LMFDB - Embedded newform 6016.2.a.o.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6016,2,Mod(1,6016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6016, 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("6016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6016 = 2^{7} \cdot 47 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6016.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:	$48.0380018560$
Analytic rank:	$0$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 4 x^{12} - 22 x^{11} + 96 x^{10} + 163 x^{9} - 840 x^{8} - 438 x^{7} + 3358 x^{6} + 5 x^{5} + \cdots - 832 $ x^13 - 4x^12 - 22x^11 + 96x^10 + 163x^9 - 840x^8 - 438x^7 + 3358x^6 + 5x^5 - 6230x^4 + 1488x^3 + 4720x^2 - 1440x - 832
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.44796$ of defining polynomial
Character	$\chi$	$=$	6016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.44796 q^{3} -2.61725 q^{5} +0.969862 q^{7} +2.99250 q^{9} +O(q^{10})$	$q-2.44796 q^{3} -2.61725 q^{5} +0.969862 q^{7} +2.99250 q^{9} -0.473992 q^{11} -6.64056 q^{13} +6.40692 q^{15} +3.15890 q^{17} -7.48180 q^{19} -2.37418 q^{21} -7.33491 q^{23} +1.85000 q^{25} +0.0183678 q^{27} -0.238789 q^{29} +0.258050 q^{31} +1.16031 q^{33} -2.53837 q^{35} -5.02125 q^{37} +16.2558 q^{39} -6.95568 q^{41} +9.54047 q^{43} -7.83211 q^{45} -1.00000 q^{47} -6.05937 q^{49} -7.73286 q^{51} -11.0106 q^{53} +1.24055 q^{55} +18.3151 q^{57} -0.263264 q^{59} +2.68534 q^{61} +2.90231 q^{63} +17.3800 q^{65} +5.43900 q^{67} +17.9556 q^{69} -4.61343 q^{71} -1.81707 q^{73} -4.52871 q^{75} -0.459707 q^{77} -12.5437 q^{79} -9.02245 q^{81} -9.15946 q^{83} -8.26764 q^{85} +0.584546 q^{87} -17.3538 q^{89} -6.44042 q^{91} -0.631695 q^{93} +19.5817 q^{95} -13.0359 q^{97} -1.41842 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q + 4 q^{3} - 6 q^{5} - 2 q^{7} + 21 q^{9}+O(q^{10}) $ 13 * q + 4 * q^3 - 6 * q^5 - 2 * q^7 + 21 * q^9	$ 13 q + 4 q^{3} - 6 q^{5} - 2 q^{7} + 21 q^{9} + 10 q^{11} - 4 q^{13} + 14 q^{15} + 10 q^{17} + 8 q^{19} - 10 q^{21} + 18 q^{23} + 23 q^{25} + 16 q^{27} - 14 q^{29} + 4 q^{31} + 14 q^{33} + 14 q^{35} - 16 q^{37} + 12 q^{39} + 10 q^{41} + 12 q^{43} - 10 q^{45} - 13 q^{47} + 9 q^{49} + 22 q^{51} - 26 q^{53} - 2 q^{55} + 20 q^{57} + 30 q^{59} - 18 q^{61} + 12 q^{63} - 4 q^{65} + 4 q^{67} - 2 q^{69} + 36 q^{71} + 10 q^{73} + 38 q^{75} - 42 q^{77} + 21 q^{81} + 12 q^{83} - 4 q^{85} + 6 q^{87} + 50 q^{89} - 4 q^{91} - 52 q^{93} + 8 q^{95} - 10 q^{97} + 34 q^{99}+O(q^{100}) $ 13 * q + 4 * q^3 - 6 * q^5 - 2 * q^7 + 21 * q^9 + 10 * q^11 - 4 * q^13 + 14 * q^15 + 10 * q^17 + 8 * q^19 - 10 * q^21 + 18 * q^23 + 23 * q^25 + 16 * q^27 - 14 * q^29 + 4 * q^31 + 14 * q^33 + 14 * q^35 - 16 * q^37 + 12 * q^39 + 10 * q^41 + 12 * q^43 - 10 * q^45 - 13 * q^47 + 9 * q^49 + 22 * q^51 - 26 * q^53 - 2 * q^55 + 20 * q^57 + 30 * q^59 - 18 * q^61 + 12 * q^63 - 4 * q^65 + 4 * q^67 - 2 * q^69 + 36 * q^71 + 10 * q^73 + 38 * q^75 - 42 * q^77 + 21 * q^81 + 12 * q^83 - 4 * q^85 + 6 * q^87 + 50 * q^89 - 4 * q^91 - 52 * q^93 + 8 * q^95 - 10 * q^97 + 34 * 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
$3$	−2.44796	−1.41333	−0.706665	−	0.707549i	$-0.749801\pi$
$3$	−2.44796	−1.41333	−0.706665	+	0.707549i	$0.749801\pi$
$4$	0	0
$5$	−2.61725	−1.17047	−0.585235	−	0.810864i	$-0.698998\pi$
$5$	−2.61725	−1.17047	−0.585235	+	0.810864i	$0.698998\pi$
$6$	0	0
$7$	0.969862	0.366573	0.183287	−	0.983060i	$-0.441326\pi$
$7$	0.969862	0.366573	0.183287	+	0.983060i	$0.441326\pi$
$8$	0	0
$9$	2.99250	0.997499
$10$	0	0
$11$	−0.473992	−0.142914	−0.0714569	−	0.997444i	$-0.522765\pi$
$11$	−0.473992	−0.142914	−0.0714569	+	0.997444i	$0.522765\pi$
$12$	0	0
$13$	−6.64056	−1.84176	−0.920880	−	0.389847i	$-0.872528\pi$
$13$	−6.64056	−1.84176	−0.920880	+	0.389847i	$0.872528\pi$
$14$	0	0
$15$	6.40692	1.65426
$16$	0	0
$17$	3.15890	0.766147	0.383073	−	0.923718i	$-0.374866\pi$
$17$	3.15890	0.766147	0.383073	+	0.923718i	$0.374866\pi$
$18$	0	0
$19$	−7.48180	−1.71644	−0.858222	−	0.513279i	$-0.828431\pi$
$19$	−7.48180	−1.71644	−0.858222	+	0.513279i	$0.828431\pi$
$20$	0	0
$21$	−2.37418	−0.518089
$22$	0	0
$23$	−7.33491	−1.52944	−0.764718	−	0.644366i	$-0.777122\pi$
$23$	−7.33491	−1.52944	−0.764718	+	0.644366i	$0.777122\pi$
$24$	0	0
$25$	1.85000	0.369999
$26$	0	0
$27$	0.0183678	0.00353489
$28$	0	0
$29$	−0.238789	−0.0443420	−0.0221710	−	0.999754i	$-0.507058\pi$
$29$	−0.238789	−0.0443420	−0.0221710	+	0.999754i	$0.507058\pi$
$30$	0	0
$31$	0.258050	0.0463471	0.0231735	−	0.999731i	$-0.492623\pi$
$31$	0.258050	0.0463471	0.0231735	+	0.999731i	$0.492623\pi$
$32$	0	0
$33$	1.16031	0.201984
$34$	0	0
$35$	−2.53837	−0.429063
$36$	0	0
$37$	−5.02125	−0.825488	−0.412744	−	0.910847i	$-0.635430\pi$
$37$	−5.02125	−0.825488	−0.412744	+	0.910847i	$0.635430\pi$
$38$	0	0
$39$	16.2558	2.60301
$40$	0	0
$41$	−6.95568	−1.08629	−0.543147	−	0.839637i	$-0.682767\pi$
