LMFDB - Embedded newform 4020.2.g.c.1609.4

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			1609.4
Character	$\chi$	$=$	4020.1609
Dual form			4020.2.g.c.1609.23

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +(-2.02556 - 0.947153i) q^{5} +2.95752i q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +(-2.02556 - 0.947153i) q^{5} +2.95752i q^{7} -1.00000 q^{9} +2.74672 q^{11} -5.48947i q^{13} +(-0.947153 + 2.02556i) q^{15} -0.836518i q^{17} -7.78037 q^{19} +2.95752 q^{21} -1.14420i q^{23} +(3.20580 + 3.83703i) q^{25} +1.00000i q^{27} +3.28812 q^{29} +5.40904 q^{31} -2.74672i q^{33} +(2.80122 - 5.99064i) q^{35} +4.63135i q^{37} -5.48947 q^{39} +1.97291 q^{41} -5.74443i q^{43} +(2.02556 + 0.947153i) q^{45} -2.67357i q^{47} -1.74693 q^{49} -0.836518 q^{51} +3.93268i q^{53} +(-5.56366 - 2.60157i) q^{55} +7.78037i q^{57} -12.2982 q^{59} -8.93068 q^{61} -2.95752i q^{63} +(-5.19937 + 11.1193i) q^{65} -1.00000i q^{67} -1.14420 q^{69} -5.94317 q^{71} +0.964685i q^{73} +(3.83703 - 3.20580i) q^{75} +8.12349i q^{77} -12.7034 q^{79} +1.00000 q^{81} -9.18552i q^{83} +(-0.792310 + 1.69442i) q^{85} -3.28812i q^{87} +8.47078 q^{89} +16.2352 q^{91} -5.40904i q^{93} +(15.7596 + 7.36920i) q^{95} -2.98952i q^{97} -2.74672 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−2.02556	−	0.947153i	−0.905859	−	0.423580i
$5$	−2.02556	−	0.947153i	−0.905859	−	0.423580i
$6$		0			0
$7$			2.95752i			1.11784i	0.829222	+	0.558919i	$0.188784\pi$
$7$			2.95752i			1.11784i	−0.829222	+	0.558919i	$0.811216\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	2.74672			0.828169			0.414084	−	0.910239i	$-0.364102\pi$
$11$	2.74672			0.828169			0.414084	+	0.910239i	$0.364102\pi$
$12$		0			0
$13$		−	5.48947i		−	1.52251i	−0.648455	−	0.761253i	$-0.724584\pi$
$13$		−	5.48947i		−	1.52251i	0.648455	−	0.761253i	$-0.275416\pi$
$14$		0			0
$15$	−0.947153	+	2.02556i	−0.244554	+	0.522998i
$16$		0			0
$17$		−	0.836518i		−	0.202885i	−0.994841	−	0.101443i	$-0.967654\pi$
$17$		−	0.836518i		−	0.202885i	0.994841	−	0.101443i	$-0.0323458\pi$
$18$		0			0
$19$	−7.78037			−1.78494			−0.892469	−	0.451108i	$-0.851029\pi$
$19$	−7.78037			−1.78494			−0.892469	+	0.451108i	$0.851029\pi$
$20$		0			0
$21$	2.95752			0.645384
$22$		0			0
$23$		−	1.14420i		−	0.238583i	−0.992859	−	0.119291i	$-0.961938\pi$
$23$		−	1.14420i		−	0.238583i	0.992859	−	0.119291i	$-0.0380623\pi$
$24$		0			0
$25$	3.20580	+	3.83703i	0.641160	+	0.767407i
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$	3.28812			0.610589			0.305295	−	0.952258i	$-0.401245\pi$
$29$	3.28812			0.610589			0.305295	+	0.952258i	$0.401245\pi$
$30$		0			0
$31$	5.40904			0.971493			0.485746	−	0.874100i	$-0.338548\pi$
$31$	5.40904			0.971493			0.485746	+	0.874100i	$0.338548\pi$
$32$		0			0
$33$		−	2.74672i		−	0.478143i
$34$		0			0
$35$	2.80122	−	5.99064i	0.473493	−	1.01260i
$36$		0			0
$37$			4.63135i			0.761389i	0.924701	+	0.380695i	$0.124315\pi$
$37$			4.63135i			0.761389i	−0.924701	+	0.380695i	$0.875685\pi$
$38$		0			0
$39$	−5.48947			−0.879019
$40$		0			0
$41$	1.97291			0.308117			0.154059	−	0.988062i	$-0.450766\pi$
$41$	1.97291			0.308117			0.154059	+	0.988062i	$0.450766\pi$
$42$		0			0
$43$		−	5.74443i		−	0.876017i	−0.898971	−	0.438009i	$-0.855684\pi$
$43$		−	5.74443i		−	0.876017i	0.898971	−	0.438009i	$-0.144316\pi$
$44$		0			0
$45$	2.02556	+	0.947153i	0.301953	+	0.141193i
$46$		0			0
$47$		−	2.67357i		−	0.389980i	−0.980805	−	0.194990i	$-0.937533\pi$
$47$		−	2.67357i		−	0.389980i	0.980805	−	0.194990i	$-0.0624675\pi$
$48$		0			0
$49$	−1.74693			−0.249561
$50$		0			0
$51$	−0.836518			−0.117136
$52$		0			0
$53$			3.93268i			0.540196i	0.962833	+	0.270098i	$0.0870561\pi$
$53$			3.93268i			0.540196i	−0.962833	+	0.270098i	$0.912944\pi$
$54$		0			0
$55$	−5.56366	−	2.60157i	−0.750204	−	0.350796i
$56$		0			0
$57$			7.78037i			1.03053i
$58$		0			0
$59$	−12.2982			−1.60109			−0.800543	−	0.599276i	$-0.795455\pi$
$59$	−12.2982			−1.60109			−0.800543	+	0.599276i	$0.795455\pi$
$60$		0			0
$61$	−8.93068			−1.14346			−0.571728	−	0.820443i	$-0.693727\pi$
$61$	−8.93068			−1.14346			−0.571728	+	0.820443i	$0.693727\pi$
$62$		0			0
$63$		−	2.95752i		−	0.372613i
$64$		0			0
$65$	−5.19937	+	11.1193i	−0.644903	+	1.37918i
$66$		0			0
$67$		−	1.00000i		−	0.122169i
$67$		−	1.00000i		−	0.122169i
$68$		0			0
$69$	−1.14420			−0.137746
$70$		0			0
$71$	−5.94317			−0.705325			−0.352662	−	0.935751i	$-0.614724\pi$
$71$	−5.94317			−0.705325			−0.352662	+	0.935751i	$0.614724\pi$
$72$		0			0
$73$			0.964685i			0.112908i	0.998405	+	0.0564539i	$0.0179794\pi$
$73$			0.964685i			0.112908i	−0.998405	+	0.0564539i	$0.982021\pi$
$74$		0			0
$75$	3.83703	−	3.20580i	0.443063	−	0.370174i
$76$		0			0
$77$			8.12349i			0.925758i
$78$		0			0
$79$	−12.7034			−1.42924			−0.714621	−	0.699512i	$-0.753401\pi$
$79$	−12.7034			−1.42924			−0.714621	+	0.699512i	$0.753401\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	9.18552i		−	1.00824i	−0.863633	−	0.504121i	$-0.831817\pi$
$83$		−	9.18552i		−	1.00824i	0.863633	−	0.504121i	$-0.168183\pi$
$84$		0			0
$85$	−0.792310	+	1.69442i	−0.0859381	+	0.183785i
$86$		0			0
$87$		−	3.28812i		−	0.352524i
$88$		0			0
$89$	8.47078			0.897901			0.448950	−	0.893557i	$-0.351798\pi$
$89$	8.47078			0.897901			0.448950	+	0.893557i	$0.351798\pi$
$90$		0			0
