LMFDB - Embedded newform 4840.2.a.bc.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4840,2,Mod(1,4840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4840, 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("4840.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4840 = 2^{3} \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4840.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:	$38.6475945783$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.22733568.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 8x^{4} - 2x^{3} + 16x^{2} + 8x - 2 $ x^6 - 8x^4 - 2x^3 + 16x^2 + 8x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-2.08589$ of defining polynomial
Character	$\chi$	$=$	4840.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.35095 q^{3} +1.00000 q^{5} +3.00679 q^{7} -1.17493 q^{9} -5.69877 q^{13} +1.35095 q^{15} -4.70191 q^{17} +3.69877 q^{19} +4.06204 q^{21} -6.83866 q^{23} +1.00000 q^{25} -5.64013 q^{27} -10.6956 q^{29} -3.50491 q^{31} +3.00679 q^{35} +5.79409 q^{37} -7.69877 q^{39} -8.39893 q^{41} -2.80716 q^{43} -1.17493 q^{45} +5.14504 q^{47} +2.04081 q^{49} -6.35205 q^{51} -12.2204 q^{53} +4.99686 q^{57} +3.30528 q^{59} -10.7990 q^{61} -3.53276 q^{63} -5.69877 q^{65} +11.5328 q^{67} -9.23871 q^{69} +2.05440 q^{71} -3.29589 q^{73} +1.35095 q^{75} +4.67984 q^{79} -4.09477 q^{81} +12.5247 q^{83} -4.70191 q^{85} -14.4493 q^{87} -4.32366 q^{89} -17.1350 q^{91} -4.73497 q^{93} +3.69877 q^{95} +2.94235 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{3} + 6 q^{5} - 4 q^{7} + 4 q^{9} - 2 q^{15} - 8 q^{17} - 12 q^{19} + 8 q^{21} - 8 q^{23} + 6 q^{25} - 14 q^{27} - 16 q^{29} - 4 q^{31} - 4 q^{35} + 8 q^{37} - 12 q^{39} - 32 q^{41} + 4 q^{43}+ \cdots - 12 q^{95}+O(q^{100}) $ 6 * q - 2 * q^3 + 6 * q^5 - 4 * q^7 + 4 * q^9 - 2 * q^15 - 8 * q^17 - 12 * q^19 + 8 * q^21 - 8 * q^23 + 6 * q^25 - 14 * q^27 - 16 * q^29 - 4 * q^31 - 4 * q^35 + 8 * q^37 - 12 * q^39 - 32 * q^41 + 4 * q^43 + 4 * q^45 - 6 * q^47 + 16 * q^49 - 40 * q^51 + 8 * q^53 + 16 * q^57 + 4 * q^59 - 16 * q^61 - 28 * q^63 - 2 * q^67 + 8 * q^69 - 28 * q^71 - 16 * q^73 - 2 * q^75 - 10 * q^81 - 12 * q^83 - 8 * q^85 + 24 * q^87 + 18 * q^89 - 24 * q^91 + 20 * q^93 - 12 * q^95

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
$3$	1.35095	0.779973	0.389987	−	0.920821i	$-0.372480\pi$
$3$	1.35095	0.779973	0.389987	+	0.920821i	$0.372480\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	3.00679	1.13646	0.568231	−	0.822869i	$-0.307628\pi$
$7$	3.00679	1.13646	0.568231	+	0.822869i	$0.307628\pi$
$8$	0	0
$9$	−1.17493	−0.391642
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−5.69877	−1.58055	−0.790277	−	0.612750i	$-0.790063\pi$
$13$	−5.69877	−1.58055	−0.790277	+	0.612750i	$0.790063\pi$
$14$	0	0
$15$	1.35095	0.348815
$16$	0	0
$17$	−4.70191	−1.14038	−0.570190	−	0.821513i	$-0.693130\pi$
$17$	−4.70191	−1.14038	−0.570190	+	0.821513i	$0.693130\pi$
$18$	0	0
$19$	3.69877	0.848556	0.424278	−	0.905532i	$-0.360528\pi$
$19$	3.69877	0.848556	0.424278	+	0.905532i	$0.360528\pi$
$20$	0	0
$21$	4.06204	0.886409
$22$	0	0
$23$	−6.83866	−1.42596	−0.712980	−	0.701185i	$-0.752655\pi$
$23$	−6.83866	−1.42596	−0.712980	+	0.701185i	$0.752655\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.64013	−1.08544
$28$	0	0
$29$	−10.6956	−1.98613	−0.993064	−	0.117573i	$-0.962489\pi$
$29$	−10.6956	−1.98613	−0.993064	+	0.117573i	$0.962489\pi$
$30$	0	0
$31$	−3.50491	−0.629501	−0.314750	−	0.949174i	$-0.601921\pi$
$31$	−3.50491	−0.629501	−0.314750	+	0.949174i	$0.601921\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	3.00679	0.508241
$36$	0	0
$37$	5.79409	0.952542	0.476271	−	0.879298i	$-0.341988\pi$
$37$	5.79409	0.952542	0.476271	+	0.879298i	$0.341988\pi$
$38$	0	0
$39$	−7.69877	−1.23279
$40$	0	0
$41$	−8.39893	−1.31169	−0.655846	−	0.754895i	$-0.727688\pi$
$41$	−8.39893	−1.31169	−0.655846	+	0.754895i	$0.727688\pi$
$42$	0	0
$43$	−2.80716	−0.428088	−0.214044	−	0.976824i	$-0.568664\pi$
$43$	−2.80716	−0.428088	−0.214044	+	0.976824i	$0.568664\pi$
$44$	0	0
$45$	−1.17493	−0.175148
$46$	0	0
$47$	5.14504	0.750481	0.375241	−	0.926927i	$-0.377560\pi$
$47$	5.14504	0.750481	0.375241	+	0.926927i	$0.377560\pi$
$48$	0	0
$49$	2.04081	0.291544
$50$	0	0
$51$	−6.35205	−0.889466
$52$	0	0
$53$	−12.2204	−1.67860	−0.839301	−	0.543667i	$-0.817035\pi$
$53$	−12.2204	−1.67860	−0.839301	+	0.543667i	$0.817035\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	4.99686	0.661851
$58$	0	0
$59$	3.30528	0.430311	0.215155	−	0.976580i	$-0.430974\pi$
$59$	3.30528	0.430311	0.215155	+	0.976580i	$0.430974\pi$
$60$	0	0
$61$	−10.7990	−1.38267	−0.691334	−	0.722536i	$-0.742976\pi$
