LMFDB - Embedded newform 4140.2.s.b.737.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4140,2,Mod(737,4140)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4140, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 2, 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("4140.737");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4140.s (of order $4$, degree $2$, 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:	$33.0580664368$
Analytic rank:	$0$
Dimension:	$44$
Relative dimension:	$22$ over $\Q(i)$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			737.1
Character	$\chi$	$=$	4140.737
Dual form			4140.2.s.b.2393.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.01776 - 0.963656i) q^{5} +(2.04752 + 2.04752i) q^{7} +O(q^{10})$	$q+(-2.01776 - 0.963656i) q^{5} +(2.04752 + 2.04752i) q^{7} -4.66935i q^{11} +(-0.347567 + 0.347567i) q^{13} +(-1.10220 + 1.10220i) q^{17} +4.66076i q^{19} +(-0.707107 - 0.707107i) q^{23} +(3.14273 + 3.88886i) q^{25} +4.98353 q^{29} +1.77811 q^{31} +(-2.15830 - 6.10450i) q^{35} +(-1.84327 - 1.84327i) q^{37} -0.953645i q^{41} +(-2.26997 + 2.26997i) q^{43} +(-2.18531 + 2.18531i) q^{47} +1.38464i q^{49} +(-2.24951 - 2.24951i) q^{53} +(-4.49965 + 9.42165i) q^{55} +13.4900 q^{59} +3.51163 q^{61} +(1.03624 - 0.366372i) q^{65} +(-0.135838 - 0.135838i) q^{67} +4.66352i q^{71} +(4.35643 - 4.35643i) q^{73} +(9.56057 - 9.56057i) q^{77} +8.72501i q^{79} +(-2.66892 - 2.66892i) q^{83} +(3.28611 - 1.16183i) q^{85} +3.24644 q^{89} -1.42330 q^{91} +(4.49137 - 9.40431i) q^{95} +(-8.18873 - 8.18873i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 44 q - 4 q^{7}+O(q^{10}) $ 44 * q - 4 * q^7	$ 44 q - 4 q^{7} - 4 q^{13} - 24 q^{25} + 32 q^{31} + 40 q^{37} - 8 q^{43} - 24 q^{55} + 64 q^{61} + 12 q^{67} - 84 q^{73} - 104 q^{85} - 48 q^{91} + 44 q^{97}+O(q^{100}) $ 44 * q - 4 * q^7 - 4 * q^13 - 24 * q^25 + 32 * q^31 + 40 * q^37 - 8 * q^43 - 24 * q^55 + 64 * q^61 + 12 * q^67 - 84 * q^73 - 104 * q^85 - 48 * q^91 + 44 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4140\mathbb{Z}\right)^\times$.

$n$	$461$	$1657$	$2071$	$3961$
$\chi(n)$	$-1$	$e\left(\frac{1}{4}\right)$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−2.01776	−	0.963656i	−0.902371	−	0.430960i
$5$	−2.01776	−	0.963656i	−0.902371	−	0.430960i
$6$		0			0
$7$	2.04752	+	2.04752i	0.773888	+	0.773888i	0.978784	−	0.204896i	$-0.0656855\pi$
$7$	2.04752	+	2.04752i	0.773888	+	0.773888i	−0.204896	+	0.978784i	$0.565686\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	4.66935i		−	1.40786i	−0.710268	−	0.703931i	$-0.751426\pi$
$11$		−	4.66935i		−	1.40786i	0.710268	−	0.703931i	$-0.248574\pi$
$12$		0			0
$13$	−0.347567	+	0.347567i	−0.0963977	+	0.0963977i	−0.753661	−	0.657263i	$-0.771714\pi$
$13$	−0.347567	+	0.347567i	−0.0963977	+	0.0963977i	0.657263	+	0.753661i	$0.271714\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.10220	+	1.10220i	−0.267322	+	0.267322i	−0.828020	−	0.560698i	$-0.810533\pi$
$17$	−1.10220	+	1.10220i	−0.267322	+	0.267322i	0.560698	+	0.828020i	$0.310533\pi$
$18$		0			0
$19$			4.66076i			1.06925i	0.845089	+	0.534626i	$0.179547\pi$
$19$			4.66076i			1.06925i	−0.845089	+	0.534626i	$0.820453\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−0.707107	−	0.707107i	−0.147442	−	0.147442i
$23$	−0.707107	−	0.707107i	−0.147442	−	0.147442i
$24$		0			0
$25$	3.14273	+	3.88886i	0.628547	+	0.777772i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	4.98353			0.925419			0.462709	−	0.886510i	$-0.346877\pi$
$29$	4.98353			0.925419			0.462709	+	0.886510i	$0.346877\pi$
$30$		0			0
$31$	1.77811			0.319358			0.159679	−	0.987169i	$-0.448954\pi$
$31$	1.77811			0.319358			0.159679	+	0.987169i	$0.448954\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−2.15830	−	6.10450i	−0.364819	−	1.03185i
$36$		0			0
$37$	−1.84327	−	1.84327i	−0.303031	−	0.303031i	0.539168	−	0.842199i	$-0.318739\pi$
$37$	−1.84327	−	1.84327i	−0.303031	−	0.303031i	−0.842199	+	0.539168i	$0.818739\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	0.953645i		−	0.148934i	−0.997223	−	0.0744672i	$-0.976274\pi$
$41$		−	0.953645i		−	0.148934i	0.997223	−	0.0744672i	$-0.0237256\pi$
$42$		0			0
$43$	−2.26997	+	2.26997i	−0.346167	+	0.346167i	−0.858680	−	0.512513i	$-0.828715\pi$
$43$	−2.26997	+	2.26997i	−0.346167	+	0.346167i	0.512513	+	0.858680i	$0.328715\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−2.18531	+	2.18531i	−0.318760	+	0.318760i	−0.848291	−	0.529531i	$-0.822368\pi$
$47$	−2.18531	+	2.18531i	−0.318760	+	0.318760i	0.529531	+	0.848291i	$0.322368\pi$
$48$		0			0
$49$			1.38464i			0.197806i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−2.24951	−	2.24951i	−0.308993	−	0.308993i	0.535526	−	0.844519i	$-0.320114\pi$
$53$	−2.24951	−	2.24951i	−0.308993	−	0.308993i	−0.844519	+	0.535526i	$0.820114\pi$
$54$		0			0
$55$	−4.49965	+	9.42165i	−0.606733	+	1.27041i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	13.4900			1.75625			0.878123	−	0.478436i	$-0.158796\pi$
$59$	13.4900			1.75625			0.878123	+	0.478436i	$0.158796\pi$
$60$		0			0
$61$	3.51163			0.449618			0.224809	−	0.974403i	$-0.427824\pi$
$61$	3.51163			0.449618			0.224809	+	0.974403i	$0.427824\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.03624	−	0.366372i	0.128530	−	0.0454429i
$66$		0			0
$67$	−0.135838	−	0.135838i	−0.0165953	−	0.0165953i	0.698760	−	0.715356i	$-0.253735\pi$
$67$	−0.135838	−	0.135838i	−0.0165953	−	0.0165953i	−0.715356	+	0.698760i	$0.753735\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			4.66352i			0.553458i	0.960948	+	0.276729i	$0.0892504\pi$
$71$			4.66352i			0.553458i	−0.960948	+	0.276729i	$0.910750\pi$
$72$		0			0
$73$	4.35643	−	4.35643i	0.509882	−	0.509882i	−0.404608	−	0.914490i	$-0.632592\pi$
