LMFDB - Embedded newform 8046.2.a.l.1.10

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:	$0$
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} - 21 x^{10} + 116 x^{9} + 106 x^{8} - 774 x^{7} - 63 x^{6} + 2013 x^{5} - 417 x^{4} + \cdots - 375 $ x^12 - 5x^11 - 21x^10 + 116x^9 + 106x^8 - 774x^7 - 63x^6 + 2013x^5 - 417x^4 - 2249x^3 + 761x^2 + 910*x - 375
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			1.10
Root			$1.99250$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +1.99250 q^{5} -0.0206178 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} +1.99250 q^{5} -0.0206178 q^{7} -1.00000 q^{8} -1.99250 q^{10} +5.26122 q^{11} +1.30783 q^{13} +0.0206178 q^{14} +1.00000 q^{16} -4.05669 q^{17} +0.295188 q^{19} +1.99250 q^{20} -5.26122 q^{22} -1.64920 q^{23} -1.02995 q^{25} -1.30783 q^{26} -0.0206178 q^{28} -3.96430 q^{29} +8.69374 q^{31} -1.00000 q^{32} +4.05669 q^{34} -0.0410810 q^{35} -8.66851 q^{37} -0.295188 q^{38} -1.99250 q^{40} -0.291717 q^{41} -0.389669 q^{43} +5.26122 q^{44} +1.64920 q^{46} +4.57012 q^{47} -6.99957 q^{49} +1.02995 q^{50} +1.30783 q^{52} +7.77757 q^{53} +10.4830 q^{55} +0.0206178 q^{56} +3.96430 q^{58} +12.8606 q^{59} +7.40709 q^{61} -8.69374 q^{62} +1.00000 q^{64} +2.60584 q^{65} +5.96178 q^{67} -4.05669 q^{68} +0.0410810 q^{70} -10.2613 q^{71} +5.21734 q^{73} +8.66851 q^{74} +0.295188 q^{76} -0.108475 q^{77} +5.69527 q^{79} +1.99250 q^{80} +0.291717 q^{82} +11.8162 q^{83} -8.08295 q^{85} +0.389669 q^{86} -5.26122 q^{88} +11.4015 q^{89} -0.0269646 q^{91} -1.64920 q^{92} -4.57012 q^{94} +0.588161 q^{95} -0.168718 q^{97} +6.99957 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{2} + 12 q^{4} + 5 q^{5} - 6 q^{7} - 12 q^{8}+O(q^{10}) $ 12 * q - 12 * q^2 + 12 * q^4 + 5 * q^5 - 6 * q^7 - 12 * q^8	$ 12 q - 12 q^{2} + 12 q^{4} + 5 q^{5} - 6 q^{7} - 12 q^{8} - 5 q^{10} + 10 q^{11} - q^{13} + 6 q^{14} + 12 q^{16} + 6 q^{17} - 10 q^{19} + 5 q^{20} - 10 q^{22} + 15 q^{23} + 7 q^{25} + q^{26} - 6 q^{28} + 33 q^{29} - 6 q^{31} - 12 q^{32} - 6 q^{34} + 16 q^{35} - 13 q^{37} + 10 q^{38} - 5 q^{40} + 20 q^{41} - 11 q^{43} + 10 q^{44} - 15 q^{46} + 15 q^{47} + 2 q^{49} - 7 q^{50} - q^{52} + 4 q^{53} - 17 q^{55} + 6 q^{56} - 33 q^{58} + 10 q^{59} - 12 q^{61} + 6 q^{62} + 12 q^{64} + 40 q^{65} - 19 q^{67} + 6 q^{68} - 16 q^{70} + 47 q^{71} - 2 q^{73} + 13 q^{74} - 10 q^{76} - 6 q^{77} - 15 q^{79} + 5 q^{80} - 20 q^{82} + 18 q^{83} - 25 q^{85} + 11 q^{86} - 10 q^{88} + 24 q^{89} - 3 q^{91} + 15 q^{92} - 15 q^{94} - 3 q^{95} - 25 q^{97} - 2 q^{98}+O(q^{100}) $ 12 * q - 12 * q^2 + 12 * q^4 + 5 * q^5 - 6 * q^7 - 12 * q^8 - 5 * q^10 + 10 * q^11 - q^13 + 6 * q^14 + 12 * q^16 + 6 * q^17 - 10 * q^19 + 5 * q^20 - 10 * q^22 + 15 * q^23 + 7 * q^25 + q^26 - 6 * q^28 + 33 * q^29 - 6 * q^31 - 12 * q^32 - 6 * q^34 + 16 * q^35 - 13 * q^37 + 10 * q^38 - 5 * q^40 + 20 * q^41 - 11 * q^43 + 10 * q^44 - 15 * q^46 + 15 * q^47 + 2 * q^49 - 7 * q^50 - q^52 + 4 * q^53 - 17 * q^55 + 6 * q^56 - 33 * q^58 + 10 * q^59 - 12 * q^61 + 6 * q^62 + 12 * q^64 + 40 * q^65 - 19 * q^67 + 6 * q^68 - 16 * q^70 + 47 * q^71 - 2 * q^73 + 13 * q^74 - 10 * q^76 - 6 * q^77 - 15 * q^79 + 5 * q^80 - 20 * q^82 + 18 * q^83 - 25 * q^85 + 11 * q^86 - 10 * q^88 + 24 * q^89 - 3 * q^91 + 15 * q^92 - 15 * q^94 - 3 * q^95 - 25 * 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
$5$	1.99250	0.891073	0.445536	−	0.895264i	$-0.353013\pi$
$5$	1.99250	0.891073	0.445536	+	0.895264i	$0.353013\pi$
$6$	0	0
$7$	−0.0206178	−0.00779281	−0.00389641	−	0.999992i	$-0.501240\pi$
$7$	−0.0206178	−0.00779281	−0.00389641	+	0.999992i	$0.501240\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−1.99250	−0.630084
$11$	5.26122	1.58632	0.793159	−	0.609015i	$-0.208435\pi$
$11$	5.26122	1.58632	0.793159	+	0.609015i	$0.208435\pi$
$12$	0	0
$13$	1.30783	0.362726	0.181363	−	0.983416i	$-0.441949\pi$
$13$	1.30783	0.362726	0.181363	+	0.983416i	$0.441949\pi$
$14$	0.0206178	0.00551035
$15$	0	0
$16$	1.00000	0.250000
$17$	−4.05669	−0.983892	−0.491946	−	0.870626i	$-0.663714\pi$
$17$	−4.05669	−0.983892	−0.491946	+	0.870626i	$0.663714\pi$
$18$	0	0
$19$	0.295188	0.0677207	0.0338604	−	0.999427i	$-0.489220\pi$
$19$	0.295188	0.0677207	0.0338604	+	0.999427i	$0.489220\pi$
$20$	1.99250	0.445536
$21$	0	0
$22$	−5.26122	−1.12170
$23$	−1.64920	−0.343882	−0.171941	−	0.985107i	$-0.555004\pi$
$23$	−1.64920	−0.343882	−0.171941	+	0.985107i	$0.555004\pi$
$24$	0	0
$25$	−1.02995	−0.205990
$26$	−1.30783	−0.256486
$27$	0	0
$28$	−0.0206178	−0.00389641
$29$	−3.96430	−0.736153	−0.368076	−	0.929796i	$-0.619984\pi$
$29$	−3.96430	−0.736153	−0.368076	+	0.929796i	$0.619984\pi$
$30$	0	0
$31$	8.69374	1.56144	0.780721	−	0.624880i	$-0.214852\pi$
$31$	8.69374	1.56144	0.780721	+	0.624880i	$0.214852\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	4.05669	0.695717
$35$	−0.0410810	−0.00694396
$36$	0	0
$37$	−8.66851	−1.42509	−0.712547	−	0.701624i	$-0.752459\pi$
$37$	−8.66851	−1.42509	−0.712547	+	0.701624i	$0.752459\pi$
$38$	−0.295188	−0.0478858
$39$	0	0
$40$	−1.99250	−0.315042
$41$	−0.291717	−0.0455585	−0.0227793	−	0.999741i	$-0.507251\pi$
$41$	−0.291717	−0.0455585	−0.0227793	+	0.999741i	$0.507251\pi$
$42$	0	0
$43$	−0.389669	−0.0594240	−0.0297120	−	0.999559i	$-0.509459\pi$
$43$	−0.389669	−0.0594240	−0.0297120	+	0.999559i	$0.509459\pi$
