LMFDB - Embedded newform 1100.2.k.e.857.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1100,2,Mod(593,1100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1100, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3, 2]))

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

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

chi := DirichletCharacter("1100.593");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1100 = 2^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1100.k (of order $4$, degree $2$, 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:	$8.78354422234$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	8.0.12745506816.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 23x^{4} + 1 $ x^8 + 23*x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			857.2
Root			$-0.323042 + 0.323042i$ of defining polynomial
Character	$\chi$	$=$	1100.857
Dual form			1100.2.k.e.593.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.323042 + 0.323042i) q^{3} +(0.901703 - 0.901703i) q^{7} +2.79129i q^{9} +O(q^{10})$	$q+(-0.323042 + 0.323042i) q^{3} +(0.901703 - 0.901703i) q^{7} +2.79129i q^{9} +(-1.79129 - 2.79129i) q^{11} +(2.51691 + 2.51691i) q^{13} +(-5.22202 + 5.22202i) q^{17} +5.00000 q^{19} +0.582576i q^{21} +(-2.83995 + 2.83995i) q^{23} +(-1.87083 - 1.87083i) q^{27} +8.37386 q^{29} +4.58258 q^{31} +(1.48036 + 0.323042i) q^{33} +(-4.96640 - 4.96640i) q^{37} -1.62614 q^{39} +5.00000i q^{41} +(6.83723 + 6.83723i) q^{43} +(0.578661 + 0.578661i) q^{47} +5.37386i q^{49} -3.37386i q^{51} +(-7.09285 + 7.09285i) q^{53} +(-1.61521 + 1.61521i) q^{57} +8.58258i q^{59} -7.79129i q^{61} +(2.51691 + 2.51691i) q^{63} +(6.19115 + 6.19115i) q^{67} -1.83485i q^{69} -0.417424 q^{71} +(0.901703 + 0.901703i) q^{73} +(-4.13212 - 0.901703i) q^{77} +7.20871 q^{79} -7.16515 q^{81} +(5.22202 + 5.22202i) q^{83} +(-2.70511 + 2.70511i) q^{87} -4.79129i q^{89} +4.53901 q^{91} +(-1.48036 + 1.48036i) q^{93} +(-3.35119 - 3.35119i) q^{97} +(7.79129 - 5.00000i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 4 q^{11} + 40 q^{19} + 12 q^{29} - 68 q^{39} - 40 q^{71} + 76 q^{79} + 16 q^{81} - 92 q^{91} + 44 q^{99}+O(q^{100}) $ 8 * q + 4 * q^11 + 40 * q^19 + 12 * q^29 - 68 * q^39 - 40 * q^71 + 76 * q^79 + 16 * q^81 - 92 * q^91 + 44 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1100\mathbb{Z}\right)^\times$.

$n$	$101$	$177$	$551$
$\chi(n)$	$-1$	$e\left(\frac{1}{4}\right)$	$1$

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
$3$	−0.323042	+	0.323042i	−0.186508	+	0.186508i	−0.794185	−	0.607676i	$-0.792102\pi$
$3$	−0.323042	+	0.323042i	−0.186508	+	0.186508i	0.607676	+	0.794185i	$0.292102\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	0.901703	−	0.901703i	0.340812	−	0.340812i	−0.515861	−	0.856672i	$-0.672528\pi$
$7$	0.901703	−	0.901703i	0.340812	−	0.340812i	0.856672	+	0.515861i	$0.172528\pi$
$8$		0			0
$9$			2.79129i			0.930429i
$10$		0			0
$11$	−1.79129	−	2.79129i	−0.540094	−	0.841605i
$11$	−1.79129	−	2.79129i	−0.540094	−	0.841605i
$12$		0			0
$13$	2.51691	+	2.51691i	0.698066	+	0.698066i	0.963993	−	0.265927i	$-0.0856781\pi$
$13$	2.51691	+	2.51691i	0.698066	+	0.698066i	−0.265927	+	0.963993i	$0.585678\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−5.22202	+	5.22202i	−1.26653	+	1.26653i	−0.318656	+	0.947871i	$0.603231\pi$
$17$	−5.22202	+	5.22202i	−1.26653	+	1.26653i	−0.947871	+	0.318656i	$0.896769\pi$
$18$		0			0
$19$	5.00000			1.14708			0.573539	−	0.819178i	$-0.305570\pi$
$19$	5.00000			1.14708			0.573539	+	0.819178i	$0.305570\pi$
$20$		0			0
$21$			0.582576i			0.127128i
$22$		0			0
$23$	−2.83995	+	2.83995i	−0.592171	+	0.592171i	−0.938218	−	0.346046i	$-0.887524\pi$
$23$	−2.83995	+	2.83995i	−0.592171	+	0.592171i	0.346046	+	0.938218i	$0.387524\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−1.87083	−	1.87083i	−0.360041	−	0.360041i
$28$		0			0
$29$	8.37386			1.55499			0.777494	−	0.628891i	$-0.216491\pi$
$29$	8.37386			1.55499			0.777494	+	0.628891i	$0.216491\pi$
$30$		0			0
$31$	4.58258			0.823055			0.411527	−	0.911397i	$-0.364995\pi$
$31$	4.58258			0.823055			0.411527	+	0.911397i	$0.364995\pi$
$32$		0			0
$33$	1.48036	+	0.323042i	0.257698	+	0.0562344i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−4.96640	−	4.96640i	−0.816472	−	0.816472i	0.169123	−	0.985595i	$-0.445906\pi$
$37$	−4.96640	−	4.96640i	−0.816472	−	0.816472i	−0.985595	+	0.169123i	$0.945906\pi$
$38$		0			0
$39$	−1.62614			−0.260390
$40$		0			0
$41$			5.00000i			0.780869i	0.920631	+	0.390434i	$0.127675\pi$
$41$			5.00000i			0.780869i	−0.920631	+	0.390434i	$0.872325\pi$
$42$		0			0
$43$	6.83723	+	6.83723i	1.04267	+	1.04267i	0.999048	+	0.0436197i	$0.0138890\pi$
$43$	6.83723	+	6.83723i	1.04267	+	1.04267i	0.0436197	+	0.999048i	$0.486111\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	0.578661	+	0.578661i	0.0844064	+	0.0844064i	0.748049	−	0.663643i	$-0.230991\pi$
$47$	0.578661	+	0.578661i	0.0844064	+	0.0844064i	−0.663643	+	0.748049i	$0.730991\pi$
$48$		0			0
$49$			5.37386i			0.767695i
$50$		0			0
$51$		−	3.37386i		−	0.472435i
$52$		0			0
$53$	−7.09285	+	7.09285i	−0.974278	+	0.974278i	−0.999677	−	0.0253995i	$-0.991914\pi$
$53$	−7.09285	+	7.09285i	−0.974278	+	0.974278i	0.0253995	+	0.999677i	$0.491914\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−1.61521	+	1.61521i	−0.213940	+	0.213940i
$58$		0			0
$59$			8.58258i			1.11736i	0.829385	+	0.558678i	$0.188691\pi$
$59$			8.58258i			1.11736i	−0.829385	+	0.558678i	$0.811309\pi$
$60$		0			0
$61$		−	7.79129i		−	0.997572i	−0.866725	−	0.498786i	$-0.833779\pi$
$61$		−	7.79129i		−	0.997572i	0.866725	−	0.498786i	$-0.166221\pi$
$62$		0			0
$63$	2.51691	+	2.51691i	0.317101	+	0.317101i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	6.19115	+	6.19115i	0.756369	+	0.756369i	0.975660	−	0.219291i	$-0.0703743\pi$
$67$	6.19115	+	6.19115i	0.756369	+	0.756369i	−0.219291	+	0.975660i	$0.570374\pi$
$68$		0			0
