LMFDB - Embedded newform 4028.2.a.d.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4028,2,Mod(1,4028)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4028, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4028.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4028 = 2^{2} \cdot 19 \cdot 53 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4028.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:	$32.1637419342$
Analytic rank:	$1$
Dimension:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{19} - 4 x^{18} - 30 x^{17} + 132 x^{16} + 332 x^{15} - 1714 x^{14} - 1598 x^{13} + 11179 x^{12} + \cdots - 49 $ x^19 - 4x^18 - 30x^17 + 132x^16 + 332x^15 - 1714x^14 - 1598x^13 + 11179x^12 + 2544x^11 - 38897x^10 + 3416x^9 + 71354x^8 - 10941x^7 - 64854x^6 + 219x^5 + 26515x^4 + 5796x^3 - 2375x^2 - 826x - 49
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$-0.365086$ of defining polynomial
Character	$\chi$	$=$	4028.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.365086 q^{3} +1.83829 q^{5} +2.27559 q^{7} -2.86671 q^{9} +O(q^{10})$	$q+0.365086 q^{3} +1.83829 q^{5} +2.27559 q^{7} -2.86671 q^{9} +2.41308 q^{11} -4.42773 q^{13} +0.671133 q^{15} -4.27528 q^{17} -1.00000 q^{19} +0.830786 q^{21} -3.42566 q^{23} -1.62070 q^{25} -2.14185 q^{27} -5.90044 q^{29} -0.273458 q^{31} +0.880982 q^{33} +4.18319 q^{35} -5.93087 q^{37} -1.61650 q^{39} -0.691610 q^{41} +5.57072 q^{43} -5.26984 q^{45} -10.2444 q^{47} -1.82169 q^{49} -1.56084 q^{51} +1.00000 q^{53} +4.43594 q^{55} -0.365086 q^{57} -4.01636 q^{59} -10.2050 q^{61} -6.52347 q^{63} -8.13944 q^{65} +2.97188 q^{67} -1.25066 q^{69} -0.559721 q^{71} -8.72913 q^{73} -0.591694 q^{75} +5.49119 q^{77} +0.654191 q^{79} +7.81818 q^{81} +3.92577 q^{83} -7.85920 q^{85} -2.15417 q^{87} +14.2283 q^{89} -10.0757 q^{91} -0.0998356 q^{93} -1.83829 q^{95} +1.00481 q^{97} -6.91762 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q - 4 q^{3} - 2 q^{5} - 11 q^{7} + 19 q^{9}+O(q^{10}) $ 19 * q - 4 * q^3 - 2 * q^5 - 11 * q^7 + 19 * q^9	$ 19 q - 4 q^{3} - 2 q^{5} - 11 q^{7} + 19 q^{9} - 11 q^{11} + q^{13} - 20 q^{15} - q^{17} - 19 q^{19} - 10 q^{21} - 16 q^{23} + 21 q^{25} - 4 q^{27} - 9 q^{31} + 7 q^{33} - 25 q^{37} - 25 q^{39} + q^{41} - 41 q^{43} - 27 q^{45} - 29 q^{47} + 14 q^{49} - 24 q^{51} + 19 q^{53} - 28 q^{55} + 4 q^{57} - 42 q^{59} + q^{61} - 41 q^{63} + 2 q^{65} - 41 q^{67} - 25 q^{69} - 20 q^{73} + 11 q^{75} - 19 q^{77} - 38 q^{79} + 23 q^{81} - 36 q^{83} - 58 q^{85} - 30 q^{87} - 25 q^{89} - 55 q^{91} - 38 q^{93} + 2 q^{95} - 13 q^{97} + q^{99}+O(q^{100}) $ 19 * q - 4 * q^3 - 2 * q^5 - 11 * q^7 + 19 * q^9 - 11 * q^11 + q^13 - 20 * q^15 - q^17 - 19 * q^19 - 10 * q^21 - 16 * q^23 + 21 * q^25 - 4 * q^27 - 9 * q^31 + 7 * q^33 - 25 * q^37 - 25 * q^39 + q^41 - 41 * q^43 - 27 * q^45 - 29 * q^47 + 14 * q^49 - 24 * q^51 + 19 * q^53 - 28 * q^55 + 4 * q^57 - 42 * q^59 + q^61 - 41 * q^63 + 2 * q^65 - 41 * q^67 - 25 * q^69 - 20 * q^73 + 11 * q^75 - 19 * q^77 - 38 * q^79 + 23 * q^81 - 36 * q^83 - 58 * q^85 - 30 * q^87 - 25 * q^89 - 55 * q^91 - 38 * q^93 + 2 * q^95 - 13 * q^97 + 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
$3$	0.365086	0.210782	0.105391	−	0.994431i	$-0.466391\pi$
$3$	0.365086	0.210782	0.105391	+	0.994431i	$0.466391\pi$
$4$	0	0
$5$	1.83829	0.822107	0.411054	−	0.911611i	$-0.365161\pi$
$5$	1.83829	0.822107	0.411054	+	0.911611i	$0.365161\pi$
$6$	0	0
$7$	2.27559	0.860093	0.430046	−	0.902807i	$-0.358497\pi$
$7$	2.27559	0.860093	0.430046	+	0.902807i	$0.358497\pi$
$8$	0	0
$9$	−2.86671	−0.955571
$10$	0	0
$11$	2.41308	0.727572	0.363786	−	0.931483i	$-0.381484\pi$
$11$	2.41308	0.727572	0.363786	+	0.931483i	$0.381484\pi$
$12$	0	0
$13$	−4.42773	−1.22803	−0.614016	−	0.789294i	$-0.710447\pi$
$13$	−4.42773	−1.22803	−0.614016	+	0.789294i	$0.710447\pi$
$14$	0	0
$15$	0.671133	0.173286
$16$	0	0
$17$	−4.27528	−1.03691	−0.518454	−	0.855105i	$-0.673492\pi$
$17$	−4.27528	−1.03691	−0.518454	+	0.855105i	$0.673492\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0.830786	0.181292
$22$	0	0
$23$	−3.42566	−0.714299	−0.357150	−	0.934047i	$-0.616251\pi$
$23$	−3.42566	−0.714299	−0.357150	+	0.934047i	$0.616251\pi$
$24$	0	0
$25$	−1.62070	−0.324140
$26$	0	0
$27$	−2.14185	−0.412200
$28$	0	0
$29$	−5.90044	−1.09568	−0.547842	−	0.836582i	$-0.684551\pi$
$29$	−5.90044	−1.09568	−0.547842	+	0.836582i	$0.684551\pi$
$30$	0	0
$31$	−0.273458	−0.0491145	−0.0245572	−	0.999698i	$-0.507818\pi$
$31$	−0.273458	−0.0491145	−0.0245572	+	0.999698i	$0.507818\pi$
$32$	0	0
$33$	0.880982	0.153359
$34$	0	0
$35$	4.18319	0.707088
$36$	0	0
$37$	−5.93087	−0.975028	−0.487514	−	0.873115i	$-0.662096\pi$
$37$	−5.93087	−0.975028	−0.487514	+	0.873115i	$0.662096\pi$
$38$	0	0
$39$	−1.61650	−0.258847
$40$	0	0
$41$	−0.691610	−0.108011	−0.0540057	−	0.998541i	$-0.517199\pi$
$41$	−0.691610	−0.108011	−0.0540057	+	0.998541i	$0.517199\pi$
