LMFDB - Embedded newform 4020.2.a.g.1.1

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:	$0$
Dimension:	$6$
Coefficient field:	6.6.195727752.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 15x^{4} - 25x^{3} - 3x^{2} + 9x + 3 $ x^6 - 15x^4 - 25x^3 - 3x^2 + 9x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.75666$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{5} -3.63834 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.00000 q^{5} -3.63834 q^{7} +1.00000 q^{9} -5.52487 q^{11} +0.707786 q^{13} -1.00000 q^{15} -1.04402 q^{17} +0.542112 q^{19} -3.63834 q^{21} +5.02008 q^{23} +1.00000 q^{25} +1.00000 q^{27} +9.47555 q^{29} -6.10764 q^{31} -5.52487 q^{33} +3.63834 q^{35} -1.83721 q^{37} +0.707786 q^{39} -6.32174 q^{41} +6.20172 q^{43} -1.00000 q^{45} -8.96782 q^{47} +6.23751 q^{49} -1.04402 q^{51} +6.97698 q^{53} +5.52487 q^{55} +0.542112 q^{57} +3.28250 q^{59} +2.03261 q^{61} -3.63834 q^{63} -0.707786 q^{65} +1.00000 q^{67} +5.02008 q^{69} +5.85059 q^{71} +6.69103 q^{73} +1.00000 q^{75} +20.1014 q^{77} -7.17626 q^{79} +1.00000 q^{81} +5.88653 q^{83} +1.04402 q^{85} +9.47555 q^{87} +16.9208 q^{89} -2.57517 q^{91} -6.10764 q^{93} -0.542112 q^{95} +2.80020 q^{97} -5.52487 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{3} - 6 q^{5} + 3 q^{7} + 6 q^{9}+O(q^{10}) $ 6 * q + 6 * q^3 - 6 * q^5 + 3 * q^7 + 6 * q^9	$ 6 q + 6 q^{3} - 6 q^{5} + 3 q^{7} + 6 q^{9} + 3 q^{11} + 3 q^{13} - 6 q^{15} + 6 q^{17} - 3 q^{19} + 3 q^{21} + 6 q^{23} + 6 q^{25} + 6 q^{27} + 21 q^{29} - 3 q^{31} + 3 q^{33} - 3 q^{35} + 3 q^{39} + 9 q^{41} - 3 q^{43} - 6 q^{45} + 21 q^{47} + 3 q^{49} + 6 q^{51} + 15 q^{53} - 3 q^{55} - 3 q^{57} + 15 q^{59} + 15 q^{61} + 3 q^{63} - 3 q^{65} + 6 q^{67} + 6 q^{69} + 18 q^{71} - 6 q^{73} + 6 q^{75} + 33 q^{77} + 9 q^{79} + 6 q^{81} + 24 q^{83} - 6 q^{85} + 21 q^{87} + 24 q^{89} - 21 q^{91} - 3 q^{93} + 3 q^{95} + 9 q^{97} + 3 q^{99}+O(q^{100}) $ 6 * q + 6 * q^3 - 6 * q^5 + 3 * q^7 + 6 * q^9 + 3 * q^11 + 3 * q^13 - 6 * q^15 + 6 * q^17 - 3 * q^19 + 3 * q^21 + 6 * q^23 + 6 * q^25 + 6 * q^27 + 21 * q^29 - 3 * q^31 + 3 * q^33 - 3 * q^35 + 3 * q^39 + 9 * q^41 - 3 * q^43 - 6 * q^45 + 21 * q^47 + 3 * q^49 + 6 * q^51 + 15 * q^53 - 3 * q^55 - 3 * q^57 + 15 * q^59 + 15 * q^61 + 3 * q^63 - 3 * q^65 + 6 * q^67 + 6 * q^69 + 18 * q^71 - 6 * q^73 + 6 * q^75 + 33 * q^77 + 9 * q^79 + 6 * q^81 + 24 * q^83 - 6 * q^85 + 21 * q^87 + 24 * q^89 - 21 * q^91 - 3 * q^93 + 3 * q^95 + 9 * q^97 + 3 * 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
$7$	−3.63834	−1.37516	−0.687582	−	0.726107i	$-0.741328\pi$
$7$	−3.63834	−1.37516	−0.687582	+	0.726107i	$0.741328\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.52487	−1.66581	−0.832906	−	0.553414i	$-0.813325\pi$
$11$	−5.52487	−1.66581	−0.832906	+	0.553414i	$0.813325\pi$
$12$	0	0
$13$	0.707786	0.196305	0.0981523	−	0.995171i	$-0.468707\pi$
$13$	0.707786	0.196305	0.0981523	+	0.995171i	$0.468707\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−1.04402	−0.253212	−0.126606	−	0.991953i	$-0.540408\pi$
$17$	−1.04402	−0.253212	−0.126606	+	0.991953i	$0.540408\pi$
$18$	0	0
$19$	0.542112	0.124369	0.0621846	−	0.998065i	$-0.480193\pi$
$19$	0.542112	0.124369	0.0621846	+	0.998065i	$0.480193\pi$
$20$	0	0
$21$	−3.63834	−0.793951
$22$	0	0
$23$	5.02008	1.04676	0.523380	−	0.852100i	$-0.324671\pi$
$23$	5.02008	1.04676	0.523380	+	0.852100i	$0.324671\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	9.47555	1.75957	0.879783	−	0.475376i	$-0.157688\pi$
$29$	9.47555	1.75957	0.879783	+	0.475376i	$0.157688\pi$
$30$	0	0
$31$	−6.10764	−1.09696	−0.548482	−	0.836162i	$-0.684794\pi$
$31$	−6.10764	−1.09696	−0.548482	+	0.836162i	$0.684794\pi$
$32$	0	0
$33$	−5.52487	−0.961757
$34$	0	0
$35$	3.63834	0.614992
$36$	0	0
$37$	−1.83721	−0.302036	−0.151018	−	0.988531i	$-0.548255\pi$
$37$	−1.83721	−0.302036	−0.151018	+	0.988531i	$0.548255\pi$
$38$	0	0
$39$	0.707786	0.113336
$40$	0	0
$41$	−6.32174	−0.987290	−0.493645	−	0.869664i	$-0.664336\pi$
$41$	−6.32174	−0.987290	−0.493645	+	0.869664i	$0.664336\pi$
$42$	0	0
$43$	6.20172	0.945753	0.472876	−	0.881129i	$-0.343216\pi$
$43$	6.20172	0.945753	0.472876	+	0.881129i	$0.343216\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−8.96782	−1.30809	−0.654045	−	0.756455i	$-0.726930\pi$
$47$	−8.96782	−1.30809	−0.654045	+	0.756455i	$0.726930\pi$
$48$	0	0
$49$	6.23751	0.891074
$50$	0	0
$51$	−1.04402	−0.146192
$52$	0	0
$53$	6.97698	0.958362	0.479181	−	0.877716i	$-0.340934\pi$
$53$	6.97698	0.958362	0.479181	+	0.877716i	$0.340934\pi$
$54$	0	0
$55$	5.52487	0.744974
$56$	0	0
$57$	0.542112	0.0718046
$58$	0	0
$59$	3.28250	0.427346	0.213673	−	0.976905i	$-0.431457\pi$
$59$	3.28250	0.427346	0.213673	+	0.976905i	$0.431457\pi$
$60$	0	0
