LMFDB - Embedded newform 4020.2.g.b.1609.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4020.1609");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$32.0998616126$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1609.18
Character	$\chi$	$=$	4020.1609
Dual form			4020.2.g.b.1609.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +(-0.880059 + 2.05560i) q^{5} -1.30492i q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +(-0.880059 + 2.05560i) q^{5} -1.30492i q^{7} -1.00000 q^{9} -1.23619 q^{11} +1.57434i q^{13} +(-2.05560 - 0.880059i) q^{15} -0.592305i q^{17} +7.53506 q^{19} +1.30492 q^{21} -6.48217i q^{23} +(-3.45099 - 3.61810i) q^{25} -1.00000i q^{27} +2.11017 q^{29} +7.06929 q^{31} -1.23619i q^{33} +(2.68239 + 1.14841i) q^{35} +0.162812i q^{37} -1.57434 q^{39} +3.68393 q^{41} -0.0937305i q^{43} +(0.880059 - 2.05560i) q^{45} +3.86807i q^{47} +5.29719 q^{49} +0.592305 q^{51} +1.93402i q^{53} +(1.08792 - 2.54111i) q^{55} +7.53506i q^{57} -5.34961 q^{59} +2.11784 q^{61} +1.30492i q^{63} +(-3.23621 - 1.38551i) q^{65} -1.00000i q^{67} +6.48217 q^{69} +3.62485 q^{71} +11.0312i q^{73} +(3.61810 - 3.45099i) q^{75} +1.61312i q^{77} +7.58300 q^{79} +1.00000 q^{81} +5.94852i q^{83} +(1.21754 + 0.521263i) q^{85} +2.11017i q^{87} -11.8934 q^{89} +2.05438 q^{91} +7.06929i q^{93} +(-6.63130 + 15.4891i) q^{95} +18.6958i q^{97} +1.23619 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 4 q^{5} - 24 q^{9}+O(q^{10}) $ 24 * q - 4 * q^5 - 24 * q^9	$ 24 q - 4 q^{5} - 24 q^{9} - 8 q^{11} - 2 q^{15} + 16 q^{19} - 20 q^{21} + 10 q^{25} + 36 q^{29} - 2 q^{35} + 4 q^{39} - 24 q^{41} + 4 q^{45} - 4 q^{51} - 4 q^{55} + 24 q^{59} - 4 q^{61} - 20 q^{65} - 4 q^{69} + 20 q^{71} - 12 q^{75} - 28 q^{79} + 24 q^{81} - 16 q^{85} + 48 q^{89} - 20 q^{91} - 4 q^{95} + 8 q^{99}+O(q^{100}) $ 24 * q - 4 * q^5 - 24 * q^9 - 8 * q^11 - 2 * q^15 + 16 * q^19 - 20 * q^21 + 10 * q^25 + 36 * q^29 - 2 * q^35 + 4 * q^39 - 24 * q^41 + 4 * q^45 - 4 * q^51 - 4 * q^55 + 24 * q^59 - 4 * q^61 - 20 * q^65 - 4 * q^69 + 20 * q^71 - 12 * q^75 - 28 * q^79 + 24 * q^81 - 16 * q^85 + 48 * q^89 - 20 * q^91 - 4 * q^95 + 8 * q^99

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$		0			0
$5$	−0.880059	+	2.05560i	−0.393574	+	0.919293i
$5$	−0.880059	+	2.05560i	−0.393574	+	0.919293i
$6$		0			0
$7$		−	1.30492i		−	0.493213i	−0.969116	−	0.246607i	$-0.920684\pi$
$7$		−	1.30492i		−	0.493213i	0.969116	−	0.246607i	$-0.0793156\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	−1.23619			−0.372724			−0.186362	−	0.982481i	$-0.559670\pi$
$11$	−1.23619			−0.372724			−0.186362	+	0.982481i	$0.559670\pi$
$12$		0			0
$13$			1.57434i			0.436642i	0.975877	+	0.218321i	$0.0700581\pi$
$13$			1.57434i			0.436642i	−0.975877	+	0.218321i	$0.929942\pi$
$14$		0			0
$15$	−2.05560	−	0.880059i	−0.530754	−	0.227230i
$16$		0			0
$17$		−	0.592305i		−	0.143655i	−0.997417	−	0.0718275i	$-0.977117\pi$
$17$		−	0.592305i		−	0.143655i	0.997417	−	0.0718275i	$-0.0228831\pi$
$18$		0			0
$19$	7.53506			1.72866			0.864330	−	0.502924i	$-0.167743\pi$
$19$	7.53506			1.72866			0.864330	+	0.502924i	$0.167743\pi$
$20$		0			0
$21$	1.30492			0.284757
$22$		0			0
$23$		−	6.48217i		−	1.35163i	−0.737073	−	0.675813i	$-0.763793\pi$
$23$		−	6.48217i		−	1.35163i	0.737073	−	0.675813i	$-0.236207\pi$
$24$		0			0
$25$	−3.45099	−	3.61810i	−0.690198	−	0.723620i
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$	2.11017			0.391849			0.195925	−	0.980619i	$-0.437229\pi$
$29$	2.11017			0.391849			0.195925	+	0.980619i	$0.437229\pi$
$30$		0			0
$31$	7.06929			1.26968			0.634841	−	0.772643i	$-0.281066\pi$
$31$	7.06929			1.26968			0.634841	+	0.772643i	$0.281066\pi$
$32$		0			0
$33$		−	1.23619i		−	0.215192i
$34$		0			0
$35$	2.68239	+	1.14841i	0.453407	+	0.194116i
$36$		0			0
$37$			0.162812i			0.0267661i	0.999910	+	0.0133830i	$0.00426008\pi$
$37$			0.162812i			0.0267661i	−0.999910	+	0.0133830i	$0.995740\pi$
$38$		0			0
$39$	−1.57434			−0.252096
$40$		0			0
$41$	3.68393			0.575334			0.287667	−	0.957731i	$-0.407120\pi$
$41$	3.68393			0.575334			0.287667	+	0.957731i	$0.407120\pi$
$42$		0			0
$43$		−	0.0937305i		−	0.0142938i	−0.999974	−	0.00714689i	$-0.997725\pi$
$43$		−	0.0937305i		−	0.0142938i	0.999974	−	0.00714689i	$-0.00227494\pi$
$44$		0			0
$45$	0.880059	−	2.05560i	0.131191	−	0.306431i
$46$		0			0
$47$			3.86807i			0.564216i	0.959383	+	0.282108i	$0.0910337\pi$
$47$			3.86807i			0.564216i	−0.959383	+	0.282108i	$0.908966\pi$
$48$		0			0
$49$	5.29719			0.756741
$50$		0			0
$51$	0.592305			0.0829393
$52$		0			0
$53$			1.93402i			0.265658i	0.991139	+	0.132829i	$0.0424061\pi$
$53$			1.93402i			0.265658i	−0.991139	+	0.132829i	$0.957594\pi$
$54$		0			0
$55$	1.08792	−	2.54111i	0.146695	−	0.342643i
$56$		0			0
$57$			7.53506i			0.998043i
$58$		0			0
$59$	−5.34961			−0.696460			−0.348230	−	0.937409i	$-0.613217\pi$
$59$	−5.34961			−0.696460			−0.348230	+	0.937409i	$0.613217\pi$
$60$		0			0
$61$	2.11784			0.271162			0.135581	−	0.990766i	$-0.456710\pi$
$61$	2.11784			0.271162			0.135581	+	0.990766i	$0.456710\pi$
$62$		0			0
$63$			1.30492i			0.164404i
$64$		0			0
$65$	−3.23621	−	1.38551i	−0.401402	−	0.171851i
$66$		0			0
$67$		−	1.00000i		−	0.122169i
$67$		−	1.00000i		−	0.122169i
$68$		0			0
$69$	6.48217			0.780361
$70$		0			0
$71$	3.62485			0.430191			0.215095	−	0.976593i	$-0.430994\pi$
$71$	3.62485			0.430191			0.215095	+	0.976593i	$0.430994\pi$
$72$		0			0
$73$			11.0312i			1.29110i	0.763717	+	0.645551i	$0.223372\pi$
$73$			11.0312i			1.29110i	−0.763717	+	0.645551i	$0.776628\pi$
