LMFDB - Embedded newform 4020.2.g.b.1609.10

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			1609.10
Character	$\chi$	$=$	4020.1609
Dual form			4020.2.g.b.1609.22

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +(1.45456 - 1.69831i) q^{5} -0.243187i q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +(1.45456 - 1.69831i) q^{5} -0.243187i q^{7} -1.00000 q^{9} +0.348249 q^{11} +2.62829i q^{13} +(-1.69831 - 1.45456i) q^{15} +1.29103i q^{17} +1.36602 q^{19} -0.243187 q^{21} -0.600572i q^{23} +(-0.768532 - 4.94058i) q^{25} +1.00000i q^{27} -3.99937 q^{29} -3.23948 q^{31} -0.348249i q^{33} +(-0.413007 - 0.353729i) q^{35} -7.96539i q^{37} +2.62829 q^{39} +2.27937 q^{41} -8.24187i q^{43} +(-1.45456 + 1.69831i) q^{45} -9.05568i q^{47} +6.94086 q^{49} +1.29103 q^{51} -10.5701i q^{53} +(0.506548 - 0.591435i) q^{55} -1.36602i q^{57} +6.00319 q^{59} -3.78086 q^{61} +0.243187i q^{63} +(4.46365 + 3.82299i) q^{65} +1.00000i q^{67} -0.600572 q^{69} -4.45872 q^{71} -8.87821i q^{73} +(-4.94058 + 0.768532i) q^{75} -0.0846895i q^{77} -16.2980 q^{79} +1.00000 q^{81} +10.3876i q^{83} +(2.19257 + 1.87788i) q^{85} +3.99937i q^{87} +8.64400 q^{89} +0.639164 q^{91} +3.23948i q^{93} +(1.98695 - 2.31992i) q^{95} +0.828410i q^{97} -0.348249 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	1.45456	−	1.69831i	0.650497	−	0.759509i
$5$	1.45456	−	1.69831i	0.650497	−	0.759509i
$6$		0			0
$7$		−	0.243187i		−	0.0919159i	−0.998943	−	0.0459580i	$-0.985366\pi$
$7$		−	0.243187i		−	0.0919159i	0.998943	−	0.0459580i	$-0.0146340\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	0.348249			0.105001			0.0525005	−	0.998621i	$-0.483281\pi$
$11$	0.348249			0.105001			0.0525005	+	0.998621i	$0.483281\pi$
$12$		0			0
$13$			2.62829i			0.728955i	0.931212	+	0.364478i	$0.118752\pi$
$13$			2.62829i			0.728955i	−0.931212	+	0.364478i	$0.881248\pi$
$14$		0			0
$15$	−1.69831	−	1.45456i	−0.438502	−	0.375565i
$16$		0			0
$17$			1.29103i			0.313121i	0.987668	+	0.156561i	$0.0500406\pi$
$17$			1.29103i			0.313121i	−0.987668	+	0.156561i	$0.949959\pi$
$18$		0			0
$19$	1.36602			0.313386			0.156693	−	0.987647i	$-0.449917\pi$
$19$	1.36602			0.313386			0.156693	+	0.987647i	$0.449917\pi$
$20$		0			0
$21$	−0.243187			−0.0530677
$22$		0			0
$23$		−	0.600572i		−	0.125228i	−0.998038	−	0.0626140i	$-0.980056\pi$
$23$		−	0.600572i		−	0.125228i	0.998038	−	0.0626140i	$-0.0199437\pi$
$24$		0			0
$25$	−0.768532	−	4.94058i	−0.153706	−	0.988117i
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$	−3.99937			−0.742664			−0.371332	−	0.928500i	$-0.621099\pi$
$29$	−3.99937			−0.742664			−0.371332	+	0.928500i	$0.621099\pi$
$30$		0			0
$31$	−3.23948			−0.581827			−0.290914	−	0.956749i	$-0.593959\pi$
$31$	−3.23948			−0.581827			−0.290914	+	0.956749i	$0.593959\pi$
$32$		0			0
$33$		−	0.348249i		−	0.0606223i
$34$		0			0
$35$	−0.413007	−	0.353729i	−0.0698109	−	0.0597910i
$36$		0			0
$37$		−	7.96539i		−	1.30950i	−0.755845	−	0.654751i	$-0.772773\pi$
$37$		−	7.96539i		−	1.30950i	0.755845	−	0.654751i	$-0.227227\pi$
$38$		0			0
$39$	2.62829			0.420863
$40$		0			0
$41$	2.27937			0.355978			0.177989	−	0.984032i	$-0.443041\pi$
$41$	2.27937			0.355978			0.177989	+	0.984032i	$0.443041\pi$
$42$		0			0
$43$		−	8.24187i		−	1.25687i	−0.777860	−	0.628437i	$-0.783695\pi$
$43$		−	8.24187i		−	1.25687i	0.777860	−	0.628437i	$-0.216305\pi$
$44$		0			0
$45$	−1.45456	+	1.69831i	−0.216832	+	0.253170i
$46$		0			0
$47$		−	9.05568i		−	1.32091i	−0.750867	−	0.660454i	$-0.770364\pi$
$47$		−	9.05568i		−	1.32091i	0.750867	−	0.660454i	$-0.229636\pi$
$48$		0			0
$49$	6.94086			0.991551
$50$		0			0
$51$	1.29103			0.180781
$52$		0			0
$53$		−	10.5701i		−	1.45192i	−0.687738	−	0.725959i	$-0.741396\pi$
$53$		−	10.5701i		−	1.45192i	0.687738	−	0.725959i	$-0.258604\pi$
$54$		0			0
$55$	0.506548	−	0.591435i	0.0683029	−	0.0797491i
$56$		0			0
$57$		−	1.36602i		−	0.180933i
$58$		0			0
$59$	6.00319			0.781549			0.390775	−	0.920486i	$-0.372207\pi$
$59$	6.00319			0.781549			0.390775	+	0.920486i	$0.372207\pi$
$60$		0			0
$61$	−3.78086			−0.484090			−0.242045	−	0.970265i	$-0.577818\pi$
$61$	−3.78086			−0.484090			−0.242045	+	0.970265i	$0.577818\pi$
$62$		0			0
$63$			0.243187i			0.0306386i
$64$		0			0
$65$	4.46365	+	3.82299i	0.553648	+	0.474184i
$66$		0			0
$67$			1.00000i			0.122169i
$67$			1.00000i			0.122169i
$68$		0			0
$69$	−0.600572			−0.0723004
$70$		0			0
$71$	−4.45872			−0.529152			−0.264576	−	0.964365i	$-0.585232\pi$
$71$	−4.45872			−0.529152			−0.264576	+	0.964365i	$0.585232\pi$
$72$		0			0
$73$		−	8.87821i		−	1.03912i	−0.854435	−	0.519558i	$-0.826097\pi$
$73$		−	8.87821i		−	1.03912i	0.854435	−	0.519558i	$-0.173903\pi$
$74$		0			0
$75$	−4.94058	+	0.768532i	−0.570489	+	0.0887425i
$76$		0			0
$77$		−	0.0846895i		−	0.00965126i
$78$		0			0
$79$	−16.2980			−1.83367			−0.916837	−	0.399263i	$-0.869266\pi$
$79$	−16.2980			−1.83367			−0.916837	+	0.399263i	$0.869266\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			10.3876i			1.14019i	0.821580	+	0.570093i	$0.193093\pi$
$83$			10.3876i			1.14019i	−0.821580	+	0.570093i	$0.806907\pi$
$84$		0			0
$85$	2.19257	+	1.87788i	0.237818	+	0.203684i
$86$		0			0
$87$			3.99937i			0.428777i
$88$		0			0
$89$	8.64400			0.916262			0.458131	−	0.888885i	$-0.348519\pi$
$89$	8.64400			0.916262			0.458131	+	0.888885i	$0.348519\pi$
