LMFDB - Embedded newform 440.2.c.b.219.1

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			219.1
Root			$-1.35246 - 0.413333i$ of defining polynomial
Character	$\chi$	$=$	440.219
Dual form			440.2.c.b.219.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.35246 - 0.413333i) q^{2} +(1.65831 + 1.11803i) q^{4} -2.23607i q^{5} -5.22173i q^{7} +(-1.78069 - 2.19753i) q^{8} +3.00000 q^{9} +(-0.924240 + 3.02420i) q^{10} -3.31662 q^{11} +3.56840i q^{13} +(-2.15831 + 7.06220i) q^{14} +(1.50000 + 3.70810i) q^{16} -4.55341 q^{17} +(-4.05739 - 1.24000i) q^{18} +(2.50000 - 3.70810i) q^{20} +(4.48561 + 1.37087i) q^{22} -5.00000 q^{25} +(1.47494 - 4.82613i) q^{26} +(5.83808 - 8.65927i) q^{28} -8.94427i q^{31} +(-0.496016 - 5.63507i) q^{32} +(6.15831 + 1.88207i) q^{34} -11.6762 q^{35} +(4.97494 + 3.35410i) q^{36} +(-4.91384 + 3.98174i) q^{40} +9.96326 q^{43} +(-5.50000 - 3.70810i) q^{44} -6.70820i q^{45} -20.2665 q^{49} +(6.76232 + 2.06666i) q^{50} +(-3.98960 + 5.91753i) q^{52} +7.41620i q^{55} +(-11.4749 + 9.29827i) q^{56} -4.00000 q^{59} +(-3.69696 + 12.0968i) q^{62} -15.6652i q^{63} +(-1.65831 + 7.82624i) q^{64} +7.97919 q^{65} +(-7.55097 - 5.09086i) q^{68} +(15.7916 + 4.82613i) q^{70} -14.8324i q^{71} +(-5.34206 - 6.59260i) q^{72} +17.0860 q^{73} +17.3185i q^{77} +(8.29156 - 3.35410i) q^{80} +9.00000 q^{81} -13.3890 q^{83} +10.1817i q^{85} +(-13.4749 - 4.11814i) q^{86} +(5.90587 + 7.28840i) q^{88} +13.2665 q^{89} +(-2.77272 + 9.07260i) q^{90} +18.6332 q^{91} +(27.4097 + 8.37680i) q^{98} -9.94987 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.35246	−	0.413333i	−0.956336	−	0.292270i
$2$	−1.35246	−	0.413333i	−0.956336	−	0.292270i
$3$		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
$7$		−	5.22173i		−	1.97363i	−0.161853	−	0.986815i	$-0.551747\pi$
$7$		−	5.22173i		−	1.97363i	0.161853	−	0.986815i	$-0.448253\pi$
$8$	−1.78069	−	2.19753i	−0.629568	−	0.776946i
$9$	3.00000			1.00000
$10$	−0.924240	+	3.02420i	−0.292270	+	0.956336i
$11$	−3.31662			−1.00000
$11$	−3.31662			−1.00000
$12$		0			0
$13$			3.56840i			0.989697i	0.868979	+	0.494848i	$0.164776\pi$
$13$			3.56840i			0.989697i	−0.868979	+	0.494848i	$0.835224\pi$
$14$	−2.15831	+	7.06220i	−0.576833	+	1.88745i
$15$		0			0
$16$	1.50000	+	3.70810i	0.375000	+	0.927025i
$17$	−4.55341			−1.10436			−0.552182	−	0.833724i	$-0.686204\pi$
$17$	−4.55341			−1.10436			−0.552182	+	0.833724i	$0.686204\pi$
$18$	−4.05739	−	1.24000i	−0.956336	−	0.292270i
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$	2.50000	−	3.70810i	0.559017	−	0.829156i
$21$		0			0
$22$	4.48561	+	1.37087i	0.956336	+	0.292270i
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$		0			0
$25$	−5.00000			−1.00000
$26$	1.47494	−	4.82613i	0.289259	−	0.946483i
$27$		0			0
$28$	5.83808	−	8.65927i	1.10329	−	1.63645i
$29$		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	$-0.703119\pi$
$31$		−	8.94427i		−	1.60644i	0.595683	−	0.803219i	$-0.296881\pi$
$32$	−0.496016	−	5.63507i	−0.0876841	−	0.996148i
$33$		0			0
$34$	6.15831	+	1.88207i	1.05614	+	0.322772i
$35$	−11.6762			−1.97363
$36$	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
$40$	−4.91384	+	3.98174i	−0.776946	+	0.629568i
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$	9.96326			1.51938			0.759691	−	0.650284i	$-0.225350\pi$
$43$	9.96326			1.51938			0.759691	+	0.650284i	$0.225350\pi$
$44$	−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
$49$	−20.2665			−2.89521
$50$	6.76232	+	2.06666i	0.956336	+	0.292270i
$51$		0			0
$52$	−3.98960	+	5.91753i	−0.553257	+	0.820613i
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$		0			0
$55$			7.41620i			1.00000i
$56$	−11.4749	+	9.29827i	−1.53340	+	1.24253i
$57$		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	$0.5\pi$
$62$	−3.69696	+	12.0968i	−0.469514	+	1.53629i
$63$		−	15.6652i		−	1.97363i
$64$	−1.65831	+	7.82624i	−0.207289	+	0.978280i
$65$	7.97919			0.989697
$66$		0			0
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$	−7.55097	−	5.09086i	−0.915689	−	0.617358i
$69$		0			0
$70$	15.7916	+	4.82613i	1.88745	+	0.576833i
$71$		−	14.8324i		−	1.76028i	−0.474713	−	0.880141i	$-0.657448\pi$
$71$		−	14.8324i		−	1.76028i	0.474713	−	0.880141i	$-0.342552\pi$
$72$	−5.34206	−	6.59260i	−0.629568	−	0.776946i