$61$	−10.7990	−1.38267	−0.691334	+	0.722536i	$0.742976\pi$
$62$	0	0
$63$	−3.53276	−0.445086
$64$	0	0
$65$	−5.69877	−0.706845
$66$	0	0
$67$	11.5328	1.40895	0.704475	−	0.709729i	$-0.251183\pi$
$67$	11.5328	1.40895	0.704475	+	0.709729i	$0.251183\pi$
$68$	0	0
$69$	−9.23871	−1.11221
$70$	0	0
$71$	2.05440	0.243812	0.121906	−	0.992542i	$-0.461099\pi$
$71$	2.05440	0.243812	0.121906	+	0.992542i	$0.461099\pi$
$72$	0	0
$73$	−3.29589	−0.385755	−0.192877	−	0.981223i	$-0.561782\pi$
$73$	−3.29589	−0.385755	−0.192877	+	0.981223i	$0.561782\pi$
$74$	0	0
$75$	1.35095	0.155995
$76$	0	0
$77$	0	0
$78$	0	0
$79$	4.67984	0.526523	0.263261	−	0.964725i	$-0.415202\pi$
$79$	4.67984	0.526523	0.263261	+	0.964725i	$0.415202\pi$
$80$	0	0
$81$	−4.09477	−0.454975
$82$	0	0
$83$	12.5247	1.37477	0.687385	−	0.726293i	$-0.258759\pi$
$83$	12.5247	1.37477	0.687385	+	0.726293i	$0.258759\pi$
$84$	0	0
$85$	−4.70191	−0.509993
$86$	0	0
$87$	−14.4493	−1.54913
$88$	0	0
$89$	−4.32366	−0.458307	−0.229153	−	0.973390i	$-0.573596\pi$
$89$	−4.32366	−0.458307	−0.229153	+	0.973390i	$0.573596\pi$
$90$	0	0
$91$	−17.1350	−1.79624
$92$	0	0
$93$	−4.73497	−0.490994
$94$	0	0
$95$	3.69877	0.379486
$96$	0	0
$97$	2.94235	0.298751	0.149375	−	0.988781i	$-0.452274\pi$
$97$	2.94235	0.298751	0.149375	+	0.988781i	$0.452274\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−2.69796	−0.268457	−0.134228	−	0.990950i	$-0.542856\pi$
$101$	−2.69796	−0.268457	−0.134228	+	0.990950i	$0.542856\pi$
$102$	0	0
$103$	16.1167	1.58803	0.794014	−	0.607900i	$-0.207988\pi$
$103$	16.1167	1.58803	0.794014	+	0.607900i	$0.207988\pi$
$104$	0	0
$105$	4.06204	0.396414
$106$	0	0
$107$	−18.7756	−1.81511	−0.907554	−	0.419935i	$-0.862053\pi$
$107$	−18.7756	−1.81511	−0.907554	+	0.419935i	$0.862053\pi$
$108$	0	0
$109$	14.8927	1.42646	0.713232	−	0.700928i	$-0.247231\pi$
$109$	14.8927	1.42646	0.713232	+	0.700928i	$0.247231\pi$
$110$	0	0
$111$	7.82754	0.742957
$112$	0	0
$113$	5.05396	0.475437	0.237718	−	0.971334i	$-0.423600\pi$
$113$	5.05396	0.475437	0.237718	+	0.971334i	$0.423600\pi$
$114$	0	0
$115$	−6.83866	−0.637709
$116$	0	0
$117$	6.69563	0.619011
$118$	0	0
$119$	−14.1377	−1.29600
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−11.3466	−1.02308
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−18.2618	−1.62047	−0.810236	−	0.586104i	$-0.800661\pi$
$127$	−18.2618	−1.62047	−0.810236	+	0.586104i	$0.800661\pi$
$128$	0	0
$129$	−3.79234	−0.333897
$130$	0	0
$131$	−9.29322	−0.811953	−0.405976	−	0.913884i	$-0.633068\pi$
$131$	−9.29322	−0.811953	−0.405976	+	0.913884i	$0.633068\pi$
$132$	0	0
$133$	11.1214	0.964351
$134$	0	0
$135$	−5.64013	−0.485425
$136$	0	0
$137$	5.14826	0.439846	0.219923	−	0.975517i	$-0.429419\pi$
$137$	5.14826	0.439846	0.219923	+	0.975517i	$0.429419\pi$
$138$	0	0
$139$	6.73066	0.570887	0.285444	−	0.958396i	$-0.407859\pi$
$139$	6.73066	0.570887	0.285444	+	0.958396i	$0.407859\pi$
$140$	0	0
$141$	6.95071	0.585355
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−10.6956	−0.888224
$146$	0	0
$147$	2.75704	0.227397
$148$	0	0
$149$	5.37053	0.439971	0.219985	−	0.975503i	$-0.429399\pi$
$149$	5.37053	0.439971	0.219985	+	0.975503i	$0.429399\pi$
$150$	0	0
$151$	1.94713	0.158455	0.0792277	−	0.996857i	$-0.474755\pi$
$151$	1.94713	0.158455	0.0792277	+	0.996857i	$0.474755\pi$
$152$	0	0
$153$	5.52439	0.446621
$154$	0	0
$155$	−3.50491	−0.281521
$156$	0	0
$157$	7.29589	0.582276	0.291138	−	0.956681i	$-0.405966\pi$
$157$	7.29589	0.582276	0.291138	+	0.956681i	$0.405966\pi$
$158$	0	0
$159$	−16.5092	−1.30926
$160$	0	0
$161$	−20.5625	−1.62055
$162$	0	0
$163$	3.96332	0.310431	0.155216	−	0.987881i	$-0.450393\pi$
$163$	3.96332	0.310431	0.155216	+	0.987881i	$0.450393\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	7.41532	0.573815	0.286907	−	0.957958i	$-0.407373\pi$
$167$	7.41532	0.573815	0.286907	+	0.957958i	$0.407373\pi$
$168$	0	0
$169$	19.4760	1.49815
$170$	0	0
$171$	−4.34578	−0.332330
$172$	0	0
$173$	−8.59979	−0.653830	−0.326915	−	0.945054i	$-0.606009\pi$
$173$	−8.59979	−0.653830	−0.326915	+	0.945054i	$0.606009\pi$
$174$	0	0
$175$	3.00679	0.227292
$176$	0	0
$177$	4.46528	0.335631
$178$	0	0
$179$	1.13004	0.0844632	0.0422316	−	0.999108i	$-0.486553\pi$
$179$	1.13004	0.0844632	0.0422316	+	0.999108i	$0.486553\pi$
$180$	0	0
$181$	9.98170	0.741934	0.370967	−	0.928646i	$-0.379026\pi$
$181$	9.98170	0.741934	0.370967	+	0.928646i	$0.379026\pi$
$182$	0	0
$183$	−14.5889	−1.07844
$184$	0	0
$185$	5.79409	0.425990
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−16.9587	−1.23356
$190$	0	0
$191$	18.4991	1.33855	0.669274	−	0.743016i	$-0.266605\pi$
$191$	18.4991	1.33855	0.669274	+	0.743016i	$0.266605\pi$
