LMFDB - Embedded newform 85.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [85,2,Mod(1,85)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(85, 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("85.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 85 = 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	85.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$0.678728417181$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	85.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

\(q+1.00000 q^{2} +2.00000 q^{3} -1.00000 q^{4} -1.00000 q^{5} +2.00000 q^{6} -2.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +2.00000 q^{11} -2.00000 q^{12} +2.00000 q^{13} -2.00000 q^{14} -2.00000 q^{15} -1.00000 q^{16} +1.00000 q^{17} +1.00000 q^{18} +1.00000 q^{20} -4.00000 q^{21} +2.00000 q^{22} +6.00000 q^{23} -6.00000 q^{24} +1.00000 q^{25} +2.00000 q^{26} -4.00000 q^{27} +2.00000 q^{28} -6.00000 q^{29} -2.00000 q^{30} -10.0000 q^{31} +5.00000 q^{32} +4.00000 q^{33} +1.00000 q^{34} +2.00000 q^{35} -1.00000 q^{36} +2.00000 q^{37} +4.00000 q^{39} +3.00000 q^{40} +10.0000 q^{41} -4.00000 q^{42} +4.00000 q^{43} -2.00000 q^{44} -1.00000 q^{45} +6.00000 q^{46} +12.0000 q^{47} -2.00000 q^{48} -3.00000 q^{49} +1.00000 q^{50} +2.00000 q^{51} -2.00000 q^{52} -10.0000 q^{53} -4.00000 q^{54} -2.00000 q^{55} +6.00000 q^{56} -6.00000 q^{58} +8.00000 q^{59} +2.00000 q^{60} -14.0000 q^{61} -10.0000 q^{62} -2.00000 q^{63} +7.00000 q^{64} -2.00000 q^{65} +4.00000 q^{66} +8.00000 q^{67} -1.00000 q^{68} +12.0000 q^{69} +2.00000 q^{70} -2.00000 q^{71} -3.00000 q^{72} -14.0000 q^{73} +2.00000 q^{74} +2.00000 q^{75} -4.00000 q^{77} +4.00000 q^{78} -14.0000 q^{79} +1.00000 q^{80} -11.0000 q^{81} +10.0000 q^{82} +4.00000 q^{83} +4.00000 q^{84} -1.00000 q^{85} +4.00000 q^{86} -12.0000 q^{87} -6.00000 q^{88} +6.00000 q^{89} -1.00000 q^{90} -4.00000 q^{91} -6.00000 q^{92} -20.0000 q^{93} +12.0000 q^{94} +10.0000 q^{96} +2.00000 q^{97} -3.00000 q^{98} +2.00000 q^{99} +O(q^{100})\)

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	0.353553	−	0.935414i	$-0.384973\pi$
$2$	1.00000	0.707107	0.353553	+	0.935414i	$0.384973\pi$
$3$	2.00000	1.15470	0.577350	−	0.816497i	$-0.304087\pi$
$3$	2.00000	1.15470	0.577350	+	0.816497i	$0.304087\pi$
$4$	−1.00000	−0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	2.00000	0.816497
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	−3.00000	−1.06066
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	−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	$0.410544\pi$
$14$	−2.00000	−0.534522
$15$	−2.00000	−0.516398
$16$	−1.00000	−0.250000
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	1.00000	0.235702
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	1.00000	0.223607
$21$	−4.00000	−0.872872
$22$	2.00000	0.426401
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	−6.00000	−1.22474
$25$	1.00000	0.200000
$26$	2.00000	0.392232
$27$	−4.00000	−0.769800
$28$	2.00000	0.377964
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	−2.00000	−0.365148
$31$	−10.0000	−1.79605	−0.898027	−	0.439941i	$-0.854999\pi$
$31$	−10.0000	−1.79605	−0.898027	+	0.439941i	$0.854999\pi$
$32$	5.00000	0.883883
$33$	4.00000	0.696311
$34$	1.00000	0.171499
$35$	2.00000	0.338062
$36$	−1.00000	−0.166667
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	0	0
$39$	4.00000	0.640513
$40$	3.00000	0.474342
$41$	10.0000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	−4.00000	−0.617213
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	−2.00000	−0.301511
$45$	−1.00000	−0.149071
$46$	6.00000	0.884652
$47$	12.0000	1.75038	0.875190	−	0.483779i	$-0.160736\pi$
$47$	12.0000	1.75038	0.875190	+	0.483779i	$0.160736\pi$
$48$	−2.00000	−0.288675
$49$	−3.00000	−0.428571
$50$	1.00000	0.141421
$51$	2.00000	0.280056
$52$	−2.00000	−0.277350
$53$	−10.0000	−1.37361	−0.686803	−	0.726844i	$-0.740986\pi$
$53$	−10.0000	−1.37361	−0.686803	+	0.726844i	$0.740986\pi$
$54$	−4.00000	−0.544331
$55$	−2.00000	−0.269680
$56$	6.00000	0.801784
$57$	0	0
$58$	−6.00000	−0.787839
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	2.00000	0.258199
$61$	−14.0000	−1.79252	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−14.0000	−1.79252	−0.896258	+	0.443533i	$0.853725\pi$
$62$	−10.0000	−1.27000
$63$	−2.00000	−0.251976
$64$	7.00000	0.875000
$65$	−2.00000	−0.248069
$66$	4.00000	0.492366
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	−1.00000	−0.121268
$69$	12.0000	1.44463
$70$	2.00000	0.239046
$71$	−2.00000	−0.237356	−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000	−0.237356	−0.118678	+	0.992933i	$0.537866\pi$
$72$	−3.00000	−0.353553
$73$	−14.0000	−1.63858	−0.819288	−	0.573382i	$-0.805631\pi$
$73$	−14.0000	−1.63858	−0.819288	+	0.573382i	$0.805631\pi$
$74$	2.00000	0.232495
$75$	2.00000	0.230940
$76$	0	0
$77$	−4.00000	−0.455842
$78$	4.00000	0.452911
$79$	−14.0000	−1.57512	−0.787562	−	0.616236i	$-0.788657\pi$
$79$	−14.0000	−1.57512	−0.787562	+	0.616236i	$0.788657\pi$
$80$	1.00000	0.111803
$81$	−11.0000	−1.22222
