LMFDB - Embedded newform 178.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [178,2,Mod(1,178)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(178, 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("178.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 178 = 2 \cdot 89 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	178.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:	$1.42133715598$
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	$\chi$	$=$	178.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.00000 q^{5} +1.00000 q^{6} -4.00000 q^{7} +1.00000 q^{8} -2.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.00000 q^{5} +1.00000 q^{6} -4.00000 q^{7} +1.00000 q^{8} -2.00000 q^{9} +3.00000 q^{10} -6.00000 q^{11} +1.00000 q^{12} +2.00000 q^{13} -4.00000 q^{14} +3.00000 q^{15} +1.00000 q^{16} +3.00000 q^{17} -2.00000 q^{18} +5.00000 q^{19} +3.00000 q^{20} -4.00000 q^{21} -6.00000 q^{22} -3.00000 q^{23} +1.00000 q^{24} +4.00000 q^{25} +2.00000 q^{26} -5.00000 q^{27} -4.00000 q^{28} +3.00000 q^{30} +5.00000 q^{31} +1.00000 q^{32} -6.00000 q^{33} +3.00000 q^{34} -12.0000 q^{35} -2.00000 q^{36} -10.0000 q^{37} +5.00000 q^{38} +2.00000 q^{39} +3.00000 q^{40} -4.00000 q^{42} -1.00000 q^{43} -6.00000 q^{44} -6.00000 q^{45} -3.00000 q^{46} +12.0000 q^{47} +1.00000 q^{48} +9.00000 q^{49} +4.00000 q^{50} +3.00000 q^{51} +2.00000 q^{52} +9.00000 q^{53} -5.00000 q^{54} -18.0000 q^{55} -4.00000 q^{56} +5.00000 q^{57} +12.0000 q^{59} +3.00000 q^{60} -10.0000 q^{61} +5.00000 q^{62} +8.00000 q^{63} +1.00000 q^{64} +6.00000 q^{65} -6.00000 q^{66} -4.00000 q^{67} +3.00000 q^{68} -3.00000 q^{69} -12.0000 q^{70} -6.00000 q^{71} -2.00000 q^{72} -1.00000 q^{73} -10.0000 q^{74} +4.00000 q^{75} +5.00000 q^{76} +24.0000 q^{77} +2.00000 q^{78} -10.0000 q^{79} +3.00000 q^{80} +1.00000 q^{81} -12.0000 q^{83} -4.00000 q^{84} +9.00000 q^{85} -1.00000 q^{86} -6.00000 q^{88} -1.00000 q^{89} -6.00000 q^{90} -8.00000 q^{91} -3.00000 q^{92} +5.00000 q^{93} +12.0000 q^{94} +15.0000 q^{95} +1.00000 q^{96} +17.0000 q^{97} +9.00000 q^{98} +12.0000 q^{99} +O(q^{100})\)

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
$3$	1.00000	0.577350	0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	1.00000	0.500000
$5$	3.00000	1.34164	0.670820	−	0.741620i	$-0.265942\pi$
$5$	3.00000	1.34164	0.670820	+	0.741620i	$0.265942\pi$
$6$	1.00000	0.408248
$7$	−4.00000	−1.51186	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000	−1.51186	−0.755929	+	0.654654i	$0.772814\pi$
$8$	1.00000	0.353553
$9$	−2.00000	−0.666667
$10$	3.00000	0.948683
$11$	−6.00000	−1.80907	−0.904534	−	0.426401i	$-0.859781\pi$
$11$	−6.00000	−1.80907	−0.904534	+	0.426401i	$0.859781\pi$
$12$	1.00000	0.288675
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	−4.00000	−1.06904
$15$	3.00000	0.774597
$16$	1.00000	0.250000
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	−2.00000	−0.471405
$19$	5.00000	1.14708	0.573539	−	0.819178i	$-0.305570\pi$
$19$	5.00000	1.14708	0.573539	+	0.819178i	$0.305570\pi$
$20$	3.00000	0.670820
$21$	−4.00000	−0.872872
$22$	−6.00000	−1.27920
$23$	−3.00000	−0.625543	−0.312772	−	0.949828i	$-0.601257\pi$
$23$	−3.00000	−0.625543	−0.312772	+	0.949828i	$0.601257\pi$
$24$	1.00000	0.204124
$25$	4.00000	0.800000
$26$	2.00000	0.392232
$27$	−5.00000	−0.962250
$28$	−4.00000	−0.755929
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	3.00000	0.547723
$31$	5.00000	0.898027	0.449013	−	0.893525i	$-0.351776\pi$
$31$	5.00000	0.898027	0.449013	+	0.893525i	$0.351776\pi$
$32$	1.00000	0.176777
$33$	−6.00000	−1.04447
$34$	3.00000	0.514496
$35$	−12.0000	−2.02837
$36$	−2.00000	−0.333333
$37$	−10.0000	−1.64399	−0.821995	−	0.569495i	$-0.807139\pi$
$37$	−10.0000	−1.64399	−0.821995	+	0.569495i	$0.807139\pi$
$38$	5.00000	0.811107
$39$	2.00000	0.320256
$40$	3.00000	0.474342
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	−4.00000	−0.617213
$43$	−1.00000	−0.152499	−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000	−0.152499	−0.0762493	+	0.997089i	$0.524294\pi$
$44$	−6.00000	−0.904534
$45$	−6.00000	−0.894427
$46$	−3.00000	−0.442326
$47$	12.0000	1.75038	0.875190	−	0.483779i	$-0.160736\pi$
$47$	12.0000	1.75038	0.875190	+	0.483779i	$0.160736\pi$
$48$	1.00000	0.144338
$49$	9.00000	1.28571
$50$	4.00000	0.565685
$51$	3.00000	0.420084
$52$	2.00000	0.277350
$53$	9.00000	1.23625	0.618123	−	0.786082i	$-0.287894\pi$
$53$	9.00000	1.23625	0.618123	+	0.786082i	$0.287894\pi$
$54$	−5.00000	−0.680414
$55$	−18.0000	−2.42712
$56$	−4.00000	−0.534522
$57$	5.00000	0.662266
$58$	0	0
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	3.00000	0.387298
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	5.00000	0.635001
$63$	8.00000	1.00791
$64$	1.00000	0.125000
$65$	6.00000	0.744208
$66$	−6.00000	−0.738549
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	3.00000	0.363803
$69$	−3.00000	−0.361158
$70$	−12.0000	−1.43427
