LMFDB - Embedded newform 4020.2.g.b.1609.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4020,2,Mod(1609,4020)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4020, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("4020.1609");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4020.g (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:	$32.0998616126$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1609.8
Character	$\chi$	$=$	4020.1609
Dual form			4020.2.g.b.1609.20

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +(0.853331 + 2.06684i) q^{5} +2.56410i q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +(0.853331 + 2.06684i) q^{5} +2.56410i q^{7} -1.00000 q^{9} +0.836073 q^{11} +5.01469i q^{13} +(2.06684 - 0.853331i) q^{15} -1.93744i q^{17} -1.56454 q^{19} +2.56410 q^{21} -2.41957i q^{23} +(-3.54365 + 3.52740i) q^{25} +1.00000i q^{27} +6.17679 q^{29} +4.81345 q^{31} -0.836073i q^{33} +(-5.29959 + 2.18803i) q^{35} -2.01556i q^{37} +5.01469 q^{39} -5.18723 q^{41} +10.6395i q^{43} +(-0.853331 - 2.06684i) q^{45} +6.76838i q^{47} +0.425387 q^{49} -1.93744 q^{51} -6.96774i q^{53} +(0.713447 + 1.72803i) q^{55} +1.56454i q^{57} +10.8471 q^{59} -7.76916 q^{61} -2.56410i q^{63} +(-10.3646 + 4.27919i) q^{65} +1.00000i q^{67} -2.41957 q^{69} -3.48188 q^{71} -2.05463i q^{73} +(3.52740 + 3.54365i) q^{75} +2.14377i q^{77} -8.11303 q^{79} +1.00000 q^{81} +12.3521i q^{83} +(4.00438 - 1.65328i) q^{85} -6.17679i q^{87} -7.76323 q^{89} -12.8582 q^{91} -4.81345i q^{93} +(-1.33507 - 3.23366i) q^{95} +2.67618i q^{97} -0.836073 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 4 q^{5} - 24 q^{9}+O(q^{10}) $ 24 * q - 4 * q^5 - 24 * q^9	$ 24 q - 4 q^{5} - 24 q^{9} - 8 q^{11} - 2 q^{15} + 16 q^{19} - 20 q^{21} + 10 q^{25} + 36 q^{29} - 2 q^{35} + 4 q^{39} - 24 q^{41} + 4 q^{45} - 4 q^{51} - 4 q^{55} + 24 q^{59} - 4 q^{61} - 20 q^{65} - 4 q^{69} + 20 q^{71} - 12 q^{75} - 28 q^{79} + 24 q^{81} - 16 q^{85} + 48 q^{89} - 20 q^{91} - 4 q^{95} + 8 q^{99}+O(q^{100}) $ 24 * q - 4 * q^5 - 24 * q^9 - 8 * q^11 - 2 * q^15 + 16 * q^19 - 20 * q^21 + 10 * q^25 + 36 * q^29 - 2 * q^35 + 4 * q^39 - 24 * q^41 + 4 * q^45 - 4 * q^51 - 4 * q^55 + 24 * q^59 - 4 * q^61 - 20 * q^65 - 4 * q^69 + 20 * q^71 - 12 * q^75 - 28 * q^79 + 24 * q^81 - 16 * q^85 + 48 * q^89 - 20 * q^91 - 4 * q^95 + 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4020\mathbb{Z}\right)^\times$.

$n$	$1141$	$2011$	$2681$	$3217$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$		−	1.00000i		−	0.577350i
$3$		−	1.00000i		−	0.577350i
$4$		0			0
$5$	0.853331	+	2.06684i	0.381621	+	0.924319i
$5$	0.853331	+	2.06684i	0.381621	+	0.924319i
$6$		0			0
$7$			2.56410i			0.969139i	0.874753	+	0.484570i	$0.161024\pi$
$7$			2.56410i			0.969139i	−0.874753	+	0.484570i	$0.838976\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	0.836073			0.252085			0.126043	−	0.992025i	$-0.459772\pi$
$11$	0.836073			0.252085			0.126043	+	0.992025i	$0.459772\pi$
$12$		0			0
$13$			5.01469i			1.39082i	0.718611	+	0.695412i	$0.244778\pi$
$13$			5.01469i			1.39082i	−0.718611	+	0.695412i	$0.755222\pi$
$14$		0			0
$15$	2.06684	−	0.853331i	0.533656	−	0.220329i
$16$		0			0
$17$		−	1.93744i		−	0.469898i	−0.972008	−	0.234949i	$-0.924508\pi$
$17$		−	1.93744i		−	0.469898i	0.972008	−	0.234949i	$-0.0754924\pi$
$18$		0			0
$19$	−1.56454			−0.358931			−0.179466	−	0.983764i	$-0.557437\pi$
$19$	−1.56454			−0.358931			−0.179466	+	0.983764i	$0.557437\pi$
$20$		0			0
$21$	2.56410			0.559533
$22$		0			0
$23$		−	2.41957i		−	0.504516i	−0.967660	−	0.252258i	$-0.918827\pi$
$23$		−	2.41957i		−	0.504516i	0.967660	−	0.252258i	$-0.0811731\pi$
$24$		0			0
$25$	−3.54365	+	3.52740i	−0.708731	+	0.705479i
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$	6.17679			1.14700			0.573500	−	0.819205i	$-0.305585\pi$
$29$	6.17679			1.14700			0.573500	+	0.819205i	$0.305585\pi$
$30$		0			0
$31$	4.81345			0.864521			0.432261	−	0.901749i	$-0.357716\pi$
$31$	4.81345			0.864521			0.432261	+	0.901749i	$0.357716\pi$
$32$		0			0
$33$		−	0.836073i		−	0.145542i
$34$		0			0
$35$	−5.29959	+	2.18803i	−0.895793	+	0.369844i
$36$		0			0
$37$		−	2.01556i		−	0.331357i	−0.986180	−	0.165678i	$-0.947019\pi$
$37$		−	2.01556i		−	0.331357i	0.986180	−	0.165678i	$-0.0529813\pi$
$38$		0			0
$39$	5.01469			0.802993
$40$		0			0
$41$	−5.18723			−0.810110			−0.405055	−	0.914292i	$-0.632748\pi$
$41$	−5.18723			−0.810110			−0.405055	+	0.914292i	$0.632748\pi$
$42$		0			0
$43$			10.6395i			1.62250i	0.584697	+	0.811252i	$0.301214\pi$
$43$			10.6395i			1.62250i	−0.584697	+	0.811252i	$0.698786\pi$
$44$		0			0
$45$	−0.853331	−	2.06684i	−0.127207	−	0.308106i
$46$		0			0
$47$			6.76838i			0.987269i	0.869669	+	0.493635i	$0.164332\pi$
$47$			6.76838i			0.987269i	−0.869669	+	0.493635i	$0.835668\pi$
$48$		0			0
$49$	0.425387			0.0607696
$50$		0			0
$51$	−1.93744			−0.271296
$52$		0			0
$53$		−	6.96774i		−	0.957093i	−0.878062	−	0.478546i	$-0.841164\pi$
$53$		−	6.96774i		−	0.957093i	0.878062	−	0.478546i	$-0.158836\pi$
$54$		0			0
$55$	0.713447	+	1.72803i	0.0962011	+	0.233007i
$56$		0			0
$57$			1.56454i			0.207229i
$58$		0			0
$59$	10.8471			1.41217			0.706084	−	0.708128i	$-0.250460\pi$
$59$	10.8471			1.41217			0.706084	+	0.708128i	$0.250460\pi$
$60$		0			0
$61$	−7.76916			−0.994739			−0.497370	−	0.867539i	$-0.665701\pi$
$61$	−7.76916			−0.994739			−0.497370	+	0.867539i	$0.665701\pi$
$62$		0			0
$63$		−	2.56410i		−	0.323046i
$64$		0			0
$65$	−10.3646	+	4.27919i	−1.28557	+	0.530768i
$66$		0			0
$67$			1.00000i			0.122169i