$44$	5.26122	0.793159
$45$	0	0
$46$	1.64920	0.243161
$47$	4.57012	0.666621	0.333310	−	0.942817i	$-0.391834\pi$
$47$	4.57012	0.666621	0.333310	+	0.942817i	$0.391834\pi$
$48$	0	0
$49$	−6.99957	−0.999939
$50$	1.02995	0.145657
$51$	0	0
$52$	1.30783	0.181363
$53$	7.77757	1.06833	0.534166	−	0.845380i	$-0.320626\pi$
$53$	7.77757	1.06833	0.534166	+	0.845380i	$0.320626\pi$
$54$	0	0
$55$	10.4830	1.41352
$56$	0.0206178	0.00275517
$57$	0	0
$58$	3.96430	0.520539
$59$	12.8606	1.67430	0.837151	−	0.546972i	$-0.184219\pi$
$59$	12.8606	1.67430	0.837151	+	0.546972i	$0.184219\pi$
$60$	0	0
$61$	7.40709	0.948381	0.474190	−	0.880422i	$-0.342741\pi$
$61$	7.40709	0.948381	0.474190	+	0.880422i	$0.342741\pi$
$62$	−8.69374	−1.10411
$63$	0	0
$64$	1.00000	0.125000
$65$	2.60584	0.323215
$66$	0	0
$67$	5.96178	0.728348	0.364174	−	0.931331i	$-0.381351\pi$
$67$	5.96178	0.728348	0.364174	+	0.931331i	$0.381351\pi$
$68$	−4.05669	−0.491946
$69$	0	0
$70$	0.0410810	0.00491012
$71$	−10.2613	−1.21779	−0.608895	−	0.793251i	$-0.708387\pi$
$71$	−10.2613	−1.21779	−0.608895	+	0.793251i	$0.708387\pi$
$72$	0	0
$73$	5.21734	0.610644	0.305322	−	0.952249i	$-0.401236\pi$
$73$	5.21734	0.610644	0.305322	+	0.952249i	$0.401236\pi$
$74$	8.66851	1.00769
$75$	0	0
$76$	0.295188	0.0338604
$77$	−0.108475	−0.0123619
$78$	0	0
$79$	5.69527	0.640768	0.320384	−	0.947288i	$-0.396188\pi$
$79$	5.69527	0.640768	0.320384	+	0.947288i	$0.396188\pi$
$80$	1.99250	0.222768
$81$	0	0
$82$	0.291717	0.0322147
$83$	11.8162	1.29700	0.648499	−	0.761215i	$-0.275397\pi$
$83$	11.8162	1.29700	0.648499	+	0.761215i	$0.275397\pi$
$84$	0	0
$85$	−8.08295	−0.876720
$86$	0.389669	0.0420191
$87$	0	0
$88$	−5.26122	−0.560848
$89$	11.4015	1.20855	0.604277	−	0.796775i	$-0.293462\pi$
$89$	11.4015	1.20855	0.604277	+	0.796775i	$0.293462\pi$
$90$	0	0
$91$	−0.0269646	−0.00282665
$92$	−1.64920	−0.171941
$93$	0	0
$94$	−4.57012	−0.471372
$95$	0.588161	0.0603441
$96$	0	0
$97$	−0.168718	−0.0171308	−0.00856538	−	0.999963i	$-0.502726\pi$
$97$	−0.168718	−0.0171308	−0.00856538	+	0.999963i	$0.502726\pi$
$98$	6.99957	0.707064
$99$	0	0
$100$	−1.02995	−0.102995
$101$	15.0043	1.49298	0.746492	−	0.665395i	$-0.231737\pi$
$101$	15.0043	1.49298	0.746492	+	0.665395i	$0.231737\pi$
$102$	0	0
$103$	−2.19991	−0.216763	−0.108382	−	0.994109i	$-0.534567\pi$
$103$	−2.19991	−0.216763	−0.108382	+	0.994109i	$0.534567\pi$
$104$	−1.30783	−0.128243
$105$	0	0
$106$	−7.77757	−0.755425
$107$	−4.80921	−0.464924	−0.232462	−	0.972605i	$-0.574678\pi$
$107$	−4.80921	−0.464924	−0.232462	+	0.972605i	$0.574678\pi$
$108$	0	0
$109$	−12.8433	−1.23016	−0.615081	−	0.788464i	$-0.710877\pi$
$109$	−12.8433	−1.23016	−0.615081	+	0.788464i	$0.710877\pi$
$110$	−10.4830	−0.999513
$111$	0	0
$112$	−0.0206178	−0.00194820
$113$	−3.21231	−0.302189	−0.151094	−	0.988519i	$-0.548280\pi$
$113$	−3.21231	−0.302189	−0.151094	+	0.988519i	$0.548280\pi$
$114$	0	0
$115$	−3.28603	−0.306424
$116$	−3.96430	−0.368076
$117$	0	0
$118$	−12.8606	−1.18391
$119$	0.0836402	0.00766729
$120$	0	0
$121$	16.6804	1.51640
$122$	−7.40709	−0.670607
$123$	0	0
$124$	8.69374	0.780721
$125$	−12.0147	−1.07462
$126$	0	0
$127$	14.1266	1.25354	0.626768	−	0.779206i	$-0.284377\pi$
$127$	14.1266	1.25354	0.626768	+	0.779206i	$0.284377\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−2.60584	−0.228548
$131$	17.7947	1.55473	0.777364	−	0.629051i	$-0.216556\pi$
$131$	17.7947	1.55473	0.777364	+	0.629051i	$0.216556\pi$
$132$	0	0
$133$	−0.00608613	−0.000527735 0
$134$	−5.96178	−0.515020
$135$	0	0
$136$	4.05669	0.347858
$137$	3.79487	0.324217	0.162109	−	0.986773i	$-0.448170\pi$
$137$	3.79487	0.324217	0.162109	+	0.986773i	$0.448170\pi$
$138$	0	0
$139$	−13.1582	−1.11607	−0.558034	−	0.829818i	$-0.688444\pi$
$139$	−13.1582	−1.11607	−0.558034	+	0.829818i	$0.688444\pi$
$140$	−0.0410810	−0.00347198
$141$	0	0
$142$	10.2613	0.861108
$143$	6.88076	0.575398
$144$	0	0
$145$	−7.89887	−0.655966
$146$	−5.21734	−0.431790
$147$	0	0
$148$	−8.66851	−0.712547
$149$	1.00000	0.0819232
$149$	1.00000	0.0819232
$150$	0	0
$151$	−13.5614	−1.10361	−0.551806	−	0.833972i	$-0.686061\pi$
$151$	−13.5614	−1.10361	−0.551806	+	0.833972i	$0.686061\pi$
$152$	−0.295188	−0.0239429
$153$	0	0
$154$	0.108475	0.00874116
$155$	17.3223	1.39136
$156$	0	0
$157$	−11.9231	−0.951564	−0.475782	−	0.879563i	$-0.657835\pi$
$157$	−11.9231	−0.951564	−0.475782	+	0.879563i	$0.657835\pi$
$158$	−5.69527	−0.453091
$159$	0	0
$160$	−1.99250	−0.157521
$161$	0.0340029	0.00267981
$162$	0	0
$163$	−2.75467	−0.215763	−0.107881	−	0.994164i	$-0.534407\pi$
$163$	−2.75467	−0.215763	−0.107881	+	0.994164i	$0.534407\pi$
$164$	−0.291717	−0.0227793
$165$	0	0
$166$	−11.8162	−0.917116
$167$	21.6245	1.67336	0.836678	−	0.547695i	$-0.184495\pi$
$167$	21.6245	1.67336	0.836678	+	0.547695i	$0.184495\pi$
$168$	0	0
$169$	−11.2896	−0.868430
$170$	8.08295	0.619934
$171$	0	0
$172$	−0.389669	−0.0297120
$173$	17.8683	1.35850	0.679252	−	0.733905i	$-0.262304\pi$
$173$	17.8683	1.35850	0.679252	+	0.733905i	$0.262304\pi$
$174$	0	0
$175$	0.0212353	0.00160524
$176$	5.26122	0.396579
$177$	0	0
$178$	−11.4015	−0.854576
$179$	14.6929	1.09820	0.549098	−	0.835758i	$-0.314971\pi$
$179$	14.6929	1.09820	0.549098	+	0.835758i	$0.314971\pi$
$180$	0	0
$181$	20.5258	1.52567	0.762834	−	0.646595i	$-0.223808\pi$
$181$	20.5258	1.52567	0.762834	+	0.646595i	$0.223808\pi$
