LMFDB - Embedded newform 4140.2.s.a.737.2

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			737.2
Character	$\chi$	$=$	4140.737
Dual form			4140.2.s.a.2393.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.21141 + 0.331134i) q^{5} +(-1.17089 - 1.17089i) q^{7} +O(q^{10})$	$q+(-2.21141 + 0.331134i) q^{5} +(-1.17089 - 1.17089i) q^{7} +0.236966i q^{11} +(0.337253 - 0.337253i) q^{13} +(0.583846 - 0.583846i) q^{17} +2.36213i q^{19} +(0.707107 + 0.707107i) q^{23} +(4.78070 - 1.46455i) q^{25} +1.54355 q^{29} +2.36020 q^{31} +(2.97704 + 2.20160i) q^{35} +(4.63463 + 4.63463i) q^{37} +3.08636i q^{41} +(5.19030 - 5.19030i) q^{43} +(-3.52866 + 3.52866i) q^{47} -4.25805i q^{49} +(-8.94118 - 8.94118i) q^{53} +(-0.0784674 - 0.524030i) q^{55} -11.6852 q^{59} +1.02874 q^{61} +(-0.634130 + 0.857482i) q^{65} +(-9.64894 - 9.64894i) q^{67} +4.90457i q^{71} +(-4.17536 + 4.17536i) q^{73} +(0.277461 - 0.277461i) q^{77} +1.47706i q^{79} +(6.21585 + 6.21585i) q^{83} +(-1.09779 + 1.48446i) q^{85} -15.7951 q^{89} -0.789771 q^{91} +(-0.782180 - 5.22365i) q^{95} +(-5.69950 - 5.69950i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 44 q - 12 q^{7}+O(q^{10}) $ 44 * q - 12 * q^7	$ 44 q - 12 q^{7} - 4 q^{13} + 24 q^{25} - 48 q^{37} + 8 q^{43} + 40 q^{55} - 96 q^{61} - 44 q^{67} + 76 q^{73} + 72 q^{85} - 48 q^{91} - 20 q^{97}+O(q^{100}) $ 44 * q - 12 * q^7 - 4 * q^13 + 24 * q^25 - 48 * q^37 + 8 * q^43 + 40 * q^55 - 96 * q^61 - 44 * q^67 + 76 * q^73 + 72 * q^85 - 48 * q^91 - 20 * 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
$5$	−2.21141	+	0.331134i	−0.988974	+	0.148087i
$5$	−2.21141	+	0.331134i	−0.988974	+	0.148087i
$6$		0			0
$7$	−1.17089	−	1.17089i	−0.442554	−	0.442554i	0.450316	−	0.892869i	$-0.351311\pi$
$7$	−1.17089	−	1.17089i	−0.442554	−	0.442554i	−0.892869	+	0.450316i	$0.851311\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			0.236966i			0.0714480i	0.999362	+	0.0357240i	$0.0113737\pi$
$11$			0.236966i			0.0714480i	−0.999362	+	0.0357240i	$0.988626\pi$
$12$		0			0
$13$	0.337253	−	0.337253i	0.0935372	−	0.0935372i	−0.658790	−	0.752327i	$-0.728931\pi$
$13$	0.337253	−	0.337253i	0.0935372	−	0.0935372i	0.752327	+	0.658790i	$0.228931\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	0.583846	−	0.583846i	0.141604	−	0.141604i	−0.632751	−	0.774355i	$-0.718075\pi$
$17$	0.583846	−	0.583846i	0.141604	−	0.141604i	0.774355	+	0.632751i	$0.218075\pi$
$18$		0			0
$19$			2.36213i			0.541910i	0.962592	+	0.270955i	$0.0873394\pi$
$19$			2.36213i			0.541910i	−0.962592	+	0.270955i	$0.912661\pi$
$20$		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
$25$	4.78070	−	1.46455i	0.956140	−	0.292909i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1.54355			0.286631			0.143315	−	0.989677i	$-0.454224\pi$
$29$	1.54355			0.286631			0.143315	+	0.989677i	$0.454224\pi$
$30$		0			0
$31$	2.36020			0.423905			0.211952	−	0.977280i	$-0.432018\pi$
$31$	2.36020			0.423905			0.211952	+	0.977280i	$0.432018\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.97704	+	2.20160i	0.503211	+	0.372138i
$36$		0			0
$37$	4.63463	+	4.63463i	0.761928	+	0.761928i	0.976671	−	0.214743i	$-0.0688913\pi$
$37$	4.63463	+	4.63463i	0.761928	+	0.761928i	−0.214743	+	0.976671i	$0.568891\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			3.08636i			0.482008i	0.970524	+	0.241004i	$0.0774767\pi$
$41$			3.08636i			0.482008i	−0.970524	+	0.241004i	$0.922523\pi$
$42$		0			0
$43$	5.19030	−	5.19030i	0.791513	−	0.791513i	−0.190227	−	0.981740i	$-0.560922\pi$
$43$	5.19030	−	5.19030i	0.791513	−	0.791513i	0.981740	+	0.190227i	$0.0609223\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−3.52866	+	3.52866i	−0.514708	+	0.514708i	−0.915965	−	0.401257i	$-0.868573\pi$
$47$	−3.52866	+	3.52866i	−0.514708	+	0.514708i	0.401257	+	0.915965i	$0.368573\pi$
$48$		0			0
$49$		−	4.25805i		−	0.608292i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−8.94118	−	8.94118i	−1.22816	−	1.22816i	−0.964657	−	0.263508i	$-0.915121\pi$
$53$	−8.94118	−	8.94118i	−1.22816	−	1.22816i	−0.263508	−	0.964657i	$-0.584879\pi$
$54$		0			0
$55$	−0.0784674	−	0.524030i	−0.0105805	−	0.0706602i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−11.6852			−1.52129			−0.760644	−	0.649169i	$-0.775117\pi$
$59$	−11.6852			−1.52129			−0.760644	+	0.649169i	$0.775117\pi$
$60$		0			0
$61$	1.02874			0.131717			0.0658584	−	0.997829i	$-0.479021\pi$
$61$	1.02874			0.131717			0.0658584	+	0.997829i	$0.479021\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−0.634130	+	0.857482i	−0.0786542	+	0.106358i
$66$		0			0
$67$	−9.64894	−	9.64894i	−1.17881	−	1.17881i	−0.980049	−	0.198757i	$-0.936310\pi$
$67$	−9.64894	−	9.64894i	−1.17881	−	1.17881i	−0.198757	−	0.980049i	$-0.563690\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			4.90457i			0.582065i	0.956713	+	0.291032i	$0.0939988\pi$
$71$			4.90457i			0.582065i	−0.956713	+	0.291032i	$0.906001\pi$
$72$		0			0
$73$	−4.17536	+	4.17536i	−0.488689	+	0.488689i	−0.907892	−	0.419203i	$-0.862309\pi$
$73$	−4.17536	+	4.17536i	−0.488689	+	0.488689i	0.419203	+	0.907892i	$0.362309\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	0.277461	−	0.277461i	0.0316196	−	0.0316196i
$78$		0			0
$79$			1.47706i			0.166182i	0.996542	+	0.0830911i	$0.0264793\pi$
$79$			1.47706i			0.166182i	−0.996542	+	0.0830911i	$0.973521\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	6.21585	+	6.21585i	0.682278	+	0.682278i	0.960513	−	0.278235i	$-0.0897493\pi$
$83$	6.21585	+	6.21585i	0.682278	+	0.682278i	−0.278235	+	0.960513i	$0.589749\pi$
$84$		0			0
$85$	−1.09779	+	1.48446i	−0.119073	+	0.161012i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−15.7951			−1.67428			−0.837140	−	0.546988i	$-0.815774\pi$
