LMFDB - Embedded newform 8550.2.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8550,2,Mod(1,8550)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8550, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8550.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8550.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:	$68.2720937282$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 190)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	8550.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})$

$q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{7} -1.00000 q^{8} +3.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} -7.00000 q^{17} -1.00000 q^{19} -5.00000 q^{23} -3.00000 q^{26} +1.00000 q^{28} +5.00000 q^{29} +10.0000 q^{31} -1.00000 q^{32} +7.00000 q^{34} -2.00000 q^{37} +1.00000 q^{38} -2.00000 q^{41} -6.00000 q^{43} +5.00000 q^{46} -6.00000 q^{49} +3.00000 q^{52} +9.00000 q^{53} -1.00000 q^{56} -5.00000 q^{58} +7.00000 q^{59} -4.00000 q^{61} -10.0000 q^{62} +1.00000 q^{64} -7.00000 q^{67} -7.00000 q^{68} +9.00000 q^{73} +2.00000 q^{74} -1.00000 q^{76} -10.0000 q^{79} +2.00000 q^{82} -2.00000 q^{83} +6.00000 q^{86} +10.0000 q^{89} +3.00000 q^{91} -5.00000 q^{92} +18.0000 q^{97} +6.00000 q^{98} +O(q^{100})$

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.00000	0.377964	0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	3.00000	0.832050	0.416025	−	0.909353i	$-0.363423\pi$
$13$	3.00000	0.832050	0.416025	+	0.909353i	$0.363423\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−7.00000	−1.69775	−0.848875	−	0.528594i	$-0.822719\pi$
$17$	−7.00000	−1.69775	−0.848875	+	0.528594i	$0.822719\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.00000	−1.04257	−0.521286	−	0.853382i	$-0.674548\pi$
$23$	−5.00000	−1.04257	−0.521286	+	0.853382i	$0.674548\pi$
$24$	0	0
$25$	0	0
$26$	−3.00000	−0.588348
$27$	0	0
$28$	1.00000	0.188982
$29$	5.00000	0.928477	0.464238	−	0.885710i	$-0.346328\pi$
$29$	5.00000	0.928477	0.464238	+	0.885710i	$0.346328\pi$
$30$	0	0
$31$	10.0000	1.79605	0.898027	−	0.439941i	$-0.145001\pi$
$31$	10.0000	1.79605	0.898027	+	0.439941i	$0.145001\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	7.00000	1.20049
$35$	0	0
$36$	0	0
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	1.00000	0.162221
$39$	0	0
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	−6.00000	−0.914991	−0.457496	−	0.889212i	$-0.651253\pi$
$43$	−6.00000	−0.914991	−0.457496	+	0.889212i	$0.651253\pi$
$44$	0	0
$45$	0	0
$46$	5.00000	0.737210
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	0	0
$51$	0	0
$52$	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	$0.287894\pi$
$54$	0	0
$55$	0	0
$56$	−1.00000	−0.133631
$57$	0	0
$58$	−5.00000	−0.656532
$59$	7.00000	0.911322	0.455661	−	0.890153i	$-0.349403\pi$
$59$	7.00000	0.911322	0.455661	+	0.890153i	$0.349403\pi$
$60$	0	0
$61$	−4.00000	−0.512148	−0.256074	−	0.966657i	$-0.582429\pi$
$61$	−4.00000	−0.512148	−0.256074	+	0.966657i	$0.582429\pi$
$62$	−10.0000	−1.27000
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−7.00000	−0.855186	−0.427593	−	0.903971i	$-0.640638\pi$
$67$	−7.00000	−0.855186	−0.427593	+	0.903971i	$0.640638\pi$
$68$	−7.00000	−0.848875
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	9.00000	1.05337	0.526685	−	0.850060i	$-0.323435\pi$
$73$	9.00000	1.05337	0.526685	+	0.850060i	$0.323435\pi$
$74$	2.00000	0.232495
$75$	0	0
$76$	−1.00000	−0.114708
$77$	0	0
$78$	0	0
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	0	0
$81$	0	0
$82$	2.00000	0.220863
$83$	−2.00000	−0.219529	−0.109764	−	0.993958i	$-0.535010\pi$
$83$	−2.00000	−0.219529	−0.109764	+	0.993958i	$0.535010\pi$
$84$	0	0
$85$	0	0
$86$	6.00000	0.646997
$87$	0	0
$88$	0	0
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	0	0
$91$	3.00000	0.314485
$92$	−5.00000	−0.521286
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	18.0000	1.82762	0.913812	−	0.406138i	$-0.133125\pi$
$97$	18.0000	1.82762	0.913812	+	0.406138i	$0.133125\pi$
$98$	6.00000	0.606092
$99$	0	0
$100$	0	0
$101$	4.00000	0.398015	0.199007	−	0.979998i	$-0.436228\pi$
