LMFDB - Embedded newform 1800.4.f.o.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 360)
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.o.649.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-34.0000i q^{7} +O(q^{10})$	$q-34.0000i q^{7} +18.0000 q^{11} +12.0000i q^{13} +106.000i q^{17} +44.0000 q^{19} +56.0000i q^{23} -270.000 q^{29} +204.000 q^{31} -120.000i q^{37} +80.0000 q^{41} +536.000i q^{43} +536.000i q^{47} -813.000 q^{49} +542.000i q^{53} +174.000 q^{59} +186.000 q^{61} -332.000i q^{67} -132.000 q^{71} -602.000i q^{73} -612.000i q^{77} +548.000 q^{79} -492.000i q^{83} +1052.00 q^{89} +408.000 q^{91} -482.000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q+O(q^{10}) $ 2 * q	$ 2 q + 36 q^{11} + 88 q^{19} - 540 q^{29} + 408 q^{31} + 160 q^{41} - 1626 q^{49} + 348 q^{59} + 372 q^{61} - 264 q^{71} + 1096 q^{79} + 2104 q^{89} + 816 q^{91}+O(q^{100}) $ 2 * q + 36 * q^11 + 88 * q^19 - 540 * q^29 + 408 * q^31 + 160 * q^41 - 1626 * q^49 + 348 * q^59 + 372 * q^61 - 264 * q^71 + 1096 * q^79 + 2104 * q^89 + 816 * 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
$7$	− 34.0000i	− 1.83583i	−0.396780	−	0.917914i	$-0.629872\pi$
$7$	− 34.0000i	− 1.83583i	0.396780	−	0.917914i	$-0.370128\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	18.0000	0.493382	0.246691	−	0.969094i	$-0.420657\pi$
$11$	18.0000	0.493382	0.246691	+	0.969094i	$0.420657\pi$
$12$	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
$17$	106.000i	1.51228i	0.654409	+	0.756140i	$0.272917\pi$
$17$	106.000i	1.51228i	−0.654409	+	0.756140i	$0.727083\pi$
$18$	0	0
$19$	44.0000	0.531279	0.265639	−	0.964072i	$-0.414417\pi$
$19$	44.0000	0.531279	0.265639	+	0.964072i	$0.414417\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	56.0000i	0.507687i	0.967245	+	0.253844i	$0.0816949\pi$
$23$	56.0000i	0.507687i	−0.967245	+	0.253844i	$0.918305\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−270.000	−1.72889	−0.864444	−	0.502729i	$-0.832329\pi$
$29$	−270.000	−1.72889	−0.864444	+	0.502729i	$0.832329\pi$
$30$	0	0
$31$	204.000	1.18192	0.590959	−	0.806701i	$-0.298749\pi$
$31$	204.000	1.18192	0.590959	+	0.806701i	$0.298749\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 120.000i	− 0.533186i	−0.963809	−	0.266593i	$-0.914102\pi$
$37$	− 120.000i	− 0.533186i	0.963809	−	0.266593i	$-0.0858979\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	80.0000	0.304729	0.152365	−	0.988324i	$-0.451311\pi$
$41$	80.0000	0.304729	0.152365	+	0.988324i	$0.451311\pi$
$42$	0	0
$43$	536.000i	1.90091i	0.310858	+	0.950456i	$0.399383\pi$
$43$	536.000i	1.90091i	−0.310858	+	0.950456i	$0.600617\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	536.000i	1.66348i	0.555164	+	0.831741i	$0.312655\pi$
$47$	536.000i	1.66348i	−0.555164	+	0.831741i	$0.687345\pi$
$48$	0	0
$49$	−813.000	−2.37026
$50$	0	0
$51$	0	0
$52$	0	0
$53$	542.000i	1.40471i	0.711829	+	0.702353i	$0.247867\pi$
$53$	542.000i	1.40471i	−0.711829	+	0.702353i	$0.752133\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	174.000	0.383947	0.191973	−	0.981400i	$-0.438511\pi$
$59$	174.000	0.383947	0.191973	+	0.981400i	$0.438511\pi$
$60$	0	0
$61$	186.000	0.390408	0.195204	−	0.980763i	$-0.437463\pi$
$61$	186.000	0.390408	0.195204	+	0.980763i	$0.437463\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 332.000i	− 0.605377i	−0.953090	−	0.302688i	$-0.902116\pi$
$67$	− 332.000i	− 0.605377i	0.953090	−	0.302688i	$-0.0978842\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−132.000	−0.220641	−0.110321	−	0.993896i	$-0.535188\pi$
$71$	−132.000	−0.220641	−0.110321	+	0.993896i	$0.535188\pi$
$72$	0	0
$73$	− 602.000i	− 0.965189i	−0.875844	−	0.482594i	$-0.839695\pi$
$73$	− 602.000i	− 0.965189i	0.875844	−	0.482594i	$-0.160305\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 612.000i	− 0.905765i
$78$	0	0
$79$	548.000	0.780441	0.390220	−	0.920721i	$-0.372399\pi$
$79$	548.000	0.780441	0.390220	+	0.920721i	$0.372399\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 492.000i	− 0.650651i	−0.945602	−	0.325325i	$-0.894526\pi$
$83$	− 492.000i	− 0.650651i	0.945602	−	0.325325i	$-0.105474\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1052.00	1.25294	0.626471	−	0.779445i	$-0.284499\pi$
