LMFDB - Embedded newform 4020.2.g.b.1609.19

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			1609.19
Character	$\chi$	$=$	4020.1609
Dual form			4020.2.g.b.1609.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +(-0.131247 - 2.23221i) q^{5} +5.13753i q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +(-0.131247 - 2.23221i) q^{5} +5.13753i q^{7} -1.00000 q^{9} -1.58948 q^{11} -2.13766i q^{13} +(2.23221 - 0.131247i) q^{15} -7.84639i q^{17} +2.19030 q^{19} -5.13753 q^{21} +5.33309i q^{23} +(-4.96555 + 0.585944i) q^{25} -1.00000i q^{27} +0.166293 q^{29} -1.09364 q^{31} -1.58948i q^{33} +(11.4681 - 0.674287i) q^{35} +2.96897i q^{37} +2.13766 q^{39} -10.9924 q^{41} -3.88353i q^{43} +(0.131247 + 2.23221i) q^{45} -6.39193i q^{47} -19.3942 q^{49} +7.84639 q^{51} -11.5030i q^{53} +(0.208615 + 3.54805i) q^{55} +2.19030i q^{57} -7.22807 q^{59} +9.90971 q^{61} -5.13753i q^{63} +(-4.77171 + 0.280562i) q^{65} -1.00000i q^{67} -5.33309 q^{69} +11.2863 q^{71} +6.60168i q^{73} +(-0.585944 - 4.96555i) q^{75} -8.16599i q^{77} -7.35082 q^{79} +1.00000 q^{81} -7.97928i q^{83} +(-17.5148 + 1.02982i) q^{85} +0.166293i q^{87} +10.6997 q^{89} +10.9823 q^{91} -1.09364i q^{93} +(-0.287471 - 4.88921i) q^{95} +1.21273i q^{97} +1.58948 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 4 q^{5} - 24 q^{9}+O(q^{10}) $ 24 * q - 4 * q^5 - 24 * q^9	$ 24 q - 4 q^{5} - 24 q^{9} - 8 q^{11} - 2 q^{15} + 16 q^{19} - 20 q^{21} + 10 q^{25} + 36 q^{29} - 2 q^{35} + 4 q^{39} - 24 q^{41} + 4 q^{45} - 4 q^{51} - 4 q^{55} + 24 q^{59} - 4 q^{61} - 20 q^{65} - 4 q^{69} + 20 q^{71} - 12 q^{75} - 28 q^{79} + 24 q^{81} - 16 q^{85} + 48 q^{89} - 20 q^{91} - 4 q^{95} + 8 q^{99}+O(q^{100}) $ 24 * q - 4 * q^5 - 24 * q^9 - 8 * q^11 - 2 * q^15 + 16 * q^19 - 20 * q^21 + 10 * q^25 + 36 * q^29 - 2 * q^35 + 4 * q^39 - 24 * q^41 + 4 * q^45 - 4 * q^51 - 4 * q^55 + 24 * q^59 - 4 * q^61 - 20 * q^65 - 4 * q^69 + 20 * q^71 - 12 * q^75 - 28 * q^79 + 24 * q^81 - 16 * q^85 + 48 * q^89 - 20 * q^91 - 4 * q^95 + 8 * q^99

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$		0			0
$5$	−0.131247	−	2.23221i	−0.0586956	−	0.998276i
$5$	−0.131247	−	2.23221i	−0.0586956	−	0.998276i
$6$		0			0
$7$			5.13753i			1.94180i	0.239478	+	0.970902i	$0.423024\pi$
$7$			5.13753i			1.94180i	−0.239478	+	0.970902i	$0.576976\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	−1.58948			−0.479245			−0.239623	−	0.970866i	$-0.577024\pi$
$11$	−1.58948			−0.479245			−0.239623	+	0.970866i	$0.577024\pi$
$12$		0			0
$13$		−	2.13766i		−	0.592880i	−0.955051	−	0.296440i	$-0.904200\pi$
$13$		−	2.13766i		−	0.592880i	0.955051	−	0.296440i	$-0.0957995\pi$
$14$		0			0
$15$	2.23221	−	0.131247i	0.576355	−	0.0338879i
$16$		0			0
$17$		−	7.84639i		−	1.90303i	−0.307605	−	0.951514i	$-0.599528\pi$
$17$		−	7.84639i		−	1.90303i	0.307605	−	0.951514i	$-0.400472\pi$
$18$		0			0
$19$	2.19030			0.502488			0.251244	−	0.967924i	$-0.419160\pi$
$19$	2.19030			0.502488			0.251244	+	0.967924i	$0.419160\pi$
$20$		0			0
$21$	−5.13753			−1.12110
$22$		0			0
$23$			5.33309i			1.11203i	0.831173	+	0.556013i	$0.187670\pi$
$23$			5.33309i			1.11203i	−0.831173	+	0.556013i	$0.812330\pi$
$24$		0			0
$25$	−4.96555	+	0.585944i	−0.993110	+	0.117189i
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$	0.166293			0.0308799			0.0154399	−	0.999881i	$-0.495085\pi$
$29$	0.166293			0.0308799			0.0154399	+	0.999881i	$0.495085\pi$
$30$		0			0
$31$	−1.09364			−0.196423			−0.0982116	−	0.995166i	$-0.531312\pi$
$31$	−1.09364			−0.196423			−0.0982116	+	0.995166i	$0.531312\pi$
$32$		0			0
$33$		−	1.58948i		−	0.276692i
$34$		0			0
$35$	11.4681	−	0.674287i	1.93846	−	0.113975i
$36$		0			0
$37$			2.96897i			0.488095i	0.969763	+	0.244048i	$0.0784753\pi$
$37$			2.96897i			0.488095i	−0.969763	+	0.244048i	$0.921525\pi$
$38$		0			0
$39$	2.13766			0.342300
$40$		0			0
$41$	−10.9924			−1.71672			−0.858362	−	0.513045i	$-0.828517\pi$
$41$	−10.9924			−1.71672			−0.858362	+	0.513045i	$0.828517\pi$
$42$		0			0
$43$		−	3.88353i		−	0.592233i	−0.955152	−	0.296117i	$-0.904308\pi$
$43$		−	3.88353i		−	0.592233i	0.955152	−	0.296117i	$-0.0956917\pi$
$44$		0			0
$45$	0.131247	+	2.23221i	0.0195652	+	0.332759i
$46$		0			0
$47$		−	6.39193i		−	0.932358i	−0.884690	−	0.466179i	$-0.845630\pi$
$47$		−	6.39193i		−	0.932358i	0.884690	−	0.466179i	$-0.154370\pi$
$48$		0			0
$49$	−19.3942			−2.77060
$50$		0			0
$51$	7.84639			1.09871
$52$		0			0
$53$		−	11.5030i		−	1.58006i	−0.613070	−	0.790029i	$-0.710066\pi$
$53$		−	11.5030i		−	1.58006i	0.613070	−	0.790029i	$-0.289934\pi$
$54$		0			0
$55$	0.208615	+	3.54805i	0.0281296	+	0.478419i
$56$		0			0
$57$			2.19030i			0.290112i
$58$		0			0
$59$	−7.22807			−0.941014			−0.470507	−	0.882396i	$-0.655929\pi$
$59$	−7.22807			−0.941014			−0.470507	+	0.882396i	$0.655929\pi$
$60$		0			0
$61$	9.90971			1.26881			0.634404	−	0.773002i	$-0.281246\pi$
$61$	9.90971			1.26881			0.634404	+	0.773002i	$0.281246\pi$
$62$		0			0
$63$		−	5.13753i		−	0.647268i
$64$		0			0
$65$	−4.77171	+	0.280562i	−0.591858	+	0.0347995i
$66$		0			0
$67$		−	1.00000i		−	0.122169i
$67$		−	1.00000i		−	0.122169i
$68$		0			0
$69$	−5.33309			−0.642029
$70$		0			0
$71$	11.2863			1.33944			0.669719	−	0.742615i	$-0.266415\pi$
$71$	11.2863			1.33944			0.669719	+	0.742615i	$0.266415\pi$
$72$		0			0
$73$			6.60168i			0.772668i	0.922359	+	0.386334i	$0.126259\pi$
$73$			6.60168i			0.772668i	−0.922359	+	0.386334i	$0.873741\pi$
