LMFDB - Embedded newform 6040.2.a.l.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6040,2,Mod(1,6040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6040, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6040.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6040 = 2^{3} \cdot 5 \cdot 151 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6040.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:	$48.2296428209$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 11x^{7} + 9x^{6} + 32x^{5} - 17x^{4} - 27x^{3} + 10x^{2} + 3x - 1 $ x^9 - x^8 - 11x^7 + 9x^6 + 32x^5 - 17x^4 - 27x^3 + 10x^2 + 3*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.41589$ of defining polynomial
Character	$\chi$	$=$	6040.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.0722232 q^{3} +1.00000 q^{5} -0.709626 q^{7} -2.99478 q^{9} +O(q^{10})$	$q+0.0722232 q^{3} +1.00000 q^{5} -0.709626 q^{7} -2.99478 q^{9} -0.425521 q^{11} +6.40956 q^{13} +0.0722232 q^{15} -4.34731 q^{17} +2.76398 q^{19} -0.0512515 q^{21} -2.37546 q^{23} +1.00000 q^{25} -0.432962 q^{27} -1.12262 q^{29} -1.10634 q^{31} -0.0307325 q^{33} -0.709626 q^{35} -7.75024 q^{37} +0.462919 q^{39} -4.05857 q^{41} -9.49867 q^{43} -2.99478 q^{45} +4.25704 q^{47} -6.49643 q^{49} -0.313977 q^{51} -5.36390 q^{53} -0.425521 q^{55} +0.199623 q^{57} +7.25935 q^{59} -3.77690 q^{61} +2.12518 q^{63} +6.40956 q^{65} +12.7000 q^{67} -0.171563 q^{69} -6.91929 q^{71} +11.6280 q^{73} +0.0722232 q^{75} +0.301961 q^{77} +10.0703 q^{79} +8.95308 q^{81} -3.42260 q^{83} -4.34731 q^{85} -0.0810794 q^{87} +1.05887 q^{89} -4.54839 q^{91} -0.0799037 q^{93} +2.76398 q^{95} -4.57137 q^{97} +1.27434 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 9 q^{5} - 2 q^{7} - 3 q^{9}+O(q^{10}) $ 9 * q + 9 * q^5 - 2 * q^7 - 3 * q^9	$ 9 q + 9 q^{5} - 2 q^{7} - 3 q^{9} - 6 q^{11} - 9 q^{13} - 2 q^{17} - 10 q^{19} - 9 q^{21} - 6 q^{23} + 9 q^{25} + 12 q^{27} - 6 q^{29} + 9 q^{31} - 11 q^{33} - 2 q^{35} - 12 q^{37} - 3 q^{39} - 20 q^{41} + q^{43} - 3 q^{45} + 22 q^{47} - 29 q^{49} + 2 q^{51} - 35 q^{53} - 6 q^{55} - 20 q^{57} + 14 q^{59} - 22 q^{61} - 12 q^{63} - 9 q^{65} + 4 q^{67} + 5 q^{69} - 22 q^{71} - 34 q^{73} - 5 q^{77} + 8 q^{79} - 31 q^{81} - 3 q^{83} - 2 q^{85} - 5 q^{89} - 7 q^{91} - 21 q^{93} - 10 q^{95} - 33 q^{97} - 15 q^{99}+O(q^{100}) $ 9 * q + 9 * q^5 - 2 * q^7 - 3 * q^9 - 6 * q^11 - 9 * q^13 - 2 * q^17 - 10 * q^19 - 9 * q^21 - 6 * q^23 + 9 * q^25 + 12 * q^27 - 6 * q^29 + 9 * q^31 - 11 * q^33 - 2 * q^35 - 12 * q^37 - 3 * q^39 - 20 * q^41 + q^43 - 3 * q^45 + 22 * q^47 - 29 * q^49 + 2 * q^51 - 35 * q^53 - 6 * q^55 - 20 * q^57 + 14 * q^59 - 22 * q^61 - 12 * q^63 - 9 * q^65 + 4 * q^67 + 5 * q^69 - 22 * q^71 - 34 * q^73 - 5 * q^77 + 8 * q^79 - 31 * q^81 - 3 * q^83 - 2 * q^85 - 5 * q^89 - 7 * q^91 - 21 * q^93 - 10 * q^95 - 33 * q^97 - 15 * 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
$3$	0.0722232	0.0416981	0.0208490	−	0.999783i	$-0.493363\pi$
$3$	0.0722232	0.0416981	0.0208490	+	0.999783i	$0.493363\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−0.709626	−0.268213	−0.134107	−	0.990967i	$-0.542816\pi$
$7$	−0.709626	−0.268213	−0.134107	+	0.990967i	$0.542816\pi$
$8$	0	0
$9$	−2.99478	−0.998261
$10$	0	0
$11$	−0.425521	−0.128299	−0.0641497	−	0.997940i	$-0.520434\pi$
$11$	−0.425521	−0.128299	−0.0641497	+	0.997940i	$0.520434\pi$
$12$	0	0
$13$	6.40956	1.77769	0.888846	−	0.458205i	$-0.151508\pi$
$13$	6.40956	1.77769	0.888846	+	0.458205i	$0.151508\pi$
$14$	0	0
$15$	0.0722232	0.0186479
$16$	0	0
$17$	−4.34731	−1.05438	−0.527189	−	0.849748i	$-0.676754\pi$
$17$	−4.34731	−1.05438	−0.527189	+	0.849748i	$0.676754\pi$
$18$	0	0
$19$	2.76398	0.634099	0.317050	−	0.948409i	$-0.397308\pi$
$19$	2.76398	0.634099	0.317050	+	0.948409i	$0.397308\pi$
$20$	0	0
$21$	−0.0512515	−0.0111840
$22$	0	0
$23$	−2.37546	−0.495317	−0.247658	−	0.968847i	$-0.579661\pi$
$23$	−2.37546	−0.495317	−0.247658	+	0.968847i	$0.579661\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−0.432962	−0.0833237
$28$	0	0
$29$	−1.12262	−0.208466	−0.104233	−	0.994553i	$-0.533239\pi$
$29$	−1.12262	−0.208466	−0.104233	+	0.994553i	$0.533239\pi$
$30$	0	0
$31$	−1.10634	−0.198705	−0.0993526	−	0.995052i	$-0.531677\pi$
$31$	−1.10634	−0.198705	−0.0993526	+	0.995052i	$0.531677\pi$
$32$	0	0
$33$	−0.0307325	−0.00534984
$34$	0	0
$35$	−0.709626	−0.119949
$36$	0	0
$37$	−7.75024	−1.27413	−0.637065	−	0.770810i	$-0.719852\pi$
$37$	−7.75024	−1.27413	−0.637065	+	0.770810i	$0.719852\pi$
$38$	0	0
$39$	0.462919	0.0741264
$40$	0	0
$41$	−4.05857	−0.633842	−0.316921	−	0.948452i	$-0.602649\pi$
$41$	−4.05857	−0.633842	−0.316921	+	0.948452i	$0.602649\pi$
$42$	0	0
$43$	−9.49867	−1.44853	−0.724267	−	0.689520i	$-0.757822\pi$
$43$	−9.49867	−1.44853	−0.724267	+	0.689520i	$0.757822\pi$
$44$	0	0
$45$	−2.99478	−0.446436
$46$	0	0
$47$	4.25704	0.620952	0.310476	−	0.950581i	$-0.399511\pi$
