LMFDB - Embedded newform 8046.2.a.j.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8046,2,Mod(1,8046)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8046, 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("8046.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8046 = 2 \cdot 3^{3} \cdot 149 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8046.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:	$64.2476334663$
Analytic rank:	$1$
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} - 3 x^{11} - 29 x^{10} + 76 x^{9} + 320 x^{8} - 724 x^{7} - 1643 x^{6} + 3265 x^{5} + 3921 x^{4} + \cdots + 423 $ x^12 - 3x^11 - 29x^10 + 76x^9 + 320x^8 - 724x^7 - 1643x^6 + 3265x^5 + 3921x^4 - 6927x^3 - 3639x^2 + 5508*x + 423
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			1.5
Root			$1.51858$ of defining polynomial
Character	$\chi$	$=$	8046.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -1.51858 q^{5} +4.35237 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -1.51858 q^{5} +4.35237 q^{7} -1.00000 q^{8} +1.51858 q^{10} -5.50747 q^{11} -0.889028 q^{13} -4.35237 q^{14} +1.00000 q^{16} +0.257830 q^{17} -3.48754 q^{19} -1.51858 q^{20} +5.50747 q^{22} +2.69314 q^{23} -2.69391 q^{25} +0.889028 q^{26} +4.35237 q^{28} +1.61241 q^{29} -0.129803 q^{31} -1.00000 q^{32} -0.257830 q^{34} -6.60944 q^{35} +9.62620 q^{37} +3.48754 q^{38} +1.51858 q^{40} -1.32208 q^{41} +8.65072 q^{43} -5.50747 q^{44} -2.69314 q^{46} -2.28763 q^{47} +11.9431 q^{49} +2.69391 q^{50} -0.889028 q^{52} +1.25633 q^{53} +8.36354 q^{55} -4.35237 q^{56} -1.61241 q^{58} +7.39850 q^{59} -3.49418 q^{61} +0.129803 q^{62} +1.00000 q^{64} +1.35006 q^{65} -3.45868 q^{67} +0.257830 q^{68} +6.60944 q^{70} -14.5730 q^{71} -3.70403 q^{73} -9.62620 q^{74} -3.48754 q^{76} -23.9705 q^{77} +9.40799 q^{79} -1.51858 q^{80} +1.32208 q^{82} +16.5111 q^{83} -0.391536 q^{85} -8.65072 q^{86} +5.50747 q^{88} -12.8672 q^{89} -3.86938 q^{91} +2.69314 q^{92} +2.28763 q^{94} +5.29612 q^{95} -11.8784 q^{97} -11.9431 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{2} + 12 q^{4} - 3 q^{5} + 6 q^{7} - 12 q^{8}+O(q^{10}) $ 12 * q - 12 * q^2 + 12 * q^4 - 3 * q^5 + 6 * q^7 - 12 * q^8	$ 12 q - 12 q^{2} + 12 q^{4} - 3 q^{5} + 6 q^{7} - 12 q^{8} + 3 q^{10} - 10 q^{11} + 5 q^{13} - 6 q^{14} + 12 q^{16} - 8 q^{17} + 2 q^{19} - 3 q^{20} + 10 q^{22} - 9 q^{23} + 7 q^{25} - 5 q^{26} + 6 q^{28} - 19 q^{29} + 10 q^{31} - 12 q^{32} + 8 q^{34} - 20 q^{35} + 11 q^{37} - 2 q^{38} + 3 q^{40} - 8 q^{41} + 13 q^{43} - 10 q^{44} + 9 q^{46} - 11 q^{47} + 2 q^{49} - 7 q^{50} + 5 q^{52} - 24 q^{53} + 3 q^{55} - 6 q^{56} + 19 q^{58} - 10 q^{59} - 10 q^{62} + 12 q^{64} - 28 q^{65} + 21 q^{67} - 8 q^{68} + 20 q^{70} - 37 q^{71} - 2 q^{73} - 11 q^{74} + 2 q^{76} - 2 q^{77} + 7 q^{79} - 3 q^{80} + 8 q^{82} - 22 q^{83} + 15 q^{85} - 13 q^{86} + 10 q^{88} - 40 q^{89} + q^{91} - 9 q^{92} + 11 q^{94} - 11 q^{95} + 7 q^{97} - 2 q^{98}+O(q^{100}) $ 12 * q - 12 * q^2 + 12 * q^4 - 3 * q^5 + 6 * q^7 - 12 * q^8 + 3 * q^10 - 10 * q^11 + 5 * q^13 - 6 * q^14 + 12 * q^16 - 8 * q^17 + 2 * q^19 - 3 * q^20 + 10 * q^22 - 9 * q^23 + 7 * q^25 - 5 * q^26 + 6 * q^28 - 19 * q^29 + 10 * q^31 - 12 * q^32 + 8 * q^34 - 20 * q^35 + 11 * q^37 - 2 * q^38 + 3 * q^40 - 8 * q^41 + 13 * q^43 - 10 * q^44 + 9 * q^46 - 11 * q^47 + 2 * q^49 - 7 * q^50 + 5 * q^52 - 24 * q^53 + 3 * q^55 - 6 * q^56 + 19 * q^58 - 10 * q^59 - 10 * q^62 + 12 * q^64 - 28 * q^65 + 21 * q^67 - 8 * q^68 + 20 * q^70 - 37 * q^71 - 2 * q^73 - 11 * q^74 + 2 * q^76 - 2 * q^77 + 7 * q^79 - 3 * q^80 + 8 * q^82 - 22 * q^83 + 15 * q^85 - 13 * q^86 + 10 * q^88 - 40 * q^89 + q^91 - 9 * q^92 + 11 * q^94 - 11 * q^95 + 7 * q^97 - 2 * 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
$5$	−1.51858	−0.679131	−0.339565	−	0.940582i	$-0.610280\pi$
$5$	−1.51858	−0.679131	−0.339565	+	0.940582i	$0.610280\pi$
$6$	0	0
$7$	4.35237	1.64504	0.822521	−	0.568735i	$-0.192567\pi$
$7$	4.35237	1.64504	0.822521	+	0.568735i	$0.192567\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	1.51858	0.480218
$11$	−5.50747	−1.66056	−0.830282	−	0.557344i	$-0.811820\pi$
$11$	−5.50747	−1.66056	−0.830282	+	0.557344i	$0.811820\pi$
$12$	0	0
$13$	−0.889028	−0.246572	−0.123286	−	0.992371i	$-0.539343\pi$
$13$	−0.889028	−0.246572	−0.123286	+	0.992371i	$0.539343\pi$
$14$	−4.35237	−1.16322
$15$	0	0
$16$	1.00000	0.250000
$17$	0.257830	0.0625330	0.0312665	−	0.999511i	$-0.490046\pi$
$17$	0.257830	0.0625330	0.0312665	+	0.999511i	$0.490046\pi$
$18$	0	0
$19$	−3.48754	−0.800097	−0.400048	−	0.916494i	$-0.631007\pi$
$19$	−3.48754	−0.800097	−0.400048	+	0.916494i	$0.631007\pi$
$20$	−1.51858	−0.339565
$21$	0	0
$22$	5.50747	1.17420
$23$	2.69314	0.561559	0.280779	−	0.959772i	$-0.409407\pi$
$23$	2.69314	0.561559	0.280779	+	0.959772i	$0.409407\pi$
$24$	0	0
$25$	−2.69391	−0.538781
$26$	0.889028	0.174353
$27$	0	0
$28$	4.35237	0.822521
$29$	1.61241	0.299418	0.149709	−	0.988730i	$-0.452166\pi$
$29$	1.61241	0.299418	0.149709	+	0.988730i	$0.452166\pi$
$30$	0	0
$31$	−0.129803	−0.0233134	−0.0116567	−	0.999932i	$-0.503711\pi$
$31$	−0.129803	−0.0233134	−0.0116567	+	0.999932i	$0.503711\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−0.257830	−0.0442175
$35$	−6.60944	−1.11720
$36$	0	0
$37$	9.62620	1.58254	0.791269	−	0.611468i	$-0.209421\pi$