$67$			1.00000i			0.122169i
$68$		0			0
$69$	−2.41957			−0.291282
$70$		0			0
$71$	−3.48188			−0.413223			−0.206611	−	0.978423i	$-0.566244\pi$
$71$	−3.48188			−0.413223			−0.206611	+	0.978423i	$0.566244\pi$
$72$		0			0
$73$		−	2.05463i		−	0.240476i	−0.992745	−	0.120238i	$-0.961634\pi$
$73$		−	2.05463i		−	0.240476i	0.992745	−	0.120238i	$-0.0383658\pi$
$74$		0			0
$75$	3.52740	+	3.54365i	0.407309	+	0.409186i
$76$		0			0
$77$			2.14377i			0.244306i
$78$		0			0
$79$	−8.11303			−0.912787			−0.456393	−	0.889778i	$-0.650859\pi$
$79$	−8.11303			−0.912787			−0.456393	+	0.889778i	$0.650859\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			12.3521i			1.35582i	0.735146	+	0.677909i	$0.237114\pi$
$83$			12.3521i			1.35582i	−0.735146	+	0.677909i	$0.762886\pi$
$84$		0			0
$85$	4.00438	−	1.65328i	0.434336	−	0.179323i
$86$		0			0
$87$		−	6.17679i		−	0.662221i
$88$		0			0
$89$	−7.76323			−0.822901			−0.411450	−	0.911432i	$-0.634978\pi$
$89$	−7.76323			−0.822901			−0.411450	+	0.911432i	$0.634978\pi$
$90$		0			0
$91$	−12.8582			−1.34790
$92$		0			0
$93$		−	4.81345i		−	0.499132i
$94$		0			0
$95$	−1.33507	−	3.23366i	−0.136976	−	0.331767i
$96$		0			0
$97$			2.67618i			0.271725i	0.990728	+	0.135862i	$0.0433805\pi$
$97$			2.67618i			0.271725i	−0.990728	+	0.135862i	$0.956620\pi$
$98$		0			0
$99$	−0.836073			−0.0840285
$100$		0			0
$101$	−12.7860			−1.27225			−0.636125	−	0.771586i	$-0.719464\pi$
$101$	−12.7860			−1.27225			−0.636125	+	0.771586i	$0.719464\pi$
$102$		0			0
$103$			5.94314i			0.585595i	0.956174	+	0.292798i	$0.0945862\pi$
$103$			5.94314i			0.585595i	−0.956174	+	0.292798i	$0.905414\pi$
$104$		0			0
$105$	2.18803	+	5.29959i	0.213530	+	0.517187i
$106$		0			0
$107$			7.12227i			0.688536i	0.938871	+	0.344268i	$0.111873\pi$
$107$			7.12227i			0.688536i	−0.938871	+	0.344268i	$0.888127\pi$
$108$		0			0
$109$	−8.43258			−0.807695			−0.403848	−	0.914826i	$-0.632327\pi$
$109$	−8.43258			−0.807695			−0.403848	+	0.914826i	$0.632327\pi$
$110$		0			0
$111$	−2.01556			−0.191309
$112$		0			0
$113$		−	0.764970i		−	0.0719623i	−0.999352	−	0.0359812i	$-0.988544\pi$
$113$		−	0.764970i		−	0.0719623i	0.999352	−	0.0359812i	$-0.0114556\pi$
$114$		0			0
$115$	5.00087	−	2.06470i	0.466333	−	0.192534i
$116$		0			0
$117$		−	5.01469i		−	0.463608i
$118$		0			0
$119$	4.96779			0.455397
$120$		0			0
$121$	−10.3010			−0.936453
$122$		0			0
$123$			5.18723i			0.467717i
$124$		0			0
$125$	−10.3145	−	4.31413i	−0.922554	−	0.385867i
$126$		0			0
$127$		−	14.1202i		−	1.25296i	−0.779436	−	0.626482i	$-0.784494\pi$
$127$		−	14.1202i		−	1.25296i	0.779436	−	0.626482i	$-0.215506\pi$
$128$		0			0
$129$	10.6395			0.936753
$130$		0			0
$131$	−1.77378			−0.154976			−0.0774881	−	0.996993i	$-0.524690\pi$
$131$	−1.77378			−0.154976			−0.0774881	+	0.996993i	$0.524690\pi$
$132$		0			0
$133$		−	4.01165i		−	0.347854i
$134$		0			0
$135$	−2.06684	+	0.853331i	−0.177885	+	0.0734430i
$136$		0			0
$137$			16.9570i			1.44874i	0.689412	+	0.724369i	$0.257869\pi$
$137$			16.9570i			1.44874i	−0.689412	+	0.724369i	$0.742131\pi$
$138$		0			0
$139$	−0.0977173			−0.00828827			−0.00414414	−	0.999991i	$-0.501319\pi$
$139$	−0.0977173			−0.00828827			−0.00414414	+	0.999991i	$0.501319\pi$
$140$		0			0
$141$	6.76838			0.570000
$142$		0			0
$143$			4.19264i			0.350607i
$144$		0			0
$145$	5.27084	+	12.7664i	0.437720	+	1.06019i
$146$		0			0
$147$		−	0.425387i		−	0.0350853i
$148$		0			0
$149$	4.35385			0.356681			0.178341	−	0.983969i	$-0.442927\pi$
$149$	4.35385			0.356681			0.178341	+	0.983969i	$0.442927\pi$
$150$		0			0
$151$	−21.7898			−1.77323			−0.886613	−	0.462512i	$-0.846948\pi$
$151$	−21.7898			−1.77323			−0.886613	+	0.462512i	$0.846948\pi$
$152$		0			0
$153$			1.93744i			0.156633i
$154$		0			0
$155$	4.10747	+	9.94863i	0.329920	+	0.799093i
$156$		0			0
$157$		−	8.04329i		−	0.641924i	−0.947092	−	0.320962i	$-0.895994\pi$
$157$		−	8.04329i		−	0.641924i	0.947092	−	0.320962i	$-0.104006\pi$
$158$		0			0
$159$	−6.96774			−0.552578
$160$		0			0
$161$	6.20403			0.488946
$162$		0			0
$163$			9.63825i			0.754926i	0.926025	+	0.377463i	$0.123203\pi$
$163$			9.63825i			0.754926i	−0.926025	+	0.377463i	$0.876797\pi$
$164$		0			0
$165$	1.72803	−	0.713447i	0.134527	−	0.0555417i
$166$		0			0
$167$		−	7.24197i		−	0.560400i	−0.959942	−	0.280200i	$-0.909599\pi$
$167$		−	7.24197i		−	0.560400i	0.959942	−	0.280200i	$-0.0904008\pi$
$168$		0			0
$169$	−12.1471			−0.934393
$170$		0			0
$171$	1.56454			0.119644
$172$		0			0
$173$			13.2691i			1.00883i	0.863461	+	0.504416i	$0.168292\pi$
$173$			13.2691i			1.00883i	−0.863461	+	0.504416i	$0.831708\pi$
$174$		0			0
$175$	−9.04460	−	9.08628i	−0.683707	−	0.686858i
$176$		0			0
$177$		−	10.8471i		−	0.815316i
$178$		0			0
$179$	2.89191			0.216151			0.108076	−	0.994143i	$-0.465531\pi$
$179$	2.89191			0.216151			0.108076	+	0.994143i	$0.465531\pi$
$180$		0			0
$181$	1.34877			0.100253			0.0501266	−	0.998743i	$-0.484038\pi$
$181$	1.34877			0.100253			0.0501266	+	0.998743i	$0.484038\pi$
$182$		0			0
$183$			7.76916i			0.574313i
$184$		0			0
$185$	4.16585	−	1.71994i	0.306279	−	0.126453i
$186$		0			0
$187$		−	1.61984i		−	0.118454i
$188$		0			0
$189$	−2.56410			−0.186511
$190$		0			0
$191$	−0.687648			−0.0497565			−0.0248782	−	0.999690i	$-0.507920\pi$
$191$	−0.687648			−0.0497565			−0.0248782	+	0.999690i	$0.507920\pi$
$192$		0			0
$193$		−	8.46127i		−	0.609055i	−0.952503	−	0.304528i	$-0.901501\pi$
$193$		−	8.46127i		−	0.609055i	0.952503	−	0.304528i	$-0.0984986\pi$
$194$		0			0
$195$	4.27919	+	10.3646i	0.306439	+	0.742221i
$196$		0			0
