LMFDB - Embedded newform 2240.2.n.b.1119.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2240,2,Mod(1119,2240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2240, 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("2240.1119");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2240 = 2^{6} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2240.n (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:	$17.8864900528$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.1731891456.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 9x^{6} + 65x^{4} - 144x^{2} + 256 $ x^8 - 9x^6 + 65x^4 - 144*x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1119.2
Root			$-2.21837 + 1.28078i$ of defining polynomial
Character	$\chi$	$=$	2240.1119
Dual form			2240.2.n.b.1119.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.56155 q^{3} +(2.00000 - 1.00000i) q^{5} +(-2.00000 + 1.73205i) q^{7} +3.56155 q^{9} +O(q^{10})$	$q-2.56155 q^{3} +(2.00000 - 1.00000i) q^{5} +(-2.00000 + 1.73205i) q^{7} +3.56155 q^{9} +0.972638 q^{11} +0.561553i q^{13} +(-5.12311 + 2.56155i) q^{15} -4.43674 q^{17} +1.12311i q^{19} +(5.12311 - 4.43674i) q^{21} +1.12311 q^{23} +(3.00000 - 4.00000i) q^{25} -1.43845 q^{27} -4.43674i q^{29} -8.87348 q^{31} -2.49146 q^{33} +(-2.26795 + 5.46410i) q^{35} +8.87348 q^{37} -1.43845i q^{39} +8.87348i q^{41} +1.94528i q^{43} +(7.12311 - 3.56155i) q^{45} -0.972638i q^{47} +(1.00000 - 6.92820i) q^{49} +11.3649 q^{51} +(1.94528 - 0.972638i) q^{55} -2.87689i q^{57} -4.00000i q^{59} -1.12311 q^{61} +(-7.12311 + 6.16879i) q^{63} +(0.561553 + 1.12311i) q^{65} -6.92820i q^{67} -2.87689 q^{69} -14.2462i q^{71} -6.92820 q^{73} +(-7.68466 + 10.2462i) q^{75} +(-1.94528 + 1.68466i) q^{77} +7.68466i q^{79} -7.00000 q^{81} +4.00000 q^{83} +(-8.87348 + 4.43674i) q^{85} +11.3649i q^{87} -8.87348i q^{89} +(-0.972638 - 1.12311i) q^{91} +22.7299 q^{93} +(1.12311 + 2.24621i) q^{95} -9.41967 q^{97} +3.46410 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{3} + 16 q^{5} - 16 q^{7} + 12 q^{9}+O(q^{10}) $ 8 * q - 4 * q^3 + 16 * q^5 - 16 * q^7 + 12 * q^9	$ 8 q - 4 q^{3} + 16 q^{5} - 16 q^{7} + 12 q^{9} - 8 q^{15} + 8 q^{21} - 24 q^{23} + 24 q^{25} - 28 q^{27} - 32 q^{35} + 24 q^{45} + 8 q^{49} + 24 q^{61} - 24 q^{63} - 12 q^{65} - 56 q^{69} - 12 q^{75} - 56 q^{81} + 32 q^{83} - 24 q^{95}+O(q^{100}) $ 8 * q - 4 * q^3 + 16 * q^5 - 16 * q^7 + 12 * q^9 - 8 * q^15 + 8 * q^21 - 24 * q^23 + 24 * q^25 - 28 * q^27 - 32 * q^35 + 24 * q^45 + 8 * q^49 + 24 * q^61 - 24 * q^63 - 12 * q^65 - 56 * q^69 - 12 * q^75 - 56 * q^81 + 32 * q^83 - 24 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2240\mathbb{Z}\right)^\times$.

$n$	$897$	$1471$	$1541$	$1921$
$\chi(n)$	$-1$	$-1$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−2.56155			−1.47891			−0.739457	−	0.673204i	$-0.764917\pi$
$3$	−2.56155			−1.47891			−0.739457	+	0.673204i	$0.764917\pi$
$4$		0			0
$5$	2.00000	−	1.00000i	0.894427	−	0.447214i
$5$	2.00000	−	1.00000i	0.894427	−	0.447214i
$6$		0			0
$7$	−2.00000	+	1.73205i	−0.755929	+	0.654654i
$7$	−2.00000	+	1.73205i	−0.755929	+	0.654654i
$8$		0			0
$9$	3.56155			1.18718
$10$		0			0
$11$	0.972638			0.293261			0.146631	−	0.989191i	$-0.453157\pi$
$11$	0.972638			0.293261			0.146631	+	0.989191i	$0.453157\pi$
$12$		0			0
$13$			0.561553i			0.155747i	0.996963	+	0.0778734i	$0.0248130\pi$
$13$			0.561553i			0.155747i	−0.996963	+	0.0778734i	$0.975187\pi$
$14$		0			0
$15$	−5.12311	+	2.56155i	−1.32278	+	0.661390i
$16$		0			0
$17$	−4.43674			−1.07607			−0.538034	−	0.842923i	$-0.680833\pi$
$17$	−4.43674			−1.07607			−0.538034	+	0.842923i	$0.680833\pi$
$18$		0			0
$19$			1.12311i			0.257658i	0.991667	+	0.128829i	$0.0411218\pi$
$19$			1.12311i			0.257658i	−0.991667	+	0.128829i	$0.958878\pi$
$20$		0			0
$21$	5.12311	−	4.43674i	1.11795	−	0.968176i
$22$		0			0
$23$	1.12311			0.234184			0.117092	−	0.993121i	$-0.462643\pi$
$23$	1.12311			0.234184			0.117092	+	0.993121i	$0.462643\pi$
$24$		0			0
$25$	3.00000	−	4.00000i	0.600000	−	0.800000i
$26$		0			0
$27$	−1.43845			−0.276829
$28$		0			0
$29$		−	4.43674i		−	0.823882i	−0.911211	−	0.411941i	$-0.864851\pi$
$29$		−	4.43674i		−	0.823882i	0.911211	−	0.411941i	$-0.135149\pi$
$30$		0			0
$31$	−8.87348			−1.59372			−0.796862	−	0.604161i	$-0.793508\pi$
$31$	−8.87348			−1.59372			−0.796862	+	0.604161i	$0.793508\pi$
$32$		0			0
$33$	−2.49146			−0.433708
$34$		0			0
$35$	−2.26795	+	5.46410i	−0.383353	+	0.923602i
$36$		0			0
$37$	8.87348			1.45879			0.729396	−	0.684092i	$-0.239801\pi$
$37$	8.87348			1.45879			0.729396	+	0.684092i	$0.239801\pi$
$38$		0			0
$39$		−	1.43845i		−	0.230336i
$40$		0			0
$41$			8.87348i			1.38580i	0.721031	+	0.692902i	$0.243668\pi$
$41$			8.87348i			1.38580i	−0.721031	+	0.692902i	$0.756332\pi$
$42$		0			0
$43$			1.94528i			0.296652i	0.988939	+	0.148326i	$0.0473885\pi$
$43$			1.94528i			0.296652i	−0.988939	+	0.148326i	$0.952612\pi$
$44$		0			0
$45$	7.12311	−	3.56155i	1.06185	−	0.530925i
$46$		0			0
$47$		−	0.972638i		−	0.141874i	−0.997481	−	0.0709369i	$-0.977401\pi$
$47$		−	0.972638i		−	0.141874i	0.997481	−	0.0709369i	$-0.0225989\pi$
$48$		0			0
$49$	1.00000	−	6.92820i	0.142857	−	0.989743i
$50$		0			0
$51$	11.3649			1.59141
$52$		0			0
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$		0			0
$55$	1.94528	−	0.972638i	0.262301	−	0.131150i
$56$		0			0
$57$		−	2.87689i		−	0.381054i
$58$		0			0
$59$		−	4.00000i		−	0.520756i	−0.965507	−	0.260378i	$-0.916153\pi$
$59$		−	4.00000i		−	0.520756i	0.965507	−	0.260378i	$-0.0838471\pi$
$60$		0			0
$61$	−1.12311			−0.143799			−0.0718995	−	0.997412i	$-0.522906\pi$
$61$	−1.12311			−0.143799			−0.0718995	+	0.997412i	$0.522906\pi$
$62$		0			0
$63$	−7.12311	+	6.16879i	−0.897427	+	0.777195i
$64$		0			0
$65$	0.561553	+	1.12311i	0.0696521	+	0.139304i
$66$		0			0