$91$	16.2352			1.70191
$92$		0			0
$93$		−	5.40904i		−	0.560892i
$94$		0			0
$95$	15.7596	+	7.36920i	1.61690	+	0.756064i
$96$		0			0
$97$		−	2.98952i		−	0.303539i	−0.988416	−	0.151770i	$-0.951503\pi$
$97$		−	2.98952i		−	0.303539i	0.988416	−	0.151770i	$-0.0484972\pi$
$98$		0			0
$99$	−2.74672			−0.276056
$100$		0			0
$101$	−8.23279			−0.819194			−0.409597	−	0.912267i	$-0.634331\pi$
$101$	−8.23279			−0.819194			−0.409597	+	0.912267i	$0.634331\pi$
$102$		0			0
$103$		−	4.19798i		−	0.413639i	−0.978379	−	0.206819i	$-0.933689\pi$
$103$		−	4.19798i		−	0.413639i	0.978379	−	0.206819i	$-0.0663113\pi$
$104$		0			0
$105$	−5.99064	−	2.80122i	−0.584627	−	0.273372i
$106$		0			0
$107$			1.91266i			0.184904i	0.995717	+	0.0924519i	$0.0294704\pi$
$107$			1.91266i			0.184904i	−0.995717	+	0.0924519i	$0.970530\pi$
$108$		0			0
$109$	−7.87248			−0.754047			−0.377024	−	0.926204i	$-0.623052\pi$
$109$	−7.87248			−0.754047			−0.377024	+	0.926204i	$0.623052\pi$
$110$		0			0
$111$	4.63135			0.439588
$112$		0			0
$113$			18.9099i			1.77890i	0.457036	+	0.889448i	$0.348911\pi$
$113$			18.9099i			1.77890i	−0.457036	+	0.889448i	$0.651089\pi$
$114$		0			0
$115$	−1.08374	+	2.31765i	−0.101059	+	0.216122i
$116$		0			0
$117$			5.48947i			0.507502i
$118$		0			0
$119$	2.47402			0.226793
$120$		0			0
$121$	−3.45550			−0.314137
$122$		0			0
$123$		−	1.97291i		−	0.177892i
$124$		0			0
$125$	−2.85929	−	10.8085i	−0.255743	−	0.966745i
$126$		0			0
$127$		−	0.152806i		−	0.0135593i	−0.999977	−	0.00677966i	$-0.997842\pi$
$127$		−	0.152806i		−	0.0135593i	0.999977	−	0.00677966i	$-0.00215805\pi$
$128$		0			0
$129$	−5.74443			−0.505769
$130$		0			0
$131$	−12.4130			−1.08453			−0.542265	−	0.840208i	$-0.682433\pi$
$131$	−12.4130			−1.08453			−0.542265	+	0.840208i	$0.682433\pi$
$132$		0			0
$133$		−	23.0106i		−	1.99527i
$134$		0			0
$135$	0.947153	−	2.02556i	0.0815180	−	0.174333i
$136$		0			0
$137$			8.99009i			0.768075i	0.923317	+	0.384038i	$0.125467\pi$
$137$			8.99009i			0.768075i	−0.923317	+	0.384038i	$0.874533\pi$
$138$		0			0
$139$	−22.8040			−1.93421			−0.967104	−	0.254380i	$-0.918128\pi$
$139$	−22.8040			−1.93421			−0.967104	+	0.254380i	$0.918128\pi$
$140$		0			0
$141$	−2.67357			−0.225155
$142$		0			0
$143$		−	15.0781i		−	1.26089i
$144$		0			0
$145$	−6.66030	−	3.11436i	−0.553108	−	0.258633i
$146$		0			0
$147$			1.74693i			0.144084i
$148$		0			0
$149$	5.72732			0.469200			0.234600	−	0.972092i	$-0.424622\pi$
$149$	5.72732			0.469200			0.234600	+	0.972092i	$0.424622\pi$
$150$		0			0
$151$	−13.3074			−1.08294			−0.541471	−	0.840720i	$-0.682132\pi$
$151$	−13.3074			−1.08294			−0.541471	+	0.840720i	$0.682132\pi$
$152$		0			0
$153$			0.836518i			0.0676284i
$154$		0			0
$155$	−10.9564	−	5.12319i	−0.880035	−	0.411505i
$156$		0			0
$157$		−	17.9141i		−	1.42970i	−0.699276	−	0.714852i	$-0.746494\pi$
$157$		−	17.9141i		−	1.42970i	0.699276	−	0.714852i	$-0.253506\pi$
$158$		0			0
$159$	3.93268			0.311882
$160$		0			0
$161$	3.38400			0.266697
$162$		0			0
$163$			13.5544i			1.06167i	0.847476	+	0.530833i	$0.178121\pi$
$163$			13.5544i			1.06167i	−0.847476	+	0.530833i	$0.821879\pi$
$164$		0			0
$165$	−2.60157	+	5.56366i	−0.202532	+	0.433130i
$166$		0			0
$167$			14.9256i			1.15498i	0.816398	+	0.577490i	$0.195968\pi$
$167$			14.9256i			1.15498i	−0.816398	+	0.577490i	$0.804032\pi$
$168$		0			0
$169$	−17.1343			−1.31802
$170$		0			0
$171$	7.78037			0.594980
$172$		0			0
$173$			15.4705i			1.17620i	0.808787	+	0.588102i	$0.200125\pi$
$173$			15.4705i			1.17620i	−0.808787	+	0.588102i	$0.799875\pi$
$174$		0			0
$175$	−11.3481	+	9.48122i	−0.857836	+	0.716713i
$176$		0			0
$177$			12.2982i			0.924387i
$178$		0			0
$179$	16.5415			1.23637			0.618185	−	0.786032i	$-0.287868\pi$
$179$	16.5415			1.23637			0.618185	+	0.786032i	$0.287868\pi$
$180$		0			0
$181$	−15.9855			−1.18819			−0.594096	−	0.804394i	$-0.702490\pi$
$181$	−15.9855			−1.18819			−0.594096	+	0.804394i	$0.702490\pi$
$182$		0			0
$183$			8.93068i			0.660175i
$184$		0			0
$185$	4.38660	−	9.38108i	0.322509	−	0.689711i
$186$		0			0
$187$		−	2.29768i		−	0.168023i
$188$		0			0
$189$	−2.95752			−0.215128
$190$		0			0
$191$	−9.89989			−0.716331			−0.358166	−	0.933658i	$-0.616598\pi$
$191$	−9.89989			−0.716331			−0.358166	+	0.933658i	$0.616598\pi$
$192$		0			0
$193$		−	13.9755i		−	1.00598i	−0.864292	−	0.502990i	$-0.832233\pi$
$193$		−	13.9755i		−	1.00598i	0.864292	−	0.502990i	$-0.167767\pi$
$194$		0			0
$195$	11.1193	+	5.19937i	0.796267	+	0.372335i
$196$		0			0
$197$			5.47194i			0.389859i	0.980817	+	0.194930i	$0.0624479\pi$
$197$			5.47194i			0.389859i	−0.980817	+	0.194930i	$0.937552\pi$
$198$		0			0
$199$	2.21401			0.156947			0.0784736	−	0.996916i	$-0.474995\pi$
$199$	2.21401			0.156947			0.0784736	+	0.996916i	$0.474995\pi$
$200$		0			0
$201$	−1.00000			−0.0705346
$202$		0			0
$203$			9.72469i			0.682540i
$204$		0			0
$205$	−3.99626	−	1.86865i	−0.279111	−	0.130512i
$206$		0			0
$207$			1.14420i			0.0795276i
$208$		0			0
$209$	−21.3705			−1.47823
$210$		0			0
$211$	−28.0506			−1.93108			−0.965542	−	0.260249i	$-0.916195\pi$
$211$	−28.0506			−1.93108			−0.965542	+	0.260249i	$0.916195\pi$
$212$		0			0
$213$			5.94317i			0.407219i
$214$		0			0
$215$	−5.44086	+	11.6357i	−0.371063	+	0.793548i
$216$		0			0
$217$			15.9974i			1.08597i
$218$		0			0
$219$	0.964685			0.0651873
