LMFDB - Embedded newform 3042.2.a.y.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3042,2,Mod(1,3042)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3042, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3042.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3042 = 2 \cdot 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3042.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:	$24.2904922949$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 78)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	3042.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +0.267949 q^{5} -0.732051 q^{7} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} +0.267949 q^{5} -0.732051 q^{7} +1.00000 q^{8} +0.267949 q^{10} +4.73205 q^{11} -0.732051 q^{14} +1.00000 q^{16} +2.26795 q^{17} -1.26795 q^{19} +0.267949 q^{20} +4.73205 q^{22} +6.19615 q^{23} -4.92820 q^{25} -0.732051 q^{28} -2.46410 q^{29} +5.46410 q^{31} +1.00000 q^{32} +2.26795 q^{34} -0.196152 q^{35} -10.4641 q^{37} -1.26795 q^{38} +0.267949 q^{40} +11.3923 q^{41} +7.66025 q^{43} +4.73205 q^{44} +6.19615 q^{46} -8.19615 q^{47} -6.46410 q^{49} -4.92820 q^{50} -0.464102 q^{53} +1.26795 q^{55} -0.732051 q^{56} -2.46410 q^{58} +8.00000 q^{59} +1.19615 q^{61} +5.46410 q^{62} +1.00000 q^{64} +11.1244 q^{67} +2.26795 q^{68} -0.196152 q^{70} +1.26795 q^{71} +9.73205 q^{73} -10.4641 q^{74} -1.26795 q^{76} -3.46410 q^{77} -9.46410 q^{79} +0.267949 q^{80} +11.3923 q^{82} +10.1962 q^{83} +0.607695 q^{85} +7.66025 q^{86} +4.73205 q^{88} -2.53590 q^{89} +6.19615 q^{92} -8.19615 q^{94} -0.339746 q^{95} +6.00000 q^{97} -6.46410 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + 4 q^{5} + 2 q^{7} + 2 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 + 4 * q^5 + 2 * q^7 + 2 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} + 4 q^{5} + 2 q^{7} + 2 q^{8} + 4 q^{10} + 6 q^{11} + 2 q^{14} + 2 q^{16} + 8 q^{17} - 6 q^{19} + 4 q^{20} + 6 q^{22} + 2 q^{23} + 4 q^{25} + 2 q^{28} + 2 q^{29} + 4 q^{31} + 2 q^{32} + 8 q^{34} + 10 q^{35} - 14 q^{37} - 6 q^{38} + 4 q^{40} + 2 q^{41} - 2 q^{43} + 6 q^{44} + 2 q^{46} - 6 q^{47} - 6 q^{49} + 4 q^{50} + 6 q^{53} + 6 q^{55} + 2 q^{56} + 2 q^{58} + 16 q^{59} - 8 q^{61} + 4 q^{62} + 2 q^{64} - 2 q^{67} + 8 q^{68} + 10 q^{70} + 6 q^{71} + 16 q^{73} - 14 q^{74} - 6 q^{76} - 12 q^{79} + 4 q^{80} + 2 q^{82} + 10 q^{83} + 22 q^{85} - 2 q^{86} + 6 q^{88} - 12 q^{89} + 2 q^{92} - 6 q^{94} - 18 q^{95} + 12 q^{97} - 6 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 4 * q^5 + 2 * q^7 + 2 * q^8 + 4 * q^10 + 6 * q^11 + 2 * q^14 + 2 * q^16 + 8 * q^17 - 6 * q^19 + 4 * q^20 + 6 * q^22 + 2 * q^23 + 4 * q^25 + 2 * q^28 + 2 * q^29 + 4 * q^31 + 2 * q^32 + 8 * q^34 + 10 * q^35 - 14 * q^37 - 6 * q^38 + 4 * q^40 + 2 * q^41 - 2 * q^43 + 6 * q^44 + 2 * q^46 - 6 * q^47 - 6 * q^49 + 4 * q^50 + 6 * q^53 + 6 * q^55 + 2 * q^56 + 2 * q^58 + 16 * q^59 - 8 * q^61 + 4 * q^62 + 2 * q^64 - 2 * q^67 + 8 * q^68 + 10 * q^70 + 6 * q^71 + 16 * q^73 - 14 * q^74 - 6 * q^76 - 12 * q^79 + 4 * q^80 + 2 * q^82 + 10 * q^83 + 22 * q^85 - 2 * q^86 + 6 * q^88 - 12 * q^89 + 2 * q^92 - 6 * q^94 - 18 * q^95 + 12 * q^97 - 6 * q^98

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0.267949	0.119831	0.0599153	−	0.998203i	$-0.480917\pi$
$5$	0.267949	0.119831	0.0599153	+	0.998203i	$0.480917\pi$
$6$	0	0
$7$	−0.732051	−0.276689	−0.138345	−	0.990384i	$-0.544178\pi$
$7$	−0.732051	−0.276689	−0.138345	+	0.990384i	$0.544178\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	0.267949	0.0847330
$11$	4.73205	1.42677	0.713384	−	0.700774i	$-0.247162\pi$
$11$	4.73205	1.42677	0.713384	+	0.700774i	$0.247162\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	−0.732051	−0.195649
$15$	0	0
$16$	1.00000	0.250000
$17$	2.26795	0.550058	0.275029	−	0.961436i	$-0.411312\pi$
$17$	2.26795	0.550058	0.275029	+	0.961436i	$0.411312\pi$
$18$	0	0
$19$	−1.26795	−0.290887	−0.145444	−	0.989367i	$-0.546461\pi$
$19$	−1.26795	−0.290887	−0.145444	+	0.989367i	$0.546461\pi$
$20$	0.267949	0.0599153
$21$	0	0
$22$	4.73205	1.00888
$23$	6.19615	1.29199	0.645994	−	0.763343i	$-0.276443\pi$
$23$	6.19615	1.29199	0.645994	+	0.763343i	$0.276443\pi$
$24$	0	0
$25$	−4.92820	−0.985641
$26$	0	0
$27$	0	0
$28$	−0.732051	−0.138345
$29$	−2.46410	−0.457572	−0.228786	−	0.973477i	$-0.573476\pi$
$29$	−2.46410	−0.457572	−0.228786	+	0.973477i	$0.573476\pi$
$30$	0	0
$31$	5.46410	0.981382	0.490691	−	0.871334i	$-0.336744\pi$
$31$	5.46410	0.981382	0.490691	+	0.871334i	$0.336744\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	2.26795	0.388950
$35$	−0.196152	−0.0331558
$36$	0	0
$37$	−10.4641	−1.72029	−0.860144	−	0.510052i	$-0.829626\pi$
$37$	−10.4641	−1.72029	−0.860144	+	0.510052i	$0.829626\pi$
$38$	−1.26795	−0.205689
$39$	0	0
$40$	0.267949	0.0423665
$41$	11.3923	1.77918	0.889590	−	0.456761i	$-0.150990\pi$
$41$	11.3923	1.77918	0.889590	+	0.456761i	$0.150990\pi$
$42$	0	0
$43$	7.66025	1.16818	0.584089	−	0.811690i	$-0.301452\pi$
$43$	7.66025	1.16818	0.584089	+	0.811690i	$0.301452\pi$
$44$	4.73205	0.713384
$45$	0	0
$46$	6.19615	0.913573
$47$	−8.19615	−1.19553	−0.597766	−	0.801671i	$-0.703945\pi$
$47$	−8.19615	−1.19553	−0.597766	+	0.801671i	$0.703945\pi$
$48$	0	0
$49$	−6.46410	−0.923443
$50$	−4.92820	−0.696953
$51$	0	0
$52$	0	0
$53$	−0.464102	−0.0637493	−0.0318746	−	0.999492i	$-0.510148\pi$