$192$	0	0
$193$	−9.38556	−0.675587	−0.337794	−	0.941220i	$-0.609681\pi$
$193$	−9.38556	−0.675587	−0.337794	+	0.941220i	$0.609681\pi$
$194$	0	0
$195$	−7.69877	−0.551320
$196$	0	0
$197$	13.4556	0.958673	0.479337	−	0.877631i	$-0.340877\pi$
$197$	13.4556	0.958673	0.479337	+	0.877631i	$0.340877\pi$
$198$	0	0
$199$	23.5214	1.66739	0.833695	−	0.552225i	$-0.186221\pi$
$199$	23.5214	1.66739	0.833695	+	0.552225i	$0.186221\pi$
$200$	0	0
$201$	15.5802	1.09894
$202$	0	0
$203$	−32.1596	−2.25716
$204$	0	0
$205$	−8.39893	−0.586606
$206$	0	0
$207$	8.03492	0.558466
$208$	0	0
$209$	0	0
$210$	0	0
$211$	5.65886	0.389572	0.194786	−	0.980846i	$-0.437599\pi$
$211$	5.65886	0.389572	0.194786	+	0.980846i	$0.437599\pi$
$212$	0	0
$213$	2.77540	0.190167
$214$	0	0
$215$	−2.80716	−0.191447
$216$	0	0
$217$	−10.5386	−0.715403
$218$	0	0
$219$	−4.45259	−0.300878
$220$	0	0
$221$	26.7951	1.80243
$222$	0	0
$223$	−18.6856	−1.25128	−0.625641	−	0.780112i	$-0.715162\pi$
$223$	−18.6856	−1.25128	−0.625641	+	0.780112i	$0.715162\pi$
$224$	0	0
$225$	−1.17493	−0.0783284
$226$	0	0
$227$	−11.5721	−0.768065	−0.384033	−	0.923319i	$-0.625465\pi$
$227$	−11.5721	−0.768065	−0.384033	+	0.923319i	$0.625465\pi$
$228$	0	0
$229$	−21.2785	−1.40612	−0.703062	−	0.711128i	$-0.748184\pi$
$229$	−21.2785	−1.40612	−0.703062	+	0.711128i	$0.748184\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	14.1659	0.928039	0.464019	−	0.885825i	$-0.346407\pi$
$233$	14.1659	0.928039	0.464019	+	0.885825i	$0.346407\pi$
$234$	0	0
$235$	5.14504	0.335626
$236$	0	0
$237$	6.32224	0.410674
$238$	0	0
$239$	7.56145	0.489110	0.244555	−	0.969635i	$-0.421358\pi$
$239$	7.56145	0.489110	0.244555	+	0.969635i	$0.421358\pi$
$240$	0	0
$241$	4.70376	0.302996	0.151498	−	0.988458i	$-0.451590\pi$
$241$	4.70376	0.302996	0.151498	+	0.988458i	$0.451590\pi$
$242$	0	0
$243$	11.3885	0.730575
$244$	0	0
$245$	2.04081	0.130383
$246$	0	0
$247$	−21.0784	−1.34119
$248$	0	0
$249$	16.9203	1.07228
$250$	0	0
$251$	29.2749	1.84781	0.923907	−	0.382616i	$-0.124977\pi$
$251$	29.2749	1.84781	0.923907	+	0.382616i	$0.124977\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−6.35205	−0.397781
$256$	0	0
$257$	11.5154	0.718312	0.359156	−	0.933278i	$-0.383065\pi$
$257$	11.5154	0.718312	0.359156	+	0.933278i	$0.383065\pi$
$258$	0	0
$259$	17.4216	1.08253
$260$	0	0
$261$	12.5666	0.777851
$262$	0	0
$263$	−4.69882	−0.289742	−0.144871	−	0.989451i	$-0.546277\pi$
$263$	−4.69882	−0.289742	−0.144871	+	0.989451i	$0.546277\pi$
$264$	0	0
$265$	−12.2204	−0.750694
$266$	0	0
$267$	−5.84106	−0.357467
$268$	0	0
$269$	−12.6594	−0.771855	−0.385927	−	0.922529i	$-0.626118\pi$
$269$	−12.6594	−0.771855	−0.385927	+	0.922529i	$0.626118\pi$
$270$	0	0
$271$	−19.9465	−1.21166	−0.605832	−	0.795593i	$-0.707160\pi$
$271$	−19.9465	−1.21166	−0.605832	+	0.795593i	$0.707160\pi$
$272$	0	0
$273$	−23.1486	−1.40102
$274$	0	0
$275$	0	0
$276$	0	0
$277$	22.6688	1.36204	0.681018	−	0.732267i	$-0.261538\pi$
$277$	22.6688	1.36204	0.681018	+	0.732267i	$0.261538\pi$
$278$	0	0
$279$	4.11801	0.246539
$280$	0	0
$281$	1.59445	0.0951171	0.0475586	−	0.998868i	$-0.484856\pi$
$281$	1.59445	0.0951171	0.0475586	+	0.998868i	$0.484856\pi$
$282$	0	0
$283$	6.09579	0.362357	0.181178	−	0.983450i	$-0.442009\pi$
$283$	6.09579	0.362357	0.181178	+	0.983450i	$0.442009\pi$
$284$	0	0
$285$	4.99686	0.295989
$286$	0	0
$287$	−25.2538	−1.49069
$288$	0	0
$289$	5.10792	0.300466
$290$	0	0
$291$	3.97498	0.233017
$292$	0	0
$293$	23.4163	1.36800	0.683999	−	0.729483i	$-0.260239\pi$
$293$	23.4163	1.36800	0.683999	+	0.729483i	$0.260239\pi$
$294$	0	0
$295$	3.30528	0.192441
$296$	0	0
$297$	0	0
$298$	0	0
$299$	38.9720	2.25381
$300$	0	0
$301$	−8.44055	−0.486505
$302$	0	0
$303$	−3.64481	−0.209389
$304$	0	0
$305$	−10.7990	−0.618348
$306$	0	0
$307$	−2.62127	−0.149604	−0.0748019	−	0.997198i	$-0.523832\pi$
$307$	−2.62127	−0.149604	−0.0748019	+	0.997198i	$0.523832\pi$
$308$	0	0
$309$	21.7729	1.23862
$310$	0	0
$311$	−30.3266	−1.71967	−0.859833	−	0.510575i	$-0.829433\pi$
$311$	−30.3266	−1.71967	−0.859833	+	0.510575i	$0.829433\pi$
$312$	0	0
$313$	−12.3246	−0.696629	−0.348314	−	0.937378i	$-0.613246\pi$
$313$	−12.3246	−0.696629	−0.348314	+	0.937378i	$0.613246\pi$
$314$	0	0
$315$	−3.53276	−0.199048
$316$	0	0
$317$	−22.3636	−1.25606	−0.628032	−	0.778187i	$-0.716139\pi$
$317$	−22.3636	−1.25606	−0.628032	+	0.778187i	$0.716139\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−25.3650	−1.41574
$322$	0	0
$323$	−17.3913	−0.967676
$324$	0	0
$325$	−5.69877	−0.316111
$326$	0	0
$327$	20.1194	1.11260
$328$	0	0
$329$	15.4701	0.852893
$330$	0	0
$331$	−28.0559	−1.54209	−0.771045	−	0.636781i	$-0.780265\pi$