$197$			17.3779i			1.23812i	0.785343	+	0.619061i	$0.212487\pi$
$197$			17.3779i			1.23812i	−0.785343	+	0.619061i	$0.787513\pi$
$198$		0			0
$199$	9.98285			0.707665			0.353833	−	0.935309i	$-0.384878\pi$
$199$	9.98285			0.707665			0.353833	+	0.935309i	$0.384878\pi$
$200$		0			0
$201$	1.00000			0.0705346
$202$		0			0
$203$			15.8379i			1.11160i
$204$		0			0
$205$	−4.42643	−	10.7212i	−0.309155	−	0.748800i
$206$		0			0
$207$			2.41957i			0.168172i
$208$		0			0
$209$	−1.30807			−0.0904813
$210$		0			0
$211$	12.4532			0.857316			0.428658	−	0.903467i	$-0.358986\pi$
$211$	12.4532			0.857316			0.428658	+	0.903467i	$0.358986\pi$
$212$		0			0
$213$			3.48188i			0.238574i
$214$		0			0
$215$	−21.9901	+	9.07899i	−1.49971	+	0.619182i
$216$		0			0
$217$			12.3422i			0.837841i
$218$		0			0
$219$	−2.05463			−0.138839
$220$		0			0
$221$	9.71566			0.653546
$222$		0			0
$223$			3.30059i			0.221024i	0.993875	+	0.110512i	$0.0352490\pi$
$223$			3.30059i			0.221024i	−0.993875	+	0.110512i	$0.964751\pi$
$224$		0			0
$225$	3.54365	−	3.52740i	0.236244	−	0.235160i
$226$		0			0
$227$		−	15.0510i		−	0.998970i	−0.866323	−	0.499485i	$-0.833523\pi$
$227$		−	15.0510i		−	0.998970i	0.866323	−	0.499485i	$-0.166477\pi$
$228$		0			0
$229$	−7.35704			−0.486167			−0.243084	−	0.970005i	$-0.578159\pi$
$229$	−7.35704			−0.486167			−0.243084	+	0.970005i	$0.578159\pi$
$230$		0			0
$231$	2.14377			0.141050
$232$		0			0
$233$		−	3.72001i		−	0.243706i	−0.992548	−	0.121853i	$-0.961116\pi$
$233$		−	3.72001i		−	0.243706i	0.992548	−	0.121853i	$-0.0388837\pi$
$234$		0			0
$235$	−13.9892	+	5.77567i	−0.912552	+	0.376763i
$236$		0			0
$237$			8.11303i			0.526998i
$238$		0			0
$239$	−13.2288			−0.855700			−0.427850	−	0.903850i	$-0.640729\pi$
$239$	−13.2288			−0.855700			−0.427850	+	0.903850i	$0.640729\pi$
$240$		0			0
$241$	3.79074			0.244183			0.122092	−	0.992519i	$-0.461040\pi$
$241$	3.79074			0.244183			0.122092	+	0.992519i	$0.461040\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$	0.362996	+	0.879206i	0.0231909	+	0.0561704i
$246$		0			0
$247$		−	7.84570i		−	0.499210i
$248$		0			0
$249$	12.3521			0.782782
$250$		0			0
$251$	11.3524			0.716555			0.358277	−	0.933615i	$-0.383364\pi$
$251$	11.3524			0.716555			0.358277	+	0.933615i	$0.383364\pi$
$252$		0			0
$253$		−	2.02294i		−	0.127181i
$254$		0			0
$255$	−1.65328	−	4.00438i	−0.103532	−	0.250764i
$256$		0			0
$257$			1.29591i			0.0808366i	0.999183	+	0.0404183i	$0.0128691\pi$
$257$			1.29591i			0.0808366i	−0.999183	+	0.0404183i	$0.987131\pi$
$258$		0			0
$259$	5.16811			0.321131
$260$		0			0
$261$	−6.17679			−0.382334
$262$		0			0
$263$			26.8074i			1.65301i	0.562928	+	0.826506i	$0.309675\pi$
$263$			26.8074i			1.65301i	−0.562928	+	0.826506i	$0.690325\pi$
$264$		0			0
$265$	14.4012	−	5.94579i	0.884659	−	0.365247i
$266$		0			0
$267$			7.76323i			0.475102i
$268$		0			0
$269$	26.0123			1.58600			0.792998	−	0.609224i	$-0.208519\pi$
$269$	26.0123			1.58600			0.792998	+	0.609224i	$0.208519\pi$
$270$		0			0
$271$	10.9537			0.665393			0.332696	−	0.943034i	$-0.392042\pi$
$271$	10.9537			0.665393			0.332696	+	0.943034i	$0.392042\pi$
$272$		0			0
$273$			12.8582i			0.778212i
$274$		0			0
$275$	−2.96275	+	2.94916i	−0.178661	+	0.177841i
$276$		0			0
$277$		−	30.8323i		−	1.85253i	−0.376870	−	0.926266i	$-0.623000\pi$
$277$		−	30.8323i		−	1.85253i	0.376870	−	0.926266i	$-0.377000\pi$
$278$		0			0
$279$	−4.81345			−0.288174
$280$		0			0
$281$	8.35683			0.498527			0.249263	−	0.968436i	$-0.419812\pi$
$281$	8.35683			0.498527			0.249263	+	0.968436i	$0.419812\pi$
$282$		0			0
$283$			13.6782i			0.813087i	0.913632	+	0.406543i	$0.133266\pi$
$283$			13.6782i			0.813087i	−0.913632	+	0.406543i	$0.866734\pi$
$284$		0			0
$285$	−3.23366	+	1.33507i	−0.191546	+	0.0790829i
$286$		0			0
$287$		−	13.3006i		−	0.785109i
$288$		0			0
$289$	13.2463			0.779196
$290$		0			0
$291$	2.67618			0.156880
$292$		0			0
$293$		−	12.2417i		−	0.715168i	−0.933881	−	0.357584i	$-0.883601\pi$
$293$		−	12.2417i		−	0.715168i	0.933881	−	0.357584i	$-0.116399\pi$
$294$		0			0
$295$	9.25614	+	22.4192i	0.538913	+	1.30529i
$296$		0			0
$297$			0.836073i			0.0485139i
$298$		0			0
$299$	12.1334			0.701693
$300$		0			0
$301$	−27.2807			−1.57243
$302$		0			0
$303$			12.7860i			0.734534i
$304$		0			0
$305$	−6.62967	−	16.0576i	−0.379614	−	0.919456i
$306$		0			0
$307$			11.0351i			0.629808i	0.949124	+	0.314904i	$0.101972\pi$
$307$			11.0351i			0.629808i	−0.949124	+	0.314904i	$0.898028\pi$
$308$		0			0
$309$	5.94314			0.338093
$310$		0			0
$311$	−0.781322			−0.0443047			−0.0221524	−	0.999755i	$-0.507052\pi$
$311$	−0.781322			−0.0443047			−0.0221524	+	0.999755i	$0.507052\pi$
$312$		0			0
$313$		−	32.4694i		−	1.83528i	−0.397412	−	0.917640i	$-0.630092\pi$
$313$		−	32.4694i		−	1.83528i	0.397412	−	0.917640i	$-0.369908\pi$
$314$		0			0
$315$	5.29959	−	2.18803i	0.298598	−	0.123281i
$316$		0			0
$317$		−	7.30229i		−	0.410138i	−0.978748	−	0.205069i	$-0.934258\pi$
$317$		−	7.30229i		−	0.410138i	0.978748	−	0.205069i	$-0.0657418\pi$
$318$		0			0
$319$	5.16424			0.289142
$320$		0			0
$321$	7.12227			0.397526
$322$		0			0
$323$			3.03121i			0.168661i
$324$		0			0
$325$	−17.6888	−	17.7703i	−0.981198	−	0.985720i
$326$		0			0
$327$			8.43258i			0.466323i
$328$		0			0
$329$	−17.3548			−0.956801
$330$		0			0
$331$	−11.7715			−0.647020			−0.323510	−	0.946225i	$-0.604863\pi$
$331$	−11.7715			−0.647020			−0.323510	+	0.946225i	$0.604863\pi$
$332$		0			0
$333$			2.01556i			0.110452i
$334$		0			0
$335$	−2.06684	+	0.853331i	−0.112924	+	0.0466224i
$336$		0			0
