LMFDB - Embedded newform 1800.4.f.s.649.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1800,4,Mod(649,1800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1800.649");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1800.f (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$106.203438010$
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:	$\Z[a_1, \ldots, a_{37}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 360)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1800.649
Dual form			1800.4.f.s.649.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+18.0000i q^{7} +O(q^{10})$	$q+18.0000i q^{7} +34.0000 q^{11} +12.0000i q^{13} +102.000i q^{17} -164.000 q^{19} +48.0000i q^{23} -146.000 q^{29} +100.000 q^{31} -328.000i q^{37} -288.000 q^{41} +120.000i q^{43} -16.0000i q^{47} +19.0000 q^{49} -126.000i q^{53} -642.000 q^{59} +602.000 q^{61} -436.000i q^{67} +652.000 q^{71} +1062.00i q^{73} +612.000i q^{77} -388.000 q^{79} -444.000i q^{83} +820.000 q^{89} -216.000 q^{91} +766.000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q+O(q^{10}) $ 2 * q	$ 2 q + 68 q^{11} - 328 q^{19} - 292 q^{29} + 200 q^{31} - 576 q^{41} + 38 q^{49} - 1284 q^{59} + 1204 q^{61} + 1304 q^{71} - 776 q^{79} + 1640 q^{89} - 432 q^{91}+O(q^{100}) $ 2 * q + 68 * q^11 - 328 * q^19 - 292 * q^29 + 200 * q^31 - 576 * q^41 + 38 * q^49 - 1284 * q^59 + 1204 * q^61 + 1304 * q^71 - 776 * q^79 + 1640 * q^89 - 432 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1800\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1001$	$1351$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	18.0000i	0.971909i	0.873984	+	0.485954i	$0.161528\pi$
$7$	18.0000i	0.971909i	−0.873984	+	0.485954i	$0.838472\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	34.0000	0.931944	0.465972	−	0.884799i	$-0.345705\pi$
$11$	34.0000	0.931944	0.465972	+	0.884799i	$0.345705\pi$
$12$	0	0
$13$	12.0000i	0.256015i	0.991773	+	0.128008i	$0.0408582\pi$
$13$	12.0000i	0.256015i	−0.991773	+	0.128008i	$0.959142\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	102.000i	1.45521i	0.685994	+	0.727607i	$0.259367\pi$
$17$	102.000i	1.45521i	−0.685994	+	0.727607i	$0.740633\pi$
$18$	0	0
$19$	−164.000	−1.98022	−0.990110	−	0.140293i	$-0.955195\pi$
$19$	−164.000	−1.98022	−0.990110	+	0.140293i	$0.955195\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	48.0000i	0.435161i	0.976042	+	0.217580i	$0.0698164\pi$
$23$	48.0000i	0.435161i	−0.976042	+	0.217580i	$0.930184\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−146.000	−0.934880	−0.467440	−	0.884025i	$-0.654824\pi$
$29$	−146.000	−0.934880	−0.467440	+	0.884025i	$0.654824\pi$
$30$	0	0
$31$	100.000	0.579372	0.289686	−	0.957122i	$-0.406449\pi$
$31$	100.000	0.579372	0.289686	+	0.957122i	$0.406449\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 328.000i	− 1.45737i	−0.684846	−	0.728687i	$-0.740131\pi$
$37$	− 328.000i	− 1.45737i	0.684846	−	0.728687i	$-0.259869\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−288.000	−1.09703	−0.548513	−	0.836142i	$-0.684806\pi$
$41$	−288.000	−1.09703	−0.548513	+	0.836142i	$0.684806\pi$
$42$	0	0
$43$	120.000i	0.425577i	0.977098	+	0.212789i	$0.0682546\pi$
$43$	120.000i	0.425577i	−0.977098	+	0.212789i	$0.931745\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 16.0000i	− 0.0496562i	−0.999692	−	0.0248281i	$-0.992096\pi$
$47$	− 16.0000i	− 0.0496562i	0.999692	−	0.0248281i	$-0.00790384\pi$
$48$	0	0
$49$	19.0000	0.0553936
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 126.000i	− 0.326555i	−0.986580	−	0.163278i	$-0.947793\pi$
$53$	− 126.000i	− 0.326555i	0.986580	−	0.163278i	$-0.0522066\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−642.000	−1.41663	−0.708316	−	0.705896i	$-0.750545\pi$
$59$	−642.000	−1.41663	−0.708316	+	0.705896i	$0.750545\pi$
$60$	0	0
$61$	602.000	1.26358	0.631789	−	0.775141i	$-0.282321\pi$
$61$	602.000	1.26358	0.631789	+	0.775141i	$0.282321\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 436.000i	− 0.795013i	−0.917599	−	0.397507i	$-0.869876\pi$
$67$	− 436.000i	− 0.795013i	0.917599	−	0.397507i	$-0.130124\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	652.000	1.08983	0.544917	−	0.838490i	$-0.316561\pi$
$71$	652.000	1.08983	0.544917	+	0.838490i	$0.316561\pi$
$72$	0	0
$73$	1062.00i	1.70271i	0.524591	+	0.851354i	$0.324218\pi$
