LMFDB - Embedded newform 960.2.d.e.289.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [960,2,Mod(289,960)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("960.289"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(960, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 960 = 2^{6} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	960.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,-8,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$7.66563859404$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.49787136.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 $ x^8 + 3x^6 + 5x^4 + 12*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			289.8
Root			$0.228425 + 1.39564i$ of defining polynomial
Character	$\chi$	$=$	960.289
Dual form			960.2.d.e.289.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +(2.18890 + 0.456850i) q^{5} +0.913701i q^{7} +1.00000 q^{9} +3.58258i q^{11} +0.913701 q^{13} +(-2.18890 - 0.456850i) q^{15} -3.58258i q^{17} +4.00000i q^{19} -0.913701i q^{21} +(4.58258 + 2.00000i) q^{25} -1.00000 q^{27} +7.84190i q^{29} -5.29150 q^{31} -3.58258i q^{33} +(-0.417424 + 2.00000i) q^{35} -7.84190 q^{37} -0.913701 q^{39} +6.00000 q^{41} +7.16515 q^{43} +(2.18890 + 0.456850i) q^{45} -6.92820i q^{47} +6.16515 q^{49} +3.58258i q^{51} +2.55040 q^{53} +(-1.63670 + 7.84190i) q^{55} -4.00000i q^{57} +7.58258i q^{59} +10.5830i q^{61} +0.913701i q^{63} +(2.00000 + 0.417424i) q^{65} +15.1652 q^{67} -6.92820 q^{71} +12.0000i q^{73} +(-4.58258 - 2.00000i) q^{75} -3.27340 q^{77} +5.29150 q^{79} +1.00000 q^{81} -11.1652 q^{83} +(1.63670 - 7.84190i) q^{85} -7.84190i q^{87} -2.00000 q^{89} +0.834849i q^{91} +5.29150 q^{93} +(-1.82740 + 8.75560i) q^{95} +7.16515i q^{97} +3.58258i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{3} + 8 q^{9} - 8 q^{27} - 40 q^{35} + 48 q^{41} - 16 q^{43} - 24 q^{49} + 16 q^{65} + 48 q^{67} + 8 q^{81} - 16 q^{83} - 16 q^{89}+O(q^{100}) $ 8 * q - 8 * q^3 + 8 * q^9 - 8 * q^27 - 40 * q^35 + 48 * q^41 - 16 * q^43 - 24 * q^49 + 16 * q^65 + 48 * q^67 + 8 * q^81 - 16 * q^83 - 16 * q^89

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/960\mathbb{Z}\right)^\times$.

$n$	$511$	$577$	$641$	$901$
$\chi(n)$	$1$	$-1$	$1$	$-1$

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
$3$	−1.00000			−0.577350
$3$	−1.00000			−0.577350
$4$		0			0
$5$	2.18890	+	0.456850i	0.978906	+	0.204310i
$5$	2.18890	+	0.456850i	0.978906	+	0.204310i
$6$		0			0
$7$			0.913701i			0.345346i	0.984979	+	0.172673i	$0.0552404\pi$
$7$			0.913701i			0.345346i	−0.984979	+	0.172673i	$0.944760\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$			3.58258i			1.08019i	0.841605	+	0.540094i	$0.181611\pi$
$11$			3.58258i			1.08019i	−0.841605	+	0.540094i	$0.818389\pi$
$12$		0			0
$13$	0.913701			0.253415			0.126707	−	0.991940i	$-0.459559\pi$
$13$	0.913701			0.253415			0.126707	+	0.991940i	$0.459559\pi$
$14$		0			0
$15$	−2.18890	−	0.456850i	−0.565172	−	0.117958i
$16$		0			0
$17$		−	3.58258i		−	0.868902i	−0.900695	−	0.434451i	$-0.856942\pi$
$17$		−	3.58258i		−	0.868902i	0.900695	−	0.434451i	$-0.143058\pi$
$18$		0			0
$19$			4.00000i			0.917663i	0.888523	+	0.458831i	$0.151732\pi$
$19$			4.00000i			0.917663i	−0.888523	+	0.458831i	$0.848268\pi$
$20$		0			0
$21$		−	0.913701i		−	0.199386i
$22$		0			0
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		0			0
$25$	4.58258	+	2.00000i	0.916515	+	0.400000i
$26$		0			0
$27$	−1.00000			−0.192450
$28$		0			0
$29$			7.84190i			1.45620i	0.685468	+	0.728102i	$0.259598\pi$
$29$			7.84190i			1.45620i	−0.685468	+	0.728102i	$0.740402\pi$
$30$		0			0
$31$	−5.29150			−0.950382			−0.475191	−	0.879883i	$-0.657621\pi$
$31$	−5.29150			−0.950382			−0.475191	+	0.879883i	$0.657621\pi$
$32$		0			0
$33$		−	3.58258i		−	0.623646i
$34$		0			0
$35$	−0.417424	+	2.00000i	−0.0705576	+	0.338062i
$36$		0			0
$37$	−7.84190			−1.28920			−0.644601	−	0.764520i	$-0.722976\pi$
$37$	−7.84190			−1.28920			−0.644601	+	0.764520i	$0.722976\pi$
$38$		0			0
$39$	−0.913701			−0.146309
$40$		0			0
$41$	6.00000			0.937043			0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000			0.937043			0.468521	+	0.883452i	$0.344787\pi$
$42$		0			0
$43$	7.16515			1.09268			0.546338	−	0.837565i	$-0.316022\pi$
$43$	7.16515			1.09268			0.546338	+	0.837565i	$0.316022\pi$
$44$		0			0
$45$	2.18890	+	0.456850i	0.326302	+	0.0681032i
$46$		0			0
$47$		−	6.92820i		−	1.01058i	−0.862949	−	0.505291i	$-0.831385\pi$
$47$		−	6.92820i		−	1.01058i	0.862949	−	0.505291i	$-0.168615\pi$
$48$		0			0
$49$	6.16515			0.880736
$50$		0			0
$51$			3.58258i			0.501661i
$52$		0			0
$53$	2.55040			0.350325			0.175162	−	0.984540i	$-0.443955\pi$
$53$	2.55040			0.350325			0.175162	+	0.984540i	$0.443955\pi$
$54$		0			0
$55$	−1.63670	+	7.84190i	−0.220693	+	1.05740i
$56$		0			0
$57$		−	4.00000i		−	0.529813i
$58$		0			0
$59$			7.58258i			0.987167i	0.869698	+	0.493584i	$0.164313\pi$
$59$			7.58258i			0.987167i	−0.869698	+	0.493584i	$0.835687\pi$
$60$		0			0
$61$			10.5830i			1.35501i	0.735516	+	0.677507i	$0.236940\pi$
$61$			10.5830i			1.35501i	−0.735516	+	0.677507i	$0.763060\pi$
$62$		0			0
$63$			0.913701i			0.115115i
$64$		0			0
$65$	2.00000	+	0.417424i	0.248069	+	0.0517751i
$66$		0			0
$67$	15.1652			1.85272			0.926359	−	0.376642i	$-0.122921\pi$
$67$	15.1652			1.85272			0.926359	+	0.376642i	$0.122921\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−6.92820			−0.822226			−0.411113	−	0.911584i	$-0.634860\pi$
$71$	−6.92820			−0.822226			−0.411113	+	0.911584i	$0.634860\pi$
$72$		0			0
$73$			12.0000i			1.40449i	0.711934	+	0.702247i	$0.247820\pi$
$73$			12.0000i			1.40449i	−0.711934	+	0.702247i	$0.752180\pi$
$74$		0			0
