LMFDB - Embedded newform 4020.2.g.c.1609.17

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:	$38$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1609.17
Character	$\chi$	$=$	4020.1609
Dual form			4020.2.g.c.1609.36

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +(2.08125 - 0.817560i) q^{5} +3.41849i q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +(2.08125 - 0.817560i) q^{5} +3.41849i q^{7} -1.00000 q^{9} +5.18352 q^{11} -5.24487i q^{13} +(-0.817560 - 2.08125i) q^{15} +7.93273i q^{17} +1.59836 q^{19} +3.41849 q^{21} +7.16283i q^{23} +(3.66319 - 3.40309i) q^{25} +1.00000i q^{27} +6.58679 q^{29} -5.96221 q^{31} -5.18352i q^{33} +(2.79482 + 7.11473i) q^{35} +10.2629i q^{37} -5.24487 q^{39} -8.07835 q^{41} +0.663360i q^{43} +(-2.08125 + 0.817560i) q^{45} +7.08264i q^{47} -4.68609 q^{49} +7.93273 q^{51} -11.5654i q^{53} +(10.7882 - 4.23784i) q^{55} -1.59836i q^{57} +1.90150 q^{59} +1.14296 q^{61} -3.41849i q^{63} +(-4.28800 - 10.9159i) q^{65} -1.00000i q^{67} +7.16283 q^{69} +1.39500 q^{71} -6.22878i q^{73} +(-3.40309 - 3.66319i) q^{75} +17.7198i q^{77} -1.75892 q^{79} +1.00000 q^{81} +12.9936i q^{83} +(6.48548 + 16.5100i) q^{85} -6.58679i q^{87} -6.29938 q^{89} +17.9296 q^{91} +5.96221i q^{93} +(3.32658 - 1.30675i) q^{95} +4.62505i q^{97} -5.18352 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 38 q - 2 q^{5} - 38 q^{9}+O(q^{10}) $ 38 * q - 2 * q^5 - 38 * q^9	$ 38 q - 2 q^{5} - 38 q^{9} + 24 q^{11} + 2 q^{15} - 16 q^{19} + 12 q^{21} + 4 q^{25} - 56 q^{29} - 4 q^{31} - 2 q^{35} + 60 q^{41} + 2 q^{45} - 70 q^{49} + 12 q^{55} - 52 q^{59} + 48 q^{61} + 16 q^{65} - 12 q^{69} + 12 q^{75} - 24 q^{79} + 38 q^{81} + 16 q^{85} - 76 q^{89} + 44 q^{91} + 36 q^{95} - 24 q^{99}+O(q^{100}) $ 38 * q - 2 * q^5 - 38 * q^9 + 24 * q^11 + 2 * q^15 - 16 * q^19 + 12 * q^21 + 4 * q^25 - 56 * q^29 - 4 * q^31 - 2 * q^35 + 60 * q^41 + 2 * q^45 - 70 * q^49 + 12 * q^55 - 52 * q^59 + 48 * q^61 + 16 * q^65 - 12 * q^69 + 12 * q^75 - 24 * q^79 + 38 * q^81 + 16 * q^85 - 76 * q^89 + 44 * q^91 + 36 * q^95 - 24 * 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
$5$	2.08125	−	0.817560i	0.930763	−	0.365624i
$5$	2.08125	−	0.817560i	0.930763	−	0.365624i
$6$		0			0
$7$			3.41849i			1.29207i	0.763308	+	0.646034i	$0.223574\pi$
$7$			3.41849i			1.29207i	−0.763308	+	0.646034i	$0.776426\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	5.18352			1.56289			0.781446	−	0.623973i	$-0.214483\pi$
$11$	5.18352			1.56289			0.781446	+	0.623973i	$0.214483\pi$
$12$		0			0
$13$		−	5.24487i		−	1.45467i	−0.686285	−	0.727333i	$-0.740760\pi$
$13$		−	5.24487i		−	1.45467i	0.686285	−	0.727333i	$-0.259240\pi$
$14$		0			0
$15$	−0.817560	−	2.08125i	−0.211093	−	0.537376i
$16$		0			0
$17$			7.93273i			1.92397i	0.273105	+	0.961984i	$0.411949\pi$
$17$			7.93273i			1.92397i	−0.273105	+	0.961984i	$0.588051\pi$
$18$		0			0
$19$	1.59836			0.366688			0.183344	−	0.983049i	$-0.441308\pi$
$19$	1.59836			0.366688			0.183344	+	0.983049i	$0.441308\pi$
$20$		0			0
$21$	3.41849			0.745976
$22$		0			0
$23$			7.16283i			1.49355i	0.665075	+	0.746777i	$0.268400\pi$
$23$			7.16283i			1.49355i	−0.665075	+	0.746777i	$0.731600\pi$
$24$		0			0
$25$	3.66319	−	3.40309i	0.732638	−	0.680619i
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$	6.58679			1.22314			0.611568	−	0.791192i	$-0.290539\pi$
$29$	6.58679			1.22314			0.611568	+	0.791192i	$0.290539\pi$
$30$		0			0
$31$	−5.96221			−1.07085			−0.535423	−	0.844584i	$-0.679848\pi$
$31$	−5.96221			−1.07085			−0.535423	+	0.844584i	$0.679848\pi$
$32$		0			0
$33$		−	5.18352i		−	0.902336i
$34$		0			0
$35$	2.79482	+	7.11473i	0.472412	+	1.20261i
$36$		0			0
$37$			10.2629i			1.68720i	0.536969	+	0.843602i	$0.319569\pi$
$37$			10.2629i			1.68720i	−0.536969	+	0.843602i	$0.680431\pi$
$38$		0			0
$39$	−5.24487			−0.839852
$40$		0			0
$41$	−8.07835			−1.26163			−0.630813	−	0.775935i	$-0.717279\pi$
$41$	−8.07835			−1.26163			−0.630813	+	0.775935i	$0.717279\pi$
$42$		0			0
$43$			0.663360i			0.101161i	0.998720	+	0.0505807i	$0.0161072\pi$
$43$			0.663360i			0.101161i	−0.998720	+	0.0505807i	$0.983893\pi$
$44$		0			0
$45$	−2.08125	+	0.817560i	−0.310254	+	0.121875i
$46$		0			0
$47$			7.08264i			1.03311i	0.856254	+	0.516555i	$0.172786\pi$
$47$			7.08264i			1.03311i	−0.856254	+	0.516555i	$0.827214\pi$
$48$		0			0
$49$	−4.68609			−0.669442
$50$		0			0
$51$	7.93273			1.11080
$52$		0			0
$53$		−	11.5654i		−	1.58863i	−0.607504	−	0.794316i	$-0.707829\pi$
$53$		−	11.5654i		−	1.58863i	0.607504	−	0.794316i	$-0.292171\pi$
$54$		0			0
$55$	10.7882	−	4.23784i	1.45468	−	0.571431i
$56$		0			0
$57$		−	1.59836i		−	0.211707i
$58$		0			0
$59$	1.90150			0.247554			0.123777	−	0.992310i	$-0.460499\pi$
$59$	1.90150			0.247554			0.123777	+	0.992310i	$0.460499\pi$
$60$		0			0
$61$	1.14296			0.146341			0.0731705	−	0.997319i	$-0.476688\pi$
$61$	1.14296			0.146341			0.0731705	+	0.997319i	$0.476688\pi$
$62$		0			0
$63$		−	3.41849i		−	0.430690i
$64$		0			0
$65$	−4.28800	−	10.9159i	−0.531861	−	1.35395i
$66$		0			0
$67$		−	1.00000i		−	0.122169i
$67$		−	1.00000i		−	0.122169i
$68$		0			0
$69$	7.16283			0.862303
$70$		0			0
$71$	1.39500			0.165556			0.0827780	−	0.996568i	$-0.473621\pi$
$71$	1.39500			0.165556			0.0827780	+	0.996568i	$0.473621\pi$
$72$		0			0
$73$		−	6.22878i		−	0.729023i	−0.931199	−	0.364512i	$-0.881236\pi$
$73$		−	6.22878i		−	0.729023i	0.931199	−	0.364512i	$-0.118764\pi$
$74$		0			0
$75$	−3.40309	−	3.66319i	−0.392955	−	0.422989i
