LMFDB - Embedded newform 2240.2.l.d.1569.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2240,2,Mod(1569,2240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2240, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2240.1569");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2240 = 2^{6} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2240.l (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$17.8864900528$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1569.1
Character	$\chi$	$=$	2240.1569
Dual form			2240.2.l.d.1569.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.11575 q^{3} +(0.660542 - 2.13628i) q^{5} +1.00000i q^{7} +6.70793 q^{9} +O(q^{10})$	$q-3.11575 q^{3} +(0.660542 - 2.13628i) q^{5} +1.00000i q^{7} +6.70793 q^{9} -0.780127i q^{11} +3.09141 q^{13} +(-2.05809 + 6.65612i) q^{15} +7.31343i q^{17} -0.207666i q^{19} -3.11575i q^{21} +1.52875i q^{23} +(-4.12737 - 2.82220i) q^{25} -11.5530 q^{27} -3.60641i q^{29} -7.94658 q^{31} +2.43068i q^{33} +(2.13628 + 0.660542i) q^{35} -3.70429 q^{37} -9.63207 q^{39} +6.24284 q^{41} +6.61428 q^{43} +(4.43087 - 14.3300i) q^{45} -2.61892i q^{47} -1.00000 q^{49} -22.7868i q^{51} -8.69448 q^{53} +(-1.66657 - 0.515307i) q^{55} +0.647035i q^{57} +14.6333i q^{59} +6.37646i q^{61} +6.70793i q^{63} +(2.04200 - 6.60411i) q^{65} -7.21281 q^{67} -4.76321i q^{69} +14.6433 q^{71} -0.502306i q^{73} +(12.8599 + 8.79329i) q^{75} +0.780127 q^{77} +11.4410 q^{79} +15.8725 q^{81} -13.8372 q^{83} +(15.6235 + 4.83083i) q^{85} +11.2367i q^{87} -16.3744 q^{89} +3.09141i q^{91} +24.7596 q^{93} +(-0.443631 - 0.137172i) q^{95} +10.2956i q^{97} -5.23304i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 4 q^{5} + 12 q^{9}+O(q^{10}) $ 24 * q + 4 * q^5 + 12 * q^9	$ 24 q + 4 q^{5} + 12 q^{9} + 4 q^{13} + 24 q^{37} + 48 q^{45} - 24 q^{49} - 88 q^{53} + 36 q^{65} - 20 q^{77} + 16 q^{81} + 56 q^{85} - 40 q^{89} - 24 q^{93}+O(q^{100}) $ 24 * q + 4 * q^5 + 12 * q^9 + 4 * q^13 + 24 * q^37 + 48 * q^45 - 24 * q^49 - 88 * q^53 + 36 * q^65 - 20 * q^77 + 16 * q^81 + 56 * q^85 - 40 * q^89 - 24 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2240\mathbb{Z}\right)^\times$.

$n$	$897$	$1471$	$1541$	$1921$
$\chi(n)$	$-1$	$1$	$-1$	$1$

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
$3$	−3.11575			−1.79888			−0.899441	−	0.437043i	$-0.856026\pi$
$3$	−3.11575			−1.79888			−0.899441	+	0.437043i	$0.856026\pi$
$4$		0			0
$5$	0.660542	−	2.13628i	0.295403	−	0.955373i
$5$	0.660542	−	2.13628i	0.295403	−	0.955373i
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		0			0
$9$	6.70793			2.23598
$10$		0			0
$11$		−	0.780127i		−	0.235217i	−0.993060	−	0.117609i	$-0.962477\pi$
$11$		−	0.780127i		−	0.235217i	0.993060	−	0.117609i	$-0.0375228\pi$
$12$		0			0
$13$	3.09141			0.857402			0.428701	−	0.903446i	$-0.358971\pi$
$13$	3.09141			0.857402			0.428701	+	0.903446i	$0.358971\pi$
$14$		0			0
$15$	−2.05809	+	6.65612i	−0.531396	+	1.71860i
$16$		0			0
$17$			7.31343i			1.77377i	0.461993	+	0.886883i	$0.347134\pi$
$17$			7.31343i			1.77377i	−0.461993	+	0.886883i	$0.652866\pi$
$18$		0			0
$19$		−	0.207666i		−	0.0476418i	−0.999716	−	0.0238209i	$-0.992417\pi$
$19$		−	0.207666i		−	0.0476418i	0.999716	−	0.0238209i	$-0.00758314\pi$
$20$		0			0
$21$		−	3.11575i		−	0.679913i
$22$		0			0
$23$			1.52875i			0.318766i	0.987217	+	0.159383i	$0.0509505\pi$
$23$			1.52875i			0.318766i	−0.987217	+	0.159383i	$0.949049\pi$
$24$		0			0
$25$	−4.12737	−	2.82220i	−0.825474	−	0.564441i
$26$		0			0
$27$	−11.5530			−2.22337
$28$		0			0
$29$		−	3.60641i		−	0.669693i	−0.942273	−	0.334846i	$-0.891316\pi$
$29$		−	3.60641i		−	0.669693i	0.942273	−	0.334846i	$-0.108684\pi$
$30$		0			0
$31$	−7.94658			−1.42725			−0.713624	−	0.700529i	$-0.752947\pi$
$31$	−7.94658			−1.42725			−0.713624	+	0.700529i	$0.752947\pi$
$32$		0			0
$33$			2.43068i			0.423128i
$34$		0			0
$35$	2.13628	+	0.660542i	0.361097	+	0.111652i
$36$		0			0
$37$	−3.70429			−0.608981			−0.304490	−	0.952515i	$-0.598486\pi$
$37$	−3.70429			−0.608981			−0.304490	+	0.952515i	$0.598486\pi$
$38$		0			0
$39$	−9.63207			−1.54237
$40$		0			0
$41$	6.24284			0.974968			0.487484	−	0.873132i	$-0.337915\pi$
$41$	6.24284			0.974968			0.487484	+	0.873132i	$0.337915\pi$
$42$		0			0
$43$	6.61428			1.00867			0.504334	−	0.863509i	$-0.331738\pi$
$43$	6.61428			1.00867			0.504334	+	0.863509i	$0.331738\pi$
$44$		0			0
$45$	4.43087	−	14.3300i	0.660515	−	2.13619i
$46$		0			0
$47$		−	2.61892i		−	0.382009i	−0.981589	−	0.191005i	$-0.938825\pi$
$47$		−	2.61892i		−	0.382009i	0.981589	−	0.191005i	$-0.0611745\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$		−	22.7868i		−	3.19080i
$52$		0			0
$53$	−8.69448			−1.19428			−0.597140	−	0.802137i	$-0.703696\pi$
$53$	−8.69448			−1.19428			−0.597140	+	0.802137i	$0.703696\pi$
$54$		0			0
$55$	−1.66657	−	0.515307i	−0.224720	−	0.0694839i
$56$		0			0
$57$			0.647035i			0.0857019i
$58$		0			0
$59$			14.6333i			1.90510i	0.304388	+	0.952548i	$0.401548\pi$
$59$			14.6333i			1.90510i	−0.304388	+	0.952548i	$0.598452\pi$
$60$		0			0
$61$			6.37646i			0.816423i	0.912887	+	0.408211i	$0.133847\pi$
$61$			6.37646i			0.816423i	−0.912887	+	0.408211i	$0.866153\pi$
$62$		0			0
$63$			6.70793i			0.845119i
$64$		0			0
$65$	2.04200	−	6.60411i	0.253279	−	0.819139i
$66$		0			0
$67$	−7.21281			−0.881185			−0.440593	−	0.897707i	$-0.645232\pi$
$67$	−7.21281			−0.881185			−0.440593	+	0.897707i	$0.645232\pi$
$68$		0			0
$69$		−	4.76321i		−	0.573423i
$70$		0			0
$71$	14.6433			1.73784			0.868921	−	0.494950i	$-0.164814\pi$
