LMFDB - Embedded newform 4140.2.s.b.737.4

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			737.4
Character	$\chi$	$=$	4140.737
Dual form			4140.2.s.b.2393.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.79713 + 1.33054i) q^{5} +(-0.390251 - 0.390251i) q^{7} +O(q^{10})$	$q+(-1.79713 + 1.33054i) q^{5} +(-0.390251 - 0.390251i) q^{7} +0.378344i q^{11} +(-3.06090 + 3.06090i) q^{13} +(5.04090 - 5.04090i) q^{17} -1.94855i q^{19} +(-0.707107 - 0.707107i) q^{23} +(1.45932 - 4.78230i) q^{25} -0.264721 q^{29} +0.305510 q^{31} +(1.22058 + 0.182086i) q^{35} +(1.13840 + 1.13840i) q^{37} +4.48650i q^{41} +(-2.13201 + 2.13201i) q^{43} +(5.21999 - 5.21999i) q^{47} -6.69541i q^{49} +(9.69083 + 9.69083i) q^{53} +(-0.503402 - 0.679931i) q^{55} +12.8410 q^{59} -11.2123 q^{61} +(1.42817 - 9.57349i) q^{65} +(-1.52032 - 1.52032i) q^{67} +11.3873i q^{71} +(-6.79549 + 6.79549i) q^{73} +(0.147649 - 0.147649i) q^{77} -3.47102i q^{79} +(-5.49333 - 5.49333i) q^{83} +(-2.35201 + 15.7662i) q^{85} +9.03991 q^{89} +2.38904 q^{91} +(2.59262 + 3.50178i) q^{95} +(0.595525 + 0.595525i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−1.79713	+	1.33054i	−0.803699	+	0.595036i
$5$	−1.79713	+	1.33054i	−0.803699	+	0.595036i
$6$		0			0
$7$	−0.390251	−	0.390251i	−0.147501	−	0.147501i	0.629500	−	0.777001i	$-0.283260\pi$
$7$	−0.390251	−	0.390251i	−0.147501	−	0.147501i	−0.777001	+	0.629500i	$0.783260\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			0.378344i			0.114075i	0.998372	+	0.0570375i	$0.0181655\pi$
$11$			0.378344i			0.114075i	−0.998372	+	0.0570375i	$0.981835\pi$
$12$		0			0
$13$	−3.06090	+	3.06090i	−0.848942	+	0.848942i	−0.990001	−	0.141059i	$-0.954949\pi$
$13$	−3.06090	+	3.06090i	−0.848942	+	0.848942i	0.141059	+	0.990001i	$0.454949\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	5.04090	−	5.04090i	1.22260	−	1.22260i	0.255892	−	0.966705i	$-0.417631\pi$
$17$	5.04090	−	5.04090i	1.22260	−	1.22260i	0.966705	−	0.255892i	$-0.0823690\pi$
$18$		0			0
$19$		−	1.94855i		−	0.447027i	−0.974701	−	0.223513i	$-0.928247\pi$
$19$		−	1.94855i		−	0.447027i	0.974701	−	0.223513i	$-0.0717527\pi$
$20$		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
$25$	1.45932	−	4.78230i	0.291865	−	0.956460i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−0.264721			−0.0491575			−0.0245787	−	0.999698i	$-0.507824\pi$
$29$	−0.264721			−0.0491575			−0.0245787	+	0.999698i	$0.507824\pi$
$30$		0			0
$31$	0.305510			0.0548713			0.0274356	−	0.999624i	$-0.491266\pi$
$31$	0.305510			0.0548713			0.0274356	+	0.999624i	$0.491266\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.22058	+	0.182086i	0.206315	+	0.0307781i
$36$		0			0
$37$	1.13840	+	1.13840i	0.187151	+	0.187151i	0.794463	−	0.607312i	$-0.207752\pi$
$37$	1.13840	+	1.13840i	0.187151	+	0.187151i	−0.607312	+	0.794463i	$0.707752\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			4.48650i			0.700673i	0.936624	+	0.350337i	$0.113933\pi$
$41$			4.48650i			0.700673i	−0.936624	+	0.350337i	$0.886067\pi$
$42$		0			0
$43$	−2.13201	+	2.13201i	−0.325128	+	0.325128i	−0.850731	−	0.525602i	$-0.823840\pi$
$43$	−2.13201	+	2.13201i	−0.325128	+	0.325128i	0.525602	+	0.850731i	$0.323840\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	5.21999	−	5.21999i	0.761414	−	0.761414i	−0.215164	−	0.976578i	$-0.569029\pi$
$47$	5.21999	−	5.21999i	0.761414	−	0.761414i	0.976578	+	0.215164i	$0.0690285\pi$
$48$		0			0
$49$		−	6.69541i		−	0.956487i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	9.69083	+	9.69083i	1.33114	+	1.33114i	0.904353	+	0.426786i	$0.140354\pi$
$53$	9.69083	+	9.69083i	1.33114	+	1.33114i	0.426786	+	0.904353i	$0.359646\pi$
$54$		0			0
$55$	−0.503402	−	0.679931i	−0.0678787	−	0.0916819i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	12.8410			1.67176			0.835879	−	0.548913i	$-0.184958\pi$
$59$	12.8410			1.67176			0.835879	+	0.548913i	$0.184958\pi$
$60$		0			0
$61$	−11.2123			−1.43558			−0.717792	−	0.696258i	$-0.754847\pi$
$61$	−11.2123			−1.43558			−0.717792	+	0.696258i	$0.754847\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.42817	−	9.57349i	0.177143	−	1.18744i
$66$		0			0
$67$	−1.52032	−	1.52032i	−0.185737	−	0.185737i	0.608113	−	0.793850i	$-0.291927\pi$
$67$	−1.52032	−	1.52032i	−0.185737	−	0.185737i	−0.793850	+	0.608113i	$0.791927\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			11.3873i			1.35142i	0.737167	+	0.675711i	$0.236163\pi$
$71$			11.3873i			1.35142i	−0.737167	+	0.675711i	$0.763837\pi$
$72$		0			0
$73$	−6.79549	+	6.79549i	−0.795352	+	0.795352i	−0.982359	−	0.187006i	$-0.940121\pi$
$73$	−6.79549	+	6.79549i	−0.795352	+	0.795352i	0.187006	+	0.982359i	$0.440121\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	0.147649	−	0.147649i	0.0168262	−	0.0168262i
$78$		0			0
$79$		−	3.47102i		−	0.390521i	−0.980751	−	0.195260i	$-0.937445\pi$
$79$		−	3.47102i		−	0.390521i	0.980751	−	0.195260i	$-0.0625552\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−5.49333	−	5.49333i	−0.602972	−	0.602972i	0.338128	−	0.941100i	$-0.390206\pi$
$83$	−5.49333	−	5.49333i	−0.602972	−	0.602972i	−0.941100	+	0.338128i	$0.890206\pi$
$84$		0			0
$85$	−2.35201	+	15.7662i	−0.255111	+	1.71009i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	9.03991			0.958228			0.479114	−	0.877753i	$-0.340958\pi$
$89$	9.03991			0.958228			0.479114	+	0.877753i	$0.340958\pi$
$90$		0			0
$91$	2.38904			0.250440
$92$		0			0
$93$		0			0
$94$		0			0
$95$	2.59262	+	3.50178i	0.265997	+	0.359275i
$96$		0			0
$97$	0.595525	+	0.595525i	0.0604664	+	0.0604664i	0.736693	−	0.676227i	$-0.236386\pi$