$75$	−4.58258	−	2.00000i	−0.529150	−	0.230940i
$76$		0			0
$77$	−3.27340			−0.373039
$78$		0			0
$79$	5.29150			0.595341			0.297670	−	0.954669i	$-0.403790\pi$
$79$	5.29150			0.595341			0.297670	+	0.954669i	$0.403790\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	−11.1652			−1.22553			−0.612767	−	0.790263i	$-0.709944\pi$
$83$	−11.1652			−1.22553			−0.612767	+	0.790263i	$0.709944\pi$
$84$		0			0
$85$	1.63670	−	7.84190i	0.177525	−	0.850574i
$86$		0			0
$87$		−	7.84190i		−	0.840740i
$88$		0			0
$89$	−2.00000			−0.212000			−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000			−0.212000			−0.106000	+	0.994366i	$0.533804\pi$
$90$		0			0
$91$			0.834849i			0.0875159i
$92$		0			0
$93$	5.29150			0.548703
$94$		0			0
$95$	−1.82740	+	8.75560i	−0.187487	+	0.898306i
$96$		0			0
$97$			7.16515i			0.727511i	0.931494	+	0.363755i	$0.118506\pi$
$97$			7.16515i			0.727511i	−0.931494	+	0.363755i	$0.881494\pi$
$98$		0			0
$99$			3.58258i			0.360062i
$100$		0			0
$101$		−	16.5975i		−	1.65151i	−0.564026	−	0.825757i	$-0.690748\pi$
$101$		−	16.5975i		−	1.65151i	0.564026	−	0.825757i	$-0.309252\pi$
$102$		0			0
$103$			16.5975i			1.63540i	0.575644	+	0.817701i	$0.304751\pi$
$103$			16.5975i			1.63540i	−0.575644	+	0.817701i	$0.695249\pi$
$104$		0			0
$105$	0.417424	−	2.00000i	0.0407364	−	0.195180i
$106$		0			0
$107$	4.00000			0.386695			0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000			0.386695			0.193347	+	0.981130i	$0.438066\pi$
$108$		0			0
$109$			8.75560i			0.838635i	0.907840	+	0.419317i	$0.137731\pi$
$109$			8.75560i			0.838635i	−0.907840	+	0.419317i	$0.862269\pi$
$110$		0			0
$111$	7.84190			0.744321
$112$		0			0
$113$		−	4.41742i		−	0.415556i	−0.978176	−	0.207778i	$-0.933377\pi$
$113$		−	4.41742i		−	0.415556i	0.978176	−	0.207778i	$-0.0666232\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	0.913701			0.0844716
$118$		0			0
$119$	3.27340			0.300072
$120$		0			0
$121$	−1.83485			−0.166804
$122$		0			0
$123$	−6.00000			−0.541002
$124$		0			0
$125$	9.11710	+	6.47135i	0.815459	+	0.578815i
$126$		0			0
$127$		−	14.7701i		−	1.31064i	−0.755354	−	0.655318i	$-0.772535\pi$
$127$		−	14.7701i		−	1.31064i	0.755354	−	0.655318i	$-0.227465\pi$
$128$		0			0
$129$	−7.16515			−0.630856
$130$		0			0
$131$		−	15.5826i		−	1.36146i	−0.732536	−	0.680728i	$-0.761664\pi$
$131$		−	15.5826i		−	1.36146i	0.732536	−	0.680728i	$-0.238336\pi$
$132$		0			0
$133$	−3.65480			−0.316912
$134$		0			0
$135$	−2.18890	−	0.456850i	−0.188391	−	0.0393194i
$136$		0			0
$137$		−	19.5826i		−	1.67305i	−0.547927	−	0.836526i	$-0.684583\pi$
$137$		−	19.5826i		−	1.67305i	0.547927	−	0.836526i	$-0.315417\pi$
$138$		0			0
$139$		−	11.1652i		−	0.947016i	−0.880790	−	0.473508i	$-0.842988\pi$
$139$		−	11.1652i		−	0.947016i	0.880790	−	0.473508i	$-0.157012\pi$
$140$		0			0
$141$			6.92820i			0.583460i
$142$		0			0
$143$			3.27340i			0.273736i
$144$		0			0
$145$	−3.58258	+	17.1652i	−0.297517	+	1.42549i
$146$		0			0
$147$	−6.16515			−0.508493
$148$		0			0
$149$		−	11.4967i		−	0.941847i	−0.882174	−	0.470923i	$-0.843921\pi$
$149$		−	11.4967i		−	0.941847i	0.882174	−	0.470923i	$-0.156079\pi$
$150$		0			0
$151$	15.8745			1.29185			0.645925	−	0.763401i	$-0.276472\pi$
$151$	15.8745			1.29185			0.645925	+	0.763401i	$0.276472\pi$
$152$		0			0
$153$		−	3.58258i		−	0.289634i
$154$		0			0
$155$	−11.5826	−	2.41742i	−0.930335	−	0.194172i
$156$		0			0
$157$	−21.6983			−1.73171			−0.865857	−	0.500292i	$-0.833226\pi$
$157$	−21.6983			−1.73171			−0.865857	+	0.500292i	$0.833226\pi$
$158$		0			0
$159$	−2.55040			−0.202260
$160$		0			0
$161$		0			0
$162$		0			0
$163$	4.00000			0.313304			0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000			0.313304			0.156652	+	0.987654i	$0.449930\pi$
$164$		0			0
$165$	1.63670	−	7.84190i	0.127417	−	0.610491i
$166$		0			0
$167$		−	22.6120i		−	1.74977i	−0.484331	−	0.874885i	$-0.660937\pi$
$167$		−	22.6120i		−	1.74977i	0.484331	−	0.874885i	$-0.339063\pi$
$168$		0			0
$169$	−12.1652			−0.935781
$170$		0			0
$171$			4.00000i			0.305888i
$172$		0			0
$173$	11.3060			0.859580			0.429790	−	0.902929i	$-0.358588\pi$
$173$	11.3060			0.859580			0.429790	+	0.902929i	$0.358588\pi$
$174$		0			0
$175$	−1.82740	+	4.18710i	−0.138139	+	0.316515i
$176$		0			0
$177$		−	7.58258i		−	0.569941i
$178$		0			0
$179$		−	7.58258i		−	0.566748i	−0.959009	−	0.283374i	$-0.908546\pi$
$179$		−	7.58258i		−	0.566748i	0.959009	−	0.283374i	$-0.0914538\pi$
$180$		0			0
$181$		−	8.75560i		−	0.650799i	−0.945577	−	0.325399i	$-0.894501\pi$
$181$		−	8.75560i		−	0.650799i	0.945577	−	0.325399i	$-0.105499\pi$
$182$		0			0
$183$		−	10.5830i		−	0.782318i
$184$		0			0
$185$	−17.1652	−	3.58258i	−1.26201	−	0.263396i
$186$		0			0
$187$	12.8348			0.938577
$188$		0			0
$189$		−	0.913701i		−	0.0664619i
$190$		0			0
$191$	−13.8564			−1.00261			−0.501307	−	0.865269i	$-0.667147\pi$
$191$	−13.8564			−1.00261			−0.501307	+	0.865269i	$0.667147\pi$
$192$		0			0
$193$			3.16515i			0.227833i	0.993490	+	0.113916i	$0.0363396\pi$
$193$			3.16515i			0.227833i	−0.993490	+	0.113916i	$0.963660\pi$
$194$		0			0
$195$	−2.00000	−	0.417424i	−0.143223	−	0.0298924i
$196$		0			0
$197$	−25.5438			−1.81992			−0.909961	−	0.414695i	$-0.863888\pi$
$197$	−25.5438			−1.81992			−0.909961	+	0.414695i	$0.863888\pi$
$198$		0			0
$199$	−1.63670			−0.116023			−0.0580113	−	0.998316i	$-0.518476\pi$
$199$	−1.63670			−0.116023			−0.0580113	+	0.998316i	$0.518476\pi$
$200$		0			0
$201$	−15.1652			−1.06967
$202$		0			0
$203$	−7.16515			−0.502895
$204$		0			0
$205$	13.1334	+	2.74110i	0.917277	+	0.191447i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−14.3303			−0.991248
$210$		0			0