$182$	0.0269646	0.00199875
$183$	0	0
$184$	1.64920	0.121581
$185$	−17.2720	−1.26986
$186$	0	0
$187$	−21.3432	−1.56077
$188$	4.57012	0.333310
$189$	0	0
$190$	−0.588161	−0.0426697
$191$	16.1527	1.16877	0.584384	−	0.811477i	$-0.301336\pi$
$191$	16.1527	1.16877	0.584384	+	0.811477i	$0.301336\pi$
$192$	0	0
$193$	0.667178	0.0480245	0.0240123	−	0.999712i	$-0.492356\pi$
$193$	0.667178	0.0480245	0.0240123	+	0.999712i	$0.492356\pi$
$194$	0.168718	0.0121133
$195$	0	0
$196$	−6.99957	−0.499970
$197$	2.10056	0.149659	0.0748295	−	0.997196i	$-0.476159\pi$
$197$	2.10056	0.149659	0.0748295	+	0.997196i	$0.476159\pi$
$198$	0	0
$199$	−17.4910	−1.23990	−0.619951	−	0.784641i	$-0.712848\pi$
$199$	−17.4910	−1.23990	−0.619951	+	0.784641i	$0.712848\pi$
$200$	1.02995	0.0728283
$201$	0	0
$202$	−15.0043	−1.05570
$203$	0.0817354	0.00573670
$204$	0	0
$205$	−0.581245	−0.0405959
$206$	2.19991	0.153275
$207$	0	0
$208$	1.30783	0.0906815
$209$	1.55305	0.107427
$210$	0	0
$211$	−6.81415	−0.469105	−0.234553	−	0.972103i	$-0.575363\pi$
$211$	−6.81415	−0.469105	−0.234553	+	0.972103i	$0.575363\pi$
$212$	7.77757	0.534166
$213$	0	0
$214$	4.80921	0.328751
$215$	−0.776416	−0.0529511
$216$	0	0
$217$	−0.179246	−0.0121680
$218$	12.8433	0.869857
$219$	0	0
$220$	10.4830	0.706762
$221$	−5.30545	−0.356883
$222$	0	0
$223$	−19.9915	−1.33873	−0.669366	−	0.742933i	$-0.733434\pi$
$223$	−19.9915	−1.33873	−0.669366	+	0.742933i	$0.733434\pi$
$224$	0.0206178	0.00137759
$225$	0	0
$226$	3.21231	0.213680
$227$	−13.5705	−0.900706	−0.450353	−	0.892851i	$-0.648702\pi$
$227$	−13.5705	−0.900706	−0.450353	+	0.892851i	$0.648702\pi$
$228$	0	0
$229$	27.0742	1.78912	0.894558	−	0.446952i	$-0.147491\pi$
$229$	27.0742	1.78912	0.894558	+	0.446952i	$0.147491\pi$
$230$	3.28603	0.216674
$231$	0	0
$232$	3.96430	0.260269
$233$	1.11238	0.0728743	0.0364371	−	0.999336i	$-0.488399\pi$
$233$	1.11238	0.0728743	0.0364371	+	0.999336i	$0.488399\pi$
$234$	0	0
$235$	9.10596	0.594007
$236$	12.8606	0.837151
$237$	0	0
$238$	−0.0836402	−0.00542159
$239$	−21.3359	−1.38011	−0.690053	−	0.723759i	$-0.742413\pi$
$239$	−21.3359	−1.38011	−0.690053	+	0.723759i	$0.742413\pi$
$240$	0	0
$241$	7.94889	0.512033	0.256016	−	0.966672i	$-0.417590\pi$
$241$	7.94889	0.512033	0.256016	+	0.966672i	$0.417590\pi$
$242$	−16.6804	−1.07226
$243$	0	0
$244$	7.40709	0.474190
$245$	−13.9466	−0.891019
$246$	0	0
$247$	0.386054	0.0245640
$248$	−8.69374	−0.552053
$249$	0	0
$250$	12.0147	0.759874
$251$	30.3459	1.91541	0.957707	−	0.287745i	$-0.0929056\pi$
$251$	30.3459	1.91541	0.957707	+	0.287745i	$0.0929056\pi$
$252$	0	0
$253$	−8.67680	−0.545506
$254$	−14.1266	−0.886383
$255$	0	0
$256$	1.00000	0.0625000
$257$	−11.3203	−0.706142	−0.353071	−	0.935596i	$-0.614863\pi$
$257$	−11.3203	−0.706142	−0.353071	+	0.935596i	$0.614863\pi$
$258$	0	0
$259$	0.178726	0.0111055
$260$	2.60584	0.161608
$261$	0	0
$262$	−17.7947	−1.09936
$263$	−0.0140866	−0.000868618 0	−0.000434309	−	1.00000i	$-0.500138\pi$
$263$	−0.0140866	−0.000868618 0	−0.000434309		1.00000i	$0.500138\pi$
$264$	0	0
$265$	15.4968	0.951961
$266$	0.00608613	0.000373165 0
$267$	0	0
$268$	5.96178	0.364174
$269$	13.4107	0.817663	0.408831	−	0.912610i	$-0.365936\pi$
$269$	13.4107	0.817663	0.408831	+	0.912610i	$0.365936\pi$
$270$	0	0
$271$	30.1331	1.83045	0.915227	−	0.402938i	$-0.132011\pi$
$271$	30.1331	1.83045	0.915227	+	0.402938i	$0.132011\pi$
$272$	−4.05669	−0.245973
$273$	0	0
$274$	−3.79487	−0.229256
$275$	−5.41878	−0.326765
$276$	0	0
$277$	−18.7209	−1.12483	−0.562414	−	0.826856i	$-0.690127\pi$
$277$	−18.7209	−1.12483	−0.562414	+	0.826856i	$0.690127\pi$
$278$	13.1582	0.789179
$279$	0	0
$280$	0.0410810	0.00245506
$281$	7.41800	0.442521	0.221260	−	0.975215i	$-0.428983\pi$
$281$	7.41800	0.442521	0.221260	+	0.975215i	$0.428983\pi$
$282$	0	0
$283$	−21.1335	−1.25625	−0.628127	−	0.778111i	$-0.716178\pi$
$283$	−21.1335	−1.25625	−0.628127	+	0.778111i	$0.716178\pi$
$284$	−10.2613	−0.608895
$285$	0	0
$286$	−6.88076	−0.406868
$287$	0.00601457	0.000355029 0
$288$	0	0
$289$	−0.543249	−0.0319558
$290$	7.89887	0.463838
$291$	0	0
$292$	5.21734	0.305322
$293$	−5.04458	−0.294707	−0.147354	−	0.989084i	$-0.547076\pi$
$293$	−5.04458	−0.294707	−0.147354	+	0.989084i	$0.547076\pi$
$294$	0	0
$295$	25.6247	1.49192
$296$	8.66851	0.503847
$297$	0	0
$298$	−1.00000	−0.0579284
$299$	−2.15687	−0.124735
$300$	0	0
$301$	0.00803414	0.000463080 0
$302$	13.5614	0.780372
$303$	0	0
$304$	0.295188	0.0169302
$305$	14.7586	0.845076
$306$	0	0
$307$	−4.06035	−0.231736	−0.115868	−	0.993265i	$-0.536965\pi$
$307$	−4.06035	−0.231736	−0.115868	+	0.993265i	$0.536965\pi$
$308$	−0.108475	−0.00618094
$309$	0	0
$310$	−17.3223	−0.983839
$311$	−30.1221	−1.70807	−0.854033	−	0.520219i	$-0.825850\pi$
$311$	−30.1221	−1.70807	−0.854033	+	0.520219i	$0.825850\pi$
$312$	0	0
$313$	5.86146	0.331309	0.165655	−	0.986184i	$-0.447026\pi$
$313$	5.86146	0.331309	0.165655	+	0.986184i	$0.447026\pi$
$314$	11.9231	0.672857
$315$	0	0
$316$	5.69527	0.320384
$317$	−13.0792	−0.734602	−0.367301	−	0.930102i	$-0.619718\pi$
$317$	−13.0792	−0.734602	−0.367301	+	0.930102i	$0.619718\pi$
$318$	0	0
$319$	−20.8571	−1.16777
$320$	1.99250	0.111384
$321$	0	0
$322$	−0.0340029	−0.00189491
$323$	−1.19749	−0.0666299
$324$	0	0
$325$	−1.34699	−0.0747177
$326$	2.75467	0.152567
$327$	0	0
$328$	0.291717	0.0161074