$220$		0			0
$221$	−4.59204			−0.308894
$222$		0			0
$223$		−	1.83158i		−	0.122652i	−0.998118	−	0.0613258i	$-0.980467\pi$
$223$		−	1.83158i		−	0.122652i	0.998118	−	0.0613258i	$-0.0195329\pi$
$224$		0			0
$225$	−3.20580	−	3.83703i	−0.213720	−	0.255802i
$226$		0			0
$227$		−	13.2018i		−	0.876232i	−0.898919	−	0.438116i	$-0.855646\pi$
$227$		−	13.2018i		−	0.876232i	0.898919	−	0.438116i	$-0.144354\pi$
$228$		0			0
$229$	−12.7004			−0.839269			−0.419634	−	0.907693i	$-0.637842\pi$
$229$	−12.7004			−0.839269			−0.419634	+	0.907693i	$0.637842\pi$
$230$		0			0
$231$	8.12349			0.534487
$232$		0			0
$233$			13.7551i			0.901129i	0.892744	+	0.450564i	$0.148777\pi$
$233$			13.7551i			0.901129i	−0.892744	+	0.450564i	$0.851223\pi$
$234$		0			0
$235$	−2.53228	+	5.41548i	−0.165188	+	0.353267i
$236$		0			0
$237$			12.7034i			0.825173i
$238$		0			0
$239$	3.29614			0.213210			0.106605	−	0.994301i	$-0.466002\pi$
$239$	3.29614			0.213210			0.106605	+	0.994301i	$0.466002\pi$
$240$		0			0
$241$	0.789127			0.0508321			0.0254161	−	0.999677i	$-0.491909\pi$
$241$	0.789127			0.0508321			0.0254161	+	0.999677i	$0.491909\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$	3.53851	+	1.65461i	0.226067	+	0.105709i
$246$		0			0
$247$			42.7101i			2.71758i
$248$		0			0
$249$	−9.18552			−0.582108
$250$		0			0
$251$	18.4247			1.16295			0.581477	−	0.813563i	$-0.302475\pi$
$251$	18.4247			1.16295			0.581477	+	0.813563i	$0.302475\pi$
$252$		0			0
$253$		−	3.14281i		−	0.197587i
$254$		0			0
$255$	1.69442	+	0.792310i	0.106109	+	0.0496164i
$256$		0			0
$257$			24.3790i			1.52072i	0.649502	+	0.760360i	$0.274977\pi$
$257$			24.3790i			1.52072i	−0.649502	+	0.760360i	$0.725023\pi$
$258$		0			0
$259$	−13.6973			−0.851109
$260$		0			0
$261$	−3.28812			−0.203530
$262$		0			0
$263$			16.1653i			0.996796i	0.866948	+	0.498398i	$0.166078\pi$
$263$			16.1653i			0.996796i	−0.866948	+	0.498398i	$0.833922\pi$
$264$		0			0
$265$	3.72485	−	7.96589i	0.228816	−	0.489341i
$266$		0			0
$267$		−	8.47078i		−	0.518403i
$268$		0			0
$269$	16.3842			0.998961			0.499481	−	0.866325i	$-0.333524\pi$
$269$	16.3842			0.998961			0.499481	+	0.866325i	$0.333524\pi$
$270$		0			0
$271$	12.4620			0.757015			0.378508	−	0.925598i	$-0.376437\pi$
$271$	12.4620			0.757015			0.378508	+	0.925598i	$0.376437\pi$
$272$		0			0
$273$		−	16.2352i		−	0.982601i
$274$		0			0
$275$	8.80545	+	10.5393i	0.530989	+	0.635542i
$276$		0			0
$277$		−	16.7278i		−	1.00507i	−0.864556	−	0.502537i	$-0.832400\pi$
$277$		−	16.7278i		−	1.00507i	0.864556	−	0.502537i	$-0.167600\pi$
$278$		0			0
$279$	−5.40904			−0.323831
$280$		0			0
$281$	−5.16809			−0.308302			−0.154151	−	0.988047i	$-0.549264\pi$
$281$	−5.16809			−0.308302			−0.154151	+	0.988047i	$0.549264\pi$
$282$		0			0
$283$		−	12.1242i		−	0.720707i	−0.932816	−	0.360354i	$-0.882656\pi$
$283$		−	12.1242i		−	0.720707i	0.932816	−	0.360354i	$-0.117344\pi$
$284$		0			0
$285$	7.36920	−	15.7596i	0.436514	−	0.933519i
$286$		0			0
$287$			5.83493i			0.344425i
$288$		0			0
$289$	16.3002			0.958838
$290$		0			0
$291$	−2.98952			−0.175248
$292$		0			0
$293$			30.9443i			1.80779i	0.427759	+	0.903893i	$0.359303\pi$
$293$			30.9443i			1.80779i	−0.427759	+	0.903893i	$0.640697\pi$
$294$		0			0
$295$	24.9107	+	11.6483i	1.45036	+	0.678188i
$296$		0			0
$297$			2.74672i			0.159381i
$298$		0			0
$299$	−6.28107			−0.363244
$300$		0			0
$301$	16.9893			0.979245
$302$		0			0
$303$			8.23279i			0.472962i
$304$		0			0
$305$	18.0896	+	8.45872i	1.03581	+	0.484345i
$306$		0			0
$307$		−	9.72403i		−	0.554980i	−0.960729	−	0.277490i	$-0.910497\pi$
$307$		−	9.72403i		−	0.554980i	0.960729	−	0.277490i	$-0.0895025\pi$
$308$		0			0
$309$	−4.19798			−0.238815
$310$		0			0
$311$	−33.3633			−1.89186			−0.945929	−	0.324373i	$-0.894847\pi$
$311$	−33.3633			−1.89186			−0.945929	+	0.324373i	$0.894847\pi$
$312$		0			0
$313$			7.42253i			0.419546i	0.977750	+	0.209773i	$0.0672726\pi$
$313$			7.42253i			0.419546i	−0.977750	+	0.209773i	$0.932727\pi$
$314$		0			0
$315$	−2.80122	+	5.99064i	−0.157831	+	0.337534i
$316$		0			0
$317$		−	33.8359i		−	1.90041i	−0.311623	−	0.950206i	$-0.600873\pi$
$317$		−	33.8359i		−	1.90041i	0.311623	−	0.950206i	$-0.399127\pi$
$318$		0			0
$319$	9.03157			0.505671
$320$		0			0
$321$	1.91266			0.106754
$322$		0			0
$323$			6.50842i			0.362138i
$324$		0			0
$325$	21.0633	−	17.5982i	1.16838	−	0.976171i
$326$		0			0
$327$			7.87248i			0.435349i
$328$		0			0
$329$	7.90713			0.435934
$330$		0			0
$331$	−10.2746			−0.564746			−0.282373	−	0.959305i	$-0.591122\pi$
$331$	−10.2746			−0.564746			−0.282373	+	0.959305i	$0.591122\pi$
$332$		0			0
$333$		−	4.63135i		−	0.253796i
$334$		0			0
$335$	−0.947153	+	2.02556i	−0.0517485	+	0.110668i
$336$		0			0
$337$		−	21.3148i		−	1.16109i	−0.814228	−	0.580546i	$-0.802839\pi$
$337$		−	21.3148i		−	1.16109i	0.814228	−	0.580546i	$-0.197161\pi$
$338$		0			0
$339$	18.9099			1.02705
$340$		0			0
$341$	14.8572			0.804560
$342$		0			0
$343$			15.5361i			0.838869i
$344$		0			0
$345$	2.31765	+	1.08374i	0.124778	+	0.0583464i
$346$		0			0
$347$			13.7129i			0.736149i	0.929796	+	0.368075i	$0.119983\pi$
$347$			13.7129i			0.736149i	−0.929796	+	0.368075i	$0.880017\pi$
$348$		0			0
$349$	−6.67969			−0.357556			−0.178778	−	0.983889i	$-0.557214\pi$
$349$	−6.67969			−0.357556			−0.178778	+	0.983889i	$0.557214\pi$
