LMFDB - Embedded newform 2016.2.s.q.289.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2016,2,Mod(289,2016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2016, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("2016.289");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$2016 = 2^{5} \cdot 3^{2} \cdot 7$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	2016.s (of order $3$ , degree $2$ , minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$16.0978410475$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} + 2x^{2} + 4$ x^4 + 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 224)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			289.2
Root			$0.707107 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	2016.289
Dual form			2016.2.s.q.865.2

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.91421 - 3.31552i) q^{5} +(-1.00000 + 2.44949i) q^{7} +(0.207107 + 0.358719i) q^{11} -2.82843 q^{13} +(-2.91421 - 5.04757i) q^{17} +(-1.79289 + 3.10538i) q^{19} +(1.62132 - 2.80821i) q^{23} +(-4.82843 - 8.36308i) q^{25} -2.82843 q^{29} +(-4.20711 - 7.28692i) q^{31} +(6.20711 + 8.00436i) q^{35} +(1.32843 - 2.30090i) q^{37} +1.17157 q^{41} -1.65685 q^{43} +(3.79289 - 6.56948i) q^{47} +(-5.00000 - 4.89898i) q^{49} +(-0.500000 - 0.866025i) q^{53} +1.58579 q^{55} +(-4.44975 - 7.70719i) q^{59} +(1.32843 - 2.30090i) q^{61} +(-5.41421 + 9.37769i) q^{65} +(5.62132 + 9.73641i) q^{67} -2.34315 q^{71} +(1.67157 + 2.89525i) q^{73} +(-1.08579 + 0.148586i) q^{77} +(4.03553 - 6.98975i) q^{79} -15.3137 q^{83} -22.3137 q^{85} +(-4.50000 + 7.79423i) q^{89} +(2.82843 - 6.92820i) q^{91} +(6.86396 + 11.8887i) q^{95} -6.82843 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q + 2 q^{5} - 4 q^{7} - 2 q^{11} - 6 q^{17} - 10 q^{19} - 2 q^{23} - 8 q^{25} - 14 q^{31} + 22 q^{35} - 6 q^{37} + 16 q^{41} + 16 q^{43} + 18 q^{47} - 20 q^{49} - 2 q^{53} + 12 q^{55} + 2 q^{59} - 6 q^{61}+ \cdots - 16 q^{97}+O(q^{100})$ 4 * q + 2 * q^5 - 4 * q^7 - 2 * q^11 - 6 * q^17 - 10 * q^19 - 2 * q^23 - 8 * q^25 - 14 * q^31 + 22 * q^35 - 6 * q^37 + 16 * q^41 + 16 * q^43 + 18 * q^47 - 20 * q^49 - 2 * q^53 + 12 * q^55 + 2 * q^59 - 6 * q^61 - 16 * q^65 + 14 * q^67 - 32 * q^71 + 18 * q^73 - 10 * q^77 + 2 * q^79 - 16 * q^83 - 44 * q^85 - 18 * q^89 + 2 * q^95 - 16 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2016\mathbb{Z}\right)^\times$ .

$n$	$127$	$577$	$1765$	$1793$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$	$1$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	1.91421	−	3.31552i	0.856062	−	1.48274i	−0.0195936	−	0.999808i	$-0.506237\pi$
$5$	1.91421	−	3.31552i	0.856062	−	1.48274i	0.875656	−	0.482935i	$-0.160429\pi$
$6$		0			0
$7$	−1.00000	+	2.44949i	−0.377964	+	0.925820i
$7$	−1.00000	+	2.44949i	−0.377964	+	0.925820i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	0.207107	+	0.358719i	0.0624450	+	0.108158i	0.895558	−	0.444945i	$-0.146777\pi$
$11$	0.207107	+	0.358719i	0.0624450	+	0.108158i	−0.833113	+	0.553103i	$0.813444\pi$
$12$		0			0
$13$	−2.82843			−0.784465			−0.392232	−	0.919866i	$-0.628297\pi$
$13$	−2.82843			−0.784465			−0.392232	+	0.919866i	$0.628297\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.91421	−	5.04757i	−0.706801	−	1.22421i	−0.966038	−	0.258401i	$-0.916804\pi$
$17$	−2.91421	−	5.04757i	−0.706801	−	1.22421i	0.259237	−	0.965814i	$-0.416529\pi$
$18$		0			0
$19$	−1.79289	+	3.10538i	−0.411318	+	0.712424i	−0.995034	−	0.0995342i	$-0.968265\pi$
$19$	−1.79289	+	3.10538i	−0.411318	+	0.712424i	0.583716	+	0.811958i	$0.301598\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	1.62132	−	2.80821i	0.338069	−	0.585552i	−0.646001	−	0.763337i	$-0.723560\pi$
$23$	1.62132	−	2.80821i	0.338069	−	0.585552i	0.984069	+	0.177785i	$0.0568931\pi$
$24$		0			0
$25$	−4.82843	−	8.36308i	−0.965685	−	1.67262i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−2.82843			−0.525226			−0.262613	−	0.964901i	$-0.584584\pi$
$29$	−2.82843			−0.525226			−0.262613	+	0.964901i	$0.584584\pi$
$30$		0			0
$31$	−4.20711	−	7.28692i	−0.755619	−	1.30877i	−0.945066	−	0.326879i	$-0.894003\pi$
$31$	−4.20711	−	7.28692i	−0.755619	−	1.30877i	0.189447	−	0.981891i	$-0.439330\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	6.20711	+	8.00436i	1.04919	+	1.35298i
$36$		0			0
$37$	1.32843	−	2.30090i	0.218392	−	0.378266i	−0.735924	−	0.677064i	$-0.763252\pi$
$37$	1.32843	−	2.30090i	0.218392	−	0.378266i	0.954317	+	0.298797i	$0.0965855\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	1.17157			0.182969			0.0914845	−	0.995807i	$-0.470839\pi$
$41$	1.17157			0.182969			0.0914845	+	0.995807i	$0.470839\pi$
$42$		0			0
$43$	−1.65685			−0.252668			−0.126334	−	0.991988i	$-0.540321\pi$
$43$	−1.65685			−0.252668			−0.126334	+	0.991988i	$0.540321\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.79289	−	6.56948i	0.553250	−	0.958258i	−0.444787	−	0.895636i	$-0.646721\pi$
$47$	3.79289	−	6.56948i	0.553250	−	0.958258i	0.998037	−	0.0626213i	$-0.0199460\pi$
$48$		0			0
$49$	−5.00000	−	4.89898i	−0.714286	−	0.699854i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−0.500000	−	0.866025i	−0.0686803	−	0.118958i	0.829640	−	0.558298i	$-0.188546\pi$
$53$	−0.500000	−	0.866025i	−0.0686803	−	0.118958i	−0.898321	+	0.439340i	$0.855212\pi$
$54$		0			0
$55$	1.58579			0.213827
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−4.44975	−	7.70719i	−0.579308	−	1.00339i	−0.995559	−	0.0941408i	$-0.969990\pi$
$59$	−4.44975	−	7.70719i	−0.579308	−	1.00339i	0.416251	−	0.909250i	$-0.363344\pi$
$60$		0			0
$61$	1.32843	−	2.30090i	0.170088	−	0.294600i	−0.768363	−	0.640015i	$-0.778928\pi$
$61$	1.32843	−	2.30090i	0.170088	−	0.294600i	0.938450	+	0.345414i	$0.112262\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−5.41421	+	9.37769i	−0.671551	+	1.16316i
$66$		0			0
$67$	5.62132	+	9.73641i	0.686754	+	1.18949i	0.972882	+	0.231301i	$0.0742982\pi$