$71$	14.6433			1.73784			0.868921	+	0.494950i	$0.164814\pi$
$72$		0			0
$73$		−	0.502306i		−	0.0587905i	−0.999568	−	0.0293952i	$-0.990642\pi$
$73$		−	0.502306i		−	0.0587905i	0.999568	−	0.0293952i	$-0.00935815\pi$
$74$		0			0
$75$	12.8599	+	8.79329i	1.48493	+	1.01536i
$76$		0			0
$77$	0.780127			0.0889037
$78$		0			0
$79$	11.4410			1.28722			0.643608	−	0.765355i	$-0.277437\pi$
$79$	11.4410			1.28722			0.643608	+	0.765355i	$0.277437\pi$
$80$		0			0
$81$	15.8725			1.76361
$82$		0			0
$83$	−13.8372			−1.51883			−0.759415	−	0.650606i	$-0.774515\pi$
$83$	−13.8372			−1.51883			−0.759415	+	0.650606i	$0.774515\pi$
$84$		0			0
$85$	15.6235	+	4.83083i	1.69461	+	0.523977i
$86$		0			0
$87$			11.2367i			1.20470i
$88$		0			0
$89$	−16.3744			−1.73568			−0.867839	−	0.496845i	$-0.834492\pi$
$89$	−16.3744			−1.73568			−0.867839	+	0.496845i	$0.834492\pi$
$90$		0			0
$91$			3.09141i			0.324068i
$92$		0			0
$93$	24.7596			2.56745
$94$		0			0
$95$	−0.443631	−	0.137172i	−0.0455156	−	0.0140735i
$96$		0			0
$97$			10.2956i			1.04536i	0.852530	+	0.522678i	$0.175067\pi$
$97$			10.2956i			1.04536i	−0.852530	+	0.522678i	$0.824933\pi$
$98$		0			0
$99$		−	5.23304i		−	0.525940i
$100$		0			0
$101$			1.51019i			0.150269i	0.997173	+	0.0751345i	$0.0239386\pi$
$101$			1.51019i			0.150269i	−0.997173	+	0.0751345i	$0.976061\pi$
$102$		0			0
$103$			12.3376i			1.21566i	0.794068	+	0.607829i	$0.207959\pi$
$103$			12.3376i			1.21566i	−0.794068	+	0.607829i	$0.792041\pi$
$104$		0			0
$105$	−6.65612	−	2.05809i	−0.649571	−	0.200849i
$106$		0			0
$107$	8.23235			0.795851			0.397925	−	0.917418i	$-0.369730\pi$
$107$	8.23235			0.795851			0.397925	+	0.917418i	$0.369730\pi$
$108$		0			0
$109$		−	7.05171i		−	0.675431i	−0.941248	−	0.337715i	$-0.890346\pi$
$109$		−	7.05171i		−	0.675431i	0.941248	−	0.337715i	$-0.109654\pi$
$110$		0			0
$111$	11.5416			1.09548
$112$		0			0
$113$		−	7.44815i		−	0.700663i	−0.936626	−	0.350332i	$-0.886069\pi$
$113$		−	7.44815i		−	0.700663i	0.936626	−	0.350332i	$-0.113931\pi$
$114$		0			0
$115$	3.26583	+	1.00980i	0.304541	+	0.0941646i
$116$		0			0
$117$	20.7369			1.91713
$118$		0			0
$119$	−7.31343			−0.670421
$120$		0			0
$121$	10.3914			0.944673
$122$		0			0
$123$	−19.4512			−1.75385
$124$		0			0
$125$	−8.75531	+	6.95302i	−0.783099	+	0.621897i
$126$		0			0
$127$			14.2620i			1.26555i	0.774336	+	0.632775i	$0.218084\pi$
$127$			14.2620i			1.26555i	−0.774336	+	0.632775i	$0.781916\pi$
$128$		0			0
$129$	−20.6085			−1.81447
$130$		0			0
$131$		−	3.78753i		−	0.330918i	−0.986217	−	0.165459i	$-0.947089\pi$
$131$		−	3.78753i		−	0.330918i	0.986217	−	0.165459i	$-0.0529105\pi$
$132$		0			0
$133$	0.207666			0.0180069
$134$		0			0
$135$	−7.63123	+	24.6804i	−0.656792	+	2.12415i
$136$		0			0
$137$			19.1281i			1.63423i	0.576477	+	0.817113i	$0.304427\pi$
$137$			19.1281i			1.63423i	−0.576477	+	0.817113i	$0.695573\pi$
$138$		0			0
$139$			13.0683i			1.10844i	0.832372	+	0.554218i	$0.186982\pi$
$139$			13.0683i			1.10844i	−0.832372	+	0.554218i	$0.813018\pi$
$140$		0			0
$141$			8.15992i			0.687189i
$142$		0			0
$143$		−	2.41169i		−	0.201676i
$144$		0			0
$145$	−7.70429	−	2.38218i	−0.639806	−	0.197829i
$146$		0			0
$147$	3.11575			0.256983
$148$		0			0
$149$			12.7098i			1.04123i	0.853793	+	0.520613i	$0.174297\pi$
$149$			12.7098i			1.04123i	−0.853793	+	0.520613i	$0.825703\pi$
$150$		0			0
$151$	20.5732			1.67423			0.837114	−	0.547029i	$-0.184241\pi$
$151$	20.5732			1.67423			0.837114	+	0.547029i	$0.184241\pi$
$152$		0			0
$153$			49.0579i			3.96610i
$154$		0			0
$155$	−5.24905	+	16.9761i	−0.421614	+	1.36355i
$156$		0			0
$157$	14.0900			1.12450			0.562251	−	0.826967i	$-0.309936\pi$
$157$	14.0900			1.12450			0.562251	+	0.826967i	$0.309936\pi$
$158$		0			0
$159$	27.0899			2.14837
$160$		0			0
$161$	−1.52875			−0.120482
$162$		0			0
$163$	−0.0610878			−0.00478477			−0.00239238	−	0.999997i	$-0.500762\pi$
$163$	−0.0610878			−0.00478477			−0.00239238	+	0.999997i	$0.500762\pi$
$164$		0			0
$165$	5.19262	+	1.60557i	0.404245	+	0.124993i
$166$		0			0
$167$		−	15.3878i		−	1.19074i	−0.803450	−	0.595372i	$-0.797005\pi$
$167$		−	15.3878i		−	1.19074i	0.803450	−	0.595372i	$-0.202995\pi$
$168$		0			0
$169$	−3.44320			−0.264861
$170$		0			0
$171$		−	1.39301i		−	0.106526i
$172$		0			0
$173$	9.82169			0.746729			0.373365	−	0.927685i	$-0.378204\pi$
$173$	9.82169			0.746729			0.373365	+	0.927685i	$0.378204\pi$
$174$		0			0
$175$	2.82220	−	4.12737i	0.213338	−	0.312000i
$176$		0			0
$177$		−	45.5938i		−	3.42704i
$178$		0			0
$179$		−	7.54793i		−	0.564159i	−0.959391	−	0.282079i	$-0.908976\pi$
$179$		−	7.54793i		−	0.564159i	0.959391	−	0.282079i	$-0.0910241\pi$
$180$		0			0
$181$			17.7551i			1.31973i	0.751386	+	0.659863i	$0.229386\pi$
$181$			17.7551i			1.31973i	−0.751386	+	0.659863i	$0.770614\pi$
$182$		0			0
$183$		−	19.8675i		−	1.46865i
$184$		0			0
$185$	−2.44684	+	7.91339i	−0.179895	+	0.581804i
$186$		0			0
$187$	5.70540			0.417221
$188$		0			0
$189$		−	11.5530i		−	0.840356i
$190$		0			0
$191$	10.5484			0.763256			0.381628	−	0.924316i	$-0.375364\pi$
$191$	10.5484			0.763256			0.381628	+	0.924316i	$0.375364\pi$
$192$		0			0
$193$			10.2499i			0.737805i	0.929468	+	0.368902i	$0.120266\pi$
$193$			10.2499i			0.737805i	−0.929468	+	0.368902i	$0.879734\pi$
$194$		0			0
$195$	−6.36238	+	20.5768i	−0.455620	+	1.47353i
$196$		0			0
$197$	12.1196			0.863487			0.431743	−	0.901996i	$-0.357899\pi$
$197$	12.1196			0.863487			0.431743	+	0.901996i	$0.357899\pi$
$198$		0			0
$199$	19.3840			1.37410			0.687048	−	0.726612i	$-0.258906\pi$