$53$	−0.464102	−0.0637493	−0.0318746	+	0.999492i	$0.510148\pi$
$54$	0	0
$55$	1.26795	0.170970
$56$	−0.732051	−0.0978244
$57$	0	0
$58$	−2.46410	−0.323552
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	0	0
$61$	1.19615	0.153152	0.0765758	−	0.997064i	$-0.475601\pi$
$61$	1.19615	0.153152	0.0765758	+	0.997064i	$0.475601\pi$
$62$	5.46410	0.693942
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	11.1244	1.35906	0.679528	−	0.733649i	$-0.262185\pi$
$67$	11.1244	1.35906	0.679528	+	0.733649i	$0.262185\pi$
$68$	2.26795	0.275029
$69$	0	0
$70$	−0.196152	−0.0234447
$71$	1.26795	0.150478	0.0752389	−	0.997166i	$-0.476028\pi$
$71$	1.26795	0.150478	0.0752389	+	0.997166i	$0.476028\pi$
$72$	0	0
$73$	9.73205	1.13905	0.569525	−	0.821974i	$-0.307127\pi$
$73$	9.73205	1.13905	0.569525	+	0.821974i	$0.307127\pi$
$74$	−10.4641	−1.21643
$75$	0	0
$76$	−1.26795	−0.145444
$77$	−3.46410	−0.394771
$78$	0	0
$79$	−9.46410	−1.06479	−0.532397	−	0.846495i	$-0.678709\pi$
$79$	−9.46410	−1.06479	−0.532397	+	0.846495i	$0.678709\pi$
$80$	0.267949	0.0299576
$81$	0	0
$82$	11.3923	1.25807
$83$	10.1962	1.11917	0.559587	−	0.828772i	$-0.310960\pi$
$83$	10.1962	1.11917	0.559587	+	0.828772i	$0.310960\pi$
$84$	0	0
$85$	0.607695	0.0659138
$86$	7.66025	0.826026
$87$	0	0
$88$	4.73205	0.504438
$89$	−2.53590	−0.268805	−0.134402	−	0.990927i	$-0.542911\pi$
$89$	−2.53590	−0.268805	−0.134402	+	0.990927i	$0.542911\pi$
$90$	0	0
$91$	0	0
$92$	6.19615	0.645994
$93$	0	0
$94$	−8.19615	−0.845369
$95$	−0.339746	−0.0348572
$96$	0	0
$97$	6.00000	0.609208	0.304604	−	0.952479i	$-0.401476\pi$
$97$	6.00000	0.609208	0.304604	+	0.952479i	$0.401476\pi$
$98$	−6.46410	−0.652973
$99$	0	0
$100$	−4.92820	−0.492820
$101$	11.9282	1.18690	0.593450	−	0.804871i	$-0.297765\pi$
$101$	11.9282	1.18690	0.593450	+	0.804871i	$0.297765\pi$
$102$	0	0
$103$	−18.7321	−1.84572	−0.922862	−	0.385131i	$-0.874156\pi$
$103$	−18.7321	−1.84572	−0.922862	+	0.385131i	$0.874156\pi$
$104$	0	0
$105$	0	0
$106$	−0.464102	−0.0450775
$107$	0.196152	0.0189628	0.00948139	−	0.999955i	$-0.496982\pi$
$107$	0.196152	0.0189628	0.00948139	+	0.999955i	$0.496982\pi$
$108$	0	0
$109$	5.46410	0.523366	0.261683	−	0.965154i	$-0.415723\pi$
$109$	5.46410	0.523366	0.261683	+	0.965154i	$0.415723\pi$
$110$	1.26795	0.120894
$111$	0	0
$112$	−0.732051	−0.0691723
$113$	18.6603	1.75541	0.877705	−	0.479202i	$-0.159074\pi$
$113$	18.6603	1.75541	0.877705	+	0.479202i	$0.159074\pi$
$114$	0	0
$115$	1.66025	0.154819
$116$	−2.46410	−0.228786
$117$	0	0
$118$	8.00000	0.736460
$119$	−1.66025	−0.152195
$120$	0	0
$121$	11.3923	1.03566
$122$	1.19615	0.108295
$123$	0	0
$124$	5.46410	0.490691
$125$	−2.66025	−0.237940
$126$	0	0
$127$	17.8564	1.58450	0.792250	−	0.610197i	$-0.208910\pi$
$127$	17.8564	1.58450	0.792250	+	0.610197i	$0.208910\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	0	0
$131$	−13.4641	−1.17636	−0.588182	−	0.808729i	$-0.700156\pi$
$131$	−13.4641	−1.17636	−0.588182	+	0.808729i	$0.700156\pi$
$132$	0	0
$133$	0.928203	0.0804854
$134$	11.1244	0.960998
$135$	0	0
$136$	2.26795	0.194475
$137$	−1.92820	−0.164738	−0.0823688	−	0.996602i	$-0.526249\pi$
$137$	−1.92820	−0.164738	−0.0823688	+	0.996602i	$0.526249\pi$
$138$	0	0
$139$	−9.85641	−0.836009	−0.418005	−	0.908445i	$-0.637270\pi$
$139$	−9.85641	−0.836009	−0.418005	+	0.908445i	$0.637270\pi$
$140$	−0.196152	−0.0165779
$141$	0	0
$142$	1.26795	0.106404
$143$	0	0
$144$	0	0
$145$	−0.660254	−0.0548311
$146$	9.73205	0.805430
$147$	0	0
$148$	−10.4641	−0.860144
$149$	2.80385	0.229700	0.114850	−	0.993383i	$-0.463361\pi$
$149$	2.80385	0.229700	0.114850	+	0.993383i	$0.463361\pi$
$150$	0	0
$151$	−3.26795	−0.265942	−0.132971	−	0.991120i	$-0.542452\pi$
$151$	−3.26795	−0.265942	−0.132971	+	0.991120i	$0.542452\pi$
$152$	−1.26795	−0.102844
$153$	0	0
$154$	−3.46410	−0.279145
$155$	1.46410	0.117599
$156$	0	0
$157$	−23.5885	−1.88256	−0.941282	−	0.337622i	$-0.890378\pi$
$157$	−23.5885	−1.88256	−0.941282	+	0.337622i	$0.890378\pi$
$158$	−9.46410	−0.752923
$159$	0	0
$160$	0.267949	0.0211832
$161$	−4.53590	−0.357479
$162$	0	0
$163$	6.53590	0.511931	0.255966	−	0.966686i	$-0.417607\pi$
$163$	6.53590	0.511931	0.255966	+	0.966686i	$0.417607\pi$
$164$	11.3923	0.889590
$165$	0	0
$166$	10.1962	0.791375
$167$	−2.53590	−0.196234	−0.0981169	−	0.995175i	$-0.531282\pi$
$167$	−2.53590	−0.196234	−0.0981169	+	0.995175i	$0.531282\pi$
$168$	0	0
$169$	0	0
$170$	0.607695	0.0466081
$171$	0	0
$172$	7.66025	0.584089
$173$	16.3923	1.24628	0.623142	−	0.782109i	$-0.285856\pi$
$173$	16.3923	1.24628	0.623142	+	0.782109i	$0.285856\pi$
$174$	0	0
$175$	3.60770	0.272716
$176$	4.73205	0.356692
$177$	0	0
$178$	−2.53590	−0.190074
$179$	−22.0526	−1.64829	−0.824143	−	0.566382i	$-0.808343\pi$
$179$	−22.0526	−1.64829	−0.824143	+	0.566382i	$0.808343\pi$
$180$	0	0
$181$	8.80385	0.654385	0.327192	−	0.944958i	$-0.393897\pi$
$181$	8.80385	0.654385	0.327192	+	0.944958i	$0.393897\pi$
$182$	0	0
$183$	0	0
$184$	6.19615	0.456786
$185$	−2.80385	−0.206143
$186$	0	0
$187$	10.7321	0.784805
$188$	−8.19615	−0.597766
$189$	0	0
$190$	−0.339746	−0.0246478
$191$	−6.92820	−0.501307	−0.250654	−	0.968077i	$-0.580646\pi$
$191$	−6.92820	−0.501307	−0.250654	+	0.968077i	$0.580646\pi$
$192$	0	0
$193$	8.26795	0.595140	0.297570	−	0.954700i	$-0.403824\pi$
$193$	8.26795	0.595140	0.297570	+	0.954700i	$0.403824\pi$
$194$	6.00000	0.430775
$195$	0	0
$196$	−6.46410	−0.461722