$329$	−0.0942260	−0.00519485
$330$	0	0
$331$	11.3159	0.621977	0.310988	−	0.950414i	$-0.399340\pi$
$331$	11.3159	0.621977	0.310988	+	0.950414i	$0.399340\pi$
$332$	11.8162	0.648499
$333$	0	0
$334$	−21.6245	−1.18324
$335$	11.8788	0.649011
$336$	0	0
$337$	3.41103	0.185810	0.0929052	−	0.995675i	$-0.470385\pi$
$337$	3.41103	0.185810	0.0929052	+	0.995675i	$0.470385\pi$
$338$	11.2896	0.614073
$339$	0	0
$340$	−8.08295	−0.438360
$341$	45.7397	2.47694
$342$	0	0
$343$	0.288641	0.0155851
$344$	0.389669	0.0210096
$345$	0	0
$346$	−17.8683	−0.960608
$347$	5.91948	0.317774	0.158887	−	0.987297i	$-0.449209\pi$
$347$	5.91948	0.317774	0.158887	+	0.987297i	$0.449209\pi$
$348$	0	0
$349$	11.2779	0.603692	0.301846	−	0.953357i	$-0.402397\pi$
$349$	11.2779	0.603692	0.301846	+	0.953357i	$0.402397\pi$
$350$	−0.0212353	−0.00113507
$351$	0	0
$352$	−5.26122	−0.280424
$353$	14.0935	0.750123	0.375061	−	0.927000i	$-0.377622\pi$
$353$	14.0935	0.750123	0.375061	+	0.927000i	$0.377622\pi$
$354$	0	0
$355$	−20.4456	−1.08514
$356$	11.4015	0.604277
$357$	0	0
$358$	−14.6929	−0.776542
$359$	3.81544	0.201371	0.100686	−	0.994918i	$-0.467896\pi$
$359$	3.81544	0.201371	0.100686	+	0.994918i	$0.467896\pi$
$360$	0	0
$361$	−18.9129	−0.995414
$362$	−20.5258	−1.07881
$363$	0	0
$364$	−0.0269646	−0.00141333
$365$	10.3955	0.544128
$366$	0	0
$367$	−16.4625	−0.859337	−0.429669	−	0.902987i	$-0.641370\pi$
$367$	−16.4625	−0.859337	−0.429669	+	0.902987i	$0.641370\pi$
$368$	−1.64920	−0.0859704
$369$	0	0
$370$	17.2720	0.897928
$371$	−0.160357	−0.00832531
$372$	0	0
$373$	20.1014	1.04081	0.520406	−	0.853919i	$-0.325781\pi$
$373$	20.1014	1.04081	0.520406	+	0.853919i	$0.325781\pi$
$374$	21.3432	1.10363
$375$	0	0
$376$	−4.57012	−0.235686
$377$	−5.18462	−0.267022
$378$	0	0
$379$	−0.0507386	−0.00260627	−0.00130313	−	0.999999i	$-0.500415\pi$
$379$	−0.0507386	−0.00260627	−0.00130313	+	0.999999i	$0.500415\pi$
$380$	0.588161	0.0301720
$381$	0	0
$382$	−16.1527	−0.826444
$383$	−24.2127	−1.23721	−0.618605	−	0.785702i	$-0.712302\pi$
$383$	−24.2127	−1.23721	−0.618605	+	0.785702i	$0.712302\pi$
$384$	0	0
$385$	−0.216136	−0.0110153
$386$	−0.667178	−0.0339585
$387$	0	0
$388$	−0.168718	−0.00856538
$389$	−16.8468	−0.854168	−0.427084	−	0.904212i	$-0.640459\pi$
$389$	−16.8468	−0.854168	−0.427084	+	0.904212i	$0.640459\pi$
$390$	0	0
$391$	6.69029	0.338343
$392$	6.99957	0.353532
$393$	0	0
$394$	−2.10056	−0.105825
$395$	11.3478	0.570971
$396$	0	0
$397$	25.7132	1.29051	0.645254	−	0.763968i	$-0.276752\pi$
$397$	25.7132	1.29051	0.645254	+	0.763968i	$0.276752\pi$
$398$	17.4910	0.876743
$399$	0	0
$400$	−1.02995	−0.0514974
$401$	−18.5979	−0.928733	−0.464366	−	0.885643i	$-0.653718\pi$
$401$	−18.5979	−0.928733	−0.464366	+	0.885643i	$0.653718\pi$
$402$	0	0
$403$	11.3699	0.566375
$404$	15.0043	0.746492
$405$	0	0
$406$	−0.0817354	−0.00405646
$407$	−45.6069	−2.26065
$408$	0	0
$409$	36.7789	1.81860	0.909300	−	0.416142i	$-0.136618\pi$
$409$	36.7789	1.81860	0.909300	+	0.416142i	$0.136618\pi$
$410$	0.581245	0.0287057
$411$	0	0
$412$	−2.19991	−0.108382
$413$	−0.265157	−0.0130475
$414$	0	0
$415$	23.5438	1.15572
$416$	−1.30783	−0.0641215
$417$	0	0
$418$	−1.55305	−0.0759620
$419$	9.76046	0.476830	0.238415	−	0.971163i	$-0.423372\pi$
$419$	9.76046	0.476830	0.238415	+	0.971163i	$0.423372\pi$
$420$	0	0
$421$	−32.0064	−1.55990	−0.779949	−	0.625843i	$-0.784755\pi$
$421$	−32.0064	−1.55990	−0.779949	+	0.625843i	$0.784755\pi$
$422$	6.81415	0.331707
$423$	0	0
$424$	−7.77757	−0.377712
$425$	4.17818	0.202672
$426$	0	0
$427$	−0.152718	−0.00739055
$428$	−4.80921	−0.232462
$429$	0	0
$430$	0.776416	0.0374421
$431$	33.5502	1.61605	0.808027	−	0.589145i	$-0.200535\pi$
$431$	33.5502	1.61605	0.808027	+	0.589145i	$0.200535\pi$
$432$	0	0
$433$	35.8412	1.72242	0.861208	−	0.508253i	$-0.169708\pi$
$433$	35.8412	1.72242	0.861208	+	0.508253i	$0.169708\pi$
$434$	0.179246	0.00860409
$435$	0	0
$436$	−12.8433	−0.615081
$437$	−0.486823	−0.0232879
$438$	0	0
$439$	−4.89036	−0.233404	−0.116702	−	0.993167i	$-0.537232\pi$
$439$	−4.89036	−0.233404	−0.116702	+	0.993167i	$0.537232\pi$
$440$	−10.4830	−0.499756
$441$	0	0
$442$	5.30545	0.252355
$443$	−6.82947	−0.324478	−0.162239	−	0.986752i	$-0.551872\pi$
$443$	−6.82947	−0.324478	−0.162239	+	0.986752i	$0.551872\pi$
$444$	0	0
$445$	22.7174	1.07691
$446$	19.9915	0.946627
$447$	0	0
$448$	−0.0206178	−0.000974101 0
$449$	29.7864	1.40571	0.702854	−	0.711334i	$-0.251909\pi$
$449$	29.7864	1.40571	0.702854	+	0.711334i	$0.251909\pi$
$450$	0	0
$451$	−1.53479	−0.0722703
$452$	−3.21231	−0.151094
$453$	0	0
$454$	13.5705	0.636895
$455$	−0.0537269	−0.00251875
$456$	0	0
$457$	4.05153	0.189522	0.0947612	−	0.995500i	$-0.469791\pi$
$457$	4.05153	0.189522	0.0947612	+	0.995500i	$0.469791\pi$
$458$	−27.0742	−1.26510
$459$	0	0
$460$	−3.28603	−0.153212
$461$	39.8347	1.85529	0.927643	−	0.373468i	$-0.121832\pi$
$461$	39.8347	1.85529	0.927643	+	0.373468i	$0.121832\pi$
$462$	0	0
$463$	−22.9167	−1.06503	−0.532515	−	0.846421i	$-0.678753\pi$
$463$	−22.9167	−1.06503	−0.532515	+	0.846421i	$0.678753\pi$
$464$	−3.96430	−0.184038
$465$	0	0
$466$	−1.11238	−0.0515299
$467$	1.53647	0.0710993	0.0355496	−	0.999368i	$-0.488682\pi$
$467$	1.53647	0.0710993	0.0355496	+	0.999368i	$0.488682\pi$
$468$	0	0
$469$	−0.122919	−0.00567588
$470$	−9.10596	−0.420027
$471$	0	0
$472$	−12.8606	−0.591955