$350$		0			0
$351$	5.48947			0.293006
$352$		0			0
$353$		−	5.46529i		−	0.290888i	−0.989367	−	0.145444i	$-0.953539\pi$
$353$		−	5.46529i		−	0.290888i	0.989367	−	0.145444i	$-0.0464610\pi$
$354$		0			0
$355$	12.0383	+	5.62909i	0.638925	+	0.298761i
$356$		0			0
$357$		−	2.47402i		−	0.130939i
$358$		0			0
$359$	−24.4078			−1.28819			−0.644097	−	0.764944i	$-0.722767\pi$
$359$	−24.4078			−1.28819			−0.644097	+	0.764944i	$0.722767\pi$
$360$		0			0
$361$	41.5341			2.18601
$362$		0			0
$363$			3.45550i			0.181367i
$364$		0			0
$365$	0.913704	−	1.95403i	0.0478255	−	0.102279i
$366$		0			0
$367$			33.8706i			1.76803i	0.467456	+	0.884016i	$0.345171\pi$
$367$			33.8706i			1.76803i	−0.467456	+	0.884016i	$0.654829\pi$
$368$		0			0
$369$	−1.97291			−0.102706
$370$		0			0
$371$	−11.6310			−0.603851
$372$		0			0
$373$		−	3.71356i		−	0.192281i	−0.995368	−	0.0961403i	$-0.969350\pi$
$373$		−	3.71356i		−	0.192281i	0.995368	−	0.0961403i	$-0.0306498\pi$
$374$		0			0
$375$	−10.8085	+	2.85929i	−0.558150	+	0.147653i
$376$		0			0
$377$		−	18.0501i		−	0.929626i
$378$		0			0
$379$	−12.9135			−0.663322			−0.331661	−	0.943399i	$-0.607609\pi$
$379$	−12.9135			−0.663322			−0.331661	+	0.943399i	$0.607609\pi$
$380$		0			0
$381$	−0.152806			−0.00782848
$382$		0			0
$383$		−	35.5864i		−	1.81838i	−0.416380	−	0.909191i	$-0.636701\pi$
$383$		−	35.5864i		−	1.81838i	0.416380	−	0.909191i	$-0.363299\pi$
$384$		0			0
$385$	7.69419	−	16.4546i	0.392132	−	0.838606i
$386$		0			0
$387$			5.74443i			0.292006i
$388$		0			0
$389$	−0.159135			−0.00806848			−0.00403424	−	0.999992i	$-0.501284\pi$
$389$	−0.159135			−0.00806848			−0.00403424	+	0.999992i	$0.501284\pi$
$390$		0			0
$391$	−0.957146			−0.0484050
$392$		0			0
$393$			12.4130i			0.626153i
$394$		0			0
$395$	25.7315	+	12.0320i	1.29469	+	0.605398i
$396$		0			0
$397$		−	19.4448i		−	0.975904i	−0.872870	−	0.487952i	$-0.837744\pi$
$397$		−	19.4448i		−	0.975904i	0.872870	−	0.487952i	$-0.162256\pi$
$398$		0			0
$399$	−23.0106			−1.15197
$400$		0			0
$401$	−9.15915			−0.457386			−0.228693	−	0.973499i	$-0.573445\pi$
$401$	−9.15915			−0.457386			−0.228693	+	0.973499i	$0.573445\pi$
$402$		0			0
$403$		−	29.6928i		−	1.47910i
$404$		0			0
$405$	−2.02556	−	0.947153i	−0.100651	−	0.0470644i
$406$		0			0
$407$			12.7210i			0.630559i
$408$		0			0
$409$	26.3811			1.30446			0.652232	−	0.758020i	$-0.273833\pi$
$409$	26.3811			1.30446			0.652232	+	0.758020i	$0.273833\pi$
$410$		0			0
$411$	8.99009			0.443448
$412$		0			0
$413$		−	36.3721i		−	1.78975i
$414$		0			0
$415$	−8.70009	+	18.6058i	−0.427071	+	0.913324i
$416$		0			0
$417$			22.8040i			1.11672i
$418$		0			0
$419$	−16.0978			−0.786430			−0.393215	−	0.919447i	$-0.628637\pi$
$419$	−16.0978			−0.786430			−0.393215	+	0.919447i	$0.628637\pi$
$420$		0			0
$421$	0.554657			0.0270323			0.0135162	−	0.999909i	$-0.495698\pi$
$421$	0.554657			0.0270323			0.0135162	+	0.999909i	$0.495698\pi$
$422$		0			0
$423$			2.67357i			0.129993i
$424$		0			0
$425$	3.20975	−	2.68171i	0.155696	−	0.130082i
$426$		0			0
$427$		−	26.4127i		−	1.27820i
$428$		0			0
$429$	−15.0781			−0.727976
$430$		0			0
$431$	6.64879			0.320261			0.160130	−	0.987096i	$-0.448809\pi$
$431$	6.64879			0.320261			0.160130	+	0.987096i	$0.448809\pi$
$432$		0			0
$433$			10.5054i			0.504856i	0.967616	+	0.252428i	$0.0812290\pi$
$433$			10.5054i			0.504856i	−0.967616	+	0.252428i	$0.918771\pi$
$434$		0			0
$435$	−3.11436	+	6.66030i	−0.149322	+	0.319337i
$436$		0			0
$437$			8.90232i			0.425856i
$438$		0			0
$439$	−7.56850			−0.361225			−0.180612	−	0.983554i	$-0.557808\pi$
$439$	−7.56850			−0.361225			−0.180612	+	0.983554i	$0.557808\pi$
$440$		0			0
$441$	1.74693			0.0831870
$442$		0			0
$443$			33.0123i			1.56846i	0.620469	+	0.784231i	$0.286942\pi$
$443$			33.0123i			1.56846i	−0.620469	+	0.784231i	$0.713058\pi$
$444$		0			0
$445$	−17.1581	−	8.02313i	−0.813371	−	0.380333i
$446$		0			0
$447$		−	5.72732i		−	0.270893i
$448$		0			0
$449$	2.62923			0.124081			0.0620405	−	0.998074i	$-0.480239\pi$
$449$	2.62923			0.124081			0.0620405	+	0.998074i	$0.480239\pi$
$450$		0			0
$451$	5.41905			0.255173
$452$		0			0
$453$			13.3074i			0.625237i
$454$		0			0
$455$	−32.8855	−	15.3773i	−1.54169	−	0.720897i
$456$		0			0
$457$		−	21.9731i		−	1.02786i	−0.857833	−	0.513929i	$-0.828190\pi$
$457$		−	21.9731i		−	1.02786i	0.857833	−	0.513929i	$-0.171810\pi$
$458$		0			0
$459$	0.836518			0.0390453
$460$		0			0
$461$	−37.4409			−1.74380			−0.871899	−	0.489686i	$-0.837111\pi$
$461$	−37.4409			−1.74380			−0.871899	+	0.489686i	$0.837111\pi$
$462$		0			0
$463$			16.7261i			0.777326i	0.921380	+	0.388663i	$0.127063\pi$
$463$			16.7261i			0.777326i	−0.921380	+	0.388663i	$0.872937\pi$
$464$		0			0
$465$	−5.12319	+	10.9564i	−0.237582	+	0.508089i
$466$		0			0
$467$		−	15.5719i		−	0.720583i	−0.932840	−	0.360291i	$-0.882677\pi$
$467$		−	15.5719i		−	0.720583i	0.932840	−	0.360291i	$-0.117323\pi$
$468$		0			0
$469$	2.95752			0.136566
$470$		0			0
$471$	−17.9141			−0.825440
$472$		0			0
$473$		−	15.7784i		−	0.725490i
$474$		0			0
$475$	−24.9423	−	29.8535i	−1.14443	−	1.36977i
$476$		0			0
$477$		−	3.93268i		−	0.180065i
$478$		0			0
$479$	−13.6312			−0.622824			−0.311412	−	0.950275i	$-0.600802\pi$
$479$	−13.6312			−0.622824			−0.311412	+	0.950275i	$0.600802\pi$
$480$		0			0
$481$	25.4237			1.15922