$89$	−15.7951			−1.67428			−0.837140	+	0.546988i	$0.815774\pi$
$90$		0			0
$91$	−0.789771			−0.0827905
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−0.782180	−	5.22365i	−0.0802500	−	0.535935i
$96$		0			0
$97$	−5.69950	−	5.69950i	−0.578696	−	0.578696i	0.355848	−	0.934544i	$-0.384192\pi$
$97$	−5.69950	−	5.69950i	−0.578696	−	0.578696i	−0.934544	+	0.355848i	$0.884192\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	4.61255i		−	0.458966i	−0.973313	−	0.229483i	$-0.926296\pi$
$101$		−	4.61255i		−	0.458966i	0.973313	−	0.229483i	$-0.0737035\pi$
$102$		0			0
$103$	8.71131	−	8.71131i	0.858351	−	0.858351i	−0.132793	−	0.991144i	$-0.542395\pi$
$103$	8.71131	−	8.71131i	0.858351	−	0.858351i	0.991144	+	0.132793i	$0.0423946\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−4.41749	+	4.41749i	−0.427055	+	0.427055i	−0.887624	−	0.460569i	$-0.847645\pi$
$107$	−4.41749	+	4.41749i	−0.427055	+	0.427055i	0.460569	+	0.887624i	$0.347645\pi$
$108$		0			0
$109$		−	8.49802i		−	0.813963i	−0.913436	−	0.406981i	$-0.866581\pi$
$109$		−	8.49802i		−	0.813963i	0.913436	−	0.406981i	$-0.133419\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	4.38951	+	4.38951i	0.412931	+	0.412931i	0.882758	−	0.469828i	$-0.155684\pi$
$113$	4.38951	+	4.38951i	0.412931	+	0.412931i	−0.469828	+	0.882758i	$0.655684\pi$
$114$		0			0
$115$	−1.79785	−	1.32956i	−0.167651	−	0.123982i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−1.36724			−0.125334
$120$		0			0
$121$	10.9438			0.994895
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−10.0871	+	4.82177i	−0.902222	+	0.431272i
$126$		0			0
$127$	−11.5192	−	11.5192i	−1.02217	−	1.02217i	−0.999749	−	0.0224174i	$-0.992864\pi$
$127$	−11.5192	−	11.5192i	−1.02217	−	1.02217i	−0.0224174	−	0.999749i	$-0.507136\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	4.49305i		−	0.392559i	−0.980548	−	0.196280i	$-0.937114\pi$
$131$		−	4.49305i		−	0.392559i	0.980548	−	0.196280i	$-0.0628860\pi$
$132$		0			0
$133$	2.76579	−	2.76579i	0.239824	−	0.239824i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−0.780533	+	0.780533i	−0.0666854	+	0.0666854i	−0.739663	−	0.672978i	$-0.765015\pi$
$137$	−0.780533	+	0.780533i	−0.0666854	+	0.0666854i	0.672978	+	0.739663i	$0.265015\pi$
$138$		0			0
$139$			6.15754i			0.522276i	0.965301	+	0.261138i	$0.0840978\pi$
$139$			6.15754i			0.522276i	−0.965301	+	0.261138i	$0.915902\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	0.0799176	+	0.0799176i	0.00668304	+	0.00668304i
$144$		0			0
$145$	−3.41344	+	0.511123i	−0.283471	+	0.0424464i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−16.3720			−1.34125			−0.670624	−	0.741797i	$-0.733974\pi$
$149$	−16.3720			−1.34125			−0.670624	+	0.741797i	$0.733974\pi$
$150$		0			0
$151$	−19.3389			−1.57377			−0.786887	−	0.617097i	$-0.788309\pi$
$151$	−19.3389			−1.57377			−0.786887	+	0.617097i	$0.788309\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−5.21938	+	0.781542i	−0.419231	+	0.0627750i
$156$		0			0
$157$	−10.0877	−	10.0877i	−0.805089	−	0.805089i	0.178797	−	0.983886i	$-0.442780\pi$
$157$	−10.0877	−	10.0877i	−0.805089	−	0.805089i	−0.983886	+	0.178797i	$0.942780\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	1.65588i		−	0.130502i
$162$		0			0
$163$	−0.608465	+	0.608465i	−0.0476586	+	0.0476586i	−0.730534	−	0.682876i	$-0.760729\pi$
$163$	−0.608465	+	0.608465i	−0.0476586	+	0.0476586i	0.682876	+	0.730534i	$0.260729\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−4.14361	+	4.14361i	−0.320642	+	0.320642i	−0.849013	−	0.528371i	$-0.822803\pi$
$167$	−4.14361	+	4.14361i	−0.320642	+	0.320642i	0.528371	+	0.849013i	$0.322803\pi$
$168$		0			0
$169$			12.7725i			0.982502i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−14.8771	−	14.8771i	−1.13108	−	1.13108i	−0.989997	−	0.141087i	$-0.954940\pi$
$173$	−14.8771	−	14.8771i	−1.13108	−	1.13108i	−0.141087	−	0.989997i	$-0.545060\pi$
$174$		0			0
$175$	−7.31248	−	3.88284i	−0.552772	−	0.293515i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	8.37688			0.626118			0.313059	−	0.949734i	$-0.398646\pi$
$179$	8.37688			0.626118			0.313059	+	0.949734i	$0.398646\pi$
$180$		0			0
$181$	−11.2938			−0.839458			−0.419729	−	0.907649i	$-0.637875\pi$
$181$	−11.2938			−0.839458			−0.419729	+	0.907649i	$0.637875\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−11.7838	−	8.71440i	−0.866359	−	0.640695i
$186$		0			0
$187$	0.138352	+	0.138352i	0.0101173	+	0.0101173i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			7.10106i			0.513815i	0.966436	+	0.256907i	$0.0827036\pi$
$191$			7.10106i			0.513815i	−0.966436	+	0.256907i	$0.917296\pi$
$192$		0			0
$193$	9.92018	−	9.92018i	0.714070	−	0.714070i	−0.253314	−	0.967384i	$-0.581520\pi$
$193$	9.92018	−	9.92018i	0.714070	−	0.714070i	0.967384	+	0.253314i	$0.0815205\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	7.84638	−	7.84638i	0.559031	−	0.559031i	−0.370000	−	0.929032i	$-0.620642\pi$
$197$	7.84638	−	7.84638i	0.559031	−	0.559031i	0.929032	+	0.370000i	$0.120642\pi$
$198$		0			0
$199$		−	5.06871i		−	0.359311i	−0.983730	−	0.179656i	$-0.942502\pi$
$199$		−	5.06871i		−	0.359311i	0.983730	−	0.179656i	$-0.0574984\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−1.80733	−	1.80733i	−0.126850	−	0.126850i
$204$		0			0
$205$	−1.02200	−	6.82521i	−0.0713793	−	0.476693i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−0.559745			−0.0387183
$210$		0			0
$211$	9.79762			0.674496			0.337248	−	0.941416i	$-0.390504\pi$
$211$	9.79762			0.674496			0.337248	+	0.941416i	$0.390504\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−9.75922	+	13.1966i	−0.665573	+	0.900000i