$42$	0	0
$43$	5.57072	0.849527	0.424763	−	0.905304i	$-0.360357\pi$
$43$	5.57072	0.849527	0.424763	+	0.905304i	$0.360357\pi$
$44$	0	0
$45$	−5.26984	−0.785582
$46$	0	0
$47$	−10.2444	−1.49429	−0.747147	−	0.664659i	$-0.768577\pi$
$47$	−10.2444	−1.49429	−0.747147	+	0.664659i	$0.768577\pi$
$48$	0	0
$49$	−1.82169	−0.260241
$50$	0	0
$51$	−1.56084	−0.218562
$52$	0	0
$53$	1.00000	0.137361
$53$	1.00000	0.137361
$54$	0	0
$55$	4.43594	0.598142
$56$	0	0
$57$	−0.365086	−0.0483568
$58$	0	0
$59$	−4.01636	−0.522886	−0.261443	−	0.965219i	$-0.584198\pi$
$59$	−4.01636	−0.522886	−0.261443	+	0.965219i	$0.584198\pi$
$60$	0	0
$61$	−10.2050	−1.30662	−0.653309	−	0.757091i	$-0.726620\pi$
$61$	−10.2050	−1.30662	−0.653309	+	0.757091i	$0.726620\pi$
$62$	0	0
$63$	−6.52347	−0.821879
$64$	0	0
$65$	−8.13944	−1.00957
$66$	0	0
$67$	2.97188	0.363073	0.181536	−	0.983384i	$-0.441893\pi$
$67$	2.97188	0.363073	0.181536	+	0.983384i	$0.441893\pi$
$68$	0	0
$69$	−1.25066	−0.150562
$70$	0	0
$71$	−0.559721	−0.0664266	−0.0332133	−	0.999448i	$-0.510574\pi$
$71$	−0.559721	−0.0664266	−0.0332133	+	0.999448i	$0.510574\pi$
$72$	0	0
$73$	−8.72913	−1.02167	−0.510834	−	0.859679i	$-0.670663\pi$
$73$	−8.72913	−1.02167	−0.510834	+	0.859679i	$0.670663\pi$
$74$	0	0
$75$	−0.591694	−0.0683230
$76$	0	0
$77$	5.49119	0.625779
$78$	0	0
$79$	0.654191	0.0736022	0.0368011	−	0.999323i	$-0.488283\pi$
$79$	0.654191	0.0736022	0.0368011	+	0.999323i	$0.488283\pi$
$80$	0	0
$81$	7.81818	0.868686
$82$	0	0
$83$	3.92577	0.430909	0.215455	−	0.976514i	$-0.430877\pi$
$83$	3.92577	0.430909	0.215455	+	0.976514i	$0.430877\pi$
$84$	0	0
$85$	−7.85920	−0.852450
$86$	0	0
$87$	−2.15417	−0.230951
$88$	0	0
$89$	14.2283	1.50819	0.754097	−	0.656763i	$-0.228075\pi$
$89$	14.2283	1.50819	0.754097	+	0.656763i	$0.228075\pi$
$90$	0	0
$91$	−10.0757	−1.05622
$92$	0	0
$93$	−0.0998356	−0.0103525
$94$	0	0
$95$	−1.83829	−0.188604
$96$	0	0
$97$	1.00481	0.102023	0.0510117	−	0.998698i	$-0.483755\pi$
$97$	1.00481	0.102023	0.0510117	+	0.998698i	$0.483755\pi$
$98$	0	0
$99$	−6.91762	−0.695247
$100$	0	0
$101$	12.3438	1.22826	0.614129	−	0.789205i	$-0.289507\pi$
$101$	12.3438	1.22826	0.614129	+	0.789205i	$0.289507\pi$
$102$	0	0
$103$	11.7699	1.15972	0.579859	−	0.814717i	$-0.303108\pi$
$103$	11.7699	1.15972	0.579859	+	0.814717i	$0.303108\pi$
$104$	0	0
$105$	1.52722	0.149042
$106$	0	0
$107$	5.20495	0.503181	0.251591	−	0.967834i	$-0.419046\pi$
$107$	5.20495	0.503181	0.251591	+	0.967834i	$0.419046\pi$
$108$	0	0
$109$	−7.37331	−0.706235	−0.353117	−	0.935579i	$-0.614878\pi$
$109$	−7.37331	−0.706235	−0.353117	+	0.935579i	$0.614878\pi$
$110$	0	0
$111$	−2.16527	−0.205519
$112$	0	0
$113$	−0.159471	−0.0150018	−0.00750088	−	0.999972i	$-0.502388\pi$
$113$	−0.159471	−0.0150018	−0.00750088	+	0.999972i	$0.502388\pi$
$114$	0	0
$115$	−6.29735	−0.587230
$116$	0	0
$117$	12.6930	1.17347
$118$	0	0
$119$	−9.72880	−0.891837
$120$	0	0
$121$	−5.17703	−0.470639
$122$	0	0
$123$	−0.252497	−0.0227669
$124$	0	0
$125$	−12.1707	−1.08858
$126$	0	0
$127$	3.00101	0.266297	0.133148	−	0.991096i	$-0.457491\pi$
$127$	3.00101	0.266297	0.133148	+	0.991096i	$0.457491\pi$
$128$	0	0
$129$	2.03379	0.179065
$130$	0	0
$131$	17.8334	1.55811	0.779055	−	0.626955i	$-0.215699\pi$
$131$	17.8334	1.55811	0.779055	+	0.626955i	$0.215699\pi$
$132$	0	0
$133$	−2.27559	−0.197319
$134$	0	0
$135$	−3.93734	−0.338872
$136$	0	0
$137$	−14.4180	−1.23181	−0.615906	−	0.787819i	$-0.711210\pi$
$137$	−14.4180	−1.23181	−0.615906	+	0.787819i	$0.711210\pi$
$138$	0	0
$139$	−4.92369	−0.417622	−0.208811	−	0.977956i	$-0.566959\pi$
$139$	−4.92369	−0.417622	−0.208811	+	0.977956i	$0.566959\pi$
$140$	0	0
$141$	−3.74007	−0.314971
$142$	0	0
$143$	−10.6845	−0.893481
$144$	0	0
$145$	−10.8467	−0.900770
$146$	0	0
$147$	−0.665071	−0.0548542
$148$	0	0
$149$	0.916035	0.0750445	0.0375222	−	0.999296i	$-0.488053\pi$
$149$	0.916035	0.0750445	0.0375222	+	0.999296i	$0.488053\pi$
$150$	0	0
$151$	22.1694	1.80412	0.902062	−	0.431607i	$-0.142053\pi$
$151$	22.1694	1.80412	0.902062	+	0.431607i	$0.142053\pi$
$152$	0	0
$153$	12.2560	0.990839
$154$	0	0
$155$	−0.502694	−0.0403774
$156$	0	0
$157$	18.2162	1.45381	0.726904	−	0.686739i	$-0.240958\pi$
$157$	18.2162	1.45381	0.726904	+	0.686739i	$0.240958\pi$
$158$	0	0
$159$	0.365086	0.0289532
$160$	0	0
$161$	−7.79540	−0.614363
$162$	0	0
$163$	−2.36979	−0.185616	−0.0928080	−	0.995684i	$-0.529584\pi$
$163$	−2.36979	−0.185616	−0.0928080	+	0.995684i	$0.529584\pi$
$164$	0	0
$165$	1.61950	0.126078
$166$	0	0
$167$	−16.2737	−1.25930	−0.629649	−	0.776880i	$-0.716801\pi$
$167$	−16.2737	−1.25930	−0.629649	+	0.776880i	$0.716801\pi$
$168$	0	0
$169$	6.60481	0.508062
$170$	0	0
$171$	2.86671	0.219223
$172$	0	0
$173$	4.09383	0.311248	0.155624	−	0.987816i	$-0.450261\pi$
$173$	4.09383	0.311248	0.155624	+	0.987816i	$0.450261\pi$
$174$	0	0
$175$	−3.68805	−0.278790
$176$	0	0
$177$	−1.46632	−0.110215
$178$	0	0