$61$	2.03261	0.260248	0.130124	−	0.991498i	$-0.458462\pi$
$61$	2.03261	0.260248	0.130124	+	0.991498i	$0.458462\pi$
$62$	0	0
$63$	−3.63834	−0.458388
$64$	0	0
$65$	−0.707786	−0.0877901
$66$	0	0
$67$	1.00000	0.122169
$67$	1.00000	0.122169
$68$	0	0
$69$	5.02008	0.604347
$70$	0	0
$71$	5.85059	0.694338	0.347169	−	0.937803i	$-0.387143\pi$
$71$	5.85059	0.694338	0.347169	+	0.937803i	$0.387143\pi$
$72$	0	0
$73$	6.69103	0.783125	0.391563	−	0.920151i	$-0.371935\pi$
$73$	6.69103	0.783125	0.391563	+	0.920151i	$0.371935\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	20.1014	2.29076
$78$	0	0
$79$	−7.17626	−0.807392	−0.403696	−	0.914893i	$-0.632275\pi$
$79$	−7.17626	−0.807392	−0.403696	+	0.914893i	$0.632275\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	5.88653	0.646131	0.323066	−	0.946377i	$-0.395287\pi$
$83$	5.88653	0.646131	0.323066	+	0.946377i	$0.395287\pi$
$84$	0	0
$85$	1.04402	0.113240
$86$	0	0
$87$	9.47555	1.01589
$88$	0	0
$89$	16.9208	1.79360	0.896801	−	0.442435i	$-0.145885\pi$
$89$	16.9208	1.79360	0.896801	+	0.442435i	$0.145885\pi$
$90$	0	0
$91$	−2.57517	−0.269951
$92$	0	0
$93$	−6.10764	−0.633333
$94$	0	0
$95$	−0.542112	−0.0556196
$96$	0	0
$97$	2.80020	0.284317	0.142159	−	0.989844i	$-0.454596\pi$
$97$	2.80020	0.284317	0.142159	+	0.989844i	$0.454596\pi$
$98$	0	0
$99$	−5.52487	−0.555271
$100$	0	0
$101$	12.3698	1.23084	0.615422	−	0.788198i	$-0.288986\pi$
$101$	12.3698	1.23084	0.615422	+	0.788198i	$0.288986\pi$
$102$	0	0
$103$	7.62700	0.751510	0.375755	−	0.926719i	$-0.377383\pi$
$103$	7.62700	0.751510	0.375755	+	0.926719i	$0.377383\pi$
$104$	0	0
$105$	3.63834	0.355066
$106$	0	0
$107$	−0.569935	−0.0550977	−0.0275488	−	0.999620i	$-0.508770\pi$
$107$	−0.569935	−0.0550977	−0.0275488	+	0.999620i	$0.508770\pi$
$108$	0	0
$109$	8.24834	0.790047	0.395024	−	0.918671i	$-0.370736\pi$
$109$	8.24834	0.790047	0.395024	+	0.918671i	$0.370736\pi$
$110$	0	0
$111$	−1.83721	−0.174380
$112$	0	0
$113$	−9.49182	−0.892915	−0.446458	−	0.894805i	$-0.647315\pi$
$113$	−9.49182	−0.892915	−0.446458	+	0.894805i	$0.647315\pi$
$114$	0	0
$115$	−5.02008	−0.468125
$116$	0	0
$117$	0.707786	0.0654349
$118$	0	0
$119$	3.79850	0.348208
$120$	0	0
$121$	19.5242	1.77493
$122$	0	0
$123$	−6.32174	−0.570012
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−0.303628	−0.0269426	−0.0134713	−	0.999909i	$-0.504288\pi$
$127$	−0.303628	−0.0269426	−0.0134713	+	0.999909i	$0.504288\pi$
$128$	0	0
$129$	6.20172	0.546031
$130$	0	0
$131$	18.3532	1.60353	0.801765	−	0.597640i	$-0.203895\pi$
$131$	18.3532	1.60353	0.801765	+	0.597640i	$0.203895\pi$
$132$	0	0
$133$	−1.97239	−0.171028
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	4.51450	0.385700	0.192850	−	0.981228i	$-0.438227\pi$
$137$	4.51450	0.385700	0.192850	+	0.981228i	$0.438227\pi$
$138$	0	0
$139$	−14.3948	−1.22095	−0.610474	−	0.792036i	$-0.709021\pi$
$139$	−14.3948	−1.22095	−0.610474	+	0.792036i	$0.709021\pi$
$140$	0	0
$141$	−8.96782	−0.755227
$142$	0	0
$143$	−3.91043	−0.327007
$144$	0	0
$145$	−9.47555	−0.786902
$146$	0	0
$147$	6.23751	0.514462
$148$	0	0
$149$	13.4883	1.10501	0.552504	−	0.833510i	$-0.313672\pi$
$149$	13.4883	1.10501	0.552504	+	0.833510i	$0.313672\pi$
$150$	0	0
$151$	−13.0330	−1.06061	−0.530304	−	0.847808i	$-0.677922\pi$
$151$	−13.0330	−1.06061	−0.530304	+	0.847808i	$0.677922\pi$
$152$	0	0
$153$	−1.04402	−0.0844039
$154$	0	0
$155$	6.10764	0.490578
$156$	0	0
$157$	7.46584	0.595839	0.297920	−	0.954591i	$-0.403707\pi$
$157$	7.46584	0.595839	0.297920	+	0.954591i	$0.403707\pi$
$158$	0	0
$159$	6.97698	0.553310
$160$	0	0
$161$	−18.2648	−1.43946
$162$	0	0
$163$	2.22232	0.174065	0.0870327	−	0.996205i	$-0.472262\pi$
$163$	2.22232	0.174065	0.0870327	+	0.996205i	$0.472262\pi$
$164$	0	0
$165$	5.52487	0.430111
$166$	0	0
$167$	17.0794	1.32165	0.660823	−	0.750541i	$-0.270207\pi$
$167$	17.0794	1.32165	0.660823	+	0.750541i	$0.270207\pi$
$168$	0	0
$169$	−12.4990	−0.961465
$170$	0	0
$171$	0.542112	0.0414564
$172$	0	0
$173$	3.12010	0.237217	0.118608	−	0.992941i	$-0.462157\pi$
$173$	3.12010	0.237217	0.118608	+	0.992941i	$0.462157\pi$
$174$	0	0
$175$	−3.63834	−0.275033
$176$	0	0
$177$	3.28250	0.246728
$178$	0	0
$179$	7.47067	0.558384	0.279192	−	0.960235i	$-0.409933\pi$
$179$	7.47067	0.558384	0.279192	+	0.960235i	$0.409933\pi$
$180$	0	0
$181$	−1.40485	−0.104422	−0.0522108	−	0.998636i	$-0.516627\pi$
$181$	−1.40485	−0.104422	−0.0522108	+	0.998636i	$0.516627\pi$
$182$	0	0
$183$	2.03261	0.150255
$184$	0	0
$185$	1.83721	0.135074
$186$	0	0
$187$	5.76807	0.421803
$188$	0	0
$189$	−3.63834	−0.264650
$190$	0	0
$191$	−4.71200	−0.340948	−0.170474	−	0.985362i	$-0.554530\pi$
$191$	−4.71200	−0.340948	−0.170474	+	0.985362i	$0.554530\pi$
$192$	0	0
$193$	8.97982	0.646382	0.323191	−	0.946334i	$-0.395244\pi$
