LMFDB - Embedded newform 1120.2.l.a.1009.18

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1120,2,Mod(1009,1120)] 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("1120.1009"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1120, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1120 = 2^{5} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1120.l (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$8.94324502638$
Analytic rank:	$0$
Dimension:	$36$
Twist minimal:	no (minimal twist has level 280)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1009.18
Character	$\chi$	$=$	1120.1009
Dual form			1120.2.l.a.1009.17

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.319826 q^{3} +(1.20181 - 1.88564i) q^{5} +1.00000i q^{7} -2.89771 q^{9} +4.31560i q^{11} +2.16506 q^{13} +(-0.384372 + 0.603077i) q^{15} +3.19581i q^{17} +5.38650i q^{19} -0.319826i q^{21} -0.947361i q^{23} +(-2.11128 - 4.53238i) q^{25} +1.88624 q^{27} +7.55412i q^{29} +1.97567 q^{31} -1.38024i q^{33} +(1.88564 + 1.20181i) q^{35} +9.35534 q^{37} -0.692441 q^{39} +8.13282 q^{41} -2.27658 q^{43} +(-3.48251 + 5.46404i) q^{45} -6.88315i q^{47} -1.00000 q^{49} -1.02210i q^{51} +6.12756 q^{53} +(8.13766 + 5.18654i) q^{55} -1.72274i q^{57} +4.30565i q^{59} -0.0705077i q^{61} -2.89771i q^{63} +(2.60200 - 4.08252i) q^{65} -10.1964 q^{67} +0.302991i q^{69} -5.61315 q^{71} +2.18227i q^{73} +(0.675244 + 1.44957i) q^{75} -4.31560 q^{77} -10.0202 q^{79} +8.08986 q^{81} +16.4283 q^{83} +(6.02615 + 3.84077i) q^{85} -2.41600i q^{87} -13.4992 q^{89} +2.16506i q^{91} -0.631870 q^{93} +(10.1570 + 6.47357i) q^{95} +11.4733i q^{97} -12.5053i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q + 36 q^{9} + 4 q^{25} + 16 q^{31} + 32 q^{39} - 8 q^{41} - 36 q^{49} + 32 q^{55} - 24 q^{65} - 56 q^{71} - 24 q^{79} + 36 q^{81} - 40 q^{89} - 40 q^{95}+O(q^{100}) $ 36 * q + 36 * q^9 + 4 * q^25 + 16 * q^31 + 32 * q^39 - 8 * q^41 - 36 * q^49 + 32 * q^55 - 24 * q^65 - 56 * q^71 - 24 * q^79 + 36 * q^81 - 40 * q^89 - 40 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1120\mathbb{Z}\right)^\times$.

$n$	$351$	$421$	$801$	$897$
$\chi(n)$	$1$	$-1$	$1$	$-1$

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
$3$	−0.319826			−0.184652			−0.0923258	−	0.995729i	$-0.529430\pi$
$3$	−0.319826			−0.184652			−0.0923258	+	0.995729i	$0.529430\pi$
$4$		0			0
$5$	1.20181	−	1.88564i	0.537468	−	0.843284i
$5$	1.20181	−	1.88564i	0.537468	−	0.843284i
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		0			0
$9$	−2.89771			−0.965904
$10$		0			0
$11$			4.31560i			1.30120i	0.759420	+	0.650600i	$0.225483\pi$
$11$			4.31560i			1.30120i	−0.759420	+	0.650600i	$0.774517\pi$
$12$		0			0
$13$	2.16506			0.600478			0.300239	−	0.953864i	$-0.402933\pi$
$13$	2.16506			0.600478			0.300239	+	0.953864i	$0.402933\pi$
$14$		0			0
$15$	−0.384372	+	0.603077i	−0.0992443	+	0.155714i
$16$		0			0
$17$			3.19581i			0.775097i	0.921849	+	0.387549i	$0.126678\pi$
$17$			3.19581i			0.775097i	−0.921849	+	0.387549i	$0.873322\pi$
$18$		0			0
$19$			5.38650i			1.23575i	0.786278	+	0.617873i	$0.212006\pi$
$19$			5.38650i			1.23575i	−0.786278	+	0.617873i	$0.787994\pi$
$20$		0			0
$21$		−	0.319826i		−	0.0697918i
$22$		0			0
$23$		−	0.947361i		−	0.197538i	−0.995110	−	0.0987692i	$-0.968509\pi$
$23$		−	0.947361i		−	0.197538i	0.995110	−	0.0987692i	$-0.0314906\pi$
$24$		0			0
$25$	−2.11128	−	4.53238i	−0.422257	−	0.906476i
$26$		0			0
$27$	1.88624			0.363007
$28$		0			0
$29$			7.55412i			1.40276i	0.712785	+	0.701382i	$0.247433\pi$
$29$			7.55412i			1.40276i	−0.712785	+	0.701382i	$0.752567\pi$
$30$		0			0
$31$	1.97567			0.354840			0.177420	−	0.984135i	$-0.443225\pi$
$31$	1.97567			0.354840			0.177420	+	0.984135i	$0.443225\pi$
$32$		0			0
$33$		−	1.38024i		−	0.240269i
$34$		0			0
$35$	1.88564	+	1.20181i	0.318732	+	0.203144i
$36$		0			0
$37$	9.35534			1.53801			0.769004	−	0.639244i	$-0.220753\pi$
$37$	9.35534			1.53801			0.769004	+	0.639244i	$0.220753\pi$
$38$		0			0
$39$	−0.692441			−0.110879
$40$		0			0
$41$	8.13282			1.27013			0.635067	−	0.772457i	$-0.280973\pi$
$41$	8.13282			1.27013			0.635067	+	0.772457i	$0.280973\pi$
$42$		0			0
$43$	−2.27658			−0.347175			−0.173587	−	0.984818i	$-0.555536\pi$
$43$	−2.27658			−0.347175			−0.173587	+	0.984818i	$0.555536\pi$
$44$		0			0
$45$	−3.48251	+	5.46404i	−0.519142	+	0.814532i
$46$		0			0
$47$		−	6.88315i		−	1.00401i	−0.864864	−	0.502006i	$-0.832596\pi$
$47$		−	6.88315i		−	1.00401i	0.864864	−	0.502006i	$-0.167404\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$		−	1.02210i		−	0.143123i
$52$		0			0
$53$	6.12756			0.841685			0.420843	−	0.907134i	$-0.361734\pi$
$53$	6.12756			0.841685			0.420843	+	0.907134i	$0.361734\pi$
$54$		0			0
$55$	8.13766	+	5.18654i	1.09728	+	0.699353i
$56$		0			0
$57$		−	1.72274i		−	0.228183i
$58$		0			0
$59$			4.30565i			0.560548i	0.959920	+	0.280274i	$0.0904253\pi$
$59$			4.30565i			0.560548i	−0.959920	+	0.280274i	$0.909575\pi$
$60$		0			0
$61$		−	0.0705077i		−	0.00902759i	−0.999990	−	0.00451380i	$-0.998563\pi$
$61$		−	0.0705077i		−	0.00902759i	0.999990	−	0.00451380i	$-0.00143679\pi$
$62$		0			0
$63$		−	2.89771i		−	0.365077i
$64$		0			0
$65$	2.60200	−	4.08252i	0.322738	−	0.506374i
$66$		0			0
$67$	−10.1964			−1.24569			−0.622845	−	0.782346i	$-0.714023\pi$
$67$	−10.1964			−1.24569			−0.622845	+	0.782346i	$0.714023\pi$
$68$		0			0
$69$			0.302991i			0.0364758i
$70$		0			0
$71$	−5.61315			−0.666159			−0.333079	−	0.942899i	$-0.608088\pi$
$71$	−5.61315			−0.666159			−0.333079	+	0.942899i	$0.608088\pi$