$73$	4.35643	−	4.35643i	0.509882	−	0.509882i	0.914490	+	0.404608i	$0.132592\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	9.56057	−	9.56057i	1.08953	−	1.08953i
$78$		0			0
$79$			8.72501i			0.981640i	0.871261	+	0.490820i	$0.163303\pi$
$79$			8.72501i			0.981640i	−0.871261	+	0.490820i	$0.836697\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−2.66892	−	2.66892i	−0.292952	−	0.292952i	0.545293	−	0.838245i	$-0.316418\pi$
$83$	−2.66892	−	2.66892i	−0.292952	−	0.292952i	−0.838245	+	0.545293i	$0.816418\pi$
$84$		0			0
$85$	3.28611	−	1.16183i	0.356429	−	0.126019i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	3.24644			0.344122			0.172061	−	0.985086i	$-0.444957\pi$
$89$	3.24644			0.344122			0.172061	+	0.985086i	$0.444957\pi$
$90$		0			0
$91$	−1.42330			−0.149202
$92$		0			0
$93$		0			0
$94$		0			0
$95$	4.49137	−	9.40431i	0.460805	−	0.964862i
$96$		0			0
$97$	−8.18873	−	8.18873i	−0.831439	−	0.831439i	0.156275	−	0.987714i	$-0.450052\pi$
$97$	−8.18873	−	8.18873i	−0.831439	−	0.831439i	−0.987714	+	0.156275i	$0.950052\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			0.134768i			0.0134099i	0.999978	+	0.00670495i	$0.00213427\pi$
$101$			0.134768i			0.0134099i	−0.999978	+	0.00670495i	$0.997866\pi$
$102$		0			0
$103$	12.0357	−	12.0357i	1.18591	−	1.18591i	0.207726	−	0.978187i	$-0.433394\pi$
$103$	12.0357	−	12.0357i	1.18591	−	1.18591i	0.978187	−	0.207726i	$-0.0666061\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	8.91360	−	8.91360i	0.861710	−	0.861710i	−0.129827	−	0.991537i	$-0.541442\pi$
$107$	8.91360	−	8.91360i	0.861710	−	0.861710i	0.991537	+	0.129827i	$0.0414421\pi$
$108$		0			0
$109$		−	3.45488i		−	0.330917i	−0.986217	−	0.165459i	$-0.947090\pi$
$109$		−	3.45488i		−	0.330917i	0.986217	−	0.165459i	$-0.0529104\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	7.45533	+	7.45533i	0.701338	+	0.701338i	0.964698	−	0.263359i	$-0.0848305\pi$
$113$	7.45533	+	7.45533i	0.701338	+	0.701338i	−0.263359	+	0.964698i	$0.584830\pi$
$114$		0			0
$115$	0.745366	+	2.10818i	0.0695058	+	0.196589i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−4.51353			−0.413755
$120$		0			0
$121$	−10.8029			−0.982078
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−2.59377	−	10.8753i	−0.231994	−	0.972717i
$126$		0			0
$127$	11.3449	+	11.3449i	1.00670	+	1.00670i	0.999977	+	0.00671833i	$0.00213853\pi$
$127$	11.3449	+	11.3449i	1.00670	+	1.00670i	0.00671833	+	0.999977i	$0.497861\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			11.0554i			0.965914i	0.875644	+	0.482957i	$0.160437\pi$
$131$			11.0554i			0.965914i	−0.875644	+	0.482957i	$0.839563\pi$
$132$		0			0
$133$	−9.54298	+	9.54298i	−0.827481	+	0.827481i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−5.68920	+	5.68920i	−0.486061	+	0.486061i	−0.907061	−	0.421000i	$-0.861679\pi$
$137$	−5.68920	+	5.68920i	−0.486061	+	0.486061i	0.421000	+	0.907061i	$0.361679\pi$
$138$		0			0
$139$		−	5.37041i		−	0.455512i	−0.973718	−	0.227756i	$-0.926861\pi$
$139$		−	5.37041i		−	0.455512i	0.973718	−	0.227756i	$-0.0731389\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	1.62291	+	1.62291i	0.135715	+	0.135715i
$144$		0			0
$145$	−10.0556	−	4.80241i	−0.835071	−	0.398819i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	12.6918			1.03976			0.519878	−	0.854241i	$-0.325977\pi$
$149$	12.6918			1.03976			0.519878	+	0.854241i	$0.325977\pi$
$150$		0			0
$151$	13.2872			1.08130			0.540650	−	0.841247i	$-0.318178\pi$
$151$	13.2872			1.08130			0.540650	+	0.841247i	$0.318178\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−3.58781	−	1.71349i	−0.288180	−	0.137631i
$156$		0			0
$157$	−2.74774	−	2.74774i	−0.219294	−	0.219294i	0.588907	−	0.808201i	$-0.299558\pi$
$157$	−2.74774	−	2.74774i	−0.219294	−	0.219294i	−0.808201	+	0.588907i	$0.799558\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	2.89562i		−	0.228207i
$162$		0			0
$163$	0.689963	−	0.689963i	0.0540421	−	0.0540421i	−0.679569	−	0.733611i	$-0.737833\pi$
$163$	0.689963	−	0.689963i	0.0540421	−	0.0540421i	0.733611	+	0.679569i	$0.237833\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1.92285	−	1.92285i	0.148795	−	0.148795i	−0.628785	−	0.777579i	$-0.716447\pi$
$167$	1.92285	−	1.92285i	0.148795	−	0.148795i	0.777579	+	0.628785i	$0.216447\pi$
$168$		0			0
$169$			12.7584i			0.981415i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	5.76223	+	5.76223i	0.438094	+	0.438094i	0.891370	−	0.453276i	$-0.149745\pi$
$173$	5.76223	+	5.76223i	0.438094	+	0.438094i	−0.453276	+	0.891370i	$0.649745\pi$
$174$		0			0
$175$	−1.52770	+	14.3973i	−0.115483	+	1.08833i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	11.0009			0.822244			0.411122	−	0.911580i	$-0.365137\pi$
$179$	11.0009			0.822244			0.411122	+	0.911580i	$0.365137\pi$
$180$		0			0
$181$	23.6773			1.75992			0.879958	−	0.475051i	$-0.157571\pi$
$181$	23.6773			1.75992			0.879958	+	0.475051i	$0.157571\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	1.94300	+	5.49555i	0.142852	+	0.404041i
$186$		0			0
$187$	5.14655	+	5.14655i	0.376353	+	0.376353i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			17.0606i			1.23446i	0.786782	+	0.617231i	$0.211746\pi$
$191$			17.0606i			1.23446i	−0.786782	+	0.617231i	$0.788254\pi$
$192$		0			0
$193$	1.25321	−	1.25321i	0.0902077	−	0.0902077i	−0.660563	−	0.750771i	$-0.729682\pi$
$193$	1.25321	−	1.25321i	0.0902077	−	0.0902077i	0.750771	+	0.660563i	$0.229682\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	2.64982	−	2.64982i	0.188792	−	0.188792i	−0.606382	−	0.795174i	$-0.707380\pi$
$197$	2.64982	−	2.64982i	0.188792	−	0.188792i	0.795174	+	0.606382i	$0.207380\pi$
$198$		0			0
$199$		−	6.47803i		−	0.459215i	−0.973283	−	0.229608i	$-0.926256\pi$
