LMFDB - Embedded newform 4020.2.a.c.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4020,2,Mod(1,4020)]

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, 0, 0]))

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

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

chi := DirichletCharacter("4020.1");

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.a (trivial)

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:	yes
Analytic conductor:	$32.0998616126$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.98117.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 8x^{2} + 4x + 1 $ x^4 - x^3 - 8x^2 + 4x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$3.12017$ of defining polynomial
Character	$\chi$	$=$	4020.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{5} +3.12017 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.00000 q^{5} +3.12017 q^{7} +1.00000 q^{9} -5.44067 q^{11} -5.41498 q^{13} -1.00000 q^{15} +0.615302 q^{17} +6.73547 q^{19} +3.12017 q^{21} -3.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} -4.12017 q^{29} -3.09448 q^{31} -5.44067 q^{33} -3.12017 q^{35} +9.24034 q^{37} -5.41498 q^{39} -9.73547 q^{41} -9.05597 q^{43} -1.00000 q^{45} -6.42472 q^{47} +2.73547 q^{49} +0.615302 q^{51} +2.09448 q^{53} +5.44067 q^{55} +6.73547 q^{57} +9.89567 q^{59} -1.26453 q^{61} +3.12017 q^{63} +5.41498 q^{65} -1.00000 q^{67} -3.00000 q^{69} -8.91011 q^{71} -6.26603 q^{73} +1.00000 q^{75} -16.9758 q^{77} -8.41498 q^{79} +1.00000 q^{81} +0.721141 q^{83} -0.615302 q^{85} -4.12017 q^{87} -5.36213 q^{89} -16.8957 q^{91} -3.09448 q^{93} -6.73547 q^{95} +7.63938 q^{97} -5.44067 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} - 4 q^{5} + q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 - 4 * q^5 + q^7 + 4 * q^9	$ 4 q + 4 q^{3} - 4 q^{5} + q^{7} + 4 q^{9} - 5 q^{11} - 5 q^{13} - 4 q^{15} - 8 q^{17} + 5 q^{19} + q^{21} - 12 q^{23} + 4 q^{25} + 4 q^{27} - 5 q^{29} - q^{31} - 5 q^{33} - q^{35} + 14 q^{37} - 5 q^{39} - 17 q^{41} - 9 q^{43} - 4 q^{45} - 7 q^{47} - 11 q^{49} - 8 q^{51} - 3 q^{53} + 5 q^{55} + 5 q^{57} - 23 q^{59} - 27 q^{61} + q^{63} + 5 q^{65} - 4 q^{67} - 12 q^{69} - 20 q^{71} - 2 q^{73} + 4 q^{75} - 23 q^{77} - 17 q^{79} + 4 q^{81} + 10 q^{83} + 8 q^{85} - 5 q^{87} - 18 q^{89} - 5 q^{91} - q^{93} - 5 q^{95} - 11 q^{97} - 5 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 - 4 * q^5 + q^7 + 4 * q^9 - 5 * q^11 - 5 * q^13 - 4 * q^15 - 8 * q^17 + 5 * q^19 + q^21 - 12 * q^23 + 4 * q^25 + 4 * q^27 - 5 * q^29 - q^31 - 5 * q^33 - q^35 + 14 * q^37 - 5 * q^39 - 17 * q^41 - 9 * q^43 - 4 * q^45 - 7 * q^47 - 11 * q^49 - 8 * q^51 - 3 * q^53 + 5 * q^55 + 5 * q^57 - 23 * q^59 - 27 * q^61 + q^63 + 5 * q^65 - 4 * q^67 - 12 * q^69 - 20 * q^71 - 2 * q^73 + 4 * q^75 - 23 * q^77 - 17 * q^79 + 4 * q^81 + 10 * q^83 + 8 * q^85 - 5 * q^87 - 18 * q^89 - 5 * q^91 - q^93 - 5 * q^95 - 11 * q^97 - 5 * q^99

Display 100 coefficients 10 coefficients

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	3.12017	1.17931	0.589657	−	0.807654i	$-0.299263\pi$
$7$	3.12017	1.17931	0.589657	+	0.807654i	$0.299263\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.44067	−1.64042	−0.820211	−	0.572060i	$-0.806144\pi$
$11$	−5.44067	−1.64042	−0.820211	+	0.572060i	$0.806144\pi$
$12$	0	0
$13$	−5.41498	−1.50185	−0.750923	−	0.660390i	$-0.770391\pi$
$13$	−5.41498	−1.50185	−0.750923	+	0.660390i	$0.770391\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	0.615302	0.149233	0.0746164	−	0.997212i	$-0.476227\pi$
$17$	0.615302	0.149233	0.0746164	+	0.997212i	$0.476227\pi$
$18$	0	0
$19$	6.73547	1.54522	0.772612	−	0.634879i	$-0.218950\pi$
$19$	6.73547	1.54522	0.772612	+	0.634879i	$0.218950\pi$
$20$	0	0
$21$	3.12017	0.680877
$22$	0	0
$23$	−3.00000	−0.625543	−0.312772	−	0.949828i	$-0.601257\pi$
$23$	−3.00000	−0.625543	−0.312772	+	0.949828i	$0.601257\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−4.12017	−0.765097	−0.382548	−	0.923935i	$-0.624953\pi$
$29$	−4.12017	−0.765097	−0.382548	+	0.923935i	$0.624953\pi$
$30$	0	0
$31$	−3.09448	−0.555786	−0.277893	−	0.960612i	$-0.589636\pi$
$31$	−3.09448	−0.555786	−0.277893	+	0.960612i	$0.589636\pi$
$32$	0	0
$33$	−5.44067	−0.947099
$34$	0	0
$35$	−3.12017	−0.527405
$36$	0	0
$37$	9.24034	1.51910	0.759552	−	0.650447i	$-0.225418\pi$
$37$	9.24034	1.51910	0.759552	+	0.650447i	$0.225418\pi$
$38$	0	0
$39$	−5.41498	−0.867091
$40$	0	0
$41$	−9.73547	−1.52043	−0.760213	−	0.649674i	$-0.774905\pi$
$41$	−9.73547	−1.52043	−0.760213	+	0.649674i	$0.774905\pi$
$42$	0	0
$43$	−9.05597	−1.38102	−0.690511	−	0.723322i	$-0.742614\pi$
$43$	−9.05597	−1.38102	−0.690511	+	0.723322i	$0.742614\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−6.42472	−0.937142	−0.468571	−	0.883426i	$-0.655231\pi$
$47$	−6.42472	−0.937142	−0.468571	+	0.883426i	$0.655231\pi$