$82$	10.0000	1.10432
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	4.00000	0.436436
$85$	−1.00000	−0.108465
$86$	4.00000	0.431331
$87$	−12.0000	−1.28654
$88$	−6.00000	−0.639602
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	−1.00000	−0.105409
$91$	−4.00000	−0.419314
$92$	−6.00000	−0.625543
$93$	−20.0000	−2.07390
$94$	12.0000	1.23771
$95$	0	0
$96$	10.0000	1.02062
$97$	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
$99$	2.00000	0.201008
$100$	−1.00000	−0.100000
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	2.00000	0.198030
$103$	12.0000	1.18240	0.591198	−	0.806527i	$-0.298655\pi$
$103$	12.0000	1.18240	0.591198	+	0.806527i	$0.298655\pi$
$104$	−6.00000	−0.588348
$105$	4.00000	0.390360
$106$	−10.0000	−0.971286
$107$	2.00000	0.193347	0.0966736	−	0.995316i	$-0.469180\pi$
$107$	2.00000	0.193347	0.0966736	+	0.995316i	$0.469180\pi$
$108$	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	$0.469465\pi$
$110$	−2.00000	−0.190693
$111$	4.00000	0.379663
$112$	2.00000	0.188982
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	−6.00000	−0.559503
$116$	6.00000	0.557086
$117$	2.00000	0.184900
$118$	8.00000	0.736460
$119$	−2.00000	−0.183340
$120$	6.00000	0.547723
$121$	−7.00000	−0.636364
$122$	−14.0000	−1.26750
$123$	20.0000	1.80334
$124$	10.0000	0.898027
$125$	−1.00000	−0.0894427
$126$	−2.00000	−0.178174
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	−3.00000	−0.265165
$129$	8.00000	0.704361
$130$	−2.00000	−0.175412
$131$	6.00000	0.524222	0.262111	−	0.965038i	$-0.415581\pi$
$131$	6.00000	0.524222	0.262111	+	0.965038i	$0.415581\pi$
$132$	−4.00000	−0.348155
$133$	0	0
$134$	8.00000	0.691095
$135$	4.00000	0.344265
$136$	−3.00000	−0.257248
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	12.0000	1.02151
$139$	14.0000	1.18746	0.593732	−	0.804663i	$-0.297654\pi$
$139$	14.0000	1.18746	0.593732	+	0.804663i	$0.297654\pi$
$140$	−2.00000	−0.169031
$141$	24.0000	2.02116
$142$	−2.00000	−0.167836
$143$	4.00000	0.334497
$144$	−1.00000	−0.0833333
$145$	6.00000	0.498273
$146$	−14.0000	−1.15865
$147$	−6.00000	−0.494872
$148$	−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
$151$	−12.0000	−0.976546	−0.488273	−	0.872691i	$-0.662373\pi$
$151$	−12.0000	−0.976546	−0.488273	+	0.872691i	$0.662373\pi$
$152$	0	0
$153$	1.00000	0.0808452
$154$	−4.00000	−0.322329
$155$	10.0000	0.803219
$156$	−4.00000	−0.320256
$157$	6.00000	0.478852	0.239426	−	0.970915i	$-0.423041\pi$
$157$	6.00000	0.478852	0.239426	+	0.970915i	$0.423041\pi$
$158$	−14.0000	−1.11378
$159$	−20.0000	−1.58610
$160$	−5.00000	−0.395285
$161$	−12.0000	−0.945732
$162$	−11.0000	−0.864242
$163$	−2.00000	−0.156652	−0.0783260	−	0.996928i	$-0.524958\pi$
$163$	−2.00000	−0.156652	−0.0783260	+	0.996928i	$0.524958\pi$
$164$	−10.0000	−0.780869
$165$	−4.00000	−0.311400
$166$	4.00000	0.310460
$167$	2.00000	0.154765	0.0773823	−	0.997001i	$-0.475344\pi$
$167$	2.00000	0.154765	0.0773823	+	0.997001i	$0.475344\pi$
$168$	12.0000	0.925820
$169$	−9.00000	−0.692308
$170$	−1.00000	−0.0766965
$171$	0	0
$172$	−4.00000	−0.304997
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	−12.0000	−0.909718
$175$	−2.00000	−0.151186
$176$	−2.00000	−0.150756
$177$	16.0000	1.20263
$178$	6.00000	0.449719
$179$	−24.0000	−1.79384	−0.896922	−	0.442189i	$-0.854202\pi$
$179$	−24.0000	−1.79384	−0.896922	+	0.442189i	$0.854202\pi$
$180$	1.00000	0.0745356
$181$	26.0000	1.93256	0.966282	−	0.257485i	$-0.0828937\pi$
$181$	26.0000	1.93256	0.966282	+	0.257485i	$0.0828937\pi$
$182$	−4.00000	−0.296500
$183$	−28.0000	−2.06982
$184$	−18.0000	−1.32698
$185$	−2.00000	−0.147043
$186$	−20.0000	−1.46647
$187$	2.00000	0.146254
$188$	−12.0000	−0.875190
$189$	8.00000	0.581914
$190$	0	0
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	14.0000	1.01036
$193$	18.0000	1.29567	0.647834	−	0.761781i	$-0.275675\pi$
$193$	18.0000	1.29567	0.647834	+	0.761781i	$0.275675\pi$
$194$	2.00000	0.143592
$195$	−4.00000	−0.286446
$196$	3.00000	0.214286
$197$	2.00000	0.142494	0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000	0.142494	0.0712470	+	0.997459i	$0.477302\pi$
$198$	2.00000	0.142134
$199$	6.00000	0.425329	0.212664	−	0.977125i	$-0.431786\pi$
$199$	6.00000	0.425329	0.212664	+	0.977125i	$0.431786\pi$
$200$	−3.00000	−0.212132
$201$	16.0000	1.12855
$202$	−6.00000	−0.422159
$203$	12.0000	0.842235
$204$	−2.00000	−0.140028
$205$	−10.0000	−0.698430
$206$	12.0000	0.836080
$207$	6.00000	0.417029
$208$	−2.00000	−0.138675
$209$	0	0
$210$	4.00000	0.276026
$211$	2.00000	0.137686	0.0688428	−	0.997628i	$-0.478069\pi$
$211$	2.00000	0.137686	0.0688428	+	0.997628i	$0.478069\pi$
$212$	10.0000	0.686803
$213$	−4.00000	−0.274075
$214$	2.00000	0.136717
$215$	−4.00000	−0.272798
$216$	12.0000	0.816497
$217$	20.0000	1.35769
$218$	2.00000	0.135457
$219$	−28.0000	−1.89206
$220$	2.00000	0.134840
$221$	2.00000	0.134535
$222$	4.00000	0.268462
$223$	16.0000	1.07144	0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000	1.07144	0.535720	+	0.844396i	$0.320040\pi$
$224$	−10.0000	−0.668153
$225$	1.00000	0.0666667
$226$	−6.00000	−0.399114