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	−2.00000	−0.235702
$73$	−1.00000	−0.117041	−0.0585206	−	0.998286i	$-0.518638\pi$
$73$	−1.00000	−0.117041	−0.0585206	+	0.998286i	$0.518638\pi$
$74$	−10.0000	−1.16248
$75$	4.00000	0.461880
$76$	5.00000	0.573539
$77$	24.0000	2.73505
$78$	2.00000	0.226455
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	3.00000	0.335410
$81$	1.00000	0.111111
$82$	0	0
$83$	−12.0000	−1.31717	−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000	−1.31717	−0.658586	+	0.752506i	$0.728845\pi$
$84$	−4.00000	−0.436436
$85$	9.00000	0.976187
$86$	−1.00000	−0.107833
$87$	0	0
$88$	−6.00000	−0.639602
$89$	−1.00000	−0.106000
$89$	−1.00000	−0.106000
$90$	−6.00000	−0.632456
$91$	−8.00000	−0.838628
$92$	−3.00000	−0.312772
$93$	5.00000	0.518476
$94$	12.0000	1.23771
$95$	15.0000	1.53897
$96$	1.00000	0.102062
$97$	17.0000	1.72609	0.863044	−	0.505128i	$-0.168555\pi$
$97$	17.0000	1.72609	0.863044	+	0.505128i	$0.168555\pi$
$98$	9.00000	0.909137
$99$	12.0000	1.20605
$100$	4.00000	0.400000
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	3.00000	0.297044
$103$	5.00000	0.492665	0.246332	−	0.969185i	$-0.420775\pi$
$103$	5.00000	0.492665	0.246332	+	0.969185i	$0.420775\pi$
$104$	2.00000	0.196116
$105$	−12.0000	−1.17108
$106$	9.00000	0.874157
$107$	18.0000	1.74013	0.870063	−	0.492941i	$-0.164078\pi$
$107$	18.0000	1.74013	0.870063	+	0.492941i	$0.164078\pi$
$108$	−5.00000	−0.481125
$109$	−7.00000	−0.670478	−0.335239	−	0.942133i	$-0.608817\pi$
$109$	−7.00000	−0.670478	−0.335239	+	0.942133i	$0.608817\pi$
$110$	−18.0000	−1.71623
$111$	−10.0000	−0.949158
$112$	−4.00000	−0.377964
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	5.00000	0.468293
$115$	−9.00000	−0.839254
$116$	0	0
$117$	−4.00000	−0.369800
$118$	12.0000	1.10469
$119$	−12.0000	−1.10004
$120$	3.00000	0.273861
$121$	25.0000	2.27273
$122$	−10.0000	−0.905357
$123$	0	0
$124$	5.00000	0.449013
$125$	−3.00000	−0.268328
$126$	8.00000	0.712697
$127$	−7.00000	−0.621150	−0.310575	−	0.950549i	$-0.600522\pi$
$127$	−7.00000	−0.621150	−0.310575	+	0.950549i	$0.600522\pi$
$128$	1.00000	0.0883883
$129$	−1.00000	−0.0880451
$130$	6.00000	0.526235
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	−6.00000	−0.522233
$133$	−20.0000	−1.73422
$134$	−4.00000	−0.345547
$135$	−15.0000	−1.29099
$136$	3.00000	0.257248
$137$	18.0000	1.53784	0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000	1.53784	0.768922	+	0.639343i	$0.220793\pi$
$138$	−3.00000	−0.255377
$139$	8.00000	0.678551	0.339276	−	0.940687i	$-0.389818\pi$
$139$	8.00000	0.678551	0.339276	+	0.940687i	$0.389818\pi$
$140$	−12.0000	−1.01419
$141$	12.0000	1.01058
$142$	−6.00000	−0.503509
$143$	−12.0000	−1.00349
$144$	−2.00000	−0.166667
$145$	0	0
$146$	−1.00000	−0.0827606
$147$	9.00000	0.742307
$148$	−10.0000	−0.821995
$149$	−12.0000	−0.983078	−0.491539	−	0.870855i	$-0.663566\pi$
$149$	−12.0000	−0.983078	−0.491539	+	0.870855i	$0.663566\pi$
$150$	4.00000	0.326599
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	5.00000	0.405554
$153$	−6.00000	−0.485071
$154$	24.0000	1.93398
$155$	15.0000	1.20483
$156$	2.00000	0.160128
$157$	14.0000	1.11732	0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000	1.11732	0.558661	+	0.829396i	$0.311315\pi$
$158$	−10.0000	−0.795557
$159$	9.00000	0.713746
$160$	3.00000	0.237171
$161$	12.0000	0.945732
$162$	1.00000	0.0785674
$163$	−1.00000	−0.0783260	−0.0391630	−	0.999233i	$-0.512469\pi$
$163$	−1.00000	−0.0783260	−0.0391630	+	0.999233i	$0.512469\pi$
$164$	0	0
$165$	−18.0000	−1.40130
$166$	−12.0000	−0.931381
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	−4.00000	−0.308607
$169$	−9.00000	−0.692308
$170$	9.00000	0.690268
$171$	−10.0000	−0.764719
$172$	−1.00000	−0.0762493
$173$	3.00000	0.228086	0.114043	−	0.993476i	$-0.463620\pi$
$173$	3.00000	0.228086	0.114043	+	0.993476i	$0.463620\pi$
$174$	0	0
$175$	−16.0000	−1.20949
$176$	−6.00000	−0.452267
$177$	12.0000	0.901975
$178$	−1.00000	−0.0749532
$179$	18.0000	1.34538	0.672692	−	0.739923i	$-0.265138\pi$
$179$	18.0000	1.34538	0.672692	+	0.739923i	$0.265138\pi$
$180$	−6.00000	−0.447214
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	−8.00000	−0.592999
$183$	−10.0000	−0.739221
$184$	−3.00000	−0.221163
$185$	−30.0000	−2.20564
$186$	5.00000	0.366618
$187$	−18.0000	−1.31629
$188$	12.0000	0.875190
$189$	20.0000	1.45479
$190$	15.0000	1.08821
$191$	−3.00000	−0.217072	−0.108536	−	0.994092i	$-0.534616\pi$
$191$	−3.00000	−0.217072	−0.108536	+	0.994092i	$0.534616\pi$
$192$	1.00000	0.0721688
$193$	−22.0000	−1.58359	−0.791797	−	0.610784i	$-0.790854\pi$
$193$	−22.0000	−1.58359	−0.791797	+	0.610784i	$0.790854\pi$
$194$	17.0000	1.22053
$195$	6.00000	0.429669
$196$	9.00000	0.642857
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	12.0000	0.852803
$199$	−10.0000	−0.708881	−0.354441	−	0.935079i	$-0.615329\pi$
$199$	−10.0000	−0.708881	−0.354441	+	0.935079i	$0.615329\pi$
$200$	4.00000	0.282843
$201$	−4.00000	−0.282138
$202$	0	0
$203$	0	0
$204$	3.00000	0.210042
