LMFDB - Embedded newform 2240.2.l.c.1569.6

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			1569.6
Character	$\chi$	$=$	2240.1569
Dual form			2240.2.l.c.1569.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.58895 q^{3} +(1.82232 + 1.29582i) q^{5} -1.00000i q^{7} -0.475223 q^{9} +O(q^{10})$	$q-1.58895 q^{3} +(1.82232 + 1.29582i) q^{5} -1.00000i q^{7} -0.475223 q^{9} +6.23324i q^{11} +2.95631 q^{13} +(-2.89559 - 2.05899i) q^{15} +5.70676i q^{17} -6.34536i q^{19} +1.58895i q^{21} -2.70071i q^{23} +(1.64172 + 4.72279i) q^{25} +5.52197 q^{27} +3.28863i q^{29} -8.81061 q^{31} -9.90434i q^{33} +(1.29582 - 1.82232i) q^{35} +0.261466 q^{37} -4.69745 q^{39} -1.95939 q^{41} -7.41660 q^{43} +(-0.866010 - 0.615802i) q^{45} +8.29054i q^{47} -1.00000 q^{49} -9.06778i q^{51} +11.1830 q^{53} +(-8.07713 + 11.3590i) q^{55} +10.0825i q^{57} +0.473341i q^{59} -10.1903i q^{61} +0.475223i q^{63} +(5.38736 + 3.83084i) q^{65} -6.57727 q^{67} +4.29131i q^{69} -0.374339 q^{71} -0.725273i q^{73} +(-2.60862 - 7.50430i) q^{75} +6.23324 q^{77} -1.28596 q^{79} -7.34849 q^{81} +10.3847 q^{83} +(-7.39491 + 10.3996i) q^{85} -5.22549i q^{87} -13.6687 q^{89} -2.95631i q^{91} +13.9997 q^{93} +(8.22242 - 11.5633i) q^{95} -5.65414i q^{97} -2.96218i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.58895			−0.917383			−0.458692	−	0.888595i	$-0.651682\pi$
$3$	−1.58895			−0.917383			−0.458692	+	0.888595i	$0.651682\pi$
$4$		0			0
$5$	1.82232	+	1.29582i	0.814968	+	0.579507i
$5$	1.82232	+	1.29582i	0.814968	+	0.579507i
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$	−0.475223			−0.158408
$10$		0			0
$11$			6.23324i			1.87939i	0.342010	+	0.939696i	$0.388892\pi$
$11$			6.23324i			1.87939i	−0.342010	+	0.939696i	$0.611108\pi$
$12$		0			0
$13$	2.95631			0.819934			0.409967	−	0.912100i	$-0.365540\pi$
$13$	2.95631			0.819934			0.409967	+	0.912100i	$0.365540\pi$
$14$		0			0
$15$	−2.89559	−	2.05899i	−0.747638	−	0.531630i
$16$		0			0
$17$			5.70676i			1.38409i	0.721853	+	0.692046i	$0.243291\pi$
$17$			5.70676i			1.38409i	−0.721853	+	0.692046i	$0.756709\pi$
$18$		0			0
$19$		−	6.34536i		−	1.45573i	−0.685723	−	0.727863i	$-0.740514\pi$
$19$		−	6.34536i		−	1.45573i	0.685723	−	0.727863i	$-0.259486\pi$
$20$		0			0
$21$			1.58895i			0.346738i
$22$		0			0
$23$		−	2.70071i		−	0.563138i	−0.959541	−	0.281569i	$-0.909145\pi$
$23$		−	2.70071i		−	0.563138i	0.959541	−	0.281569i	$-0.0908548\pi$
$24$		0			0
$25$	1.64172	+	4.72279i	0.328344	+	0.944558i
$26$		0			0
$27$	5.52197			1.06270
$28$		0			0
$29$			3.28863i			0.610684i	0.952243	+	0.305342i	$0.0987708\pi$
$29$			3.28863i			0.610684i	−0.952243	+	0.305342i	$0.901229\pi$
$30$		0			0
$31$	−8.81061			−1.58243			−0.791216	−	0.611537i	$-0.790552\pi$
$31$	−8.81061			−1.58243			−0.791216	+	0.611537i	$0.790552\pi$
$32$		0			0
$33$		−	9.90434i		−	1.72412i
$34$		0			0
$35$	1.29582	−	1.82232i	0.219033	−	0.308029i
$36$		0			0
$37$	0.261466			0.0429847			0.0214923	−	0.999769i	$-0.493158\pi$
$37$	0.261466			0.0429847			0.0214923	+	0.999769i	$0.493158\pi$
$38$		0			0
$39$	−4.69745			−0.752194
$40$		0			0
$41$	−1.95939			−0.306006			−0.153003	−	0.988226i	$-0.548894\pi$
$41$	−1.95939			−0.306006			−0.153003	+	0.988226i	$0.548894\pi$
$42$		0			0
$43$	−7.41660			−1.13102			−0.565511	−	0.824741i	$-0.691321\pi$
$43$	−7.41660			−1.13102			−0.565511	+	0.824741i	$0.691321\pi$
$44$		0			0
$45$	−0.866010	−	0.615802i	−0.129097	−	0.0917983i
$46$		0			0
$47$			8.29054i			1.20930i	0.796491	+	0.604650i	$0.206687\pi$
$47$			8.29054i			1.20930i	−0.796491	+	0.604650i	$0.793313\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$		−	9.06778i		−	1.26974i
$52$		0			0
$53$	11.1830			1.53611			0.768054	−	0.640385i	$-0.221225\pi$
$53$	11.1830			1.53611			0.768054	+	0.640385i	$0.221225\pi$
$54$		0			0
$55$	−8.07713	+	11.3590i	−1.08912	+	1.53164i
$56$		0			0
$57$			10.0825i			1.33546i
$58$		0			0
$59$			0.473341i			0.0616237i	0.999525	+	0.0308119i	$0.00980927\pi$
$59$			0.473341i			0.0616237i	−0.999525	+	0.0308119i	$0.990191\pi$
$60$		0			0
$61$		−	10.1903i		−	1.30474i	−0.757901	−	0.652370i	$-0.773775\pi$
$61$		−	10.1903i		−	1.30474i	0.757901	−	0.652370i	$-0.226225\pi$
$62$		0			0
$63$			0.475223i			0.0598725i
$64$		0			0
$65$	5.38736	+	3.83084i	0.668219	+	0.475157i
$66$		0			0
$67$	−6.57727			−0.803541			−0.401771	−	0.915740i	$-0.631605\pi$
$67$	−6.57727			−0.803541			−0.401771	+	0.915740i	$0.631605\pi$
$68$		0			0
$69$			4.29131i			0.516613i
$70$		0			0
$71$	−0.374339			−0.0444258			−0.0222129	−	0.999753i	$-0.507071\pi$
$71$	−0.374339			−0.0444258			−0.0222129	+	0.999753i	$0.507071\pi$
$72$		0			0
$73$		−	0.725273i		−	0.0848868i	−0.999099	−	0.0424434i	$-0.986486\pi$
$73$		−	0.725273i		−	0.0848868i	0.999099	−	0.0424434i	$-0.0135142\pi$
$74$		0			0
$75$	−2.60862	−	7.50430i	−0.301217	−	0.866522i
$76$		0			0
$77$	6.23324			0.710344
$78$		0			0
$79$	−1.28596			−0.144682			−0.0723412	−	0.997380i	$-0.523047\pi$
$79$	−1.28596			−0.144682			−0.0723412	+	0.997380i	$0.523047\pi$
$80$		0			0
$81$	−7.34849			−0.816499
$82$		0			0
$83$	10.3847			1.13987			0.569936	−	0.821689i	$-0.306968\pi$
$83$	10.3847			1.13987			0.569936	+	0.821689i	$0.306968\pi$
$84$		0			0
$85$	−7.39491	+	10.3996i	−0.802091	+	1.12799i
$86$		0			0
$87$		−	5.22549i		−	0.560231i
$88$		0			0
$89$	−13.6687			−1.44888			−0.724442	−	0.689336i	$-0.757902\pi$
$89$	−13.6687			−1.44888			−0.724442	+	0.689336i	$0.757902\pi$