$48$	0	0
$49$	2.73547	0.390782
$50$	0	0
$51$	0.615302	0.0861596
$52$	0	0
$53$	2.09448	0.287700	0.143850	−	0.989600i	$-0.454052\pi$
$53$	2.09448	0.287700	0.143850	+	0.989600i	$0.454052\pi$
$54$	0	0
$55$	5.44067	0.733619
$56$	0	0
$57$	6.73547	0.892135
$58$	0	0
$59$	9.89567	1.28831	0.644153	−	0.764897i	$-0.277210\pi$
$59$	9.89567	1.28831	0.644153	+	0.764897i	$0.277210\pi$
$60$	0	0
$61$	−1.26453	−0.161906	−0.0809529	−	0.996718i	$-0.525796\pi$
$61$	−1.26453	−0.161906	−0.0809529	+	0.996718i	$0.525796\pi$
$62$	0	0
$63$	3.12017	0.393105
$64$	0	0
$65$	5.41498	0.671646
$66$	0	0
$67$	−1.00000	−0.122169
$67$	−1.00000	−0.122169
$68$	0	0
$69$	−3.00000	−0.361158
$70$	0	0
$71$	−8.91011	−1.05744	−0.528718	−	0.848798i	$-0.677327\pi$
$71$	−8.91011	−1.05744	−0.528718	+	0.848798i	$0.677327\pi$
$72$	0	0
$73$	−6.26603	−0.733384	−0.366692	−	0.930342i	$-0.619510\pi$
$73$	−6.26603	−0.733384	−0.366692	+	0.930342i	$0.619510\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−16.9758	−1.93457
$78$	0	0
$79$	−8.41498	−0.946759	−0.473380	−	0.880859i	$-0.656966\pi$
$79$	−8.41498	−0.946759	−0.473380	+	0.880859i	$0.656966\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	0.721141	0.0791555	0.0395778	−	0.999216i	$-0.487399\pi$
$83$	0.721141	0.0791555	0.0395778	+	0.999216i	$0.487399\pi$
$84$	0	0
$85$	−0.615302	−0.0667389
$86$	0	0
$87$	−4.12017	−0.441729
$88$	0	0
$89$	−5.36213	−0.568385	−0.284192	−	0.958767i	$-0.591725\pi$
$89$	−5.36213	−0.568385	−0.284192	+	0.958767i	$0.591725\pi$
$90$	0	0
$91$	−16.8957	−1.77115
$92$	0	0
$93$	−3.09448	−0.320883
$94$	0	0
$95$	−6.73547	−0.691045
$96$	0	0
$97$	7.63938	0.775661	0.387831	−	0.921731i	$-0.373225\pi$
$97$	7.63938	0.775661	0.387831	+	0.921731i	$0.373225\pi$
$98$	0	0
$99$	−5.44067	−0.546808
$100$	0	0
$101$	−0.948624	−0.0943916	−0.0471958	−	0.998886i	$-0.515028\pi$
$101$	−0.948624	−0.0943916	−0.0471958	+	0.998886i	$0.515028\pi$
$102$	0	0
$103$	−13.2963	−1.31012	−0.655062	−	0.755575i	$-0.727358\pi$
$103$	−13.2963	−1.31012	−0.655062	+	0.755575i	$0.727358\pi$
$104$	0	0
$105$	−3.12017	−0.304498
$106$	0	0
$107$	−2.10584	−0.203579	−0.101790	−	0.994806i	$-0.532457\pi$
$107$	−2.10584	−0.203579	−0.101790	+	0.994806i	$0.532457\pi$
$108$	0	0
$109$	−5.27886	−0.505623	−0.252811	−	0.967516i	$-0.581355\pi$
$109$	−5.27886	−0.505623	−0.252811	+	0.967516i	$0.581355\pi$
$110$	0	0
$111$	9.24034	0.877055
$112$	0	0
$113$	−16.8315	−1.58337	−0.791686	−	0.610929i	$-0.790796\pi$
$113$	−16.8315	−1.58337	−0.791686	+	0.610929i	$0.790796\pi$
$114$	0	0
$115$	3.00000	0.279751
$116$	0	0
$117$	−5.41498	−0.500615
$118$	0	0
$119$	1.91985	0.175992
$120$	0	0
$121$	18.6009	1.69099
$122$	0	0
$123$	−9.73547	−0.877818
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−12.1761	−1.08046	−0.540229	−	0.841518i	$-0.681662\pi$
$127$	−12.1761	−1.08046	−0.540229	+	0.841518i	$0.681662\pi$
$128$	0	0
$129$	−9.05597	−0.797334
$130$	0	0
$131$	−13.4134	−1.17193	−0.585966	−	0.810336i	$-0.699285\pi$
$131$	−13.4134	−1.17193	−0.585966	+	0.810336i	$0.699285\pi$
$132$	0	0
$133$	21.0158	1.82230
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	21.5110	1.83781	0.918903	−	0.394483i	$-0.129076\pi$
$137$	21.5110	1.83781	0.918903	+	0.394483i	$0.129076\pi$
$138$	0	0
$139$	14.2419	1.20798	0.603989	−	0.796993i	$-0.293577\pi$
$139$	14.2419	1.20798	0.603989	+	0.796993i	$0.293577\pi$
$140$	0	0
$141$	−6.42472	−0.541059
$142$	0	0
$143$	29.4611	2.46366
$144$	0	0
$145$	4.12017	0.342162
$146$	0	0
$147$	2.73547	0.225618
$148$	0	0
$149$	−3.85414	−0.315743	−0.157872	−	0.987460i	$-0.550463\pi$
$149$	−3.85414	−0.315743	−0.157872	+	0.987460i	$0.550463\pi$
$150$	0	0
$151$	15.3492	1.24910	0.624549	−	0.780986i	$-0.285283\pi$
$151$	15.3492	1.24910	0.624549	+	0.780986i	$0.285283\pi$
$152$	0	0
$153$	0.615302	0.0497442
$154$	0	0
$155$	3.09448	0.248555
$156$	0	0
$157$	−8.12017	−0.648060	−0.324030	−	0.946047i	$-0.605038\pi$
$157$	−8.12017	−0.648060	−0.324030	+	0.946047i	$0.605038\pi$
$158$	0	0
$159$	2.09448	0.166103
$160$	0	0
$161$	−9.36052	−0.737712
$162$	0	0
$163$	−3.18588	−0.249538	−0.124769	−	0.992186i	$-0.539819\pi$
$163$	−3.18588	−0.249538	−0.124769	+	0.992186i	$0.539819\pi$
$164$	0	0
$165$	5.44067	0.423555
$166$	0	0
$167$	−10.2948	−0.796636	−0.398318	−	0.917247i	$-0.630406\pi$
$167$	−10.2948	−0.796636	−0.398318	+	0.917247i	$0.630406\pi$
$168$	0	0
$169$	16.3220	1.25554
$170$	0	0
$171$	6.73547	0.515075
$172$	0	0
$173$	7.93418	0.603225	0.301612	−	0.953431i	$-0.402475\pi$
$173$	7.93418	0.603225	0.301612	+	0.953431i	$0.402475\pi$
$174$	0	0
$175$	3.12017	0.235863
$176$	0	0
$177$	9.89567	0.743804
$178$	0	0
$179$	−3.28860	−0.245801	−0.122901	−	0.992419i	$-0.539220\pi$
$179$	−3.28860	−0.245801	−0.122901	+	0.992419i	$0.539220\pi$
$180$	0	0
$181$	19.7801	1.47024	0.735121	−	0.677936i	$-0.237125\pi$