$97$	0.595525	+	0.595525i	0.0604664	+	0.0604664i	−0.676227	+	0.736693i	$0.736386\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			9.10005i			0.905489i	0.891640	+	0.452744i	$0.149555\pi$
$101$			9.10005i			0.905489i	−0.891640	+	0.452744i	$0.850445\pi$
$102$		0			0
$103$	−7.45740	+	7.45740i	−0.734799	+	0.734799i	−0.971566	−	0.236767i	$-0.923912\pi$
$103$	−7.45740	+	7.45740i	−0.734799	+	0.734799i	0.236767	+	0.971566i	$0.423912\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−12.9285	+	12.9285i	−1.24985	+	1.24985i	−0.294061	+	0.955787i	$0.595007\pi$
$107$	−12.9285	+	12.9285i	−1.24985	+	1.24985i	−0.955787	+	0.294061i	$0.904993\pi$
$108$		0			0
$109$			2.66333i			0.255100i	0.991832	+	0.127550i	$0.0407114\pi$
$109$			2.66333i			0.255100i	−0.991832	+	0.127550i	$0.959289\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	8.00681	+	8.00681i	0.753217	+	0.753217i	0.975078	−	0.221861i	$-0.0712132\pi$
$113$	8.00681	+	8.00681i	0.753217	+	0.753217i	−0.221861	+	0.975078i	$0.571213\pi$
$114$		0			0
$115$	2.21159	+	0.329926i	0.206232	+	0.0307657i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−3.93443			−0.360669
$120$		0			0
$121$	10.8569			0.986987
$122$		0			0
$123$		0			0
$124$		0			0
$125$	3.74046	+	10.5361i	0.334557	+	0.942376i
$126$		0			0
$127$	13.0117	+	13.0117i	1.15461	+	1.15461i	0.985618	+	0.168988i	$0.0540498\pi$
$127$	13.0117	+	13.0117i	1.15461	+	1.15461i	0.168988	+	0.985618i	$0.445950\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	18.5763i		−	1.62302i	−0.584340	−	0.811509i	$-0.698647\pi$
$131$		−	18.5763i		−	1.62302i	0.584340	−	0.811509i	$-0.301353\pi$
$132$		0			0
$133$	−0.760423	+	0.760423i	−0.0659370	+	0.0659370i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	2.91585	−	2.91585i	0.249118	−	0.249118i	−0.571491	−	0.820608i	$-0.693635\pi$
$137$	2.91585	−	2.91585i	0.249118	−	0.249118i	0.820608	+	0.571491i	$0.193635\pi$
$138$		0			0
$139$			17.6430i			1.49646i	0.663438	+	0.748231i	$0.269097\pi$
$139$			17.6430i			1.49646i	−0.663438	+	0.748231i	$0.730903\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1.15807	−	1.15807i	−0.0968430	−	0.0968430i
$144$		0			0
$145$	0.475737	−	0.352222i	0.0395078	−	0.0292505i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	1.68085			0.137701			0.0688504	−	0.997627i	$-0.478067\pi$
$149$	1.68085			0.137701			0.0688504	+	0.997627i	$0.478067\pi$
$150$		0			0
$151$	−2.42425			−0.197282			−0.0986412	−	0.995123i	$-0.531450\pi$
$151$	−2.42425			−0.197282			−0.0986412	+	0.995123i	$0.531450\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−0.549041	+	0.406494i	−0.0441000	+	0.0326504i
$156$		0			0
$157$	6.83622	+	6.83622i	0.545590	+	0.545590i	0.925162	−	0.379572i	$-0.123929\pi$
$157$	6.83622	+	6.83622i	0.545590	+	0.545590i	−0.379572	+	0.925162i	$0.623929\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			0.551899i			0.0434957i
$162$		0			0
$163$	−14.6273	+	14.6273i	−1.14570	+	1.14570i	−0.158311	+	0.987389i	$0.550605\pi$
$163$	−14.6273	+	14.6273i	−1.14570	+	1.14570i	−0.987389	+	0.158311i	$0.949395\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	12.3506	−	12.3506i	0.955721	−	0.955721i	−0.0433391	−	0.999060i	$-0.513800\pi$
$167$	12.3506	−	12.3506i	0.955721	−	0.955721i	0.999060	+	0.0433391i	$0.0137996\pi$
$168$		0			0
$169$		−	5.73826i		−	0.441404i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1.05676	−	1.05676i	−0.0803443	−	0.0803443i	0.665793	−	0.746137i	$-0.268094\pi$
$173$	−1.05676	−	1.05676i	−0.0803443	−	0.0803443i	−0.746137	+	0.665793i	$0.768094\pi$
$174$		0			0
$175$	−2.43580	+	1.29680i	−0.184129	+	0.0980286i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	21.8585			1.63378			0.816891	−	0.576791i	$-0.195695\pi$
$179$	21.8585			1.63378			0.816891	+	0.576791i	$0.195695\pi$
$180$		0			0
$181$	22.2558			1.65426			0.827131	−	0.562009i	$-0.189971\pi$
$181$	22.2558			1.65426			0.827131	+	0.562009i	$0.189971\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−3.56053	−	0.531159i	−0.261775	−	0.0390516i
$186$		0			0
$187$	1.90719	+	1.90719i	0.139468	+	0.139468i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	5.59932i		−	0.405152i	−0.979267	−	0.202576i	$-0.935069\pi$
$191$		−	5.59932i		−	0.405152i	0.979267	−	0.202576i	$-0.0649314\pi$
$192$		0			0
$193$	8.98826	−	8.98826i	0.646989	−	0.646989i	−0.305275	−	0.952264i	$-0.598748\pi$
$193$	8.98826	−	8.98826i	0.646989	−	0.646989i	0.952264	+	0.305275i	$0.0987485\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−12.8838	+	12.8838i	−0.917935	+	0.917935i	−0.996879	−	0.0789441i	$-0.974845\pi$
$197$	−12.8838	+	12.8838i	−0.917935	+	0.917935i	0.0789441	+	0.996879i	$0.474845\pi$
$198$		0			0
$199$			0.592145i			0.0419761i	0.999780	+	0.0209880i	$0.00668119\pi$
$199$			0.592145i			0.0419761i	−0.999780	+	0.0209880i	$0.993319\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	0.103308	+	0.103308i	0.00725079	+	0.00725079i
$204$		0			0
$205$	−5.96947	−	8.06280i	−0.416926	−	0.563130i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	0.737220			0.0509946
$210$		0			0
$211$	−17.8701			−1.23023			−0.615114	−	0.788438i	$-0.710890\pi$
$211$	−17.8701			−1.23023			−0.615114	+	0.788438i	$0.710890\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	0.994764	−	6.66822i	0.0678424	−	0.454769i
$216$		0			0
$217$	−0.119226	−	0.119226i	−0.00809358	−	0.00809358i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			30.8594i			2.07583i
$222$		0			0
$223$	4.20692	−	4.20692i	0.281717	−	0.281717i	−0.552077	−	0.833793i	$-0.686164\pi$