$89$	1052.00	1.25294	0.626471	+	0.779445i	$0.284499\pi$
$90$	0	0
$91$	408.000	0.470000
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 482.000i	− 0.504533i	−0.967658	−	0.252266i	$-0.918824\pi$
$97$	− 482.000i	− 0.504533i	0.967658	−	0.252266i	$-0.0811759\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1214.00	1.19601	0.598007	−	0.801491i	$-0.295959\pi$
$101$	1214.00	1.19601	0.598007	+	0.801491i	$0.295959\pi$
$102$	0	0
$103$	898.000i	0.859054i	0.903054	+	0.429527i	$0.141320\pi$
$103$	898.000i	0.859054i	−0.903054	+	0.429527i	$0.858680\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 1364.00i	− 1.23236i	−0.787604	−	0.616182i	$-0.788679\pi$
$107$	− 1364.00i	− 1.23236i	0.787604	−	0.616182i	$-0.211321\pi$
$108$	0	0
$109$	−218.000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−218.000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1386.00i	1.15384i	0.816801	+	0.576920i	$0.195746\pi$
$113$	1386.00i	1.15384i	−0.816801	+	0.576920i	$0.804254\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	3604.00	2.77629
$120$	0	0
$121$	−1007.00	−0.756574
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	814.000i	0.568747i	0.958714	+	0.284373i	$0.0917855\pi$
$127$	814.000i	0.568747i	−0.958714	+	0.284373i	$0.908214\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−1282.00	−0.855029	−0.427515	−	0.904008i	$-0.640611\pi$
$131$	−1282.00	−0.855029	−0.427515	+	0.904008i	$0.640611\pi$
$132$	0	0
$133$	− 1496.00i	− 0.975336i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 3066.00i	− 1.91202i	−0.293342	−	0.956008i	$-0.594768\pi$
$137$	− 3066.00i	− 1.91202i	0.293342	−	0.956008i	$-0.405232\pi$
$138$	0	0
$139$	1332.00	0.812797	0.406398	−	0.913696i	$-0.366784\pi$
$139$	1332.00	0.812797	0.406398	+	0.913696i	$0.366784\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	216.000i	0.126313i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1470.00	0.808236	0.404118	−	0.914707i	$-0.367579\pi$
$149$	1470.00	0.808236	0.404118	+	0.914707i	$0.367579\pi$
$150$	0	0
$151$	−2592.00	−1.39691	−0.698457	−	0.715652i	$-0.746130\pi$
$151$	−2592.00	−1.39691	−0.698457	+	0.715652i	$0.746130\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	3332.00i	1.69377i	0.531773	+	0.846887i	$0.321526\pi$
$157$	3332.00i	1.69377i	−0.531773	+	0.846887i	$0.678474\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1904.00	0.932026
$162$	0	0
$163$	− 748.000i	− 0.359435i	−0.983718	−	0.179717i	$-0.942482\pi$
$163$	− 748.000i	− 0.359435i	0.983718	−	0.179717i	$-0.0575183\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2560.00i	1.18622i	0.805121	+	0.593110i	$0.202100\pi$
$167$	2560.00i	1.18622i	−0.805121	+	0.593110i	$0.797900\pi$
$168$	0	0
$169$	2053.00	0.934456
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1206.00i	0.530003i	0.964248	+	0.265001i	$0.0853724\pi$
$173$	1206.00i	0.530003i	−0.964248	+	0.265001i	$0.914628\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	1694.00	0.707349	0.353675	−	0.935369i	$-0.384932\pi$
$179$	1694.00	0.707349	0.353675	+	0.935369i	$0.384932\pi$
$180$	0	0
$181$	3722.00	1.52848	0.764238	−	0.644935i	$-0.223115\pi$
$181$	3722.00	1.52848	0.764238	+	0.644935i	$0.223115\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	1908.00i	0.746133i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2836.00	1.07438	0.537188	−	0.843463i	$-0.319487\pi$
$191$	2836.00	1.07438	0.537188	+	0.843463i	$0.319487\pi$
$192$	0	0
$193$	− 234.000i	− 0.0872730i	−0.999047	−	0.0436365i	$-0.986106\pi$
$193$	− 234.000i	− 0.0872730i	0.999047	−	0.0436365i	$-0.0138943\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3814.00i	1.37937i	0.724109	+	0.689686i	$0.242251\pi$
$197$	3814.00i	1.37937i	−0.724109	+	0.689686i	$0.757749\pi$
$198$	0	0
$199$	−2352.00	−0.837833	−0.418917	−	0.908025i	$-0.637590\pi$
$199$	−2352.00	−0.837833	−0.418917	+	0.908025i	$0.637590\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	9180.00i	3.17394i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	792.000	0.262123
$210$	0	0
$211$	−3660.00	−1.19415	−0.597073	−	0.802187i	$-0.703670\pi$
$211$	−3660.00	−1.19415	−0.597073	+	0.802187i	$0.703670\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 6936.00i	− 2.16980i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−1272.00	−0.387167
