LMFDB - Embedded newform 4020.2.a.d.1.2

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:	$\Q(\zeta_{15})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 4x^{2} + 4x + 1 $ x^4 - x^3 - 4x^2 + 4x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.33826$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +1.00000 q^{5} -1.40898 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.00000 q^{5} -1.40898 q^{7} +1.00000 q^{9} +3.19236 q^{11} -7.12165 q^{13} +1.00000 q^{15} +5.07794 q^{17} -5.01478 q^{19} -1.40898 q^{21} -3.97766 q^{23} +1.00000 q^{25} +1.00000 q^{27} -9.15780 q^{29} -7.41056 q^{31} +3.19236 q^{33} -1.40898 q^{35} -5.97766 q^{37} -7.12165 q^{39} +0.574329 q^{41} -0.608897 q^{43} +1.00000 q^{45} +2.70353 q^{47} -5.01478 q^{49} +5.07794 q^{51} +3.64662 q^{53} +3.19236 q^{55} -5.01478 q^{57} +4.84497 q^{59} -0.942488 q^{61} -1.40898 q^{63} -7.12165 q^{65} +1.00000 q^{67} -3.97766 q^{69} +4.36121 q^{71} -9.56519 q^{73} +1.00000 q^{75} -4.49797 q^{77} -6.26308 q^{79} +1.00000 q^{81} +2.76105 q^{83} +5.07794 q^{85} -9.15780 q^{87} +17.4042 q^{89} +10.0342 q^{91} -7.41056 q^{93} -5.01478 q^{95} +7.08261 q^{97} +3.19236 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 4 q^{5} - 3 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 + 4 * q^5 - 3 * q^7 + 4 * q^9	$ 4 q + 4 q^{3} + 4 q^{5} - 3 q^{7} + 4 q^{9} - 5 q^{11} - 9 q^{13} + 4 q^{15} - 8 q^{17} - 7 q^{19} - 3 q^{21} - 12 q^{23} + 4 q^{25} + 4 q^{27} - 9 q^{29} - 9 q^{31} - 5 q^{33} - 3 q^{35} - 20 q^{37} - 9 q^{39} - 11 q^{41} + q^{43} + 4 q^{45} - 5 q^{47} - 7 q^{49} - 8 q^{51} - 15 q^{53} - 5 q^{55} - 7 q^{57} + 7 q^{59} - 7 q^{61} - 3 q^{63} - 9 q^{65} + 4 q^{67} - 12 q^{69} - 26 q^{73} + 4 q^{75} - 15 q^{77} - 9 q^{79} + 4 q^{81} - 8 q^{83} - 8 q^{85} - 9 q^{87} + 10 q^{89} + 3 q^{91} - 9 q^{93} - 7 q^{95} - 11 q^{97} - 5 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 + 4 * q^5 - 3 * q^7 + 4 * q^9 - 5 * q^11 - 9 * q^13 + 4 * q^15 - 8 * q^17 - 7 * q^19 - 3 * q^21 - 12 * q^23 + 4 * q^25 + 4 * q^27 - 9 * q^29 - 9 * q^31 - 5 * q^33 - 3 * q^35 - 20 * q^37 - 9 * q^39 - 11 * q^41 + q^43 + 4 * q^45 - 5 * q^47 - 7 * q^49 - 8 * q^51 - 15 * q^53 - 5 * q^55 - 7 * q^57 + 7 * q^59 - 7 * q^61 - 3 * q^63 - 9 * q^65 + 4 * q^67 - 12 * q^69 - 26 * q^73 + 4 * q^75 - 15 * q^77 - 9 * q^79 + 4 * q^81 - 8 * q^83 - 8 * q^85 - 9 * q^87 + 10 * q^89 + 3 * q^91 - 9 * q^93 - 7 * 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
$7$	−1.40898	−0.532543	−0.266272	−	0.963898i	$-0.585792\pi$
$7$	−1.40898	−0.532543	−0.266272	+	0.963898i	$0.585792\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.19236	0.962534	0.481267	−	0.876574i	$-0.340177\pi$
$11$	3.19236	0.962534	0.481267	+	0.876574i	$0.340177\pi$
$12$	0	0
$13$	−7.12165	−1.97519	−0.987595	−	0.157024i	$-0.949810\pi$
$13$	−7.12165	−1.97519	−0.987595	+	0.157024i	$0.949810\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	5.07794	1.23158	0.615791	−	0.787910i	$-0.288837\pi$
$17$	5.07794	1.23158	0.615791	+	0.787910i	$0.288837\pi$
$18$	0	0
$19$	−5.01478	−1.15047	−0.575235	−	0.817988i	$-0.695089\pi$
$19$	−5.01478	−1.15047	−0.575235	+	0.817988i	$0.695089\pi$
$20$	0	0
$21$	−1.40898	−0.307464
$22$	0	0
$23$	−3.97766	−0.829399	−0.414700	−	0.909958i	$-0.636113\pi$
$23$	−3.97766	−0.829399	−0.414700	+	0.909958i	$0.636113\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−9.15780	−1.70056	−0.850280	−	0.526331i	$-0.823567\pi$
$29$	−9.15780	−1.70056	−0.850280	+	0.526331i	$0.823567\pi$
$30$	0	0
$31$	−7.41056	−1.33098	−0.665488	−	0.746409i	$-0.731776\pi$
$31$	−7.41056	−1.33098	−0.665488	+	0.746409i	$0.731776\pi$
$32$	0	0
$33$	3.19236	0.555719
$34$	0	0
$35$	−1.40898	−0.238161
$36$	0	0
$37$	−5.97766	−0.982721	−0.491361	−	0.870956i	$-0.663500\pi$
$37$	−5.97766	−0.982721	−0.491361	+	0.870956i	$0.663500\pi$
$38$	0	0
$39$	−7.12165	−1.14038
$40$	0	0
$41$	0.574329	0.0896952	0.0448476	−	0.998994i	$-0.485720\pi$
$41$	0.574329	0.0896952	0.0448476	+	0.998994i	$0.485720\pi$
$42$	0	0
$43$	−0.608897	−0.0928560	−0.0464280	−	0.998922i	$-0.514784\pi$
$43$	−0.608897	−0.0928560	−0.0464280	+	0.998922i	$0.514784\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	2.70353	0.394351	0.197175	−	0.980368i	$-0.436823\pi$
$47$	2.70353	0.394351	0.197175	+	0.980368i	$0.436823\pi$
$48$	0	0
$49$	−5.01478	−0.716398
$50$	0	0
$51$	5.07794	0.711054
$52$	0	0
$53$	3.64662	0.500902	0.250451	−	0.968129i	$-0.419421\pi$
$53$	3.64662	0.500902	0.250451	+	0.968129i	$0.419421\pi$
$54$	0	0
$55$	3.19236	0.430458
$56$	0	0
$57$	−5.01478	−0.664224
$58$	0	0
$59$	4.84497	0.630761	0.315380	−	0.948965i	$-0.397868\pi$
$59$	4.84497	0.630761	0.315380	+	0.948965i	$0.397868\pi$
$60$	0	0
$61$	−0.942488	−0.120673	−0.0603366	−	0.998178i	$-0.519217\pi$