$73$	1062.00i	1.70271i	−0.524591	+	0.851354i	$0.675782\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	612.000i	0.905765i
$78$	0	0
$79$	−388.000	−0.552575	−0.276287	−	0.961075i	$-0.589104\pi$
$79$	−388.000	−0.552575	−0.276287	+	0.961075i	$0.589104\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 444.000i	− 0.587173i	−0.955933	−	0.293586i	$-0.905151\pi$
$83$	− 444.000i	− 0.587173i	0.955933	−	0.293586i	$-0.0948488\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	820.000	0.976627	0.488314	−	0.872668i	$-0.337612\pi$
$89$	820.000	0.976627	0.488314	+	0.872668i	$0.337612\pi$
$90$	0	0
$91$	−216.000	−0.248824
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	766.000i	0.801809i	0.916120	+	0.400905i	$0.131304\pi$
$97$	766.000i	0.801809i	−0.916120	+	0.400905i	$0.868696\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−798.000	−0.786178	−0.393089	−	0.919500i	$-0.628594\pi$
$101$	−798.000	−0.786178	−0.393089	+	0.919500i	$0.628594\pi$
$102$	0	0
$103$	− 402.000i	− 0.384565i	−0.981340	−	0.192283i	$-0.938411\pi$
$103$	− 402.000i	− 0.384565i	0.981340	−	0.192283i	$-0.0615891\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 1444.00i	− 1.30464i	−0.757943	−	0.652321i	$-0.773795\pi$
$107$	− 1444.00i	− 1.30464i	0.757943	−	0.652321i	$-0.226205\pi$
$108$	0	0
$109$	198.000	0.173990	0.0869952	−	0.996209i	$-0.472274\pi$
$109$	198.000	0.173990	0.0869952	+	0.996209i	$0.472274\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 2010.00i	− 1.67332i	−0.547724	−	0.836659i	$-0.684506\pi$
$113$	− 2010.00i	− 1.67332i	0.547724	−	0.836659i	$-0.315494\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1836.00	−1.41433
$120$	0	0
$121$	−175.000	−0.131480
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	866.000i	0.605079i	0.953137	+	0.302540i	$0.0978345\pi$
$127$	866.000i	0.605079i	−0.953137	+	0.302540i	$0.902166\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−2098.00	−1.39926	−0.699630	−	0.714505i	$-0.746652\pi$
$131$	−2098.00	−1.39926	−0.699630	+	0.714505i	$0.746652\pi$
$132$	0	0
$133$	− 2952.00i	− 1.92459i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 886.000i	− 0.552526i	−0.961082	−	0.276263i	$-0.910904\pi$
$137$	− 886.000i	− 0.552526i	0.961082	−	0.276263i	$-0.0890961\pi$
$138$	0	0
$139$	500.000	0.305104	0.152552	−	0.988295i	$-0.451251\pi$
$139$	500.000	0.305104	0.152552	+	0.988295i	$0.451251\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	408.000i	0.238592i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2302.00	−1.26569	−0.632843	−	0.774280i	$-0.718112\pi$
$149$	−2302.00	−1.26569	−0.632843	+	0.774280i	$0.718112\pi$
$150$	0	0
$151$	−2384.00	−1.28482	−0.642408	−	0.766363i	$-0.722064\pi$
$151$	−2384.00	−1.28482	−0.642408	+	0.766363i	$0.722064\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 1452.00i	− 0.738103i	−0.929409	−	0.369052i	$-0.879683\pi$
$157$	− 1452.00i	− 0.738103i	0.929409	−	0.369052i	$-0.120317\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−864.000	−0.422936
$162$	0	0
$163$	604.000i	0.290239i	0.989414	+	0.145119i	$0.0463566\pi$
$163$	604.000i	0.290239i	−0.989414	+	0.145119i	$0.953643\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	664.000i	0.307676i	0.988096	+	0.153838i	$0.0491634\pi$
$167$	664.000i	0.307676i	−0.988096	+	0.153838i	$0.950837\pi$
$168$	0	0
$169$	2053.00	0.934456
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 4118.00i	− 1.80974i	−0.425684	−	0.904872i	$-0.639966\pi$
$173$	− 4118.00i	− 1.80974i	0.425684	−	0.904872i	$-0.360034\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−1746.00	−0.729062	−0.364531	−	0.931191i	$-0.618771\pi$
$179$	−1746.00	−0.729062	−0.364531	+	0.931191i	$0.618771\pi$
$180$	0	0
$181$	−1270.00	−0.521538	−0.260769	−	0.965401i	$-0.583976\pi$
$181$	−1270.00	−0.521538	−0.260769	+	0.965401i	$0.583976\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	3468.00i	1.35618i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2676.00	1.01376	0.506881	−	0.862016i	$-0.330798\pi$
$191$	2676.00	1.01376	0.506881	+	0.862016i	$0.330798\pi$
$192$	0	0
$193$	− 3146.00i	− 1.17334i	−0.809827	−	0.586668i	$-0.800439\pi$
$193$	− 3146.00i	− 1.17334i	0.809827	−	0.586668i	$-0.199561\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3674.00i	1.32874i	0.747404	+	0.664370i	$0.231300\pi$
$197$	3674.00i	1.32874i	−0.747404	+	0.664370i	$0.768700\pi$