$47$	4.25704	0.620952	0.310476	+	0.950581i	$0.399511\pi$
$48$	0	0
$49$	−6.49643	−0.928062
$50$	0	0
$51$	−0.313977	−0.0439655
$52$	0	0
$53$	−5.36390	−0.736789	−0.368395	−	0.929670i	$-0.620092\pi$
$53$	−5.36390	−0.736789	−0.368395	+	0.929670i	$0.620092\pi$
$54$	0	0
$55$	−0.425521	−0.0573773
$56$	0	0
$57$	0.199623	0.0264407
$58$	0	0
$59$	7.25935	0.945087	0.472544	−	0.881307i	$-0.343336\pi$
$59$	7.25935	0.945087	0.472544	+	0.881307i	$0.343336\pi$
$60$	0	0
$61$	−3.77690	−0.483582	−0.241791	−	0.970328i	$-0.577735\pi$
$61$	−3.77690	−0.483582	−0.241791	+	0.970328i	$0.577735\pi$
$62$	0	0
$63$	2.12518	0.267747
$64$	0	0
$65$	6.40956	0.795008
$66$	0	0
$67$	12.7000	1.55155	0.775773	−	0.631012i	$-0.217360\pi$
$67$	12.7000	1.55155	0.775773	+	0.631012i	$0.217360\pi$
$68$	0	0
$69$	−0.171563	−0.0206538
$70$	0	0
$71$	−6.91929	−0.821169	−0.410584	−	0.911823i	$-0.634675\pi$
$71$	−6.91929	−0.821169	−0.410584	+	0.911823i	$0.634675\pi$
$72$	0	0
$73$	11.6280	1.36096	0.680478	−	0.732768i	$-0.261772\pi$
$73$	11.6280	1.36096	0.680478	+	0.732768i	$0.261772\pi$
$74$	0	0
$75$	0.0722232	0.00833962
$76$	0	0
$77$	0.301961	0.0344116
$78$	0	0
$79$	10.0703	1.13299	0.566497	−	0.824064i	$-0.308298\pi$
$79$	10.0703	1.13299	0.566497	+	0.824064i	$0.308298\pi$
$80$	0	0
$81$	8.95308	0.994787
$82$	0	0
$83$	−3.42260	−0.375680	−0.187840	−	0.982200i	$-0.560149\pi$
$83$	−3.42260	−0.375680	−0.187840	+	0.982200i	$0.560149\pi$
$84$	0	0
$85$	−4.34731	−0.471532
$86$	0	0
$87$	−0.0810794	−0.00869263
$88$	0	0
$89$	1.05887	0.112240	0.0561199	−	0.998424i	$-0.482127\pi$
$89$	1.05887	0.112240	0.0561199	+	0.998424i	$0.482127\pi$
$90$	0	0
$91$	−4.54839	−0.476801
$92$	0	0
$93$	−0.0799037	−0.00828562
$94$	0	0
$95$	2.76398	0.283578
$96$	0	0
$97$	−4.57137	−0.464152	−0.232076	−	0.972698i	$-0.574552\pi$
$97$	−4.57137	−0.464152	−0.232076	+	0.972698i	$0.574552\pi$
$98$	0	0
$99$	1.27434	0.128076
$100$	0	0
$101$	−14.6098	−1.45373	−0.726866	−	0.686779i	$-0.759024\pi$
$101$	−14.6098	−1.45373	−0.726866	+	0.686779i	$0.759024\pi$
$102$	0	0
$103$	6.19587	0.610497	0.305249	−	0.952273i	$-0.401260\pi$
$103$	6.19587	0.610497	0.305249	+	0.952273i	$0.401260\pi$
$104$	0	0
$105$	−0.0512515	−0.00500163
$106$	0	0
$107$	−14.3924	−1.39137	−0.695684	−	0.718348i	$-0.744899\pi$
$107$	−14.3924	−1.39137	−0.695684	+	0.718348i	$0.744899\pi$
$108$	0	0
$109$	−5.52240	−0.528950	−0.264475	−	0.964393i	$-0.585199\pi$
$109$	−5.52240	−0.528950	−0.264475	+	0.964393i	$0.585199\pi$
$110$	0	0
$111$	−0.559747	−0.0531288
$112$	0	0
$113$	14.9331	1.40479	0.702396	−	0.711787i	$-0.252114\pi$
$113$	14.9331	1.40479	0.702396	+	0.711787i	$0.252114\pi$
$114$	0	0
$115$	−2.37546	−0.221512
$116$	0	0
$117$	−19.1953	−1.77460
$118$	0	0
$119$	3.08496	0.282798
$120$	0	0
$121$	−10.8189	−0.983539
$122$	0	0
$123$	−0.293123	−0.0264300
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−8.72098	−0.773862	−0.386931	−	0.922109i	$-0.626465\pi$
$127$	−8.72098	−0.773862	−0.386931	+	0.922109i	$0.626465\pi$
$128$	0	0
$129$	−0.686024	−0.0604011
$130$	0	0
$131$	−15.1693	−1.32535	−0.662676	−	0.748906i	$-0.730579\pi$
$131$	−15.1693	−1.32535	−0.662676	+	0.748906i	$0.730579\pi$
$132$	0	0
$133$	−1.96139	−0.170074
$134$	0	0
$135$	−0.432962	−0.0372635
$136$	0	0
$137$	−15.4998	−1.32424	−0.662118	−	0.749400i	$-0.730342\pi$
$137$	−15.4998	−1.32424	−0.662118	+	0.749400i	$0.730342\pi$
$138$	0	0
$139$	6.44923	0.547017	0.273508	−	0.961870i	$-0.411816\pi$
$139$	6.44923	0.547017	0.273508	+	0.961870i	$0.411816\pi$
$140$	0	0
$141$	0.307457	0.0258925
$142$	0	0
$143$	−2.72740	−0.228077
$144$	0	0
$145$	−1.12262	−0.0932287
$146$	0	0
$147$	−0.469193	−0.0386984
$148$	0	0
$149$	−9.80041	−0.802881	−0.401441	−	0.915885i	$-0.631490\pi$
$149$	−9.80041	−0.802881	−0.401441	+	0.915885i	$0.631490\pi$
$150$	0	0
$151$	−1.00000	−0.0813788
$151$	−1.00000	−0.0813788
$152$	0	0
$153$	13.0193	1.05254
$154$	0	0
$155$	−1.10634	−0.0888636
$156$	0	0
$157$	1.30208	0.103917	0.0519585	−	0.998649i	$-0.483454\pi$
$157$	1.30208	0.103917	0.0519585	+	0.998649i	$0.483454\pi$
$158$	0	0
$159$	−0.387398	−0.0307227
$160$	0	0
$161$	1.68569	0.132851
$162$	0	0
$163$	−1.15240	−0.0902627	−0.0451314	−	0.998981i	$-0.514371\pi$
$163$	−1.15240	−0.0902627	−0.0451314	+	0.998981i	$0.514371\pi$
$164$	0	0
$165$	−0.0307325	−0.00239252
$166$	0	0
$167$	−20.7287	−1.60403	−0.802016	−	0.597302i	$-0.796239\pi$
$167$	−20.7287	−1.60403	−0.802016	+	0.597302i	$0.796239\pi$
$168$	0	0
$169$	28.0825	2.16019
$170$	0	0
$171$	−8.27751	−0.632997
$172$	0	0
$173$	−19.9420	−1.51616	−0.758080	−	0.652161i	$-0.773863\pi$
$173$	−19.9420	−1.51616	−0.758080	+	0.652161i	$0.773863\pi$
$174$	0	0
$175$	−0.709626	−0.0536427
$176$	0	0
$177$	0.524294	0.0394083
$178$	0	0
$179$	1.27486	0.0952877	0.0476439	−	0.998864i	$-0.484829\pi$
$179$	1.27486	0.0952877	0.0476439	+	0.998864i	$0.484829\pi$
$180$	0	0
