LMFDB - Embedded newform 4020.2.g.c.1609.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4020,2,Mod(1609,4020)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4020, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))

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

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

chi := DirichletCharacter("4020.1609");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$32.0998616126$
Analytic rank:	$0$
Dimension:	$38$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1609.2
Character	$\chi$	$=$	4020.1609
Dual form			4020.2.g.c.1609.21

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +(-2.22744 + 0.196195i) q^{5} +0.128273i q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +(-2.22744 + 0.196195i) q^{5} +0.128273i q^{7} -1.00000 q^{9} +0.0389925 q^{11} -3.81852i q^{13} +(0.196195 + 2.22744i) q^{15} +6.63903i q^{17} +7.30512 q^{19} +0.128273 q^{21} -3.64921i q^{23} +(4.92301 - 0.874027i) q^{25} +1.00000i q^{27} -8.25219 q^{29} -7.07939 q^{31} -0.0389925i q^{33} +(-0.0251666 - 0.285721i) q^{35} -7.37252i q^{37} -3.81852 q^{39} +1.87351 q^{41} -7.72398i q^{43} +(2.22744 - 0.196195i) q^{45} +9.62406i q^{47} +6.98355 q^{49} +6.63903 q^{51} -3.82384i q^{53} +(-0.0868535 + 0.00765013i) q^{55} -7.30512i q^{57} +9.48802 q^{59} -6.69509 q^{61} -0.128273i q^{63} +(0.749174 + 8.50553i) q^{65} -1.00000i q^{67} -3.64921 q^{69} +4.88025 q^{71} -0.309819i q^{73} +(-0.874027 - 4.92301i) q^{75} +0.00500168i q^{77} -5.72909 q^{79} +1.00000 q^{81} -14.6840i q^{83} +(-1.30255 - 14.7881i) q^{85} +8.25219i q^{87} -11.6793 q^{89} +0.489813 q^{91} +7.07939i q^{93} +(-16.2718 + 1.43323i) q^{95} +7.10494i q^{97} -0.0389925 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 38 q - 2 q^{5} - 38 q^{9}+O(q^{10}) $ 38 * q - 2 * q^5 - 38 * q^9	$ 38 q - 2 q^{5} - 38 q^{9} + 24 q^{11} + 2 q^{15} - 16 q^{19} + 12 q^{21} + 4 q^{25} - 56 q^{29} - 4 q^{31} - 2 q^{35} + 60 q^{41} + 2 q^{45} - 70 q^{49} + 12 q^{55} - 52 q^{59} + 48 q^{61} + 16 q^{65} - 12 q^{69} + 12 q^{75} - 24 q^{79} + 38 q^{81} + 16 q^{85} - 76 q^{89} + 44 q^{91} + 36 q^{95} - 24 q^{99}+O(q^{100}) $ 38 * q - 2 * q^5 - 38 * q^9 + 24 * q^11 + 2 * q^15 - 16 * q^19 + 12 * q^21 + 4 * q^25 - 56 * q^29 - 4 * q^31 - 2 * q^35 + 60 * q^41 + 2 * q^45 - 70 * q^49 + 12 * q^55 - 52 * q^59 + 48 * q^61 + 16 * q^65 - 12 * q^69 + 12 * q^75 - 24 * q^79 + 38 * q^81 + 16 * q^85 - 76 * q^89 + 44 * q^91 + 36 * q^95 - 24 * q^99

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		−	1.00000i		−	0.577350i
$3$		−	1.00000i		−	0.577350i
$4$		0			0
$5$	−2.22744	+	0.196195i	−0.996143	+	0.0877411i
$5$	−2.22744	+	0.196195i	−0.996143	+	0.0877411i
$6$		0			0
$7$			0.128273i			0.0484827i	0.999706	+	0.0242413i	$0.00771701\pi$
$7$			0.128273i			0.0484827i	−0.999706	+	0.0242413i	$0.992283\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	0.0389925			0.0117567			0.00587834	−	0.999983i	$-0.498129\pi$
$11$	0.0389925			0.0117567			0.00587834	+	0.999983i	$0.498129\pi$
$12$		0			0
$13$		−	3.81852i		−	1.05907i	−0.848289	−	0.529533i	$-0.822367\pi$
$13$		−	3.81852i		−	1.05907i	0.848289	−	0.529533i	$-0.177633\pi$
$14$		0			0
$15$	0.196195	+	2.22744i	0.0506573	+	0.575124i
$16$		0			0
$17$			6.63903i			1.61020i	0.593137	+	0.805101i	$0.297889\pi$
$17$			6.63903i			1.61020i	−0.593137	+	0.805101i	$0.702111\pi$
$18$		0			0
$19$	7.30512			1.67591			0.837955	−	0.545739i	$-0.183751\pi$
$19$	7.30512			1.67591			0.837955	+	0.545739i	$0.183751\pi$
$20$		0			0
$21$	0.128273			0.0279915
$22$		0			0
$23$		−	3.64921i		−	0.760914i	−0.924799	−	0.380457i	$-0.875767\pi$
$23$		−	3.64921i		−	0.760914i	0.924799	−	0.380457i	$-0.124233\pi$
$24$		0			0
$25$	4.92301	−	0.874027i	0.984603	−	0.174805i
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$	−8.25219			−1.53239			−0.766197	−	0.642606i	$-0.777853\pi$
$29$	−8.25219			−1.53239			−0.766197	+	0.642606i	$0.777853\pi$
$30$		0			0
$31$	−7.07939			−1.27150			−0.635748	−	0.771896i	$-0.719308\pi$
$31$	−7.07939			−1.27150			−0.635748	+	0.771896i	$0.719308\pi$
$32$		0			0
$33$		−	0.0389925i		−	0.00678772i
$34$		0			0
$35$	−0.0251666	−	0.285721i	−0.00425392	−	0.0482957i
$36$		0			0
$37$		−	7.37252i		−	1.21204i	−0.795451	−	0.606018i	$-0.792766\pi$
$37$		−	7.37252i		−	1.21204i	0.795451	−	0.606018i	$-0.207234\pi$
$38$		0			0
$39$	−3.81852			−0.611452
$40$		0			0
$41$	1.87351			0.292593			0.146296	−	0.989241i	$-0.453265\pi$
$41$	1.87351			0.292593			0.146296	+	0.989241i	$0.453265\pi$
$42$		0			0
$43$		−	7.72398i		−	1.17790i	−0.808171	−	0.588948i	$-0.799542\pi$
$43$		−	7.72398i		−	1.17790i	0.808171	−	0.588948i	$-0.200458\pi$
$44$		0			0
$45$	2.22744	−	0.196195i	0.332048	−	0.0292470i
$46$		0			0
$47$			9.62406i			1.40381i	0.712269	+	0.701907i	$0.247668\pi$
$47$			9.62406i			1.40381i	−0.712269	+	0.701907i	$0.752332\pi$
$48$		0			0
$49$	6.98355			0.997649
$50$		0			0
$51$	6.63903			0.929651
$52$		0			0
$53$		−	3.82384i		−	0.525244i	−0.964899	−	0.262622i	$-0.915413\pi$
$53$		−	3.82384i		−	0.525244i	0.964899	−	0.262622i	$-0.0845872\pi$
$54$		0			0
$55$	−0.0868535	+	0.00765013i	−0.0117113	+	0.00103154i
$56$		0			0
$57$		−	7.30512i		−	0.967587i
$58$		0			0
$59$	9.48802			1.23524			0.617618	−	0.786478i	$-0.288098\pi$
$59$	9.48802			1.23524			0.617618	+	0.786478i	$0.288098\pi$
$60$		0			0
$61$	−6.69509			−0.857218			−0.428609	−	0.903490i	$-0.640996\pi$
$61$	−6.69509			−0.857218			−0.428609	+	0.903490i	$0.640996\pi$
$62$		0			0
$63$		−	0.128273i		−	0.0161609i
$64$		0			0
$65$	0.749174	+	8.50553i	0.0929236	+	1.05498i
$66$		0			0