$90$		0			0
$91$	0.639164			0.0670026
$92$		0			0
$93$			3.23948i			0.335918i
$94$		0			0
$95$	1.98695	−	2.31992i	0.203857	−	0.238019i
$96$		0			0
$97$			0.828410i			0.0841123i	0.999115	+	0.0420562i	$0.0133908\pi$
$97$			0.828410i			0.0841123i	−0.999115	+	0.0420562i	$0.986609\pi$
$98$		0			0
$99$	−0.348249			−0.0350003
$100$		0			0
$101$	−3.67000			−0.365178			−0.182589	−	0.983189i	$-0.558448\pi$
$101$	−3.67000			−0.365178			−0.182589	+	0.983189i	$0.558448\pi$
$102$		0			0
$103$		−	18.2295i		−	1.79620i	−0.439788	−	0.898102i	$-0.644946\pi$
$103$		−	18.2295i		−	1.79620i	0.439788	−	0.898102i	$-0.355054\pi$
$104$		0			0
$105$	−0.353729	+	0.413007i	−0.0345204	+	0.0403053i
$106$		0			0
$107$		−	19.0797i		−	1.84450i	−0.386590	−	0.922252i	$-0.626347\pi$
$107$		−	19.0797i		−	1.84450i	0.386590	−	0.922252i	$-0.373653\pi$
$108$		0			0
$109$	7.89028			0.755752			0.377876	−	0.925856i	$-0.376655\pi$
$109$	7.89028			0.755752			0.377876	+	0.925856i	$0.376655\pi$
$110$		0			0
$111$	−7.96539			−0.756041
$112$		0			0
$113$			5.54739i			0.521854i	0.965359	+	0.260927i	$0.0840282\pi$
$113$			5.54739i			0.521854i	−0.965359	+	0.260927i	$0.915972\pi$
$114$		0			0
$115$	−1.01996	−	0.873566i	−0.0951117	−	0.0814604i
$116$		0			0
$117$		−	2.62829i		−	0.242985i
$118$		0			0
$119$	0.313962			0.0287808
$120$		0			0
$121$	−10.8787			−0.988975
$122$		0			0
$123$		−	2.27937i		−	0.205524i
$124$		0			0
$125$	−9.50853	−	5.88115i	−0.850469	−	0.526026i
$126$		0			0
$127$		−	11.3276i		−	1.00516i	−0.864530	−	0.502581i	$-0.832384\pi$
$127$		−	11.3276i		−	1.00516i	0.864530	−	0.502581i	$-0.167616\pi$
$128$		0			0
$129$	−8.24187			−0.725657
$130$		0			0
$131$	−8.32162			−0.727064			−0.363532	−	0.931582i	$-0.618429\pi$
$131$	−8.32162			−0.727064			−0.363532	+	0.931582i	$0.618429\pi$
$132$		0			0
$133$		−	0.332197i		−	0.0288051i
$134$		0			0
$135$	1.69831	+	1.45456i	0.146167	+	0.125188i
$136$		0			0
$137$			13.4140i			1.14603i	0.819544	+	0.573016i	$0.194227\pi$
$137$			13.4140i			1.14603i	−0.819544	+	0.573016i	$0.805773\pi$
$138$		0			0
$139$	−13.2582			−1.12454			−0.562272	−	0.826952i	$-0.690073\pi$
$139$	−13.2582			−1.12454			−0.562272	+	0.826952i	$0.690073\pi$
$140$		0			0
$141$	−9.05568			−0.762626
$142$		0			0
$143$			0.915298i			0.0765410i
$144$		0			0
$145$	−5.81731	+	6.79218i	−0.483101	+	0.564060i
$146$		0			0
$147$		−	6.94086i		−	0.572473i
$148$		0			0
$149$	20.1304			1.64914			0.824572	−	0.565757i	$-0.191416\pi$
$149$	20.1304			1.64914			0.824572	+	0.565757i	$0.191416\pi$
$150$		0			0
$151$	23.0072			1.87230			0.936150	−	0.351602i	$-0.114363\pi$
$151$	23.0072			1.87230			0.936150	+	0.351602i	$0.114363\pi$
$152$		0			0
$153$		−	1.29103i		−	0.104374i
$154$		0			0
$155$	−4.71200	+	5.50164i	−0.378477	+	0.441903i
$156$		0			0
$157$			2.04826i			0.163469i	0.996654	+	0.0817345i	$0.0260460\pi$
$157$			2.04826i			0.163469i	−0.996654	+	0.0817345i	$0.973954\pi$
$158$		0			0
$159$	−10.5701			−0.838266
$160$		0			0
$161$	−0.146051			−0.0115104
$162$		0			0
$163$			10.8378i			0.848880i	0.905456	+	0.424440i	$0.139529\pi$
$163$			10.8378i			0.848880i	−0.905456	+	0.424440i	$0.860471\pi$
$164$		0			0
$165$	−0.591435	−	0.506548i	−0.0460432	−	0.0394347i
$166$		0			0
$167$			15.7540i			1.21908i	0.792755	+	0.609541i	$0.208646\pi$
$167$			15.7540i			1.21908i	−0.792755	+	0.609541i	$0.791354\pi$
$168$		0			0
$169$	6.09211			0.468624
$170$		0			0
$171$	−1.36602			−0.104462
$172$		0			0
$173$			9.79373i			0.744604i	0.928112	+	0.372302i	$0.121431\pi$
$173$			9.79373i			0.744604i	−0.928112	+	0.372302i	$0.878569\pi$
$174$		0			0
$175$	−1.20148	+	0.186897i	−0.0908236	+	0.0141281i
$176$		0			0
$177$		−	6.00319i		−	0.451228i
$178$		0			0
$179$	−5.38131			−0.402218			−0.201109	−	0.979569i	$-0.564455\pi$
$179$	−5.38131			−0.402218			−0.201109	+	0.979569i	$0.564455\pi$
$180$		0			0
$181$	−3.76426			−0.279796			−0.139898	−	0.990166i	$-0.544677\pi$
$181$	−3.76426			−0.279796			−0.139898	+	0.990166i	$0.544677\pi$
$182$		0			0
$183$			3.78086i			0.279490i
$184$		0			0
$185$	−13.5277	−	11.5861i	−0.994578	−	0.851828i
$186$		0			0
$187$			0.449600i			0.0328780i
$188$		0			0
$189$	0.243187			0.0176892
$190$		0			0
$191$	−22.6591			−1.63956			−0.819778	−	0.572682i	$-0.805903\pi$
$191$	−22.6591			−1.63956			−0.819778	+	0.572682i	$0.805903\pi$
$192$		0			0
$193$		−	2.56499i		−	0.184632i	−0.995730	−	0.0923160i	$-0.970573\pi$
$193$		−	2.56499i		−	0.184632i	0.995730	−	0.0923160i	$-0.0294270\pi$
$194$		0			0
$195$	3.82299	−	4.46365i	0.273770	−	0.319649i
$196$		0			0
$197$			0.115117i			0.00820173i	0.999992	+	0.00410087i	$0.00130535\pi$
$197$			0.115117i			0.00820173i	−0.999992	+	0.00410087i	$0.998695\pi$
$198$		0			0
$199$	7.63561			0.541274			0.270637	−	0.962681i	$-0.412766\pi$
$199$	7.63561			0.541274			0.270637	+	0.962681i	$0.412766\pi$
$200$		0			0
$201$	1.00000			0.0705346
$202$		0			0
$203$			0.972593i			0.0682626i
$204$		0			0
$205$	3.31547	−	3.87109i	0.231563	−	0.270368i
$206$		0			0
$207$			0.600572i			0.0417426i
$208$		0			0
$209$	0.475714			0.0329058
$210$		0			0
$211$	0.669505			0.0460906			0.0230453	−	0.999734i	$-0.492664\pi$
$211$	0.669505			0.0460906			0.0230453	+	0.999734i	$0.492664\pi$
$212$		0			0
$213$			4.45872i			0.305506i
$214$		0			0
$215$	−13.9973	−	11.9883i	−0.954607	−	0.817593i
$216$		0			0
$217$			0.787797i			0.0534792i