$181$	19.7801	1.47024	0.735121	+	0.677936i	$0.237125\pi$
$182$	0	0
$183$	−1.26453	−0.0934764
$184$	0	0
$185$	−9.24034	−0.679364
$186$	0	0
$187$	−3.34766	−0.244805
$188$	0	0
$189$	3.12017	0.226959
$190$	0	0
$191$	9.91676	0.717552	0.358776	−	0.933424i	$-0.383194\pi$
$191$	9.91676	0.717552	0.358776	+	0.933424i	$0.383194\pi$
$192$	0	0
$193$	−23.9311	−1.72260	−0.861299	−	0.508098i	$-0.830349\pi$
$193$	−23.9311	−1.72260	−0.861299	+	0.508098i	$0.830349\pi$
$194$	0	0
$195$	5.41498	0.387775
$196$	0	0
$197$	−16.2193	−1.15557	−0.577787	−	0.816188i	$-0.696084\pi$
$197$	−16.2193	−1.15557	−0.577787	+	0.816188i	$0.696084\pi$
$198$	0	0
$199$	−2.00151	−0.141883	−0.0709415	−	0.997480i	$-0.522600\pi$
$199$	−2.00151	−0.141883	−0.0709415	+	0.997480i	$0.522600\pi$
$200$	0	0
$201$	−1.00000	−0.0705346
$202$	0	0
$203$	−12.8556	−0.902290
$204$	0	0
$205$	9.73547	0.679955
$206$	0	0
$207$	−3.00000	−0.208514
$208$	0	0
$209$	−36.6455	−2.53482
$210$	0	0
$211$	13.1186	0.903119	0.451559	−	0.892241i	$-0.350868\pi$
$211$	13.1186	0.903119	0.451559	+	0.892241i	$0.350868\pi$
$212$	0	0
$213$	−8.91011	−0.610511
$214$	0	0
$215$	9.05597	0.617612
$216$	0	0
$217$	−9.65532	−0.655446
$218$	0	0
$219$	−6.26603	−0.423419
$220$	0	0
$221$	−3.33185	−0.224124
$222$	0	0
$223$	−8.93418	−0.598277	−0.299139	−	0.954210i	$-0.596699\pi$
$223$	−8.93418	−0.598277	−0.299139	+	0.954210i	$0.596699\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	17.6614	1.17223	0.586115	−	0.810228i	$-0.300657\pi$
$227$	17.6614	1.17223	0.586115	+	0.810228i	$0.300657\pi$
$228$	0	0
$229$	7.58800	0.501429	0.250715	−	0.968061i	$-0.419334\pi$
$229$	7.58800	0.501429	0.250715	+	0.968061i	$0.419334\pi$
$230$	0	0
$231$	−16.9758	−1.11693
$232$	0	0
$233$	19.0560	1.24840	0.624199	−	0.781265i	$-0.285425\pi$
$233$	19.0560	1.24840	0.624199	+	0.781265i	$0.285425\pi$
$234$	0	0
$235$	6.42472	0.419102
$236$	0	0
$237$	−8.41498	−0.546612
$238$	0	0
$239$	−2.25479	−0.145850	−0.0729250	−	0.997337i	$-0.523233\pi$
$239$	−2.25479	−0.145850	−0.0729250	+	0.997337i	$0.523233\pi$
$240$	0	0
$241$	−18.1329	−1.16804	−0.584021	−	0.811738i	$-0.698521\pi$
$241$	−18.1329	−1.16804	−0.584021	+	0.811738i	$0.698521\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−2.73547	−0.174763
$246$	0	0
$247$	−36.4725	−2.32069
$248$	0	0
$249$	0.721141	0.0457005
$250$	0	0
$251$	15.1442	0.955896	0.477948	−	0.878388i	$-0.341381\pi$
$251$	15.1442	0.955896	0.477948	+	0.878388i	$0.341381\pi$
$252$	0	0
$253$	16.3220	1.02616
$254$	0	0
$255$	−0.615302	−0.0385317
$256$	0	0
$257$	−14.7883	−0.922470	−0.461235	−	0.887278i	$-0.652594\pi$
$257$	−14.7883	−0.922470	−0.461235	+	0.887278i	$0.652594\pi$
$258$	0	0
$259$	28.8315	1.79150
$260$	0	0
$261$	−4.12017	−0.255032
$262$	0	0
$263$	4.37646	0.269864	0.134932	−	0.990855i	$-0.456918\pi$
$263$	4.37646	0.269864	0.134932	+	0.990855i	$0.456918\pi$
$264$	0	0
$265$	−2.09448	−0.128663
$266$	0	0
$267$	−5.36213	−0.328157
$268$	0	0
$269$	31.5782	1.92536	0.962678	−	0.270649i	$-0.0872384\pi$
$269$	31.5782	1.92536	0.962678	+	0.270649i	$0.0872384\pi$
$270$	0	0
$271$	−21.1314	−1.28364	−0.641821	−	0.766854i	$-0.721821\pi$
$271$	−21.1314	−1.28364	−0.641821	+	0.766854i	$0.721821\pi$
$272$	0	0
$273$	−16.8957	−1.02257
$274$	0	0
$275$	−5.44067	−0.328085
$276$	0	0
$277$	26.5957	1.59798	0.798991	−	0.601343i	$-0.205368\pi$
$277$	26.5957	1.59798	0.798991	+	0.601343i	$0.205368\pi$
$278$	0	0
$279$	−3.09448	−0.185262
$280$	0	0
$281$	−15.8526	−0.945684	−0.472842	−	0.881147i	$-0.656772\pi$
$281$	−15.8526	−0.945684	−0.472842	+	0.881147i	$0.656772\pi$
$282$	0	0
$283$	22.1345	1.31576	0.657880	−	0.753123i	$-0.271453\pi$
$283$	22.1345	1.31576	0.657880	+	0.753123i	$0.271453\pi$
$284$	0	0
$285$	−6.73547	−0.398975
$286$	0	0
$287$	−30.3764	−1.79306
$288$	0	0
$289$	−16.6214	−0.977730
$290$	0	0
$291$	7.63938	0.447828
$292$	0	0
$293$	−13.5238	−0.790069	−0.395034	−	0.918666i	$-0.629267\pi$
$293$	−13.5238	−0.790069	−0.395034	+	0.918666i	$0.629267\pi$
$294$	0	0
$295$	−9.89567	−0.576148
$296$	0	0
$297$	−5.44067	−0.315700
$298$	0	0
$299$	16.2449	0.939469
$300$	0	0
$301$	−28.2562	−1.62866
$302$	0	0
$303$	−0.948624	−0.0544970
$304$	0	0
$305$	1.26453	0.0724065
$306$	0	0
$307$	−28.0173	−1.59903	−0.799517	−	0.600643i	$-0.794911\pi$
$307$	−28.0173	−1.59903	−0.799517	+	0.600643i	$0.794911\pi$
$308$	0	0
$309$	−13.2963	−0.756401
$310$	0	0
$311$	−26.4950	−1.50239	−0.751197	−	0.660078i	$-0.770523\pi$
$311$	−26.4950	−1.50239	−0.751197	+	0.660078i	$0.770523\pi$
$312$	0	0
$313$	11.5654	0.653717	0.326858	−	0.945073i	$-0.394010\pi$
$313$	11.5654	0.653717	0.326858	+	0.945073i	$0.394010\pi$
$314$	0	0
$315$	−3.12017	−0.175802
$316$	0	0
$317$	−17.2449	−0.968572	−0.484286	−	0.874910i	$-0.660921\pi$
$317$	−17.2449	−0.968572	−0.484286	+	0.874910i	$0.660921\pi$
$318$	0	0
$319$	22.4165	1.25508
$320$	0	0
$321$	−2.10584	−0.117536
