LMFDB - Embedded newform 440.2.c.b.219.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [440,2,Mod(219,440)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(440, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 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("440.219");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 440 = 2^{3} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	440.c (of order $2$, degree $1$, 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:	$3.51341768894$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.599695360000.19
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{4} + 16 $ x^8 - 3*x^4 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			219.5
Root			$0.413333 - 1.35246i$ of defining polynomial
Character	$\chi$	$=$	440.219
Dual form			440.2.c.b.219.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.413333 - 1.35246i) q^{2} +(-1.65831 - 1.11803i) q^{4} +2.23607i q^{5} +0.856447i q^{7} +(-2.19753 + 1.78069i) q^{8} +3.00000 q^{9} +(3.02420 + 0.924240i) q^{10} +3.31662 q^{11} -6.26630i q^{13} +(1.15831 + 0.353997i) q^{14} +(1.50000 + 3.70810i) q^{16} +6.87506 q^{17} +(1.24000 - 4.05739i) q^{18} +(2.50000 - 3.70810i) q^{20} +(1.37087 - 4.48561i) q^{22} -5.00000 q^{25} +(-8.47494 - 2.59007i) q^{26} +(0.957536 - 1.42026i) q^{28} +8.94427i q^{31} +(5.63507 - 0.496016i) q^{32} +(2.84169 - 9.29827i) q^{34} -1.91507 q^{35} +(-4.97494 - 3.35410i) q^{36} +(-3.98174 - 4.91384i) q^{40} -8.52839 q^{43} +(-5.50000 - 3.70810i) q^{44} +6.70820i q^{45} +6.26650 q^{49} +(-2.06666 + 6.76232i) q^{50} +(-7.00593 + 10.3915i) q^{52} +7.41620i q^{55} +(-1.52506 - 1.88207i) q^{56} -4.00000 q^{59} +(12.0968 + 3.69696i) q^{62} +2.56934i q^{63} +(1.65831 - 7.82624i) q^{64} +14.0119 q^{65} +(-11.4010 - 7.68655i) q^{68} +(-0.791562 + 2.59007i) q^{70} -14.8324i q^{71} +(-6.59260 + 5.34206i) q^{72} +0.261743 q^{73} +2.84051i q^{77} +(-8.29156 + 3.35410i) q^{80} +9.00000 q^{81} -12.3585 q^{83} +15.3731i q^{85} +(-3.52506 + 11.5343i) q^{86} +(-7.28840 + 5.90587i) q^{88} -13.2665 q^{89} +(9.07260 + 2.77272i) q^{90} +5.36675 q^{91} +(2.59015 - 8.47521i) q^{98} +9.94987 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 24 q^{9} - 4 q^{14} + 12 q^{16} + 20 q^{20} - 40 q^{25} - 28 q^{26} + 36 q^{34} - 44 q^{44} - 56 q^{49} - 52 q^{56} - 32 q^{59} + 60 q^{70} + 72 q^{81} - 68 q^{86} + 96 q^{91}+O(q^{100}) $ 8 * q + 24 * q^9 - 4 * q^14 + 12 * q^16 + 20 * q^20 - 40 * q^25 - 28 * q^26 + 36 * q^34 - 44 * q^44 - 56 * q^49 - 52 * q^56 - 32 * q^59 + 60 * q^70 + 72 * q^81 - 68 * q^86 + 96 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/440\mathbb{Z}\right)^\times$.

$n$	$111$	$177$	$221$	$321$
$\chi(n)$	$-1$	$-1$	$-1$	$-1$

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 $

$2$	0.413333	−	1.35246i	0.292270	−	0.956336i
$2$	0.413333	−	1.35246i	0.292270	−	0.956336i
$3$		0			0		1.00000			$0$
$3$		0			0		−1.00000			$\pi$
$4$	−1.65831	−	1.11803i	−0.829156	−	0.559017i
$5$			2.23607i			1.00000i
$5$			2.23607i			1.00000i
$6$		0			0
$7$			0.856447i			0.323706i	0.986815	+	0.161853i	$0.0517471\pi$
$7$			0.856447i			0.323706i	−0.986815	+	0.161853i	$0.948253\pi$
$8$	−2.19753	+	1.78069i	−0.776946	+	0.629568i
$9$	3.00000			1.00000
$10$	3.02420	+	0.924240i	0.956336	+	0.292270i
$11$	3.31662			1.00000
$11$	3.31662			1.00000
$12$		0			0
$13$		−	6.26630i		−	1.73796i	−0.494848	−	0.868979i	$-0.664776\pi$
$13$		−	6.26630i		−	1.73796i	0.494848	−	0.868979i	$-0.335224\pi$
$14$	1.15831	+	0.353997i	0.309572	+	0.0946098i
$15$		0			0
$16$	1.50000	+	3.70810i	0.375000	+	0.927025i
$17$	6.87506			1.66745			0.833724	−	0.552182i	$-0.186204\pi$
$17$	6.87506			1.66745			0.833724	+	0.552182i	$0.186204\pi$
$18$	1.24000	−	4.05739i	0.292270	−	0.956336i
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$	2.50000	−	3.70810i	0.559017	−	0.829156i
$21$		0			0
$22$	1.37087	−	4.48561i	0.292270	−	0.956336i
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$		0			0
$25$	−5.00000			−1.00000
$26$	−8.47494	−	2.59007i	−1.66207	−	0.507954i
$27$		0			0
$28$	0.957536	−	1.42026i	0.180957	−	0.268403i
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$			8.94427i			1.60644i	0.595683	+	0.803219i	$0.296881\pi$
$31$			8.94427i			1.60644i	−0.595683	+	0.803219i	$0.703119\pi$
$32$	5.63507	−	0.496016i	0.996148	−	0.0876841i
$33$		0			0
$34$	2.84169	−	9.29827i	0.487345	−	1.59464i
$35$	−1.91507			−0.323706
$36$	−4.97494	−	3.35410i	−0.829156	−	0.559017i
$37$		0			0			−	1.00000i	$-0.5\pi$
$37$		0			0				1.00000i	$0.5\pi$
$38$		0			0
$39$		0			0
$40$	−3.98174	−	4.91384i	−0.629568	−	0.776946i
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$	−8.52839			−1.30057			−0.650284	−	0.759691i	$-0.725350\pi$
$43$	−8.52839			−1.30057			−0.650284	+	0.759691i	$0.725350\pi$
$44$	−5.50000	−	3.70810i	−0.829156	−	0.559017i
$45$			6.70820i			1.00000i
$46$		0			0
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$		0			0
$49$	6.26650			0.895214
$50$	−2.06666	+	6.76232i	−0.292270	+	0.956336i