$198$	0	0
$199$	1392.00	0.495861	0.247930	−	0.968778i	$-0.420250\pi$
$199$	1392.00	0.495861	0.247930	+	0.968778i	$0.420250\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 2628.00i	− 0.908618i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−5576.00	−1.84545
$210$	0	0
$211$	−540.000	−0.176185	−0.0880927	−	0.996112i	$-0.528077\pi$
$211$	−540.000	−0.176185	−0.0880927	+	0.996112i	$0.528077\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	1800.00i	0.563097i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−1224.00	−0.372557
$222$	0	0
$223$	4166.00i	1.25101i	0.780219	+	0.625507i	$0.215108\pi$
$223$	4166.00i	1.25101i	−0.780219	+	0.625507i	$0.784892\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	5024.00i	1.46896i	0.678629	+	0.734481i	$0.262575\pi$
$227$	5024.00i	1.46896i	−0.678629	+	0.734481i	$0.737425\pi$
$228$	0	0
$229$	−4454.00	−1.28528	−0.642639	−	0.766169i	$-0.722160\pi$
$229$	−4454.00	−1.28528	−0.642639	+	0.766169i	$0.722160\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 1526.00i	− 0.429063i	−0.976717	−	0.214531i	$-0.931178\pi$
$233$	− 1526.00i	− 0.429063i	0.976717	−	0.214531i	$-0.0688224\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−6828.00	−1.84798	−0.923989	−	0.382420i	$-0.875091\pi$
$239$	−6828.00	−1.84798	−0.923989	+	0.382420i	$0.875091\pi$
$240$	0	0
$241$	5782.00	1.54544	0.772721	−	0.634746i	$-0.218895\pi$
$241$	5782.00	1.54544	0.772721	+	0.634746i	$0.218895\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 1968.00i	− 0.506967i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−6394.00	−1.60791	−0.803956	−	0.594689i	$-0.797275\pi$
$251$	−6394.00	−1.60791	−0.803956	+	0.594689i	$0.797275\pi$
$252$	0	0
$253$	1632.00i	0.405545i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 1862.00i	− 0.451939i	−0.974134	−	0.225970i	$-0.927445\pi$
$257$	− 1862.00i	− 0.451939i	0.974134	−	0.225970i	$-0.0725550\pi$
$258$	0	0
$259$	5904.00	1.41644
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 6504.00i	− 1.52492i	−0.647036	−	0.762460i	$-0.723992\pi$
$263$	− 6504.00i	− 1.52492i	0.647036	−	0.762460i	$-0.276008\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−8298.00	−1.88081	−0.940405	−	0.340056i	$-0.889554\pi$
$269$	−8298.00	−1.88081	−0.940405	+	0.340056i	$0.889554\pi$
$270$	0	0
$271$	1848.00	0.414236	0.207118	−	0.978316i	$-0.433592\pi$
$271$	1848.00	0.414236	0.207118	+	0.978316i	$0.433592\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 2824.00i	− 0.612555i	−0.951942	−	0.306277i	$-0.900916\pi$
$277$	− 2824.00i	− 0.612555i	0.951942	−	0.306277i	$-0.0990835\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−1940.00	−0.411853	−0.205927	−	0.978567i	$-0.566021\pi$
$281$	−1940.00	−0.411853	−0.205927	+	0.978567i	$0.566021\pi$
$282$	0	0
$283$	6548.00i	1.37540i	0.725995	+	0.687700i	$0.241380\pi$
$283$	6548.00i	1.37540i	−0.725995	+	0.687700i	$0.758620\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 5184.00i	− 1.06621i
$288$	0	0
$289$	−5491.00	−1.11765
$290$	0	0
$291$	0	0
$292$	0	0
$293$	6566.00i	1.30918i	0.755984	+	0.654590i	$0.227159\pi$
$293$	6566.00i	1.30918i	−0.755984	+	0.654590i	$0.772841\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−576.000	−0.111408
$300$	0	0
$301$	−2160.00	−0.413622
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 8432.00i	− 1.56756i	−0.621041	−	0.783778i	$-0.713290\pi$
$307$	− 8432.00i	− 1.56756i	0.621041	−	0.783778i	$-0.286710\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−4916.00	−0.896337	−0.448168	−	0.893949i	$-0.647924\pi$
$311$	−4916.00	−0.896337	−0.448168	+	0.893949i	$0.647924\pi$
$312$	0	0
$313$	− 10106.0i	− 1.82500i	−0.409077	−	0.912500i	$-0.634149\pi$
$313$	− 10106.0i	− 1.82500i	0.409077	−	0.912500i	$-0.365851\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 3382.00i	− 0.599218i	−0.954062	−	0.299609i	$-0.903144\pi$
$317$	− 3382.00i	− 0.599218i	0.954062	−	0.299609i	$-0.0968562\pi$
$318$	0	0
$319$	−4964.00	−0.871256
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 16728.0i	− 2.88164i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	288.000	0.0482613
$330$	0	0
$331$	−6460.00	−1.07273	−0.536365	−	0.843986i	$-0.680203\pi$
$331$	−6460.00	−1.07273	−0.536365	+	0.843986i	$0.680203\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 5294.00i	− 0.855735i	−0.903842	−	0.427867i	$-0.859265\pi$
