LMFDB - Embedded newform 8046.2.a.f.1.2

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:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 17x^{6} - 2x^{5} + 71x^{4} - 18x^{3} - 81x^{2} + 36x + 9 $ x^8 - 17x^6 - 2x^5 + 71x^4 - 18x^3 - 81x^2 + 36x + 9
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			1.2
Root			$1.40004$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -1.40004 q^{5} -2.13611 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} -1.40004 q^{5} -2.13611 q^{7} +1.00000 q^{8} -1.40004 q^{10} -2.83452 q^{11} +5.71214 q^{13} -2.13611 q^{14} +1.00000 q^{16} +5.87290 q^{17} -3.20948 q^{19} -1.40004 q^{20} -2.83452 q^{22} -3.97313 q^{23} -3.03987 q^{25} +5.71214 q^{26} -2.13611 q^{28} +0.100627 q^{29} -5.43523 q^{31} +1.00000 q^{32} +5.87290 q^{34} +2.99065 q^{35} +6.63998 q^{37} -3.20948 q^{38} -1.40004 q^{40} -1.02469 q^{41} -3.45987 q^{43} -2.83452 q^{44} -3.97313 q^{46} +4.46817 q^{47} -2.43702 q^{49} -3.03987 q^{50} +5.71214 q^{52} +9.91571 q^{53} +3.96845 q^{55} -2.13611 q^{56} +0.100627 q^{58} -9.99566 q^{59} -0.434578 q^{61} -5.43523 q^{62} +1.00000 q^{64} -7.99726 q^{65} -9.79274 q^{67} +5.87290 q^{68} +2.99065 q^{70} +11.9919 q^{71} -11.4593 q^{73} +6.63998 q^{74} -3.20948 q^{76} +6.05484 q^{77} -0.914747 q^{79} -1.40004 q^{80} -1.02469 q^{82} -9.47992 q^{83} -8.22232 q^{85} -3.45987 q^{86} -2.83452 q^{88} +8.06705 q^{89} -12.2018 q^{91} -3.97313 q^{92} +4.46817 q^{94} +4.49342 q^{95} +0.417684 q^{97} -2.43702 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{2} + 8 q^{4} - 5 q^{7} + 8 q^{8}+O(q^{10}) $ 8 * q + 8 * q^2 + 8 * q^4 - 5 * q^7 + 8 * q^8	$ 8 q + 8 q^{2} + 8 q^{4} - 5 q^{7} + 8 q^{8} - 6 q^{11} - 8 q^{13} - 5 q^{14} + 8 q^{16} + 5 q^{17} - 14 q^{19} - 6 q^{22} - 21 q^{23} - 6 q^{25} - 8 q^{26} - 5 q^{28} - 3 q^{29} - 4 q^{31} + 8 q^{32} + 5 q^{34} + 2 q^{35} - 3 q^{37} - 14 q^{38} - 7 q^{41} - 12 q^{43} - 6 q^{44} - 21 q^{46} - 25 q^{47} - 7 q^{49} - 6 q^{50} - 8 q^{52} - 3 q^{53} - 9 q^{55} - 5 q^{56} - 3 q^{58} - 2 q^{59} - 17 q^{61} - 4 q^{62} + 8 q^{64} - 32 q^{65} - 14 q^{67} + 5 q^{68} + 2 q^{70} - 7 q^{71} - 10 q^{73} - 3 q^{74} - 14 q^{76} - 12 q^{77} - 33 q^{79} - 7 q^{82} - 13 q^{83} - 33 q^{85} - 12 q^{86} - 6 q^{88} + 22 q^{89} - 22 q^{91} - 21 q^{92} - 25 q^{94} + 14 q^{95} - 11 q^{97} - 7 q^{98}+O(q^{100}) $ 8 * q + 8 * q^2 + 8 * q^4 - 5 * q^7 + 8 * q^8 - 6 * q^11 - 8 * q^13 - 5 * q^14 + 8 * q^16 + 5 * q^17 - 14 * q^19 - 6 * q^22 - 21 * q^23 - 6 * q^25 - 8 * q^26 - 5 * q^28 - 3 * q^29 - 4 * q^31 + 8 * q^32 + 5 * q^34 + 2 * q^35 - 3 * q^37 - 14 * q^38 - 7 * q^41 - 12 * q^43 - 6 * q^44 - 21 * q^46 - 25 * q^47 - 7 * q^49 - 6 * q^50 - 8 * q^52 - 3 * q^53 - 9 * q^55 - 5 * q^56 - 3 * q^58 - 2 * q^59 - 17 * q^61 - 4 * q^62 + 8 * q^64 - 32 * q^65 - 14 * q^67 + 5 * q^68 + 2 * q^70 - 7 * q^71 - 10 * q^73 - 3 * q^74 - 14 * q^76 - 12 * q^77 - 33 * q^79 - 7 * q^82 - 13 * q^83 - 33 * q^85 - 12 * q^86 - 6 * q^88 + 22 * q^89 - 22 * q^91 - 21 * q^92 - 25 * q^94 + 14 * q^95 - 11 * q^97 - 7 * 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
$5$	−1.40004	−0.626119	−0.313060	−	0.949733i	$-0.601354\pi$