$473$	−2.05014	−0.0942654
$474$	0	0
$475$	−0.304028	−0.0139498
$476$	0.0836402	0.00383364
$477$	0	0
$478$	21.3359	0.975882
$479$	22.1043	1.00997	0.504985	−	0.863128i	$-0.331498\pi$
$479$	22.1043	1.00997	0.504985	+	0.863128i	$0.331498\pi$
$480$	0	0
$481$	−11.3369	−0.516918
$482$	−7.94889	−0.362062
$483$	0	0
$484$	16.6804	0.758202
$485$	−0.336171	−0.0152648
$486$	0	0
$487$	−34.8299	−1.57829	−0.789147	−	0.614204i	$-0.789477\pi$
$487$	−34.8299	−1.57829	−0.789147	+	0.614204i	$0.789477\pi$
$488$	−7.40709	−0.335303
$489$	0	0
$490$	13.9466	0.630045
$491$	−12.6656	−0.571589	−0.285795	−	0.958291i	$-0.592257\pi$
$491$	−12.6656	−0.571589	−0.285795	+	0.958291i	$0.592257\pi$
$492$	0	0
$493$	16.0820	0.724295
$494$	−0.386054	−0.0173694
$495$	0	0
$496$	8.69374	0.390360
$497$	0.211566	0.00949001
$498$	0	0
$499$	17.1306	0.766872	0.383436	−	0.923567i	$-0.374741\pi$
$499$	17.1306	0.766872	0.383436	+	0.923567i	$0.374741\pi$
$500$	−12.0147	−0.537312
$501$	0	0
$502$	−30.3459	−1.35440
$503$	8.08515	0.360499	0.180250	−	0.983621i	$-0.442309\pi$
$503$	8.08515	0.360499	0.180250	+	0.983621i	$0.442309\pi$
$504$	0	0
$505$	29.8960	1.33036
$506$	8.67680	0.385731
$507$	0	0
$508$	14.1266	0.626768
$509$	−27.1112	−1.20168	−0.600842	−	0.799368i	$-0.705168\pi$
$509$	−27.1112	−1.20168	−0.600842	+	0.799368i	$0.705168\pi$
$510$	0	0
$511$	−0.107570	−0.00475863
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	11.3203	0.499318
$515$	−4.38331	−0.193152
$516$	0	0
$517$	24.0444	1.05747
$518$	−0.178726	−0.00785277
$519$	0	0
$520$	−2.60584	−0.114274
$521$	−4.65077	−0.203754	−0.101877	−	0.994797i	$-0.532485\pi$
$521$	−4.65077	−0.203754	−0.101877	+	0.994797i	$0.532485\pi$
$522$	0	0
$523$	−11.6457	−0.509232	−0.254616	−	0.967042i	$-0.581949\pi$
$523$	−11.6457	−0.509232	−0.254616	+	0.967042i	$0.581949\pi$
$524$	17.7947	0.777364
$525$	0	0
$526$	0.0140866	0.000614206 0
$527$	−35.2678	−1.53629
$528$	0	0
$529$	−20.2801	−0.881745
$530$	−15.4968	−0.673138
$531$	0	0
$532$	−0.00608613	−0.000263867 0
$533$	−0.381515	−0.0165252
$534$	0	0
$535$	−9.58235	−0.414281
$536$	−5.96178	−0.257510
$537$	0	0
$538$	−13.4107	−0.578175
$539$	−36.8263	−1.58622
$540$	0	0
$541$	−8.80776	−0.378675	−0.189338	−	0.981912i	$-0.560634\pi$
$541$	−8.80776	−0.378675	−0.189338	+	0.981912i	$0.560634\pi$
$542$	−30.1331	−1.29433
$543$	0	0
$544$	4.05669	0.173929
$545$	−25.5902	−1.09616
$546$	0	0
$547$	−22.6940	−0.970325	−0.485162	−	0.874424i	$-0.661239\pi$
$547$	−22.6940	−0.970325	−0.485162	+	0.874424i	$0.661239\pi$
$548$	3.79487	0.162109
$549$	0	0
$550$	5.41878	0.231058
$551$	−1.17021	−0.0498528
$552$	0	0
$553$	−0.117424	−0.00499338
$554$	18.7209	0.795373
$555$	0	0
$556$	−13.1582	−0.558034
$557$	−20.2130	−0.856454	−0.428227	−	0.903671i	$-0.640862\pi$
$557$	−20.2130	−0.856454	−0.428227	+	0.903671i	$0.640862\pi$
$558$	0	0
$559$	−0.509620	−0.0215546
$560$	−0.0410810	−0.00173599
$561$	0	0
$562$	−7.41800	−0.312909
$563$	3.95448	0.166662	0.0833308	−	0.996522i	$-0.473444\pi$
$563$	3.95448	0.166662	0.0833308	+	0.996522i	$0.473444\pi$
$564$	0	0
$565$	−6.40052	−0.269272
$566$	21.1335	0.888305
$567$	0	0
$568$	10.2613	0.430554
$569$	−17.7277	−0.743182	−0.371591	−	0.928396i	$-0.621188\pi$
$569$	−17.7277	−0.743182	−0.371591	+	0.928396i	$0.621188\pi$
$570$	0	0
$571$	−4.10744	−0.171891	−0.0859455	−	0.996300i	$-0.527391\pi$
$571$	−4.10744	−0.171891	−0.0859455	+	0.996300i	$0.527391\pi$
$572$	6.88076	0.287699
$573$	0	0
$574$	−0.00601457	−0.000251043 0
$575$	1.69859	0.0708361
$576$	0	0
$577$	−38.2521	−1.59245	−0.796227	−	0.604998i	$-0.793174\pi$
$577$	−38.2521	−1.59245	−0.796227	+	0.604998i	$0.793174\pi$
$578$	0.543249	0.0225962
$579$	0	0
$580$	−7.89887	−0.327983
$581$	−0.243625	−0.0101073
$582$	0	0
$583$	40.9195	1.69471
$584$	−5.21734	−0.215895
$585$	0	0
$586$	5.04458	0.208390
$587$	26.6533	1.10010	0.550050	−	0.835131i	$-0.314609\pi$
$587$	26.6533	1.10010	0.550050	+	0.835131i	$0.314609\pi$
$588$	0	0
$589$	2.56629	0.105742
$590$	−25.6247	−1.05495
$591$	0	0
$592$	−8.66851	−0.356274
$593$	−32.6922	−1.34251	−0.671254	−	0.741228i	$-0.734244\pi$
$593$	−32.6922	−1.34251	−0.671254	+	0.741228i	$0.734244\pi$
$594$	0	0
$595$	0.166653	0.00683211
$596$	1.00000	0.0409616
$597$	0	0
$598$	2.15687	0.0882008
$599$	−39.6511	−1.62010	−0.810050	−	0.586361i	$-0.800560\pi$
$599$	−39.6511	−1.62010	−0.810050	+	0.586361i	$0.800560\pi$
$600$	0	0
$601$	−22.2768	−0.908688	−0.454344	−	0.890826i	$-0.650126\pi$
$601$	−22.2768	−0.908688	−0.454344	+	0.890826i	$0.650126\pi$
$602$	−0.00803414	−0.000327447 0
$603$	0	0
$604$	−13.5614	−0.551806
$605$	33.2358	1.35123
$606$	0	0
$607$	−0.442899	−0.0179767	−0.00898836	−	0.999960i	$-0.502861\pi$
$607$	−0.442899	−0.0179767	−0.00898836	+	0.999960i	$0.502861\pi$
$608$	−0.295188	−0.0119714
$609$	0	0
$610$	−14.7586	−0.597559
$611$	5.97693	0.241801
$612$	0	0
$613$	23.1291	0.934177	0.467089	−	0.884211i	$-0.345303\pi$
$613$	23.1291	0.934177	0.467089	+	0.884211i	$0.345303\pi$
$614$	4.06035	0.163862
$615$	0	0
$616$	0.108475	0.00437058
$617$	18.6553	0.751033	0.375517	−	0.926816i	$-0.377465\pi$
$617$	18.6553	0.751033	0.375517	+	0.926816i	$0.377465\pi$
$618$	0	0
$619$	−17.4442	−0.701142	−0.350571	−	0.936536i	$-0.614012\pi$
$619$	−17.4442	−0.701142	−0.350571	+	0.936536i	$0.614012\pi$
$620$	17.3223	0.695679
$621$	0	0
$622$	30.1221	1.20778