$482$		0			0
$483$		−	3.38400i		−	0.153978i
$484$		0			0
$485$	−2.83153	+	6.05545i	−0.128573	+	0.274964i
$486$		0			0
$487$		−	22.3372i		−	1.01219i	−0.862477	−	0.506097i	$-0.831088\pi$
$487$		−	22.3372i		−	1.01219i	0.862477	−	0.506097i	$-0.168912\pi$
$488$		0			0
$489$	13.5544			0.612953
$490$		0			0
$491$	19.4758			0.878928			0.439464	−	0.898260i	$-0.355168\pi$
$491$	19.4758			0.878928			0.439464	+	0.898260i	$0.355168\pi$
$492$		0			0
$493$		−	2.75057i		−	0.123880i
$494$		0			0
$495$	5.56366	+	2.60157i	0.250068	+	0.116932i
$496$		0			0
$497$		−	17.5771i		−	0.788438i
$498$		0			0
$499$	13.1507			0.588708			0.294354	−	0.955696i	$-0.404895\pi$
$499$	13.1507			0.588708			0.294354	+	0.955696i	$0.404895\pi$
$500$		0			0
$501$	14.9256			0.666828
$502$		0			0
$503$		−	9.16789i		−	0.408776i	−0.978890	−	0.204388i	$-0.934480\pi$
$503$		−	9.16789i		−	0.408776i	0.978890	−	0.204388i	$-0.0655204\pi$
$504$		0			0
$505$	16.6760	+	7.79772i	0.742074	+	0.346994i
$506$		0			0
$507$			17.1343i			0.760962i
$508$		0			0
$509$	0.688601			0.0305217			0.0152609	−	0.999884i	$-0.495142\pi$
$509$	0.688601			0.0305217			0.0152609	+	0.999884i	$0.495142\pi$
$510$		0			0
$511$	−2.85307			−0.126213
$512$		0			0
$513$		−	7.78037i		−	0.343512i
$514$		0			0
$515$	−3.97613	+	8.50326i	−0.175209	+	0.374698i
$516$		0			0
$517$		−	7.34356i		−	0.322969i
$518$		0			0
$519$	15.4705			0.679082
$520$		0			0
$521$	38.5420			1.68856			0.844278	−	0.535905i	$-0.180029\pi$
$521$	38.5420			1.68856			0.844278	+	0.535905i	$0.180029\pi$
$522$		0			0
$523$			14.7842i			0.646468i	0.946319	+	0.323234i	$0.104770\pi$
$523$			14.7842i			0.646468i	−0.946319	+	0.323234i	$0.895230\pi$
$524$		0			0
$525$	9.48122	+	11.3481i	0.413795	+	0.495272i
$526$		0			0
$527$		−	4.52476i		−	0.197102i
$528$		0			0
$529$	21.6908			0.943078
$530$		0			0
$531$	12.2982			0.533695
$532$		0			0
$533$		−	10.8303i		−	0.469110i
$534$		0			0
$535$	1.81158	−	3.87421i	0.0783215	−	0.167497i
$536$		0			0
$537$		−	16.5415i		−	0.713819i
$538$		0			0
$539$	−4.79833			−0.206679
$540$		0			0
$541$	−23.1951			−0.997235			−0.498617	−	0.866822i	$-0.666159\pi$
$541$	−23.1951			−0.997235			−0.498617	+	0.866822i	$0.666159\pi$
$542$		0			0
$543$			15.9855i			0.686003i
$544$		0			0
$545$	15.9462	+	7.45645i	0.683060	+	0.319399i
$546$		0			0
$547$		−	27.5999i		−	1.18009i	−0.807371	−	0.590044i	$-0.799110\pi$
$547$		−	27.5999i		−	1.18009i	0.807371	−	0.590044i	$-0.200890\pi$
$548$		0			0
$549$	8.93068			0.381152
$550$		0			0
$551$	−25.5828			−1.08986
$552$		0			0
$553$		−	37.5705i		−	1.59766i
$554$		0			0
$555$	−9.38108	−	4.38660i	−0.398205	−	0.186201i
$556$		0			0
$557$			2.27768i			0.0965086i	0.998835	+	0.0482543i	$0.0153658\pi$
$557$			2.27768i			0.0965086i	−0.998835	+	0.0482543i	$0.984634\pi$
$558$		0			0
$559$	−31.5339			−1.33374
$560$		0			0
$561$	−2.29768			−0.0970083
$562$		0			0
$563$		−	29.1148i		−	1.22704i	−0.789679	−	0.613521i	$-0.789753\pi$
$563$		−	29.1148i		−	1.22704i	0.789679	−	0.613521i	$-0.210247\pi$
$564$		0			0
$565$	17.9106	−	38.3032i	0.753504	−	1.61143i
$566$		0			0
$567$			2.95752i			0.124204i
$568$		0			0
$569$	−37.2631			−1.56215			−0.781075	−	0.624437i	$-0.785329\pi$
$569$	−37.2631			−1.56215			−0.781075	+	0.624437i	$0.785329\pi$
$570$		0			0
$571$	−7.77045			−0.325183			−0.162592	−	0.986693i	$-0.551985\pi$
$571$	−7.77045			−0.325183			−0.162592	+	0.986693i	$0.551985\pi$
$572$		0			0
$573$			9.89989i			0.413574i
$574$		0			0
$575$	4.39035	−	3.66809i	0.183090	−	0.152970i
$576$		0			0
$577$		−	39.9697i		−	1.66396i	−0.554806	−	0.831980i	$-0.687207\pi$
$577$		−	39.9697i		−	1.66396i	0.554806	−	0.831980i	$-0.312793\pi$
$578$		0			0
$579$	−13.9755			−0.580802
$580$		0			0
$581$	27.1664			1.12705
$582$		0			0
$583$			10.8020i			0.447373i
$584$		0			0
$585$	5.19937	−	11.1193i	0.214968	−	0.459725i
$586$		0			0
$587$			32.4241i			1.33829i	0.743134	+	0.669143i	$0.233339\pi$
$587$			32.4241i			1.33829i	−0.743134	+	0.669143i	$0.766661\pi$
$588$		0			0
$589$	−42.0844			−1.73406
$590$		0			0
$591$	5.47194			0.225085
$592$		0			0
$593$			27.3374i			1.12261i	0.827608	+	0.561306i	$0.189701\pi$
$593$			27.3374i			1.12261i	−0.827608	+	0.561306i	$0.810299\pi$
$594$		0			0
$595$	−5.01128	−	2.34327i	−0.205442	−	0.0960649i
$596$		0			0
$597$		−	2.21401i		−	0.0906135i
$598$		0			0
$599$	−44.1243			−1.80287			−0.901435	−	0.432915i	$-0.857485\pi$
$599$	−44.1243			−1.80287			−0.901435	+	0.432915i	$0.857485\pi$
$600$		0			0
$601$	28.9423			1.18058			0.590290	−	0.807191i	$-0.299014\pi$
$601$	28.9423			1.18058			0.590290	+	0.807191i	$0.299014\pi$
$602$		0			0
$603$			1.00000i			0.0407231i
$604$		0			0
$605$	6.99934	+	3.27289i	0.284563	+	0.133062i
$606$		0			0
$607$		−	28.5954i		−	1.16065i	−0.814384	−	0.580326i	$-0.802925\pi$
$607$		−	28.5954i		−	1.16065i	0.814384	−	0.580326i	$-0.197075\pi$
$608$		0			0
$609$	9.72469			0.394064
$610$		0			0
$611$	−14.6765			−0.593747
$612$		0			0
$613$			21.3445i			0.862096i	0.902329	+	0.431048i	$0.141856\pi$
$613$			21.3445i			0.862096i	−0.902329	+	0.431048i	$0.858144\pi$
$614$		0			0
$615$	−1.86865	+	3.99626i	−0.0753513	+	0.161145i
$616$		0			0
$617$		−	13.0558i		−	0.525608i	−0.964849	−	0.262804i	$-0.915353\pi$
$617$		−	13.0558i		−	0.525608i	0.964849	−	0.262804i	$-0.0846473\pi$
$618$		0			0