$67$		−	6.92820i		−	0.846415i	−0.906033	−	0.423207i	$-0.860904\pi$
$67$		−	6.92820i		−	0.846415i	0.906033	−	0.423207i	$-0.139096\pi$
$68$		0			0
$69$	−2.87689			−0.346337
$70$		0			0
$71$		−	14.2462i		−	1.69071i	−0.534202	−	0.845357i	$-0.679388\pi$
$71$		−	14.2462i		−	1.69071i	0.534202	−	0.845357i	$-0.320612\pi$
$72$		0			0
$73$	−6.92820			−0.810885			−0.405442	−	0.914121i	$-0.632883\pi$
$73$	−6.92820			−0.810885			−0.405442	+	0.914121i	$0.632883\pi$
$74$		0			0
$75$	−7.68466	+	10.2462i	−0.887348	+	1.18313i
$76$		0			0
$77$	−1.94528	+	1.68466i	−0.221685	+	0.191985i
$78$		0			0
$79$			7.68466i			0.864592i	0.901732	+	0.432296i	$0.142296\pi$
$79$			7.68466i			0.864592i	−0.901732	+	0.432296i	$0.857704\pi$
$80$		0			0
$81$	−7.00000			−0.777778
$82$		0			0
$83$	4.00000			0.439057			0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000			0.439057			0.219529	+	0.975606i	$0.429548\pi$
$84$		0			0
$85$	−8.87348	+	4.43674i	−0.962464	+	0.481232i
$86$		0			0
$87$			11.3649i			1.21845i
$88$		0			0
$89$		−	8.87348i		−	0.940587i	−0.882510	−	0.470293i	$-0.844148\pi$
$89$		−	8.87348i		−	0.940587i	0.882510	−	0.470293i	$-0.155852\pi$
$90$		0			0
$91$	−0.972638	−	1.12311i	−0.101960	−	0.117733i
$92$		0			0
$93$	22.7299			2.35698
$94$		0			0
$95$	1.12311	+	2.24621i	0.115228	+	0.230456i
$96$		0			0
$97$	−9.41967			−0.956422			−0.478211	−	0.878245i	$-0.658715\pi$
$97$	−9.41967			−0.956422			−0.478211	+	0.878245i	$0.658715\pi$
$98$		0			0
$99$	3.46410			0.348155
$100$		0			0
$101$	−11.3693			−1.13129			−0.565645	−	0.824649i	$-0.691373\pi$
$101$	−11.3693			−1.13129			−0.565645	+	0.824649i	$0.691373\pi$
$102$		0			0
$103$			14.8290i			1.46115i	0.682833	+	0.730575i	$0.260748\pi$
$103$			14.8290i			1.46115i	−0.682833	+	0.730575i	$0.739252\pi$
$104$		0			0
$105$	5.80947	−	13.9966i	0.566946	−	1.36593i
$106$		0			0
$107$		−	10.8188i		−	1.04589i	−0.852367	−	0.522944i	$-0.824834\pi$
$107$		−	10.8188i		−	1.04589i	0.852367	−	0.522944i	$-0.175166\pi$
$108$		0			0
$109$			18.2931i			1.75217i	0.482161	+	0.876083i	$0.339852\pi$
$109$			18.2931i			1.75217i	−0.482161	+	0.876083i	$0.660148\pi$
$110$		0			0
$111$	−22.7299			−2.15743
$112$		0			0
$113$		−	15.3693i		−	1.44582i	−0.690940	−	0.722912i	$-0.742803\pi$
$113$		−	15.3693i		−	1.44582i	0.690940	−	0.722912i	$-0.257197\pi$
$114$		0			0
$115$	2.24621	−	1.12311i	0.209460	−	0.104730i
$116$		0			0
$117$			2.00000i			0.184900i
$118$		0			0
$119$	8.87348	−	7.68466i	0.813431	−	0.704451i
$120$		0			0
$121$	−10.0540			−0.913998
$122$		0			0
$123$		−	22.7299i		−	2.04948i
$124$		0			0
$125$	2.00000	−	11.0000i	0.178885	−	0.983870i
$126$		0			0
$127$	−12.0000			−1.06483			−0.532414	−	0.846484i	$-0.678715\pi$
$127$	−12.0000			−1.06483			−0.532414	+	0.846484i	$0.678715\pi$
$128$		0			0
$129$		−	4.98293i		−	0.438722i
$130$		0			0
$131$		−	19.3693i		−	1.69231i	−0.532941	−	0.846153i	$-0.678913\pi$
$131$		−	19.3693i		−	1.69231i	0.532941	−	0.846153i	$-0.321087\pi$
$132$		0			0
$133$	−1.94528	−	2.24621i	−0.168677	−	0.194771i
$134$		0			0
$135$	−2.87689	+	1.43845i	−0.247604	+	0.123802i
$136$		0			0
$137$		−	13.1231i		−	1.12118i	−0.828093	−	0.560591i	$-0.810574\pi$
$137$		−	13.1231i		−	1.12118i	0.828093	−	0.560591i	$-0.189426\pi$
$138$		0			0
$139$		−	14.2462i		−	1.20835i	−0.796852	−	0.604174i	$-0.793503\pi$
$139$		−	14.2462i		−	1.20835i	0.796852	−	0.604174i	$-0.206497\pi$
$140$		0			0
$141$			2.49146i			0.209819i
$142$		0			0
$143$			0.546188i			0.0456745i
$144$		0			0
$145$	−4.43674	−	8.87348i	−0.368451	−	0.736902i
$146$		0			0
$147$	−2.56155	+	17.7470i	−0.211273	+	1.46374i
$148$		0			0
$149$			10.8188i			0.886307i	0.896446	+	0.443153i	$0.146140\pi$
$149$			10.8188i			0.886307i	−0.896446	+	0.443153i	$0.853860\pi$
$150$		0			0
$151$		−	15.0540i		−	1.22508i	−0.790441	−	0.612538i	$-0.790149\pi$
$151$		−	15.0540i		−	1.22508i	0.790441	−	0.612538i	$-0.209851\pi$
$152$		0			0
$153$	−15.8017			−1.27749
$154$		0			0
$155$	−17.7470	+	8.87348i	−1.42547	+	0.712735i
$156$		0			0
$157$		−	20.2462i		−	1.61582i	−0.589303	−	0.807912i	$-0.700598\pi$
$157$		−	20.2462i		−	1.61582i	0.589303	−	0.807912i	$-0.299402\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−2.24621	+	1.94528i	−0.177026	+	0.153309i
$162$		0			0
$163$		−	24.6752i		−	1.93271i	−0.257216	−	0.966354i	$-0.582805\pi$
$163$		−	24.6752i		−	1.93271i	0.257216	−	0.966354i	$-0.417195\pi$
$164$		0			0
$165$	−4.98293	+	2.49146i	−0.387920	+	0.193960i
$166$		0			0
$167$			2.91791i			0.225795i	0.993607	+	0.112897i	$0.0360132\pi$
$167$			2.91791i			0.225795i	−0.993607	+	0.112897i	$0.963987\pi$
$168$		0			0
$169$	12.6847			0.975743
$170$		0			0
$171$			4.00000i			0.305888i
$172$		0			0
$173$		−	9.68466i		−	0.736311i	−0.929764	−	0.368155i	$-0.879989\pi$
$173$		−	9.68466i		−	0.736311i	0.929764	−	0.368155i	$-0.120011\pi$
$174$		0			0
$175$	0.928203	+	13.1962i	0.0701656	+	0.997535i
$176$		0			0
$177$			10.2462i			0.770152i
$178$		0			0
$179$	−21.2111			−1.58539			−0.792695	−	0.609619i	$-0.791322\pi$
$179$	−21.2111			−1.58539			−0.792695	+	0.609619i	$0.791322\pi$
$180$		0			0
$181$	−14.2462			−1.05891			−0.529456	−	0.848337i	$-0.677604\pi$
$181$	−14.2462			−1.05891			−0.529456	+	0.848337i	$0.677604\pi$
$182$		0			0
$183$	2.87689			0.212666
$184$		0			0
$185$	17.7470	−	8.87348i	1.30478	−	0.652391i
$186$		0			0
$187$	−4.31534			−0.315569
$188$		0			0
$189$	2.87689	−	2.49146i	0.209263	−	0.181227i
$190$		0			0
$191$		−	5.43845i		−	0.393512i	−0.980452	−	0.196756i	$-0.936959\pi$
$191$		−	5.43845i		−	0.393512i	0.980452	−	0.196756i	$-0.0630407\pi$
$192$		0			0
$193$			7.36932i			0.530455i	0.964186	+	0.265228i	$0.0854471\pi$
$193$			7.36932i			0.530455i	−0.964186	+	0.265228i	$0.914553\pi$
$194$		0			0
$195$	−1.43845	−	2.87689i	−0.103009	−	0.206019i
$196$		0			0
$197$	−22.7299			−1.61944			−0.809719	−	0.586818i	$-0.800380\pi$