$5$	−1.40004	−0.626119	−0.313060	+	0.949733i	$0.601354\pi$
$6$	0	0
$7$	−2.13611	−0.807375	−0.403687	−	0.914897i	$-0.632272\pi$
$7$	−2.13611	−0.807375	−0.403687	+	0.914897i	$0.632272\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−1.40004	−0.442733
$11$	−2.83452	−0.854638	−0.427319	−	0.904101i	$-0.640542\pi$
$11$	−2.83452	−0.854638	−0.427319	+	0.904101i	$0.640542\pi$
$12$	0	0
$13$	5.71214	1.58426	0.792132	−	0.610350i	$-0.208971\pi$
$13$	5.71214	1.58426	0.792132	+	0.610350i	$0.208971\pi$
$14$	−2.13611	−0.570900
$15$	0	0
$16$	1.00000	0.250000
$17$	5.87290	1.42439	0.712193	−	0.701983i	$-0.247702\pi$
$17$	5.87290	1.42439	0.712193	+	0.701983i	$0.247702\pi$
$18$	0	0
$19$	−3.20948	−0.736306	−0.368153	−	0.929765i	$-0.620010\pi$
$19$	−3.20948	−0.736306	−0.368153	+	0.929765i	$0.620010\pi$
$20$	−1.40004	−0.313060
$21$	0	0
$22$	−2.83452	−0.604321
$23$	−3.97313	−0.828455	−0.414228	−	0.910173i	$-0.635948\pi$
$23$	−3.97313	−0.828455	−0.414228	+	0.910173i	$0.635948\pi$
$24$	0	0
$25$	−3.03987	−0.607975
$26$	5.71214	1.12024
$27$	0	0
$28$	−2.13611	−0.403687
$29$	0.100627	0.0186859	0.00934295	−	0.999956i	$-0.497026\pi$
$29$	0.100627	0.0186859	0.00934295	+	0.999956i	$0.497026\pi$
$30$	0	0
$31$	−5.43523	−0.976197	−0.488098	−	0.872789i	$-0.662309\pi$
$31$	−5.43523	−0.976197	−0.488098	+	0.872789i	$0.662309\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	5.87290	1.00719
$35$	2.99065	0.505513
$36$	0	0
$37$	6.63998	1.09161	0.545803	−	0.837913i	$-0.316225\pi$
$37$	6.63998	1.09161	0.545803	+	0.837913i	$0.316225\pi$
$38$	−3.20948	−0.520647
$39$	0	0
$40$	−1.40004	−0.221367
$41$	−1.02469	−0.160030	−0.0800148	−	0.996794i	$-0.525497\pi$
$41$	−1.02469	−0.160030	−0.0800148	+	0.996794i	$0.525497\pi$
$42$	0	0
$43$	−3.45987	−0.527626	−0.263813	−	0.964574i	$-0.584980\pi$
$43$	−3.45987	−0.527626	−0.263813	+	0.964574i	$0.584980\pi$
$44$	−2.83452	−0.427319
$45$	0	0
$46$	−3.97313	−0.585806
$47$	4.46817	0.651749	0.325875	−	0.945413i	$-0.394341\pi$
$47$	4.46817	0.651749	0.325875	+	0.945413i	$0.394341\pi$
$48$	0	0
$49$	−2.43702	−0.348146
$50$	−3.03987	−0.429903
$51$	0	0
$52$	5.71214	0.792132
$53$	9.91571	1.36203	0.681014	−	0.732270i	$-0.261539\pi$
$53$	9.91571	1.36203	0.681014	+	0.732270i	$0.261539\pi$
$54$	0	0
$55$	3.96845	0.535105
$56$	−2.13611	−0.285450
$57$	0	0
$58$	0.100627	0.0132129
$59$	−9.99566	−1.30132	−0.650662	−	0.759368i	$-0.725508\pi$
$59$	−9.99566	−1.30132	−0.650662	+	0.759368i	$0.725508\pi$
$60$	0	0
$61$	−0.434578	−0.0556420	−0.0278210	−	0.999613i	$-0.508857\pi$
$61$	−0.434578	−0.0556420	−0.0278210	+	0.999613i	$0.508857\pi$
$62$	−5.43523	−0.690275
$63$	0	0
$64$	1.00000	0.125000
$65$	−7.99726	−0.991938
$66$	0	0
$67$	−9.79274	−1.19637	−0.598187	−	0.801356i	$-0.704112\pi$
$67$	−9.79274	−1.19637	−0.598187	+	0.801356i	$0.704112\pi$
$68$	5.87290	0.712193
$69$	0	0
$70$	2.99065	0.357451
$71$	11.9919	1.42318	0.711590	−	0.702595i	$-0.247975\pi$
$71$	11.9919	1.42318	0.711590	+	0.702595i	$0.247975\pi$
$72$	0	0
$73$	−11.4593	−1.34121	−0.670603	−	0.741816i	$-0.733965\pi$
$73$	−11.4593	−1.34121	−0.670603	+	0.741816i	$0.733965\pi$
$74$	6.63998	0.771882
$75$	0	0