$197$	9.85641	0.702240	0.351120	−	0.936330i	$-0.385801\pi$
$197$	9.85641	0.702240	0.351120	+	0.936330i	$0.385801\pi$
$198$	0	0
$199$	−3.80385	−0.269648	−0.134824	−	0.990870i	$-0.543047\pi$
$199$	−3.80385	−0.269648	−0.134824	+	0.990870i	$0.543047\pi$
$200$	−4.92820	−0.348477
$201$	0	0
$202$	11.9282	0.839265
$203$	1.80385	0.126605
$204$	0	0
$205$	3.05256	0.213200
$206$	−18.7321	−1.30512
$207$	0	0
$208$	0	0
$209$	−6.00000	−0.415029
$210$	0	0
$211$	−4.39230	−0.302379	−0.151189	−	0.988505i	$-0.548310\pi$
$211$	−4.39230	−0.302379	−0.151189	+	0.988505i	$0.548310\pi$
$212$	−0.464102	−0.0318746
$213$	0	0
$214$	0.196152	0.0134087
$215$	2.05256	0.139983
$216$	0	0
$217$	−4.00000	−0.271538
$218$	5.46410	0.370076
$219$	0	0
$220$	1.26795	0.0854851
$221$	0	0
$222$	0	0
$223$	13.0718	0.875352	0.437676	−	0.899133i	$-0.355802\pi$
$223$	13.0718	0.875352	0.437676	+	0.899133i	$0.355802\pi$
$224$	−0.732051	−0.0489122
$225$	0	0
$226$	18.6603	1.24126
$227$	−1.80385	−0.119726	−0.0598628	−	0.998207i	$-0.519066\pi$
$227$	−1.80385	−0.119726	−0.0598628	+	0.998207i	$0.519066\pi$
$228$	0	0
$229$	−15.8564	−1.04782	−0.523910	−	0.851773i	$-0.675527\pi$
$229$	−15.8564	−1.04782	−0.523910	+	0.851773i	$0.675527\pi$
$230$	1.66025	0.109474
$231$	0	0
$232$	−2.46410	−0.161776
$233$	−19.8564	−1.30084	−0.650418	−	0.759576i	$-0.725406\pi$
$233$	−19.8564	−1.30084	−0.650418	+	0.759576i	$0.725406\pi$
$234$	0	0
$235$	−2.19615	−0.143261
$236$	8.00000	0.520756
$237$	0	0
$238$	−1.66025	−0.107618
$239$	−9.66025	−0.624870	−0.312435	−	0.949939i	$-0.601145\pi$
$239$	−9.66025	−0.624870	−0.312435	+	0.949939i	$0.601145\pi$
$240$	0	0
$241$	17.5885	1.13297	0.566486	−	0.824071i	$-0.308302\pi$
$241$	17.5885	1.13297	0.566486	+	0.824071i	$0.308302\pi$
$242$	11.3923	0.732325
$243$	0	0
$244$	1.19615	0.0765758
$245$	−1.73205	−0.110657
$246$	0	0
$247$	0	0
$248$	5.46410	0.346971
$249$	0	0
$250$	−2.66025	−0.168249
$251$	−6.53590	−0.412542	−0.206271	−	0.978495i	$-0.566133\pi$
$251$	−6.53590	−0.412542	−0.206271	+	0.978495i	$0.566133\pi$
$252$	0	0
$253$	29.3205	1.84336
$254$	17.8564	1.12041
$255$	0	0
$256$	1.00000	0.0625000
$257$	−26.6603	−1.66302	−0.831510	−	0.555509i	$-0.812523\pi$
$257$	−26.6603	−1.66302	−0.831510	+	0.555509i	$0.812523\pi$
$258$	0	0
$259$	7.66025	0.475985
$260$	0	0
$261$	0	0
$262$	−13.4641	−0.831815
$263$	−28.0526	−1.72979	−0.864897	−	0.501949i	$-0.832617\pi$
$263$	−28.0526	−1.72979	−0.864897	+	0.501949i	$0.832617\pi$
$264$	0	0
$265$	−0.124356	−0.00763911
$266$	0.928203	0.0569118
$267$	0	0
$268$	11.1244	0.679528
$269$	1.46410	0.0892679	0.0446339	−	0.999003i	$-0.485788\pi$
$269$	1.46410	0.0892679	0.0446339	+	0.999003i	$0.485788\pi$
$270$	0	0
$271$	−5.85641	−0.355751	−0.177876	−	0.984053i	$-0.556922\pi$
$271$	−5.85641	−0.355751	−0.177876	+	0.984053i	$0.556922\pi$
$272$	2.26795	0.137515
$273$	0	0
$274$	−1.92820	−0.116487
$275$	−23.3205	−1.40628
$276$	0	0
$277$	−2.26795	−0.136268	−0.0681339	−	0.997676i	$-0.521705\pi$
$277$	−2.26795	−0.136268	−0.0681339	+	0.997676i	$0.521705\pi$
$278$	−9.85641	−0.591148
$279$	0	0
$280$	−0.196152	−0.0117223
$281$	−22.3205	−1.33153	−0.665765	−	0.746162i	$-0.731895\pi$
$281$	−22.3205	−1.33153	−0.665765	+	0.746162i	$0.731895\pi$
$282$	0	0
$283$	8.33975	0.495746	0.247873	−	0.968792i	$-0.420268\pi$
$283$	8.33975	0.495746	0.247873	+	0.968792i	$0.420268\pi$
$284$	1.26795	0.0752389
$285$	0	0
$286$	0	0
$287$	−8.33975	−0.492280
$288$	0	0
$289$	−11.8564	−0.697436
$290$	−0.660254	−0.0387715
$291$	0	0
$292$	9.73205	0.569525
$293$	−14.5167	−0.848072	−0.424036	−	0.905645i	$-0.639387\pi$
$293$	−14.5167	−0.848072	−0.424036	+	0.905645i	$0.639387\pi$
$294$	0	0
$295$	2.14359	0.124805
$296$	−10.4641	−0.608214
$297$	0	0
$298$	2.80385	0.162423
$299$	0	0
$300$	0	0
$301$	−5.60770	−0.323222
$302$	−3.26795	−0.188049
$303$	0	0
$304$	−1.26795	−0.0727219
$305$	0.320508	0.0183522
$306$	0	0
$307$	−8.58846	−0.490169	−0.245085	−	0.969502i	$-0.578816\pi$
$307$	−8.58846	−0.490169	−0.245085	+	0.969502i	$0.578816\pi$
$308$	−3.46410	−0.197386
$309$	0	0
$310$	1.46410	0.0831554
$311$	−15.6603	−0.888012	−0.444006	−	0.896024i	$-0.646443\pi$
$311$	−15.6603	−0.888012	−0.444006	+	0.896024i	$0.646443\pi$
$312$	0	0
$313$	13.4641	0.761036	0.380518	−	0.924774i	$-0.375746\pi$
$313$	13.4641	0.761036	0.380518	+	0.924774i	$0.375746\pi$
$314$	−23.5885	−1.33117
$315$	0	0
$316$	−9.46410	−0.532397
$317$	3.33975	0.187579	0.0937894	−	0.995592i	$-0.470102\pi$
$317$	3.33975	0.187579	0.0937894	+	0.995592i	$0.470102\pi$
$318$	0	0
$319$	−11.6603	−0.652849
$320$	0.267949	0.0149788
$321$	0	0
$322$	−4.53590	−0.252776
$323$	−2.87564	−0.160005
$324$	0	0
$325$	0	0
$326$	6.53590	0.361990
$327$	0	0
$328$	11.3923	0.629035
$329$	6.00000	0.330791
$330$	0	0
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	10.1962	0.559587
$333$	0	0
$334$	−2.53590	−0.138758
$335$	2.98076	0.162856
$336$	0	0
$337$	6.85641	0.373492	0.186746	−	0.982408i	$-0.440206\pi$
$337$	6.85641	0.373492	0.186746	+	0.982408i	$0.440206\pi$
$338$	0	0
$339$	0	0
$340$	0.607695	0.0329569
$341$	25.8564	1.40020
$342$	0	0
$343$	9.85641	0.532196
$344$	7.66025	0.413013
$345$	0	0
$346$	16.3923	0.881256
$347$	−8.87564	−0.476470	−0.238235	−	0.971208i	$-0.576569\pi$
$347$	−8.87564	−0.476470	−0.238235	+	0.971208i	$0.576569\pi$
$348$	0	0
$349$	19.3205	1.03420	0.517102	−	0.855924i	$-0.327011\pi$
$349$	19.3205	1.03420	0.517102	+	0.855924i	$0.327011\pi$
