LMFDB - Embedded newform 4020.2.f.a.401.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4020,2,Mod(401,4020)]

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, 1, 0, 1]))

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

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

chi := DirichletCharacter("4020.401");

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.f (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:	$46$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			401.17
Character	$\chi$	$=$	4020.401
Dual form			4020.2.f.a.401.18

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.574646 - 1.63395i) q^{3} -1.00000 q^{5} -4.28520i q^{7} +(-2.33956 + 1.87788i) q^{9} +O(q^{10})$	\(q+(-0.574646 - 1.63395i) q^{3} -1.00000 q^{5} -4.28520i q^{7} +(-2.33956 + 1.87788i) q^{9} +1.89841 q^{11} +0.406163i q^{13} +(0.574646 + 1.63395i) q^{15} +4.17771i q^{17} -5.36381 q^{19} +(-7.00179 + 2.46248i) q^{21} +6.34204i q^{23} +1.00000 q^{25} +(4.41278 + 2.74360i) q^{27} -3.90840i q^{29} +3.07537i q^{31} +(-1.09091 - 3.10190i) q^{33} +4.28520i q^{35} +7.35727 q^{37} +(0.663648 - 0.233400i) q^{39} +6.75844 q^{41} +7.33793i q^{43} +(2.33956 - 1.87788i) q^{45} +0.438204i q^{47} -11.3630 q^{49} +(6.82616 - 2.40071i) q^{51} -5.23027 q^{53} -1.89841 q^{55} +(3.08230 + 8.76419i) q^{57} +12.3982i q^{59} +7.71806i q^{61} +(8.04711 + 10.0255i) q^{63} -0.406163i q^{65} +(-1.79086 + 7.98704i) q^{67} +(10.3626 - 3.64443i) q^{69} +2.28010i q^{71} +0.538920 q^{73} +(-0.574646 - 1.63395i) q^{75} -8.13507i q^{77} -1.67546i q^{79} +(1.94711 - 8.78685i) q^{81} -3.18158i q^{83} -4.17771i q^{85} +(-6.38612 + 2.24595i) q^{87} -10.7644i q^{89} +1.74049 q^{91} +(5.02499 - 1.76725i) q^{93} +5.36381 q^{95} -7.69321i q^{97} +(-4.44145 + 3.56499i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 46 q - 46 q^{5} - 4 q^{9}+O(q^{10}) $ 46 * q - 46 * q^5 - 4 * q^9	$ 46 q - 46 q^{5} - 4 q^{9} + 8 q^{19} + 46 q^{25} - 18 q^{27} + 4 q^{33} - 8 q^{37} - 12 q^{39} - 4 q^{41} + 4 q^{45} - 62 q^{49} - 8 q^{51} - 8 q^{53} - 12 q^{57} - 10 q^{63} - 14 q^{67} - 40 q^{73} - 12 q^{81} - 4 q^{91} + 2 q^{93} - 8 q^{95}+O(q^{100}) $ 46 * q - 46 * q^5 - 4 * q^9 + 8 * q^19 + 46 * q^25 - 18 * q^27 + 4 * q^33 - 8 * q^37 - 12 * q^39 - 4 * q^41 + 4 * q^45 - 62 * q^49 - 8 * q^51 - 8 * q^53 - 12 * q^57 - 10 * q^63 - 14 * q^67 - 40 * q^73 - 12 * q^81 - 4 * q^91 + 2 * q^93 - 8 * q^95

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
$3$	−0.574646	−	1.63395i	−0.331772	−	0.943360i
$3$	−0.574646	−	1.63395i	−0.331772	−	0.943360i
$4$		0			0
$5$	−1.00000			−0.447214
$5$	−1.00000			−0.447214
$6$		0			0
$7$		−	4.28520i		−	1.61965i	−0.586669	−	0.809827i	$-0.699561\pi$
$7$		−	4.28520i		−	1.61965i	0.586669	−	0.809827i	$-0.300439\pi$
$8$		0			0
$9$	−2.33956	+	1.87788i	−0.779854	+	0.625961i
$10$		0			0
$11$	1.89841			0.572392			0.286196	−	0.958171i	$-0.407609\pi$
$11$	1.89841			0.572392			0.286196	+	0.958171i	$0.407609\pi$
$12$		0			0
$13$			0.406163i			0.112649i	0.998413	+	0.0563246i	$0.0179382\pi$
$13$			0.406163i			0.112649i	−0.998413	+	0.0563246i	$0.982062\pi$
$14$		0			0
$15$	0.574646	+	1.63395i	0.148373	+	0.421883i
$16$		0			0
$17$			4.17771i			1.01324i	0.862168	+	0.506622i	$0.169106\pi$
$17$			4.17771i			1.01324i	−0.862168	+	0.506622i	$0.830894\pi$
$18$		0			0
$19$	−5.36381			−1.23054			−0.615272	−	0.788315i	$-0.710954\pi$
$19$	−5.36381			−1.23054			−0.615272	+	0.788315i	$0.710954\pi$
$20$		0			0
$21$	−7.00179	+	2.46248i	−1.52792	+	0.537356i
$22$		0			0
$23$			6.34204i			1.32241i	0.750207	+	0.661203i	$0.229954\pi$
$23$			6.34204i			1.32241i	−0.750207	+	0.661203i	$0.770046\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	4.41278	+	2.74360i	0.849240	+	0.528006i
$28$		0			0
$29$		−	3.90840i		−	0.725772i	−0.931834	−	0.362886i	$-0.881791\pi$
$29$		−	3.90840i		−	0.725772i	0.931834	−	0.362886i	$-0.118209\pi$
$30$		0			0
$31$			3.07537i			0.552353i	0.961107	+	0.276176i	$0.0890674\pi$
$31$			3.07537i			0.552353i	−0.961107	+	0.276176i	$0.910933\pi$
$32$		0			0
$33$	−1.09091	−	3.10190i	−0.189904	−	0.539971i
$34$		0			0
$35$			4.28520i			0.724331i
$36$		0			0
$37$	7.35727			1.20953			0.604764	−	0.796405i	$-0.293268\pi$
$37$	7.35727			1.20953			0.604764	+	0.796405i	$0.293268\pi$
$38$		0			0
$39$	0.663648	−	0.233400i	0.106269	−	0.0373739i
$40$		0			0
$41$	6.75844			1.05549			0.527745	−	0.849403i	$-0.323037\pi$
$41$	6.75844			1.05549			0.527745	+	0.849403i	$0.323037\pi$
$42$		0			0
$43$			7.33793i			1.11902i	0.828822	+	0.559512i	$0.189012\pi$
$43$			7.33793i			1.11902i	−0.828822	+	0.559512i	$0.810988\pi$
$44$		0			0
$45$	2.33956	−	1.87788i	0.348761	−	0.279938i
$46$		0			0
$47$			0.438204i			0.0639187i	0.999489	+	0.0319593i	$0.0101747\pi$
$47$			0.438204i			0.0639187i	−0.999489	+	0.0319593i	$0.989825\pi$
$48$		0			0
$49$	−11.3630			−1.62328
$50$		0			0
$51$	6.82616	−	2.40071i	0.955853	−	0.336166i
$52$		0			0
$53$	−5.23027			−0.718432			−0.359216	−	0.933254i	$-0.616956\pi$
$53$	−5.23027			−0.718432			−0.359216	+	0.933254i	$0.616956\pi$
$54$		0			0
$55$	−1.89841			−0.255981
$56$		0			0
$57$	3.08230	+	8.76419i	0.408260	+	1.16084i
$58$		0			0
$59$			12.3982i			1.61410i	0.590480	+	0.807052i	$0.298938\pi$
$59$			12.3982i			1.61410i	−0.590480	+	0.807052i	$0.701062\pi$
$60$		0			0
$61$			7.71806i			0.988196i	0.869406	+	0.494098i	$0.164502\pi$
$61$			7.71806i			0.988196i	−0.869406	+	0.494098i	$0.835498\pi$
$62$		0			0
$63$	8.04711	+	10.0255i	1.01384	+	1.26309i
$64$		0			0
$65$		−	0.406163i		−	0.0503783i
$66$		0			0
$67$	−1.79086	+	7.98704i	−0.218789	+	0.975772i
$67$	−1.79086	+	7.98704i	−0.218789	+	0.975772i
$68$		0			0
$69$	10.3626	−	3.64443i	1.24750	−	0.438738i
$70$		0			0