$90$		0			0
$91$		−	2.95631i		−	0.309906i
$92$		0			0
$93$	13.9997			1.45170
$94$		0			0
$95$	8.22242	−	11.5633i	0.843602	−	1.18637i
$96$		0			0
$97$		−	5.65414i		−	0.574091i	−0.957917	−	0.287046i	$-0.907327\pi$
$97$		−	5.65414i		−	0.574091i	0.957917	−	0.287046i	$-0.0926731\pi$
$98$		0			0
$99$		−	2.96218i		−	0.297710i
$100$		0			0
$101$			16.4599i			1.63783i	0.573918	+	0.818913i	$0.305423\pi$
$101$			16.4599i			1.63783i	−0.573918	+	0.818913i	$0.694577\pi$
$102$		0			0
$103$			9.86432i			0.971961i	0.873970	+	0.485980i	$0.161537\pi$
$103$			9.86432i			0.971961i	−0.873970	+	0.485980i	$0.838463\pi$
$104$		0			0
$105$	−2.05899	+	2.89559i	−0.200937	+	0.282580i
$106$		0			0
$107$	−11.5824			−1.11971			−0.559854	−	0.828591i	$-0.689143\pi$
$107$	−11.5824			−1.11971			−0.559854	+	0.828591i	$0.689143\pi$
$108$		0			0
$109$			15.5670i			1.49105i	0.666476	+	0.745526i	$0.267802\pi$
$109$			15.5670i			1.49105i	−0.666476	+	0.745526i	$0.732198\pi$
$110$		0			0
$111$	−0.415457			−0.0394334
$112$		0			0
$113$			5.66822i			0.533222i	0.963804	+	0.266611i	$0.0859038\pi$
$113$			5.66822i			0.533222i	−0.963804	+	0.266611i	$0.914096\pi$
$114$		0			0
$115$	3.49963	−	4.92157i	0.326342	−	0.458939i
$116$		0			0
$117$	−1.40491			−0.129884
$118$		0			0
$119$	5.70676			0.523138
$120$		0			0
$121$	−27.8533			−2.53212
$122$		0			0
$123$	3.11339			0.280725
$124$		0			0
$125$	−3.12812	+	10.7338i	−0.279788	+	0.960062i
$126$		0			0
$127$			4.75682i			0.422099i	0.977475	+	0.211050i	$0.0676882\pi$
$127$			4.75682i			0.422099i	−0.977475	+	0.211050i	$0.932312\pi$
$128$		0			0
$129$	11.7846			1.03758
$130$		0			0
$131$			15.9643i			1.39480i	0.716680	+	0.697402i	$0.245661\pi$
$131$			15.9643i			1.39480i	−0.716680	+	0.697402i	$0.754339\pi$
$132$		0			0
$133$	−6.34536			−0.550212
$134$		0			0
$135$	10.0628	+	7.15546i	0.866069	+	0.615844i
$136$		0			0
$137$		−	9.67550i		−	0.826633i	−0.910587	−	0.413317i	$-0.864370\pi$
$137$		−	9.67550i		−	0.826633i	0.910587	−	0.413317i	$-0.135630\pi$
$138$		0			0
$139$		−	0.679086i		−	0.0575993i	−0.999585	−	0.0287996i	$-0.990832\pi$
$139$		−	0.679086i		−	0.0575993i	0.999585	−	0.0287996i	$-0.00916848\pi$
$140$		0			0
$141$		−	13.1733i		−	1.10939i
$142$		0			0
$143$			18.4274i			1.54098i
$144$		0			0
$145$	−4.26147	+	5.99295i	−0.353895	+	0.497688i
$146$		0			0
$147$	1.58895			0.131055
$148$		0			0
$149$			4.51929i			0.370235i	0.982716	+	0.185118i	$0.0592666\pi$
$149$			4.51929i			0.370235i	−0.982716	+	0.185118i	$0.940733\pi$
$150$		0			0
$151$	−19.0927			−1.55374			−0.776872	−	0.629659i	$-0.783195\pi$
$151$	−19.0927			−1.55374			−0.776872	+	0.629659i	$0.783195\pi$
$152$		0			0
$153$		−	2.71198i		−	0.219251i
$154$		0			0
$155$	−16.0558	−	11.4169i	−1.28963	−	0.917030i
$156$		0			0
$157$	8.92471			0.712269			0.356135	−	0.934435i	$-0.384094\pi$
$157$	8.92471			0.712269			0.356135	+	0.934435i	$0.384094\pi$
$158$		0			0
$159$	−17.7693			−1.40920
$160$		0			0
$161$	−2.70071			−0.212846
$162$		0			0
$163$	18.8182			1.47396			0.736978	−	0.675916i	$-0.236252\pi$
$163$	18.8182			1.47396			0.736978	+	0.675916i	$0.236252\pi$
$164$		0			0
$165$	12.8342	−	18.0489i	0.999141	−	1.40510i
$166$		0			0
$167$			3.01048i			0.232958i	0.993193	+	0.116479i	$0.0371608\pi$
$167$			3.01048i			0.232958i	−0.993193	+	0.116479i	$0.962839\pi$
$168$		0			0
$169$	−4.26021			−0.327709
$170$		0			0
$171$			3.01546i			0.230598i
$172$		0			0
$173$	−17.8738			−1.35892			−0.679459	−	0.733714i	$-0.737785\pi$
$173$	−17.8738			−1.35892			−0.679459	+	0.733714i	$0.737785\pi$
$174$		0			0
$175$	4.72279	−	1.64172i	0.357009	−	0.124102i
$176$		0			0
$177$		−	0.752117i		−	0.0565326i
$178$		0			0
$179$			2.09671i			0.156716i	0.996925	+	0.0783578i	$0.0249677\pi$
$179$			2.09671i			0.156716i	−0.996925	+	0.0783578i	$0.975032\pi$
$180$		0			0
$181$			9.21217i			0.684735i	0.939566	+	0.342368i	$0.111229\pi$
$181$			9.21217i			0.684735i	−0.939566	+	0.342368i	$0.888771\pi$
$182$		0			0
$183$			16.1920i			1.19695i
$184$		0			0
$185$	0.476475	+	0.338811i	0.0350311	+	0.0249099i
$186$		0			0
$187$	−35.5716			−2.60125
$188$		0			0
$189$		−	5.52197i		−	0.401664i
$190$		0			0
$191$	−13.6525			−0.987859			−0.493929	−	0.869502i	$-0.664440\pi$
$191$	−13.6525			−0.987859			−0.493929	+	0.869502i	$0.664440\pi$
$192$		0			0
$193$		−	24.4528i		−	1.76015i	−0.474830	−	0.880077i	$-0.657491\pi$
$193$		−	24.4528i		−	1.76015i	0.474830	−	0.880077i	$-0.342509\pi$
$194$		0			0
$195$	−8.56027	−	6.08703i	−0.613013	−	0.435901i
$196$		0			0
$197$	−20.9522			−1.49278			−0.746391	−	0.665508i	$-0.768215\pi$
$197$	−20.9522			−1.49278			−0.746391	+	0.665508i	$0.768215\pi$
$198$		0			0
$199$	14.4769			1.02624			0.513120	−	0.858317i	$-0.328490\pi$
$199$	14.4769			1.02624			0.513120	+	0.858317i	$0.328490\pi$
$200$		0			0
$201$	10.4510			0.737155
$202$		0			0
$203$	3.28863			0.230817
$204$		0			0
$205$	−3.57065	−	2.53901i	−0.249385	−	0.177332i
$206$		0			0
$207$			1.28344i			0.0892054i
$208$		0			0
$209$	39.5521			2.73588
$210$		0			0
$211$		−	5.46438i		−	0.376183i	−0.982151	−	0.188092i	$-0.939770\pi$
$211$		−	5.46438i		−	0.376183i	0.982151	−	0.188092i	$-0.0602302\pi$
$212$		0			0
$213$	0.594807			0.0407555
$214$		0			0
$215$	−13.5154	−	9.61055i	−0.921746	−	0.655434i
$216$		0			0
$217$			8.81061i			0.598103i
$218$		0			0
$219$			1.15243i			0.0778737i
$220$		0			0
$221$			16.8710i			1.13486i