$74$		0			0
$75$	3.61810	−	3.45099i	0.417782	−	0.398486i
$76$		0			0
$77$			1.61312i			0.183832i
$78$		0			0
$79$	7.58300			0.853155			0.426577	−	0.904451i	$-0.359719\pi$
$79$	7.58300			0.853155			0.426577	+	0.904451i	$0.359719\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			5.94852i			0.652935i	0.945208	+	0.326467i	$0.105858\pi$
$83$			5.94852i			0.652935i	−0.945208	+	0.326467i	$0.894142\pi$
$84$		0			0
$85$	1.21754	+	0.521263i	0.132061	+	0.0565390i
$86$		0			0
$87$			2.11017i			0.226234i
$88$		0			0
$89$	−11.8934			−1.26070			−0.630350	−	0.776311i	$-0.717089\pi$
$89$	−11.8934			−1.26070			−0.630350	+	0.776311i	$0.717089\pi$
$90$		0			0
$91$	2.05438			0.215358
$92$		0			0
$93$			7.06929i			0.733051i
$94$		0			0
$95$	−6.63130	+	15.4891i	−0.680357	+	1.58915i
$96$		0			0
$97$			18.6958i			1.89827i	0.314874	+	0.949133i	$0.398038\pi$
$97$			18.6958i			1.89827i	−0.314874	+	0.949133i	$0.601962\pi$
$98$		0			0
$99$	1.23619			0.124241
$100$		0			0
$101$	−5.56043			−0.553284			−0.276642	−	0.960973i	$-0.589222\pi$
$101$	−5.56043			−0.553284			−0.276642	+	0.960973i	$0.589222\pi$
$102$		0			0
$103$		−	2.21921i		−	0.218665i	−0.994005	−	0.109333i	$-0.965129\pi$
$103$		−	2.21921i		−	0.218665i	0.994005	−	0.109333i	$-0.0348714\pi$
$104$		0			0
$105$	−1.14841	+	2.68239i	−0.112073	+	0.261775i
$106$		0			0
$107$		−	5.31490i		−	0.513810i	−0.966437	−	0.256905i	$-0.917297\pi$
$107$		−	5.31490i		−	0.513810i	0.966437	−	0.256905i	$-0.0827028\pi$
$108$		0			0
$109$	2.24639			0.215165			0.107582	−	0.994196i	$-0.465689\pi$
$109$	2.24639			0.215165			0.107582	+	0.994196i	$0.465689\pi$
$110$		0			0
$111$	−0.162812			−0.0154534
$112$		0			0
$113$		−	20.7190i		−	1.94908i	−0.224223	−	0.974538i	$-0.571984\pi$
$113$		−	20.7190i		−	1.94908i	0.224223	−	0.974538i	$-0.428016\pi$
$114$		0			0
$115$	13.3248	+	5.70469i	1.24254	+	0.531965i
$116$		0			0
$117$		−	1.57434i		−	0.145547i
$118$		0			0
$119$	−0.772910			−0.0708526
$120$		0			0
$121$	−9.47184			−0.861077
$122$		0			0
$123$			3.68393i			0.332169i
$124$		0			0
$125$	10.4745	−	3.90972i	0.936863	−	0.349696i
$126$		0			0
$127$			16.1439i			1.43254i	0.697822	+	0.716271i	$0.254153\pi$
$127$			16.1439i			1.43254i	−0.697822	+	0.716271i	$0.745847\pi$
$128$		0			0
$129$	0.0937305			0.00825251
$130$		0			0
$131$	19.2491			1.68180			0.840900	−	0.541191i	$-0.182026\pi$
$131$	19.2491			1.68180			0.840900	+	0.541191i	$0.182026\pi$
$132$		0			0
$133$		−	9.83264i		−	0.852598i
$134$		0			0
$135$	2.05560	+	0.880059i	0.176918	+	0.0757434i
$136$		0			0
$137$			3.27970i			0.280204i	0.990137	+	0.140102i	$0.0447430\pi$
$137$			3.27970i			0.280204i	−0.990137	+	0.140102i	$0.955257\pi$
$138$		0			0
$139$	8.38014			0.710794			0.355397	−	0.934715i	$-0.384346\pi$
$139$	8.38014			0.710794			0.355397	+	0.934715i	$0.384346\pi$
$140$		0			0
$141$	−3.86807			−0.325750
$142$		0			0
$143$		−	1.94617i		−	0.162747i
$144$		0			0
$145$	−1.85708	+	4.33767i	−0.154222	+	0.360224i
$146$		0			0
$147$			5.29719i			0.436904i
$148$		0			0
$149$	2.27506			0.186380			0.0931901	−	0.995648i	$-0.470294\pi$
$149$	2.27506			0.186380			0.0931901	+	0.995648i	$0.470294\pi$
$150$		0			0
$151$	−0.289865			−0.0235888			−0.0117944	−	0.999930i	$-0.503754\pi$
$151$	−0.289865			−0.0235888			−0.0117944	+	0.999930i	$0.503754\pi$
$152$		0			0
$153$			0.592305i			0.0478850i
$154$		0			0
$155$	−6.22139	+	14.5316i	−0.499714	+	1.16721i
$156$		0			0
$157$			11.7437i			0.937247i	0.883398	+	0.468623i	$0.155250\pi$
$157$			11.7437i			0.937247i	−0.883398	+	0.468623i	$0.844750\pi$
$158$		0			0
$159$	−1.93402			−0.153378
$160$		0			0
$161$	−8.45871			−0.666640
$162$		0			0
$163$		−	0.452956i		−	0.0354783i	−0.999843	−	0.0177391i	$-0.994353\pi$
$163$		−	0.452956i		−	0.0354783i	0.999843	−	0.0177391i	$-0.00564684\pi$
$164$		0			0
$165$	2.54111	+	1.08792i	0.197825	+	0.0846942i
$166$		0			0
$167$			8.10079i			0.626858i	0.949612	+	0.313429i	$0.101478\pi$
$167$			8.10079i			0.626858i	−0.949612	+	0.313429i	$0.898522\pi$
$168$		0			0
$169$	10.5215			0.809343
$170$		0			0
$171$	−7.53506			−0.576220
$172$		0			0
$173$			4.88616i			0.371488i	0.982598	+	0.185744i	$0.0594695\pi$
$173$			4.88616i			0.371488i	−0.982598	+	0.185744i	$0.940530\pi$
$174$		0			0
$175$	−4.72133	+	4.50327i	−0.356899	+	0.340415i
$176$		0			0
$177$		−	5.34961i		−	0.402101i
$178$		0			0
$179$	12.5004			0.934325			0.467162	−	0.884172i	$-0.345276\pi$
$179$	12.5004			0.934325			0.467162	+	0.884172i	$0.345276\pi$
$180$		0			0
$181$	7.49590			0.557166			0.278583	−	0.960412i	$-0.410135\pi$
$181$	7.49590			0.557166			0.278583	+	0.960412i	$0.410135\pi$
$182$		0			0
$183$			2.11784i			0.156555i
$184$		0			0
$185$	−0.334676	−	0.143284i	−0.0246059	−	0.0105344i
$186$		0			0
$187$			0.732199i			0.0535437i
$188$		0			0
$189$	−1.30492			−0.0949189
$190$		0			0
$191$	−22.1237			−1.60081			−0.800406	−	0.599458i	$-0.795383\pi$
$191$	−22.1237			−1.60081			−0.800406	+	0.599458i	$0.795383\pi$
$192$		0			0
$193$		−	14.6187i		−	1.05227i	−0.850400	−	0.526137i	$-0.823640\pi$
$193$		−	14.6187i		−	1.05227i	0.850400	−	0.526137i	$-0.176360\pi$
$194$		0			0
$195$	1.38551	−	3.23621i	0.0992184	−	0.231750i
$196$		0			0
$197$		−	12.6226i		−	0.899326i	−0.893198	−	0.449663i	$-0.851544\pi$
$197$		−	12.6226i		−	0.899326i	0.893198	−	0.449663i	$-0.148456\pi$
$198$		0			0
$199$	−6.02683			−0.427231			−0.213615	−	0.976918i	$-0.568524\pi$
$199$	−6.02683			−0.427231			−0.213615	+	0.976918i	$0.568524\pi$
$200$		0			0
$201$	1.00000			0.0705346
$202$		0			0
