LMFDB - Embedded newform 1760.2.b.f.1409.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1760,2,Mod(1409,1760)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1760, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1760.1409");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$1760 = 2^{5} \cdot 5 \cdot 11$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1760.b (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:	$14.0536707557$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{14} + 16x^{12} + 96x^{10} + 272x^{8} + 372x^{6} + 225x^{4} + 56x^{2} + 4$ x^14 + 16x^12 + 96x^10 + 272x^8 + 372x^6 + 225x^4 + 56x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$2^{13}$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1409.3
Root			$-1.51372i$ of defining polynomial
Character	$\chi$	$=$	1760.1409
Dual form			1760.2.b.f.1409.12

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.13376i q^{3} +(2.12532 - 0.694983i) q^{5} +1.20437i q^{7} -1.55295 q^{9} +1.00000 q^{11} +4.14048i q^{13} +(-1.48293 - 4.53494i) q^{15} -0.621669i q^{17} +4.76215 q^{19} +2.56983 q^{21} -4.66983i q^{23} +(4.03400 - 2.95413i) q^{25} -3.08766i q^{27} -2.10643 q^{29} +2.08954 q^{31} -2.13376i q^{33} +(0.837014 + 2.55967i) q^{35} +8.90878i q^{37} +8.83481 q^{39} +9.62524 q^{41} -10.2003i q^{43} +(-3.30052 + 1.07927i) q^{45} -6.90511i q^{47} +5.54950 q^{49} -1.32650 q^{51} +3.32273i q^{53} +(2.12532 - 0.694983i) q^{55} -10.1613i q^{57} -10.7689 q^{59} +9.40465 q^{61} -1.87032i q^{63} +(2.87756 + 8.79986i) q^{65} +5.38505i q^{67} -9.96433 q^{69} -8.42895 q^{71} -8.46268i q^{73} +(-6.30341 - 8.60760i) q^{75} +1.20437i q^{77} +6.44641 q^{79} -11.2472 q^{81} -0.675980i q^{83} +(-0.432050 - 1.32125i) q^{85} +4.49462i q^{87} -15.6891 q^{89} -4.98666 q^{91} -4.45860i q^{93} +(10.1211 - 3.30961i) q^{95} +10.5470i q^{97} -1.55295 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$14 q + 2 q^{5} - 12 q^{9} + 14 q^{11} + 12 q^{15} - 14 q^{19} + 26 q^{21} - 4 q^{25} + 22 q^{29} - 22 q^{31} - 2 q^{35} - 8 q^{39} + 8 q^{41} - 8 q^{45} - 32 q^{49} - 14 q^{51} + 2 q^{55} - 52 q^{59} - 10 q^{61}+ \cdots - 12 q^{99}+O(q^{100})$ 14 * q + 2 * q^5 - 12 * q^9 + 14 * q^11 + 12 * q^15 - 14 * q^19 + 26 * q^21 - 4 * q^25 + 22 * q^29 - 22 * q^31 - 2 * q^35 - 8 * q^39 + 8 * q^41 - 8 * q^45 - 32 * q^49 - 14 * q^51 + 2 * q^55 - 52 * q^59 - 10 * q^61 + 8 * q^65 - 4 * q^69 + 6 * q^71 + 10 * q^75 + 64 * q^79 - 2 * q^81 - 34 * q^85 - 22 * q^89 + 16 * q^91 - 28 * q^95 - 12 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1760\mathbb{Z}\right)^\times$ .

$n$	$321$	$991$	$1057$	$1541$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$3$		−	2.13376i		−	1.23193i	−0.787774	−	0.615965i	$-0.788766\pi$
$3$		−	2.13376i		−	1.23193i	0.787774	−	0.615965i	$-0.211234\pi$
$4$		0			0
$5$	2.12532	−	0.694983i	0.950473	−	0.310806i
$5$	2.12532	−	0.694983i	0.950473	−	0.310806i
$6$		0			0
$7$			1.20437i			0.455208i	0.973754	+	0.227604i	$0.0730891\pi$
$7$			1.20437i			0.455208i	−0.973754	+	0.227604i	$0.926911\pi$
$8$		0			0
$9$	−1.55295			−0.517651
$10$		0			0
$11$	1.00000			0.301511
$11$	1.00000			0.301511
$12$		0			0
$13$			4.14048i			1.14836i	0.818728	+	0.574181i	$0.194680\pi$
$13$			4.14048i			1.14836i	−0.818728	+	0.574181i	$0.805320\pi$
$14$		0			0
$15$	−1.48293	−	4.53494i	−0.382891	−	1.17092i
$16$		0			0
$17$		−	0.621669i		−	0.150777i	−0.997154	−	0.0753885i	$-0.975980\pi$
$17$		−	0.621669i		−	0.150777i	0.997154	−	0.0753885i	$-0.0240197\pi$
$18$		0			0
$19$	4.76215			1.09251			0.546256	−	0.837618i	$-0.316053\pi$
$19$	4.76215			1.09251			0.546256	+	0.837618i	$0.316053\pi$
$20$		0			0
$21$	2.56983			0.560784
$22$		0			0
$23$		−	4.66983i		−	0.973728i	−0.873478	−	0.486864i	$-0.838141\pi$
$23$		−	4.66983i		−	0.973728i	0.873478	−	0.486864i	$-0.161859\pi$
$24$		0			0
$25$	4.03400	−	2.95413i	0.806800	−	0.590825i
$26$		0			0
$27$		−	3.08766i		−	0.594221i
$28$		0			0
$29$	−2.10643			−0.391154			−0.195577	−	0.980688i	$-0.562658\pi$
$29$	−2.10643			−0.391154			−0.195577	+	0.980688i	$0.562658\pi$
$30$		0			0
$31$	2.08954			0.375293			0.187647	−	0.982237i	$-0.439914\pi$
$31$	2.08954			0.375293			0.187647	+	0.982237i	$0.439914\pi$
$32$		0			0
$33$		−	2.13376i		−	0.371441i
$34$		0			0
$35$	0.837014	+	2.55967i	0.141481	+	0.432663i
$36$		0			0
$37$			8.90878i			1.46459i	0.680985	+	0.732297i	$0.261552\pi$
$37$			8.90878i			1.46459i	−0.680985	+	0.732297i	$0.738448\pi$
$38$		0			0
$39$	8.83481			1.41470
$40$		0			0
$41$	9.62524			1.50321			0.751605	−	0.659614i	$-0.229280\pi$
$41$	9.62524			1.50321			0.751605	+	0.659614i	$0.229280\pi$
$42$		0			0
$43$		−	10.2003i		−	1.55553i	−0.628556	−	0.777764i	$-0.716354\pi$
$43$		−	10.2003i		−	1.55553i	0.628556	−	0.777764i	$-0.283646\pi$
$44$		0			0
$45$	−3.30052	+	1.07927i	−0.492013	+	0.160889i
$46$		0			0
$47$		−	6.90511i		−	1.00721i	−0.863933	−	0.503607i	$-0.832006\pi$
$47$		−	6.90511i		−	1.00721i	0.863933	−	0.503607i	$-0.167994\pi$
$48$		0			0
$49$	5.54950			0.792786
$50$		0			0
$51$	−1.32650			−0.185747
$52$		0			0
$53$			3.32273i			0.456412i	0.973613	+	0.228206i	$0.0732859\pi$
$53$			3.32273i			0.456412i	−0.973613	+	0.228206i	$0.926714\pi$
$54$		0			0
$55$	2.12532	−	0.694983i	0.286579	−	0.0937115i
$56$		0			0
$57$		−	10.1613i		−	1.34590i
$58$		0			0
$59$	−10.7689			−1.40200			−0.700998	−	0.713163i	$-0.747262\pi$
$59$	−10.7689			−1.40200			−0.700998	+	0.713163i	$0.747262\pi$