$193$	8.97982	0.646382	0.323191	+	0.946334i	$0.395244\pi$
$194$	0	0
$195$	−0.707786	−0.0506856
$196$	0	0
$197$	−20.5496	−1.46410	−0.732048	−	0.681253i	$-0.761435\pi$
$197$	−20.5496	−1.46410	−0.732048	+	0.681253i	$0.761435\pi$
$198$	0	0
$199$	11.5962	0.822035	0.411018	−	0.911627i	$-0.365173\pi$
$199$	11.5962	0.822035	0.411018	+	0.911627i	$0.365173\pi$
$200$	0	0
$201$	1.00000	0.0705346
$202$	0	0
$203$	−34.4753	−2.41969
$204$	0	0
$205$	6.32174	0.441529
$206$	0	0
$207$	5.02008	0.348920
$208$	0	0
$209$	−2.99510	−0.207176
$210$	0	0
$211$	9.41415	0.648097	0.324048	−	0.946041i	$-0.394956\pi$
$211$	9.41415	0.648097	0.324048	+	0.946041i	$0.394956\pi$
$212$	0	0
$213$	5.85059	0.400876
$214$	0	0
$215$	−6.20172	−0.422953
$216$	0	0
$217$	22.2217	1.50851
$218$	0	0
$219$	6.69103	0.452138
$220$	0	0
$221$	−0.738942	−0.0497066
$222$	0	0
$223$	−13.1857	−0.882980	−0.441490	−	0.897266i	$-0.645550\pi$
$223$	−13.1857	−0.882980	−0.441490	+	0.897266i	$0.645550\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−13.0097	−0.863486	−0.431743	−	0.901997i	$-0.642101\pi$
$227$	−13.0097	−0.863486	−0.431743	+	0.901997i	$0.642101\pi$
$228$	0	0
$229$	19.4004	1.28201	0.641007	−	0.767535i	$-0.278517\pi$
$229$	19.4004	1.28201	0.641007	+	0.767535i	$0.278517\pi$
$230$	0	0
$231$	20.1014	1.32257
$232$	0	0
$233$	−3.99955	−0.262019	−0.131010	−	0.991381i	$-0.541822\pi$
$233$	−3.99955	−0.262019	−0.131010	+	0.991381i	$0.541822\pi$
$234$	0	0
$235$	8.96782	0.584996
$236$	0	0
$237$	−7.17626	−0.466148
$238$	0	0
$239$	10.9867	0.710670	0.355335	−	0.934739i	$-0.384367\pi$
$239$	10.9867	0.710670	0.355335	+	0.934739i	$0.384367\pi$
$240$	0	0
$241$	22.8869	1.47427	0.737137	−	0.675743i	$-0.236177\pi$
$241$	22.8869	1.47427	0.737137	+	0.675743i	$0.236177\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−6.23751	−0.398500
$246$	0	0
$247$	0.383700	0.0244142
$248$	0	0
$249$	5.88653	0.373044
$250$	0	0
$251$	−8.84674	−0.558401	−0.279201	−	0.960233i	$-0.590069\pi$
$251$	−8.84674	−0.558401	−0.279201	+	0.960233i	$0.590069\pi$
$252$	0	0
$253$	−27.7353	−1.74370
$254$	0	0
$255$	1.04402	0.0653790
$256$	0	0
$257$	−12.0126	−0.749326	−0.374663	−	0.927161i	$-0.622242\pi$
$257$	−12.0126	−0.749326	−0.374663	+	0.927161i	$0.622242\pi$
$258$	0	0
$259$	6.68440	0.415348
$260$	0	0
$261$	9.47555	0.586522
$262$	0	0
$263$	5.70871	0.352014	0.176007	−	0.984389i	$-0.443682\pi$
$263$	5.70871	0.352014	0.176007	+	0.984389i	$0.443682\pi$
$264$	0	0
$265$	−6.97698	−0.428592
$266$	0	0
$267$	16.9208	1.03554
$268$	0	0
$269$	19.8427	1.20983	0.604915	−	0.796290i	$-0.293207\pi$
$269$	19.8427	1.20983	0.604915	+	0.796290i	$0.293207\pi$
$270$	0	0
$271$	3.44605	0.209333	0.104666	−	0.994507i	$-0.466622\pi$
$271$	3.44605	0.209333	0.104666	+	0.994507i	$0.466622\pi$
$272$	0	0
$273$	−2.57517	−0.155856
$274$	0	0
$275$	−5.52487	−0.333162
$276$	0	0
$277$	−12.1064	−0.727401	−0.363700	−	0.931516i	$-0.618487\pi$
$277$	−12.1064	−0.727401	−0.363700	+	0.931516i	$0.618487\pi$
$278$	0	0
$279$	−6.10764	−0.365655
$280$	0	0
$281$	−18.8539	−1.12473	−0.562365	−	0.826889i	$-0.690108\pi$
$281$	−18.8539	−1.12473	−0.562365	+	0.826889i	$0.690108\pi$
$282$	0	0
$283$	−21.2288	−1.26192	−0.630959	−	0.775816i	$-0.717338\pi$
$283$	−21.2288	−1.26192	−0.630959	+	0.775816i	$0.717338\pi$
$284$	0	0
$285$	−0.542112	−0.0321120
$286$	0	0
$287$	23.0006	1.35768
$288$	0	0
$289$	−15.9100	−0.935884
$290$	0	0
$291$	2.80020	0.164151
$292$	0	0
$293$	−20.7611	−1.21288	−0.606438	−	0.795130i	$-0.707402\pi$
$293$	−20.7611	−1.21288	−0.606438	+	0.795130i	$0.707402\pi$
$294$	0	0
$295$	−3.28250	−0.191115
$296$	0	0
$297$	−5.52487	−0.320586
$298$	0	0
$299$	3.55314	0.205484
$300$	0	0
$301$	−22.5639	−1.30056
$302$	0	0
$303$	12.3698	0.710628
$304$	0	0
$305$	−2.03261	−0.116387
$306$	0	0
$307$	−11.7465	−0.670407	−0.335203	−	0.942146i	$-0.608805\pi$
$307$	−11.7465	−0.670407	−0.335203	+	0.942146i	$0.608805\pi$
$308$	0	0
$309$	7.62700	0.433885
$310$	0	0
$311$	28.6890	1.62680	0.813401	−	0.581703i	$-0.197614\pi$
$311$	28.6890	1.62680	0.813401	+	0.581703i	$0.197614\pi$
$312$	0	0
$313$	2.24581	0.126941	0.0634703	−	0.997984i	$-0.479783\pi$
$313$	2.24581	0.126941	0.0634703	+	0.997984i	$0.479783\pi$
$314$	0	0
$315$	3.63834	0.204997
$316$	0	0
$317$	−7.62618	−0.428329	−0.214165	−	0.976798i	$-0.568703\pi$
$317$	−7.62618	−0.428329	−0.214165	+	0.976798i	$0.568703\pi$
$318$	0	0
$319$	−52.3512	−2.93111
$320$	0	0
$321$	−0.569935	−0.0318106
$322$	0	0
$323$	−0.565976	−0.0314917
$324$	0	0
$325$	0.707786	0.0392609
$326$	0	0
$327$	8.24834	0.456134
$328$	0	0
$329$	32.6280	1.79884
$330$	0	0
$331$	−2.45389	−0.134878	−0.0674390	−	0.997723i	$-0.521483\pi$
$331$	−2.45389	−0.134878	−0.0674390	+	0.997723i	$0.521483\pi$
$332$	0	0
$333$	−1.83721	−0.100679
$334$	0	0