$203$		−	2.75361i		−	0.193265i
$204$		0			0
$205$	−3.24208	+	7.57270i	−0.226437	+	0.528900i
$206$		0			0
$207$			6.48217i			0.450542i
$208$		0			0
$209$	−9.31474			−0.644314
$210$		0			0
$211$	10.4818			0.721599			0.360800	−	0.932643i	$-0.382504\pi$
$211$	10.4818			0.721599			0.360800	+	0.932643i	$0.382504\pi$
$212$		0			0
$213$			3.62485i			0.248371i
$214$		0			0
$215$	0.192673	+	0.0824884i	0.0131402	+	0.00562566i
$216$		0			0
$217$		−	9.22485i		−	0.626224i
$218$		0			0
$219$	−11.0312			−0.745419
$220$		0			0
$221$	0.932487			0.0627259
$222$		0			0
$223$			12.2257i			0.818693i	0.912379	+	0.409346i	$0.134243\pi$
$223$			12.2257i			0.818693i	−0.912379	+	0.409346i	$0.865757\pi$
$224$		0			0
$225$	3.45099	+	3.61810i	0.230066	+	0.241207i
$226$		0			0
$227$			1.01465i			0.0673444i	0.999433	+	0.0336722i	$0.0107202\pi$
$227$			1.01465i			0.0673444i	−0.999433	+	0.0336722i	$0.989280\pi$
$228$		0			0
$229$	28.3803			1.87542			0.937712	−	0.347412i	$-0.112940\pi$
$229$	28.3803			1.87542			0.937712	+	0.347412i	$0.112940\pi$
$230$		0			0
$231$	−1.61312			−0.106136
$232$		0			0
$233$		−	6.93103i		−	0.454067i	−0.973887	−	0.227033i	$-0.927097\pi$
$233$		−	6.93103i		−	0.454067i	0.973887	−	0.227033i	$-0.0729027\pi$
$234$		0			0
$235$	−7.95121	−	3.40413i	−0.518680	−	0.222061i
$236$		0			0
$237$			7.58300i			0.492569i
$238$		0			0
$239$	−7.71985			−0.499356			−0.249678	−	0.968329i	$-0.580325\pi$
$239$	−7.71985			−0.499356			−0.249678	+	0.968329i	$0.580325\pi$
$240$		0			0
$241$	−29.7181			−1.91431			−0.957155	−	0.289576i	$-0.906486\pi$
$241$	−29.7181			−1.91431			−0.957155	+	0.289576i	$0.906486\pi$
$242$		0			0
$243$			1.00000i			0.0641500i
$244$		0			0
$245$	−4.66184	+	10.8889i	−0.297834	+	0.695666i
$246$		0			0
$247$			11.8627i			0.754807i
$248$		0			0
$249$	−5.94852			−0.376972
$250$		0			0
$251$	−10.6919			−0.674865			−0.337432	−	0.941350i	$-0.609558\pi$
$251$	−10.6919			−0.674865			−0.337432	+	0.941350i	$0.609558\pi$
$252$		0			0
$253$			8.01317i			0.503784i
$254$		0			0
$255$	−0.521263	+	1.21754i	−0.0326428	+	0.0762455i
$256$		0			0
$257$			25.4556i			1.58788i	0.607998	+	0.793939i	$0.291973\pi$
$257$			25.4556i			1.58788i	−0.607998	+	0.793939i	$0.708027\pi$
$258$		0			0
$259$	0.212456			0.0132014
$260$		0			0
$261$	−2.11017			−0.130616
$262$		0			0
$263$			28.9432i			1.78472i	0.451328	+	0.892358i	$0.350951\pi$
$263$			28.9432i			1.78472i	−0.451328	+	0.892358i	$0.649049\pi$
$264$		0			0
$265$	−3.97557	−	1.70205i	−0.244217	−	0.104556i
$266$		0			0
$267$		−	11.8934i		−	0.727866i
$268$		0			0
$269$	24.4214			1.48900			0.744499	−	0.667623i	$-0.232688\pi$
$269$	24.4214			1.48900			0.744499	+	0.667623i	$0.232688\pi$
$270$		0			0
$271$	0.114801			0.00697369			0.00348685	−	0.999994i	$-0.498890\pi$
$271$	0.114801			0.00697369			0.00348685	+	0.999994i	$0.498890\pi$
$272$		0			0
$273$			2.05438i			0.124337i
$274$		0			0
$275$	4.26607	+	4.47265i	0.257254	+	0.269711i
$276$		0			0
$277$		−	11.1579i		−	0.670415i	−0.942144	−	0.335207i	$-0.891194\pi$
$277$		−	11.1579i		−	0.670415i	0.942144	−	0.335207i	$-0.108806\pi$
$278$		0			0
$279$	−7.06929			−0.423227
$280$		0			0
$281$	−28.9061			−1.72439			−0.862196	−	0.506576i	$-0.830911\pi$
$281$	−28.9061			−1.72439			−0.862196	+	0.506576i	$0.830911\pi$
$282$		0			0
$283$			6.60958i			0.392899i	0.980514	+	0.196449i	$0.0629412\pi$
$283$			6.60958i			0.392899i	−0.980514	+	0.196449i	$0.937059\pi$
$284$		0			0
$285$	−15.4891	−	6.63130i	−0.917494	−	0.392804i
$286$		0			0
$287$		−	4.80724i		−	0.283762i
$288$		0			0
$289$	16.6492			0.979363
$290$		0			0
$291$	−18.6958			−1.09596
$292$		0			0
$293$			16.4874i			0.963204i	0.876390	+	0.481602i	$0.159945\pi$
$293$			16.4874i			0.963204i	−0.876390	+	0.481602i	$0.840055\pi$
$294$		0			0
$295$	4.70797	−	10.9967i	0.274109	−	0.640250i
$296$		0			0
$297$			1.23619i			0.0717308i
$298$		0			0
$299$	10.2051			0.590177
$300$		0			0
$301$	−0.122311			−0.00704988
$302$		0			0
$303$		−	5.56043i		−	0.319438i
$304$		0			0
$305$	−1.86383	+	4.35344i	−0.106722	+	0.249277i
$306$		0			0
$307$			14.7640i			0.842625i	0.906916	+	0.421312i	$0.138430\pi$
$307$			14.7640i			0.842625i	−0.906916	+	0.421312i	$0.861570\pi$
$308$		0			0
$309$	2.21921			0.126247
$310$		0			0
$311$	18.9466			1.07437			0.537183	−	0.843466i	$-0.319489\pi$
$311$	18.9466			1.07437			0.537183	+	0.843466i	$0.319489\pi$
$312$		0			0
$313$		−	3.12572i		−	0.176676i	−0.996091	−	0.0883381i	$-0.971844\pi$
$313$		−	3.12572i		−	0.176676i	0.996091	−	0.0883381i	$-0.0281556\pi$
$314$		0			0
$315$	−2.68239	−	1.14841i	−0.151136	−	0.0647054i
$316$		0			0
$317$			25.8863i			1.45392i	0.686681	+	0.726959i	$0.259067\pi$
$317$			25.8863i			1.45392i	−0.686681	+	0.726959i	$0.740933\pi$
$318$		0			0
$319$	−2.60857			−0.146052
$320$		0			0
$321$	5.31490			0.296649
$322$		0			0
$323$		−	4.46305i		−	0.248331i
$324$		0			0
$325$	5.69611	−	5.43302i	0.315963	−	0.301370i
$326$		0			0
$327$			2.24639i			0.124225i
$328$		0			0
$329$	5.04752			0.278279
$330$		0			0
$331$	8.97633			0.493384			0.246692	−	0.969094i	$-0.420656\pi$
$331$	8.97633			0.493384			0.246692	+	0.969094i	$0.420656\pi$
$332$		0			0
$333$		−	0.162812i		−	0.00892202i
$334$		0			0
$335$	2.05560	+	0.880059i	0.112309	+	0.0480828i
$336$		0			0
$337$		−	20.7894i		−	1.13247i	−0.824243	−	0.566236i	$-0.808399\pi$
$337$		−	20.7894i		−	1.13247i	0.824243	−	0.566236i	$-0.191601\pi$
$338$		0			0
$339$	20.7190			1.12530
$340$		0			0
$341$	−8.73895			−0.473241
$342$		0			0
$343$		−	16.0468i		−	0.866448i
$344$		0			0