$41$	−6.95568	−1.08629	−0.543147	+	0.839637i	$0.682767\pi$
$42$	0	0
$43$	9.54047	1.45491	0.727454	−	0.686156i	$-0.240703\pi$
$43$	9.54047	1.45491	0.727454	+	0.686156i	$0.240703\pi$
$44$	0	0
$45$	−7.83211	−1.16754
$46$	0	0
$47$	−1.00000	−0.145865
$47$	−1.00000	−0.145865
$48$	0	0
$49$	−6.05937	−0.865624
$50$	0	0
$51$	−7.73286	−1.08282
$52$	0	0
$53$	−11.0106	−1.51242	−0.756208	−	0.654331i	$-0.772950\pi$
$53$	−11.0106	−1.51242	−0.756208	+	0.654331i	$0.772950\pi$
$54$	0	0
$55$	1.24055	0.167276
$56$	0	0
$57$	18.3151	2.42590
$58$	0	0
$59$	−0.263264	−0.0342740	−0.0171370	−	0.999853i	$-0.505455\pi$
$59$	−0.263264	−0.0342740	−0.0171370	+	0.999853i	$0.505455\pi$
$60$	0	0
$61$	2.68534	0.343822	0.171911	−	0.985112i	$-0.445006\pi$
$61$	2.68534	0.343822	0.171911	+	0.985112i	$0.445006\pi$
$62$	0	0
$63$	2.90231	0.365657
$64$	0	0
$65$	17.3800	2.15572
$66$	0	0
$67$	5.43900	0.664479	0.332240	−	0.943195i	$-0.392196\pi$
$67$	5.43900	0.664479	0.332240	+	0.943195i	$0.392196\pi$
$68$	0	0
$69$	17.9556	2.16160
$70$	0	0
$71$	−4.61343	−0.547514	−0.273757	−	0.961799i	$-0.588266\pi$
$71$	−4.61343	−0.547514	−0.273757	+	0.961799i	$0.588266\pi$
$72$	0	0
$73$	−1.81707	−0.212672	−0.106336	−	0.994330i	$-0.533912\pi$
$73$	−1.81707	−0.212672	−0.106336	+	0.994330i	$0.533912\pi$
$74$	0	0
$75$	−4.52871	−0.522931
$76$	0	0
$77$	−0.459707	−0.0523884
$78$	0	0
$79$	−12.5437	−1.41128	−0.705640	−	0.708571i	$-0.749340\pi$
$79$	−12.5437	−1.41128	−0.705640	+	0.708571i	$0.749340\pi$
$80$	0	0
$81$	−9.02245	−1.00249
$82$	0	0
$83$	−9.15946	−1.00538	−0.502691	−	0.864466i	$-0.667657\pi$
$83$	−9.15946	−1.00538	−0.502691	+	0.864466i	$0.667657\pi$
$84$	0	0
$85$	−8.26764	−0.896751
$86$	0	0
$87$	0.584546	0.0626699
$88$	0	0
$89$	−17.3538	−1.83950	−0.919748	−	0.392510i	$-0.871607\pi$
$89$	−17.3538	−1.83950	−0.919748	+	0.392510i	$0.871607\pi$
$90$	0	0
$91$	−6.44042	−0.675140
$92$	0	0
$93$	−0.631695	−0.0655037
$94$	0	0
$95$	19.5817	2.00905
$96$	0	0
$97$	−13.0359	−1.32359	−0.661797	−	0.749684i	$-0.730206\pi$
$97$	−13.0359	−1.32359	−0.661797	+	0.749684i	$0.730206\pi$
$98$	0	0
$99$	−1.41842	−0.142556
$100$	0	0
$101$	2.41297	0.240099	0.120050	−	0.992768i	$-0.461695\pi$
$101$	2.41297	0.240099	0.120050	+	0.992768i	$0.461695\pi$
$102$	0	0
$103$	−11.5964	−1.14263	−0.571313	−	0.820732i	$-0.693566\pi$
$103$	−11.5964	−1.14263	−0.571313	+	0.820732i	$0.693566\pi$
$104$	0	0
$105$	6.21382	0.606407
$106$	0	0
$107$	−4.45510	−0.430691	−0.215345	−	0.976538i	$-0.569088\pi$
$107$	−4.45510	−0.430691	−0.215345	+	0.976538i	$0.569088\pi$
$108$	0	0
$109$	−10.4174	−0.997811	−0.498905	−	0.866656i	$-0.666264\pi$
$109$	−10.4174	−0.997811	−0.498905	+	0.866656i	$0.666264\pi$
$110$	0	0
$111$	12.2918	1.16669
$112$	0	0
$113$	0.605469	0.0569577	0.0284789	−	0.999594i	$-0.490934\pi$
$113$	0.605469	0.0569577	0.0284789	+	0.999594i	$0.490934\pi$
$114$	0	0
$115$	19.1973	1.79016
$116$	0	0
$117$	−19.8718	−1.83715
$118$	0	0
$119$	3.06370	0.280849
$120$	0	0
$121$	−10.7753	−0.979576
$122$	0	0
$123$	17.0272	1.53529
$124$	0	0
$125$	8.24435	0.737397
$126$	0	0
$127$	13.4447	1.19302	0.596511	−	0.802605i	$-0.296553\pi$
$127$	13.4447	1.19302	0.596511	+	0.802605i	$0.296553\pi$
$128$	0	0
$129$	−23.3547	−2.05626
$130$	0	0
$131$	8.61395	0.752604	0.376302	−	0.926497i	$-0.377195\pi$
$131$	8.61395	0.752604	0.376302	+	0.926497i	$0.377195\pi$
$132$	0	0
$133$	−7.25632	−0.629203
$134$	0	0
$135$	−0.0480731	−0.00413748
$136$	0	0
$137$	1.19715	0.102280	0.0511398	−	0.998692i	$-0.483715\pi$
$137$	1.19715	0.102280	0.0511398	+	0.998692i	$0.483715\pi$
$138$	0	0
$139$	13.2473	1.12362	0.561810	−	0.827266i	$-0.310105\pi$
$139$	13.2473	1.12362	0.561810	+	0.827266i	$0.310105\pi$
$140$	0	0
$141$	2.44796	0.206155
$142$	0	0
$143$	3.14757	0.263213
$144$	0	0
$145$	0.624971	0.0519010
$146$	0	0
$147$	14.8331	1.22341
$148$	0	0
$149$	−20.5131	−1.68050	−0.840248	−	0.542202i	$-0.817591\pi$
$149$	−20.5131	−1.68050	−0.840248	+	0.542202i	$0.817591\pi$
$150$	0	0
$151$	−12.1418	−0.988090	−0.494045	−	0.869436i	$-0.664482\pi$
$151$	−12.1418	−0.988090	−0.494045	+	0.869436i	$0.664482\pi$
$152$	0	0
$153$	9.45301	0.764230
$154$	0	0
$155$	−0.675380	−0.0542479
$156$	0	0
$157$	−5.71193	−0.455862	−0.227931	−	0.973677i	$-0.573196\pi$
$157$	−5.71193	−0.455862	−0.227931	+	0.973677i	$0.573196\pi$
$158$	0	0
$159$	26.9534	2.13754
$160$	0	0
$161$	−7.11385	−0.560650
$162$	0	0
$163$	−10.3082	−0.807398	−0.403699	−	0.914892i	$-0.632276\pi$
$163$	−10.3082	−0.807398	−0.403699	+	0.914892i	$0.632276\pi$
$164$	0	0
$165$	−3.03683	−0.236417
$166$	0	0
$167$	9.57667	0.741065	0.370533	−	0.928819i	$-0.379175\pi$
$167$	9.57667	0.741065	0.370533	+	0.928819i	$0.379175\pi$
$168$	0	0
$169$	31.0970	2.39208
$170$	0	0
$171$	−22.3893	−1.71215
$172$	0	0
$173$	−25.2499	−1.91971	−0.959857	−	0.280491i	$-0.909503\pi$
$173$	−25.2499	−1.91971	−0.959857	+	0.280491i	$0.909503\pi$
$174$	0	0
$175$	1.79424	0.135632
$176$	0	0
$177$	0.644459	0.0484405
$178$	0	0
