LMFDB - Embedded newform 6024.2.a.k.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6024,2,Mod(1,6024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6024, 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("6024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6024 = 2^{3} \cdot 3 \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6024.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:	$48.1018821776$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 10x^{6} + 25x^{5} + 5x^{4} - 36x^{3} + 11x^{2} + 3x - 1 $ x^8 - 2x^7 - 10x^6 + 25x^5 + 5x^4 - 36x^3 + 11x^2 + 3*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.38731$ of defining polynomial
Character	$\chi$	$=$	6024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.69055 q^{5} +2.31718 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.69055 q^{5} +2.31718 q^{7} +1.00000 q^{9} +0.271673 q^{11} +0.381008 q^{13} -1.69055 q^{15} -3.91589 q^{17} -7.86989 q^{19} +2.31718 q^{21} +6.38788 q^{23} -2.14202 q^{25} +1.00000 q^{27} +1.34844 q^{29} -3.51880 q^{31} +0.271673 q^{33} -3.91732 q^{35} -3.32734 q^{37} +0.381008 q^{39} +4.64567 q^{41} +3.26210 q^{43} -1.69055 q^{45} -8.46786 q^{47} -1.63068 q^{49} -3.91589 q^{51} +0.738345 q^{53} -0.459278 q^{55} -7.86989 q^{57} -5.38569 q^{59} +2.58968 q^{61} +2.31718 q^{63} -0.644116 q^{65} +4.19823 q^{67} +6.38788 q^{69} -15.3838 q^{71} +0.142034 q^{73} -2.14202 q^{75} +0.629516 q^{77} +3.31119 q^{79} +1.00000 q^{81} -10.2715 q^{83} +6.62002 q^{85} +1.34844 q^{87} -7.10516 q^{89} +0.882865 q^{91} -3.51880 q^{93} +13.3045 q^{95} -17.7783 q^{97} +0.271673 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{3} - 5 q^{5} + q^{7} + 8 q^{9}+O(q^{10}) $ 8 * q + 8 * q^3 - 5 * q^5 + q^7 + 8 * q^9	$ 8 q + 8 q^{3} - 5 q^{5} + q^{7} + 8 q^{9} - 6 q^{11} - 4 q^{13} - 5 q^{15} - 12 q^{17} - 10 q^{19} + q^{21} - 6 q^{23} - 7 q^{25} + 8 q^{27} - 2 q^{29} - 3 q^{31} - 6 q^{33} + 4 q^{35} - 16 q^{37} - 4 q^{39} + 2 q^{41} - 2 q^{43} - 5 q^{45} - 15 q^{47} - 21 q^{49} - 12 q^{51} + 13 q^{53} - 33 q^{55} - 10 q^{57} + 8 q^{59} - 38 q^{61} + q^{63} + 16 q^{65} - 31 q^{67} - 6 q^{69} + 5 q^{71} - 26 q^{73} - 7 q^{75} - 24 q^{77} - 25 q^{79} + 8 q^{81} - 7 q^{83} - 18 q^{85} - 2 q^{87} - 14 q^{89} - 6 q^{91} - 3 q^{93} - q^{95} - 51 q^{97} - 6 q^{99}+O(q^{100}) $ 8 * q + 8 * q^3 - 5 * q^5 + q^7 + 8 * q^9 - 6 * q^11 - 4 * q^13 - 5 * q^15 - 12 * q^17 - 10 * q^19 + q^21 - 6 * q^23 - 7 * q^25 + 8 * q^27 - 2 * q^29 - 3 * q^31 - 6 * q^33 + 4 * q^35 - 16 * q^37 - 4 * q^39 + 2 * q^41 - 2 * q^43 - 5 * q^45 - 15 * q^47 - 21 * q^49 - 12 * q^51 + 13 * q^53 - 33 * q^55 - 10 * q^57 + 8 * q^59 - 38 * q^61 + q^63 + 16 * q^65 - 31 * q^67 - 6 * q^69 + 5 * q^71 - 26 * q^73 - 7 * q^75 - 24 * q^77 - 25 * q^79 + 8 * q^81 - 7 * q^83 - 18 * q^85 - 2 * q^87 - 14 * q^89 - 6 * q^91 - 3 * q^93 - q^95 - 51 * q^97 - 6 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−1.69055	−0.756039	−0.378020	−	0.925798i	$-0.623395\pi$
$5$	−1.69055	−0.756039	−0.378020	+	0.925798i	$0.623395\pi$
$6$	0	0
$7$	2.31718	0.875812	0.437906	−	0.899021i	$-0.355720\pi$
$7$	2.31718	0.875812	0.437906	+	0.899021i	$0.355720\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0.271673	0.0819125	0.0409563	−	0.999161i	$-0.486960\pi$
$11$	0.271673	0.0819125	0.0409563	+	0.999161i	$0.486960\pi$
$12$	0	0
$13$	0.381008	0.105673	0.0528364	−	0.998603i	$-0.483174\pi$
$13$	0.381008	0.105673	0.0528364	+	0.998603i	$0.483174\pi$
$14$	0	0
$15$	−1.69055	−0.436499
$16$	0	0
$17$	−3.91589	−0.949742	−0.474871	−	0.880055i	$-0.657505\pi$
$17$	−3.91589	−0.949742	−0.474871	+	0.880055i	$0.657505\pi$
$18$	0	0
$19$	−7.86989	−1.80548	−0.902738	−	0.430190i	$-0.858446\pi$
$19$	−7.86989	−1.80548	−0.902738	+	0.430190i	$0.858446\pi$
$20$	0	0
$21$	2.31718	0.505650
$22$	0	0
$23$	6.38788	1.33196	0.665982	−	0.745968i	$-0.268013\pi$
$23$	6.38788	1.33196	0.665982	+	0.745968i	$0.268013\pi$
$24$	0	0
$25$	−2.14202	−0.428405
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	1.34844	0.250398	0.125199	−	0.992132i	$-0.460043\pi$
$29$	1.34844	0.250398	0.125199	+	0.992132i	$0.460043\pi$
$30$	0	0
$31$	−3.51880	−0.631995	−0.315998	−	0.948760i	$-0.602339\pi$
$31$	−3.51880	−0.631995	−0.315998	+	0.948760i	$0.602339\pi$
$32$	0	0
$33$	0.271673	0.0472922
$34$	0	0
$35$	−3.91732	−0.662148
$36$	0	0
$37$	−3.32734	−0.547011	−0.273505	−	0.961870i	$-0.588183\pi$
$37$	−3.32734	−0.547011	−0.273505	+	0.961870i	$0.588183\pi$
$38$	0	0
$39$	0.381008	0.0610102
$40$	0	0
$41$	4.64567	0.725532	0.362766	−	0.931880i	$-0.381832\pi$
$41$	4.64567	0.725532	0.362766	+	0.931880i	$0.381832\pi$
$42$	0	0
$43$	3.26210	0.497465	0.248733	−	0.968572i	$-0.419986\pi$
$43$	3.26210	0.497465	0.248733	+	0.968572i	$0.419986\pi$
$44$	0	0
$45$	−1.69055	−0.252013
$46$	0	0
$47$	−8.46786	−1.23516	−0.617582	−	0.786506i	$-0.711888\pi$
