i	$0.532301\pi$
$878$	−7.00000	−0.236239
$879$	−87.0000	−2.93444
$880$	10.0000	0.337100
$881$	−36.0000	−1.21287	−0.606435	−	0.795133i	$-0.707401\pi$
$881$	−36.0000	−1.21287	−0.606435	+	0.795133i	$0.707401\pi$
$882$	0	0
$883$	−20.0000	−0.673054	−0.336527	−	0.941674i	$-0.609252\pi$
$883$	−20.0000	−0.673054	−0.336527	+	0.941674i	$0.609252\pi$
$884$	3.00000	0.100901
$885$	−24.0000	−0.806751
$886$	0	0
$887$	−35.0000	−1.17518	−0.587592	−	0.809157i	$-0.699924\pi$
$887$	−35.0000	−1.17518	−0.587592	+	0.809157i	$0.699924\pi$
$888$	−24.0000	−0.805387
$889$	0	0
$890$	18.0000	0.603361
$891$	−45.0000	−1.50756
$892$	−12.0000	−0.401790
$893$	20.0000	0.669274
$894$	51.0000	1.70570
$895$	−8.00000	−0.267411
$896$	0	0
$897$	72.0000	2.40401
$898$	6.00000	0.200223
$899$	24.0000	0.800445
$900$	−6.00000	−0.200000
$901$	−9.00000	−0.299833
$902$	30.0000	0.998891
$903$	0	0
$904$	−16.0000	−0.532152
$905$	24.0000	0.797787
$906$	−6.00000	−0.199337
$907$	48.0000	1.59381	0.796907	−	0.604102i	$-0.206468\pi$
$907$	48.0000	1.59381	0.796907	+	0.604102i	$0.206468\pi$
$908$	3.00000	0.0995585
$909$	−12.0000	−0.398015
$910$	0	0
$911$	−20.0000	−0.662630	−0.331315	−	0.943520i	$-0.607492\pi$
$911$	−20.0000	−0.662630	−0.331315	+	0.943520i	$0.607492\pi$
$912$	−6.00000	−0.198680
$913$	60.0000	1.98571
$914$	10.0000	0.330771
$915$	−24.0000	−0.793416
$916$	2.00000	0.0660819
$917$	0	0
$918$	9.00000	0.297044
$919$	14.0000	0.461817	0.230909	−	0.972975i	$-0.425830\pi$
$919$	14.0000	0.461817	0.230909	+	0.972975i	$0.425830\pi$
$920$	−16.0000	−0.527504
$921$	−78.0000	−2.57019
$922$	11.0000	0.362266
$923$	15.0000	0.493731
$924$	0	0
$925$	−8.00000	−0.263038
$926$	34.0000	1.11731
$927$	84.0000	2.75892
$928$	6.00000	0.196960
$929$	−2.00000	−0.0656179	−0.0328089	−	0.999462i	$-0.510445\pi$
$929$	−2.00000	−0.0656179	−0.0328089	+	0.999462i	$0.510445\pi$
$930$	−24.0000	−0.786991
$931$	0	0
$932$	−8.00000	−0.262049
$933$	45.0000	1.47323
$934$	6.00000	0.196326
$935$	−10.0000	−0.327035
$936$	18.0000	0.588348
$937$	2.00000	0.0653372	0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000	0.0653372	0.0326686	+	0.999466i	$0.489599\pi$
$938$	0	0
$939$	18.0000	0.587408
$940$	20.0000	0.652328
$941$	−30.0000	−0.977972	−0.488986	−	0.872292i	$-0.662633\pi$
$941$	−30.0000	−0.977972	−0.488986	+	0.872292i	$0.662633\pi$
$942$	−33.0000	−1.07520
$943$	−48.0000	−1.56310
$944$	4.00000	0.130189
$945$	0	0
$946$	20.0000	0.650256
$947$	3.00000	0.0974869	0.0487435	−	0.998811i	$-0.484478\pi$
$947$	3.00000	0.0974869	0.0487435	+	0.998811i	$0.484478\pi$
$948$	−3.00000	−0.0974355
$949$	−6.00000	−0.194768
$950$	−2.00000	−0.0648886
$951$	36.0000	1.16738
$952$	0	0
$953$	−15.0000	−0.485898	−0.242949	−	0.970039i	$-0.578115\pi$
$953$	−15.0000	−0.485898	−0.242949	+	0.970039i	$0.578115\pi$
$954$	−54.0000	−1.74831
$955$	48.0000	1.55324
$956$	−4.00000	−0.129369
$957$	90.0000	2.90929
$958$	0	0
$959$	0	0
$960$	−6.00000	−0.193649
$961$	−15.0000	−0.483871
$962$	24.0000	0.773791
$963$	−54.0000	−1.74013
$964$	18.0000	0.579741
$965$	32.0000	1.03012
$966$	0	0
$967$	14.0000	0.450210	0.225105	−	0.974335i	$-0.427728\pi$
$967$	14.0000	0.450210	0.225105	+	0.974335i	$0.427728\pi$
$968$	−14.0000	−0.449977
$969$	6.00000	0.192748
$970$	8.00000	0.256865
$971$	−30.0000	−0.962746	−0.481373	−	0.876516i	$-0.659862\pi$
$971$	−30.0000	−0.962746	−0.481373	+	0.876516i	$0.659862\pi$
$972$	0	0
$973$	0	0
$974$	−11.0000	−0.352463
$975$	9.00000	0.288231
$976$	4.00000	0.128037
$977$	10.0000	0.319928	0.159964	−	0.987123i	$-0.448862\pi$
$977$	10.0000	0.319928	0.159964	+	0.987123i	$0.448862\pi$
$978$	−36.0000	−1.15115
$979$	−45.0000	−1.43821
$980$	0	0
$981$	−96.0000	−3.06504
$982$	38.0000	1.21263
$983$	−37.0000	−1.18012	−0.590058	−	0.807361i	$-0.700895\pi$
$983$	−37.0000	−1.18012	−0.590058	+	0.807361i	$0.700895\pi$
$984$	−18.0000	−0.573819
$985$	4.00000	0.127451
$986$	−6.00000	−0.191079
$987$	0	0
$988$	6.00000	0.190885
$989$	−32.0000	−1.01754
$990$	−60.0000	−1.90693
$991$	−47.0000	−1.49300	−0.746502	−	0.665383i	$-0.768268\pi$
$991$	−47.0000	−1.49300	−0.746502	+	0.665383i	$0.768268\pi$
$992$	4.00000	0.127000
$993$	−24.0000	−0.761617
$994$	0	0
$995$	30.0000	0.951064
$996$	−36.0000	−1.14070
$997$	18.0000	0.570066	0.285033	−	0.958518i	$-0.407995\pi$
$997$	18.0000	0.570066	0.285033	+	0.958518i	$0.407995\pi$
$998$	11.0000	0.348199
$999$	72.0000	2.27798

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1666.2.a.h.1.1		1
7.2	even	3		238.2.e.b.137.1	✓	2
7.4	even	3		238.2.e.b.205.1	yes	2
7.6	odd	2		1666.2.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
238.2.e.b.137.1	✓	2	7.2	even	3
238.2.e.b.205.1	yes	2	7.4	even	3
1666.2.a.a.1.1		1	7.6	odd	2
1666.2.a.h.1.1		1	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1666 = 2 \cdot 7^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1666.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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