$322$	0	0
$323$	4.14435	0.230598
$324$	0	0
$325$	−5.41498	−0.300369
$326$	0	0
$327$	−5.27886	−0.291922
$328$	0	0
$329$	−20.0462	−1.10518
$330$	0	0
$331$	−22.2017	−1.22032	−0.610159	−	0.792279i	$-0.708894\pi$
$331$	−22.2017	−1.22032	−0.610159	+	0.792279i	$0.708894\pi$
$332$	0	0
$333$	9.24034	0.506368
$334$	0	0
$335$	1.00000	0.0546358
$336$	0	0
$337$	−4.34104	−0.236471	−0.118236	−	0.992986i	$-0.537724\pi$
$337$	−4.34104	−0.236471	−0.118236	+	0.992986i	$0.537724\pi$
$338$	0	0
$339$	−16.8315	−0.914160
$340$	0	0
$341$	16.8361	0.911724
$342$	0	0
$343$	−13.3061	−0.718459
$344$	0	0
$345$	3.00000	0.161515
$346$	0	0
$347$	−20.7627	−1.11460	−0.557299	−	0.830312i	$-0.688162\pi$
$347$	−20.7627	−1.11460	−0.557299	+	0.830312i	$0.688162\pi$
$348$	0	0
$349$	−19.7082	−1.05495	−0.527477	−	0.849569i	$-0.676862\pi$
$349$	−19.7082	−1.05495	−0.527477	+	0.849569i	$0.676862\pi$
$350$	0	0
$351$	−5.41498	−0.289030
$352$	0	0
$353$	10.2948	0.547937	0.273969	−	0.961739i	$-0.411664\pi$
$353$	10.2948	0.547937	0.273969	+	0.961739i	$0.411664\pi$
$354$	0	0
$355$	8.91011	0.472900
$356$	0	0
$357$	1.91985	0.101609
$358$	0	0
$359$	3.47757	0.183539	0.0917695	−	0.995780i	$-0.470748\pi$
$359$	3.47757	0.183539	0.0917695	+	0.995780i	$0.470748\pi$
$360$	0	0
$361$	26.3666	1.38772
$362$	0	0
$363$	18.6009	0.976292
$364$	0	0
$365$	6.26603	0.327979
$366$	0	0
$367$	−7.67950	−0.400867	−0.200433	−	0.979707i	$-0.564235\pi$
$367$	−7.67950	−0.400867	−0.200433	+	0.979707i	$0.564235\pi$
$368$	0	0
$369$	−9.73547	−0.506809
$370$	0	0
$371$	6.53515	0.339288
$372$	0	0
$373$	−11.6374	−0.602559	−0.301280	−	0.953536i	$-0.597414\pi$
$373$	−11.6374	−0.602559	−0.301280	+	0.953536i	$0.597414\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	22.3106	1.14906
$378$	0	0
$379$	−19.0190	−0.976938	−0.488469	−	0.872581i	$-0.662445\pi$
$379$	−19.0190	−0.976938	−0.488469	+	0.872581i	$0.662445\pi$
$380$	0	0
$381$	−12.1761	−0.623803
$382$	0	0
$383$	10.7452	0.549055	0.274527	−	0.961579i	$-0.411479\pi$
$383$	10.7452	0.549055	0.274527	+	0.961579i	$0.411479\pi$
$384$	0	0
$385$	16.9758	0.865168
$386$	0	0
$387$	−9.05597	−0.460341
$388$	0	0
$389$	−23.1520	−1.17385	−0.586926	−	0.809641i	$-0.699662\pi$
$389$	−23.1520	−1.17385	−0.586926	+	0.809641i	$0.699662\pi$
$390$	0	0
$391$	−1.84591	−0.0933515
$392$	0	0
$393$	−13.4134	−0.676615
$394$	0	0
$395$	8.41498	0.423404
$396$	0	0
$397$	20.9151	1.04970	0.524851	−	0.851194i	$-0.324121\pi$
$397$	20.9151	1.04970	0.524851	+	0.851194i	$0.324121\pi$
$398$	0	0
$399$	21.0158	1.05211
$400$	0	0
$401$	15.6970	0.783869	0.391934	−	0.919993i	$-0.371806\pi$
$401$	15.6970	0.783869	0.391934	+	0.919993i	$0.371806\pi$
$402$	0	0
$403$	16.7566	0.834704
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−50.2736	−2.49197
$408$	0	0
$409$	−21.5926	−1.06769	−0.533843	−	0.845583i	$-0.679253\pi$
$409$	−21.5926	−1.06769	−0.533843	+	0.845583i	$0.679253\pi$
$410$	0	0
$411$	21.5110	1.06106
$412$	0	0
$413$	30.8762	1.51932
$414$	0	0
$415$	−0.721141	−0.0353994
$416$	0	0
$417$	14.2419	0.697426
$418$	0	0
$419$	25.8747	1.26406	0.632031	−	0.774943i	$-0.282221\pi$
$419$	25.8747	1.26406	0.632031	+	0.774943i	$0.282221\pi$
$420$	0	0
$421$	13.0831	0.637633	0.318816	−	0.947817i	$-0.396715\pi$
$421$	13.0831	0.637633	0.318816	+	0.947817i	$0.396715\pi$
$422$	0	0
$423$	−6.42472	−0.312381
$424$	0	0
$425$	0.615302	0.0298465
$426$	0	0
$427$	−3.94554	−0.190938
$428$	0	0
$429$	29.4611	1.42240
$430$	0	0
$431$	33.9403	1.63485	0.817423	−	0.576038i	$-0.195402\pi$
$431$	33.9403	1.63485	0.817423	+	0.576038i	$0.195402\pi$
$432$	0	0
$433$	24.1986	1.16291	0.581455	−	0.813578i	$-0.302484\pi$
$433$	24.1986	1.16291	0.581455	+	0.813578i	$0.302484\pi$
$434$	0	0
$435$	4.12017	0.197547
$436$	0	0
$437$	−20.2064	−0.966604
$438$	0	0
$439$	25.3924	1.21191	0.605957	−	0.795497i	$-0.292790\pi$
$439$	25.3924	1.21191	0.605957	+	0.795497i	$0.292790\pi$
$440$	0	0
$441$	2.73547	0.130261
$442$	0	0
$443$	36.5541	1.73674	0.868369	−	0.495918i	$-0.165168\pi$
$443$	36.5541	1.73674	0.868369	+	0.495918i	$0.165168\pi$
$444$	0	0
$445$	5.36213	0.254189
$446$	0	0
$447$	−3.85414	−0.182295
$448$	0	0
$449$	−19.5865	−0.924345	−0.462173	−	0.886790i	$-0.652930\pi$
$449$	−19.5865	−0.924345	−0.462173	+	0.886790i	$0.652930\pi$
$450$	0	0
$451$	52.9675	2.49414
$452$	0	0
$453$	15.3492	0.721167
$454$	0	0
$455$	16.8957	0.792081
$456$	0	0
$457$	22.5860	1.05653	0.528264	−	0.849080i	$-0.322843\pi$
$457$	22.5860	1.05653	0.528264	+	0.849080i	$0.322843\pi$
$458$	0	0
$459$	0.615302	0.0287199
$460$	0	0
$461$	−25.8895	−1.20579	−0.602896	−	0.797820i	$-0.705987\pi$
$461$	−25.8895	−1.20579	−0.602896	+	0.797820i	$0.705987\pi$
$462$	0	0
$463$	39.7801	1.84874	0.924370	−	0.381498i	$-0.124592\pi$
$463$	39.7801	1.84874	0.924370	+	0.381498i	$0.124592\pi$
$464$	0	0
$465$	3.09448	0.143503
$466$	0	0