$51$		0			0
$52$	−7.00593	+	10.3915i	−0.971548	+	1.44104i
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$		0			0
$55$			7.41620i			1.00000i
$56$	−1.52506	−	1.88207i	−0.203795	−	0.251502i
$57$		0			0
$58$		0			0
$59$	−4.00000			−0.520756			−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000			−0.520756			−0.260378	+	0.965507i	$0.583847\pi$
$60$		0			0
$61$		0			0			−	1.00000i	$-0.5\pi$
$61$		0			0				1.00000i	$0.5\pi$
$62$	12.0968	+	3.69696i	1.53629	+	0.469514i
$63$			2.56934i			0.323706i
$64$	1.65831	−	7.82624i	0.207289	−	0.978280i
$65$	14.0119			1.73796
$66$		0			0
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$	−11.4010	−	7.68655i	−1.38257	−	0.932132i
$69$		0			0
$70$	−0.791562	+	2.59007i	−0.0946098	+	0.309572i
$71$		−	14.8324i		−	1.76028i	−0.474713	−	0.880141i	$-0.657448\pi$
$71$		−	14.8324i		−	1.76028i	0.474713	−	0.880141i	$-0.342552\pi$
$72$	−6.59260	+	5.34206i	−0.776946	+	0.629568i
$73$	0.261743			0.0306347			0.0153173	−	0.999883i	$-0.495124\pi$
$73$	0.261743			0.0306347			0.0153173	+	0.999883i	$0.495124\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			2.84051i			0.323706i
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$	−8.29156	+	3.35410i	−0.927025	+	0.375000i
$81$	9.00000			1.00000
$82$		0			0
$83$	−12.3585			−1.35653			−0.678263	−	0.734819i	$-0.737267\pi$
$83$	−12.3585			−1.35653			−0.678263	+	0.734819i	$0.737267\pi$
$84$		0			0
$85$			15.3731i			1.66745i
$86$	−3.52506	+	11.5343i	−0.380117	+	1.24378i
$87$		0			0
$88$	−7.28840	+	5.90587i	−0.776946	+	0.629568i
$89$	−13.2665			−1.40625			−0.703123	−	0.711068i	$-0.748212\pi$
$89$	−13.2665			−1.40625			−0.703123	+	0.711068i	$0.748212\pi$
$90$	9.07260	+	2.77272i	0.956336	+	0.292270i
$91$	5.36675			0.562588
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$	2.59015	−	8.47521i	0.261644	−	0.856125i
$99$	9.94987			1.00000
$100$	8.29156	+	5.59017i	0.829156	+	0.559017i
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$		0			0
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$	11.1583	+	13.7704i	1.09416	+	1.35030i
$105$		0			0
$106$		0			0
$107$	−18.9719			−1.83408			−0.917039	−	0.398796i	$-0.869428\pi$
$107$	−18.9719			−1.83408			−0.917039	+	0.398796i	$0.869428\pi$
$108$		0			0
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$	10.0301	+	3.06536i	0.956336	+	0.292270i
$111$		0			0
$112$	−3.17579	+	1.28467i	−0.300084	+	0.121390i
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		−	18.7989i		−	1.73796i
$118$	−1.65333	+	5.40985i	−0.152201	+	0.498017i
$119$			5.88813i			0.539764i
$120$		0			0
$121$	11.0000			1.00000
$122$		0			0
$123$		0			0
$124$	10.0000	−	14.8324i	0.898027	−	1.33199i
$125$		−	11.1803i		−	1.00000i
$126$	3.47494	+	1.06199i	0.309572	+	0.0946098i
$127$			22.4959i			1.99618i	0.0617417	+	0.998092i	$0.480334\pi$
$127$			22.4959i			1.99618i	−0.0617417	+	0.998092i	$0.519666\pi$
$128$	−9.89926	−	5.47765i	−0.874979	−	0.484160i
$129$		0			0
$130$	5.79156	−	18.9505i	0.507954	−	1.66207i
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	−15.1082	+	12.2423i	−1.29552	+	1.04977i
$137$		0			0		1.00000			$0$
$137$		0			0		−1.00000			$\pi$
$138$		0			0
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$	3.17579	+	2.14112i	0.268403	+	0.180957i
$141$		0			0
$142$	−20.0603	−	6.13071i	−1.68342	−	0.514478i
$143$		−	20.7830i		−	1.73796i
$144$	4.50000	+	11.1243i	0.375000	+	0.927025i
$145$		0			0
$146$	0.108187	−	0.353997i	0.00895360	−	0.0292970i
$147$		0			0
$148$		0			0
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$		0			0
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$		0			0
$153$	20.6252			1.66745
$154$	3.84169	+	1.17408i	0.309572	+	0.0946098i
$155$	−20.0000			−1.60644
$156$		0			0
$157$		0			0			−	1.00000i	$-0.5\pi$
$157$		0			0				1.00000i	$0.5\pi$
$158$		0			0
$159$		0			0
$160$	1.10913	+	12.6004i	0.0876841	+	0.996148i
$161$		0			0
$162$	3.71999	−	12.1722i	0.292270	−	0.956336i
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$		0			0
$165$		0			0
$166$	−5.10819	+	16.7145i	−0.396472	+	1.29729i
$167$			24.2087i			1.87333i	0.350228	+	0.936665i	$0.386104\pi$
$167$			24.2087i			1.87333i	−0.350228	+	0.936665i	$0.613896\pi$
$168$		0			0
$169$	−26.2665			−2.02050
$170$	20.7916	+	6.35421i	1.59464	+	0.487345i
$171$		0			0
$172$	14.1427	+	9.53503i	1.07837	+	0.727040i
$173$			9.69209i			0.736876i	0.929652	+	0.368438i	$0.120107\pi$
$173$			9.69209i			0.736876i	−0.929652	+	0.368438i	$0.879893\pi$
$174$		0			0
$175$		−	4.28223i		−	0.323706i
$176$	4.97494	+	12.2984i	0.375000	+	0.927025i
$177$		0			0
$178$	−5.48348	+	17.9424i	−0.411004	+	1.34484i
$179$	−19.8997			−1.48738			−0.743689	−	0.668526i	$-0.766925\pi$
$179$	−19.8997			−1.48738			−0.743689	+	0.668526i	$0.766925\pi$
$180$	7.50000	−	11.1243i	0.559017	−	0.829156i
$181$		−	4.47214i		−	0.332411i	−0.986091	−	0.166206i	$-0.946848\pi$
$181$		−	4.47214i		−	0.332411i	0.986091	−	0.166206i	$-0.0531515\pi$
$182$	2.21825	−	7.25833i	0.164428	−	0.538023i
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	22.8020			1.66745