$67$		−	1.00000i		−	0.122169i
$67$		−	1.00000i		−	0.122169i
$68$		0			0
$69$	−3.64921			−0.439314
$70$		0			0
$71$	4.88025			0.579179			0.289590	−	0.957151i	$-0.406481\pi$
$71$	4.88025			0.579179			0.289590	+	0.957151i	$0.406481\pi$
$72$		0			0
$73$		−	0.309819i		−	0.0362615i	−0.999836	−	0.0181308i	$-0.994228\pi$
$73$		−	0.309819i		−	0.0362615i	0.999836	−	0.0181308i	$-0.00577152\pi$
$74$		0			0
$75$	−0.874027	−	4.92301i	−0.100924	−	0.568461i
$76$		0			0
$77$			0.00500168i			0.000569995i
$78$		0			0
$79$	−5.72909			−0.644573			−0.322286	−	0.946642i	$-0.604451\pi$
$79$	−5.72909			−0.644573			−0.322286	+	0.946642i	$0.604451\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	14.6840i		−	1.61178i	−0.592066	−	0.805889i	$-0.701688\pi$
$83$		−	14.6840i		−	1.61178i	0.592066	−	0.805889i	$-0.298312\pi$
$84$		0			0
$85$	−1.30255	−	14.7881i	−0.141281	−	1.60399i
$86$		0			0
$87$			8.25219i			0.884728i
$88$		0			0
$89$	−11.6793			−1.23800			−0.619000	−	0.785391i	$-0.712462\pi$
$89$	−11.6793			−1.23800			−0.619000	+	0.785391i	$0.712462\pi$
$90$		0			0
$91$	0.489813			0.0513464
$92$		0			0
$93$			7.07939i			0.734099i
$94$		0			0
$95$	−16.2718	+	1.43323i	−1.66945	+	0.147046i
$96$		0			0
$97$			7.10494i			0.721397i	0.932682	+	0.360699i	$0.117462\pi$
$97$			7.10494i			0.721397i	−0.932682	+	0.360699i	$0.882538\pi$
$98$		0			0
$99$	−0.0389925			−0.00391889
$100$		0			0
$101$	−10.7135			−1.06604			−0.533018	−	0.846104i	$-0.678942\pi$
$101$	−10.7135			−1.06604			−0.533018	+	0.846104i	$0.678942\pi$
$102$		0			0
$103$			14.2892i			1.40796i	0.710220	+	0.703980i	$0.248595\pi$
$103$			14.2892i			1.40796i	−0.710220	+	0.703980i	$0.751405\pi$
$104$		0			0
$105$	−0.285721	+	0.0251666i	−0.0278835	+	0.00245600i
$106$		0			0
$107$			8.57849i			0.829314i	0.909978	+	0.414657i	$0.136098\pi$
$107$			8.57849i			0.829314i	−0.909978	+	0.414657i	$0.863902\pi$
$108$		0			0
$109$	−19.4768			−1.86554			−0.932771	−	0.360470i	$-0.882616\pi$
$109$	−19.4768			−1.86554			−0.932771	+	0.360470i	$0.882616\pi$
$110$		0			0
$111$	−7.37252			−0.699769
$112$		0			0
$113$		−	13.6083i		−	1.28016i	−0.768307	−	0.640082i	$-0.778900\pi$
$113$		−	13.6083i		−	1.28016i	0.768307	−	0.640082i	$-0.221100\pi$
$114$		0			0
$115$	0.715958	+	8.12842i	0.0667634	+	0.757979i
$116$		0			0
$117$			3.81852i			0.353022i
$118$		0			0
$119$	−0.851610			−0.0780669
$120$		0			0
$121$	−10.9985			−0.999862
$122$		0			0
$123$		−	1.87351i		−	0.168929i
$124$		0			0
$125$	−10.7943	+	2.91272i	−0.965468	+	0.260521i
$126$		0			0
$127$		−	13.3524i		−	1.18483i	−0.805631	−	0.592417i	$-0.798174\pi$
$127$		−	13.3524i		−	1.18483i	0.805631	−	0.592417i	$-0.201826\pi$
$128$		0			0
$129$	−7.72398			−0.680058
$130$		0			0
$131$	−10.7666			−0.940684			−0.470342	−	0.882484i	$-0.655869\pi$
$131$	−10.7666			−0.940684			−0.470342	+	0.882484i	$0.655869\pi$
$132$		0			0
$133$			0.937051i			0.0812526i
$134$		0			0
$135$	−0.196195	−	2.22744i	−0.0168858	−	0.191708i
$136$		0			0
$137$			8.39755i			0.717451i	0.933443	+	0.358726i	$0.116789\pi$
$137$			8.39755i			0.717451i	−0.933443	+	0.358726i	$0.883211\pi$
$138$		0			0
$139$	−20.3819			−1.72877			−0.864387	−	0.502827i	$-0.832293\pi$
$139$	−20.3819			−1.72877			−0.864387	+	0.502827i	$0.832293\pi$
$140$		0			0
$141$	9.62406			0.810492
$142$		0			0
$143$		−	0.148893i		−	0.0124511i
$144$		0			0
$145$	18.3813	−	1.61904i	1.52648	−	0.134454i
$146$		0			0
$147$		−	6.98355i		−	0.575993i
$148$		0			0
$149$	−8.64717			−0.708404			−0.354202	−	0.935169i	$-0.615247\pi$
$149$	−8.64717			−0.708404			−0.354202	+	0.935169i	$0.615247\pi$
$150$		0			0
$151$	−11.9792			−0.974854			−0.487427	−	0.873164i	$-0.662065\pi$
$151$	−11.9792			−0.974854			−0.487427	+	0.873164i	$0.662065\pi$
$152$		0			0
$153$		−	6.63903i		−	0.536734i
$154$		0			0
$155$	15.7690	−	1.38894i	1.26659	−	0.111563i
$156$		0			0
$157$		−	8.35135i		−	0.666510i	−0.942837	−	0.333255i	$-0.891853\pi$
$157$		−	8.35135i		−	0.666510i	0.942837	−	0.333255i	$-0.108147\pi$
$158$		0			0
$159$	−3.82384			−0.303250
$160$		0			0
$161$	0.468096			0.0368911
$162$		0			0
$163$		−	5.06915i		−	0.397046i	−0.980096	−	0.198523i	$-0.936385\pi$
$163$		−	5.06915i		−	0.397046i	0.980096	−	0.198523i	$-0.0636145\pi$
$164$		0			0
$165$	0.00765013	+	0.0868535i	0.000595562	+	0.00676154i
$166$		0			0
$167$			0.558745i			0.0432370i	0.999766	+	0.0216185i	$0.00688191\pi$
$167$			0.558745i			0.0432370i	−0.999766	+	0.0216185i	$0.993118\pi$
$168$		0			0
$169$	−1.58107			−0.121621
$170$		0			0
$171$	−7.30512			−0.558637
$172$		0			0
$173$			13.7845i			1.04802i	0.851714	+	0.524008i	$0.175564\pi$
$173$			13.7845i			1.04802i	−0.851714	+	0.524008i	$0.824436\pi$
$174$		0			0
$175$	0.112114	+	0.631490i	0.00847504	+	0.0477362i
$176$		0			0
$177$		−	9.48802i		−	0.713163i
$178$		0			0
$179$	8.05610			0.602141			0.301071	−	0.953602i	$-0.402656\pi$
$179$	8.05610			0.602141			0.301071	+	0.953602i	$0.402656\pi$
$180$		0			0
$181$	14.8933			1.10701			0.553506	−	0.832845i	$-0.313289\pi$
$181$	14.8933			1.10701			0.553506	+	0.832845i	$0.313289\pi$
$182$		0			0
$183$			6.69509i			0.494915i
$184$		0			0
$185$	1.44645	+	16.4219i	0.106345	+	1.20736i
$186$		0			0
$187$			0.258872i			0.0189306i
$188$		0			0
$189$	−0.128273			−0.00933050
$190$		0			0
$191$	3.66876			0.265462			0.132731	−	0.991152i	$-0.457625\pi$
$191$	3.66876			0.265462			0.132731	+	0.991152i	$0.457625\pi$
$192$		0			0
$193$		−	12.1084i		−	0.871585i	−0.900047	−	0.435793i	$-0.856468\pi$
$193$		−	12.1084i		−	0.871585i	0.900047	−	0.435793i	$-0.143532\pi$
$194$		0			0
$195$	8.50553	−	0.749174i	0.609094	−	0.0536495i
$196$		0			0