$222$	0	0
$223$	− 2646.00i	− 0.794571i	−0.917695	−	0.397285i	$-0.869952\pi$
$223$	− 2646.00i	− 0.794571i	0.917695	−	0.397285i	$-0.130048\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 240.000i	− 0.0701734i	−0.999384	−	0.0350867i	$-0.988829\pi$
$227$	− 240.000i	− 0.0701734i	0.999384	−	0.0350867i	$-0.0111707\pi$
$228$	0	0
$229$	4698.00	1.35569	0.677844	−	0.735206i	$-0.262914\pi$
$229$	4698.00	1.35569	0.677844	+	0.735206i	$0.262914\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	3814.00i	1.07238i	0.844099	+	0.536188i	$0.180136\pi$
$233$	3814.00i	1.07238i	−0.844099	+	0.536188i	$0.819864\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	2148.00	0.581350	0.290675	−	0.956822i	$-0.406120\pi$
$239$	2148.00	0.581350	0.290675	+	0.956822i	$0.406120\pi$
$240$	0	0
$241$	−3370.00	−0.900750	−0.450375	−	0.892839i	$-0.648710\pi$
$241$	−3370.00	−0.900750	−0.450375	+	0.892839i	$0.648710\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	528.000i	0.136016i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	6134.00	1.54253	0.771264	−	0.636515i	$-0.219625\pi$
$251$	6134.00	1.54253	0.771264	+	0.636515i	$0.219625\pi$
$252$	0	0
$253$	1008.00i	0.250484i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4566.00i	1.10825i	0.832435	+	0.554123i	$0.186946\pi$
$257$	4566.00i	1.10825i	−0.832435	+	0.554123i	$0.813054\pi$
$258$	0	0
$259$	−4080.00	−0.978837
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 1920.00i	− 0.450161i	−0.974340	−	0.225080i	$-0.927736\pi$
$263$	− 1920.00i	− 0.450161i	0.974340	−	0.225080i	$-0.0722645\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	5802.00	1.31507	0.657536	−	0.753423i	$-0.271599\pi$
$269$	5802.00	1.31507	0.657536	+	0.753423i	$0.271599\pi$
$270$	0	0
$271$	1640.00	0.367612	0.183806	−	0.982963i	$-0.441158\pi$
$271$	1640.00	0.367612	0.183806	+	0.982963i	$0.441158\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	2792.00i	0.605614i	0.953052	+	0.302807i	$0.0979237\pi$
$277$	2792.00i	0.605614i	−0.953052	+	0.302807i	$0.902076\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	1108.00	0.235223	0.117612	−	0.993060i	$-0.462476\pi$
$281$	1108.00	0.235223	0.117612	+	0.993060i	$0.462476\pi$
$282$	0	0
$283$	6028.00i	1.26617i	0.774080	+	0.633087i	$0.218213\pi$
$283$	6028.00i	1.26617i	−0.774080	+	0.633087i	$0.781787\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 2720.00i	− 0.559430i
$288$	0	0
$289$	−6323.00	−1.28699
$290$	0	0
$291$	0	0
$292$	0	0
$293$	7994.00i	1.59391i	0.604041	+	0.796953i	$0.293556\pi$
$293$	7994.00i	1.59391i	−0.604041	+	0.796953i	$0.706444\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−672.000	−0.129976
$300$	0	0
$301$	18224.0	3.48975
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 736.000i	− 0.136827i	−0.997657	−	0.0684133i	$-0.978206\pi$
$307$	− 736.000i	− 0.136827i	0.997657	−	0.0684133i	$-0.0217936\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−5380.00	−0.980938	−0.490469	−	0.871459i	$-0.663175\pi$
$311$	−5380.00	−0.980938	−0.490469	+	0.871459i	$0.663175\pi$
$312$	0	0
$313$	− 1370.00i	− 0.247402i	−0.992320	−	0.123701i	$-0.960524\pi$
$313$	− 1370.00i	− 0.247402i	0.992320	−	0.123701i	$-0.0394764\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 5770.00i	− 1.02232i	−0.859486	−	0.511160i	$-0.829216\pi$
$317$	− 5770.00i	− 1.02232i	0.859486	−	0.511160i	$-0.170784\pi$
$318$	0	0
$319$	−4860.00	−0.853002
$320$	0	0
$321$	0	0
$322$	0	0
$323$	4664.00i	0.803442i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	18224.0	3.05387
$330$	0	0
$331$	−4172.00	−0.692791	−0.346396	−	0.938089i	$-0.612594\pi$
$331$	−4172.00	−0.692791	−0.346396	+	0.938089i	$0.612594\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 8206.00i	− 1.32644i	−0.748426	−	0.663219i	$-0.769190\pi$
$337$	− 8206.00i	− 1.32644i	0.748426	−	0.663219i	$-0.230810\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	3672.00	0.583138
$342$	0	0
$343$	15980.0i	2.51557i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 10848.0i	− 1.67825i	−0.543942	−	0.839123i	$-0.683069\pi$
$347$	− 10848.0i	− 1.67825i	0.543942	−	0.839123i	$-0.316931\pi$
$348$	0	0
$349$	1694.00	0.259822	0.129911	−	0.991526i	$-0.458531\pi$
