LMFDB - Embedded newform 1336.2.a.e.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1336,2,Mod(1,1336)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1336, 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("1336.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1336 = 2^{3} \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1336.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:	$10.6680137100$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 5 x^{11} - 15 x^{10} + 100 x^{9} + 36 x^{8} - 641 x^{7} + 129 x^{6} + 1804 x^{5} - 433 x^{4} + \cdots + 224 $ x^12 - 5x^11 - 15x^10 + 100x^9 + 36x^8 - 641x^7 + 129x^6 + 1804x^5 - 433x^4 - 2399x^3 + 148x^2 + 1224*x + 224
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$1.18487$ of defining polynomial
Character	$\chi$	$=$	1336.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.18487 q^{3} -0.696822 q^{5} +2.51795 q^{7} -1.59609 q^{9} +O(q^{10})$	$q+1.18487 q^{3} -0.696822 q^{5} +2.51795 q^{7} -1.59609 q^{9} -0.459192 q^{11} +3.40070 q^{13} -0.825643 q^{15} +5.32018 q^{17} +0.179773 q^{19} +2.98344 q^{21} +1.56958 q^{23} -4.51444 q^{25} -5.44576 q^{27} +5.66009 q^{29} -1.09227 q^{31} -0.544083 q^{33} -1.75457 q^{35} +7.02496 q^{37} +4.02939 q^{39} +6.41186 q^{41} +4.43815 q^{43} +1.11219 q^{45} +11.5404 q^{47} -0.659908 q^{49} +6.30371 q^{51} -12.6446 q^{53} +0.319975 q^{55} +0.213008 q^{57} +13.9426 q^{59} -9.38289 q^{61} -4.01887 q^{63} -2.36968 q^{65} -8.91858 q^{67} +1.85975 q^{69} -5.22285 q^{71} +9.43107 q^{73} -5.34902 q^{75} -1.15623 q^{77} -8.98229 q^{79} -1.66425 q^{81} +15.2480 q^{83} -3.70722 q^{85} +6.70646 q^{87} +7.32969 q^{89} +8.56281 q^{91} -1.29420 q^{93} -0.125270 q^{95} -10.8571 q^{97} +0.732911 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 5 q^{3} - 2 q^{5} + 10 q^{7} + 19 q^{9}+O(q^{10}) $ 12 * q + 5 * q^3 - 2 * q^5 + 10 * q^7 + 19 * q^9	$ 12 q + 5 q^{3} - 2 q^{5} + 10 q^{7} + 19 q^{9} + 14 q^{11} - 9 q^{13} + 8 q^{15} + 4 q^{17} + 9 q^{19} + 7 q^{21} + 15 q^{23} + 18 q^{25} + 20 q^{27} + 17 q^{29} + 11 q^{31} + 8 q^{33} + 19 q^{35} - 29 q^{37} + 26 q^{39} + 14 q^{41} + 15 q^{43} - 26 q^{45} + 17 q^{47} + 20 q^{49} + 36 q^{51} - 7 q^{53} + 13 q^{55} + 3 q^{57} + 32 q^{59} - 8 q^{61} + 50 q^{63} + 37 q^{65} + 39 q^{67} + q^{69} + 35 q^{71} - 12 q^{73} + 29 q^{75} + 13 q^{77} + 36 q^{79} + 44 q^{81} + 25 q^{83} - 38 q^{85} + 14 q^{87} + 23 q^{89} - 2 q^{91} - 25 q^{93} + 38 q^{95} - 2 q^{97} + 43 q^{99}+O(q^{100}) $ 12 * q + 5 * q^3 - 2 * q^5 + 10 * q^7 + 19 * q^9 + 14 * q^11 - 9 * q^13 + 8 * q^15 + 4 * q^17 + 9 * q^19 + 7 * q^21 + 15 * q^23 + 18 * q^25 + 20 * q^27 + 17 * q^29 + 11 * q^31 + 8 * q^33 + 19 * q^35 - 29 * q^37 + 26 * q^39 + 14 * q^41 + 15 * q^43 - 26 * q^45 + 17 * q^47 + 20 * q^49 + 36 * q^51 - 7 * q^53 + 13 * q^55 + 3 * q^57 + 32 * q^59 - 8 * q^61 + 50 * q^63 + 37 * q^65 + 39 * q^67 + q^69 + 35 * q^71 - 12 * q^73 + 29 * q^75 + 13 * q^77 + 36 * q^79 + 44 * q^81 + 25 * q^83 - 38 * q^85 + 14 * q^87 + 23 * q^89 - 2 * q^91 - 25 * q^93 + 38 * q^95 - 2 * q^97 + 43 * 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
$3$	1.18487	0.684084	0.342042	−	0.939685i	$-0.388881\pi$
$3$	1.18487	0.684084	0.342042	+	0.939685i	$0.388881\pi$
$4$	0	0
$5$	−0.696822	−0.311628	−0.155814	−	0.987786i	$-0.549800\pi$
$5$	−0.696822	−0.311628	−0.155814	+	0.987786i	$0.549800\pi$
$6$	0	0
$7$	2.51795	0.951697	0.475849	−	0.879527i	$-0.342141\pi$
$7$	2.51795	0.951697	0.475849	+	0.879527i	$0.342141\pi$
$8$	0	0
$9$	−1.59609	−0.532029
$10$	0	0
$11$	−0.459192	−0.138452	−0.0692259	−	0.997601i	$-0.522053\pi$
$11$	−0.459192	−0.138452	−0.0692259	+	0.997601i	$0.522053\pi$
$12$	0	0
$13$	3.40070	0.943185	0.471593	−	0.881817i	$-0.343679\pi$
$13$	3.40070	0.943185	0.471593	+	0.881817i	$0.343679\pi$
$14$	0	0
$15$	−0.825643	−0.213180
$16$	0	0
$17$	5.32018	1.29033	0.645166	−	0.764042i	$-0.276788\pi$
$17$	5.32018	1.29033	0.645166	+	0.764042i	$0.276788\pi$
$18$	0	0
$19$	0.179773	0.0412428	0.0206214	−	0.999787i	$-0.493436\pi$
$19$	0.179773	0.0412428	0.0206214	+	0.999787i	$0.493436\pi$
$20$	0	0
$21$	2.98344	0.651041
$22$	0	0
$23$	1.56958	0.327280	0.163640	−	0.986520i	$-0.447676\pi$
$23$	1.56958	0.327280	0.163640	+	0.986520i	$0.447676\pi$
$24$	0	0
$25$	−4.51444	−0.902888
$26$	0	0
$27$	−5.44576	−1.04804
$28$	0	0
$29$	5.66009	1.05105	0.525526	−	0.850777i	$-0.323868\pi$
$29$	5.66009	1.05105	0.525526	+	0.850777i	$0.323868\pi$
$30$	0	0
$31$	−1.09227	−0.196178	−0.0980890	−	0.995178i	$-0.531273\pi$
$31$	−1.09227	−0.196178	−0.0980890	+	0.995178i	$0.531273\pi$
$32$	0	0
$33$	−0.544083	−0.0947126
$34$	0	0
$35$	−1.75457	−0.296576
$36$	0	0
$37$	7.02496	1.15490	0.577448	−	0.816427i	$-0.304049\pi$
$37$	7.02496	1.15490	0.577448	+	0.816427i	$0.304049\pi$
$38$	0	0
$39$	4.02939	0.645218
$40$	0	0
$41$	6.41186	1.00136	0.500682	−	0.865631i	$-0.333083\pi$