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	26.8328i		−	1.94155i	−0.239983	−	0.970777i	$-0.577142\pi$
$191$		−	26.8328i		−	1.94155i	0.239983	−	0.970777i	$-0.422858\pi$
$192$		0			0
$193$	−27.7620			−1.99835			−0.999176	−	0.0405839i	$-0.987078\pi$
$193$	−27.7620			−1.99835			−0.999176	+	0.0405839i	$0.987078\pi$
$194$		0			0
$195$		0			0
$196$	−10.3918	−	7.00616i	−0.742272	−	0.500440i
$197$		−	27.9057i		−	1.98820i	−0.108471	−	0.994100i	$-0.534595\pi$
$197$		−	27.9057i		−	1.98820i	0.108471	−	0.994100i	$-0.465405\pi$
$198$	4.11261	−	13.4568i	0.292270	−	0.956336i
$199$		−	14.8324i		−	1.05144i	−0.850657	−	0.525720i	$-0.823796\pi$
$199$		−	14.8324i		−	1.05144i	0.850657	−	0.525720i	$-0.176204\pi$
$200$	10.9877	−	8.90343i	0.776946	−	0.629568i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	23.2361	−	9.39945i	1.61113	−	0.651734i
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$		0			0
$214$	−7.84169	+	25.6587i	−0.536047	+	1.75400i
$215$		−	19.0701i		−	1.30057i
$216$		0			0
$217$	−7.66029			−0.520014
$218$		0			0
$219$		0			0
$220$	8.29156	−	12.2984i	0.559017	−	0.829156i
$221$		−	43.0812i		−	2.89796i
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$	0.424812	+	4.82613i	0.0283839	+	0.322460i
$225$	−15.0000			−1.00000
$226$		0			0
$227$	29.4153			1.95236			0.976182	−	0.216954i	$-0.0696120\pi$
$227$	29.4153			1.95236			0.976182	+	0.216954i	$0.0696120\pi$
$228$		0			0
$229$			29.6648i			1.96030i	0.198246	+	0.980152i	$0.436476\pi$
$229$			29.6648i			1.96030i	−0.198246	+	0.980152i	$0.563524\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	7.39855			0.484695			0.242348	−	0.970190i	$-0.422083\pi$
$233$	7.39855			0.484695			0.242348	+	0.970190i	$0.422083\pi$
$234$	−25.4248	−	7.77020i	−1.66207	−	0.507954i
$235$		0			0
$236$	6.63325	+	4.47214i	0.431788	+	0.291111i
$237$		0			0
$238$	7.96347	+	2.43375i	0.516195	+	0.157757i
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$	4.54666	−	14.8771i	0.292270	−	0.956336i
$243$		0			0
$244$		0			0
$245$			14.0123i			0.895214i
$246$		0			0
$247$		0			0
$248$	−15.9269	−	19.6553i	−1.01136	−	1.24812i
$249$		0			0
$250$	−15.1210	−	4.62120i	−0.956336	−	0.292270i
$251$	28.0000			1.76734			0.883672	−	0.468106i	$-0.155064\pi$
$251$	28.0000			1.76734			0.883672	+	0.468106i	$0.155064\pi$
$252$	2.87261	−	4.26077i	0.180957	−	0.268403i
$253$		0			0
$254$	30.4248	+	9.29827i	1.90902	+	0.583425i
$255$		0			0
$256$	−11.5000	+	11.1243i	−0.718750	+	0.695269i
$257$		0			0		1.00000			$0$
$257$		0			0		−1.00000			$\pi$
$258$		0			0
$259$		0			0
$260$	−23.2361	−	15.6657i	−1.44104	−	0.971548i
$261$		0			0
$262$		0			0
$263$		−	25.9216i		−	1.59840i	−0.601067	−	0.799198i	$-0.705258\pi$
$263$		−	25.9216i		−	1.59840i	0.601067	−	0.799198i	$-0.294742\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	29.6648i		−	1.80869i	−0.426798	−	0.904347i	$-0.640358\pi$
$269$		−	29.6648i		−	1.80869i	0.426798	−	0.904347i	$-0.359642\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$	10.3126	+	25.4934i	0.625293	+	1.54577i
$273$		0			0
$274$		0			0
$275$	−16.5831			−1.00000
$276$		0			0
$277$		−	11.9473i		−	0.717845i	−0.933367	−	0.358923i	$-0.883144\pi$
$277$		−	11.9473i		−	0.717845i	0.933367	−	0.358923i	$-0.116856\pi$
$278$		0			0
$279$			26.8328i			1.60644i
$280$	4.20844	−	3.41014i	0.251502	−	0.203795i
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$	4.69825			0.279282			0.139641	−	0.990202i	$-0.455405\pi$
$283$	4.69825			0.279282			0.139641	+	0.990202i	$0.455405\pi$
$284$	−16.5831	+	24.5967i	−0.984027	+	1.45955i
$285$		0			0
$286$	−28.1082	−	8.59027i	−1.66207	−	0.507954i
$287$		0			0
$288$	16.9052	−	1.48805i	0.996148	−	0.0876841i
$289$	30.2665			1.78038
$290$		0			0
$291$		0			0
$292$	−0.434051	−	0.292637i	−0.0254009	−	0.0171253i
$293$			24.4799i			1.43013i	0.699057	+	0.715066i	$0.253603\pi$
$293$			24.4799i			1.43013i	−0.699057	+	0.715066i	$0.746397\pi$
$294$		0			0
$295$		−	8.94427i		−	0.520756i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		−	7.30411i		−	0.421002i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$	8.52506	−	27.8948i	0.487345	−	1.59464i
$307$	−15.1417			−0.864183			−0.432092	−	0.901830i	$-0.642224\pi$
$307$	−15.1417			−0.864183			−0.432092	+	0.901830i	$0.642224\pi$
$308$	3.17579	−	4.71046i	0.180957	−	0.268403i
$309$		0			0
$310$	−8.26665	+	27.0493i	−0.469514	+	1.53629i
$311$			14.8324i			0.841068i	0.907277	+	0.420534i	$0.138157\pi$
$311$			14.8324i			0.841068i	−0.907277	+	0.420534i	$0.861843\pi$
$312$		0			0
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$		0			0
$315$	−5.74522			−0.323706
$316$		0			0
$317$		0			0			−	1.00000i	$-0.5\pi$
$317$		0			0				1.00000i	$0.5\pi$
$318$		0			0
$319$		0			0
$320$	17.5000	+	3.70810i	0.978280	+	0.207289i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−14.9248	−	10.0623i	−0.829156	−	0.559017i
$325$			31.3315i			1.73796i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	6.63325			0.364596			0.182298	−	0.983243i	$-0.441646\pi$