$76$		0			0
$77$			17.7198i			2.01936i
$78$		0			0
$79$	−1.75892			−0.197894			−0.0989469	−	0.995093i	$-0.531547\pi$
$79$	−1.75892			−0.197894			−0.0989469	+	0.995093i	$0.531547\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			12.9936i			1.42624i	0.701044	+	0.713118i	$0.252718\pi$
$83$			12.9936i			1.42624i	−0.701044	+	0.713118i	$0.747282\pi$
$84$		0			0
$85$	6.48548	+	16.5100i	0.703449	+	1.79076i
$86$		0			0
$87$		−	6.58679i		−	0.706178i
$88$		0			0
$89$	−6.29938			−0.667733			−0.333867	−	0.942620i	$-0.608354\pi$
$89$	−6.29938			−0.667733			−0.333867	+	0.942620i	$0.608354\pi$
$90$		0			0
$91$	17.9296			1.87953
$92$		0			0
$93$			5.96221i			0.618253i
$94$		0			0
$95$	3.32658	−	1.30675i	0.341300	−	0.134070i
$96$		0			0
$97$			4.62505i			0.469603i	0.972043	+	0.234801i	$0.0754440\pi$
$97$			4.62505i			0.469603i	−0.972043	+	0.234801i	$0.924556\pi$
$98$		0			0
$99$	−5.18352			−0.520964
$100$		0			0
$101$	9.37470			0.932818			0.466409	−	0.884569i	$-0.345548\pi$
$101$	9.37470			0.932818			0.466409	+	0.884569i	$0.345548\pi$
$102$		0			0
$103$			7.81662i			0.770194i	0.922876	+	0.385097i	$0.125832\pi$
$103$			7.81662i			0.770194i	−0.922876	+	0.385097i	$0.874168\pi$
$104$		0			0
$105$	7.11473	−	2.79482i	0.694327	−	0.272747i
$106$		0			0
$107$		−	9.25657i		−	0.894867i	−0.894317	−	0.447433i	$-0.852338\pi$
$107$		−	9.25657i		−	0.894867i	0.894317	−	0.447433i	$-0.147662\pi$
$108$		0			0
$109$	−16.7849			−1.60770			−0.803849	−	0.594834i	$-0.797218\pi$
$109$	−16.7849			−1.60770			−0.803849	+	0.594834i	$0.797218\pi$
$110$		0			0
$111$	10.2629			0.974108
$112$		0			0
$113$			0.985409i			0.0926995i	0.998925	+	0.0463497i	$0.0147589\pi$
$113$			0.985409i			0.0926995i	−0.998925	+	0.0463497i	$0.985241\pi$
$114$		0			0
$115$	5.85605	+	14.9076i	0.546079	+	1.39014i
$116$		0			0
$117$			5.24487i			0.484889i
$118$		0			0
$119$	−27.1180			−2.48590
$120$		0			0
$121$	15.8689			1.44263
$122$		0			0
$123$			8.07835i			0.728400i
$124$		0			0
$125$	4.84177	−	10.0776i	0.433061	−	0.901364i
$126$		0			0
$127$		−	3.93033i		−	0.348760i	−0.984678	−	0.174380i	$-0.944208\pi$
$127$		−	3.93033i		−	0.348760i	0.984678	−	0.174380i	$-0.0557921\pi$
$128$		0			0
$129$	0.663360			0.0584056
$130$		0			0
$131$	−17.9128			−1.56504			−0.782522	−	0.622622i	$-0.786067\pi$
$131$	−17.9128			−1.56504			−0.782522	+	0.622622i	$0.786067\pi$
$132$		0			0
$133$			5.46397i			0.473786i
$134$		0			0
$135$	0.817560	+	2.08125i	0.0703644	+	0.179125i
$136$		0			0
$137$			5.44440i			0.465146i	0.972579	+	0.232573i	$0.0747145\pi$
$137$			5.44440i			0.465146i	−0.972579	+	0.232573i	$0.925286\pi$
$138$		0			0
$139$	12.3886			1.05079			0.525395	−	0.850858i	$-0.323917\pi$
$139$	12.3886			1.05079			0.525395	+	0.850858i	$0.323917\pi$
$140$		0			0
$141$	7.08264			0.596466
$142$		0			0
$143$		−	27.1869i		−	2.27349i
$144$		0			0
$145$	13.7088	−	5.38510i	1.13845	−	0.447208i
$146$		0			0
$147$			4.68609i			0.386502i
$148$		0			0
$149$	10.4378			0.855094			0.427547	−	0.903993i	$-0.359378\pi$
$149$	10.4378			0.855094			0.427547	+	0.903993i	$0.359378\pi$
$150$		0			0
$151$	2.58130			0.210064			0.105032	−	0.994469i	$-0.466506\pi$
$151$	2.58130			0.210064			0.105032	+	0.994469i	$0.466506\pi$
$152$		0			0
$153$		−	7.93273i		−	0.641323i
$154$		0			0
$155$	−12.4088	+	4.87447i	−0.996703	+	0.391527i
$156$		0			0
$157$			2.37987i			0.189934i	0.995480	+	0.0949672i	$0.0302746\pi$
$157$			2.37987i			0.189934i	−0.995480	+	0.0949672i	$0.969725\pi$
$158$		0			0
$159$	−11.5654			−0.917197
$160$		0			0
$161$	−24.4861			−1.92977
$162$		0			0
$163$		−	8.42295i		−	0.659737i	−0.944027	−	0.329868i	$-0.892996\pi$
$163$		−	8.42295i		−	0.659737i	0.944027	−	0.329868i	$-0.107004\pi$
$164$		0			0
$165$	−4.23784	−	10.7882i	−0.329916	−	0.839860i
$166$		0			0
$167$			5.31683i			0.411428i	0.978612	+	0.205714i	$0.0659517\pi$
$167$			5.31683i			0.411428i	−0.978612	+	0.205714i	$0.934048\pi$
$168$		0			0
$169$	−14.5087			−1.11605
$170$		0			0
$171$	−1.59836			−0.122229
$172$		0			0
$173$		−	15.2214i		−	1.15726i	−0.815591	−	0.578629i	$-0.803588\pi$
$173$		−	15.2214i		−	1.15726i	0.815591	−	0.578629i	$-0.196412\pi$
$174$		0			0
$175$	11.6334	+	12.5226i	0.879406	+	0.946619i
$176$		0			0
$177$		−	1.90150i		−	0.142925i
$178$		0			0
$179$	13.6194			1.01796			0.508980	−	0.860779i	$-0.330023\pi$
$179$	13.6194			1.01796			0.508980	+	0.860779i	$0.330023\pi$
$180$		0			0
$181$	−5.42287			−0.403079			−0.201539	−	0.979480i	$-0.564594\pi$
$181$	−5.42287			−0.403079			−0.201539	+	0.979480i	$0.564594\pi$
$182$		0			0
$183$		−	1.14296i		−	0.0844900i
$184$		0			0
$185$	8.39051	+	21.3596i	0.616883	+	1.57039i
$186$		0			0
$187$			41.1195i			3.00695i
$188$		0			0
$189$	−3.41849			−0.248659
$190$		0			0
$191$	3.71059			0.268489			0.134244	−	0.990948i	$-0.457139\pi$
$191$	3.71059			0.268489			0.134244	+	0.990948i	$0.457139\pi$
$192$		0			0
$193$			4.66970i			0.336133i	0.985776	+	0.168066i	$0.0537523\pi$
$193$			4.66970i			0.336133i	−0.985776	+	0.168066i	$0.946248\pi$
$194$		0			0
$195$	−10.9159	+	4.28800i	−0.781703	+	0.307070i
$196$		0			0
$197$			21.7808i			1.55182i	0.630844	+	0.775909i	$0.282709\pi$
$197$			21.7808i			1.55182i	−0.630844	+	0.775909i	$0.717291\pi$
$198$		0			0
$199$	26.0873			1.84928			0.924641	−	0.380840i	$-0.124365\pi$
$199$	26.0873			1.84928			0.924641	+	0.380840i	$0.124365\pi$
$200$		0			0
$201$	−1.00000			−0.0705346
$202$		0			0
$203$			22.5169i			1.58038i
$204$		0			0
$205$	−16.8131	+	6.60454i	−1.17427	+	0.461281i