$71$			2.28010i			0.270598i	0.990805	+	0.135299i	$0.0431995\pi$
$71$			2.28010i			0.270598i	−0.990805	+	0.135299i	$0.956801\pi$
$72$		0			0
$73$	0.538920			0.0630758			0.0315379	−	0.999503i	$-0.489960\pi$
$73$	0.538920			0.0630758			0.0315379	+	0.999503i	$0.489960\pi$
$74$		0			0
$75$	−0.574646	−	1.63395i	−0.0663545	−	0.188672i
$76$		0			0
$77$		−	8.13507i		−	0.927077i
$78$		0			0
$79$		−	1.67546i		−	0.188504i	−0.995548	−	0.0942521i	$-0.969954\pi$
$79$		−	1.67546i		−	0.188504i	0.995548	−	0.0942521i	$-0.0300460\pi$
$80$		0			0
$81$	1.94711	−	8.78685i	0.216346	−	0.976317i
$82$		0			0
$83$		−	3.18158i		−	0.349224i	−0.984637	−	0.174612i	$-0.944133\pi$
$83$		−	3.18158i		−	0.349224i	0.984637	−	0.174612i	$-0.0558671\pi$
$84$		0			0
$85$		−	4.17771i		−	0.453137i
$86$		0			0
$87$	−6.38612	+	2.24595i	−0.684664	+	0.240791i
$88$		0			0
$89$		−	10.7644i		−	1.14102i	−0.821290	−	0.570511i	$-0.806745\pi$
$89$		−	10.7644i		−	1.14102i	0.821290	−	0.570511i	$-0.193255\pi$
$90$		0			0
$91$	1.74049			0.182453
$92$		0			0
$93$	5.02499	−	1.76725i	0.521067	−	0.183255i
$94$		0			0
$95$	5.36381			0.550316
$96$		0			0
$97$		−	7.69321i		−	0.781127i	−0.920576	−	0.390563i	$-0.872280\pi$
$97$		−	7.69321i		−	0.781127i	0.920576	−	0.390563i	$-0.127720\pi$
$98$		0			0
$99$	−4.44145	+	3.56499i	−0.446382	+	0.358295i
$100$		0			0
$101$	−3.45264			−0.343551			−0.171775	−	0.985136i	$-0.554950\pi$
$101$	−3.45264			−0.343551			−0.171775	+	0.985136i	$0.554950\pi$
$102$		0			0
$103$	−8.82052			−0.869112			−0.434556	−	0.900645i	$-0.643095\pi$
$103$	−8.82052			−0.869112			−0.434556	+	0.900645i	$0.643095\pi$
$104$		0			0
$105$	7.00179	−	2.46248i	0.683305	−	0.240313i
$106$		0			0
$107$			12.6047i			1.21854i	0.792964	+	0.609269i	$0.208537\pi$
$107$			12.6047i			1.21854i	−0.792964	+	0.609269i	$0.791463\pi$
$108$		0			0
$109$		−	15.8640i		−	1.51949i	−0.650218	−	0.759747i	$-0.725323\pi$
$109$		−	15.8640i		−	1.51949i	0.650218	−	0.759747i	$-0.274677\pi$
$110$		0			0
$111$	−4.22783	−	12.0214i	−0.401288	−	1.14102i
$112$		0			0
$113$	−2.38152			−0.224034			−0.112017	−	0.993706i	$-0.535731\pi$
$113$	−2.38152			−0.224034			−0.112017	+	0.993706i	$0.535731\pi$
$114$		0			0
$115$		−	6.34204i		−	0.591398i
$116$		0			0
$117$	−0.762726	−	0.950243i	−0.0705140	−	0.0878500i
$118$		0			0
$119$	17.9023			1.64111
$120$		0			0
$121$	−7.39604			−0.672368
$122$		0			0
$123$	−3.88371	−	11.0429i	−0.350183	−	0.995707i
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	8.48764			0.753156			0.376578	−	0.926385i	$-0.377101\pi$
$127$	8.48764			0.753156			0.376578	+	0.926385i	$0.377101\pi$
$128$		0			0
$129$	11.9898	−	4.21672i	1.05564	−	0.371261i
$130$		0			0
$131$			4.67326i			0.408305i	0.978939	+	0.204152i	$0.0654438\pi$
$131$			4.67326i			0.408305i	−0.978939	+	0.204152i	$0.934556\pi$
$132$		0			0
$133$			22.9850i			1.99306i
$134$		0			0
$135$	−4.41278	−	2.74360i	−0.379792	−	0.236132i
$136$		0			0
$137$	0.666604			0.0569518			0.0284759	−	0.999594i	$-0.490935\pi$
$137$	0.666604			0.0569518			0.0284759	+	0.999594i	$0.490935\pi$
$138$		0			0
$139$			3.55441i			0.301481i	0.988573	+	0.150741i	$0.0481658\pi$
$139$			3.55441i			0.301481i	−0.988573	+	0.150741i	$0.951834\pi$
$140$		0			0
$141$	0.716003	−	0.251813i	0.0602983	−	0.0212064i
$142$		0			0
$143$			0.771063i			0.0644795i
$144$		0			0
$145$			3.90840i			0.324575i
$146$		0			0
$147$	6.52968	+	18.5665i	0.538559	+	1.53134i
$148$		0			0
$149$		−	19.8184i		−	1.62359i	−0.583945	−	0.811793i	$-0.698492\pi$
$149$		−	19.8184i		−	1.62359i	0.583945	−	0.811793i	$-0.301508\pi$
$150$		0			0
$151$	6.37528			0.518813			0.259406	−	0.965768i	$-0.416473\pi$
$151$	6.37528			0.518813			0.259406	+	0.965768i	$0.416473\pi$
$152$		0			0
$153$	−7.84526	−	9.77402i	−0.634251	−	0.790183i
$154$		0			0
$155$		−	3.07537i		−	0.247020i
$156$		0			0
$157$	13.4683			1.07489			0.537443	−	0.843300i	$-0.319390\pi$
$157$	13.4683			1.07489			0.537443	+	0.843300i	$0.319390\pi$
$158$		0			0
$159$	3.00555	+	8.54597i	0.238356	+	0.677740i
$160$		0			0
$161$	27.1769			2.14184
$162$		0			0
$163$	−10.5652			−0.827530			−0.413765	−	0.910384i	$-0.635787\pi$
$163$	−10.5652			−0.827530			−0.413765	+	0.910384i	$0.635787\pi$
$164$		0			0
$165$	1.09091	+	3.10190i	0.0849275	+	0.241483i
$166$		0			0
$167$			23.2643i			1.80025i	0.435633	+	0.900124i	$0.356524\pi$
$167$			23.2643i			1.80025i	−0.435633	+	0.900124i	$0.643476\pi$
$168$		0			0
$169$	12.8350			0.987310
$170$		0			0
$171$	12.5490	−	10.0726i	0.959645	−	0.770272i
$172$		0			0
$173$		−	1.53218i		−	0.116490i	−0.998302	−	0.0582448i	$-0.981450\pi$
$173$		−	1.53218i		−	0.116490i	0.998302	−	0.0582448i	$-0.0185504\pi$
$174$		0			0
$175$		−	4.28520i		−	0.323931i
$176$		0			0
$177$	20.2579	−	7.12456i	1.52268	−	0.535515i
$178$		0			0
$179$	13.0231			0.973395			0.486698	−	0.873570i	$-0.338201\pi$
$179$	13.0231			0.973395			0.486698	+	0.873570i	$0.338201\pi$
$180$		0			0
$181$	17.4814			1.29939			0.649693	−	0.760197i	$-0.274898\pi$
$181$	17.4814			1.29939			0.649693	+	0.760197i	$0.274898\pi$
$182$		0			0
$183$	12.6109	−	4.43515i	0.932224	−	0.327856i
$184$		0			0
$185$	−7.35727			−0.540917
$186$		0			0
$187$			7.93101i			0.579973i
$188$		0			0
$189$	11.7569	−	18.9097i	0.855188	−	1.37548i
$190$		0			0
$191$	23.9071			1.72986			0.864929	−	0.501894i	$-0.167363\pi$
$191$	23.9071			1.72986			0.864929	+	0.501894i	$0.167363\pi$
$192$		0			0
$193$	−22.0317			−1.58588			−0.792938	−	0.609302i	$-0.791450\pi$
$193$	−22.0317			−1.58588			−0.792938	+	0.609302i	$0.791450\pi$
$194$		0			0
$195$	−0.663648	+	0.233400i	−0.0475248	+	0.0167141i
$196$		0			0
$197$	−7.87770			−0.561263			−0.280631	−	0.959816i	$-0.590544\pi$
$197$	−7.87770			−0.561263			−0.280631	+	0.959816i	$0.590544\pi$
$198$		0			0
$199$	−15.0814			−1.06909			−0.534547	−	0.845139i	$-0.679518\pi$
