LMFDB - Embedded newform 4020.2.g.c.1609.27

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4020,2,Mod(1609,4020)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [38] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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			1609.27
Character	$\chi$	$=$	4020.1609
Dual form			4020.2.g.c.1609.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +(-0.676225 + 2.13137i) q^{5} +4.23231i q^{7} -1.00000 q^{9} -1.39022 q^{11} +6.36038i q^{13} +(-2.13137 - 0.676225i) q^{15} -4.10855i q^{17} -5.15323 q^{19} -4.23231 q^{21} -1.91335i q^{23} +(-4.08544 - 2.88257i) q^{25} -1.00000i q^{27} -5.40553 q^{29} +0.479723 q^{31} -1.39022i q^{33} +(-9.02060 - 2.86199i) q^{35} +2.52485i q^{37} -6.36038 q^{39} +4.14785 q^{41} +0.604550i q^{43} +(0.676225 - 2.13137i) q^{45} -0.216591i q^{47} -10.9124 q^{49} +4.10855 q^{51} +2.89586i q^{53} +(0.940102 - 2.96307i) q^{55} -5.15323i q^{57} -0.893364 q^{59} +2.90540 q^{61} -4.23231i q^{63} +(-13.5563 - 4.30105i) q^{65} +1.00000i q^{67} +1.91335 q^{69} +8.26764 q^{71} +12.6582i q^{73} +(2.88257 - 4.08544i) q^{75} -5.88384i q^{77} +13.7111 q^{79} +1.00000 q^{81} -12.0898i q^{83} +(8.75681 + 2.77830i) q^{85} -5.40553i q^{87} +17.0123 q^{89} -26.9191 q^{91} +0.479723i q^{93} +(3.48474 - 10.9834i) q^{95} -0.636187i q^{97} +1.39022 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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}+ \cdots - 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

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
$5$	−0.676225	+	2.13137i	−0.302417	+	0.953176i
$5$	−0.676225	+	2.13137i	−0.302417	+	0.953176i
$6$		0			0
$7$			4.23231i			1.59966i	0.600225	+	0.799831i	$0.295078\pi$
$7$			4.23231i			1.59966i	−0.600225	+	0.799831i	$0.704922\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	−1.39022			−0.419167			−0.209584	−	0.977791i	$-0.567211\pi$
$11$	−1.39022			−0.419167			−0.209584	+	0.977791i	$0.567211\pi$
$12$		0			0
$13$			6.36038i			1.76405i	0.471201	+	0.882026i	$0.343821\pi$
$13$			6.36038i			1.76405i	−0.471201	+	0.882026i	$0.656179\pi$
$14$		0			0
$15$	−2.13137	−	0.676225i	−0.550316	−	0.174601i
$16$		0			0
$17$		−	4.10855i		−	0.996469i	−0.867042	−	0.498234i	$-0.833982\pi$
$17$		−	4.10855i		−	0.996469i	0.867042	−	0.498234i	$-0.166018\pi$
$18$		0			0
$19$	−5.15323			−1.18223			−0.591116	−	0.806587i	$-0.701312\pi$
$19$	−5.15323			−1.18223			−0.591116	+	0.806587i	$0.701312\pi$
$20$		0			0
$21$	−4.23231			−0.923566
$22$		0			0
$23$		−	1.91335i		−	0.398962i	−0.979902	−	0.199481i	$-0.936074\pi$
$23$		−	1.91335i		−	0.398962i	0.979902	−	0.199481i	$-0.0639256\pi$
$24$		0			0
$25$	−4.08544	−	2.88257i	−0.817088	−	0.576513i
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$	−5.40553			−1.00378			−0.501891	−	0.864931i	$-0.667362\pi$
$29$	−5.40553			−1.00378			−0.501891	+	0.864931i	$0.667362\pi$
$30$		0			0
$31$	0.479723			0.0861608			0.0430804	−	0.999072i	$-0.486283\pi$
$31$	0.479723			0.0861608			0.0430804	+	0.999072i	$0.486283\pi$
$32$		0			0
$33$		−	1.39022i		−	0.242006i
$34$		0			0
$35$	−9.02060	−	2.86199i	−1.52476	−	0.483765i
$36$		0			0
$37$			2.52485i			0.415083i	0.978226	+	0.207541i	$0.0665462\pi$
$37$			2.52485i			0.415083i	−0.978226	+	0.207541i	$0.933454\pi$
$38$		0			0
$39$	−6.36038			−1.01848
$40$		0			0
$41$	4.14785			0.647785			0.323893	−	0.946094i	$-0.395008\pi$
$41$	4.14785			0.647785			0.323893	+	0.946094i	$0.395008\pi$
$42$		0			0
$43$			0.604550i			0.0921930i	0.998937	+	0.0460965i	$0.0146782\pi$
$43$			0.604550i			0.0921930i	−0.998937	+	0.0460965i	$0.985322\pi$
$44$		0			0
$45$	0.676225	−	2.13137i	0.100806	−	0.317725i
$46$		0			0
$47$		−	0.216591i		−	0.0315931i	−0.999875	−	0.0157965i	$-0.994972\pi$
$47$		−	0.216591i		−	0.0315931i	0.999875	−	0.0157965i	$-0.00502841\pi$
$48$		0			0
$49$	−10.9124			−1.55892
$50$		0			0
$51$	4.10855			0.575312
$52$		0			0
$53$			2.89586i			0.397777i	0.980022	+	0.198889i	$0.0637332\pi$
$53$			2.89586i			0.397777i	−0.980022	+	0.198889i	$0.936267\pi$
$54$		0			0
$55$	0.940102	−	2.96307i	0.126763	−	0.399540i
$56$		0			0
$57$		−	5.15323i		−	0.682562i
$58$		0			0
$59$	−0.893364			−0.116306			−0.0581530	−	0.998308i	$-0.518521\pi$
$59$	−0.893364			−0.116306			−0.0581530	+	0.998308i	$0.518521\pi$
$60$		0			0
$61$	2.90540			0.371998			0.185999	−	0.982550i	$-0.440448\pi$
$61$	2.90540			0.371998			0.185999	+	0.982550i	$0.440448\pi$
$62$		0			0
$63$		−	4.23231i		−	0.533221i
$64$		0			0
$65$	−13.5563	−	4.30105i	−1.68145	−	0.533479i
$66$		0			0
$67$			1.00000i			0.122169i
$67$			1.00000i			0.122169i
$68$		0			0
$69$	1.91335			0.230341
$70$		0			0
$71$	8.26764			0.981188			0.490594	−	0.871388i	$-0.336780\pi$
$71$	8.26764			0.981188			0.490594	+	0.871388i	$0.336780\pi$
$72$		0			0
$73$			12.6582i			1.48153i	0.671764	+	0.740765i	$0.265537\pi$
$73$			12.6582i			1.48153i	−0.671764	+	0.740765i	$0.734463\pi$
$74$		0			0
$75$	2.88257	−	4.08544i	0.332850	−	0.471746i
$76$		0			0
$77$		−	5.88384i		−	0.670526i
$78$		0			0
$79$	13.7111			1.54262			0.771308	−	0.636462i	$-0.219603\pi$
$79$	13.7111			1.54262			0.771308	+	0.636462i	$0.219603\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	12.0898i		−	1.32703i	−0.748163	−	0.663514i	$-0.769064\pi$
$83$		−	12.0898i		−	1.32703i	0.748163	−	0.663514i	$-0.230936\pi$
$84$		0			0
$85$	8.75681	+	2.77830i	0.949810	+	0.301349i
