LMFDB - Embedded newform 700.4.e.c.449.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [700,4,Mod(449,700)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(700, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("700.449"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 700 = 2^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	700.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,0,0,-74,0,56] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$41.3013370040$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 140)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	700.449
Dual form			700.4.e.c.449.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+8.00000i q^{3} -7.00000i q^{7} -37.0000 q^{9} +28.0000 q^{11} +82.0000i q^{13} +46.0000i q^{17} -8.00000 q^{19} +56.0000 q^{21} -128.000i q^{23} -80.0000i q^{27} -174.000 q^{29} -152.000 q^{31} +224.000i q^{33} +290.000i q^{37} -656.000 q^{39} +50.0000 q^{41} +396.000i q^{43} +296.000i q^{47} -49.0000 q^{49} -368.000 q^{51} -570.000i q^{53} -64.0000i q^{57} +272.000 q^{59} -662.000 q^{61} +259.000i q^{63} -876.000i q^{67} +1024.00 q^{69} -880.000 q^{71} -638.000i q^{73} -196.000i q^{77} +600.000 q^{79} -359.000 q^{81} +624.000i q^{83} -1392.00i q^{87} -698.000 q^{89} +574.000 q^{91} -1216.00i q^{93} -754.000i q^{97} -1036.00 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 74 q^{9} + 56 q^{11} - 16 q^{19} + 112 q^{21} - 348 q^{29} - 304 q^{31} - 1312 q^{39} + 100 q^{41} - 98 q^{49} - 736 q^{51} + 544 q^{59} - 1324 q^{61} + 2048 q^{69} - 1760 q^{71} + 1200 q^{79} - 718 q^{81}+ \cdots - 2072 q^{99}+O(q^{100}) $ 2 * q - 74 * q^9 + 56 * q^11 - 16 * q^19 + 112 * q^21 - 348 * q^29 - 304 * q^31 - 1312 * q^39 + 100 * q^41 - 98 * q^49 - 736 * q^51 + 544 * q^59 - 1324 * q^61 + 2048 * q^69 - 1760 * q^71 + 1200 * q^79 - 718 * q^81 - 1396 * q^89 + 1148 * q^91 - 2072 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/700\mathbb{Z}\right)^\times$.

$n$	$101$	$351$	$477$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	0	0
$2$	0	0
$3$	8.00000i	1.53960i	0.638285	+	0.769800i	$0.279644\pi$
$3$	8.00000i	1.53960i	−0.638285	+	0.769800i	$0.720356\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 7.00000i	− 0.377964i
$7$	− 7.00000i	− 0.377964i
$8$	0	0
$9$	−37.0000	−1.37037
$10$	0	0
$11$	28.0000	0.767483	0.383742	−	0.923440i	$-0.374635\pi$
$11$	28.0000	0.767483	0.383742	+	0.923440i	$0.374635\pi$
$12$	0	0
$13$	82.0000i	1.74944i	0.484629	+	0.874720i	$0.338954\pi$
$13$	82.0000i	1.74944i	−0.484629	+	0.874720i	$0.661046\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	46.0000i	0.656273i	0.944630	+	0.328136i	$0.106421\pi$
$17$	46.0000i	0.656273i	−0.944630	+	0.328136i	$0.893579\pi$
$18$	0	0
$19$	−8.00000	−0.0965961	−0.0482980	−	0.998833i	$-0.515380\pi$
$19$	−8.00000	−0.0965961	−0.0482980	+	0.998833i	$0.515380\pi$
$20$	0	0
$21$	56.0000	0.581914
$22$	0	0
$23$	− 128.000i	− 1.16043i	−0.814464	−	0.580214i	$-0.802969\pi$
$23$	− 128.000i	− 1.16043i	0.814464	−	0.580214i	$-0.197031\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 80.0000i	− 0.570222i
$28$	0	0
$29$	−174.000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−174.000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	−152.000	−0.880645	−0.440323	−	0.897840i	$-0.645136\pi$
$31$	−152.000	−0.880645	−0.440323	+	0.897840i	$0.645136\pi$
$32$	0	0
$33$	224.000i	1.18162i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	290.000i	1.28853i	0.764801	+	0.644266i	$0.222837\pi$
$37$	290.000i	1.28853i	−0.764801	+	0.644266i	$0.777163\pi$
$38$	0	0
$39$	−656.000	−2.69344
$40$	0	0
$41$	50.0000	0.190456	0.0952279	−	0.995455i	$-0.469642\pi$
$41$	50.0000	0.190456	0.0952279	+	0.995455i	$0.469642\pi$
$42$	0	0
$43$	396.000i	1.40441i	0.711977	+	0.702203i	$0.247800\pi$
$43$	396.000i	1.40441i	−0.711977	+	0.702203i	$0.752200\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	296.000i	0.918639i	0.888271	+	0.459320i	$0.151907\pi$
$47$	296.000i	0.918639i	−0.888271	+	0.459320i	$0.848093\pi$
$48$	0	0
$49$	−49.0000	−0.142857
$50$	0	0
$51$	−368.000	−1.01040
$52$	0	0
$53$	− 570.000i	− 1.47727i	−0.674103	−	0.738637i	$-0.735470\pi$
$53$	− 570.000i	− 1.47727i	0.674103	−	0.738637i	$-0.264530\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 64.0000i	− 0.148719i
$58$	0	0
$59$	272.000	0.600193	0.300096	−	0.953909i	$-0.402981\pi$
$59$	272.000	0.600193	0.300096	+	0.953909i	$0.402981\pi$
$60$	0	0
$61$	−662.000	−1.38951	−0.694757	−	0.719244i	$-0.744488\pi$
$61$	−662.000	−1.38951	−0.694757	+	0.719244i	$0.744488\pi$
$62$	0	0
$63$	259.000i	0.517951i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 876.000i	− 1.59732i	−0.601783	−	0.798660i	$-0.705543\pi$
$67$	− 876.000i	− 1.59732i	0.601783	−	0.798660i	$-0.294457\pi$
$68$	0	0
$69$	1024.00	1.78660
$70$	0	0