$199$	−15.0814			−1.06909			−0.534547	+	0.845139i	$0.679518\pi$
$200$		0			0
$201$	14.0795	−	1.66355i	0.993092	−	0.117338i
$202$		0			0
$203$	−16.7483			−1.17550
$204$		0			0
$205$	−6.75844			−0.472030
$206$		0			0
$207$	−11.9096	−	14.8376i	−0.827775	−	1.03128i
$208$		0			0
$209$	−10.1827			−0.704353
$210$		0			0
$211$	10.3829			0.714786			0.357393	−	0.933954i	$-0.383666\pi$
$211$	10.3829			0.714786			0.357393	+	0.933954i	$0.383666\pi$
$212$		0			0
$213$	3.72556	−	1.31025i	0.255271	−	0.0897769i
$214$		0			0
$215$		−	7.33793i		−	0.500443i
$216$		0			0
$217$	13.1786			0.894621
$218$		0			0
$219$	−0.309688	−	0.880566i	−0.0209268	−	0.0595031i
$220$		0			0
$221$	−1.69683			−0.114141
$222$		0			0
$223$	20.3552			1.36308			0.681541	−	0.731780i	$-0.261310\pi$
$223$	20.3552			1.36308			0.681541	+	0.731780i	$0.261310\pi$
$224$		0			0
$225$	−2.33956	+	1.87788i	−0.155971	+	0.125192i
$226$		0			0
$227$			13.4575i			0.893208i	0.894732	+	0.446604i	$0.147367\pi$
$227$			13.4575i			0.893208i	−0.894732	+	0.446604i	$0.852633\pi$
$228$		0			0
$229$			13.8909i			0.917938i	0.888452	+	0.458969i	$0.151781\pi$
$229$			13.8909i			0.917938i	−0.888452	+	0.458969i	$0.848219\pi$
$230$		0			0
$231$	−13.2923	+	4.67479i	−0.874567	+	0.307578i
$232$		0			0
$233$	26.1014			1.70996			0.854980	−	0.518661i	$-0.173569\pi$
$233$	26.1014			1.70996			0.854980	+	0.518661i	$0.173569\pi$
$234$		0			0
$235$		−	0.438204i		−	0.0285853i
$236$		0			0
$237$	−2.73761	+	0.962798i	−0.177827	+	0.0625405i
$238$		0			0
$239$	9.98603			0.645942			0.322971	−	0.946409i	$-0.395318\pi$
$239$	9.98603			0.645942			0.322971	+	0.946409i	$0.395318\pi$
$240$		0			0
$241$	−8.91315			−0.574146			−0.287073	−	0.957909i	$-0.592682\pi$
$241$	−8.91315			−0.574146			−0.287073	+	0.957909i	$0.592682\pi$
$242$		0			0
$243$	−15.4761	+	1.86786i	−0.992795	+	0.119823i
$244$		0			0
$245$	11.3630			0.725953
$246$		0			0
$247$		−	2.17858i		−	0.138620i
$248$		0			0
$249$	−5.19853	+	1.82828i	−0.329444	+	0.115863i
$250$		0			0
$251$	−5.67604			−0.358268			−0.179134	−	0.983825i	$-0.557330\pi$
$251$	−5.67604			−0.358268			−0.179134	+	0.983825i	$0.557330\pi$
$252$		0			0
$253$			12.0398i			0.756935i
$254$		0			0
$255$	−6.82616	+	2.40071i	−0.427471	+	0.150338i
$256$		0			0
$257$			19.6314i			1.22457i	0.790635	+	0.612287i	$0.209750\pi$
$257$			19.6314i			1.22457i	−0.790635	+	0.612287i	$0.790250\pi$
$258$		0			0
$259$		−	31.5274i		−	1.95902i
$260$		0			0
$261$	7.33952	+	9.14395i	0.454305	+	0.565996i
$262$		0			0
$263$			21.8339i			1.34634i	0.739489	+	0.673168i	$0.235067\pi$
$263$			21.8339i			1.34634i	−0.739489	+	0.673168i	$0.764933\pi$
$264$		0			0
$265$	5.23027			0.321293
$266$		0			0
$267$	−17.5884	+	6.18572i	−1.07639	+	0.378560i
$268$		0			0
$269$			25.3812i			1.54752i	0.633478	+	0.773761i	$0.281627\pi$
$269$			25.3812i			1.54752i	−0.633478	+	0.773761i	$0.718373\pi$
$270$		0			0
$271$		−	22.4238i		−	1.36215i	−0.732215	−	0.681074i	$-0.761513\pi$
$271$		−	22.4238i		−	1.36215i	0.732215	−	0.681074i	$-0.238487\pi$
$272$		0			0
$273$	−1.00017	−	2.84387i	−0.0605328	−	0.172119i
$274$		0			0
$275$	1.89841			0.114478
$276$		0			0
$277$	−4.63766			−0.278650			−0.139325	−	0.990247i	$-0.544493\pi$
$277$	−4.63766			−0.278650			−0.139325	+	0.990247i	$0.544493\pi$
$278$		0			0
$279$	−5.77519	−	7.19502i	−0.345751	−	0.430755i
$280$		0			0
$281$	6.81797			0.406726			0.203363	−	0.979103i	$-0.434813\pi$
$281$	6.81797			0.406726			0.203363	+	0.979103i	$0.434813\pi$
$282$		0			0
$283$	−29.4962			−1.75337			−0.876684	−	0.481067i	$-0.840249\pi$
$283$	−29.4962			−1.75337			−0.876684	+	0.481067i	$0.840249\pi$
$284$		0			0
$285$	−3.08230	−	8.76419i	−0.182580	−	0.519146i
$286$		0			0
$287$		−	28.9613i		−	1.70953i
$288$		0			0
$289$	−0.453281			−0.0266636
$290$		0			0
$291$	−12.5703	+	4.42087i	−0.736883	+	0.259156i
$292$		0			0
$293$		−	8.47788i		−	0.495283i	−0.968852	−	0.247642i	$-0.920344\pi$
$293$		−	8.47788i		−	0.495283i	0.968852	−	0.247642i	$-0.0796555\pi$
$294$		0			0
$295$		−	12.3982i		−	0.721849i
$296$		0			0
$297$	8.37727	+	5.20848i	0.486098	+	0.302227i
$298$		0			0
$299$	−2.57590			−0.148968
$300$		0			0
$301$	31.4445			1.81243
$302$		0			0
$303$	1.98405	+	5.64143i	0.113981	+	0.324092i
$304$		0			0
$305$		−	7.71806i		−	0.441935i
$306$		0			0
$307$	27.3618			1.56162			0.780812	−	0.624767i	$-0.214806\pi$
$307$	27.3618			1.56162			0.780812	+	0.624767i	$0.214806\pi$
$308$		0			0
$309$	5.06868	+	14.4123i	0.288347	+	0.819885i
$310$		0			0
$311$	16.3539			0.927345			0.463672	−	0.886007i	$-0.346531\pi$
$311$	16.3539			0.927345			0.463672	+	0.886007i	$0.346531\pi$
$312$		0			0
$313$		−	10.7965i		−	0.610254i	−0.952312	−	0.305127i	$-0.901301\pi$
$313$		−	10.7965i		−	0.610254i	0.952312	−	0.305127i	$-0.0986988\pi$
$314$		0			0
$315$	−8.04711	−	10.0255i	−0.453403	−	0.564873i
$316$		0			0
$317$		−	18.6391i		−	1.04687i	−0.852064	−	0.523437i	$-0.824650\pi$
$317$		−	18.6391i		−	1.04687i	0.852064	−	0.523437i	$-0.175350\pi$
$318$		0			0
$319$		−	7.41974i		−	0.415426i
$320$		0			0
$321$	20.5953	−	7.24322i	1.14952	−	0.404277i
$322$		0			0
$323$		−	22.4085i		−	1.24684i
$324$		0			0
$325$			0.406163i			0.0225299i
$326$		0			0
$327$	−25.9209	+	9.11619i	−1.43343	+	0.504126i
$328$		0			0
$329$	1.87779			0.103526
$330$		0			0
$331$		−	29.6048i		−	1.62723i	−0.581407	−	0.813613i	$-0.697498\pi$
$331$		−	29.6048i		−	1.62723i	0.581407	−	0.813613i	$-0.302502\pi$
$332$		0			0
$333$	−17.2128	+	13.8161i	−0.943255	+	0.757117i
$334$		0			0
$335$	1.79086	−	7.98704i	0.0978454	−	0.436379i
$336$		0			0
$337$			23.8591i			1.29969i	0.760067	+	0.649845i	$0.225166\pi$
$337$			23.8591i			1.29969i	−0.760067	+	0.649845i	$0.774834\pi$
$338$		0			0
$339$	1.36853	+	3.89127i	0.0743283	+	0.211345i