$86$		0			0
$87$		−	5.40553i		−	0.579534i
$88$		0			0
$89$	17.0123			1.80330			0.901648	−	0.432471i	$-0.142358\pi$
$89$	17.0123			1.80330			0.901648	+	0.432471i	$0.142358\pi$
$90$		0			0
$91$	−26.9191			−2.82189
$92$		0			0
$93$			0.479723i			0.0497449i
$94$		0			0
$95$	3.48474	−	10.9834i	0.357527	−	1.12687i
$96$		0			0
$97$		−	0.636187i		−	0.0645950i	−0.999478	−	0.0322975i	$-0.989718\pi$
$97$		−	0.636187i		−	0.0645950i	0.999478	−	0.0322975i	$-0.0102824\pi$
$98$		0			0
$99$	1.39022			0.139722
$100$		0			0
$101$	−1.71282			−0.170432			−0.0852162	−	0.996362i	$-0.527158\pi$
$101$	−1.71282			−0.170432			−0.0852162	+	0.996362i	$0.527158\pi$
$102$		0			0
$103$		−	6.41050i		−	0.631645i	−0.948818	−	0.315823i	$-0.897720\pi$
$103$		−	6.41050i		−	0.631645i	0.948818	−	0.315823i	$-0.102280\pi$
$104$		0			0
$105$	2.86199	−	9.02060i	0.279302	−	0.880320i
$106$		0			0
$107$			18.9246i			1.82951i	0.404008	+	0.914755i	$0.367617\pi$
$107$			18.9246i			1.82951i	−0.404008	+	0.914755i	$0.632383\pi$
$108$		0			0
$109$	4.89694			0.469042			0.234521	−	0.972111i	$-0.424648\pi$
$109$	4.89694			0.469042			0.234521	+	0.972111i	$0.424648\pi$
$110$		0			0
$111$	−2.52485			−0.239648
$112$		0			0
$113$			2.88818i			0.271697i	0.990730	+	0.135848i	$0.0433760\pi$
$113$			2.88818i			0.271697i	−0.990730	+	0.135848i	$0.956624\pi$
$114$		0			0
$115$	4.07806	+	1.29386i	0.380281	+	0.120653i
$116$		0			0
$117$		−	6.36038i		−	0.588017i
$118$		0			0
$119$	17.3886			1.59401
$120$		0			0
$121$	−9.06729			−0.824299
$122$		0			0
$123$			4.14785i			0.373999i
$124$		0			0
$125$	8.90648	−	6.75830i	0.796620	−	0.604481i
$126$		0			0
$127$		−	3.09910i		−	0.275001i	−0.990502	−	0.137500i	$-0.956093\pi$
$127$		−	3.09910i		−	0.275001i	0.990502	−	0.137500i	$-0.0439068\pi$
$128$		0			0
$129$	−0.604550			−0.0532276
$130$		0			0
$131$	2.52174			0.220326			0.110163	−	0.993914i	$-0.464863\pi$
$131$	2.52174			0.220326			0.110163	+	0.993914i	$0.464863\pi$
$132$		0			0
$133$		−	21.8101i		−	1.89117i
$134$		0			0
$135$	2.13137	+	0.676225i	0.183439	+	0.0582002i
$136$		0			0
$137$			10.3830i			0.887077i	0.896255	+	0.443538i	$0.146277\pi$
$137$			10.3830i			0.887077i	−0.896255	+	0.443538i	$0.853723\pi$
$138$		0			0
$139$	−14.7427			−1.25046			−0.625230	−	0.780440i	$-0.714995\pi$
$139$	−14.7427			−1.25046			−0.625230	+	0.780440i	$0.714995\pi$
$140$		0			0
$141$	0.216591			0.0182403
$142$		0			0
$143$		−	8.84233i		−	0.739433i
$144$		0			0
$145$	3.65536	−	11.5212i	0.303561	−	0.956781i
$146$		0			0
$147$		−	10.9124i		−	0.900044i
$148$		0			0
$149$	−8.09056			−0.662804			−0.331402	−	0.943490i	$-0.607522\pi$
$149$	−8.09056			−0.662804			−0.331402	+	0.943490i	$0.607522\pi$
$150$		0			0
$151$	7.26776			0.591442			0.295721	−	0.955274i	$-0.404440\pi$
$151$	7.26776			0.591442			0.295721	+	0.955274i	$0.404440\pi$
$152$		0			0
$153$			4.10855i			0.332156i
$154$		0			0
$155$	−0.324401	+	1.02246i	−0.0260565	+	0.0821263i
$156$		0			0
$157$		−	19.9836i		−	1.59486i	−0.603410	−	0.797431i	$-0.706192\pi$
$157$		−	19.9836i		−	1.59486i	0.603410	−	0.797431i	$-0.293808\pi$
$158$		0			0
$159$	−2.89586			−0.229657
$160$		0			0
$161$	8.09791			0.638205
$162$		0			0
$163$			13.3663i			1.04693i	0.852048	+	0.523464i	$0.175361\pi$
$163$			13.3663i			1.04693i	−0.852048	+	0.523464i	$0.824639\pi$
$164$		0			0
$165$	2.96307	+	0.940102i	0.230674	+	0.0731868i
$166$		0			0
$167$		−	10.4531i		−	0.808886i	−0.914563	−	0.404443i	$-0.867465\pi$
$167$		−	10.4531i		−	0.808886i	0.914563	−	0.404443i	$-0.132535\pi$
$168$		0			0
$169$	−27.4544			−2.11188
$170$		0			0
$171$	5.15323			0.394077
$172$		0			0
$173$			17.7535i			1.34977i	0.737922	+	0.674886i	$0.235807\pi$
$173$			17.7535i			1.34977i	−0.737922	+	0.674886i	$0.764193\pi$
$174$		0			0
$175$	12.1999	−	17.2908i	0.922227	−	1.30707i
$176$		0			0
$177$		−	0.893364i		−	0.0671493i
$178$		0			0
$179$	−16.2777			−1.21665			−0.608327	−	0.793687i	$-0.708159\pi$
$179$	−16.2777			−1.21665			−0.608327	+	0.793687i	$0.708159\pi$
$180$		0			0
$181$	−17.6664			−1.31314			−0.656568	−	0.754267i	$-0.727993\pi$
$181$	−17.6664			−1.31314			−0.656568	+	0.754267i	$0.727993\pi$
$182$		0			0
$183$			2.90540i			0.214773i
$184$		0			0
$185$	−5.38138	−	1.70737i	−0.395647	−	0.125528i
$186$		0			0
$187$			5.71178i			0.417687i
$188$		0			0
$189$	4.23231			0.307855
$190$		0			0
$191$	20.0200			1.44859			0.724297	−	0.689488i	$-0.242164\pi$
$191$	20.0200			1.44859			0.724297	+	0.689488i	$0.242164\pi$
$192$		0			0
$193$		−	1.15297i		−	0.0829923i	−0.999139	−	0.0414962i	$-0.986788\pi$
$193$		−	1.15297i		−	0.0829923i	0.999139	−	0.0414962i	$-0.0132124\pi$
$194$		0			0
$195$	4.30105	−	13.5563i	0.308004	−	0.970786i
$196$		0			0
$197$			3.53115i			0.251584i	0.992057	+	0.125792i	$0.0401472\pi$
$197$			3.53115i			0.251584i	−0.992057	+	0.125792i	$0.959853\pi$
$198$		0			0
$199$	−3.63461			−0.257651			−0.128825	−	0.991667i	$-0.541121\pi$
$199$	−3.63461			−0.257651			−0.128825	+	0.991667i	$0.541121\pi$
$200$		0			0
$201$	−1.00000			−0.0705346
$202$		0			0
$203$		−	22.8779i		−	1.60571i
$204$		0			0
$205$	−2.80488	+	8.84058i	−0.195901	+	0.617453i
$206$		0			0
$207$			1.91335i			0.132987i
$208$		0			0
$209$	7.16412			0.495553
$210$		0			0
$211$	19.9563			1.37385			0.686925	−	0.726728i	$-0.258960\pi$
$211$	19.9563			1.37385			0.686925	+	0.726728i	$0.258960\pi$
$212$		0			0
$213$			8.26764i			0.566489i