$222$		0			0
$223$			18.0755i			1.21043i	0.796063	+	0.605214i	$0.206912\pi$
$223$			18.0755i			1.21043i	−0.796063	+	0.605214i	$0.793088\pi$
$224$		0			0
$225$	−0.780184	−	2.24438i	−0.0520122	−	0.149625i
$226$		0			0
$227$	9.86391			0.654691			0.327345	−	0.944905i	$-0.393846\pi$
$227$	9.86391			0.654691			0.327345	+	0.944905i	$0.393846\pi$
$228$		0			0
$229$			15.9215i			1.05212i	0.850446	+	0.526062i	$0.176332\pi$
$229$			15.9215i			1.05212i	−0.850446	+	0.526062i	$0.823668\pi$
$230$		0			0
$231$	−9.90434			−0.651657
$232$		0			0
$233$			12.7939i			0.838155i	0.907951	+	0.419077i	$0.137646\pi$
$233$			12.7939i			0.838155i	−0.907951	+	0.419077i	$0.862354\pi$
$234$		0			0
$235$	−10.7430	+	15.1080i	−0.700797	+	0.985540i
$236$		0			0
$237$	2.04334			0.132729
$238$		0			0
$239$	5.61020			0.362893			0.181447	−	0.983401i	$-0.441922\pi$
$239$	5.61020			0.362893			0.181447	+	0.983401i	$0.441922\pi$
$240$		0			0
$241$	−8.52628			−0.549226			−0.274613	−	0.961555i	$-0.588550\pi$
$241$	−8.52628			−0.549226			−0.274613	+	0.961555i	$0.588550\pi$
$242$		0			0
$243$	−4.88949			−0.313661
$244$		0			0
$245$	−1.82232	−	1.29582i	−0.116424	−	0.0827867i
$246$		0			0
$247$		−	18.7589i		−	1.19360i
$248$		0			0
$249$	−16.5009			−1.04570
$250$		0			0
$251$			22.3295i			1.40943i	0.709492	+	0.704713i	$0.248924\pi$
$251$			22.3295i			1.40943i	−0.709492	+	0.704713i	$0.751076\pi$
$252$		0			0
$253$	16.8342			1.05836
$254$		0			0
$255$	11.7502	−	16.5244i	0.735825	−	1.03480i
$256$		0			0
$257$		−	6.39040i		−	0.398623i	−0.979936	−	0.199311i	$-0.936130\pi$
$257$		−	6.39040i		−	0.398623i	0.979936	−	0.199311i	$-0.0638705\pi$
$258$		0			0
$259$		−	0.261466i		−	0.0162467i
$260$		0			0
$261$		−	1.56284i		−	0.0967371i
$262$		0			0
$263$		−	0.441717i		−	0.0272375i	−0.999907	−	0.0136187i	$-0.995665\pi$
$263$		−	0.441717i		−	0.0272375i	0.999907	−	0.0136187i	$-0.00433511\pi$
$264$		0			0
$265$	20.3791	+	14.4912i	1.25188	+	0.890185i
$266$		0			0
$267$	21.7190			1.32918
$268$		0			0
$269$		−	10.2778i		−	0.626649i	−0.949646	−	0.313324i	$-0.898557\pi$
$269$		−	10.2778i		−	0.626649i	0.949646	−	0.313324i	$-0.101443\pi$
$270$		0			0
$271$	25.8313			1.56914			0.784569	−	0.620042i	$-0.212884\pi$
$271$	25.8313			1.56914			0.784569	+	0.620042i	$0.212884\pi$
$272$		0			0
$273$			4.69745i			0.284302i
$274$		0			0
$275$	−29.4383	+	10.2332i	−1.77520	+	0.617088i
$276$		0			0
$277$	20.3285			1.22142			0.610711	−	0.791854i	$-0.290884\pi$
$277$	20.3285			1.22142			0.610711	+	0.791854i	$0.290884\pi$
$278$		0			0
$279$	4.18700			0.250669
$280$		0			0
$281$	−10.3713			−0.618700			−0.309350	−	0.950948i	$-0.600111\pi$
$281$	−10.3713			−0.618700			−0.309350	+	0.950948i	$0.600111\pi$
$282$		0			0
$283$	33.0666			1.96560			0.982802	−	0.184663i	$-0.0591194\pi$
$283$	33.0666			1.96560			0.982802	+	0.184663i	$0.0591194\pi$
$284$		0			0
$285$	−13.0651	+	18.3735i	−0.773907	+	1.08836i
$286$		0			0
$287$			1.95939i			0.115659i
$288$		0			0
$289$	−15.5671			−0.915713
$290$		0			0
$291$			8.98418i			0.526662i
$292$		0			0
$293$	0.193644			0.0113128			0.00565641	−	0.999984i	$-0.498199\pi$
$293$	0.193644			0.0113128			0.00565641	+	0.999984i	$0.498199\pi$
$294$		0			0
$295$	−0.613363	+	0.862580i	−0.0357113	+	0.0502213i
$296$		0			0
$297$			34.4198i			1.99724i
$298$		0			0
$299$		−	7.98416i		−	0.461736i
$300$		0			0
$301$			7.41660i			0.427486i
$302$		0			0
$303$		−	26.1541i		−	1.50251i
$304$		0			0
$305$	13.2048	−	18.5701i	0.756105	−	1.06332i
$306$		0			0
$307$	−3.55685			−0.203000			−0.101500	−	0.994836i	$-0.532364\pi$
$307$	−3.55685			−0.203000			−0.101500	+	0.994836i	$0.532364\pi$
$308$		0			0
$309$		−	15.6740i		−	0.891661i
$310$		0			0
$311$	2.72886			0.154739			0.0773696	−	0.997002i	$-0.475348\pi$
$311$	2.72886			0.154739			0.0773696	+	0.997002i	$0.475348\pi$
$312$		0			0
$313$		−	4.32641i		−	0.244543i	−0.992497	−	0.122272i	$-0.960982\pi$
$313$		−	4.32641i		−	0.244543i	0.992497	−	0.122272i	$-0.0390179\pi$
$314$		0			0
$315$	−0.615802	+	0.866010i	−0.0346965	+	0.0487941i
$316$		0			0
$317$	−32.0595			−1.80064			−0.900322	−	0.435225i	$-0.856669\pi$
$317$	−32.0595			−1.80064			−0.900322	+	0.435225i	$0.856669\pi$
$318$		0			0
$319$	−20.4988			−1.14772
$320$		0			0
$321$	18.4038			1.02720
$322$		0			0
$323$	36.2115			2.01486
$324$		0			0
$325$	4.85344	+	13.9620i	0.269220	+	0.774475i
$326$		0			0
$327$		−	24.7353i		−	1.36787i
$328$		0			0
$329$	8.29054			0.457072
$330$		0			0
$331$		−	4.01303i		−	0.220576i	−0.993900	−	0.110288i	$-0.964823\pi$
$331$		−	4.01303i		−	0.220576i	0.993900	−	0.110288i	$-0.0351773\pi$
$332$		0			0
$333$	−0.124255			−0.00680911
$334$		0			0
$335$	−11.9859	−	8.52293i	−0.654860	−	0.465657i
$336$		0			0
$337$		−	3.81549i		−	0.207843i	−0.994585	−	0.103922i	$-0.966861\pi$
$337$		−	3.81549i		−	0.207843i	0.994585	−	0.103922i	$-0.0331391\pi$
$338$		0			0
$339$		−	9.00655i		−	0.489169i
$340$		0			0
$341$		−	54.9186i		−	2.97401i
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$	−5.56075	+	7.82016i	−0.299381	+	0.421023i
$346$		0			0
$347$	0.325379			0.0174672			0.00873362	−	0.999962i	$-0.497220\pi$
$347$	0.325379			0.0174672			0.00873362	+	0.999962i	$0.497220\pi$
$348$		0			0
$349$		−	2.90621i		−	0.155566i	−0.996970	−	0.0777829i	$-0.975216\pi$
$349$		−	2.90621i		−	0.155566i	0.996970	−	0.0777829i	$-0.0247841\pi$
$350$		0			0
$351$	16.3247			0.871347
$352$		0			0