$181$	−2.32402	−0.172743	−0.0863716	−	0.996263i	$-0.527527\pi$
$181$	−2.32402	−0.172743	−0.0863716	+	0.996263i	$0.527527\pi$
$182$	0	0
$183$	−0.272779	−0.0201644
$184$	0	0
$185$	−7.75024	−0.569809
$186$	0	0
$187$	1.84987	0.135276
$188$	0	0
$189$	0.307241	0.0223485
$190$	0	0
$191$	21.5672	1.56055	0.780275	−	0.625437i	$-0.215079\pi$
$191$	21.5672	1.56055	0.780275	+	0.625437i	$0.215079\pi$
$192$	0	0
$193$	−23.0265	−1.65749	−0.828743	−	0.559629i	$-0.810944\pi$
$193$	−23.0265	−1.65749	−0.828743	+	0.559629i	$0.810944\pi$
$194$	0	0
$195$	0.462919	0.0331503
$196$	0	0
$197$	11.1640	0.795400	0.397700	−	0.917516i	$-0.369808\pi$
$197$	11.1640	0.795400	0.397700	+	0.917516i	$0.369808\pi$
$198$	0	0
$199$	13.2879	0.941955	0.470977	−	0.882145i	$-0.343901\pi$
$199$	13.2879	0.941955	0.470977	+	0.882145i	$0.343901\pi$
$200$	0	0
$201$	0.917232	0.0646965
$202$	0	0
$203$	0.796642	0.0559133
$204$	0	0
$205$	−4.05857	−0.283463
$206$	0	0
$207$	7.11398	0.494456
$208$	0	0
$209$	−1.17613	−0.0813546
$210$	0	0
$211$	10.3780	0.714449	0.357225	−	0.934018i	$-0.383723\pi$
$211$	10.3780	0.714449	0.357225	+	0.934018i	$0.383723\pi$
$212$	0	0
$213$	−0.499733	−0.0342412
$214$	0	0
$215$	−9.49867	−0.647804
$216$	0	0
$217$	0.785090	0.0532954
$218$	0	0
$219$	0.839813	0.0567493
$220$	0	0
$221$	−27.8644	−1.87436
$222$	0	0
$223$	−16.2413	−1.08760	−0.543799	−	0.839215i	$-0.683015\pi$
$223$	−16.2413	−1.08760	−0.543799	+	0.839215i	$0.683015\pi$
$224$	0	0
$225$	−2.99478	−0.199652
$226$	0	0
$227$	15.8156	1.04972	0.524860	−	0.851189i	$-0.324118\pi$
$227$	15.8156	1.04972	0.524860	+	0.851189i	$0.324118\pi$
$228$	0	0
$229$	−26.4248	−1.74620	−0.873101	−	0.487540i	$-0.837894\pi$
$229$	−26.4248	−1.74620	−0.873101	+	0.487540i	$0.837894\pi$
$230$	0	0
$231$	0.0218086	0.00143490
$232$	0	0
$233$	2.08510	0.136599	0.0682996	−	0.997665i	$-0.478243\pi$
$233$	2.08510	0.136599	0.0682996	+	0.997665i	$0.478243\pi$
$234$	0	0
$235$	4.25704	0.277698
$236$	0	0
$237$	0.727307	0.0472437
$238$	0	0
$239$	−14.1081	−0.912580	−0.456290	−	0.889831i	$-0.650822\pi$
$239$	−14.1081	−0.912580	−0.456290	+	0.889831i	$0.650822\pi$
$240$	0	0
$241$	7.38114	0.475461	0.237730	−	0.971331i	$-0.423597\pi$
$241$	7.38114	0.475461	0.237730	+	0.971331i	$0.423597\pi$
$242$	0	0
$243$	1.94551	0.124804
$244$	0	0
$245$	−6.49643	−0.415042
$246$	0	0
$247$	17.7159	1.12723
$248$	0	0
$249$	−0.247191	−0.0156651
$250$	0	0
$251$	−17.2947	−1.09163	−0.545817	−	0.837904i	$-0.683781\pi$
$251$	−17.2947	−1.09163	−0.545817	+	0.837904i	$0.683781\pi$
$252$	0	0
$253$	1.01081	0.0635489
$254$	0	0
$255$	−0.313977	−0.0196620
$256$	0	0
$257$	0.978328	0.0610264	0.0305132	−	0.999534i	$-0.490286\pi$
$257$	0.978328	0.0610264	0.0305132	+	0.999534i	$0.490286\pi$
$258$	0	0
$259$	5.49977	0.341739
$260$	0	0
$261$	3.36201	0.208103
$262$	0	0
$263$	−25.8209	−1.59218	−0.796092	−	0.605175i	$-0.793103\pi$
$263$	−25.8209	−1.59218	−0.796092	+	0.605175i	$0.793103\pi$
$264$	0	0
$265$	−5.36390	−0.329502
$266$	0	0
$267$	0.0764748	0.00468018
$268$	0	0
$269$	6.91986	0.421911	0.210956	−	0.977496i	$-0.432342\pi$
$269$	6.91986	0.421911	0.210956	+	0.977496i	$0.432342\pi$
$270$	0	0
$271$	16.0232	0.973338	0.486669	−	0.873586i	$-0.338212\pi$
$271$	16.0232	0.973338	0.486669	+	0.873586i	$0.338212\pi$
$272$	0	0
$273$	−0.328499	−0.0198817
$274$	0	0
$275$	−0.425521	−0.0256599
$276$	0	0
$277$	−19.9286	−1.19739	−0.598696	−	0.800977i	$-0.704314\pi$
$277$	−19.9286	−1.19739	−0.598696	+	0.800977i	$0.704314\pi$
$278$	0	0
$279$	3.31326	0.198360
$280$	0	0
$281$	−18.0361	−1.07594	−0.537971	−	0.842963i	$-0.680809\pi$
$281$	−18.0361	−1.07594	−0.537971	+	0.842963i	$0.680809\pi$
$282$	0	0
$283$	−9.38500	−0.557880	−0.278940	−	0.960308i	$-0.589983\pi$
$283$	−9.38500	−0.557880	−0.278940	+	0.960308i	$0.589983\pi$
$284$	0	0
$285$	0.199623	0.0118247
$286$	0	0
$287$	2.88007	0.170005
$288$	0	0
$289$	1.89911	0.111712
$290$	0	0
$291$	−0.330159	−0.0193542
$292$	0	0
$293$	−0.207112	−0.0120996	−0.00604982	−	0.999982i	$-0.501926\pi$
$293$	−0.207112	−0.0120996	−0.00604982	+	0.999982i	$0.501926\pi$
$294$	0	0
$295$	7.25935	0.422656
$296$	0	0
$297$	0.184235	0.0106904
$298$	0	0
$299$	−15.2256	−0.880521
$300$	0	0
$301$	6.74050	0.388516
$302$	0	0
$303$	−1.05517	−0.0606178
$304$	0	0
$305$	−3.77690	−0.216264
$306$	0	0
$307$	4.53418	0.258779	0.129390	−	0.991594i	$-0.458698\pi$
$307$	4.53418	0.258779	0.129390	+	0.991594i	$0.458698\pi$
$308$	0	0
$309$	0.447485	0.0254566
$310$	0	0
$311$	−20.3605	−1.15454	−0.577269	−	0.816554i	$-0.695882\pi$
$311$	−20.3605	−1.15454	−0.577269	+	0.816554i	$0.695882\pi$
$312$	0	0
$313$	−7.35555	−0.415760	−0.207880	−	0.978154i	$-0.566656\pi$
$313$	−7.35555	−0.415760	−0.207880	+	0.978154i	$0.566656\pi$
$314$	0	0
$315$	2.12518	0.119740
$316$	0	0
$317$	13.4717	0.756645	0.378322	−	0.925674i	$-0.376501\pi$
$317$	13.4717	0.756645	0.378322	+	0.925674i	$0.376501\pi$
$318$	0	0
$319$	0.477700	0.0267461
$320$	0	0