$69$		−	1.83485i		−	0.220890i
$70$		0			0
$71$	−0.417424			−0.0495392			−0.0247696	−	0.999693i	$-0.507885\pi$
$71$	−0.417424			−0.0495392			−0.0247696	+	0.999693i	$0.507885\pi$
$72$		0			0
$73$	0.901703	+	0.901703i	0.105536	+	0.105536i	0.757903	−	0.652367i	$-0.226224\pi$
$73$	0.901703	+	0.901703i	0.105536	+	0.105536i	−0.652367	+	0.757903i	$0.726224\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−4.13212	−	0.901703i	−0.470899	−	0.102759i
$78$		0			0
$79$	7.20871			0.811043			0.405522	−	0.914085i	$-0.367090\pi$
$79$	7.20871			0.811043			0.405522	+	0.914085i	$0.367090\pi$
$80$		0			0
$81$	−7.16515			−0.796128
$82$		0			0
$83$	5.22202	+	5.22202i	0.573191	+	0.573191i	0.933019	−	0.359828i	$-0.117164\pi$
$83$	5.22202	+	5.22202i	0.573191	+	0.573191i	−0.359828	+	0.933019i	$0.617164\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−2.70511	+	2.70511i	−0.290018	+	0.290018i
$88$		0			0
$89$		−	4.79129i		−	0.507875i	−0.967221	−	0.253938i	$-0.918274\pi$
$89$		−	4.79129i		−	0.507875i	0.967221	−	0.253938i	$-0.0817258\pi$
$90$		0			0
$91$	4.53901			0.475818
$92$		0			0
$93$	−1.48036	+	1.48036i	−0.153507	+	0.153507i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−3.35119	−	3.35119i	−0.340262	−	0.340262i	0.516204	−	0.856466i	$-0.327345\pi$
$97$	−3.35119	−	3.35119i	−0.340262	−	0.340262i	−0.856466	+	0.516204i	$0.827345\pi$
$98$		0			0
$99$	7.79129	−	5.00000i	0.783054	−	0.502519i
$100$		0			0
$101$			7.79129i			0.775262i	0.921815	+	0.387631i	$0.126707\pi$
$101$			7.79129i			0.775262i	−0.921815	+	0.387631i	$0.873293\pi$
$102$		0			0
$103$	−9.60976	+	9.60976i	−0.946878	+	0.946878i	−0.998658	−	0.0517804i	$-0.983510\pi$
$103$	−9.60976	+	9.60976i	−0.946878	+	0.946878i	0.0517804	+	0.998658i	$0.483510\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−0.713507	+	0.713507i	−0.0689773	+	0.0689773i	−0.740754	−	0.671777i	$-0.765532\pi$
$107$	−0.713507	+	0.713507i	−0.0689773	+	0.0689773i	0.671777	+	0.740754i	$0.265532\pi$
$108$		0			0
$109$	−1.62614			−0.155756			−0.0778778	−	0.996963i	$-0.524814\pi$
$109$	−1.62614			−0.155756			−0.0778778	+	0.996963i	$0.524814\pi$
$110$		0			0
$111$	3.20871			0.304557
$112$		0			0
$113$	12.4497	−	12.4497i	1.17117	−	1.17117i	0.189240	−	0.981931i	$-0.439398\pi$
$113$	12.4497	−	12.4497i	1.17117	−	1.17117i	0.981931	−	0.189240i	$-0.0606024\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−7.02543	+	7.02543i	−0.649501	+	0.649501i
$118$		0			0
$119$			9.41742i			0.863294i
$120$		0			0
$121$	−4.58258	+	10.0000i	−0.416598	+	0.909091i
$122$		0			0
$123$	−1.61521	−	1.61521i	−0.145639	−	0.145639i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	10.2558	−	10.2558i	0.910059	−	0.910059i	−0.0862172	−	0.996276i	$-0.527478\pi$
$127$	10.2558	−	10.2558i	0.910059	−	0.910059i	0.996276	+	0.0862172i	$0.0274779\pi$
$128$		0			0
$129$	−4.41742			−0.388933
$130$		0			0
$131$		−	3.37386i		−	0.294776i	−0.989079	−	0.147388i	$-0.952913\pi$
$131$		−	3.37386i		−	0.294776i	0.989079	−	0.147388i	$-0.0470866\pi$
$132$		0			0
$133$	4.50851	−	4.50851i	0.390938	−	0.390938i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−4.06470	−	4.06470i	−0.347271	−	0.347271i	0.511821	−	0.859092i	$-0.328971\pi$
$137$	−4.06470	−	4.06470i	−0.347271	−	0.347271i	−0.859092	+	0.511821i	$0.828971\pi$
$138$		0			0
$139$	−21.7477			−1.84462			−0.922309	−	0.386453i	$-0.873700\pi$
$139$	−21.7477			−1.84462			−0.922309	+	0.386453i	$0.873700\pi$
$140$		0			0
$141$	−0.373864			−0.0314850
$142$		0			0
$143$	2.51691	−	11.5339i	0.210475	−	0.964517i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−1.73598	−	1.73598i	−0.143181	−	0.143181i
$148$		0			0
$149$	4.41742			0.361889			0.180945	−	0.983493i	$-0.442084\pi$
$149$	4.41742			0.361889			0.180945	+	0.983493i	$0.442084\pi$
$150$		0			0
$151$		−	20.5826i		−	1.67499i	−0.546448	−	0.837493i	$-0.684020\pi$
$151$		−	20.5826i		−	1.67499i	0.546448	−	0.837493i	$-0.315980\pi$
$152$		0			0
$153$	−14.5762	−	14.5762i	−1.17841	−	1.17841i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	9.79796	+	9.79796i	0.781962	+	0.781962i	0.980162	−	0.198199i	$-0.0635094\pi$
$157$	9.79796	+	9.79796i	0.781962	+	0.781962i	−0.198199	+	0.980162i	$0.563509\pi$
$158$		0			0
$159$		−	4.58258i		−	0.363422i
$160$		0			0
$161$			5.12159i			0.403638i
$162$		0			0
$163$	−10.6997	+	10.6997i	−0.838062	+	0.838062i	−0.988604	−	0.150541i	$-0.951898\pi$
$163$	−10.6997	+	10.6997i	−0.838062	+	0.838062i	0.150541	+	0.988604i	$0.451898\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	11.1575	−	11.1575i	0.863397	−	0.863397i	−0.128334	−	0.991731i	$-0.540963\pi$
$167$	11.1575	−	11.1575i	0.863397	−	0.863397i	0.991731	+	0.128334i	$0.0409629\pi$
$168$		0			0
$169$		−	0.330303i		−	0.0254079i
$170$		0			0
$171$			13.9564i			1.06728i
$172$		0			0
$173$	−0.713507	−	0.713507i	−0.0542469	−	0.0542469i	0.679463	−	0.733710i	$-0.262213\pi$
$173$	−0.713507	−	0.713507i	−0.0542469	−	0.0542469i	−0.733710	+	0.679463i	$0.762213\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−2.77253	−	2.77253i	−0.208396	−	0.208396i
$178$		0			0
$179$		−	14.2087i		−	1.06201i	−0.847369	−	0.531005i	$-0.821815\pi$
$179$		−	14.2087i		−	1.06201i	0.847369	−	0.531005i	$-0.178185\pi$
$180$		0			0
$181$	7.37386			0.548095			0.274047	−	0.961716i	$-0.411637\pi$
$181$	7.37386			0.548095			0.274047	+	0.961716i	$0.411637\pi$
$182$		0			0
$183$	2.51691	+	2.51691i	0.186056	+	0.186056i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	23.9303	+	5.22202i	1.74996	+	0.381872i
$188$		0			0
$189$	−3.37386			−0.245412
$190$		0			0
$191$	11.2087			0.811034			0.405517	−	0.914088i	$-0.367092\pi$
$191$	11.2087			0.811034			0.405517	+	0.914088i	$0.367092\pi$
$192$		0			0
$193$	−10.4440	−	10.4440i	−0.751779	−	0.751779i	0.223032	−	0.974811i	$-0.428404\pi$
$193$	−10.4440	−	10.4440i	−0.751779	−	0.751779i	−0.974811	+	0.223032i	$0.928404\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−5.93553	+	5.93553i	−0.422889	+	0.422889i	−0.886197	−	0.463308i	$-0.846662\pi$