$61$	−0.942488	−0.120673	−0.0603366	+	0.998178i	$0.519217\pi$
$62$	0	0
$63$	−1.40898	−0.177514
$64$	0	0
$65$	−7.12165	−0.883332
$66$	0	0
$67$	1.00000	0.122169
$67$	1.00000	0.122169
$68$	0	0
$69$	−3.97766	−0.478854
$70$	0	0
$71$	4.36121	0.517580	0.258790	−	0.965934i	$-0.416676\pi$
$71$	4.36121	0.517580	0.258790	+	0.965934i	$0.416676\pi$
$72$	0	0
$73$	−9.56519	−1.11952	−0.559761	−	0.828654i	$-0.689107\pi$
$73$	−9.56519	−1.11952	−0.559761	+	0.828654i	$0.689107\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−4.49797	−0.512591
$78$	0	0
$79$	−6.26308	−0.704651	−0.352326	−	0.935877i	$-0.614609\pi$
$79$	−6.26308	−0.704651	−0.352326	+	0.935877i	$0.614609\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	2.76105	0.303064	0.151532	−	0.988452i	$-0.451579\pi$
$83$	2.76105	0.303064	0.151532	+	0.988452i	$0.451579\pi$
$84$	0	0
$85$	5.07794	0.550780
$86$	0	0
$87$	−9.15780	−0.981819
$88$	0	0
$89$	17.4042	1.84484	0.922420	−	0.386189i	$-0.126209\pi$
$89$	17.4042	1.84484	0.922420	+	0.386189i	$0.126209\pi$
$90$	0	0
$91$	10.0342	1.05187
$92$	0	0
$93$	−7.41056	−0.768439
$94$	0	0
$95$	−5.01478	−0.514506
$96$	0	0
$97$	7.08261	0.719130	0.359565	−	0.933120i	$-0.382925\pi$
$97$	7.08261	0.719130	0.359565	+	0.933120i	$0.382925\pi$
$98$	0	0
$99$	3.19236	0.320845
$100$	0	0
$101$	7.76011	0.772160	0.386080	−	0.922465i	$-0.373829\pi$
$101$	7.76011	0.772160	0.386080	+	0.922465i	$0.373829\pi$
$102$	0	0
$103$	−12.4077	−1.22256	−0.611282	−	0.791413i	$-0.709346\pi$
$103$	−12.4077	−1.22256	−0.611282	+	0.791413i	$0.709346\pi$
$104$	0	0
$105$	−1.40898	−0.137502
$106$	0	0
$107$	−16.4526	−1.59053	−0.795264	−	0.606263i	$-0.792668\pi$
$107$	−16.4526	−1.59053	−0.795264	+	0.606263i	$0.792668\pi$
$108$	0	0
$109$	−7.07348	−0.677516	−0.338758	−	0.940874i	$-0.610007\pi$
$109$	−7.07348	−0.677516	−0.338758	+	0.940874i	$0.610007\pi$
$110$	0	0
$111$	−5.97766	−0.567374
$112$	0	0
$113$	−13.7510	−1.29358	−0.646791	−	0.762667i	$-0.723890\pi$
$113$	−13.7510	−1.29358	−0.646791	+	0.762667i	$0.723890\pi$
$114$	0	0
$115$	−3.97766	−0.370919
$116$	0	0
$117$	−7.12165	−0.658397
$118$	0	0
$119$	−7.15470	−0.655871
$120$	0	0
$121$	−0.808817	−0.0735288
$122$	0	0
$123$	0.574329	0.0517855
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	5.05654	0.448695	0.224347	−	0.974509i	$-0.427975\pi$
$127$	5.05654	0.448695	0.224347	+	0.974509i	$0.427975\pi$
$128$	0	0
$129$	−0.608897	−0.0536104
$130$	0	0
$131$	10.6151	0.927450	0.463725	−	0.885979i	$-0.346513\pi$
$131$	10.6151	0.927450	0.463725	+	0.885979i	$0.346513\pi$
$132$	0	0
$133$	7.06572	0.612675
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	−7.42843	−0.634654	−0.317327	−	0.948316i	$-0.602785\pi$
$137$	−7.42843	−0.634654	−0.317327	+	0.948316i	$0.602785\pi$
$138$	0	0
$139$	4.06723	0.344978	0.172489	−	0.985011i	$-0.444819\pi$
$139$	4.06723	0.344978	0.172489	+	0.985011i	$0.444819\pi$
$140$	0	0
$141$	2.70353	0.227679
$142$	0	0
$143$	−22.7349	−1.90119
$144$	0	0
$145$	−9.15780	−0.760513
$146$	0	0
$147$	−5.01478	−0.413612
$148$	0	0
$149$	−5.76978	−0.472679	−0.236340	−	0.971671i	$-0.575948\pi$
$149$	−5.76978	−0.472679	−0.236340	+	0.971671i	$0.575948\pi$
$150$	0	0
$151$	−12.2872	−0.999918	−0.499959	−	0.866049i	$-0.666652\pi$
$151$	−12.2872	−0.999918	−0.499959	+	0.866049i	$0.666652\pi$
$152$	0	0
$153$	5.07794	0.410527
$154$	0	0
$155$	−7.41056	−0.595230
$156$	0	0
$157$	3.90345	0.311530	0.155765	−	0.987794i	$-0.450216\pi$
$157$	3.90345	0.311530	0.155765	+	0.987794i	$0.450216\pi$
$158$	0	0
$159$	3.64662	0.289196
$160$	0	0
$161$	5.60443	0.441691
$162$	0	0
$163$	16.6366	1.30308	0.651538	−	0.758616i	$-0.274124\pi$
$163$	16.6366	1.30308	0.651538	+	0.758616i	$0.274124\pi$
$164$	0	0
$165$	3.19236	0.248525
$166$	0	0
$167$	−8.70162	−0.673352	−0.336676	−	0.941621i	$-0.609303\pi$
$167$	−8.70162	−0.673352	−0.336676	+	0.941621i	$0.609303\pi$
$168$	0	0
$169$	37.7179	2.90137
$170$	0	0
$171$	−5.01478	−0.383490
$172$	0	0
$173$	−11.5203	−0.875873	−0.437936	−	0.899006i	$-0.644290\pi$
$173$	−11.5203	−0.875873	−0.437936	+	0.899006i	$0.644290\pi$
$174$	0	0
$175$	−1.40898	−0.106509
$176$	0	0
$177$	4.84497	0.364170
$178$	0	0
$179$	22.4854	1.68064	0.840319	−	0.542092i	$-0.182368\pi$
$179$	22.4854	1.68064	0.840319	+	0.542092i	$0.182368\pi$
$180$	0	0
$181$	−4.63400	−0.344442	−0.172221	−	0.985058i	$-0.555094\pi$
$181$	−4.63400	−0.344442	−0.172221	+	0.985058i	$0.555094\pi$
$182$	0	0
$183$	−0.942488	−0.0696707
$184$	0	0
$185$	−5.97766	−0.439486
$186$	0	0
$187$	16.2106	1.18544
$188$	0	0
$189$	−1.40898	−0.102488
$190$	0	0
$191$	−4.11540	−0.297780	−0.148890	−	0.988854i	$-0.547570\pi$
$191$	−4.11540	−0.297780	−0.148890	+	0.988854i	$0.547570\pi$
$192$	0	0
$193$	−23.3799	−1.68292	−0.841462	−	0.540316i	$-0.818305\pi$