$47$	−8.46786	−1.23516	−0.617582	+	0.786506i	$0.711888\pi$
$48$	0	0
$49$	−1.63068	−0.232954
$50$	0	0
$51$	−3.91589	−0.548334
$52$	0	0
$53$	0.738345	0.101419	0.0507097	−	0.998713i	$-0.483852\pi$
$53$	0.738345	0.101419	0.0507097	+	0.998713i	$0.483852\pi$
$54$	0	0
$55$	−0.459278	−0.0619291
$56$	0	0
$57$	−7.86989	−1.04239
$58$	0	0
$59$	−5.38569	−0.701157	−0.350578	−	0.936533i	$-0.614015\pi$
$59$	−5.38569	−0.701157	−0.350578	+	0.936533i	$0.614015\pi$
$60$	0	0
$61$	2.58968	0.331574	0.165787	−	0.986162i	$-0.446984\pi$
$61$	2.58968	0.331574	0.165787	+	0.986162i	$0.446984\pi$
$62$	0	0
$63$	2.31718	0.291937
$64$	0	0
$65$	−0.644116	−0.0798927
$66$	0	0
$67$	4.19823	0.512895	0.256447	−	0.966558i	$-0.417448\pi$
$67$	4.19823	0.512895	0.256447	+	0.966558i	$0.417448\pi$
$68$	0	0
$69$	6.38788	0.769010
$70$	0	0
$71$	−15.3838	−1.82572	−0.912862	−	0.408269i	$-0.866133\pi$
$71$	−15.3838	−1.82572	−0.912862	+	0.408269i	$0.866133\pi$
$72$	0	0
$73$	0.142034	0.0166238	0.00831192	−	0.999965i	$-0.497354\pi$
$73$	0.142034	0.0166238	0.00831192	+	0.999965i	$0.497354\pi$
$74$	0	0
$75$	−2.14202	−0.247340
$76$	0	0
$77$	0.629516	0.0717400
$78$	0	0
$79$	3.31119	0.372538	0.186269	−	0.982499i	$-0.440360\pi$
$79$	3.31119	0.372538	0.186269	+	0.982499i	$0.440360\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−10.2715	−1.12744	−0.563722	−	0.825965i	$-0.690631\pi$
$83$	−10.2715	−1.12744	−0.563722	+	0.825965i	$0.690631\pi$
$84$	0	0
$85$	6.62002	0.718042
$86$	0	0
$87$	1.34844	0.144567
$88$	0	0
$89$	−7.10516	−0.753146	−0.376573	−	0.926387i	$-0.622898\pi$
$89$	−7.10516	−0.753146	−0.376573	+	0.926387i	$0.622898\pi$
$90$	0	0
$91$	0.882865	0.0925494
$92$	0	0
$93$	−3.51880	−0.364883
$94$	0	0
$95$	13.3045	1.36501
$96$	0	0
$97$	−17.7783	−1.80511	−0.902555	−	0.430574i	$-0.858311\pi$
$97$	−17.7783	−1.80511	−0.902555	+	0.430574i	$0.858311\pi$
$98$	0	0
$99$	0.271673	0.0273042
$100$	0	0
$101$	12.0054	1.19459	0.597293	−	0.802023i	$-0.296243\pi$
$101$	12.0054	1.19459	0.597293	+	0.802023i	$0.296243\pi$
$102$	0	0
$103$	−8.44574	−0.832183	−0.416092	−	0.909323i	$-0.636600\pi$
$103$	−8.44574	−0.832183	−0.416092	+	0.909323i	$0.636600\pi$
$104$	0	0
$105$	−3.91732	−0.382291
$106$	0	0
$107$	−1.24465	−0.120325	−0.0601625	−	0.998189i	$-0.519162\pi$
$107$	−1.24465	−0.120325	−0.0601625	+	0.998189i	$0.519162\pi$
$108$	0	0
$109$	−18.9472	−1.81481	−0.907405	−	0.420257i	$-0.861940\pi$
$109$	−18.9472	−1.81481	−0.907405	+	0.420257i	$0.861940\pi$
$110$	0	0
$111$	−3.32734	−0.315817
$112$	0	0
$113$	6.51359	0.612747	0.306374	−	0.951911i	$-0.400884\pi$
$113$	6.51359	0.612747	0.306374	+	0.951911i	$0.400884\pi$
$114$	0	0
$115$	−10.7991	−1.00702
$116$	0	0
$117$	0.381008	0.0352242
$118$	0	0
$119$	−9.07381	−0.831795
$120$	0	0
$121$	−10.9262	−0.993290
$122$	0	0
$123$	4.64567	0.418886
$124$	0	0
$125$	12.0740	1.07993
$126$	0	0
$127$	−3.40748	−0.302365	−0.151182	−	0.988506i	$-0.548308\pi$
$127$	−3.40748	−0.302365	−0.151182	+	0.988506i	$0.548308\pi$
$128$	0	0
$129$	3.26210	0.287212
$130$	0	0
$131$	22.4689	1.96312	0.981559	−	0.191158i	$-0.0612243\pi$
$131$	22.4689	1.96312	0.981559	+	0.191158i	$0.0612243\pi$
$132$	0	0
$133$	−18.2359	−1.58126
$134$	0	0
$135$	−1.69055	−0.145500
$136$	0	0
$137$	13.6668	1.16763	0.583816	−	0.811886i	$-0.301559\pi$
$137$	13.6668	1.16763	0.583816	+	0.811886i	$0.301559\pi$
$138$	0	0
$139$	−4.49936	−0.381631	−0.190815	−	0.981626i	$-0.561113\pi$
$139$	−4.49936	−0.381631	−0.190815	+	0.981626i	$0.561113\pi$
$140$	0	0
$141$	−8.46786	−0.713123
$142$	0	0
$143$	0.103510	0.00865592
$144$	0	0
$145$	−2.27960	−0.189311
$146$	0	0
$147$	−1.63068	−0.134496
$148$	0	0
$149$	18.3579	1.50393	0.751967	−	0.659201i	$-0.229105\pi$
$149$	18.3579	1.50393	0.751967	+	0.659201i	$0.229105\pi$
$150$	0	0
$151$	−16.0768	−1.30831	−0.654157	−	0.756359i	$-0.726976\pi$
$151$	−16.0768	−1.30831	−0.654157	+	0.756359i	$0.726976\pi$
$152$	0	0
$153$	−3.91589	−0.316581
$154$	0	0
$155$	5.94873	0.477813
$156$	0	0
$157$	4.16584	0.332471	0.166235	−	0.986086i	$-0.446839\pi$
$157$	4.16584	0.332471	0.166235	+	0.986086i	$0.446839\pi$
$158$	0	0
$159$	0.738345	0.0585545
$160$	0	0
$161$	14.8019	1.16655
$162$	0	0
$163$	4.54340	0.355866	0.177933	−	0.984043i	$-0.443059\pi$
$163$	4.54340	0.355866	0.177933	+	0.984043i	$0.443059\pi$
$164$	0	0
$165$	−0.459278	−0.0357548
$166$	0	0
$167$	−9.16157	−0.708944	−0.354472	−	0.935067i	$-0.615339\pi$
$167$	−9.16157	−0.708944	−0.354472	+	0.935067i	$0.615339\pi$
$168$	0	0
$169$	−12.8548	−0.988833
$170$	0	0
$171$	−7.86989	−0.601826
$172$	0	0
$173$	6.63437	0.504402	0.252201	−	0.967675i	$-0.418846\pi$
$173$	6.63437	0.504402	0.252201	+	0.967675i	$0.418846\pi$
$174$	0	0
$175$	−4.96346	−0.375202
$176$	0	0
$177$	−5.38569	−0.404813
$178$	0	0
$179$	4.04454	0.302303	0.151152	−	0.988511i	$-0.451702\pi$
$179$	4.04454	0.302303	0.151152	+	0.988511i	$0.451702\pi$
$180$	0	0