$206$		0			0
$207$		−	7.16283i		−	0.497851i
$208$		0			0
$209$	8.28512			0.573094
$210$		0			0
$211$	25.5255			1.75725			0.878626	−	0.477511i	$-0.158461\pi$
$211$	25.5255			1.75725			0.878626	+	0.477511i	$0.158461\pi$
$212$		0			0
$213$		−	1.39500i		−	0.0955838i
$214$		0			0
$215$	0.542337	+	1.38062i	0.0369871	+	0.0941573i
$216$		0			0
$217$		−	20.3818i		−	1.38361i
$218$		0			0
$219$	−6.22878			−0.420902
$220$		0			0
$221$	41.6061			2.79873
$222$		0			0
$223$		−	17.1864i		−	1.15089i	−0.817842	−	0.575443i	$-0.804830\pi$
$223$		−	17.1864i		−	1.15089i	0.817842	−	0.575443i	$-0.195170\pi$
$224$		0			0
$225$	−3.66319	+	3.40309i	−0.244213	+	0.226873i
$226$		0			0
$227$		−	10.5003i		−	0.696926i	−0.937322	−	0.348463i	$-0.886704\pi$
$227$		−	10.5003i		−	0.696926i	0.937322	−	0.348463i	$-0.113296\pi$
$228$		0			0
$229$	6.29601			0.416052			0.208026	−	0.978123i	$-0.433296\pi$
$229$	6.29601			0.416052			0.208026	+	0.978123i	$0.433296\pi$
$230$		0			0
$231$	17.7198			1.16588
$232$		0			0
$233$		−	27.9039i		−	1.82804i	−0.405664	−	0.914022i	$-0.632960\pi$
$233$		−	27.9039i		−	1.82804i	0.405664	−	0.914022i	$-0.367040\pi$
$234$		0			0
$235$	5.79049	+	14.7407i	0.377730	+	0.961580i
$236$		0			0
$237$			1.75892i			0.114254i
$238$		0			0
$239$	23.3664			1.51145			0.755725	−	0.654889i	$-0.227285\pi$
$239$	23.3664			1.51145			0.755725	+	0.654889i	$0.227285\pi$
$240$		0			0
$241$	−20.9579			−1.35002			−0.675009	−	0.737810i	$-0.735860\pi$
$241$	−20.9579			−1.35002			−0.675009	+	0.737810i	$0.735860\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$	−9.75292	+	3.83116i	−0.623091	+	0.244764i
$246$		0			0
$247$		−	8.38318i		−	0.533409i
$248$		0			0
$249$	12.9936			0.823438
$250$		0			0
$251$	−11.0003			−0.694336			−0.347168	−	0.937803i	$-0.612857\pi$
$251$	−11.0003			−0.694336			−0.347168	+	0.937803i	$0.612857\pi$
$252$		0			0
$253$			37.1287i			2.33426i
$254$		0			0
$255$	16.5100	−	6.48548i	1.03389	−	0.406137i
$256$		0			0
$257$		−	22.4819i		−	1.40238i	−0.712974	−	0.701190i	$-0.752652\pi$
$257$		−	22.4819i		−	1.40238i	0.712974	−	0.701190i	$-0.247348\pi$
$258$		0			0
$259$	−35.0835			−2.17998
$260$		0			0
$261$	−6.58679			−0.407712
$262$		0			0
$263$		−	19.3224i		−	1.19147i	−0.803181	−	0.595735i	$-0.796861\pi$
$263$		−	19.3224i		−	1.19147i	0.803181	−	0.595735i	$-0.203139\pi$
$264$		0			0
$265$	−9.45543	−	24.0705i	−0.580842	−	1.47864i
$266$		0			0
$267$			6.29938i			0.385516i
$268$		0			0
$269$	−3.71965			−0.226791			−0.113396	−	0.993550i	$-0.536173\pi$
$269$	−3.71965			−0.226791			−0.113396	+	0.993550i	$0.536173\pi$
$270$		0			0
$271$	15.3434			0.932043			0.466022	−	0.884773i	$-0.345687\pi$
$271$	15.3434			0.932043			0.466022	+	0.884773i	$0.345687\pi$
$272$		0			0
$273$		−	17.9296i		−	1.08515i
$274$		0			0
$275$	18.9882	−	17.6400i	1.14503	−	1.06373i
$276$		0			0
$277$		−	25.6005i		−	1.53818i	−0.639138	−	0.769092i	$-0.720709\pi$
$277$		−	25.6005i		−	1.53818i	0.639138	−	0.769092i	$-0.279291\pi$
$278$		0			0
$279$	5.96221			0.356948
$280$		0			0
$281$	24.1995			1.44362			0.721811	−	0.692090i	$-0.243310\pi$
$281$	24.1995			1.44362			0.721811	+	0.692090i	$0.243310\pi$
$282$		0			0
$283$			10.3634i			0.616037i	0.951380	+	0.308019i	$0.0996659\pi$
$283$			10.3634i			0.616037i	−0.951380	+	0.308019i	$0.900334\pi$
$284$		0			0
$285$	−1.30675	−	3.32658i	−0.0774054	−	0.197049i
$286$		0			0
$287$		−	27.6158i		−	1.63011i
$288$		0			0
$289$	−45.9281			−2.70166
$290$		0			0
$291$	4.62505			0.271125
$292$		0			0
$293$			4.91431i			0.287097i	0.989643	+	0.143549i	$0.0458513\pi$
$293$			4.91431i			0.287097i	−0.989643	+	0.143549i	$0.954149\pi$
$294$		0			0
$295$	3.95749	−	1.55459i	0.230414	−	0.0905116i
$296$		0			0
$297$			5.18352i			0.300779i
$298$		0			0
$299$	37.5681			2.17262
$300$		0			0
$301$	−2.26769			−0.130708
$302$		0			0
$303$		−	9.37470i		−	0.538563i
$304$		0			0
$305$	2.37878	−	0.934439i	0.136209	−	0.0535058i
$306$		0			0
$307$		−	2.87657i		−	0.164175i	−0.996625	−	0.0820873i	$-0.973841\pi$
$307$		−	2.87657i		−	0.164175i	0.996625	−	0.0820873i	$-0.0261586\pi$
$308$		0			0
$309$	7.81662			0.444672
$310$		0			0
$311$	6.15003			0.348736			0.174368	−	0.984681i	$-0.444212\pi$
$311$	6.15003			0.348736			0.174368	+	0.984681i	$0.444212\pi$
$312$		0			0
$313$			4.46096i			0.252149i	0.992021	+	0.126074i	$0.0402378\pi$
$313$			4.46096i			0.252149i	−0.992021	+	0.126074i	$0.959762\pi$
$314$		0			0
$315$	−2.79482	−	7.11473i	−0.157471	−	0.400870i
$316$		0			0
$317$		−	14.7678i		−	0.829444i	−0.909948	−	0.414722i	$-0.863879\pi$
$317$		−	14.7678i		−	0.829444i	0.909948	−	0.414722i	$-0.136121\pi$
$318$		0			0
$319$	34.1428			1.91163
$320$		0			0
$321$	−9.25657			−0.516652
$322$		0			0
$323$			12.6793i			0.705496i
$324$		0			0
$325$	−17.8488	−	19.2130i	−0.990073	−	1.06574i
$326$		0			0
$327$			16.7849i			0.928205i
$328$		0			0
$329$	−24.2120			−1.33485
$330$		0			0
$331$	13.0188			0.715578			0.357789	−	0.933802i	$-0.383531\pi$
$331$	13.0188			0.715578			0.357789	+	0.933802i	$0.383531\pi$
$332$		0			0
$333$		−	10.2629i		−	0.562401i
$334$		0			0
$335$	−0.817560	−	2.08125i	−0.0446681	−	0.113711i
$336$		0			0
$337$			5.35694i			0.291811i	0.989299	+	0.145906i	$0.0466096\pi$
$337$			5.35694i			0.291811i	−0.989299	+	0.145906i	$0.953390\pi$
$338$		0			0
$339$	0.985409			0.0535201
$340$		0			0
$341$	−30.9053			−1.67362
$342$		0			0
$343$			7.91008i			0.427104i
$344$		0			0
$345$	14.9076	−	5.85605i	0.802600	−	0.315279i
$346$		0			0