$74$		0			0
$75$	−0.585944	−	4.96555i	−0.0676590	−	0.573372i
$76$		0			0
$77$		−	8.16599i		−	0.930600i
$78$		0			0
$79$	−7.35082			−0.827032			−0.413516	−	0.910497i	$-0.635699\pi$
$79$	−7.35082			−0.827032			−0.413516	+	0.910497i	$0.635699\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	7.97928i		−	0.875839i	−0.899014	−	0.437920i	$-0.855715\pi$
$83$		−	7.97928i		−	0.875839i	0.899014	−	0.437920i	$-0.144285\pi$
$84$		0			0
$85$	−17.5148	+	1.02982i	−1.89975	+	0.111699i
$86$		0			0
$87$			0.166293i			0.0178285i
$88$		0			0
$89$	10.6997			1.13416			0.567082	−	0.823661i	$-0.308072\pi$
$89$	10.6997			1.13416			0.567082	+	0.823661i	$0.308072\pi$
$90$		0			0
$91$	10.9823			1.15126
$92$		0			0
$93$		−	1.09364i		−	0.113405i
$94$		0			0
$95$	−0.287471	−	4.88921i	−0.0294939	−	0.501622i
$96$		0			0
$97$			1.21273i			0.123134i	0.998103	+	0.0615670i	$0.0196098\pi$
$97$			1.21273i			0.123134i	−0.998103	+	0.0615670i	$0.980390\pi$
$98$		0			0
$99$	1.58948			0.159748
$100$		0			0
$101$	0.301193			0.0299698			0.0149849	−	0.999888i	$-0.495230\pi$
$101$	0.301193			0.0299698			0.0149849	+	0.999888i	$0.495230\pi$
$102$		0			0
$103$		−	0.803026i		−	0.0791245i	−0.999217	−	0.0395623i	$-0.987404\pi$
$103$		−	0.803026i		−	0.0791245i	0.999217	−	0.0395623i	$-0.0125963\pi$
$104$		0			0
$105$	0.674287	+	11.4681i	0.0658037	+	1.11917i
$106$		0			0
$107$		−	14.4533i		−	1.39725i	−0.715489	−	0.698624i	$-0.753796\pi$
$107$		−	14.4533i		−	1.39725i	0.715489	−	0.698624i	$-0.246204\pi$
$108$		0			0
$109$	−0.720132			−0.0689761			−0.0344881	−	0.999405i	$-0.510980\pi$
$109$	−0.720132			−0.0689761			−0.0344881	+	0.999405i	$0.510980\pi$
$110$		0			0
$111$	−2.96897			−0.281802
$112$		0			0
$113$		−	6.73412i		−	0.633493i	−0.948510	−	0.316746i	$-0.897410\pi$
$113$		−	6.73412i		−	0.633493i	0.948510	−	0.316746i	$-0.102590\pi$
$114$		0			0
$115$	11.9046	−	0.699954i	1.11011	−	0.0652711i
$116$		0			0
$117$			2.13766i			0.197627i
$118$		0			0
$119$	40.3110			3.69531
$120$		0			0
$121$	−8.47356			−0.770324
$122$		0			0
$123$		−	10.9924i		−	0.991151i
$124$		0			0
$125$	1.95967	+	11.0073i	0.175278	+	0.984519i
$126$		0			0
$127$		−	3.46495i		−	0.307465i	−0.988113	−	0.153732i	$-0.950871\pi$
$127$		−	3.46495i		−	0.307465i	0.988113	−	0.153732i	$-0.0491293\pi$
$128$		0			0
$129$	3.88353			0.341926
$130$		0			0
$131$	−13.8300			−1.20834			−0.604168	−	0.796857i	$-0.706494\pi$
$131$	−13.8300			−1.20834			−0.604168	+	0.796857i	$0.706494\pi$
$132$		0			0
$133$			11.2527i			0.975733i
$134$		0			0
$135$	−2.23221	+	0.131247i	−0.192118	+	0.0112960i
$136$		0			0
$137$		−	8.75541i		−	0.748025i	−0.927424	−	0.374013i	$-0.877982\pi$
$137$		−	8.75541i		−	0.748025i	0.927424	−	0.374013i	$-0.122018\pi$
$138$		0			0
$139$	6.79685			0.576501			0.288251	−	0.957555i	$-0.406926\pi$
$139$	6.79685			0.576501			0.288251	+	0.957555i	$0.406926\pi$
$140$		0			0
$141$	6.39193			0.538297
$142$		0			0
$143$			3.39776i			0.284135i
$144$		0			0
$145$	−0.0218256	−	0.371202i	−0.00181251	−	0.0308266i
$146$		0			0
$147$		−	19.3942i		−	1.59961i
$148$		0			0
$149$	14.2220			1.16511			0.582554	−	0.812792i	$-0.302053\pi$
$149$	14.2220			1.16511			0.582554	+	0.812792i	$0.302053\pi$
$150$		0			0
$151$	6.96765			0.567019			0.283510	−	0.958969i	$-0.408501\pi$
$151$	6.96765			0.567019			0.283510	+	0.958969i	$0.408501\pi$
$152$		0			0
$153$			7.84639i			0.634343i
$154$		0			0
$155$	0.143537	+	2.44123i	0.0115292	+	0.196085i
$156$		0			0
$157$		−	23.5394i		−	1.87865i	−0.343027	−	0.939325i	$-0.611452\pi$
$157$		−	23.5394i		−	1.87865i	0.343027	−	0.939325i	$-0.388548\pi$
$158$		0			0
$159$	11.5030			0.912247
$160$		0			0
$161$	−27.3989			−2.15934
$162$		0			0
$163$			7.48265i			0.586086i	0.956099	+	0.293043i	$0.0946679\pi$
$163$			7.48265i			0.586086i	−0.956099	+	0.293043i	$0.905332\pi$
$164$		0			0
$165$	−3.54805	+	0.208615i	−0.276215	+	0.0162406i
$166$		0			0
$167$		−	9.48930i		−	0.734304i	−0.930161	−	0.367152i	$-0.880333\pi$
$167$		−	9.48930i		−	0.734304i	0.930161	−	0.367152i	$-0.119667\pi$
$168$		0			0
$169$	8.43041			0.648493
$170$		0			0
$171$	−2.19030			−0.167496
$172$		0			0
$173$		−	21.0437i		−	1.59992i	−0.600051	−	0.799962i	$-0.704853\pi$
$173$		−	21.0437i		−	1.59992i	0.600051	−	0.799962i	$-0.295147\pi$
$174$		0			0
$175$	−3.01031	−	25.5106i	−0.227558	−	1.92842i
$176$		0			0
$177$		−	7.22807i		−	0.543295i
$178$		0			0
$179$	5.55392			0.415120			0.207560	−	0.978222i	$-0.433448\pi$
$179$	5.55392			0.415120			0.207560	+	0.978222i	$0.433448\pi$
$180$		0			0
$181$	−0.126497			−0.00940243			−0.00470121	−	0.999989i	$-0.501496\pi$
$181$	−0.126497			−0.00940243			−0.00470121	+	0.999989i	$0.501496\pi$
$182$		0			0
$183$			9.90971i			0.732547i
$184$		0			0
$185$	6.62737	−	0.389669i	0.487254	−	0.0286491i
$186$		0			0
$187$			12.4717i			0.912018i
$188$		0			0
$189$	5.13753			0.373700
$190$		0			0
$191$	−12.8499			−0.929784			−0.464892	−	0.885367i	$-0.653907\pi$
$191$	−12.8499			−0.929784			−0.464892	+	0.885367i	$0.653907\pi$
$192$		0			0
$193$		−	22.9422i		−	1.65141i	−0.564100	−	0.825707i	$-0.690777\pi$
$193$		−	22.9422i		−	1.65141i	0.564100	−	0.825707i	$-0.309223\pi$
$194$		0			0
$195$	−0.280562	−	4.77171i	−0.0200915	−	0.341709i
$196$		0			0
$197$		−	7.57232i		−	0.539505i	−0.962930	−	0.269753i	$-0.913058\pi$
$197$		−	7.57232i		−	0.539505i	0.962930	−	0.269753i	$-0.0869419\pi$
$198$		0			0
$199$	−15.8138			−1.12101			−0.560505	−	0.828151i	$-0.689393\pi$