$335$	−1.00000	−0.0546358
$336$	0	0
$337$	2.38243	0.129779	0.0648897	−	0.997892i	$-0.479330\pi$
$337$	2.38243	0.129779	0.0648897	+	0.997892i	$0.479330\pi$
$338$	0	0
$339$	−9.49182	−0.515525
$340$	0	0
$341$	33.7439	1.82734
$342$	0	0
$343$	2.77418	0.149792
$344$	0	0
$345$	−5.02008	−0.270272
$346$	0	0
$347$	−21.2106	−1.13864	−0.569322	−	0.822114i	$-0.692794\pi$
$347$	−21.2106	−1.13864	−0.569322	+	0.822114i	$0.692794\pi$
$348$	0	0
$349$	−25.4762	−1.36371	−0.681856	−	0.731486i	$-0.738827\pi$
$349$	−25.4762	−1.36371	−0.681856	+	0.731486i	$0.738827\pi$
$350$	0	0
$351$	0.707786	0.0377788
$352$	0	0
$353$	8.62164	0.458884	0.229442	−	0.973322i	$-0.426310\pi$
$353$	8.62164	0.458884	0.229442	+	0.973322i	$0.426310\pi$
$354$	0	0
$355$	−5.85059	−0.310517
$356$	0	0
$357$	3.79850	0.201038
$358$	0	0
$359$	−0.612333	−0.0323177	−0.0161588	−	0.999869i	$-0.505144\pi$
$359$	−0.612333	−0.0323177	−0.0161588	+	0.999869i	$0.505144\pi$
$360$	0	0
$361$	−18.7061	−0.984532
$362$	0	0
$363$	19.5242	1.02476
$364$	0	0
$365$	−6.69103	−0.350224
$366$	0	0
$367$	20.2638	1.05776	0.528881	−	0.848696i	$-0.322612\pi$
$367$	20.2638	1.05776	0.528881	+	0.848696i	$0.322612\pi$
$368$	0	0
$369$	−6.32174	−0.329097
$370$	0	0
$371$	−25.3846	−1.31790
$372$	0	0
$373$	−13.4802	−0.697980	−0.348990	−	0.937127i	$-0.613475\pi$
$373$	−13.4802	−0.697980	−0.348990	+	0.937127i	$0.613475\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	6.70666	0.345411
$378$	0	0
$379$	−16.1407	−0.829091	−0.414546	−	0.910029i	$-0.636059\pi$
$379$	−16.1407	−0.829091	−0.414546	+	0.910029i	$0.636059\pi$
$380$	0	0
$381$	−0.303628	−0.0155553
$382$	0	0
$383$	27.9678	1.42909	0.714545	−	0.699589i	$-0.246634\pi$
$383$	27.9678	1.42909	0.714545	+	0.699589i	$0.246634\pi$
$384$	0	0
$385$	−20.1014	−1.02446
$386$	0	0
$387$	6.20172	0.315251
$388$	0	0
$389$	4.45571	0.225913	0.112957	−	0.993600i	$-0.463968\pi$
$389$	4.45571	0.225913	0.112957	+	0.993600i	$0.463968\pi$
$390$	0	0
$391$	−5.24106	−0.265052
$392$	0	0
$393$	18.3532	0.925798
$394$	0	0
$395$	7.17626	0.361077
$396$	0	0
$397$	11.7809	0.591265	0.295632	−	0.955302i	$-0.404470\pi$
$397$	11.7809	0.591265	0.295632	+	0.955302i	$0.404470\pi$
$398$	0	0
$399$	−1.97239	−0.0987430
$400$	0	0
$401$	29.2695	1.46165	0.730823	−	0.682567i	$-0.239136\pi$
$401$	29.2695	1.46165	0.730823	+	0.682567i	$0.239136\pi$
$402$	0	0
$403$	−4.32290	−0.215339
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	10.1504	0.503135
$408$	0	0
$409$	2.37570	0.117471	0.0587355	−	0.998274i	$-0.481293\pi$
$409$	2.37570	0.117471	0.0587355	+	0.998274i	$0.481293\pi$
$410$	0	0
$411$	4.51450	0.222684
$412$	0	0
$413$	−11.9429	−0.587670
$414$	0	0
$415$	−5.88653	−0.288959
$416$	0	0
$417$	−14.3948	−0.704915
$418$	0	0
$419$	−0.391817	−0.0191415	−0.00957076	−	0.999954i	$-0.503047\pi$
$419$	−0.391817	−0.0191415	−0.00957076	+	0.999954i	$0.503047\pi$
$420$	0	0
$421$	0.376692	0.0183589	0.00917943	−	0.999958i	$-0.497078\pi$
$421$	0.376692	0.0183589	0.00917943	+	0.999958i	$0.497078\pi$
$422$	0	0
$423$	−8.96782	−0.436030
$424$	0	0
$425$	−1.04402	−0.0506424
$426$	0	0
$427$	−7.39531	−0.357884
$428$	0	0
$429$	−3.91043	−0.188797
$430$	0	0
$431$	34.4639	1.66007	0.830035	−	0.557712i	$-0.188321\pi$
$431$	34.4639	1.66007	0.830035	+	0.557712i	$0.188321\pi$
$432$	0	0
$433$	0.268295	0.0128934	0.00644672	−	0.999979i	$-0.497948\pi$
$433$	0.268295	0.0128934	0.00644672	+	0.999979i	$0.497948\pi$
$434$	0	0
$435$	−9.47555	−0.454318
$436$	0	0
$437$	2.72145	0.130185
$438$	0	0
$439$	−11.1188	−0.530673	−0.265337	−	0.964156i	$-0.585483\pi$
$439$	−11.1188	−0.530673	−0.265337	+	0.964156i	$0.585483\pi$
$440$	0	0
$441$	6.23751	0.297025
$442$	0	0
$443$	−17.7189	−0.841849	−0.420925	−	0.907096i	$-0.638294\pi$
$443$	−17.7189	−0.841849	−0.420925	+	0.907096i	$0.638294\pi$
$444$	0	0
$445$	−16.9208	−0.802123
$446$	0	0
$447$	13.4883	0.637977
$448$	0	0
$449$	−2.61798	−0.123550	−0.0617750	−	0.998090i	$-0.519676\pi$
$449$	−2.61798	−0.123550	−0.0617750	+	0.998090i	$0.519676\pi$
$450$	0	0
$451$	34.9268	1.64464
$452$	0	0
$453$	−13.0330	−0.612342
$454$	0	0
$455$	2.57517	0.120726
$456$	0	0
$457$	0.949945	0.0444365	0.0222183	−	0.999753i	$-0.492927\pi$
$457$	0.949945	0.0444365	0.0222183	+	0.999753i	$0.492927\pi$
$458$	0	0
$459$	−1.04402	−0.0487306
$460$	0	0
$461$	−1.27588	−0.0594236	−0.0297118	−	0.999559i	$-0.509459\pi$
$461$	−1.27588	−0.0594236	−0.0297118	+	0.999559i	$0.509459\pi$
$462$	0	0
$463$	−28.8357	−1.34011	−0.670055	−	0.742312i	$-0.733729\pi$
$463$	−28.8357	−1.34011	−0.670055	+	0.742312i	$0.733729\pi$
$464$	0	0
$465$	6.10764	0.283235
$466$	0	0
$467$	31.2759	1.44728	0.723639	−	0.690179i	$-0.242468\pi$
$467$	31.2759	1.44728	0.723639	+	0.690179i	$0.242468\pi$
$468$	0	0
$469$	−3.63834	−0.168003
$470$	0	0
$471$	7.46584	0.344008