$467$	25.2321	1.16760	0.583801	−	0.811896i	$-0.301565\pi$
$467$	25.2321	1.16760	0.583801	+	0.811896i	$0.301565\pi$
$468$	0	0
$469$	−3.12017	−0.144076
$470$	0	0
$471$	−8.12017	−0.374158
$472$	0	0
$473$	49.2705	2.26546
$474$	0	0
$475$	6.73547	0.309045
$476$	0	0
$477$	2.09448	0.0958998
$478$	0	0
$479$	−42.4112	−1.93782	−0.968909	−	0.247416i	$-0.920419\pi$
$479$	−42.4112	−1.93782	−0.968909	+	0.247416i	$0.920419\pi$
$480$	0	0
$481$	−50.0363	−2.28146
$482$	0	0
$483$	−9.36052	−0.425918
$484$	0	0
$485$	−7.63938	−0.346886
$486$	0	0
$487$	12.1505	0.550590	0.275295	−	0.961360i	$-0.411225\pi$
$487$	12.1505	0.550590	0.275295	+	0.961360i	$0.411225\pi$
$488$	0	0
$489$	−3.18588	−0.144071
$490$	0	0
$491$	4.31429	0.194701	0.0973505	−	0.995250i	$-0.468963\pi$
$491$	4.31429	0.194701	0.0973505	+	0.995250i	$0.468963\pi$
$492$	0	0
$493$	−2.53515	−0.114177
$494$	0	0
$495$	5.44067	0.244540
$496$	0	0
$497$	−27.8011	−1.24705
$498$	0	0
$499$	−13.0302	−0.583311	−0.291655	−	0.956523i	$-0.594206\pi$
$499$	−13.0302	−0.583311	−0.291655	+	0.956523i	$0.594206\pi$
$500$	0	0
$501$	−10.2948	−0.459938
$502$	0	0
$503$	−8.23681	−0.367261	−0.183631	−	0.982995i	$-0.558785\pi$
$503$	−8.23681	−0.367261	−0.183631	+	0.982995i	$0.558785\pi$
$504$	0	0
$505$	0.948624	0.0422132
$506$	0	0
$507$	16.3220	0.724886
$508$	0	0
$509$	−1.13153	−0.0501540	−0.0250770	−	0.999686i	$-0.507983\pi$
$509$	−1.13153	−0.0501540	−0.0250770	+	0.999686i	$0.507983\pi$
$510$	0	0
$511$	−19.5511	−0.864890
$512$	0	0
$513$	6.73547	0.297378
$514$	0	0
$515$	13.2963	0.585906
$516$	0	0
$517$	34.9548	1.53731
$518$	0	0
$519$	7.93418	0.348272
$520$	0	0
$521$	−27.3975	−1.20030	−0.600152	−	0.799886i	$-0.704893\pi$
$521$	−27.3975	−1.20030	−0.600152	+	0.799886i	$0.704893\pi$
$522$	0	0
$523$	8.78534	0.384156	0.192078	−	0.981380i	$-0.438477\pi$
$523$	8.78534	0.384156	0.192078	+	0.981380i	$0.438477\pi$
$524$	0	0
$525$	3.12017	0.136175
$526$	0	0
$527$	−1.90404	−0.0829414
$528$	0	0
$529$	−14.0000	−0.608696
$530$	0	0
$531$	9.89567	0.429435
$532$	0	0
$533$	52.7174	2.28344
$534$	0	0
$535$	2.10584	0.0910433
$536$	0	0
$537$	−3.28860	−0.141914
$538$	0	0
$539$	−14.8828	−0.641048
$540$	0	0
$541$	−36.8392	−1.58384	−0.791920	−	0.610625i	$-0.790918\pi$
$541$	−36.8392	−1.58384	−0.791920	+	0.610625i	$0.790918\pi$
$542$	0	0
$543$	19.7801	0.848845
$544$	0	0
$545$	5.27886	0.226121
$546$	0	0
$547$	3.73239	0.159585	0.0797927	−	0.996811i	$-0.474574\pi$
$547$	3.73239	0.159585	0.0797927	+	0.996811i	$0.474574\pi$
$548$	0	0
$549$	−1.26453	−0.0539686
$550$	0	0
$551$	−27.7513	−1.18225
$552$	0	0
$553$	−26.2562	−1.11653
$554$	0	0
$555$	−9.24034	−0.392231
$556$	0	0
$557$	44.1339	1.87001	0.935006	−	0.354631i	$-0.115394\pi$
$557$	44.1339	1.87001	0.935006	+	0.354631i	$0.115394\pi$
$558$	0	0
$559$	49.0379	2.07408
$560$	0	0
$561$	−3.34766	−0.141338
$562$	0	0
$563$	43.8493	1.84803	0.924014	−	0.382357i	$-0.124888\pi$
$563$	43.8493	1.84803	0.924014	+	0.382357i	$0.124888\pi$
$564$	0	0
$565$	16.8315	0.708105
$566$	0	0
$567$	3.12017	0.131035
$568$	0	0
$569$	−20.5352	−0.860878	−0.430439	−	0.902620i	$-0.641641\pi$
$569$	−20.5352	−0.860878	−0.430439	+	0.902620i	$0.641641\pi$
$570$	0	0
$571$	37.1376	1.55416	0.777081	−	0.629401i	$-0.216700\pi$
$571$	37.1376	1.55416	0.777081	+	0.629401i	$0.216700\pi$
$572$	0	0
$573$	9.91676	0.414279
$574$	0	0
$575$	−3.00000	−0.125109
$576$	0	0
$577$	1.49204	0.0621146	0.0310573	−	0.999518i	$-0.490113\pi$
$577$	1.49204	0.0621146	0.0310573	+	0.999518i	$0.490113\pi$
$578$	0	0
$579$	−23.9311	−0.994542
$580$	0	0
$581$	2.25008	0.0933492
$582$	0	0
$583$	−11.3954	−0.471949
$584$	0	0
$585$	5.41498	0.223882
$586$	0	0
$587$	−16.9722	−0.700517	−0.350258	−	0.936653i	$-0.613906\pi$
$587$	−16.9722	−0.700517	−0.350258	+	0.936653i	$0.613906\pi$
$588$	0	0
$589$	−20.8428	−0.858813
$590$	0	0
$591$	−16.2193	−0.667171
$592$	0	0
$593$	−6.88593	−0.282771	−0.141386	−	0.989955i	$-0.545156\pi$
$593$	−6.88593	−0.282771	−0.141386	+	0.989955i	$0.545156\pi$
$594$	0	0
$595$	−1.91985	−0.0787061
$596$	0	0
$597$	−2.00151	−0.0819162
$598$	0	0
$599$	−22.1360	−0.904453	−0.452226	−	0.891903i	$-0.649370\pi$
$599$	−22.1360	−0.904453	−0.452226	+	0.891903i	$0.649370\pi$
$600$	0	0
$601$	−42.7462	−1.74365	−0.871826	−	0.489815i	$-0.837064\pi$
$601$	−42.7462	−1.74365	−0.871826	+	0.489815i	$0.837064\pi$
$602$	0	0
$603$	−1.00000	−0.0407231
$604$	0	0
$605$	−18.6009	−0.756233
$606$	0	0
$607$	35.3953	1.43665	0.718325	−	0.695708i	$-0.244909\pi$
$607$	35.3953	1.43665	0.718325	+	0.695708i	$0.244909\pi$
$608$	0	0
$609$	−12.8556	−0.520937
$610$	0	0
$611$	34.7897	1.40744
$612$	0	0
$613$	−0.361989	−0.0146206	−0.00731030	−	0.999973i	$-0.502327\pi$
$613$	−0.361989	−0.0146206	−0.00731030	+	0.999973i	$0.502327\pi$
$614$	0	0
$615$	9.73547	0.392572
$616$	0	0
$617$	12.0333	0.484442	0.242221	−	0.970221i	$-0.422124\pi$
