LMFDB - Embedded newform 1800.4.f.b.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 72)
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.b.649.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-12.0000i q^{7} +O(q^{10})$	$q-12.0000i q^{7} -64.0000 q^{11} -58.0000i q^{13} -32.0000i q^{17} +136.000 q^{19} -128.000i q^{23} -144.000 q^{29} +20.0000 q^{31} -18.0000i q^{37} +288.000 q^{41} +200.000i q^{43} -384.000i q^{47} +199.000 q^{49} +496.000i q^{53} -128.000 q^{59} -458.000 q^{61} -496.000i q^{67} -512.000 q^{71} +602.000i q^{73} +768.000i q^{77} -1108.00 q^{79} +704.000i q^{83} -960.000 q^{89} -696.000 q^{91} +206.000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q+O(q^{10}) $ 2 * q	$ 2 q - 128 q^{11} + 272 q^{19} - 288 q^{29} + 40 q^{31} + 576 q^{41} + 398 q^{49} - 256 q^{59} - 916 q^{61} - 1024 q^{71} - 2216 q^{79} - 1920 q^{89} - 1392 q^{91}+O(q^{100}) $ 2 * q - 128 * q^11 + 272 * q^19 - 288 * q^29 + 40 * q^31 + 576 * q^41 + 398 * q^49 - 256 * q^59 - 916 * q^61 - 1024 * q^71 - 2216 * q^79 - 1920 * q^89 - 1392 * 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
$7$	− 12.0000i	− 0.647939i	−0.946068	−	0.323970i	$-0.894982\pi$
$7$	− 12.0000i	− 0.647939i	0.946068	−	0.323970i	$-0.105018\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−64.0000	−1.75425	−0.877124	−	0.480264i	$-0.840541\pi$
$11$	−64.0000	−1.75425	−0.877124	+	0.480264i	$0.840541\pi$
$12$	0	0
$13$	− 58.0000i	− 1.23741i	−0.785624	−	0.618704i	$-0.787658\pi$
$13$	− 58.0000i	− 1.23741i	0.785624	−	0.618704i	$-0.212342\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 32.0000i	− 0.456538i	−0.973598	−	0.228269i	$-0.926693\pi$
$17$	− 32.0000i	− 0.456538i	0.973598	−	0.228269i	$-0.0733065\pi$
$18$	0	0
$19$	136.000	1.64213	0.821067	−	0.570832i	$-0.193379\pi$
$19$	136.000	1.64213	0.821067	+	0.570832i	$0.193379\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 128.000i	− 1.16043i	−0.814464	−	0.580214i	$-0.802969\pi$
$23$	− 128.000i	− 1.16043i	0.814464	−	0.580214i	$-0.197031\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−144.000	−0.922073	−0.461037	−	0.887381i	$-0.652522\pi$
$29$	−144.000	−0.922073	−0.461037	+	0.887381i	$0.652522\pi$
$30$	0	0
$31$	20.0000	0.115874	0.0579372	−	0.998320i	$-0.481548\pi$
$31$	20.0000	0.115874	0.0579372	+	0.998320i	$0.481548\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 18.0000i	− 0.0799779i	−0.999200	−	0.0399889i	$-0.987268\pi$
$37$	− 18.0000i	− 0.0799779i	0.999200	−	0.0399889i	$-0.0127323\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	288.000	1.09703	0.548513	−	0.836142i	$-0.315194\pi$
$41$	288.000	1.09703	0.548513	+	0.836142i	$0.315194\pi$
$42$	0	0
$43$	200.000i	0.709296i	0.935000	+	0.354648i	$0.115399\pi$
$43$	200.000i	0.709296i	−0.935000	+	0.354648i	$0.884601\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 384.000i	− 1.19175i	−0.803078	−	0.595874i	$-0.796806\pi$
$47$	− 384.000i	− 1.19175i	0.803078	−	0.595874i	$-0.203194\pi$
$48$	0	0
$49$	199.000	0.580175
$50$	0	0
$51$	0	0
$52$	0	0
$53$	496.000i	1.28549i	0.766081	+	0.642744i	$0.222204\pi$
$53$	496.000i	1.28549i	−0.766081	+	0.642744i	$0.777796\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−128.000	−0.282444	−0.141222	−	0.989978i	$-0.545103\pi$
$59$	−128.000	−0.282444	−0.141222	+	0.989978i	$0.545103\pi$
$60$	0	0
$61$	−458.000	−0.961326	−0.480663	−	0.876905i	$-0.659604\pi$
$61$	−458.000	−0.961326	−0.480663	+	0.876905i	$0.659604\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 496.000i	− 0.904419i	−0.891912	−	0.452209i	$-0.850636\pi$
$67$	− 496.000i	− 0.904419i	0.891912	−	0.452209i	$-0.149364\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−512.000	−0.855820	−0.427910	−	0.903821i	$-0.640750\pi$
$71$	−512.000	−0.855820	−0.427910	+	0.903821i	$0.640750\pi$
$72$	0	0
$73$	602.000i	0.965189i	0.875844	+	0.482594i	$0.160305\pi$
$73$	602.000i	0.965189i	−0.875844	+	0.482594i	$0.839695\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	768.000i	1.13665i
$78$	0	0
$79$	−1108.00	−1.57797	−0.788986	−	0.614412i	$-0.789393\pi$
$79$	−1108.00	−1.57797	−0.788986	+	0.614412i	$0.789393\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	704.000i	0.931013i	0.885045	+	0.465506i	$0.154128\pi$
$83$	704.000i	0.931013i	−0.885045	+	0.465506i	$0.845872\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−960.000	−1.14337	−0.571684	−	0.820474i	$-0.693710\pi$