$205$	0	0
$206$	5.00000	0.348367
$207$	6.00000	0.417029
$208$	2.00000	0.138675
$209$	−30.0000	−2.07514
$210$	−12.0000	−0.828079
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	9.00000	0.618123
$213$	−6.00000	−0.411113
$214$	18.0000	1.23045
$215$	−3.00000	−0.204598
$216$	−5.00000	−0.340207
$217$	−20.0000	−1.35769
$218$	−7.00000	−0.474100
$219$	−1.00000	−0.0675737
$220$	−18.0000	−1.21356
$221$	6.00000	0.403604
$222$	−10.0000	−0.671156
$223$	8.00000	0.535720	0.267860	−	0.963458i	$-0.413684\pi$
$223$	8.00000	0.535720	0.267860	+	0.963458i	$0.413684\pi$
$224$	−4.00000	−0.267261
$225$	−8.00000	−0.533333
$226$	−6.00000	−0.399114
$227$	24.0000	1.59294	0.796468	−	0.604681i	$-0.206699\pi$
$227$	24.0000	1.59294	0.796468	+	0.604681i	$0.206699\pi$
$228$	5.00000	0.331133
$229$	−4.00000	−0.264327	−0.132164	−	0.991228i	$-0.542192\pi$
$229$	−4.00000	−0.264327	−0.132164	+	0.991228i	$0.542192\pi$
$230$	−9.00000	−0.593442
$231$	24.0000	1.57908
$232$	0	0
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	−4.00000	−0.261488
$235$	36.0000	2.34838
$236$	12.0000	0.781133
$237$	−10.0000	−0.649570
$238$	−12.0000	−0.777844
$239$	15.0000	0.970269	0.485135	−	0.874439i	$-0.338771\pi$
$239$	15.0000	0.970269	0.485135	+	0.874439i	$0.338771\pi$
$240$	3.00000	0.193649
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	25.0000	1.60706
$243$	16.0000	1.02640
$244$	−10.0000	−0.640184
$245$	27.0000	1.72497
$246$	0	0
$247$	10.0000	0.636285
$248$	5.00000	0.317500
$249$	−12.0000	−0.760469
$250$	−3.00000	−0.189737
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	8.00000	0.503953
$253$	18.0000	1.13165
$254$	−7.00000	−0.439219
$255$	9.00000	0.563602
$256$	1.00000	0.0625000
$257$	−3.00000	−0.187135	−0.0935674	−	0.995613i	$-0.529827\pi$
$257$	−3.00000	−0.187135	−0.0935674	+	0.995613i	$0.529827\pi$
$258$	−1.00000	−0.0622573
$259$	40.0000	2.48548
$260$	6.00000	0.372104
$261$	0	0
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	−6.00000	−0.369274
$265$	27.0000	1.65860
$266$	−20.0000	−1.22628
$267$	−1.00000	−0.0611990
$268$	−4.00000	−0.244339
$269$	15.0000	0.914566	0.457283	−	0.889321i	$-0.348823\pi$
$269$	15.0000	0.914566	0.457283	+	0.889321i	$0.348823\pi$
$270$	−15.0000	−0.912871
$271$	−22.0000	−1.33640	−0.668202	−	0.743980i	$-0.732936\pi$
$271$	−22.0000	−1.33640	−0.668202	+	0.743980i	$0.732936\pi$
$272$	3.00000	0.181902
$273$	−8.00000	−0.484182
$274$	18.0000	1.08742
$275$	−24.0000	−1.44725
$276$	−3.00000	−0.180579
$277$	−1.00000	−0.0600842	−0.0300421	−	0.999549i	$-0.509564\pi$
$277$	−1.00000	−0.0600842	−0.0300421	+	0.999549i	$0.509564\pi$
$278$	8.00000	0.479808
$279$	−10.0000	−0.598684
$280$	−12.0000	−0.717137
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	12.0000	0.714590
$283$	14.0000	0.832214	0.416107	−	0.909316i	$-0.363394\pi$
$283$	14.0000	0.832214	0.416107	+	0.909316i	$0.363394\pi$
$284$	−6.00000	−0.356034
$285$	15.0000	0.888523
$286$	−12.0000	−0.709575
$287$	0	0
$288$	−2.00000	−0.117851
$289$	−8.00000	−0.470588
$290$	0	0
$291$	17.0000	0.996558
$292$	−1.00000	−0.0585206
$293$	−18.0000	−1.05157	−0.525786	−	0.850617i	$-0.676229\pi$
$293$	−18.0000	−1.05157	−0.525786	+	0.850617i	$0.676229\pi$
$294$	9.00000	0.524891
$295$	36.0000	2.09600
$296$	−10.0000	−0.581238
$297$	30.0000	1.74078
$298$	−12.0000	−0.695141
$299$	−6.00000	−0.346989
$300$	4.00000	0.230940
$301$	4.00000	0.230556
$302$	−16.0000	−0.920697
$303$	0	0
$304$	5.00000	0.286770
$305$	−30.0000	−1.71780
$306$	−6.00000	−0.342997
$307$	2.00000	0.114146	0.0570730	−	0.998370i	$-0.481823\pi$
$307$	2.00000	0.114146	0.0570730	+	0.998370i	$0.481823\pi$
$308$	24.0000	1.36753
$309$	5.00000	0.284440
$310$	15.0000	0.851943
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	2.00000	0.113228
$313$	20.0000	1.13047	0.565233	−	0.824931i	$-0.308786\pi$
$313$	20.0000	1.13047	0.565233	+	0.824931i	$0.308786\pi$
$314$	14.0000	0.790066
$315$	24.0000	1.35225
$316$	−10.0000	−0.562544
$317$	−27.0000	−1.51647	−0.758236	−	0.651981i	$-0.773938\pi$
$317$	−27.0000	−1.51647	−0.758236	+	0.651981i	$0.773938\pi$
$318$	9.00000	0.504695
$319$	0	0
$320$	3.00000	0.167705
$321$	18.0000	1.00466
$322$	12.0000	0.668734
$323$	15.0000	0.834622
$324$	1.00000	0.0555556
$325$	8.00000	0.443760
$326$	−1.00000	−0.0553849
$327$	−7.00000	−0.387101
$328$	0	0
$329$	−48.0000	−2.64633
$330$	−18.0000	−0.990867
$331$	−34.0000	−1.86881	−0.934405	−	0.356214i	$-0.884068\pi$
$331$	−34.0000	−1.86881	−0.934405	+	0.356214i	$0.884068\pi$
$332$	−12.0000	−0.658586
$333$	20.0000	1.09599
$334$	−12.0000	−0.656611
$335$	−12.0000	−0.655630
$336$	−4.00000	−0.218218
$337$	8.00000	0.435788	0.217894	−	0.975972i	$-0.430081\pi$
$337$	8.00000	0.435788	0.217894	+	0.975972i	$0.430081\pi$
$338$	−9.00000	−0.489535
$339$	−6.00000	−0.325875
$340$	9.00000	0.488094
$341$	−30.0000	−1.62459
$342$	−10.0000	−0.540738
$343$	−8.00000	−0.431959
$344$	−1.00000	−0.0539164
$345$	−9.00000	−0.484544
$346$	3.00000	0.161281
$347$	18.0000	0.966291	0.483145	−	0.875540i	$-0.339494\pi$
