LMFDB - Embedded newform 1440.2.bi.e.847.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1440,2,Mod(847,1440)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1440, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 2, 0, 1]))

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

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

chi := DirichletCharacter("1440.847");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1440 = 2^{5} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1440.bi (of order $4$, degree $2$, not 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:	$11.4984578911$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$12$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			847.4
Character	$\chi$	$=$	1440.847
Dual form			1440.2.bi.e.1423.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.28903 + 1.82713i) q^{5} +(1.45533 - 1.45533i) q^{7} +O(q^{10})$	$q+(-1.28903 + 1.82713i) q^{5} +(1.45533 - 1.45533i) q^{7} +0.725670 q^{11} +(-4.57738 - 4.57738i) q^{13} +(-2.36535 - 2.36535i) q^{17} +6.41350i q^{19} +(-1.35791 - 1.35791i) q^{23} +(-1.67680 - 4.71045i) q^{25} -2.91898 q^{29} -5.71240i q^{31} +(0.783110 + 4.53503i) q^{35} +(-2.65700 + 2.65700i) q^{37} -1.02625 q^{41} +(7.38725 - 7.38725i) q^{43} +(-1.22848 + 1.22848i) q^{47} +2.76404i q^{49} +(-9.48969 - 9.48969i) q^{53} +(-0.935410 + 1.32589i) q^{55} -6.43011i q^{59} -1.18105i q^{61} +(14.2638 - 2.46308i) q^{65} +(-3.02625 - 3.02625i) q^{67} -6.55658i q^{71} +(4.38725 - 4.38725i) q^{73} +(1.05609 - 1.05609i) q^{77} -6.75224 q^{79} +(1.37207 - 1.37207i) q^{83} +(7.37079 - 1.27279i) q^{85} -4.63060i q^{89} -13.3232 q^{91} +(-11.7183 - 8.26720i) q^{95} +(3.27977 + 3.27977i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q - 8 q^{17} - 8 q^{25} - 48 q^{35} + 32 q^{43} + 8 q^{65} - 48 q^{67} - 40 q^{73} + 80 q^{83} - 64 q^{91} - 24 q^{97}+O(q^{100}) $ 24 * q - 8 * q^17 - 8 * q^25 - 48 * q^35 + 32 * q^43 + 8 * q^65 - 48 * q^67 - 40 * q^73 + 80 * q^83 - 64 * q^91 - 24 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1440\mathbb{Z}\right)^\times$.

$n$	$577$	$641$	$901$	$991$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$1$	$-1$	$-1$

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			0
$3$		0			0
$4$		0			0
$5$	−1.28903	+	1.82713i	−0.576472	+	0.817117i
$5$	−1.28903	+	1.82713i	−0.576472	+	0.817117i
$6$		0			0
$7$	1.45533	−	1.45533i	0.550062	−	0.550062i	−0.376397	−	0.926459i	$-0.622837\pi$
$7$	1.45533	−	1.45533i	0.550062	−	0.550062i	0.926459	+	0.376397i	$0.122837\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	0.725670			0.218798			0.109399	−	0.993998i	$-0.465107\pi$
$11$	0.725670			0.218798			0.109399	+	0.993998i	$0.465107\pi$
$12$		0			0
$13$	−4.57738	−	4.57738i	−1.26954	−	1.26954i	−0.946328	−	0.323208i	$-0.895239\pi$
$13$	−4.57738	−	4.57738i	−1.26954	−	1.26954i	−0.323208	−	0.946328i	$-0.604761\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.36535	−	2.36535i	−0.573681	−	0.573681i	0.359474	−	0.933155i	$-0.382956\pi$
$17$	−2.36535	−	2.36535i	−0.573681	−	0.573681i	−0.933155	+	0.359474i	$0.882956\pi$
$18$		0			0
$19$			6.41350i			1.47136i	0.677330	+	0.735679i	$0.263137\pi$
$19$			6.41350i			1.47136i	−0.677330	+	0.735679i	$0.736863\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−1.35791	−	1.35791i	−0.283144	−	0.283144i	0.551217	−	0.834362i	$-0.314164\pi$
$23$	−1.35791	−	1.35791i	−0.283144	−	0.283144i	−0.834362	+	0.551217i	$0.814164\pi$
$24$		0			0
$25$	−1.67680	−	4.71045i	−0.335360	−	0.942090i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−2.91898			−0.542042			−0.271021	−	0.962573i	$-0.587361\pi$
$29$	−2.91898			−0.542042			−0.271021	+	0.962573i	$0.587361\pi$
$30$		0			0
$31$		−	5.71240i		−	1.02598i	−0.858395	−	0.512989i	$-0.828538\pi$
$31$		−	5.71240i		−	1.02598i	0.858395	−	0.512989i	$-0.171462\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	0.783110	+	4.53503i	0.132370	+	0.766560i
$36$		0			0
$37$	−2.65700	+	2.65700i	−0.436808	+	0.436808i	−0.890936	−	0.454128i	$-0.849951\pi$
$37$	−2.65700	+	2.65700i	−0.436808	+	0.436808i	0.454128	+	0.890936i	$0.349951\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−1.02625			−0.160274			−0.0801368	−	0.996784i	$-0.525536\pi$
$41$	−1.02625			−0.160274			−0.0801368	+	0.996784i	$0.525536\pi$
$42$		0			0
$43$	7.38725	−	7.38725i	1.12655	−	1.12655i	0.135810	−	0.990735i	$-0.456636\pi$
$43$	7.38725	−	7.38725i	1.12655	−	1.12655i	0.990735	−	0.135810i	$-0.0433638\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−1.22848	+	1.22848i	−0.179192	+	0.179192i	−0.791004	−	0.611812i	$-0.790441\pi$
$47$	−1.22848	+	1.22848i	−0.179192	+	0.179192i	0.611812	+	0.791004i	$0.290441\pi$
$48$		0			0
$49$			2.76404i			0.394864i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−9.48969	−	9.48969i	−1.30351	−	1.30351i	−0.926010	−	0.377500i	$-0.876784\pi$
$53$	−9.48969	−	9.48969i	−1.30351	−	1.30351i	−0.377500	−	0.926010i	$-0.623216\pi$
$54$		0			0
$55$	−0.935410	+	1.32589i	−0.126131	+	0.178783i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	6.43011i		−	0.837129i	−0.908187	−	0.418564i	$-0.862533\pi$
$59$		−	6.43011i		−	0.837129i	0.908187	−	0.418564i	$-0.137467\pi$
$60$		0			0
$61$		−	1.18105i		−	0.151218i	−0.997138	−	0.0756089i	$-0.975910\pi$
$61$		−	1.18105i		−	0.151218i	0.997138	−	0.0756089i	$-0.0240901\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	14.2638	−	2.46308i	1.76921	−	0.305508i
$66$		0			0
$67$	−3.02625	−	3.02625i	−0.369716	−	0.369716i	0.497658	−	0.867373i	$-0.334194\pi$
$67$	−3.02625	−	3.02625i	−0.369716	−	0.369716i	−0.867373	+	0.497658i	$0.834194\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	6.55658i		−	0.778123i	−0.921212	−	0.389062i	$-0.872799\pi$
$71$		−	6.55658i		−	0.778123i	0.921212	−	0.389062i	$-0.127201\pi$
$72$		0			0
$73$	4.38725	−	4.38725i	0.513489	−	0.513489i	−0.402105	−	0.915594i	$-0.631721\pi$
