LMFDB - Embedded newform 1800.4.f.d.649.1

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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-16.0000i q^{7} +O(q^{10})$	$q-16.0000i q^{7} -36.0000 q^{11} -42.0000i q^{13} -110.000i q^{17} +116.000 q^{19} -16.0000i q^{23} +198.000 q^{29} +240.000 q^{31} +258.000i q^{37} -442.000 q^{41} -292.000i q^{43} +392.000i q^{47} +87.0000 q^{49} -142.000i q^{53} -348.000 q^{59} -570.000 q^{61} -692.000i q^{67} -168.000 q^{71} -134.000i q^{73} +576.000i q^{77} -784.000 q^{79} -564.000i q^{83} +1034.00 q^{89} -672.000 q^{91} +382.000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q+O(q^{10}) $ 2 * q	$ 2 q - 72 q^{11} + 232 q^{19} + 396 q^{29} + 480 q^{31} - 884 q^{41} + 174 q^{49} - 696 q^{59} - 1140 q^{61} - 336 q^{71} - 1568 q^{79} + 2068 q^{89} - 1344 q^{91}+O(q^{100}) $ 2 * q - 72 * q^11 + 232 * q^19 + 396 * q^29 + 480 * q^31 - 884 * q^41 + 174 * q^49 - 696 * q^59 - 1140 * q^61 - 336 * q^71 - 1568 * q^79 + 2068 * q^89 - 1344 * 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
$7$	− 16.0000i	− 0.863919i	−0.901893	−	0.431959i	$-0.857822\pi$
$7$	− 16.0000i	− 0.863919i	0.901893	−	0.431959i	$-0.142178\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−36.0000	−0.986764	−0.493382	−	0.869813i	$-0.664240\pi$
$11$	−36.0000	−0.986764	−0.493382	+	0.869813i	$0.664240\pi$
$12$	0	0
$13$	− 42.0000i	− 0.896054i	−0.894020	−	0.448027i	$-0.852127\pi$
$13$	− 42.0000i	− 0.896054i	0.894020	−	0.448027i	$-0.147873\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 110.000i	− 1.56935i	−0.619909	−	0.784674i	$-0.712830\pi$
$17$	− 110.000i	− 1.56935i	0.619909	−	0.784674i	$-0.287170\pi$
$18$	0	0
$19$	116.000	1.40064	0.700322	−	0.713827i	$-0.253040\pi$
$19$	116.000	1.40064	0.700322	+	0.713827i	$0.253040\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 16.0000i	− 0.145054i	−0.997366	−	0.0725268i	$-0.976894\pi$
$23$	− 16.0000i	− 0.145054i	0.997366	−	0.0725268i	$-0.0231063\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	198.000	1.26785	0.633925	−	0.773394i	$-0.281443\pi$
$29$	198.000	1.26785	0.633925	+	0.773394i	$0.281443\pi$
$30$	0	0
$31$	240.000	1.39049	0.695246	−	0.718772i	$-0.255295\pi$
$31$	240.000	1.39049	0.695246	+	0.718772i	$0.255295\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	258.000i	1.14635i	0.819433	+	0.573175i	$0.194288\pi$
$37$	258.000i	1.14635i	−0.819433	+	0.573175i	$0.805712\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−442.000	−1.68363	−0.841815	−	0.539767i	$-0.818512\pi$
$41$	−442.000	−1.68363	−0.841815	+	0.539767i	$0.818512\pi$
$42$	0	0
$43$	− 292.000i	− 1.03557i	−0.855510	−	0.517786i	$-0.826756\pi$
$43$	− 292.000i	− 1.03557i	0.855510	−	0.517786i	$-0.173244\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	392.000i	1.21658i	0.793716	+	0.608288i	$0.208143\pi$
$47$	392.000i	1.21658i	−0.793716	+	0.608288i	$0.791857\pi$
$48$	0	0
$49$	87.0000	0.253644
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 142.000i	− 0.368023i	−0.982924	−	0.184011i	$-0.941092\pi$
$53$	− 142.000i	− 0.368023i	0.982924	−	0.184011i	$-0.0589083\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−348.000	−0.767894	−0.383947	−	0.923355i	$-0.625435\pi$
$59$	−348.000	−0.767894	−0.383947	+	0.923355i	$0.625435\pi$
$60$	0	0
$61$	−570.000	−1.19641	−0.598205	−	0.801343i	$-0.704119\pi$
$61$	−570.000	−1.19641	−0.598205	+	0.801343i	$0.704119\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 692.000i	− 1.26181i	−0.775860	−	0.630905i	$-0.782684\pi$
$67$	− 692.000i	− 1.26181i	0.775860	−	0.630905i	$-0.217316\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−168.000	−0.280816	−0.140408	−	0.990094i	$-0.544841\pi$
$71$	−168.000	−0.280816	−0.140408	+	0.990094i	$0.544841\pi$
$72$	0	0
$73$	− 134.000i	− 0.214843i	−0.994214	−	0.107421i	$-0.965741\pi$
$73$	− 134.000i	− 0.214843i	0.994214	−	0.107421i	$-0.0342594\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	576.000i	0.852484i
$78$	0	0
$79$	−784.000	−1.11654	−0.558271	−	0.829658i	$-0.688535\pi$
$79$	−784.000	−1.11654	−0.558271	+	0.829658i	$0.688535\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 564.000i	− 0.745868i	−0.927858	−	0.372934i	$-0.878352\pi$
$83$	− 564.000i	− 0.745868i	0.927858	−	0.372934i	$-0.121648\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1034.00	1.23150	0.615752	−	0.787940i	$-0.288852\pi$