$214$		0			0
$215$	−1.28852	−	0.408812i	−0.0878761	−	0.0278807i
$216$		0			0
$217$			2.03034i			0.137828i
$218$		0			0
$219$	−12.6582			−0.855361
$220$		0			0
$221$	26.1319			1.75782
$222$		0			0
$223$		−	21.2283i		−	1.42155i	−0.703417	−	0.710777i	$-0.748343\pi$
$223$		−	21.2283i		−	1.42155i	0.703417	−	0.710777i	$-0.251657\pi$
$224$		0			0
$225$	4.08544	+	2.88257i	0.272363	+	0.192171i
$226$		0			0
$227$			11.5637i			0.767507i	0.923436	+	0.383754i	$0.125369\pi$
$227$			11.5637i			0.767507i	−0.923436	+	0.383754i	$0.874631\pi$
$228$		0			0
$229$	−12.9167			−0.853560			−0.426780	−	0.904355i	$-0.640352\pi$
$229$	−12.9167			−0.853560			−0.426780	+	0.904355i	$0.640352\pi$
$230$		0			0
$231$	5.88384			0.387128
$232$		0			0
$233$		−	15.9294i		−	1.04357i	−0.853078	−	0.521784i	$-0.825267\pi$
$233$		−	15.9294i		−	1.04357i	0.853078	−	0.521784i	$-0.174733\pi$
$234$		0			0
$235$	0.461635	+	0.146464i	0.0301138	+	0.00955429i
$236$		0			0
$237$			13.7111i			0.890629i
$238$		0			0
$239$	−9.80764			−0.634403			−0.317202	−	0.948358i	$-0.602743\pi$
$239$	−9.80764			−0.634403			−0.317202	+	0.948358i	$0.602743\pi$
$240$		0			0
$241$	−18.9392			−1.21998			−0.609989	−	0.792410i	$-0.708826\pi$
$241$	−18.9392			−1.21998			−0.609989	+	0.792410i	$0.708826\pi$
$242$		0			0
$243$			1.00000i			0.0641500i
$244$		0			0
$245$	7.37927	−	23.2584i	0.471444	−	1.48593i
$246$		0			0
$247$		−	32.7765i		−	2.08552i
$248$		0			0
$249$	12.0898			0.766160
$250$		0			0
$251$	−13.4401			−0.848334			−0.424167	−	0.905584i	$-0.639433\pi$
$251$	−13.4401			−0.848334			−0.424167	+	0.905584i	$0.639433\pi$
$252$		0			0
$253$			2.65998i			0.167232i
$254$		0			0
$255$	−2.77830	+	8.75681i	−0.173984	+	0.548373i
$256$		0			0
$257$			6.47817i			0.404097i	0.979376	+	0.202049i	$0.0647599\pi$
$257$			6.47817i			0.404097i	−0.979376	+	0.202049i	$0.935240\pi$
$258$		0			0
$259$	−10.6860			−0.663993
$260$		0			0
$261$	5.40553			0.334594
$262$		0			0
$263$		−	21.0096i		−	1.29551i	−0.761850	−	0.647753i	$-0.775709\pi$
$263$		−	21.0096i		−	1.29551i	0.761850	−	0.647753i	$-0.224291\pi$
$264$		0			0
$265$	−6.17214	−	1.95825i	−0.379151	−	0.120295i
$266$		0			0
$267$			17.0123i			1.04113i
$268$		0			0
$269$	−14.6527			−0.893393			−0.446697	−	0.894686i	$-0.647400\pi$
$269$	−14.6527			−0.893393			−0.446697	+	0.894686i	$0.647400\pi$
$270$		0			0
$271$	−15.3153			−0.930337			−0.465169	−	0.885222i	$-0.654006\pi$
$271$	−15.3153			−0.930337			−0.465169	+	0.885222i	$0.654006\pi$
$272$		0			0
$273$		−	26.9191i		−	1.62922i
$274$		0			0
$275$	5.67966	+	4.00740i	0.342496	+	0.241655i
$276$		0			0
$277$			14.6763i			0.881815i	0.897553	+	0.440907i	$0.145343\pi$
$277$			14.6763i			0.881815i	−0.897553	+	0.440907i	$0.854657\pi$
$278$		0			0
$279$	−0.479723			−0.0287203
$280$		0			0
$281$	16.9400			1.01056			0.505278	−	0.862957i	$-0.331390\pi$
$281$	16.9400			1.01056			0.505278	+	0.862957i	$0.331390\pi$
$282$		0			0
$283$		−	17.7694i		−	1.05628i	−0.849158	−	0.528139i	$-0.822890\pi$
$283$		−	17.7694i		−	1.05628i	0.849158	−	0.528139i	$-0.177110\pi$
$284$		0			0
$285$	10.9834	+	3.48474i	0.650602	+	0.206418i
$286$		0			0
$287$			17.5550i			1.03624i
$288$		0			0
$289$	0.119848			0.00704986
$290$		0			0
$291$	0.636187			0.0372939
$292$		0			0
$293$			12.4037i			0.724630i	0.932056	+	0.362315i	$0.118014\pi$
$293$			12.4037i			0.724630i	−0.932056	+	0.362315i	$0.881986\pi$
$294$		0			0
$295$	0.604115	−	1.90408i	0.0351729	−	0.110860i
$296$		0			0
$297$			1.39022i			0.0806687i
$298$		0			0
$299$	12.1697			0.703790
$300$		0			0
$301$	−2.55864			−0.147478
$302$		0			0
$303$		−	1.71282i		−	0.0983992i
$304$		0			0
$305$	−1.96470	+	6.19246i	−0.112499	+	0.354579i
$306$		0			0
$307$		−	2.43370i		−	0.138899i	−0.997585	−	0.0694494i	$-0.977876\pi$
$307$		−	2.43370i		−	0.138899i	0.997585	−	0.0694494i	$-0.0221242\pi$
$308$		0			0
$309$	6.41050			0.364681
$310$		0			0
$311$	−15.6408			−0.886909			−0.443455	−	0.896297i	$-0.646247\pi$
$311$	−15.6408			−0.886909			−0.443455	+	0.896297i	$0.646247\pi$
$312$		0			0
$313$			34.9072i			1.97307i	0.163538	+	0.986537i	$0.447710\pi$
$313$			34.9072i			1.97307i	−0.163538	+	0.986537i	$0.552290\pi$
$314$		0			0
$315$	9.02060	+	2.86199i	0.508253	+	0.161255i
$316$		0			0
$317$			9.84170i			0.552765i	0.961048	+	0.276382i	$0.0891356\pi$
$317$			9.84170i			0.552765i	−0.961048	+	0.276382i	$0.910864\pi$
$318$		0			0
$319$	7.51488			0.420752
$320$		0			0
$321$	−18.9246			−1.05627
$322$		0			0
$323$			21.1723i			1.17806i
$324$		0			0
$325$	18.3342	−	25.9849i	1.01700	−	1.44139i
$326$		0			0
$327$			4.89694i			0.270802i
$328$		0			0
$329$	0.916682			0.0505383
$330$		0			0
$331$	16.9414			0.931182			0.465591	−	0.885000i	$-0.345842\pi$
$331$	16.9414			0.931182			0.465591	+	0.885000i	$0.345842\pi$
$332$		0			0
$333$		−	2.52485i		−	0.138361i
$334$		0			0
$335$	−2.13137	−	0.676225i	−0.116449	−	0.0369461i
$336$		0			0
$337$		−	27.6820i		−	1.50793i	−0.656913	−	0.753966i	$-0.728138\pi$
$337$		−	27.6820i		−	1.50793i	0.656913	−	0.753966i	$-0.271862\pi$
$338$		0			0
$339$	−2.88818			−0.156864
$340$		0			0
$341$	−0.666920			−0.0361158
$342$		0			0
$343$		−	16.5587i		−	0.894086i
$344$		0			0
$345$	−1.29386	+	4.07806i	−0.0696590	+	0.219555i
$346$		0			0
$347$		−	30.2355i		−	1.62312i	−0.584266	−	0.811562i	$-0.698618\pi$
$347$		−	30.2355i		−	1.62312i	0.584266	−	0.811562i	$-0.301382\pi$
$348$		0			0