$199$	−15.8138			−1.12101			−0.560505	+	0.828151i	$0.689393\pi$
$200$		0			0
$201$	1.00000			0.0705346
$202$		0			0
$203$			0.854336i			0.0599627i
$204$		0			0
$205$	1.44272	+	24.5374i	0.100764	+	1.71376i
$206$		0			0
$207$		−	5.33309i		−	0.370676i
$208$		0			0
$209$	−3.48143			−0.240815
$210$		0			0
$211$	−17.8626			−1.22971			−0.614855	−	0.788640i	$-0.710786\pi$
$211$	−17.8626			−1.22971			−0.614855	+	0.788640i	$0.710786\pi$
$212$		0			0
$213$			11.2863i			0.773325i
$214$		0			0
$215$	−8.66888	+	0.509704i	−0.591212	+	0.0347615i
$216$		0			0
$217$		−	5.61860i		−	0.381415i
$218$		0			0
$219$	−6.60168			−0.446100
$220$		0			0
$221$	−16.7729			−1.12827
$222$		0			0
$223$		−	4.23944i		−	0.283894i	−0.989874	−	0.141947i	$-0.954664\pi$
$223$		−	4.23944i		−	0.283894i	0.989874	−	0.141947i	$-0.0453362\pi$
$224$		0			0
$225$	4.96555	−	0.585944i	0.331037	−	0.0390629i
$226$		0			0
$227$			9.50951i			0.631168i	0.948898	+	0.315584i	$0.102200\pi$
$227$			9.50951i			0.631168i	−0.948898	+	0.315584i	$0.897800\pi$
$228$		0			0
$229$	11.1809			0.738857			0.369429	−	0.929259i	$-0.379553\pi$
$229$	11.1809			0.738857			0.369429	+	0.929259i	$0.379553\pi$
$230$		0			0
$231$	8.16599			0.537282
$232$		0			0
$233$			1.05644i			0.0692096i	0.999401	+	0.0346048i	$0.0110173\pi$
$233$			1.05644i			0.0692096i	−0.999401	+	0.0346048i	$0.988983\pi$
$234$		0			0
$235$	−14.2681	+	0.838923i	−0.930751	+	0.0547253i
$236$		0			0
$237$		−	7.35082i		−	0.477487i
$238$		0			0
$239$	−6.35097			−0.410810			−0.205405	−	0.978677i	$-0.565851\pi$
$239$	−6.35097			−0.410810			−0.205405	+	0.978677i	$0.565851\pi$
$240$		0			0
$241$	−5.56297			−0.358342			−0.179171	−	0.983818i	$-0.557342\pi$
$241$	−5.56297			−0.358342			−0.179171	+	0.983818i	$0.557342\pi$
$242$		0			0
$243$			1.00000i			0.0641500i
$244$		0			0
$245$	2.54544	+	43.2920i	0.162622	+	2.76582i
$246$		0			0
$247$		−	4.68211i		−	0.297915i
$248$		0			0
$249$	7.97928			0.505666
$250$		0			0
$251$	6.13053			0.386955			0.193478	−	0.981105i	$-0.438023\pi$
$251$	6.13053			0.386955			0.193478	+	0.981105i	$0.438023\pi$
$252$		0			0
$253$		−	8.47683i		−	0.532934i
$254$		0			0
$255$	−1.02982	−	17.5148i	−0.0644897	−	1.09682i
$256$		0			0
$257$			21.0238i			1.31143i	0.755010	+	0.655713i	$0.227632\pi$
$257$			21.0238i			1.31143i	−0.755010	+	0.655713i	$0.772368\pi$
$258$		0			0
$259$	−15.2532			−0.947785
$260$		0			0
$261$	−0.166293			−0.0102933
$262$		0			0
$263$			27.2397i			1.67967i	0.542841	+	0.839836i	$0.317349\pi$
$263$			27.2397i			1.67967i	−0.542841	+	0.839836i	$0.682651\pi$
$264$		0			0
$265$	−25.6771	+	1.50974i	−1.57733	+	0.0927424i
$266$		0			0
$267$			10.6997i			0.654810i
$268$		0			0
$269$	−6.61800			−0.403507			−0.201753	−	0.979436i	$-0.564664\pi$
$269$	−6.61800			−0.403507			−0.201753	+	0.979436i	$0.564664\pi$
$270$		0			0
$271$	11.5286			0.700310			0.350155	−	0.936692i	$-0.386129\pi$
$271$	11.5286			0.700310			0.350155	+	0.936692i	$0.386129\pi$
$272$		0			0
$273$			10.9823i			0.664679i
$274$		0			0
$275$	7.89263	−	0.931345i	0.475943	−	0.0561622i
$276$		0			0
$277$		−	27.2244i		−	1.63576i	−0.575391	−	0.817879i	$-0.695150\pi$
$277$		−	27.2244i		−	1.63576i	0.575391	−	0.817879i	$-0.304850\pi$
$278$		0			0
$279$	1.09364			0.0654744
$280$		0			0
$281$	−13.9886			−0.834489			−0.417244	−	0.908794i	$-0.637004\pi$
$281$	−13.9886			−0.834489			−0.417244	+	0.908794i	$0.637004\pi$
$282$		0			0
$283$			21.1526i			1.25739i	0.777651	+	0.628696i	$0.216411\pi$
$283$			21.1526i			1.25739i	−0.777651	+	0.628696i	$0.783589\pi$
$284$		0			0
$285$	4.88921	−	0.287471i	0.289612	−	0.0170283i
$286$		0			0
$287$		−	56.4737i		−	3.33354i
$288$		0			0
$289$	−44.5658			−2.62152
$290$		0			0
$291$	−1.21273			−0.0710914
$292$		0			0
$293$		−	3.54350i		−	0.207013i	−0.994629	−	0.103507i	$-0.966994\pi$
$293$		−	3.54350i		−	0.207013i	0.994629	−	0.103507i	$-0.0330063\pi$
$294$		0			0
$295$	0.948665	+	16.1346i	0.0552334	+	0.939392i
$296$		0			0
$297$			1.58948i			0.0922308i
$298$		0			0
$299$	11.4003			0.659299
$300$		0			0
$301$	19.9518			1.15000
$302$		0			0
$303$			0.301193i			0.0173031i
$304$		0			0
$305$	−1.30062	−	22.1206i	−0.0744735	−	1.26662i
$306$		0			0
$307$			0.211948i			0.0120965i	0.999982	+	0.00604825i	$0.00192523\pi$
$307$			0.211948i			0.0120965i	−0.999982	+	0.00604825i	$0.998075\pi$
$308$		0			0
$309$	0.803026			0.0456826
$310$		0			0
$311$	−8.73795			−0.495483			−0.247742	−	0.968826i	$-0.579688\pi$
$311$	−8.73795			−0.495483			−0.247742	+	0.968826i	$0.579688\pi$
$312$		0			0
$313$		−	23.6370i		−	1.33604i	−0.744144	−	0.668020i	$-0.767142\pi$
$313$		−	23.6370i		−	1.33604i	0.744144	−	0.668020i	$-0.232858\pi$
$314$		0			0
$315$	−11.4681	+	0.674287i	−0.646152	+	0.0379918i
$316$		0			0
$317$			18.0553i			1.01409i	0.861921	+	0.507043i	$0.169262\pi$
$317$			18.0553i			1.01409i	−0.861921	+	0.507043i	$0.830738\pi$
$318$		0			0
$319$	−0.264319			−0.0147990
$320$		0			0
$321$	14.4533			0.806702
$322$		0			0
$323$		−	17.1859i		−	0.956249i
$324$		0			0
$325$	1.25255	+	10.6147i	0.0694789	+	0.588795i
$326$		0			0
$327$		−	0.720132i		−	0.0398234i
$328$		0			0
$329$	32.8387			1.81046
$330$		0			0
$331$	−18.9544			−1.04183			−0.520913	−	0.853610i	$-0.674408\pi$
$331$	−18.9544			−1.04183			−0.520913	+	0.853610i	$0.674408\pi$
$332$		0			0
$333$		−	2.96897i		−	0.162698i
$334$		0			0
$335$	−2.23221	+	0.131247i	−0.121959	+	0.00717081i
$336$		0			0