$347$	18.0000	0.966291	0.483145	+	0.875540i	$0.339494\pi$
$348$	0	0
$349$	−16.0000	−0.856460	−0.428230	−	0.903670i	$-0.640863\pi$
$349$	−16.0000	−0.856460	−0.428230	+	0.903670i	$0.640863\pi$
$350$	−16.0000	−0.855236
$351$	−10.0000	−0.533761
$352$	−6.00000	−0.319801
$353$	6.00000	0.319348	0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000	0.319348	0.159674	+	0.987170i	$0.448956\pi$
$354$	12.0000	0.637793
$355$	−18.0000	−0.955341
$356$	−1.00000	−0.0529999
$357$	−12.0000	−0.635107
$358$	18.0000	0.951330
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	−6.00000	−0.316228
$361$	6.00000	0.315789
$362$	−10.0000	−0.525588
$363$	25.0000	1.31216
$364$	−8.00000	−0.419314
$365$	−3.00000	−0.157027
$366$	−10.0000	−0.522708
$367$	−16.0000	−0.835193	−0.417597	−	0.908633i	$-0.637127\pi$
$367$	−16.0000	−0.835193	−0.417597	+	0.908633i	$0.637127\pi$
$368$	−3.00000	−0.156386
$369$	0	0
$370$	−30.0000	−1.55963
$371$	−36.0000	−1.86903
$372$	5.00000	0.259238
$373$	−25.0000	−1.29445	−0.647225	−	0.762299i	$-0.724071\pi$
$373$	−25.0000	−1.29445	−0.647225	+	0.762299i	$0.724071\pi$
$374$	−18.0000	−0.930758
$375$	−3.00000	−0.154919
$376$	12.0000	0.618853
$377$	0	0
$378$	20.0000	1.02869
$379$	−1.00000	−0.0513665	−0.0256833	−	0.999670i	$-0.508176\pi$
$379$	−1.00000	−0.0513665	−0.0256833	+	0.999670i	$0.508176\pi$
$380$	15.0000	0.769484
$381$	−7.00000	−0.358621
$382$	−3.00000	−0.153493
$383$	15.0000	0.766464	0.383232	−	0.923652i	$-0.374811\pi$
$383$	15.0000	0.766464	0.383232	+	0.923652i	$0.374811\pi$
$384$	1.00000	0.0510310
$385$	72.0000	3.66946
$386$	−22.0000	−1.11977
$387$	2.00000	0.101666
$388$	17.0000	0.863044
$389$	36.0000	1.82527	0.912636	−	0.408773i	$-0.134043\pi$
$389$	36.0000	1.82527	0.912636	+	0.408773i	$0.134043\pi$
$390$	6.00000	0.303822
$391$	−9.00000	−0.455150
$392$	9.00000	0.454569
$393$	0	0
$394$	0	0
$395$	−30.0000	−1.50946
$396$	12.0000	0.603023
$397$	20.0000	1.00377	0.501886	−	0.864934i	$-0.332640\pi$
$397$	20.0000	1.00377	0.501886	+	0.864934i	$0.332640\pi$
$398$	−10.0000	−0.501255
$399$	−20.0000	−1.00125
$400$	4.00000	0.200000
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	−4.00000	−0.199502
$403$	10.0000	0.498135
$404$	0	0
$405$	3.00000	0.149071
$406$	0	0
$407$	60.0000	2.97409
$408$	3.00000	0.148522
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	0	0
$411$	18.0000	0.887875
$412$	5.00000	0.246332
$413$	−48.0000	−2.36193
$414$	6.00000	0.294884
$415$	−36.0000	−1.76717
$416$	2.00000	0.0980581
$417$	8.00000	0.391762
$418$	−30.0000	−1.46735
$419$	9.00000	0.439679	0.219839	−	0.975536i	$-0.429447\pi$
$419$	9.00000	0.439679	0.219839	+	0.975536i	$0.429447\pi$
$420$	−12.0000	−0.585540
$421$	8.00000	0.389896	0.194948	−	0.980814i	$-0.437546\pi$
$421$	8.00000	0.389896	0.194948	+	0.980814i	$0.437546\pi$
$422$	−4.00000	−0.194717
$423$	−24.0000	−1.16692
$424$	9.00000	0.437079
$425$	12.0000	0.582086
$426$	−6.00000	−0.290701
$427$	40.0000	1.93574
$428$	18.0000	0.870063
$429$	−12.0000	−0.579365
$430$	−3.00000	−0.144673
$431$	−15.0000	−0.722525	−0.361262	−	0.932464i	$-0.617654\pi$
$431$	−15.0000	−0.722525	−0.361262	+	0.932464i	$0.617654\pi$
$432$	−5.00000	−0.240563
$433$	−16.0000	−0.768911	−0.384455	−	0.923144i	$-0.625611\pi$
$433$	−16.0000	−0.768911	−0.384455	+	0.923144i	$0.625611\pi$
$434$	−20.0000	−0.960031
$435$	0	0
$436$	−7.00000	−0.335239
$437$	−15.0000	−0.717547
$438$	−1.00000	−0.0477818
$439$	−1.00000	−0.0477274	−0.0238637	−	0.999715i	$-0.507597\pi$
$439$	−1.00000	−0.0477274	−0.0238637	+	0.999715i	$0.507597\pi$
$440$	−18.0000	−0.858116
$441$	−18.0000	−0.857143
$442$	6.00000	0.285391
$443$	6.00000	0.285069	0.142534	−	0.989790i	$-0.454475\pi$
$443$	6.00000	0.285069	0.142534	+	0.989790i	$0.454475\pi$
$444$	−10.0000	−0.474579
$445$	−3.00000	−0.142214
$446$	8.00000	0.378811
$447$	−12.0000	−0.567581
$448$	−4.00000	−0.188982
$449$	39.0000	1.84052	0.920262	−	0.391303i	$-0.127976\pi$
$449$	39.0000	1.84052	0.920262	+	0.391303i	$0.127976\pi$
$450$	−8.00000	−0.377124
$451$	0	0
$452$	−6.00000	−0.282216
$453$	−16.0000	−0.751746
$454$	24.0000	1.12638
$455$	−24.0000	−1.12514
$456$	5.00000	0.234146
$457$	−10.0000	−0.467780	−0.233890	−	0.972263i	$-0.575146\pi$
$457$	−10.0000	−0.467780	−0.233890	+	0.972263i	$0.575146\pi$
$458$	−4.00000	−0.186908
$459$	−15.0000	−0.700140
$460$	−9.00000	−0.419627
$461$	18.0000	0.838344	0.419172	−	0.907907i	$-0.362320\pi$
$461$	18.0000	0.838344	0.419172	+	0.907907i	$0.362320\pi$
$462$	24.0000	1.11658
$463$	20.0000	0.929479	0.464739	−	0.885448i	$-0.346148\pi$
$463$	20.0000	0.929479	0.464739	+	0.885448i	$0.346148\pi$
$464$	0	0
$465$	15.0000	0.695608
$466$	6.00000	0.277945
$467$	18.0000	0.832941	0.416470	−	0.909149i	$-0.363267\pi$
$467$	18.0000	0.832941	0.416470	+	0.909149i	$0.363267\pi$
$468$	−4.00000	−0.184900
$469$	16.0000	0.738811
$470$	36.0000	1.66056
$471$	14.0000	0.645086
$472$	12.0000	0.552345
$473$	6.00000	0.275880
$474$	−10.0000	−0.459315
$475$	20.0000	0.917663