$218$		0			0
$219$	−8.87821			−0.599934
$220$		0			0
$221$	−3.39320			−0.228251
$222$		0			0
$223$			26.2148i			1.75547i	0.479144	+	0.877736i	$0.340947\pi$
$223$			26.2148i			1.75547i	−0.479144	+	0.877736i	$0.659053\pi$
$224$		0			0
$225$	0.768532	+	4.94058i	0.0512355	+	0.329372i
$226$		0			0
$227$		−	14.2115i		−	0.943248i	−0.881800	−	0.471624i	$-0.843668\pi$
$227$		−	14.2115i		−	0.943248i	0.881800	−	0.471624i	$-0.156332\pi$
$228$		0			0
$229$	6.41350			0.423816			0.211908	−	0.977290i	$-0.432032\pi$
$229$	6.41350			0.423816			0.211908	+	0.977290i	$0.432032\pi$
$230$		0			0
$231$	−0.0846895			−0.00557216
$232$		0			0
$233$			11.2402i			0.736372i	0.929752	+	0.368186i	$0.120021\pi$
$233$			11.2402i			0.736372i	−0.929752	+	0.368186i	$0.879979\pi$
$234$		0			0
$235$	−15.3794	−	13.1720i	−1.00324	−	0.859247i
$236$		0			0
$237$			16.2980i			1.05867i
$238$		0			0
$239$	10.7654			0.696354			0.348177	−	0.937429i	$-0.386801\pi$
$239$	10.7654			0.696354			0.348177	+	0.937429i	$0.386801\pi$
$240$		0			0
$241$	−0.229596			−0.0147896			−0.00739478	−	0.999973i	$-0.502354\pi$
$241$	−0.229596			−0.0147896			−0.00739478	+	0.999973i	$0.502354\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$	10.0959	−	11.7878i	0.645002	−	0.753092i
$246$		0			0
$247$			3.59028i			0.228444i
$248$		0			0
$249$	10.3876			0.658287
$250$		0			0
$251$	−5.71115			−0.360484			−0.180242	−	0.983622i	$-0.557688\pi$
$251$	−5.71115			−0.360484			−0.180242	+	0.983622i	$0.557688\pi$
$252$		0			0
$253$		−	0.209149i		−	0.0131491i
$254$		0			0
$255$	1.87788	−	2.19257i	0.117597	−	0.137304i
$256$		0			0
$257$			12.8726i			0.802968i	0.915866	+	0.401484i	$0.131506\pi$
$257$			12.8726i			0.802968i	−0.915866	+	0.401484i	$0.868494\pi$
$258$		0			0
$259$	−1.93708			−0.120364
$260$		0			0
$261$	3.99937			0.247555
$262$		0			0
$263$		−	21.7586i		−	1.34169i	−0.741598	−	0.670845i	$-0.765932\pi$
$263$		−	21.7586i		−	1.34169i	0.741598	−	0.670845i	$-0.234068\pi$
$264$		0			0
$265$	−17.9514	−	15.3748i	−1.10274	−	0.944469i
$266$		0			0
$267$		−	8.64400i		−	0.529004i
$268$		0			0
$269$	−30.9686			−1.88819			−0.944096	−	0.329672i	$-0.893062\pi$
$269$	−30.9686			−1.88819			−0.944096	+	0.329672i	$0.893062\pi$
$270$		0			0
$271$	18.6020			1.12999			0.564996	−	0.825094i	$-0.308878\pi$
$271$	18.6020			1.12999			0.564996	+	0.825094i	$0.308878\pi$
$272$		0			0
$273$		−	0.639164i		−	0.0386840i
$274$		0			0
$275$	−0.267641	−	1.72055i	−0.0161393	−	0.103753i
$276$		0			0
$277$			4.97556i			0.298952i	0.988765	+	0.149476i	$0.0477587\pi$
$277$			4.97556i			0.298952i	−0.988765	+	0.149476i	$0.952241\pi$
$278$		0			0
$279$	3.23948			0.193942
$280$		0			0
$281$	10.6486			0.635242			0.317621	−	0.948218i	$-0.397116\pi$
$281$	10.6486			0.635242			0.317621	+	0.948218i	$0.397116\pi$
$282$		0			0
$283$		−	20.1137i		−	1.19563i	−0.801633	−	0.597817i	$-0.796035\pi$
$283$		−	20.1137i		−	1.19563i	0.801633	−	0.597817i	$-0.203965\pi$
$284$		0			0
$285$	−2.31992	−	1.98695i	−0.137420	−	0.117697i
$286$		0			0
$287$		−	0.554313i		−	0.0327200i
$288$		0			0
$289$	15.3332			0.901955
$290$		0			0
$291$	0.828410			0.0485623
$292$		0			0
$293$		−	5.53165i		−	0.323163i	−0.986859	−	0.161581i	$-0.948341\pi$
$293$		−	5.53165i		−	0.323163i	0.986859	−	0.161581i	$-0.0516594\pi$
$294$		0			0
$295$	8.73198	−	10.1953i	0.508396	−	0.593593i
$296$		0			0
$297$			0.348249i			0.0202074i
$298$		0			0
$299$	1.57847			0.0912856
$300$		0			0
$301$	−2.00431			−0.115527
$302$		0			0
$303$			3.67000i			0.210836i
$304$		0			0
$305$	−5.49948	+	6.42109i	−0.314899	+	0.367671i
$306$		0			0
$307$		−	30.4742i		−	1.73926i	−0.493708	−	0.869628i	$-0.664359\pi$
$307$		−	30.4742i		−	1.73926i	0.493708	−	0.869628i	$-0.335641\pi$
$308$		0			0
$309$	−18.2295			−1.03704
$310$		0			0
$311$	−24.1197			−1.36770			−0.683851	−	0.729622i	$-0.739696\pi$
$311$	−24.1197			−1.36770			−0.683851	+	0.729622i	$0.739696\pi$
$312$		0			0
$313$			3.01994i			0.170697i	0.996351	+	0.0853485i	$0.0272004\pi$
$313$			3.01994i			0.170697i	−0.996351	+	0.0853485i	$0.972800\pi$
$314$		0			0
$315$	0.413007	+	0.353729i	0.0232703	+	0.0199303i
$316$		0			0
$317$			34.4860i			1.93693i	0.249158	+	0.968463i	$0.419846\pi$
$317$			34.4860i			1.93693i	−0.249158	+	0.968463i	$0.580154\pi$
$318$		0			0
$319$	−1.39278			−0.0779804
$320$		0			0
$321$	−19.0797			−1.06492
$322$		0			0
$323$			1.76357i			0.0981277i
$324$		0			0
$325$	12.9853	−	2.01992i	0.720293	−	0.112045i
$326$		0			0
$327$		−	7.89028i		−	0.436334i
$328$		0			0
$329$	−2.20222			−0.121412
$330$		0			0
$331$	−20.2865			−1.11505			−0.557523	−	0.830162i	$-0.688248\pi$
$331$	−20.2865			−1.11505			−0.557523	+	0.830162i	$0.688248\pi$
$332$		0			0
$333$			7.96539i			0.436501i
$334$		0			0
$335$	1.69831	+	1.45456i	0.0927887	+	0.0794709i
$336$		0			0
$337$		−	17.7605i		−	0.967474i	−0.875213	−	0.483737i	$-0.839279\pi$
$337$		−	17.7605i		−	0.967474i	0.875213	−	0.483737i	$-0.160721\pi$
$338$		0			0
$339$	5.54739			0.301293
$340$		0			0
$341$	−1.12814			−0.0610924
$342$		0			0
$343$		−	3.39023i		−	0.183055i
$344$		0			0
$345$	−0.873566	+	1.01996i	−0.0470312	+	0.0549128i
$346$		0			0
$347$		−	5.05750i		−	0.271501i	−0.990743	−	0.135750i	$-0.956655\pi$
$347$		−	5.05750i		−	0.271501i	0.990743	−	0.135750i	$-0.0433446\pi$
$348$		0			0
$349$	15.9805			0.855415			0.427707	−	0.903917i	$-0.359321\pi$