$193$	−23.3799	−1.68292	−0.841462	+	0.540316i	$0.818305\pi$
$194$	0	0
$195$	−7.12165	−0.509992
$196$	0	0
$197$	−0.283840	−0.0202227	−0.0101114	−	0.999949i	$-0.503219\pi$
$197$	−0.283840	−0.0202227	−0.0101114	+	0.999949i	$0.503219\pi$
$198$	0	0
$199$	12.9851	0.920489	0.460244	−	0.887792i	$-0.347762\pi$
$199$	12.9851	0.920489	0.460244	+	0.887792i	$0.347762\pi$
$200$	0	0
$201$	1.00000	0.0705346
$202$	0	0
$203$	12.9031	0.905622
$204$	0	0
$205$	0.574329	0.0401129
$206$	0	0
$207$	−3.97766	−0.276466
$208$	0	0
$209$	−16.0090	−1.10737
$210$	0	0
$211$	6.78745	0.467268	0.233634	−	0.972325i	$-0.424938\pi$
$211$	6.78745	0.467268	0.233634	+	0.972325i	$0.424938\pi$
$212$	0	0
$213$	4.36121	0.298825
$214$	0	0
$215$	−0.608897	−0.0415264
$216$	0	0
$217$	10.4413	0.708802
$218$	0	0
$219$	−9.56519	−0.646356
$220$	0	0
$221$	−36.1633	−2.43261
$222$	0	0
$223$	−15.6896	−1.05065	−0.525327	−	0.850901i	$-0.676057\pi$
$223$	−15.6896	−1.05065	−0.525327	+	0.850901i	$0.676057\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	0.492320	0.0326764	0.0163382	−	0.999867i	$-0.494799\pi$
$227$	0.492320	0.0326764	0.0163382	+	0.999867i	$0.494799\pi$
$228$	0	0
$229$	8.23704	0.544319	0.272160	−	0.962252i	$-0.412262\pi$
$229$	8.23704	0.544319	0.272160	+	0.962252i	$0.412262\pi$
$230$	0	0
$231$	−4.49797	−0.295944
$232$	0	0
$233$	−18.8164	−1.23271	−0.616353	−	0.787470i	$-0.711391\pi$
$233$	−18.8164	−1.23271	−0.616353	+	0.787470i	$0.711391\pi$
$234$	0	0
$235$	2.70353	0.176359
$236$	0	0
$237$	−6.26308	−0.406831
$238$	0	0
$239$	−25.6164	−1.65699	−0.828493	−	0.559999i	$-0.810802\pi$
$239$	−25.6164	−1.65699	−0.828493	+	0.559999i	$0.810802\pi$
$240$	0	0
$241$	−6.34931	−0.408995	−0.204497	−	0.978867i	$-0.565556\pi$
$241$	−6.34931	−0.408995	−0.204497	+	0.978867i	$0.565556\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−5.01478	−0.320383
$246$	0	0
$247$	35.7135	2.27240
$248$	0	0
$249$	2.76105	0.174974
$250$	0	0
$251$	−6.22753	−0.393079	−0.196539	−	0.980496i	$-0.562970\pi$
$251$	−6.22753	−0.393079	−0.196539	+	0.980496i	$0.562970\pi$
$252$	0	0
$253$	−12.6981	−0.798325
$254$	0	0
$255$	5.07794	0.317993
$256$	0	0
$257$	6.69691	0.417742	0.208871	−	0.977943i	$-0.433021\pi$
$257$	6.69691	0.417742	0.208871	+	0.977943i	$0.433021\pi$
$258$	0	0
$259$	8.42238	0.523342
$260$	0	0
$261$	−9.15780	−0.566853
$262$	0	0
$263$	2.35167	0.145010	0.0725051	−	0.997368i	$-0.476901\pi$
$263$	2.35167	0.145010	0.0725051	+	0.997368i	$0.476901\pi$
$264$	0	0
$265$	3.64662	0.224010
$266$	0	0
$267$	17.4042	1.06512
$268$	0	0
$269$	21.9663	1.33931	0.669654	−	0.742673i	$-0.266442\pi$
$269$	21.9663	1.33931	0.669654	+	0.742673i	$0.266442\pi$
$270$	0	0
$271$	13.3758	0.812522	0.406261	−	0.913757i	$-0.366832\pi$
$271$	13.3758	0.812522	0.406261	+	0.913757i	$0.366832\pi$
$272$	0	0
$273$	10.0342	0.607300
$274$	0	0
$275$	3.19236	0.192507
$276$	0	0
$277$	−7.09333	−0.426197	−0.213098	−	0.977031i	$-0.568356\pi$
$277$	−7.09333	−0.426197	−0.213098	+	0.977031i	$0.568356\pi$
$278$	0	0
$279$	−7.41056	−0.443658
$280$	0	0
$281$	−27.9710	−1.66861	−0.834304	−	0.551305i	$-0.814130\pi$
$281$	−27.9710	−1.66861	−0.834304	+	0.551305i	$0.814130\pi$
$282$	0	0
$283$	11.6233	0.690934	0.345467	−	0.938431i	$-0.387721\pi$
$283$	11.6233	0.690934	0.345467	+	0.938431i	$0.387721\pi$
$284$	0	0
$285$	−5.01478	−0.297050
$286$	0	0
$287$	−0.809217	−0.0477665
$288$	0	0
$289$	8.78550	0.516794
$290$	0	0
$291$	7.08261	0.415190
$292$	0	0
$293$	−17.2213	−1.00608	−0.503039	−	0.864264i	$-0.667785\pi$
$293$	−17.2213	−1.00608	−0.503039	+	0.864264i	$0.667785\pi$
$294$	0	0
$295$	4.84497	0.282085
$296$	0	0
$297$	3.19236	0.185240
$298$	0	0
$299$	28.3275	1.63822
$300$	0	0
$301$	0.857922	0.0494498
$302$	0	0
$303$	7.76011	0.445807
$304$	0	0
$305$	−0.942488	−0.0539667
$306$	0	0
$307$	−6.94276	−0.396244	−0.198122	−	0.980177i	$-0.563484\pi$
$307$	−6.94276	−0.396244	−0.198122	+	0.980177i	$0.563484\pi$
$308$	0	0
$309$	−12.4077	−0.705848
$310$	0	0
$311$	11.2723	0.639191	0.319596	−	0.947554i	$-0.396453\pi$
$311$	11.2723	0.639191	0.319596	+	0.947554i	$0.396453\pi$
$312$	0	0
$313$	5.18067	0.292829	0.146414	−	0.989223i	$-0.453227\pi$
$313$	5.18067	0.292829	0.146414	+	0.989223i	$0.453227\pi$
$314$	0	0
$315$	−1.40898	−0.0793869
$316$	0	0
$317$	14.4454	0.811333	0.405666	−	0.914021i	$-0.367039\pi$
$317$	14.4454	0.811333	0.405666	+	0.914021i	$0.367039\pi$
$318$	0	0
$319$	−29.2350	−1.63685
$320$	0	0
$321$	−16.4526	−0.918292
$322$	0	0
$323$	−25.4648	−1.41690
$324$	0	0
$325$	−7.12165	−0.395038
$326$	0	0
$327$	−7.07348	−0.391164
$328$	0	0
$329$	−3.80922	−0.210009
$330$	0	0
$331$	3.92364	0.215663	0.107831	−	0.994169i	$-0.465609\pi$
$331$	3.92364	0.215663	0.107831	+	0.994169i	$0.465609\pi$
$332$	0	0
$333$	−5.97766	−0.327574
$334$	0	0
$335$	1.00000	0.0546358