$623$	−0.235074	−0.00941803
$624$	0	0
$625$	−18.7895	−0.751579
$626$	−5.86146	−0.234271
$627$	0	0
$628$	−11.9231	−0.475782
$629$	35.1655	1.40214
$630$	0	0
$631$	37.4513	1.49091	0.745456	−	0.666555i	$-0.232232\pi$
$631$	37.4513	1.49091	0.745456	+	0.666555i	$0.232232\pi$
$632$	−5.69527	−0.226546
$633$	0	0
$634$	13.0792	0.519442
$635$	28.1473	1.11699
$636$	0	0
$637$	−9.15423	−0.362704
$638$	20.8571	0.825740
$639$	0	0
$640$	−1.99250	−0.0787604
$641$	−5.73239	−0.226416	−0.113208	−	0.993571i	$-0.536113\pi$
$641$	−5.73239	−0.226416	−0.113208	+	0.993571i	$0.536113\pi$
$642$	0	0
$643$	−5.32139	−0.209855	−0.104928	−	0.994480i	$-0.533461\pi$
$643$	−5.32139	−0.209855	−0.104928	+	0.994480i	$0.533461\pi$
$644$	0.0340029	0.00133990
$645$	0	0
$646$	1.19749	0.0471144
$647$	27.7330	1.09030	0.545148	−	0.838340i	$-0.316474\pi$
$647$	27.7330	1.09030	0.545148	+	0.838340i	$0.316474\pi$
$648$	0	0
$649$	67.6622	2.65598
$650$	1.34699	0.0528334
$651$	0	0
$652$	−2.75467	−0.107881
$653$	10.3642	0.405584	0.202792	−	0.979222i	$-0.434998\pi$
$653$	10.3642	0.405584	0.202792	+	0.979222i	$0.434998\pi$
$654$	0	0
$655$	35.4559	1.38538
$656$	−0.291717	−0.0113896
$657$	0	0
$658$	0.0942260	0.00367331
$659$	−36.1988	−1.41011	−0.705053	−	0.709155i	$-0.749077\pi$
$659$	−36.1988	−1.41011	−0.705053	+	0.709155i	$0.749077\pi$
$660$	0	0
$661$	43.6839	1.69911	0.849553	−	0.527503i	$-0.176872\pi$
$661$	43.6839	1.69911	0.849553	+	0.527503i	$0.176872\pi$
$662$	−11.3159	−0.439804
$663$	0	0
$664$	−11.8162	−0.458558
$665$	−0.0121266	−0.000470250 0
$666$	0	0
$667$	6.53793	0.253150
$668$	21.6245	0.836678
$669$	0	0
$670$	−11.8788	−0.458920
$671$	38.9703	1.50443
$672$	0	0
$673$	−40.3949	−1.55711	−0.778555	−	0.627576i	$-0.784047\pi$
$673$	−40.3949	−1.55711	−0.778555	+	0.627576i	$0.784047\pi$
$674$	−3.41103	−0.131388
$675$	0	0
$676$	−11.2896	−0.434215
$677$	14.3060	0.549823	0.274912	−	0.961469i	$-0.411351\pi$
$677$	14.3060	0.549823	0.274912	+	0.961469i	$0.411351\pi$
$678$	0	0
$679$	0.00347861	0.000133497 0
$680$	8.08295	0.309967
$681$	0	0
$682$	−45.7397	−1.75146
$683$	−2.76087	−0.105642	−0.0528209	−	0.998604i	$-0.516821\pi$
$683$	−2.76087	−0.105642	−0.0528209	+	0.998604i	$0.516821\pi$
$684$	0	0
$685$	7.56127	0.288901
$686$	−0.288641	−0.0110204
$687$	0	0
$688$	−0.389669	−0.0148560
$689$	10.1717	0.387512
$690$	0	0
$691$	1.81589	0.0690797	0.0345399	−	0.999403i	$-0.489003\pi$
$691$	1.81589	0.0690797	0.0345399	+	0.999403i	$0.489003\pi$
$692$	17.8683	0.679252
$693$	0	0
$694$	−5.91948	−0.224700
$695$	−26.2178	−0.994497
$696$	0	0
$697$	1.18340	0.0448247
$698$	−11.2779	−0.426875
$699$	0	0
$700$	0.0212353	0.000802619 0
$701$	−44.8307	−1.69323	−0.846617	−	0.532203i	$-0.821364\pi$
$701$	−44.8307	−1.69323	−0.846617	+	0.532203i	$0.821364\pi$
$702$	0	0
$703$	−2.55884	−0.0965084
$704$	5.26122	0.198290
$705$	0	0
$706$	−14.0935	−0.530417
$707$	−0.309356	−0.0116345
$708$	0	0
$709$	19.6903	0.739485	0.369742	−	0.929134i	$-0.379446\pi$
$709$	19.6903	0.739485	0.369742	+	0.929134i	$0.379446\pi$
$710$	20.4456	0.767310
$711$	0	0
$712$	−11.4015	−0.427288
$713$	−14.3377	−0.536951
$714$	0	0
$715$	13.7099	0.512722
$716$	14.6929	0.549098
$717$	0	0
$718$	−3.81544	−0.142391
$719$	38.2650	1.42704	0.713522	−	0.700632i	$-0.247099\pi$
$719$	38.2650	1.42704	0.713522	+	0.700632i	$0.247099\pi$
$720$	0	0
$721$	0.0453573	0.00168920
$722$	18.9129	0.703864
$723$	0	0
$724$	20.5258	0.762834
$725$	4.08303	0.151640
$726$	0	0
$727$	−27.4811	−1.01922	−0.509610	−	0.860406i	$-0.670210\pi$
$727$	−27.4811	−1.01922	−0.509610	+	0.860406i	$0.670210\pi$
$728$	0.0269646	0.000999373 0
$729$	0	0
$730$	−10.3955	−0.384756
$731$	1.58077	0.0584668
$732$	0	0
$733$	10.4364	0.385477	0.192738	−	0.981250i	$-0.438263\pi$
$733$	10.4364	0.385477	0.192738	+	0.981250i	$0.438263\pi$
$734$	16.4625	0.607643
$735$	0	0
$736$	1.64920	0.0607903
$737$	31.3663	1.15539
$738$	0	0
$739$	27.0604	0.995432	0.497716	−	0.867340i	$-0.334172\pi$
$739$	27.0604	0.995432	0.497716	+	0.867340i	$0.334172\pi$
$740$	−17.2720	−0.634931
$741$	0	0
$742$	0.160357	0.00588688
$743$	−8.47771	−0.311017	−0.155508	−	0.987835i	$-0.549702\pi$
$743$	−8.47771	−0.311017	−0.155508	+	0.987835i	$0.549702\pi$
$744$	0	0
$745$	1.99250	0.0729995
$746$	−20.1014	−0.735965
$747$	0	0
$748$	−21.3432	−0.780383
$749$	0.0991556	0.00362307
$750$	0	0
$751$	0.633305	0.0231096	0.0115548	−	0.999933i	$-0.496322\pi$
$751$	0.633305	0.0231096	0.0115548	+	0.999933i	$0.496322\pi$
$752$	4.57012	0.166655
$753$	0	0
$754$	5.18462	0.188813
$755$	−27.0211	−0.983399
$756$	0	0
$757$	−16.7917	−0.610304	−0.305152	−	0.952304i	$-0.598707\pi$
$757$	−16.7917	−0.610304	−0.305152	+	0.952304i	$0.598707\pi$
$758$	0.0507386	0.00184291
$759$	0	0
$760$	−0.588161	−0.0213348
$761$	15.9360	0.577681	0.288841	−	0.957377i	$-0.406730\pi$
$761$	15.9360	0.577681	0.288841	+	0.957377i	$0.406730\pi$
$762$	0	0
$763$	0.264801	0.00958643
$764$	16.1527	0.584384
$765$	0	0
$766$	24.2127	0.874840
$767$	16.8194	0.607313
$768$	0	0
$769$	16.2867	0.587315	0.293657	−	0.955911i	$-0.405127\pi$
$769$	16.2867	0.587315	0.293657	+	0.955911i	$0.405127\pi$
$770$	0.216136	0.00778901
$771$	0	0
$772$	0.667178	0.0240123
$773$	47.5479	1.71018	0.855090	−	0.518480i	$-0.173502\pi$
$773$	47.5479	1.71018	0.855090	+	0.518480i	$0.173502\pi$
$774$	0	0
$775$	−8.95410	−0.321641
$776$	0.168718	0.00605664
$777$	0	0