$321$	−1.03947	−0.0580174
$322$	0	0
$323$	−12.0159	−0.668580
$324$	0	0
$325$	6.40956	0.355539
$326$	0	0
$327$	−0.398846	−0.0220562
$328$	0	0
$329$	−3.02090	−0.166548
$330$	0	0
$331$	−5.06564	−0.278433	−0.139216	−	0.990262i	$-0.544458\pi$
$331$	−5.06564	−0.278433	−0.139216	+	0.990262i	$0.544458\pi$
$332$	0	0
$333$	23.2103	1.27192
$334$	0	0
$335$	12.7000	0.693873
$336$	0	0
$337$	5.66912	0.308817	0.154408	−	0.988007i	$-0.450653\pi$
$337$	5.66912	0.308817	0.154408	+	0.988007i	$0.450653\pi$
$338$	0	0
$339$	1.07852	0.0585771
$340$	0	0
$341$	0.470773	0.0254938
$342$	0	0
$343$	9.57742	0.517132
$344$	0	0
$345$	−0.171563	−0.00923664
$346$	0	0
$347$	−10.1912	−0.547091	−0.273546	−	0.961859i	$-0.588196\pi$
$347$	−10.1912	−0.547091	−0.273546	+	0.961859i	$0.588196\pi$
$348$	0	0
$349$	−18.8306	−1.00798	−0.503990	−	0.863710i	$-0.668135\pi$
$349$	−18.8306	−1.00798	−0.503990	+	0.863710i	$0.668135\pi$
$350$	0	0
$351$	−2.77510	−0.148124
$352$	0	0
$353$	8.92794	0.475187	0.237593	−	0.971365i	$-0.423641\pi$
$353$	8.92794	0.475187	0.237593	+	0.971365i	$0.423641\pi$
$354$	0	0
$355$	−6.91929	−0.367238
$356$	0	0
$357$	0.222806	0.0117921
$358$	0	0
$359$	20.4790	1.08084	0.540419	−	0.841396i	$-0.318266\pi$
$359$	20.4790	1.08084	0.540419	+	0.841396i	$0.318266\pi$
$360$	0	0
$361$	−11.3604	−0.597918
$362$	0	0
$363$	−0.781378	−0.0410117
$364$	0	0
$365$	11.6280	0.608638
$366$	0	0
$367$	12.4178	0.648203	0.324101	−	0.946022i	$-0.394938\pi$
$367$	12.4178	0.648203	0.324101	+	0.946022i	$0.394938\pi$
$368$	0	0
$369$	12.1545	0.632740
$370$	0	0
$371$	3.80637	0.197617
$372$	0	0
$373$	−24.4196	−1.26440	−0.632198	−	0.774807i	$-0.717847\pi$
$373$	−24.4196	−1.26440	−0.632198	+	0.774807i	$0.717847\pi$
$374$	0	0
$375$	0.0722232	0.00372959
$376$	0	0
$377$	−7.19552	−0.370588
$378$	0	0
$379$	19.9830	1.02646	0.513228	−	0.858252i	$-0.328449\pi$
$379$	19.9830	1.02646	0.513228	+	0.858252i	$0.328449\pi$
$380$	0	0
$381$	−0.629857	−0.0322686
$382$	0	0
$383$	−2.98342	−0.152446	−0.0762228	−	0.997091i	$-0.524286\pi$
$383$	−2.98342	−0.152446	−0.0762228	+	0.997091i	$0.524286\pi$
$384$	0	0
$385$	0.301961	0.0153893
$386$	0	0
$387$	28.4465	1.44602
$388$	0	0
$389$	9.79740	0.496748	0.248374	−	0.968664i	$-0.420104\pi$
$389$	9.79740	0.496748	0.248374	+	0.968664i	$0.420104\pi$
$390$	0	0
$391$	10.3268	0.522251
$392$	0	0
$393$	−1.09558	−0.0552646
$394$	0	0
$395$	10.0703	0.506691
$396$	0	0
$397$	0.268968	0.0134991	0.00674957	−	0.999977i	$-0.497852\pi$
$397$	0.268968	0.0134991	0.00674957	+	0.999977i	$0.497852\pi$
$398$	0	0
$399$	−0.141658	−0.00709176
$400$	0	0
$401$	−27.5671	−1.37663	−0.688316	−	0.725411i	$-0.741650\pi$
$401$	−27.5671	−1.37663	−0.688316	+	0.725411i	$0.741650\pi$
$402$	0	0
$403$	−7.09118	−0.353237
$404$	0	0
$405$	8.95308	0.444882
$406$	0	0
$407$	3.29789	0.163470
$408$	0	0
$409$	−6.35099	−0.314036	−0.157018	−	0.987596i	$-0.550188\pi$
$409$	−6.35099	−0.314036	−0.157018	+	0.987596i	$0.550188\pi$
$410$	0	0
$411$	−1.11944	−0.0552181
$412$	0	0
$413$	−5.15143	−0.253485
$414$	0	0
$415$	−3.42260	−0.168009
$416$	0	0
$417$	0.465784	0.0228096
$418$	0	0
$419$	−37.8847	−1.85079	−0.925393	−	0.379008i	$-0.876265\pi$
$419$	−37.8847	−1.85079	−0.925393	+	0.379008i	$0.876265\pi$
$420$	0	0
$421$	−0.950439	−0.0463216	−0.0231608	−	0.999732i	$-0.507373\pi$
$421$	−0.950439	−0.0463216	−0.0231608	+	0.999732i	$0.507373\pi$
$422$	0	0
$423$	−12.7489	−0.619873
$424$	0	0
$425$	−4.34731	−0.210876
$426$	0	0
$427$	2.68018	0.129703
$428$	0	0
$429$	−0.196982	−0.00951038
$430$	0	0
$431$	−2.49509	−0.120184	−0.0600921	−	0.998193i	$-0.519139\pi$
$431$	−2.49509	−0.120184	−0.0600921	+	0.998193i	$0.519139\pi$
$432$	0	0
$433$	9.35197	0.449427	0.224713	−	0.974425i	$-0.427855\pi$
$433$	9.35197	0.449427	0.224713	+	0.974425i	$0.427855\pi$
$434$	0	0
$435$	−0.0810794	−0.00388746
$436$	0	0
$437$	−6.56570	−0.314080
$438$	0	0
$439$	−16.1962	−0.773002	−0.386501	−	0.922289i	$-0.626316\pi$
$439$	−16.1962	−0.773002	−0.386501	+	0.922289i	$0.626316\pi$
$440$	0	0
$441$	19.4554	0.926448
$442$	0	0
$443$	22.2330	1.05632	0.528161	−	0.849144i	$-0.322882\pi$
$443$	22.2330	1.05632	0.528161	+	0.849144i	$0.322882\pi$
$444$	0	0
$445$	1.05887	0.0501951
$446$	0	0
$447$	−0.707817	−0.0334786
$448$	0	0
$449$	24.4972	1.15609	0.578047	−	0.816004i	$-0.303815\pi$
$449$	24.4972	1.15609	0.578047	+	0.816004i	$0.303815\pi$
$450$	0	0
$451$	1.72701	0.0813216
$452$	0	0
$453$	−0.0722232	−0.00339334
$454$	0	0
$455$	−4.54839	−0.213232
$456$	0	0
$457$	18.1628	0.849621	0.424810	−	0.905282i	$-0.360341\pi$
$457$	18.1628	0.849621	0.424810	+	0.905282i	$0.360341\pi$
$458$	0	0
$459$	1.88222	0.0878546
$460$	0	0
$461$	−7.11272	−0.331272	−0.165636	−	0.986187i	$-0.552968\pi$
$461$	−7.11272	−0.331272	−0.165636	+	0.986187i	$0.552968\pi$
$462$	0	0
$463$	9.63466	0.447761	0.223880	−	0.974617i	$-0.428128\pi$
$463$	9.63466	0.447761	0.223880	+	0.974617i	$0.428128\pi$
$464$	0	0