$472$	0	0
$473$	−34.2637	−1.57545
$474$	0	0
$475$	0.542112	0.0248738
$476$	0	0
$477$	6.97698	0.319454
$478$	0	0
$479$	24.5671	1.12250	0.561251	−	0.827646i	$-0.310320\pi$
$479$	24.5671	1.12250	0.561251	+	0.827646i	$0.310320\pi$
$480$	0	0
$481$	−1.30035	−0.0592910
$482$	0	0
$483$	−18.2648	−0.831075
$484$	0	0
$485$	−2.80020	−0.127151
$486$	0	0
$487$	−2.25934	−0.102380	−0.0511902	−	0.998689i	$-0.516301\pi$
$487$	−2.25934	−0.102380	−0.0511902	+	0.998689i	$0.516301\pi$
$488$	0	0
$489$	2.22232	0.100497
$490$	0	0
$491$	9.62489	0.434365	0.217183	−	0.976131i	$-0.430313\pi$
$491$	9.62489	0.434365	0.217183	+	0.976131i	$0.430313\pi$
$492$	0	0
$493$	−9.89266	−0.445543
$494$	0	0
$495$	5.52487	0.248325
$496$	0	0
$497$	−21.2865	−0.954828
$498$	0	0
$499$	−28.1419	−1.25980	−0.629901	−	0.776675i	$-0.716905\pi$
$499$	−28.1419	−1.25980	−0.629901	+	0.776675i	$0.716905\pi$
$500$	0	0
$501$	17.0794	0.763053
$502$	0	0
$503$	23.7662	1.05968	0.529842	−	0.848097i	$-0.322251\pi$
$503$	23.7662	1.05968	0.529842	+	0.848097i	$0.322251\pi$
$504$	0	0
$505$	−12.3698	−0.550450
$506$	0	0
$507$	−12.4990	−0.555102
$508$	0	0
$509$	36.6161	1.62298	0.811490	−	0.584367i	$-0.198657\pi$
$509$	36.6161	1.62298	0.811490	+	0.584367i	$0.198657\pi$
$510$	0	0
$511$	−24.3442	−1.07693
$512$	0	0
$513$	0.542112	0.0239349
$514$	0	0
$515$	−7.62700	−0.336086
$516$	0	0
$517$	49.5461	2.17903
$518$	0	0
$519$	3.12010	0.136957
$520$	0	0
$521$	−12.5894	−0.551550	−0.275775	−	0.961222i	$-0.588935\pi$
$521$	−12.5894	−0.551550	−0.275775	+	0.961222i	$0.588935\pi$
$522$	0	0
$523$	−20.6857	−0.904520	−0.452260	−	0.891886i	$-0.649382\pi$
$523$	−20.6857	−0.904520	−0.452260	+	0.891886i	$0.649382\pi$
$524$	0	0
$525$	−3.63834	−0.158790
$526$	0	0
$527$	6.37649	0.277764
$528$	0	0
$529$	2.20122	0.0957053
$530$	0	0
$531$	3.28250	0.142449
$532$	0	0
$533$	−4.47444	−0.193810
$534$	0	0
$535$	0.569935	0.0246404
$536$	0	0
$537$	7.47067	0.322383
$538$	0	0
$539$	−34.4615	−1.48436
$540$	0	0
$541$	0.600853	0.0258327	0.0129163	−	0.999917i	$-0.495888\pi$
$541$	0.600853	0.0258327	0.0129163	+	0.999917i	$0.495888\pi$
$542$	0	0
$543$	−1.40485	−0.0602879
$544$	0	0
$545$	−8.24834	−0.353320
$546$	0	0
$547$	9.64936	0.412577	0.206289	−	0.978491i	$-0.433861\pi$
$547$	9.64936	0.412577	0.206289	+	0.978491i	$0.433861\pi$
$548$	0	0
$549$	2.03261	0.0867495
$550$	0	0
$551$	5.13681	0.218836
$552$	0	0
$553$	26.1097	1.11030
$554$	0	0
$555$	1.83721	0.0779853
$556$	0	0
$557$	13.6057	0.576492	0.288246	−	0.957556i	$-0.406928\pi$
$557$	13.6057	0.576492	0.288246	+	0.957556i	$0.406928\pi$
$558$	0	0
$559$	4.38949	0.185656
$560$	0	0
$561$	5.76807	0.243528
$562$	0	0
$563$	−9.37202	−0.394984	−0.197492	−	0.980305i	$-0.563280\pi$
$563$	−9.37202	−0.394984	−0.197492	+	0.980305i	$0.563280\pi$
$564$	0	0
$565$	9.49182	0.399324
$566$	0	0
$567$	−3.63834	−0.152796
$568$	0	0
$569$	5.62066	0.235630	0.117815	−	0.993036i	$-0.462411\pi$
$569$	5.62066	0.235630	0.117815	+	0.993036i	$0.462411\pi$
$570$	0	0
$571$	23.5098	0.983854	0.491927	−	0.870636i	$-0.336293\pi$
$571$	23.5098	0.983854	0.491927	+	0.870636i	$0.336293\pi$
$572$	0	0
$573$	−4.71200	−0.196847
$574$	0	0
$575$	5.02008	0.209352
$576$	0	0
$577$	−29.0863	−1.21088	−0.605439	−	0.795892i	$-0.707002\pi$
$577$	−29.0863	−1.21088	−0.605439	+	0.795892i	$0.707002\pi$
$578$	0	0
$579$	8.97982	0.373189
$580$	0	0
$581$	−21.4172	−0.888536
$582$	0	0
$583$	−38.5469	−1.59645
$584$	0	0
$585$	−0.707786	−0.0292634
$586$	0	0
$587$	16.1094	0.664908	0.332454	−	0.943119i	$-0.392123\pi$
$587$	16.1094	0.664908	0.332454	+	0.943119i	$0.392123\pi$
$588$	0	0
$589$	−3.31103	−0.136429
$590$	0	0
$591$	−20.5496	−0.845297
$592$	0	0
$593$	45.4055	1.86458	0.932291	−	0.361709i	$-0.117807\pi$
$593$	45.4055	1.86458	0.932291	+	0.361709i	$0.117807\pi$
$594$	0	0
$595$	−3.79850	−0.155723
$596$	0	0
$597$	11.5962	0.474602
$598$	0	0
$599$	39.1297	1.59880	0.799399	−	0.600801i	$-0.205151\pi$
$599$	39.1297	1.59880	0.799399	+	0.600801i	$0.205151\pi$
$600$	0	0
$601$	−35.8300	−1.46154	−0.730768	−	0.682626i	$-0.760838\pi$
$601$	−35.8300	−1.46154	−0.730768	+	0.682626i	$0.760838\pi$
$602$	0	0
$603$	1.00000	0.0407231
$604$	0	0
$605$	−19.5242	−0.793773
$606$	0	0
$607$	−4.52806	−0.183788	−0.0918941	−	0.995769i	$-0.529292\pi$
$607$	−4.52806	−0.183788	−0.0918941	+	0.995769i	$0.529292\pi$
$608$	0	0
$609$	−34.4753	−1.39701
$610$	0	0
$611$	−6.34730	−0.256784
$612$	0	0
$613$	28.2147	1.13958	0.569791	−	0.821789i	$-0.307024\pi$
$613$	28.2147	1.13958	0.569791	+	0.821789i	$0.307024\pi$
$614$	0	0
$615$	6.32174	0.254917
$616$	0	0
$617$	1.17774	0.0474139	0.0237069	−	0.999719i	$-0.492453\pi$
$617$	1.17774	0.0474139	0.0237069	+	0.999719i	$0.492453\pi$
$618$	0	0
$619$	−1.39439	−0.0560453	−0.0280227	−	0.999607i	$-0.508921\pi$