$197$	−22.7299			−1.61944			−0.809719	+	0.586818i	$0.800380\pi$
$198$		0			0
$199$	17.7470			1.25805			0.629024	−	0.777386i	$-0.283455\pi$
$199$	17.7470			1.25805			0.629024	+	0.777386i	$0.283455\pi$
$200$		0			0
$201$			17.7470i			1.25177i
$202$		0			0
$203$	7.68466	+	8.87348i	0.539357	+	0.622796i
$204$		0			0
$205$	8.87348	+	17.7470i	0.619751	+	1.23950i
$206$		0			0
$207$	4.00000			0.278019
$208$		0			0
$209$			1.09238i			0.0755612i
$210$		0			0
$211$	−14.8290			−1.02087			−0.510437	−	0.859915i	$-0.670516\pi$
$211$	−14.8290			−1.02087			−0.510437	+	0.859915i	$0.670516\pi$
$212$		0			0
$213$			36.4924i			2.50042i
$214$		0			0
$215$	1.94528	+	3.89055i	0.132667	+	0.265333i
$216$		0			0
$217$	17.7470	−	15.3693i	1.20474	−	1.04334i
$218$		0			0
$219$	17.7470			1.19923
$220$		0			0
$221$		−	2.49146i		−	0.167594i
$222$		0			0
$223$		−	0.972638i		−	0.0651327i	−0.999470	−	0.0325663i	$-0.989632\pi$
$223$		−	0.972638i		−	0.0651327i	0.999470	−	0.0325663i	$-0.0103680\pi$
$224$		0			0
$225$	10.6847	−	14.2462i	0.712311	−	0.949747i
$226$		0			0
$227$	−15.6847			−1.04103			−0.520514	−	0.853853i	$-0.674260\pi$
$227$	−15.6847			−1.04103			−0.520514	+	0.853853i	$0.674260\pi$
$228$		0			0
$229$	1.12311			0.0742169			0.0371085	−	0.999311i	$-0.488185\pi$
$229$	1.12311			0.0742169			0.0371085	+	0.999311i	$0.488185\pi$
$230$		0			0
$231$	4.98293	−	4.31534i	0.327853	−	0.283929i
$232$		0			0
$233$			28.4924i			1.86660i	0.359097	+	0.933300i	$0.383085\pi$
$233$			28.4924i			1.86660i	−0.359097	+	0.933300i	$0.616915\pi$
$234$		0			0
$235$	−0.972638	−	1.94528i	−0.0634479	−	0.126896i
$236$		0			0
$237$		−	19.6847i		−	1.27866i
$238$		0			0
$239$			7.68466i			0.497079i	0.968622	+	0.248540i	$0.0799506\pi$
$239$			7.68466i			0.497079i	−0.968622	+	0.248540i	$0.920049\pi$
$240$		0			0
$241$		−	4.98293i		−	0.320979i	−0.987038	−	0.160489i	$-0.948693\pi$
$241$		−	4.98293i		−	0.320979i	0.987038	−	0.160489i	$-0.0513072\pi$
$242$		0			0
$243$	22.2462			1.42710
$244$		0			0
$245$	−4.92820	−	14.8564i	−0.314851	−	0.949141i
$246$		0			0
$247$	−0.630683			−0.0401294
$248$		0			0
$249$	−10.2462			−0.649327
$250$		0			0
$251$		−	20.0000i		−	1.26239i	−0.775625	−	0.631194i	$-0.782565\pi$
$251$		−	20.0000i		−	1.26239i	0.775625	−	0.631194i	$-0.217435\pi$
$252$		0			0
$253$	1.09238			0.0686770
$254$		0			0
$255$	22.7299	−	11.3649i	1.42340	−	0.711700i
$256$		0			0
$257$	6.92820			0.432169			0.216085	−	0.976375i	$-0.430671\pi$
$257$	6.92820			0.432169			0.216085	+	0.976375i	$0.430671\pi$
$258$		0			0
$259$	−17.7470	+	15.3693i	−1.10274	+	0.955003i
$260$		0			0
$261$		−	15.8017i		−	0.978100i
$262$		0			0
$263$	12.0000			0.739952			0.369976	−	0.929041i	$-0.379366\pi$
$263$	12.0000			0.739952			0.369976	+	0.929041i	$0.379366\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$			22.7299i			1.39105i
$268$		0			0
$269$	20.0000			1.21942			0.609711	−	0.792624i	$-0.291286\pi$
$269$	20.0000			1.21942			0.609711	+	0.792624i	$0.291286\pi$
$270$		0			0
$271$	−22.7299			−1.38074			−0.690371	−	0.723455i	$-0.742553\pi$
$271$	−22.7299			−1.38074			−0.690371	+	0.723455i	$0.742553\pi$
$272$		0			0
$273$	2.49146	+	2.87689i	0.150790	+	0.174118i
$274$		0			0
$275$	2.91791	−	3.89055i	0.175957	−	0.234609i
$276$		0			0
$277$	−13.8564			−0.832551			−0.416275	−	0.909239i	$-0.636665\pi$
$277$	−13.8564			−0.832551			−0.416275	+	0.909239i	$0.636665\pi$
$278$		0			0
$279$	−31.6034			−1.89204
$280$		0			0
$281$	−25.6847			−1.53222			−0.766109	−	0.642711i	$-0.777810\pi$
$281$	−25.6847			−1.53222			−0.766109	+	0.642711i	$0.777810\pi$
$282$		0			0
$283$	5.43845			0.323282			0.161641	−	0.986850i	$-0.448321\pi$
$283$	5.43845			0.323282			0.161641	+	0.986850i	$0.448321\pi$
$284$		0			0
$285$	−2.87689	−	5.75379i	−0.170413	−	0.340825i
$286$		0			0
$287$	−15.3693	−	17.7470i	−0.907222	−	1.04757i
$288$		0			0
$289$	2.68466			0.157921
$290$		0			0
$291$	24.1290			1.41447
$292$		0			0
$293$			14.3153i			0.836311i	0.908375	+	0.418156i	$0.137323\pi$
$293$			14.3153i			0.836311i	−0.908375	+	0.418156i	$0.862677\pi$
$294$		0			0
$295$	−4.00000	−	8.00000i	−0.232889	−	0.465778i
$296$		0			0
$297$	−1.39909			−0.0811833
$298$		0			0
$299$			0.630683i			0.0364733i
$300$		0			0
$301$	−3.36932	−	3.89055i	−0.194204	−	0.224248i
$302$		0			0
$303$	29.1231			1.67308
$304$		0			0
$305$	−2.24621	+	1.12311i	−0.128618	+	0.0643088i
$306$		0			0
$307$	−9.93087			−0.566785			−0.283392	−	0.959004i	$-0.591460\pi$
$307$	−9.93087			−0.566785			−0.283392	+	0.959004i	$0.591460\pi$
$308$		0			0
$309$		−	37.9854i		−	2.16091i
$310$		0			0
$311$	18.8393			1.06828			0.534140	−	0.845396i	$-0.320635\pi$
$311$	18.8393			1.06828			0.534140	+	0.845396i	$0.320635\pi$
$312$		0			0
$313$	−18.2931			−1.03399			−0.516995	−	0.855988i	$-0.672949\pi$
$313$	−18.2931			−1.03399			−0.516995	+	0.855988i	$0.672949\pi$
$314$		0			0
$315$	−8.07742	+	19.4607i	−0.455111	+	1.09649i
$316$		0			0
$317$	17.7470			0.996768			0.498384	−	0.866956i	$-0.333927\pi$
$317$	17.7470			0.996768			0.498384	+	0.866956i	$0.333927\pi$
$318$		0			0
$319$		−	4.31534i		−	0.241613i
$320$		0			0
$321$			27.7128i			1.54678i
$322$		0			0
$323$		−	4.98293i		−	0.277257i
$324$		0			0
$325$	2.24621	+	1.68466i	0.124597	+	0.0934480i
$326$		0			0
$327$		−	46.8589i		−	2.59130i
$328$		0			0
$329$	1.68466	+	1.94528i	0.0928782	+	0.107247i
$330$		0			0
$331$	17.3205			0.952021			0.476011	−	0.879440i	$-0.342082\pi$
$331$	17.3205			0.952021			0.476011	+	0.879440i	$0.342082\pi$
$332$		0			0
$333$	31.6034			1.73185
$334$		0			0
$335$	−6.92820	−	13.8564i	−0.378528	−	0.757056i
$336$		0			0
$337$		−	22.7386i		−	1.23865i	−0.785134	−	0.619326i	$-0.787406\pi$
$337$		−	22.7386i		−	1.23865i	0.785134	−	0.619326i	$-0.212594\pi$
$338$		0			0
$339$			39.3693i			2.13825i
$340$		0			0
$341$	−8.63068			−0.467378
$342$		0			0
$343$	10.0000	+	15.5885i	0.539949	+	0.841698i