$465$	−0.0799037	−0.00370544
$466$	0	0
$467$	8.59139	0.397562	0.198781	−	0.980044i	$-0.436302\pi$
$467$	8.59139	0.397562	0.198781	+	0.980044i	$0.436302\pi$
$468$	0	0
$469$	−9.01222	−0.416146
$470$	0	0
$471$	0.0940401	0.00433314
$472$	0	0
$473$	4.04189	0.185846
$474$	0	0
$475$	2.76398	0.126820
$476$	0	0
$477$	16.0637	0.735508
$478$	0	0
$479$	24.6154	1.12471	0.562354	−	0.826896i	$-0.309896\pi$
$479$	24.6154	1.12471	0.562354	+	0.826896i	$0.309896\pi$
$480$	0	0
$481$	−49.6756	−2.26501
$482$	0	0
$483$	0.121746	0.00553962
$484$	0	0
$485$	−4.57137	−0.207575
$486$	0	0
$487$	31.6149	1.43261	0.716305	−	0.697788i	$-0.245832\pi$
$487$	31.6149	1.43261	0.716305	+	0.697788i	$0.245832\pi$
$488$	0	0
$489$	−0.0832298	−0.00376378
$490$	0	0
$491$	29.4660	1.32978	0.664890	−	0.746941i	$-0.268478\pi$
$491$	29.4660	1.32978	0.664890	+	0.746941i	$0.268478\pi$
$492$	0	0
$493$	4.88039	0.219802
$494$	0	0
$495$	1.27434	0.0572775
$496$	0	0
$497$	4.91011	0.220248
$498$	0	0
$499$	−17.2934	−0.774159	−0.387079	−	0.922046i	$-0.626516\pi$
$499$	−17.2934	−0.774159	−0.387079	+	0.922046i	$0.626516\pi$
$500$	0	0
$501$	−1.49709	−0.0668851
$502$	0	0
$503$	21.9170	0.977229	0.488615	−	0.872500i	$-0.337502\pi$
$503$	21.9170	0.977229	0.488615	+	0.872500i	$0.337502\pi$
$504$	0	0
$505$	−14.6098	−0.650129
$506$	0	0
$507$	2.02821	0.0900759
$508$	0	0
$509$	−20.3000	−0.899783	−0.449891	−	0.893083i	$-0.648537\pi$
$509$	−20.3000	−0.899783	−0.449891	+	0.893083i	$0.648537\pi$
$510$	0	0
$511$	−8.25154	−0.365027
$512$	0	0
$513$	−1.19670	−0.0528355
$514$	0	0
$515$	6.19587	0.273023
$516$	0	0
$517$	−1.81146	−0.0796679
$518$	0	0
$519$	−1.44027	−0.0632210
$520$	0	0
$521$	−41.0111	−1.79673	−0.898364	−	0.439253i	$-0.855243\pi$
$521$	−41.0111	−1.79673	−0.898364	+	0.439253i	$0.855243\pi$
$522$	0	0
$523$	2.08941	0.0913636	0.0456818	−	0.998956i	$-0.485454\pi$
$523$	2.08941	0.0913636	0.0456818	+	0.998956i	$0.485454\pi$
$524$	0	0
$525$	−0.0512515	−0.00223680
$526$	0	0
$527$	4.80962	0.209510
$528$	0	0
$529$	−17.3572	−0.754661
$530$	0	0
$531$	−21.7402	−0.943444
$532$	0	0
$533$	−26.0136	−1.12678
$534$	0	0
$535$	−14.3924	−0.622239
$536$	0	0
$537$	0.0920747	0.00397332
$538$	0	0
$539$	2.76437	0.119070
$540$	0	0
$541$	31.1115	1.33759	0.668795	−	0.743447i	$-0.266810\pi$
$541$	31.1115	1.33759	0.668795	+	0.743447i	$0.266810\pi$
$542$	0	0
$543$	−0.167848	−0.00720306
$544$	0	0
$545$	−5.52240	−0.236554
$546$	0	0
$547$	27.0295	1.15570	0.577849	−	0.816144i	$-0.303892\pi$
$547$	27.0295	1.15570	0.577849	+	0.816144i	$0.303892\pi$
$548$	0	0
$549$	11.3110	0.482741
$550$	0	0
$551$	−3.10290	−0.132188
$552$	0	0
$553$	−7.14613	−0.303884
$554$	0	0
$555$	−0.559747	−0.0237599
$556$	0	0
$557$	−7.37891	−0.312655	−0.156327	−	0.987705i	$-0.549965\pi$
$557$	−7.37891	−0.312655	−0.156327	+	0.987705i	$0.549965\pi$
$558$	0	0
$559$	−60.8823	−2.57505
$560$	0	0
$561$	0.133604	0.00564075
$562$	0	0
$563$	15.0319	0.633517	0.316759	−	0.948506i	$-0.397405\pi$
$563$	15.0319	0.633517	0.316759	+	0.948506i	$0.397405\pi$
$564$	0	0
$565$	14.9331	0.628242
$566$	0	0
$567$	−6.35334	−0.266815
$568$	0	0
$569$	3.11822	0.130722	0.0653612	−	0.997862i	$-0.479180\pi$
$569$	3.11822	0.130722	0.0653612	+	0.997862i	$0.479180\pi$
$570$	0	0
$571$	−3.83446	−0.160467	−0.0802335	−	0.996776i	$-0.525567\pi$
$571$	−3.83446	−0.160467	−0.0802335	+	0.996776i	$0.525567\pi$
$572$	0	0
$573$	1.55765	0.0650719
$574$	0	0
$575$	−2.37546	−0.0990634
$576$	0	0
$577$	−3.66270	−0.152480	−0.0762401	−	0.997089i	$-0.524292\pi$
$577$	−3.66270	−0.152480	−0.0762401	+	0.997089i	$0.524292\pi$
$578$	0	0
$579$	−1.66305	−0.0691140
$580$	0	0
$581$	2.42877	0.100762
$582$	0	0
$583$	2.28246	0.0945296
$584$	0	0
$585$	−19.1953	−0.793626
$586$	0	0
$587$	5.97122	0.246459	0.123229	−	0.992378i	$-0.460675\pi$
$587$	5.97122	0.246459	0.123229	+	0.992378i	$0.460675\pi$
$588$	0	0
$589$	−3.05791	−0.125999
$590$	0	0
$591$	0.806298	0.0331667
$592$	0	0
$593$	4.17424	0.171416	0.0857078	−	0.996320i	$-0.472685\pi$
$593$	4.17424	0.171416	0.0857078	+	0.996320i	$0.472685\pi$
$594$	0	0
$595$	3.08496	0.126471
$596$	0	0
$597$	0.959695	0.0392777
$598$	0	0
$599$	−4.48902	−0.183416	−0.0917082	−	0.995786i	$-0.529233\pi$
$599$	−4.48902	−0.183416	−0.0917082	+	0.995786i	$0.529233\pi$
$600$	0	0
$601$	−42.0849	−1.71668	−0.858339	−	0.513082i	$-0.828504\pi$
$601$	−42.0849	−1.71668	−0.858339	+	0.513082i	$0.828504\pi$
$602$	0	0
$603$	−38.0336	−1.54885
$604$	0	0
$605$	−10.8189	−0.439852
$606$	0	0
$607$	33.7422	1.36955	0.684777	−	0.728752i	$-0.259899\pi$
$607$	33.7422	1.36955	0.684777	+	0.728752i	$0.259899\pi$
$608$	0	0
$609$	0.0575360	0.00233148
$610$	0	0
$611$	27.2857	1.10386
$612$	0	0
$613$	15.3190	0.618729	0.309365	−	0.950944i	$-0.399884\pi$
$613$	15.3190	0.618729	0.309365	+	0.950944i	$0.399884\pi$
$614$	0	0
$615$	−0.293123	−0.0118199
$616$	0	0
$617$	23.3517	0.940106	0.470053	−	0.882638i	$-0.344235\pi$