$340$		0			0
$341$			5.83831i			0.316162i
$342$		0			0
$343$			18.6962i			1.00950i
$344$		0			0
$345$	−10.3626	+	3.64443i	−0.557901	+	0.196210i
$346$		0			0
$347$	−5.91221			−0.317384			−0.158692	−	0.987328i	$-0.550728\pi$
$347$	−5.91221			−0.317384			−0.158692	+	0.987328i	$0.550728\pi$
$348$		0			0
$349$	−7.81712			−0.418441			−0.209220	−	0.977869i	$-0.567093\pi$
$349$	−7.81712			−0.418441			−0.209220	+	0.977869i	$0.567093\pi$
$350$		0			0
$351$	−1.11435	+	1.79231i	−0.0594795	+	0.0956663i
$352$		0			0
$353$	9.20893			0.490142			0.245071	−	0.969505i	$-0.421189\pi$
$353$	9.20893			0.490142			0.245071	+	0.969505i	$0.421189\pi$
$354$		0			0
$355$		−	2.28010i		−	0.121015i
$356$		0			0
$357$	−10.2875	−	29.2515i	−0.544473	−	1.54815i
$358$		0			0
$359$			11.5858i			0.611473i	0.952116	+	0.305737i	$0.0989027\pi$
$359$			11.5858i			0.611473i	−0.952116	+	0.305737i	$0.901097\pi$
$360$		0			0
$361$	9.77051			0.514237
$362$		0			0
$363$	4.25011	+	12.0847i	0.223073	+	0.634284i
$364$		0			0
$365$	−0.538920			−0.0282084
$366$		0			0
$367$			22.3207i			1.16513i	0.812784	+	0.582565i	$0.197951\pi$
$367$			22.3207i			1.16513i	−0.812784	+	0.582565i	$0.802049\pi$
$368$		0			0
$369$	−15.8118	+	12.6916i	−0.823129	+	0.660696i
$370$		0			0
$371$			22.4127i			1.16361i
$372$		0			0
$373$			3.68064i			0.190576i	0.995450	+	0.0952881i	$0.0303772\pi$
$373$			3.68064i			0.190576i	−0.995450	+	0.0952881i	$0.969623\pi$
$374$		0			0
$375$	0.574646	+	1.63395i	0.0296746	+	0.0843766i
$376$		0			0
$377$	1.58745			0.0817577
$378$		0			0
$379$			22.9156i			1.17709i	0.808463	+	0.588546i	$0.200300\pi$
$379$			22.9156i			1.17709i	−0.808463	+	0.588546i	$0.799700\pi$
$380$		0			0
$381$	−4.87739	−	13.8683i	−0.249876	−	0.710497i
$382$		0			0
$383$	−12.5331			−0.640413			−0.320206	−	0.947348i	$-0.603752\pi$
$383$	−12.5331			−0.640413			−0.320206	+	0.947348i	$0.603752\pi$
$384$		0			0
$385$			8.13507i			0.414601i
$386$		0			0
$387$	−13.7798	−	17.1676i	−0.700466	−	0.872676i
$388$		0			0
$389$			9.00019i			0.456328i	0.973623	+	0.228164i	$0.0732722\pi$
$389$			9.00019i			0.456328i	−0.973623	+	0.228164i	$0.926728\pi$
$390$		0			0
$391$	−26.4952			−1.33992
$392$		0			0
$393$	7.63586	−	2.68547i	0.385178	−	0.135464i
$394$		0			0
$395$			1.67546i			0.0843016i
$396$		0			0
$397$	−19.0545			−0.956320			−0.478160	−	0.878273i	$-0.658696\pi$
$397$	−19.0545			−0.956320			−0.478160	+	0.878273i	$0.658696\pi$
$398$		0			0
$399$	37.5563	−	13.2083i	1.88017	−	0.661240i
$400$		0			0
$401$	−8.30887			−0.414925			−0.207463	−	0.978243i	$-0.566521\pi$
$401$	−8.30887			−0.414925			−0.207463	+	0.978243i	$0.566521\pi$
$402$		0			0
$403$	−1.24910			−0.0622221
$404$		0			0
$405$	−1.94711	+	8.78685i	−0.0967527	+	0.436622i
$406$		0			0
$407$	13.9671			0.692324
$408$		0			0
$409$			19.2556i			0.952128i	0.879411	+	0.476064i	$0.157937\pi$
$409$			19.2556i			0.952128i	−0.879411	+	0.476064i	$0.842063\pi$
$410$		0			0
$411$	−0.383062	−	1.08920i	−0.0188950	−	0.0537260i
$412$		0			0
$413$	53.1287			2.61429
$414$		0			0
$415$			3.18158i			0.156178i
$416$		0			0
$417$	5.80772	−	2.04253i	0.284405	−	0.100023i
$418$		0			0
$419$			13.2345i			0.646550i	0.946305	+	0.323275i	$0.104784\pi$
$419$			13.2345i			0.646550i	−0.946305	+	0.323275i	$0.895216\pi$
$420$		0			0
$421$	−27.8802			−1.35880			−0.679400	−	0.733768i	$-0.737760\pi$
$421$	−27.8802			−1.35880			−0.679400	+	0.733768i	$0.737760\pi$
$422$		0			0
$423$	−0.822897	−	1.02521i	−0.0400106	−	0.0498473i
$424$		0			0
$425$			4.17771i			0.202649i
$426$		0			0
$427$	33.0734			1.60054
$428$		0			0
$429$	1.25988	−	0.443089i	0.0608274	−	0.0213925i
$430$		0			0
$431$		−	12.7170i		−	0.612558i	−0.951942	−	0.306279i	$-0.900916\pi$
$431$		−	12.7170i		−	0.612558i	0.951942	−	0.306279i	$-0.0990841\pi$
$432$		0			0
$433$		−	13.6665i		−	0.656770i	−0.944544	−	0.328385i	$-0.893496\pi$
$433$		−	13.6665i		−	0.656770i	0.944544	−	0.328385i	$-0.106504\pi$
$434$		0			0
$435$	6.38612	−	2.24595i	0.306191	−	0.107685i
$436$		0			0
$437$		−	34.0175i		−	1.62728i
$438$		0			0
$439$	−0.194743			−0.00929459			−0.00464729	−	0.999989i	$-0.501479\pi$
$439$	−0.194743			−0.00929459			−0.00464729	+	0.999989i	$0.501479\pi$
$440$		0			0
$441$	26.5844	−	21.3383i	1.26592	−	1.01611i
$442$		0			0
$443$	13.5078			0.641775			0.320888	−	0.947117i	$-0.396019\pi$
$443$	13.5078			0.641775			0.320888	+	0.947117i	$0.396019\pi$
$444$		0			0
$445$			10.7644i			0.510281i
$446$		0			0
$447$	−32.3822	+	11.3886i	−1.53163	+	0.538661i
$448$		0			0
$449$		−	13.5782i		−	0.640796i	−0.947283	−	0.320398i	$-0.896183\pi$
$449$		−	13.5782i		−	0.640796i	0.947283	−	0.320398i	$-0.103817\pi$
$450$		0			0
$451$	12.8303			0.604154
$452$		0			0
$453$	−3.66353	−	10.4169i	−0.172128	−	0.489427i
$454$		0			0
$455$	−1.74049			−0.0815954
$456$		0			0
$457$	7.62393			0.356632			0.178316	−	0.983973i	$-0.442935\pi$
$457$	7.62393			0.356632			0.178316	+	0.983973i	$0.442935\pi$
$458$		0			0
$459$	−11.4620	+	18.4353i	−0.534999	+	0.860488i
$460$		0			0
$461$			14.0257i			0.653240i	0.945156	+	0.326620i	$0.105910\pi$
$461$			14.0257i			0.653240i	−0.945156	+	0.326620i	$0.894090\pi$
$462$		0			0
$463$			17.4953i			0.813073i	0.913634	+	0.406537i	$0.133264\pi$
$463$			17.4953i			0.813073i	−0.913634	+	0.406537i	$0.866736\pi$
$464$		0			0
$465$	−5.02499	+	1.76725i	−0.233028	+	0.0819543i
$466$		0			0
$467$			13.0377i			0.603312i	0.953417	+	0.301656i	$0.0975394\pi$
$467$			13.0377i			0.603312i	−0.953417	+	0.301656i	$0.902461\pi$
$468$		0			0
$469$	34.2261	+	7.67422i	1.58041	+	0.354362i
$470$		0			0
$471$	−7.73950	−	22.0064i	−0.356617	−	1.01400i
$472$		0			0
$473$			13.9304i			0.640521i
$474$		0			0
$475$	−5.36381			−0.246109
$476$		0			0
$477$	12.2365	−	9.82183i	0.560272	−	0.449711i
$478$		0			0