$89$	−960.000	−1.14337	−0.571684	+	0.820474i	$0.693710\pi$
$90$	0	0
$91$	−696.000	−0.801765
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	206.000i	0.215630i	0.994171	+	0.107815i	$0.0343855\pi$
$97$	206.000i	0.215630i	−0.994171	+	0.107815i	$0.965615\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−432.000	−0.425600	−0.212800	−	0.977096i	$-0.568258\pi$
$101$	−432.000	−0.425600	−0.212800	+	0.977096i	$0.568258\pi$
$102$	0	0
$103$	68.0000i	0.0650509i	0.999471	+	0.0325254i	$0.0103550\pi$
$103$	68.0000i	0.0650509i	−0.999471	+	0.0325254i	$0.989645\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	384.000i	0.346941i	0.984839	+	0.173470i	$0.0554981\pi$
$107$	384.000i	0.346941i	−0.984839	+	0.173470i	$0.944502\pi$
$108$	0	0
$109$	518.000	0.455187	0.227594	−	0.973756i	$-0.426914\pi$
$109$	518.000	0.455187	0.227594	+	0.973756i	$0.426914\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 960.000i	− 0.799196i	−0.916690	−	0.399598i	$-0.869150\pi$
$113$	− 960.000i	− 0.799196i	0.916690	−	0.399598i	$-0.130850\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−384.000	−0.295809
$120$	0	0
$121$	2765.00	2.07739
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	796.000i	0.556170i	0.960556	+	0.278085i	$0.0896997\pi$
$127$	796.000i	0.556170i	−0.960556	+	0.278085i	$0.910300\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−512.000	−0.341478	−0.170739	−	0.985316i	$-0.554616\pi$
$131$	−512.000	−0.341478	−0.170739	+	0.985316i	$0.554616\pi$
$132$	0	0
$133$	− 1632.00i	− 1.06400i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 1824.00i	− 1.13748i	−0.822517	−	0.568740i	$-0.807431\pi$
$137$	− 1824.00i	− 1.13748i	0.822517	−	0.568740i	$-0.192569\pi$
$138$	0	0
$139$	2160.00	1.31805	0.659024	−	0.752121i	$-0.270969\pi$
$139$	2160.00	1.31805	0.659024	+	0.752121i	$0.270969\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	3712.00i	2.17072i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−688.000	−0.378276	−0.189138	−	0.981950i	$-0.560569\pi$
$149$	−688.000	−0.378276	−0.189138	+	0.981950i	$0.560569\pi$
$150$	0	0
$151$	−844.000	−0.454859	−0.227430	−	0.973795i	$-0.573032\pi$
$151$	−844.000	−0.454859	−0.227430	+	0.973795i	$0.573032\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	118.000i	0.0599836i	0.999550	+	0.0299918i	$0.00954812\pi$
$157$	118.000i	0.0599836i	−0.999550	+	0.0299918i	$0.990452\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−1536.00	−0.751887
$162$	0	0
$163$	− 3576.00i	− 1.71837i	−0.511667	−	0.859184i	$-0.670972\pi$
$163$	− 3576.00i	− 1.71837i	0.511667	−	0.859184i	$-0.329028\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 384.000i	− 0.177933i	−0.996035	−	0.0889665i	$-0.971644\pi$
$167$	− 384.000i	− 0.177933i	0.996035	−	0.0889665i	$-0.0283564\pi$
$168$	0	0
$169$	−1167.00	−0.531179
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2448.00i	1.07583i	0.843000	+	0.537913i	$0.180787\pi$
$173$	2448.00i	1.07583i	−0.843000	+	0.537913i	$0.819213\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−4224.00	−1.76378	−0.881890	−	0.471455i	$-0.843729\pi$
$179$	−4224.00	−1.76378	−0.881890	+	0.471455i	$0.843729\pi$
$180$	0	0
$181$	−510.000	−0.209436	−0.104718	−	0.994502i	$-0.533394\pi$
$181$	−510.000	−0.209436	−0.104718	+	0.994502i	$0.533394\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	2048.00i	0.800880i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	384.000	0.145473	0.0727363	−	0.997351i	$-0.476827\pi$
$191$	384.000	0.145473	0.0727363	+	0.997351i	$0.476827\pi$
$192$	0	0
$193$	3454.00i	1.28821i	0.764937	+	0.644105i	$0.222770\pi$
$193$	3454.00i	1.28821i	−0.764937	+	0.644105i	$0.777230\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3216.00i	1.16310i	0.813511	+	0.581550i	$0.197553\pi$
$197$	3216.00i	1.16310i	−0.813511	+	0.581550i	$0.802447\pi$
$198$	0	0
$199$	−1708.00	−0.608427	−0.304213	−	0.952604i	$-0.598394\pi$
$199$	−1708.00	−0.608427	−0.304213	+	0.952604i	$0.598394\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1728.00i	0.597447i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−8704.00	−2.88071
$210$	0	0
$211$	2320.00	0.756945	0.378472	−	0.925613i	$-0.376449\pi$
$211$	2320.00	0.756945	0.378472	+	0.925613i	$0.376449\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 240.000i	− 0.0750795i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−1856.00	−0.564923