$227$	−2.00000	−0.132745	−0.0663723	−	0.997795i	$-0.521143\pi$
$227$	−2.00000	−0.132745	−0.0663723	+	0.997795i	$0.521143\pi$
$228$	0	0
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	−6.00000	−0.395628
$231$	−8.00000	−0.526361
$232$	18.0000	1.18176
$233$	−14.0000	−0.917170	−0.458585	−	0.888650i	$-0.651644\pi$
$233$	−14.0000	−0.917170	−0.458585	+	0.888650i	$0.651644\pi$
$234$	2.00000	0.130744
$235$	−12.0000	−0.782794
$236$	−8.00000	−0.520756
$237$	−28.0000	−1.81880
$238$	−2.00000	−0.129641
$239$	−8.00000	−0.517477	−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000	−0.517477	−0.258738	+	0.965947i	$0.583307\pi$
$240$	2.00000	0.129099
$241$	−6.00000	−0.386494	−0.193247	−	0.981150i	$-0.561902\pi$
$241$	−6.00000	−0.386494	−0.193247	+	0.981150i	$0.561902\pi$
$242$	−7.00000	−0.449977
$243$	−10.0000	−0.641500
$244$	14.0000	0.896258
$245$	3.00000	0.191663
$246$	20.0000	1.27515
$247$	0	0
$248$	30.0000	1.90500
$249$	8.00000	0.506979
$250$	−1.00000	−0.0632456
$251$	28.0000	1.76734	0.883672	−	0.468106i	$-0.155064\pi$
$251$	28.0000	1.76734	0.883672	+	0.468106i	$0.155064\pi$
$252$	2.00000	0.125988
$253$	12.0000	0.754434
$254$	8.00000	0.501965
$255$	−2.00000	−0.125245
$256$	−17.0000	−1.06250
$257$	−2.00000	−0.124757	−0.0623783	−	0.998053i	$-0.519869\pi$
$257$	−2.00000	−0.124757	−0.0623783	+	0.998053i	$0.519869\pi$
$258$	8.00000	0.498058
$259$	−4.00000	−0.248548
$260$	2.00000	0.124035
$261$	−6.00000	−0.371391
$262$	6.00000	0.370681
$263$	−8.00000	−0.493301	−0.246651	−	0.969104i	$-0.579330\pi$
$263$	−8.00000	−0.493301	−0.246651	+	0.969104i	$0.579330\pi$
$264$	−12.0000	−0.738549
$265$	10.0000	0.614295
$266$	0	0
$267$	12.0000	0.734388
$268$	−8.00000	−0.488678
$269$	10.0000	0.609711	0.304855	−	0.952399i	$-0.401392\pi$
$269$	10.0000	0.609711	0.304855	+	0.952399i	$0.401392\pi$
$270$	4.00000	0.243432
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	−1.00000	−0.0606339
$273$	−8.00000	−0.484182
$274$	−2.00000	−0.120824
$275$	2.00000	0.120605
$276$	−12.0000	−0.722315
$277$	−30.0000	−1.80253	−0.901263	−	0.433273i	$-0.857359\pi$
$277$	−30.0000	−1.80253	−0.901263	+	0.433273i	$0.857359\pi$
$278$	14.0000	0.839664
$279$	−10.0000	−0.598684
$280$	−6.00000	−0.358569
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	24.0000	1.42918
$283$	6.00000	0.356663	0.178331	−	0.983970i	$-0.442930\pi$
$283$	6.00000	0.356663	0.178331	+	0.983970i	$0.442930\pi$
$284$	2.00000	0.118678
$285$	0	0
$286$	4.00000	0.236525
$287$	−20.0000	−1.18056
$288$	5.00000	0.294628
$289$	1.00000	0.0588235
$290$	6.00000	0.352332
$291$	4.00000	0.234484
$292$	14.0000	0.819288
$293$	30.0000	1.75262	0.876309	−	0.481749i	$-0.159998\pi$
$293$	30.0000	1.75262	0.876309	+	0.481749i	$0.159998\pi$
$294$	−6.00000	−0.349927
$295$	−8.00000	−0.465778
$296$	−6.00000	−0.348743
$297$	−8.00000	−0.464207
$298$	6.00000	0.347571
$299$	12.0000	0.693978
$300$	−2.00000	−0.115470
$301$	−8.00000	−0.461112
$302$	−12.0000	−0.690522
$303$	−12.0000	−0.689382
$304$	0	0
$305$	14.0000	0.801638
$306$	1.00000	0.0571662
$307$	−16.0000	−0.913168	−0.456584	−	0.889680i	$-0.650927\pi$
$307$	−16.0000	−0.913168	−0.456584	+	0.889680i	$0.650927\pi$
$308$	4.00000	0.227921
$309$	24.0000	1.36531
$310$	10.0000	0.567962
$311$	6.00000	0.340229	0.170114	−	0.985424i	$-0.445586\pi$
$311$	6.00000	0.340229	0.170114	+	0.985424i	$0.445586\pi$
$312$	−12.0000	−0.679366
$313$	2.00000	0.113047	0.0565233	−	0.998401i	$-0.481998\pi$
$313$	2.00000	0.113047	0.0565233	+	0.998401i	$0.481998\pi$
$314$	6.00000	0.338600
$315$	2.00000	0.112687
$316$	14.0000	0.787562
$317$	10.0000	0.561656	0.280828	−	0.959758i	$-0.409391\pi$
$317$	10.0000	0.561656	0.280828	+	0.959758i	$0.409391\pi$
$318$	−20.0000	−1.12154
$319$	−12.0000	−0.671871
$320$	−7.00000	−0.391312
$321$	4.00000	0.223258
$322$	−12.0000	−0.668734
$323$	0	0
$324$	11.0000	0.611111
$325$	2.00000	0.110940
$326$	−2.00000	−0.110770
$327$	4.00000	0.221201
$328$	−30.0000	−1.65647
$329$	−24.0000	−1.32316
$330$	−4.00000	−0.220193
$331$	16.0000	0.879440	0.439720	−	0.898135i	$-0.355078\pi$
$331$	16.0000	0.879440	0.439720	+	0.898135i	$0.355078\pi$
$332$	−4.00000	−0.219529
$333$	2.00000	0.109599
$334$	2.00000	0.109435
$335$	−8.00000	−0.437087
$336$	4.00000	0.218218
$337$	−6.00000	−0.326841	−0.163420	−	0.986557i	$-0.552253\pi$
$337$	−6.00000	−0.326841	−0.163420	+	0.986557i	$0.552253\pi$
$338$	−9.00000	−0.489535
$339$	−12.0000	−0.651751
$340$	1.00000	0.0542326
$341$	−20.0000	−1.08306
$342$	0	0
$343$	20.0000	1.07990
$344$	−12.0000	−0.646997
$345$	−12.0000	−0.646058
$346$	−6.00000	−0.322562
$347$	−26.0000	−1.39575	−0.697877	−	0.716218i	$-0.745872\pi$
$347$	−26.0000	−1.39575	−0.697877	+	0.716218i	$0.745872\pi$
$348$	12.0000	0.643268
$349$	22.0000	1.17763	0.588817	−	0.808267i	$-0.299594\pi$
$349$	22.0000	1.17763	0.588817	+	0.808267i	$0.299594\pi$
$350$	−2.00000	−0.106904
$351$	−8.00000	−0.427008
$352$	10.0000	0.533002
$353$	−14.0000	−0.745145	−0.372572	−	0.928003i	$-0.621524\pi$
$353$	−14.0000	−0.745145	−0.372572	+	0.928003i	$0.621524\pi$
$354$	16.0000	0.850390
$355$	2.00000	0.106149
$356$	−6.00000	−0.317999
