LMFDB - Embedded newform 7200.2.o.k.7199.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7200,2,Mod(7199,7200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7200, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("7200.7199");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7200 = 2^{5} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7200.o (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:	$57.4922894553$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1440)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			7199.1
Root			$0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	7200.7199
Dual form			7200.2.o.k.7199.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.585786 q^{7} +O(q^{10})$	$q+0.585786 q^{7} +1.41421 q^{11} -2.24264i q^{13} -1.17157 q^{17} +5.65685i q^{19} -3.17157i q^{23} -2.00000i q^{29} +3.17157i q^{31} +4.58579i q^{37} +8.24264i q^{41} -6.82843 q^{43} +8.00000i q^{47} -6.65685 q^{49} +6.82843 q^{53} -5.41421 q^{59} -3.17157 q^{61} +11.3137 q^{67} +13.6569 q^{71} -0.828427i q^{73} +0.828427 q^{77} -6.48528i q^{79} +1.17157i q^{83} -0.928932i q^{89} -1.31371i q^{91} +10.4853i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{7}+O(q^{10}) $ 4 * q + 8 * q^7	$ 4 q + 8 q^{7} - 16 q^{17} - 16 q^{43} - 4 q^{49} + 16 q^{53} - 16 q^{59} - 24 q^{61} + 32 q^{71} - 8 q^{77}+O(q^{100}) $ 4 * q + 8 * q^7 - 16 * q^17 - 16 * q^43 - 4 * q^49 + 16 * q^53 - 16 * q^59 - 24 * q^61 + 32 * q^71 - 8 * q^77

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/7200\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$6401$	$6751$
$\chi(n)$	$-1$	$1$	$-1$	$-1$

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
$7$	0.585786	0.221406	0.110703	−	0.993854i	$-0.464690\pi$
$7$	0.585786	0.221406	0.110703	+	0.993854i	$0.464690\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.41421	0.426401	0.213201	−	0.977008i	$-0.431611\pi$
$11$	1.41421	0.426401	0.213201	+	0.977008i	$0.431611\pi$
$12$	0	0
$13$	− 2.24264i	− 0.621997i	−0.950410	−	0.310998i	$-0.899337\pi$
$13$	− 2.24264i	− 0.621997i	0.950410	−	0.310998i	$-0.100663\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.17157	−0.284148	−0.142074	−	0.989856i	$-0.545377\pi$
$17$	−1.17157	−0.284148	−0.142074	+	0.989856i	$0.545377\pi$
$18$	0	0
$19$	5.65685i	1.29777i	0.760886	+	0.648886i	$0.224765\pi$
$19$	5.65685i	1.29777i	−0.760886	+	0.648886i	$0.775235\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 3.17157i	− 0.661319i	−0.943750	−	0.330659i	$-0.892729\pi$
$23$	− 3.17157i	− 0.661319i	0.943750	−	0.330659i	$-0.107271\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 2.00000i	− 0.371391i	−0.982607	−	0.185695i	$-0.940546\pi$
$29$	− 2.00000i	− 0.371391i	0.982607	−	0.185695i	$-0.0594537\pi$
$30$	0	0
$31$	3.17157i	0.569631i	0.958582	+	0.284816i	$0.0919324\pi$
$31$	3.17157i	0.569631i	−0.958582	+	0.284816i	$0.908068\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	4.58579i	0.753899i	0.926234	+	0.376949i	$0.123027\pi$
$37$	4.58579i	0.753899i	−0.926234	+	0.376949i	$0.876973\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	8.24264i	1.28728i	0.765327	+	0.643642i	$0.222577\pi$
$41$	8.24264i	1.28728i	−0.765327	+	0.643642i	$0.777423\pi$
$42$	0	0
$43$	−6.82843	−1.04133	−0.520663	−	0.853762i	$-0.674315\pi$
$43$	−6.82843	−1.04133	−0.520663	+	0.853762i	$0.674315\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.00000i	1.16692i	0.812142	+	0.583460i	$0.198301\pi$
$47$	8.00000i	1.16692i	−0.812142	+	0.583460i	$0.801699\pi$
$48$	0	0
$49$	−6.65685	−0.950979
$50$	0	0
$51$	0	0
$52$	0	0
$53$	6.82843	0.937957	0.468978	−	0.883210i	$-0.344622\pi$
$53$	6.82843	0.937957	0.468978	+	0.883210i	$0.344622\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−5.41421	−0.704871	−0.352435	−	0.935836i	$-0.614646\pi$
$59$	−5.41421	−0.704871	−0.352435	+	0.935836i	$0.614646\pi$
$60$	0	0
$61$	−3.17157	−0.406078	−0.203039	−	0.979171i	$-0.565082\pi$
$61$	−3.17157	−0.406078	−0.203039	+	0.979171i	$0.565082\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	11.3137	1.38219	0.691095	−	0.722764i	$-0.257129\pi$
$67$	11.3137	1.38219	0.691095	+	0.722764i	$0.257129\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	13.6569	1.62077	0.810385	−	0.585897i	$-0.199258\pi$
$71$	13.6569	1.62077	0.810385	+	0.585897i	$0.199258\pi$
$72$	0	0
$73$	− 0.828427i	− 0.0969601i	−0.998824	−	0.0484800i	$-0.984562\pi$