$211$		−	11.1652i		−	0.768641i	−0.923200	−	0.384320i	$-0.874436\pi$
$211$		−	11.1652i		−	0.768641i	0.923200	−	0.384320i	$-0.125564\pi$
$212$		0			0
$213$	6.92820			0.474713
$214$		0			0
$215$	15.6838	+	3.27340i	1.06963	+	0.223244i
$216$		0			0
$217$		−	4.83485i		−	0.328211i
$218$		0			0
$219$		−	12.0000i		−	0.810885i
$220$		0			0
$221$		−	3.27340i		−	0.220193i
$222$		0			0
$223$			0.913701i			0.0611859i	0.999532	+	0.0305930i	$0.00973956\pi$
$223$			0.913701i			0.0611859i	−0.999532	+	0.0305930i	$0.990260\pi$
$224$		0			0
$225$	4.58258	+	2.00000i	0.305505	+	0.133333i
$226$		0			0
$227$	−3.16515			−0.210078			−0.105039	−	0.994468i	$-0.533497\pi$
$227$	−3.16515			−0.210078			−0.105039	+	0.994468i	$0.533497\pi$
$228$		0			0
$229$		−	22.6120i		−	1.49424i	−0.664687	−	0.747122i	$-0.731435\pi$
$229$		−	22.6120i		−	1.49424i	0.664687	−	0.747122i	$-0.268565\pi$
$230$		0			0
$231$	3.27340			0.215374
$232$		0			0
$233$			19.5826i			1.28290i	0.767166	+	0.641449i	$0.221666\pi$
$233$			19.5826i			1.28290i	−0.767166	+	0.641449i	$0.778334\pi$
$234$		0			0
$235$	3.16515	−	15.1652i	0.206472	−	0.989265i
$236$		0			0
$237$	−5.29150			−0.343720
$238$		0			0
$239$	21.1660			1.36912			0.684558	−	0.728959i	$-0.259995\pi$
$239$	21.1660			1.36912			0.684558	+	0.728959i	$0.259995\pi$
$240$		0			0
$241$	−10.0000			−0.644157			−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000			−0.644157			−0.322078	+	0.946713i	$0.604381\pi$
$242$		0			0
$243$	−1.00000			−0.0641500
$244$		0			0
$245$	13.4949	+	2.81655i	0.862158	+	0.179943i
$246$		0			0
$247$			3.65480i			0.232549i
$248$		0			0
$249$	11.1652			0.707563
$250$		0			0
$251$		−	11.5826i		−	0.731086i	−0.930795	−	0.365543i	$-0.880883\pi$
$251$		−	11.5826i		−	0.731086i	0.930795	−	0.365543i	$-0.119117\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	−1.63670	+	7.84190i	−0.102494	+	0.491079i
$256$		0			0
$257$			2.74773i			0.171399i	0.996321	+	0.0856993i	$0.0273124\pi$
$257$			2.74773i			0.171399i	−0.996321	+	0.0856993i	$0.972688\pi$
$258$		0			0
$259$		−	7.16515i		−	0.445221i
$260$		0			0
$261$			7.84190i			0.485402i
$262$		0			0
$263$		−	8.75560i		−	0.539894i	−0.962875	−	0.269947i	$-0.912994\pi$
$263$		−	8.75560i		−	0.539894i	0.962875	−	0.269947i	$-0.0870061\pi$
$264$		0			0
$265$	5.58258	+	1.16515i	0.342935	+	0.0715747i
$266$		0			0
$267$	2.00000			0.122398
$268$		0			0
$269$			9.66930i			0.589548i	0.955567	+	0.294774i	$0.0952444\pi$
$269$			9.66930i			0.589548i	−0.955567	+	0.294774i	$0.904756\pi$
$270$		0			0
$271$	26.0761			1.58401			0.792006	−	0.610514i	$-0.209037\pi$
$271$	26.0761			1.58401			0.792006	+	0.610514i	$0.209037\pi$
$272$		0			0
$273$		−	0.834849i		−	0.0505273i
$274$		0			0
$275$	−7.16515	+	16.4174i	−0.432075	+	0.990008i
$276$		0			0
$277$	−14.7701			−0.887450			−0.443725	−	0.896163i	$-0.646343\pi$
$277$	−14.7701			−0.887450			−0.443725	+	0.896163i	$0.646343\pi$
$278$		0			0
$279$	−5.29150			−0.316794
$280$		0			0
$281$	−4.33030			−0.258324			−0.129162	−	0.991623i	$-0.541229\pi$
$281$	−4.33030			−0.258324			−0.129162	+	0.991623i	$0.541229\pi$
$282$		0			0
$283$	−7.16515			−0.425924			−0.212962	−	0.977060i	$-0.568311\pi$
$283$	−7.16515			−0.425924			−0.212962	+	0.977060i	$0.568311\pi$
$284$		0			0
$285$	1.82740	−	8.75560i	0.108246	−	0.518637i
$286$		0			0
$287$			5.48220i			0.323604i
$288$		0			0
$289$	4.16515			0.245009
$290$		0			0
$291$		−	7.16515i		−	0.420029i
$292$		0			0
$293$	4.37780			0.255754			0.127877	−	0.991790i	$-0.459184\pi$
$293$	4.37780			0.255754			0.127877	+	0.991790i	$0.459184\pi$
$294$		0			0
$295$	−3.46410	+	16.5975i	−0.201688	+	0.966344i
$296$		0			0
$297$		−	3.58258i		−	0.207882i
$298$		0			0
$299$		0			0
$300$		0			0
$301$			6.54680i			0.377351i
$302$		0			0
$303$			16.5975i			0.953502i
$304$		0			0
$305$	−4.83485	+	23.1652i	−0.276843	+	1.32643i
$306$		0			0
$307$	4.00000			0.228292			0.114146	−	0.993464i	$-0.463587\pi$
$307$	4.00000			0.228292			0.114146	+	0.993464i	$0.463587\pi$
$308$		0			0
$309$		−	16.5975i		−	0.944199i
$310$		0			0
$311$	−17.5112			−0.992970			−0.496485	−	0.868045i	$-0.665376\pi$
$311$	−17.5112			−0.992970			−0.496485	+	0.868045i	$0.665376\pi$
$312$		0			0
$313$			19.1652i			1.08328i	0.840611	+	0.541639i	$0.182196\pi$
$313$			19.1652i			1.08328i	−0.840611	+	0.541639i	$0.817804\pi$
$314$		0			0
$315$	−0.417424	+	2.00000i	−0.0235192	+	0.112687i
$316$		0			0
$317$	2.93180			0.164666			0.0823332	−	0.996605i	$-0.473763\pi$
$317$	2.93180			0.164666			0.0823332	+	0.996605i	$0.473763\pi$
$318$		0			0
$319$	−28.0942			−1.57297
$320$		0			0
$321$	−4.00000			−0.223258
$322$		0			0
$323$	14.3303			0.797359
$324$		0			0
$325$	4.18710	+	1.82740i	0.232259	+	0.101366i
$326$		0			0
$327$		−	8.75560i		−	0.484186i
$328$		0			0
$329$	6.33030			0.349001
$330$		0			0
$331$			26.3303i			1.44724i	0.690196	+	0.723622i	$0.257524\pi$
$331$			26.3303i			1.44724i	−0.690196	+	0.723622i	$0.742476\pi$
$332$		0			0
$333$	−7.84190			−0.429734
$334$		0			0
$335$	33.1950	+	6.92820i	1.81364	+	0.378528i
$336$		0			0
$337$		−	10.3303i		−	0.562727i	−0.959601	−	0.281364i	$-0.909213\pi$
$337$		−	10.3303i		−	0.562727i	0.959601	−	0.281364i	$-0.0907867\pi$
$338$		0			0
$339$			4.41742i			0.239922i
$340$		0			0
$341$		−	18.9572i		−	1.02659i
$342$		0			0
$343$			12.0290i			0.649505i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	19.1652			1.02884			0.514420	−	0.857539i	$-0.328007\pi$
$347$	19.1652			1.02884			0.514420	+	0.857539i	$0.328007\pi$
$348$		0			0
$349$		−	3.27340i		−	0.175221i	−0.996155	−	0.0876106i	$-0.972077\pi$
$349$		−	3.27340i		−	0.175221i	0.996155	−	0.0876106i	$-0.0279231\pi$
$350$		0			0
$351$	−0.913701			−0.0487697
$352$		0			0