$41$	6.41186	1.00136	0.500682	+	0.865631i	$0.333083\pi$
$42$	0	0
$43$	4.43815	0.676811	0.338406	−	0.941000i	$-0.390112\pi$
$43$	4.43815	0.676811	0.338406	+	0.941000i	$0.390112\pi$
$44$	0	0
$45$	1.11219	0.165795
$46$	0	0
$47$	11.5404	1.68333	0.841667	−	0.539996i	$-0.181574\pi$
$47$	11.5404	1.68333	0.841667	+	0.539996i	$0.181574\pi$
$48$	0	0
$49$	−0.659908	−0.0942726
$50$	0	0
$51$	6.30371	0.882696
$52$	0	0
$53$	−12.6446	−1.73687	−0.868437	−	0.495800i	$-0.834875\pi$
$53$	−12.6446	−1.73687	−0.868437	+	0.495800i	$0.834875\pi$
$54$	0	0
$55$	0.319975	0.0431455
$56$	0	0
$57$	0.213008	0.0282136
$58$	0	0
$59$	13.9426	1.81518	0.907588	−	0.419863i	$-0.137922\pi$
$59$	13.9426	1.81518	0.907588	+	0.419863i	$0.137922\pi$
$60$	0	0
$61$	−9.38289	−1.20136	−0.600678	−	0.799491i	$-0.705103\pi$
$61$	−9.38289	−1.20136	−0.600678	+	0.799491i	$0.705103\pi$
$62$	0	0
$63$	−4.01887	−0.506330
$64$	0	0
$65$	−2.36968	−0.293923
$66$	0	0
$67$	−8.91858	−1.08958	−0.544789	−	0.838573i	$-0.683390\pi$
$67$	−8.91858	−1.08958	−0.544789	+	0.838573i	$0.683390\pi$
$68$	0	0
$69$	1.85975	0.223887
$70$	0	0
$71$	−5.22285	−0.619838	−0.309919	−	0.950763i	$-0.600302\pi$
$71$	−5.22285	−0.619838	−0.309919	+	0.950763i	$0.600302\pi$
$72$	0	0
$73$	9.43107	1.10382	0.551912	−	0.833902i	$-0.313898\pi$
$73$	9.43107	1.10382	0.551912	+	0.833902i	$0.313898\pi$
$74$	0	0
$75$	−5.34902	−0.617651
$76$	0	0
$77$	−1.15623	−0.131764
$78$	0	0
$79$	−8.98229	−1.01059	−0.505293	−	0.862948i	$-0.668616\pi$
$79$	−8.98229	−1.01059	−0.505293	+	0.862948i	$0.668616\pi$
$80$	0	0
$81$	−1.66425	−0.184917
$82$	0	0
$83$	15.2480	1.67368	0.836840	−	0.547447i	$-0.184400\pi$
$83$	15.2480	1.67368	0.836840	+	0.547447i	$0.184400\pi$
$84$	0	0
$85$	−3.70722	−0.402104
$86$	0	0
$87$	6.70646	0.719008
$88$	0	0
$89$	7.32969	0.776946	0.388473	−	0.921460i	$-0.373003\pi$
$89$	7.32969	0.776946	0.388473	+	0.921460i	$0.373003\pi$
$90$	0	0
$91$	8.56281	0.897627
$92$	0	0
$93$	−1.29420	−0.134202
$94$	0	0
$95$	−0.125270	−0.0128524
$96$	0	0
$97$	−10.8571	−1.10238	−0.551188	−	0.834381i	$-0.685825\pi$
$97$	−10.8571	−1.10238	−0.551188	+	0.834381i	$0.685825\pi$
$98$	0	0
$99$	0.732911	0.0736603
$100$	0	0
$101$	−17.9552	−1.78661	−0.893303	−	0.449455i	$-0.851618\pi$
$101$	−17.9552	−1.78661	−0.893303	+	0.449455i	$0.851618\pi$
$102$	0	0
$103$	17.1537	1.69021	0.845103	−	0.534604i	$-0.179539\pi$
$103$	17.1537	1.69021	0.845103	+	0.534604i	$0.179539\pi$
$104$	0	0
$105$	−2.07893	−0.202883
$106$	0	0
$107$	17.4959	1.69139	0.845696	−	0.533665i	$-0.179186\pi$
$107$	17.4959	1.69139	0.845696	+	0.533665i	$0.179186\pi$
$108$	0	0
$109$	−11.3698	−1.08903	−0.544516	−	0.838750i	$-0.683287\pi$
$109$	−11.3698	−1.08903	−0.544516	+	0.838750i	$0.683287\pi$
$110$	0	0
$111$	8.32366	0.790046
$112$	0	0
$113$	9.89342	0.930695	0.465347	−	0.885128i	$-0.345929\pi$
$113$	9.89342	0.930695	0.465347	+	0.885128i	$0.345929\pi$
$114$	0	0
$115$	−1.09372	−0.101990
$116$	0	0
$117$	−5.42782	−0.501802
$118$	0	0
$119$	13.3960	1.22801
$120$	0	0
$121$	−10.7891	−0.980831
$122$	0	0
$123$	7.59721	0.685017
$124$	0	0
$125$	6.62987	0.592994
$126$	0	0
$127$	−18.9379	−1.68047	−0.840234	−	0.542224i	$-0.817582\pi$
$127$	−18.9379	−1.68047	−0.840234	+	0.542224i	$0.817582\pi$
$128$	0	0
$129$	5.25862	0.462996
$130$	0	0
$131$	−18.9082	−1.65202	−0.826010	−	0.563656i	$-0.809394\pi$
$131$	−18.9082	−1.65202	−0.826010	+	0.563656i	$0.809394\pi$
$132$	0	0
$133$	0.452661	0.0392507
$134$	0	0
$135$	3.79472	0.326598
$136$	0	0
$137$	6.37158	0.544361	0.272180	−	0.962246i	$-0.412255\pi$
$137$	6.37158	0.544361	0.272180	+	0.962246i	$0.412255\pi$
$138$	0	0
$139$	−12.3511	−1.04761	−0.523803	−	0.851840i	$-0.675487\pi$
$139$	−12.3511	−1.04761	−0.523803	+	0.851840i	$0.675487\pi$
$140$	0	0
$141$	13.6738	1.15154
$142$	0	0
$143$	−1.56158	−0.130586
$144$	0	0
$145$	−3.94408	−0.327538
$146$	0	0
$147$	−0.781905	−0.0644904
$148$	0	0
$149$	−17.6969	−1.44978	−0.724892	−	0.688862i	$-0.758110\pi$
$149$	−17.6969	−1.44978	−0.724892	+	0.688862i	$0.758110\pi$
$150$	0	0
$151$	11.5226	0.937697	0.468848	−	0.883279i	$-0.344669\pi$
$151$	11.5226	0.937697	0.468848	+	0.883279i	$0.344669\pi$
$152$	0	0
$153$	−8.49146	−0.686494
$154$	0	0
$155$	0.761120	0.0611346
$156$	0	0
$157$	1.24650	0.0994816	0.0497408	−	0.998762i	$-0.484160\pi$
$157$	1.24650	0.0994816	0.0497408	+	0.998762i	$0.484160\pi$
$158$	0	0
$159$	−14.9822	−1.18817
$160$	0	0
$161$	3.95213	0.311472
$162$	0	0
$163$	16.5214	1.29405	0.647026	−	0.762468i	$-0.276012\pi$
$163$	16.5214	1.29405	0.647026	+	0.762468i	$0.276012\pi$
$164$	0	0
$165$	0.379129	0.0295151
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	−1.43522	−0.110402
$170$	0	0
$171$	−0.286934	−0.0219424
$172$	0	0
$173$	10.2446	0.778881	0.389441	−	0.921052i	$-0.372668\pi$
$173$	10.2446	0.778881	0.389441	+	0.921052i	$0.372668\pi$
$174$	0	0
$175$	−11.3671	−0.859276
$176$	0	0
$177$	16.5202	1.24173
$178$	0	0
$179$	−19.2931	−1.44203	−0.721017	−	0.692917i	$-0.756325\pi$
