LMFDB - Embedded newform 4028.2.a.f.1.13

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:	$0$
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} + 124 x^{16} + 364 x^{15} - 1554 x^{14} - 2310 x^{13} + 10113 x^{12} + \cdots + 139 $ x^19 - 4x^18 - 30x^17 + 124x^16 + 364x^15 - 1554x^14 - 2310x^13 + 10113x^12 + 8368x^11 - 36567x^10 - 18074x^9 + 72868x^8 + 23819x^7 - 73816x^6 - 17877x^5 + 30973x^4 + 5074x^3 - 4299x^2 - 454x + 139
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			1.13
Root			$1.62054$ 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)

$f(q)$	$=$	$q+1.62054 q^{3} +4.01966 q^{5} +2.03129 q^{7} -0.373838 q^{9} +O(q^{10})$	$q+1.62054 q^{3} +4.01966 q^{5} +2.03129 q^{7} -0.373838 q^{9} +1.99641 q^{11} -2.41207 q^{13} +6.51404 q^{15} +7.86048 q^{17} -1.00000 q^{19} +3.29179 q^{21} -2.69673 q^{23} +11.1577 q^{25} -5.46745 q^{27} -2.77292 q^{29} +1.68916 q^{31} +3.23527 q^{33} +8.16509 q^{35} +1.67524 q^{37} -3.90886 q^{39} -5.75764 q^{41} -0.427957 q^{43} -1.50270 q^{45} +8.77616 q^{47} -2.87387 q^{49} +12.7383 q^{51} -1.00000 q^{53} +8.02490 q^{55} -1.62054 q^{57} +6.30717 q^{59} -11.0305 q^{61} -0.759373 q^{63} -9.69571 q^{65} +2.31789 q^{67} -4.37016 q^{69} +4.02928 q^{71} -1.68391 q^{73} +18.0815 q^{75} +4.05529 q^{77} +11.2928 q^{79} -7.73873 q^{81} -13.1222 q^{83} +31.5965 q^{85} -4.49364 q^{87} -2.43850 q^{89} -4.89961 q^{91} +2.73736 q^{93} -4.01966 q^{95} -9.81320 q^{97} -0.746335 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q + 4 q^{3} + 4 q^{5} + 13 q^{7} + 19 q^{9}+O(q^{10}) $ 19 * q + 4 * q^3 + 4 * q^5 + 13 * q^7 + 19 * q^9	$ 19 q + 4 q^{3} + 4 q^{5} + 13 q^{7} + 19 q^{9} + q^{11} - q^{13} + 8 q^{15} + 3 q^{17} - 19 q^{19} + 8 q^{21} + 10 q^{23} + 21 q^{25} + 28 q^{27} + 2 q^{29} + 25 q^{31} + q^{33} + 20 q^{35} + 19 q^{37} + 37 q^{39} - 9 q^{41} + 35 q^{43} + 37 q^{45} + 23 q^{47} + 30 q^{49} + 34 q^{51} - 19 q^{53} + 40 q^{55} - 4 q^{57} + 16 q^{59} + 21 q^{61} + 3 q^{63} - 10 q^{65} + 67 q^{67} + 23 q^{69} + 18 q^{71} - 20 q^{73} + 33 q^{75} + 37 q^{77} + 2 q^{79} + 23 q^{81} + 38 q^{83} + 8 q^{85} + 18 q^{87} - q^{89} - 9 q^{91} + 14 q^{93} - 4 q^{95} - 21 q^{97} + 33 q^{99}+O(q^{100}) $ 19 * q + 4 * q^3 + 4 * q^5 + 13 * q^7 + 19 * q^9 + q^11 - q^13 + 8 * q^15 + 3 * q^17 - 19 * q^19 + 8 * q^21 + 10 * q^23 + 21 * q^25 + 28 * q^27 + 2 * q^29 + 25 * q^31 + q^33 + 20 * q^35 + 19 * q^37 + 37 * q^39 - 9 * q^41 + 35 * q^43 + 37 * q^45 + 23 * q^47 + 30 * q^49 + 34 * q^51 - 19 * q^53 + 40 * q^55 - 4 * q^57 + 16 * q^59 + 21 * q^61 + 3 * q^63 - 10 * q^65 + 67 * q^67 + 23 * q^69 + 18 * q^71 - 20 * q^73 + 33 * q^75 + 37 * q^77 + 2 * q^79 + 23 * q^81 + 38 * q^83 + 8 * q^85 + 18 * q^87 - q^89 - 9 * q^91 + 14 * q^93 - 4 * q^95 - 21 * q^97 + 33 * 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
$3$	1.62054	0.935621	0.467811	−	0.883829i	$-0.345043\pi$
$3$	1.62054	0.935621	0.467811	+	0.883829i	$0.345043\pi$
$4$	0	0
$5$	4.01966	1.79765	0.898824	−	0.438310i	$-0.144423\pi$
$5$	4.01966	1.79765	0.898824	+	0.438310i	$0.144423\pi$
$6$	0	0
$7$	2.03129	0.767754	0.383877	−	0.923384i	$-0.374589\pi$
$7$	2.03129	0.767754	0.383877	+	0.923384i	$0.374589\pi$
$8$	0	0
$9$	−0.373838	−0.124613
$10$	0	0
$11$	1.99641	0.601941	0.300970	−	0.953633i	$-0.402689\pi$
$11$	1.99641	0.601941	0.300970	+	0.953633i	$0.402689\pi$
$12$	0	0
$13$	−2.41207	−0.668988	−0.334494	−	0.942398i	$-0.608565\pi$
$13$	−2.41207	−0.668988	−0.334494	+	0.942398i	$0.608565\pi$
$14$	0	0
$15$	6.51404	1.68192
$16$	0	0
$17$	7.86048	1.90645	0.953224	−	0.302266i	$-0.0977431\pi$
$17$	7.86048	1.90645	0.953224	+	0.302266i	$0.0977431\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	3.29179	0.718327
$22$	0	0
$23$	−2.69673	−0.562306	−0.281153	−	0.959663i	$-0.590717\pi$
$23$	−2.69673	−0.562306	−0.281153	+	0.959663i	$0.590717\pi$
$24$	0	0
$25$	11.1577	2.23154
$26$	0	0
$27$	−5.46745	−1.05221
$28$	0	0
$29$	−2.77292	−0.514919	−0.257459	−	0.966289i	$-0.582885\pi$
$29$	−2.77292	−0.514919	−0.257459	+	0.966289i	$0.582885\pi$
$30$	0	0
$31$	1.68916	0.303382	0.151691	−	0.988428i	$-0.451528\pi$
$31$	1.68916	0.303382	0.151691	+	0.988428i	$0.451528\pi$
$32$	0	0
$33$	3.23527	0.563189
$34$	0	0
$35$	8.16509	1.38015
$36$	0	0
$37$	1.67524	0.275407	0.137704	−	0.990473i	$-0.456028\pi$
$37$	1.67524	0.275407	0.137704	+	0.990473i	$0.456028\pi$
$38$	0	0
$39$	−3.90886	−0.625919
$40$	0	0
$41$	−5.75764	−0.899192	−0.449596	−	0.893232i	$-0.648432\pi$
$41$	−5.75764	−0.899192	−0.449596	+	0.893232i	$0.648432\pi$
$42$	0	0
$43$	−0.427957	−0.0652629	−0.0326314	−	0.999467i	$-0.510389\pi$
$43$	−0.427957	−0.0652629	−0.0326314	+	0.999467i	$0.510389\pi$
$44$	0	0
$45$	−1.50270	−0.224010
$46$	0	0
$47$	8.77616	1.28013	0.640067	−	0.768319i	$-0.278907\pi$
$47$	8.77616	1.28013	0.640067	+	0.768319i	$0.278907\pi$
$48$	0	0
$49$	−2.87387	−0.410553