$476$	−12.0000	−0.550019
$477$	−18.0000	−0.824163
$478$	15.0000	0.686084
$479$	−6.00000	−0.274147	−0.137073	−	0.990561i	$-0.543770\pi$
$479$	−6.00000	−0.274147	−0.137073	+	0.990561i	$0.543770\pi$
$480$	3.00000	0.136931
$481$	−20.0000	−0.911922
$482$	−22.0000	−1.00207
$483$	12.0000	0.546019
$484$	25.0000	1.13636
$485$	51.0000	2.31579
$486$	16.0000	0.725775
$487$	−4.00000	−0.181257	−0.0906287	−	0.995885i	$-0.528888\pi$
$487$	−4.00000	−0.181257	−0.0906287	+	0.995885i	$0.528888\pi$
$488$	−10.0000	−0.452679
$489$	−1.00000	−0.0452216
$490$	27.0000	1.21974
$491$	−15.0000	−0.676941	−0.338470	−	0.940977i	$-0.609909\pi$
$491$	−15.0000	−0.676941	−0.338470	+	0.940977i	$0.609909\pi$
$492$	0	0
$493$	0	0
$494$	10.0000	0.449921
$495$	36.0000	1.61808
$496$	5.00000	0.224507
$497$	24.0000	1.07655
$498$	−12.0000	−0.537733
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	−3.00000	−0.134164
$501$	−12.0000	−0.536120
$502$	−12.0000	−0.535586
$503$	24.0000	1.07011	0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000	1.07011	0.535054	+	0.844818i	$0.320291\pi$
$504$	8.00000	0.356348
$505$	0	0
$506$	18.0000	0.800198
$507$	−9.00000	−0.399704
$508$	−7.00000	−0.310575
$509$	−33.0000	−1.46270	−0.731350	−	0.682003i	$-0.761109\pi$
$509$	−33.0000	−1.46270	−0.731350	+	0.682003i	$0.761109\pi$
$510$	9.00000	0.398527
$511$	4.00000	0.176950
$512$	1.00000	0.0441942
$513$	−25.0000	−1.10378
$514$	−3.00000	−0.132324
$515$	15.0000	0.660979
$516$	−1.00000	−0.0440225
$517$	−72.0000	−3.16656
$518$	40.0000	1.75750
$519$	3.00000	0.131685
$520$	6.00000	0.263117
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	0	0
$523$	−4.00000	−0.174908	−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000	−0.174908	−0.0874539	+	0.996169i	$0.527873\pi$
$524$	0	0
$525$	−16.0000	−0.698297
$526$	0	0
$527$	15.0000	0.653410
$528$	−6.00000	−0.261116
$529$	−14.0000	−0.608696
$530$	27.0000	1.17281
$531$	−24.0000	−1.04151
$532$	−20.0000	−0.867110
$533$	0	0
$534$	−1.00000	−0.0432742
$535$	54.0000	2.33462
$536$	−4.00000	−0.172774
$537$	18.0000	0.776757
$538$	15.0000	0.646696
$539$	−54.0000	−2.32594
$540$	−15.0000	−0.645497
$541$	38.0000	1.63375	0.816874	−	0.576816i	$-0.195705\pi$
$541$	38.0000	1.63375	0.816874	+	0.576816i	$0.195705\pi$
$542$	−22.0000	−0.944981
$543$	−10.0000	−0.429141
$544$	3.00000	0.128624
$545$	−21.0000	−0.899541
$546$	−8.00000	−0.342368
$547$	−28.0000	−1.19719	−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000	−1.19719	−0.598597	+	0.801050i	$0.704275\pi$
$548$	18.0000	0.768922
$549$	20.0000	0.853579
$550$	−24.0000	−1.02336
$551$	0	0
$552$	−3.00000	−0.127688
$553$	40.0000	1.70097
$554$	−1.00000	−0.0424859
$555$	−30.0000	−1.27343
$556$	8.00000	0.339276
$557$	−24.0000	−1.01691	−0.508456	−	0.861088i	$-0.669784\pi$
$557$	−24.0000	−1.01691	−0.508456	+	0.861088i	$0.669784\pi$
$558$	−10.0000	−0.423334
$559$	−2.00000	−0.0845910
$560$	−12.0000	−0.507093
$561$	−18.0000	−0.759961
$562$	−6.00000	−0.253095
$563$	−21.0000	−0.885044	−0.442522	−	0.896758i	$-0.645916\pi$
$563$	−21.0000	−0.885044	−0.442522	+	0.896758i	$0.645916\pi$
$564$	12.0000	0.505291
$565$	−18.0000	−0.757266
$566$	14.0000	0.588464
$567$	−4.00000	−0.167984
$568$	−6.00000	−0.251754
$569$	−18.0000	−0.754599	−0.377300	−	0.926091i	$-0.623147\pi$
$569$	−18.0000	−0.754599	−0.377300	+	0.926091i	$0.623147\pi$
$570$	15.0000	0.628281
$571$	47.0000	1.96689	0.983444	−	0.181210i	$-0.0580014\pi$
$571$	47.0000	1.96689	0.983444	+	0.181210i	$0.0580014\pi$
$572$	−12.0000	−0.501745
$573$	−3.00000	−0.125327
$574$	0	0
$575$	−12.0000	−0.500435
$576$	−2.00000	−0.0833333
$577$	44.0000	1.83174	0.915872	−	0.401470i	$-0.131501\pi$
$577$	44.0000	1.83174	0.915872	+	0.401470i	$0.131501\pi$
$578$	−8.00000	−0.332756
$579$	−22.0000	−0.914289
$580$	0	0
$581$	48.0000	1.99138
$582$	17.0000	0.704673
$583$	−54.0000	−2.23645
$584$	−1.00000	−0.0413803
$585$	−12.0000	−0.496139
$586$	−18.0000	−0.743573
$587$	−18.0000	−0.742940	−0.371470	−	0.928445i	$-0.621146\pi$
$587$	−18.0000	−0.742940	−0.371470	+	0.928445i	$0.621146\pi$
$588$	9.00000	0.371154
$589$	25.0000	1.03011
$590$	36.0000	1.48210
$591$	0	0
$592$	−10.0000	−0.410997
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	30.0000	1.23091
$595$	−36.0000	−1.47586
$596$	−12.0000	−0.491539
$597$	−10.0000	−0.409273
$598$	−6.00000	−0.245358
$599$	−3.00000	−0.122577	−0.0612883	−	0.998120i	$-0.519521\pi$
$599$	−3.00000	−0.122577	−0.0612883	+	0.998120i	$0.519521\pi$
$600$	4.00000	0.163299
$601$	35.0000	1.42768	0.713840	−	0.700309i	$-0.246954\pi$
$601$	35.0000	1.42768	0.713840	+	0.700309i	$0.246954\pi$
$602$	4.00000	0.163028
$603$	8.00000	0.325785
$604$	−16.0000	−0.651031
$605$	75.0000	3.04918
$606$	0	0
$607$	2.00000	0.0811775	0.0405887	−	0.999176i	$-0.487077\pi$
$607$	2.00000	0.0811775	0.0405887	+	0.999176i	$0.487077\pi$
$608$	5.00000	0.202777
$609$	0	0
$610$	−30.0000	−1.21466
$611$	24.0000	0.970936
$612$	−6.00000	−0.242536
$613$	41.0000	1.65597	0.827987	−	0.560747i	$-0.189486\pi$