$344$		0			0
$345$	−5.75379	+	2.87689i	−0.309774	+	0.154887i
$346$		0			0
$347$			15.8017i			0.848279i	0.905597	+	0.424139i	$0.139423\pi$
$347$			15.8017i			0.848279i	−0.905597	+	0.424139i	$0.860577\pi$
$348$		0			0
$349$	1.12311			0.0601185			0.0300592	−	0.999548i	$-0.490430\pi$
$349$	1.12311			0.0601185			0.0300592	+	0.999548i	$0.490430\pi$
$350$		0			0
$351$		−	0.807764i		−	0.0431153i
$352$		0			0
$353$	18.2931			0.973646			0.486823	−	0.873501i	$-0.338156\pi$
$353$	18.2931			0.973646			0.486823	+	0.873501i	$0.338156\pi$
$354$		0			0
$355$	−14.2462	−	28.4924i	−0.756110	−	1.51222i
$356$		0			0
$357$	−22.7299	+	19.6847i	−1.20299	+	1.04182i
$358$		0			0
$359$			12.0000i			0.633336i	0.948536	+	0.316668i	$0.102564\pi$
$359$			12.0000i			0.633336i	−0.948536	+	0.316668i	$0.897436\pi$
$360$		0			0
$361$	17.7386			0.933612
$362$		0			0
$363$	25.7538			1.35172
$364$		0			0
$365$	−13.8564	+	6.92820i	−0.725277	+	0.362639i
$366$		0			0
$367$			36.4666i			1.90354i	0.306814	+	0.951769i	$0.400737\pi$
$367$			36.4666i			1.90354i	−0.306814	+	0.951769i	$0.599263\pi$
$368$		0			0
$369$			31.6034i			1.64521i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	26.6204			1.37835			0.689177	−	0.724593i	$-0.257972\pi$
$373$	26.6204			1.37835			0.689177	+	0.724593i	$0.257972\pi$
$374$		0			0
$375$	−5.12311	+	28.1771i	−0.264556	+	1.45506i
$376$		0			0
$377$	2.49146			0.128317
$378$		0			0
$379$	28.1393			1.44542			0.722708	−	0.691153i	$-0.242897\pi$
$379$	28.1393			1.44542			0.722708	+	0.691153i	$0.242897\pi$
$380$		0			0
$381$	30.7386			1.57479
$382$		0			0
$383$			0.426450i			0.0217906i	0.999941	+	0.0108953i	$0.00346815\pi$
$383$			0.426450i			0.0217906i	−0.999941	+	0.0108953i	$0.996532\pi$
$384$		0			0
$385$	−2.20589	+	5.31459i	−0.112423	+	0.270857i
$386$		0			0
$387$			6.92820i			0.352180i
$388$		0			0
$389$		−	0.546188i		−	0.0276928i	−0.999904	−	0.0138464i	$-0.995592\pi$
$389$		−	0.546188i		−	0.0276928i	0.999904	−	0.0138464i	$-0.00440759\pi$
$390$		0			0
$391$	−4.98293			−0.251997
$392$		0			0
$393$			49.6155i			2.50277i
$394$		0			0
$395$	7.68466	+	15.3693i	0.386657	+	0.773314i
$396$		0			0
$397$			3.93087i			0.197285i	0.995123	+	0.0986423i	$0.0314500\pi$
$397$			3.93087i			0.197285i	−0.995123	+	0.0986423i	$0.968550\pi$
$398$		0			0
$399$	4.98293	+	5.75379i	0.249458	+	0.288050i
$400$		0			0
$401$	−2.80776			−0.140213			−0.0701065	−	0.997540i	$-0.522334\pi$
$401$	−2.80776			−0.140213			−0.0701065	+	0.997540i	$0.522334\pi$
$402$		0			0
$403$		−	4.98293i		−	0.248217i
$404$		0			0
$405$	−14.0000	+	7.00000i	−0.695666	+	0.347833i
$406$		0			0
$407$	8.63068			0.427807
$408$		0			0
$409$		−	31.6034i		−	1.56269i	−0.624102	−	0.781343i	$-0.714535\pi$
$409$		−	31.6034i		−	1.56269i	0.624102	−	0.781343i	$-0.285465\pi$
$410$		0			0
$411$			33.6155i			1.65813i
$412$		0			0
$413$	6.92820	+	8.00000i	0.340915	+	0.393654i
$414$		0			0
$415$	8.00000	−	4.00000i	0.392705	−	0.196352i
$416$		0			0
$417$			36.4924i			1.78704i
$418$		0			0
$419$			26.7386i			1.30627i	0.757242	+	0.653134i	$0.226546\pi$
$419$			26.7386i			1.30627i	−0.757242	+	0.653134i	$0.773454\pi$
$420$		0			0
$421$			4.43674i			0.216233i	0.994138	+	0.108117i	$0.0344820\pi$
$421$			4.43674i			0.216233i	−0.994138	+	0.108117i	$0.965518\pi$
$422$		0			0
$423$		−	3.46410i		−	0.168430i
$424$		0			0
$425$	−13.3102	+	17.7470i	−0.645640	+	0.860854i
$426$		0			0
$427$	2.24621	−	1.94528i	0.108702	−	0.0941385i
$428$		0			0
$429$		−	1.39909i		−	0.0675486i
$430$		0			0
$431$			16.3153i			0.785882i	0.919564	+	0.392941i	$0.128542\pi$
$431$			16.3153i			0.785882i	−0.919564	+	0.392941i	$0.871458\pi$
$432$		0			0
$433$	−3.03765			−0.145980			−0.0729901	−	0.997333i	$-0.523254\pi$
$433$	−3.03765			−0.145980			−0.0729901	+	0.997333i	$0.523254\pi$
$434$		0			0
$435$	11.3649	+	22.7299i	0.544907	+	1.08981i
$436$		0			0
$437$			1.26137i			0.0603393i
$438$		0			0
$439$	−13.8564			−0.661330			−0.330665	−	0.943748i	$-0.607273\pi$
$439$	−13.8564			−0.661330			−0.330665	+	0.943748i	$0.607273\pi$
$440$		0			0
$441$	3.56155	−	24.6752i	0.169598	−	1.17501i
$442$		0			0
$443$		−	29.6581i		−	1.40910i	−0.709655	−	0.704549i	$-0.751149\pi$
$443$		−	29.6581i		−	1.40910i	0.709655	−	0.704549i	$-0.248851\pi$
$444$		0			0
$445$	−8.87348	−	17.7470i	−0.420643	−	0.841287i
$446$		0			0
$447$		−	27.7128i		−	1.31077i
$448$		0			0
$449$	29.0540			1.37114			0.685571	−	0.728006i	$-0.259553\pi$
$449$	29.0540			1.37114			0.685571	+	0.728006i	$0.259553\pi$
$450$		0			0
$451$			8.63068i			0.406403i
$452$		0			0
$453$			38.5616i			1.81178i
$454$		0			0
$455$	−3.06838	−	1.27357i	−0.143848	−	0.0597060i
$456$		0			0
$457$		−	0.630683i		−	0.0295021i	−0.999891	−	0.0147511i	$-0.995304\pi$
$457$		−	0.630683i		−	0.0295021i	0.999891	−	0.0147511i	$-0.00469558\pi$
$458$		0			0
$459$	6.38202			0.297887
$460$		0			0
$461$	−20.0000			−0.931493			−0.465746	−	0.884918i	$-0.654214\pi$
$461$	−20.0000			−0.931493			−0.465746	+	0.884918i	$0.654214\pi$
$462$		0			0
$463$	−11.3693			−0.528377			−0.264188	−	0.964471i	$-0.585104\pi$
$463$	−11.3693			−0.528377			−0.264188	+	0.964471i	$0.585104\pi$
$464$		0			0
$465$	45.4598	−	22.7299i	2.10815	−	1.05407i
$466$		0			0
$467$	31.0540			1.43701			0.718503	−	0.695524i	$-0.244828\pi$
$467$	31.0540			1.43701			0.718503	+	0.695524i	$0.244828\pi$
$468$		0			0
$469$	12.0000	+	13.8564i	0.554109	+	0.639829i
$470$		0			0
$471$			51.8617i			2.38966i
$472$		0			0
$473$			1.89205i			0.0869965i
$474$		0			0
$475$	4.49242	+	3.36932i	0.206126	+	0.154595i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	18.8393			0.860791			0.430396	−	0.902640i	$-0.358374\pi$
$479$	18.8393			0.860791			0.430396	+	0.902640i	$0.358374\pi$
$480$		0			0
$481$			4.98293i			0.227202i
$482$		0			0
$483$	5.75379	−	4.98293i	0.261806	−	0.226731i
$484$		0			0
$485$	−18.8393	+	9.41967i	−0.855450	+	0.427725i
