LMFDB - Embedded newform 2240.2.l.b.1569.5

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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 3x^{10} + 11x^{8} + 22x^{6} + 99x^{4} + 243x^{2} + 729 $ x^12 + 3x^10 + 11x^8 + 22x^6 + 99x^4 + 243*x^2 + 729
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1569.5
Root			$0.517456 + 1.65295i$ of defining polynomial
Character	$\chi$	$=$	2240.1569
Dual form			2240.2.l.b.1569.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.03491 q^{3} +(0.188470 - 2.22811i) q^{5} +1.00000i q^{7} -1.92896 q^{9} +O(q^{10})$	$q-1.03491 q^{3} +(0.188470 - 2.22811i) q^{5} +1.00000i q^{7} -1.92896 q^{9} -3.68284i q^{11} -5.30590 q^{13} +(-0.195049 + 2.30590i) q^{15} +5.49113i q^{17} +1.62306i q^{19} -1.03491i q^{21} +1.24612i q^{23} +(-4.92896 - 0.839862i) q^{25} +5.10104 q^{27} +7.17086i q^{29} +2.06982 q^{31} +3.81141i q^{33} +(2.22811 + 0.188470i) q^{35} +5.85792 q^{37} +5.49113 q^{39} +2.00000 q^{41} +10.5922 q^{43} +(-0.363550 + 4.29793i) q^{45} -0.928958i q^{47} -1.00000 q^{49} -5.68284i q^{51} -6.61180 q^{53} +(-8.20577 - 0.694102i) q^{55} -1.67972i q^{57} -4.23486i q^{59} -8.59587i q^{61} -1.92896i q^{63} +(-1.00000 + 11.8221i) q^{65} +9.30254 q^{67} -1.28963i q^{69} +3.74955 q^{71} +10.5922i q^{73} +(5.10104 + 0.869183i) q^{75} +3.68284 q^{77} -9.63078 q^{79} +0.507756 q^{81} +4.84632 q^{83} +(12.2349 + 1.03491i) q^{85} -7.42120i q^{87} +12.6118 q^{89} -5.30590i q^{91} -2.14208 q^{93} +(3.61636 + 0.305898i) q^{95} +12.3338i q^{97} +7.10404i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 4 q^{5} + 24 q^{9}+O(q^{10}) $ 12 * q + 4 * q^5 + 24 * q^9	$ 12 q + 4 q^{5} + 24 q^{9} - 20 q^{13} - 12 q^{25} - 24 q^{37} + 24 q^{41} + 48 q^{45} - 12 q^{49} + 8 q^{53} - 12 q^{65} + 4 q^{77} + 20 q^{81} + 56 q^{85} + 64 q^{89} - 120 q^{93}+O(q^{100}) $ 12 * q + 4 * q^5 + 24 * q^9 - 20 * q^13 - 12 * q^25 - 24 * q^37 + 24 * q^41 + 48 * q^45 - 12 * q^49 + 8 * q^53 - 12 * q^65 + 4 * q^77 + 20 * q^81 + 56 * q^85 + 64 * q^89 - 120 * 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
$3$	−1.03491			−0.597506			−0.298753	−	0.954330i	$-0.596571\pi$
$3$	−1.03491			−0.597506			−0.298753	+	0.954330i	$0.596571\pi$
$4$		0			0
$5$	0.188470	−	2.22811i	0.0842861	−	0.996442i
$5$	0.188470	−	2.22811i	0.0842861	−	0.996442i
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		0			0
$9$	−1.92896			−0.642986
$10$		0			0
$11$		−	3.68284i		−	1.11042i	−0.831711	−	0.555209i	$-0.812638\pi$
$11$		−	3.68284i		−	1.11042i	0.831711	−	0.555209i	$-0.187362\pi$
$12$		0			0
$13$	−5.30590			−1.47159			−0.735796	−	0.677204i	$-0.763192\pi$
$13$	−5.30590			−1.47159			−0.735796	+	0.677204i	$0.763192\pi$
$14$		0			0
$15$	−0.195049	+	2.30590i	−0.0503615	+	0.595380i
$16$		0			0
$17$			5.49113i			1.33180i	0.746043	+	0.665898i	$0.231951\pi$
$17$			5.49113i			1.33180i	−0.746043	+	0.665898i	$0.768049\pi$
$18$		0			0
$19$			1.62306i			0.372356i	0.982516	+	0.186178i	$0.0596101\pi$
$19$			1.62306i			0.372356i	−0.982516	+	0.186178i	$0.940390\pi$
$20$		0			0
$21$		−	1.03491i		−	0.225836i
$22$		0			0
$23$			1.24612i			0.259834i	0.991525	+	0.129917i	$0.0414711\pi$
$23$			1.24612i			0.259834i	−0.991525	+	0.129917i	$0.958529\pi$
$24$		0			0
$25$	−4.92896	−	0.839862i	−0.985792	−	0.167972i
$26$		0			0
$27$	5.10104			0.981695
$28$		0			0
$29$			7.17086i			1.33159i	0.746133	+	0.665797i	$0.231909\pi$
$29$			7.17086i			1.33159i	−0.746133	+	0.665797i	$0.768091\pi$
$30$		0			0
$31$	2.06982			0.371751			0.185876	−	0.982573i	$-0.440488\pi$
$31$	2.06982			0.371751			0.185876	+	0.982573i	$0.440488\pi$
$32$		0			0
$33$			3.81141i			0.663481i
$34$		0			0
$35$	2.22811	+	0.188470i	0.376620	+	0.0318572i
$36$		0			0
$37$	5.85792			0.963036			0.481518	−	0.876436i	$-0.340086\pi$
$37$	5.85792			0.963036			0.481518	+	0.876436i	$0.340086\pi$
$38$		0			0
$39$	5.49113			0.879285
$40$		0			0
$41$	2.00000			0.312348			0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000			0.312348			0.156174	+	0.987730i	$0.450084\pi$
$42$		0			0
$43$	10.5922			1.61529			0.807645	−	0.589669i	$-0.200742\pi$
$43$	10.5922			1.61529			0.807645	+	0.589669i	$0.200742\pi$
$44$		0			0
$45$	−0.363550	+	4.29793i	−0.0541948	+	0.640698i
$46$		0			0
$47$		−	0.928958i		−	0.135503i	−0.997702	−	0.0677513i	$-0.978418\pi$
$47$		−	0.928958i		−	0.135503i	0.997702	−	0.0677513i	$-0.0215824\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$		−	5.68284i		−	0.795756i
$52$		0			0
$53$	−6.61180			−0.908200			−0.454100	−	0.890951i	$-0.650039\pi$
$53$	−6.61180			−0.908200			−0.454100	+	0.890951i	$0.650039\pi$
$54$		0			0
$55$	−8.20577	−	0.694102i	−1.10647	−	0.0935928i
$56$		0			0
$57$		−	1.67972i		−	0.222485i
$58$		0			0
$59$		−	4.23486i		−	0.551331i	−0.961254	−	0.275666i	$-0.911102\pi$
$59$		−	4.23486i		−	0.551331i	0.961254	−	0.275666i	$-0.0888983\pi$
$60$		0			0
$61$		−	8.59587i		−	1.10059i	−0.834971	−	0.550294i	$-0.814516\pi$
$61$		−	8.59587i		−	1.10059i	0.834971	−	0.550294i	$-0.185484\pi$
$62$		0			0
$63$		−	1.92896i		−	0.243026i
$64$		0			0
$65$	−1.00000	+	11.8221i	−0.124035	+	1.46635i
$66$		0			0
$67$	9.30254			1.13649			0.568243	−	0.822861i	$-0.307623\pi$
$67$	9.30254			1.13649			0.568243	+	0.822861i	$0.307623\pi$
$68$		0			0
$69$		−	1.28963i		−	0.155253i
$70$		0			0
$71$	3.74955			0.444989			0.222495	−	0.974934i	$-0.428580\pi$
$71$	3.74955			0.444989			0.222495	+	0.974934i	$0.428580\pi$
$72$		0			0
$73$			10.5922i			1.23972i	0.784713	+	0.619860i	$0.212811\pi$
$73$			10.5922i			1.23972i	−0.784713	+	0.619860i	$0.787189\pi$