$72$		0			0
$73$			2.18227i			0.255415i	0.991812	+	0.127708i	$0.0407619\pi$
$73$			2.18227i			0.255415i	−0.991812	+	0.127708i	$0.959238\pi$
$74$		0			0
$75$	0.675244	+	1.44957i	0.0779704	+	0.167382i
$76$		0			0
$77$	−4.31560			−0.491808
$78$		0			0
$79$	−10.0202			−1.12737			−0.563683	−	0.825991i	$-0.690616\pi$
$79$	−10.0202			−1.12737			−0.563683	+	0.825991i	$0.690616\pi$
$80$		0			0
$81$	8.08986			0.898874
$82$		0			0
$83$	16.4283			1.80324			0.901620	−	0.432530i	$-0.142379\pi$
$83$	16.4283			1.80324			0.901620	+	0.432530i	$0.142379\pi$
$84$		0			0
$85$	6.02615	+	3.84077i	0.653627	+	0.416590i
$86$		0			0
$87$		−	2.41600i		−	0.259023i
$88$		0			0
$89$	−13.4992			−1.43091			−0.715455	−	0.698659i	$-0.753781\pi$
$89$	−13.4992			−1.43091			−0.715455	+	0.698659i	$0.753781\pi$
$90$		0			0
$91$			2.16506i			0.226960i
$92$		0			0
$93$	−0.631870			−0.0655218
$94$		0			0
$95$	10.1570	+	6.47357i	1.04209	+	0.664174i
$96$		0			0
$97$			11.4733i			1.16494i	0.812853	+	0.582469i	$0.197913\pi$
$97$			11.4733i			1.16494i	−0.812853	+	0.582469i	$0.802087\pi$
$98$		0			0
$99$		−	12.5053i		−	1.25683i
$100$		0			0
$101$			10.0737i			1.00237i	0.865341	+	0.501184i	$0.167102\pi$
$101$			10.0737i			1.00237i	−0.865341	+	0.501184i	$0.832898\pi$
$102$		0			0
$103$			9.84605i			0.970160i	0.874470	+	0.485080i	$0.161210\pi$
$103$			9.84605i			0.970160i	−0.874470	+	0.485080i	$0.838790\pi$
$104$		0			0
$105$	−0.603077	−	0.384372i	−0.0588543	−	0.0375108i
$106$		0			0
$107$	18.8737			1.82459			0.912297	−	0.409530i	$-0.134307\pi$
$107$	18.8737			1.82459			0.912297	+	0.409530i	$0.134307\pi$
$108$		0			0
$109$			2.69541i			0.258174i	0.991633	+	0.129087i	$0.0412046\pi$
$109$			2.69541i			0.258174i	−0.991633	+	0.129087i	$0.958795\pi$
$110$		0			0
$111$	−2.99208			−0.283996
$112$		0			0
$113$		−	5.00235i		−	0.470581i	−0.971925	−	0.235291i	$-0.924396\pi$
$113$		−	5.00235i		−	0.470581i	0.971925	−	0.235291i	$-0.0756042\pi$
$114$		0			0
$115$	−1.78638	−	1.13855i	−0.166581	−	0.106171i
$116$		0			0
$117$	−6.27371			−0.580004
$118$		0			0
$119$	−3.19581			−0.292959
$120$		0			0
$121$	−7.62436			−0.693124
$122$		0			0
$123$	−2.60109			−0.234532
$124$		0			0
$125$	−11.0838	−	1.46596i	−0.991367	−	0.131119i
$126$		0			0
$127$		−	6.20713i		−	0.550794i	−0.961331	−	0.275397i	$-0.911191\pi$
$127$		−	6.20713i		−	0.550794i	0.961331	−	0.275397i	$-0.0888093\pi$
$128$		0			0
$129$	0.728108			0.0641064
$130$		0			0
$131$			13.6430i			1.19199i	0.802987	+	0.595997i	$0.203243\pi$
$131$			13.6430i			1.19199i	−0.802987	+	0.595997i	$0.796757\pi$
$132$		0			0
$133$	−5.38650			−0.467068
$134$		0			0
$135$	2.26691	−	3.55677i	0.195105	−	0.306118i
$136$		0			0
$137$			16.3488i			1.39677i	0.715723	+	0.698384i	$0.246097\pi$
$137$			16.3488i			1.39677i	−0.715723	+	0.698384i	$0.753903\pi$
$138$		0			0
$139$		−	1.91419i		−	0.162359i	−0.996699	−	0.0811796i	$-0.974131\pi$
$139$		−	1.91419i		−	0.162359i	0.996699	−	0.0811796i	$-0.0258687\pi$
$140$		0			0
$141$			2.20141i			0.185392i
$142$		0			0
$143$			9.34350i			0.781343i
$144$		0			0
$145$	14.2444	+	9.07865i	1.18293	+	0.753941i
$146$		0			0
$147$	0.319826			0.0263788
$148$		0			0
$149$		−	14.9281i		−	1.22296i	−0.791260	−	0.611480i	$-0.790575\pi$
$149$		−	14.9281i		−	1.22296i	0.791260	−	0.611480i	$-0.209425\pi$
$150$		0			0
$151$	−0.106712			−0.00868413			−0.00434207	−	0.999991i	$-0.501382\pi$
$151$	−0.106712			−0.00868413			−0.00434207	+	0.999991i	$0.501382\pi$
$152$		0			0
$153$		−	9.26053i		−	0.748669i
$154$		0			0
$155$	2.37438	−	3.72540i	0.190715	−	0.299231i
$156$		0			0
$157$	−10.7375			−0.856946			−0.428473	−	0.903555i	$-0.640948\pi$
$157$	−10.7375			−0.856946			−0.428473	+	0.903555i	$0.640948\pi$
$158$		0			0
$159$	−1.95975			−0.155419
$160$		0			0
$161$	0.947361			0.0746625
$162$		0			0
$163$	3.76600			0.294976			0.147488	−	0.989064i	$-0.452881\pi$
$163$	3.76600			0.294976			0.147488	+	0.989064i	$0.452881\pi$
$164$		0			0
$165$	−2.60264	−	1.65879i	−0.202615	−	0.129137i
$166$		0			0
$167$		−	24.5491i		−	1.89967i	−0.312754	−	0.949834i	$-0.601252\pi$
$167$		−	24.5491i		−	1.89967i	0.312754	−	0.949834i	$-0.398748\pi$
$168$		0			0
$169$	−8.31253			−0.639426
$170$		0			0
$171$		−	15.6085i		−	1.19361i
$172$		0			0
$173$	−18.9785			−1.44291			−0.721453	−	0.692464i	$-0.756525\pi$
$173$	−18.9785			−1.44291			−0.721453	+	0.692464i	$0.756525\pi$
$174$		0			0
$175$	4.53238	−	2.11128i	0.342616	−	0.159598i
$176$		0			0
$177$		−	1.37706i		−	0.103506i
$178$		0			0
$179$			1.39807i			0.104497i	0.998634	+	0.0522483i	$0.0166387\pi$
$179$			1.39807i			0.104497i	−0.998634	+	0.0522483i	$0.983361\pi$
$180$		0			0
$181$		−	19.9912i		−	1.48593i	−0.669329	−	0.742966i	$-0.733418\pi$
$181$		−	19.9912i		−	1.48593i	0.669329	−	0.742966i	$-0.266582\pi$
$182$		0			0
$183$			0.0225502i			0.00166696i
$184$		0			0
$185$	11.2434	−	17.6408i	0.826630	−	1.29698i
$186$		0			0
$187$	−13.7918			−1.00856
$188$		0			0
$189$			1.88624i			0.137204i
$190$		0			0
$191$	16.1124			1.16585			0.582925	−	0.812526i	$-0.301908\pi$
$191$	16.1124			1.16585			0.582925	+	0.812526i	$0.301908\pi$
$192$		0			0
$193$		−	12.9801i		−	0.934329i	−0.884170	−	0.467165i	$-0.845276\pi$
$193$		−	12.9801i		−	0.934329i	0.884170	−	0.467165i	$-0.154724\pi$
$194$		0			0
$195$	−0.832186	+	1.30570i	−0.0595941	+	0.0935028i
$196$		0			0
$197$	16.5028			1.17577			0.587887	−	0.808943i	$-0.299960\pi$
$197$	16.5028			1.17577			0.587887	+	0.808943i	$0.299960\pi$
$198$		0			0
$199$	−21.6505			−1.53477			−0.767383	−	0.641189i	$-0.778441\pi$