$73$	4.38725	−	4.38725i	0.513489	−	0.513489i	0.915594	+	0.402105i	$0.131721\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	1.05609	−	1.05609i	0.120352	−	0.120352i
$78$		0			0
$79$	−6.75224			−0.759686			−0.379843	−	0.925051i	$-0.624022\pi$
$79$	−6.75224			−0.759686			−0.379843	+	0.925051i	$0.624022\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	1.37207	−	1.37207i	0.150604	−	0.150604i	−0.627784	−	0.778388i	$-0.716038\pi$
$83$	1.37207	−	1.37207i	0.150604	−	0.150604i	0.778388	+	0.627784i	$0.216038\pi$
$84$		0			0
$85$	7.37079	−	1.27279i	0.799475	−	0.138053i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	4.63060i		−	0.490843i	−0.969416	−	0.245421i	$-0.921074\pi$
$89$		−	4.63060i		−	0.490843i	0.969416	−	0.245421i	$-0.0789263\pi$
$90$		0			0
$91$	−13.3232			−1.39665
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−11.7183	−	8.26720i	−1.20227	−	0.848197i
$96$		0			0
$97$	3.27977	+	3.27977i	0.333010	+	0.333010i	0.853728	−	0.520719i	$-0.174336\pi$
$97$	3.27977	+	3.27977i	0.333010	+	0.333010i	−0.520719	+	0.853728i	$0.674336\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	13.5791i		−	1.35117i	−0.737284	−	0.675583i	$-0.763892\pi$
$101$		−	13.5791i		−	1.35117i	0.737284	−	0.675583i	$-0.236108\pi$
$102$		0			0
$103$	7.90300	+	7.90300i	0.778706	+	0.778706i	0.979611	−	0.200905i	$-0.0643882\pi$
$103$	7.90300	+	7.90300i	0.778706	+	0.778706i	−0.200905	+	0.979611i	$0.564388\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−11.1728	−	11.1728i	−1.08012	−	1.08012i	−0.996498	−	0.0836196i	$-0.973352\pi$
$107$	−11.1728	−	11.1728i	−1.08012	−	1.08012i	−0.0836196	−	0.996498i	$-0.526648\pi$
$108$		0			0
$109$	−12.4480			−1.19230			−0.596150	−	0.802873i	$-0.703304\pi$
$109$	−12.4480			−1.19230			−0.596150	+	0.802873i	$0.703304\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−9.43541	+	9.43541i	−0.887609	+	0.887609i	−0.994293	−	0.106684i	$-0.965977\pi$
$113$	−9.43541	+	9.43541i	−0.887609	+	0.887609i	0.106684	+	0.994293i	$0.465977\pi$
$114$		0			0
$115$	4.23147	−	0.730691i	0.394587	−	0.0681373i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−6.88470			−0.631120
$120$		0			0
$121$	−10.4734			−0.952128
$122$		0			0
$123$		0			0
$124$		0			0
$125$	10.7680	+	3.00818i	0.963124	+	0.269060i
$126$		0			0
$127$	−9.88355	+	9.88355i	−0.877024	+	0.877024i	−0.993226	−	0.116202i	$-0.962928\pi$
$127$	−9.88355	+	9.88355i	−0.877024	+	0.877024i	0.116202	+	0.993226i	$0.462928\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	14.0012			1.22329			0.611647	−	0.791131i	$-0.290507\pi$
$131$	14.0012			1.22329			0.611647	+	0.791131i	$0.290507\pi$
$132$		0			0
$133$	9.33375	+	9.33375i	0.809338	+	0.809338i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	4.31776	+	4.31776i	0.368891	+	0.368891i	0.867073	−	0.498182i	$-0.165999\pi$
$137$	4.31776	+	4.31776i	0.368891	+	0.368891i	−0.498182	+	0.867073i	$0.665999\pi$
$138$		0			0
$139$			10.2753i			0.871538i	0.900058	+	0.435769i	$0.143524\pi$
$139$			10.2753i			0.871538i	−0.900058	+	0.435769i	$0.856476\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−3.32166	−	3.32166i	−0.277772	−	0.277772i
$144$		0			0
$145$	3.76266	−	5.33336i	0.312472	−	0.442912i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−3.03780			−0.248867			−0.124433	−	0.992228i	$-0.539711\pi$
$149$	−3.03780			−0.248867			−0.124433	+	0.992228i	$0.539711\pi$
$150$		0			0
$151$		−	5.52994i		−	0.450020i	−0.974356	−	0.225010i	$-0.927758\pi$
$151$		−	5.52994i		−	0.450020i	0.974356	−	0.225010i	$-0.0722415\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	10.4373	+	7.36346i	0.838344	+	0.591447i
$156$		0			0
$157$	4.36202	−	4.36202i	0.348127	−	0.348127i	−0.511284	−	0.859412i	$-0.670830\pi$
$157$	4.36202	−	4.36202i	0.348127	−	0.348127i	0.859412	+	0.511284i	$0.170830\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−3.95241			−0.311494
$162$		0			0
$163$	9.10371	−	9.10371i	0.713058	−	0.713058i	−0.254116	−	0.967174i	$-0.581785\pi$
$163$	9.10371	−	9.10371i	0.713058	−	0.713058i	0.967174	+	0.254116i	$0.0817845\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	15.2102	−	15.2102i	1.17700	−	1.17700i	0.196497	−	0.980504i	$-0.437043\pi$
$167$	15.2102	−	15.2102i	1.17700	−	1.17700i	0.980504	−	0.196497i	$-0.0629566\pi$
$168$		0			0
$169$			28.9048i			2.22344i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	5.23564	+	5.23564i	0.398058	+	0.398058i	0.877548	−	0.479489i	$-0.159178\pi$
$173$	5.23564	+	5.23564i	0.398058	+	0.398058i	−0.479489	+	0.877548i	$0.659178\pi$
$174$		0			0
$175$	−9.29554	−	4.41495i	−0.702677	−	0.333739i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			16.3444i			1.22164i	0.791771	+	0.610819i	$0.209160\pi$
$179$			16.3444i			1.22164i	−0.791771	+	0.610819i	$0.790840\pi$
$180$		0			0
$181$			1.33892i			0.0995215i	0.998761	+	0.0497608i	$0.0158459\pi$
$181$			1.33892i			0.0995215i	−0.998761	+	0.0497608i	$0.984154\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−1.42973	−	8.27963i	−0.105116	−	0.608731i
$186$		0			0
$187$	−1.71646	−	1.71646i	−0.125520	−	0.125520i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			18.5684i			1.34356i	0.740750	+	0.671780i	$0.234470\pi$
$191$			18.5684i			1.34356i	−0.740750	+	0.671780i	$0.765530\pi$
$192$		0			0
$193$	2.10748	−	2.10748i	0.151700	−	0.151700i	−0.627177	−	0.778877i	$-0.715790\pi$
$193$	2.10748	−	2.10748i	0.151700	−	0.151700i	0.778877	+	0.627177i	$0.215790\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−8.63291	+	8.63291i	−0.615069	+	0.615069i	−0.944263	−	0.329193i	$-0.893223\pi$
$197$	−8.63291	+	8.63291i	−0.615069	+	0.615069i	0.329193	+	0.944263i	$0.393223\pi$
$198$		0			0
$199$	−2.36127			−0.167386			−0.0836932	−	0.996492i	$-0.526672\pi$
$199$	−2.36127			−0.167386			−0.0836932	+	0.996492i	$0.526672\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−4.24808	+	4.24808i	−0.298157	+	0.298157i