$73$	− 0.828427i	− 0.0969601i	0.998824	−	0.0484800i	$-0.0154377\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.828427	0.0944080
$78$	0	0
$79$	− 6.48528i	− 0.729651i	−0.931076	−	0.364826i	$-0.881129\pi$
$79$	− 6.48528i	− 0.729651i	0.931076	−	0.364826i	$-0.118871\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	1.17157i	0.128597i	0.997931	+	0.0642984i	$0.0204810\pi$
$83$	1.17157i	0.128597i	−0.997931	+	0.0642984i	$0.979519\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 0.928932i	− 0.0984666i	−0.998787	−	0.0492333i	$-0.984322\pi$
$89$	− 0.928932i	− 0.0984666i	0.998787	−	0.0492333i	$-0.0156778\pi$
$90$	0	0
$91$	− 1.31371i	− 0.137714i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	10.4853i	1.06462i	0.846550	+	0.532310i	$0.178676\pi$
$97$	10.4853i	1.06462i	−0.846550	+	0.532310i	$0.821324\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	11.1716i	1.11161i	0.831312	+	0.555807i	$0.187590\pi$
$101$	11.1716i	1.11161i	−0.831312	+	0.555807i	$0.812410\pi$
$102$	0	0
$103$	2.24264	0.220974	0.110487	−	0.993878i	$-0.464759\pi$
$103$	2.24264	0.220974	0.110487	+	0.993878i	$0.464759\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 18.1421i	− 1.75387i	−0.480612	−	0.876933i	$-0.659586\pi$
$107$	− 18.1421i	− 1.75387i	0.480612	−	0.876933i	$-0.340414\pi$
$108$	0	0
$109$	−8.82843	−0.845610	−0.422805	−	0.906221i	$-0.638954\pi$
$109$	−8.82843	−0.845610	−0.422805	+	0.906221i	$0.638954\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	4.00000	0.376288	0.188144	−	0.982141i	$-0.439753\pi$
$113$	4.00000	0.376288	0.188144	+	0.982141i	$0.439753\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−0.686292	−0.0629122
$120$	0	0
$121$	−9.00000	−0.818182
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	12.5858	1.11681	0.558404	−	0.829569i	$-0.311414\pi$
$127$	12.5858	1.11681	0.558404	+	0.829569i	$0.311414\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	5.89949	0.515441	0.257721	−	0.966219i	$-0.417029\pi$
$131$	5.89949	0.515441	0.257721	+	0.966219i	$0.417029\pi$
$132$	0	0
$133$	3.31371i	0.287335i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	5.65685	0.483298	0.241649	−	0.970364i	$-0.422312\pi$
$137$	5.65685	0.483298	0.241649	+	0.970364i	$0.422312\pi$
$138$	0	0
$139$	19.6569i	1.66727i	0.552314	+	0.833636i	$0.313745\pi$
$139$	19.6569i	1.66727i	−0.552314	+	0.833636i	$0.686255\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 3.17157i	− 0.265220i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 0.828427i	− 0.0678674i	−0.999424	−	0.0339337i	$-0.989196\pi$
$149$	− 0.828427i	− 0.0678674i	0.999424	−	0.0339337i	$-0.0108035\pi$
$150$	0	0
$151$	22.9706i	1.86932i	0.355546	+	0.934659i	$0.384295\pi$
$151$	22.9706i	1.86932i	−0.355546	+	0.934659i	$0.615705\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	19.8995i	1.58815i	0.607818	+	0.794076i	$0.292045\pi$
$157$	19.8995i	1.58815i	−0.607818	+	0.794076i	$0.707955\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 1.85786i	− 0.146420i
$162$	0	0
$163$	−8.48528	−0.664619	−0.332309	−	0.943170i	$-0.607828\pi$
$163$	−8.48528	−0.664619	−0.332309	+	0.943170i	$0.607828\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2.48528i	0.192317i	0.995366	+	0.0961584i	$0.0306555\pi$
$167$	2.48528i	0.192317i	−0.995366	+	0.0961584i	$0.969344\pi$
$168$	0	0
$169$	7.97056	0.613120
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−16.4853	−1.25335	−0.626676	−	0.779280i	$-0.715585\pi$
$173$	−16.4853	−1.25335	−0.626676	+	0.779280i	$0.715585\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	10.5858	0.791219	0.395609	−	0.918419i	$-0.370533\pi$
$179$	10.5858	0.791219	0.395609	+	0.918419i	$0.370533\pi$
$180$	0	0
$181$	−19.1716	−1.42501	−0.712506	−	0.701666i	$-0.752440\pi$
$181$	−19.1716	−1.42501	−0.712506	+	0.701666i	$0.752440\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−1.65685	−0.121161
$188$	0	0
$189$	0	0
$190$	0	0
$191$	6.82843	0.494088	0.247044	−	0.969004i	$-0.420541\pi$
$191$	6.82843	0.494088	0.247044	+	0.969004i	$0.420541\pi$
$192$	0	0
$193$	− 15.6569i	− 1.12701i	−0.826114	−	0.563503i	$-0.809454\pi$
$193$	− 15.6569i	− 1.12701i	0.826114	−	0.563503i	$-0.190546\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	0	0
$199$	− 4.34315i	− 0.307877i	−0.988080	−	0.153939i	$-0.950804\pi$
$199$	− 4.34315i	− 0.307877i	0.988080	−	0.153939i	$-0.0491958\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 1.17157i	− 0.0822283i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	8.00000i	0.553372i