$179$	−19.2931	−1.44203	−0.721017	+	0.692917i	$0.756325\pi$
$180$	0	0
$181$	−11.8099	−0.877821	−0.438911	−	0.898531i	$-0.644635\pi$
$181$	−11.8099	−0.877821	−0.438911	+	0.898531i	$0.644635\pi$
$182$	0	0
$183$	−11.1175	−0.821828
$184$	0	0
$185$	−4.89515	−0.359898
$186$	0	0
$187$	−2.44299	−0.178649
$188$	0	0
$189$	−13.7122	−0.997413
$190$	0	0
$191$	7.46192	0.539926	0.269963	−	0.962871i	$-0.412989\pi$
$191$	7.46192	0.539926	0.269963	+	0.962871i	$0.412989\pi$
$192$	0	0
$193$	−9.25752	−0.666371	−0.333185	−	0.942861i	$-0.608124\pi$
$193$	−9.25752	−0.666371	−0.333185	+	0.942861i	$0.608124\pi$
$194$	0	0
$195$	−2.80777	−0.201068
$196$	0	0
$197$	2.42312	0.172640	0.0863201	−	0.996267i	$-0.472489\pi$
$197$	2.42312	0.172640	0.0863201	+	0.996267i	$0.472489\pi$
$198$	0	0
$199$	9.10574	0.645489	0.322744	−	0.946486i	$-0.395395\pi$
$199$	9.10574	0.645489	0.322744	+	0.946486i	$0.395395\pi$
$200$	0	0
$201$	−10.5673	−0.745363
$202$	0	0
$203$	14.2518	1.00028
$204$	0	0
$205$	−4.46792	−0.312053
$206$	0	0
$207$	−2.50519	−0.174123
$208$	0	0
$209$	−0.0825506	−0.00571014
$210$	0	0
$211$	8.55686	0.589079	0.294539	−	0.955639i	$-0.404834\pi$
$211$	8.55686	0.589079	0.294539	+	0.955639i	$0.404834\pi$
$212$	0	0
$213$	−6.18839	−0.424022
$214$	0	0
$215$	−3.09260	−0.210914
$216$	0	0
$217$	−2.75029	−0.186702
$218$	0	0
$219$	11.1746	0.755108
$220$	0	0
$221$	18.0923	1.21702
$222$	0	0
$223$	0.865704	0.0579718	0.0289859	−	0.999580i	$-0.490772\pi$
$223$	0.865704	0.0579718	0.0289859	+	0.999580i	$0.490772\pi$
$224$	0	0
$225$	7.20543	0.480362
$226$	0	0
$227$	−6.28540	−0.417177	−0.208588	−	0.978004i	$-0.566887\pi$
$227$	−6.28540	−0.417177	−0.208588	+	0.978004i	$0.566887\pi$
$228$	0	0
$229$	−4.31659	−0.285248	−0.142624	−	0.989777i	$-0.545554\pi$
$229$	−4.31659	−0.285248	−0.142624	+	0.989777i	$0.545554\pi$
$230$	0	0
$231$	−1.36998	−0.0901377
$232$	0	0
$233$	5.23284	0.342815	0.171408	−	0.985200i	$-0.445169\pi$
$233$	5.23284	0.342815	0.171408	+	0.985200i	$0.445169\pi$
$234$	0	0
$235$	−8.04158	−0.524575
$236$	0	0
$237$	−10.6428	−0.691326
$238$	0	0
$239$	−20.4748	−1.32440	−0.662201	−	0.749326i	$-0.730378\pi$
$239$	−20.4748	−1.32440	−0.662201	+	0.749326i	$0.730378\pi$
$240$	0	0
$241$	−21.7321	−1.39989	−0.699943	−	0.714198i	$-0.746791\pi$
$241$	−21.7321	−1.39989	−0.699943	+	0.714198i	$0.746791\pi$
$242$	0	0
$243$	14.3654	0.921538
$244$	0	0
$245$	0.459839	0.0293780
$246$	0	0
$247$	0.611356	0.0388996
$248$	0	0
$249$	18.0668	1.14494
$250$	0	0
$251$	−20.6191	−1.30146	−0.650732	−	0.759307i	$-0.725538\pi$
$251$	−20.6191	−1.30146	−0.650732	+	0.759307i	$0.725538\pi$
$252$	0	0
$253$	−0.720740	−0.0453125
$254$	0	0
$255$	−4.39257	−0.275073
$256$	0	0
$257$	9.86253	0.615208	0.307604	−	0.951514i	$-0.400473\pi$
$257$	9.86253	0.615208	0.307604	+	0.951514i	$0.400473\pi$
$258$	0	0
$259$	17.6885	1.09911
$260$	0	0
$261$	−9.03400	−0.559190
$262$	0	0
$263$	−4.22550	−0.260556	−0.130278	−	0.991478i	$-0.541587\pi$
$263$	−4.22550	−0.260556	−0.130278	+	0.991478i	$0.541587\pi$
$264$	0	0
$265$	8.81106	0.541259
$266$	0	0
$267$	8.68472	0.531496
$268$	0	0
$269$	−17.9365	−1.09361	−0.546805	−	0.837260i	$-0.684156\pi$
$269$	−17.9365	−1.09361	−0.546805	+	0.837260i	$0.684156\pi$
$270$	0	0
$271$	18.4862	1.12296	0.561479	−	0.827491i	$-0.310233\pi$
$271$	18.4862	1.12296	0.561479	+	0.827491i	$0.310233\pi$
$272$	0	0
$273$	10.1458	0.614052
$274$	0	0
$275$	2.07300	0.125006
$276$	0	0
$277$	−25.3811	−1.52500	−0.762502	−	0.646986i	$-0.776029\pi$
$277$	−25.3811	−1.52500	−0.762502	+	0.646986i	$0.776029\pi$
$278$	0	0
$279$	1.74336	0.104372
$280$	0	0
$281$	−28.4398	−1.69657	−0.848287	−	0.529537i	$-0.822366\pi$
$281$	−28.4398	−1.69657	−0.848287	+	0.529537i	$0.822366\pi$
$282$	0	0
$283$	8.66264	0.514940	0.257470	−	0.966286i	$-0.417111\pi$
$283$	8.66264	0.514940	0.257470	+	0.966286i	$0.417111\pi$
$284$	0	0
$285$	−0.148429	−0.00879215
$286$	0	0
$287$	16.1448	0.952995
$288$	0	0
$289$	11.3043	0.664958
$290$	0	0
$291$	−12.8643	−0.754118
$292$	0	0
$293$	−24.9423	−1.45715	−0.728573	−	0.684968i	$-0.759816\pi$
$293$	−24.9423	−1.45715	−0.728573	+	0.684968i	$0.759816\pi$
$294$	0	0
$295$	−9.71553	−0.565660
$296$	0	0
$297$	2.50065	0.145102
$298$	0	0
$299$	5.33768	0.308686
$300$	0	0
$301$	11.1751	0.644119
$302$	0	0
$303$	−21.2745	−1.22219
$304$	0	0
$305$	6.53820	0.374376
$306$	0	0
$307$	1.84177	0.105115	0.0525576	−	0.998618i	$-0.483263\pi$
$307$	1.84177	0.105115	0.0525576	+	0.998618i	$0.483263\pi$
$308$	0	0
$309$	20.3249	1.15624
$310$	0	0
$311$	1.77283	0.100528	0.0502641	−	0.998736i	$-0.483994\pi$
$311$	1.77283	0.100528	0.0502641	+	0.998736i	$0.483994\pi$
$312$	0	0
$313$	32.8627	1.85751	0.928756	−	0.370692i	$-0.120879\pi$
$313$	32.8627	1.85751	0.928756	+	0.370692i	$0.120879\pi$
$314$	0	0
$315$	2.80044	0.157787
$316$	0	0
$317$	−25.0848	−1.40890	−0.704450	−	0.709753i	$-0.748806\pi$
$317$	−25.0848	−1.40890	−0.704450	+	0.709753i	$0.748806\pi$