$73$	17.0860			1.99977			0.999883	−	0.0153173i	$-0.00487585\pi$
$73$	17.0860			1.99977			0.999883	+	0.0153173i	$0.00487585\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			17.3185i			1.97363i
$78$		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
$83$	−13.3890			−1.46964			−0.734819	−	0.678263i	$-0.762733\pi$
$83$	−13.3890			−1.46964			−0.734819	+	0.678263i	$0.762733\pi$
$84$		0			0
$85$			10.1817i			1.10436i
$86$	−13.4749	−	4.11814i	−1.45304	−	0.444070i
$87$		0			0
$88$	5.90587	+	7.28840i	0.629568	+	0.776946i
$89$	13.2665			1.40625			0.703123	−	0.711068i	$-0.251788\pi$
$89$	13.2665			1.40625			0.703123	+	0.711068i	$0.251788\pi$
$90$	−2.77272	+	9.07260i	−0.292270	+	0.956336i
$91$	18.6332			1.95330
$92$		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			$\pi$
$98$	27.4097	+	8.37680i	2.76880	+	0.846185i
$99$	−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	$0.5\pi$
$104$	7.84169	−	6.35421i	0.768941	−	0.623081i
$105$		0			0
$106$		0			0
$107$	8.25036			0.797593			0.398796	−	0.917039i	$-0.369428\pi$
$107$	8.25036			0.797593			0.398796	+	0.917039i	$0.369428\pi$
$108$		0			0
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$	3.06536	−	10.0301i	0.292270	−	0.956336i
$111$		0			0
$112$	19.3627	−	7.83260i	1.82960	−	0.740111i
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$			10.7052i			0.989697i
$118$	5.40985	+	1.65333i	0.498017	+	0.152201i
$119$			23.7767i			2.17960i
$120$		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
$126$	−6.47494	+	21.1866i	−0.576833	+	1.88745i
$127$			1.39159i			0.123483i	0.998092	+	0.0617417i	$0.0196655\pi$
$127$			1.39159i			0.123483i	−0.998092	+	0.0617417i	$0.980334\pi$
$128$	5.47765	−	9.89926i	0.484160	−	0.874979i
$129$		0			0
$130$	−10.7916	−	3.29806i	−0.946483	−	0.289259i
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	8.10819	+	10.0063i	0.695271	+	0.858030i
$137$		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			$\pi$
$140$	−19.3627	−	13.0543i	−1.63645	−	1.10329i
$141$		0			0
$142$	−6.13071	+	20.0603i	−0.514478	+	1.68342i
$143$		−	11.8351i		−	0.989697i
$144$	4.50000	+	11.1243i	0.375000	+	0.927025i
$145$		0			0
$146$	−23.1082	−	7.06220i	−1.91245	−	0.584472i
$147$		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
$153$	−13.6602			−1.10436
$154$	7.15831	−	23.4227i	0.576833	−	1.88745i
$155$	−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
$160$	−12.6004	+	1.10913i	−0.996148	+	0.0876841i
$161$		0			0
$162$	−12.1722	−	3.71999i	−0.956336	−	0.292270i
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$		0			0
$165$		0			0
$166$	18.1082	+	5.53413i	1.40547	+	0.429532i
$167$		−	9.05188i		−	0.700455i	−0.936665	−	0.350228i	$-0.886104\pi$
$167$		−	9.05188i		−	0.700455i	0.936665	−	0.350228i	$-0.113896\pi$
$168$		0			0
$169$	0.266499			0.0204999
$170$	4.20844	−	13.7704i	0.322772	−	1.05614i
$171$		0			0
$172$	16.5222	+	11.1393i	1.25981	+	0.849361i
$173$		−	24.4553i		−	1.85930i	−0.368438	−	0.929652i	$-0.620107\pi$
$173$		−	24.4553i		−	1.85930i	0.368438	−	0.929652i	$-0.379893\pi$
$174$		0			0
$175$			26.1087i			1.97363i
$176$	−4.97494	−	12.2984i	−0.375000	−	0.927025i
$177$		0			0
$178$	−17.9424	−	5.48348i	−1.34484	−	0.411004i
$179$	19.8997			1.48738			0.743689	−	0.668526i	$-0.233075\pi$
$179$	19.8997			1.48738			0.743689	+	0.668526i	$0.233075\pi$
$180$	7.50000	−	11.1243i	0.559017	−	0.829156i
$181$			4.47214i			0.332411i	0.986091	+	0.166206i	$0.0531515\pi$
$181$			4.47214i			0.332411i	−0.986091	+	0.166206i	$0.946848\pi$
$182$	−25.2008	−	7.70173i	−1.86801	−	0.570890i
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	15.1019			1.10436
$188$		0			0
$189$		0			0
$190$		0			0
$191$			26.8328i			1.94155i	0.239983	+	0.970777i	$0.422858\pi$
$191$			26.8328i			1.94155i	−0.239983	+	0.970777i	$0.577142\pi$
$192$		0			0
$193$	1.12762			0.0811678			0.0405839	−	0.999176i	$-0.487078\pi$
$193$	1.12762			0.0811678			0.0405839	+	0.999176i	$0.487078\pi$
$194$		0			0
$195$		0			0
$196$	−33.6082	−	22.6586i	−2.40058	−	1.61847i
$197$		−	3.04492i		−	0.216941i	−0.994100	−	0.108471i	$-0.965405\pi$
$197$		−	3.04492i		−	0.216941i	0.994100	−	0.108471i	$-0.0345954\pi$
$198$	13.4568	+	4.11261i	0.956336	+	0.292270i