$71$	−880.000	−1.47094	−0.735470	−	0.677557i	$-0.763039\pi$
$71$	−880.000	−1.47094	−0.735470	+	0.677557i	$0.763039\pi$
$72$	0	0
$73$	− 638.000i	− 1.02291i	−0.859311	−	0.511454i	$-0.829107\pi$
$73$	− 638.000i	− 1.02291i	0.859311	−	0.511454i	$-0.170893\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 196.000i	− 0.290081i
$78$	0	0
$79$	600.000	0.854497	0.427249	−	0.904134i	$-0.359483\pi$
$79$	600.000	0.854497	0.427249	+	0.904134i	$0.359483\pi$
$80$	0	0
$81$	−359.000	−0.492455
$82$	0	0
$83$	624.000i	0.825216i	0.910909	+	0.412608i	$0.135382\pi$
$83$	624.000i	0.825216i	−0.910909	+	0.412608i	$0.864618\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 1392.00i	− 1.71538i
$88$	0	0
$89$	−698.000	−0.831324	−0.415662	−	0.909519i	$-0.636450\pi$
$89$	−698.000	−0.831324	−0.415662	+	0.909519i	$0.636450\pi$
$90$	0	0
$91$	574.000	0.661226
$92$	0	0
$93$	− 1216.00i	− 1.35584i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 754.000i	− 0.789248i	−0.918843	−	0.394624i	$-0.870875\pi$
$97$	− 754.000i	− 0.789248i	0.918843	−	0.394624i	$-0.129125\pi$
$98$	0	0
$99$	−1036.00	−1.05174
$100$	0	0
$101$	354.000	0.348756	0.174378	−	0.984679i	$-0.444209\pi$
$101$	354.000	0.348756	0.174378	+	0.984679i	$0.444209\pi$
$102$	0	0
$103$	1904.00i	1.82142i	0.413042	+	0.910712i	$0.364466\pi$
$103$	1904.00i	1.82142i	−0.413042	+	0.910712i	$0.635534\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 1108.00i	− 1.00107i	−0.865717	−	0.500535i	$-0.833137\pi$
$107$	− 1108.00i	− 1.00107i	0.865717	−	0.500535i	$-0.166863\pi$
$108$	0	0
$109$	146.000	0.128296	0.0641480	−	0.997940i	$-0.479567\pi$
$109$	146.000	0.128296	0.0641480	+	0.997940i	$0.479567\pi$
$110$	0	0
$111$	−2320.00	−1.98383
$112$	0	0
$113$	− 2078.00i	− 1.72993i	−0.501834	−	0.864964i	$-0.667341\pi$
$113$	− 2078.00i	− 1.72993i	0.501834	−	0.864964i	$-0.332659\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 3034.00i	− 2.39738i
$118$	0	0
$119$	322.000	0.248048
$120$	0	0
$121$	−547.000	−0.410969
$122$	0	0
$123$	400.000i	0.293226i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	600.000i	0.419224i	0.977785	+	0.209612i	$0.0672200\pi$
$127$	600.000i	0.419224i	−0.977785	+	0.209612i	$0.932780\pi$
$128$	0	0
$129$	−3168.00	−2.16222
$130$	0	0
$131$	−168.000	−0.112048	−0.0560238	−	0.998429i	$-0.517842\pi$
$131$	−168.000	−0.112048	−0.0560238	+	0.998429i	$0.517842\pi$
$132$	0	0
$133$	56.0000i	0.0365099i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 394.000i	− 0.245706i	−0.992425	−	0.122853i	$-0.960796\pi$
$137$	− 394.000i	− 0.245706i	0.992425	−	0.122853i	$-0.0392043\pi$
$138$	0	0
$139$	472.000	0.288018	0.144009	−	0.989576i	$-0.454001\pi$
$139$	472.000	0.288018	0.144009	+	0.989576i	$0.454001\pi$
$140$	0	0
$141$	−2368.00	−1.41434
$142$	0	0
$143$	2296.00i	1.34267i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 392.000i	− 0.219943i
$148$	0	0
$149$	2658.00	1.46142	0.730711	−	0.682687i	$-0.239189\pi$
$149$	2658.00	1.46142	0.730711	+	0.682687i	$0.239189\pi$
$150$	0	0
$151$	1256.00	0.676900	0.338450	−	0.940984i	$-0.390097\pi$
$151$	1256.00	0.676900	0.338450	+	0.940984i	$0.390097\pi$
$152$	0	0
$153$	− 1702.00i	− 0.899337i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 3266.00i	− 1.66022i	−0.557597	−	0.830112i	$-0.688276\pi$
$157$	− 3266.00i	− 1.66022i	0.557597	−	0.830112i	$-0.311724\pi$
$158$	0	0
$159$	4560.00	2.27441
$160$	0	0
$161$	−896.000	−0.438601
$162$	0	0
$163$	1716.00i	0.824586i	0.911051	+	0.412293i	$0.135272\pi$
$163$	1716.00i	0.824586i	−0.911051	+	0.412293i	$0.864728\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2272.00i	1.05277i	0.850246	+	0.526385i	$0.176453\pi$
$167$	2272.00i	1.05277i	−0.850246	+	0.526385i	$0.823547\pi$
$168$	0	0
$169$	−4527.00	−2.06054
$170$	0	0
$171$	296.000	0.132372
$172$	0	0
$173$	4226.00i	1.85721i	0.371073	+	0.928604i	$0.378990\pi$
$173$	4226.00i	1.85721i	−0.371073	+	0.928604i	$0.621010\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	2176.00i	0.924057i
$178$	0	0
$179$	−988.000	−0.412551	−0.206275	−	0.978494i	$-0.566134\pi$
$179$	−988.000	−0.412551	−0.206275	+	0.978494i	$0.566134\pi$
$180$	0	0
$181$	2138.00	0.877991	0.438995	−	0.898489i	$-0.355334\pi$
$181$	2138.00	0.877991	0.438995	+	0.898489i	$0.355334\pi$
$182$	0	0
$183$	− 5296.00i	− 2.13930i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	1288.00i	0.503679i
$188$	0	0
$189$	−560.000	−0.215524
$190$	0	0
$191$	−3832.00	−1.45170	−0.725848	−	0.687856i	$-0.758552\pi$
$191$	−3832.00	−1.45170	−0.725848	+	0.687856i	$0.758552\pi$
$192$	0	0
$193$	1378.00i	0.513941i	0.966419	+	0.256970i	$0.0827243\pi$
$193$	1378.00i	0.513941i	−0.966419	+	0.256970i	$0.917276\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4266.00i	1.54284i	0.636325	+	0.771421i	$0.280454\pi$
$197$	4266.00i	1.54284i	−0.636325	+	0.771421i	$0.719546\pi$