$318$	0	0
$319$	−2.59907	−0.145520
$320$	0	0
$321$	20.7303	1.15705
$322$	0	0
$323$	0.956426	0.0532170
$324$	0	0
$325$	−15.3523	−0.851590
$326$	0	0
$327$	−13.4718	−0.744990
$328$	0	0
$329$	29.0581	1.60202
$330$	0	0
$331$	1.31692	0.0723847	0.0361923	−	0.999345i	$-0.488477\pi$
$331$	1.31692	0.0723847	0.0361923	+	0.999345i	$0.488477\pi$
$332$	0	0
$333$	−11.2124	−0.614438
$334$	0	0
$335$	6.21466	0.339543
$336$	0	0
$337$	−26.3252	−1.43403	−0.717013	−	0.697060i	$-0.754491\pi$
$337$	−26.3252	−1.43403	−0.717013	+	0.697060i	$0.754491\pi$
$338$	0	0
$339$	11.7224	0.636673
$340$	0	0
$341$	0.501563	0.0271612
$342$	0	0
$343$	−19.2873	−1.04142
$344$	0	0
$345$	−1.29591	−0.0697696
$346$	0	0
$347$	14.0794	0.755821	0.377911	−	0.925842i	$-0.376643\pi$
$347$	14.0794	0.755821	0.377911	+	0.925842i	$0.376643\pi$
$348$	0	0
$349$	12.7893	0.684597	0.342299	−	0.939591i	$-0.388795\pi$
$349$	12.7893	0.684597	0.342299	+	0.939591i	$0.388795\pi$
$350$	0	0
$351$	−18.5194	−0.988493
$352$	0	0
$353$	−1.36900	−0.0728648	−0.0364324	−	0.999336i	$-0.511599\pi$
$353$	−1.36900	−0.0728648	−0.0364324	+	0.999336i	$0.511599\pi$
$354$	0	0
$355$	3.63940	0.193159
$356$	0	0
$357$	15.8725	0.840059
$358$	0	0
$359$	−14.4337	−0.761783	−0.380892	−	0.924620i	$-0.624383\pi$
$359$	−14.4337	−0.761783	−0.380892	+	0.924620i	$0.624383\pi$
$360$	0	0
$361$	−18.9677	−0.998299
$362$	0	0
$363$	−12.7837	−0.670971
$364$	0	0
$365$	−6.57178	−0.343983
$366$	0	0
$367$	−5.73176	−0.299196	−0.149598	−	0.988747i	$-0.547798\pi$
$367$	−5.73176	−0.299196	−0.149598	+	0.988747i	$0.547798\pi$
$368$	0	0
$369$	−10.2339	−0.532754
$370$	0	0
$371$	−31.8386	−1.65298
$372$	0	0
$373$	6.23994	0.323092	0.161546	−	0.986865i	$-0.448352\pi$
$373$	6.23994	0.323092	0.161546	+	0.986865i	$0.448352\pi$
$374$	0	0
$375$	7.85553	0.405658
$376$	0	0
$377$	19.2483	0.991337
$378$	0	0
$379$	−32.8768	−1.68877	−0.844384	−	0.535739i	$-0.820033\pi$
$379$	−32.8768	−1.68877	−0.844384	+	0.535739i	$0.820033\pi$
$380$	0	0
$381$	−22.4389	−1.14958
$382$	0	0
$383$	−3.36511	−0.171949	−0.0859746	−	0.996297i	$-0.527400\pi$
$383$	−3.36511	−0.171949	−0.0859746	+	0.996297i	$0.527400\pi$
$384$	0	0
$385$	0.805683	0.0410614
$386$	0	0
$387$	−7.08367	−0.360083
$388$	0	0
$389$	−22.0548	−1.11822	−0.559111	−	0.829092i	$-0.688858\pi$
$389$	−22.0548	−1.11822	−0.559111	+	0.829092i	$0.688858\pi$
$390$	0	0
$391$	8.35045	0.422300
$392$	0	0
$393$	−22.4038	−1.13012
$394$	0	0
$395$	6.25905	0.314927
$396$	0	0
$397$	−27.6900	−1.38972	−0.694862	−	0.719144i	$-0.744534\pi$
$397$	−27.6900	−1.38972	−0.694862	+	0.719144i	$0.744534\pi$
$398$	0	0
$399$	0.536344	0.0268508
$400$	0	0
$401$	27.8419	1.39036	0.695179	−	0.718837i	$-0.255325\pi$
$401$	27.8419	1.39036	0.695179	+	0.718837i	$0.255325\pi$
$402$	0	0
$403$	−3.71449	−0.185032
$404$	0	0
$405$	1.15969	0.0576252
$406$	0	0
$407$	−3.22581	−0.159897
$408$	0	0
$409$	13.3125	0.658259	0.329129	−	0.944285i	$-0.393245\pi$
$409$	13.3125	0.658259	0.329129	+	0.944285i	$0.393245\pi$
$410$	0	0
$411$	7.54948	0.372389
$412$	0	0
$413$	35.1069	1.72750
$414$	0	0
$415$	−10.6251	−0.521566
$416$	0	0
$417$	−14.6344	−0.716650
$418$	0	0
$419$	6.65220	0.324981	0.162491	−	0.986710i	$-0.448047\pi$
$419$	6.65220	0.324981	0.162491	+	0.986710i	$0.448047\pi$
$420$	0	0
$421$	12.7102	0.619459	0.309730	−	0.950825i	$-0.399761\pi$
$421$	12.7102	0.619459	0.309730	+	0.950825i	$0.399761\pi$
$422$	0	0
$423$	−18.4194	−0.895582
$424$	0	0
$425$	−24.0176	−1.16503
$426$	0	0
$427$	−23.6257	−1.14333
$428$	0	0
$429$	−1.85026	−0.0893316
$430$	0	0
$431$	−4.13580	−0.199215	−0.0996073	−	0.995027i	$-0.531759\pi$
$431$	−4.13580	−0.199215	−0.0996073	+	0.995027i	$0.531759\pi$
$432$	0	0
$433$	−22.2120	−1.06744	−0.533720	−	0.845661i	$-0.679207\pi$
$433$	−22.2120	−1.06744	−0.533720	+	0.845661i	$0.679207\pi$
$434$	0	0
$435$	−4.67321	−0.224063
$436$	0	0
$437$	0.282169	0.0134980
$438$	0	0
$439$	9.52476	0.454592	0.227296	−	0.973826i	$-0.427012\pi$
$439$	9.52476	0.454592	0.227296	+	0.973826i	$0.427012\pi$
$440$	0	0
$441$	1.05327	0.0501558
$442$	0	0
$443$	−3.20660	−0.152350	−0.0761751	−	0.997094i	$-0.524271\pi$
$443$	−3.20660	−0.152350	−0.0761751	+	0.997094i	$0.524271\pi$
$444$	0	0
$445$	−5.10749	−0.242118
$446$	0	0
$447$	−20.9685	−0.991774
$448$	0	0
$449$	27.0145	1.27489	0.637446	−	0.770495i	$-0.279991\pi$
$449$	27.0145	1.27489	0.637446	+	0.770495i	$0.279991\pi$
$450$	0	0
$451$	−2.94428	−0.138641
$452$	0	0
$453$	13.6528	0.641463
$454$	0	0
$455$	−5.96676	−0.279726
$456$	0	0
$457$	11.8385	0.553783	0.276892	−	0.960901i	$-0.410696\pi$
$457$	11.8385	0.553783	0.276892	+	0.960901i	$0.410696\pi$
$458$	0	0
$459$	−28.9724	−1.35232
$460$	0	0
$461$	−20.4078	−0.950484	−0.475242	−	0.879855i	$-0.657640\pi$
$461$	−20.4078	−0.950484	−0.475242	+	0.879855i	$0.657640\pi$
$462$	0	0
$463$	−18.9452	−0.880458	−0.440229	−	0.897886i	$-0.645103\pi$
$463$	−18.9452	−0.880458	−0.440229	+	0.897886i	$0.645103\pi$