$199$		−	6.47803i		−	0.459215i	0.973283	−	0.229608i	$-0.0737443\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	10.2039	+	10.2039i	0.716171	+	0.716171i
$204$		0			0
$205$	−0.918986	+	1.92423i	−0.0641848	+	0.134394i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	21.7627			1.50536
$210$		0			0
$211$	21.0269			1.44755			0.723776	−	0.690035i	$-0.242405\pi$
$211$	21.0269			1.44755			0.723776	+	0.690035i	$0.242405\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	6.76773	−	2.39279i	0.461555	−	0.163187i
$216$		0			0
$217$	3.64071	+	3.64071i	0.247147	+	0.247147i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	0.766174i		−	0.0515384i
$222$		0			0
$223$	−1.42860	+	1.42860i	−0.0956660	+	0.0956660i	−0.753320	−	0.657654i	$-0.771549\pi$
$223$	−1.42860	+	1.42860i	−0.0956660	+	0.0956660i	0.657654	+	0.753320i	$0.271549\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	7.74548	−	7.74548i	0.514086	−	0.514086i	−0.401690	−	0.915776i	$-0.631577\pi$
$227$	7.74548	−	7.74548i	0.514086	−	0.514086i	0.915776	+	0.401690i	$0.131577\pi$
$228$		0			0
$229$			1.45523i			0.0961644i	0.998843	+	0.0480822i	$0.0153109\pi$
$229$			1.45523i			0.0961644i	−0.998843	+	0.0480822i	$0.984689\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−19.4063	−	19.4063i	−1.27135	−	1.27135i	−0.945382	−	0.325964i	$-0.894311\pi$
$233$	−19.4063	−	19.4063i	−1.27135	−	1.27135i	−0.325964	−	0.945382i	$-0.605689\pi$
$234$		0			0
$235$	6.51531	−	2.30355i	0.425012	−	0.150267i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−19.7942			−1.28038			−0.640189	−	0.768218i	$-0.721144\pi$
$239$	−19.7942			−1.28038			−0.640189	+	0.768218i	$0.721144\pi$
$240$		0			0
$241$	17.5696			1.13176			0.565879	−	0.824488i	$-0.308537\pi$
$241$	17.5696			1.13176			0.565879	+	0.824488i	$0.308537\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	1.33432	−	2.79387i	0.0852463	−	0.178494i
$246$		0			0
$247$	−1.61993	−	1.61993i	−0.103073	−	0.103073i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	23.1304i		−	1.45998i	−0.683460	−	0.729988i	$-0.739526\pi$
$251$		−	23.1304i		−	1.45998i	0.683460	−	0.729988i	$-0.260474\pi$
$252$		0			0
$253$	−3.30173	+	3.30173i	−0.207578	+	0.207578i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	12.2788	−	12.2788i	0.765931	−	0.765931i	−0.211456	−	0.977387i	$-0.567821\pi$
$257$	12.2788	−	12.2788i	0.765931	−	0.765931i	0.977387	+	0.211456i	$0.0678206\pi$
$258$		0			0
$259$		−	7.54823i		−	0.469024i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−10.4185	−	10.4185i	−0.642431	−	0.642431i	0.308721	−	0.951153i	$-0.400099\pi$
$263$	−10.4185	−	10.4185i	−0.642431	−	0.642431i	−0.951153	+	0.308721i	$0.900099\pi$
$264$		0			0
$265$	2.37122	+	6.70672i	0.145663	+	0.411990i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	14.6377			0.892473			0.446237	−	0.894915i	$-0.352764\pi$
$269$	14.6377			0.892473			0.446237	+	0.894915i	$0.352764\pi$
$270$		0			0
$271$	9.68766			0.588483			0.294242	−	0.955731i	$-0.404933\pi$
$271$	9.68766			0.588483			0.294242	+	0.955731i	$0.404933\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	18.1585	−	14.6745i	1.09500	−	0.884908i
$276$		0			0
$277$	5.76219	+	5.76219i	0.346216	+	0.346216i	0.858698	−	0.512482i	$-0.171274\pi$
$277$	5.76219	+	5.76219i	0.346216	+	0.346216i	−0.512482	+	0.858698i	$0.671274\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			21.0107i			1.25339i	0.779263	+	0.626697i	$0.215594\pi$
$281$			21.0107i			1.25339i	−0.779263	+	0.626697i	$0.784406\pi$
$282$		0			0
$283$	3.48458	−	3.48458i	0.207137	−	0.207137i	−0.595912	−	0.803049i	$-0.703209\pi$
$283$	3.48458	−	3.48458i	0.207137	−	0.207137i	0.803049	+	0.595912i	$0.203209\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	1.95260	−	1.95260i	0.115259	−	0.115259i
$288$		0			0
$289$			14.5703i			0.857078i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	7.06392	+	7.06392i	0.412679	+	0.412679i	0.882671	−	0.469992i	$-0.155743\pi$
$293$	7.06392	+	7.06392i	0.412679	+	0.412679i	−0.469992	+	0.882671i	$0.655743\pi$
$294$		0			0
$295$	−27.2196	−	12.9997i	−1.58478	−	0.756872i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	0.491534			0.0284261
$300$		0			0
$301$	−9.29560			−0.535789
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−7.08563	−	3.38400i	−0.405722	−	0.193767i
$306$		0			0
$307$	14.8176	+	14.8176i	0.845686	+	0.845686i	0.989591	−	0.143905i	$-0.0459660\pi$
$307$	14.8176	+	14.8176i	0.845686	+	0.845686i	−0.143905	+	0.989591i	$0.545966\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	24.5727i		−	1.39339i	−0.717368	−	0.696694i	$-0.754654\pi$
$311$		−	24.5727i		−	1.39339i	0.717368	−	0.696694i	$-0.245346\pi$
$312$		0			0
$313$	−16.9696	+	16.9696i	−0.959176	+	0.959176i	−0.999199	−	0.0400229i	$-0.987257\pi$
$313$	−16.9696	+	16.9696i	−0.959176	+	0.959176i	0.0400229	+	0.999199i	$0.487257\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	16.9976	−	16.9976i	0.954682	−	0.954682i	−0.0443351	−	0.999017i	$-0.514117\pi$
$317$	16.9976	−	16.9976i	0.954682	−	0.954682i	0.999017	+	0.0443351i	$0.0141169\pi$
$318$		0			0
$319$		−	23.2699i		−	1.30286i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−5.13708	−	5.13708i	−0.285835	−	0.285835i
$324$		0			0
$325$	−2.44395	−	0.259328i	−0.135566	−	0.0143849i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−8.94890			−0.493369
$330$		0			0
$331$	−5.58897			−0.307198			−0.153599	−	0.988133i	$-0.549086\pi$
$331$	−5.58897			−0.307198			−0.153599	+	0.988133i	$0.549086\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	0.143188	+	0.404991i	0.00782321	+	0.0221270i
$336$		0			0
$337$	10.3348	+	10.3348i	0.562974	+	0.562974i	0.930151	−	0.367177i	$-0.119676\pi$