$617$	12.0333	0.484442	0.242221	+	0.970221i	$0.422124\pi$
$618$	0	0
$619$	27.8751	1.12040	0.560198	−	0.828359i	$-0.310725\pi$
$619$	27.8751	1.12040	0.560198	+	0.828359i	$0.310725\pi$
$620$	0	0
$621$	−3.00000	−0.120386
$622$	0	0
$623$	−16.7308	−0.670304
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−36.6455	−1.46348
$628$	0	0
$629$	5.68561	0.226700
$630$	0	0
$631$	−9.61218	−0.382655	−0.191327	−	0.981526i	$-0.561279\pi$
$631$	−9.61218	−0.382655	−0.191327	+	0.981526i	$0.561279\pi$
$632$	0	0
$633$	13.1186	0.521416
$634$	0	0
$635$	12.1761	0.483195
$636$	0	0
$637$	−14.8125	−0.586894
$638$	0	0
$639$	−8.91011	−0.352478
$640$	0	0
$641$	34.4842	1.36204	0.681022	−	0.732263i	$-0.261536\pi$
$641$	34.4842	1.36204	0.681022	+	0.732263i	$0.261536\pi$
$642$	0	0
$643$	8.98714	0.354418	0.177209	−	0.984173i	$-0.443293\pi$
$643$	8.98714	0.354418	0.177209	+	0.984173i	$0.443293\pi$
$644$	0	0
$645$	9.05597	0.356578
$646$	0	0
$647$	3.30146	0.129794	0.0648969	−	0.997892i	$-0.479328\pi$
$647$	3.30146	0.129794	0.0648969	+	0.997892i	$0.479328\pi$
$648$	0	0
$649$	−53.8390	−2.11337
$650$	0	0
$651$	−9.65532	−0.378422
$652$	0	0
$653$	−16.6969	−0.653398	−0.326699	−	0.945128i	$-0.605936\pi$
$653$	−16.6969	−0.653398	−0.326699	+	0.945128i	$0.605936\pi$
$654$	0	0
$655$	13.4134	0.524104
$656$	0	0
$657$	−6.26603	−0.244461
$658$	0	0
$659$	−11.3878	−0.443606	−0.221803	−	0.975091i	$-0.571194\pi$
$659$	−11.3878	−0.443606	−0.221803	+	0.975091i	$0.571194\pi$
$660$	0	0
$661$	−13.8701	−0.539484	−0.269742	−	0.962933i	$-0.586938\pi$
$661$	−13.8701	−0.539484	−0.269742	+	0.962933i	$0.586938\pi$
$662$	0	0
$663$	−3.33185	−0.129398
$664$	0	0
$665$	−21.0158	−0.814959
$666$	0	0
$667$	12.3605	0.478601
$668$	0	0
$669$	−8.93418	−0.345415
$670$	0	0
$671$	6.87986	0.265594
$672$	0	0
$673$	0.268057	0.0103328	0.00516641	−	0.999987i	$-0.498355\pi$
$673$	0.268057	0.0103328	0.00516641	+	0.999987i	$0.498355\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	−11.5845	−0.445229	−0.222614	−	0.974907i	$-0.571459\pi$
$677$	−11.5845	−0.445229	−0.222614	+	0.974907i	$0.571459\pi$
$678$	0	0
$679$	23.8362	0.914748
$680$	0	0
$681$	17.6614	0.676787
$682$	0	0
$683$	−24.5330	−0.938730	−0.469365	−	0.883004i	$-0.655517\pi$
$683$	−24.5330	−0.938730	−0.469365	+	0.883004i	$0.655517\pi$
$684$	0	0
$685$	−21.5110	−0.821892
$686$	0	0
$687$	7.58800	0.289500
$688$	0	0
$689$	−11.3416	−0.432080
$690$	0	0
$691$	40.6810	1.54758	0.773789	−	0.633443i	$-0.218359\pi$
$691$	40.6810	1.54758	0.773789	+	0.633443i	$0.218359\pi$
$692$	0	0
$693$	−16.9758	−0.644858
$694$	0	0
$695$	−14.2419	−0.540224
$696$	0	0
$697$	−5.99026	−0.226897
$698$	0	0
$699$	19.0560	0.720763
$700$	0	0
$701$	20.8634	0.787998	0.393999	−	0.919111i	$-0.371091\pi$
$701$	20.8634	0.787998	0.393999	+	0.919111i	$0.371091\pi$
$702$	0	0
$703$	62.2381	2.34735
$704$	0	0
$705$	6.42472	0.241969
$706$	0	0
$707$	−2.95987	−0.111317
$708$	0	0
$709$	31.5474	1.18479	0.592393	−	0.805649i	$-0.298183\pi$
$709$	31.5474	1.18479	0.592393	+	0.805649i	$0.298183\pi$
$710$	0	0
$711$	−8.41498	−0.315586
$712$	0	0
$713$	9.28345	0.347668
$714$	0	0
$715$	−29.4611	−1.10178
$716$	0	0
$717$	−2.25479	−0.0842065
$718$	0	0
$719$	15.7802	0.588502	0.294251	−	0.955728i	$-0.404930\pi$
$719$	15.7802	0.588502	0.294251	+	0.955728i	$0.404930\pi$
$720$	0	0
$721$	−41.4868	−1.54505
$722$	0	0
$723$	−18.1329	−0.674369
$724$	0	0
$725$	−4.12017	−0.153019
$726$	0	0
$727$	7.95024	0.294858	0.147429	−	0.989073i	$-0.452900\pi$
$727$	7.95024	0.294858	0.147429	+	0.989073i	$0.452900\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−5.57216	−0.206094
$732$	0	0
$733$	−12.3908	−0.457665	−0.228832	−	0.973466i	$-0.573491\pi$
$733$	−12.3908	−0.457665	−0.228832	+	0.973466i	$0.573491\pi$
$734$	0	0
$735$	−2.73547	−0.100899
$736$	0	0
$737$	5.44067	0.200410
$738$	0	0
$739$	19.0114	0.699344	0.349672	−	0.936872i	$-0.386293\pi$
$739$	19.0114	0.699344	0.349672	+	0.936872i	$0.386293\pi$
$740$	0	0
$741$	−36.4725	−1.33985
$742$	0	0
$743$	−47.9244	−1.75817	−0.879087	−	0.476661i	$-0.841847\pi$
$743$	−47.9244	−1.75817	−0.879087	+	0.476661i	$0.841847\pi$
$744$	0	0
$745$	3.85414	0.141205
$746$	0	0
$747$	0.721141	0.0263852
$748$	0	0
$749$	−6.57058	−0.240084
$750$	0	0
$751$	7.55569	0.275711	0.137856	−	0.990452i	$-0.455979\pi$
$751$	7.55569	0.275711	0.137856	+	0.990452i	$0.455979\pi$
$752$	0	0
$753$	15.1442	0.551887
$754$	0	0
$755$	−15.3492	−0.558613
$756$	0	0
$757$	−21.5140	−0.781939	−0.390970	−	0.920404i	$-0.627860\pi$
$757$	−21.5140	−0.781939	−0.390970	+	0.920404i	$0.627860\pi$
$758$	0	0
$759$	16.3220	0.592451
$760$	0	0
$761$	−16.8716	−0.611595	−0.305797	−	0.952097i	$-0.598923\pi$
$761$	−16.8716	−0.611595	−0.305797	+	0.952097i	$0.598923\pi$
$762$	0	0
$763$	−16.4709	−0.596288
$764$	0	0
$765$	−0.615302	−0.0222463
$766$	0	0
$767$	−53.5848	−1.93484
$768$	0	0