$74$		0			0
$75$	5.10104	+	0.869183i	0.589017	+	0.100365i
$76$		0			0
$77$	3.68284			0.419698
$78$		0			0
$79$	−9.63078			−1.08355			−0.541774	−	0.840524i	$-0.682247\pi$
$79$	−9.63078			−1.08355			−0.541774	+	0.840524i	$0.682247\pi$
$80$		0			0
$81$	0.507756			0.0564174
$82$		0			0
$83$	4.84632			0.531953			0.265976	−	0.963980i	$-0.414306\pi$
$83$	4.84632			0.531953			0.265976	+	0.963980i	$0.414306\pi$
$84$		0			0
$85$	12.2349	+	1.03491i	1.32706	+	0.112252i
$86$		0			0
$87$		−	7.42120i		−	0.795636i
$88$		0			0
$89$	12.6118			1.33685			0.668424	−	0.743781i	$-0.266969\pi$
$89$	12.6118			1.33685			0.668424	+	0.743781i	$0.266969\pi$
$90$		0			0
$91$		−	5.30590i		−	0.556209i
$92$		0			0
$93$	−2.14208			−0.222124
$94$		0			0
$95$	3.61636	+	0.305898i	0.371031	+	0.0313844i
$96$		0			0
$97$			12.3338i			1.25230i	0.779701	+	0.626152i	$0.215371\pi$
$97$			12.3338i			1.25230i	−0.779701	+	0.626152i	$0.784629\pi$
$98$		0			0
$99$			7.10404i			0.713983i
$100$		0			0
$101$			0.316577i			0.0315006i	0.999876	+	0.0157503i	$0.00501368\pi$
$101$			0.316577i			0.0315006i	−0.999876	+	0.0157503i	$0.994986\pi$
$102$		0			0
$103$			2.31716i			0.228317i	0.993463	+	0.114158i	$0.0364172\pi$
$103$			2.31716i			0.228317i	−0.993463	+	0.114158i	$0.963583\pi$
$104$		0			0
$105$	−2.30590	−	0.195049i	−0.225033	−	0.0190349i
$106$		0			0
$107$	−5.81937			−0.562580			−0.281290	−	0.959623i	$-0.590762\pi$
$107$	−5.81937			−0.562580			−0.281290	+	0.959623i	$0.590762\pi$
$108$		0			0
$109$		−	14.0135i		−	1.34225i	−0.741345	−	0.671124i	$-0.765812\pi$
$109$		−	14.0135i		−	1.34225i	0.741345	−	0.671124i	$-0.234188\pi$
$110$		0			0
$111$	−6.06242			−0.575420
$112$		0			0
$113$			8.52235i			0.801715i	0.916140	+	0.400857i	$0.131288\pi$
$113$			8.52235i			0.801715i	−0.916140	+	0.400857i	$0.868712\pi$
$114$		0			0
$115$	2.77650	+	0.234856i	0.258910	+	0.0219004i
$116$		0			0
$117$	10.2349			0.946213
$118$		0			0
$119$	−5.49113			−0.503371
$120$		0			0
$121$	−2.56329			−0.233026
$122$		0			0
$123$	−2.06982			−0.186630
$124$		0			0
$125$	−2.80026	+	10.8240i	−0.250463	+	0.968126i
$126$		0			0
$127$		−	19.9775i		−	1.77271i	−0.463003	−	0.886357i	$-0.653228\pi$
$127$		−	19.9775i		−	1.77271i	0.463003	−	0.886357i	$-0.346772\pi$
$128$		0			0
$129$	−10.9620			−0.965146
$130$		0			0
$131$			21.4584i			1.87483i	0.348210	+	0.937417i	$0.386790\pi$
$131$			21.4584i			1.87483i	−0.348210	+	0.937417i	$0.613210\pi$
$132$		0			0
$133$	−1.62306			−0.140737
$134$		0			0
$135$	0.961390	−	11.3657i	0.0827432	−	0.978201i
$136$		0			0
$137$			3.35945i			0.287017i	0.989649	+	0.143509i	$0.0458384\pi$
$137$			3.35945i			0.287017i	−0.989649	+	0.143509i	$0.954162\pi$
$138$		0			0
$139$			6.72710i			0.570585i	0.958440	+	0.285293i	$0.0920908\pi$
$139$			6.72710i			0.570585i	−0.958440	+	0.285293i	$0.907909\pi$
$140$		0			0
$141$			0.961390i			0.0809636i
$142$		0			0
$143$			19.5408i			1.63408i
$144$		0			0
$145$	15.9775	+	1.35149i	1.32686	+	0.112235i
$146$		0			0
$147$	1.03491			0.0853581
$148$		0			0
$149$			3.35945i			0.275217i	0.990487	+	0.137608i	$0.0439415\pi$
$149$			3.35945i			0.275217i	−0.990487	+	0.137608i	$0.956058\pi$
$150$		0			0
$151$	16.8635			1.37233			0.686166	−	0.727445i	$-0.259292\pi$
$151$	16.8635			1.37233			0.686166	+	0.727445i	$0.259292\pi$
$152$		0			0
$153$		−	10.5922i		−	0.856326i
$154$		0			0
$155$	0.390099	−	4.61180i	0.0313335	−	0.370428i
$156$		0			0
$157$	−5.60053			−0.446971			−0.223485	−	0.974707i	$-0.571744\pi$
$157$	−5.60053			−0.446971			−0.223485	+	0.974707i	$0.571744\pi$
$158$		0			0
$159$	6.84262			0.542655
$160$		0			0
$161$	−1.24612			−0.0982082
$162$		0			0
$163$	22.3546			1.75095			0.875475	−	0.483263i	$-0.160549\pi$
$163$	22.3546			1.75095			0.875475	+	0.483263i	$0.160549\pi$
$164$		0			0
$165$	8.49224	+	0.718335i	0.661120	+	0.0559223i
$166$		0			0
$167$			20.7643i			1.60679i	0.595444	+	0.803397i	$0.296976\pi$
$167$			20.7643i			1.60679i	−0.595444	+	0.803397i	$0.703024\pi$
$168$		0			0
$169$	15.1525			1.16558
$170$		0			0
$171$		−	3.13082i		−	0.239420i
$172$		0			0
$173$	−10.6941			−0.813058			−0.406529	−	0.913638i	$-0.633261\pi$
$173$	−10.6941			−0.813058			−0.406529	+	0.913638i	$0.633261\pi$
$174$		0			0
$175$	0.839862	−	4.92896i	0.0634876	−	0.372594i
$176$		0			0
$177$			4.38270i			0.329424i
$178$		0			0
$179$		−	6.63433i		−	0.495873i	−0.968776	−	0.247936i	$-0.920248\pi$
$179$		−	6.63433i		−	0.495873i	0.968776	−	0.247936i	$-0.0797524\pi$
$180$		0			0
$181$			12.0790i			0.897828i	0.893575	+	0.448914i	$0.148189\pi$
$181$			12.0790i			0.897828i	−0.893575	+	0.448914i	$0.851811\pi$
$182$		0			0
$183$			8.89596i			0.657608i
$184$		0			0
$185$	1.10404	−	13.0521i	0.0811705	−	0.959609i
$186$		0			0
$187$	20.2229			1.47885
$188$		0			0
$189$			5.10104i			0.371046i
$190$		0			0
$191$	12.9902			0.939940			0.469970	−	0.882682i	$-0.344265\pi$
$191$	12.9902			0.939940			0.469970	+	0.882682i	$0.344265\pi$
$192$		0			0
$193$			19.5046i			1.40397i	0.712190	+	0.701986i	$0.247703\pi$
$193$			19.5046i			1.40397i	−0.712190	+	0.701986i	$0.752297\pi$
$194$		0			0
$195$	1.03491	−	12.2349i	0.0741115	−	0.876156i
$196$		0			0
$197$	−17.1040			−1.21861			−0.609306	−	0.792935i	$-0.708552\pi$
$197$	−17.1040			−1.21861			−0.609306	+	0.792935i	$0.708552\pi$
$198$		0			0
$199$	−6.84262			−0.485061			−0.242530	−	0.970144i	$-0.577977\pi$
$199$	−6.84262			−0.485061			−0.242530	+	0.970144i	$0.577977\pi$
$200$		0			0
$201$	−9.62731			−0.679058
$202$		0			0
$203$	−7.17086			−0.503296
$204$		0			0