$37$	9.62620	1.58254	0.791269	+	0.611468i	$0.209421\pi$
$38$	3.48754	0.565754
$39$	0	0
$40$	1.51858	0.240109
$41$	−1.32208	−0.206474	−0.103237	−	0.994657i	$-0.532920\pi$
$41$	−1.32208	−0.206474	−0.103237	+	0.994657i	$0.532920\pi$
$42$	0	0
$43$	8.65072	1.31922	0.659612	−	0.751607i	$-0.270721\pi$
$43$	8.65072	1.31922	0.659612	+	0.751607i	$0.270721\pi$
$44$	−5.50747	−0.830282
$45$	0	0
$46$	−2.69314	−0.397082
$47$	−2.28763	−0.333684	−0.166842	−	0.985984i	$-0.553357\pi$
$47$	−2.28763	−0.333684	−0.166842	+	0.985984i	$0.553357\pi$
$48$	0	0
$49$	11.9431	1.70616
$50$	2.69391	0.380976
$51$	0	0
$52$	−0.889028	−0.123286
$53$	1.25633	0.172570	0.0862851	−	0.996270i	$-0.472500\pi$
$53$	1.25633	0.172570	0.0862851	+	0.996270i	$0.472500\pi$
$54$	0	0
$55$	8.36354	1.12774
$56$	−4.35237	−0.581610
$57$	0	0
$58$	−1.61241	−0.211720
$59$	7.39850	0.963202	0.481601	−	0.876390i	$-0.340055\pi$
$59$	7.39850	0.963202	0.481601	+	0.876390i	$0.340055\pi$
$60$	0	0
$61$	−3.49418	−0.447384	−0.223692	−	0.974660i	$-0.571811\pi$
$61$	−3.49418	−0.447384	−0.223692	+	0.974660i	$0.571811\pi$
$62$	0.129803	0.0164851
$63$	0	0
$64$	1.00000	0.125000
$65$	1.35006	0.167455
$66$	0	0
$67$	−3.45868	−0.422545	−0.211272	−	0.977427i	$-0.567761\pi$
$67$	−3.45868	−0.422545	−0.211272	+	0.977427i	$0.567761\pi$
$68$	0.257830	0.0312665
$69$	0	0
$70$	6.60944	0.789979
$71$	−14.5730	−1.72950	−0.864751	−	0.502201i	$-0.832524\pi$
$71$	−14.5730	−1.72950	−0.864751	+	0.502201i	$0.832524\pi$
$72$	0	0
$73$	−3.70403	−0.433524	−0.216762	−	0.976224i	$-0.569550\pi$
$73$	−3.70403	−0.433524	−0.216762	+	0.976224i	$0.569550\pi$
$74$	−9.62620	−1.11902
$75$	0	0
$76$	−3.48754	−0.400048
$77$	−23.9705	−2.73170
$78$	0	0
$79$	9.40799	1.05848	0.529241	−	0.848472i	$-0.322477\pi$
$79$	9.40799	1.05848	0.529241	+	0.848472i	$0.322477\pi$
$80$	−1.51858	−0.169783
$81$	0	0
$82$	1.32208	0.145999
$83$	16.5111	1.81233	0.906165	−	0.422924i	$-0.138996\pi$
$83$	16.5111	1.81233	0.906165	+	0.422924i	$0.138996\pi$
$84$	0	0
$85$	−0.391536	−0.0424681
$86$	−8.65072	−0.932832
$87$	0	0
$88$	5.50747	0.587098
$89$	−12.8672	−1.36392	−0.681960	−	0.731390i	$-0.738872\pi$
$89$	−12.8672	−1.36392	−0.681960	+	0.731390i	$0.738872\pi$
$90$	0	0
$91$	−3.86938	−0.405621
$92$	2.69314	0.280779
$93$	0	0
$94$	2.28763	0.235951
$95$	5.29612	0.543370
$96$	0	0
$97$	−11.8784	−1.20607	−0.603037	−	0.797713i	$-0.706043\pi$
$97$	−11.8784	−1.20607	−0.603037	+	0.797713i	$0.706043\pi$
$98$	−11.9431	−1.20644
$99$	0	0
$100$	−2.69391	−0.269391
$101$	6.88141	0.684726	0.342363	−	0.939568i	$-0.388773\pi$
$101$	6.88141	0.684726	0.342363	+	0.939568i	$0.388773\pi$
$102$	0	0
$103$	−19.5670	−1.92800	−0.963998	−	0.265911i	$-0.914327\pi$
$103$	−19.5670	−1.92800	−0.963998	+	0.265911i	$0.914327\pi$
$104$	0.889028	0.0871764
$105$	0	0
$106$	−1.25633	−0.122026
$107$	−19.5947	−1.89429	−0.947147	−	0.320799i	$-0.896049\pi$
$107$	−19.5947	−1.89429	−0.947147	+	0.320799i	$0.896049\pi$
$108$	0	0
$109$	15.2902	1.46454	0.732269	−	0.681015i	$-0.238461\pi$
$109$	15.2902	1.46454	0.732269	+	0.681015i	$0.238461\pi$
$110$	−8.36354	−0.797432
$111$	0	0
$112$	4.35237	0.411261
$113$	−3.27705	−0.308279	−0.154140	−	0.988049i	$-0.549261\pi$
$113$	−3.27705	−0.308279	−0.154140	+	0.988049i	$0.549261\pi$
$114$	0	0
$115$	−4.08976	−0.381372
$116$	1.61241	0.149709
$117$	0	0
$118$	−7.39850	−0.681087
$119$	1.12217	0.102869
$120$	0	0
$121$	19.3322	1.75747
$122$	3.49418	0.316348
$123$	0	0
$124$	−0.129803	−0.0116567
$125$	11.6838	1.04503
$126$	0	0
$127$	−2.37983	−0.211176	−0.105588	−	0.994410i	$-0.533672\pi$
$127$	−2.37983	−0.211176	−0.105588	+	0.994410i	$0.533672\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−1.35006	−0.118408
$131$	9.93489	0.868015	0.434008	−	0.900909i	$-0.357099\pi$
$131$	9.93489	0.868015	0.434008	+	0.900909i	$0.357099\pi$
$132$	0	0
$133$	−15.1791	−1.31619
$134$	3.45868	0.298784
$135$	0	0
$136$	−0.257830	−0.0221087
$137$	−21.3464	−1.82374	−0.911872	−	0.410474i	$-0.865363\pi$
$137$	−21.3464	−1.82374	−0.911872	+	0.410474i	$0.865363\pi$
$138$	0	0
$139$	−4.04927	−0.343455	−0.171727	−	0.985145i	$-0.554935\pi$
$139$	−4.04927	−0.343455	−0.171727	+	0.985145i	$0.554935\pi$
$140$	−6.60944	−0.558599
$141$	0	0
$142$	14.5730	1.22294
$143$	4.89629	0.409449
$144$	0	0
$145$	−2.44858	−0.203344
$146$	3.70403	0.306548
$147$	0	0
$148$	9.62620	0.791269
$149$	−1.00000	−0.0819232
$149$	−1.00000	−0.0819232
$150$	0	0
$151$	−9.65060	−0.785355	−0.392677	−	0.919676i	$-0.628451\pi$
$151$	−9.65060	−0.785355	−0.392677	+	0.919676i	$0.628451\pi$
$152$	3.48754	0.282877
$153$	0	0
$154$	23.9705	1.93160
$155$	0.197117	0.0158328
$156$	0	0
$157$	−21.5773	−1.72205	−0.861027	−	0.508559i	$-0.830179\pi$
$157$	−21.5773	−1.72205	−0.861027	+	0.508559i	$0.830179\pi$
$158$	−9.40799	−0.748459
$159$	0	0
$160$	1.51858	0.120054
$161$	11.7216	0.923788
$162$	0	0
$163$	13.4786	1.05573	0.527865	−	0.849329i	$-0.322993\pi$
$163$	13.4786	1.05573	0.527865	+	0.849329i	$0.322993\pi$
$164$	−1.32208	−0.103237
$165$	0	0
$166$	−16.5111	−1.28151
$167$	1.95084	0.150961	0.0754804	−	0.997147i	$-0.475951\pi$
$167$	1.95084	0.150961	0.0754804	+	0.997147i	$0.475951\pi$
$168$	0	0
$169$	−12.2096	−0.939202
$170$	0.391536	0.0300295
$171$	0	0
$172$	8.65072	0.659612
$173$	0.332976	0.0253157	0.0126578	−	0.999920i	$-0.495971\pi$
$173$	0.332976	0.0253157	0.0126578	+	0.999920i	$0.495971\pi$
$174$	0	0
$175$	−11.7249	−0.886318