$179$	6.23383	0.465938	0.232969	−	0.972484i	$-0.425156\pi$
$179$	6.23383	0.465938	0.232969	+	0.972484i	$0.425156\pi$
$180$	0	0
$181$	5.64159	0.419336	0.209668	−	0.977773i	$-0.432762\pi$
$181$	5.64159	0.419336	0.209668	+	0.977773i	$0.432762\pi$
$182$	0	0
$183$	−6.57359	−0.485934
$184$	0	0
$185$	13.1419	0.966208
$186$	0	0
$187$	−1.49729	−0.109493
$188$	0	0
$189$	0.0178142	0.00129580
$190$	0	0
$191$	25.3832	1.83666	0.918332	−	0.395810i	$-0.129536\pi$
$191$	25.3832	1.83666	0.918332	+	0.395810i	$0.129536\pi$
$192$	0	0
$193$	−11.7467	−0.845542	−0.422771	−	0.906236i	$-0.638943\pi$
$193$	−11.7467	−0.845542	−0.422771	+	0.906236i	$0.638943\pi$
$194$	0	0
$195$	−42.5455	−3.04675
$196$	0	0
$197$	5.46326	0.389241	0.194621	−	0.980879i	$-0.437652\pi$
$197$	5.46326	0.389241	0.194621	+	0.980879i	$0.437652\pi$
$198$	0	0
$199$	7.58006	0.537336	0.268668	−	0.963233i	$-0.413417\pi$
$199$	7.58006	0.537336	0.268668	+	0.963233i	$0.413417\pi$
$200$	0	0
$201$	−13.3144	−0.939128
$202$	0	0
$203$	−0.231593	−0.0162546
$204$	0	0
$205$	18.2047	1.27147
$206$	0	0
$207$	−21.9497	−1.52561
$208$	0	0
$209$	3.54631	0.245304
$210$	0	0
$211$	11.1427	0.767093	0.383546	−	0.923522i	$-0.374703\pi$
$211$	11.1427	0.767093	0.383546	+	0.923522i	$0.374703\pi$
$212$	0	0
$213$	11.2935	0.773817
$214$	0	0
$215$	−24.9698	−1.70293
$216$	0	0
$217$	0.250273	0.0169896
$218$	0	0
$219$	4.44811	0.300575
$220$	0	0
$221$	−20.9769	−1.41106
$222$	0	0
$223$	−25.1287	−1.68274	−0.841372	−	0.540457i	$-0.818252\pi$
$223$	−25.1287	−1.68274	−0.841372	+	0.540457i	$0.818252\pi$
$224$	0	0
$225$	5.53611	0.369074
$226$	0	0
$227$	−12.0240	−0.798059	−0.399029	−	0.916938i	$-0.630653\pi$
$227$	−12.0240	−0.798059	−0.399029	+	0.916938i	$0.630653\pi$
$228$	0	0
$229$	10.8831	0.719177	0.359588	−	0.933111i	$-0.382917\pi$
$229$	10.8831	0.719177	0.359588	+	0.933111i	$0.382917\pi$
$230$	0	0
$231$	1.12534	0.0740421
$232$	0	0
$233$	24.6380	1.61409	0.807045	−	0.590490i	$-0.201066\pi$
$233$	24.6380	1.61409	0.807045	+	0.590490i	$0.201066\pi$
$234$	0	0
$235$	2.61725	0.170731
$236$	0	0
$237$	30.7065	1.99460
$238$	0	0
$239$	24.3852	1.57735	0.788675	−	0.614811i	$-0.210768\pi$
$239$	24.3852	1.57735	0.788675	+	0.614811i	$0.210768\pi$
$240$	0	0
$241$	−7.75449	−0.499511	−0.249755	−	0.968309i	$-0.580350\pi$
$241$	−7.75449	−0.499511	−0.249755	+	0.968309i	$0.580350\pi$
$242$	0	0
$243$	22.0315	1.41332
$244$	0	0
$245$	15.8589	1.01319
$246$	0	0
$247$	49.6833	3.16128
$248$	0	0
$249$	22.4220	1.42094
$250$	0	0
$251$	−8.96064	−0.565590	−0.282795	−	0.959180i	$-0.591262\pi$
$251$	−8.96064	−0.565590	−0.282795	+	0.959180i	$0.591262\pi$
$252$	0	0
$253$	3.47669	0.218578
$254$	0	0
$255$	20.2388	1.26740
$256$	0	0
$257$	−1.28197	−0.0799672	−0.0399836	−	0.999200i	$-0.512731\pi$
$257$	−1.28197	−0.0799672	−0.0399836	+	0.999200i	$0.512731\pi$
$258$	0	0
$259$	−4.86992	−0.302602
$260$	0	0
$261$	−0.714576	−0.0442311
$262$	0	0
$263$	−7.18958	−0.443328	−0.221664	−	0.975123i	$-0.571149\pi$
$263$	−7.18958	−0.443328	−0.221664	+	0.975123i	$0.571149\pi$
$264$	0	0
$265$	28.8174	1.77024
$266$	0	0
$267$	42.4813	2.59981
$268$	0	0
$269$	−3.67075	−0.223809	−0.111905	−	0.993719i	$-0.535695\pi$
$269$	−3.67075	−0.223809	−0.111905	+	0.993719i	$0.535695\pi$
$270$	0	0
$271$	8.89230	0.540169	0.270084	−	0.962837i	$-0.412948\pi$
$271$	8.89230	0.540169	0.270084	+	0.962837i	$0.412948\pi$
$272$	0	0
$273$	15.7659	0.954195
$274$	0	0
$275$	−0.876883	−0.0528780
$276$	0	0
$277$	−11.1653	−0.670860	−0.335430	−	0.942065i	$-0.608882\pi$
$277$	−11.1653	−0.670860	−0.335430	+	0.942065i	$0.608882\pi$
$278$	0	0
$279$	0.772213	0.0462312
$280$	0	0
$281$	20.1746	1.20352	0.601759	−	0.798678i	$-0.294467\pi$
$281$	20.1746	1.20352	0.601759	+	0.798678i	$0.294467\pi$
$282$	0	0
$283$	26.2804	1.56221	0.781105	−	0.624400i	$-0.214656\pi$
$283$	26.2804	1.56221	0.781105	+	0.624400i	$0.214656\pi$
$284$	0	0
$285$	−47.9353	−2.83944
$286$	0	0
$287$	−6.74605	−0.398207
$288$	0	0
$289$	−7.02133	−0.413020
$290$	0	0
$291$	31.9113	1.87067
$292$	0	0
$293$	0.635812	0.0371445	0.0185723	−	0.999828i	$-0.494088\pi$
$293$	0.635812	0.0371445	0.0185723	+	0.999828i	$0.494088\pi$
$294$	0	0
$295$	0.689027	0.0401167
$296$	0	0
$297$	−0.00870619	−0.000505184 0
$298$	0	0
$299$	48.7079	2.81685
$300$	0	0
$301$	9.25294	0.533331
$302$	0	0
$303$	−5.90684	−0.339339
$304$	0	0
$305$	−7.02820	−0.402433
$306$	0	0
$307$	−12.7918	−0.730065	−0.365032	−	0.930995i	$-0.618942\pi$
$307$	−12.7918	−0.730065	−0.365032	+	0.930995i	$0.618942\pi$
$308$	0	0
$309$	28.3875	1.61491
$310$	0	0
$311$	7.63812	0.433118	0.216559	−	0.976270i	$-0.430517\pi$
$311$	7.63812	0.433118	0.216559	+	0.976270i	$0.430517\pi$
$312$	0	0
$313$	−6.28882	−0.355465	−0.177733	−	0.984079i	$-0.556876\pi$
$313$	−6.28882	−0.355465	−0.177733	+	0.984079i	$0.556876\pi$
$314$	0	0
$315$	−7.59607	−0.427990
$316$	0	0
$317$	−3.14229	−0.176489	−0.0882443	−	0.996099i	$-0.528126\pi$
$317$	−3.14229	−0.176489	−0.0882443	+	0.996099i	$0.528126\pi$