$101$	4.00000	0.398015	0.199007	+	0.979998i	$0.436228\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	−3.00000	−0.294174
$105$	0	0
$106$	−9.00000	−0.874157
$107$	9.00000	0.870063	0.435031	−	0.900415i	$-0.356737\pi$
$107$	9.00000	0.870063	0.435031	+	0.900415i	$0.356737\pi$
$108$	0	0
$109$	13.0000	1.24517	0.622587	−	0.782551i	$-0.286082\pi$
$109$	13.0000	1.24517	0.622587	+	0.782551i	$0.286082\pi$
$110$	0	0
$111$	0	0
$112$	1.00000	0.0944911
$113$	8.00000	0.752577	0.376288	−	0.926503i	$-0.377200\pi$
$113$	8.00000	0.752577	0.376288	+	0.926503i	$0.377200\pi$
$114$	0	0
$115$	0	0
$116$	5.00000	0.464238
$117$	0	0
$118$	−7.00000	−0.644402
$119$	−7.00000	−0.641689
$120$	0	0
$121$	−11.0000	−1.00000
$122$	4.00000	0.362143
$123$	0	0
$124$	10.0000	0.898027
$125$	0	0
$126$	0	0
$127$	6.00000	0.532414	0.266207	−	0.963916i	$-0.414230\pi$
$127$	6.00000	0.532414	0.266207	+	0.963916i	$0.414230\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	0	0
$131$	20.0000	1.74741	0.873704	−	0.486458i	$-0.161711\pi$
$131$	20.0000	1.74741	0.873704	+	0.486458i	$0.161711\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	7.00000	0.604708
$135$	0	0
$136$	7.00000	0.600245
$137$	−3.00000	−0.256307	−0.128154	−	0.991754i	$-0.540905\pi$
$137$	−3.00000	−0.256307	−0.128154	+	0.991754i	$0.540905\pi$
$138$	0	0
$139$	12.0000	1.01783	0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000	1.01783	0.508913	+	0.860818i	$0.330047\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	−9.00000	−0.744845
$147$	0	0
$148$	−2.00000	−0.164399
$149$	−4.00000	−0.327693	−0.163846	−	0.986486i	$-0.552390\pi$
$149$	−4.00000	−0.327693	−0.163846	+	0.986486i	$0.552390\pi$
$150$	0	0
$151$	−6.00000	−0.488273	−0.244137	−	0.969741i	$-0.578505\pi$
$151$	−6.00000	−0.488273	−0.244137	+	0.969741i	$0.578505\pi$
$152$	1.00000	0.0811107
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−18.0000	−1.43656	−0.718278	−	0.695756i	$-0.755069\pi$
$157$	−18.0000	−1.43656	−0.718278	+	0.695756i	$0.755069\pi$
$158$	10.0000	0.795557
$159$	0	0
$160$	0	0
$161$	−5.00000	−0.394055
$162$	0	0
$163$	18.0000	1.40987	0.704934	−	0.709273i	$-0.250976\pi$
$163$	18.0000	1.40987	0.704934	+	0.709273i	$0.250976\pi$
$164$	−2.00000	−0.156174
$165$	0	0
$166$	2.00000	0.155230
$167$	14.0000	1.08335	0.541676	−	0.840587i	$-0.317790\pi$
$167$	14.0000	1.08335	0.541676	+	0.840587i	$0.317790\pi$
$168$	0	0
$169$	−4.00000	−0.307692
$170$	0	0
$171$	0	0
$172$	−6.00000	−0.457496
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	−10.0000	−0.749532
$179$	−24.0000	−1.79384	−0.896922	−	0.442189i	$-0.854202\pi$
$179$	−24.0000	−1.79384	−0.896922	+	0.442189i	$0.854202\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	−3.00000	−0.222375
$183$	0	0
$184$	5.00000	0.368605
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	7.00000	0.506502	0.253251	−	0.967401i	$-0.418500\pi$
$191$	7.00000	0.506502	0.253251	+	0.967401i	$0.418500\pi$
$192$	0	0
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	−18.0000	−1.29232
$195$	0	0
$196$	−6.00000	−0.428571
$197$	−10.0000	−0.712470	−0.356235	−	0.934396i	$-0.615940\pi$
$197$	−10.0000	−0.712470	−0.356235	+	0.934396i	$0.615940\pi$
$198$	0	0
$199$	17.0000	1.20510	0.602549	−	0.798082i	$-0.294152\pi$
$199$	17.0000	1.20510	0.602549	+	0.798082i	$0.294152\pi$
$200$	0	0
$201$	0	0
$202$	−4.00000	−0.281439
$203$	5.00000	0.350931
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	3.00000	0.208013
$209$	0	0
$210$	0	0
$211$	5.00000	0.344214	0.172107	−	0.985078i	$-0.444942\pi$
$211$	5.00000	0.344214	0.172107	+	0.985078i	$0.444942\pi$
$212$	9.00000	0.618123
$213$	0	0
$214$	−9.00000	−0.615227
$215$	0	0
$216$	0	0
$217$	10.0000	0.678844
$218$	−13.0000	−0.880471
$219$	0	0
$220$	0	0
$221$	−21.0000	−1.41261
$222$	0	0
$223$	22.0000	1.47323	0.736614	−	0.676313i	$-0.236423\pi$
$223$	22.0000	1.47323	0.736614	+	0.676313i	$0.236423\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	−8.00000	−0.532152
$227$	−25.0000	−1.65931	−0.829654	−	0.558278i	$-0.811462\pi$
$227$	−25.0000	−1.65931	−0.829654	+	0.558278i	$0.811462\pi$
$228$	0	0
$229$	−18.0000	−1.18947	−0.594737	−	0.803921i	$-0.702744\pi$
$229$	−18.0000	−1.18947	−0.594737	+	0.803921i	$0.702744\pi$
$230$	0	0
$231$	0	0