$76$	−3.20948	−0.368153
$77$	6.05484	0.690013
$78$	0	0
$79$	−0.914747	−0.102917	−0.0514585	−	0.998675i	$-0.516387\pi$
$79$	−0.914747	−0.102917	−0.0514585	+	0.998675i	$0.516387\pi$
$80$	−1.40004	−0.156530
$81$	0	0
$82$	−1.02469	−0.113158
$83$	−9.47992	−1.04056	−0.520278	−	0.853997i	$-0.674172\pi$
$83$	−9.47992	−1.04056	−0.520278	+	0.853997i	$0.674172\pi$
$84$	0	0
$85$	−8.22232	−0.891836
$86$	−3.45987	−0.373088
$87$	0	0
$88$	−2.83452	−0.302160
$89$	8.06705	0.855106	0.427553	−	0.903990i	$-0.359376\pi$
$89$	8.06705	0.855106	0.427553	+	0.903990i	$0.359376\pi$
$90$	0	0
$91$	−12.2018	−1.27909
$92$	−3.97313	−0.414228
$93$	0	0
$94$	4.46817	0.460856
$95$	4.49342	0.461015
$96$	0	0
$97$	0.417684	0.0424094	0.0212047	−	0.999775i	$-0.493250\pi$
$97$	0.417684	0.0424094	0.0212047	+	0.999775i	$0.493250\pi$
$98$	−2.43702	−0.246177
$99$	0	0
$100$	−3.03987	−0.303987
$101$	2.99840	0.298352	0.149176	−	0.988811i	$-0.452338\pi$
$101$	2.99840	0.298352	0.149176	+	0.988811i	$0.452338\pi$
$102$	0	0
$103$	13.6270	1.34271	0.671354	−	0.741137i	$-0.265713\pi$
$103$	13.6270	1.34271	0.671354	+	0.741137i	$0.265713\pi$
$104$	5.71214	0.560122
$105$	0	0
$106$	9.91571	0.963099
$107$	−13.5914	−1.31393	−0.656963	−	0.753923i	$-0.728159\pi$
$107$	−13.5914	−1.31393	−0.656963	+	0.753923i	$0.728159\pi$
$108$	0	0
$109$	−5.23427	−0.501353	−0.250676	−	0.968071i	$-0.580653\pi$
$109$	−5.23427	−0.501353	−0.250676	+	0.968071i	$0.580653\pi$
$110$	3.96845	0.378377
$111$	0	0
$112$	−2.13611	−0.201844
$113$	−7.25732	−0.682712	−0.341356	−	0.939934i	$-0.610886\pi$
$113$	−7.25732	−0.682712	−0.341356	+	0.939934i	$0.610886\pi$
$114$	0	0
$115$	5.56256	0.518712
$116$	0.100627	0.00934295
$117$	0	0
$118$	−9.99566	−0.920175
$119$	−12.5452	−1.15001
$120$	0	0
$121$	−2.96552	−0.269593
$122$	−0.434578	−0.0393448
$123$	0	0
$124$	−5.43523	−0.488098
$125$	11.2562	1.00678
$126$	0	0
$127$	−14.4025	−1.27802	−0.639008	−	0.769200i	$-0.720655\pi$
$127$	−14.4025	−1.27802	−0.639008	+	0.769200i	$0.720655\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−7.99726	−0.701406
$131$	−9.19483	−0.803356	−0.401678	−	0.915781i	$-0.631573\pi$
$131$	−9.19483	−0.803356	−0.401678	+	0.915781i	$0.631573\pi$
$132$	0	0
$133$	6.85581	0.594474
$134$	−9.79274	−0.845964
$135$	0	0
$136$	5.87290	0.503597
$137$	7.48889	0.639819	0.319909	−	0.947448i	$-0.396348\pi$
$137$	7.48889	0.639819	0.319909	+	0.947448i	$0.396348\pi$
$138$	0	0
$139$	−5.25571	−0.445784	−0.222892	−	0.974843i	$-0.571550\pi$
$139$	−5.25571	−0.445784	−0.222892	+	0.974843i	$0.571550\pi$
$140$	2.99065	0.252756
$141$	0	0
$142$	11.9919	1.00634
$143$	−16.1912	−1.35397
$144$	0	0
$145$	−0.140882	−0.0116996
$146$	−11.4593	−0.948376
$147$	0	0
$148$	6.63998	0.545803
$149$	1.00000	0.0819232
$149$	1.00000	0.0819232
$150$	0	0
$151$	4.12197	0.335442	0.167721	−	0.985835i	$-0.446359\pi$
$151$	4.12197	0.335442	0.167721	+	0.985835i	$0.446359\pi$
$152$	−3.20948	−0.260323
$153$	0	0
$154$	6.05484	0.487913
$155$	7.60957	0.611215
$156$	0	0
$157$	−16.0196	−1.27850	−0.639252	−	0.768998i	$-0.720756\pi$
$157$	−16.0196	−1.27850	−0.639252	+	0.768998i	$0.720756\pi$
$158$	−0.914747	−0.0727733
$159$	0	0
$160$	−1.40004	−0.110683