$331$	−28.0559	−1.54209	−0.771045	+	0.636781i	$0.780265\pi$
$332$	0	0
$333$	−6.80762	−0.373056
$334$	0	0
$335$	11.5328	0.630102
$336$	0	0
$337$	−30.7483	−1.67496	−0.837482	−	0.546464i	$-0.815973\pi$
$337$	−30.7483	−1.67496	−0.837482	+	0.546464i	$0.815973\pi$
$338$	0	0
$339$	6.82766	0.370828
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−14.9113	−0.805132
$344$	0	0
$345$	−9.23871	−0.497396
$346$	0	0
$347$	−21.4212	−1.14995	−0.574974	−	0.818172i	$-0.694988\pi$
$347$	−21.4212	−1.14995	−0.574974	+	0.818172i	$0.694988\pi$
$348$	0	0
$349$	−22.9134	−1.22653	−0.613264	−	0.789878i	$-0.710144\pi$
$349$	−22.9134	−1.22653	−0.613264	+	0.789878i	$0.710144\pi$
$350$	0	0
$351$	32.1418	1.71560
$352$	0	0
$353$	−1.25101	−0.0665843	−0.0332922	−	0.999446i	$-0.510599\pi$
$353$	−1.25101	−0.0665843	−0.0332922	+	0.999446i	$0.510599\pi$
$354$	0	0
$355$	2.05440	0.109036
$356$	0	0
$357$	−19.0993	−1.01084
$358$	0	0
$359$	−36.1951	−1.91030	−0.955152	−	0.296115i	$-0.904309\pi$
$359$	−36.1951	−1.91030	−0.955152	+	0.296115i	$0.904309\pi$
$360$	0	0
$361$	−5.31912	−0.279953
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−3.29589	−0.172515
$366$	0	0
$367$	−38.2633	−1.99733	−0.998664	−	0.0516727i	$-0.983545\pi$
$367$	−38.2633	−1.99733	−0.998664	+	0.0516727i	$0.983545\pi$
$368$	0	0
$369$	9.86811	0.513714
$370$	0	0
$371$	−36.7442	−1.90767
$372$	0	0
$373$	26.8965	1.39265	0.696324	−	0.717728i	$-0.254818\pi$
$373$	26.8965	1.39265	0.696324	+	0.717728i	$0.254818\pi$
$374$	0	0
$375$	1.35095	0.0697629
$376$	0	0
$377$	60.9519	3.13918
$378$	0	0
$379$	12.2189	0.627644	0.313822	−	0.949482i	$-0.398390\pi$
$379$	12.2189	0.627644	0.313822	+	0.949482i	$0.398390\pi$
$380$	0	0
$381$	−24.6708	−1.26392
$382$	0	0
$383$	−5.43228	−0.277577	−0.138788	−	0.990322i	$-0.544321\pi$
$383$	−5.43228	−0.277577	−0.138788	+	0.990322i	$0.544321\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	3.29820	0.167657
$388$	0	0
$389$	−6.49665	−0.329393	−0.164697	−	0.986344i	$-0.552664\pi$
$389$	−6.49665	−0.329393	−0.164697	+	0.986344i	$0.552664\pi$
$390$	0	0
$391$	32.1547	1.62614
$392$	0	0
$393$	−12.5547	−0.633301
$394$	0	0
$395$	4.67984	0.235468
$396$	0	0
$397$	−19.1186	−0.959536	−0.479768	−	0.877395i	$-0.659279\pi$
$397$	−19.1186	−0.959536	−0.479768	+	0.877395i	$0.659279\pi$
$398$	0	0
$399$	15.0245	0.752168
$400$	0	0
$401$	−12.9874	−0.648562	−0.324281	−	0.945961i	$-0.605122\pi$
$401$	−12.9874	−0.648562	−0.324281	+	0.945961i	$0.605122\pi$
$402$	0	0
$403$	19.9737	0.994960
$404$	0	0
$405$	−4.09477	−0.203471
$406$	0	0
$407$	0	0
$408$	0	0
$409$	2.03820	0.100783	0.0503913	−	0.998730i	$-0.483953\pi$
$409$	2.03820	0.100783	0.0503913	+	0.998730i	$0.483953\pi$
$410$	0	0
$411$	6.95506	0.343068
$412$	0	0
$413$	9.93829	0.489031
$414$	0	0
$415$	12.5247	0.614816
$416$	0	0
$417$	9.09281	0.445277
$418$	0	0
$419$	21.8002	1.06501	0.532504	−	0.846427i	$-0.321251\pi$
$419$	21.8002	1.06501	0.532504	+	0.846427i	$0.321251\pi$
$420$	0	0
$421$	−16.3277	−0.795763	−0.397882	−	0.917437i	$-0.630254\pi$
$421$	−16.3277	−0.795763	−0.397882	+	0.917437i	$0.630254\pi$
$422$	0	0
$423$	−6.04504	−0.293920
$424$	0	0
$425$	−4.70191	−0.228076
$426$	0	0
$427$	−32.4703	−1.57135
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−0.582599	−0.0280628	−0.0140314	−	0.999902i	$-0.504466\pi$
$431$	−0.582599	−0.0280628	−0.0140314	+	0.999902i	$0.504466\pi$
$432$	0	0
$433$	20.0423	0.963170	0.481585	−	0.876399i	$-0.340061\pi$
$433$	20.0423	0.963170	0.481585	+	0.876399i	$0.340061\pi$
$434$	0	0
$435$	−14.4493	−0.692791
$436$	0	0
$437$	−25.2946	−1.21001
$438$	0	0
$439$	−15.9036	−0.759038	−0.379519	−	0.925184i	$-0.623911\pi$
$439$	−15.9036	−0.759038	−0.379519	+	0.925184i	$0.623911\pi$
$440$	0	0
$441$	−2.39780	−0.114181
$442$	0	0
$443$	−12.1974	−0.579514	−0.289757	−	0.957100i	$-0.593575\pi$
$443$	−12.1974	−0.579514	−0.289757	+	0.957100i	$0.593575\pi$
$444$	0	0
$445$	−4.32366	−0.204961
$446$	0	0
$447$	7.25533	0.343165
$448$	0	0
$449$	15.5212	0.732489	0.366245	−	0.930519i	$-0.380643\pi$
$449$	15.5212	0.732489	0.366245	+	0.930519i	$0.380643\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	2.63048	0.123591
$454$	0	0
$455$	−17.1350	−0.803302
$456$	0	0
$457$	−20.1078	−0.940604	−0.470302	−	0.882506i	$-0.655855\pi$
$457$	−20.1078	−0.940604	−0.470302	+	0.882506i	$0.655855\pi$
$458$	0	0
$459$	26.5194	1.23782
$460$	0	0
$461$	−13.2592	−0.617541	−0.308770	−	0.951137i	$-0.599917\pi$
$461$	−13.2592	−0.617541	−0.308770	+	0.951137i	$0.599917\pi$
$462$	0	0
$463$	−8.73477	−0.405939	−0.202970	−	0.979185i	$-0.565059\pi$
$463$	−8.73477	−0.405939	−0.202970	+	0.979185i	$0.565059\pi$
$464$	0	0
$465$	−4.73497	−0.219579
$466$	0	0