$197$		−	3.59577i		−	0.256188i	−0.991762	−	0.128094i	$-0.959114\pi$
$197$		−	3.59577i		−	0.256188i	0.991762	−	0.128094i	$-0.0408859\pi$
$198$		0			0
$199$	−7.80795			−0.553491			−0.276745	−	0.960943i	$-0.589256\pi$
$199$	−7.80795			−0.553491			−0.276745	+	0.960943i	$0.589256\pi$
$200$		0			0
$201$	−1.00000			−0.0705346
$202$		0			0
$203$		−	1.05853i		−	0.0742946i
$204$		0			0
$205$	−4.17314	+	0.367573i	−0.291464	+	0.0256724i
$206$		0			0
$207$			3.64921i			0.253638i
$208$		0			0
$209$	0.284845			0.0197031
$210$		0			0
$211$	−23.8266			−1.64029			−0.820146	−	0.572154i	$-0.806108\pi$
$211$	−23.8266			−1.64029			−0.820146	+	0.572154i	$0.806108\pi$
$212$		0			0
$213$		−	4.88025i		−	0.334389i
$214$		0			0
$215$	1.51541	+	17.2047i	0.103350	+	1.17335i
$216$		0			0
$217$		−	0.908096i		−	0.0616456i
$218$		0			0
$219$	−0.309819			−0.0209356
$220$		0			0
$221$	25.3513			1.70531
$222$		0			0
$223$		−	8.63279i		−	0.578094i	−0.957315	−	0.289047i	$-0.906662\pi$
$223$		−	8.63279i		−	0.578094i	0.957315	−	0.289047i	$-0.0933384\pi$
$224$		0			0
$225$	−4.92301	+	0.874027i	−0.328201	+	0.0582685i
$226$		0			0
$227$		−	8.58160i		−	0.569581i	−0.958590	−	0.284790i	$-0.908076\pi$
$227$		−	8.58160i		−	0.569581i	0.958590	−	0.284790i	$-0.0919240\pi$
$228$		0			0
$229$	−3.20564			−0.211835			−0.105917	−	0.994375i	$-0.533778\pi$
$229$	−3.20564			−0.211835			−0.105917	+	0.994375i	$0.533778\pi$
$230$		0			0
$231$	0.00500168			0.000329087
$232$		0			0
$233$			11.2934i			0.739854i	0.929061	+	0.369927i	$0.120617\pi$
$233$			11.2934i			0.739854i	−0.929061	+	0.369927i	$0.879383\pi$
$234$		0			0
$235$	−1.88819	−	21.4371i	−0.123172	−	1.39840i
$236$		0			0
$237$			5.72909i			0.372144i
$238$		0			0
$239$	−21.5672			−1.39507			−0.697534	−	0.716552i	$-0.745719\pi$
$239$	−21.5672			−1.39507			−0.697534	+	0.716552i	$0.745719\pi$
$240$		0			0
$241$	9.36087			0.602987			0.301493	−	0.953468i	$-0.402515\pi$
$241$	9.36087			0.602987			0.301493	+	0.953468i	$0.402515\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$	−15.5555	+	1.37014i	−0.993802	+	0.0875349i
$246$		0			0
$247$		−	27.8947i		−	1.77490i
$248$		0			0
$249$	−14.6840			−0.930561
$250$		0			0
$251$	−10.7667			−0.679589			−0.339794	−	0.940500i	$-0.610358\pi$
$251$	−10.7667			−0.679589			−0.339794	+	0.940500i	$0.610358\pi$
$252$		0			0
$253$		−	0.142292i		−	0.00894581i
$254$		0			0
$255$	−14.7881	+	1.30255i	−0.926065	+	0.0815686i
$256$		0			0
$257$		−	22.0839i		−	1.37756i	−0.724972	−	0.688779i	$-0.758147\pi$
$257$		−	22.0839i		−	1.37756i	0.724972	−	0.688779i	$-0.241853\pi$
$258$		0			0
$259$	0.945697			0.0587627
$260$		0			0
$261$	8.25219			0.510798
$262$		0			0
$263$			21.5995i			1.33188i	0.746006	+	0.665940i	$0.231969\pi$
$263$			21.5995i			1.33188i	−0.746006	+	0.665940i	$0.768031\pi$
$264$		0			0
$265$	0.750218	+	8.51738i	0.0460855	+	0.523218i
$266$		0			0
$267$			11.6793i			0.714759i
$268$		0			0
$269$	−6.86649			−0.418657			−0.209329	−	0.977845i	$-0.567128\pi$
$269$	−6.86649			−0.418657			−0.209329	+	0.977845i	$0.567128\pi$
$270$		0			0
$271$	−13.7804			−0.837101			−0.418550	−	0.908194i	$-0.637462\pi$
$271$	−13.7804			−0.837101			−0.418550	+	0.908194i	$0.637462\pi$
$272$		0			0
$273$		−	0.489813i		−	0.0296448i
$274$		0			0
$275$	0.191960	−	0.0340805i	0.0115757	−	0.00205513i
$276$		0			0
$277$			13.0593i			0.784660i	0.919825	+	0.392330i	$0.128331\pi$
$277$			13.0593i			0.784660i	−0.919825	+	0.392330i	$0.871669\pi$
$278$		0			0
$279$	7.07939			0.423832
$280$		0			0
$281$	−1.32410			−0.0789894			−0.0394947	−	0.999220i	$-0.512575\pi$
$281$	−1.32410			−0.0789894			−0.0394947	+	0.999220i	$0.512575\pi$
$282$		0			0
$283$			5.70161i			0.338926i	0.985537	+	0.169463i	$0.0542033\pi$
$283$			5.70161i			0.338926i	−0.985537	+	0.169463i	$0.945797\pi$
$284$		0			0
$285$	1.43323	+	16.2718i	0.0848972	+	0.963855i
$286$		0			0
$287$			0.240321i			0.0141857i
$288$		0			0
$289$	−27.0768			−1.59275
$290$		0			0
$291$	7.10494			0.416499
$292$		0			0
$293$		−	20.4885i		−	1.19695i	−0.801142	−	0.598475i	$-0.795774\pi$
$293$		−	20.4885i		−	1.19695i	0.801142	−	0.598475i	$-0.204226\pi$
$294$		0			0
$295$	−21.1340	+	1.86150i	−1.23047	+	0.108381i
$296$		0			0
$297$			0.0389925i			0.00226257i
$298$		0			0
$299$	−13.9346			−0.805858
$300$		0			0
$301$	0.990778			0.0571075
$302$		0			0
$303$			10.7135i			0.615476i
$304$		0			0
$305$	14.9129	−	1.31354i	0.853912	−	0.0752132i
$306$		0			0
$307$			30.2141i			1.72441i	0.506559	+	0.862205i	$0.330917\pi$
$307$			30.2141i			1.72441i	−0.506559	+	0.862205i	$0.669083\pi$
$308$		0			0
$309$	14.2892			0.812886
$310$		0			0
$311$	−4.73813			−0.268675			−0.134337	−	0.990936i	$-0.542891\pi$
$311$	−4.73813			−0.268675			−0.134337	+	0.990936i	$0.542891\pi$
$312$		0			0
$313$		−	28.2789i		−	1.59842i	−0.601053	−	0.799209i	$-0.705252\pi$
$313$		−	28.2789i		−	1.59842i	0.601053	−	0.799209i	$-0.294748\pi$
$314$		0			0
$315$	0.0251666	+	0.285721i	0.00141797	+	0.0160986i
$316$		0			0
$317$			17.4748i			0.981484i	0.871305	+	0.490742i	$0.163274\pi$
$317$			17.4748i			0.981484i	−0.871305	+	0.490742i	$0.836726\pi$
$318$		0			0
$319$	−0.321773			−0.0180159
$320$		0			0
$321$	8.57849			0.478805
$322$		0			0
$323$			48.4990i			2.69855i
$324$		0			0
$325$	−3.33749	−	18.7986i	−0.185131	−	1.04276i
$326$		0			0
$327$			19.4768i			1.07707i
$328$		0			0
$329$	−1.23451			−0.0680607
$330$		0			0
$331$	14.5657			0.800602			0.400301	−	0.916384i	$-0.368906\pi$
$331$	14.5657			0.800602			0.400301	+	0.916384i	$0.368906\pi$
$332$		0			0
$333$			7.37252i			0.404012i
$334$		0			0
$335$	0.196195	+	2.22744i	0.0107193	+	0.121698i
$336$		0			0