$464$	0	0
$465$	0.901827	0.0418212
$466$	0	0
$467$	19.7189	0.912483	0.456242	−	0.889856i	$-0.349195\pi$
$467$	19.7189	0.912483	0.456242	+	0.889856i	$0.349195\pi$
$468$	0	0
$469$	−22.4566	−1.03695
$470$	0	0
$471$	1.47694	0.0680538
$472$	0	0
$473$	−2.03796	−0.0937057
$474$	0	0
$475$	−0.811576	−0.0372377
$476$	0	0
$477$	20.1819	0.924067
$478$	0	0
$479$	7.08744	0.323833	0.161917	−	0.986804i	$-0.448232\pi$
$479$	7.08744	0.323833	0.161917	+	0.986804i	$0.448232\pi$
$480$	0	0
$481$	23.8898	1.08928
$482$	0	0
$483$	4.68276	0.213073
$484$	0	0
$485$	7.56550	0.343532
$486$	0	0
$487$	19.6712	0.891387	0.445694	−	0.895186i	$-0.352957\pi$
$487$	19.6712	0.891387	0.445694	+	0.895186i	$0.352957\pi$
$488$	0	0
$489$	19.5756	0.885241
$490$	0	0
$491$	15.9513	0.719874	0.359937	−	0.932977i	$-0.382798\pi$
$491$	15.9513	0.719874	0.359937	+	0.932977i	$0.382798\pi$
$492$	0	0
$493$	30.1127	1.35621
$494$	0	0
$495$	−0.510708	−0.0229546
$496$	0	0
$497$	−13.1509	−0.589898
$498$	0	0
$499$	−7.55102	−0.338030	−0.169015	−	0.985613i	$-0.554059\pi$
$499$	−7.55102	−0.338030	−0.169015	+	0.985613i	$0.554059\pi$
$500$	0	0
$501$	1.18487	0.0529360
$502$	0	0
$503$	4.79671	0.213875	0.106937	−	0.994266i	$-0.465896\pi$
$503$	4.79671	0.213875	0.106937	+	0.994266i	$0.465896\pi$
$504$	0	0
$505$	12.5116	0.556757
$506$	0	0
$507$	−1.70055	−0.0755240
$508$	0	0
$509$	20.1267	0.892099	0.446050	−	0.895008i	$-0.352831\pi$
$509$	20.1267	0.892099	0.446050	+	0.895008i	$0.352831\pi$
$510$	0	0
$511$	23.7470	1.05051
$512$	0	0
$513$	−0.979003	−0.0432240
$514$	0	0
$515$	−11.9531	−0.526716
$516$	0	0
$517$	−5.29925	−0.233061
$518$	0	0
$519$	12.1385	0.532820
$520$	0	0
$521$	27.9867	1.22612	0.613059	−	0.790037i	$-0.289939\pi$
$521$	27.9867	1.22612	0.613059	+	0.790037i	$0.289939\pi$
$522$	0	0
$523$	34.1581	1.49363	0.746815	−	0.665032i	$-0.231582\pi$
$523$	34.1581	1.49363	0.746815	+	0.665032i	$0.231582\pi$
$524$	0	0
$525$	−13.4686	−0.587817
$526$	0	0
$527$	−5.81108	−0.253135
$528$	0	0
$529$	−20.5364	−0.892888
$530$	0	0
$531$	−22.2536	−0.965725
$532$	0	0
$533$	21.8048	0.944471
$534$	0	0
$535$	−12.1915	−0.527086
$536$	0	0
$537$	−22.8598	−0.986473
$538$	0	0
$539$	0.303025	0.0130522
$540$	0	0
$541$	6.18979	0.266120	0.133060	−	0.991108i	$-0.457520\pi$
$541$	6.18979	0.266120	0.133060	+	0.991108i	$0.457520\pi$
$542$	0	0
$543$	−13.9931	−0.600504
$544$	0	0
$545$	7.92275	0.339373
$546$	0	0
$547$	−32.7706	−1.40117	−0.700584	−	0.713570i	$-0.747077\pi$
$547$	−32.7706	−1.40117	−0.700584	+	0.713570i	$0.747077\pi$
$548$	0	0
$549$	14.9759	0.639156
$550$	0	0
$551$	1.01753	0.0433484
$552$	0	0
$553$	−22.6170	−0.961772
$554$	0	0
$555$	−5.80011	−0.246201
$556$	0	0
$557$	20.6931	0.876796	0.438398	−	0.898781i	$-0.355546\pi$
$557$	20.6931	0.876796	0.438398	+	0.898781i	$0.355546\pi$
$558$	0	0
$559$	15.0928	0.638359
$560$	0	0
$561$	−2.89462	−0.122211
$562$	0	0
$563$	32.2086	1.35743	0.678716	−	0.734401i	$-0.262537\pi$
$563$	32.2086	1.35743	0.678716	+	0.734401i	$0.262537\pi$
$564$	0	0
$565$	−6.89395	−0.290031
$566$	0	0
$567$	−4.19050	−0.175985
$568$	0	0
$569$	38.9742	1.63388	0.816941	−	0.576721i	$-0.195668\pi$
$569$	38.9742	1.63388	0.816941	+	0.576721i	$0.195668\pi$
$570$	0	0
$571$	−11.0366	−0.461867	−0.230933	−	0.972970i	$-0.574178\pi$
$571$	−11.0366	−0.461867	−0.230933	+	0.972970i	$0.574178\pi$
$572$	0	0
$573$	8.84139	0.369355
$574$	0	0
$575$	−7.08578	−0.295497
$576$	0	0
$577$	29.4269	1.22506	0.612528	−	0.790449i	$-0.290153\pi$
$577$	29.4269	1.22506	0.612528	+	0.790449i	$0.290153\pi$
$578$	0	0
$579$	−10.9689	−0.455854
$580$	0	0
$581$	38.3937	1.59284
$582$	0	0
$583$	5.80632	0.240473
$584$	0	0
$585$	3.78222	0.156376
$586$	0	0
$587$	−25.2038	−1.04027	−0.520136	−	0.854083i	$-0.674119\pi$
$587$	−25.2038	−1.04027	−0.520136	+	0.854083i	$0.674119\pi$
$588$	0	0
$589$	−0.196362	−0.00809094
$590$	0	0
$591$	2.87108	0.118100
$592$	0	0
$593$	41.1809	1.69110	0.845548	−	0.533900i	$-0.179274\pi$
$593$	41.1809	1.69110	0.845548	+	0.533900i	$0.179274\pi$
$594$	0	0
$595$	−9.33460	−0.382681
$596$	0	0
$597$	10.7891	0.441569
$598$	0	0
$599$	−16.7618	−0.684867	−0.342434	−	0.939542i	$-0.611251\pi$
$599$	−16.7618	−0.684867	−0.342434	+	0.939542i	$0.611251\pi$
$600$	0	0
$601$	25.8803	1.05568	0.527840	−	0.849344i	$-0.323002\pi$
$601$	25.8803	1.05568	0.527840	+	0.849344i	$0.323002\pi$
$602$	0	0
$603$	14.2348	0.579687
$604$	0	0
$605$	7.51811	0.305655
$606$	0	0
$607$	40.1120	1.62810	0.814048	−	0.580797i	$-0.197259\pi$
$607$	40.1120	1.62810	0.814048	+	0.580797i	$0.197259\pi$
$608$	0	0
$609$	16.8866	0.684278
$610$	0	0
$611$	39.2453	1.58770
$612$	0	0
$613$	−11.8969	−0.480512	−0.240256	−	0.970710i	$-0.577231\pi$
$613$	−11.8969	−0.480512	−0.240256	+	0.970710i	$0.577231\pi$
$614$	0	0
$615$	−5.29390	−0.213471
$616$	0	0
$617$	−25.3077	−1.01885	−0.509425	−	0.860515i	$-0.670142\pi$
$617$	−25.3077	−1.01885	−0.509425	+	0.860515i	$0.670142\pi$