$345$	−5.70469	+	13.3248i	−0.307130	+	0.717381i
$346$		0			0
$347$			7.71805i			0.414327i	0.978306	+	0.207163i	$0.0664232\pi$
$347$			7.71805i			0.414327i	−0.978306	+	0.207163i	$0.933577\pi$
$348$		0			0
$349$	26.6170			1.42478			0.712388	−	0.701786i	$-0.247614\pi$
$349$	26.6170			1.42478			0.712388	+	0.701786i	$0.247614\pi$
$350$		0			0
$351$	1.57434			0.0840319
$352$		0			0
$353$			4.27287i			0.227422i	0.993514	+	0.113711i	$0.0362738\pi$
$353$			4.27287i			0.227422i	−0.993514	+	0.113711i	$0.963726\pi$
$354$		0			0
$355$	−3.19009	+	7.45125i	−0.169312	+	0.395471i
$356$		0			0
$357$		−	0.772910i		−	0.0409067i
$358$		0			0
$359$	−30.2397			−1.59599			−0.797994	−	0.602665i	$-0.794106\pi$
$359$	−30.2397			−1.59599			−0.797994	+	0.602665i	$0.794106\pi$
$360$		0			0
$361$	37.7771			1.98827
$362$		0			0
$363$		−	9.47184i		−	0.497143i
$364$		0			0
$365$	−22.6757	−	9.70810i	−1.18690	−	0.508145i
$366$		0			0
$367$			12.4654i			0.650688i	0.945596	+	0.325344i	$0.105480\pi$
$367$			12.4654i			0.650688i	−0.945596	+	0.325344i	$0.894520\pi$
$368$		0			0
$369$	−3.68393			−0.191778
$370$		0			0
$371$	2.52374			0.131026
$372$		0			0
$373$		−	10.5777i		−	0.547691i	−0.961774	−	0.273846i	$-0.911704\pi$
$373$		−	10.5777i		−	0.547691i	0.961774	−	0.273846i	$-0.0882958\pi$
$374$		0			0
$375$	3.90972	+	10.4745i	0.201897	+	0.540898i
$376$		0			0
$377$			3.32212i			0.171098i
$378$		0			0
$379$	2.74645			0.141076			0.0705378	−	0.997509i	$-0.477528\pi$
$379$	2.74645			0.141076			0.0705378	+	0.997509i	$0.477528\pi$
$380$		0			0
$381$	−16.1439			−0.827079
$382$		0			0
$383$		−	6.90931i		−	0.353049i	−0.984296	−	0.176525i	$-0.943515\pi$
$383$		−	6.90931i		−	0.353049i	0.984296	−	0.176525i	$-0.0564855\pi$
$384$		0			0
$385$	−3.31594	−	1.41964i	−0.168996	−	0.0723518i
$386$		0			0
$387$			0.0937305i			0.00476459i
$388$		0			0
$389$	6.19305			0.314000			0.157000	−	0.987599i	$-0.449818\pi$
$389$	6.19305			0.314000			0.157000	+	0.987599i	$0.449818\pi$
$390$		0			0
$391$	−3.83942			−0.194168
$392$		0			0
$393$			19.2491i			0.970987i
$394$		0			0
$395$	−6.67349	+	15.5876i	−0.335780	+	0.784299i
$396$		0			0
$397$		−	7.16458i		−	0.359580i	−0.983705	−	0.179790i	$-0.942458\pi$
$397$		−	7.16458i		−	0.359580i	0.983705	−	0.179790i	$-0.0575418\pi$
$398$		0			0
$399$	9.83264			0.492248
$400$		0			0
$401$	16.8598			0.841936			0.420968	−	0.907076i	$-0.361690\pi$
$401$	16.8598			0.841936			0.420968	+	0.907076i	$0.361690\pi$
$402$		0			0
$403$			11.1294i			0.554397i
$404$		0			0
$405$	−0.880059	+	2.05560i	−0.0437305	+	0.102144i
$406$		0			0
$407$		−	0.201266i		−	0.00997636i
$408$		0			0
$409$	22.4182			1.10851			0.554254	−	0.832347i	$-0.313004\pi$
$409$	22.4182			1.10851			0.554254	+	0.832347i	$0.313004\pi$
$410$		0			0
$411$	−3.27970			−0.161776
$412$		0			0
$413$			6.98081i			0.343503i
$414$		0			0
$415$	−12.2278	−	5.23505i	−0.600238	−	0.256979i
$416$		0			0
$417$			8.38014i			0.410377i
$418$		0			0
$419$	30.7268			1.50110			0.750552	−	0.660812i	$-0.229788\pi$
$419$	30.7268			1.50110			0.750552	+	0.660812i	$0.229788\pi$
$420$		0			0
$421$	3.28105			0.159909			0.0799544	−	0.996799i	$-0.474523\pi$
$421$	3.28105			0.159909			0.0799544	+	0.996799i	$0.474523\pi$
$422$		0			0
$423$		−	3.86807i		−	0.188072i
$424$		0			0
$425$	−2.14302	+	2.04404i	−0.103952	+	0.0991504i
$426$		0			0
$427$		−	2.76361i		−	0.133741i
$428$		0			0
$429$	1.94617			0.0939621
$430$		0			0
$431$	17.1165			0.824471			0.412235	−	0.911077i	$-0.364748\pi$
$431$	17.1165			0.824471			0.412235	+	0.911077i	$0.364748\pi$
$432$		0			0
$433$		−	31.4276i		−	1.51031i	−0.655545	−	0.755156i	$-0.727561\pi$
$433$		−	31.4276i		−	1.51031i	0.655545	−	0.755156i	$-0.272439\pi$
$434$		0			0
$435$	−4.33767	−	1.85708i	−0.207976	−	0.0890401i
$436$		0			0
$437$		−	48.8435i		−	2.33650i
$438$		0			0
$439$	15.6652			0.747658			0.373829	−	0.927498i	$-0.378045\pi$
$439$	15.6652			0.747658			0.373829	+	0.927498i	$0.378045\pi$
$440$		0			0
$441$	−5.29719			−0.252247
$442$		0			0
$443$		−	15.2049i		−	0.722404i	−0.932488	−	0.361202i	$-0.882366\pi$
$443$		−	15.2049i		−	0.722404i	0.932488	−	0.361202i	$-0.117634\pi$
$444$		0			0
$445$	10.4669	−	24.4481i	0.496180	−	1.15895i
$446$		0			0
$447$			2.27506i			0.107607i
$448$		0			0
$449$	−16.5501			−0.781050			−0.390525	−	0.920592i	$-0.627706\pi$
$449$	−16.5501			−0.781050			−0.390525	+	0.920592i	$0.627706\pi$
$450$		0			0
$451$	−4.55403			−0.214441
$452$		0			0
$453$		−	0.289865i		−	0.0136190i
$454$		0			0
$455$	−1.80798	+	4.22299i	−0.0847593	+	0.197977i
$456$		0			0
$457$		−	38.5015i		−	1.80103i	−0.434830	−	0.900513i	$-0.643192\pi$
$457$		−	38.5015i		−	1.80103i	0.434830	−	0.900513i	$-0.356808\pi$
$458$		0			0
$459$	−0.592305			−0.0276464
$460$		0			0
$461$	−12.5513			−0.584570			−0.292285	−	0.956331i	$-0.594416\pi$
$461$	−12.5513			−0.584570			−0.292285	+	0.956331i	$0.594416\pi$
$462$		0			0
$463$			12.1269i			0.563586i	0.959475	+	0.281793i	$0.0909292\pi$
$463$			12.1269i			0.563586i	−0.959475	+	0.281793i	$0.909071\pi$
$464$		0			0
$465$	−14.5316	−	6.22139i	−0.673888	−	0.288510i
$466$		0			0
$467$		−	0.227492i		−	0.0105271i	−0.999986	−	0.00526353i	$-0.998325\pi$
$467$		−	0.227492i		−	0.0105271i	0.999986	−	0.00526353i	$-0.00167544\pi$
$468$		0			0
$469$	−1.30492			−0.0602556
$470$		0			0
$471$	−11.7437			−0.541120
$472$		0			0
$473$			0.115868i			0.00532763i
$474$		0			0
$475$	−26.0034	−	27.2626i	−1.19312	−	1.25089i
$476$		0			0
$477$		−	1.93402i		−	0.0885526i
$478$		0			0
$479$	14.1305			0.645641			0.322820	−	0.946460i	$-0.395369\pi$