$89$	1034.00	1.23150	0.615752	+	0.787940i	$0.288852\pi$
$90$	0	0
$91$	−672.000	−0.774118
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	382.000i	0.399858i	0.979810	+	0.199929i	$0.0640711\pi$
$97$	382.000i	0.399858i	−0.979810	+	0.199929i	$0.935929\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	674.000	0.664015	0.332007	−	0.943277i	$-0.392274\pi$
$101$	674.000	0.664015	0.332007	+	0.943277i	$0.392274\pi$
$102$	0	0
$103$	− 992.000i	− 0.948977i	−0.880262	−	0.474489i	$-0.842633\pi$
$103$	− 992.000i	− 0.948977i	0.880262	−	0.474489i	$-0.157367\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 500.000i	− 0.451746i	−0.974157	−	0.225873i	$-0.927477\pi$
$107$	− 500.000i	− 0.451746i	0.974157	−	0.225873i	$-0.0725234\pi$
$108$	0	0
$109$	−1046.00	−0.919162	−0.459581	−	0.888136i	$-0.652000\pi$
$109$	−1046.00	−0.919162	−0.459581	+	0.888136i	$0.652000\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	558.000i	0.464533i	0.972652	+	0.232266i	$0.0746141\pi$
$113$	558.000i	0.464533i	−0.972652	+	0.232266i	$0.925386\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1760.00	−1.35579
$120$	0	0
$121$	−35.0000	−0.0262960
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	328.000i	0.229176i	0.993413	+	0.114588i	$0.0365547\pi$
$127$	328.000i	0.229176i	−0.993413	+	0.114588i	$0.963445\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	212.000	0.141393	0.0706967	−	0.997498i	$-0.477478\pi$
$131$	212.000	0.141393	0.0706967	+	0.997498i	$0.477478\pi$
$132$	0	0
$133$	− 1856.00i	− 1.21004i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1434.00i	0.894269i	0.894467	+	0.447135i	$0.147556\pi$
$137$	1434.00i	0.894269i	−0.894467	+	0.447135i	$0.852444\pi$
$138$	0	0
$139$	−2196.00	−1.34002	−0.670008	−	0.742354i	$-0.733709\pi$
$139$	−2196.00	−1.34002	−0.670008	+	0.742354i	$0.733709\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1512.00i	0.884194i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2418.00	−1.32946	−0.664732	−	0.747081i	$-0.731454\pi$
$149$	−2418.00	−1.32946	−0.664732	+	0.747081i	$0.731454\pi$
$150$	0	0
$151$	3672.00	1.97896	0.989481	−	0.144666i	$-0.0462108\pi$
$151$	3672.00	1.97896	0.989481	+	0.144666i	$0.0462108\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 358.000i	− 0.181984i	−0.995852	−	0.0909921i	$-0.970996\pi$
$157$	− 358.000i	− 0.181984i	0.995852	−	0.0909921i	$-0.0290038\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−256.000	−0.125314
$162$	0	0
$163$	2564.00i	1.23207i	0.787717	+	0.616037i	$0.211263\pi$
$163$	2564.00i	1.23207i	−0.787717	+	0.616037i	$0.788737\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 3056.00i	− 1.41605i	−0.706187	−	0.708025i	$-0.749586\pi$
$167$	− 3056.00i	− 1.41605i	0.706187	−	0.708025i	$-0.250414\pi$
$168$	0	0
$169$	433.000	0.197087
$170$	0	0
$171$	0	0
$172$	0	0
$173$	234.000i	0.102836i	0.998677	+	0.0514182i	$0.0163741\pi$
$173$	234.000i	0.102836i	−0.998677	+	0.0514182i	$0.983626\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	524.000	0.218802	0.109401	−	0.993998i	$-0.465107\pi$
$179$	524.000	0.218802	0.109401	+	0.993998i	$0.465107\pi$
$180$	0	0
$181$	−1138.00	−0.467331	−0.233665	−	0.972317i	$-0.575072\pi$
$181$	−1138.00	−0.467331	−0.233665	+	0.972317i	$0.575072\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	3960.00i	1.54858i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1520.00	−0.575829	−0.287915	−	0.957656i	$-0.592962\pi$
$191$	−1520.00	−0.575829	−0.287915	+	0.957656i	$0.592962\pi$
$192$	0	0
$193$	− 2142.00i	− 0.798884i	−0.916759	−	0.399442i	$-0.869204\pi$
$193$	− 2142.00i	− 0.798884i	0.916759	−	0.399442i	$-0.130796\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 2306.00i	− 0.833988i	−0.908909	−	0.416994i	$-0.863084\pi$
$197$	− 2306.00i	− 0.833988i	0.908909	−	0.416994i	$-0.136916\pi$
$198$	0	0
$199$	−3288.00	−1.17126	−0.585628	−	0.810580i	$-0.699152\pi$
$199$	−3288.00	−1.17126	−0.585628	+	0.810580i	$0.699152\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 3168.00i	− 1.09532i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−4176.00	−1.38211
$210$	0	0
$211$	−3876.00	−1.26462	−0.632310	−	0.774715i	$-0.717893\pi$
$211$	−3876.00	−1.26462	−0.632310	+	0.774715i	$0.717893\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 3840.00i	− 1.20127i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−4620.00	−1.40622
$222$	0	0
$223$	− 5688.00i	− 1.70806i	−0.520226	−	0.854028i	$-0.674152\pi$