$618$	0	0
$619$	−13.9850	−0.562105	−0.281053	−	0.959692i	$-0.590684\pi$
$619$	−13.9850	−0.562105	−0.281053	+	0.959692i	$0.590684\pi$
$620$	0	0
$621$	−8.54756	−0.343002
$622$	0	0
$623$	18.4558	0.739417
$624$	0	0
$625$	17.9524	0.718094
$626$	0	0
$627$	−0.0978116	−0.00390622
$628$	0	0
$629$	37.3740	1.49020
$630$	0	0
$631$	38.9532	1.55070	0.775352	−	0.631529i	$-0.217572\pi$
$631$	38.9532	1.55070	0.775352	+	0.631529i	$0.217572\pi$
$632$	0	0
$633$	10.1388	0.402979
$634$	0	0
$635$	13.1964	0.523681
$636$	0	0
$637$	−2.24415	−0.0889166
$638$	0	0
$639$	8.33612	0.329772
$640$	0	0
$641$	−0.116710	−0.00460978	−0.00230489	−	0.999997i	$-0.500734\pi$
$641$	−0.116710	−0.00460978	−0.00230489	+	0.999997i	$0.500734\pi$
$642$	0	0
$643$	−6.01446	−0.237187	−0.118594	−	0.992943i	$-0.537839\pi$
$643$	−6.01446	−0.237187	−0.118594	+	0.992943i	$0.537839\pi$
$644$	0	0
$645$	−3.66433	−0.144283
$646$	0	0
$647$	−12.8979	−0.507069	−0.253534	−	0.967326i	$-0.581593\pi$
$647$	−12.8979	−0.507069	−0.253534	+	0.967326i	$0.581593\pi$
$648$	0	0
$649$	−6.40235	−0.251314
$650$	0	0
$651$	−3.25873	−0.127720
$652$	0	0
$653$	27.3349	1.06970	0.534848	−	0.844948i	$-0.320369\pi$
$653$	27.3349	1.06970	0.534848	+	0.844948i	$0.320369\pi$
$654$	0	0
$655$	13.1757	0.514816
$656$	0	0
$657$	−15.0528	−0.587266
$658$	0	0
$659$	−7.02189	−0.273534	−0.136767	−	0.990603i	$-0.543671\pi$
$659$	−7.02189	−0.273534	−0.136767	+	0.990603i	$0.543671\pi$
$660$	0	0
$661$	−16.8998	−0.657327	−0.328663	−	0.944447i	$-0.606598\pi$
$661$	−16.8998	−0.657327	−0.328663	+	0.944447i	$0.606598\pi$
$662$	0	0
$663$	21.4370	0.832546
$664$	0	0
$665$	−0.315424	−0.0122316
$666$	0	0
$667$	8.88397	0.343989
$668$	0	0
$669$	1.02575	0.0396576
$670$	0	0
$671$	4.30855	0.166330
$672$	0	0
$673$	−31.7726	−1.22475	−0.612373	−	0.790569i	$-0.709785\pi$
$673$	−31.7726	−1.22475	−0.612373	+	0.790569i	$0.709785\pi$
$674$	0	0
$675$	24.5845	0.946260
$676$	0	0
$677$	−29.6341	−1.13893	−0.569465	−	0.822015i	$-0.692850\pi$
$677$	−29.6341	−1.13893	−0.569465	+	0.822015i	$0.692850\pi$
$678$	0	0
$679$	−27.3378	−1.04913
$680$	0	0
$681$	−7.44737	−0.285384
$682$	0	0
$683$	2.83295	0.108400	0.0541999	−	0.998530i	$-0.482739\pi$
$683$	2.83295	0.108400	0.0541999	+	0.998530i	$0.482739\pi$
$684$	0	0
$685$	−4.43986	−0.169638
$686$	0	0
$687$	−5.11459	−0.195134
$688$	0	0
$689$	−43.0006	−1.63819
$690$	0	0
$691$	−28.0283	−1.06625	−0.533123	−	0.846038i	$-0.678982\pi$
$691$	−28.0283	−1.06625	−0.533123	+	0.846038i	$0.678982\pi$
$692$	0	0
$693$	1.84544	0.0701023
$694$	0	0
$695$	8.60651	0.326464
$696$	0	0
$697$	34.1122	1.29209
$698$	0	0
$699$	6.20023	0.234514
$700$	0	0
$701$	7.08043	0.267424	0.133712	−	0.991020i	$-0.457310\pi$
$701$	7.08043	0.267424	0.133712	+	0.991020i	$0.457310\pi$
$702$	0	0
$703$	1.26290	0.0476312
$704$	0	0
$705$	−9.52821	−0.358853
$706$	0	0
$707$	−45.2103	−1.70031
$708$	0	0
$709$	5.27218	0.198001	0.0990004	−	0.995087i	$-0.468435\pi$
$709$	5.27218	0.198001	0.0990004	+	0.995087i	$0.468435\pi$
$710$	0	0
$711$	14.3365	0.537661
$712$	0	0
$713$	−1.71441	−0.0642052
$714$	0	0
$715$	1.08814	0.0406942
$716$	0	0
$717$	−24.2599	−0.906002
$718$	0	0
$719$	−48.3910	−1.80468	−0.902339	−	0.431026i	$-0.858152\pi$
$719$	−48.3910	−1.80468	−0.902339	+	0.431026i	$0.858152\pi$
$720$	0	0
$721$	43.1923	1.60856
$722$	0	0
$723$	−25.7497	−0.957641
$724$	0	0
$725$	−25.5521	−0.948983
$726$	0	0
$727$	25.1502	0.932769	0.466385	−	0.884582i	$-0.345556\pi$
$727$	25.1502	0.932769	0.466385	+	0.884582i	$0.345556\pi$
$728$	0	0
$729$	22.0138	0.815326
$730$	0	0
$731$	23.6117	0.873312
$732$	0	0
$733$	−13.9872	−0.516630	−0.258315	−	0.966061i	$-0.583167\pi$
$733$	−13.9872	−0.516630	−0.258315	+	0.966061i	$0.583167\pi$
$734$	0	0
$735$	0.544849	0.0200970
$736$	0	0
$737$	4.09534	0.150854
$738$	0	0
$739$	20.3041	0.746898	0.373449	−	0.927651i	$-0.378175\pi$
$739$	20.3041	0.746898	0.373449	+	0.927651i	$0.378175\pi$
$740$	0	0
$741$	0.724376	0.0266106
$742$	0	0
$743$	38.7071	1.42003	0.710013	−	0.704189i	$-0.248689\pi$
$743$	38.7071	1.42003	0.710013	+	0.704189i	$0.248689\pi$
$744$	0	0
$745$	12.3316	0.451794
$746$	0	0
$747$	−24.3371	−0.890446
$748$	0	0
$749$	44.0539	1.60969
$750$	0	0
$751$	35.9545	1.31200	0.655998	−	0.754762i	$-0.272248\pi$
$751$	35.9545	1.31200	0.655998	+	0.754762i	$0.272248\pi$
$752$	0	0
$753$	−24.4309	−0.890312
$754$	0	0
$755$	−8.02921	−0.292213
$756$	0	0
$757$	−44.6851	−1.62411	−0.812054	−	0.583582i	$-0.801651\pi$
$757$	−44.6851	−1.62411	−0.812054	+	0.583582i	$0.801651\pi$
$758$	0	0
$759$	−0.853982	−0.0309976
$760$	0	0
$761$	42.1570	1.52819	0.764095	−	0.645104i	$-0.223186\pi$
$761$	42.1570	1.52819	0.764095	+	0.645104i	$0.223186\pi$
$762$	0	0
$763$	−28.6287	−1.03643
$764$	0	0
$765$	5.91704	0.213931
$766$	0	0
$767$	47.4147	1.71205
$768$	0	0
$769$	21.2089	0.764811	0.382406	−	0.923995i	$-0.375096\pi$
$769$	21.2089	0.764811	0.382406	+	0.923995i	$0.375096\pi$
$770$	0	0