$204$		0			0
$205$	1.32287	−	1.87510i	0.0923932	−	0.130962i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			4.65409i			0.321930i
$210$		0			0
$211$	3.38157			0.232797			0.116398	−	0.993203i	$-0.462865\pi$
$211$	3.38157			0.232797			0.116398	+	0.993203i	$0.462865\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	3.97507	+	23.0199i	0.271098	+	1.56994i
$216$		0			0
$217$	−8.31341	−	8.31341i	−0.564351	−	0.564351i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			21.6542i			1.45662i
$222$		0			0
$223$	−9.32012	−	9.32012i	−0.624122	−	0.624122i	0.322461	−	0.946583i	$-0.395490\pi$
$223$	−9.32012	−	9.32012i	−0.624122	−	0.624122i	−0.946583	+	0.322461i	$0.895490\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−14.9247	−	14.9247i	−0.990590	−	0.990590i	0.00936616	−	0.999956i	$-0.497019\pi$
$227$	−14.9247	−	14.9247i	−0.990590	−	0.990590i	−0.999956	+	0.00936616i	$0.997019\pi$
$228$		0			0
$229$	13.5299			0.894082			0.447041	−	0.894514i	$-0.352478\pi$
$229$	13.5299			0.894082			0.447041	+	0.894514i	$0.352478\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−4.26526	+	4.26526i	−0.279426	+	0.279426i	−0.832880	−	0.553454i	$-0.813310\pi$
$233$	−4.26526	+	4.26526i	−0.279426	+	0.279426i	0.553454	+	0.832880i	$0.313310\pi$
$234$		0			0
$235$	−0.661043	−	3.82813i	−0.0431217	−	0.249720i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	2.73939			0.177196			0.0885981	−	0.996067i	$-0.471761\pi$
$239$	2.73939			0.177196			0.0885981	+	0.996067i	$0.471761\pi$
$240$		0			0
$241$	−16.4833			−1.06178			−0.530890	−	0.847440i	$-0.678142\pi$
$241$	−16.4833			−1.06178			−0.530890	+	0.847440i	$0.678142\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−5.05027	−	3.56294i	−0.322650	−	0.227628i
$246$		0			0
$247$	29.3570	−	29.3570i	1.86794	−	1.86794i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−14.0488			−0.886754			−0.443377	−	0.896335i	$-0.646220\pi$
$251$	−14.0488			−0.886754			−0.443377	+	0.896335i	$0.646220\pi$
$252$		0			0
$253$	−0.985396	−	0.985396i	−0.0619513	−	0.0619513i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−4.43264	−	4.43264i	−0.276501	−	0.276501i	0.555210	−	0.831710i	$-0.312638\pi$
$257$	−4.43264	−	4.43264i	−0.276501	−	0.276501i	−0.831710	+	0.555210i	$0.812638\pi$
$258$		0			0
$259$			7.73360i			0.480543i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	19.3432	+	19.3432i	1.19275	+	1.19275i	0.976292	+	0.216458i	$0.0694505\pi$
$263$	19.3432	+	19.3432i	1.19275	+	1.19275i	0.216458	+	0.976292i	$0.430549\pi$
$264$		0			0
$265$	29.5714	−	5.10639i	1.81656	−	0.313683i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	24.0508			1.46640			0.733201	−	0.680012i	$-0.238026\pi$
$269$	24.0508			1.46640			0.733201	+	0.680012i	$0.238026\pi$
$270$		0			0
$271$			24.3905i			1.48162i	0.671716	+	0.740809i	$0.265558\pi$
$271$			24.3905i			1.48162i	−0.671716	+	0.740809i	$0.734442\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1.21680	−	3.41823i	−0.0733761	−	0.206127i
$276$		0			0
$277$	6.76755	−	6.76755i	0.406623	−	0.406623i	−0.473936	−	0.880559i	$-0.657167\pi$
$277$	6.76755	−	6.76755i	0.406623	−	0.406623i	0.880559	+	0.473936i	$0.157167\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−4.79740			−0.286189			−0.143094	−	0.989709i	$-0.545705\pi$
$281$	−4.79740			−0.286189			−0.143094	+	0.989709i	$0.545705\pi$
$282$		0			0
$283$	1.57491	−	1.57491i	0.0936188	−	0.0936188i	−0.658746	−	0.752365i	$-0.728913\pi$
$283$	1.57491	−	1.57491i	0.0936188	−	0.0936188i	0.752365	+	0.658746i	$0.228913\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−1.49353	+	1.49353i	−0.0881604	+	0.0881604i
$288$		0			0
$289$		−	5.81028i		−	0.341781i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	16.8458	+	16.8458i	0.984141	+	0.984141i	0.999876	−	0.0157353i	$-0.00500890\pi$
$293$	16.8458	+	16.8458i	0.984141	+	0.984141i	−0.0157353	+	0.999876i	$0.505009\pi$
$294$		0			0
$295$	11.7486	+	8.28861i	0.684032	+	0.482581i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			12.4314i			0.718924i
$300$		0			0
$301$		−	21.5017i		−	1.23934i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	2.15793	+	1.52241i	0.123563	+	0.0871728i
$306$		0			0
$307$	−13.5999	−	13.5999i	−0.776185	−	0.776185i	0.202994	−	0.979180i	$-0.434933\pi$
$307$	−13.5999	−	13.5999i	−0.776185	−	0.776185i	−0.979180	+	0.202994i	$0.934933\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			6.98730i			0.396213i	0.980180	+	0.198107i	$0.0634792\pi$
$311$			6.98730i			0.396213i	−0.980180	+	0.198107i	$0.936521\pi$
$312$		0			0
$313$	13.3659	−	13.3659i	0.755486	−	0.755486i	−0.220011	−	0.975497i	$-0.570609\pi$
$313$	13.3659	−	13.3659i	0.755486	−	0.755486i	0.975497	+	0.220011i	$0.0706093\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	2.24312	−	2.24312i	0.125986	−	0.125986i	−0.641302	−	0.767288i	$-0.721605\pi$
$317$	2.24312	−	2.24312i	0.125986	−	0.125986i	0.767288	+	0.641302i	$0.221605\pi$
$318$		0			0
$319$	−2.11822			−0.118597
$320$		0			0
$321$		0			0
$322$		0			0
$323$	15.1702	−	15.1702i	0.844090	−	0.844090i
$324$		0			0
$325$	−13.8861	+	29.2369i	−0.770265	+	1.62177i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			3.57567i			0.197133i
$330$		0			0
$331$	0.360999			0.0198423			0.00992116	−	0.999951i	$-0.496842\pi$
$331$	0.360999			0.0198423			0.00992116	+	0.999951i	$0.496842\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	9.43028	−	1.62842i	0.515231	−	0.0889702i
$336$		0			0
$337$	−0.546946	−	0.546946i	−0.0297941	−	0.0297941i	0.692053	−	0.721847i	$-0.256707\pi$
$337$	−0.546946	−	0.546946i	−0.0297941	−	0.0297941i	−0.721847	+	0.692053i	$0.756707\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	4.14532i		−	0.224482i
$342$		0			0
$343$	14.2099	+	14.2099i	0.767261	+	0.767261i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	11.5591	+	11.5591i	0.620526	+	0.620526i	0.945666	−	0.325140i	$-0.105411\pi$