$205$	0.376939	−	4.45622i	0.0263266	−	0.311236i
$206$		0			0
$207$		−	2.40372i		−	0.167070i
$208$		0			0
$209$	5.97747			0.413470
$210$		0			0
$211$			10.2946i			0.708712i	0.935111	+	0.354356i	$0.115300\pi$
$211$			10.2946i			0.708712i	−0.935111	+	0.354356i	$0.884700\pi$
$212$		0			0
$213$	−3.88045			−0.265884
$214$		0			0
$215$	1.99630	−	23.6005i	0.136147	−	1.60954i
$216$		0			0
$217$			2.06982i			0.140509i
$218$		0			0
$219$		−	10.9620i		−	0.740740i
$220$		0			0
$221$		−	29.1354i		−	1.95986i
$222$		0			0
$223$			24.4142i			1.63489i	0.576003	+	0.817447i	$0.304611\pi$
$223$			24.4142i			1.63489i	−0.576003	+	0.817447i	$0.695389\pi$
$224$		0			0
$225$	9.50776	+	1.62006i	0.633850	+	0.108004i
$226$		0			0
$227$	16.1568			1.07237			0.536183	−	0.844102i	$-0.319866\pi$
$227$	16.1568			1.07237			0.536183	+	0.844102i	$0.319866\pi$
$228$		0			0
$229$			0.973049i			0.0643009i	0.999483	+	0.0321504i	$0.0102356\pi$
$229$			0.973049i			0.0643009i	−0.999483	+	0.0321504i	$0.989764\pi$
$230$		0			0
$231$	−3.81141			−0.250772
$232$		0			0
$233$		−	17.8249i		−	1.16775i	−0.811845	−	0.583874i	$-0.801536\pi$
$233$		−	17.8249i		−	1.16775i	0.811845	−	0.583874i	$-0.198464\pi$
$234$		0			0
$235$	−2.06982	−	0.175080i	−0.135020	−	0.0114210i
$236$		0			0
$237$	9.96700			0.647426
$238$		0			0
$239$	−21.0031			−1.35858			−0.679290	−	0.733870i	$-0.737712\pi$
$239$	−21.0031			−1.35858			−0.679290	+	0.733870i	$0.737712\pi$
$240$		0			0
$241$	11.9775			0.771537			0.385768	−	0.922596i	$-0.373936\pi$
$241$	11.9775			0.771537			0.385768	+	0.922596i	$0.373936\pi$
$242$		0			0
$243$	−15.8286			−1.01540
$244$		0			0
$245$	−0.188470	+	2.22811i	−0.0120409	+	0.142349i
$246$		0			0
$247$		−	8.61180i		−	0.547955i
$248$		0			0
$249$	−5.01551			−0.317845
$250$		0			0
$251$			20.9887i			1.32480i	0.749152	+	0.662399i	$0.230461\pi$
$251$			20.9887i			1.32480i	−0.749152	+	0.662399i	$0.769539\pi$
$252$		0			0
$253$	4.58926			0.288525
$254$		0			0
$255$	−12.6620	−	1.07104i	−0.792925	−	0.0670712i
$256$		0			0
$257$			2.45992i			0.153446i	0.997052	+	0.0767228i	$0.0244457\pi$
$257$			2.45992i			0.153446i	−0.997052	+	0.0767228i	$0.975554\pi$
$258$		0			0
$259$			5.85792i			0.363993i
$260$		0			0
$261$		−	13.8323i		−	0.856197i
$262$		0			0
$263$			27.7158i			1.70903i	0.519425	+	0.854516i	$0.326146\pi$
$263$			27.7158i			1.70903i	−0.519425	+	0.854516i	$0.673854\pi$
$264$		0			0
$265$	−1.24612	+	14.7318i	−0.0765487	+	0.904968i
$266$		0			0
$267$	−13.0521			−0.798775
$268$		0			0
$269$			16.2187i			0.988871i	0.869214	+	0.494435i	$0.164625\pi$
$269$			16.2187i			0.988871i	−0.869214	+	0.494435i	$0.835375\pi$
$270$		0			0
$271$	25.3240			1.53832			0.769161	−	0.639055i	$-0.220674\pi$
$271$	25.3240			1.53832			0.769161	+	0.639055i	$0.220674\pi$
$272$		0			0
$273$			5.49113i			0.332339i
$274$		0			0
$275$	−3.09307	+	18.1525i	−0.186519	+	1.09464i
$276$		0			0
$277$	−17.3431			−1.04205			−0.521024	−	0.853542i	$-0.674450\pi$
$277$	−17.3431			−1.04205			−0.521024	+	0.853542i	$0.674450\pi$
$278$		0			0
$279$	−3.99260			−0.239031
$280$		0			0
$281$	−21.6603			−1.29215			−0.646073	−	0.763276i	$-0.723590\pi$
$281$	−21.6603			−1.29215			−0.646073	+	0.763276i	$0.723590\pi$
$282$		0			0
$283$	−3.10473			−0.184557			−0.0922786	−	0.995733i	$-0.529415\pi$
$283$	−3.10473			−0.184557			−0.0922786	+	0.995733i	$0.529415\pi$
$284$		0			0
$285$	−3.74261	−	0.316577i	−0.221693	−	0.0187524i
$286$		0			0
$287$			2.00000i			0.118056i
$288$		0			0
$289$	−13.1525			−0.773679
$290$		0			0
$291$		−	12.7643i		−	0.748259i
$292$		0			0
$293$	−27.6335			−1.61437			−0.807184	−	0.590300i	$-0.799009\pi$
$293$	−27.6335			−1.61437			−0.807184	+	0.590300i	$0.799009\pi$
$294$		0			0
$295$	−9.43573	−	0.798141i	−0.549369	−	0.0464696i
$296$		0			0
$297$		−	18.7863i		−	1.09009i
$298$		0			0
$299$		−	6.61180i		−	0.382370i
$300$		0			0
$301$			10.5922i			0.610522i
$302$		0			0
$303$		−	0.327629i		−	0.0188218i
$304$		0			0
$305$	−19.1525	−	1.62006i	−1.09667	−	0.0927643i
$306$		0			0
$307$	−28.4287			−1.62251			−0.811256	−	0.584691i	$-0.801216\pi$
$307$	−28.4287			−1.62251			−0.811256	+	0.584691i	$0.801216\pi$
$308$		0			0
$309$		−	2.39806i		−	0.136421i
$310$		0			0
$311$	−30.0968			−1.70663			−0.853316	−	0.521394i	$-0.825412\pi$
$311$	−30.0968			−1.70663			−0.853316	+	0.521394i	$0.825412\pi$
$312$		0			0
$313$		−	7.56096i		−	0.427371i	−0.976903	−	0.213685i	$-0.931453\pi$
$313$		−	7.56096i		−	0.427371i	0.976903	−	0.213685i	$-0.0685467\pi$
$314$		0			0
$315$	−4.29793	−	0.363550i	−0.242161	−	0.0204837i
$316$		0			0
$317$	16.7539			0.940992			0.470496	−	0.882402i	$-0.344075\pi$
$317$	16.7539			0.940992			0.470496	+	0.882402i	$0.344075\pi$
$318$		0			0
$319$	26.4091			1.47863
$320$		0			0
$321$	6.02253			0.336145
$322$		0			0
$323$	−8.91244			−0.495902
$324$		0			0
$325$	26.1525	+	4.45622i	1.45068	+	0.247187i
$326$		0			0
$327$			14.5027i			0.802002i
$328$		0			0
$329$	0.928958			0.0512151
$330$		0			0
$331$		−	22.1196i		−	1.21580i	−0.794013	−	0.607900i	$-0.792012\pi$
$331$		−	22.1196i		−	1.21580i	0.794013	−	0.607900i	$-0.207988\pi$
$332$		0			0
$333$	−11.2997			−0.619219
$334$		0			0
$335$	1.75325	−	20.7271i	0.0957901	−	1.13244i
$336$		0			0
$337$			8.52235i			0.464242i	0.972687	+	0.232121i	$0.0745665\pi$
$337$			8.52235i			0.464242i	−0.972687	+	0.232121i	$0.925434\pi$
$338$		0			0
$339$		−	8.81987i		−	0.479030i
$340$		0			0
$341$		−	7.62282i		−	0.412799i
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$	−2.87343	−	0.243055i	−0.154700	−	0.0130856i
$346$		0			0