$222$	0	0
$223$	116.000i	0.0348338i	0.999848	+	0.0174169i	$0.00554425\pi$
$223$	116.000i	0.0348338i	−0.999848	+	0.0174169i	$0.994456\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 1344.00i	− 0.392971i	−0.980507	−	0.196485i	$-0.937047\pi$
$227$	− 1344.00i	− 0.392971i	0.980507	−	0.196485i	$-0.0629529\pi$
$228$	0	0
$229$	−4594.00	−1.32568	−0.662839	−	0.748762i	$-0.730648\pi$
$229$	−4594.00	−1.32568	−0.662839	+	0.748762i	$0.730648\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	5056.00i	1.42159i	0.703401	+	0.710793i	$0.251664\pi$
$233$	5056.00i	1.42159i	−0.703401	+	0.710793i	$0.748336\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−3712.00	−1.00464	−0.502321	−	0.864681i	$-0.667520\pi$
$239$	−3712.00	−1.00464	−0.502321	+	0.864681i	$0.667520\pi$
$240$	0	0
$241$	−978.000	−0.261405	−0.130702	−	0.991422i	$-0.541723\pi$
$241$	−978.000	−0.261405	−0.130702	+	0.991422i	$0.541723\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 7888.00i	− 2.03199i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−1856.00	−0.466732	−0.233366	−	0.972389i	$-0.574974\pi$
$251$	−1856.00	−0.466732	−0.233366	+	0.972389i	$0.574974\pi$
$252$	0	0
$253$	8192.00i	2.03568i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 7808.00i	− 1.89513i	−0.319556	−	0.947567i	$-0.603534\pi$
$257$	− 7808.00i	− 1.89513i	0.319556	−	0.947567i	$-0.396466\pi$
$258$	0	0
$259$	−216.000	−0.0518208
$260$	0	0
$261$	0	0
$262$	0	0
$263$	1024.00i	0.240086i	0.992769	+	0.120043i	$0.0383032\pi$
$263$	1024.00i	0.240086i	−0.992769	+	0.120043i	$0.961697\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	1328.00	0.301002	0.150501	−	0.988610i	$-0.451911\pi$
$269$	1328.00	0.301002	0.150501	+	0.988610i	$0.451911\pi$
$270$	0	0
$271$	−5812.00	−1.30278	−0.651391	−	0.758742i	$-0.725814\pi$
$271$	−5812.00	−1.30278	−0.651391	+	0.758742i	$0.725814\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	8386.00i	1.81901i	0.415692	+	0.909505i	$0.363539\pi$
$277$	8386.00i	1.81901i	−0.415692	+	0.909505i	$0.636461\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	640.000	0.135869	0.0679345	−	0.997690i	$-0.478359\pi$
$281$	640.000	0.135869	0.0679345	+	0.997690i	$0.478359\pi$
$282$	0	0
$283$	− 4832.00i	− 1.01496i	−0.861665	−	0.507478i	$-0.830578\pi$
$283$	− 4832.00i	− 1.01496i	0.861665	−	0.507478i	$-0.169422\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 3456.00i	− 0.710806i
$288$	0	0
$289$	3889.00	0.791573
$290$	0	0
$291$	0	0
$292$	0	0
$293$	6384.00i	1.27289i	0.771321	+	0.636446i	$0.219596\pi$
$293$	6384.00i	1.27289i	−0.771321	+	0.636446i	$0.780404\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−7424.00	−1.43592
$300$	0	0
$301$	2400.00	0.459580
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 3312.00i	− 0.615719i	−0.951432	−	0.307860i	$-0.900387\pi$
$307$	− 3312.00i	− 0.615719i	0.951432	−	0.307860i	$-0.0996127\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−9984.00	−1.82039	−0.910194	−	0.414182i	$-0.864068\pi$
$311$	−9984.00	−1.82039	−0.910194	+	0.414182i	$0.864068\pi$
$312$	0	0
$313$	− 2586.00i	− 0.466995i	−0.972357	−	0.233497i	$-0.924983\pi$
$313$	− 2586.00i	− 0.466995i	0.972357	−	0.233497i	$-0.0750170\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2832.00i	0.501770i	0.968017	+	0.250885i	$0.0807215\pi$
$317$	2832.00i	0.501770i	−0.968017	+	0.250885i	$0.919278\pi$
$318$	0	0
$319$	9216.00	1.61755
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 4352.00i	− 0.749696i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−4608.00	−0.772180
$330$	0	0
$331$	−5920.00	−0.983059	−0.491530	−	0.870861i	$-0.663562\pi$
$331$	−5920.00	−0.983059	−0.491530	+	0.870861i	$0.663562\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 4674.00i	− 0.755516i	−0.925904	−	0.377758i	$-0.876695\pi$
$337$	− 4674.00i	− 0.755516i	0.925904	−	0.377758i	$-0.123305\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−1280.00	−0.203272
$342$	0	0
$343$	− 6504.00i	− 1.02386i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	9024.00i	1.39606i	0.716067	+	0.698031i	$0.245940\pi$
$347$	9024.00i	1.39606i	−0.716067	+	0.698031i	$0.754060\pi$
$348$	0	0
$349$	4362.00	0.669033	0.334516	−	0.942390i	$-0.391427\pi$
$349$	4362.00	0.669033	0.334516	+	0.942390i	$0.391427\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	8768.00i	1.32202i	0.750376	+	0.661011i	$0.229872\pi$