$613$	41.0000	1.65597	0.827987	+	0.560747i	$0.189486\pi$
$614$	2.00000	0.0807134
$615$	0	0
$616$	24.0000	0.966988
$617$	−36.0000	−1.44931	−0.724653	−	0.689114i	$-0.758000\pi$
$617$	−36.0000	−1.44931	−0.724653	+	0.689114i	$0.758000\pi$
$618$	5.00000	0.201129
$619$	44.0000	1.76851	0.884255	−	0.467005i	$-0.154667\pi$
$619$	44.0000	1.76851	0.884255	+	0.467005i	$0.154667\pi$
$620$	15.0000	0.602414
$621$	15.0000	0.601929
$622$	12.0000	0.481156
$623$	4.00000	0.160257
$624$	2.00000	0.0800641
$625$	−29.0000	−1.16000
$626$	20.0000	0.799361
$627$	−30.0000	−1.19808
$628$	14.0000	0.558661
$629$	−30.0000	−1.19618
$630$	24.0000	0.956183
$631$	−40.0000	−1.59237	−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000	−1.59237	−0.796187	+	0.605050i	$0.793153\pi$
$632$	−10.0000	−0.397779
$633$	−4.00000	−0.158986
$634$	−27.0000	−1.07231
$635$	−21.0000	−0.833360
$636$	9.00000	0.356873
$637$	18.0000	0.713186
$638$	0	0
$639$	12.0000	0.474713
$640$	3.00000	0.118585
$641$	9.00000	0.355479	0.177739	−	0.984078i	$-0.443122\pi$
$641$	9.00000	0.355479	0.177739	+	0.984078i	$0.443122\pi$
$642$	18.0000	0.710403
$643$	32.0000	1.26196	0.630978	−	0.775800i	$-0.282654\pi$
$643$	32.0000	1.26196	0.630978	+	0.775800i	$0.282654\pi$
$644$	12.0000	0.472866
$645$	−3.00000	−0.118125
$646$	15.0000	0.590167
$647$	24.0000	0.943537	0.471769	−	0.881722i	$-0.343616\pi$
$647$	24.0000	0.943537	0.471769	+	0.881722i	$0.343616\pi$
$648$	1.00000	0.0392837
$649$	−72.0000	−2.82625
$650$	8.00000	0.313786
$651$	−20.0000	−0.783862
$652$	−1.00000	−0.0391630
$653$	48.0000	1.87839	0.939193	−	0.343391i	$-0.111576\pi$
$653$	48.0000	1.87839	0.939193	+	0.343391i	$0.111576\pi$
$654$	−7.00000	−0.273722
$655$	0	0
$656$	0	0
$657$	2.00000	0.0780274
$658$	−48.0000	−1.87123
$659$	24.0000	0.934907	0.467454	−	0.884018i	$-0.345171\pi$
$659$	24.0000	0.934907	0.467454	+	0.884018i	$0.345171\pi$
$660$	−18.0000	−0.700649
$661$	2.00000	0.0777910	0.0388955	−	0.999243i	$-0.487616\pi$
$661$	2.00000	0.0777910	0.0388955	+	0.999243i	$0.487616\pi$
$662$	−34.0000	−1.32145
$663$	6.00000	0.233021
$664$	−12.0000	−0.465690
$665$	−60.0000	−2.32670
$666$	20.0000	0.774984
$667$	0	0
$668$	−12.0000	−0.464294
$669$	8.00000	0.309298
$670$	−12.0000	−0.463600
$671$	60.0000	2.31627
$672$	−4.00000	−0.154303
$673$	29.0000	1.11787	0.558934	−	0.829212i	$-0.311211\pi$
$673$	29.0000	1.11787	0.558934	+	0.829212i	$0.311211\pi$
$674$	8.00000	0.308148
$675$	−20.0000	−0.769800
$676$	−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	$0.695581\pi$
$678$	−6.00000	−0.230429
$679$	−68.0000	−2.60960
$680$	9.00000	0.345134
$681$	24.0000	0.919682
$682$	−30.0000	−1.14876
$683$	15.0000	0.573959	0.286980	−	0.957937i	$-0.407349\pi$
$683$	15.0000	0.573959	0.286980	+	0.957937i	$0.407349\pi$
$684$	−10.0000	−0.382360
$685$	54.0000	2.06323
$686$	−8.00000	−0.305441
$687$	−4.00000	−0.152610
$688$	−1.00000	−0.0381246
$689$	18.0000	0.685745
$690$	−9.00000	−0.342624
$691$	−10.0000	−0.380418	−0.190209	−	0.981744i	$-0.560917\pi$
$691$	−10.0000	−0.380418	−0.190209	+	0.981744i	$0.560917\pi$
$692$	3.00000	0.114043
$693$	−48.0000	−1.82337
$694$	18.0000	0.683271
$695$	24.0000	0.910372
$696$	0	0
$697$	0	0
$698$	−16.0000	−0.605609
$699$	6.00000	0.226941
$700$	−16.0000	−0.604743
$701$	−15.0000	−0.566542	−0.283271	−	0.959040i	$-0.591420\pi$
$701$	−15.0000	−0.566542	−0.283271	+	0.959040i	$0.591420\pi$
$702$	−10.0000	−0.377426
$703$	−50.0000	−1.88579
$704$	−6.00000	−0.226134
$705$	36.0000	1.35584
$706$	6.00000	0.225813
$707$	0	0
$708$	12.0000	0.450988
$709$	−22.0000	−0.826227	−0.413114	−	0.910679i	$-0.635559\pi$
$709$	−22.0000	−0.826227	−0.413114	+	0.910679i	$0.635559\pi$
$710$	−18.0000	−0.675528
$711$	20.0000	0.750059
$712$	−1.00000	−0.0374766
$713$	−15.0000	−0.561754
$714$	−12.0000	−0.449089
$715$	−36.0000	−1.34632
$716$	18.0000	0.672692
$717$	15.0000	0.560185
$718$	−24.0000	−0.895672
$719$	21.0000	0.783168	0.391584	−	0.920142i	$-0.371927\pi$
$719$	21.0000	0.783168	0.391584	+	0.920142i	$0.371927\pi$
$720$	−6.00000	−0.223607
$721$	−20.0000	−0.744839
$722$	6.00000	0.223297
$723$	−22.0000	−0.818189
$724$	−10.0000	−0.371647
$725$	0	0
$726$	25.0000	0.927837
$727$	−1.00000	−0.0370879	−0.0185440	−	0.999828i	$-0.505903\pi$
$727$	−1.00000	−0.0370879	−0.0185440	+	0.999828i	$0.505903\pi$
$728$	−8.00000	−0.296500
$729$	13.0000	0.481481
$730$	−3.00000	−0.111035
$731$	−3.00000	−0.110959
$732$	−10.0000	−0.369611
$733$	5.00000	0.184679	0.0923396	−	0.995728i	$-0.470565\pi$
$733$	5.00000	0.184679	0.0923396	+	0.995728i	$0.470565\pi$
$734$	−16.0000	−0.590571
$735$	27.0000	0.995910
$736$	−3.00000	−0.110581
$737$	24.0000	0.884051
$738$	0	0
$739$	−7.00000	−0.257499	−0.128750	−	0.991677i	$-0.541096\pi$
$739$	−7.00000	−0.257499	−0.128750	+	0.991677i	$0.541096\pi$
$740$	−30.0000	−1.10282
$741$	10.0000	0.367359
$742$	−36.0000	−1.32160
$743$	−3.00000	−0.110059	−0.0550297	−	0.998485i	$-0.517525\pi$
$743$	−3.00000	−0.110059	−0.0550297	+	0.998485i	$0.517525\pi$
$744$	5.00000	0.183309