$337$		−	11.5522i		−	0.629286i	−0.949210	−	0.314643i	$-0.898115\pi$
$337$		−	11.5522i		−	0.629286i	0.949210	−	0.314643i	$-0.101885\pi$
$338$		0			0
$339$	−13.6083			−0.739103
$340$		0			0
$341$	−0.276043			−0.0149486
$342$		0			0
$343$			1.79371i			0.0968514i
$344$		0			0
$345$	8.12842	−	0.715958i	0.437619	−	0.0385459i
$346$		0			0
$347$		−	35.1190i		−	1.88529i	−0.333802	−	0.942643i	$-0.608332\pi$
$347$		−	35.1190i		−	1.88529i	0.333802	−	0.942643i	$-0.391668\pi$
$348$		0			0
$349$	13.5558			0.725625			0.362813	−	0.931862i	$-0.381817\pi$
$349$	13.5558			0.725625			0.362813	+	0.931862i	$0.381817\pi$
$350$		0			0
$351$	3.81852			0.203817
$352$		0			0
$353$		−	2.02952i		−	0.108021i	−0.998540	−	0.0540103i	$-0.982800\pi$
$353$		−	2.02952i		−	0.108021i	0.998540	−	0.0540103i	$-0.0172004\pi$
$354$		0			0
$355$	−10.8705	+	0.957482i	−0.576946	+	0.0508178i
$356$		0			0
$357$			0.851610i			0.0450720i
$358$		0			0
$359$	−7.15157			−0.377445			−0.188723	−	0.982030i	$-0.560435\pi$
$359$	−7.15157			−0.377445			−0.188723	+	0.982030i	$0.560435\pi$
$360$		0			0
$361$	34.3648			1.80867
$362$		0			0
$363$			10.9985i			0.577270i
$364$		0			0
$365$	0.0607849	+	0.690104i	0.00318163	+	0.0361217i
$366$		0			0
$367$		−	28.1192i		−	1.46781i	−0.679251	−	0.733906i	$-0.737695\pi$
$367$		−	28.1192i		−	1.46781i	0.679251	−	0.733906i	$-0.262305\pi$
$368$		0			0
$369$	−1.87351			−0.0975309
$370$		0			0
$371$	0.490495			0.0254652
$372$		0			0
$373$		−	33.3825i		−	1.72848i	−0.503079	−	0.864240i	$-0.667800\pi$
$373$		−	33.3825i		−	1.72848i	0.503079	−	0.864240i	$-0.332200\pi$
$374$		0			0
$375$	2.91272	+	10.7943i	0.150412	+	0.557413i
$376$		0			0
$377$			31.5111i			1.62291i
$378$		0			0
$379$	6.25718			0.321410			0.160705	−	0.987002i	$-0.448623\pi$
$379$	6.25718			0.321410			0.160705	+	0.987002i	$0.448623\pi$
$380$		0			0
$381$	−13.3524			−0.684064
$382$		0			0
$383$			21.9335i			1.12075i	0.828239	+	0.560375i	$0.189343\pi$
$383$			21.9335i			1.12075i	−0.828239	+	0.560375i	$0.810657\pi$
$384$		0			0
$385$	−0.000981306	−	0.0111410i	−5.00120e−5	−	0.000567797i
$386$		0			0
$387$			7.72398i			0.392632i
$388$		0			0
$389$	−15.3986			−0.780739			−0.390369	−	0.920658i	$-0.627653\pi$
$389$	−15.3986			−0.780739			−0.390369	+	0.920658i	$0.627653\pi$
$390$		0			0
$391$	24.2273			1.22523
$392$		0			0
$393$			10.7666i			0.543104i
$394$		0			0
$395$	12.7612	−	1.12402i	0.642087	−	0.0565555i
$396$		0			0
$397$			3.69910i			0.185652i	0.995682	+	0.0928262i	$0.0295901\pi$
$397$			3.69910i			0.185652i	−0.995682	+	0.0928262i	$0.970410\pi$
$398$		0			0
$399$	0.937051			0.0469112
$400$		0			0
$401$	−25.1620			−1.25653			−0.628265	−	0.777999i	$-0.716235\pi$
$401$	−25.1620			−1.25653			−0.628265	+	0.777999i	$0.716235\pi$
$402$		0			0
$403$			27.0328i			1.34660i
$404$		0			0
$405$	−2.22744	+	0.196195i	−0.110683	+	0.00974901i
$406$		0			0
$407$		−	0.287473i		−	0.0142495i
$408$		0			0
$409$	−13.9226			−0.688430			−0.344215	−	0.938891i	$-0.611855\pi$
$409$	−13.9226			−0.688430			−0.344215	+	0.938891i	$0.611855\pi$
$410$		0			0
$411$	8.39755			0.414221
$412$		0			0
$413$			1.21706i			0.0598875i
$414$		0			0
$415$	2.88093	+	32.7078i	0.141419	+	1.60556i
$416$		0			0
$417$			20.3819i			0.998108i
$418$		0			0
$419$	6.28480			0.307033			0.153516	−	0.988146i	$-0.450940\pi$
$419$	6.28480			0.307033			0.153516	+	0.988146i	$0.450940\pi$
$420$		0			0
$421$	30.2864			1.47607			0.738035	−	0.674763i	$-0.235754\pi$
$421$	30.2864			1.47607			0.738035	+	0.674763i	$0.235754\pi$
$422$		0			0
$423$		−	9.62406i		−	0.467938i
$424$		0			0
$425$	5.80270	+	32.6841i	0.281472	+	1.58541i
$426$		0			0
$427$		−	0.858799i		−	0.0415602i
$428$		0			0
$429$	−0.148893			−0.00718864
$430$		0			0
$431$	7.21790			0.347674			0.173837	−	0.984774i	$-0.444383\pi$
$431$	7.21790			0.347674			0.173837	+	0.984774i	$0.444383\pi$
$432$		0			0
$433$			21.0149i			1.00991i	0.863145	+	0.504957i	$0.168492\pi$
$433$			21.0149i			1.00991i	−0.863145	+	0.504957i	$0.831508\pi$
$434$		0			0
$435$	−1.61904	−	18.3813i	−0.0776270	−	0.881316i
$436$		0			0
$437$		−	26.6580i		−	1.27522i
$438$		0			0
$439$	37.4029			1.78514			0.892572	−	0.450905i	$-0.148899\pi$
$439$	37.4029			1.78514			0.892572	+	0.450905i	$0.148899\pi$
$440$		0			0
$441$	−6.98355			−0.332550
$442$		0			0
$443$		−	27.5717i		−	1.30997i	−0.755641	−	0.654986i	$-0.772675\pi$
$443$		−	27.5717i		−	1.30997i	0.755641	−	0.654986i	$-0.227325\pi$
$444$		0			0
$445$	26.0149	−	2.29141i	1.23322	−	0.108623i
$446$		0			0
$447$			8.64717i			0.408997i
$448$		0			0
$449$	7.43854			0.351046			0.175523	−	0.984475i	$-0.443838\pi$
$449$	7.43854			0.351046			0.175523	+	0.984475i	$0.443838\pi$
$450$		0			0
$451$	0.0730527			0.00343992
$452$		0			0
$453$			11.9792i			0.562832i
$454$		0			0
$455$	−1.09103	+	0.0960989i	−0.0511483	+	0.00450519i
$456$		0			0
$457$		−	16.6425i		−	0.778505i	−0.921131	−	0.389253i	$-0.872733\pi$
$457$		−	16.6425i		−	0.778505i	0.921131	−	0.389253i	$-0.127267\pi$
$458$		0			0
$459$	−6.63903			−0.309884
$460$		0			0
$461$	23.7119			1.10438			0.552188	−	0.833720i	$-0.313793\pi$
$461$	23.7119			1.10438			0.552188	+	0.833720i	$0.313793\pi$
$462$		0			0
$463$		−	27.3481i		−	1.27097i	−0.772112	−	0.635486i	$-0.780800\pi$
$463$		−	27.3481i		−	1.27097i	0.772112	−	0.635486i	$-0.219200\pi$
$464$		0			0
$465$	−1.38894	−	15.7690i	−0.0644106	−	0.731268i
$466$		0			0
$467$		−	17.6793i		−	0.818102i	−0.912512	−	0.409051i	$-0.865860\pi$
$467$		−	17.6793i		−	0.818102i	0.912512	−	0.409051i	$-0.134140\pi$
$468$		0			0
$469$	0.128273			0.00592310
$470$		0			0
$471$	−8.35135			−0.384810
$472$		0			0