$318$	0	0
$319$	0.113184	0.00633709
$320$	0	0
$321$	10.9059	0.608707
$322$	0	0
$323$	−23.6343	−1.31505
$324$	0	0
$325$	−12.2850	−0.681449
$326$	0	0
$327$	25.5015	1.41023
$328$	0	0
$329$	−0.969862	−0.0534702
$330$	0	0
$331$	10.2535	0.563585	0.281793	−	0.959475i	$-0.409071\pi$
$331$	10.2535	0.563585	0.281793	+	0.959475i	$0.409071\pi$
$332$	0	0
$333$	−15.0261	−0.823423
$334$	0	0
$335$	−14.2352	−0.777753
$336$	0	0
$337$	26.2950	1.43238	0.716191	−	0.697904i	$-0.245884\pi$
$337$	26.2950	1.43238	0.716191	+	0.697904i	$0.245884\pi$
$338$	0	0
$339$	−1.48216	−0.0805000
$340$	0	0
$341$	−0.122313	−0.00662364
$342$	0	0
$343$	−12.6658	−0.683888
$344$	0	0
$345$	−46.9942	−2.53008
$346$	0	0
$347$	24.6248	1.32193	0.660965	−	0.750417i	$-0.270147\pi$
$347$	24.6248	1.32193	0.660965	+	0.750417i	$0.270147\pi$
$348$	0	0
$349$	14.7813	0.791223	0.395612	−	0.918418i	$-0.370533\pi$
$349$	14.7813	0.791223	0.395612	+	0.918418i	$0.370533\pi$
$350$	0	0
$351$	−0.121973	−0.00651041
$352$	0	0
$353$	−19.2714	−1.02571	−0.512857	−	0.858474i	$-0.671413\pi$
$353$	−19.2714	−1.02571	−0.512857	+	0.858474i	$0.671413\pi$
$354$	0	0
$355$	12.0745	0.640848
$356$	0	0
$357$	−7.49981	−0.396932
$358$	0	0
$359$	−13.5190	−0.713506	−0.356753	−	0.934199i	$-0.616116\pi$
$359$	−13.5190	−0.713506	−0.356753	+	0.934199i	$0.616116\pi$
$360$	0	0
$361$	36.9774	1.94618
$362$	0	0
$363$	26.3776	1.38446
$364$	0	0
$365$	4.75573	0.248926
$366$	0	0
$367$	−26.7317	−1.39538	−0.697691	−	0.716399i	$-0.745789\pi$
$367$	−26.7317	−1.39538	−0.697691	+	0.716399i	$0.745789\pi$
$368$	0	0
$369$	−20.8148	−1.08358
$370$	0	0
$371$	−10.6787	−0.554411
$372$	0	0
$373$	1.87039	0.0968450	0.0484225	−	0.998827i	$-0.484581\pi$
$373$	1.87039	0.0968450	0.0484225	+	0.998827i	$0.484581\pi$
$374$	0	0
$375$	−20.1818	−1.04218
$376$	0	0
$377$	1.58569	0.0816674
$378$	0	0
$379$	−11.2384	−0.577276	−0.288638	−	0.957438i	$-0.593202\pi$
$379$	−11.2384	−0.577276	−0.288638	+	0.957438i	$0.593202\pi$
$380$	0	0
$381$	−32.9120	−1.68613
$382$	0	0
$383$	−27.9753	−1.42947	−0.714735	−	0.699395i	$-0.753453\pi$
$383$	−27.9753	−1.42947	−0.714735	+	0.699395i	$0.753453\pi$
$384$	0	0
$385$	1.20317	0.0613191
$386$	0	0
$387$	28.5498	1.45127
$388$	0	0
$389$	18.9643	0.961528	0.480764	−	0.876850i	$-0.340359\pi$
$389$	18.9643	0.961528	0.480764	+	0.876850i	$0.340359\pi$
$390$	0	0
$391$	−23.1703	−1.17177
$392$	0	0
$393$	−21.0866	−1.06368
$394$	0	0
$395$	32.8301	1.65186
$396$	0	0
$397$	−39.3775	−1.97630	−0.988150	−	0.153491i	$-0.950948\pi$
$397$	−39.3775	−1.97630	−0.988150	+	0.153491i	$0.950948\pi$
$398$	0	0
$399$	17.7632	0.889270
$400$	0	0
$401$	33.4629	1.67106	0.835528	−	0.549448i	$-0.185162\pi$
$401$	33.4629	1.67106	0.835528	+	0.549448i	$0.185162\pi$
$402$	0	0
$403$	−1.71359	−0.0853602
$404$	0	0
$405$	23.6140	1.17339
$406$	0	0
$407$	2.38003	0.117974
$408$	0	0
$409$	25.6708	1.26934	0.634670	−	0.772783i	$-0.281136\pi$
$409$	25.6708	1.26934	0.634670	+	0.772783i	$0.281136\pi$
$410$	0	0
$411$	−2.93058	−0.144555
$412$	0	0
$413$	−0.255330	−0.0125639
$414$	0	0
$415$	23.9726	1.17677
$416$	0	0
$417$	−32.4288	−1.58805
$418$	0	0
$419$	27.1293	1.32536	0.662678	−	0.748905i	$-0.269420\pi$
$419$	27.1293	1.32536	0.662678	+	0.748905i	$0.269420\pi$
$420$	0	0
$421$	−5.99319	−0.292090	−0.146045	−	0.989278i	$-0.546654\pi$
$421$	−5.99319	−0.292090	−0.146045	+	0.989278i	$0.546654\pi$
$422$	0	0
$423$	−2.99250	−0.145500
$424$	0	0
$425$	5.84396	0.283474
$426$	0	0
$427$	2.60441	0.126036
$428$	0	0
$429$	−7.70512	−0.372006
$430$	0	0
$431$	16.8113	0.809772	0.404886	−	0.914367i	$-0.367311\pi$
$431$	16.8113	0.809772	0.404886	+	0.914367i	$0.367311\pi$
$432$	0	0
$433$	−13.2229	−0.635450	−0.317725	−	0.948183i	$-0.602919\pi$
$433$	−13.2229	−0.635450	−0.317725	+	0.948183i	$0.602919\pi$
$434$	0	0
$435$	−1.52990	−0.0733532
$436$	0	0
$437$	54.8784	2.62519
$438$	0	0
$439$	−5.74513	−0.274200	−0.137100	−	0.990557i	$-0.543778\pi$
$439$	−5.74513	−0.274200	−0.137100	+	0.990557i	$0.543778\pi$
$440$	0	0
$441$	−18.1326	−0.863459
$442$	0	0
$443$	13.7162	0.651677	0.325839	−	0.945425i	$-0.394353\pi$
$443$	13.7162	0.651677	0.325839	+	0.945425i	$0.394353\pi$
$444$	0	0
$445$	45.4191	2.15307
$446$	0	0
$447$	50.2151	2.37509
$448$	0	0
$449$	21.6893	1.02358	0.511791	−	0.859110i	$-0.328982\pi$
$449$	21.6893	1.02358	0.511791	+	0.859110i	$0.328982\pi$
$450$	0	0
$451$	3.29693	0.155247
$452$	0	0
$453$	29.7227	1.39650
$454$	0	0
$455$	16.8562	0.790231
$456$	0	0
$457$	37.9152	1.77360	0.886799	−	0.462156i	$-0.152924\pi$
$457$	37.9152	1.77360	0.886799	+	0.462156i	$0.152924\pi$
$458$	0	0
$459$	0.0580221	0.00270824
$460$	0	0
$461$	−37.6645	−1.75421	−0.877106	−	0.480296i	$-0.840529\pi$
$461$	−37.6645	−1.75421	−0.877106	+	0.480296i	$0.840529\pi$
$462$	0	0
$463$	−10.0809	−0.468497	−0.234248	−	0.972177i	$-0.575263\pi$
$463$	−10.0809	−0.468497	−0.234248	+	0.972177i	$0.575263\pi$
$464$	0	0
$465$	1.65330	0.0766701
$466$	0	0