$60$		0			0
$61$	9.40465			1.20414			0.602071	−	0.798443i	$-0.294342\pi$
$61$	9.40465			1.20414			0.602071	+	0.798443i	$0.294342\pi$
$62$		0			0
$63$		−	1.87032i		−	0.235639i
$64$		0			0
$65$	2.87756	+	8.79986i	0.356918	+	1.09149i
$66$		0			0
$67$			5.38505i			0.657888i	0.944349	+	0.328944i	$0.106693\pi$
$67$			5.38505i			0.657888i	−0.944349	+	0.328944i	$0.893307\pi$
$68$		0			0
$69$	−9.96433			−1.19956
$70$		0			0
$71$	−8.42895			−1.00033			−0.500166	−	0.865929i	$-0.666728\pi$
$71$	−8.42895			−1.00033			−0.500166	+	0.865929i	$0.666728\pi$
$72$		0			0
$73$		−	8.46268i		−	0.990482i	−0.868756	−	0.495241i	$-0.835080\pi$
$73$		−	8.46268i		−	0.990482i	0.868756	−	0.495241i	$-0.164920\pi$
$74$		0			0
$75$	−6.30341	−	8.60760i	−0.727855	−	0.993920i
$76$		0			0
$77$			1.20437i			0.137250i
$78$		0			0
$79$	6.44641			0.725278			0.362639	−	0.931930i	$-0.381876\pi$
$79$	6.44641			0.725278			0.362639	+	0.931930i	$0.381876\pi$
$80$		0			0
$81$	−11.2472			−1.24969
$82$		0			0
$83$		−	0.675980i		−	0.0741984i	−0.999312	−	0.0370992i	$-0.988188\pi$
$83$		−	0.675980i		−	0.0741984i	0.999312	−	0.0370992i	$-0.0118118\pi$
$84$		0			0
$85$	−0.432050	−	1.32125i	−0.0468624	−	0.143309i
$86$		0			0
$87$			4.49462i			0.481874i
$88$		0			0
$89$	−15.6891			−1.66304			−0.831518	−	0.555497i	$-0.812528\pi$
$89$	−15.6891			−1.66304			−0.831518	+	0.555497i	$0.812528\pi$
$90$		0			0
$91$	−4.98666			−0.522744
$92$		0			0
$93$		−	4.45860i		−	0.462335i
$94$		0			0
$95$	10.1211	−	3.30961i	1.03840	−	0.339559i
$96$		0			0
$97$			10.5470i			1.07088i	0.844572	+	0.535442i	$0.179855\pi$
$97$			10.5470i			1.07088i	−0.844572	+	0.535442i	$0.820145\pi$
$98$		0			0
$99$	−1.55295			−0.156078
$100$		0			0
$101$	−1.27495			−0.126862			−0.0634311	−	0.997986i	$-0.520204\pi$
$101$	−1.27495			−0.126862			−0.0634311	+	0.997986i	$0.520204\pi$
$102$		0			0
$103$			3.58766i			0.353503i	0.984256	+	0.176751i	$0.0565589\pi$
$103$			3.58766i			0.353503i	−0.984256	+	0.176751i	$0.943441\pi$
$104$		0			0
$105$	5.46173	−	1.78599i	0.533010	−	0.174295i
$106$		0			0
$107$			4.97478i			0.480930i	0.970658	+	0.240465i	$0.0772999\pi$
$107$			4.97478i			0.480930i	−0.970658	+	0.240465i	$0.922700\pi$
$108$		0			0
$109$	−6.63873			−0.635875			−0.317937	−	0.948112i	$-0.602990\pi$
$109$	−6.63873			−0.635875			−0.317937	+	0.948112i	$0.602990\pi$
$110$		0			0
$111$	19.0092			1.80428
$112$		0			0
$113$		−	4.16340i		−	0.391660i	−0.980638	−	0.195830i	$-0.937260\pi$
$113$		−	4.16340i		−	0.391660i	0.980638	−	0.195830i	$-0.0627400\pi$
$114$		0			0
$115$	−3.24545	−	9.92491i	−0.302640	−	0.925502i
$116$		0			0
$117$		−	6.42997i		−	0.594451i
$118$		0			0
$119$	0.748718			0.0686348
$120$		0			0
$121$	1.00000			0.0909091
$122$		0			0
$123$		−	20.5380i		−	1.85185i
$124$		0			0
$125$	6.52048	−	9.08203i	0.583210	−	0.812322i
$126$		0			0
$127$			4.75874i			0.422270i	0.977457	+	0.211135i	$0.0677160\pi$
$127$			4.75874i			0.422270i	−0.977457	+	0.211135i	$0.932284\pi$
$128$		0			0
$129$	−21.7650			−1.91630
$130$		0			0
$131$	10.8864			0.951146			0.475573	−	0.879676i	$-0.342241\pi$
$131$	10.8864			0.951146			0.475573	+	0.879676i	$0.342241\pi$
$132$		0			0
$133$			5.73537i			0.497320i
$134$		0			0
$135$	−2.14587	−	6.56228i	−0.184687	−	0.564791i
$136$		0			0
$137$		−	10.0795i		−	0.861146i	−0.902556	−	0.430573i	$-0.858312\pi$
$137$		−	10.0795i		−	0.861146i	0.902556	−	0.430573i	$-0.141688\pi$
$138$		0			0
$139$	−20.5862			−1.74610			−0.873048	−	0.487634i	$-0.837860\pi$
$139$	−20.5862			−1.74610			−0.873048	+	0.487634i	$0.837860\pi$
$140$		0			0
$141$	−14.7339			−1.24082
$142$		0			0
$143$			4.14048i			0.346244i
$144$		0			0
$145$	−4.47684	+	1.46393i	−0.371781	+	0.121573i
$146$		0			0
$147$		−	11.8413i		−	0.976656i
$148$		0			0
$149$	−6.65848			−0.545484			−0.272742	−	0.962087i	$-0.587931\pi$
$149$	−6.65848			−0.545484			−0.272742	+	0.962087i	$0.587931\pi$
$150$		0			0
$151$	20.8504			1.69678			0.848389	−	0.529373i	$-0.177573\pi$
$151$	20.8504			1.69678			0.848389	+	0.529373i	$0.177573\pi$
$152$		0			0
$153$			0.965422i			0.0780498i
$154$		0			0
$155$	4.44096	−	1.45220i	0.356706	−	0.116643i
$156$		0			0
$157$		−	15.8397i		−	1.26415i	−0.774909	−	0.632073i	$-0.782204\pi$
$157$		−	15.8397i		−	1.26415i	0.774909	−	0.632073i	$-0.217796\pi$
$158$		0			0
$159$	7.08992			0.562267
$160$		0			0
$161$	5.62419			0.443248
$162$		0			0
$163$		−	1.86398i		−	0.145998i	−0.997332	−	0.0729990i	$-0.976743\pi$
$163$		−	1.86398i		−	0.145998i	0.997332	−	0.0729990i	$-0.0232570\pi$
$164$		0			0
$165$	−1.48293	−	4.53494i	−0.115446	−	0.353045i
$166$		0			0
$167$			14.7054i			1.13794i	0.822359	+	0.568969i	$0.192658\pi$
$167$			14.7054i			1.13794i	−0.822359	+	0.568969i	$0.807342\pi$
$168$		0			0
$169$	−4.14358			−0.318737
$170$		0			0
$171$	−7.39539			−0.565540
$172$		0			0
$173$			11.9226i			0.906462i	0.891393	+	0.453231i	$0.149729\pi$
$173$			11.9226i			0.906462i	−0.891393	+	0.453231i	$0.850271\pi$
$174$		0			0
$175$	3.55785	+	4.85841i	0.268948	+	0.367261i
$176$		0			0
$177$			22.9784i			1.72716i
$178$		0			0
$179$	−10.9996			−0.822151			−0.411075	−	0.911601i	$-0.634847\pi$
$179$	−10.9996			−0.822151			−0.411075	+	0.911601i	$0.634847\pi$
$180$		0			0
$181$	−0.0996441			−0.00740649			−0.00370324	−	0.999993i	$-0.501179\pi$
$181$	−0.0996441			−0.00740649			−0.00370324	+	0.999993i	$0.501179\pi$
$182$		0			0
$183$		−	20.0673i		−	1.48342i
