LMFDB - Embedded newform 4028.2.a.d.1.10

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			1.10
Root			$-0.0800487$ 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)

$f(q)$	$=$	$q+0.0800487 q^{3} +0.442813 q^{5} +2.60518 q^{7} -2.99359 q^{9} +O(q^{10})$	$q+0.0800487 q^{3} +0.442813 q^{5} +2.60518 q^{7} -2.99359 q^{9} -2.05333 q^{11} +2.33968 q^{13} +0.0354466 q^{15} +2.04003 q^{17} -1.00000 q^{19} +0.208542 q^{21} -6.75394 q^{23} -4.80392 q^{25} -0.479779 q^{27} -0.161496 q^{29} -7.92315 q^{31} -0.164366 q^{33} +1.15361 q^{35} +3.74579 q^{37} +0.187289 q^{39} -5.17031 q^{41} -4.07281 q^{43} -1.32560 q^{45} +2.84920 q^{47} -0.213017 q^{49} +0.163302 q^{51} +1.00000 q^{53} -0.909239 q^{55} -0.0800487 q^{57} +8.26122 q^{59} -4.20321 q^{61} -7.79886 q^{63} +1.03604 q^{65} -10.7790 q^{67} -0.540644 q^{69} -4.25972 q^{71} +2.99568 q^{73} -0.384547 q^{75} -5.34930 q^{77} +1.77749 q^{79} +8.94237 q^{81} +13.1358 q^{83} +0.903350 q^{85} -0.0129275 q^{87} -8.31561 q^{89} +6.09530 q^{91} -0.634237 q^{93} -0.442813 q^{95} -13.1455 q^{97} +6.14683 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	0.0800487	0.0462161	0.0231081	−	0.999733i	$-0.492644\pi$
$3$	0.0800487	0.0462161	0.0231081	+	0.999733i	$0.492644\pi$
$4$	0	0
$5$	0.442813	0.198032	0.0990159	−	0.995086i	$-0.468431\pi$
$5$	0.442813	0.198032	0.0990159	+	0.995086i	$0.468431\pi$
$6$	0	0
$7$	2.60518	0.984667	0.492333	−	0.870407i	$-0.336144\pi$
$7$	2.60518	0.984667	0.492333	+	0.870407i	$0.336144\pi$
$8$	0	0
$9$	−2.99359	−0.997864
$10$	0	0
$11$	−2.05333	−0.619102	−0.309551	−	0.950883i	$-0.600179\pi$
$11$	−2.05333	−0.619102	−0.309551	+	0.950883i	$0.600179\pi$
$12$	0	0
$13$	2.33968	0.648911	0.324456	−	0.945901i	$-0.394819\pi$
$13$	2.33968	0.648911	0.324456	+	0.945901i	$0.394819\pi$
$14$	0	0
$15$	0.0354466	0.00915226
$16$	0	0
$17$	2.04003	0.494779	0.247390	−	0.968916i	$-0.420427\pi$
$17$	2.04003	0.494779	0.247390	+	0.968916i	$0.420427\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0.208542	0.0455075
$22$	0	0
$23$	−6.75394	−1.40829	−0.704147	−	0.710054i	$-0.748670\pi$
$23$	−6.75394	−1.40829	−0.704147	+	0.710054i	$0.748670\pi$
$24$	0	0
$25$	−4.80392	−0.960783
$26$	0	0
$27$	−0.479779	−0.0923335
$28$	0	0
$29$	−0.161496	−0.0299890	−0.0149945	−	0.999888i	$-0.504773\pi$
$29$	−0.161496	−0.0299890	−0.0149945	+	0.999888i	$0.504773\pi$
$30$	0	0
$31$	−7.92315	−1.42304	−0.711520	−	0.702666i	$-0.751993\pi$
$31$	−7.92315	−1.42304	−0.711520	+	0.702666i	$0.751993\pi$
$32$	0	0
$33$	−0.164366	−0.0286125
$34$	0	0
$35$	1.15361	0.194995
$36$	0	0
$37$	3.74579	0.615804	0.307902	−	0.951418i	$-0.400373\pi$
$37$	3.74579	0.615804	0.307902	+	0.951418i	$0.400373\pi$
$38$	0	0
$39$	0.187289	0.0299902
$40$	0	0
$41$	−5.17031	−0.807467	−0.403734	−	0.914877i	$-0.632288\pi$
$41$	−5.17031	−0.807467	−0.403734	+	0.914877i	$0.632288\pi$
$42$	0	0
$43$	−4.07281	−0.621097	−0.310549	−	0.950557i	$-0.600513\pi$
$43$	−4.07281	−0.621097	−0.310549	+	0.950557i	$0.600513\pi$
$44$	0	0
$45$	−1.32560	−0.197609
$46$	0	0
$47$	2.84920	0.415598	0.207799	−	0.978172i	$-0.433370\pi$
$47$	2.84920	0.415598	0.207799	+	0.978172i	$0.433370\pi$
$48$	0	0
$49$	−0.213017	−0.0304310
$50$	0	0
$51$	0.163302	0.0228668
$52$	0	0
$53$	1.00000	0.137361
$53$	1.00000	0.137361
$54$	0	0
$55$	−0.909239	−0.122602
$56$	0	0
$57$	−0.0800487	−0.0106027
$58$	0	0
$59$	8.26122	1.07552	0.537759	−	0.843098i	$-0.319271\pi$
$59$	8.26122	1.07552	0.537759	+	0.843098i	$0.319271\pi$
$60$	0	0
$61$	−4.20321	−0.538165	−0.269083	−	0.963117i	$-0.586721\pi$
$61$	−4.20321	−0.538165	−0.269083	+	0.963117i	$0.586721\pi$
$62$	0	0
$63$	−7.79886	−0.982564
$64$	0	0
$65$	1.03604	0.128505
$66$	0	0
$67$	−10.7790	−1.31687	−0.658434	−	0.752638i	$-0.728781\pi$
$67$	−10.7790	−1.31687	−0.658434	+	0.752638i	$0.728781\pi$
$68$	0	0
$69$	−0.540644	−0.0650859
$70$	0	0
$71$	−4.25972	−0.505535	−0.252768	−	0.967527i	$-0.581341\pi$
$71$	−4.25972	−0.505535	−0.252768	+	0.967527i	$0.581341\pi$
$72$	0	0
$73$	2.99568	0.350618	0.175309	−	0.984513i	$-0.443908\pi$
$73$	2.99568	0.350618	0.175309	+	0.984513i	$0.443908\pi$
$74$	0	0
$75$	−0.384547	−0.0444037
$76$	0	0
$77$	−5.34930	−0.609609
$78$	0	0
$79$	1.77749	0.199983	0.0999915	−	0.994988i	$-0.468118\pi$
$79$	1.77749	0.199983	0.0999915	+	0.994988i	$0.468118\pi$
$80$	0	0
$81$	8.94237	0.993597
$82$	0	0
$83$	13.1358	1.44184	0.720920	−	0.693018i	$-0.243719\pi$
$83$	13.1358	1.44184	0.720920	+	0.693018i	$0.243719\pi$
$84$	0	0
$85$	0.903350	0.0979821
$86$	0	0
$87$	−0.0129275	−0.00138597