$617$	23.3517	0.940106	0.470053	+	0.882638i	$0.344235\pi$
$618$	0	0
$619$	−8.96970	−0.360523	−0.180261	−	0.983619i	$-0.557694\pi$
$619$	−8.96970	−0.360523	−0.180261	+	0.983619i	$0.557694\pi$
$620$	0	0
$621$	1.02848	0.0412716
$622$	0	0
$623$	−0.751400	−0.0301042
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−0.0849439	−0.00339233
$628$	0	0
$629$	33.6927	1.34342
$630$	0	0
$631$	−21.0985	−0.839919	−0.419959	−	0.907543i	$-0.637956\pi$
$631$	−21.0985	−0.839919	−0.419959	+	0.907543i	$0.637956\pi$
$632$	0	0
$633$	0.749531	0.0297912
$634$	0	0
$635$	−8.72098	−0.346082
$636$	0	0
$637$	−41.6393	−1.64981
$638$	0	0
$639$	20.7218	0.819741
$640$	0	0
$641$	16.9207	0.668328	0.334164	−	0.942515i	$-0.391546\pi$
$641$	16.9207	0.668328	0.334164	+	0.942515i	$0.391546\pi$
$642$	0	0
$643$	3.47485	0.137035	0.0685173	−	0.997650i	$-0.478173\pi$
$643$	3.47485	0.137035	0.0685173	+	0.997650i	$0.478173\pi$
$644$	0	0
$645$	−0.686024	−0.0270122
$646$	0	0
$647$	−32.7346	−1.28693	−0.643465	−	0.765476i	$-0.722504\pi$
$647$	−32.7346	−1.28693	−0.643465	+	0.765476i	$0.722504\pi$
$648$	0	0
$649$	−3.08901	−0.121254
$650$	0	0
$651$	0.0567017	0.00222232
$652$	0	0
$653$	7.51048	0.293908	0.146954	−	0.989143i	$-0.453053\pi$
$653$	7.51048	0.293908	0.146954	+	0.989143i	$0.453053\pi$
$654$	0	0
$655$	−15.1693	−0.592715
$656$	0	0
$657$	−34.8234	−1.35859
$658$	0	0
$659$	−11.5988	−0.451824	−0.225912	−	0.974148i	$-0.572536\pi$
$659$	−11.5988	−0.451824	−0.225912	+	0.974148i	$0.572536\pi$
$660$	0	0
$661$	13.9039	0.540797	0.270399	−	0.962748i	$-0.412845\pi$
$661$	13.9039	0.540797	0.270399	+	0.962748i	$0.412845\pi$
$662$	0	0
$663$	−2.01245	−0.0781572
$664$	0	0
$665$	−1.96139	−0.0760594
$666$	0	0
$667$	2.66674	0.103257
$668$	0	0
$669$	−1.17300	−0.0453508
$670$	0	0
$671$	1.60715	0.0620433
$672$	0	0
$673$	−39.5867	−1.52595	−0.762977	−	0.646425i	$-0.776263\pi$
$673$	−39.5867	−1.52595	−0.762977	+	0.646425i	$0.776263\pi$
$674$	0	0
$675$	−0.432962	−0.0166647
$676$	0	0
$677$	29.2265	1.12327	0.561633	−	0.827387i	$-0.310173\pi$
$677$	29.2265	1.12327	0.561633	+	0.827387i	$0.310173\pi$
$678$	0	0
$679$	3.24396	0.124492
$680$	0	0
$681$	1.14225	0.0437713
$682$	0	0
$683$	10.6734	0.408405	0.204203	−	0.978929i	$-0.434540\pi$
$683$	10.6734	0.408405	0.204203	+	0.978929i	$0.434540\pi$
$684$	0	0
$685$	−15.4998	−0.592216
$686$	0	0
$687$	−1.90849	−0.0728133
$688$	0	0
$689$	−34.3803	−1.30978
$690$	0	0
$691$	−5.74934	−0.218715	−0.109358	−	0.994002i	$-0.534879\pi$
$691$	−5.74934	−0.218715	−0.109358	+	0.994002i	$0.534879\pi$
$692$	0	0
$693$	−0.904307	−0.0343518
$694$	0	0
$695$	6.44923	0.244633
$696$	0	0
$697$	17.6439	0.668309
$698$	0	0
$699$	0.150592	0.00569593
$700$	0	0
$701$	−7.14573	−0.269890	−0.134945	−	0.990853i	$-0.543086\pi$
$701$	−7.14573	−0.269890	−0.134945	+	0.990853i	$0.543086\pi$
$702$	0	0
$703$	−21.4215	−0.807926
$704$	0	0
$705$	0.307457	0.0115795
$706$	0	0
$707$	10.3675	0.389910
$708$	0	0
$709$	3.10751	0.116705	0.0583526	−	0.998296i	$-0.481415\pi$
$709$	3.10751	0.116705	0.0583526	+	0.998296i	$0.481415\pi$
$710$	0	0
$711$	−30.1583	−1.13102
$712$	0	0
$713$	2.62807	0.0984220
$714$	0	0
$715$	−2.72740	−0.101999
$716$	0	0
$717$	−1.01894	−0.0380528
$718$	0	0
$719$	−0.755404	−0.0281718	−0.0140859	−	0.999901i	$-0.504484\pi$
$719$	−0.755404	−0.0281718	−0.0140859	+	0.999901i	$0.504484\pi$
$720$	0	0
$721$	−4.39675	−0.163743
$722$	0	0
$723$	0.533089	0.0198258
$724$	0	0
$725$	−1.12262	−0.0416932
$726$	0	0
$727$	−35.3433	−1.31081	−0.655405	−	0.755278i	$-0.727502\pi$
$727$	−35.3433	−1.31081	−0.655405	+	0.755278i	$0.727502\pi$
$728$	0	0
$729$	−26.7187	−0.989583
$730$	0	0
$731$	41.2937	1.52730
$732$	0	0
$733$	−18.3054	−0.676127	−0.338064	−	0.941123i	$-0.609772\pi$
$733$	−18.3054	−0.676127	−0.338064	+	0.941123i	$0.609772\pi$
$734$	0	0
$735$	−0.469193	−0.0173064
$736$	0	0
$737$	−5.40410	−0.199063
$738$	0	0
$739$	39.9731	1.47044	0.735218	−	0.677831i	$-0.237080\pi$
$739$	39.9731	1.47044	0.735218	+	0.677831i	$0.237080\pi$
$740$	0	0
$741$	1.27950	0.0470035
$742$	0	0
$743$	49.6574	1.82175	0.910876	−	0.412679i	$-0.135407\pi$
$743$	49.6574	1.82175	0.910876	+	0.412679i	$0.135407\pi$
$744$	0	0
$745$	−9.80041	−0.359059
$746$	0	0
$747$	10.2500	0.375026
$748$	0	0
$749$	10.2132	0.373183
$750$	0	0
$751$	−52.4001	−1.91211	−0.956053	−	0.293193i	$-0.905282\pi$
$751$	−52.4001	−1.91211	−0.956053	+	0.293193i	$0.905282\pi$
$752$	0	0
$753$	−1.24908	−0.0455190
$754$	0	0
$755$	−1.00000	−0.0363937
$756$	0	0
$757$	−0.176781	−0.00642521	−0.00321261	−	0.999995i	$-0.501023\pi$
$757$	−0.176781	−0.00642521	−0.00321261	+	0.999995i	$0.501023\pi$
$758$	0	0
$759$	0.0730037	0.00264987
$760$	0	0
$761$	−4.65090	−0.168595	−0.0842975	−	0.996441i	$-0.526865\pi$
$761$	−4.65090	−0.168595	−0.0842975	+	0.996441i	$0.526865\pi$
$762$	0	0
$763$	3.91884	0.141872
$764$	0	0
$765$	13.0193	0.470712
$766$	0	0
$767$	46.5293	1.68007
$768$	0	0