$479$		−	33.7559i		−	1.54234i	−0.636627	−	0.771172i	$-0.719671\pi$
$479$		−	33.7559i		−	1.54234i	0.636627	−	0.771172i	$-0.280329\pi$
$480$		0			0
$481$			2.98825i			0.136252i
$482$		0			0
$483$	−15.6171	−	44.4056i	−0.710604	−	2.02053i
$484$		0			0
$485$			7.69321i			0.349331i
$486$		0			0
$487$			0.157730i			0.00714745i	0.999994	+	0.00357372i	$0.00113755\pi$
$487$			0.157730i			0.00714745i	−0.999994	+	0.00357372i	$0.998862\pi$
$488$		0			0
$489$	6.07125	+	17.2630i	0.274552	+	0.780658i
$490$		0			0
$491$			8.50705i			0.383918i	0.981403	+	0.191959i	$0.0614840\pi$
$491$			8.50705i			0.383918i	−0.981403	+	0.191959i	$0.938516\pi$
$492$		0			0
$493$	16.3282			0.735384
$494$		0			0
$495$	4.44145	−	3.56499i	0.199628	−	0.160234i
$496$		0			0
$497$	9.77068			0.438275
$498$		0			0
$499$			7.36676i			0.329782i	0.986312	+	0.164891i	$0.0527272\pi$
$499$			7.36676i			0.329782i	−0.986312	+	0.164891i	$0.947273\pi$
$500$		0			0
$501$	38.0127	−	13.3688i	1.69828	−	0.597273i
$502$		0			0
$503$	42.3909			1.89012			0.945058	−	0.326903i	$-0.106005\pi$
$503$	42.3909			1.89012			0.945058	+	0.326903i	$0.106005\pi$
$504$		0			0
$505$	3.45264			0.153640
$506$		0			0
$507$	−7.37561	−	20.9718i	−0.327562	−	0.931388i
$508$		0			0
$509$			10.8503i			0.480931i	0.970658	+	0.240466i	$0.0773001\pi$
$509$			10.8503i			0.480931i	−0.970658	+	0.240466i	$0.922700\pi$
$510$		0			0
$511$		−	2.30938i		−	0.102161i
$512$		0			0
$513$	−23.6693	−	14.7162i	−1.04503	−	0.649735i
$514$		0			0
$515$	8.82052			0.388679
$516$		0			0
$517$			0.831891i			0.0365865i
$518$		0			0
$519$	−2.50350	+	0.880462i	−0.109892	+	0.0386480i
$520$		0			0
$521$	−15.1740			−0.664784			−0.332392	−	0.943141i	$-0.607856\pi$
$521$	−15.1740			−0.664784			−0.332392	+	0.943141i	$0.607856\pi$
$522$		0			0
$523$	−33.6521			−1.47151			−0.735753	−	0.677250i	$-0.763171\pi$
$523$	−33.6521			−1.47151			−0.735753	+	0.677250i	$0.763171\pi$
$524$		0			0
$525$	−7.00179	+	2.46248i	−0.305583	+	0.107471i
$526$		0			0
$527$	−12.8480			−0.559668
$528$		0			0
$529$	−17.2215			−0.748759
$530$		0			0
$531$	−23.2823	−	29.0063i	−1.01037	−	1.25877i
$532$		0			0
$533$			2.74503i			0.118900i
$534$		0			0
$535$		−	12.6047i		−	0.544947i
$536$		0			0
$537$	−7.48370	−	21.2791i	−0.322946	−	0.918262i
$538$		0			0
$539$	−21.5715			−0.929152
$540$		0			0
$541$		−	0.948628i		−	0.0407847i	−0.999792	−	0.0203924i	$-0.993508\pi$
$541$		−	0.948628i		−	0.0407847i	0.999792	−	0.0203924i	$-0.00649154\pi$
$542$		0			0
$543$	−10.0456	−	28.5637i	−0.431100	−	1.22579i
$544$		0			0
$545$			15.8640i			0.679539i
$546$		0			0
$547$			12.6294i			0.539995i	0.962861	+	0.269998i	$0.0870229\pi$
$547$			12.6294i			0.539995i	−0.962861	+	0.269998i	$0.912977\pi$
$548$		0			0
$549$	−14.4936	−	18.0569i	−0.618572	−	0.770649i
$550$		0			0
$551$			20.9639i			0.893094i
$552$		0			0
$553$	−7.17969			−0.305312
$554$		0			0
$555$	4.22783	+	12.0214i	0.179461	+	0.510279i
$556$		0			0
$557$			28.4069i			1.20364i	0.798633	+	0.601819i	$0.205557\pi$
$557$			28.4069i			1.20364i	−0.798633	+	0.601819i	$0.794443\pi$
$558$		0			0
$559$	−2.98040			−0.126057
$560$		0			0
$561$	12.9588	−	4.55753i	0.547123	−	0.192419i
$562$		0			0
$563$	−43.8324			−1.84731			−0.923657	−	0.383220i	$-0.874815\pi$
$563$	−43.8324			−1.84731			−0.923657	+	0.383220i	$0.874815\pi$
$564$		0			0
$565$	2.38152			0.100191
$566$		0			0
$567$	−37.6534	−	8.34376i	−1.58130	−	0.350405i
$568$		0			0
$569$			35.6890i			1.49616i	0.663608	+	0.748080i	$0.269024\pi$
$569$			35.6890i			1.49616i	−0.663608	+	0.748080i	$0.730976\pi$
$570$		0			0
$571$	19.1578			0.801728			0.400864	−	0.916138i	$-0.368710\pi$
$571$	19.1578			0.801728			0.400864	+	0.916138i	$0.368710\pi$
$572$		0			0
$573$	−13.7381	−	39.0630i	−0.573919	−	1.63188i
$574$		0			0
$575$			6.34204i			0.264481i
$576$		0			0
$577$		−	34.3077i		−	1.42825i	−0.700020	−	0.714124i	$-0.746825\pi$
$577$		−	34.3077i		−	1.42825i	0.700020	−	0.714124i	$-0.253175\pi$
$578$		0			0
$579$	12.6604	+	35.9986i	0.526150	+	1.49605i
$580$		0			0
$581$	−13.6337			−0.565622
$582$		0			0
$583$	−9.92918			−0.411225
$584$		0			0
$585$	0.762726	+	0.950243i	0.0315348	+	0.0392877i
$586$		0			0
$587$	10.2259			0.422067			0.211034	−	0.977479i	$-0.432317\pi$
$587$	10.2259			0.422067			0.211034	+	0.977479i	$0.432317\pi$
$588$		0			0
$589$		−	16.4957i		−	0.679694i
$590$		0			0
$591$	4.52689	+	12.8717i	0.186211	+	0.529472i
$592$		0			0
$593$	28.8955			1.18660			0.593298	−	0.804983i	$-0.297826\pi$
$593$	28.8955			1.18660			0.593298	+	0.804983i	$0.297826\pi$
$594$		0			0
$595$	−17.9023			−0.733925
$596$		0			0
$597$	8.66648	+	24.6422i	0.354695	+	1.00854i
$598$		0			0
$599$	−46.0631			−1.88209			−0.941044	−	0.338283i	$-0.890154\pi$
$599$	−46.0631			−1.88209			−0.941044	+	0.338283i	$0.890154\pi$
$600$		0			0
$601$	−3.76408			−0.153540			−0.0767701	−	0.997049i	$-0.524461\pi$
$601$	−3.76408			−0.153540			−0.0767701	+	0.997049i	$0.524461\pi$
$602$		0			0
$603$	−10.8089	−	22.0492i	−0.440172	−	0.897914i
$604$		0			0
$605$	7.39604			0.300692
$606$		0			0
$607$	33.2910			1.35124			0.675620	−	0.737250i	$-0.263876\pi$
$607$	33.2910			1.35124			0.675620	+	0.737250i	$0.263876\pi$
$608$		0			0
$609$	9.62434	+	27.3658i	0.389998	+	1.10892i
$610$		0			0
$611$	−0.177982			−0.00720039
$612$		0			0
$613$	−22.8220			−0.921772			−0.460886	−	0.887459i	$-0.652468\pi$
$613$	−22.8220			−0.921772			−0.460886	+	0.887459i	$0.652468\pi$
$614$		0			0
$615$	3.88371	+	11.0429i	0.156606	+	0.445294i
$616$		0			0
$617$			3.74796i			0.150887i	0.997150	+	0.0754435i	$0.0240373\pi$
$617$			3.74796i			0.150887i	−0.997150	+	0.0754435i	$0.975963\pi$
$618$		0			0
$619$	−29.4861			−1.18515			−0.592573	−	0.805516i	$-0.701888\pi$
$619$	−29.4861			−1.18515			−0.592573	+	0.805516i	$0.701888\pi$