$769$	35.8407	1.29245	0.646224	−	0.763148i	$-0.276347\pi$
$769$	35.8407	1.29245	0.646224	+	0.763148i	$0.276347\pi$
$770$	0	0
$771$	−14.7883	−0.532588
$772$	0	0
$773$	33.8346	1.21695	0.608473	−	0.793575i	$-0.291782\pi$
$773$	33.8346	1.21695	0.608473	+	0.793575i	$0.291782\pi$
$774$	0	0
$775$	−3.09448	−0.111157
$776$	0	0
$777$	28.8315	1.03432
$778$	0	0
$779$	−65.5730	−2.34940
$780$	0	0
$781$	48.4769	1.73464
$782$	0	0
$783$	−4.12017	−0.147243
$784$	0	0
$785$	8.12017	0.289821
$786$	0	0
$787$	38.8139	1.38357	0.691784	−	0.722105i	$-0.256825\pi$
$787$	38.8139	1.38357	0.691784	+	0.722105i	$0.256825\pi$
$788$	0	0
$789$	4.37646	0.155806
$790$	0	0
$791$	−52.5171	−1.86729
$792$	0	0
$793$	6.84738	0.243158
$794$	0	0
$795$	−2.09448	−0.0742837
$796$	0	0
$797$	−32.1956	−1.14043	−0.570214	−	0.821496i	$-0.693140\pi$
$797$	−32.1956	−1.14043	−0.570214	+	0.821496i	$0.693140\pi$
$798$	0	0
$799$	−3.95314	−0.139852
$800$	0	0
$801$	−5.36213	−0.189462
$802$	0	0
$803$	34.0914	1.20306
$804$	0	0
$805$	9.36052	0.329915
$806$	0	0
$807$	31.5782	1.11160
$808$	0	0
$809$	−26.9774	−0.948475	−0.474237	−	0.880397i	$-0.657276\pi$
$809$	−26.9774	−0.948475	−0.474237	+	0.880397i	$0.657276\pi$
$810$	0	0
$811$	4.24652	0.149115	0.0745577	−	0.997217i	$-0.476245\pi$
$811$	4.24652	0.149115	0.0745577	+	0.997217i	$0.476245\pi$
$812$	0	0
$813$	−21.1314	−0.741111
$814$	0	0
$815$	3.18588	0.111597
$816$	0	0
$817$	−60.9963	−2.13399
$818$	0	0
$819$	−16.8957	−0.590382
$820$	0	0
$821$	−18.5926	−0.648887	−0.324444	−	0.945905i	$-0.605177\pi$
$821$	−18.5926	−0.648887	−0.324444	+	0.945905i	$0.605177\pi$
$822$	0	0
$823$	47.9593	1.67176	0.835878	−	0.548915i	$-0.184959\pi$
$823$	47.9593	1.67176	0.835878	+	0.548915i	$0.184959\pi$
$824$	0	0
$825$	−5.44067	−0.189420
$826$	0	0
$827$	23.7997	0.827597	0.413799	−	0.910368i	$-0.364202\pi$
$827$	23.7997	0.827597	0.413799	+	0.910368i	$0.364202\pi$
$828$	0	0
$829$	−26.2255	−0.910849	−0.455425	−	0.890274i	$-0.650513\pi$
$829$	−26.2255	−0.910849	−0.455425	+	0.890274i	$0.650513\pi$
$830$	0	0
$831$	26.5957	0.922595
$832$	0	0
$833$	1.68314	0.0583175
$834$	0	0
$835$	10.2948	0.356266
$836$	0	0
$837$	−3.09448	−0.106961
$838$	0	0
$839$	23.5063	0.811527	0.405763	−	0.913978i	$-0.367006\pi$
$839$	23.5063	0.811527	0.405763	+	0.913978i	$0.367006\pi$
$840$	0	0
$841$	−12.0242	−0.414627
$842$	0	0
$843$	−15.8526	−0.545991
$844$	0	0
$845$	−16.3220	−0.561494
$846$	0	0
$847$	58.0379	1.99421
$848$	0	0
$849$	22.1345	0.759654
$850$	0	0
$851$	−27.7210	−0.950265
$852$	0	0
$853$	55.1658	1.88884	0.944420	−	0.328741i	$-0.106624\pi$
$853$	55.1658	1.88884	0.944420	+	0.328741i	$0.106624\pi$
$854$	0	0
$855$	−6.73547	−0.230348
$856$	0	0
$857$	19.0272	0.649957	0.324978	−	0.945721i	$-0.394643\pi$
$857$	19.0272	0.649957	0.324978	+	0.945721i	$0.394643\pi$
$858$	0	0
$859$	8.61527	0.293949	0.146975	−	0.989140i	$-0.453046\pi$
$859$	8.61527	0.293949	0.146975	+	0.989140i	$0.453046\pi$
$860$	0	0
$861$	−30.3764	−1.03522
$862$	0	0
$863$	12.1525	0.413676	0.206838	−	0.978375i	$-0.433683\pi$
$863$	12.1525	0.413676	0.206838	+	0.978375i	$0.433683\pi$
$864$	0	0
$865$	−7.93418	−0.269770
$866$	0	0
$867$	−16.6214	−0.564492
$868$	0	0
$869$	45.7831	1.55309
$870$	0	0
$871$	5.41498	0.183480
$872$	0	0
$873$	7.63938	0.258554
$874$	0	0
$875$	−3.12017	−0.105481
$876$	0	0
$877$	3.89886	0.131655	0.0658276	−	0.997831i	$-0.479031\pi$
$877$	3.89886	0.131655	0.0658276	+	0.997831i	$0.479031\pi$
$878$	0	0
$879$	−13.5238	−0.456146
$880$	0	0
$881$	35.0976	1.18247	0.591234	−	0.806500i	$-0.298641\pi$
$881$	35.0976	1.18247	0.591234	+	0.806500i	$0.298641\pi$
$882$	0	0
$883$	−18.1197	−0.609775	−0.304887	−	0.952388i	$-0.598619\pi$
$883$	−18.1197	−0.609775	−0.304887	+	0.952388i	$0.598619\pi$
$884$	0	0
$885$	−9.89567	−0.332639
$886$	0	0
$887$	33.5859	1.12770	0.563852	−	0.825876i	$-0.309319\pi$
$887$	33.5859	1.12770	0.563852	+	0.825876i	$0.309319\pi$
$888$	0	0
$889$	−37.9917	−1.27420
$890$	0	0
$891$	−5.44067	−0.182269
$892$	0	0
$893$	−43.2735	−1.44809
$894$	0	0
$895$	3.28860	0.109926
$896$	0	0
$897$	16.2449	0.542403
$898$	0	0
$899$	12.7498	0.425230
$900$	0	0
$901$	1.28874	0.0429342
$902$	0	0
$903$	−28.2562	−0.940307
$904$	0	0
$905$	−19.7801	−0.657512
$906$	0	0
$907$	37.4482	1.24345	0.621724	−	0.783236i	$-0.286432\pi$
$907$	37.4482	1.24345	0.621724	+	0.783236i	$0.286432\pi$
$908$	0	0
$909$	−0.948624	−0.0314639
$910$	0	0
$911$	11.6985	0.387590	0.193795	−	0.981042i	$-0.437920\pi$
$911$	11.6985	0.387590	0.193795	+	0.981042i	$0.437920\pi$
$912$	0	0
$913$	−3.92349	−0.129849
$914$	0	0
$915$	1.26453	0.0418039
$916$	0	0
$917$	−41.8520	−1.38208
$918$	0	0
$919$	33.7997	1.11495	0.557474	−	0.830194i	$-0.311771\pi$
$919$	33.7997	1.11495	0.557474	+	0.830194i	$0.311771\pi$
$920$	0	0
$921$	−28.0173	−0.923203
$922$	0	0
$923$	48.2481	1.58810
$924$	0	0
$925$	9.24034	0.303821
$926$	0	0