$336$	0	0
$337$	−13.1244	−0.714932	−0.357466	−	0.933926i	$-0.616359\pi$
$337$	−13.1244	−0.714932	−0.357466	+	0.933926i	$0.616359\pi$
$338$	0	0
$339$	−13.7510	−0.746850
$340$	0	0
$341$	−23.6572	−1.28111
$342$	0	0
$343$	16.9286	0.914056
$344$	0	0
$345$	−3.97766	−0.214150
$346$	0	0
$347$	8.47002	0.454695	0.227347	−	0.973814i	$-0.426995\pi$
$347$	8.47002	0.454695	0.227347	+	0.973814i	$0.426995\pi$
$348$	0	0
$349$	30.6970	1.64317	0.821587	−	0.570083i	$-0.193089\pi$
$349$	30.6970	1.64317	0.821587	+	0.570083i	$0.193089\pi$
$350$	0	0
$351$	−7.12165	−0.380125
$352$	0	0
$353$	−20.5785	−1.09528	−0.547641	−	0.836714i	$-0.684474\pi$
$353$	−20.5785	−1.09528	−0.547641	+	0.836714i	$0.684474\pi$
$354$	0	0
$355$	4.36121	0.231469
$356$	0	0
$357$	−7.15470	−0.378667
$358$	0	0
$359$	33.1675	1.75051	0.875256	−	0.483661i	$-0.160693\pi$
$359$	33.1675	1.75051	0.875256	+	0.483661i	$0.160693\pi$
$360$	0	0
$361$	6.14805	0.323582
$362$	0	0
$363$	−0.808817	−0.0424519
$364$	0	0
$365$	−9.56519	−0.500665
$366$	0	0
$367$	−12.8295	−0.669696	−0.334848	−	0.942272i	$-0.608685\pi$
$367$	−12.8295	−0.669696	−0.334848	+	0.942272i	$0.608685\pi$
$368$	0	0
$369$	0.574329	0.0298984
$370$	0	0
$371$	−5.13801	−0.266752
$372$	0	0
$373$	26.4487	1.36946	0.684732	−	0.728795i	$-0.259919\pi$
$373$	26.4487	1.36946	0.684732	+	0.728795i	$0.259919\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	65.2186	3.35893
$378$	0	0
$379$	−2.05942	−0.105785	−0.0528927	−	0.998600i	$-0.516844\pi$
$379$	−2.05942	−0.105785	−0.0528927	+	0.998600i	$0.516844\pi$
$380$	0	0
$381$	5.05654	0.259054
$382$	0	0
$383$	−9.65451	−0.493323	−0.246661	−	0.969102i	$-0.579334\pi$
$383$	−9.65451	−0.493323	−0.246661	+	0.969102i	$0.579334\pi$
$384$	0	0
$385$	−4.49797	−0.229238
$386$	0	0
$387$	−0.608897	−0.0309520
$388$	0	0
$389$	17.2130	0.872734	0.436367	−	0.899769i	$-0.356265\pi$
$389$	17.2130	0.872734	0.436367	+	0.899769i	$0.356265\pi$
$390$	0	0
$391$	−20.1983	−1.02147
$392$	0	0
$393$	10.6151	0.535463
$394$	0	0
$395$	−6.26308	−0.315130
$396$	0	0
$397$	7.94172	0.398583	0.199292	−	0.979940i	$-0.436136\pi$
$397$	7.94172	0.398583	0.199292	+	0.979940i	$0.436136\pi$
$398$	0	0
$399$	7.06572	0.353728
$400$	0	0
$401$	21.6216	1.07973	0.539867	−	0.841751i	$-0.318475\pi$
$401$	21.6216	1.07973	0.539867	+	0.841751i	$0.318475\pi$
$402$	0	0
$403$	52.7754	2.62893
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	−19.0829	−0.945902
$408$	0	0
$409$	−25.9292	−1.28211	−0.641057	−	0.767493i	$-0.721504\pi$
$409$	−25.9292	−1.28211	−0.641057	+	0.767493i	$0.721504\pi$
$410$	0	0
$411$	−7.42843	−0.366417
$412$	0	0
$413$	−6.82644	−0.335907
$414$	0	0
$415$	2.76105	0.135534
$416$	0	0
$417$	4.06723	0.199173
$418$	0	0
$419$	−40.0322	−1.95570	−0.977851	−	0.209305i	$-0.932880\pi$
$419$	−40.0322	−1.95570	−0.977851	+	0.209305i	$0.932880\pi$
$420$	0	0
$421$	−35.6473	−1.73735	−0.868673	−	0.495386i	$-0.835026\pi$
$421$	−35.6473	−1.73735	−0.868673	+	0.495386i	$0.835026\pi$
$422$	0	0
$423$	2.70353	0.131450
$424$	0	0
$425$	5.07794	0.246316
$426$	0	0
$427$	1.32794	0.0642637
$428$	0	0
$429$	−22.7349	−1.09765
$430$	0	0
$431$	32.2527	1.55356	0.776779	−	0.629773i	$-0.216852\pi$
$431$	32.2527	1.55356	0.776779	+	0.629773i	$0.216852\pi$
$432$	0	0
$433$	−37.9201	−1.82232	−0.911162	−	0.412049i	$-0.864813\pi$
$433$	−37.9201	−1.82232	−0.911162	+	0.412049i	$0.864813\pi$
$434$	0	0
$435$	−9.15780	−0.439083
$436$	0	0
$437$	19.9471	0.954199
$438$	0	0
$439$	−25.1467	−1.20019	−0.600094	−	0.799929i	$-0.704870\pi$
$439$	−25.1467	−1.20019	−0.600094	+	0.799929i	$0.704870\pi$
$440$	0	0
$441$	−5.01478	−0.238799
$442$	0	0
$443$	−11.7090	−0.556310	−0.278155	−	0.960536i	$-0.589723\pi$
$443$	−11.7090	−0.556310	−0.278155	+	0.960536i	$0.589723\pi$
$444$	0	0
$445$	17.4042	0.825037
$446$	0	0
$447$	−5.76978	−0.272901
$448$	0	0
$449$	−4.00592	−0.189051	−0.0945255	−	0.995522i	$-0.530133\pi$
$449$	−4.00592	−0.189051	−0.0945255	+	0.995522i	$0.530133\pi$
$450$	0	0
$451$	1.83347	0.0863346
$452$	0	0
$453$	−12.2872	−0.577303
$454$	0	0
$455$	10.0342	0.470412
$456$	0	0
$457$	−1.99765	−0.0934460	−0.0467230	−	0.998908i	$-0.514878\pi$
$457$	−1.99765	−0.0934460	−0.0467230	+	0.998908i	$0.514878\pi$
$458$	0	0
$459$	5.07794	0.237018
$460$	0	0
$461$	11.9412	0.556156	0.278078	−	0.960558i	$-0.410303\pi$
$461$	11.9412	0.556156	0.278078	+	0.960558i	$0.410303\pi$
$462$	0	0
$463$	14.1301	0.656679	0.328340	−	0.944560i	$-0.393511\pi$
$463$	14.1301	0.656679	0.328340	+	0.944560i	$0.393511\pi$
$464$	0	0
$465$	−7.41056	−0.343656
$466$	0	0
$467$	19.3624	0.895986	0.447993	−	0.894037i	$-0.352139\pi$
$467$	19.3624	0.895986	0.447993	+	0.894037i	$0.352139\pi$
$468$	0	0
$469$	−1.40898	−0.0650605
$470$	0	0
$471$	3.90345	0.179862
$472$	0	0
$473$	−1.94382	−0.0893770
$474$	0	0