$467$	11.0862	0.513008	0.256504	−	0.966543i	$-0.417429\pi$
$467$	11.0862	0.513008	0.256504	+	0.966543i	$0.417429\pi$
$468$	0	0
$469$	5.27508	0.243580
$470$	0	0
$471$	13.9826	0.644282
$472$	0	0
$473$	−4.52210	−0.207927
$474$	0	0
$475$	−13.8413	−0.635083
$476$	0	0
$477$	−32.9490	−1.50863
$478$	0	0
$479$	−9.90668	−0.452648	−0.226324	−	0.974052i	$-0.572671\pi$
$479$	−9.90668	−0.452648	−0.226324	+	0.974052i	$0.572671\pi$
$480$	0	0
$481$	33.3439	1.52035
$482$	0	0
$483$	17.4144	0.792383
$484$	0	0
$485$	34.1182	1.54923
$486$	0	0
$487$	−39.5056	−1.79017	−0.895085	−	0.445896i	$-0.852885\pi$
$487$	−39.5056	−1.79017	−0.895085	+	0.445896i	$0.852885\pi$
$488$	0	0
$489$	25.2340	1.14112
$490$	0	0
$491$	8.34518	0.376613	0.188306	−	0.982110i	$-0.439700\pi$
$491$	8.34518	0.376613	0.188306	+	0.982110i	$0.439700\pi$
$492$	0	0
$493$	−0.754312	−0.0339725
$494$	0	0
$495$	3.71236	0.166858
$496$	0	0
$497$	−4.47439	−0.200704
$498$	0	0
$499$	−20.8745	−0.934473	−0.467237	−	0.884132i	$-0.654750\pi$
$499$	−20.8745	−0.934473	−0.467237	+	0.884132i	$0.654750\pi$
$500$	0	0
$501$	−23.4433	−1.04737
$502$	0	0
$503$	20.9180	0.932687	0.466344	−	0.884604i	$-0.345571\pi$
$503$	20.9180	0.932687	0.466344	+	0.884604i	$0.345571\pi$
$504$	0	0
$505$	−6.31534	−0.281029
$506$	0	0
$507$	−76.1241	−3.38079
$508$	0	0
$509$	−21.3760	−0.947475	−0.473738	−	0.880666i	$-0.657096\pi$
$509$	−21.3760	−0.947475	−0.473738	+	0.880666i	$0.657096\pi$
$510$	0	0
$511$	−1.76231	−0.0779599
$512$	0	0
$513$	−0.137424	−0.00606743
$514$	0	0
$515$	30.3507	1.33741
$516$	0	0
$517$	0.473992	0.0208461
$518$	0	0
$519$	61.8107	2.71319
$520$	0	0
$521$	6.90431	0.302483	0.151242	−	0.988497i	$-0.451673\pi$
$521$	6.90431	0.302483	0.151242	+	0.988497i	$0.451673\pi$
$522$	0	0
$523$	−35.8100	−1.56586	−0.782930	−	0.622110i	$-0.786276\pi$
$523$	−35.8100	−1.56586	−0.782930	+	0.622110i	$0.786276\pi$
$524$	0	0
$525$	−4.39222	−0.191692
$526$	0	0
$527$	0.815154	0.0355087
$528$	0	0
$529$	30.8010	1.33917
$530$	0	0
$531$	−0.787816	−0.0341883
$532$	0	0
$533$	46.1896	2.00069
$534$	0	0
$535$	11.6601	0.504110
$536$	0	0
$537$	−15.2601	−0.658524
$538$	0	0
$539$	2.87209	0.123710
$540$	0	0
$541$	−12.7144	−0.546636	−0.273318	−	0.961924i	$-0.588121\pi$
$541$	−12.7144	−0.546636	−0.273318	+	0.961924i	$0.588121\pi$
$542$	0	0
$543$	−13.8104	−0.592659
$544$	0	0
$545$	27.2651	1.16791
$546$	0	0
$547$	−12.2429	−0.523469	−0.261735	−	0.965140i	$-0.584294\pi$
$547$	−12.2429	−0.523469	−0.261735	+	0.965140i	$0.584294\pi$
$548$	0	0
$549$	8.03586	0.342962
$550$	0	0
$551$	1.78657	0.0761106
$552$	0	0
$553$	−12.1657	−0.517337
$554$	0	0
$555$	−32.1707	−1.36557
$556$	0	0
$557$	5.18209	0.219572	0.109786	−	0.993955i	$-0.464983\pi$
$557$	5.18209	0.219572	0.109786	+	0.993955i	$0.464983\pi$
$558$	0	0
$559$	−63.3540	−2.67959
$560$	0	0
$561$	3.66531	0.154750
$562$	0	0
$563$	−11.7674	−0.495937	−0.247969	−	0.968768i	$-0.579763\pi$
$563$	−11.7674	−0.495937	−0.247969	+	0.968768i	$0.579763\pi$
$564$	0	0
$565$	−1.58466	−0.0666673
$566$	0	0
$567$	−8.75053	−0.367488
$568$	0	0
$569$	−18.8119	−0.788637	−0.394318	−	0.918974i	$-0.629019\pi$
$569$	−18.8119	−0.788637	−0.394318	+	0.918974i	$0.629019\pi$
$570$	0	0
$571$	7.29865	0.305439	0.152720	−	0.988270i	$-0.451197\pi$
$571$	7.29865	0.305439	0.152720	+	0.988270i	$0.451197\pi$
$572$	0	0
$573$	−62.1370	−2.59581
$574$	0	0
$575$	−13.5696	−0.565890
$576$	0	0
$577$	−5.51273	−0.229498	−0.114749	−	0.993395i	$-0.536606\pi$
$577$	−5.51273	−0.229498	−0.114749	+	0.993395i	$0.536606\pi$
$578$	0	0
$579$	28.7553	1.19503
$580$	0	0
$581$	−8.88342	−0.368546
$582$	0	0
$583$	5.21891	0.216145
$584$	0	0
$585$	52.0096	2.15033
$586$	0	0
$587$	−8.89828	−0.367271	−0.183636	−	0.982994i	$-0.558787\pi$
$587$	−8.89828	−0.367271	−0.183636	+	0.982994i	$0.558787\pi$
$588$	0	0
$589$	−1.93068	−0.0795522
$590$	0	0
$591$	−13.3738	−0.550126
$592$	0	0
$593$	39.2058	1.60999	0.804995	−	0.593282i	$-0.202168\pi$
$593$	39.2058	1.60999	0.804995	+	0.593282i	$0.202168\pi$
$594$	0	0
$595$	−8.01847	−0.328725
$596$	0	0
$597$	−18.5557	−0.759433
$598$	0	0
$599$	22.3275	0.912278	0.456139	−	0.889909i	$-0.349232\pi$
$599$	22.3275	0.912278	0.456139	+	0.889909i	$0.349232\pi$
$600$	0	0
$601$	−35.0325	−1.42901	−0.714503	−	0.699633i	$-0.753347\pi$
$601$	−35.0325	−1.42901	−0.714503	+	0.699633i	$0.753347\pi$
$602$	0	0
$603$	16.2762	0.662817
$604$	0	0
$605$	28.2017	1.14656
$606$	0	0
$607$	−8.76360	−0.355704	−0.177852	−	0.984057i	$-0.556915\pi$
$607$	−8.76360	−0.355704	−0.177852	+	0.984057i	$0.556915\pi$
$608$	0	0
$609$	0.566929	0.0229731
$610$	0	0
$611$	6.64056	0.268648
$612$	0	0
$613$	−38.8724	−1.57004	−0.785021	−	0.619470i	$-0.787348\pi$
$613$	−38.8724	−1.57004	−0.785021	+	0.619470i	$0.787348\pi$
$614$	0	0
$615$	−44.5645	−1.79701
$616$	0	0
$617$	23.1363	0.931431	0.465716	−	0.884934i	$-0.345797\pi$
$617$	23.1363	0.931431	0.465716	+	0.884934i	$0.345797\pi$
$618$	0	0
$619$	28.6487	1.15149	0.575744	−	0.817630i	$-0.304712\pi$