$479$	14.1305			0.645641			0.322820	+	0.946460i	$0.395369\pi$
$480$		0			0
$481$	−0.256320			−0.0116872
$482$		0			0
$483$		−	8.45871i		−	0.384885i
$484$		0			0
$485$	−38.4310	−	16.4534i	−1.74506	−	0.747109i
$486$		0			0
$487$			16.9555i			0.768329i	0.923265	+	0.384164i	$0.125510\pi$
$487$			16.9555i			0.768329i	−0.923265	+	0.384164i	$0.874490\pi$
$488$		0			0
$489$	0.452956			0.0204834
$490$		0			0
$491$	14.8656			0.670877			0.335438	−	0.942062i	$-0.391116\pi$
$491$	14.8656			0.670877			0.335438	+	0.942062i	$0.391116\pi$
$492$		0			0
$493$		−	1.24987i		−	0.0562911i
$494$		0			0
$495$	−1.08792	+	2.54111i	−0.0488982	+	0.114214i
$496$		0			0
$497$		−	4.73014i		−	0.212176i
$498$		0			0
$499$	−34.9128			−1.56291			−0.781456	−	0.623960i	$-0.785523\pi$
$499$	−34.9128			−1.56291			−0.781456	+	0.623960i	$0.785523\pi$
$500$		0			0
$501$	−8.10079			−0.361916
$502$		0			0
$503$			17.1124i			0.763006i	0.924368	+	0.381503i	$0.124593\pi$
$503$			17.1124i			0.763006i	−0.924368	+	0.381503i	$0.875407\pi$
$504$		0			0
$505$	4.89351	−	11.4300i	0.217758	−	0.508630i
$506$		0			0
$507$			10.5215i			0.467275i
$508$		0			0
$509$	34.4109			1.52524			0.762618	−	0.646850i	$-0.223914\pi$
$509$	34.4109			1.52524			0.762618	+	0.646850i	$0.223914\pi$
$510$		0			0
$511$	14.3948			0.636789
$512$		0			0
$513$		−	7.53506i		−	0.332681i
$514$		0			0
$515$	4.56181	+	1.95304i	0.201017	+	0.0860611i
$516$		0			0
$517$		−	4.78166i		−	0.210297i
$518$		0			0
$519$	−4.88616			−0.214479
$520$		0			0
$521$	12.6956			0.556205			0.278102	−	0.960551i	$-0.410295\pi$
$521$	12.6956			0.556205			0.278102	+	0.960551i	$0.410295\pi$
$522$		0			0
$523$		−	19.7757i		−	0.864733i	−0.901698	−	0.432367i	$-0.857679\pi$
$523$		−	19.7757i		−	0.864733i	0.901698	−	0.432367i	$-0.142321\pi$
$524$		0			0
$525$	−4.50327	−	4.72133i	−0.196539	−	0.206056i
$526$		0			0
$527$		−	4.18717i		−	0.182396i
$528$		0			0
$529$	−19.0185			−0.826892
$530$		0			0
$531$	5.34961			0.232153
$532$		0			0
$533$			5.79975i			0.251215i
$534$		0			0
$535$	10.9253	+	4.67742i	0.472342	+	0.202223i
$536$		0			0
$537$			12.5004i			0.539433i
$538$		0			0
$539$	−6.54831			−0.282056
$540$		0			0
$541$	−26.7233			−1.14893			−0.574463	−	0.818531i	$-0.694789\pi$
$541$	−26.7233			−1.14893			−0.574463	+	0.818531i	$0.694789\pi$
$542$		0			0
$543$			7.49590i			0.321680i
$544$		0			0
$545$	−1.97695	+	4.61768i	−0.0846834	+	0.197799i
$546$		0			0
$547$		−	30.0531i		−	1.28498i	−0.766295	−	0.642489i	$-0.777902\pi$
$547$		−	30.0531i		−	1.28498i	0.766295	−	0.642489i	$-0.222098\pi$
$548$		0			0
$549$	−2.11784			−0.0903873
$550$		0			0
$551$	15.9003			0.677375
$552$		0			0
$553$		−	9.89521i		−	0.420787i
$554$		0			0
$555$	0.143284	−	0.334676i	0.00608206	−	0.0142062i
$556$		0			0
$557$		−	30.5660i		−	1.29512i	−0.762014	−	0.647561i	$-0.775789\pi$
$557$		−	30.5660i		−	1.29512i	0.762014	−	0.647561i	$-0.224211\pi$
$558$		0			0
$559$	0.147563			0.00624127
$560$		0			0
$561$	−0.732199			−0.0309135
$562$		0			0
$563$			13.9746i			0.588959i	0.955658	+	0.294479i	$0.0951462\pi$
$563$			13.9746i			0.588959i	−0.955658	+	0.294479i	$0.904854\pi$
$564$		0			0
$565$	42.5899	+	18.2339i	1.79177	+	0.767106i
$566$		0			0
$567$		−	1.30492i		−	0.0548015i
$568$		0			0
$569$	−16.3060			−0.683583			−0.341791	−	0.939776i	$-0.611034\pi$
$569$	−16.3060			−0.683583			−0.341791	+	0.939776i	$0.611034\pi$
$570$		0			0
$571$	23.9588			1.00265			0.501323	−	0.865260i	$-0.332847\pi$
$571$	23.9588			1.00265			0.501323	+	0.865260i	$0.332847\pi$
$572$		0			0
$573$		−	22.1237i		−	0.924229i
$574$		0			0
$575$	−23.4531	+	22.3699i	−0.978064	+	0.932890i
$576$		0			0
$577$		−	19.5651i		−	0.814507i	−0.913315	−	0.407254i	$-0.866486\pi$
$577$		−	19.5651i		−	0.814507i	0.913315	−	0.407254i	$-0.133514\pi$
$578$		0			0
$579$	14.6187			0.607531
$580$		0			0
$581$	7.76234			0.322036
$582$		0			0
$583$		−	2.39081i		−	0.0990171i
$584$		0			0
$585$	3.23621	+	1.38551i	0.133801	+	0.0572838i
$586$		0			0
$587$			19.9745i			0.824435i	0.911085	+	0.412218i	$0.135246\pi$
$587$			19.9745i			0.824435i	−0.911085	+	0.412218i	$0.864754\pi$
$588$		0			0
$589$	53.2675			2.19485
$590$		0			0
$591$	12.6226			0.519226
$592$		0			0
$593$			34.2195i			1.40523i	0.711572	+	0.702613i	$0.247984\pi$
$593$			34.2195i			1.40523i	−0.711572	+	0.702613i	$0.752016\pi$
$594$		0			0
$595$	0.680207	−	1.58879i	0.0278858	−	0.0651342i
$596$		0			0
$597$		−	6.02683i		−	0.246662i
$598$		0			0
$599$	9.75053			0.398396			0.199198	−	0.979959i	$-0.436166\pi$
$599$	9.75053			0.398396			0.199198	+	0.979959i	$0.436166\pi$
$600$		0			0
$601$	28.8348			1.17620			0.588098	−	0.808790i	$-0.299877\pi$
$601$	28.8348			1.17620			0.588098	+	0.808790i	$0.299877\pi$
$602$		0			0
$603$			1.00000i			0.0407231i
$604$		0			0
$605$	8.33578	−	19.4703i	0.338898	−	0.791582i
$606$		0			0
$607$			32.8932i			1.33509i	0.744567	+	0.667547i	$0.232656\pi$
$607$			32.8932i			1.33509i	−0.744567	+	0.667547i	$0.767344\pi$
$608$		0			0
$609$	2.75361			0.111582
$610$		0			0
$611$	−6.08965			−0.246361
$612$		0			0
$613$		−	32.5057i		−	1.31289i	−0.754373	−	0.656446i	$-0.772059\pi$
$613$		−	32.5057i		−	1.31289i	0.754373	−	0.656446i	$-0.227941\pi$
$614$		0			0
$615$	−7.57270	−	3.24208i	−0.305361	−	0.130733i
$616$		0			0
$617$		−	13.6881i		−	0.551063i	−0.961292	−	0.275531i	$-0.911146\pi$
$617$		−	13.6881i		−	0.551063i	0.961292	−	0.275531i	$-0.0888538\pi$
$618$		0			0
$619$	−27.9120			−1.12188			−0.560940	−	0.827857i	$-0.689560\pi$
$619$	−27.9120			−1.12188			−0.560940	+	0.827857i	$0.689560\pi$