$232$	−5.00000	−0.328266
$233$	26.0000	1.70332	0.851658	−	0.524097i	$-0.175597\pi$
$233$	26.0000	1.70332	0.851658	+	0.524097i	$0.175597\pi$
$234$	0	0
$235$	0	0
$236$	7.00000	0.455661
$237$	0	0
$238$	7.00000	0.453743
$239$	27.0000	1.74648	0.873242	−	0.487286i	$-0.162013\pi$
$239$	27.0000	1.74648	0.873242	+	0.487286i	$0.162013\pi$
$240$	0	0
$241$	−28.0000	−1.80364	−0.901819	−	0.432113i	$-0.857768\pi$
$241$	−28.0000	−1.80364	−0.901819	+	0.432113i	$0.857768\pi$
$242$	11.0000	0.707107
$243$	0	0
$244$	−4.00000	−0.256074
$245$	0	0
$246$	0	0
$247$	−3.00000	−0.190885
$248$	−10.0000	−0.635001
$249$	0	0
$250$	0	0
$251$	−4.00000	−0.252478	−0.126239	−	0.992000i	$-0.540291\pi$
$251$	−4.00000	−0.252478	−0.126239	+	0.992000i	$0.540291\pi$
$252$	0	0
$253$	0	0
$254$	−6.00000	−0.376473
$255$	0	0
$256$	1.00000	0.0625000
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	0	0
$259$	−2.00000	−0.124274
$260$	0	0
$261$	0	0
$262$	−20.0000	−1.23560
$263$	−8.00000	−0.493301	−0.246651	−	0.969104i	$-0.579330\pi$
$263$	−8.00000	−0.493301	−0.246651	+	0.969104i	$0.579330\pi$
$264$	0	0
$265$	0	0
$266$	1.00000	0.0613139
$267$	0	0
$268$	−7.00000	−0.427593
$269$	10.0000	0.609711	0.304855	−	0.952399i	$-0.401392\pi$
$269$	10.0000	0.609711	0.304855	+	0.952399i	$0.401392\pi$
$270$	0	0
$271$	5.00000	0.303728	0.151864	−	0.988401i	$-0.451472\pi$
$271$	5.00000	0.303728	0.151864	+	0.988401i	$0.451472\pi$
$272$	−7.00000	−0.424437
$273$	0	0
$274$	3.00000	0.181237
$275$	0	0
$276$	0	0
$277$	−28.0000	−1.68236	−0.841178	−	0.540758i	$-0.818138\pi$
$277$	−28.0000	−1.68236	−0.841178	+	0.540758i	$0.818138\pi$
$278$	−12.0000	−0.719712
$279$	0	0
$280$	0	0
$281$	26.0000	1.55103	0.775515	−	0.631329i	$-0.217490\pi$
$281$	26.0000	1.55103	0.775515	+	0.631329i	$0.217490\pi$
$282$	0	0
$283$	26.0000	1.54554	0.772770	−	0.634686i	$-0.218871\pi$
$283$	26.0000	1.54554	0.772770	+	0.634686i	$0.218871\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2.00000	−0.118056
$288$	0	0
$289$	32.0000	1.88235
$290$	0	0
$291$	0	0
$292$	9.00000	0.526685
$293$	31.0000	1.81104	0.905520	−	0.424304i	$-0.139481\pi$
$293$	31.0000	1.81104	0.905520	+	0.424304i	$0.139481\pi$
$294$	0	0
$295$	0	0
$296$	2.00000	0.116248
$297$	0	0
$298$	4.00000	0.231714
$299$	−15.0000	−0.867472
$300$	0	0
$301$	−6.00000	−0.345834
$302$	6.00000	0.345261
$303$	0	0
$304$	−1.00000	−0.0573539
$305$	0	0
$306$	0	0
$307$	−4.00000	−0.228292	−0.114146	−	0.993464i	$-0.536413\pi$
$307$	−4.00000	−0.228292	−0.114146	+	0.993464i	$0.536413\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−33.0000	−1.87126	−0.935629	−	0.352985i	$-0.885167\pi$
$311$	−33.0000	−1.87126	−0.935629	+	0.352985i	$0.885167\pi$
$312$	0	0
$313$	29.0000	1.63918	0.819588	−	0.572953i	$-0.194202\pi$
$313$	29.0000	1.63918	0.819588	+	0.572953i	$0.194202\pi$
$314$	18.0000	1.01580
$315$	0	0
$316$	−10.0000	−0.562544
$317$	3.00000	0.168497	0.0842484	−	0.996445i	$-0.473151\pi$
$317$	3.00000	0.168497	0.0842484	+	0.996445i	$0.473151\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	5.00000	0.278639
$323$	7.00000	0.389490
$324$	0	0
$325$	0	0
$326$	−18.0000	−0.996928
$327$	0	0
$328$	2.00000	0.110432
$329$	0	0
$330$	0	0
$331$	−17.0000	−0.934405	−0.467202	−	0.884150i	$-0.654738\pi$
$331$	−17.0000	−0.934405	−0.467202	+	0.884150i	$0.654738\pi$
$332$	−2.00000	−0.109764
$333$	0	0
$334$	−14.0000	−0.766046
$335$	0	0
$336$	0	0
$337$	14.0000	0.762629	0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000	0.762629	0.381314	+	0.924445i	$0.375472\pi$
$338$	4.00000	0.217571
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−13.0000	−0.701934
$344$	6.00000	0.323498
$345$	0	0
$346$	−6.00000	−0.322562
$347$	−18.0000	−0.966291	−0.483145	−	0.875540i	$-0.660506\pi$
$347$	−18.0000	−0.966291	−0.483145	+	0.875540i	$0.660506\pi$
$348$	0	0
$349$	10.0000	0.535288	0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000	0.535288	0.267644	+	0.963518i	$0.413755\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	27.0000	1.43706	0.718532	−	0.695493i	$-0.244814\pi$
$353$	27.0000	1.43706	0.718532	+	0.695493i	$0.244814\pi$
$354$	0	0
$355$	0	0
$356$	10.0000	0.529999
$357$	0	0
$358$	24.0000	1.26844