$184$		0			0
$185$	6.19145	+	18.9340i	0.455204	+	1.39206i
$186$		0			0
$187$		−	0.621669i		−	0.0454610i
$188$		0			0
$189$	3.71867			0.270494
$190$		0			0
$191$	−20.5239			−1.48506			−0.742530	−	0.669813i	$-0.766374\pi$
$191$	−20.5239			−1.48506			−0.742530	+	0.669813i	$0.766374\pi$
$192$		0			0
$193$		−	16.8714i		−	1.21443i	−0.794539	−	0.607214i	$-0.792287\pi$
$193$		−	16.8714i		−	1.21443i	0.794539	−	0.607214i	$-0.207713\pi$
$194$		0			0
$195$	18.7768	−	6.14004i	1.34464	−	0.439698i
$196$		0			0
$197$			1.36235i			0.0970631i	0.998822	+	0.0485316i	$0.0154541\pi$
$197$			1.36235i			0.0970631i	−0.998822	+	0.0485316i	$0.984546\pi$
$198$		0			0
$199$	20.1085			1.42546			0.712728	−	0.701440i	$-0.247459\pi$
$199$	20.1085			1.42546			0.712728	+	0.701440i	$0.247459\pi$
$200$		0			0
$201$	11.4904			0.810472
$202$		0			0
$203$		−	2.53691i		−	0.178056i
$204$		0			0
$205$	20.4567	−	6.68938i	1.42876	−	0.467206i
$206$		0			0
$207$			7.25203i			0.504051i
$208$		0			0
$209$	4.76215			0.329405
$210$		0			0
$211$	−6.77219			−0.466217			−0.233109	−	0.972451i	$-0.574890\pi$
$211$	−6.77219			−0.466217			−0.233109	+	0.972451i	$0.574890\pi$
$212$		0			0
$213$			17.9854i			1.23234i
$214$		0			0
$215$	−7.08902	−	21.6789i	−0.483467	−	1.47849i
$216$		0			0
$217$			2.51658i			0.170836i
$218$		0			0
$219$	−18.0574			−1.22020
$220$		0			0
$221$	2.57401			0.173147
$222$		0			0
$223$			2.11259i			0.141470i	0.997495	+	0.0707349i	$0.0225344\pi$
$223$			2.11259i			0.141470i	−0.997495	+	0.0707349i	$0.977466\pi$
$224$		0			0
$225$	−6.26460	+	4.58762i	−0.417640	+	0.305841i
$226$		0			0
$227$			7.64802i			0.507617i	0.967254	+	0.253808i	$0.0816833\pi$
$227$			7.64802i			0.507617i	−0.967254	+	0.253808i	$0.918317\pi$
$228$		0			0
$229$	−12.2471			−0.809310			−0.404655	−	0.914469i	$-0.632608\pi$
$229$	−12.2471			−0.809310			−0.404655	+	0.914469i	$0.632608\pi$
$230$		0			0
$231$	2.56983			0.169083
$232$		0			0
$233$		−	15.9502i		−	1.04493i	−0.852659	−	0.522467i	$-0.825012\pi$
$233$		−	15.9502i		−	1.04493i	0.852659	−	0.522467i	$-0.174988\pi$
$234$		0			0
$235$	−4.79893	−	14.6756i	−0.313048	−	0.957330i
$236$		0			0
$237$		−	13.7551i		−	0.893492i
$238$		0			0
$239$	28.5217			1.84491			0.922456	−	0.386101i	$-0.126178\pi$
$239$	28.5217			1.84491			0.922456	+	0.386101i	$0.126178\pi$
$240$		0			0
$241$	4.51452			0.290806			0.145403	−	0.989373i	$-0.453552\pi$
$241$	4.51452			0.290806			0.145403	+	0.989373i	$0.453552\pi$
$242$		0			0
$243$			14.7359i			0.945308i
$244$		0			0
$245$	11.7945	−	3.85681i	0.753522	−	0.246402i
$246$		0			0
$247$			19.7176i			1.25460i
$248$		0			0
$249$	−1.44238			−0.0914072
$250$		0			0
$251$	−14.1643			−0.894044			−0.447022	−	0.894523i	$-0.647515\pi$
$251$	−14.1643			−0.894044			−0.447022	+	0.894523i	$0.647515\pi$
$252$		0			0
$253$		−	4.66983i		−	0.293590i
$254$		0			0
$255$	−2.81923	+	0.921892i	−0.176547	+	0.0577311i
$256$		0			0
$257$			3.25007i			0.202734i	0.994849	+	0.101367i	$0.0323216\pi$
$257$			3.25007i			0.202734i	−0.994849	+	0.101367i	$0.967678\pi$
$258$		0			0
$259$	−10.7294			−0.666695
$260$		0			0
$261$	3.27118			0.202481
$262$		0			0
$263$			5.56822i			0.343351i	0.985154	+	0.171676i	$0.0549181\pi$
$263$			5.56822i			0.343351i	−0.985154	+	0.171676i	$0.945082\pi$
$264$		0			0
$265$	2.30924	+	7.06187i	0.141855	+	0.433807i
$266$		0			0
$267$			33.4768i			2.04874i
$268$		0			0
$269$	14.2454			0.868557			0.434279	−	0.900779i	$-0.357003\pi$
$269$	14.2454			0.868557			0.434279	+	0.900779i	$0.357003\pi$
$270$		0			0
$271$	−16.5866			−1.00757			−0.503783	−	0.863830i	$-0.668059\pi$
$271$	−16.5866			−1.00757			−0.503783	+	0.863830i	$0.668059\pi$
$272$		0			0
$273$			10.6404i			0.643983i
$274$		0			0
$275$	4.03400	−	2.95413i	0.243259	−	0.178141i
$276$		0			0
$277$		−	3.39484i		−	0.203976i	−0.994786	−	0.101988i	$-0.967480\pi$
$277$		−	3.39484i		−	0.203976i	0.994786	−	0.101988i	$-0.0325204\pi$
$278$		0			0
$279$	−3.24496			−0.194271
$280$		0			0
$281$	−17.2362			−1.02823			−0.514113	−	0.857723i	$-0.671879\pi$
$281$	−17.2362			−1.02823			−0.514113	+	0.857723i	$0.671879\pi$
$282$		0			0
$283$			31.9797i			1.90100i	0.310732	+	0.950498i	$0.399426\pi$
$283$			31.9797i			1.90100i	−0.310732	+	0.950498i	$0.600574\pi$
$284$		0			0
$285$	−7.06194	−	21.5961i	−0.418313	−	1.27924i
$286$		0			0
$287$			11.5923i			0.684273i
$288$		0			0
$289$	16.6135			0.977266
$290$		0			0
$291$	22.5048			1.31925
$292$		0			0
$293$			10.1889i			0.595240i	0.954684	+	0.297620i	$0.0961928\pi$
$293$			10.1889i			0.595240i	−0.954684	+	0.297620i	$0.903807\pi$
$294$		0			0
$295$	−22.8875	+	7.48423i	−1.33256	+	0.435748i
$296$		0			0
$297$		−	3.08766i		−	0.179164i
$298$		0			0
$299$	19.3354			1.11819
$300$		0			0
$301$	12.2849			0.708088
$302$		0			0
$303$			2.72044i			0.156285i
$304$		0			0
$305$	19.9879	−	6.53607i	1.14450	−	0.374254i
$306$		0			0
$307$			20.3727i			1.16273i	0.813643	+	0.581365i	$0.197481\pi$
$307$			20.3727i			1.16273i	−0.813643	+	0.581365i	$0.802519\pi$
$308$		0			0
$309$	7.65523			0.435491
$310$		0			0
$311$	−4.63809			−0.263002			−0.131501	−	0.991316i	$-0.541980\pi$
$311$	−4.63809			−0.263002			−0.131501	+	0.991316i	$0.541980\pi$
$312$		0			0
$313$			25.9376i			1.46608i	0.680185	+	0.733040i	$0.261899\pi$
$313$			25.9376i			1.46608i	−0.680185	+	0.733040i	$0.738101\pi$
$314$		0			0
$315$	−1.29984	−	3.97504i	−0.0732378	−	0.223968i
$316$		0			0