$181$	19.9515	1.48299	0.741493	−	0.670961i	$-0.234118\pi$
$181$	19.9515	1.48299	0.741493	+	0.670961i	$0.234118\pi$
$182$	0	0
$183$	2.58968	0.191435
$184$	0	0
$185$	5.62504	0.413561
$186$	0	0
$187$	−1.06384	−0.0777958
$188$	0	0
$189$	2.31718	0.168550
$190$	0	0
$191$	10.2627	0.742580	0.371290	−	0.928517i	$-0.378916\pi$
$191$	10.2627	0.742580	0.371290	+	0.928517i	$0.378916\pi$
$192$	0	0
$193$	17.0416	1.22668	0.613342	−	0.789817i	$-0.289825\pi$
$193$	17.0416	1.22668	0.613342	+	0.789817i	$0.289825\pi$
$194$	0	0
$195$	−0.644116	−0.0461261
$196$	0	0
$197$	−24.6333	−1.75505	−0.877524	−	0.479533i	$-0.840806\pi$
$197$	−24.6333	−1.75505	−0.877524	+	0.479533i	$0.840806\pi$
$198$	0	0
$199$	−7.85511	−0.556834	−0.278417	−	0.960460i	$-0.589810\pi$
$199$	−7.85511	−0.556834	−0.278417	+	0.960460i	$0.589810\pi$
$200$	0	0
$201$	4.19823	0.296120
$202$	0	0
$203$	3.12457	0.219302
$204$	0	0
$205$	−7.85377	−0.548531
$206$	0	0
$207$	6.38788	0.443988
$208$	0	0
$209$	−2.13804	−0.147891
$210$	0	0
$211$	11.3322	0.780140	0.390070	−	0.920785i	$-0.372451\pi$
$211$	11.3322	0.780140	0.390070	+	0.920785i	$0.372451\pi$
$212$	0	0
$213$	−15.3838	−1.05408
$214$	0	0
$215$	−5.51476	−0.376103
$216$	0	0
$217$	−8.15370	−0.553509
$218$	0	0
$219$	0.142034	0.00959778
$220$	0	0
$221$	−1.49199	−0.100362
$222$	0	0
$223$	7.00723	0.469239	0.234619	−	0.972087i	$-0.424616\pi$
$223$	7.00723	0.469239	0.234619	+	0.972087i	$0.424616\pi$
$224$	0	0
$225$	−2.14202	−0.142802
$226$	0	0
$227$	−19.8081	−1.31471	−0.657354	−	0.753582i	$-0.728324\pi$
$227$	−19.8081	−1.31471	−0.657354	+	0.753582i	$0.728324\pi$
$228$	0	0
$229$	−4.95843	−0.327662	−0.163831	−	0.986488i	$-0.552385\pi$
$229$	−4.95843	−0.327662	−0.163831	+	0.986488i	$0.552385\pi$
$230$	0	0
$231$	0.629516	0.0414191
$232$	0	0
$233$	−11.0512	−0.723986	−0.361993	−	0.932181i	$-0.617904\pi$
$233$	−11.0512	−0.723986	−0.361993	+	0.932181i	$0.617904\pi$
$234$	0	0
$235$	14.3154	0.933833
$236$	0	0
$237$	3.31119	0.215085
$238$	0	0
$239$	−3.09225	−0.200021	−0.100011	−	0.994986i	$-0.531888\pi$
$239$	−3.09225	−0.200021	−0.100011	+	0.994986i	$0.531888\pi$
$240$	0	0
$241$	−15.8274	−1.01953	−0.509765	−	0.860314i	$-0.670268\pi$
$241$	−15.8274	−1.01953	−0.509765	+	0.860314i	$0.670268\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	2.75675	0.176122
$246$	0	0
$247$	−2.99849	−0.190790
$248$	0	0
$249$	−10.2715	−0.650930
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	1.73541	0.109105
$254$	0	0
$255$	6.62002	0.414562
$256$	0	0
$257$	−1.59557	−0.0995292	−0.0497646	−	0.998761i	$-0.515847\pi$
$257$	−1.59557	−0.0995292	−0.0497646	+	0.998761i	$0.515847\pi$
$258$	0	0
$259$	−7.71004	−0.479078
$260$	0	0
$261$	1.34844	0.0834660
$262$	0	0
$263$	12.4749	0.769238	0.384619	−	0.923075i	$-0.374333\pi$
$263$	12.4749	0.769238	0.384619	+	0.923075i	$0.374333\pi$
$264$	0	0
$265$	−1.24821	−0.0766771
$266$	0	0
$267$	−7.10516	−0.434829
$268$	0	0
$269$	22.4387	1.36811	0.684055	−	0.729430i	$-0.260215\pi$
$269$	22.4387	1.36811	0.684055	+	0.729430i	$0.260215\pi$
$270$	0	0
$271$	−10.3201	−0.626903	−0.313452	−	0.949604i	$-0.601485\pi$
$271$	−10.3201	−0.626903	−0.313452	+	0.949604i	$0.601485\pi$
$272$	0	0
$273$	0.882865	0.0534334
$274$	0	0
$275$	−0.581931	−0.0350917
$276$	0	0
$277$	−12.3691	−0.743187	−0.371594	−	0.928395i	$-0.621189\pi$
$277$	−12.3691	−0.743187	−0.371594	+	0.928395i	$0.621189\pi$
$278$	0	0
$279$	−3.51880	−0.210665
$280$	0	0
$281$	−7.64222	−0.455897	−0.227948	−	0.973673i	$-0.573202\pi$
$281$	−7.64222	−0.455897	−0.227948	+	0.973673i	$0.573202\pi$
$282$	0	0
$283$	−17.9283	−1.06573	−0.532863	−	0.846202i	$-0.678884\pi$
$283$	−17.9283	−1.06573	−0.532863	+	0.846202i	$0.678884\pi$
$284$	0	0
$285$	13.3045	0.788089
$286$	0	0
$287$	10.7649	0.635430
$288$	0	0
$289$	−1.66584	−0.0979906
$290$	0	0
$291$	−17.7783	−1.04218
$292$	0	0
$293$	−26.7747	−1.56420	−0.782098	−	0.623156i	$-0.785850\pi$
$293$	−26.7747	−1.56420	−0.782098	+	0.623156i	$0.785850\pi$
$294$	0	0
$295$	9.10480	0.530102
$296$	0	0
$297$	0.271673	0.0157641
$298$	0	0
$299$	2.43383	0.140752
$300$	0	0
$301$	7.55887	0.435686
$302$	0	0
$303$	12.0054	0.689694
$304$	0	0
$305$	−4.37799	−0.250683
$306$	0	0
$307$	4.76248	0.271809	0.135905	−	0.990722i	$-0.456606\pi$
$307$	4.76248	0.271809	0.135905	+	0.990722i	$0.456606\pi$
$308$	0	0
$309$	−8.44574	−0.480461
$310$	0	0
$311$	−19.5015	−1.10583	−0.552916	−	0.833237i	$-0.686485\pi$
$311$	−19.5015	−1.10583	−0.552916	+	0.833237i	$0.686485\pi$
$312$	0	0
$313$	−15.7418	−0.889778	−0.444889	−	0.895586i	$-0.646757\pi$
$313$	−15.7418	−0.889778	−0.444889	+	0.895586i	$0.646757\pi$
$314$	0	0
$315$	−3.91732	−0.220716
$316$	0	0
$317$	−6.06785	−0.340804	−0.170402	−	0.985375i	$-0.554507\pi$
$317$	−6.06785	−0.340804	−0.170402	+	0.985375i	$0.554507\pi$
$318$	0	0
$319$	0.366334	0.0205107
$320$	0	0
$321$	−1.24465	−0.0694697
$322$	0	0
$323$	30.8176	1.71474
$324$	0	0