$67$	5.62132	+	9.73641i	0.686754	+	1.18949i	−0.286129	+	0.958191i	$0.592368\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−2.34315			−0.278080			−0.139040	−	0.990287i	$-0.544402\pi$
$71$	−2.34315			−0.278080			−0.139040	+	0.990287i	$0.544402\pi$
$72$		0			0
$73$	1.67157	+	2.89525i	0.195643	+	0.338863i	0.947111	−	0.320906i	$-0.103987\pi$
$73$	1.67157	+	2.89525i	0.195643	+	0.338863i	−0.751468	+	0.659769i	$0.770654\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−1.08579	+	0.148586i	−0.123737	+	0.0169330i
$78$		0			0
$79$	4.03553	−	6.98975i	0.454033	−	0.786408i	−0.544599	−	0.838697i	$-0.683318\pi$
$79$	4.03553	−	6.98975i	0.454033	−	0.786408i	0.998632	+	0.0522883i	$0.0166515\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−15.3137			−1.68090			−0.840449	−	0.541891i	$-0.817709\pi$
$83$	−15.3137			−1.68090			−0.840449	+	0.541891i	$0.817709\pi$
$84$		0			0
$85$	−22.3137			−2.42026
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−4.50000	+	7.79423i	−0.476999	+	0.826187i	−0.999653	−	0.0263586i	$-0.991609\pi$
$89$	−4.50000	+	7.79423i	−0.476999	+	0.826187i	0.522654	+	0.852545i	$0.324942\pi$
$90$		0			0
$91$	2.82843	−	6.92820i	0.296500	−	0.726273i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	6.86396	+	11.8887i	0.704228	+	1.21976i
$96$		0			0
$97$	−6.82843			−0.693322			−0.346661	−	0.937991i	$-0.612685\pi$
$97$	−6.82843			−0.693322			−0.346661	+	0.937991i	$0.612685\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−1.74264	−	3.01834i	−0.173399	−	0.300336i	0.766207	−	0.642594i	$-0.222142\pi$
$101$	−1.74264	−	3.01834i	−0.173399	−	0.300336i	−0.939606	+	0.342258i	$0.888808\pi$
$102$		0			0
$103$	−2.79289	+	4.83743i	−0.275192	+	0.476646i	−0.970184	−	0.242371i	$-0.922075\pi$
$103$	−2.79289	+	4.83743i	−0.275192	+	0.476646i	0.694992	+	0.719018i	$0.255408\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	3.44975	−	5.97514i	0.333500	−	0.577638i	−0.649696	−	0.760194i	$-0.725104\pi$
$107$	3.44975	−	5.97514i	0.333500	−	0.577638i	0.983195	+	0.182556i	$0.0584371\pi$
$108$		0			0
$109$	6.91421	+	11.9758i	0.662262	+	1.14707i	0.980020	+	0.198899i	$0.0637366\pi$
$109$	6.91421	+	11.9758i	0.662262	+	1.14707i	−0.317758	+	0.948172i	$0.602930\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	10.1421			0.954092			0.477046	−	0.878878i	$-0.341708\pi$
$113$	10.1421			0.954092			0.477046	+	0.878878i	$0.341708\pi$
$114$		0			0
$115$	−6.20711	−	10.7510i	−0.578816	−	1.00254i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	15.2782	−	2.09077i	1.40055	−	0.191661i
$120$		0			0
$121$	5.41421	−	9.37769i	0.492201	−	0.852518i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−17.8284			−1.59462
$126$		0			0
$127$	5.65685			0.501965			0.250982	−	0.967992i	$-0.419246\pi$
$127$	5.65685			0.501965			0.250982	+	0.967992i	$0.419246\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−8.86396	+	15.3528i	−0.774448	+	1.34138i	0.160656	+	0.987010i	$0.448639\pi$
$131$	−8.86396	+	15.3528i	−0.774448	+	1.34138i	−0.935104	+	0.354373i	$0.884694\pi$
$132$		0			0
$133$	−5.81371	−	7.49706i	−0.504112	−	0.650077i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−8.57107	−	14.8455i	−0.732276	−	1.26834i	−0.955908	−	0.293665i	$-0.905125\pi$
$137$	−8.57107	−	14.8455i	−0.732276	−	1.26834i	0.223633	−	0.974674i	$-0.428208\pi$
$138$		0			0
$139$	−7.31371			−0.620341			−0.310170	−	0.950681i	$-0.600386\pi$
$139$	−7.31371			−0.620341			−0.310170	+	0.950681i	$0.600386\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−0.585786	−	1.01461i	−0.0489859	−	0.0848461i
$144$		0			0
$145$	−5.41421	+	9.37769i	−0.449626	+	0.778775i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	5.91421	−	10.2437i	0.484511	−	0.839198i	−0.515330	−	0.856992i	$-0.672331\pi$
$149$	5.91421	−	10.2437i	0.484511	−	0.839198i	0.999842	+	0.0177935i	$0.00566413\pi$
$150$		0			0
$151$	9.44975	+	16.3674i	0.769010	+	1.33196i	0.938101	+	0.346363i	$0.112583\pi$
$151$	9.44975	+	16.3674i	0.769010	+	1.33196i	−0.169091	+	0.985600i	$0.554083\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−32.2132			−2.58743
$156$		0			0
$157$	−10.3284	−	17.8894i	−0.824298	−	1.42773i	−0.902454	−	0.430785i	$-0.858237\pi$
$157$	−10.3284	−	17.8894i	−0.824298	−	1.42773i	0.0781562	−	0.996941i	$-0.475097\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	5.25736	+	6.77962i	0.414338	+	0.534309i
$162$		0			0
$163$	−7.62132	+	13.2005i	−0.596948	+	1.03394i	0.396321	+	0.918112i	$0.370287\pi$
$163$	−7.62132	+	13.2005i	−0.596948	+	1.03394i	−0.993269	+	0.115832i	$0.963047\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−2.00000			−0.154765			−0.0773823	−	0.997001i	$-0.524656\pi$
$167$	−2.00000			−0.154765			−0.0773823	+	0.997001i	$0.524656\pi$
$168$		0			0
$169$	−5.00000			−0.384615
$170$		0			0
$171$		0			0
$172$		0			0
$173$	5.50000	−	9.52628i	0.418157	−	0.724270i	−0.577597	−	0.816322i	$-0.696009\pi$
$173$	5.50000	−	9.52628i	0.418157	−	0.724270i	0.995754	+	0.0920525i	$0.0293428\pi$
$174$		0			0
$175$	25.3137	−	3.46410i	1.91354	−	0.261861i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	4.20711	+	7.28692i	0.314454	+	0.544650i	0.979321	−	0.202311i	$-0.0648453\pi$
$179$	4.20711	+	7.28692i	0.314454	+	0.544650i	−0.664867	+	0.746961i	$0.731512\pi$
$180$		0			0
$181$	13.3137			0.989600			0.494800	−	0.869007i	$-0.335241\pi$
$181$	13.3137			0.989600			0.494800	+	0.869007i	$0.335241\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−5.08579	−	8.80884i	−0.373914	−	0.647639i
$186$		0			0
$187$	1.20711	−	2.09077i	0.0882724	−	0.152892i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	5.44975	−	9.43924i	0.394330	−	0.682999i	−0.598686	−	0.800984i	$-0.704310\pi$
$191$	5.44975	−	9.43924i	0.394330	−	0.682999i	0.993015	+	0.117985i	$0.0376434\pi$
$192$		0			0
$193$	−9.57107	−	16.5776i	−0.688941	−	1.19328i	−0.972181	−	0.234231i	$-0.924743\pi$