$353$	8768.00i	1.32202i	−0.750376	+	0.661011i	$0.770128\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−6144.00	−0.903253	−0.451627	−	0.892207i	$-0.649156\pi$
$359$	−6144.00	−0.903253	−0.451627	+	0.892207i	$0.649156\pi$
$360$	0	0
$361$	11637.0	1.69660
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	4564.00i	0.649152i	0.945860	+	0.324576i	$0.105222\pi$
$367$	4564.00i	0.649152i	−0.945860	+	0.324576i	$0.894778\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	5952.00	0.832918
$372$	0	0
$373$	8770.00i	1.21741i	0.793397	+	0.608704i	$0.208310\pi$
$373$	8770.00i	1.21741i	−0.793397	+	0.608704i	$0.791690\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	8352.00i	1.14098i
$378$	0	0
$379$	1096.00	0.148543	0.0742714	−	0.997238i	$-0.476337\pi$
$379$	1096.00	0.148543	0.0742714	+	0.997238i	$0.476337\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	10368.0i	1.38324i	0.722263	+	0.691619i	$0.243102\pi$
$383$	10368.0i	1.38324i	−0.722263	+	0.691619i	$0.756898\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−3248.00	−0.423342	−0.211671	−	0.977341i	$-0.567891\pi$
$389$	−3248.00	−0.423342	−0.211671	+	0.977341i	$0.567891\pi$
$390$	0	0
$391$	−4096.00	−0.529779
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 6106.00i	− 0.771918i	−0.922516	−	0.385959i	$-0.873871\pi$
$397$	− 6106.00i	− 0.771918i	0.922516	−	0.385959i	$-0.126129\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−7008.00	−0.872725	−0.436363	−	0.899771i	$-0.643734\pi$
$401$	−7008.00	−0.872725	−0.436363	+	0.899771i	$0.643734\pi$
$402$	0	0
$403$	− 1160.00i	− 0.143384i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1152.00i	0.140301i
$408$	0	0
$409$	−1590.00	−0.192226	−0.0961130	−	0.995370i	$-0.530641\pi$
$409$	−1590.00	−0.192226	−0.0961130	+	0.995370i	$0.530641\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	1536.00i	0.183006i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−192.000	−0.0223862	−0.0111931	−	0.999937i	$-0.503563\pi$
$419$	−192.000	−0.0223862	−0.0111931	+	0.999937i	$0.503563\pi$
$420$	0	0
$421$	9074.00	1.05045	0.525225	−	0.850963i	$-0.323981\pi$
$421$	9074.00	1.05045	0.525225	+	0.850963i	$0.323981\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	5496.00i	0.622881i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	5248.00	0.586513	0.293257	−	0.956034i	$-0.405261\pi$
$431$	5248.00	0.586513	0.293257	+	0.956034i	$0.405261\pi$
$432$	0	0
$433$	8222.00i	0.912527i	0.889845	+	0.456263i	$0.150813\pi$
$433$	8222.00i	0.912527i	−0.889845	+	0.456263i	$0.849187\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 17408.0i	− 1.90558i
$438$	0	0
$439$	−16236.0	−1.76515	−0.882576	−	0.470169i	$-0.844193\pi$
$439$	−16236.0	−1.76515	−0.882576	+	0.470169i	$0.844193\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	14528.0i	1.55812i	0.626951	+	0.779059i	$0.284303\pi$
$443$	14528.0i	1.55812i	−0.626951	+	0.779059i	$0.715697\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	6304.00	0.662593	0.331296	−	0.943527i	$-0.392514\pi$
$449$	6304.00	0.662593	0.331296	+	0.943527i	$0.392514\pi$
$450$	0	0
$451$	−18432.0	−1.92445
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	1958.00i	0.200419i	0.994966	+	0.100209i	$0.0319513\pi$
$457$	1958.00i	0.200419i	−0.994966	+	0.100209i	$0.968049\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−4048.00	−0.408968	−0.204484	−	0.978870i	$-0.565552\pi$
$461$	−4048.00	−0.408968	−0.204484	+	0.978870i	$0.565552\pi$
$462$	0	0
$463$	− 16988.0i	− 1.70518i	−0.522579	−	0.852591i	$-0.675030\pi$
$463$	− 16988.0i	− 1.70518i	0.522579	−	0.852591i	$-0.324970\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 6720.00i	− 0.665877i	−0.942949	−	0.332938i	$-0.891960\pi$
$467$	− 6720.00i	− 0.665877i	0.942949	−	0.332938i	$-0.108040\pi$
$468$	0	0
$469$	−5952.00	−0.586008
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 12800.0i	− 1.24428i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	9728.00	0.927941	0.463970	−	0.885851i	$-0.346424\pi$
$479$	9728.00	0.927941	0.463970	+	0.885851i	$0.346424\pi$
$480$	0	0
$481$	−1044.00	−0.0989653
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 8444.00i	− 0.785696i	−0.919603	−	0.392848i	$-0.871490\pi$
$487$	− 8444.00i	− 0.785696i	0.919603	−	0.392848i	$-0.128510\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	15360.0	1.41179	0.705893	−	0.708318i	$-0.250546\pi$