$337$	10.3348	+	10.3348i	0.562974	+	0.562974i	−0.367177	+	0.930151i	$0.619676\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	8.30263i		−	0.449612i
$342$		0			0
$343$	11.4975	−	11.4975i	0.620809	−	0.620809i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−10.3846	+	10.3846i	−0.557473	+	0.557473i	−0.928587	−	0.371114i	$-0.878976\pi$
$347$	−10.3846	+	10.3846i	−0.557473	+	0.557473i	0.371114	+	0.928587i	$0.378976\pi$
$348$		0			0
$349$		−	7.33445i		−	0.392604i	−0.980543	−	0.196302i	$-0.937107\pi$
$349$		−	7.33445i		−	0.392604i	0.980543	−	0.196302i	$-0.0628933\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−7.09829	−	7.09829i	−0.377804	−	0.377804i	0.492505	−	0.870309i	$-0.336081\pi$
$353$	−7.09829	−	7.09829i	−0.377804	−	0.377804i	−0.870309	+	0.492505i	$0.836081\pi$
$354$		0			0
$355$	4.49403	−	9.40987i	0.238518	−	0.499424i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−0.403320			−0.0212864			−0.0106432	−	0.999943i	$-0.503388\pi$
$359$	−0.403320			−0.0212864			−0.0106432	+	0.999943i	$0.503388\pi$
$360$		0			0
$361$	−2.72269			−0.143299
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−12.9884	+	4.59215i	−0.679841	+	0.240364i
$366$		0			0
$367$	23.0609	+	23.0609i	1.20377	+	1.20377i	0.973011	+	0.230759i	$0.0741208\pi$
$367$	23.0609	+	23.0609i	1.20377	+	1.20377i	0.230759	+	0.973011i	$0.425879\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	9.21179i		−	0.478253i
$372$		0			0
$373$	−19.4933	+	19.4933i	−1.00932	+	1.00932i	−0.00936843	+	0.999956i	$0.502982\pi$
$373$	−19.4933	+	19.4933i	−1.00932	+	1.00932i	−0.999956	+	0.00936843i	$0.997018\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−1.73211	+	1.73211i	−0.0892082	+	0.0892082i
$378$		0			0
$379$		−	26.8546i		−	1.37943i	−0.724083	−	0.689713i	$-0.757737\pi$
$379$		−	26.8546i		−	1.37943i	0.724083	−	0.689713i	$-0.242263\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	1.38629	+	1.38629i	0.0708363	+	0.0708363i	0.741637	−	0.670801i	$-0.234050\pi$
$383$	1.38629	+	1.38629i	0.0708363	+	0.0708363i	−0.670801	+	0.741637i	$0.734050\pi$
$384$		0			0
$385$	−28.5041	+	10.0779i	−1.45270	+	0.513616i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	21.3529			1.08263			0.541317	−	0.840819i	$-0.317926\pi$
$389$	21.3529			1.08263			0.541317	+	0.840819i	$0.317926\pi$
$390$		0			0
$391$	1.55874			0.0788290
$392$		0			0
$393$		0			0
$394$		0			0
$395$	8.40791	−	17.6050i	0.423048	−	0.885804i
$396$		0			0
$397$	−2.97782	−	2.97782i	−0.149453	−	0.149453i	0.628421	−	0.777874i	$-0.283702\pi$
$397$	−2.97782	−	2.97782i	−0.149453	−	0.149453i	−0.777874	+	0.628421i	$0.783702\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	20.8540i		−	1.04140i	−0.853740	−	0.520699i	$-0.825671\pi$
$401$		−	20.8540i		−	1.04140i	0.853740	−	0.520699i	$-0.174329\pi$
$402$		0			0
$403$	−0.618012	+	0.618012i	−0.0307854	+	0.0307854i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−8.60686	+	8.60686i	−0.426626	+	0.426626i
$408$		0			0
$409$		−	9.09630i		−	0.449783i	−0.974384	−	0.224892i	$-0.927797\pi$
$409$		−	9.09630i		−	0.449783i	0.974384	−	0.224892i	$-0.0722028\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	27.6209	+	27.6209i	1.35914	+	1.35914i
$414$		0			0
$415$	2.81333	+	7.95717i	0.138101	+	0.390602i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−14.1032			−0.688985			−0.344493	−	0.938789i	$-0.611949\pi$
$419$	−14.1032			−0.688985			−0.344493	+	0.938789i	$0.611949\pi$
$420$		0			0
$421$	7.58693			0.369765			0.184882	−	0.982761i	$-0.440810\pi$
$421$	7.58693			0.369765			0.184882	+	0.982761i	$0.440810\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−7.75020	−	0.822376i	−0.375940	−	0.0398911i
$426$		0			0
$427$	7.19011	+	7.19011i	0.347954	+	0.347954i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			11.2708i			0.542897i	0.962453	+	0.271448i	$0.0875026\pi$
$431$			11.2708i			0.542897i	−0.962453	+	0.271448i	$0.912497\pi$
$432$		0			0
$433$	−18.2651	+	18.2651i	−0.877766	+	0.877766i	−0.993303	−	0.115537i	$-0.963141\pi$
$433$	−18.2651	+	18.2651i	−0.877766	+	0.877766i	0.115537	+	0.993303i	$0.463141\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	3.29566	−	3.29566i	0.157653	−	0.157653i
$438$		0			0
$439$		−	9.98401i		−	0.476511i	−0.971203	−	0.238256i	$-0.923424\pi$
$439$		−	9.98401i		−	0.476511i	0.971203	−	0.238256i	$-0.0765756\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−22.1973	−	22.1973i	−1.05463	−	1.05463i	−0.998419	−	0.0562070i	$-0.982099\pi$
$443$	−22.1973	−	22.1973i	−1.05463	−	1.05463i	−0.0562070	−	0.998419i	$-0.517901\pi$
$444$		0			0
$445$	−6.55054	−	3.12845i	−0.310526	−	0.148303i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	7.79350			0.367798			0.183899	−	0.982945i	$-0.441128\pi$
$449$	7.79350			0.367798			0.183899	+	0.982945i	$0.441128\pi$
$450$		0			0
$451$	−4.45291			−0.209679
$452$		0			0
$453$		0			0
$454$		0			0
$455$	2.87187	+	1.37157i	0.134636	+	0.0643001i
$456$		0			0
$457$	−13.7092	−	13.7092i	−0.641290	−	0.641290i	0.309583	−	0.950873i	$-0.399811\pi$
$457$	−13.7092	−	13.7092i	−0.641290	−	0.641290i	−0.950873	+	0.309583i	$0.899811\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			18.8657i			0.878664i	0.898325	+	0.439332i	$0.144785\pi$
$461$			18.8657i			0.878664i	−0.898325	+	0.439332i	$0.855215\pi$
$462$		0			0
$463$	−12.6818	+	12.6818i	−0.589372	+	0.589372i	−0.937461	−	0.348089i	$-0.886831\pi$
$463$	−12.6818	+	12.6818i	−0.589372	+	0.589372i	0.348089	+	0.937461i	$0.386831\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−6.32146	+	6.32146i	−0.292522	+	0.292522i	−0.838076	−	0.545554i	$-0.816319\pi$
$467$	−6.32146	+	6.32146i	−0.292522	+	0.292522i	0.545554	+	0.838076i	$0.316319\pi$
$468$		0			0