$223$	− 5688.00i	− 1.70806i	0.520226	−	0.854028i	$-0.325848\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 2796.00i	− 0.817520i	−0.912642	−	0.408760i	$-0.865961\pi$
$227$	− 2796.00i	− 0.817520i	0.912642	−	0.408760i	$-0.134039\pi$
$228$	0	0
$229$	−4446.00	−1.28297	−0.641485	−	0.767136i	$-0.721681\pi$
$229$	−4446.00	−1.28297	−0.641485	+	0.767136i	$0.721681\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 2522.00i	− 0.709106i	−0.935036	−	0.354553i	$-0.884633\pi$
$233$	− 2522.00i	− 0.709106i	0.935036	−	0.354553i	$-0.115367\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	816.000	0.220848	0.110424	−	0.993885i	$-0.464779\pi$
$239$	816.000	0.220848	0.110424	+	0.993885i	$0.464779\pi$
$240$	0	0
$241$	−5422.00	−1.44922	−0.724609	−	0.689160i	$-0.757980\pi$
$241$	−5422.00	−1.44922	−0.724609	+	0.689160i	$0.757980\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 4872.00i	− 1.25505i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	5900.00	1.48368	0.741842	−	0.670575i	$-0.233952\pi$
$251$	5900.00	1.48368	0.741842	+	0.670575i	$0.233952\pi$
$252$	0	0
$253$	576.000i	0.143134i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	5250.00i	1.27426i	0.770754	+	0.637132i	$0.219880\pi$
$257$	5250.00i	1.27426i	−0.770754	+	0.637132i	$0.780120\pi$
$258$	0	0
$259$	4128.00	0.990353
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 6240.00i	− 1.46302i	−0.681829	−	0.731511i	$-0.738815\pi$
$263$	− 6240.00i	− 1.46302i	0.681829	−	0.731511i	$-0.261185\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−714.000	−0.161834	−0.0809170	−	0.996721i	$-0.525785\pi$
$269$	−714.000	−0.161834	−0.0809170	+	0.996721i	$0.525785\pi$
$270$	0	0
$271$	2144.00	0.480586	0.240293	−	0.970700i	$-0.422757\pi$
$271$	2144.00	0.480586	0.240293	+	0.970700i	$0.422757\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	4466.00i	0.968722i	0.874868	+	0.484361i	$0.160948\pi$
$277$	4466.00i	0.968722i	−0.874868	+	0.484361i	$0.839052\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	5302.00	1.12559	0.562795	−	0.826596i	$-0.309726\pi$
$281$	5302.00	1.12559	0.562795	+	0.826596i	$0.309726\pi$
$282$	0	0
$283$	− 6932.00i	− 1.45606i	−0.685546	−	0.728029i	$-0.740436\pi$
$283$	− 6932.00i	− 1.45606i	0.685546	−	0.728029i	$-0.259564\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	7072.00i	1.45452i
$288$	0	0
$289$	−7187.00	−1.46285
$290$	0	0
$291$	0	0
$292$	0	0
$293$	4034.00i	0.804330i	0.915567	+	0.402165i	$0.131742\pi$
$293$	4034.00i	0.804330i	−0.915567	+	0.402165i	$0.868258\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−672.000	−0.129976
$300$	0	0
$301$	−4672.00	−0.894650
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	3836.00i	0.713134i	0.934270	+	0.356567i	$0.116053\pi$
$307$	3836.00i	0.713134i	−0.934270	+	0.356567i	$0.883947\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−664.000	−0.121067	−0.0605337	−	0.998166i	$-0.519280\pi$
$311$	−664.000	−0.121067	−0.0605337	+	0.998166i	$0.519280\pi$
$312$	0	0
$313$	2986.00i	0.539229i	0.962968	+	0.269615i	$0.0868963\pi$
$313$	2986.00i	0.539229i	−0.962968	+	0.269615i	$0.913104\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2726.00i	0.482989i	0.970402	+	0.241494i	$0.0776375\pi$
$317$	2726.00i	0.482989i	−0.970402	+	0.241494i	$0.922362\pi$
$318$	0	0
$319$	−7128.00	−1.25107
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 12760.0i	− 2.19810i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	6272.00	1.05102
$330$	0	0
$331$	−9212.00	−1.52972	−0.764860	−	0.644197i	$-0.777192\pi$
$331$	−9212.00	−1.52972	−0.764860	+	0.644197i	$0.777192\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	3278.00i	0.529864i	0.964267	+	0.264932i	$0.0853494\pi$
$337$	3278.00i	0.529864i	−0.964267	+	0.264932i	$0.914651\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−8640.00	−1.37209
$342$	0	0
$343$	− 6880.00i	− 1.08305i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	4956.00i	0.766721i	0.923599	+	0.383360i	$0.125233\pi$
$347$	4956.00i	0.766721i	−0.923599	+	0.383360i	$0.874767\pi$
$348$	0	0
$349$	−4678.00	−0.717500	−0.358750	−	0.933434i	$-0.616797\pi$
$349$	−4678.00	−0.717500	−0.358750	+	0.933434i	$0.616797\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 1890.00i	− 0.284970i	−0.989797	−	0.142485i	$-0.954491\pi$