$199$	−21.6505			−1.53477			−0.767383	+	0.641189i	$0.778441\pi$
$200$		0			0
$201$	3.26108			0.230019
$202$		0			0
$203$	−7.55412			−0.530195
$204$		0			0
$205$	9.77414	−	15.3356i	0.682656	−	1.07108i
$206$		0			0
$207$			2.74518i			0.190803i
$208$		0			0
$209$	−23.2459			−1.60795
$210$		0			0
$211$		−	24.5315i		−	1.68882i	−0.535698	−	0.844409i	$-0.679952\pi$
$211$		−	24.5315i		−	1.68882i	0.535698	−	0.844409i	$-0.320048\pi$
$212$		0			0
$213$	1.79523			0.123007
$214$		0			0
$215$	−2.73602	+	4.29280i	−0.186595	+	0.292767i
$216$		0			0
$217$			1.97567i			0.134117i
$218$		0			0
$219$		−	0.697946i		−	0.0471628i
$220$		0			0
$221$			6.91910i			0.465429i
$222$		0			0
$223$			7.24853i			0.485398i	0.970102	+	0.242699i	$0.0780327\pi$
$223$			7.24853i			0.485398i	−0.970102	+	0.242699i	$0.921967\pi$
$224$		0			0
$225$	6.11789	+	13.1335i	0.407860	+	0.875569i
$226$		0			0
$227$	−24.2854			−1.61188			−0.805939	−	0.591998i	$-0.798339\pi$
$227$	−24.2854			−1.61188			−0.805939	+	0.591998i	$0.798339\pi$
$228$		0			0
$229$		−	6.92485i		−	0.457607i	−0.973473	−	0.228803i	$-0.926519\pi$
$229$		−	6.92485i		−	0.457607i	0.973473	−	0.228803i	$-0.0734813\pi$
$230$		0			0
$231$	1.38024			0.0908131
$232$		0			0
$233$			10.1782i			0.666796i	0.942786	+	0.333398i	$0.108195\pi$
$233$			10.1782i			0.666796i	−0.942786	+	0.333398i	$0.891805\pi$
$234$		0			0
$235$	−12.9792	−	8.27227i	−0.846667	−	0.539624i
$236$		0			0
$237$	3.20473			0.208170
$238$		0			0
$239$	−0.483092			−0.0312487			−0.0156243	−	0.999878i	$-0.504974\pi$
$239$	−0.483092			−0.0312487			−0.0156243	+	0.999878i	$0.504974\pi$
$240$		0			0
$241$	−14.0241			−0.903369			−0.451685	−	0.892178i	$-0.649177\pi$
$241$	−14.0241			−0.903369			−0.451685	+	0.892178i	$0.649177\pi$
$242$		0			0
$243$	−8.24607			−0.528986
$244$		0			0
$245$	−1.20181	+	1.88564i	−0.0767811	+	0.120469i
$246$		0			0
$247$			11.6621i			0.742039i
$248$		0			0
$249$	−5.25420			−0.332971
$250$		0			0
$251$			4.58585i			0.289456i	0.989471	+	0.144728i	$0.0462308\pi$
$251$			4.58585i			0.289456i	−0.989471	+	0.144728i	$0.953769\pi$
$252$		0			0
$253$	4.08843			0.257037
$254$		0			0
$255$	−1.92732	−	1.22838i	−0.120693	−	0.0769240i
$256$		0			0
$257$		−	19.6305i		−	1.22452i	−0.790657	−	0.612260i	$-0.790261\pi$
$257$		−	19.6305i		−	1.22452i	0.790657	−	0.612260i	$-0.209739\pi$
$258$		0			0
$259$			9.35534i			0.581312i
$260$		0			0
$261$		−	21.8897i		−	1.35494i
$262$		0			0
$263$		−	13.7976i		−	0.850796i	−0.905006	−	0.425398i	$-0.860134\pi$
$263$		−	13.7976i		−	0.850796i	0.905006	−	0.425398i	$-0.139866\pi$
$264$		0			0
$265$	7.36419	−	11.5544i	0.452379	−	0.709780i
$266$		0			0
$267$	4.31739			0.264220
$268$		0			0
$269$			16.3189i			0.994981i	0.867469	+	0.497491i	$0.165745\pi$
$269$			16.3189i			0.994981i	−0.867469	+	0.497491i	$0.834255\pi$
$270$		0			0
$271$	12.5470			0.762176			0.381088	−	0.924539i	$-0.375549\pi$
$271$	12.5470			0.762176			0.381088	+	0.924539i	$0.375549\pi$
$272$		0			0
$273$		−	0.692441i		−	0.0419085i
$274$		0			0
$275$	19.5599	−	9.11145i	1.17951	−	0.549441i
$276$		0			0
$277$	−1.71783			−0.103214			−0.0516071	−	0.998667i	$-0.516434\pi$
$277$	−1.71783			−0.103214			−0.0516071	+	0.998667i	$0.516434\pi$
$278$		0			0
$279$	−5.72491			−0.342741
$280$		0			0
$281$	20.0452			1.19579			0.597897	−	0.801573i	$-0.296003\pi$
$281$	20.0452			1.19579			0.597897	+	0.801573i	$0.296003\pi$
$282$		0			0
$283$	−16.3366			−0.971108			−0.485554	−	0.874207i	$-0.661382\pi$
$283$	−16.3366			−0.971108			−0.485554	+	0.874207i	$0.661382\pi$
$284$		0			0
$285$	−3.24847	−	2.07042i	−0.192423	−	0.122641i
$286$		0			0
$287$			8.13282i			0.480065i
$288$		0			0
$289$	6.78681			0.399224
$290$		0			0
$291$		−	3.66946i		−	0.215108i
$292$		0			0
$293$	18.2850			1.06822			0.534109	−	0.845415i	$-0.320647\pi$
$293$	18.2850			1.06822			0.534109	+	0.845415i	$0.320647\pi$
$294$		0			0
$295$	8.11891	+	5.17459i	0.472701	+	0.301277i
$296$		0			0
$297$			8.14026i			0.472346i
$298$		0			0
$299$		−	2.05109i		−	0.118618i
$300$		0			0
$301$		−	2.27658i		−	0.131220i
$302$		0			0
$303$		−	3.22182i		−	0.185089i
$304$		0			0
$305$	−0.132952	−	0.0847372i	−0.00761283	−	0.00485204i
$306$		0			0
$307$	−0.296292			−0.0169103			−0.00845513	−	0.999964i	$-0.502691\pi$
$307$	−0.296292			−0.0169103			−0.00845513	+	0.999964i	$0.502691\pi$
$308$		0			0
$309$		−	3.14902i		−	0.179142i
$310$		0			0
$311$	25.9792			1.47315			0.736573	−	0.676358i	$-0.236443\pi$
$311$	25.9792			1.47315			0.736573	+	0.676358i	$0.236443\pi$
$312$		0			0
$313$			19.2606i			1.08867i	0.838867	+	0.544336i	$0.183218\pi$
$313$			19.2606i			1.08867i	−0.838867	+	0.544336i	$0.816782\pi$
$314$		0			0
$315$	−5.46404	−	3.48251i	−0.307864	−	0.196217i
$316$		0			0
$317$	1.29196			0.0725638			0.0362819	−	0.999342i	$-0.488449\pi$
$317$	1.29196			0.0725638			0.0362819	+	0.999342i	$0.488449\pi$
$318$		0			0
$319$	−32.6005			−1.82528
$320$		0			0
$321$	−6.03631			−0.336914
$322$		0			0
$323$	−17.2142			−0.957824
$324$		0			0
$325$	−4.57105	−	9.81286i	−0.253556	−	0.544319i
$326$		0			0
$327$		−	0.862063i		−	0.0476722i
$328$		0			0
$329$	6.88315			0.379481
$330$		0			0
$331$		−	14.5727i		−	0.800986i	−0.916300	−	0.400493i	$-0.868839\pi$
$331$		−	14.5727i		−	0.800986i	0.916300	−	0.400493i	$-0.131161\pi$
$332$		0			0
$333$	−27.1091			−1.48557
$334$		0			0
$335$	−12.2542	+	19.2268i	−0.669518	+	1.05047i
$336$		0			0
$337$		−	14.6600i		−	0.798580i	−0.916825	−	0.399290i	$-0.869257\pi$
$337$		−	14.6600i		−	0.798580i	0.916825	−	0.399290i	$-0.130743\pi$
$338$		0			0
$339$			1.59988i			0.0868936i
$340$		0			0