$473$		−	0.301177i		−	0.0138481i
$474$		0			0
$475$	35.9632	−	6.38488i	1.65011	−	0.292958i
$476$		0			0
$477$			3.82384i			0.175081i
$478$		0			0
$479$	−21.5193			−0.983242			−0.491621	−	0.870809i	$-0.663595\pi$
$479$	−21.5193			−0.983242			−0.491621	+	0.870809i	$0.663595\pi$
$480$		0			0
$481$	−28.1521			−1.28363
$482$		0			0
$483$		−	0.468096i		−	0.0212991i
$484$		0			0
$485$	−1.39395	−	15.8259i	−0.0632962	−	0.718615i
$486$		0			0
$487$			35.6523i			1.61556i	0.589485	+	0.807780i	$0.299331\pi$
$487$			35.6523i			1.61556i	−0.589485	+	0.807780i	$0.700669\pi$
$488$		0			0
$489$	−5.06915			−0.229235
$490$		0			0
$491$	−37.7609			−1.70413			−0.852064	−	0.523438i	$-0.824649\pi$
$491$	−37.7609			−1.70413			−0.852064	+	0.523438i	$0.824649\pi$
$492$		0			0
$493$		−	54.7866i		−	2.46746i
$494$		0			0
$495$	0.0868535	−	0.00765013i	0.00390378	−	0.000343848i
$496$		0			0
$497$			0.626005i			0.0280802i
$498$		0			0
$499$	−16.8673			−0.755084			−0.377542	−	0.925992i	$-0.623231\pi$
$499$	−16.8673			−0.755084			−0.377542	+	0.925992i	$0.623231\pi$
$500$		0			0
$501$	0.558745			0.0249629
$502$		0			0
$503$		−	20.4352i		−	0.911162i	−0.890194	−	0.455581i	$-0.849432\pi$
$503$		−	20.4352i		−	0.911162i	0.890194	−	0.455581i	$-0.150568\pi$
$504$		0			0
$505$	23.8638	−	2.10194i	1.06192	−	0.0935351i
$506$		0			0
$507$			1.58107i			0.0702180i
$508$		0			0
$509$	37.1112			1.64492			0.822462	−	0.568820i	$-0.192600\pi$
$509$	37.1112			1.64492			0.822462	+	0.568820i	$0.192600\pi$
$510$		0			0
$511$	0.0397414			0.00175806
$512$		0			0
$513$			7.30512i			0.322529i
$514$		0			0
$515$	−2.80348	−	31.8285i	−0.123536	−	1.40253i
$516$		0			0
$517$			0.375266i			0.0165042i
$518$		0			0
$519$	13.7845			0.605072
$520$		0			0
$521$	6.14633			0.269276			0.134638	−	0.990895i	$-0.457013\pi$
$521$	6.14633			0.269276			0.134638	+	0.990895i	$0.457013\pi$
$522$		0			0
$523$		−	22.5742i		−	0.987103i	−0.869717	−	0.493551i	$-0.835699\pi$
$523$		−	22.5742i		−	0.987103i	0.869717	−	0.493551i	$-0.164301\pi$
$524$		0			0
$525$	0.631490	−	0.112114i	0.0275605	−	0.00489306i
$526$		0			0
$527$		−	47.0003i		−	2.04737i
$528$		0			0
$529$	9.68324			0.421010
$530$		0			0
$531$	−9.48802			−0.411745
$532$		0			0
$533$		−	7.15402i		−	0.309875i
$534$		0			0
$535$	−1.68306	−	19.1081i	−0.0727649	−	0.826116i
$536$		0			0
$537$		−	8.05610i		−	0.347646i
$538$		0			0
$539$	0.272306			0.0117290
$540$		0			0
$541$	10.5657			0.454254			0.227127	−	0.973865i	$-0.427067\pi$
$541$	10.5657			0.454254			0.227127	+	0.973865i	$0.427067\pi$
$542$		0			0
$543$		−	14.8933i		−	0.639134i
$544$		0			0
$545$	43.3835	−	3.82126i	1.85835	−	0.163685i
$546$		0			0
$547$			38.7727i			1.65780i	0.559397	+	0.828900i	$0.311033\pi$
$547$			38.7727i			1.65780i	−0.559397	+	0.828900i	$0.688967\pi$
$548$		0			0
$549$	6.69509			0.285739
$550$		0			0
$551$	−60.2833			−2.56815
$552$		0			0
$553$		−	0.734888i		−	0.0312506i
$554$		0			0
$555$	16.4219	−	1.44645i	0.697070	−	0.0613985i
$556$		0			0
$557$		−	16.3818i		−	0.694119i	−0.937843	−	0.347059i	$-0.887180\pi$
$557$		−	16.3818i		−	0.694119i	0.937843	−	0.347059i	$-0.112820\pi$
$558$		0			0
$559$	−29.4941			−1.24747
$560$		0			0
$561$	0.258872			0.0109296
$562$		0			0
$563$			14.7878i			0.623231i	0.950208	+	0.311616i	$0.100870\pi$
$563$			14.7878i			0.623231i	−0.950208	+	0.311616i	$0.899130\pi$
$564$		0			0
$565$	2.66989	+	30.3118i	0.112323	+	1.27523i
$566$		0			0
$567$			0.128273i			0.00538696i
$568$		0			0
$569$	35.0863			1.47089			0.735447	−	0.677582i	$-0.236972\pi$
$569$	35.0863			1.47089			0.735447	+	0.677582i	$0.236972\pi$
$570$		0			0
$571$	10.3369			0.432585			0.216292	−	0.976329i	$-0.430604\pi$
$571$	10.3369			0.432585			0.216292	+	0.976329i	$0.430604\pi$
$572$		0			0
$573$		−	3.66876i		−	0.153265i
$574$		0			0
$575$	−3.18951	−	17.9651i	−0.133012	−	0.749198i
$576$		0			0
$577$		−	1.76255i		−	0.0733760i	−0.999327	−	0.0366880i	$-0.988319\pi$
$577$		−	1.76255i		−	0.0733760i	0.999327	−	0.0366880i	$-0.0116808\pi$
$578$		0			0
$579$	−12.1084			−0.503210
$580$		0			0
$581$	1.88356			0.0781434
$582$		0			0
$583$		−	0.149101i		−	0.00617512i
$584$		0			0
$585$	−0.749174	−	8.50553i	−0.0309745	−	0.351661i
$586$		0			0
$587$			4.55117i			0.187847i	0.995579	+	0.0939235i	$0.0299409\pi$
$587$			4.55117i			0.187847i	−0.995579	+	0.0939235i	$0.970059\pi$
$588$		0			0
$589$	−51.7158			−2.13091
$590$		0			0
$591$	−3.59577			−0.147910
$592$		0			0
$593$			2.67480i			0.109841i	0.998491	+	0.0549203i	$0.0174905\pi$
$593$			2.67480i			0.109841i	−0.998491	+	0.0549203i	$0.982510\pi$
$594$		0			0
$595$	1.89691	−	0.167082i	0.0777658	−	0.00684968i
$596$		0			0
$597$			7.80795i			0.319558i
$598$		0			0
$599$	−26.0897			−1.06600			−0.532998	−	0.846116i	$-0.678935\pi$
$599$	−26.0897			−1.06600			−0.532998	+	0.846116i	$0.678935\pi$
$600$		0			0
$601$	−37.9540			−1.54818			−0.774088	−	0.633078i	$-0.781791\pi$
$601$	−37.9540			−1.54818			−0.774088	+	0.633078i	$0.781791\pi$
$602$		0			0
$603$			1.00000i			0.0407231i
$604$		0			0
$605$	24.4985	−	2.15785i	0.996006	−	0.0877290i
$606$		0			0
$607$			14.7399i			0.598273i	0.954210	+	0.299137i	$0.0966986\pi$
$607$			14.7399i			0.598273i	−0.954210	+	0.299137i	$0.903301\pi$
$608$		0			0
$609$	−1.05853			−0.0428940
$610$		0			0
$611$	36.7497			1.48673
$612$		0			0
$613$			13.6655i			0.551944i	0.961166	+	0.275972i	$0.0889996\pi$
$613$			13.6655i			0.551944i	−0.961166	+	0.275972i	$0.911000\pi$
$614$		0			0
$615$	0.367573	+	4.17314i	0.0148220	+	0.168277i
$616$		0			0
$617$			40.4183i			1.62718i	0.581438	+	0.813590i	$0.302490\pi$