$88$	0	0
$89$	−8.31561	−0.881453	−0.440727	−	0.897641i	$-0.645279\pi$
$89$	−8.31561	−0.881453	−0.440727	+	0.897641i	$0.645279\pi$
$90$	0	0
$91$	6.09530	0.638962
$92$	0	0
$93$	−0.634237	−0.0657674
$94$	0	0
$95$	−0.442813	−0.0454316
$96$	0	0
$97$	−13.1455	−1.33472	−0.667360	−	0.744736i	$-0.732576\pi$
$97$	−13.1455	−1.33472	−0.667360	+	0.744736i	$0.732576\pi$
$98$	0	0
$99$	6.14683	0.617779
$100$	0	0
$101$	0.0172236	0.00171381	0.000856907	−	1.00000i	$-0.499727\pi$
$101$	0.0172236	0.00171381	0.000856907		1.00000i	$0.499727\pi$
$102$	0	0
$103$	4.31935	0.425598	0.212799	−	0.977096i	$-0.431742\pi$
$103$	4.31935	0.425598	0.212799	+	0.977096i	$0.431742\pi$
$104$	0	0
$105$	0.0923448	0.00901193
$106$	0	0
$107$	−13.5615	−1.31104	−0.655519	−	0.755179i	$-0.727550\pi$
$107$	−13.5615	−1.31104	−0.655519	+	0.755179i	$0.727550\pi$
$108$	0	0
$109$	4.87340	0.466787	0.233394	−	0.972382i	$-0.425017\pi$
$109$	4.87340	0.466787	0.233394	+	0.972382i	$0.425017\pi$
$110$	0	0
$111$	0.299846	0.0284601
$112$	0	0
$113$	−10.5045	−0.988184	−0.494092	−	0.869410i	$-0.664499\pi$
$113$	−10.5045	−0.988184	−0.494092	+	0.869410i	$0.664499\pi$
$114$	0	0
$115$	−2.99073	−0.278887
$116$	0	0
$117$	−7.00406	−0.647525
$118$	0	0
$119$	5.31465	0.487193
$120$	0	0
$121$	−6.78385	−0.616713
$122$	0	0
$123$	−0.413877	−0.0373180
$124$	0	0
$125$	−4.34130	−0.388298
$126$	0	0
$127$	2.11653	0.187812	0.0939058	−	0.995581i	$-0.470065\pi$
$127$	2.11653	0.187812	0.0939058	+	0.995581i	$0.470065\pi$
$128$	0	0
$129$	−0.326023	−0.0287047
$130$	0	0
$131$	−5.17455	−0.452102	−0.226051	−	0.974115i	$-0.572582\pi$
$131$	−5.17455	−0.452102	−0.226051	+	0.974115i	$0.572582\pi$
$132$	0	0
$133$	−2.60518	−0.225898
$134$	0	0
$135$	−0.212452	−0.0182850
$136$	0	0
$137$	11.5877	0.990004	0.495002	−	0.868892i	$-0.335167\pi$
$137$	11.5877	0.990004	0.495002	+	0.868892i	$0.335167\pi$
$138$	0	0
$139$	−22.1807	−1.88135	−0.940673	−	0.339315i	$-0.889805\pi$
$139$	−22.1807	−1.88135	−0.940673	+	0.339315i	$0.889805\pi$
$140$	0	0
$141$	0.228074	0.0192073
$142$	0	0
$143$	−4.80414	−0.401742
$144$	0	0
$145$	−0.0715123	−0.00593877
$146$	0	0
$147$	−0.0170517	−0.00140640
$148$	0	0
$149$	11.1654	0.914703	0.457352	−	0.889286i	$-0.348798\pi$
$149$	11.1654	0.914703	0.457352	+	0.889286i	$0.348798\pi$
$150$	0	0
$151$	−21.8788	−1.78047	−0.890237	−	0.455497i	$-0.849461\pi$
$151$	−21.8788	−1.78047	−0.890237	+	0.455497i	$0.849461\pi$
$152$	0	0
$153$	−6.10701	−0.493723
$154$	0	0
$155$	−3.50847	−0.281807
$156$	0	0
$157$	−7.15969	−0.571405	−0.285703	−	0.958318i	$-0.592227\pi$
$157$	−7.15969	−0.571405	−0.285703	+	0.958318i	$0.592227\pi$
$158$	0	0
$159$	0.0800487	0.00634827
$160$	0	0
$161$	−17.5953	−1.38670
$162$	0	0
$163$	17.1676	1.34467	0.672334	−	0.740248i	$-0.265292\pi$
$163$	17.1676	1.34467	0.672334	+	0.740248i	$0.265292\pi$
$164$	0	0
$165$	−0.0727834	−0.00566618
$166$	0	0
$167$	17.2803	1.33719	0.668595	−	0.743627i	$-0.266896\pi$
$167$	17.2803	1.33719	0.668595	+	0.743627i	$0.266896\pi$
$168$	0	0
$169$	−7.52588	−0.578914
$170$	0	0
$171$	2.99359	0.228926
$172$	0	0
$173$	13.1836	1.00233	0.501164	−	0.865352i	$-0.332905\pi$
$173$	13.1836	1.00233	0.501164	+	0.865352i	$0.332905\pi$
$174$	0	0
$175$	−12.5151	−0.946052
$176$	0	0
$177$	0.661299	0.0497063
$178$	0	0
$179$	5.00811	0.374324	0.187162	−	0.982329i	$-0.440071\pi$
$179$	5.00811	0.374324	0.187162	+	0.982329i	$0.440071\pi$
$180$	0	0
$181$	2.15207	0.159962	0.0799810	−	0.996796i	$-0.474514\pi$
$181$	2.15207	0.159962	0.0799810	+	0.996796i	$0.474514\pi$
$182$	0	0
$183$	−0.336461	−0.0248719
$184$	0	0
$185$	1.65868	0.121949
$186$	0	0
$187$	−4.18884	−0.306319
$188$	0	0
$189$	−1.24991	−0.0909178
$190$	0	0
$191$	21.9716	1.58981	0.794905	−	0.606734i	$-0.207521\pi$
$191$	21.9716	1.58981	0.794905	+	0.606734i	$0.207521\pi$
$192$	0	0
$193$	3.78614	0.272533	0.136266	−	0.990672i	$-0.456490\pi$
$193$	3.78614	0.272533	0.136266	+	0.990672i	$0.456490\pi$
$194$	0	0
$195$	0.0829337	0.00593901
$196$	0	0
$197$	7.05465	0.502623	0.251312	−	0.967906i	$-0.419138\pi$
$197$	7.05465	0.502623	0.251312	+	0.967906i	$0.419138\pi$
$198$	0	0
$199$	−17.2692	−1.22418	−0.612091	−	0.790788i	$-0.709671\pi$
$199$	−17.2692	−1.22418	−0.612091	+	0.790788i	$0.709671\pi$
$200$	0	0
$201$	−0.862847	−0.0608606
$202$	0	0
$203$	−0.420726	−0.0295292
$204$	0	0
$205$	−2.28948	−0.159904
$206$	0	0
$207$	20.2185	1.40529
$208$	0	0
$209$	2.05333	0.142032
$210$	0	0
$211$	−23.5576	−1.62177	−0.810886	−	0.585204i	$-0.801014\pi$
$211$	−23.5576	−1.62177	−0.810886	+	0.585204i	$0.801014\pi$