$199$		−	14.8324i		−	1.05144i	−0.850657	−	0.525720i	$-0.823796\pi$
$199$		−	14.8324i		−	1.05144i	0.850657	−	0.525720i	$-0.176204\pi$
$200$	8.90343	+	10.9877i	0.629568	+	0.776946i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−13.2320	+	5.35260i	−0.917474	+	0.371136i
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$		0			0
$214$	−11.1583	−	3.41014i	−0.762767	−	0.233113i
$215$		−	22.2785i		−	1.51938i
$216$		0			0
$217$	−46.7046			−3.17052
$218$		0			0
$219$		0			0
$220$	−8.29156	+	12.2984i	−0.559017	+	0.829156i
$221$		−	16.2484i		−	1.09298i
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$	−29.4248	+	2.59007i	−1.96603	+	0.173056i
$225$	−15.0000			−1.00000
$226$		0			0
$227$	−6.53747			−0.433907			−0.216954	−	0.976182i	$-0.569612\pi$
$227$	−6.53747			−0.433907			−0.216954	+	0.976182i	$0.569612\pi$
$228$		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
$233$	29.6186			1.94038			0.970190	−	0.242348i	$-0.0779174\pi$
$233$	29.6186			1.94038			0.970190	+	0.242348i	$0.0779174\pi$
$234$	4.42481	−	14.4784i	0.289259	−	0.946483i
$235$		0			0
$236$	−6.63325	−	4.47214i	−0.431788	−	0.291111i
$237$		0			0
$238$	9.82767	−	32.1571i	0.637033	−	2.08443i
$239$		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			$\pi$
$242$	−14.8771	−	4.54666i	−0.956336	−	0.292270i
$243$		0			0
$244$		0			0
$245$			45.3173i			2.89521i
$246$		0			0
$247$		0			0
$248$	−19.6553	+	15.9269i	−1.24812	+	1.01136i
$249$		0			0
$250$	4.62120	−	15.1210i	0.292270	−	0.956336i
$251$	28.0000			1.76734			0.883672	−	0.468106i	$-0.155064\pi$
$251$	28.0000			1.76734			0.883672	+	0.468106i	$0.155064\pi$
$252$	17.5142	−	25.9778i	1.10329	−	1.63645i
$253$		0			0
$254$	0.575188	−	1.88207i	0.0360905	−	0.118092i
$255$		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
$260$	13.2320	+	8.92101i	0.820613	+	0.553257i
$261$		0			0
$262$		0			0
$263$			19.4953i			1.20213i	0.799198	+	0.601067i	$0.205258\pi$
$263$			19.4953i			1.20213i	−0.799198	+	0.601067i	$0.794742\pi$
$264$		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	$0.5\pi$
$272$	−6.83011	−	16.8845i	−0.414136	−	1.02377i
$273$		0			0
$274$		0			0
$275$	16.5831			1.00000
$276$		0			0
$277$		−	31.0687i		−	1.86673i	−0.358923	−	0.933367i	$-0.616856\pi$
$277$		−	31.0687i		−	1.86673i	0.358923	−	0.933367i	$-0.383144\pi$
$278$		0			0
$279$		−	26.8328i		−	1.60644i
$280$	20.7916	+	25.6587i	1.24253	+	1.53340i
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$	−33.3156			−1.98040			−0.990202	−	0.139641i	$-0.955405\pi$
$283$	−33.3156			−1.98040			−0.990202	+	0.139641i	$0.955405\pi$
$284$	16.5831	−	24.5967i	0.984027	−	1.45955i
$285$		0			0
$286$	−4.89181	+	16.0065i	−0.289259	+	0.946483i
$287$		0			0
$288$	−1.48805	−	16.9052i	−0.0876841	−	0.996148i
$289$	3.73350			0.219618
$290$		0			0
$291$		0			0
$292$	28.3339	+	19.1027i	1.65812	+	1.11790i
$293$			23.9319i			1.39811i	0.715066	+	0.699057i	$0.246397\pi$
$293$			23.9319i			1.39811i	−0.715066	+	0.699057i	$0.753603\pi$
$294$		0			0
$295$			8.94427i			0.520756i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		−	52.0255i		−	2.99870i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$	18.4749	+	5.64621i	1.05614	+	0.322772i
$307$	31.6027			1.80366			0.901830	−	0.432092i	$-0.142224\pi$
$307$	31.6027			1.80366			0.901830	+	0.432092i	$0.142224\pi$
$308$	−19.3627	+	28.7195i	−1.10329	+	1.63645i
$309$		0			0
$310$	27.0493	+	8.26665i	1.53629	+	0.469514i
$311$			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
$315$	−35.0285			−1.97363
$316$		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
$325$		−	17.8420i		−	0.989697i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−6.63325			−0.364596			−0.182298	−	0.983243i	$-0.558354\pi$
$331$	−6.63325			−0.364596			−0.182298	+	0.983243i	$0.558354\pi$
$332$	−22.2032	−	14.9694i	−1.21856	−	0.821553i
$333$		0			0
$334$	−3.74144	+	12.2423i	−0.204722	+	0.669871i
$335$		0			0
$336$		0			0
$337$	−11.4050			−0.621269			−0.310634	−	0.950529i	$-0.600541\pi$
$337$	−11.4050			−0.621269			−0.310634	+	0.950529i	$0.600541\pi$
$338$	−0.360430	−	0.110153i	−0.0196048	−	0.00599152i
$339$		0			0
$340$	−11.3835	+	16.8845i	−0.617358	+	0.915689i
$341$			29.6648i			1.60644i
$342$		0			0
$343$			69.2741i			3.74045i
$344$	−17.7414	−	21.8946i	−0.956554	−	1.18048i