$353$		−	33.9129i		−	1.80500i	−0.430689	−	0.902500i	$-0.641730\pi$
$353$		−	33.9129i		−	1.80500i	0.430689	−	0.902500i	$-0.358270\pi$
$354$		0			0
$355$	−15.1652	−	3.16515i	−0.804883	−	0.167989i
$356$		0			0
$357$	−3.27340			−0.173247
$358$		0			0
$359$	−6.92820			−0.365657			−0.182828	−	0.983145i	$-0.558525\pi$
$359$	−6.92820			−0.365657			−0.182828	+	0.983145i	$0.558525\pi$
$360$		0			0
$361$	3.00000			0.157895
$362$		0			0
$363$	1.83485			0.0963046
$364$		0			0
$365$	−5.48220	+	26.2668i	−0.286952	+	1.37487i
$366$		0			0
$367$		−	35.9361i		−	1.87585i	−0.346838	−	0.937925i	$-0.612745\pi$
$367$		−	35.9361i		−	1.87585i	0.346838	−	0.937925i	$-0.387255\pi$
$368$		0			0
$369$	6.00000			0.312348
$370$		0			0
$371$			2.33030i			0.120983i
$372$		0			0
$373$	6.01450			0.311419			0.155710	−	0.987803i	$-0.450234\pi$
$373$	6.01450			0.311419			0.155710	+	0.987803i	$0.450234\pi$
$374$		0			0
$375$	−9.11710	−	6.47135i	−0.470805	−	0.334179i
$376$		0			0
$377$			7.16515i			0.369024i
$378$		0			0
$379$		−	4.83485i		−	0.248349i	−0.992260	−	0.124175i	$-0.960372\pi$
$379$		−	4.83485i		−	0.248349i	0.992260	−	0.124175i	$-0.0396283\pi$
$380$		0			0
$381$			14.7701i			0.756696i
$382$		0			0
$383$			22.9934i			1.17491i	0.809257	+	0.587454i	$0.199870\pi$
$383$			22.9934i			1.17491i	−0.809257	+	0.587454i	$0.800130\pi$
$384$		0			0
$385$	−7.16515	−	1.49545i	−0.365170	−	0.0762154i
$386$		0			0
$387$	7.16515			0.364225
$388$		0			0
$389$			18.4249i			0.934180i	0.884210	+	0.467090i	$0.154698\pi$
$389$			18.4249i			0.934180i	−0.884210	+	0.467090i	$0.845302\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$			15.5826i			0.786037i
$394$		0			0
$395$	11.5826	+	2.41742i	0.582783	+	0.121634i
$396$		0			0
$397$	7.84190			0.393574			0.196787	−	0.980446i	$-0.436949\pi$
$397$	7.84190			0.393574			0.196787	+	0.980446i	$0.436949\pi$
$398$		0			0
$399$	3.65480			0.182969
$400$		0			0
$401$	22.0000			1.09863			0.549314	−	0.835616i	$-0.314889\pi$
$401$	22.0000			1.09863			0.549314	+	0.835616i	$0.314889\pi$
$402$		0			0
$403$	−4.83485			−0.240841
$404$		0			0
$405$	2.18890	+	0.456850i	0.108767	+	0.0227011i
$406$		0			0
$407$		−	28.0942i		−	1.39258i
$408$		0			0
$409$	18.8348			0.931323			0.465662	−	0.884963i	$-0.345816\pi$
$409$	18.8348			0.931323			0.465662	+	0.884963i	$0.345816\pi$
$410$		0			0
$411$			19.5826i			0.965937i
$412$		0			0
$413$	−6.92820			−0.340915
$414$		0			0
$415$	−24.4394	−	5.10080i	−1.19968	−	0.250389i
$416$		0			0
$417$			11.1652i			0.546760i
$418$		0			0
$419$			10.7477i			0.525061i	0.964924	+	0.262530i	$0.0845570\pi$
$419$			10.7477i			0.525061i	−0.964924	+	0.262530i	$0.915443\pi$
$420$		0			0
$421$			18.9572i			0.923918i	0.886901	+	0.461959i	$0.152853\pi$
$421$			18.9572i			0.923918i	−0.886901	+	0.461959i	$0.847147\pi$
$422$		0			0
$423$		−	6.92820i		−	0.336861i
$424$		0			0
$425$	7.16515	−	16.4174i	0.347561	−	0.796362i
$426$		0			0
$427$	−9.66970			−0.467949
$428$		0			0
$429$		−	3.27340i		−	0.158041i
$430$		0			0
$431$	−17.1298			−0.825114			−0.412557	−	0.910932i	$-0.635364\pi$
$431$	−17.1298			−0.825114			−0.412557	+	0.910932i	$0.635364\pi$
$432$		0			0
$433$		−	38.3303i		−	1.84204i	−0.389519	−	0.921018i	$-0.627359\pi$
$433$		−	38.3303i		−	1.84204i	0.389519	−	0.921018i	$-0.372641\pi$
$434$		0			0
$435$	3.58258	−	17.1652i	0.171771	−	0.823006i
$436$		0			0
$437$		0			0
$438$		0			0
$439$	2.01810			0.0963187			0.0481594	−	0.998840i	$-0.484664\pi$
$439$	2.01810			0.0963187			0.0481594	+	0.998840i	$0.484664\pi$
$440$		0			0
$441$	6.16515			0.293579
$442$		0			0
$443$	−12.8348			−0.609802			−0.304901	−	0.952384i	$-0.598623\pi$
$443$	−12.8348			−0.609802			−0.304901	+	0.952384i	$0.598623\pi$
$444$		0			0
$445$	−4.37780	−	0.913701i	−0.207528	−	0.0433136i
$446$		0			0
$447$			11.4967i			0.543776i
$448$		0			0
$449$	8.33030			0.393131			0.196566	−	0.980491i	$-0.437021\pi$
$449$	8.33030			0.393131			0.196566	+	0.980491i	$0.437021\pi$
$450$		0			0
$451$			21.4955i			1.01218i
$452$		0			0
$453$	−15.8745			−0.745849
$454$		0			0
$455$	−0.381401	+	1.82740i	−0.0178803	+	0.0856699i
$456$		0			0
$457$		−	23.1652i		−	1.08362i	−0.840501	−	0.541810i	$-0.817739\pi$
$457$		−	23.1652i		−	1.08362i	0.840501	−	0.541810i	$-0.182261\pi$
$458$		0			0
$459$			3.58258i			0.167220i
$460$		0			0
$461$		−	25.3531i		−	1.18081i	−0.807106	−	0.590406i	$-0.798968\pi$
$461$		−	25.3531i		−	1.18081i	0.807106	−	0.590406i	$-0.201032\pi$
$462$		0			0
$463$		−	39.5909i		−	1.83995i	−0.391982	−	0.919973i	$-0.628210\pi$
$463$		−	39.5909i		−	1.83995i	0.391982	−	0.919973i	$-0.371790\pi$
$464$		0			0
$465$	11.5826	+	2.41742i	0.537129	+	0.112105i
$466$		0			0
$467$	19.1652			0.886857			0.443429	−	0.896310i	$-0.353762\pi$
$467$	19.1652			0.886857			0.443429	+	0.896310i	$0.353762\pi$
$468$		0			0
$469$			13.8564i			0.639829i
$470$		0			0
$471$	21.6983			0.999805
$472$		0			0
$473$			25.6697i			1.18029i
$474$		0			0
$475$	−8.00000	+	18.3303i	−0.367065	+	0.841052i
$476$		0			0
$477$	2.55040			0.116775
$478$		0			0
$479$	−31.7490			−1.45065			−0.725325	−	0.688407i	$-0.758310\pi$
$479$	−31.7490			−1.45065			−0.725325	+	0.688407i	$0.758310\pi$
$480$		0			0
$481$	−7.16515			−0.326703
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−3.27340	+	15.6838i	−0.148637	+	0.712165i
$486$		0			0
$487$			20.2523i			0.917720i	0.888509	+	0.458860i	$0.151742\pi$
$487$			20.2523i			0.917720i	−0.888509	+	0.458860i	$0.848258\pi$
$488$		0			0
$489$	−4.00000			−0.180886
$490$		0			0
$491$		−	14.7477i		−	0.665556i	−0.943005	−	0.332778i	$-0.892014\pi$
$491$		−	14.7477i		−	0.665556i	0.943005	−	0.332778i	$-0.107986\pi$
$492$		0			0
$493$	28.0942			1.26530
$494$		0			0
$495$	−1.63670	+	7.84190i	−0.0735642	+	0.352467i