$337$	− 5294.00i	− 0.855735i	0.903842	−	0.427867i	$-0.140735\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	3400.00	0.539942
$342$	0	0
$343$	6516.00i	1.02575i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	12096.0i	1.87132i	0.352906	+	0.935659i	$0.385194\pi$
$347$	12096.0i	1.87132i	−0.352906	+	0.935659i	$0.614806\pi$
$348$	0	0
$349$	862.000	0.132211	0.0661057	−	0.997813i	$-0.478943\pi$
$349$	862.000	0.132211	0.0661057	+	0.997813i	$0.478943\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 6878.00i	− 1.03705i	−0.855062	−	0.518525i	$-0.826481\pi$
$353$	− 6878.00i	− 1.03705i	0.855062	−	0.518525i	$-0.173519\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−6216.00	−0.913838	−0.456919	−	0.889508i	$-0.651047\pi$
$359$	−6216.00	−0.913838	−0.456919	+	0.889508i	$0.651047\pi$
$360$	0	0
$361$	20037.0	2.92127
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	13274.0i	1.88800i	0.329941	+	0.944002i	$0.392971\pi$
$367$	13274.0i	1.88800i	−0.329941	+	0.944002i	$0.607029\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	2268.00	0.317382
$372$	0	0
$373$	1300.00i	0.180460i	0.995921	+	0.0902298i	$0.0287602\pi$
$373$	1300.00i	0.180460i	−0.995921	+	0.0902298i	$0.971240\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 1752.00i	− 0.239344i
$378$	0	0
$379$	−13324.0	−1.80583	−0.902913	−	0.429824i	$-0.858576\pi$
$379$	−13324.0	−1.80583	−0.902913	+	0.429824i	$0.858576\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	6192.00i	0.826100i	0.910708	+	0.413050i	$0.135537\pi$
$383$	6192.00i	0.826100i	−0.910708	+	0.413050i	$0.864463\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−2022.00	−0.263546	−0.131773	−	0.991280i	$-0.542067\pi$
$389$	−2022.00	−0.263546	−0.131773	+	0.991280i	$0.542067\pi$
$390$	0	0
$391$	−4896.00	−0.633252
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 7856.00i	− 0.993152i	−0.867993	−	0.496576i	$-0.834590\pi$
$397$	− 7856.00i	− 0.993152i	0.867993	−	0.496576i	$-0.165410\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	1148.00	0.142964	0.0714818	−	0.997442i	$-0.477227\pi$
$401$	1148.00	0.142964	0.0714818	+	0.997442i	$0.477227\pi$
$402$	0	0
$403$	1200.00i	0.148328i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 11152.0i	− 1.35819i
$408$	0	0
$409$	6310.00	0.762859	0.381430	−	0.924398i	$-0.375432\pi$
$409$	6310.00	0.762859	0.381430	+	0.924398i	$0.375432\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 11556.0i	− 1.37684i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	13362.0	1.55794	0.778969	−	0.627062i	$-0.215743\pi$
$419$	13362.0	1.55794	0.778969	+	0.627062i	$0.215743\pi$
$420$	0	0
$421$	−5146.00	−0.595726	−0.297863	−	0.954609i	$-0.596274\pi$
$421$	−5146.00	−0.595726	−0.297863	+	0.954609i	$0.596274\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	10836.0i	1.22808i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−6368.00	−0.711684	−0.355842	−	0.934546i	$-0.615806\pi$
$431$	−6368.00	−0.711684	−0.355842	+	0.934546i	$0.615806\pi$
$432$	0	0
$433$	− 6138.00i	− 0.681232i	−0.940202	−	0.340616i	$-0.889364\pi$
$433$	− 6138.00i	− 0.681232i	0.940202	−	0.340616i	$-0.110636\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 7872.00i	− 0.861714i
$438$	0	0
$439$	4424.00	0.480970	0.240485	−	0.970653i	$-0.422693\pi$
$439$	4424.00	0.480970	0.240485	+	0.970653i	$0.422693\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 488.000i	− 0.0523377i	−0.999658	−	0.0261688i	$-0.991669\pi$
$443$	− 488.000i	− 0.0523377i	0.999658	−	0.0261688i	$-0.00833075\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−16884.0	−1.77462	−0.887311	−	0.461172i	$-0.847429\pi$
$449$	−16884.0	−1.77462	−0.887311	+	0.461172i	$0.847429\pi$
$450$	0	0
$451$	−9792.00	−1.02237
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	5398.00i	0.552533i	0.961081	+	0.276267i	$0.0890973\pi$
$457$	5398.00i	0.552533i	−0.961081	+	0.276267i	$0.910903\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−6122.00	−0.618503	−0.309252	−	0.950980i	$-0.600079\pi$
$461$	−6122.00	−0.618503	−0.309252	+	0.950980i	$0.600079\pi$
$462$	0	0
$463$	8162.00i	0.819266i	0.912250	+	0.409633i	$0.134343\pi$
$463$	8162.00i	0.819266i	−0.912250	+	0.409633i	$0.865657\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	2660.00i	0.263576i	0.991278	+	0.131788i	$0.0420719\pi$
$467$	2660.00i	0.263576i	−0.991278	+	0.131788i	$0.957928\pi$