$50$	0	0
$51$	12.7383	1.78371
$52$	0	0
$53$	−1.00000	−0.137361
$53$	−1.00000	−0.137361
$54$	0	0
$55$	8.02490	1.08208
$56$	0	0
$57$	−1.62054	−0.214646
$58$	0	0
$59$	6.30717	0.821123	0.410562	−	0.911833i	$-0.365333\pi$
$59$	6.30717	0.821123	0.410562	+	0.911833i	$0.365333\pi$
$60$	0	0
$61$	−11.0305	−1.41231	−0.706156	−	0.708056i	$-0.749572\pi$
$61$	−11.0305	−1.41231	−0.706156	+	0.708056i	$0.749572\pi$
$62$	0	0
$63$	−0.759373	−0.0956720
$64$	0	0
$65$	−9.69571	−1.20260
$66$	0	0
$67$	2.31789	0.283175	0.141587	−	0.989926i	$-0.454779\pi$
$67$	2.31789	0.283175	0.141587	+	0.989926i	$0.454779\pi$
$68$	0	0
$69$	−4.37016	−0.526106
$70$	0	0
$71$	4.02928	0.478187	0.239094	−	0.970997i	$-0.423150\pi$
$71$	4.02928	0.478187	0.239094	+	0.970997i	$0.423150\pi$
$72$	0	0
$73$	−1.68391	−0.197087	−0.0985433	−	0.995133i	$-0.531418\pi$
$73$	−1.68391	−0.197087	−0.0985433	+	0.995133i	$0.531418\pi$
$74$	0	0
$75$	18.0815	2.08787
$76$	0	0
$77$	4.05529	0.462143
$78$	0	0
$79$	11.2928	1.27053	0.635267	−	0.772293i	$-0.280890\pi$
$79$	11.2928	1.27053	0.635267	+	0.772293i	$0.280890\pi$
$80$	0	0
$81$	−7.73873	−0.859859
$82$	0	0
$83$	−13.1222	−1.44035	−0.720177	−	0.693791i	$-0.755939\pi$
$83$	−13.1222	−1.44035	−0.720177	+	0.693791i	$0.755939\pi$
$84$	0	0
$85$	31.5965	3.42712
$86$	0	0
$87$	−4.49364	−0.481769
$88$	0	0
$89$	−2.43850	−0.258481	−0.129240	−	0.991613i	$-0.541254\pi$
$89$	−2.43850	−0.258481	−0.129240	+	0.991613i	$0.541254\pi$
$90$	0	0
$91$	−4.89961	−0.513618
$92$	0	0
$93$	2.73736	0.283851
$94$	0	0
$95$	−4.01966	−0.412409
$96$	0	0
$97$	−9.81320	−0.996379	−0.498190	−	0.867068i	$-0.666002\pi$
$97$	−9.81320	−0.996379	−0.498190	+	0.867068i	$0.666002\pi$
$98$	0	0
$99$	−0.746335	−0.0750095
$100$	0	0
$101$	5.71752	0.568914	0.284457	−	0.958689i	$-0.408187\pi$
$101$	5.71752	0.568914	0.284457	+	0.958689i	$0.408187\pi$
$102$	0	0
$103$	−14.8981	−1.46795	−0.733977	−	0.679174i	$-0.762338\pi$
$103$	−14.8981	−1.46795	−0.733977	+	0.679174i	$0.762338\pi$
$104$	0	0
$105$	13.2319	1.29130
$106$	0	0
$107$	11.7069	1.13175	0.565873	−	0.824493i	$-0.308539\pi$
$107$	11.7069	1.13175	0.565873	+	0.824493i	$0.308539\pi$
$108$	0	0
$109$	7.50671	0.719012	0.359506	−	0.933143i	$-0.382945\pi$
$109$	7.50671	0.719012	0.359506	+	0.933143i	$0.382945\pi$
$110$	0	0
$111$	2.71479	0.257677
$112$	0	0
$113$	8.09760	0.761758	0.380879	−	0.924625i	$-0.375621\pi$
$113$	8.09760	0.761758	0.380879	+	0.924625i	$0.375621\pi$
$114$	0	0
$115$	−10.8399	−1.01083
$116$	0	0
$117$	0.901724	0.0833644
$118$	0	0
$119$	15.9669	1.46368
$120$	0	0
$121$	−7.01434	−0.637667
$122$	0	0
$123$	−9.33050	−0.841303
$124$	0	0
$125$	24.7518	2.21387
$126$	0	0
$127$	1.23444	0.109538	0.0547692	−	0.998499i	$-0.482558\pi$
$127$	1.23444	0.109538	0.0547692	+	0.998499i	$0.482558\pi$
$128$	0	0
$129$	−0.693523	−0.0610613
$130$	0	0
$131$	−0.750499	−0.0655714	−0.0327857	−	0.999462i	$-0.510438\pi$
$131$	−0.750499	−0.0655714	−0.0327857	+	0.999462i	$0.510438\pi$
$132$	0	0
$133$	−2.03129	−0.176135
$134$	0	0
$135$	−21.9773	−1.89151
$136$	0	0
$137$	11.8848	1.01539	0.507693	−	0.861538i	$-0.330498\pi$
$137$	11.8848	1.01539	0.507693	+	0.861538i	$0.330498\pi$
$138$	0	0
$139$	−7.75940	−0.658144	−0.329072	−	0.944305i	$-0.606736\pi$
$139$	−7.75940	−0.658144	−0.329072	+	0.944305i	$0.606736\pi$
$140$	0	0
$141$	14.2221	1.19772
$142$	0	0
$143$	−4.81548	−0.402691
$144$	0	0
$145$	−11.1462	−0.925642
$146$	0	0
$147$	−4.65723	−0.384122
$148$	0	0
$149$	16.4708	1.34934	0.674670	−	0.738119i	$-0.264286\pi$
$149$	16.4708	1.34934	0.674670	+	0.738119i	$0.264286\pi$
$150$	0	0
$151$	−12.1833	−0.991461	−0.495731	−	0.868476i	$-0.665100\pi$
$151$	−12.1833	−0.991461	−0.495731	+	0.868476i	$0.665100\pi$
$152$	0	0
$153$	−2.93855	−0.237568
$154$	0	0
$155$	6.78986	0.545375
$156$	0	0
$157$	−11.4401	−0.913021	−0.456511	−	0.889718i	$-0.650901\pi$
$157$	−11.4401	−0.913021	−0.456511	+	0.889718i	$0.650901\pi$
$158$	0	0
$159$	−1.62054	−0.128517
$160$	0	0
$161$	−5.47783	−0.431713
$162$	0	0
$163$	11.0853	0.868271	0.434136	−	0.900848i	$-0.357054\pi$
$163$	11.0853	0.868271	0.434136	+	0.900848i	$0.357054\pi$
$164$	0	0
$165$	13.0047	1.01241
$166$	0	0
$167$	−11.2062	−0.867165	−0.433582	−	0.901114i	$-0.642751\pi$
$167$	−11.2062	−0.867165	−0.433582	+	0.901114i	$0.642751\pi$
$168$	0	0
$169$	−7.18192	−0.552455
$170$	0	0
$171$	0.373838	0.0285881
$172$	0	0
$173$	8.35342	0.635099	0.317549	−	0.948242i	$-0.397140\pi$
$173$	8.35342	0.635099	0.317549	+	0.948242i	$0.397140\pi$
$174$	0	0
$175$	22.6645	1.71327
$176$	0	0
$177$	10.2210	0.768261
$178$	0	0
$179$	−16.2421	−1.21399	−0.606994	−	0.794706i	$-0.707625\pi$
$179$	−16.2421	−1.21399	−0.606994	+	0.794706i	$0.707625\pi$
$180$	0	0
$181$	−4.71611	−0.350546	−0.175273	−	0.984520i	$-0.556081\pi$
$181$	−4.71611	−0.350546	−0.175273	+	0.984520i	$0.556081\pi$
$182$	0	0
$183$	−17.8754	−1.32139