$353$		−	28.9596i		−	1.54136i	−0.637221	−	0.770681i	$-0.719916\pi$
$353$		−	28.9596i		−	1.54136i	0.637221	−	0.770681i	$-0.280084\pi$
$354$		0			0
$355$	−0.682166	−	0.485074i	−0.0362056	−	0.0257451i
$356$		0			0
$357$	−9.06778			−0.479918
$358$		0			0
$359$	15.9936			0.844112			0.422056	−	0.906570i	$-0.361309\pi$
$359$	15.9936			0.844112			0.422056	+	0.906570i	$0.361309\pi$
$360$		0			0
$361$	−21.2636			−1.11914
$362$		0			0
$363$	44.2576			2.32292
$364$		0			0
$365$	0.939821	−	1.32168i	0.0491925	−	0.0691800i
$366$		0			0
$367$			26.8127i			1.39961i	0.714334	+	0.699805i	$0.246730\pi$
$367$			26.8127i			1.39961i	−0.714334	+	0.699805i	$0.753270\pi$
$368$		0			0
$369$	0.931149			0.0484737
$370$		0			0
$371$		−	11.1830i		−	0.580594i
$372$		0			0
$373$	27.4943			1.42360			0.711801	−	0.702382i	$-0.247880\pi$
$373$	27.4943			1.42360			0.711801	+	0.702382i	$0.247880\pi$
$374$		0			0
$375$	4.97045	−	17.0555i	0.256673	−	0.880745i
$376$		0			0
$377$			9.72223i			0.500720i
$378$		0			0
$379$		−	21.9708i		−	1.12856i	−0.825582	−	0.564282i	$-0.809153\pi$
$379$		−	21.9708i		−	1.12856i	0.825582	−	0.564282i	$-0.190847\pi$
$380$		0			0
$381$		−	7.55837i		−	0.387227i
$382$		0			0
$383$		−	3.93986i		−	0.201317i	−0.994921	−	0.100659i	$-0.967905\pi$
$383$		−	3.93986i		−	0.201317i	0.994921	−	0.100659i	$-0.0320950\pi$
$384$		0			0
$385$	11.3590	+	8.07713i	0.578907	+	0.411649i
$386$		0			0
$387$	3.52454			0.179162
$388$		0			0
$389$		−	38.1520i		−	1.93438i	−0.254052	−	0.967191i	$-0.581763\pi$
$389$		−	38.1520i		−	1.93438i	0.254052	−	0.967191i	$-0.418237\pi$
$390$		0			0
$391$	15.4123			0.779435
$392$		0			0
$393$		−	25.3665i		−	1.27957i
$394$		0			0
$395$	−2.34344	−	1.66637i	−0.117911	−	0.0838444i
$396$		0			0
$397$	27.4692			1.37864			0.689320	−	0.724457i	$-0.257909\pi$
$397$	27.4692			1.37864			0.689320	+	0.724457i	$0.257909\pi$
$398$		0			0
$399$	10.0825			0.504756
$400$		0			0
$401$	35.6100			1.77828			0.889139	−	0.457637i	$-0.151304\pi$
$401$	35.6100			1.77828			0.889139	+	0.457637i	$0.151304\pi$
$402$		0			0
$403$	−26.0469			−1.29749
$404$		0			0
$405$	−13.3913	−	9.52230i	−0.665420	−	0.473167i
$406$		0			0
$407$			1.62978i			0.0807851i
$408$		0			0
$409$	6.63883			0.328269			0.164135	−	0.986438i	$-0.447517\pi$
$409$	6.63883			0.328269			0.164135	+	0.986438i	$0.447517\pi$
$410$		0			0
$411$			15.3739i			0.758340i
$412$		0			0
$413$	0.473341			0.0232916
$414$		0			0
$415$	18.9243	+	13.4567i	0.928958	+	0.660563i
$416$		0			0
$417$			1.07904i			0.0528406i
$418$		0			0
$419$			7.96211i			0.388974i	0.980905	+	0.194487i	$0.0623043\pi$
$419$			7.96211i			0.388974i	−0.980905	+	0.194487i	$0.937696\pi$
$420$		0			0
$421$		−	25.4629i		−	1.24099i	−0.784212	−	0.620493i	$-0.786933\pi$
$421$		−	25.4629i		−	1.24099i	0.784212	−	0.620493i	$-0.213067\pi$
$422$		0			0
$423$		−	3.93986i		−	0.191562i
$424$		0			0
$425$	−26.9518	+	9.36891i	−1.30736	+	0.454459i
$426$		0			0
$427$	−10.1903			−0.493145
$428$		0			0
$429$		−	29.2803i		−	1.41367i
$430$		0			0
$431$	−1.36627			−0.0658109			−0.0329054	−	0.999458i	$-0.510476\pi$
$431$	−1.36627			−0.0658109			−0.0329054	+	0.999458i	$0.510476\pi$
$432$		0			0
$433$		−	4.08717i		−	0.196417i	−0.995166	−	0.0982085i	$-0.968689\pi$
$433$		−	4.08717i		−	0.196417i	0.995166	−	0.0982085i	$-0.0313112\pi$
$434$		0			0
$435$	6.77128	−	9.52253i	0.324658	−	0.456570i
$436$		0			0
$437$	−17.1370			−0.819774
$438$		0			0
$439$	−4.17954			−0.199479			−0.0997393	−	0.995014i	$-0.531801\pi$
$439$	−4.17954			−0.199479			−0.0997393	+	0.995014i	$0.531801\pi$
$440$		0			0
$441$	0.475223			0.0226297
$442$		0			0
$443$	−29.4612			−1.39974			−0.699872	−	0.714268i	$-0.746760\pi$
$443$	−29.4612			−1.39974			−0.699872	+	0.714268i	$0.746760\pi$
$444$		0			0
$445$	−24.9089	−	17.7122i	−1.18079	−	0.839638i
$446$		0			0
$447$		−	7.18095i		−	0.339647i
$448$		0			0
$449$	0.418490			0.0197498			0.00987488	−	0.999951i	$-0.496857\pi$
$449$	0.418490			0.0197498			0.00987488	+	0.999951i	$0.496857\pi$
$450$		0			0
$451$		−	12.2134i		−	0.575105i
$452$		0			0
$453$	30.3375			1.42538
$454$		0			0
$455$	3.83084	−	5.38736i	0.179592	−	0.252563i
$456$		0			0
$457$		−	18.3799i		−	0.859774i	−0.902883	−	0.429887i	$-0.858553\pi$
$457$		−	18.3799i		−	0.859774i	0.902883	−	0.429887i	$-0.141447\pi$
$458$		0			0
$459$			31.5126i			1.47088i
$460$		0			0
$461$			15.9341i			0.742124i	0.928608	+	0.371062i	$0.121006\pi$
$461$			15.9341i			0.742124i	−0.928608	+	0.371062i	$0.878994\pi$
$462$		0			0
$463$		−	12.3100i		−	0.572093i	−0.958216	−	0.286046i	$-0.907659\pi$
$463$		−	12.3100i		−	0.572093i	0.958216	−	0.286046i	$-0.0923412\pi$
$464$		0			0
$465$	25.5119	+	18.1410i	1.18309	+	0.841268i
$466$		0			0
$467$	7.37820			0.341422			0.170711	−	0.985321i	$-0.445393\pi$
$467$	7.37820			0.341422			0.170711	+	0.985321i	$0.445393\pi$
$468$		0			0
$469$			6.57727i			0.303710i
$470$		0			0
$471$	−14.1810			−0.653424
$472$		0			0
$473$		−	46.2295i		−	2.12563i
$474$		0			0
$475$	29.9678	−	10.4173i	1.37502	−	0.477979i
$476$		0			0
$477$	−5.31444			−0.243331
$478$		0			0
$479$	−5.35158			−0.244520			−0.122260	−	0.992498i	$-0.539014\pi$
$479$	−5.35158			−0.244520			−0.122260	+	0.992498i	$0.539014\pi$
$480$		0			0
$481$	0.772974			0.0352446
$482$		0			0
$483$	4.29131			0.195261
$484$		0			0
$485$	7.32673	−	10.3037i	0.332690	−	0.467866i
$486$		0			0
$487$		−	27.5479i		−	1.24831i	−0.781299	−	0.624157i	$-0.785442\pi$