$216$		0			0
$217$	−2.76353	−	2.76353i	−0.187601	−	0.187601i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	0.393808i		−	0.0264904i
$222$		0			0
$223$	4.34774	−	4.34774i	0.291146	−	0.291146i	−0.546387	−	0.837533i	$-0.683997\pi$
$223$	4.34774	−	4.34774i	0.291146	−	0.291146i	0.837533	+	0.546387i	$0.183997\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	10.1341	−	10.1341i	0.672627	−	0.672627i	−0.285694	−	0.958321i	$-0.592224\pi$
$227$	10.1341	−	10.1341i	0.672627	−	0.672627i	0.958321	+	0.285694i	$0.0922241\pi$
$228$		0			0
$229$		−	9.62245i		−	0.635869i	−0.948113	−	0.317935i	$-0.897011\pi$
$229$		−	9.62245i		−	0.635869i	0.948113	−	0.317935i	$-0.102989\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−1.95843	−	1.95843i	−0.128301	−	0.128301i	0.640040	−	0.768341i	$-0.278918\pi$
$233$	−1.95843	−	1.95843i	−0.128301	−	0.128301i	−0.768341	+	0.640040i	$0.778918\pi$
$234$		0			0
$235$	6.63487	−	8.97179i	0.432811	−	0.585255i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−15.1744			−0.981552			−0.490776	−	0.871286i	$-0.663287\pi$
$239$	−15.1744			−0.981552			−0.490776	+	0.871286i	$0.663287\pi$
$240$		0			0
$241$	−5.14639			−0.331508			−0.165754	−	0.986167i	$-0.553006\pi$
$241$	−5.14639			−0.331508			−0.165754	+	0.986167i	$0.553006\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	1.40998	+	9.41630i	0.0900804	+	0.601585i
$246$		0			0
$247$	0.796635	+	0.796635i	0.0506887	+	0.0506887i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	15.5179i		−	0.979484i	−0.871867	−	0.489742i	$-0.837091\pi$
$251$		−	15.5179i		−	0.979484i	0.871867	−	0.489742i	$-0.162909\pi$
$252$		0			0
$253$	−0.167560	+	0.167560i	−0.0105344	+	0.0105344i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−8.24210	+	8.24210i	−0.514128	+	0.514128i	−0.915789	−	0.401661i	$-0.868433\pi$
$257$	−8.24210	+	8.24210i	−0.514128	+	0.514128i	0.401661	+	0.915789i	$0.368433\pi$
$258$		0			0
$259$		−	10.8533i		−	0.674388i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	2.81839	+	2.81839i	0.173789	+	0.173789i	0.788642	−	0.614853i	$-0.210785\pi$
$263$	2.81839	+	2.81839i	0.173789	+	0.173789i	−0.614853	+	0.788642i	$0.710785\pi$
$264$		0			0
$265$	22.7334	+	16.8119i	1.39650	+	1.03275i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−15.4961			−0.944816			−0.472408	−	0.881380i	$-0.656615\pi$
$269$	−15.4961			−0.944816			−0.472408	+	0.881380i	$0.656615\pi$
$270$		0			0
$271$	−29.8582			−1.81376			−0.906879	−	0.421391i	$-0.861542\pi$
$271$	−29.8582			−1.81376			−0.906879	+	0.421391i	$0.861542\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	0.347048	+	1.13286i	0.0209278	+	0.0683143i
$276$		0			0
$277$	1.45706	+	1.45706i	0.0875460	+	0.0875460i	0.749524	−	0.661978i	$-0.230283\pi$
$277$	1.45706	+	1.45706i	0.0875460	+	0.0875460i	−0.661978	+	0.749524i	$0.730283\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	22.4344i		−	1.33832i	−0.743117	−	0.669162i	$-0.766653\pi$
$281$		−	22.4344i		−	1.33832i	0.743117	−	0.669162i	$-0.233347\pi$
$282$		0			0
$283$	−6.64556	+	6.64556i	−0.395038	+	0.395038i	−0.876479	−	0.481441i	$-0.840114\pi$
$283$	−6.64556	+	6.64556i	−0.395038	+	0.395038i	0.481441	+	0.876479i	$0.340114\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	3.61378	−	3.61378i	0.213314	−	0.213314i
$288$		0			0
$289$			16.3182i			0.959897i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−10.7572	−	10.7572i	−0.628442	−	0.628442i	0.319234	−	0.947676i	$-0.396574\pi$
$293$	−10.7572	−	10.7572i	−0.628442	−	0.628442i	−0.947676	+	0.319234i	$0.896574\pi$
$294$		0			0
$295$	25.8409	−	3.86937i	1.50451	−	0.225284i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	0.476948			0.0275826
$300$		0			0
$301$	−12.1545			−0.700575
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−2.27497	+	0.340651i	−0.130265	+	0.0195056i
$306$		0			0
$307$	−7.63849	−	7.63849i	−0.435951	−	0.435951i	0.454696	−	0.890647i	$-0.349748\pi$
$307$	−7.63849	−	7.63849i	−0.435951	−	0.435951i	−0.890647	+	0.454696i	$0.849748\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			19.9368i			1.13051i	0.824915	+	0.565257i	$0.191223\pi$
$311$			19.9368i			1.13051i	−0.824915	+	0.565257i	$0.808777\pi$
$312$		0			0
$313$	11.5969	−	11.5969i	0.655497	−	0.655497i	−0.298814	−	0.954311i	$-0.596591\pi$
$313$	11.5969	−	11.5969i	0.655497	−	0.655497i	0.954311	+	0.298814i	$0.0965911\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	1.75980	−	1.75980i	0.0988400	−	0.0988400i	−0.655958	−	0.754798i	$-0.727735\pi$
$317$	1.75980	−	1.75980i	0.0988400	−	0.0988400i	0.754798	+	0.655958i	$0.227735\pi$
$318$		0			0
$319$			0.365770i			0.0204792i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	1.37912	+	1.37912i	0.0767363	+	0.0767363i
$324$		0			0
$325$	1.11838	−	2.10623i	0.0620368	−	0.116833i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	8.26333			0.455572
$330$		0			0
$331$	5.43515			0.298743			0.149372	−	0.988781i	$-0.452275\pi$
$331$	5.43515			0.298743			0.149372	+	0.988781i	$0.452275\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	24.5329	+	18.1427i	1.34037	+	0.991242i
$336$		0			0
$337$	3.01361	+	3.01361i	0.164162	+	0.164162i	0.784408	−	0.620246i	$-0.212967\pi$
$337$	3.01361	+	3.01361i	0.164162	+	0.164162i	−0.620246	+	0.784408i	$0.712967\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			0.559288i			0.0302871i
$342$		0			0
$343$	−13.1819	+	13.1819i	−0.711756	+	0.711756i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−12.2232	+	12.2232i	−0.656177	+	0.656177i	−0.954473	−	0.298296i	$-0.903582\pi$
$347$	−12.2232	+	12.2232i	−0.656177	+	0.656177i	0.298296	+	0.954473i	$0.403582\pi$