$184$	0	0
$185$	6.73389	0.495085
$186$	0	0
$187$	15.6928	1.14757
$188$	0	0
$189$	−11.1060	−0.807840
$190$	0	0
$191$	−11.8121	−0.854695	−0.427347	−	0.904087i	$-0.640552\pi$
$191$	−11.8121	−0.854695	−0.427347	+	0.904087i	$0.640552\pi$
$192$	0	0
$193$	−21.9459	−1.57970	−0.789851	−	0.613299i	$-0.789842\pi$
$193$	−21.9459	−1.57970	−0.789851	+	0.613299i	$0.789842\pi$
$194$	0	0
$195$	−15.7123	−1.12518
$196$	0	0
$197$	−11.9390	−0.850622	−0.425311	−	0.905047i	$-0.639835\pi$
$197$	−11.9390	−0.850622	−0.425311	+	0.905047i	$0.639835\pi$
$198$	0	0
$199$	9.35892	0.663436	0.331718	−	0.943379i	$-0.392372\pi$
$199$	9.35892	0.663436	0.331718	+	0.943379i	$0.392372\pi$
$200$	0	0
$201$	3.75623	0.264944
$202$	0	0
$203$	−5.63260	−0.395331
$204$	0	0
$205$	−23.1438	−1.61643
$206$	0	0
$207$	1.00814	0.0700706
$208$	0	0
$209$	−1.99641	−0.138095
$210$	0	0
$211$	6.73761	0.463836	0.231918	−	0.972735i	$-0.425500\pi$
$211$	6.73761	0.463836	0.231918	+	0.972735i	$0.425500\pi$
$212$	0	0
$213$	6.52962	0.447402
$214$	0	0
$215$	−1.72024	−0.117320
$216$	0	0
$217$	3.43117	0.232923
$218$	0	0
$219$	−2.72885	−0.184398
$220$	0	0
$221$	−18.9600	−1.27539
$222$	0	0
$223$	−21.1357	−1.41535	−0.707677	−	0.706536i	$-0.750257\pi$
$223$	−21.1357	−1.41535	−0.707677	+	0.706536i	$0.750257\pi$
$224$	0	0
$225$	−4.17117	−0.278078
$226$	0	0
$227$	−6.11089	−0.405594	−0.202797	−	0.979221i	$-0.565003\pi$
$227$	−6.11089	−0.405594	−0.202797	+	0.979221i	$0.565003\pi$
$228$	0	0
$229$	8.71257	0.575743	0.287871	−	0.957669i	$-0.407052\pi$
$229$	8.71257	0.575743	0.287871	+	0.957669i	$0.407052\pi$
$230$	0	0
$231$	6.57177	0.432391
$232$	0	0
$233$	6.91475	0.453000	0.226500	−	0.974011i	$-0.427272\pi$
$233$	6.91475	0.453000	0.226500	+	0.974011i	$0.427272\pi$
$234$	0	0
$235$	35.2772	2.30123
$236$	0	0
$237$	18.3004	1.18874
$238$	0	0
$239$	22.9376	1.48371	0.741855	−	0.670560i	$-0.233946\pi$
$239$	22.9376	1.48371	0.741855	+	0.670560i	$0.233946\pi$
$240$	0	0
$241$	−22.7914	−1.46812	−0.734061	−	0.679083i	$-0.762377\pi$
$241$	−22.7914	−1.46812	−0.734061	+	0.679083i	$0.762377\pi$
$242$	0	0
$243$	3.86141	0.247709
$244$	0	0
$245$	−11.5520	−0.738030
$246$	0	0
$247$	2.41207	0.153476
$248$	0	0
$249$	−21.2652	−1.34763
$250$	0	0
$251$	16.0697	1.01431	0.507156	−	0.861854i	$-0.330697\pi$
$251$	16.0697	1.01431	0.507156	+	0.861854i	$0.330697\pi$
$252$	0	0
$253$	−5.38377	−0.338475
$254$	0	0
$255$	51.2035	3.20649
$256$	0	0
$257$	−11.3284	−0.706648	−0.353324	−	0.935501i	$-0.614949\pi$
$257$	−11.3284	−0.706648	−0.353324	+	0.935501i	$0.614949\pi$
$258$	0	0
$259$	3.40289	0.211445
$260$	0	0
$261$	1.03662	0.0641655
$262$	0	0
$263$	−0.0715187	−0.00441003	−0.00220502	−	0.999998i	$-0.500702\pi$
$263$	−0.0715187	−0.00441003	−0.00220502	+	0.999998i	$0.500702\pi$
$264$	0	0
$265$	−4.01966	−0.246926
$266$	0	0
$267$	−3.95170	−0.241840
$268$	0	0
$269$	−0.233965	−0.0142651	−0.00713255	−	0.999975i	$-0.502270\pi$
$269$	−0.233965	−0.0142651	−0.00713255	+	0.999975i	$0.502270\pi$
$270$	0	0
$271$	−25.8106	−1.56788	−0.783942	−	0.620834i	$-0.786794\pi$
$271$	−25.8106	−1.56788	−0.783942	+	0.620834i	$0.786794\pi$
$272$	0	0
$273$	−7.94003	−0.480552
$274$	0	0
$275$	22.2753	1.34325
$276$	0	0
$277$	−19.1134	−1.14841	−0.574207	−	0.818710i	$-0.694689\pi$
$277$	−19.1134	−1.14841	−0.574207	+	0.818710i	$0.694689\pi$
$278$	0	0
$279$	−0.631473	−0.0378053
$280$	0	0
$281$	7.13658	0.425732	0.212866	−	0.977081i	$-0.431720\pi$
$281$	7.13658	0.425732	0.212866	+	0.977081i	$0.431720\pi$
$282$	0	0
$283$	−9.14200	−0.543436	−0.271718	−	0.962377i	$-0.587592\pi$
$283$	−9.14200	−0.543436	−0.271718	+	0.962377i	$0.587592\pi$
$284$	0	0
$285$	−6.51404	−0.385858
$286$	0	0
$287$	−11.6954	−0.690359
$288$	0	0
$289$	44.7872	2.63454
$290$	0	0
$291$	−15.9027	−0.932234
$292$	0	0
$293$	−25.6850	−1.50053	−0.750266	−	0.661136i	$-0.770075\pi$
$293$	−25.6850	−1.50053	−0.750266	+	0.661136i	$0.770075\pi$
$294$	0	0
$295$	25.3527	1.47609
$296$	0	0
$297$	−10.9153	−0.633369
$298$	0	0
$299$	6.50469	0.376176
$300$	0	0
$301$	−0.869304	−0.0501059
$302$	0	0
$303$	9.26548	0.532288
$304$	0	0
$305$	−44.3389	−2.53884
$306$	0	0
$307$	13.9852	0.798180	0.399090	−	0.916912i	$-0.369326\pi$
$307$	13.9852	0.798180	0.399090	+	0.916912i	$0.369326\pi$
$308$	0	0
$309$	−24.1430	−1.37345
$310$	0	0
$311$	5.85232	0.331855	0.165927	−	0.986138i	$-0.446938\pi$
$311$	5.85232	0.331855	0.165927	+	0.986138i	$0.446938\pi$
$312$	0	0
$313$	22.1103	1.24975	0.624873	−	0.780727i	$-0.285151\pi$
$313$	22.1103	1.24975	0.624873	+	0.780727i	$0.285151\pi$
$314$	0	0
$315$	−3.05242	−0.171985
$316$	0	0
$317$	−6.03931	−0.339201	−0.169601	−	0.985513i	$-0.554248\pi$
$317$	−6.03931	−0.339201	−0.169601	+	0.985513i	$0.554248\pi$
$318$	0	0
$319$	−5.53589	−0.309950
$320$	0	0
$321$	18.9715	1.05889
$322$	0	0
$323$	−7.86048	−0.437369
$324$	0	0
$325$	−26.9131	−1.49287
$326$	0	0
$327$	12.1650	0.672723