$210$	0	0
$211$	6.00000i	0.413057i	0.978441	+	0.206529i	$0.0662166\pi$
$211$	6.00000i	0.413057i	−0.978441	+	0.206529i	$0.933783\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	1.85786i	0.126120i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	2.62742i	0.176739i
$222$	0	0
$223$	23.8995	1.60043	0.800214	−	0.599714i	$-0.204719\pi$
$223$	23.8995	1.60043	0.800214	+	0.599714i	$0.204719\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 28.2843i	− 1.87729i	−0.344881	−	0.938647i	$-0.612081\pi$
$227$	− 28.2843i	− 1.87729i	0.344881	−	0.938647i	$-0.387919\pi$
$228$	0	0
$229$	3.65685	0.241652	0.120826	−	0.992674i	$-0.461446\pi$
$229$	3.65685	0.241652	0.120826	+	0.992674i	$0.461446\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−4.48528	−0.293841	−0.146920	−	0.989148i	$-0.546936\pi$
$233$	−4.48528	−0.293841	−0.146920	+	0.989148i	$0.546936\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	22.1421	1.43226	0.716128	−	0.697969i	$-0.245913\pi$
$239$	22.1421	1.43226	0.716128	+	0.697969i	$0.245913\pi$
$240$	0	0
$241$	−25.6569	−1.65270	−0.826352	−	0.563154i	$-0.809588\pi$
$241$	−25.6569	−1.65270	−0.826352	+	0.563154i	$0.809588\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	12.6863	0.807209
$248$	0	0
$249$	0	0
$250$	0	0
$251$	4.72792	0.298424	0.149212	−	0.988805i	$-0.452326\pi$
$251$	4.72792	0.298424	0.149212	+	0.988805i	$0.452326\pi$
$252$	0	0
$253$	− 4.48528i	− 0.281987i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	24.9706	1.55762	0.778810	−	0.627259i	$-0.215823\pi$
$257$	24.9706	1.55762	0.778810	+	0.627259i	$0.215823\pi$
$258$	0	0
$259$	2.68629i	0.166918i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	19.1716i	1.18217i	0.806609	+	0.591085i	$0.201300\pi$
$263$	19.1716i	1.18217i	−0.806609	+	0.591085i	$0.798700\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	22.9706i	1.40054i	0.713878	+	0.700270i	$0.246937\pi$
$269$	22.9706i	1.40054i	−0.713878	+	0.700270i	$0.753063\pi$
$270$	0	0
$271$	21.3137i	1.29472i	0.762186	+	0.647358i	$0.224126\pi$
$271$	21.3137i	1.29472i	−0.762186	+	0.647358i	$0.775874\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 20.8701i	− 1.25396i	−0.779035	−	0.626980i	$-0.784291\pi$
$277$	− 20.8701i	− 1.25396i	0.779035	−	0.626980i	$-0.215709\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	17.8995i	1.06779i	0.845549	+	0.533897i	$0.179273\pi$
$281$	17.8995i	1.06779i	−0.845549	+	0.533897i	$0.820727\pi$
$282$	0	0
$283$	−7.31371	−0.434755	−0.217377	−	0.976088i	$-0.569750\pi$
$283$	−7.31371	−0.434755	−0.217377	+	0.976088i	$0.569750\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4.82843i	0.285013i
$288$	0	0
$289$	−15.6274	−0.919260
$290$	0	0
$291$	0	0
$292$	0	0
$293$	2.00000	0.116841	0.0584206	−	0.998292i	$-0.481394\pi$
$293$	2.00000	0.116841	0.0584206	+	0.998292i	$0.481394\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−7.11270	−0.411338
$300$	0	0
$301$	−4.00000	−0.230556
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−3.31371	−0.189123	−0.0945617	−	0.995519i	$-0.530145\pi$
$307$	−3.31371	−0.189123	−0.0945617	+	0.995519i	$0.530145\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−11.5147	−0.652940	−0.326470	−	0.945208i	$-0.605859\pi$
$311$	−11.5147	−0.652940	−0.326470	+	0.945208i	$0.605859\pi$
$312$	0	0
$313$	34.2843i	1.93786i	0.247332	+	0.968931i	$0.420446\pi$
$313$	34.2843i	1.93786i	−0.247332	+	0.968931i	$0.579554\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	17.3137	0.972435	0.486217	−	0.873838i	$-0.338376\pi$
$317$	17.3137	0.972435	0.486217	+	0.873838i	$0.338376\pi$
$318$	0	0
$319$	− 2.82843i	− 0.158362i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 6.62742i	− 0.368759i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	4.68629i	0.258364i
$330$	0	0
$331$	31.3137i	1.72116i	0.509318	+	0.860579i	$0.329898\pi$
$331$	31.3137i	1.72116i	−0.509318	+	0.860579i	$0.670102\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	16.8284i	0.916703i	0.888771	+	0.458351i	$0.151560\pi$
$337$	16.8284i	0.916703i	−0.888771	+	0.458351i	$0.848440\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	4.48528i	0.242892i
$342$	0	0
$343$	−8.00000	−0.431959
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 20.9706i	− 1.12576i	−0.826539	−	0.562879i	$-0.809694\pi$
$347$	− 20.9706i	− 1.12576i	0.826539	−	0.562879i	$-0.190306\pi$
$348$	0	0
$349$	−8.82843	−0.472575	−0.236287	−	0.971683i	$-0.575931\pi$