$197$	−5.93553	+	5.93553i	−0.422889	+	0.422889i	0.463308	+	0.886197i	$0.346662\pi$
$198$		0			0
$199$		−	15.5390i		−	1.10153i	−0.834660	−	0.550766i	$-0.814336\pi$
$199$		−	15.5390i		−	1.10153i	0.834660	−	0.550766i	$-0.185664\pi$
$200$		0			0
$201$	−4.00000			−0.282138
$202$		0			0
$203$	7.55074	−	7.55074i	0.529958	−	0.529958i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−7.92713	−	7.92713i	−0.550974	−	0.550974i
$208$		0			0
$209$	−8.95644	−	13.9564i	−0.619530	−	0.965387i
$210$		0			0
$211$		−	17.3303i		−	1.19307i	−0.802588	−	0.596534i	$-0.796544\pi$
$211$		−	17.3303i		−	1.19307i	0.802588	−	0.596534i	$-0.203456\pi$
$212$		0			0
$213$	0.134846	−	0.134846i	0.00923946	−	0.00923946i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	4.13212	−	4.13212i	0.280507	−	0.280507i
$218$		0			0
$219$	−0.582576			−0.0393668
$220$		0			0
$221$	−26.2867			−1.76824
$222$		0			0
$223$	17.8599	−	17.8599i	1.19599	−	1.19599i	0.220633	−	0.975357i	$-0.429188\pi$
$223$	17.8599	−	17.8599i	1.19599	−	1.19599i	0.975357	−	0.220633i	$-0.0708122\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	3.41862	−	3.41862i	0.226901	−	0.226901i	−0.584495	−	0.811397i	$-0.698708\pi$
$227$	3.41862	−	3.41862i	0.226901	−	0.226901i	0.811397	+	0.584495i	$0.198708\pi$
$228$		0			0
$229$			26.3739i			1.74283i	0.490543	+	0.871417i	$0.336798\pi$
$229$			26.3739i			1.74283i	−0.490543	+	0.871417i	$0.663202\pi$
$230$		0			0
$231$	1.62614	−	1.04356i	0.106992	−	0.0686613i
$232$		0			0
$233$	−14.5762	−	14.5762i	−0.954916	−	0.954916i	0.0441104	−	0.999027i	$-0.485955\pi$
$233$	−14.5762	−	14.5762i	−0.954916	−	0.954916i	−0.999027	+	0.0441104i	$0.985955\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−2.32872	+	2.32872i	−0.151266	+	0.151266i
$238$		0			0
$239$	−20.1216			−1.30156			−0.650779	−	0.759267i	$-0.725557\pi$
$239$	−20.1216			−1.30156			−0.650779	+	0.759267i	$0.725557\pi$
$240$		0			0
$241$			1.62614i			0.104749i	0.998628	+	0.0523743i	$0.0166789\pi$
$241$			1.62614i			0.104749i	−0.998628	+	0.0523743i	$0.983321\pi$
$242$		0			0
$243$	7.92713	−	7.92713i	0.508526	−	0.508526i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	12.5846	+	12.5846i	0.800737	+	0.800737i
$248$		0			0
$249$	−3.37386			−0.213810
$250$		0			0
$251$	2.37386			0.149837			0.0749185	−	0.997190i	$-0.476130\pi$
$251$	2.37386			0.149837			0.0749185	+	0.997190i	$0.476130\pi$
$252$		0			0
$253$	13.0143	+	2.83995i	0.818202	+	0.178546i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	1.29217	+	1.29217i	0.0806032	+	0.0806032i	0.746259	−	0.665656i	$-0.231848\pi$
$257$	1.29217	+	1.29217i	0.0806032	+	0.0806032i	−0.665656	+	0.746259i	$0.731848\pi$
$258$		0			0
$259$	−8.95644			−0.556526
$260$		0			0
$261$			23.3739i			1.44681i
$262$		0			0
$263$	−5.74733	−	5.74733i	−0.354396	−	0.354396i	0.507346	−	0.861742i	$-0.330626\pi$
$263$	−5.74733	−	5.74733i	−0.354396	−	0.354396i	−0.861742	+	0.507346i	$0.830626\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	1.54779	+	1.54779i	0.0947230	+	0.0947230i
$268$		0			0
$269$			6.79129i			0.414072i	0.978333	+	0.207036i	$0.0663817\pi$
$269$			6.79129i			0.414072i	−0.978333	+	0.207036i	$0.933618\pi$
$270$		0			0
$271$			3.83485i			0.232950i	0.993194	+	0.116475i	$0.0371596\pi$
$271$			3.83485i			0.232950i	−0.993194	+	0.116475i	$0.962840\pi$
$272$		0			0
$273$	−1.46629	+	1.46629i	−0.0887440	+	0.0887440i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−6.12372	+	6.12372i	−0.367939	+	0.367939i	−0.866725	−	0.498786i	$-0.833779\pi$
$277$	−6.12372	+	6.12372i	−0.367939	+	0.367939i	0.498786	+	0.866725i	$0.333779\pi$
$278$		0			0
$279$			12.7913i			0.765794i
$280$		0			0
$281$			10.5826i			0.631304i	0.948875	+	0.315652i	$0.102223\pi$
$281$			10.5826i			0.631304i	−0.948875	+	0.315652i	$0.897777\pi$
$282$		0			0
$283$	−13.8627	−	13.8627i	−0.824049	−	0.824049i	0.162637	−	0.986686i	$-0.448000\pi$
$283$	−13.8627	−	13.8627i	−0.824049	−	0.824049i	−0.986686	+	0.162637i	$0.948000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	4.50851	+	4.50851i	0.266129	+	0.266129i
$288$		0			0
$289$		−	37.5390i		−	2.20818i
$290$		0			0
$291$	2.16515			0.126923
$292$		0			0
$293$	−12.2474	−	12.2474i	−0.715504	−	0.715504i	0.252177	−	0.967681i	$-0.418853\pi$
$293$	−12.2474	−	12.2474i	−0.715504	−	0.715504i	−0.967681	+	0.252177i	$0.918853\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−1.87083	+	8.57321i	−0.108556	+	0.497468i
$298$		0			0
$299$	−14.2958			−0.826749
$300$		0			0
$301$	12.3303			0.710707
$302$		0			0
$303$	−2.51691	−	2.51691i	−0.144593	−	0.144593i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	22.5033	−	22.5033i	1.28433	−	1.28433i	0.346153	−	0.938178i	$-0.387488\pi$
$307$	22.5033	−	22.5033i	1.28433	−	1.28433i	0.938178	−	0.346153i	$-0.112512\pi$
$308$		0			0
$309$		−	6.20871i		−	0.353201i
$310$		0			0
$311$	−13.0000			−0.737162			−0.368581	−	0.929596i	$-0.620156\pi$
$311$	−13.0000			−0.737162			−0.368581	+	0.929596i	$0.620156\pi$
$312$		0			0
$313$	−9.15188	+	9.15188i	−0.517295	+	0.517295i	−0.916752	−	0.399457i	$-0.869199\pi$
$313$	−9.15188	+	9.15188i	−0.517295	+	0.517295i	0.399457	+	0.916752i	$0.369199\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−23.1494	−	23.1494i	−1.30020	−	1.30020i	−0.928256	−	0.371943i	$-0.878692\pi$
$317$	−23.1494	−	23.1494i	−1.30020	−	1.30020i	−0.371943	−	0.928256i	$-0.621308\pi$
$318$		0			0
$319$	−15.0000	−	23.3739i	−0.839839	−	1.30869i
$320$		0			0
$321$		−	0.460985i		−	0.0257297i
$322$		0			0
$323$	−26.1101	+	26.1101i	−1.45281	+	1.45281i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	0.525310	−	0.525310i	0.0290497	−	0.0290497i
$328$		0			0
$329$	1.04356			0.0575334
$330$		0			0
$331$	−1.00000			−0.0549650			−0.0274825	−	0.999622i	$-0.508749\pi$
$331$	−1.00000			−0.0549650			−0.0274825	+	0.999622i	$0.508749\pi$
$332$		0			0
$333$	13.8627	−	13.8627i	0.759669	−	0.759669i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	20.8881	−	20.8881i	1.13785	−	1.13785i	0.149011	−	0.988836i	$-0.452391\pi$