$345$		0			0
$346$	−10.1082	+	33.0749i	−0.543419	+	1.77812i
$347$	16.8148			0.902667			0.451334	−	0.892355i	$-0.350948\pi$
$347$	16.8148			0.902667			0.451334	+	0.892355i	$0.350948\pi$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$	10.7916	−	35.3110i	0.576833	−	1.88745i
$351$		0			0
$352$	1.64510	+	18.6894i	0.0876841	+	0.996148i
$353$		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
$358$	−26.9137	−	8.22521i	−1.42243	−	0.434716i
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$	−14.7415	+	11.9452i	−0.776946	+	0.629568i
$361$	19.0000			1.00000
$362$	1.84848	−	6.04840i	0.0971539	−	0.317897i
$363$		0			0
$364$	30.8997	+	20.8326i	1.61959	+	1.09193i
$365$		−	38.2055i		−	1.99977i
$366$		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
$373$		−	31.5921i		−	1.63578i	−0.575375	−	0.817890i	$-0.695144\pi$
$373$		−	31.5921i		−	1.63578i	0.575375	−	0.817890i	$-0.304856\pi$
$374$	−20.4248	−	6.24212i	−1.05614	−	0.322772i
$375$		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
$382$	11.0909	−	36.2904i	0.567459	−	1.85678i
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$		0			0
$385$	38.7254			1.97363
$386$	−1.52506	−	0.466082i	−0.0776237	−	0.0237229i
$387$	29.8898			1.51938
$388$		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
$392$	36.0883	+	44.5363i	1.82273	+	2.24942i
$393$		0			0
$394$	−1.25856	+	4.11814i	−0.0634055	+	0.207469i
$395$		0			0
$396$	−16.5000	−	11.1243i	−0.829156	−	0.559017i
$397$		0			0			−	1.00000i	$-0.5\pi$
$397$		0			0				1.00000i	$0.5\pi$
$398$	−6.13071	+	20.0603i	−0.307305	+	1.00553i
$399$		0			0
$400$	−7.50000	−	18.5405i	−0.375000	−	0.927025i
$401$	39.7995			1.98749			0.993746	−	0.111664i	$-0.0356180\pi$
$401$	39.7995			1.98749			0.993746	+	0.111664i	$0.0356180\pi$
$402$		0			0
$403$	31.9168			1.58989
$404$		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
$413$			20.8869i			1.02778i
$414$		0			0
$415$			29.9388i			1.46964i
$416$	20.1082	−	1.76999i	0.985885	−	0.0867807i
$417$		0			0
$418$		0			0
$419$	−19.8997			−0.972166			−0.486083	−	0.873913i	$-0.661575\pi$
$419$	−19.8997			−0.972166			−0.486083	+	0.873913i	$0.661575\pi$
$420$		0			0
$421$		−	31.3050i		−	1.52571i	−0.646570	−	0.762855i	$-0.723797\pi$
$421$		−	31.3050i		−	1.52571i	0.646570	−	0.762855i	$-0.276203\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	22.7670			1.10436
$426$		0			0
$427$		0			0
$428$	13.6817	+	9.22419i	0.661329	+	0.445868i
$429$		0			0
$430$	−9.20844	+	30.1309i	−0.444070	+	1.45304i
$431$		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			$\pi$
$434$	63.1662	+	19.3045i	3.03208	+	0.926647i
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$	16.2973	−	13.2059i	0.776946	−	0.629568i
$441$	−60.7995			−2.89521
$442$	−6.71599	+	21.9753i	−0.319447	+	1.04526i
$443$		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
$448$	40.8665	+	8.65927i	1.93076	+	0.409112i
$449$	−13.2665			−0.626085			−0.313042	−	0.949739i	$-0.601348\pi$
$449$	−13.2665			−0.626085			−0.313042	+	0.949739i	$0.601348\pi$
$450$	20.2869	+	6.19999i	0.956336	+	0.292270i
$451$		0			0
$452$		0			0
$453$		0			0
$454$	8.84169	+	2.70215i	0.414961	+	0.126818i
$455$		−	41.6652i		−	1.95330i
$456$		0			0
$457$	10.2344			0.478746			0.239373	−	0.970928i	$-0.423058\pi$
$457$	10.2344			0.478746			0.239373	+	0.970928i	$0.423058\pi$
$458$	12.2614	−	40.1205i	0.572939	−	1.87471i
$459$		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
$466$	−40.0581	−	12.2423i	−1.85565	−	0.567115i
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$	−11.9688	+	17.7526i	−0.553257	+	0.820613i
$469$		0			0
$470$		0			0
$471$		0			0
$472$	7.12275	+	8.79014i	0.327851	+	0.404599i
$473$	−33.0444			−1.51938
$474$		0			0
$475$		0			0
$476$	−26.5831	+	39.4291i	−1.21844	+	1.80723i
$477$		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
$490$	18.7311	−	61.2899i	0.846185	−	2.76880i
$491$		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
$497$	−77.4508			−3.47414
$498$		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
$502$	−37.8690	−	11.5733i	−1.69017	−	0.516542i
$503$			8.00491i			0.356921i	0.983947	+	0.178461i	$0.0571117\pi$
$503$			8.00491i			0.356921i	−0.983947	+	0.178461i	$0.942888\pi$