$193$	−9.57107	−	16.5776i	−0.688941	−	1.19328i	0.283240	−	0.959049i	$-0.408591\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	13.1716			0.938436			0.469218	−	0.883082i	$-0.344536\pi$
$197$	13.1716			0.938436			0.469218	+	0.883082i	$0.344536\pi$
$198$		0			0
$199$	−1.37868	−	2.38794i	−0.0977320	−	0.169277i	0.813014	−	0.582245i	$-0.197825\pi$
$199$	−1.37868	−	2.38794i	−0.0977320	−	0.169277i	−0.910746	+	0.412968i	$0.864492\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	2.82843	−	6.92820i	0.198517	−	0.486265i
$204$		0			0
$205$	2.24264	−	3.88437i	0.156633	−	0.271296i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−1.48528			−0.102739
$210$		0			0
$211$	−12.0000			−0.826114			−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000			−0.826114			−0.413057	+	0.910705i	$0.635539\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−3.17157	+	5.49333i	−0.216299	+	0.374642i
$216$		0			0
$217$	22.0563	−	3.01834i	1.49728	−	0.204898i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	8.24264	+	14.2767i	0.554460	+	0.960353i
$222$		0			0
$223$	13.6569			0.914531			0.457265	−	0.889330i	$-0.348829\pi$
$223$	13.6569			0.914531			0.457265	+	0.889330i	$0.348829\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−0.964466	−	1.67050i	−0.0640139	−	0.110875i	0.832242	−	0.554412i	$-0.187057\pi$
$227$	−0.964466	−	1.67050i	−0.0640139	−	0.110875i	−0.896256	+	0.443537i	$0.853724\pi$
$228$		0			0
$229$	4.91421	−	8.51167i	0.324740	−	0.562467i	−0.656719	−	0.754135i	$-0.728056\pi$
$229$	4.91421	−	8.51167i	0.324740	−	0.562467i	0.981460	+	0.191668i	$0.0613897\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	3.91421	−	6.77962i	0.256429	−	0.444147i	−0.708854	−	0.705355i	$-0.750787\pi$
$233$	3.91421	−	6.77962i	0.256429	−	0.444147i	0.965283	+	0.261208i	$0.0841208\pi$
$234$		0			0
$235$	−14.5208	−	25.1508i	−0.947234	−	1.64066i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	21.3137			1.37867			0.689335	−	0.724443i	$-0.257903\pi$
$239$	21.3137			1.37867			0.689335	+	0.724443i	$0.257903\pi$
$240$		0			0
$241$	6.15685	+	10.6640i	0.396598	+	0.686928i	0.993304	−	0.115532i	$-0.0368574\pi$
$241$	6.15685	+	10.6640i	0.396598	+	0.686928i	−0.596706	+	0.802460i	$0.703524\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−25.8137	+	7.19988i	−1.64918	+	0.459984i
$246$		0			0
$247$	5.07107	−	8.78335i	0.322664	−	0.558871i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−2.97056			−0.187500			−0.0937501	−	0.995596i	$-0.529885\pi$
$251$	−2.97056			−0.187500			−0.0937501	+	0.995596i	$0.529885\pi$
$252$		0			0
$253$	1.34315			0.0844428
$254$		0			0
$255$		0			0
$256$		0			0
$257$	10.7426	−	18.6068i	0.670108	−	1.16066i	−0.307766	−	0.951462i	$-0.599581\pi$
$257$	10.7426	−	18.6068i	0.670108	−	1.16066i	0.977873	−	0.209198i	$-0.0670854\pi$
$258$		0			0
$259$	4.30761	+	5.55487i	0.267662	+	0.345163i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−4.44975	−	7.70719i	−0.274383	−	0.475246i	0.695596	−	0.718433i	$-0.255140\pi$
$263$	−4.44975	−	7.70719i	−0.274383	−	0.475246i	−0.969979	+	0.243187i	$0.921807\pi$
$264$		0			0
$265$	−3.82843			−0.235178
$266$		0			0
$267$		0			0
$268$		0			0
$269$	0.671573	+	1.16320i	0.0409465	+	0.0709215i	0.885772	−	0.464120i	$-0.153629\pi$
$269$	0.671573	+	1.16320i	0.0409465	+	0.0709215i	−0.844826	+	0.535041i	$0.820296\pi$
$270$		0			0
$271$	−6.10660	+	10.5769i	−0.370950	+	0.642504i	−0.989712	−	0.143075i	$-0.954301\pi$
$271$	−6.10660	+	10.5769i	−0.370950	+	0.642504i	0.618762	+	0.785578i	$0.287634\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	2.00000	−	3.46410i	0.120605	−	0.208893i
$276$		0			0
$277$	6.15685	+	10.6640i	0.369930	+	0.640737i	0.989554	−	0.144162i	$-0.0460485\pi$
$277$	6.15685	+	10.6640i	0.369930	+	0.640737i	−0.619625	+	0.784898i	$0.712715\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−4.48528			−0.267569			−0.133785	−	0.991010i	$-0.542713\pi$
$281$	−4.48528			−0.267569			−0.133785	+	0.991010i	$0.542713\pi$
$282$		0			0
$283$	−4.20711	−	7.28692i	−0.250087	−	0.433163i	0.713463	−	0.700693i	$-0.247126\pi$
$283$	−4.20711	−	7.28692i	−0.250087	−	0.433163i	−0.963549	+	0.267530i	$0.913792\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−1.17157	+	2.86976i	−0.0691558	+	0.169396i
$288$		0			0
$289$	−8.48528	+	14.6969i	−0.499134	+	0.864526i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	16.6274			0.971384			0.485692	−	0.874130i	$-0.338568\pi$
$293$	16.6274			0.971384			0.485692	+	0.874130i	$0.338568\pi$
$294$		0			0
$295$	−34.0711			−1.98369
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−4.58579	+	7.94282i	−0.265203	+	0.459345i
$300$		0			0
$301$	1.65685	−	4.05845i	0.0954995	−	0.233925i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−5.08579	−	8.80884i	−0.291211	−	0.504393i
$306$		0			0
$307$	25.6569			1.46431			0.732157	−	0.681136i	$-0.238514\pi$
$307$	25.6569			1.46431			0.732157	+	0.681136i	$0.238514\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−13.6213	−	23.5928i	−0.772394	−	1.33783i	−0.936247	−	0.351341i	$-0.885726\pi$
$311$	−13.6213	−	23.5928i	−0.772394	−	1.33783i	0.163853	−	0.986485i	$-0.447608\pi$
$312$		0			0
$313$	−13.6421	+	23.6289i	−0.771099	+	1.33558i	0.165862	+	0.986149i	$0.446959\pi$
$313$	−13.6421	+	23.6289i	−0.771099	+	1.33558i	−0.936961	+	0.349434i	$0.886374\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−16.1569	+	27.9845i	−0.907459	+	1.57177i	−0.0898778	+	0.995953i	$0.528648\pi$
$317$	−16.1569	+	27.9845i	−0.907459	+	1.57177i	−0.817582	+	0.575813i	$0.804686\pi$
$318$		0			0
$319$	−0.585786	−	1.01461i	−0.0327977	−	0.0568074i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	20.8995			1.16288
$324$		0			0
$325$	13.6569	+	23.6544i	0.757546	+	1.31211i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	12.2990	+	15.8601i	0.678065	+	0.874398i
$330$		0			0
$331$	−12.7929	+	22.1579i	−0.703161	+	1.21791i	0.264190	+	0.964471i	$0.414895\pi$