$347$	−14.0753			−0.755604			−0.377802	−	0.925886i	$-0.623320\pi$
$347$	−14.0753			−0.755604			−0.377802	+	0.925886i	$0.623320\pi$
$348$		0			0
$349$			26.4208i			1.41427i	0.707078	+	0.707135i	$0.250013\pi$
$349$			26.4208i			1.41427i	−0.707078	+	0.707135i	$0.749987\pi$
$350$		0			0
$351$	−27.0656			−1.44465
$352$		0			0
$353$		−	10.9204i		−	0.581234i	−0.956839	−	0.290617i	$-0.906139\pi$
$353$		−	10.9204i		−	0.581234i	0.956839	−	0.290617i	$-0.0938606\pi$
$354$		0			0
$355$	0.706675	−	8.35441i	0.0375064	−	0.443406i
$356$		0			0
$357$	5.68284			0.300768
$358$		0			0
$359$	36.3063			1.91617			0.958085	−	0.286483i	$-0.0924862\pi$
$359$	36.3063			1.91617			0.958085	+	0.286483i	$0.0924862\pi$
$360$		0			0
$361$	16.3657			0.861351
$362$		0			0
$363$	2.65277			0.139234
$364$		0			0
$365$	23.6005	+	1.99630i	1.23531	+	0.104491i
$366$		0			0
$367$		−	11.8249i		−	0.617256i	−0.951183	−	0.308628i	$-0.900130\pi$
$367$		−	11.8249i		−	0.617256i	0.951183	−	0.308628i	$-0.0998698\pi$
$368$		0			0
$369$	−3.85792			−0.200835
$370$		0			0
$371$		−	6.61180i		−	0.343267i
$372$		0			0
$373$	−13.7384			−0.711346			−0.355673	−	0.934611i	$-0.615748\pi$
$373$	−13.7384			−0.711346			−0.355673	+	0.934611i	$0.615748\pi$
$374$		0			0
$375$	2.89803	−	11.2019i	0.149653	−	0.578461i
$376$		0			0
$377$		−	38.0478i		−	1.95956i
$378$		0			0
$379$			3.10404i			0.159444i	0.996817	+	0.0797219i	$0.0254032\pi$
$379$			3.10404i			0.159444i	−0.996817	+	0.0797219i	$0.974597\pi$
$380$		0			0
$381$			20.6749i			1.05921i
$382$		0			0
$383$		−	2.73135i		−	0.139565i	−0.997562	−	0.0697826i	$-0.977769\pi$
$383$		−	2.73135i		−	0.139565i	0.997562	−	0.0697826i	$-0.0222306\pi$
$384$		0			0
$385$	0.694102	−	8.20577i	0.0353747	−	0.418205i
$386$		0			0
$387$	−20.4319			−1.03861
$388$		0			0
$389$		−	16.7165i		−	0.847558i	−0.905766	−	0.423779i	$-0.860703\pi$
$389$		−	16.7165i		−	0.847558i	0.905766	−	0.423779i	$-0.139297\pi$
$390$		0			0
$391$	−6.84262			−0.346046
$392$		0			0
$393$		−	22.2076i		−	1.12022i
$394$		0			0
$395$	−1.81511	+	21.4584i	−0.0913280	+	1.07969i
$396$		0			0
$397$	32.2679			1.61948			0.809739	−	0.586791i	$-0.199609\pi$
$397$	32.2679			1.61948			0.809739	+	0.586791i	$0.199609\pi$
$398$		0			0
$399$	1.67972			0.0840914
$400$		0			0
$401$	14.1525			0.706745			0.353372	−	0.935483i	$-0.385035\pi$
$401$	14.1525			0.706745			0.353372	+	0.935483i	$0.385035\pi$
$402$		0			0
$403$	−10.9823			−0.547066
$404$		0			0
$405$	0.0956966	−	1.13134i	0.00475520	−	0.0562166i
$406$		0			0
$407$		−	21.5738i		−	1.06937i
$408$		0			0
$409$	−30.3051			−1.49849			−0.749245	−	0.662293i	$-0.769584\pi$
$409$	−30.3051			−1.49849			−0.749245	+	0.662293i	$0.769584\pi$
$410$		0			0
$411$		−	3.47673i		−	0.171495i
$412$		0			0
$413$	4.23486			0.208384
$414$		0			0
$415$	0.913384	−	10.7981i	0.0448362	−	0.530060i
$416$		0			0
$417$		−	6.96195i		−	0.340928i
$418$		0			0
$419$		−	23.4810i		−	1.14712i	−0.819163	−	0.573560i	$-0.805562\pi$
$419$		−	23.4810i		−	1.14712i	0.819163	−	0.573560i	$-0.194438\pi$
$420$		0			0
$421$			1.74159i			0.0848797i	0.999099	+	0.0424399i	$0.0135131\pi$
$421$			1.74159i			0.0848797i	−0.999099	+	0.0424399i	$0.986487\pi$
$422$		0			0
$423$			1.79192i			0.0871262i
$424$		0			0
$425$	4.61180	−	27.0656i	0.223705	−	1.31287i
$426$		0			0
$427$	8.59587			0.415983
$428$		0			0
$429$		−	20.2229i		−	0.976373i
$430$		0			0
$431$	30.5487			1.47148			0.735741	−	0.677263i	$-0.236834\pi$
$431$	30.5487			1.47148			0.735741	+	0.677263i	$0.236834\pi$
$432$		0			0
$433$			14.7318i			0.707966i	0.935252	+	0.353983i	$0.115173\pi$
$433$			14.7318i			0.707966i	−0.935252	+	0.353983i	$0.884827\pi$
$434$		0			0
$435$	−16.5353	−	1.39867i	−0.792805	−	0.0670611i
$436$		0			0
$437$	−2.02253			−0.0967508
$438$		0			0
$439$	1.28963			0.0615505			0.0307752	−	0.999526i	$-0.490202\pi$
$439$	1.28963			0.0615505			0.0307752	+	0.999526i	$0.490202\pi$
$440$		0			0
$441$	1.92896			0.0918552
$442$		0			0
$443$	−12.0055			−0.570399			−0.285200	−	0.958468i	$-0.592060\pi$
$443$	−12.0055			−0.570399			−0.285200	+	0.958468i	$0.592060\pi$
$444$		0			0
$445$	2.37694	−	28.1005i	0.112678	−	1.33209i
$446$		0			0
$447$		−	3.47673i		−	0.164444i
$448$		0			0
$449$	−2.57880			−0.121701			−0.0608505	−	0.998147i	$-0.519381\pi$
$449$	−2.57880			−0.121701			−0.0608505	+	0.998147i	$0.519381\pi$
$450$		0			0
$451$		−	7.36567i		−	0.346836i
$452$		0			0
$453$	−17.4522			−0.819977
$454$		0			0
$455$	−11.8221	−	1.00000i	−0.554230	−	0.0468807i
$456$		0			0
$457$		−	25.0809i		−	1.17324i	−0.809864	−	0.586618i	$-0.800459\pi$
$457$		−	25.0809i		−	1.17324i	0.809864	−	0.586618i	$-0.199541\pi$
$458$		0			0
$459$			28.0105i			1.30742i
$460$		0			0
$461$			13.3687i			0.622641i	0.950305	+	0.311320i	$0.100771\pi$
$461$			13.3687i			0.622641i	−0.950305	+	0.311320i	$0.899229\pi$
$462$		0			0
$463$			35.7158i			1.65986i	0.557871	+	0.829928i	$0.311618\pi$
$463$			35.7158i			1.65986i	−0.557871	+	0.829928i	$0.688382\pi$
$464$		0			0
$465$	−0.403717	+	4.77280i	−0.0187219	+	0.221333i
$466$		0			0
$467$	−21.5628			−0.997806			−0.498903	−	0.866658i	$-0.666264\pi$
$467$	−21.5628			−0.997806			−0.498903	+	0.866658i	$0.666264\pi$
$468$		0			0
$469$			9.30254i			0.429552i
$470$		0			0
$471$	5.79605			0.267068
$472$		0			0
$473$		−	39.0092i		−	1.79365i
$474$		0			0
$475$	1.36315	−	8.00000i	0.0625455	−	0.367065i
$476$		0			0
$477$	12.7539			0.583960
$478$		0			0
$479$	−31.5335			−1.44080			−0.720400	−	0.693559i	$-0.756042\pi$
$479$	−31.5335			−1.44080			−0.720400	+	0.693559i	$0.756042\pi$
$480$		0			0