$325$	−0.816129	−0.0452707
$326$	0	0
$327$	−18.9472	−1.04778
$328$	0	0
$329$	−19.6216	−1.08177
$330$	0	0
$331$	−0.0963604	−0.00529645	−0.00264822	−	0.999996i	$-0.500843\pi$
$331$	−0.0963604	−0.00529645	−0.00264822	+	0.999996i	$0.500843\pi$
$332$	0	0
$333$	−3.32734	−0.182337
$334$	0	0
$335$	−7.09733	−0.387769
$336$	0	0
$337$	−29.3530	−1.59896	−0.799481	−	0.600691i	$-0.794892\pi$
$337$	−29.3530	−1.59896	−0.799481	+	0.600691i	$0.794892\pi$
$338$	0	0
$339$	6.51359	0.353770
$340$	0	0
$341$	−0.955964	−0.0517684
$342$	0	0
$343$	−19.9988	−1.07984
$344$	0	0
$345$	−10.7991	−0.581402
$346$	0	0
$347$	5.75760	0.309084	0.154542	−	0.987986i	$-0.450610\pi$
$347$	5.75760	0.309084	0.154542	+	0.987986i	$0.450610\pi$
$348$	0	0
$349$	−13.9088	−0.744518	−0.372259	−	0.928129i	$-0.621417\pi$
$349$	−13.9088	−0.744518	−0.372259	+	0.928129i	$0.621417\pi$
$350$	0	0
$351$	0.381008	0.0203367
$352$	0	0
$353$	−28.5271	−1.51835	−0.759173	−	0.650889i	$-0.774396\pi$
$353$	−28.5271	−1.51835	−0.759173	+	0.650889i	$0.774396\pi$
$354$	0	0
$355$	26.0072	1.38032
$356$	0	0
$357$	−9.07381	−0.480237
$358$	0	0
$359$	−25.1497	−1.32735	−0.663675	−	0.748021i	$-0.731004\pi$
$359$	−25.1497	−1.32735	−0.663675	+	0.748021i	$0.731004\pi$
$360$	0	0
$361$	42.9352	2.25975
$362$	0	0
$363$	−10.9262	−0.573476
$364$	0	0
$365$	−0.240116	−0.0125683
$366$	0	0
$367$	−0.490018	−0.0255787	−0.0127894	−	0.999918i	$-0.504071\pi$
$367$	−0.490018	−0.0255787	−0.0127894	+	0.999918i	$0.504071\pi$
$368$	0	0
$369$	4.64567	0.241844
$370$	0	0
$371$	1.71088	0.0888243
$372$	0	0
$373$	−7.83264	−0.405559	−0.202779	−	0.979224i	$-0.564997\pi$
$373$	−7.83264	−0.405559	−0.202779	+	0.979224i	$0.564997\pi$
$374$	0	0
$375$	12.0740	0.623498
$376$	0	0
$377$	0.513765	0.0264602
$378$	0	0
$379$	−11.5566	−0.593623	−0.296812	−	0.954936i	$-0.595923\pi$
$379$	−11.5566	−0.593623	−0.296812	+	0.954936i	$0.595923\pi$
$380$	0	0
$381$	−3.40748	−0.174570
$382$	0	0
$383$	30.6654	1.56693	0.783464	−	0.621438i	$-0.213451\pi$
$383$	30.6654	1.56693	0.783464	+	0.621438i	$0.213451\pi$
$384$	0	0
$385$	−1.06423	−0.0542382
$386$	0	0
$387$	3.26210	0.165822
$388$	0	0
$389$	−3.88754	−0.197106	−0.0985530	−	0.995132i	$-0.531421\pi$
$389$	−3.88754	−0.197106	−0.0985530	+	0.995132i	$0.531421\pi$
$390$	0	0
$391$	−25.0142	−1.26502
$392$	0	0
$393$	22.4689	1.13341
$394$	0	0
$395$	−5.59775	−0.281654
$396$	0	0
$397$	−24.8960	−1.24950	−0.624748	−	0.780826i	$-0.714798\pi$
$397$	−24.8960	−1.24950	−0.624748	+	0.780826i	$0.714798\pi$
$398$	0	0
$399$	−18.2359	−0.912939
$400$	0	0
$401$	14.5928	0.728727	0.364364	−	0.931257i	$-0.381287\pi$
$401$	14.5928	0.728727	0.364364	+	0.931257i	$0.381287\pi$
$402$	0	0
$403$	−1.34069	−0.0667847
$404$	0	0
$405$	−1.69055	−0.0840043
$406$	0	0
$407$	−0.903948	−0.0448070
$408$	0	0
$409$	−4.57000	−0.225972	−0.112986	−	0.993597i	$-0.536042\pi$
$409$	−4.57000	−0.225972	−0.112986	+	0.993597i	$0.536042\pi$
$410$	0	0
$411$	13.6668	0.674132
$412$	0	0
$413$	−12.4796	−0.614081
$414$	0	0
$415$	17.3645	0.852392
$416$	0	0
$417$	−4.49936	−0.220335
$418$	0	0
$419$	20.8605	1.01910	0.509552	−	0.860440i	$-0.329811\pi$
$419$	20.8605	1.01910	0.509552	+	0.860440i	$0.329811\pi$
$420$	0	0
$421$	−14.5772	−0.710448	−0.355224	−	0.934781i	$-0.615595\pi$
$421$	−14.5772	−0.710448	−0.355224	+	0.934781i	$0.615595\pi$
$422$	0	0
$423$	−8.46786	−0.411722
$424$	0	0
$425$	8.38792	0.406874
$426$	0	0
$427$	6.00075	0.290397
$428$	0	0
$429$	0.103510	0.00499750
$430$	0	0
$431$	12.3084	0.592873	0.296437	−	0.955053i	$-0.404202\pi$
$431$	12.3084	0.592873	0.296437	+	0.955053i	$0.404202\pi$
$432$	0	0
$433$	−5.76955	−0.277267	−0.138633	−	0.990344i	$-0.544271\pi$
$433$	−5.76955	−0.277267	−0.138633	+	0.990344i	$0.544271\pi$
$434$	0	0
$435$	−2.27960	−0.109299
$436$	0	0
$437$	−50.2719	−2.40483
$438$	0	0
$439$	−29.5511	−1.41040	−0.705199	−	0.709009i	$-0.749143\pi$
$439$	−29.5511	−1.41040	−0.705199	+	0.709009i	$0.749143\pi$
$440$	0	0
$441$	−1.63068	−0.0776514
$442$	0	0
$443$	5.81296	0.276182	0.138091	−	0.990420i	$-0.455903\pi$
$443$	5.81296	0.276182	0.138091	+	0.990420i	$0.455903\pi$
$444$	0	0
$445$	12.0117	0.569407
$446$	0	0
$447$	18.3579	0.868297
$448$	0	0
$449$	−19.5267	−0.921522	−0.460761	−	0.887524i	$-0.652424\pi$
$449$	−19.5267	−0.921522	−0.460761	+	0.887524i	$0.652424\pi$
$450$	0	0
$451$	1.26211	0.0594302
$452$	0	0
$453$	−16.0768	−0.755355
$454$	0	0
$455$	−1.49253	−0.0699710
$456$	0	0
$457$	27.0483	1.26526	0.632632	−	0.774452i	$-0.281975\pi$
$457$	27.0483	1.26526	0.632632	+	0.774452i	$0.281975\pi$
$458$	0	0
$459$	−3.91589	−0.182778
$460$	0	0
$461$	−38.2015	−1.77922	−0.889610	−	0.456721i	$-0.849024\pi$
$461$	−38.2015	−1.77922	−0.889610	+	0.456721i	$0.849024\pi$
$462$	0	0
$463$	20.4776	0.951673	0.475837	−	0.879534i	$-0.342145\pi$
$463$	20.4776	0.951673	0.475837	+	0.879534i	$0.342145\pi$
$464$	0	0
$465$	5.94873	0.275866
$466$	0	0