$223$	4.20692	−	4.20692i	0.281717	−	0.281717i	0.833793	+	0.552077i	$0.186164\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	10.8073	−	10.8073i	0.717303	−	0.717303i	−0.250749	−	0.968052i	$-0.580677\pi$
$227$	10.8073	−	10.8073i	0.717303	−	0.717303i	0.968052	+	0.250749i	$0.0806769\pi$
$228$		0			0
$229$			23.9329i			1.58153i	0.612120	+	0.790765i	$0.290317\pi$
$229$			23.9329i			1.58153i	−0.612120	+	0.790765i	$0.709683\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	4.95547	+	4.95547i	0.324644	+	0.324644i	0.850545	−	0.525902i	$-0.176272\pi$
$233$	4.95547	+	4.95547i	0.324644	+	0.324644i	−0.525902	+	0.850545i	$0.676272\pi$
$234$		0			0
$235$	−2.43557	+	16.3264i	−0.158879	+	1.06502i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	19.8752			1.28562			0.642810	−	0.766025i	$-0.277768\pi$
$239$	19.8752			1.28562			0.642810	+	0.766025i	$0.277768\pi$
$240$		0			0
$241$	−0.195792			−0.0126121			−0.00630605	−	0.999980i	$-0.502007\pi$
$241$	−0.195792			−0.0126121			−0.00630605	+	0.999980i	$0.502007\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	8.90851	+	12.0325i	0.569144	+	0.768728i
$246$		0			0
$247$	5.96431	+	5.96431i	0.379500	+	0.379500i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			0.592680i			0.0374097i	0.999825	+	0.0187048i	$0.00595428\pi$
$251$			0.592680i			0.0374097i	−0.999825	+	0.0187048i	$0.994046\pi$
$252$		0			0
$253$	0.267529	−	0.267529i	0.0168194	−	0.0168194i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−17.9835	+	17.9835i	−1.12178	+	1.12178i	−0.130305	+	0.991474i	$0.541596\pi$
$257$	−17.9835	+	17.9835i	−1.12178	+	1.12178i	−0.991474	+	0.130305i	$0.958404\pi$
$258$		0			0
$259$		−	0.888523i		−	0.0552101i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−0.211954	−	0.211954i	−0.0130696	−	0.0130696i	0.700542	−	0.713611i	$-0.252942\pi$
$263$	−0.211954	−	0.211954i	−0.0130696	−	0.0130696i	−0.713611	+	0.700542i	$0.752942\pi$
$264$		0			0
$265$	−30.3097	−	4.52160i	−1.86191	−	0.277760i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	12.1274			0.739419			0.369710	−	0.929147i	$-0.379457\pi$
$269$	12.1274			0.739419			0.369710	+	0.929147i	$0.379457\pi$
$270$		0			0
$271$	−15.7837			−0.958793			−0.479396	−	0.877598i	$-0.659144\pi$
$271$	−15.7837			−0.958793			−0.479396	+	0.877598i	$0.659144\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.80935	+	0.552126i	0.109108	+	0.0332944i
$276$		0			0
$277$	−12.3108	−	12.3108i	−0.739683	−	0.739683i	0.232834	−	0.972517i	$-0.425200\pi$
$277$	−12.3108	−	12.3108i	−0.739683	−	0.739683i	−0.972517	+	0.232834i	$0.925200\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	11.1437i		−	0.664778i	−0.943142	−	0.332389i	$-0.892145\pi$
$281$		−	11.1437i		−	0.664778i	0.943142	−	0.332389i	$-0.107855\pi$
$282$		0			0
$283$	16.9336	−	16.9336i	1.00660	−	1.00660i	0.00661907	−	0.999978i	$-0.497893\pi$
$283$	16.9336	−	16.9336i	1.00660	−	1.00660i	0.999978	−	0.00661907i	$-0.00210693\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	1.75086	−	1.75086i	0.103350	−	0.103350i
$288$		0			0
$289$		−	33.8213i		−	1.98949i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	6.64771	+	6.64771i	0.388363	+	0.388363i	0.874103	−	0.485740i	$-0.161450\pi$
$293$	6.64771	+	6.64771i	0.388363	+	0.388363i	−0.485740	+	0.874103i	$0.661450\pi$
$294$		0			0
$295$	−23.0769	+	17.0855i	−1.34359	+	0.994757i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	4.32877			0.250339
$300$		0			0
$301$	1.66404			0.0959137
$302$		0			0
$303$		0			0
$304$		0			0
$305$	20.1499	−	14.9184i	1.15378	−	0.854224i
$306$		0			0
$307$	22.1708	+	22.1708i	1.26535	+	1.26535i	0.948462	+	0.316890i	$0.102639\pi$
$307$	22.1708	+	22.1708i	1.26535	+	1.26535i	0.316890	+	0.948462i	$0.397361\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			9.08384i			0.515097i	0.966265	+	0.257549i	$0.0829148\pi$
$311$			9.08384i			0.515097i	−0.966265	+	0.257549i	$0.917085\pi$
$312$		0			0
$313$	−22.6175	+	22.6175i	−1.27842	+	1.27842i	−0.336865	+	0.941553i	$0.609366\pi$
$313$	−22.6175	+	22.6175i	−1.27842	+	1.27842i	−0.941553	+	0.336865i	$0.890634\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−21.7782	+	21.7782i	−1.22318	+	1.22318i	−0.256689	+	0.966494i	$0.582632\pi$
$317$	−21.7782	+	21.7782i	−1.22318	+	1.22318i	−0.966494	+	0.256689i	$0.917368\pi$
$318$		0			0
$319$		−	0.100156i		−	0.00560764i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−9.82241	−	9.82241i	−0.546534	−	0.546534i
$324$		0			0
$325$	10.1713	+	19.1050i	0.564203	+	1.05975i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−4.07422			−0.224619
$330$		0			0
$331$	30.1036			1.65464			0.827320	−	0.561730i	$-0.189864\pi$
$331$	30.1036			1.65464			0.827320	+	0.561730i	$0.189864\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	4.75506	+	0.709360i	0.259797	+	0.0387565i
$336$		0			0
$337$	−12.0825	−	12.0825i	−0.658178	−	0.658178i	0.296771	−	0.954949i	$-0.404090\pi$
$337$	−12.0825	−	12.0825i	−0.658178	−	0.658178i	−0.954949	+	0.296771i	$0.904090\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			0.115588i			0.00625944i
$342$		0			0
$343$	−5.34465	+	5.34465i	−0.288584	+	0.288584i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−8.56567	+	8.56567i	−0.459829	+	0.459829i	−0.898599	−	0.438770i	$-0.855414\pi$
$347$	−8.56567	+	8.56567i	−0.459829	+	0.459829i	0.438770	+	0.898599i	$0.355414\pi$
$348$		0			0
$349$			17.3738i			0.929997i	0.885311	+	0.464999i	$0.153945\pi$
$349$			17.3738i			0.929997i	−0.885311	+	0.464999i	$0.846055\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	7.10209	+	7.10209i	0.378006	+	0.378006i	0.870382	−	0.492376i	$-0.163872\pi$