$317$			6.99840i			0.393069i	0.980497	+	0.196535i	$0.0629688\pi$
$317$			6.99840i			0.393069i	−0.980497	+	0.196535i	$0.937031\pi$
$318$		0			0
$319$	−2.10643			−0.117937
$320$		0			0
$321$	10.6150			0.592472
$322$		0			0
$323$		−	2.96048i		−	0.164726i
$324$		0			0
$325$	12.2315	+	16.7027i	0.678482	+	0.926499i
$326$		0			0
$327$			14.1655i			0.783353i
$328$		0			0
$329$	8.31628			0.458491
$330$		0			0
$331$	−6.73753			−0.370328			−0.185164	−	0.982708i	$-0.559282\pi$
$331$	−6.73753			−0.370328			−0.185164	+	0.982708i	$0.559282\pi$
$332$		0			0
$333$		−	13.8349i		−	0.758148i
$334$		0			0
$335$	3.74252	+	11.4450i	0.204475	+	0.625305i
$336$		0			0
$337$			3.20655i			0.174672i	0.996179	+	0.0873361i	$0.0278354\pi$
$337$			3.20655i			0.174672i	−0.996179	+	0.0873361i	$0.972165\pi$
$338$		0			0
$339$	−8.88371			−0.482497
$340$		0			0
$341$	2.08954			0.113155
$342$		0			0
$343$			15.1142i			0.816090i
$344$		0			0
$345$	−21.1774	+	6.92504i	−1.14015	+	0.372831i
$346$		0			0
$347$			12.6613i			0.679696i	0.940480	+	0.339848i	$0.110376\pi$
$347$			12.6613i			0.679696i	−0.940480	+	0.339848i	$0.889624\pi$
$348$		0			0
$349$	0.596140			0.0319106			0.0159553	−	0.999873i	$-0.494921\pi$
$349$	0.596140			0.0319106			0.0159553	+	0.999873i	$0.494921\pi$
$350$		0			0
$351$	12.7844			0.682381
$352$		0			0
$353$			21.4134i			1.13972i	0.821742	+	0.569860i	$0.193003\pi$
$353$			21.4134i			1.13972i	−0.821742	+	0.569860i	$0.806997\pi$
$354$		0			0
$355$	−17.9142	+	5.85798i	−0.950789	+	0.310909i
$356$		0			0
$357$		−	1.59759i		−	0.0845533i
$358$		0			0
$359$	−17.2379			−0.909781			−0.454891	−	0.890547i	$-0.650322\pi$
$359$	−17.2379			−0.909781			−0.454891	+	0.890547i	$0.650322\pi$
$360$		0			0
$361$	3.67808			0.193583
$362$		0			0
$363$		−	2.13376i		−	0.111994i
$364$		0			0
$365$	−5.88142	−	17.9859i	−0.307848	−	0.941427i
$366$		0			0
$367$			31.2870i			1.63317i	0.577228	+	0.816583i	$0.304134\pi$
$367$			31.2870i			1.63317i	−0.577228	+	0.816583i	$0.695866\pi$
$368$		0			0
$369$	−14.9475			−0.778137
$370$		0			0
$371$	−4.00178			−0.207762
$372$		0			0
$373$			18.6441i			0.965355i	0.875798	+	0.482677i	$0.160336\pi$
$373$			18.6441i			0.965355i	−0.875798	+	0.482677i	$0.839664\pi$
$374$		0			0
$375$	−19.3789	−	13.9132i	−1.00072	−	0.718473i
$376$		0			0
$377$		−	8.72162i		−	0.449186i
$378$		0			0
$379$	−33.4682			−1.71915			−0.859574	−	0.511011i	$-0.829271\pi$
$379$	−33.4682			−1.71915			−0.859574	+	0.511011i	$0.829271\pi$
$380$		0			0
$381$	10.1540			0.520207
$382$		0			0
$383$		−	32.4451i		−	1.65787i	−0.559347	−	0.828934i	$-0.688948\pi$
$383$		−	32.4451i		−	1.65787i	0.559347	−	0.828934i	$-0.311052\pi$
$384$		0			0
$385$	0.837014	+	2.55967i	0.0426582	+	0.130453i
$386$		0			0
$387$			15.8405i			0.805220i
$388$		0			0
$389$	17.2080			0.872481			0.436241	−	0.899830i	$-0.356310\pi$
$389$	17.2080			0.872481			0.436241	+	0.899830i	$0.356310\pi$
$390$		0			0
$391$	−2.90309			−0.146816
$392$		0			0
$393$		−	23.2289i		−	1.17174i
$394$		0			0
$395$	13.7007	−	4.48015i	0.689358	−	0.225421i
$396$		0			0
$397$			24.8359i			1.24648i	0.782032	+	0.623238i	$0.214183\pi$
$397$			24.8359i			1.24648i	−0.782032	+	0.623238i	$0.785817\pi$
$398$		0			0
$399$	12.2379			0.612663
$400$		0			0
$401$	−11.6749			−0.583017			−0.291509	−	0.956568i	$-0.594157\pi$
$401$	−11.6749			−0.583017			−0.291509	+	0.956568i	$0.594157\pi$
$402$		0			0
$403$			8.65172i			0.430973i
$404$		0			0
$405$	−23.9039	+	7.81661i	−1.18780	+	0.388410i
$406$		0			0
$407$			8.90878i			0.441592i
$408$		0			0
$409$	32.7457			1.61917			0.809584	−	0.587003i	$-0.199692\pi$
$409$	32.7457			1.61917			0.809584	+	0.587003i	$0.199692\pi$
$410$		0			0
$411$	−21.5072			−1.06087
$412$		0			0
$413$		−	12.9697i		−	0.638199i
$414$		0			0
$415$	−0.469794	−	1.43668i	−0.0230613	−	0.0705236i
$416$		0			0
$417$			43.9260i			2.15107i
$418$		0			0
$419$	40.3114			1.96934			0.984670	−	0.174429i	$-0.0558079\pi$
$419$	40.3114			1.96934			0.984670	+	0.174429i	$0.0558079\pi$
$420$		0			0
$421$	−13.2601			−0.646256			−0.323128	−	0.946355i	$-0.604734\pi$
$421$	−13.2601			−0.646256			−0.323128	+	0.946355i	$0.604734\pi$
$422$		0			0
$423$			10.7233i			0.521385i
$424$		0			0
$425$	−1.83649	−	2.50781i	−0.0890828	−	0.121647i
$426$		0			0
$427$			11.3266i			0.548135i
$428$		0			0
$429$	8.83481			0.426549
$430$		0			0
$431$	−0.602928			−0.0290420			−0.0145210	−	0.999895i	$-0.504622\pi$
$431$	−0.602928			−0.0290420			−0.0145210	+	0.999895i	$0.504622\pi$
$432$		0			0
$433$		−	35.5966i		−	1.71066i	−0.518080	−	0.855332i	$-0.673353\pi$
$433$		−	35.5966i		−	1.71066i	0.518080	−	0.855332i	$-0.326647\pi$
$434$		0			0
$435$	3.12368	+	9.55252i	0.149769	+	0.458008i
$436$		0			0
$437$		−	22.2384i		−	1.06381i
$438$		0			0
$439$	−11.2346			−0.536198			−0.268099	−	0.963391i	$-0.586395\pi$
$439$	−11.2346			−0.536198			−0.268099	+	0.963391i	$0.586395\pi$
$440$		0			0
$441$	−8.61811			−0.410386
$442$		0			0
$443$		−	24.5417i		−	1.16601i	−0.812468	−	0.583006i	$-0.801877\pi$
$443$		−	24.5417i		−	1.16601i	0.812468	−	0.583006i	$-0.198123\pi$
$444$		0			0
$445$	−33.3443	+	10.9036i	−1.58067	+	0.516882i
$446$		0			0
$447$			14.2076i			0.671998i
$448$		0			0
$449$	11.7896			0.556384			0.278192	−	0.960526i	$-0.410265\pi$
$449$	11.7896			0.556384			0.278192	+	0.960526i	$0.410265\pi$
$450$		0			0
$451$	9.62524			0.453235
$452$		0			0
$453$		−	44.4898i		−	2.09031i
$454$		0			0
$455$	−10.5983	+	3.46564i	−0.496854	+	0.162472i