$349$	−15.3093			−0.819486			−0.409743	−	0.912201i	$-0.634382\pi$
$349$	−15.3093			−0.819486			−0.409743	+	0.912201i	$0.634382\pi$
$350$		0			0
$351$	6.36038			0.339492
$352$		0			0
$353$			15.8352i			0.842824i	0.906869	+	0.421412i	$0.138465\pi$
$353$			15.8352i			0.842824i	−0.906869	+	0.421412i	$0.861535\pi$
$354$		0			0
$355$	−5.59079	+	17.6214i	−0.296728	+	0.935245i
$356$		0			0
$357$			17.3886i			0.920305i
$358$		0			0
$359$	10.6388			0.561496			0.280748	−	0.959782i	$-0.409417\pi$
$359$	10.6388			0.561496			0.280748	+	0.959782i	$0.409417\pi$
$360$		0			0
$361$	7.55578			0.397673
$362$		0			0
$363$		−	9.06729i		−	0.475909i
$364$		0			0
$365$	−26.9792	−	8.55979i	−1.41216	−	0.448040i
$366$		0			0
$367$			21.5187i			1.12327i	0.827386	+	0.561633i	$0.189827\pi$
$367$			21.5187i			1.12327i	−0.827386	+	0.561633i	$0.810173\pi$
$368$		0			0
$369$	−4.14785			−0.215928
$370$		0			0
$371$	−12.2562			−0.636309
$372$		0			0
$373$		−	28.9322i		−	1.49805i	−0.662540	−	0.749027i	$-0.730521\pi$
$373$		−	28.9322i		−	1.49805i	0.662540	−	0.749027i	$-0.269479\pi$
$374$		0			0
$375$	6.75830	+	8.90648i	0.348997	+	0.459929i
$376$		0			0
$377$		−	34.3812i		−	1.77072i
$378$		0			0
$379$	−18.3578			−0.942976			−0.471488	−	0.881873i	$-0.656283\pi$
$379$	−18.3578			−0.942976			−0.471488	+	0.881873i	$0.656283\pi$
$380$		0			0
$381$	3.09910			0.158772
$382$		0			0
$383$		−	6.14772i		−	0.314134i	−0.987588	−	0.157067i	$-0.949796\pi$
$383$		−	6.14772i		−	0.314134i	0.987588	−	0.157067i	$-0.0502039\pi$
$384$		0			0
$385$	12.5406	+	3.97880i	0.639129	+	0.202778i
$386$		0			0
$387$		−	0.604550i		−	0.0307310i
$388$		0			0
$389$	−10.1069			−0.512439			−0.256219	−	0.966619i	$-0.582477\pi$
$389$	−10.1069			−0.512439			−0.256219	+	0.966619i	$0.582477\pi$
$390$		0			0
$391$	−7.86111			−0.397553
$392$		0			0
$393$			2.52174i			0.127205i
$394$		0			0
$395$	−9.27177	+	29.2233i	−0.466513	+	1.47038i
$396$		0			0
$397$		−	7.09727i		−	0.356202i	−0.984012	−	0.178101i	$-0.943005\pi$
$397$		−	7.09727i		−	0.356202i	0.984012	−	0.178101i	$-0.0569953\pi$
$398$		0			0
$399$	21.8101			1.09187
$400$		0			0
$401$	4.82791			0.241094			0.120547	−	0.992708i	$-0.461535\pi$
$401$	4.82791			0.241094			0.120547	+	0.992708i	$0.461535\pi$
$402$		0			0
$403$			3.05122i			0.151992i
$404$		0			0
$405$	−0.676225	+	2.13137i	−0.0336019	+	0.105908i
$406$		0			0
$407$		−	3.51010i		−	0.173989i
$408$		0			0
$409$	−34.0608			−1.68420			−0.842098	−	0.539324i	$-0.818680\pi$
$409$	−34.0608			−1.68420			−0.842098	+	0.539324i	$0.818680\pi$
$410$		0			0
$411$	−10.3830			−0.512154
$412$		0			0
$413$		−	3.78099i		−	0.186050i
$414$		0			0
$415$	25.7678	+	8.17543i	1.26489	+	0.401316i
$416$		0			0
$417$		−	14.7427i		−	0.721954i
$418$		0			0
$419$	−30.2974			−1.48013			−0.740063	−	0.672538i	$-0.765204\pi$
$419$	−30.2974			−1.48013			−0.740063	+	0.672538i	$0.765204\pi$
$420$		0			0
$421$	−13.6514			−0.665327			−0.332663	−	0.943046i	$-0.607947\pi$
$421$	−13.6514			−0.665327			−0.332663	+	0.943046i	$0.607947\pi$
$422$		0			0
$423$			0.216591i			0.0105310i
$424$		0			0
$425$	−11.8432	+	16.7852i	−0.574477	+	0.814203i
$426$		0			0
$427$			12.2965i			0.595071i
$428$		0			0
$429$	8.84233			0.426912
$430$		0			0
$431$	−4.97781			−0.239773			−0.119886	−	0.992788i	$-0.538253\pi$
$431$	−4.97781			−0.239773			−0.119886	+	0.992788i	$0.538253\pi$
$432$		0			0
$433$			29.9998i			1.44170i	0.693093	+	0.720848i	$0.256248\pi$
$433$			29.9998i			1.44170i	−0.693093	+	0.720848i	$0.743752\pi$
$434$		0			0
$435$	11.5212	+	3.65536i	0.552398	+	0.175261i
$436$		0			0
$437$			9.85996i			0.471666i
$438$		0			0
$439$	−4.68934			−0.223810			−0.111905	−	0.993719i	$-0.535695\pi$
$439$	−4.68934			−0.223810			−0.111905	+	0.993719i	$0.535695\pi$
$440$		0			0
$441$	10.9124			0.519640
$442$		0			0
$443$		−	24.5335i		−	1.16562i	−0.812608	−	0.582811i	$-0.801953\pi$
$443$		−	24.5335i		−	1.16562i	0.812608	−	0.582811i	$-0.198047\pi$
$444$		0			0
$445$	−11.5041	+	36.2594i	−0.545347	+	1.71886i
$446$		0			0
$447$		−	8.09056i		−	0.382670i
$448$		0			0
$449$	−16.8758			−0.796417			−0.398208	−	0.917295i	$-0.630368\pi$
$449$	−16.8758			−0.796417			−0.398208	+	0.917295i	$0.630368\pi$
$450$		0			0
$451$	−5.76642			−0.271530
$452$		0			0
$453$			7.26776i			0.341469i
$454$		0			0
$455$	18.2034	−	57.3744i	0.853387	−	2.68976i
$456$		0			0
$457$		−	31.4653i		−	1.47188i	−0.677045	−	0.735942i	$-0.736740\pi$
$457$		−	31.4653i		−	1.47188i	0.677045	−	0.735942i	$-0.263260\pi$
$458$		0			0
$459$	−4.10855			−0.191771
$460$		0			0
$461$	33.7846			1.57350			0.786752	−	0.617270i	$-0.211761\pi$
$461$	33.7846			1.57350			0.786752	+	0.617270i	$0.211761\pi$
$462$		0			0
$463$			21.4179i			0.995373i	0.867357	+	0.497687i	$0.165817\pi$
$463$			21.4179i			0.995373i	−0.867357	+	0.497687i	$0.834183\pi$
$464$		0			0
$465$	−1.02246	−	0.324401i	−0.0474157	−	0.0150437i
$466$		0			0
$467$		−	26.8173i		−	1.24096i	−0.784224	−	0.620478i	$-0.786938\pi$
$467$		−	26.8173i		−	1.24096i	0.784224	−	0.620478i	$-0.213062\pi$
$468$		0			0
$469$	−4.23231			−0.195430
$470$		0			0
$471$	19.9836			0.920794
$472$		0			0
$473$		−	0.840457i		−	0.0386443i
$474$		0			0
$475$	21.0532	+	14.8545i	0.965988	+	0.681572i
$476$		0			0
$477$		−	2.89586i		−	0.132592i
$478$		0			0
$479$	−16.9878			−0.776194			−0.388097	−	0.921619i	$-0.626867\pi$
$479$	−16.9878			−0.776194			−0.388097	+	0.921619i	$0.626867\pi$