$357$	−4.00000	−0.211702
$358$	−24.0000	−1.26844
$359$	20.0000	1.05556	0.527780	−	0.849381i	$-0.323025\pi$
$359$	20.0000	1.05556	0.527780	+	0.849381i	$0.323025\pi$
$360$	3.00000	0.158114
$361$	−19.0000	−1.00000
$362$	26.0000	1.36653
$363$	−14.0000	−0.734809
$364$	4.00000	0.209657
$365$	14.0000	0.732793
$366$	−28.0000	−1.46358
$367$	−34.0000	−1.77479	−0.887393	−	0.461014i	$-0.847486\pi$
$367$	−34.0000	−1.77479	−0.887393	+	0.461014i	$0.847486\pi$
$368$	−6.00000	−0.312772
$369$	10.0000	0.520579
$370$	−2.00000	−0.103975
$371$	20.0000	1.03835
$372$	20.0000	1.03695
$373$	10.0000	0.517780	0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000	0.517780	0.258890	+	0.965907i	$0.416643\pi$
$374$	2.00000	0.103418
$375$	−2.00000	−0.103280
$376$	−36.0000	−1.85656
$377$	−12.0000	−0.618031
$378$	8.00000	0.411476
$379$	−2.00000	−0.102733	−0.0513665	−	0.998680i	$-0.516358\pi$
$379$	−2.00000	−0.102733	−0.0513665	+	0.998680i	$0.516358\pi$
$380$	0	0
$381$	16.0000	0.819705
$382$	8.00000	0.409316
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	−6.00000	−0.306186
$385$	4.00000	0.203859
$386$	18.0000	0.916176
$387$	4.00000	0.203331
$388$	−2.00000	−0.101535
$389$	26.0000	1.31825	0.659126	−	0.752032i	$-0.270926\pi$
$389$	26.0000	1.31825	0.659126	+	0.752032i	$0.270926\pi$
$390$	−4.00000	−0.202548
$391$	6.00000	0.303433
$392$	9.00000	0.454569
$393$	12.0000	0.605320
$394$	2.00000	0.100759
$395$	14.0000	0.704416
$396$	−2.00000	−0.100504
$397$	−14.0000	−0.702640	−0.351320	−	0.936255i	$-0.614267\pi$
$397$	−14.0000	−0.702640	−0.351320	+	0.936255i	$0.614267\pi$
$398$	6.00000	0.300753
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	16.0000	0.798007
$403$	−20.0000	−0.996271
$404$	6.00000	0.298511
$405$	11.0000	0.546594
$406$	12.0000	0.595550
$407$	4.00000	0.198273
$408$	−6.00000	−0.297044
$409$	−38.0000	−1.87898	−0.939490	−	0.342578i	$-0.888700\pi$
$409$	−38.0000	−1.87898	−0.939490	+	0.342578i	$0.888700\pi$
$410$	−10.0000	−0.493865
$411$	−4.00000	−0.197305
$412$	−12.0000	−0.591198
$413$	−16.0000	−0.787309
$414$	6.00000	0.294884
$415$	−4.00000	−0.196352
$416$	10.0000	0.490290
$417$	28.0000	1.37117
$418$	0	0
$419$	18.0000	0.879358	0.439679	−	0.898155i	$-0.355092\pi$
$419$	18.0000	0.879358	0.439679	+	0.898155i	$0.355092\pi$
$420$	−4.00000	−0.195180
$421$	26.0000	1.26716	0.633581	−	0.773676i	$-0.281584\pi$
$421$	26.0000	1.26716	0.633581	+	0.773676i	$0.281584\pi$
$422$	2.00000	0.0973585
$423$	12.0000	0.583460
$424$	30.0000	1.45693
$425$	1.00000	0.0485071
$426$	−4.00000	−0.193801
$427$	28.0000	1.35501
$428$	−2.00000	−0.0966736
$429$	8.00000	0.386244
$430$	−4.00000	−0.192897
$431$	−38.0000	−1.83040	−0.915198	−	0.403005i	$-0.867966\pi$
$431$	−38.0000	−1.83040	−0.915198	+	0.403005i	$0.867966\pi$
$432$	4.00000	0.192450
$433$	−34.0000	−1.63394	−0.816968	−	0.576683i	$-0.804347\pi$
$433$	−34.0000	−1.63394	−0.816968	+	0.576683i	$0.804347\pi$
$434$	20.0000	0.960031
$435$	12.0000	0.575356
$436$	−2.00000	−0.0957826
$437$	0	0
$438$	−28.0000	−1.33789
$439$	6.00000	0.286364	0.143182	−	0.989696i	$-0.454267\pi$
$439$	6.00000	0.286364	0.143182	+	0.989696i	$0.454267\pi$
$440$	6.00000	0.286039
$441$	−3.00000	−0.142857
$442$	2.00000	0.0951303
$443$	24.0000	1.14027	0.570137	−	0.821549i	$-0.306890\pi$
$443$	24.0000	1.14027	0.570137	+	0.821549i	$0.306890\pi$
$444$	−4.00000	−0.189832
$445$	−6.00000	−0.284427
$446$	16.0000	0.757622
$447$	12.0000	0.567581
$448$	−14.0000	−0.661438
$449$	18.0000	0.849473	0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000	0.849473	0.424736	+	0.905317i	$0.360367\pi$
$450$	1.00000	0.0471405
$451$	20.0000	0.941763
$452$	6.00000	0.282216
$453$	−24.0000	−1.12762
$454$	−2.00000	−0.0938647
$455$	4.00000	0.187523
$456$	0	0
$457$	6.00000	0.280668	0.140334	−	0.990104i	$-0.455182\pi$
$457$	6.00000	0.280668	0.140334	+	0.990104i	$0.455182\pi$
$458$	10.0000	0.467269
$459$	−4.00000	−0.186704
$460$	6.00000	0.279751
$461$	−26.0000	−1.21094	−0.605470	−	0.795868i	$-0.707015\pi$
$461$	−26.0000	−1.21094	−0.605470	+	0.795868i	$0.707015\pi$
$462$	−8.00000	−0.372194
$463$	−36.0000	−1.67306	−0.836531	−	0.547920i	$-0.815420\pi$
$463$	−36.0000	−1.67306	−0.836531	+	0.547920i	$0.815420\pi$
$464$	6.00000	0.278543
$465$	20.0000	0.927478
$466$	−14.0000	−0.648537
$467$	28.0000	1.29569	0.647843	−	0.761774i	$-0.275671\pi$
$467$	28.0000	1.29569	0.647843	+	0.761774i	$0.275671\pi$
$468$	−2.00000	−0.0924500
$469$	−16.0000	−0.738811
$470$	−12.0000	−0.553519
$471$	12.0000	0.552931
$472$	−24.0000	−1.10469
$473$	8.00000	0.367840
$474$	−28.0000	−1.28608
$475$	0	0
$476$	2.00000	0.0916698
$477$	−10.0000	−0.457869
$478$	−8.00000	−0.365911
$479$	−18.0000	−0.822441	−0.411220	−	0.911536i	$-0.634897\pi$
$479$	−18.0000	−0.822441	−0.411220	+	0.911536i	$0.634897\pi$
$480$	−10.0000	−0.456435
$481$	4.00000	0.182384
$482$	−6.00000	−0.273293
$483$	−24.0000	−1.09204
$484$	7.00000	0.318182
$485$	−2.00000	−0.0908153
$486$	−10.0000	−0.453609
$487$	22.0000	0.996915	0.498458	−	0.866914i	$-0.333900\pi$
$487$	22.0000	0.996915	0.498458	+	0.866914i	$0.333900\pi$
$488$	42.0000	1.90125