$347$	11.5591	+	11.5591i	0.620526	+	0.620526i	−0.325140	+	0.945666i	$0.605411\pi$
$348$		0			0
$349$	−10.5842			−0.566562			−0.283281	−	0.959037i	$-0.591423\pi$
$349$	−10.5842			−0.566562			−0.283281	+	0.959037i	$0.591423\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	16.2334	−	16.2334i	0.864016	−	0.864016i	−0.127785	−	0.991802i	$-0.540787\pi$
$353$	16.2334	−	16.2334i	0.864016	−	0.864016i	0.991802	+	0.127785i	$0.0407869\pi$
$354$		0			0
$355$	11.9797	+	8.45163i	0.635818	+	0.448566i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	6.79961			0.358869			0.179435	−	0.983770i	$-0.442573\pi$
$359$	6.79961			0.358869			0.179435	+	0.983770i	$0.442573\pi$
$360$		0			0
$361$	−22.1330			−1.16490
$362$		0			0
$363$		0			0
$364$		0			0
$365$	2.36078	+	13.6714i	0.123569	+	0.715592i
$366$		0			0
$367$	−3.23043	+	3.23043i	−0.168627	+	0.168627i	−0.786376	−	0.617748i	$-0.788045\pi$
$367$	−3.23043	+	3.23043i	−0.168627	+	0.168627i	0.617748	+	0.786376i	$0.288045\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−27.6212			−1.43402
$372$		0			0
$373$	2.12448	+	2.12448i	0.110001	+	0.110001i	0.759965	−	0.649964i	$-0.225216\pi$
$373$	2.12448	+	2.12448i	0.110001	+	0.110001i	−0.649964	+	0.759965i	$0.725216\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	13.3613	+	13.3613i	0.688142	+	0.688142i
$378$		0			0
$379$			19.0820i			0.980175i	0.871673	+	0.490087i	$0.163035\pi$
$379$			19.0820i			0.980175i	−0.871673	+	0.490087i	$0.836965\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−2.96986	−	2.96986i	−0.151753	−	0.151753i	0.627148	−	0.778900i	$-0.284222\pi$
$383$	−2.96986	−	2.96986i	−0.151753	−	0.151753i	−0.778900	+	0.627148i	$0.784222\pi$
$384$		0			0
$385$	0.568279	+	3.29094i	0.0289622	+	0.167722i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	0.831983			0.0421832			0.0210916	−	0.999778i	$-0.493286\pi$
$389$	0.831983			0.0421832			0.0210916	+	0.999778i	$0.493286\pi$
$390$		0			0
$391$			6.42387i			0.324869i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	8.70384	−	12.3372i	0.437938	−	0.620752i
$396$		0			0
$397$	−11.7504	+	11.7504i	−0.589736	+	0.589736i	−0.937560	−	0.347824i	$-0.886921\pi$
$397$	−11.7504	+	11.7504i	−0.589736	+	0.589736i	0.347824	+	0.937560i	$0.386921\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−27.1058			−1.35360			−0.676799	−	0.736168i	$-0.736633\pi$
$401$	−27.1058			−1.35360			−0.676799	+	0.736168i	$0.736633\pi$
$402$		0			0
$403$	−26.1478	+	26.1478i	−1.30252	+	1.30252i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−1.92810	+	1.92810i	−0.0955725	+	0.0955725i
$408$		0			0
$409$			13.2732i			0.656319i	0.944622	+	0.328159i	$0.106428\pi$
$409$			13.2732i			0.656319i	−0.944622	+	0.328159i	$0.893572\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−9.35791	−	9.35791i	−0.460473	−	0.460473i
$414$		0			0
$415$	0.738307	+	4.27558i	0.0362421	+	0.209880i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			24.3069i			1.18747i	0.804662	+	0.593734i	$0.202347\pi$
$419$			24.3069i			1.18747i	−0.804662	+	0.593734i	$0.797653\pi$
$420$		0			0
$421$			1.60486i			0.0782162i	0.999235	+	0.0391081i	$0.0124517\pi$
$421$			1.60486i			0.0782162i	−0.999235	+	0.0391081i	$0.987548\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−7.17563	+	15.1081i	−0.348069	+	0.732848i
$426$		0			0
$427$	−1.71881	−	1.71881i	−0.0831792	−	0.0831792i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	16.0008i		−	0.770733i	−0.922764	−	0.385367i	$-0.874075\pi$
$431$		−	16.0008i		−	0.770733i	0.922764	−	0.385367i	$-0.125925\pi$
$432$		0			0
$433$	−17.6673	+	17.6673i	−0.849037	+	0.849037i	−0.990013	−	0.140976i	$-0.954976\pi$
$433$	−17.6673	+	17.6673i	−0.849037	+	0.849037i	0.140976	+	0.990013i	$0.454976\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	8.70898	−	8.70898i	0.416607	−	0.416607i
$438$		0			0
$439$	34.5085			1.64700			0.823500	−	0.567316i	$-0.192018\pi$
$439$	34.5085			1.64700			0.823500	+	0.567316i	$0.192018\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−2.03044	+	2.03044i	−0.0964691	+	0.0964691i	−0.753694	−	0.657225i	$-0.771730\pi$
$443$	−2.03044	+	2.03044i	−0.0964691	+	0.0964691i	0.657225	+	0.753694i	$0.271730\pi$
$444$		0			0
$445$	8.46071	+	5.96899i	0.401076	+	0.282957i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			19.7303i			0.931129i	0.885014	+	0.465565i	$0.154149\pi$
$449$			19.7303i			0.931129i	−0.885014	+	0.465565i	$0.845851\pi$
$450$		0			0
$451$	−0.744720			−0.0350675
$452$		0			0
$453$		0			0
$454$		0			0
$455$	17.1740	−	24.3431i	0.805128	−	1.14122i
$456$		0			0
$457$	−22.2146	−	22.2146i	−1.03915	−	1.03915i	−0.999202	−	0.0399513i	$-0.987280\pi$
$457$	−22.2146	−	22.2146i	−1.03915	−	1.03915i	−0.0399513	−	0.999202i	$-0.512720\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	20.6988i		−	0.964039i	−0.876161	−	0.482019i	$-0.839904\pi$
$461$		−	20.6988i		−	0.964039i	0.876161	−	0.482019i	$-0.160096\pi$
$462$		0			0
$463$	−5.67502	−	5.67502i	−0.263741	−	0.263741i	0.562831	−	0.826572i	$-0.309712\pi$
$463$	−5.67502	−	5.67502i	−0.263741	−	0.263741i	−0.826572	+	0.562831i	$0.809712\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	3.78863	+	3.78863i	0.175317	+	0.175317i	0.789311	−	0.613994i	$-0.210438\pi$
$467$	3.78863	+	3.78863i	0.175317	+	0.175317i	−0.613994	+	0.789311i	$0.710438\pi$
$468$		0			0
$469$	−8.80837			−0.406733
$470$		0			0
$471$		0			0
$472$		0			0
$473$	5.36071	−	5.36071i	0.246485	−	0.246485i
$474$		0			0
$475$	30.2105	−	10.7542i	1.38615	−	0.493435i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−11.8931			−0.543409			−0.271705	−	0.962381i	$-0.587587\pi$
$479$	−11.8931			−0.543409			−0.271705	+	0.962381i	$0.587587\pi$
$480$		0			0
$481$	24.3242			1.10909
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−10.2203	+	1.76484i	−0.464079	+	0.0801372i
$486$		0			0
$487$	−22.1137	+	22.1137i	−1.00207	+	1.00207i	−0.00207135	+	0.999998i	$0.500659\pi$