$337$		−	14.1789i		−	0.772375i	−0.922420	−	0.386187i	$-0.873792\pi$
$337$		−	14.1789i		−	0.772375i	0.922420	−	0.386187i	$-0.126208\pi$
$338$		0			0
$339$	6.73412			0.365747
$340$		0			0
$341$	1.73831			0.0941350
$342$		0			0
$343$		−	63.6756i		−	3.43816i
$344$		0			0
$345$	0.699954	+	11.9046i	0.0376843	+	0.640922i
$346$		0			0
$347$			7.86111i			0.422006i	0.977485	+	0.211003i	$0.0676730\pi$
$347$			7.86111i			0.422006i	−0.977485	+	0.211003i	$0.932327\pi$
$348$		0			0
$349$	13.9328			0.745808			0.372904	−	0.927870i	$-0.378362\pi$
$349$	13.9328			0.745808			0.372904	+	0.927870i	$0.378362\pi$
$350$		0			0
$351$	−2.13766			−0.114100
$352$		0			0
$353$		−	16.1430i		−	0.859206i	−0.903018	−	0.429603i	$-0.858654\pi$
$353$		−	16.1430i		−	0.859206i	0.903018	−	0.429603i	$-0.141346\pi$
$354$		0			0
$355$	−1.48130	−	25.1934i	−0.0786191	−	1.33713i
$356$		0			0
$357$			40.3110i			2.13349i
$358$		0			0
$359$	−2.11711			−0.111737			−0.0558684	−	0.998438i	$-0.517793\pi$
$359$	−2.11711			−0.111737			−0.0558684	+	0.998438i	$0.517793\pi$
$360$		0			0
$361$	−14.2026			−0.747506
$362$		0			0
$363$		−	8.47356i		−	0.444747i
$364$		0			0
$365$	14.7363	−	0.866453i	0.771336	−	0.0453522i
$366$		0			0
$367$		−	30.8434i		−	1.61001i	−0.593267	−	0.805006i	$-0.702162\pi$
$367$		−	30.8434i		−	1.61001i	0.593267	−	0.805006i	$-0.297838\pi$
$368$		0			0
$369$	10.9924			0.572241
$370$		0			0
$371$	59.0970			3.06816
$372$		0			0
$373$			2.51060i			0.129994i	0.997885	+	0.0649969i	$0.0207038\pi$
$373$			2.51060i			0.129994i	−0.997885	+	0.0649969i	$0.979296\pi$
$374$		0			0
$375$	−11.0073	+	1.95967i	−0.568412	+	0.101197i
$376$		0			0
$377$		−	0.355478i		−	0.0183081i
$378$		0			0
$379$	21.7842			1.11898			0.559489	−	0.828838i	$-0.310997\pi$
$379$	21.7842			1.11898			0.559489	+	0.828838i	$0.310997\pi$
$380$		0			0
$381$	3.46495			0.177515
$382$		0			0
$383$		−	6.84768i		−	0.349900i	−0.984577	−	0.174950i	$-0.944024\pi$
$383$		−	6.84768i		−	0.349900i	0.984577	−	0.174950i	$-0.0559764\pi$
$384$		0			0
$385$	−18.2282	+	1.07176i	−0.928996	+	0.0546222i
$386$		0			0
$387$			3.88353i			0.197411i
$388$		0			0
$389$	21.1047			1.07005			0.535025	−	0.844836i	$-0.320302\pi$
$389$	21.1047			1.07005			0.535025	+	0.844836i	$0.320302\pi$
$390$		0			0
$391$	41.8455			2.11622
$392$		0			0
$393$		−	13.8300i		−	0.697633i
$394$		0			0
$395$	0.964776	+	16.4086i	0.0485431	+	0.825606i
$396$		0			0
$397$			20.8970i			1.04879i	0.851475	+	0.524395i	$0.175708\pi$
$397$			20.8970i			1.04879i	−0.851475	+	0.524395i	$0.824292\pi$
$398$		0			0
$399$	−11.2527			−0.563340
$400$		0			0
$401$	−2.02781			−0.101264			−0.0506319	−	0.998717i	$-0.516124\pi$
$401$	−2.02781			−0.101264			−0.0506319	+	0.998717i	$0.516124\pi$
$402$		0			0
$403$			2.33783i			0.116456i
$404$		0			0
$405$	−0.131247	−	2.23221i	−0.00652173	−	0.110920i
$406$		0			0
$407$		−	4.71911i		−	0.233917i
$408$		0			0
$409$	28.2065			1.39472			0.697362	−	0.716720i	$-0.254357\pi$
$409$	28.2065			1.39472			0.697362	+	0.716720i	$0.254357\pi$
$410$		0			0
$411$	8.75541			0.431873
$412$		0			0
$413$		−	37.1344i		−	1.82726i
$414$		0			0
$415$	−17.8114	+	1.04726i	−0.874329	+	0.0514079i
$416$		0			0
$417$			6.79685i			0.332843i
$418$		0			0
$419$	29.1725			1.42517			0.712584	−	0.701587i	$-0.247525\pi$
$419$	29.1725			1.42517			0.712584	+	0.701587i	$0.247525\pi$
$420$		0			0
$421$	−35.4645			−1.72843			−0.864216	−	0.503121i	$-0.832185\pi$
$421$	−35.4645			−1.72843			−0.864216	+	0.503121i	$0.832185\pi$
$422$		0			0
$423$			6.39193i			0.310786i
$424$		0			0
$425$	4.59754	+	38.9616i	0.223014	+	1.88992i
$426$		0			0
$427$			50.9114i			2.46378i
$428$		0			0
$429$	−3.39776			−0.164046
$430$		0			0
$431$	−27.7109			−1.33479			−0.667394	−	0.744704i	$-0.732590\pi$
$431$	−27.7109			−1.33479			−0.667394	+	0.744704i	$0.732590\pi$
$432$		0			0
$433$		−	7.00944i		−	0.336852i	−0.985714	−	0.168426i	$-0.946132\pi$
$433$		−	7.00944i		−	0.336852i	0.985714	−	0.168426i	$-0.0538684\pi$
$434$		0			0
$435$	0.371202	−	0.0218256i	0.0177978	−	0.00104646i
$436$		0			0
$437$			11.6811i			0.558780i
$438$		0			0
$439$	7.31691			0.349217			0.174608	−	0.984638i	$-0.444134\pi$
$439$	7.31691			0.349217			0.174608	+	0.984638i	$0.444134\pi$
$440$		0			0
$441$	19.3942			0.923534
$442$		0			0
$443$			15.4169i			0.732480i	0.930520	+	0.366240i	$0.119355\pi$
$443$			15.4169i			0.732480i	−0.930520	+	0.366240i	$0.880645\pi$
$444$		0			0
$445$	−1.40430	−	23.8840i	−0.0665704	−	1.13221i
$446$		0			0
$447$			14.2220i			0.672676i
$448$		0			0
$449$	12.8020			0.604165			0.302082	−	0.953282i	$-0.402318\pi$
$449$	12.8020			0.604165			0.302082	+	0.953282i	$0.402318\pi$
$450$		0			0
$451$	17.4722			0.822732
$452$		0			0
$453$			6.96765i			0.327369i
$454$		0			0
$455$	−1.44140	−	24.5148i	−0.0675737	−	1.14927i
$456$		0			0
$457$		−	20.3179i		−	0.950431i	−0.879869	−	0.475215i	$-0.842370\pi$
$457$		−	20.3179i		−	0.950431i	0.879869	−	0.475215i	$-0.157630\pi$
$458$		0			0
$459$	−7.84639			−0.366238
$460$		0			0
$461$	0.656940			0.0305967			0.0152984	−	0.999883i	$-0.495130\pi$
$461$	0.656940			0.0305967			0.0152984	+	0.999883i	$0.495130\pi$
$462$		0			0
$463$			9.92569i			0.461286i	0.973038	+	0.230643i	$0.0740829\pi$
$463$			9.92569i			0.461286i	−0.973038	+	0.230643i	$0.925917\pi$
$464$		0			0
$465$	−2.44123	+	0.143537i	−0.113210	+	0.00665638i
$466$		0			0
$467$			10.7341i			0.496713i	0.968669	+	0.248357i	$0.0798905\pi$
$467$			10.7341i			0.496713i	−0.968669	+	0.248357i	$0.920110\pi$