$341$			8.52618i			0.461718i
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$	0.571332	+	0.364139i	0.0307595	+	0.0196046i
$346$		0			0
$347$	2.43803			0.130880			0.0654400	−	0.997857i	$-0.479155\pi$
$347$	2.43803			0.130880			0.0654400	+	0.997857i	$0.479155\pi$
$348$		0			0
$349$		−	22.0454i		−	1.18007i	−0.807379	−	0.590033i	$-0.799115\pi$
$349$		−	22.0454i		−	1.18007i	0.807379	−	0.590033i	$-0.200885\pi$
$350$		0			0
$351$	4.08382			0.217978
$352$		0			0
$353$			21.9292i			1.16717i	0.812050	+	0.583587i	$0.198351\pi$
$353$			21.9292i			1.16717i	−0.812050	+	0.583587i	$0.801649\pi$
$354$		0			0
$355$	−6.74597	+	10.5844i	−0.358039	+	0.561761i
$356$		0			0
$357$	1.02210			0.0540954
$358$		0			0
$359$	−25.8696			−1.36535			−0.682674	−	0.730724i	$-0.739183\pi$
$359$	−25.8696			−1.36535			−0.682674	+	0.730724i	$0.739183\pi$
$360$		0			0
$361$	−10.0143			−0.527070
$362$		0			0
$363$	2.43847			0.127986
$364$		0			0
$365$	4.11497	+	2.62268i	0.215388	+	0.137277i
$366$		0			0
$367$		−	8.32030i		−	0.434316i	−0.976136	−	0.217158i	$-0.930321\pi$
$367$		−	8.32030i		−	0.434316i	0.976136	−	0.217158i	$-0.0696787\pi$
$368$		0			0
$369$	−23.5666			−1.22683
$370$		0			0
$371$			6.12756i			0.318127i
$372$		0			0
$373$	31.1510			1.61294			0.806468	−	0.591277i	$-0.201376\pi$
$373$	31.1510			1.61294			0.806468	+	0.591277i	$0.201376\pi$
$374$		0			0
$375$	3.54489	+	0.468851i	0.183057	+	0.0242114i
$376$		0			0
$377$			16.3551i			0.842330i
$378$		0			0
$379$			6.70705i			0.344518i	0.985052	+	0.172259i	$0.0551066\pi$
$379$			6.70705i			0.344518i	−0.985052	+	0.172259i	$0.944893\pi$
$380$		0			0
$381$			1.98520i			0.101705i
$382$		0			0
$383$		−	9.09458i		−	0.464711i	−0.972631	−	0.232356i	$-0.925357\pi$
$383$		−	9.09458i		−	0.464711i	0.972631	−	0.232356i	$-0.0746433\pi$
$384$		0			0
$385$	−5.18654	+	8.13766i	−0.264331	+	0.414734i
$386$		0			0
$387$	6.59686			0.335337
$388$		0			0
$389$			3.44036i			0.174433i	0.996189	+	0.0872167i	$0.0277973\pi$
$389$			3.44036i			0.174433i	−0.996189	+	0.0872167i	$0.972203\pi$
$390$		0			0
$391$	3.02758			0.153111
$392$		0			0
$393$		−	4.36338i		−	0.220104i
$394$		0			0
$395$	−12.0425	+	18.8946i	−0.605922	+	0.950689i
$396$		0			0
$397$	−22.8353			−1.14607			−0.573035	−	0.819531i	$-0.694234\pi$
$397$	−22.8353			−1.14607			−0.573035	+	0.819531i	$0.694234\pi$
$398$		0			0
$399$	1.72274			0.0862449
$400$		0			0
$401$	2.13911			0.106822			0.0534111	−	0.998573i	$-0.482991\pi$
$401$	2.13911			0.106822			0.0534111	+	0.998573i	$0.482991\pi$
$402$		0			0
$403$	4.27743			0.213074
$404$		0			0
$405$	9.72252	−	15.2546i	0.483116	−	0.758006i
$406$		0			0
$407$			40.3739i			2.00126i
$408$		0			0
$409$	4.74000			0.234378			0.117189	−	0.993110i	$-0.462612\pi$
$409$	4.74000			0.234378			0.117189	+	0.993110i	$0.462612\pi$
$410$		0			0
$411$		−	5.22876i		−	0.257916i
$412$		0			0
$413$	−4.30565			−0.211867
$414$		0			0
$415$	19.7438	−	30.9779i	0.969183	−	1.52064i
$416$		0			0
$417$			0.612207i			0.0299799i
$418$		0			0
$419$			1.06086i			0.0518265i	0.999664	+	0.0259133i	$0.00824937\pi$
$419$			1.06086i			0.0518265i	−0.999664	+	0.0259133i	$0.991751\pi$
$420$		0			0
$421$		−	11.4474i		−	0.557912i	−0.960304	−	0.278956i	$-0.910012\pi$
$421$		−	11.4474i		−	0.557912i	0.960304	−	0.278956i	$-0.0899884\pi$
$422$		0			0
$423$			19.9454i			0.969778i
$424$		0			0
$425$	14.4846	−	6.74726i	0.702607	−	0.327290i
$426$		0			0
$427$	0.0705077			0.00341211
$428$		0			0
$429$		−	2.98830i		−	0.144276i
$430$		0			0
$431$	2.17956			0.104986			0.0524929	−	0.998621i	$-0.483283\pi$
$431$	2.17956			0.104986			0.0524929	+	0.998621i	$0.483283\pi$
$432$		0			0
$433$		−	19.2652i		−	0.925827i	−0.886403	−	0.462914i	$-0.846804\pi$
$433$		−	19.2652i		−	0.925827i	0.886403	−	0.462914i	$-0.153196\pi$
$434$		0			0
$435$	−4.55572	−	2.90359i	−0.218430	−	0.139216i
$436$		0			0
$437$	5.10295			0.244107
$438$		0			0
$439$	36.2154			1.72847			0.864233	−	0.503091i	$-0.167804\pi$
$439$	36.2154			1.72847			0.864233	+	0.503091i	$0.167804\pi$
$440$		0			0
$441$	2.89771			0.137986
$442$		0			0
$443$	36.5965			1.73875			0.869377	−	0.494150i	$-0.164521\pi$
$443$	36.5965			1.73875			0.869377	+	0.494150i	$0.164521\pi$
$444$		0			0
$445$	−16.2235	+	25.4546i	−0.769068	+	1.20666i
$446$		0			0
$447$			4.77440i			0.225821i
$448$		0			0
$449$	−30.2438			−1.42729			−0.713647	−	0.700506i	$-0.752958\pi$
$449$	−30.2438			−1.42729			−0.713647	+	0.700506i	$0.752958\pi$
$450$		0			0
$451$			35.0980i			1.65270i
$452$		0			0
$453$	0.0341294			0.00160354
$454$		0			0
$455$	4.08252	+	2.60200i	0.191391	+	0.121983i
$456$		0			0
$457$		−	19.8067i		−	0.926517i	−0.886223	−	0.463258i	$-0.846680\pi$
$457$		−	19.8067i		−	0.926517i	0.886223	−	0.463258i	$-0.153320\pi$
$458$		0			0
$459$			6.02807i			0.281366i
$460$		0			0
$461$			24.5182i			1.14193i	0.820976	+	0.570963i	$0.193430\pi$
$461$			24.5182i			1.14193i	−0.820976	+	0.570963i	$0.806570\pi$
$462$		0			0
$463$		−	29.3342i		−	1.36328i	−0.731689	−	0.681638i	$-0.761268\pi$
$463$		−	29.3342i		−	1.36328i	0.731689	−	0.681638i	$-0.238732\pi$
$464$		0			0
$465$	−0.759390	+	1.19148i	−0.0352159	+	0.0552535i
$466$		0			0
$467$	34.1899			1.58212			0.791060	−	0.611738i	$-0.209530\pi$
$467$	34.1899			1.58212			0.791060	+	0.611738i	$0.209530\pi$
$468$		0			0
$469$		−	10.1964i		−	0.470826i
$470$		0			0
$471$	3.43413			0.158237
$472$		0			0
$473$		−	9.82478i		−	0.451744i
$474$		0			0
$475$	24.4136	−	11.3724i	1.12018	−	0.521803i
$476$		0			0
$477$	−17.7559			−0.812987
$478$		0			0
$479$	−8.33951			−0.381042			−0.190521	−	0.981683i	$-0.561018\pi$