$489$	−4.00000	−0.180886
$490$	3.00000	0.135526
$491$	−16.0000	−0.722070	−0.361035	−	0.932552i	$-0.617576\pi$
$491$	−16.0000	−0.722070	−0.361035	+	0.932552i	$0.617576\pi$
$492$	−20.0000	−0.901670
$493$	−6.00000	−0.270226
$494$	0	0
$495$	−2.00000	−0.0898933
$496$	10.0000	0.449013
$497$	4.00000	0.179425
$498$	8.00000	0.358489
$499$	−14.0000	−0.626726	−0.313363	−	0.949633i	$-0.601456\pi$
$499$	−14.0000	−0.626726	−0.313363	+	0.949633i	$0.601456\pi$
$500$	1.00000	0.0447214
$501$	4.00000	0.178707
$502$	28.0000	1.24970
$503$	6.00000	0.267527	0.133763	−	0.991013i	$-0.457294\pi$
$503$	6.00000	0.267527	0.133763	+	0.991013i	$0.457294\pi$
$504$	6.00000	0.267261
$505$	6.00000	0.266996
$506$	12.0000	0.533465
$507$	−18.0000	−0.799408
$508$	−8.00000	−0.354943
$509$	−34.0000	−1.50702	−0.753512	−	0.657434i	$-0.771642\pi$
$509$	−34.0000	−1.50702	−0.753512	+	0.657434i	$0.771642\pi$
$510$	−2.00000	−0.0885615
$511$	28.0000	1.23865
$512$	−11.0000	−0.486136
$513$	0	0
$514$	−2.00000	−0.0882162
$515$	−12.0000	−0.528783
$516$	−8.00000	−0.352180
$517$	24.0000	1.05552
$518$	−4.00000	−0.175750
$519$	−12.0000	−0.526742
$520$	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	$0.728243\pi$
$522$	−6.00000	−0.262613
$523$	32.0000	1.39926	0.699631	−	0.714504i	$-0.253348\pi$
$523$	32.0000	1.39926	0.699631	+	0.714504i	$0.253348\pi$
$524$	−6.00000	−0.262111
$525$	−4.00000	−0.174574
$526$	−8.00000	−0.348817
$527$	−10.0000	−0.435607
$528$	−4.00000	−0.174078
$529$	13.0000	0.565217
$530$	10.0000	0.434372
$531$	8.00000	0.347170
$532$	0	0
$533$	20.0000	0.866296
$534$	12.0000	0.519291
$535$	−2.00000	−0.0864675
$536$	−24.0000	−1.03664
$537$	−48.0000	−2.07135
$538$	10.0000	0.431131
$539$	−6.00000	−0.258438
$540$	−4.00000	−0.172133
$541$	−14.0000	−0.601907	−0.300954	−	0.953639i	$-0.597305\pi$
$541$	−14.0000	−0.601907	−0.300954	+	0.953639i	$0.597305\pi$
$542$	−16.0000	−0.687259
$543$	52.0000	2.23153
$544$	5.00000	0.214373
$545$	−2.00000	−0.0856706
$546$	−8.00000	−0.342368
$547$	−22.0000	−0.940652	−0.470326	−	0.882493i	$-0.655864\pi$
$547$	−22.0000	−0.940652	−0.470326	+	0.882493i	$0.655864\pi$
$548$	2.00000	0.0854358
$549$	−14.0000	−0.597505
$550$	2.00000	0.0852803
$551$	0	0
$552$	−36.0000	−1.53226
$553$	28.0000	1.19068
$554$	−30.0000	−1.27458
$555$	−4.00000	−0.169791
$556$	−14.0000	−0.593732
$557$	2.00000	0.0847427	0.0423714	−	0.999102i	$-0.486509\pi$
$557$	2.00000	0.0847427	0.0423714	+	0.999102i	$0.486509\pi$
$558$	−10.0000	−0.423334
$559$	8.00000	0.338364
$560$	−2.00000	−0.0845154
$561$	4.00000	0.168880
$562$	−6.00000	−0.253095
$563$	4.00000	0.168580	0.0842900	−	0.996441i	$-0.473138\pi$
$563$	4.00000	0.168580	0.0842900	+	0.996441i	$0.473138\pi$
$564$	−24.0000	−1.01058
$565$	6.00000	0.252422
$566$	6.00000	0.252199
$567$	22.0000	0.923913
$568$	6.00000	0.251754
$569$	−38.0000	−1.59304	−0.796521	−	0.604610i	$-0.793329\pi$
$569$	−38.0000	−1.59304	−0.796521	+	0.604610i	$0.793329\pi$
$570$	0	0
$571$	18.0000	0.753277	0.376638	−	0.926360i	$-0.377080\pi$
$571$	18.0000	0.753277	0.376638	+	0.926360i	$0.377080\pi$
$572$	−4.00000	−0.167248
$573$	16.0000	0.668410
$574$	−20.0000	−0.834784
$575$	6.00000	0.250217
$576$	7.00000	0.291667
$577$	6.00000	0.249783	0.124892	−	0.992170i	$-0.460142\pi$
$577$	6.00000	0.249783	0.124892	+	0.992170i	$0.460142\pi$
$578$	1.00000	0.0415945
$579$	36.0000	1.49611
$580$	−6.00000	−0.249136
$581$	−8.00000	−0.331896
$582$	4.00000	0.165805
$583$	−20.0000	−0.828315
$584$	42.0000	1.73797
$585$	−2.00000	−0.0826898
$586$	30.0000	1.23929
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	6.00000	0.247436
$589$	0	0
$590$	−8.00000	−0.329355
$591$	4.00000	0.164538
$592$	−2.00000	−0.0821995
$593$	−30.0000	−1.23195	−0.615976	−	0.787765i	$-0.711238\pi$
$593$	−30.0000	−1.23195	−0.615976	+	0.787765i	$0.711238\pi$
$594$	−8.00000	−0.328244
$595$	2.00000	0.0819920
$596$	−6.00000	−0.245770
$597$	12.0000	0.491127
$598$	12.0000	0.490716
$599$	16.0000	0.653742	0.326871	−	0.945069i	$-0.394006\pi$
$599$	16.0000	0.653742	0.326871	+	0.945069i	$0.394006\pi$
$600$	−6.00000	−0.244949
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	−8.00000	−0.326056
$603$	8.00000	0.325785
$604$	12.0000	0.488273
$605$	7.00000	0.284590
$606$	−12.0000	−0.487467
$607$	18.0000	0.730597	0.365299	−	0.930890i	$-0.380967\pi$
$607$	18.0000	0.730597	0.365299	+	0.930890i	$0.380967\pi$
$608$	0	0
$609$	24.0000	0.972529
$610$	14.0000	0.566843
$611$	24.0000	0.970936
$612$	−1.00000	−0.0404226
$613$	46.0000	1.85792	0.928961	−	0.370177i	$-0.120703\pi$
$613$	46.0000	1.85792	0.928961	+	0.370177i	$0.120703\pi$
$614$	−16.0000	−0.645707
$615$	−20.0000	−0.806478
$616$	12.0000	0.483494
$617$	42.0000	1.69086	0.845428	−	0.534089i	$-0.179345\pi$
$617$	42.0000	1.69086	0.845428	+	0.534089i	$0.179345\pi$
$618$	24.0000	0.965422
$619$	−14.0000	−0.562708	−0.281354	−	0.959604i	$-0.590783\pi$
$619$	−14.0000	−0.562708	−0.281354	+	0.959604i	$0.590783\pi$
$620$	−10.0000	−0.401610
$621$	−24.0000	−0.963087
$622$	6.00000	0.240578
$623$	−12.0000	−0.480770
$624$	−4.00000	−0.160128
$625$	1.00000	0.0400000