$468$		0			0
$469$	5.13753			0.237229
$470$		0			0
$471$	23.5394			1.08464
$472$		0			0
$473$			6.17279i			0.283825i
$474$		0			0
$475$	−10.8760	+	1.28339i	−0.499026	+	0.0588860i
$476$		0			0
$477$			11.5030i			0.526686i
$478$		0			0
$479$	−18.2098			−0.832027			−0.416013	−	0.909358i	$-0.636573\pi$
$479$	−18.2098			−0.832027			−0.416013	+	0.909358i	$0.636573\pi$
$480$		0			0
$481$	6.34664			0.289382
$482$		0			0
$483$		−	27.3989i		−	1.24669i
$484$		0			0
$485$	2.70707	−	0.159168i	0.122922	−	0.00722742i
$486$		0			0
$487$		−	29.8666i		−	1.35338i	−0.736267	−	0.676692i	$-0.763413\pi$
$487$		−	29.8666i		−	1.35338i	0.736267	−	0.676692i	$-0.236587\pi$
$488$		0			0
$489$	−7.48265			−0.338377
$490$		0			0
$491$	−28.9497			−1.30648			−0.653242	−	0.757150i	$-0.726591\pi$
$491$	−28.9497			−1.30648			−0.653242	+	0.757150i	$0.726591\pi$
$492$		0			0
$493$		−	1.30480i		−	0.0587653i
$494$		0			0
$495$	−0.208615	−	3.54805i	−0.00937653	−	0.159473i
$496$		0			0
$497$			57.9837i			2.60092i
$498$		0			0
$499$	16.9517			0.758863			0.379432	−	0.925220i	$-0.376120\pi$
$499$	16.9517			0.758863			0.379432	+	0.925220i	$0.376120\pi$
$500$		0			0
$501$	9.48930			0.423951
$502$		0			0
$503$			29.1064i			1.29779i	0.760877	+	0.648896i	$0.224769\pi$
$503$			29.1064i			1.29779i	−0.760877	+	0.648896i	$0.775231\pi$
$504$		0			0
$505$	−0.0395308	−	0.672327i	−0.00175910	−	0.0299181i
$506$		0			0
$507$			8.43041i			0.374408i
$508$		0			0
$509$	5.75308			0.255001			0.127500	−	0.991839i	$-0.459305\pi$
$509$	5.75308			0.255001			0.127500	+	0.991839i	$0.459305\pi$
$510$		0			0
$511$	−33.9163			−1.50037
$512$		0			0
$513$		−	2.19030i		−	0.0967039i
$514$		0			0
$515$	−1.79253	+	0.105395i	−0.0789881	+	0.00464426i
$516$		0			0
$517$			10.1598i			0.446828i
$518$		0			0
$519$	21.0437			0.923716
$520$		0			0
$521$	−1.12822			−0.0494281			−0.0247140	−	0.999695i	$-0.507868\pi$
$521$	−1.12822			−0.0494281			−0.0247140	+	0.999695i	$0.507868\pi$
$522$		0			0
$523$			5.52130i			0.241429i	0.992687	+	0.120715i	$0.0385186\pi$
$523$			5.52130i			0.241429i	−0.992687	+	0.120715i	$0.961481\pi$
$524$		0			0
$525$	25.5106	−	3.01031i	1.11338	−	0.131380i
$526$		0			0
$527$			8.58111i			0.373799i
$528$		0			0
$529$	−5.44188			−0.236604
$530$		0			0
$531$	7.22807			0.313671
$532$		0			0
$533$			23.4980i			1.01781i
$534$		0			0
$535$	−32.2627	+	1.89695i	−1.39484	+	0.0820124i
$536$		0			0
$537$			5.55392i			0.239669i
$538$		0			0
$539$	30.8266			1.32780
$540$		0			0
$541$	−34.7228			−1.49285			−0.746424	−	0.665471i	$-0.768231\pi$
$541$	−34.7228			−1.49285			−0.746424	+	0.665471i	$0.768231\pi$
$542$		0			0
$543$		−	0.126497i		−	0.00542850i
$544$		0			0
$545$	0.0945154	+	1.60749i	0.00404860	+	0.0688572i
$546$		0			0
$547$			37.9636i			1.62321i	0.584209	+	0.811603i	$0.301405\pi$
$547$			37.9636i			1.62321i	−0.584209	+	0.811603i	$0.698595\pi$
$548$		0			0
$549$	−9.90971			−0.422936
$550$		0			0
$551$	0.364231			0.0155168
$552$		0			0
$553$		−	37.7651i		−	1.60593i
$554$		0			0
$555$	0.389669	+	6.62737i	0.0165405	+	0.281316i
$556$		0			0
$557$			16.1346i			0.683647i	0.939764	+	0.341823i	$0.111044\pi$
$557$			16.1346i			0.683647i	−0.939764	+	0.341823i	$0.888956\pi$
$558$		0			0
$559$	−8.30168			−0.351124
$560$		0			0
$561$	−12.4717			−0.526554
$562$		0			0
$563$			0.748928i			0.0315635i	0.999875	+	0.0157818i	$0.00502370\pi$
$563$			0.748928i			0.0315635i	−0.999875	+	0.0157818i	$0.994976\pi$
$564$		0			0
$565$	−15.0320	+	0.883836i	−0.632401	+	0.0371833i
$566$		0			0
$567$			5.13753i			0.215756i
$568$		0			0
$569$	16.2395			0.680797			0.340399	−	0.940281i	$-0.389438\pi$
$569$	16.2395			0.680797			0.340399	+	0.940281i	$0.389438\pi$
$570$		0			0
$571$	−36.9827			−1.54768			−0.773839	−	0.633382i	$-0.781666\pi$
$571$	−36.9827			−1.54768			−0.773839	+	0.633382i	$0.781666\pi$
$572$		0			0
$573$		−	12.8499i		−	0.536811i
$574$		0			0
$575$	−3.12489	−	26.4817i	−0.130317	−	1.10436i
$576$		0			0
$577$			37.4555i			1.55929i	0.626219	+	0.779647i	$0.284602\pi$
$577$			37.4555i			1.55929i	−0.626219	+	0.779647i	$0.715398\pi$
$578$		0			0
$579$	22.9422			0.953444
$580$		0			0
$581$	40.9938			1.70071
$582$		0			0
$583$			18.2837i			0.757235i
$584$		0			0
$585$	4.77171	−	0.280562i	0.197286	−	0.0115998i
$586$		0			0
$587$		−	39.3877i		−	1.62570i	−0.582471	−	0.812852i	$-0.697914\pi$
$587$		−	39.3877i		−	1.62570i	0.582471	−	0.812852i	$-0.302086\pi$
$588$		0			0
$589$	−2.39539			−0.0987004
$590$		0			0
$591$	7.57232			0.311483
$592$		0			0
$593$			13.4835i			0.553699i	0.960913	+	0.276849i	$0.0892903\pi$
$593$			13.4835i			0.553699i	−0.960913	+	0.276849i	$0.910710\pi$
$594$		0			0
$595$	−5.29072	−	89.9828i	−0.216898	−	3.68894i
$596$		0			0
$597$		−	15.8138i		−	0.647215i
$598$		0			0
$599$	−30.3879			−1.24161			−0.620807	−	0.783963i	$-0.713195\pi$
$599$	−30.3879			−1.24161			−0.620807	+	0.783963i	$0.713195\pi$
$600$		0			0
$601$	14.5247			0.592473			0.296237	−	0.955115i	$-0.404268\pi$
$601$	14.5247			0.592473			0.296237	+	0.955115i	$0.404268\pi$
$602$		0			0
$603$			1.00000i			0.0407231i
$604$		0			0
$605$	1.11213	+	18.9148i	0.0452146	+	0.768996i
$606$		0			0
$607$			38.1735i			1.54941i	0.632321	+	0.774707i	$0.282102\pi$
$607$			38.1735i			1.54941i	−0.632321	+	0.774707i	$0.717898\pi$
$608$		0			0
$609$	−0.854336			−0.0346195
$610$		0			0
$611$	−13.6638			−0.552777
$612$		0			0
$613$		−	22.0977i		−	0.892517i	−0.894904	−	0.446258i	$-0.852756\pi$