$198$	0	0
$199$	−5120.00	−1.82386	−0.911928	−	0.410351i	$-0.865406\pi$
$199$	−5120.00	−1.82386	−0.911928	+	0.410351i	$0.865406\pi$
$200$	0	0
$201$	7008.00	2.45923
$202$	0	0
$203$	1218.00i	0.421117i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	4736.00i	1.59022i
$208$	0	0
$209$	−224.000	−0.0741359
$210$	0	0
$211$	−1780.00	−0.580759	−0.290380	−	0.956911i	$-0.593782\pi$
$211$	−1780.00	−0.580759	−0.290380	+	0.956911i	$0.593782\pi$
$212$	0	0
$213$	− 7040.00i	− 2.26466i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	1064.00i	0.332853i
$218$	0	0
$219$	5104.00	1.57487
$220$	0	0
$221$	−3772.00	−1.14811
$222$	0	0
$223$	4896.00i	1.47023i	0.677945	+	0.735113i	$0.262871\pi$
$223$	4896.00i	1.47023i	−0.677945	+	0.735113i	$0.737129\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3504.00i	1.02453i	0.858827	+	0.512266i	$0.171194\pi$
$227$	3504.00i	1.02453i	−0.858827	+	0.512266i	$0.828806\pi$
$228$	0	0
$229$	−826.000	−0.238356	−0.119178	−	0.992873i	$-0.538026\pi$
$229$	−826.000	−0.238356	−0.119178	+	0.992873i	$0.538026\pi$
$230$	0	0
$231$	1568.00	0.446610
$232$	0	0
$233$	− 278.000i	− 0.0781647i	−0.999236	−	0.0390824i	$-0.987557\pi$
$233$	− 278.000i	− 0.0781647i	0.999236	−	0.0390824i	$-0.0124435\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	4800.00i	1.31558i
$238$	0	0
$239$	−4736.00	−1.28178	−0.640892	−	0.767631i	$-0.721435\pi$
$239$	−4736.00	−1.28178	−0.640892	+	0.767631i	$0.721435\pi$
$240$	0	0
$241$	−3990.00	−1.06647	−0.533233	−	0.845968i	$-0.679023\pi$
$241$	−3990.00	−1.06647	−0.533233	+	0.845968i	$0.679023\pi$
$242$	0	0
$243$	− 5032.00i	− 1.32841i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 656.000i	− 0.168989i
$248$	0	0
$249$	−4992.00	−1.27050
$250$	0	0
$251$	2296.00	0.577379	0.288690	−	0.957423i	$-0.406780\pi$
$251$	2296.00	0.577379	0.288690	+	0.957423i	$0.406780\pi$
$252$	0	0
$253$	− 3584.00i	− 0.890609i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 4058.00i	− 0.984946i	−0.870328	−	0.492473i	$-0.836093\pi$
$257$	− 4058.00i	− 0.984946i	0.870328	−	0.492473i	$-0.163907\pi$
$258$	0	0
$259$	2030.00	0.487020
$260$	0	0
$261$	6438.00	1.52683
$262$	0	0
$263$	5640.00i	1.32235i	0.750233	+	0.661174i	$0.229941\pi$
$263$	5640.00i	1.32235i	−0.750233	+	0.661174i	$0.770059\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 5584.00i	− 1.27991i
$268$	0	0
$269$	7758.00	1.75841	0.879207	−	0.476439i	$-0.158073\pi$
$269$	7758.00	1.75841	0.879207	+	0.476439i	$0.158073\pi$
$270$	0	0
$271$	3264.00	0.731638	0.365819	−	0.930686i	$-0.380789\pi$
$271$	3264.00	0.731638	0.365819	+	0.930686i	$0.380789\pi$
$272$	0	0
$273$	4592.00i	1.01802i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 566.000i	− 0.122771i	−0.998114	−	0.0613856i	$-0.980448\pi$
$277$	− 566.000i	− 0.122771i	0.998114	−	0.0613856i	$-0.0195519\pi$
$278$	0	0
$279$	5624.00	1.20681
$280$	0	0
$281$	4234.00	0.898859	0.449429	−	0.893316i	$-0.351627\pi$
$281$	4234.00	0.898859	0.449429	+	0.893316i	$0.351627\pi$
$282$	0	0
$283$	728.000i	0.152916i	0.997073	+	0.0764578i	$0.0243610\pi$
$283$	728.000i	0.152916i	−0.997073	+	0.0764578i	$0.975639\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 350.000i	− 0.0719855i
$288$	0	0
$289$	2797.00	0.569306
$290$	0	0
$291$	6032.00	1.21513
$292$	0	0
$293$	5490.00i	1.09464i	0.836924	+	0.547319i	$0.184352\pi$
$293$	5490.00i	1.09464i	−0.836924	+	0.547319i	$0.815648\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 2240.00i	− 0.437636i
$298$	0	0
$299$	10496.0	2.03010
$300$	0	0
$301$	2772.00	0.530815
$302$	0	0
$303$	2832.00i	0.536944i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 1664.00i	− 0.309347i	−0.987966	−	0.154673i	$-0.950567\pi$
$307$	− 1664.00i	− 0.309347i	0.987966	−	0.154673i	$-0.0494325\pi$
$308$	0	0
$309$	−15232.0	−2.80427
$310$	0	0
$311$	1688.00	0.307774	0.153887	−	0.988088i	$-0.450821\pi$
$311$	1688.00	0.307774	0.153887	+	0.988088i	$0.450821\pi$
$312$	0	0
$313$	− 2294.00i	− 0.414264i	−0.978313	−	0.207132i	$-0.933587\pi$
$313$	− 2294.00i	− 0.414264i	0.978313	−	0.207132i	$-0.0664129\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1426.00i	0.252657i	0.991988	+	0.126328i	$0.0403193\pi$
$317$	1426.00i	0.252657i	−0.991988	+	0.126328i	$0.959681\pi$
$318$	0	0
$319$	−4872.00	−0.855109
$320$	0	0
$321$	8864.00	1.54125
$322$	0	0
$323$	− 368.000i	− 0.0633934i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	1168.00i	0.197525i
$328$	0	0
$329$	2072.00	0.347213
$330$	0	0
$331$	−740.000	−0.122882	−0.0614412	−	0.998111i	$-0.519570\pi$
$331$	−740.000	−0.122882	−0.0614412	+	0.998111i	$0.519570\pi$
$332$	0	0
$333$	− 10730.0i	− 1.76577i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	11086.0i	1.79197i	0.444087	+	0.895984i	$0.353528\pi$
$337$	11086.0i	1.79197i	−0.444087	+	0.895984i	$0.646472\pi$
$338$	0	0
$339$	16624.0	2.66340
$340$	0	0