$331$	6.63325			0.364596			0.182298	+	0.983243i	$0.441646\pi$
$332$	20.4943	+	13.8173i	1.12477	+	0.758321i
$333$		0			0
$334$	32.7414	+	10.0063i	1.79153	+	0.547518i
$335$		0			0
$336$		0			0
$337$	−34.8988			−1.90106			−0.950529	−	0.310634i	$-0.899459\pi$
$337$	−34.8988			−1.90106			−0.950529	+	0.310634i	$0.899459\pi$
$338$	−10.8568	+	35.5245i	−0.590532	+	1.93228i
$339$		0			0
$340$	17.1877	−	25.4934i	0.932132	−	1.38257i
$341$			29.6648i			1.60644i
$342$		0			0
$343$			11.3620i			0.613493i
$344$	18.7414	−	15.1864i	1.01047	−	0.818796i
$345$		0			0
$346$	13.1082	+	4.00605i	0.704700	+	0.215367i
$347$	33.2455			1.78471			0.892355	−	0.451334i	$-0.149052\pi$
$347$	33.2455			1.78471			0.892355	+	0.451334i	$0.149052\pi$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$	−5.79156	−	1.76999i	−0.309572	−	0.0946098i
$351$		0			0
$352$	18.6894	−	1.64510i	0.996148	−	0.0876841i
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$		0			0
$355$	33.1662			1.76028
$356$	22.0000	+	14.8324i	1.16600	+	0.786115i
$357$		0			0
$358$	−8.22521	+	26.9137i	−0.434716	+	1.42243i
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$	−11.9452	−	14.7415i	−0.629568	−	0.776946i
$361$	19.0000			1.00000
$362$	−6.04840	−	1.84848i	−0.317897	−	0.0971539i
$363$		0			0
$364$	−8.89975	−	6.00021i	−0.466474	−	0.314496i
$365$			0.585274i			0.0306347i
$366$		0			0
$367$		0			0			−	1.00000i	$-0.5\pi$
$367$		0			0				1.00000i	$0.5\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$			22.2247i			1.15075i	0.817890	+	0.575375i	$0.195144\pi$
$373$			22.2247i			1.15075i	−0.817890	+	0.575375i	$0.804856\pi$
$374$	9.42481	−	30.8389i	0.487345	−	1.59464i
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−36.0000			−1.84920			−0.924598	−	0.380945i	$-0.875599\pi$
$379$	−36.0000			−1.84920			−0.924598	+	0.380945i	$0.875599\pi$
$380$		0			0
$381$		0			0
$382$	−36.2904	−	11.0909i	−1.85678	−	0.567459i
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$		0			0
$385$	−6.35158			−0.323706
$386$	−11.4749	+	37.5471i	−0.584059	+	1.91110i
$387$	−25.5852			−1.30057
$388$		0			0
$389$			29.6648i			1.50406i	0.659126	+	0.752032i	$0.270926\pi$
$389$			29.6648i			1.50406i	−0.659126	+	0.752032i	$0.729074\pi$
$390$		0			0
$391$		0			0
$392$	−13.7708	+	11.1587i	−0.695533	+	0.563598i
$393$		0			0
$394$	−37.7414	−	11.5343i	−1.90139	−	0.581092i
$395$		0			0
$396$	−16.5000	−	11.1243i	−0.829156	−	0.559017i
$397$		0			0			−	1.00000i	$-0.5\pi$
$397$		0			0				1.00000i	$0.5\pi$
$398$	−20.0603	−	6.13071i	−1.00553	−	0.307305i
$399$		0			0
$400$	−7.50000	−	18.5405i	−0.375000	−	0.927025i
$401$	−39.7995			−1.98749			−0.993746	−	0.111664i	$-0.964382\pi$
$401$	−39.7995			−1.98749			−0.993746	+	0.111664i	$0.964382\pi$
$402$		0			0
$403$	56.0475			2.79192
$404$		0			0
$405$			20.1246i			1.00000i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	3.42579i		−	0.168572i
$414$		0			0
$415$		−	27.6345i		−	1.35653i
$416$	−3.10819	−	35.3110i	−0.152391	−	1.73126i
$417$		0			0
$418$		0			0
$419$	19.8997			0.972166			0.486083	−	0.873913i	$-0.338425\pi$
$419$	19.8997			0.972166			0.486083	+	0.873913i	$0.338425\pi$
$420$		0			0
$421$			31.3050i			1.52571i	0.646570	+	0.762855i	$0.276203\pi$
$421$			31.3050i			1.52571i	−0.646570	+	0.762855i	$0.723797\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−34.3753			−1.66745
$426$		0			0
$427$		0			0
$428$	31.4613	+	21.2112i	1.52074	+	1.02528i
$429$		0			0
$430$	−25.7916	−	7.88228i	−1.24378	−	0.380117i
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		0			0
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$	−3.16625	+	10.3603i	−0.151985	+	0.497308i
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$	−13.2059	−	16.2973i	−0.629568	−	0.776946i
$441$	18.7995			0.895214
$442$	−58.2657	−	17.8069i	−2.77142	−	0.846986i
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$		0			0
$445$		−	29.6648i		−	1.40625i
$446$		0			0
$447$		0			0
$448$	6.70276	+	1.42026i	0.316675	+	0.0671008i
$449$	13.2665			0.626085			0.313042	−	0.949739i	$-0.398652\pi$
$449$	13.2665			0.626085			0.313042	+	0.949739i	$0.398652\pi$
$450$	−6.19999	+	20.2869i	−0.292270	+	0.956336i
$451$		0			0
$452$		0			0
$453$		0			0
$454$	12.1583	−	39.7831i	0.570618	−	1.86712i
$455$			12.0004i			0.562588i
$456$		0			0
$457$	−41.5121			−1.94186			−0.970928	−	0.239373i	$-0.923058\pi$
$457$	−41.5121			−1.94186			−0.970928	+	0.239373i	$0.923058\pi$
$458$	40.1205	+	12.2614i	1.87471	+	0.572939i
$459$		0			0
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$		0			0			−	1.00000i	$-0.5\pi$
$463$		0			0				1.00000i	$0.5\pi$
$464$		0			0
$465$		0			0
$466$	3.05806	−	10.0063i	0.141662	−	0.463531i
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$	−21.0178	+	31.1744i	−0.971548	+	1.44104i
$469$		0			0
$470$		0			0
$471$		0			0
$472$	8.79014	−	7.12275i	0.404599	−	0.327851i
$473$	−28.2855			−1.30057
$474$		0			0
$475$		0			0
$476$	6.58312	−	9.76435i	0.301737	−	0.447548i
$477$		0			0
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−18.2414	−	12.2984i	−0.829156	−	0.559017i