$347$		−	28.3748i		−	1.52324i	−0.648025	−	0.761619i	$-0.724405\pi$
$347$		−	28.3748i		−	1.52324i	0.648025	−	0.761619i	$-0.275595\pi$
$348$		0			0
$349$	−18.6940			−1.00067			−0.500335	−	0.865832i	$-0.666790\pi$
$349$	−18.6940			−1.00067			−0.500335	+	0.865832i	$0.666790\pi$
$350$		0			0
$351$	5.24487			0.279951
$352$		0			0
$353$		−	15.5467i		−	0.827465i	−0.910399	−	0.413732i	$-0.864225\pi$
$353$		−	15.5467i		−	0.827465i	0.910399	−	0.413732i	$-0.135775\pi$
$354$		0			0
$355$	2.90334	−	1.14050i	0.154093	−	0.0605313i
$356$		0			0
$357$			27.1180i			1.43523i
$358$		0			0
$359$	34.8793			1.84086			0.920430	−	0.390908i	$-0.127839\pi$
$359$	34.8793			1.84086			0.920430	+	0.390908i	$0.127839\pi$
$360$		0			0
$361$	−16.4453			−0.865540
$362$		0			0
$363$		−	15.8689i		−	0.832903i
$364$		0			0
$365$	−5.09240	−	12.9636i	−0.266549	−	0.678548i
$366$		0			0
$367$		−	12.4800i		−	0.651452i	−0.945464	−	0.325726i	$-0.894391\pi$
$367$		−	12.4800i		−	0.651452i	0.945464	−	0.325726i	$-0.105609\pi$
$368$		0			0
$369$	8.07835			0.420542
$370$		0			0
$371$	39.5363			2.05262
$372$		0			0
$373$			5.61837i			0.290908i	0.989365	+	0.145454i	$0.0464643\pi$
$373$			5.61837i			0.290908i	−0.989365	+	0.145454i	$0.953536\pi$
$374$		0			0
$375$	−10.0776	−	4.84177i	−0.520403	−	0.250028i
$376$		0			0
$377$		−	34.5469i		−	1.77926i
$378$		0			0
$379$	12.6866			0.651669			0.325834	−	0.945427i	$-0.394355\pi$
$379$	12.6866			0.651669			0.325834	+	0.945427i	$0.394355\pi$
$380$		0			0
$381$	−3.93033			−0.201357
$382$		0			0
$383$			22.8508i			1.16762i	0.811889	+	0.583811i	$0.198439\pi$
$383$			22.8508i			1.16762i	−0.811889	+	0.583811i	$0.801561\pi$
$384$		0			0
$385$	14.4870	+	36.8794i	0.738328	+	1.87955i
$386$		0			0
$387$		−	0.663360i		−	0.0337205i
$388$		0			0
$389$	37.2207			1.88716			0.943581	−	0.331142i	$-0.107434\pi$
$389$	37.2207			1.88716			0.943581	+	0.331142i	$0.107434\pi$
$390$		0			0
$391$	−56.8208			−2.87355
$392$		0			0
$393$			17.9128i			0.903579i
$394$		0			0
$395$	−3.66075	+	1.43802i	−0.184192	+	0.0723548i
$396$		0			0
$397$		−	0.301013i		−	0.0151074i	−0.999971	−	0.00755370i	$-0.997596\pi$
$397$		−	0.301013i		−	0.0151074i	0.999971	−	0.00755370i	$-0.00240444\pi$
$398$		0			0
$399$	5.46397			0.273541
$400$		0			0
$401$	0.0301904			0.00150764			0.000753818	−	1.00000i	$-0.499760\pi$
$401$	0.0301904			0.00150764			0.000753818		1.00000i	$0.499760\pi$
$402$		0			0
$403$			31.2711i			1.55772i
$404$		0			0
$405$	2.08125	−	0.817560i	0.103418	−	0.0406249i
$406$		0			0
$407$			53.1978i			2.63692i
$408$		0			0
$409$	−9.13879			−0.451884			−0.225942	−	0.974141i	$-0.572546\pi$
$409$	−9.13879			−0.451884			−0.225942	+	0.974141i	$0.572546\pi$
$410$		0			0
$411$	5.44440			0.268552
$412$		0			0
$413$			6.50025i			0.319856i
$414$		0			0
$415$	10.6231	+	27.0430i	0.521466	+	1.32749i
$416$		0			0
$417$		−	12.3886i		−	0.606674i
$418$		0			0
$419$	0.473003			0.0231077			0.0115539	−	0.999933i	$-0.496322\pi$
$419$	0.473003			0.0231077			0.0115539	+	0.999933i	$0.496322\pi$
$420$		0			0
$421$	−31.6592			−1.54297			−0.771487	−	0.636245i	$-0.780487\pi$
$421$	−31.6592			−1.54297			−0.771487	+	0.636245i	$0.780487\pi$
$422$		0			0
$423$		−	7.08264i		−	0.344370i
$424$		0			0
$425$	26.9958	+	29.0591i	1.30949	+	1.40957i
$426$		0			0
$427$			3.90720i			0.189083i
$428$		0			0
$429$	−27.1869			−1.31260
$430$		0			0
$431$	−15.9666			−0.769084			−0.384542	−	0.923108i	$-0.625641\pi$
$431$	−15.9666			−0.769084			−0.384542	+	0.923108i	$0.625641\pi$
$432$		0			0
$433$			11.5387i			0.554516i	0.960795	+	0.277258i	$0.0894257\pi$
$433$			11.5387i			0.554516i	−0.960795	+	0.277258i	$0.910574\pi$
$434$		0			0
$435$	−5.38510	−	13.7088i	−0.258196	−	0.657284i
$436$		0			0
$437$			11.4488i			0.547668i
$438$		0			0
$439$	−18.1762			−0.867501			−0.433750	−	0.901033i	$-0.642810\pi$
$439$	−18.1762			−0.867501			−0.433750	+	0.901033i	$0.642810\pi$
$440$		0			0
$441$	4.68609			0.223147
$442$		0			0
$443$		−	15.0286i		−	0.714032i	−0.934098	−	0.357016i	$-0.883794\pi$
$443$		−	15.0286i		−	0.714032i	0.934098	−	0.357016i	$-0.116206\pi$
$444$		0			0
$445$	−13.1106	+	5.15013i	−0.621501	+	0.244139i
$446$		0			0
$447$		−	10.4378i		−	0.493689i
$448$		0			0
$449$	−6.19596			−0.292405			−0.146203	−	0.989255i	$-0.546705\pi$
$449$	−6.19596			−0.292405			−0.146203	+	0.989255i	$0.546705\pi$
$450$		0			0
$451$	−41.8743			−1.97178
$452$		0			0
$453$		−	2.58130i		−	0.121280i
$454$		0			0
$455$	37.3159	−	14.6585i	1.74939	−	0.687201i
$456$		0			0
$457$			2.21675i			0.103695i	0.998655	+	0.0518476i	$0.0165110\pi$
$457$			2.21675i			0.103695i	−0.998655	+	0.0518476i	$0.983489\pi$
$458$		0			0
$459$	−7.93273			−0.370268
$460$		0			0
$461$	−7.01529			−0.326735			−0.163367	−	0.986565i	$-0.552236\pi$
$461$	−7.01529			−0.326735			−0.163367	+	0.986565i	$0.552236\pi$
$462$		0			0
$463$			6.89526i			0.320450i	0.987081	+	0.160225i	$0.0512219\pi$
$463$			6.89526i			0.320450i	−0.987081	+	0.160225i	$0.948778\pi$
$464$		0			0
$465$	4.87447	+	12.4088i	0.226048	+	0.575447i
$466$		0			0
$467$			2.92697i			0.135444i	0.997704	+	0.0677220i	$0.0215731\pi$
$467$			2.92697i			0.135444i	−0.997704	+	0.0677220i	$0.978427\pi$
$468$		0			0
$469$	3.41849			0.157851
$470$		0			0
$471$	2.37987			0.109659
$472$		0			0
$473$			3.43854i			0.158104i
$474$		0			0
$475$	5.85508	−	5.43936i	0.268650	−	0.249575i
$476$		0			0
$477$			11.5654i			0.529544i
$478$		0			0
$479$	−28.3826			−1.29684			−0.648418	−	0.761285i	$-0.724569\pi$
$479$	−28.3826			−1.29684			−0.648418	+	0.761285i	$0.724569\pi$