$353$	7.10209	+	7.10209i	0.378006	+	0.378006i	−0.492376	+	0.870382i	$0.663872\pi$
$354$		0			0
$355$	−15.1512	−	20.4644i	−0.804144	−	1.08614i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	0.778254			0.0410747			0.0205373	−	0.999789i	$-0.493462\pi$
$359$	0.778254			0.0410747			0.0205373	+	0.999789i	$0.493462\pi$
$360$		0			0
$361$	15.2032			0.800167
$362$		0			0
$363$		0			0
$364$		0			0
$365$	3.17068	−	21.2540i	0.165961	−	1.11249i
$366$		0			0
$367$	14.8512	+	14.8512i	0.775228	+	0.775228i	0.979015	−	0.203787i	$-0.0653250\pi$
$367$	14.8512	+	14.8512i	0.775228	+	0.775228i	−0.203787	+	0.979015i	$0.565325\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	7.56373i		−	0.392689i
$372$		0			0
$373$	−2.72301	+	2.72301i	−0.140992	+	0.140992i	−0.774080	−	0.633088i	$-0.781787\pi$
$373$	−2.72301	+	2.72301i	−0.140992	+	0.140992i	0.633088	+	0.774080i	$0.281787\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	0.810286	−	0.810286i	0.0417318	−	0.0417318i
$378$		0			0
$379$		−	10.2608i		−	0.527062i	−0.964651	−	0.263531i	$-0.915113\pi$
$379$		−	10.2608i		−	0.527062i	0.964651	−	0.263531i	$-0.0848871\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	14.9668	+	14.9668i	0.764768	+	0.764768i	0.977180	−	0.212412i	$-0.0681319\pi$
$383$	14.9668	+	14.9668i	0.764768	+	0.764768i	−0.212412	+	0.977180i	$0.568132\pi$
$384$		0			0
$385$	−0.0688909	+	0.461798i	−0.00351101	+	0.0235354i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−1.98218			−0.100500			−0.0502502	−	0.998737i	$-0.516002\pi$
$389$	−1.98218			−0.100500			−0.0502502	+	0.998737i	$0.516002\pi$
$390$		0			0
$391$	−7.12890			−0.360524
$392$		0			0
$393$		0			0
$394$		0			0
$395$	4.61834	+	6.23787i	0.232374	+	0.313861i
$396$		0			0
$397$	−5.27555	−	5.27555i	−0.264772	−	0.264772i	0.562217	−	0.826990i	$-0.309948\pi$
$397$	−5.27555	−	5.27555i	−0.264772	−	0.264772i	−0.826990	+	0.562217i	$0.809948\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			15.5574i			0.776900i	0.921470	+	0.388450i	$0.126989\pi$
$401$			15.5574i			0.776900i	−0.921470	+	0.388450i	$0.873011\pi$
$402$		0			0
$403$	−0.935138	+	0.935138i	−0.0465825	+	0.0465825i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−0.430706	+	0.430706i	−0.0213493	+	0.0213493i
$408$		0			0
$409$		−	34.0115i		−	1.68176i	−0.541220	−	0.840881i	$-0.682037\pi$
$409$		−	34.0115i		−	1.68176i	0.541220	−	0.840881i	$-0.317963\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−5.01123	−	5.01123i	−0.246586	−	0.246586i
$414$		0			0
$415$	17.1813	+	2.56311i	0.843397	+	0.125818i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	13.7236			0.670442			0.335221	−	0.942140i	$-0.391189\pi$
$419$	13.7236			0.670442			0.335221	+	0.942140i	$0.391189\pi$
$420$		0			0
$421$	−15.9539			−0.777547			−0.388773	−	0.921333i	$-0.627101\pi$
$421$	−15.9539			−0.777547			−0.388773	+	0.921333i	$0.627101\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−16.7508	−	31.4634i	−0.812532	−	1.52620i
$426$		0			0
$427$	4.37560	+	4.37560i	0.211750	+	0.211750i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	33.3406i		−	1.60596i	−0.596005	−	0.802981i	$-0.703246\pi$
$431$		−	33.3406i		−	1.60596i	0.596005	−	0.802981i	$-0.296754\pi$
$432$		0			0
$433$	5.93869	−	5.93869i	0.285395	−	0.285395i	−0.549861	−	0.835256i	$-0.685319\pi$
$433$	5.93869	−	5.93869i	0.285395	−	0.285395i	0.835256	+	0.549861i	$0.185319\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−1.37783	+	1.37783i	−0.0659105	+	0.0659105i
$438$		0			0
$439$		−	11.0639i		−	0.528053i	−0.964515	−	0.264026i	$-0.914949\pi$
$439$		−	11.0639i		−	0.528053i	0.964515	−	0.264026i	$-0.0850506\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	9.62107	+	9.62107i	0.457111	+	0.457111i	0.897706	−	0.440595i	$-0.145232\pi$
$443$	9.62107	+	9.62107i	0.457111	+	0.457111i	−0.440595	+	0.897706i	$0.645232\pi$
$444$		0			0
$445$	−16.2458	+	12.0280i	−0.770127	+	0.570180i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−21.0299			−0.992462			−0.496231	−	0.868191i	$-0.665283\pi$
$449$	−21.0299			−0.992462			−0.496231	+	0.868191i	$0.665283\pi$
$450$		0			0
$451$	−1.69744			−0.0799293
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−4.29341	+	3.17872i	−0.201278	+	0.149021i
$456$		0			0
$457$	−8.72296	−	8.72296i	−0.408043	−	0.408043i	0.473013	−	0.881056i	$-0.343166\pi$
$457$	−8.72296	−	8.72296i	−0.408043	−	0.408043i	−0.881056	+	0.473013i	$0.843166\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			4.09765i			0.190846i	0.995437	+	0.0954232i	$0.0304204\pi$
$461$			4.09765i			0.190846i	−0.995437	+	0.0954232i	$0.969580\pi$
$462$		0			0
$463$	14.6742	−	14.6742i	0.681966	−	0.681966i	−0.278477	−	0.960443i	$-0.589830\pi$
$463$	14.6742	−	14.6742i	0.681966	−	0.681966i	0.960443	+	0.278477i	$0.0898296\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	11.3781	−	11.3781i	0.526513	−	0.526513i	−0.393018	−	0.919531i	$-0.628569\pi$
$467$	11.3781	−	11.3781i	0.526513	−	0.526513i	0.919531	+	0.393018i	$0.128569\pi$
$468$		0			0
$469$			1.18662i			0.0547928i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−0.806633	−	0.806633i	−0.0370890	−	0.0370890i
$474$		0			0
$475$	−9.31852	−	2.84356i	−0.427563	−	0.130471i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	18.2134			0.832190			0.416095	−	0.909321i	$-0.363398\pi$
$479$	18.2134			0.832190			0.416095	+	0.909321i	$0.363398\pi$
$480$		0			0
$481$	−6.96905			−0.317761