$486$		0			0
$487$	−12.0000			−0.543772			−0.271886	−	0.962329i	$-0.587647\pi$
$487$	−12.0000			−0.543772			−0.271886	+	0.962329i	$0.587647\pi$
$488$		0			0
$489$			63.2067i			2.85831i
$490$		0			0
$491$	−28.6855			−1.29456			−0.647278	−	0.762254i	$-0.724093\pi$
$491$	−28.6855			−1.29456			−0.647278	+	0.762254i	$0.724093\pi$
$492$		0			0
$493$			19.6847i			0.886552i
$494$		0			0
$495$	6.92820	−	3.46410i	0.311400	−	0.155700i
$496$		0			0
$497$	24.6752	+	28.4924i	1.10683	+	1.27806i
$498$		0			0
$499$	−14.8290			−0.663839			−0.331920	−	0.943308i	$-0.607696\pi$
$499$	−14.8290			−0.663839			−0.331920	+	0.943308i	$0.607696\pi$
$500$		0			0
$501$		−	7.47439i		−	0.333931i
$502$		0			0
$503$		−	30.6307i		−	1.36576i	−0.730532	−	0.682878i	$-0.760728\pi$
$503$		−	30.6307i		−	1.36576i	0.730532	−	0.682878i	$-0.239272\pi$
$504$		0			0
$505$	−22.7386	+	11.3693i	−1.01186	+	0.505928i
$506$		0			0
$507$	−32.4924			−1.44304
$508$		0			0
$509$	−4.00000			−0.177297			−0.0886484	−	0.996063i	$-0.528255\pi$
$509$	−4.00000			−0.177297			−0.0886484	+	0.996063i	$0.528255\pi$
$510$		0			0
$511$	13.8564	−	12.0000i	0.612971	−	0.530849i
$512$		0			0
$513$		−	1.61553i		−	0.0713273i
$514$		0			0
$515$	14.8290	+	29.6581i	0.653446	+	1.30689i
$516$		0			0
$517$		−	0.946025i		−	0.0416061i
$518$		0			0
$519$			24.8078i			1.08894i
$520$		0			0
$521$		−	22.7299i		−	0.995814i	−0.867230	−	0.497907i	$-0.834102\pi$
$521$		−	22.7299i		−	0.995814i	0.867230	−	0.497907i	$-0.165898\pi$
$522$		0			0
$523$	7.50758			0.328283			0.164142	−	0.986437i	$-0.447515\pi$
$523$	7.50758			0.328283			0.164142	+	0.986437i	$0.447515\pi$
$524$		0			0
$525$	−2.37764	−	33.8026i	−0.103769	−	1.47527i
$526$		0			0
$527$	39.3693			1.71495
$528$		0			0
$529$	−21.7386			−0.945158
$530$		0			0
$531$		−	14.2462i		−	0.618233i
$532$		0			0
$533$	−4.98293			−0.215835
$534$		0			0
$535$	−10.8188	−	21.6375i	−0.467736	−	0.935471i
$536$		0			0
$537$	54.3333			2.34465
$538$		0			0
$539$	0.972638	−	6.73863i	0.0418945	−	0.290254i
$540$		0			0
$541$			8.32729i			0.358018i	0.983847	+	0.179009i	$0.0572892\pi$
$541$			8.32729i			0.358018i	−0.983847	+	0.179009i	$0.942711\pi$
$542$		0			0
$543$	36.4924			1.56604
$544$		0			0
$545$	18.2931	+	36.5863i	0.783592	+	1.56718i
$546$		0			0
$547$			15.8017i			0.675631i	0.941212	+	0.337816i	$0.109688\pi$
$547$			15.8017i			0.675631i	−0.941212	+	0.337816i	$0.890312\pi$
$548$		0			0
$549$	−4.00000			−0.170716
$550$		0			0
$551$	4.98293			0.212280
$552$		0			0
$553$	−13.3102	−	15.3693i	−0.566008	−	0.653570i
$554$		0			0
$555$	−45.4598	+	22.7299i	−1.92966	+	0.964830i
$556$		0			0
$557$	13.8564			0.587115			0.293557	−	0.955941i	$-0.405161\pi$
$557$	13.8564			0.587115			0.293557	+	0.955941i	$0.405161\pi$
$558$		0			0
$559$	−1.09238			−0.0462025
$560$		0			0
$561$	11.0540			0.466699
$562$		0			0
$563$	20.0000			0.842900			0.421450	−	0.906852i	$-0.361521\pi$
$563$	20.0000			0.842900			0.421450	+	0.906852i	$0.361521\pi$
$564$		0			0
$565$	−15.3693	−	30.7386i	−0.646592	−	1.29318i
$566$		0			0
$567$	14.0000	−	12.1244i	0.587945	−	0.509175i
$568$		0			0
$569$	−32.2462			−1.35183			−0.675916	−	0.736979i	$-0.736252\pi$
$569$	−32.2462			−1.35183			−0.675916	+	0.736979i	$0.736252\pi$
$570$		0			0
$571$	−7.35465			−0.307783			−0.153891	−	0.988088i	$-0.549181\pi$
$571$	−7.35465			−0.307783			−0.153891	+	0.988088i	$0.549181\pi$
$572$		0			0
$573$			13.9309i			0.581970i
$574$		0			0
$575$	3.36932	−	4.49242i	0.140510	−	0.187347i
$576$		0			0
$577$	−39.9307			−1.66233			−0.831167	−	0.556022i	$-0.812327\pi$
$577$	−39.9307			−1.66233			−0.831167	+	0.556022i	$0.812327\pi$
$578$		0			0
$579$		−	18.8769i		−	0.784497i
$580$		0			0
$581$	−8.00000	+	6.92820i	−0.331896	+	0.287430i
$582$		0			0
$583$		0			0
$584$		0			0
$585$	2.00000	+	4.00000i	0.0826898	+	0.165380i
$586$		0			0
$587$	28.0000			1.15568			0.577842	−	0.816149i	$-0.303895\pi$
$587$	28.0000			1.15568			0.577842	+	0.816149i	$0.303895\pi$
$588$		0			0
$589$		−	9.96585i		−	0.410636i
$590$		0			0
$591$	58.2238			2.39501
$592$		0			0
$593$	0.546188			0.0224292			0.0112146	−	0.999937i	$-0.496430\pi$
$593$	0.546188			0.0224292			0.0112146	+	0.999937i	$0.496430\pi$
$594$		0			0
$595$	10.0623	−	24.2428i	0.412514	−	0.993858i
$596$		0			0
$597$	−45.4598			−1.86054
$598$		0			0
$599$			5.43845i			0.222209i	0.993809	+	0.111104i	$0.0354388\pi$
$599$			5.43845i			0.222209i	−0.993809	+	0.111104i	$0.964561\pi$
$600$		0			0
$601$			9.96585i			0.406516i	0.979125	+	0.203258i	$0.0651530\pi$
$601$			9.96585i			0.406516i	−0.979125	+	0.203258i	$0.934847\pi$
$602$		0			0
$603$		−	24.6752i		−	1.00485i
$604$		0			0
$605$	−20.1080	+	10.0540i	−0.817504	+	0.408752i
$606$		0			0
$607$		−	16.7743i		−	0.680849i	−0.940272	−	0.340424i	$-0.889429\pi$
$607$		−	16.7743i		−	0.680849i	0.940272	−	0.340424i	$-0.110571\pi$
$608$		0			0
$609$	−19.6847	−	22.7299i	−0.797663	−	0.921061i
$610$		0			0
$611$	0.546188			0.0220964
$612$		0			0
$613$	9.96585			0.402517			0.201259	−	0.979538i	$-0.435497\pi$
$613$	9.96585			0.402517			0.201259	+	0.979538i	$0.435497\pi$
$614$		0			0
$615$	−22.7299	−	45.4598i	−0.916557	−	1.83311i
$616$		0			0
$617$			30.7386i			1.23749i	0.785592	+	0.618745i	$0.212359\pi$
$617$			30.7386i			1.23749i	−0.785592	+	0.618745i	$0.787641\pi$
$618$		0			0
$619$		−	29.6155i		−	1.19035i	−0.803597	−	0.595174i	$-0.797083\pi$
$619$		−	29.6155i		−	1.19035i	0.803597	−	0.595174i	$-0.202917\pi$
$620$		0			0
$621$	−1.61553			−0.0648289
$622$		0			0
$623$	15.3693	+	17.7470i	0.615759	+	0.711017i
$624$		0			0
$625$	−7.00000	−	24.0000i	−0.280000	−	0.960000i
$626$		0			0
$627$		−	2.79818i		−	0.111748i
$628$		0			0
$629$	−39.3693			−1.56976
$630$		0			0
$631$		−	16.3153i		−	0.649503i	−0.945799	−	0.324752i	$-0.894719\pi$
$631$		−	16.3153i		−	0.649503i	0.945799	−	0.324752i	$-0.105281\pi$
$632$		0			0