$349$	15.9805			0.855415			0.427707	+	0.903917i	$0.359321\pi$
$350$		0			0
$351$	−2.62829			−0.140288
$352$		0			0
$353$			20.2131i			1.07583i	0.842998	+	0.537916i	$0.180788\pi$
$353$			20.2131i			1.07583i	−0.842998	+	0.537916i	$0.819212\pi$
$354$		0			0
$355$	−6.48545	+	7.57230i	−0.344212	+	0.401896i
$356$		0			0
$357$		−	0.313962i		−	0.0166166i
$358$		0			0
$359$	1.11759			0.0589840			0.0294920	−	0.999565i	$-0.490611\pi$
$359$	1.11759			0.0589840			0.0294920	+	0.999565i	$0.490611\pi$
$360$		0			0
$361$	−17.1340			−0.901789
$362$		0			0
$363$			10.8787i			0.570985i
$364$		0			0
$365$	−15.0780	−	12.9139i	−0.789217	−	0.675942i
$366$		0			0
$367$		−	16.5732i		−	0.865113i	−0.901607	−	0.432556i	$-0.857612\pi$
$367$		−	16.5732i		−	0.865113i	0.901607	−	0.432556i	$-0.142388\pi$
$368$		0			0
$369$	−2.27937			−0.118659
$370$		0			0
$371$	−2.57051			−0.133454
$372$		0			0
$373$		−	11.1433i		−	0.576977i	−0.957483	−	0.288489i	$-0.906847\pi$
$373$		−	11.1433i		−	0.576977i	0.957483	−	0.288489i	$-0.0931527\pi$
$374$		0			0
$375$	−5.88115	+	9.50853i	−0.303701	+	0.491018i
$376$		0			0
$377$		−	10.5115i		−	0.541369i
$378$		0			0
$379$	33.3738			1.71430			0.857148	−	0.515070i	$-0.172234\pi$
$379$	33.3738			1.71430			0.857148	+	0.515070i	$0.172234\pi$
$380$		0			0
$381$	−11.3276			−0.580330
$382$		0			0
$383$		−	14.6887i		−	0.750557i	−0.926912	−	0.375279i	$-0.877547\pi$
$383$		−	14.6887i		−	0.750557i	0.926912	−	0.375279i	$-0.122453\pi$
$384$		0			0
$385$	−0.143829	−	0.123186i	−0.00733021	−	0.00627812i
$386$		0			0
$387$			8.24187i			0.418958i
$388$		0			0
$389$	35.7047			1.81030			0.905149	−	0.425094i	$-0.139759\pi$
$389$	35.7047			1.81030			0.905149	+	0.425094i	$0.139759\pi$
$390$		0			0
$391$	0.775357			0.0392115
$392$		0			0
$393$			8.32162i			0.419770i
$394$		0			0
$395$	−23.7064	+	27.6792i	−1.19280	+	1.39269i
$396$		0			0
$397$		−	19.8878i		−	0.998139i	−0.866562	−	0.499070i	$-0.833675\pi$
$397$		−	19.8878i		−	0.998139i	0.866562	−	0.499070i	$-0.166325\pi$
$398$		0			0
$399$	−0.332197			−0.0166307
$400$		0			0
$401$	32.0058			1.59829			0.799147	−	0.601136i	$-0.205285\pi$
$401$	32.0058			1.59829			0.799147	+	0.601136i	$0.205285\pi$
$402$		0			0
$403$		−	8.51427i		−	0.424126i
$404$		0			0
$405$	1.45456	−	1.69831i	0.0722775	−	0.0843898i
$406$		0			0
$407$		−	2.77394i		−	0.137499i
$408$		0			0
$409$	17.5485			0.867715			0.433858	−	0.900981i	$-0.357152\pi$
$409$	17.5485			0.867715			0.433858	+	0.900981i	$0.357152\pi$
$410$		0			0
$411$	13.4140			0.661662
$412$		0			0
$413$		−	1.45990i		−	0.0718368i
$414$		0			0
$415$	17.6414	+	15.1093i	0.865981	+	0.741688i
$416$		0			0
$417$			13.2582i			0.649256i
$418$		0			0
$419$	−9.89826			−0.483562			−0.241781	−	0.970331i	$-0.577731\pi$
$419$	−9.89826			−0.483562			−0.241781	+	0.970331i	$0.577731\pi$
$420$		0			0
$421$	−2.12038			−0.103341			−0.0516705	−	0.998664i	$-0.516455\pi$
$421$	−2.12038			−0.103341			−0.0516705	+	0.998664i	$0.516455\pi$
$422$		0			0
$423$			9.05568i			0.440302i
$424$		0			0
$425$	6.37845	−	0.992199i	0.309400	−	0.0481287i
$426$		0			0
$427$			0.919456i			0.0444956i
$428$		0			0
$429$	0.915298			0.0441910
$430$		0			0
$431$	−18.5843			−0.895173			−0.447587	−	0.894241i	$-0.647716\pi$
$431$	−18.5843			−0.895173			−0.447587	+	0.894241i	$0.647716\pi$
$432$		0			0
$433$			26.8648i			1.29104i	0.763743	+	0.645520i	$0.223359\pi$
$433$			26.8648i			1.29104i	−0.763743	+	0.645520i	$0.776641\pi$
$434$		0			0
$435$	6.79218	+	5.81731i	0.325660	+	0.278918i
$436$		0			0
$437$		−	0.820392i		−	0.0392447i
$438$		0			0
$439$	−12.5212			−0.597606			−0.298803	−	0.954315i	$-0.596587\pi$
$439$	−12.5212			−0.597606			−0.298803	+	0.954315i	$0.596587\pi$
$440$		0			0
$441$	−6.94086			−0.330517
$442$		0			0
$443$		−	18.2936i		−	0.869155i	−0.900634	−	0.434578i	$-0.856898\pi$
$443$		−	18.2936i		−	0.869155i	0.900634	−	0.434578i	$-0.143102\pi$
$444$		0			0
$445$	12.5732	−	14.6802i	0.596026	−	0.695909i
$446$		0			0
$447$		−	20.1304i		−	0.952134i
$448$		0			0
$449$	−16.7717			−0.791505			−0.395752	−	0.918357i	$-0.629516\pi$
$449$	−16.7717			−0.791505			−0.395752	+	0.918357i	$0.629516\pi$
$450$		0			0
$451$	0.793789			0.0373780
$452$		0			0
$453$		−	23.0072i		−	1.08097i
$454$		0			0
$455$	0.929700	−	1.08550i	0.0435850	−	0.0508890i
$456$		0			0
$457$		−	34.3782i		−	1.60815i	−0.594531	−	0.804073i	$-0.702662\pi$
$457$		−	34.3782i		−	1.60815i	0.594531	−	0.804073i	$-0.297338\pi$
$458$		0			0
$459$	−1.29103			−0.0602602
$460$		0			0
$461$	−35.5010			−1.65344			−0.826722	−	0.562610i	$-0.809797\pi$
$461$	−35.5010			−1.65344			−0.826722	+	0.562610i	$0.809797\pi$
$462$		0			0
$463$			12.2881i			0.571079i	0.958367	+	0.285539i	$0.0921727\pi$
$463$			12.2881i			0.571079i	−0.958367	+	0.285539i	$0.907827\pi$
$464$		0			0
$465$	5.50164	+	4.71200i	0.255133	+	0.218514i
$466$		0			0
$467$			8.76116i			0.405418i	0.979239	+	0.202709i	$0.0649746\pi$
$467$			8.76116i			0.405418i	−0.979239	+	0.202709i	$0.935025\pi$
$468$		0			0
$469$	0.243187			0.0112293
$470$		0			0
$471$	2.04826			0.0943789
$472$		0			0
$473$		−	2.87022i		−	0.131973i
$474$		0			0
$475$	−1.04983	−	6.74892i	−0.0481694	−	0.309662i
$476$		0			0
$477$			10.5701i			0.483973i
$478$		0			0
$479$	−26.8588			−1.22721			−0.613604	−	0.789614i	$-0.710281\pi$
$479$	−26.8588			−1.22721			−0.613604	+	0.789614i	$0.710281\pi$