$467$	1.78201	0.0824618	0.0412309	−	0.999150i	$-0.486872\pi$
$467$	1.78201	0.0824618	0.0412309	+	0.999150i	$0.486872\pi$
$468$	0	0
$469$	9.72804	0.449199
$470$	0	0
$471$	4.16584	0.191952
$472$	0	0
$473$	0.886225	0.0407487
$474$	0	0
$475$	16.8575	0.773475
$476$	0	0
$477$	0.738345	0.0338065
$478$	0	0
$479$	10.0359	0.458552	0.229276	−	0.973361i	$-0.426364\pi$
$479$	10.0359	0.458552	0.229276	+	0.973361i	$0.426364\pi$
$480$	0	0
$481$	−1.26774	−0.0578041
$482$	0	0
$483$	14.8019	0.673508
$484$	0	0
$485$	30.0552	1.36473
$486$	0	0
$487$	−15.2830	−0.692539	−0.346270	−	0.938135i	$-0.612552\pi$
$487$	−15.2830	−0.692539	−0.346270	+	0.938135i	$0.612552\pi$
$488$	0	0
$489$	4.54340	0.205460
$490$	0	0
$491$	17.9769	0.811288	0.405644	−	0.914031i	$-0.367047\pi$
$491$	17.9769	0.811288	0.405644	+	0.914031i	$0.367047\pi$
$492$	0	0
$493$	−5.28032	−0.237814
$494$	0	0
$495$	−0.459278	−0.0206430
$496$	0	0
$497$	−35.6471	−1.59899
$498$	0	0
$499$	39.7994	1.78167	0.890833	−	0.454331i	$-0.150122\pi$
$499$	39.7994	1.78167	0.890833	+	0.454331i	$0.150122\pi$
$500$	0	0
$501$	−9.16157	−0.409309
$502$	0	0
$503$	−8.16608	−0.364108	−0.182054	−	0.983289i	$-0.558275\pi$
$503$	−8.16608	−0.364108	−0.182054	+	0.983289i	$0.558275\pi$
$504$	0	0
$505$	−20.2958	−0.903153
$506$	0	0
$507$	−12.8548	−0.570903
$508$	0	0
$509$	4.13058	0.183085	0.0915424	−	0.995801i	$-0.470820\pi$
$509$	4.13058	0.183085	0.0915424	+	0.995801i	$0.470820\pi$
$510$	0	0
$511$	0.329119	0.0145593
$512$	0	0
$513$	−7.86989	−0.347464
$514$	0	0
$515$	14.2780	0.629163
$516$	0	0
$517$	−2.30049	−0.101176
$518$	0	0
$519$	6.63437	0.291216
$520$	0	0
$521$	20.3654	0.892226	0.446113	−	0.894977i	$-0.352808\pi$
$521$	20.3654	0.892226	0.446113	+	0.894977i	$0.352808\pi$
$522$	0	0
$523$	30.9097	1.35159	0.675793	−	0.737092i	$-0.263801\pi$
$523$	30.9097	1.35159	0.675793	+	0.737092i	$0.263801\pi$
$524$	0	0
$525$	−4.96346	−0.216623
$526$	0	0
$527$	13.7792	0.600232
$528$	0	0
$529$	17.8050	0.774129
$530$	0	0
$531$	−5.38569	−0.233719
$532$	0	0
$533$	1.77004	0.0766690
$534$	0	0
$535$	2.10415	0.0909704
$536$	0	0
$537$	4.04454	0.174535
$538$	0	0
$539$	−0.443012	−0.0190819
$540$	0	0
$541$	−4.05861	−0.174493	−0.0872467	−	0.996187i	$-0.527807\pi$
$541$	−4.05861	−0.174493	−0.0872467	+	0.996187i	$0.527807\pi$
$542$	0	0
$543$	19.9515	0.856202
$544$	0	0
$545$	32.0312	1.37207
$546$	0	0
$547$	38.6740	1.65358	0.826790	−	0.562511i	$-0.190165\pi$
$547$	38.6740	1.65358	0.826790	+	0.562511i	$0.190165\pi$
$548$	0	0
$549$	2.58968	0.110525
$550$	0	0
$551$	−10.6120	−0.452088
$552$	0	0
$553$	7.67263	0.326273
$554$	0	0
$555$	5.62504	0.238770
$556$	0	0
$557$	−5.21169	−0.220827	−0.110413	−	0.993886i	$-0.535217\pi$
$557$	−5.21169	−0.220827	−0.110413	+	0.993886i	$0.535217\pi$
$558$	0	0
$559$	1.24289	0.0525685
$560$	0	0
$561$	−1.06384	−0.0449154
$562$	0	0
$563$	3.48122	0.146716	0.0733579	−	0.997306i	$-0.476628\pi$
$563$	3.48122	0.146716	0.0733579	+	0.997306i	$0.476628\pi$
$564$	0	0
$565$	−11.0116	−0.463261
$566$	0	0
$567$	2.31718	0.0973124
$568$	0	0
$569$	30.3445	1.27211	0.636053	−	0.771645i	$-0.280566\pi$
$569$	30.3445	1.27211	0.636053	+	0.771645i	$0.280566\pi$
$570$	0	0
$571$	−7.29820	−0.305420	−0.152710	−	0.988271i	$-0.548800\pi$
$571$	−7.29820	−0.305420	−0.152710	+	0.988271i	$0.548800\pi$
$572$	0	0
$573$	10.2627	0.428729
$574$	0	0
$575$	−13.6830	−0.570620
$576$	0	0
$577$	−9.39767	−0.391230	−0.195615	−	0.980681i	$-0.562670\pi$
$577$	−9.39767	−0.391230	−0.195615	+	0.980681i	$0.562670\pi$
$578$	0	0
$579$	17.0416	0.708226
$580$	0	0
$581$	−23.8009	−0.987429
$582$	0	0
$583$	0.200588	0.00830753
$584$	0	0
$585$	−0.644116	−0.0266309
$586$	0	0
$587$	39.8732	1.64574	0.822872	−	0.568227i	$-0.192370\pi$
$587$	39.8732	1.64574	0.822872	+	0.568227i	$0.192370\pi$
$588$	0	0
$589$	27.6926	1.14105
$590$	0	0
$591$	−24.6333	−1.01328
$592$	0	0
$593$	36.5532	1.50106	0.750530	−	0.660836i	$-0.229798\pi$
$593$	36.5532	1.50106	0.750530	+	0.660836i	$0.229798\pi$
$594$	0	0
$595$	15.3398	0.628869
$596$	0	0
$597$	−7.85511	−0.321488
$598$	0	0
$599$	33.9952	1.38901	0.694504	−	0.719489i	$-0.255624\pi$
$599$	33.9952	1.38901	0.694504	+	0.719489i	$0.255624\pi$
$600$	0	0
$601$	2.11066	0.0860957	0.0430479	−	0.999073i	$-0.486293\pi$
$601$	2.11066	0.0860957	0.0430479	+	0.999073i	$0.486293\pi$
$602$	0	0
$603$	4.19823	0.170965
$604$	0	0
$605$	18.4713	0.750966
$606$	0	0
$607$	−9.07594	−0.368381	−0.184191	−	0.982891i	$-0.558966\pi$
$607$	−9.07594	−0.368381	−0.184191	+	0.982891i	$0.558966\pi$
$608$	0	0
$609$	3.12457	0.126614
$610$	0	0
$611$	−3.22633	−0.130523
$612$	0	0
$613$	−29.0681	−1.17405	−0.587024	−	0.809570i	$-0.699701\pi$
$613$	−29.0681	−1.17405	−0.587024	+	0.809570i	$0.699701\pi$
$614$	0	0
$615$	−7.85377	−0.316694
$616$	0	0
$617$	−42.2273	−1.70001	−0.850004	−	0.526776i	$-0.823401\pi$
$617$	−42.2273	−1.70001	−0.850004	+	0.526776i	$0.823401\pi$