$469$		−	0.556262i		−	0.0256858i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	10.5993	+	10.5993i	0.487356	+	0.487356i
$474$		0			0
$475$	−18.1250	+	14.6475i	−0.831634	+	0.672075i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−19.8847			−0.908556			−0.454278	−	0.890860i	$-0.650103\pi$
$479$	−19.8847			−0.908556			−0.454278	+	0.890860i	$0.650103\pi$
$480$		0			0
$481$	1.28132			0.0584230
$482$		0			0
$483$		0			0
$484$		0			0
$485$	8.63179	+	24.4140i	0.391950	+	1.10858i
$486$		0			0
$487$	10.7323	+	10.7323i	0.486326	+	0.486326i	0.907145	−	0.420819i	$-0.138257\pi$
$487$	10.7323	+	10.7323i	0.486326	+	0.486326i	−0.420819	+	0.907145i	$0.638257\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	26.3441i		−	1.18889i	−0.804135	−	0.594446i	$-0.797371\pi$
$491$		−	26.3441i		−	1.18889i	0.804135	−	0.594446i	$-0.202629\pi$
$492$		0			0
$493$	−5.49283	+	5.49283i	−0.247385	+	0.247385i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−9.54863	+	9.54863i	−0.428314	+	0.428314i
$498$		0			0
$499$		−	15.2995i		−	0.684898i	−0.939536	−	0.342449i	$-0.888744\pi$
$499$		−	15.2995i		−	0.684898i	0.939536	−	0.342449i	$-0.111256\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	5.51060	+	5.51060i	0.245706	+	0.245706i	0.819206	−	0.573500i	$-0.194415\pi$
$503$	5.51060	+	5.51060i	0.245706	+	0.245706i	−0.573500	+	0.819206i	$0.694415\pi$
$504$		0			0
$505$	0.129870	−	0.271929i	0.00577913	−	0.0121007i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	1.53836			0.0681867			0.0340934	−	0.999419i	$-0.489146\pi$
$509$	1.53836			0.0681867			0.0340934	+	0.999419i	$0.489146\pi$
$510$		0			0
$511$	17.8397			0.789183
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−35.8835	+	12.6869i	−1.58121	+	0.559052i
$516$		0			0
$517$	10.2040	+	10.2040i	0.448770	+	0.448770i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			18.2143i			0.797984i	0.916954	+	0.398992i	$0.130640\pi$
$521$			18.2143i			0.797984i	−0.916954	+	0.398992i	$0.869360\pi$
$522$		0			0
$523$	25.7937	−	25.7937i	1.12788	−	1.12788i	0.137357	−	0.990522i	$-0.456139\pi$
$523$	25.7937	−	25.7937i	1.12788	−	1.12788i	0.990522	−	0.137357i	$-0.0438607\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−1.95983	+	1.95983i	−0.0853714	+	0.0853714i
$528$		0			0
$529$			1.00000i			0.0434783i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	0.331455	+	0.331455i	0.0143569	+	0.0143569i
$534$		0			0
$535$	−26.5752	+	9.39588i	−1.14894	+	0.406219i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	6.46537			0.278483
$540$		0			0
$541$	18.0315			0.775236			0.387618	−	0.921820i	$-0.373298\pi$
$541$	18.0315			0.775236			0.387618	+	0.921820i	$0.373298\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−3.32931	+	6.97112i	−0.142612	+	0.298610i
$546$		0			0
$547$	13.0149	+	13.0149i	0.556477	+	0.556477i	0.928303	−	0.371825i	$-0.121268\pi$
$547$	13.0149	+	13.0149i	0.556477	+	0.556477i	−0.371825	+	0.928303i	$0.621268\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			23.2271i			0.989506i
$552$		0			0
$553$	−17.8646	+	17.8646i	−0.759680	+	0.759680i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−31.8958	+	31.8958i	−1.35147	+	1.35147i	−0.467450	+	0.884020i	$0.654827\pi$
$557$	−31.8958	+	31.8958i	−1.35147	+	1.35147i	−0.884020	+	0.467450i	$0.845173\pi$
$558$		0			0
$559$		−	1.57793i		−	0.0667394i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−17.7334	−	17.7334i	−0.747373	−	0.747373i	0.226612	−	0.973985i	$-0.427235\pi$
$563$	−17.7334	−	17.7334i	−0.747373	−	0.747373i	−0.973985	+	0.226612i	$0.927235\pi$
$564$		0			0
$565$	−7.85871	−	22.2275i	−0.330619	−	0.935116i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−19.2449			−0.806787			−0.403393	−	0.915027i	$-0.632169\pi$
$569$	−19.2449			−0.806787			−0.403393	+	0.915027i	$0.632169\pi$
$570$		0			0
$571$	−33.5166			−1.40263			−0.701313	−	0.712854i	$-0.747402\pi$
$571$	−33.5166			−1.40263			−0.701313	+	0.712854i	$0.747402\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	0.527590	−	4.97209i	0.0220020	−	0.207350i
$576$		0			0
$577$	25.3770	+	25.3770i	1.05646	+	1.05646i	0.998308	+	0.0581507i	$0.0185204\pi$
$577$	25.3770	+	25.3770i	1.05646	+	1.05646i	0.0581507	+	0.998308i	$0.481480\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	10.9293i		−	0.453424i
$582$		0			0
$583$	−10.5037	+	10.5037i	−0.435020	+	0.435020i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−26.8461	+	26.8461i	−1.10806	+	1.10806i	−0.114652	+	0.993406i	$0.536575\pi$
$587$	−26.8461	+	26.8461i	−1.10806	+	1.10806i	−0.993406	+	0.114652i	$0.963425\pi$
$588$		0			0
$589$			8.28735i			0.341474i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−5.10222	−	5.10222i	−0.209523	−	0.209523i	0.594542	−	0.804065i	$-0.297333\pi$
$593$	−5.10222	−	5.10222i	−0.209523	−	0.209523i	−0.804065	+	0.594542i	$0.797333\pi$
$594$		0			0
$595$	9.10723	+	4.34949i	0.373360	+	0.178312i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	4.15959			0.169956			0.0849780	−	0.996383i	$-0.472918\pi$
$599$	4.15959			0.169956			0.0849780	+	0.996383i	$0.472918\pi$
$600$		0			0
$601$	−3.93790			−0.160630			−0.0803152	−	0.996770i	$-0.525593\pi$
$601$	−3.93790			−0.160630			−0.0803152	+	0.996770i	$0.525593\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	21.7976	+	10.4102i	0.886198	+	0.423236i
$606$		0			0
$607$	−9.65515	−	9.65515i	−0.391890	−	0.391890i	0.483470	−	0.875361i	$-0.339376\pi$
$607$	−9.65515	−	9.65515i	−0.391890	−	0.391890i	−0.875361	+	0.483470i	$0.839376\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	1.51908i		−	0.0614554i
$612$		0			0
$613$	−4.52738	+	4.52738i	−0.182859	+	0.182859i	−0.792601	−	0.609741i	$-0.791273\pi$
$613$	−4.52738	+	4.52738i	−0.182859	+	0.182859i	0.609741	+	0.792601i	$0.291273\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−34.5332	+	34.5332i	−1.39025	+	1.39025i	−0.565519	+	0.824735i	$0.691324\pi$