$176$	−5.50747	−0.415141
$177$	0	0
$178$	12.8672	0.964437
$179$	−8.14115	−0.608498	−0.304249	−	0.952592i	$-0.598406\pi$
$179$	−8.14115	−0.608498	−0.304249	+	0.952592i	$0.598406\pi$
$180$	0	0
$181$	−7.87618	−0.585432	−0.292716	−	0.956199i	$-0.594559\pi$
$181$	−7.87618	−0.585432	−0.292716	+	0.956199i	$0.594559\pi$
$182$	3.86938	0.286818
$183$	0	0
$184$	−2.69314	−0.198541
$185$	−14.6182	−1.07475
$186$	0	0
$187$	−1.41999	−0.103840
$188$	−2.28763	−0.166842
$189$	0	0
$190$	−5.29612	−0.384221
$191$	20.2488	1.46515	0.732577	−	0.680684i	$-0.238317\pi$
$191$	20.2488	1.46515	0.732577	+	0.680684i	$0.238317\pi$
$192$	0	0
$193$	−13.2272	−0.952115	−0.476058	−	0.879414i	$-0.657935\pi$
$193$	−13.2272	−0.952115	−0.476058	+	0.879414i	$0.657935\pi$
$194$	11.8784	0.852823
$195$	0	0
$196$	11.9431	0.853082
$197$	−10.5773	−0.753600	−0.376800	−	0.926295i	$-0.622976\pi$
$197$	−10.5773	−0.753600	−0.376800	+	0.926295i	$0.622976\pi$
$198$	0	0
$199$	−14.0343	−0.994863	−0.497431	−	0.867503i	$-0.665723\pi$
$199$	−14.0343	−0.994863	−0.497431	+	0.867503i	$0.665723\pi$
$200$	2.69391	0.190488
$201$	0	0
$202$	−6.88141	−0.484174
$203$	7.01783	0.492555
$204$	0	0
$205$	2.00768	0.140223
$206$	19.5670	1.36330
$207$	0	0
$208$	−0.889028	−0.0616430
$209$	19.2075	1.32861
$210$	0	0
$211$	−25.8982	−1.78290	−0.891452	−	0.453115i	$-0.850313\pi$
$211$	−25.8982	−1.78290	−0.891452	+	0.453115i	$0.850313\pi$
$212$	1.25633	0.0862851
$213$	0	0
$214$	19.5947	1.33947
$215$	−13.1368	−0.895925
$216$	0	0
$217$	−0.564953	−0.0383515
$218$	−15.2902	−1.03559
$219$	0	0
$220$	8.36354	0.563870
$221$	−0.229218	−0.0154189
$222$	0	0
$223$	−25.9086	−1.73497	−0.867483	−	0.497467i	$-0.834264\pi$
$223$	−25.9086	−1.73497	−0.867483	+	0.497467i	$0.834264\pi$
$224$	−4.35237	−0.290805
$225$	0	0
$226$	3.27705	0.217986
$227$	19.7476	1.31069	0.655345	−	0.755329i	$-0.272523\pi$
$227$	19.7476	1.31069	0.655345	+	0.755329i	$0.272523\pi$
$228$	0	0
$229$	22.8704	1.51132	0.755658	−	0.654966i	$-0.227317\pi$
$229$	22.8704	1.51132	0.755658	+	0.654966i	$0.227317\pi$
$230$	4.08976	0.269671
$231$	0	0
$232$	−1.61241	−0.105860
$233$	−1.79710	−0.117732	−0.0588661	−	0.998266i	$-0.518749\pi$
$233$	−1.79710	−0.117732	−0.0588661	+	0.998266i	$0.518749\pi$
$234$	0	0
$235$	3.47395	0.226615
$236$	7.39850	0.481601
$237$	0	0
$238$	−1.12217	−0.0727396
$239$	1.88863	0.122165	0.0610826	−	0.998133i	$-0.480545\pi$
$239$	1.88863	0.122165	0.0610826	+	0.998133i	$0.480545\pi$
$240$	0	0
$241$	8.85516	0.570411	0.285205	−	0.958466i	$-0.407938\pi$
$241$	8.85516	0.570411	0.285205	+	0.958466i	$0.407938\pi$
$242$	−19.3322	−1.24272
$243$	0	0
$244$	−3.49418	−0.223692
$245$	−18.1366	−1.15871
$246$	0	0
$247$	3.10052	0.197281
$248$	0.129803	0.00824253
$249$	0	0
$250$	−11.6838	−0.738950
$251$	−17.6848	−1.11626	−0.558128	−	0.829755i	$-0.688480\pi$
$251$	−17.6848	−1.11626	−0.558128	+	0.829755i	$0.688480\pi$
$252$	0	0
$253$	−14.8324	−0.932504
$254$	2.37983	0.149324
$255$	0	0
$256$	1.00000	0.0625000
$257$	21.8389	1.36227	0.681136	−	0.732157i	$-0.261486\pi$
$257$	21.8389	1.36227	0.681136	+	0.732157i	$0.261486\pi$
$258$	0	0
$259$	41.8968	2.60334
$260$	1.35006	0.0837274
$261$	0	0
$262$	−9.93489	−0.613779
$263$	−15.4275	−0.951302	−0.475651	−	0.879634i	$-0.657787\pi$
$263$	−15.4275	−0.951302	−0.475651	+	0.879634i	$0.657787\pi$
$264$	0	0
$265$	−1.90784	−0.117198
$266$	15.1791	0.930689
$267$	0	0
$268$	−3.45868	−0.211272
$269$	−8.22690	−0.501603	−0.250801	−	0.968039i	$-0.580694\pi$
$269$	−8.22690	−0.501603	−0.250801	+	0.968039i	$0.580694\pi$
$270$	0	0
$271$	9.80919	0.595866	0.297933	−	0.954587i	$-0.403703\pi$
$271$	9.80919	0.595866	0.297933	+	0.954587i	$0.403703\pi$
$272$	0.257830	0.0156332
$273$	0	0
$274$	21.3464	1.28958
$275$	14.8366	0.894681
$276$	0	0
$277$	1.50186	0.0902379	0.0451189	−	0.998982i	$-0.485633\pi$
$277$	1.50186	0.0902379	0.0451189	+	0.998982i	$0.485633\pi$
$278$	4.04927	0.242859
$279$	0	0
$280$	6.60944	0.394989
$281$	6.05893	0.361446	0.180723	−	0.983534i	$-0.442156\pi$
$281$	6.05893	0.361446	0.180723	+	0.983534i	$0.442156\pi$
$282$	0	0
$283$	14.4994	0.861898	0.430949	−	0.902376i	$-0.358179\pi$
$283$	14.4994	0.861898	0.430949	+	0.902376i	$0.358179\pi$
$284$	−14.5730	−0.864751
$285$	0	0
$286$	−4.89629	−0.289524
$287$	−5.75417	−0.339658
$288$	0	0
$289$	−16.9335	−0.996090
$290$	2.44858	0.143786
$291$	0	0
$292$	−3.70403	−0.216762
$293$	−8.49809	−0.496464	−0.248232	−	0.968701i	$-0.579850\pi$
$293$	−8.49809	−0.496464	−0.248232	+	0.968701i	$0.579850\pi$
$294$	0	0
$295$	−11.2352	−0.654140
$296$	−9.62620	−0.559512
$297$	0	0
$298$	1.00000	0.0579284
$299$	−2.39428	−0.138465
$300$	0	0
$301$	37.6512	2.17018
$302$	9.65060	0.555330
$303$	0	0
$304$	−3.48754	−0.200024
$305$	5.30620	0.303832
$306$	0	0
$307$	−2.56218	−0.146231	−0.0731157	−	0.997323i	$-0.523294\pi$
$307$	−2.56218	−0.146231	−0.0731157	+	0.997323i	$0.523294\pi$
$308$	−23.9705	−1.36585
$309$	0	0
$310$	−0.197117	−0.0111955
$311$	−5.29938	−0.300500	−0.150250	−	0.988648i	$-0.548008\pi$
$311$	−5.29938	−0.300500	−0.150250	+	0.988648i	$0.548008\pi$
$312$	0	0
$313$	27.8375	1.57347	0.786735	−	0.617291i	$-0.211770\pi$
$313$	27.8375	1.57347	0.786735	+	0.617291i	$0.211770\pi$
$314$	21.5773	1.21768
$315$	0	0
$316$	9.40799	0.529241
$317$	−23.2519	−1.30596	−0.652979	−	0.757376i	$-0.726481\pi$
$317$	−23.2519	−1.30596	−0.652979	+	0.757376i	$0.726481\pi$