$481$	−31.0815			−1.41719
$482$		0			0
$483$	1.28963			0.0586800
$484$		0			0
$485$	27.4810	+	2.32454i	1.24785	+	0.105552i
$486$		0			0
$487$		−	27.9775i		−	1.26778i	−0.773423	−	0.633890i	$-0.781457\pi$
$487$		−	27.9775i		−	1.26778i	0.773423	−	0.633890i	$-0.218543\pi$
$488$		0			0
$489$	−23.1351			−1.04620
$490$		0			0
$491$		−	31.6293i		−	1.42741i	−0.700447	−	0.713705i	$-0.747016\pi$
$491$		−	31.6293i		−	1.42741i	0.700447	−	0.713705i	$-0.252984\pi$
$492$		0			0
$493$	−39.3761			−1.77341
$494$		0			0
$495$	15.8286	+	1.33889i	0.711442	+	0.0601788i
$496$		0			0
$497$			3.74955i			0.168190i
$498$		0			0
$499$			26.2946i			1.17711i	0.808457	+	0.588555i	$0.200303\pi$
$499$			26.2946i			1.17711i	−0.808457	+	0.588555i	$0.799697\pi$
$500$		0			0
$501$		−	21.4893i		−	0.960069i
$502$		0			0
$503$		−	4.29463i		−	0.191488i	−0.995406	−	0.0957441i	$-0.969477\pi$
$503$		−	4.29463i		−	0.191488i	0.995406	−	0.0957441i	$-0.0305230\pi$
$504$		0			0
$505$	0.705368	+	0.0596651i	0.0313885	+	0.00265506i
$506$		0			0
$507$	−15.6815			−0.696442
$508$		0			0
$509$		−	40.7625i		−	1.80676i	−0.428836	−	0.903382i	$-0.641076\pi$
$509$		−	40.7625i		−	1.80676i	0.428836	−	0.903382i	$-0.358924\pi$
$510$		0			0
$511$	−10.5922			−0.468570
$512$		0			0
$513$			8.27929i			0.365540i
$514$		0			0
$515$	5.16290	+	0.436715i	0.227504	+	0.0192439i
$516$		0			0
$517$	−3.42120			−0.150464
$518$		0			0
$519$	11.0674			0.485807
$520$		0			0
$521$	30.0660			1.31722			0.658608	−	0.752487i	$-0.271146\pi$
$521$	30.0660			1.31722			0.658608	+	0.752487i	$0.271146\pi$
$522$		0			0
$523$	−2.65277			−0.115998			−0.0579988	−	0.998317i	$-0.518472\pi$
$523$	−2.65277			−0.115998			−0.0579988	+	0.998317i	$0.518472\pi$
$524$		0			0
$525$	−0.869183	+	5.10104i	−0.0379342	+	0.222627i
$526$		0			0
$527$			11.3657i			0.495096i
$528$		0			0
$529$	21.4472			0.932486
$530$		0			0
$531$			8.16886i			0.354498i
$532$		0			0
$533$	−10.6118			−0.459648
$534$		0			0
$535$	−1.09677	+	12.9662i	−0.0474177	+	0.560578i
$536$		0			0
$537$			6.86594i			0.296287i
$538$		0			0
$539$			3.68284i			0.158631i
$540$		0			0
$541$		−	5.10104i		−	0.219311i	−0.993970	−	0.109655i	$-0.965025\pi$
$541$		−	5.10104i		−	0.219311i	0.993970	−	0.109655i	$-0.0349747\pi$
$542$		0			0
$543$		−	12.5007i		−	0.536458i
$544$		0			0
$545$	−31.2236	−	2.64111i	−1.33747	−	0.113133i
$546$		0			0
$547$	28.4171			1.21503			0.607513	−	0.794310i	$-0.292167\pi$
$547$	28.4171			1.21503			0.607513	+	0.794310i	$0.292167\pi$
$548$		0			0
$549$			16.5811i			0.707663i
$550$		0			0
$551$	−11.6387			−0.495827
$552$		0			0
$553$		−	9.63078i		−	0.409542i
$554$		0			0
$555$	−1.14258	+	13.5078i	−0.0484999	+	0.573372i
$556$		0			0
$557$	17.4542			0.739558			0.369779	−	0.929120i	$-0.379433\pi$
$557$	17.4542			0.739558			0.369779	+	0.929120i	$0.379433\pi$
$558$		0			0
$559$	−56.2010			−2.37705
$560$		0			0
$561$	−20.9290			−0.883621
$562$		0			0
$563$	−35.7233			−1.50556			−0.752779	−	0.658274i	$-0.771287\pi$
$563$	−35.7233			−1.50556			−0.752779	+	0.658274i	$0.771287\pi$
$564$		0			0
$565$	18.9887	+	1.60620i	0.798862	+	0.0675734i
$566$		0			0
$567$			0.507756i			0.0213238i
$568$		0			0
$569$	17.8579			0.748643			0.374321	−	0.927299i	$-0.377876\pi$
$569$	17.8579			0.748643			0.374321	+	0.927299i	$0.377876\pi$
$570$		0			0
$571$			26.8199i			1.12238i	0.827688	+	0.561188i	$0.189656\pi$
$571$			26.8199i			1.12238i	−0.827688	+	0.561188i	$0.810344\pi$
$572$		0			0
$573$	−13.4437			−0.561620
$574$		0			0
$575$	1.04657	−	6.14208i	0.0436450	−	0.256143i
$576$		0			0
$577$			40.3840i			1.68121i	0.541649	+	0.840605i	$0.317800\pi$
$577$			40.3840i			1.68121i	−0.541649	+	0.840605i	$0.682200\pi$
$578$		0			0
$579$		−	20.1855i		−	0.838883i
$580$		0			0
$581$			4.84632i			0.201059i
$582$		0			0
$583$			24.3502i			1.00848i
$584$		0			0
$585$	1.92896	−	22.8044i	0.0797526	−	0.942846i
$586$		0			0
$587$	0.706675			0.0291676			0.0145838	−	0.999894i	$-0.495358\pi$
$587$	0.706675			0.0291676			0.0145838	+	0.999894i	$0.495358\pi$
$588$		0			0
$589$			3.35945i			0.138424i
$590$		0			0
$591$	17.7012			0.728129
$592$		0			0
$593$			10.1402i			0.416408i	0.978085	+	0.208204i	$0.0667619\pi$
$593$			10.1402i			0.416408i	−0.978085	+	0.208204i	$0.933238\pi$
$594$		0			0
$595$	−1.03491	+	12.2349i	−0.0424272	+	0.501580i
$596$		0			0
$597$	7.08151			0.289827
$598$		0			0
$599$	28.2359			1.15369			0.576843	−	0.816855i	$-0.304284\pi$
$599$	28.2359			1.15369			0.576843	+	0.816855i	$0.304284\pi$
$600$		0			0
$601$	24.6118			1.00394			0.501968	−	0.864886i	$-0.332610\pi$
$601$	24.6118			1.00394			0.501968	+	0.864886i	$0.332610\pi$
$602$		0			0
$603$	−17.9442			−0.730745
$604$		0			0
$605$	−0.483101	+	5.71128i	−0.0196409	+	0.232197i
$606$		0			0
$607$		−	14.9064i		−	0.605033i	−0.953144	−	0.302517i	$-0.902173\pi$
$607$		−	14.9064i		−	0.605033i	0.953144	−	0.302517i	$-0.0978268\pi$
$608$		0			0
$609$	7.42120			0.300722
$610$		0			0
$611$			4.92896i			0.199404i
$612$		0			0
$613$	−22.7313			−0.918110			−0.459055	−	0.888408i	$-0.651812\pi$
$613$	−22.7313			−0.918110			−0.459055	+	0.888408i	$0.651812\pi$
$614$		0			0
$615$	−0.390099	+	4.61180i	−0.0157303	+	0.185966i
$616$		0			0
$617$		−	32.0473i		−	1.29017i	−0.764109	−	0.645087i	$-0.776821\pi$
$617$		−	32.0473i		−	1.29017i	0.764109	−	0.645087i	$-0.223179\pi$
$618$		0			0
$619$		−	9.21934i		−	0.370557i	−0.982686	−	0.185278i	$-0.940681\pi$
$619$		−	9.21934i		−	0.370557i	0.982686	−	0.185278i	$-0.0593187\pi$
$620$		0			0
$621$			6.35651i			0.255078i
$622$		0			0
$623$			12.6118i			0.505281i