$618$	0	0
$619$	33.7194	1.35530	0.677649	−	0.735386i	$-0.262999\pi$
$619$	33.7194	1.35530	0.677649	+	0.735386i	$0.262999\pi$
$620$	0	0
$621$	6.38788	0.256337
$622$	0	0
$623$	−16.4639	−0.659614
$624$	0	0
$625$	−9.70161	−0.388064
$626$	0	0
$627$	−2.13804	−0.0853850
$628$	0	0
$629$	13.0295	0.519519
$630$	0	0
$631$	10.2239	0.407006	0.203503	−	0.979074i	$-0.434767\pi$
$631$	10.2239	0.407006	0.203503	+	0.979074i	$0.434767\pi$
$632$	0	0
$633$	11.3322	0.450414
$634$	0	0
$635$	5.76053	0.228600
$636$	0	0
$637$	−0.621302	−0.0246169
$638$	0	0
$639$	−15.3838	−0.608575
$640$	0	0
$641$	40.9528	1.61754	0.808770	−	0.588126i	$-0.200134\pi$
$641$	40.9528	1.61754	0.808770	+	0.588126i	$0.200134\pi$
$642$	0	0
$643$	6.12477	0.241537	0.120769	−	0.992681i	$-0.461464\pi$
$643$	6.12477	0.241537	0.120769	+	0.992681i	$0.461464\pi$
$644$	0	0
$645$	−5.51476	−0.217143
$646$	0	0
$647$	−45.1887	−1.77655	−0.888276	−	0.459310i	$-0.848097\pi$
$647$	−45.1887	−1.77655	−0.888276	+	0.459310i	$0.848097\pi$
$648$	0	0
$649$	−1.46315	−0.0574335
$650$	0	0
$651$	−8.15370	−0.319569
$652$	0	0
$653$	14.0469	0.549699	0.274849	−	0.961487i	$-0.411372\pi$
$653$	14.0469	0.549699	0.274849	+	0.961487i	$0.411372\pi$
$654$	0	0
$655$	−37.9849	−1.48419
$656$	0	0
$657$	0.142034	0.00554128
$658$	0	0
$659$	−38.2660	−1.49063	−0.745316	−	0.666712i	$-0.767701\pi$
$659$	−38.2660	−1.49063	−0.745316	+	0.666712i	$0.767701\pi$
$660$	0	0
$661$	−22.8825	−0.890027	−0.445013	−	0.895524i	$-0.646801\pi$
$661$	−22.8825	−0.890027	−0.445013	+	0.895524i	$0.646801\pi$
$662$	0	0
$663$	−1.49199	−0.0579439
$664$	0	0
$665$	30.8289	1.19549
$666$	0	0
$667$	8.61364	0.333521
$668$	0	0
$669$	7.00723	0.270915
$670$	0	0
$671$	0.703546	0.0271601
$672$	0	0
$673$	−20.9288	−0.806745	−0.403372	−	0.915036i	$-0.632162\pi$
$673$	−20.9288	−0.806745	−0.403372	+	0.915036i	$0.632162\pi$
$674$	0	0
$675$	−2.14202	−0.0824466
$676$	0	0
$677$	24.6579	0.947679	0.473839	−	0.880611i	$-0.342868\pi$
$677$	24.6579	0.947679	0.473839	+	0.880611i	$0.342868\pi$
$678$	0	0
$679$	−41.1955	−1.58094
$680$	0	0
$681$	−19.8081	−0.759047
$682$	0	0
$683$	16.7140	0.639544	0.319772	−	0.947495i	$-0.396394\pi$
$683$	16.7140	0.639544	0.319772	+	0.947495i	$0.396394\pi$
$684$	0	0
$685$	−23.1044	−0.882775
$686$	0	0
$687$	−4.95843	−0.189176
$688$	0	0
$689$	0.281316	0.0107173
$690$	0	0
$691$	37.3596	1.42123	0.710614	−	0.703582i	$-0.248417\pi$
$691$	37.3596	1.42123	0.710614	+	0.703582i	$0.248417\pi$
$692$	0	0
$693$	0.629516	0.0239133
$694$	0	0
$695$	7.60642	0.288528
$696$	0	0
$697$	−18.1919	−0.689068
$698$	0	0
$699$	−11.0512	−0.417994
$700$	0	0
$701$	−37.3030	−1.40891	−0.704457	−	0.709747i	$-0.748809\pi$
$701$	−37.3030	−1.40891	−0.704457	+	0.709747i	$0.748809\pi$
$702$	0	0
$703$	26.1858	0.987615
$704$	0	0
$705$	14.3154	0.539149
$706$	0	0
$707$	27.8187	1.04623
$708$	0	0
$709$	25.3164	0.950777	0.475389	−	0.879776i	$-0.342307\pi$
$709$	25.3164	0.950777	0.475389	+	0.879776i	$0.342307\pi$
$710$	0	0
$711$	3.31119	0.124179
$712$	0	0
$713$	−22.4777	−0.841795
$714$	0	0
$715$	−0.174989	−0.00654421
$716$	0	0
$717$	−3.09225	−0.115482
$718$	0	0
$719$	25.6333	0.955961	0.477981	−	0.878370i	$-0.341369\pi$
$719$	25.6333	0.955961	0.477981	+	0.878370i	$0.341369\pi$
$720$	0	0
$721$	−19.5703	−0.728836
$722$	0	0
$723$	−15.8274	−0.588626
$724$	0	0
$725$	−2.88838	−0.107272
$726$	0	0
$727$	−10.5135	−0.389923	−0.194962	−	0.980811i	$-0.562458\pi$
$727$	−10.5135	−0.389923	−0.194962	+	0.980811i	$0.562458\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−12.7740	−0.472464
$732$	0	0
$733$	−48.9913	−1.80954	−0.904768	−	0.425905i	$-0.859956\pi$
$733$	−48.9913	−1.80954	−0.904768	+	0.425905i	$0.859956\pi$
$734$	0	0
$735$	2.75675	0.101684
$736$	0	0
$737$	1.14055	0.0420125
$738$	0	0
$739$	−8.24620	−0.303341	−0.151671	−	0.988431i	$-0.548465\pi$
$739$	−8.24620	−0.303341	−0.151671	+	0.988431i	$0.548465\pi$
$740$	0	0
$741$	−2.99849	−0.110152
$742$	0	0
$743$	30.8574	1.13205	0.566024	−	0.824388i	$-0.308481\pi$
$743$	30.8574	1.13205	0.566024	+	0.824388i	$0.308481\pi$
$744$	0	0
$745$	−31.0350	−1.13703
$746$	0	0
$747$	−10.2715	−0.375815
$748$	0	0
$749$	−2.88408	−0.105382
$750$	0	0
$751$	−34.0186	−1.24136	−0.620678	−	0.784066i	$-0.713142\pi$
$751$	−34.0186	−1.24136	−0.620678	+	0.784066i	$0.713142\pi$
$752$	0	0
$753$	1.00000	0.0364420
$754$	0	0
$755$	27.1787	0.989136
$756$	0	0
$757$	47.8579	1.73943	0.869713	−	0.493559i	$-0.164304\pi$
$757$	47.8579	1.73943	0.869713	+	0.493559i	$0.164304\pi$
$758$	0	0
$759$	1.73541	0.0629916
$760$	0	0
$761$	16.7770	0.608165	0.304083	−	0.952646i	$-0.401650\pi$
$761$	16.7770	0.608165	0.304083	+	0.952646i	$0.401650\pi$
$762$	0	0
$763$	−43.9040	−1.58943
$764$	0	0
$765$	6.62002	0.239347
$766$	0	0
$767$	−2.05199	−0.0740931
$768$	0	0
$769$	31.3730	1.13134	0.565671	−	0.824631i	$-0.308617\pi$
$769$	31.3730	1.13134	0.565671	+	0.824631i	$0.308617\pi$