$199$	19.3840			1.37410			0.687048	+	0.726612i	$0.258906\pi$
$200$		0			0
$201$	22.4734			1.58515
$202$		0			0
$203$	3.60641			0.253120
$204$		0			0
$205$	4.12366	−	13.3364i	0.288009	−	0.931457i
$206$		0			0
$207$			10.2547i			0.712754i
$208$		0			0
$209$	−0.162006			−0.0112062
$210$		0			0
$211$		−	6.74843i		−	0.464581i	−0.972646	−	0.232291i	$-0.925378\pi$
$211$		−	6.74843i		−	0.464581i	0.972646	−	0.232291i	$-0.0746221\pi$
$212$		0			0
$213$	−45.6250			−3.12617
$214$		0			0
$215$	4.36901	−	14.1299i	0.297964	−	0.963654i
$216$		0			0
$217$		−	7.94658i		−	0.539449i
$218$		0			0
$219$			1.56506i			0.105757i
$220$		0			0
$221$			22.6088i			1.52083i
$222$		0			0
$223$		−	18.7700i		−	1.25694i	−0.777836	−	0.628468i	$-0.783682\pi$
$223$		−	18.7700i		−	1.25694i	0.777836	−	0.628468i	$-0.216318\pi$
$224$		0			0
$225$	−27.6861	−	18.9311i	−1.84574	−	1.26207i
$226$		0			0
$227$	−2.45729			−0.163096			−0.0815481	−	0.996669i	$-0.525986\pi$
$227$	−2.45729			−0.163096			−0.0815481	+	0.996669i	$0.525986\pi$
$228$		0			0
$229$			16.3838i			1.08267i	0.840806	+	0.541336i	$0.182081\pi$
$229$			16.3838i			1.08267i	−0.840806	+	0.541336i	$0.817919\pi$
$230$		0			0
$231$	−2.43068			−0.159927
$232$		0			0
$233$			18.7227i			1.22656i	0.789864	+	0.613282i	$0.210151\pi$
$233$			18.7227i			1.22656i	−0.789864	+	0.613282i	$0.789849\pi$
$234$		0			0
$235$	−5.59475	−	1.72991i	−0.364961	−	0.112847i
$236$		0			0
$237$	−35.6474			−2.31555
$238$		0			0
$239$	−19.8849			−1.28625			−0.643123	−	0.765763i	$-0.722362\pi$
$239$	−19.8849			−1.28625			−0.643123	+	0.765763i	$0.722362\pi$
$240$		0			0
$241$	22.7523			1.46561			0.732803	−	0.680441i	$-0.238212\pi$
$241$	22.7523			1.46561			0.732803	+	0.680441i	$0.238212\pi$
$242$		0			0
$243$	−14.7958			−0.949153
$244$		0			0
$245$	−0.660542	+	2.13628i	−0.0422005	+	0.136482i
$246$		0			0
$247$		−	0.641979i		−	0.0408482i
$248$		0			0
$249$	43.1133			2.73220
$250$		0			0
$251$			3.48168i			0.219762i	0.993945	+	0.109881i	$0.0350469\pi$
$251$			3.48168i			0.219762i	−0.993945	+	0.109881i	$0.964953\pi$
$252$		0			0
$253$	1.19262			0.0749793
$254$		0			0
$255$	−48.6790	−	15.0517i	−3.04840	−	0.942572i
$256$		0			0
$257$		−	21.5841i		−	1.34638i	−0.739469	−	0.673191i	$-0.764923\pi$
$257$		−	21.5841i		−	1.34638i	0.739469	−	0.673191i	$-0.235077\pi$
$258$		0			0
$259$		−	3.70429i		−	0.230173i
$260$		0			0
$261$		−	24.1915i		−	1.49742i
$262$		0			0
$263$		−	9.07710i		−	0.559718i	−0.960041	−	0.279859i	$-0.909712\pi$
$263$		−	9.07710i		−	0.559718i	0.960041	−	0.279859i	$-0.0902878\pi$
$264$		0			0
$265$	−5.74307	+	18.5738i	−0.352794	+	1.14098i
$266$		0			0
$267$	51.0185			3.12228
$268$		0			0
$269$			31.6554i			1.93006i	0.262136	+	0.965031i	$0.415573\pi$
$269$			31.6554i			1.93006i	−0.262136	+	0.965031i	$0.584427\pi$
$270$		0			0
$271$	10.1433			0.616165			0.308082	−	0.951360i	$-0.400313\pi$
$271$	10.1433			0.616165			0.308082	+	0.951360i	$0.400313\pi$
$272$		0			0
$273$		−	9.63207i		−	0.582959i
$274$		0			0
$275$	−2.20168	+	3.21987i	−0.132766	+	0.194166i
$276$		0			0
$277$	25.0993			1.50807			0.754036	−	0.656833i	$-0.228104\pi$
$277$	25.0993			1.50807			0.754036	+	0.656833i	$0.228104\pi$
$278$		0			0
$279$	−53.3051			−3.19129
$280$		0			0
$281$	−12.2025			−0.727938			−0.363969	−	0.931411i	$-0.618579\pi$
$281$	−12.2025			−0.727938			−0.363969	+	0.931411i	$0.618579\pi$
$282$		0			0
$283$	−26.6338			−1.58322			−0.791608	−	0.611029i	$-0.790756\pi$
$283$	−26.6338			−1.58322			−0.791608	+	0.611029i	$0.790756\pi$
$284$		0			0
$285$	1.38225	+	0.427394i	0.0818772	+	0.0253166i
$286$		0			0
$287$			6.24284i			0.368503i
$288$		0			0
$289$	−36.4862			−2.14625
$290$		0			0
$291$		−	32.0785i		−	1.88047i
$292$		0			0
$293$	−11.5580			−0.675228			−0.337614	−	0.941285i	$-0.609620\pi$
$293$	−11.5580			−0.675228			−0.337614	+	0.941285i	$0.609620\pi$
$294$		0			0
$295$	31.2608	+	9.66592i	1.82008	+	0.562772i
$296$		0			0
$297$			9.01280i			0.522976i
$298$		0			0
$299$			4.72599i			0.273311i
$300$		0			0
$301$			6.61428i			0.381241i
$302$		0			0
$303$		−	4.70537i		−	0.270316i
$304$		0			0
$305$	13.6219	+	4.21192i	0.779988	+	0.241174i
$306$		0			0
$307$	5.28984			0.301907			0.150954	−	0.988541i	$-0.451766\pi$
$307$	5.28984			0.301907			0.150954	+	0.988541i	$0.451766\pi$
$308$		0			0
$309$		−	38.4409i		−	2.18682i
$310$		0			0
$311$	8.05081			0.456519			0.228260	−	0.973600i	$-0.426696\pi$
$311$	8.05081			0.456519			0.228260	+	0.973600i	$0.426696\pi$
$312$		0			0
$313$		−	3.21760i		−	0.181869i	−0.995857	−	0.0909347i	$-0.971015\pi$
$313$		−	3.21760i		−	0.181869i	0.995857	−	0.0909347i	$-0.0289855\pi$
$314$		0			0
$315$	14.3300	+	4.43087i	0.807404	+	0.249651i
$316$		0			0
$317$	−19.3949			−1.08932			−0.544662	−	0.838656i	$-0.683342\pi$
$317$	−19.3949			−1.08932			−0.544662	+	0.838656i	$0.683342\pi$
$318$		0			0
$319$	−2.81346			−0.157523
$320$		0			0
$321$	−25.6500			−1.43164
$322$		0			0
$323$	1.51875			0.0845054
$324$		0			0
$325$	−12.7594	−	8.72458i	−0.707763	−	0.483953i
$326$		0			0
$327$			21.9714i			1.21502i
$328$		0			0
$329$	2.61892			0.144386
$330$		0			0
$331$			17.1320i			0.941662i	0.882223	+	0.470831i	$0.156046\pi$
$331$			17.1320i			0.941662i	−0.882223	+	0.470831i	$0.843954\pi$
$332$		0			0
$333$	−24.8481			−1.36167
$334$		0			0
$335$	−4.76436	+	15.4086i	−0.260305	+	0.841860i
$336$		0			0
$337$		−	11.5830i		−	0.630965i	−0.948931	−	0.315482i	$-0.897834\pi$
$337$		−	11.5830i		−	0.630965i	0.948931	−	0.315482i	$-0.102166\pi$
$338$		0			0
$339$			23.2066i			1.26041i
$340$		0			0
$341$			6.19934i			0.335713i