$179$	10.0618	0.752056	0.376028	−	0.926608i	$-0.377290\pi$
$179$	10.0618	0.752056	0.376028	+	0.926608i	$0.377290\pi$
$180$	0	0
$181$	7.04187	0.523418	0.261709	−	0.965147i	$-0.415714\pi$
$181$	7.04187	0.523418	0.261709	+	0.965147i	$0.415714\pi$
$182$	0	0
$183$	−3.72571	−0.275412
$184$	0	0
$185$	−10.9026	−0.801578
$186$	0	0
$187$	−10.3166	−0.754425
$188$	0	0
$189$	−4.87398	−0.354530
$190$	0	0
$191$	−10.8187	−0.782815	−0.391408	−	0.920217i	$-0.628012\pi$
$191$	−10.8187	−0.782815	−0.391408	+	0.920217i	$0.628012\pi$
$192$	0	0
$193$	7.23220	0.520585	0.260293	−	0.965530i	$-0.416181\pi$
$193$	7.23220	0.520585	0.260293	+	0.965530i	$0.416181\pi$
$194$	0	0
$195$	−2.97160	−0.212800
$196$	0	0
$197$	−10.3616	−0.738231	−0.369116	−	0.929384i	$-0.620339\pi$
$197$	−10.3616	−0.738231	−0.369116	+	0.929384i	$0.620339\pi$
$198$	0	0
$199$	3.63058	0.257365	0.128683	−	0.991686i	$-0.458925\pi$
$199$	3.63058	0.257365	0.128683	+	0.991686i	$0.458925\pi$
$200$	0	0
$201$	1.08499	0.0765293
$202$	0	0
$203$	−13.4270	−0.942390
$204$	0	0
$205$	−1.27138	−0.0887969
$206$	0	0
$207$	9.82038	0.682563
$208$	0	0
$209$	−2.41308	−0.166916
$210$	0	0
$211$	18.3740	1.26492	0.632459	−	0.774593i	$-0.282045\pi$
$211$	18.3740	1.26492	0.632459	+	0.774593i	$0.282045\pi$
$212$	0	0
$213$	−0.204346	−0.0140016
$214$	0	0
$215$	10.2406	0.698402
$216$	0	0
$217$	−0.622278	−0.0422430
$218$	0	0
$219$	−3.18688	−0.215350
$220$	0	0
$221$	18.9298	1.27336
$222$	0	0
$223$	−4.42784	−0.296510	−0.148255	−	0.988949i	$-0.547366\pi$
$223$	−4.42784	−0.296510	−0.148255	+	0.988949i	$0.547366\pi$
$224$	0	0
$225$	4.64608	0.309739
$226$	0	0
$227$	−16.5773	−1.10027	−0.550136	−	0.835075i	$-0.685424\pi$
$227$	−16.5773	−1.10027	−0.550136	+	0.835075i	$0.685424\pi$
$228$	0	0
$229$	3.81084	0.251828	0.125914	−	0.992041i	$-0.459814\pi$
$229$	3.81084	0.251828	0.125914	+	0.992041i	$0.459814\pi$
$230$	0	0
$231$	2.00476	0.131903
$232$	0	0
$233$	16.7584	1.09788	0.548938	−	0.835863i	$-0.315032\pi$
$233$	16.7584	1.09788	0.548938	+	0.835863i	$0.315032\pi$
$234$	0	0
$235$	−18.8321	−1.22847
$236$	0	0
$237$	0.238836	0.0155141
$238$	0	0
$239$	−19.2842	−1.24739	−0.623696	−	0.781667i	$-0.714370\pi$
$239$	−19.2842	−1.24739	−0.623696	+	0.781667i	$0.714370\pi$
$240$	0	0
$241$	1.26067	0.0812068	0.0406034	−	0.999175i	$-0.487072\pi$
$241$	1.26067	0.0812068	0.0406034	+	0.999175i	$0.487072\pi$
$242$	0	0
$243$	9.27986	0.595304
$244$	0	0
$245$	−3.34878	−0.213946
$246$	0	0
$247$	4.42773	0.281730
$248$	0	0
$249$	1.43324	0.0908281
$250$	0	0
$251$	−22.5871	−1.42569	−0.712843	−	0.701324i	$-0.752593\pi$
$251$	−22.5871	−1.42569	−0.712843	+	0.701324i	$0.752593\pi$
$252$	0	0
$253$	−8.26640	−0.519704
$254$	0	0
$255$	−2.86928	−0.179681
$256$	0	0
$257$	−14.5451	−0.907300	−0.453650	−	0.891180i	$-0.649878\pi$
$257$	−14.5451	−0.907300	−0.453650	+	0.891180i	$0.649878\pi$
$258$	0	0
$259$	−13.4962	−0.838615
$260$	0	0
$261$	16.9149	1.04700
$262$	0	0
$263$	−18.9001	−1.16543	−0.582717	−	0.812675i	$-0.698010\pi$
$263$	−18.9001	−1.16543	−0.582717	+	0.812675i	$0.698010\pi$
$264$	0	0
$265$	1.83829	0.112925
$266$	0	0
$267$	5.19454	0.317901
$268$	0	0
$269$	17.3478	1.05772	0.528858	−	0.848710i	$-0.322620\pi$
$269$	17.3478	1.05772	0.528858	+	0.848710i	$0.322620\pi$
$270$	0	0
$271$	−2.20084	−0.133691	−0.0668456	−	0.997763i	$-0.521293\pi$
$271$	−2.20084	−0.133691	−0.0668456	+	0.997763i	$0.521293\pi$
$272$	0	0
$273$	−3.67850	−0.222633
$274$	0	0
$275$	−3.91088	−0.235835
$276$	0	0
$277$	−5.93069	−0.356341	−0.178170	−	0.984000i	$-0.557018\pi$
$277$	−5.93069	−0.356341	−0.178170	+	0.984000i	$0.557018\pi$
$278$	0	0
$279$	0.783925	0.0469324
$280$	0	0
$281$	13.7347	0.819346	0.409673	−	0.912232i	$-0.365643\pi$
$281$	13.7347	0.819346	0.409673	+	0.912232i	$0.365643\pi$
$282$	0	0
$283$	−17.7038	−1.05238	−0.526191	−	0.850367i	$-0.676380\pi$
$283$	−17.7038	−1.05238	−0.526191	+	0.850367i	$0.676380\pi$
$284$	0	0
$285$	−0.671133	−0.0397545
$286$	0	0
$287$	−1.57382	−0.0928998
$288$	0	0
$289$	1.27804	0.0751790
$290$	0	0
$291$	0.366843	0.0215047
$292$	0	0
$293$	7.75305	0.452938	0.226469	−	0.974018i	$-0.427282\pi$
$293$	7.75305	0.452938	0.226469	+	0.974018i	$0.427282\pi$
$294$	0	0
$295$	−7.38323	−0.429868
$296$	0	0
$297$	−5.16847	−0.299905
$298$	0	0
$299$	15.1679	0.877182
$300$	0	0
$301$	12.6767	0.730672
$302$	0	0
$303$	4.50656	0.258895
$304$	0	0
$305$	−18.7598	−1.07418
$306$	0	0
$307$	−17.1912	−0.981154	−0.490577	−	0.871398i	$-0.663214\pi$
$307$	−17.1912	−0.981154	−0.490577	+	0.871398i	$0.663214\pi$
$308$	0	0
$309$	4.29701	0.244448
$310$	0	0
$311$	3.14829	0.178523	0.0892615	−	0.996008i	$-0.471549\pi$
$311$	3.14829	0.178523	0.0892615	+	0.996008i	$0.471549\pi$
$312$	0	0
$313$	15.5784	0.880543	0.440271	−	0.897865i	$-0.354882\pi$
$313$	15.5784	0.880543	0.440271	+	0.897865i	$0.354882\pi$
$314$	0	0
$315$	−11.9920	−0.675673
$316$	0	0
$317$	10.4030	0.584292	0.292146	−	0.956374i	$-0.405631\pi$