$769$	−9.07879	−0.327390	−0.163695	−	0.986511i	$-0.552341\pi$
$769$	−9.07879	−0.327390	−0.163695	+	0.986511i	$0.552341\pi$
$770$	0	0
$771$	0.0706580	0.00254468
$772$	0	0
$773$	28.3926	1.02121	0.510605	−	0.859815i	$-0.329421\pi$
$773$	28.3926	1.02121	0.510605	+	0.859815i	$0.329421\pi$
$774$	0	0
$775$	−1.10634	−0.0397410
$776$	0	0
$777$	0.397211	0.0142499
$778$	0	0
$779$	−11.2178	−0.401919
$780$	0	0
$781$	2.94431	0.105356
$782$	0	0
$783$	0.486054	0.0173701
$784$	0	0
$785$	1.30208	0.0464731
$786$	0	0
$787$	35.8995	1.27968	0.639839	−	0.768509i	$-0.279001\pi$
$787$	35.8995	1.27968	0.639839	+	0.768509i	$0.279001\pi$
$788$	0	0
$789$	−1.86487	−0.0663910
$790$	0	0
$791$	−10.5969	−0.376784
$792$	0	0
$793$	−24.2083	−0.859660
$794$	0	0
$795$	−0.387398	−0.0137396
$796$	0	0
$797$	5.72995	0.202965	0.101483	−	0.994837i	$-0.467641\pi$
$797$	5.72995	0.202965	0.101483	+	0.994837i	$0.467641\pi$
$798$	0	0
$799$	−18.5067	−0.654718
$800$	0	0
$801$	−3.17108	−0.112045
$802$	0	0
$803$	−4.94797	−0.174610
$804$	0	0
$805$	1.68569	0.0594126
$806$	0	0
$807$	0.499775	0.0175929
$808$	0	0
$809$	−19.2161	−0.675603	−0.337802	−	0.941217i	$-0.609683\pi$
$809$	−19.2161	−0.675603	−0.337802	+	0.941217i	$0.609683\pi$
$810$	0	0
$811$	−10.8110	−0.379627	−0.189814	−	0.981820i	$-0.560788\pi$
$811$	−10.8110	−0.379627	−0.189814	+	0.981820i	$0.560788\pi$
$812$	0	0
$813$	1.15724	0.0405863
$814$	0	0
$815$	−1.15240	−0.0403667
$816$	0	0
$817$	−26.2541	−0.918515
$818$	0	0
$819$	13.6214	0.475972
$820$	0	0
$821$	−7.62772	−0.266209	−0.133105	−	0.991102i	$-0.542495\pi$
$821$	−7.62772	−0.266209	−0.133105	+	0.991102i	$0.542495\pi$
$822$	0	0
$823$	15.5370	0.541587	0.270794	−	0.962637i	$-0.412714\pi$
$823$	15.5370	0.541587	0.270794	+	0.962637i	$0.412714\pi$
$824$	0	0
$825$	−0.0307325	−0.00106997
$826$	0	0
$827$	−4.22755	−0.147006	−0.0735031	−	0.997295i	$-0.523418\pi$
$827$	−4.22755	−0.147006	−0.0735031	+	0.997295i	$0.523418\pi$
$828$	0	0
$829$	18.9122	0.656849	0.328424	−	0.944530i	$-0.393482\pi$
$829$	18.9122	0.656849	0.328424	+	0.944530i	$0.393482\pi$
$830$	0	0
$831$	−1.43930	−0.0499289
$832$	0	0
$833$	28.2420	0.978527
$834$	0	0
$835$	−20.7287	−0.717345
$836$	0	0
$837$	0.479005	0.0165568
$838$	0	0
$839$	−25.4110	−0.877284	−0.438642	−	0.898662i	$-0.644540\pi$
$839$	−25.4110	−0.877284	−0.438642	+	0.898662i	$0.644540\pi$
$840$	0	0
$841$	−27.7397	−0.956542
$842$	0	0
$843$	−1.30262	−0.0448648
$844$	0	0
$845$	28.0825	0.966067
$846$	0	0
$847$	7.67739	0.263798
$848$	0	0
$849$	−0.677815	−0.0232625
$850$	0	0
$851$	18.4103	0.631099
$852$	0	0
$853$	21.2801	0.728615	0.364308	−	0.931279i	$-0.381306\pi$
$853$	21.2801	0.728615	0.364308	+	0.931279i	$0.381306\pi$
$854$	0	0
$855$	−8.27751	−0.283085
$856$	0	0
$857$	40.9808	1.39988	0.699939	−	0.714202i	$-0.253210\pi$
$857$	40.9808	1.39988	0.699939	+	0.714202i	$0.253210\pi$
$858$	0	0
$859$	−48.7945	−1.66485	−0.832424	−	0.554140i	$-0.813047\pi$
$859$	−48.7945	−1.66485	−0.832424	+	0.554140i	$0.813047\pi$
$860$	0	0
$861$	0.208008	0.00708888
$862$	0	0
$863$	10.5779	0.360075	0.180037	−	0.983660i	$-0.442378\pi$
$863$	10.5779	0.360075	0.180037	+	0.983660i	$0.442378\pi$
$864$	0	0
$865$	−19.9420	−0.678048
$866$	0	0
$867$	0.137160	0.00465819
$868$	0	0
$869$	−4.28512	−0.145363
$870$	0	0
$871$	81.4012	2.75817
$872$	0	0
$873$	13.6903	0.463345
$874$	0	0
$875$	−0.709626	−0.0239897
$876$	0	0
$877$	43.0020	1.45207	0.726037	−	0.687656i	$-0.241360\pi$
$877$	43.0020	1.45207	0.726037	+	0.687656i	$0.241360\pi$
$878$	0	0
$879$	−0.0149583	−0.000504532 0
$880$	0	0
$881$	−34.9536	−1.17762	−0.588809	−	0.808273i	$-0.700403\pi$
$881$	−34.9536	−1.17762	−0.588809	+	0.808273i	$0.700403\pi$
$882$	0	0
$883$	54.2117	1.82437	0.912185	−	0.409778i	$-0.134394\pi$
$883$	54.2117	1.82437	0.912185	+	0.409778i	$0.134394\pi$
$884$	0	0
$885$	0.524294	0.0176239
$886$	0	0
$887$	15.5465	0.522002	0.261001	−	0.965339i	$-0.415947\pi$
$887$	15.5465	0.522002	0.261001	+	0.965339i	$0.415947\pi$
$888$	0	0
$889$	6.18864	0.207560
$890$	0	0
$891$	−3.80973	−0.127631
$892$	0	0
$893$	11.7663	0.393746
$894$	0	0
$895$	1.27486	0.0426140
$896$	0	0
$897$	−1.09964	−0.0367160
$898$	0	0
$899$	1.24201	0.0414232
$900$	0	0
$901$	23.3186	0.776854
$902$	0	0
$903$	0.486821	0.0162004
$904$	0	0
$905$	−2.32402	−0.0772531
$906$	0	0
$907$	−47.3249	−1.57140	−0.785698	−	0.618610i	$-0.787696\pi$
$907$	−47.3249	−1.57140	−0.785698	+	0.618610i	$0.787696\pi$
$908$	0	0
$909$	43.7533	1.45120
$910$	0	0
$911$	−22.7348	−0.753236	−0.376618	−	0.926369i	$-0.622913\pi$
$911$	−22.7348	−0.753236	−0.376618	+	0.926369i	$0.622913\pi$
$912$	0	0
$913$	1.45639	0.0481995
$914$	0	0
$915$	−0.272779	−0.00901781
$916$	0	0
$917$	10.7646	0.355477
$918$	0	0
$919$	12.5632	0.414422	0.207211	−	0.978296i	$-0.433561\pi$
$919$	12.5632	0.414422	0.207211	+	0.978296i	$0.433561\pi$
$920$	0	0
$921$	0.327473	0.0107906
$922$	0	0
$923$	−44.3496	−1.45979
$924$	0	0
$925$	−7.75024	−0.254826