$624$		0			0
$625$	23.5893	+	8.27929i	0.943571	+	0.331172i
$626$		0			0
$627$	−6.18615			−0.247051
$628$		0			0
$629$			32.1666i			1.28257i
$630$		0			0
$631$	11.5536			0.459940			0.229970	−	0.973198i	$-0.426137\pi$
$631$	11.5536			0.459940			0.229970	+	0.973198i	$0.426137\pi$
$632$		0			0
$633$		−	10.6540i		−	0.423460i
$634$		0			0
$635$	−44.5120	−	3.76514i	−1.76641	−	0.149415i
$636$		0			0
$637$	5.30590			0.210227
$638$		0			0
$639$	−7.23272			−0.286122
$640$		0			0
$641$	21.8579			0.863336			0.431668	−	0.902033i	$-0.357925\pi$
$641$	21.8579			0.863336			0.431668	+	0.902033i	$0.357925\pi$
$642$		0			0
$643$	6.58791			0.259802			0.129901	−	0.991527i	$-0.458534\pi$
$643$	6.58791			0.259802			0.129901	+	0.991527i	$0.458534\pi$
$644$		0			0
$645$	−2.06599	+	24.4245i	−0.0813485	+	0.961712i
$646$		0			0
$647$			12.7539i			0.501407i	0.968064	+	0.250703i	$0.0806619\pi$
$647$			12.7539i			0.501407i	−0.968064	+	0.250703i	$0.919338\pi$
$648$		0			0
$649$	−15.5963			−0.612208
$650$		0			0
$651$		−	2.14208i		−	0.0839549i
$652$		0			0
$653$	−37.4627			−1.46603			−0.733014	−	0.680213i	$-0.761887\pi$
$653$	−37.4627			−1.46603			−0.733014	+	0.680213i	$0.761887\pi$
$654$		0			0
$655$	47.8118	+	4.04426i	1.86816	+	0.158022i
$656$		0			0
$657$		−	20.4319i		−	0.797123i
$658$		0			0
$659$			1.19059i			0.0463789i	0.999731	+	0.0231895i	$0.00738209\pi$
$659$			1.19059i			0.0463789i	−0.999731	+	0.0231895i	$0.992618\pi$
$660$		0			0
$661$			24.2272i			0.942329i	0.882045	+	0.471165i	$0.156166\pi$
$661$			24.2272i			0.942329i	−0.882045	+	0.471165i	$0.843834\pi$
$662$		0			0
$663$			30.1525i			1.17103i
$664$		0			0
$665$	−0.305898	+	3.61636i	−0.0118622	+	0.140236i
$666$		0			0
$667$	−8.93576			−0.345994
$668$		0			0
$669$		−	25.2665i		−	0.976860i
$670$		0			0
$671$	−31.6572			−1.22211
$672$		0			0
$673$		−	10.8585i		−	0.418566i	−0.977855	−	0.209283i	$-0.932887\pi$
$673$		−	10.8585i		−	0.418566i	0.977855	−	0.209283i	$-0.0671130\pi$
$674$		0			0
$675$	−25.1428	−	4.28417i	−0.967746	−	0.164898i
$676$		0			0
$677$	−28.5520			−1.09734			−0.548672	−	0.836038i	$-0.684866\pi$
$677$	−28.5520			−1.09734			−0.548672	+	0.836038i	$0.684866\pi$
$678$		0			0
$679$	−12.3338			−0.473326
$680$		0			0
$681$	−16.7209			−0.640746
$682$		0			0
$683$	2.45992			0.0941263			0.0470631	−	0.998892i	$-0.485014\pi$
$683$	2.45992			0.0941263			0.0470631	+	0.998892i	$0.485014\pi$
$684$		0			0
$685$	7.48522	+	0.633154i	0.285996	+	0.0241916i
$686$		0			0
$687$		−	1.00702i		−	0.0384202i
$688$		0			0
$689$	35.0815			1.33650
$690$		0			0
$691$		−	13.9072i		−	0.529056i	−0.964378	−	0.264528i	$-0.914784\pi$
$691$		−	13.9072i		−	0.529056i	0.964378	−	0.264528i	$-0.0852161\pi$
$692$		0			0
$693$	−7.10404			−0.269860
$694$		0			0
$695$	14.9887	+	1.26785i	0.568555	+	0.0480924i
$696$		0			0
$697$			10.9823i			0.415983i
$698$		0			0
$699$			18.4472i			0.697736i
$700$		0			0
$701$			43.4771i			1.64211i	0.570851	+	0.821054i	$0.306613\pi$
$701$			43.4771i			1.64211i	−0.570851	+	0.821054i	$0.693387\pi$
$702$		0			0
$703$			9.50776i			0.358592i
$704$		0			0
$705$	2.14208	+	0.181193i	0.0806755	+	0.00682411i
$706$		0			0
$707$	−0.316577			−0.0119061
$708$		0			0
$709$		−	17.3729i		−	0.652454i	−0.945291	−	0.326227i	$-0.894223\pi$
$709$		−	17.3729i		−	0.652454i	0.945291	−	0.326227i	$-0.105777\pi$
$710$		0			0
$711$	18.5774			0.696706
$712$		0			0
$713$			2.57925i			0.0965937i
$714$		0			0
$715$	43.5390	+	3.68284i	1.62827	+	0.137730i
$716$		0			0
$717$	21.7364			0.811760
$718$		0			0
$719$	33.5800			1.25232			0.626161	−	0.779694i	$-0.284625\pi$
$719$	33.5800			1.25232			0.626161	+	0.779694i	$0.284625\pi$
$720$		0			0
$721$	−2.31716			−0.0862957
$722$		0			0
$723$	−12.3956			−0.460998
$724$		0			0
$725$	6.02253	−	35.3449i	0.223671	−	1.31268i
$726$		0			0
$727$		−	25.6933i		−	0.952912i	−0.879198	−	0.476456i	$-0.841921\pi$
$727$		−	25.6933i		−	0.952912i	0.879198	−	0.476456i	$-0.158079\pi$
$728$		0			0
$729$	14.8579			0.550293
$730$		0			0
$731$			58.1630i			2.15124i
$732$		0			0
$733$	−28.1483			−1.03968			−0.519841	−	0.854263i	$-0.674009\pi$
$733$	−28.1483			−1.03968			−0.519841	+	0.854263i	$0.674009\pi$
$734$		0			0
$735$	0.195049	−	2.30590i	0.00719450	−	0.0850543i
$736$		0			0
$737$		−	34.2597i		−	1.26197i
$738$		0			0
$739$		−	0.840432i		−	0.0309158i	−0.999881	−	0.0154579i	$-0.995079\pi$
$739$		−	0.840432i		−	0.0309158i	0.999881	−	0.0154579i	$-0.00492060\pi$
$740$		0			0
$741$			8.91244i			0.327407i
$742$		0			0
$743$		−	9.24612i		−	0.339207i	−0.985512	−	0.169604i	$-0.945751\pi$
$743$		−	9.24612i		−	0.339207i	0.985512	−	0.169604i	$-0.0542488\pi$
$744$		0			0
$745$	7.48522	+	0.633154i	0.274237	+	0.0231970i
$746$		0			0
$747$	−9.34835			−0.342038
$748$		0			0
$749$		−	5.81937i		−	0.212635i
$750$		0			0
$751$	−45.5469			−1.66203			−0.831016	−	0.556249i	$-0.812240\pi$
$751$	−45.5469			−1.66203			−0.831016	+	0.556249i	$0.812240\pi$
$752$		0			0
$753$		−	21.7215i		−	0.791575i
$754$		0			0
$755$	3.17826	−	37.5738i	0.115669	−	1.36745i
$756$		0			0
$757$	−4.23910			−0.154073			−0.0770364	−	0.997028i	$-0.524546\pi$
$757$	−4.23910			−0.154073			−0.0770364	+	0.997028i	$0.524546\pi$
$758$		0			0
$759$	−4.74948			−0.172395
$760$		0			0
$761$	−31.2236			−1.13185			−0.565927	−	0.824455i	$-0.691482\pi$
$761$	−31.2236			−1.13185			−0.565927	+	0.824455i	$0.691482\pi$
$762$		0			0
$763$	14.0135			0.507322
$764$		0			0
$765$	−23.6005	−	1.99630i	−0.853279	−	0.0721764i
$766$		0			0
$767$			22.4697i			0.811334i
$768$		0			0