$353$	− 1890.00i	− 0.284970i	0.989797	−	0.142485i	$-0.0455093\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−6472.00	−0.951474	−0.475737	−	0.879588i	$-0.657819\pi$
$359$	−6472.00	−0.951474	−0.475737	+	0.879588i	$0.657819\pi$
$360$	0	0
$361$	6597.00	0.961802
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 1960.00i	− 0.278777i	−0.990238	−	0.139389i	$-0.955486\pi$
$367$	− 1960.00i	− 0.278777i	0.990238	−	0.139389i	$-0.0445137\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−2272.00	−0.317942
$372$	0	0
$373$	8750.00i	1.21463i	0.794460	+	0.607316i	$0.207754\pi$
$373$	8750.00i	1.21463i	−0.794460	+	0.607316i	$0.792246\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 8316.00i	− 1.13606i
$378$	0	0
$379$	380.000	0.0515021	0.0257510	−	0.999668i	$-0.491802\pi$
$379$	380.000	0.0515021	0.0257510	+	0.999668i	$0.491802\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	9688.00i	1.29252i	0.763119	+	0.646258i	$0.223667\pi$
$383$	9688.00i	1.29252i	−0.763119	+	0.646258i	$0.776333\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	3870.00	0.504413	0.252207	−	0.967673i	$-0.418844\pi$
$389$	3870.00	0.504413	0.252207	+	0.967673i	$0.418844\pi$
$390$	0	0
$391$	−1760.00	−0.227639
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 1622.00i	− 0.205053i	−0.994730	−	0.102526i	$-0.967307\pi$
$397$	− 1622.00i	− 0.205053i	0.994730	−	0.102526i	$-0.0326926\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−9906.00	−1.23362	−0.616811	−	0.787112i	$-0.711576\pi$
$401$	−9906.00	−1.23362	−0.616811	+	0.787112i	$0.711576\pi$
$402$	0	0
$403$	− 10080.0i	− 1.24596i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 9288.00i	− 1.13118i
$408$	0	0
$409$	4214.00	0.509459	0.254730	−	0.967012i	$-0.418014\pi$
$409$	4214.00	0.509459	0.254730	+	0.967012i	$0.418014\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	5568.00i	0.663398i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−7012.00	−0.817562	−0.408781	−	0.912632i	$-0.634046\pi$
$419$	−7012.00	−0.817562	−0.408781	+	0.912632i	$0.634046\pi$
$420$	0	0
$421$	−1602.00	−0.185455	−0.0927277	−	0.995692i	$-0.529559\pi$
$421$	−1602.00	−0.185455	−0.0927277	+	0.995692i	$0.529559\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	9120.00i	1.03360i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	3584.00	0.400546	0.200273	−	0.979740i	$-0.435817\pi$
$431$	3584.00	0.400546	0.200273	+	0.979740i	$0.435817\pi$
$432$	0	0
$433$	− 3470.00i	− 0.385121i	−0.981285	−	0.192561i	$-0.938321\pi$
$433$	− 3470.00i	− 0.385121i	0.981285	−	0.192561i	$-0.0616792\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 1856.00i	− 0.203168i
$438$	0	0
$439$	3416.00	0.371382	0.185691	−	0.982608i	$-0.440548\pi$
$439$	3416.00	0.371382	0.185691	+	0.982608i	$0.440548\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 9708.00i	− 1.04118i	−0.853808	−	0.520588i	$-0.825713\pi$
$443$	− 9708.00i	− 1.04118i	0.853808	−	0.520588i	$-0.174287\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−10366.0	−1.08954	−0.544768	−	0.838587i	$-0.683382\pi$
$449$	−10366.0	−1.08954	−0.544768	+	0.838587i	$0.683382\pi$
$450$	0	0
$451$	15912.0	1.66135
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	16742.0i	1.71369i	0.515572	+	0.856847i	$0.327580\pi$
$457$	16742.0i	1.71369i	−0.515572	+	0.856847i	$0.672420\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	1258.00	0.127095	0.0635476	−	0.997979i	$-0.479759\pi$
$461$	1258.00	0.127095	0.0635476	+	0.997979i	$0.479759\pi$
$462$	0	0
$463$	13528.0i	1.35788i	0.734193	+	0.678941i	$0.237561\pi$
$463$	13528.0i	1.35788i	−0.734193	+	0.678941i	$0.762439\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	6916.00i	0.685298i	0.939463	+	0.342649i	$0.111324\pi$
$467$	6916.00i	0.685298i	−0.939463	+	0.342649i	$0.888676\pi$
$468$	0	0
$469$	−11072.0	−1.09010
$470$	0	0
$471$	0	0
$472$	0	0
$473$	10512.0i	1.02187i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	1728.00	0.164832	0.0824158	−	0.996598i	$-0.473736\pi$
$479$	1728.00	0.164832	0.0824158	+	0.996598i	$0.473736\pi$
$480$	0	0
$481$	10836.0	1.02719
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 16656.0i	− 1.54981i	−0.632080	−	0.774903i	$-0.717799\pi$
$487$	− 16656.0i	− 1.54981i	0.632080	−	0.774903i	$-0.282201\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	1084.00	0.0996339	0.0498169	−	0.998758i	$-0.484136\pi$
$491$	1084.00	0.0996339	0.0498169	+	0.998758i	$0.484136\pi$
$492$	0	0