$337$			21.4825i			1.17022i	0.810952	+	0.585112i	$0.198950\pi$
$337$			21.4825i			1.17022i	−0.810952	+	0.585112i	$0.801050\pi$
$338$		0			0
$339$	−0.764970			−0.0415475
$340$		0			0
$341$	4.02439			0.217933
$342$		0			0
$343$			19.0394i			1.02803i
$344$		0			0
$345$	−2.06470	−	5.00087i	−0.111159	−	0.269238i
$346$		0			0
$347$			7.66709i			0.411591i	0.978595	+	0.205796i	$0.0659782\pi$
$347$			7.66709i			0.411591i	−0.978595	+	0.205796i	$0.934022\pi$
$348$		0			0
$349$	−6.11411			−0.327281			−0.163640	−	0.986520i	$-0.552324\pi$
$349$	−6.11411			−0.327281			−0.163640	+	0.986520i	$0.552324\pi$
$350$		0			0
$351$	−5.01469			−0.267664
$352$		0			0
$353$		−	4.11216i		−	0.218868i	−0.993994	−	0.109434i	$-0.965096\pi$
$353$		−	4.11216i		−	0.218868i	0.993994	−	0.109434i	$-0.0349039\pi$
$354$		0			0
$355$	−2.97119	−	7.19648i	−0.157695	−	0.381950i
$356$		0			0
$357$		−	4.96779i		−	0.262923i
$358$		0			0
$359$	4.71720			0.248964			0.124482	−	0.992222i	$-0.460273\pi$
$359$	4.71720			0.248964			0.124482	+	0.992222i	$0.460273\pi$
$360$		0			0
$361$	−16.5522			−0.871169
$362$		0			0
$363$			10.3010i			0.540661i
$364$		0			0
$365$	4.24659	−	1.75328i	0.222277	−	0.0917707i
$366$		0			0
$367$			22.7730i			1.18874i	0.804191	+	0.594372i	$0.202599\pi$
$367$			22.7730i			1.18874i	−0.804191	+	0.594372i	$0.797401\pi$
$368$		0			0
$369$	5.18723			0.270037
$370$		0			0
$371$	17.8660			0.927556
$372$		0			0
$373$			19.3438i			1.00158i	0.865568	+	0.500791i	$0.166957\pi$
$373$			19.3438i			1.00158i	−0.865568	+	0.500791i	$0.833043\pi$
$374$		0			0
$375$	−4.31413	+	10.3145i	−0.222781	+	0.532637i
$376$		0			0
$377$			30.9747i			1.59528i
$378$		0			0
$379$	34.5937			1.77696			0.888481	−	0.458914i	$-0.151761\pi$
$379$	34.5937			1.77696			0.888481	+	0.458914i	$0.151761\pi$
$380$		0			0
$381$	−14.1202			−0.723400
$382$		0			0
$383$		−	7.47483i		−	0.381946i	−0.981595	−	0.190973i	$-0.938836\pi$
$383$		−	7.47483i		−	0.381946i	0.981595	−	0.190973i	$-0.0611643\pi$
$384$		0			0
$385$	−4.43084	+	1.82935i	−0.225816	+	0.0932323i
$386$		0			0
$387$		−	10.6395i		−	0.540835i
$388$		0			0
$389$	5.97737			0.303065			0.151532	−	0.988452i	$-0.451579\pi$
$389$	5.97737			0.303065			0.151532	+	0.988452i	$0.451579\pi$
$390$		0			0
$391$	−4.68778			−0.237071
$392$		0			0
$393$			1.77378i			0.0894756i
$394$		0			0
$395$	−6.92310	−	16.7683i	−0.348339	−	0.843706i
$396$		0			0
$397$			11.0358i			0.553873i	0.960888	+	0.276937i	$0.0893192\pi$
$397$			11.0358i			0.553873i	−0.960888	+	0.276937i	$0.910681\pi$
$398$		0			0
$399$	−4.01165			−0.200834
$400$		0			0
$401$	19.1180			0.954705			0.477353	−	0.878712i	$-0.341596\pi$
$401$	19.1180			0.954705			0.477353	+	0.878712i	$0.341596\pi$
$402$		0			0
$403$			24.1380i			1.20240i
$404$		0			0
$405$	0.853331	+	2.06684i	0.0424024	+	0.102702i
$406$		0			0
$407$		−	1.68516i		−	0.0835301i
$408$		0			0
$409$	13.4234			0.663743			0.331872	−	0.943325i	$-0.392320\pi$
$409$	13.4234			0.663743			0.331872	+	0.943325i	$0.392320\pi$
$410$		0			0
$411$	16.9570			0.836429
$412$		0			0
$413$			27.8130i			1.36859i
$414$		0			0
$415$	−25.5298	+	10.5404i	−1.25321	+	0.517409i
$416$		0			0
$417$			0.0977173i			0.00478524i
$418$		0			0
$419$	39.1932			1.91471			0.957357	−	0.288909i	$-0.0932924\pi$
$419$	39.1932			1.91471			0.957357	+	0.288909i	$0.0932924\pi$
$420$		0			0
$421$	−27.5557			−1.34298			−0.671491	−	0.741013i	$-0.734346\pi$
$421$	−27.5557			−1.34298			−0.671491	+	0.741013i	$0.734346\pi$
$422$		0			0
$423$		−	6.76838i		−	0.329090i
$424$		0			0
$425$	6.83412	+	6.86561i	0.331503	+	0.333031i
$426$		0			0
$427$		−	19.9209i		−	0.964041i
$428$		0			0
$429$	4.19264			0.202423
$430$		0			0
$431$	12.5008			0.602142			0.301071	−	0.953602i	$-0.402656\pi$
$431$	12.5008			0.602142			0.301071	+	0.953602i	$0.402656\pi$
$432$		0			0
$433$		−	16.9731i		−	0.815674i	−0.913055	−	0.407837i	$-0.866283\pi$
$433$		−	16.9731i		−	0.815674i	0.913055	−	0.407837i	$-0.133717\pi$
$434$		0			0
$435$	12.7664	−	5.27084i	0.612104	−	0.252718i
$436$		0			0
$437$			3.78553i			0.181086i
$438$		0			0
$439$	−7.69317			−0.367175			−0.183588	−	0.983003i	$-0.558771\pi$
$439$	−7.69317			−0.367175			−0.183588	+	0.983003i	$0.558771\pi$
$440$		0			0
$441$	−0.425387			−0.0202565
$442$		0			0
$443$			34.3020i			1.62974i	0.579647	+	0.814868i	$0.303190\pi$
$443$			34.3020i			1.62974i	−0.579647	+	0.814868i	$0.696810\pi$
$444$		0			0
$445$	−6.62460	−	16.0453i	−0.314036	−	0.760622i
$446$		0			0
$447$		−	4.35385i		−	0.205930i
$448$		0			0
$449$	25.3131			1.19460			0.597299	−	0.802019i	$-0.296241\pi$
$449$	25.3131			1.19460			0.597299	+	0.802019i	$0.296241\pi$
$450$		0			0
$451$	−4.33690			−0.204217
$452$		0			0
$453$			21.7898i			1.02377i
$454$		0			0
$455$	−10.9723	−	26.5758i	−0.514388	−	1.24589i
$456$		0			0
$457$		−	36.0544i		−	1.68655i	−0.537479	−	0.843277i	$-0.680623\pi$
$457$		−	36.0544i		−	1.68655i	0.537479	−	0.843277i	$-0.319377\pi$
$458$		0			0
$459$	1.93744			0.0904320
$460$		0			0
$461$	13.0726			0.608851			0.304425	−	0.952536i	$-0.401536\pi$
$461$	13.0726			0.608851			0.304425	+	0.952536i	$0.401536\pi$
$462$		0			0
$463$		−	1.61691i		−	0.0751444i	−0.999294	−	0.0375722i	$-0.988038\pi$
$463$		−	1.61691i		−	0.0751444i	0.999294	−	0.0375722i	$-0.0119624\pi$
$464$		0			0
$465$	9.94863	−	4.10747i	0.461357	−	0.190479i
$466$		0			0
$467$			9.09778i			0.420995i	0.977594	+	0.210498i	$0.0675084\pi$
$467$			9.09778i			0.420995i	−0.977594	+	0.210498i	$0.932492\pi$
$468$		0			0
$469$	−2.56410			−0.118399
$470$		0			0
$471$	−8.04329			−0.370615
$472$		0			0
$473$			8.89537i			0.409010i