$337$	20.8881	−	20.8881i	1.13785	−	1.13785i	0.988836	−	0.149011i	$-0.0476090\pi$
$338$		0			0
$339$			8.04356i			0.436866i
$340$		0			0
$341$	−8.20871	−	12.7913i	−0.444527	−	0.692687i
$342$		0			0
$343$	11.1575	+	11.1575i	0.602451	+	0.602451i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−2.70511	+	2.70511i	−0.145218	+	0.145218i	−0.775978	−	0.630760i	$-0.782743\pi$
$347$	−2.70511	+	2.70511i	−0.145218	+	0.145218i	0.630760	+	0.775978i	$0.282743\pi$
$348$		0			0
$349$	21.7477			1.16413			0.582065	−	0.813143i	$-0.302245\pi$
$349$	21.7477			1.16413			0.582065	+	0.813143i	$0.302245\pi$
$350$		0			0
$351$		−	9.41742i		−	0.502665i
$352$		0			0
$353$	22.8938	−	22.8938i	1.21851	−	1.21851i	0.250359	−	0.968153i	$-0.419451\pi$
$353$	22.8938	−	22.8938i	1.21851	−	1.21851i	0.968153	−	0.250359i	$-0.0805486\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−3.04222	−	3.04222i	−0.161012	−	0.161012i
$358$		0			0
$359$	26.7477			1.41169			0.705846	−	0.708366i	$-0.250567\pi$
$359$	26.7477			1.41169			0.705846	+	0.708366i	$0.250567\pi$
$360$		0			0
$361$	6.00000			0.315789
$362$		0			0
$363$	−1.75006	−	4.71078i	−0.0918541	−	0.247252i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−0.120774	−	0.120774i	−0.00630433	−	0.00630433i	0.703948	−	0.710252i	$-0.251419\pi$
$367$	−0.120774	−	0.120774i	−0.00630433	−	0.00630433i	−0.710252	+	0.703948i	$0.751419\pi$
$368$		0			0
$369$	−13.9564			−0.726543
$370$		0			0
$371$			12.7913i			0.664091i
$372$		0			0
$373$	17.9948	+	17.9948i	0.931734	+	0.931734i	0.997814	−	0.0660799i	$-0.0210492\pi$
$373$	17.9948	+	17.9948i	0.931734	+	0.931734i	−0.0660799	+	0.997814i	$0.521049\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	21.0763	+	21.0763i	1.08548	+	1.08548i
$378$		0			0
$379$			4.16515i			0.213949i	0.994262	+	0.106975i	$0.0341164\pi$
$379$			4.16515i			0.213949i	−0.994262	+	0.106975i	$0.965884\pi$
$380$		0			0
$381$			6.62614i			0.339467i
$382$		0			0
$383$	−17.7925	+	17.7925i	−0.909155	+	0.909155i	−0.996204	−	0.0870491i	$-0.972256\pi$
$383$	−17.7925	+	17.7925i	−0.909155	+	0.909155i	0.0870491	+	0.996204i	$0.472256\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−19.0847	+	19.0847i	−0.970129	+	0.970129i
$388$		0			0
$389$			18.0000i			0.912636i	0.889817	+	0.456318i	$0.150832\pi$
$389$			18.0000i			0.912636i	−0.889817	+	0.456318i	$0.849168\pi$
$390$		0			0
$391$		−	29.6606i		−	1.50000i
$392$		0			0
$393$	1.08990	+	1.08990i	0.0549781	+	0.0549781i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	1.10397	+	1.10397i	0.0554067	+	0.0554067i	0.734267	−	0.678861i	$-0.237526\pi$
$397$	1.10397	+	1.10397i	0.0554067	+	0.0554067i	−0.678861	+	0.734267i	$0.737526\pi$
$398$		0			0
$399$			2.91288i			0.145826i
$400$		0			0
$401$	2.00000			0.0998752			0.0499376	−	0.998752i	$-0.484098\pi$
$401$	2.00000			0.0998752			0.0499376	+	0.998752i	$0.484098\pi$
$402$		0			0
$403$	11.5339	+	11.5339i	0.574547	+	0.574547i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−4.96640	+	22.7589i	−0.246175	+	1.12812i
$408$		0			0
$409$	−11.7477			−0.580888			−0.290444	−	0.956892i	$-0.593803\pi$
$409$	−11.7477			−0.580888			−0.290444	+	0.956892i	$0.593803\pi$
$410$		0			0
$411$	2.62614			0.129538
$412$		0			0
$413$	7.73893	+	7.73893i	0.380808	+	0.380808i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	7.02543	−	7.02543i	0.344037	−	0.344037i
$418$		0			0
$419$		−	16.5826i		−	0.810112i	−0.914292	−	0.405056i	$-0.867252\pi$
$419$		−	16.5826i		−	0.810112i	0.914292	−	0.405056i	$-0.132748\pi$
$420$		0			0
$421$	10.9564			0.533984			0.266992	−	0.963699i	$-0.413970\pi$
$421$	10.9564			0.533984			0.266992	+	0.963699i	$0.413970\pi$
$422$		0			0
$423$	−1.61521	+	1.61521i	−0.0785342	+	0.0785342i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−7.02543	−	7.02543i	−0.339984	−	0.339984i
$428$		0			0
$429$	2.91288	+	4.53901i	0.140635	+	0.219146i
$430$		0			0
$431$			32.9129i			1.58536i	0.609640	+	0.792679i	$0.291314\pi$
$431$			32.9129i			1.58536i	−0.609640	+	0.792679i	$0.708686\pi$
$432$		0			0
$433$	−14.8992	+	14.8992i	−0.716010	+	0.716010i	−0.967786	−	0.251775i	$-0.918986\pi$
$433$	−14.8992	+	14.8992i	−0.716010	+	0.716010i	0.251775	+	0.967786i	$0.418986\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−14.1998	+	14.1998i	−0.679267	+	0.679267i
$438$		0			0
$439$	−3.95644			−0.188831			−0.0944153	−	0.995533i	$-0.530098\pi$
$439$	−3.95644			−0.188831			−0.0944153	+	0.995533i	$0.530098\pi$
$440$		0			0
$441$	−15.0000			−0.714286
$442$		0			0
$443$	18.9498	−	18.9498i	0.900334	−	0.900334i	−0.0951310	−	0.995465i	$-0.530327\pi$
$443$	18.9498	−	18.9498i	0.900334	−	0.900334i	0.995465	+	0.0951310i	$0.0303270\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−1.42701	+	1.42701i	−0.0674954	+	0.0674954i
$448$		0			0
$449$			36.9564i			1.74408i	0.489433	+	0.872041i	$0.337204\pi$
$449$			36.9564i			1.74408i	−0.489433	+	0.872041i	$0.662796\pi$
$450$		0			0
$451$	13.9564	−	8.95644i	0.657183	−	0.421742i
$452$		0			0
$453$	6.64903	+	6.64903i	0.312399	+	0.312399i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	0.901703	−	0.901703i	0.0421799	−	0.0421799i	−0.685702	−	0.727882i	$-0.740505\pi$
$457$	0.901703	−	0.901703i	0.0421799	−	0.0421799i	0.727882	+	0.685702i	$0.240505\pi$
$458$		0			0
$459$	19.5390			0.912003
$460$		0			0
$461$			20.5826i			0.958626i	0.877644	+	0.479313i	$0.159114\pi$
$461$			20.5826i			0.958626i	−0.877644	+	0.479313i	$0.840886\pi$
$462$		0			0
$463$	−6.25857	+	6.25857i	−0.290860	+	0.290860i	−0.837420	−	0.546560i	$-0.815937\pi$
$463$	−6.25857	+	6.25857i	−0.290860	+	0.290860i	0.546560	+	0.837420i	$0.315937\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−20.8207	−	20.8207i	−0.963465	−	0.963465i	0.0358904	−	0.999356i	$-0.488573\pi$
$467$	−20.8207	−	20.8207i	−0.963465	−	0.963465i	−0.999356	+	0.0358904i	$0.988573\pi$
$468$		0			0
$469$	11.1652			0.515559
$470$		0			0