$349$	1694.00	0.259822	0.129911	+	0.991526i	$0.458531\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 6642.00i	− 1.00147i	−0.865601	−	0.500734i	$-0.833064\pi$
$353$	− 6642.00i	− 1.00147i	0.865601	−	0.500734i	$-0.166936\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	10376.0	1.52542	0.762708	−	0.646743i	$-0.223869\pi$
$359$	10376.0	1.52542	0.762708	+	0.646743i	$0.223869\pi$
$360$	0	0
$361$	−4923.00	−0.717743
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	2198.00i	0.312629i	0.987707	+	0.156314i	$0.0499613\pi$
$367$	2198.00i	0.312629i	−0.987707	+	0.156314i	$0.950039\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	18428.0	2.57880
$372$	0	0
$373$	− 12220.0i	− 1.69632i	−0.529740	−	0.848160i	$-0.677710\pi$
$373$	− 12220.0i	− 1.69632i	0.529740	−	0.848160i	$-0.322290\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 3240.00i	− 0.442622i
$378$	0	0
$379$	10388.0	1.40790	0.703952	−	0.710247i	$-0.251417\pi$
$379$	10388.0	1.40790	0.703952	+	0.710247i	$0.251417\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	10552.0i	1.40779i	0.710306	+	0.703893i	$0.248557\pi$
$383$	10552.0i	1.40779i	−0.710306	+	0.703893i	$0.751443\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	8262.00	1.07686	0.538432	−	0.842669i	$-0.319017\pi$
$389$	8262.00	1.07686	0.538432	+	0.842669i	$0.319017\pi$
$390$	0	0
$391$	−5936.00	−0.767766
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 2864.00i	− 0.362066i	−0.983477	−	0.181033i	$-0.942056\pi$
$397$	− 2864.00i	− 0.362066i	0.983477	−	0.181033i	$-0.0579440\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−12588.0	−1.56762	−0.783809	−	0.621002i	$-0.786726\pi$
$401$	−12588.0	−1.56762	−0.783809	+	0.621002i	$0.786726\pi$
$402$	0	0
$403$	2448.00i	0.302589i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 2160.00i	− 0.263064i
$408$	0	0
$409$	−10330.0	−1.24886	−0.624432	−	0.781079i	$-0.714670\pi$
$409$	−10330.0	−1.24886	−0.624432	+	0.781079i	$0.714670\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 5916.00i	− 0.704860i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	1250.00	0.145743	0.0728717	−	0.997341i	$-0.476784\pi$
$419$	1250.00	0.145743	0.0728717	+	0.997341i	$0.476784\pi$
$420$	0	0
$421$	5670.00	0.656387	0.328193	−	0.944611i	$-0.393560\pi$
$421$	5670.00	0.656387	0.328193	+	0.944611i	$0.393560\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 6324.00i	− 0.716721i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−12976.0	−1.45019	−0.725095	−	0.688649i	$-0.758204\pi$
$431$	−12976.0	−1.45019	−0.725095	+	0.688649i	$0.758204\pi$
$432$	0	0
$433$	− 9050.00i	− 1.00442i	−0.864745	−	0.502212i	$-0.832520\pi$
$433$	− 9050.00i	− 1.00442i	0.864745	−	0.502212i	$-0.167480\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2464.00i	0.269723i
$438$	0	0
$439$	17528.0	1.90562	0.952808	−	0.303572i	$-0.0981794\pi$
$439$	17528.0	1.90562	0.952808	+	0.303572i	$0.0981794\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	2568.00i	0.275416i	0.990473	+	0.137708i	$0.0439736\pi$
$443$	2568.00i	0.275416i	−0.990473	+	0.137708i	$0.956026\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−12652.0	−1.32981	−0.664905	−	0.746928i	$-0.731528\pi$
$449$	−12652.0	−1.32981	−0.664905	+	0.746928i	$0.731528\pi$
$450$	0	0
$451$	1440.00	0.150348
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	6230.00i	0.637696i	0.947806	+	0.318848i	$0.103296\pi$
$457$	6230.00i	0.637696i	−0.947806	+	0.318848i	$0.896704\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	5290.00	0.534447	0.267223	−	0.963635i	$-0.413894\pi$
$461$	5290.00	0.534447	0.267223	+	0.963635i	$0.413894\pi$
$462$	0	0
$463$	8110.00i	0.814047i	0.913418	+	0.407023i	$0.133433\pi$
$463$	8110.00i	0.814047i	−0.913418	+	0.407023i	$0.866567\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	2020.00i	0.200159i	0.994979	+	0.100080i	$0.0319098\pi$
$467$	2020.00i	0.200159i	−0.994979	+	0.100080i	$0.968090\pi$
$468$	0	0
$469$	−11288.0	−1.11137
$470$	0	0
$471$	0	0
$472$	0	0
$473$	9648.00i	0.937876i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	9684.00	0.923744	0.461872	−	0.886947i	$-0.347178\pi$
$479$	9684.00	0.923744	0.461872	+	0.886947i	$0.347178\pi$
$480$	0	0
$481$	1440.00	0.136504
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	18426.0i	1.71450i	0.514900	+	0.857250i	$0.327829\pi$