$617$	−34.5332	+	34.5332i	−1.39025	+	1.39025i	−0.824735	+	0.565519i	$0.808676\pi$
$618$		0			0
$619$		−	16.7040i		−	0.671392i	−0.941970	−	0.335696i	$-0.891029\pi$
$619$		−	16.7040i		−	0.671392i	0.941970	−	0.335696i	$-0.108971\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	6.64713	+	6.64713i	0.266312	+	0.266312i
$624$		0			0
$625$	−5.24644	+	24.4433i	−0.209858	+	0.977732i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	4.06328			0.162014
$630$		0			0
$631$	−33.8233			−1.34648			−0.673242	−	0.739423i	$-0.735099\pi$
$631$	−33.8233			−1.34648			−0.673242	+	0.739423i	$0.735099\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−11.9587	−	33.8239i	−0.474567	−	1.34226i
$636$		0			0
$637$	−0.481255	−	0.481255i	−0.0190680	−	0.0190680i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			39.5728i			1.56303i	0.623886	+	0.781515i	$0.285553\pi$
$641$			39.5728i			1.56303i	−0.623886	+	0.781515i	$0.714447\pi$
$642$		0			0
$643$	18.4780	−	18.4780i	0.728701	−	0.728701i	−0.241660	−	0.970361i	$-0.577692\pi$
$643$	18.4780	−	18.4780i	0.728701	−	0.728701i	0.970361	+	0.241660i	$0.0776920\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−10.2416	+	10.2416i	−0.402638	+	0.402638i	−0.879162	−	0.476523i	$-0.841897\pi$
$647$	−10.2416	+	10.2416i	−0.402638	+	0.402638i	0.476523	+	0.879162i	$0.341897\pi$
$648$		0			0
$649$		−	62.9895i		−	2.47255i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−3.88722	−	3.88722i	−0.152119	−	0.152119i	0.626945	−	0.779064i	$-0.284305\pi$
$653$	−3.88722	−	3.88722i	−0.152119	−	0.152119i	−0.779064	+	0.626945i	$0.784305\pi$
$654$		0			0
$655$	10.6536	−	22.3072i	0.416270	−	0.871613i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	36.9716			1.44021			0.720105	−	0.693865i	$-0.244094\pi$
$659$	36.9716			1.44021			0.720105	+	0.693865i	$0.244094\pi$
$660$		0			0
$661$	−7.95719			−0.309499			−0.154749	−	0.987954i	$-0.549457\pi$
$661$	−7.95719			−0.309499			−0.154749	+	0.987954i	$0.549457\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	28.4516	−	10.0593i	1.10331	−	0.390084i
$666$		0			0
$667$	−3.52389	−	3.52389i	−0.136446	−	0.136446i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	16.3970i		−	0.633000i
$672$		0			0
$673$	−27.4019	+	27.4019i	−1.05627	+	1.05627i	−0.0579460	+	0.998320i	$0.518455\pi$
$673$	−27.4019	+	27.4019i	−1.05627	+	1.05627i	−0.998320	+	0.0579460i	$0.981545\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−3.87793	+	3.87793i	−0.149041	+	0.149041i	−0.777690	−	0.628648i	$-0.783608\pi$
$677$	−3.87793	+	3.87793i	−0.149041	+	0.149041i	0.628648	+	0.777690i	$0.283608\pi$
$678$		0			0
$679$		−	33.5331i		−	1.28688i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−18.6263	−	18.6263i	−0.712717	−	0.712717i	0.254385	−	0.967103i	$-0.418127\pi$
$683$	−18.6263	−	18.6263i	−0.712717	−	0.712717i	−0.967103	+	0.254385i	$0.918127\pi$
$684$		0			0
$685$	16.9619	−	5.99703i	0.648080	−	0.229135i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	1.56371			0.0595725
$690$		0			0
$691$	−16.1640			−0.614907			−0.307453	−	0.951563i	$-0.599477\pi$
$691$	−16.1640			−0.614907			−0.307453	+	0.951563i	$0.599477\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−5.17523	+	10.8362i	−0.196308	+	0.411041i
$696$		0			0
$697$	1.05110	+	1.05110i	0.0398134	+	0.0398134i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			25.3770i			0.958477i	0.877685	+	0.479239i	$0.159087\pi$
$701$			25.3770i			0.958477i	−0.877685	+	0.479239i	$0.840913\pi$
$702$		0			0
$703$	8.59102	−	8.59102i	0.324016	−	0.324016i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−0.275939	+	0.275939i	−0.0103778	+	0.0103778i
$708$		0			0
$709$		−	8.65257i		−	0.324954i	−0.986712	−	0.162477i	$-0.948052\pi$
$709$		−	8.65257i		−	0.324954i	0.986712	−	0.162477i	$-0.0519483\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−1.25731	−	1.25731i	−0.0470868	−	0.0470868i
$714$		0			0
$715$	−1.71072	−	4.83858i	−0.0639774	−	0.180953i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	8.68120			0.323754			0.161877	−	0.986811i	$-0.448245\pi$
$719$	8.68120			0.323754			0.161877	+	0.986811i	$0.448245\pi$
$720$		0			0
$721$	49.2866			1.83553
$722$		0			0
$723$		0			0
$724$		0			0
$725$	15.6619	+	19.3803i	0.581669	+	0.719765i
$726$		0			0
$727$	−15.3157	−	15.3157i	−0.568028	−	0.568028i	0.363548	−	0.931576i	$-0.381565\pi$
$727$	−15.3157	−	15.3157i	−0.568028	−	0.568028i	−0.931576	+	0.363548i	$0.881565\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	5.00391i		−	0.185076i
$732$		0			0
$733$	30.8625	−	30.8625i	1.13993	−	1.13993i	0.151472	−	0.988462i	$-0.451599\pi$
$733$	30.8625	−	30.8625i	1.13993	−	1.13993i	0.988462	−	0.151472i	$-0.0484012\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−0.634277	+	0.634277i	−0.0233639	+	0.0233639i
$738$		0			0
$739$		−	11.6605i		−	0.428938i	−0.976731	−	0.214469i	$-0.931198\pi$
$739$		−	11.6605i		−	0.428938i	0.976731	−	0.214469i	$-0.0688021\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−11.6313	−	11.6313i	−0.426711	−	0.426711i	0.460795	−	0.887506i	$-0.347564\pi$
$743$	−11.6313	−	11.6313i	−0.426711	−	0.426711i	−0.887506	+	0.460795i	$0.847564\pi$
$744$		0			0
$745$	−25.6091	−	12.2306i	−0.938246	−	0.448093i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	36.5015			1.33373
$750$		0			0
$751$	−6.96469			−0.254145			−0.127073	−	0.991893i	$-0.540558\pi$
$751$	−6.96469			−0.254145			−0.127073	+	0.991893i	$0.540558\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−26.8105	−	12.8043i	−0.975734	−	0.465997i
$756$		0			0
$757$	7.18426	+	7.18426i	0.261116	+	0.261116i	0.825508	−	0.564391i	$-0.190889\pi$