$348$		0			0
$349$		−	4.48682i		−	0.240174i	−0.992763	−	0.120087i	$-0.961683\pi$
$349$		−	4.48682i		−	0.240174i	0.992763	−	0.120087i	$-0.0383174\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	8.71775	+	8.71775i	0.463999	+	0.463999i	0.899964	−	0.435965i	$-0.143593\pi$
$353$	8.71775	+	8.71775i	0.463999	+	0.463999i	−0.435965	+	0.899964i	$0.643593\pi$
$354$		0			0
$355$	−1.62407	−	10.8460i	−0.0861965	−	0.575647i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−34.2789			−1.80917			−0.904586	−	0.426292i	$-0.859820\pi$
$359$	−34.2789			−1.80917			−0.904586	+	0.426292i	$0.859820\pi$
$360$		0			0
$361$	13.4203			0.706334
$362$		0			0
$363$		0			0
$364$		0			0
$365$	7.85085	−	10.6161i	0.410932	−	0.555670i
$366$		0			0
$367$	5.67345	+	5.67345i	0.296152	+	0.296152i	0.839504	−	0.543353i	$-0.182845\pi$
$367$	5.67345	+	5.67345i	0.296152	+	0.296152i	−0.543353	+	0.839504i	$0.682845\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			20.9382i			1.08706i
$372$		0			0
$373$	7.75276	−	7.75276i	0.401423	−	0.401423i	−0.477311	−	0.878734i	$-0.658389\pi$
$373$	7.75276	−	7.75276i	0.401423	−	0.401423i	0.878734	+	0.477311i	$0.158389\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	0.520569	−	0.520569i	0.0268106	−	0.0268106i
$378$		0			0
$379$			4.85705i			0.249490i	0.992189	+	0.124745i	$0.0398112\pi$
$379$			4.85705i			0.249490i	−0.992189	+	0.124745i	$0.960189\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	14.0151	+	14.0151i	0.716136	+	0.716136i	0.967812	−	0.251676i	$-0.0809817\pi$
$383$	14.0151	+	14.0151i	0.716136	+	0.716136i	−0.251676	+	0.967812i	$0.580982\pi$
$384$		0			0
$385$	−0.521704	+	0.705457i	−0.0265885	+	0.0359534i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	14.3761			0.728897			0.364448	−	0.931224i	$-0.381258\pi$
$389$	14.3761			0.728897			0.364448	+	0.931224i	$0.381258\pi$
$390$		0			0
$391$	0.825683			0.0417566
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−0.489104	−	3.26639i	−0.0246095	−	0.164350i
$396$		0			0
$397$	−6.50564	−	6.50564i	−0.326509	−	0.326509i	0.524749	−	0.851257i	$-0.324159\pi$
$397$	−6.50564	−	6.50564i	−0.326509	−	0.326509i	−0.851257	+	0.524749i	$0.824159\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	26.0037i		−	1.29856i	−0.760548	−	0.649281i	$-0.775070\pi$
$401$		−	26.0037i		−	1.29856i	0.760548	−	0.649281i	$-0.224930\pi$
$402$		0			0
$403$	0.795985	−	0.795985i	0.0396509	−	0.0396509i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−1.09825	+	1.09825i	−0.0544382	+	0.0544382i
$408$		0			0
$409$			8.00847i			0.395993i	0.980203	+	0.197997i	$0.0634435\pi$
$409$			8.00847i			0.395993i	−0.980203	+	0.197997i	$0.936556\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	13.6821	+	13.6821i	0.673252	+	0.673252i
$414$		0			0
$415$	−15.8041	−	11.6875i	−0.775792	−	0.573719i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−19.0015			−0.928284			−0.464142	−	0.885761i	$-0.653637\pi$
$419$	−19.0015			−0.928284			−0.464142	+	0.885761i	$0.653637\pi$
$420$		0			0
$421$	15.5269			0.756735			0.378368	−	0.925655i	$-0.376485\pi$
$421$	15.5269			0.756735			0.378368	+	0.925655i	$0.376485\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	1.93612	−	3.64626i	0.0939158	−	0.176870i
$426$		0			0
$427$	−1.20454	−	1.20454i	−0.0582918	−	0.0582918i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	14.6282i		−	0.704616i	−0.935884	−	0.352308i	$-0.885397\pi$
$431$		−	14.6282i		−	0.704616i	0.935884	−	0.352308i	$-0.114603\pi$
$432$		0			0
$433$	15.2479	−	15.2479i	0.732766	−	0.732766i	−0.238401	−	0.971167i	$-0.576623\pi$
$433$	15.2479	−	15.2479i	0.732766	−	0.732766i	0.971167	+	0.238401i	$0.0766232\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−1.67028	+	1.67028i	−0.0799002	+	0.0799002i
$438$		0			0
$439$			36.6993i			1.75156i	0.482710	+	0.875780i	$0.339652\pi$
$439$			36.6993i			1.75156i	−0.482710	+	0.875780i	$0.660348\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−1.09271	−	1.09271i	−0.0519163	−	0.0519163i	0.680672	−	0.732588i	$-0.261688\pi$
$443$	−1.09271	−	1.09271i	−0.0519163	−	0.0519163i	−0.732588	+	0.680672i	$0.761688\pi$
$444$		0			0
$445$	34.9296	−	5.23030i	1.65582	−	0.247940i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−3.52290			−0.166256			−0.0831280	−	0.996539i	$-0.526491\pi$
$449$	−3.52290			−0.166256			−0.0831280	+	0.996539i	$0.526491\pi$
$450$		0			0
$451$	−0.731362			−0.0344385
$452$		0			0
$453$		0			0
$454$		0			0
$455$	1.74651	−	0.261520i	0.0818776	−	0.0122602i
$456$		0			0
$457$	−5.80574	−	5.80574i	−0.271581	−	0.271581i	0.558156	−	0.829736i	$-0.311509\pi$
$457$	−5.80574	−	5.80574i	−0.271581	−	0.271581i	−0.829736	+	0.558156i	$0.811509\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	13.3076i		−	0.619798i	−0.950770	−	0.309899i	$-0.899705\pi$
$461$		−	13.3076i		−	0.619798i	0.950770	−	0.309899i	$-0.100295\pi$
$462$		0			0
$463$	20.9153	−	20.9153i	0.972017	−	0.972017i	−0.0276023	−	0.999619i	$-0.508787\pi$
$463$	20.9153	−	20.9153i	0.972017	−	0.972017i	0.999619	+	0.0276023i	$0.00878719\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	3.67470	−	3.67470i	0.170045	−	0.170045i	−0.616954	−	0.786999i	$-0.711634\pi$
$467$	3.67470	−	3.67470i	0.170045	−	0.170045i	0.786999	+	0.616954i	$0.211634\pi$
$468$		0			0
$469$			22.5956i			1.04337i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	1.22993	+	1.22993i	0.0565520	+	0.0565520i
$474$		0			0
$475$	3.45945	+	11.2926i	0.158730	+	0.518142i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−23.1913			−1.05964			−0.529820	−	0.848110i	$-0.677740\pi$
$479$	−23.1913			−1.05964			−0.529820	+	0.848110i	$0.677740\pi$