$467$	−36.0072	−1.66621	−0.833107	−	0.553111i	$-0.813440\pi$
$467$	−36.0072	−1.66621	−0.833107	+	0.553111i	$0.813440\pi$
$468$	0	0
$469$	34.6766	1.60122
$470$	0	0
$471$	9.85641	0.454159
$472$	0	0
$473$	0	0
$474$	0	0
$475$	3.69877	0.169711
$476$	0	0
$477$	14.3581	0.657411
$478$	0	0
$479$	−12.5656	−0.574139	−0.287069	−	0.957910i	$-0.592681\pi$
$479$	−12.5656	−0.574139	−0.287069	+	0.957910i	$0.592681\pi$
$480$	0	0
$481$	−33.0192	−1.50554
$482$	0	0
$483$	−27.7789	−1.26398
$484$	0	0
$485$	2.94235	0.133605
$486$	0	0
$487$	29.8003	1.35038	0.675189	−	0.737644i	$-0.264062\pi$
$487$	29.8003	1.35038	0.675189	+	0.737644i	$0.264062\pi$
$488$	0	0
$489$	5.35426	0.242128
$490$	0	0
$491$	−8.62233	−0.389120	−0.194560	−	0.980891i	$-0.562328\pi$
$491$	−8.62233	−0.389120	−0.194560	+	0.980891i	$0.562328\pi$
$492$	0	0
$493$	50.2898	2.26494
$494$	0	0
$495$	0	0
$496$	0	0
$497$	6.17716	0.277083
$498$	0	0
$499$	33.2734	1.48952	0.744761	−	0.667332i	$-0.232564\pi$
$499$	33.2734	1.48952	0.744761	+	0.667332i	$0.232564\pi$
$500$	0	0
$501$	10.0178	0.447560
$502$	0	0
$503$	9.56522	0.426492	0.213246	−	0.976999i	$-0.431596\pi$
$503$	9.56522	0.426492	0.213246	+	0.976999i	$0.431596\pi$
$504$	0	0
$505$	−2.69796	−0.120057
$506$	0	0
$507$	26.3111	1.16852
$508$	0	0
$509$	−21.4598	−0.951187	−0.475593	−	0.879665i	$-0.657767\pi$
$509$	−21.4598	−0.951187	−0.475593	+	0.879665i	$0.657767\pi$
$510$	0	0
$511$	−9.91007	−0.438396
$512$	0	0
$513$	−20.8615	−0.921059
$514$	0	0
$515$	16.1167	0.710187
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−11.6179	−0.509970
$520$	0	0
$521$	12.3687	0.541881	0.270941	−	0.962596i	$-0.412665\pi$
$521$	12.3687	0.541881	0.270941	+	0.962596i	$0.412665\pi$
$522$	0	0
$523$	13.9337	0.609280	0.304640	−	0.952468i	$-0.401464\pi$
$523$	13.9337	0.609280	0.304640	+	0.952468i	$0.401464\pi$
$524$	0	0
$525$	4.06204	0.177282
$526$	0	0
$527$	16.4798	0.717870
$528$	0	0
$529$	23.7673	1.03336
$530$	0	0
$531$	−3.88346	−0.168528
$532$	0	0
$533$	47.8635	2.07320
$534$	0	0
$535$	−18.7756	−0.811741
$536$	0	0
$537$	1.52663	0.0658790
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−38.0571	−1.63620	−0.818100	−	0.575076i	$-0.804973\pi$
$541$	−38.0571	−1.63620	−0.818100	+	0.575076i	$0.804973\pi$
$542$	0	0
$543$	13.4848	0.578688
$544$	0	0
$545$	14.8927	0.637934
$546$	0	0
$547$	33.5537	1.43465	0.717325	−	0.696738i	$-0.245366\pi$
$547$	33.5537	1.43465	0.717325	+	0.696738i	$0.245366\pi$
$548$	0	0
$549$	12.6880	0.541510
$550$	0	0
$551$	−39.5607	−1.68534
$552$	0	0
$553$	14.0713	0.598373
$554$	0	0
$555$	7.82754	0.332261
$556$	0	0
$557$	45.9285	1.94605	0.973025	−	0.230698i	$-0.0741008\pi$
$557$	45.9285	1.94605	0.973025	+	0.230698i	$0.0741008\pi$
$558$	0	0
$559$	15.9973	0.676616
$560$	0	0
$561$	0	0
$562$	0	0
$563$	19.1695	0.807898	0.403949	−	0.914781i	$-0.367637\pi$
$563$	19.1695	0.807898	0.403949	+	0.914781i	$0.367637\pi$
$564$	0	0
$565$	5.05396	0.212622
$566$	0	0
$567$	−12.3121	−0.517061
$568$	0	0
$569$	16.7995	0.704271	0.352136	−	0.935949i	$-0.385456\pi$
$569$	16.7995	0.704271	0.352136	+	0.935949i	$0.385456\pi$
$570$	0	0
$571$	−33.5389	−1.40356	−0.701780	−	0.712394i	$-0.747611\pi$
$571$	−33.5389	−1.40356	−0.701780	+	0.712394i	$0.747611\pi$
$572$	0	0
$573$	24.9914	1.04403
$574$	0	0
$575$	−6.83866	−0.285192
$576$	0	0
$577$	18.7364	0.780008	0.390004	−	0.920813i	$-0.372474\pi$
$577$	18.7364	0.780008	0.390004	+	0.920813i	$0.372474\pi$
$578$	0	0
$579$	−12.6795	−0.526940
$580$	0	0
$581$	37.6593	1.56237
$582$	0	0
$583$	0	0
$584$	0	0
$585$	6.69563	0.276830
$586$	0	0
$587$	6.76096	0.279055	0.139527	−	0.990218i	$-0.455442\pi$
$587$	6.76096	0.279055	0.139527	+	0.990218i	$0.455442\pi$
$588$	0	0
$589$	−12.9639	−0.534167
$590$	0	0
$591$	18.1779	0.747739
$592$	0	0
$593$	15.3645	0.630946	0.315473	−	0.948935i	$-0.397837\pi$
$593$	15.3645	0.630946	0.315473	+	0.948935i	$0.397837\pi$
$594$	0	0
$595$	−14.1377	−0.579588
$596$	0	0
$597$	31.7764	1.30052
$598$	0	0
$599$	−4.76071	−0.194517	−0.0972587	−	0.995259i	$-0.531007\pi$
$599$	−4.76071	−0.194517	−0.0972587	+	0.995259i	$0.531007\pi$
$600$	0	0
$601$	2.84016	0.115853	0.0579263	−	0.998321i	$-0.481551\pi$
$601$	2.84016	0.115853	0.0579263	+	0.998321i	$0.481551\pi$
$602$	0	0
$603$	−13.5501	−0.551804
$604$	0	0
$605$	0	0
$606$	0	0
$607$	22.3497	0.907148	0.453574	−	0.891219i	$-0.350149\pi$
$607$	22.3497	0.907148	0.453574	+	0.891219i	$0.350149\pi$
$608$	0	0
$609$	−43.4461	−1.76052
$610$	0	0
$611$	−29.3204	−1.18618
$612$	0	0
$613$	19.2315	0.776754	0.388377	−	0.921501i	$-0.373036\pi$
$613$	19.2315	0.776754	0.388377	+	0.921501i	$0.373036\pi$
$614$	0	0
$615$	−11.3466	−0.457537
$616$	0	0