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$	−10.1755	−	3.14630i	−0.547832	−	0.169391i
$346$		0			0
$347$	−16.7866			−0.901149			−0.450575	−	0.892739i	$-0.648781\pi$
$347$	−16.7866			−0.901149			−0.450575	+	0.892739i	$0.648781\pi$
$348$		0			0
$349$		−	29.8563i		−	1.59817i	−0.601218	−	0.799085i	$-0.705318\pi$
$349$		−	29.8563i		−	1.59817i	0.601218	−	0.799085i	$-0.294682\pi$
$350$		0			0
$351$	−35.7150			−1.90633
$352$		0			0
$353$			13.4892i			0.717957i	0.933346	+	0.358979i	$0.116875\pi$
$353$			13.4892i			0.717957i	−0.933346	+	0.358979i	$0.883125\pi$
$354$		0			0
$355$	9.67253	−	31.2822i	0.513364	−	1.66029i
$356$		0			0
$357$	22.7868			1.20601
$358$		0			0
$359$	−12.7437			−0.672589			−0.336295	−	0.941757i	$-0.609174\pi$
$359$	−12.7437			−0.672589			−0.336295	+	0.941757i	$0.609174\pi$
$360$		0			0
$361$	18.9569			0.997730
$362$		0			0
$363$	−32.3771			−1.69935
$364$		0			0
$365$	−1.07307	−	0.331794i	−0.0561668	−	0.0173669i
$366$		0			0
$367$			19.6342i			1.02490i	0.858717	+	0.512449i	$0.171262\pi$
$367$			19.6342i			1.02490i	−0.858717	+	0.512449i	$0.828738\pi$
$368$		0			0
$369$	41.8765			2.18000
$370$		0			0
$371$		−	8.69448i		−	0.451395i
$372$		0			0
$373$	−0.0932918			−0.00483046			−0.00241523	−	0.999997i	$-0.500769\pi$
$373$	−0.0932918			−0.00483046			−0.00241523	+	0.999997i	$0.500769\pi$
$374$		0			0
$375$	27.2794	−	21.6639i	1.40870	−	1.11872i
$376$		0			0
$377$		−	11.1489i		−	0.574196i
$378$		0			0
$379$			8.35354i			0.429093i	0.976714	+	0.214546i	$0.0688273\pi$
$379$			8.35354i			0.429093i	−0.976714	+	0.214546i	$0.931173\pi$
$380$		0			0
$381$		−	44.4369i		−	2.27657i
$382$		0			0
$383$		−	17.5675i		−	0.897659i	−0.893617	−	0.448830i	$-0.851841\pi$
$383$		−	17.5675i		−	0.897659i	0.893617	−	0.448830i	$-0.148159\pi$
$384$		0			0
$385$	0.515307	−	1.66657i	0.0262625	−	0.0849362i
$386$		0			0
$387$	44.3681			2.25536
$388$		0			0
$389$		−	19.7337i		−	1.00054i	−0.865870	−	0.500270i	$-0.833234\pi$
$389$		−	19.7337i		−	1.00054i	0.865870	−	0.500270i	$-0.166766\pi$
$390$		0			0
$391$	−11.1804			−0.565417
$392$		0			0
$393$			11.8010i			0.595282i
$394$		0			0
$395$	7.55728	−	24.4412i	0.380248	−	1.22977i
$396$		0			0
$397$	27.4713			1.37874			0.689372	−	0.724408i	$-0.257887\pi$
$397$	27.4713			1.37874			0.689372	+	0.724408i	$0.257887\pi$
$398$		0			0
$399$	−0.647035			−0.0323923
$400$		0			0
$401$	8.20403			0.409690			0.204845	−	0.978794i	$-0.434331\pi$
$401$	8.20403			0.409690			0.204845	+	0.978794i	$0.434331\pi$
$402$		0			0
$403$	−24.5661			−1.22373
$404$		0			0
$405$	10.4844	−	33.9081i	0.520976	−	1.68490i
$406$		0			0
$407$			2.88981i			0.143243i
$408$		0			0
$409$	−21.5743			−1.06678			−0.533389	−	0.845870i	$-0.679082\pi$
$409$	−21.5743			−1.06678			−0.533389	+	0.845870i	$0.679082\pi$
$410$		0			0
$411$		−	59.5986i		−	2.93978i
$412$		0			0
$413$	−14.6333			−0.720059
$414$		0			0
$415$	−9.14005	+	29.5601i	−0.448668	+	1.45105i
$416$		0			0
$417$		−	40.7175i		−	1.99394i
$418$		0			0
$419$			18.4499i			0.901336i	0.892692	+	0.450668i	$0.148814\pi$
$419$			18.4499i			0.901336i	−0.892692	+	0.450668i	$0.851186\pi$
$420$		0			0
$421$		−	8.05800i		−	0.392723i	−0.980532	−	0.196362i	$-0.937087\pi$
$421$		−	8.05800i		−	0.392723i	0.980532	−	0.196362i	$-0.0629126\pi$
$422$		0			0
$423$		−	17.5675i		−	0.854163i
$424$		0			0
$425$	20.6400	−	30.1852i	1.00119	−	1.46420i
$426$		0			0
$427$	−6.37646			−0.308579
$428$		0			0
$429$			7.51424i			0.362791i
$430$		0			0
$431$	1.24841			0.0601340			0.0300670	−	0.999548i	$-0.490428\pi$
$431$	1.24841			0.0601340			0.0300670	+	0.999548i	$0.490428\pi$
$432$		0			0
$433$			9.23559i			0.443834i	0.975066	+	0.221917i	$0.0712314\pi$
$433$			9.23559i			0.443834i	−0.975066	+	0.221917i	$0.928769\pi$
$434$		0			0
$435$	24.0047	+	7.42229i	1.15094	+	0.355872i
$436$		0			0
$437$	0.317469			0.0151866
$438$		0			0
$439$	−18.2320			−0.870166			−0.435083	−	0.900390i	$-0.643281\pi$
$439$	−18.2320			−0.870166			−0.435083	+	0.900390i	$0.643281\pi$
$440$		0			0
$441$	−6.70793			−0.319425
$442$		0			0
$443$	23.5857			1.12059			0.560296	−	0.828292i	$-0.310687\pi$
$443$	23.5857			1.12059			0.560296	+	0.828292i	$0.310687\pi$
$444$		0			0
$445$	−10.8159	+	34.9802i	−0.512725	+	1.65822i
$446$		0			0
$447$		−	39.6006i		−	1.87304i
$448$		0			0
$449$	1.62083			0.0764917			0.0382458	−	0.999268i	$-0.487823\pi$
$449$	1.62083			0.0764917			0.0382458	+	0.999268i	$0.487823\pi$
$450$		0			0
$451$		−	4.87021i		−	0.229329i
$452$		0			0
$453$	−64.1012			−3.01174
$454$		0			0
$455$	6.60411	+	2.04200i	0.309605	+	0.0957306i
$456$		0			0
$457$		−	17.4779i		−	0.817581i	−0.912628	−	0.408790i	$-0.865951\pi$
$457$		−	17.4779i		−	0.817581i	0.912628	−	0.408790i	$-0.134049\pi$
$458$		0			0
$459$		−	84.4919i		−	3.94375i
$460$		0			0
$461$			12.5433i			0.584200i	0.956388	+	0.292100i	$0.0943540\pi$
$461$			12.5433i			0.584200i	−0.956388	+	0.292100i	$0.905646\pi$
$462$		0			0
$463$			25.1797i			1.17020i	0.810961	+	0.585101i	$0.198945\pi$
$463$			25.1797i			1.17020i	−0.810961	+	0.585101i	$0.801055\pi$
$464$		0			0
$465$	16.3547	−	52.8934i	0.758433	−	2.45287i
$466$		0			0
$467$	−4.98968			−0.230895			−0.115447	−	0.993314i	$-0.536830\pi$
$467$	−4.98968			−0.230895			−0.115447	+	0.993314i	$0.536830\pi$
$468$		0			0
$469$		−	7.21281i		−	0.333057i
$470$		0			0
$471$	−43.9009			−2.02284
$472$		0			0
$473$		−	5.15998i		−	0.237256i
$474$		0			0
$475$	−0.586074	+	0.857112i	−0.0268909	+	0.0393270i
$476$		0			0
$477$	−58.3219			−2.67038
$478$		0			0
$479$	33.8772			1.54789			0.773945	−	0.633253i	$-0.218281\pi$