$491$	15360.0	1.41179	0.705893	+	0.708318i	$0.250546\pi$
$492$	0	0
$493$	4608.00i	0.420961i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	6144.00i	0.554519i
$498$	0	0
$499$	−6624.00	−0.594250	−0.297125	−	0.954839i	$-0.596028\pi$
$499$	−6624.00	−0.594250	−0.297125	+	0.954839i	$0.596028\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 6912.00i	− 0.612705i	−0.951918	−	0.306353i	$-0.900891\pi$
$503$	− 6912.00i	− 0.612705i	0.951918	−	0.306353i	$-0.0991087\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−19920.0	−1.73465	−0.867327	−	0.497739i	$-0.834164\pi$
$509$	−19920.0	−1.73465	−0.867327	+	0.497739i	$0.834164\pi$
$510$	0	0
$511$	7224.00	0.625383
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	24576.0i	2.09062i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−3680.00	−0.309451	−0.154725	−	0.987958i	$-0.549449\pi$
$521$	−3680.00	−0.309451	−0.154725	+	0.987958i	$0.549449\pi$
$522$	0	0
$523$	11720.0i	0.979885i	0.871755	+	0.489942i	$0.162982\pi$
$523$	11720.0i	0.979885i	−0.871755	+	0.489942i	$0.837018\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 640.000i	− 0.0529010i
$528$	0	0
$529$	−4217.00	−0.346593
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 16704.0i	− 1.35747i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−12736.0	−1.01777
$540$	0	0
$541$	11754.0	0.934092	0.467046	−	0.884233i	$-0.345318\pi$
$541$	11754.0	0.934092	0.467046	+	0.884233i	$0.345318\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 18904.0i	− 1.47765i	−0.673895	−	0.738827i	$-0.735380\pi$
$547$	− 18904.0i	− 1.47765i	0.673895	−	0.738827i	$-0.264620\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−19584.0	−1.51417
$552$	0	0
$553$	13296.0i	1.02243i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 3088.00i	− 0.234906i	−0.993078	−	0.117453i	$-0.962527\pi$
$557$	− 3088.00i	− 0.234906i	0.993078	−	0.117453i	$-0.0374730\pi$
$558$	0	0
$559$	11600.0	0.877688
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 21440.0i	− 1.60495i	−0.596684	−	0.802476i	$-0.703515\pi$
$563$	− 21440.0i	− 1.60495i	0.596684	−	0.802476i	$-0.296485\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	22624.0	1.66687	0.833434	−	0.552620i	$-0.186372\pi$
$569$	22624.0	1.66687	0.833434	+	0.552620i	$0.186372\pi$
$570$	0	0
$571$	−6000.00	−0.439741	−0.219871	−	0.975529i	$-0.570564\pi$
$571$	−6000.00	−0.439741	−0.219871	+	0.975529i	$0.570564\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	19922.0i	1.43737i	0.695335	+	0.718686i	$0.255256\pi$
$577$	19922.0i	1.43737i	−0.695335	+	0.718686i	$0.744744\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	8448.00	0.603239
$582$	0	0
$583$	− 31744.0i	− 2.25506i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	3584.00i	0.252006i	0.992030	+	0.126003i	$0.0402149\pi$
$587$	3584.00i	0.252006i	−0.992030	+	0.126003i	$0.959785\pi$
$588$	0	0
$589$	2720.00	0.190281
$590$	0	0
$591$	0	0
$592$	0	0
$593$	1984.00i	0.137391i	0.997638	+	0.0686957i	$0.0218838\pi$
$593$	1984.00i	0.137391i	−0.997638	+	0.0686957i	$0.978116\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	14976.0	1.02154	0.510770	−	0.859717i	$-0.329360\pi$
$599$	14976.0	1.02154	0.510770	+	0.859717i	$0.329360\pi$
$600$	0	0
$601$	25738.0	1.74688	0.873440	−	0.486932i	$-0.161884\pi$
$601$	25738.0	1.74688	0.873440	+	0.486932i	$0.161884\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 8548.00i	− 0.571586i	−0.958291	−	0.285793i	$-0.907743\pi$
$607$	− 8548.00i	− 0.571586i	0.958291	−	0.285793i	$-0.0922570\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−22272.0	−1.47468
$612$	0	0
$613$	− 8558.00i	− 0.563873i	−0.959433	−	0.281937i	$-0.909023\pi$
$613$	− 8558.00i	− 0.563873i	0.959433	−	0.281937i	$-0.0909768\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	10368.0i	0.676499i	0.941056	+	0.338250i	$0.109835\pi$
$617$	10368.0i	0.676499i	−0.941056	+	0.338250i	$0.890165\pi$
$618$	0	0
$619$	13088.0	0.849840	0.424920	−	0.905231i	$-0.360302\pi$
$619$	13088.0	0.849840	0.424920	+	0.905231i	$0.360302\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	11520.0i	0.740833i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−576.000	−0.0365129
$630$	0	0
$631$	−4412.00	−0.278350	−0.139175	−	0.990268i	$-0.544445\pi$