$619$	−1.39439	−0.0560453	−0.0280227	+	0.999607i	$0.508921\pi$
$620$	0	0
$621$	5.02008	0.201449
$622$	0	0
$623$	−61.5636	−2.46649
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−2.99510	−0.119613
$628$	0	0
$629$	1.91808	0.0764790
$630$	0	0
$631$	6.48665	0.258230	0.129115	−	0.991630i	$-0.458786\pi$
$631$	6.48665	0.258230	0.129115	+	0.991630i	$0.458786\pi$
$632$	0	0
$633$	9.41415	0.374179
$634$	0	0
$635$	0.303628	0.0120491
$636$	0	0
$637$	4.41483	0.174922
$638$	0	0
$639$	5.85059	0.231446
$640$	0	0
$641$	−28.2310	−1.11506	−0.557529	−	0.830157i	$-0.688251\pi$
$641$	−28.2310	−1.11506	−0.557529	+	0.830157i	$0.688251\pi$
$642$	0	0
$643$	−43.0394	−1.69731	−0.848655	−	0.528947i	$-0.822587\pi$
$643$	−43.0394	−1.69731	−0.848655	+	0.528947i	$0.822587\pi$
$644$	0	0
$645$	−6.20172	−0.244192
$646$	0	0
$647$	−40.9451	−1.60972	−0.804860	−	0.593465i	$-0.797760\pi$
$647$	−40.9451	−1.60972	−0.804860	+	0.593465i	$0.797760\pi$
$648$	0	0
$649$	−18.1354	−0.711878
$650$	0	0
$651$	22.2217	0.870936
$652$	0	0
$653$	−25.2640	−0.988658	−0.494329	−	0.869275i	$-0.664586\pi$
$653$	−25.2640	−0.988658	−0.494329	+	0.869275i	$0.664586\pi$
$654$	0	0
$655$	−18.3532	−0.717120
$656$	0	0
$657$	6.69103	0.261042
$658$	0	0
$659$	−28.8495	−1.12382	−0.561909	−	0.827199i	$-0.689933\pi$
$659$	−28.8495	−1.12382	−0.561909	+	0.827199i	$0.689933\pi$
$660$	0	0
$661$	39.5072	1.53665	0.768326	−	0.640058i	$-0.221090\pi$
$661$	39.5072	1.53665	0.768326	+	0.640058i	$0.221090\pi$
$662$	0	0
$663$	−0.738942	−0.0286981
$664$	0	0
$665$	1.97239	0.0764860
$666$	0	0
$667$	47.5680	1.84184
$668$	0	0
$669$	−13.1857	−0.509789
$670$	0	0
$671$	−11.2299	−0.433525
$672$	0	0
$673$	45.0247	1.73557	0.867787	−	0.496936i	$-0.165542\pi$
$673$	45.0247	1.73557	0.867787	+	0.496936i	$0.165542\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	29.4786	1.13296	0.566478	−	0.824077i	$-0.308306\pi$
$677$	29.4786	1.13296	0.566478	+	0.824077i	$0.308306\pi$
$678$	0	0
$679$	−10.1881	−0.390983
$680$	0	0
$681$	−13.0097	−0.498534
$682$	0	0
$683$	11.3308	0.433560	0.216780	−	0.976220i	$-0.430445\pi$
$683$	11.3308	0.433560	0.216780	+	0.976220i	$0.430445\pi$
$684$	0	0
$685$	−4.51450	−0.172490
$686$	0	0
$687$	19.4004	0.740171
$688$	0	0
$689$	4.93821	0.188131
$690$	0	0
$691$	−21.4723	−0.816844	−0.408422	−	0.912793i	$-0.633921\pi$
$691$	−21.4723	−0.816844	−0.408422	+	0.912793i	$0.633921\pi$
$692$	0	0
$693$	20.1014	0.763588
$694$	0	0
$695$	14.3948	0.546024
$696$	0	0
$697$	6.60002	0.249993
$698$	0	0
$699$	−3.99955	−0.151277
$700$	0	0
$701$	49.5837	1.87275	0.936376	−	0.350998i	$-0.114158\pi$
$701$	49.5837	1.87275	0.936376	+	0.350998i	$0.114158\pi$
$702$	0	0
$703$	−0.995975	−0.0375639
$704$	0	0
$705$	8.96782	0.337748
$706$	0	0
$707$	−45.0056	−1.69261
$708$	0	0
$709$	23.2428	0.872901	0.436451	−	0.899728i	$-0.356235\pi$
$709$	23.2428	0.872901	0.436451	+	0.899728i	$0.356235\pi$
$710$	0	0
$711$	−7.17626	−0.269131
$712$	0	0
$713$	−30.6609	−1.14826
$714$	0	0
$715$	3.91043	0.146242
$716$	0	0
$717$	10.9867	0.410305
$718$	0	0
$719$	0.362860	0.0135324	0.00676620	−	0.999977i	$-0.497846\pi$
$719$	0.362860	0.0135324	0.00676620	+	0.999977i	$0.497846\pi$
$720$	0	0
$721$	−27.7496	−1.03345
$722$	0	0
$723$	22.8869	0.851173
$724$	0	0
$725$	9.47555	0.351913
$726$	0	0
$727$	−21.2462	−0.787978	−0.393989	−	0.919115i	$-0.628905\pi$
$727$	−21.2462	−0.787978	−0.393989	+	0.919115i	$0.628905\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−6.47471	−0.239476
$732$	0	0
$733$	37.6784	1.39168	0.695841	−	0.718196i	$-0.255032\pi$
$733$	37.6784	1.39168	0.695841	+	0.718196i	$0.255032\pi$
$734$	0	0
$735$	−6.23751	−0.230074
$736$	0	0
$737$	−5.52487	−0.203511
$738$	0	0
$739$	−26.9030	−0.989641	−0.494821	−	0.868995i	$-0.664766\pi$
$739$	−26.9030	−0.989641	−0.494821	+	0.868995i	$0.664766\pi$
$740$	0	0
$741$	0.383700	0.0140956
$742$	0	0
$743$	37.1991	1.36470	0.682351	−	0.731024i	$-0.260957\pi$
$743$	37.1991	1.36470	0.682351	+	0.731024i	$0.260957\pi$
$744$	0	0
$745$	−13.4883	−0.494175
$746$	0	0
$747$	5.88653	0.215377
$748$	0	0
$749$	2.07362	0.0757683
$750$	0	0
$751$	39.1211	1.42755	0.713774	−	0.700376i	$-0.246984\pi$
$751$	39.1211	1.42755	0.713774	+	0.700376i	$0.246984\pi$
$752$	0	0
$753$	−8.84674	−0.322393
$754$	0	0
$755$	13.0330	0.474318
$756$	0	0
$757$	34.2602	1.24521	0.622604	−	0.782537i	$-0.286075\pi$
$757$	34.2602	1.24521	0.622604	+	0.782537i	$0.286075\pi$
$758$	0	0
$759$	−27.7353	−1.00673
$760$	0	0
$761$	49.8076	1.80552	0.902762	−	0.430140i	$-0.141536\pi$
$761$	49.8076	1.80552	0.902762	+	0.430140i	$0.141536\pi$
$762$	0	0
$763$	−30.0102	−1.08644
$764$	0	0
$765$	1.04402	0.0377466
$766$	0	0
$767$	2.32331	0.0838899
$768$	0	0
$769$	−17.4523	−0.629347	−0.314674	−	0.949200i	$-0.601895\pi$
$769$	−17.4523	−0.629347	−0.314674	+	0.949200i	$0.601895\pi$