$456$		0			0
$457$		−	20.9828i		−	0.981534i	−0.871291	−	0.490767i	$-0.836717\pi$
$457$		−	20.9828i		−	0.981534i	0.871291	−	0.490767i	$-0.163283\pi$
$458$		0			0
$459$	−1.91950			−0.0895948
$460$		0			0
$461$	−29.8844			−1.39186			−0.695928	−	0.718111i	$-0.745007\pi$
$461$	−29.8844			−1.39186			−0.695928	+	0.718111i	$0.745007\pi$
$462$		0			0
$463$			24.4425i			1.13594i	0.823049	+	0.567970i	$0.192271\pi$
$463$			24.4425i			1.13594i	−0.823049	+	0.567970i	$0.807729\pi$
$464$		0			0
$465$	−3.09865	−	9.47596i	−0.143696	−	0.439437i
$466$		0			0
$467$		−	3.13953i		−	0.145280i	−0.997358	−	0.0726401i	$-0.976858\pi$
$467$		−	3.13953i		−	0.145280i	0.997358	−	0.0726401i	$-0.0231425\pi$
$468$		0			0
$469$	−6.48557			−0.299476
$470$		0			0
$471$	−33.7982			−1.55734
$472$		0			0
$473$		−	10.2003i		−	0.469009i
$474$		0			0
$475$	19.2105	−	14.0680i	0.881438	−	0.645484i
$476$		0			0
$477$		−	5.16004i		−	0.236262i
$478$		0			0
$479$	−12.5709			−0.574380			−0.287190	−	0.957874i	$-0.592721\pi$
$479$	−12.5709			−0.574380			−0.287190	+	0.957874i	$0.592721\pi$
$480$		0			0
$481$	−36.8866			−1.68189
$482$		0			0
$483$		−	12.0007i		−	0.546051i
$484$		0			0
$485$	7.32997	+	22.4157i	0.332837	+	1.01785i
$486$		0			0
$487$		−	13.2520i		−	0.600503i	−0.953860	−	0.300252i	$-0.902929\pi$
$487$		−	13.2520i		−	0.600503i	0.953860	−	0.300252i	$-0.0970707\pi$
$488$		0			0
$489$	−3.97729			−0.179859
$490$		0			0
$491$	11.3070			0.510278			0.255139	−	0.966904i	$-0.417879\pi$
$491$	11.3070			0.510278			0.255139	+	0.966904i	$0.417879\pi$
$492$		0			0
$493$			1.30950i			0.0589770i
$494$		0			0
$495$	−3.30052	+	1.07927i	−0.148348	+	0.0485098i
$496$		0			0
$497$		−	10.1515i		−	0.455359i
$498$		0			0
$499$	6.06431			0.271476			0.135738	−	0.990745i	$-0.456659\pi$
$499$	6.06431			0.271476			0.135738	+	0.990745i	$0.456659\pi$
$500$		0			0
$501$	31.3779			1.40186
$502$		0			0
$503$		−	39.7943i		−	1.77434i	−0.461444	−	0.887169i	$-0.652669\pi$
$503$		−	39.7943i		−	1.77434i	0.461444	−	0.887169i	$-0.347331\pi$
$504$		0			0
$505$	−2.70968	+	0.886068i	−0.120579	+	0.0394295i
$506$		0			0
$507$			8.84143i			0.392662i
$508$		0			0
$509$	−11.1058			−0.492258			−0.246129	−	0.969237i	$-0.579159\pi$
$509$	−11.1058			−0.492258			−0.246129	+	0.969237i	$0.579159\pi$
$510$		0			0
$511$	10.1922			0.450875
$512$		0			0
$513$		−	14.7039i		−	0.649193i
$514$		0			0
$515$	2.49336	+	7.62494i	0.109871	+	0.335995i
$516$		0			0
$517$		−	6.90511i		−	0.303686i
$518$		0			0
$519$	25.4401			1.11670
$520$		0			0
$521$	−8.57831			−0.375822			−0.187911	−	0.982186i	$-0.560172\pi$
$521$	−8.57831			−0.375822			−0.187911	+	0.982186i	$0.560172\pi$
$522$		0			0
$523$			25.6845i			1.12311i	0.827441	+	0.561553i	$0.189796\pi$
$523$			25.6845i			1.12311i	−0.827441	+	0.561553i	$0.810204\pi$
$524$		0			0
$525$	10.3667	−	7.59162i	0.452440	−	0.331325i
$526$		0			0
$527$		−	1.29901i		−	0.0565856i
$528$		0			0
$529$	1.19266			0.0518546
$530$		0			0
$531$	16.7236			0.725744
$532$		0			0
$533$			39.8531i			1.72623i
$534$		0			0
$535$	3.45739	+	10.5730i	0.149476	+	0.457111i
$536$		0			0
$537$			23.4706i			1.01283i
$538$		0			0
$539$	5.54950			0.239034
$540$		0			0
$541$	−5.62955			−0.242033			−0.121017	−	0.992650i	$-0.538615\pi$
$541$	−5.62955			−0.242033			−0.121017	+	0.992650i	$0.538615\pi$
$542$		0			0
$543$			0.212617i			0.00912427i
$544$		0			0
$545$	−14.1094	+	4.61380i	−0.604382	+	0.197634i
$546$		0			0
$547$			44.0338i			1.88275i	0.337365	+	0.941374i	$0.390464\pi$
$547$			44.0338i			1.88275i	−0.337365	+	0.941374i	$0.609536\pi$
$548$		0			0
$549$	−14.6050			−0.623325
$550$		0			0
$551$	−10.0311			−0.427340
$552$		0			0
$553$			7.76384i			0.330152i
$554$		0			0
$555$	40.4008	−	13.2111i	1.71492	−	0.560780i
$556$		0			0
$557$			40.5943i			1.72003i	0.510265	+	0.860017i	$0.329547\pi$
$557$			40.5943i			1.72003i	−0.510265	+	0.860017i	$0.670453\pi$
$558$		0			0
$559$	42.2341			1.78631
$560$		0			0
$561$	−1.32650			−0.0560047
$562$		0			0
$563$			6.63333i			0.279562i	0.990182	+	0.139781i	$0.0446398\pi$
$563$			6.63333i			0.279562i	−0.990182	+	0.139781i	$0.955360\pi$
$564$		0			0
$565$	−2.89349	−	8.84857i	−0.121730	−	0.372262i
$566$		0			0
$567$		−	13.5457i		−	0.568868i
$568$		0			0
$569$	18.3346			0.768625			0.384313	−	0.923203i	$-0.374438\pi$
$569$	18.3346			0.768625			0.384313	+	0.923203i	$0.374438\pi$
$570$		0			0
$571$	9.13047			0.382098			0.191049	−	0.981580i	$-0.438811\pi$
$571$	9.13047			0.382098			0.191049	+	0.981580i	$0.438811\pi$
$572$		0			0
$573$			43.7932i			1.82949i
$574$		0			0
$575$	−13.7953	−	18.8381i	−0.575303	−	0.785603i
$576$		0			0
$577$		−	9.93656i		−	0.413665i	−0.978376	−	0.206832i	$-0.933685\pi$
$577$		−	9.93656i		−	0.413665i	0.978376	−	0.206832i	$-0.0663154\pi$
$578$		0			0
$579$	−35.9995			−1.49609
$580$		0			0
$581$	0.814127			0.0337757
$582$		0			0
$583$			3.32273i			0.137613i
$584$		0			0
$585$	−4.46872	−	13.6658i	−0.184759	−	0.565010i
$586$		0			0
$587$			1.66282i			0.0686318i	0.999411	+	0.0343159i	$0.0109252\pi$
$587$			1.66282i			0.0686318i	−0.999411	+	0.0343159i	$0.989075\pi$
$588$		0			0
$589$	9.95073			0.410013
$590$		0			0
$591$	2.90693			0.119575
$592$		0			0
$593$		−	22.3866i		−	0.919307i	−0.888098	−	0.459653i	$-0.847974\pi$
$593$		−	22.3866i		−	0.919307i	0.888098	−	0.459653i	$-0.152026\pi$
$594$		0			0
$595$	1.59127	−	0.520346i	0.0652356	−	0.0213321i
$596$		0			0