$474$		0			0
$475$	5.54420	−	5.51877i	0.254385	−	0.253218i
$476$		0			0
$477$			6.96774i			0.319031i
$478$		0			0
$479$	−1.26328			−0.0577207			−0.0288604	−	0.999583i	$-0.509188\pi$
$479$	−1.26328			−0.0577207			−0.0288604	+	0.999583i	$0.509188\pi$
$480$		0			0
$481$	10.1074			0.460859
$482$		0			0
$483$		−	6.20403i		−	0.282293i
$484$		0			0
$485$	−5.53123	+	2.28367i	−0.251160	+	0.103696i
$486$		0			0
$487$			14.2590i			0.646135i	0.946376	+	0.323067i	$0.104714\pi$
$487$			14.2590i			0.646135i	−0.946376	+	0.323067i	$0.895286\pi$
$488$		0			0
$489$	9.63825			0.435857
$490$		0			0
$491$	17.8082			0.803674			0.401837	−	0.915711i	$-0.368372\pi$
$491$	17.8082			0.803674			0.401837	+	0.915711i	$0.368372\pi$
$492$		0			0
$493$		−	11.9672i		−	0.538974i
$494$		0			0
$495$	−0.713447	−	1.72803i	−0.0320670	−	0.0776691i
$496$		0			0
$497$		−	8.92789i		−	0.400470i
$498$		0			0
$499$	−9.53077			−0.426656			−0.213328	−	0.976981i	$-0.568430\pi$
$499$	−9.53077			−0.426656			−0.213328	+	0.976981i	$0.568430\pi$
$500$		0			0
$501$	−7.24197			−0.323547
$502$		0			0
$503$			21.1836i			0.944532i	0.881456	+	0.472266i	$0.156564\pi$
$503$			21.1836i			0.944532i	−0.881456	+	0.472266i	$0.843436\pi$
$504$		0			0
$505$	−10.9107	−	26.4265i	−0.485518	−	1.17596i
$506$		0			0
$507$			12.1471i			0.539472i
$508$		0			0
$509$	2.51154			0.111322			0.0556610	−	0.998450i	$-0.482273\pi$
$509$	2.51154			0.111322			0.0556610	+	0.998450i	$0.482273\pi$
$510$		0			0
$511$	5.26827			0.233055
$512$		0			0
$513$		−	1.56454i		−	0.0690763i
$514$		0			0
$515$	−12.2835	+	5.07147i	−0.541277	+	0.223475i
$516$		0			0
$517$			5.65886i			0.248876i
$518$		0			0
$519$	13.2691			0.582450
$520$		0			0
$521$	35.1358			1.53933			0.769664	−	0.638449i	$-0.220424\pi$
$521$	35.1358			1.53933			0.769664	+	0.638449i	$0.220424\pi$
$522$		0			0
$523$			17.8915i			0.782339i	0.920319	+	0.391170i	$0.127929\pi$
$523$			17.8915i			0.782339i	−0.920319	+	0.391170i	$0.872071\pi$
$524$		0			0
$525$	−9.08628	+	9.04460i	−0.396558	+	0.394739i
$526$		0			0
$527$		−	9.32577i		−	0.406237i
$528$		0			0
$529$	17.1457			0.745464
$530$		0			0
$531$	−10.8471			−0.470723
$532$		0			0
$533$		−	26.0124i		−	1.12672i
$534$		0			0
$535$	−14.7206	+	6.07765i	−0.636427	+	0.262760i
$536$		0			0
$537$		−	2.89191i		−	0.124795i
$538$		0			0
$539$	0.355654			0.0153191
$540$		0			0
$541$	−4.40995			−0.189598			−0.0947992	−	0.995496i	$-0.530221\pi$
$541$	−4.40995			−0.189598			−0.0947992	+	0.995496i	$0.530221\pi$
$542$		0			0
$543$		−	1.34877i		−	0.0578812i
$544$		0			0
$545$	−7.19578	−	17.4288i	−0.308234	−	0.746568i
$546$		0			0
$547$			7.34000i			0.313836i	0.987612	+	0.156918i	$0.0501558\pi$
$547$			7.34000i			0.313836i	−0.987612	+	0.156918i	$0.949844\pi$
$548$		0			0
$549$	7.76916			0.331580
$550$		0			0
$551$	−9.66386			−0.411694
$552$		0			0
$553$		−	20.8026i		−	0.884617i
$554$		0			0
$555$	−1.71994	−	4.16585i	−0.0730075	−	0.176830i
$556$		0			0
$557$		−	15.0988i		−	0.639758i	−0.947458	−	0.319879i	$-0.896358\pi$
$557$		−	15.0988i		−	0.639758i	0.947458	−	0.319879i	$-0.103642\pi$
$558$		0			0
$559$	−53.3536			−2.25662
$560$		0			0
$561$	−1.61984			−0.0683897
$562$		0			0
$563$		−	46.0012i		−	1.93872i	−0.245644	−	0.969360i	$-0.579000\pi$
$563$		−	46.0012i		−	1.93872i	0.245644	−	0.969360i	$-0.421000\pi$
$564$		0			0
$565$	1.58107	−	0.652772i	0.0665161	−	0.0274623i
$566$		0			0
$567$			2.56410i			0.107682i
$568$		0			0
$569$	8.18195			0.343005			0.171503	−	0.985184i	$-0.445138\pi$
$569$	8.18195			0.343005			0.171503	+	0.985184i	$0.445138\pi$
$570$		0			0
$571$	−32.9221			−1.37775			−0.688875	−	0.724881i	$-0.741895\pi$
$571$	−32.9221			−1.37775			−0.688875	+	0.724881i	$0.741895\pi$
$572$		0			0
$573$			0.687648i			0.0287269i
$574$		0			0
$575$	8.53479	+	8.57413i	0.355925	+	0.357566i
$576$		0			0
$577$			12.6328i			0.525911i	0.964808	+	0.262955i	$0.0846972\pi$
$577$			12.6328i			0.525911i	−0.964808	+	0.262955i	$0.915303\pi$
$578$		0			0
$579$	−8.46127			−0.351638
$580$		0			0
$581$	−31.6720			−1.31398
$582$		0			0
$583$		−	5.82554i		−	0.241269i
$584$		0			0
$585$	10.3646	−	4.27919i	0.428522	−	0.176923i
$586$		0			0
$587$		−	23.6043i		−	0.974254i	−0.873331	−	0.487127i	$-0.838045\pi$
$587$		−	23.6043i		−	0.974254i	0.873331	−	0.487127i	$-0.161955\pi$
$588$		0			0
$589$	−7.53086			−0.310303
$590$		0			0
$591$	17.3779			0.714830
$592$		0			0
$593$		−	11.6470i		−	0.478286i	−0.970984	−	0.239143i	$-0.923134\pi$
$593$		−	11.6470i		−	0.478286i	0.970984	−	0.239143i	$-0.0768664\pi$
$594$		0			0
$595$	4.23917	+	10.2676i	0.173789	+	0.420932i
$596$		0			0
$597$		−	9.98285i		−	0.408571i
$598$		0			0
$599$	1.46578			0.0598902			0.0299451	−	0.999552i	$-0.490467\pi$
$599$	1.46578			0.0598902			0.0299451	+	0.999552i	$0.490467\pi$
$600$		0			0
$601$	−31.8985			−1.30117			−0.650584	−	0.759434i	$-0.725476\pi$
$601$	−31.8985			−1.30117			−0.650584	+	0.759434i	$0.725476\pi$
$602$		0			0
$603$		−	1.00000i		−	0.0407231i
$604$		0			0
$605$	−8.79015	−	21.2905i	−0.357370	−	0.865581i
$606$		0			0
$607$		−	39.4160i		−	1.59985i	−0.600103	−	0.799923i	$-0.704874\pi$
$607$		−	39.4160i		−	1.59985i	0.600103	−	0.799923i	$-0.295126\pi$
$608$		0			0
$609$	15.8379			0.641784
$610$		0			0
$611$	−33.9413			−1.37312
$612$		0			0
$613$		−	23.7620i		−	0.959738i	−0.877340	−	0.479869i	$-0.840684\pi$
$613$		−	23.7620i		−	0.959738i	0.877340	−	0.479869i	$-0.159316\pi$
$614$		0			0
$615$	−10.7212	+	4.42643i	−0.432320	+	0.178491i
$616$		0			0
$617$			23.9343i			0.963559i	0.876293	+	0.481779i	$0.160009\pi$
$617$			23.9343i			0.963559i	−0.876293	+	0.481779i	$0.839991\pi$