$471$	−6.33030			−0.291685
$472$		0			0
$473$	6.83723	−	31.3321i	0.314376	−	1.44065i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−19.7982	−	19.7982i	−0.906497	−	0.906497i
$478$		0			0
$479$	31.7477			1.45059			0.725295	−	0.688438i	$-0.241703\pi$
$479$	31.7477			1.45059			0.725295	+	0.688438i	$0.241703\pi$
$480$		0			0
$481$		−	25.0000i		−	1.13990i
$482$		0			0
$483$	−1.65449	−	1.65449i	−0.0752818	−	0.0752818i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	0.0674228	+	0.0674228i	0.00305522	+	0.00305522i	0.708633	−	0.705578i	$-0.249312\pi$
$487$	0.0674228	+	0.0674228i	0.00305522	+	0.00305522i	−0.705578	+	0.708633i	$0.749312\pi$
$488$		0			0
$489$		−	6.91288i		−	0.312611i
$490$		0			0
$491$		−	15.0000i		−	0.676941i	−0.940977	−	0.338470i	$-0.890091\pi$
$491$		−	15.0000i		−	0.676941i	0.940977	−	0.338470i	$-0.109909\pi$
$492$		0			0
$493$	−43.7285	+	43.7285i	−1.96943	+	1.96943i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−0.376393	+	0.376393i	−0.0168835	+	0.0168835i
$498$		0			0
$499$			34.1216i			1.52749i	0.645517	+	0.763746i	$0.276642\pi$
$499$			34.1216i			1.52749i	−0.645517	+	0.763746i	$0.723358\pi$
$500$		0			0
$501$			7.20871i			0.322062i
$502$		0			0
$503$	16.9049	+	16.9049i	0.753751	+	0.753751i	0.975177	−	0.221426i	$-0.0710711\pi$
$503$	16.9049	+	16.9049i	0.753751	+	0.753751i	−0.221426	+	0.975177i	$0.571071\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	0.106702	+	0.106702i	0.00473879	+	0.00473879i
$508$		0			0
$509$		−	18.6261i		−	0.825589i	−0.910824	−	0.412794i	$-0.864553\pi$
$509$		−	18.6261i		−	0.825589i	0.910824	−	0.412794i	$-0.135447\pi$
$510$		0			0
$511$	1.62614			0.0719360
$512$		0			0
$513$	−9.35414	−	9.35414i	−0.412996	−	0.412996i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	0.578661	−	2.65176i	0.0254495	−	0.116624i
$518$		0			0
$519$	0.460985			0.0202350
$520$		0			0
$521$	−8.33030			−0.364957			−0.182479	−	0.983210i	$-0.558412\pi$
$521$	−8.33030			−0.364957			−0.182479	+	0.983210i	$0.558412\pi$
$522$		0			0
$523$	3.23042	+	3.23042i	0.141256	+	0.141256i	0.774199	−	0.632942i	$-0.218153\pi$
$523$	3.23042	+	3.23042i	0.141256	+	0.141256i	−0.632942	+	0.774199i	$0.718153\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−23.9303	+	23.9303i	−1.04242	+	1.04242i
$528$		0			0
$529$			6.86932i			0.298666i
$530$		0			0
$531$	−23.9564			−1.03962
$532$		0			0
$533$	−12.5846	+	12.5846i	−0.545098	+	0.545098i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	4.59001	+	4.59001i	0.198074	+	0.198074i
$538$		0			0
$539$	15.0000	−	9.62614i	0.646096	−	0.414627i
$540$		0			0
$541$		−	37.7913i		−	1.62477i	−0.583118	−	0.812387i	$-0.698168\pi$
$541$		−	37.7913i		−	1.62477i	0.583118	−	0.812387i	$-0.301832\pi$
$542$		0			0
$543$	−2.38207	+	2.38207i	−0.102224	+	0.102224i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	14.5762	−	14.5762i	0.623232	−	0.623232i	−0.323125	−	0.946356i	$-0.604733\pi$
$547$	14.5762	−	14.5762i	0.623232	−	0.623232i	0.946356	+	0.323125i	$0.104733\pi$
$548$		0			0
$549$	21.7477			0.928170
$550$		0			0
$551$	41.8693			1.78369
$552$		0			0
$553$	6.50012	−	6.50012i	0.276413	−	0.276413i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−27.0118	+	27.0118i	−1.14453	+	1.14453i	−0.156915	+	0.987612i	$0.550155\pi$
$557$	−27.0118	+	27.0118i	−1.14453	+	1.14453i	−0.987612	+	0.156915i	$0.949845\pi$
$558$		0			0
$559$			34.4174i			1.45570i
$560$		0			0
$561$	−9.41742	+	6.04356i	−0.397604	+	0.255159i
$562$		0			0
$563$	14.9526	+	14.9526i	0.630175	+	0.630175i	0.948112	−	0.317937i	$-0.102990\pi$
$563$	14.9526	+	14.9526i	0.630175	+	0.630175i	−0.317937	+	0.948112i	$0.602990\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−6.46084	+	6.46084i	−0.271330	+	0.271330i
$568$		0			0
$569$	−27.7913			−1.16507			−0.582536	−	0.812805i	$-0.697939\pi$
$569$	−27.7913			−1.16507			−0.582536	+	0.812805i	$0.697939\pi$
$570$		0			0
$571$			15.7042i			0.657199i	0.944469	+	0.328599i	$0.106577\pi$
$571$			15.7042i			0.657199i	−0.944469	+	0.328599i	$0.893423\pi$
$572$		0			0
$573$	−3.62088	+	3.62088i	−0.151265	+	0.151265i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	11.4132	+	11.4132i	0.475136	+	0.475136i	0.903572	−	0.428436i	$-0.140935\pi$
$577$	11.4132	+	11.4132i	0.475136	+	0.475136i	−0.428436	+	0.903572i	$0.640935\pi$
$578$		0			0
$579$	6.74773			0.280426
$580$		0			0
$581$	9.41742			0.390701
$582$		0			0
$583$	32.5035	+	7.09285i	1.34616	+	0.293756i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−12.1941	−	12.1941i	−0.503304	−	0.503304i	0.409159	−	0.912463i	$-0.365822\pi$
$587$	−12.1941	−	12.1941i	−0.503304	−	0.503304i	−0.912463	+	0.409159i	$0.865822\pi$
$588$		0			0
$589$	22.9129			0.944109
$590$		0			0
$591$		−	3.83485i		−	0.157745i
$592$		0			0
$593$	23.4050	+	23.4050i	0.961128	+	0.961128i	0.999272	−	0.0381442i	$-0.0121446\pi$
$593$	23.4050	+	23.4050i	0.961128	+	0.961128i	−0.0381442	+	0.999272i	$0.512145\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	5.01975	+	5.01975i	0.205445	+	0.205445i
$598$		0			0
$599$			24.7042i			1.00938i	0.863299	+	0.504692i	$0.168394\pi$
$599$			24.7042i			1.00938i	−0.863299	+	0.504692i	$0.831606\pi$
$600$		0			0
$601$		−	26.8693i		−	1.09602i	−0.836471	−	0.548011i	$-0.815385\pi$
$601$		−	26.8693i		−	1.09602i	0.836471	−	0.548011i	$-0.184615\pi$
$602$		0			0
$603$	−17.2813	+	17.2813i	−0.703748	+	0.703748i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	20.8881	−	20.8881i	0.847821	−	0.847821i	−0.142040	−	0.989861i	$-0.545366\pi$
$607$	20.8881	−	20.8881i	0.847821	−	0.847821i	0.989861	+	0.142040i	$0.0453661\pi$
$608$		0			0
$609$			4.87841i			0.197683i
$610$		0			0
$611$			2.91288i			0.117842i
$612$		0			0
$613$	20.3235	+	20.3235i	0.820858	+	0.820858i	0.986231	−	0.165373i	$-0.0528827\pi$
$613$	20.3235	+	20.3235i	0.820858	+	0.820858i	−0.165373	+	0.986231i	$0.552883\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	9.55641	+	9.55641i	0.384727	+	0.384727i	0.872802	−	0.488075i	$-0.162301\pi$