$504$	−34.4248	+	27.8948i	−1.53340	+	1.24253i
$505$		0			0
$506$		0			0
$507$		0			0
$508$	−1.55584	+	2.30769i	−0.0690293	+	0.102387i
$509$		−	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
$511$		−	89.2186i		−	3.94680i
$512$	20.1514	−	10.2919i	0.890573	−	0.454841i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	−14.2084	−	17.5345i	−0.623081	−	0.768941i
$521$	−39.7995			−1.74365			−0.871824	−	0.489820i	$-0.837063\pi$
$521$	−39.7995			−1.74365			−0.871824	+	0.489820i	$0.837063\pi$
$522$		0			0
$523$	−28.1769			−1.23209			−0.616044	−	0.787711i	$-0.711266\pi$
$523$	−28.1769			−1.23209			−0.616044	+	0.787711i	$0.711266\pi$
$524$		0			0
$525$		0			0
$526$	8.05806	−	26.3667i	0.351348	−	1.14964i
$527$			40.7269i			1.77409i
$528$		0			0
$529$	−23.0000			−1.00000
$530$		0			0
$531$	−12.0000			−0.520756
$532$		0			0
$533$		0			0
$534$		0			0
$535$		−	18.4484i		−	0.797593i
$536$		0			0
$537$		0			0
$538$	−12.2614	+	40.1205i	−0.528628	+	1.72972i
$539$	67.2164			2.89521
$540$		0			0
$541$		0			0			−	1.00000i	$-0.5\pi$
$541$		0			0				1.00000i	$0.5\pi$
$542$		0			0
$543$		0			0
$544$	2.25856	+	25.6587i	0.0968351	+	1.10011i
$545$		0			0
$546$		0			0
$547$	38.4542			1.64418			0.822092	−	0.569354i	$-0.192807\pi$
$547$	38.4542			1.64418			0.822092	+	0.569354i	$0.192807\pi$
$548$		0			0
$549$		0			0
$550$	−22.4281	−	6.85435i	−0.956336	−	0.292270i
$551$		0			0
$552$		0			0
$553$		0			0
$554$	−12.8417	+	42.0192i	−0.545591	+	1.78523i
$555$		0			0
$556$		0			0
$557$			45.3423i			1.92121i	0.277911	+	0.960607i	$0.410358\pi$
$557$			45.3423i			1.92121i	−0.277911	+	0.960607i	$0.589642\pi$
$558$	−11.0909	+	36.2904i	−0.469514	+	1.53629i
$559$			35.5529i			1.50373i
$560$	−17.5142	−	43.2963i	−0.740111	−	1.82960i
$561$		0			0
$562$		0			0
$563$	−18.5277			−0.780850			−0.390425	−	0.920635i	$-0.627672\pi$
$563$	−18.5277			−0.780850			−0.390425	+	0.920635i	$0.627672\pi$
$564$		0			0
$565$		0			0
$566$	45.0581	+	13.7704i	1.89393	+	0.578813i
$567$		−	46.9956i		−	1.97363i
$568$	−32.5947	+	26.4118i	−1.36764	+	1.10822i
$569$		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			$\pi$
$572$	13.2320	−	19.6262i	0.553257	−	0.820613i
$573$		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			$\pi$
$578$	−5.04942	−	1.54318i	−0.210028	−	0.0641877i
$579$		0			0
$580$		0			0
$581$			69.9140i			2.90052i
$582$		0			0
$583$		0			0
$584$	−30.4248	−	37.5471i	−1.25899	−	1.55371i
$585$	23.9376			0.989697
$586$	9.89181	−	32.3669i	0.408627	−	1.33707i
$587$		0			0		1.00000			$0$
$587$		0			0		−1.00000			$\pi$
$588$		0			0
$589$		0			0
$590$	3.69696	−	12.0968i	0.152201	−	0.498017i
$591$		0			0
$592$		0			0
$593$	2.29817			0.0943744			0.0471872	−	0.998886i	$-0.484974\pi$
$593$	2.29817			0.0943744			0.0471872	+	0.998886i	$0.484974\pi$
$594$		0			0
$595$	53.1662			2.17960
$596$		0			0
$597$		0			0
$598$		0			0
$599$			44.7214i			1.82727i	0.406541	+	0.913633i	$0.366735\pi$
$599$			44.7214i			1.82727i	−0.406541	+	0.913633i	$0.633265\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$	−21.5038	+	70.3625i	−0.876430	+	2.86776i
$603$		0			0
$604$		0			0
$605$		−	24.5967i		−	1.00000i
$606$		0			0
$607$			12.8820i			0.522865i	0.965222	+	0.261433i	$0.0841949\pi$
$607$			12.8820i			0.522865i	−0.965222	+	0.261433i	$0.915805\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$	−22.6529	−	15.2726i	−0.915689	−	0.617358i
$613$		−	44.8188i		−	1.81021i	−0.425184	−	0.905107i	$-0.639791\pi$
$613$		−	44.8188i		−	1.81021i	0.425184	−	0.905107i	$-0.360209\pi$
$614$	−42.7414	−	13.0624i	−1.72490	−	0.527156i
$615$		0			0
$616$	38.0581	−	30.8389i	1.53340	−	1.24253i
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$	−46.4327			−1.86629			−0.933145	−	0.359501i	$-0.882947\pi$
$619$	−46.4327			−1.86629			−0.933145	+	0.359501i	$0.882947\pi$
$620$	−33.1662	−	22.3607i	−1.33199	−	0.898027i
$621$		0			0
$622$	6.13071	−	20.0603i	0.245819	−	0.804343i
$623$		−	69.2741i		−	2.77541i
$624$		0			0
$625$	25.0000			1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$	47.3747	+	14.4784i	1.88745	+	0.576833i
$631$			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
$635$	3.11168			0.123483
$636$		0			0
$637$		−	72.3190i		−	2.86538i
$638$		0			0
$639$		−	44.4972i		−	1.76028i