$926$	0	0
$927$	−18.5553	−0.609436
$928$	0	0
$929$	27.9530	0.917108	0.458554	−	0.888666i	$-0.348367\pi$
$929$	27.9530	0.917108	0.458554	+	0.888666i	$0.348367\pi$
$930$	0	0
$931$	−17.9560	−0.588483
$932$	0	0
$933$	−1.47050	−0.0481420
$934$	0	0
$935$	1.84987	0.0604973
$936$	0	0
$937$	4.67569	0.152748	0.0763740	−	0.997079i	$-0.475666\pi$
$937$	4.67569	0.152748	0.0763740	+	0.997079i	$0.475666\pi$
$938$	0	0
$939$	−0.531241	−0.0173364
$940$	0	0
$941$	3.59574	0.117218	0.0586088	−	0.998281i	$-0.481334\pi$
$941$	3.59574	0.117218	0.0586088	+	0.998281i	$0.481334\pi$
$942$	0	0
$943$	9.64095	0.313953
$944$	0	0
$945$	0.307241	0.00999456
$946$	0	0
$947$	−42.9876	−1.39691	−0.698454	−	0.715655i	$-0.746128\pi$
$947$	−42.9876	−1.39691	−0.698454	+	0.715655i	$0.746128\pi$
$948$	0	0
$949$	74.5305	2.41936
$950$	0	0
$951$	0.972968	0.0315506
$952$	0	0
$953$	−33.4247	−1.08273	−0.541365	−	0.840787i	$-0.682092\pi$
$953$	−33.4247	−1.08273	−0.541365	+	0.840787i	$0.682092\pi$
$954$	0	0
$955$	21.5672	0.697899
$956$	0	0
$957$	0.0345010	0.00111526
$958$	0	0
$959$	10.9990	0.355178
$960$	0	0
$961$	−29.7760	−0.960516
$962$	0	0
$963$	43.1022	1.38895
$964$	0	0
$965$	−23.0265	−0.741250
$966$	0	0
$967$	57.6070	1.85252	0.926259	−	0.376888i	$-0.123006\pi$
$967$	57.6070	1.85252	0.926259	+	0.376888i	$0.123006\pi$
$968$	0	0
$969$	−0.867824	−0.0278785
$970$	0	0
$971$	−29.7096	−0.953426	−0.476713	−	0.879059i	$-0.658172\pi$
$971$	−29.7096	−0.953426	−0.476713	+	0.879059i	$0.658172\pi$
$972$	0	0
$973$	−4.57654	−0.146717
$974$	0	0
$975$	0.462919	0.0148253
$976$	0	0
$977$	40.2087	1.28639	0.643195	−	0.765702i	$-0.277608\pi$
$977$	40.2087	1.28639	0.643195	+	0.765702i	$0.277608\pi$
$978$	0	0
$979$	−0.450571	−0.0144003
$980$	0	0
$981$	16.5384	0.528031
$982$	0	0
$983$	29.5098	0.941216	0.470608	−	0.882342i	$-0.344035\pi$
$983$	29.5098	0.941216	0.470608	+	0.882342i	$0.344035\pi$
$984$	0	0
$985$	11.1640	0.355714
$986$	0	0
$987$	−0.218179	−0.00694472
$988$	0	0
$989$	22.5637	0.717483
$990$	0	0
$991$	27.7388	0.881151	0.440576	−	0.897716i	$-0.354774\pi$
$991$	27.7388	0.881151	0.440576	+	0.897716i	$0.354774\pi$
$992$	0	0
$993$	−0.365857	−0.0116101
$994$	0	0
$995$	13.2879	0.421255
$996$	0	0
$997$	39.6362	1.25529	0.627646	−	0.778499i	$-0.284019\pi$
$997$	39.6362	1.25529	0.627646	+	0.778499i	$0.284019\pi$
$998$	0	0
$999$	3.35556	0.106165

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6040.2.a.l.1.5	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6040.2.a.l.1.5	✓	9	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:	\( N \)	\(=\)	\( 6040 = 2^{3} \cdot 5 \cdot 151 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6040.a (trivial)

\(f(q)\)	\(=\)	\(q+0.0722232 q^{3} +1.00000 q^{5} -0.709626 q^{7} -2.99478 q^{9} +O(q^{10})\)	\(q+0.0722232 q^{3} +1.00000 q^{5} -0.709626 q^{7} -2.99478 q^{9} -0.425521 q^{11} +6.40956 q^{13} +0.0722232 q^{15} -4.34731 q^{17} +2.76398 q^{19} -0.0512515 q^{21} -2.37546 q^{23} +1.00000 q^{25} -0.432962 q^{27} -1.12262 q^{29} -1.10634 q^{31} -0.0307325 q^{33} -0.709626 q^{35} -7.75024 q^{37} +0.462919 q^{39} -4.05857 q^{41} -9.49867 q^{43} -2.99478 q^{45} +4.25704 q^{47} -6.49643 q^{49} -0.313977 q^{51} -5.36390 q^{53} -0.425521 q^{55} +0.199623 q^{57} +7.25935 q^{59} -3.77690 q^{61} +2.12518 q^{63} +6.40956 q^{65} +12.7000 q^{67} -0.171563 q^{69} -6.91929 q^{71} +11.6280 q^{73} +0.0722232 q^{75} +0.301961 q^{77} +10.0703 q^{79} +8.95308 q^{81} -3.42260 q^{83} -4.34731 q^{85} -0.0810794 q^{87} +1.05887 q^{89} -4.54839 q^{91} -0.0799037 q^{93} +2.76398 q^{95} -4.57137 q^{97} +1.27434 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 9 q^{5} - 2 q^{7} - 3 q^{9}+O(q^{10}) \) 9 * q + 9 * q^5 - 2 * q^7 - 3 * q^9	\( 9 q + 9 q^{5} - 2 q^{7} - 3 q^{9} - 6 q^{11} - 9 q^{13} - 2 q^{17} - 10 q^{19} - 9 q^{21} - 6 q^{23} + 9 q^{25} + 12 q^{27} - 6 q^{29} + 9 q^{31} - 11 q^{33} - 2 q^{35} - 12 q^{37} - 3 q^{39} - 20 q^{41} + q^{43} - 3 q^{45} + 22 q^{47} - 29 q^{49} + 2 q^{51} - 35 q^{53} - 6 q^{55} - 20 q^{57} + 14 q^{59} - 22 q^{61} - 12 q^{63} - 9 q^{65} + 4 q^{67} + 5 q^{69} - 22 q^{71} - 34 q^{73} - 5 q^{77} + 8 q^{79} - 31 q^{81} - 3 q^{83} - 2 q^{85} - 5 q^{89} - 7 q^{91} - 21 q^{93} - 10 q^{95} - 33 q^{97} - 15 q^{99}+O(q^{100}) \) 9 * q + 9 * q^5 - 2 * q^7 - 3 * q^9 - 6 * q^11 - 9 * q^13 - 2 * q^17 - 10 * q^19 - 9 * q^21 - 6 * q^23 + 9 * q^25 + 12 * q^27 - 6 * q^29 + 9 * q^31 - 11 * q^33 - 2 * q^35 - 12 * q^37 - 3 * q^39 - 20 * q^41 + q^43 - 3 * q^45 + 22 * q^47 - 29 * q^49 + 2 * q^51 - 35 * q^53 - 6 * q^55 - 20 * q^57 + 14 * q^59 - 22 * q^61 - 12 * q^63 - 9 * q^65 + 4 * q^67 + 5 * q^69 - 22 * q^71 - 34 * q^73 - 5 * q^77 + 8 * q^79 - 31 * q^81 - 3 * q^83 - 2 * q^85 - 5 * q^89 - 7 * q^91 - 21 * q^93 - 10 * q^95 - 33 * q^97 - 15 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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