$328$	0	0
$329$	17.8269	0.982828
$330$	0	0
$331$	11.3832	0.625676	0.312838	−	0.949806i	$-0.398720\pi$
$331$	11.3832	0.625676	0.312838	+	0.949806i	$0.398720\pi$
$332$	0	0
$333$	−0.626268	−0.0343193
$334$	0	0
$335$	9.31712	0.509049
$336$	0	0
$337$	16.1151	0.877847	0.438923	−	0.898524i	$-0.355360\pi$
$337$	16.1151	0.877847	0.438923	+	0.898524i	$0.355360\pi$
$338$	0	0
$339$	13.1225	0.712717
$340$	0	0
$341$	3.37226	0.182618
$342$	0	0
$343$	−20.0567	−1.08296
$344$	0	0
$345$	−17.5666	−0.945753
$346$	0	0
$347$	22.2382	1.19381	0.596906	−	0.802312i	$-0.296397\pi$
$347$	22.2382	1.19381	0.596906	+	0.802312i	$0.296397\pi$
$348$	0	0
$349$	10.6050	0.567674	0.283837	−	0.958873i	$-0.408393\pi$
$349$	10.6050	0.567674	0.283837	+	0.958873i	$0.408393\pi$
$350$	0	0
$351$	13.1879	0.703917
$352$	0	0
$353$	5.76897	0.307051	0.153526	−	0.988145i	$-0.450937\pi$
$353$	5.76897	0.307051	0.153526	+	0.988145i	$0.450937\pi$
$354$	0	0
$355$	16.1963	0.859612
$356$	0	0
$357$	25.8751	1.36945
$358$	0	0
$359$	15.2719	0.806021	0.403010	−	0.915195i	$-0.367964\pi$
$359$	15.2719	0.806021	0.403010	+	0.915195i	$0.367964\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−11.3670	−0.596615
$364$	0	0
$365$	−6.76875	−0.354292
$366$	0	0
$367$	−21.6591	−1.13060	−0.565299	−	0.824886i	$-0.691239\pi$
$367$	−21.6591	−1.13060	−0.565299	+	0.824886i	$0.691239\pi$
$368$	0	0
$369$	2.15243	0.112051
$370$	0	0
$371$	−2.03129	−0.105459
$372$	0	0
$373$	19.5149	1.01044	0.505222	−	0.862989i	$-0.331411\pi$
$373$	19.5149	1.01044	0.505222	+	0.862989i	$0.331411\pi$
$374$	0	0
$375$	40.1114	2.07135
$376$	0	0
$377$	6.68848	0.344474
$378$	0	0
$379$	37.3640	1.91926	0.959629	−	0.281268i	$-0.0907550\pi$
$379$	37.3640	1.91926	0.959629	+	0.281268i	$0.0907550\pi$
$380$	0	0
$381$	2.00046	0.102486
$382$	0	0
$383$	24.1323	1.23310	0.616552	−	0.787314i	$-0.288529\pi$
$383$	24.1323	1.23310	0.616552	+	0.787314i	$0.288529\pi$
$384$	0	0
$385$	16.3009	0.830770
$386$	0	0
$387$	0.159987	0.00813259
$388$	0	0
$389$	−16.3909	−0.831052	−0.415526	−	0.909581i	$-0.636402\pi$
$389$	−16.3909	−0.831052	−0.415526	+	0.909581i	$0.636402\pi$
$390$	0	0
$391$	−21.1976	−1.07201
$392$	0	0
$393$	−1.21622	−0.0613500
$394$	0	0
$395$	45.3931	2.28397
$396$	0	0
$397$	32.7902	1.64569	0.822847	−	0.568264i	$-0.192385\pi$
$397$	32.7902	1.64569	0.822847	+	0.568264i	$0.192385\pi$
$398$	0	0
$399$	−3.29179	−0.164796
$400$	0	0
$401$	4.48302	0.223871	0.111936	−	0.993715i	$-0.464295\pi$
$401$	4.48302	0.223871	0.111936	+	0.993715i	$0.464295\pi$
$402$	0	0
$403$	−4.07437	−0.202959
$404$	0	0
$405$	−31.1071	−1.54572
$406$	0	0
$407$	3.34446	0.165779
$408$	0	0
$409$	−34.4145	−1.70169	−0.850844	−	0.525418i	$-0.823909\pi$
$409$	−34.4145	−1.70169	−0.850844	+	0.525418i	$0.823909\pi$
$410$	0	0
$411$	19.2598	0.950017
$412$	0	0
$413$	12.8117	0.630421
$414$	0	0
$415$	−52.7470	−2.58925
$416$	0	0
$417$	−12.5745	−0.615774
$418$	0	0
$419$	27.4555	1.34129	0.670644	−	0.741779i	$-0.266018\pi$
$419$	27.4555	1.34129	0.670644	+	0.741779i	$0.266018\pi$
$420$	0	0
$421$	−3.78789	−0.184610	−0.0923052	−	0.995731i	$-0.529424\pi$
$421$	−3.78789	−0.184610	−0.0923052	+	0.995731i	$0.529424\pi$
$422$	0	0
$423$	−3.28086	−0.159521
$424$	0	0
$425$	87.7048	4.25431
$426$	0	0
$427$	−22.4061	−1.08431
$428$	0	0
$429$	−7.80370	−0.376766
$430$	0	0
$431$	21.0534	1.01411	0.507054	−	0.861914i	$-0.330734\pi$
$431$	21.0534	1.01411	0.507054	+	0.861914i	$0.330734\pi$
$432$	0	0
$433$	−23.7308	−1.14043	−0.570215	−	0.821495i	$-0.693140\pi$
$433$	−23.7308	−1.14043	−0.570215	+	0.821495i	$0.693140\pi$
$434$	0	0
$435$	−18.0629	−0.866051
$436$	0	0
$437$	2.69673	0.129002
$438$	0	0
$439$	−30.3007	−1.44617	−0.723087	−	0.690757i	$-0.757277\pi$
$439$	−30.3007	−1.44617	−0.723087	+	0.690757i	$0.757277\pi$
$440$	0	0
$441$	1.07436	0.0511602
$442$	0	0
$443$	1.65673	0.0787136	0.0393568	−	0.999225i	$-0.487469\pi$
$443$	1.65673	0.0787136	0.0393568	+	0.999225i	$0.487469\pi$
$444$	0	0
$445$	−9.80196	−0.464657
$446$	0	0
$447$	26.6916	1.26247
$448$	0	0
$449$	−36.8920	−1.74104	−0.870520	−	0.492133i	$-0.836217\pi$
$449$	−36.8920	−1.74104	−0.870520	+	0.492133i	$0.836217\pi$
$450$	0	0
$451$	−11.4946	−0.541260
$452$	0	0
$453$	−19.7435	−0.927632
$454$	0	0
$455$	−19.6948	−0.923305
$456$	0	0
$457$	37.6024	1.75896	0.879482	−	0.475931i	$-0.157889\pi$
$457$	37.6024	1.75896	0.879482	+	0.475931i	$0.157889\pi$
$458$	0	0
$459$	−42.9768	−2.00599
$460$	0	0
$461$	−19.8431	−0.924188	−0.462094	−	0.886831i	$-0.652902\pi$
$461$	−19.8431	−0.924188	−0.462094	+	0.886831i	$0.652902\pi$
$462$	0	0
$463$	9.59478	0.445907	0.222954	−	0.974829i	$-0.428430\pi$
$463$	9.59478	0.445907	0.222954	+	0.974829i	$0.428430\pi$
$464$	0	0
$465$	11.0033	0.510264
$466$	0	0
$467$	−36.1548	−1.67304	−0.836522	−	0.547933i	$-0.815415\pi$
$467$	−36.1548	−1.67304	−0.836522	+	0.547933i	$0.815415\pi$