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−1.86260	−	0.277863i	−0.0845765	−	0.0126171i
$486$		0			0
$487$	7.72112	+	7.72112i	0.349877	+	0.349877i	0.860064	−	0.510187i	$-0.170424\pi$
$487$	7.72112	+	7.72112i	0.349877	+	0.349877i	−0.510187	+	0.860064i	$0.670424\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	5.14636i		−	0.232252i	−0.993234	−	0.116126i	$-0.962952\pi$
$491$		−	5.14636i		−	0.232252i	0.993234	−	0.116126i	$-0.0370477\pi$
$492$		0			0
$493$	−1.33443	+	1.33443i	−0.0600998	+	0.0600998i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	4.44390	−	4.44390i	0.199336	−	0.199336i
$498$		0			0
$499$		−	4.79931i		−	0.214847i	−0.994213	−	0.107423i	$-0.965740\pi$
$499$		−	4.79931i		−	0.214847i	0.994213	−	0.107423i	$-0.0342600\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−27.0724	−	27.0724i	−1.20710	−	1.20710i	−0.971962	−	0.235139i	$-0.924446\pi$
$503$	−27.0724	−	27.0724i	−1.20710	−	1.20710i	−0.235139	−	0.971962i	$-0.575554\pi$
$504$		0			0
$505$	−12.1080	−	16.3539i	−0.538798	−	0.727741i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	17.3756			0.770160			0.385080	−	0.922883i	$-0.374174\pi$
$509$	17.3756			0.770160			0.385080	+	0.922883i	$0.374174\pi$
$510$		0			0
$511$	5.30390			0.234631
$512$		0			0
$513$		0			0
$514$		0			0
$515$	3.47951	−	23.3243i	0.153326	−	1.02779i
$516$		0			0
$517$	1.97495	+	1.97495i	0.0868583	+	0.0868583i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			19.4972i			0.854189i	0.904207	+	0.427094i	$0.140463\pi$
$521$			19.4972i			0.854189i	−0.904207	+	0.427094i	$0.859537\pi$
$522$		0			0
$523$	15.7092	−	15.7092i	0.686913	−	0.686913i	−0.274635	−	0.961549i	$-0.588557\pi$
$523$	15.7092	−	15.7092i	0.686913	−	0.686913i	0.961549	+	0.274635i	$0.0885571\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	1.54005	−	1.54005i	0.0670855	−	0.0670855i
$528$		0			0
$529$			1.00000i			0.0434783i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−13.7327	−	13.7327i	−0.594831	−	0.594831i
$534$		0			0
$535$	6.03226	−	40.4361i	0.260797	−	1.74821i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	2.53317			0.109111
$540$		0			0
$541$	19.8334			0.852705			0.426352	−	0.904557i	$-0.359798\pi$
$541$	19.8334			0.852705			0.426352	+	0.904557i	$0.359798\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−3.54367	−	4.78633i	−0.151794	−	0.205024i
$546$		0			0
$547$	−16.7996	−	16.7996i	−0.718298	−	0.718298i	0.249959	−	0.968257i	$-0.419583\pi$
$547$	−16.7996	−	16.7996i	−0.718298	−	0.718298i	−0.968257	+	0.249959i	$0.919583\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			0.515821i			0.0219747i
$552$		0			0
$553$	−1.35457	+	1.35457i	−0.0576023	+	0.0576023i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	2.18914	−	2.18914i	0.0927568	−	0.0927568i	−0.659206	−	0.751963i	$-0.729107\pi$
$557$	2.18914	−	2.18914i	0.0927568	−	0.0927568i	0.751963	+	0.659206i	$0.229107\pi$
$558$		0			0
$559$		−	13.0518i		−	0.552030i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	6.06369	+	6.06369i	0.255554	+	0.255554i	0.823243	−	0.567689i	$-0.192162\pi$
$563$	6.06369	+	6.06369i	0.255554	+	0.255554i	−0.567689	+	0.823243i	$0.692162\pi$
$564$		0			0
$565$	−25.0426	−	3.73586i	−1.05355	−	0.157169i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	1.41183			0.0591872			0.0295936	−	0.999562i	$-0.490579\pi$
$569$	1.41183			0.0591872			0.0295936	+	0.999562i	$0.490579\pi$
$570$		0			0
$571$	18.0732			0.756341			0.378170	−	0.925736i	$-0.376553\pi$
$571$	18.0732			0.756341			0.378170	+	0.925736i	$0.376553\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−4.41349	+	2.34970i	−0.184055	+	0.0979892i
$576$		0			0
$577$	9.43120	+	9.43120i	0.392626	+	0.392626i	0.875622	−	0.482996i	$-0.160452\pi$
$577$	9.43120	+	9.43120i	0.392626	+	0.392626i	−0.482996	+	0.875622i	$0.660452\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			4.28756i			0.177878i
$582$		0			0
$583$	−3.66647	+	3.66647i	−0.151850	+	0.151850i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	9.35721	−	9.35721i	0.386213	−	0.386213i	−0.487121	−	0.873334i	$-0.661953\pi$
$587$	9.35721	−	9.35721i	0.386213	−	0.386213i	0.873334	+	0.487121i	$0.161953\pi$
$588$		0			0
$589$		−	0.595301i		−	0.0245289i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	1.93801	+	1.93801i	0.0795847	+	0.0795847i	0.745779	−	0.666194i	$-0.232078\pi$
$593$	1.93801	+	1.93801i	0.0795847	+	0.0795847i	−0.666194	+	0.745779i	$0.732078\pi$
$594$		0			0
$595$	7.07067	−	5.23493i	0.289869	−	0.214611i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	32.7542			1.33830			0.669151	−	0.743126i	$-0.266658\pi$
$599$	32.7542			1.33830			0.669151	+	0.743126i	$0.266658\pi$
$600$		0			0
$601$	0.394168			0.0160785			0.00803923	−	0.999968i	$-0.497441\pi$
$601$	0.394168			0.0160785			0.00803923	+	0.999968i	$0.497441\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−19.5111	+	14.4455i	−0.793241	+	0.587293i
$606$		0			0
$607$	15.0119	+	15.0119i	0.609314	+	0.609314i	0.942767	−	0.333453i	$-0.108214\pi$
$607$	15.0119	+	15.0119i	0.609314	+	0.609314i	−0.333453	+	0.942767i	$0.608214\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			31.9558i			1.29279i
$612$		0			0
$613$	−14.1990	+	14.1990i	−0.573494	+	0.573494i	−0.933103	−	0.359609i	$-0.882910\pi$
$613$	−14.1990	+	14.1990i	−0.573494	+	0.573494i	0.359609	+	0.933103i	$0.382910\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	1.73614	−	1.73614i	0.0698942	−	0.0698942i	−0.671296	−	0.741190i	$-0.734262\pi$
$617$	1.73614	−	1.73614i	0.0698942	−	0.0698942i	0.741190	+	0.671296i	$0.234262\pi$
$618$		0			0