$331$	−12.7929	+	22.1579i	−0.703161	+	1.21791i	−0.967351	+	0.253440i	$0.918438\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	43.0416			2.35162
$336$		0			0
$337$	14.8284			0.807756			0.403878	−	0.914813i	$-0.367662\pi$
$337$	14.8284			0.807756			0.403878	+	0.914813i	$0.367662\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	1.74264	−	3.01834i	0.0943693	−	0.163452i
$342$		0			0
$343$	17.0000	−	7.34847i	0.917914	−	0.396780i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	7.55025	+	13.0774i	0.405319	+	0.702033i	0.994359	−	0.106071i	$-0.0338272\pi$
$347$	7.55025	+	13.0774i	0.405319	+	0.702033i	−0.589040	+	0.808104i	$0.700494\pi$
$348$		0			0
$349$	10.8284			0.579632			0.289816	−	0.957082i	$-0.406406\pi$
$349$	10.8284			0.579632			0.289816	+	0.957082i	$0.406406\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	3.91421	+	6.77962i	0.208333	+	0.360843i	0.951189	−	0.308608i	$-0.0998630\pi$
$353$	3.91421	+	6.77962i	0.208333	+	0.360843i	−0.742857	+	0.669450i	$0.766530\pi$
$354$		0			0
$355$	−4.48528	+	7.76874i	−0.238054	+	0.412322i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−12.6924	+	21.9839i	−0.669879	+	1.16026i	0.308059	+	0.951367i	$0.400321\pi$
$359$	−12.6924	+	21.9839i	−0.669879	+	1.16026i	−0.977938	+	0.208897i	$0.933013\pi$
$360$		0			0
$361$	3.07107	+	5.31925i	0.161635	+	0.279960i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	12.7990			0.669930
$366$		0			0
$367$	−14.8640	−	25.7451i	−0.775892	−	1.34389i	−0.934292	−	0.356510i	$-0.883967\pi$
$367$	−14.8640	−	25.7451i	−0.775892	−	1.34389i	0.158399	−	0.987375i	$-0.449367\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	2.62132	−	0.358719i	0.136092	−	0.0186238i
$372$		0			0
$373$	−7.15685	+	12.3960i	−0.370568	+	0.641842i	−0.989653	−	0.143482i	$-0.954170\pi$
$373$	−7.15685	+	12.3960i	−0.370568	+	0.641842i	0.619085	+	0.785324i	$0.287504\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	8.00000			0.412021
$378$		0			0
$379$	−1.02944			−0.0528786			−0.0264393	−	0.999650i	$-0.508417\pi$
$379$	−1.02944			−0.0528786			−0.0264393	+	0.999650i	$0.508417\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−1.37868	+	2.38794i	−0.0704472	+	0.122018i	−0.899097	−	0.437749i	$-0.855776\pi$
$383$	−1.37868	+	2.38794i	−0.0704472	+	0.122018i	0.828650	+	0.559767i	$0.189109\pi$
$384$		0			0
$385$	−1.58579	+	3.88437i	−0.0808192	+	0.197966i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	11.9142	+	20.6360i	0.604075	+	1.04629i	0.992197	+	0.124680i	$0.0397904\pi$
$389$	11.9142	+	20.6360i	0.604075	+	1.04629i	−0.388122	+	0.921608i	$0.626876\pi$
$390$		0			0
$391$	−18.8995			−0.955789
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−15.4497	−	26.7597i	−0.777361	−	1.34643i
$396$		0			0
$397$	4.91421	−	8.51167i	0.246637	−	0.427188i	−0.715953	−	0.698148i	$-0.754008\pi$
$397$	4.91421	−	8.51167i	0.246637	−	0.427188i	0.962591	+	0.270960i	$0.0873410\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	1.08579	−	1.88064i	0.0542216	−	0.0939145i	−0.837641	−	0.546222i	$-0.816066\pi$
$401$	1.08579	−	1.88064i	0.0542216	−	0.0939145i	0.891862	+	0.452307i	$0.149399\pi$
$402$		0			0
$403$	11.8995	+	20.6105i	0.592756	+	1.02668i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	1.10051			0.0545500
$408$		0			0
$409$	18.2279	+	31.5717i	0.901313	+	1.56112i	0.825792	+	0.563975i	$0.190729\pi$
$409$	18.2279	+	31.5717i	0.901313	+	1.56112i	0.0755210	+	0.997144i	$0.475938\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	23.3284	−	3.19242i	1.14792	−	0.157089i
$414$		0			0
$415$	−29.3137	+	50.7728i	−1.43895	+	2.49234i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	1.65685			0.0809426			0.0404713	−	0.999181i	$-0.487114\pi$
$419$	1.65685			0.0809426			0.0404713	+	0.999181i	$0.487114\pi$
$420$		0			0
$421$	0.485281			0.0236512			0.0118256	−	0.999930i	$-0.496236\pi$
$421$	0.485281			0.0236512			0.0118256	+	0.999930i	$0.496236\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−28.1421	+	48.7436i	−1.36509	+	2.36441i
$426$		0			0
$427$	4.30761	+	5.55487i	0.208460	+	0.268819i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−10.6213	−	18.3967i	−0.511611	−	0.886136i	−0.999909	−	0.0134595i	$-0.995716\pi$
$431$	−10.6213	−	18.3967i	−0.511611	−	0.886136i	0.488298	−	0.872677i	$-0.337618\pi$
$432$		0			0
$433$	28.4853			1.36892			0.684458	−	0.729053i	$-0.260039\pi$
$433$	28.4853			1.36892			0.684458	+	0.729053i	$0.260039\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	5.81371	+	10.0696i	0.278107	+	0.481696i
$438$		0			0
$439$	18.3492	−	31.7818i	0.875762	−	1.51686i	0.0198123	−	0.999804i	$-0.493693\pi$
$439$	18.3492	−	31.7818i	0.875762	−	1.51686i	0.855949	−	0.517060i	$-0.172974\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−13.0355	+	22.5782i	−0.619337	+	1.07272i	0.370270	+	0.928924i	$0.379265\pi$
$443$	−13.0355	+	22.5782i	−0.619337	+	1.07272i	−0.989607	+	0.143799i	$0.954068\pi$
$444$		0			0
$445$	17.2279	+	29.8396i	0.816682	+	1.41453i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−6.82843			−0.322253			−0.161127	−	0.986934i	$-0.551513\pi$
$449$	−6.82843			−0.322253			−0.161127	+	0.986934i	$0.551513\pi$
$450$		0			0
$451$	0.242641	+	0.420266i	0.0114255	+	0.0197896i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−17.5563	−	22.6398i	−0.823054	−	1.06137i
$456$		0			0
$457$	18.6421	−	32.2891i	0.872042	−	1.51042i	0.0121619	−	0.999926i	$-0.496129\pi$
$457$	18.6421	−	32.2891i	0.872042	−	1.51042i	0.859880	−	0.510496i	$-0.170538\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−25.4558			−1.18560			−0.592798	−	0.805351i	$-0.701977\pi$
$461$	−25.4558			−1.18560			−0.592798	+	0.805351i	$0.701977\pi$
$462$		0			0
$463$	11.3137			0.525793			0.262896	−	0.964824i	$-0.415322\pi$
$463$	11.3137			0.525793			0.262896	+	0.964824i	$0.415322\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	9.27817	−	16.0703i	0.429343	−	0.743643i	−0.567472	−	0.823393i	$-0.692079\pi$