$212$	0	0
$213$	−0.340985	−0.0233639
$214$	0	0
$215$	−1.80349	−0.122997
$216$	0	0
$217$	−20.6413	−1.40122
$218$	0	0
$219$	0.239800	0.0162042
$220$	0	0
$221$	4.77302	0.321068
$222$	0	0
$223$	17.2802	1.15717	0.578584	−	0.815623i	$-0.303605\pi$
$223$	17.2802	1.15717	0.578584	+	0.815623i	$0.303605\pi$
$224$	0	0
$225$	14.3810	0.958731
$226$	0	0
$227$	−0.177530	−0.0117831	−0.00589155	−	0.999983i	$-0.501875\pi$
$227$	−0.177530	−0.0117831	−0.00589155	+	0.999983i	$0.501875\pi$
$228$	0	0
$229$	−11.8364	−0.782171	−0.391086	−	0.920354i	$-0.627900\pi$
$229$	−11.8364	−0.782171	−0.391086	+	0.920354i	$0.627900\pi$
$230$	0	0
$231$	−0.428204	−0.0281738
$232$	0	0
$233$	21.5076	1.40901	0.704505	−	0.709699i	$-0.251169\pi$
$233$	21.5076	1.40901	0.704505	+	0.709699i	$0.251169\pi$
$234$	0	0
$235$	1.26166	0.0823016
$236$	0	0
$237$	0.142286	0.00924244
$238$	0	0
$239$	−8.78149	−0.568027	−0.284014	−	0.958820i	$-0.591666\pi$
$239$	−8.78149	−0.568027	−0.284014	+	0.958820i	$0.591666\pi$
$240$	0	0
$241$	−21.2194	−1.36686	−0.683432	−	0.730014i	$-0.739513\pi$
$241$	−21.2194	−1.36686	−0.683432	+	0.730014i	$0.739513\pi$
$242$	0	0
$243$	2.15516	0.138254
$244$	0	0
$245$	−0.0943265	−0.00602630
$246$	0	0
$247$	−2.33968	−0.148870
$248$	0	0
$249$	1.05150	0.0666363
$250$	0	0
$251$	−30.5509	−1.92836	−0.964179	−	0.265252i	$-0.914545\pi$
$251$	−30.5509	−1.92836	−0.964179	+	0.265252i	$0.914545\pi$
$252$	0	0
$253$	13.8681	0.871877
$254$	0	0
$255$	0.0723120	0.00452835
$256$	0	0
$257$	−11.2946	−0.704535	−0.352268	−	0.935899i	$-0.614589\pi$
$257$	−11.2946	−0.704535	−0.352268	+	0.935899i	$0.614589\pi$
$258$	0	0
$259$	9.75848	0.606362
$260$	0	0
$261$	0.483452	0.0299249
$262$	0	0
$263$	−0.0775000	−0.00477885	−0.00238943	−	0.999997i	$-0.500761\pi$
$263$	−0.0775000	−0.00477885	−0.00238943	+	0.999997i	$0.500761\pi$
$264$	0	0
$265$	0.442813	0.0272018
$266$	0	0
$267$	−0.665654	−0.0407374
$268$	0	0
$269$	−21.5966	−1.31677	−0.658385	−	0.752681i	$-0.728760\pi$
$269$	−21.5966	−1.31677	−0.658385	+	0.752681i	$0.728760\pi$
$270$	0	0
$271$	6.03169	0.366399	0.183199	−	0.983076i	$-0.441355\pi$
$271$	6.03169	0.366399	0.183199	+	0.983076i	$0.441355\pi$
$272$	0	0
$273$	0.487921	0.0295303
$274$	0	0
$275$	9.86401	0.594822
$276$	0	0
$277$	27.7533	1.66753	0.833766	−	0.552117i	$-0.186180\pi$
$277$	27.7533	1.66753	0.833766	+	0.552117i	$0.186180\pi$
$278$	0	0
$279$	23.7187	1.42000
$280$	0	0
$281$	12.4658	0.743647	0.371824	−	0.928303i	$-0.378733\pi$
$281$	12.4658	0.743647	0.371824	+	0.928303i	$0.378733\pi$
$282$	0	0
$283$	28.6112	1.70076	0.850378	−	0.526172i	$-0.176373\pi$
$283$	28.6112	1.70076	0.850378	+	0.526172i	$0.176373\pi$
$284$	0	0
$285$	−0.0354466	−0.00209967
$286$	0	0
$287$	−13.4696	−0.795086
$288$	0	0
$289$	−12.8383	−0.755193
$290$	0	0
$291$	−1.05228	−0.0616856
$292$	0	0
$293$	1.23617	0.0722178	0.0361089	−	0.999348i	$-0.488504\pi$
$293$	1.23617	0.0722178	0.0361089	+	0.999348i	$0.488504\pi$
$294$	0	0
$295$	3.65817	0.212987
$296$	0	0
$297$	0.985144	0.0571638
$298$	0	0
$299$	−15.8021	−0.913858
$300$	0	0
$301$	−10.6104	−0.611574
$302$	0	0
$303$	0.00137873	7.92059e−5 0
$304$	0	0
$305$	−1.86123	−0.106574
$306$	0	0
$307$	−31.5868	−1.80275	−0.901376	−	0.433038i	$-0.857442\pi$
$307$	−31.5868	−1.80275	−0.901376	+	0.433038i	$0.857442\pi$
$308$	0	0
$309$	0.345758	0.0196695
$310$	0	0
$311$	−26.1076	−1.48043	−0.740213	−	0.672373i	$-0.765275\pi$
$311$	−26.1076	−1.48043	−0.740213	+	0.672373i	$0.765275\pi$
$312$	0	0
$313$	11.9883	0.677618	0.338809	−	0.940855i	$-0.389976\pi$
$313$	11.9883	0.677618	0.338809	+	0.940855i	$0.389976\pi$
$314$	0	0
$315$	−3.45343	−0.194579
$316$	0	0
$317$	11.2848	0.633817	0.316909	−	0.948456i	$-0.397355\pi$
$317$	11.2848	0.633817	0.316909	+	0.948456i	$0.397355\pi$
$318$	0	0
$319$	0.331603	0.0185662
$320$	0	0
$321$	−1.08558	−0.0605911
$322$	0	0
$323$	−2.04003	−0.113510
$324$	0	0
$325$	−11.2396	−0.623463
$326$	0	0
$327$	0.390109	0.0215731
$328$	0	0
$329$	7.42268	0.409226
$330$	0	0
$331$	−33.5123	−1.84200	−0.921002	−	0.389558i	$-0.872628\pi$
$331$	−33.5123	−1.84200	−0.921002	+	0.389558i	$0.872628\pi$
$332$	0	0
$333$	−11.2134	−0.614489
$334$	0	0
$335$	−4.77309	−0.260782
$336$	0	0
$337$	4.52540	0.246514	0.123257	−	0.992375i	$-0.460666\pi$
$337$	4.52540	0.246514	0.123257	+	0.992375i	$0.460666\pi$
$338$	0	0
$339$	−0.840875	−0.0456701
$340$	0	0
$341$	16.2688	0.881006
$342$	0	0
$343$	−18.7912	−1.01463
$344$	0	0
$345$	−0.239404	−0.0128891
$346$	0	0
$347$	−34.4767	−1.85081	−0.925403	−	0.378985i	$-0.876273\pi$