$617$			40.4183i			1.62718i	−0.581438	+	0.813590i	$0.697510\pi$
$618$		0			0
$619$	−4.32869			−0.173985			−0.0869923	−	0.996209i	$-0.527726\pi$
$619$	−4.32869			−0.173985			−0.0869923	+	0.996209i	$0.527726\pi$
$620$		0			0
$621$	3.64921			0.146438
$622$		0			0
$623$		−	1.49814i		−	0.0600215i
$624$		0			0
$625$	23.4722	−	8.60570i	0.938886	−	0.344228i
$626$		0			0
$627$		−	0.284845i		−	0.0113756i
$628$		0			0
$629$	48.9464			1.95162
$630$		0			0
$631$	−13.0385			−0.519054			−0.259527	−	0.965736i	$-0.583567\pi$
$631$	−13.0385			−0.519054			−0.259527	+	0.965736i	$0.583567\pi$
$632$		0			0
$633$			23.8266i			0.947023i
$634$		0			0
$635$	2.61968	+	29.7417i	0.103959	+	1.18026i
$636$		0			0
$637$		−	26.6668i		−	1.05658i
$638$		0			0
$639$	−4.88025			−0.193060
$640$		0			0
$641$	17.0995			0.675391			0.337695	−	0.941255i	$-0.390353\pi$
$641$	17.0995			0.675391			0.337695	+	0.941255i	$0.390353\pi$
$642$		0			0
$643$			0.255024i			0.0100571i	0.999987	+	0.00502857i	$0.00160065\pi$
$643$			0.255024i			0.0100571i	−0.999987	+	0.00502857i	$0.998399\pi$
$644$		0			0
$645$	17.2047	−	1.51541i	0.677435	−	0.0596691i
$646$		0			0
$647$			31.4978i			1.23831i	0.785270	+	0.619153i	$0.212524\pi$
$647$			31.4978i			1.23831i	−0.785270	+	0.619153i	$0.787476\pi$
$648$		0			0
$649$	0.369961			0.0145223
$650$		0			0
$651$	−0.908096			−0.0355911
$652$		0			0
$653$			24.0739i			0.942083i	0.882111	+	0.471042i	$0.156122\pi$
$653$			24.0739i			0.942083i	−0.882111	+	0.471042i	$0.843878\pi$
$654$		0			0
$655$	23.9820	−	2.11236i	0.937056	−	0.0825367i
$656$		0			0
$657$			0.309819i			0.0120872i
$658$		0			0
$659$	−48.0291			−1.87095			−0.935474	−	0.353397i	$-0.885027\pi$
$659$	−48.0291			−1.87095			−0.935474	+	0.353397i	$0.885027\pi$
$660$		0			0
$661$	24.6120			0.957295			0.478647	−	0.878007i	$-0.341127\pi$
$661$	24.6120			0.957295			0.478647	+	0.878007i	$0.341127\pi$
$662$		0			0
$663$		−	25.3513i		−	0.984562i
$664$		0			0
$665$	−0.183845	−	2.08723i	−0.00712919	−	0.0809392i
$666$		0			0
$667$			30.1140i			1.16602i
$668$		0			0
$669$	−8.63279			−0.333763
$670$		0			0
$671$	−0.261058			−0.0100780
$672$		0			0
$673$		−	27.8417i		−	1.07322i	−0.843831	−	0.536609i	$-0.819705\pi$
$673$		−	27.8417i		−	1.07322i	0.843831	−	0.536609i	$-0.180295\pi$
$674$		0			0
$675$	0.874027	+	4.92301i	0.0336413	+	0.189487i
$676$		0			0
$677$			43.7995i			1.68335i	0.539983	+	0.841676i	$0.318431\pi$
$677$			43.7995i			1.68335i	−0.539983	+	0.841676i	$0.681569\pi$
$678$		0			0
$679$	−0.911372			−0.0349753
$680$		0			0
$681$	−8.58160			−0.328848
$682$		0			0
$683$			2.92819i			0.112044i	0.998430	+	0.0560221i	$0.0178417\pi$
$683$			2.92819i			0.112044i	−0.998430	+	0.0560221i	$0.982158\pi$
$684$		0			0
$685$	−1.64756	−	18.7051i	−0.0629500	−	0.714684i
$686$		0			0
$687$			3.20564i			0.122303i
$688$		0			0
$689$	−14.6014			−0.556268
$690$		0			0
$691$	5.44751			0.207233			0.103617	−	0.994617i	$-0.466959\pi$
$691$	5.44751			0.207233			0.103617	+	0.994617i	$0.466959\pi$
$692$		0			0
$693$		−	0.00500168i		−	0.000189998i
$694$		0			0
$695$	45.3997	−	3.99884i	1.72211	−	0.151685i
$696$		0			0
$697$			12.4383i			0.471134i
$698$		0			0
$699$	11.2934			0.427155
$700$		0			0
$701$	40.7153			1.53780			0.768898	−	0.639372i	$-0.220806\pi$
$701$	40.7153			1.53780			0.768898	+	0.639372i	$0.220806\pi$
$702$		0			0
$703$		−	53.8572i		−	2.03126i
$704$		0			0
$705$	−21.4371	+	1.88819i	−0.807367	+	0.0711135i
$706$		0			0
$707$		−	1.37426i		−	0.0516843i
$708$		0			0
$709$	32.6966			1.22795			0.613974	−	0.789326i	$-0.289570\pi$
$709$	32.6966			1.22795			0.613974	+	0.789326i	$0.289570\pi$
$710$		0			0
$711$	5.72909			0.214858
$712$		0			0
$713$			25.8342i			0.967499i
$714$		0			0
$715$	0.0292122	+	0.331652i	0.00109247	+	0.0124031i
$716$		0			0
$717$			21.5672i			0.805442i
$718$		0			0
$719$	−25.1366			−0.937436			−0.468718	−	0.883348i	$-0.655284\pi$
$719$	−25.1366			−0.937436			−0.468718	+	0.883348i	$0.655284\pi$
$720$		0			0
$721$	−1.83292			−0.0682616
$722$		0			0
$723$		−	9.36087i		−	0.348135i
$724$		0			0
$725$	−40.6257	+	7.21264i	−1.50880	+	0.267871i
$726$		0			0
$727$		−	15.0875i		−	0.559564i	−0.960064	−	0.279782i	$-0.909738\pi$
$727$		−	15.0875i		−	0.559564i	0.960064	−	0.279782i	$-0.0902622\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	51.2797			1.89665
$732$		0			0
$733$		−	21.1707i		−	0.781959i	−0.920399	−	0.390980i	$-0.872136\pi$
$733$		−	21.1707i		−	0.781959i	0.920399	−	0.390980i	$-0.127864\pi$
$734$		0			0
$735$	1.37014	+	15.5555i	0.0505383	+	0.573772i
$736$		0			0
$737$		−	0.0389925i		−	0.00143631i
$738$		0			0
$739$	−33.3401			−1.22643			−0.613217	−	0.789914i	$-0.710125\pi$
$739$	−33.3401			−1.22643			−0.613217	+	0.789914i	$0.710125\pi$
$740$		0			0
$741$	−27.8947			−1.02474
$742$		0			0
$743$		−	16.8561i		−	0.618391i	−0.950998	−	0.309196i	$-0.899940\pi$
$743$		−	16.8561i		−	0.618391i	0.950998	−	0.309196i	$-0.100060\pi$
$744$		0			0
$745$	19.2611	−	1.69653i	0.705672	−	0.0621561i
$746$		0			0
$747$			14.6840i			0.537260i
$748$		0			0
$749$	−1.10039			−0.0402074
$750$		0			0
$751$	9.45818			0.345134			0.172567	−	0.984998i	$-0.444794\pi$
$751$	9.45818			0.345134			0.172567	+	0.984998i	$0.444794\pi$
$752$		0			0
$753$			10.7667i			0.392361i
$754$		0			0
$755$	26.6830	−	2.35026i	0.971095	−	0.0855348i
$756$		0			0
$757$			37.9574i			1.37959i	0.724007	+	0.689793i	$0.242298\pi$
$757$			37.9574i			1.37959i	−0.724007	+	0.689793i	$0.757702\pi$
$758$		0			0
$759$	−0.142292			−0.00516487
$760$		0			0
$761$	29.9280			1.08489			0.542444	−	0.840092i	$-0.317499\pi$
$761$	29.9280			1.08489			0.542444	+	0.840092i	$0.317499\pi$