$487$	−22.1137	+	22.1137i	−1.00207	+	1.00207i	−0.999998	+	0.00207135i	$0.999341\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	17.8731			0.806600			0.403300	−	0.915068i	$-0.367863\pi$
$491$	17.8731			0.806600			0.403300	+	0.915068i	$0.367863\pi$
$492$		0			0
$493$	6.90441	+	6.90441i	0.310959	+	0.310959i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−9.54197	−	9.54197i	−0.428016	−	0.428016i
$498$		0			0
$499$			10.6090i			0.474923i	0.971397	+	0.237461i	$0.0763153\pi$
$499$			10.6090i			0.474923i	−0.971397	+	0.237461i	$0.923685\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	8.19221	+	8.19221i	0.365273	+	0.365273i	0.865750	−	0.500477i	$-0.166842\pi$
$503$	8.19221	+	8.19221i	0.365273	+	0.365273i	−0.500477	+	0.865750i	$0.666842\pi$
$504$		0			0
$505$	24.8107	+	17.5038i	1.10406	+	0.778909i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−35.0769			−1.55476			−0.777379	−	0.629033i	$-0.783451\pi$
$509$	−35.0769			−1.55476			−0.777379	+	0.629033i	$0.783451\pi$
$510$		0			0
$511$		−	12.7698i		−	0.564902i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−24.6270	+	4.25260i	−1.08520	+	0.187392i
$516$		0			0
$517$	−0.891469	+	0.891469i	−0.0392068	+	0.0392068i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	13.5092			0.591848			0.295924	−	0.955211i	$-0.404372\pi$
$521$	13.5092			0.591848			0.295924	+	0.955211i	$0.404372\pi$
$522$		0			0
$523$	−3.12911	+	3.12911i	−0.136827	+	0.136827i	−0.772203	−	0.635376i	$-0.780845\pi$
$523$	−3.12911	+	3.12911i	−0.136827	+	0.136827i	0.635376	+	0.772203i	$0.280845\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−13.5118	+	13.5118i	−0.588583	+	0.588583i
$528$		0			0
$529$		−	19.3121i		−	0.839659i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	4.69754	+	4.69754i	0.203473	+	0.203473i
$534$		0			0
$535$	34.8163	−	6.01208i	1.50524	−	0.259925i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			2.00578i			0.0863952i
$540$		0			0
$541$			39.9149i			1.71608i	0.513585	+	0.858038i	$0.328317\pi$
$541$			39.9149i			1.71608i	−0.513585	+	0.858038i	$0.671683\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	16.0458	−	22.7441i	0.687327	−	0.974249i
$546$		0			0
$547$	−24.0087	−	24.0087i	−1.02654	−	1.02654i	−0.999638	−	0.0269017i	$-0.991436\pi$
$547$	−24.0087	−	24.0087i	−1.02654	−	1.02654i	−0.0269017	−	0.999638i	$-0.508564\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	18.7209i		−	0.797538i
$552$		0			0
$553$	−9.82671	+	9.82671i	−0.417874	+	0.417874i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−5.38621	+	5.38621i	−0.228221	+	0.228221i	−0.811949	−	0.583728i	$-0.801593\pi$
$557$	−5.38621	+	5.38621i	−0.228221	+	0.228221i	0.583728	+	0.811949i	$0.301593\pi$
$558$		0			0
$559$	−67.6285			−2.86038
$560$		0			0
$561$		0			0
$562$		0			0
$563$	11.0962	−	11.0962i	0.467648	−	0.467648i	−0.433504	−	0.901152i	$-0.642723\pi$
$563$	11.0962	−	11.0962i	0.467648	−	0.467648i	0.901152	+	0.433504i	$0.142723\pi$
$564$		0			0
$565$	−5.07718	−	29.4022i	−0.213599	−	1.23696i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	13.7723i		−	0.577366i	−0.957425	−	0.288683i	$-0.906783\pi$
$569$		−	13.7723i		−	0.577366i	0.957425	−	0.288683i	$-0.0932173\pi$
$570$		0			0
$571$	2.65223			0.110992			0.0554962	−	0.998459i	$-0.482326\pi$
$571$	2.65223			0.110992			0.0554962	+	0.998459i	$0.482326\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−4.11943	+	8.67333i	−0.171792	+	0.361703i
$576$		0			0
$577$	−15.4206	−	15.4206i	−0.641968	−	0.641968i	0.309071	−	0.951039i	$-0.399982\pi$
$577$	−15.4206	−	15.4206i	−0.641968	−	0.641968i	−0.951039	+	0.309071i	$0.899982\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	3.99361i		−	0.165683i
$582$		0			0
$583$	−6.88638	−	6.88638i	−0.285205	−	0.285205i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	24.4859	+	24.4859i	1.01064	+	1.01064i	0.999943	+	0.0107002i	$0.00340605\pi$
$587$	24.4859	+	24.4859i	1.01064	+	1.01064i	0.0107002	+	0.999943i	$0.496594\pi$
$588$		0			0
$589$	36.6365			1.50958
$590$		0			0
$591$		0			0
$592$		0			0
$593$	29.7519	−	29.7519i	1.22176	−	1.22176i	0.254759	−	0.967005i	$-0.418004\pi$
$593$	29.7519	−	29.7519i	1.22176	−	1.22176i	0.967005	−	0.254759i	$-0.0819961\pi$
$594$		0			0
$595$	8.87459	−	12.5792i	0.363823	−	0.515699i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	2.67797			0.109419			0.0547094	−	0.998502i	$-0.482577\pi$
$599$	2.67797			0.109419			0.0547094	+	0.998502i	$0.482577\pi$
$600$		0			0
$601$	14.2758			0.582321			0.291161	−	0.956674i	$-0.405959\pi$
$601$	14.2758			0.582321			0.291161	+	0.956674i	$0.405959\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	13.5005	−	19.1363i	0.548875	−	0.778000i
$606$		0			0
$607$	4.79670	−	4.79670i	0.194692	−	0.194692i	−0.603028	−	0.797720i	$-0.706039\pi$
$607$	4.79670	−	4.79670i	0.194692	−	0.194692i	0.797720	+	0.603028i	$0.206039\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	11.2464			0.454981
$612$		0			0
$613$	−8.18940	−	8.18940i	−0.330767	−	0.330767i	0.522111	−	0.852878i	$-0.325145\pi$
$613$	−8.18940	−	8.18940i	−0.330767	−	0.330767i	−0.852878	+	0.522111i	$0.825145\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−19.8311	−	19.8311i	−0.798368	−	0.798368i	0.184470	−	0.982838i	$-0.440943\pi$
$617$	−19.8311	−	19.8311i	−0.798368	−	0.798368i	−0.982838	+	0.184470i	$0.940943\pi$
$618$		0			0
$619$		−	14.8818i		−	0.598149i	−0.954230	−	0.299075i	$-0.903322\pi$
$619$		−	14.8818i		−	0.598149i	0.954230	−	0.299075i	$-0.0966780\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−6.73904	−	6.73904i	−0.269994	−	0.269994i
$624$		0			0
$625$	−19.3767	+	15.7970i	−0.775067	+	0.631879i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	12.5694			0.501176
$630$		0			0
$631$		−	25.6770i		−	1.02218i	−0.859526	−	0.511092i	$-0.829241\pi$
$631$		−	25.6770i		−	1.02218i	0.859526	−	0.511092i	$-0.170759\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−5.31833	−	30.7987i	−0.211051	−	1.22221i