$468$	0	0
$469$	4.70829	0.217409
$470$	0	0
$471$	−18.5392	−0.854242
$472$	0	0
$473$	−0.854379	−0.0392844
$474$	0	0
$475$	−11.1577	−0.511950
$476$	0	0
$477$	0.373838	0.0171169
$478$	0	0
$479$	34.9855	1.59853	0.799265	−	0.600979i	$-0.205222\pi$
$479$	34.9855	1.59853	0.799265	+	0.600979i	$0.205222\pi$
$480$	0	0
$481$	−4.04079	−0.184244
$482$	0	0
$483$	−8.87705	−0.403920
$484$	0	0
$485$	−39.4457	−1.79114
$486$	0	0
$487$	15.0742	0.683076	0.341538	−	0.939868i	$-0.389052\pi$
$487$	15.0742	0.683076	0.341538	+	0.939868i	$0.389052\pi$
$488$	0	0
$489$	17.9643	0.812373
$490$	0	0
$491$	5.05424	0.228095	0.114047	−	0.993475i	$-0.463618\pi$
$491$	5.05424	0.228095	0.114047	+	0.993475i	$0.463618\pi$
$492$	0	0
$493$	−21.7965	−0.981665
$494$	0	0
$495$	−3.00002	−0.134841
$496$	0	0
$497$	8.18462	0.367130
$498$	0	0
$499$	5.53043	0.247576	0.123788	−	0.992309i	$-0.460496\pi$
$499$	5.53043	0.247576	0.123788	+	0.992309i	$0.460496\pi$
$500$	0	0
$501$	−18.1602	−0.811338
$502$	0	0
$503$	27.1579	1.21091	0.605456	−	0.795879i	$-0.292991\pi$
$503$	27.1579	1.21091	0.605456	+	0.795879i	$0.292991\pi$
$504$	0	0
$505$	22.9825	1.02271
$506$	0	0
$507$	−11.6386	−0.516889
$508$	0	0
$509$	−27.8507	−1.23446	−0.617231	−	0.786782i	$-0.711746\pi$
$509$	−27.8507	−1.23446	−0.617231	+	0.786782i	$0.711746\pi$
$510$	0	0
$511$	−3.42050	−0.151314
$512$	0	0
$513$	5.46745	0.241394
$514$	0	0
$515$	−59.8854	−2.63887
$516$	0	0
$517$	17.5208	0.770565
$518$	0	0
$519$	13.5371	0.594212
$520$	0	0
$521$	−40.5555	−1.77677	−0.888385	−	0.459099i	$-0.848172\pi$
$521$	−40.5555	−1.77677	−0.888385	+	0.459099i	$0.848172\pi$
$522$	0	0
$523$	28.4210	1.24276	0.621382	−	0.783508i	$-0.286572\pi$
$523$	28.4210	1.24276	0.621382	+	0.783508i	$0.286572\pi$
$524$	0	0
$525$	36.7288	1.60297
$526$	0	0
$527$	13.2776	0.578382
$528$	0	0
$529$	−15.7277	−0.683812
$530$	0	0
$531$	−2.35786	−0.102322
$532$	0	0
$533$	13.8878	0.601548
$534$	0	0
$535$	47.0577	2.03448
$536$	0	0
$537$	−26.3210	−1.13583
$538$	0	0
$539$	−5.73743	−0.247129
$540$	0	0
$541$	23.2611	1.00007	0.500037	−	0.866004i	$-0.333320\pi$
$541$	23.2611	1.00007	0.500037	+	0.866004i	$0.333320\pi$
$542$	0	0
$543$	−7.64266	−0.327978
$544$	0	0
$545$	30.1744	1.29253
$546$	0	0
$547$	−34.4176	−1.47159	−0.735796	−	0.677203i	$-0.763192\pi$
$547$	−34.4176	−1.47159	−0.735796	+	0.677203i	$0.763192\pi$
$548$	0	0
$549$	4.12363	0.175992
$550$	0	0
$551$	2.77292	0.118130
$552$	0	0
$553$	22.9388	0.975458
$554$	0	0
$555$	10.9126	0.463212
$556$	0	0
$557$	−29.9078	−1.26723	−0.633616	−	0.773647i	$-0.718430\pi$
$557$	−29.9078	−1.26723	−0.633616	+	0.773647i	$0.718430\pi$
$558$	0	0
$559$	1.03226	0.0436601
$560$	0	0
$561$	25.4308	1.07369
$562$	0	0
$563$	25.3710	1.06926	0.534630	−	0.845086i	$-0.320451\pi$
$563$	25.3710	1.06926	0.534630	+	0.845086i	$0.320451\pi$
$564$	0	0
$565$	32.5496	1.36937
$566$	0	0
$567$	−15.7196	−0.660160
$568$	0	0
$569$	−31.0964	−1.30363	−0.651814	−	0.758379i	$-0.725992\pi$
$569$	−31.0964	−1.30363	−0.651814	+	0.758379i	$0.725992\pi$
$570$	0	0
$571$	31.0093	1.29770	0.648849	−	0.760917i	$-0.275251\pi$
$571$	31.0093	1.29770	0.648849	+	0.760917i	$0.275251\pi$
$572$	0	0
$573$	−19.1421	−0.799671
$574$	0	0
$575$	−30.0892	−1.25481
$576$	0	0
$577$	−2.57806	−0.107326	−0.0536630	−	0.998559i	$-0.517090\pi$
$577$	−2.57806	−0.107326	−0.0536630	+	0.998559i	$0.517090\pi$
$578$	0	0
$579$	−35.5643	−1.47800
$580$	0	0
$581$	−26.6550	−1.10584
$582$	0	0
$583$	−1.99641	−0.0826829
$584$	0	0
$585$	3.62463	0.149860
$586$	0	0
$587$	−19.1813	−0.791699	−0.395849	−	0.918315i	$-0.629550\pi$
$587$	−19.1813	−0.791699	−0.395849	+	0.918315i	$0.629550\pi$
$588$	0	0
$589$	−1.68916	−0.0696007
$590$	0	0
$591$	−19.3477	−0.795860
$592$	0	0
$593$	−32.2265	−1.32338	−0.661692	−	0.749776i	$-0.730161\pi$
$593$	−32.2265	−1.32338	−0.661692	+	0.749776i	$0.730161\pi$
$594$	0	0
$595$	64.1816	2.63119
$596$	0	0
$597$	15.1665	0.620725
$598$	0	0
$599$	27.8580	1.13825	0.569124	−	0.822252i	$-0.307283\pi$
$599$	27.8580	1.13825	0.569124	+	0.822252i	$0.307283\pi$
$600$	0	0
$601$	12.1758	0.496660	0.248330	−	0.968675i	$-0.420118\pi$
$601$	12.1758	0.496660	0.248330	+	0.968675i	$0.420118\pi$
$602$	0	0
$603$	−0.866515	−0.0352872
$604$	0	0
$605$	−28.1953	−1.14630
$606$	0	0
$607$	5.45148	0.221269	0.110634	−	0.993861i	$-0.464712\pi$
$607$	5.45148	0.221269	0.110634	+	0.993861i	$0.464712\pi$
$608$	0	0
$609$	−9.12788	−0.369880
$610$	0	0
$611$	−21.1687	−0.856394
$612$	0	0
$613$	−27.8418	−1.12452	−0.562259	−	0.826961i	$-0.690068\pi$
$613$	−27.8418	−1.12452	−0.562259	+	0.826961i	$0.690068\pi$
$614$	0	0
$615$	−37.5055	−1.51237
$616$	0	0
$617$	30.4676	1.22658	0.613289	−	0.789859i	$-0.289846\pi$
$617$	30.4676	1.22658	0.613289	+	0.789859i	$0.289846\pi$
$618$	0	0
$619$	−29.1747	−1.17263	−0.586316	−	0.810082i	$-0.699422\pi$
$619$	−29.1747	−1.17263	−0.586316	+	0.810082i	$0.699422\pi$