$619$	−0.0190092			−0.000764043			−0.000382021	−	1.00000i	$-0.500122\pi$
$619$	−0.0190092			−0.000764043			−0.000382021		1.00000i	$0.500122\pi$
$620$		0			0
$621$	1.14420			0.0459153
$622$		0			0
$623$			25.0525i			1.00371i
$624$		0			0
$625$	−4.44567	+	24.6015i	−0.177827	+	0.984062i
$626$		0			0
$627$			21.3705i			0.853457i
$628$		0			0
$629$	3.87421			0.154475
$630$		0			0
$631$	12.3875			0.493138			0.246569	−	0.969125i	$-0.420697\pi$
$631$	12.3875			0.493138			0.246569	+	0.969125i	$0.420697\pi$
$632$		0			0
$633$			28.0506i			1.11491i
$634$		0			0
$635$	−0.144730	+	0.309517i	−0.00574345	+	0.0122828i
$636$		0			0
$637$			9.58971i			0.379958i
$638$		0			0
$639$	5.94317			0.235108
$640$		0			0
$641$	26.3167			1.03945			0.519723	−	0.854335i	$-0.326035\pi$
$641$	26.3167			1.03945			0.519723	+	0.854335i	$0.326035\pi$
$642$		0			0
$643$		−	42.6412i		−	1.68161i	−0.541342	−	0.840803i	$-0.682084\pi$
$643$		−	42.6412i		−	1.68161i	0.541342	−	0.840803i	$-0.317916\pi$
$644$		0			0
$645$	11.6357	+	5.44086i	0.458155	+	0.214233i
$646$		0			0
$647$		−	46.1030i		−	1.81250i	−0.422747	−	0.906248i	$-0.638934\pi$
$647$		−	46.1030i		−	1.81250i	0.422747	−	0.906248i	$-0.361066\pi$
$648$		0			0
$649$	−33.7797			−1.32597
$650$		0			0
$651$	15.9974			0.626986
$652$		0			0
$653$		−	47.8854i		−	1.87390i	−0.349460	−	0.936951i	$-0.613635\pi$
$653$		−	47.8854i		−	1.87390i	0.349460	−	0.936951i	$-0.386365\pi$
$654$		0			0
$655$	25.1433	+	11.7570i	0.982430	+	0.459385i
$656$		0			0
$657$		−	0.964685i		−	0.0376359i
$658$		0			0
$659$	41.9626			1.63463			0.817315	−	0.576192i	$-0.195462\pi$
$659$	41.9626			1.63463			0.817315	+	0.576192i	$0.195462\pi$
$660$		0			0
$661$	3.99221			0.155279			0.0776394	−	0.996982i	$-0.475262\pi$
$661$	3.99221			0.155279			0.0776394	+	0.996982i	$0.475262\pi$
$662$		0			0
$663$			4.59204i			0.178340i
$664$		0			0
$665$	−21.7946	+	46.6094i	−0.845157	+	1.80743i
$666$		0			0
$667$		−	3.76228i		−	0.145676i
$668$		0			0
$669$	−1.83158			−0.0708129
$670$		0			0
$671$	−24.5301			−0.946974
$672$		0			0
$673$		−	18.6406i		−	0.718543i	−0.933233	−	0.359271i	$-0.883025\pi$
$673$		−	18.6406i		−	0.718543i	0.933233	−	0.359271i	$-0.116975\pi$
$674$		0			0
$675$	−3.83703	+	3.20580i	−0.147688	+	0.123391i
$676$		0			0
$677$		−	14.6577i		−	0.563342i	−0.959511	−	0.281671i	$-0.909111\pi$
$677$		−	14.6577i		−	0.563342i	0.959511	−	0.281671i	$-0.0908887\pi$
$678$		0			0
$679$	8.84155			0.339308
$680$		0			0
$681$	−13.2018			−0.505893
$682$		0			0
$683$		−	36.0897i		−	1.38094i	−0.723363	−	0.690468i	$-0.757405\pi$
$683$		−	36.0897i		−	1.38094i	0.723363	−	0.690468i	$-0.242595\pi$
$684$		0			0
$685$	8.51499	−	18.2100i	0.325341	−	0.695768i
$686$		0			0
$687$			12.7004i			0.484552i
$688$		0			0
$689$	21.5884			0.822451
$690$		0			0
$691$	15.5459			0.591396			0.295698	−	0.955282i	$-0.404448\pi$
$691$	15.5459			0.591396			0.295698	+	0.955282i	$0.404448\pi$
$692$		0			0
$693$		−	8.12349i		−	0.308586i
$694$		0			0
$695$	46.1909	+	21.5989i	1.75212	+	0.819292i
$696$		0			0
$697$		−	1.65038i		−	0.0625125i
$698$		0			0
$699$	13.7551			0.520267
$700$		0			0
$701$	36.1326			1.36471			0.682355	−	0.731021i	$-0.260956\pi$
$701$	36.1326			1.36471			0.682355	+	0.731021i	$0.260956\pi$
$702$		0			0
$703$		−	36.0336i		−	1.35903i
$704$		0			0
$705$	5.41548	+	2.53228i	0.203959	+	0.0953712i
$706$		0			0
$707$		−	24.3487i		−	0.915725i
$708$		0			0
$709$	−42.2485			−1.58668			−0.793338	−	0.608781i	$-0.791659\pi$
$709$	−42.2485			−1.58668			−0.793338	+	0.608781i	$0.791659\pi$
$710$		0			0
$711$	12.7034			0.476414
$712$		0			0
$713$		−	6.18904i		−	0.231782i
$714$		0			0
$715$	−14.2812	+	30.5416i	−0.534088	+	1.14219i
$716$		0			0
$717$		−	3.29614i		−	0.123097i
$718$		0			0
$719$	13.7436			0.512551			0.256275	−	0.966604i	$-0.417505\pi$
$719$	13.7436			0.512551			0.256275	+	0.966604i	$0.417505\pi$
$720$		0			0
$721$	12.4156			0.462381
$722$		0			0
$723$		−	0.789127i		−	0.0293479i
$724$		0			0
$725$	10.5411	+	12.6166i	0.391486	+	0.468570i
$726$		0			0
$727$			23.6271i			0.876281i	0.898907	+	0.438140i	$0.144363\pi$
$727$			23.6271i			0.876281i	−0.898907	+	0.438140i	$0.855637\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	−4.80532			−0.177731
$732$		0			0
$733$			41.9518i			1.54953i	0.632252	+	0.774763i	$0.282131\pi$
$733$			41.9518i			1.54953i	−0.632252	+	0.774763i	$0.717869\pi$
$734$		0			0
$735$	1.65461	−	3.53851i	0.0610311	−	0.130520i
$736$		0			0
$737$		−	2.74672i		−	0.101177i
$738$		0			0
$739$	19.3488			0.711758			0.355879	−	0.934532i	$-0.384181\pi$
$739$	19.3488			0.711758			0.355879	+	0.934532i	$0.384181\pi$
$740$		0			0
$741$	42.7101			1.56900
$742$		0			0
$743$			8.54217i			0.313382i	0.987648	+	0.156691i	$0.0500827\pi$
$743$			8.54217i			0.313382i	−0.987648	+	0.156691i	$0.949917\pi$
$744$		0			0
$745$	−11.6010	−	5.42465i	−0.425029	−	0.198744i
$746$		0			0
$747$			9.18552i			0.336080i
$748$		0			0
$749$	−5.65673			−0.206692
$750$		0			0
$751$	9.22485			0.336620			0.168310	−	0.985734i	$-0.446169\pi$
$751$	9.22485			0.336620			0.168310	+	0.985734i	$0.446169\pi$
$752$		0			0
$753$		−	18.4247i		−	0.671432i
$754$		0			0
$755$	26.9550	+	12.6042i	0.980992	+	0.458712i
$756$		0			0
$757$			16.6768i			0.606129i	0.952970	+	0.303064i	$0.0980097\pi$
$757$			16.6768i			0.606129i	−0.952970	+	0.303064i	$0.901990\pi$