$778$	16.8468	0.603988
$779$	−0.0861112	−0.00308525
$780$	0	0
$781$	−53.9869	−1.93180
$782$	−6.69029	−0.239244
$783$	0	0
$784$	−6.99957	−0.249985
$785$	−23.7567	−0.847912
$786$	0	0
$787$	49.1312	1.75134	0.875668	−	0.482913i	$-0.160421\pi$
$787$	49.1312	1.75134	0.875668	+	0.482913i	$0.160421\pi$
$788$	2.10056	0.0748295
$789$	0	0
$790$	−11.3478	−0.403737
$791$	0.0662309	0.00235490
$792$	0	0
$793$	9.68719	0.344002
$794$	−25.7132	−0.912527
$795$	0	0
$796$	−17.4910	−0.619951
$797$	41.7653	1.47940	0.739701	−	0.672936i	$-0.234967\pi$
$797$	41.7653	1.47940	0.739701	+	0.672936i	$0.234967\pi$
$798$	0	0
$799$	−18.5396	−0.655883
$800$	1.02995	0.0364142
$801$	0	0
$802$	18.5979	0.656713
$803$	27.4496	0.968675
$804$	0	0
$805$	0.0677508	0.00238790
$806$	−11.3699	−0.400488
$807$	0	0
$808$	−15.0043	−0.527849
$809$	33.4974	1.17771	0.588853	−	0.808240i	$-0.299580\pi$
$809$	33.4974	1.17771	0.588853	+	0.808240i	$0.299580\pi$
$810$	0	0
$811$	24.2531	0.851641	0.425821	−	0.904808i	$-0.359985\pi$
$811$	24.2531	0.851641	0.425821	+	0.904808i	$0.359985\pi$
$812$	0.0817354	0.00286835
$813$	0	0
$814$	45.6069	1.59852
$815$	−5.48869	−0.192260
$816$	0	0
$817$	−0.115026	−0.00402424
$818$	−36.7789	−1.28594
$819$	0	0
$820$	−0.581245	−0.0202980
$821$	32.0387	1.11816	0.559080	−	0.829114i	$-0.311154\pi$
$821$	32.0387	1.11816	0.559080	+	0.829114i	$0.311154\pi$
$822$	0	0
$823$	19.6719	0.685718	0.342859	−	0.939387i	$-0.388605\pi$
$823$	19.6719	0.685718	0.342859	+	0.939387i	$0.388605\pi$
$824$	2.19991	0.0766374
$825$	0	0
$826$	0.265157	0.00922599
$827$	31.4913	1.09506	0.547529	−	0.836786i	$-0.315568\pi$
$827$	31.4913	1.09506	0.547529	+	0.836786i	$0.315568\pi$
$828$	0	0
$829$	11.9019	0.413371	0.206686	−	0.978407i	$-0.433732\pi$
$829$	11.9019	0.413371	0.206686	+	0.978407i	$0.433732\pi$
$830$	−23.5438	−0.817217
$831$	0	0
$832$	1.30783	0.0453407
$833$	28.3951	0.983833
$834$	0	0
$835$	43.0868	1.49108
$836$	1.55305	0.0537133
$837$	0	0
$838$	−9.76046	−0.337170
$839$	12.7598	0.440517	0.220258	−	0.975442i	$-0.429310\pi$
$839$	12.7598	0.440517	0.220258	+	0.975442i	$0.429310\pi$
$840$	0	0
$841$	−13.2843	−0.458079
$842$	32.0064	1.10301
$843$	0	0
$844$	−6.81415	−0.234553
$845$	−22.4945	−0.773834
$846$	0	0
$847$	−0.343915	−0.0118170
$848$	7.77757	0.267083
$849$	0	0
$850$	−4.17818	−0.143310
$851$	14.2961	0.490064
$852$	0	0
$853$	32.8050	1.12322	0.561612	−	0.827401i	$-0.310182\pi$
$853$	32.8050	1.12322	0.561612	+	0.827401i	$0.310182\pi$
$854$	0.152718	0.00522591
$855$	0	0
$856$	4.80921	0.164375
$857$	−7.56418	−0.258388	−0.129194	−	0.991619i	$-0.541239\pi$
$857$	−7.56418	−0.258388	−0.129194	+	0.991619i	$0.541239\pi$
$858$	0	0
$859$	−52.5155	−1.79181	−0.895904	−	0.444249i	$-0.853471\pi$
$859$	−52.5155	−1.79181	−0.895904	+	0.444249i	$0.853471\pi$
$860$	−0.776416	−0.0264756
$861$	0	0
$862$	−33.5502	−1.14272
$863$	35.4413	1.20644	0.603218	−	0.797576i	$-0.293885\pi$
$863$	35.4413	1.20644	0.603218	+	0.797576i	$0.293885\pi$
$864$	0	0
$865$	35.6026	1.21053
$866$	−35.8412	−1.21793
$867$	0	0
$868$	−0.179246	−0.00608401
$869$	29.9641	1.01646
$870$	0	0
$871$	7.79698	0.264191
$872$	12.8433	0.434928
$873$	0	0
$874$	0.486823	0.0164670
$875$	0.247716	0.00837434
$876$	0	0
$877$	36.4135	1.22960	0.614799	−	0.788684i	$-0.289237\pi$
$877$	36.4135	1.22960	0.614799	+	0.788684i	$0.289237\pi$
$878$	4.89036	0.165042
$879$	0	0
$880$	10.4830	0.353381
$881$	−56.1916	−1.89314	−0.946572	−	0.322492i	$-0.895479\pi$
$881$	−56.1916	−1.89314	−0.946572	+	0.322492i	$0.895479\pi$
$882$	0	0
$883$	26.3628	0.887180	0.443590	−	0.896230i	$-0.353705\pi$
$883$	26.3628	0.887180	0.443590	+	0.896230i	$0.353705\pi$
$884$	−5.30545	−0.178442
$885$	0	0
$886$	6.82947	0.229440
$887$	−51.2228	−1.71989	−0.859947	−	0.510383i	$-0.829504\pi$
$887$	−51.2228	−1.71989	−0.859947	+	0.510383i	$0.829504\pi$
$888$	0	0
$889$	−0.291261	−0.00976857
$890$	−22.7174	−0.761489
$891$	0	0
$892$	−19.9915	−0.669366
$893$	1.34904	0.0451440
$894$	0	0
$895$	29.2755	0.978573
$896$	0.0206178	0.000688794 0
$897$	0	0
$898$	−29.7864	−0.993986
$899$	−34.4646	−1.14946
$900$	0	0
$901$	−31.5512	−1.05112
$902$	1.53479	0.0511028
$903$	0	0
$904$	3.21231	0.106840
$905$	40.8975	1.35948
$906$	0	0
$907$	−12.8286	−0.425966	−0.212983	−	0.977056i	$-0.568318\pi$
$907$	−12.8286	−0.425966	−0.212983	+	0.977056i	$0.568318\pi$
$908$	−13.5705	−0.450353
$909$	0	0
$910$	0.0537269	0.00178103
$911$	8.07943	0.267683	0.133842	−	0.991003i	$-0.457269\pi$
$911$	8.07943	0.267683	0.133842	+	0.991003i	$0.457269\pi$
$912$	0	0
$913$	62.1677	2.05745
$914$	−4.05153	−0.134013
$915$	0	0
$916$	27.0742	0.894558
$917$	−0.366888	−0.0121157
$918$	0	0
$919$	−39.1276	−1.29070	−0.645350	−	0.763887i	$-0.723288\pi$
$919$	−39.1276	−1.29070	−0.645350	+	0.763887i	$0.723288\pi$
$920$	3.28603	0.108337
$921$	0	0
$922$	−39.8347	−1.31189
$923$	−13.4200	−0.441724
$924$	0	0
$925$	8.92811	0.293555
$926$	22.9167	0.753090
$927$	0	0
$928$	3.96430	0.130135
$929$	27.9350	0.916518	0.458259	−	0.888819i	$-0.348473\pi$
$929$	27.9350	0.916518	0.458259	+	0.888819i	$0.348473\pi$
$930$	0	0
$931$	−2.06619	−0.0677166
$932$	1.11238	0.0364371
$933$	0	0
$934$	−1.53647	−0.0502748
$935$	−42.5262	−1.39076
$936$	0	0
$937$	−13.7977	−0.450751	−0.225375	−	0.974272i	$-0.572361\pi$
$937$	−13.7977	−0.450751	−0.225375	+	0.974272i	$0.572361\pi$
$938$	0.122919	0.00401345