$475$	−5.01478	−0.230094
$476$	0	0
$477$	3.64662	0.166967
$478$	0	0
$479$	26.3178	1.20249	0.601245	−	0.799065i	$-0.294672\pi$
$479$	26.3178	1.20249	0.601245	+	0.799065i	$0.294672\pi$
$480$	0	0
$481$	42.5708	1.94106
$482$	0	0
$483$	5.60443	0.255010
$484$	0	0
$485$	7.08261	0.321605
$486$	0	0
$487$	31.5711	1.43062	0.715312	−	0.698805i	$-0.246285\pi$
$487$	31.5711	1.43062	0.715312	+	0.698805i	$0.246285\pi$
$488$	0	0
$489$	16.6366	0.752331
$490$	0	0
$491$	−6.50301	−0.293477	−0.146738	−	0.989175i	$-0.546878\pi$
$491$	−6.50301	−0.293477	−0.146738	+	0.989175i	$0.546878\pi$
$492$	0	0
$493$	−46.5028	−2.09438
$494$	0	0
$495$	3.19236	0.143486
$496$	0	0
$497$	−6.14484	−0.275634
$498$	0	0
$499$	−21.3230	−0.954550	−0.477275	−	0.878754i	$-0.658375\pi$
$499$	−21.3230	−0.954550	−0.477275	+	0.878754i	$0.658375\pi$
$500$	0	0
$501$	−8.70162	−0.388760
$502$	0	0
$503$	34.7064	1.54748	0.773740	−	0.633503i	$-0.218384\pi$
$503$	34.7064	1.54748	0.773740	+	0.633503i	$0.218384\pi$
$504$	0	0
$505$	7.76011	0.345320
$506$	0	0
$507$	37.7179	1.67511
$508$	0	0
$509$	−7.45057	−0.330241	−0.165120	−	0.986273i	$-0.552801\pi$
$509$	−7.45057	−0.330241	−0.165120	+	0.986273i	$0.552801\pi$
$510$	0	0
$511$	13.4771	0.596193
$512$	0	0
$513$	−5.01478	−0.221408
$514$	0	0
$515$	−12.4077	−0.546747
$516$	0	0
$517$	8.63066	0.379576
$518$	0	0
$519$	−11.5203	−0.505685
$520$	0	0
$521$	−41.2982	−1.80931	−0.904654	−	0.426146i	$-0.859871\pi$
$521$	−41.2982	−1.80931	−0.904654	+	0.426146i	$0.859871\pi$
$522$	0	0
$523$	21.2562	0.929467	0.464734	−	0.885451i	$-0.346150\pi$
$523$	21.2562	0.929467	0.464734	+	0.885451i	$0.346150\pi$
$524$	0	0
$525$	−1.40898	−0.0614928
$526$	0	0
$527$	−37.6304	−1.63921
$528$	0	0
$529$	−7.17823	−0.312097
$530$	0	0
$531$	4.84497	0.210254
$532$	0	0
$533$	−4.09017	−0.177165
$534$	0	0
$535$	−16.4526	−0.711306
$536$	0	0
$537$	22.4854	0.970317
$538$	0	0
$539$	−16.0090	−0.689557
$540$	0	0
$541$	15.5886	0.670205	0.335103	−	0.942182i	$-0.391229\pi$
$541$	15.5886	0.670205	0.335103	+	0.942182i	$0.391229\pi$
$542$	0	0
$543$	−4.63400	−0.198864
$544$	0	0
$545$	−7.07348	−0.302994
$546$	0	0
$547$	−9.08171	−0.388306	−0.194153	−	0.980971i	$-0.562196\pi$
$547$	−9.08171	−0.388306	−0.194153	+	0.980971i	$0.562196\pi$
$548$	0	0
$549$	−0.942488	−0.0402244
$550$	0	0
$551$	45.9244	1.95644
$552$	0	0
$553$	8.82453	0.375257
$554$	0	0
$555$	−5.97766	−0.253738
$556$	0	0
$557$	−39.9587	−1.69311	−0.846553	−	0.532305i	$-0.821326\pi$
$557$	−39.9587	−1.69311	−0.846553	+	0.532305i	$0.821326\pi$
$558$	0	0
$559$	4.33635	0.183408
$560$	0	0
$561$	16.2106	0.684414
$562$	0	0
$563$	−3.87802	−0.163439	−0.0817196	−	0.996655i	$-0.526041\pi$
$563$	−3.87802	−0.163439	−0.0817196	+	0.996655i	$0.526041\pi$
$564$	0	0
$565$	−13.7510	−0.578508
$566$	0	0
$567$	−1.40898	−0.0591715
$568$	0	0
$569$	14.5621	0.610473	0.305237	−	0.952277i	$-0.401264\pi$
$569$	14.5621	0.610473	0.305237	+	0.952277i	$0.401264\pi$
$570$	0	0
$571$	31.7077	1.32692	0.663462	−	0.748210i	$-0.269086\pi$
$571$	31.7077	1.32692	0.663462	+	0.748210i	$0.269086\pi$
$572$	0	0
$573$	−4.11540	−0.171923
$574$	0	0
$575$	−3.97766	−0.165880
$576$	0	0
$577$	−41.9141	−1.74491	−0.872454	−	0.488697i	$-0.837472\pi$
$577$	−41.9141	−1.74491	−0.872454	+	0.488697i	$0.837472\pi$
$578$	0	0
$579$	−23.3799	−0.971637
$580$	0	0
$581$	−3.89025	−0.161395
$582$	0	0
$583$	11.6414	0.482135
$584$	0	0
$585$	−7.12165	−0.294444
$586$	0	0
$587$	−21.1381	−0.872462	−0.436231	−	0.899835i	$-0.643687\pi$
$587$	−21.1381	−0.872462	−0.436231	+	0.899835i	$0.643687\pi$
$588$	0	0
$589$	37.1623	1.53125
$590$	0	0
$591$	−0.283840	−0.0116756
$592$	0	0
$593$	−8.24651	−0.338644	−0.169322	−	0.985561i	$-0.554158\pi$
$593$	−8.24651	−0.338644	−0.169322	+	0.985561i	$0.554158\pi$
$594$	0	0
$595$	−7.15470	−0.293314
$596$	0	0
$597$	12.9851	0.531444
$598$	0	0
$599$	5.12676	0.209474	0.104737	−	0.994500i	$-0.466600\pi$
$599$	5.12676	0.209474	0.104737	+	0.994500i	$0.466600\pi$
$600$	0	0
$601$	8.97949	0.366281	0.183140	−	0.983087i	$-0.441374\pi$
$601$	8.97949	0.366281	0.183140	+	0.983087i	$0.441374\pi$
$602$	0	0
$603$	1.00000	0.0407231
$604$	0	0
$605$	−0.808817	−0.0328831
$606$	0	0
$607$	−2.28663	−0.0928113	−0.0464057	−	0.998923i	$-0.514777\pi$
$607$	−2.28663	−0.0928113	−0.0464057	+	0.998923i	$0.514777\pi$
$608$	0	0
$609$	12.9031	0.522861
$610$	0	0
$611$	−19.2536	−0.778918
$612$	0	0
$613$	19.4831	0.786913	0.393457	−	0.919343i	$-0.371279\pi$
$613$	19.4831	0.786913	0.393457	+	0.919343i	$0.371279\pi$
$614$	0	0
$615$	0.574329	0.0231592
$616$	0	0
$617$	20.8578	0.839705	0.419852	−	0.907592i	$-0.362082\pi$
$617$	20.8578	0.839705	0.419852	+	0.907592i	$0.362082\pi$
$618$	0	0
$619$	−32.3846	−1.30165	−0.650823	−	0.759229i	$-0.725576\pi$
$619$	−32.3846	−1.30165	−0.650823	+	0.759229i	$0.725576\pi$