$350$	3.60770	0.192839
$351$	0	0
$352$	4.73205	0.252219
$353$	19.7846	1.05303	0.526514	−	0.850166i	$-0.323499\pi$
$353$	19.7846	1.05303	0.526514	+	0.850166i	$0.323499\pi$
$354$	0	0
$355$	0.339746	0.0180318
$356$	−2.53590	−0.134402
$357$	0	0
$358$	−22.0526	−1.16551
$359$	23.1244	1.22046	0.610228	−	0.792226i	$-0.291078\pi$
$359$	23.1244	1.22046	0.610228	+	0.792226i	$0.291078\pi$
$360$	0	0
$361$	−17.3923	−0.915384
$362$	8.80385	0.462720
$363$	0	0
$364$	0	0
$365$	2.60770	0.136493
$366$	0	0
$367$	−14.7321	−0.769007	−0.384503	−	0.923124i	$-0.625627\pi$
$367$	−14.7321	−0.769007	−0.384503	+	0.923124i	$0.625627\pi$
$368$	6.19615	0.322997
$369$	0	0
$370$	−2.80385	−0.145765
$371$	0.339746	0.0176387
$372$	0	0
$373$	−10.2679	−0.531654	−0.265827	−	0.964021i	$-0.585645\pi$
$373$	−10.2679	−0.531654	−0.265827	+	0.964021i	$0.585645\pi$
$374$	10.7321	0.554941
$375$	0	0
$376$	−8.19615	−0.422684
$377$	0	0
$378$	0	0
$379$	−1.46410	−0.0752058	−0.0376029	−	0.999293i	$-0.511972\pi$
$379$	−1.46410	−0.0752058	−0.0376029	+	0.999293i	$0.511972\pi$
$380$	−0.339746	−0.0174286
$381$	0	0
$382$	−6.92820	−0.354478
$383$	5.46410	0.279203	0.139601	−	0.990208i	$-0.455418\pi$
$383$	5.46410	0.279203	0.139601	+	0.990208i	$0.455418\pi$
$384$	0	0
$385$	−0.928203	−0.0473056
$386$	8.26795	0.420828
$387$	0	0
$388$	6.00000	0.304604
$389$	29.7846	1.51014	0.755070	−	0.655644i	$-0.227603\pi$
$389$	29.7846	1.51014	0.755070	+	0.655644i	$0.227603\pi$
$390$	0	0
$391$	14.0526	0.710668
$392$	−6.46410	−0.326486
$393$	0	0
$394$	9.85641	0.496559
$395$	−2.53590	−0.127595
$396$	0	0
$397$	0.392305	0.0196892	0.00984461	−	0.999952i	$-0.496866\pi$
$397$	0.392305	0.0196892	0.00984461	+	0.999952i	$0.496866\pi$
$398$	−3.80385	−0.190670
$399$	0	0
$400$	−4.92820	−0.246410
$401$	21.9282	1.09504	0.547521	−	0.836792i	$-0.315572\pi$
$401$	21.9282	1.09504	0.547521	+	0.836792i	$0.315572\pi$
$402$	0	0
$403$	0	0
$404$	11.9282	0.593450
$405$	0	0
$406$	1.80385	0.0895235
$407$	−49.5167	−2.45445
$408$	0	0
$409$	−14.2679	−0.705505	−0.352752	−	0.935717i	$-0.614754\pi$
$409$	−14.2679	−0.705505	−0.352752	+	0.935717i	$0.614754\pi$
$410$	3.05256	0.150755
$411$	0	0
$412$	−18.7321	−0.922862
$413$	−5.85641	−0.288175
$414$	0	0
$415$	2.73205	0.134111
$416$	0	0
$417$	0	0
$418$	−6.00000	−0.293470
$419$	10.5359	0.514712	0.257356	−	0.966317i	$-0.417149\pi$
$419$	10.5359	0.514712	0.257356	+	0.966317i	$0.417149\pi$
$420$	0	0
$421$	−32.7128	−1.59432	−0.797162	−	0.603765i	$-0.793667\pi$
$421$	−32.7128	−1.59432	−0.797162	+	0.603765i	$0.793667\pi$
$422$	−4.39230	−0.213814
$423$	0	0
$424$	−0.464102	−0.0225388
$425$	−11.1769	−0.542160
$426$	0	0
$427$	−0.875644	−0.0423754
$428$	0.196152	0.00948139
$429$	0	0
$430$	2.05256	0.0989832
$431$	−11.1244	−0.535841	−0.267921	−	0.963441i	$-0.586337\pi$
$431$	−11.1244	−0.535841	−0.267921	+	0.963441i	$0.586337\pi$
$432$	0	0
$433$	−14.8564	−0.713953	−0.356977	−	0.934113i	$-0.616192\pi$
$433$	−14.8564	−0.713953	−0.356977	+	0.934113i	$0.616192\pi$
$434$	−4.00000	−0.192006
$435$	0	0
$436$	5.46410	0.261683
$437$	−7.85641	−0.375823
$438$	0	0
$439$	−17.6603	−0.842878	−0.421439	−	0.906857i	$-0.638475\pi$
$439$	−17.6603	−0.842878	−0.421439	+	0.906857i	$0.638475\pi$
$440$	1.26795	0.0604471
$441$	0	0
$442$	0	0
$443$	−36.3923	−1.72905	−0.864525	−	0.502589i	$-0.832381\pi$
$443$	−36.3923	−1.72905	−0.864525	+	0.502589i	$0.832381\pi$
$444$	0	0
$445$	−0.679492	−0.0322110
$446$	13.0718	0.618968
$447$	0	0
$448$	−0.732051	−0.0345861
$449$	−23.3205	−1.10056	−0.550281	−	0.834979i	$-0.685480\pi$
$449$	−23.3205	−1.10056	−0.550281	+	0.834979i	$0.685480\pi$
$450$	0	0
$451$	53.9090	2.53847
$452$	18.6603	0.877705
$453$	0	0
$454$	−1.80385	−0.0846588
$455$	0	0
$456$	0	0
$457$	−18.6603	−0.872890	−0.436445	−	0.899731i	$-0.643763\pi$
$457$	−18.6603	−0.872890	−0.436445	+	0.899731i	$0.643763\pi$
$458$	−15.8564	−0.740921
$459$	0	0
$460$	1.66025	0.0774097
$461$	25.7321	1.19846	0.599231	−	0.800577i	$-0.295473\pi$
$461$	25.7321	1.19846	0.599231	+	0.800577i	$0.295473\pi$
$462$	0	0
$463$	28.0526	1.30371	0.651856	−	0.758342i	$-0.273990\pi$
$463$	28.0526	1.30371	0.651856	+	0.758342i	$0.273990\pi$
$464$	−2.46410	−0.114393
$465$	0	0
$466$	−19.8564	−0.919830
$467$	12.5885	0.582524	0.291262	−	0.956643i	$-0.405925\pi$
$467$	12.5885	0.582524	0.291262	+	0.956643i	$0.405925\pi$
$468$	0	0
$469$	−8.14359	−0.376036
$470$	−2.19615	−0.101301
$471$	0	0
$472$	8.00000	0.368230
$473$	36.2487	1.66672
$474$	0	0
$475$	6.24871	0.286711
$476$	−1.66025	−0.0760976
$477$	0	0
$478$	−9.66025	−0.441850
$479$	26.5359	1.21246	0.606228	−	0.795291i	$-0.292682\pi$
$479$	26.5359	1.21246	0.606228	+	0.795291i	$0.292682\pi$
$480$	0	0
$481$	0	0
$482$	17.5885	0.801132
$483$	0	0
$484$	11.3923	0.517832
$485$	1.60770	0.0730017
$486$	0	0
$487$	−21.1244	−0.957236	−0.478618	−	0.878023i	$-0.658862\pi$
$487$	−21.1244	−0.957236	−0.478618	+	0.878023i	$0.658862\pi$
$488$	1.19615	0.0541473
$489$	0	0
$490$	−1.73205	−0.0782461
$491$	−5.26795	−0.237739	−0.118870	−	0.992910i	$-0.537927\pi$
$491$	−5.26795	−0.237739	−0.118870	+	0.992910i	$0.537927\pi$
$492$	0	0
$493$	−5.58846	−0.251691
$494$	0	0
$495$	0	0
$496$	5.46410	0.245345
$497$	−0.928203	−0.0416356
$498$	0	0
$499$	−32.0000	−1.43252	−0.716258	−	0.697835i	$-0.754147\pi$
$499$	−32.0000	−1.43252	−0.716258	+	0.697835i	$0.754147\pi$
$500$	−2.66025	−0.118970
$501$	0	0
$502$	−6.53590	−0.291711