$468$	0	0
$469$	7848.00	0.772680
$470$	0	0
$471$	0	0
$472$	0	0
$473$	4080.00i	0.396614i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−9788.00	−0.933664	−0.466832	−	0.884346i	$-0.654605\pi$
$479$	−9788.00	−0.933664	−0.466832	+	0.884346i	$0.654605\pi$
$480$	0	0
$481$	3936.00	0.373111
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 4714.00i	− 0.438628i	−0.975654	−	0.219314i	$-0.929618\pi$
$487$	− 4714.00i	− 0.438628i	0.975654	−	0.219314i	$-0.0703819\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−6690.00	−0.614899	−0.307450	−	0.951564i	$-0.599476\pi$
$491$	−6690.00	−0.614899	−0.307450	+	0.951564i	$0.599476\pi$
$492$	0	0
$493$	− 14892.0i	− 1.36045i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	11736.0i	1.05922i
$498$	0	0
$499$	20636.0	1.85129	0.925646	−	0.378392i	$-0.123523\pi$
$499$	20636.0	1.85129	0.925646	+	0.378392i	$0.123523\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	15952.0i	1.41404i	0.707191	+	0.707022i	$0.249962\pi$
$503$	15952.0i	1.41404i	−0.707191	+	0.707022i	$0.750038\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	2230.00	0.194191	0.0970953	−	0.995275i	$-0.469045\pi$
$509$	2230.00	0.194191	0.0970953	+	0.995275i	$0.469045\pi$
$510$	0	0
$511$	−19116.0	−1.65488
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 544.000i	− 0.0462768i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	1260.00	0.105953	0.0529766	−	0.998596i	$-0.483129\pi$
$521$	1260.00	0.105953	0.0529766	+	0.998596i	$0.483129\pi$
$522$	0	0
$523$	2900.00i	0.242463i	0.992624	+	0.121231i	$0.0386843\pi$
$523$	2900.00i	0.242463i	−0.992624	+	0.121231i	$0.961316\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	10200.0i	0.843110i
$528$	0	0
$529$	9863.00	0.810635
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 3456.00i	− 0.280855i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	646.000	0.0516237
$540$	0	0
$541$	19554.0	1.55396	0.776980	−	0.629526i	$-0.216751\pi$
$541$	19554.0	1.55396	0.776980	+	0.629526i	$0.216751\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 2664.00i	− 0.208235i	−0.994565	−	0.104117i	$-0.966798\pi$
$547$	− 2664.00i	− 0.208235i	0.994565	−	0.104117i	$-0.0332018\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	23944.0	1.85127
$552$	0	0
$553$	− 6984.00i	− 0.537052i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	11358.0i	0.864011i	0.901871	+	0.432005i	$0.142194\pi$
$557$	11358.0i	0.864011i	−0.901871	+	0.432005i	$0.857806\pi$
$558$	0	0
$559$	−1440.00	−0.108954
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 2440.00i	− 0.182653i	−0.995821	−	0.0913266i	$-0.970889\pi$
$563$	− 2440.00i	− 0.182653i	0.995821	−	0.0913266i	$-0.0291107\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	24156.0	1.77974	0.889870	−	0.456214i	$-0.150795\pi$
$569$	24156.0	1.77974	0.889870	+	0.456214i	$0.150795\pi$
$570$	0	0
$571$	−2220.00	−0.162704	−0.0813521	−	0.996685i	$-0.525924\pi$
$571$	−2220.00	−0.162704	−0.0813521	+	0.996685i	$0.525924\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	5782.00i	0.417171i	0.978004	+	0.208586i	$0.0668860\pi$
$577$	5782.00i	0.417171i	−0.978004	+	0.208586i	$0.933114\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	7992.00	0.570678
$582$	0	0
$583$	− 4284.00i	− 0.304331i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 1684.00i	− 0.118409i	−0.998246	−	0.0592045i	$-0.981144\pi$
$587$	− 1684.00i	− 0.118409i	0.998246	−	0.0592045i	$-0.0188564\pi$
$588$	0	0
$589$	−16400.0	−1.14728
$590$	0	0
$591$	0	0
$592$	0	0
$593$	15246.0i	1.05578i	0.849312	+	0.527891i	$0.177017\pi$
$593$	15246.0i	1.05578i	−0.849312	+	0.527891i	$0.822983\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−9016.00	−0.614998	−0.307499	−	0.951548i	$-0.599492\pi$
$599$	−9016.00	−0.614998	−0.307499	+	0.951548i	$0.599492\pi$
$600$	0	0
$601$	−18682.0	−1.26798	−0.633989	−	0.773342i	$-0.718584\pi$
$601$	−18682.0	−1.26798	−0.633989	+	0.773342i	$0.718584\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	22022.0i	1.47256i	0.676676	+	0.736281i	$0.263420\pi$
$607$	22022.0i	1.47256i	−0.676676	+	0.736281i	$0.736580\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	192.000	0.0127127
$612$	0	0
$613$	− 22808.0i	− 1.50278i	−0.659856	−	0.751392i	$-0.729383\pi$
$613$	− 22808.0i	− 1.50278i	0.659856	−	0.751392i	$-0.270617\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	9422.00i	0.614774i	0.951585	+	0.307387i	$0.0994546\pi$