$493$	− 21780.0i	− 1.98970i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	2688.00i	0.242602i
$498$	0	0
$499$	−5804.00	−0.520687	−0.260343	−	0.965516i	$-0.583836\pi$
$499$	−5804.00	−0.520687	−0.260343	+	0.965516i	$0.583836\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 10512.0i	− 0.931823i	−0.884831	−	0.465911i	$-0.845727\pi$
$503$	− 10512.0i	− 0.931823i	0.884831	−	0.465911i	$-0.154273\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−4314.00	−0.375667	−0.187834	−	0.982201i	$-0.560147\pi$
$509$	−4314.00	−0.375667	−0.187834	+	0.982201i	$0.560147\pi$
$510$	0	0
$511$	−2144.00	−0.185607
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 14112.0i	− 1.20047i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	1190.00	0.100067	0.0500334	−	0.998748i	$-0.484067\pi$
$521$	1190.00	0.100067	0.0500334	+	0.998748i	$0.484067\pi$
$522$	0	0
$523$	− 3780.00i	− 0.316038i	−0.987436	−	0.158019i	$-0.949489\pi$
$523$	− 3780.00i	− 0.316038i	0.987436	−	0.158019i	$-0.0505107\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 26400.0i	− 2.18217i
$528$	0	0
$529$	11911.0	0.978959
$530$	0	0
$531$	0	0
$532$	0	0
$533$	18564.0i	1.50862i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−3132.00	−0.250287
$540$	0	0
$541$	−11002.0	−0.874331	−0.437165	−	0.899381i	$-0.644018\pi$
$541$	−11002.0	−0.874331	−0.437165	+	0.899381i	$0.644018\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 5908.00i	− 0.461806i	−0.972977	−	0.230903i	$-0.925832\pi$
$547$	− 5908.00i	− 0.461806i	0.972977	−	0.230903i	$-0.0741680\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	22968.0	1.77581
$552$	0	0
$553$	12544.0i	0.964602i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	14806.0i	1.12630i	0.826354	+	0.563151i	$0.190411\pi$
$557$	14806.0i	1.12630i	−0.826354	+	0.563151i	$0.809589\pi$
$558$	0	0
$559$	−12264.0	−0.927928
$560$	0	0
$561$	0	0
$562$	0	0
$563$	684.000i	0.0512028i	0.999672	+	0.0256014i	$0.00815007\pi$
$563$	684.000i	0.0512028i	−0.999672	+	0.0256014i	$0.991850\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−2582.00	−0.190234	−0.0951169	−	0.995466i	$-0.530323\pi$
$569$	−2582.00	−0.190234	−0.0951169	+	0.995466i	$0.530323\pi$
$570$	0	0
$571$	−2540.00	−0.186157	−0.0930785	−	0.995659i	$-0.529671\pi$
$571$	−2540.00	−0.186157	−0.0930785	+	0.995659i	$0.529671\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 22786.0i	− 1.64401i	−0.569480	−	0.822005i	$-0.692856\pi$
$577$	− 22786.0i	− 1.64401i	0.569480	−	0.822005i	$-0.307144\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−9024.00	−0.644369
$582$	0	0
$583$	5112.00i	0.363152i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	7884.00i	0.554357i	0.960818	+	0.277178i	$0.0893993\pi$
$587$	7884.00i	0.554357i	−0.960818	+	0.277178i	$0.910601\pi$
$588$	0	0
$589$	27840.0	1.94758
$590$	0	0
$591$	0	0
$592$	0	0
$593$	21902.0i	1.51671i	0.651843	+	0.758354i	$0.273996\pi$
$593$	21902.0i	1.51671i	−0.651843	+	0.758354i	$0.726004\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	15080.0	1.02863	0.514317	−	0.857600i	$-0.328045\pi$
$599$	15080.0	1.02863	0.514317	+	0.857600i	$0.328045\pi$
$600$	0	0
$601$	−19702.0	−1.33721	−0.668603	−	0.743619i	$-0.733108\pi$
$601$	−19702.0	−1.33721	−0.668603	+	0.743619i	$0.733108\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 7320.00i	− 0.489472i	−0.969590	−	0.244736i	$-0.921299\pi$
$607$	− 7320.00i	− 0.489472i	0.969590	−	0.244736i	$-0.0787013\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	16464.0	1.09012
$612$	0	0
$613$	24350.0i	1.60438i	0.597066	+	0.802192i	$0.296333\pi$
$613$	24350.0i	1.60438i	−0.597066	+	0.802192i	$0.703667\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	19546.0i	1.27535i	0.770305	+	0.637676i	$0.220104\pi$
$617$	19546.0i	1.27535i	−0.770305	+	0.637676i	$0.779896\pi$
$618$	0	0
$619$	−3476.00	−0.225706	−0.112853	−	0.993612i	$-0.535999\pi$
$619$	−3476.00	−0.225706	−0.112853	+	0.993612i	$0.535999\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 16544.0i	− 1.06392i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	28380.0	1.79902
$630$	0	0
$631$	21880.0	1.38039	0.690197	−	0.723621i	$-0.257524\pi$
$631$	21880.0	1.38039	0.690197	+	0.723621i	$0.257524\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 3654.00i	− 0.227279i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−20994.0	−1.29362	−0.646812	−	0.762649i	$-0.723898\pi$