$496$		0			0
$497$		−	6.33030i		−	0.283953i
$498$		0			0
$499$			18.3303i			0.820577i	0.911956	+	0.410289i	$0.134572\pi$
$499$			18.3303i			0.820577i	−0.911956	+	0.410289i	$0.865428\pi$
$500$		0			0
$501$			22.6120i			1.01023i
$502$		0			0
$503$			14.2378i			0.634832i	0.948286	+	0.317416i	$0.102815\pi$
$503$			14.2378i			0.634832i	−0.948286	+	0.317416i	$0.897185\pi$
$504$		0			0
$505$	7.58258	−	36.3303i	0.337420	−	1.61668i
$506$		0			0
$507$	12.1652			0.540273
$508$		0			0
$509$		−	8.22330i		−	0.364492i	−0.983253	−	0.182246i	$-0.941663\pi$
$509$		−	8.22330i		−	0.364492i	0.983253	−	0.182246i	$-0.0583367\pi$
$510$		0			0
$511$	−10.9644			−0.485037
$512$		0			0
$513$		−	4.00000i		−	0.176604i
$514$		0			0
$515$	−7.58258	+	36.3303i	−0.334128	+	1.60090i
$516$		0			0
$517$	24.8208			1.09162
$518$		0			0
$519$	−11.3060			−0.496279
$520$		0			0
$521$	38.6606			1.69375			0.846876	−	0.531791i	$-0.178481\pi$
$521$	38.6606			1.69375			0.846876	+	0.531791i	$0.178481\pi$
$522$		0			0
$523$	−18.3303			−0.801528			−0.400764	−	0.916181i	$-0.631255\pi$
$523$	−18.3303			−0.801528			−0.400764	+	0.916181i	$0.631255\pi$
$524$		0			0
$525$	1.82740	−	4.18710i	0.0797543	−	0.182740i
$526$		0			0
$527$			18.9572i			0.825789i
$528$		0			0
$529$	23.0000			1.00000
$530$		0			0
$531$			7.58258i			0.329056i
$532$		0			0
$533$	5.48220			0.237461
$534$		0			0
$535$	8.75560	+	1.82740i	0.378538	+	0.0790054i
$536$		0			0
$537$			7.58258i			0.327212i
$538$		0			0
$539$			22.0871i			0.951360i
$540$		0			0
$541$			16.0652i			0.690697i	0.938474	+	0.345349i	$0.112239\pi$
$541$			16.0652i			0.690697i	−0.938474	+	0.345349i	$0.887761\pi$
$542$		0			0
$543$			8.75560i			0.375739i
$544$		0			0
$545$	−4.00000	+	19.1652i	−0.171341	+	0.820945i
$546$		0			0
$547$	−29.4955			−1.26113			−0.630567	−	0.776135i	$-0.717178\pi$
$547$	−29.4955			−1.26113			−0.630567	+	0.776135i	$0.717178\pi$
$548$		0			0
$549$			10.5830i			0.451672i
$550$		0			0
$551$	−31.3676			−1.33631
$552$		0			0
$553$			4.83485i			0.205599i
$554$		0			0
$555$	17.1652	+	3.58258i	0.728620	+	0.152072i
$556$		0			0
$557$	−20.0616			−0.850038			−0.425019	−	0.905185i	$-0.639733\pi$
$557$	−20.0616			−0.850038			−0.425019	+	0.905185i	$0.639733\pi$
$558$		0			0
$559$	6.54680			0.276900
$560$		0			0
$561$	−12.8348			−0.541888
$562$		0			0
$563$	25.4955			1.07451			0.537253	−	0.843421i	$-0.319462\pi$
$563$	25.4955			1.07451			0.537253	+	0.843421i	$0.319462\pi$
$564$		0			0
$565$	2.01810	−	9.66930i	0.0849022	−	0.406791i
$566$		0			0
$567$			0.913701i			0.0383718i
$568$		0			0
$569$	−0.330303			−0.0138470			−0.00692351	−	0.999976i	$-0.502204\pi$
$569$	−0.330303			−0.0138470			−0.00692351	+	0.999976i	$0.502204\pi$
$570$		0			0
$571$			20.0000i			0.836974i	0.908223	+	0.418487i	$0.137439\pi$
$571$			20.0000i			0.836974i	−0.908223	+	0.418487i	$0.862561\pi$
$572$		0			0
$573$	13.8564			0.578860
$574$		0			0
$575$		0			0
$576$		0			0
$577$		−	3.16515i		−	0.131767i	−0.997827	−	0.0658835i	$-0.979013\pi$
$577$		−	3.16515i		−	0.131767i	0.997827	−	0.0658835i	$-0.0209866\pi$
$578$		0			0
$579$		−	3.16515i		−	0.131539i
$580$		0			0
$581$		−	10.2016i		−	0.423234i
$582$		0			0
$583$			9.13701i			0.378416i
$584$		0			0
$585$	2.00000	+	0.417424i	0.0826898	+	0.0172584i
$586$		0			0
$587$	−42.3303			−1.74716			−0.873579	−	0.486682i	$-0.838207\pi$
$587$	−42.3303			−1.74716			−0.873579	+	0.486682i	$0.838207\pi$
$588$		0			0
$589$		−	21.1660i		−	0.872130i
$590$		0			0
$591$	25.5438			1.05073
$592$		0			0
$593$			9.91288i			0.407073i	0.979067	+	0.203537i	$0.0652436\pi$
$593$			9.91288i			0.407073i	−0.979067	+	0.203537i	$0.934756\pi$
$594$		0			0
$595$	7.16515	+	1.49545i	0.293743	+	0.0613076i
$596$		0			0
$597$	1.63670			0.0669857
$598$		0			0
$599$	28.0942			1.14790			0.573949	−	0.818891i	$-0.305411\pi$
$599$	28.0942			1.14790			0.573949	+	0.818891i	$0.305411\pi$
$600$		0			0
$601$	−5.16515			−0.210691			−0.105345	−	0.994436i	$-0.533595\pi$
$601$	−5.16515			−0.210691			−0.105345	+	0.994436i	$0.533595\pi$
$602$		0			0
$603$	15.1652			0.617573
$604$		0			0
$605$	−4.01630	−	0.838251i	−0.163286	−	0.0340798i
$606$		0			0
$607$		−	9.28790i		−	0.376984i	−0.982075	−	0.188492i	$-0.939640\pi$
$607$		−	9.28790i		−	0.376984i	0.982075	−	0.188492i	$-0.0603600\pi$
$608$		0			0
$609$	7.16515			0.290347
$610$		0			0
$611$		−	6.33030i		−	0.256097i
$612$		0			0
$613$	15.1515			0.611964			0.305982	−	0.952037i	$-0.401015\pi$
$613$	15.1515			0.611964			0.305982	+	0.952037i	$0.401015\pi$
$614$		0			0
$615$	−13.1334	−	2.74110i	−0.529590	−	0.110532i
$616$		0			0
$617$			4.41742i			0.177839i	0.996039	+	0.0889194i	$0.0283414\pi$
$617$			4.41742i			0.177839i	−0.996039	+	0.0889194i	$0.971659\pi$
$618$		0			0
$619$			9.49545i			0.381655i	0.981624	+	0.190827i	$0.0611170\pi$
$619$			9.49545i			0.381655i	−0.981624	+	0.190827i	$0.938883\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	1.82740i		−	0.0732133i
$624$		0			0
$625$	17.0000	+	18.3303i	0.680000	+	0.733212i
$626$		0			0
$627$	14.3303			0.572297
$628$		0			0
$629$			28.0942i			1.12019i
$630$		0			0
$631$	1.63670			0.0651560			0.0325780	−	0.999469i	$-0.489628\pi$
$631$	1.63670			0.0651560			0.0325780	+	0.999469i	$0.489628\pi$
$632$		0			0
$633$			11.1652i			0.443775i
$634$		0			0
$635$	6.74773	−	32.3303i	0.267775	−	1.28299i
$636$		0			0
$637$	5.63310			0.223192
$638$		0			0
$639$	−6.92820			−0.274075
$640$		0			0
$641$	32.3303			1.27697			0.638485	−	0.769634i	$-0.279561\pi$
$641$	32.3303			1.27697			0.638485	+	0.769634i	$0.279561\pi$
$642$		0			0
$643$	−18.3303			−0.722877			−0.361438	−	0.932396i	$-0.617714\pi$
$643$	−18.3303			−0.722877			−0.361438	+	0.932396i	$0.617714\pi$