$617$	−14.8714	−0.598700	−0.299350	−	0.954143i	$-0.596770\pi$
$617$	−14.8714	−0.598700	−0.299350	+	0.954143i	$0.596770\pi$
$618$	0	0
$619$	−34.7781	−1.39785	−0.698925	−	0.715195i	$-0.746338\pi$
$619$	−34.7781	−1.39785	−0.698925	+	0.715195i	$0.746338\pi$
$620$	0	0
$621$	38.5709	1.54780
$622$	0	0
$623$	−13.0003	−0.520848
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−27.2433	−1.08626
$630$	0	0
$631$	−19.2646	−0.766913	−0.383456	−	0.923559i	$-0.625266\pi$
$631$	−19.2646	−0.766913	−0.383456	+	0.923559i	$0.625266\pi$
$632$	0	0
$633$	7.64486	0.303856
$634$	0	0
$635$	−18.2618	−0.724697
$636$	0	0
$637$	−11.6301	−0.460802
$638$	0	0
$639$	−2.41377	−0.0954872
$640$	0	0
$641$	−41.6297	−1.64428	−0.822138	−	0.569289i	$-0.807219\pi$
$641$	−41.6297	−1.64428	−0.822138	+	0.569289i	$0.807219\pi$
$642$	0	0
$643$	12.9034	0.508862	0.254431	−	0.967091i	$-0.418112\pi$
$643$	12.9034	0.508862	0.254431	+	0.967091i	$0.418112\pi$
$644$	0	0
$645$	−3.79234	−0.149323
$646$	0	0
$647$	−8.73477	−0.343399	−0.171700	−	0.985149i	$-0.554926\pi$
$647$	−8.73477	−0.343399	−0.171700	+	0.985149i	$0.554926\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−14.2371	−0.557995
$652$	0	0
$653$	−41.5489	−1.62594	−0.812968	−	0.582308i	$-0.802150\pi$
$653$	−41.5489	−1.62594	−0.812968	+	0.582308i	$0.802150\pi$
$654$	0	0
$655$	−9.29322	−0.363116
$656$	0	0
$657$	3.87243	0.151078
$658$	0	0
$659$	−34.8683	−1.35827	−0.679137	−	0.734011i	$-0.737646\pi$
$659$	−34.8683	−1.35827	−0.679137	+	0.734011i	$0.737646\pi$
$660$	0	0
$661$	−3.05492	−0.118823	−0.0594113	−	0.998234i	$-0.518922\pi$
$661$	−3.05492	−0.118823	−0.0594113	+	0.998234i	$0.518922\pi$
$662$	0	0
$663$	36.1989	1.40585
$664$	0	0
$665$	11.1214	0.431271
$666$	0	0
$667$	73.1438	2.83214
$668$	0	0
$669$	−25.2434	−0.975966
$670$	0	0
$671$	0	0
$672$	0	0
$673$	6.56543	0.253079	0.126539	−	0.991962i	$-0.459613\pi$
$673$	6.56543	0.253079	0.126539	+	0.991962i	$0.459613\pi$
$674$	0	0
$675$	−5.64013	−0.217089
$676$	0	0
$677$	−8.92088	−0.342857	−0.171429	−	0.985197i	$-0.554838\pi$
$677$	−8.92088	−0.342857	−0.171429	+	0.985197i	$0.554838\pi$
$678$	0	0
$679$	8.84705	0.339519
$680$	0	0
$681$	−15.6333	−0.599070
$682$	0	0
$683$	−9.77717	−0.374113	−0.187057	−	0.982349i	$-0.559895\pi$
$683$	−9.77717	−0.374113	−0.187057	+	0.982349i	$0.559895\pi$
$684$	0	0
$685$	5.14826	0.196705
$686$	0	0
$687$	−28.7463	−1.09674
$688$	0	0
$689$	69.6413	2.65312
$690$	0	0
$691$	−6.19944	−0.235838	−0.117919	−	0.993023i	$-0.537622\pi$
$691$	−6.19944	−0.235838	−0.117919	+	0.993023i	$0.537622\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	6.73066	0.255309
$696$	0	0
$697$	39.4910	1.49583
$698$	0	0
$699$	19.1375	0.723845
$700$	0	0
$701$	−18.7398	−0.707791	−0.353896	−	0.935285i	$-0.615143\pi$
$701$	−18.7398	−0.707791	−0.353896	+	0.935285i	$0.615143\pi$
$702$	0	0
$703$	21.4310	0.808285
$704$	0	0
$705$	6.95071	0.261779
$706$	0	0
$707$	−8.11220	−0.305091
$708$	0	0
$709$	−48.8505	−1.83462	−0.917309	−	0.398176i	$-0.869643\pi$
$709$	−48.8505	−1.83462	−0.917309	+	0.398176i	$0.869643\pi$
$710$	0	0
$711$	−5.49846	−0.206208
$712$	0	0
$713$	23.9689	0.897643
$714$	0	0
$715$	0	0
$716$	0	0
$717$	10.2152	0.381492
$718$	0	0
$719$	−10.4572	−0.389988	−0.194994	−	0.980804i	$-0.562469\pi$
$719$	−10.4572	−0.389988	−0.194994	+	0.980804i	$0.562469\pi$
$720$	0	0
$721$	48.4597	1.80473
$722$	0	0
$723$	6.35456	0.236329
$724$	0	0
$725$	−10.6956	−0.397226
$726$	0	0
$727$	−32.6965	−1.21265	−0.606323	−	0.795219i	$-0.707356\pi$
$727$	−32.6965	−1.21265	−0.606323	+	0.795219i	$0.707356\pi$
$728$	0	0
$729$	27.6697	1.02480
$730$	0	0
$731$	13.1990	0.488183
$732$	0	0
$733$	−4.32167	−0.159624	−0.0798122	−	0.996810i	$-0.525432\pi$
$733$	−4.32167	−0.159624	−0.0798122	+	0.996810i	$0.525432\pi$
$734$	0	0
$735$	2.75704	0.101695
$736$	0	0
$737$	0	0
$738$	0	0
$739$	8.71832	0.320709	0.160354	−	0.987060i	$-0.448736\pi$
$739$	8.71832	0.320709	0.160354	+	0.987060i	$0.448736\pi$
$740$	0	0
$741$	−28.4760	−1.04609
$742$	0	0
$743$	−32.8875	−1.20653	−0.603263	−	0.797542i	$-0.706133\pi$
$743$	−32.8875	−1.20653	−0.603263	+	0.797542i	$0.706133\pi$
$744$	0	0
$745$	5.37053	0.196761
$746$	0	0
$747$	−14.7157	−0.538418
$748$	0	0
$749$	−56.4544	−2.06280
$750$	0	0
$751$	−43.9740	−1.60464	−0.802318	−	0.596897i	$-0.796400\pi$
$751$	−43.9740	−1.60464	−0.802318	+	0.596897i	$0.796400\pi$
$752$	0	0
$753$	39.5490	1.44125
$754$	0	0
$755$	1.94713	0.0708634
$756$	0	0
$757$	−21.9285	−0.797004	−0.398502	−	0.917168i	$-0.630470\pi$
$757$	−21.9285	−0.797004	−0.398502	+	0.917168i	$0.630470\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−24.3364	−0.882195	−0.441097	−	0.897459i	$-0.645411\pi$
$761$	−24.3364	−0.882195	−0.441097	+	0.897459i	$0.645411\pi$