$341$	−4256.00	−0.675881
$342$	0	0
$343$	343.000i	0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	6420.00i	0.993209i	0.867977	+	0.496605i	$0.165420\pi$
$347$	6420.00i	0.993209i	−0.867977	+	0.496605i	$0.834580\pi$
$348$	0	0
$349$	6430.00	0.986218	0.493109	−	0.869968i	$-0.335860\pi$
$349$	6430.00	0.986218	0.493109	+	0.869968i	$0.335860\pi$
$350$	0	0
$351$	6560.00	0.997570
$352$	0	0
$353$	− 5910.00i	− 0.891098i	−0.895258	−	0.445549i	$-0.853009\pi$
$353$	− 5910.00i	− 0.891098i	0.895258	−	0.445549i	$-0.146991\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	2576.00i	0.381895i
$358$	0	0
$359$	5720.00	0.840919	0.420460	−	0.907311i	$-0.361869\pi$
$359$	5720.00	0.840919	0.420460	+	0.907311i	$0.361869\pi$
$360$	0	0
$361$	−6795.00	−0.990669
$362$	0	0
$363$	− 4376.00i	− 0.632728i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 8320.00i	− 1.18338i	−0.806166	−	0.591690i	$-0.798461\pi$
$367$	− 8320.00i	− 1.18338i	0.806166	−	0.591690i	$-0.201539\pi$
$368$	0	0
$369$	−1850.00	−0.260995
$370$	0	0
$371$	−3990.00	−0.558357
$372$	0	0
$373$	1718.00i	0.238484i	0.992865	+	0.119242i	$0.0380465\pi$
$373$	1718.00i	0.238484i	−0.992865	+	0.119242i	$0.961954\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 14268.0i	− 1.94918i
$378$	0	0
$379$	−364.000	−0.0493336	−0.0246668	−	0.999696i	$-0.507852\pi$
$379$	−364.000	−0.0493336	−0.0246668	+	0.999696i	$0.507852\pi$
$380$	0	0
$381$	−4800.00	−0.645437
$382$	0	0
$383$	2088.00i	0.278569i	0.990252	+	0.139284i	$0.0444802\pi$
$383$	2088.00i	0.278569i	−0.990252	+	0.139284i	$0.955520\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 14652.0i	− 1.92456i
$388$	0	0
$389$	−1446.00	−0.188471	−0.0942354	−	0.995550i	$-0.530041\pi$
$389$	−1446.00	−0.188471	−0.0942354	+	0.995550i	$0.530041\pi$
$390$	0	0
$391$	5888.00	0.761557
$392$	0	0
$393$	− 1344.00i	− 0.172508i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	10910.0i	1.37924i	0.724173	+	0.689619i	$0.242222\pi$
$397$	10910.0i	1.37924i	−0.724173	+	0.689619i	$0.757778\pi$
$398$	0	0
$399$	−448.000	−0.0562107
$400$	0	0
$401$	3090.00	0.384806	0.192403	−	0.981316i	$-0.438372\pi$
$401$	3090.00	0.384806	0.192403	+	0.981316i	$0.438372\pi$
$402$	0	0
$403$	− 12464.0i	− 1.54064i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	8120.00i	0.988927i
$408$	0	0
$409$	−2194.00	−0.265248	−0.132624	−	0.991166i	$-0.542340\pi$
$409$	−2194.00	−0.265248	−0.132624	+	0.991166i	$0.542340\pi$
$410$	0	0
$411$	3152.00	0.378289
$412$	0	0
$413$	− 1904.00i	− 0.226852i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	3776.00i	0.443433i
$418$	0	0
$419$	−3744.00	−0.436531	−0.218265	−	0.975889i	$-0.570040\pi$
$419$	−3744.00	−0.436531	−0.218265	+	0.975889i	$0.570040\pi$
$420$	0	0
$421$	−13106.0	−1.51721	−0.758607	−	0.651548i	$-0.774120\pi$
$421$	−13106.0	−1.51721	−0.758607	+	0.651548i	$0.774120\pi$
$422$	0	0
$423$	− 10952.0i	− 1.25888i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	4634.00i	0.525187i
$428$	0	0
$429$	−18368.0	−2.06717
$430$	0	0
$431$	12240.0	1.36794	0.683968	−	0.729512i	$-0.260253\pi$
$431$	12240.0	1.36794	0.683968	+	0.729512i	$0.260253\pi$
$432$	0	0
$433$	12402.0i	1.37645i	0.725498	+	0.688224i	$0.241609\pi$
$433$	12402.0i	1.37645i	−0.725498	+	0.688224i	$0.758391\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1024.00i	0.112093i
$438$	0	0
$439$	12280.0	1.33506	0.667531	−	0.744582i	$-0.267351\pi$
$439$	12280.0	1.33506	0.667531	+	0.744582i	$0.267351\pi$
$440$	0	0
$441$	1813.00	0.195767
$442$	0	0
$443$	3092.00i	0.331615i	0.986158	+	0.165807i	$0.0530230\pi$
$443$	3092.00i	0.331615i	−0.986158	+	0.165807i	$0.946977\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	21264.0i	2.25001i
$448$	0	0
$449$	14750.0	1.55032	0.775162	−	0.631762i	$-0.217668\pi$
$449$	14750.0	1.55032	0.775162	+	0.631762i	$0.217668\pi$
$450$	0	0
$451$	1400.00	0.146172
$452$	0	0
$453$	10048.0i	1.04216i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 6394.00i	− 0.654483i	−0.944941	−	0.327241i	$-0.893881\pi$
$457$	− 6394.00i	− 0.654483i	0.944941	−	0.327241i	$-0.106119\pi$
$458$	0	0
$459$	3680.00	0.374222
$460$	0	0
$461$	−14590.0	−1.47402	−0.737011	−	0.675881i	$-0.763763\pi$
$461$	−14590.0	−1.47402	−0.737011	+	0.675881i	$0.763763\pi$
$462$	0	0
$463$	13664.0i	1.37153i	0.727821	+	0.685767i	$0.240533\pi$
$463$	13664.0i	1.37153i	−0.727821	+	0.685767i	$0.759467\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 4960.00i	− 0.491481i	−0.969336	−	0.245740i	$-0.920969\pi$
$467$	− 4960.00i	− 0.491481i	0.969336	−	0.245740i	$-0.0790310\pi$
$468$	0	0
$469$	−6132.00	−0.603730
$470$	0	0
$471$	26128.0	2.55608
$472$	0	0
$473$	11088.0i	1.07786i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	21090.0i	2.02441i
$478$	0	0
$479$	−5304.00	−0.505941	−0.252971	−	0.967474i	$-0.581408\pi$