$349$	−8.82843	−0.472575	−0.236287	+	0.971683i	$0.575931\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−15.5147	−0.825765	−0.412883	−	0.910784i	$-0.635478\pi$
$353$	−15.5147	−0.825765	−0.412883	+	0.910784i	$0.635478\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	31.1127	1.64207	0.821033	−	0.570881i	$-0.193398\pi$
$359$	31.1127	1.64207	0.821033	+	0.570881i	$0.193398\pi$
$360$	0	0
$361$	−13.0000	−0.684211
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	18.9289	0.988082	0.494041	−	0.869439i	$-0.335519\pi$
$367$	18.9289	0.988082	0.494041	+	0.869439i	$0.335519\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	4.00000	0.207670
$372$	0	0
$373$	− 2.92893i	− 0.151654i	−0.997121	−	0.0758272i	$-0.975840\pi$
$373$	− 2.92893i	− 0.151654i	0.997121	−	0.0758272i	$-0.0241597\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−4.48528	−0.231004
$378$	0	0
$379$	6.68629i	0.343452i	0.985145	+	0.171726i	$0.0549343\pi$
$379$	6.68629i	0.343452i	−0.985145	+	0.171726i	$0.945066\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	30.6274i	1.56499i	0.622658	+	0.782494i	$0.286053\pi$
$383$	30.6274i	1.56499i	−0.622658	+	0.782494i	$0.713947\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	3.65685i	0.185410i	0.995694	+	0.0927049i	$0.0295513\pi$
$389$	3.65685i	0.185410i	−0.995694	+	0.0927049i	$0.970449\pi$
$390$	0	0
$391$	3.71573i	0.187912i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	20.3848i	1.02308i	0.859259	+	0.511541i	$0.170925\pi$
$397$	20.3848i	1.02308i	−0.859259	+	0.511541i	$0.829075\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 8.72792i	− 0.435852i	−0.975965	−	0.217926i	$-0.930071\pi$
$401$	− 8.72792i	− 0.435852i	0.975965	−	0.217926i	$-0.0699291\pi$
$402$	0	0
$403$	7.11270	0.354309
$404$	0	0
$405$	0	0
$406$	0	0
$407$	6.48528i	0.321463i
$408$	0	0
$409$	−0.343146	−0.0169675	−0.00848373	−	0.999964i	$-0.502700\pi$
$409$	−0.343146	−0.0169675	−0.00848373	+	0.999964i	$0.502700\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−3.17157	−0.156063
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−27.7574	−1.35604	−0.678018	−	0.735045i	$-0.737161\pi$
$419$	−27.7574	−1.35604	−0.678018	+	0.735045i	$0.737161\pi$
$420$	0	0
$421$	−39.9411	−1.94661	−0.973306	−	0.229513i	$-0.926287\pi$
$421$	−39.9411	−1.94661	−0.973306	+	0.229513i	$0.926287\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−1.85786	−0.0899084
$428$	0	0
$429$	0	0
$430$	0	0
$431$	4.48528	0.216048	0.108024	−	0.994148i	$-0.465548\pi$
$431$	4.48528	0.216048	0.108024	+	0.994148i	$0.465548\pi$
$432$	0	0
$433$	− 12.8284i	− 0.616495i	−0.951306	−	0.308247i	$-0.900258\pi$
$433$	− 12.8284i	− 0.616495i	0.951306	−	0.308247i	$-0.0997425\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	17.9411	0.858240
$438$	0	0
$439$	− 15.6569i	− 0.747261i	−0.927578	−	0.373630i	$-0.878113\pi$
$439$	− 15.6569i	− 0.747261i	0.927578	−	0.373630i	$-0.121887\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	7.31371i	0.347485i	0.984791	+	0.173742i	$0.0555860\pi$
$443$	7.31371i	0.347485i	−0.984791	+	0.173742i	$0.944414\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	25.2132i	1.18988i	0.803768	+	0.594942i	$0.202825\pi$
$449$	25.2132i	1.18988i	−0.803768	+	0.594942i	$0.797175\pi$
$450$	0	0
$451$	11.6569i	0.548900i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	3.85786i	0.180463i	0.995921	+	0.0902316i	$0.0287607\pi$
$457$	3.85786i	0.180463i	−0.995921	+	0.0902316i	$0.971239\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	5.79899i	0.270086i	0.990840	+	0.135043i	$0.0431172\pi$
$461$	5.79899i	0.270086i	−0.990840	+	0.135043i	$0.956883\pi$
$462$	0	0
$463$	−16.3848	−0.761465	−0.380733	−	0.924685i	$-0.624328\pi$
$463$	−16.3848	−0.761465	−0.380733	+	0.924685i	$0.624328\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 27.7990i	− 1.28638i	−0.765705	−	0.643192i	$-0.777610\pi$
$467$	− 27.7990i	− 1.28638i	0.765705	−	0.643192i	$-0.222390\pi$
$468$	0	0
$469$	6.62742	0.306026
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−9.65685	−0.444023
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	8.00000	0.365529	0.182765	−	0.983157i	$-0.441495\pi$
$479$	8.00000	0.365529	0.182765	+	0.983157i	$0.441495\pi$
$480$	0	0
$481$	10.2843	0.468922
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	42.0416	1.90509	0.952544	−	0.304401i	$-0.0984562\pi$
$487$	42.0416	1.90509	0.952544	+	0.304401i	$0.0984562\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	27.0711	1.22170	0.610850	−	0.791746i	$-0.290828\pi$