$762$	0	0
$763$	44.7793	1.62112
$764$	0	0
$765$	5.52439	0.199735
$766$	0	0
$767$	−18.8360	−0.680129
$768$	0	0
$769$	−28.4041	−1.02428	−0.512138	−	0.858903i	$-0.671146\pi$
$769$	−28.4041	−1.02428	−0.512138	+	0.858903i	$0.671146\pi$
$770$	0	0
$771$	15.5568	0.560264
$772$	0	0
$773$	−6.15809	−0.221491	−0.110746	−	0.993849i	$-0.535324\pi$
$773$	−6.15809	−0.221491	−0.110746	+	0.993849i	$0.535324\pi$
$774$	0	0
$775$	−3.50491	−0.125900
$776$	0	0
$777$	23.5358	0.844342
$778$	0	0
$779$	−31.0657	−1.11304
$780$	0	0
$781$	0	0
$782$	0	0
$783$	60.3247	2.15583
$784$	0	0
$785$	7.29589	0.260402
$786$	0	0
$787$	14.4250	0.514195	0.257098	−	0.966385i	$-0.417234\pi$
$787$	14.4250	0.514195	0.257098	+	0.966385i	$0.417234\pi$
$788$	0	0
$789$	−6.34789	−0.225991
$790$	0	0
$791$	15.1962	0.540315
$792$	0	0
$793$	61.5409	2.18538
$794$	0	0
$795$	−16.5092	−0.585521
$796$	0	0
$797$	13.7588	0.487362	0.243681	−	0.969855i	$-0.421645\pi$
$797$	13.7588	0.487362	0.243681	+	0.969855i	$0.421645\pi$
$798$	0	0
$799$	−24.1915	−0.855834
$800$	0	0
$801$	5.07998	0.179492
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−20.5625	−0.724731
$806$	0	0
$807$	−17.1022	−0.602026
$808$	0	0
$809$	9.68668	0.340566	0.170283	−	0.985395i	$-0.445532\pi$
$809$	9.68668	0.340566	0.170283	+	0.985395i	$0.445532\pi$
$810$	0	0
$811$	−6.12820	−0.215190	−0.107595	−	0.994195i	$-0.534315\pi$
$811$	−6.12820	−0.215190	−0.107595	+	0.994195i	$0.534315\pi$
$812$	0	0
$813$	−26.9468	−0.945065
$814$	0	0
$815$	3.96332	0.138829
$816$	0	0
$817$	−10.3830	−0.363256
$818$	0	0
$819$	20.1324	0.703482
$820$	0	0
$821$	23.8825	0.833505	0.416753	−	0.909020i	$-0.363168\pi$
$821$	23.8825	0.833505	0.416753	+	0.909020i	$0.363168\pi$
$822$	0	0
$823$	−34.2553	−1.19406	−0.597032	−	0.802217i	$-0.703654\pi$
$823$	−34.2553	−1.19406	−0.597032	+	0.802217i	$0.703654\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	6.94363	0.241454	0.120727	−	0.992686i	$-0.461477\pi$
$827$	6.94363	0.241454	0.120727	+	0.992686i	$0.461477\pi$
$828$	0	0
$829$	30.1998	1.04888	0.524442	−	0.851446i	$-0.324274\pi$
$829$	30.1998	1.04888	0.524442	+	0.851446i	$0.324274\pi$
$830$	0	0
$831$	30.6245	1.06235
$832$	0	0
$833$	−9.59570	−0.332471
$834$	0	0
$835$	7.41532	0.256618
$836$	0	0
$837$	19.7682	0.683288
$838$	0	0
$839$	50.0106	1.72656	0.863280	−	0.504726i	$-0.168407\pi$
$839$	50.0106	1.72656	0.863280	+	0.504726i	$0.168407\pi$
$840$	0	0
$841$	85.3965	2.94471
$842$	0	0
$843$	2.15403	0.0741888
$844$	0	0
$845$	19.4760	0.669993
$846$	0	0
$847$	0	0
$848$	0	0
$849$	8.23512	0.282629
$850$	0	0
$851$	−39.6238	−1.35829
$852$	0	0
$853$	−40.9697	−1.40278	−0.701388	−	0.712780i	$-0.747436\pi$
$853$	−40.9697	−1.40278	−0.701388	+	0.712780i	$0.747436\pi$
$854$	0	0
$855$	−4.34578	−0.148622
$856$	0	0
$857$	−45.8893	−1.56755	−0.783774	−	0.621046i	$-0.786708\pi$
$857$	−45.8893	−1.56755	−0.783774	+	0.621046i	$0.786708\pi$
$858$	0	0
$859$	−5.42402	−0.185065	−0.0925327	−	0.995710i	$-0.529496\pi$
$859$	−5.42402	−0.185065	−0.0925327	+	0.995710i	$0.529496\pi$
$860$	0	0
$861$	−34.1168	−1.16270
$862$	0	0
$863$	24.7119	0.841202	0.420601	−	0.907246i	$-0.361819\pi$
$863$	24.7119	0.841202	0.420601	+	0.907246i	$0.361819\pi$
$864$	0	0
$865$	−8.59979	−0.292402
$866$	0	0
$867$	6.90056	0.234355
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−65.7225	−2.22692
$872$	0	0
$873$	−3.45705	−0.117003
$874$	0	0
$875$	3.00679	0.101648
$876$	0	0
$877$	33.0006	1.11435	0.557175	−	0.830395i	$-0.311885\pi$
$877$	33.0006	1.11435	0.557175	+	0.830395i	$0.311885\pi$
$878$	0	0
$879$	31.6344	1.06700
$880$	0	0
$881$	−24.5289	−0.826401	−0.413200	−	0.910640i	$-0.635589\pi$
$881$	−24.5289	−0.826401	−0.413200	+	0.910640i	$0.635589\pi$
$882$	0	0
$883$	24.7421	0.832639	0.416320	−	0.909218i	$-0.363320\pi$
$883$	24.7421	0.832639	0.416320	+	0.909218i	$0.363320\pi$
$884$	0	0
$885$	4.46528	0.150099
$886$	0	0
$887$	41.1605	1.38203	0.691017	−	0.722838i	$-0.257163\pi$
$887$	41.1605	1.38203	0.691017	+	0.722838i	$0.257163\pi$
$888$	0	0
$889$	−54.9095	−1.84160
$890$	0	0
$891$	0	0
$892$	0	0
$893$	19.0303	0.636825
$894$	0	0
$895$	1.13004	0.0377731
$896$	0	0
$897$	52.6493	1.75791
$898$	0	0
$899$	37.4872	1.25027
$900$	0	0
$901$	57.4592	1.91424
$902$	0	0
$903$	−11.4028	−0.379461
$904$	0	0
$905$	9.98170	0.331803
$906$	0	0
$907$	27.4276	0.910717	0.455358	−	0.890308i	$-0.349511\pi$
$907$	27.4276	0.910717	0.455358	+	0.890308i	$0.349511\pi$
$908$	0	0
$909$	3.16990	0.105139
$910$	0	0
$911$	−3.80200	−0.125966	−0.0629830	−	0.998015i	$-0.520061\pi$
$911$	−3.80200	−0.125966	−0.0629830	+	0.998015i	$0.520061\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−14.5889	−0.482294