$317$	10.4030	0.584292	0.292146	+	0.956374i	$0.405631\pi$
$318$	0	0
$319$	−14.2383	−0.797189
$320$	0	0
$321$	1.90025	0.106062
$322$	0	0
$323$	4.27528	0.237883
$324$	0	0
$325$	7.17602	0.398054
$326$	0	0
$327$	−2.69189	−0.148862
$328$	0	0
$329$	−23.3120	−1.28523
$330$	0	0
$331$	−13.9480	−0.766653	−0.383326	−	0.923613i	$-0.625222\pi$
$331$	−13.9480	−0.766653	−0.383326	+	0.923613i	$0.625222\pi$
$332$	0	0
$333$	17.0021	0.931708
$334$	0	0
$335$	5.46317	0.298485
$336$	0	0
$337$	−20.5402	−1.11890	−0.559449	−	0.828865i	$-0.688987\pi$
$337$	−20.5402	−1.11890	−0.559449	+	0.828865i	$0.688987\pi$
$338$	0	0
$339$	−0.0582205	−0.00316211
$340$	0	0
$341$	−0.659877	−0.0357343
$342$	0	0
$343$	−20.0745	−1.08392
$344$	0	0
$345$	−2.29907	−0.123778
$346$	0	0
$347$	−7.07054	−0.379567	−0.189783	−	0.981826i	$-0.560779\pi$
$347$	−7.07054	−0.379567	−0.189783	+	0.981826i	$0.560779\pi$
$348$	0	0
$349$	−12.8202	−0.686252	−0.343126	−	0.939289i	$-0.611486\pi$
$349$	−12.8202	−0.686252	−0.343126	+	0.939289i	$0.611486\pi$
$350$	0	0
$351$	9.48355	0.506195
$352$	0	0
$353$	4.87465	0.259451	0.129726	−	0.991550i	$-0.458590\pi$
$353$	4.87465	0.259451	0.129726	+	0.991550i	$0.458590\pi$
$354$	0	0
$355$	−1.02893	−0.0546098
$356$	0	0
$357$	−3.55184	−0.187984
$358$	0	0
$359$	3.96853	0.209451	0.104726	−	0.994501i	$-0.466604\pi$
$359$	3.96853	0.209451	0.104726	+	0.994501i	$0.466604\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−1.89006	−0.0992024
$364$	0	0
$365$	−16.0467	−0.839921
$366$	0	0
$367$	−0.171291	−0.00894131	−0.00447066	−	0.999990i	$-0.501423\pi$
$367$	−0.171291	−0.00894131	−0.00447066	+	0.999990i	$0.501423\pi$
$368$	0	0
$369$	1.98265	0.103212
$370$	0	0
$371$	2.27559	0.118143
$372$	0	0
$373$	−17.7392	−0.918502	−0.459251	−	0.888307i	$-0.651882\pi$
$373$	−17.7392	−0.918502	−0.459251	+	0.888307i	$0.651882\pi$
$374$	0	0
$375$	−4.44337	−0.229454
$376$	0	0
$377$	26.1256	1.34554
$378$	0	0
$379$	−28.5638	−1.46722	−0.733612	−	0.679568i	$-0.762167\pi$
$379$	−28.5638	−1.46722	−0.733612	+	0.679568i	$0.762167\pi$
$380$	0	0
$381$	1.09563	0.0561307
$382$	0	0
$383$	9.44208	0.482468	0.241234	−	0.970467i	$-0.422448\pi$
$383$	9.44208	0.482468	0.241234	+	0.970467i	$0.422448\pi$
$384$	0	0
$385$	10.0944	0.514458
$386$	0	0
$387$	−15.9697	−0.811783
$388$	0	0
$389$	6.93596	0.351667	0.175834	−	0.984420i	$-0.443738\pi$
$389$	6.93596	0.351667	0.175834	+	0.984420i	$0.443738\pi$
$390$	0	0
$391$	14.6457	0.740663
$392$	0	0
$393$	6.51072	0.328422
$394$	0	0
$395$	1.20259	0.0605089
$396$	0	0
$397$	9.30878	0.467194	0.233597	−	0.972333i	$-0.424950\pi$
$397$	9.30878	0.467194	0.233597	+	0.972333i	$0.424950\pi$
$398$	0	0
$399$	−0.830786	−0.0415913
$400$	0	0
$401$	−9.58048	−0.478427	−0.239213	−	0.970967i	$-0.576890\pi$
$401$	−9.58048	−0.478427	−0.239213	+	0.970967i	$0.576890\pi$
$402$	0	0
$403$	1.21080	0.0603141
$404$	0	0
$405$	14.3721	0.714153
$406$	0	0
$407$	−14.3117	−0.709403
$408$	0	0
$409$	−34.6263	−1.71216	−0.856081	−	0.516841i	$-0.827108\pi$
$409$	−34.6263	−1.71216	−0.856081	+	0.516841i	$0.827108\pi$
$410$	0	0
$411$	−5.26381	−0.259644
$412$	0	0
$413$	−9.13959	−0.449730
$414$	0	0
$415$	7.21670	0.354254
$416$	0	0
$417$	−1.79757	−0.0880273
$418$	0	0
$419$	8.12708	0.397034	0.198517	−	0.980097i	$-0.436388\pi$
$419$	8.12708	0.397034	0.198517	+	0.980097i	$0.436388\pi$
$420$	0	0
$421$	−18.8936	−0.920819	−0.460410	−	0.887707i	$-0.652297\pi$
$421$	−18.8936	−0.920819	−0.460410	+	0.887707i	$0.652297\pi$
$422$	0	0
$423$	29.3676	1.42790
$424$	0	0
$425$	6.92895	0.336103
$426$	0	0
$427$	−23.2224	−1.12381
$428$	0	0
$429$	−3.90075	−0.188330
$430$	0	0
$431$	−2.09780	−0.101048	−0.0505238	−	0.998723i	$-0.516089\pi$
$431$	−2.09780	−0.101048	−0.0505238	+	0.998723i	$0.516089\pi$
$432$	0	0
$433$	36.9969	1.77796	0.888980	−	0.457947i	$-0.151415\pi$
$433$	36.9969	1.77796	0.888980	+	0.457947i	$0.151415\pi$
$434$	0	0
$435$	−3.95998	−0.189866
$436$	0	0
$437$	3.42566	0.163871
$438$	0	0
$439$	−4.05435	−0.193504	−0.0967519	−	0.995309i	$-0.530845\pi$
$439$	−4.05435	−0.193504	−0.0967519	+	0.995309i	$0.530845\pi$
$440$	0	0
$441$	5.22225	0.248678
$442$	0	0
$443$	27.0595	1.28563	0.642817	−	0.766020i	$-0.277765\pi$
$443$	27.0595	1.28563	0.642817	+	0.766020i	$0.277765\pi$
$444$	0	0
$445$	26.1557	1.23990
$446$	0	0
$447$	0.334431	0.0158181
$448$	0	0
$449$	28.5046	1.34521	0.672607	−	0.740000i	$-0.265175\pi$
$449$	28.5046	1.34521	0.672607	+	0.740000i	$0.265175\pi$
$450$	0	0
$451$	−1.66891	−0.0785860
$452$	0	0
$453$	8.09375	0.380277
$454$	0	0
$455$	−18.5220	−0.868327
$456$	0	0
$457$	−12.9074	−0.603782	−0.301891	−	0.953342i	$-0.597618\pi$
$457$	−12.9074	−0.603782	−0.301891	+	0.953342i	$0.597618\pi$
$458$	0	0
$459$	9.15703	0.427413
$460$	0	0
$461$	14.8318	0.690787	0.345393	−	0.938458i	$-0.387746\pi$
$461$	14.8318	0.690787	0.345393	+	0.938458i	$0.387746\pi$
$462$	0	0
$463$	−19.7785	−0.919187	−0.459593	−	0.888129i	$-0.652005\pi$