$771$	11.6858	0.420854
$772$	0	0
$773$	−22.7642	−0.818770	−0.409385	−	0.912362i	$-0.634257\pi$
$773$	−22.7642	−0.818770	−0.409385	+	0.912362i	$0.634257\pi$
$774$	0	0
$775$	4.93100	0.177127
$776$	0	0
$777$	20.9586	0.751885
$778$	0	0
$779$	1.15268	0.0412991
$780$	0	0
$781$	2.39829	0.0858177
$782$	0	0
$783$	−30.8235	−1.10154
$784$	0	0
$785$	−0.868590	−0.0310013
$786$	0	0
$787$	−41.9767	−1.49631	−0.748153	−	0.663526i	$-0.769059\pi$
$787$	−41.9767	−1.49631	−0.748153	+	0.663526i	$0.769059\pi$
$788$	0	0
$789$	−5.00666	−0.178242
$790$	0	0
$791$	24.9112	0.885739
$792$	0	0
$793$	−31.9084	−1.13310
$794$	0	0
$795$	10.4399	0.370267
$796$	0	0
$797$	−7.20016	−0.255043	−0.127521	−	0.991836i	$-0.540702\pi$
$797$	−7.20016	−0.255043	−0.127521	+	0.991836i	$0.540702\pi$
$798$	0	0
$799$	61.3968	2.17206
$800$	0	0
$801$	−11.6988	−0.413357
$802$	0	0
$803$	−4.33068	−0.152826
$804$	0	0
$805$	−2.75393	−0.0970634
$806$	0	0
$807$	−21.2524	−0.748121
$808$	0	0
$809$	6.47714	0.227724	0.113862	−	0.993497i	$-0.463678\pi$
$809$	6.47714	0.227724	0.113862	+	0.993497i	$0.463678\pi$
$810$	0	0
$811$	31.2647	1.09785	0.548926	−	0.835871i	$-0.315037\pi$
$811$	31.2647	1.09785	0.548926	+	0.835871i	$0.315037\pi$
$812$	0	0
$813$	21.9037	0.768197
$814$	0	0
$815$	−11.5124	−0.403264
$816$	0	0
$817$	0.797861	0.0279136
$818$	0	0
$819$	−13.6670	−0.477563
$820$	0	0
$821$	37.2856	1.30128	0.650638	−	0.759388i	$-0.274502\pi$
$821$	37.2856	1.30128	0.650638	+	0.759388i	$0.274502\pi$
$822$	0	0
$823$	31.6995	1.10498	0.552488	−	0.833521i	$-0.313678\pi$
$823$	31.6995	1.10498	0.552488	+	0.833521i	$0.313678\pi$
$824$	0	0
$825$	2.45623	0.0855149
$826$	0	0
$827$	0.273829	0.00952195	0.00476098	−	0.999989i	$-0.498485\pi$
$827$	0.273829	0.00952195	0.00476098	+	0.999989i	$0.498485\pi$
$828$	0	0
$829$	29.7037	1.03165	0.515827	−	0.856693i	$-0.327485\pi$
$829$	29.7037	1.03165	0.515827	+	0.856693i	$0.327485\pi$
$830$	0	0
$831$	−30.0733	−1.04323
$832$	0	0
$833$	−3.51083	−0.121643
$834$	0	0
$835$	−0.696822	−0.0241145
$836$	0	0
$837$	5.94825	0.205602
$838$	0	0
$839$	−27.0492	−0.933843	−0.466921	−	0.884299i	$-0.654637\pi$
$839$	−27.0492	−0.933843	−0.466921	+	0.884299i	$0.654637\pi$
$840$	0	0
$841$	3.03664	0.104712
$842$	0	0
$843$	−33.6974	−1.16060
$844$	0	0
$845$	1.00009	0.0344043
$846$	0	0
$847$	−27.1666	−0.933454
$848$	0	0
$849$	10.2641	0.352263
$850$	0	0
$851$	11.0262	0.377975
$852$	0	0
$853$	−41.6200	−1.42504	−0.712521	−	0.701651i	$-0.752446\pi$
$853$	−41.6200	−1.42504	−0.712521	+	0.701651i	$0.752446\pi$
$854$	0	0
$855$	0.199942	0.00683787
$856$	0	0
$857$	47.6335	1.62713	0.813564	−	0.581475i	$-0.197524\pi$
$857$	47.6335	1.62713	0.813564	+	0.581475i	$0.197524\pi$
$858$	0	0
$859$	4.70655	0.160586	0.0802928	−	0.996771i	$-0.474414\pi$
$859$	4.70655	0.160586	0.0802928	+	0.996771i	$0.474414\pi$
$860$	0	0
$861$	19.1294	0.651929
$862$	0	0
$863$	41.1479	1.40069	0.700346	−	0.713804i	$-0.253029\pi$
$863$	41.1479	1.40069	0.700346	+	0.713804i	$0.253029\pi$
$864$	0	0
$865$	−7.13865	−0.242722
$866$	0	0
$867$	13.3941	0.454887
$868$	0	0
$869$	4.12460	0.139917
$870$	0	0
$871$	−30.3294	−1.02767
$872$	0	0
$873$	17.3289	0.586496
$874$	0	0
$875$	16.6937	0.564350
$876$	0	0
$877$	−38.9107	−1.31392	−0.656960	−	0.753925i	$-0.728158\pi$
$877$	−38.9107	−1.31392	−0.656960	+	0.753925i	$0.728158\pi$
$878$	0	0
$879$	−29.5534	−0.996811
$880$	0	0
$881$	22.0765	0.743777	0.371889	−	0.928277i	$-0.378710\pi$
$881$	22.0765	0.743777	0.371889	+	0.928277i	$0.378710\pi$
$882$	0	0
$883$	29.5204	0.993440	0.496720	−	0.867911i	$-0.334538\pi$
$883$	29.5204	0.993440	0.496720	+	0.867911i	$0.334538\pi$
$884$	0	0
$885$	−11.5116	−0.386959
$886$	0	0
$887$	−2.61398	−0.0877689	−0.0438844	−	0.999037i	$-0.513973\pi$
$887$	−2.61398	−0.0877689	−0.0438844	+	0.999037i	$0.513973\pi$
$888$	0	0
$889$	−47.6848	−1.59930
$890$	0	0
$891$	0.764210	0.0256020
$892$	0	0
$893$	2.07465	0.0694255
$894$	0	0
$895$	13.4439	0.449379
$896$	0	0
$897$	6.32445	0.211167
$898$	0	0
$899$	−6.18236	−0.206193
$900$	0	0
$901$	−67.2717	−2.24114
$902$	0	0
$903$	13.2410	0.440632
$904$	0	0
$905$	8.22938	0.273554
$906$	0	0
$907$	28.5516	0.948040	0.474020	−	0.880514i	$-0.342802\pi$
$907$	28.5516	0.948040	0.474020	+	0.880514i	$0.342802\pi$
$908$	0	0
$909$	28.6580	0.950526
$910$	0	0
$911$	−12.3652	−0.409677	−0.204839	−	0.978796i	$-0.565667\pi$
$911$	−12.3652	−0.409677	−0.204839	+	0.978796i	$0.565667\pi$
$912$	0	0
$913$	−7.00175	−0.231724
$914$	0	0
$915$	7.74691	0.256105
$916$	0	0
$917$	−47.6100	−1.57222
$918$	0	0
$919$	−27.1304	−0.894948	−0.447474	−	0.894297i	$-0.647676\pi$
$919$	−27.1304	−0.894948	−0.447474	+	0.894297i	$0.647676\pi$
$920$	0	0
$921$	2.18225	0.0719076
$922$	0	0
$923$	−17.7614	−0.584622
$924$	0	0
$925$	−31.7138	−1.04274
$926$	0	0
$927$	−27.3788	−0.899238
$928$	0	0
$929$	0.466140	0.0152936	0.00764678	−	0.999971i	$-0.497566\pi$
$929$	0.466140	0.0152936	0.00764678	+	0.999971i	$0.497566\pi$