$619$		−	44.8347i		−	1.80206i	−0.433757	−	0.901030i	$-0.642812\pi$
$619$		−	44.8347i		−	1.80206i	0.433757	−	0.901030i	$-0.357188\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−3.52784	−	3.52784i	−0.141340	−	0.141340i
$624$		0			0
$625$	−20.7408	−	13.9578i	−0.829630	−	0.558313i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	11.4771			0.457621
$630$		0			0
$631$	24.5581			0.977641			0.488820	−	0.872384i	$-0.337427\pi$
$631$	24.5581			0.977641			0.488820	+	0.872384i	$0.337427\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−40.6964	−	6.07109i	−1.61499	−	0.240924i
$636$		0			0
$637$	20.4940	+	20.4940i	0.812002	+	0.812002i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			29.7042i			1.17324i	0.809861	+	0.586622i	$0.199543\pi$
$641$			29.7042i			1.17324i	−0.809861	+	0.586622i	$0.800457\pi$
$642$		0			0
$643$	−29.7459	+	29.7459i	−1.17306	+	1.17306i	−0.191588	+	0.981475i	$0.561364\pi$
$643$	−29.7459	+	29.7459i	−1.17306	+	1.17306i	−0.981475	+	0.191588i	$0.938636\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−21.8748	+	21.8748i	−0.859989	+	0.859989i	−0.991336	−	0.131347i	$-0.958070\pi$
$647$	−21.8748	+	21.8748i	−0.859989	+	0.859989i	0.131347	+	0.991336i	$0.458070\pi$
$648$		0			0
$649$			4.85832i			0.190706i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−17.3774	−	17.3774i	−0.680029	−	0.680029i	0.279978	−	0.960007i	$-0.409673\pi$
$653$	−17.3774	−	17.3774i	−0.680029	−	0.680029i	−0.960007	+	0.279978i	$0.909673\pi$
$654$		0			0
$655$	24.7165	+	33.3839i	0.965754	+	1.30442i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−45.3952			−1.76834			−0.884172	−	0.467161i	$-0.845277\pi$
$659$	−45.3952			−1.76834			−0.884172	+	0.467161i	$0.845277\pi$
$660$		0			0
$661$	−25.1430			−0.977949			−0.488975	−	0.872298i	$-0.662629\pi$
$661$	−25.1430			−0.977949			−0.488975	+	0.872298i	$0.662629\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	0.354802	−	2.37835i	0.0137586	−	0.0922284i
$666$		0			0
$667$	0.187186	+	0.187186i	0.00724787	+	0.00724787i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	4.24209i		−	0.163764i
$672$		0			0
$673$	−16.1263	+	16.1263i	−0.621623	+	0.621623i	−0.945946	−	0.324323i	$-0.894863\pi$
$673$	−16.1263	+	16.1263i	−0.621623	+	0.621623i	0.324323	+	0.945946i	$0.394863\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	20.9685	−	20.9685i	0.805885	−	0.805885i	−0.178123	−	0.984008i	$-0.557003\pi$
$677$	20.9685	−	20.9685i	0.805885	−	0.805885i	0.984008	+	0.178123i	$0.0570026\pi$
$678$		0			0
$679$		−	0.464809i		−	0.0178377i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−13.2992	−	13.2992i	−0.508879	−	0.508879i	0.405303	−	0.914182i	$-0.367166\pi$
$683$	−13.2992	−	13.2992i	−0.508879	−	0.508879i	−0.914182	+	0.405303i	$0.867166\pi$
$684$		0			0
$685$	−1.36049	+	9.11979i	−0.0519817	+	0.348449i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−59.3254			−2.26012
$690$		0			0
$691$	23.7767			0.904507			0.452254	−	0.891889i	$-0.350620\pi$
$691$	23.7767			0.904507			0.452254	+	0.891889i	$0.350620\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−23.4748	−	31.7068i	−0.890449	−	1.20271i
$696$		0			0
$697$	22.6160	+	22.6160i	0.856641	+	0.856641i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	4.72768i		−	0.178562i	−0.996006	−	0.0892811i	$-0.971543\pi$
$701$		−	4.72768i		−	0.178562i	0.996006	−	0.0892811i	$-0.0284569\pi$
$702$		0			0
$703$	2.21822	−	2.21822i	0.0836617	−	0.0836617i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	3.55131	−	3.55131i	0.133561	−	0.133561i
$708$		0			0
$709$		−	41.0895i		−	1.54315i	−0.636139	−	0.771574i	$-0.719470\pi$
$709$		−	41.0895i		−	1.54315i	0.636139	−	0.771574i	$-0.280530\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−0.216028	−	0.216028i	−0.00809033	−	0.00809033i
$714$		0			0
$715$	3.62207	+	0.540340i	0.135458	+	0.0202076i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−19.3520			−0.721708			−0.360854	−	0.932622i	$-0.617515\pi$
$719$	−19.3520			−0.721708			−0.360854	+	0.932622i	$0.617515\pi$
$720$		0			0
$721$	5.82052			0.216768
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−0.386314	+	1.26598i	−0.0143473	+	0.0470171i
$726$		0			0
$727$	2.58374	+	2.58374i	0.0958258	+	0.0958258i	0.753395	−	0.657569i	$-0.228415\pi$
$727$	2.58374	+	2.58374i	0.0958258	+	0.0958258i	−0.657569	+	0.753395i	$0.728415\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			21.4945i			0.795002i
$732$		0			0
$733$	−15.5843	+	15.5843i	−0.575618	+	0.575618i	−0.933693	−	0.358075i	$-0.883433\pi$
$733$	−15.5843	+	15.5843i	−0.575618	+	0.575618i	0.358075	+	0.933693i	$0.383433\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	0.575204	−	0.575204i	0.0211879	−	0.0211879i
$738$		0			0
$739$			15.9119i			0.585330i	0.956215	+	0.292665i	$0.0945422\pi$
$739$			15.9119i			0.585330i	−0.956215	+	0.292665i	$0.905458\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	13.5394	+	13.5394i	0.496714	+	0.496714i	0.910414	−	0.413699i	$-0.135764\pi$
$743$	13.5394	+	13.5394i	0.496714	+	0.496714i	−0.413699	+	0.910414i	$0.635764\pi$
$744$		0			0
$745$	−3.02071	+	2.23644i	−0.110670	+	0.0819370i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	10.0907			0.368708
$750$		0			0
$751$	32.2409			1.17649			0.588244	−	0.808684i	$-0.299820\pi$
$751$	32.2409			1.17649			0.588244	+	0.808684i	$0.299820\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	4.35667	−	3.22556i	0.158556	−	0.117390i
$756$		0			0
$757$	−18.4185	−	18.4185i	−0.669432	−	0.669432i	0.288153	−	0.957585i	$-0.406959\pi$