$491$	27.0711	1.22170	0.610850	+	0.791746i	$0.290828\pi$
$492$	0	0
$493$	2.34315i	0.105530i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	8.00000	0.358849
$498$	0	0
$499$	20.9706i	0.938771i	0.882993	+	0.469386i	$0.155525\pi$
$499$	20.9706i	0.938771i	−0.882993	+	0.469386i	$0.844475\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 16.0000i	− 0.713405i	−0.934218	−	0.356702i	$-0.883901\pi$
$503$	− 16.0000i	− 0.713405i	0.934218	−	0.356702i	$-0.116099\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 8.14214i	− 0.360894i	−0.983585	−	0.180447i	$-0.942246\pi$
$509$	− 8.14214i	− 0.360894i	0.983585	−	0.180447i	$-0.0577544\pi$
$510$	0	0
$511$	− 0.485281i	− 0.0214676i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	11.3137i	0.497576i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	32.7279i	1.43384i	0.697157	+	0.716918i	$0.254448\pi$
$521$	32.7279i	1.43384i	−0.697157	+	0.716918i	$0.745552\pi$
$522$	0	0
$523$	11.7990	0.515934	0.257967	−	0.966154i	$-0.416947\pi$
$523$	11.7990	0.515934	0.257967	+	0.966154i	$0.416947\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 3.71573i	− 0.161860i
$528$	0	0
$529$	12.9411	0.562658
$530$	0	0
$531$	0	0
$532$	0	0
$533$	18.4853	0.800686
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−9.41421	−0.405499
$540$	0	0
$541$	24.1421	1.03795	0.518976	−	0.854789i	$-0.326313\pi$
$541$	24.1421	1.03795	0.518976	+	0.854789i	$0.326313\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−43.7990	−1.87271	−0.936355	−	0.351055i	$-0.885823\pi$
$547$	−43.7990	−1.87271	−0.936355	+	0.351055i	$0.885823\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	11.3137	0.481980
$552$	0	0
$553$	− 3.79899i	− 0.161549i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	3.51472	0.148923	0.0744617	−	0.997224i	$-0.476276\pi$
$557$	3.51472	0.148923	0.0744617	+	0.997224i	$0.476276\pi$
$558$	0	0
$559$	15.3137i	0.647701i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 33.6569i	− 1.41847i	−0.704974	−	0.709234i	$-0.749041\pi$
$563$	− 33.6569i	− 1.41847i	0.704974	−	0.709234i	$-0.250959\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 37.4142i	− 1.56849i	−0.620454	−	0.784243i	$-0.713052\pi$
$569$	− 37.4142i	− 1.56849i	0.620454	−	0.784243i	$-0.286948\pi$
$570$	0	0
$571$	10.6274i	0.444744i	0.974962	+	0.222372i	$0.0713799\pi$
$571$	10.6274i	0.444744i	−0.974962	+	0.222372i	$0.928620\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	23.4558i	0.976480i	0.872710	+	0.488240i	$0.162361\pi$
$577$	23.4558i	0.976480i	−0.872710	+	0.488240i	$0.837639\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0.686292i	0.0284722i
$582$	0	0
$583$	9.65685	0.399946
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 13.4558i	− 0.555382i	−0.960670	−	0.277691i	$-0.910431\pi$
$587$	− 13.4558i	− 0.555382i	0.960670	−	0.277691i	$-0.0895692\pi$
$588$	0	0
$589$	−17.9411	−0.739251
$590$	0	0
$591$	0	0
$592$	0	0
$593$	30.6274	1.25772	0.628859	−	0.777520i	$-0.283522\pi$
$593$	30.6274	1.25772	0.628859	+	0.777520i	$0.283522\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	42.1421	1.72188	0.860940	−	0.508706i	$-0.169876\pi$
$599$	42.1421	1.72188	0.860940	+	0.508706i	$0.169876\pi$
$600$	0	0
$601$	−21.9411	−0.894997	−0.447499	−	0.894285i	$-0.647685\pi$
$601$	−21.9411	−0.894997	−0.447499	+	0.894285i	$0.647685\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	38.0416	1.54406	0.772031	−	0.635585i	$-0.219241\pi$
$607$	38.0416	1.54406	0.772031	+	0.635585i	$0.219241\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	17.9411	0.725820
$612$	0	0
$613$	6.92893i	0.279857i	0.990162	+	0.139928i	$0.0446873\pi$
$613$	6.92893i	0.279857i	−0.990162	+	0.139928i	$0.955313\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−18.8284	−0.758004	−0.379002	−	0.925396i	$-0.623733\pi$
$617$	−18.8284	−0.758004	−0.379002	+	0.925396i	$0.623733\pi$
$618$	0	0
$619$	− 6.97056i	− 0.280171i	−0.990139	−	0.140085i	$-0.955262\pi$
$619$	− 6.97056i	− 0.280171i	0.990139	−	0.140085i	$-0.0447377\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 0.544156i	− 0.0218011i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 5.37258i	− 0.214219i
$630$	0	0
$631$	− 12.1421i	− 0.483371i	−0.970355	−	0.241685i	$-0.922300\pi$
$631$	− 12.1421i	− 0.483371i	0.970355	−	0.241685i	$-0.0777002\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	14.9289i	0.591506i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 38.8701i	− 1.53527i	−0.640884	−	0.767637i	$-0.721432\pi$