$757$	7.18426	+	7.18426i	0.261116	+	0.261116i	−0.564391	+	0.825508i	$0.690889\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			4.94432i			0.179231i	0.995976	+	0.0896157i	$0.0285639\pi$
$761$			4.94432i			0.179231i	−0.995976	+	0.0896157i	$0.971436\pi$
$762$		0			0
$763$	7.07392	−	7.07392i	0.256093	−	0.256093i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−4.68867	+	4.68867i	−0.169298	+	0.169298i
$768$		0			0
$769$		−	51.4564i		−	1.85557i	−0.373120	−	0.927783i	$-0.621712\pi$
$769$		−	51.4564i		−	1.85557i	0.373120	−	0.927783i	$-0.378288\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	8.38564	+	8.38564i	0.301611	+	0.301611i	0.841644	−	0.540033i	$-0.181588\pi$
$773$	8.38564	+	8.38564i	0.301611	+	0.301611i	−0.540033	+	0.841644i	$0.681588\pi$
$774$		0			0
$775$	5.58813	+	6.91482i	0.200732	+	0.248388i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	4.44471			0.159248
$780$		0			0
$781$	21.7756			0.779193
$782$		0			0
$783$		0			0
$784$		0			0
$785$	2.89642	+	8.19218i	0.103378	+	0.292391i
$786$		0			0
$787$	−17.9538	−	17.9538i	−0.639983	−	0.639983i	0.310568	−	0.950551i	$-0.399481\pi$
$787$	−17.9538	−	17.9538i	−0.639983	−	0.639983i	−0.950551	+	0.310568i	$0.899481\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			30.5298i			1.08551i
$792$		0			0
$793$	−1.22053	+	1.22053i	−0.0433421	+	0.0433421i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	5.20365	−	5.20365i	0.184323	−	0.184323i	−0.608914	−	0.793236i	$-0.708394\pi$
$797$	5.20365	−	5.20365i	0.184323	−	0.184323i	0.793236	+	0.608914i	$0.208394\pi$
$798$		0			0
$799$		−	4.81728i		−	0.170423i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−20.3417	−	20.3417i	−0.717844	−	0.717844i
$804$		0			0
$805$	−2.79039	+	5.84268i	−0.0983482	+	0.205928i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−25.3517			−0.891317			−0.445659	−	0.895203i	$-0.647030\pi$
$809$	−25.3517			−0.891317			−0.445659	+	0.895203i	$0.647030\pi$
$810$		0			0
$811$	−48.1579			−1.69105			−0.845527	−	0.533933i	$-0.820714\pi$
$811$	−48.1579			−1.69105			−0.845527	+	0.533933i	$0.820714\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−2.05707	+	0.727295i	−0.0720560	+	0.0254760i
$816$		0			0
$817$	−10.5798	−	10.5798i	−0.370140	−	0.370140i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	2.51008i		−	0.0876024i	−0.999040	−	0.0438012i	$-0.986053\pi$
$821$		−	2.51008i		−	0.0876024i	0.999040	−	0.0438012i	$-0.0139468\pi$
$822$		0			0
$823$	−17.8611	+	17.8611i	−0.622597	+	0.622597i	−0.946195	−	0.323597i	$-0.895108\pi$
$823$	−17.8611	+	17.8611i	−0.622597	+	0.622597i	0.323597	+	0.946195i	$0.395108\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−3.43483	+	3.43483i	−0.119441	+	0.119441i	−0.764301	−	0.644860i	$-0.776916\pi$
$827$	−3.43483	+	3.43483i	−0.119441	+	0.119441i	0.644860	+	0.764301i	$0.276916\pi$
$828$		0			0
$829$		−	56.3207i		−	1.95610i	−0.208375	−	0.978049i	$-0.566817\pi$
$829$		−	56.3207i		−	1.95610i	0.208375	−	0.978049i	$-0.433183\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−1.52615	−	1.52615i	−0.0528778	−	0.0528778i
$834$		0			0
$835$	−5.73283	+	2.02689i	−0.198393	+	0.0701435i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−30.0830			−1.03858			−0.519291	−	0.854597i	$-0.673804\pi$
$839$	−30.0830			−1.03858			−0.519291	+	0.854597i	$0.673804\pi$
$840$		0			0
$841$	−4.16440			−0.143600
$842$		0			0
$843$		0			0
$844$		0			0
$845$	12.2947	−	25.7434i	0.422951	−	0.885600i
$846$		0			0
$847$	−22.1190	−	22.1190i	−0.760018	−	0.760018i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			2.60677i			0.0893590i
$852$		0			0
$853$	−25.1144	+	25.1144i	−0.859899	+	0.859899i	−0.991326	−	0.131427i	$-0.958044\pi$
$853$	−25.1144	+	25.1144i	−0.859899	+	0.859899i	0.131427	+	0.991326i	$0.458044\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	14.4269	−	14.4269i	0.492815	−	0.492815i	−0.416377	−	0.909192i	$-0.636700\pi$
$857$	14.4269	−	14.4269i	0.492815	−	0.492815i	0.909192	+	0.416377i	$0.136700\pi$
$858$		0			0
$859$			27.8635i			0.950691i	0.879799	+	0.475345i	$0.157677\pi$
$859$			27.8635i			0.950691i	−0.879799	+	0.475345i	$0.842323\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−7.86037	−	7.86037i	−0.267570	−	0.267570i	0.560550	−	0.828120i	$-0.310590\pi$
$863$	−7.86037	−	7.86037i	−0.267570	−	0.267570i	−0.828120	+	0.560550i	$0.810590\pi$
$864$		0			0
$865$	−6.07401	−	17.1796i	−0.206523	−	0.584125i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	40.7401			1.38201
$870$		0			0
$871$	0.0944258			0.00319950
$872$		0			0
$873$		0			0
$874$		0			0
$875$	16.9566	−	27.5782i	0.573237	−	0.932312i
$876$		0			0
$877$	38.9064	+	38.9064i	1.31377	+	1.31377i	0.918611	+	0.395164i	$0.129312\pi$
$877$	38.9064	+	38.9064i	1.31377	+	1.31377i	0.395164	+	0.918611i	$0.370688\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			47.5594i			1.60232i	0.598452	+	0.801158i	$0.295783\pi$
$881$			47.5594i			1.60232i	−0.598452	+	0.801158i	$0.704217\pi$
$882$		0			0
$883$	38.6165	−	38.6165i	1.29955	−	1.29955i	0.370859	−	0.928689i	$-0.379063\pi$
$883$	38.6165	−	38.6165i	1.29955	−	1.29955i	0.928689	−	0.370859i	$-0.120937\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	7.63544	−	7.63544i	0.256373	−	0.256373i	−0.567204	−	0.823577i	$-0.691975\pi$
$887$	7.63544	−	7.63544i	0.256373	−	0.256373i	0.823577	+	0.567204i	$0.191975\pi$
$888$		0			0
$889$			46.4577i			1.55814i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−10.1852	−	10.1852i	−0.340834	−	0.340834i
$894$		0			0
$895$	−22.1972	−	10.6011i	−0.741970	−	0.354354i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	8.86127			0.295540
$900$		0			0
$901$	4.95880			0.165201
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−47.7751	−	22.8167i	−1.58810	−	0.758454i