$318$	0	0
$319$	−8.88032	−0.497202
$320$	−1.51858	−0.0848914
$321$	0	0
$322$	−11.7216	−0.653217
$323$	−0.899192	−0.0500324
$324$	0	0
$325$	2.39496	0.132848
$326$	−13.4786	−0.746513
$327$	0	0
$328$	1.32208	0.0729994
$329$	−9.95660	−0.548925
$330$	0	0
$331$	17.5832	0.966458	0.483229	−	0.875494i	$-0.339464\pi$
$331$	17.5832	0.966458	0.483229	+	0.875494i	$0.339464\pi$
$332$	16.5111	0.906165
$333$	0	0
$334$	−1.95084	−0.106745
$335$	5.25229	0.286963
$336$	0	0
$337$	3.34833	0.182395	0.0911975	−	0.995833i	$-0.470931\pi$
$337$	3.34833	0.182395	0.0911975	+	0.995833i	$0.470931\pi$
$338$	12.2096	0.664116
$339$	0	0
$340$	−0.391536	−0.0212340
$341$	0.714888	0.0387133
$342$	0	0
$343$	21.5144	1.16167
$344$	−8.65072	−0.466416
$345$	0	0
$346$	−0.332976	−0.0179009
$347$	−10.0699	−0.540578	−0.270289	−	0.962779i	$-0.587119\pi$
$347$	−10.0699	−0.540578	−0.270289	+	0.962779i	$0.587119\pi$
$348$	0	0
$349$	17.0001	0.909996	0.454998	−	0.890492i	$-0.349640\pi$
$349$	17.0001	0.909996	0.454998	+	0.890492i	$0.349640\pi$
$350$	11.7249	0.626721
$351$	0	0
$352$	5.50747	0.293549
$353$	−2.55240	−0.135850	−0.0679252	−	0.997690i	$-0.521638\pi$
$353$	−2.55240	−0.135850	−0.0679252	+	0.997690i	$0.521638\pi$
$354$	0	0
$355$	22.1304	1.17456
$356$	−12.8672	−0.681960
$357$	0	0
$358$	8.14115	0.430273
$359$	11.4760	0.605683	0.302841	−	0.953041i	$-0.402065\pi$
$359$	11.4760	0.605683	0.302841	+	0.953041i	$0.402065\pi$
$360$	0	0
$361$	−6.83706	−0.359845
$362$	7.87618	0.413963
$363$	0	0
$364$	−3.86938	−0.202811
$365$	5.62488	0.294420
$366$	0	0
$367$	−1.93896	−0.101213	−0.0506065	−	0.998719i	$-0.516115\pi$
$367$	−1.93896	−0.101213	−0.0506065	+	0.998719i	$0.516115\pi$
$368$	2.69314	0.140390
$369$	0	0
$370$	14.6182	0.759963
$371$	5.46801	0.283885
$372$	0	0
$373$	−15.1484	−0.784352	−0.392176	−	0.919890i	$-0.628278\pi$
$373$	−15.1484	−0.784352	−0.392176	+	0.919890i	$0.628278\pi$
$374$	1.41999	0.0734259
$375$	0	0
$376$	2.28763	0.117975
$377$	−1.43348	−0.0738281
$378$	0	0
$379$	−19.9218	−1.02332	−0.511658	−	0.859189i	$-0.670969\pi$
$379$	−19.9218	−1.02332	−0.511658	+	0.859189i	$0.670969\pi$
$380$	5.29612	0.271685
$381$	0	0
$382$	−20.2488	−1.03602
$383$	−34.5336	−1.76459	−0.882293	−	0.470700i	$-0.844002\pi$
$383$	−34.5336	−1.76459	−0.882293	+	0.470700i	$0.844002\pi$
$384$	0	0
$385$	36.4012	1.85518
$386$	13.2272	0.673247
$387$	0	0
$388$	−11.8784	−0.603037
$389$	5.95485	0.301923	0.150961	−	0.988540i	$-0.451763\pi$
$389$	5.95485	0.301923	0.150961	+	0.988540i	$0.451763\pi$
$390$	0	0
$391$	0.694373	0.0351159
$392$	−11.9431	−0.603220
$393$	0	0
$394$	10.5773	0.532876
$395$	−14.2868	−0.718847
$396$	0	0
$397$	−26.6111	−1.33557	−0.667786	−	0.744353i	$-0.732758\pi$
$397$	−26.6111	−1.33557	−0.667786	+	0.744353i	$0.732758\pi$
$398$	14.0343	0.703474
$399$	0	0
$400$	−2.69391	−0.134695
$401$	15.2785	0.762970	0.381485	−	0.924375i	$-0.375413\pi$
$401$	15.2785	0.762970	0.381485	+	0.924375i	$0.375413\pi$
$402$	0	0
$403$	0.115399	0.00574843
$404$	6.88141	0.342363
$405$	0	0
$406$	−7.01783	−0.348289
$407$	−53.0160	−2.62790
$408$	0	0
$409$	9.09083	0.449512	0.224756	−	0.974415i	$-0.427841\pi$
$409$	9.09083	0.449512	0.224756	+	0.974415i	$0.427841\pi$
$410$	−2.00768	−0.0991523
$411$	0	0
$412$	−19.5670	−0.963998
$413$	32.2010	1.58451
$414$	0	0
$415$	−25.0735	−1.23081
$416$	0.889028	0.0435882
$417$	0	0
$418$	−19.2075	−0.939470
$419$	−17.4245	−0.851242	−0.425621	−	0.904902i	$-0.639944\pi$
$419$	−17.4245	−0.851242	−0.425621	+	0.904902i	$0.639944\pi$
$420$	0	0
$421$	−12.1905	−0.594126	−0.297063	−	0.954858i	$-0.596007\pi$
$421$	−12.1905	−0.594126	−0.297063	+	0.954858i	$0.596007\pi$
$422$	25.8982	1.26070
$423$	0	0
$424$	−1.25633	−0.0610128
$425$	−0.694570	−0.0336916
$426$	0	0
$427$	−15.2080	−0.735966
$428$	−19.5947	−0.947147
$429$	0	0
$430$	13.1368	0.633515
$431$	11.9045	0.573421	0.286711	−	0.958017i	$-0.407438\pi$
$431$	11.9045	0.573421	0.286711	+	0.958017i	$0.407438\pi$
$432$	0	0
$433$	6.49961	0.312351	0.156176	−	0.987729i	$-0.450083\pi$
$433$	6.49961	0.312351	0.156176	+	0.987729i	$0.450083\pi$
$434$	0.564953	0.0271186
$435$	0	0
$436$	15.2902	0.732269
$437$	−9.39244	−0.449301
$438$	0	0
$439$	−18.5085	−0.883363	−0.441682	−	0.897172i	$-0.645618\pi$
$439$	−18.5085	−0.883363	−0.441682	+	0.897172i	$0.645618\pi$
$440$	−8.36354	−0.398716
$441$	0	0
$442$	0.229218	0.0109028
$443$	19.5275	0.927780	0.463890	−	0.885893i	$-0.346453\pi$
$443$	19.5275	0.927780	0.463890	+	0.885893i	$0.346453\pi$
$444$	0	0
$445$	19.5399	0.926280
$446$	25.9086	1.22681
$447$	0	0
$448$	4.35237	0.205630
$449$	−41.5708	−1.96185	−0.980923	−	0.194396i	$-0.937725\pi$
$449$	−41.5708	−1.96185	−0.980923	+	0.194396i	$0.937725\pi$
$450$	0	0
$451$	7.28129	0.342862
$452$	−3.27705	−0.154140
$453$	0	0
$454$	−19.7476	−0.926798
$455$	5.87598	0.275470
$456$	0	0
$457$	−14.8990	−0.696944	−0.348472	−	0.937319i	$-0.613299\pi$
$457$	−14.8990	−0.696944	−0.348472	+	0.937319i	$0.613299\pi$
$458$	−22.8704	−1.06866
$459$	0	0
$460$	−4.08976	−0.190686
$461$	17.6770	0.823298	0.411649	−	0.911342i	$-0.364953\pi$
$461$	17.6770	0.823298	0.411649	+	0.911342i	$0.364953\pi$
$462$	0	0
$463$	−8.16756	−0.379579	−0.189789	−	0.981825i	$-0.560781\pi$
$463$	−8.16756	−0.379579	−0.189789	+	0.981825i	$0.560781\pi$
$464$	1.61241	0.0748545
$465$	0	0
$466$	1.79710	0.0832492
$467$	−21.4875	−0.994324	−0.497162	−	0.867658i	$-0.665625\pi$