$770$	0	0
$771$	−12.0126	−0.432624
$772$	0	0
$773$	−14.1853	−0.510210	−0.255105	−	0.966913i	$-0.582110\pi$
$773$	−14.1853	−0.510210	−0.255105	+	0.966913i	$0.582110\pi$
$774$	0	0
$775$	−6.10764	−0.219393
$776$	0	0
$777$	6.68440	0.239802
$778$	0	0
$779$	−3.42709	−0.122788
$780$	0	0
$781$	−32.3238	−1.15664
$782$	0	0
$783$	9.47555	0.338629
$784$	0	0
$785$	−7.46584	−0.266467
$786$	0	0
$787$	−40.9772	−1.46068	−0.730340	−	0.683084i	$-0.760638\pi$
$787$	−40.9772	−1.46068	−0.730340	+	0.683084i	$0.760638\pi$
$788$	0	0
$789$	5.70871	0.203236
$790$	0	0
$791$	34.5345	1.22790
$792$	0	0
$793$	1.43865	0.0510880
$794$	0	0
$795$	−6.97698	−0.247448
$796$	0	0
$797$	24.7488	0.876649	0.438324	−	0.898817i	$-0.355572\pi$
$797$	24.7488	0.876649	0.438324	+	0.898817i	$0.355572\pi$
$798$	0	0
$799$	9.36258	0.331224
$800$	0	0
$801$	16.9208	0.597867
$802$	0	0
$803$	−36.9671	−1.30454
$804$	0	0
$805$	18.2648	0.643748
$806$	0	0
$807$	19.8427	0.698496
$808$	0	0
$809$	−13.5955	−0.477993	−0.238997	−	0.971020i	$-0.576818\pi$
$809$	−13.5955	−0.477993	−0.238997	+	0.971020i	$0.576818\pi$
$810$	0	0
$811$	6.68038	0.234580	0.117290	−	0.993098i	$-0.462579\pi$
$811$	6.68038	0.234580	0.117290	+	0.993098i	$0.462579\pi$
$812$	0	0
$813$	3.44605	0.120858
$814$	0	0
$815$	−2.22232	−0.0778444
$816$	0	0
$817$	3.36203	0.117622
$818$	0	0
$819$	−2.57517	−0.0899836
$820$	0	0
$821$	−10.3362	−0.360737	−0.180368	−	0.983599i	$-0.557729\pi$
$821$	−10.3362	−0.360737	−0.180368	+	0.983599i	$0.557729\pi$
$822$	0	0
$823$	−28.4095	−0.990292	−0.495146	−	0.868810i	$-0.664886\pi$
$823$	−28.4095	−0.990292	−0.495146	+	0.868810i	$0.664886\pi$
$824$	0	0
$825$	−5.52487	−0.192351
$826$	0	0
$827$	5.63983	0.196116	0.0980580	−	0.995181i	$-0.468737\pi$
$827$	5.63983	0.196116	0.0980580	+	0.995181i	$0.468737\pi$
$828$	0	0
$829$	43.2799	1.50317	0.751587	−	0.659634i	$-0.229289\pi$
$829$	43.2799	1.50317	0.751587	+	0.659634i	$0.229289\pi$
$830$	0	0
$831$	−12.1064	−0.419965
$832$	0	0
$833$	−6.51208	−0.225630
$834$	0	0
$835$	−17.0794	−0.591058
$836$	0	0
$837$	−6.10764	−0.211111
$838$	0	0
$839$	−25.6622	−0.885957	−0.442978	−	0.896532i	$-0.646078\pi$
$839$	−25.6622	−0.885957	−0.442978	+	0.896532i	$0.646078\pi$
$840$	0	0
$841$	60.7861	2.09607
$842$	0	0
$843$	−18.8539	−0.649363
$844$	0	0
$845$	12.4990	0.429980
$846$	0	0
$847$	−71.0358	−2.44082
$848$	0	0
$849$	−21.2288	−0.728569
$850$	0	0
$851$	−9.22295	−0.316159
$852$	0	0
$853$	−23.9798	−0.821054	−0.410527	−	0.911849i	$-0.634655\pi$
$853$	−23.9798	−0.821054	−0.410527	+	0.911849i	$0.634655\pi$
$854$	0	0
$855$	−0.542112	−0.0185399
$856$	0	0
$857$	33.3424	1.13895	0.569477	−	0.822007i	$-0.307146\pi$
$857$	33.3424	1.13895	0.569477	+	0.822007i	$0.307146\pi$
$858$	0	0
$859$	−32.4041	−1.10561	−0.552807	−	0.833309i	$-0.686443\pi$
$859$	−32.4041	−1.10561	−0.552807	+	0.833309i	$0.686443\pi$
$860$	0	0
$861$	23.0006	0.783860
$862$	0	0
$863$	25.7812	0.877601	0.438801	−	0.898584i	$-0.355403\pi$
$863$	25.7812	0.877601	0.438801	+	0.898584i	$0.355403\pi$
$864$	0	0
$865$	−3.12010	−0.106086
$866$	0	0
$867$	−15.9100	−0.540333
$868$	0	0
$869$	39.6479	1.34496
$870$	0	0
$871$	0.707786	0.0239824
$872$	0	0
$873$	2.80020	0.0947724
$874$	0	0
$875$	3.63834	0.122998
$876$	0	0
$877$	−20.2061	−0.682311	−0.341156	−	0.940007i	$-0.610818\pi$
$877$	−20.2061	−0.682311	−0.341156	+	0.940007i	$0.610818\pi$
$878$	0	0
$879$	−20.7611	−0.700255
$880$	0	0
$881$	23.8803	0.804549	0.402274	−	0.915519i	$-0.368220\pi$
$881$	23.8803	0.804549	0.402274	+	0.915519i	$0.368220\pi$
$882$	0	0
$883$	−5.80589	−0.195384	−0.0976919	−	0.995217i	$-0.531146\pi$
$883$	−5.80589	−0.195384	−0.0976919	+	0.995217i	$0.531146\pi$
$884$	0	0
$885$	−3.28250	−0.110340
$886$	0	0
$887$	7.93356	0.266383	0.133192	−	0.991090i	$-0.457477\pi$
$887$	7.93356	0.266383	0.133192	+	0.991090i	$0.457477\pi$
$888$	0	0
$889$	1.10470	0.0370505
$890$	0	0
$891$	−5.52487	−0.185090
$892$	0	0
$893$	−4.86157	−0.162686
$894$	0	0
$895$	−7.47067	−0.249717
$896$	0	0
$897$	3.55314	0.118636
$898$	0	0
$899$	−57.8733	−1.93018
$900$	0	0
$901$	−7.28410	−0.242669
$902$	0	0
$903$	−22.5639	−0.750881
$904$	0	0
$905$	1.40485	0.0466988
$906$	0	0
$907$	2.35513	0.0782009	0.0391005	−	0.999235i	$-0.487551\pi$
$907$	2.35513	0.0782009	0.0391005	+	0.999235i	$0.487551\pi$
$908$	0	0
$909$	12.3698	0.410281
$910$	0	0
$911$	−59.2051	−1.96155	−0.980776	−	0.195139i	$-0.937484\pi$
$911$	−59.2051	−1.96155	−0.980776	+	0.195139i	$0.937484\pi$
$912$	0	0
$913$	−32.5224	−1.07633
$914$	0	0
$915$	−2.03261	−0.0671959
$916$	0	0
$917$	−66.7753	−2.20511
$918$	0	0
$919$	58.0152	1.91375	0.956873	−	0.290507i	$-0.0938240\pi$
$919$	58.0152	1.91375	0.956873	+	0.290507i	$0.0938240\pi$
$920$	0	0
$921$	−11.7465	−0.387059
$922$	0	0
$923$	4.14097	0.136302