$161$	8.48706	0.668874
$162$	0	0
$163$	4.47853	0.350785	0.175393	−	0.984499i	$-0.443881\pi$
$163$	4.47853	0.350785	0.175393	+	0.984499i	$0.443881\pi$
$164$	−1.02469	−0.0800148
$165$	0	0
$166$	−9.47992	−0.735784
$167$	−15.7335	−1.21749	−0.608746	−	0.793365i	$-0.708327\pi$
$167$	−15.7335	−1.21749	−0.608746	+	0.793365i	$0.708327\pi$
$168$	0	0
$169$	19.6286	1.50989
$170$	−8.22232	−0.630623
$171$	0	0
$172$	−3.45987	−0.263813
$173$	15.7737	1.19926	0.599628	−	0.800279i	$-0.295315\pi$
$173$	15.7737	1.19926	0.599628	+	0.800279i	$0.295315\pi$
$174$	0	0
$175$	6.49351	0.490863
$176$	−2.83452	−0.213660
$177$	0	0
$178$	8.06705	0.604651
$179$	1.28621	0.0961361	0.0480680	−	0.998844i	$-0.484694\pi$
$179$	1.28621	0.0961361	0.0480680	+	0.998844i	$0.484694\pi$
$180$	0	0
$181$	5.65131	0.420058	0.210029	−	0.977695i	$-0.432644\pi$
$181$	5.65131	0.420058	0.210029	+	0.977695i	$0.432644\pi$
$182$	−12.2018	−0.904456
$183$	0	0
$184$	−3.97313	−0.292903
$185$	−9.29627	−0.683476
$186$	0	0
$187$	−16.6468	−1.21734
$188$	4.46817	0.325875
$189$	0	0
$190$	4.49342	0.325987
$191$	−19.9838	−1.44598	−0.722990	−	0.690858i	$-0.757233\pi$
$191$	−19.9838	−1.44598	−0.722990	+	0.690858i	$0.757233\pi$
$192$	0	0
$193$	−10.7945	−0.777007	−0.388503	−	0.921447i	$-0.627008\pi$
$193$	−10.7945	−0.777007	−0.388503	+	0.921447i	$0.627008\pi$
$194$	0.417684	0.0299880
$195$	0	0
$196$	−2.43702	−0.174073
$197$	−5.90747	−0.420890	−0.210445	−	0.977606i	$-0.567491\pi$
$197$	−5.90747	−0.420890	−0.210445	+	0.977606i	$0.567491\pi$
$198$	0	0
$199$	−10.6822	−0.757244	−0.378622	−	0.925551i	$-0.623602\pi$
$199$	−10.6822	−0.757244	−0.378622	+	0.925551i	$0.623602\pi$
$200$	−3.03987	−0.214952
$201$	0	0
$202$	2.99840	0.210967
$203$	−0.214950	−0.0150865
$204$	0	0
$205$	1.43461	0.100198
$206$	13.6270	0.949437
$207$	0	0
$208$	5.71214	0.396066
$209$	9.09732	0.629275
$210$	0	0
$211$	−9.02578	−0.621360	−0.310680	−	0.950515i	$-0.600557\pi$
$211$	−9.02578	−0.621360	−0.310680	+	0.950515i	$0.600557\pi$
$212$	9.91571	0.681014
$213$	0	0
$214$	−13.5914	−0.929086
$215$	4.84398	0.330357
$216$	0	0
$217$	11.6103	0.788157
$218$	−5.23427	−0.354510
$219$	0	0
$220$	3.96845	0.267553
$221$	33.5468	2.25660
$222$	0	0
$223$	−24.8499	−1.66407	−0.832037	−	0.554719i	$-0.812826\pi$
$223$	−24.8499	−1.66407	−0.832037	+	0.554719i	$0.812826\pi$
$224$	−2.13611	−0.142725
$225$	0	0
$226$	−7.25732	−0.482750
$227$	−21.0159	−1.39487	−0.697437	−	0.716646i	$-0.745676\pi$
$227$	−21.0159	−1.39487	−0.697437	+	0.716646i	$0.745676\pi$
$228$	0	0
$229$	−6.46347	−0.427118	−0.213559	−	0.976930i	$-0.568506\pi$
$229$	−6.46347	−0.427118	−0.213559	+	0.976930i	$0.568506\pi$
$230$	5.56256	0.366785
$231$	0	0
$232$	0.100627	0.00660646
$233$	−22.3440	−1.46381	−0.731904	−	0.681408i	$-0.761368\pi$
$233$	−22.3440	−1.46381	−0.731904	+	0.681408i	$0.761368\pi$
$234$	0	0
$235$	−6.25564	−0.408073
$236$	−9.99566	−0.650662
$237$	0	0
$238$	−12.5452	−0.813182
$239$	9.89922	0.640327	0.320164	−	0.947362i	$-0.396262\pi$
$239$	9.89922	0.640327	0.320164	+	0.947362i	$0.396262\pi$
$240$	0	0
$241$	−16.2922	−1.04947	−0.524735	−	0.851266i	$-0.675836\pi$
$241$	−16.2922	−1.04947	−0.524735	+	0.851266i	$0.675836\pi$