$641$	−20994.0	−1.29362	−0.646812	+	0.762649i	$0.723898\pi$
$642$	0	0
$643$	− 18204.0i	− 1.11648i	−0.829680	−	0.558239i	$-0.811477\pi$
$643$	− 18204.0i	− 1.11648i	0.829680	−	0.558239i	$-0.188523\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 2064.00i	− 0.125416i	−0.998032	−	0.0627080i	$-0.980026\pi$
$647$	− 2064.00i	− 0.125416i	0.998032	−	0.0627080i	$-0.0199737\pi$
$648$	0	0
$649$	12528.0	0.757730
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 9942.00i	− 0.595805i	−0.954596	−	0.297902i	$-0.903713\pi$
$653$	− 9942.00i	− 0.595805i	0.954596	−	0.297902i	$-0.0962870\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	24236.0	1.43263	0.716313	−	0.697779i	$-0.245828\pi$
$659$	24236.0	1.43263	0.716313	+	0.697779i	$0.245828\pi$
$660$	0	0
$661$	17614.0	1.03647	0.518234	−	0.855239i	$-0.326590\pi$
$661$	17614.0	1.03647	0.518234	+	0.855239i	$0.326590\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 3168.00i	− 0.183906i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	20520.0	1.18057
$672$	0	0
$673$	13058.0i	0.747918i	0.927445	+	0.373959i	$0.122000\pi$
$673$	13058.0i	0.747918i	−0.927445	+	0.373959i	$0.878000\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 33186.0i	− 1.88396i	−0.335668	−	0.941980i	$-0.608962\pi$
$677$	− 33186.0i	− 1.88396i	0.335668	−	0.941980i	$-0.391038\pi$
$678$	0	0
$679$	6112.00	0.345445
$680$	0	0
$681$	0	0
$682$	0	0
$683$	31716.0i	1.77684i	0.459035	+	0.888418i	$0.348195\pi$
$683$	31716.0i	1.77684i	−0.459035	+	0.888418i	$0.651805\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−5964.00	−0.329768
$690$	0	0
$691$	−2084.00	−0.114731	−0.0573655	−	0.998353i	$-0.518270\pi$
$691$	−2084.00	−0.114731	−0.0573655	+	0.998353i	$0.518270\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	48620.0i	2.64220i
$698$	0	0
$699$	0	0
$700$	0	0
$701$	7418.00	0.399678	0.199839	−	0.979829i	$-0.435958\pi$
$701$	7418.00	0.399678	0.199839	+	0.979829i	$0.435958\pi$
$702$	0	0
$703$	29928.0i	1.60563i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 10784.0i	− 0.573655i
$708$	0	0
$709$	18242.0	0.966280	0.483140	−	0.875543i	$-0.339496\pi$
$709$	18242.0	0.966280	0.483140	+	0.875543i	$0.339496\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 3840.00i	− 0.201696i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	3024.00	0.156851	0.0784257	−	0.996920i	$-0.475011\pi$
$719$	3024.00	0.156851	0.0784257	+	0.996920i	$0.475011\pi$
$720$	0	0
$721$	−15872.0	−0.819839
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	26176.0i	1.33537i	0.744444	+	0.667685i	$0.232715\pi$
$727$	26176.0i	1.33537i	−0.744444	+	0.667685i	$0.767285\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−32120.0	−1.62517
$732$	0	0
$733$	− 17818.0i	− 0.897848i	−0.893570	−	0.448924i	$-0.851807\pi$
$733$	− 17818.0i	− 0.897848i	0.893570	−	0.448924i	$-0.148193\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	24912.0i	1.24511i
$738$	0	0
$739$	22052.0	1.09769	0.548847	−	0.835923i	$-0.315067\pi$
$739$	22052.0	1.09769	0.548847	+	0.835923i	$0.315067\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 15840.0i	− 0.782117i	−0.920366	−	0.391059i	$-0.872109\pi$
$743$	− 15840.0i	− 0.782117i	0.920366	−	0.391059i	$-0.127891\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−8000.00	−0.390272
$750$	0	0
$751$	21024.0	1.02154	0.510770	−	0.859717i	$-0.329360\pi$
$751$	21024.0	1.02154	0.510770	+	0.859717i	$0.329360\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	38034.0i	1.82612i	0.407831	+	0.913058i	$0.366285\pi$
$757$	38034.0i	1.82612i	−0.407831	+	0.913058i	$0.633715\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−37802.0	−1.80069	−0.900343	−	0.435182i	$-0.856684\pi$
$761$	−37802.0	−1.80069	−0.900343	+	0.435182i	$0.856684\pi$
$762$	0	0
$763$	16736.0i	0.794081i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	14616.0i	0.688075i
$768$	0	0
$769$	−15042.0	−0.705369	−0.352684	−	0.935742i	$-0.614731\pi$
$769$	−15042.0	−0.705369	−0.352684	+	0.935742i	$0.614731\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 5950.00i	− 0.276852i	−0.990373	−	0.138426i	$-0.955796\pi$
$773$	− 5950.00i	− 0.276852i	0.990373	−	0.138426i	$-0.0442043\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−51272.0	−2.35816
$780$	0	0
$781$	6048.00	0.277099
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 23364.0i	− 1.05824i	−0.848546	−	0.529121i	$-0.822522\pi$