$620$		0			0
$621$	−17.4000	+	27.9860i	−0.698239	+	1.12304i
$622$		0			0
$623$	−46.1276			−1.84806
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	5.85146	+	16.6380i	0.233685	+	0.664458i
$628$		0			0
$629$			30.7365i			1.22555i
$630$		0			0
$631$		−	24.5356i		−	0.976746i	−0.872635	−	0.488373i	$-0.837591\pi$
$631$		−	24.5356i		−	0.976746i	0.872635	−	0.488373i	$-0.162409\pi$
$632$		0			0
$633$	−5.96648	−	16.9650i	−0.237146	−	0.674300i
$634$		0			0
$635$	−8.48764			−0.336822
$636$		0			0
$637$		−	4.61521i		−	0.182861i
$638$		0			0
$639$	−4.28176	−	5.33443i	−0.169384	−	0.211027i
$640$		0			0
$641$	−17.3857			−0.686695			−0.343348	−	0.939208i	$-0.611561\pi$
$641$	−17.3857			−0.686695			−0.343348	+	0.939208i	$0.611561\pi$
$642$		0			0
$643$	−12.1810			−0.480370			−0.240185	−	0.970727i	$-0.577208\pi$
$643$	−12.1810			−0.480370			−0.240185	+	0.970727i	$0.577208\pi$
$644$		0			0
$645$	−11.9898	+	4.21672i	−0.472098	+	0.166033i
$646$		0			0
$647$	−40.3480			−1.58624			−0.793122	−	0.609063i	$-0.791546\pi$
$647$	−40.3480			−1.58624			−0.793122	+	0.609063i	$0.791546\pi$
$648$		0			0
$649$			23.5368i			0.923900i
$650$		0			0
$651$	−7.57303	−	21.5331i	−0.296810	−	0.843949i
$652$		0			0
$653$	−43.0958			−1.68647			−0.843235	−	0.537545i	$-0.819352\pi$
$653$	−43.0958			−1.68647			−0.843235	+	0.537545i	$0.819352\pi$
$654$		0			0
$655$		−	4.67326i		−	0.182599i
$656$		0			0
$657$	−1.26084	+	1.01203i	−0.0491899	+	0.0394830i
$658$		0			0
$659$		−	1.55631i		−	0.0606252i	−0.999540	−	0.0303126i	$-0.990350\pi$
$659$		−	1.55631i		−	0.0606252i	0.999540	−	0.0303126i	$-0.00965028\pi$
$660$		0			0
$661$		−	13.5157i		−	0.525700i	−0.964837	−	0.262850i	$-0.915337\pi$
$661$		−	13.5157i		−	0.525700i	0.964837	−	0.262850i	$-0.0846625\pi$
$662$		0			0
$663$	0.975078	+	2.77253i	0.0378689	+	0.107676i
$664$		0			0
$665$		−	22.9850i		−	0.891321i
$666$		0			0
$667$	24.7872			0.959765
$668$		0			0
$669$	−11.6970	−	33.2593i	−0.452233	−	1.28588i
$670$		0			0
$671$			14.6520i			0.565635i
$672$		0			0
$673$		−	29.9958i		−	1.15625i	−0.815947	−	0.578127i	$-0.803784\pi$
$673$		−	29.9958i		−	1.15625i	0.815947	−	0.578127i	$-0.196216\pi$
$674$		0			0
$675$	4.41278	+	2.74360i	0.169848	+	0.105601i
$676$		0			0
$677$	37.3115			1.43400			0.716999	−	0.697075i	$-0.245515\pi$
$677$	37.3115			1.43400			0.716999	+	0.697075i	$0.245515\pi$
$678$		0			0
$679$	−32.9669			−1.26516
$680$		0			0
$681$	21.9889	−	7.73333i	0.842617	−	0.296342i
$682$		0			0
$683$	10.4018			0.398016			0.199008	−	0.979998i	$-0.436228\pi$
$683$	10.4018			0.398016			0.199008	+	0.979998i	$0.436228\pi$
$684$		0			0
$685$	−0.666604			−0.0254696
$686$		0			0
$687$	22.6970	−	7.98237i	0.865946	−	0.304547i
$688$		0			0
$689$		−	2.12434i		−	0.0809309i
$690$		0			0
$691$	25.8840			0.984673			0.492337	−	0.870405i	$-0.336143\pi$
$691$	25.8840			0.984673			0.492337	+	0.870405i	$0.336143\pi$
$692$		0			0
$693$	15.2767	+	19.0325i	0.580314	+	0.722985i
$694$		0			0
$695$		−	3.55441i		−	0.134827i
$696$		0			0
$697$			28.2348i			1.06947i
$698$		0			0
$699$	−14.9991	−	42.6483i	−0.567317	−	1.61311i
$700$		0			0
$701$	13.9653			0.527461			0.263731	−	0.964596i	$-0.415047\pi$
$701$	13.9653			0.527461			0.263731	+	0.964596i	$0.415047\pi$
$702$		0			0
$703$	−39.4630			−1.48838
$704$		0			0
$705$	−0.716003	+	0.251813i	−0.0269662	+	0.00948381i
$706$		0			0
$707$			14.7953i			0.556433i
$708$		0			0
$709$	17.3853			0.652919			0.326460	−	0.945211i	$-0.394144\pi$
$709$	17.3853			0.652919			0.326460	+	0.945211i	$0.394144\pi$
$710$		0			0
$711$	3.14632	+	3.91985i	0.117996	+	0.147006i
$712$		0			0
$713$	−19.5041			−0.730435
$714$		0			0
$715$		−	0.771063i		−	0.0288361i
$716$		0			0
$717$	−5.73844	−	16.3166i	−0.214306	−	0.609356i
$718$		0			0
$719$			27.4685i			1.02440i	0.858866	+	0.512201i	$0.171170\pi$
$719$			27.4685i			1.02440i	−0.858866	+	0.512201i	$0.828830\pi$
$720$		0			0
$721$			37.7977i			1.40766i
$722$		0			0
$723$	5.12191	+	14.5636i	0.190486	+	0.541626i
$724$		0			0
$725$		−	3.90840i		−	0.145154i
$726$		0			0
$727$		−	7.49280i		−	0.277892i	−0.990300	−	0.138946i	$-0.955628\pi$
$727$		−	7.49280i		−	0.277892i	0.990300	−	0.138946i	$-0.0443715\pi$
$728$		0			0
$729$	11.9453	+	24.2138i	0.442418	+	0.896809i
$730$		0			0
$731$	−30.6558			−1.13384
$732$		0			0
$733$		−	6.72234i		−	0.248295i	−0.992264	−	0.124148i	$-0.960380\pi$
$733$		−	6.72234i		−	0.248295i	0.992264	−	0.124148i	$-0.0396196\pi$
$734$		0			0
$735$	−6.52968	−	18.5665i	−0.240851	−	0.684835i
$736$		0			0
$737$	−3.39979	+	15.1627i	−0.125233	+	0.558524i
$738$		0			0
$739$		−	17.8932i		−	0.658213i	−0.944293	−	0.329107i	$-0.893252\pi$
$739$		−	17.8932i		−	0.658213i	0.944293	−	0.329107i	$-0.106748\pi$
$740$		0			0
$741$	−3.55969	+	1.25191i	−0.130768	+	0.0459902i
$742$		0			0
$743$			19.7347i			0.723995i	0.932179	+	0.361998i	$0.117905\pi$
$743$			19.7347i			0.723995i	−0.932179	+	0.361998i	$0.882095\pi$
$744$		0			0
$745$			19.8184i			0.726090i
$746$		0			0
$747$	5.97464	+	7.44351i	0.218600	+	0.272344i
$748$		0			0
$749$	54.0135			1.97361
$750$		0			0
$751$	50.7953			1.85355			0.926774	−	0.375619i	$-0.122570\pi$
$751$	50.7953			1.85355			0.926774	+	0.375619i	$0.122570\pi$
$752$		0			0
$753$	3.26172	+	9.27434i	0.118864	+	0.337976i
$754$		0			0
$755$	−6.37528			−0.232020
$756$		0			0
$757$		−	14.0521i		−	0.510731i	−0.966845	−	0.255366i	$-0.917804\pi$
$757$		−	14.0521i		−	0.510731i	0.966845	−	0.255366i	$-0.0821958\pi$
$758$		0			0
$759$	19.6724	−	6.91862i	0.714062	−	0.251130i
$760$		0			0
$761$			19.3009i			0.699657i	0.936814	+	0.349828i	$0.113760\pi$
$761$			19.3009i			0.699657i	−0.936814	+	0.349828i	$0.886240\pi$
$762$		0			0
$763$	−67.9804			−2.46106
$764$		0			0
$765$	7.84526	+	9.77402i	0.283646	+	0.353380i