$485$		0			0
$486$		0			0
$487$		0			0			−	1.00000i	$-0.5\pi$
$487$		0			0				1.00000i	$0.5\pi$
$488$		0			0
$489$		0			0
$490$	18.9511	+	5.79175i	0.856125	+	0.261644i
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$			22.2486i			1.00000i
$496$	−33.1662	+	13.4164i	−1.48921	+	0.602414i
$497$	12.7032			0.569814
$498$		0			0
$499$	4.00000			0.179065			0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000			0.179065			0.0895323	+	0.995984i	$0.471463\pi$
$500$	−12.5000	+	18.5405i	−0.559017	+	0.829156i
$501$		0			0
$502$	11.5733	−	37.8690i	0.516542	−	1.69017i
$503$			44.1353i			1.96789i	0.178461	+	0.983947i	$0.442888\pi$
$503$			44.1353i			1.96789i	−0.178461	+	0.983947i	$0.557112\pi$
$504$	−4.57519	−	5.64621i	−0.203795	−	0.251502i
$505$		0			0
$506$		0			0
$507$		0			0
$508$	25.1511	−	37.3052i	1.11590	−	1.65515i
$509$		−	29.6648i		−	1.31487i	−0.753512	−	0.657434i	$-0.771642\pi$
$509$		−	29.6648i		−	1.31487i	0.753512	−	0.657434i	$-0.228358\pi$
$510$		0			0
$511$			0.224169i			0.00991664i
$512$	10.2919	+	20.1514i	0.454841	+	0.890573i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	−30.7916	+	24.9507i	−1.35030	+	1.09416i
$521$	39.7995			1.74365			0.871824	−	0.489820i	$-0.162937\pi$
$521$	39.7995			1.74365			0.871824	+	0.489820i	$0.162937\pi$
$522$		0			0
$523$	36.0286			1.57542			0.787711	−	0.616044i	$-0.211266\pi$
$523$	36.0286			1.57542			0.787711	+	0.616044i	$0.211266\pi$
$524$		0			0
$525$		0			0
$526$	−35.0581	−	10.7143i	−1.52860	−	0.467164i
$527$			61.4924i			2.67865i
$528$		0			0
$529$	−23.0000			−1.00000
$530$		0			0
$531$	−12.0000			−0.520756
$532$		0			0
$533$		0			0
$534$		0			0
$535$		−	42.4224i		−	1.83408i
$536$		0			0
$537$		0			0
$538$	−40.1205	−	12.2614i	−1.72972	−	0.528628i
$539$	20.7836			0.895214
$540$		0			0
$541$		0			0			−	1.00000i	$-0.5\pi$
$541$		0			0				1.00000i	$0.5\pi$
$542$		0			0
$543$		0			0
$544$	38.7414	−	3.41014i	1.66103	−	0.146209i
$545$		0			0
$546$		0			0
$547$	26.6322			1.13871			0.569354	−	0.822092i	$-0.307193\pi$
$547$	26.6322			1.13871			0.569354	+	0.822092i	$0.307193\pi$
$548$		0			0
$549$		0			0
$550$	−6.85435	+	22.4281i	−0.292270	+	0.956336i
$551$		0			0
$552$		0			0
$553$		0			0
$554$	−16.1583	−	4.93822i	−0.686501	−	0.209805i
$555$		0			0
$556$		0			0
$557$		−	13.1179i		−	0.555822i	−0.960607	−	0.277911i	$-0.910358\pi$
$557$		−	13.1179i		−	0.555822i	0.960607	−	0.277911i	$-0.0896420\pi$
$558$	36.2904	+	11.0909i	1.53629	+	0.469514i
$559$			53.4415i			2.26033i
$560$	−2.87261	−	7.10128i	−0.121390	−	0.300084i
$561$		0			0
$562$		0			0
$563$	−43.6889			−1.84127			−0.920635	−	0.390425i	$-0.872328\pi$
$563$	−43.6889			−1.84127			−0.920635	+	0.390425i	$0.872328\pi$
$564$		0			0
$565$		0			0
$566$	1.94194	−	6.35421i	0.0816258	−	0.267087i
$567$			7.70802i			0.323706i
$568$	26.4118	+	32.5947i	1.10822	+	1.36764i
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$	−23.2361	+	34.4646i	−0.971548	+	1.44104i
$573$		0			0
$574$		0			0
$575$		0			0
$576$	4.97494	−	23.4787i	0.207289	−	0.978280i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	12.5101	−	40.9343i	0.520353	−	1.70264i
$579$		0			0
$580$		0			0
$581$		−	10.5844i		−	0.439116i
$582$		0			0
$583$		0			0
$584$	−0.575188	+	0.466082i	−0.0238015	+	0.0192866i
$585$	42.0356			1.73796
$586$	33.1082	+	10.1183i	1.36769	+	0.417985i
$587$		0			0		1.00000			$0$
$587$		0			0		−1.00000			$\pi$
$588$		0			0
$589$		0			0
$590$	−12.0968	−	3.69696i	−0.498017	−	0.152201i
$591$		0			0
$592$		0			0
$593$	48.6489			1.99777			0.998886	−	0.0471872i	$-0.0150257\pi$
$593$	48.6489			1.99777			0.998886	+	0.0471872i	$0.0150257\pi$
$594$		0			0
$595$	−13.1662			−0.539764
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	44.7214i		−	1.82727i	−0.406541	−	0.913633i	$-0.633265\pi$
$599$		−	44.7214i		−	1.82727i	0.406541	−	0.913633i	$-0.366735\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$	−9.87854	−	3.01903i	−0.402619	−	0.123046i
$603$		0			0
$604$		0			0
$605$			24.5967i			1.00000i
$606$		0			0
$607$		−	47.5610i		−	1.93044i	−0.261433	−	0.965222i	$-0.584195\pi$
$607$		−	47.5610i		−	1.93044i	0.261433	−	0.965222i	$-0.415805\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$	−34.2030	−	23.0597i	−1.38257	−	0.932132i
$613$		−	21.0541i		−	0.850368i	−0.905107	−	0.425184i	$-0.860209\pi$
$613$		−	21.0541i		−	0.850368i	0.905107	−	0.425184i	$-0.139791\pi$
$614$	−6.25856	+	20.4786i	−0.252575	+	0.826449i
$615$		0			0
$616$	−5.05806	−	6.24212i	−0.203795	−	0.251502i
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$	46.4327			1.86629			0.933145	−	0.359501i	$-0.117053\pi$
$619$	46.4327			1.86629			0.933145	+	0.359501i	$0.117053\pi$
$620$	33.1662	+	22.3607i	1.33199	+	0.898027i
$621$		0			0
$622$	20.0603	+	6.13071i	0.804343	+	0.245819i
$623$		−	11.3620i		−	0.455211i
$624$		0			0
$625$	25.0000			1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$	−2.37469	+	7.77020i	−0.0946098	+	0.309572i
$631$			14.8324i			0.590468i	0.955425	+	0.295234i	$0.0953977\pi$