$641$	− 38.8701i	− 1.53527i	0.640884	−	0.767637i	$-0.278568\pi$
$642$	0	0
$643$	−6.82843	−0.269287	−0.134643	−	0.990894i	$-0.542989\pi$
$643$	−6.82843	−0.269287	−0.134643	+	0.990894i	$0.542989\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	48.9706i	1.92523i	0.270871	+	0.962616i	$0.412688\pi$
$647$	48.9706i	1.92523i	−0.270871	+	0.962616i	$0.587312\pi$
$648$	0	0
$649$	−7.65685	−0.300558
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−26.2843	−1.02858	−0.514292	−	0.857615i	$-0.671945\pi$
$653$	−26.2843	−1.02858	−0.514292	+	0.857615i	$0.671945\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−44.0416	−1.71562	−0.857809	−	0.513968i	$-0.828175\pi$
$659$	−44.0416	−1.71562	−0.857809	+	0.513968i	$0.828175\pi$
$660$	0	0
$661$	0.142136	0.00552844	0.00276422	−	0.999996i	$-0.499120\pi$
$661$	0.142136	0.00552844	0.00276422	+	0.999996i	$0.499120\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−6.34315	−0.245608
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−4.48528	−0.173152
$672$	0	0
$673$	− 30.7696i	− 1.18608i	−0.805173	−	0.593040i	$-0.797928\pi$
$673$	− 30.7696i	− 1.18608i	0.805173	−	0.593040i	$-0.202072\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−29.3137	−1.12662	−0.563309	−	0.826247i	$-0.690472\pi$
$677$	−29.3137	−1.12662	−0.563309	+	0.826247i	$0.690472\pi$
$678$	0	0
$679$	6.14214i	0.235714i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 19.7990i	− 0.757587i	−0.925481	−	0.378794i	$-0.876339\pi$
$683$	− 19.7990i	− 0.757587i	0.925481	−	0.378794i	$-0.123661\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 15.3137i	− 0.583406i
$690$	0	0
$691$	18.3431i	0.697806i	0.937159	+	0.348903i	$0.113446\pi$
$691$	18.3431i	0.697806i	−0.937159	+	0.348903i	$0.886554\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 9.65685i	− 0.365779i
$698$	0	0
$699$	0	0
$700$	0	0
$701$	38.4853i	1.45357i	0.686866	+	0.726785i	$0.258986\pi$
$701$	38.4853i	1.45357i	−0.686866	+	0.726785i	$0.741014\pi$
$702$	0	0
$703$	−25.9411	−0.978388
$704$	0	0
$705$	0	0
$706$	0	0
$707$	6.54416i	0.246118i
$708$	0	0
$709$	20.3431	0.764003	0.382001	−	0.924162i	$-0.375235\pi$
$709$	20.3431	0.764003	0.382001	+	0.924162i	$0.375235\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	10.0589	0.376708
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	11.7990	0.440028	0.220014	−	0.975497i	$-0.429390\pi$
$719$	11.7990	0.440028	0.220014	+	0.975497i	$0.429390\pi$
$720$	0	0
$721$	1.31371	0.0489251
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	18.2426	0.676582	0.338291	−	0.941042i	$-0.390151\pi$
$727$	18.2426	0.676582	0.338291	+	0.941042i	$0.390151\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	8.00000	0.295891
$732$	0	0
$733$	− 6.24264i	− 0.230577i	−0.993332	−	0.115289i	$-0.963221\pi$
$733$	− 6.24264i	− 0.230577i	0.993332	−	0.115289i	$-0.0367793\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	16.0000	0.589368
$738$	0	0
$739$	12.0000i	0.441427i	0.975339	+	0.220714i	$0.0708386\pi$
$739$	12.0000i	0.441427i	−0.975339	+	0.220714i	$0.929161\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	40.9706i	1.50306i	0.659697	+	0.751532i	$0.270685\pi$
$743$	40.9706i	1.50306i	−0.659697	+	0.751532i	$0.729315\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 10.6274i	− 0.388317i
$750$	0	0
$751$	− 17.7990i	− 0.649494i	−0.945801	−	0.324747i	$-0.894721\pi$
$751$	− 17.7990i	− 0.649494i	0.945801	−	0.324747i	$-0.105279\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 0.585786i	− 0.0212908i	−0.999943	−	0.0106454i	$-0.996611\pi$
$757$	− 0.585786i	− 0.0212908i	0.999943	−	0.0106454i	$-0.00338860\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	14.8701i	0.539039i	0.962995	+	0.269520i	$0.0868649\pi$
$761$	14.8701i	0.539039i	−0.962995	+	0.269520i	$0.913135\pi$
$762$	0	0
$763$	−5.17157	−0.187224
$764$	0	0
$765$	0	0
$766$	0	0
$767$	12.1421i	0.438427i
$768$	0	0
$769$	−47.5980	−1.71643	−0.858214	−	0.513293i	$-0.828425\pi$
$769$	−47.5980	−1.71643	−0.858214	+	0.513293i	$0.828425\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	47.2548	1.69964	0.849819	−	0.527074i	$-0.176711\pi$
$773$	47.2548	1.69964	0.849819	+	0.527074i	$0.176711\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−46.6274	−1.67060
$780$	0	0
$781$	19.3137	0.691099
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−2.14214	−0.0763589	−0.0381794	−	0.999271i	$-0.512156\pi$