$479$	33.8772			1.54789			0.773945	+	0.633253i	$0.218281\pi$
$480$		0			0
$481$	−11.4515			−0.522142
$482$		0			0
$483$	4.76321			0.216733
$484$		0			0
$485$	21.9942	+	6.80065i	0.998705	+	0.308802i
$486$		0			0
$487$		−	26.2197i		−	1.18813i	−0.804419	−	0.594063i	$-0.797523\pi$
$487$		−	26.2197i		−	1.18813i	0.804419	−	0.594063i	$-0.202477\pi$
$488$		0			0
$489$	0.190335			0.00860723
$490$		0			0
$491$		−	0.912260i		−	0.0411697i	−0.999788	−	0.0205849i	$-0.993447\pi$
$491$		−	0.912260i		−	0.0411697i	0.999788	−	0.0205849i	$-0.00655283\pi$
$492$		0			0
$493$	26.3752			1.18788
$494$		0			0
$495$	−11.1792	−	3.45664i	−0.502469	−	0.155364i
$496$		0			0
$497$			14.6433i			0.656843i
$498$		0			0
$499$			9.73724i			0.435899i	0.975960	+	0.217949i	$0.0699368\pi$
$499$			9.73724i			0.435899i	−0.975960	+	0.217949i	$0.930063\pi$
$500$		0			0
$501$			47.9446i			2.14201i
$502$		0			0
$503$		−	11.5199i		−	0.513646i	−0.966458	−	0.256823i	$-0.917324\pi$
$503$		−	11.5199i		−	0.513646i	0.966458	−	0.256823i	$-0.0826757\pi$
$504$		0			0
$505$	3.22618	+	0.997541i	0.143563	+	0.0443900i
$506$		0			0
$507$	10.7282			0.476454
$508$		0			0
$509$			10.5269i			0.466598i	0.972405	+	0.233299i	$0.0749520\pi$
$509$			10.5269i			0.466598i	−0.972405	+	0.233299i	$0.925048\pi$
$510$		0			0
$511$	0.502306			0.0222207
$512$		0			0
$513$			2.39916i			0.105925i
$514$		0			0
$515$	26.3565	+	8.14949i	1.16141	+	0.359109i
$516$		0			0
$517$	−2.04309			−0.0898551
$518$		0			0
$519$	−30.6020			−1.34328
$520$		0			0
$521$	−20.2935			−0.889075			−0.444538	−	0.895760i	$-0.646632\pi$
$521$	−20.2935			−0.889075			−0.444538	+	0.895760i	$0.646632\pi$
$522$		0			0
$523$	18.0756			0.790392			0.395196	−	0.918597i	$-0.370677\pi$
$523$	18.0756			0.790392			0.395196	+	0.918597i	$0.370677\pi$
$524$		0			0
$525$	−8.79329	+	12.8599i	−0.383771	+	0.561251i
$526$		0			0
$527$		−	58.1167i		−	2.53161i
$528$		0			0
$529$	20.6629			0.898388
$530$		0			0
$531$			98.1592i			4.25975i
$532$		0			0
$533$	19.2992			0.835939
$534$		0			0
$535$	5.43781	−	17.5866i	0.235097	−	0.760334i
$536$		0			0
$537$			23.5175i			1.01485i
$538$		0			0
$539$			0.780127i			0.0336025i
$540$		0			0
$541$		−	0.588349i		−	0.0252951i	−0.999920	−	0.0126475i	$-0.995974\pi$
$541$		−	0.588349i		−	0.0252951i	0.999920	−	0.0126475i	$-0.00402595\pi$
$542$		0			0
$543$		−	55.3206i		−	2.37403i
$544$		0			0
$545$	−15.0644	−	4.65795i	−0.645288	−	0.199525i
$546$		0			0
$547$	−10.5454			−0.450890			−0.225445	−	0.974256i	$-0.572384\pi$
$547$	−10.5454			−0.450890			−0.225445	+	0.974256i	$0.572384\pi$
$548$		0			0
$549$			42.7728i			1.82550i
$550$		0			0
$551$	−0.748926			−0.0319053
$552$		0			0
$553$			11.4410i			0.486522i
$554$		0			0
$555$	7.62374	−	24.6562i	0.323610	−	1.04660i
$556$		0			0
$557$	−38.3739			−1.62596			−0.812978	−	0.582295i	$-0.802155\pi$
$557$	−38.3739			−1.62596			−0.812978	+	0.582295i	$0.802155\pi$
$558$		0			0
$559$	20.4474			0.864834
$560$		0			0
$561$	−17.7766			−0.750530
$562$		0			0
$563$	16.4307			0.692473			0.346237	−	0.938147i	$-0.387459\pi$
$563$	16.4307			0.692473			0.346237	+	0.938147i	$0.387459\pi$
$564$		0			0
$565$	−15.9113	−	4.91982i	−0.669395	−	0.206978i
$566$		0			0
$567$			15.8725i			0.666582i
$568$		0			0
$569$	−18.1733			−0.761866			−0.380933	−	0.924603i	$-0.624397\pi$
$569$	−18.1733			−0.761866			−0.380933	+	0.924603i	$0.624397\pi$
$570$		0			0
$571$			12.4832i			0.522406i	0.965284	+	0.261203i	$0.0841192\pi$
$571$			12.4832i			0.522406i	−0.965284	+	0.261203i	$0.915881\pi$
$572$		0			0
$573$	−32.8662			−1.37301
$574$		0			0
$575$	4.31444	−	6.30971i	0.179925	−	0.263133i
$576$		0			0
$577$			19.0366i			0.792505i	0.918142	+	0.396252i	$0.129689\pi$
$577$			19.0366i			0.792505i	−0.918142	+	0.396252i	$0.870311\pi$
$578$		0			0
$579$		−	31.9362i		−	1.32722i
$580$		0			0
$581$		−	13.8372i		−	0.574064i
$582$		0			0
$583$			6.78280i			0.280915i
$584$		0			0
$585$	13.6976	−	44.2999i	0.566327	−	1.83157i
$586$		0			0
$587$	−4.45396			−0.183835			−0.0919173	−	0.995767i	$-0.529300\pi$
$587$	−4.45396			−0.183835			−0.0919173	+	0.995767i	$0.529300\pi$
$588$		0			0
$589$			1.65023i			0.0679966i
$590$		0			0
$591$	−37.7618			−1.55331
$592$		0			0
$593$		−	35.4961i		−	1.45765i	−0.684699	−	0.728826i	$-0.740066\pi$
$593$		−	35.4961i		−	1.45765i	0.684699	−	0.728826i	$-0.259934\pi$
$594$		0			0
$595$	−4.83083	+	15.6235i	−0.198045	+	0.640502i
$596$		0			0
$597$	−60.3958			−2.47183
$598$		0			0
$599$	−1.92977			−0.0788483			−0.0394242	−	0.999223i	$-0.512552\pi$
$599$	−1.92977			−0.0788483			−0.0394242	+	0.999223i	$0.512552\pi$
$600$		0			0
$601$	−41.5646			−1.69546			−0.847728	−	0.530430i	$-0.822030\pi$
$601$	−41.5646			−1.69546			−0.847728	+	0.530430i	$0.822030\pi$
$602$		0			0
$603$	−48.3830			−1.97031
$604$		0			0
$605$	6.86396	−	22.1989i	0.279060	−	0.902515i
$606$		0			0
$607$			31.1119i			1.26279i	0.775460	+	0.631396i	$0.217518\pi$
$607$			31.1119i			1.26279i	−0.775460	+	0.631396i	$0.782482\pi$
$608$		0			0
$609$	−11.2367			−0.455333
$610$		0			0
$611$		−	8.09616i		−	0.327535i
$612$		0			0
$613$	28.7427			1.16091			0.580453	−	0.814293i	$-0.302875\pi$
$613$	28.7427			1.16091			0.580453	+	0.814293i	$0.302875\pi$
$614$		0			0
$615$	−12.8483	+	41.5531i	−0.518094	+	1.67558i
$616$		0			0
$617$		−	3.31564i		−	0.133483i	−0.997770	−	0.0667414i	$-0.978740\pi$
$617$		−	3.31564i		−	0.133483i	0.997770	−	0.0667414i	$-0.0212602\pi$
$618$		0			0
$619$			21.0326i			0.845374i	0.906276	+	0.422687i	$0.138913\pi$
$619$			21.0326i			0.845374i	−0.906276	+	0.422687i	$0.861087\pi$