$242$	−2.96552	−0.190631
$243$	0	0
$244$	−0.434578	−0.0278210
$245$	3.41194	0.217981
$246$	0	0
$247$	−18.3330	−1.16650
$248$	−5.43523	−0.345138
$249$	0	0
$250$	11.2562	0.711904
$251$	14.3914	0.908377	0.454188	−	0.890906i	$-0.349929\pi$
$251$	14.3914	0.908377	0.454188	+	0.890906i	$0.349929\pi$
$252$	0	0
$253$	11.2619	0.708030
$254$	−14.4025	−0.903694
$255$	0	0
$256$	1.00000	0.0625000
$257$	−2.59865	−0.162099	−0.0810495	−	0.996710i	$-0.525827\pi$
$257$	−2.59865	−0.162099	−0.0810495	+	0.996710i	$0.525827\pi$
$258$	0	0
$259$	−14.1837	−0.881335
$260$	−7.99726	−0.495969
$261$	0	0
$262$	−9.19483	−0.568058
$263$	−2.01653	−0.124345	−0.0621724	−	0.998065i	$-0.519803\pi$
$263$	−2.01653	−0.124345	−0.0621724	+	0.998065i	$0.519803\pi$
$264$	0	0
$265$	−13.8824	−0.852792
$266$	6.85581	0.420357
$267$	0	0
$268$	−9.79274	−0.598187
$269$	−24.0715	−1.46766	−0.733832	−	0.679331i	$-0.762270\pi$
$269$	−24.0715	−1.46766	−0.733832	+	0.679331i	$0.762270\pi$
$270$	0	0
$271$	−0.879519	−0.0534270	−0.0267135	−	0.999643i	$-0.508504\pi$
$271$	−0.879519	−0.0534270	−0.0267135	+	0.999643i	$0.508504\pi$
$272$	5.87290	0.356097
$273$	0	0
$274$	7.48889	0.452420
$275$	8.61657	0.519599
$276$	0	0
$277$	2.71892	0.163364	0.0816820	−	0.996658i	$-0.473971\pi$
$277$	2.71892	0.163364	0.0816820	+	0.996658i	$0.473971\pi$
$278$	−5.25571	−0.315217
$279$	0	0
$280$	2.99065	0.178726
$281$	3.95321	0.235829	0.117914	−	0.993024i	$-0.462379\pi$
$281$	3.95321	0.235829	0.117914	+	0.993024i	$0.462379\pi$
$282$	0	0
$283$	−32.6314	−1.93974	−0.969869	−	0.243627i	$-0.921663\pi$
$283$	−32.6314	−1.93974	−0.969869	+	0.243627i	$0.921663\pi$
$284$	11.9919	0.711590
$285$	0	0
$286$	−16.1912	−0.957403
$287$	2.18885	0.129204
$288$	0	0
$289$	17.4909	1.02888
$290$	−0.140882	−0.00827287
$291$	0	0
$292$	−11.4593	−0.670603
$293$	−19.8773	−1.16125	−0.580623	−	0.814172i	$-0.697191\pi$
$293$	−19.8773	−1.16125	−0.580623	+	0.814172i	$0.697191\pi$
$294$	0	0
$295$	13.9944	0.814784
$296$	6.63998	0.385941
$297$	0	0
$298$	1.00000	0.0579284
$299$	−22.6951	−1.31249
$300$	0	0
$301$	7.39068	0.425992
$302$	4.12197	0.237193
$303$	0	0
$304$	−3.20948	−0.184076
$305$	0.608429	0.0348385
$306$	0	0
$307$	1.31921	0.0752913	0.0376457	−	0.999291i	$-0.488014\pi$
$307$	1.31921	0.0752913	0.0376457	+	0.999291i	$0.488014\pi$
$308$	6.05484	0.345007
$309$	0	0
$310$	7.60957	0.432195
$311$	8.78774	0.498307	0.249153	−	0.968464i	$-0.419848\pi$
$311$	8.78774	0.498307	0.249153	+	0.968464i	$0.419848\pi$
$312$	0	0
$313$	−11.8713	−0.671004	−0.335502	−	0.942040i	$-0.608906\pi$
$313$	−11.8713	−0.671004	−0.335502	+	0.942040i	$0.608906\pi$
$314$	−16.0196	−0.904038
$315$	0	0
$316$	−0.914747	−0.0514585
$317$	9.89669	0.555853	0.277927	−	0.960602i	$-0.410353\pi$
$317$	9.89669	0.555853	0.277927	+	0.960602i	$0.410353\pi$
$318$	0	0
$319$	−0.285228	−0.0159697
$320$	−1.40004	−0.0782649
$321$	0	0
$322$	8.48706	0.472965
$323$	−18.8490	−1.04878
$324$	0	0
$325$	−17.3642	−0.963192
$326$	4.47853	0.248043
$327$	0	0
$328$	−1.02469	−0.0565790
$329$	−9.54451	−0.526206
$330$	0	0
$331$	2.86701	0.157585	0.0787924	−	0.996891i	$-0.474894\pi$
$331$	2.86701	0.157585	0.0787924	+	0.996891i	$0.474894\pi$