$479$	−5304.00	−0.505941	−0.252971	+	0.967474i	$0.581408\pi$
$480$	0	0
$481$	−23780.0	−2.25421
$482$	0	0
$483$	− 7168.00i	− 0.675270i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	21120.0i	1.96517i	0.185810	+	0.982586i	$0.440509\pi$
$487$	21120.0i	1.96517i	−0.185810	+	0.982586i	$0.559491\pi$
$488$	0	0
$489$	−13728.0	−1.26953
$490$	0	0
$491$	10500.0	0.965088	0.482544	−	0.875872i	$-0.339713\pi$
$491$	10500.0	0.965088	0.482544	+	0.875872i	$0.339713\pi$
$492$	0	0
$493$	− 8004.00i	− 0.731201i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	6160.00i	0.555963i
$498$	0	0
$499$	3620.00	0.324756	0.162378	−	0.986729i	$-0.448084\pi$
$499$	3620.00	0.324756	0.162378	+	0.986729i	$0.448084\pi$
$500$	0	0
$501$	−18176.0	−1.62085
$502$	0	0
$503$	11176.0i	0.990682i	0.868699	+	0.495341i	$0.164957\pi$
$503$	11176.0i	0.990682i	−0.868699	+	0.495341i	$0.835043\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 36216.0i	− 3.17240i
$508$	0	0
$509$	−3362.00	−0.292766	−0.146383	−	0.989228i	$-0.546763\pi$
$509$	−3362.00	−0.292766	−0.146383	+	0.989228i	$0.546763\pi$
$510$	0	0
$511$	−4466.00	−0.386623
$512$	0	0
$513$	640.000i	0.0550813i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	8288.00i	0.705040i
$518$	0	0
$519$	−33808.0	−2.85936
$520$	0	0
$521$	16530.0	1.39000	0.695002	−	0.719007i	$-0.255403\pi$
$521$	16530.0	1.39000	0.695002	+	0.719007i	$0.255403\pi$
$522$	0	0
$523$	6440.00i	0.538435i	0.963079	+	0.269218i	$0.0867651\pi$
$523$	6440.00i	0.538435i	−0.963079	+	0.269218i	$0.913235\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 6992.00i	− 0.577944i
$528$	0	0
$529$	−4217.00	−0.346593
$530$	0	0
$531$	−10064.0	−0.822487
$532$	0	0
$533$	4100.00i	0.333191i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 7904.00i	− 0.635163i
$538$	0	0
$539$	−1372.00	−0.109640
$540$	0	0
$541$	−7210.00	−0.572980	−0.286490	−	0.958083i	$-0.592488\pi$
$541$	−7210.00	−0.572980	−0.286490	+	0.958083i	$0.592488\pi$
$542$	0	0
$543$	17104.0i	1.35175i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 21796.0i	− 1.70371i	−0.523777	−	0.851855i	$-0.675478\pi$
$547$	− 21796.0i	− 1.70371i	0.523777	−	0.851855i	$-0.324522\pi$
$548$	0	0
$549$	24494.0	1.90415
$550$	0	0
$551$	1392.00	0.107625
$552$	0	0
$553$	− 4200.00i	− 0.322970i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 10030.0i	− 0.762989i	−0.924371	−	0.381494i	$-0.875410\pi$
$557$	− 10030.0i	− 0.762989i	0.924371	−	0.381494i	$-0.124590\pi$
$558$	0	0
$559$	−32472.0	−2.45692
$560$	0	0
$561$	−10304.0	−0.775464
$562$	0	0
$563$	9144.00i	0.684500i	0.939609	+	0.342250i	$0.111189\pi$
$563$	9144.00i	0.684500i	−0.939609	+	0.342250i	$0.888811\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	2513.00i	0.186131i
$568$	0	0
$569$	−11946.0	−0.880145	−0.440072	−	0.897962i	$-0.645047\pi$
$569$	−11946.0	−0.880145	−0.440072	+	0.897962i	$0.645047\pi$
$570$	0	0
$571$	11596.0	0.849873	0.424937	−	0.905223i	$-0.360296\pi$
$571$	11596.0	0.849873	0.424937	+	0.905223i	$0.360296\pi$
$572$	0	0
$573$	− 30656.0i	− 2.23503i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 1754.00i	− 0.126551i	−0.997996	−	0.0632755i	$-0.979845\pi$
$577$	− 1754.00i	− 0.126551i	0.997996	−	0.0632755i	$-0.0201547\pi$
$578$	0	0
$579$	−11024.0	−0.791264
$580$	0	0
$581$	4368.00	0.311902
$582$	0	0
$583$	− 15960.0i	− 1.13378i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 13152.0i	− 0.924772i	−0.886679	−	0.462386i	$-0.846993\pi$
$587$	− 13152.0i	− 0.924772i	0.886679	−	0.462386i	$-0.153007\pi$
$588$	0	0
$589$	1216.00	0.0850669
$590$	0	0
$591$	−34128.0	−2.37536
$592$	0	0
$593$	− 13014.0i	− 0.901216i	−0.892722	−	0.450608i	$-0.851207\pi$
$593$	− 13014.0i	− 0.901216i	0.892722	−	0.450608i	$-0.148793\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 40960.0i	− 2.80801i
$598$	0	0
$599$	19936.0	1.35987	0.679936	−	0.733272i	$-0.262008\pi$
$599$	19936.0	1.35987	0.679936	+	0.733272i	$0.262008\pi$
$600$	0	0
$601$	15578.0	1.05730	0.528652	−	0.848839i	$-0.322698\pi$
$601$	15578.0	1.05730	0.528652	+	0.848839i	$0.322698\pi$
$602$	0	0
$603$	32412.0i	2.18892i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	3792.00i	0.253563i	0.991931	+	0.126781i	$0.0404646\pi$
$607$	3792.00i	0.253563i	−0.991931	+	0.126781i	$0.959535\pi$
$608$	0	0
$609$	−9744.00	−0.648353
$610$	0	0
$611$	−24272.0	−1.60710
$612$	0	0
$613$	− 5074.00i	− 0.334318i	−0.985930	−	0.167159i	$-0.946541\pi$
$613$	− 5074.00i	− 0.334318i	0.985930	−	0.167159i	$-0.0534593\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	13670.0i	0.891951i	0.895045	+	0.445975i	$0.147143\pi$
$617$	13670.0i	0.891951i	−0.895045	+	0.445975i	$0.852857\pi$
$618$	0	0
$619$	18560.0	1.20515	0.602576	−	0.798061i	$-0.294141\pi$
$619$	18560.0	1.20515	0.602576	+	0.798061i	$0.294141\pi$
$620$	0	0
$621$	−10240.0	−0.661702
$622$	0	0