$503$	10.9808	0.489608	0.244804	−	0.969573i	$-0.421276\pi$
$503$	10.9808	0.489608	0.244804	+	0.969573i	$0.421276\pi$
$504$	0	0
$505$	3.19615	0.142227
$506$	29.3205	1.30346
$507$	0	0
$508$	17.8564	0.792250
$509$	10.2679	0.455119	0.227559	−	0.973764i	$-0.426925\pi$
$509$	10.2679	0.455119	0.227559	+	0.973764i	$0.426925\pi$
$510$	0	0
$511$	−7.12436	−0.315163
$512$	1.00000	0.0441942
$513$	0	0
$514$	−26.6603	−1.17593
$515$	−5.01924	−0.221174
$516$	0	0
$517$	−38.7846	−1.70575
$518$	7.66025	0.336572
$519$	0	0
$520$	0	0
$521$	17.4449	0.764273	0.382137	−	0.924106i	$-0.375188\pi$
$521$	17.4449	0.764273	0.382137	+	0.924106i	$0.375188\pi$
$522$	0	0
$523$	36.4449	1.59362	0.796811	−	0.604228i	$-0.206518\pi$
$523$	36.4449	1.59362	0.796811	+	0.604228i	$0.206518\pi$
$524$	−13.4641	−0.588182
$525$	0	0
$526$	−28.0526	−1.22315
$527$	12.3923	0.539817
$528$	0	0
$529$	15.3923	0.669231
$530$	−0.124356	−0.00540166
$531$	0	0
$532$	0.928203	0.0402427
$533$	0	0
$534$	0	0
$535$	0.0525589	0.00227232
$536$	11.1244	0.480499
$537$	0	0
$538$	1.46410	0.0631219
$539$	−30.5885	−1.31754
$540$	0	0
$541$	−40.3205	−1.73351	−0.866757	−	0.498731i	$-0.833800\pi$
$541$	−40.3205	−1.73351	−0.866757	+	0.498731i	$0.833800\pi$
$542$	−5.85641	−0.251554
$543$	0	0
$544$	2.26795	0.0972375
$545$	1.46410	0.0627152
$546$	0	0
$547$	6.19615	0.264928	0.132464	−	0.991188i	$-0.457711\pi$
$547$	6.19615	0.264928	0.132464	+	0.991188i	$0.457711\pi$
$548$	−1.92820	−0.0823688
$549$	0	0
$550$	−23.3205	−0.994390
$551$	3.12436	0.133102
$552$	0	0
$553$	6.92820	0.294617
$554$	−2.26795	−0.0963559
$555$	0	0
$556$	−9.85641	−0.418005
$557$	−30.3731	−1.28695	−0.643474	−	0.765468i	$-0.722508\pi$
$557$	−30.3731	−1.28695	−0.643474	+	0.765468i	$0.722508\pi$
$558$	0	0
$559$	0	0
$560$	−0.196152	−0.00828895
$561$	0	0
$562$	−22.3205	−0.941534
$563$	−21.0718	−0.888070	−0.444035	−	0.896009i	$-0.646453\pi$
$563$	−21.0718	−0.888070	−0.444035	+	0.896009i	$0.646453\pi$
$564$	0	0
$565$	5.00000	0.210352
$566$	8.33975	0.350546
$567$	0	0
$568$	1.26795	0.0532020
$569$	38.6410	1.61992	0.809958	−	0.586488i	$-0.199490\pi$
$569$	38.6410	1.61992	0.809958	+	0.586488i	$0.199490\pi$
$570$	0	0
$571$	−24.0526	−1.00657	−0.503284	−	0.864121i	$-0.667875\pi$
$571$	−24.0526	−1.00657	−0.503284	+	0.864121i	$0.667875\pi$
$572$	0	0
$573$	0	0
$574$	−8.33975	−0.348094
$575$	−30.5359	−1.27343
$576$	0	0
$577$	−0.267949	−0.0111549	−0.00557744	−	0.999984i	$-0.501775\pi$
$577$	−0.267949	−0.0111549	−0.00557744	+	0.999984i	$0.501775\pi$
$578$	−11.8564	−0.493161
$579$	0	0
$580$	−0.660254	−0.0274156
$581$	−7.46410	−0.309663
$582$	0	0
$583$	−2.19615	−0.0909553
$584$	9.73205	0.402715
$585$	0	0
$586$	−14.5167	−0.599678
$587$	16.0000	0.660391	0.330195	−	0.943913i	$-0.392885\pi$
$587$	16.0000	0.660391	0.330195	+	0.943913i	$0.392885\pi$
$588$	0	0
$589$	−6.92820	−0.285472
$590$	2.14359	0.0882503
$591$	0	0
$592$	−10.4641	−0.430072
$593$	−36.8564	−1.51351	−0.756756	−	0.653698i	$-0.773217\pi$
$593$	−36.8564	−1.51351	−0.756756	+	0.653698i	$0.773217\pi$
$594$	0	0
$595$	−0.444864	−0.0182376
$596$	2.80385	0.114850
$597$	0	0
$598$	0	0
$599$	9.46410	0.386693	0.193346	−	0.981131i	$-0.438066\pi$
$599$	9.46410	0.386693	0.193346	+	0.981131i	$0.438066\pi$
$600$	0	0
$601$	−5.92820	−0.241816	−0.120908	−	0.992664i	$-0.538581\pi$
$601$	−5.92820	−0.241816	−0.120908	+	0.992664i	$0.538581\pi$
$602$	−5.60770	−0.228553
$603$	0	0
$604$	−3.26795	−0.132971
$605$	3.05256	0.124104
$606$	0	0
$607$	0.784610	0.0318463	0.0159232	−	0.999873i	$-0.494931\pi$
$607$	0.784610	0.0318463	0.0159232	+	0.999873i	$0.494931\pi$
$608$	−1.26795	−0.0514221
$609$	0	0
$610$	0.320508	0.0129770
$611$	0	0
$612$	0	0
$613$	−11.3923	−0.460131	−0.230065	−	0.973175i	$-0.573894\pi$
$613$	−11.3923	−0.460131	−0.230065	+	0.973175i	$0.573894\pi$
$614$	−8.58846	−0.346602
$615$	0	0
$616$	−3.46410	−0.139573
$617$	35.2487	1.41906	0.709530	−	0.704675i	$-0.248907\pi$
$617$	35.2487	1.41906	0.709530	+	0.704675i	$0.248907\pi$
$618$	0	0
$619$	−10.5359	−0.423474	−0.211737	−	0.977327i	$-0.567912\pi$
$619$	−10.5359	−0.423474	−0.211737	+	0.977327i	$0.567912\pi$
$620$	1.46410	0.0587997
$621$	0	0
$622$	−15.6603	−0.627919
$623$	1.85641	0.0743754
$624$	0	0
$625$	23.9282	0.957128
$626$	13.4641	0.538134
$627$	0	0
$628$	−23.5885	−0.941282
$629$	−23.7321	−0.946259
$630$	0	0
$631$	−47.7128	−1.89942	−0.949709	−	0.313135i	$-0.898621\pi$
$631$	−47.7128	−1.89942	−0.949709	+	0.313135i	$0.898621\pi$
$632$	−9.46410	−0.376462
$633$	0	0
$634$	3.33975	0.132638
$635$	4.78461	0.189871
$636$	0	0
$637$	0	0
$638$	−11.6603	−0.461634
$639$	0	0
$640$	0.267949	0.0105916
$641$	−25.9808	−1.02618	−0.513089	−	0.858335i	$-0.671499\pi$
$641$	−25.9808	−1.02618	−0.513089	+	0.858335i	$0.671499\pi$
$642$	0	0
$643$	−13.8564	−0.546443	−0.273222	−	0.961951i	$-0.588089\pi$
$643$	−13.8564	−0.546443	−0.273222	+	0.961951i	$0.588089\pi$
$644$	−4.53590	−0.178739
$645$	0	0
$646$	−2.87564	−0.113141
$647$	−26.2487	−1.03194	−0.515972	−	0.856606i	$-0.672569\pi$
$647$	−26.2487	−1.03194	−0.515972	+	0.856606i	$0.672569\pi$
$648$	0	0
$649$	37.8564	1.48599
$650$	0	0
$651$	0	0
$652$	6.53590	0.255966
$653$	10.5359	0.412302	0.206151	−	0.978520i	$-0.433906\pi$
$653$	10.5359	0.412302	0.206151	+	0.978520i	$0.433906\pi$
$654$	0	0
$655$	−3.60770	−0.140964
$656$	11.3923	0.444795
$657$	0	0
$658$	6.00000	0.233904
$659$	−38.2487	−1.48996	−0.744979	−	0.667088i	$-0.767541\pi$