$644$		0			0
$645$	−15.6838	−	3.27340i	−0.617549	−	0.128890i
$646$		0			0
$647$			45.6054i			1.79293i	0.443110	+	0.896467i	$0.353875\pi$
$647$			45.6054i			1.79293i	−0.443110	+	0.896467i	$0.646125\pi$
$648$		0			0
$649$	−27.1652			−1.06633
$650$		0			0
$651$			4.83485i			0.189493i
$652$		0			0
$653$	−25.5438			−0.999607			−0.499803	−	0.866139i	$-0.666594\pi$
$653$	−25.5438			−0.999607			−0.499803	+	0.866139i	$0.666594\pi$
$654$		0			0
$655$	7.11890	−	34.1087i	0.278159	−	1.33274i
$656$		0			0
$657$			12.0000i			0.468165i
$658$		0			0
$659$		−	16.4174i		−	0.639532i	−0.947497	−	0.319766i	$-0.896396\pi$
$659$		−	16.4174i		−	0.639532i	0.947497	−	0.319766i	$-0.103604\pi$
$660$		0			0
$661$			45.6054i			1.77385i	0.461918	+	0.886923i	$0.347161\pi$
$661$			45.6054i			1.77385i	−0.461918	+	0.886923i	$0.652839\pi$
$662$		0			0
$663$			3.27340i			0.127128i
$664$		0			0
$665$	−8.00000	−	1.66970i	−0.310227	−	0.0647481i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		−	0.913701i		−	0.0353257i
$670$		0			0
$671$	−37.9144			−1.46367
$672$		0			0
$673$		−	40.0000i		−	1.54189i	−0.636904	−	0.770943i	$-0.719785\pi$
$673$		−	40.0000i		−	1.54189i	0.636904	−	0.770943i	$-0.280215\pi$
$674$		0			0
$675$	−4.58258	−	2.00000i	−0.176383	−	0.0769800i
$676$		0			0
$677$	23.3350			0.896837			0.448419	−	0.893824i	$-0.351987\pi$
$677$	23.3350			0.896837			0.448419	+	0.893824i	$0.351987\pi$
$678$		0			0
$679$	−6.54680			−0.251243
$680$		0			0
$681$	3.16515			0.121289
$682$		0			0
$683$	49.4955			1.89389			0.946945	−	0.321394i	$-0.104151\pi$
$683$	49.4955			1.89389			0.946945	+	0.321394i	$0.104151\pi$
$684$		0			0
$685$	8.94630	−	42.8643i	0.341821	−	1.63776i
$686$		0			0
$687$			22.6120i			0.862702i
$688$		0			0
$689$	2.33030			0.0887775
$690$		0			0
$691$		−	32.6606i		−	1.24247i	−0.783625	−	0.621234i	$-0.786632\pi$
$691$		−	32.6606i		−	1.24247i	0.783625	−	0.621234i	$-0.213368\pi$
$692$		0			0
$693$	−3.27340			−0.124346
$694$		0			0
$695$	5.10080	−	24.4394i	0.193484	−	0.927040i
$696$		0			0
$697$		−	21.4955i		−	0.814198i
$698$		0			0
$699$		−	19.5826i		−	0.740681i
$700$		0			0
$701$		−	40.6555i		−	1.53554i	−0.640727	−	0.767769i	$-0.721367\pi$
$701$		−	40.6555i		−	1.53554i	0.640727	−	0.767769i	$-0.278633\pi$
$702$		0			0
$703$		−	31.3676i		−	1.18305i
$704$		0			0
$705$	−3.16515	+	15.1652i	−0.119206	+	0.571153i
$706$		0			0
$707$	15.1652			0.570344
$708$		0			0
$709$			5.10080i			0.191565i	0.995402	+	0.0957823i	$0.0305353\pi$
$709$			5.10080i			0.191565i	−0.995402	+	0.0957823i	$0.969465\pi$
$710$		0			0
$711$	5.29150			0.198447
$712$		0			0
$713$		0			0
$714$		0			0
$715$	−1.49545	+	7.16515i	−0.0559268	+	0.267961i
$716$		0			0
$717$	−21.1660			−0.790459
$718$		0			0
$719$	52.1522			1.94495			0.972475	−	0.233008i	$-0.0748567\pi$
$719$	52.1522			1.94495			0.972475	+	0.233008i	$0.0748567\pi$
$720$		0			0
$721$	−15.1652			−0.564780
$722$		0			0
$723$	10.0000			0.371904
$724$		0			0
$725$	−15.6838	+	35.9361i	−0.582482	+	1.33463i
$726$		0			0
$727$			23.1443i			0.858375i	0.903215	+	0.429187i	$0.141200\pi$
$727$			23.1443i			0.858375i	−0.903215	+	0.429187i	$0.858800\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$		−	25.6697i		−	0.949428i
$732$		0			0
$733$	−39.5909			−1.46232			−0.731162	−	0.682204i	$-0.761022\pi$
$733$	−39.5909			−1.46232			−0.731162	+	0.682204i	$0.761022\pi$
$734$		0			0
$735$	−13.4949	−	2.81655i	−0.497767	−	0.103890i
$736$		0			0
$737$			54.3303i			2.00128i
$738$		0			0
$739$			20.0000i			0.735712i	0.929883	+	0.367856i	$0.119908\pi$
$739$			20.0000i			0.735712i	−0.929883	+	0.367856i	$0.880092\pi$
$740$		0			0
$741$		−	3.65480i		−	0.134263i
$742$		0			0
$743$		−	26.6482i		−	0.977628i	−0.872388	−	0.488814i	$-0.837430\pi$
$743$		−	26.6482i		−	0.977628i	0.872388	−	0.488814i	$-0.162570\pi$
$744$		0			0
$745$	5.25227	−	25.1652i	0.192428	−	0.921980i
$746$		0			0
$747$	−11.1652			−0.408512
$748$		0			0
$749$			3.65480i			0.133544i
$750$		0			0
$751$	22.4213			0.818165			0.409083	−	0.912497i	$-0.365849\pi$
$751$	22.4213			0.818165			0.409083	+	0.912497i	$0.365849\pi$
$752$		0			0
$753$			11.5826i			0.422093i
$754$		0			0
$755$	34.7477	+	7.25227i	1.26460	+	0.263937i
$756$		0			0
$757$	12.9427			0.470411			0.235205	−	0.971946i	$-0.424424\pi$
$757$	12.9427			0.470411			0.235205	+	0.971946i	$0.424424\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−22.0000			−0.797499			−0.398750	−	0.917060i	$-0.630556\pi$
$761$	−22.0000			−0.797499			−0.398750	+	0.917060i	$0.630556\pi$
$762$		0			0
$763$	−8.00000			−0.289619
$764$		0			0
$765$	1.63670	−	7.84190i	0.0591750	−	0.283525i
$766$		0			0
$767$			6.92820i			0.250163i
$768$		0			0
$769$	−31.4955			−1.13576			−0.567878	−	0.823113i	$-0.692235\pi$
$769$	−31.4955			−1.13576			−0.567878	+	0.823113i	$0.692235\pi$
$770$		0			0
$771$		−	2.74773i		−	0.0989570i
$772$		0			0
$773$	−14.5794			−0.524385			−0.262192	−	0.965016i	$-0.584446\pi$
$773$	−14.5794			−0.524385			−0.262192	+	0.965016i	$0.584446\pi$
$774$		0			0
$775$	−24.2487	−	10.5830i	−0.871039	−	0.380153i
$776$		0			0
$777$			7.16515i			0.257048i
$778$		0			0
$779$			24.0000i			0.859889i
$780$		0			0
$781$		−	24.8208i		−	0.888159i
$782$		0			0
$783$		−	7.84190i		−	0.280247i
$784$		0			0
$785$	−47.4955	−	9.91288i	−1.69519	−	0.353806i
$786$		0			0
$787$	−36.0000			−1.28326			−0.641631	−	0.767014i	$-0.721742\pi$
$787$	−36.0000			−1.28326			−0.641631	+	0.767014i	$0.721742\pi$
$788$		0			0
$789$			8.75560i			0.311708i
$790$		0			0
$791$	4.03620			0.143511
$792$		0			0
$793$			9.66970i			0.343381i
$794$		0			0
$795$	−5.58258	−	1.16515i	−0.197994	−	0.0413237i
$796$		0			0
$797$	26.6084			0.942518			0.471259	−	0.881995i	$-0.343800\pi$