$758$		0			0
$759$	−3.14281			−0.114077
$760$		0			0
$761$	37.6354			1.36428			0.682142	−	0.731220i	$-0.261049\pi$
$761$	37.6354			1.36428			0.682142	+	0.731220i	$0.261049\pi$
$762$		0			0
$763$		−	23.2830i		−	0.842902i
$764$		0			0
$765$	0.792310	−	1.69442i	0.0286460	−	0.0612618i
$766$		0			0
$767$			67.5105i			2.43766i
$768$		0			0
$769$	−42.7260			−1.54074			−0.770369	−	0.637598i	$-0.779928\pi$
$769$	−42.7260			−1.54074			−0.770369	+	0.637598i	$0.779928\pi$
$770$		0			0
$771$	24.3790			0.877988
$772$		0			0
$773$		−	1.98882i		−	0.0715330i	−0.999360	−	0.0357665i	$-0.988613\pi$
$773$		−	1.98882i		−	0.0715330i	0.999360	−	0.0357665i	$-0.0113873\pi$
$774$		0			0
$775$	17.3403	+	20.7547i	0.622883	+	0.745530i
$776$		0			0
$777$			13.6973i			0.491388i
$778$		0			0
$779$	−15.3500			−0.549970
$780$		0			0
$781$	−16.3243			−0.584128
$782$		0			0
$783$			3.28812i			0.117508i
$784$		0			0
$785$	−16.9674	+	36.2862i	−0.605594	+	1.29511i
$786$		0			0
$787$		−	0.0279170i		−	0.000995132i	−1.00000		0.000497566i	$-0.999842\pi$
$787$		−	0.0279170i		−	0.000995132i	1.00000		0.000497566i	$-0.000158380\pi$
$788$		0			0
$789$	16.1653			0.575500
$790$		0			0
$791$	−55.9265			−1.98852
$792$		0			0
$793$			49.0247i			1.74092i
$794$		0			0
$795$	−7.96589	−	3.72485i	−0.282521	−	0.132107i
$796$		0			0
$797$		−	16.5883i		−	0.587589i	−0.955869	−	0.293794i	$-0.905082\pi$
$797$		−	16.5883i		−	0.587589i	0.955869	−	0.293794i	$-0.0949181\pi$
$798$		0			0
$799$	−2.23649			−0.0791213
$800$		0			0
$801$	−8.47078			−0.299300
$802$		0			0
$803$			2.64972i			0.0935067i
$804$		0			0
$805$	−6.85451	−	3.20517i	−0.241590	−	0.112967i
$806$		0			0
$807$		−	16.3842i		−	0.576751i
$808$		0			0
$809$	19.4267			0.683007			0.341503	−	0.939881i	$-0.389064\pi$
$809$	19.4267			0.683007			0.341503	+	0.939881i	$0.389064\pi$
$810$		0			0
$811$	29.0189			1.01899			0.509495	−	0.860474i	$-0.329832\pi$
$811$	29.0189			1.01899			0.509495	+	0.860474i	$0.329832\pi$
$812$		0			0
$813$		−	12.4620i		−	0.437063i
$814$		0			0
$815$	12.8381	−	27.4554i	0.449700	−	0.961720i
$816$		0			0
$817$			44.6938i			1.56364i
$818$		0			0
$819$	−16.2352			−0.567305
$820$		0			0
$821$	−2.29024			−0.0799300			−0.0399650	−	0.999201i	$-0.512725\pi$
$821$	−2.29024			−0.0799300			−0.0399650	+	0.999201i	$0.512725\pi$
$822$		0			0
$823$		−	18.9274i		−	0.659769i	−0.944021	−	0.329884i	$-0.892990\pi$
$823$		−	18.9274i		−	0.659769i	0.944021	−	0.329884i	$-0.107010\pi$
$824$		0			0
$825$	10.5393	−	8.80545i	0.366931	−	0.306567i
$826$		0			0
$827$			4.13628i			0.143832i	0.997411	+	0.0719162i	$0.0229114\pi$
$827$			4.13628i			0.143832i	−0.997411	+	0.0719162i	$0.977089\pi$
$828$		0			0
$829$	25.0018			0.868348			0.434174	−	0.900829i	$-0.357040\pi$
$829$	25.0018			0.868348			0.434174	+	0.900829i	$0.357040\pi$
$830$		0			0
$831$	−16.7278			−0.580280
$832$		0			0
$833$			1.46133i			0.0506323i
$834$		0			0
$835$	14.1369	−	30.2328i	0.489226	−	1.04625i
$836$		0			0
$837$			5.40904i			0.186964i
$838$		0			0
$839$	9.72466			0.335733			0.167866	−	0.985810i	$-0.446312\pi$
$839$	9.72466			0.335733			0.167866	+	0.985810i	$0.446312\pi$
$840$		0			0
$841$	−18.1882			−0.627181
$842$		0			0
$843$			5.16809i			0.177998i
$844$		0			0
$845$	34.7066	+	16.2288i	1.19394	+	0.558289i
$846$		0			0
$847$		−	10.2197i		−	0.351154i
$848$		0			0
$849$	−12.1242			−0.416101
$850$		0			0
$851$	5.29920			0.181654
$852$		0			0
$853$		−	23.4876i		−	0.804201i	−0.915596	−	0.402100i	$-0.868280\pi$
$853$		−	23.4876i		−	0.804201i	0.915596	−	0.402100i	$-0.131720\pi$
$854$		0			0
$855$	−15.7596	−	7.36920i	−0.538968	−	0.252021i
$856$		0			0
$857$		−	15.5598i		−	0.531512i	−0.964040	−	0.265756i	$-0.914378\pi$
$857$		−	15.5598i		−	0.531512i	0.964040	−	0.265756i	$-0.0856216\pi$
$858$		0			0
$859$	−49.0903			−1.67494			−0.837470	−	0.546484i	$-0.815966\pi$
$859$	−49.0903			−1.67494			−0.837470	+	0.546484i	$0.815966\pi$
$860$		0			0
$861$	5.83493			0.198854
$862$		0			0
$863$		−	21.5921i		−	0.735002i	−0.930023	−	0.367501i	$-0.880213\pi$
$863$		−	21.5921i		−	0.735002i	0.930023	−	0.367501i	$-0.119787\pi$
$864$		0			0
$865$	14.6530	−	31.3366i	0.498216	−	1.06547i
$866$		0			0
$867$		−	16.3002i		−	0.553585i
$868$		0			0
$869$	−34.8927			−1.18365
$870$		0			0
$871$	−5.48947			−0.186004
$872$		0			0
$873$			2.98952i			0.101180i
$874$		0			0
$875$	31.9665	−	8.45641i	1.08066	−	0.285879i
$876$		0			0
$877$			2.27099i			0.0766857i	0.999265	+	0.0383429i	$0.0122079\pi$
$877$			2.27099i			0.0766857i	−0.999265	+	0.0383429i	$0.987792\pi$
$878$		0			0
$879$	30.9443			1.04373
$880$		0			0
$881$	29.3030			0.987242			0.493621	−	0.869677i	$-0.335673\pi$
$881$	29.3030			0.987242			0.493621	+	0.869677i	$0.335673\pi$
$882$		0			0
$883$			45.7790i			1.54059i	0.637690	+	0.770293i	$0.279890\pi$
$883$			45.7790i			1.54059i	−0.637690	+	0.770293i	$0.720110\pi$
$884$		0			0
$885$	11.6483	−	24.9107i	0.391552	−	0.837364i
$886$		0			0
$887$		−	32.7876i		−	1.10090i	−0.834868	−	0.550450i	$-0.814456\pi$
$887$		−	32.7876i		−	1.10090i	0.834868	−	0.550450i	$-0.185544\pi$
$888$		0			0
$889$	0.451926			0.0151571
$890$		0			0
$891$	2.74672			0.0920187
$892$		0			0
$893$			20.8014i			0.696091i
$894$		0			0
$895$	−33.5059	−	15.6673i	−1.11998	−	0.523702i
$896$		0			0
$897$			6.28107i			0.209719i
$898$		0			0
$899$	17.7856			0.593183