$631$			14.8324i			0.590468i	−0.955425	+	0.295234i	$0.904602\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−50.3023			−1.99618
$636$		0			0
$637$		−	39.2678i		−	1.55585i
$638$		0			0
$639$		−	44.4972i		−	1.76028i
$640$	12.2484	−	22.1354i	0.484160	−	0.874979i
$641$	−39.7995			−1.57199			−0.785993	−	0.618236i	$-0.787848\pi$
$641$	−39.7995			−1.57199			−0.785993	+	0.618236i	$0.787848\pi$
$642$		0			0
$643$		0			0		1.00000			$0$
$643$		0			0		−1.00000			$\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$	−19.7778	+	16.0262i	−0.776946	+	0.629568i
$649$	−13.2665			−0.520756
$650$	42.3747	+	12.9503i	1.66207	+	0.507954i
$651$		0			0
$652$		0			0
$653$		0			0			−	1.00000i	$-0.5\pi$
$653$		0			0				1.00000i	$0.5\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	0.785228			0.0306347
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$			29.6648i			1.15383i	0.816805	+	0.576913i	$0.195743\pi$
$661$			29.6648i			1.15383i	−0.816805	+	0.576913i	$0.804257\pi$
$662$	2.74174	−	8.97122i	0.106561	−	0.348677i
$663$		0			0
$664$	27.1583	−	22.0067i	1.05395	−	0.854025i
$665$		0			0
$666$		0			0
$667$		0			0
$668$	27.0662	−	40.1457i	1.04722	−	1.55328i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	20.1017			0.774864			0.387432	−	0.921898i	$-0.373362\pi$
$673$	20.1017			0.774864			0.387432	+	0.921898i	$0.373362\pi$
$674$	−14.4248	+	47.1993i	−0.555623	+	1.81805i
$675$		0			0
$676$	43.5581	+	29.3668i	1.67531	+	1.12949i
$677$		−	34.7573i		−	1.33583i	−0.744237	−	0.667915i	$-0.767187\pi$
$677$		−	34.7573i		−	1.33583i	0.744237	−	0.667915i	$-0.232813\pi$
$678$		0			0
$679$		0			0
$680$	−27.3747	−	33.7829i	−1.04977	−	1.29552i
$681$		0			0
$682$	40.1205	+	12.2614i	1.53629	+	0.469514i
$683$		0			0		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$		0			0
$685$		0			0
$686$	15.3668	+	4.69630i	0.586705	+	0.179306i
$687$		0			0
$688$	−12.7926	−	31.6241i	−0.487713	−	1.20566i
$689$		0			0
$690$		0			0
$691$	−28.0000			−1.06517			−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000			−1.06517			−0.532585	+	0.846376i	$0.678779\pi$
$692$	10.8361	−	16.0725i	0.411926	−	0.610985i
$693$			8.52154i			0.323706i
$694$	13.7414	−	44.9633i	0.521618	−	1.70678i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−4.78768	+	7.10128i	−0.180957	+	0.268403i
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$		0			0
$704$	5.50000	−	25.9567i	0.207289	−	0.978280i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$			4.47214i			0.167955i	0.996468	+	0.0839773i	$0.0267623\pi$
$709$			4.47214i			0.167955i	−0.996468	+	0.0839773i	$0.973238\pi$
$710$	13.7087	−	44.8561i	0.514478	−	1.68342i
$711$		0			0
$712$	29.1536	−	23.6235i	1.09258	−	0.885327i
$713$		0			0
$714$		0			0
$715$	46.4721			1.73796
$716$	33.0000	+	22.2486i	1.23327	+	0.831469i
$717$		0			0
$718$		0			0
$719$		−	26.8328i		−	1.00070i	−0.865825	−	0.500348i	$-0.833206\pi$
$719$		−	26.8328i		−	1.00070i	0.865825	−	0.500348i	$-0.166794\pi$
$720$	−24.8747	+	10.0623i	−0.927025	+	0.375000i
$721$		0			0
$722$	7.85332	−	25.6968i	0.292270	−	0.956336i
$723$		0			0
$724$	−5.00000	+	7.41620i	−0.185824	+	0.275621i
$725$		0			0
$726$		0			0
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$	−11.7936	+	9.55650i	−0.437101	+	0.354187i
$729$	27.0000			1.00000
$730$	0.791562	+	0.241913i	0.0292970	+	0.00895360i
$731$	−58.6332			−2.16863
$732$		0			0
$733$			52.9709i			1.95652i	0.207371	+	0.978262i	$0.433509\pi$
$733$			52.9709i			1.95652i	−0.207371	+	0.978262i	$0.566491\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		1.00000			$0$
$739$		0			0		−1.00000			$\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	49.2739i		−	1.80769i	−0.427865	−	0.903843i	$-0.640734\pi$
$743$		−	49.2739i		−	1.80769i	0.427865	−	0.903843i	$-0.359266\pi$
$744$		0			0
$745$		0			0
$746$	30.0581	+	9.18618i	1.10050	+	0.336330i
$747$	−37.0756			−1.35653
$748$	−37.8128	−	25.4934i	−1.38257	−	0.932132i
$749$		−	16.2484i		−	0.593703i
$750$		0			0
$751$			44.4972i			1.62373i	0.583848	+	0.811863i	$0.301546\pi$
$751$			44.4972i			1.62373i	−0.583848	+	0.811863i	$0.698454\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0			−	1.00000i	$-0.5\pi$
$757$		0			0				1.00000i	$0.5\pi$
$758$	−14.8800	+	48.6887i	−0.540465	+	1.76845i
$759$		0			0
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$		0			0
$764$	−30.0000	+	44.4972i	−1.08536	+	1.60985i
$765$			46.1193i			1.66745i
$766$		0			0
$767$			25.0652i			0.905052i
$768$		0			0
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$	−2.62531	+	8.59027i	−0.0946098	+	0.309572i
$771$		0			0
$772$	46.0381	+	31.0389i	1.65695	+	1.11711i
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$	−10.5752	+	34.6030i	−0.380117	+	1.24378i
$775$		−	44.7214i		−	1.60644i
$776$		0			0
$777$		0			0
$778$	40.1205	+	12.2614i	1.43839	+	0.439593i
$779$		0			0
$780$		0			0
$781$		−	49.1935i		−	1.76028i
$782$		0			0
$783$		0			0
$784$	9.39975	+	23.2368i	0.335705	+	0.829886i
$785$		0			0
$786$		0			0
$787$	11.3116			0.403214			0.201607	−	0.979467i	$-0.435384\pi$