$631$	−4412.00	−0.278350	−0.139175	+	0.990268i	$0.544445\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 11542.0i	− 0.717913i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	30176.0	1.85941	0.929704	−	0.368308i	$-0.120063\pi$
$641$	30176.0	1.85941	0.929704	+	0.368308i	$0.120063\pi$
$642$	0	0
$643$	− 21288.0i	− 1.30562i	−0.757520	−	0.652812i	$-0.773589\pi$
$643$	− 21288.0i	− 1.30562i	0.757520	−	0.652812i	$-0.226411\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 17024.0i	− 1.03444i	−0.855853	−	0.517220i	$-0.826967\pi$
$647$	− 17024.0i	− 1.03444i	0.855853	−	0.517220i	$-0.173033\pi$
$648$	0	0
$649$	8192.00	0.495476
$650$	0	0
$651$	0	0
$652$	0	0
$653$	2256.00i	0.135198i	0.997713	+	0.0675989i	$0.0215338\pi$
$653$	2256.00i	0.135198i	−0.997713	+	0.0675989i	$0.978466\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	23808.0	1.40733	0.703663	−	0.710534i	$-0.251546\pi$
$659$	23808.0	1.40733	0.703663	+	0.710534i	$0.251546\pi$
$660$	0	0
$661$	−26242.0	−1.54417	−0.772084	−	0.635520i	$-0.780786\pi$
$661$	−26242.0	−1.54417	−0.772084	+	0.635520i	$0.780786\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	18432.0i	1.07000i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	29312.0	1.68640
$672$	0	0
$673$	24590.0i	1.40843i	0.709986	+	0.704216i	$0.248701\pi$
$673$	24590.0i	1.40843i	−0.709986	+	0.704216i	$0.751299\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	2864.00i	0.162589i	0.996690	+	0.0812943i	$0.0259054\pi$
$677$	2864.00i	0.162589i	−0.996690	+	0.0812943i	$0.974095\pi$
$678$	0	0
$679$	2472.00	0.139715
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 7616.00i	− 0.426674i	−0.976979	−	0.213337i	$-0.931567\pi$
$683$	− 7616.00i	− 0.426674i	0.976979	−	0.213337i	$-0.0684332\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	28768.0	1.59067
$690$	0	0
$691$	2168.00	0.119355	0.0596777	−	0.998218i	$-0.480993\pi$
$691$	2168.00	0.119355	0.0596777	+	0.998218i	$0.480993\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 9216.00i	− 0.500833i
$698$	0	0
$699$	0	0
$700$	0	0
$701$	18000.0	0.969830	0.484915	−	0.874561i	$-0.338851\pi$
$701$	18000.0	0.969830	0.484915	+	0.874561i	$0.338851\pi$
$702$	0	0
$703$	− 2448.00i	− 0.131334i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	5184.00i	0.275763i
$708$	0	0
$709$	−3506.00	−0.185713	−0.0928566	−	0.995679i	$-0.529600\pi$
$709$	−3506.00	−0.185713	−0.0928566	+	0.995679i	$0.529600\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 2560.00i	− 0.134464i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	15616.0	0.809984	0.404992	−	0.914320i	$-0.367274\pi$
$719$	15616.0	0.809984	0.404992	+	0.914320i	$0.367274\pi$
$720$	0	0
$721$	816.000	0.0421490
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 15036.0i	− 0.767062i	−0.923528	−	0.383531i	$-0.874708\pi$
$727$	− 15036.0i	− 0.767062i	0.923528	−	0.383531i	$-0.125292\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	6400.00	0.323820
$732$	0	0
$733$	19126.0i	0.963758i	0.876238	+	0.481879i	$0.160046\pi$
$733$	19126.0i	0.963758i	−0.876238	+	0.481879i	$0.839954\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	31744.0i	1.58657i
$738$	0	0
$739$	17392.0	0.865731	0.432865	−	0.901459i	$-0.357503\pi$
$739$	17392.0	0.865731	0.432865	+	0.901459i	$0.357503\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 32384.0i	− 1.59900i	−0.600669	−	0.799498i	$-0.705099\pi$
$743$	− 32384.0i	− 1.59900i	0.600669	−	0.799498i	$-0.294901\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	4608.00	0.224797
$750$	0	0
$751$	−27708.0	−1.34631	−0.673155	−	0.739501i	$-0.735061\pi$
$751$	−27708.0	−1.34631	−0.673155	+	0.739501i	$0.735061\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 37246.0i	− 1.78828i	−0.447786	−	0.894141i	$-0.647787\pi$
$757$	− 37246.0i	− 1.78828i	0.447786	−	0.894141i	$-0.352213\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	4192.00	0.199684	0.0998422	−	0.995003i	$-0.468166\pi$
$761$	4192.00	0.199684	0.0998422	+	0.995003i	$0.468166\pi$
$762$	0	0
$763$	− 6216.00i	− 0.294934i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	7424.00i	0.349498i
$768$	0	0
$769$	−26882.0	−1.26058	−0.630292	−	0.776358i	$-0.717065\pi$
$769$	−26882.0	−1.26058	−0.630292	+	0.776358i	$0.717065\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 17232.0i	− 0.801801i	−0.916122	−	0.400900i	$-0.868697\pi$
$773$	− 17232.0i	− 0.801801i	0.916122	−	0.400900i	$-0.131303\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	39168.0	1.80146