$757$	−18.4185	−	18.4185i	−0.669432	−	0.669432i	−0.957585	+	0.288153i	$0.906959\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			46.8459i			1.69816i	0.528263	+	0.849081i	$0.322843\pi$
$761$			46.8459i			1.69816i	−0.528263	+	0.849081i	$0.677157\pi$
$762$		0			0
$763$	1.03937	−	1.03937i	0.0376276	−	0.0376276i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−39.3051	+	39.3051i	−1.41923	+	1.41923i
$768$		0			0
$769$		−	10.6033i		−	0.382364i	−0.981555	−	0.191182i	$-0.938768\pi$
$769$		−	10.6033i		−	0.382364i	0.981555	−	0.191182i	$-0.0612321\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−5.69554	−	5.69554i	−0.204854	−	0.204854i	0.597222	−	0.802076i	$-0.296271\pi$
$773$	−5.69554	−	5.69554i	−0.204854	−	0.204854i	−0.802076	+	0.597222i	$0.796271\pi$
$774$		0			0
$775$	0.445838	−	1.46104i	0.0160150	−	0.0524822i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	8.74214			0.313220
$780$		0			0
$781$	−4.30831			−0.154163
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−21.3814	−	3.18968i	−0.763135	−	0.113844i
$786$		0			0
$787$	33.6861	+	33.6861i	1.20078	+	1.20078i	0.973929	+	0.226851i	$0.0728430\pi$
$787$	33.6861	+	33.6861i	1.20078	+	1.20078i	0.226851	+	0.973929i	$0.427157\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	6.24934i		−	0.222201i
$792$		0			0
$793$	34.3197	−	34.3197i	1.21873	−	1.21873i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	0.0466887	−	0.0466887i	0.00165380	−	0.00165380i	−0.706279	−	0.707933i	$-0.749628\pi$
$797$	0.0466887	−	0.0466887i	0.00165380	−	0.00165380i	0.707933	+	0.706279i	$0.249628\pi$
$798$		0			0
$799$		−	52.6269i		−	1.86181i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−2.57103	−	2.57103i	−0.0907298	−	0.0907298i
$804$		0			0
$805$	−0.734324	−	0.991832i	−0.0258815	−	0.0349575i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−10.4158			−0.366201			−0.183101	−	0.983094i	$-0.558613\pi$
$809$	−10.4158			−0.366201			−0.183101	+	0.983094i	$0.558613\pi$
$810$		0			0
$811$	35.1939			1.23582			0.617912	−	0.786247i	$-0.287979\pi$
$811$	35.1939			1.23582			0.617912	+	0.786247i	$0.287979\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	6.82489	−	45.7494i	0.239065	−	1.60253i
$816$		0			0
$817$	4.15432	+	4.15432i	0.145341	+	0.145341i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			19.1595i			0.668671i	0.942454	+	0.334336i	$0.108512\pi$
$821$			19.1595i			0.668671i	−0.942454	+	0.334336i	$0.891488\pi$
$822$		0			0
$823$	−4.10453	+	4.10453i	−0.143075	+	0.143075i	−0.775016	−	0.631941i	$-0.782258\pi$
$823$	−4.10453	+	4.10453i	−0.143075	+	0.143075i	0.631941	+	0.775016i	$0.282258\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−22.5903	+	22.5903i	−0.785541	+	0.785541i	−0.980760	−	0.195218i	$-0.937458\pi$
$827$	−22.5903	+	22.5903i	−0.785541	+	0.785541i	0.195218	+	0.980760i	$0.437458\pi$
$828$		0			0
$829$			3.14301i			0.109161i	0.998509	+	0.0545806i	$0.0173822\pi$
$829$			3.14301i			0.109161i	−0.998509	+	0.0545806i	$0.982618\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−33.7509	−	33.7509i	−1.16940	−	1.16940i
$834$		0			0
$835$	−5.76263	+	38.6287i	−0.199424	+	1.33680i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	26.5815			0.917696			0.458848	−	0.888515i	$-0.348262\pi$
$839$	26.5815			0.917696			0.458848	+	0.888515i	$0.348262\pi$
$840$		0			0
$841$	−28.9299			−0.997584
$842$		0			0
$843$		0			0
$844$		0			0
$845$	7.63499	+	10.3124i	0.262652	+	0.354756i
$846$		0			0
$847$	−4.23690	−	4.23690i	−0.145582	−	0.145582i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	1.60994i		−	0.0551879i
$852$		0			0
$853$	−11.9932	+	11.9932i	−0.410639	+	0.410639i	−0.881961	−	0.471322i	$-0.843777\pi$
$853$	−11.9932	+	11.9932i	−0.410639	+	0.410639i	0.471322	+	0.881961i	$0.343777\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−3.94081	+	3.94081i	−0.134616	+	0.134616i	−0.771204	−	0.636588i	$-0.780345\pi$
$857$	−3.94081	+	3.94081i	−0.134616	+	0.134616i	0.636588	+	0.771204i	$0.280345\pi$
$858$		0			0
$859$		−	55.4201i		−	1.89091i	−0.325756	−	0.945454i	$-0.605619\pi$
$859$		−	55.4201i		−	1.89091i	0.325756	−	0.945454i	$-0.394381\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−14.2660	−	14.2660i	−0.485621	−	0.485621i	0.421300	−	0.906921i	$-0.361574\pi$
$863$	−14.2660	−	14.2660i	−0.485621	−	0.485621i	−0.906921	+	0.421300i	$0.861574\pi$
$864$		0			0
$865$	3.30521	+	0.493070i	0.112380	+	0.0167649i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	1.31324			0.0445486
$870$		0			0
$871$	9.30712			0.315360
$872$		0			0
$873$		0			0
$874$		0			0
$875$	2.65200	−	5.57144i	0.0896541	−	0.188349i
$876$		0			0
$877$	−18.7475	−	18.7475i	−0.633058	−	0.633058i	0.315776	−	0.948834i	$-0.397735\pi$
$877$	−18.7475	−	18.7475i	−0.633058	−	0.633058i	−0.948834	+	0.315776i	$0.897735\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	51.3210i		−	1.72905i	−0.502593	−	0.864523i	$-0.667620\pi$
$881$		−	51.3210i		−	1.72905i	0.502593	−	0.864523i	$-0.332380\pi$
$882$		0			0
$883$	−32.6631	+	32.6631i	−1.09920	+	1.09920i	−0.104697	+	0.994504i	$0.533387\pi$
$883$	−32.6631	+	32.6631i	−1.09920	+	1.09920i	−0.994504	+	0.104697i	$0.966613\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−8.80786	+	8.80786i	−0.295739	+	0.295739i	−0.839342	−	0.543603i	$-0.817060\pi$
$887$	−8.80786	+	8.80786i	−0.295739	+	0.295739i	0.543603	+	0.839342i	$0.317060\pi$
$888$		0			0
$889$		−	10.1557i		−	0.340611i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−10.1714	−	10.1714i	−0.340373	−	0.340373i
$894$		0			0