$359$	−27.0000	−1.42501	−0.712503	−	0.701669i	$-0.752438\pi$
$359$	−27.0000	−1.42501	−0.712503	+	0.701669i	$0.752438\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	10.0000	0.525588
$363$	0	0
$364$	3.00000	0.157243
$365$	0	0
$366$	0	0
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	−5.00000	−0.260643
$369$	0	0
$370$	0	0
$371$	9.00000	0.467257
$372$	0	0
$373$	−3.00000	−0.155334	−0.0776671	−	0.996979i	$-0.524747\pi$
$373$	−3.00000	−0.155334	−0.0776671	+	0.996979i	$0.524747\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	15.0000	0.772539
$378$	0	0
$379$	−7.00000	−0.359566	−0.179783	−	0.983706i	$-0.557540\pi$
$379$	−7.00000	−0.359566	−0.179783	+	0.983706i	$0.557540\pi$
$380$	0	0
$381$	0	0
$382$	−7.00000	−0.358151
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	0	0
$386$	−2.00000	−0.101797
$387$	0	0
$388$	18.0000	0.913812
$389$	−24.0000	−1.21685	−0.608424	−	0.793612i	$-0.708198\pi$
$389$	−24.0000	−1.21685	−0.608424	+	0.793612i	$0.708198\pi$
$390$	0	0
$391$	35.0000	1.77003
$392$	6.00000	0.303046
$393$	0	0
$394$	10.0000	0.503793
$395$	0	0
$396$	0	0
$397$	20.0000	1.00377	0.501886	−	0.864934i	$-0.332640\pi$
$397$	20.0000	1.00377	0.501886	+	0.864934i	$0.332640\pi$
$398$	−17.0000	−0.852133
$399$	0	0
$400$	0	0
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	0	0
$403$	30.0000	1.49441
$404$	4.00000	0.199007
$405$	0	0
$406$	−5.00000	−0.248146
$407$	0	0
$408$	0	0
$409$	26.0000	1.28562	0.642809	−	0.766027i	$-0.277769\pi$
$409$	26.0000	1.28562	0.642809	+	0.766027i	$0.277769\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	7.00000	0.344447
$414$	0	0
$415$	0	0
$416$	−3.00000	−0.147087
$417$	0	0
$418$	0	0
$419$	26.0000	1.27018	0.635092	−	0.772437i	$-0.280962\pi$
$419$	26.0000	1.27018	0.635092	+	0.772437i	$0.280962\pi$
$420$	0	0
$421$	31.0000	1.51085	0.755424	−	0.655237i	$-0.227431\pi$
$421$	31.0000	1.51085	0.755424	+	0.655237i	$0.227431\pi$
$422$	−5.00000	−0.243396
$423$	0	0
$424$	−9.00000	−0.437079
$425$	0	0
$426$	0	0
$427$	−4.00000	−0.193574
$428$	9.00000	0.435031
$429$	0	0
$430$	0	0
$431$	−12.0000	−0.578020	−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000	−0.578020	−0.289010	+	0.957326i	$0.593326\pi$
$432$	0	0
$433$	−12.0000	−0.576683	−0.288342	−	0.957528i	$-0.593104\pi$
$433$	−12.0000	−0.576683	−0.288342	+	0.957528i	$0.593104\pi$
$434$	−10.0000	−0.480015
$435$	0	0
$436$	13.0000	0.622587
$437$	5.00000	0.239182
$438$	0	0
$439$	10.0000	0.477274	0.238637	−	0.971109i	$-0.423299\pi$
$439$	10.0000	0.477274	0.238637	+	0.971109i	$0.423299\pi$
$440$	0	0
$441$	0	0
$442$	21.0000	0.998868
$443$	24.0000	1.14027	0.570137	−	0.821549i	$-0.306890\pi$
$443$	24.0000	1.14027	0.570137	+	0.821549i	$0.306890\pi$
$444$	0	0
$445$	0	0
$446$	−22.0000	−1.04173
$447$	0	0
$448$	1.00000	0.0472456
$449$	−14.0000	−0.660701	−0.330350	−	0.943858i	$-0.607167\pi$
$449$	−14.0000	−0.660701	−0.330350	+	0.943858i	$0.607167\pi$
$450$	0	0
$451$	0	0
$452$	8.00000	0.376288
$453$	0	0
$454$	25.0000	1.17331
$455$	0	0
$456$	0	0
$457$	41.0000	1.91790	0.958950	−	0.283577i	$-0.0915211\pi$
$457$	41.0000	1.91790	0.958950	+	0.283577i	$0.0915211\pi$
$458$	18.0000	0.841085
$459$	0	0
$460$	0	0
$461$	10.0000	0.465746	0.232873	−	0.972507i	$-0.425187\pi$
$461$	10.0000	0.465746	0.232873	+	0.972507i	$0.425187\pi$
$462$	0	0
$463$	−16.0000	−0.743583	−0.371792	−	0.928316i	$-0.621256\pi$
$463$	−16.0000	−0.743583	−0.371792	+	0.928316i	$0.621256\pi$
$464$	5.00000	0.232119
$465$	0	0
$466$	−26.0000	−1.20443
$467$	−8.00000	−0.370196	−0.185098	−	0.982720i	$-0.559260\pi$
$467$	−8.00000	−0.370196	−0.185098	+	0.982720i	$0.559260\pi$
$468$	0	0
$469$	−7.00000	−0.323230
$470$	0	0
$471$	0	0
$472$	−7.00000	−0.322201
$473$	0	0
$474$	0	0
$475$	0	0
$476$	−7.00000	−0.320844
$477$	0	0
$478$	−27.0000	−1.23495
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	0	0
$481$	−6.00000	−0.273576
$482$	28.0000	1.27537
$483$	0	0
$484$	−11.0000	−0.500000
$485$	0	0
$486$	0	0
$487$	2.00000	0.0906287	0.0453143	−	0.998973i	$-0.485571\pi$
$487$	2.00000	0.0906287	0.0453143	+	0.998973i	$0.485571\pi$
$488$	4.00000	0.181071
$489$	0	0
$490$	0	0
$491$	−14.0000	−0.631811	−0.315906	−	0.948791i	$-0.602308\pi$