$619$	28.6487	1.15149	0.575744	+	0.817630i	$0.304712\pi$
$620$	0	0
$621$	−0.134726	−0.00540638
$622$	0	0
$623$	−16.8308	−0.674310
$624$	0	0
$625$	−30.8275	−1.23310
$626$	0	0
$627$	−8.68122	−0.346695
$628$	0	0
$629$	−15.8616	−0.632445
$630$	0	0
$631$	22.0666	0.878458	0.439229	−	0.898375i	$-0.355252\pi$
$631$	22.0666	0.878458	0.439229	+	0.898375i	$0.355252\pi$
$632$	0	0
$633$	−27.2768	−1.08415
$634$	0	0
$635$	−35.1881	−1.39640
$636$	0	0
$637$	40.2376	1.59427
$638$	0	0
$639$	−13.8057	−0.546144
$640$	0	0
$641$	−44.2817	−1.74902	−0.874510	−	0.485007i	$-0.838817\pi$
$641$	−44.2817	−1.74902	−0.874510	+	0.485007i	$0.838817\pi$
$642$	0	0
$643$	−50.1565	−1.97798	−0.988989	−	0.147986i	$-0.952721\pi$
$643$	−50.1565	−1.97798	−0.988989	+	0.147986i	$0.952721\pi$
$644$	0	0
$645$	61.1250	2.40679
$646$	0	0
$647$	−30.5698	−1.20182	−0.600911	−	0.799316i	$-0.705195\pi$
$647$	−30.5698	−1.20182	−0.600911	+	0.799316i	$0.705195\pi$
$648$	0	0
$649$	0.124785	0.00489823
$650$	0	0
$651$	−0.612657	−0.0240119
$652$	0	0
$653$	37.3989	1.46353	0.731766	−	0.681556i	$-0.238696\pi$
$653$	37.3989	1.46353	0.731766	+	0.681556i	$0.238696\pi$
$654$	0	0
$655$	−22.5449	−0.880900
$656$	0	0
$657$	−5.43758	−0.212140
$658$	0	0
$659$	49.1759	1.91562	0.957811	−	0.287400i	$-0.0927910\pi$
$659$	49.1759	1.91562	0.957811	+	0.287400i	$0.0927910\pi$
$660$	0	0
$661$	−3.21428	−0.125021	−0.0625105	−	0.998044i	$-0.519911\pi$
$661$	−3.21428	−0.125021	−0.0625105	+	0.998044i	$0.519911\pi$
$662$	0	0
$663$	51.3505	1.99429
$664$	0	0
$665$	18.9916	0.736462
$666$	0	0
$667$	1.75150	0.0678183
$668$	0	0
$669$	61.5141	2.37827
$670$	0	0
$671$	−1.27283	−0.0491370
$672$	0	0
$673$	44.8373	1.72835	0.864175	−	0.503192i	$-0.167841\pi$
$673$	44.8373	1.72835	0.864175	+	0.503192i	$0.167841\pi$
$674$	0	0
$675$	0.0339804	0.00130791
$676$	0	0
$677$	9.54909	0.367001	0.183501	−	0.983020i	$-0.441257\pi$
$677$	9.54909	0.367001	0.183501	+	0.983020i	$0.441257\pi$
$678$	0	0
$679$	−12.6430	−0.485194
$680$	0	0
$681$	29.4342	1.12792
$682$	0	0
$683$	−20.7226	−0.792927	−0.396463	−	0.918051i	$-0.629763\pi$
$683$	−20.7226	−0.792927	−0.396463	+	0.918051i	$0.629763\pi$
$684$	0	0
$685$	−3.13325	−0.119715
$686$	0	0
$687$	−26.6414	−1.01643
$688$	0	0
$689$	73.1162	2.78551
$690$	0	0
$691$	−29.6289	−1.12714	−0.563569	−	0.826069i	$-0.690572\pi$
$691$	−29.6289	−1.12714	−0.563569	+	0.826069i	$0.690572\pi$
$692$	0	0
$693$	−1.37567	−0.0522574
$694$	0	0
$695$	−34.6715	−1.31516
$696$	0	0
$697$	−21.9723	−0.832261
$698$	0	0
$699$	−60.3128	−2.28124
$700$	0	0
$701$	45.1068	1.70366	0.851830	−	0.523818i	$-0.175493\pi$
$701$	45.1068	1.70366	0.851830	+	0.523818i	$0.175493\pi$
$702$	0	0
$703$	37.5680	1.41690
$704$	0	0
$705$	−6.40692	−0.241298
$706$	0	0
$707$	2.34025	0.0880140
$708$	0	0
$709$	0.546678	0.0205309	0.0102655	−	0.999947i	$-0.496732\pi$
$709$	0.546678	0.0205309	0.0102655	+	0.999947i	$0.496732\pi$
$710$	0	0
$711$	−37.5371	−1.40775
$712$	0	0
$713$	−1.89277	−0.0708849
$714$	0	0
$715$	−8.23797	−0.308083
$716$	0	0
$717$	−59.6940	−2.22931
$718$	0	0
$719$	−4.96610	−0.185204	−0.0926022	−	0.995703i	$-0.529518\pi$
$719$	−4.96610	−0.185204	−0.0926022	+	0.995703i	$0.529518\pi$
$720$	0	0
$721$	−11.2469	−0.418857
$722$	0	0
$723$	18.9827	0.705973
$724$	0	0
$725$	−0.441759	−0.0164065
$726$	0	0
$727$	−21.4446	−0.795336	−0.397668	−	0.917529i	$-0.630180\pi$
$727$	−21.4446	−0.795336	−0.397668	+	0.917529i	$0.630180\pi$
$728$	0	0
$729$	−26.8648	−0.994992
$730$	0	0
$731$	30.1374	1.11467
$732$	0	0
$733$	−1.49069	−0.0550598	−0.0275299	−	0.999621i	$-0.508764\pi$
$733$	−1.49069	−0.0550598	−0.0275299	+	0.999621i	$0.508764\pi$
$734$	0	0
$735$	−38.8219	−1.43197
$736$	0	0
$737$	−2.57804	−0.0949633
$738$	0	0
$739$	15.8949	0.584703	0.292351	−	0.956311i	$-0.405562\pi$
$739$	15.8949	0.584703	0.292351	+	0.956311i	$0.405562\pi$
$740$	0	0
$741$	−121.623	−4.46792
$742$	0	0
$743$	5.27473	0.193511	0.0967555	−	0.995308i	$-0.469153\pi$
$743$	5.27473	0.193511	0.0967555	+	0.995308i	$0.469153\pi$
$744$	0	0
$745$	53.6878	1.96697
$746$	0	0
$747$	−27.4097	−1.00287
$748$	0	0
$749$	−4.32083	−0.157880
$750$	0	0
$751$	−27.5739	−1.00618	−0.503092	−	0.864233i	$-0.667804\pi$
$751$	−27.5739	−1.00618	−0.503092	+	0.864233i	$0.667804\pi$
$752$	0	0
$753$	21.9353	0.799365
$754$	0	0
$755$	31.7782	1.15653
$756$	0	0
$757$	25.3115	0.919962	0.459981	−	0.887929i	$-0.347856\pi$
$757$	25.3115	0.919962	0.459981	+	0.887929i	$0.347856\pi$
$758$	0	0
$759$	−8.51079	−0.308922
$760$	0	0
$761$	−16.3431	−0.592435	−0.296218	−	0.955120i	$-0.595725\pi$
$761$	−16.3431	−0.592435	−0.296218	+	0.955120i	$0.595725\pi$
$762$	0	0
$763$	−10.1035	−0.365771
$764$	0	0
$765$	−24.7409	−0.894508
$766$	0	0
$767$	1.74822	0.0631245
$768$	0	0
$769$	−23.0793	−0.832262	−0.416131	−	0.909305i	$-0.636614\pi$
$769$	−23.0793	−0.832262	−0.416131	+	0.909305i	$0.636614\pi$
$770$	0	0
$771$	3.13821	0.113020
$772$	0	0
$773$	−11.1054	−0.399434	−0.199717	−	0.979854i	$-0.564002\pi$