$927$	−13.2963	−0.436708
$928$	0	0
$929$	34.5490	1.13351	0.566757	−	0.823885i	$-0.308198\pi$
$929$	34.5490	1.13351	0.566757	+	0.823885i	$0.308198\pi$
$930$	0	0
$931$	18.4247	0.603846
$932$	0	0
$933$	−26.4950	−0.867408
$934$	0	0
$935$	3.34766	0.109480
$936$	0	0
$937$	−27.8766	−0.910690	−0.455345	−	0.890315i	$-0.650484\pi$
$937$	−27.8766	−0.910690	−0.455345	+	0.890315i	$0.650484\pi$
$938$	0	0
$939$	11.5654	0.377424
$940$	0	0
$941$	12.2782	0.400259	0.200129	−	0.979769i	$-0.435864\pi$
$941$	12.2782	0.400259	0.200129	+	0.979769i	$0.435864\pi$
$942$	0	0
$943$	29.2064	0.951092
$944$	0	0
$945$	−3.12017	−0.101499
$946$	0	0
$947$	16.2080	0.526689	0.263345	−	0.964702i	$-0.415174\pi$
$947$	16.2080	0.526689	0.263345	+	0.964702i	$0.415174\pi$
$948$	0	0
$949$	33.9304	1.10143
$950$	0	0
$951$	−17.2449	−0.559205
$952$	0	0
$953$	−33.3409	−1.08002	−0.540009	−	0.841659i	$-0.681579\pi$
$953$	−33.3409	−1.08002	−0.540009	+	0.841659i	$0.681579\pi$
$954$	0	0
$955$	−9.91676	−0.320899
$956$	0	0
$957$	22.4165	0.724622
$958$	0	0
$959$	67.1179	2.16735
$960$	0	0
$961$	−21.4242	−0.691102
$962$	0	0
$963$	−2.10584	−0.0678597
$964$	0	0
$965$	23.9311	0.770369
$966$	0	0
$967$	43.4725	1.39798	0.698990	−	0.715131i	$-0.253633\pi$
$967$	43.4725	1.39798	0.698990	+	0.715131i	$0.253633\pi$
$968$	0	0
$969$	4.14435	0.133136
$970$	0	0
$971$	−19.5994	−0.628974	−0.314487	−	0.949262i	$-0.601833\pi$
$971$	−19.5994	−0.628974	−0.314487	+	0.949262i	$0.601833\pi$
$972$	0	0
$973$	44.4370	1.42459
$974$	0	0
$975$	−5.41498	−0.173418
$976$	0	0
$977$	−37.5381	−1.20095	−0.600476	−	0.799643i	$-0.705022\pi$
$977$	−37.5381	−1.20095	−0.600476	+	0.799643i	$0.705022\pi$
$978$	0	0
$979$	29.1736	0.932391
$980$	0	0
$981$	−5.27886	−0.168541
$982$	0	0
$983$	−7.78674	−0.248359	−0.124179	−	0.992260i	$-0.539630\pi$
$983$	−7.78674	−0.248359	−0.124179	+	0.992260i	$0.539630\pi$
$984$	0	0
$985$	16.2193	0.516788
$986$	0	0
$987$	−20.0462	−0.638079
$988$	0	0
$989$	27.1679	0.863889
$990$	0	0
$991$	41.5110	1.31864	0.659320	−	0.751862i	$-0.270844\pi$
$991$	41.5110	1.31864	0.659320	+	0.751862i	$0.270844\pi$
$992$	0	0
$993$	−22.2017	−0.704550
$994$	0	0
$995$	2.00151	0.0634520
$996$	0	0
$997$	−11.4936	−0.364007	−0.182003	−	0.983298i	$-0.558258\pi$
$997$	−11.4936	−0.364007	−0.182003	+	0.983298i	$0.558258\pi$
$998$	0	0
$999$	9.24034	0.292352

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4020.2.a.c.1.4	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4020.2.a.c.1.4	✓	4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4020.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{5} +3.12017 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.00000 q^{5} +3.12017 q^{7} +1.00000 q^{9} -5.44067 q^{11} -5.41498 q^{13} -1.00000 q^{15} +0.615302 q^{17} +6.73547 q^{19} +3.12017 q^{21} -3.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} -4.12017 q^{29} -3.09448 q^{31} -5.44067 q^{33} -3.12017 q^{35} +9.24034 q^{37} -5.41498 q^{39} -9.73547 q^{41} -9.05597 q^{43} -1.00000 q^{45} -6.42472 q^{47} +2.73547 q^{49} +0.615302 q^{51} +2.09448 q^{53} +5.44067 q^{55} +6.73547 q^{57} +9.89567 q^{59} -1.26453 q^{61} +3.12017 q^{63} +5.41498 q^{65} -1.00000 q^{67} -3.00000 q^{69} -8.91011 q^{71} -6.26603 q^{73} +1.00000 q^{75} -16.9758 q^{77} -8.41498 q^{79} +1.00000 q^{81} +0.721141 q^{83} -0.615302 q^{85} -4.12017 q^{87} -5.36213 q^{89} -16.8957 q^{91} -3.09448 q^{93} -6.73547 q^{95} +7.63938 q^{97} -5.44067 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} - 4 q^{5} + q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 - 4 * q^5 + q^7 + 4 * q^9	\( 4 q + 4 q^{3} - 4 q^{5} + q^{7} + 4 q^{9} - 5 q^{11} - 5 q^{13} - 4 q^{15} - 8 q^{17} + 5 q^{19} + q^{21} - 12 q^{23} + 4 q^{25} + 4 q^{27} - 5 q^{29} - q^{31} - 5 q^{33} - q^{35} + 14 q^{37} - 5 q^{39} - 17 q^{41} - 9 q^{43} - 4 q^{45} - 7 q^{47} - 11 q^{49} - 8 q^{51} - 3 q^{53} + 5 q^{55} + 5 q^{57} - 23 q^{59} - 27 q^{61} + q^{63} + 5 q^{65} - 4 q^{67} - 12 q^{69} - 20 q^{71} - 2 q^{73} + 4 q^{75} - 23 q^{77} - 17 q^{79} + 4 q^{81} + 10 q^{83} + 8 q^{85} - 5 q^{87} - 18 q^{89} - 5 q^{91} - q^{93} - 5 q^{95} - 11 q^{97} - 5 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 - 4 * q^5 + q^7 + 4 * q^9 - 5 * q^11 - 5 * q^13 - 4 * q^15 - 8 * q^17 + 5 * q^19 + q^21 - 12 * q^23 + 4 * q^25 + 4 * q^27 - 5 * q^29 - q^31 - 5 * q^33 - q^35 + 14 * q^37 - 5 * q^39 - 17 * q^41 - 9 * q^43 - 4 * q^45 - 7 * q^47 - 11 * q^49 - 8 * q^51 - 3 * q^53 + 5 * q^55 + 5 * q^57 - 23 * q^59 - 27 * q^61 + q^63 + 5 * q^65 - 4 * q^67 - 12 * q^69 - 20 * q^71 - 2 * q^73 + 4 * q^75 - 23 * q^77 - 17 * q^79 + 4 * q^81 + 10 * q^83 + 8 * q^85 - 5 * q^87 - 18 * q^89 - 5 * q^91 - q^93 - 5 * q^95 - 11 * q^97 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

Properties

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