$895$	−39.2825	+	29.0837i	−1.31307	+	0.972159i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−0.0808750			−0.00269733
$900$		0			0
$901$	97.7010			3.25489
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−39.9965	+	29.6123i	−1.32953	+	0.984345i
$906$		0			0
$907$	−12.1956	−	12.1956i	−0.404950	−	0.404950i	0.475023	−	0.879973i	$-0.342440\pi$
$907$	−12.1956	−	12.1956i	−0.404950	−	0.404950i	−0.879973	+	0.475023i	$0.842440\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	31.3821i		−	1.03973i	−0.854247	−	0.519867i	$-0.825981\pi$
$911$		−	31.3821i		−	1.03973i	0.854247	−	0.519867i	$-0.174019\pi$
$912$		0			0
$913$	2.07837	−	2.07837i	0.0687839	−	0.0687839i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−7.24942	+	7.24942i	−0.239397	+	0.239397i
$918$		0			0
$919$		−	27.5489i		−	0.908753i	−0.890810	−	0.454376i	$-0.849862\pi$
$919$		−	27.5489i		−	0.908753i	0.890810	−	0.454376i	$-0.150138\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−34.8554	−	34.8554i	−1.14728	−	1.14728i
$924$		0			0
$925$	7.10545	−	3.78287i	0.233626	−	0.124380i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−45.1369			−1.48090			−0.740448	−	0.672114i	$-0.765386\pi$
$929$	−45.1369			−1.48090			−0.740448	+	0.672114i	$0.765386\pi$
$930$		0			0
$931$	−13.0463			−0.427575
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−5.96506	−	0.889868i	−0.195078	−	0.0291018i
$936$		0			0
$937$	11.7767	+	11.7767i	0.384729	+	0.384729i	0.872803	−	0.488074i	$-0.162300\pi$
$937$	11.7767	+	11.7767i	0.384729	+	0.384729i	−0.488074	+	0.872803i	$0.662300\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	0.396097i		−	0.0129124i	−0.999979	−	0.00645619i	$-0.997945\pi$
$941$		−	0.396097i		−	0.0129124i	0.999979	−	0.00645619i	$-0.00205508\pi$
$942$		0			0
$943$	3.17243	−	3.17243i	0.103309	−	0.103309i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	35.1802	−	35.1802i	1.14320	−	1.14320i	0.155343	−	0.987861i	$-0.450352\pi$
$947$	35.1802	−	35.1802i	1.14320	−	1.14320i	0.987861	−	0.155343i	$-0.0496483\pi$
$948$		0			0
$949$		−	41.6007i		−	1.35042i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−4.74912	−	4.74912i	−0.153839	−	0.153839i	0.625991	−	0.779830i	$-0.284695\pi$
$953$	−4.74912	−	4.74912i	−0.153839	−	0.153839i	−0.779830	+	0.625991i	$0.784695\pi$
$954$		0			0
$955$	7.45012	+	10.0627i	0.241080	+	0.325621i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−2.27583			−0.0734903
$960$		0			0
$961$	−30.9067			−0.996989
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−4.19379	+	28.1123i	−0.135003	+	0.904966i
$966$		0			0
$967$	−15.6993	−	15.6993i	−0.504855	−	0.504855i	0.408088	−	0.912943i	$-0.366196\pi$
$967$	−15.6993	−	15.6993i	−0.504855	−	0.504855i	−0.912943	+	0.408088i	$0.866196\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	25.6278i		−	0.822434i	−0.911537	−	0.411217i	$-0.865104\pi$
$971$		−	25.6278i		−	0.822434i	0.911537	−	0.411217i	$-0.134896\pi$
$972$		0			0
$973$	6.88522	−	6.88522i	0.220730	−	0.220730i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−2.83000	+	2.83000i	−0.0905397	+	0.0905397i	−0.750926	−	0.660386i	$-0.770393\pi$
$977$	−2.83000	+	2.83000i	−0.0905397	+	0.0905397i	0.660386	+	0.750926i	$0.270393\pi$
$978$		0			0
$979$			3.42019i			0.109310i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	3.15281	+	3.15281i	0.100559	+	0.100559i	0.755596	−	0.655037i	$-0.227347\pi$
$983$	3.15281	+	3.15281i	0.100559	+	0.100559i	−0.655037	+	0.755596i	$0.727347\pi$
$984$		0			0
$985$	6.01140	−	40.2963i	0.191539	−	1.28395i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	3.01512			0.0958752
$990$		0			0
$991$	4.64796			0.147647			0.0738237	−	0.997271i	$-0.476480\pi$
$991$	4.64796			0.147647			0.0738237	+	0.997271i	$0.476480\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−0.787873	−	1.06416i	−0.0249773	−	0.0337361i
$996$		0			0
$997$	−0.504569	−	0.504569i	−0.0159798	−	0.0159798i	0.699072	−	0.715052i	$-0.253597\pi$
$997$	−0.504569	−	0.504569i	−0.0159798	−	0.0159798i	−0.715052	+	0.699072i	$0.753597\pi$
$998$		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.737.4	✓	44
3.2	odd	2	inner	4140.2.s.b.737.19	yes	44
5.3	odd	4	inner	4140.2.s.b.2393.19	yes	44
15.8	even	4	inner	4140.2.s.b.2393.4	yes	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4140.2.s.b.737.4	✓	44	1.1	even	1	trivial
4140.2.s.b.737.19	yes	44	3.2	odd	2	inner
4140.2.s.b.2393.4	yes	44	15.8	even	4	inner
4140.2.s.b.2393.19	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)

\(f(q)\)	\(=\)	\(q+(-1.79713 + 1.33054i) q^{5} +(-0.390251 - 0.390251i) q^{7} +O(q^{10})\)	\(q+(-1.79713 + 1.33054i) q^{5} +(-0.390251 - 0.390251i) q^{7} +0.378344i q^{11} +(-3.06090 + 3.06090i) q^{13} +(5.04090 - 5.04090i) q^{17} -1.94855i q^{19} +(-0.707107 - 0.707107i) q^{23} +(1.45932 - 4.78230i) q^{25} -0.264721 q^{29} +0.305510 q^{31} +(1.22058 + 0.182086i) q^{35} +(1.13840 + 1.13840i) q^{37} +4.48650i q^{41} +(-2.13201 + 2.13201i) q^{43} +(5.21999 - 5.21999i) q^{47} -6.69541i q^{49} +(9.69083 + 9.69083i) q^{53} +(-0.503402 - 0.679931i) q^{55} +12.8410 q^{59} -11.2123 q^{61} +(1.42817 - 9.57349i) q^{65} +(-1.52032 - 1.52032i) q^{67} +11.3873i q^{71} +(-6.79549 + 6.79549i) q^{73} +(0.147649 - 0.147649i) q^{77} -3.47102i q^{79} +(-5.49333 - 5.49333i) q^{83} +(-2.35201 + 15.7662i) q^{85} +9.03991 q^{89} +2.38904 q^{91} +(2.59262 + 3.50178i) q^{95} +(0.595525 + 0.595525i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 4 q^{7}+O(q^{10}) \) 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

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