$467$	−21.4875	−0.994324	−0.497162	+	0.867658i	$0.665625\pi$
$468$	0	0
$469$	−15.0534	−0.695103
$470$	−3.47395	−0.160241
$471$	0	0
$472$	−7.39850	−0.340543
$473$	−47.6436	−2.19065
$474$	0	0
$475$	9.39511	0.431077
$476$	1.12217	0.0514347
$477$	0	0
$478$	−1.88863	−0.0863839
$479$	−16.0082	−0.731433	−0.365716	−	0.930726i	$-0.619176\pi$
$479$	−16.0082	−0.731433	−0.365716	+	0.930726i	$0.619176\pi$
$480$	0	0
$481$	−8.55797	−0.390210
$482$	−8.85516	−0.403341
$483$	0	0
$484$	19.3322	0.878735
$485$	18.0384	0.819082
$486$	0	0
$487$	15.3252	0.694452	0.347226	−	0.937782i	$-0.387124\pi$
$487$	15.3252	0.694452	0.347226	+	0.937782i	$0.387124\pi$
$488$	3.49418	0.158174
$489$	0	0
$490$	18.1366	0.819330
$491$	−8.53751	−0.385293	−0.192646	−	0.981268i	$-0.561707\pi$
$491$	−8.53751	−0.385293	−0.192646	+	0.981268i	$0.561707\pi$
$492$	0	0
$493$	0.415729	0.0187235
$494$	−3.10052	−0.139499
$495$	0	0
$496$	−0.129803	−0.00582835
$497$	−63.4273	−2.84510
$498$	0	0
$499$	−10.1643	−0.455018	−0.227509	−	0.973776i	$-0.573058\pi$
$499$	−10.1643	−0.455018	−0.227509	+	0.973776i	$0.573058\pi$
$500$	11.6838	0.522517
$501$	0	0
$502$	17.6848	0.789312
$503$	−18.8216	−0.839215	−0.419608	−	0.907706i	$-0.637832\pi$
$503$	−18.8216	−0.839215	−0.419608	+	0.907706i	$0.637832\pi$
$504$	0	0
$505$	−10.4500	−0.465018
$506$	14.8324	0.659380
$507$	0	0
$508$	−2.37983	−0.105588
$509$	27.8511	1.23448	0.617239	−	0.786776i	$-0.288251\pi$
$509$	27.8511	1.23448	0.617239	+	0.786776i	$0.288251\pi$
$510$	0	0
$511$	−16.1213	−0.713165
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−21.8389	−0.963272
$515$	29.7141	1.30936
$516$	0	0
$517$	12.5990	0.554104
$518$	−41.8968	−1.84084
$519$	0	0
$520$	−1.35006	−0.0592042
$521$	−24.9165	−1.09161	−0.545805	−	0.837912i	$-0.683776\pi$
$521$	−24.9165	−1.09161	−0.545805	+	0.837912i	$0.683776\pi$
$522$	0	0
$523$	−37.3882	−1.63487	−0.817436	−	0.576019i	$-0.804605\pi$
$523$	−37.3882	−1.63487	−0.817436	+	0.576019i	$0.804605\pi$
$524$	9.93489	0.434008
$525$	0	0
$526$	15.4275	0.672672
$527$	−0.0334672	−0.00145785
$528$	0	0
$529$	−15.7470	−0.684652
$530$	1.90784	0.0828713
$531$	0	0
$532$	−15.1791	−0.658096
$533$	1.17536	0.0509106
$534$	0	0
$535$	29.7562	1.28647
$536$	3.45868	0.149392
$537$	0	0
$538$	8.22690	0.354687
$539$	−65.7764	−2.83319
$540$	0	0
$541$	11.8136	0.507908	0.253954	−	0.967216i	$-0.418269\pi$
$541$	11.8136	0.507908	0.253954	+	0.967216i	$0.418269\pi$
$542$	−9.80919	−0.421341
$543$	0	0
$544$	−0.257830	−0.0110544
$545$	−23.2195	−0.994613
$546$	0	0
$547$	−19.7249	−0.843375	−0.421688	−	0.906741i	$-0.638562\pi$
$547$	−19.7249	−0.843375	−0.421688	+	0.906741i	$0.638562\pi$
$548$	−21.3464	−0.911872
$549$	0	0
$550$	−14.8366	−0.632635
$551$	−5.62336	−0.239563
$552$	0	0
$553$	40.9471	1.74125
$554$	−1.50186	−0.0638078
$555$	0	0
$556$	−4.04927	−0.171727
$557$	14.2105	0.602120	0.301060	−	0.953605i	$-0.402660\pi$
$557$	14.2105	0.602120	0.301060	+	0.953605i	$0.402660\pi$
$558$	0	0
$559$	−7.69074	−0.325284
$560$	−6.60944	−0.279300
$561$	0	0
$562$	−6.05893	−0.255581
$563$	24.2364	1.02144	0.510721	−	0.859746i	$-0.329378\pi$
$563$	24.2364	1.02144	0.510721	+	0.859746i	$0.329378\pi$
$564$	0	0
$565$	4.97648	0.209362
$566$	−14.4994	−0.609454
$567$	0	0
$568$	14.5730	0.611471
$569$	−9.53601	−0.399770	−0.199885	−	0.979819i	$-0.564057\pi$
$569$	−9.53601	−0.399770	−0.199885	+	0.979819i	$0.564057\pi$
$570$	0	0
$571$	−0.958077	−0.0400943	−0.0200471	−	0.999799i	$-0.506382\pi$
$571$	−0.958077	−0.0400943	−0.0200471	+	0.999799i	$0.506382\pi$
$572$	4.89629	0.204724
$573$	0	0
$574$	5.75417	0.240174
$575$	−7.25507	−0.302557
$576$	0	0
$577$	24.3502	1.01371	0.506856	−	0.862031i	$-0.330807\pi$
$577$	24.3502	1.01371	0.506856	+	0.862031i	$0.330807\pi$
$578$	16.9335	0.704342
$579$	0	0
$580$	−2.44858	−0.101672
$581$	71.8625	2.98136
$582$	0	0
$583$	−6.91919	−0.286564
$584$	3.70403	0.153274
$585$	0	0
$586$	8.49809	0.351053
$587$	40.5631	1.67422	0.837108	−	0.547037i	$-0.184244\pi$
$587$	40.5631	1.67422	0.837108	+	0.547037i	$0.184244\pi$
$588$	0	0
$589$	0.452695	0.0186530
$590$	11.2352	0.462547
$591$	0	0
$592$	9.62620	0.395635
$593$	−1.78740	−0.0733996	−0.0366998	−	0.999326i	$-0.511685\pi$
$593$	−1.78740	−0.0733996	−0.0366998	+	0.999326i	$0.511685\pi$
$594$	0	0
$595$	−1.70411	−0.0698617
$596$	−1.00000	−0.0409616
$597$	0	0
$598$	2.39428	0.0979094
$599$	−22.9380	−0.937222	−0.468611	−	0.883405i	$-0.655245\pi$
$599$	−22.9380	−0.937222	−0.468611	+	0.883405i	$0.655245\pi$
$600$	0	0
$601$	7.77077	0.316976	0.158488	−	0.987361i	$-0.449338\pi$
$601$	7.77077	0.316976	0.158488	+	0.987361i	$0.449338\pi$
$602$	−37.6512	−1.53455
$603$	0	0
$604$	−9.65060	−0.392677
$605$	−29.3575	−1.19355
$606$	0	0
$607$	1.31561	0.0533990	0.0266995	−	0.999644i	$-0.491500\pi$
$607$	1.31561	0.0533990	0.0266995	+	0.999644i	$0.491500\pi$
$608$	3.48754	0.141438
$609$	0	0
$610$	−5.30620	−0.214842
$611$	2.03376	0.0822773
$612$	0	0
$613$	−45.8574	−1.85216	−0.926082	−	0.377321i	$-0.876845\pi$
$613$	−45.8574	−1.85216	−0.926082	+	0.377321i	$0.876845\pi$
$614$	2.56218	0.103401
$615$	0	0
$616$	23.9705	0.965801
$617$	23.6340	0.951470	0.475735	−	0.879589i	$-0.342182\pi$
$617$	23.6340	0.951470	0.475735	+	0.879589i	$0.342182\pi$
$618$	0	0
$619$	−25.4446	−1.02270	−0.511352	−	0.859371i	$-0.670855\pi$
$619$	−25.4446	−1.02270	−0.511352	+	0.859371i	$0.670855\pi$
$620$	0.197117	0.00791642