$491$	−14.0000	−0.631811	−0.315906	+	0.948791i	$0.602308\pi$
$492$	0	0
$493$	−35.0000	−1.57632
$494$	3.00000	0.134976
$495$	0	0
$496$	10.0000	0.449013
$497$	0	0
$498$	0	0
$499$	42.0000	1.88018	0.940089	−	0.340929i	$-0.110742\pi$
$499$	42.0000	1.88018	0.940089	+	0.340929i	$0.110742\pi$
$500$	0	0
$501$	0	0
$502$	4.00000	0.178529
$503$	−17.0000	−0.757993	−0.378996	−	0.925398i	$-0.623731\pi$
$503$	−17.0000	−0.757993	−0.378996	+	0.925398i	$0.623731\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	6.00000	0.266207
$509$	42.0000	1.86162	0.930809	−	0.365507i	$-0.119104\pi$
$509$	42.0000	1.86162	0.930809	+	0.365507i	$0.119104\pi$
$510$	0	0
$511$	9.00000	0.398137
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	6.00000	0.264649
$515$	0	0
$516$	0	0
$517$	0	0
$518$	2.00000	0.0878750
$519$	0	0
$520$	0	0
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	0	0
$523$	19.0000	0.830812	0.415406	−	0.909636i	$-0.363640\pi$
$523$	19.0000	0.830812	0.415406	+	0.909636i	$0.363640\pi$
$524$	20.0000	0.873704
$525$	0	0
$526$	8.00000	0.348817
$527$	−70.0000	−3.04925
$528$	0	0
$529$	2.00000	0.0869565
$530$	0	0
$531$	0	0
$532$	−1.00000	−0.0433555
$533$	−6.00000	−0.259889
$534$	0	0
$535$	0	0
$536$	7.00000	0.302354
$537$	0	0
$538$	−10.0000	−0.431131
$539$	0	0
$540$	0	0
$541$	−28.0000	−1.20381	−0.601907	−	0.798566i	$-0.705592\pi$
$541$	−28.0000	−1.20381	−0.601907	+	0.798566i	$0.705592\pi$
$542$	−5.00000	−0.214768
$543$	0	0
$544$	7.00000	0.300123
$545$	0	0
$546$	0	0
$547$	20.0000	0.855138	0.427569	−	0.903983i	$-0.359370\pi$
$547$	20.0000	0.855138	0.427569	+	0.903983i	$0.359370\pi$
$548$	−3.00000	−0.128154
$549$	0	0
$550$	0	0
$551$	−5.00000	−0.213007
$552$	0	0
$553$	−10.0000	−0.425243
$554$	28.0000	1.18961
$555$	0	0
$556$	12.0000	0.508913
$557$	8.00000	0.338971	0.169485	−	0.985533i	$-0.445789\pi$
$557$	8.00000	0.338971	0.169485	+	0.985533i	$0.445789\pi$
$558$	0	0
$559$	−18.0000	−0.761319
$560$	0	0
$561$	0	0
$562$	−26.0000	−1.09674
$563$	−44.0000	−1.85438	−0.927189	−	0.374593i	$-0.877783\pi$
$563$	−44.0000	−1.85438	−0.927189	+	0.374593i	$0.877783\pi$
$564$	0	0
$565$	0	0
$566$	−26.0000	−1.09286
$567$	0	0
$568$	0	0
$569$	8.00000	0.335377	0.167689	−	0.985840i	$-0.446370\pi$
$569$	8.00000	0.335377	0.167689	+	0.985840i	$0.446370\pi$
$570$	0	0
$571$	10.0000	0.418487	0.209243	−	0.977864i	$-0.432900\pi$
$571$	10.0000	0.418487	0.209243	+	0.977864i	$0.432900\pi$
$572$	0	0
$573$	0	0
$574$	2.00000	0.0834784
$575$	0	0
$576$	0	0
$577$	11.0000	0.457936	0.228968	−	0.973434i	$-0.426465\pi$
$577$	11.0000	0.457936	0.228968	+	0.973434i	$0.426465\pi$
$578$	−32.0000	−1.33102
$579$	0	0
$580$	0	0
$581$	−2.00000	−0.0829740
$582$	0	0
$583$	0	0
$584$	−9.00000	−0.372423
$585$	0	0
$586$	−31.0000	−1.28060
$587$	30.0000	1.23823	0.619116	−	0.785299i	$-0.287491\pi$
$587$	30.0000	1.23823	0.619116	+	0.785299i	$0.287491\pi$
$588$	0	0
$589$	−10.0000	−0.412043
$590$	0	0
$591$	0	0
$592$	−2.00000	−0.0821995
$593$	−34.0000	−1.39621	−0.698106	−	0.715994i	$-0.745974\pi$
$593$	−34.0000	−1.39621	−0.698106	+	0.715994i	$0.745974\pi$
$594$	0	0
$595$	0	0
$596$	−4.00000	−0.163846
$597$	0	0
$598$	15.0000	0.613396
$599$	−30.0000	−1.22577	−0.612883	−	0.790173i	$-0.709990\pi$
$599$	−30.0000	−1.22577	−0.612883	+	0.790173i	$0.709990\pi$
$600$	0	0
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	6.00000	0.244542
$603$	0	0
$604$	−6.00000	−0.244137
$605$	0	0
$606$	0	0
$607$	−22.0000	−0.892952	−0.446476	−	0.894795i	$-0.647321\pi$
$607$	−22.0000	−0.892952	−0.446476	+	0.894795i	$0.647321\pi$
$608$	1.00000	0.0405554
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	16.0000	0.646234	0.323117	−	0.946359i	$-0.395269\pi$
$613$	16.0000	0.646234	0.323117	+	0.946359i	$0.395269\pi$
$614$	4.00000	0.161427
$615$	0	0
$616$	0	0
$617$	−2.00000	−0.0805170	−0.0402585	−	0.999189i	$-0.512818\pi$
$617$	−2.00000	−0.0805170	−0.0402585	+	0.999189i	$0.512818\pi$
$618$	0	0
$619$	−8.00000	−0.321547	−0.160774	−	0.986991i	$-0.551399\pi$
$619$	−8.00000	−0.321547	−0.160774	+	0.986991i	$0.551399\pi$
$620$	0	0
$621$	0	0
$622$	33.0000	1.32318
$623$	10.0000	0.400642
$624$	0	0
$625$	0	0
$626$	−29.0000	−1.15907
$627$	0	0
$628$	−18.0000	−0.718278