$597$		−	42.9069i		−	1.75606i
$598$		0			0
$599$	−36.9636			−1.51029			−0.755147	−	0.655556i	$-0.772434\pi$
$599$	−36.9636			−1.51029			−0.755147	+	0.655556i	$0.772434\pi$
$600$		0			0
$601$	−1.08085			−0.0440887			−0.0220443	−	0.999757i	$-0.507017\pi$
$601$	−1.08085			−0.0440887			−0.0220443	+	0.999757i	$0.507017\pi$
$602$		0			0
$603$		−	8.36272i		−	0.340556i
$604$		0			0
$605$	2.12532	−	0.694983i	0.0864067	−	0.0282551i
$606$		0			0
$607$			15.9873i			0.648903i	0.945902	+	0.324451i	$0.105180\pi$
$607$			15.9873i			0.648903i	−0.945902	+	0.324451i	$0.894820\pi$
$608$		0			0
$609$	−5.41317			−0.219353
$610$		0			0
$611$	28.5905			1.15665
$612$		0			0
$613$			13.6367i			0.550781i	0.961332	+	0.275390i	$0.0888071\pi$
$613$			13.6367i			0.550781i	−0.961332	+	0.275390i	$0.911193\pi$
$614$		0			0
$615$	−14.2736	−	43.6499i	−0.575565	−	1.76013i
$616$		0			0
$617$		−	40.9104i		−	1.64699i	−0.567323	−	0.823496i	$-0.692021\pi$
$617$		−	40.9104i		−	1.64699i	0.567323	−	0.823496i	$-0.307979\pi$
$618$		0			0
$619$	−42.1598			−1.69454			−0.847272	−	0.531159i	$-0.821757\pi$
$619$	−42.1598			−1.69454			−0.847272	+	0.531159i	$0.821757\pi$
$620$		0			0
$621$	−14.4189			−0.578609
$622$		0			0
$623$		−	18.8954i		−	0.757027i
$624$		0			0
$625$	7.54627	−	23.8339i	0.301851	−	0.953355i
$626$		0			0
$627$		−	10.1613i		−	0.405804i
$628$		0			0
$629$	5.53831			0.220827
$630$		0			0
$631$	−29.6308			−1.17958			−0.589792	−	0.807555i	$-0.700790\pi$
$631$	−29.6308			−1.17958			−0.589792	+	0.807555i	$0.700790\pi$
$632$		0			0
$633$			14.4503i			0.574347i
$634$		0			0
$635$	3.30724	+	10.1139i	0.131244	+	0.401356i
$636$		0			0
$637$			22.9776i			0.910406i
$638$		0			0
$639$	13.0898			0.517823
$640$		0			0
$641$	36.4240			1.43866			0.719330	−	0.694668i	$-0.244449\pi$
$641$	36.4240			1.43866			0.719330	+	0.694668i	$0.244449\pi$
$642$		0			0
$643$		−	26.9899i		−	1.06438i	−0.846626	−	0.532188i	$-0.821370\pi$
$643$		−	26.9899i		−	1.06438i	0.846626	−	0.532188i	$-0.178630\pi$
$644$		0			0
$645$	−46.2577	+	15.1263i	−1.82139	+	0.595598i
$646$		0			0
$647$			30.6852i			1.20636i	0.797605	+	0.603181i	$0.206100\pi$
$647$			30.6852i			1.20636i	−0.797605	+	0.603181i	$0.793900\pi$
$648$		0			0
$649$	−10.7689			−0.422718
$650$		0			0
$651$	5.36978			0.210458
$652$		0			0
$653$			21.9274i			0.858085i	0.903284	+	0.429042i	$0.141149\pi$
$653$			21.9274i			0.858085i	−0.903284	+	0.429042i	$0.858851\pi$
$654$		0			0
$655$	23.1370	−	7.56583i	0.904039	−	0.295622i
$656$		0			0
$657$			13.1421i			0.512724i
$658$		0			0
$659$	−19.1287			−0.745148			−0.372574	−	0.928003i	$-0.621525\pi$
$659$	−19.1287			−0.745148			−0.372574	+	0.928003i	$0.621525\pi$
$660$		0			0
$661$	−50.9975			−1.98357			−0.991786	−	0.127912i	$-0.959172\pi$
$661$	−50.9975			−1.98357			−0.991786	+	0.127912i	$0.959172\pi$
$662$		0			0
$663$		−	5.49233i		−	0.213304i
$664$		0			0
$665$	3.98599	+	12.1895i	0.154570	+	0.472689i
$666$		0			0
$667$			9.83667i			0.380877i
$668$		0			0
$669$	4.50778			0.174281
$670$		0			0
$671$	9.40465			0.363062
$672$		0			0
$673$		−	23.3223i		−	0.899010i	−0.893278	−	0.449505i	$-0.851600\pi$
$673$		−	23.3223i		−	0.899010i	0.893278	−	0.449505i	$-0.148400\pi$
$674$		0			0
$675$	−9.12134	−	12.4556i	−0.351081	−	0.479417i
$676$		0			0
$677$		−	6.30482i		−	0.242314i	−0.992633	−	0.121157i	$-0.961340\pi$
$677$		−	6.30482i		−	0.242314i	0.992633	−	0.121157i	$-0.0386605\pi$
$678$		0			0
$679$	−12.7024			−0.487475
$680$		0			0
$681$	16.3191			0.625348
$682$		0			0
$683$		−	36.8716i		−	1.41085i	−0.708783	−	0.705426i	$-0.750756\pi$
$683$		−	36.8716i		−	1.41085i	0.708783	−	0.705426i	$-0.249244\pi$
$684$		0			0
$685$	−7.00505	−	21.4221i	−0.267649	−	0.818496i
$686$		0			0
$687$			26.1324i			0.997013i
$688$		0			0
$689$	−13.7577			−0.524126
$690$		0			0
$691$	35.5795			1.35351			0.676754	−	0.736209i	$-0.263386\pi$
$691$	35.5795			1.35351			0.676754	+	0.736209i	$0.263386\pi$
$692$		0			0
$693$		−	1.87032i		−	0.0710477i
$694$		0			0
$695$	−43.7523	+	14.3070i	−1.65962	+	0.542697i
$696$		0			0
$697$		−	5.98372i		−	0.226649i
$698$		0			0
$699$	−34.0340			−1.28729
$700$		0			0
$701$	12.1069			0.457271			0.228636	−	0.973512i	$-0.426574\pi$
$701$	12.1069			0.457271			0.228636	+	0.973512i	$0.426574\pi$
$702$		0			0
$703$			42.4249i			1.60009i
$704$		0			0
$705$	−31.3142	+	10.2398i	−1.17936	+	0.385653i
$706$		0			0
$707$		−	1.53551i		−	0.0577486i
$708$		0			0
$709$	31.1844			1.17115			0.585577	−	0.810617i	$-0.300868\pi$
$709$	31.1844			1.17115			0.585577	+	0.810617i	$0.300868\pi$
$710$		0			0
$711$	−10.0110			−0.375441
$712$		0			0
$713$		−	9.75783i		−	0.365433i
$714$		0			0
$715$	2.87756	+	8.79986i	0.107615	+	0.329096i
$716$		0			0
$717$		−	60.8585i		−	2.27280i
$718$		0			0
$719$	44.1954			1.64821			0.824106	−	0.566436i	$-0.191678\pi$
$719$	44.1954			1.64821			0.824106	+	0.566436i	$0.191678\pi$
$720$		0			0
$721$	−4.32086			−0.160917
$722$		0			0
$723$		−	9.63292i		−	0.358252i
$724$		0			0
$725$	−8.49732	+	6.22265i	−0.315583	+	0.231104i
$726$		0			0
$727$		−	0.419917i		−	0.0155739i	−0.999970	−	0.00778694i	$-0.997521\pi$
$727$		−	0.419917i		−	0.0155739i	0.999970	−	0.00778694i	$-0.00247868\pi$
$728$		0			0
$729$	−2.29867			−0.0851359
$730$		0			0
$731$	−6.34120			−0.234538
$732$		0			0
$733$		−	15.6107i		−	0.576595i	−0.957541	−	0.288297i	$-0.906911\pi$
$733$		−	15.6107i		−	0.576595i	0.957541	−	0.288297i	$-0.0930892\pi$
$734$		0			0