$636$		0			0
$637$	12.6521	−	12.6521i	0.501293	−	0.501293i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−21.1942			−0.837122			−0.418561	−	0.908189i	$-0.637465\pi$
$641$	−21.1942			−0.837122			−0.418561	+	0.908189i	$0.637465\pi$
$642$		0			0
$643$	21.5857	−	21.5857i	0.851256	−	0.851256i	−0.139032	−	0.990288i	$-0.544399\pi$
$643$	21.5857	−	21.5857i	0.851256	−	0.851256i	0.990288	+	0.139032i	$0.0443992\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	4.64626	−	4.64626i	0.182663	−	0.182663i	−0.609852	−	0.792515i	$-0.708771\pi$
$647$	4.64626	−	4.64626i	0.182663	−	0.182663i	0.792515	+	0.609852i	$0.208771\pi$
$648$		0			0
$649$		−	4.66614i		−	0.183162i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−5.98212	−	5.98212i	−0.234098	−	0.234098i	0.580303	−	0.814401i	$-0.302934\pi$
$653$	−5.98212	−	5.98212i	−0.234098	−	0.234098i	−0.814401	+	0.580303i	$0.802934\pi$
$654$		0			0
$655$	−18.0480	+	25.5821i	−0.705195	+	0.999575i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	41.9465i		−	1.63400i	−0.576635	−	0.817002i	$-0.695635\pi$
$659$		−	41.9465i		−	1.63400i	0.576635	−	0.817002i	$-0.304365\pi$
$660$		0			0
$661$		−	49.8515i		−	1.93900i	−0.245095	−	0.969499i	$-0.578819\pi$
$661$		−	49.8515i		−	1.93900i	0.245095	−	0.969499i	$-0.421181\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−29.0854	+	5.02248i	−1.12789	+	0.194763i
$666$		0			0
$667$	3.96373	+	3.96373i	0.153476	+	0.153476i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	0.857052i		−	0.0330861i
$672$		0			0
$673$	14.3656	−	14.3656i	0.553754	−	0.553754i	−0.373768	−	0.927522i	$-0.621934\pi$
$673$	14.3656	−	14.3656i	0.553754	−	0.553754i	0.927522	+	0.373768i	$0.121934\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−1.59459	+	1.59459i	−0.0612852	+	0.0612852i	−0.737085	−	0.675800i	$-0.763798\pi$
$677$	−1.59459	+	1.59459i	−0.0612852	+	0.0612852i	0.675800	+	0.737085i	$0.263798\pi$
$678$		0			0
$679$	9.54627			0.366352
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−3.56279	+	3.56279i	−0.136326	+	0.136326i	−0.771977	−	0.635651i	$-0.780732\pi$
$683$	−3.56279	+	3.56279i	−0.136326	+	0.136326i	0.635651	+	0.771977i	$0.280732\pi$
$684$		0			0
$685$	−13.4548	+	2.32338i	−0.514083	+	0.0887719i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			86.8758i			3.30970i
$690$		0			0
$691$	−5.67883			−0.216033			−0.108016	−	0.994149i	$-0.534450\pi$
$691$	−5.67883			−0.216033			−0.108016	+	0.994149i	$0.534450\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−18.7743	−	13.2452i	−0.712149	−	0.502417i
$696$		0			0
$697$	2.42744	+	2.42744i	0.0919459	+	0.0919459i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			14.1700i			0.535194i	0.963531	+	0.267597i	$0.0862296\pi$
$701$			14.1700i			0.535194i	−0.963531	+	0.267597i	$0.913770\pi$
$702$		0			0
$703$	−17.0407	−	17.0407i	−0.642701	−	0.642701i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−19.7620	−	19.7620i	−0.743225	−	0.743225i
$708$		0			0
$709$	−14.5473			−0.546337			−0.273168	−	0.961966i	$-0.588072\pi$
$709$	−14.5473			−0.546337			−0.273168	+	0.961966i	$0.588072\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−7.75694	+	7.75694i	−0.290500	+	0.290500i
$714$		0			0
$715$	10.3508	−	1.78738i	0.387099	−	0.0668444i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−41.3534			−1.54222			−0.771110	−	0.636702i	$-0.780298\pi$
$719$	−41.3534			−1.54222			−0.771110	+	0.636702i	$0.780298\pi$
$720$		0			0
$721$	23.0029			0.856673
$722$		0			0
$723$		0			0
$724$		0			0
$725$	4.89456	+	13.7497i	0.181779	+	0.510652i
$726$		0			0
$727$	17.1262	−	17.1262i	0.635177	−	0.635177i	−0.314185	−	0.949362i	$-0.601731\pi$
$727$	17.1262	−	17.1262i	0.635177	−	0.635177i	0.949362	+	0.314185i	$0.101731\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−34.9468			−1.29255
$732$		0			0
$733$	−10.8014	−	10.8014i	−0.398958	−	0.398958i	0.478907	−	0.877865i	$-0.341033\pi$
$733$	−10.8014	−	10.8014i	−0.398958	−	0.398958i	−0.877865	+	0.478907i	$0.841033\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−2.19606	−	2.19606i	−0.0808929	−	0.0808929i
$738$		0			0
$739$		−	28.4640i		−	1.04707i	−0.852006	−	0.523533i	$-0.824614\pi$
$739$		−	28.4640i		−	1.04707i	0.852006	−	0.523533i	$-0.175386\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−37.4663	−	37.4663i	−1.37451	−	1.37451i	−0.853634	−	0.520873i	$-0.825607\pi$
$743$	−37.4663	−	37.4663i	−1.37451	−	1.37451i	−0.520873	−	0.853634i	$-0.674393\pi$
$744$		0			0
$745$	3.91582	−	5.55046i	0.143465	−	0.203353i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−32.5202			−1.18826
$750$		0			0
$751$		−	33.7409i		−	1.23122i	−0.788050	−	0.615611i	$-0.788909\pi$
$751$		−	33.7409i		−	1.23122i	0.788050	−	0.615611i	$-0.211091\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	10.1039	+	7.12826i	0.367719	+	0.259424i
$756$		0			0
$757$	3.61554	−	3.61554i	0.131409	−	0.131409i	−0.638343	−	0.769752i	$-0.720380\pi$
$757$	3.61554	−	3.61554i	0.131409	−	0.131409i	0.769752	+	0.638343i	$0.220380\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	53.5331			1.94057			0.970286	−	0.241959i	$-0.0777901\pi$
$761$	53.5331			1.94057			0.970286	+	0.241959i	$0.0777901\pi$
$762$		0			0
$763$	−18.1159	+	18.1159i	−0.655839	+	0.655839i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−29.4330	+	29.4330i	−1.06277	+	1.06277i
$768$		0			0
$769$			0.470664i			0.0169726i	0.999964	+	0.00848630i	$0.00270130\pi$
$769$			0.470664i			0.0169726i	−0.999964	+	0.00848630i	$0.997299\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	13.2975	+	13.2975i	0.478279	+	0.478279i	0.904581	−	0.426302i	$-0.140184\pi$
$773$	13.2975	+	13.2975i	0.478279	+	0.478279i	−0.426302	+	0.904581i	$0.640184\pi$
$774$		0			0
$775$	−26.9080	+	9.57857i	−0.966563	+	0.344072i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	6.58187i		−	0.235820i