$924$	0	0
$925$	−1.83721	−0.0604071
$926$	0	0
$927$	7.62700	0.250503
$928$	0	0
$929$	−47.9002	−1.57156	−0.785778	−	0.618509i	$-0.787737\pi$
$929$	−47.9002	−1.57156	−0.785778	+	0.618509i	$0.787737\pi$
$930$	0	0
$931$	3.38143	0.110822
$932$	0	0
$933$	28.6890	0.939235
$934$	0	0
$935$	−5.76807	−0.188636
$936$	0	0
$937$	−5.86053	−0.191455	−0.0957275	−	0.995408i	$-0.530518\pi$
$937$	−5.86053	−0.191455	−0.0957275	+	0.995408i	$0.530518\pi$
$938$	0	0
$939$	2.24581	0.0732891
$940$	0	0
$941$	−48.5043	−1.58119	−0.790597	−	0.612336i	$-0.790230\pi$
$941$	−48.5043	−1.58119	−0.790597	+	0.612336i	$0.790230\pi$
$942$	0	0
$943$	−31.7357	−1.03346
$944$	0	0
$945$	3.63834	0.118355
$946$	0	0
$947$	−50.0234	−1.62554	−0.812771	−	0.582583i	$-0.802042\pi$
$947$	−50.0234	−1.62554	−0.812771	+	0.582583i	$0.802042\pi$
$948$	0	0
$949$	4.73582	0.153731
$950$	0	0
$951$	−7.62618	−0.247296
$952$	0	0
$953$	17.9603	0.581791	0.290896	−	0.956755i	$-0.406047\pi$
$953$	17.9603	0.581791	0.290896	+	0.956755i	$0.406047\pi$
$954$	0	0
$955$	4.71200	0.152477
$956$	0	0
$957$	−52.3512	−1.69227
$958$	0	0
$959$	−16.4253	−0.530400
$960$	0	0
$961$	6.30328	0.203332
$962$	0	0
$963$	−0.569935	−0.0183659
$964$	0	0
$965$	−8.97982	−0.289071
$966$	0	0
$967$	−48.1520	−1.54846	−0.774231	−	0.632903i	$-0.781863\pi$
$967$	−48.1520	−1.54846	−0.774231	+	0.632903i	$0.781863\pi$
$968$	0	0
$969$	−0.565976	−0.0181818
$970$	0	0
$971$	33.8934	1.08769	0.543846	−	0.839185i	$-0.316968\pi$
$971$	33.8934	1.08769	0.543846	+	0.839185i	$0.316968\pi$
$972$	0	0
$973$	52.3730	1.67900
$974$	0	0
$975$	0.707786	0.0226673
$976$	0	0
$977$	−17.9496	−0.574260	−0.287130	−	0.957892i	$-0.592701\pi$
$977$	−17.9496	−0.574260	−0.287130	+	0.957892i	$0.592701\pi$
$978$	0	0
$979$	−93.4853	−2.98780
$980$	0	0
$981$	8.24834	0.263349
$982$	0	0
$983$	9.48206	0.302431	0.151215	−	0.988501i	$-0.451681\pi$
$983$	9.48206	0.302431	0.151215	+	0.988501i	$0.451681\pi$
$984$	0	0
$985$	20.5496	0.654764
$986$	0	0
$987$	32.6280	1.03856
$988$	0	0
$989$	31.1331	0.989976
$990$	0	0
$991$	27.3569	0.869021	0.434511	−	0.900667i	$-0.356921\pi$
$991$	27.3569	0.869021	0.434511	+	0.900667i	$0.356921\pi$
$992$	0	0
$993$	−2.45389	−0.0778719
$994$	0	0
$995$	−11.5962	−0.367625
$996$	0	0
$997$	7.21089	0.228371	0.114186	−	0.993459i	$-0.463574\pi$
$997$	7.21089	0.228371	0.114186	+	0.993459i	$0.463574\pi$
$998$	0	0
$999$	−1.83721	−0.0581268

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4020.2.a.g.1.1	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4020.2.a.g.1.1	✓	6	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{5} -3.63834 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.00000 q^{5} -3.63834 q^{7} +1.00000 q^{9} -5.52487 q^{11} +0.707786 q^{13} -1.00000 q^{15} -1.04402 q^{17} +0.542112 q^{19} -3.63834 q^{21} +5.02008 q^{23} +1.00000 q^{25} +1.00000 q^{27} +9.47555 q^{29} -6.10764 q^{31} -5.52487 q^{33} +3.63834 q^{35} -1.83721 q^{37} +0.707786 q^{39} -6.32174 q^{41} +6.20172 q^{43} -1.00000 q^{45} -8.96782 q^{47} +6.23751 q^{49} -1.04402 q^{51} +6.97698 q^{53} +5.52487 q^{55} +0.542112 q^{57} +3.28250 q^{59} +2.03261 q^{61} -3.63834 q^{63} -0.707786 q^{65} +1.00000 q^{67} +5.02008 q^{69} +5.85059 q^{71} +6.69103 q^{73} +1.00000 q^{75} +20.1014 q^{77} -7.17626 q^{79} +1.00000 q^{81} +5.88653 q^{83} +1.04402 q^{85} +9.47555 q^{87} +16.9208 q^{89} -2.57517 q^{91} -6.10764 q^{93} -0.542112 q^{95} +2.80020 q^{97} -5.52487 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{3} - 6 q^{5} + 3 q^{7} + 6 q^{9}+O(q^{10}) \) 6 * q + 6 * q^3 - 6 * q^5 + 3 * q^7 + 6 * q^9	\( 6 q + 6 q^{3} - 6 q^{5} + 3 q^{7} + 6 q^{9} + 3 q^{11} + 3 q^{13} - 6 q^{15} + 6 q^{17} - 3 q^{19} + 3 q^{21} + 6 q^{23} + 6 q^{25} + 6 q^{27} + 21 q^{29} - 3 q^{31} + 3 q^{33} - 3 q^{35} + 3 q^{39} + 9 q^{41} - 3 q^{43} - 6 q^{45} + 21 q^{47} + 3 q^{49} + 6 q^{51} + 15 q^{53} - 3 q^{55} - 3 q^{57} + 15 q^{59} + 15 q^{61} + 3 q^{63} - 3 q^{65} + 6 q^{67} + 6 q^{69} + 18 q^{71} - 6 q^{73} + 6 q^{75} + 33 q^{77} + 9 q^{79} + 6 q^{81} + 24 q^{83} - 6 q^{85} + 21 q^{87} + 24 q^{89} - 21 q^{91} - 3 q^{93} + 3 q^{95} + 9 q^{97} + 3 q^{99}+O(q^{100}) \) 6 * q + 6 * q^3 - 6 * q^5 + 3 * q^7 + 6 * q^9 + 3 * q^11 + 3 * q^13 - 6 * q^15 + 6 * q^17 - 3 * q^19 + 3 * q^21 + 6 * q^23 + 6 * q^25 + 6 * q^27 + 21 * q^29 - 3 * q^31 + 3 * q^33 - 3 * q^35 + 3 * q^39 + 9 * q^41 - 3 * q^43 - 6 * q^45 + 21 * q^47 + 3 * q^49 + 6 * q^51 + 15 * q^53 - 3 * q^55 - 3 * q^57 + 15 * q^59 + 15 * q^61 + 3 * q^63 - 3 * q^65 + 6 * q^67 + 6 * q^69 + 18 * q^71 - 6 * q^73 + 6 * q^75 + 33 * q^77 + 9 * q^79 + 6 * q^81 + 24 * q^83 - 6 * q^85 + 21 * q^87 + 24 * q^89 - 21 * q^91 - 3 * q^93 + 3 * q^95 + 9 * q^97 + 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more