$487$		−	27.5479i		−	1.24831i	0.781299	−	0.624157i	$-0.214558\pi$
$488$		0			0
$489$	−29.9013			−1.35218
$490$		0			0
$491$		−	7.81875i		−	0.352855i	−0.984314	−	0.176428i	$-0.943546\pi$
$491$		−	7.81875i		−	0.352855i	0.984314	−	0.176428i	$-0.0564542\pi$
$492$		0			0
$493$	−18.7675			−0.845243
$494$		0			0
$495$	3.83844	−	5.39805i	0.172525	−	0.242624i
$496$		0			0
$497$			0.374339i			0.0167914i
$498$		0			0
$499$		−	5.38316i		−	0.240983i	−0.992714	−	0.120492i	$-0.961553\pi$
$499$		−	5.38316i		−	0.240983i	0.992714	−	0.120492i	$-0.0384471\pi$
$500$		0			0
$501$		−	4.78352i		−	0.213712i
$502$		0			0
$503$			3.86422i			0.172297i	0.996282	+	0.0861484i	$0.0274559\pi$
$503$			3.86422i			0.172297i	−0.996282	+	0.0861484i	$0.972544\pi$
$504$		0			0
$505$	−21.3291	+	29.9953i	−0.949131	+	1.33477i
$506$		0			0
$507$	6.76929			0.300635
$508$		0			0
$509$		−	8.51030i		−	0.377213i	−0.982053	−	0.188606i	$-0.939603\pi$
$509$		−	8.51030i		−	0.377213i	0.982053	−	0.188606i	$-0.0603970\pi$
$510$		0			0
$511$	−0.725273			−0.0320842
$512$		0			0
$513$		−	35.0389i		−	1.54701i
$514$		0			0
$515$	−12.7824	+	17.9760i	−0.563258	+	0.792116i
$516$		0			0
$517$	−51.6769			−2.27275
$518$		0			0
$519$	28.4006			1.24665
$520$		0			0
$521$	13.9240			0.610022			0.305011	−	0.952349i	$-0.401340\pi$
$521$	13.9240			0.610022			0.305011	+	0.952349i	$0.401340\pi$
$522$		0			0
$523$	15.2888			0.668531			0.334265	−	0.942479i	$-0.391512\pi$
$523$	15.2888			0.668531			0.334265	+	0.942479i	$0.391512\pi$
$524$		0			0
$525$	−7.50430	+	2.60862i	−0.327515	+	0.113850i
$526$		0			0
$527$		−	50.2800i		−	2.19023i
$528$		0			0
$529$	15.7061			0.682876
$530$		0			0
$531$		−	0.224942i		−	0.00976167i
$532$		0			0
$533$	−5.79258			−0.250904
$534$		0			0
$535$	−21.1068	−	15.0086i	−0.912526	−	0.648878i
$536$		0			0
$537$		−	3.33158i		−	0.143768i
$538$		0			0
$539$		−	6.23324i		−	0.268485i
$540$		0			0
$541$			3.65408i			0.157101i	0.996910	+	0.0785506i	$0.0250292\pi$
$541$			3.65408i			0.157101i	−0.996910	+	0.0785506i	$0.974971\pi$
$542$		0			0
$543$		−	14.6377i		−	0.628165i
$544$		0			0
$545$	−20.1720	+	28.3682i	−0.864075	+	1.21516i
$546$		0			0
$547$	−16.6263			−0.710888			−0.355444	−	0.934698i	$-0.615670\pi$
$547$	−16.6263			−0.710888			−0.355444	+	0.934698i	$0.615670\pi$
$548$		0			0
$549$			4.84268i			0.206681i
$550$		0			0
$551$	20.8676			0.888988
$552$		0			0
$553$			1.28596i			0.0546848i
$554$		0			0
$555$	−0.757097	−	0.538356i	−0.0321370	−	0.0228519i
$556$		0			0
$557$	36.7145			1.55564			0.777821	−	0.628486i	$-0.216325\pi$
$557$	36.7145			1.55564			0.777821	+	0.628486i	$0.216325\pi$
$558$		0			0
$559$	−21.9258			−0.927362
$560$		0			0
$561$	56.5217			2.38635
$562$		0			0
$563$	14.5328			0.612483			0.306241	−	0.951954i	$-0.400929\pi$
$563$	14.5328			0.612483			0.306241	+	0.951954i	$0.400929\pi$
$564$		0			0
$565$	−7.34498	+	10.3293i	−0.309005	+	0.434558i
$566$		0			0
$567$			7.34849i			0.308608i
$568$		0			0
$569$	24.1018			1.01040			0.505200	−	0.863002i	$-0.331419\pi$
$569$	24.1018			1.01040			0.505200	+	0.863002i	$0.331419\pi$
$570$		0			0
$571$			17.1735i			0.718689i	0.933205	+	0.359344i	$0.117000\pi$
$571$			17.1735i			0.718689i	−0.933205	+	0.359344i	$0.883000\pi$
$572$		0			0
$573$	21.6932			0.906245
$574$		0			0
$575$	12.7549	−	4.43382i	0.531916	−	0.184903i
$576$		0			0
$577$		−	2.73128i		−	0.113705i	−0.998383	−	0.0568523i	$-0.981894\pi$
$577$		−	2.73128i		−	0.113705i	0.998383	−	0.0568523i	$-0.0181064\pi$
$578$		0			0
$579$			38.8545i			1.61474i
$580$		0			0
$581$		−	10.3847i		−	0.430831i
$582$		0			0
$583$			69.7066i			2.88695i
$584$		0			0
$585$	−2.56020	−	1.82050i	−0.105851	−	0.0752685i
$586$		0			0
$587$	25.9982			1.07306			0.536530	−	0.843882i	$-0.319735\pi$
$587$	25.9982			1.07306			0.536530	+	0.843882i	$0.319735\pi$
$588$		0			0
$589$			55.9065i			2.30359i
$590$		0			0
$591$	33.2921			1.36945
$592$		0			0
$593$		−	45.3821i		−	1.86362i	−0.362947	−	0.931810i	$-0.618230\pi$
$593$		−	45.3821i		−	1.86362i	0.362947	−	0.931810i	$-0.381770\pi$
$594$		0			0
$595$	10.3996	+	7.39491i	0.426340	+	0.303162i
$596$		0			0
$597$	−23.0031			−0.941456
$598$		0			0
$599$	28.9588			1.18322			0.591612	−	0.806223i	$-0.298492\pi$
$599$	28.9588			1.18322			0.591612	+	0.806223i	$0.298492\pi$
$600$		0			0
$601$	13.8766			0.566040			0.283020	−	0.959114i	$-0.408664\pi$
$601$	13.8766			0.566040			0.283020	+	0.959114i	$0.408664\pi$
$602$		0			0
$603$	3.12567			0.127287
$604$		0			0
$605$	−50.7577	−	36.0927i	−2.06359	−	1.46738i
$606$		0			0
$607$		−	12.8984i		−	0.523529i	−0.965132	−	0.261765i	$-0.915696\pi$
$607$		−	12.8984i		−	0.523529i	0.965132	−	0.261765i	$-0.0843044\pi$
$608$		0			0
$609$	−5.22549			−0.211748
$610$		0			0
$611$			24.5094i			0.991546i
$612$		0			0
$613$	0.580955			0.0234645			0.0117323	−	0.999931i	$-0.496265\pi$
$613$	0.580955			0.0234645			0.0117323	+	0.999931i	$0.496265\pi$
$614$		0			0
$615$	5.67360	+	4.03438i	0.228781	+	0.162682i
$616$		0			0
$617$			47.7064i			1.92059i	0.278993	+	0.960293i	$0.410000\pi$
$617$			47.7064i			1.92059i	−0.278993	+	0.960293i	$0.590000\pi$
$618$		0			0
$619$			5.14344i			0.206732i	0.994643	+	0.103366i	$0.0329614\pi$
$619$			5.14344i			0.206732i	−0.994643	+	0.103366i	$0.967039\pi$
$620$		0			0
$621$		−	14.9133i		−	0.598449i
$622$		0			0
$623$			13.6687i			0.547627i
$624$		0			0
$625$	−19.6095	+	15.5070i	−0.784380	+	0.620280i
$626$		0			0