$620$		0			0
$621$		−	17.6616i		−	0.708736i
$622$		0			0
$623$		−	16.3744i		−	0.656025i
$624$		0			0
$625$	9.07035	+	23.2965i	0.362814	+	0.931862i
$626$		0			0
$627$	0.504770			0.0201586
$628$		0			0
$629$		−	27.0910i		−	1.08019i
$630$		0			0
$631$	5.91397			0.235432			0.117716	−	0.993047i	$-0.462443\pi$
$631$	5.91397			0.235432			0.117716	+	0.993047i	$0.462443\pi$
$632$		0			0
$633$			21.0265i			0.835727i
$634$		0			0
$635$	30.4676	+	9.42066i	1.20907	+	0.373847i
$636$		0			0
$637$	−3.09141			−0.122486
$638$		0			0
$639$	98.2263			3.88577
$640$		0			0
$641$	−4.64380			−0.183419			−0.0917095	−	0.995786i	$-0.529233\pi$
$641$	−4.64380			−0.183419			−0.0917095	+	0.995786i	$0.529233\pi$
$642$		0			0
$643$	18.9654			0.747922			0.373961	−	0.927444i	$-0.377999\pi$
$643$	18.9654			0.747922			0.373961	+	0.927444i	$0.377999\pi$
$644$		0			0
$645$	−13.6128	+	44.0254i	−0.536002	+	1.73350i
$646$		0			0
$647$		−	9.18211i		−	0.360986i	−0.983576	−	0.180493i	$-0.942231\pi$
$647$		−	9.18211i		−	0.360986i	0.983576	−	0.180493i	$-0.0577693\pi$
$648$		0			0
$649$	11.4159			0.448111
$650$		0			0
$651$			24.7596i			0.970405i
$652$		0			0
$653$	−4.20199			−0.164437			−0.0822184	−	0.996614i	$-0.526200\pi$
$653$	−4.20199			−0.164437			−0.0822184	+	0.996614i	$0.526200\pi$
$654$		0			0
$655$	−8.09121	−	2.50182i	−0.316150	−	0.0977542i
$656$		0			0
$657$		−	3.36943i		−	0.131454i
$658$		0			0
$659$		−	5.48955i		−	0.213843i	−0.994267	−	0.106921i	$-0.965901\pi$
$659$		−	5.48955i		−	0.213843i	0.994267	−	0.106921i	$-0.0340993\pi$
$660$		0			0
$661$			10.5752i			0.411329i	0.978623	+	0.205664i	$0.0659355\pi$
$661$			10.5752i			0.411329i	−0.978623	+	0.205664i	$0.934065\pi$
$662$		0			0
$663$		−	70.4434i		−	2.73580i
$664$		0			0
$665$	0.137172	−	0.443631i	0.00531930	−	0.0172033i
$666$		0			0
$667$	5.51329			0.213475
$668$		0			0
$669$			58.4828i			2.26108i
$670$		0			0
$671$	4.97445			0.192037
$672$		0			0
$673$			42.0174i			1.61965i	0.586670	+	0.809826i	$0.300439\pi$
$673$			42.0174i			1.61965i	−0.586670	+	0.809826i	$0.699561\pi$
$674$		0			0
$675$	47.6834	+	32.6049i	1.83534	+	1.25496i
$676$		0			0
$677$	3.86589			0.148578			0.0742891	−	0.997237i	$-0.476331\pi$
$677$	3.86589			0.148578			0.0742891	+	0.997237i	$0.476331\pi$
$678$		0			0
$679$	−10.2956			−0.395108
$680$		0			0
$681$	7.65632			0.293391
$682$		0			0
$683$	−8.61529			−0.329655			−0.164828	−	0.986322i	$-0.552707\pi$
$683$	−8.61529			−0.329655			−0.164828	+	0.986322i	$0.552707\pi$
$684$		0			0
$685$	40.8630	+	12.6349i	1.56130	+	0.482756i
$686$		0			0
$687$		−	51.0479i		−	1.94760i
$688$		0			0
$689$	−26.8782			−1.02398
$690$		0			0
$691$			35.2087i			1.33940i	0.742631	+	0.669701i	$0.233578\pi$
$691$			35.2087i			1.33940i	−0.742631	+	0.669701i	$0.766422\pi$
$692$		0			0
$693$	5.23304			0.198787
$694$		0			0
$695$	27.9174	+	8.63213i	1.05897	+	0.327435i
$696$		0			0
$697$			45.6566i			1.72937i
$698$		0			0
$699$		−	58.3353i		−	2.20644i
$700$		0			0
$701$		−	34.9933i		−	1.32168i	−0.750527	−	0.660840i	$-0.770200\pi$
$701$		−	34.9933i		−	1.32168i	0.750527	−	0.660840i	$-0.229800\pi$
$702$		0			0
$703$			0.769253i			0.0290129i
$704$		0			0
$705$	17.4319	+	5.38997i	0.656522	+	0.202998i
$706$		0			0
$707$	−1.51019			−0.0567964
$708$		0			0
$709$			36.3304i			1.36442i	0.731158	+	0.682209i	$0.238980\pi$
$709$			36.3304i			1.36442i	−0.731158	+	0.682209i	$0.761020\pi$
$710$		0			0
$711$	76.7456			2.87818
$712$		0			0
$713$		−	12.1483i		−	0.454959i
$714$		0			0
$715$	−5.15204	−	1.59302i	−0.192676	−	0.0595757i
$716$		0			0
$717$	61.9564			2.31380
$718$		0			0
$719$	10.5471			0.393341			0.196670	−	0.980470i	$-0.436987\pi$
$719$	10.5471			0.393341			0.196670	+	0.980470i	$0.436987\pi$
$720$		0			0
$721$	−12.3376			−0.459475
$722$		0			0
$723$	−70.8906			−2.63645
$724$		0			0
$725$	−10.1780	+	14.8850i	−0.378002	+	0.552814i
$726$		0			0
$727$			30.5631i			1.13352i	0.823883	+	0.566761i	$0.191804\pi$
$727$			30.5631i			1.13352i	−0.823883	+	0.566761i	$0.808196\pi$
$728$		0			0
$729$	−1.51731			−0.0561968
$730$		0			0
$731$			48.3731i			1.78914i
$732$		0			0
$733$	−8.32197			−0.307379			−0.153690	−	0.988119i	$-0.549116\pi$
$733$	−8.32197			−0.307379			−0.153690	+	0.988119i	$0.549116\pi$
$734$		0			0
$735$	2.05809	−	6.65612i	0.0759137	−	0.245515i
$736$		0			0
$737$			5.62691i			0.207270i
$738$		0			0
$739$			20.2933i			0.746502i	0.927730	+	0.373251i	$0.121757\pi$
$739$			20.2933i			0.746502i	−0.927730	+	0.373251i	$0.878243\pi$
$740$		0			0
$741$			2.00025i			0.0734810i
$742$		0			0
$743$			36.2809i			1.33102i	0.746391	+	0.665508i	$0.231785\pi$
$743$			36.2809i			1.33102i	−0.746391	+	0.665508i	$0.768215\pi$
$744$		0			0
$745$	27.1516	+	8.39535i	0.994759	+	0.307582i
$746$		0			0
$747$	−92.8189			−3.39607
$748$		0			0
$749$			8.23235i			0.300803i
$750$		0			0
$751$	−2.28827			−0.0835001			−0.0417501	−	0.999128i	$-0.513293\pi$
$751$	−2.28827			−0.0835001			−0.0417501	+	0.999128i	$0.513293\pi$
$752$		0			0
$753$		−	10.8481i		−	0.395325i
$754$		0			0
$755$	13.5895	−	43.9502i	0.494572	−	1.59951i
$756$		0			0
$757$	22.3190			0.811198			0.405599	−	0.914051i	$-0.367063\pi$
$757$	22.3190			0.811198			0.405599	+	0.914051i	$0.367063\pi$
$758$		0			0
$759$	−3.71591			−0.134879
$760$		0			0
$761$	2.99267			0.108484			0.0542421	−	0.998528i	$-0.482726\pi$
$761$	2.99267			0.108484			0.0542421	+	0.998528i	$0.482726\pi$
$762$		0			0
$763$	7.05171			0.255289
$764$		0			0
$765$	104.801	+	32.4048i	3.78910	+	1.17160i
$766$		0			0
$767$			45.2376i			1.63343i
$768$		0			0