$659$	−38.2487	−1.48996	−0.744979	+	0.667088i	$0.767541\pi$
$660$	0	0
$661$	9.39230	0.365318	0.182659	−	0.983176i	$-0.441530\pi$
$661$	9.39230	0.365318	0.182659	+	0.983176i	$0.441530\pi$
$662$	−20.0000	−0.777322
$663$	0	0
$664$	10.1962	0.395687
$665$	0.248711	0.00964461
$666$	0	0
$667$	−15.2679	−0.591177
$668$	−2.53590	−0.0981169
$669$	0	0
$670$	2.98076	0.115157
$671$	5.66025	0.218512
$672$	0	0
$673$	14.0718	0.542428	0.271214	−	0.962519i	$-0.412575\pi$
$673$	14.0718	0.542428	0.271214	+	0.962519i	$0.412575\pi$
$674$	6.85641	0.264099
$675$	0	0
$676$	0	0
$677$	38.5359	1.48105	0.740527	−	0.672026i	$-0.234576\pi$
$677$	38.5359	1.48105	0.740527	+	0.672026i	$0.234576\pi$
$678$	0	0
$679$	−4.39230	−0.168561
$680$	0.607695	0.0233040
$681$	0	0
$682$	25.8564	0.990093
$683$	−37.8564	−1.44854	−0.724268	−	0.689519i	$-0.757822\pi$
$683$	−37.8564	−1.44854	−0.724268	+	0.689519i	$0.757822\pi$
$684$	0	0
$685$	−0.516660	−0.0197406
$686$	9.85641	0.376319
$687$	0	0
$688$	7.66025	0.292044
$689$	0	0
$690$	0	0
$691$	−26.3397	−1.00201	−0.501006	−	0.865444i	$-0.667036\pi$
$691$	−26.3397	−1.00201	−0.501006	+	0.865444i	$0.667036\pi$
$692$	16.3923	0.623142
$693$	0	0
$694$	−8.87564	−0.336915
$695$	−2.64102	−0.100179
$696$	0	0
$697$	25.8372	0.978653
$698$	19.3205	0.731292
$699$	0	0
$700$	3.60770	0.136358
$701$	−31.3205	−1.18296	−0.591480	−	0.806320i	$-0.701456\pi$
$701$	−31.3205	−1.18296	−0.591480	+	0.806320i	$0.701456\pi$
$702$	0	0
$703$	13.2679	0.500410
$704$	4.73205	0.178346
$705$	0	0
$706$	19.7846	0.744604
$707$	−8.73205	−0.328403
$708$	0	0
$709$	−40.8564	−1.53439	−0.767197	−	0.641411i	$-0.778349\pi$
$709$	−40.8564	−1.53439	−0.767197	+	0.641411i	$0.778349\pi$
$710$	0.339746	0.0127504
$711$	0	0
$712$	−2.53590	−0.0950368
$713$	33.8564	1.26793
$714$	0	0
$715$	0	0
$716$	−22.0526	−0.824143
$717$	0	0
$718$	23.1244	0.862993
$719$	22.5359	0.840447	0.420224	−	0.907421i	$-0.361952\pi$
$719$	22.5359	0.840447	0.420224	+	0.907421i	$0.361952\pi$
$720$	0	0
$721$	13.7128	0.510692
$722$	−17.3923	−0.647275
$723$	0	0
$724$	8.80385	0.327192
$725$	12.1436	0.451002
$726$	0	0
$727$	20.9808	0.778133	0.389067	−	0.921210i	$-0.372798\pi$
$727$	20.9808	0.778133	0.389067	+	0.921210i	$0.372798\pi$
$728$	0	0
$729$	0	0
$730$	2.60770	0.0965151
$731$	17.3731	0.642566
$732$	0	0
$733$	−19.0000	−0.701781	−0.350891	−	0.936416i	$-0.614121\pi$
$733$	−19.0000	−0.701781	−0.350891	+	0.936416i	$0.614121\pi$
$734$	−14.7321	−0.543770
$735$	0	0
$736$	6.19615	0.228393
$737$	52.6410	1.93906
$738$	0	0
$739$	10.9282	0.402000	0.201000	−	0.979591i	$-0.435581\pi$
$739$	10.9282	0.402000	0.201000	+	0.979591i	$0.435581\pi$
$740$	−2.80385	−0.103071
$741$	0	0
$742$	0.339746	0.0124725
$743$	−27.6077	−1.01283	−0.506414	−	0.862290i	$-0.669029\pi$
$743$	−27.6077	−1.01283	−0.506414	+	0.862290i	$0.669029\pi$
$744$	0	0
$745$	0.751289	0.0275251
$746$	−10.2679	−0.375936
$747$	0	0
$748$	10.7321	0.392403
$749$	−0.143594	−0.00524679
$750$	0	0
$751$	15.9090	0.580526	0.290263	−	0.956947i	$-0.406257\pi$
$751$	15.9090	0.580526	0.290263	+	0.956947i	$0.406257\pi$
$752$	−8.19615	−0.298883
$753$	0	0
$754$	0	0
$755$	−0.875644	−0.0318680
$756$	0	0
$757$	7.07180	0.257029	0.128514	−	0.991708i	$-0.458979\pi$
$757$	7.07180	0.257029	0.128514	+	0.991708i	$0.458979\pi$
$758$	−1.46410	−0.0531786
$759$	0	0
$760$	−0.339746	−0.0123239
$761$	−23.3205	−0.845368	−0.422684	−	0.906277i	$-0.638912\pi$
$761$	−23.3205	−0.845368	−0.422684	+	0.906277i	$0.638912\pi$
$762$	0	0
$763$	−4.00000	−0.144810
$764$	−6.92820	−0.250654
$765$	0	0
$766$	5.46410	0.197426
$767$	0	0
$768$	0	0
$769$	16.1436	0.582153	0.291076	−	0.956700i	$-0.405987\pi$
$769$	16.1436	0.582153	0.291076	+	0.956700i	$0.405987\pi$
$770$	−0.928203	−0.0334501
$771$	0	0
$772$	8.26795	0.297570
$773$	−35.0718	−1.26144	−0.630722	−	0.776009i	$-0.717241\pi$
$773$	−35.0718	−1.26144	−0.630722	+	0.776009i	$0.717241\pi$
$774$	0	0
$775$	−26.9282	−0.967290
$776$	6.00000	0.215387
$777$	0	0
$778$	29.7846	1.06783
$779$	−14.4449	−0.517541
$780$	0	0
$781$	6.00000	0.214697
$782$	14.0526	0.502518
$783$	0	0
$784$	−6.46410	−0.230861
$785$	−6.32051	−0.225589
$786$	0	0
$787$	−39.3205	−1.40162	−0.700812	−	0.713346i	$-0.747179\pi$
$787$	−39.3205	−1.40162	−0.700812	+	0.713346i	$0.747179\pi$
$788$	9.85641	0.351120
$789$	0	0
$790$	−2.53590	−0.0902232
$791$	−13.6603	−0.485703
$792$	0	0
$793$	0	0
$794$	0.392305	0.0139224
$795$	0	0
$796$	−3.80385	−0.134824
$797$	34.0000	1.20434	0.602171	−	0.798367i	$-0.294303\pi$
$797$	34.0000	1.20434	0.602171	+	0.798367i	$0.294303\pi$
$798$	0	0
$799$	−18.5885	−0.657612
$800$	−4.92820	−0.174238
$801$	0	0
$802$	21.9282	0.774312
$803$	46.0526	1.62516
$804$	0	0
$805$	−1.21539	−0.0428369
$806$	0	0
$807$	0	0
$808$	11.9282	0.419633
$809$	22.4115	0.787948	0.393974	−	0.919122i	$-0.371100\pi$
$809$	22.4115	0.787948	0.393974	+	0.919122i	$0.371100\pi$
$810$	0	0
$811$	45.1769	1.58638	0.793188	−	0.608977i	$-0.208420\pi$
$811$	45.1769	1.58638	0.793188	+	0.608977i	$0.208420\pi$
$812$	1.80385	0.0633026
$813$	0	0
$814$	−49.5167	−1.73556
$815$	1.75129	0.0613450
$816$	0	0
$817$	−9.71281	−0.339808
$818$	−14.2679	−0.498867
$819$	0	0
$820$	3.05256	0.106600
$821$	12.9282	0.451197	0.225599	−	0.974220i	$-0.427566\pi$
$821$	12.9282	0.451197	0.225599	+	0.974220i	$0.427566\pi$
$822$	0	0
$823$	−41.5692	−1.44901	−0.724506	−	0.689269i	$-0.757932\pi$