$735$	−8.22952	−	25.1667i	−0.303550	−	0.928286i
$736$		0			0
$737$			5.38505i			0.198361i
$738$		0			0
$739$	12.5531			0.461773			0.230887	−	0.972981i	$-0.425837\pi$
$739$	12.5531			0.461773			0.230887	+	0.972981i	$0.425837\pi$
$740$		0			0
$741$	42.0727			1.54558
$742$		0			0
$743$		−	41.3322i		−	1.51633i	−0.652061	−	0.758166i	$-0.726096\pi$
$743$		−	41.3322i		−	1.51633i	0.652061	−	0.758166i	$-0.273904\pi$
$744$		0			0
$745$	−14.1514	+	4.62753i	−0.518468	+	0.169540i
$746$		0			0
$747$			1.04976i			0.0384088i
$748$		0			0
$749$	−5.99146			−0.218923
$750$		0			0
$751$	−13.6104			−0.496652			−0.248326	−	0.968677i	$-0.579880\pi$
$751$	−13.6104			−0.496652			−0.248326	+	0.968677i	$0.579880\pi$
$752$		0			0
$753$			30.2233i			1.10140i
$754$		0			0
$755$	44.3138	−	14.4906i	1.61274	−	0.527369i
$756$		0			0
$757$			15.8830i			0.577279i	0.957438	+	0.288640i	$0.0932029\pi$
$757$			15.8830i			0.577279i	−0.957438	+	0.288640i	$0.906797\pi$
$758$		0			0
$759$	−9.96433			−0.361682
$760$		0			0
$761$	−0.731594			−0.0265203			−0.0132601	−	0.999912i	$-0.504221\pi$
$761$	−0.731594			−0.0265203			−0.0132601	+	0.999912i	$0.504221\pi$
$762$		0			0
$763$		−	7.99546i		−	0.289455i
$764$		0			0
$765$	0.670952	+	2.05183i	0.0242583	+	0.0741842i
$766$		0			0
$767$		−	44.5886i		−	1.61000i
$768$		0			0
$769$	28.1471			1.01501			0.507505	−	0.861649i	$-0.330568\pi$
$769$	28.1471			1.01501			0.507505	+	0.861649i	$0.330568\pi$
$770$		0			0
$771$	6.93488			0.249753
$772$		0			0
$773$		−	27.3629i		−	0.984176i	−0.870545	−	0.492088i	$-0.836234\pi$
$773$		−	27.3629i		−	0.984176i	0.870545	−	0.492088i	$-0.163766\pi$
$774$		0			0
$775$	8.42922	−	6.17278i	0.302786	−	0.221733i
$776$		0			0
$777$			22.8941i			0.821321i
$778$		0			0
$779$	45.8368			1.64228
$780$		0			0
$781$	−8.42895			−0.301612
$782$		0			0
$783$			6.50393i			0.232432i
$784$		0			0
$785$	−11.0083	−	33.6645i	−0.392904	−	1.20154i
$786$		0			0
$787$			15.3954i			0.548788i	0.961617	+	0.274394i	$0.0884773\pi$
$787$			15.3954i			0.548788i	−0.961617	+	0.274394i	$0.911523\pi$
$788$		0			0
$789$	11.8813			0.422985
$790$		0			0
$791$	5.01426			0.178286
$792$		0			0
$793$			38.9398i			1.38279i
$794$		0			0
$795$	15.0684	−	4.92737i	0.534420	−	0.174756i
$796$		0			0
$797$			48.3762i			1.71357i	0.515673	+	0.856786i	$0.327542\pi$
$797$			48.3762i			1.71357i	−0.515673	+	0.856786i	$0.672458\pi$
$798$		0			0
$799$	−4.29269			−0.151865
$800$		0			0
$801$	24.3644			0.860872
$802$		0			0
$803$		−	8.46268i		−	0.298642i
$804$		0			0
$805$	11.9532	−	3.90872i	0.421296	−	0.137764i
$806$		0			0
$807$		−	30.3963i		−	1.07000i
$808$		0			0
$809$	34.1856			1.20190			0.600950	−	0.799287i	$-0.294789\pi$
$809$	34.1856			1.20190			0.600950	+	0.799287i	$0.294789\pi$
$810$		0			0
$811$	−28.5790			−1.00354			−0.501771	−	0.865000i	$-0.667318\pi$
$811$	−28.5790			−1.00354			−0.501771	+	0.865000i	$0.667318\pi$
$812$		0			0
$813$			35.3920i			1.24125i
$814$		0			0
$815$	−1.29543	−	3.96155i	−0.0453770	−	0.138767i
$816$		0			0
$817$		−	48.5753i		−	1.69943i
$818$		0			0
$819$	7.74404			0.270599
$820$		0			0
$821$	−48.9846			−1.70957			−0.854787	−	0.518978i	$-0.826313\pi$
$821$	−48.9846			−1.70957			−0.854787	+	0.518978i	$0.826313\pi$
$822$		0			0
$823$			51.9200i			1.80982i	0.425605	+	0.904909i	$0.360061\pi$
$823$			51.9200i			1.80982i	−0.425605	+	0.904909i	$0.639939\pi$
$824$		0			0
$825$	−6.30341	−	8.60760i	−0.219457	−	0.299678i
$826$		0			0
$827$		−	55.7187i		−	1.93753i	−0.247980	−	0.968765i	$-0.579767\pi$
$827$		−	55.7187i		−	1.93753i	0.247980	−	0.968765i	$-0.420233\pi$
$828$		0			0
$829$	55.7521			1.93635			0.968175	−	0.250274i	$-0.0805206\pi$
$829$	55.7521			1.93635			0.968175	+	0.250274i	$0.0805206\pi$
$830$		0			0
$831$	−7.24379			−0.251284
$832$		0			0
$833$		−	3.44995i		−	0.119534i
$834$		0			0
$835$	10.2200	+	31.2537i	0.353678	+	1.08158i
$836$		0			0
$837$		−	6.45180i		−	0.223007i
$838$		0			0
$839$	−19.2438			−0.664369			−0.332184	−	0.943214i	$-0.607786\pi$
$839$	−19.2438			−0.664369			−0.332184	+	0.943214i	$0.607786\pi$
$840$		0			0
$841$	−24.5630			−0.846999
$842$		0			0
$843$			36.7780i			1.26670i
$844$		0			0
$845$	−8.80645	+	2.87972i	−0.302951	+	0.0990654i
$846$		0			0
$847$			1.20437i			0.0413825i
$848$		0			0
$849$	68.2371			2.34189
$850$		0			0
$851$	41.6025			1.42612
$852$		0			0
$853$		−	33.8108i		−	1.15766i	−0.815448	−	0.578830i	$-0.803509\pi$
$853$		−	33.8108i		−	1.15766i	0.815448	−	0.578830i	$-0.196491\pi$
$854$		0			0
$855$	−15.7176	+	5.13967i	−0.537530	+	0.175773i
$856$		0			0
$857$			58.4040i			1.99504i	0.0703617	+	0.997522i	$0.477585\pi$
$857$			58.4040i			1.99504i	−0.0703617	+	0.997522i	$0.522415\pi$
$858$		0			0
$859$	−23.8572			−0.813998			−0.406999	−	0.913429i	$-0.633425\pi$
$859$	−23.8572			−0.813998			−0.406999	+	0.913429i	$0.633425\pi$
$860$		0			0
$861$	24.7353			0.842976
$862$		0			0
$863$		−	4.37835i		−	0.149041i	−0.997219	−	0.0745204i	$-0.976257\pi$
$863$		−	4.37835i		−	0.149041i	0.997219	−	0.0745204i	$-0.0237426\pi$
$864$		0			0
$865$	8.28604	+	25.3395i	0.281734	+	0.861568i
$866$		0			0
$867$		−	35.4494i		−	1.20392i
$868$		0			0
$869$	6.44641			0.218680
$870$		0			0
$871$	−22.2967			−0.755494
$872$		0			0
$873$		−	16.3790i		−	0.554344i
$874$		0			0
$875$	10.9381	+	7.85305i	0.369775	+	0.265482i
$876$		0			0
$877$			46.8329i			1.58143i	0.612182	+	0.790717i	$0.290292\pi$
$877$			46.8329i			1.58143i	−0.612182	+	0.790717i	$0.709708\pi$