$613$		−	22.0977i		−	0.892517i	0.894904	−	0.446258i	$-0.147244\pi$
$614$		0			0
$615$	−24.5374	+	1.44272i	−0.989442	+	0.0581762i
$616$		0			0
$617$		−	26.6287i		−	1.07203i	−0.844208	−	0.536016i	$-0.819929\pi$
$617$		−	26.6287i		−	1.07203i	0.844208	−	0.536016i	$-0.180071\pi$
$618$		0			0
$619$	33.3643			1.34102			0.670512	−	0.741899i	$-0.266075\pi$
$619$	33.3643			1.34102			0.670512	+	0.741899i	$0.266075\pi$
$620$		0			0
$621$	5.33309			0.214010
$622$		0			0
$623$			54.9699i			2.20232i
$624$		0			0
$625$	24.3133	−	5.81907i	0.972534	−	0.232763i
$626$		0			0
$627$		−	3.48143i		−	0.139035i
$628$		0			0
$629$	23.2957			0.928859
$630$		0			0
$631$	−19.1516			−0.762412			−0.381206	−	0.924490i	$-0.624491\pi$
$631$	−19.1516			−0.762412			−0.381206	+	0.924490i	$0.624491\pi$
$632$		0			0
$633$		−	17.8626i		−	0.709974i
$634$		0			0
$635$	−7.73451	+	0.454766i	−0.306935	+	0.0180468i
$636$		0			0
$637$			41.4582i			1.64263i
$638$		0			0
$639$	−11.2863			−0.446479
$640$		0			0
$641$	38.5893			1.52419			0.762094	−	0.647467i	$-0.224172\pi$
$641$	38.5893			1.52419			0.762094	+	0.647467i	$0.224172\pi$
$642$		0			0
$643$		−	33.9692i		−	1.33961i	−0.742535	−	0.669807i	$-0.766377\pi$
$643$		−	33.9692i		−	1.33961i	0.742535	−	0.669807i	$-0.233623\pi$
$644$		0			0
$645$	−0.509704	−	8.66888i	−0.0200696	−	0.341337i
$646$		0			0
$647$			41.6986i			1.63934i	0.572835	+	0.819670i	$0.305843\pi$
$647$			41.6986i			1.63934i	−0.572835	+	0.819670i	$0.694157\pi$
$648$		0			0
$649$	11.4888			0.450977
$650$		0			0
$651$	5.61860			0.220210
$652$		0			0
$653$			46.2684i			1.81062i	0.424750	+	0.905311i	$0.360362\pi$
$653$			46.2684i			1.81062i	−0.424750	+	0.905311i	$0.639638\pi$
$654$		0			0
$655$	1.81515	+	30.8716i	0.0709240	+	1.20625i
$656$		0			0
$657$		−	6.60168i		−	0.257556i
$658$		0			0
$659$	−13.6714			−0.532561			−0.266280	−	0.963896i	$-0.585795\pi$
$659$	−13.6714			−0.532561			−0.266280	+	0.963896i	$0.585795\pi$
$660$		0			0
$661$	−31.9138			−1.24130			−0.620652	−	0.784086i	$-0.713132\pi$
$661$	−31.9138			−1.24130			−0.620652	+	0.784086i	$0.713132\pi$
$662$		0			0
$663$		−	16.7729i		−	0.651406i
$664$		0			0
$665$	25.1184	−	1.47689i	0.974051	−	0.0572713i
$666$		0			0
$667$			0.886857i			0.0343393i
$668$		0			0
$669$	4.23944			0.163906
$670$		0			0
$671$	−15.7513			−0.608071
$672$		0			0
$673$			36.0200i			1.38847i	0.719750	+	0.694234i	$0.244257\pi$
$673$			36.0200i			1.38847i	−0.719750	+	0.694234i	$0.755743\pi$
$674$		0			0
$675$	0.585944	+	4.96555i	0.0225530	+	0.191124i
$676$		0			0
$677$			39.4383i			1.51574i	0.652408	+	0.757868i	$0.273759\pi$
$677$			39.4383i			1.51574i	−0.652408	+	0.757868i	$0.726241\pi$
$678$		0			0
$679$	−6.23043			−0.239102
$680$		0			0
$681$	−9.50951			−0.364405
$682$		0			0
$683$		−	1.89018i		−	0.0723258i	−0.999346	−	0.0361629i	$-0.988486\pi$
$683$		−	1.89018i		−	0.0723258i	0.999346	−	0.0361629i	$-0.0115135\pi$
$684$		0			0
$685$	−19.5439	+	1.14912i	−0.746736	+	0.0439058i
$686$		0			0
$687$			11.1809i			0.426579i
$688$		0			0
$689$	−24.5895			−0.936785
$690$		0			0
$691$	42.5705			1.61946			0.809730	−	0.586803i	$-0.199614\pi$
$691$	42.5705			1.61946			0.809730	+	0.586803i	$0.199614\pi$
$692$		0			0
$693$			8.16599i			0.310200i
$694$		0			0
$695$	−0.892068	−	15.1720i	−0.0338381	−	0.575507i
$696$		0			0
$697$			86.2506i			3.26697i
$698$		0			0
$699$	−1.05644			−0.0399582
$700$		0			0
$701$	−30.6231			−1.15662			−0.578309	−	0.815818i	$-0.696287\pi$
$701$	−30.6231			−1.15662			−0.578309	+	0.815818i	$0.696287\pi$
$702$		0			0
$703$			6.50292i			0.245262i
$704$		0			0
$705$	−0.838923	−	14.2681i	−0.0315957	−	0.537369i
$706$		0			0
$707$			1.54739i			0.0581955i
$708$		0			0
$709$	9.00019			0.338009			0.169005	−	0.985615i	$-0.445945\pi$
$709$	9.00019			0.338009			0.169005	+	0.985615i	$0.445945\pi$
$710$		0			0
$711$	7.35082			0.275677
$712$		0			0
$713$		−	5.83248i		−	0.218428i
$714$		0			0
$715$	7.58453	−	0.445947i	0.283645	−	0.0166775i
$716$		0			0
$717$		−	6.35097i		−	0.237181i
$718$		0			0
$719$	−38.6949			−1.44308			−0.721538	−	0.692374i	$-0.756565\pi$
$719$	−38.6949			−1.44308			−0.721538	+	0.692374i	$0.756565\pi$
$720$		0			0
$721$	4.12557			0.153644
$722$		0			0
$723$		−	5.56297i		−	0.206889i
$724$		0			0
$725$	−0.825737	+	0.0974386i	−0.0306671	+	0.00361878i
$726$		0			0
$727$		−	31.7793i		−	1.17863i	−0.807903	−	0.589315i	$-0.799398\pi$
$727$		−	31.7793i		−	1.17863i	0.807903	−	0.589315i	$-0.200602\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	−30.4717			−1.12704
$732$		0			0
$733$		−	0.0764341i		−	0.00282316i	−0.999999	−	0.00141158i	$-0.999551\pi$
$733$		−	0.0764341i		−	0.00282316i	0.999999	−	0.00141158i	$-0.000449320\pi$
$734$		0			0
$735$	−43.2920	+	2.54544i	−1.59685	+	0.0938899i
$736$		0			0
$737$			1.58948i			0.0585492i
$738$		0			0
$739$	−4.44098			−0.163364			−0.0816820	−	0.996658i	$-0.526029\pi$
$739$	−4.44098			−0.163364			−0.0816820	+	0.996658i	$0.526029\pi$
$740$		0			0
$741$	4.68211			0.172002
$742$		0			0
$743$		−	49.5586i		−	1.81813i	−0.416656	−	0.909064i	$-0.636798\pi$
$743$		−	49.5586i		−	1.81813i	0.416656	−	0.909064i	$-0.363202\pi$
$744$		0			0
$745$	−1.86660	−	31.7464i	−0.0683868	−	1.16310i
$746$		0			0
$747$			7.97928i			0.291946i
$748$		0			0
$749$	74.2540			2.71318
$750$		0			0
$751$	−26.4764			−0.966139			−0.483069	−	0.875582i	$-0.660478\pi$
$751$	−26.4764			−0.966139			−0.483069	+	0.875582i	$0.660478\pi$