$480$		0			0
$481$	3.12608			0.142537
$482$		0			0
$483$		0			0
$484$		0			0
$485$	14.4912	+	10.7167i	0.658013	+	0.486618i
$486$		0			0
$487$	−9.64681	−	9.64681i	−0.437139	−	0.437139i	0.453909	−	0.891048i	$-0.350029\pi$
$487$	−9.64681	−	9.64681i	−0.437139	−	0.437139i	−0.891048	+	0.453909i	$0.850029\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			11.6279i			0.524759i	0.964965	+	0.262380i	$0.0845073\pi$
$491$			11.6279i			0.524759i	−0.964965	+	0.262380i	$0.915493\pi$
$492$		0			0
$493$	0.901199	−	0.901199i	0.0405879	−	0.0405879i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	5.74269	−	5.74269i	0.257595	−	0.257595i
$498$		0			0
$499$			9.95123i			0.445478i	0.974878	+	0.222739i	$0.0714998\pi$
$499$			9.95123i			0.445478i	−0.974878	+	0.222739i	$0.928500\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−8.99713	−	8.99713i	−0.401162	−	0.401162i	0.477480	−	0.878642i	$-0.341550\pi$
$503$	−8.99713	−	8.99713i	−0.401162	−	0.401162i	−0.878642	+	0.477480i	$0.841550\pi$
$504$		0			0
$505$	1.52737	+	10.2003i	0.0679671	+	0.453906i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−14.6707			−0.650268			−0.325134	−	0.945668i	$-0.605409\pi$
$509$	−14.6707			−0.650268			−0.325134	+	0.945668i	$0.605409\pi$
$510$		0			0
$511$	9.77776			0.432542
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−16.3797	+	22.1489i	−0.721776	+	0.975997i
$516$		0			0
$517$	−0.836173	−	0.836173i	−0.0367748	−	0.0367748i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			21.9484i			0.961575i	0.876837	+	0.480788i	$0.159649\pi$
$521$			21.9484i			0.961575i	−0.876837	+	0.480788i	$0.840351\pi$
$522$		0			0
$523$	−1.32692	+	1.32692i	−0.0580220	+	0.0580220i	−0.735522	−	0.677500i	$-0.763063\pi$
$523$	−1.32692	+	1.32692i	−0.0580220	+	0.0580220i	0.677500	+	0.735522i	$0.263063\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	1.37799	−	1.37799i	0.0600264	−	0.0600264i
$528$		0			0
$529$			1.00000i			0.0434783i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	1.04088	+	1.04088i	0.0450857	+	0.0450857i
$534$		0			0
$535$	8.30612	−	11.2317i	0.359105	−	0.485588i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	1.00901			0.0434613
$540$		0			0
$541$	−36.9363			−1.58802			−0.794008	−	0.607907i	$-0.792009\pi$
$541$	−36.9363			−1.58802			−0.794008	+	0.607907i	$0.792009\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	2.81398	+	18.7926i	0.120538	+	0.804988i
$546$		0			0
$547$	−7.65309	−	7.65309i	−0.327222	−	0.327222i	0.524307	−	0.851529i	$-0.324324\pi$
$547$	−7.65309	−	7.65309i	−0.327222	−	0.327222i	−0.851529	+	0.524307i	$0.824324\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			3.64608i			0.155328i
$552$		0			0
$553$	1.72947	−	1.72947i	0.0735446	−	0.0735446i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	0.477072	−	0.477072i	0.0202142	−	0.0202142i	−0.696927	−	0.717142i	$-0.745450\pi$
$557$	0.477072	−	0.477072i	0.0202142	−	0.0202142i	0.717142	+	0.696927i	$0.245450\pi$
$558$		0			0
$559$		−	3.50089i		−	0.148072i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−5.06690	−	5.06690i	−0.213544	−	0.213544i	0.592227	−	0.805771i	$-0.298249\pi$
$563$	−5.06690	−	5.06690i	−0.213544	−	0.213544i	−0.805771	+	0.592227i	$0.798249\pi$
$564$		0			0
$565$	−11.1605	−	8.25351i	−0.469528	−	0.347228i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	21.4094			0.897528			0.448764	−	0.893650i	$-0.351864\pi$
$569$	21.4094			0.897528			0.448764	+	0.893650i	$0.351864\pi$
$570$		0			0
$571$	−26.6495			−1.11525			−0.557623	−	0.830094i	$-0.688287\pi$
$571$	−26.6495			−1.11525			−0.557623	+	0.830094i	$0.688287\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	4.41606	+	2.34488i	0.184162	+	0.0977881i
$576$		0			0
$577$	12.8069	+	12.8069i	0.533157	+	0.533157i	0.921510	−	0.388354i	$-0.126956\pi$
$577$	12.8069	+	12.8069i	0.533157	+	0.533157i	−0.388354	+	0.921510i	$0.626956\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	14.5561i		−	0.603890i
$582$		0			0
$583$	2.11876	−	2.11876i	0.0877499	−	0.0877499i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	14.5530	−	14.5530i	0.600665	−	0.600665i	−0.339824	−	0.940489i	$-0.610367\pi$
$587$	14.5530	−	14.5530i	0.600665	−	0.600665i	0.940489	+	0.339824i	$0.110367\pi$
$588$		0			0
$589$			5.57510i			0.229718i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	21.5050	+	21.5050i	0.883107	+	0.883107i	0.993849	−	0.110743i	$-0.0353229\pi$
$593$	21.5050	+	21.5050i	0.883107	+	0.883107i	−0.110743	+	0.993849i	$0.535323\pi$
$594$		0			0
$595$	3.02353	−	0.452738i	0.123952	−	0.0185604i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	22.0407			0.900560			0.450280	−	0.892887i	$-0.351324\pi$
$599$	22.0407			0.900560			0.450280	+	0.892887i	$0.351324\pi$
$600$		0			0
$601$	25.2303			1.02917			0.514583	−	0.857440i	$-0.327947\pi$
$601$	25.2303			1.02917			0.514583	+	0.857440i	$0.327947\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−24.2014	+	3.62387i	−0.983926	+	0.147331i
$606$		0			0
$607$	8.47387	+	8.47387i	0.343944	+	0.343944i	0.857848	−	0.513904i	$-0.171801\pi$
$607$	8.47387	+	8.47387i	0.343944	+	0.343944i	−0.513904	+	0.857848i	$0.671801\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			2.38010i			0.0962887i
$612$		0			0
$613$	−21.4859	+	21.4859i	−0.867807	+	0.867807i	−0.992229	−	0.124422i	$-0.960292\pi$
$613$	−21.4859	+	21.4859i	−0.867807	+	0.867807i	0.124422	+	0.992229i	$0.460292\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	26.3205	−	26.3205i	1.05962	−	1.05962i	0.0615174	−	0.998106i	$-0.480406\pi$
$617$	26.3205	−	26.3205i	1.05962	−	1.05962i	0.998106	−	0.0615174i	$-0.0195940\pi$
$618$		0			0