$623$	4886.00i	0.314211i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 1792.00i	− 0.114140i
$628$	0	0
$629$	−13340.0	−0.845629
$630$	0	0
$631$	−8400.00	−0.529950	−0.264975	−	0.964255i	$-0.585364\pi$
$631$	−8400.00	−0.529950	−0.264975	+	0.964255i	$0.585364\pi$
$632$	0	0
$633$	− 14240.0i	− 0.894138i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 4018.00i	− 0.249920i
$638$	0	0
$639$	32560.0	2.01573
$640$	0	0
$641$	16770.0	1.03335	0.516673	−	0.856183i	$-0.327170\pi$
$641$	16770.0	1.03335	0.516673	+	0.856183i	$0.327170\pi$
$642$	0	0
$643$	− 12152.0i	− 0.745300i	−0.927972	−	0.372650i	$-0.878449\pi$
$643$	− 12152.0i	− 0.745300i	0.927972	−	0.372650i	$-0.121551\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 14896.0i	− 0.905135i	−0.891730	−	0.452567i	$-0.850508\pi$
$647$	− 14896.0i	− 0.905135i	0.891730	−	0.452567i	$-0.149492\pi$
$648$	0	0
$649$	7616.00	0.460638
$650$	0	0
$651$	−8512.00	−0.512460
$652$	0	0
$653$	11942.0i	0.715661i	0.933787	+	0.357830i	$0.116483\pi$
$653$	11942.0i	0.715661i	−0.933787	+	0.357830i	$0.883517\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	23606.0i	1.40176i
$658$	0	0
$659$	31772.0	1.87809	0.939045	−	0.343794i	$-0.111712\pi$
$659$	31772.0	1.87809	0.939045	+	0.343794i	$0.111712\pi$
$660$	0	0
$661$	−9502.00	−0.559130	−0.279565	−	0.960127i	$-0.590190\pi$
$661$	−9502.00	−0.559130	−0.279565	+	0.960127i	$0.590190\pi$
$662$	0	0
$663$	− 30176.0i	− 1.76763i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	22272.0i	1.29292i
$668$	0	0
$669$	−39168.0	−2.26356
$670$	0	0
$671$	−18536.0	−1.06643
$672$	0	0
$673$	− 4750.00i	− 0.272064i	−0.990704	−	0.136032i	$-0.956565\pi$
$673$	− 4750.00i	− 0.272064i	0.990704	−	0.136032i	$-0.0434350\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 23610.0i	− 1.34033i	−0.742210	−	0.670167i	$-0.766223\pi$
$677$	− 23610.0i	− 1.34033i	0.742210	−	0.670167i	$-0.233777\pi$
$678$	0	0
$679$	−5278.00	−0.298308
$680$	0	0
$681$	−28032.0	−1.57737
$682$	0	0
$683$	26852.0i	1.50434i	0.658970	+	0.752169i	$0.270992\pi$
$683$	26852.0i	1.50434i	−0.658970	+	0.752169i	$0.729008\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 6608.00i	− 0.366974i
$688$	0	0
$689$	46740.0	2.58440
$690$	0	0
$691$	−2648.00	−0.145781	−0.0728905	−	0.997340i	$-0.523222\pi$
$691$	−2648.00	−0.145781	−0.0728905	+	0.997340i	$0.523222\pi$
$692$	0	0
$693$	7252.00i	0.397519i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	2300.00i	0.124991i
$698$	0	0
$699$	2224.00	0.120342
$700$	0	0
$701$	29118.0	1.56886	0.784431	−	0.620217i	$-0.212955\pi$
$701$	29118.0	1.56886	0.784431	+	0.620217i	$0.212955\pi$
$702$	0	0
$703$	− 2320.00i	− 0.124467i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 2478.00i	− 0.131817i
$708$	0	0
$709$	14474.0	0.766689	0.383344	−	0.923605i	$-0.374772\pi$
$709$	14474.0	0.766689	0.383344	+	0.923605i	$0.374772\pi$
$710$	0	0
$711$	−22200.0	−1.17098
$712$	0	0
$713$	19456.0i	1.02193i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 37888.0i	− 1.97344i
$718$	0	0
$719$	−10968.0	−0.568898	−0.284449	−	0.958691i	$-0.591811\pi$
$719$	−10968.0	−0.568898	−0.284449	+	0.958691i	$0.591811\pi$
$720$	0	0
$721$	13328.0	0.688434
$722$	0	0
$723$	− 31920.0i	− 1.64193i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 18624.0i	− 0.950104i	−0.879958	−	0.475052i	$-0.842429\pi$
$727$	− 18624.0i	− 0.950104i	0.879958	−	0.475052i	$-0.157571\pi$
$728$	0	0
$729$	30563.0	1.55276
$730$	0	0
$731$	−18216.0	−0.921673
$732$	0	0
$733$	− 37014.0i	− 1.86513i	−0.360997	−	0.932567i	$-0.617563\pi$
$733$	− 37014.0i	− 1.86513i	0.360997	−	0.932567i	$-0.382437\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 24528.0i	− 1.22592i
$738$	0	0
$739$	−1284.00	−0.0639143	−0.0319572	−	0.999489i	$-0.510174\pi$
$739$	−1284.00	−0.0639143	−0.0319572	+	0.999489i	$0.510174\pi$
$740$	0	0
$741$	5248.00	0.260176
$742$	0	0
$743$	25584.0i	1.26324i	0.775279	+	0.631619i	$0.217609\pi$
$743$	25584.0i	1.26324i	−0.775279	+	0.631619i	$0.782391\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 23088.0i	− 1.13085i
$748$	0	0
$749$	−7756.00	−0.378369
$750$	0	0
$751$	−3872.00	−0.188138	−0.0940688	−	0.995566i	$-0.529987\pi$
$751$	−3872.00	−0.188138	−0.0940688	+	0.995566i	$0.529987\pi$
$752$	0	0
$753$	18368.0i	0.888934i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 22238.0i	− 1.06771i	−0.845577	−	0.533853i	$-0.820743\pi$
$757$	− 22238.0i	− 1.06771i	0.845577	−	0.533853i	$-0.179257\pi$
$758$	0	0
$759$	28672.0	1.37118
$760$	0	0
$761$	10178.0	0.484826	0.242413	−	0.970173i	$-0.422061\pi$
$761$	10178.0	0.484826	0.242413	+	0.970173i	$0.422061\pi$
$762$	0	0
$763$	− 1022.00i	− 0.0484913i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	22304.0i	1.05000i
$768$	0	0
$769$	−31498.0	−1.47704	−0.738522	−	0.674229i	$-0.764476\pi$
$769$	−31498.0	−1.47704	−0.738522	+	0.674229i	$0.764476\pi$
$770$	0	0