$620$		0			0
$621$	−6.48217			−0.260120
$622$		0			0
$623$			15.5200i			0.621794i
$624$		0			0
$625$	−1.18132	+	24.9721i	−0.0472527	+	0.998883i
$626$		0			0
$627$		−	9.31474i		−	0.371995i
$628$		0			0
$629$	0.0964341			0.00384508
$630$		0			0
$631$	−0.0300462			−0.00119612			−0.000598060	−	1.00000i	$-0.500190\pi$
$631$	−0.0300462			−0.00119612			−0.000598060		1.00000i	$0.500190\pi$
$632$		0			0
$633$			10.4818i			0.416615i
$634$		0			0
$635$	−33.1855	−	14.2076i	−1.31693	−	0.563812i
$636$		0			0
$637$			8.33955i			0.330425i
$638$		0			0
$639$	−3.62485			−0.143397
$640$		0			0
$641$	−9.76797			−0.385812			−0.192906	−	0.981217i	$-0.561791\pi$
$641$	−9.76797			−0.385812			−0.192906	+	0.981217i	$0.561791\pi$
$642$		0			0
$643$			23.3098i			0.919250i	0.888113	+	0.459625i	$0.152016\pi$
$643$			23.3098i			0.919250i	−0.888113	+	0.459625i	$0.847984\pi$
$644$		0			0
$645$	−0.0824884	+	0.192673i	−0.00324798	+	0.00758647i
$646$		0			0
$647$		−	16.8313i		−	0.661705i	−0.943682	−	0.330852i	$-0.892664\pi$
$647$		−	16.8313i		−	0.661705i	0.943682	−	0.330852i	$-0.107336\pi$
$648$		0			0
$649$	6.61311			0.259587
$650$		0			0
$651$	9.22485			0.361550
$652$		0			0
$653$		−	12.0118i		−	0.470057i	−0.971988	−	0.235029i	$-0.924482\pi$
$653$		−	12.0118i		−	0.470057i	0.971988	−	0.235029i	$-0.0755184\pi$
$654$		0			0
$655$	−16.9403	+	39.5684i	−0.661913	+	1.54607i
$656$		0			0
$657$		−	11.0312i		−	0.430368i
$658$		0			0
$659$	1.52770			0.0595107			0.0297553	−	0.999557i	$-0.490527\pi$
$659$	1.52770			0.0595107			0.0297553	+	0.999557i	$0.490527\pi$
$660$		0			0
$661$	−32.5270			−1.26516			−0.632578	−	0.774497i	$-0.718003\pi$
$661$	−32.5270			−1.26516			−0.632578	+	0.774497i	$0.718003\pi$
$662$		0			0
$663$			0.932487i			0.0362148i
$664$		0			0
$665$	20.2120	+	8.65331i	0.783788	+	0.335561i
$666$		0			0
$667$		−	13.6785i		−	0.529634i
$668$		0			0
$669$	−12.2257			−0.472673
$670$		0			0
$671$	−2.61805			−0.101069
$672$		0			0
$673$			0.0149889i			0.000577778i	1.00000		0.000288889i	$9.19562e-5\pi$
$673$			0.0149889i			0.000577778i	−1.00000		0.000288889i	$0.999908\pi$
$674$		0			0
$675$	−3.61810	+	3.45099i	−0.139261	+	0.132829i
$676$		0			0
$677$			1.93040i			0.0741914i	0.999312	+	0.0370957i	$0.0118106\pi$
$677$			1.93040i			0.0741914i	−0.999312	+	0.0370957i	$0.988189\pi$
$678$		0			0
$679$	24.3965			0.936250
$680$		0			0
$681$	−1.01465			−0.0388813
$682$		0			0
$683$		−	12.0554i		−	0.461287i	−0.973038	−	0.230643i	$-0.925917\pi$
$683$		−	12.0554i		−	0.461287i	0.973038	−	0.230643i	$-0.0740831\pi$
$684$		0			0
$685$	−6.74176	−	2.88633i	−0.257589	−	0.110281i
$686$		0			0
$687$			28.3803i			1.08278i
$688$		0			0
$689$	−3.04480			−0.115997
$690$		0			0
$691$	0.226926			0.00863269			0.00431634	−	0.999991i	$-0.498626\pi$
$691$	0.226926			0.00863269			0.00431634	+	0.999991i	$0.498626\pi$
$692$		0			0
$693$		−	1.61312i		−	0.0612775i
$694$		0			0
$695$	−7.37502	+	17.2262i	−0.279750	+	0.653428i
$696$		0			0
$697$		−	2.18201i		−	0.0826496i
$698$		0			0
$699$	6.93103			0.262156
$700$		0			0
$701$	−17.2451			−0.651340			−0.325670	−	0.945484i	$-0.605590\pi$
$701$	−17.2451			−0.651340			−0.325670	+	0.945484i	$0.605590\pi$
$702$		0			0
$703$			1.22680i			0.0462695i
$704$		0			0
$705$	3.40413	−	7.95121i	0.128207	−	0.299460i
$706$		0			0
$707$			7.25592i			0.272887i
$708$		0			0
$709$	11.2177			0.421289			0.210644	−	0.977563i	$-0.432444\pi$
$709$	11.2177			0.421289			0.210644	+	0.977563i	$0.432444\pi$
$710$		0			0
$711$	−7.58300			−0.284385
$712$		0			0
$713$		−	45.8243i		−	1.71613i
$714$		0			0
$715$	4.00056	+	1.71275i	0.149612	+	0.0640531i
$716$		0			0
$717$		−	7.71985i		−	0.288303i
$718$		0			0
$719$	41.0928			1.53250			0.766251	−	0.642541i	$-0.222120\pi$
$719$	41.0928			1.53250			0.766251	+	0.642541i	$0.222120\pi$
$720$		0			0
$721$	−2.89589			−0.107849
$722$		0			0
$723$		−	29.7181i		−	1.10523i
$724$		0			0
$725$	−7.28219	−	7.63482i	−0.270454	−	0.283550i
$726$		0			0
$727$		−	27.0300i		−	1.00249i	−0.865307	−	0.501243i	$-0.832876\pi$
$727$		−	27.0300i		−	1.00249i	0.865307	−	0.501243i	$-0.167124\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	−0.0555170			−0.00205337
$732$		0			0
$733$		−	44.1973i		−	1.63247i	−0.577723	−	0.816233i	$-0.696059\pi$
$733$		−	44.1973i		−	1.63247i	0.577723	−	0.816233i	$-0.303941\pi$
$734$		0			0
$735$	−10.8889	−	4.66184i	−0.401643	−	0.171954i
$736$		0			0
$737$			1.23619i			0.0455355i
$738$		0			0
$739$	−7.33310			−0.269753			−0.134876	−	0.990862i	$-0.543064\pi$
$739$	−7.33310			−0.269753			−0.134876	+	0.990862i	$0.543064\pi$
$740$		0			0
$741$	−11.8627			−0.435788
$742$		0			0
$743$			9.52091i			0.349288i	0.984632	+	0.174644i	$0.0558775\pi$
$743$			9.52091i			0.349288i	−0.984632	+	0.174644i	$0.944122\pi$
$744$		0			0
$745$	−2.00219	+	4.67661i	−0.0733545	+	0.171338i
$746$		0			0
$747$		−	5.94852i		−	0.217645i
$748$		0			0
$749$	−6.93551			−0.253418
$750$		0			0
$751$	28.3616			1.03493			0.517464	−	0.855705i	$-0.326876\pi$
$751$	28.3616			1.03493			0.517464	+	0.855705i	$0.326876\pi$
$752$		0			0
$753$		−	10.6919i		−	0.389633i
$754$		0			0
$755$	0.255098	−	0.595846i	0.00928397	−	0.0216851i
$756$		0			0
$757$		−	28.3033i		−	1.02870i	−0.857580	−	0.514350i	$-0.828033\pi$
$757$		−	28.3033i		−	1.02870i	0.857580	−	0.514350i	$-0.171967\pi$
$758$		0			0
$759$	−8.01317			−0.290860
$760$		0			0
$761$	−24.8758			−0.901747			−0.450873	−	0.892588i	$-0.648887\pi$
$761$	−24.8758			−0.901747			−0.450873	+	0.892588i	$0.648887\pi$
$762$		0			0