$769$	22.3587			0.806274			0.403137	−	0.915140i	$-0.367920\pi$
$769$	22.3587			0.806274			0.403137	+	0.915140i	$0.367920\pi$
$770$		0			0
$771$		−	2.54580i		−	0.0916848i
$772$		0			0
$773$	26.0147			0.935684			0.467842	−	0.883812i	$-0.345032\pi$
$773$	26.0147			0.935684			0.467842	+	0.883812i	$0.345032\pi$
$774$		0			0
$775$	−10.2021	−	1.73837i	−0.366469	−	0.0624439i
$776$		0			0
$777$		−	6.06242i		−	0.217488i
$778$		0			0
$779$			3.24612i			0.116304i
$780$		0			0
$781$		−	13.8090i		−	0.494124i
$782$		0			0
$783$			36.5788i			1.30722i
$784$		0			0
$785$	−1.05553	+	12.4786i	−0.0376734	+	0.445380i
$786$		0			0
$787$	18.2266			0.649710			0.324855	−	0.945764i	$-0.394685\pi$
$787$	18.2266			0.649710			0.324855	+	0.945764i	$0.394685\pi$
$788$		0			0
$789$		−	28.6834i		−	1.02116i
$790$		0			0
$791$	−8.52235			−0.303020
$792$		0			0
$793$			45.6088i			1.61962i
$794$		0			0
$795$	1.28963	−	15.2461i	0.0457383	−	0.540724i
$796$		0			0
$797$	41.9612			1.48634			0.743170	−	0.669102i	$-0.233321\pi$
$797$	41.9612			1.48634			0.743170	+	0.669102i	$0.233321\pi$
$798$		0			0
$799$	5.10104			0.180462
$800$		0			0
$801$	−24.3276			−0.859575
$802$		0			0
$803$	39.0092			1.37661
$804$		0			0
$805$	−0.234856	+	2.77650i	−0.00827759	+	0.0978587i
$806$		0			0
$807$		−	16.7849i		−	0.590857i
$808$		0			0
$809$	−28.2946			−0.994786			−0.497393	−	0.867525i	$-0.665709\pi$
$809$	−28.2946			−0.994786			−0.497393	+	0.867525i	$0.665709\pi$
$810$		0			0
$811$			37.9732i			1.33342i	0.745317	+	0.666710i	$0.232298\pi$
$811$			37.9732i			1.33342i	−0.745317	+	0.666710i	$0.767702\pi$
$812$		0			0
$813$	−26.2081			−0.919157
$814$		0			0
$815$	4.21317	−	49.8086i	0.147581	−	1.74472i
$816$		0			0
$817$			17.1917i			0.601463i
$818$		0			0
$819$			10.2349i			0.357635i
$820$		0			0
$821$			7.31790i			0.255397i	0.991813	+	0.127698i	$0.0407589\pi$
$821$			7.31790i			0.255397i	−0.991813	+	0.127698i	$0.959241\pi$
$822$		0			0
$823$			39.0815i			1.36230i	0.732146	+	0.681148i	$0.238519\pi$
$823$			39.0815i			1.36230i	−0.732146	+	0.681148i	$0.761481\pi$
$824$		0			0
$825$	3.20106	−	18.7863i	0.111447	−	0.654054i
$826$		0			0
$827$	−42.1256			−1.46485			−0.732426	−	0.680847i	$-0.761612\pi$
$827$	−42.1256			−1.46485			−0.732426	+	0.680847i	$0.761612\pi$
$828$		0			0
$829$			20.3583i			0.707074i	0.935421	+	0.353537i	$0.115021\pi$
$829$			20.3583i			0.707074i	−0.935421	+	0.353537i	$0.884979\pi$
$830$		0			0
$831$	17.9486			0.622631
$832$		0			0
$833$		−	5.49113i		−	0.190257i
$834$		0			0
$835$	46.2653	+	3.91345i	1.60108	+	0.135430i
$836$		0			0
$837$	10.5582			0.364946
$838$		0			0
$839$	−11.7625			−0.406085			−0.203043	−	0.979170i	$-0.565083\pi$
$839$	−11.7625			−0.406085			−0.203043	+	0.979170i	$0.565083\pi$
$840$		0			0
$841$	−22.4212			−0.773145
$842$		0			0
$843$	22.4165			0.772065
$844$		0			0
$845$	2.85579	−	33.7616i	0.0982423	−	1.16143i
$846$		0			0
$847$		−	2.56329i		−	0.0880755i
$848$		0			0
$849$	3.21312			0.110274
$850$		0			0
$851$			7.29968i			0.250230i
$852$		0			0
$853$	37.7861			1.29377			0.646885	−	0.762587i	$-0.276071\pi$
$853$	37.7861			1.29377			0.646885	+	0.762587i	$0.276071\pi$
$854$		0			0
$855$	−6.97581	−	0.590064i	−0.238568	−	0.0201798i
$856$		0			0
$857$			7.37976i			0.252088i	0.992025	+	0.126044i	$0.0402280\pi$
$857$			7.37976i			0.252088i	−0.992025	+	0.126044i	$0.959772\pi$
$858$		0			0
$859$		−	6.37694i		−	0.217578i	−0.994065	−	0.108789i	$-0.965303\pi$
$859$		−	6.37694i		−	0.217578i	0.994065	−	0.108789i	$-0.0346973\pi$
$860$		0			0
$861$		−	2.06982i		−	0.0705394i
$862$		0			0
$863$		−	7.41074i		−	0.252264i	−0.992013	−	0.126132i	$-0.959744\pi$
$863$		−	7.41074i		−	0.252264i	0.992013	−	0.126132i	$-0.0402564\pi$
$864$		0			0
$865$	−2.01551	+	23.8276i	−0.0685295	+	0.810164i
$866$		0			0
$867$	13.6117			0.462278
$868$		0			0
$869$			35.4686i			1.20319i
$870$		0			0
$871$	−49.3583			−1.67244
$872$		0			0
$873$		−	23.7913i		−	0.805214i
$874$		0			0
$875$	−10.8240	−	2.80026i	−0.365917	−	0.0946662i
$876$		0			0
$877$	−15.9549			−0.538760			−0.269380	−	0.963034i	$-0.586819\pi$
$877$	−15.9549			−0.538760			−0.269380	+	0.963034i	$0.586819\pi$
$878$		0			0
$879$	28.5983			0.964595
$880$		0			0
$881$	16.1421			0.543841			0.271920	−	0.962320i	$-0.412341\pi$
$881$	16.1421			0.543841			0.271920	+	0.962320i	$0.412341\pi$
$882$		0			0
$883$	1.67972			0.0565272			0.0282636	−	0.999601i	$-0.491002\pi$
$883$	1.67972			0.0565272			0.0282636	+	0.999601i	$0.491002\pi$
$884$		0			0
$885$	9.76514	+	0.826006i	0.328252	+	0.0277659i
$886$		0			0
$887$		−	34.9169i		−	1.17239i	−0.810168	−	0.586197i	$-0.800624\pi$
$887$		−	34.9169i		−	1.17239i	0.810168	−	0.586197i	$-0.199376\pi$
$888$		0			0
$889$	19.9775			0.670023
$890$		0			0
$891$		−	1.86998i		−	0.0626468i
$892$		0			0
$893$	1.50776			0.0504551
$894$		0			0
$895$	−14.7820	−	1.25037i	−0.494108	−	0.0417952i
$896$		0			0
$897$			6.84262i			0.228468i
$898$		0			0
$899$			14.8424i			0.495022i
$900$		0			0
$901$		−	36.3063i		−	1.20954i
$902$		0			0
$903$		−	10.9620i		−	0.364791i
$904$		0			0
$905$	26.9134	+	2.27653i	0.894633	+	0.0756745i
$906$		0			0
$907$	55.8109			1.85317			0.926585	−	0.376086i	$-0.122730\pi$
$907$	55.8109			1.85317			0.926585	+	0.376086i	$0.122730\pi$
$908$		0			0
$909$		−	0.610664i		−	0.0202544i
$910$		0			0
$911$	−28.0270			−0.928575			−0.464287	−	0.885685i	$-0.653689\pi$
$911$	−28.0270			−0.928575			−0.464287	+	0.885685i	$0.653689\pi$
$912$		0			0
$913$		−	17.8482i		−	0.590689i
$914$		0			0
$915$	19.8212	+	1.67662i	0.655268	+	0.0554273i
$916$		0			0