$620$	0	0
$621$	14.7442	0.591665
$622$	0	0
$623$	−4.95330	−0.198450
$624$	0	0
$625$	43.7056	1.74822
$626$	0	0
$627$	−3.23527	−0.129204
$628$	0	0
$629$	13.1682	0.525049
$630$	0	0
$631$	24.9353	0.992658	0.496329	−	0.868134i	$-0.334681\pi$
$631$	24.9353	0.992658	0.496329	+	0.868134i	$0.334681\pi$
$632$	0	0
$633$	10.9186	0.433975
$634$	0	0
$635$	4.96201	0.196912
$636$	0	0
$637$	6.93198	0.274655
$638$	0	0
$639$	−1.50630	−0.0595883
$640$	0	0
$641$	−38.3754	−1.51574	−0.757869	−	0.652407i	$-0.773759\pi$
$641$	−38.3754	−1.51574	−0.757869	+	0.652407i	$0.773759\pi$
$642$	0	0
$643$	−3.25862	−0.128507	−0.0642537	−	0.997934i	$-0.520467\pi$
$643$	−3.25862	−0.128507	−0.0642537	+	0.997934i	$0.520467\pi$
$644$	0	0
$645$	−2.78773	−0.109767
$646$	0	0
$647$	−27.6897	−1.08859	−0.544297	−	0.838893i	$-0.683204\pi$
$647$	−27.6897	−1.08859	−0.544297	+	0.838893i	$0.683204\pi$
$648$	0	0
$649$	12.5917	0.494268
$650$	0	0
$651$	5.56036	0.217928
$652$	0	0
$653$	11.4194	0.446876	0.223438	−	0.974718i	$-0.428272\pi$
$653$	11.4194	0.446876	0.223438	+	0.974718i	$0.428272\pi$
$654$	0	0
$655$	−3.01675	−0.117874
$656$	0	0
$657$	0.629510	0.0245595
$658$	0	0
$659$	−3.10703	−0.121033	−0.0605163	−	0.998167i	$-0.519275\pi$
$659$	−3.10703	−0.121033	−0.0605163	+	0.998167i	$0.519275\pi$
$660$	0	0
$661$	21.7813	0.847195	0.423597	−	0.905851i	$-0.360767\pi$
$661$	21.7813	0.847195	0.423597	+	0.905851i	$0.360767\pi$
$662$	0	0
$663$	−30.7256	−1.19328
$664$	0	0
$665$	−8.16509	−0.316629
$666$	0	0
$667$	7.47781	0.289542
$668$	0	0
$669$	−34.2514	−1.32423
$670$	0	0
$671$	−22.0214	−0.850128
$672$	0	0
$673$	0.149431	0.00576014	0.00288007	−	0.999996i	$-0.499083\pi$
$673$	0.149431	0.00576014	0.00288007	+	0.999996i	$0.499083\pi$
$674$	0	0
$675$	−61.0041	−2.34805
$676$	0	0
$677$	−34.2308	−1.31560	−0.657799	−	0.753194i	$-0.728512\pi$
$677$	−34.2308	−1.31560	−0.657799	+	0.753194i	$0.728512\pi$
$678$	0	0
$679$	−19.9334	−0.764975
$680$	0	0
$681$	−9.90297	−0.379483
$682$	0	0
$683$	−18.6833	−0.714897	−0.357449	−	0.933933i	$-0.616353\pi$
$683$	−18.6833	−0.714897	−0.357449	+	0.933933i	$0.616353\pi$
$684$	0	0
$685$	47.7728	1.82531
$686$	0	0
$687$	14.1191	0.538677
$688$	0	0
$689$	2.41207	0.0918925
$690$	0	0
$691$	−10.0526	−0.382420	−0.191210	−	0.981549i	$-0.561241\pi$
$691$	−10.0526	−0.382420	−0.191210	+	0.981549i	$0.561241\pi$
$692$	0	0
$693$	−1.51602	−0.0575889
$694$	0	0
$695$	−31.1902	−1.18311
$696$	0	0
$697$	−45.2578	−1.71426
$698$	0	0
$699$	11.2057	0.423837
$700$	0	0
$701$	31.5608	1.19203	0.596017	−	0.802972i	$-0.296749\pi$
$701$	31.5608	1.19203	0.596017	+	0.802972i	$0.296749\pi$
$702$	0	0
$703$	−1.67524	−0.0631827
$704$	0	0
$705$	57.1682	2.15308
$706$	0	0
$707$	11.6139	0.436786
$708$	0	0
$709$	0.195919	0.00735788	0.00367894	−	0.999993i	$-0.498829\pi$
$709$	0.195919	0.00735788	0.00367894	+	0.999993i	$0.498829\pi$
$710$	0	0
$711$	−4.22166	−0.158325
$712$	0	0
$713$	−4.55520	−0.170594
$714$	0	0
$715$	−19.3566	−0.723897
$716$	0	0
$717$	37.1714	1.38819
$718$	0	0
$719$	16.1168	0.601055	0.300527	−	0.953773i	$-0.402837\pi$
$719$	16.1168	0.601055	0.300527	+	0.953773i	$0.402837\pi$
$720$	0	0
$721$	−30.2623	−1.12703
$722$	0	0
$723$	−36.9344	−1.37361
$724$	0	0
$725$	−30.9394	−1.14906
$726$	0	0
$727$	23.8779	0.885581	0.442791	−	0.896625i	$-0.353988\pi$
$727$	23.8779	0.885581	0.442791	+	0.896625i	$0.353988\pi$
$728$	0	0
$729$	29.4738	1.09162
$730$	0	0
$731$	−3.36395	−0.124420
$732$	0	0
$733$	−53.7576	−1.98558	−0.992791	−	0.119861i	$-0.961755\pi$
$733$	−53.7576	−1.98558	−0.992791	+	0.119861i	$0.961755\pi$
$734$	0	0
$735$	−18.7205	−0.690517
$736$	0	0
$737$	4.62745	0.170454
$738$	0	0
$739$	−32.3793	−1.19109	−0.595546	−	0.803321i	$-0.703064\pi$
$739$	−32.3793	−1.19109	−0.595546	+	0.803321i	$0.703064\pi$
$740$	0	0
$741$	3.90886	0.143596
$742$	0	0
$743$	−22.3845	−0.821207	−0.410604	−	0.911814i	$-0.634682\pi$
$743$	−22.3845	−0.821207	−0.410604	+	0.911814i	$0.634682\pi$
$744$	0	0
$745$	66.2070	2.42564
$746$	0	0
$747$	4.90560	0.179486
$748$	0	0
$749$	23.7800	0.868903
$750$	0	0
$751$	41.5843	1.51743	0.758716	−	0.651421i	$-0.225827\pi$
$751$	41.5843	1.51743	0.758716	+	0.651421i	$0.225827\pi$
$752$	0	0
$753$	26.0417	0.949012
$754$	0	0
$755$	−48.9727	−1.78230
$756$	0	0
$757$	−13.4951	−0.490488	−0.245244	−	0.969461i	$-0.578868\pi$
$757$	−13.4951	−0.490488	−0.245244	+	0.969461i	$0.578868\pi$
$758$	0	0
$759$	−8.72464	−0.316684
$760$	0	0
$761$	4.61922	0.167447	0.0837233	−	0.996489i	$-0.473319\pi$
$761$	4.61922	0.167447	0.0837233	+	0.996489i	$0.473319\pi$
$762$	0	0
$763$	15.2483	0.552025
$764$	0	0
$765$	−11.8120	−0.427063
$766$	0	0
$767$	−15.2133	−0.549322
$768$	0	0
$769$	−25.4299	−0.917024	−0.458512	−	0.888688i	$-0.651617\pi$
$769$	−25.4299	−0.917024	−0.458512	+	0.888688i	$0.651617\pi$
$770$	0	0
$771$	−18.3582	−0.661155
$772$	0	0
$773$	13.9254	0.500864	0.250432	−	0.968134i	$-0.419427\pi$