$618$		0			0
$619$	1.10939			0.0445903			0.0222951	−	0.999751i	$-0.492903\pi$
$619$	1.10939			0.0445903			0.0222951	+	0.999751i	$0.492903\pi$
$620$		0			0
$621$	2.41957			0.0970941
$622$		0			0
$623$		−	19.9057i		−	0.797505i
$624$		0			0
$625$	0.114952	−	24.9997i	0.00459807	−	0.999989i
$626$		0			0
$627$			1.30807i			0.0522394i
$628$		0			0
$629$	−3.90503			−0.155704
$630$		0			0
$631$	−8.61277			−0.342869			−0.171434	−	0.985196i	$-0.554840\pi$
$631$	−8.61277			−0.342869			−0.171434	+	0.985196i	$0.554840\pi$
$632$		0			0
$633$		−	12.4532i		−	0.494972i
$634$		0			0
$635$	29.1842	−	12.0492i	1.15814	−	0.478158i
$636$		0			0
$637$			2.13318i			0.0845198i
$638$		0			0
$639$	3.48188			0.137741
$640$		0			0
$641$	−26.5073			−1.04697			−0.523487	−	0.852034i	$-0.675369\pi$
$641$	−26.5073			−1.04697			−0.523487	+	0.852034i	$0.675369\pi$
$642$		0			0
$643$		−	8.48031i		−	0.334431i	−0.985920	−	0.167215i	$-0.946522\pi$
$643$		−	8.48031i		−	0.334431i	0.985920	−	0.167215i	$-0.0534775\pi$
$644$		0			0
$645$	9.07899	+	21.9901i	0.357485	+	0.865859i
$646$		0			0
$647$			49.4330i			1.94341i	0.236192	+	0.971706i	$0.424101\pi$
$647$			49.4330i			1.94341i	−0.236192	+	0.971706i	$0.575899\pi$
$648$		0			0
$649$	9.06894			0.355987
$650$		0			0
$651$	12.3422			0.483728
$652$		0			0
$653$		−	2.88169i		−	0.112769i	−0.998409	−	0.0563847i	$-0.982043\pi$
$653$		−	2.88169i		−	0.112769i	0.998409	−	0.0563847i	$-0.0179573\pi$
$654$		0			0
$655$	−1.51362	−	3.66613i	−0.0591422	−	0.143247i
$656$		0			0
$657$			2.05463i			0.0801587i
$658$		0			0
$659$	−19.1083			−0.744354			−0.372177	−	0.928162i	$-0.621389\pi$
$659$	−19.1083			−0.744354			−0.372177	+	0.928162i	$0.621389\pi$
$660$		0			0
$661$	15.9195			0.619195			0.309598	−	0.950868i	$-0.399806\pi$
$661$	15.9195			0.619195			0.309598	+	0.950868i	$0.399806\pi$
$662$		0			0
$663$		−	9.71566i		−	0.377325i
$664$		0			0
$665$	8.29144	−	3.42326i	0.321528	−	0.132748i
$666$		0			0
$667$		−	14.9452i		−	0.578680i
$668$		0			0
$669$	3.30059			0.127608
$670$		0			0
$671$	−6.49558			−0.250759
$672$		0			0
$673$			8.71045i			0.335763i	0.985807	+	0.167882i	$0.0536927\pi$
$673$			8.71045i			0.335763i	−0.985807	+	0.167882i	$0.946307\pi$
$674$		0			0
$675$	−3.52740	−	3.54365i	−0.135770	−	0.136395i
$676$		0			0
$677$		−	27.6099i		−	1.06113i	−0.847643	−	0.530567i	$-0.821979\pi$
$677$		−	27.6099i		−	1.06113i	0.847643	−	0.530567i	$-0.178021\pi$
$678$		0			0
$679$	−6.86199			−0.263339
$680$		0			0
$681$	−15.0510			−0.576755
$682$		0			0
$683$		−	29.4785i		−	1.12796i	−0.825788	−	0.563981i	$-0.809269\pi$
$683$		−	29.4785i		−	1.12796i	0.825788	−	0.563981i	$-0.190731\pi$
$684$		0			0
$685$	−35.0475	+	14.4700i	−1.33910	+	0.552869i
$686$		0			0
$687$			7.35704i			0.280689i
$688$		0			0
$689$	34.9411			1.33115
$690$		0			0
$691$	13.8055			0.525185			0.262593	−	0.964907i	$-0.415422\pi$
$691$	13.8055			0.525185			0.262593	+	0.964907i	$0.415422\pi$
$692$		0			0
$693$		−	2.14377i		−	0.0814353i
$694$		0			0
$695$	−0.0833852	−	0.201966i	−0.00316298	−	0.00766101i
$696$		0			0
$697$			10.0500i			0.380669i
$698$		0			0
$699$	−3.72001			−0.140704
$700$		0			0
$701$	−34.1033			−1.28807			−0.644033	−	0.764998i	$-0.722740\pi$
$701$	−34.1033			−1.28807			−0.644033	+	0.764998i	$0.722740\pi$
$702$		0			0
$703$			3.15344i			0.118934i
$704$		0			0
$705$	5.77567	+	13.9892i	0.217524	+	0.526862i
$706$		0			0
$707$		−	32.7845i		−	1.23299i
$708$		0			0
$709$	24.1254			0.906047			0.453024	−	0.891499i	$-0.350345\pi$
$709$	24.1254			0.906047			0.453024	+	0.891499i	$0.350345\pi$
$710$		0			0
$711$	8.11303			0.304262
$712$		0			0
$713$		−	11.6465i		−	0.436165i
$714$		0			0
$715$	−8.66552	+	3.57771i	−0.324072	+	0.133799i
$716$		0			0
$717$			13.2288i			0.494038i
$718$		0			0
$719$	35.3365			1.31783			0.658915	−	0.752217i	$-0.271016\pi$
$719$	35.3365			1.31783			0.658915	+	0.752217i	$0.271016\pi$
$720$		0			0
$721$	−15.2388			−0.567523
$722$		0			0
$723$		−	3.79074i		−	0.140979i
$724$		0			0
$725$	−21.8884	+	21.7880i	−0.812915	+	0.809185i
$726$		0			0
$727$			3.09392i			0.114747i	0.998353	+	0.0573736i	$0.0182726\pi$
$727$			3.09392i			0.114747i	−0.998353	+	0.0573736i	$0.981727\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	20.6133			0.762412
$732$		0			0
$733$		−	29.3644i		−	1.08460i	−0.840185	−	0.542300i	$-0.817554\pi$
$733$		−	29.3644i		−	1.08460i	0.840185	−	0.542300i	$-0.182446\pi$
$734$		0			0
$735$	0.879206	−	0.362996i	0.0324300	−	0.0133893i
$736$		0			0
$737$			0.836073i			0.0307971i
$738$		0			0
$739$	−40.8127			−1.50132			−0.750659	−	0.660689i	$-0.770264\pi$
$739$	−40.8127			−1.50132			−0.750659	+	0.660689i	$0.770264\pi$
$740$		0			0
$741$	−7.84570			−0.288219
$742$		0			0
$743$		−	39.4342i		−	1.44670i	−0.690482	−	0.723349i	$-0.742602\pi$
$743$		−	39.4342i		−	1.44670i	0.690482	−	0.723349i	$-0.257398\pi$
$744$		0			0
$745$	3.71528	+	8.99871i	0.136117	+	0.329687i
$746$		0			0
$747$		−	12.3521i		−	0.451939i
$748$		0			0
$749$	−18.2622			−0.667287
$750$		0			0
$751$	32.8882			1.20011			0.600053	−	0.799960i	$-0.295146\pi$
$751$	32.8882			1.20011			0.600053	+	0.799960i	$0.295146\pi$
$752$		0			0
$753$		−	11.3524i		−	0.413703i
$754$		0			0
$755$	−18.5939	−	45.0360i	−0.676701	−	1.63903i
$756$		0			0
$757$		−	22.7310i		−	0.826173i	−0.910692	−	0.413086i	$-0.864451\pi$
$757$		−	22.7310i		−	0.826173i	0.910692	−	0.413086i	$-0.135549\pi$
$758$		0			0
$759$	−2.02294			−0.0734280
$760$		0			0
$761$	−38.2083			−1.38505			−0.692525	−	0.721394i	$-0.743502\pi$