$480$		0			0
$481$	20.9353			0.954569
$482$		0			0
$483$			0.146051i			0.00664555i
$484$		0			0
$485$	1.40690	+	1.20497i	0.0638840	+	0.0547148i
$486$		0			0
$487$			24.2493i			1.09884i	0.835547	+	0.549419i	$0.185151\pi$
$487$			24.2493i			1.09884i	−0.835547	+	0.549419i	$0.814849\pi$
$488$		0			0
$489$	10.8378			0.490101
$490$		0			0
$491$	−3.62812			−0.163735			−0.0818675	−	0.996643i	$-0.526088\pi$
$491$	−3.62812			−0.163735			−0.0818675	+	0.996643i	$0.526088\pi$
$492$		0			0
$493$		−	5.16331i		−	0.232544i
$494$		0			0
$495$	−0.506548	+	0.591435i	−0.0227676	+	0.0265830i
$496$		0			0
$497$			1.08430i			0.0486375i
$498$		0			0
$499$	−6.73744			−0.301609			−0.150805	−	0.988564i	$-0.548186\pi$
$499$	−6.73744			−0.301609			−0.150805	+	0.988564i	$0.548186\pi$
$500$		0			0
$501$	15.7540			0.703837
$502$		0			0
$503$			9.52703i			0.424789i	0.977184	+	0.212395i	$0.0681262\pi$
$503$			9.52703i			0.424789i	−0.977184	+	0.212395i	$0.931874\pi$
$504$		0			0
$505$	−5.33822	+	6.23280i	−0.237548	+	0.277356i
$506$		0			0
$507$		−	6.09211i		−	0.270560i
$508$		0			0
$509$	14.5611			0.645408			0.322704	−	0.946500i	$-0.395408\pi$
$509$	14.5611			0.645408			0.322704	+	0.946500i	$0.395408\pi$
$510$		0			0
$511$	−2.15906			−0.0955113
$512$		0			0
$513$			1.36602i			0.0603111i
$514$		0			0
$515$	−30.9593	−	26.5158i	−1.36423	−	1.16843i
$516$		0			0
$517$		−	3.15363i		−	0.138697i
$518$		0			0
$519$	9.79373			0.429897
$520$		0			0
$521$	24.0602			1.05410			0.527048	−	0.849836i	$-0.323299\pi$
$521$	24.0602			1.05410			0.527048	+	0.849836i	$0.323299\pi$
$522$		0			0
$523$			22.8445i			0.998919i	0.866337	+	0.499460i	$0.166468\pi$
$523$			22.8445i			0.998919i	−0.866337	+	0.499460i	$0.833532\pi$
$524$		0			0
$525$	0.186897	+	1.20148i	0.00815684	+	0.0524370i
$526$		0			0
$527$		−	4.18227i		−	0.182182i
$528$		0			0
$529$	22.6393			0.984318
$530$		0			0
$531$	−6.00319			−0.260516
$532$		0			0
$533$			5.99084i			0.259492i
$534$		0			0
$535$	−32.4033	−	27.7525i	−1.40092	−	1.19984i
$536$		0			0
$537$			5.38131i			0.232221i
$538$		0			0
$539$	2.41715			0.104114
$540$		0			0
$541$	38.8419			1.66994			0.834971	−	0.550294i	$-0.185484\pi$
$541$	38.8419			1.66994			0.834971	+	0.550294i	$0.185484\pi$
$542$		0			0
$543$			3.76426i			0.161540i
$544$		0			0
$545$	11.4769	−	13.4002i	0.491615	−	0.574000i
$546$		0			0
$547$		−	25.4692i		−	1.08899i	−0.838765	−	0.544493i	$-0.816722\pi$
$547$		−	25.4692i		−	1.08899i	0.838765	−	0.544493i	$-0.183278\pi$
$548$		0			0
$549$	3.78086			0.161363
$550$		0			0
$551$	−5.46321			−0.232740
$552$		0			0
$553$			3.96347i			0.168544i
$554$		0			0
$555$	−11.5861	+	13.5277i	−0.491803	+	0.574220i
$556$		0			0
$557$		−	5.56886i		−	0.235960i	−0.993016	−	0.117980i	$-0.962358\pi$
$557$		−	5.56886i		−	0.235960i	0.993016	−	0.117980i	$-0.0376419\pi$
$558$		0			0
$559$	21.6620			0.916205
$560$		0			0
$561$	0.449600			0.0189821
$562$		0			0
$563$		−	3.52200i		−	0.148435i	−0.997242	−	0.0742173i	$-0.976354\pi$
$563$		−	3.52200i		−	0.148435i	0.997242	−	0.0742173i	$-0.0236458\pi$
$564$		0			0
$565$	9.42120	+	8.06898i	0.396353	+	0.339465i
$566$		0			0
$567$		−	0.243187i		−	0.0102129i
$568$		0			0
$569$	12.7892			0.536151			0.268076	−	0.963398i	$-0.413612\pi$
$569$	12.7892			0.536151			0.268076	+	0.963398i	$0.413612\pi$
$570$		0			0
$571$	32.0643			1.34185			0.670924	−	0.741526i	$-0.265898\pi$
$571$	32.0643			1.34185			0.670924	+	0.741526i	$0.265898\pi$
$572$		0			0
$573$			22.6591i			0.946598i
$574$		0			0
$575$	−2.96718	+	0.461559i	−0.123740	+	0.0192483i
$576$		0			0
$577$		−	7.73038i		−	0.321820i	−0.986969	−	0.160910i	$-0.948557\pi$
$577$		−	7.73038i		−	0.321820i	0.986969	−	0.160910i	$-0.0514428\pi$
$578$		0			0
$579$	−2.56499			−0.106597
$580$		0			0
$581$	2.52612			0.104801
$582$		0			0
$583$		−	3.68104i		−	0.152453i
$584$		0			0
$585$	−4.46365	−	3.82299i	−0.184549	−	0.158061i
$586$		0			0
$587$			32.7972i			1.35368i	0.736128	+	0.676842i	$0.236652\pi$
$587$			32.7972i			1.35368i	−0.736128	+	0.676842i	$0.763348\pi$
$588$		0			0
$589$	−4.42518			−0.182336
$590$		0			0
$591$	0.115117			0.00473527
$592$		0			0
$593$			17.1119i			0.702702i	0.936244	+	0.351351i	$0.114278\pi$
$593$			17.1119i			0.702702i	−0.936244	+	0.351351i	$0.885722\pi$
$594$		0			0
$595$	0.456675	−	0.533205i	0.0187218	−	0.0218593i
$596$		0			0
$597$		−	7.63561i		−	0.312505i
$598$		0			0
$599$	9.88400			0.403849			0.201925	−	0.979401i	$-0.435280\pi$
$599$	9.88400			0.403849			0.201925	+	0.979401i	$0.435280\pi$
$600$		0			0
$601$	8.34260			0.340302			0.170151	−	0.985418i	$-0.445575\pi$
$601$	8.34260			0.340302			0.170151	+	0.985418i	$0.445575\pi$
$602$		0			0
$603$		−	1.00000i		−	0.0407231i
$604$		0			0
$605$	−15.8237	+	18.4755i	−0.643325	+	0.751135i
$606$		0			0
$607$		−	9.58194i		−	0.388919i	−0.980911	−	0.194460i	$-0.937705\pi$
$607$		−	9.58194i		−	0.388919i	0.980911	−	0.194460i	$-0.0622953\pi$
$608$		0			0
$609$	0.972593			0.0394114
$610$		0			0
$611$	23.8009			0.962882
$612$		0			0
$613$		−	43.4482i		−	1.75486i	−0.479709	−	0.877428i	$-0.659258\pi$
$613$		−	43.4482i		−	1.75486i	0.479709	−	0.877428i	$-0.340742\pi$
$614$		0			0
$615$	−3.87109	−	3.31547i	−0.156097	−	0.133693i
$616$		0			0
$617$			6.09240i			0.245271i	0.992452	+	0.122635i	$0.0391346\pi$
$617$			6.09240i			0.245271i	−0.992452	+	0.122635i	$0.960865\pi$