$633$	37.9854			1.50978
$634$		0			0
$635$	−24.0000	+	12.0000i	−0.952411	+	0.476205i
$636$		0			0
$637$	3.89055	+	0.561553i	0.154149	+	0.0222495i
$638$		0			0
$639$		−	50.7386i		−	2.00719i
$640$		0			0
$641$	8.24621			0.325706			0.162853	−	0.986650i	$-0.447930\pi$
$641$	8.24621			0.325706			0.162853	+	0.986650i	$0.447930\pi$
$642$		0			0
$643$	25.3002			0.997742			0.498871	−	0.866676i	$-0.333748\pi$
$643$	25.3002			0.997742			0.498871	+	0.866676i	$0.333748\pi$
$644$		0			0
$645$	−4.98293	−	9.96585i	−0.196203	−	0.392405i
$646$		0			0
$647$			32.0298i			1.25922i	0.776911	+	0.629611i	$0.216786\pi$
$647$			32.0298i			1.25922i	−0.776911	+	0.629611i	$0.783214\pi$
$648$		0			0
$649$		−	3.89055i		−	0.152718i
$650$		0			0
$651$	−45.4598	+	39.3693i	−1.78171	+	1.54301i
$652$		0			0
$653$	−12.7640			−0.499495			−0.249748	−	0.968311i	$-0.580348\pi$
$653$	−12.7640			−0.499495			−0.249748	+	0.968311i	$0.580348\pi$
$654$		0			0
$655$	−19.3693	−	38.7386i	−0.756822	−	1.51364i
$656$		0			0
$657$	−24.6752			−0.962670
$658$		0			0
$659$	−2.91791			−0.113666			−0.0568329	−	0.998384i	$-0.518100\pi$
$659$	−2.91791			−0.113666			−0.0568329	+	0.998384i	$0.518100\pi$
$660$		0			0
$661$	38.2462			1.48761			0.743803	−	0.668399i	$-0.233020\pi$
$661$	38.2462			1.48761			0.743803	+	0.668399i	$0.233020\pi$
$662$		0			0
$663$			6.38202i			0.247857i
$664$		0			0
$665$	−6.13676	−	2.54715i	−0.237973	−	0.0987741i
$666$		0			0
$667$		−	4.98293i		−	0.192940i
$668$		0			0
$669$			2.49146i			0.0963255i
$670$		0			0
$671$	−1.09238			−0.0421707
$672$		0			0
$673$		−	15.3693i		−	0.592444i	−0.955119	−	0.296222i	$-0.904273\pi$
$673$		−	15.3693i		−	0.592444i	0.955119	−	0.296222i	$-0.0957268\pi$
$674$		0			0
$675$	−4.31534	+	5.75379i	−0.166098	+	0.221463i
$676$		0			0
$677$			17.6847i			0.679677i	0.940484	+	0.339838i	$0.110372\pi$
$677$			17.6847i			0.679677i	−0.940484	+	0.339838i	$0.889628\pi$
$678$		0			0
$679$	18.8393	−	16.3153i	0.722987	−	0.626125i
$680$		0			0
$681$	40.1771			1.53959
$682$		0			0
$683$			29.6581i			1.13484i	0.823430	+	0.567418i	$0.192057\pi$
$683$			29.6581i			1.13484i	−0.823430	+	0.567418i	$0.807943\pi$
$684$		0			0
$685$	−13.1231	−	26.2462i	−0.501408	−	1.00282i
$686$		0			0
$687$	−2.87689			−0.109760
$688$		0			0
$689$		0			0
$690$		0			0
$691$		−	7.50758i		−	0.285602i	−0.989751	−	0.142801i	$-0.954389\pi$
$691$		−	7.50758i		−	0.285602i	0.989751	−	0.142801i	$-0.0456108\pi$
$692$		0			0
$693$	−6.92820	+	6.00000i	−0.263181	+	0.227921i
$694$		0			0
$695$	−14.2462	−	28.4924i	−0.540390	−	1.08078i
$696$		0			0
$697$		−	39.3693i		−	1.49122i
$698$		0			0
$699$		−	72.9848i		−	2.76054i
$700$		0			0
$701$			41.0230i			1.54942i	0.632318	+	0.774709i	$0.282104\pi$
$701$			41.0230i			1.54942i	−0.632318	+	0.774709i	$0.717896\pi$
$702$		0			0
$703$			9.96585i			0.375869i
$704$		0			0
$705$	2.49146	+	4.98293i	0.0938339	+	0.187668i
$706$		0			0
$707$	22.7386	−	19.6922i	0.855174	−	0.740603i
$708$		0			0
$709$		−	0.546188i		−	0.0205125i	−0.999947	−	0.0102563i	$-0.996735\pi$
$709$		−	0.546188i		−	0.0205125i	0.999947	−	0.0102563i	$-0.00326472\pi$
$710$		0			0
$711$			27.3693i			1.02643i
$712$		0			0
$713$	−9.96585			−0.373224
$714$		0			0
$715$	0.546188	+	1.09238i	0.0204263	+	0.0408525i
$716$		0			0
$717$		−	19.6847i		−	0.735137i
$718$		0			0
$719$	8.87348			0.330925			0.165462	−	0.986216i	$-0.447088\pi$
$719$	8.87348			0.330925			0.165462	+	0.986216i	$0.447088\pi$
$720$		0			0
$721$	−25.6847	−	29.6581i	−0.956547	−	1.10452i
$722$		0			0
$723$			12.7640i			0.474699i
$724$		0			0
$725$	−17.7470	−	13.3102i	−0.659105	−	0.494329i
$726$		0			0
$727$			14.2829i			0.529722i	0.964287	+	0.264861i	$0.0853260\pi$
$727$			14.2829i			0.529722i	−0.964287	+	0.264861i	$0.914674\pi$
$728$		0			0
$729$	−35.9848			−1.33277
$730$		0			0
$731$		−	8.63068i		−	0.319217i
$732$		0			0
$733$		−	31.3002i		−	1.15610i	−0.816002	−	0.578049i	$-0.803814\pi$
$733$		−	31.3002i		−	1.15610i	0.816002	−	0.578049i	$-0.196186\pi$
$734$		0			0
$735$	12.6239	+	38.0555i	0.465638	+	1.40370i
$736$		0			0
$737$		−	6.73863i		−	0.248221i
$738$		0			0
$739$	−20.6649			−0.760170			−0.380085	−	0.924952i	$-0.624105\pi$
$739$	−20.6649			−0.760170			−0.380085	+	0.924952i	$0.624105\pi$
$740$		0			0
$741$	1.61553			0.0593479
$742$		0			0
$743$	38.2462			1.40312			0.701559	−	0.712612i	$-0.252488\pi$
$743$	38.2462			1.40312			0.701559	+	0.712612i	$0.252488\pi$
$744$		0			0
$745$	10.8188	+	21.6375i	0.396369	+	0.792737i
$746$		0			0
$747$	14.2462			0.521242
$748$		0			0
$749$	18.7386	+	21.6375i	0.684695	+	0.790617i
$750$		0			0
$751$		−	15.6847i		−	0.572341i	−0.958179	−	0.286171i	$-0.907618\pi$
$751$		−	15.6847i		−	0.572341i	0.958179	−	0.286171i	$-0.0923824\pi$
$752$		0			0
$753$			51.2311i			1.86696i
$754$		0			0
$755$	−15.0540	−	30.1080i	−0.547870	−	1.09574i
$756$		0			0
$757$	13.8564			0.503620			0.251810	−	0.967777i	$-0.418974\pi$
$757$	13.8564			0.503620			0.251810	+	0.967777i	$0.418974\pi$
$758$		0			0
$759$	−2.79818			−0.101567
$760$		0			0
$761$		−	30.5110i		−	1.10602i	−0.833174	−	0.553011i	$-0.813479\pi$
$761$		−	30.5110i		−	1.10602i	0.833174	−	0.553011i	$-0.186521\pi$
$762$		0			0
$763$	−31.6847	−	36.5863i	−1.14706	−	1.32451i
$764$		0			0
$765$	−31.6034	+	15.8017i	−1.14262	+	0.571311i
$766$		0			0
$767$	2.24621			0.0811060
$768$		0			0
$769$			3.89055i			0.140297i	0.997537	+	0.0701484i	$0.0223473\pi$
$769$			3.89055i			0.140297i	−0.997537	+	0.0701484i	$0.977653\pi$
$770$		0			0
$771$	−17.7470			−0.639141
$772$		0			0
$773$			2.94602i			0.105961i	0.998596	+	0.0529806i	$0.0168721\pi$
$773$			2.94602i			0.105961i	−0.998596	+	0.0529806i	$0.983128\pi$
$774$		0			0
$775$	−26.6204	+	35.4939i	−0.956234	+	1.27498i
$776$		0			0
$777$	45.4598	−	39.3693i	1.63086	−	1.41237i
$778$		0			0
$779$	−9.96585			−0.357064
$780$		0			0
$781$		−	13.8564i		−	0.495821i