$629$	14.0000	0.558217
$630$	0	0
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	10.0000	0.397779
$633$	0	0
$634$	−3.00000	−0.119145
$635$	0	0
$636$	0	0
$637$	−18.0000	−0.713186
$638$	0	0
$639$	0	0
$640$	0	0
$641$	12.0000	0.473972	0.236986	−	0.971513i	$-0.423841\pi$
$641$	12.0000	0.473972	0.236986	+	0.971513i	$0.423841\pi$
$642$	0	0
$643$	2.00000	0.0788723	0.0394362	−	0.999222i	$-0.487444\pi$
$643$	2.00000	0.0788723	0.0394362	+	0.999222i	$0.487444\pi$
$644$	−5.00000	−0.197028
$645$	0	0
$646$	−7.00000	−0.275411
$647$	−9.00000	−0.353827	−0.176913	−	0.984226i	$-0.556611\pi$
$647$	−9.00000	−0.353827	−0.176913	+	0.984226i	$0.556611\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	18.0000	0.704934
$653$	−12.0000	−0.469596	−0.234798	−	0.972044i	$-0.575443\pi$
$653$	−12.0000	−0.469596	−0.234798	+	0.972044i	$0.575443\pi$
$654$	0	0
$655$	0	0
$656$	−2.00000	−0.0780869
$657$	0	0
$658$	0	0
$659$	−15.0000	−0.584317	−0.292159	−	0.956370i	$-0.594373\pi$
$659$	−15.0000	−0.584317	−0.292159	+	0.956370i	$0.594373\pi$
$660$	0	0
$661$	17.0000	0.661223	0.330612	−	0.943767i	$-0.392745\pi$
$661$	17.0000	0.661223	0.330612	+	0.943767i	$0.392745\pi$
$662$	17.0000	0.660724
$663$	0	0
$664$	2.00000	0.0776151
$665$	0	0
$666$	0	0
$667$	−25.0000	−0.968004
$668$	14.0000	0.541676
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	28.0000	1.07932	0.539660	−	0.841883i	$-0.318553\pi$
$673$	28.0000	1.07932	0.539660	+	0.841883i	$0.318553\pi$
$674$	−14.0000	−0.539260
$675$	0	0
$676$	−4.00000	−0.153846
$677$	19.0000	0.730229	0.365115	−	0.930963i	$-0.381030\pi$
$677$	19.0000	0.730229	0.365115	+	0.930963i	$0.381030\pi$
$678$	0	0
$679$	18.0000	0.690777
$680$	0	0
$681$	0	0
$682$	0	0
$683$	36.0000	1.37750	0.688751	−	0.724998i	$-0.258159\pi$
$683$	36.0000	1.37750	0.688751	+	0.724998i	$0.258159\pi$
$684$	0	0
$685$	0	0
$686$	13.0000	0.496342
$687$	0	0
$688$	−6.00000	−0.228748
$689$	27.0000	1.02862
$690$	0	0
$691$	10.0000	0.380418	0.190209	−	0.981744i	$-0.439083\pi$
$691$	10.0000	0.380418	0.190209	+	0.981744i	$0.439083\pi$
$692$	6.00000	0.228086
$693$	0	0
$694$	18.0000	0.683271
$695$	0	0
$696$	0	0
$697$	14.0000	0.530288
$698$	−10.0000	−0.378506
$699$	0	0
$700$	0	0
$701$	−24.0000	−0.906467	−0.453234	−	0.891392i	$-0.649730\pi$
$701$	−24.0000	−0.906467	−0.453234	+	0.891392i	$0.649730\pi$
$702$	0	0
$703$	2.00000	0.0754314
$704$	0	0
$705$	0	0
$706$	−27.0000	−1.01616
$707$	4.00000	0.150435
$708$	0	0
$709$	−34.0000	−1.27690	−0.638448	−	0.769665i	$-0.720423\pi$
$709$	−34.0000	−1.27690	−0.638448	+	0.769665i	$0.720423\pi$
$710$	0	0
$711$	0	0
$712$	−10.0000	−0.374766
$713$	−50.0000	−1.87251
$714$	0	0
$715$	0	0
$716$	−24.0000	−0.896922
$717$	0	0
$718$	27.0000	1.00763
$719$	−29.0000	−1.08152	−0.540759	−	0.841178i	$-0.681863\pi$
$719$	−29.0000	−1.08152	−0.540759	+	0.841178i	$0.681863\pi$
$720$	0	0
$721$	0	0
$722$	−1.00000	−0.0372161
$723$	0	0
$724$	−10.0000	−0.371647
$725$	0	0
$726$	0	0
$727$	−11.0000	−0.407967	−0.203984	−	0.978974i	$-0.565389\pi$
$727$	−11.0000	−0.407967	−0.203984	+	0.978974i	$0.565389\pi$
$728$	−3.00000	−0.111187
$729$	0	0
$730$	0	0
$731$	42.0000	1.55343
$732$	0	0
$733$	−24.0000	−0.886460	−0.443230	−	0.896408i	$-0.646168\pi$
$733$	−24.0000	−0.886460	−0.443230	+	0.896408i	$0.646168\pi$
$734$	8.00000	0.295285
$735$	0	0
$736$	5.00000	0.184302
$737$	0	0
$738$	0	0
$739$	50.0000	1.83928	0.919640	−	0.392763i	$-0.128481\pi$
$739$	50.0000	1.83928	0.919640	+	0.392763i	$0.128481\pi$
$740$	0	0
$741$	0	0
$742$	−9.00000	−0.330400
$743$	34.0000	1.24734	0.623670	−	0.781688i	$-0.285641\pi$
$743$	34.0000	1.24734	0.623670	+	0.781688i	$0.285641\pi$
$744$	0	0
$745$	0	0
$746$	3.00000	0.109838
$747$	0	0
$748$	0	0
$749$	9.00000	0.328853
$750$	0	0
$751$	46.0000	1.67856	0.839282	−	0.543696i	$-0.182976\pi$
$751$	46.0000	1.67856	0.839282	+	0.543696i	$0.182976\pi$
$752$	0	0
$753$	0	0
$754$	−15.0000	−0.546268
$755$	0	0
$756$	0	0
$757$	−22.0000	−0.799604	−0.399802	−	0.916602i	$-0.630921\pi$
$757$	−22.0000	−0.799604	−0.399802	+	0.916602i	$0.630921\pi$
$758$	7.00000	0.254251
$759$	0	0
$760$	0	0
$761$	45.0000	1.63125	0.815624	−	0.578582i	$-0.196394\pi$
$761$	45.0000	1.63125	0.815624	+	0.578582i	$0.196394\pi$