$823$	−41.5692	−1.44901	−0.724506	+	0.689269i	$0.757932\pi$
$824$	−18.7321	−0.652562
$825$	0	0
$826$	−5.85641	−0.203770
$827$	33.4641	1.16366	0.581830	−	0.813310i	$-0.302337\pi$
$827$	33.4641	1.16366	0.581830	+	0.813310i	$0.302337\pi$
$828$	0	0
$829$	12.1244	0.421096	0.210548	−	0.977583i	$-0.432475\pi$
$829$	12.1244	0.421096	0.210548	+	0.977583i	$0.432475\pi$
$830$	2.73205	0.0948309
$831$	0	0
$832$	0	0
$833$	−14.6603	−0.507948
$834$	0	0
$835$	−0.679492	−0.0235148
$836$	−6.00000	−0.207514
$837$	0	0
$838$	10.5359	0.363957
$839$	14.1436	0.488291	0.244146	−	0.969739i	$-0.421493\pi$
$839$	14.1436	0.488291	0.244146	+	0.969739i	$0.421493\pi$
$840$	0	0
$841$	−22.9282	−0.790628
$842$	−32.7128	−1.12736
$843$	0	0
$844$	−4.39230	−0.151189
$845$	0	0
$846$	0	0
$847$	−8.33975	−0.286557
$848$	−0.464102	−0.0159373
$849$	0	0
$850$	−11.1769	−0.383365
$851$	−64.8372	−2.22259
$852$	0	0
$853$	−8.17691	−0.279972	−0.139986	−	0.990153i	$-0.544706\pi$
$853$	−8.17691	−0.279972	−0.139986	+	0.990153i	$0.544706\pi$
$854$	−0.875644	−0.0299639
$855$	0	0
$856$	0.196152	0.00670435
$857$	19.4449	0.664224	0.332112	−	0.943240i	$-0.392239\pi$
$857$	19.4449	0.664224	0.332112	+	0.943240i	$0.392239\pi$
$858$	0	0
$859$	−22.8756	−0.780507	−0.390253	−	0.920707i	$-0.627613\pi$
$859$	−22.8756	−0.780507	−0.390253	+	0.920707i	$0.627613\pi$
$860$	2.05256	0.0699917
$861$	0	0
$862$	−11.1244	−0.378897
$863$	7.12436	0.242516	0.121258	−	0.992621i	$-0.461307\pi$
$863$	7.12436	0.242516	0.121258	+	0.992621i	$0.461307\pi$
$864$	0	0
$865$	4.39230	0.149343
$866$	−14.8564	−0.504841
$867$	0	0
$868$	−4.00000	−0.135769
$869$	−44.7846	−1.51921
$870$	0	0
$871$	0	0
$872$	5.46410	0.185038
$873$	0	0
$874$	−7.85641	−0.265747
$875$	1.94744	0.0658355
$876$	0	0
$877$	−10.0718	−0.340100	−0.170050	−	0.985435i	$-0.554393\pi$
$877$	−10.0718	−0.340100	−0.170050	+	0.985435i	$0.554393\pi$
$878$	−17.6603	−0.596005
$879$	0	0
$880$	1.26795	0.0427426
$881$	−51.8372	−1.74644	−0.873219	−	0.487327i	$-0.837972\pi$
$881$	−51.8372	−1.74644	−0.873219	+	0.487327i	$0.837972\pi$
$882$	0	0
$883$	29.0718	0.978344	0.489172	−	0.872187i	$-0.337299\pi$
$883$	29.0718	0.978344	0.489172	+	0.872187i	$0.337299\pi$
$884$	0	0
$885$	0	0
$886$	−36.3923	−1.22262
$887$	−10.1436	−0.340589	−0.170294	−	0.985393i	$-0.554472\pi$
$887$	−10.1436	−0.340589	−0.170294	+	0.985393i	$0.554472\pi$
$888$	0	0
$889$	−13.0718	−0.438414
$890$	−0.679492	−0.0227766
$891$	0	0
$892$	13.0718	0.437676
$893$	10.3923	0.347765
$894$	0	0
$895$	−5.90897	−0.197515
$896$	−0.732051	−0.0244561
$897$	0	0
$898$	−23.3205	−0.778215
$899$	−13.4641	−0.449053
$900$	0	0
$901$	−1.05256	−0.0350658
$902$	53.9090	1.79497
$903$	0	0
$904$	18.6603	0.620631
$905$	2.35898	0.0784153
$906$	0	0
$907$	15.6077	0.518245	0.259123	−	0.965844i	$-0.416567\pi$
$907$	15.6077	0.518245	0.259123	+	0.965844i	$0.416567\pi$
$908$	−1.80385	−0.0598628
$909$	0	0
$910$	0	0
$911$	9.46410	0.313560	0.156780	−	0.987634i	$-0.449889\pi$
$911$	9.46410	0.313560	0.156780	+	0.987634i	$0.449889\pi$
$912$	0	0
$913$	48.2487	1.59680
$914$	−18.6603	−0.617226
$915$	0	0
$916$	−15.8564	−0.523910
$917$	9.85641	0.325487
$918$	0	0
$919$	−57.9615	−1.91197	−0.955987	−	0.293409i	$-0.905210\pi$
$919$	−57.9615	−1.91197	−0.955987	+	0.293409i	$0.905210\pi$
$920$	1.66025	0.0547370
$921$	0	0
$922$	25.7321	0.847440
$923$	0	0
$924$	0	0
$925$	51.5692	1.69559
$926$	28.0526	0.921864
$927$	0	0
$928$	−2.46410	−0.0808881
$929$	9.24871	0.303440	0.151720	−	0.988423i	$-0.451519\pi$
$929$	9.24871	0.303440	0.151720	+	0.988423i	$0.451519\pi$
$930$	0	0
$931$	8.19615	0.268618
$932$	−19.8564	−0.650418
$933$	0	0
$934$	12.5885	0.411907
$935$	2.87564	0.0940436
$936$	0	0
$937$	43.2487	1.41287	0.706437	−	0.707776i	$-0.250301\pi$
$937$	43.2487	1.41287	0.706437	+	0.707776i	$0.250301\pi$
$938$	−8.14359	−0.265898
$939$	0	0
$940$	−2.19615	−0.0716306
$941$	−56.6410	−1.84644	−0.923222	−	0.384267i	$-0.874454\pi$
$941$	−56.6410	−1.84644	−0.923222	+	0.384267i	$0.874454\pi$
$942$	0	0
$943$	70.5885	2.29868
$944$	8.00000	0.260378
$945$	0	0
$946$	36.2487	1.17855
$947$	34.9282	1.13501	0.567507	−	0.823369i	$-0.307908\pi$
$947$	34.9282	1.13501	0.567507	+	0.823369i	$0.307908\pi$
$948$	0	0
$949$	0	0
$950$	6.24871	0.202735
$951$	0	0
$952$	−1.66025	−0.0538091
$953$	41.5692	1.34656	0.673280	−	0.739388i	$-0.264885\pi$
$953$	41.5692	1.34656	0.673280	+	0.739388i	$0.264885\pi$
$954$	0	0
$955$	−1.85641	−0.0600719
$956$	−9.66025	−0.312435
$957$	0	0
$958$	26.5359	0.857336
$959$	1.41154	0.0455811
$960$	0	0
$961$	−1.14359	−0.0368901
$962$	0	0
$963$	0	0
$964$	17.5885	0.566486
$965$	2.21539	0.0713159
$966$	0	0
$967$	18.8756	0.607000	0.303500	−	0.952831i	$-0.401845\pi$
$967$	18.8756	0.607000	0.303500	+	0.952831i	$0.401845\pi$
$968$	11.3923	0.366163
$969$	0	0
$970$	1.60770	0.0516200
$971$	18.2487	0.585629	0.292815	−	0.956169i	$-0.405408\pi$
$971$	18.2487	0.585629	0.292815	+	0.956169i	$0.405408\pi$
$972$	0	0
$973$	7.21539	0.231315
$974$	−21.1244	−0.676868
$975$	0	0
$976$	1.19615	0.0382879
$977$	32.0718	1.02607	0.513034	−	0.858368i	$-0.328522\pi$
$977$	32.0718	1.02607	0.513034	+	0.858368i	$0.328522\pi$
$978$	0	0
$979$	−12.0000	−0.383522
$980$	−1.73205	−0.0553283
$981$	0	0
$982$	−5.26795	−0.168107
$983$	−20.7846	−0.662926	−0.331463	−	0.943468i	$-0.607542\pi$
$983$	−20.7846	−0.662926	−0.331463	+	0.943468i	$0.607542\pi$
$984$	0	0
$985$	2.64102	0.0841498
$986$	−5.58846	−0.177973