$761$	−38.2083			−1.38505			−0.692525	+	0.721394i	$0.743502\pi$
$762$		0			0
$763$		−	21.6220i		−	0.782769i
$764$		0			0
$765$	−4.00438	+	1.65328i	−0.144779	+	0.0597744i
$766$		0			0
$767$			54.3947i			1.96408i
$768$		0			0
$769$	3.98893			0.143844			0.0719222	−	0.997410i	$-0.477087\pi$
$769$	3.98893			0.143844			0.0719222	+	0.997410i	$0.477087\pi$
$770$		0			0
$771$	1.29591			0.0466711
$772$		0			0
$773$		−	44.7744i		−	1.61042i	−0.592988	−	0.805211i	$-0.702052\pi$
$773$		−	44.7744i		−	1.61042i	0.592988	−	0.805211i	$-0.297948\pi$
$774$		0			0
$775$	−17.0572	+	16.9789i	−0.612713	+	0.609902i
$776$		0			0
$777$		−	5.16811i		−	0.185405i
$778$		0			0
$779$	8.11566			0.290774
$780$		0			0
$781$	−2.91110			−0.104167
$782$		0			0
$783$			6.17679i			0.220740i
$784$		0			0
$785$	16.6242	−	6.86359i	0.593343	−	0.244972i
$786$		0			0
$787$			22.6203i			0.806326i	0.915128	+	0.403163i	$0.132089\pi$
$787$			22.6203i			0.806326i	−0.915128	+	0.403163i	$0.867911\pi$
$788$		0			0
$789$	26.8074			0.954367
$790$		0			0
$791$	1.96146			0.0697415
$792$		0			0
$793$		−	38.9599i		−	1.38351i
$794$		0			0
$795$	−5.94579	−	14.4012i	−0.210875	−	0.510758i
$796$		0			0
$797$			11.2746i			0.399365i	0.979861	+	0.199683i	$0.0639911\pi$
$797$			11.2746i			0.399365i	−0.979861	+	0.199683i	$0.936009\pi$
$798$		0			0
$799$	13.1133			0.463916
$800$		0			0
$801$	7.76323			0.274300
$802$		0			0
$803$		−	1.71782i		−	0.0606205i
$804$		0			0
$805$	5.29409	+	12.8227i	0.186592	+	0.451942i
$806$		0			0
$807$		−	26.0123i		−	0.915675i
$808$		0			0
$809$	10.5263			0.370085			0.185043	−	0.982731i	$-0.440758\pi$
$809$	10.5263			0.370085			0.185043	+	0.982731i	$0.440758\pi$
$810$		0			0
$811$	−25.9168			−0.910063			−0.455032	−	0.890475i	$-0.650372\pi$
$811$	−25.9168			−0.910063			−0.455032	+	0.890475i	$0.650372\pi$
$812$		0			0
$813$		−	10.9537i		−	0.384165i
$814$		0			0
$815$	−19.9207	+	8.22461i	−0.697792	+	0.288096i
$816$		0			0
$817$		−	16.6459i		−	0.582367i
$818$		0			0
$819$	12.8582			0.449301
$820$		0			0
$821$	54.2916			1.89479			0.947395	−	0.320066i	$-0.103705\pi$
$821$	54.2916			1.89479			0.947395	+	0.320066i	$0.103705\pi$
$822$		0			0
$823$		−	14.7325i		−	0.513543i	−0.966472	−	0.256772i	$-0.917341\pi$
$823$		−	14.7325i		−	0.513543i	0.966472	−	0.256772i	$-0.0826588\pi$
$824$		0			0
$825$	2.94916	+	2.96275i	0.102677	+	0.103150i
$826$		0			0
$827$		−	1.72313i		−	0.0599189i	−0.999551	−	0.0299595i	$-0.990462\pi$
$827$		−	1.72313i		−	0.0599189i	0.999551	−	0.0299595i	$-0.00953782\pi$
$828$		0			0
$829$	−23.6018			−0.819726			−0.409863	−	0.912147i	$-0.634423\pi$
$829$	−23.6018			−0.819726			−0.409863	+	0.912147i	$0.634423\pi$
$830$		0			0
$831$	−30.8323			−1.06956
$832$		0			0
$833$		−	0.824162i		−	0.0285555i
$834$		0			0
$835$	14.9680	−	6.17979i	0.517988	−	0.213861i
$836$		0			0
$837$			4.81345i			0.166377i
$838$		0			0
$839$	47.0461			1.62421			0.812106	−	0.583511i	$-0.198321\pi$
$839$	47.0461			1.62421			0.812106	+	0.583511i	$0.198321\pi$
$840$		0			0
$841$	9.15272			0.315611
$842$		0			0
$843$		−	8.35683i		−	0.287825i
$844$		0			0
$845$	−10.3655	−	25.1061i	−0.356584	−	0.863677i
$846$		0			0
$847$		−	26.4128i		−	0.907553i
$848$		0			0
$849$	13.6782			0.469436
$850$		0			0
$851$	−4.87680			−0.167175
$852$		0			0
$853$		−	44.3086i		−	1.51710i	−0.651615	−	0.758550i	$-0.725908\pi$
$853$		−	44.3086i		−	1.51710i	0.651615	−	0.758550i	$-0.274092\pi$
$854$		0			0
$855$	1.33507	+	3.23366i	0.0456586	+	0.110589i
$856$		0			0
$857$			48.1710i			1.64549i	0.568410	+	0.822745i	$0.307559\pi$
$857$			48.1710i			1.64549i	−0.568410	+	0.822745i	$0.692441\pi$
$858$		0			0
$859$	26.6206			0.908283			0.454142	−	0.890930i	$-0.349946\pi$
$859$	26.6206			0.908283			0.454142	+	0.890930i	$0.349946\pi$
$860$		0			0
$861$	−13.3006			−0.453283
$862$		0			0
$863$		−	8.98076i		−	0.305709i	−0.988249	−	0.152854i	$-0.951153\pi$
$863$		−	8.98076i		−	0.305709i	0.988249	−	0.152854i	$-0.0488465\pi$
$864$		0			0
$865$	−27.4252	+	11.3230i	−0.932483	+	0.384992i
$866$		0			0
$867$		−	13.2463i		−	0.449869i
$868$		0			0
$869$	−6.78308			−0.230100
$870$		0			0
$871$	−5.01469			−0.169916
$872$		0			0
$873$		−	2.67618i		−	0.0905749i
$874$		0			0
$875$	11.0619	−	26.4473i	0.373959	−	0.894083i
$876$		0			0
$877$		−	48.0966i		−	1.62411i	−0.583583	−	0.812054i	$-0.698350\pi$
$877$		−	48.0966i		−	1.62411i	0.583583	−	0.812054i	$-0.301650\pi$
$878$		0			0
$879$	−12.2417			−0.412902
$880$		0			0
$881$	44.7308			1.50702			0.753509	−	0.657437i	$-0.228359\pi$
$881$	44.7308			1.50702			0.753509	+	0.657437i	$0.228359\pi$
$882$		0			0
$883$			32.2537i			1.08542i	0.839919	+	0.542711i	$0.182602\pi$
$883$			32.2537i			1.08542i	−0.839919	+	0.542711i	$0.817398\pi$
$884$		0			0
$885$	22.4192	−	9.25614i	0.753612	−	0.311142i
$886$		0			0
$887$			21.9114i			0.735713i	0.929883	+	0.367856i	$0.119908\pi$
$887$			21.9114i			0.735713i	−0.929883	+	0.367856i	$0.880092\pi$
$888$		0			0
$889$	36.2056			1.21430
$890$		0			0
$891$	0.836073			0.0280095
$892$		0			0
$893$		−	10.5894i		−	0.354362i
$894$		0			0
$895$	2.46775	+	5.97711i	0.0824879	+	0.199793i
$896$		0			0
$897$		−	12.1334i		−	0.405123i
$898$		0			0
$899$	29.7317			0.991607
$900$		0			0
$901$	−13.4996			−0.449736
$902$		0			0
$903$			27.2807i			0.907844i
$904$		0			0
$905$	1.15095	+	2.78769i	0.0382587	+	0.0926659i
$906$		0			0
$907$			35.2055i			1.16898i	0.811401	+	0.584489i	$0.198705\pi$
$907$			35.2055i			1.16898i	−0.811401	+	0.584489i	$0.801295\pi$
$908$		0			0
$909$	12.7860			0.424083
$910$		0			0
$911$	33.5713			1.11227			0.556133	−	0.831093i	$-0.312284\pi$