$797$	26.6084			0.942518			0.471259	+	0.881995i	$0.343800\pi$
$798$		0			0
$799$	−24.8208			−0.878097
$800$		0			0
$801$	−2.00000			−0.0706665
$802$		0			0
$803$	−42.9909			−1.51712
$804$		0			0
$805$		0			0
$806$		0			0
$807$		−	9.66930i		−	0.340376i
$808$		0			0
$809$	−30.0000			−1.05474			−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000			−1.05474			−0.527372	+	0.849635i	$0.676823\pi$
$810$		0			0
$811$		−	19.1652i		−	0.672979i	−0.941687	−	0.336490i	$-0.890760\pi$
$811$		−	19.1652i		−	0.672979i	0.941687	−	0.336490i	$-0.109240\pi$
$812$		0			0
$813$	−26.0761			−0.914529
$814$		0			0
$815$	8.75560	+	1.82740i	0.306695	+	0.0640111i
$816$		0			0
$817$			28.6606i			1.00271i
$818$		0			0
$819$			0.834849i			0.0291720i
$820$		0			0
$821$		−	9.66930i		−	0.337461i	−0.985662	−	0.168731i	$-0.946033\pi$
$821$		−	9.66930i		−	0.337461i	0.985662	−	0.168731i	$-0.0539668\pi$
$822$		0			0
$823$			14.7701i			0.514854i	0.966298	+	0.257427i	$0.0828746\pi$
$823$			14.7701i			0.514854i	−0.966298	+	0.257427i	$0.917125\pi$
$824$		0			0
$825$	7.16515	−	16.4174i	0.249459	−	0.571581i
$826$		0			0
$827$	28.0000			0.973655			0.486828	−	0.873498i	$-0.338154\pi$
$827$	28.0000			0.973655			0.486828	+	0.873498i	$0.338154\pi$
$828$		0			0
$829$		−	16.0652i		−	0.557968i	−0.960296	−	0.278984i	$-0.910002\pi$
$829$		−	16.0652i		−	0.557968i	0.960296	−	0.278984i	$-0.0899976\pi$
$830$		0			0
$831$	14.7701			0.512369
$832$		0			0
$833$		−	22.0871i		−	0.765273i
$834$		0			0
$835$	10.3303	−	49.4955i	0.357495	−	1.71286i
$836$		0			0
$837$	5.29150			0.182901
$838$		0			0
$839$	13.4750			0.465209			0.232604	−	0.972571i	$-0.425275\pi$
$839$	13.4750			0.465209			0.232604	+	0.972571i	$0.425275\pi$
$840$		0			0
$841$	−32.4955			−1.12053
$842$		0			0
$843$	4.33030			0.149144
$844$		0			0
$845$	−26.6283	−	5.55765i	−0.916042	−	0.191189i
$846$		0			0
$847$		−	1.67650i		−	0.0576053i
$848$		0			0
$849$	7.16515			0.245907
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−23.5257			−0.805505			−0.402753	−	0.915309i	$-0.631946\pi$
$853$	−23.5257			−0.805505			−0.402753	+	0.915309i	$0.631946\pi$
$854$		0			0
$855$	−1.82740	+	8.75560i	−0.0624958	+	0.299435i
$856$		0			0
$857$		−	20.4174i		−	0.697446i	−0.937226	−	0.348723i	$-0.886615\pi$
$857$		−	20.4174i		−	0.697446i	0.937226	−	0.348723i	$-0.113385\pi$
$858$		0			0
$859$		−	43.1652i		−	1.47278i	−0.676559	−	0.736388i	$-0.736530\pi$
$859$		−	43.1652i		−	1.47278i	0.676559	−	0.736388i	$-0.263470\pi$
$860$		0			0
$861$		−	5.48220i		−	0.186833i
$862$		0			0
$863$			2.20880i			0.0751885i	0.999293	+	0.0375942i	$0.0119694\pi$
$863$			2.20880i			0.0751885i	−0.999293	+	0.0375942i	$0.988031\pi$
$864$		0			0
$865$	24.7477	+	5.16515i	0.841448	+	0.175620i
$866$		0			0
$867$	−4.16515			−0.141456
$868$		0			0
$869$			18.9572i			0.643079i
$870$		0			0
$871$	13.8564			0.469506
$872$		0			0
$873$			7.16515i			0.242504i
$874$		0			0
$875$	−5.91288	+	8.33030i	−0.199892	+	0.281616i
$876$		0			0
$877$	54.8933			1.85362			0.926808	−	0.375536i	$-0.122541\pi$
$877$	54.8933			1.85362			0.926808	+	0.375536i	$0.122541\pi$
$878$		0			0
$879$	−4.37780			−0.147660
$880$		0			0
$881$	−0.330303			−0.0111282			−0.00556409	−	0.999985i	$-0.501771\pi$
$881$	−0.330303			−0.0111282			−0.00556409	+	0.999985i	$0.501771\pi$
$882$		0			0
$883$	−2.33030			−0.0784209			−0.0392105	−	0.999231i	$-0.512484\pi$
$883$	−2.33030			−0.0784209			−0.0392105	+	0.999231i	$0.512484\pi$
$884$		0			0
$885$	3.46410	−	16.5975i	0.116445	−	0.557919i
$886$		0			0
$887$			15.3024i			0.513805i	0.966437	+	0.256902i	$0.0827018\pi$
$887$			15.3024i			0.513805i	−0.966437	+	0.256902i	$0.917298\pi$
$888$		0			0
$889$	13.4955			0.452623
$890$		0			0
$891$			3.58258i			0.120021i
$892$		0			0
$893$	27.7128			0.927374
$894$		0			0
$895$	3.46410	−	16.5975i	0.115792	−	0.554794i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	41.4955i		−	1.38395i
$900$		0			0
$901$		−	9.13701i		−	0.304398i
$902$		0			0
$903$		−	6.54680i		−	0.217864i
$904$		0			0
$905$	4.00000	−	19.1652i	0.132964	−	0.637071i
$906$		0			0
$907$	−37.4955			−1.24502			−0.622508	−	0.782613i	$-0.713886\pi$
$907$	−37.4955			−1.24502			−0.622508	+	0.782613i	$0.713886\pi$
$908$		0			0
$909$		−	16.5975i		−	0.550505i
$910$		0			0
$911$	35.0224			1.16034			0.580172	−	0.814494i	$-0.302985\pi$
$911$	35.0224			1.16034			0.580172	+	0.814494i	$0.302985\pi$
$912$		0			0
$913$		−	40.0000i		−	1.32381i
$914$		0			0
$915$	4.83485	−	23.1652i	0.159835	−	0.765816i
$916$		0			0
$917$	14.2378			0.470174
$918$		0			0
$919$	43.5873			1.43781			0.718907	−	0.695107i	$-0.244643\pi$
$919$	43.5873			1.43781			0.718907	+	0.695107i	$0.244643\pi$
$920$		0			0
$921$	−4.00000			−0.131804
$922$		0			0
$923$	−6.33030			−0.208364
$924$		0			0
$925$	−35.9361	−	15.6838i	−1.18157	−	0.515680i
$926$		0			0
$927$			16.5975i			0.545134i
$928$		0			0
$929$	−42.6606			−1.39965			−0.699825	−	0.714315i	$-0.746738\pi$
$929$	−42.6606			−1.39965			−0.699825	+	0.714315i	$0.746738\pi$
$930$		0			0
$931$			24.6606i			0.808219i
$932$		0			0
$933$	17.5112			0.573291
$934$		0			0
$935$	28.0942	+	5.86360i	0.918779	+	0.191760i
$936$		0			0
$937$			22.3303i			0.729499i	0.931106	+	0.364750i	$0.118845\pi$
$937$			22.3303i			0.729499i	−0.931106	+	0.364750i	$0.881155\pi$
$938$		0			0
$939$		−	19.1652i		−	0.625431i
$940$		0			0
$941$		−	44.3103i		−	1.44448i	−0.691645	−	0.722238i	$-0.743114\pi$
$941$		−	44.3103i		−	1.44448i	0.691645	−	0.722238i	$-0.256886\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$	0.417424	−	2.00000i	0.0135788	−	0.0650600i
$946$		0			0
$947$	−52.0000			−1.68977			−0.844886	−	0.534946i	$-0.820332\pi$