$621$	0	0
$622$	5.29938	0.212486
$623$	−56.0028	−2.24370
$624$	0	0
$625$	−4.27333	−0.170933
$626$	−27.8375	−1.11261
$627$	0	0
$628$	−21.5773	−0.861027
$629$	2.48192	0.0989608
$630$	0	0
$631$	−44.4428	−1.76924	−0.884620	−	0.466313i	$-0.845582\pi$
$631$	−44.4428	−1.76924	−0.884620	+	0.466313i	$0.845582\pi$
$632$	−9.40799	−0.374230
$633$	0	0
$634$	23.2519	0.923451
$635$	3.61398	0.143416
$636$	0	0
$637$	−10.6178	−0.420692
$638$	8.88032	0.351575
$639$	0	0
$640$	1.51858	0.0600272
$641$	41.7490	1.64899	0.824493	−	0.565872i	$-0.191460\pi$
$641$	41.7490	1.64899	0.824493	+	0.565872i	$0.191460\pi$
$642$	0	0
$643$	−8.60191	−0.339226	−0.169613	−	0.985511i	$-0.554252\pi$
$643$	−8.60191	−0.339226	−0.169613	+	0.985511i	$0.554252\pi$
$644$	11.7216	0.461894
$645$	0	0
$646$	0.899192	0.0353783
$647$	−1.39668	−0.0549093	−0.0274546	−	0.999623i	$-0.508740\pi$
$647$	−1.39668	−0.0549093	−0.0274546	+	0.999623i	$0.508740\pi$
$648$	0	0
$649$	−40.7470	−1.59946
$650$	−2.39496	−0.0939380
$651$	0	0
$652$	13.4786	0.527865
$653$	6.77316	0.265054	0.132527	−	0.991179i	$-0.457691\pi$
$653$	6.77316	0.265054	0.132527	+	0.991179i	$0.457691\pi$
$654$	0	0
$655$	−15.0869	−0.589496
$656$	−1.32208	−0.0516184
$657$	0	0
$658$	9.95660	0.388149
$659$	19.0173	0.740809	0.370405	−	0.928871i	$-0.379219\pi$
$659$	19.0173	0.740809	0.370405	+	0.928871i	$0.379219\pi$
$660$	0	0
$661$	−18.7130	−0.727851	−0.363925	−	0.931428i	$-0.618564\pi$
$661$	−18.7130	−0.727851	−0.363925	+	0.931428i	$0.618564\pi$
$662$	−17.5832	−0.683389
$663$	0	0
$664$	−16.5111	−0.640756
$665$	23.0507	0.893867
$666$	0	0
$667$	4.34246	0.168141
$668$	1.95084	0.0754804
$669$	0	0
$670$	−5.25229	−0.202913
$671$	19.2441	0.742910
$672$	0	0
$673$	9.16966	0.353464	0.176732	−	0.984259i	$-0.443447\pi$
$673$	9.16966	0.353464	0.176732	+	0.984259i	$0.443447\pi$
$674$	−3.34833	−0.128973
$675$	0	0
$676$	−12.2096	−0.469601
$677$	−28.5767	−1.09829	−0.549145	−	0.835727i	$-0.685047\pi$
$677$	−28.5767	−1.09829	−0.549145	+	0.835727i	$0.685047\pi$
$678$	0	0
$679$	−51.6994	−1.98404
$680$	0.391536	0.0150147
$681$	0	0
$682$	−0.714888	−0.0273745
$683$	−0.346634	−0.0132636	−0.00663179	−	0.999978i	$-0.502111\pi$
$683$	−0.346634	−0.0132636	−0.00663179	+	0.999978i	$0.502111\pi$
$684$	0	0
$685$	32.4162	1.23856
$686$	−21.5144	−0.821423
$687$	0	0
$688$	8.65072	0.329806
$689$	−1.11691	−0.0425510
$690$	0	0
$691$	7.90123	0.300577	0.150289	−	0.988642i	$-0.451980\pi$
$691$	7.90123	0.300577	0.150289	+	0.988642i	$0.451980\pi$
$692$	0.332976	0.0126578
$693$	0	0
$694$	10.0699	0.382247
$695$	6.14915	0.233251
$696$	0	0
$697$	−0.340871	−0.0129114
$698$	−17.0001	−0.643464
$699$	0	0
$700$	−11.7249	−0.443159
$701$	−44.4543	−1.67901	−0.839507	−	0.543349i	$-0.817156\pi$
$701$	−44.4543	−1.67901	−0.839507	+	0.543349i	$0.817156\pi$
$702$	0	0
$703$	−33.5718	−1.26618
$704$	−5.50747	−0.207570
$705$	0	0
$706$	2.55240	0.0960607
$707$	29.9505	1.12640
$708$	0	0
$709$	13.0815	0.491287	0.245644	−	0.969360i	$-0.421001\pi$
$709$	13.0815	0.491287	0.245644	+	0.969360i	$0.421001\pi$
$710$	−22.1304	−0.830538
$711$	0	0
$712$	12.8672	0.482218
$713$	−0.349579	−0.0130918
$714$	0	0
$715$	−7.43543	−0.278069
$716$	−8.14115	−0.304249
$717$	0	0
$718$	−11.4760	−0.428282
$719$	−16.1980	−0.604085	−0.302042	−	0.953295i	$-0.597668\pi$
$719$	−16.1980	−0.604085	−0.302042	+	0.953295i	$0.597668\pi$
$720$	0	0
$721$	−85.1629	−3.17163
$722$	6.83706	0.254449
$723$	0	0
$724$	−7.87618	−0.292716
$725$	−4.34369	−0.161321
$726$	0	0
$727$	42.9835	1.59417	0.797086	−	0.603866i	$-0.206374\pi$
$727$	42.9835	1.59417	0.797086	+	0.603866i	$0.206374\pi$
$728$	3.86938	0.143409
$729$	0	0
$730$	−5.62488	−0.208186
$731$	2.23042	0.0824949
$732$	0	0
$733$	−11.9049	−0.439719	−0.219859	−	0.975532i	$-0.570560\pi$
$733$	−11.9049	−0.439719	−0.219859	+	0.975532i	$0.570560\pi$
$734$	1.93896	0.0715685
$735$	0	0
$736$	−2.69314	−0.0992705
$737$	19.0485	0.701662
$738$	0	0
$739$	7.31009	0.268906	0.134453	−	0.990920i	$-0.457072\pi$
$739$	7.31009	0.268906	0.134453	+	0.990920i	$0.457072\pi$
$740$	−14.6182	−0.537375
$741$	0	0
$742$	−5.46801	−0.200737
$743$	19.0299	0.698138	0.349069	−	0.937097i	$-0.386498\pi$
$743$	19.0299	0.698138	0.349069	+	0.937097i	$0.386498\pi$
$744$	0	0
$745$	1.51858	0.0556366
$746$	15.1484	0.554621
$747$	0	0
$748$	−1.41999	−0.0519200
$749$	−85.2836	−3.11619
$750$	0	0
$751$	9.70708	0.354216	0.177108	−	0.984191i	$-0.443326\pi$
$751$	9.70708	0.354216	0.177108	+	0.984191i	$0.443326\pi$
$752$	−2.28763	−0.0834211
$753$	0	0
$754$	1.43348	0.0522043
$755$	14.6552	0.533358
$756$	0	0
$757$	−30.8281	−1.12047	−0.560233	−	0.828335i	$-0.689288\pi$
$757$	−30.8281	−1.12047	−0.560233	+	0.828335i	$0.689288\pi$
$758$	19.9218	0.723594
$759$	0	0
$760$	−5.29612	−0.192110
$761$	35.9356	1.30266	0.651332	−	0.758793i	$-0.274211\pi$
$761$	35.9356	1.30266	0.651332	+	0.758793i	$0.274211\pi$
$762$	0	0
$763$	66.5488	2.40923
$764$	20.2488	0.732577
$765$	0	0
$766$	34.5336	1.24775
$767$	−6.57747	−0.237499
$768$	0	0
$769$	51.3229	1.85075	0.925376	−	0.379051i	$-0.123750\pi$
$769$	51.3229	1.85075	0.925376	+	0.379051i	$0.123750\pi$
$770$	−36.4012	−1.31181
$771$	0	0
$772$	−13.2272	−0.476058
$773$	−42.7957	−1.53925	−0.769627	−	0.638493i	$-0.779558\pi$
$773$	−42.7957	−1.53925	−0.769627	+	0.638493i	$0.779558\pi$
$774$	0	0
$775$	0.349678	0.0125608
$776$	11.8784	0.426411
$777$	0	0