$762$	0	0
$763$	13.0000	0.470632
$764$	7.00000	0.253251
$765$	0	0
$766$	0	0
$767$	21.0000	0.758266
$768$	0	0
$769$	49.0000	1.76699	0.883493	−	0.468445i	$-0.155186\pi$
$769$	49.0000	1.76699	0.883493	+	0.468445i	$0.155186\pi$
$770$	0	0
$771$	0	0
$772$	2.00000	0.0719816
$773$	15.0000	0.539513	0.269756	−	0.962929i	$-0.413057\pi$
$773$	15.0000	0.539513	0.269756	+	0.962929i	$0.413057\pi$
$774$	0	0
$775$	0	0
$776$	−18.0000	−0.646162
$777$	0	0
$778$	24.0000	0.860442
$779$	2.00000	0.0716574
$780$	0	0
$781$	0	0
$782$	−35.0000	−1.25160
$783$	0	0
$784$	−6.00000	−0.214286
$785$	0	0
$786$	0	0
$787$	−11.0000	−0.392108	−0.196054	−	0.980593i	$-0.562813\pi$
$787$	−11.0000	−0.392108	−0.196054	+	0.980593i	$0.562813\pi$
$788$	−10.0000	−0.356235
$789$	0	0
$790$	0	0
$791$	8.00000	0.284447
$792$	0	0
$793$	−12.0000	−0.426132
$794$	−20.0000	−0.709773
$795$	0	0
$796$	17.0000	0.602549
$797$	5.00000	0.177109	0.0885545	−	0.996071i	$-0.471775\pi$
$797$	5.00000	0.177109	0.0885545	+	0.996071i	$0.471775\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	18.0000	0.635602
$803$	0	0
$804$	0	0
$805$	0	0
$806$	−30.0000	−1.05670
$807$	0	0
$808$	−4.00000	−0.140720
$809$	39.0000	1.37117	0.685583	−	0.727994i	$-0.259547\pi$
$809$	39.0000	1.37117	0.685583	+	0.727994i	$0.259547\pi$
$810$	0	0
$811$	11.0000	0.386262	0.193131	−	0.981173i	$-0.438136\pi$
$811$	11.0000	0.386262	0.193131	+	0.981173i	$0.438136\pi$
$812$	5.00000	0.175466
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	6.00000	0.209913
$818$	−26.0000	−0.909069
$819$	0	0
$820$	0	0
$821$	56.0000	1.95441	0.977207	−	0.212290i	$-0.0680921\pi$
$821$	56.0000	1.95441	0.977207	+	0.212290i	$0.0680921\pi$
$822$	0	0
$823$	31.0000	1.08059	0.540296	−	0.841475i	$-0.318312\pi$
$823$	31.0000	1.08059	0.540296	+	0.841475i	$0.318312\pi$
$824$	0	0
$825$	0	0
$826$	−7.00000	−0.243561
$827$	−9.00000	−0.312961	−0.156480	−	0.987681i	$-0.550015\pi$
$827$	−9.00000	−0.312961	−0.156480	+	0.987681i	$0.550015\pi$
$828$	0	0
$829$	−51.0000	−1.77130	−0.885652	−	0.464350i	$-0.846288\pi$
$829$	−51.0000	−1.77130	−0.885652	+	0.464350i	$0.846288\pi$
$830$	0	0
$831$	0	0
$832$	3.00000	0.104006
$833$	42.0000	1.45521
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	−26.0000	−0.898155
$839$	48.0000	1.65714	0.828572	−	0.559883i	$-0.189154\pi$
$839$	48.0000	1.65714	0.828572	+	0.559883i	$0.189154\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	−31.0000	−1.06833
$843$	0	0
$844$	5.00000	0.172107
$845$	0	0
$846$	0	0
$847$	−11.0000	−0.377964
$848$	9.00000	0.309061
$849$	0	0
$850$	0	0
$851$	10.0000	0.342796
$852$	0	0
$853$	46.0000	1.57501	0.787505	−	0.616308i	$-0.211372\pi$
$853$	46.0000	1.57501	0.787505	+	0.616308i	$0.211372\pi$
$854$	4.00000	0.136877
$855$	0	0
$856$	−9.00000	−0.307614
$857$	24.0000	0.819824	0.409912	−	0.912125i	$-0.365559\pi$
$857$	24.0000	0.819824	0.409912	+	0.912125i	$0.365559\pi$
$858$	0	0
$859$	56.0000	1.91070	0.955348	−	0.295484i	$-0.0954809\pi$
$859$	56.0000	1.91070	0.955348	+	0.295484i	$0.0954809\pi$
$860$	0	0
$861$	0	0
$862$	12.0000	0.408722
$863$	24.0000	0.816970	0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000	0.816970	0.408485	+	0.912765i	$0.366057\pi$
$864$	0	0
$865$	0	0
$866$	12.0000	0.407777
$867$	0	0
$868$	10.0000	0.339422
$869$	0	0
$870$	0	0
$871$	−21.0000	−0.711558
$872$	−13.0000	−0.440236
$873$	0	0
$874$	−5.00000	−0.169128
$875$	0	0
$876$	0	0
$877$	5.00000	0.168838	0.0844190	−	0.996430i	$-0.473097\pi$
$877$	5.00000	0.168838	0.0844190	+	0.996430i	$0.473097\pi$
$878$	−10.0000	−0.337484
$879$	0	0
$880$	0	0
$881$	−2.00000	−0.0673817	−0.0336909	−	0.999432i	$-0.510726\pi$
$881$	−2.00000	−0.0673817	−0.0336909	+	0.999432i	$0.510726\pi$
$882$	0	0
$883$	−26.0000	−0.874970	−0.437485	−	0.899226i	$-0.644131\pi$
$883$	−26.0000	−0.874970	−0.437485	+	0.899226i	$0.644131\pi$
$884$	−21.0000	−0.706306
$885$	0	0
$886$	−24.0000	−0.806296
$887$	52.0000	1.74599	0.872995	−	0.487730i	$-0.162175\pi$
$887$	52.0000	1.74599	0.872995	+	0.487730i	$0.162175\pi$
$888$	0	0
$889$	6.00000	0.201234
$890$	0	0
$891$	0	0
$892$	22.0000	0.736614
$893$	0	0
$894$	0	0
$895$	0	0
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	14.0000	0.467186