$878$		0			0
$879$	21.7406			0.733293
$880$		0			0
$881$	2.72179			0.0916993			0.0458496	−	0.998948i	$-0.485400\pi$
$881$	2.72179			0.0916993			0.0458496	+	0.998948i	$0.485400\pi$
$882$		0			0
$883$			22.8773i			0.769881i	0.922941	+	0.384941i	$0.125778\pi$
$883$			22.8773i			0.769881i	−0.922941	+	0.384941i	$0.874222\pi$
$884$		0			0
$885$	15.9696	+	48.8365i	0.536811	+	1.64162i
$886$		0			0
$887$		−	37.0004i		−	1.24235i	−0.783672	−	0.621175i	$-0.786656\pi$
$887$		−	37.0004i		−	1.24235i	0.783672	−	0.621175i	$-0.213344\pi$
$888$		0			0
$889$	−5.73127			−0.192221
$890$		0			0
$891$	−11.2472			−0.376795
$892$		0			0
$893$		−	32.8832i		−	1.10039i
$894$		0			0
$895$	−23.3778	+	7.64455i	−0.781432	+	0.255529i
$896$		0			0
$897$		−	41.2571i		−	1.37753i
$898$		0			0
$899$	−4.40147			−0.146797
$900$		0			0
$901$	2.06564			0.0688164
$902$		0			0
$903$		−	26.2130i		−	0.872315i
$904$		0			0
$905$	−0.211776	+	0.0692510i	−0.00703967	+	0.00230198i
$906$		0			0
$907$		−	6.01498i		−	0.199724i	−0.995001	−	0.0998621i	$-0.968160\pi$
$907$		−	6.01498i		−	0.199724i	0.995001	−	0.0998621i	$-0.0318402\pi$
$908$		0			0
$909$	1.97993			0.0656703
$910$		0			0
$911$	−9.98170			−0.330708			−0.165354	−	0.986234i	$-0.552877\pi$
$911$	−9.98170			−0.330708			−0.165354	+	0.986234i	$0.552877\pi$
$912$		0			0
$913$		−	0.675980i		−	0.0223717i
$914$		0			0
$915$	−13.9464	−	42.6495i	−0.461055	−	1.40995i
$916$		0			0
$917$			13.1112i			0.432969i
$918$		0			0
$919$	36.3004			1.19744			0.598719	−	0.800959i	$-0.295677\pi$
$919$	36.3004			1.19744			0.598719	+	0.800959i	$0.295677\pi$
$920$		0			0
$921$	43.4705			1.43240
$922$		0			0
$923$		−	34.8999i		−	1.14874i
$924$		0			0
$925$	26.3177	+	35.9380i	0.865319	+	1.18163i
$926$		0			0
$927$		−	5.57147i		−	0.182991i
$928$		0			0
$929$	−25.3480			−0.831640			−0.415820	−	0.909447i	$-0.636505\pi$
$929$	−25.3480			−0.831640			−0.415820	+	0.909447i	$0.636505\pi$
$930$		0			0
$931$	26.4276			0.866128
$932$		0			0
$933$			9.89659i			0.324000i
$934$		0			0
$935$	−0.432050	−	1.32125i	−0.0141295	−	0.0432094i
$936$		0			0
$937$		−	36.8662i		−	1.20437i	−0.798358	−	0.602183i	$-0.794298\pi$
$937$		−	36.8662i		−	1.20437i	0.798358	−	0.602183i	$-0.205702\pi$
$938$		0			0
$939$	55.3448			1.80611
$940$		0			0
$941$	10.2385			0.333764			0.166882	−	0.985977i	$-0.446630\pi$
$941$	10.2385			0.333764			0.166882	+	0.985977i	$0.446630\pi$
$942$		0			0
$943$		−	44.9483i		−	1.46372i
$944$		0			0
$945$	7.90339	−	2.58442i	0.257097	−	0.0840710i
$946$		0			0
$947$			16.4386i			0.534183i	0.963671	+	0.267092i	$0.0860626\pi$
$947$			16.4386i			0.534183i	−0.963671	+	0.267092i	$0.913937\pi$
$948$		0			0
$949$	35.0396			1.13743
$950$		0			0
$951$	14.9329			0.484234
$952$		0			0
$953$			57.7188i			1.86969i	0.355050	+	0.934847i	$0.384464\pi$
$953$			57.7188i			1.86969i	−0.355050	+	0.934847i	$0.615536\pi$
$954$		0			0
$955$	−43.6200	+	14.2638i	−1.41151	+	0.461565i
$956$		0			0
$957$			4.49462i			0.145290i
$958$		0			0
$959$	12.1394			0.392000
$960$		0			0
$961$	−26.6338			−0.859155
$962$		0			0
$963$		−	7.72559i		−	0.248954i
$964$		0			0
$965$	−11.7253	−	35.8571i	−0.377451	−	1.15428i
$966$		0			0
$967$		−	31.9836i		−	1.02852i	−0.857633	−	0.514262i	$-0.828066\pi$
$967$		−	31.9836i		−	1.02852i	0.857633	−	0.514262i	$-0.171934\pi$
$968$		0			0
$969$	−6.31697			−0.202930
$970$		0			0
$971$	47.6080			1.52781			0.763906	−	0.645327i	$-0.223279\pi$
$971$	47.6080			1.52781			0.763906	+	0.645327i	$0.223279\pi$
$972$		0			0
$973$		−	24.7933i		−	0.794837i
$974$		0			0
$975$	35.6396	−	26.0992i	1.14138	−	0.835842i
$976$		0			0
$977$			5.75975i			0.184271i	0.995746	+	0.0921353i	$0.0293692\pi$
$977$			5.75975i			0.184271i	−0.995746	+	0.0921353i	$0.970631\pi$
$978$		0			0
$979$	−15.6891			−0.501425
$980$		0			0
$981$	10.3096			0.329161
$982$		0			0
$983$		−	31.8921i		−	1.01720i	−0.861003	−	0.508601i	$-0.830163\pi$
$983$		−	31.8921i		−	1.01720i	0.861003	−	0.508601i	$-0.169837\pi$
$984$		0			0
$985$	0.946807	+	2.89543i	0.0301678	+	0.0922559i
$986$		0			0
$987$		−	17.7450i		−	0.564829i
$988$		0			0
$989$	−47.6336			−1.51466
$990$		0			0
$991$	−43.6276			−1.38588			−0.692938	−	0.720997i	$-0.743684\pi$
$991$	−43.6276			−1.38588			−0.692938	+	0.720997i	$0.743684\pi$
$992$		0			0
$993$			14.3763i			0.456218i
$994$		0			0
$995$	42.7372	−	13.9751i	1.35486	−	0.443040i
$996$		0			0
$997$		−	39.0472i		−	1.23664i	−0.785927	−	0.618319i	$-0.787814\pi$
$997$		−	39.0472i		−	1.23664i	0.785927	−	0.618319i	$-0.212186\pi$
$998$		0			0
$999$	27.5073			0.870292

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1760.2.b.f.1409.3	yes	14
4.3	odd	2		1760.2.b.e.1409.12	yes	14
5.2	odd	4		8800.2.a.cd.1.2		7
5.3	odd	4		8800.2.a.cj.1.6		7
5.4	even	2	inner	1760.2.b.f.1409.12	yes	14
20.3	even	4		8800.2.a.cc.1.2		7
20.7	even	4		8800.2.a.ci.1.6		7
20.19	odd	2		1760.2.b.e.1409.3	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1760.2.b.e.1409.3	✓	14	20.19	odd	2
1760.2.b.e.1409.12	yes	14	4.3	odd	2
1760.2.b.f.1409.3	yes	14	1.1	even	1	trivial
1760.2.b.f.1409.12	yes	14	5.4	even	2	inner
8800.2.a.cc.1.2		7	20.3	even	4
8800.2.a.cd.1.2		7	5.2	odd	4
8800.2.a.ci.1.6		7	20.7	even	4
8800.2.a.cj.1.6		7	5.3	odd	4

Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1760.2.b.f.1409.3

Level	$1760$
Weight	$2$
Character	1760.1409
Analytic conductor	$14.054$
Analytic rank	$0$
Dimension	$14$
Inner twists	$2$