$617$	9.55641	+	9.55641i	0.384727	+	0.384727i	−0.488075	+	0.872802i	$0.662301\pi$
$618$		0			0
$619$		−	23.7477i		−	0.954502i	−0.878767	−	0.477251i	$-0.841633\pi$
$619$		−	23.7477i		−	0.954502i	0.878767	−	0.477251i	$-0.158367\pi$
$620$		0			0
$621$	10.6261			0.426412
$622$		0			0
$623$	−4.32032	−	4.32032i	−0.173090	−	0.173090i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	7.40182	+	1.61521i	0.295600	+	0.0645053i
$628$		0			0
$629$	51.8693			2.06817
$630$		0			0
$631$	1.79129			0.0713100			0.0356550	−	0.999364i	$-0.488648\pi$
$631$	1.79129			0.0713100			0.0356550	+	0.999364i	$0.488648\pi$
$632$		0			0
$633$	5.59841	+	5.59841i	0.222517	+	0.222517i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−13.5255	+	13.5255i	−0.535902	+	0.535902i
$638$		0			0
$639$		−	1.16515i		−	0.0460927i
$640$		0			0
$641$	27.8348			1.09941			0.549705	−	0.835359i	$-0.314740\pi$
$641$	27.8348			1.09941			0.549705	+	0.835359i	$0.314740\pi$
$642$		0			0
$643$	−1.93825	+	1.93825i	−0.0764372	+	0.0764372i	−0.744292	−	0.667855i	$-0.767213\pi$
$643$	−1.93825	+	1.93825i	−0.0764372	+	0.0764372i	0.667855	+	0.744292i	$0.267213\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−34.1187	−	34.1187i	−1.34135	−	1.34135i	−0.894722	−	0.446624i	$-0.852626\pi$
$647$	−34.1187	−	34.1187i	−1.34135	−	1.34135i	−0.446624	−	0.894722i	$-0.647374\pi$
$648$		0			0
$649$	23.9564	−	15.3739i	0.940372	−	0.603477i
$650$		0			0
$651$			2.66970i			0.104634i
$652$		0			0
$653$	−16.1773	+	16.1773i	−0.633067	+	0.633067i	−0.948836	−	0.315769i	$-0.897737\pi$
$653$	−16.1773	+	16.1773i	−0.633067	+	0.633067i	0.315769	+	0.948836i	$0.397737\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−2.51691	+	2.51691i	−0.0981941	+	0.0981941i
$658$		0			0
$659$	5.12159			0.199509			0.0997544	−	0.995012i	$-0.468194\pi$
$659$	5.12159			0.199509			0.0997544	+	0.995012i	$0.468194\pi$
$660$		0			0
$661$	6.91288			0.268880			0.134440	−	0.990922i	$-0.457076\pi$
$661$	6.91288			0.268880			0.134440	+	0.990922i	$0.457076\pi$
$662$		0			0
$663$	8.49172	−	8.49172i	0.329791	−	0.329791i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−23.7814	+	23.7814i	−0.920819	+	0.920819i
$668$		0			0
$669$			11.5390i			0.446124i
$670$		0			0
$671$	−21.7477	+	13.9564i	−0.839562	+	0.538782i
$672$		0			0
$673$	−1.08990	−	1.08990i	−0.0420125	−	0.0420125i	0.685788	−	0.727801i	$-0.259458\pi$
$673$	−1.08990	−	1.08990i	−0.0420125	−	0.0420125i	−0.727801	+	0.685788i	$0.759458\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	19.0847	−	19.0847i	0.733484	−	0.733484i	−0.237824	−	0.971308i	$-0.576434\pi$
$677$	19.0847	−	19.0847i	0.733484	−	0.733484i	0.971308	+	0.237824i	$0.0764343\pi$
$678$		0			0
$679$	−6.04356			−0.231931
$680$		0			0
$681$			2.20871i			0.0846380i
$682$		0			0
$683$	3.16300	−	3.16300i	0.121029	−	0.121029i	−0.643998	−	0.765027i	$-0.722726\pi$
$683$	3.16300	−	3.16300i	0.121029	−	0.121029i	0.765027	+	0.643998i	$0.222726\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−8.51986	−	8.51986i	−0.325053	−	0.325053i
$688$		0			0
$689$	−35.7042			−1.36022
$690$		0			0
$691$	42.0345			1.59907			0.799533	−	0.600622i	$-0.205080\pi$
$691$	42.0345			1.59907			0.799533	+	0.600622i	$0.205080\pi$
$692$		0			0
$693$	2.51691	−	11.5339i	0.0956096	−	0.438138i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−26.1101	−	26.1101i	−0.988991	−	0.988991i
$698$		0			0
$699$	9.41742			0.356200
$700$		0			0
$701$		−	29.0780i		−	1.09826i	−0.835736	−	0.549131i	$-0.814959\pi$
$701$		−	29.0780i		−	1.09826i	0.835736	−	0.549131i	$-0.185041\pi$
$702$		0			0
$703$	−24.8320	−	24.8320i	−0.936557	−	0.936557i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	7.02543	+	7.02543i	0.264218	+	0.264218i
$708$		0			0
$709$			3.58258i			0.134546i	0.997735	+	0.0672732i	$0.0214299\pi$
$709$			3.58258i			0.134546i	−0.997735	+	0.0672732i	$0.978570\pi$
$710$		0			0
$711$			20.1216i			0.754619i
$712$		0			0
$713$	−13.0143	+	13.0143i	−0.487390	+	0.487390i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	6.50012	−	6.50012i	0.242751	−	0.242751i
$718$		0			0
$719$			10.9129i			0.406982i	0.979077	+	0.203491i	$0.0652287\pi$
$719$			10.9129i			0.406982i	−0.979077	+	0.203491i	$0.934771\pi$
$720$		0			0
$721$			17.3303i			0.645414i
$722$		0			0
$723$	−0.525310	−	0.525310i	−0.0195365	−	0.0195365i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	30.4978	+	30.4978i	1.13110	+	1.13110i	0.989994	+	0.141108i	$0.0450666\pi$
$727$	30.4978	+	30.4978i	1.13110	+	1.13110i	0.141108	+	0.989994i	$0.454933\pi$
$728$		0			0
$729$		−	16.3739i		−	0.606439i
$730$		0			0
$731$	−71.4083			−2.64113
$732$		0			0
$733$	34.8997	+	34.8997i	1.28905	+	1.28905i	0.935365	+	0.353683i	$0.115071\pi$
$733$	34.8997	+	34.8997i	1.28905	+	1.28905i	0.353683	+	0.935365i	$0.384929\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	6.19115	−	28.3714i	0.228054	−	1.04507i
$738$		0			0
$739$	−14.5390			−0.534826			−0.267413	−	0.963582i	$-0.586169\pi$
$739$	−14.5390			−0.534826			−0.267413	+	0.963582i	$0.586169\pi$
$740$		0			0
$741$	−8.13068			−0.298688
$742$		0			0
$743$	−19.9864	−	19.9864i	−0.733229	−	0.733229i	0.238029	−	0.971258i	$-0.423499\pi$
$743$	−19.9864	−	19.9864i	−0.733229	−	0.733229i	−0.971258	+	0.238029i	$0.923499\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−14.5762	+	14.5762i	−0.533314	+	0.533314i
$748$		0			0
$749$			1.28674i			0.0470165i
$750$		0			0
$751$	−46.7042			−1.70426			−0.852130	−	0.523331i	$-0.824689\pi$
$751$	−46.7042			−1.70426			−0.852130	+	0.523331i	$0.824689\pi$
$752$		0			0
$753$	−0.766857	+	0.766857i	−0.0279458	+	0.0279458i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	8.84291	+	8.84291i	0.321401	+	0.321401i	0.849304	−	0.527904i	$-0.177022\pi$
$757$	8.84291	+	8.84291i	0.321401	+	0.321401i	−0.527904	+	0.849304i	$0.677022\pi$
$758$		0			0
$759$	−5.12159	+	3.28674i	−0.185902	+	0.119301i
$760$		0			0
$761$			7.66970i			0.278026i	0.990291	+	0.139013i	$0.0443930\pi$