$745$	−36.0000	−1.31894
$746$	−25.0000	−0.915315
$747$	24.0000	0.878114
$748$	−18.0000	−0.658145
$749$	−72.0000	−2.63082
$750$	−3.00000	−0.109545
$751$	−34.0000	−1.24068	−0.620339	−	0.784334i	$-0.713005\pi$
$751$	−34.0000	−1.24068	−0.620339	+	0.784334i	$0.713005\pi$
$752$	12.0000	0.437595
$753$	−12.0000	−0.437304
$754$	0	0
$755$	−48.0000	−1.74690
$756$	20.0000	0.727393
$757$	−1.00000	−0.0363456	−0.0181728	−	0.999835i	$-0.505785\pi$
$757$	−1.00000	−0.0363456	−0.0181728	+	0.999835i	$0.505785\pi$
$758$	−1.00000	−0.0363216
$759$	18.0000	0.653359
$760$	15.0000	0.544107
$761$	−33.0000	−1.19625	−0.598125	−	0.801403i	$-0.704087\pi$
$761$	−33.0000	−1.19625	−0.598125	+	0.801403i	$0.704087\pi$
$762$	−7.00000	−0.253583
$763$	28.0000	1.01367
$764$	−3.00000	−0.108536
$765$	−18.0000	−0.650791
$766$	15.0000	0.541972
$767$	24.0000	0.866590
$768$	1.00000	0.0360844
$769$	5.00000	0.180305	0.0901523	−	0.995928i	$-0.471265\pi$
$769$	5.00000	0.180305	0.0901523	+	0.995928i	$0.471265\pi$
$770$	72.0000	2.59470
$771$	−3.00000	−0.108042
$772$	−22.0000	−0.791797
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	2.00000	0.0718885
$775$	20.0000	0.718421
$776$	17.0000	0.610264
$777$	40.0000	1.43499
$778$	36.0000	1.29066
$779$	0	0
$780$	6.00000	0.214834
$781$	36.0000	1.28818
$782$	−9.00000	−0.321839
$783$	0	0
$784$	9.00000	0.321429
$785$	42.0000	1.49904
$786$	0	0
$787$	−1.00000	−0.0356462	−0.0178231	−	0.999841i	$-0.505674\pi$
$787$	−1.00000	−0.0356462	−0.0178231	+	0.999841i	$0.505674\pi$
$788$	0	0
$789$	0	0
$790$	−30.0000	−1.06735
$791$	24.0000	0.853342
$792$	12.0000	0.426401
$793$	−20.0000	−0.710221
$794$	20.0000	0.709773
$795$	27.0000	0.957591
$796$	−10.0000	−0.354441
$797$	−18.0000	−0.637593	−0.318796	−	0.947823i	$-0.603279\pi$
$797$	−18.0000	−0.637593	−0.318796	+	0.947823i	$0.603279\pi$
$798$	−20.0000	−0.707992
$799$	36.0000	1.27359
$800$	4.00000	0.141421
$801$	2.00000	0.0706665
$802$	−18.0000	−0.635602
$803$	6.00000	0.211735
$804$	−4.00000	−0.141069
$805$	36.0000	1.26883
$806$	10.0000	0.352235
$807$	15.0000	0.528025
$808$	0	0
$809$	15.0000	0.527372	0.263686	−	0.964609i	$-0.415062\pi$
$809$	15.0000	0.527372	0.263686	+	0.964609i	$0.415062\pi$
$810$	3.00000	0.105409
$811$	−16.0000	−0.561836	−0.280918	−	0.959732i	$-0.590639\pi$
$811$	−16.0000	−0.561836	−0.280918	+	0.959732i	$0.590639\pi$
$812$	0	0
$813$	−22.0000	−0.771574
$814$	60.0000	2.10300
$815$	−3.00000	−0.105085
$816$	3.00000	0.105021
$817$	−5.00000	−0.174928
$818$	14.0000	0.489499
$819$	16.0000	0.559085
$820$	0	0
$821$	30.0000	1.04701	0.523504	−	0.852023i	$-0.324625\pi$
$821$	30.0000	1.04701	0.523504	+	0.852023i	$0.324625\pi$
$822$	18.0000	0.627822
$823$	8.00000	0.278862	0.139431	−	0.990232i	$-0.455473\pi$
$823$	8.00000	0.278862	0.139431	+	0.990232i	$0.455473\pi$
$824$	5.00000	0.174183
$825$	−24.0000	−0.835573
$826$	−48.0000	−1.67013
$827$	15.0000	0.521601	0.260801	−	0.965393i	$-0.416014\pi$
$827$	15.0000	0.521601	0.260801	+	0.965393i	$0.416014\pi$
$828$	6.00000	0.208514
$829$	20.0000	0.694629	0.347314	−	0.937749i	$-0.387094\pi$
$829$	20.0000	0.694629	0.347314	+	0.937749i	$0.387094\pi$
$830$	−36.0000	−1.24958
$831$	−1.00000	−0.0346896
$832$	2.00000	0.0693375
$833$	27.0000	0.935495
$834$	8.00000	0.277017
$835$	−36.0000	−1.24583
$836$	−30.0000	−1.03757
$837$	−25.0000	−0.864126
$838$	9.00000	0.310900
$839$	15.0000	0.517858	0.258929	−	0.965896i	$-0.416631\pi$
$839$	15.0000	0.517858	0.258929	+	0.965896i	$0.416631\pi$
$840$	−12.0000	−0.414039
$841$	−29.0000	−1.00000
$842$	8.00000	0.275698
$843$	−6.00000	−0.206651
$844$	−4.00000	−0.137686
$845$	−27.0000	−0.928828
$846$	−24.0000	−0.825137
$847$	−100.000	−3.43604
$848$	9.00000	0.309061
$849$	14.0000	0.480479
$850$	12.0000	0.411597
$851$	30.0000	1.02839
$852$	−6.00000	−0.205557
$853$	56.0000	1.91740	0.958702	−	0.284413i	$-0.0917988\pi$
$853$	56.0000	1.91740	0.958702	+	0.284413i	$0.0917988\pi$
$854$	40.0000	1.36877
$855$	−30.0000	−1.02598
$856$	18.0000	0.615227
$857$	12.0000	0.409912	0.204956	−	0.978771i	$-0.434295\pi$
$857$	12.0000	0.409912	0.204956	+	0.978771i	$0.434295\pi$
$858$	−12.0000	−0.409673
$859$	−52.0000	−1.77422	−0.887109	−	0.461561i	$-0.847290\pi$
$859$	−52.0000	−1.77422	−0.887109	+	0.461561i	$0.847290\pi$
$860$	−3.00000	−0.102299
$861$	0	0
$862$	−15.0000	−0.510902
$863$	33.0000	1.12333	0.561667	−	0.827364i	$-0.310160\pi$
$863$	33.0000	1.12333	0.561667	+	0.827364i	$0.310160\pi$
$864$	−5.00000	−0.170103
$865$	9.00000	0.306009
$866$	−16.0000	−0.543702
$867$	−8.00000	−0.271694
$868$	−20.0000	−0.678844
$869$	60.0000	2.03536
$870$	0	0
$871$	−8.00000	−0.271070
$872$	−7.00000	−0.237050
$873$	−34.0000	−1.15073
$874$	−15.0000	−0.507383
$875$	12.0000	0.405674
$876$	−1.00000	−0.0337869
$877$	−4.00000	−0.135070	−0.0675352	−	0.997717i	$-0.521513\pi$
$877$	−4.00000	−0.135070	−0.0675352	+	0.997717i	$0.521513\pi$
$878$	−1.00000	−0.0337484
$879$	−18.0000	−0.607125
$880$	−18.0000	−0.606780
$881$	6.00000	0.202145	0.101073	−	0.994879i	$-0.467773\pi$