$911$	33.5713			1.11227			0.556133	+	0.831093i	$0.312284\pi$
$912$		0			0
$913$			10.3272i			0.341782i
$914$		0			0
$915$	−16.0576	+	6.62967i	−0.530848	+	0.219170i
$916$		0			0
$917$		−	4.54816i		−	0.150194i
$918$		0			0
$919$	32.5789			1.07468			0.537340	−	0.843366i	$-0.319429\pi$
$919$	32.5789			1.07468			0.537340	+	0.843366i	$0.319429\pi$
$920$		0			0
$921$	11.0351			0.363620
$922$		0			0
$923$		−	17.4605i		−	0.574721i
$924$		0			0
$925$	7.10969	+	7.14246i	0.233765	+	0.234842i
$926$		0			0
$927$		−	5.94314i		−	0.195198i
$928$		0			0
$929$	6.25505			0.205222			0.102611	−	0.994722i	$-0.467280\pi$
$929$	6.25505			0.205222			0.102611	+	0.994722i	$0.467280\pi$
$930$		0			0
$931$	−0.665536			−0.0218121
$932$		0			0
$933$			0.781322i			0.0255793i
$934$		0			0
$935$	3.34795	−	1.38226i	0.109490	−	0.0452047i
$936$		0			0
$937$			32.5705i			1.06403i	0.846735	+	0.532016i	$0.178565\pi$
$937$			32.5705i			1.06403i	−0.846735	+	0.532016i	$0.821435\pi$
$938$		0			0
$939$	−32.4694			−1.05960
$940$		0			0
$941$	28.5639			0.931157			0.465579	−	0.885007i	$-0.345846\pi$
$941$	28.5639			0.931157			0.465579	+	0.885007i	$0.345846\pi$
$942$		0			0
$943$			12.5509i			0.408713i
$944$		0			0
$945$	−2.18803	−	5.29959i	−0.0711765	−	0.172396i
$946$		0			0
$947$			39.5061i			1.28378i	0.766798	+	0.641889i	$0.221849\pi$
$947$			39.5061i			1.28378i	−0.766798	+	0.641889i	$0.778151\pi$
$948$		0			0
$949$	10.3033			0.334460
$950$		0			0
$951$	−7.30229			−0.236793
$952$		0			0
$953$			46.4590i			1.50495i	0.658618	+	0.752477i	$0.271141\pi$
$953$			46.4590i			1.50495i	−0.658618	+	0.752477i	$0.728859\pi$
$954$		0			0
$955$	−0.586792	−	1.42126i	−0.0189881	−	0.0459909i
$956$		0			0
$957$		−	5.16424i		−	0.166936i
$958$		0			0
$959$	−43.4796			−1.40403
$960$		0			0
$961$	−7.83070			−0.252603
$962$		0			0
$963$		−	7.12227i		−	0.229512i
$964$		0			0
$965$	17.4881	−	7.22026i	0.562961	−	0.232428i
$966$		0			0
$967$		−	32.3580i		−	1.04056i	−0.853995	−	0.520282i	$-0.825827\pi$
$967$		−	32.3580i		−	1.04056i	0.853995	−	0.520282i	$-0.174173\pi$
$968$		0			0
$969$	3.03121			0.0973765
$970$		0			0
$971$	−14.2713			−0.457989			−0.228994	−	0.973428i	$-0.573544\pi$
$971$	−14.2713			−0.457989			−0.228994	+	0.973428i	$0.573544\pi$
$972$		0			0
$973$		−	0.250557i		−	0.00803249i
$974$		0			0
$975$	−17.7703	+	17.6888i	−0.569106	+	0.566495i
$976$		0			0
$977$		−	18.0347i		−	0.576982i	−0.957483	−	0.288491i	$-0.906847\pi$
$977$		−	18.0347i		−	0.576982i	0.957483	−	0.288491i	$-0.0931535\pi$
$978$		0			0
$979$	−6.49062			−0.207441
$980$		0			0
$981$	8.43258			0.269232
$982$		0			0
$983$			3.49474i			0.111465i	0.998446	+	0.0557325i	$0.0177494\pi$
$983$			3.49474i			0.111465i	−0.998446	+	0.0557325i	$0.982251\pi$
$984$		0			0
$985$	−35.9173	+	14.8291i	−1.14442	+	0.472494i
$986$		0			0
$987$			17.3548i			0.552409i
$988$		0			0
$989$	25.7430			0.818579
$990$		0			0
$991$	−4.53276			−0.143988			−0.0719939	−	0.997405i	$-0.522936\pi$
$991$	−4.53276			−0.143988			−0.0719939	+	0.997405i	$0.522936\pi$
$992$		0			0
$993$			11.7715i			0.373557i
$994$		0			0
$995$	8.51867	+	20.6329i	0.270060	+	0.654108i
$996$		0			0
$997$		−	27.2215i		−	0.862113i	−0.902325	−	0.431057i	$-0.858141\pi$
$997$		−	27.2215i		−	0.862113i	0.902325	−	0.431057i	$-0.141859\pi$
$998$		0			0
$999$	2.01556			0.0637696

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4020.2.g.b.1609.8	✓	24
5.4	even	2	inner	4020.2.g.b.1609.20	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4020.2.g.b.1609.8	✓	24	1.1	even	1	trivial
4020.2.g.b.1609.20	yes	24	5.4	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4020.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +(0.853331 + 2.06684i) q^{5} +2.56410i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +(0.853331 + 2.06684i) q^{5} +2.56410i q^{7} -1.00000 q^{9} +0.836073 q^{11} +5.01469i q^{13} +(2.06684 - 0.853331i) q^{15} -1.93744i q^{17} -1.56454 q^{19} +2.56410 q^{21} -2.41957i q^{23} +(-3.54365 + 3.52740i) q^{25} +1.00000i q^{27} +6.17679 q^{29} +4.81345 q^{31} -0.836073i q^{33} +(-5.29959 + 2.18803i) q^{35} -2.01556i q^{37} +5.01469 q^{39} -5.18723 q^{41} +10.6395i q^{43} +(-0.853331 - 2.06684i) q^{45} +6.76838i q^{47} +0.425387 q^{49} -1.93744 q^{51} -6.96774i q^{53} +(0.713447 + 1.72803i) q^{55} +1.56454i q^{57} +10.8471 q^{59} -7.76916 q^{61} -2.56410i q^{63} +(-10.3646 + 4.27919i) q^{65} +1.00000i q^{67} -2.41957 q^{69} -3.48188 q^{71} -2.05463i q^{73} +(3.52740 + 3.54365i) q^{75} +2.14377i q^{77} -8.11303 q^{79} +1.00000 q^{81} +12.3521i q^{83} +(4.00438 - 1.65328i) q^{85} -6.17679i q^{87} -7.76323 q^{89} -12.8582 q^{91} -4.81345i q^{93} +(-1.33507 - 3.23366i) q^{95} +2.67618i q^{97} -0.836073 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 4 q^{5} - 24 q^{9}+O(q^{10}) \) 24 * q - 4 * q^5 - 24 * q^9	\( 24 q - 4 q^{5} - 24 q^{9} - 8 q^{11} - 2 q^{15} + 16 q^{19} - 20 q^{21} + 10 q^{25} + 36 q^{29} - 2 q^{35} + 4 q^{39} - 24 q^{41} + 4 q^{45} - 4 q^{51} - 4 q^{55} + 24 q^{59} - 4 q^{61} - 20 q^{65} - 4 q^{69} + 20 q^{71} - 12 q^{75} - 28 q^{79} + 24 q^{81} - 16 q^{85} + 48 q^{89} - 20 q^{91} - 4 q^{95} + 8 q^{99}+O(q^{100}) \) 24 * q - 4 * q^5 - 24 * q^9 - 8 * q^11 - 2 * q^15 + 16 * q^19 - 20 * q^21 + 10 * q^25 + 36 * q^29 - 2 * q^35 + 4 * q^39 - 24 * q^41 + 4 * q^45 - 4 * q^51 - 4 * q^55 + 24 * q^59 - 4 * q^61 - 20 * q^65 - 4 * q^69 + 20 * q^71 - 12 * q^75 - 28 * q^79 + 24 * q^81 - 16 * q^85 + 48 * q^89 - 20 * q^91 - 4 * q^95 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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