$770$	0	0
$771$	−1.59557	−0.0574632
$772$	0	0
$773$	−28.5427	−1.02661	−0.513304	−	0.858207i	$-0.671579\pi$
$773$	−28.5427	−1.02661	−0.513304	+	0.858207i	$0.671579\pi$
$774$	0	0
$775$	7.53736	0.270750
$776$	0	0
$777$	−7.71004	−0.276596
$778$	0	0
$779$	−36.5609	−1.30993
$780$	0	0
$781$	−4.17937	−0.149550
$782$	0	0
$783$	1.34844	0.0481891
$784$	0	0
$785$	−7.04259	−0.251361
$786$	0	0
$787$	46.3960	1.65384	0.826919	−	0.562322i	$-0.190092\pi$
$787$	46.3960	1.65384	0.826919	+	0.562322i	$0.190092\pi$
$788$	0	0
$789$	12.4749	0.444120
$790$	0	0
$791$	15.0932	0.536651
$792$	0	0
$793$	0.986689	0.0350384
$794$	0	0
$795$	−1.24821	−0.0442695
$796$	0	0
$797$	3.88186	0.137503	0.0687513	−	0.997634i	$-0.478099\pi$
$797$	3.88186	0.137503	0.0687513	+	0.997634i	$0.478099\pi$
$798$	0	0
$799$	33.1592	1.17309
$800$	0	0
$801$	−7.10516	−0.251049
$802$	0	0
$803$	0.0385869	0.00136170
$804$	0	0
$805$	−25.0234	−0.881957
$806$	0	0
$807$	22.4387	0.789879
$808$	0	0
$809$	34.3791	1.20871	0.604353	−	0.796717i	$-0.293432\pi$
$809$	34.3791	1.20871	0.604353	+	0.796717i	$0.293432\pi$
$810$	0	0
$811$	35.2602	1.23815	0.619077	−	0.785330i	$-0.287507\pi$
$811$	35.2602	1.23815	0.619077	+	0.785330i	$0.287507\pi$
$812$	0	0
$813$	−10.3201	−0.361943
$814$	0	0
$815$	−7.68087	−0.269049
$816$	0	0
$817$	−25.6724	−0.898162
$818$	0	0
$819$	0.882865	0.0308498
$820$	0	0
$821$	31.4188	1.09652	0.548262	−	0.836306i	$-0.315289\pi$
$821$	31.4188	1.09652	0.548262	+	0.836306i	$0.315289\pi$
$822$	0	0
$823$	30.1332	1.05038	0.525190	−	0.850985i	$-0.323994\pi$
$823$	30.1332	1.05038	0.525190	+	0.850985i	$0.323994\pi$
$824$	0	0
$825$	−0.581931	−0.0202602
$826$	0	0
$827$	29.0770	1.01111	0.505554	−	0.862795i	$-0.331288\pi$
$827$	29.0770	1.01111	0.505554	+	0.862795i	$0.331288\pi$
$828$	0	0
$829$	−39.5031	−1.37200	−0.685999	−	0.727602i	$-0.740635\pi$
$829$	−39.5031	−1.37200	−0.685999	+	0.727602i	$0.740635\pi$
$830$	0	0
$831$	−12.3691	−0.429079
$832$	0	0
$833$	6.38555	0.221246
$834$	0	0
$835$	15.4881	0.535989
$836$	0	0
$837$	−3.51880	−0.121628
$838$	0	0
$839$	−18.7348	−0.646797	−0.323398	−	0.946263i	$-0.604825\pi$
$839$	−18.7348	−0.646797	−0.323398	+	0.946263i	$0.604825\pi$
$840$	0	0
$841$	−27.1817	−0.937301
$842$	0	0
$843$	−7.64222	−0.263212
$844$	0	0
$845$	21.7318	0.747597
$846$	0	0
$847$	−25.3180	−0.869935
$848$	0	0
$849$	−17.9283	−0.615297
$850$	0	0
$851$	−21.2546	−0.728599
$852$	0	0
$853$	−5.32226	−0.182231	−0.0911154	−	0.995840i	$-0.529043\pi$
$853$	−5.32226	−0.182231	−0.0911154	+	0.995840i	$0.529043\pi$
$854$	0	0
$855$	13.3045	0.455004
$856$	0	0
$857$	−34.3248	−1.17251	−0.586256	−	0.810126i	$-0.699399\pi$
$857$	−34.3248	−1.17251	−0.586256	+	0.810126i	$0.699399\pi$
$858$	0	0
$859$	32.6614	1.11439	0.557197	−	0.830380i	$-0.311877\pi$
$859$	32.6614	1.11439	0.557197	+	0.830380i	$0.311877\pi$
$860$	0	0
$861$	10.7649	0.366865
$862$	0	0
$863$	−25.7980	−0.878174	−0.439087	−	0.898445i	$-0.644698\pi$
$863$	−25.7980	−0.878174	−0.439087	+	0.898445i	$0.644698\pi$
$864$	0	0
$865$	−11.2158	−0.381347
$866$	0	0
$867$	−1.66584	−0.0565749
$868$	0	0
$869$	0.899563	0.0305156
$870$	0	0
$871$	1.59956	0.0541990
$872$	0	0
$873$	−17.7783	−0.601704
$874$	0	0
$875$	27.9776	0.945815
$876$	0	0
$877$	49.0747	1.65713	0.828567	−	0.559890i	$-0.189157\pi$
$877$	49.0747	1.65713	0.828567	+	0.559890i	$0.189157\pi$
$878$	0	0
$879$	−26.7747	−0.903088
$880$	0	0
$881$	27.8633	0.938739	0.469369	−	0.883002i	$-0.344481\pi$
$881$	27.8633	0.938739	0.469369	+	0.883002i	$0.344481\pi$
$882$	0	0
$883$	−26.4527	−0.890204	−0.445102	−	0.895480i	$-0.646832\pi$
$883$	−26.4527	−0.890204	−0.445102	+	0.895480i	$0.646832\pi$
$884$	0	0
$885$	9.10480	0.306055
$886$	0	0
$887$	−37.7146	−1.26633	−0.633166	−	0.774016i	$-0.718245\pi$
$887$	−37.7146	−1.26633	−0.633166	+	0.774016i	$0.718245\pi$
$888$	0	0
$889$	−7.89574	−0.264815
$890$	0	0
$891$	0.271673	0.00910139
$892$	0	0
$893$	66.6412	2.23006
$894$	0	0
$895$	−6.83752	−0.228553
$896$	0	0
$897$	2.43383	0.0812634
$898$	0	0
$899$	−4.74488	−0.158250
$900$	0	0
$901$	−2.89127	−0.0963223
$902$	0	0
$903$	7.55887	0.251543
$904$	0	0
$905$	−33.7292	−1.12120
$906$	0	0
$907$	21.9567	0.729061	0.364530	−	0.931191i	$-0.381230\pi$
$907$	21.9567	0.729061	0.364530	+	0.931191i	$0.381230\pi$
$908$	0	0
$909$	12.0054	0.398195
$910$	0	0
$911$	24.1789	0.801081	0.400541	−	0.916279i	$-0.368822\pi$
$911$	24.1789	0.801081	0.400541	+	0.916279i	$0.368822\pi$
$912$	0	0
$913$	−2.79049	−0.0923518
$914$	0	0
$915$	−4.37799	−0.144732
$916$	0	0
$917$	52.0645	1.71932
$918$	0	0
$919$	59.0567	1.94810	0.974050	−	0.226331i	$-0.0726732\pi$
$919$	59.0567	1.94810	0.974050	+	0.226331i	$0.0726732\pi$
$920$	0	0
$921$	4.76248	0.156929
$922$	0	0
$923$	−5.86136	−0.192929
$924$	0	0
$925$	7.12724	0.234342
$926$	0	0
$927$	−8.44574	−0.277394
$928$	0	0
$929$	24.1319	0.791743	0.395871	−	0.918306i	$-0.370443\pi$