$617$	9422.00i	0.614774i	−0.951585	+	0.307387i	$0.900545\pi$
$618$	0	0
$619$	−4172.00	−0.270900	−0.135450	−	0.990784i	$-0.543248\pi$
$619$	−4172.00	−0.270900	−0.135450	+	0.990784i	$0.543248\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	14760.0i	0.949192i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	33456.0	2.12079
$630$	0	0
$631$	−11572.0	−0.730070	−0.365035	−	0.930994i	$-0.618943\pi$
$631$	−11572.0	−0.730070	−0.365035	+	0.930994i	$0.618943\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	228.000i	0.0141816i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−936.000	−0.0576752	−0.0288376	−	0.999584i	$-0.509181\pi$
$641$	−936.000	−0.0576752	−0.0288376	+	0.999584i	$0.509181\pi$
$642$	0	0
$643$	15892.0i	0.974680i	0.873212	+	0.487340i	$0.162033\pi$
$643$	15892.0i	0.974680i	−0.873212	+	0.487340i	$0.837967\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 25056.0i	− 1.52249i	−0.648463	−	0.761247i	$-0.724588\pi$
$647$	− 25056.0i	− 1.52249i	0.648463	−	0.761247i	$-0.275412\pi$
$648$	0	0
$649$	−21828.0	−1.32022
$650$	0	0
$651$	0	0
$652$	0	0
$653$	4054.00i	0.242948i	0.992595	+	0.121474i	$0.0387622\pi$
$653$	4054.00i	0.242948i	−0.992595	+	0.121474i	$0.961238\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−6758.00	−0.399475	−0.199738	−	0.979849i	$-0.564009\pi$
$659$	−6758.00	−0.399475	−0.199738	+	0.979849i	$0.564009\pi$
$660$	0	0
$661$	25098.0	1.47685	0.738426	−	0.674335i	$-0.235569\pi$
$661$	25098.0	1.47685	0.738426	+	0.674335i	$0.235569\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 7008.00i	− 0.406823i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	20468.0	1.17758
$672$	0	0
$673$	2830.00i	0.162093i	0.996710	+	0.0810464i	$0.0258262\pi$
$673$	2830.00i	0.162093i	−0.996710	+	0.0810464i	$0.974174\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 10654.0i	− 0.604825i	−0.953177	−	0.302412i	$-0.902208\pi$
$677$	− 10654.0i	− 0.604825i	0.953177	−	0.302412i	$-0.0977920\pi$
$678$	0	0
$679$	−13788.0	−0.779286
$680$	0	0
$681$	0	0
$682$	0	0
$683$	17156.0i	0.961136i	0.876957	+	0.480568i	$0.159570\pi$
$683$	17156.0i	0.961136i	−0.876957	+	0.480568i	$0.840430\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1512.00	0.0836032
$690$	0	0
$691$	−812.000	−0.0447032	−0.0223516	−	0.999750i	$-0.507115\pi$
$691$	−812.000	−0.0447032	−0.0223516	+	0.999750i	$0.507115\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 29376.0i	− 1.59641i
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−30270.0	−1.63093	−0.815465	−	0.578806i	$-0.803519\pi$
$701$	−30270.0	−1.63093	−0.815465	+	0.578806i	$0.803519\pi$
$702$	0	0
$703$	53792.0i	2.88592i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 14364.0i	− 0.764093i
$708$	0	0
$709$	394.000	0.0208702	0.0104351	−	0.999946i	$-0.496678\pi$
$709$	394.000	0.0208702	0.0104351	+	0.999946i	$0.496678\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	4800.00i	0.252120i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	37224.0	1.93077	0.965383	−	0.260836i	$-0.0839982\pi$
$719$	37224.0	1.93077	0.965383	+	0.260836i	$0.0839982\pi$
$720$	0	0
$721$	7236.00	0.373762
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	12614.0i	0.643504i	0.946824	+	0.321752i	$0.104272\pi$
$727$	12614.0i	0.643504i	−0.946824	+	0.321752i	$0.895728\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−12240.0	−0.619306
$732$	0	0
$733$	− 25664.0i	− 1.29321i	−0.762826	−	0.646604i	$-0.776189\pi$
$733$	− 25664.0i	− 1.29321i	0.762826	−	0.646604i	$-0.223811\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 14824.0i	− 0.740908i
$738$	0	0
$739$	18772.0	0.934424	0.467212	−	0.884145i	$-0.345259\pi$
$739$	18772.0	0.934424	0.467212	+	0.884145i	$0.345259\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 19376.0i	− 0.956711i	−0.878166	−	0.478356i	$-0.841233\pi$
$743$	− 19376.0i	− 0.956711i	0.878166	−	0.478356i	$-0.158767\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	25992.0	1.26799
$750$	0	0
$751$	30092.0	1.46215	0.731074	−	0.682299i	$-0.239020\pi$
$751$	30092.0	1.46215	0.731074	+	0.682299i	$0.239020\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 18136.0i	− 0.870758i	−0.900247	−	0.435379i	$-0.856614\pi$