$479$	−8.33951			−0.381042			−0.190521	+	0.981683i	$0.561018\pi$
$480$		0			0
$481$	20.2548			0.923541
$482$		0			0
$483$	−0.302991			−0.0137866
$484$		0			0
$485$	21.6345	+	13.7888i	0.982373	+	0.626116i
$486$		0			0
$487$		−	11.4033i		−	0.516731i	−0.966047	−	0.258366i	$-0.916816\pi$
$487$		−	11.4033i		−	0.516731i	0.966047	−	0.258366i	$-0.0831840\pi$
$488$		0			0
$489$	−1.20447			−0.0544678
$490$		0			0
$491$			29.0913i			1.31287i	0.754381	+	0.656437i	$0.227937\pi$
$491$			29.0913i			1.31287i	−0.754381	+	0.656437i	$0.772063\pi$
$492$		0			0
$493$	−24.1415			−1.08728
$494$		0			0
$495$	−23.5806	−	15.0291i	−1.05987	−	0.675508i
$496$		0			0
$497$		−	5.61315i		−	0.251784i
$498$		0			0
$499$		−	35.8988i		−	1.60705i	−0.595270	−	0.803526i	$-0.702955\pi$
$499$		−	35.8988i		−	1.60705i	0.595270	−	0.803526i	$-0.297045\pi$
$500$		0			0
$501$			7.85145i			0.350777i
$502$		0			0
$503$		−	24.3630i		−	1.08629i	−0.839639	−	0.543145i	$-0.817233\pi$
$503$		−	24.3630i		−	1.08629i	0.839639	−	0.543145i	$-0.182767\pi$
$504$		0			0
$505$	18.9953	+	12.1067i	0.845281	+	0.538741i
$506$		0			0
$507$	2.65856			0.118071
$508$		0			0
$509$			42.2864i			1.87431i	0.348911	+	0.937156i	$0.386551\pi$
$509$			42.2864i			1.87431i	−0.348911	+	0.937156i	$0.613449\pi$
$510$		0			0
$511$	−2.18227			−0.0965378
$512$		0			0
$513$			10.1602i			0.448585i
$514$		0			0
$515$	18.5661	+	11.8331i	0.818121	+	0.521430i
$516$		0			0
$517$	29.7049			1.30642
$518$		0			0
$519$	6.06981			0.266435
$520$		0			0
$521$	−4.28566			−0.187758			−0.0938790	−	0.995584i	$-0.529927\pi$
$521$	−4.28566			−0.187758			−0.0938790	+	0.995584i	$0.529927\pi$
$522$		0			0
$523$	−4.71705			−0.206262			−0.103131	−	0.994668i	$-0.532886\pi$
$523$	−4.71705			−0.206262			−0.103131	+	0.994668i	$0.532886\pi$
$524$		0			0
$525$	−1.44957	+	0.675244i	−0.0632646	+	0.0294701i
$526$		0			0
$527$			6.31385i			0.275036i
$528$		0			0
$529$	22.1025			0.960979
$530$		0			0
$531$		−	12.4765i		−	0.541436i
$532$		0			0
$533$	17.6080			0.762688
$534$		0			0
$535$	22.6827	−	35.5891i	0.980660	−	1.53865i
$536$		0			0
$537$		−	0.447139i		−	0.0192955i
$538$		0			0
$539$		−	4.31560i		−	0.185886i
$540$		0			0
$541$			25.9673i			1.11642i	0.829700	+	0.558210i	$0.188512\pi$
$541$			25.9673i			1.11642i	−0.829700	+	0.558210i	$0.811488\pi$
$542$		0			0
$543$			6.39370i			0.274380i
$544$		0			0
$545$	5.08258	+	3.23938i	0.217714	+	0.138760i
$546$		0			0
$547$	5.39594			0.230714			0.115357	−	0.993324i	$-0.463199\pi$
$547$	5.39594			0.230714			0.115357	+	0.993324i	$0.463199\pi$
$548$		0			0
$549$			0.204311i			0.00871978i
$550$		0			0
$551$	−40.6902			−1.73346
$552$		0			0
$553$		−	10.0202i		−	0.426104i
$554$		0			0
$555$	−3.59593	+	5.64199i	−0.152639	+	0.239489i
$556$		0			0
$557$	−5.03653			−0.213405			−0.106702	−	0.994291i	$-0.534029\pi$
$557$	−5.03653			−0.213405			−0.106702	+	0.994291i	$0.534029\pi$
$558$		0			0
$559$	−4.92891			−0.208471
$560$		0			0
$561$	4.41098			0.186232
$562$		0			0
$563$	0.717513			0.0302396			0.0151198	−	0.999886i	$-0.495187\pi$
$563$	0.717513			0.0302396			0.0151198	+	0.999886i	$0.495187\pi$
$564$		0			0
$565$	−9.43263	−	6.01189i	−0.396834	−	0.252922i
$566$		0			0
$567$			8.08986i			0.339742i
$568$		0			0
$569$	16.4900			0.691296			0.345648	−	0.938364i	$-0.387659\pi$
$569$	16.4900			0.691296			0.345648	+	0.938364i	$0.387659\pi$
$570$		0			0
$571$		−	1.12858i		−	0.0472298i	−0.999721	−	0.0236149i	$-0.992482\pi$
$571$		−	1.12858i		−	0.0472298i	0.999721	−	0.0236149i	$-0.00751755\pi$
$572$		0			0
$573$	−5.15315			−0.215276
$574$		0			0
$575$	−4.29380	+	2.00015i	−0.179064	+	0.0834119i
$576$		0			0
$577$		−	36.2807i		−	1.51039i	−0.655503	−	0.755193i	$-0.727543\pi$
$577$		−	36.2807i		−	1.51039i	0.655503	−	0.755193i	$-0.272457\pi$
$578$		0			0
$579$			4.15138i			0.172525i
$580$		0			0
$581$			16.4283i			0.681560i
$582$		0			0
$583$			26.4441i			1.09520i
$584$		0			0
$585$	−7.53983	+	11.8300i	−0.311734	+	0.489109i
$586$		0			0
$587$	0.344317			0.0142115			0.00710573	−	0.999975i	$-0.497738\pi$
$587$	0.344317			0.0142115			0.00710573	+	0.999975i	$0.497738\pi$
$588$		0			0
$589$			10.6419i			0.438493i
$590$		0			0
$591$	−5.27802			−0.217109
$592$		0			0
$593$			22.9480i			0.942362i	0.882036	+	0.471181i	$0.156172\pi$
$593$			22.9480i			0.942362i	−0.882036	+	0.471181i	$0.843828\pi$
$594$		0			0
$595$	−3.84077	+	6.02615i	−0.157456	+	0.247048i
$596$		0			0
$597$	6.92440			0.283397
$598$		0			0
$599$	−3.90767			−0.159663			−0.0798315	−	0.996808i	$-0.525438\pi$
$599$	−3.90767			−0.159663			−0.0798315	+	0.996808i	$0.525438\pi$
$600$		0			0
$601$	−13.2081			−0.538769			−0.269384	−	0.963033i	$-0.586820\pi$
$601$	−13.2081			−0.538769			−0.269384	+	0.963033i	$0.586820\pi$
$602$		0			0
$603$	29.5462			1.20322
$604$		0			0
$605$	−9.16307	+	14.3768i	−0.372532	+	0.584500i
$606$		0			0
$607$			31.9342i			1.29617i	0.761568	+	0.648085i	$0.224430\pi$
$607$			31.9342i			1.29617i	−0.761568	+	0.648085i	$0.775570\pi$
$608$		0			0
$609$	2.41600			0.0979014
$610$		0			0
$611$		−	14.9024i		−	0.602887i
$612$		0			0
$613$	−20.5483			−0.829937			−0.414969	−	0.909836i	$-0.636207\pi$
$613$	−20.5483			−0.829937			−0.414969	+	0.909836i	$0.636207\pi$
$614$		0			0
$615$	−3.12603	+	4.90472i	−0.126054	+	0.197777i
$616$		0			0
$617$			0.543748i			0.0218905i	0.999940	+	0.0109452i	$0.00348405\pi$
$617$			0.543748i			0.0218905i	−0.999940	+	0.0109452i	$0.996516\pi$
$618$		0			0
$619$			14.5163i			0.583460i	0.956501	+	0.291730i	$0.0942308\pi$