$480$		0			0
$481$	−16.0590			−0.732228
$482$		0			0
$483$			8.09791i			0.368468i
$484$		0			0
$485$	1.35595	+	0.430205i	0.0615704	+	0.0195346i
$486$		0			0
$487$			39.5253i			1.79106i	0.444999	+	0.895531i	$0.353204\pi$
$487$			39.5253i			1.79106i	−0.444999	+	0.895531i	$0.646796\pi$
$488$		0			0
$489$	−13.3663			−0.604445
$490$		0			0
$491$	−19.0417			−0.859339			−0.429669	−	0.902986i	$-0.641370\pi$
$491$	−19.0417			−0.859339			−0.429669	+	0.902986i	$0.641370\pi$
$492$		0			0
$493$			22.2089i			1.00024i
$494$		0			0
$495$	−0.940102	+	2.96307i	−0.0422544	+	0.133180i
$496$		0			0
$497$			34.9912i			1.56957i
$498$		0			0
$499$	−20.3317			−0.910170			−0.455085	−	0.890448i	$-0.650391\pi$
$499$	−20.3317			−0.910170			−0.455085	+	0.890448i	$0.650391\pi$
$500$		0			0
$501$	10.4531			0.467011
$502$		0			0
$503$		−	32.9475i		−	1.46906i	−0.678578	−	0.734528i	$-0.737404\pi$
$503$		−	32.9475i		−	1.46906i	0.678578	−	0.734528i	$-0.262596\pi$
$504$		0			0
$505$	1.15825	−	3.65065i	0.0515416	−	0.162452i
$506$		0			0
$507$		−	27.4544i		−	1.21929i
$508$		0			0
$509$	35.7348			1.58392			0.791959	−	0.610574i	$-0.209061\pi$
$509$	35.7348			1.58392			0.791959	+	0.610574i	$0.209061\pi$
$510$		0			0
$511$	−53.5734			−2.36995
$512$		0			0
$513$			5.15323i			0.227521i
$514$		0			0
$515$	13.6631	+	4.33494i	0.602069	+	0.191020i
$516$		0			0
$517$			0.301110i			0.0132428i
$518$		0			0
$519$	−17.7535			−0.779292
$520$		0			0
$521$	44.8076			1.96305			0.981527	−	0.191322i	$-0.0612774\pi$
$521$	44.8076			1.96305			0.981527	+	0.191322i	$0.0612774\pi$
$522$		0			0
$523$		−	13.8543i		−	0.605806i	−0.953021	−	0.302903i	$-0.902044\pi$
$523$		−	13.8543i		−	0.605806i	0.953021	−	0.302903i	$-0.0979558\pi$
$524$		0			0
$525$	17.2908	+	12.1999i	0.754634	+	0.532448i
$526$		0			0
$527$		−	1.97096i		−	0.0858565i
$528$		0			0
$529$	19.3391			0.840829
$530$		0			0
$531$	0.893364			0.0387687
$532$		0			0
$533$			26.3819i			1.14273i
$534$		0			0
$535$	−40.3353	−	12.7973i	−1.74385	−	0.553275i
$536$		0			0
$537$		−	16.2777i		−	0.702435i
$538$		0			0
$539$	15.1707			0.653449
$540$		0			0
$541$	3.29139			0.141508			0.0707540	−	0.997494i	$-0.477459\pi$
$541$	3.29139			0.141508			0.0707540	+	0.997494i	$0.477459\pi$
$542$		0			0
$543$		−	17.6664i		−	0.758140i
$544$		0			0
$545$	−3.31143	+	10.4372i	−0.141846	+	0.447079i
$546$		0			0
$547$		−	7.38046i		−	0.315566i	−0.987474	−	0.157783i	$-0.949565\pi$
$547$		−	7.38046i		−	0.315566i	0.987474	−	0.157783i	$-0.0504346\pi$
$548$		0			0
$549$	−2.90540			−0.123999
$550$		0			0
$551$	27.8559			1.18670
$552$		0			0
$553$			58.0295i			2.46766i
$554$		0			0
$555$	1.70737	−	5.38138i	0.0724737	−	0.228427i
$556$		0			0
$557$		−	1.83073i		−	0.0775705i	−0.999248	−	0.0387852i	$-0.987651\pi$
$557$		−	1.83073i		−	0.0775705i	0.999248	−	0.0387852i	$-0.0123488\pi$
$558$		0			0
$559$	−3.84517			−0.162633
$560$		0			0
$561$	−5.71178			−0.241152
$562$		0			0
$563$		−	19.4344i		−	0.819063i	−0.912296	−	0.409532i	$-0.865692\pi$
$563$		−	19.4344i		−	0.819063i	0.912296	−	0.409532i	$-0.134308\pi$
$564$		0			0
$565$	−6.15576	−	1.95306i	−0.258975	−	0.0821657i
$566$		0			0
$567$			4.23231i			0.177740i
$568$		0			0
$569$	−14.5151			−0.608506			−0.304253	−	0.952591i	$-0.598407\pi$
$569$	−14.5151			−0.608506			−0.304253	+	0.952591i	$0.598407\pi$
$570$		0			0
$571$	−12.3594			−0.517225			−0.258613	−	0.965981i	$-0.583265\pi$
$571$	−12.3594			−0.517225			−0.258613	+	0.965981i	$0.583265\pi$
$572$		0			0
$573$			20.0200i			0.836346i
$574$		0			0
$575$	−5.51537	+	7.81690i	−0.230007	+	0.325987i
$576$		0			0
$577$			32.1837i			1.33983i	0.742439	+	0.669913i	$0.233669\pi$
$577$			32.1837i			1.33983i	−0.742439	+	0.669913i	$0.766331\pi$
$578$		0			0
$579$	1.15297			0.0479157
$580$		0			0
$581$	51.1678			2.12280
$582$		0			0
$583$		−	4.02588i		−	0.166735i
$584$		0			0
$585$	13.5563	+	4.30105i	0.560484	+	0.177826i
$586$		0			0
$587$		−	36.8765i		−	1.52206i	−0.648718	−	0.761029i	$-0.724695\pi$
$587$		−	36.8765i		−	1.52206i	0.648718	−	0.761029i	$-0.275305\pi$
$588$		0			0
$589$	−2.47212			−0.101862
$590$		0			0
$591$	−3.53115			−0.145252
$592$		0			0
$593$			31.4549i			1.29170i	0.763465	+	0.645849i	$0.223496\pi$
$593$			31.4549i			1.29170i	−0.763465	+	0.645849i	$0.776504\pi$
$594$		0			0
$595$	−11.7586	+	37.0616i	−0.482057	+	1.51938i
$596$		0			0
$597$		−	3.63461i		−	0.148755i
$598$		0			0
$599$	8.68221			0.354746			0.177373	−	0.984144i	$-0.443240\pi$
$599$	8.68221			0.354746			0.177373	+	0.984144i	$0.443240\pi$
$600$		0			0
$601$	11.1262			0.453848			0.226924	−	0.973913i	$-0.427133\pi$
$601$	11.1262			0.453848			0.226924	+	0.973913i	$0.427133\pi$
$602$		0			0
$603$		−	1.00000i		−	0.0407231i
$604$		0			0
$605$	6.13153	−	19.3257i	0.249282	−	0.785702i
$606$		0			0
$607$		−	27.8119i		−	1.12885i	−0.825485	−	0.564424i	$-0.809098\pi$
$607$		−	27.8119i		−	1.12885i	0.825485	−	0.564424i	$-0.190902\pi$
$608$		0			0
$609$	22.8779			0.927059
$610$		0			0
$611$	1.37760			0.0557319
$612$		0			0
$613$			27.6331i			1.11609i	0.829811	+	0.558045i	$0.188448\pi$
$613$			27.6331i			1.11609i	−0.829811	+	0.558045i	$0.811552\pi$
$614$		0			0
$615$	−8.84058	−	2.80488i	−0.356487	−	0.113104i
$616$		0			0
$617$			17.6017i			0.708618i	0.935128	+	0.354309i	$0.115284\pi$
$617$			17.6017i			0.708618i	−0.935128	+	0.354309i	$0.884716\pi$
$618$		0			0