$620$	0	0
$621$	−3.97766	−0.159618
$622$	0	0
$623$	−24.5221	−0.982457
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−16.0090	−0.639338
$628$	0	0
$629$	−30.3542	−1.21030
$630$	0	0
$631$	36.1789	1.44026	0.720129	−	0.693840i	$-0.244083\pi$
$631$	36.1789	1.44026	0.720129	+	0.693840i	$0.244083\pi$
$632$	0	0
$633$	6.78745	0.269777
$634$	0	0
$635$	5.05654	0.200662
$636$	0	0
$637$	35.7135	1.41502
$638$	0	0
$639$	4.36121	0.172527
$640$	0	0
$641$	44.8883	1.77298	0.886490	−	0.462748i	$-0.153137\pi$
$641$	44.8883	1.77298	0.886490	+	0.462748i	$0.153137\pi$
$642$	0	0
$643$	−35.2853	−1.39152	−0.695759	−	0.718275i	$-0.744932\pi$
$643$	−35.2853	−1.39152	−0.695759	+	0.718275i	$0.744932\pi$
$644$	0	0
$645$	−0.608897	−0.0239753
$646$	0	0
$647$	−20.3420	−0.799726	−0.399863	−	0.916575i	$-0.630942\pi$
$647$	−20.3420	−0.799726	−0.399863	+	0.916575i	$0.630942\pi$
$648$	0	0
$649$	15.4669	0.607128
$650$	0	0
$651$	10.4413	0.409227
$652$	0	0
$653$	1.11494	0.0436310	0.0218155	−	0.999762i	$-0.493055\pi$
$653$	1.11494	0.0436310	0.0218155	+	0.999762i	$0.493055\pi$
$654$	0	0
$655$	10.6151	0.414768
$656$	0	0
$657$	−9.56519	−0.373174
$658$	0	0
$659$	−0.240781	−0.00937950	−0.00468975	−	0.999989i	$-0.501493\pi$
$659$	−0.240781	−0.00937950	−0.00468975	+	0.999989i	$0.501493\pi$
$660$	0	0
$661$	1.25430	0.0487866	0.0243933	−	0.999702i	$-0.492235\pi$
$661$	1.25430	0.0487866	0.0243933	+	0.999702i	$0.492235\pi$
$662$	0	0
$663$	−36.1633	−1.40447
$664$	0	0
$665$	7.06572	0.273997
$666$	0	0
$667$	36.4266	1.41044
$668$	0	0
$669$	−15.6896	−0.606595
$670$	0	0
$671$	−3.00876	−0.116152
$672$	0	0
$673$	−34.9585	−1.34755	−0.673776	−	0.738936i	$-0.735329\pi$
$673$	−34.9585	−1.34755	−0.673776	+	0.738936i	$0.735329\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	23.5337	0.904474	0.452237	−	0.891898i	$-0.350626\pi$
$677$	23.5337	0.904474	0.452237	+	0.891898i	$0.350626\pi$
$678$	0	0
$679$	−9.97924	−0.382968
$680$	0	0
$681$	0.492320	0.0188657
$682$	0	0
$683$	−25.4885	−0.975290	−0.487645	−	0.873042i	$-0.662144\pi$
$683$	−25.4885	−0.975290	−0.487645	+	0.873042i	$0.662144\pi$
$684$	0	0
$685$	−7.42843	−0.283826
$686$	0	0
$687$	8.23704	0.314263
$688$	0	0
$689$	−25.9700	−0.989377
$690$	0	0
$691$	−4.43882	−0.168861	−0.0844303	−	0.996429i	$-0.526907\pi$
$691$	−4.43882	−0.168861	−0.0844303	+	0.996429i	$0.526907\pi$
$692$	0	0
$693$	−4.49797	−0.170864
$694$	0	0
$695$	4.06723	0.154279
$696$	0	0
$697$	2.91641	0.110467
$698$	0	0
$699$	−18.8164	−0.711703
$700$	0	0
$701$	0.121248	0.00457947	0.00228974	−	0.999997i	$-0.499271\pi$
$701$	0.121248	0.00457947	0.00228974	+	0.999997i	$0.499271\pi$
$702$	0	0
$703$	29.9767	1.13059
$704$	0	0
$705$	2.70353	0.101821
$706$	0	0
$707$	−10.9338	−0.411209
$708$	0	0
$709$	23.1327	0.868768	0.434384	−	0.900728i	$-0.356966\pi$
$709$	23.1327	0.868768	0.434384	+	0.900728i	$0.356966\pi$
$710$	0	0
$711$	−6.26308	−0.234884
$712$	0	0
$713$	29.4767	1.10391
$714$	0	0
$715$	−22.7349	−0.850236
$716$	0	0
$717$	−25.6164	−0.956662
$718$	0	0
$719$	−12.4597	−0.464669	−0.232334	−	0.972636i	$-0.574636\pi$
$719$	−12.4597	−0.464669	−0.232334	+	0.972636i	$0.574636\pi$
$720$	0	0
$721$	17.4821	0.651068
$722$	0	0
$723$	−6.34931	−0.236133
$724$	0	0
$725$	−9.15780	−0.340112
$726$	0	0
$727$	53.3852	1.97995	0.989974	−	0.141251i	$-0.0451125\pi$
$727$	53.3852	1.97995	0.989974	+	0.141251i	$0.0451125\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−3.09194	−0.114360
$732$	0	0
$733$	−23.0134	−0.850021	−0.425010	−	0.905189i	$-0.639730\pi$
$733$	−23.0134	−0.850021	−0.425010	+	0.905189i	$0.639730\pi$
$734$	0	0
$735$	−5.01478	−0.184973
$736$	0	0
$737$	3.19236	0.117592
$738$	0	0
$739$	50.7352	1.86633	0.933163	−	0.359454i	$-0.117037\pi$
$739$	50.7352	1.86633	0.933163	+	0.359454i	$0.117037\pi$
$740$	0	0
$741$	35.7135	1.31197
$742$	0	0
$743$	5.71341	0.209605	0.104802	−	0.994493i	$-0.466579\pi$
$743$	5.71341	0.209605	0.104802	+	0.994493i	$0.466579\pi$
$744$	0	0
$745$	−5.76978	−0.211388
$746$	0	0
$747$	2.76105	0.101021
$748$	0	0
$749$	23.1813	0.847025
$750$	0	0
$751$	46.5856	1.69993	0.849967	−	0.526836i	$-0.176622\pi$
$751$	46.5856	1.69993	0.849967	+	0.526836i	$0.176622\pi$
$752$	0	0
$753$	−6.22753	−0.226944
$754$	0	0
$755$	−12.2872	−0.447177
$756$	0	0
$757$	−30.3230	−1.10211	−0.551054	−	0.834470i	$-0.685774\pi$
$757$	−30.3230	−1.10211	−0.551054	+	0.834470i	$0.685774\pi$
$758$	0	0
$759$	−12.6981	−0.460913
$760$	0	0
$761$	−21.8682	−0.792720	−0.396360	−	0.918095i	$-0.629727\pi$
$761$	−21.8682	−0.792720	−0.396360	+	0.918095i	$0.629727\pi$
$762$	0	0
$763$	9.96637	0.360807
$764$	0	0
$765$	5.07794	0.183593
$766$	0	0
$767$	−34.5041	−1.24587
$768$	0	0
$769$	−12.0091	−0.433061	−0.216530	−	0.976276i	$-0.569474\pi$
$769$	−12.0091	−0.433061	−0.216530	+	0.976276i	$0.569474\pi$