$487$	18426.0i	1.71450i	−0.514900	+	0.857250i	$0.672171\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	4558.00	0.418940	0.209470	−	0.977815i	$-0.432826\pi$
$491$	4558.00	0.418940	0.209470	+	0.977815i	$0.432826\pi$
$492$	0	0
$493$	− 28620.0i	− 2.61456i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	4488.00i	0.405059i
$498$	0	0
$499$	460.000	0.0412674	0.0206337	−	0.999787i	$-0.493432\pi$
$499$	460.000	0.0412674	0.0206337	+	0.999787i	$0.493432\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 8568.00i	− 0.759499i	−0.925089	−	0.379750i	$-0.876010\pi$
$503$	− 8568.00i	− 0.759499i	0.925089	−	0.379750i	$-0.123990\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−16374.0	−1.42586	−0.712932	−	0.701233i	$-0.752633\pi$
$509$	−16374.0	−1.42586	−0.712932	+	0.701233i	$0.752633\pi$
$510$	0	0
$511$	−20468.0	−1.77192
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	9648.00i	0.820732i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	21620.0	1.81802	0.909011	−	0.416772i	$-0.136839\pi$
$521$	21620.0	1.81802	0.909011	+	0.416772i	$0.136839\pi$
$522$	0	0
$523$	16524.0i	1.38154i	0.723076	+	0.690769i	$0.242728\pi$
$523$	16524.0i	1.38154i	−0.723076	+	0.690769i	$0.757272\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	21624.0i	1.78739i
$528$	0	0
$529$	9031.00	0.742254
$530$	0	0
$531$	0	0
$532$	0	0
$533$	960.000i	0.0780154i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−14634.0	−1.16945
$540$	0	0
$541$	−4990.00	−0.396556	−0.198278	−	0.980146i	$-0.563535\pi$
$541$	−4990.00	−0.396556	−0.198278	+	0.980146i	$0.563535\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	15224.0i	1.19000i	0.803725	+	0.595001i	$0.202848\pi$
$547$	15224.0i	1.19000i	−0.803725	+	0.595001i	$0.797152\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−11880.0	−0.918521
$552$	0	0
$553$	− 18632.0i	− 1.43275i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	5698.00i	0.433451i	0.976233	+	0.216725i	$0.0695376\pi$
$557$	5698.00i	0.433451i	−0.976233	+	0.216725i	$0.930462\pi$
$558$	0	0
$559$	−6432.00	−0.486663
$560$	0	0
$561$	0	0
$562$	0	0
$563$	5976.00i	0.447351i	0.974664	+	0.223675i	$0.0718055\pi$
$563$	5976.00i	0.447351i	−0.974664	+	0.223675i	$0.928194\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−16460.0	−1.21272	−0.606361	−	0.795189i	$-0.707371\pi$
$569$	−16460.0	−1.21272	−0.606361	+	0.795189i	$0.707371\pi$
$570$	0	0
$571$	−18236.0	−1.33652	−0.668260	−	0.743928i	$-0.732961\pi$
$571$	−18236.0	−1.33652	−0.668260	+	0.743928i	$0.732961\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 20842.0i	− 1.50375i	−0.659306	−	0.751875i	$-0.729150\pi$
$577$	− 20842.0i	− 1.50375i	0.659306	−	0.751875i	$-0.270850\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−16728.0	−1.19448
$582$	0	0
$583$	9756.00i	0.693057i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	11772.0i	0.827738i	0.910336	+	0.413869i	$0.135823\pi$
$587$	11772.0i	0.827738i	−0.910336	+	0.413869i	$0.864177\pi$
$588$	0	0
$589$	8976.00	0.627928
$590$	0	0
$591$	0	0
$592$	0	0
$593$	4514.00i	0.312593i	0.987710	+	0.156297i	$0.0499556\pi$
$593$	4514.00i	0.312593i	−0.987710	+	0.156297i	$0.950044\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−25096.0	−1.71184	−0.855922	−	0.517105i	$-0.827010\pi$
$599$	−25096.0	−1.71184	−0.855922	+	0.517105i	$0.827010\pi$
$600$	0	0
$601$	16262.0	1.10373	0.551864	−	0.833934i	$-0.313917\pi$
$601$	16262.0	1.10373	0.551864	+	0.833934i	$0.313917\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 2262.00i	− 0.151255i	−0.997136	−	0.0756275i	$-0.975904\pi$
$607$	− 2262.00i	− 0.151255i	0.997136	−	0.0756275i	$-0.0240960\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−6432.00	−0.425877
$612$	0	0
$613$	14216.0i	0.936670i	0.883551	+	0.468335i	$0.155146\pi$
$613$	14216.0i	0.936670i	−0.883551	+	0.468335i	$0.844854\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 2558.00i	− 0.166906i	−0.996512	−	0.0834532i	$-0.973405\pi$
$617$	− 2558.00i	− 0.166906i	0.996512	−	0.0834532i	$-0.0265949\pi$
$618$	0	0
$619$	17044.0	1.10671	0.553357	−	0.832944i	$-0.313346\pi$
$619$	17044.0	1.10671	0.553357	+	0.832944i	$0.313346\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 35768.0i	− 2.30018i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	12720.0	0.806327