$763$		−	2.93135i		−	0.106122i
$764$		0			0
$765$	−1.21754	−	0.521263i	−0.0440203	−	0.0188463i
$766$		0			0
$767$		−	8.42208i		−	0.304104i
$768$		0			0
$769$	5.54759			0.200051			0.100026	−	0.994985i	$-0.468108\pi$
$769$	5.54759			0.200051			0.100026	+	0.994985i	$0.468108\pi$
$770$		0			0
$771$	−25.4556			−0.916762
$772$		0			0
$773$			3.11314i			0.111972i	0.998432	+	0.0559858i	$0.0178302\pi$
$773$			3.11314i			0.111972i	−0.998432	+	0.0559858i	$0.982170\pi$
$774$		0			0
$775$	−24.3960	−	25.5774i	−0.876332	−	0.918767i
$776$		0			0
$777$			0.212456i			0.00762182i
$778$		0			0
$779$	27.7586			0.994557
$780$		0			0
$781$	−4.48099			−0.160343
$782$		0			0
$783$		−	2.11017i		−	0.0754114i
$784$		0			0
$785$	−24.1403	−	10.3351i	−0.861604	−	0.368876i
$786$		0			0
$787$			42.4624i			1.51362i	0.653634	+	0.756811i	$0.273244\pi$
$787$			42.4624i			1.51362i	−0.653634	+	0.756811i	$0.726756\pi$
$788$		0			0
$789$	−28.9432			−1.03041
$790$		0			0
$791$	−27.0366			−0.961310
$792$		0			0
$793$			3.33420i			0.118401i
$794$		0			0
$795$	1.70205	−	3.97557i	0.0603655	−	0.140999i
$796$		0			0
$797$			10.5642i			0.374202i	0.982341	+	0.187101i	$0.0599091\pi$
$797$			10.5642i			0.374202i	−0.982341	+	0.187101i	$0.940091\pi$
$798$		0			0
$799$	2.29108			0.0810525
$800$		0			0
$801$	11.8934			0.420234
$802$		0			0
$803$		−	13.6366i		−	0.481225i
$804$		0			0
$805$	7.44417	−	17.3877i	0.262372	−	0.612837i
$806$		0			0
$807$			24.4214i			0.859673i
$808$		0			0
$809$	−34.5813			−1.21581			−0.607906	−	0.794009i	$-0.707990\pi$
$809$	−34.5813			−1.21581			−0.607906	+	0.794009i	$0.707990\pi$
$810$		0			0
$811$	−48.9854			−1.72011			−0.860055	−	0.510201i	$-0.829571\pi$
$811$	−48.9854			−1.72011			−0.860055	+	0.510201i	$0.829571\pi$
$812$		0			0
$813$			0.114801i			0.00402626i
$814$		0			0
$815$	0.931098	+	0.398629i	0.0326149	+	0.0139633i
$816$		0			0
$817$		−	0.706265i		−	0.0247091i
$818$		0			0
$819$	−2.05438			−0.0717859
$820$		0			0
$821$	−15.4895			−0.540587			−0.270294	−	0.962778i	$-0.587121\pi$
$821$	−15.4895			−0.540587			−0.270294	+	0.962778i	$0.587121\pi$
$822$		0			0
$823$		−	25.8631i		−	0.901532i	−0.892642	−	0.450766i	$-0.851151\pi$
$823$		−	25.8631i		−	0.901532i	0.892642	−	0.450766i	$-0.148849\pi$
$824$		0			0
$825$	−4.47265	+	4.26607i	−0.155718	+	0.148525i
$826$		0			0
$827$		−	16.9253i		−	0.588550i	−0.955721	−	0.294275i	$-0.904922\pi$
$827$		−	16.9253i		−	0.588550i	0.955721	−	0.294275i	$-0.0950782\pi$
$828$		0			0
$829$	−13.1239			−0.455811			−0.227906	−	0.973683i	$-0.573188\pi$
$829$	−13.1239			−0.455811			−0.227906	+	0.973683i	$0.573188\pi$
$830$		0			0
$831$	11.1579			0.387064
$832$		0			0
$833$		−	3.13755i		−	0.108710i
$834$		0			0
$835$	−16.6520	−	7.12917i	−0.576266	−	0.246715i
$836$		0			0
$837$		−	7.06929i		−	0.244350i
$838$		0			0
$839$	6.22286			0.214837			0.107419	−	0.994214i	$-0.465742\pi$
$839$	6.22286			0.214837			0.107419	+	0.994214i	$0.465742\pi$
$840$		0			0
$841$	−24.5472			−0.846454
$842$		0			0
$843$		−	28.9061i		−	0.995578i
$844$		0			0
$845$	−9.25951	+	21.6279i	−0.318537	+	0.744023i
$846$		0			0
$847$			12.3600i			0.424694i
$848$		0			0
$849$	−6.60958			−0.226840
$850$		0			0
$851$	1.05537			0.0361777
$852$		0			0
$853$			13.4403i			0.460186i	0.973169	+	0.230093i	$0.0739030\pi$
$853$			13.4403i			0.460186i	−0.973169	+	0.230093i	$0.926097\pi$
$854$		0			0
$855$	6.63130	−	15.4891i	0.226786	−	0.529715i
$856$		0			0
$857$		−	51.3309i		−	1.75343i	−0.481011	−	0.876715i	$-0.659730\pi$
$857$		−	51.3309i		−	1.75343i	0.481011	−	0.876715i	$-0.340270\pi$
$858$		0			0
$859$	11.4338			0.390116			0.195058	−	0.980792i	$-0.437510\pi$
$859$	11.4338			0.390116			0.195058	+	0.980792i	$0.437510\pi$
$860$		0			0
$861$	4.80724			0.163830
$862$		0			0
$863$			32.8696i			1.11889i	0.828866	+	0.559447i	$0.188986\pi$
$863$			32.8696i			1.11889i	−0.828866	+	0.559447i	$0.811014\pi$
$864$		0			0
$865$	−10.0440	−	4.30011i	−0.341506	−	0.146208i
$866$		0			0
$867$			16.6492i			0.565436i
$868$		0			0
$869$	−9.37400			−0.317991
$870$		0			0
$871$	1.57434			0.0533444
$872$		0			0
$873$		−	18.6958i		−	0.632756i
$874$		0			0
$875$	−5.10187	−	13.6683i	−0.172475	−	0.462073i
$876$		0			0
$877$		−	27.7686i		−	0.937678i	−0.883284	−	0.468839i	$-0.844672\pi$
$877$		−	27.7686i		−	0.937678i	0.883284	−	0.468839i	$-0.155328\pi$
$878$		0			0
$879$	−16.4874			−0.556106
$880$		0			0
$881$	3.19038			0.107487			0.0537434	−	0.998555i	$-0.482885\pi$
$881$	3.19038			0.107487			0.0537434	+	0.998555i	$0.482885\pi$
$882$		0			0
$883$		−	20.0164i		−	0.673605i	−0.941575	−	0.336802i	$-0.890655\pi$
$883$		−	20.0164i		−	0.673605i	0.941575	−	0.336802i	$-0.109345\pi$
$884$		0			0
$885$	10.9967	+	4.70797i	0.369649	+	0.158257i
$886$		0			0
$887$			0.490396i			0.0164659i	0.999966	+	0.00823295i	$0.00262066\pi$
$887$			0.490396i			0.0164659i	−0.999966	+	0.00823295i	$0.997379\pi$
$888$		0			0
$889$	21.0665			0.706549
$890$		0			0
$891$	−1.23619			−0.0414138
$892$		0			0
$893$			29.1461i			0.975338i
$894$		0			0
$895$	−11.0011	+	25.6959i	−0.367726	+	0.858918i
$896$		0			0
$897$			10.2051i			0.340739i
$898$		0			0
$899$	14.9174			0.497524
$900$		0			0
$901$	1.14553			0.0381631
$902$		0			0
$903$		−	0.122311i		−	0.00407025i
$904$		0			0
$905$	−6.59683	+	15.4086i	−0.219286	+	0.512198i
$906$		0			0
$907$			9.67301i			0.321187i	0.987021	+	0.160593i	$0.0513408\pi$
$907$			9.67301i			0.321187i	−0.987021	+	0.160593i	$0.948659\pi$
$908$		0			0
$909$	5.56043			0.184428
$910$		0			0
$911$	−22.4548			−0.743962			−0.371981	−	0.928240i	$-0.621321\pi$