$782$		0			0
$783$			6.38202i			0.228075i
$784$		0			0
$785$	−20.2462	−	40.4924i	−0.722618	−	1.44524i
$786$		0			0
$787$	−44.8078			−1.59722			−0.798612	−	0.601846i	$-0.794432\pi$
$787$	−44.8078			−1.59722			−0.798612	+	0.601846i	$0.794432\pi$
$788$		0			0
$789$	−30.7386			−1.09432
$790$		0			0
$791$	26.6204	+	30.7386i	0.946514	+	1.09294i
$792$		0			0
$793$		−	0.630683i		−	0.0223962i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	21.0540i		−	0.745770i	−0.927878	−	0.372885i	$-0.878369\pi$
$797$		−	21.0540i		−	0.745770i	0.927878	−	0.372885i	$-0.121631\pi$
$798$		0			0
$799$			4.31534i			0.152666i
$800$		0			0
$801$		−	31.6034i		−	1.11665i
$802$		0			0
$803$	−6.73863			−0.237801
$804$		0			0
$805$	−2.54715	+	6.13676i	−0.0897751	+	0.216292i
$806$		0			0
$807$	−51.2311			−1.80342
$808$		0			0
$809$	31.3002			1.10046			0.550228	−	0.835014i	$-0.314541\pi$
$809$	31.3002			1.10046			0.550228	+	0.835014i	$0.314541\pi$
$810$		0			0
$811$			14.2462i			0.500252i	0.968213	+	0.250126i	$0.0804721\pi$
$811$			14.2462i			0.500252i	−0.968213	+	0.250126i	$0.919528\pi$
$812$		0			0
$813$	58.2238			2.04200
$814$		0			0
$815$	−24.6752	−	49.3503i	−0.864333	−	1.72867i
$816$		0			0
$817$	−2.18475			−0.0764347
$818$		0			0
$819$	−3.46410	−	4.00000i	−0.121046	−	0.139771i
$820$		0			0
$821$		−	18.2931i		−	0.638435i	−0.947681	−	0.319218i	$-0.896580\pi$
$821$		−	18.2931i		−	0.638435i	0.947681	−	0.319218i	$-0.103420\pi$
$822$		0			0
$823$	−35.3693			−1.23290			−0.616448	−	0.787395i	$-0.711429\pi$
$823$	−35.3693			−1.23290			−0.616448	+	0.787395i	$0.711429\pi$
$824$		0			0
$825$	−7.47439	+	9.96585i	−0.260225	+	0.346967i
$826$		0			0
$827$		−	34.6410i		−	1.20459i	−0.798275	−	0.602293i	$-0.794254\pi$
$827$		−	34.6410i		−	1.20459i	0.798275	−	0.602293i	$-0.205746\pi$
$828$		0			0
$829$	−29.6155			−1.02859			−0.514295	−	0.857613i	$-0.671946\pi$
$829$	−29.6155			−1.02859			−0.514295	+	0.857613i	$0.671946\pi$
$830$		0			0
$831$	35.4939			1.23127
$832$		0			0
$833$	−4.43674	+	30.7386i	−0.153724	+	1.06503i
$834$		0			0
$835$	2.91791	+	5.83583i	0.100979	+	0.201957i
$836$		0			0
$837$	12.7640			0.441189
$838$		0			0
$839$	−32.6957			−1.12878			−0.564391	−	0.825507i	$-0.690889\pi$
$839$	−32.6957			−1.12878			−0.564391	+	0.825507i	$0.690889\pi$
$840$		0			0
$841$	9.31534			0.321219
$842$		0			0
$843$	65.7926			2.26602
$844$		0			0
$845$	25.3693	−	12.6847i	0.872731	−	0.436366i
$846$		0			0
$847$	20.1080	−	17.4140i	0.690917	−	0.598352i
$848$		0			0
$849$	−13.9309			−0.478106
$850$		0			0
$851$	9.96585			0.341625
$852$		0			0
$853$		−	22.4924i		−	0.770126i	−0.922890	−	0.385063i	$-0.874180\pi$
$853$		−	22.4924i		−	0.770126i	0.922890	−	0.385063i	$-0.125820\pi$
$854$		0			0
$855$	4.00000	+	8.00000i	0.136797	+	0.273594i
$856$		0			0
$857$	−32.4563			−1.10868			−0.554342	−	0.832289i	$-0.687030\pi$
$857$	−32.4563			−1.10868			−0.554342	+	0.832289i	$0.687030\pi$
$858$		0			0
$859$		−	9.75379i		−	0.332795i	−0.986059	−	0.166397i	$-0.946787\pi$
$859$		−	9.75379i		−	0.332795i	0.986059	−	0.166397i	$-0.0532135\pi$
$860$		0			0
$861$	39.3693	+	45.4598i	1.34170	+	1.54926i
$862$		0			0
$863$	−3.36932			−0.114693			−0.0573464	−	0.998354i	$-0.518264\pi$
$863$	−3.36932			−0.114693			−0.0573464	+	0.998354i	$0.518264\pi$
$864$		0			0
$865$	−9.68466	−	19.3693i	−0.329288	−	0.658577i
$866$		0			0
$867$	−6.87689			−0.233552
$868$		0			0
$869$			7.47439i			0.253551i
$870$		0			0
$871$	3.89055			0.131826
$872$		0			0
$873$	−33.5486			−1.13545
$874$		0			0
$875$	15.0526	+	25.4641i	0.508869	+	0.860844i
$876$		0			0
$877$	−26.6204			−0.898908			−0.449454	−	0.893303i	$-0.648381\pi$
$877$	−26.6204			−0.898908			−0.449454	+	0.893303i	$0.648381\pi$
$878$		0			0
$879$		−	36.6695i		−	1.23683i
$880$		0			0
$881$			13.8564i			0.466834i	0.972377	+	0.233417i	$0.0749907\pi$
$881$			13.8564i			0.466834i	−0.972377	+	0.233417i	$0.925009\pi$
$882$		0			0
$883$		−	15.8017i		−	0.531769i	−0.964005	−	0.265884i	$-0.914336\pi$
$883$		−	15.8017i		−	0.531769i	0.964005	−	0.265884i	$-0.0856640\pi$
$884$		0			0
$885$	10.2462	+	20.4924i	0.344423	+	0.688845i
$886$		0			0
$887$		−	32.0298i		−	1.07546i	−0.843118	−	0.537728i	$-0.819283\pi$
$887$		−	32.0298i		−	1.07546i	0.843118	−	0.537728i	$-0.180717\pi$
$888$		0			0
$889$	24.0000	−	20.7846i	0.804934	−	0.697093i
$890$		0			0
$891$	−6.80847			−0.228092
$892$		0			0
$893$	1.09238			0.0365549
$894$		0			0
$895$	−42.4221	+	21.2111i	−1.41802	+	0.709008i
$896$		0			0
$897$		−	1.61553i		−	0.0539409i
$898$		0			0
$899$			39.3693i			1.31304i
$900$		0			0
$901$		0			0
$902$		0			0
$903$	8.63068	+	9.96585i	0.287211	+	0.331643i
$904$		0			0
$905$	−28.4924	+	14.2462i	−0.947120	+	0.473560i
$906$		0			0
$907$			33.5486i			1.11396i	0.830524	+	0.556982i	$0.188041\pi$
$907$			33.5486i			1.11396i	−0.830524	+	0.556982i	$0.811959\pi$
$908$		0			0
$909$	−40.4924			−1.34305
$910$		0			0
$911$			18.7386i			0.620839i	0.950600	+	0.310419i	$0.100469\pi$
$911$			18.7386i			0.620839i	−0.950600	+	0.310419i	$0.899531\pi$
$912$		0			0
$913$	3.89055			0.128758
$914$		0			0
$915$	5.75379	−	2.87689i	0.190214	−	0.0951072i
$916$		0			0
$917$	33.5486	+	38.7386i	1.10787	+	1.27926i
$918$		0			0
$919$		−	7.68466i		−	0.253493i	−0.991935	−	0.126747i	$-0.959546\pi$
$919$		−	7.68466i		−	0.253493i	0.991935	−	0.126747i	$-0.0404536\pi$
$920$		0			0
$921$	25.4384			0.838225
$922$		0			0
$923$	8.00000			0.263323
$924$		0			0
$925$	26.6204	−	35.4939i	0.875275	−	1.16703i
$926$		0			0
$927$			52.8144i			1.73465i
$928$		0			0
$929$		−	9.96585i		−	0.326969i	−0.986546	−	0.163485i	$-0.947727\pi$
$929$		−	9.96585i		−	0.326969i	0.986546	−	0.163485i	$-0.0522734\pi$
$930$		0			0
$931$	7.78110	+	1.12311i	0.255015	+	0.0368083i
$932$		0			0
$933$	−48.2579			−1.57989
$934$		0			0
$935$	−8.63068	+	4.31534i	−0.282254	+	0.141127i
$936$		0			0