$766$		0			0
$767$	−5.03567			−0.181828
$768$		0			0
$769$			41.0289i			1.47954i	0.672859	+	0.739770i	$0.265066\pi$
$769$			41.0289i			1.47954i	−0.672859	+	0.739770i	$0.734934\pi$
$770$		0			0
$771$	32.0767	−	11.2811i	1.15521	−	0.406280i
$772$		0			0
$773$			28.2906i			1.01754i	0.860902	+	0.508771i	$0.169900\pi$
$773$			28.2906i			1.01754i	−0.860902	+	0.508771i	$0.830100\pi$
$774$		0			0
$775$			3.07537i			0.110471i
$776$		0			0
$777$	−51.5141	+	18.1171i	−1.84806	+	0.649947i
$778$		0			0
$779$	−36.2510			−1.29883
$780$		0			0
$781$			4.32856i			0.154888i
$782$		0			0
$783$	10.7231	−	17.2469i	0.383212	−	0.616355i
$784$		0			0
$785$	−13.4683			−0.480703
$786$		0			0
$787$		−	42.6684i		−	1.52096i	−0.649359	−	0.760482i	$-0.724963\pi$
$787$		−	42.6684i		−	1.52096i	0.649359	−	0.760482i	$-0.275037\pi$
$788$		0			0
$789$	35.6754	−	12.5468i	1.27008	−	0.446677i
$790$		0			0
$791$			10.2053i			0.362858i
$792$		0			0
$793$	−3.13479			−0.111320
$794$		0			0
$795$	−3.00555	−	8.54597i	−0.106596	−	0.303094i
$796$		0			0
$797$			19.5800i			0.693559i	0.937947	+	0.346780i	$0.112725\pi$
$797$			19.5800i			0.693559i	−0.937947	+	0.346780i	$0.887275\pi$
$798$		0			0
$799$	−1.83069			−0.0647652
$800$		0			0
$801$	20.2143	+	25.1840i	0.714236	+	0.889831i
$802$		0			0
$803$	1.02309			0.0361041
$804$		0			0
$805$	−27.1769			−0.957861
$806$		0			0
$807$	41.4716	−	14.5852i	1.45987	−	0.513425i
$808$		0			0
$809$	10.0149			0.352105			0.176052	−	0.984381i	$-0.443667\pi$
$809$	10.0149			0.352105			0.176052	+	0.984381i	$0.443667\pi$
$810$		0			0
$811$		−	9.40288i		−	0.330180i	−0.986279	−	0.165090i	$-0.947209\pi$
$811$		−	9.40288i		−	0.330180i	0.986279	−	0.165090i	$-0.0527914\pi$
$812$		0			0
$813$	−36.6393	+	12.8857i	−1.28500	+	0.451923i
$814$		0			0
$815$	10.5652			0.370083
$816$		0			0
$817$		−	39.3593i		−	1.37701i
$818$		0			0
$819$	−4.07198	+	3.26844i	−0.142287	+	0.114208i
$820$		0			0
$821$		−	13.8487i		−	0.483323i	−0.970361	−	0.241661i	$-0.922308\pi$
$821$		−	13.8487i		−	0.483323i	0.970361	−	0.241661i	$-0.0776923\pi$
$822$		0			0
$823$	0.748367			0.0260864			0.0130432	−	0.999915i	$-0.495848\pi$
$823$	0.748367			0.0260864			0.0130432	+	0.999915i	$0.495848\pi$
$824$		0			0
$825$	−1.09091	−	3.10190i	−0.0379808	−	0.107994i
$826$		0			0
$827$			29.5643i			1.02805i	0.857775	+	0.514026i	$0.171846\pi$
$827$			29.5643i			1.02805i	−0.857775	+	0.514026i	$0.828154\pi$
$828$		0			0
$829$	−31.5162			−1.09460			−0.547301	−	0.836936i	$-0.684345\pi$
$829$	−31.5162			−1.09460			−0.547301	+	0.836936i	$0.684345\pi$
$830$		0			0
$831$	2.66502	+	7.57769i	0.0924484	+	0.262867i
$832$		0			0
$833$		−	47.4712i		−	1.64478i
$834$		0			0
$835$		−	23.2643i		−	0.805096i
$836$		0			0
$837$	−8.43759	+	13.5709i	−0.291646	+	0.469080i
$838$		0			0
$839$			50.2758i			1.73571i	0.496814	+	0.867857i	$0.334503\pi$
$839$			50.2758i			1.73571i	−0.496814	+	0.867857i	$0.665497\pi$
$840$		0			0
$841$	13.7244			0.473255
$842$		0			0
$843$	−3.91792	−	11.1402i	−0.134940	−	0.383689i
$844$		0			0
$845$	−12.8350			−0.441539
$846$		0			0
$847$			31.6935i			1.08900i
$848$		0			0
$849$	16.9499	+	48.1952i	0.581719	+	1.65406i
$850$		0			0
$851$			46.6601i			1.59949i
$852$		0			0
$853$	27.3332			0.935872			0.467936	−	0.883762i	$-0.344998\pi$
$853$	27.3332			0.935872			0.467936	+	0.883762i	$0.344998\pi$
$854$		0			0
$855$	−12.5490	+	10.0726i	−0.429166	+	0.344476i
$856$		0			0
$857$	−47.0378			−1.60678			−0.803391	−	0.595452i	$-0.796973\pi$
$857$	−47.0378			−1.60678			−0.803391	+	0.595452i	$0.796973\pi$
$858$		0			0
$859$	44.2069			1.50832			0.754160	−	0.656691i	$-0.228044\pi$
$859$	44.2069			1.50832			0.754160	+	0.656691i	$0.228044\pi$
$860$		0			0
$861$	−47.3212	+	16.6425i	−1.61270	+	0.567175i
$862$		0			0
$863$		−	1.15449i		−	0.0392994i	−0.999807	−	0.0196497i	$-0.993745\pi$
$863$		−	1.15449i		−	0.0392994i	0.999807	−	0.0196497i	$-0.00625509\pi$
$864$		0			0
$865$			1.53218i			0.0520957i
$866$		0			0
$867$	0.260476	+	0.740638i	0.00884625	+	0.0251534i
$868$		0			0
$869$		−	3.18071i		−	0.107898i
$870$		0			0
$871$	−3.24404	−	0.727382i	−0.109920	−	0.0246464i
$872$		0			0
$873$	14.4469	+	17.9987i	0.488955	+	0.609165i
$874$		0			0
$875$			4.28520i			0.144866i
$876$		0			0
$877$	−54.2860			−1.83311			−0.916553	−	0.399913i	$-0.869040\pi$
$877$	−54.2860			−1.83311			−0.916553	+	0.399913i	$0.869040\pi$
$878$		0			0
$879$	−13.8524	+	4.87178i	−0.467230	+	0.164321i
$880$		0			0
$881$		−	23.3680i		−	0.787286i	−0.919263	−	0.393643i	$-0.871215\pi$
$881$		−	23.3680i		−	0.787286i	0.919263	−	0.393643i	$-0.128785\pi$
$882$		0			0
$883$			53.8253i			1.81137i	0.423955	+	0.905683i	$0.360641\pi$
$883$			53.8253i			1.81137i	−0.423955	+	0.905683i	$0.639359\pi$
$884$		0			0
$885$	−20.2579	+	7.12456i	−0.680963	+	0.239490i
$886$		0			0
$887$			22.7411i			0.763572i	0.924251	+	0.381786i	$0.124691\pi$
$887$			22.7411i			0.763572i	−0.924251	+	0.381786i	$0.875309\pi$
$888$		0			0
$889$		−	36.3712i		−	1.21985i
$890$		0			0
$891$	3.69641	−	16.6810i	0.123834	−	0.558836i
$892$		0			0
$893$		−	2.35045i		−	0.0786547i
$894$		0			0
$895$	−13.0231			−0.435316
$896$		0			0
$897$	1.48023	+	4.20888i	0.0494235	+	0.140530i
$898$		0			0
$899$	12.0198			0.400882
$900$		0			0
$901$		−	21.8505i		−	0.727947i
$902$		0			0
$903$	−18.0695	−	51.3787i	−0.601315	−	1.70978i
$904$		0			0
$905$	−17.4814			−0.581103
$906$		0			0
$907$	−21.9429			−0.728603			−0.364301	−	0.931281i	$-0.618692\pi$
$907$	−21.9429			−0.728603			−0.364301	+	0.931281i	$0.618692\pi$
$908$		0			0
$909$	8.07767	−	6.48365i	0.267919	−	0.215049i
$910$		0			0
$911$		−	31.8147i		−	1.05407i	−0.849844	−	0.527034i	$-0.823304\pi$
$911$		−	31.8147i		−	1.05407i	0.849844	−	0.527034i	$-0.176696\pi$
$912$		0			0
$913$		−	6.03994i		−	0.199893i