$463$	−19.7785	−0.919187	−0.459593	+	0.888129i	$0.652005\pi$
$464$	0	0
$465$	−0.183526	−0.00851084
$466$	0	0
$467$	−32.3974	−1.49917	−0.749586	−	0.661907i	$-0.769747\pi$
$467$	−32.3974	−1.49917	−0.749586	+	0.661907i	$0.769747\pi$
$468$	0	0
$469$	6.76278	0.312276
$470$	0	0
$471$	6.65047	0.306437
$472$	0	0
$473$	13.4426	0.618092
$474$	0	0
$475$	1.62070	0.0743628
$476$	0	0
$477$	−2.86671	−0.131258
$478$	0	0
$479$	−0.910503	−0.0416019	−0.0208010	−	0.999784i	$-0.506622\pi$
$479$	−0.910503	−0.0416019	−0.0208010	+	0.999784i	$0.506622\pi$
$480$	0	0
$481$	26.2603	1.19737
$482$	0	0
$483$	−2.84599	−0.129497
$484$	0	0
$485$	1.84714	0.0838741
$486$	0	0
$487$	12.7737	0.578830	0.289415	−	0.957204i	$-0.406539\pi$
$487$	12.7737	0.578830	0.289415	+	0.957204i	$0.406539\pi$
$488$	0	0
$489$	−0.865175	−0.0391246
$490$	0	0
$491$	−14.7255	−0.664554	−0.332277	−	0.943182i	$-0.607817\pi$
$491$	−14.7255	−0.664554	−0.332277	+	0.943182i	$0.607817\pi$
$492$	0	0
$493$	25.2261	1.13612
$494$	0	0
$495$	−12.7166	−0.571567
$496$	0	0
$497$	−1.27370	−0.0571331
$498$	0	0
$499$	−18.7657	−0.840066	−0.420033	−	0.907509i	$-0.637982\pi$
$499$	−18.7657	−0.840066	−0.420033	+	0.907509i	$0.637982\pi$
$500$	0	0
$501$	−5.94130	−0.265438
$502$	0	0
$503$	−7.27932	−0.324569	−0.162284	−	0.986744i	$-0.551886\pi$
$503$	−7.27932	−0.324569	−0.162284	+	0.986744i	$0.551886\pi$
$504$	0	0
$505$	22.6915	1.00976
$506$	0	0
$507$	2.41132	0.107091
$508$	0	0
$509$	15.2015	0.673795	0.336898	−	0.941541i	$-0.390622\pi$
$509$	15.2015	0.673795	0.336898	+	0.941541i	$0.390622\pi$
$510$	0	0
$511$	−19.8639	−0.878729
$512$	0	0
$513$	2.14185	0.0945651
$514$	0	0
$515$	21.6364	0.953413
$516$	0	0
$517$	−24.7205	−1.08721
$518$	0	0
$519$	1.49460	0.0656057
$520$	0	0
$521$	−33.0490	−1.44790	−0.723952	−	0.689850i	$-0.757676\pi$
$521$	−33.0490	−1.44790	−0.723952	+	0.689850i	$0.757676\pi$
$522$	0	0
$523$	5.66824	0.247855	0.123927	−	0.992291i	$-0.460451\pi$
$523$	5.66824	0.247855	0.123927	+	0.992291i	$0.460451\pi$
$524$	0	0
$525$	−1.34645	−0.0587641
$526$	0	0
$527$	1.16911	0.0509272
$528$	0	0
$529$	−11.2649	−0.489777
$530$	0	0
$531$	11.5138	0.499654
$532$	0	0
$533$	3.06226	0.132641
$534$	0	0
$535$	9.56819	0.413669
$536$	0	0
$537$	3.67343	0.158520
$538$	0	0
$539$	−4.39588	−0.189344
$540$	0	0
$541$	−25.9925	−1.11750	−0.558752	−	0.829335i	$-0.688720\pi$
$541$	−25.9925	−1.11750	−0.558752	+	0.829335i	$0.688720\pi$
$542$	0	0
$543$	2.57089	0.110327
$544$	0	0
$545$	−13.5543	−0.580601
$546$	0	0
$547$	−9.23811	−0.394993	−0.197496	−	0.980304i	$-0.563281\pi$
$547$	−9.23811	−0.394993	−0.197496	+	0.980304i	$0.563281\pi$
$548$	0	0
$549$	29.2548	1.24857
$550$	0	0
$551$	5.90044	0.251367
$552$	0	0
$553$	1.48867	0.0633047
$554$	0	0
$555$	−3.98040	−0.168958
$556$	0	0
$557$	−15.5011	−0.656801	−0.328401	−	0.944539i	$-0.606510\pi$
$557$	−15.5011	−0.656801	−0.328401	+	0.944539i	$0.606510\pi$
$558$	0	0
$559$	−24.6657	−1.04325
$560$	0	0
$561$	−3.76645	−0.159020
$562$	0	0
$563$	−15.1780	−0.639678	−0.319839	−	0.947472i	$-0.603629\pi$
$563$	−15.1780	−0.639678	−0.319839	+	0.947472i	$0.603629\pi$
$564$	0	0
$565$	−0.293153	−0.0123331
$566$	0	0
$567$	17.7910	0.747151
$568$	0	0
$569$	−22.3694	−0.937773	−0.468887	−	0.883258i	$-0.655345\pi$
$569$	−22.3694	−0.937773	−0.468887	+	0.883258i	$0.655345\pi$
$570$	0	0
$571$	29.8261	1.24818	0.624091	−	0.781351i	$-0.285469\pi$
$571$	29.8261	1.24818	0.624091	+	0.781351i	$0.285469\pi$
$572$	0	0
$573$	−3.94976	−0.165004
$574$	0	0
$575$	5.55196	0.231533
$576$	0	0
$577$	−19.8886	−0.827974	−0.413987	−	0.910283i	$-0.635864\pi$
$577$	−19.8886	−0.827974	−0.413987	+	0.910283i	$0.635864\pi$
$578$	0	0
$579$	2.64037	0.109730
$580$	0	0
$581$	8.93345	0.370622
$582$	0	0
$583$	2.41308	0.0999397
$584$	0	0
$585$	23.3334	0.964719
$586$	0	0
$587$	−1.00191	−0.0413533	−0.0206766	−	0.999786i	$-0.506582\pi$
$587$	−1.00191	−0.0413533	−0.0206766	+	0.999786i	$0.506582\pi$
$588$	0	0
$589$	0.273458	0.0112676
$590$	0	0
$591$	−3.78286	−0.155606
$592$	0	0
$593$	−0.251773	−0.0103391	−0.00516954	−	0.999987i	$-0.501646\pi$
$593$	−0.251773	−0.0103391	−0.00516954	+	0.999987i	$0.501646\pi$
$594$	0	0
$595$	−17.8843	−0.733186
$596$	0	0
$597$	1.32547	0.0542480
$598$	0	0
$599$	18.1401	0.741186	0.370593	−	0.928795i	$-0.379154\pi$
$599$	18.1401	0.741186	0.370593	+	0.928795i	$0.379154\pi$
$600$	0	0
$601$	28.5244	1.16354	0.581768	−	0.813355i	$-0.302361\pi$
$601$	28.5244	1.16354	0.581768	+	0.813355i	$0.302361\pi$
$602$	0	0
$603$	−8.51952	−0.346942
$604$	0	0
$605$	−9.51687	−0.386916
$606$	0	0
$607$	−0.241729	−0.00981147	−0.00490574	−	0.999988i	$-0.501562\pi$
$607$	−0.241729	−0.00981147	−0.00490574	+	0.999988i	$0.501562\pi$
$608$	0	0
$609$	−4.90200	−0.198639
$610$	0	0
$611$	45.3593	1.83504
$612$	0	0
$613$	8.05012	0.325141	0.162571	−	0.986697i	$-0.448021\pi$
$613$	8.05012	0.325141	0.162571	+	0.986697i	$0.448021\pi$
$614$	0	0