$640$	−22.1354	−	12.2484i	−0.874979	−	0.484160i
$641$	39.7995			1.57199			0.785993	−	0.618236i	$-0.212152\pi$
$641$	39.7995			1.57199			0.785993	+	0.618236i	$0.212152\pi$
$642$		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	$0.5\pi$
$648$	−16.0262	−	19.7778i	−0.629568	−	0.776946i
$649$	13.2665			0.520756
$650$	−7.37469	+	24.1307i	−0.289259	+	0.946483i
$651$		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
$657$	51.2580			1.99977
$658$		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	$0.804257\pi$
$662$	8.97122	+	2.74174i	0.348677	+	0.106561i
$663$		0			0
$664$	23.8417	+	29.4229i	0.925237	+	1.14183i
$665$		0			0
$666$		0			0
$667$		0			0
$668$	10.1203	−	15.0108i	0.391566	−	0.580787i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−47.8322			−1.84380			−0.921898	−	0.387432i	$-0.873362\pi$
$673$	−47.8322			−1.84380			−0.921898	+	0.387432i	$0.873362\pi$
$674$	15.4248	+	4.71405i	0.594142	+	0.181578i
$675$		0			0
$676$	0.441939	+	0.297955i	0.0169976	+	0.0114598i
$677$			38.7289i			1.48847i	0.667915	+	0.744237i	$0.267187\pi$
$677$			38.7289i			1.48847i	−0.667915	+	0.744237i	$0.732813\pi$
$678$		0			0
$679$		0			0
$680$	22.3747	−	18.1305i	0.858030	−	0.695271i
$681$		0			0
$682$	12.2614	−	40.1205i	0.469514	−	1.53629i
$683$		0			0		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$		0			0
$685$		0			0
$686$	28.6332	−	93.6907i	1.09322	−	3.57713i
$687$		0			0
$688$	14.9449	+	36.9447i	0.569768	+	1.40851i
$689$		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	$0.678779\pi$
$692$	27.3419	−	40.5546i	1.03938	−	1.54165i
$693$			51.9556i			1.97363i
$694$	−22.7414	−	6.95012i	−0.863253	−	0.263823i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−29.1904	+	43.2963i	−1.10329	+	1.63645i
$701$		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	$-0.973238\pi$
$709$		−	4.47214i		−	0.167955i	0.996468	−	0.0839773i	$-0.0267623\pi$
$710$	44.8561	+	13.7087i	1.68342	+	0.514478i
$711$		0			0
$712$	−23.6235	−	29.1536i	−0.885327	−	1.09258i
$713$		0			0
$714$		0			0
$715$	−26.4640			−0.989697
$716$	33.0000	+	22.2486i	1.23327	+	0.831469i
$717$		0			0
$718$		0			0
$719$			26.8328i			1.00070i	0.865825	+	0.500348i	$0.166794\pi$
$719$			26.8328i			1.00070i	−0.865825	+	0.500348i	$0.833206\pi$
$720$	24.8747	−	10.0623i	0.927025	−	0.375000i
$721$		0			0
$722$	−25.6968	−	7.85332i	−0.956336	−	0.292270i
$723$		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	$0.5\pi$
$728$	−33.1800	−	40.9472i	−1.22973	−	1.51760i
$729$	27.0000			1.00000
$730$	−15.7916	+	51.6715i	−0.584472	+	1.91245i
$731$	−45.3668			−1.67795
$732$		0			0
$733$		−	11.2287i		−	0.414741i	−0.978262	−	0.207371i	$-0.933509\pi$
$733$		−	11.2287i		−	0.414741i	0.978262	−	0.207371i	$-0.0664906\pi$
$734$		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
$743$			23.3255i			0.855729i	0.903843	+	0.427865i	$0.140734\pi$
$743$			23.3255i			0.855729i	−0.903843	+	0.427865i	$0.859266\pi$
$744$		0			0
$745$		0			0
$746$	−13.0581	+	42.7272i	−0.478090	+	1.56435i
$747$	−40.1671			−1.46964
$748$	25.0437	+	16.8845i	0.915689	+	0.617358i
$749$		−	43.0812i		−	1.57415i
$750$		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	$0.5\pi$
$758$	48.6887	+	14.8800i	1.76845	+	0.540465i
$759$		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
$765$			30.5452i			1.10436i
$766$		0			0
$767$		−	14.2736i		−	0.515390i
$768$		0			0
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$	−52.3747	−	16.0065i	−1.88745	−	0.576833i
$771$		0			0
$772$	1.86994	+	1.26072i	0.0673008	+	0.0453742i
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$	−40.4248	−	12.3544i	−1.45304	−	0.444070i
$775$			44.7214i			1.60644i
$776$		0			0
$777$		0			0
$778$	12.2614	−	40.1205i	0.439593	−	1.43839i
$779$		0			0
$780$		0			0
$781$			49.1935i			1.76028i
$782$		0			0
$783$		0			0
$784$	−30.3997	−	75.1502i	−1.08571	−	2.68394i
$785$		0			0
$786$		0			0
$787$	−54.9550			−1.95893			−0.979467	−	0.201607i	$-0.935384\pi$
$787$	−54.9550			−1.95893			−0.979467	+	0.201607i	$0.935384\pi$
$788$	3.40432	−	5.04942i	0.121274	−	0.179878i
$789$		0			0
$790$		0			0
$791$		0			0
$792$	17.7176	+	21.8652i	0.629568	+	0.776946i
$793$		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
$800$	2.48008	+	28.1753i	0.0876841	+	0.996148i