$914$		0			0
$915$	−12.6109	+	4.43515i	−0.416903	+	0.146622i
$916$		0			0
$917$	20.0259			0.661313
$918$		0			0
$919$		−	16.7785i		−	0.553471i	−0.960946	−	0.276735i	$-0.910747\pi$
$919$		−	16.7785i		−	0.553471i	0.960946	−	0.276735i	$-0.0892526\pi$
$920$		0			0
$921$	−15.7234	−	44.7078i	−0.518103	−	1.47317i
$922$		0			0
$923$	−0.926091			−0.0304827
$924$		0			0
$925$	7.35727			0.241905
$926$		0			0
$927$	20.6362	−	16.5639i	0.677781	−	0.544030i
$928$		0			0
$929$	−52.8126			−1.73272			−0.866362	−	0.499417i	$-0.833548\pi$
$929$	−52.8126			−1.73272			−0.866362	+	0.499417i	$0.833548\pi$
$930$		0			0
$931$	60.9488			1.99752
$932$		0			0
$933$	−9.39771	−	26.7214i	−0.307667	−	0.874820i
$934$		0			0
$935$		−	7.93101i		−	0.259372i
$936$		0			0
$937$			8.62558i			0.281785i	0.990025	+	0.140893i	$0.0449972\pi$
$937$			8.62558i			0.281785i	−0.990025	+	0.140893i	$0.955003\pi$
$938$		0			0
$939$	−17.6409	+	6.20416i	−0.575688	+	0.202465i
$940$		0			0
$941$	−24.8573			−0.810324			−0.405162	−	0.914245i	$-0.632785\pi$
$941$	−24.8573			−0.810324			−0.405162	+	0.914245i	$0.632785\pi$
$942$		0			0
$943$			42.8623i			1.39579i
$944$		0			0
$945$	−11.7569	+	18.9097i	−0.382452	+	0.615131i
$946$		0			0
$947$			57.9140i			1.88195i	0.338473	+	0.940976i	$0.390089\pi$
$947$			57.9140i			1.88195i	−0.338473	+	0.940976i	$0.609911\pi$
$948$		0			0
$949$			0.218889i			0.00710544i
$950$		0			0
$951$	−30.4552	+	10.7109i	−0.987578	+	0.347324i
$952$		0			0
$953$			33.7763i			1.09412i	0.837093	+	0.547061i	$0.184253\pi$
$953$			33.7763i			1.09412i	−0.837093	+	0.547061i	$0.815747\pi$
$954$		0			0
$955$	−23.9071			−0.773616
$956$		0			0
$957$	−12.1235	+	4.26373i	−0.391896	+	0.137827i
$958$		0			0
$959$		−	2.85653i		−	0.0922423i
$960$		0			0
$961$	21.5421			0.694906
$962$		0			0
$963$	−23.6701	−	29.4894i	−0.762757	−	0.950282i
$964$		0			0
$965$	22.0317			0.709225
$966$		0			0
$967$	40.9195			1.31588			0.657942	−	0.753069i	$-0.271427\pi$
$967$	40.9195			1.31588			0.657942	+	0.753069i	$0.271427\pi$
$968$		0			0
$969$	−36.6143	+	12.8769i	−1.17622	+	0.413667i
$970$		0			0
$971$			34.3084i			1.10101i	0.834832	+	0.550504i	$0.185565\pi$
$971$			34.3084i			1.10101i	−0.834832	+	0.550504i	$0.814435\pi$
$972$		0			0
$973$	15.2314			0.488295
$974$		0			0
$975$	0.663648	−	0.233400i	0.0212537	−	0.00747478i
$976$		0			0
$977$		−	33.1908i		−	1.06187i	−0.847413	−	0.530935i	$-0.821841\pi$
$977$		−	33.1908i		−	1.06187i	0.847413	−	0.530935i	$-0.178159\pi$
$978$		0			0
$979$		−	20.4352i		−	0.653112i
$980$		0			0
$981$	29.7907	+	37.1148i	0.951145	+	1.18498i
$982$		0			0
$983$	−38.4206			−1.22543			−0.612713	−	0.790306i	$-0.709922\pi$
$983$	−38.4206			−1.22543			−0.612713	+	0.790306i	$0.709922\pi$
$984$		0			0
$985$	7.87770			0.251004
$986$		0			0
$987$	−1.07907	−	3.06822i	−0.0343471	−	0.0976624i
$988$		0			0
$989$	−46.5375			−1.47981
$990$		0			0
$991$		−	33.9572i		−	1.07869i	−0.842086	−	0.539344i	$-0.818672\pi$
$991$		−	33.9572i		−	1.07869i	0.842086	−	0.539344i	$-0.181328\pi$
$992$		0			0
$993$	−48.3726	+	17.0123i	−1.53506	+	0.539868i
$994$		0			0
$995$	15.0814			0.478113
$996$		0			0
$997$	14.3745			0.455246			0.227623	−	0.973749i	$-0.426905\pi$
$997$	14.3745			0.455246			0.227623	+	0.973749i	$0.426905\pi$
$998$		0			0
$999$	32.4660	+	20.1854i	1.02718	+	0.638638i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4020.2.f.a.401.17	✓	46
3.2	odd	2		4020.2.f.b.401.29	yes	46
67.66	odd	2		4020.2.f.b.401.30	yes	46
201.200	even	2	inner	4020.2.f.a.401.18	yes	46

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4020.2.f.a.401.17	✓	46	1.1	even	1	trivial
4020.2.f.a.401.18	yes	46	201.200	even	2	inner
4020.2.f.b.401.29	yes	46	3.2	odd	2
4020.2.f.b.401.30	yes	46	67.66	odd	2

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.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.574646 - 1.63395i) q^{3} -1.00000 q^{5} -4.28520i q^{7} +(-2.33956 + 1.87788i) q^{9} +O(q^{10})\)	\(q+(-0.574646 - 1.63395i) q^{3} -1.00000 q^{5} -4.28520i q^{7} +(-2.33956 + 1.87788i) q^{9} +1.89841 q^{11} +0.406163i q^{13} +(0.574646 + 1.63395i) q^{15} +4.17771i q^{17} -5.36381 q^{19} +(-7.00179 + 2.46248i) q^{21} +6.34204i q^{23} +1.00000 q^{25} +(4.41278 + 2.74360i) q^{27} -3.90840i q^{29} +3.07537i q^{31} +(-1.09091 - 3.10190i) q^{33} +4.28520i q^{35} +7.35727 q^{37} +(0.663648 - 0.233400i) q^{39} +6.75844 q^{41} +7.33793i q^{43} +(2.33956 - 1.87788i) q^{45} +0.438204i q^{47} -11.3630 q^{49} +(6.82616 - 2.40071i) q^{51} -5.23027 q^{53} -1.89841 q^{55} +(3.08230 + 8.76419i) q^{57} +12.3982i q^{59} +7.71806i q^{61} +(8.04711 + 10.0255i) q^{63} -0.406163i q^{65} +(-1.79086 + 7.98704i) q^{67} +(10.3626 - 3.64443i) q^{69} +2.28010i q^{71} +0.538920 q^{73} +(-0.574646 - 1.63395i) q^{75} -8.13507i q^{77} -1.67546i q^{79} +(1.94711 - 8.78685i) q^{81} -3.18158i q^{83} -4.17771i q^{85} +(-6.38612 + 2.24595i) q^{87} -10.7644i q^{89} +1.74049 q^{91} +(5.02499 - 1.76725i) q^{93} +5.36381 q^{95} -7.69321i q^{97} +(-4.44145 + 3.56499i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q - 46 q^{5} - 4 q^{9}+O(q^{10}) \) 46 * q - 46 * q^5 - 4 * q^9	\( 46 q - 46 q^{5} - 4 q^{9} + 8 q^{19} + 46 q^{25} - 18 q^{27} + 4 q^{33} - 8 q^{37} - 12 q^{39} - 4 q^{41} + 4 q^{45} - 62 q^{49} - 8 q^{51} - 8 q^{53} - 12 q^{57} - 10 q^{63} - 14 q^{67} - 40 q^{73} - 12 q^{81} - 4 q^{91} + 2 q^{93} - 8 q^{95}+O(q^{100}) \) 46 * q - 46 * q^5 - 4 * q^9 + 8 * q^19 + 46 * q^25 - 18 * q^27 + 4 * q^33 - 8 * q^37 - 12 * q^39 - 4 * q^41 + 4 * q^45 - 62 * q^49 - 8 * q^51 - 8 * q^53 - 12 * q^57 - 10 * q^63 - 14 * q^67 - 40 * q^73 - 12 * q^81 - 4 * q^91 + 2 * q^93 - 8 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

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