$619$			14.5163i			0.583460i	−0.956501	+	0.291730i	$0.905769\pi$
$620$		0			0
$621$		−	1.78695i		−	0.0717079i
$622$		0			0
$623$		−	13.4992i		−	0.540833i
$624$		0			0
$625$	−16.0850	+	19.1383i	−0.643398	+	0.765532i
$626$		0			0
$627$	7.43465			0.296912
$628$		0			0
$629$			29.8979i			1.19211i
$630$		0			0
$631$	−23.5413			−0.937165			−0.468582	−	0.883420i	$-0.655235\pi$
$631$	−23.5413			−0.937165			−0.468582	+	0.883420i	$0.655235\pi$
$632$		0			0
$633$			7.84582i			0.311843i
$634$		0			0
$635$	−11.7044	−	7.45982i	−0.464476	−	0.296034i
$636$		0			0
$637$	−2.16506			−0.0857826
$638$		0			0
$639$	16.2653			0.643445
$640$		0			0
$641$	−7.64593			−0.301996			−0.150998	−	0.988534i	$-0.548249\pi$
$641$	−7.64593			−0.301996			−0.150998	+	0.988534i	$0.548249\pi$
$642$		0			0
$643$	−26.9759			−1.06383			−0.531914	−	0.846799i	$-0.678527\pi$
$643$	−26.9759			−1.06383			−0.531914	+	0.846799i	$0.678527\pi$
$644$		0			0
$645$	0.875051	−	1.37295i	0.0344551	−	0.0540599i
$646$		0			0
$647$		−	10.6009i		−	0.416763i	−0.978048	−	0.208382i	$-0.933180\pi$
$647$		−	10.6009i		−	0.416763i	0.978048	−	0.208382i	$-0.0668196\pi$
$648$		0			0
$649$	−18.5815			−0.729386
$650$		0			0
$651$		−	0.631870i		−	0.0247649i
$652$		0			0
$653$	13.6159			0.532831			0.266416	−	0.963858i	$-0.414161\pi$
$653$	13.6159			0.532831			0.266416	+	0.963858i	$0.414161\pi$
$654$		0			0
$655$	25.7258	+	16.3963i	1.00519	+	0.640658i
$656$		0			0
$657$		−	6.32358i		−	0.246706i
$658$		0			0
$659$			15.6129i			0.608193i	0.952641	+	0.304096i	$0.0983545\pi$
$659$			15.6129i			0.608193i	−0.952641	+	0.304096i	$0.901646\pi$
$660$		0			0
$661$		−	5.45451i		−	0.212156i	−0.994358	−	0.106078i	$-0.966171\pi$
$661$		−	5.45451i		−	0.212156i	0.994358	−	0.106078i	$-0.0338293\pi$
$662$		0			0
$663$		−	2.21291i		−	0.0859423i
$664$		0			0
$665$	−6.47357	+	10.1570i	−0.251034	+	0.393871i
$666$		0			0
$667$	7.15648			0.277100
$668$		0			0
$669$		−	2.31827i		−	0.0896295i
$670$		0			0
$671$	0.304283			0.0117467
$672$		0			0
$673$		−	1.98399i		−	0.0764773i	−0.999269	−	0.0382386i	$-0.987825\pi$
$673$		−	1.98399i		−	0.0764773i	0.999269	−	0.0382386i	$-0.0121747\pi$
$674$		0			0
$675$	−3.98239	−	8.54917i	−0.153282	−	0.329058i
$676$		0			0
$677$	19.7354			0.758495			0.379247	−	0.925295i	$-0.376183\pi$
$677$	19.7354			0.758495			0.379247	+	0.925295i	$0.376183\pi$
$678$		0			0
$679$	−11.4733			−0.440305
$680$		0			0
$681$	7.76710			0.297636
$682$		0			0
$683$	2.75254			0.105323			0.0526615	−	0.998612i	$-0.483230\pi$
$683$	2.75254			0.105323			0.0526615	+	0.998612i	$0.483230\pi$
$684$		0			0
$685$	30.8279	+	19.6482i	1.17787	+	0.750718i
$686$		0			0
$687$			2.21475i			0.0844979i
$688$		0			0
$689$	13.2665			0.505414
$690$		0			0
$691$			17.8656i			0.679640i	0.940491	+	0.339820i	$0.110366\pi$
$691$			17.8656i			0.679640i	−0.940491	+	0.339820i	$0.889634\pi$
$692$		0			0
$693$	12.5053			0.475039
$694$		0			0
$695$	−3.60947	−	2.30050i	−0.136915	−	0.0872629i
$696$		0			0
$697$			25.9909i			0.984477i
$698$		0			0
$699$		−	3.25525i		−	0.123125i
$700$		0			0
$701$		−	21.3058i		−	0.804710i	−0.915484	−	0.402355i	$-0.868192\pi$
$701$		−	21.3058i		−	0.804710i	0.915484	−	0.402355i	$-0.131808\pi$
$702$		0			0
$703$			50.3925i			1.90059i
$704$		0			0
$705$	4.15107	+	2.64569i	0.156338	+	0.0996424i
$706$		0			0
$707$	−10.0737			−0.378860
$708$		0			0
$709$		−	3.57791i		−	0.134371i	−0.997740	−	0.0671857i	$-0.978598\pi$
$709$		−	3.57791i		−	0.134371i	0.997740	−	0.0671857i	$-0.0214020\pi$
$710$		0			0
$711$	29.0358			1.08893
$712$		0			0
$713$		−	1.87167i		−	0.0700945i
$714$		0			0
$715$	17.6185	+	11.2292i	0.658894	+	0.419947i
$716$		0			0
$717$	0.154506			0.00577012
$718$		0			0
$719$	31.1438			1.16147			0.580735	−	0.814093i	$-0.302765\pi$
$719$	31.1438			1.16147			0.580735	+	0.814093i	$0.302765\pi$
$720$		0			0
$721$	−9.84605			−0.366686
$722$		0			0
$723$	4.48526			0.166809
$724$		0			0
$725$	34.2381	−	15.9489i	1.27157	−	0.592327i
$726$		0			0
$727$		−	5.71789i		−	0.212065i	−0.994363	−	0.106032i	$-0.966185\pi$
$727$		−	5.71789i		−	0.212065i	0.994363	−	0.106032i	$-0.0338147\pi$
$728$		0			0
$729$	−21.6323			−0.801196
$730$		0			0
$731$		−	7.27550i		−	0.269094i
$732$		0			0
$733$	22.7095			0.838795			0.419398	−	0.907803i	$-0.362241\pi$
$733$	22.7095			0.838795			0.419398	+	0.907803i	$0.362241\pi$
$734$		0			0
$735$	0.384372	−	0.603077i	0.0141778	−	0.0222448i
$736$		0			0
$737$		−	44.0036i		−	1.62089i
$738$		0			0
$739$			28.1649i			1.03606i	0.855362	+	0.518031i	$0.173335\pi$
$739$			28.1649i			1.03606i	−0.855362	+	0.518031i	$0.826665\pi$
$740$		0			0
$741$		−	3.72983i		−	0.137019i
$742$		0			0
$743$		−	41.3914i		−	1.51850i	−0.650797	−	0.759252i	$-0.725565\pi$
$743$		−	41.3914i		−	1.51850i	0.650797	−	0.759252i	$-0.274435\pi$
$744$		0			0
$745$	−28.1491	−	17.9408i	−1.03130	−	0.657301i
$746$		0			0
$747$	−47.6044			−1.74176
$748$		0			0
$749$			18.8737i			0.689631i
$750$		0			0
$751$	−46.0020			−1.67864			−0.839318	−	0.543641i	$-0.817045\pi$
$751$	−46.0020			−1.67864			−0.839318	+	0.543641i	$0.817045\pi$
$752$		0			0
$753$		−	1.46668i		−	0.0534486i
$754$		0			0
$755$	−0.128248	+	0.201221i	−0.00466744	+	0.00732319i
$756$		0			0
$757$	−17.9355			−0.651877			−0.325938	−	0.945391i	$-0.605680\pi$
$757$	−17.9355			−0.651877			−0.325938	+	0.945391i	$0.605680\pi$
$758$		0			0
$759$	−1.30759			−0.0474623
$760$		0			0
$761$	13.0061			0.471471			0.235736	−	0.971817i	$-0.424250\pi$
$761$	13.0061			0.471471			0.235736	+	0.971817i	$0.424250\pi$
$762$		0			0
$763$	−2.69541			−0.0975804
$764$		0			0