$780$	0	0
$781$	32768.0	1.50132
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	31816.0i	1.44106i	0.693421	+	0.720532i	$0.256102\pi$
$787$	31816.0i	1.44106i	−0.693421	+	0.720532i	$0.743898\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−11520.0	−0.517831
$792$	0	0
$793$	26564.0i	1.18955i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 8272.00i	− 0.367640i	−0.982960	−	0.183820i	$-0.941154\pi$
$797$	− 8272.00i	− 0.367640i	0.982960	−	0.183820i	$-0.0588464\pi$
$798$	0	0
$799$	−12288.0	−0.544078
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 38528.0i	− 1.69318i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−9184.00	−0.399125	−0.199563	−	0.979885i	$-0.563952\pi$
$809$	−9184.00	−0.399125	−0.199563	+	0.979885i	$0.563952\pi$
$810$	0	0
$811$	−19832.0	−0.858688	−0.429344	−	0.903141i	$-0.641255\pi$
$811$	−19832.0	−0.858688	−0.429344	+	0.903141i	$0.641255\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	27200.0i	1.16476i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−15216.0	−0.646823	−0.323412	−	0.946258i	$-0.604830\pi$
$821$	−15216.0	−0.646823	−0.323412	+	0.946258i	$0.604830\pi$
$822$	0	0
$823$	39772.0i	1.68453i	0.539067	+	0.842263i	$0.318777\pi$
$823$	39772.0i	1.68453i	−0.539067	+	0.842263i	$0.681223\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	18304.0i	0.769640i	0.922992	+	0.384820i	$0.125737\pi$
$827$	18304.0i	0.769640i	−0.922992	+	0.384820i	$0.874263\pi$
$828$	0	0
$829$	−4906.00	−0.205540	−0.102770	−	0.994705i	$-0.532771\pi$
$829$	−4906.00	−0.205540	−0.102770	+	0.994705i	$0.532771\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 6368.00i	− 0.264872i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	15360.0	0.632045	0.316023	−	0.948752i	$-0.397652\pi$
$839$	15360.0	0.632045	0.316023	+	0.948752i	$0.397652\pi$
$840$	0	0
$841$	−3653.00	−0.149781
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 33180.0i	− 1.34602i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−2304.00	−0.0928086
$852$	0	0
$853$	24802.0i	0.995550i	0.867306	+	0.497775i	$0.165850\pi$
$853$	24802.0i	0.995550i	−0.867306	+	0.497775i	$0.834150\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 15072.0i	− 0.600758i	−0.953820	−	0.300379i	$-0.902887\pi$
$857$	− 15072.0i	− 0.600758i	0.953820	−	0.300379i	$-0.0971132\pi$
$858$	0	0
$859$	−1800.00	−0.0714962	−0.0357481	−	0.999361i	$-0.511381\pi$
$859$	−1800.00	−0.0714962	−0.0357481	+	0.999361i	$0.511381\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	7552.00i	0.297883i	0.988846	+	0.148942i	$0.0475866\pi$
$863$	7552.00i	0.297883i	−0.988846	+	0.148942i	$0.952413\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	70912.0	2.76815
$870$	0	0
$871$	−28768.0	−1.11913
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	20838.0i	0.802337i	0.916004	+	0.401168i	$0.131396\pi$
$877$	20838.0i	0.802337i	−0.916004	+	0.401168i	$0.868604\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−47744.0	−1.82581	−0.912904	−	0.408175i	$-0.866165\pi$
$881$	−47744.0	−1.82581	−0.912904	+	0.408175i	$0.866165\pi$
$882$	0	0
$883$	− 28280.0i	− 1.07780i	−0.842370	−	0.538900i	$-0.818840\pi$
$883$	− 28280.0i	− 1.07780i	0.842370	−	0.538900i	$-0.181160\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 7424.00i	− 0.281030i	−0.990079	−	0.140515i	$-0.955124\pi$
$887$	− 7424.00i	− 0.281030i	0.990079	−	0.140515i	$-0.0448758\pi$
$888$	0	0
$889$	9552.00	0.360364
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 52224.0i	− 1.95701i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−2880.00	−0.106845
$900$	0	0
$901$	15872.0	0.586873
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 5912.00i	− 0.216433i	−0.994127	−	0.108217i	$-0.965486\pi$
$907$	− 5912.00i	− 0.216433i	0.994127	−	0.108217i	$-0.0345140\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	26240.0	0.954303	0.477151	−	0.878821i	$-0.341669\pi$
$911$	26240.0	0.954303	0.477151	+	0.878821i	$0.341669\pi$
$912$	0	0
$913$	− 45056.0i	− 1.63323i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	6144.00i	0.221257i
$918$	0	0
$919$	−35620.0	−1.27856	−0.639279	−	0.768975i	$-0.720767\pi$
$919$	−35620.0	−1.27856	−0.639279	+	0.768975i	$0.720767\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	29696.0i	1.05900i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−3232.00	−0.114143	−0.0570713	−	0.998370i	$-0.518176\pi$