$626$	2.00000	0.0799361
$627$	0	0
$628$	−6.00000	−0.239426
$629$	2.00000	0.0797452
$630$	2.00000	0.0796819
$631$	−4.00000	−0.159237	−0.0796187	−	0.996825i	$-0.525370\pi$
$631$	−4.00000	−0.159237	−0.0796187	+	0.996825i	$0.525370\pi$
$632$	42.0000	1.67067
$633$	4.00000	0.158986
$634$	10.0000	0.397151
$635$	−8.00000	−0.317470
$636$	20.0000	0.793052
$637$	−6.00000	−0.237729
$638$	−12.0000	−0.475085
$639$	−2.00000	−0.0791188
$640$	3.00000	0.118585
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	4.00000	0.157867
$643$	−14.0000	−0.552106	−0.276053	−	0.961142i	$-0.589027\pi$
$643$	−14.0000	−0.552106	−0.276053	+	0.961142i	$0.589027\pi$
$644$	12.0000	0.472866
$645$	−8.00000	−0.315000
$646$	0	0
$647$	28.0000	1.10079	0.550397	−	0.834903i	$-0.314476\pi$
$647$	28.0000	1.10079	0.550397	+	0.834903i	$0.314476\pi$
$648$	33.0000	1.29636
$649$	16.0000	0.628055
$650$	2.00000	0.0784465
$651$	40.0000	1.56772
$652$	2.00000	0.0783260
$653$	−14.0000	−0.547862	−0.273931	−	0.961749i	$-0.588324\pi$
$653$	−14.0000	−0.547862	−0.273931	+	0.961749i	$0.588324\pi$
$654$	4.00000	0.156412
$655$	−6.00000	−0.234439
$656$	−10.0000	−0.390434
$657$	−14.0000	−0.546192
$658$	−24.0000	−0.935617
$659$	−44.0000	−1.71400	−0.856998	−	0.515319i	$-0.827673\pi$
$659$	−44.0000	−1.71400	−0.856998	+	0.515319i	$0.827673\pi$
$660$	4.00000	0.155700
$661$	46.0000	1.78919	0.894596	−	0.446875i	$-0.147463\pi$
$661$	46.0000	1.78919	0.894596	+	0.446875i	$0.147463\pi$
$662$	16.0000	0.621858
$663$	4.00000	0.155347
$664$	−12.0000	−0.465690
$665$	0	0
$666$	2.00000	0.0774984
$667$	−36.0000	−1.39393
$668$	−2.00000	−0.0773823
$669$	32.0000	1.23719
$670$	−8.00000	−0.309067
$671$	−28.0000	−1.08093
$672$	−20.0000	−0.771517
$673$	10.0000	0.385472	0.192736	−	0.981251i	$-0.438264\pi$
$673$	10.0000	0.385472	0.192736	+	0.981251i	$0.438264\pi$
$674$	−6.00000	−0.231111
$675$	−4.00000	−0.153960
$676$	9.00000	0.346154
$677$	−30.0000	−1.15299	−0.576497	−	0.817099i	$-0.695581\pi$
$677$	−30.0000	−1.15299	−0.576497	+	0.817099i	$0.695581\pi$
$678$	−12.0000	−0.460857
$679$	−4.00000	−0.153506
$680$	3.00000	0.115045
$681$	−4.00000	−0.153280
$682$	−20.0000	−0.765840
$683$	18.0000	0.688751	0.344375	−	0.938832i	$-0.388091\pi$
$683$	18.0000	0.688751	0.344375	+	0.938832i	$0.388091\pi$
$684$	0	0
$685$	2.00000	0.0764161
$686$	20.0000	0.763604
$687$	20.0000	0.763048
$688$	−4.00000	−0.152499
$689$	−20.0000	−0.761939
$690$	−12.0000	−0.456832
$691$	2.00000	0.0760836	0.0380418	−	0.999276i	$-0.487888\pi$
$691$	2.00000	0.0760836	0.0380418	+	0.999276i	$0.487888\pi$
$692$	6.00000	0.228086
$693$	−4.00000	−0.151947
$694$	−26.0000	−0.986947
$695$	−14.0000	−0.531050
$696$	36.0000	1.36458
$697$	10.0000	0.378777
$698$	22.0000	0.832712
$699$	−28.0000	−1.05906
$700$	2.00000	0.0755929
$701$	−30.0000	−1.13308	−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000	−1.13308	−0.566542	+	0.824033i	$0.691719\pi$
$702$	−8.00000	−0.301941
$703$	0	0
$704$	14.0000	0.527645
$705$	−24.0000	−0.903892
$706$	−14.0000	−0.526897
$707$	12.0000	0.451306
$708$	−16.0000	−0.601317
$709$	−6.00000	−0.225335	−0.112667	−	0.993633i	$-0.535939\pi$
$709$	−6.00000	−0.225335	−0.112667	+	0.993633i	$0.535939\pi$
$710$	2.00000	0.0750587
$711$	−14.0000	−0.525041
$712$	−18.0000	−0.674579
$713$	−60.0000	−2.24702
$714$	−4.00000	−0.149696
$715$	−4.00000	−0.149592
$716$	24.0000	0.896922
$717$	−16.0000	−0.597531
$718$	20.0000	0.746393
$719$	14.0000	0.522112	0.261056	−	0.965324i	$-0.415929\pi$
$719$	14.0000	0.522112	0.261056	+	0.965324i	$0.415929\pi$
$720$	1.00000	0.0372678
$721$	−24.0000	−0.893807
$722$	−19.0000	−0.707107
$723$	−12.0000	−0.446285
$724$	−26.0000	−0.966282
$725$	−6.00000	−0.222834
$726$	−14.0000	−0.519589
$727$	−12.0000	−0.445055	−0.222528	−	0.974926i	$-0.571431\pi$
$727$	−12.0000	−0.445055	−0.222528	+	0.974926i	$0.571431\pi$
$728$	12.0000	0.444750
$729$	13.0000	0.481481
$730$	14.0000	0.518163
$731$	4.00000	0.147945
$732$	28.0000	1.03491
$733$	−2.00000	−0.0738717	−0.0369358	−	0.999318i	$-0.511760\pi$
$733$	−2.00000	−0.0738717	−0.0369358	+	0.999318i	$0.511760\pi$
$734$	−34.0000	−1.25496
$735$	6.00000	0.221313
$736$	30.0000	1.10581
$737$	16.0000	0.589368
$738$	10.0000	0.368105
$739$	−8.00000	−0.294285	−0.147142	−	0.989115i	$-0.547008\pi$
$739$	−8.00000	−0.294285	−0.147142	+	0.989115i	$0.547008\pi$
$740$	2.00000	0.0735215
$741$	0	0
$742$	20.0000	0.734223
$743$	−6.00000	−0.220119	−0.110059	−	0.993925i	$-0.535104\pi$
$743$	−6.00000	−0.220119	−0.110059	+	0.993925i	$0.535104\pi$
$744$	60.0000	2.19971
$745$	−6.00000	−0.219823
$746$	10.0000	0.366126
$747$	4.00000	0.146352
$748$	−2.00000	−0.0731272
$749$	−4.00000	−0.146157
$750$	−2.00000	−0.0730297
$751$	−2.00000	−0.0729810	−0.0364905	−	0.999334i	$-0.511618\pi$
$751$	−2.00000	−0.0729810	−0.0364905	+	0.999334i	$0.511618\pi$
$752$	−12.0000	−0.437595
$753$	56.0000	2.04075
$754$	−12.0000	−0.437014
$755$	12.0000	0.436725
$756$	−8.00000	−0.290957
$757$	−22.0000	−0.799604	−0.399802	−	0.916602i	$-0.630921\pi$
$757$	−22.0000	−0.799604	−0.399802	+	0.916602i	$0.630921\pi$
$758$	−2.00000	−0.0726433
$759$	24.0000	0.871145