$618$		0			0
$619$	−0.0361608			−0.00145343			−0.000726713	−	1.00000i	$-0.500231\pi$
$619$	−0.0361608			−0.00145343			−0.000726713		1.00000i	$0.500231\pi$
$620$		0			0
$621$	0.600572			0.0241001
$622$		0			0
$623$		−	2.10210i		−	0.0842190i
$624$		0			0
$625$	−23.8187	+	7.59400i	−0.952749	+	0.303760i
$626$		0			0
$627$		−	0.475714i		−	0.0189982i
$628$		0			0
$629$	10.2836			0.410033
$630$		0			0
$631$	−40.0856			−1.59578			−0.797890	−	0.602802i	$-0.794051\pi$
$631$	−40.0856			−1.59578			−0.797890	+	0.602802i	$0.794051\pi$
$632$		0			0
$633$		−	0.669505i		−	0.0266104i
$634$		0			0
$635$	−19.2378	−	16.4766i	−0.763429	−	0.653855i
$636$		0			0
$637$			18.2426i			0.722797i
$638$		0			0
$639$	4.45872			0.176384
$640$		0			0
$641$	−46.1783			−1.82393			−0.911967	−	0.410263i	$-0.865437\pi$
$641$	−46.1783			−1.82393			−0.911967	+	0.410263i	$0.865437\pi$
$642$		0			0
$643$			24.4979i			0.966104i	0.875592	+	0.483052i	$0.160472\pi$
$643$			24.4979i			0.966104i	−0.875592	+	0.483052i	$0.839528\pi$
$644$		0			0
$645$	−11.9883	+	13.9973i	−0.472038	+	0.551142i
$646$		0			0
$647$			37.6660i			1.48080i	0.672166	+	0.740401i	$0.265364\pi$
$647$			37.6660i			1.48080i	−0.672166	+	0.740401i	$0.734636\pi$
$648$		0			0
$649$	2.09061			0.0820634
$650$		0			0
$651$	0.787797			0.0308762
$652$		0			0
$653$			13.1227i			0.513530i	0.966474	+	0.256765i	$0.0826566\pi$
$653$			13.1227i			0.513530i	−0.966474	+	0.256765i	$0.917343\pi$
$654$		0			0
$655$	−12.1043	+	14.1327i	−0.472953	+	0.552211i
$656$		0			0
$657$			8.87821i			0.346372i
$658$		0			0
$659$	22.7196			0.885029			0.442514	−	0.896761i	$-0.354087\pi$
$659$	22.7196			0.885029			0.442514	+	0.896761i	$0.354087\pi$
$660$		0			0
$661$	46.6427			1.81419			0.907095	−	0.420926i	$-0.138295\pi$
$661$	46.6427			1.81419			0.907095	+	0.420926i	$0.138295\pi$
$662$		0			0
$663$			3.39320i			0.131781i
$664$		0			0
$665$	−0.564175	−	0.483199i	−0.0218778	−	0.0187377i
$666$		0			0
$667$			2.40191i			0.0930023i
$668$		0			0
$669$	26.2148			1.01352
$670$		0			0
$671$	−1.31668			−0.0508299
$672$		0			0
$673$			16.3729i			0.631129i	0.948904	+	0.315565i	$0.102194\pi$
$673$			16.3729i			0.631129i	−0.948904	+	0.315565i	$0.897806\pi$
$674$		0			0
$675$	4.94058	−	0.768532i	0.190163	−	0.0295808i
$676$		0			0
$677$			23.1803i			0.890893i	0.895309	+	0.445446i	$0.146955\pi$
$677$			23.1803i			0.890893i	−0.895309	+	0.445446i	$0.853045\pi$
$678$		0			0
$679$	0.201458			0.00773126
$680$		0			0
$681$	−14.2115			−0.544585
$682$		0			0
$683$		−	9.81125i		−	0.375417i	−0.982225	−	0.187709i	$-0.939894\pi$
$683$		−	9.81125i		−	0.375417i	0.982225	−	0.187709i	$-0.0601060\pi$
$684$		0			0
$685$	22.7811	+	19.5114i	0.870421	+	0.745491i
$686$		0			0
$687$		−	6.41350i		−	0.244690i
$688$		0			0
$689$	27.7813			1.05838
$690$		0			0
$691$	15.8801			0.604107			0.302054	−	0.953291i	$-0.402328\pi$
$691$	15.8801			0.604107			0.302054	+	0.953291i	$0.402328\pi$
$692$		0			0
$693$			0.0846895i			0.00321709i
$694$		0			0
$695$	−19.2848	+	22.5165i	−0.731513	+	0.854101i
$696$		0			0
$697$			2.94274i			0.111464i
$698$		0			0
$699$	11.2402			0.425145
$700$		0			0
$701$	28.7019			1.08405			0.542027	−	0.840361i	$-0.317657\pi$
$701$	28.7019			1.08405			0.542027	+	0.840361i	$0.317657\pi$
$702$		0			0
$703$		−	10.8809i		−	0.410380i
$704$		0			0
$705$	−13.1720	+	15.3794i	−0.496086	+	0.579221i
$706$		0			0
$707$			0.892494i			0.0335657i
$708$		0			0
$709$	36.0371			1.35340			0.676702	−	0.736257i	$-0.263409\pi$
$709$	36.0371			1.35340			0.676702	+	0.736257i	$0.263409\pi$
$710$		0			0
$711$	16.2980			0.611224
$712$		0			0
$713$			1.94554i			0.0728610i
$714$		0			0
$715$	1.55446	+	1.33135i	0.0581336	+	0.0497897i
$716$		0			0
$717$		−	10.7654i		−	0.402040i
$718$		0			0
$719$	10.8888			0.406084			0.203042	−	0.979170i	$-0.434917\pi$
$719$	10.8888			0.406084			0.203042	+	0.979170i	$0.434917\pi$
$720$		0			0
$721$	−4.43316			−0.165100
$722$		0			0
$723$			0.229596i			0.00853876i
$724$		0			0
$725$	3.07364	+	19.7592i	0.114152	+	0.733839i
$726$		0			0
$727$			13.1718i			0.488515i	0.969710	+	0.244258i	$0.0785443\pi$
$727$			13.1718i			0.488515i	−0.969710	+	0.244258i	$0.921456\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	10.6405			0.393554
$732$		0			0
$733$		−	8.79199i		−	0.324740i	−0.986730	−	0.162370i	$-0.948086\pi$
$733$		−	8.79199i		−	0.324740i	0.986730	−	0.162370i	$-0.0519138\pi$
$734$		0			0
$735$	−11.7878	−	10.0959i	−0.434798	−	0.372392i
$736$		0			0
$737$			0.348249i			0.0128279i
$738$		0			0
$739$	48.8812			1.79813			0.899063	−	0.437819i	$-0.144249\pi$
$739$	48.8812			1.79813			0.899063	+	0.437819i	$0.144249\pi$
$740$		0			0
$741$	3.59028			0.131892
$742$		0			0
$743$			34.5643i			1.26804i	0.773316	+	0.634021i	$0.218597\pi$
$743$			34.5643i			1.26804i	−0.773316	+	0.634021i	$0.781403\pi$
$744$		0			0
$745$	29.2808	−	34.1877i	1.07276	−	1.25254i
$746$		0			0
$747$		−	10.3876i		−	0.380062i
$748$		0			0
$749$	−4.63993			−0.169539
$750$		0			0
$751$	−50.8582			−1.85584			−0.927921	−	0.372776i	$-0.878406\pi$
$751$	−50.8582			−1.85584			−0.927921	+	0.372776i	$0.878406\pi$
$752$		0			0
$753$			5.71115i			0.208126i
$754$		0			0
$755$	33.4653	−	39.0734i	1.21793	−	1.42203i
$756$		0			0
$757$			15.0671i			0.547621i	0.961784	+	0.273811i	$0.0882842\pi$
$757$			15.0671i			0.547621i	−0.961784	+	0.273811i	$0.911716\pi$