$987$	0	0
$988$	0	0
$989$	47.4641	1.50927
$990$	0	0
$991$	−8.58846	−0.272821	−0.136411	−	0.990652i	$-0.543557\pi$
$991$	−8.58846	−0.272821	−0.136411	+	0.990652i	$0.543557\pi$
$992$	5.46410	0.173485
$993$	0	0
$994$	−0.928203	−0.0294408
$995$	−1.01924	−0.0323120
$996$	0	0
$997$	38.6603	1.22438	0.612191	−	0.790710i	$-0.290288\pi$
$997$	38.6603	1.22438	0.612191	+	0.790710i	$0.290288\pi$
$998$	−32.0000	−1.01294
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3042.2.a.y.1.1		2
3.2	odd	2		1014.2.a.i.1.2		2
12.11	even	2		8112.2.a.bj.1.2		2
13.5	odd	4		3042.2.b.i.1351.2		4
13.6	odd	12		234.2.l.c.127.1		4
13.8	odd	4		3042.2.b.i.1351.3		4
13.11	odd	12		234.2.l.c.199.1		4
13.12	even	2		3042.2.a.p.1.2		2
39.2	even	12		1014.2.i.a.823.1		4
39.5	even	4		1014.2.b.e.337.3		4
39.8	even	4		1014.2.b.e.337.2		4
39.11	even	12		78.2.i.a.43.2	✓	4
39.17	odd	6		1014.2.e.g.991.1		4
39.20	even	12		1014.2.i.a.361.1		4
39.23	odd	6		1014.2.e.g.529.1		4
39.29	odd	6		1014.2.e.i.529.2		4
39.32	even	12		78.2.i.a.49.2	yes	4
39.35	odd	6		1014.2.e.i.991.2		4
39.38	odd	2		1014.2.a.k.1.1		2
52.11	even	12		1872.2.by.h.433.2		4
52.19	even	12		1872.2.by.h.1297.1		4
156.11	odd	12		624.2.bv.e.433.1		4
156.71	odd	12		624.2.bv.e.49.2		4
156.155	even	2		8112.2.a.bp.1.1		2
195.32	odd	12		1950.2.y.g.49.1		4
195.89	even	12		1950.2.bc.d.901.1		4
195.128	odd	12		1950.2.y.g.199.1		4
195.149	even	12		1950.2.bc.d.751.1		4
195.167	odd	12		1950.2.y.b.199.2		4
195.188	odd	12		1950.2.y.b.49.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
78.2.i.a.43.2	✓	4	39.11	even	12
78.2.i.a.49.2	yes	4	39.32	even	12
234.2.l.c.127.1		4	13.6	odd	12
234.2.l.c.199.1		4	13.11	odd	12
624.2.bv.e.49.2		4	156.71	odd	12
624.2.bv.e.433.1		4	156.11	odd	12
1014.2.a.i.1.2		2	3.2	odd	2
1014.2.a.k.1.1		2	39.38	odd	2
1014.2.b.e.337.2		4	39.8	even	4
1014.2.b.e.337.3		4	39.5	even	4
1014.2.e.g.529.1		4	39.23	odd	6
1014.2.e.g.991.1		4	39.17	odd	6
1014.2.e.i.529.2		4	39.29	odd	6
1014.2.e.i.991.2		4	39.35	odd	6
1014.2.i.a.361.1		4	39.20	even	12
1014.2.i.a.823.1		4	39.2	even	12
1872.2.by.h.433.2		4	52.11	even	12
1872.2.by.h.1297.1		4	52.19	even	12
1950.2.y.b.49.2		4	195.188	odd	12
1950.2.y.b.199.2		4	195.167	odd	12
1950.2.y.g.49.1		4	195.32	odd	12
1950.2.y.g.199.1		4	195.128	odd	12
1950.2.bc.d.751.1		4	195.149	even	12
1950.2.bc.d.901.1		4	195.89	even	12
3042.2.a.p.1.2		2	13.12	even	2
3042.2.a.y.1.1		2	1.1	even	1	trivial
3042.2.b.i.1351.2		4	13.5	odd	4
3042.2.b.i.1351.3		4	13.8	odd	4
8112.2.a.bj.1.2		2	12.11	even	2
8112.2.a.bp.1.1		2	156.155	even	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:	\( N \)	\(=\)	\( 3042 = 2 \cdot 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3042.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +0.267949 q^{5} -0.732051 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +0.267949 q^{5} -0.732051 q^{7} +1.00000 q^{8} +0.267949 q^{10} +4.73205 q^{11} -0.732051 q^{14} +1.00000 q^{16} +2.26795 q^{17} -1.26795 q^{19} +0.267949 q^{20} +4.73205 q^{22} +6.19615 q^{23} -4.92820 q^{25} -0.732051 q^{28} -2.46410 q^{29} +5.46410 q^{31} +1.00000 q^{32} +2.26795 q^{34} -0.196152 q^{35} -10.4641 q^{37} -1.26795 q^{38} +0.267949 q^{40} +11.3923 q^{41} +7.66025 q^{43} +4.73205 q^{44} +6.19615 q^{46} -8.19615 q^{47} -6.46410 q^{49} -4.92820 q^{50} -0.464102 q^{53} +1.26795 q^{55} -0.732051 q^{56} -2.46410 q^{58} +8.00000 q^{59} +1.19615 q^{61} +5.46410 q^{62} +1.00000 q^{64} +11.1244 q^{67} +2.26795 q^{68} -0.196152 q^{70} +1.26795 q^{71} +9.73205 q^{73} -10.4641 q^{74} -1.26795 q^{76} -3.46410 q^{77} -9.46410 q^{79} +0.267949 q^{80} +11.3923 q^{82} +10.1962 q^{83} +0.607695 q^{85} +7.66025 q^{86} +4.73205 q^{88} -2.53590 q^{89} +6.19615 q^{92} -8.19615 q^{94} -0.339746 q^{95} +6.00000 q^{97} -6.46410 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 4 q^{5} + 2 q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 + 4 * q^5 + 2 * q^7 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} + 4 q^{5} + 2 q^{7} + 2 q^{8} + 4 q^{10} + 6 q^{11} + 2 q^{14} + 2 q^{16} + 8 q^{17} - 6 q^{19} + 4 q^{20} + 6 q^{22} + 2 q^{23} + 4 q^{25} + 2 q^{28} + 2 q^{29} + 4 q^{31} + 2 q^{32} + 8 q^{34} + 10 q^{35} - 14 q^{37} - 6 q^{38} + 4 q^{40} + 2 q^{41} - 2 q^{43} + 6 q^{44} + 2 q^{46} - 6 q^{47} - 6 q^{49} + 4 q^{50} + 6 q^{53} + 6 q^{55} + 2 q^{56} + 2 q^{58} + 16 q^{59} - 8 q^{61} + 4 q^{62} + 2 q^{64} - 2 q^{67} + 8 q^{68} + 10 q^{70} + 6 q^{71} + 16 q^{73} - 14 q^{74} - 6 q^{76} - 12 q^{79} + 4 q^{80} + 2 q^{82} + 10 q^{83} + 22 q^{85} - 2 q^{86} + 6 q^{88} - 12 q^{89} + 2 q^{92} - 6 q^{94} - 18 q^{95} + 12 q^{97} - 6 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 4 * q^5 + 2 * q^7 + 2 * q^8 + 4 * q^10 + 6 * q^11 + 2 * q^14 + 2 * q^16 + 8 * q^17 - 6 * q^19 + 4 * q^20 + 6 * q^22 + 2 * q^23 + 4 * q^25 + 2 * q^28 + 2 * q^29 + 4 * q^31 + 2 * q^32 + 8 * q^34 + 10 * q^35 - 14 * q^37 - 6 * q^38 + 4 * q^40 + 2 * q^41 - 2 * q^43 + 6 * q^44 + 2 * q^46 - 6 * q^47 - 6 * q^49 + 4 * q^50 + 6 * q^53 + 6 * q^55 + 2 * q^56 + 2 * q^58 + 16 * q^59 - 8 * q^61 + 4 * q^62 + 2 * q^64 - 2 * q^67 + 8 * q^68 + 10 * q^70 + 6 * q^71 + 16 * q^73 - 14 * q^74 - 6 * q^76 - 12 * q^79 + 4 * q^80 + 2 * q^82 + 10 * q^83 + 22 * q^85 - 2 * q^86 + 6 * q^88 - 12 * q^89 + 2 * q^92 - 6 * q^94 - 18 * q^95 + 12 * q^97 - 6 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more