$347$	−34.4767	−1.85081	−0.925403	+	0.378985i	$0.876273\pi$
$348$	0	0
$349$	15.3968	0.824173	0.412086	−	0.911145i	$-0.364800\pi$
$349$	15.3968	0.824173	0.412086	+	0.911145i	$0.364800\pi$
$350$	0	0
$351$	−1.12253	−0.0599163
$352$	0	0
$353$	−7.39183	−0.393427	−0.196714	−	0.980461i	$-0.563027\pi$
$353$	−7.39183	−0.393427	−0.196714	+	0.980461i	$0.563027\pi$
$354$	0	0
$355$	−1.88626	−0.100112
$356$	0	0
$357$	0.425430	0.0225162
$358$	0	0
$359$	−31.8118	−1.67896	−0.839481	−	0.543389i	$-0.817141\pi$
$359$	−31.8118	−1.67896	−0.839481	+	0.543389i	$0.817141\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−0.543038	−0.0285021
$364$	0	0
$365$	1.32653	0.0694335
$366$	0	0
$367$	−2.60800	−0.136136	−0.0680682	−	0.997681i	$-0.521684\pi$
$367$	−2.60800	−0.136136	−0.0680682	+	0.997681i	$0.521684\pi$
$368$	0	0
$369$	15.4778	0.805743
$370$	0	0
$371$	2.60518	0.135254
$372$	0	0
$373$	0.546587	0.0283012	0.0141506	−	0.999900i	$-0.495496\pi$
$373$	0.546587	0.0283012	0.0141506	+	0.999900i	$0.495496\pi$
$374$	0	0
$375$	−0.347515	−0.0179456
$376$	0	0
$377$	−0.377848	−0.0194602
$378$	0	0
$379$	−34.5170	−1.77302	−0.886510	−	0.462709i	$-0.846878\pi$
$379$	−34.5170	−1.77302	−0.886510	+	0.462709i	$0.846878\pi$
$380$	0	0
$381$	0.169425	0.00867993
$382$	0	0
$383$	10.5252	0.537813	0.268906	−	0.963166i	$-0.413338\pi$
$383$	10.5252	0.537813	0.268906	+	0.963166i	$0.413338\pi$
$384$	0	0
$385$	−2.36874	−0.120722
$386$	0	0
$387$	12.1923	0.619771
$388$	0	0
$389$	−35.1600	−1.78268	−0.891342	−	0.453331i	$-0.850236\pi$
$389$	−35.1600	−1.78268	−0.891342	+	0.453331i	$0.850236\pi$
$390$	0	0
$391$	−13.7782	−0.696795
$392$	0	0
$393$	−0.414216	−0.0208944
$394$	0	0
$395$	0.787094	0.0396030
$396$	0	0
$397$	−28.6583	−1.43832	−0.719158	−	0.694846i	$-0.755472\pi$
$397$	−28.6583	−1.43832	−0.719158	+	0.694846i	$0.755472\pi$
$398$	0	0
$399$	−0.208542	−0.0104401
$400$	0	0
$401$	32.8079	1.63835	0.819175	−	0.573544i	$-0.194432\pi$
$401$	32.8079	1.63835	0.819175	+	0.573544i	$0.194432\pi$
$402$	0	0
$403$	−18.5377	−0.923426
$404$	0	0
$405$	3.95979	0.196764
$406$	0	0
$407$	−7.69134	−0.381245
$408$	0	0
$409$	−16.2118	−0.801620	−0.400810	−	0.916161i	$-0.631271\pi$
$409$	−16.2118	−0.801620	−0.400810	+	0.916161i	$0.631271\pi$
$410$	0	0
$411$	0.927580	0.0457541
$412$	0	0
$413$	21.5220	1.05903
$414$	0	0
$415$	5.81670	0.285530
$416$	0	0
$417$	−1.77554	−0.0869485
$418$	0	0
$419$	−14.8854	−0.727200	−0.363600	−	0.931555i	$-0.618452\pi$
$419$	−14.8854	−0.727200	−0.363600	+	0.931555i	$0.618452\pi$
$420$	0	0
$421$	24.6578	1.20175	0.600874	−	0.799344i	$-0.294819\pi$
$421$	24.6578	1.20175	0.600874	+	0.799344i	$0.294819\pi$
$422$	0	0
$423$	−8.52933	−0.414710
$424$	0	0
$425$	−9.80012	−0.475376
$426$	0	0
$427$	−10.9501	−0.529914
$428$	0	0
$429$	−0.384565	−0.0185670
$430$	0	0
$431$	37.3628	1.79970	0.899850	−	0.436199i	$-0.143676\pi$
$431$	37.3628	1.79970	0.899850	+	0.436199i	$0.143676\pi$
$432$	0	0
$433$	8.55750	0.411247	0.205624	−	0.978631i	$-0.434078\pi$
$433$	8.55750	0.411247	0.205624	+	0.978631i	$0.434078\pi$
$434$	0	0
$435$	−0.00572446	−0.000274467 0
$436$	0	0
$437$	6.75394	0.323085
$438$	0	0
$439$	34.9645	1.66876	0.834382	−	0.551187i	$-0.185825\pi$
$439$	34.9645	1.66876	0.834382	+	0.551187i	$0.185825\pi$
$440$	0	0
$441$	0.637685	0.0303660
$442$	0	0
$443$	22.2168	1.05555	0.527777	−	0.849383i	$-0.323026\pi$
$443$	22.2168	1.05555	0.527777	+	0.849383i	$0.323026\pi$
$444$	0	0
$445$	−3.68226	−0.174556
$446$	0	0
$447$	0.893774	0.0422740
$448$	0	0
$449$	12.2221	0.576795	0.288397	−	0.957511i	$-0.406878\pi$
$449$	12.2221	0.576795	0.288397	+	0.957511i	$0.406878\pi$
$450$	0	0
$451$	10.6163	0.499904
$452$	0	0
$453$	−1.75137	−0.0822866
$454$	0	0
$455$	2.69908	0.126535
$456$	0	0
$457$	19.3167	0.903597	0.451798	−	0.892120i	$-0.350783\pi$
$457$	19.3167	0.903597	0.451798	+	0.892120i	$0.350783\pi$
$458$	0	0
$459$	−0.978763	−0.0456847
$460$	0	0
$461$	7.75002	0.360954	0.180477	−	0.983579i	$-0.442236\pi$
$461$	7.75002	0.360954	0.180477	+	0.983579i	$0.442236\pi$
$462$	0	0
$463$	−12.6871	−0.589622	−0.294811	−	0.955556i	$-0.595257\pi$
$463$	−12.6871	−0.589622	−0.294811	+	0.955556i	$0.595257\pi$
$464$	0	0
$465$	−0.280848	−0.0130240
$466$	0	0
$467$	9.81309	0.454095	0.227048	−	0.973884i	$-0.427093\pi$
$467$	9.81309	0.454095	0.227048	+	0.973884i	$0.427093\pi$
$468$	0	0
$469$	−28.0814	−1.29668
$470$	0	0
$471$	−0.573123	−0.0264081
$472$	0	0
$473$	8.36281	0.384522
$474$	0	0
$475$	4.80392	0.220419
$476$	0	0
$477$	−2.99359	−0.137067
$478$	0	0