$480$		0			0
$481$	53.8274			2.45432
$482$		0			0
$483$			24.4861i			1.11416i
$484$		0			0
$485$	3.78126	+	9.62588i	0.171698	+	0.437089i
$486$		0			0
$487$		−	20.0285i		−	0.907579i	−0.891109	−	0.453789i	$-0.850072\pi$
$487$		−	20.0285i		−	0.907579i	0.891109	−	0.453789i	$-0.149928\pi$
$488$		0			0
$489$	−8.42295			−0.380899
$490$		0			0
$491$	5.61015			0.253183			0.126591	−	0.991955i	$-0.459596\pi$
$491$	5.61015			0.253183			0.126591	+	0.991955i	$0.459596\pi$
$492$		0			0
$493$			52.2512i			2.35328i
$494$		0			0
$495$	−10.7882	+	4.23784i	−0.484894	+	0.190477i
$496$		0			0
$497$			4.76880i			0.213910i
$498$		0			0
$499$	−14.1619			−0.633974			−0.316987	−	0.948430i	$-0.602671\pi$
$499$	−14.1619			−0.633974			−0.316987	+	0.948430i	$0.602671\pi$
$500$		0			0
$501$	5.31683			0.237538
$502$		0			0
$503$		−	0.705378i		−	0.0314513i	−0.999876	−	0.0157256i	$-0.994994\pi$
$503$		−	0.705378i		−	0.0314513i	0.999876	−	0.0157256i	$-0.00500583\pi$
$504$		0			0
$505$	19.5111	−	7.66439i	0.868232	−	0.341061i
$506$		0			0
$507$			14.5087i			0.644354i
$508$		0			0
$509$	40.4309			1.79207			0.896035	−	0.443984i	$-0.146435\pi$
$509$	40.4309			1.79207			0.896035	+	0.443984i	$0.146435\pi$
$510$		0			0
$511$	21.2930			0.941948
$512$		0			0
$513$			1.59836i			0.0705692i
$514$		0			0
$515$	6.39056	+	16.2683i	0.281602	+	0.716868i
$516$		0			0
$517$			36.7131i			1.61464i
$518$		0			0
$519$	−15.2214			−0.668143
$520$		0			0
$521$	−21.4333			−0.939011			−0.469506	−	0.882929i	$-0.655568\pi$
$521$	−21.4333			−0.939011			−0.469506	+	0.882929i	$0.655568\pi$
$522$		0			0
$523$		−	34.4820i		−	1.50779i	−0.656994	−	0.753895i	$-0.728172\pi$
$523$		−	34.4820i		−	1.50779i	0.656994	−	0.753895i	$-0.271828\pi$
$524$		0			0
$525$	12.5226	−	11.6334i	0.546530	−	0.507725i
$526$		0			0
$527$		−	47.2966i		−	2.06027i
$528$		0			0
$529$	−28.3061			−1.23070
$530$		0			0
$531$	−1.90150			−0.0825179
$532$		0			0
$533$			42.3699i			1.83524i
$534$		0			0
$535$	−7.56781	−	19.2652i	−0.327185	−	0.832908i
$536$		0			0
$537$		−	13.6194i		−	0.587719i
$538$		0			0
$539$	−24.2905			−1.04626
$540$		0			0
$541$	−7.74511			−0.332988			−0.166494	−	0.986042i	$-0.553245\pi$
$541$	−7.74511			−0.332988			−0.166494	+	0.986042i	$0.553245\pi$
$542$		0			0
$543$			5.42287i			0.232718i
$544$		0			0
$545$	−34.9335	+	13.7226i	−1.49638	+	0.587813i
$546$		0			0
$547$			5.58452i			0.238777i	0.992848	+	0.119388i	$0.0380934\pi$
$547$			5.58452i			0.238777i	−0.992848	+	0.119388i	$0.961907\pi$
$548$		0			0
$549$	−1.14296			−0.0487803
$550$		0			0
$551$	10.5280			0.448510
$552$		0			0
$553$		−	6.01285i		−	0.255692i
$554$		0			0
$555$	21.3596	−	8.39051i	0.906663	−	0.356157i
$556$		0			0
$557$			4.65576i			0.197271i	0.995124	+	0.0986355i	$0.0314478\pi$
$557$			4.65576i			0.197271i	−0.995124	+	0.0986355i	$0.968552\pi$
$558$		0			0
$559$	3.47924			0.147156
$560$		0			0
$561$	41.1195			1.73607
$562$		0			0
$563$			7.52137i			0.316988i	0.987360	+	0.158494i	$0.0506639\pi$
$563$			7.52137i			0.316988i	−0.987360	+	0.158494i	$0.949336\pi$
$564$		0			0
$565$	0.805631	+	2.05088i	0.0338932	+	0.0862812i
$566$		0			0
$567$			3.41849i			0.143563i
$568$		0			0
$569$	−47.1101			−1.97496			−0.987479	−	0.157752i	$-0.949575\pi$
$569$	−47.1101			−1.97496			−0.987479	+	0.157752i	$0.949575\pi$
$570$		0			0
$571$	47.2241			1.97627			0.988133	−	0.153600i	$-0.0490868\pi$
$571$	47.2241			1.97627			0.988133	+	0.153600i	$0.0490868\pi$
$572$		0			0
$573$		−	3.71059i		−	0.155012i
$574$		0			0
$575$	24.3758	+	26.2388i	1.01654	+	1.09423i
$576$		0			0
$577$			31.2367i			1.30040i	0.759762	+	0.650201i	$0.225315\pi$
$577$			31.2367i			1.30040i	−0.759762	+	0.650201i	$0.774685\pi$
$578$		0			0
$579$	4.66970			0.194066
$580$		0			0
$581$	−44.4186			−1.84279
$582$		0			0
$583$		−	59.9496i		−	2.48286i
$584$		0			0
$585$	4.28800	+	10.9159i	0.177287	+	0.451316i
$586$		0			0
$587$		−	14.6525i		−	0.604772i	−0.953186	−	0.302386i	$-0.902217\pi$
$587$		−	14.6525i		−	0.604772i	0.953186	−	0.302386i	$-0.0977832\pi$
$588$		0			0
$589$	−9.52975			−0.392666
$590$		0			0
$591$	21.7808			0.895943
$592$		0			0
$593$			43.6008i			1.79047i	0.445593	+	0.895236i	$0.352993\pi$
$593$			43.6008i			1.79047i	−0.445593	+	0.895236i	$0.647007\pi$
$594$		0			0
$595$	−56.4392	+	22.1706i	−2.31378	+	0.908905i
$596$		0			0
$597$		−	26.0873i		−	1.06768i
$598$		0			0
$599$	−22.8903			−0.935271			−0.467636	−	0.883921i	$-0.654894\pi$
$599$	−22.8903			−0.935271			−0.467636	+	0.883921i	$0.654894\pi$
$600$		0			0
$601$	16.4901			0.672645			0.336322	−	0.941747i	$-0.390817\pi$
$601$	16.4901			0.672645			0.336322	+	0.941747i	$0.390817\pi$
$602$		0			0
$603$			1.00000i			0.0407231i
$604$		0			0
$605$	33.0272	−	12.9738i	1.34275	−	0.527460i
$606$		0			0
$607$			10.0851i			0.409344i	0.978831	+	0.204672i	$0.0656127\pi$
$607$			10.0851i			0.409344i	−0.978831	+	0.204672i	$0.934387\pi$
$608$		0			0
$609$	22.5169			0.912431
$610$		0			0
$611$	37.1476			1.50283
$612$		0			0
$613$			23.8535i			0.963434i	0.876327	+	0.481717i	$0.159987\pi$
$613$			23.8535i			0.963434i	−0.876327	+	0.481717i	$0.840013\pi$
$614$		0			0
$615$	6.60454	+	16.8131i	0.266321	+	0.677968i
$616$		0			0
$617$		−	0.706932i		−	0.0284600i	−0.999899	−	0.0142300i	$-0.995470\pi$
$617$		−	0.706932i		−	0.0284600i	0.999899	−	0.0142300i	$-0.00452971\pi$
$618$		0			0
$619$	36.6278			1.47220			0.736099	−	0.676874i	$-0.236666\pi$
$619$	36.6278			1.47220			0.736099	+	0.676874i	$0.236666\pi$
$620$		0			0