$758$		0			0
$759$	−0.209149			−0.00759161
$760$		0			0
$761$	−30.8857			−1.11961			−0.559803	−	0.828626i	$-0.689123\pi$
$761$	−30.8857			−1.11961			−0.559803	+	0.828626i	$0.689123\pi$
$762$		0			0
$763$		−	1.91881i		−	0.0694656i
$764$		0			0
$765$	−2.19257	−	1.87788i	−0.0792727	−	0.0678948i
$766$		0			0
$767$			15.7781i			0.569715i
$768$		0			0
$769$	23.7448			0.856258			0.428129	−	0.903718i	$-0.359173\pi$
$769$	23.7448			0.856258			0.428129	+	0.903718i	$0.359173\pi$
$770$		0			0
$771$	12.8726			0.463594
$772$		0			0
$773$			19.2043i			0.690730i	0.938469	+	0.345365i	$0.112245\pi$
$773$			19.2043i			0.690730i	−0.938469	+	0.345365i	$0.887755\pi$
$774$		0			0
$775$	2.48964	+	16.0049i	0.0894306	+	0.574913i
$776$		0			0
$777$			1.93708i			0.0694922i
$778$		0			0
$779$	3.11366			0.111559
$780$		0			0
$781$	−1.55274			−0.0555615
$782$		0			0
$783$		−	3.99937i		−	0.142926i
$784$		0			0
$785$	3.47859	+	2.97931i	0.124156	+	0.106336i
$786$		0			0
$787$			21.5866i			0.769478i	0.923025	+	0.384739i	$0.125709\pi$
$787$			21.5866i			0.769478i	−0.923025	+	0.384739i	$0.874291\pi$
$788$		0			0
$789$	−21.7586			−0.774625
$790$		0			0
$791$	1.34905			0.0479667
$792$		0			0
$793$		−	9.93719i		−	0.352880i
$794$		0			0
$795$	−15.3748	+	17.9514i	−0.545290	+	0.636670i
$796$		0			0
$797$		−	0.628606i		−	0.0222664i	−0.999938	−	0.0111332i	$-0.996456\pi$
$797$		−	0.628606i		−	0.0222664i	0.999938	−	0.0111332i	$-0.00354388\pi$
$798$		0			0
$799$	11.6912			0.413604
$800$		0			0
$801$	−8.64400			−0.305421
$802$		0			0
$803$		−	3.09183i		−	0.109108i
$804$		0			0
$805$	−0.212439	+	0.248040i	−0.00748751	+	0.00874228i
$806$		0			0
$807$			30.9686i			1.09015i
$808$		0			0
$809$	−6.66456			−0.234314			−0.117157	−	0.993113i	$-0.537378\pi$
$809$	−6.66456			−0.234314			−0.117157	+	0.993113i	$0.537378\pi$
$810$		0			0
$811$	15.1579			0.532266			0.266133	−	0.963936i	$-0.414254\pi$
$811$	15.1579			0.532266			0.266133	+	0.963936i	$0.414254\pi$
$812$		0			0
$813$		−	18.6020i		−	0.652401i
$814$		0			0
$815$	18.4059	+	15.7642i	0.644732	+	0.552194i
$816$		0			0
$817$		−	11.2585i		−	0.393887i
$818$		0			0
$819$	−0.639164			−0.0223342
$820$		0			0
$821$	−3.66243			−0.127820			−0.0639099	−	0.997956i	$-0.520357\pi$
$821$	−3.66243			−0.127820			−0.0639099	+	0.997956i	$0.520357\pi$
$822$		0			0
$823$		−	32.5049i		−	1.13305i	−0.824044	−	0.566526i	$-0.808287\pi$
$823$		−	32.5049i		−	1.13305i	0.824044	−	0.566526i	$-0.191713\pi$
$824$		0			0
$825$	−1.72055	+	0.267641i	−0.0599019	+	0.00931805i
$826$		0			0
$827$		−	11.2718i		−	0.391958i	−0.980608	−	0.195979i	$-0.937212\pi$
$827$		−	11.2718i		−	0.391958i	0.980608	−	0.195979i	$-0.0627884\pi$
$828$		0			0
$829$	40.7842			1.41649			0.708247	−	0.705965i	$-0.249487\pi$
$829$	40.7842			1.41649			0.708247	+	0.705965i	$0.249487\pi$
$830$		0			0
$831$	4.97556			0.172600
$832$		0			0
$833$			8.96087i			0.310476i
$834$		0			0
$835$	26.7552	+	22.9151i	0.925903	+	0.793009i
$836$		0			0
$837$		−	3.23948i		−	0.111973i
$838$		0			0
$839$	−27.6090			−0.953170			−0.476585	−	0.879129i	$-0.658125\pi$
$839$	−27.6090			−0.953170			−0.476585	+	0.879129i	$0.658125\pi$
$840$		0			0
$841$	−13.0051			−0.448450
$842$		0			0
$843$		−	10.6486i		−	0.366757i
$844$		0			0
$845$	8.86132	−	10.3463i	0.304839	−	0.355924i
$846$		0			0
$847$			2.64556i			0.0909025i
$848$		0			0
$849$	−20.1137			−0.690299
$850$		0			0
$851$	−4.78379			−0.163986
$852$		0			0
$853$			10.4575i			0.358056i	0.983844	+	0.179028i	$0.0572953\pi$
$853$			10.4575i			0.358056i	−0.983844	+	0.179028i	$0.942705\pi$
$854$		0			0
$855$	−1.98695	+	2.31992i	−0.0679522	+	0.0793398i
$856$		0			0
$857$			1.36756i			0.0467149i	0.999727	+	0.0233574i	$0.00743558\pi$
$857$			1.36756i			0.0467149i	−0.999727	+	0.0233574i	$0.992564\pi$
$858$		0			0
$859$	32.6188			1.11294			0.556470	−	0.830868i	$-0.312156\pi$
$859$	32.6188			1.11294			0.556470	+	0.830868i	$0.312156\pi$
$860$		0			0
$861$	−0.554313			−0.0188909
$862$		0			0
$863$		−	17.9528i		−	0.611119i	−0.952173	−	0.305559i	$-0.901157\pi$
$863$		−	17.9528i		−	0.611119i	0.952173	−	0.305559i	$-0.0988435\pi$
$864$		0			0
$865$	16.6328	+	14.2455i	0.565533	+	0.484363i
$866$		0			0
$867$		−	15.3332i		−	0.520744i
$868$		0			0
$869$	−5.67577			−0.192537
$870$		0			0
$871$	−2.62829			−0.0890561
$872$		0			0
$873$		−	0.828410i		−	0.0280374i
$874$		0			0
$875$	−1.43022	+	2.31235i	−0.0483501	+	0.0781716i
$876$		0			0
$877$			47.5206i			1.60466i	0.596884	+	0.802328i	$0.296405\pi$
$877$			47.5206i			1.60466i	−0.596884	+	0.802328i	$0.703595\pi$
$878$		0			0
$879$	−5.53165			−0.186578
$880$		0			0
$881$	17.2506			0.581189			0.290594	−	0.956846i	$-0.406147\pi$
$881$	17.2506			0.581189			0.290594	+	0.956846i	$0.406147\pi$
$882$		0			0
$883$			12.2789i			0.413218i	0.978424	+	0.206609i	$0.0662428\pi$
$883$			12.2789i			0.413218i	−0.978424	+	0.206609i	$0.933757\pi$
$884$		0			0
$885$	−10.1953	−	8.73198i	−0.342711	−	0.293522i
$886$		0			0
$887$			13.2768i			0.445792i	0.974842	+	0.222896i	$0.0715510\pi$
$887$			13.2768i			0.445792i	−0.974842	+	0.222896i	$0.928449\pi$
$888$		0			0
$889$	−2.75472			−0.0923904
$890$		0			0
$891$	0.348249			0.0116668
$892$		0			0
$893$		−	12.3702i		−	0.413954i
$894$		0			0
$895$	−7.82742	+	9.13915i	−0.261642	+	0.305488i
$896$		0			0
$897$		−	1.57847i		−	0.0527037i