$615$	−0.464162	−0.0187168
$616$	0	0
$617$	−28.9280	−1.16460	−0.582299	−	0.812975i	$-0.697847\pi$
$617$	−28.9280	−1.16460	−0.582299	+	0.812975i	$0.697847\pi$
$618$	0	0
$619$	32.7646	1.31692	0.658460	−	0.752615i	$-0.271208\pi$
$619$	32.7646	1.31692	0.658460	+	0.752615i	$0.271208\pi$
$620$	0	0
$621$	7.33726	0.294434
$622$	0	0
$623$	32.3777	1.29719
$624$	0	0
$625$	−14.2698	−0.570794
$626$	0	0
$627$	−0.880982	−0.0351830
$628$	0	0
$629$	25.3561	1.01101
$630$	0	0
$631$	−36.8516	−1.46704	−0.733519	−	0.679669i	$-0.762124\pi$
$631$	−36.8516	−1.46704	−0.733519	+	0.679669i	$0.762124\pi$
$632$	0	0
$633$	6.70809	0.266623
$634$	0	0
$635$	5.51673	0.218925
$636$	0	0
$637$	8.06593	0.319584
$638$	0	0
$639$	1.60456	0.0634754
$640$	0	0
$641$	−6.60568	−0.260909	−0.130454	−	0.991454i	$-0.541644\pi$
$641$	−6.60568	−0.260909	−0.130454	+	0.991454i	$0.541644\pi$
$642$	0	0
$643$	42.7741	1.68685	0.843424	−	0.537249i	$-0.180536\pi$
$643$	42.7741	1.68685	0.843424	+	0.537249i	$0.180536\pi$
$644$	0	0
$645$	3.73869	0.147211
$646$	0	0
$647$	8.15034	0.320423	0.160211	−	0.987083i	$-0.448782\pi$
$647$	8.15034	0.320423	0.160211	+	0.987083i	$0.448782\pi$
$648$	0	0
$649$	−9.69181	−0.380437
$650$	0	0
$651$	−0.227185	−0.00890408
$652$	0	0
$653$	−4.60838	−0.180340	−0.0901700	−	0.995926i	$-0.528741\pi$
$653$	−4.60838	−0.180340	−0.0901700	+	0.995926i	$0.528741\pi$
$654$	0	0
$655$	32.7829	1.28093
$656$	0	0
$657$	25.0239	0.976276
$658$	0	0
$659$	44.4681	1.73223	0.866116	−	0.499843i	$-0.166609\pi$
$659$	44.4681	1.73223	0.866116	+	0.499843i	$0.166609\pi$
$660$	0	0
$661$	12.0881	0.470173	0.235087	−	0.971974i	$-0.424463\pi$
$661$	12.0881	0.470173	0.235087	+	0.971974i	$0.424463\pi$
$662$	0	0
$663$	6.91100	0.268401
$664$	0	0
$665$	−4.18319	−0.162217
$666$	0	0
$667$	20.2129	0.782647
$668$	0	0
$669$	−1.61654	−0.0624991
$670$	0	0
$671$	−24.6256	−0.950659
$672$	0	0
$673$	19.5988	0.755476	0.377738	−	0.925912i	$-0.376702\pi$
$673$	19.5988	0.755476	0.377738	+	0.925912i	$0.376702\pi$
$674$	0	0
$675$	3.47130	0.133610
$676$	0	0
$677$	12.8702	0.494643	0.247322	−	0.968933i	$-0.420450\pi$
$677$	12.8702	0.494643	0.247322	+	0.968933i	$0.420450\pi$
$678$	0	0
$679$	2.28654	0.0877495
$680$	0	0
$681$	−6.05213	−0.231918
$682$	0	0
$683$	−1.10598	−0.0423193	−0.0211596	−	0.999776i	$-0.506736\pi$
$683$	−1.10598	−0.0423193	−0.0211596	+	0.999776i	$0.506736\pi$
$684$	0	0
$685$	−26.5044	−1.01268
$686$	0	0
$687$	1.39129	0.0530808
$688$	0	0
$689$	−4.42773	−0.168683
$690$	0	0
$691$	−19.1982	−0.730336	−0.365168	−	0.930942i	$-0.618988\pi$
$691$	−19.1982	−0.730336	−0.365168	+	0.930942i	$0.618988\pi$
$692$	0	0
$693$	−15.7417	−0.597976
$694$	0	0
$695$	−9.05116	−0.343330
$696$	0	0
$697$	2.95683	0.111998
$698$	0	0
$699$	6.11824	0.231413
$700$	0	0
$701$	13.4882	0.509442	0.254721	−	0.967015i	$-0.418016\pi$
$701$	13.4882	0.509442	0.254721	+	0.967015i	$0.418016\pi$
$702$	0	0
$703$	5.93087	0.223687
$704$	0	0
$705$	−6.87532	−0.258940
$706$	0	0
$707$	28.0895	1.05642
$708$	0	0
$709$	−32.4390	−1.21827	−0.609136	−	0.793066i	$-0.708484\pi$
$709$	−32.4390	−1.21827	−0.609136	+	0.793066i	$0.708484\pi$
$710$	0	0
$711$	−1.87538	−0.0703322
$712$	0	0
$713$	0.936773	0.0350824
$714$	0	0
$715$	−19.6412	−0.734538
$716$	0	0
$717$	−7.04040	−0.262928
$718$	0	0
$719$	12.3977	0.462358	0.231179	−	0.972911i	$-0.425742\pi$
$719$	12.3977	0.462358	0.231179	+	0.972911i	$0.425742\pi$
$720$	0	0
$721$	26.7834	0.997465
$722$	0	0
$723$	0.460252	0.0171170
$724$	0	0
$725$	9.56284	0.355155
$726$	0	0
$727$	−4.21713	−0.156405	−0.0782024	−	0.996938i	$-0.524918\pi$
$727$	−4.21713	−0.156405	−0.0782024	+	0.996938i	$0.524918\pi$
$728$	0	0
$729$	−20.0666	−0.743207
$730$	0	0
$731$	−23.8164	−0.880882
$732$	0	0
$733$	31.9424	1.17982	0.589910	−	0.807469i	$-0.299163\pi$
$733$	31.9424	1.17982	0.589910	+	0.807469i	$0.299163\pi$
$734$	0	0
$735$	−1.22259	−0.0450960
$736$	0	0
$737$	7.17139	0.264162
$738$	0	0
$739$	32.4801	1.19480	0.597399	−	0.801944i	$-0.296201\pi$
$739$	32.4801	1.19480	0.597399	+	0.801944i	$0.296201\pi$
$740$	0	0
$741$	1.61650	0.0593837
$742$	0	0
$743$	22.5413	0.826959	0.413479	−	0.910514i	$-0.364313\pi$
$743$	22.5413	0.826959	0.413479	+	0.910514i	$0.364313\pi$
$744$	0	0
$745$	1.68393	0.0616946
$746$	0	0
$747$	−11.2541	−0.411764
$748$	0	0
$749$	11.8443	0.432783
$750$	0	0
$751$	−39.7138	−1.44918	−0.724589	−	0.689181i	$-0.757971\pi$
$751$	−39.7138	−1.44918	−0.724589	+	0.689181i	$0.757971\pi$
$752$	0	0
$753$	−8.24623	−0.300509
$754$	0	0
$755$	40.7538	1.48318
$756$	0	0
$757$	5.77538	0.209910	0.104955	−	0.994477i	$-0.466530\pi$
$757$	5.77538	0.209910	0.104955	+	0.994477i	$0.466530\pi$
$758$	0	0
$759$	−3.01794	−0.109544
$760$	0	0
$761$	45.3770	1.64492	0.822458	−	0.568826i	$-0.192602\pi$
$761$	45.3770	1.64492	0.822458	+	0.568826i	$0.192602\pi$
$762$	0	0
$763$	−16.7786	−0.607427
$764$	0	0
$765$	22.5301	0.814576
$766$	0	0
$767$	17.7834	0.642120
$768$	0	0