$467$	9.27817	−	16.0703i	0.429343	−	0.743643i	0.996815	+	0.0797491i	$0.0254119\pi$
$468$		0			0
$469$	−29.4706	+	4.03295i	−1.36082	+	0.186225i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−0.343146	−	0.594346i	−0.0157779	−	0.0273281i
$474$		0			0
$475$	34.6274			1.58881
$476$		0			0
$477$		0			0
$478$		0			0
$479$	15.3492	+	26.5857i	0.701325	+	1.21473i	0.968002	+	0.250944i	$0.0807410\pi$
$479$	15.3492	+	26.5857i	0.701325	+	1.21473i	−0.266677	+	0.963786i	$0.585926\pi$
$480$		0			0
$481$	−3.75736	+	6.50794i	−0.171321	+	0.296736i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−13.0711	+	22.6398i	−0.593527	+	1.02802i
$486$		0			0
$487$	−6.86396	−	11.8887i	−0.311036	−	0.538730i	0.667551	−	0.744564i	$-0.267343\pi$
$487$	−6.86396	−	11.8887i	−0.311036	−	0.538730i	−0.978587	+	0.205834i	$0.934009\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	7.65685			0.345549			0.172774	−	0.984961i	$-0.444727\pi$
$491$	7.65685			0.345549			0.172774	+	0.984961i	$0.444727\pi$
$492$		0			0
$493$	8.24264	+	14.2767i	0.371230	+	0.642989i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	2.34315	−	5.73951i	0.105104	−	0.257452i
$498$		0			0
$499$	−13.2782	+	22.9985i	−0.594413	+	1.02955i	0.399217	+	0.916857i	$0.369282\pi$
$499$	−13.2782	+	22.9985i	−0.594413	+	1.02955i	−0.993629	+	0.112696i	$0.964051\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	21.6569			0.965631			0.482816	−	0.875722i	$-0.339614\pi$
$503$	21.6569			0.965631			0.482816	+	0.875722i	$0.339614\pi$
$504$		0			0
$505$	−13.3431			−0.593762
$506$		0			0
$507$		0			0
$508$		0			0
$509$	12.2574	−	21.2304i	0.543298	−	0.941020i	−0.455414	−	0.890280i	$-0.650509\pi$
$509$	12.2574	−	21.2304i	0.543298	−	0.941020i	0.998712	−	0.0507398i	$-0.0161579\pi$
$510$		0			0
$511$	−8.76346	+	1.19925i	−0.387672	+	0.0530518i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	10.6924	+	18.5198i	0.471163	+	0.816078i
$516$		0			0
$517$	3.14214			0.138191
$518$		0			0
$519$		0			0
$520$		0			0
$521$	3.50000	+	6.06218i	0.153338	+	0.265589i	0.932453	−	0.361293i	$-0.117664\pi$
$521$	3.50000	+	6.06218i	0.153338	+	0.265589i	−0.779115	+	0.626881i	$0.784331\pi$
$522$		0			0
$523$	3.86396	−	6.69258i	0.168959	−	0.292646i	−0.769095	−	0.639134i	$-0.779293\pi$
$523$	3.86396	−	6.69258i	0.168959	−	0.292646i	0.938054	+	0.346489i	$0.112626\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−24.5208	+	42.4713i	−1.06814	+	1.85008i
$528$		0			0
$529$	6.24264	+	10.8126i	0.271419	+	0.470112i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−3.31371			−0.143533
$534$		0			0
$535$	−13.2071	−	22.8754i	−0.570993	−	0.988989i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	0.721825	−	2.80821i	0.0310912	−	0.120958i
$540$		0			0
$541$	10.0858	−	17.4691i	0.433622	−	0.751055i	−0.563560	−	0.826075i	$-0.690569\pi$
$541$	10.0858	−	17.4691i	0.433622	−	0.751055i	0.997182	+	0.0750200i	$0.0239021\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	52.9411			2.26775
$546$		0			0
$547$	−22.9706			−0.982150			−0.491075	−	0.871117i	$-0.663396\pi$
$547$	−22.9706			−0.982150			−0.491075	+	0.871117i	$0.663396\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	5.07107	−	8.78335i	0.216035	−	0.374183i
$552$		0			0
$553$	13.0858	+	16.8747i	0.556464	+	0.717587i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−2.91421	−	5.04757i	−0.123479	−	0.213872i	0.797658	−	0.603110i	$-0.206072\pi$
$557$	−2.91421	−	5.04757i	−0.123479	−	0.213872i	−0.921137	+	0.389237i	$0.872739\pi$
$558$		0			0
$559$	4.68629			0.198209
$560$		0			0
$561$		0			0
$562$		0			0
$563$	0.0355339	+	0.0615465i	0.00149758	+	0.00259388i	0.866773	−	0.498703i	$-0.166190\pi$
$563$	0.0355339	+	0.0615465i	0.00149758	+	0.00259388i	−0.865276	+	0.501296i	$0.832857\pi$
$564$		0			0
$565$	19.4142	−	33.6264i	0.816762	−	1.41467i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	12.6716	−	21.9478i	0.531220	−	0.920100i	−0.468116	−	0.883667i	$-0.655067\pi$
$569$	12.6716	−	21.9478i	0.531220	−	0.920100i	0.999336	−	0.0364330i	$-0.0115996\pi$
$570$		0			0
$571$	7.79289	+	13.4977i	0.326122	+	0.564861i	0.981739	−	0.190233i	$-0.0609245\pi$
$571$	7.79289	+	13.4977i	0.326122	+	0.564861i	−0.655616	+	0.755094i	$0.727591\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−31.3137			−1.30587
$576$		0			0
$577$	−19.5000	−	33.7750i	−0.811796	−	1.40607i	−0.911606	−	0.411065i	$-0.865157\pi$
$577$	−19.5000	−	33.7750i	−0.811796	−	1.40607i	0.0998105	−	0.995006i	$-0.468176\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	15.3137	−	37.5108i	0.635320	−	1.55621i
$582$		0			0
$583$	0.207107	−	0.358719i	0.00857749	−	0.0148566i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	4.97056			0.205157			0.102579	−	0.994725i	$-0.467291\pi$
$587$	4.97056			0.205157			0.102579	+	0.994725i	$0.467291\pi$
$588$		0			0
$589$	30.1716			1.24320
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−5.74264	+	9.94655i	−0.235822	+	0.408456i	−0.959511	−	0.281670i	$-0.909112\pi$
$593$	−5.74264	+	9.94655i	−0.235822	+	0.408456i	0.723689	+	0.690126i	$0.242445\pi$
$594$		0			0
$595$	22.3137	−	54.6572i	0.914773	−	2.24073i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−8.93503	−	15.4759i	−0.365075	−	0.632329i	0.623713	−	0.781654i	$-0.285623\pi$
$599$	−8.93503	−	15.4759i	−0.365075	−	0.632329i	−0.988788	+	0.149324i	$0.952290\pi$
$600$		0			0
$601$	−2.14214			−0.0873795			−0.0436898	−	0.999045i	$-0.513911\pi$
$601$	−2.14214			−0.0873795			−0.0436898	+	0.999045i	$0.513911\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−20.7279	−	35.9018i	−0.842710	−	1.45962i
$606$		0			0
$607$	0.692388	−	1.19925i	0.0281032	−	0.0486761i	−0.851632	−	0.524141i	$-0.824387\pi$
$607$	0.692388	−	1.19925i	0.0281032	−	0.0486761i	0.879735	+	0.475464i	$0.157720\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−10.7279	+	18.5813i	−0.434005	+	0.751719i
$612$		0			0
$613$	−10.7426	−	18.6068i	−0.433891	−	0.751522i	0.563313	−	0.826243i	$-0.309526\pi$