$898$		0			0
$899$	12.9559			0.432102
$900$		0			0
$901$	13.6464			0.454626
$902$		0			0
$903$			2.00431i			0.0666994i
$904$		0			0
$905$	−5.47533	+	6.39290i	−0.182006	+	0.212507i
$906$		0			0
$907$		−	26.5941i		−	0.883043i	−0.897251	−	0.441521i	$-0.854439\pi$
$907$		−	26.5941i		−	0.883043i	0.897251	−	0.441521i	$-0.145561\pi$
$908$		0			0
$909$	3.67000			0.121726
$910$		0			0
$911$	10.3603			0.343253			0.171627	−	0.985162i	$-0.445098\pi$
$911$	10.3603			0.343253			0.171627	+	0.985162i	$0.445098\pi$
$912$		0			0
$913$			3.61747i			0.119721i
$914$		0			0
$915$	6.42109	+	5.49948i	0.212275	+	0.181807i
$916$		0			0
$917$			2.02371i			0.0668287i
$918$		0			0
$919$	21.2996			0.702610			0.351305	−	0.936261i	$-0.385738\pi$
$919$	21.2996			0.702610			0.351305	+	0.936261i	$0.385738\pi$
$920$		0			0
$921$	−30.4742			−1.00416
$922$		0			0
$923$		−	11.7188i		−	0.385728i
$924$		0			0
$925$	−39.3537	+	6.12166i	−1.29394	+	0.201279i
$926$		0			0
$927$			18.2295i			0.598734i
$928$		0			0
$929$	6.73373			0.220927			0.110463	−	0.993880i	$-0.464767\pi$
$929$	6.73373			0.220927			0.110463	+	0.993880i	$0.464767\pi$
$930$		0			0
$931$	9.48134			0.310738
$932$		0			0
$933$			24.1197i			0.789643i
$934$		0			0
$935$	0.763562	+	0.653969i	0.0249711	+	0.0213871i
$936$		0			0
$937$			56.0134i			1.82988i	0.403592	+	0.914939i	$0.367761\pi$
$937$			56.0134i			1.82988i	−0.403592	+	0.914939i	$0.632239\pi$
$938$		0			0
$939$	3.01994			0.0985520
$940$		0			0
$941$	4.54236			0.148077			0.0740384	−	0.997255i	$-0.476411\pi$
$941$	4.54236			0.148077			0.0740384	+	0.997255i	$0.476411\pi$
$942$		0			0
$943$		−	1.36893i		−	0.0445784i
$944$		0			0
$945$	0.353729	−	0.413007i	0.0115068	−	0.0134351i
$946$		0			0
$947$			25.5803i			0.831247i	0.909537	+	0.415623i	$0.136437\pi$
$947$			25.5803i			0.831247i	−0.909537	+	0.415623i	$0.863563\pi$
$948$		0			0
$949$	23.3345			0.757469
$950$		0			0
$951$	34.4860			1.11828
$952$		0			0
$953$			37.2314i			1.20604i	0.797725	+	0.603022i	$0.206037\pi$
$953$			37.2314i			1.20604i	−0.797725	+	0.603022i	$0.793963\pi$
$954$		0			0
$955$	−32.9589	+	38.4822i	−1.06653	+	1.24526i
$956$		0			0
$957$			1.39278i			0.0450220i
$958$		0			0
$959$	3.26210			0.105339
$960$		0			0
$961$	−20.5058			−0.661477
$962$		0			0
$963$			19.0797i			0.614835i
$964$		0			0
$965$	−4.35616	−	3.73092i	−0.140230	−	0.120103i
$966$		0			0
$967$		−	36.0430i		−	1.15907i	−0.814949	−	0.579533i	$-0.803235\pi$
$967$		−	36.0430i		−	1.15907i	0.814949	−	0.579533i	$-0.196765\pi$
$968$		0			0
$969$	1.76357			0.0566541
$970$		0			0
$971$	−42.3875			−1.36028			−0.680140	−	0.733082i	$-0.738081\pi$
$971$	−42.3875			−1.36028			−0.680140	+	0.733082i	$0.738081\pi$
$972$		0			0
$973$			3.22421i			0.103363i
$974$		0			0
$975$	−2.01992	−	12.9853i	−0.0646893	−	0.415861i
$976$		0			0
$977$			39.1608i			1.25286i	0.779476	+	0.626432i	$0.215485\pi$
$977$			39.1608i			1.25286i	−0.779476	+	0.626432i	$0.784515\pi$
$978$		0			0
$979$	3.01026			0.0962084
$980$		0			0
$981$	−7.89028			−0.251917
$982$		0			0
$983$		−	20.7388i		−	0.661466i	−0.943724	−	0.330733i	$-0.892704\pi$
$983$		−	20.7388i		−	0.661466i	0.943724	−	0.330733i	$-0.107296\pi$
$984$		0			0
$985$	0.195504	+	0.167444i	0.00622929	+	0.00533520i
$986$		0			0
$987$			2.20222i			0.0700975i
$988$		0			0
$989$	−4.94984			−0.157396
$990$		0			0
$991$	2.90436			0.0922601			0.0461301	−	0.998935i	$-0.485311\pi$
$991$	2.90436			0.0922601			0.0461301	+	0.998935i	$0.485311\pi$
$992$		0			0
$993$			20.2865i			0.643772i
$994$		0			0
$995$	11.1064	−	12.9677i	0.352097	−	0.411102i
$996$		0			0
$997$			9.01155i			0.285399i	0.989766	+	0.142699i	$0.0455782\pi$
$997$			9.01155i			0.285399i	−0.989766	+	0.142699i	$0.954422\pi$
$998$		0			0
$999$	7.96539			0.252014

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4020.2.g.b.1609.10	✓	24
5.4	even	2	inner	4020.2.g.b.1609.22	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4020.2.g.b.1609.10	✓	24	1.1	even	1	trivial
4020.2.g.b.1609.22	yes	24	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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +(1.45456 - 1.69831i) q^{5} -0.243187i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +(1.45456 - 1.69831i) q^{5} -0.243187i q^{7} -1.00000 q^{9} +0.348249 q^{11} +2.62829i q^{13} +(-1.69831 - 1.45456i) q^{15} +1.29103i q^{17} +1.36602 q^{19} -0.243187 q^{21} -0.600572i q^{23} +(-0.768532 - 4.94058i) q^{25} +1.00000i q^{27} -3.99937 q^{29} -3.23948 q^{31} -0.348249i q^{33} +(-0.413007 - 0.353729i) q^{35} -7.96539i q^{37} +2.62829 q^{39} +2.27937 q^{41} -8.24187i q^{43} +(-1.45456 + 1.69831i) q^{45} -9.05568i q^{47} +6.94086 q^{49} +1.29103 q^{51} -10.5701i q^{53} +(0.506548 - 0.591435i) q^{55} -1.36602i q^{57} +6.00319 q^{59} -3.78086 q^{61} +0.243187i q^{63} +(4.46365 + 3.82299i) q^{65} +1.00000i q^{67} -0.600572 q^{69} -4.45872 q^{71} -8.87821i q^{73} +(-4.94058 + 0.768532i) q^{75} -0.0846895i q^{77} -16.2980 q^{79} +1.00000 q^{81} +10.3876i q^{83} +(2.19257 + 1.87788i) q^{85} +3.99937i q^{87} +8.64400 q^{89} +0.639164 q^{91} +3.23948i q^{93} +(1.98695 - 2.31992i) q^{95} +0.828410i q^{97} -0.348249 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 4 q^{5} - 24 q^{9}+O(q^{10}) \) 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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