$787$	−2.14214	−0.0763589	−0.0381794	+	0.999271i	$0.512156\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	2.34315	0.0833127
$792$	0	0
$793$	7.11270i	0.252579i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−16.4853	−0.583939	−0.291969	−	0.956428i	$-0.594311\pi$
$797$	−16.4853	−0.583939	−0.291969	+	0.956428i	$0.594311\pi$
$798$	0	0
$799$	− 9.37258i	− 0.331578i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 1.17157i	− 0.0413439i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	29.8995i	1.05121i	0.850729	+	0.525605i	$0.176161\pi$
$809$	29.8995i	1.05121i	−0.850729	+	0.525605i	$0.823839\pi$
$810$	0	0
$811$	− 32.6274i	− 1.14570i	−0.819659	−	0.572852i	$-0.805837\pi$
$811$	− 32.6274i	− 1.14570i	0.819659	−	0.572852i	$-0.194163\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 38.6274i	− 1.35140i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 17.3137i	− 0.604253i	−0.953268	−	0.302126i	$-0.902304\pi$
$821$	− 17.3137i	− 0.604253i	0.953268	−	0.302126i	$-0.0976964\pi$
$822$	0	0
$823$	−16.8701	−0.588053	−0.294027	−	0.955797i	$-0.594995\pi$
$823$	−16.8701	−0.588053	−0.294027	+	0.955797i	$0.594995\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	5.37258i	0.186823i	0.995628	+	0.0934115i	$0.0297772\pi$
$827$	5.37258i	0.186823i	−0.995628	+	0.0934115i	$0.970223\pi$
$828$	0	0
$829$	−30.4853	−1.05880	−0.529399	−	0.848373i	$-0.677582\pi$
$829$	−30.4853	−1.05880	−0.529399	+	0.848373i	$0.677582\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	7.79899	0.270219
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−7.51472	−0.259437	−0.129718	−	0.991551i	$-0.541407\pi$
$839$	−7.51472	−0.259437	−0.129718	+	0.991551i	$0.541407\pi$
$840$	0	0
$841$	25.0000	0.862069
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−5.27208	−0.181151
$848$	0	0
$849$	0	0
$850$	0	0
$851$	14.5442	0.498567
$852$	0	0
$853$	− 4.78680i	− 0.163897i	−0.996637	−	0.0819484i	$-0.973886\pi$
$853$	− 4.78680i	− 0.163897i	0.996637	−	0.0819484i	$-0.0261143\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−31.5147	−1.07652	−0.538261	−	0.842778i	$-0.680919\pi$
$857$	−31.5147	−1.07652	−0.538261	+	0.842778i	$0.680919\pi$
$858$	0	0
$859$	10.0000i	0.341196i	0.985341	+	0.170598i	$0.0545699\pi$
$859$	10.0000i	0.341196i	−0.985341	+	0.170598i	$0.945430\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 22.7696i	− 0.775085i	−0.921852	−	0.387542i	$-0.873324\pi$
$863$	− 22.7696i	− 0.775085i	0.921852	−	0.387542i	$-0.126676\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 9.17157i	− 0.311124i
$870$	0	0
$871$	− 25.3726i	− 0.859717i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	7.41421i	0.250360i	0.992134	+	0.125180i	$0.0399509\pi$
$877$	7.41421i	0.250360i	−0.992134	+	0.125180i	$0.960049\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	22.5858i	0.760934i	0.924794	+	0.380467i	$0.124237\pi$
$881$	22.5858i	0.760934i	−0.924794	+	0.380467i	$0.875763\pi$
$882$	0	0
$883$	−17.4558	−0.587436	−0.293718	−	0.955892i	$-0.594893\pi$
$883$	−17.4558	−0.587436	−0.293718	+	0.955892i	$0.594893\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0.142136i	0.00477245i	0.999997	+	0.00238622i	$0.000759559\pi$
$887$	0.142136i	0.00477245i	−0.999997	+	0.00238622i	$0.999240\pi$
$888$	0	0
$889$	7.37258	0.247268
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−45.2548	−1.51440
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	6.34315	0.211556
$900$	0	0
$901$	−8.00000	−0.266519
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−31.1127	−1.03308	−0.516540	−	0.856263i	$-0.672780\pi$
$907$	−31.1127	−1.03308	−0.516540	+	0.856263i	$0.672780\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−20.2843	−0.672048	−0.336024	−	0.941853i	$-0.609082\pi$
$911$	−20.2843	−0.672048	−0.336024	+	0.941853i	$0.609082\pi$
$912$	0	0
$913$	1.65685i	0.0548339i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	3.45584	0.114122
$918$	0	0
$919$	− 9.51472i	− 0.313862i	−0.987610	−	0.156931i	$-0.949840\pi$
$919$	− 9.51472i	− 0.313862i	0.987610	−	0.156931i	$-0.0501600\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 30.6274i	− 1.00811i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	7.27208i	0.238589i	0.992859	+	0.119295i	$0.0380633\pi$
$929$	7.27208i	0.238589i	−0.992859	+	0.119295i	$0.961937\pi$
$930$	0	0
$931$	− 37.6569i	− 1.23415i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	14.2843i	0.466647i	0.972399	+	0.233323i	$0.0749601\pi$