$619$			23.1401i			0.930077i	0.885290	+	0.465039i	$0.153960\pi$
$619$			23.1401i			0.930077i	−0.885290	+	0.465039i	$0.846040\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	18.4943	+	18.4943i	0.740959	+	0.740959i
$624$		0			0
$625$	20.7102	−	14.0031i	0.828408	−	0.560125i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	5.41182			0.215783
$630$		0			0
$631$	−20.2505			−0.806161			−0.403081	−	0.915165i	$-0.632061\pi$
$631$	−20.2505			−0.806161			−0.403081	+	0.915165i	$0.632061\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	29.2882	+	21.6594i	1.16227	+	0.859526i
$636$		0			0
$637$	−1.43604	−	1.43604i	−0.0568979	−	0.0568979i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	14.0582i		−	0.555264i	−0.960688	−	0.277632i	$-0.910450\pi$
$641$		−	14.0582i		−	0.555264i	0.960688	−	0.277632i	$-0.0895496\pi$
$642$		0			0
$643$	−31.7723	+	31.7723i	−1.25298	+	1.25298i	−0.298598	+	0.954379i	$0.596519\pi$
$643$	−31.7723	+	31.7723i	−1.25298	+	1.25298i	−0.954379	+	0.298598i	$0.903481\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−16.2341	+	16.2341i	−0.638229	+	0.638229i	−0.950118	−	0.311889i	$-0.899038\pi$
$647$	−16.2341	+	16.2341i	−0.638229	+	0.638229i	0.311889	+	0.950118i	$0.399038\pi$
$648$		0			0
$649$		−	2.76900i		−	0.108693i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−24.9835	−	24.9835i	−0.977680	−	0.977680i	0.0220765	−	0.999756i	$-0.492972\pi$
$653$	−24.9835	−	24.9835i	−0.977680	−	0.977680i	−0.999756	+	0.0220765i	$0.992972\pi$
$654$		0			0
$655$	1.48780	+	9.93598i	0.0581331	+	0.388231i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	26.4445			1.03013			0.515065	−	0.857151i	$-0.327768\pi$
$659$	26.4445			1.03013			0.515065	+	0.857151i	$0.327768\pi$
$660$		0			0
$661$	−18.8424			−0.732883			−0.366442	−	0.930441i	$-0.619424\pi$
$661$	−18.8424			−0.732883			−0.366442	+	0.930441i	$0.619424\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−5.20046	+	7.03214i	−0.201665	+	0.272695i
$666$		0			0
$667$	1.09146	+	1.09146i	0.0422614	+	0.0422614i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			0.243777i			0.00941090i
$672$		0			0
$673$	4.06135	−	4.06135i	0.156553	−	0.156553i	−0.624484	−	0.781038i	$-0.714691\pi$
$673$	4.06135	−	4.06135i	0.156553	−	0.156553i	0.781038	+	0.624484i	$0.214691\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	22.7091	−	22.7091i	0.872783	−	0.872783i	−0.119992	−	0.992775i	$-0.538287\pi$
$677$	22.7091	−	22.7091i	0.872783	−	0.872783i	0.992775	+	0.119992i	$0.0382869\pi$
$678$		0			0
$679$			13.3469i			0.512208i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−2.56553	−	2.56553i	−0.0981673	−	0.0981673i	0.656317	−	0.754485i	$-0.272113\pi$
$683$	−2.56553	−	2.56553i	−0.0981673	−	0.0981673i	−0.754485	+	0.656317i	$0.772113\pi$
$684$		0			0
$685$	1.46762	−	1.98454i	0.0560749	−	0.0758254i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−6.03088			−0.229758
$690$		0			0
$691$	14.0232			0.533467			0.266733	−	0.963770i	$-0.414056\pi$
$691$	14.0232			0.533467			0.266733	+	0.963770i	$0.414056\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−2.03897	−	13.6169i	−0.0773425	−	0.516517i
$696$		0			0
$697$	1.80196	+	1.80196i	0.0682540	+	0.0682540i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	39.1199i		−	1.47754i	−0.673958	−	0.738769i	$-0.735407\pi$
$701$		−	39.1199i		−	1.47754i	0.673958	−	0.738769i	$-0.264593\pi$
$702$		0			0
$703$	−10.9476	+	10.9476i	−0.412896	+	0.412896i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−5.40078	+	5.40078i	−0.203117	+	0.203117i
$708$		0			0
$709$		−	3.91662i		−	0.147092i	−0.997292	−	0.0735458i	$-0.976568\pi$
$709$		−	3.91662i		−	0.147092i	0.997292	−	0.0735458i	$-0.0234315\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	1.66891	+	1.66891i	0.0625014	+	0.0625014i
$714$		0			0
$715$	−0.203194	−	0.150267i	−0.00759903	−	0.00561968i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	5.22620			0.194904			0.0974522	−	0.995240i	$-0.468931\pi$
$719$	5.22620			0.194904			0.0974522	+	0.995240i	$0.468931\pi$
$720$		0			0
$721$	−20.3999			−0.759733
$722$		0			0
$723$		0			0
$724$		0			0
$725$	7.37927	−	2.26061i	0.274059	−	0.0839569i
$726$		0			0
$727$	−5.51969	−	5.51969i	−0.204714	−	0.204714i	0.597302	−	0.802016i	$-0.296239\pi$
$727$	−5.51969	−	5.51969i	−0.204714	−	0.204714i	−0.802016	+	0.597302i	$0.796239\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	6.06067i		−	0.224162i
$732$		0			0
$733$	4.68832	−	4.68832i	0.173167	−	0.173167i	−0.615202	−	0.788369i	$-0.710926\pi$
$733$	4.68832	−	4.68832i	0.173167	−	0.173167i	0.788369	+	0.615202i	$0.210926\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	2.28647	−	2.28647i	0.0842233	−	0.0842233i
$738$		0			0
$739$			43.0964i			1.58532i	0.609661	+	0.792662i	$0.291306\pi$
$739$			43.0964i			1.58532i	−0.609661	+	0.792662i	$0.708694\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	1.77780	+	1.77780i	0.0652211	+	0.0652211i	0.738965	−	0.673744i	$-0.235315\pi$
$743$	1.77780	+	1.77780i	0.0652211	+	0.0652211i	−0.673744	+	0.738965i	$0.735315\pi$
$744$		0			0
$745$	36.2053	−	5.42133i	1.32646	−	0.198622i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	10.3448			0.377990
$750$		0			0
$751$	−8.44200			−0.308053			−0.154026	−	0.988067i	$-0.549224\pi$
$751$	−8.44200			−0.308053			−0.154026	+	0.988067i	$0.549224\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	42.7662	−	6.40375i	1.55642	−	0.233056i
$756$		0			0
$757$	−32.2981	−	32.2981i	−1.17390	−	1.17390i	−0.981273	−	0.192622i	$-0.938301\pi$
$757$	−32.2981	−	32.2981i	−1.17390	−	1.17390i	−0.192622	−	0.981273i	$-0.561699\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			6.13799i			0.222502i	0.993792	+	0.111251i	$0.0354858\pi$