$773$	13.9254	0.500864	0.250432	+	0.968134i	$0.419427\pi$
$774$	0	0
$775$	18.8471	0.677009
$776$	0	0
$777$	5.51453	0.197833
$778$	0	0
$779$	5.75764	0.206289
$780$	0	0
$781$	8.04410	0.287840
$782$	0	0
$783$	15.1608	0.541803
$784$	0	0
$785$	−45.9854	−1.64129
$786$	0	0
$787$	30.2950	1.07990	0.539949	−	0.841697i	$-0.318443\pi$
$787$	30.2950	1.07990	0.539949	+	0.841697i	$0.318443\pi$
$788$	0	0
$789$	−0.115899	−0.00412612
$790$	0	0
$791$	16.4486	0.584843
$792$	0	0
$793$	26.6064	0.944819
$794$	0	0
$795$	−6.51404	−0.231029
$796$	0	0
$797$	−43.1204	−1.52740	−0.763702	−	0.645569i	$-0.776620\pi$
$797$	−43.1204	−1.52740	−0.763702	+	0.645569i	$0.776620\pi$
$798$	0	0
$799$	68.9848	2.44051
$800$	0	0
$801$	0.911606	0.0322100
$802$	0	0
$803$	−3.36177	−0.118634
$804$	0	0
$805$	−22.0190	−0.776068
$806$	0	0
$807$	−0.379150	−0.0133467
$808$	0	0
$809$	17.1966	0.604601	0.302300	−	0.953213i	$-0.402245\pi$
$809$	17.1966	0.604601	0.302300	+	0.953213i	$0.402245\pi$
$810$	0	0
$811$	−36.9222	−1.29651	−0.648257	−	0.761421i	$-0.724502\pi$
$811$	−36.9222	−1.29651	−0.648257	+	0.761421i	$0.724502\pi$
$812$	0	0
$813$	−41.8273	−1.46695
$814$	0	0
$815$	44.5594	1.56085
$816$	0	0
$817$	0.427957	0.0149723
$818$	0	0
$819$	1.83166	0.0640034
$820$	0	0
$821$	33.3448	1.16374	0.581872	−	0.813281i	$-0.302321\pi$
$821$	33.3448	1.16374	0.581872	+	0.813281i	$0.302321\pi$
$822$	0	0
$823$	35.6142	1.24143	0.620716	−	0.784035i	$-0.286842\pi$
$823$	35.6142	1.24143	0.620716	+	0.784035i	$0.286842\pi$
$824$	0	0
$825$	36.0982	1.25678
$826$	0	0
$827$	−4.98138	−0.173219	−0.0866097	−	0.996242i	$-0.527603\pi$
$827$	−4.98138	−0.173219	−0.0866097	+	0.996242i	$0.527603\pi$
$828$	0	0
$829$	−22.1672	−0.769898	−0.384949	−	0.922938i	$-0.625781\pi$
$829$	−22.1672	−0.769898	−0.384949	+	0.922938i	$0.625781\pi$
$830$	0	0
$831$	−30.9741	−1.07448
$832$	0	0
$833$	−22.5900	−0.782698
$834$	0	0
$835$	−45.0453	−1.55886
$836$	0	0
$837$	−9.23541	−0.319222
$838$	0	0
$839$	14.0980	0.486719	0.243359	−	0.969936i	$-0.421751\pi$
$839$	14.0980	0.486719	0.243359	+	0.969936i	$0.421751\pi$
$840$	0	0
$841$	−21.3109	−0.734859
$842$	0	0
$843$	11.5651	0.398324
$844$	0	0
$845$	−28.8689	−0.993120
$846$	0	0
$847$	−14.2481	−0.489572
$848$	0	0
$849$	−14.8150	−0.508450
$850$	0	0
$851$	−4.51765	−0.154863
$852$	0	0
$853$	−8.50358	−0.291157	−0.145578	−	0.989347i	$-0.546504\pi$
$853$	−8.50358	−0.291157	−0.145578	+	0.989347i	$0.546504\pi$
$854$	0	0
$855$	1.50270	0.0513914
$856$	0	0
$857$	−9.05042	−0.309157	−0.154578	−	0.987981i	$-0.549402\pi$
$857$	−9.05042	−0.309157	−0.154578	+	0.987981i	$0.549402\pi$
$858$	0	0
$859$	23.1311	0.789223	0.394611	−	0.918848i	$-0.370879\pi$
$859$	23.1311	0.789223	0.394611	+	0.918848i	$0.370879\pi$
$860$	0	0
$861$	−18.9529	−0.645914
$862$	0	0
$863$	16.2859	0.554380	0.277190	−	0.960815i	$-0.410597\pi$
$863$	16.2859	0.554380	0.277190	+	0.960815i	$0.410597\pi$
$864$	0	0
$865$	33.5779	1.14168
$866$	0	0
$867$	72.5796	2.46493
$868$	0	0
$869$	22.5450	0.764786
$870$	0	0
$871$	−5.59090	−0.189440
$872$	0	0
$873$	3.66855	0.124162
$874$	0	0
$875$	50.2781	1.69971
$876$	0	0
$877$	−5.88051	−0.198571	−0.0992854	−	0.995059i	$-0.531656\pi$
$877$	−5.88051	−0.198571	−0.0992854	+	0.995059i	$0.531656\pi$
$878$	0	0
$879$	−41.6236	−1.40393
$880$	0	0
$881$	5.82136	0.196127	0.0980633	−	0.995180i	$-0.468735\pi$
$881$	5.82136	0.196127	0.0980633	+	0.995180i	$0.468735\pi$
$882$	0	0
$883$	28.4002	0.955744	0.477872	−	0.878429i	$-0.341408\pi$
$883$	28.4002	0.955744	0.477872	+	0.878429i	$0.341408\pi$
$884$	0	0
$885$	41.0851	1.38106
$886$	0	0
$887$	−52.3884	−1.75903	−0.879515	−	0.475870i	$-0.842133\pi$
$887$	−52.3884	−1.75903	−0.879515	+	0.475870i	$0.842133\pi$
$888$	0	0
$889$	2.50749	0.0840986
$890$	0	0
$891$	−15.4497	−0.517584
$892$	0	0
$893$	−8.77616	−0.293683
$894$	0	0
$895$	−65.2876	−2.18232
$896$	0	0
$897$	10.5411	0.351958
$898$	0	0
$899$	−4.68391	−0.156217
$900$	0	0
$901$	−7.86048	−0.261871
$902$	0	0
$903$	−1.40875	−0.0468801
$904$	0	0
$905$	−18.9572	−0.630157
$906$	0	0
$907$	−30.8588	−1.02465	−0.512324	−	0.858792i	$-0.671215\pi$
$907$	−30.8588	−1.02465	−0.512324	+	0.858792i	$0.671215\pi$
$908$	0	0
$909$	−2.13743	−0.0708940
$910$	0	0
$911$	45.0949	1.49406	0.747031	−	0.664790i	$-0.231479\pi$
$911$	45.0949	1.49406	0.747031	+	0.664790i	$0.231479\pi$
$912$	0	0
$913$	−26.1974	−0.867007
$914$	0	0
$915$	−71.8532	−2.37539
$916$	0	0
$917$	−1.52448	−0.0503428
$918$	0	0
$919$	5.53999	0.182747	0.0913737	−	0.995817i	$-0.470874\pi$
$919$	5.53999	0.182747	0.0913737	+	0.995817i	$0.470874\pi$
$920$	0	0
$921$	22.6637	0.746794
$922$	0	0
$923$	−9.71890	−0.319901
$924$	0	0
$925$	18.6918	0.614582
$926$	0	0
$927$	5.56949	0.182926
$928$	0	0
$929$	−12.2324	−0.401333	−0.200666	−	0.979660i	$-0.564311\pi$
$929$	−12.2324	−0.401333	−0.200666	+	0.979660i	$0.564311\pi$