$765$	−17.4620	−	11.1294i	−0.631341	−	0.402386i
$766$		0			0
$767$			9.32198i			0.336597i
$768$		0			0
$769$	33.2033			1.19734			0.598671	−	0.800995i	$-0.295696\pi$
$769$	33.2033			1.19734			0.598671	+	0.800995i	$0.295696\pi$
$770$		0			0
$771$			6.27836i			0.226110i
$772$		0			0
$773$	3.75488			0.135053			0.0675267	−	0.997717i	$-0.478489\pi$
$773$	3.75488			0.135053			0.0675267	+	0.997717i	$0.478489\pi$
$774$		0			0
$775$	−4.17119	−	8.95447i	−0.149834	−	0.321654i
$776$		0			0
$777$		−	2.99208i		−	0.107340i
$778$		0			0
$779$			43.8074i			1.56956i
$780$		0			0
$781$		−	24.2241i		−	0.866807i
$782$		0			0
$783$			14.2489i			0.509214i
$784$		0			0
$785$	−12.9045	+	20.2471i	−0.460581	+	0.722649i
$786$		0			0
$787$	42.6751			1.52120			0.760602	−	0.649218i	$-0.224904\pi$
$787$	42.6751			1.52120			0.760602	+	0.649218i	$0.224904\pi$
$788$		0			0
$789$			4.41283i			0.157101i
$790$		0			0
$791$	5.00235			0.177863
$792$		0			0
$793$		−	0.152653i		−	0.00542087i
$794$		0			0
$795$	−2.35526	+	3.69539i	−0.0835325	+	0.131062i
$796$		0			0
$797$	−6.50695			−0.230488			−0.115244	−	0.993337i	$-0.536765\pi$
$797$	−6.50695			−0.230488			−0.115244	+	0.993337i	$0.536765\pi$
$798$		0			0
$799$	21.9972			0.778206
$800$		0			0
$801$	39.1167			1.38212
$802$		0			0
$803$	−9.41778			−0.332346
$804$		0			0
$805$	1.13855	−	1.78638i	0.0401287	−	0.0629617i
$806$		0			0
$807$		−	5.21921i		−	0.183725i
$808$		0			0
$809$	31.1366			1.09470			0.547352	−	0.836903i	$-0.315636\pi$
$809$	31.1366			1.09470			0.547352	+	0.836903i	$0.315636\pi$
$810$		0			0
$811$			13.3805i			0.469854i	0.972013	+	0.234927i	$0.0754851\pi$
$811$			13.3805i			0.469854i	−0.972013	+	0.234927i	$0.924515\pi$
$812$		0			0
$813$	−4.01286			−0.140737
$814$		0			0
$815$	4.52603	−	7.10133i	0.158540	−	0.248749i
$816$		0			0
$817$		−	12.2628i		−	0.429020i
$818$		0			0
$819$		−	6.27371i		−	0.219221i
$820$		0			0
$821$		−	20.9626i		−	0.731600i	−0.930693	−	0.365800i	$-0.880795\pi$
$821$		−	20.9626i		−	0.731600i	0.930693	−	0.365800i	$-0.119205\pi$
$822$		0			0
$823$		−	13.8301i		−	0.482086i	−0.970514	−	0.241043i	$-0.922510\pi$
$823$		−	13.8301i		−	0.482086i	0.970514	−	0.241043i	$-0.0774896\pi$
$824$		0			0
$825$	−6.25577	+	2.91408i	−0.217798	+	0.101455i
$826$		0			0
$827$	12.1584			0.422790			0.211395	−	0.977401i	$-0.432199\pi$
$827$	12.1584			0.422790			0.211395	+	0.977401i	$0.432199\pi$
$828$		0			0
$829$		−	35.0277i		−	1.21656i	−0.793722	−	0.608281i	$-0.791859\pi$
$829$		−	35.0277i		−	1.21656i	0.793722	−	0.608281i	$-0.208141\pi$
$830$		0			0
$831$	0.549406			0.0190587
$832$		0			0
$833$		−	3.19581i		−	0.110728i
$834$		0			0
$835$	−46.2908	−	29.5035i	−1.60196	−	1.02101i
$836$		0			0
$837$	3.72658			0.128810
$838$		0			0
$839$	54.0162			1.86485			0.932423	−	0.361367i	$-0.117690\pi$
$839$	54.0162			1.86485			0.932423	+	0.361367i	$0.117690\pi$
$840$		0			0
$841$	−28.0647			−0.967748
$842$		0			0
$843$	−6.41097			−0.220805
$844$		0			0
$845$	−9.99012	+	15.6745i	−0.343671	+	0.539218i
$846$		0			0
$847$		−	7.62436i		−	0.261976i
$848$		0			0
$849$	5.22486			0.179317
$850$		0			0
$851$		−	8.86288i		−	0.303816i
$852$		0			0
$853$	−32.8962			−1.12634			−0.563171	−	0.826340i	$-0.690419\pi$
$853$	−32.8962			−1.12634			−0.563171	+	0.826340i	$0.690419\pi$
$854$		0			0
$855$	−29.4320	−	18.7585i	−1.00655	−	0.641528i
$856$		0			0
$857$		−	10.8426i		−	0.370375i	−0.982703	−	0.185187i	$-0.940711\pi$
$857$		−	10.8426i		−	0.370375i	0.982703	−	0.185187i	$-0.0592892\pi$
$858$		0			0
$859$		−	30.4434i		−	1.03872i	−0.854557	−	0.519358i	$-0.826171\pi$
$859$		−	30.4434i		−	1.03872i	0.854557	−	0.519358i	$-0.173829\pi$
$860$		0			0
$861$		−	2.60109i		−	0.0886449i
$862$		0			0
$863$		−	6.17946i		−	0.210351i	−0.994454	−	0.105176i	$-0.966460\pi$
$863$		−	6.17946i		−	0.210351i	0.994454	−	0.105176i	$-0.0335405\pi$
$864$		0			0
$865$	−22.8086	+	35.7866i	−0.775515	+	1.21678i
$866$		0			0
$867$	−2.17060			−0.0737174
$868$		0			0
$869$		−	43.2433i		−	1.46693i
$870$		0			0
$871$	−22.0758			−0.748009
$872$		0			0
$873$		−	33.2463i		−	1.12522i
$874$		0			0
$875$	1.46596	−	11.0838i	0.0495583	−	0.374701i
$876$		0			0
$877$	5.84128			0.197246			0.0986229	−	0.995125i	$-0.468556\pi$
$877$	5.84128			0.197246			0.0986229	+	0.995125i	$0.468556\pi$
$878$		0			0
$879$	−5.84801			−0.197248
$880$		0			0
$881$	42.6160			1.43577			0.717886	−	0.696161i	$-0.245110\pi$
$881$	42.6160			1.43577			0.717886	+	0.696161i	$0.245110\pi$
$882$		0			0
$883$	34.4349			1.15883			0.579413	−	0.815034i	$-0.303282\pi$
$883$	34.4349			1.15883			0.579413	+	0.815034i	$0.303282\pi$
$884$		0			0
$885$	−2.59664	−	1.65497i	−0.0872851	−	0.0556312i
$886$		0			0
$887$			15.0372i			0.504899i	0.967610	+	0.252450i	$0.0812362\pi$
$887$			15.0372i			0.504899i	−0.967610	+	0.252450i	$0.918764\pi$
$888$		0			0
$889$	6.20713			0.208180
$890$		0			0
$891$			34.9126i			1.16962i
$892$		0			0
$893$	37.0761			1.24070
$894$		0			0
$895$	2.63626	+	1.68022i	0.0881204	+	0.0561636i
$896$		0			0
$897$			0.655992i			0.0219029i
$898$		0			0
$899$			14.9244i			0.497757i
$900$		0			0
$901$			19.5825i			0.652388i
$902$		0			0
$903$			0.728108i			0.0242299i
$904$		0			0
$905$	−37.6962	−	24.0257i	−1.25306	−	0.798641i
$906$		0			0
$907$	−24.7673			−0.822384			−0.411192	−	0.911549i	$-0.634887\pi$
$907$	−24.7673			−0.822384			−0.411192	+	0.911549i	$0.634887\pi$
$908$		0			0
$909$		−	29.1906i		−	0.968191i
$910$		0			0
$911$	17.6510			0.584804			0.292402	−	0.956295i	$-0.405545\pi$
$911$	17.6510			0.584804			0.292402	+	0.956295i	$0.405545\pi$
$912$		0			0