$613$	−10.7426	−	18.6068i	−0.433891	−	0.751522i	−0.997204	+	0.0747219i	$0.976193\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	43.1127			1.73565			0.867826	−	0.496868i	$-0.165517\pi$
$617$	43.1127			1.73565			0.867826	+	0.496868i	$0.165517\pi$
$618$		0			0
$619$	−16.0355	−	27.7744i	−0.644523	−	1.11635i	−0.984412	−	0.175880i	$-0.943723\pi$
$619$	−16.0355	−	27.7744i	−0.644523	−	1.11635i	0.339889	−	0.940466i	$-0.389610\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−14.5919	−	18.8169i	−0.584611	−	0.753884i
$624$		0			0
$625$	−9.98528	+	17.2950i	−0.399411	+	0.691801i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−15.4853			−0.617439
$630$		0			0
$631$	18.3431			0.730229			0.365115	−	0.930963i	$-0.381030\pi$
$631$	18.3431			0.730229			0.365115	+	0.930963i	$0.381030\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	10.8284	−	18.7554i	0.429713	−	0.744285i
$636$		0			0
$637$	14.1421	+	13.8564i	0.560332	+	0.549011i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−8.50000	−	14.7224i	−0.335730	−	0.581501i	0.647895	−	0.761730i	$-0.275650\pi$
$641$	−8.50000	−	14.7224i	−0.335730	−	0.581501i	−0.983625	+	0.180229i	$0.942316\pi$
$642$		0			0
$643$	44.9706			1.77347			0.886733	−	0.462282i	$-0.152969\pi$
$643$	44.9706			1.77347			0.886733	+	0.462282i	$0.152969\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−12.9350	−	22.4041i	−0.508528	−	0.880797i	−0.999951	−	0.00987597i	$-0.996856\pi$
$647$	−12.9350	−	22.4041i	−0.508528	−	0.880797i	0.491423	−	0.870921i	$-0.336477\pi$
$648$		0			0
$649$	1.84315	−	3.19242i	0.0723498	−	0.125314i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	14.4706	−	25.0637i	0.566277	−	0.980820i	−0.430653	−	0.902518i	$-0.641717\pi$
$653$	14.4706	−	25.0637i	0.566277	−	0.980820i	0.996930	−	0.0783026i	$-0.0249500\pi$
$654$		0			0
$655$	33.9350	+	58.7772i	1.32595	+	2.29662i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	10.6274			0.413985			0.206993	−	0.978342i	$-0.433632\pi$
$659$	10.6274			0.413985			0.206993	+	0.978342i	$0.433632\pi$
$660$		0			0
$661$	9.67157	+	16.7517i	0.376181	+	0.651564i	0.990503	−	0.137491i	$-0.0439039\pi$
$661$	9.67157	+	16.7517i	0.376181	+	0.651564i	−0.614322	+	0.789055i	$0.710571\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−35.9853	+	4.92447i	−1.39545	+	0.190963i
$666$		0			0
$667$	−4.58579	+	7.94282i	−0.177562	+	0.307547i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	1.10051			0.0424845
$672$		0			0
$673$	26.1421			1.00771			0.503853	−	0.863790i	$-0.331915\pi$
$673$	26.1421			1.00771			0.503853	+	0.863790i	$0.331915\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	10.3995	−	18.0125i	0.399685	−	0.692275i	−0.594002	−	0.804464i	$-0.702453\pi$
$677$	10.3995	−	18.0125i	0.399685	−	0.692275i	0.993687	+	0.112189i	$0.0357862\pi$
$678$		0			0
$679$	6.82843	−	16.7262i	0.262051	−	0.641891i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−24.2782	−	42.0510i	−0.928979	−	1.60904i	−0.785034	−	0.619452i	$-0.787355\pi$
$683$	−24.2782	−	42.0510i	−0.928979	−	1.60904i	−0.143944	−	0.989586i	$-0.545979\pi$
$684$		0			0
$685$	−65.6274			−2.50749
$686$		0			0
$687$		0			0
$688$		0			0
$689$	1.41421	+	2.44949i	0.0538772	+	0.0933181i
$690$		0			0
$691$	10.5208	−	18.2226i	0.400231	−	0.693220i	−0.593523	−	0.804817i	$-0.702263\pi$
$691$	10.5208	−	18.2226i	0.400231	−	0.693220i	0.993754	+	0.111597i	$0.0355967\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−14.0000	+	24.2487i	−0.531050	+	0.919806i
$696$		0			0
$697$	−3.41421	−	5.91359i	−0.129323	−	0.223993i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−14.0000			−0.528773			−0.264386	−	0.964417i	$-0.585169\pi$
$701$	−14.0000			−0.528773			−0.264386	+	0.964417i	$0.585169\pi$
$702$		0			0
$703$	4.76346	+	8.25055i	0.179657	+	0.311175i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	9.13604	−	1.25024i	0.343596	−	0.0470200i
$708$		0			0
$709$	−21.2990	+	36.8909i	−0.799900	+	1.38547i	0.119780	+	0.992800i	$0.461781\pi$
$709$	−21.2990	+	36.8909i	−0.799900	+	1.38547i	−0.919680	+	0.392668i	$0.871552\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−27.2843			−1.02180
$714$		0			0
$715$	−4.48528			−0.167740
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−8.52082	+	14.7585i	−0.317773	+	0.550399i	−0.980023	−	0.198884i	$-0.936268\pi$
$719$	−8.52082	+	14.7585i	−0.317773	+	0.550399i	0.662250	+	0.749283i	$0.269602\pi$
$720$		0			0
$721$	−9.05635	−	11.6786i	−0.337276	−	0.434934i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	13.6569	+	23.6544i	0.507203	+	0.878501i
$726$		0			0
$727$	45.6569			1.69332			0.846659	−	0.532135i	$-0.178610\pi$
$727$	45.6569			1.69332			0.846659	+	0.532135i	$0.178610\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	4.82843	+	8.36308i	0.178586	+	0.309320i
$732$		0			0
$733$	−23.1569	+	40.1088i	−0.855318	+	1.48145i	0.0210318	+	0.999779i	$0.493305\pi$
$733$	−23.1569	+	40.1088i	−0.855318	+	1.48145i	−0.876350	+	0.481675i	$0.840028\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−2.32843	+	4.03295i	−0.0857687	+	0.148556i
$738$		0			0
$739$	−26.3492	−	45.6382i	−0.969273	−	1.67883i	−0.697669	−	0.716420i	$-0.745779\pi$
$739$	−26.3492	−	45.6382i	−0.969273	−	1.67883i	−0.271603	−	0.962409i	$-0.587554\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−37.6569			−1.38150			−0.690748	−	0.723096i	$-0.742719\pi$
$743$	−37.6569			−1.38150			−0.690748	+	0.723096i	$0.742719\pi$
$744$		0			0
$745$	−22.6421	−	39.2173i	−0.829544	−	1.43681i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	11.1863	+	14.4253i	0.408738	+	0.527087i
$750$		0			0
$751$	1.86396	−	3.22848i	0.0680169	−	0.117809i	−0.830011	−	0.557746i	$-0.811666\pi$
$751$	1.86396	−	3.22848i	0.0680169	−	0.117809i	0.898028	+	0.439938i	$0.144999\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	72.3553			2.63328
$756$		0			0
$757$	22.1421			0.804770			0.402385	−	0.915471i	$-0.368181\pi$