$769$	−0.713327			−0.0257232			−0.0128616	−	0.999917i	$-0.504094\pi$
$769$	−0.713327			−0.0257232			−0.0128616	+	0.999917i	$0.504094\pi$
$770$		0			0
$771$			67.2509i			2.42198i
$772$		0			0
$773$	−25.6727			−0.923381			−0.461691	−	0.887041i	$-0.652757\pi$
$773$	−25.6727			−0.923381			−0.461691	+	0.887041i	$0.652757\pi$
$774$		0			0
$775$	32.7985	+	22.4269i	1.17816	+	0.805597i
$776$		0			0
$777$			11.5416i			0.414054i
$778$		0			0
$779$		−	1.29642i		−	0.0464492i
$780$		0			0
$781$		−	11.4237i		−	0.408770i
$782$		0			0
$783$			41.6648i			1.48898i
$784$		0			0
$785$	9.30701	−	30.1001i	0.332181	−	1.07432i
$786$		0			0
$787$	−41.9846			−1.49659			−0.748294	−	0.663367i	$-0.769127\pi$
$787$	−41.9846			−1.49659			−0.748294	+	0.663367i	$0.769127\pi$
$788$		0			0
$789$			28.2820i			1.00687i
$790$		0			0
$791$	7.44815			0.264826
$792$		0			0
$793$			19.7123i			0.700003i
$794$		0			0
$795$	17.8940	−	57.8715i	0.634635	−	2.05249i
$796$		0			0
$797$	−22.5201			−0.797705			−0.398852	−	0.917015i	$-0.630591\pi$
$797$	−22.5201			−0.797705			−0.398852	+	0.917015i	$0.630591\pi$
$798$		0			0
$799$	19.1533			0.677595
$800$		0			0
$801$	−109.838			−3.88093
$802$		0			0
$803$	−0.391863			−0.0138285
$804$		0			0
$805$	−1.00980	+	3.26583i	−0.0355909	+	0.115106i
$806$		0			0
$807$		−	98.6304i		−	3.47195i
$808$		0			0
$809$	9.11472			0.320457			0.160228	−	0.987080i	$-0.448777\pi$
$809$	9.11472			0.320457			0.160228	+	0.987080i	$0.448777\pi$
$810$		0			0
$811$		−	9.04562i		−	0.317635i	−0.987308	−	0.158817i	$-0.949232\pi$
$811$		−	9.04562i		−	0.317635i	0.987308	−	0.158817i	$-0.0507681\pi$
$812$		0			0
$813$	−31.6042			−1.10841
$814$		0			0
$815$	−0.0403511	+	0.130501i	−0.00141344	+	0.00457123i
$816$		0			0
$817$		−	1.37356i		−	0.0480547i
$818$		0			0
$819$			20.7369i			0.724607i
$820$		0			0
$821$			36.8007i			1.28435i	0.766557	+	0.642177i	$0.221968\pi$
$821$			36.8007i			1.28435i	−0.766557	+	0.642177i	$0.778032\pi$
$822$		0			0
$823$			35.1816i			1.22635i	0.789946	+	0.613177i	$0.210109\pi$
$823$			35.1816i			1.22635i	−0.789946	+	0.613177i	$0.789891\pi$
$824$		0			0
$825$	6.85988	−	10.0323i	0.238831	−	0.349281i
$826$		0			0
$827$	16.7117			0.581123			0.290561	−	0.956856i	$-0.406158\pi$
$827$	16.7117			0.581123			0.290561	+	0.956856i	$0.406158\pi$
$828$		0			0
$829$		−	40.9691i		−	1.42292i	−0.702729	−	0.711458i	$-0.748035\pi$
$829$		−	40.9691i		−	1.42292i	0.702729	−	0.711458i	$-0.251965\pi$
$830$		0			0
$831$	−78.2033			−2.71284
$832$		0			0
$833$		−	7.31343i		−	0.253395i
$834$		0			0
$835$	−32.8726	−	10.1643i	−1.13760	−	0.351750i
$836$		0			0
$837$	91.8067			3.17330
$838$		0			0
$839$	−33.1392			−1.14409			−0.572046	−	0.820221i	$-0.693850\pi$
$839$	−33.1392			−1.14409			−0.572046	+	0.820221i	$0.693850\pi$
$840$		0			0
$841$	15.9938			0.551512
$842$		0			0
$843$	38.0199			1.30947
$844$		0			0
$845$	−2.27438	+	7.35563i	−0.0782409	+	0.253041i
$846$		0			0
$847$			10.3914i			0.357053i
$848$		0			0
$849$	82.9844			2.84802
$850$		0			0
$851$		−	5.66293i		−	0.194123i
$852$		0			0
$853$	24.6954			0.845554			0.422777	−	0.906234i	$-0.361055\pi$
$853$	24.6954			0.845554			0.422777	+	0.906234i	$0.361055\pi$
$854$		0			0
$855$	−2.97585	−	0.920138i	−0.101772	−	0.0314681i
$856$		0			0
$857$			31.8936i			1.08946i	0.838610	+	0.544732i	$0.183368\pi$
$857$			31.8936i			1.08946i	−0.838610	+	0.544732i	$0.816632\pi$
$858$		0			0
$859$			25.9029i			0.883794i	0.897066	+	0.441897i	$0.145694\pi$
$859$			25.9029i			0.883794i	−0.897066	+	0.441897i	$0.854306\pi$
$860$		0			0
$861$		−	19.4512i		−	0.662893i
$862$		0			0
$863$			1.68267i			0.0572788i	0.999590	+	0.0286394i	$0.00911745\pi$
$863$			1.68267i			0.0572788i	−0.999590	+	0.0286394i	$0.990883\pi$
$864$		0			0
$865$	6.48764	−	20.9819i	0.220586	−	0.713405i
$866$		0			0
$867$	113.682			3.86085
$868$		0			0
$869$		−	8.92546i		−	0.302775i
$870$		0			0
$871$	−22.2977			−0.755530
$872$		0			0
$873$			69.0619i			2.33739i
$874$		0			0
$875$	−6.95302	−	8.75531i	−0.235055	−	0.295983i
$876$		0			0
$877$	−18.1199			−0.611864			−0.305932	−	0.952053i	$-0.598968\pi$
$877$	−18.1199			−0.611864			−0.305932	+	0.952053i	$0.598968\pi$
$878$		0			0
$879$	36.0120			1.21465
$880$		0			0
$881$	−23.0698			−0.777242			−0.388621	−	0.921398i	$-0.627048\pi$
$881$	−23.0698			−0.777242			−0.388621	+	0.921398i	$0.627048\pi$
$882$		0			0
$883$	−26.3787			−0.887715			−0.443857	−	0.896097i	$-0.646390\pi$
$883$	−26.3787			−0.887715			−0.443857	+	0.896097i	$0.646390\pi$
$884$		0			0
$885$	−97.4011	−	30.1166i	−3.27410	−	1.01236i
$886$		0			0
$887$			16.5725i			0.556450i	0.960516	+	0.278225i	$0.0897460\pi$
$887$			16.5725i			0.556450i	−0.960516	+	0.278225i	$0.910254\pi$
$888$		0			0
$889$	−14.2620			−0.478333
$890$		0			0
$891$		−	12.3826i		−	0.414831i
$892$		0			0
$893$	−0.543860			−0.0181996
$894$		0			0
$895$	−16.1245	−	4.98572i	−0.538982	−	0.166654i
$896$		0			0
$897$		−	14.7250i		−	0.491654i
$898$		0			0
$899$			28.6586i			0.955818i
$900$		0			0
$901$		−	63.5865i		−	2.11837i
$902$		0			0
$903$		−	20.6085i		−	0.685807i
$904$		0			0
$905$	37.9298	+	11.7280i	1.26083	+	0.389852i
$906$		0			0
$907$	17.6465			0.585941			0.292971	−	0.956121i	$-0.405356\pi$
$907$	17.6465			0.585941			0.292971	+	0.956121i	$0.405356\pi$
$908$		0			0
$909$			10.1302i			0.335998i
$910$		0			0
$911$	−17.6942			−0.586235			−0.293117	−	0.956076i	$-0.594693\pi$
$911$	−17.6942			−0.586235			−0.293117	+	0.956076i	$0.594693\pi$
$912$		0			0
$913$			10.7948i			0.357255i
$914$		0			0
$915$	−42.4425	−	13.1233i	−1.40311	−	0.433843i
$916$		0			0
$917$	3.78753			0.125075