$479$	−29.7465	−1.35915	−0.679576	−	0.733605i	$-0.737836\pi$
$479$	−29.7465	−1.35915	−0.679576	+	0.733605i	$0.737836\pi$
$480$	0	0
$481$	8.76396	0.399602
$482$	0	0
$483$	−1.40848	−0.0640879
$484$	0	0
$485$	−5.82098	−0.264317
$486$	0	0
$487$	20.1548	0.913299	0.456650	−	0.889647i	$-0.349049\pi$
$487$	20.1548	0.913299	0.456650	+	0.889647i	$0.349049\pi$
$488$	0	0
$489$	1.37424	0.0621454
$490$	0	0
$491$	−25.9467	−1.17096	−0.585480	−	0.810687i	$-0.699094\pi$
$491$	−25.9467	−1.17096	−0.585480	+	0.810687i	$0.699094\pi$
$492$	0	0
$493$	−0.329455	−0.0148379
$494$	0	0
$495$	2.72189	0.122340
$496$	0	0
$497$	−11.0973	−0.497784
$498$	0	0
$499$	41.1611	1.84262	0.921312	−	0.388823i	$-0.127118\pi$
$499$	41.1611	1.84262	0.921312	+	0.388823i	$0.127118\pi$
$500$	0	0
$501$	1.38327	0.0617997
$502$	0	0
$503$	−0.0598213	−0.00266730	−0.00133365	−	0.999999i	$-0.500425\pi$
$503$	−0.0598213	−0.00266730	−0.00133365	+	0.999999i	$0.500425\pi$
$504$	0	0
$505$	0.00762684	0.000339390 0
$506$	0	0
$507$	−0.602437	−0.0267552
$508$	0	0
$509$	40.9387	1.81458	0.907289	−	0.420508i	$-0.138148\pi$
$509$	40.9387	1.81458	0.907289	+	0.420508i	$0.138148\pi$
$510$	0	0
$511$	7.80430	0.345242
$512$	0	0
$513$	0.479779	0.0211828
$514$	0	0
$515$	1.91266	0.0842820
$516$	0	0
$517$	−5.85033	−0.257297
$518$	0	0
$519$	1.05533	0.0463237
$520$	0	0
$521$	41.3009	1.80943	0.904714	−	0.426021i	$-0.140085\pi$
$521$	41.3009	1.80943	0.904714	+	0.426021i	$0.140085\pi$
$522$	0	0
$523$	−2.06733	−0.0903981	−0.0451991	−	0.998978i	$-0.514392\pi$
$523$	−2.06733	−0.0903981	−0.0451991	+	0.998978i	$0.514392\pi$
$524$	0	0
$525$	−1.00182	−0.0437228
$526$	0	0
$527$	−16.1634	−0.704090
$528$	0	0
$529$	22.6157	0.983293
$530$	0	0
$531$	−24.7307	−1.07322
$532$	0	0
$533$	−12.0969	−0.523975
$534$	0	0
$535$	−6.00520	−0.259627
$536$	0	0
$537$	0.400893	0.0172998
$538$	0	0
$539$	0.437393	0.0188399
$540$	0	0
$541$	−24.6525	−1.05989	−0.529946	−	0.848031i	$-0.677788\pi$
$541$	−24.6525	−1.05989	−0.529946	+	0.848031i	$0.677788\pi$
$542$	0	0
$543$	0.172270	0.00739282
$544$	0	0
$545$	2.15800	0.0924387
$546$	0	0
$547$	−15.6190	−0.667819	−0.333909	−	0.942605i	$-0.608368\pi$
$547$	−15.6190	−0.667819	−0.333909	+	0.942605i	$0.608368\pi$
$548$	0	0
$549$	12.5827	0.537016
$550$	0	0
$551$	0.161496	0.00687994
$552$	0	0
$553$	4.63068	0.196917
$554$	0	0
$555$	0.132775	0.00563600
$556$	0	0
$557$	28.0983	1.19056	0.595281	−	0.803518i	$-0.297041\pi$
$557$	28.0983	1.19056	0.595281	+	0.803518i	$0.297041\pi$
$558$	0	0
$559$	−9.52908	−0.403037
$560$	0	0
$561$	−0.335311	−0.0141569
$562$	0	0
$563$	1.62749	0.0685904	0.0342952	−	0.999412i	$-0.489081\pi$
$563$	1.62749	0.0685904	0.0342952	+	0.999412i	$0.489081\pi$
$564$	0	0
$565$	−4.65155	−0.195692
$566$	0	0
$567$	23.2965	0.978362
$568$	0	0
$569$	−26.9361	−1.12922	−0.564610	−	0.825358i	$-0.690973\pi$
$569$	−26.9361	−1.12922	−0.564610	+	0.825358i	$0.690973\pi$
$570$	0	0
$571$	−39.2736	−1.64355	−0.821775	−	0.569811i	$-0.807016\pi$
$571$	−39.2736	−1.64355	−0.821775	+	0.569811i	$0.807016\pi$
$572$	0	0
$573$	1.75880	0.0734748
$574$	0	0
$575$	32.4454	1.35307
$576$	0	0
$577$	14.9871	0.623923	0.311961	−	0.950095i	$-0.399014\pi$
$577$	14.9871	0.623923	0.311961	+	0.950095i	$0.399014\pi$
$578$	0	0
$579$	0.303076	0.0125954
$580$	0	0
$581$	34.2212	1.41973
$582$	0	0
$583$	−2.05333	−0.0850401
$584$	0	0
$585$	−3.10148	−0.128231
$586$	0	0
$587$	38.9297	1.60680	0.803400	−	0.595439i	$-0.203022\pi$
$587$	38.9297	1.60680	0.803400	+	0.595439i	$0.203022\pi$
$588$	0	0
$589$	7.92315	0.326468
$590$	0	0
$591$	0.564716	0.0232293
$592$	0	0
$593$	−23.6009	−0.969174	−0.484587	−	0.874743i	$-0.661030\pi$
$593$	−23.6009	−0.969174	−0.484587	+	0.874743i	$0.661030\pi$
$594$	0	0
$595$	2.35339	0.0964797
$596$	0	0
$597$	−1.38238	−0.0565769
$598$	0	0
$599$	−36.9916	−1.51144	−0.755718	−	0.654897i	$-0.772712\pi$
$599$	−36.9916	−1.51144	−0.755718	+	0.654897i	$0.772712\pi$
$600$	0	0
$601$	39.7323	1.62071	0.810357	−	0.585936i	$-0.199273\pi$
$601$	39.7323	1.62071	0.810357	+	0.585936i	$0.199273\pi$
$602$	0	0
$603$	32.2680	1.31406
$604$	0	0
$605$	−3.00397	−0.122129
$606$	0	0
$607$	42.9832	1.74464	0.872318	−	0.488939i	$-0.162616\pi$
$607$	42.9832	1.74464	0.872318	+	0.488939i	$0.162616\pi$
$608$	0	0
$609$	−0.0336785	−0.00136472
$610$	0	0
$611$	6.66622	0.269686
$612$	0	0
$613$	20.2633	0.818428	0.409214	−	0.912438i	$-0.365803\pi$
$613$	20.2633	0.818428	0.409214	+	0.912438i	$0.365803\pi$
$614$	0	0
$615$	−0.183270	−0.00739015
$616$	0	0
$617$	43.4290	1.74839	0.874193	−	0.485579i	$-0.161391\pi$