$757$	22.1421			0.804770			0.402385	+	0.915471i	$0.368181\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	10.0563	−	17.4181i	0.364542	−	0.631406i	−0.624160	−	0.781296i	$-0.714559\pi$
$761$	10.0563	−	17.4181i	0.364542	−	0.631406i	0.988703	+	0.149890i	$0.0478921\pi$
$762$		0			0
$763$	−36.2487	+	4.96053i	−1.31229	+	0.179583i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	12.5858	+	21.7992i	0.454446	+	0.787124i
$768$		0			0
$769$	42.1421			1.51968			0.759842	−	0.650108i	$-0.225276\pi$
$769$	42.1421			1.51968			0.759842	+	0.650108i	$0.225276\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	9.57107	+	16.5776i	0.344247	+	0.596254i	0.985217	−	0.171313i	$-0.0548008\pi$
$773$	9.57107	+	16.5776i	0.344247	+	0.596254i	−0.640969	+	0.767566i	$0.721467\pi$
$774$		0			0
$775$	−40.6274	+	70.3688i	−1.45938	+	2.52772i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−2.10051	+	3.63818i	−0.0752584	+	0.130351i
$780$		0			0
$781$	−0.485281	−	0.840532i	−0.0173647	−	0.0300766i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−79.0833			−2.82260
$786$		0			0
$787$	−1.89340	−	3.27946i	−0.0674924	−	0.116900i	0.830305	−	0.557310i	$-0.188166\pi$
$787$	−1.89340	−	3.27946i	−0.0674924	−	0.116900i	−0.897797	+	0.440410i	$0.854833\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−10.1421	+	24.8431i	−0.360613	+	0.883317i
$792$		0			0
$793$	−3.75736	+	6.50794i	−0.133428	+	0.231104i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	15.1127			0.535319			0.267660	−	0.963514i	$-0.413750\pi$
$797$	15.1127			0.535319			0.267660	+	0.963514i	$0.413750\pi$
$798$		0			0
$799$	−44.2132			−1.56415
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−0.692388	+	1.19925i	−0.0244338	+	0.0423207i
$804$		0			0
$805$	32.5416	−	4.45322i	1.14694	−	0.156955i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	7.98528	+	13.8309i	0.280748	+	0.486269i	0.971569	−	0.236756i	$-0.0760843\pi$
$809$	7.98528	+	13.8309i	0.280748	+	0.486269i	−0.690822	+	0.723025i	$0.742751\pi$
$810$		0			0
$811$	−6.34315			−0.222738			−0.111369	−	0.993779i	$-0.535524\pi$
$811$	−6.34315			−0.222738			−0.111369	+	0.993779i	$0.535524\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	29.1777	+	50.5372i	1.02205	+	1.77024i
$816$		0			0
$817$	2.97056	−	5.14517i	0.103927	−	0.180007i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−6.57107	+	11.3814i	−0.229332	+	0.397214i	−0.957610	−	0.288067i	$-0.906987\pi$
$821$	−6.57107	+	11.3814i	−0.229332	+	0.397214i	0.728278	+	0.685281i	$0.240321\pi$
$822$		0			0
$823$	−17.0061	−	29.4554i	−0.592795	−	1.02675i	−0.993854	−	0.110699i	$-0.964691\pi$
$823$	−17.0061	−	29.4554i	−0.592795	−	1.02675i	0.401059	−	0.916052i	$-0.368642\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−30.3431			−1.05513			−0.527567	−	0.849513i	$-0.676896\pi$
$827$	−30.3431			−1.05513			−0.527567	+	0.849513i	$0.676896\pi$
$828$		0			0
$829$	−20.3995	−	35.3330i	−0.708504	−	1.22716i	−0.965412	−	0.260729i	$-0.916037\pi$
$829$	−20.3995	−	35.3330i	−0.708504	−	1.22716i	0.256908	−	0.966436i	$-0.417296\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−10.1569	+	39.5145i	−0.351914	+	1.36910i
$834$		0			0
$835$	−3.82843	+	6.63103i	−0.132488	+	0.229476i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−19.3137			−0.666783			−0.333392	−	0.942788i	$-0.608193\pi$
$839$	−19.3137			−0.666783			−0.333392	+	0.942788i	$0.608193\pi$
$840$		0			0
$841$	−21.0000			−0.724138
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−9.57107	+	16.5776i	−0.329255	+	0.570286i
$846$		0			0
$847$	17.5563	+	22.6398i	0.603243	+	0.777911i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−4.30761	−	7.46100i	−0.147663	−	0.255760i
$852$		0			0
$853$	−23.5147			−0.805129			−0.402564	−	0.915392i	$-0.631881\pi$
$853$	−23.5147			−0.805129			−0.402564	+	0.915392i	$0.631881\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	18.3284	+	31.7458i	0.626087	+	1.08441i	0.988330	+	0.152331i	$0.0486779\pi$
$857$	18.3284	+	31.7458i	0.626087	+	1.08441i	−0.362242	+	0.932084i	$0.617989\pi$
$858$		0			0
$859$	−18.2782	+	31.6587i	−0.623643	+	1.08018i	0.365158	+	0.930945i	$0.381015\pi$
$859$	−18.2782	+	31.6587i	−0.623643	+	1.08018i	−0.988802	+	0.149236i	$0.952318\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−1.69239	+	2.93130i	−0.0576096	+	0.0997827i	−0.893392	−	0.449278i	$-0.851681\pi$
$863$	−1.69239	+	2.93130i	−0.0576096	+	0.0997827i	0.835782	+	0.549061i	$0.185015\pi$
$864$		0			0
$865$	−21.0563	−	36.4707i	−0.715937	−	1.24004i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	3.34315			0.113408
$870$		0			0
$871$	−15.8995	−	27.5387i	−0.538734	−	0.933114i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	17.8284	−	43.6705i	0.602711	−	1.47633i
$876$		0			0
$877$	16.7132	−	28.9481i	0.564365	−	0.977508i	−0.432744	−	0.901517i	$-0.642454\pi$
$877$	16.7132	−	28.9481i	0.564365	−	0.977508i	0.997108	−	0.0759915i	$-0.0242122\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	26.0000			0.875962			0.437981	−	0.898984i	$-0.355694\pi$
$881$	26.0000			0.875962			0.437981	+	0.898984i	$0.355694\pi$
$882$		0			0
$883$	49.2548			1.65756			0.828779	−	0.559577i	$-0.189036\pi$
$883$	49.2548			1.65756			0.828779	+	0.559577i	$0.189036\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	25.4497	−	44.0803i	0.854519	−	1.48007i	−0.0225717	−	0.999745i	$-0.507185\pi$
$887$	25.4497	−	44.0803i	0.854519	−	1.48007i	0.877091	−	0.480325i	$-0.159481\pi$
$888$		0			0
$889$	−5.65685	+	13.8564i	−0.189725	+	0.464729i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	13.6005	+	23.5568i	0.455124	+	0.788297i
$894$		0			0
$895$	32.2132			1.07677
$896$		0			0
$897$		0			0
$898$		0			0
$899$	11.8995	+	20.6105i	0.396870	+	0.687400i
$900$		0			0
$901$	−2.91421	+	5.04757i	−0.0970865	+	0.168159i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	25.4853	−	44.1418i	0.847159	−	1.46732i
$906$		0			0