$762$		0			0
$763$		−	2.49835i		−	0.0904465i
$764$		0			0
$765$	1.30255	+	14.7881i	0.0470936	+	0.534664i
$766$		0			0
$767$		−	36.2302i		−	1.30820i
$768$		0			0
$769$	5.03061			0.181408			0.0907042	−	0.995878i	$-0.471088\pi$
$769$	5.03061			0.181408			0.0907042	+	0.995878i	$0.471088\pi$
$770$		0			0
$771$	−22.0839			−0.795333
$772$		0			0
$773$		−	18.4372i		−	0.663141i	−0.943430	−	0.331571i	$-0.892421\pi$
$773$		−	18.4372i		−	0.663141i	0.943430	−	0.331571i	$-0.107579\pi$
$774$		0			0
$775$	−34.8520	+	6.18758i	−1.25192	+	0.222264i
$776$		0			0
$777$		−	0.945697i		−	0.0339267i
$778$		0			0
$779$	13.6862			0.490359
$780$		0			0
$781$	0.190293			0.00680922
$782$		0			0
$783$		−	8.25219i		−	0.294909i
$784$		0			0
$785$	1.63849	+	18.6022i	0.0584804	+	0.663940i
$786$		0			0
$787$		−	35.9859i		−	1.28276i	−0.767224	−	0.641379i	$-0.778363\pi$
$787$		−	35.9859i		−	1.28276i	0.767224	−	0.641379i	$-0.221637\pi$
$788$		0			0
$789$	21.5995			0.768961
$790$		0			0
$791$	1.74558			0.0620658
$792$		0			0
$793$			25.5653i			0.907850i
$794$		0			0
$795$	8.51738	−	0.750218i	0.302080	−	0.0266075i
$796$		0			0
$797$		−	37.9803i		−	1.34533i	−0.739947	−	0.672665i	$-0.765149\pi$
$797$		−	37.9803i		−	1.34533i	0.739947	−	0.672665i	$-0.234851\pi$
$798$		0			0
$799$	−63.8945			−2.26043
$800$		0			0
$801$	11.6793			0.412666
$802$		0			0
$803$		−	0.0120806i		−	0.000426315i
$804$		0			0
$805$	−1.04266	+	0.0918381i	−0.0367489	+	0.00323687i
$806$		0			0
$807$			6.86649i			0.241712i
$808$		0			0
$809$	−11.6319			−0.408954			−0.204477	−	0.978871i	$-0.565549\pi$
$809$	−11.6319			−0.408954			−0.204477	+	0.978871i	$0.565549\pi$
$810$		0			0
$811$	−1.24573			−0.0437436			−0.0218718	−	0.999761i	$-0.506963\pi$
$811$	−1.24573			−0.0437436			−0.0218718	+	0.999761i	$0.506963\pi$
$812$		0			0
$813$			13.7804i			0.483300i
$814$		0			0
$815$	0.994542	+	11.2912i	0.0348373	+	0.395515i
$816$		0			0
$817$		−	56.4246i		−	1.97405i
$818$		0			0
$819$	−0.489813			−0.0171155
$820$		0			0
$821$	24.3315			0.849176			0.424588	−	0.905387i	$-0.360419\pi$
$821$	24.3315			0.849176			0.424588	+	0.905387i	$0.360419\pi$
$822$		0			0
$823$		−	1.47183i		−	0.0513049i	−0.999671	−	0.0256524i	$-0.991834\pi$
$823$		−	1.47183i		−	0.0513049i	0.999671	−	0.0256524i	$-0.00816632\pi$
$824$		0			0
$825$	−0.0340805	−	0.191960i	−0.00118653	−	0.00668321i
$826$		0			0
$827$		−	35.9758i		−	1.25100i	−0.780223	−	0.625501i	$-0.784895\pi$
$827$		−	35.9758i		−	1.25100i	0.780223	−	0.625501i	$-0.215105\pi$
$828$		0			0
$829$	0.293129			0.0101808			0.00509040	−	0.999987i	$-0.498380\pi$
$829$	0.293129			0.0101808			0.00509040	+	0.999987i	$0.498380\pi$
$830$		0			0
$831$	13.0593			0.453024
$832$		0			0
$833$			46.3640i			1.60642i
$834$		0			0
$835$	−0.109623	−	1.24457i	−0.00379366	−	0.0430702i
$836$		0			0
$837$		−	7.07939i		−	0.244700i
$838$		0			0
$839$	−51.0206			−1.76143			−0.880713	−	0.473651i	$-0.842936\pi$
$839$	−51.0206			−1.76143			−0.880713	+	0.473651i	$0.842936\pi$
$840$		0			0
$841$	39.0987			1.34823
$842$		0			0
$843$			1.32410i			0.0456046i
$844$		0			0
$845$	3.52175	−	0.310199i	0.121152	−	0.0106712i
$846$		0			0
$847$		−	1.41081i		−	0.0484760i
$848$		0			0
$849$	5.70161			0.195679
$850$		0			0
$851$	−26.9039			−0.922254
$852$		0			0
$853$		−	1.28002i		−	0.0438272i	−0.999760	−	0.0219136i	$-0.993024\pi$
$853$		−	1.28002i		−	0.0438272i	0.999760	−	0.0219136i	$-0.00697587\pi$
$854$		0			0
$855$	16.2718	−	1.43323i	0.556482	−	0.0490154i
$856$		0			0
$857$			20.9886i			0.716957i	0.933538	+	0.358479i	$0.116704\pi$
$857$			20.9886i			0.716957i	−0.933538	+	0.358479i	$0.883296\pi$
$858$		0			0
$859$	53.4651			1.82421			0.912103	−	0.409960i	$-0.134457\pi$
$859$	53.4651			1.82421			0.912103	+	0.409960i	$0.134457\pi$
$860$		0			0
$861$	0.240321			0.00819011
$862$		0			0
$863$		−	30.2543i		−	1.02987i	−0.857230	−	0.514934i	$-0.827817\pi$
$863$		−	30.2543i		−	1.02987i	0.857230	−	0.514934i	$-0.172183\pi$
$864$		0			0
$865$	−2.70445	−	30.7042i	−0.0919540	−	1.04397i
$866$		0			0
$867$			27.0768i			0.919576i
$868$		0			0
$869$	−0.223391			−0.00757803
$870$		0			0
$871$	−3.81852			−0.129386
$872$		0			0
$873$		−	7.10494i		−	0.240466i
$874$		0			0
$875$	−0.373623	−	1.38461i	−0.0126308	−	0.0468085i
$876$		0			0
$877$		−	56.9298i		−	1.92238i	−0.275882	−	0.961192i	$-0.588970\pi$
$877$		−	56.9298i		−	1.92238i	0.275882	−	0.961192i	$-0.411030\pi$
$878$		0			0
$879$	−20.4885			−0.691059
$880$		0			0
$881$	−20.1820			−0.679949			−0.339975	−	0.940435i	$-0.610418\pi$
$881$	−20.1820			−0.679949			−0.339975	+	0.940435i	$0.610418\pi$
$882$		0			0
$883$		−	12.3122i		−	0.414339i	−0.978305	−	0.207170i	$-0.933575\pi$
$883$		−	12.3122i		−	0.414339i	0.978305	−	0.207170i	$-0.0664252\pi$
$884$		0			0
$885$	1.86150	+	21.1340i	0.0625738	+	0.710413i
$886$		0			0
$887$		−	46.8050i		−	1.57156i	−0.618507	−	0.785779i	$-0.712262\pi$
$887$		−	46.8050i		−	1.57156i	0.618507	−	0.785779i	$-0.287738\pi$
$888$		0			0
$889$	1.71275			0.0574439
$890$		0			0
$891$	0.0389925			0.00130630
$892$		0			0
$893$			70.3050i			2.35267i
$894$		0			0
$895$	−17.9445	+	1.58057i	−0.599819	+	0.0528325i
$896$		0			0
$897$			13.9346i			0.465262i
$898$		0			0
$899$	58.4205			1.94843
$900$		0			0
$901$	25.3866			0.845749
$902$		0			0
$903$		−	0.990778i		−	0.0329710i
$904$		0			0
$905$	−33.1741	+	2.92200i	−1.10274	+	0.0971305i
$906$		0			0
$907$			38.5478i			1.27996i	0.768393	+	0.639979i	$0.221057\pi$
$907$			38.5478i			1.27996i	−0.768393	+	0.639979i	$0.778943\pi$
$908$		0			0
$909$	10.7135			0.355345
$910$		0			0
$911$	−5.15058			−0.170646			−0.0853231	−	0.996353i	$-0.527192\pi$