$937$	14.2843i	0.466647i	−0.972399	+	0.233323i	$0.925040\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	13.0294i	0.424748i	0.977189	+	0.212374i	$0.0681194\pi$
$941$	13.0294i	0.424748i	−0.977189	+	0.212374i	$0.931881\pi$
$942$	0	0
$943$	26.1421	0.851305
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 11.0294i	− 0.358409i	−0.983812	−	0.179204i	$-0.942648\pi$
$947$	− 11.0294i	− 0.358409i	0.983812	−	0.179204i	$-0.0573523\pi$
$948$	0	0
$949$	−1.85786	−0.0603088
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−13.6569	−0.442389	−0.221194	−	0.975230i	$-0.570996\pi$
$953$	−13.6569	−0.442389	−0.221194	+	0.975230i	$0.570996\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	3.31371	0.107005
$960$	0	0
$961$	20.9411	0.675520
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	0.100505	0.00323202	0.00161601	−	0.999999i	$-0.499486\pi$
$967$	0.100505	0.00323202	0.00161601	+	0.999999i	$0.499486\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−29.6985	−0.953070	−0.476535	−	0.879156i	$-0.658107\pi$
$971$	−29.6985	−0.953070	−0.476535	+	0.879156i	$0.658107\pi$
$972$	0	0
$973$	11.5147i	0.369145i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−7.51472	−0.240417	−0.120209	−	0.992749i	$-0.538356\pi$
$977$	−7.51472	−0.240417	−0.120209	+	0.992749i	$0.538356\pi$
$978$	0	0
$979$	− 1.31371i	− 0.0419863i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	3.31371i	0.105691i	0.998603	+	0.0528454i	$0.0168291\pi$
$983$	3.31371i	0.105691i	−0.998603	+	0.0528454i	$0.983171\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	21.6569i	0.688648i
$990$	0	0
$991$	11.6569i	0.370292i	0.982711	+	0.185146i	$0.0592758\pi$
$991$	11.6569i	0.370292i	−0.982711	+	0.185146i	$0.940724\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 25.7574i	− 0.815744i	−0.913039	−	0.407872i	$-0.866271\pi$
$997$	− 25.7574i	− 0.815744i	0.913039	−	0.407872i	$-0.133729\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7200.2.o.k.7199.1		4
3.2	odd	2		7200.2.o.l.7199.1		4
4.3	odd	2		7200.2.o.c.7199.3		4
5.2	odd	4		7200.2.h.e.1151.3		4
5.3	odd	4		1440.2.h.c.1151.4	yes	4
5.4	even	2		7200.2.o.d.7199.4		4
12.11	even	2		7200.2.o.d.7199.3		4
15.2	even	4		7200.2.h.f.1151.3		4
15.8	even	4		1440.2.h.b.1151.2	✓	4
15.14	odd	2		7200.2.o.c.7199.4		4
20.3	even	4		1440.2.h.b.1151.3	yes	4
20.7	even	4		7200.2.h.f.1151.2		4
20.19	odd	2		7200.2.o.l.7199.2		4
40.3	even	4		2880.2.h.b.1151.1		4
40.13	odd	4		2880.2.h.c.1151.2		4
60.23	odd	4		1440.2.h.c.1151.1	yes	4
60.47	odd	4		7200.2.h.e.1151.2		4
60.59	even	2	inner	7200.2.o.k.7199.2		4
120.53	even	4		2880.2.h.b.1151.4		4
120.83	odd	4		2880.2.h.c.1151.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1440.2.h.b.1151.2	✓	4	15.8	even	4
1440.2.h.b.1151.3	yes	4	20.3	even	4
1440.2.h.c.1151.1	yes	4	60.23	odd	4
1440.2.h.c.1151.4	yes	4	5.3	odd	4
2880.2.h.b.1151.1		4	40.3	even	4
2880.2.h.b.1151.4		4	120.53	even	4
2880.2.h.c.1151.2		4	40.13	odd	4
2880.2.h.c.1151.3		4	120.83	odd	4
7200.2.h.e.1151.2		4	60.47	odd	4
7200.2.h.e.1151.3		4	5.2	odd	4
7200.2.h.f.1151.2		4	20.7	even	4
7200.2.h.f.1151.3		4	15.2	even	4
7200.2.o.c.7199.3		4	4.3	odd	2
7200.2.o.c.7199.4		4	15.14	odd	2
7200.2.o.d.7199.3		4	12.11	even	2
7200.2.o.d.7199.4		4	5.4	even	2
7200.2.o.k.7199.1		4	1.1	even	1	trivial
7200.2.o.k.7199.2		4	60.59	even	2	inner
7200.2.o.l.7199.1		4	3.2	odd	2
7200.2.o.l.7199.2		4	20.19	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:	\( N \)	\(=\)	\( 7200 = 2^{5} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7200.o (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.585786 q^{7} +O(q^{10})\)	\(q+0.585786 q^{7} +1.41421 q^{11} -2.24264i q^{13} -1.17157 q^{17} +5.65685i q^{19} -3.17157i q^{23} -2.00000i q^{29} +3.17157i q^{31} +4.58579i q^{37} +8.24264i q^{41} -6.82843 q^{43} +8.00000i q^{47} -6.65685 q^{49} +6.82843 q^{53} -5.41421 q^{59} -3.17157 q^{61} +11.3137 q^{67} +13.6569 q^{71} -0.828427i q^{73} +0.828427 q^{77} -6.48528i q^{79} +1.17157i q^{83} -0.928932i q^{89} -1.31371i q^{91} +10.4853i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{7}+O(q^{10}) \) 4 * q + 8 * q^7	\( 4 q + 8 q^{7} - 16 q^{17} - 16 q^{43} - 4 q^{49} + 16 q^{53} - 16 q^{59} - 24 q^{61} + 32 q^{71} - 8 q^{77}+O(q^{100}) \) 4 * q + 8 * q^7 - 16 * q^17 - 16 * q^43 - 4 * q^49 + 16 * q^53 - 16 * q^59 - 24 * q^61 + 32 * q^71 - 8 * q^77

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