$752$		0			0
$753$			6.13053i			0.223409i
$754$		0			0
$755$	−0.914486	−	15.5533i	−0.0332815	−	0.566042i
$756$		0			0
$757$			8.41467i			0.305836i	0.988239	+	0.152918i	$0.0488671\pi$
$757$			8.41467i			0.305836i	−0.988239	+	0.152918i	$0.951133\pi$
$758$		0			0
$759$	8.47683			0.307689
$760$		0			0
$761$	−34.9896			−1.26837			−0.634186	−	0.773180i	$-0.718665\pi$
$761$	−34.9896			−1.26837			−0.634186	+	0.773180i	$0.718665\pi$
$762$		0			0
$763$		−	3.69970i		−	0.133938i
$764$		0			0
$765$	17.5148	−	1.02982i	0.633249	−	0.0372331i
$766$		0			0
$767$			15.4511i			0.557909i
$768$		0			0
$769$	20.6147			0.743386			0.371693	−	0.928356i	$-0.378777\pi$
$769$	20.6147			0.743386			0.371693	+	0.928356i	$0.378777\pi$
$770$		0			0
$771$	−21.0238			−0.757152
$772$		0			0
$773$		−	4.23263i		−	0.152237i	−0.997099	−	0.0761185i	$-0.975747\pi$
$773$		−	4.23263i		−	0.152237i	0.997099	−	0.0761185i	$-0.0242527\pi$
$774$		0			0
$775$	5.43052	−	0.640811i	0.195070	−	0.0230186i
$776$		0			0
$777$		−	15.2532i		−	0.547204i
$778$		0			0
$779$	−24.0766			−0.862633
$780$		0			0
$781$	−17.9393			−0.641919
$782$		0			0
$783$		−	0.166293i		−	0.00594284i
$784$		0			0
$785$	−52.5450	+	3.08949i	−1.87541	+	0.110269i
$786$		0			0
$787$			16.1107i			0.574283i	0.957888	+	0.287142i	$0.0927051\pi$
$787$			16.1107i			0.574283i	−0.957888	+	0.287142i	$0.907295\pi$
$788$		0			0
$789$	−27.2397			−0.969759
$790$		0			0
$791$	34.5967			1.23012
$792$		0			0
$793$		−	21.1836i		−	0.752251i
$794$		0			0
$795$	−1.50974	−	25.6771i	−0.0535449	−	0.910674i
$796$		0			0
$797$		−	35.9384i		−	1.27300i	−0.771276	−	0.636501i	$-0.780381\pi$
$797$		−	35.9384i		−	1.27300i	0.771276	−	0.636501i	$-0.219619\pi$
$798$		0			0
$799$	−50.1535			−1.77430
$800$		0			0
$801$	−10.6997			−0.378055
$802$		0			0
$803$		−	10.4932i		−	0.370297i
$804$		0			0
$805$	3.59604	+	61.1602i	0.126744	+	2.15561i
$806$		0			0
$807$		−	6.61800i		−	0.232965i
$808$		0			0
$809$	−2.60283			−0.0915106			−0.0457553	−	0.998953i	$-0.514569\pi$
$809$	−2.60283			−0.0915106			−0.0457553	+	0.998953i	$0.514569\pi$
$810$		0			0
$811$	12.1447			0.426458			0.213229	−	0.977002i	$-0.431602\pi$
$811$	12.1447			0.426458			0.213229	+	0.977002i	$0.431602\pi$
$812$		0			0
$813$			11.5286i			0.404324i
$814$		0			0
$815$	16.7029	−	0.982078i	0.585076	−	0.0344007i
$816$		0			0
$817$		−	8.50609i		−	0.297590i
$818$		0			0
$819$	−10.9823			−0.383752
$820$		0			0
$821$	−27.4149			−0.956786			−0.478393	−	0.878146i	$-0.658781\pi$
$821$	−27.4149			−0.956786			−0.478393	+	0.878146i	$0.658781\pi$
$822$		0			0
$823$			34.8864i			1.21606i	0.793913	+	0.608031i	$0.208041\pi$
$823$			34.8864i			1.21606i	−0.793913	+	0.608031i	$0.791959\pi$
$824$		0			0
$825$	0.931345	+	7.89263i	0.0324253	+	0.274786i
$826$		0			0
$827$		−	37.7690i		−	1.31336i	−0.754171	−	0.656678i	$-0.771961\pi$
$827$		−	37.7690i		−	1.31336i	0.754171	−	0.656678i	$-0.228039\pi$
$828$		0			0
$829$	−36.2440			−1.25881			−0.629404	−	0.777079i	$-0.716701\pi$
$829$	−36.2440			−1.25881			−0.629404	+	0.777079i	$0.716701\pi$
$830$		0			0
$831$	27.2244			0.944405
$832$		0			0
$833$			152.174i			5.27253i
$834$		0			0
$835$	−21.1821	+	1.24545i	−0.733038	+	0.0431004i
$836$		0			0
$837$			1.09364i			0.0378017i
$838$		0			0
$839$	−25.2110			−0.870380			−0.435190	−	0.900339i	$-0.643319\pi$
$839$	−25.2110			−0.870380			−0.435190	+	0.900339i	$0.643319\pi$
$840$		0			0
$841$	−28.9723			−0.999046
$842$		0			0
$843$		−	13.9886i		−	0.481792i
$844$		0			0
$845$	−1.10647	−	18.8185i	−0.0380637	−	0.647375i
$846$		0			0
$847$		−	43.5332i		−	1.49582i
$848$		0			0
$849$	−21.1526			−0.725955
$850$		0			0
$851$	−15.8338			−0.542775
$852$		0			0
$853$			41.3678i			1.41641i	0.706009	+	0.708203i	$0.250494\pi$
$853$			41.3678i			1.41641i	−0.706009	+	0.708203i	$0.749506\pi$
$854$		0			0
$855$	0.287471	+	4.88921i	0.00983129	+	0.167207i
$856$		0			0
$857$			40.5195i			1.38412i	0.721841	+	0.692059i	$0.243296\pi$
$857$			40.5195i			1.38412i	−0.721841	+	0.692059i	$0.756704\pi$
$858$		0			0
$859$	−26.7240			−0.911810			−0.455905	−	0.890028i	$-0.650684\pi$
$859$	−26.7240			−0.911810			−0.455905	+	0.890028i	$0.650684\pi$
$860$		0			0
$861$	56.4737			1.92462
$862$		0			0
$863$		−	34.0616i		−	1.15947i	−0.814805	−	0.579735i	$-0.803156\pi$
$863$		−	34.0616i		−	1.15947i	0.814805	−	0.579735i	$-0.196844\pi$
$864$		0			0
$865$	−46.9740	+	2.76193i	−1.59717	+	0.0939085i
$866$		0			0
$867$		−	44.5658i		−	1.51353i
$868$		0			0
$869$	11.6840			0.396351
$870$		0			0
$871$	−2.13766			−0.0724319
$872$		0			0
$873$		−	1.21273i		−	0.0410447i
$874$		0			0
$875$	−56.5501	+	10.0678i	−1.91174	+	0.340355i
$876$		0			0
$877$			29.2015i			0.986065i	0.870011	+	0.493033i	$0.164112\pi$
$877$			29.2015i			0.986065i	−0.870011	+	0.493033i	$0.835888\pi$
$878$		0			0
$879$	3.54350			0.119519
$880$		0			0
$881$	51.2971			1.72824			0.864121	−	0.503284i	$-0.167875\pi$
$881$	51.2971			1.72824			0.864121	+	0.503284i	$0.167875\pi$
$882$		0			0
$883$			25.5324i			0.859234i	0.903011	+	0.429617i	$0.141351\pi$
$883$			25.5324i			0.859234i	−0.903011	+	0.429617i	$0.858649\pi$
$884$		0			0
$885$	−16.1346	+	0.948665i	−0.542358	+	0.0318890i
$886$		0			0
$887$			6.53683i			0.219485i	0.993960	+	0.109743i	$0.0350027\pi$
$887$			6.53683i			0.219485i	−0.993960	+	0.109743i	$0.964997\pi$
$888$		0			0
$889$	17.8013			0.597036
$890$		0			0
$891$	−1.58948			−0.0532495
$892$		0			0
$893$		−	14.0002i		−	0.468499i