$761$			7.66970i			0.278026i	−0.990291	+	0.139013i	$0.955607\pi$
$762$		0			0
$763$	−1.46629	+	1.46629i	−0.0530833	+	0.0530833i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−21.6016	+	21.6016i	−0.779988	+	0.779988i
$768$		0			0
$769$	−21.7477			−0.784243			−0.392122	−	0.919913i	$-0.628259\pi$
$769$	−21.7477			−0.784243			−0.392122	+	0.919913i	$0.628259\pi$
$770$		0			0
$771$	−0.834849			−0.0300663
$772$		0			0
$773$	−15.3571	+	15.3571i	−0.552356	+	0.552356i	−0.927120	−	0.374764i	$-0.877724\pi$
$773$	−15.3571	+	15.3571i	−0.552356	+	0.552356i	0.374764	+	0.927120i	$0.377724\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	2.89331	−	2.89331i	0.103797	−	0.103797i
$778$		0			0
$779$			25.0000i			0.895718i
$780$		0			0
$781$	0.747727	+	1.16515i	0.0267558	+	0.0416924i
$782$		0			0
$783$	−15.6661	−	15.6661i	−0.559859	−	0.559859i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−7.92713	+	7.92713i	−0.282572	+	0.282572i	−0.834134	−	0.551562i	$-0.814032\pi$
$787$	−7.92713	+	7.92713i	−0.282572	+	0.282572i	0.551562	+	0.834134i	$0.314032\pi$
$788$		0			0
$789$	3.71326			0.132195
$790$		0			0
$791$		−	22.4519i		−	0.798297i
$792$		0			0
$793$	19.6100	−	19.6100i	0.696371	−	0.696371i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	2.39614	+	2.39614i	0.0848756	+	0.0848756i	0.748270	−	0.663394i	$-0.230885\pi$
$797$	2.39614	+	2.39614i	0.0848756	+	0.0848756i	−0.663394	+	0.748270i	$0.730885\pi$
$798$		0			0
$799$	−6.04356			−0.213806
$800$		0			0
$801$	13.3739			0.472542
$802$		0			0
$803$	0.901703	−	4.13212i	0.0318204	−	0.145819i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−2.19387	−	2.19387i	−0.0772279	−	0.0772279i
$808$		0			0
$809$	−28.8348			−1.01378			−0.506890	−	0.862011i	$-0.669205\pi$
$809$	−28.8348			−1.01378			−0.506890	+	0.862011i	$0.669205\pi$
$810$		0			0
$811$			25.0000i			0.877869i	0.898519	+	0.438934i	$0.144644\pi$
$811$			25.0000i			0.877869i	−0.898519	+	0.438934i	$0.855356\pi$
$812$		0			0
$813$	−1.23882	−	1.23882i	−0.0434472	−	0.0434472i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	34.1862	+	34.1862i	1.19602	+	1.19602i
$818$		0			0
$819$			12.6697i			0.442715i
$820$		0			0
$821$			3.49545i			0.121992i	0.998138	+	0.0609961i	$0.0194277\pi$
$821$			3.49545i			0.121992i	−0.998138	+	0.0609961i	$0.980572\pi$
$822$		0			0
$823$	−9.52827	+	9.52827i	−0.332135	+	0.332135i	−0.853397	−	0.521262i	$-0.825461\pi$
$823$	−9.52827	+	9.52827i	−0.332135	+	0.332135i	0.521262	+	0.853397i	$0.325461\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−13.2981	+	13.2981i	−0.462419	+	0.462419i	−0.899448	−	0.437028i	$-0.856031\pi$
$827$	−13.2981	+	13.2981i	−0.462419	+	0.462419i	0.437028	+	0.899448i	$0.356031\pi$
$828$		0			0
$829$			45.4519i			1.57861i	0.614002	+	0.789305i	$0.289559\pi$
$829$			45.4519i			1.57861i	−0.614002	+	0.789305i	$0.710441\pi$
$830$		0			0
$831$		−	3.95644i		−	0.137247i
$832$		0			0
$833$	−28.0624	−	28.0624i	−0.972306	−	0.972306i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−8.57321	−	8.57321i	−0.296334	−	0.296334i
$838$		0			0
$839$		−	51.1996i		−	1.76761i	−0.467858	−	0.883804i	$-0.654974\pi$
$839$		−	51.1996i		−	1.76761i	0.467858	−	0.883804i	$-0.345026\pi$
$840$		0			0
$841$	41.1216			1.41799
$842$		0			0
$843$	−3.41862	−	3.41862i	−0.117743	−	0.117743i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	4.88491	+	13.1492i	0.167847	+	0.451810i
$848$		0			0
$849$	8.95644			0.307384
$850$		0			0
$851$	28.2087			0.966982
$852$		0			0
$853$	34.3744	+	34.3744i	1.17696	+	1.17696i	0.980516	+	0.196439i	$0.0629379\pi$
$853$	34.3744	+	34.3744i	1.17696	+	1.17696i	0.196439	+	0.980516i	$0.437062\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	32.0456	−	32.0456i	1.09466	−	1.09466i	0.0996339	−	0.995024i	$-0.468233\pi$
$857$	32.0456	−	32.0456i	1.09466	−	1.09466i	0.995024	−	0.0996339i	$-0.0317672\pi$
$858$		0			0
$859$		−	44.3303i		−	1.51253i	−0.654265	−	0.756265i	$-0.727022\pi$
$859$		−	44.3303i		−	1.51253i	0.654265	−	0.756265i	$-0.272978\pi$
$860$		0			0
$861$	−2.91288			−0.0992706
$862$		0			0
$863$	10.3766	−	10.3766i	0.353224	−	0.353224i	−0.508084	−	0.861308i	$-0.669646\pi$
$863$	10.3766	−	10.3766i	0.353224	−	0.353224i	0.861308	+	0.508084i	$0.169646\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	12.1267	+	12.1267i	0.411843	+	0.411843i
$868$		0			0
$869$	−12.9129	−	20.1216i	−0.438039	−	0.682578i
$870$		0			0
$871$			31.1652i			1.05599i
$872$		0			0
$873$	9.35414	−	9.35414i	0.316590	−	0.316590i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	11.8711	−	11.8711i	0.400857	−	0.400857i	−0.477678	−	0.878535i	$-0.658521\pi$
$877$	11.8711	−	11.8711i	0.400857	−	0.400857i	0.878535	+	0.477678i	$0.158521\pi$
$878$		0			0
$879$	7.91288			0.266895
$880$		0			0
$881$	−7.04356			−0.237304			−0.118652	−	0.992936i	$-0.537857\pi$
$881$	−7.04356			−0.237304			−0.118652	+	0.992936i	$0.537857\pi$
$882$		0			0
$883$	31.2647	−	31.2647i	1.05214	−	1.05214i	0.0535774	−	0.998564i	$-0.482938\pi$
$883$	31.2647	−	31.2647i	1.05214	−	1.05214i	0.998564	−	0.0535774i	$-0.0170624\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	37.0795	−	37.0795i	1.24501	−	1.24501i	0.287108	−	0.957898i	$-0.407306\pi$
$887$	37.0795	−	37.0795i	1.24501	−	1.24501i	0.957898	−	0.287108i	$-0.0926940\pi$
$888$		0			0
$889$		−	18.4955i		−	0.620318i
$890$		0			0
$891$	12.8348	+	20.0000i	0.429984	+	0.670025i
$892$		0			0
$893$	2.89331	+	2.89331i	0.0968208	+	0.0968208i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	4.61815	−	4.61815i	0.154196	−	0.154196i
$898$		0			0
$899$	38.3739			1.27984
$900$		0			0
$901$		−	74.0780i		−	2.46790i
$902$		0			0
$903$	−3.98320	+	3.98320i	−0.132553	+	0.132553i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−22.8263	−	22.8263i	−0.757936	−	0.757936i	0.218010	−	0.975946i	$-0.430043\pi$
$907$	−22.8263	−	22.8263i	−0.757936	−	0.757936i	−0.975946	+	0.218010i	$0.930043\pi$
$908$		0			0
$909$	−21.7477			−0.721327