$332$	−9.47992	−0.520278
$333$	0	0
$334$	−15.7335	−0.860897
$335$	13.7103	0.749073
$336$	0	0
$337$	−26.9675	−1.46901	−0.734506	−	0.678602i	$-0.762586\pi$
$337$	−26.9675	−1.46901	−0.734506	+	0.678602i	$0.762586\pi$
$338$	19.6286	1.06765
$339$	0	0
$340$	−8.22232	−0.445918
$341$	15.4063	0.834295
$342$	0	0
$343$	20.1585	1.08846
$344$	−3.45987	−0.186544
$345$	0	0
$346$	15.7737	0.848001
$347$	2.86874	0.154002	0.0770010	−	0.997031i	$-0.475466\pi$
$347$	2.86874	0.154002	0.0770010	+	0.997031i	$0.475466\pi$
$348$	0	0
$349$	−19.6662	−1.05271	−0.526355	−	0.850265i	$-0.676442\pi$
$349$	−19.6662	−1.05271	−0.526355	+	0.850265i	$0.676442\pi$
$350$	6.49351	0.347093
$351$	0	0
$352$	−2.83452	−0.151080
$353$	−6.58385	−0.350423	−0.175211	−	0.984531i	$-0.556061\pi$
$353$	−6.58385	−0.350423	−0.175211	+	0.984531i	$0.556061\pi$
$354$	0	0
$355$	−16.7892	−0.891080
$356$	8.06705	0.427553
$357$	0	0
$358$	1.28621	0.0679785
$359$	−7.90476	−0.417197	−0.208599	−	0.978001i	$-0.566890\pi$
$359$	−7.90476	−0.417197	−0.208599	+	0.978001i	$0.566890\pi$
$360$	0	0
$361$	−8.69923	−0.457854
$362$	5.65131	0.297026
$363$	0	0
$364$	−12.2018	−0.639547
$365$	16.0435	0.839755
$366$	0	0
$367$	8.11501	0.423600	0.211800	−	0.977313i	$-0.432067\pi$
$367$	8.11501	0.423600	0.211800	+	0.977313i	$0.432067\pi$
$368$	−3.97313	−0.207114
$369$	0	0
$370$	−9.29627	−0.483290
$371$	−21.1811	−1.09967
$372$	0	0
$373$	−4.08805	−0.211671	−0.105836	−	0.994384i	$-0.533752\pi$
$373$	−4.08805	−0.211671	−0.105836	+	0.994384i	$0.533752\pi$
$374$	−16.6468	−0.860786
$375$	0	0
$376$	4.46817	0.230428
$377$	0.574794	0.0296034
$378$	0	0
$379$	16.2729	0.835884	0.417942	−	0.908474i	$-0.362752\pi$
$379$	16.2729	0.835884	0.417942	+	0.908474i	$0.362752\pi$
$380$	4.49342	0.230507
$381$	0	0
$382$	−19.9838	−1.02246
$383$	−3.40986	−0.174236	−0.0871179	−	0.996198i	$-0.527766\pi$
$383$	−3.40986	−0.174236	−0.0871179	+	0.996198i	$0.527766\pi$
$384$	0	0
$385$	−8.47705	−0.432031
$386$	−10.7945	−0.549427
$387$	0	0
$388$	0.417684	0.0212047
$389$	9.04552	0.458626	0.229313	−	0.973353i	$-0.426352\pi$
$389$	9.04552	0.458626	0.229313	+	0.973353i	$0.426352\pi$
$390$	0	0
$391$	−23.3338	−1.18004
$392$	−2.43702	−0.123088
$393$	0	0
$394$	−5.90747	−0.297614
$395$	1.28069	0.0644383
$396$	0	0
$397$	−24.1560	−1.21235	−0.606177	−	0.795330i	$-0.707298\pi$
$397$	−24.1560	−1.21235	−0.606177	+	0.795330i	$0.707298\pi$
$398$	−10.6822	−0.535452
$399$	0	0
$400$	−3.03987	−0.151994
$401$	18.6368	0.930679	0.465340	−	0.885132i	$-0.345932\pi$
$401$	18.6368	0.930679	0.465340	+	0.885132i	$0.345932\pi$
$402$	0	0
$403$	−31.0468	−1.54655
$404$	2.99840	0.149176
$405$	0	0
$406$	−0.214950	−0.0106678
$407$	−18.8211	−0.932929
$408$	0	0
$409$	25.2949	1.25075	0.625377	−	0.780323i	$-0.284945\pi$
$409$	25.2949	1.25075	0.625377	+	0.780323i	$0.284945\pi$
$410$	1.43461	0.0708504
$411$	0	0
$412$	13.6270	0.671354
$413$	21.3518	1.05066
$414$	0	0
$415$	13.2723	0.651512
$416$	5.71214	0.280061
$417$	0	0
$418$	9.09732	0.444965
$419$	23.3242	1.13946	0.569731	−	0.821831i	$-0.307047\pi$
$419$	23.3242	1.13946	0.569731	+	0.821831i	$0.307047\pi$
$420$	0	0