$787$	11.3116			0.403214			0.201607	+	0.979467i	$0.435384\pi$
$788$	−31.1995	+	46.2764i	−1.11144	+	1.64853i
$789$		0			0
$790$		0			0
$791$		0			0
$792$	−21.8652	+	17.7176i	−0.776946	+	0.629568i
$793$		0			0
$794$		0			0
$795$		0			0
$796$	−16.5831	+	24.5967i	−0.587773	+	0.871809i
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$	−28.1753	+	2.48008i	−0.996148	+	0.0876841i
$801$	−39.7995			−1.40625
$802$	−16.4504	+	53.8273i	−0.580885	+	1.90071i
$803$	0.868102			0.0306347
$804$		0			0
$805$		0			0
$806$	23.1662	−	75.8021i	0.815996	−	2.67002i
$807$		0			0
$808$		0			0
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$	27.2178	+	8.31816i	0.956336	+	0.292270i
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$	16.1003			0.562588
$820$		0			0
$821$		0			0			−	1.00000i	$-0.5\pi$
$821$		0			0				1.00000i	$0.5\pi$
$822$		0			0
$823$		0			0			−	1.00000i	$-0.5\pi$
$823$		0			0				1.00000i	$0.5\pi$
$824$		0			0
$825$		0			0
$826$	−4.63325	−	1.41599i	−0.161211	−	0.0492686i
$827$	−21.7550			−0.756497			−0.378248	−	0.925704i	$-0.623473\pi$
$827$	−21.7550			−0.756497			−0.378248	+	0.925704i	$0.623473\pi$
$828$		0			0
$829$		−	22.3607i		−	0.776619i	−0.921529	−	0.388309i	$-0.873059\pi$
$829$		−	22.3607i		−	0.776619i	0.921529	−	0.388309i	$-0.126941\pi$
$830$	−37.3747	−	11.4223i	−1.29729	−	0.396472i
$831$		0			0
$832$	−49.0415	−	10.3915i	−1.70021	−	0.360260i
$833$	43.0826			1.49272
$834$		0			0
$835$	−54.1324			−1.87333
$836$		0			0
$837$		0			0
$838$	8.22521	−	26.9137i	0.284135	−	0.929717i
$839$		−	14.8324i		−	0.512071i	−0.966667	−	0.256036i	$-0.917584\pi$
$839$		−	14.8324i		−	0.512071i	0.966667	−	0.256036i	$-0.0824164\pi$
$840$		0			0
$841$	−29.0000			−1.00000
$842$	42.3388	+	12.9394i	1.45909	+	0.445920i
$843$		0			0
$844$		0			0
$845$		−	58.7337i		−	2.02050i
$846$		0			0
$847$			9.42091i			0.323706i
$848$		0			0
$849$		0			0
$850$	−14.2084	+	46.4913i	−0.487345	+	1.59464i
$851$		0			0
$852$		0			0
$853$		−	25.6505i		−	0.878255i	−0.898425	−	0.439128i	$-0.855288\pi$
$853$		−	25.6505i		−	0.878255i	0.898425	−	0.439128i	$-0.144712\pi$
$854$		0			0
$855$		0			0
$856$	41.6913	−	33.7829i	1.42498	−	1.15468i
$857$	55.2623			1.88772			0.943861	−	0.330342i	$-0.107164\pi$
$857$	55.2623			1.88772			0.943861	+	0.330342i	$0.107164\pi$
$858$		0			0
$859$	−46.4327			−1.58426			−0.792132	−	0.610349i	$-0.791029\pi$
$859$	−46.4327			−1.58426			−0.792132	+	0.610349i	$0.791029\pi$
$860$	−21.3210	+	31.6241i	−0.727040	+	1.07837i
$861$		0			0
$862$		0			0
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$		0			0
$865$	−21.6722			−0.736876
$866$		0			0
$867$		0			0
$868$	12.7032	+	8.56447i	0.431173	+	0.290697i
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$	9.57536			0.323706
$876$		0			0
$877$		−	4.01106i		−	0.135444i	−0.997704	−	0.0677220i	$-0.978427\pi$
$877$		−	4.01106i		−	0.135444i	0.997704	−	0.0677220i	$-0.0215731\pi$
$878$		0			0
$879$		0			0
$880$	−27.5000	+	11.1243i	−0.927025	+	0.375000i
$881$	−2.00000			−0.0673817			−0.0336909	−	0.999432i	$-0.510726\pi$
$881$	−2.00000			−0.0673817			−0.0336909	+	0.999432i	$0.510726\pi$
$882$	7.77044	−	25.4256i	0.261644	−	0.856125i
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$	−48.1662	+	71.4421i	−1.62001	+	2.40286i
$885$		0			0
$886$		0			0
$887$			31.0603i			1.04290i	0.853281	+	0.521452i	$0.174609\pi$
$887$			31.0603i			1.04290i	−0.853281	+	0.521452i	$0.825391\pi$
$888$		0			0
$889$	−19.2665			−0.646178
$890$	−40.1205	−	12.2614i	−1.34484	−	0.411004i
$891$	29.8496			1.00000
$892$		0			0
$893$		0			0
$894$		0			0
$895$		−	44.4972i		−	1.48738i
$896$	4.69131	−	8.47819i	0.156726	−	0.283236i
$897$		0			0
$898$	5.48348	−	17.9424i	0.182986	−	0.598747i
$899$		0			0
$900$	24.8747	+	16.7705i	0.829156	+	0.559017i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	10.0000			0.332411
$906$		0			0
$907$		0			0		1.00000			$0$
$907$		0			0		−1.00000			$\pi$
$908$	−48.7798	−	32.8873i	−1.61881	−	1.09140i
$909$		0			0
$910$	16.2301	+	4.96016i	0.538023	+	0.164428i
$911$			8.94427i			0.296337i	0.988962	+	0.148168i	$0.0473378\pi$
$911$			8.94427i			0.296337i	−0.988962	+	0.148168i	$0.952662\pi$
$912$		0			0
$913$	−40.9886			−1.35653
$914$	−17.1583	+	56.1436i	−0.567547	+	1.85707i
$915$		0			0
$916$	33.1662	−	49.1935i	1.09584	−	1.62540i
$917$		0			0
$918$		0			0
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−92.9442			−3.05930
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	14.0000			0.459325			0.229663	−	0.973270i	$-0.426238\pi$
$929$	14.0000			0.459325			0.229663	+	0.973270i	$0.426238\pi$
$930$		0			0
$931$		0			0
$932$	−12.2691	−	8.27183i	−0.401888	−	0.270953i
$933$		0			0
$934$		0			0
$935$			50.9868i			1.66745i
$936$	33.4749	+	41.3112i	1.09416	+	1.35030i
$937$	49.1724			1.60639			0.803196	−	0.595714i	$-0.203131\pi$
$937$	49.1724			1.60639			0.803196	+	0.595714i	$0.203131\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$		0			0
$943$		0			0
$944$	−6.00000	−	14.8324i	−0.195283	−	0.482753i
$945$		0			0