$929$	−3232.00	−0.114143	−0.0570713	+	0.998370i	$0.518176\pi$
$930$	0	0
$931$	27064.0	0.952725
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 11478.0i	− 0.400181i	−0.979777	−	0.200091i	$-0.935876\pi$
$937$	− 11478.0i	− 0.400181i	0.979777	−	0.200091i	$-0.0641237\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	39984.0	1.38517	0.692583	−	0.721338i	$-0.256473\pi$
$941$	39984.0	1.38517	0.692583	+	0.721338i	$0.256473\pi$
$942$	0	0
$943$	− 36864.0i	− 1.27302i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 24192.0i	− 0.830131i	−0.909791	−	0.415066i	$-0.863759\pi$
$947$	− 24192.0i	− 0.830131i	0.909791	−	0.415066i	$-0.136241\pi$
$948$	0	0
$949$	34916.0	1.19433
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 39456.0i	− 1.34114i	−0.741847	−	0.670569i	$-0.766050\pi$
$953$	− 39456.0i	− 1.34114i	0.741847	−	0.670569i	$-0.233950\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−21888.0	−0.737018
$960$	0	0
$961$	−29391.0	−0.986573
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 41668.0i	− 1.38568i	−0.721091	−	0.692840i	$-0.756359\pi$
$967$	− 41668.0i	− 1.38568i	0.721091	−	0.692840i	$-0.243641\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	51648.0	1.70697	0.853483	−	0.521121i	$-0.174486\pi$
$971$	51648.0	1.70697	0.853483	+	0.521121i	$0.174486\pi$
$972$	0	0
$973$	− 25920.0i	− 0.854015i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 55776.0i	− 1.82644i	−0.407466	−	0.913220i	$-0.633588\pi$
$977$	− 55776.0i	− 1.82644i	0.407466	−	0.913220i	$-0.366412\pi$
$978$	0	0
$979$	61440.0	2.00575
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 36096.0i	− 1.17119i	−0.810602	−	0.585597i	$-0.800860\pi$
$983$	− 36096.0i	− 1.17119i	0.810602	−	0.585597i	$-0.199140\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	25600.0	0.823087
$990$	0	0
$991$	42532.0	1.36334	0.681672	−	0.731658i	$-0.261253\pi$
$991$	42532.0	1.36334	0.681672	+	0.731658i	$0.261253\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	29806.0i	0.946806i	0.880846	+	0.473403i	$0.156975\pi$
$997$	29806.0i	0.946806i	−0.880846	+	0.473403i	$0.843025\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1800.4.f.b.649.1		2
3.2	odd	2		1800.4.f.x.649.1		2
5.2	odd	4		1800.4.a.z.1.1		1
5.3	odd	4		72.4.a.a.1.1	✓	1
5.4	even	2	inner	1800.4.f.b.649.2		2
15.2	even	4		1800.4.a.ba.1.1		1
15.8	even	4		72.4.a.d.1.1	yes	1
15.14	odd	2		1800.4.f.x.649.2		2
20.3	even	4		144.4.a.a.1.1		1
40.3	even	4		576.4.a.x.1.1		1
40.13	odd	4		576.4.a.w.1.1		1
45.13	odd	12		648.4.i.l.217.1		2
45.23	even	12		648.4.i.a.217.1		2
45.38	even	12		648.4.i.a.433.1		2
45.43	odd	12		648.4.i.l.433.1		2
60.23	odd	4		144.4.a.f.1.1		1
120.53	even	4		576.4.a.c.1.1		1
120.83	odd	4		576.4.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.4.a.a.1.1	✓	1	5.3	odd	4
72.4.a.d.1.1	yes	1	15.8	even	4
144.4.a.a.1.1		1	20.3	even	4
144.4.a.f.1.1		1	60.23	odd	4
576.4.a.c.1.1		1	120.53	even	4
576.4.a.d.1.1		1	120.83	odd	4
576.4.a.w.1.1		1	40.13	odd	4
576.4.a.x.1.1		1	40.3	even	4
648.4.i.a.217.1		2	45.23	even	12
648.4.i.a.433.1		2	45.38	even	12
648.4.i.l.217.1		2	45.13	odd	12
648.4.i.l.433.1		2	45.43	odd	12
1800.4.a.z.1.1		1	5.2	odd	4
1800.4.a.ba.1.1		1	15.2	even	4
1800.4.f.b.649.1		2	1.1	even	1	trivial
1800.4.f.b.649.2		2	5.4	even	2	inner
1800.4.f.x.649.1		2	3.2	odd	2
1800.4.f.x.649.2		2	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-12.0000i q^{7} +O(q^{10})\)	\(q-12.0000i q^{7} -64.0000 q^{11} -58.0000i q^{13} -32.0000i q^{17} +136.000 q^{19} -128.000i q^{23} -144.000 q^{29} +20.0000 q^{31} -18.0000i q^{37} +288.000 q^{41} +200.000i q^{43} -384.000i q^{47} +199.000 q^{49} +496.000i q^{53} -128.000 q^{59} -458.000 q^{61} -496.000i q^{67} -512.000 q^{71} +602.000i q^{73} +768.000i q^{77} -1108.00 q^{79} +704.000i q^{83} -960.000 q^{89} -696.000 q^{91} +206.000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q+O(q^{10}) \) 2 * q	\( 2 q - 128 q^{11} + 272 q^{19} - 288 q^{29} + 40 q^{31} + 576 q^{41} + 398 q^{49} - 256 q^{59} - 916 q^{61} - 1024 q^{71} - 2216 q^{79} - 1920 q^{89} - 1392 q^{91}+O(q^{100}) \) 2 * q - 128 * q^11 + 272 * q^19 - 288 * q^29 + 40 * q^31 + 576 * q^41 + 398 * q^49 - 256 * q^59 - 916 * q^61 - 1024 * q^71 - 2216 * q^79 - 1920 * q^89 - 1392 * q^91

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