$621$	−7.16283			−0.287434
$622$		0			0
$623$		−	21.5344i		−	0.862757i
$624$		0			0
$625$	1.83792	−	24.9323i	0.0735167	−	0.997294i
$626$		0			0
$627$		−	8.28512i		−	0.330876i
$628$		0			0
$629$	−81.4125			−3.24613
$630$		0			0
$631$	−36.4962			−1.45289			−0.726445	−	0.687225i	$-0.758829\pi$
$631$	−36.4962			−1.45289			−0.726445	+	0.687225i	$0.758829\pi$
$632$		0			0
$633$		−	25.5255i		−	1.01455i
$634$		0			0
$635$	−3.21328	−	8.17999i	−0.127515	−	0.324613i
$636$		0			0
$637$			24.5780i			0.973814i
$638$		0			0
$639$	−1.39500			−0.0551853
$640$		0			0
$641$	5.86090			0.231492			0.115746	−	0.993279i	$-0.463074\pi$
$641$	5.86090			0.231492			0.115746	+	0.993279i	$0.463074\pi$
$642$		0			0
$643$		−	14.6775i		−	0.578825i	−0.957205	−	0.289412i	$-0.906540\pi$
$643$		−	14.6775i		−	0.578825i	0.957205	−	0.289412i	$-0.0934599\pi$
$644$		0			0
$645$	1.38062	−	0.542337i	0.0543617	−	0.0213545i
$646$		0			0
$647$		−	17.5924i		−	0.691630i	−0.938303	−	0.345815i	$-0.887603\pi$
$647$		−	17.5924i		−	0.691630i	0.938303	−	0.345815i	$-0.112397\pi$
$648$		0			0
$649$	9.85645			0.386900
$650$		0			0
$651$	−20.3818			−0.798825
$652$		0			0
$653$		−	9.97398i		−	0.390312i	−0.980772	−	0.195156i	$-0.937479\pi$
$653$		−	9.97398i		−	0.390312i	0.980772	−	0.195156i	$-0.0625213\pi$
$654$		0			0
$655$	−37.2809	+	14.6448i	−1.45669	+	0.572218i
$656$		0			0
$657$			6.22878i			0.243008i
$658$		0			0
$659$	−12.5427			−0.488595			−0.244298	−	0.969700i	$-0.578557\pi$
$659$	−12.5427			−0.488595			−0.244298	+	0.969700i	$0.578557\pi$
$660$		0			0
$661$	8.99415			0.349832			0.174916	−	0.984583i	$-0.444035\pi$
$661$	8.99415			0.349832			0.174916	+	0.984583i	$0.444035\pi$
$662$		0			0
$663$		−	41.6061i		−	1.61585i
$664$		0			0
$665$	4.46713	+	11.3719i	0.173228	+	0.440983i
$666$		0			0
$667$			47.1801i			1.82682i
$668$		0			0
$669$	−17.1864			−0.664464
$670$		0			0
$671$	5.92456			0.228715
$672$		0			0
$673$		−	0.182820i		−	0.00704719i	−0.999994	−	0.00352360i	$-0.998878\pi$
$673$		−	0.182820i		−	0.00704719i	0.999994	−	0.00352360i	$-0.00112160\pi$
$674$		0			0
$675$	3.40309	+	3.66319i	0.130985	+	0.140996i
$676$		0			0
$677$		−	48.2910i		−	1.85598i	−0.372610	−	0.927988i	$-0.621537\pi$
$677$		−	48.2910i		−	1.85598i	0.372610	−	0.927988i	$-0.378463\pi$
$678$		0			0
$679$	−15.8107			−0.606759
$680$		0			0
$681$	−10.5003			−0.402371
$682$		0			0
$683$		−	45.2104i		−	1.72993i	−0.501833	−	0.864965i	$-0.667341\pi$
$683$		−	45.2104i		−	1.72993i	0.501833	−	0.864965i	$-0.332659\pi$
$684$		0			0
$685$	4.45112	+	11.3311i	0.170069	+	0.432941i
$686$		0			0
$687$		−	6.29601i		−	0.240208i
$688$		0			0
$689$	−60.6592			−2.31093
$690$		0			0
$691$	−26.6390			−1.01339			−0.506697	−	0.862124i	$-0.669134\pi$
$691$	−26.6390			−1.01339			−0.506697	+	0.862124i	$0.669134\pi$
$692$		0			0
$693$		−	17.7198i		−	0.673121i
$694$		0			0
$695$	25.7838	−	10.1285i	0.978037	−	0.384194i
$696$		0			0
$697$		−	64.0833i		−	2.42733i
$698$		0			0
$699$	−27.9039			−1.05542
$700$		0			0
$701$	−36.7808			−1.38919			−0.694596	−	0.719400i	$-0.744417\pi$
$701$	−36.7808			−1.38919			−0.694596	+	0.719400i	$0.744417\pi$
$702$		0			0
$703$			16.4037i			0.618678i
$704$		0			0
$705$	14.7407	−	5.79049i	0.555168	−	0.218082i
$706$		0			0
$707$			32.0474i			1.20526i
$708$		0			0
$709$	−8.39744			−0.315372			−0.157686	−	0.987489i	$-0.550403\pi$
$709$	−8.39744			−0.315372			−0.157686	+	0.987489i	$0.550403\pi$
$710$		0			0
$711$	1.75892			0.0659646
$712$		0			0
$713$		−	42.7063i		−	1.59936i
$714$		0			0
$715$	−22.2270	−	56.5828i	−0.831241	−	2.11608i
$716$		0			0
$717$		−	23.3664i		−	0.872636i
$718$		0			0
$719$	−44.8620			−1.67307			−0.836536	−	0.547913i	$-0.815423\pi$
$719$	−44.8620			−1.67307			−0.836536	+	0.547913i	$0.815423\pi$
$720$		0			0
$721$	−26.7210			−0.995144
$722$		0			0
$723$			20.9579i			0.779433i
$724$		0			0
$725$	24.1287	−	22.4155i	0.896116	−	0.832490i
$726$		0			0
$727$			39.2720i			1.45652i	0.685301	+	0.728260i	$0.259671\pi$
$727$			39.2720i			1.45652i	−0.685301	+	0.728260i	$0.740329\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	−5.26225			−0.194631
$732$		0			0
$733$			20.9749i			0.774726i	0.921927	+	0.387363i	$0.126614\pi$
$733$			20.9749i			0.774726i	−0.921927	+	0.387363i	$0.873386\pi$
$734$		0			0
$735$	3.83116	+	9.75292i	0.141315	+	0.359742i
$736$		0			0
$737$		−	5.18352i		−	0.190938i
$738$		0			0
$739$	−1.13014			−0.0415728			−0.0207864	−	0.999784i	$-0.506617\pi$
$739$	−1.13014			−0.0415728			−0.0207864	+	0.999784i	$0.506617\pi$
$740$		0			0
$741$	−8.38318			−0.307964
$742$		0			0
$743$			48.0720i			1.76359i	0.471633	+	0.881795i	$0.343665\pi$
$743$			48.0720i			1.76359i	−0.471633	+	0.881795i	$0.656335\pi$
$744$		0			0
$745$	21.7236	−	8.53349i	0.795889	−	0.312643i
$746$		0			0
$747$		−	12.9936i		−	0.475412i
$748$		0			0
$749$	31.6435			1.15623
$750$		0			0
$751$	34.0475			1.24241			0.621206	−	0.783647i	$-0.286643\pi$
$751$	34.0475			1.24241			0.621206	+	0.783647i	$0.286643\pi$
$752$		0			0
$753$			11.0003i			0.400875i
$754$		0			0
$755$	5.37233	−	2.11037i	0.195519	−	0.0768043i
$756$		0			0
$757$			1.62146i			0.0589329i	0.999566	+	0.0294665i	$0.00938082\pi$
$757$			1.62146i			0.0589329i	−0.999566	+	0.0294665i	$0.990619\pi$
$758$		0			0
$759$	37.1287			1.34769
$760$		0			0
$761$	−9.62522			−0.348914			−0.174457	−	0.984665i	$-0.555817\pi$
$761$	−9.62522			−0.348914			−0.174457	+	0.984665i	$0.555817\pi$
$762$		0			0
$763$		−	57.3789i		−	2.07726i
$764$		0			0