$801$	39.7995			1.40625
$802$	−53.8273	−	16.4504i	−1.90071	−	0.580885i
$803$	−56.6679			−1.99977
$804$		0			0
$805$		0			0
$806$	−43.1662	−	13.1922i	−1.52047	−	0.464677i
$807$		0			0
$808$		0			0
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$	−8.31816	+	27.2178i	−0.292270	+	0.956336i
$811$		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
$819$	55.8997			1.95330
$820$		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
$826$	8.63325	−	28.2488i	0.300389	−	0.982902i
$827$	53.2421			1.85141			0.925704	−	0.378248i	$-0.123473\pi$
$827$	53.2421			1.85141			0.925704	+	0.378248i	$0.123473\pi$
$828$		0			0
$829$			22.3607i			0.776619i	0.921529	+	0.388309i	$0.126941\pi$
$829$			22.3607i			0.776619i	−0.921529	+	0.388309i	$0.873059\pi$
$830$	12.3747	−	40.4911i	0.429532	−	1.40547i
$831$		0			0
$832$	−27.9272	−	5.91753i	−0.968200	−	0.205153i
$833$	92.2816			3.19737
$834$		0			0
$835$	−20.2406			−0.700455
$836$		0			0
$837$		0			0
$838$	26.9137	+	8.22521i	0.929717	+	0.284135i
$839$		−	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
$842$	−12.9394	+	42.3388i	−0.445920	+	1.45909i
$843$		0			0
$844$		0			0
$845$		−	0.595910i		−	0.0204999i
$846$		0			0
$847$		−	57.4391i		−	1.97363i
$848$		0			0
$849$		0			0
$850$	−30.7916	−	9.41035i	−1.05614	−	0.322772i
$851$		0			0
$852$		0			0
$853$			52.4791i			1.79685i	0.439128	+	0.898425i	$0.355288\pi$
$853$			52.4791i			1.79685i	−0.439128	+	0.898425i	$0.644712\pi$
$854$		0			0
$855$		0			0
$856$	−14.6913	−	18.1305i	−0.502139	−	0.619686i
$857$	−19.3412			−0.660684			−0.330342	−	0.943861i	$-0.607164\pi$
$857$	−19.3412			−0.660684			−0.330342	+	0.943861i	$0.607164\pi$
$858$		0			0
$859$	46.4327			1.58426			0.792132	−	0.610349i	$-0.208971\pi$
$859$	46.4327			1.58426			0.792132	+	0.610349i	$0.208971\pi$
$860$	24.9081	−	36.9447i	0.849361	−	1.25981i
$861$		0			0
$862$		0			0
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$		0			0
$865$	−54.6838			−1.85930
$866$		0			0
$867$		0			0
$868$	−77.4508	−	52.2173i	−2.62885	−	1.77237i
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$	58.3808			1.97363
$876$		0			0
$877$			59.0924i			1.99541i	0.0677220	+	0.997704i	$0.478427\pi$
$877$			59.0924i			1.99541i	−0.0677220	+	0.997704i	$0.521573\pi$
$878$		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	$0.510726\pi$
$882$	82.2291	+	25.1304i	2.76880	+	0.846185i
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$	18.1662	−	26.9449i	0.610997	−	0.906255i
$885$		0			0
$886$		0			0
$887$		−	50.8257i		−	1.70656i	−0.521452	−	0.853281i	$-0.674609\pi$
$887$		−	50.8257i		−	1.70656i	0.521452	−	0.853281i	$-0.325391\pi$
$888$		0			0
$889$	7.26650			0.243711
$890$	−12.2614	+	40.1205i	−0.411004	+	1.34484i
$891$	−29.8496			−1.00000
$892$		0			0
$893$		0			0
$894$		0			0
$895$		−	44.4972i		−	1.48738i
$896$	−51.6913	−	28.6028i	−1.72689	−	0.955553i
$897$		0			0
$898$	17.9424	+	5.48348i	0.598747	+	0.182986i
$899$		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			$\pi$
$908$	−10.8412	−	7.30911i	−0.359777	−	0.242561i
$909$		0			0
$910$	−17.2216	+	56.3507i	−0.570890	+	1.86801i
$911$		−	8.94427i		−	0.296337i	−0.988962	−	0.148168i	$-0.952662\pi$
$911$		−	8.94427i		−	0.296337i	0.988962	−	0.148168i	$-0.0473378\pi$
$912$		0			0
$913$	44.4064			1.46964
$914$	−13.8417	−	4.23022i	−0.457842	−	0.139923i
$915$		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
$923$	52.9280			1.74215
$924$		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
$932$	49.1169	+	33.1146i	1.60888	+	1.08470i
$933$		0			0
$934$		0			0
$935$		−	33.7690i		−	1.10436i
$936$	23.5251	−	19.0626i	0.768941	−	0.623081i
$937$	36.4702			1.19143			0.595714	−	0.803196i	$-0.296869\pi$
$937$	36.4702			1.19143			0.595714	+	0.803196i	$0.296869\pi$
$938$		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
$946$	44.6913	+	13.6583i	1.45304	+	0.444070i
$947$		0			0		1.00000			$0$
$947$		0			0		−1.00000			$\pi$
$948$		0			0
$949$			60.9697i			1.97916i
$950$		0			0
$951$		0			0
$952$	52.2500	−	42.3388i	1.69343	−	1.37221i
$953$	60.3648			1.95541			0.977704	−	0.209987i	$-0.0673422\pi$
$953$	60.3648			1.95541			0.977704	+	0.209987i	$0.0673422\pi$
$954$		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
$963$	24.7511			0.797593