$770$	0	0
$771$	6.69691	0.241183
$772$	0	0
$773$	−13.0084	−0.467878	−0.233939	−	0.972251i	$-0.575162\pi$
$773$	−13.0084	−0.467878	−0.233939	+	0.972251i	$0.575162\pi$
$774$	0	0
$775$	−7.41056	−0.266195
$776$	0	0
$777$	8.42238	0.302151
$778$	0	0
$779$	−2.88014	−0.103192
$780$	0	0
$781$	13.9226	0.498188
$782$	0	0
$783$	−9.15780	−0.327273
$784$	0	0
$785$	3.90345	0.139320
$786$	0	0
$787$	38.9971	1.39010	0.695048	−	0.718963i	$-0.255383\pi$
$787$	38.9971	1.39010	0.695048	+	0.718963i	$0.255383\pi$
$788$	0	0
$789$	2.35167	0.0837216
$790$	0	0
$791$	19.3748	0.688889
$792$	0	0
$793$	6.71207	0.238352
$794$	0	0
$795$	3.64662	0.129332
$796$	0	0
$797$	−33.3873	−1.18264	−0.591320	−	0.806437i	$-0.701393\pi$
$797$	−33.3873	−1.18264	−0.591320	+	0.806437i	$0.701393\pi$
$798$	0	0
$799$	13.7284	0.485675
$800$	0	0
$801$	17.4042	0.614946
$802$	0	0
$803$	−30.5356	−1.07758
$804$	0	0
$805$	5.60443	0.197530
$806$	0	0
$807$	21.9663	0.773250
$808$	0	0
$809$	8.29680	0.291700	0.145850	−	0.989307i	$-0.453408\pi$
$809$	8.29680	0.291700	0.145850	+	0.989307i	$0.453408\pi$
$810$	0	0
$811$	−30.7078	−1.07830	−0.539148	−	0.842211i	$-0.681254\pi$
$811$	−30.7078	−1.07830	−0.539148	+	0.842211i	$0.681254\pi$
$812$	0	0
$813$	13.3758	0.469110
$814$	0	0
$815$	16.6366	0.582753
$816$	0	0
$817$	3.05349	0.106828
$818$	0	0
$819$	10.0342	0.350625
$820$	0	0
$821$	50.0162	1.74558	0.872788	−	0.488100i	$-0.162310\pi$
$821$	50.0162	1.74558	0.872788	+	0.488100i	$0.162310\pi$
$822$	0	0
$823$	−15.7692	−0.549681	−0.274841	−	0.961490i	$-0.588625\pi$
$823$	−15.7692	−0.549681	−0.274841	+	0.961490i	$0.588625\pi$
$824$	0	0
$825$	3.19236	0.111144
$826$	0	0
$827$	0.404919	0.0140804	0.00704021	−	0.999975i	$-0.497759\pi$
$827$	0.404919	0.0140804	0.00704021	+	0.999975i	$0.497759\pi$
$828$	0	0
$829$	−32.6858	−1.13522	−0.567612	−	0.823296i	$-0.692133\pi$
$829$	−32.6858	−1.13522	−0.567612	+	0.823296i	$0.692133\pi$
$830$	0	0
$831$	−7.09333	−0.246065
$832$	0	0
$833$	−25.4648	−0.882302
$834$	0	0
$835$	−8.70162	−0.301132
$836$	0	0
$837$	−7.41056	−0.256146
$838$	0	0
$839$	11.8959	0.410691	0.205346	−	0.978690i	$-0.434168\pi$
$839$	11.8959	0.410691	0.205346	+	0.978690i	$0.434168\pi$
$840$	0	0
$841$	54.8652	1.89190
$842$	0	0
$843$	−27.9710	−0.963371
$844$	0	0
$845$	37.7179	1.29753
$846$	0	0
$847$	1.13961	0.0391573
$848$	0	0
$849$	11.6233	0.398911
$850$	0	0
$851$	23.7771	0.815068
$852$	0	0
$853$	51.9217	1.77776	0.888882	−	0.458136i	$-0.151483\pi$
$853$	51.9217	1.77776	0.888882	+	0.458136i	$0.151483\pi$
$854$	0	0
$855$	−5.01478	−0.171502
$856$	0	0
$857$	27.3591	0.934571	0.467285	−	0.884107i	$-0.345232\pi$
$857$	27.3591	0.934571	0.467285	+	0.884107i	$0.345232\pi$
$858$	0	0
$859$	−17.4530	−0.595487	−0.297744	−	0.954646i	$-0.596234\pi$
$859$	−17.4530	−0.595487	−0.297744	+	0.954646i	$0.596234\pi$
$860$	0	0
$861$	−0.809217	−0.0275780
$862$	0	0
$863$	15.7344	0.535606	0.267803	−	0.963474i	$-0.413702\pi$
$863$	15.7344	0.535606	0.267803	+	0.963474i	$0.413702\pi$
$864$	0	0
$865$	−11.5203	−0.391702
$866$	0	0
$867$	8.78550	0.298371
$868$	0	0
$869$	−19.9940	−0.678251
$870$	0	0
$871$	−7.12165	−0.241308
$872$	0	0
$873$	7.08261	0.239710
$874$	0	0
$875$	−1.40898	−0.0476321
$876$	0	0
$877$	−40.1725	−1.35653	−0.678265	−	0.734818i	$-0.737268\pi$
$877$	−40.1725	−1.35653	−0.678265	+	0.734818i	$0.737268\pi$
$878$	0	0
$879$	−17.2213	−0.580860
$880$	0	0
$881$	−16.2135	−0.546248	−0.273124	−	0.961979i	$-0.588057\pi$
$881$	−16.2135	−0.546248	−0.273124	+	0.961979i	$0.588057\pi$
$882$	0	0
$883$	33.0586	1.11251	0.556255	−	0.831012i	$-0.312238\pi$
$883$	33.0586	1.11251	0.556255	+	0.831012i	$0.312238\pi$
$884$	0	0
$885$	4.84497	0.162862
$886$	0	0
$887$	32.0379	1.07573	0.537863	−	0.843032i	$-0.319232\pi$
$887$	32.0379	1.07573	0.537863	+	0.843032i	$0.319232\pi$
$888$	0	0
$889$	−7.12454	−0.238949
$890$	0	0
$891$	3.19236	0.106948
$892$	0	0
$893$	−13.5576	−0.453689
$894$	0	0
$895$	22.4854	0.751604
$896$	0	0
$897$	28.3275	0.945827
$898$	0	0
$899$	67.8644	2.26340
$900$	0	0
$901$	18.5174	0.616902
$902$	0	0
$903$	0.857922	0.0285499
$904$	0	0
$905$	−4.63400	−0.154039
$906$	0	0
$907$	32.6299	1.08346	0.541729	−	0.840553i	$-0.317770\pi$
$907$	32.6299	1.08346	0.541729	+	0.840553i	$0.317770\pi$
$908$	0	0
$909$	7.76011	0.257387
$910$	0	0
$911$	−12.7094	−0.421082	−0.210541	−	0.977585i	$-0.567522\pi$
$911$	−12.7094	−0.421082	−0.210541	+	0.977585i	$0.567522\pi$
$912$	0	0
$913$	8.81426	0.291709
$914$	0	0
$915$	−0.942488	−0.0311577
$916$	0	0
$917$	−14.9565	−0.493907
$918$	0	0
$919$	9.31275	0.307199	0.153600	−	0.988133i	$-0.450913\pi$
$919$	9.31275	0.307199	0.153600	+	0.988133i	$0.450913\pi$
$920$	0	0
$921$	−6.94276	−0.228772
$922$	0	0
$923$	−31.0590	−1.02232
$924$	0	0
$925$	−5.97766	−0.196544