$900$		0			0
$901$	3.28976			0.109598
$902$		0			0
$903$		−	16.9893i		−	0.565368i
$904$		0			0
$905$	32.3796	+	15.1407i	1.07633	+	0.503294i
$906$		0			0
$907$			21.1709i			0.702970i	0.936194	+	0.351485i	$0.114323\pi$
$907$			21.1709i			0.702970i	−0.936194	+	0.351485i	$0.885677\pi$
$908$		0			0
$909$	8.23279			0.273065
$910$		0			0
$911$	−10.3820			−0.343970			−0.171985	−	0.985100i	$-0.555018\pi$
$911$	−10.3820			−0.343970			−0.171985	+	0.985100i	$0.555018\pi$
$912$		0			0
$913$		−	25.2301i		−	0.834994i
$914$		0			0
$915$	8.45872	−	18.0896i	0.279637	−	0.598025i
$916$		0			0
$917$		−	36.7117i		−	1.21233i
$918$		0			0
$919$	−40.7376			−1.34381			−0.671905	−	0.740637i	$-0.734524\pi$
$919$	−40.7376			−1.34381			−0.671905	+	0.740637i	$0.734524\pi$
$920$		0			0
$921$	−9.72403			−0.320418
$922$		0			0
$923$			32.6249i			1.07386i
$924$		0			0
$925$	−17.7706	+	14.8472i	−0.584295	+	0.488172i
$926$		0			0
$927$			4.19798i			0.137880i
$928$		0			0
$929$	−6.29322			−0.206474			−0.103237	−	0.994657i	$-0.532920\pi$
$929$	−6.29322			−0.206474			−0.103237	+	0.994657i	$0.532920\pi$
$930$		0			0
$931$	13.5917			0.445451
$932$		0			0
$933$			33.3633i			1.09226i
$934$		0			0
$935$	−2.17626	+	4.65410i	−0.0711713	+	0.152205i
$936$		0			0
$937$		−	14.6773i		−	0.479487i	−0.970836	−	0.239744i	$-0.922937\pi$
$937$		−	14.6773i		−	0.479487i	0.970836	−	0.239744i	$-0.0770634\pi$
$938$		0			0
$939$	7.42253			0.242225
$940$		0			0
$941$	−24.5443			−0.800120			−0.400060	−	0.916489i	$-0.631011\pi$
$941$	−24.5443			−0.800120			−0.400060	+	0.916489i	$0.631011\pi$
$942$		0			0
$943$		−	2.25741i		−	0.0735115i
$944$		0			0
$945$	5.99064	+	2.80122i	0.194876	+	0.0911239i
$946$		0			0
$947$		−	10.2631i		−	0.333505i	−0.985999	−	0.166753i	$-0.946672\pi$
$947$		−	10.2631i		−	0.333505i	0.985999	−	0.166753i	$-0.0533281\pi$
$948$		0			0
$949$	5.29561			0.171903
$950$		0			0
$951$	−33.8359			−1.09720
$952$		0			0
$953$			15.5361i			0.503263i	0.967823	+	0.251631i	$0.0809670\pi$
$953$			15.5361i			0.503263i	−0.967823	+	0.251631i	$0.919033\pi$
$954$		0			0
$955$	20.0528	+	9.37671i	0.648895	+	0.303423i
$956$		0			0
$957$		−	9.03157i		−	0.291949i
$958$		0			0
$959$	−26.5884			−0.858583
$960$		0			0
$961$	−1.74225			−0.0562016
$962$		0			0
$963$		−	1.91266i		−	0.0616346i
$964$		0			0
$965$	−13.2370	+	28.3083i	−0.426113	+	0.911275i
$966$		0			0
$967$			52.2910i			1.68157i	0.541372	+	0.840783i	$0.317905\pi$
$967$			52.2910i			1.68157i	−0.541372	+	0.840783i	$0.682095\pi$
$968$		0			0
$969$	6.50842			0.209080
$970$		0			0
$971$	58.4265			1.87499			0.937497	−	0.347993i	$-0.113137\pi$
$971$	58.4265			1.87499			0.937497	+	0.347993i	$0.113137\pi$
$972$		0			0
$973$		−	67.4432i		−	2.16213i
$974$		0			0
$975$	−17.5982	−	21.0633i	−0.563592	−	0.674566i
$976$		0			0
$977$		−	43.4260i		−	1.38932i	−0.719338	−	0.694660i	$-0.755555\pi$
$977$		−	43.4260i		−	1.38932i	0.719338	−	0.694660i	$-0.244445\pi$
$978$		0			0
$979$	23.2669			0.743613
$980$		0			0
$981$	7.87248			0.251349
$982$		0			0
$983$			14.4108i			0.459632i	0.973234	+	0.229816i	$0.0738124\pi$
$983$			14.4108i			0.459632i	−0.973234	+	0.229816i	$0.926188\pi$
$984$		0			0
$985$	5.18276	−	11.0837i	0.165137	−	0.353158i
$986$		0			0
$987$		−	7.90713i		−	0.251687i
$988$		0			0
$989$	−6.57280			−0.209003
$990$		0			0
$991$	−42.8742			−1.36194			−0.680972	−	0.732310i	$-0.738442\pi$
$991$	−42.8742			−1.36194			−0.680972	+	0.732310i	$0.738442\pi$
$992$		0			0
$993$			10.2746i			0.326056i
$994$		0			0
$995$	−4.48462	−	2.09701i	−0.142172	−	0.0664797i
$996$		0			0
$997$		−	7.02882i		−	0.222605i	−0.993787	−	0.111303i	$-0.964498\pi$
$997$		−	7.02882i		−	0.222605i	0.993787	−	0.111303i	$-0.0355023\pi$
$998$		0			0
$999$	−4.63135			−0.146529

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4020.2.g.c.1609.4	✓	38
5.4	even	2	inner	4020.2.g.c.1609.23	yes	38

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4020.2.g.c.1609.4	✓	38	1.1	even	1	trivial
4020.2.g.c.1609.23	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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +(-2.02556 - 0.947153i) q^{5} +2.95752i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +(-2.02556 - 0.947153i) q^{5} +2.95752i q^{7} -1.00000 q^{9} +2.74672 q^{11} -5.48947i q^{13} +(-0.947153 + 2.02556i) q^{15} -0.836518i q^{17} -7.78037 q^{19} +2.95752 q^{21} -1.14420i q^{23} +(3.20580 + 3.83703i) q^{25} +1.00000i q^{27} +3.28812 q^{29} +5.40904 q^{31} -2.74672i q^{33} +(2.80122 - 5.99064i) q^{35} +4.63135i q^{37} -5.48947 q^{39} +1.97291 q^{41} -5.74443i q^{43} +(2.02556 + 0.947153i) q^{45} -2.67357i q^{47} -1.74693 q^{49} -0.836518 q^{51} +3.93268i q^{53} +(-5.56366 - 2.60157i) q^{55} +7.78037i q^{57} -12.2982 q^{59} -8.93068 q^{61} -2.95752i q^{63} +(-5.19937 + 11.1193i) q^{65} -1.00000i q^{67} -1.14420 q^{69} -5.94317 q^{71} +0.964685i q^{73} +(3.83703 - 3.20580i) q^{75} +8.12349i q^{77} -12.7034 q^{79} +1.00000 q^{81} -9.18552i q^{83} +(-0.792310 + 1.69442i) q^{85} -3.28812i q^{87} +8.47078 q^{89} +16.2352 q^{91} -5.40904i q^{93} +(15.7596 + 7.36920i) q^{95} -2.98952i q^{97} -2.74672 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 38 q - 2 q^{5} - 38 q^{9}+O(q^{10}) \) 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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