$630$	0	0
$631$	20980.0	1.32361	0.661807	−	0.749674i	$-0.269790\pi$
$631$	20980.0	1.32361	0.661807	+	0.749674i	$0.269790\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 9756.00i	− 0.606824i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	7176.00	0.442176	0.221088	−	0.975254i	$-0.429039\pi$
$641$	7176.00	0.442176	0.221088	+	0.975254i	$0.429039\pi$
$642$	0	0
$643$	− 2724.00i	− 0.167067i	−0.996505	−	0.0835335i	$-0.973379\pi$
$643$	− 2724.00i	− 0.167067i	0.996505	−	0.0835335i	$-0.0266206\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	10392.0i	0.631455i	0.948850	+	0.315728i	$0.102249\pi$
$647$	10392.0i	0.631455i	−0.948850	+	0.315728i	$0.897751\pi$
$648$	0	0
$649$	3132.00	0.189433
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 11958.0i	− 0.716620i	−0.933603	−	0.358310i	$-0.883353\pi$
$653$	− 11958.0i	− 0.716620i	0.933603	−	0.358310i	$-0.116647\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−13366.0	−0.790084	−0.395042	−	0.918663i	$-0.629270\pi$
$659$	−13366.0	−0.790084	−0.395042	+	0.918663i	$0.629270\pi$
$660$	0	0
$661$	14698.0	0.864880	0.432440	−	0.901663i	$-0.357653\pi$
$661$	14698.0	0.864880	0.432440	+	0.901663i	$0.357653\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 15120.0i	− 0.877734i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	3348.00	0.192620
$672$	0	0
$673$	− 7570.00i	− 0.433584i	−0.976218	−	0.216792i	$-0.930441\pi$
$673$	− 7570.00i	− 0.433584i	0.976218	−	0.216792i	$-0.0695593\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 21378.0i	− 1.21362i	−0.794845	−	0.606812i	$-0.792448\pi$
$677$	− 21378.0i	− 1.21362i	0.794845	−	0.606812i	$-0.207552\pi$
$678$	0	0
$679$	−16388.0	−0.926235
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 15804.0i	− 0.885393i	−0.896672	−	0.442696i	$-0.854022\pi$
$683$	− 15804.0i	− 0.885393i	0.896672	−	0.442696i	$-0.145978\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−6504.00	−0.359627
$690$	0	0
$691$	−22028.0	−1.21271	−0.606356	−	0.795193i	$-0.707370\pi$
$691$	−22028.0	−1.21271	−0.606356	+	0.795193i	$0.707370\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	8480.00i	0.460836i
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−1762.00	−0.0949356	−0.0474678	−	0.998873i	$-0.515115\pi$
$701$	−1762.00	−0.0949356	−0.0474678	+	0.998873i	$0.515115\pi$
$702$	0	0
$703$	− 5280.00i	− 0.283270i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 41276.0i	− 2.19568i
$708$	0	0
$709$	2474.00	0.131048	0.0655240	−	0.997851i	$-0.479128\pi$
$709$	2474.00	0.131048	0.0655240	+	0.997851i	$0.479128\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	11424.0i	0.600045i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	32040.0	1.66188	0.830939	−	0.556363i	$-0.187804\pi$
$719$	32040.0	1.66188	0.830939	+	0.556363i	$0.187804\pi$
$720$	0	0
$721$	30532.0	1.57708
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	12874.0i	0.656768i	0.944544	+	0.328384i	$0.106504\pi$
$727$	12874.0i	0.656768i	−0.944544	+	0.328384i	$0.893496\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−56816.0	−2.87471
$732$	0	0
$733$	28208.0i	1.42140i	0.703495	+	0.710700i	$0.251622\pi$
$733$	28208.0i	1.42140i	−0.703495	+	0.710700i	$0.748378\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 5976.00i	− 0.298682i
$738$	0	0
$739$	−29068.0	−1.44693	−0.723467	−	0.690359i	$-0.757452\pi$
$739$	−29068.0	−1.44693	−0.723467	+	0.690359i	$0.757452\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 28152.0i	− 1.39004i	−0.718992	−	0.695018i	$-0.755396\pi$
$743$	− 28152.0i	− 1.39004i	0.718992	−	0.695018i	$-0.244604\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−46376.0	−2.26241
$750$	0	0
$751$	−29916.0	−1.45360	−0.726798	−	0.686851i	$-0.758992\pi$
$751$	−29916.0	−1.45360	−0.726798	+	0.686851i	$0.758992\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 32904.0i	− 1.57981i	−0.613229	−	0.789905i	$-0.710130\pi$
$757$	− 32904.0i	− 1.57981i	0.613229	−	0.789905i	$-0.289870\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−21764.0	−1.03672	−0.518360	−	0.855162i	$-0.673457\pi$
$761$	−21764.0	−1.03672	−0.518360	+	0.855162i	$0.673457\pi$
$762$	0	0
$763$	7412.00i	0.351681i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2088.00i	0.0982964i