$881$	6.00000	0.202145	0.101073	+	0.994879i	$0.467773\pi$
$882$	−18.0000	−0.606092
$883$	−19.0000	−0.639401	−0.319700	−	0.947519i	$-0.603582\pi$
$883$	−19.0000	−0.639401	−0.319700	+	0.947519i	$0.603582\pi$
$884$	6.00000	0.201802
$885$	36.0000	1.21013
$886$	6.00000	0.201574
$887$	57.0000	1.91387	0.956936	−	0.290298i	$-0.0937544\pi$
$887$	57.0000	1.91387	0.956936	+	0.290298i	$0.0937544\pi$
$888$	−10.0000	−0.335578
$889$	28.0000	0.939090
$890$	−3.00000	−0.100560
$891$	−6.00000	−0.201008
$892$	8.00000	0.267860
$893$	60.0000	2.00782
$894$	−12.0000	−0.401340
$895$	54.0000	1.80502
$896$	−4.00000	−0.133631
$897$	−6.00000	−0.200334
$898$	39.0000	1.30145
$899$	0	0
$900$	−8.00000	−0.266667
$901$	27.0000	0.899500
$902$	0	0
$903$	4.00000	0.133112
$904$	−6.00000	−0.199557
$905$	−30.0000	−0.997234
$906$	−16.0000	−0.531564
$907$	38.0000	1.26177	0.630885	−	0.775877i	$-0.282692\pi$
$907$	38.0000	1.26177	0.630885	+	0.775877i	$0.282692\pi$
$908$	24.0000	0.796468
$909$	0	0
$910$	−24.0000	−0.795592
$911$	−48.0000	−1.59031	−0.795155	−	0.606406i	$-0.792611\pi$
$911$	−48.0000	−1.59031	−0.795155	+	0.606406i	$0.792611\pi$
$912$	5.00000	0.165567
$913$	72.0000	2.38285
$914$	−10.0000	−0.330771
$915$	−30.0000	−0.991769
$916$	−4.00000	−0.132164
$917$	0	0
$918$	−15.0000	−0.495074
$919$	29.0000	0.956622	0.478311	−	0.878191i	$-0.341249\pi$
$919$	29.0000	0.956622	0.478311	+	0.878191i	$0.341249\pi$
$920$	−9.00000	−0.296721
$921$	2.00000	0.0659022
$922$	18.0000	0.592798
$923$	−12.0000	−0.394985
$924$	24.0000	0.789542
$925$	−40.0000	−1.31519
$926$	20.0000	0.657241
$927$	−10.0000	−0.328443
$928$	0	0
$929$	15.0000	0.492134	0.246067	−	0.969253i	$-0.420862\pi$
$929$	15.0000	0.492134	0.246067	+	0.969253i	$0.420862\pi$
$930$	15.0000	0.491869
$931$	45.0000	1.47482
$932$	6.00000	0.196537
$933$	12.0000	0.392862
$934$	18.0000	0.588978
$935$	−54.0000	−1.76599
$936$	−4.00000	−0.130744
$937$	−7.00000	−0.228680	−0.114340	−	0.993442i	$-0.536475\pi$
$937$	−7.00000	−0.228680	−0.114340	+	0.993442i	$0.536475\pi$
$938$	16.0000	0.522419
$939$	20.0000	0.652675
$940$	36.0000	1.17419
$941$	36.0000	1.17357	0.586783	−	0.809744i	$-0.300394\pi$
$941$	36.0000	1.17357	0.586783	+	0.809744i	$0.300394\pi$
$942$	14.0000	0.456145
$943$	0	0
$944$	12.0000	0.390567
$945$	60.0000	1.95180
$946$	6.00000	0.195077
$947$	−48.0000	−1.55979	−0.779895	−	0.625910i	$-0.784728\pi$
$947$	−48.0000	−1.55979	−0.779895	+	0.625910i	$0.784728\pi$
$948$	−10.0000	−0.324785
$949$	−2.00000	−0.0649227
$950$	20.0000	0.648886
$951$	−27.0000	−0.875535
$952$	−12.0000	−0.388922
$953$	−36.0000	−1.16615	−0.583077	−	0.812417i	$-0.698151\pi$
$953$	−36.0000	−1.16615	−0.583077	+	0.812417i	$0.698151\pi$
$954$	−18.0000	−0.582772
$955$	−9.00000	−0.291233
$956$	15.0000	0.485135
$957$	0	0
$958$	−6.00000	−0.193851
$959$	−72.0000	−2.32500
$960$	3.00000	0.0968246
$961$	−6.00000	−0.193548
$962$	−20.0000	−0.644826
$963$	−36.0000	−1.16008
$964$	−22.0000	−0.708572
$965$	−66.0000	−2.12462
$966$	12.0000	0.386094
$967$	−1.00000	−0.0321578	−0.0160789	−	0.999871i	$-0.505118\pi$
$967$	−1.00000	−0.0321578	−0.0160789	+	0.999871i	$0.505118\pi$
$968$	25.0000	0.803530
$969$	15.0000	0.481869
$970$	51.0000	1.63751
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	16.0000	0.513200
$973$	−32.0000	−1.02587
$974$	−4.00000	−0.128168
$975$	8.00000	0.256205
$976$	−10.0000	−0.320092
$977$	−9.00000	−0.287936	−0.143968	−	0.989582i	$-0.545986\pi$
$977$	−9.00000	−0.287936	−0.143968	+	0.989582i	$0.545986\pi$
$978$	−1.00000	−0.0319765
$979$	6.00000	0.191761
$980$	27.0000	0.862483
$981$	14.0000	0.446986
$982$	−15.0000	−0.478669
$983$	24.0000	0.765481	0.382741	−	0.923856i	$-0.374980\pi$
$983$	24.0000	0.765481	0.382741	+	0.923856i	$0.374980\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−48.0000	−1.52786
$988$	10.0000	0.318142
$989$	3.00000	0.0953945
$990$	36.0000	1.14416
$991$	−19.0000	−0.603555	−0.301777	−	0.953378i	$-0.597580\pi$
$991$	−19.0000	−0.603555	−0.301777	+	0.953378i	$0.597580\pi$
$992$	5.00000	0.158750
$993$	−34.0000	−1.07896
$994$	24.0000	0.761234
$995$	−30.0000	−0.951064
$996$	−12.0000	−0.380235
$997$	26.0000	0.823428	0.411714	−	0.911313i	$-0.364930\pi$
$997$	26.0000	0.823428	0.411714	+	0.911313i	$0.364930\pi$
$998$	−4.00000	−0.126618
$999$	50.0000	1.58193

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	178.2.a.b.1.1	✓	1
3.2	odd	2		1602.2.a.a.1.1		1
4.3	odd	2		1424.2.a.c.1.1		1
5.4	even	2		4450.2.a.c.1.1		1
7.6	odd	2		8722.2.a.m.1.1		1
8.3	odd	2		5696.2.a.j.1.1		1
8.5	even	2		5696.2.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
178.2.a.b.1.1	✓	1	1.1	even	1	trivial
1424.2.a.c.1.1		1	4.3	odd	2
1602.2.a.a.1.1		1	3.2	odd	2
4450.2.a.c.1.1		1	5.4	even	2
5696.2.a.e.1.1		1	8.5	even	2
5696.2.a.j.1.1		1	8.3	odd	2
8722.2.a.m.1.1		1	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 178 = 2 \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	178.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more