$627$	−62.8466			−2.50985
$628$		0			0
$629$			1.49212i			0.0594948i
$630$		0			0
$631$	4.26275			0.169697			0.0848486	−	0.996394i	$-0.472959\pi$
$631$	4.26275			0.169697			0.0848486	+	0.996394i	$0.472959\pi$
$632$		0			0
$633$			8.68265i			0.345104i
$634$		0			0
$635$	−6.16396	+	8.66846i	−0.244609	+	0.343997i
$636$		0			0
$637$	−2.95631			−0.117133
$638$		0			0
$639$	0.177894			0.00703739
$640$		0			0
$641$	−33.1477			−1.30925			−0.654627	−	0.755952i	$-0.727174\pi$
$641$	−33.1477			−1.30925			−0.654627	+	0.755952i	$0.727174\pi$
$642$		0			0
$643$	5.00734			0.197470			0.0987351	−	0.995114i	$-0.468520\pi$
$643$	5.00734			0.197470			0.0987351	+	0.995114i	$0.468520\pi$
$644$		0			0
$645$	21.4754	+	15.2707i	0.845594	+	0.601285i
$646$		0			0
$647$			43.4421i			1.70789i	0.520367	+	0.853943i	$0.325795\pi$
$647$			43.4421i			1.70789i	−0.520367	+	0.853943i	$0.674205\pi$
$648$		0			0
$649$	−2.95045			−0.115815
$650$		0			0
$651$		−	13.9997i		−	0.548690i
$652$		0			0
$653$	−18.7101			−0.732181			−0.366090	−	0.930579i	$-0.619304\pi$
$653$	−18.7101			−0.732181			−0.366090	+	0.930579i	$0.619304\pi$
$654$		0			0
$655$	−20.6868	+	29.0920i	−0.808298	+	1.13672i
$656$		0			0
$657$			0.344667i			0.0134467i
$658$		0			0
$659$			27.4636i			1.06983i	0.844906	+	0.534915i	$0.179656\pi$
$659$			27.4636i			1.06983i	−0.844906	+	0.534915i	$0.820344\pi$
$660$		0			0
$661$			37.8678i			1.47289i	0.676499	+	0.736443i	$0.263496\pi$
$661$			37.8678i			1.47289i	−0.676499	+	0.736443i	$0.736504\pi$
$662$		0			0
$663$		−	26.8072i		−	1.04111i
$664$		0			0
$665$	−11.5633	−	8.22242i	−0.448405	−	0.318852i
$666$		0			0
$667$	8.88166			0.343899
$668$		0			0
$669$		−	28.7212i		−	1.11043i
$670$		0			0
$671$	63.5188			2.45212
$672$		0			0
$673$			6.80095i			0.262157i	0.991372	+	0.131079i	$0.0418440\pi$
$673$			6.80095i			0.262157i	−0.991372	+	0.131079i	$0.958156\pi$
$674$		0			0
$675$	9.06554	+	26.0791i	0.348933	+	1.00379i
$676$		0			0
$677$	17.9072			0.688228			0.344114	−	0.938928i	$-0.388179\pi$
$677$	17.9072			0.688228			0.344114	+	0.938928i	$0.388179\pi$
$678$		0			0
$679$	−5.65414			−0.216986
$680$		0			0
$681$	−15.6733			−0.600603
$682$		0			0
$683$	7.56125			0.289323			0.144662	−	0.989481i	$-0.453791\pi$
$683$	7.56125			0.289323			0.144662	+	0.989481i	$0.453791\pi$
$684$		0			0
$685$	12.5377	−	17.6319i	0.479040	−	0.673679i
$686$		0			0
$687$		−	25.2986i		−	0.965202i
$688$		0			0
$689$	33.0606			1.25951
$690$		0			0
$691$		−	25.3828i		−	0.965607i	−0.875729	−	0.482804i	$-0.839618\pi$
$691$		−	25.3828i		−	0.965607i	0.875729	−	0.482804i	$-0.160382\pi$
$692$		0			0
$693$	−2.96218			−0.112524
$694$		0			0
$695$	0.879970	−	1.23751i	0.0333792	−	0.0469415i
$696$		0			0
$697$		−	11.1818i		−	0.423540i
$698$		0			0
$699$		−	20.3289i		−	0.768909i
$700$		0			0
$701$			20.6973i			0.781727i	0.920449	+	0.390863i	$0.127823\pi$
$701$			20.6973i			0.781727i	−0.920449	+	0.390863i	$0.872177\pi$
$702$		0			0
$703$		−	1.65909i		−	0.0625739i
$704$		0			0
$705$	17.0702	−	24.0060i	0.642900	−	0.904118i
$706$		0			0
$707$	16.4599			0.619040
$708$		0			0
$709$			6.03521i			0.226657i	0.993558	+	0.113328i	$0.0361512\pi$
$709$			6.03521i			0.226657i	−0.993558	+	0.113328i	$0.963849\pi$
$710$		0			0
$711$	0.611120			0.0229188
$712$		0			0
$713$			23.7949i			0.891127i
$714$		0			0
$715$	−23.8785	+	33.5807i	−0.893007	+	1.25585i
$716$		0			0
$717$	−8.91435			−0.332912
$718$		0			0
$719$	25.8542			0.964200			0.482100	−	0.876116i	$-0.339874\pi$
$719$	25.8542			0.964200			0.482100	+	0.876116i	$0.339874\pi$
$720$		0			0
$721$	9.86432			0.367367
$722$		0			0
$723$	13.5479			0.503851
$724$		0			0
$725$	−15.5315	+	5.39902i	−0.576827	+	0.200515i
$726$		0			0
$727$			11.4864i			0.426006i	0.977052	+	0.213003i	$0.0683244\pi$
$727$			11.4864i			0.426006i	−0.977052	+	0.213003i	$0.931676\pi$
$728$		0			0
$729$	29.8147			1.10425
$730$		0			0
$731$		−	42.3248i		−	1.56544i
$732$		0			0
$733$	21.0982			0.779278			0.389639	−	0.920968i	$-0.372600\pi$
$733$	21.0982			0.779278			0.389639	+	0.920968i	$0.372600\pi$
$734$		0			0
$735$	2.89559	+	2.05899i	0.106805	+	0.0759471i
$736$		0			0
$737$		−	40.9977i		−	1.51017i
$738$		0			0
$739$			31.2344i			1.14897i	0.818514	+	0.574487i	$0.194798\pi$
$739$			31.2344i			1.14897i	−0.818514	+	0.574487i	$0.805202\pi$
$740$		0			0
$741$			29.8070i			1.09499i
$742$		0			0
$743$		−	25.8880i		−	0.949738i	−0.880056	−	0.474869i	$-0.842495\pi$
$743$		−	25.8880i		−	0.949738i	0.880056	−	0.474869i	$-0.157505\pi$
$744$		0			0
$745$	−5.85617	+	8.23561i	−0.214554	+	0.301730i
$746$		0			0
$747$	−4.93506			−0.180564
$748$		0			0
$749$			11.5824i			0.423210i
$750$		0			0
$751$	−51.8160			−1.89079			−0.945396	−	0.325923i	$-0.894325\pi$
$751$	−51.8160			−1.89079			−0.945396	+	0.325923i	$0.894325\pi$
$752$		0			0
$753$		−	35.4806i		−	1.29298i
$754$		0			0
$755$	−34.7931	−	24.7407i	−1.26625	−	0.900405i
$756$		0			0
$757$	−32.2858			−1.17345			−0.586723	−	0.809788i	$-0.699582\pi$
$757$	−32.2858			−1.17345			−0.586723	+	0.809788i	$0.699582\pi$
$758$		0			0
$759$	−26.7488			−0.970919
$760$		0			0
$761$	−12.1707			−0.441188			−0.220594	−	0.975366i	$-0.570800\pi$
$761$	−12.1707			−0.441188			−0.220594	+	0.975366i	$0.570800\pi$
$762$		0			0
$763$	15.5670			0.563565
$764$		0			0
$765$	3.51423	−	4.94211i	0.127057	−	0.178682i
$766$		0			0
$767$			1.39934i			0.0505274i
$768$		0			0