$617$	43.4290	1.74839	0.874193	+	0.485579i	$0.161391\pi$
$618$	0	0
$619$	−28.6938	−1.15330	−0.576651	−	0.816991i	$-0.695641\pi$
$619$	−28.6938	−1.15330	−0.576651	+	0.816991i	$0.695641\pi$
$620$	0	0
$621$	3.24040	0.130033
$622$	0	0
$623$	−21.6637	−0.867938
$624$	0	0
$625$	22.0972	0.883888
$626$	0	0
$627$	0.164366	0.00656415
$628$	0	0
$629$	7.64152	0.304687
$630$	0	0
$631$	34.0787	1.35665	0.678326	−	0.734761i	$-0.262706\pi$
$631$	34.0787	1.35665	0.678326	+	0.734761i	$0.262706\pi$
$632$	0	0
$633$	−1.88575	−0.0749520
$634$	0	0
$635$	0.937226	0.0371927
$636$	0	0
$637$	−0.498392	−0.0197470
$638$	0	0
$639$	12.7519	0.504456
$640$	0	0
$641$	49.8468	1.96883	0.984414	−	0.175864i	$-0.0562720\pi$
$641$	49.8468	1.96883	0.984414	+	0.175864i	$0.0562720\pi$
$642$	0	0
$643$	−35.7487	−1.40979	−0.704896	−	0.709311i	$-0.749006\pi$
$643$	−35.7487	−1.40979	−0.704896	+	0.709311i	$0.749006\pi$
$644$	0	0
$645$	−0.144367	−0.00568445
$646$	0	0
$647$	−26.6225	−1.04664	−0.523319	−	0.852137i	$-0.675307\pi$
$647$	−26.6225	−1.04664	−0.523319	+	0.852137i	$0.675307\pi$
$648$	0	0
$649$	−16.9630	−0.665855
$650$	0	0
$651$	−1.65231	−0.0647589
$652$	0	0
$653$	−16.5839	−0.648977	−0.324488	−	0.945890i	$-0.605192\pi$
$653$	−16.5839	−0.648977	−0.324488	+	0.945890i	$0.605192\pi$
$654$	0	0
$655$	−2.29136	−0.0895307
$656$	0	0
$657$	−8.96785	−0.349869
$658$	0	0
$659$	19.7302	0.768580	0.384290	−	0.923212i	$-0.374446\pi$
$659$	19.7302	0.768580	0.384290	+	0.923212i	$0.374446\pi$
$660$	0	0
$661$	24.3031	0.945283	0.472641	−	0.881255i	$-0.343301\pi$
$661$	24.3031	0.945283	0.472641	+	0.881255i	$0.343301\pi$
$662$	0	0
$663$	0.382074	0.0148385
$664$	0	0
$665$	−1.15361	−0.0447350
$666$	0	0
$667$	1.09073	0.0422333
$668$	0	0
$669$	1.38326	0.0534798
$670$	0	0
$671$	8.63056	0.333179
$672$	0	0
$673$	−8.92628	−0.344083	−0.172041	−	0.985090i	$-0.555036\pi$
$673$	−8.92628	−0.344083	−0.172041	+	0.985090i	$0.555036\pi$
$674$	0	0
$675$	2.30482	0.0887125
$676$	0	0
$677$	−26.3563	−1.01295	−0.506477	−	0.862254i	$-0.669052\pi$
$677$	−26.3563	−1.01295	−0.506477	+	0.862254i	$0.669052\pi$
$678$	0	0
$679$	−34.2463	−1.31425
$680$	0	0
$681$	−0.0142111	−0.000544569 0
$682$	0	0
$683$	4.89120	0.187157	0.0935783	−	0.995612i	$-0.470169\pi$
$683$	4.89120	0.187157	0.0935783	+	0.995612i	$0.470169\pi$
$684$	0	0
$685$	5.13118	0.196052
$686$	0	0
$687$	−0.947488	−0.0361489
$688$	0	0
$689$	2.33968	0.0891348
$690$	0	0
$691$	18.2422	0.693965	0.346982	−	0.937872i	$-0.387206\pi$
$691$	18.2422	0.693965	0.346982	+	0.937872i	$0.387206\pi$
$692$	0	0
$693$	16.0136	0.608307
$694$	0	0
$695$	−9.82191	−0.372566
$696$	0	0
$697$	−10.5476	−0.399518
$698$	0	0
$699$	1.72166	0.0651190
$700$	0	0
$701$	33.4734	1.26427	0.632136	−	0.774857i	$-0.282178\pi$
$701$	33.4734	1.26427	0.632136	+	0.774857i	$0.282178\pi$
$702$	0	0
$703$	−3.74579	−0.141275
$704$	0	0
$705$	0.100994	0.00380366
$706$	0	0
$707$	0.0448707	0.00168754
$708$	0	0
$709$	7.77292	0.291918	0.145959	−	0.989291i	$-0.453373\pi$
$709$	7.77292	0.291918	0.145959	+	0.989291i	$0.453373\pi$
$710$	0	0
$711$	−5.32108	−0.199556
$712$	0	0
$713$	53.5125	2.00406
$714$	0	0
$715$	−2.12733	−0.0795577
$716$	0	0
$717$	−0.702947	−0.0262520
$718$	0	0
$719$	−6.23898	−0.232675	−0.116337	−	0.993210i	$-0.537115\pi$
$719$	−6.23898	−0.232675	−0.116337	+	0.993210i	$0.537115\pi$
$720$	0	0
$721$	11.2527	0.419072
$722$	0	0
$723$	−1.69859	−0.0631711
$724$	0	0
$725$	0.775811	0.0288129
$726$	0	0
$727$	14.6584	0.543651	0.271826	−	0.962347i	$-0.412373\pi$
$727$	14.6584	0.543651	0.271826	+	0.962347i	$0.412373\pi$
$728$	0	0
$729$	−26.6546	−0.987207
$730$	0	0
$731$	−8.30864	−0.307306
$732$	0	0
$733$	6.06986	0.224195	0.112098	−	0.993697i	$-0.464243\pi$
$733$	6.06986	0.224195	0.112098	+	0.993697i	$0.464243\pi$
$734$	0	0
$735$	−0.00755071	−0.000278512 0
$736$	0	0
$737$	22.1329	0.815275
$738$	0	0
$739$	4.49102	0.165205	0.0826025	−	0.996583i	$-0.473677\pi$
$739$	4.49102	0.165205	0.0826025	+	0.996583i	$0.473677\pi$
$740$	0	0
$741$	−0.187289	−0.00688022
$742$	0	0
$743$	−10.7290	−0.393607	−0.196804	−	0.980443i	$-0.563056\pi$
$743$	−10.7290	−0.393607	−0.196804	+	0.980443i	$0.563056\pi$
$744$	0	0
$745$	4.94417	0.181140
$746$	0	0
$747$	−39.3232	−1.43876
$748$	0	0
$749$	−35.3302	−1.29094
$750$	0	0
$751$	48.8939	1.78416	0.892081	−	0.451875i	$-0.149245\pi$
$751$	48.8939	1.78416	0.892081	+	0.451875i	$0.149245\pi$
$752$	0	0
$753$	−2.44556	−0.0891213
$754$	0	0
$755$	−9.68822	−0.352591
$756$	0	0