$899$	50.0000	1.66759
$900$	0	0
$901$	−63.0000	−2.09883
$902$	0	0
$903$	0	0
$904$	−8.00000	−0.266076
$905$	0	0
$906$	0	0
$907$	−5.00000	−0.166022	−0.0830111	−	0.996549i	$-0.526454\pi$
$907$	−5.00000	−0.166022	−0.0830111	+	0.996549i	$0.526454\pi$
$908$	−25.0000	−0.829654
$909$	0	0
$910$	0	0
$911$	32.0000	1.06021	0.530104	−	0.847933i	$-0.322153\pi$
$911$	32.0000	1.06021	0.530104	+	0.847933i	$0.322153\pi$
$912$	0	0
$913$	0	0
$914$	−41.0000	−1.35616
$915$	0	0
$916$	−18.0000	−0.594737
$917$	20.0000	0.660458
$918$	0	0
$919$	11.0000	0.362857	0.181428	−	0.983404i	$-0.441928\pi$
$919$	11.0000	0.362857	0.181428	+	0.983404i	$0.441928\pi$
$920$	0	0
$921$	0	0
$922$	−10.0000	−0.329332
$923$	0	0
$924$	0	0
$925$	0	0
$926$	16.0000	0.525793
$927$	0	0
$928$	−5.00000	−0.164133
$929$	−21.0000	−0.688988	−0.344494	−	0.938789i	$-0.611949\pi$
$929$	−21.0000	−0.688988	−0.344494	+	0.938789i	$0.611949\pi$
$930$	0	0
$931$	6.00000	0.196642
$932$	26.0000	0.851658
$933$	0	0
$934$	8.00000	0.261768
$935$	0	0
$936$	0	0
$937$	13.0000	0.424691	0.212346	−	0.977195i	$-0.431890\pi$
$937$	13.0000	0.424691	0.212346	+	0.977195i	$0.431890\pi$
$938$	7.00000	0.228558
$939$	0	0
$940$	0	0
$941$	37.0000	1.20617	0.603083	−	0.797679i	$-0.293939\pi$
$941$	37.0000	1.20617	0.603083	+	0.797679i	$0.293939\pi$
$942$	0	0
$943$	10.0000	0.325645
$944$	7.00000	0.227831
$945$	0	0
$946$	0	0
$947$	−28.0000	−0.909878	−0.454939	−	0.890523i	$-0.650339\pi$
$947$	−28.0000	−0.909878	−0.454939	+	0.890523i	$0.650339\pi$
$948$	0	0
$949$	27.0000	0.876457
$950$	0	0
$951$	0	0
$952$	7.00000	0.226871
$953$	44.0000	1.42530	0.712650	−	0.701520i	$-0.247495\pi$
$953$	44.0000	1.42530	0.712650	+	0.701520i	$0.247495\pi$
$954$	0	0
$955$	0	0
$956$	27.0000	0.873242
$957$	0	0
$958$	−24.0000	−0.775405
$959$	−3.00000	−0.0968751
$960$	0	0
$961$	69.0000	2.22581
$962$	6.00000	0.193448
$963$	0	0
$964$	−28.0000	−0.901819
$965$	0	0
$966$	0	0
$967$	−12.0000	−0.385894	−0.192947	−	0.981209i	$-0.561805\pi$
$967$	−12.0000	−0.385894	−0.192947	+	0.981209i	$0.561805\pi$
$968$	11.0000	0.353553
$969$	0	0
$970$	0	0
$971$	−36.0000	−1.15529	−0.577647	−	0.816286i	$-0.696029\pi$
$971$	−36.0000	−1.15529	−0.577647	+	0.816286i	$0.696029\pi$
$972$	0	0
$973$	12.0000	0.384702
$974$	−2.00000	−0.0640841
$975$	0	0
$976$	−4.00000	−0.128037
$977$	−18.0000	−0.575871	−0.287936	−	0.957650i	$-0.592969\pi$
$977$	−18.0000	−0.575871	−0.287936	+	0.957650i	$0.592969\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	14.0000	0.446758
$983$	−18.0000	−0.574111	−0.287055	−	0.957914i	$-0.592676\pi$
$983$	−18.0000	−0.574111	−0.287055	+	0.957914i	$0.592676\pi$
$984$	0	0
$985$	0	0
$986$	35.0000	1.11463
$987$	0	0
$988$	−3.00000	−0.0954427
$989$	30.0000	0.953945
$990$	0	0
$991$	−46.0000	−1.46124	−0.730619	−	0.682785i	$-0.760768\pi$
$991$	−46.0000	−1.46124	−0.730619	+	0.682785i	$0.760768\pi$
$992$	−10.0000	−0.317500
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−14.0000	−0.443384	−0.221692	−	0.975117i	$-0.571158\pi$
$997$	−14.0000	−0.443384	−0.221692	+	0.975117i	$0.571158\pi$
$998$	−42.0000	−1.32949
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8550.2.a.l.1.1		1
3.2	odd	2		950.2.a.e.1.1		1
5.4	even	2		1710.2.a.r.1.1		1
12.11	even	2		7600.2.a.g.1.1		1
15.2	even	4		950.2.b.d.799.2		2
15.8	even	4		950.2.b.d.799.1		2
15.14	odd	2		190.2.a.a.1.1	✓	1
60.59	even	2		1520.2.a.g.1.1		1
105.104	even	2		9310.2.a.i.1.1		1
120.29	odd	2		6080.2.a.r.1.1		1
120.59	even	2		6080.2.a.i.1.1		1
285.284	even	2		3610.2.a.h.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
190.2.a.a.1.1	✓	1	15.14	odd	2
950.2.a.e.1.1		1	3.2	odd	2
950.2.b.d.799.1		2	15.8	even	4
950.2.b.d.799.2		2	15.2	even	4
1520.2.a.g.1.1		1	60.59	even	2
1710.2.a.r.1.1		1	5.4	even	2
3610.2.a.h.1.1		1	285.284	even	2
6080.2.a.i.1.1		1	120.59	even	2
6080.2.a.r.1.1		1	120.29	odd	2
7600.2.a.g.1.1		1	12.11	even	2
8550.2.a.l.1.1		1	1.1	even	1	trivial
9310.2.a.i.1.1		1	105.104	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:	\( N \)	\(=\)	\( 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8550.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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