$918$		0			0
$919$	−30.5650			−1.00825			−0.504123	−	0.863632i	$-0.668184\pi$
$919$	−30.5650			−1.00825			−0.504123	+	0.863632i	$0.668184\pi$
$920$		0			0
$921$	−16.4818			−0.543095
$922$		0			0
$923$	45.2685			1.49003
$924$		0			0
$925$	15.2890	+	10.4542i	0.502698	+	0.343733i
$926$		0			0
$927$			82.7596i			2.71818i
$928$		0			0
$929$	−57.4220			−1.88396			−0.941978	−	0.335676i	$-0.891035\pi$
$929$	−57.4220			−1.88396			−0.941978	+	0.335676i	$0.891035\pi$
$930$		0			0
$931$			0.207666i			0.00680596i
$932$		0			0
$933$	−25.0843			−0.821224
$934$		0			0
$935$	3.76866	−	12.1883i	0.123248	−	0.398601i
$936$		0			0
$937$		−	57.8853i		−	1.89103i	−0.325580	−	0.945515i	$-0.605559\pi$
$937$		−	57.8853i		−	1.89103i	0.325580	−	0.945515i	$-0.394441\pi$
$938$		0			0
$939$			10.0253i			0.327162i
$940$		0			0
$941$			20.5689i			0.670529i	0.942124	+	0.335264i	$0.108826\pi$
$941$			20.5689i			0.670529i	−0.942124	+	0.335264i	$0.891174\pi$
$942$		0			0
$943$			9.54374i			0.310787i
$944$		0			0
$945$	−24.6804	−	7.63123i	−0.802853	−	0.248244i
$946$		0			0
$947$	−2.78499			−0.0905001			−0.0452500	−	0.998976i	$-0.514408\pi$
$947$	−2.78499			−0.0905001			−0.0452500	+	0.998976i	$0.514408\pi$
$948$		0			0
$949$		−	1.55283i		−	0.0504071i
$950$		0			0
$951$	60.4296			1.95956
$952$		0			0
$953$		−	13.3339i		−	0.431929i	−0.976401	−	0.215964i	$-0.930710\pi$
$953$		−	13.3339i		−	0.431929i	0.976401	−	0.215964i	$-0.0692895\pi$
$954$		0			0
$955$	6.96766	−	22.5343i	0.225468	−	0.729194i
$956$		0			0
$957$	8.76604			0.283366
$958$		0			0
$959$	−19.1281			−0.617680
$960$		0			0
$961$	32.1481			1.03704
$962$		0			0
$963$	55.2220			1.77950
$964$		0			0
$965$	21.8967	+	6.77050i	0.704879	+	0.217950i
$966$		0			0
$967$		−	3.48530i		−	0.112080i	−0.998429	−	0.0560399i	$-0.982153\pi$
$967$		−	3.48530i		−	0.112080i	0.998429	−	0.0560399i	$-0.0178474\pi$
$968$		0			0
$969$	−4.73204			−0.152015
$970$		0			0
$971$		−	29.9970i		−	0.962651i	−0.876542	−	0.481325i	$-0.840156\pi$
$971$		−	29.9970i		−	0.962651i	0.876542	−	0.481325i	$-0.159844\pi$
$972$		0			0
$973$	−13.0683			−0.418949
$974$		0			0
$975$	39.7551	+	27.1836i	1.27318	+	0.870573i
$976$		0			0
$977$		−	31.8837i		−	1.02005i	−0.860160	−	0.510025i	$-0.829636\pi$
$977$		−	31.8837i		−	1.02005i	0.860160	−	0.510025i	$-0.170364\pi$
$978$		0			0
$979$			12.7741i			0.408261i
$980$		0			0
$981$		−	47.3023i		−	1.51025i
$982$		0			0
$983$		−	52.0621i		−	1.66052i	−0.557373	−	0.830262i	$-0.688191\pi$
$983$		−	52.0621i		−	1.66052i	0.557373	−	0.830262i	$-0.311809\pi$
$984$		0			0
$985$	8.00552	−	25.8909i	0.255077	−	0.824952i
$986$		0			0
$987$	−8.15992			−0.259733
$988$		0			0
$989$			10.1116i			0.321529i
$990$		0			0
$991$	17.1169			0.543737			0.271869	−	0.962334i	$-0.412358\pi$
$991$	17.1169			0.543737			0.271869	+	0.962334i	$0.412358\pi$
$992$		0			0
$993$		−	53.3792i		−	1.69394i
$994$		0			0
$995$	12.8039	−	41.4096i	0.405912	−	1.31277i
$996$		0			0
$997$	−18.5800			−0.588435			−0.294217	−	0.955738i	$-0.595059\pi$
$997$	−18.5800			−0.588435			−0.294217	+	0.955738i	$0.595059\pi$
$998$		0			0
$999$	42.7956			1.35399

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.2.l.d.1569.1	yes	24
4.3	odd	2	inner	2240.2.l.d.1569.24	yes	24
5.4	even	2		2240.2.l.c.1569.24	yes	24
8.3	odd	2		2240.2.l.c.1569.2	yes	24
8.5	even	2		2240.2.l.c.1569.23	yes	24
20.19	odd	2		2240.2.l.c.1569.1	✓	24
40.19	odd	2	inner	2240.2.l.d.1569.23	yes	24
40.29	even	2	inner	2240.2.l.d.1569.2	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2240.2.l.c.1569.1	✓	24	20.19	odd	2
2240.2.l.c.1569.2	yes	24	8.3	odd	2
2240.2.l.c.1569.23	yes	24	8.5	even	2
2240.2.l.c.1569.24	yes	24	5.4	even	2
2240.2.l.d.1569.1	yes	24	1.1	even	1	trivial
2240.2.l.d.1569.2	yes	24	40.29	even	2	inner
2240.2.l.d.1569.23	yes	24	40.19	odd	2	inner
2240.2.l.d.1569.24	yes	24	4.3	odd	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2240.l (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.11575 q^{3} +(0.660542 - 2.13628i) q^{5} +1.00000i q^{7} +6.70793 q^{9} +O(q^{10})\)	\(q-3.11575 q^{3} +(0.660542 - 2.13628i) q^{5} +1.00000i q^{7} +6.70793 q^{9} -0.780127i q^{11} +3.09141 q^{13} +(-2.05809 + 6.65612i) q^{15} +7.31343i q^{17} -0.207666i q^{19} -3.11575i q^{21} +1.52875i q^{23} +(-4.12737 - 2.82220i) q^{25} -11.5530 q^{27} -3.60641i q^{29} -7.94658 q^{31} +2.43068i q^{33} +(2.13628 + 0.660542i) q^{35} -3.70429 q^{37} -9.63207 q^{39} +6.24284 q^{41} +6.61428 q^{43} +(4.43087 - 14.3300i) q^{45} -2.61892i q^{47} -1.00000 q^{49} -22.7868i q^{51} -8.69448 q^{53} +(-1.66657 - 0.515307i) q^{55} +0.647035i q^{57} +14.6333i q^{59} +6.37646i q^{61} +6.70793i q^{63} +(2.04200 - 6.60411i) q^{65} -7.21281 q^{67} -4.76321i q^{69} +14.6433 q^{71} -0.502306i q^{73} +(12.8599 + 8.79329i) q^{75} +0.780127 q^{77} +11.4410 q^{79} +15.8725 q^{81} -13.8372 q^{83} +(15.6235 + 4.83083i) q^{85} +11.2367i q^{87} -16.3744 q^{89} +3.09141i q^{91} +24.7596 q^{93} +(-0.443631 - 0.137172i) q^{95} +10.2956i q^{97} -5.23304i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 4 q^{5} + 12 q^{9}+O(q^{10}) \) 24 * q + 4 * q^5 + 12 * q^9	\( 24 q + 4 q^{5} + 12 q^{9} + 4 q^{13} + 24 q^{37} + 48 q^{45} - 24 q^{49} - 88 q^{53} + 36 q^{65} - 20 q^{77} + 16 q^{81} + 56 q^{85} - 40 q^{89} - 24 q^{93}+O(q^{100}) \) 24 * q + 4 * q^5 + 12 * q^9 + 4 * q^13 + 24 * q^37 + 48 * q^45 - 24 * q^49 - 88 * q^53 + 36 * q^65 - 20 * q^77 + 16 * q^81 + 56 * q^85 - 40 * q^89 - 24 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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