$911$	−5.15058			−0.170646			−0.0853231	+	0.996353i	$0.527192\pi$
$912$		0			0
$913$		−	0.572566i		−	0.0189492i
$914$		0			0
$915$	−1.31354	−	14.9129i	−0.0434244	−	0.493006i
$916$		0			0
$917$		−	1.38107i		−	0.0456069i
$918$		0			0
$919$	−23.1528			−0.763741			−0.381870	−	0.924216i	$-0.624720\pi$
$919$	−23.1528			−0.763741			−0.381870	+	0.924216i	$0.624720\pi$
$920$		0			0
$921$	30.2141			0.995589
$922$		0			0
$923$		−	18.6353i		−	0.613389i
$924$		0			0
$925$	−6.44379	−	36.2950i	−0.211870	−	1.19337i
$926$		0			0
$927$		−	14.2892i		−	0.469320i
$928$		0			0
$929$	3.27644			0.107496			0.0537482	−	0.998555i	$-0.482883\pi$
$929$	3.27644			0.107496			0.0537482	+	0.998555i	$0.482883\pi$
$930$		0			0
$931$	51.0157			1.67197
$932$		0			0
$933$			4.73813i			0.155120i
$934$		0			0
$935$	−0.0507895	−	0.576624i	−0.00166099	−	0.0188576i
$936$		0			0
$937$		−	15.8785i		−	0.518727i	−0.965780	−	0.259364i	$-0.916487\pi$
$937$		−	15.8785i		−	0.518727i	0.965780	−	0.259364i	$-0.0835128\pi$
$938$		0			0
$939$	−28.2789			−0.922847
$940$		0			0
$941$	32.1579			1.04832			0.524159	−	0.851620i	$-0.324380\pi$
$941$	32.1579			1.04832			0.524159	+	0.851620i	$0.324380\pi$
$942$		0			0
$943$		−	6.83683i		−	0.222638i
$944$		0			0
$945$	0.285721	−	0.0251666i	0.00929451	−	0.000818668i
$946$		0			0
$947$			27.2041i			0.884016i	0.897011	+	0.442008i	$0.145734\pi$
$947$			27.2041i			0.884016i	−0.897011	+	0.442008i	$0.854266\pi$
$948$		0			0
$949$	−1.18305			−0.0384034
$950$		0			0
$951$	17.4748			0.566660
$952$		0			0
$953$		−	32.0736i		−	1.03896i	−0.854481	−	0.519482i	$-0.826125\pi$
$953$		−	32.0736i		−	1.03896i	0.854481	−	0.519482i	$-0.173875\pi$
$954$		0			0
$955$	−8.17195	+	0.719792i	−0.264438	+	0.0232919i
$956$		0			0
$957$			0.321773i			0.0104015i
$958$		0			0
$959$	−1.07718			−0.0347840
$960$		0			0
$961$	19.1178			0.616703
$962$		0			0
$963$		−	8.57849i		−	0.276438i
$964$		0			0
$965$	2.37562	+	26.9709i	0.0764738	+	0.868224i
$966$		0			0
$967$		−	24.0799i		−	0.774358i	−0.922005	−	0.387179i	$-0.873450\pi$
$967$		−	24.0799i		−	0.774358i	0.922005	−	0.387179i	$-0.126550\pi$
$968$		0			0
$969$	48.4990			1.55801
$970$		0			0
$971$	−4.76129			−0.152797			−0.0763985	−	0.997077i	$-0.524342\pi$
$971$	−4.76129			−0.152797			−0.0763985	+	0.997077i	$0.524342\pi$
$972$		0			0
$973$		−	2.61446i		−	0.0838156i
$974$		0			0
$975$	−18.7986	+	3.33749i	−0.602038	+	0.106885i
$976$		0			0
$977$		−	6.80650i		−	0.217759i	−0.994055	−	0.108880i	$-0.965274\pi$
$977$		−	6.80650i		−	0.217759i	0.994055	−	0.108880i	$-0.0347263\pi$
$978$		0			0
$979$	−0.455403			−0.0145548
$980$		0			0
$981$	19.4768			0.621847
$982$		0			0
$983$			30.9051i			0.985719i	0.870109	+	0.492859i	$0.164048\pi$
$983$			30.9051i			0.985719i	−0.870109	+	0.492859i	$0.835952\pi$
$984$		0			0
$985$	0.705472	+	8.00937i	0.0224782	+	0.255200i
$986$		0			0
$987$			1.23451i			0.0392948i
$988$		0			0
$989$	−28.1864			−0.896277
$990$		0			0
$991$	−1.30741			−0.0415313			−0.0207657	−	0.999784i	$-0.506610\pi$
$991$	−1.30741			−0.0415313			−0.0207657	+	0.999784i	$0.506610\pi$
$992$		0			0
$993$		−	14.5657i		−	0.462228i
$994$		0			0
$995$	17.3918	−	1.53188i	0.551356	−	0.0485639i
$996$		0			0
$997$		−	58.9319i		−	1.86639i	−0.359370	−	0.933195i	$-0.617008\pi$
$997$		−	58.9319i		−	1.86639i	0.359370	−	0.933195i	$-0.382992\pi$
$998$		0			0
$999$	7.37252			0.233256

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4020.2.g.c.1609.2	✓	38
5.4	even	2	inner	4020.2.g.c.1609.21	yes	38

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4020.2.g.c.1609.2	✓	38	1.1	even	1	trivial
4020.2.g.c.1609.21	yes	38	5.4	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +(-2.22744 + 0.196195i) q^{5} +0.128273i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +(-2.22744 + 0.196195i) q^{5} +0.128273i q^{7} -1.00000 q^{9} +0.0389925 q^{11} -3.81852i q^{13} +(0.196195 + 2.22744i) q^{15} +6.63903i q^{17} +7.30512 q^{19} +0.128273 q^{21} -3.64921i q^{23} +(4.92301 - 0.874027i) q^{25} +1.00000i q^{27} -8.25219 q^{29} -7.07939 q^{31} -0.0389925i q^{33} +(-0.0251666 - 0.285721i) q^{35} -7.37252i q^{37} -3.81852 q^{39} +1.87351 q^{41} -7.72398i q^{43} +(2.22744 - 0.196195i) q^{45} +9.62406i q^{47} +6.98355 q^{49} +6.63903 q^{51} -3.82384i q^{53} +(-0.0868535 + 0.00765013i) q^{55} -7.30512i q^{57} +9.48802 q^{59} -6.69509 q^{61} -0.128273i q^{63} +(0.749174 + 8.50553i) q^{65} -1.00000i q^{67} -3.64921 q^{69} +4.88025 q^{71} -0.309819i q^{73} +(-0.874027 - 4.92301i) q^{75} +0.00500168i q^{77} -5.72909 q^{79} +1.00000 q^{81} -14.6840i q^{83} +(-1.30255 - 14.7881i) q^{85} +8.25219i q^{87} -11.6793 q^{89} +0.489813 q^{91} +7.07939i q^{93} +(-16.2718 + 1.43323i) q^{95} +7.10494i q^{97} -0.0389925 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 38 q - 2 q^{5} - 38 q^{9}+O(q^{10}) \) 38 * q - 2 * q^5 - 38 * q^9	\( 38 q - 2 q^{5} - 38 q^{9} + 24 q^{11} + 2 q^{15} - 16 q^{19} + 12 q^{21} + 4 q^{25} - 56 q^{29} - 4 q^{31} - 2 q^{35} + 60 q^{41} + 2 q^{45} - 70 q^{49} + 12 q^{55} - 52 q^{59} + 48 q^{61} + 16 q^{65} - 12 q^{69} + 12 q^{75} - 24 q^{79} + 38 q^{81} + 16 q^{85} - 76 q^{89} + 44 q^{91} + 36 q^{95} - 24 q^{99}+O(q^{100}) \) 38 * q - 2 * q^5 - 38 * q^9 + 24 * q^11 + 2 * q^15 - 16 * q^19 + 12 * q^21 + 4 * q^25 - 56 * q^29 - 4 * q^31 - 2 * q^35 + 60 * q^41 + 2 * q^45 - 70 * q^49 + 12 * q^55 - 52 * q^59 + 48 * q^61 + 16 * q^65 - 12 * q^69 + 12 * q^75 - 24 * q^79 + 38 * q^81 + 16 * q^85 - 76 * q^89 + 44 * q^91 + 36 * q^95 - 24 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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