$946$	−11.6913	+	38.2551i	−0.380117	+	1.24378i
$947$		0			0		1.00000			$0$
$947$		0			0		−1.00000			$\pi$
$948$		0			0
$949$		−	1.64016i		−	0.0532418i
$950$		0			0
$951$		0			0
$952$	−10.4849	−	12.9394i	−0.339818	−	0.419367i
$953$	−12.9649			−0.419974			−0.209987	−	0.977704i	$-0.567342\pi$
$953$	−12.9649			−0.419974			−0.209987	+	0.977704i	$0.567342\pi$
$954$		0			0
$955$	60.0000			1.94155
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−49.0000			−1.58065
$962$		0			0
$963$	−56.9156			−1.83408
$964$		0			0
$965$		−	62.0777i		−	1.99835i
$966$		0			0
$967$			37.2837i			1.19896i	0.800389	+	0.599481i	$0.204626\pi$
$967$			37.2837i			1.19896i	−0.800389	+	0.599481i	$0.795374\pi$
$968$	−24.1729	+	19.5875i	−0.776946	+	0.629568i
$969$		0			0
$970$		0			0
$971$	−59.6992			−1.91584			−0.957920	−	0.287035i	$-0.907330\pi$
$971$	−59.6992			−1.91584			−0.957920	+	0.287035i	$0.907330\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$		0			0		1.00000			$0$
$977$		0			0		−1.00000			$\pi$
$978$		0			0
$979$	−44.0000			−1.40625
$980$	15.6662	−	23.2368i	0.500440	−	0.742272i
$981$		0			0
$982$		0			0
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$		0			0
$985$	62.3991			1.98820
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$	30.0904	+	9.19607i	0.956336	+	0.292270i
$991$		−	62.6099i		−	1.98887i	−0.105356	−	0.994435i	$-0.533598\pi$
$991$		−	62.6099i		−	1.98887i	0.105356	−	0.994435i	$-0.466402\pi$
$992$	4.43651	+	50.4016i	0.140859	+	1.60025i
$993$		0			0
$994$	5.25063	−	17.1805i	0.166540	−	0.544934i
$995$	33.1662			1.05144
$996$		0			0
$997$			58.6519i			1.85753i	0.370675	+	0.928763i	$0.379126\pi$
$997$			58.6519i			1.85753i	−0.370675	+	0.928763i	$0.620874\pi$
$998$	1.65333	−	5.40985i	0.0523353	−	0.171246i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	440.2.c.b.219.5	yes	8
4.3	odd	2		1760.2.c.b.879.6		8
5.4	even	2	inner	440.2.c.b.219.4	yes	8
8.3	odd	2	inner	440.2.c.b.219.6	yes	8
8.5	even	2		1760.2.c.b.879.3		8
11.10	odd	2	inner	440.2.c.b.219.4	yes	8
20.19	odd	2		1760.2.c.b.879.7		8
40.19	odd	2	inner	440.2.c.b.219.3	✓	8
40.29	even	2		1760.2.c.b.879.2		8
44.43	even	2		1760.2.c.b.879.7		8
55.54	odd	2	CM	440.2.c.b.219.5	yes	8
88.21	odd	2		1760.2.c.b.879.2		8
88.43	even	2	inner	440.2.c.b.219.3	✓	8
220.219	even	2		1760.2.c.b.879.6		8
440.109	odd	2		1760.2.c.b.879.3		8
440.219	even	2	inner	440.2.c.b.219.6	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
440.2.c.b.219.3	✓	8	40.19	odd	2	inner
440.2.c.b.219.3	✓	8	88.43	even	2	inner
440.2.c.b.219.4	yes	8	5.4	even	2	inner
440.2.c.b.219.4	yes	8	11.10	odd	2	inner
440.2.c.b.219.5	yes	8	1.1	even	1	trivial
440.2.c.b.219.5	yes	8	55.54	odd	2	CM
440.2.c.b.219.6	yes	8	8.3	odd	2	inner
440.2.c.b.219.6	yes	8	440.219	even	2	inner
1760.2.c.b.879.2		8	40.29	even	2
1760.2.c.b.879.2		8	88.21	odd	2
1760.2.c.b.879.3		8	8.5	even	2
1760.2.c.b.879.3		8	440.109	odd	2
1760.2.c.b.879.6		8	4.3	odd	2
1760.2.c.b.879.6		8	220.219	even	2
1760.2.c.b.879.7		8	20.19	odd	2
1760.2.c.b.879.7		8	44.43	even	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:	\( N \)	\(=\)	\( 440 = 2^{3} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	440.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(0.413333 - 1.35246i) q^{2} +(-1.65831 - 1.11803i) q^{4} +2.23607i q^{5} +0.856447i q^{7} +(-2.19753 + 1.78069i) q^{8} +3.00000 q^{9} +(3.02420 + 0.924240i) q^{10} +3.31662 q^{11} -6.26630i q^{13} +(1.15831 + 0.353997i) q^{14} +(1.50000 + 3.70810i) q^{16} +6.87506 q^{17} +(1.24000 - 4.05739i) q^{18} +(2.50000 - 3.70810i) q^{20} +(1.37087 - 4.48561i) q^{22} -5.00000 q^{25} +(-8.47494 - 2.59007i) q^{26} +(0.957536 - 1.42026i) q^{28} +8.94427i q^{31} +(5.63507 - 0.496016i) q^{32} +(2.84169 - 9.29827i) q^{34} -1.91507 q^{35} +(-4.97494 - 3.35410i) q^{36} +(-3.98174 - 4.91384i) q^{40} -8.52839 q^{43} +(-5.50000 - 3.70810i) q^{44} +6.70820i q^{45} +6.26650 q^{49} +(-2.06666 + 6.76232i) q^{50} +(-7.00593 + 10.3915i) q^{52} +7.41620i q^{55} +(-1.52506 - 1.88207i) q^{56} -4.00000 q^{59} +(12.0968 + 3.69696i) q^{62} +2.56934i q^{63} +(1.65831 - 7.82624i) q^{64} +14.0119 q^{65} +(-11.4010 - 7.68655i) q^{68} +(-0.791562 + 2.59007i) q^{70} -14.8324i q^{71} +(-6.59260 + 5.34206i) q^{72} +0.261743 q^{73} +2.84051i q^{77} +(-8.29156 + 3.35410i) q^{80} +9.00000 q^{81} -12.3585 q^{83} +15.3731i q^{85} +(-3.52506 + 11.5343i) q^{86} +(-7.28840 + 5.90587i) q^{88} -13.2665 q^{89} +(9.07260 + 2.77272i) q^{90} +5.36675 q^{91} +(2.59015 - 8.47521i) q^{98} +9.94987 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 24 q^{9} - 4 q^{14} + 12 q^{16} + 20 q^{20} - 40 q^{25} - 28 q^{26} + 36 q^{34} - 44 q^{44} - 56 q^{49} - 52 q^{56} - 32 q^{59} + 60 q^{70} + 72 q^{81} - 68 q^{86} + 96 q^{91}+O(q^{100}) \) 8 * q + 24 * q^9 - 4 * q^14 + 12 * q^16 + 20 * q^20 - 40 * q^25 - 28 * q^26 + 36 * q^34 - 44 * q^44 - 56 * q^49 - 52 * q^56 - 32 * q^59 + 60 * q^70 + 72 * q^81 - 68 * q^86 + 96 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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