$947$	−52.0000			−1.68977			−0.844886	+	0.534946i	$0.820332\pi$
$948$		0			0
$949$			10.9644i			0.355920i
$950$		0			0
$951$	−2.93180			−0.0950702
$952$		0			0
$953$			56.2432i			1.82190i	0.412522	+	0.910948i	$0.364648\pi$
$953$			56.2432i			1.82190i	−0.412522	+	0.910948i	$0.635352\pi$
$954$		0			0
$955$	−30.3303	−	6.33030i	−0.981466	−	0.204844i
$956$		0			0
$957$	28.0942			0.908157
$958$		0			0
$959$	17.8926			0.577782
$960$		0			0
$961$	−3.00000			−0.0967742
$962$		0			0
$963$	4.00000			0.128898
$964$		0			0
$965$	−1.44600	+	6.92820i	−0.0465484	+	0.223027i
$966$		0			0
$967$			11.1153i			0.357444i	0.983900	+	0.178722i	$0.0571963\pi$
$967$			11.1153i			0.357444i	−0.983900	+	0.178722i	$0.942804\pi$
$968$		0			0
$969$	−14.3303			−0.460356
$970$		0			0
$971$		−	49.9129i		−	1.60178i	−0.598811	−	0.800890i	$-0.704360\pi$
$971$		−	49.9129i		−	1.60178i	0.598811	−	0.800890i	$-0.295640\pi$
$972$		0			0
$973$	10.2016			0.327048
$974$		0			0
$975$	−4.18710	−	1.82740i	−0.134095	−	0.0585237i
$976$		0			0
$977$			34.7477i			1.11168i	0.831290	+	0.555839i	$0.187603\pi$
$977$			34.7477i			1.11168i	−0.831290	+	0.555839i	$0.812397\pi$
$978$		0			0
$979$		−	7.16515i		−	0.228999i
$980$		0			0
$981$			8.75560i			0.279545i
$982$		0			0
$983$		−	50.7062i		−	1.61728i	−0.588306	−	0.808639i	$-0.700205\pi$
$983$		−	50.7062i		−	1.61728i	0.588306	−	0.808639i	$-0.299795\pi$
$984$		0			0
$985$	−55.9129	−	11.6697i	−1.78153	−	0.371827i
$986$		0			0
$987$	−6.33030			−0.201496
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−43.5873			−1.38460			−0.692298	−	0.721611i	$-0.743402\pi$
$991$	−43.5873			−1.38460			−0.692298	+	0.721611i	$0.743402\pi$
$992$		0			0
$993$		−	26.3303i		−	0.835567i
$994$		0			0
$995$	−3.58258	−	0.747727i	−0.113575	−	0.0237045i
$996$		0			0
$997$	39.2095			1.24178			0.620889	−	0.783899i	$-0.286772\pi$
$997$	39.2095			1.24178			0.620889	+	0.783899i	$0.286772\pi$
$998$		0			0
$999$	7.84190			0.248107

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	960.2.d.e.289.8	yes	8
3.2	odd	2		2880.2.d.j.289.1		8
4.3	odd	2		960.2.d.f.289.8	yes	8
5.2	odd	4		4800.2.k.q.2401.6		8
5.3	odd	4		4800.2.k.r.2401.3		8
5.4	even	2		960.2.d.f.289.2	yes	8
8.3	odd	2	inner	960.2.d.e.289.1	✓	8
8.5	even	2		960.2.d.f.289.1	yes	8
12.11	even	2		2880.2.d.i.289.1		8
15.14	odd	2		2880.2.d.i.289.7		8
16.3	odd	4		3840.2.f.k.769.7		8
16.5	even	4		3840.2.f.k.769.6		8
16.11	odd	4		3840.2.f.i.769.2		8
16.13	even	4		3840.2.f.i.769.3		8
20.3	even	4		4800.2.k.r.2401.6		8
20.7	even	4		4800.2.k.q.2401.3		8
20.19	odd	2	inner	960.2.d.e.289.2	yes	8
24.5	odd	2		2880.2.d.i.289.8		8
24.11	even	2		2880.2.d.j.289.8		8
40.3	even	4		4800.2.k.r.2401.2		8
40.13	odd	4		4800.2.k.r.2401.7		8
40.19	odd	2		960.2.d.f.289.7	yes	8
40.27	even	4		4800.2.k.q.2401.7		8
40.29	even	2	inner	960.2.d.e.289.7	yes	8
40.37	odd	4		4800.2.k.q.2401.2		8
60.59	even	2		2880.2.d.j.289.7		8
80.19	odd	4		3840.2.f.k.769.3		8
80.29	even	4		3840.2.f.i.769.7		8
80.59	odd	4		3840.2.f.i.769.6		8
80.69	even	4		3840.2.f.k.769.2		8
120.29	odd	2		2880.2.d.j.289.2		8
120.59	even	2		2880.2.d.i.289.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
960.2.d.e.289.1	✓	8	8.3	odd	2	inner
960.2.d.e.289.2	yes	8	20.19	odd	2	inner
960.2.d.e.289.7	yes	8	40.29	even	2	inner
960.2.d.e.289.8	yes	8	1.1	even	1	trivial
960.2.d.f.289.1	yes	8	8.5	even	2
960.2.d.f.289.2	yes	8	5.4	even	2
960.2.d.f.289.7	yes	8	40.19	odd	2
960.2.d.f.289.8	yes	8	4.3	odd	2
2880.2.d.i.289.1		8	12.11	even	2
2880.2.d.i.289.2		8	120.59	even	2
2880.2.d.i.289.7		8	15.14	odd	2
2880.2.d.i.289.8		8	24.5	odd	2
2880.2.d.j.289.1		8	3.2	odd	2
2880.2.d.j.289.2		8	120.29	odd	2
2880.2.d.j.289.7		8	60.59	even	2
2880.2.d.j.289.8		8	24.11	even	2
3840.2.f.i.769.2		8	16.11	odd	4
3840.2.f.i.769.3		8	16.13	even	4
3840.2.f.i.769.6		8	80.59	odd	4
3840.2.f.i.769.7		8	80.29	even	4
3840.2.f.k.769.2		8	80.69	even	4
3840.2.f.k.769.3		8	80.19	odd	4
3840.2.f.k.769.6		8	16.5	even	4
3840.2.f.k.769.7		8	16.3	odd	4
4800.2.k.q.2401.2		8	40.37	odd	4
4800.2.k.q.2401.3		8	20.7	even	4
4800.2.k.q.2401.6		8	5.2	odd	4
4800.2.k.q.2401.7		8	40.27	even	4
4800.2.k.r.2401.2		8	40.3	even	4
4800.2.k.r.2401.3		8	5.3	odd	4
4800.2.k.r.2401.6		8	20.3	even	4
4800.2.k.r.2401.7		8	40.13	odd	4

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	960.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +(2.18890 + 0.456850i) q^{5} +0.913701i q^{7} +1.00000 q^{9} +3.58258i q^{11} +0.913701 q^{13} +(-2.18890 - 0.456850i) q^{15} -3.58258i q^{17} +4.00000i q^{19} -0.913701i q^{21} +(4.58258 + 2.00000i) q^{25} -1.00000 q^{27} +7.84190i q^{29} -5.29150 q^{31} -3.58258i q^{33} +(-0.417424 + 2.00000i) q^{35} -7.84190 q^{37} -0.913701 q^{39} +6.00000 q^{41} +7.16515 q^{43} +(2.18890 + 0.456850i) q^{45} -6.92820i q^{47} +6.16515 q^{49} +3.58258i q^{51} +2.55040 q^{53} +(-1.63670 + 7.84190i) q^{55} -4.00000i q^{57} +7.58258i q^{59} +10.5830i q^{61} +0.913701i q^{63} +(2.00000 + 0.417424i) q^{65} +15.1652 q^{67} -6.92820 q^{71} +12.0000i q^{73} +(-4.58258 - 2.00000i) q^{75} -3.27340 q^{77} +5.29150 q^{79} +1.00000 q^{81} -11.1652 q^{83} +(1.63670 - 7.84190i) q^{85} -7.84190i q^{87} -2.00000 q^{89} +0.834849i q^{91} +5.29150 q^{93} +(-1.82740 + 8.75560i) q^{95} +7.16515i q^{97} +3.58258i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{3} + 8 q^{9} - 8 q^{27} - 40 q^{35} + 48 q^{41} - 16 q^{43} - 24 q^{49} + 16 q^{65} + 48 q^{67} + 8 q^{81} - 16 q^{83} - 16 q^{89}+O(q^{100}) \) 8 * q - 8 * q^3 + 8 * q^9 - 8 * q^27 - 40 * q^35 + 48 * q^41 - 16 * q^43 - 24 * q^49 + 16 * q^65 + 48 * q^67 + 8 * q^81 - 16 * q^83 - 16 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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