$937$	−31.0572			−1.01459			−0.507297	−	0.861771i	$-0.669355\pi$
$937$	−31.0572			−1.01459			−0.507297	+	0.861771i	$0.669355\pi$
$938$		0			0
$939$	46.8589			1.52918
$940$		0			0
$941$	−19.3693			−0.631422			−0.315711	−	0.948855i	$-0.602243\pi$
$941$	−19.3693			−0.631422			−0.315711	+	0.948855i	$0.602243\pi$
$942$		0			0
$943$			9.96585i			0.324533i
$944$		0			0
$945$	3.26233	−	7.85982i	0.106123	−	0.255680i
$946$		0			0
$947$			19.6922i			0.639912i	0.947432	+	0.319956i	$0.103668\pi$
$947$			19.6922i			0.639912i	−0.947432	+	0.319956i	$0.896332\pi$
$948$		0			0
$949$		−	3.89055i		−	0.126293i
$950$		0			0
$951$	−45.4598			−1.47413
$952$		0			0
$953$			17.6155i			0.570623i	0.958435	+	0.285311i	$0.0920970\pi$
$953$			17.6155i			0.570623i	−0.958435	+	0.285311i	$0.907903\pi$
$954$		0			0
$955$	−5.43845	−	10.8769i	−0.175984	−	0.351968i
$956$		0			0
$957$			11.0540i			0.357324i
$958$		0			0
$959$	22.7299	+	26.2462i	0.733986	+	0.847534i
$960$		0			0
$961$	47.7386			1.53996
$962$		0			0
$963$		−	38.5316i		−	1.24166i
$964$		0			0
$965$	7.36932	+	14.7386i	0.237227	+	0.474453i
$966$		0			0
$967$	−5.26137			−0.169194			−0.0845971	−	0.996415i	$-0.526960\pi$
$967$	−5.26137			−0.169194			−0.0845971	+	0.996415i	$0.526960\pi$
$968$		0			0
$969$			12.7640i			0.410040i
$970$		0			0
$971$			12.6307i			0.405338i	0.979247	+	0.202669i	$0.0649615\pi$
$971$			12.6307i			0.405338i	−0.979247	+	0.202669i	$0.935038\pi$
$972$		0			0
$973$	24.6752	+	28.4924i	0.791049	+	0.913425i
$974$		0			0
$975$	−5.75379	−	4.31534i	−0.184269	−	0.138202i
$976$		0			0
$977$			37.1231i			1.18767i	0.804586	+	0.593837i	$0.202387\pi$
$977$			37.1231i			1.18767i	−0.804586	+	0.593837i	$0.797613\pi$
$978$		0			0
$979$		−	8.63068i		−	0.275838i
$980$		0			0
$981$			65.1520i			2.08014i
$982$		0			0
$983$		−	4.86319i		−	0.155112i	−0.996988	−	0.0775558i	$-0.975288\pi$
$983$		−	4.86319i		−	0.155112i	0.996988	−	0.0775558i	$-0.0247116\pi$
$984$		0			0
$985$	−45.4598	+	22.7299i	−1.44847	+	0.724234i
$986$		0			0
$987$	−4.31534	−	4.98293i	−0.137359	−	0.158608i
$988$		0			0
$989$			2.18475i			0.0694710i
$990$		0			0
$991$			49.4773i			1.57170i	0.618419	+	0.785849i	$0.287774\pi$
$991$			49.4773i			1.57170i	−0.618419	+	0.785849i	$0.712226\pi$
$992$		0			0
$993$	−44.3674			−1.40796
$994$		0			0
$995$	35.4939	−	17.7470i	1.12523	−	0.562616i
$996$		0			0
$997$		−	54.1771i		−	1.71581i	−0.513812	−	0.857903i	$-0.671767\pi$
$997$		−	54.1771i		−	1.71581i	0.513812	−	0.857903i	$-0.328233\pi$
$998$		0			0
$999$	−12.7640			−0.403836

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.2.n.b.1119.2	yes	8
4.3	odd	2		2240.2.n.h.1119.5	yes	8
5.4	even	2		2240.2.n.h.1119.7	yes	8
7.6	odd	2		2240.2.n.g.1119.7	yes	8
8.3	odd	2		2240.2.n.a.1119.3	yes	8
8.5	even	2		2240.2.n.g.1119.8	yes	8
20.19	odd	2	inner	2240.2.n.b.1119.4	yes	8
28.27	even	2		2240.2.n.a.1119.4	yes	8
35.34	odd	2		2240.2.n.a.1119.2	yes	8
40.19	odd	2		2240.2.n.g.1119.6	yes	8
40.29	even	2		2240.2.n.a.1119.1	✓	8
56.13	odd	2	inner	2240.2.n.b.1119.1	yes	8
56.27	even	2		2240.2.n.h.1119.6	yes	8
140.139	even	2		2240.2.n.g.1119.5	yes	8
280.69	odd	2		2240.2.n.h.1119.8	yes	8
280.139	even	2	inner	2240.2.n.b.1119.3	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2240.2.n.a.1119.1	✓	8	40.29	even	2
2240.2.n.a.1119.2	yes	8	35.34	odd	2
2240.2.n.a.1119.3	yes	8	8.3	odd	2
2240.2.n.a.1119.4	yes	8	28.27	even	2
2240.2.n.b.1119.1	yes	8	56.13	odd	2	inner
2240.2.n.b.1119.2	yes	8	1.1	even	1	trivial
2240.2.n.b.1119.3	yes	8	280.139	even	2	inner
2240.2.n.b.1119.4	yes	8	20.19	odd	2	inner
2240.2.n.g.1119.5	yes	8	140.139	even	2
2240.2.n.g.1119.6	yes	8	40.19	odd	2
2240.2.n.g.1119.7	yes	8	7.6	odd	2
2240.2.n.g.1119.8	yes	8	8.5	even	2
2240.2.n.h.1119.5	yes	8	4.3	odd	2
2240.2.n.h.1119.6	yes	8	56.27	even	2
2240.2.n.h.1119.7	yes	8	5.4	even	2
2240.2.n.h.1119.8	yes	8	280.69	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:	\( N \)	\(=\)	\( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2240.n (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.56155 q^{3} +(2.00000 - 1.00000i) q^{5} +(-2.00000 + 1.73205i) q^{7} +3.56155 q^{9} +O(q^{10})\)	\(q-2.56155 q^{3} +(2.00000 - 1.00000i) q^{5} +(-2.00000 + 1.73205i) q^{7} +3.56155 q^{9} +0.972638 q^{11} +0.561553i q^{13} +(-5.12311 + 2.56155i) q^{15} -4.43674 q^{17} +1.12311i q^{19} +(5.12311 - 4.43674i) q^{21} +1.12311 q^{23} +(3.00000 - 4.00000i) q^{25} -1.43845 q^{27} -4.43674i q^{29} -8.87348 q^{31} -2.49146 q^{33} +(-2.26795 + 5.46410i) q^{35} +8.87348 q^{37} -1.43845i q^{39} +8.87348i q^{41} +1.94528i q^{43} +(7.12311 - 3.56155i) q^{45} -0.972638i q^{47} +(1.00000 - 6.92820i) q^{49} +11.3649 q^{51} +(1.94528 - 0.972638i) q^{55} -2.87689i q^{57} -4.00000i q^{59} -1.12311 q^{61} +(-7.12311 + 6.16879i) q^{63} +(0.561553 + 1.12311i) q^{65} -6.92820i q^{67} -2.87689 q^{69} -14.2462i q^{71} -6.92820 q^{73} +(-7.68466 + 10.2462i) q^{75} +(-1.94528 + 1.68466i) q^{77} +7.68466i q^{79} -7.00000 q^{81} +4.00000 q^{83} +(-8.87348 + 4.43674i) q^{85} +11.3649i q^{87} -8.87348i q^{89} +(-0.972638 - 1.12311i) q^{91} +22.7299 q^{93} +(1.12311 + 2.24621i) q^{95} -9.41967 q^{97} +3.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{3} + 16 q^{5} - 16 q^{7} + 12 q^{9}+O(q^{10}) \) 8 * q - 4 * q^3 + 16 * q^5 - 16 * q^7 + 12 * q^9	\( 8 q - 4 q^{3} + 16 q^{5} - 16 q^{7} + 12 q^{9} - 8 q^{15} + 8 q^{21} - 24 q^{23} + 24 q^{25} - 28 q^{27} - 32 q^{35} + 24 q^{45} + 8 q^{49} + 24 q^{61} - 24 q^{63} - 12 q^{65} - 56 q^{69} - 12 q^{75} - 56 q^{81} + 32 q^{83} - 24 q^{95}+O(q^{100}) \) 8 * q - 4 * q^3 + 16 * q^5 - 16 * q^7 + 12 * q^9 - 8 * q^15 + 8 * q^21 - 24 * q^23 + 24 * q^25 - 28 * q^27 - 32 * q^35 + 24 * q^45 + 8 * q^49 + 24 * q^61 - 24 * q^63 - 12 * q^65 - 56 * q^69 - 12 * q^75 - 56 * q^81 + 32 * q^83 - 24 * q^95

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