$911$	−22.4548			−0.743962			−0.371981	+	0.928240i	$0.621321\pi$
$912$		0			0
$913$		−	7.35348i		−	0.243365i
$914$		0			0
$915$	−4.35344	−	1.86383i	−0.143920	−	0.0616162i
$916$		0			0
$917$		−	25.1185i		−	0.829486i
$918$		0			0
$919$	43.3615			1.43037			0.715183	−	0.698938i	$-0.246344\pi$
$919$	43.3615			1.43037			0.715183	+	0.698938i	$0.246344\pi$
$920$		0			0
$921$	−14.7640			−0.486490
$922$		0			0
$923$			5.70674i			0.187840i
$924$		0			0
$925$	0.589069	−	0.561862i	0.0193685	−	0.0184739i
$926$		0			0
$927$			2.21921i			0.0728885i
$928$		0			0
$929$	16.1920			0.531242			0.265621	−	0.964078i	$-0.414423\pi$
$929$	16.1920			0.531242			0.265621	+	0.964078i	$0.414423\pi$
$930$		0			0
$931$	39.9146			1.30815
$932$		0			0
$933$			18.9466i			0.620285i
$934$		0			0
$935$	−1.50511	−	0.644379i	−0.0492223	−	0.0210734i
$936$		0			0
$937$			20.9083i			0.683044i	0.939874	+	0.341522i	$0.110942\pi$
$937$			20.9083i			0.683044i	−0.939874	+	0.341522i	$0.889058\pi$
$938$		0			0
$939$	3.12572			0.102004
$940$		0			0
$941$	−7.92418			−0.258321			−0.129160	−	0.991624i	$-0.541228\pi$
$941$	−7.92418			−0.258321			−0.129160	+	0.991624i	$0.541228\pi$
$942$		0			0
$943$		−	23.8799i		−	0.777636i
$944$		0			0
$945$	1.14841	−	2.68239i	0.0373577	−	0.0872583i
$946$		0			0
$947$		−	36.1312i		−	1.17411i	−0.809549	−	0.587053i	$-0.800288\pi$
$947$		−	36.1312i		−	1.17411i	0.809549	−	0.587053i	$-0.199712\pi$
$948$		0			0
$949$	−17.3668			−0.563750
$950$		0			0
$951$	−25.8863			−0.839420
$952$		0			0
$953$		−	18.0300i		−	0.584049i	−0.956411	−	0.292025i	$-0.905671\pi$
$953$		−	18.0300i		−	0.584049i	0.956411	−	0.292025i	$-0.0943289\pi$
$954$		0			0
$955$	19.4701	−	45.4774i	0.630039	−	1.47161i
$956$		0			0
$957$		−	2.60857i		−	0.0843230i
$958$		0			0
$959$	4.27975			0.138200
$960$		0			0
$961$	18.9748			0.612090
$962$		0			0
$963$			5.31490i			0.171270i
$964$		0			0
$965$	30.0501	+	12.8653i	0.967348	+	0.414148i
$966$		0			0
$967$			57.0853i			1.83574i	0.396881	+	0.917870i	$0.370092\pi$
$967$			57.0853i			1.83574i	−0.396881	+	0.917870i	$0.629908\pi$
$968$		0			0
$969$	4.46305			0.143374
$970$		0			0
$971$	−52.7922			−1.69418			−0.847091	−	0.531448i	$-0.821648\pi$
$971$	−52.7922			−1.69418			−0.847091	+	0.531448i	$0.821648\pi$
$972$		0			0
$973$		−	10.9354i		−	0.350573i
$974$		0			0
$975$	5.43302	+	5.69611i	0.173996	+	0.182422i
$976$		0			0
$977$		−	40.3290i		−	1.29024i	−0.764082	−	0.645119i	$-0.776808\pi$
$977$		−	40.3290i		−	1.29024i	0.764082	−	0.645119i	$-0.223192\pi$
$978$		0			0
$979$	14.7025			0.469894
$980$		0			0
$981$	−2.24639			−0.0717216
$982$		0			0
$983$			51.8308i			1.65315i	0.562830	+	0.826573i	$0.309713\pi$
$983$			51.8308i			1.65315i	−0.562830	+	0.826573i	$0.690287\pi$
$984$		0			0
$985$	25.9471	+	11.1087i	0.826744	+	0.353952i
$986$		0			0
$987$			5.04752i			0.160664i
$988$		0			0
$989$	−0.607577			−0.0193198
$990$		0			0
$991$	−26.5090			−0.842087			−0.421044	−	0.907040i	$-0.638336\pi$
$991$	−26.5090			−0.842087			−0.421044	+	0.907040i	$0.638336\pi$
$992$		0			0
$993$			8.97633i			0.284855i
$994$		0			0
$995$	5.30397	−	12.3888i	0.168147	−	0.392750i
$996$		0			0
$997$			4.65120i			0.147305i	0.997284	+	0.0736525i	$0.0234656\pi$
$997$			4.65120i			0.147305i	−0.997284	+	0.0736525i	$0.976534\pi$
$998$		0			0
$999$	0.162812			0.00515113

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4020.2.g.b.1609.18	yes	24
5.4	even	2	inner	4020.2.g.b.1609.6	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4020.2.g.b.1609.6	✓	24	5.4	even	2	inner
4020.2.g.b.1609.18	yes	24	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +(-0.880059 + 2.05560i) q^{5} -1.30492i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +(-0.880059 + 2.05560i) q^{5} -1.30492i q^{7} -1.00000 q^{9} -1.23619 q^{11} +1.57434i q^{13} +(-2.05560 - 0.880059i) q^{15} -0.592305i q^{17} +7.53506 q^{19} +1.30492 q^{21} -6.48217i q^{23} +(-3.45099 - 3.61810i) q^{25} -1.00000i q^{27} +2.11017 q^{29} +7.06929 q^{31} -1.23619i q^{33} +(2.68239 + 1.14841i) q^{35} +0.162812i q^{37} -1.57434 q^{39} +3.68393 q^{41} -0.0937305i q^{43} +(0.880059 - 2.05560i) q^{45} +3.86807i q^{47} +5.29719 q^{49} +0.592305 q^{51} +1.93402i q^{53} +(1.08792 - 2.54111i) q^{55} +7.53506i q^{57} -5.34961 q^{59} +2.11784 q^{61} +1.30492i q^{63} +(-3.23621 - 1.38551i) q^{65} -1.00000i q^{67} +6.48217 q^{69} +3.62485 q^{71} +11.0312i q^{73} +(3.61810 - 3.45099i) q^{75} +1.61312i q^{77} +7.58300 q^{79} +1.00000 q^{81} +5.94852i q^{83} +(1.21754 + 0.521263i) q^{85} +2.11017i q^{87} -11.8934 q^{89} +2.05438 q^{91} +7.06929i q^{93} +(-6.63130 + 15.4891i) q^{95} +18.6958i q^{97} +1.23619 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 4 q^{5} - 24 q^{9}+O(q^{10}) \) 24 * q - 4 * q^5 - 24 * q^9	\( 24 q - 4 q^{5} - 24 q^{9} - 8 q^{11} - 2 q^{15} + 16 q^{19} - 20 q^{21} + 10 q^{25} + 36 q^{29} - 2 q^{35} + 4 q^{39} - 24 q^{41} + 4 q^{45} - 4 q^{51} - 4 q^{55} + 24 q^{59} - 4 q^{61} - 20 q^{65} - 4 q^{69} + 20 q^{71} - 12 q^{75} - 28 q^{79} + 24 q^{81} - 16 q^{85} + 48 q^{89} - 20 q^{91} - 4 q^{95} + 8 q^{99}+O(q^{100}) \) 24 * q - 4 * q^5 - 24 * q^9 - 8 * q^11 - 2 * q^15 + 16 * q^19 - 20 * q^21 + 10 * q^25 + 36 * q^29 - 2 * q^35 + 4 * q^39 - 24 * q^41 + 4 * q^45 - 4 * q^51 - 4 * q^55 + 24 * q^59 - 4 * q^61 - 20 * q^65 - 4 * q^69 + 20 * q^71 - 12 * q^75 - 28 * q^79 + 24 * q^81 - 16 * q^85 + 48 * q^89 - 20 * q^91 - 4 * q^95 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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