$906$		0			0
$907$	35.9442	+	35.9442i	1.19351	+	1.19351i	0.976076	+	0.217430i	$0.0697674\pi$
$907$	35.9442	+	35.9442i	1.19351	+	1.19351i	0.217430	+	0.976076i	$0.430233\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			2.63872i			0.0874248i	0.999044	+	0.0437124i	$0.0139185\pi$
$911$			2.63872i			0.0874248i	−0.999044	+	0.0437124i	$0.986081\pi$
$912$		0			0
$913$	−12.4621	+	12.4621i	−0.412437	+	0.412437i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−22.6361	+	22.6361i	−0.747510	+	0.747510i
$918$		0			0
$919$		−	29.2930i		−	0.966288i	−0.875541	−	0.483144i	$-0.839495\pi$
$919$		−	29.2930i		−	0.966288i	0.875541	−	0.483144i	$-0.160505\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−1.62088	−	1.62088i	−0.0533520	−	0.0533520i
$924$		0			0
$925$	1.37531	−	12.9611i	0.0452198	−	0.426158i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	45.0200			1.47706			0.738529	−	0.674222i	$-0.235521\pi$
$929$	45.0200			1.47706			0.738529	+	0.674222i	$0.235521\pi$
$930$		0			0
$931$	−6.45347			−0.211504
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−5.42501	−	15.3440i	−0.177417	−	0.501803i
$936$		0			0
$937$	−17.6031	−	17.6031i	−0.575069	−	0.575069i	0.358472	−	0.933540i	$-0.383298\pi$
$937$	−17.6031	−	17.6031i	−0.575069	−	0.575069i	−0.933540	+	0.358472i	$0.883298\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			44.8077i			1.46069i	0.683079	+	0.730345i	$0.260641\pi$
$941$			44.8077i			1.46069i	−0.683079	+	0.730345i	$0.739359\pi$
$942$		0			0
$943$	−0.674329	+	0.674329i	−0.0219592	+	0.0219592i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−34.6112	+	34.6112i	−1.12471	+	1.12471i	−0.133691	+	0.991023i	$0.542683\pi$
$947$	−34.6112	+	34.6112i	−1.12471	+	1.12471i	−0.991023	+	0.133691i	$0.957317\pi$
$948$		0			0
$949$			3.02830i			0.0983029i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−5.99667	−	5.99667i	−0.194251	−	0.194251i	0.603279	−	0.797530i	$-0.293861\pi$
$953$	−5.99667	−	5.99667i	−0.194251	−	0.194251i	−0.797530	+	0.603279i	$0.793861\pi$
$954$		0			0
$955$	16.4406	−	34.4242i	0.532004	−	1.11394i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−23.2975			−0.752314
$960$		0			0
$961$	−27.8383			−0.898010
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−3.73633	+	1.32101i	−0.120277	+	0.0425249i
$966$		0			0
$967$	2.11408	+	2.11408i	0.0679842	+	0.0679842i	0.740281	−	0.672297i	$-0.234692\pi$
$967$	2.11408	+	2.11408i	0.0679842	+	0.0679842i	−0.672297	+	0.740281i	$0.734692\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			2.51425i			0.0806860i	0.999186	+	0.0403430i	$0.0128451\pi$
$971$			2.51425i			0.0806860i	−0.999186	+	0.0403430i	$0.987155\pi$
$972$		0			0
$973$	10.9960	−	10.9960i	0.352516	−	0.352516i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−40.8574	+	40.8574i	−1.30714	+	1.30714i	−0.383678	+	0.923467i	$0.625343\pi$
$977$	−40.8574	+	40.8574i	−1.30714	+	1.30714i	−0.923467	+	0.383678i	$0.874657\pi$
$978$		0			0
$979$		−	15.1588i		−	0.484476i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−3.81510	−	3.81510i	−0.121683	−	0.121683i	0.643643	−	0.765326i	$-0.277422\pi$
$983$	−3.81510	−	3.81510i	−0.121683	−	0.121683i	−0.765326	+	0.643643i	$0.777422\pi$
$984$		0			0
$985$	−7.90022	+	2.79319i	−0.251722	+	0.0889985i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	3.21022			0.102079
$990$		0			0
$991$	53.6497			1.70424			0.852119	−	0.523347i	$-0.175317\pi$
$991$	53.6497			1.70424			0.852119	+	0.523347i	$0.175317\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−6.24259	+	13.0711i	−0.197903	+	0.414382i
$996$		0			0
$997$	−16.1478	−	16.1478i	−0.511406	−	0.511406i	0.403551	−	0.914957i	$-0.367776\pi$
$997$	−16.1478	−	16.1478i	−0.511406	−	0.511406i	−0.914957	+	0.403551i	$0.867776\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4140.2.s.b.737.1	✓	44
3.2	odd	2	inner	4140.2.s.b.737.22	yes	44
5.3	odd	4	inner	4140.2.s.b.2393.22	yes	44
15.8	even	4	inner	4140.2.s.b.2393.1	yes	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4140.2.s.b.737.1	✓	44	1.1	even	1	trivial
4140.2.s.b.737.22	yes	44	3.2	odd	2	inner
4140.2.s.b.2393.1	yes	44	15.8	even	4	inner
4140.2.s.b.2393.22	yes	44	5.3	odd	4	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:	\( N \)	\(=\)	\( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4140.s (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.01776 - 0.963656i) q^{5} +(2.04752 + 2.04752i) q^{7} +O(q^{10})\)	\(q+(-2.01776 - 0.963656i) q^{5} +(2.04752 + 2.04752i) q^{7} -4.66935i q^{11} +(-0.347567 + 0.347567i) q^{13} +(-1.10220 + 1.10220i) q^{17} +4.66076i q^{19} +(-0.707107 - 0.707107i) q^{23} +(3.14273 + 3.88886i) q^{25} +4.98353 q^{29} +1.77811 q^{31} +(-2.15830 - 6.10450i) q^{35} +(-1.84327 - 1.84327i) q^{37} -0.953645i q^{41} +(-2.26997 + 2.26997i) q^{43} +(-2.18531 + 2.18531i) q^{47} +1.38464i q^{49} +(-2.24951 - 2.24951i) q^{53} +(-4.49965 + 9.42165i) q^{55} +13.4900 q^{59} +3.51163 q^{61} +(1.03624 - 0.366372i) q^{65} +(-0.135838 - 0.135838i) q^{67} +4.66352i q^{71} +(4.35643 - 4.35643i) q^{73} +(9.56057 - 9.56057i) q^{77} +8.72501i q^{79} +(-2.66892 - 2.66892i) q^{83} +(3.28611 - 1.16183i) q^{85} +3.24644 q^{89} -1.42330 q^{91} +(4.49137 - 9.40431i) q^{95} +(-8.18873 - 8.18873i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 4 q^{7}+O(q^{10}) \) 44 * q - 4 * q^7	\( 44 q - 4 q^{7} - 4 q^{13} - 24 q^{25} + 32 q^{31} + 40 q^{37} - 8 q^{43} - 24 q^{55} + 64 q^{61} + 12 q^{67} - 84 q^{73} - 104 q^{85} - 48 q^{91} + 44 q^{97}+O(q^{100}) \) 44 * q - 4 * q^7 - 4 * q^13 - 24 * q^25 + 32 * q^31 + 40 * q^37 - 8 * q^43 - 24 * q^55 + 64 * q^61 + 12 * q^67 - 84 * q^73 - 104 * q^85 - 48 * q^91 + 44 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more