$619$	−16.8586			−0.677604			−0.338802	−	0.940858i	$-0.610022\pi$
$619$	−16.8586			−0.677604			−0.338802	+	0.940858i	$0.610022\pi$
$620$		0			0
$621$	−1.91335			−0.0767803
$622$		0			0
$623$			72.0012i			2.88467i
$624$		0			0
$625$	8.38163	+	23.5531i	0.335265	+	0.942124i
$626$		0			0
$627$			7.16412i			0.286108i
$628$		0			0
$629$	10.3735			0.413617
$630$		0			0
$631$	41.7361			1.66149			0.830744	−	0.556654i	$-0.187915\pi$
$631$	41.7361			1.66149			0.830744	+	0.556654i	$0.187915\pi$
$632$		0			0
$633$			19.9563i			0.793192i
$634$		0			0
$635$	6.60532	+	2.09569i	0.262124	+	0.0831649i
$636$		0			0
$637$		−	69.4073i		−	2.75002i
$638$		0			0
$639$	−8.26764			−0.327063
$640$		0			0
$641$	12.2358			0.483286			0.241643	−	0.970365i	$-0.422314\pi$
$641$	12.2358			0.483286			0.241643	+	0.970365i	$0.422314\pi$
$642$		0			0
$643$			0.694878i			0.0274033i	0.999906	+	0.0137017i	$0.00436151\pi$
$643$			0.694878i			0.0274033i	−0.999906	+	0.0137017i	$0.995638\pi$
$644$		0			0
$645$	0.408812	−	1.28852i	0.0160969	−	0.0507353i
$646$		0			0
$647$		−	2.15047i		−	0.0845437i	−0.999106	−	0.0422719i	$-0.986540\pi$
$647$		−	2.15047i		−	0.0845437i	0.999106	−	0.0422719i	$-0.0134596\pi$
$648$		0			0
$649$	1.24197			0.0487517
$650$		0			0
$651$	−2.03034			−0.0795751
$652$		0			0
$653$		−	35.3611i		−	1.38379i	−0.722000	−	0.691893i	$-0.756777\pi$
$653$		−	35.3611i		−	1.38379i	0.722000	−	0.691893i	$-0.243223\pi$
$654$		0			0
$655$	−1.70526	+	5.37475i	−0.0666302	+	0.210009i
$656$		0			0
$657$		−	12.6582i		−	0.493843i
$658$		0			0
$659$	−5.37707			−0.209461			−0.104730	−	0.994501i	$-0.533398\pi$
$659$	−5.37707			−0.209461			−0.104730	+	0.994501i	$0.533398\pi$
$660$		0			0
$661$	11.5561			0.449481			0.224741	−	0.974419i	$-0.427846\pi$
$661$	11.5561			0.449481			0.224741	+	0.974419i	$0.427846\pi$
$662$		0			0
$663$			26.1319i			1.01488i
$664$		0			0
$665$	46.4852	+	14.7485i	1.80262	+	0.571923i
$666$		0			0
$667$			10.3427i			0.400471i
$668$		0			0
$669$	21.2283			0.820735
$670$		0			0
$671$	−4.03914			−0.155929
$672$		0			0
$673$			15.1408i			0.583634i	0.956474	+	0.291817i	$0.0942598\pi$
$673$			15.1408i			0.583634i	−0.956474	+	0.291817i	$0.905740\pi$
$674$		0			0
$675$	−2.88257	+	4.08544i	−0.110950	+	0.157249i
$676$		0			0
$677$			40.1536i			1.54323i	0.636092	+	0.771613i	$0.280550\pi$
$677$			40.1536i			1.54323i	−0.636092	+	0.771613i	$0.719450\pi$
$678$		0			0
$679$	2.69254			0.103330
$680$		0			0
$681$	−11.5637			−0.443121
$682$		0			0
$683$		−	39.1608i		−	1.49844i	−0.662319	−	0.749222i	$-0.730427\pi$
$683$		−	39.1608i		−	1.49844i	0.662319	−	0.749222i	$-0.269573\pi$
$684$		0			0
$685$	−22.1299	−	7.02122i	−0.845540	−	0.268267i
$686$		0			0
$687$		−	12.9167i		−	0.492803i
$688$		0			0
$689$	−18.4188			−0.701699
$690$		0			0
$691$	36.7016			1.39620			0.698098	−	0.716002i	$-0.254030\pi$
$691$	36.7016			1.39620			0.698098	+	0.716002i	$0.254030\pi$
$692$		0			0
$693$			5.88384i			0.223509i
$694$		0			0
$695$	9.96939	−	31.4221i	0.378161	−	1.19191i
$696$		0			0
$697$		−	17.0416i		−	0.645498i
$698$		0			0
$699$	15.9294			0.602504
$700$		0			0
$701$	−39.1254			−1.47775			−0.738873	−	0.673844i	$-0.764642\pi$
$701$	−39.1254			−1.47775			−0.738873	+	0.673844i	$0.764642\pi$
$702$		0			0
$703$		−	13.0111i		−	0.490724i
$704$		0			0
$705$	−0.146464	+	0.461635i	−0.00551617	+	0.0173862i
$706$		0			0
$707$		−	7.24920i		−	0.272634i
$708$		0			0
$709$	5.88538			0.221030			0.110515	−	0.993874i	$-0.464750\pi$
$709$	5.88538			0.221030			0.110515	+	0.993874i	$0.464750\pi$
$710$		0			0
$711$	−13.7111			−0.514205
$712$		0			0
$713$		−	0.917880i		−	0.0343749i
$714$		0			0
$715$	18.8462	+	5.97940i	0.704809	+	0.223617i
$716$		0			0
$717$		−	9.80764i		−	0.366273i
$718$		0			0
$719$	−8.40620			−0.313498			−0.156749	−	0.987638i	$-0.550101\pi$
$719$	−8.40620			−0.313498			−0.156749	+	0.987638i	$0.550101\pi$
$720$		0			0
$721$	27.1312			1.01042
$722$		0			0
$723$		−	18.9392i		−	0.704355i
$724$		0			0
$725$	22.0840	+	15.5818i	0.820178	+	0.578694i
$726$		0			0
$727$		−	17.9152i		−	0.664439i	−0.943202	−	0.332220i	$-0.892203\pi$
$727$		−	17.9152i		−	0.664439i	0.943202	−	0.332220i	$-0.107797\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	2.48382			0.0918674
$732$		0			0
$733$		−	22.1553i		−	0.818325i	−0.912462	−	0.409162i	$-0.865821\pi$
$733$		−	22.1553i		−	0.818325i	0.912462	−	0.409162i	$-0.134179\pi$
$734$		0			0
$735$	23.2584	+	7.37927i	0.857900	+	0.272189i
$736$		0			0
$737$		−	1.39022i		−	0.0512094i
$738$		0			0
$739$	23.3616			0.859371			0.429686	−	0.902979i	$-0.358624\pi$
$739$	23.3616			0.859371			0.429686	+	0.902979i	$0.358624\pi$
$740$		0			0
$741$	32.7765			1.20407
$742$		0			0
$743$			13.2885i			0.487508i	0.969837	+	0.243754i	$0.0783789\pi$
$743$			13.2885i			0.487508i	−0.969837	+	0.243754i	$0.921621\pi$
$744$		0			0
$745$	5.47104	−	17.2439i	0.200443	−	0.631769i
$746$		0			0
$747$			12.0898i			0.442343i
$748$		0			0
$749$	−80.0948			−2.92660
$750$		0			0
$751$	−35.4870			−1.29494			−0.647470	−	0.762091i	$-0.724173\pi$
$751$	−35.4870			−1.29494			−0.647470	+	0.762091i	$0.724173\pi$
$752$		0			0
$753$		−	13.4401i		−	0.489786i
$754$		0			0
$755$	−4.91464	+	15.4903i	−0.178862	+	0.563748i
$756$		0			0
$757$			37.0646i			1.34714i	0.739125	+	0.673568i	$0.235239\pi$
$757$			37.0646i			1.34714i	−0.739125	+	0.673568i	$0.764761\pi$
$758$		0			0
$759$	−2.65998			−0.0965513