$765$	−6.48548	−	16.5100i	−0.234483	−	0.596919i
$766$		0			0
$767$		−	9.97311i		−	0.360108i
$768$		0			0
$769$	−43.3671			−1.56386			−0.781929	−	0.623367i	$-0.785764\pi$
$769$	−43.3671			−1.56386			−0.781929	+	0.623367i	$0.785764\pi$
$770$		0			0
$771$	−22.4819			−0.809665
$772$		0			0
$773$		−	23.1785i		−	0.833672i	−0.908982	−	0.416836i	$-0.863139\pi$
$773$		−	23.1785i		−	0.833672i	0.908982	−	0.416836i	$-0.136861\pi$
$774$		0			0
$775$	−21.8407	+	20.2900i	−0.784542	+	0.728837i
$776$		0			0
$777$			35.0835i			1.25861i
$778$		0			0
$779$	−12.9121			−0.462623
$780$		0			0
$781$	7.23102			0.258746
$782$		0			0
$783$			6.58679i			0.235393i
$784$		0			0
$785$	1.94569	+	4.95310i	0.0694446	+	0.176784i
$786$		0			0
$787$			23.5640i			0.839967i	0.907532	+	0.419984i	$0.137964\pi$
$787$			23.5640i			0.839967i	−0.907532	+	0.419984i	$0.862036\pi$
$788$		0			0
$789$	−19.3224			−0.687895
$790$		0			0
$791$	−3.36861			−0.119774
$792$		0			0
$793$		−	5.99468i		−	0.212877i
$794$		0			0
$795$	−24.0705	+	9.45543i	−0.853693	+	0.335350i
$796$		0			0
$797$			1.39253i			0.0493258i	0.999696	+	0.0246629i	$0.00785124\pi$
$797$			1.39253i			0.0493258i	−0.999696	+	0.0246629i	$0.992149\pi$
$798$		0			0
$799$	−56.1847			−1.98767
$800$		0			0
$801$	6.29938			0.222578
$802$		0			0
$803$		−	32.2870i		−	1.13938i
$804$		0			0
$805$	−50.9616	+	20.0188i	−1.79616	+	0.705572i
$806$		0			0
$807$			3.71965i			0.130938i
$808$		0			0
$809$	31.3071			1.10070			0.550350	−	0.834934i	$-0.314494\pi$
$809$	31.3071			1.10070			0.550350	+	0.834934i	$0.314494\pi$
$810$		0			0
$811$	24.7145			0.867844			0.433922	−	0.900950i	$-0.357129\pi$
$811$	24.7145			0.867844			0.433922	+	0.900950i	$0.357129\pi$
$812$		0			0
$813$		−	15.3434i		−	0.538115i
$814$		0			0
$815$	−6.88627	−	17.5303i	−0.241216	−	0.614058i
$816$		0			0
$817$			1.06029i			0.0370947i
$818$		0			0
$819$	−17.9296			−0.626510
$820$		0			0
$821$	44.0688			1.53801			0.769006	−	0.639242i	$-0.220752\pi$
$821$	44.0688			1.53801			0.769006	+	0.639242i	$0.220752\pi$
$822$		0			0
$823$		−	31.3790i		−	1.09380i	−0.837197	−	0.546902i	$-0.815807\pi$
$823$		−	31.3790i		−	1.09380i	0.837197	−	0.546902i	$-0.184193\pi$
$824$		0			0
$825$	−17.6400	−	18.9882i	−0.614146	−	0.661085i
$826$		0			0
$827$		−	41.9888i		−	1.46009i	−0.683397	−	0.730047i	$-0.739498\pi$
$827$		−	41.9888i		−	1.46009i	0.683397	−	0.730047i	$-0.260502\pi$
$828$		0			0
$829$	1.45124			0.0504037			0.0252018	−	0.999682i	$-0.491977\pi$
$829$	1.45124			0.0504037			0.0252018	+	0.999682i	$0.491977\pi$
$830$		0			0
$831$	−25.6005			−0.888071
$832$		0			0
$833$		−	37.1735i		−	1.28798i
$834$		0			0
$835$	4.34683	+	11.0656i	0.150428	+	0.382942i
$836$		0			0
$837$		−	5.96221i		−	0.206084i
$838$		0			0
$839$	15.9161			0.549487			0.274743	−	0.961518i	$-0.411407\pi$
$839$	15.9161			0.549487			0.274743	+	0.961518i	$0.411407\pi$
$840$		0			0
$841$	14.3858			0.496063
$842$		0			0
$843$		−	24.1995i		−	0.833476i
$844$		0			0
$845$	−30.1962	+	11.8617i	−1.03878	+	0.408056i
$846$		0			0
$847$			54.2478i			1.86398i
$848$		0			0
$849$	10.3634			0.355669
$850$		0			0
$851$	−73.5111			−2.51993
$852$		0			0
$853$			27.1330i			0.929015i	0.885569	+	0.464507i	$0.153769\pi$
$853$			27.1330i			0.929015i	−0.885569	+	0.464507i	$0.846231\pi$
$854$		0			0
$855$	−3.32658	+	1.30675i	−0.113767	+	0.0446900i
$856$		0			0
$857$		−	41.8254i		−	1.42873i	−0.699774	−	0.714364i	$-0.746716\pi$
$857$		−	41.8254i		−	1.42873i	0.699774	−	0.714364i	$-0.253284\pi$
$858$		0			0
$859$	25.8125			0.880710			0.440355	−	0.897824i	$-0.354853\pi$
$859$	25.8125			0.880710			0.440355	+	0.897824i	$0.354853\pi$
$860$		0			0
$861$	−27.6158			−0.941143
$862$		0			0
$863$			2.31036i			0.0786455i	0.999227	+	0.0393227i	$0.0125200\pi$
$863$			2.31036i			0.0786455i	−0.999227	+	0.0393227i	$0.987480\pi$
$864$		0			0
$865$	−12.4444	−	31.6794i	−0.423121	−	1.07713i
$866$		0			0
$867$			45.9281i			1.55980i
$868$		0			0
$869$	−9.11740			−0.309287
$870$		0			0
$871$	−5.24487			−0.177716
$872$		0			0
$873$		−	4.62505i		−	0.156534i
$874$		0			0
$875$	34.4501	+	16.5516i	1.16462	+	0.559545i
$876$		0			0
$877$		−	39.6911i		−	1.34027i	−0.742237	−	0.670137i	$-0.766235\pi$
$877$		−	39.6911i		−	1.34027i	0.742237	−	0.670137i	$-0.233765\pi$
$878$		0			0
$879$	4.91431			0.165756
$880$		0			0
$881$	−22.8795			−0.770830			−0.385415	−	0.922743i	$-0.625942\pi$
$881$	−22.8795			−0.770830			−0.385415	+	0.922743i	$0.625942\pi$
$882$		0			0
$883$		−	24.8762i		−	0.837152i	−0.908182	−	0.418576i	$-0.862529\pi$
$883$		−	24.8762i		−	0.837152i	0.908182	−	0.418576i	$-0.137471\pi$
$884$		0			0
$885$	−1.55459	−	3.95749i	−0.0522569	−	0.133029i
$886$		0			0
$887$			49.3187i			1.65596i	0.560759	+	0.827979i	$0.310509\pi$
$887$			49.3187i			1.65596i	−0.560759	+	0.827979i	$0.689491\pi$
$888$		0			0
$889$	13.4358			0.450622
$890$		0			0
$891$	5.18352			0.173655
$892$		0			0
$893$			11.3206i			0.378829i
$894$		0			0
$895$	28.3453	−	11.1347i	0.947478	−	0.372190i
$896$		0			0
$897$		−	37.5681i		−	1.25436i
$898$		0			0
$899$	−39.2719			−1.30979
$900$		0			0
$901$	91.7453			3.05648
$902$		0			0
$903$			2.26769i			0.0754640i
$904$		0			0
$905$	−11.2863	+	4.43353i	−0.375171	+	0.147375i
$906$		0			0
$907$			15.1653i			0.503556i	0.967785	+	0.251778i	$0.0810153\pi$
$907$			15.1653i			0.503556i	−0.967785	+	0.251778i	$0.918985\pi$
$908$		0			0
$909$	−9.37470			−0.310939
$910$		0			0
$911$	−27.8443			−0.922524			−0.461262	−	0.887264i	$-0.652603\pi$