$780$		0			0
$781$		−	4.75791i		−	0.170252i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	2.34720	+	13.5928i	0.0837751	+	0.485146i
$786$		0			0
$787$	13.2592	+	13.2592i	0.472639	+	0.472639i	0.902768	−	0.430128i	$-0.141532\pi$
$787$	13.2592	+	13.2592i	0.472639	+	0.472639i	−0.430128	+	0.902768i	$0.641532\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			27.4632i			0.976480i
$792$		0			0
$793$	−5.40611	+	5.40611i	−0.191977	+	0.191977i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	17.1774	−	17.1774i	0.608453	−	0.608453i	−0.334089	−	0.942542i	$-0.608428\pi$
$797$	17.1774	−	17.1774i	0.608453	−	0.608453i	0.942542	+	0.334089i	$0.108428\pi$
$798$		0			0
$799$	5.81155			0.205598
$800$		0			0
$801$		0			0
$802$		0			0
$803$	3.18370	−	3.18370i	0.112350	−	0.112350i
$804$		0			0
$805$	5.09478	−	7.22157i	0.179567	−	0.254527i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	41.7364i		−	1.46737i	−0.679488	−	0.733686i	$-0.737798\pi$
$809$		−	41.7364i		−	1.46737i	0.679488	−	0.733686i	$-0.262202\pi$
$810$		0			0
$811$	−34.4392			−1.20932			−0.604661	−	0.796483i	$-0.706692\pi$
$811$	−34.4392			−1.20932			−0.604661	+	0.796483i	$0.706692\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	4.89870	+	28.3686i	0.171594	+	0.993709i
$816$		0			0
$817$	47.3782	+	47.3782i	1.65755	+	1.65755i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			12.0902i			0.421952i	0.977491	+	0.210976i	$0.0676642\pi$
$821$			12.0902i			0.421952i	−0.977491	+	0.210976i	$0.932336\pi$
$822$		0			0
$823$	−10.0784	−	10.0784i	−0.351310	−	0.351310i	0.509287	−	0.860597i	$-0.329909\pi$
$823$	−10.0784	−	10.0784i	−0.351310	−	0.351310i	−0.860597	+	0.509287i	$0.829909\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	22.1941	+	22.1941i	0.771763	+	0.771763i	0.978415	−	0.206652i	$-0.0662567\pi$
$827$	22.1941	+	22.1941i	0.771763	+	0.771763i	−0.206652	+	0.978415i	$0.566257\pi$
$828$		0			0
$829$	2.71154			0.0941755			0.0470878	−	0.998891i	$-0.485006\pi$
$829$	2.71154			0.0941755			0.0470878	+	0.998891i	$0.485006\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	6.53792	−	6.53792i	0.226526	−	0.226526i
$834$		0			0
$835$	8.18460	+	47.3974i	0.283240	+	1.64026i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	53.2192			1.83733			0.918666	−	0.395036i	$-0.129268\pi$
$839$	53.2192			1.83733			0.918666	+	0.395036i	$0.129268\pi$
$840$		0			0
$841$	−20.4795			−0.706191
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−52.8127	−	37.2591i	−1.81681	−	1.28175i
$846$		0			0
$847$	−15.2422	+	15.2422i	−0.523729	+	0.523729i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	7.21594			0.247359
$852$		0			0
$853$	−18.5528	−	18.5528i	−0.635236	−	0.635236i	0.314141	−	0.949376i	$-0.398284\pi$
$853$	−18.5528	−	18.5528i	−0.635236	−	0.635236i	−0.949376	+	0.314141i	$0.898284\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	12.8811	+	12.8811i	0.440009	+	0.440009i	0.892015	−	0.452006i	$-0.149291\pi$
$857$	12.8811	+	12.8811i	0.440009	+	0.440009i	−0.452006	+	0.892015i	$0.649291\pi$
$858$		0			0
$859$		−	3.82914i		−	0.130648i	−0.997864	−	0.0653242i	$-0.979192\pi$
$859$		−	3.82914i		−	0.130648i	0.997864	−	0.0653242i	$-0.0208082\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−16.5089	−	16.5089i	−0.561970	−	0.561970i	0.367897	−	0.929867i	$-0.380078\pi$
$863$	−16.5089	−	16.5089i	−0.561970	−	0.561970i	−0.929867	+	0.367897i	$0.880078\pi$
$864$		0			0
$865$	−16.3151	+	2.81729i	−0.554730	+	0.0957908i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−4.89989			−0.166218
$870$		0			0
$871$			27.7046i			0.938734i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	20.0489	−	11.2932i	0.677777	−	0.381778i
$876$		0			0
$877$	27.6423	−	27.6423i	0.933416	−	0.933416i	−0.0645020	−	0.997918i	$-0.520546\pi$
$877$	27.6423	−	27.6423i	0.933416	−	0.933416i	0.997918	+	0.0645020i	$0.0205459\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	14.6973			0.495165			0.247582	−	0.968867i	$-0.420364\pi$
$881$	14.6973			0.495165			0.247582	+	0.968867i	$0.420364\pi$
$882$		0			0
$883$	31.5345	−	31.5345i	1.06122	−	1.06122i	0.0632202	−	0.998000i	$-0.479863\pi$
$883$	31.5345	−	31.5345i	1.06122	−	1.06122i	0.998000	−	0.0632202i	$-0.0201370\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−18.5981	+	18.5981i	−0.624464	+	0.624464i	−0.946670	−	0.322206i	$-0.895576\pi$
$887$	−18.5981	+	18.5981i	−0.624464	+	0.624464i	0.322206	+	0.946670i	$0.395576\pi$
$888$		0			0
$889$			28.7676i			0.964835i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−7.87885	−	7.87885i	−0.263656	−	0.263656i
$894$		0			0
$895$	−29.8633	−	21.0684i	−0.998221	−	0.704240i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			16.6744i			0.556123i
$900$		0			0
$901$			44.8928i			1.49560i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−2.44639	−	1.72591i	−0.0813207	−	0.0573713i
$906$		0			0
$907$	15.1848	+	15.1848i	0.504203	+	0.504203i	0.912741	−	0.408538i	$-0.133961\pi$
$907$	15.1848	+	15.1848i	0.504203	+	0.504203i	−0.408538	+	0.912741i	$0.633961\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	51.3671i		−	1.70187i	−0.525272	−	0.850934i	$-0.676036\pi$
$911$		−	51.3671i		−	1.70187i	0.525272	−	0.850934i	$-0.323964\pi$
$912$		0			0
$913$	0.995667	−	0.995667i	0.0329518	−	0.0329518i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	20.3764	−	20.3764i	0.672888	−	0.672888i
$918$		0			0
$919$	31.4287			1.03674			0.518369	−	0.855157i	$-0.326539\pi$
$919$	31.4287			1.03674			0.518369	+	0.855157i	$0.326539\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−30.0120	+	30.0120i	−0.987855	+	0.987855i
$924$		0			0
$925$	16.9709	+	8.06040i	0.558000	+	0.265024i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	17.2221i		−	0.565038i	−0.959262	−	0.282519i	$-0.908830\pi$
$929$		−	17.2221i		−	0.565038i	0.959262	−	0.282519i	$-0.0911700\pi$
$930$		0			0
$931$	−17.7272			−0.580986