$787$	− 23364.0i	− 1.05824i	0.848546	−	0.529121i	$-0.177478\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	8928.00	0.401319
$792$	0	0
$793$	23940.0i	1.07205i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	19846.0i	0.882034i	0.897499	+	0.441017i	$0.145382\pi$
$797$	19846.0i	0.882034i	−0.897499	+	0.441017i	$0.854618\pi$
$798$	0	0
$799$	43120.0	1.90923
$800$	0	0
$801$	0	0
$802$	0	0
$803$	4824.00i	0.211999i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	24762.0	1.07613	0.538063	−	0.842905i	$-0.319156\pi$
$809$	24762.0	1.07613	0.538063	+	0.842905i	$0.319156\pi$
$810$	0	0
$811$	16644.0	0.720653	0.360327	−	0.932826i	$-0.382665\pi$
$811$	16644.0	0.720653	0.360327	+	0.932826i	$0.382665\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 33872.0i	− 1.45047i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−3182.00	−0.135265	−0.0676325	−	0.997710i	$-0.521545\pi$
$821$	−3182.00	−0.135265	−0.0676325	+	0.997710i	$0.521545\pi$
$822$	0	0
$823$	− 7504.00i	− 0.317829i	−0.987292	−	0.158914i	$-0.949201\pi$
$823$	− 7504.00i	− 0.317829i	0.987292	−	0.158914i	$-0.0507994\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	12604.0i	0.529969i	0.964253	+	0.264984i	$0.0853668\pi$
$827$	12604.0i	0.529969i	−0.964253	+	0.264984i	$0.914633\pi$
$828$	0	0
$829$	−12230.0	−0.512383	−0.256191	−	0.966626i	$-0.582468\pi$
$829$	−12230.0	−0.512383	−0.256191	+	0.966626i	$0.582468\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 9570.00i	− 0.398056i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	9656.00	0.397333	0.198666	−	0.980067i	$-0.436339\pi$
$839$	9656.00	0.397333	0.198666	+	0.980067i	$0.436339\pi$
$840$	0	0
$841$	14815.0	0.607446
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	560.000i	0.0227176i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	4128.00	0.166282
$852$	0	0
$853$	5806.00i	0.233052i	0.993188	+	0.116526i	$0.0371759\pi$
$853$	5806.00i	0.233052i	−0.993188	+	0.116526i	$0.962824\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 39094.0i	− 1.55826i	−0.626865	−	0.779128i	$-0.715662\pi$
$857$	− 39094.0i	− 1.55826i	0.626865	−	0.779128i	$-0.284338\pi$
$858$	0	0
$859$	18876.0	0.749756	0.374878	−	0.927074i	$-0.377685\pi$
$859$	18876.0	0.749756	0.374878	+	0.927074i	$0.377685\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 32296.0i	− 1.27389i	−0.770909	−	0.636946i	$-0.780197\pi$
$863$	− 32296.0i	− 1.27389i	0.770909	−	0.636946i	$-0.219803\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	28224.0	1.10176
$870$	0	0
$871$	−29064.0	−1.13065
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	9578.00i	0.368787i	0.982853	+	0.184393i	$0.0590321\pi$
$877$	9578.00i	0.368787i	−0.982853	+	0.184393i	$0.940968\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	41710.0	1.59506	0.797529	−	0.603281i	$-0.206140\pi$
$881$	41710.0	1.59506	0.797529	+	0.603281i	$0.206140\pi$
$882$	0	0
$883$	2260.00i	0.0861326i	0.999072	+	0.0430663i	$0.0137127\pi$
$883$	2260.00i	0.0861326i	−0.999072	+	0.0430663i	$0.986287\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 33696.0i	− 1.27554i	−0.770228	−	0.637768i	$-0.779858\pi$
$887$	− 33696.0i	− 1.27554i	0.770228	−	0.637768i	$-0.220142\pi$
$888$	0	0
$889$	5248.00	0.197989
$890$	0	0
$891$	0	0
$892$	0	0
$893$	45472.0i	1.70399i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	47520.0	1.76294
$900$	0	0
$901$	−15620.0	−0.577556
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 7756.00i	− 0.283940i	−0.989871	−	0.141970i	$-0.954656\pi$
$907$	− 7756.00i	− 0.283940i	0.989871	−	0.141970i	$-0.0453437\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	5312.00	0.193188	0.0965941	−	0.995324i	$-0.469205\pi$
$911$	5312.00	0.193188	0.0965941	+	0.995324i	$0.469205\pi$
$912$	0	0
$913$	20304.0i	0.735996i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 3392.00i	− 0.122152i
$918$	0	0
$919$	23576.0	0.846246	0.423123	−	0.906072i	$-0.360934\pi$
$919$	23576.0	0.846246	0.423123	+	0.906072i	$0.360934\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	7056.00i	0.251626i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−19038.0	−0.672354	−0.336177	−	0.941799i	$-0.609134\pi$
$929$	−19038.0	−0.672354	−0.336177	+	0.941799i	$0.609134\pi$
$930$	0	0
$931$	10092.0	0.355265
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 20570.0i	− 0.717175i	−0.933496	−	0.358587i	$-0.883259\pi$