$911$	−27.8443			−0.922524			−0.461262	+	0.887264i	$0.652603\pi$
$912$		0			0
$913$			67.3528i			2.22905i
$914$		0			0
$915$	−0.934439	−	2.37878i	−0.0308916	−	0.0786401i
$916$		0			0
$917$		−	61.2346i		−	2.02215i
$918$		0			0
$919$	−50.9981			−1.68227			−0.841135	−	0.540825i	$-0.818112\pi$
$919$	−50.9981			−1.68227			−0.841135	+	0.540825i	$0.818112\pi$
$920$		0			0
$921$	−2.87657			−0.0947862
$922$		0			0
$923$		−	7.31660i		−	0.240829i
$924$		0			0
$925$	34.9255	+	37.5948i	1.14834	+	1.23611i
$926$		0			0
$927$		−	7.81662i		−	0.256731i
$928$		0			0
$929$	−42.0324			−1.37904			−0.689519	−	0.724268i	$-0.742178\pi$
$929$	−42.0324			−1.37904			−0.689519	+	0.724268i	$0.742178\pi$
$930$		0			0
$931$	−7.49004			−0.245476
$932$		0			0
$933$		−	6.15003i		−	0.201343i
$934$		0			0
$935$	33.6177	+	85.5798i	1.09942	+	2.79876i
$936$		0			0
$937$			1.16094i			0.0379264i	0.999820	+	0.0189632i	$0.00603653\pi$
$937$			1.16094i			0.0379264i	−0.999820	+	0.0189632i	$0.993963\pi$
$938$		0			0
$939$	4.46096			0.145578
$940$		0			0
$941$	−48.5340			−1.58216			−0.791082	−	0.611710i	$-0.790482\pi$
$941$	−48.5340			−1.58216			−0.791082	+	0.611710i	$0.790482\pi$
$942$		0			0
$943$		−	57.8638i		−	1.88431i
$944$		0			0
$945$	−7.11473	+	2.79482i	−0.231442	+	0.0909156i
$946$		0			0
$947$		−	50.3681i		−	1.63674i	−0.574689	−	0.818372i	$-0.694877\pi$
$947$		−	50.3681i		−	1.63674i	0.574689	−	0.818372i	$-0.305123\pi$
$948$		0			0
$949$	−32.6692			−1.06049
$950$		0			0
$951$	−14.7678			−0.478880
$952$		0			0
$953$		−	55.4655i		−	1.79670i	−0.439277	−	0.898352i	$-0.644765\pi$
$953$		−	55.4655i		−	1.79670i	0.439277	−	0.898352i	$-0.355235\pi$
$954$		0			0
$955$	7.72266	−	3.03363i	0.249899	−	0.0981660i
$956$		0			0
$957$		−	34.1428i		−	1.10368i
$958$		0			0
$959$	−18.6116			−0.601001
$960$		0			0
$961$	4.54801			0.146710
$962$		0			0
$963$			9.25657i			0.298289i
$964$		0			0
$965$	3.81777	+	9.71881i	0.122898	+	0.312860i
$966$		0			0
$967$			53.8961i			1.73318i	0.499019	+	0.866591i	$0.333694\pi$
$967$			53.8961i			1.73318i	−0.499019	+	0.866591i	$0.666306\pi$
$968$		0			0
$969$	12.6793			0.407319
$970$		0			0
$971$	10.3151			0.331028			0.165514	−	0.986207i	$-0.447072\pi$
$971$	10.3151			0.331028			0.165514	+	0.986207i	$0.447072\pi$
$972$		0			0
$973$			42.3505i			1.35769i
$974$		0			0
$975$	−19.2130	+	17.8488i	−0.615307	+	0.571619i
$976$		0			0
$977$		−	35.3713i		−	1.13163i	−0.824533	−	0.565814i	$-0.808562\pi$
$977$		−	35.3713i		−	1.13163i	0.824533	−	0.565814i	$-0.191438\pi$
$978$		0			0
$979$	−32.6530			−1.04359
$980$		0			0
$981$	16.7849			0.535899
$982$		0			0
$983$			31.7285i			1.01198i	0.862539	+	0.505991i	$0.168873\pi$
$983$			31.7285i			1.01198i	−0.862539	+	0.505991i	$0.831127\pi$
$984$		0			0
$985$	17.8071	+	45.3313i	0.567383	+	1.44438i
$986$		0			0
$987$			24.2120i			0.770675i
$988$		0			0
$989$	−4.75153			−0.151090
$990$		0			0
$991$	14.7206			0.467617			0.233808	−	0.972283i	$-0.424881\pi$
$991$	14.7206			0.467617			0.233808	+	0.972283i	$0.424881\pi$
$992$		0			0
$993$		−	13.0188i		−	0.413139i
$994$		0			0
$995$	54.2942	−	21.3280i	1.72124	−	0.676142i
$996$		0			0
$997$			17.8245i			0.564506i	0.959340	+	0.282253i	$0.0910818\pi$
$997$			17.8245i			0.564506i	−0.959340	+	0.282253i	$0.908918\pi$
$998$		0			0
$999$	−10.2629			−0.324703

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4020.2.g.c.1609.17	✓	38
5.4	even	2	inner	4020.2.g.c.1609.36	yes	38

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4020.2.g.c.1609.17	✓	38	1.1	even	1	trivial
4020.2.g.c.1609.36	yes	38	5.4	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +(2.08125 - 0.817560i) q^{5} +3.41849i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +(2.08125 - 0.817560i) q^{5} +3.41849i q^{7} -1.00000 q^{9} +5.18352 q^{11} -5.24487i q^{13} +(-0.817560 - 2.08125i) q^{15} +7.93273i q^{17} +1.59836 q^{19} +3.41849 q^{21} +7.16283i q^{23} +(3.66319 - 3.40309i) q^{25} +1.00000i q^{27} +6.58679 q^{29} -5.96221 q^{31} -5.18352i q^{33} +(2.79482 + 7.11473i) q^{35} +10.2629i q^{37} -5.24487 q^{39} -8.07835 q^{41} +0.663360i q^{43} +(-2.08125 + 0.817560i) q^{45} +7.08264i q^{47} -4.68609 q^{49} +7.93273 q^{51} -11.5654i q^{53} +(10.7882 - 4.23784i) q^{55} -1.59836i q^{57} +1.90150 q^{59} +1.14296 q^{61} -3.41849i q^{63} +(-4.28800 - 10.9159i) q^{65} -1.00000i q^{67} +7.16283 q^{69} +1.39500 q^{71} -6.22878i q^{73} +(-3.40309 - 3.66319i) q^{75} +17.7198i q^{77} -1.75892 q^{79} +1.00000 q^{81} +12.9936i q^{83} +(6.48548 + 16.5100i) q^{85} -6.58679i q^{87} -6.29938 q^{89} +17.9296 q^{91} +5.96221i q^{93} +(3.32658 - 1.30675i) q^{95} +4.62505i q^{97} -5.18352 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 38 q - 2 q^{5} - 38 q^{9}+O(q^{10}) \) 38 * q - 2 * q^5 - 38 * q^9	\( 38 q - 2 q^{5} - 38 q^{9} + 24 q^{11} + 2 q^{15} - 16 q^{19} + 12 q^{21} + 4 q^{25} - 56 q^{29} - 4 q^{31} - 2 q^{35} + 60 q^{41} + 2 q^{45} - 70 q^{49} + 12 q^{55} - 52 q^{59} + 48 q^{61} + 16 q^{65} - 12 q^{69} + 12 q^{75} - 24 q^{79} + 38 q^{81} + 16 q^{85} - 76 q^{89} + 44 q^{91} + 36 q^{95} - 24 q^{99}+O(q^{100}) \) 38 * q - 2 * q^5 - 38 * q^9 + 24 * q^11 + 2 * q^15 - 16 * q^19 + 12 * q^21 + 4 * q^25 - 56 * q^29 - 4 * q^31 - 2 * q^35 + 60 * q^41 + 2 * q^45 - 70 * q^49 + 12 * q^55 - 52 * q^59 + 48 * q^61 + 16 * q^65 - 12 * q^69 + 12 * q^75 - 24 * q^79 + 38 * q^81 + 16 * q^85 - 76 * q^89 + 44 * q^91 + 36 * q^95 - 24 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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