$769$	14.1326			0.509635			0.254817	−	0.966989i	$-0.417985\pi$
$769$	14.1326			0.509635			0.254817	+	0.966989i	$0.417985\pi$
$770$		0			0
$771$			10.1541i			0.365690i
$772$		0			0
$773$	0.0573455			0.00206257			0.00103129	−	0.999999i	$-0.499672\pi$
$773$	0.0573455			0.00206257			0.00103129	+	0.999999i	$0.499672\pi$
$774$		0			0
$775$	−14.4646	−	41.6107i	−0.519582	−	1.49470i
$776$		0			0
$777$			0.415457i			0.0149044i
$778$		0			0
$779$			12.4331i			0.445460i
$780$		0			0
$781$		−	2.33334i		−	0.0834936i
$782$		0			0
$783$			18.1597i			0.648976i
$784$		0			0
$785$	16.2637	+	11.5648i	0.580476	+	0.412765i
$786$		0			0
$787$	−48.7645			−1.73827			−0.869133	−	0.494578i	$-0.835323\pi$
$787$	−48.7645			−1.73827			−0.869133	+	0.494578i	$0.835323\pi$
$788$		0			0
$789$			0.701869i			0.0249872i
$790$		0			0
$791$	5.66822			0.201539
$792$		0			0
$793$		−	30.1258i		−	1.06980i
$794$		0			0
$795$	−32.3815	−	23.0258i	−1.14845	−	0.816641i
$796$		0			0
$797$	−3.05093			−0.108069			−0.0540347	−	0.998539i	$-0.517208\pi$
$797$	−3.05093			−0.108069			−0.0540347	+	0.998539i	$0.517208\pi$
$798$		0			0
$799$	−47.3121			−1.67378
$800$		0			0
$801$	6.49570			0.229514
$802$		0			0
$803$	4.52080			0.159536
$804$		0			0
$805$	−4.92157	−	3.49963i	−0.173463	−	0.123346i
$806$		0			0
$807$			16.3310i			0.574877i
$808$		0			0
$809$	−40.4379			−1.42172			−0.710860	−	0.703333i	$-0.751694\pi$
$809$	−40.4379			−1.42172			−0.710860	+	0.703333i	$0.751694\pi$
$810$		0			0
$811$		−	29.0307i		−	1.01941i	−0.860350	−	0.509704i	$-0.829755\pi$
$811$		−	29.0307i		−	1.01941i	0.860350	−	0.509704i	$-0.170245\pi$
$812$		0			0
$813$	−41.0447			−1.43950
$814$		0			0
$815$	34.2929	+	24.3850i	1.20123	+	0.854168i
$816$		0			0
$817$			47.0610i			1.64646i
$818$		0			0
$819$			1.40491i			0.0490915i
$820$		0			0
$821$		−	15.4134i		−	0.537930i	−0.963150	−	0.268965i	$-0.913318\pi$
$821$		−	15.4134i		−	0.537930i	0.963150	−	0.268965i	$-0.0866817\pi$
$822$		0			0
$823$			7.20617i			0.251191i	0.992082	+	0.125596i	$0.0400842\pi$
$823$			7.20617i			0.251191i	−0.992082	+	0.125596i	$0.959916\pi$
$824$		0			0
$825$	46.7761	−	16.2602i	1.62853	−	0.566106i
$826$		0			0
$827$	21.8258			0.758957			0.379479	−	0.925201i	$-0.376103\pi$
$827$	21.8258			0.758957			0.379479	+	0.925201i	$0.376103\pi$
$828$		0			0
$829$		−	8.62004i		−	0.299386i	−0.988732	−	0.149693i	$-0.952171\pi$
$829$		−	8.62004i		−	0.299386i	0.988732	−	0.149693i	$-0.0478286\pi$
$830$		0			0
$831$	−32.3011			−1.12051
$832$		0			0
$833$		−	5.70676i		−	0.197728i
$834$		0			0
$835$	−3.90103	+	5.48607i	−0.135001	+	0.189853i
$836$		0			0
$837$	−48.6519			−1.68166
$838$		0			0
$839$	19.8747			0.686151			0.343075	−	0.939308i	$-0.388531\pi$
$839$	19.8747			0.686151			0.343075	+	0.939308i	$0.388531\pi$
$840$		0			0
$841$	18.1849			0.627065
$842$		0			0
$843$	16.4795			0.567585
$844$		0			0
$845$	−7.76349	−	5.52045i	−0.267072	−	0.189909i
$846$		0			0
$847$			27.8533i			0.957050i
$848$		0			0
$849$	−52.5413			−1.80321
$850$		0			0
$851$		−	0.706144i		−	0.0242063i
$852$		0			0
$853$	5.61999			0.192425			0.0962124	−	0.995361i	$-0.469327\pi$
$853$	5.61999			0.192425			0.0962124	+	0.995361i	$0.469327\pi$
$854$		0			0
$855$	−3.90748	+	5.49514i	−0.133633	+	0.187930i
$856$		0			0
$857$		−	2.51499i		−	0.0859104i	−0.999077	−	0.0429552i	$-0.986323\pi$
$857$		−	2.51499i		−	0.0859104i	0.999077	−	0.0429552i	$-0.0136773\pi$
$858$		0			0
$859$			17.3568i			0.592207i	0.955156	+	0.296103i	$0.0956873\pi$
$859$			17.3568i			0.592207i	−0.955156	+	0.296103i	$0.904313\pi$
$860$		0			0
$861$		−	3.11339i		−	0.106104i
$862$		0			0
$863$			47.2715i			1.60914i	0.593858	+	0.804570i	$0.297604\pi$
$863$			47.2715i			1.60914i	−0.593858	+	0.804570i	$0.702396\pi$
$864$		0			0
$865$	−32.5718	−	23.1611i	−1.10747	−	0.787502i
$866$		0			0
$867$	24.7355			0.840060
$868$		0			0
$869$		−	8.01573i		−	0.271915i
$870$		0			0
$871$	−19.4445			−0.658851
$872$		0			0
$873$			2.68698i			0.0909405i
$874$		0			0
$875$	10.7338	+	3.12812i	0.362869	+	0.105750i
$876$		0			0
$877$	16.4369			0.555034			0.277517	−	0.960721i	$-0.410488\pi$
$877$	16.4369			0.555034			0.277517	+	0.960721i	$0.410488\pi$
$878$		0			0
$879$	−0.307692			−0.0103782
$880$		0			0
$881$	−21.0317			−0.708575			−0.354287	−	0.935137i	$-0.615277\pi$
$881$	−21.0317			−0.708575			−0.354287	+	0.935137i	$0.615277\pi$
$882$		0			0
$883$	19.9676			0.671964			0.335982	−	0.941868i	$-0.390932\pi$
$883$	19.9676			0.671964			0.335982	+	0.941868i	$0.390932\pi$
$884$		0			0
$885$	0.974605	−	1.37060i	0.0327610	−	0.0460722i
$886$		0			0
$887$			36.9013i			1.23902i	0.784987	+	0.619512i	$0.212669\pi$
$887$			36.9013i			1.23902i	−0.784987	+	0.619512i	$0.787331\pi$
$888$		0			0
$889$	4.75682			0.159539
$890$		0			0
$891$		−	45.8049i		−	1.53452i
$892$		0			0
$893$	52.6065			1.76041
$894$		0			0
$895$	−2.71695	+	3.82089i	−0.0908177	+	0.127718i
$896$		0			0
$897$			12.6865i			0.423589i
$898$		0			0
$899$		−	28.9749i		−	0.966366i
$900$		0			0
$901$			63.8189i			2.12612i
$902$		0			0
$903$		−	11.7846i		−	0.392168i
$904$		0			0
$905$	−11.9373	+	16.7875i	−0.396809	+	0.558037i
$906$		0			0
$907$	−51.1687			−1.69903			−0.849515	−	0.527565i	$-0.823105\pi$
$907$	−51.1687			−1.69903			−0.849515	+	0.527565i	$0.823105\pi$
$908$		0			0
$909$		−	7.82214i		−	0.259444i
$910$		0			0
$911$	−33.9887			−1.12610			−0.563048	−	0.826424i	$-0.690372\pi$
$911$	−33.9887			−1.12610			−0.563048	+	0.826424i	$0.690372\pi$