$964$		0			0
$965$		−	2.52143i		−	0.0811678i
$966$		0			0
$967$			49.7788i			1.60078i	0.599481	+	0.800389i	$0.295374\pi$
$967$			49.7788i			1.60078i	−0.599481	+	0.800389i	$0.704626\pi$
$968$	−19.5875	−	24.1729i	−0.629568	−	0.776946i
$969$		0			0
$970$		0			0
$971$	59.6992			1.91584			0.957920	−	0.287035i	$-0.0926697\pi$
$971$	59.6992			1.91584			0.957920	+	0.287035i	$0.0926697\pi$
$972$		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
$980$	−50.6662	+	75.1502i	−1.61847	+	2.40058i
$981$		0			0
$982$		0			0
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$		0			0
$985$	−6.80864			−0.216941
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$	9.19607	−	30.0904i	0.292270	−	0.956336i
$991$			62.6099i			1.98887i	0.105356	+	0.994435i	$0.466402\pi$
$991$			62.6099i			1.98887i	−0.105356	+	0.994435i	$0.533598\pi$
$992$	−50.4016	+	4.43651i	−1.60025	+	0.140859i
$993$		0			0
$994$	104.749	+	32.0129i	3.32245	+	1.01539i
$995$	−33.1662			−1.05144
$996$		0			0
$997$			23.4084i			0.741350i	0.928763	+	0.370675i	$0.120874\pi$
$997$			23.4084i			0.741350i	−0.928763	+	0.370675i	$0.879126\pi$
$998$	−5.40985	−	1.65333i	−0.171246	−	0.0523353i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	440.2.c.b.219.1	✓	8
4.3	odd	2		1760.2.c.b.879.4		8
5.4	even	2	inner	440.2.c.b.219.8	yes	8
8.3	odd	2	inner	440.2.c.b.219.2	yes	8
8.5	even	2		1760.2.c.b.879.5		8
11.10	odd	2	inner	440.2.c.b.219.8	yes	8
20.19	odd	2		1760.2.c.b.879.1		8
40.19	odd	2	inner	440.2.c.b.219.7	yes	8
40.29	even	2		1760.2.c.b.879.8		8
44.43	even	2		1760.2.c.b.879.1		8
55.54	odd	2	CM	440.2.c.b.219.1	✓	8
88.21	odd	2		1760.2.c.b.879.8		8
88.43	even	2	inner	440.2.c.b.219.7	yes	8
220.219	even	2		1760.2.c.b.879.4		8
440.109	odd	2		1760.2.c.b.879.5		8
440.219	even	2	inner	440.2.c.b.219.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
440.2.c.b.219.1	✓	8	1.1	even	1	trivial
440.2.c.b.219.1	✓	8	55.54	odd	2	CM
440.2.c.b.219.2	yes	8	8.3	odd	2	inner
440.2.c.b.219.2	yes	8	440.219	even	2	inner
440.2.c.b.219.7	yes	8	40.19	odd	2	inner
440.2.c.b.219.7	yes	8	88.43	even	2	inner
440.2.c.b.219.8	yes	8	5.4	even	2	inner
440.2.c.b.219.8	yes	8	11.10	odd	2	inner
1760.2.c.b.879.1		8	20.19	odd	2
1760.2.c.b.879.1		8	44.43	even	2
1760.2.c.b.879.4		8	4.3	odd	2
1760.2.c.b.879.4		8	220.219	even	2
1760.2.c.b.879.5		8	8.5	even	2
1760.2.c.b.879.5		8	440.109	odd	2
1760.2.c.b.879.8		8	40.29	even	2
1760.2.c.b.879.8		8	88.21	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:	\( N \)	\(=\)	\( 440 = 2^{3} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	440.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.35246 - 0.413333i) q^{2} +(1.65831 + 1.11803i) q^{4} -2.23607i q^{5} -5.22173i q^{7} +(-1.78069 - 2.19753i) q^{8} +3.00000 q^{9} +(-0.924240 + 3.02420i) q^{10} -3.31662 q^{11} +3.56840i q^{13} +(-2.15831 + 7.06220i) q^{14} +(1.50000 + 3.70810i) q^{16} -4.55341 q^{17} +(-4.05739 - 1.24000i) q^{18} +(2.50000 - 3.70810i) q^{20} +(4.48561 + 1.37087i) q^{22} -5.00000 q^{25} +(1.47494 - 4.82613i) q^{26} +(5.83808 - 8.65927i) q^{28} -8.94427i q^{31} +(-0.496016 - 5.63507i) q^{32} +(6.15831 + 1.88207i) q^{34} -11.6762 q^{35} +(4.97494 + 3.35410i) q^{36} +(-4.91384 + 3.98174i) q^{40} +9.96326 q^{43} +(-5.50000 - 3.70810i) q^{44} -6.70820i q^{45} -20.2665 q^{49} +(6.76232 + 2.06666i) q^{50} +(-3.98960 + 5.91753i) q^{52} +7.41620i q^{55} +(-11.4749 + 9.29827i) q^{56} -4.00000 q^{59} +(-3.69696 + 12.0968i) q^{62} -15.6652i q^{63} +(-1.65831 + 7.82624i) q^{64} +7.97919 q^{65} +(-7.55097 - 5.09086i) q^{68} +(15.7916 + 4.82613i) q^{70} -14.8324i q^{71} +(-5.34206 - 6.59260i) q^{72} +17.0860 q^{73} +17.3185i q^{77} +(8.29156 - 3.35410i) q^{80} +9.00000 q^{81} -13.3890 q^{83} +10.1817i q^{85} +(-13.4749 - 4.11814i) q^{86} +(5.90587 + 7.28840i) q^{88} +13.2665 q^{89} +(-2.77272 + 9.07260i) q^{90} +18.6332 q^{91} +(27.4097 + 8.37680i) q^{98} -9.94987 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 24 q^{9} - 4 q^{14} + 12 q^{16} + 20 q^{20} - 40 q^{25} - 28 q^{26} + 36 q^{34} - 44 q^{44} - 56 q^{49} - 52 q^{56} - 32 q^{59} + 60 q^{70} + 72 q^{81} - 68 q^{86} + 96 q^{91}+O(q^{100}) \) 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

Properties

Related objects

Downloads

Learn more