$421$	−24.6935	−1.20349	−0.601743	−	0.798690i	$-0.705527\pi$
$421$	−24.6935	−1.20349	−0.601743	+	0.798690i	$0.705527\pi$
$422$	−9.02578	−0.439368
$423$	0	0
$424$	9.91571	0.481550
$425$	−17.8529	−0.865991
$426$	0	0
$427$	0.928307	0.0449239
$428$	−13.5914	−0.656963
$429$	0	0
$430$	4.84398	0.233597
$431$	15.8410	0.763036	0.381518	−	0.924361i	$-0.375401\pi$
$431$	15.8410	0.763036	0.381518	+	0.924361i	$0.375401\pi$
$432$	0	0
$433$	29.0445	1.39579	0.697894	−	0.716201i	$-0.254120\pi$
$433$	29.0445	1.39579	0.697894	+	0.716201i	$0.254120\pi$
$434$	11.6103	0.557311
$435$	0	0
$436$	−5.23427	−0.250676
$437$	12.7517	0.609996
$438$	0	0
$439$	−17.2149	−0.821623	−0.410812	−	0.911720i	$-0.634755\pi$
$439$	−17.2149	−0.821623	−0.410812	+	0.911720i	$0.634755\pi$
$440$	3.96845	0.189188
$441$	0	0
$442$	33.5468	1.59566
$443$	−1.17623	−0.0558843	−0.0279421	−	0.999610i	$-0.508895\pi$
$443$	−1.17623	−0.0558843	−0.0279421	+	0.999610i	$0.508895\pi$
$444$	0	0
$445$	−11.2942	−0.535398
$446$	−24.8499	−1.17668
$447$	0	0
$448$	−2.13611	−0.100922
$449$	1.59665	0.0753504	0.0376752	−	0.999290i	$-0.488005\pi$
$449$	1.59665	0.0753504	0.0376752	+	0.999290i	$0.488005\pi$
$450$	0	0
$451$	2.90450	0.136767
$452$	−7.25732	−0.341356
$453$	0	0
$454$	−21.0159	−0.986325
$455$	17.0830	0.800865
$456$	0	0
$457$	11.8057	0.552248	0.276124	−	0.961122i	$-0.410950\pi$
$457$	11.8057	0.552248	0.276124	+	0.961122i	$0.410950\pi$
$458$	−6.46347	−0.302018
$459$	0	0
$460$	5.56256	0.259356
$461$	32.8342	1.52924	0.764621	−	0.644480i	$-0.222926\pi$
$461$	32.8342	1.52924	0.764621	+	0.644480i	$0.222926\pi$
$462$	0	0
$463$	33.9261	1.57668	0.788340	−	0.615240i	$-0.210941\pi$
$463$	33.9261	1.57668	0.788340	+	0.615240i	$0.210941\pi$
$464$	0.100627	0.00467148
$465$	0	0
$466$	−22.3440	−1.03507
$467$	10.5076	0.486232	0.243116	−	0.969997i	$-0.421830\pi$
$467$	10.5076	0.486232	0.243116	+	0.969997i	$0.421830\pi$
$468$	0	0
$469$	20.9184	0.965922
$470$	−6.25564	−0.288551
$471$	0	0
$472$	−9.99566	−0.460087
$473$	9.80706	0.450929
$474$	0	0
$475$	9.75642	0.447655
$476$	−12.5452	−0.575007
$477$	0	0
$478$	9.89922	0.452780
$479$	12.5213	0.572111	0.286056	−	0.958213i	$-0.407656\pi$
$479$	12.5213	0.572111	0.286056	+	0.958213i	$0.407656\pi$
$480$	0	0
$481$	37.9285	1.72939
$482$	−16.2922	−0.742088
$483$	0	0
$484$	−2.96552	−0.134797
$485$	−0.584776	−0.0265533
$486$	0	0
$487$	25.7221	1.16558	0.582791	−	0.812622i	$-0.301961\pi$
$487$	25.7221	1.16558	0.582791	+	0.812622i	$0.301961\pi$
$488$	−0.434578	−0.0196724
$489$	0	0
$490$	3.41194	0.154136
$491$	19.0581	0.860080	0.430040	−	0.902810i	$-0.358499\pi$
$491$	19.0581	0.860080	0.430040	+	0.902810i	$0.358499\pi$
$492$	0	0
$493$	0.590970	0.0266159
$494$	−18.3330	−0.824842
$495$	0	0
$496$	−5.43523	−0.244049
$497$	−25.6161	−1.14904
$498$	0	0
$499$	32.8816	1.47198	0.735991	−	0.676992i	$-0.236717\pi$
$499$	32.8816	1.47198	0.735991	+	0.676992i	$0.236717\pi$
$500$	11.2562	0.503392
$501$	0	0
$502$	14.3914	0.642319
$503$	−12.6670	−0.564792	−0.282396	−	0.959298i	$-0.591129\pi$
$503$	−12.6670	−0.564792	−0.282396	+	0.959298i	$0.591129\pi$
$504$	0	0
$505$	−4.19790	−0.186804
$506$	11.2619	0.500653
$507$	0	0