$916$	0	0
$917$	−27.9428	−0.922753
$918$	0	0
$919$	27.4379	0.905093	0.452546	−	0.891741i	$-0.350516\pi$
$919$	27.4379	0.905093	0.452546	+	0.891741i	$0.350516\pi$
$920$	0	0
$921$	−3.54121	−0.116687
$922$	0	0
$923$	−11.7075	−0.385359
$924$	0	0
$925$	5.79409	0.190508
$926$	0	0
$927$	−18.9359	−0.621938
$928$	0	0
$929$	5.96209	0.195610	0.0978048	−	0.995206i	$-0.468818\pi$
$929$	5.96209	0.195610	0.0978048	+	0.995206i	$0.468818\pi$
$930$	0	0
$931$	7.54849	0.247392
$932$	0	0
$933$	−40.9699	−1.34129
$934$	0	0
$935$	0	0
$936$	0	0
$937$	23.5196	0.768353	0.384176	−	0.923260i	$-0.374485\pi$
$937$	23.5196	0.768353	0.384176	+	0.923260i	$0.374485\pi$
$938$	0	0
$939$	−16.6500	−0.543352
$940$	0	0
$941$	−45.3577	−1.47862	−0.739309	−	0.673366i	$-0.764848\pi$
$941$	−45.3577	−1.47862	−0.739309	+	0.673366i	$0.764848\pi$
$942$	0	0
$943$	57.4374	1.87042
$944$	0	0
$945$	−16.9587	−0.551667
$946$	0	0
$947$	52.0311	1.69078	0.845391	−	0.534148i	$-0.179367\pi$
$947$	52.0311	1.69078	0.845391	+	0.534148i	$0.179367\pi$
$948$	0	0
$949$	18.7825	0.609706
$950$	0	0
$951$	−30.2122	−0.979697
$952$	0	0
$953$	−26.5884	−0.861284	−0.430642	−	0.902523i	$-0.641713\pi$
$953$	−26.5884	−0.861284	−0.430642	+	0.902523i	$0.641713\pi$
$954$	0	0
$955$	18.4991	0.598617
$956$	0	0
$957$	0	0
$958$	0	0
$959$	15.4798	0.499868
$960$	0	0
$961$	−18.7156	−0.603729
$962$	0	0
$963$	22.0600	0.710872
$964$	0	0
$965$	−9.38556	−0.302132
$966$	0	0
$967$	14.5187	0.466890	0.233445	−	0.972370i	$-0.425000\pi$
$967$	14.5187	0.466890	0.233445	+	0.972370i	$0.425000\pi$
$968$	0	0
$969$	−23.4948	−0.754761
$970$	0	0
$971$	55.0093	1.76533	0.882667	−	0.469999i	$-0.155746\pi$
$971$	55.0093	1.76533	0.882667	+	0.469999i	$0.155746\pi$
$972$	0	0
$973$	20.2377	0.648791
$974$	0	0
$975$	−7.69877	−0.246558
$976$	0	0
$977$	37.7764	1.20857	0.604287	−	0.796767i	$-0.293458\pi$
$977$	37.7764	1.20857	0.604287	+	0.796767i	$0.293458\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−17.4978	−0.558663
$982$	0	0
$983$	−20.0610	−0.639845	−0.319922	−	0.947444i	$-0.603657\pi$
$983$	−20.0610	−0.639845	−0.319922	+	0.947444i	$0.603657\pi$
$984$	0	0
$985$	13.4556	0.428732
$986$	0	0
$987$	20.8994	0.665234
$988$	0	0
$989$	19.1972	0.610436
$990$	0	0
$991$	−56.3026	−1.78851	−0.894255	−	0.447557i	$-0.852294\pi$
$991$	−56.3026	−1.78851	−0.894255	+	0.447557i	$0.852294\pi$
$992$	0	0
$993$	−37.9021	−1.20279
$994$	0	0
$995$	23.5214	0.745680
$996$	0	0
$997$	−27.9469	−0.885087	−0.442544	−	0.896747i	$-0.645924\pi$
$997$	−27.9469	−0.885087	−0.442544	+	0.896747i	$0.645924\pi$
$998$	0	0
$999$	−32.6794	−1.03393

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4840.2.a.bc.1.5	✓	6
4.3	odd	2		9680.2.a.db.1.2		6
11.10	odd	2		4840.2.a.bd.1.5	yes	6
44.43	even	2		9680.2.a.da.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4840.2.a.bc.1.5	✓	6	1.1	even	1	trivial
4840.2.a.bd.1.5	yes	6	11.10	odd	2
9680.2.a.da.1.2		6	44.43	even	2
9680.2.a.db.1.2		6	4.3	odd	2

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:	\( N \)	\(=\)	\( 4840 = 2^{3} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4840.a (trivial)

\(f(q)\)	\(=\)	\(q+1.35095 q^{3} +1.00000 q^{5} +3.00679 q^{7} -1.17493 q^{9} -5.69877 q^{13} +1.35095 q^{15} -4.70191 q^{17} +3.69877 q^{19} +4.06204 q^{21} -6.83866 q^{23} +1.00000 q^{25} -5.64013 q^{27} -10.6956 q^{29} -3.50491 q^{31} +3.00679 q^{35} +5.79409 q^{37} -7.69877 q^{39} -8.39893 q^{41} -2.80716 q^{43} -1.17493 q^{45} +5.14504 q^{47} +2.04081 q^{49} -6.35205 q^{51} -12.2204 q^{53} +4.99686 q^{57} +3.30528 q^{59} -10.7990 q^{61} -3.53276 q^{63} -5.69877 q^{65} +11.5328 q^{67} -9.23871 q^{69} +2.05440 q^{71} -3.29589 q^{73} +1.35095 q^{75} +4.67984 q^{79} -4.09477 q^{81} +12.5247 q^{83} -4.70191 q^{85} -14.4493 q^{87} -4.32366 q^{89} -17.1350 q^{91} -4.73497 q^{93} +3.69877 q^{95} +2.94235 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{3} + 6 q^{5} - 4 q^{7} + 4 q^{9} - 2 q^{15} - 8 q^{17} - 12 q^{19} + 8 q^{21} - 8 q^{23} + 6 q^{25} - 14 q^{27} - 16 q^{29} - 4 q^{31} - 4 q^{35} + 8 q^{37} - 12 q^{39} - 32 q^{41} + 4 q^{43}+ \cdots - 12 q^{95}+O(q^{100}) \) 6 * q - 2 * q^3 + 6 * q^5 - 4 * q^7 + 4 * q^9 - 2 * q^15 - 8 * q^17 - 12 * q^19 + 8 * q^21 - 8 * q^23 + 6 * q^25 - 14 * q^27 - 16 * q^29 - 4 * q^31 - 4 * q^35 + 8 * q^37 - 12 * q^39 - 32 * q^41 + 4 * q^43 + 4 * q^45 - 6 * q^47 + 16 * q^49 - 40 * q^51 + 8 * q^53 + 16 * q^57 + 4 * q^59 - 16 * q^61 - 28 * q^63 - 2 * q^67 + 8 * q^69 - 28 * q^71 - 16 * q^73 - 2 * q^75 - 10 * q^81 - 12 * q^83 - 8 * q^85 + 24 * q^87 + 18 * q^89 - 24 * q^91 + 20 * q^93 - 12 * q^95

Introduction

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