$929$	24.1319	0.791743	0.395871	+	0.918306i	$0.370443\pi$
$930$	0	0
$931$	12.8333	0.420593
$932$	0	0
$933$	−19.5015	−0.638452
$934$	0	0
$935$	1.79848	0.0588166
$936$	0	0
$937$	40.5270	1.32396	0.661979	−	0.749522i	$-0.269717\pi$
$937$	40.5270	1.32396	0.661979	+	0.749522i	$0.269717\pi$
$938$	0	0
$939$	−15.7418	−0.513713
$940$	0	0
$941$	−22.9152	−0.747014	−0.373507	−	0.927627i	$-0.621845\pi$
$941$	−22.9152	−0.747014	−0.373507	+	0.927627i	$0.621845\pi$
$942$	0	0
$943$	29.6760	0.966383
$944$	0	0
$945$	−3.91732	−0.127430
$946$	0	0
$947$	1.68290	0.0546868	0.0273434	−	0.999626i	$-0.491295\pi$
$947$	1.68290	0.0546868	0.0273434	+	0.999626i	$0.491295\pi$
$948$	0	0
$949$	0.0541162	0.00175669
$950$	0	0
$951$	−6.06785	−0.196763
$952$	0	0
$953$	31.5311	1.02139	0.510696	−	0.859762i	$-0.329388\pi$
$953$	31.5311	1.02139	0.510696	+	0.859762i	$0.329388\pi$
$954$	0	0
$955$	−17.3496	−0.561420
$956$	0	0
$957$	0.366334	0.0118419
$958$	0	0
$959$	31.6684	1.02263
$960$	0	0
$961$	−18.6180	−0.600582
$962$	0	0
$963$	−1.24465	−0.0401083
$964$	0	0
$965$	−28.8098	−0.927421
$966$	0	0
$967$	22.9439	0.737826	0.368913	−	0.929464i	$-0.379730\pi$
$967$	22.9439	0.737826	0.368913	+	0.929464i	$0.379730\pi$
$968$	0	0
$969$	30.8176	0.990004
$970$	0	0
$971$	−28.8900	−0.927123	−0.463562	−	0.886065i	$-0.653429\pi$
$971$	−28.8900	−0.927123	−0.463562	+	0.886065i	$0.653429\pi$
$972$	0	0
$973$	−10.4258	−0.334237
$974$	0	0
$975$	−0.816129	−0.0261371
$976$	0	0
$977$	5.08079	0.162549	0.0812744	−	0.996692i	$-0.474101\pi$
$977$	5.08079	0.162549	0.0812744	+	0.996692i	$0.474101\pi$
$978$	0	0
$979$	−1.93028	−0.0616921
$980$	0	0
$981$	−18.9472	−0.604937
$982$	0	0
$983$	−33.3983	−1.06524	−0.532621	−	0.846354i	$-0.678793\pi$
$983$	−33.3983	−1.06524	−0.532621	+	0.846354i	$0.678793\pi$
$984$	0	0
$985$	41.6439	1.32688
$986$	0	0
$987$	−19.6216	−0.624561
$988$	0	0
$989$	20.8379	0.662606
$990$	0	0
$991$	46.4888	1.47676	0.738382	−	0.674382i	$-0.235590\pi$
$991$	46.4888	1.47676	0.738382	+	0.674382i	$0.235590\pi$
$992$	0	0
$993$	−0.0963604	−0.00305791
$994$	0	0
$995$	13.2795	0.420988
$996$	0	0
$997$	5.10913	0.161808	0.0809038	−	0.996722i	$-0.474219\pi$
$997$	5.10913	0.161808	0.0809038	+	0.996722i	$0.474219\pi$
$998$	0	0
$999$	−3.32734	−0.105272

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6024.2.a.k.1.3	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6024.2.a.k.1.3	✓	8	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:	\( N \)	\(=\)	\( 6024 = 2^{3} \cdot 3 \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6024.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.69055 q^{5} +2.31718 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.69055 q^{5} +2.31718 q^{7} +1.00000 q^{9} +0.271673 q^{11} +0.381008 q^{13} -1.69055 q^{15} -3.91589 q^{17} -7.86989 q^{19} +2.31718 q^{21} +6.38788 q^{23} -2.14202 q^{25} +1.00000 q^{27} +1.34844 q^{29} -3.51880 q^{31} +0.271673 q^{33} -3.91732 q^{35} -3.32734 q^{37} +0.381008 q^{39} +4.64567 q^{41} +3.26210 q^{43} -1.69055 q^{45} -8.46786 q^{47} -1.63068 q^{49} -3.91589 q^{51} +0.738345 q^{53} -0.459278 q^{55} -7.86989 q^{57} -5.38569 q^{59} +2.58968 q^{61} +2.31718 q^{63} -0.644116 q^{65} +4.19823 q^{67} +6.38788 q^{69} -15.3838 q^{71} +0.142034 q^{73} -2.14202 q^{75} +0.629516 q^{77} +3.31119 q^{79} +1.00000 q^{81} -10.2715 q^{83} +6.62002 q^{85} +1.34844 q^{87} -7.10516 q^{89} +0.882865 q^{91} -3.51880 q^{93} +13.3045 q^{95} -17.7783 q^{97} +0.271673 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{3} - 5 q^{5} + q^{7} + 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^3 - 5 * q^5 + q^7 + 8 * q^9	\( 8 q + 8 q^{3} - 5 q^{5} + q^{7} + 8 q^{9} - 6 q^{11} - 4 q^{13} - 5 q^{15} - 12 q^{17} - 10 q^{19} + q^{21} - 6 q^{23} - 7 q^{25} + 8 q^{27} - 2 q^{29} - 3 q^{31} - 6 q^{33} + 4 q^{35} - 16 q^{37} - 4 q^{39} + 2 q^{41} - 2 q^{43} - 5 q^{45} - 15 q^{47} - 21 q^{49} - 12 q^{51} + 13 q^{53} - 33 q^{55} - 10 q^{57} + 8 q^{59} - 38 q^{61} + q^{63} + 16 q^{65} - 31 q^{67} - 6 q^{69} + 5 q^{71} - 26 q^{73} - 7 q^{75} - 24 q^{77} - 25 q^{79} + 8 q^{81} - 7 q^{83} - 18 q^{85} - 2 q^{87} - 14 q^{89} - 6 q^{91} - 3 q^{93} - q^{95} - 51 q^{97} - 6 q^{99}+O(q^{100}) \) 8 * q + 8 * q^3 - 5 * q^5 + q^7 + 8 * q^9 - 6 * q^11 - 4 * q^13 - 5 * q^15 - 12 * q^17 - 10 * q^19 + q^21 - 6 * q^23 - 7 * q^25 + 8 * q^27 - 2 * q^29 - 3 * q^31 - 6 * q^33 + 4 * q^35 - 16 * q^37 - 4 * q^39 + 2 * q^41 - 2 * q^43 - 5 * q^45 - 15 * q^47 - 21 * q^49 - 12 * q^51 + 13 * q^53 - 33 * q^55 - 10 * q^57 + 8 * q^59 - 38 * q^61 + q^63 + 16 * q^65 - 31 * q^67 - 6 * q^69 + 5 * q^71 - 26 * q^73 - 7 * q^75 - 24 * q^77 - 25 * q^79 + 8 * q^81 - 7 * q^83 - 18 * q^85 - 2 * q^87 - 14 * q^89 - 6 * q^91 - 3 * q^93 - q^95 - 51 * q^97 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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