$761$			6.13799i			0.222502i	−0.993792	+	0.111251i	$0.964514\pi$
$762$		0			0
$763$	−9.95022	+	9.95022i	−0.360222	+	0.360222i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−3.94088	+	3.94088i	−0.142297	+	0.142297i
$768$		0			0
$769$		−	14.3459i		−	0.517328i	−0.965967	−	0.258664i	$-0.916718\pi$
$769$		−	14.3459i		−	0.517328i	0.965967	−	0.258664i	$-0.0832822\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	18.0775	+	18.0775i	0.650203	+	0.650203i	0.953042	−	0.302839i	$-0.0979344\pi$
$773$	18.0775	+	18.0775i	0.650203	+	0.650203i	−0.302839	+	0.953042i	$0.597934\pi$
$774$		0			0
$775$	11.2834	−	3.45663i	0.405312	−	0.124166i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−7.29037			−0.261205
$780$		0			0
$781$	−1.16222			−0.0415874
$782$		0			0
$783$		0			0
$784$		0			0
$785$	25.6486	+	18.9678i	0.915436	+	0.676989i
$786$		0			0
$787$	9.86175	+	9.86175i	0.351534	+	0.351534i	0.860680	−	0.509146i	$-0.170039\pi$
$787$	9.86175	+	9.86175i	0.351534	+	0.351534i	−0.509146	+	0.860680i	$0.670039\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	10.2793i		−	0.365488i
$792$		0			0
$793$	0.346946	−	0.346946i	0.0123204	−	0.0123204i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	16.0431	−	16.0431i	0.568277	−	0.568277i	−0.363369	−	0.931645i	$-0.618373\pi$
$797$	16.0431	−	16.0431i	0.568277	−	0.568277i	0.931645	+	0.363369i	$0.118373\pi$
$798$		0			0
$799$			4.12039i			0.145769i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−0.989419	−	0.989419i	−0.0349158	−	0.0349158i
$804$		0			0
$805$	0.548319	+	3.66185i	0.0193257	+	0.129063i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	16.2710			0.572059			0.286030	−	0.958221i	$-0.407664\pi$
$809$	16.2710			0.572059			0.286030	+	0.958221i	$0.407664\pi$
$810$		0			0
$811$	−34.6563			−1.21695			−0.608474	−	0.793574i	$-0.708218\pi$
$811$	−34.6563			−1.21695			−0.608474	+	0.793574i	$0.708218\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	1.14408	−	1.54705i	0.0400755	−	0.0541908i
$816$		0			0
$817$	12.2602	+	12.2602i	0.428929	+	0.428929i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	26.2710i		−	0.916864i	−0.888729	−	0.458432i	$-0.848411\pi$
$821$		−	26.2710i		−	0.916864i	0.888729	−	0.458432i	$-0.151589\pi$
$822$		0			0
$823$	−15.6854	+	15.6854i	−0.546759	+	0.546759i	−0.925502	−	0.378743i	$-0.876356\pi$
$823$	−15.6854	+	15.6854i	−0.546759	+	0.546759i	0.378743	+	0.925502i	$0.376356\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−5.91071	+	5.91071i	−0.205536	+	0.205536i	−0.802367	−	0.596831i	$-0.796426\pi$
$827$	−5.91071	+	5.91071i	−0.205536	+	0.205536i	0.596831	+	0.802367i	$0.296426\pi$
$828$		0			0
$829$			5.36041i			0.186175i	0.995658	+	0.0930873i	$0.0296736\pi$
$829$			5.36041i			0.186175i	−0.995658	+	0.0930873i	$0.970326\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−2.48604	−	2.48604i	−0.0861363	−	0.0861363i
$834$		0			0
$835$	7.79114	−	10.5353i	0.269624	−	0.364590i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−21.6812			−0.748520			−0.374260	−	0.927324i	$-0.622103\pi$
$839$	−21.6812			−0.748520			−0.374260	+	0.927324i	$0.622103\pi$
$840$		0			0
$841$	−26.6174			−0.917843
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−4.22941	−	28.2453i	−0.145496	−	0.971669i
$846$		0			0
$847$	−12.8140	−	12.8140i	−0.440295	−	0.440295i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			6.55435i			0.224680i
$852$		0			0
$853$	10.2226	−	10.2226i	0.350015	−	0.350015i	−0.510100	−	0.860115i	$-0.670392\pi$
$853$	10.2226	−	10.2226i	0.350015	−	0.350015i	0.860115	+	0.510100i	$0.170392\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	38.3231	−	38.3231i	1.30909	−	1.30909i	0.387024	−	0.922070i	$-0.373503\pi$
$857$	38.3231	−	38.3231i	1.30909	−	1.30909i	0.922070	−	0.387024i	$-0.126497\pi$
$858$		0			0
$859$			34.9238i			1.19158i	0.803138	+	0.595792i	$0.203162\pi$
$859$			34.9238i			1.19158i	−0.803138	+	0.595792i	$0.796838\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	10.8808	+	10.8808i	0.370388	+	0.370388i	0.867619	−	0.497230i	$-0.165650\pi$
$863$	10.8808	+	10.8808i	0.370388	+	0.370388i	−0.497230	+	0.867619i	$0.665650\pi$
$864$		0			0
$865$	37.8257	+	27.9731i	1.28611	+	0.951114i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−0.350013			−0.0118734
$870$		0			0
$871$	−6.50827			−0.220524
$872$		0			0
$873$		0			0
$874$		0			0
$875$	17.4567	+	6.16517i	0.590143	+	0.208421i
$876$		0			0
$877$	27.1631	+	27.1631i	0.917234	+	0.917234i	0.996827	−	0.0795937i	$-0.0253623\pi$
$877$	27.1631	+	27.1631i	0.917234	+	0.917234i	−0.0795937	+	0.996827i	$0.525362\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	19.0510i		−	0.641843i	−0.947106	−	0.320921i	$-0.896007\pi$
$881$		−	19.0510i		−	0.641843i	0.947106	−	0.320921i	$-0.103993\pi$
$882$		0			0
$883$	22.4321	−	22.4321i	0.754901	−	0.754901i	−0.220489	−	0.975390i	$-0.570765\pi$
$883$	22.4321	−	22.4321i	0.754901	−	0.754901i	0.975390	+	0.220489i	$0.0707652\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	23.7311	−	23.7311i	0.796813	−	0.796813i	−0.185779	−	0.982592i	$-0.559481\pi$
$887$	23.7311	−	23.7311i	0.796813	−	0.796813i	0.982592	+	0.185779i	$0.0594808\pi$
$888$		0			0
$889$			26.9754i			0.904727i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−8.33515	−	8.33515i	−0.278925	−	0.278925i
$894$		0			0
$895$	−18.5248	+	2.77387i	−0.619214	+	0.0927201i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	3.64310			0.121504
$900$		0			0
$901$	−10.4405			−0.347825
$902$		0			0
$903$		0			0
$904$		0			0
$905$	24.9752	−	3.73974i	0.830203	−	0.124313i
$906$		0			0