$926$	0	0
$927$	−12.4077	−0.407521
$928$	0	0
$929$	43.1930	1.41712	0.708558	−	0.705652i	$-0.249346\pi$
$929$	43.1930	1.41712	0.708558	+	0.705652i	$0.249346\pi$
$930$	0	0
$931$	25.1481	0.824194
$932$	0	0
$933$	11.2723	0.369037
$934$	0	0
$935$	16.2106	0.530145
$936$	0	0
$937$	15.7629	0.514953	0.257476	−	0.966285i	$-0.417109\pi$
$937$	15.7629	0.514953	0.257476	+	0.966285i	$0.417109\pi$
$938$	0	0
$939$	5.18067	0.169065
$940$	0	0
$941$	23.1363	0.754223	0.377111	−	0.926168i	$-0.376917\pi$
$941$	23.1363	0.754223	0.377111	+	0.926168i	$0.376917\pi$
$942$	0	0
$943$	−2.28449	−0.0743931
$944$	0	0
$945$	−1.40898	−0.0458340
$946$	0	0
$947$	13.2913	0.431908	0.215954	−	0.976404i	$-0.430714\pi$
$947$	13.2913	0.431908	0.215954	+	0.976404i	$0.430714\pi$
$948$	0	0
$949$	68.1199	2.21127
$950$	0	0
$951$	14.4454	0.468423
$952$	0	0
$953$	31.0911	1.00714	0.503570	−	0.863954i	$-0.332019\pi$
$953$	31.0911	1.00714	0.503570	+	0.863954i	$0.332019\pi$
$954$	0	0
$955$	−4.11540	−0.133171
$956$	0	0
$957$	−29.2350	−0.945034
$958$	0	0
$959$	10.4665	0.337981
$960$	0	0
$961$	23.9164	0.771495
$962$	0	0
$963$	−16.4526	−0.530176
$964$	0	0
$965$	−23.3799	−0.752627
$966$	0	0
$967$	19.6421	0.631648	0.315824	−	0.948818i	$-0.397719\pi$
$967$	19.6421	0.631648	0.315824	+	0.948818i	$0.397719\pi$
$968$	0	0
$969$	−25.4648	−0.818047
$970$	0	0
$971$	13.7237	0.440414	0.220207	−	0.975453i	$-0.429327\pi$
$971$	13.7237	0.440414	0.220207	+	0.975453i	$0.429327\pi$
$972$	0	0
$973$	−5.73063	−0.183715
$974$	0	0
$975$	−7.12165	−0.228075
$976$	0	0
$977$	0.748298	0.0239402	0.0119701	−	0.999928i	$-0.496190\pi$
$977$	0.748298	0.0239402	0.0119701	+	0.999928i	$0.496190\pi$
$978$	0	0
$979$	55.5605	1.77572
$980$	0	0
$981$	−7.07348	−0.225839
$982$	0	0
$983$	−10.3407	−0.329818	−0.164909	−	0.986309i	$-0.552733\pi$
$983$	−10.3407	−0.329818	−0.164909	+	0.986309i	$0.552733\pi$
$984$	0	0
$985$	−0.283840	−0.00904389
$986$	0	0
$987$	−3.80922	−0.121249
$988$	0	0
$989$	2.42199	0.0770147
$990$	0	0
$991$	38.9132	1.23612	0.618059	−	0.786132i	$-0.287919\pi$
$991$	38.9132	1.23612	0.618059	+	0.786132i	$0.287919\pi$
$992$	0	0
$993$	3.92364	0.124513
$994$	0	0
$995$	12.9851	0.411655
$996$	0	0
$997$	−56.1626	−1.77869	−0.889343	−	0.457240i	$-0.848838\pi$
$997$	−56.1626	−1.77869	−0.889343	+	0.457240i	$0.848838\pi$
$998$	0	0
$999$	−5.97766	−0.189125

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4020.2.a.d.1.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4020.2.a.d.1.2	✓	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.00000 q^{5} -1.40898 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.00000 q^{5} -1.40898 q^{7} +1.00000 q^{9} +3.19236 q^{11} -7.12165 q^{13} +1.00000 q^{15} +5.07794 q^{17} -5.01478 q^{19} -1.40898 q^{21} -3.97766 q^{23} +1.00000 q^{25} +1.00000 q^{27} -9.15780 q^{29} -7.41056 q^{31} +3.19236 q^{33} -1.40898 q^{35} -5.97766 q^{37} -7.12165 q^{39} +0.574329 q^{41} -0.608897 q^{43} +1.00000 q^{45} +2.70353 q^{47} -5.01478 q^{49} +5.07794 q^{51} +3.64662 q^{53} +3.19236 q^{55} -5.01478 q^{57} +4.84497 q^{59} -0.942488 q^{61} -1.40898 q^{63} -7.12165 q^{65} +1.00000 q^{67} -3.97766 q^{69} +4.36121 q^{71} -9.56519 q^{73} +1.00000 q^{75} -4.49797 q^{77} -6.26308 q^{79} +1.00000 q^{81} +2.76105 q^{83} +5.07794 q^{85} -9.15780 q^{87} +17.4042 q^{89} +10.0342 q^{91} -7.41056 q^{93} -5.01478 q^{95} +7.08261 q^{97} +3.19236 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 4 q^{5} - 3 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 + 4 * q^5 - 3 * q^7 + 4 * q^9	\( 4 q + 4 q^{3} + 4 q^{5} - 3 q^{7} + 4 q^{9} - 5 q^{11} - 9 q^{13} + 4 q^{15} - 8 q^{17} - 7 q^{19} - 3 q^{21} - 12 q^{23} + 4 q^{25} + 4 q^{27} - 9 q^{29} - 9 q^{31} - 5 q^{33} - 3 q^{35} - 20 q^{37} - 9 q^{39} - 11 q^{41} + q^{43} + 4 q^{45} - 5 q^{47} - 7 q^{49} - 8 q^{51} - 15 q^{53} - 5 q^{55} - 7 q^{57} + 7 q^{59} - 7 q^{61} - 3 q^{63} - 9 q^{65} + 4 q^{67} - 12 q^{69} - 26 q^{73} + 4 q^{75} - 15 q^{77} - 9 q^{79} + 4 q^{81} - 8 q^{83} - 8 q^{85} - 9 q^{87} + 10 q^{89} + 3 q^{91} - 9 q^{93} - 7 q^{95} - 11 q^{97} - 5 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 + 4 * q^5 - 3 * q^7 + 4 * q^9 - 5 * q^11 - 9 * q^13 + 4 * q^15 - 8 * q^17 - 7 * q^19 - 3 * q^21 - 12 * q^23 + 4 * q^25 + 4 * q^27 - 9 * q^29 - 9 * q^31 - 5 * q^33 - 3 * q^35 - 20 * q^37 - 9 * q^39 - 11 * q^41 + q^43 + 4 * q^45 - 5 * q^47 - 7 * q^49 - 8 * q^51 - 15 * q^53 - 5 * q^55 - 7 * q^57 + 7 * q^59 - 7 * q^61 - 3 * q^63 - 9 * q^65 + 4 * q^67 - 12 * q^69 - 26 * q^73 + 4 * q^75 - 15 * q^77 - 9 * q^79 + 4 * q^81 - 8 * q^83 - 8 * q^85 - 9 * q^87 + 10 * q^89 + 3 * q^91 - 9 * q^93 - 7 * q^95 - 11 * q^97 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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