$773$	−11.1054	−0.399434	−0.199717	+	0.979854i	$0.564002\pi$
$774$	0	0
$775$	0.477391	0.0171484
$776$	0	0
$777$	11.9213	0.427676
$778$	0	0
$779$	52.0410	1.86456
$780$	0	0
$781$	2.18673	0.0782473
$782$	0	0
$783$	−0.00438603	−0.000156744 0
$784$	0	0
$785$	14.9495	0.533572
$786$	0	0
$787$	−38.5494	−1.37414	−0.687069	−	0.726592i	$-0.741103\pi$
$787$	−38.5494	−1.37414	−0.687069	+	0.726592i	$0.741103\pi$
$788$	0	0
$789$	17.5998	0.626569
$790$	0	0
$791$	0.587221	0.0208792
$792$	0	0
$793$	−17.8321	−0.633238
$794$	0	0
$795$	−70.5437	−2.50193
$796$	0	0
$797$	−30.6336	−1.08510	−0.542549	−	0.840024i	$-0.682541\pi$
$797$	−30.6336	−1.08510	−0.542549	+	0.840024i	$0.682541\pi$
$798$	0	0
$799$	−3.15890	−0.111754
$800$	0	0
$801$	−51.9311	−1.83489
$802$	0	0
$803$	0.861276	0.0303938
$804$	0	0
$805$	18.6187	0.656224
$806$	0	0
$807$	8.98583	0.316316
$808$	0	0
$809$	0.546656	0.0192194	0.00960970	−	0.999954i	$-0.496941\pi$
$809$	0.546656	0.0192194	0.00960970	+	0.999954i	$0.496941\pi$
$810$	0	0
$811$	13.0179	0.457120	0.228560	−	0.973530i	$-0.426598\pi$
$811$	13.0179	0.457120	0.228560	+	0.973530i	$0.426598\pi$
$812$	0	0
$813$	−21.7680	−0.763436
$814$	0	0
$815$	26.9790	0.945035
$816$	0	0
$817$	−71.3799	−2.49727
$818$	0	0
$819$	−19.2729	−0.673451
$820$	0	0
$821$	−18.6959	−0.652492	−0.326246	−	0.945285i	$-0.605784\pi$
$821$	−18.6959	−0.652492	−0.326246	+	0.945285i	$0.605784\pi$
$822$	0	0
$823$	0.553054	0.0192783	0.00963913	−	0.999954i	$-0.496932\pi$
$823$	0.553054	0.0192783	0.00963913	+	0.999954i	$0.496932\pi$
$824$	0	0
$825$	2.14657	0.0747340
$826$	0	0
$827$	−20.4130	−0.709828	−0.354914	−	0.934899i	$-0.615490\pi$
$827$	−20.4130	−0.709828	−0.354914	+	0.934899i	$0.615490\pi$
$828$	0	0
$829$	−3.11497	−0.108187	−0.0540937	−	0.998536i	$-0.517227\pi$
$829$	−3.11497	−0.108187	−0.0540937	+	0.998536i	$0.517227\pi$
$830$	0	0
$831$	27.3323	0.948147
$832$	0	0
$833$	−19.1410	−0.663195
$834$	0	0
$835$	−25.0645	−0.867394
$836$	0	0
$837$	0.00473981	0.000163832 0
$838$	0	0
$839$	−15.2173	−0.525361	−0.262681	−	0.964883i	$-0.584607\pi$
$839$	−15.2173	−0.525361	−0.262681	+	0.964883i	$0.584607\pi$
$840$	0	0
$841$	−28.9430	−0.998034
$842$	0	0
$843$	−49.3867	−1.70097
$844$	0	0
$845$	−81.3886	−2.79985
$846$	0	0
$847$	−10.4506	−0.359086
$848$	0	0
$849$	−64.3334	−2.20792
$850$	0	0
$851$	36.8304	1.26253
$852$	0	0
$853$	22.2277	0.761063	0.380531	−	0.924768i	$-0.375741\pi$
$853$	22.2277	0.761063	0.380531	+	0.924768i	$0.375741\pi$
$854$	0	0
$855$	58.5983	2.00402
$856$	0	0
$857$	−15.9729	−0.545626	−0.272813	−	0.962067i	$-0.587954\pi$
$857$	−15.9729	−0.545626	−0.272813	+	0.962067i	$0.587954\pi$
$858$	0	0
$859$	6.39658	0.218249	0.109124	−	0.994028i	$-0.465195\pi$
$859$	6.39658	0.218249	0.109124	+	0.994028i	$0.465195\pi$
$860$	0	0
$861$	16.5140	0.562797
$862$	0	0
$863$	19.7248	0.671441	0.335721	−	0.941962i	$-0.391020\pi$
$863$	19.7248	0.671441	0.335721	+	0.941962i	$0.391020\pi$
$864$	0	0
$865$	66.0853	2.24697
$866$	0	0
$867$	17.1879	0.583732
$868$	0	0
$869$	5.94562	0.201691
$870$	0	0
$871$	−36.1180	−1.22381
$872$	0	0
$873$	−39.0098	−1.32028
$874$	0	0
$875$	7.99588	0.270310
$876$	0	0
$877$	21.1196	0.713158	0.356579	−	0.934265i	$-0.383943\pi$
$877$	21.1196	0.713158	0.356579	+	0.934265i	$0.383943\pi$
$878$	0	0
$879$	−1.55644	−0.0524974
$880$	0	0
$881$	−19.0553	−0.641989	−0.320994	−	0.947081i	$-0.604017\pi$
$881$	−19.0553	−0.641989	−0.320994	+	0.947081i	$0.604017\pi$
$882$	0	0
$883$	−46.2771	−1.55735	−0.778674	−	0.627429i	$-0.784107\pi$
$883$	−46.2771	−1.55735	−0.778674	+	0.627429i	$0.784107\pi$
$884$	0	0
$885$	−1.68671	−0.0566981
$886$	0	0
$887$	51.1963	1.71900	0.859502	−	0.511132i	$-0.170774\pi$
$887$	51.1963	1.71900	0.859502	+	0.511132i	$0.170774\pi$
$888$	0	0
$889$	13.0395	0.437330
$890$	0	0
$891$	4.27657	0.143270
$892$	0	0
$893$	7.48180	0.250369
$894$	0	0
$895$	−16.3155	−0.545367
$896$	0	0
$897$	−119.235	−3.98114
$898$	0	0
$899$	−0.0616195	−0.00205512
$900$	0	0
$901$	−34.7813	−1.15873
$902$	0	0
$903$	−22.6508	−0.753772
$904$	0	0
$905$	−14.7654	−0.490820
$906$	0	0
$907$	−3.84218	−0.127577	−0.0637887	−	0.997963i	$-0.520318\pi$
$907$	−3.84218	−0.127577	−0.0637887	+	0.997963i	$0.520318\pi$
$908$	0	0
$909$	7.22080	0.239499
$910$	0	0
$911$	−15.0954	−0.500133	−0.250066	−	0.968229i	$-0.580452\pi$
$911$	−15.0954	−0.500133	−0.250066	+	0.968229i	$0.580452\pi$
$912$	0	0
$913$	4.34151	0.143683
$914$	0	0
$915$	17.2047	0.568771
$916$	0	0
$917$	8.35434	0.275885
$918$	0	0
$919$	−25.6466	−0.846003	−0.423001	−	0.906129i	$-0.639023\pi$
$919$	−25.6466	−0.846003	−0.423001	+	0.906129i	$0.639023\pi$
$920$	0	0
$921$	31.3137	1.03182
$922$	0	0
$923$	30.6358	1.00839
$924$	0	0
$925$	−9.28928	−0.305430
$926$	0	0
$927$	−34.7022	−1.13977
$928$	0	0
$929$	1.45167	0.0476278	0.0238139	−	0.999716i	$-0.492419\pi$
$929$	1.45167	0.0476278	0.0238139	+	0.999716i	$0.492419\pi$
$930$	0	0
$931$	45.3350	1.48579
$932$	0	0
$933$	−18.6978	−0.612139