$760$	0	0
$761$	6.00000	0.217500	0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000	0.217500	0.108750	+	0.994069i	$0.465315\pi$
$762$	16.0000	0.579619
$763$	−4.00000	−0.144810
$764$	−8.00000	−0.289430
$765$	−1.00000	−0.0361551
$766$	0	0
$767$	16.0000	0.577727
$768$	−34.0000	−1.22687
$769$	−18.0000	−0.649097	−0.324548	−	0.945869i	$-0.605212\pi$
$769$	−18.0000	−0.649097	−0.324548	+	0.945869i	$0.605212\pi$
$770$	4.00000	0.144150
$771$	−4.00000	−0.144056
$772$	−18.0000	−0.647834
$773$	−6.00000	−0.215805	−0.107903	−	0.994161i	$-0.534413\pi$
$773$	−6.00000	−0.215805	−0.107903	+	0.994161i	$0.534413\pi$
$774$	4.00000	0.143777
$775$	−10.0000	−0.359211
$776$	−6.00000	−0.215387
$777$	−8.00000	−0.286998
$778$	26.0000	0.932145
$779$	0	0
$780$	4.00000	0.143223
$781$	−4.00000	−0.143131
$782$	6.00000	0.214560
$783$	24.0000	0.857690
$784$	3.00000	0.107143
$785$	−6.00000	−0.214149
$786$	12.0000	0.428026
$787$	−6.00000	−0.213877	−0.106938	−	0.994266i	$-0.534105\pi$
$787$	−6.00000	−0.213877	−0.106938	+	0.994266i	$0.534105\pi$
$788$	−2.00000	−0.0712470
$789$	−16.0000	−0.569615
$790$	14.0000	0.498098
$791$	12.0000	0.426671
$792$	−6.00000	−0.213201
$793$	−28.0000	−0.994309
$794$	−14.0000	−0.496841
$795$	20.0000	0.709327
$796$	−6.00000	−0.212664
$797$	−2.00000	−0.0708436	−0.0354218	−	0.999372i	$-0.511277\pi$
$797$	−2.00000	−0.0708436	−0.0354218	+	0.999372i	$0.511277\pi$
$798$	0	0
$799$	12.0000	0.424529
$800$	5.00000	0.176777
$801$	6.00000	0.212000
$802$	18.0000	0.635602
$803$	−28.0000	−0.988099
$804$	−16.0000	−0.564276
$805$	12.0000	0.422944
$806$	−20.0000	−0.704470
$807$	20.0000	0.704033
$808$	18.0000	0.633238
$809$	26.0000	0.914111	0.457056	−	0.889438i	$-0.348904\pi$
$809$	26.0000	0.914111	0.457056	+	0.889438i	$0.348904\pi$
$810$	11.0000	0.386501
$811$	−2.00000	−0.0702295	−0.0351147	−	0.999383i	$-0.511180\pi$
$811$	−2.00000	−0.0702295	−0.0351147	+	0.999383i	$0.511180\pi$
$812$	−12.0000	−0.421117
$813$	−32.0000	−1.12229
$814$	4.00000	0.140200
$815$	2.00000	0.0700569
$816$	−2.00000	−0.0700140
$817$	0	0
$818$	−38.0000	−1.32864
$819$	−4.00000	−0.139771
$820$	10.0000	0.349215
$821$	2.00000	0.0698005	0.0349002	−	0.999391i	$-0.488889\pi$
$821$	2.00000	0.0698005	0.0349002	+	0.999391i	$0.488889\pi$
$822$	−4.00000	−0.139516
$823$	−34.0000	−1.18517	−0.592583	−	0.805510i	$-0.701892\pi$
$823$	−34.0000	−1.18517	−0.592583	+	0.805510i	$0.701892\pi$
$824$	−36.0000	−1.25412
$825$	4.00000	0.139262
$826$	−16.0000	−0.556711
$827$	−42.0000	−1.46048	−0.730242	−	0.683189i	$-0.760592\pi$
$827$	−42.0000	−1.46048	−0.730242	+	0.683189i	$0.760592\pi$
$828$	−6.00000	−0.208514
$829$	−2.00000	−0.0694629	−0.0347314	−	0.999397i	$-0.511058\pi$
$829$	−2.00000	−0.0694629	−0.0347314	+	0.999397i	$0.511058\pi$
$830$	−4.00000	−0.138842
$831$	−60.0000	−2.08138
$832$	14.0000	0.485363
$833$	−3.00000	−0.103944
$834$	28.0000	0.969561
$835$	−2.00000	−0.0692129
$836$	0	0
$837$	40.0000	1.38260
$838$	18.0000	0.621800
$839$	26.0000	0.897620	0.448810	−	0.893627i	$-0.351848\pi$
$839$	26.0000	0.897620	0.448810	+	0.893627i	$0.351848\pi$
$840$	−12.0000	−0.414039
$841$	7.00000	0.241379
$842$	26.0000	0.896019
$843$	−12.0000	−0.413302
$844$	−2.00000	−0.0688428
$845$	9.00000	0.309609
$846$	12.0000	0.412568
$847$	14.0000	0.481046
$848$	10.0000	0.343401
$849$	12.0000	0.411839
$850$	1.00000	0.0342997
$851$	12.0000	0.411355
$852$	4.00000	0.137038
$853$	34.0000	1.16414	0.582069	−	0.813139i	$-0.302243\pi$
$853$	34.0000	1.16414	0.582069	+	0.813139i	$0.302243\pi$
$854$	28.0000	0.958140
$855$	0	0
$856$	−6.00000	−0.205076
$857$	34.0000	1.16142	0.580709	−	0.814111i	$-0.302775\pi$
$857$	34.0000	1.16142	0.580709	+	0.814111i	$0.302775\pi$
$858$	8.00000	0.273115
$859$	−24.0000	−0.818869	−0.409435	−	0.912339i	$-0.634274\pi$
$859$	−24.0000	−0.818869	−0.409435	+	0.912339i	$0.634274\pi$
$860$	4.00000	0.136399
$861$	−40.0000	−1.36320
$862$	−38.0000	−1.29429
$863$	−4.00000	−0.136162	−0.0680808	−	0.997680i	$-0.521688\pi$
$863$	−4.00000	−0.136162	−0.0680808	+	0.997680i	$0.521688\pi$
$864$	−20.0000	−0.680414
$865$	6.00000	0.204006
$866$	−34.0000	−1.15537
$867$	2.00000	0.0679236
$868$	−20.0000	−0.678844
$869$	−28.0000	−0.949835
$870$	12.0000	0.406838
$871$	16.0000	0.542139
$872$	−6.00000	−0.203186
$873$	2.00000	0.0676897
$874$	0	0
$875$	2.00000	0.0676123
$876$	28.0000	0.946032
$877$	−14.0000	−0.472746	−0.236373	−	0.971662i	$-0.575959\pi$
$877$	−14.0000	−0.472746	−0.236373	+	0.971662i	$0.575959\pi$
$878$	6.00000	0.202490
$879$	60.0000	2.02375
$880$	2.00000	0.0674200
$881$	42.0000	1.41502	0.707508	−	0.706705i	$-0.249819\pi$
$881$	42.0000	1.41502	0.707508	+	0.706705i	$0.249819\pi$
$882$	−3.00000	−0.101015
$883$	16.0000	0.538443	0.269221	−	0.963078i	$-0.413234\pi$
$883$	16.0000	0.538443	0.269221	+	0.963078i	$0.413234\pi$
$884$	−2.00000	−0.0672673
$885$	−16.0000	−0.537834
$886$	24.0000	0.806296
$887$	−22.0000	−0.738688	−0.369344	−	0.929293i	$-0.620418\pi$
$887$	−22.0000	−0.738688	−0.369344	+	0.929293i	$0.620418\pi$
$888$	−12.0000	−0.402694
$889$	−16.0000	−0.536623
$890$	−6.00000	−0.201120