$917$	−21.4584			−0.708620
$918$		0			0
$919$	20.7368			0.684043			0.342021	−	0.939692i	$-0.388888\pi$
$919$	20.7368			0.684043			0.342021	+	0.939692i	$0.388888\pi$
$920$		0			0
$921$	29.4212			0.969462
$922$		0			0
$923$	−19.8947			−0.654842
$924$		0			0
$925$	−28.8734	−	4.91984i	−0.949353	−	0.161763i
$926$		0			0
$927$		−	4.46971i		−	0.146805i
$928$		0			0
$929$	−3.97747			−0.130496			−0.0652482	−	0.997869i	$-0.520784\pi$
$929$	−3.97747			−0.130496			−0.0652482	+	0.997869i	$0.520784\pi$
$930$		0			0
$931$		−	1.62306i		−	0.0531937i
$932$		0			0
$933$	31.1475			1.01972
$934$		0			0
$935$	3.81141	−	45.0590i	0.124646	−	1.47359i
$936$		0			0
$937$			48.7637i			1.59304i	0.604611	+	0.796521i	$0.293329\pi$
$937$			48.7637i			1.59304i	−0.604611	+	0.796521i	$0.706671\pi$
$938$		0			0
$939$			7.82492i			0.255357i
$940$		0			0
$941$			1.87697i			0.0611875i	0.999532	+	0.0305938i	$0.00973981\pi$
$941$			1.87697i			0.0611875i	−0.999532	+	0.0305938i	$0.990260\pi$
$942$		0			0
$943$			2.49224i			0.0811586i
$944$		0			0
$945$	11.3657	+	0.961390i	0.369725	+	0.0312740i
$946$		0			0
$947$	−38.1330			−1.23916			−0.619578	−	0.784935i	$-0.712696\pi$
$947$	−38.1330			−1.23916			−0.619578	+	0.784935i	$0.712696\pi$
$948$		0			0
$949$		−	56.2010i		−	1.82436i
$950$		0			0
$951$	−17.3388			−0.562248
$952$		0			0
$953$			48.7252i			1.57836i	0.614160	+	0.789182i	$0.289495\pi$
$953$			48.7252i			1.57836i	−0.614160	+	0.789182i	$0.710505\pi$
$954$		0			0
$955$	2.44826	−	28.9437i	0.0792239	−	0.936595i
$956$		0			0
$957$	−27.3311			−0.883488
$958$		0			0
$959$	−3.35945			−0.108482
$960$		0			0
$961$	−26.7158			−0.861801
$962$		0			0
$963$	11.2253			0.361731
$964$		0			0
$965$	43.4584	+	3.67603i	1.39898	+	0.118335i
$966$		0			0
$967$			37.2671i			1.19843i	0.800589	+	0.599214i	$0.204520\pi$
$967$			37.2671i			1.19843i	−0.800589	+	0.599214i	$0.795480\pi$
$968$		0			0
$969$	9.22359			0.296304
$970$		0			0
$971$		−	60.3753i		−	1.93754i	−0.247967	−	0.968768i	$-0.579763\pi$
$971$		−	60.3753i		−	1.93754i	0.247967	−	0.968768i	$-0.420237\pi$
$972$		0			0
$973$	−6.72710			−0.215661
$974$		0			0
$975$	−27.0656	−	4.61180i	−0.866792	−	0.147696i
$976$		0			0
$977$			32.8231i			1.05010i	0.851070	+	0.525052i	$0.175954\pi$
$977$			32.8231i			1.05010i	−0.851070	+	0.525052i	$0.824046\pi$
$978$		0			0
$979$		−	46.4472i		−	1.48446i
$980$		0			0
$981$			27.0314i			0.863047i
$982$		0			0
$983$			14.5563i			0.464273i	0.972683	+	0.232136i	$0.0745716\pi$
$983$			14.5563i			0.464273i	−0.972683	+	0.232136i	$0.925428\pi$
$984$		0			0
$985$	−3.22359	+	38.1097i	−0.102712	+	1.21428i
$986$		0			0
$987$	−0.961390			−0.0306014
$988$		0			0
$989$			13.1991i			0.419708i
$990$		0			0
$991$	32.6804			1.03813			0.519064	−	0.854735i	$-0.326281\pi$
$991$	32.6804			1.03813			0.519064	+	0.854735i	$0.326281\pi$
$992$		0			0
$993$			22.8918i			0.726449i
$994$		0			0
$995$	−1.28963	+	15.2461i	−0.0408839	+	0.483335i
$996$		0			0
$997$	16.9107			0.535566			0.267783	−	0.963479i	$-0.413709\pi$
$997$	16.9107			0.535566			0.267783	+	0.963479i	$0.413709\pi$
$998$		0			0
$999$	29.8814			0.945407

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.2.l.b.1569.5	yes	12
4.3	odd	2	inner	2240.2.l.b.1569.7	yes	12
5.4	even	2		2240.2.l.a.1569.7	yes	12
8.3	odd	2		2240.2.l.a.1569.6	yes	12
8.5	even	2		2240.2.l.a.1569.8	yes	12
20.19	odd	2		2240.2.l.a.1569.5	✓	12
40.19	odd	2	inner	2240.2.l.b.1569.8	yes	12
40.29	even	2	inner	2240.2.l.b.1569.6	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2240.2.l.a.1569.5	✓	12	20.19	odd	2
2240.2.l.a.1569.6	yes	12	8.3	odd	2
2240.2.l.a.1569.7	yes	12	5.4	even	2
2240.2.l.a.1569.8	yes	12	8.5	even	2
2240.2.l.b.1569.5	yes	12	1.1	even	1	trivial
2240.2.l.b.1569.6	yes	12	40.29	even	2	inner
2240.2.l.b.1569.7	yes	12	4.3	odd	2	inner
2240.2.l.b.1569.8	yes	12	40.19	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)

\(f(q)\)	\(=\)	\(q-1.03491 q^{3} +(0.188470 - 2.22811i) q^{5} +1.00000i q^{7} -1.92896 q^{9} +O(q^{10})\)	\(q-1.03491 q^{3} +(0.188470 - 2.22811i) q^{5} +1.00000i q^{7} -1.92896 q^{9} -3.68284i q^{11} -5.30590 q^{13} +(-0.195049 + 2.30590i) q^{15} +5.49113i q^{17} +1.62306i q^{19} -1.03491i q^{21} +1.24612i q^{23} +(-4.92896 - 0.839862i) q^{25} +5.10104 q^{27} +7.17086i q^{29} +2.06982 q^{31} +3.81141i q^{33} +(2.22811 + 0.188470i) q^{35} +5.85792 q^{37} +5.49113 q^{39} +2.00000 q^{41} +10.5922 q^{43} +(-0.363550 + 4.29793i) q^{45} -0.928958i q^{47} -1.00000 q^{49} -5.68284i q^{51} -6.61180 q^{53} +(-8.20577 - 0.694102i) q^{55} -1.67972i q^{57} -4.23486i q^{59} -8.59587i q^{61} -1.92896i q^{63} +(-1.00000 + 11.8221i) q^{65} +9.30254 q^{67} -1.28963i q^{69} +3.74955 q^{71} +10.5922i q^{73} +(5.10104 + 0.869183i) q^{75} +3.68284 q^{77} -9.63078 q^{79} +0.507756 q^{81} +4.84632 q^{83} +(12.2349 + 1.03491i) q^{85} -7.42120i q^{87} +12.6118 q^{89} -5.30590i q^{91} -2.14208 q^{93} +(3.61636 + 0.305898i) q^{95} +12.3338i q^{97} +7.10404i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 4 q^{5} + 24 q^{9}+O(q^{10}) \) 12 * q + 4 * q^5 + 24 * q^9	\( 12 q + 4 q^{5} + 24 q^{9} - 20 q^{13} - 12 q^{25} - 24 q^{37} + 24 q^{41} + 48 q^{45} - 12 q^{49} + 8 q^{53} - 12 q^{65} + 4 q^{77} + 20 q^{81} + 56 q^{85} + 64 q^{89} - 120 q^{93}+O(q^{100}) \) 12 * q + 4 * q^5 + 24 * q^9 - 20 * q^13 - 12 * q^25 - 24 * q^37 + 24 * q^41 + 48 * q^45 - 12 * q^49 + 8 * q^53 - 12 * q^65 + 4 * q^77 + 20 * q^81 + 56 * q^85 + 64 * q^89 - 120 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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