$757$	−11.7495	−0.427041	−0.213521	−	0.976939i	$-0.568493\pi$
$757$	−11.7495	−0.427041	−0.213521	+	0.976939i	$0.568493\pi$
$758$	0	0
$759$	1.11012	0.0402948
$760$	0	0
$761$	−9.70454	−0.351789	−0.175895	−	0.984409i	$-0.556282\pi$
$761$	−9.70454	−0.351789	−0.175895	+	0.984409i	$0.556282\pi$
$762$	0	0
$763$	12.6961	0.459630
$764$	0	0
$765$	−2.70426	−0.0977728
$766$	0	0
$767$	19.3286	0.697916
$768$	0	0
$769$	−14.8869	−0.536834	−0.268417	−	0.963303i	$-0.586500\pi$
$769$	−14.8869	−0.536834	−0.268417	+	0.963303i	$0.586500\pi$
$770$	0	0
$771$	−0.904114	−0.0325609
$772$	0	0
$773$	−2.74773	−0.0988289	−0.0494145	−	0.998778i	$-0.515736\pi$
$773$	−2.74773	−0.0988289	−0.0494145	+	0.998778i	$0.515736\pi$
$774$	0	0
$775$	38.0621	1.36723
$776$	0	0
$777$	0.781153	0.0280237
$778$	0	0
$779$	5.17031	0.185246
$780$	0	0
$781$	8.74660	0.312978
$782$	0	0
$783$	0.0774822	0.00276899
$784$	0	0
$785$	−3.17040	−0.113156
$786$	0	0
$787$	−4.24788	−0.151421	−0.0757103	−	0.997130i	$-0.524122\pi$
$787$	−4.24788	−0.151421	−0.0757103	+	0.997130i	$0.524122\pi$
$788$	0	0
$789$	−0.00620377	−0.000220860 0
$790$	0	0
$791$	−27.3663	−0.973033
$792$	0	0
$793$	−9.83417	−0.349222
$794$	0	0
$795$	0.0354466	0.00125716
$796$	0	0
$797$	3.42079	0.121171	0.0605854	−	0.998163i	$-0.480703\pi$
$797$	3.42079	0.121171	0.0605854	+	0.998163i	$0.480703\pi$
$798$	0	0
$799$	5.81244	0.205629
$800$	0	0
$801$	24.8936	0.879571
$802$	0	0
$803$	−6.15111	−0.217068
$804$	0	0
$805$	−7.79140	−0.274611
$806$	0	0
$807$	−1.72878	−0.0608560
$808$	0	0
$809$	20.2668	0.712543	0.356272	−	0.934382i	$-0.384048\pi$
$809$	20.2668	0.712543	0.356272	+	0.934382i	$0.384048\pi$
$810$	0	0
$811$	−16.6631	−0.585121	−0.292561	−	0.956247i	$-0.594507\pi$
$811$	−16.6631	−0.585121	−0.292561	+	0.956247i	$0.594507\pi$
$812$	0	0
$813$	0.482829	0.0169335
$814$	0	0
$815$	7.60202	0.266287
$816$	0	0
$817$	4.07281	0.142489
$818$	0	0
$819$	−18.2469	−0.637597
$820$	0	0
$821$	40.5886	1.41655	0.708277	−	0.705935i	$-0.249473\pi$
$821$	40.5886	1.41655	0.708277	+	0.705935i	$0.249473\pi$
$822$	0	0
$823$	28.3801	0.989267	0.494634	−	0.869102i	$-0.335302\pi$
$823$	28.3801	0.989267	0.494634	+	0.869102i	$0.335302\pi$
$824$	0	0
$825$	0.789601	0.0274904
$826$	0	0
$827$	−38.6329	−1.34340	−0.671699	−	0.740824i	$-0.734435\pi$
$827$	−38.6329	−1.34340	−0.671699	+	0.740824i	$0.734435\pi$
$828$	0	0
$829$	−24.0546	−0.835449	−0.417725	−	0.908574i	$-0.637172\pi$
$829$	−24.0546	−0.835449	−0.417725	+	0.908574i	$0.637172\pi$
$830$	0	0
$831$	2.22161	0.0770669
$832$	0	0
$833$	−0.434560	−0.0150566
$834$	0	0
$835$	7.65194	0.264806
$836$	0	0
$837$	3.80136	0.131394
$838$	0	0
$839$	−12.5322	−0.432660	−0.216330	−	0.976320i	$-0.569409\pi$
$839$	−12.5322	−0.432660	−0.216330	+	0.976320i	$0.569409\pi$
$840$	0	0
$841$	−28.9739	−0.999101
$842$	0	0
$843$	0.997871	0.0343685
$844$	0	0
$845$	−3.33256	−0.114643
$846$	0	0
$847$	−17.6732	−0.607257
$848$	0	0
$849$	2.29029	0.0786024
$850$	0	0
$851$	−25.2989	−0.867234
$852$	0	0
$853$	18.0175	0.616908	0.308454	−	0.951239i	$-0.400188\pi$
$853$	18.0175	0.616908	0.308454	+	0.951239i	$0.400188\pi$
$854$	0	0
$855$	1.32560	0.0453346
$856$	0	0
$857$	−27.4522	−0.937748	−0.468874	−	0.883265i	$-0.655340\pi$
$857$	−27.4522	−0.937748	−0.468874	+	0.883265i	$0.655340\pi$
$858$	0	0
$859$	35.8818	1.22427	0.612136	−	0.790753i	$-0.290311\pi$
$859$	35.8818	1.22427	0.612136	+	0.790753i	$0.290311\pi$
$860$	0	0
$861$	−1.07823	−0.0367458
$862$	0	0
$863$	35.0943	1.19463	0.597313	−	0.802008i	$-0.296235\pi$
$863$	35.0943	1.19463	0.597313	+	0.802008i	$0.296235\pi$
$864$	0	0
$865$	5.83785	0.198493
$866$	0	0
$867$	−1.02769	−0.0349021
$868$	0	0
$869$	−3.64977	−0.123810
$870$	0	0
$871$	−25.2195	−0.854531
$872$	0	0
$873$	39.3522	1.33187
$874$	0	0
$875$	−11.3099	−0.382344
$876$	0	0
$877$	32.5063	1.09766	0.548829	−	0.835934i	$-0.315074\pi$
$877$	32.5063	1.09766	0.548829	+	0.835934i	$0.315074\pi$
$878$	0	0
$879$	0.0989537	0.00333763
$880$	0	0
$881$	−28.6801	−0.966259	−0.483129	−	0.875549i	$-0.660500\pi$
$881$	−28.6801	−0.966259	−0.483129	+	0.875549i	$0.660500\pi$
$882$	0	0
$883$	−6.16251	−0.207385	−0.103692	−	0.994609i	$-0.533066\pi$
$883$	−6.16251	−0.207385	−0.103692	+	0.994609i	$0.533066\pi$
$884$	0	0
$885$	0.292832	0.00984343
$886$	0	0
$887$	0.262178	0.00880309	0.00440154	−	0.999990i	$-0.498599\pi$
$887$	0.262178	0.00880309	0.00440154	+	0.999990i	$0.498599\pi$
$888$	0	0
$889$	5.51395	0.184932
$890$	0	0
$891$	−18.3616	−0.615137
$892$	0	0
$893$	−2.84920	−0.0953447
$894$	0	0
$895$	2.21765	0.0741280
$896$	0	0