$891$	−22.0000	−0.737028
$892$	−16.0000	−0.535720
$893$	0	0
$894$	12.0000	0.401340
$895$	24.0000	0.802232
$896$	6.00000	0.200446
$897$	24.0000	0.801337
$898$	18.0000	0.600668
$899$	60.0000	2.00111
$900$	−1.00000	−0.0333333
$901$	−10.0000	−0.333148
$902$	20.0000	0.665927
$903$	−16.0000	−0.532447
$904$	18.0000	0.598671
$905$	−26.0000	−0.864269
$906$	−24.0000	−0.797347
$907$	30.0000	0.996134	0.498067	−	0.867139i	$-0.334043\pi$
$907$	30.0000	0.996134	0.498067	+	0.867139i	$0.334043\pi$
$908$	2.00000	0.0663723
$909$	−6.00000	−0.199007
$910$	4.00000	0.132599
$911$	−38.0000	−1.25900	−0.629498	−	0.777002i	$-0.716739\pi$
$911$	−38.0000	−1.25900	−0.629498	+	0.777002i	$0.716739\pi$
$912$	0	0
$913$	8.00000	0.264761
$914$	6.00000	0.198462
$915$	28.0000	0.925651
$916$	−10.0000	−0.330409
$917$	−12.0000	−0.396275
$918$	−4.00000	−0.132020
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	18.0000	0.593442
$921$	−32.0000	−1.05444
$922$	−26.0000	−0.856264
$923$	−4.00000	−0.131662
$924$	8.00000	0.263181
$925$	2.00000	0.0657596
$926$	−36.0000	−1.18303
$927$	12.0000	0.394132
$928$	−30.0000	−0.984798
$929$	58.0000	1.90292	0.951459	−	0.307775i	$-0.0995844\pi$
$929$	58.0000	1.90292	0.951459	+	0.307775i	$0.0995844\pi$
$930$	20.0000	0.655826
$931$	0	0
$932$	14.0000	0.458585
$933$	12.0000	0.392862
$934$	28.0000	0.916188
$935$	−2.00000	−0.0654070
$936$	−6.00000	−0.196116
$937$	−38.0000	−1.24141	−0.620703	−	0.784046i	$-0.713153\pi$
$937$	−38.0000	−1.24141	−0.620703	+	0.784046i	$0.713153\pi$
$938$	−16.0000	−0.522419
$939$	4.00000	0.130535
$940$	12.0000	0.391397
$941$	34.0000	1.10837	0.554184	−	0.832394i	$-0.313030\pi$
$941$	34.0000	1.10837	0.554184	+	0.832394i	$0.313030\pi$
$942$	12.0000	0.390981
$943$	60.0000	1.95387
$944$	−8.00000	−0.260378
$945$	−8.00000	−0.260240
$946$	8.00000	0.260102
$947$	26.0000	0.844886	0.422443	−	0.906389i	$-0.361173\pi$
$947$	26.0000	0.844886	0.422443	+	0.906389i	$0.361173\pi$
$948$	28.0000	0.909398
$949$	−28.0000	−0.908918
$950$	0	0
$951$	20.0000	0.648544
$952$	6.00000	0.194461
$953$	−18.0000	−0.583077	−0.291539	−	0.956559i	$-0.594167\pi$
$953$	−18.0000	−0.583077	−0.291539	+	0.956559i	$0.594167\pi$
$954$	−10.0000	−0.323762
$955$	−8.00000	−0.258874
$956$	8.00000	0.258738
$957$	−24.0000	−0.775810
$958$	−18.0000	−0.581554
$959$	4.00000	0.129167
$960$	−14.0000	−0.451848
$961$	69.0000	2.22581
$962$	4.00000	0.128965
$963$	2.00000	0.0644491
$964$	6.00000	0.193247
$965$	−18.0000	−0.579441
$966$	−24.0000	−0.772187
$967$	32.0000	1.02905	0.514525	−	0.857475i	$-0.327968\pi$
$967$	32.0000	1.02905	0.514525	+	0.857475i	$0.327968\pi$
$968$	21.0000	0.674966
$969$	0	0
$970$	−2.00000	−0.0642161
$971$	−8.00000	−0.256732	−0.128366	−	0.991727i	$-0.540973\pi$
$971$	−8.00000	−0.256732	−0.128366	+	0.991727i	$0.540973\pi$
$972$	10.0000	0.320750
$973$	−28.0000	−0.897639
$974$	22.0000	0.704925
$975$	4.00000	0.128103
$976$	14.0000	0.448129
$977$	34.0000	1.08776	0.543878	−	0.839164i	$-0.316955\pi$
$977$	34.0000	1.08776	0.543878	+	0.839164i	$0.316955\pi$
$978$	−4.00000	−0.127906
$979$	12.0000	0.383522
$980$	−3.00000	−0.0958315
$981$	2.00000	0.0638551
$982$	−16.0000	−0.510581
$983$	−6.00000	−0.191370	−0.0956851	−	0.995412i	$-0.530504\pi$
$983$	−6.00000	−0.191370	−0.0956851	+	0.995412i	$0.530504\pi$
$984$	−60.0000	−1.91273
$985$	−2.00000	−0.0637253
$986$	−6.00000	−0.191079
$987$	−48.0000	−1.52786
$988$	0	0
$989$	24.0000	0.763156
$990$	−2.00000	−0.0635642
$991$	−58.0000	−1.84243	−0.921215	−	0.389053i	$-0.872802\pi$
$991$	−58.0000	−1.84243	−0.921215	+	0.389053i	$0.872802\pi$
$992$	−50.0000	−1.58750
$993$	32.0000	1.01549
$994$	4.00000	0.126872
$995$	−6.00000	−0.190213
$996$	−8.00000	−0.253490
$997$	−62.0000	−1.96356	−0.981780	−	0.190022i	$-0.939144\pi$
$997$	−62.0000	−1.96356	−0.981780	+	0.190022i	$0.939144\pi$
$998$	−14.0000	−0.443162
$999$	−8.00000	−0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	85.2.a.a.1.1	✓	1
3.2	odd	2		765.2.a.a.1.1		1
4.3	odd	2		1360.2.a.b.1.1		1
5.2	odd	4		425.2.b.c.324.2		2
5.3	odd	4		425.2.b.c.324.1		2
5.4	even	2		425.2.a.a.1.1		1
7.6	odd	2		4165.2.a.l.1.1		1
8.3	odd	2		5440.2.a.x.1.1		1
8.5	even	2		5440.2.a.e.1.1		1
15.14	odd	2		3825.2.a.l.1.1		1
17.4	even	4		1445.2.d.a.866.1		2
17.13	even	4		1445.2.d.a.866.2		2
17.16	even	2		1445.2.a.c.1.1		1
20.19	odd	2		6800.2.a.v.1.1		1
85.84	even	2		7225.2.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.2.a.a.1.1	✓	1	1.1	even	1	trivial
425.2.a.a.1.1		1	5.4	even	2
425.2.b.c.324.1		2	5.3	odd	4
425.2.b.c.324.2		2	5.2	odd	4
765.2.a.a.1.1		1	3.2	odd	2
1360.2.a.b.1.1		1	4.3	odd	2
1445.2.a.c.1.1		1	17.16	even	2
1445.2.d.a.866.1		2	17.4	even	4
1445.2.d.a.866.2		2	17.13	even	4
3825.2.a.l.1.1		1	15.14	odd	2
4165.2.a.l.1.1		1	7.6	odd	2
5440.2.a.e.1.1		1	8.5	even	2
5440.2.a.x.1.1		1	8.3	odd	2
6800.2.a.v.1.1		1	20.19	odd	2
7225.2.a.d.1.1		1	85.84	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:	\( N \)	\(=\)	\( 85 = 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	85.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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