$768$	0	0
$769$	3570.00	0.167409	0.0837045	−	0.996491i	$-0.473325\pi$
$769$	3570.00	0.167409	0.0837045	+	0.996491i	$0.473325\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 19486.0i	− 0.906679i	−0.891338	−	0.453339i	$-0.850233\pi$
$773$	− 19486.0i	− 0.906679i	0.891338	−	0.453339i	$-0.149767\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	3520.00	0.161896
$780$	0	0
$781$	−2376.00	−0.108860
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 19764.0i	− 0.895185i	−0.894238	−	0.447592i	$-0.852282\pi$
$787$	− 19764.0i	− 0.895185i	0.894238	−	0.447592i	$-0.147718\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	47124.0	2.11825
$792$	0	0
$793$	2232.00i	0.0999504i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 14390.0i	− 0.639548i	−0.947494	−	0.319774i	$-0.896393\pi$
$797$	− 14390.0i	− 0.639548i	0.947494	−	0.319774i	$-0.103607\pi$
$798$	0	0
$799$	−56816.0	−2.51565
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 10836.0i	− 0.476207i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−28536.0	−1.24014	−0.620069	−	0.784547i	$-0.712896\pi$
$809$	−28536.0	−1.24014	−0.620069	+	0.784547i	$0.712896\pi$
$810$	0	0
$811$	27732.0	1.20074	0.600371	−	0.799721i	$-0.295019\pi$
$811$	27732.0	1.20074	0.600371	+	0.799721i	$0.295019\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	23584.0i	1.00991i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	8086.00	0.343731	0.171866	−	0.985120i	$-0.445021\pi$
$821$	8086.00	0.343731	0.171866	+	0.985120i	$0.445021\pi$
$822$	0	0
$823$	39854.0i	1.68800i	0.536344	+	0.843999i	$0.319805\pi$
$823$	39854.0i	1.68800i	−0.536344	+	0.843999i	$0.680195\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	17752.0i	0.746430i	0.927745	+	0.373215i	$0.121745\pi$
$827$	17752.0i	0.746430i	−0.927745	+	0.373215i	$0.878255\pi$
$828$	0	0
$829$	−23858.0	−0.999545	−0.499772	−	0.866157i	$-0.666583\pi$
$829$	−23858.0	−0.999545	−0.499772	+	0.866157i	$0.666583\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 86178.0i	− 3.58450i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−13888.0	−0.571474	−0.285737	−	0.958308i	$-0.592238\pi$
$839$	−13888.0	−0.571474	−0.285737	+	0.958308i	$0.592238\pi$
$840$	0	0
$841$	48511.0	1.98905
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	34238.0i	1.38894i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	6720.00	0.270692
$852$	0	0
$853$	− 16568.0i	− 0.665038i	−0.943097	−	0.332519i	$-0.892101\pi$
$853$	− 16568.0i	− 0.665038i	0.943097	−	0.332519i	$-0.107899\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	13034.0i	0.519525i	0.965673	+	0.259763i	$0.0836443\pi$
$857$	13034.0i	0.519525i	−0.965673	+	0.259763i	$0.916356\pi$
$858$	0	0
$859$	34356.0	1.36462	0.682312	−	0.731061i	$-0.260975\pi$
$859$	34356.0	1.36462	0.682312	+	0.731061i	$0.260975\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	16016.0i	0.631739i	0.948803	+	0.315870i	$0.102296\pi$
$863$	16016.0i	0.631739i	−0.948803	+	0.315870i	$0.897704\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	9864.00	0.385056
$870$	0	0
$871$	3984.00	0.154986
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 19780.0i	− 0.761600i	−0.924657	−	0.380800i	$-0.875649\pi$
$877$	− 19780.0i	− 0.761600i	0.924657	−	0.380800i	$-0.124351\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−41036.0	−1.56928	−0.784641	−	0.619950i	$-0.787153\pi$
$881$	−41036.0	−1.56928	−0.784641	+	0.619950i	$0.787153\pi$
$882$	0	0
$883$	− 35108.0i	− 1.33803i	−0.743250	−	0.669014i	$-0.766717\pi$
$883$	− 35108.0i	− 1.33803i	0.743250	−	0.669014i	$-0.233283\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 18648.0i	− 0.705906i	−0.935641	−	0.352953i	$-0.885178\pi$
$887$	− 18648.0i	− 0.705906i	0.935641	−	0.352953i	$-0.114822\pi$
$888$	0	0
$889$	27676.0	1.04412
$890$	0	0
$891$	0	0
$892$	0	0
$893$	23584.0i	0.883772i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−55080.0	−2.04340
$900$	0	0
$901$	−57452.0	−2.12431
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 21688.0i	− 0.793978i	−0.917823	−	0.396989i	$-0.870055\pi$
$907$	− 21688.0i	− 0.793978i	0.917823	−	0.396989i	$-0.129945\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−42064.0	−1.52979	−0.764897	−	0.644153i	$-0.777210\pi$