$930$	0	0
$931$	2.87387	0.0941873
$932$	0	0
$933$	9.48395	0.310490
$934$	0	0
$935$	63.0796	2.06292
$936$	0	0
$937$	−52.2703	−1.70760	−0.853799	−	0.520603i	$-0.825707\pi$
$937$	−52.2703	−1.70760	−0.853799	+	0.520603i	$0.825707\pi$
$938$	0	0
$939$	35.8306	1.16929
$940$	0	0
$941$	−25.8949	−0.844150	−0.422075	−	0.906561i	$-0.638698\pi$
$941$	−25.8949	−0.844150	−0.422075	+	0.906561i	$0.638698\pi$
$942$	0	0
$943$	15.5268	0.505621
$944$	0	0
$945$	−44.6422	−1.45221
$946$	0	0
$947$	−18.0399	−0.586217	−0.293108	−	0.956079i	$-0.594690\pi$
$947$	−18.0399	−0.586217	−0.293108	+	0.956079i	$0.594690\pi$
$948$	0	0
$949$	4.06170	0.131849
$950$	0	0
$951$	−9.78696	−0.317364
$952$	0	0
$953$	−31.0181	−1.00477	−0.502387	−	0.864643i	$-0.667545\pi$
$953$	−31.0181	−1.00477	−0.502387	+	0.864643i	$0.667545\pi$
$954$	0	0
$955$	−47.4807	−1.53644
$956$	0	0
$957$	−8.97116	−0.289996
$958$	0	0
$959$	24.1414	0.779567
$960$	0	0
$961$	−28.1467	−0.907959
$962$	0	0
$963$	−4.37648	−0.141030
$964$	0	0
$965$	−88.2152	−2.83975
$966$	0	0
$967$	−9.40500	−0.302444	−0.151222	−	0.988500i	$-0.548321\pi$
$967$	−9.40500	−0.302444	−0.151222	+	0.988500i	$0.548321\pi$
$968$	0	0
$969$	−12.7383	−0.409212
$970$	0	0
$971$	−2.68509	−0.0861686	−0.0430843	−	0.999071i	$-0.513718\pi$
$971$	−2.68509	−0.0861686	−0.0430843	+	0.999071i	$0.513718\pi$
$972$	0	0
$973$	−15.7616	−0.505293
$974$	0	0
$975$	−43.6139	−1.39676
$976$	0	0
$977$	27.3431	0.874782	0.437391	−	0.899271i	$-0.355903\pi$
$977$	27.3431	0.874782	0.437391	+	0.899271i	$0.355903\pi$
$978$	0	0
$979$	−4.86825	−0.155590
$980$	0	0
$981$	−2.80630	−0.0895982
$982$	0	0
$983$	−4.43737	−0.141530	−0.0707651	−	0.997493i	$-0.522544\pi$
$983$	−4.43737	−0.141530	−0.0707651	+	0.997493i	$0.522544\pi$
$984$	0	0
$985$	−47.9909	−1.52912
$986$	0	0
$987$	28.8893	0.919555
$988$	0	0
$989$	1.15408	0.0366977
$990$	0	0
$991$	4.34268	0.137950	0.0689748	−	0.997618i	$-0.478027\pi$
$991$	4.34268	0.137950	0.0689748	+	0.997618i	$0.478027\pi$
$992$	0	0
$993$	18.4469	0.585396
$994$	0	0
$995$	37.6197	1.19262
$996$	0	0
$997$	18.0848	0.572753	0.286376	−	0.958117i	$-0.407549\pi$
$997$	18.0848	0.572753	0.286376	+	0.958117i	$0.407549\pi$
$998$	0	0
$999$	−9.15928	−0.289787

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4028.2.a.f.1.13	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4028.2.a.f.1.13	✓	19	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:	\( N \)	\(=\)	\( 4028 = 2^{2} \cdot 19 \cdot 53 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4028.a (trivial)

\(f(q)\)	\(=\)	\(q+1.62054 q^{3} +4.01966 q^{5} +2.03129 q^{7} -0.373838 q^{9} +O(q^{10})\)	\(q+1.62054 q^{3} +4.01966 q^{5} +2.03129 q^{7} -0.373838 q^{9} +1.99641 q^{11} -2.41207 q^{13} +6.51404 q^{15} +7.86048 q^{17} -1.00000 q^{19} +3.29179 q^{21} -2.69673 q^{23} +11.1577 q^{25} -5.46745 q^{27} -2.77292 q^{29} +1.68916 q^{31} +3.23527 q^{33} +8.16509 q^{35} +1.67524 q^{37} -3.90886 q^{39} -5.75764 q^{41} -0.427957 q^{43} -1.50270 q^{45} +8.77616 q^{47} -2.87387 q^{49} +12.7383 q^{51} -1.00000 q^{53} +8.02490 q^{55} -1.62054 q^{57} +6.30717 q^{59} -11.0305 q^{61} -0.759373 q^{63} -9.69571 q^{65} +2.31789 q^{67} -4.37016 q^{69} +4.02928 q^{71} -1.68391 q^{73} +18.0815 q^{75} +4.05529 q^{77} +11.2928 q^{79} -7.73873 q^{81} -13.1222 q^{83} +31.5965 q^{85} -4.49364 q^{87} -2.43850 q^{89} -4.89961 q^{91} +2.73736 q^{93} -4.01966 q^{95} -9.81320 q^{97} -0.746335 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q + 4 q^{3} + 4 q^{5} + 13 q^{7} + 19 q^{9}+O(q^{10}) \) 19 * q + 4 * q^3 + 4 * q^5 + 13 * q^7 + 19 * q^9	\( 19 q + 4 q^{3} + 4 q^{5} + 13 q^{7} + 19 q^{9} + q^{11} - q^{13} + 8 q^{15} + 3 q^{17} - 19 q^{19} + 8 q^{21} + 10 q^{23} + 21 q^{25} + 28 q^{27} + 2 q^{29} + 25 q^{31} + q^{33} + 20 q^{35} + 19 q^{37} + 37 q^{39} - 9 q^{41} + 35 q^{43} + 37 q^{45} + 23 q^{47} + 30 q^{49} + 34 q^{51} - 19 q^{53} + 40 q^{55} - 4 q^{57} + 16 q^{59} + 21 q^{61} + 3 q^{63} - 10 q^{65} + 67 q^{67} + 23 q^{69} + 18 q^{71} - 20 q^{73} + 33 q^{75} + 37 q^{77} + 2 q^{79} + 23 q^{81} + 38 q^{83} + 8 q^{85} + 18 q^{87} - q^{89} - 9 q^{91} + 14 q^{93} - 4 q^{95} - 21 q^{97} + 33 q^{99}+O(q^{100}) \) 19 * q + 4 * q^3 + 4 * q^5 + 13 * q^7 + 19 * q^9 + q^11 - q^13 + 8 * q^15 + 3 * q^17 - 19 * q^19 + 8 * q^21 + 10 * q^23 + 21 * q^25 + 28 * q^27 + 2 * q^29 + 25 * q^31 + q^33 + 20 * q^35 + 19 * q^37 + 37 * q^39 - 9 * q^41 + 35 * q^43 + 37 * q^45 + 23 * q^47 + 30 * q^49 + 34 * q^51 - 19 * q^53 + 40 * q^55 - 4 * q^57 + 16 * q^59 + 21 * q^61 + 3 * q^63 - 10 * q^65 + 67 * q^67 + 23 * q^69 + 18 * q^71 - 20 * q^73 + 33 * q^75 + 37 * q^77 + 2 * q^79 + 23 * q^81 + 38 * q^83 + 8 * q^85 + 18 * q^87 - q^89 - 9 * q^91 + 14 * q^93 - 4 * q^95 - 21 * q^97 + 33 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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