$932$		0			0
$933$		0			0
$934$		0			0
$935$	5.34876	−	0.923625i	0.174923	−	0.0302058i
$936$		0			0
$937$	−29.3915	−	29.3915i	−0.960177	−	0.960177i	0.0390597	−	0.999237i	$-0.487564\pi$
$937$	−29.3915	−	29.3915i	−0.960177	−	0.960177i	−0.999237	+	0.0390597i	$0.987564\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	40.1614i		−	1.30922i	−0.755966	−	0.654611i	$-0.772832\pi$
$941$		−	40.1614i		−	1.30922i	0.755966	−	0.654611i	$-0.227168\pi$
$942$		0			0
$943$	1.39356	+	1.39356i	0.0453806	+	0.0453806i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	20.5497	+	20.5497i	0.667777	+	0.667777i	0.957201	−	0.289424i	$-0.0934638\pi$
$947$	20.5497	+	20.5497i	0.667777	+	0.667777i	−0.289424	+	0.957201i	$0.593464\pi$
$948$		0			0
$949$	−40.1642			−1.30379
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−14.0425	+	14.0425i	−0.454882	+	0.454882i	−0.896971	−	0.442089i	$-0.854238\pi$
$953$	−14.0425	+	14.0425i	−0.454882	+	0.454882i	0.442089	+	0.896971i	$0.354238\pi$
$954$		0			0
$955$	−33.9268	−	23.9352i	−1.09785	−	0.774525i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	12.5675			0.405826
$960$		0			0
$961$	−1.63154			−0.0526302
$962$		0			0
$963$		0			0
$964$		0			0
$965$	1.13404	+	6.56726i	0.0365059	+	0.211408i
$966$		0			0
$967$	0.168063	−	0.168063i	0.00540454	−	0.00540454i	−0.704399	−	0.709804i	$-0.748784\pi$
$967$	0.168063	−	0.168063i	0.00540454	−	0.00540454i	0.709804	+	0.704399i	$0.248784\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−45.2762			−1.45298			−0.726492	−	0.687175i	$-0.758851\pi$
$971$	−45.2762			−1.45298			−0.726492	+	0.687175i	$0.758851\pi$
$972$		0			0
$973$	14.9539	+	14.9539i	0.479400	+	0.479400i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−33.1509	−	33.1509i	−1.06059	−	1.06059i	−0.998042	−	0.0625482i	$-0.980077\pi$
$977$	−33.1509	−	33.1509i	−1.06059	−	1.06059i	−0.0625482	−	0.998042i	$-0.519923\pi$
$978$		0			0
$979$		−	3.36029i		−	0.107395i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−3.16720	−	3.16720i	−0.101018	−	0.101018i	0.654792	−	0.755809i	$-0.272756\pi$
$983$	−3.16720	−	3.16720i	−0.101018	−	0.101018i	−0.755809	+	0.654792i	$0.772756\pi$
$984$		0			0
$985$	−4.64536	−	26.9015i	−0.148013	−	0.857154i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−20.0625			−0.637950
$990$		0			0
$991$			28.3462i			0.900447i	0.892916	+	0.450224i	$0.148656\pi$
$991$			28.3462i			0.900447i	−0.892916	+	0.450224i	$0.851344\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	3.04375	−	4.31435i	0.0964935	−	0.136774i
$996$		0			0
$997$	−28.3453	+	28.3453i	−0.897705	+	0.897705i	−0.995233	−	0.0975279i	$-0.968906\pi$
$997$	−28.3453	+	28.3453i	−0.897705	+	0.897705i	0.0975279	+	0.995233i	$0.468906\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1440.2.bi.e.847.4		24
3.2	odd	2		480.2.bh.a.367.10		24
4.3	odd	2		360.2.w.e.307.2		24
5.3	odd	4	inner	1440.2.bi.e.1423.9		24
8.3	odd	2	inner	1440.2.bi.e.847.9		24
8.5	even	2		360.2.w.e.307.4		24
12.11	even	2		120.2.v.a.67.11	yes	24
15.2	even	4		2400.2.bh.b.943.2		24
15.8	even	4		480.2.bh.a.463.9		24
15.14	odd	2		2400.2.bh.b.1807.1		24
20.3	even	4		360.2.w.e.163.4		24
24.5	odd	2		120.2.v.a.67.9	yes	24
24.11	even	2		480.2.bh.a.367.9		24
40.3	even	4	inner	1440.2.bi.e.1423.4		24
40.13	odd	4		360.2.w.e.163.2		24
60.23	odd	4		120.2.v.a.43.9	✓	24
60.47	odd	4		600.2.v.b.43.4		24
60.59	even	2		600.2.v.b.307.2		24
120.29	odd	2		600.2.v.b.307.4		24
120.53	even	4		120.2.v.a.43.11	yes	24
120.59	even	2		2400.2.bh.b.1807.2		24
120.77	even	4		600.2.v.b.43.2		24
120.83	odd	4		480.2.bh.a.463.10		24
120.107	odd	4		2400.2.bh.b.943.1		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.2.v.a.43.9	✓	24	60.23	odd	4
120.2.v.a.43.11	yes	24	120.53	even	4
120.2.v.a.67.9	yes	24	24.5	odd	2
120.2.v.a.67.11	yes	24	12.11	even	2
360.2.w.e.163.2		24	40.13	odd	4
360.2.w.e.163.4		24	20.3	even	4
360.2.w.e.307.2		24	4.3	odd	2
360.2.w.e.307.4		24	8.5	even	2
480.2.bh.a.367.9		24	24.11	even	2
480.2.bh.a.367.10		24	3.2	odd	2
480.2.bh.a.463.9		24	15.8	even	4
480.2.bh.a.463.10		24	120.83	odd	4
600.2.v.b.43.2		24	120.77	even	4
600.2.v.b.43.4		24	60.47	odd	4
600.2.v.b.307.2		24	60.59	even	2
600.2.v.b.307.4		24	120.29	odd	2
1440.2.bi.e.847.4		24	1.1	even	1	trivial
1440.2.bi.e.847.9		24	8.3	odd	2	inner
1440.2.bi.e.1423.4		24	40.3	even	4	inner
1440.2.bi.e.1423.9		24	5.3	odd	4	inner
2400.2.bh.b.943.1		24	120.107	odd	4
2400.2.bh.b.943.2		24	15.2	even	4
2400.2.bh.b.1807.1		24	15.14	odd	2
2400.2.bh.b.1807.2		24	120.59	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1440.bi (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.28903 + 1.82713i) q^{5} +(1.45533 - 1.45533i) q^{7} +O(q^{10})\)	\(q+(-1.28903 + 1.82713i) q^{5} +(1.45533 - 1.45533i) q^{7} +0.725670 q^{11} +(-4.57738 - 4.57738i) q^{13} +(-2.36535 - 2.36535i) q^{17} +6.41350i q^{19} +(-1.35791 - 1.35791i) q^{23} +(-1.67680 - 4.71045i) q^{25} -2.91898 q^{29} -5.71240i q^{31} +(0.783110 + 4.53503i) q^{35} +(-2.65700 + 2.65700i) q^{37} -1.02625 q^{41} +(7.38725 - 7.38725i) q^{43} +(-1.22848 + 1.22848i) q^{47} +2.76404i q^{49} +(-9.48969 - 9.48969i) q^{53} +(-0.935410 + 1.32589i) q^{55} -6.43011i q^{59} -1.18105i q^{61} +(14.2638 - 2.46308i) q^{65} +(-3.02625 - 3.02625i) q^{67} -6.55658i q^{71} +(4.38725 - 4.38725i) q^{73} +(1.05609 - 1.05609i) q^{77} -6.75224 q^{79} +(1.37207 - 1.37207i) q^{83} +(7.37079 - 1.27279i) q^{85} -4.63060i q^{89} -13.3232 q^{91} +(-11.7183 - 8.26720i) q^{95} +(3.27977 + 3.27977i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q - 8 q^{17} - 8 q^{25} - 48 q^{35} + 32 q^{43} + 8 q^{65} - 48 q^{67} - 40 q^{73} + 80 q^{83} - 64 q^{91} - 24 q^{97}+O(q^{100}) \) 24 * q - 8 * q^17 - 8 * q^25 - 48 * q^35 + 32 * q^43 + 8 * q^65 - 48 * q^67 - 40 * q^73 + 80 * q^83 - 64 * q^91 - 24 * q^97

Introduction

L-functions

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