$930$	0	0
$931$	−0.118634	−0.00388807
$932$	0	0
$933$	2.10057	0.0687697
$934$	0	0
$935$	1.70233	0.0556720
$936$	0	0
$937$	46.4866	1.51865	0.759325	−	0.650711i	$-0.225529\pi$
$937$	46.4866	1.51865	0.759325	+	0.650711i	$0.225529\pi$
$938$	0	0
$939$	38.9380	1.27069
$940$	0	0
$941$	−42.3007	−1.37896	−0.689482	−	0.724302i	$-0.742162\pi$
$941$	−42.3007	−1.37896	−0.689482	+	0.724302i	$0.742162\pi$
$942$	0	0
$943$	10.0639	0.327727
$944$	0	0
$945$	9.55494	0.310822
$946$	0	0
$947$	12.2133	0.396880	0.198440	−	0.980113i	$-0.436412\pi$
$947$	12.2133	0.396880	0.198440	+	0.980113i	$0.436412\pi$
$948$	0	0
$949$	32.0723	1.04111
$950$	0	0
$951$	−29.7221	−0.963807
$952$	0	0
$953$	−42.6783	−1.38249	−0.691243	−	0.722622i	$-0.742937\pi$
$953$	−42.6783	−1.38249	−0.691243	+	0.722622i	$0.742937\pi$
$954$	0	0
$955$	−5.19963	−0.168256
$956$	0	0
$957$	−3.07956	−0.0995480
$958$	0	0
$959$	16.0433	0.518067
$960$	0	0
$961$	−29.8069	−0.961514
$962$	0	0
$963$	−27.9250	−0.899869
$964$	0	0
$965$	6.45085	0.207660
$966$	0	0
$967$	29.5956	0.951729	0.475865	−	0.879519i	$-0.342135\pi$
$967$	29.5956	0.951729	0.475865	+	0.879519i	$0.342135\pi$
$968$	0	0
$969$	1.13324	0.0364049
$970$	0	0
$971$	−42.5380	−1.36511	−0.682555	−	0.730834i	$-0.739131\pi$
$971$	−42.5380	−1.36511	−0.682555	+	0.730834i	$0.739131\pi$
$972$	0	0
$973$	−31.0995	−0.997003
$974$	0	0
$975$	−18.1904	−0.582560
$976$	0	0
$977$	−13.9942	−0.447715	−0.223857	−	0.974622i	$-0.571865\pi$
$977$	−13.9942	−0.447715	−0.223857	+	0.974622i	$0.571865\pi$
$978$	0	0
$979$	−3.36574	−0.107569
$980$	0	0
$981$	18.1472	0.579397
$982$	0	0
$983$	47.5911	1.51792	0.758960	−	0.651137i	$-0.225708\pi$
$983$	47.5911	1.51792	0.758960	+	0.651137i	$0.225708\pi$
$984$	0	0
$985$	−1.68848	−0.0537996
$986$	0	0
$987$	34.4300	1.09592
$988$	0	0
$989$	6.96604	0.221507
$990$	0	0
$991$	−25.7156	−0.816885	−0.408442	−	0.912784i	$-0.633928\pi$
$991$	−25.7156	−0.816885	−0.408442	+	0.912784i	$0.633928\pi$
$992$	0	0
$993$	1.56038	0.0495172
$994$	0	0
$995$	−6.34508	−0.201153
$996$	0	0
$997$	−28.1221	−0.890636	−0.445318	−	0.895373i	$-0.646909\pi$
$997$	−28.1221	−0.890636	−0.445318	+	0.895373i	$0.646909\pi$
$998$	0	0
$999$	−38.2562	−1.21037

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1336.2.a.e.1.7	✓	12
4.3	odd	2		2672.2.a.n.1.6		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1336.2.a.e.1.7	✓	12	1.1	even	1	trivial
2672.2.a.n.1.6		12	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1336 = 2^{3} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1336.a (trivial)

\(f(q)\)	\(=\)	\(q+1.18487 q^{3} -0.696822 q^{5} +2.51795 q^{7} -1.59609 q^{9} +O(q^{10})\)	\(q+1.18487 q^{3} -0.696822 q^{5} +2.51795 q^{7} -1.59609 q^{9} -0.459192 q^{11} +3.40070 q^{13} -0.825643 q^{15} +5.32018 q^{17} +0.179773 q^{19} +2.98344 q^{21} +1.56958 q^{23} -4.51444 q^{25} -5.44576 q^{27} +5.66009 q^{29} -1.09227 q^{31} -0.544083 q^{33} -1.75457 q^{35} +7.02496 q^{37} +4.02939 q^{39} +6.41186 q^{41} +4.43815 q^{43} +1.11219 q^{45} +11.5404 q^{47} -0.659908 q^{49} +6.30371 q^{51} -12.6446 q^{53} +0.319975 q^{55} +0.213008 q^{57} +13.9426 q^{59} -9.38289 q^{61} -4.01887 q^{63} -2.36968 q^{65} -8.91858 q^{67} +1.85975 q^{69} -5.22285 q^{71} +9.43107 q^{73} -5.34902 q^{75} -1.15623 q^{77} -8.98229 q^{79} -1.66425 q^{81} +15.2480 q^{83} -3.70722 q^{85} +6.70646 q^{87} +7.32969 q^{89} +8.56281 q^{91} -1.29420 q^{93} -0.125270 q^{95} -10.8571 q^{97} +0.732911 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 5 q^{3} - 2 q^{5} + 10 q^{7} + 19 q^{9}+O(q^{10}) \) 12 * q + 5 * q^3 - 2 * q^5 + 10 * q^7 + 19 * q^9	\( 12 q + 5 q^{3} - 2 q^{5} + 10 q^{7} + 19 q^{9} + 14 q^{11} - 9 q^{13} + 8 q^{15} + 4 q^{17} + 9 q^{19} + 7 q^{21} + 15 q^{23} + 18 q^{25} + 20 q^{27} + 17 q^{29} + 11 q^{31} + 8 q^{33} + 19 q^{35} - 29 q^{37} + 26 q^{39} + 14 q^{41} + 15 q^{43} - 26 q^{45} + 17 q^{47} + 20 q^{49} + 36 q^{51} - 7 q^{53} + 13 q^{55} + 3 q^{57} + 32 q^{59} - 8 q^{61} + 50 q^{63} + 37 q^{65} + 39 q^{67} + q^{69} + 35 q^{71} - 12 q^{73} + 29 q^{75} + 13 q^{77} + 36 q^{79} + 44 q^{81} + 25 q^{83} - 38 q^{85} + 14 q^{87} + 23 q^{89} - 2 q^{91} - 25 q^{93} + 38 q^{95} - 2 q^{97} + 43 q^{99}+O(q^{100}) \) 12 * q + 5 * q^3 - 2 * q^5 + 10 * q^7 + 19 * q^9 + 14 * q^11 - 9 * q^13 + 8 * q^15 + 4 * q^17 + 9 * q^19 + 7 * q^21 + 15 * q^23 + 18 * q^25 + 20 * q^27 + 17 * q^29 + 11 * q^31 + 8 * q^33 + 19 * q^35 - 29 * q^37 + 26 * q^39 + 14 * q^41 + 15 * q^43 - 26 * q^45 + 17 * q^47 + 20 * q^49 + 36 * q^51 - 7 * q^53 + 13 * q^55 + 3 * q^57 + 32 * q^59 - 8 * q^61 + 50 * q^63 + 37 * q^65 + 39 * q^67 + q^69 + 35 * q^71 - 12 * q^73 + 29 * q^75 + 13 * q^77 + 36 * q^79 + 44 * q^81 + 25 * q^83 - 38 * q^85 + 14 * q^87 + 23 * q^89 - 2 * q^91 - 25 * q^93 + 38 * q^95 - 2 * q^97 + 43 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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