$757$	− 18136.0i	− 0.870758i	0.900247	−	0.435379i	$-0.143386\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	10948.0	0.521504	0.260752	−	0.965406i	$-0.416029\pi$
$761$	10948.0	0.521504	0.260752	+	0.965406i	$0.416029\pi$
$762$	0	0
$763$	3564.00i	0.169103i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 7704.00i	− 0.362680i
$768$	0	0
$769$	−1422.00	−0.0666822	−0.0333411	−	0.999444i	$-0.510615\pi$
$769$	−1422.00	−0.0666822	−0.0333411	+	0.999444i	$0.510615\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	26142.0i	1.21638i	0.793791	+	0.608190i	$0.208104\pi$
$773$	26142.0i	1.21638i	−0.793791	+	0.608190i	$0.791896\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	47232.0	2.17235
$780$	0	0
$781$	22168.0	1.01566
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 23404.0i	− 1.06005i	−0.847981	−	0.530027i	$-0.822182\pi$
$787$	− 23404.0i	− 1.06005i	0.847981	−	0.530027i	$-0.177818\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	36180.0	1.62631
$792$	0	0
$793$	7224.00i	0.323495i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 1418.00i	− 0.0630215i	−0.999503	−	0.0315108i	$-0.989968\pi$
$797$	− 1418.00i	− 0.0630215i	0.999503	−	0.0315108i	$-0.0100318\pi$
$798$	0	0
$799$	1632.00	0.0722603
$800$	0	0
$801$	0	0
$802$	0	0
$803$	36108.0i	1.58683i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	17304.0	0.752010	0.376005	−	0.926618i	$-0.377298\pi$
$809$	17304.0	0.752010	0.376005	+	0.926618i	$0.377298\pi$
$810$	0	0
$811$	−28012.0	−1.21287	−0.606433	−	0.795135i	$-0.707400\pi$
$811$	−28012.0	−1.21287	−0.606433	+	0.795135i	$0.707400\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 19680.0i	− 0.842737i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	32266.0	1.37161	0.685805	−	0.727786i	$-0.259450\pi$
$821$	32266.0	1.37161	0.685805	+	0.727786i	$0.259450\pi$
$822$	0	0
$823$	4962.00i	0.210163i	0.994464	+	0.105082i	$0.0335104\pi$
$823$	4962.00i	0.210163i	−0.994464	+	0.105082i	$0.966490\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 5064.00i	− 0.212929i	−0.994316	−	0.106465i	$-0.966047\pi$
$827$	− 5064.00i	− 0.212929i	0.994316	−	0.106465i	$-0.0339531\pi$
$828$	0	0
$829$	8174.00	0.342454	0.171227	−	0.985232i	$-0.445227\pi$
$829$	8174.00	0.342454	0.171227	+	0.985232i	$0.445227\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1938.00i	0.0806095i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	28240.0	1.16204	0.581021	−	0.813889i	$-0.302653\pi$
$839$	28240.0	1.16204	0.581021	+	0.813889i	$0.302653\pi$
$840$	0	0
$841$	−3073.00	−0.125999
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 3150.00i	− 0.127787i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	15744.0	0.634192
$852$	0	0
$853$	10472.0i	0.420345i	0.977664	+	0.210173i	$0.0674026\pi$
$853$	10472.0i	0.420345i	−0.977664	+	0.210173i	$0.932597\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	32102.0i	1.27956i	0.768558	+	0.639780i	$0.220975\pi$
$857$	32102.0i	1.27956i	−0.768558	+	0.639780i	$0.779025\pi$
$858$	0	0
$859$	11060.0	0.439304	0.219652	−	0.975578i	$-0.429508\pi$
$859$	11060.0	0.439304	0.219652	+	0.975578i	$0.429508\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	36088.0i	1.42346i	0.702451	+	0.711732i	$0.252089\pi$
$863$	36088.0i	1.42346i	−0.702451	+	0.711732i	$0.747911\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−13192.0	−0.514969
$870$	0	0
$871$	5232.00	0.203536
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	34508.0i	1.32868i	0.747431	+	0.664340i	$0.231287\pi$
$877$	34508.0i	1.32868i	−0.747431	+	0.664340i	$0.768713\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−6596.00	−0.252242	−0.126121	−	0.992015i	$-0.540253\pi$
$881$	−6596.00	−0.252242	−0.126121	+	0.992015i	$0.540253\pi$
$882$	0	0
$883$	17620.0i	0.671529i	0.941946	+	0.335765i	$0.108995\pi$
$883$	17620.0i	0.671529i	−0.941946	+	0.335765i	$0.891005\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	50784.0i	1.92239i	0.275870	+	0.961195i	$0.411034\pi$
$887$	50784.0i	1.92239i	−0.275870	+	0.961195i	$0.588966\pi$
$888$	0	0
$889$	−15588.0	−0.588082
$890$	0	0
$891$	0	0
$892$	0	0
$893$	2624.00i	0.0983301i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−14600.0	−0.541643
$900$	0	0
$901$	12852.0	0.475208
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 16072.0i	− 0.588381i	−0.955747	−	0.294191i	$-0.904950\pi$
$907$	− 16072.0i	− 0.588381i	0.955747	−	0.294191i	$-0.0950501\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−41760.0	−1.51874	−0.759369	−	0.650660i	$-0.774492\pi$