$912$		0			0
$913$			64.7305i			2.14227i
$914$		0			0
$915$	−20.9818	+	29.5070i	−0.693638	+	0.975472i
$916$		0			0
$917$	15.9643			0.527187
$918$		0			0
$919$	31.9957			1.05544			0.527721	−	0.849418i	$-0.323047\pi$
$919$	31.9957			1.05544			0.527721	+	0.849418i	$0.323047\pi$
$920$		0			0
$921$	5.65167			0.186229
$922$		0			0
$923$	−1.10666			−0.0364262
$924$		0			0
$925$	0.429254	+	1.23485i	0.0141138	+	0.0406015i
$926$		0			0
$927$		−	4.68775i		−	0.153966i
$928$		0			0
$929$	−37.1250			−1.21803			−0.609016	−	0.793158i	$-0.708435\pi$
$929$	−37.1250			−1.21803			−0.609016	+	0.793158i	$0.708435\pi$
$930$		0			0
$931$			6.34536i			0.207961i
$932$		0			0
$933$	−4.33603			−0.141955
$934$		0			0
$935$	−64.8230	−	46.0943i	−2.11994	−	1.50744i
$936$		0			0
$937$			36.7905i			1.20189i	0.799289	+	0.600947i	$0.205210\pi$
$937$			36.7905i			1.20189i	−0.799289	+	0.600947i	$0.794790\pi$
$938$		0			0
$939$			6.87447i			0.224340i
$940$		0			0
$941$		−	11.1507i		−	0.363504i	−0.983344	−	0.181752i	$-0.941823\pi$
$941$		−	11.1507i		−	0.363504i	0.983344	−	0.181752i	$-0.0581767\pi$
$942$		0			0
$943$			5.29176i			0.172323i
$944$		0			0
$945$	7.15546	−	10.0628i	0.232767	−	0.327343i
$946$		0			0
$947$	34.0733			1.10723			0.553616	−	0.832772i	$-0.313247\pi$
$947$	34.0733			1.10723			0.553616	+	0.832772i	$0.313247\pi$
$948$		0			0
$949$		−	2.14413i		−	0.0696016i
$950$		0			0
$951$	50.9412			1.65188
$952$		0			0
$953$			25.0287i			0.810758i	0.914149	+	0.405379i	$0.132860\pi$
$953$			25.0287i			0.810758i	−0.914149	+	0.405379i	$0.867140\pi$
$954$		0			0
$955$	−24.8792	−	17.6911i	−0.805073	−	0.572471i
$956$		0			0
$957$	32.5717			1.05289
$958$		0			0
$959$	−9.67550			−0.312438
$960$		0			0
$961$	46.6268			1.50409
$962$		0			0
$963$	5.50420			0.177370
$964$		0			0
$965$	31.6864	−	44.5610i	1.02002	−	1.43447i
$966$		0			0
$967$			10.7710i			0.346373i	0.984889	+	0.173186i	$0.0554063\pi$
$967$			10.7710i			0.346373i	−0.984889	+	0.173186i	$0.944594\pi$
$968$		0			0
$969$	−57.5384			−1.84840
$970$		0			0
$971$			1.47986i			0.0474909i	0.999718	+	0.0237454i	$0.00755912\pi$
$971$			1.47986i			0.0474909i	−0.999718	+	0.0237454i	$0.992441\pi$
$972$		0			0
$973$	−0.679086			−0.0217705
$974$		0			0
$975$	−7.71190	−	22.1851i	−0.246978	−	0.710491i
$976$		0			0
$977$		−	32.5367i		−	1.04094i	−0.853879	−	0.520471i	$-0.825756\pi$
$977$		−	32.5367i		−	1.04094i	0.853879	−	0.520471i	$-0.174244\pi$
$978$		0			0
$979$		−	85.2006i		−	2.72302i
$980$		0			0
$981$		−	7.39782i		−	0.236194i
$982$		0			0
$983$		−	28.4844i		−	0.908510i	−0.890872	−	0.454255i	$-0.849906\pi$
$983$		−	28.4844i		−	0.908510i	0.890872	−	0.454255i	$-0.150094\pi$
$984$		0			0
$985$	−38.1816	−	27.1502i	−1.21657	−	0.865077i
$986$		0			0
$987$	−13.1733			−0.419311
$988$		0			0
$989$			20.0301i			0.636921i
$990$		0			0
$991$	10.8762			0.345493			0.172746	−	0.984966i	$-0.444736\pi$
$991$	10.8762			0.345493			0.172746	+	0.984966i	$0.444736\pi$
$992$		0			0
$993$			6.37652i			0.202353i
$994$		0			0
$995$	26.3816	+	18.7594i	0.836352	+	0.594713i
$996$		0			0
$997$	48.4491			1.53440			0.767199	−	0.641409i	$-0.221650\pi$
$997$	48.4491			1.53440			0.767199	+	0.641409i	$0.221650\pi$
$998$		0			0
$999$	1.44381			0.0456800

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.2.l.c.1569.6	yes	24
4.3	odd	2	inner	2240.2.l.c.1569.19	yes	24
5.4	even	2		2240.2.l.d.1569.19	yes	24
8.3	odd	2		2240.2.l.d.1569.5	yes	24
8.5	even	2		2240.2.l.d.1569.20	yes	24
20.19	odd	2		2240.2.l.d.1569.6	yes	24
40.19	odd	2	inner	2240.2.l.c.1569.20	yes	24
40.29	even	2	inner	2240.2.l.c.1569.5	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2240.2.l.c.1569.5	✓	24	40.29	even	2	inner
2240.2.l.c.1569.6	yes	24	1.1	even	1	trivial
2240.2.l.c.1569.19	yes	24	4.3	odd	2	inner
2240.2.l.c.1569.20	yes	24	40.19	odd	2	inner
2240.2.l.d.1569.5	yes	24	8.3	odd	2
2240.2.l.d.1569.6	yes	24	20.19	odd	2
2240.2.l.d.1569.19	yes	24	5.4	even	2
2240.2.l.d.1569.20	yes	24	8.5	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:	\( N \)	\(=\)	\( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2240.l (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.58895 q^{3} +(1.82232 + 1.29582i) q^{5} -1.00000i q^{7} -0.475223 q^{9} +O(q^{10})\)	\(q-1.58895 q^{3} +(1.82232 + 1.29582i) q^{5} -1.00000i q^{7} -0.475223 q^{9} +6.23324i q^{11} +2.95631 q^{13} +(-2.89559 - 2.05899i) q^{15} +5.70676i q^{17} -6.34536i q^{19} +1.58895i q^{21} -2.70071i q^{23} +(1.64172 + 4.72279i) q^{25} +5.52197 q^{27} +3.28863i q^{29} -8.81061 q^{31} -9.90434i q^{33} +(1.29582 - 1.82232i) q^{35} +0.261466 q^{37} -4.69745 q^{39} -1.95939 q^{41} -7.41660 q^{43} +(-0.866010 - 0.615802i) q^{45} +8.29054i q^{47} -1.00000 q^{49} -9.06778i q^{51} +11.1830 q^{53} +(-8.07713 + 11.3590i) q^{55} +10.0825i q^{57} +0.473341i q^{59} -10.1903i q^{61} +0.475223i q^{63} +(5.38736 + 3.83084i) q^{65} -6.57727 q^{67} +4.29131i q^{69} -0.374339 q^{71} -0.725273i q^{73} +(-2.60862 - 7.50430i) q^{75} +6.23324 q^{77} -1.28596 q^{79} -7.34849 q^{81} +10.3847 q^{83} +(-7.39491 + 10.3996i) q^{85} -5.22549i q^{87} -13.6687 q^{89} -2.95631i q^{91} +13.9997 q^{93} +(8.22242 - 11.5633i) q^{95} -5.65414i q^{97} -2.96218i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 4 q^{5} + 12 q^{9}+O(q^{10}) \) 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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