$769$	−31.1945	−1.12490	−0.562450	−	0.826831i	$-0.690141\pi$
$769$	−31.1945	−1.12490	−0.562450	+	0.826831i	$0.690141\pi$
$770$	0	0
$771$	−5.31022	−0.191243
$772$	0	0
$773$	−20.9001	−0.751724	−0.375862	−	0.926676i	$-0.622653\pi$
$773$	−20.9001	−0.751724	−0.375862	+	0.926676i	$0.622653\pi$
$774$	0	0
$775$	0.443193	0.0159200
$776$	0	0
$777$	−4.92728	−0.176765
$778$	0	0
$779$	0.691610	0.0247795
$780$	0	0
$781$	−1.35065	−0.0483302
$782$	0	0
$783$	12.6379	0.451641
$784$	0	0
$785$	33.4866	1.19519
$786$	0	0
$787$	8.58949	0.306182	0.153091	−	0.988212i	$-0.451077\pi$
$787$	8.58949	0.306182	0.153091	+	0.988212i	$0.451077\pi$
$788$	0	0
$789$	−6.90018	−0.245653
$790$	0	0
$791$	−0.362890	−0.0129029
$792$	0	0
$793$	45.1851	1.60457
$794$	0	0
$795$	0.671133	0.0238026
$796$	0	0
$797$	−1.70559	−0.0604149	−0.0302075	−	0.999544i	$-0.509617\pi$
$797$	−1.70559	−0.0604149	−0.0302075	+	0.999544i	$0.509617\pi$
$798$	0	0
$799$	43.7975	1.54945
$800$	0	0
$801$	−40.7884	−1.44119
$802$	0	0
$803$	−21.0641	−0.743337
$804$	0	0
$805$	−14.3302	−0.505073
$806$	0	0
$807$	6.33345	0.222948
$808$	0	0
$809$	49.2670	1.73214	0.866068	−	0.499926i	$-0.166639\pi$
$809$	49.2670	1.73214	0.866068	+	0.499926i	$0.166639\pi$
$810$	0	0
$811$	43.8455	1.53962	0.769812	−	0.638270i	$-0.220350\pi$
$811$	43.8455	1.53962	0.769812	+	0.638270i	$0.220350\pi$
$812$	0	0
$813$	−0.803494	−0.0281798
$814$	0	0
$815$	−4.35635	−0.152596
$816$	0	0
$817$	−5.57072	−0.194895
$818$	0	0
$819$	28.8842	1.00929
$820$	0	0
$821$	−8.27610	−0.288838	−0.144419	−	0.989517i	$-0.546131\pi$
$821$	−8.27610	−0.288838	−0.144419	+	0.989517i	$0.546131\pi$
$822$	0	0
$823$	31.7299	1.10603	0.553017	−	0.833170i	$-0.313477\pi$
$823$	31.7299	1.10603	0.553017	+	0.833170i	$0.313477\pi$
$824$	0	0
$825$	−1.42781	−0.0497099
$826$	0	0
$827$	−3.99508	−0.138923	−0.0694613	−	0.997585i	$-0.522128\pi$
$827$	−3.99508	−0.138923	−0.0694613	+	0.997585i	$0.522128\pi$
$828$	0	0
$829$	−32.9878	−1.14571	−0.572857	−	0.819655i	$-0.694165\pi$
$829$	−32.9878	−1.14571	−0.572857	+	0.819655i	$0.694165\pi$
$830$	0	0
$831$	−2.16521	−0.0751103
$832$	0	0
$833$	7.78822	0.269846
$834$	0	0
$835$	−29.9158	−1.03528
$836$	0	0
$837$	0.585707	0.0202450
$838$	0	0
$839$	0.191877	0.00662434	0.00331217	−	0.999995i	$-0.498946\pi$
$839$	0.191877	0.00662434	0.00331217	+	0.999995i	$0.498946\pi$
$840$	0	0
$841$	5.81522	0.200525
$842$	0	0
$843$	5.01436	0.172704
$844$	0	0
$845$	12.1415	0.417681
$846$	0	0
$847$	−11.7808	−0.404793
$848$	0	0
$849$	−6.46340	−0.221823
$850$	0	0
$851$	20.3171	0.696462
$852$	0	0
$853$	−53.2103	−1.82189	−0.910943	−	0.412532i	$-0.864644\pi$
$853$	−53.2103	−1.82189	−0.910943	+	0.412532i	$0.864644\pi$
$854$	0	0
$855$	5.26984	0.180225
$856$	0	0
$857$	−11.5394	−0.394179	−0.197090	−	0.980385i	$-0.563149\pi$
$857$	−11.5394	−0.394179	−0.197090	+	0.980385i	$0.563149\pi$
$858$	0	0
$859$	−48.1549	−1.64303	−0.821513	−	0.570190i	$-0.806870\pi$
$859$	−48.1549	−1.64303	−0.821513	+	0.570190i	$0.806870\pi$
$860$	0	0
$861$	−0.574580	−0.0195816
$862$	0	0
$863$	−1.82690	−0.0621885	−0.0310942	−	0.999516i	$-0.509899\pi$
$863$	−1.82690	−0.0621885	−0.0310942	+	0.999516i	$0.509899\pi$
$864$	0	0
$865$	7.52564	0.255880
$866$	0	0
$867$	0.466595	0.0158464
$868$	0	0
$869$	1.57862	0.0535509
$870$	0	0
$871$	−13.1587	−0.445865
$872$	0	0
$873$	−2.88051	−0.0974905
$874$	0	0
$875$	−27.6956	−0.936284
$876$	0	0
$877$	−11.5880	−0.391300	−0.195650	−	0.980674i	$-0.562682\pi$
$877$	−11.5880	−0.391300	−0.195650	+	0.980674i	$0.562682\pi$
$878$	0	0
$879$	2.83053	0.0954714
$880$	0	0
$881$	54.2666	1.82829	0.914145	−	0.405388i	$-0.132864\pi$
$881$	54.2666	1.82829	0.914145	+	0.405388i	$0.132864\pi$
$882$	0	0
$883$	−28.4020	−0.955803	−0.477902	−	0.878413i	$-0.658602\pi$
$883$	−28.4020	−0.955803	−0.477902	+	0.878413i	$0.658602\pi$
$884$	0	0
$885$	−2.69551	−0.0906086
$886$	0	0
$887$	−47.4924	−1.59464	−0.797319	−	0.603559i	$-0.793749\pi$
$887$	−47.4924	−1.59464	−0.797319	+	0.603559i	$0.793749\pi$
$888$	0	0
$889$	6.82908	0.229040
$890$	0	0
$891$	18.8659	0.632032
$892$	0	0
$893$	10.2444	0.342814
$894$	0	0
$895$	18.4965	0.618271
$896$	0	0
$897$	5.53758	0.184895
$898$	0	0
$899$	1.61352	0.0538140
$900$	0	0
$901$	−4.27528	−0.142430
$902$	0	0
$903$	4.62808	0.154013
$904$	0	0
$905$	12.9450	0.430306
$906$	0	0
$907$	22.2406	0.738487	0.369243	−	0.929333i	$-0.379617\pi$
$907$	22.2406	0.738487	0.369243	+	0.929333i	$0.379617\pi$
$908$	0	0
$909$	−35.3863	−1.17369
$910$	0	0
$911$	40.4905	1.34151	0.670755	−	0.741679i	$-0.265970\pi$
$911$	40.4905	1.34151	0.670755	+	0.741679i	$0.265970\pi$
$912$	0	0
$913$	9.47321	0.313518
$914$	0	0
$915$	−6.84892	−0.226418
$916$	0	0
$917$	40.5815	1.34012
$918$	0	0
$919$	32.8769	1.08451	0.542254	−	0.840215i	$-0.317571\pi$
$919$	32.8769	1.08451	0.542254	+	0.840215i	$0.317571\pi$
$920$	0	0
$921$	−6.27627	−0.206810
$922$	0	0
$923$	2.47829	0.0815740
$924$	0	0
$925$	9.61215	0.316045