$934$	0	0
$935$	3.91879	0.128158
$936$	0	0
$937$	−16.9144	−0.552571	−0.276285	−	0.961076i	$-0.589103\pi$
$937$	−16.9144	−0.552571	−0.276285	+	0.961076i	$0.589103\pi$
$938$	0	0
$939$	15.3948	0.502389
$940$	0	0
$941$	−0.774438	−0.0252460	−0.0126230	−	0.999920i	$-0.504018\pi$
$941$	−0.774438	−0.0252460	−0.0126230	+	0.999920i	$0.504018\pi$
$942$	0	0
$943$	51.0193	1.66142
$944$	0	0
$945$	−0.0466243	−0.00151669
$946$	0	0
$947$	18.0249	0.585730	0.292865	−	0.956154i	$-0.405391\pi$
$947$	18.0249	0.585730	0.292865	+	0.956154i	$0.405391\pi$
$948$	0	0
$949$	12.0664	0.391690
$950$	0	0
$951$	7.69219	0.249436
$952$	0	0
$953$	−29.4350	−0.953492	−0.476746	−	0.879041i	$-0.658184\pi$
$953$	−29.4350	−0.953492	−0.476746	+	0.879041i	$0.658184\pi$
$954$	0	0
$955$	−66.4342	−2.14976
$956$	0	0
$957$	−0.277070	−0.00895640
$958$	0	0
$959$	1.16107	0.0374930
$960$	0	0
$961$	−30.9334	−0.997852
$962$	0	0
$963$	−13.3319	−0.429613
$964$	0	0
$965$	30.7439	0.989682
$966$	0	0
$967$	35.0867	1.12831	0.564156	−	0.825669i	$-0.309202\pi$
$967$	35.0867	1.12831	0.564156	+	0.825669i	$0.309202\pi$
$968$	0	0
$969$	57.8558	1.85859
$970$	0	0
$971$	17.1707	0.551034	0.275517	−	0.961296i	$-0.411151\pi$
$971$	17.1707	0.551034	0.275517	+	0.961296i	$0.411151\pi$
$972$	0	0
$973$	12.8480	0.411889
$974$	0	0
$975$	30.0732	0.963112
$976$	0	0
$977$	−11.8116	−0.377886	−0.188943	−	0.981988i	$-0.560506\pi$
$977$	−11.8116	−0.377886	−0.188943	+	0.981988i	$0.560506\pi$
$978$	0	0
$979$	8.22554	0.262889
$980$	0	0
$981$	−31.1742	−0.995315
$982$	0	0
$983$	37.3881	1.19249	0.596247	−	0.802801i	$-0.296658\pi$
$983$	37.3881	1.19249	0.596247	+	0.802801i	$0.296658\pi$
$984$	0	0
$985$	−14.2987	−0.455595
$986$	0	0
$987$	2.37418	0.0755710
$988$	0	0
$989$	−69.9785	−2.22519
$990$	0	0
$991$	16.8807	0.536232	0.268116	−	0.963387i	$-0.413599\pi$
$991$	16.8807	0.536232	0.268116	+	0.963387i	$0.413599\pi$
$992$	0	0
$993$	−25.1002	−0.796532
$994$	0	0
$995$	−19.8389	−0.628936
$996$	0	0
$997$	2.81569	0.0891738	0.0445869	−	0.999006i	$-0.485803\pi$
$997$	2.81569	0.0891738	0.0445869	+	0.999006i	$0.485803\pi$
$998$	0	0
$999$	−0.0922293	−0.00291801

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6016.2.a.o.1.2	yes	13
4.3	odd	2		6016.2.a.m.1.12	✓	13
8.3	odd	2		6016.2.a.p.1.2	yes	13
8.5	even	2		6016.2.a.n.1.12	yes	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6016.2.a.m.1.12	✓	13	4.3	odd	2
6016.2.a.n.1.12	yes	13	8.5	even	2
6016.2.a.o.1.2	yes	13	1.1	even	1	trivial
6016.2.a.p.1.2	yes	13	8.3	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:	\( N \)	\(=\)	\( 6016 = 2^{7} \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6016.a (trivial)

\(f(q)\)	\(=\)	\(q-2.44796 q^{3} -2.61725 q^{5} +0.969862 q^{7} +2.99250 q^{9} +O(q^{10})\)	\(q-2.44796 q^{3} -2.61725 q^{5} +0.969862 q^{7} +2.99250 q^{9} -0.473992 q^{11} -6.64056 q^{13} +6.40692 q^{15} +3.15890 q^{17} -7.48180 q^{19} -2.37418 q^{21} -7.33491 q^{23} +1.85000 q^{25} +0.0183678 q^{27} -0.238789 q^{29} +0.258050 q^{31} +1.16031 q^{33} -2.53837 q^{35} -5.02125 q^{37} +16.2558 q^{39} -6.95568 q^{41} +9.54047 q^{43} -7.83211 q^{45} -1.00000 q^{47} -6.05937 q^{49} -7.73286 q^{51} -11.0106 q^{53} +1.24055 q^{55} +18.3151 q^{57} -0.263264 q^{59} +2.68534 q^{61} +2.90231 q^{63} +17.3800 q^{65} +5.43900 q^{67} +17.9556 q^{69} -4.61343 q^{71} -1.81707 q^{73} -4.52871 q^{75} -0.459707 q^{77} -12.5437 q^{79} -9.02245 q^{81} -9.15946 q^{83} -8.26764 q^{85} +0.584546 q^{87} -17.3538 q^{89} -6.44042 q^{91} -0.631695 q^{93} +19.5817 q^{95} -13.0359 q^{97} -1.41842 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q + 4 q^{3} - 6 q^{5} - 2 q^{7} + 21 q^{9}+O(q^{10}) \) 13 * q + 4 * q^3 - 6 * q^5 - 2 * q^7 + 21 * q^9	\( 13 q + 4 q^{3} - 6 q^{5} - 2 q^{7} + 21 q^{9} + 10 q^{11} - 4 q^{13} + 14 q^{15} + 10 q^{17} + 8 q^{19} - 10 q^{21} + 18 q^{23} + 23 q^{25} + 16 q^{27} - 14 q^{29} + 4 q^{31} + 14 q^{33} + 14 q^{35} - 16 q^{37} + 12 q^{39} + 10 q^{41} + 12 q^{43} - 10 q^{45} - 13 q^{47} + 9 q^{49} + 22 q^{51} - 26 q^{53} - 2 q^{55} + 20 q^{57} + 30 q^{59} - 18 q^{61} + 12 q^{63} - 4 q^{65} + 4 q^{67} - 2 q^{69} + 36 q^{71} + 10 q^{73} + 38 q^{75} - 42 q^{77} + 21 q^{81} + 12 q^{83} - 4 q^{85} + 6 q^{87} + 50 q^{89} - 4 q^{91} - 52 q^{93} + 8 q^{95} - 10 q^{97} + 34 q^{99}+O(q^{100}) \) 13 * q + 4 * q^3 - 6 * q^5 - 2 * q^7 + 21 * q^9 + 10 * q^11 - 4 * q^13 + 14 * q^15 + 10 * q^17 + 8 * q^19 - 10 * q^21 + 18 * q^23 + 23 * q^25 + 16 * q^27 - 14 * q^29 + 4 * q^31 + 14 * q^33 + 14 * q^35 - 16 * q^37 + 12 * q^39 + 10 * q^41 + 12 * q^43 - 10 * q^45 - 13 * q^47 + 9 * q^49 + 22 * q^51 - 26 * q^53 - 2 * q^55 + 20 * q^57 + 30 * q^59 - 18 * q^61 + 12 * q^63 - 4 * q^65 + 4 * q^67 - 2 * q^69 + 36 * q^71 + 10 * q^73 + 38 * q^75 - 42 * q^77 + 21 * q^81 + 12 * q^83 - 4 * q^85 + 6 * q^87 + 50 * q^89 - 4 * q^91 - 52 * q^93 + 8 * q^95 - 10 * q^97 + 34 * q^99

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