$760$		0			0
$761$	18.9718			0.687726			0.343863	−	0.939020i	$-0.388264\pi$
$761$	18.9718			0.687726			0.343863	+	0.939020i	$0.388264\pi$
$762$		0			0
$763$			20.7254i			0.750309i
$764$		0			0
$765$	−8.75681	−	2.77830i	−0.316603	−	0.100450i
$766$		0			0
$767$		−	5.68213i		−	0.205170i
$768$		0			0
$769$	−17.7057			−0.638483			−0.319241	−	0.947673i	$-0.603428\pi$
$769$	−17.7057			−0.638483			−0.319241	+	0.947673i	$0.603428\pi$
$770$		0			0
$771$	−6.47817			−0.233306
$772$		0			0
$773$			28.7012i			1.03231i	0.856495	+	0.516155i	$0.172637\pi$
$773$			28.7012i			1.03231i	−0.856495	+	0.516155i	$0.827363\pi$
$774$		0			0
$775$	−1.95988	−	1.38283i	−0.0704009	−	0.0496728i
$776$		0			0
$777$		−	10.6860i		−	0.383356i
$778$		0			0
$779$	−21.3748			−0.765832
$780$		0			0
$781$	−11.4938			−0.411282
$782$		0			0
$783$			5.40553i			0.193178i
$784$		0			0
$785$	42.5923	+	13.5134i	1.52018	+	0.482313i
$786$		0			0
$787$			29.9880i			1.06896i	0.845182	+	0.534478i	$0.179492\pi$
$787$			29.9880i			1.06896i	−0.845182	+	0.534478i	$0.820508\pi$
$788$		0			0
$789$	21.0096			0.747961
$790$		0			0
$791$	−12.2237			−0.434623
$792$		0			0
$793$			18.4794i			0.656224i
$794$		0			0
$795$	1.95825	−	6.17214i	0.0694521	−	0.218903i
$796$		0			0
$797$			33.4386i			1.18446i	0.805770	+	0.592228i	$0.201752\pi$
$797$			33.4386i			1.18446i	−0.805770	+	0.592228i	$0.798248\pi$
$798$		0			0
$799$	−0.889875			−0.0314815
$800$		0			0
$801$	−17.0123			−0.601099
$802$		0			0
$803$		−	17.5977i		−	0.621008i
$804$		0			0
$805$	−5.47601	+	17.2596i	−0.193004	+	0.608321i
$806$		0			0
$807$		−	14.6527i		−	0.515801i
$808$		0			0
$809$	−29.7628			−1.04640			−0.523201	−	0.852209i	$-0.675262\pi$
$809$	−29.7628			−1.04640			−0.523201	+	0.852209i	$0.675262\pi$
$810$		0			0
$811$	−29.1877			−1.02492			−0.512459	−	0.858712i	$-0.671265\pi$
$811$	−29.1877			−1.02492			−0.512459	+	0.858712i	$0.671265\pi$
$812$		0			0
$813$		−	15.3153i		−	0.537130i
$814$		0			0
$815$	−28.4885	−	9.03862i	−0.997907	−	0.316609i
$816$		0			0
$817$		−	3.11538i		−	0.108994i
$818$		0			0
$819$	26.9191			0.940629
$820$		0			0
$821$	30.3728			1.06002			0.530008	−	0.847992i	$-0.322189\pi$
$821$	30.3728			1.06002			0.530008	+	0.847992i	$0.322189\pi$
$822$		0			0
$823$			43.0909i			1.50206i	0.660271	+	0.751028i	$0.270442\pi$
$823$			43.0909i			1.50206i	−0.660271	+	0.751028i	$0.729558\pi$
$824$		0			0
$825$	−4.00740	+	5.67966i	−0.139520	+	0.197740i
$826$		0			0
$827$		−	36.1128i		−	1.25577i	−0.778308	−	0.627883i	$-0.783922\pi$
$827$		−	36.1128i		−	1.25577i	0.778308	−	0.627883i	$-0.216078\pi$
$828$		0			0
$829$	−20.6499			−0.717202			−0.358601	−	0.933491i	$-0.616746\pi$
$829$	−20.6499			−0.717202			−0.358601	+	0.933491i	$0.616746\pi$
$830$		0			0
$831$	−14.6763			−0.509116
$832$		0			0
$833$			44.8343i			1.55342i
$834$		0			0
$835$	22.2794	+	7.06865i	0.771010	+	0.244621i
$836$		0			0
$837$		−	0.479723i		−	0.0165816i
$838$		0			0
$839$	−10.7406			−0.370806			−0.185403	−	0.982663i	$-0.559359\pi$
$839$	−10.7406			−0.370806			−0.185403	+	0.982663i	$0.559359\pi$
$840$		0			0
$841$	0.219775			0.00757845
$842$		0			0
$843$			16.9400i			0.583445i
$844$		0			0
$845$	18.5654	−	58.5154i	0.638668	−	2.01299i
$846$		0			0
$847$		−	38.3756i		−	1.31860i
$848$		0			0
$849$	17.7694			0.609843
$850$		0			0
$851$	4.83094			0.165602
$852$		0			0
$853$			27.9205i			0.955978i	0.878366	+	0.477989i	$0.158634\pi$
$853$			27.9205i			0.955978i	−0.878366	+	0.477989i	$0.841366\pi$
$854$		0			0
$855$	−3.48474	+	10.9834i	−0.119176	+	0.375625i
$856$		0			0
$857$			40.7854i			1.39320i	0.717458	+	0.696602i	$0.245306\pi$
$857$			40.7854i			1.39320i	−0.717458	+	0.696602i	$0.754694\pi$
$858$		0			0
$859$	−38.7575			−1.32239			−0.661195	−	0.750214i	$-0.729950\pi$
$859$	−38.7575			−1.32239			−0.661195	+	0.750214i	$0.729950\pi$
$860$		0			0
$861$	−17.5550			−0.598272
$862$		0			0
$863$			38.2235i			1.30114i	0.759445	+	0.650572i	$0.225471\pi$
$863$			38.2235i			1.30114i	−0.759445	+	0.650572i	$0.774529\pi$
$864$		0			0
$865$	−37.8392	−	12.0054i	−1.28657	−	0.408194i
$866$		0			0
$867$			0.119848i			0.00407024i
$868$		0			0
$869$	−19.0614			−0.646614
$870$		0			0
$871$	−6.36038			−0.215513
$872$		0			0
$873$			0.636187i			0.0215317i
$874$		0			0
$875$	28.6032	+	37.6950i	0.966966	+	1.27432i
$876$		0			0
$877$			30.0536i			1.01484i	0.861700	+	0.507419i	$0.169400\pi$
$877$			30.0536i			1.01484i	−0.861700	+	0.507419i	$0.830600\pi$
$878$		0			0
$879$	−12.4037			−0.418365
$880$		0			0
$881$	11.7852			0.397054			0.198527	−	0.980095i	$-0.436384\pi$
$881$	11.7852			0.397054			0.198527	+	0.980095i	$0.436384\pi$
$882$		0			0
$883$			29.9106i			1.00657i	0.864120	+	0.503287i	$0.167876\pi$
$883$			29.9106i			1.00657i	−0.864120	+	0.503287i	$0.832124\pi$
$884$		0			0
$885$	1.90408	+	0.604115i	0.0640051	+	0.0203071i
$886$		0			0
$887$		−	37.5542i		−	1.26095i	−0.776212	−	0.630473i	$-0.782861\pi$
$887$		−	37.5542i		−	1.26095i	0.776212	−	0.630473i	$-0.217139\pi$
$888$		0			0
$889$	13.1164			0.439909
$890$		0			0
$891$	−1.39022			−0.0465741
$892$		0			0
$893$			1.11614i			0.0373504i
$894$		0			0
$895$	11.0074	−	34.6938i	0.367937	−	1.15968i
$896$		0			0
$897$			12.1697i			0.406333i
$898$		0			0
$899$	−2.59316			−0.0864866
$900$		0			0
$901$	11.8978			0.396372
$902$		0			0
$903$		−	2.55864i		−	0.0851463i
$904$		0			0
$905$	11.9465	−	37.6537i	0.397115	−	1.25165i