$911$	−42064.0	−1.52979	−0.764897	+	0.644153i	$0.777210\pi$
$912$	0	0
$913$	− 8856.00i	− 0.321020i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	43588.0i	1.56969i
$918$	0	0
$919$	44420.0	1.59443	0.797215	−	0.603696i	$-0.206306\pi$
$919$	44420.0	1.59443	0.797215	+	0.603696i	$0.206306\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 1584.00i	− 0.0564875i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	17124.0	0.604758	0.302379	−	0.953188i	$-0.402219\pi$
$929$	17124.0	0.604758	0.302379	+	0.953188i	$0.402219\pi$
$930$	0	0
$931$	−35772.0	−1.25927
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	11110.0i	0.387351i	0.981066	+	0.193675i	$0.0620409\pi$
$937$	11110.0i	0.387351i	−0.981066	+	0.193675i	$0.937959\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	12962.0	0.449043	0.224521	−	0.974469i	$-0.427918\pi$
$941$	12962.0	0.449043	0.224521	+	0.974469i	$0.427918\pi$
$942$	0	0
$943$	4480.00i	0.154707i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	25672.0i	0.880916i	0.897773	+	0.440458i	$0.145184\pi$
$947$	25672.0i	0.880916i	−0.897773	+	0.440458i	$0.854816\pi$
$948$	0	0
$949$	7224.00	0.247103
$950$	0	0
$951$	0	0
$952$	0	0
$953$	2082.00i	0.0707687i	0.999374	+	0.0353844i	$0.0112655\pi$
$953$	2082.00i	0.0707687i	−0.999374	+	0.0353844i	$0.988734\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−104244.	−3.51013
$960$	0	0
$961$	11825.0	0.396932
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 5666.00i	− 0.188424i	−0.995552	−	0.0942121i	$-0.969967\pi$
$967$	− 5666.00i	− 0.188424i	0.995552	−	0.0942121i	$-0.0300332\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	28622.0	0.945956	0.472978	−	0.881074i	$-0.343179\pi$
$971$	28622.0	0.945956	0.472978	+	0.881074i	$0.343179\pi$
$972$	0	0
$973$	− 45288.0i	− 1.49215i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	24586.0i	0.805093i	0.915400	+	0.402546i	$0.131875\pi$
$977$	24586.0i	0.805093i	−0.915400	+	0.402546i	$0.868125\pi$
$978$	0	0
$979$	18936.0	0.618179
$980$	0	0
$981$	0	0
$982$	0	0
$983$	40632.0i	1.31837i	0.751980	+	0.659186i	$0.229099\pi$
$983$	40632.0i	1.31837i	−0.751980	+	0.659186i	$0.770901\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−30016.0	−0.965069
$990$	0	0
$991$	8768.00	0.281054	0.140527	−	0.990077i	$-0.455120\pi$
$991$	8768.00	0.281054	0.140527	+	0.990077i	$0.455120\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 37212.0i	− 1.18206i	−0.806649	−	0.591031i	$-0.798721\pi$
$997$	− 37212.0i	− 1.18206i	0.806649	−	0.591031i	$-0.201279\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1800.4.f.o.649.1		2
3.2	odd	2		1800.4.f.i.649.1		2
5.2	odd	4		360.4.a.g.1.1	✓	1
5.3	odd	4		1800.4.a.b.1.1		1
5.4	even	2	inner	1800.4.f.o.649.2		2
15.2	even	4		360.4.a.o.1.1	yes	1
15.8	even	4		1800.4.a.a.1.1		1
15.14	odd	2		1800.4.f.i.649.2		2
20.7	even	4		720.4.a.a.1.1		1
60.47	odd	4		720.4.a.p.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.4.a.g.1.1	✓	1	5.2	odd	4
360.4.a.o.1.1	yes	1	15.2	even	4
720.4.a.a.1.1		1	20.7	even	4
720.4.a.p.1.1		1	60.47	odd	4
1800.4.a.a.1.1		1	15.8	even	4
1800.4.a.b.1.1		1	5.3	odd	4
1800.4.f.i.649.1		2	3.2	odd	2
1800.4.f.i.649.2		2	15.14	odd	2
1800.4.f.o.649.1		2	1.1	even	1	trivial
1800.4.f.o.649.2		2	5.4	even	2	inner

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)

\(f(q)\)	\(=\)	\(q-34.0000i q^{7} +O(q^{10})\)	\(q-34.0000i q^{7} +18.0000 q^{11} +12.0000i q^{13} +106.000i q^{17} +44.0000 q^{19} +56.0000i q^{23} -270.000 q^{29} +204.000 q^{31} -120.000i q^{37} +80.0000 q^{41} +536.000i q^{43} +536.000i q^{47} -813.000 q^{49} +542.000i q^{53} +174.000 q^{59} +186.000 q^{61} -332.000i q^{67} -132.000 q^{71} -602.000i q^{73} -612.000i q^{77} +548.000 q^{79} -492.000i q^{83} +1052.00 q^{89} +408.000 q^{91} -482.000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q+O(q^{10}) \) 2 * q	\( 2 q + 36 q^{11} + 88 q^{19} - 540 q^{29} + 408 q^{31} + 160 q^{41} - 1626 q^{49} + 348 q^{59} + 372 q^{61} - 264 q^{71} + 1096 q^{79} + 2104 q^{89} + 816 q^{91}+O(q^{100}) \) 2 * q + 36 * q^11 + 88 * q^19 - 540 * q^29 + 408 * q^31 + 160 * q^41 - 1626 * q^49 + 348 * q^59 + 372 * q^61 - 264 * q^71 + 1096 * q^79 + 2104 * q^89 + 816 * q^91

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