$937$	− 20570.0i	− 0.717175i	0.933496	−	0.358587i	$-0.116741\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	21386.0	0.740875	0.370438	−	0.928857i	$-0.379208\pi$
$941$	21386.0	0.740875	0.370438	+	0.928857i	$0.379208\pi$
$942$	0	0
$943$	7072.00i	0.244216i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	38020.0i	1.30463i	0.757948	+	0.652315i	$0.226202\pi$
$947$	38020.0i	1.30463i	−0.757948	+	0.652315i	$0.773798\pi$
$948$	0	0
$949$	−5628.00	−0.192511
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 20202.0i	− 0.686681i	−0.939211	−	0.343340i	$-0.888442\pi$
$953$	− 20202.0i	− 0.686681i	0.939211	−	0.343340i	$-0.111558\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	22944.0	0.772576
$960$	0	0
$961$	27809.0	0.933470
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 29840.0i	− 0.992337i	−0.868226	−	0.496168i	$-0.834740\pi$
$967$	− 29840.0i	− 0.992337i	0.868226	−	0.496168i	$-0.165260\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	12476.0	0.412332	0.206166	−	0.978517i	$-0.433901\pi$
$971$	12476.0	0.412332	0.206166	+	0.978517i	$0.433901\pi$
$972$	0	0
$973$	35136.0i	1.15767i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 36974.0i	− 1.21075i	−0.795940	−	0.605375i	$-0.793023\pi$
$977$	− 36974.0i	− 1.21075i	0.795940	−	0.605375i	$-0.206977\pi$
$978$	0	0
$979$	−37224.0	−1.21520
$980$	0	0
$981$	0	0
$982$	0	0
$983$	16368.0i	0.531087i	0.964099	+	0.265543i	$0.0855513\pi$
$983$	16368.0i	0.531087i	−0.964099	+	0.265543i	$0.914449\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−4672.00	−0.150213
$990$	0	0
$991$	−49552.0	−1.58837	−0.794183	−	0.607678i	$-0.792101\pi$
$991$	−49552.0	−1.58837	−0.794183	+	0.607678i	$0.792101\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 24414.0i	− 0.775526i	−0.921759	−	0.387763i	$-0.873248\pi$
$997$	− 24414.0i	− 0.775526i	0.921759	−	0.387763i	$-0.126752\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1800.4.f.d.649.1		2
3.2	odd	2		200.4.c.f.49.2		2
5.2	odd	4		360.4.a.f.1.1		1
5.3	odd	4		1800.4.a.h.1.1		1
5.4	even	2	inner	1800.4.f.d.649.2		2
12.11	even	2		400.4.c.h.49.1		2
15.2	even	4		40.4.a.b.1.1	✓	1
15.8	even	4		200.4.a.d.1.1		1
15.14	odd	2		200.4.c.f.49.1		2
20.7	even	4		720.4.a.d.1.1		1
60.23	odd	4		400.4.a.p.1.1		1
60.47	odd	4		80.4.a.b.1.1		1
60.59	even	2		400.4.c.h.49.2		2
105.62	odd	4		1960.4.a.e.1.1		1
120.53	even	4		1600.4.a.bk.1.1		1
120.77	even	4		320.4.a.e.1.1		1
120.83	odd	4		1600.4.a.q.1.1		1
120.107	odd	4		320.4.a.j.1.1		1
240.77	even	4		1280.4.d.d.641.2		2
240.107	odd	4		1280.4.d.m.641.2		2
240.197	even	4		1280.4.d.d.641.1		2
240.227	odd	4		1280.4.d.m.641.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.4.a.b.1.1	✓	1	15.2	even	4
80.4.a.b.1.1		1	60.47	odd	4
200.4.a.d.1.1		1	15.8	even	4
200.4.c.f.49.1		2	15.14	odd	2
200.4.c.f.49.2		2	3.2	odd	2
320.4.a.e.1.1		1	120.77	even	4
320.4.a.j.1.1		1	120.107	odd	4
360.4.a.f.1.1		1	5.2	odd	4
400.4.a.p.1.1		1	60.23	odd	4
400.4.c.h.49.1		2	12.11	even	2
400.4.c.h.49.2		2	60.59	even	2
720.4.a.d.1.1		1	20.7	even	4
1280.4.d.d.641.1		2	240.197	even	4
1280.4.d.d.641.2		2	240.77	even	4
1280.4.d.m.641.1		2	240.227	odd	4
1280.4.d.m.641.2		2	240.107	odd	4
1600.4.a.q.1.1		1	120.83	odd	4
1600.4.a.bk.1.1		1	120.53	even	4
1800.4.a.h.1.1		1	5.3	odd	4
1800.4.f.d.649.1		2	1.1	even	1	trivial
1800.4.f.d.649.2		2	5.4	even	2	inner
1960.4.a.e.1.1		1	105.62	odd	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 $\F_{q}$

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)

\(f(q)\)	\(=\)	\(q-16.0000i q^{7} +O(q^{10})\)	\(q-16.0000i q^{7} -36.0000 q^{11} -42.0000i q^{13} -110.000i q^{17} +116.000 q^{19} -16.0000i q^{23} +198.000 q^{29} +240.000 q^{31} +258.000i q^{37} -442.000 q^{41} -292.000i q^{43} +392.000i q^{47} +87.0000 q^{49} -142.000i q^{53} -348.000 q^{59} -570.000 q^{61} -692.000i q^{67} -168.000 q^{71} -134.000i q^{73} +576.000i q^{77} -784.000 q^{79} -564.000i q^{83} +1034.00 q^{89} -672.000 q^{91} +382.000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q+O(q^{10}) \) 2 * q	\( 2 q - 72 q^{11} + 232 q^{19} + 396 q^{29} + 480 q^{31} - 884 q^{41} + 174 q^{49} - 696 q^{59} - 1140 q^{61} - 336 q^{71} - 1568 q^{79} + 2068 q^{89} - 1344 q^{91}+O(q^{100}) \) 2 * q - 72 * q^11 + 232 * q^19 + 396 * q^29 + 480 * q^31 - 884 * q^41 + 174 * q^49 - 696 * q^59 - 1140 * q^61 - 336 * q^71 - 1568 * q^79 + 2068 * q^89 - 1344 * q^91

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