$906$		0			0
$907$			11.1361i			0.369767i	0.982760	+	0.184883i	$0.0591907\pi$
$907$			11.1361i			0.369767i	−0.982760	+	0.184883i	$0.940809\pi$
$908$		0			0
$909$	1.71282			0.0568108
$910$		0			0
$911$	−12.0528			−0.399326			−0.199663	−	0.979865i	$-0.563985\pi$
$911$	−12.0528			−0.399326			−0.199663	+	0.979865i	$0.563985\pi$
$912$		0			0
$913$			16.8075i			0.556247i
$914$		0			0
$915$	−6.19246	−	1.96470i	−0.204717	−	0.0649511i
$916$		0			0
$917$			10.6728i			0.352447i
$918$		0			0
$919$	47.7133			1.57392			0.786958	−	0.617007i	$-0.211655\pi$
$919$	47.7133			1.57392			0.786958	+	0.617007i	$0.211655\pi$
$920$		0			0
$921$	2.43370			0.0801932
$922$		0			0
$923$			52.5853i			1.73087i
$924$		0			0
$925$	7.27805	−	10.3151i	0.239301	−	0.339159i
$926$		0			0
$927$			6.41050i			0.210548i
$928$		0			0
$929$	−22.7529			−0.746497			−0.373249	−	0.927731i	$-0.621756\pi$
$929$	−22.7529			−0.746497			−0.373249	+	0.927731i	$0.621756\pi$
$930$		0			0
$931$	56.2344			1.84301
$932$		0			0
$933$		−	15.6408i		−	0.512057i
$934$		0			0
$935$	−12.1739	−	3.86245i	−0.398129	−	0.126316i
$936$		0			0
$937$		−	6.73047i		−	0.219875i	−0.993939	−	0.109937i	$-0.964935\pi$
$937$		−	6.73047i		−	0.219875i	0.993939	−	0.109937i	$-0.0350650\pi$
$938$		0			0
$939$	−34.9072			−1.13915
$940$		0			0
$941$	−22.4839			−0.732954			−0.366477	−	0.930427i	$-0.619436\pi$
$941$	−22.4839			−0.732954			−0.366477	+	0.930427i	$0.619436\pi$
$942$		0			0
$943$		−	7.93631i		−	0.258442i
$944$		0			0
$945$	−2.86199	+	9.02060i	−0.0931007	+	0.293440i
$946$		0			0
$947$			29.8044i			0.968514i	0.874926	+	0.484257i	$0.160910\pi$
$947$			29.8044i			0.968514i	−0.874926	+	0.484257i	$0.839090\pi$
$948$		0			0
$949$	−80.5109			−2.61349
$950$		0			0
$951$	−9.84170			−0.319139
$952$		0			0
$953$		−	43.4121i		−	1.40626i	−0.711063	−	0.703128i	$-0.751786\pi$
$953$		−	43.4121i		−	1.40626i	0.711063	−	0.703128i	$-0.248214\pi$
$954$		0			0
$955$	−13.5380	+	42.6699i	−0.438079	+	1.38076i
$956$		0			0
$957$			7.51488i			0.242922i
$958$		0			0
$959$	−43.9439			−1.41902
$960$		0			0
$961$	−30.7699			−0.992576
$962$		0			0
$963$		−	18.9246i		−	0.609837i
$964$		0			0
$965$	2.45739	+	0.779665i	0.0791063	+	0.0250983i
$966$		0			0
$967$		−	23.2974i		−	0.749195i	−0.927187	−	0.374598i	$-0.877781\pi$
$967$		−	23.2974i		−	0.749195i	0.927187	−	0.374598i	$-0.122219\pi$
$968$		0			0
$969$	−21.1723			−0.680152
$970$		0			0
$971$	−31.9394			−1.02498			−0.512492	−	0.858692i	$-0.671278\pi$
$971$	−31.9394			−1.02498			−0.512492	+	0.858692i	$0.671278\pi$
$972$		0			0
$973$		−	62.3957i		−	2.00032i
$974$		0			0
$975$	25.9849	+	18.3342i	0.832184	+	0.587165i
$976$		0			0
$977$			36.5084i			1.16801i	0.811751	+	0.584004i	$0.198515\pi$
$977$			36.5084i			1.16801i	−0.811751	+	0.584004i	$0.801485\pi$
$978$		0			0
$979$	−23.6508			−0.755882
$980$		0			0
$981$	−4.89694			−0.156347
$982$		0			0
$983$			25.4785i			0.812637i	0.913732	+	0.406318i	$0.133188\pi$
$983$			25.4785i			0.812637i	−0.913732	+	0.406318i	$0.866812\pi$
$984$		0			0
$985$	−7.52618	−	2.38785i	−0.239804	−	0.0760833i
$986$		0			0
$987$			0.916682i			0.0291783i
$988$		0			0
$989$	1.15672			0.0367815
$990$		0			0
$991$	9.65306			0.306640			0.153320	−	0.988177i	$-0.451004\pi$
$991$	9.65306			0.306640			0.153320	+	0.988177i	$0.451004\pi$
$992$		0			0
$993$			16.9414i			0.537618i
$994$		0			0
$995$	2.45782	−	7.74669i	0.0779180	−	0.245587i
$996$		0			0
$997$			53.2290i			1.68578i	0.538086	+	0.842890i	$0.319148\pi$
$997$			53.2290i			1.68578i	−0.538086	+	0.842890i	$0.680852\pi$
$998$		0			0
$999$	2.52485			0.0798827

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4020.2.g.c.1609.27	yes	38
5.4	even	2	inner	4020.2.g.c.1609.8	✓	38

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4020.2.g.c.1609.8	✓	38	5.4	even	2	inner
4020.2.g.c.1609.27	yes	38	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 $\F_{q}$
Belyi maps

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)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +(-0.676225 + 2.13137i) q^{5} +4.23231i q^{7} -1.00000 q^{9} -1.39022 q^{11} +6.36038i q^{13} +(-2.13137 - 0.676225i) q^{15} -4.10855i q^{17} -5.15323 q^{19} -4.23231 q^{21} -1.91335i q^{23} +(-4.08544 - 2.88257i) q^{25} -1.00000i q^{27} -5.40553 q^{29} +0.479723 q^{31} -1.39022i q^{33} +(-9.02060 - 2.86199i) q^{35} +2.52485i q^{37} -6.36038 q^{39} +4.14785 q^{41} +0.604550i q^{43} +(0.676225 - 2.13137i) q^{45} -0.216591i q^{47} -10.9124 q^{49} +4.10855 q^{51} +2.89586i q^{53} +(0.940102 - 2.96307i) q^{55} -5.15323i q^{57} -0.893364 q^{59} +2.90540 q^{61} -4.23231i q^{63} +(-13.5563 - 4.30105i) q^{65} +1.00000i q^{67} +1.91335 q^{69} +8.26764 q^{71} +12.6582i q^{73} +(2.88257 - 4.08544i) q^{75} -5.88384i q^{77} +13.7111 q^{79} +1.00000 q^{81} -12.0898i q^{83} +(8.75681 + 2.77830i) q^{85} -5.40553i q^{87} +17.0123 q^{89} -26.9191 q^{91} +0.479723i q^{93} +(3.48474 - 10.9834i) q^{95} -0.636187i q^{97} +1.39022 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 38 q - 2 q^{5} - 38 q^{9} + 24 q^{11} + 2 q^{15} - 16 q^{19} + 12 q^{21} + 4 q^{25} - 56 q^{29} - 4 q^{31} - 2 q^{35} + 60 q^{41} + 2 q^{45} - 70 q^{49} + 12 q^{55} - 52 q^{59} + 48 q^{61} + 16 q^{65} - 12 q^{69}+ \cdots - 24 q^{99}+O(q^{100}) \) 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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