$911$	−41760.0	−1.51874	−0.759369	+	0.650660i	$0.774492\pi$
$912$	0	0
$913$	− 15096.0i	− 0.547212i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 37764.0i	− 1.35995i
$918$	0	0
$919$	−34100.0	−1.22400	−0.612000	−	0.790858i	$-0.709635\pi$
$919$	−34100.0	−1.22400	−0.612000	+	0.790858i	$0.709635\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	7824.00i	0.279014i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	22812.0	0.805638	0.402819	−	0.915280i	$-0.368030\pi$
$929$	22812.0	0.805638	0.402819	+	0.915280i	$0.368030\pi$
$930$	0	0
$931$	−3116.00	−0.109691
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	38982.0i	1.35911i	0.733624	+	0.679555i	$0.237827\pi$
$937$	38982.0i	1.35911i	−0.733624	+	0.679555i	$0.762173\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	52766.0	1.82797	0.913986	−	0.405745i	$-0.132988\pi$
$941$	52766.0	1.82797	0.913986	+	0.405745i	$0.132988\pi$
$942$	0	0
$943$	− 13824.0i	− 0.477382i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 13608.0i	− 0.466949i	−0.972363	−	0.233474i	$-0.924990\pi$
$947$	− 13608.0i	− 0.466949i	0.972363	−	0.233474i	$-0.0750095\pi$
$948$	0	0
$949$	−12744.0	−0.435920
$950$	0	0
$951$	0	0
$952$	0	0
$953$	6446.00i	0.219104i	0.993981	+	0.109552i	$0.0349417\pi$
$953$	6446.00i	0.219104i	−0.993981	+	0.109552i	$0.965058\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	15948.0	0.537005
$960$	0	0
$961$	−19791.0	−0.664328
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	42642.0i	1.41807i	0.705173	+	0.709035i	$0.250869\pi$
$967$	42642.0i	1.41807i	−0.705173	+	0.709035i	$0.749131\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−19938.0	−0.658950	−0.329475	−	0.944164i	$-0.606872\pi$
$971$	−19938.0	−0.658950	−0.329475	+	0.944164i	$0.606872\pi$
$972$	0	0
$973$	9000.00i	0.296533i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 49754.0i	− 1.62924i	−0.579992	−	0.814622i	$-0.696944\pi$
$977$	− 49754.0i	− 1.62924i	0.579992	−	0.814622i	$-0.303056\pi$
$978$	0	0
$979$	27880.0	0.910162
$980$	0	0
$981$	0	0
$982$	0	0
$983$	7936.00i	0.257497i	0.991677	+	0.128748i	$0.0410959\pi$
$983$	7936.00i	0.257497i	−0.991677	+	0.128748i	$0.958904\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−5760.00	−0.185194
$990$	0	0
$991$	−33248.0	−1.06575	−0.532875	−	0.846194i	$-0.678888\pi$
$991$	−33248.0	−1.06575	−0.532875	+	0.846194i	$0.678888\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	20196.0i	0.641538i	0.947157	+	0.320769i	$0.103941\pi$
$997$	20196.0i	0.641538i	−0.947157	+	0.320769i	$0.896059\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1800.4.f.s.649.2		2
3.2	odd	2		1800.4.f.e.649.2		2
5.2	odd	4		360.4.a.j.1.1	yes	1
5.3	odd	4		1800.4.a.be.1.1		1
5.4	even	2	inner	1800.4.f.s.649.1		2
15.2	even	4		360.4.a.a.1.1	✓	1
15.8	even	4		1800.4.a.bc.1.1		1
15.14	odd	2		1800.4.f.e.649.1		2
20.7	even	4		720.4.a.z.1.1		1
60.47	odd	4		720.4.a.m.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.4.a.a.1.1	✓	1	15.2	even	4
360.4.a.j.1.1	yes	1	5.2	odd	4
720.4.a.m.1.1		1	60.47	odd	4
720.4.a.z.1.1		1	20.7	even	4
1800.4.a.bc.1.1		1	15.8	even	4
1800.4.a.be.1.1		1	5.3	odd	4
1800.4.f.e.649.1		2	15.14	odd	2
1800.4.f.e.649.2		2	3.2	odd	2
1800.4.f.s.649.1		2	5.4	even	2	inner
1800.4.f.s.649.2		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1800.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+18.0000i q^{7} +O(q^{10})\)	\(q+18.0000i q^{7} +34.0000 q^{11} +12.0000i q^{13} +102.000i q^{17} -164.000 q^{19} +48.0000i q^{23} -146.000 q^{29} +100.000 q^{31} -328.000i q^{37} -288.000 q^{41} +120.000i q^{43} -16.0000i q^{47} +19.0000 q^{49} -126.000i q^{53} -642.000 q^{59} +602.000 q^{61} -436.000i q^{67} +652.000 q^{71} +1062.00i q^{73} +612.000i q^{77} -388.000 q^{79} -444.000i q^{83} +820.000 q^{89} -216.000 q^{91} +766.000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q+O(q^{10}) \) 2 * q	\( 2 q + 68 q^{11} - 328 q^{19} - 292 q^{29} + 200 q^{31} - 576 q^{41} + 38 q^{49} - 1284 q^{59} + 1204 q^{61} + 1304 q^{71} - 776 q^{79} + 1640 q^{89} - 432 q^{91}+O(q^{100}) \) 2 * q + 68 * q^11 - 328 * q^19 - 292 * q^29 + 200 * q^31 - 576 * q^41 + 38 * q^49 - 1284 * q^59 + 1204 * q^61 + 1304 * q^71 - 776 * q^79 + 1640 * q^89 - 432 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more