$913$			70.8979i			2.34638i
$914$		0			0
$915$	0.0425216	+	0.0271012i	0.00140572	+	0.000895937i
$916$		0			0
$917$	−13.6430			−0.450531
$918$		0			0
$919$	−21.1056			−0.696211			−0.348105	−	0.937455i	$-0.613175\pi$
$919$	−21.1056			−0.696211			−0.348105	+	0.937455i	$0.613175\pi$
$920$		0			0
$921$	0.0947618			0.00312251
$922$		0			0
$923$	−12.1528			−0.400014
$924$		0			0
$925$	−19.7518	−	42.4020i	−0.649435	−	1.39417i
$926$		0			0
$927$		−	28.5310i		−	0.937081i
$928$		0			0
$929$	−29.7053			−0.974601			−0.487300	−	0.873234i	$-0.662018\pi$
$929$	−29.7053			−0.974601			−0.487300	+	0.873234i	$0.662018\pi$
$930$		0			0
$931$		−	5.38650i		−	0.176535i
$932$		0			0
$933$	−8.30883			−0.272019
$934$		0			0
$935$	−16.5752	+	26.0064i	−0.542067	+	0.850501i
$936$		0			0
$937$			53.4101i			1.74483i	0.488765	+	0.872415i	$0.337447\pi$
$937$			53.4101i			1.74483i	−0.488765	+	0.872415i	$0.662553\pi$
$938$		0			0
$939$		−	6.16003i		−	0.201025i
$940$		0			0
$941$		−	5.54752i		−	0.180844i	−0.995904	−	0.0904220i	$-0.971178\pi$
$941$		−	5.54752i		−	0.180844i	0.995904	−	0.0904220i	$-0.0288216\pi$
$942$		0			0
$943$		−	7.70472i		−	0.250900i
$944$		0			0
$945$	3.55677	+	2.26691i	0.115702	+	0.0737427i
$946$		0			0
$947$	−19.2133			−0.624350			−0.312175	−	0.950025i	$-0.601057\pi$
$947$	−19.2133			−0.624350			−0.312175	+	0.950025i	$0.601057\pi$
$948$		0			0
$949$			4.72473i			0.153371i
$950$		0			0
$951$	−0.413203			−0.0133990
$952$		0			0
$953$		−	36.8866i		−	1.19488i	−0.801915	−	0.597438i	$-0.796186\pi$
$953$		−	36.8866i		−	1.19488i	0.801915	−	0.597438i	$-0.203814\pi$
$954$		0			0
$955$	19.3641	−	30.3821i	0.626606	−	0.983142i
$956$		0			0
$957$	10.4265			0.337041
$958$		0			0
$959$	−16.3488			−0.527929
$960$		0			0
$961$	−27.0967			−0.874088
$962$		0			0
$963$	−54.6906			−1.76238
$964$		0			0
$965$	−24.4758	−	15.5997i	−0.787905	−	0.502172i
$966$		0			0
$967$			23.1225i			0.743569i	0.928319	+	0.371785i	$0.121254\pi$
$967$			23.1225i			0.743569i	−0.928319	+	0.371785i	$0.878746\pi$
$968$		0			0
$969$	5.50555			0.176864
$970$		0			0
$971$		−	34.1621i		−	1.09632i	−0.836375	−	0.548158i	$-0.815329\pi$
$971$		−	34.1621i		−	1.09632i	0.836375	−	0.548158i	$-0.184671\pi$
$972$		0			0
$973$	1.91419			0.0613660
$974$		0			0
$975$	1.46194	+	3.13841i	0.0468196	+	0.100509i
$976$		0			0
$977$		−	41.5157i		−	1.32820i	−0.747642	−	0.664102i	$-0.768814\pi$
$977$		−	41.5157i		−	1.32820i	0.747642	−	0.664102i	$-0.231186\pi$
$978$		0			0
$979$		−	58.2570i		−	1.86190i
$980$		0			0
$981$		−	7.81052i		−	0.249371i
$982$		0			0
$983$			27.0085i			0.861438i	0.902486	+	0.430719i	$0.141740\pi$
$983$			27.0085i			0.861438i	−0.902486	+	0.430719i	$0.858260\pi$
$984$		0			0
$985$	19.8333	−	31.1183i	0.631941	−	0.991512i
$986$		0			0
$987$	−2.20141			−0.0700717
$988$		0			0
$989$			2.15674i			0.0685803i
$990$		0			0
$991$	−13.7895			−0.438037			−0.219019	−	0.975721i	$-0.570286\pi$
$991$	−13.7895			−0.438037			−0.219019	+	0.975721i	$0.570286\pi$
$992$		0			0
$993$			4.66072i			0.147903i
$994$		0			0
$995$	−26.0199	+	40.8251i	−0.824887	+	1.29424i
$996$		0			0
$997$	−21.0429			−0.666435			−0.333217	−	0.942850i	$-0.608134\pi$
$997$	−21.0429			−0.666435			−0.333217	+	0.942850i	$0.608134\pi$
$998$		0			0
$999$	17.6464			0.558308

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1120.2.l.a.1009.18		36
4.3	odd	2		280.2.l.a.29.19	yes	36
5.4	even	2	inner	1120.2.l.a.1009.20		36
8.3	odd	2		280.2.l.a.29.17	✓	36
8.5	even	2	inner	1120.2.l.a.1009.19		36
20.19	odd	2		280.2.l.a.29.18	yes	36
40.19	odd	2		280.2.l.a.29.20	yes	36
40.29	even	2	inner	1120.2.l.a.1009.17		36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.2.l.a.29.17	✓	36	8.3	odd	2
280.2.l.a.29.18	yes	36	20.19	odd	2
280.2.l.a.29.19	yes	36	4.3	odd	2
280.2.l.a.29.20	yes	36	40.19	odd	2
1120.2.l.a.1009.17		36	40.29	even	2	inner
1120.2.l.a.1009.18		36	1.1	even	1	trivial
1120.2.l.a.1009.19		36	8.5	even	2	inner
1120.2.l.a.1009.20		36	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 1120 = 2^{5} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1120.l (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.319826 q^{3} +(1.20181 - 1.88564i) q^{5} +1.00000i q^{7} -2.89771 q^{9} +4.31560i q^{11} +2.16506 q^{13} +(-0.384372 + 0.603077i) q^{15} +3.19581i q^{17} +5.38650i q^{19} -0.319826i q^{21} -0.947361i q^{23} +(-2.11128 - 4.53238i) q^{25} +1.88624 q^{27} +7.55412i q^{29} +1.97567 q^{31} -1.38024i q^{33} +(1.88564 + 1.20181i) q^{35} +9.35534 q^{37} -0.692441 q^{39} +8.13282 q^{41} -2.27658 q^{43} +(-3.48251 + 5.46404i) q^{45} -6.88315i q^{47} -1.00000 q^{49} -1.02210i q^{51} +6.12756 q^{53} +(8.13766 + 5.18654i) q^{55} -1.72274i q^{57} +4.30565i q^{59} -0.0705077i q^{61} -2.89771i q^{63} +(2.60200 - 4.08252i) q^{65} -10.1964 q^{67} +0.302991i q^{69} -5.61315 q^{71} +2.18227i q^{73} +(0.675244 + 1.44957i) q^{75} -4.31560 q^{77} -10.0202 q^{79} +8.08986 q^{81} +16.4283 q^{83} +(6.02615 + 3.84077i) q^{85} -2.41600i q^{87} -13.4992 q^{89} +2.16506i q^{91} -0.631870 q^{93} +(10.1570 + 6.47357i) q^{95} +11.4733i q^{97} -12.5053i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 36 q^{9} + 4 q^{25} + 16 q^{31} + 32 q^{39} - 8 q^{41} - 36 q^{49} + 32 q^{55} - 24 q^{65} - 56 q^{71} - 24 q^{79} + 36 q^{81} - 40 q^{89} - 40 q^{95}+O(q^{100}) \) 36 * q + 36 * q^9 + 4 * q^25 + 16 * q^31 + 32 * q^39 - 8 * q^41 - 36 * q^49 + 32 * q^55 - 24 * q^65 - 56 * q^71 - 24 * q^79 + 36 * q^81 - 40 * q^89 - 40 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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