$939$	0	0
$940$	9.10596	0.297004
$941$	40.3166	1.31428	0.657142	−	0.753767i	$-0.271765\pi$
$941$	40.3166	1.31428	0.657142	+	0.753767i	$0.271765\pi$
$942$	0	0
$943$	0.481099	0.0156667
$944$	12.8606	0.418576
$945$	0	0
$946$	2.05014	0.0666557
$947$	5.20950	0.169286	0.0846431	−	0.996411i	$-0.473025\pi$
$947$	5.20950	0.169286	0.0846431	+	0.996411i	$0.473025\pi$
$948$	0	0
$949$	6.82338	0.221496
$950$	0.304028	0.00986397
$951$	0	0
$952$	−0.0836402	−0.00271080
$953$	13.5440	0.438733	0.219367	−	0.975642i	$-0.429601\pi$
$953$	13.5440	0.438733	0.219367	+	0.975642i	$0.429601\pi$
$954$	0	0
$955$	32.1843	1.04146
$956$	−21.3359	−0.690053
$957$	0	0
$958$	−22.1043	−0.714157
$959$	−0.0782419	−0.00252656
$960$	0	0
$961$	44.5811	1.43810
$962$	11.3369	0.365517
$963$	0	0
$964$	7.94889	0.256016
$965$	1.32935	0.0427933
$966$	0	0
$967$	−29.4315	−0.946452	−0.473226	−	0.880941i	$-0.656911\pi$
$967$	−29.4315	−0.946452	−0.473226	+	0.880941i	$0.656911\pi$
$968$	−16.6804	−0.536130
$969$	0	0
$970$	0.336171	0.0107938
$971$	−37.1081	−1.19086	−0.595428	−	0.803409i	$-0.703017\pi$
$971$	−37.1081	−1.19086	−0.595428	+	0.803409i	$0.703017\pi$
$972$	0	0
$973$	0.271294	0.00869730
$974$	34.8299	1.11602
$975$	0	0
$976$	7.40709	0.237095
$977$	−44.0988	−1.41084	−0.705422	−	0.708787i	$-0.749243\pi$
$977$	−44.0988	−1.41084	−0.705422	+	0.708787i	$0.749243\pi$
$978$	0	0
$979$	59.9856	1.91715
$980$	−13.9466	−0.445509
$981$	0	0
$982$	12.6656	0.404175
$983$	−57.9838	−1.84940	−0.924699	−	0.380700i	$-0.875683\pi$
$983$	−57.9838	−1.84940	−0.924699	+	0.380700i	$0.875683\pi$
$984$	0	0
$985$	4.18537	0.133357
$986$	−16.0820	−0.512154
$987$	0	0
$988$	0.386054	0.0122820
$989$	0.642642	0.0204348
$990$	0	0
$991$	−0.591718	−0.0187965	−0.00939827	−	0.999956i	$-0.502992\pi$
$991$	−0.591718	−0.0187965	−0.00939827	+	0.999956i	$0.502992\pi$
$992$	−8.69374	−0.276027
$993$	0	0
$994$	−0.211566	−0.00671045
$995$	−34.8507	−1.10484
$996$	0	0
$997$	9.30402	0.294661	0.147331	−	0.989087i	$-0.452932\pi$
$997$	9.30402	0.294661	0.147331	+	0.989087i	$0.452932\pi$
$998$	−17.1306	−0.542260
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.l.1.10	✓	12
3.2	odd	2		8046.2.a.m.1.3	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.l.1.10	✓	12	1.1	even	1	trivial
8046.2.a.m.1.3	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.99250 q^{5} -0.0206178 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.99250 q^{5} -0.0206178 q^{7} -1.00000 q^{8} -1.99250 q^{10} +5.26122 q^{11} +1.30783 q^{13} +0.0206178 q^{14} +1.00000 q^{16} -4.05669 q^{17} +0.295188 q^{19} +1.99250 q^{20} -5.26122 q^{22} -1.64920 q^{23} -1.02995 q^{25} -1.30783 q^{26} -0.0206178 q^{28} -3.96430 q^{29} +8.69374 q^{31} -1.00000 q^{32} +4.05669 q^{34} -0.0410810 q^{35} -8.66851 q^{37} -0.295188 q^{38} -1.99250 q^{40} -0.291717 q^{41} -0.389669 q^{43} +5.26122 q^{44} +1.64920 q^{46} +4.57012 q^{47} -6.99957 q^{49} +1.02995 q^{50} +1.30783 q^{52} +7.77757 q^{53} +10.4830 q^{55} +0.0206178 q^{56} +3.96430 q^{58} +12.8606 q^{59} +7.40709 q^{61} -8.69374 q^{62} +1.00000 q^{64} +2.60584 q^{65} +5.96178 q^{67} -4.05669 q^{68} +0.0410810 q^{70} -10.2613 q^{71} +5.21734 q^{73} +8.66851 q^{74} +0.295188 q^{76} -0.108475 q^{77} +5.69527 q^{79} +1.99250 q^{80} +0.291717 q^{82} +11.8162 q^{83} -8.08295 q^{85} +0.389669 q^{86} -5.26122 q^{88} +11.4015 q^{89} -0.0269646 q^{91} -1.64920 q^{92} -4.57012 q^{94} +0.588161 q^{95} -0.168718 q^{97} +6.99957 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{2} + 12 q^{4} + 5 q^{5} - 6 q^{7} - 12 q^{8}+O(q^{10}) \) 12 * q - 12 * q^2 + 12 * q^4 + 5 * q^5 - 6 * q^7 - 12 * q^8	\( 12 q - 12 q^{2} + 12 q^{4} + 5 q^{5} - 6 q^{7} - 12 q^{8} - 5 q^{10} + 10 q^{11} - q^{13} + 6 q^{14} + 12 q^{16} + 6 q^{17} - 10 q^{19} + 5 q^{20} - 10 q^{22} + 15 q^{23} + 7 q^{25} + q^{26} - 6 q^{28} + 33 q^{29} - 6 q^{31} - 12 q^{32} - 6 q^{34} + 16 q^{35} - 13 q^{37} + 10 q^{38} - 5 q^{40} + 20 q^{41} - 11 q^{43} + 10 q^{44} - 15 q^{46} + 15 q^{47} + 2 q^{49} - 7 q^{50} - q^{52} + 4 q^{53} - 17 q^{55} + 6 q^{56} - 33 q^{58} + 10 q^{59} - 12 q^{61} + 6 q^{62} + 12 q^{64} + 40 q^{65} - 19 q^{67} + 6 q^{68} - 16 q^{70} + 47 q^{71} - 2 q^{73} + 13 q^{74} - 10 q^{76} - 6 q^{77} - 15 q^{79} + 5 q^{80} - 20 q^{82} + 18 q^{83} - 25 q^{85} + 11 q^{86} - 10 q^{88} + 24 q^{89} - 3 q^{91} + 15 q^{92} - 15 q^{94} - 3 q^{95} - 25 q^{97} - 2 q^{98}+O(q^{100}) \) 12 * q - 12 * q^2 + 12 * q^4 + 5 * q^5 - 6 * q^7 - 12 * q^8 - 5 * q^10 + 10 * q^11 - q^13 + 6 * q^14 + 12 * q^16 + 6 * q^17 - 10 * q^19 + 5 * q^20 - 10 * q^22 + 15 * q^23 + 7 * q^25 + q^26 - 6 * q^28 + 33 * q^29 - 6 * q^31 - 12 * q^32 - 6 * q^34 + 16 * q^35 - 13 * q^37 + 10 * q^38 - 5 * q^40 + 20 * q^41 - 11 * q^43 + 10 * q^44 - 15 * q^46 + 15 * q^47 + 2 * q^49 - 7 * q^50 - q^52 + 4 * q^53 - 17 * q^55 + 6 * q^56 - 33 * q^58 + 10 * q^59 - 12 * q^61 + 6 * q^62 + 12 * q^64 + 40 * q^65 - 19 * q^67 + 6 * q^68 - 16 * q^70 + 47 * q^71 - 2 * q^73 + 13 * q^74 - 10 * q^76 - 6 * q^77 - 15 * q^79 + 5 * q^80 - 20 * q^82 + 18 * q^83 - 25 * q^85 + 11 * q^86 - 10 * q^88 + 24 * q^89 - 3 * q^91 + 15 * q^92 - 15 * q^94 - 3 * q^95 - 25 * q^97 - 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

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Database

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