$907$	−4.74628	−	4.74628i	−0.157598	−	0.157598i	0.623904	−	0.781501i	$-0.285546\pi$
$907$	−4.74628	−	4.74628i	−0.157598	−	0.157598i	−0.781501	+	0.623904i	$0.785546\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			4.43833i			0.147049i	0.997293	+	0.0735243i	$0.0234246\pi$
$911$			4.43833i			0.147049i	−0.997293	+	0.0735243i	$0.976575\pi$
$912$		0			0
$913$	−1.47295	+	1.47295i	−0.0487474	+	0.0487474i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−5.26085	+	5.26085i	−0.173729	+	0.173729i
$918$		0			0
$919$			44.6146i			1.47170i	0.677144	+	0.735850i	$0.263217\pi$
$919$			44.6146i			1.47170i	−0.677144	+	0.735850i	$0.736783\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	1.65408	+	1.65408i	0.0544447	+	0.0544447i
$924$		0			0
$925$	28.9444	+	15.3691i	0.951686	+	0.505334i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−4.49631			−0.147519			−0.0737595	−	0.997276i	$-0.523500\pi$
$929$	−4.49631			−0.147519			−0.0737595	+	0.997276i	$0.523500\pi$
$930$		0			0
$931$	10.0581			0.329639
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−0.351766	−	0.260140i	−0.0115040	−	0.00850749i
$936$		0			0
$937$	−18.9327	−	18.9327i	−0.618504	−	0.618504i	0.326643	−	0.945148i	$-0.394082\pi$
$937$	−18.9327	−	18.9327i	−0.618504	−	0.618504i	−0.945148	+	0.326643i	$0.894082\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	19.2039i		−	0.626030i	−0.949748	−	0.313015i	$-0.898661\pi$
$941$		−	19.2039i		−	0.626030i	0.949748	−	0.313015i	$-0.101339\pi$
$942$		0			0
$943$	−2.18238	+	2.18238i	−0.0710682	+	0.0710682i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1.78118	−	1.78118i	0.0578806	−	0.0578806i	−0.677574	−	0.735455i	$-0.736969\pi$
$947$	1.78118	−	1.78118i	0.0578806	−	0.0578806i	0.735455	+	0.677574i	$0.236969\pi$
$948$		0			0
$949$			2.81631i			0.0914212i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	2.74767	+	2.74767i	0.0890057	+	0.0890057i	0.750208	−	0.661202i	$-0.229953\pi$
$953$	2.74767	+	2.74767i	0.0890057	+	0.0890057i	−0.661202	+	0.750208i	$0.729953\pi$
$954$		0			0
$955$	−2.35140	−	15.7034i	−0.0760895	−	0.508150i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	1.82783			0.0590238
$960$		0			0
$961$	−25.4294			−0.820305
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−18.6527	+	25.2225i	−0.600452	+	0.811942i
$966$		0			0
$967$	−28.2804	−	28.2804i	−0.909437	−	0.909437i	0.0867898	−	0.996227i	$-0.472339\pi$
$967$	−28.2804	−	28.2804i	−0.909437	−	0.909437i	−0.996227	+	0.0867898i	$0.972339\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	0.597597i		−	0.0191778i	−0.999954	−	0.00958890i	$-0.996948\pi$
$971$		−	0.597597i		−	0.0191778i	0.999954	−	0.00958890i	$-0.00305229\pi$
$972$		0			0
$973$	7.20979	−	7.20979i	0.231135	−	0.231135i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	2.87856	−	2.87856i	0.0920933	−	0.0920933i	−0.659559	−	0.751653i	$-0.729257\pi$
$977$	2.87856	−	2.87856i	0.0920933	−	0.0920933i	0.751653	+	0.659559i	$0.229257\pi$
$978$		0			0
$979$		−	3.74291i		−	0.119624i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	15.3133	+	15.3133i	0.488418	+	0.488418i	0.907807	−	0.419389i	$-0.137756\pi$
$983$	15.3133	+	15.3133i	0.488418	+	0.488418i	−0.419389	+	0.907807i	$0.637756\pi$
$984$		0			0
$985$	−14.7534	+	19.9498i	−0.470082	+	0.635653i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	7.34019			0.233405
$990$		0			0
$991$	12.8447			0.408025			0.204013	−	0.978968i	$-0.434602\pi$
$991$	12.8447			0.408025			0.204013	+	0.978968i	$0.434602\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	1.67842	+	11.2090i	0.0532095	+	0.355350i
$996$		0			0
$997$	37.1287	+	37.1287i	1.17588	+	1.17588i	0.980785	+	0.195092i	$0.0625006\pi$
$997$	37.1287	+	37.1287i	1.17588	+	1.17588i	0.195092	+	0.980785i	$0.437499\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4140.2.s.a.737.2	✓	44
3.2	odd	2	inner	4140.2.s.a.737.21	yes	44
5.3	odd	4	inner	4140.2.s.a.2393.21	yes	44
15.8	even	4	inner	4140.2.s.a.2393.2	yes	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4140.2.s.a.737.2	✓	44	1.1	even	1	trivial
4140.2.s.a.737.21	yes	44	3.2	odd	2	inner
4140.2.s.a.2393.2	yes	44	15.8	even	4	inner
4140.2.s.a.2393.21	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)

\(f(q)\)	\(=\)	\(q+(-2.21141 + 0.331134i) q^{5} +(-1.17089 - 1.17089i) q^{7} +O(q^{10})\)	\(q+(-2.21141 + 0.331134i) q^{5} +(-1.17089 - 1.17089i) q^{7} +0.236966i q^{11} +(0.337253 - 0.337253i) q^{13} +(0.583846 - 0.583846i) q^{17} +2.36213i q^{19} +(0.707107 + 0.707107i) q^{23} +(4.78070 - 1.46455i) q^{25} +1.54355 q^{29} +2.36020 q^{31} +(2.97704 + 2.20160i) q^{35} +(4.63463 + 4.63463i) q^{37} +3.08636i q^{41} +(5.19030 - 5.19030i) q^{43} +(-3.52866 + 3.52866i) q^{47} -4.25805i q^{49} +(-8.94118 - 8.94118i) q^{53} +(-0.0784674 - 0.524030i) q^{55} -11.6852 q^{59} +1.02874 q^{61} +(-0.634130 + 0.857482i) q^{65} +(-9.64894 - 9.64894i) q^{67} +4.90457i q^{71} +(-4.17536 + 4.17536i) q^{73} +(0.277461 - 0.277461i) q^{77} +1.47706i q^{79} +(6.21585 + 6.21585i) q^{83} +(-1.09779 + 1.48446i) q^{85} -15.7951 q^{89} -0.789771 q^{91} +(-0.782180 - 5.22365i) q^{95} +(-5.69950 - 5.69950i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 12 q^{7}+O(q^{10}) \) 44 * q - 12 * q^7	\( 44 q - 12 q^{7} - 4 q^{13} + 24 q^{25} - 48 q^{37} + 8 q^{43} + 40 q^{55} - 96 q^{61} - 44 q^{67} + 76 q^{73} + 72 q^{85} - 48 q^{91} - 20 q^{97}+O(q^{100}) \) 44 * q - 12 * q^7 - 4 * q^13 + 24 * q^25 - 48 * q^37 + 8 * q^43 + 40 * q^55 - 96 * q^61 - 44 * q^67 + 76 * q^73 + 72 * q^85 - 48 * q^91 - 20 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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