$778$	−5.95485	−0.213492
$779$	4.61079	0.165199
$780$	0	0
$781$	80.2605	2.87195
$782$	−0.694373	−0.0248307
$783$	0	0
$784$	11.9431	0.426541
$785$	32.7669	1.16950
$786$	0	0
$787$	54.5692	1.94518	0.972590	−	0.232526i	$-0.0746990\pi$
$787$	54.5692	1.94518	0.972590	+	0.232526i	$0.0746990\pi$
$788$	−10.5773	−0.376800
$789$	0	0
$790$	14.2868	0.508302
$791$	−14.2630	−0.507132
$792$	0	0
$793$	3.10643	0.110312
$794$	26.6111	0.944392
$795$	0	0
$796$	−14.0343	−0.497431
$797$	−32.8397	−1.16324	−0.581621	−	0.813460i	$-0.697581\pi$
$797$	−32.8397	−1.16324	−0.581621	+	0.813460i	$0.697581\pi$
$798$	0	0
$799$	−0.589818	−0.0208663
$800$	2.69391	0.0952440
$801$	0	0
$802$	−15.2785	−0.539501
$803$	20.3998	0.719894
$804$	0	0
$805$	−17.8001	−0.627373
$806$	−0.115399	−0.00406475
$807$	0	0
$808$	−6.88141	−0.242087
$809$	−12.8541	−0.451926	−0.225963	−	0.974136i	$-0.572553\pi$
$809$	−12.8541	−0.451926	−0.225963	+	0.974136i	$0.572553\pi$
$810$	0	0
$811$	−17.6585	−0.620074	−0.310037	−	0.950724i	$-0.600341\pi$
$811$	−17.6585	−0.620074	−0.310037	+	0.950724i	$0.600341\pi$
$812$	7.01783	0.246277
$813$	0	0
$814$	53.0160	1.85821
$815$	−20.4684	−0.716978
$816$	0	0
$817$	−30.1697	−1.05551
$818$	−9.09083	−0.317853
$819$	0	0
$820$	2.00768	0.0701113
$821$	12.1988	0.425740	0.212870	−	0.977081i	$-0.431719\pi$
$821$	12.1988	0.425740	0.212870	+	0.977081i	$0.431719\pi$
$822$	0	0
$823$	−9.87571	−0.344245	−0.172123	−	0.985076i	$-0.555063\pi$
$823$	−9.87571	−0.344245	−0.172123	+	0.985076i	$0.555063\pi$
$824$	19.5670	0.681649
$825$	0	0
$826$	−32.2010	−1.12042
$827$	30.5112	1.06098	0.530489	−	0.847692i	$-0.322008\pi$
$827$	30.5112	1.06098	0.530489	+	0.847692i	$0.322008\pi$
$828$	0	0
$829$	−46.7400	−1.62335	−0.811674	−	0.584111i	$-0.801443\pi$
$829$	−46.7400	−1.62335	−0.811674	+	0.584111i	$0.801443\pi$
$830$	25.0735	0.870314
$831$	0	0
$832$	−0.889028	−0.0308215
$833$	3.07930	0.106691
$834$	0	0
$835$	−2.96252	−0.102522
$836$	19.2075	0.664306
$837$	0	0
$838$	17.4245	0.601919
$839$	21.5539	0.744123	0.372062	−	0.928208i	$-0.378651\pi$
$839$	21.5539	0.744123	0.372062	+	0.928208i	$0.378651\pi$
$840$	0	0
$841$	−26.4001	−0.910349
$842$	12.1905	0.420111
$843$	0	0
$844$	−25.8982	−0.891452
$845$	18.5413	0.637841
$846$	0	0
$847$	84.1408	2.89111
$848$	1.25633	0.0431425
$849$	0	0
$850$	0.694570	0.0238236
$851$	25.9247	0.888688
$852$	0	0
$853$	−11.8218	−0.404770	−0.202385	−	0.979306i	$-0.564869\pi$
$853$	−11.8218	−0.404770	−0.202385	+	0.979306i	$0.564869\pi$
$854$	15.2080	0.520406
$855$	0	0
$856$	19.5947	0.669734
$857$	15.6430	0.534355	0.267178	−	0.963647i	$-0.413909\pi$
$857$	15.6430	0.534355	0.267178	+	0.963647i	$0.413909\pi$
$858$	0	0
$859$	11.9286	0.406998	0.203499	−	0.979075i	$-0.434769\pi$
$859$	11.9286	0.406998	0.203499	+	0.979075i	$0.434769\pi$
$860$	−13.1368	−0.447963
$861$	0	0
$862$	−11.9045	−0.405470
$863$	−43.6027	−1.48425	−0.742127	−	0.670260i	$-0.766183\pi$
$863$	−43.6027	−1.48425	−0.742127	+	0.670260i	$0.766183\pi$
$864$	0	0
$865$	−0.505651	−0.0171927
$866$	−6.49961	−0.220866
$867$	0	0
$868$	−0.564953	−0.0191757
$869$	−51.8142	−1.75768
$870$	0	0
$871$	3.07486	0.104188
$872$	−15.2902	−0.517793
$873$	0	0
$874$	9.39244	0.317704
$875$	50.8524	1.71912
$876$	0	0
$877$	−22.3603	−0.755053	−0.377527	−	0.925999i	$-0.623225\pi$
$877$	−22.3603	−0.755053	−0.377527	+	0.925999i	$0.623225\pi$
$878$	18.5085	0.624632
$879$	0	0
$880$	8.36354	0.281935
$881$	30.2108	1.01783	0.508914	−	0.860817i	$-0.330047\pi$
$881$	30.2108	1.01783	0.508914	+	0.860817i	$0.330047\pi$
$882$	0	0
$883$	39.4746	1.32843	0.664213	−	0.747544i	$-0.268767\pi$
$883$	39.4746	1.32843	0.664213	+	0.747544i	$0.268767\pi$
$884$	−0.229218	−0.00770944
$885$	0	0
$886$	−19.5275	−0.656039
$887$	12.9706	0.435511	0.217755	−	0.976003i	$-0.430126\pi$
$887$	12.9706	0.435511	0.217755	+	0.976003i	$0.430126\pi$
$888$	0	0
$889$	−10.3579	−0.347394
$890$	−19.5399	−0.654979
$891$	0	0
$892$	−25.9086	−0.867483
$893$	7.97818	0.266980
$894$	0	0
$895$	12.3630	0.413250
$896$	−4.35237	−0.145403
$897$	0	0
$898$	41.5708	1.38723
$899$	−0.209297	−0.00698044
$900$	0	0
$901$	0.323919	0.0107913
$902$	−7.28129	−0.242440
$903$	0	0
$904$	3.27705	0.108993
$905$	11.9606	0.397585
$906$	0	0
$907$	−6.49042	−0.215511	−0.107755	−	0.994177i	$-0.534366\pi$
$907$	−6.49042	−0.215511	−0.107755	+	0.994177i	$0.534366\pi$
$908$	19.7476	0.655345
$909$	0	0
$910$	−5.87598	−0.194787
$911$	−13.1414	−0.435394	−0.217697	−	0.976016i	$-0.569854\pi$
$911$	−13.1414	−0.435394	−0.217697	+	0.976016i	$0.569854\pi$
$912$	0	0
$913$	−90.9344	−3.00949
$914$	14.8990	0.492814
$915$	0	0
$916$	22.8704	0.755658
$917$	43.2403	1.42792
$918$	0	0
$919$	−29.8602	−0.984997	−0.492498	−	0.870313i	$-0.663916\pi$
$919$	−29.8602	−0.984997	−0.492498	+	0.870313i	$0.663916\pi$
$920$	4.08976	0.134835
$921$	0	0
$922$	−17.6770	−0.582160
$923$	12.9558	0.426447
$924$	0	0
$925$	−25.9321	−0.852642
$926$	8.16756	0.268403
$927$	0	0
$928$	−1.61241	−0.0529301
$929$	−51.0056	−1.67344	−0.836721	−	0.547630i	$-0.815530\pi$
$929$	−51.0056	−1.67344	−0.836721	+	0.547630i	$0.815530\pi$
$930$	0	0
$931$	−41.6522	−1.36510
$932$	−1.79710	−0.0588661
$933$	0	0
$934$	21.4875	0.703093
$935$	2.15637	0.0705209
$936$	0	0
$937$	−43.1681	−1.41024	−0.705120	−	0.709088i	$-0.749107\pi$
$937$	−43.1681	−1.41024	−0.705120	+	0.709088i	$0.749107\pi$
$938$	15.0534	0.491512
$939$	0	0