$771$	32464.0	1.51642
$772$	0	0
$773$	19090.0i	0.888253i	0.895964	+	0.444127i	$0.146486\pi$
$773$	19090.0i	0.888253i	−0.895964	+	0.444127i	$0.853514\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	16240.0i	0.749816i
$778$	0	0
$779$	−400.000	−0.0183973
$780$	0	0
$781$	−24640.0	−1.12892
$782$	0	0
$783$	13920.0i	0.635326i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 23360.0i	− 1.05806i	−0.848603	−	0.529031i	$-0.822556\pi$
$787$	− 23360.0i	− 1.05806i	0.848603	−	0.529031i	$-0.177444\pi$
$788$	0	0
$789$	−45120.0	−2.03589
$790$	0	0
$791$	−14546.0	−0.653851
$792$	0	0
$793$	− 54284.0i	− 2.43087i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	28686.0i	1.27492i	0.770484	+	0.637459i	$0.220015\pi$
$797$	28686.0i	1.27492i	−0.770484	+	0.637459i	$0.779985\pi$
$798$	0	0
$799$	−13616.0	−0.602878
$800$	0	0
$801$	25826.0	1.13922
$802$	0	0
$803$	− 17864.0i	− 0.785065i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	62064.0i	2.70726i
$808$	0	0
$809$	−33226.0	−1.44396	−0.721980	−	0.691914i	$-0.756768\pi$
$809$	−33226.0	−1.44396	−0.721980	+	0.691914i	$0.756768\pi$
$810$	0	0
$811$	−33024.0	−1.42988	−0.714938	−	0.699188i	$-0.753545\pi$
$811$	−33024.0	−1.42988	−0.714938	+	0.699188i	$0.753545\pi$
$812$	0	0
$813$	26112.0i	1.12643i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 3168.00i	− 0.135660i
$818$	0	0
$819$	−21238.0	−0.906124
$820$	0	0
$821$	−17090.0	−0.726486	−0.363243	−	0.931694i	$-0.618331\pi$
$821$	−17090.0	−0.726486	−0.363243	+	0.931694i	$0.618331\pi$
$822$	0	0
$823$	− 11720.0i	− 0.496396i	−0.968709	−	0.248198i	$-0.920162\pi$
$823$	− 11720.0i	− 0.496396i	0.968709	−	0.248198i	$-0.0798383\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	34764.0i	1.46174i	0.682514	+	0.730872i	$0.260886\pi$
$827$	34764.0i	1.46174i	−0.682514	+	0.730872i	$0.739114\pi$
$828$	0	0
$829$	−650.000	−0.0272321	−0.0136161	−	0.999907i	$-0.504334\pi$
$829$	−650.000	−0.0272321	−0.0136161	+	0.999907i	$0.504334\pi$
$830$	0	0
$831$	4528.00	0.189019
$832$	0	0
$833$	− 2254.00i	− 0.0937533i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	12160.0i	0.502164i
$838$	0	0
$839$	33888.0	1.39445	0.697225	−	0.716852i	$-0.254418\pi$
$839$	33888.0	1.39445	0.697225	+	0.716852i	$0.254418\pi$
$840$	0	0
$841$	5887.00	0.241379
$842$	0	0
$843$	33872.0i	1.38388i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	3829.00i	0.155332i
$848$	0	0
$849$	−5824.00	−0.235429
$850$	0	0
$851$	37120.0	1.49525
$852$	0	0
$853$	− 14246.0i	− 0.571833i	−0.958255	−	0.285917i	$-0.907702\pi$
$853$	− 14246.0i	− 0.571833i	0.958255	−	0.285917i	$-0.0922980\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 362.000i	− 0.0144290i	−0.999974	−	0.00721452i	$-0.997704\pi$
$857$	− 362.000i	− 0.0144290i	0.999974	−	0.00721452i	$-0.00229647\pi$
$858$	0	0
$859$	−25544.0	−1.01461	−0.507305	−	0.861767i	$-0.669358\pi$
$859$	−25544.0	−1.01461	−0.507305	+	0.861767i	$0.669358\pi$
$860$	0	0
$861$	2800.00	0.110829
$862$	0	0
$863$	18576.0i	0.732717i	0.930474	+	0.366358i	$0.119396\pi$
$863$	18576.0i	0.732717i	−0.930474	+	0.366358i	$0.880604\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	22376.0i	0.876504i
$868$	0	0
$869$	16800.0	0.655812
$870$	0	0
$871$	71832.0	2.79441
$872$	0	0
$873$	27898.0i	1.08156i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 29606.0i	− 1.13994i	−0.821667	−	0.569968i	$-0.806956\pi$
$877$	− 29606.0i	− 1.13994i	0.821667	−	0.569968i	$-0.193044\pi$
$878$	0	0
$879$	−43920.0	−1.68531
$880$	0	0
$881$	45378.0	1.73533	0.867664	−	0.497151i	$-0.165621\pi$
$881$	45378.0	1.73533	0.867664	+	0.497151i	$0.165621\pi$
$882$	0	0
$883$	22540.0i	0.859039i	0.903058	+	0.429519i	$0.141317\pi$
$883$	22540.0i	0.859039i	−0.903058	+	0.429519i	$0.858683\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	20336.0i	0.769804i	0.922957	+	0.384902i	$0.125765\pi$
$887$	20336.0i	0.769804i	−0.922957	+	0.384902i	$0.874235\pi$
$888$	0	0
$889$	4200.00	0.158452
$890$	0	0
$891$	−10052.0	−0.377951
$892$	0	0
$893$	− 2368.00i	− 0.0887370i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	83968.0i	3.12554i
$898$	0	0
$899$	26448.0	0.981190
$900$	0	0
$901$	26220.0	0.969495
$902$	0	0
$903$	22176.0i	0.817244i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	34764.0i	1.27268i	0.771409	+	0.636339i	$0.219552\pi$
$907$	34764.0i	1.27268i	−0.771409	+	0.636339i	$0.780448\pi$
$908$	0	0
$909$	−13098.0	−0.477924
$910$	0	0
$911$	−240.000	−0.00872838	−0.00436419	−	0.999990i	$-0.501389\pi$
$911$	−240.000	−0.00872838	−0.00436419	+	0.999990i	$0.501389\pi$
$912$	0	0
$913$	17472.0i	0.633339i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1176.00i	0.0423500i
$918$	0	0
$919$	−37264.0	−1.33757	−0.668785	−	0.743456i	$-0.733185\pi$
$919$	−37264.0	−1.33757	−0.668785	+	0.743456i	$0.733185\pi$
$920$	0	0
$921$	13312.0	0.476271
$922$	0	0