$926$	0	0
$927$	−33.7408	−1.10819
$928$	0	0
$929$	8.85144	0.290406	0.145203	−	0.989402i	$-0.453616\pi$
$929$	8.85144	0.290406	0.145203	+	0.989402i	$0.453616\pi$
$930$	0	0
$931$	1.82169	0.0597033
$932$	0	0
$933$	1.14940	0.0376295
$934$	0	0
$935$	−18.9649	−0.620219
$936$	0	0
$937$	43.5859	1.42389	0.711944	−	0.702236i	$-0.247815\pi$
$937$	43.5859	1.42389	0.711944	+	0.702236i	$0.247815\pi$
$938$	0	0
$939$	5.68745	0.185603
$940$	0	0
$941$	34.7750	1.13363	0.566816	−	0.823844i	$-0.308175\pi$
$941$	34.7750	1.13363	0.566816	+	0.823844i	$0.308175\pi$
$942$	0	0
$943$	2.36922	0.0771524
$944$	0	0
$945$	−8.95978	−0.291462
$946$	0	0
$947$	8.61235	0.279864	0.139932	−	0.990161i	$-0.455312\pi$
$947$	8.61235	0.279864	0.139932	+	0.990161i	$0.455312\pi$
$948$	0	0
$949$	38.6503	1.25464
$950$	0	0
$951$	3.79799	0.123158
$952$	0	0
$953$	32.9224	1.06646	0.533231	−	0.845970i	$-0.320978\pi$
$953$	32.9224	1.06646	0.533231	+	0.845970i	$0.320978\pi$
$954$	0	0
$955$	−19.8879	−0.643558
$956$	0	0
$957$	−5.19819	−0.168033
$958$	0	0
$959$	−32.8095	−1.05947
$960$	0	0
$961$	−30.9252	−0.997588
$962$	0	0
$963$	−14.9211	−0.480825
$964$	0	0
$965$	13.2949	0.427977
$966$	0	0
$967$	−29.4130	−0.945857	−0.472929	−	0.881101i	$-0.656803\pi$
$967$	−29.4130	−0.945857	−0.472929	+	0.881101i	$0.656803\pi$
$968$	0	0
$969$	1.56084	0.0501416
$970$	0	0
$971$	−55.8139	−1.79115	−0.895576	−	0.444908i	$-0.853236\pi$
$971$	−55.8139	−1.79115	−0.895576	+	0.444908i	$0.853236\pi$
$972$	0	0
$973$	−11.2043	−0.359193
$974$	0	0
$975$	2.61986	0.0839028
$976$	0	0
$977$	52.6188	1.68342	0.841712	−	0.539927i	$-0.181548\pi$
$977$	52.6188	1.68342	0.841712	+	0.539927i	$0.181548\pi$
$978$	0	0
$979$	34.3340	1.09732
$980$	0	0
$981$	21.1372	0.674857
$982$	0	0
$983$	23.0219	0.734284	0.367142	−	0.930165i	$-0.380336\pi$
$983$	23.0219	0.734284	0.367142	+	0.930165i	$0.380336\pi$
$984$	0	0
$985$	−19.0475	−0.606905
$986$	0	0
$987$	−8.51087	−0.270904
$988$	0	0
$989$	−19.0834	−0.606816
$990$	0	0
$991$	−44.8664	−1.42523	−0.712614	−	0.701557i	$-0.752489\pi$
$991$	−44.8664	−1.42523	−0.712614	+	0.701557i	$0.752489\pi$
$992$	0	0
$993$	−5.09223	−0.161597
$994$	0	0
$995$	6.67405	0.211582
$996$	0	0
$997$	−57.3091	−1.81500	−0.907498	−	0.420056i	$-0.862011\pi$
$997$	−57.3091	−1.81500	−0.907498	+	0.420056i	$0.862011\pi$
$998$	0	0
$999$	12.7030	0.401906

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4028.2.a.d.1.12	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4028.2.a.d.1.12	✓	19	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 4028 = 2^{2} \cdot 19 \cdot 53 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4028.a (trivial)

\(f(q)\)	\(=\)	\(q+0.365086 q^{3} +1.83829 q^{5} +2.27559 q^{7} -2.86671 q^{9} +O(q^{10})\)	\(q+0.365086 q^{3} +1.83829 q^{5} +2.27559 q^{7} -2.86671 q^{9} +2.41308 q^{11} -4.42773 q^{13} +0.671133 q^{15} -4.27528 q^{17} -1.00000 q^{19} +0.830786 q^{21} -3.42566 q^{23} -1.62070 q^{25} -2.14185 q^{27} -5.90044 q^{29} -0.273458 q^{31} +0.880982 q^{33} +4.18319 q^{35} -5.93087 q^{37} -1.61650 q^{39} -0.691610 q^{41} +5.57072 q^{43} -5.26984 q^{45} -10.2444 q^{47} -1.82169 q^{49} -1.56084 q^{51} +1.00000 q^{53} +4.43594 q^{55} -0.365086 q^{57} -4.01636 q^{59} -10.2050 q^{61} -6.52347 q^{63} -8.13944 q^{65} +2.97188 q^{67} -1.25066 q^{69} -0.559721 q^{71} -8.72913 q^{73} -0.591694 q^{75} +5.49119 q^{77} +0.654191 q^{79} +7.81818 q^{81} +3.92577 q^{83} -7.85920 q^{85} -2.15417 q^{87} +14.2283 q^{89} -10.0757 q^{91} -0.0998356 q^{93} -1.83829 q^{95} +1.00481 q^{97} -6.91762 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 4 q^{3} - 2 q^{5} - 11 q^{7} + 19 q^{9}+O(q^{10}) \) 19 * q - 4 * q^3 - 2 * q^5 - 11 * q^7 + 19 * q^9	\( 19 q - 4 q^{3} - 2 q^{5} - 11 q^{7} + 19 q^{9} - 11 q^{11} + q^{13} - 20 q^{15} - q^{17} - 19 q^{19} - 10 q^{21} - 16 q^{23} + 21 q^{25} - 4 q^{27} - 9 q^{31} + 7 q^{33} - 25 q^{37} - 25 q^{39} + q^{41} - 41 q^{43} - 27 q^{45} - 29 q^{47} + 14 q^{49} - 24 q^{51} + 19 q^{53} - 28 q^{55} + 4 q^{57} - 42 q^{59} + q^{61} - 41 q^{63} + 2 q^{65} - 41 q^{67} - 25 q^{69} - 20 q^{73} + 11 q^{75} - 19 q^{77} - 38 q^{79} + 23 q^{81} - 36 q^{83} - 58 q^{85} - 30 q^{87} - 25 q^{89} - 55 q^{91} - 38 q^{93} + 2 q^{95} - 13 q^{97} + q^{99}+O(q^{100}) \) 19 * q - 4 * q^3 - 2 * q^5 - 11 * q^7 + 19 * q^9 - 11 * q^11 + q^13 - 20 * q^15 - q^17 - 19 * q^19 - 10 * q^21 - 16 * q^23 + 21 * q^25 - 4 * q^27 - 9 * q^31 + 7 * q^33 - 25 * q^37 - 25 * q^39 + q^41 - 41 * q^43 - 27 * q^45 - 29 * q^47 + 14 * q^49 - 24 * q^51 + 19 * q^53 - 28 * q^55 + 4 * q^57 - 42 * q^59 + q^61 - 41 * q^63 + 2 * q^65 - 41 * q^67 - 25 * q^69 - 20 * q^73 + 11 * q^75 - 19 * q^77 - 38 * q^79 + 23 * q^81 - 36 * q^83 - 58 * q^85 - 30 * q^87 - 25 * q^89 - 55 * q^91 - 38 * q^93 + 2 * q^95 - 13 * q^97 + q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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