$897$	−1.26494	−0.0422350
$898$	0	0
$899$	1.27955	0.0426755
$900$	0	0
$901$	2.04003	0.0679632
$902$	0	0
$903$	−0.849349	−0.0282646
$904$	0	0
$905$	0.952963	0.0316775
$906$	0	0
$907$	38.5866	1.28125	0.640624	−	0.767855i	$-0.278676\pi$
$907$	38.5866	1.28125	0.640624	+	0.767855i	$0.278676\pi$
$908$	0	0
$909$	−0.0515605	−0.00171015
$910$	0	0
$911$	11.7598	0.389619	0.194810	−	0.980841i	$-0.437591\pi$
$911$	11.7598	0.389619	0.194810	+	0.980841i	$0.437591\pi$
$912$	0	0
$913$	−26.9721	−0.892646
$914$	0	0
$915$	−0.148989	−0.00492543
$916$	0	0
$917$	−13.4807	−0.445170
$918$	0	0
$919$	−11.4742	−0.378500	−0.189250	−	0.981929i	$-0.560606\pi$
$919$	−11.4742	−0.378500	−0.189250	+	0.981929i	$0.560606\pi$
$920$	0	0
$921$	−2.52848	−0.0833162
$922$	0	0
$923$	−9.96639	−0.328048
$924$	0	0
$925$	−17.9945	−0.591655
$926$	0	0
$927$	−12.9304	−0.424689
$928$	0	0
$929$	26.1220	0.857035	0.428517	−	0.903534i	$-0.359036\pi$
$929$	26.1220	0.857035	0.428517	+	0.903534i	$0.359036\pi$
$930$	0	0
$931$	0.213017	0.00698134
$932$	0	0
$933$	−2.08988	−0.0684195
$934$	0	0
$935$	−1.85487	−0.0606608
$936$	0	0
$937$	3.58835	0.117226	0.0586131	−	0.998281i	$-0.481332\pi$
$937$	3.58835	0.117226	0.0586131	+	0.998281i	$0.481332\pi$
$938$	0	0
$939$	0.959646	0.0313169
$940$	0	0
$941$	−55.1725	−1.79857	−0.899287	−	0.437359i	$-0.855914\pi$
$941$	−55.1725	−1.79857	−0.899287	+	0.437359i	$0.855914\pi$
$942$	0	0
$943$	34.9200	1.13715
$944$	0	0
$945$	−0.553477	−0.0180046
$946$	0	0
$947$	26.9388	0.875393	0.437697	−	0.899123i	$-0.355794\pi$
$947$	26.9388	0.875393	0.437697	+	0.899123i	$0.355794\pi$
$948$	0	0
$949$	7.00894	0.227520
$950$	0	0
$951$	0.903333	0.0292926
$952$	0	0
$953$	−41.4776	−1.34359	−0.671796	−	0.740737i	$-0.734477\pi$
$953$	−41.4776	−1.34359	−0.671796	+	0.740737i	$0.734477\pi$
$954$	0	0
$955$	9.72930	0.314833
$956$	0	0
$957$	0.0265444	0.000858059 0
$958$	0	0
$959$	30.1881	0.974824
$960$	0	0
$961$	31.7763	1.02504
$962$	0	0
$963$	40.5975	1.30824
$964$	0	0
$965$	1.67655	0.0539701
$966$	0	0
$967$	−32.2835	−1.03817	−0.519084	−	0.854723i	$-0.673727\pi$
$967$	−32.2835	−1.03817	−0.519084	+	0.854723i	$0.673727\pi$
$968$	0	0
$969$	−0.163302	−0.00524600
$970$	0	0
$971$	1.53014	0.0491045	0.0245522	−	0.999699i	$-0.492184\pi$
$971$	1.53014	0.0491045	0.0245522	+	0.999699i	$0.492184\pi$
$972$	0	0
$973$	−57.7849	−1.85250
$974$	0	0
$975$	−0.899719	−0.0288141
$976$	0	0
$977$	46.5952	1.49071	0.745357	−	0.666666i	$-0.232279\pi$
$977$	46.5952	1.49071	0.745357	+	0.666666i	$0.232279\pi$
$978$	0	0
$979$	17.0747	0.545709
$980$	0	0
$981$	−14.5890	−0.465790
$982$	0	0
$983$	20.0055	0.638076	0.319038	−	0.947742i	$-0.396640\pi$
$983$	20.0055	0.638076	0.319038	+	0.947742i	$0.396640\pi$
$984$	0	0
$985$	3.12389	0.0995354
$986$	0	0
$987$	0.594176	0.0189128
$988$	0	0
$989$	27.5075	0.874688
$990$	0	0
$991$	8.00418	0.254261	0.127131	−	0.991886i	$-0.459423\pi$
$991$	8.00418	0.254261	0.127131	+	0.991886i	$0.459423\pi$
$992$	0	0
$993$	−2.68262	−0.0851303
$994$	0	0
$995$	−7.64702	−0.242427
$996$	0	0
$997$	−3.35634	−0.106296	−0.0531482	−	0.998587i	$-0.516926\pi$
$997$	−3.35634	−0.106296	−0.0531482	+	0.998587i	$0.516926\pi$
$998$	0	0
$999$	−1.79715	−0.0568594

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4028.2.a.d.1.10	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4028.2.a.d.1.10	✓	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)

\(f(q)\)	\(=\)	\(q+0.0800487 q^{3} +0.442813 q^{5} +2.60518 q^{7} -2.99359 q^{9} +O(q^{10})\)	\(q+0.0800487 q^{3} +0.442813 q^{5} +2.60518 q^{7} -2.99359 q^{9} -2.05333 q^{11} +2.33968 q^{13} +0.0354466 q^{15} +2.04003 q^{17} -1.00000 q^{19} +0.208542 q^{21} -6.75394 q^{23} -4.80392 q^{25} -0.479779 q^{27} -0.161496 q^{29} -7.92315 q^{31} -0.164366 q^{33} +1.15361 q^{35} +3.74579 q^{37} +0.187289 q^{39} -5.17031 q^{41} -4.07281 q^{43} -1.32560 q^{45} +2.84920 q^{47} -0.213017 q^{49} +0.163302 q^{51} +1.00000 q^{53} -0.909239 q^{55} -0.0800487 q^{57} +8.26122 q^{59} -4.20321 q^{61} -7.79886 q^{63} +1.03604 q^{65} -10.7790 q^{67} -0.540644 q^{69} -4.25972 q^{71} +2.99568 q^{73} -0.384547 q^{75} -5.34930 q^{77} +1.77749 q^{79} +8.94237 q^{81} +13.1358 q^{83} +0.903350 q^{85} -0.0129275 q^{87} -8.31561 q^{89} +6.09530 q^{91} -0.634237 q^{93} -0.442813 q^{95} -13.1455 q^{97} +6.14683 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 4 q^{3} - 2 q^{5} - 11 q^{7} + 19 q^{9}+O(q^{10}) \) 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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