$910$		0			0
$911$	−34.7477			−1.15124			−0.575622	−	0.817716i	$-0.695240\pi$
$911$	−34.7477			−1.15124			−0.575622	+	0.817716i	$0.695240\pi$
$912$		0			0
$913$	5.22202	−	23.9303i	0.172824	−	0.791978i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−3.04222	−	3.04222i	−0.100463	−	0.100463i
$918$		0			0
$919$	−42.3303			−1.39635			−0.698174	−	0.715928i	$-0.746004\pi$
$919$	−42.3303			−1.39635			−0.698174	+	0.715928i	$0.746004\pi$
$920$		0			0
$921$			14.5390i			0.479077i
$922$		0			0
$923$	−1.05062	−	1.05062i	−0.0345816	−	0.0345816i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−26.8236	−	26.8236i	−0.881003	−	0.881003i
$928$		0			0
$929$		−	33.3303i		−	1.09353i	−0.837286	−	0.546766i	$-0.815859\pi$
$929$		−	33.3303i		−	1.09353i	0.837286	−	0.546766i	$-0.184141\pi$
$930$		0			0
$931$			26.8693i			0.880606i
$932$		0			0
$933$	4.19954	−	4.19954i	0.137487	−	0.137487i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	5.03383	−	5.03383i	0.164448	−	0.164448i	−0.620086	−	0.784534i	$-0.712902\pi$
$937$	5.03383	−	5.03383i	0.164448	−	0.164448i	0.784534	+	0.620086i	$0.212902\pi$
$938$		0			0
$939$		−	5.91288i		−	0.192959i
$940$		0			0
$941$		−	20.5826i		−	0.670973i	−0.942045	−	0.335486i	$-0.891099\pi$
$941$		−	20.5826i		−	0.670973i	0.942045	−	0.335486i	$-0.108901\pi$
$942$		0			0
$943$	−14.1998	−	14.1998i	−0.462408	−	0.462408i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	12.6913	+	12.6913i	0.412411	+	0.412411i	0.882578	−	0.470167i	$-0.155806\pi$
$947$	12.6913	+	12.6913i	0.412411	+	0.412411i	−0.470167	+	0.882578i	$0.655806\pi$
$948$		0			0
$949$			4.53901i			0.147343i
$950$		0			0
$951$	14.9564			0.484996
$952$		0			0
$953$	−5.41022	−	5.41022i	−0.175254	−	0.175254i	0.614029	−	0.789283i	$-0.289548\pi$
$953$	−5.41022	−	5.41022i	−0.175254	−	0.175254i	−0.789283	+	0.614029i	$0.789548\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	12.3964	+	2.70511i	0.400718	+	0.0874438i
$958$		0			0
$959$	−7.33030			−0.236708
$960$		0			0
$961$	−10.0000			−0.322581
$962$		0			0
$963$	−1.99160	−	1.99160i	−0.0641785	−	0.0641785i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	22.1269	−	22.1269i	0.711553	−	0.711553i	−0.255307	−	0.966860i	$-0.582176\pi$
$967$	22.1269	−	22.1269i	0.711553	−	0.711553i	0.966860	+	0.255307i	$0.0821765\pi$
$968$		0			0
$969$		−	16.8693i		−	0.541921i
$970$		0			0
$971$	−10.5390			−0.338213			−0.169107	−	0.985598i	$-0.554088\pi$
$971$	−10.5390			−0.338213			−0.169107	+	0.985598i	$0.554088\pi$
$972$		0			0
$973$	−19.6100	+	19.6100i	−0.628667	+	0.628667i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−0.308970	−	0.308970i	−0.00988482	−	0.00988482i	0.702147	−	0.712032i	$-0.252225\pi$
$977$	−0.308970	−	0.308970i	−0.00988482	−	0.00988482i	−0.712032	+	0.702147i	$0.752225\pi$
$978$		0			0
$979$	−13.3739	+	8.58258i	−0.427431	+	0.274300i
$980$		0			0
$981$		−	4.53901i		−	0.144920i
$982$		0			0
$983$	0.915775	−	0.915775i	0.0292087	−	0.0292087i	−0.692352	−	0.721560i	$-0.743425\pi$
$983$	0.915775	−	0.915775i	0.0292087	−	0.0292087i	0.721560	+	0.692352i	$0.243425\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−0.337114	+	0.337114i	−0.0107305	+	0.0107305i
$988$		0			0
$989$	−38.8348			−1.23488
$990$		0			0
$991$	9.20871			0.292524			0.146262	−	0.989246i	$-0.453276\pi$
$991$	9.20871			0.292524			0.146262	+	0.989246i	$0.453276\pi$
$992$		0			0
$993$	0.323042	−	0.323042i	0.0102514	−	0.0102514i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	22.8797	−	22.8797i	0.724607	−	0.724607i	−0.244933	−	0.969540i	$-0.578766\pi$
$997$	22.8797	−	22.8797i	0.724607	−	0.724607i	0.969540	+	0.244933i	$0.0787660\pi$
$998$		0			0
$999$			18.5826i			0.587927i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1100.2.k.e.857.2	yes	8
5.2	odd	4		1100.2.k.d.593.3	yes	8
5.3	odd	4		1100.2.k.d.593.2	✓	8
5.4	even	2	inner	1100.2.k.e.857.3	yes	8
11.10	odd	2		1100.2.k.d.857.2	yes	8
55.32	even	4	inner	1100.2.k.e.593.3	yes	8
55.43	even	4	inner	1100.2.k.e.593.2	yes	8
55.54	odd	2		1100.2.k.d.857.3	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1100.2.k.d.593.2	✓	8	5.3	odd	4
1100.2.k.d.593.3	yes	8	5.2	odd	4
1100.2.k.d.857.2	yes	8	11.10	odd	2
1100.2.k.d.857.3	yes	8	55.54	odd	2
1100.2.k.e.593.2	yes	8	55.43	even	4	inner
1100.2.k.e.593.3	yes	8	55.32	even	4	inner
1100.2.k.e.857.2	yes	8	1.1	even	1	trivial
1100.2.k.e.857.3	yes	8	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:	\( N \)	\(=\)	\( 1100 = 2^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1100.k (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.323042 + 0.323042i) q^{3} +(0.901703 - 0.901703i) q^{7} +2.79129i q^{9} +O(q^{10})\)	\(q+(-0.323042 + 0.323042i) q^{3} +(0.901703 - 0.901703i) q^{7} +2.79129i q^{9} +(-1.79129 - 2.79129i) q^{11} +(2.51691 + 2.51691i) q^{13} +(-5.22202 + 5.22202i) q^{17} +5.00000 q^{19} +0.582576i q^{21} +(-2.83995 + 2.83995i) q^{23} +(-1.87083 - 1.87083i) q^{27} +8.37386 q^{29} +4.58258 q^{31} +(1.48036 + 0.323042i) q^{33} +(-4.96640 - 4.96640i) q^{37} -1.62614 q^{39} +5.00000i q^{41} +(6.83723 + 6.83723i) q^{43} +(0.578661 + 0.578661i) q^{47} +5.37386i q^{49} -3.37386i q^{51} +(-7.09285 + 7.09285i) q^{53} +(-1.61521 + 1.61521i) q^{57} +8.58258i q^{59} -7.79129i q^{61} +(2.51691 + 2.51691i) q^{63} +(6.19115 + 6.19115i) q^{67} -1.83485i q^{69} -0.417424 q^{71} +(0.901703 + 0.901703i) q^{73} +(-4.13212 - 0.901703i) q^{77} +7.20871 q^{79} -7.16515 q^{81} +(5.22202 + 5.22202i) q^{83} +(-2.70511 + 2.70511i) q^{87} -4.79129i q^{89} +4.53901 q^{91} +(-1.48036 + 1.48036i) q^{93} +(-3.35119 - 3.35119i) q^{97} +(7.79129 - 5.00000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 4 q^{11} + 40 q^{19} + 12 q^{29} - 68 q^{39} - 40 q^{71} + 76 q^{79} + 16 q^{81} - 92 q^{91} + 44 q^{99}+O(q^{100}) \) 8 * q + 4 * q^11 + 40 * q^19 + 12 * q^29 - 68 * q^39 - 40 * q^71 + 76 * q^79 + 16 * q^81 - 92 * q^91 + 44 * q^99

Introduction

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