$894$		0			0
$895$	−0.728937	−	12.3975i	−0.0243657	−	0.414404i
$896$		0			0
$897$			11.4003i			0.380646i
$898$		0			0
$899$	−0.181865			−0.00606553
$900$		0			0
$901$	−90.2569			−3.00689
$902$		0			0
$903$			19.9518i			0.663953i
$904$		0			0
$905$	0.0166024	+	0.282368i	0.000551881	+	0.00938622i
$906$		0			0
$907$			28.2281i			0.937299i	0.883384	+	0.468650i	$0.155259\pi$
$907$			28.2281i			0.937299i	−0.883384	+	0.468650i	$0.844741\pi$
$908$		0			0
$909$	−0.301193			−0.00998994
$910$		0			0
$911$	35.2819			1.16894			0.584470	−	0.811415i	$-0.301302\pi$
$911$	35.2819			1.16894			0.584470	+	0.811415i	$0.301302\pi$
$912$		0			0
$913$			12.6829i			0.419742i
$914$		0			0
$915$	22.1206	−	1.30062i	0.731284	−	0.0429973i
$916$		0			0
$917$		−	71.0522i		−	2.34635i
$918$		0			0
$919$	−7.62615			−0.251563			−0.125782	−	0.992058i	$-0.540144\pi$
$919$	−7.62615			−0.251563			−0.125782	+	0.992058i	$0.540144\pi$
$920$		0			0
$921$	−0.211948			−0.00698392
$922$		0			0
$923$		−	24.1263i		−	0.794126i
$924$		0			0
$925$	−1.73965	−	14.7426i	−0.0571993	−	0.484732i
$926$		0			0
$927$			0.803026i			0.0263748i
$928$		0			0
$929$	46.8223			1.53619			0.768095	−	0.640336i	$-0.221205\pi$
$929$	46.8223			1.53619			0.768095	+	0.640336i	$0.221205\pi$
$930$		0			0
$931$	−42.4790			−1.39219
$932$		0			0
$933$		−	8.73795i		−	0.286068i
$934$		0			0
$935$	27.8394	−	1.63687i	0.910445	−	0.0535314i
$936$		0			0
$937$			58.9956i			1.92730i	0.267161	+	0.963652i	$0.413914\pi$
$937$			58.9956i			1.92730i	−0.267161	+	0.963652i	$0.586086\pi$
$938$		0			0
$939$	23.6370			0.771363
$940$		0			0
$941$	−20.4639			−0.667104			−0.333552	−	0.942732i	$-0.608247\pi$
$941$	−20.4639			−0.667104			−0.333552	+	0.942732i	$0.608247\pi$
$942$		0			0
$943$		−	58.6235i		−	1.90904i
$944$		0			0
$945$	−0.674287	−	11.4681i	−0.0219346	−	0.373056i
$946$		0			0
$947$			55.8815i			1.81590i	0.419075	+	0.907952i	$0.362355\pi$
$947$			55.8815i			1.81590i	−0.419075	+	0.907952i	$0.637645\pi$
$948$		0			0
$949$	14.1121			0.458099
$950$		0			0
$951$	−18.0553			−0.585483
$952$		0			0
$953$		−	5.68604i		−	0.184189i	−0.995750	−	0.0920945i	$-0.970644\pi$
$953$		−	5.68604i		−	0.184189i	0.995750	−	0.0920945i	$-0.0293562\pi$
$954$		0			0
$955$	1.68651	+	28.6837i	0.0545743	+	0.928181i
$956$		0			0
$957$		−	0.264319i		−	0.00854423i
$958$		0			0
$959$	44.9812			1.45252
$960$		0			0
$961$	−29.8040			−0.961418
$962$		0			0
$963$			14.4533i			0.465750i
$964$		0			0
$965$	−51.2118	+	3.01110i	−1.64857	+	0.0969307i
$966$		0			0
$967$			3.84847i			0.123759i	0.998084	+	0.0618793i	$0.0197094\pi$
$967$			3.84847i			0.123759i	−0.998084	+	0.0618793i	$0.980291\pi$
$968$		0			0
$969$	17.1859			0.552091
$970$		0			0
$971$	−40.5115			−1.30008			−0.650039	−	0.759901i	$-0.725247\pi$
$971$	−40.5115			−1.30008			−0.650039	+	0.759901i	$0.725247\pi$
$972$		0			0
$973$			34.9190i			1.11945i
$974$		0			0
$975$	−10.6147	+	1.25255i	−0.339941	+	0.0401137i
$976$		0			0
$977$		−	39.9866i		−	1.27928i	−0.768673	−	0.639642i	$-0.779083\pi$
$977$		−	39.9866i		−	1.27928i	0.768673	−	0.639642i	$-0.220917\pi$
$978$		0			0
$979$	−17.0069			−0.543543
$980$		0			0
$981$	0.720132			0.0229920
$982$		0			0
$983$			12.6615i			0.403840i	0.979402	+	0.201920i	$0.0647181\pi$
$983$			12.6615i			0.403840i	−0.979402	+	0.201920i	$0.935282\pi$
$984$		0			0
$985$	−16.9030	+	0.993847i	−0.538575	+	0.0316666i
$986$		0			0
$987$			32.8387i			1.04527i
$988$		0			0
$989$	20.7113			0.658580
$990$		0			0
$991$	−1.35522			−0.0430501			−0.0215250	−	0.999768i	$-0.506852\pi$
$991$	−1.35522			−0.0430501			−0.0215250	+	0.999768i	$0.506852\pi$
$992$		0			0
$993$		−	18.9544i		−	0.601498i
$994$		0			0
$995$	2.07552	+	35.2997i	0.0657983	+	1.11908i
$996$		0			0
$997$		−	8.85396i		−	0.280408i	−0.990123	−	0.140204i	$-0.955224\pi$
$997$		−	8.85396i		−	0.280408i	0.990123	−	0.140204i	$-0.0447758\pi$
$998$		0			0
$999$	2.96897			0.0939340

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4020.2.g.b.1609.19	yes	24
5.4	even	2	inner	4020.2.g.b.1609.7	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4020.2.g.b.1609.7	✓	24	5.4	even	2	inner
4020.2.g.b.1609.19	yes	24	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +(-0.131247 - 2.23221i) q^{5} +5.13753i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +(-0.131247 - 2.23221i) q^{5} +5.13753i q^{7} -1.00000 q^{9} -1.58948 q^{11} -2.13766i q^{13} +(2.23221 - 0.131247i) q^{15} -7.84639i q^{17} +2.19030 q^{19} -5.13753 q^{21} +5.33309i q^{23} +(-4.96555 + 0.585944i) q^{25} -1.00000i q^{27} +0.166293 q^{29} -1.09364 q^{31} -1.58948i q^{33} +(11.4681 - 0.674287i) q^{35} +2.96897i q^{37} +2.13766 q^{39} -10.9924 q^{41} -3.88353i q^{43} +(0.131247 + 2.23221i) q^{45} -6.39193i q^{47} -19.3942 q^{49} +7.84639 q^{51} -11.5030i q^{53} +(0.208615 + 3.54805i) q^{55} +2.19030i q^{57} -7.22807 q^{59} +9.90971 q^{61} -5.13753i q^{63} +(-4.77171 + 0.280562i) q^{65} -1.00000i q^{67} -5.33309 q^{69} +11.2863 q^{71} +6.60168i q^{73} +(-0.585944 - 4.96555i) q^{75} -8.16599i q^{77} -7.35082 q^{79} +1.00000 q^{81} -7.97928i q^{83} +(-17.5148 + 1.02982i) q^{85} +0.166293i q^{87} +10.6997 q^{89} +10.9823 q^{91} -1.09364i q^{93} +(-0.287471 - 4.88921i) q^{95} +1.21273i q^{97} +1.58948 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 4 q^{5} - 24 q^{9}+O(q^{10}) \) 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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