$923$	− 72160.0i	− 2.57332i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 70448.0i	− 2.49603i
$928$	0	0
$929$	13286.0	0.469214	0.234607	−	0.972090i	$-0.424620\pi$
$929$	13286.0	0.469214	0.234607	+	0.972090i	$0.424620\pi$
$930$	0	0
$931$	392.000	0.0137994
$932$	0	0
$933$	13504.0i	0.473849i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	13614.0i	0.474653i	0.971430	+	0.237327i	$0.0762711\pi$
$937$	13614.0i	0.474653i	−0.971430	+	0.237327i	$0.923729\pi$
$938$	0	0
$939$	18352.0	0.637801
$940$	0	0
$941$	36570.0	1.26689	0.633447	−	0.773786i	$-0.281639\pi$
$941$	36570.0	1.26689	0.633447	+	0.773786i	$0.281639\pi$
$942$	0	0
$943$	− 6400.00i	− 0.221010i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 8772.00i	− 0.301005i	−0.988610	−	0.150502i	$-0.951911\pi$
$947$	− 8772.00i	− 0.301005i	0.988610	−	0.150502i	$-0.0480892\pi$
$948$	0	0
$949$	52316.0	1.78951
$950$	0	0
$951$	−11408.0	−0.388990
$952$	0	0
$953$	− 8742.00i	− 0.297147i	−0.988901	−	0.148574i	$-0.952532\pi$
$953$	− 8742.00i	− 0.297147i	0.988901	−	0.148574i	$-0.0474682\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 38976.0i	− 1.31653i
$958$	0	0
$959$	−2758.00	−0.0928681
$960$	0	0
$961$	−6687.00	−0.224464
$962$	0	0
$963$	40996.0i	1.37184i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	51248.0i	1.70427i	0.523326	+	0.852133i	$0.324691\pi$
$967$	51248.0i	1.70427i	−0.523326	+	0.852133i	$0.675309\pi$
$968$	0	0
$969$	2944.00	0.0976005
$970$	0	0
$971$	41376.0	1.36748	0.683738	−	0.729728i	$-0.260353\pi$
$971$	41376.0	1.36748	0.683738	+	0.729728i	$0.260353\pi$
$972$	0	0
$973$	− 3304.00i	− 0.108861i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 15234.0i	− 0.498852i	−0.968394	−	0.249426i	$-0.919758\pi$
$977$	− 15234.0i	− 0.498852i	0.968394	−	0.249426i	$-0.0802420\pi$
$978$	0	0
$979$	−19544.0	−0.638028
$980$	0	0
$981$	−5402.00	−0.175813
$982$	0	0
$983$	− 22384.0i	− 0.726286i	−0.931733	−	0.363143i	$-0.881704\pi$
$983$	− 22384.0i	− 0.726286i	0.931733	−	0.363143i	$-0.118296\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	16576.0i	0.534569i
$988$	0	0
$989$	50688.0	1.62971
$990$	0	0
$991$	−3656.00	−0.117191	−0.0585957	−	0.998282i	$-0.518662\pi$
$991$	−3656.00	−0.117191	−0.0585957	+	0.998282i	$0.518662\pi$
$992$	0	0
$993$	− 5920.00i	− 0.189190i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	15406.0i	0.489381i	0.969601	+	0.244691i	$0.0786864\pi$
$997$	15406.0i	0.489381i	−0.969601	+	0.244691i	$0.921314\pi$
$998$	0	0
$999$	23200.0	0.734750

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	700.4.e.c.449.2		2
5.2	odd	4		140.4.a.e.1.1	✓	1
5.3	odd	4		700.4.a.b.1.1		1
5.4	even	2	inner	700.4.e.c.449.1		2
15.2	even	4		1260.4.a.j.1.1		1
20.7	even	4		560.4.a.b.1.1		1
35.2	odd	12		980.4.i.b.361.1		2
35.12	even	12		980.4.i.q.361.1		2
35.17	even	12		980.4.i.q.961.1		2
35.27	even	4		980.4.a.b.1.1		1
35.32	odd	12		980.4.i.b.961.1		2
40.27	even	4		2240.4.a.bj.1.1		1
40.37	odd	4		2240.4.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.4.a.e.1.1	✓	1	5.2	odd	4
560.4.a.b.1.1		1	20.7	even	4
700.4.a.b.1.1		1	5.3	odd	4
700.4.e.c.449.1		2	5.4	even	2	inner
700.4.e.c.449.2		2	1.1	even	1	trivial
980.4.a.b.1.1		1	35.27	even	4
980.4.i.b.361.1		2	35.2	odd	12
980.4.i.b.961.1		2	35.32	odd	12
980.4.i.q.361.1		2	35.12	even	12
980.4.i.q.961.1		2	35.17	even	12
1260.4.a.j.1.1		1	15.2	even	4
2240.4.a.c.1.1		1	40.37	odd	4
2240.4.a.bj.1.1		1	40.27	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	700.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+8.00000i q^{3} -7.00000i q^{7} -37.0000 q^{9} +28.0000 q^{11} +82.0000i q^{13} +46.0000i q^{17} -8.00000 q^{19} +56.0000 q^{21} -128.000i q^{23} -80.0000i q^{27} -174.000 q^{29} -152.000 q^{31} +224.000i q^{33} +290.000i q^{37} -656.000 q^{39} +50.0000 q^{41} +396.000i q^{43} +296.000i q^{47} -49.0000 q^{49} -368.000 q^{51} -570.000i q^{53} -64.0000i q^{57} +272.000 q^{59} -662.000 q^{61} +259.000i q^{63} -876.000i q^{67} +1024.00 q^{69} -880.000 q^{71} -638.000i q^{73} -196.000i q^{77} +600.000 q^{79} -359.000 q^{81} +624.000i q^{83} -1392.00i q^{87} -698.000 q^{89} +574.000 q^{91} -1216.00i q^{93} -754.000i q^{97} -1036.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 74 q^{9} + 56 q^{11} - 16 q^{19} + 112 q^{21} - 348 q^{29} - 304 q^{31} - 1312 q^{39} + 100 q^{41} - 98 q^{49} - 736 q^{51} + 544 q^{59} - 1324 q^{61} + 2048 q^{69} - 1760 q^{71} + 1200 q^{79} - 718 q^{81}+ \cdots - 2072 q^{99}+O(q^{100}) \) 2 * q - 74 * q^9 + 56 * q^11 - 16 * q^19 + 112 * q^21 - 348 * q^29 - 304 * q^31 - 1312 * q^39 + 100 * q^41 - 98 * q^49 - 736 * q^51 + 544 * q^59 - 1324 * q^61 + 2048 * q^69 - 1760 * q^71 + 1200 * q^79 - 718 * q^81 - 1396 * q^89 + 1148 * q^91 - 2072 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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