$907$	19.7635	+	34.2313i	0.656235	+	1.13663i	0.981583	+	0.191038i	$0.0611852\pi$
$907$	19.7635	+	34.2313i	0.656235	+	1.13663i	−0.325348	+	0.945594i	$0.605481\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−8.00000			−0.265052			−0.132526	−	0.991180i	$-0.542309\pi$
$911$	−8.00000			−0.265052			−0.132526	+	0.991180i	$0.542309\pi$
$912$		0			0
$913$	−3.17157	−	5.49333i	−0.104964	−	0.181803i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−28.7426	−	37.0650i	−0.949166	−	1.22399i
$918$		0			0
$919$	−12.2782	+	21.2664i	−0.405020	+	0.701515i	−0.994324	−	0.106397i	$-0.966069\pi$
$919$	−12.2782	+	21.2664i	−0.405020	+	0.701515i	0.589304	+	0.807911i	$0.299402\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	6.62742			0.218144
$924$		0			0
$925$	−25.6569			−0.843592
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−15.3284	+	26.5496i	−0.502909	+	0.871065i	0.497085	+	0.867702i	$0.334404\pi$
$929$	−15.3284	+	26.5496i	−0.502909	+	0.871065i	−0.999994	+	0.00336273i	$0.998930\pi$
$930$		0			0
$931$	24.1777	−	6.74356i	0.792391	−	0.221011i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−4.62132	−	8.00436i	−0.151133	−	0.261771i
$936$		0			0
$937$	−28.6274			−0.935217			−0.467608	−	0.883936i	$-0.654884\pi$
$937$	−28.6274			−0.935217			−0.467608	+	0.883936i	$0.654884\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	26.1274	+	45.2540i	0.851729	+	1.47524i	0.879646	+	0.475628i	$0.157779\pi$
$941$	26.1274	+	45.2540i	0.851729	+	1.47524i	−0.0279168	+	0.999610i	$0.508887\pi$
$942$		0			0
$943$	1.89949	−	3.29002i	0.0618561	−	0.107138i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	12.1066	−	20.9692i	0.393412	−	0.681409i	−0.599485	−	0.800386i	$-0.704628\pi$
$947$	12.1066	−	20.9692i	0.393412	−	0.681409i	0.992897	+	0.118977i	$0.0379614\pi$
$948$		0			0
$949$	−4.72792	−	8.18900i	−0.153475	−	0.265826i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−35.1127			−1.13741			−0.568706	−	0.822541i	$-0.692556\pi$
$953$	−35.1127			−1.13741			−0.568706	+	0.822541i	$0.692556\pi$
$954$		0			0
$955$	−20.8640	−	36.1374i	−0.675142	−	1.16938i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	44.9350	−	6.14922i	1.45103	−	0.198569i
$960$		0			0
$961$	−19.8995	+	34.4669i	−0.641919	+	1.11184i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−73.2843			−2.35910
$966$		0			0
$967$	−29.6569			−0.953700			−0.476850	−	0.878985i	$-0.658222\pi$
$967$	−29.6569			−0.953700			−0.476850	+	0.878985i	$0.658222\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−19.2071	+	33.2677i	−0.616385	+	1.06761i	0.373754	+	0.927528i	$0.378070\pi$
$971$	−19.2071	+	33.2677i	−0.616385	+	1.06761i	−0.990140	+	0.140083i	$0.955263\pi$
$972$		0			0
$973$	7.31371	−	17.9149i	0.234467	−	0.574324i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−25.7426	−	44.5876i	−0.823580	−	1.42648i	−0.903000	−	0.429641i	$-0.858640\pi$
$977$	−25.7426	−	44.5876i	−0.823580	−	1.42648i	0.0794196	−	0.996841i	$-0.474693\pi$
$978$		0			0
$979$	−3.72792			−0.119145
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−30.4203	−	52.6895i	−0.970257	−	1.68053i	−0.694773	−	0.719229i	$-0.744495\pi$
$983$	−30.4203	−	52.6895i	−0.970257	−	1.68053i	−0.275484	−	0.961306i	$-0.588838\pi$
$984$		0			0
$985$	25.2132	−	43.6705i	0.803359	−	1.39146i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−2.68629	+	4.65279i	−0.0854191	+	0.147950i
$990$		0			0
$991$	−7.34924	−	12.7293i	−0.233456	−	0.404358i	0.725367	−	0.688363i	$-0.241670\pi$
$991$	−7.34924	−	12.7293i	−0.233456	−	0.404358i	−0.958823	+	0.284004i	$0.908337\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−10.5563			−0.334659
$996$		0			0
$997$	−5.64214	−	9.77247i	−0.178688	−	0.309497i	0.762743	−	0.646701i	$-0.223852\pi$
$997$	−5.64214	−	9.77247i	−0.178688	−	0.309497i	−0.941431	+	0.337204i	$0.890519\pi$
$998$		0			0
$999$		0			0

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	2016.2.s.q.289.2		4
3.2	odd	2		224.2.i.a.65.1	✓	4
4.3	odd	2		2016.2.s.s.289.2		4
7.4	even	3	inner	2016.2.s.q.865.2		4
12.11	even	2		224.2.i.d.65.2	yes	4
21.2	odd	6		1568.2.a.w.1.2		2
21.5	even	6		1568.2.a.j.1.1		2
21.11	odd	6		224.2.i.a.193.1	yes	4
21.17	even	6		1568.2.i.x.1537.2		4
21.20	even	2		1568.2.i.x.961.2		4
24.5	odd	2		448.2.i.j.65.2		4
24.11	even	2		448.2.i.g.65.1		4
28.11	odd	6		2016.2.s.s.865.2		4
84.11	even	6		224.2.i.d.193.2	yes	4
84.23	even	6		1568.2.a.l.1.1		2
84.47	odd	6		1568.2.a.u.1.2		2
84.59	odd	6		1568.2.i.o.1537.1		4
84.83	odd	2		1568.2.i.o.961.1		4
168.5	even	6		3136.2.a.bx.1.2		2
168.11	even	6		448.2.i.g.193.1		4
168.53	odd	6		448.2.i.j.193.2		4
168.107	even	6		3136.2.a.bw.1.2		2
168.131	odd	6		3136.2.a.be.1.1		2
168.149	odd	6		3136.2.a.bd.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.2.i.a.65.1	✓	4	3.2	odd	2
224.2.i.a.193.1	yes	4	21.11	odd	6
224.2.i.d.65.2	yes	4	12.11	even	2
224.2.i.d.193.2	yes	4	84.11	even	6
448.2.i.g.65.1		4	24.11	even	2
448.2.i.g.193.1		4	168.11	even	6
448.2.i.j.65.2		4	24.5	odd	2
448.2.i.j.193.2		4	168.53	odd	6
1568.2.a.j.1.1		2	21.5	even	6
1568.2.a.l.1.1		2	84.23	even	6
1568.2.a.u.1.2		2	84.47	odd	6
1568.2.a.w.1.2		2	21.2	odd	6
1568.2.i.o.961.1		4	84.83	odd	2
1568.2.i.o.1537.1		4	84.59	odd	6
1568.2.i.x.961.2		4	21.20	even	2
1568.2.i.x.1537.2		4	21.17	even	6
2016.2.s.q.289.2		4	1.1	even	1	trivial
2016.2.s.q.865.2		4	7.4	even	3	inner
2016.2.s.s.289.2		4	4.3	odd	2
2016.2.s.s.865.2		4	28.11	odd	6
3136.2.a.bd.1.1		2	168.149	odd	6
3136.2.a.be.1.1		2	168.131	odd	6
3136.2.a.bw.1.2		2	168.107	even	6
3136.2.a.bx.1.2		2	168.5	even	6

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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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Learn more

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	2016.2.s.q.289.2

Level	$2016$
Weight	$2$
Character	2016.289
Analytic conductor	$16.098$
Analytic rank	$0$
Dimension	$4$
Inner twists	$2$