$940$	3.47395	0.113308
$941$	19.2726	0.628270	0.314135	−	0.949378i	$-0.398286\pi$
$941$	19.2726	0.628270	0.314135	+	0.949378i	$0.398286\pi$
$942$	0	0
$943$	−3.56054	−0.115947
$944$	7.39850	0.240801
$945$	0	0
$946$	47.6436	1.54903
$947$	−8.94475	−0.290665	−0.145333	−	0.989383i	$-0.546425\pi$
$947$	−8.94475	−0.290665	−0.145333	+	0.989383i	$0.546425\pi$
$948$	0	0
$949$	3.29299	0.106895
$950$	−9.39511	−0.304818
$951$	0	0
$952$	−1.12217	−0.0363698
$953$	−21.0275	−0.681146	−0.340573	−	0.940218i	$-0.610621\pi$
$953$	−21.0275	−0.681146	−0.340573	+	0.940218i	$0.610621\pi$
$954$	0	0
$955$	−30.7495	−0.995032
$956$	1.88863	0.0610826
$957$	0	0
$958$	16.0082	0.517201
$959$	−92.9074	−3.00014
$960$	0	0
$961$	−30.9832	−0.999456
$962$	8.55797	0.275920
$963$	0	0
$964$	8.85516	0.285205
$965$	20.0866	0.646611
$966$	0	0
$967$	−36.1986	−1.16407	−0.582034	−	0.813164i	$-0.697743\pi$
$967$	−36.1986	−1.16407	−0.582034	+	0.813164i	$0.697743\pi$
$968$	−19.3322	−0.621360
$969$	0	0
$970$	−18.0384	−0.579178
$971$	−47.3120	−1.51831	−0.759157	−	0.650908i	$-0.774388\pi$
$971$	−47.3120	−1.51831	−0.759157	+	0.650908i	$0.774388\pi$
$972$	0	0
$973$	−17.6239	−0.564998
$974$	−15.3252	−0.491052
$975$	0	0
$976$	−3.49418	−0.111846
$977$	38.5262	1.23256	0.616282	−	0.787526i	$-0.288638\pi$
$977$	38.5262	1.23256	0.616282	+	0.787526i	$0.288638\pi$
$978$	0	0
$979$	70.8656	2.26487
$980$	−18.1366	−0.579354
$981$	0	0
$982$	8.53751	0.272443
$983$	−38.6141	−1.23160	−0.615799	−	0.787904i	$-0.711167\pi$
$983$	−38.6141	−1.23160	−0.615799	+	0.787904i	$0.711167\pi$
$984$	0	0
$985$	16.0625	0.511793
$986$	−0.415729	−0.0132395
$987$	0	0
$988$	3.10052	0.0986407
$989$	23.2976	0.740821
$990$	0	0
$991$	0.397942	0.0126410	0.00632052	−	0.999980i	$-0.497988\pi$
$991$	0.397942	0.0126410	0.00632052	+	0.999980i	$0.497988\pi$
$992$	0.129803	0.00412126
$993$	0	0
$994$	63.4273	2.01179
$995$	21.3122	0.675642
$996$	0	0
$997$	36.2177	1.14703	0.573513	−	0.819196i	$-0.305580\pi$
$997$	36.2177	1.14703	0.573513	+	0.819196i	$0.305580\pi$
$998$	10.1643	0.321746
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.j.1.5	✓	12
3.2	odd	2		8046.2.a.o.1.8	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.j.1.5	✓	12	1.1	even	1	trivial
8046.2.a.o.1.8	yes	12	3.2	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:	\( N \)	\(=\)	\( 8046 = 2 \cdot 3^{3} \cdot 149 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8046.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.51858 q^{5} +4.35237 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.51858 q^{5} +4.35237 q^{7} -1.00000 q^{8} +1.51858 q^{10} -5.50747 q^{11} -0.889028 q^{13} -4.35237 q^{14} +1.00000 q^{16} +0.257830 q^{17} -3.48754 q^{19} -1.51858 q^{20} +5.50747 q^{22} +2.69314 q^{23} -2.69391 q^{25} +0.889028 q^{26} +4.35237 q^{28} +1.61241 q^{29} -0.129803 q^{31} -1.00000 q^{32} -0.257830 q^{34} -6.60944 q^{35} +9.62620 q^{37} +3.48754 q^{38} +1.51858 q^{40} -1.32208 q^{41} +8.65072 q^{43} -5.50747 q^{44} -2.69314 q^{46} -2.28763 q^{47} +11.9431 q^{49} +2.69391 q^{50} -0.889028 q^{52} +1.25633 q^{53} +8.36354 q^{55} -4.35237 q^{56} -1.61241 q^{58} +7.39850 q^{59} -3.49418 q^{61} +0.129803 q^{62} +1.00000 q^{64} +1.35006 q^{65} -3.45868 q^{67} +0.257830 q^{68} +6.60944 q^{70} -14.5730 q^{71} -3.70403 q^{73} -9.62620 q^{74} -3.48754 q^{76} -23.9705 q^{77} +9.40799 q^{79} -1.51858 q^{80} +1.32208 q^{82} +16.5111 q^{83} -0.391536 q^{85} -8.65072 q^{86} +5.50747 q^{88} -12.8672 q^{89} -3.86938 q^{91} +2.69314 q^{92} +2.28763 q^{94} +5.29612 q^{95} -11.8784 q^{97} -11.9431 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{2} + 12 q^{4} - 3 q^{5} + 6 q^{7} - 12 q^{8}+O(q^{10}) \) 12 * q - 12 * q^2 + 12 * q^4 - 3 * q^5 + 6 * q^7 - 12 * q^8	\( 12 q - 12 q^{2} + 12 q^{4} - 3 q^{5} + 6 q^{7} - 12 q^{8} + 3 q^{10} - 10 q^{11} + 5 q^{13} - 6 q^{14} + 12 q^{16} - 8 q^{17} + 2 q^{19} - 3 q^{20} + 10 q^{22} - 9 q^{23} + 7 q^{25} - 5 q^{26} + 6 q^{28} - 19 q^{29} + 10 q^{31} - 12 q^{32} + 8 q^{34} - 20 q^{35} + 11 q^{37} - 2 q^{38} + 3 q^{40} - 8 q^{41} + 13 q^{43} - 10 q^{44} + 9 q^{46} - 11 q^{47} + 2 q^{49} - 7 q^{50} + 5 q^{52} - 24 q^{53} + 3 q^{55} - 6 q^{56} + 19 q^{58} - 10 q^{59} - 10 q^{62} + 12 q^{64} - 28 q^{65} + 21 q^{67} - 8 q^{68} + 20 q^{70} - 37 q^{71} - 2 q^{73} - 11 q^{74} + 2 q^{76} - 2 q^{77} + 7 q^{79} - 3 q^{80} + 8 q^{82} - 22 q^{83} + 15 q^{85} - 13 q^{86} + 10 q^{88} - 40 q^{89} + q^{91} - 9 q^{92} + 11 q^{94} - 11 q^{95} + 7 q^{97} - 2 q^{98}+O(q^{100}) \) 12 * q - 12 * q^2 + 12 * q^4 - 3 * q^5 + 6 * q^7 - 12 * q^8 + 3 * q^10 - 10 * q^11 + 5 * q^13 - 6 * q^14 + 12 * q^16 - 8 * q^17 + 2 * q^19 - 3 * q^20 + 10 * q^22 - 9 * q^23 + 7 * q^25 - 5 * q^26 + 6 * q^28 - 19 * q^29 + 10 * q^31 - 12 * q^32 + 8 * q^34 - 20 * q^35 + 11 * q^37 - 2 * q^38 + 3 * q^40 - 8 * q^41 + 13 * q^43 - 10 * q^44 + 9 * q^46 - 11 * q^47 + 2 * q^49 - 7 * q^50 + 5 * q^52 - 24 * q^53 + 3 * q^55 - 6 * q^56 + 19 * q^58 - 10 * q^59 - 10 * q^62 + 12 * q^64 - 28 * q^65 + 21 * q^67 - 8 * q^68 + 20 * q^70 - 37 * q^71 - 2 * q^73 - 11 * q^74 + 2 * q^76 - 2 * q^77 + 7 * q^79 - 3 * q^80 + 8 * q^82 - 22 * q^83 + 15 * q^85 - 13 * q^86 + 10 * q^88 - 40 * q^89 + q^91 - 9 * q^92 + 11 * q^94 - 11 * q^95 + 7 * q^97 - 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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