$508$	−14.4025	−0.639008
$509$	−2.72517	−0.120791	−0.0603954	−	0.998175i	$-0.519236\pi$
$509$	−2.72517	−0.120791	−0.0603954	+	0.998175i	$0.519236\pi$
$510$	0	0
$511$	24.4783	1.08286
$512$	1.00000	0.0441942
$513$	0	0
$514$	−2.59865	−0.114621
$515$	−19.0784	−0.840695
$516$	0	0
$517$	−12.6651	−0.557010
$518$	−14.1837	−0.623198
$519$	0	0
$520$	−7.99726	−0.350703
$521$	7.74356	0.339252	0.169626	−	0.985509i	$-0.445744\pi$
$521$	7.74356	0.339252	0.169626	+	0.985509i	$0.445744\pi$
$522$	0	0
$523$	19.2524	0.841850	0.420925	−	0.907095i	$-0.361706\pi$
$523$	19.2524	0.841850	0.420925	+	0.907095i	$0.361706\pi$
$524$	−9.19483	−0.401678
$525$	0	0
$526$	−2.01653	−0.0879250
$527$	−31.9206	−1.39048
$528$	0	0
$529$	−7.21423	−0.313662
$530$	−13.8824	−0.603015
$531$	0	0
$532$	6.85581	0.297237
$533$	−5.85317	−0.253529
$534$	0	0
$535$	19.0285	0.822674
$536$	−9.79274	−0.422982
$537$	0	0
$538$	−24.0715	−1.03779
$539$	6.90778	0.297539
$540$	0	0
$541$	28.6463	1.23160	0.615800	−	0.787903i	$-0.288833\pi$
$541$	28.6463	1.23160	0.615800	+	0.787903i	$0.288833\pi$
$542$	−0.879519	−0.0377786
$543$	0	0
$544$	5.87290	0.251798
$545$	7.32822	0.313906
$546$	0	0
$547$	23.5378	1.00640	0.503202	−	0.864169i	$-0.332155\pi$
$547$	23.5378	1.00640	0.503202	+	0.864169i	$0.332155\pi$
$548$	7.48889	0.319909
$549$	0	0
$550$	8.61657	0.367412
$551$	−0.322959	−0.0137585
$552$	0	0
$553$	1.95400	0.0830926
$554$	2.71892	0.115516
$555$	0	0
$556$	−5.25571	−0.222892
$557$	33.9075	1.43670	0.718352	−	0.695679i	$-0.244897\pi$
$557$	33.9075	1.43670	0.718352	+	0.695679i	$0.244897\pi$
$558$	0	0
$559$	−19.7633	−0.835898
$560$	2.99065	0.126378
$561$	0	0
$562$	3.95321	0.166756
$563$	31.3723	1.32219	0.661093	−	0.750304i	$-0.270093\pi$
$563$	31.3723	1.32219	0.661093	+	0.750304i	$0.270093\pi$
$564$	0	0
$565$	10.1606	0.427459
$566$	−32.6314	−1.37160
$567$	0	0
$568$	11.9919	0.503170
$569$	−22.7184	−0.952405	−0.476202	−	0.879336i	$-0.657987\pi$
$569$	−22.7184	−0.952405	−0.476202	+	0.879336i	$0.657987\pi$
$570$	0	0
$571$	−30.1486	−1.26168	−0.630839	−	0.775914i	$-0.717289\pi$
$571$	−30.1486	−1.26168	−0.630839	+	0.775914i	$0.717289\pi$
$572$	−16.1912	−0.676986
$573$	0	0
$574$	2.18885	0.0913609
$575$	12.0778	0.503680
$576$	0	0
$577$	18.0000	0.749348	0.374674	−	0.927157i	$-0.377755\pi$
$577$	18.0000	0.749348	0.374674	+	0.927157i	$0.377755\pi$
$578$	17.4909	0.727526
$579$	0	0
$580$	−0.140882	−0.00584980
$581$	20.2502	0.840118
$582$	0	0
$583$	−28.1062	−1.16404
$584$	−11.4593	−0.474188
$585$	0	0
$586$	−19.8773	−0.821125
$587$	19.3067	0.796874	0.398437	−	0.917196i	$-0.369553\pi$
$587$	19.3067	0.796874	0.398437	+	0.917196i	$0.369553\pi$
$588$	0	0
$589$	17.4443	0.718779
$590$	13.9944	0.576139
$591$	0	0
$592$	6.63998	0.272902
$593$	21.9933	0.903155	0.451578	−	0.892232i	$-0.350861\pi$
$593$	21.9933	0.903155	0.451578	+	0.892232i	$0.350861\pi$
$594$	0	0
$595$	17.5638	0.720045
$596$	1.00000	0.0409616
$597$	0	0
$598$	−22.6951	−0.928072
$599$	−4.80300	−0.196245	−0.0981227	−	0.995174i	$-0.531284\pi$
$599$	−4.80300	−0.196245	−0.0981227	+	0.995174i	$0.531284\pi$
$600$	0	0
$601$	7.91148	0.322716	0.161358	−	0.986896i	$-0.448413\pi$
$601$	7.91148	0.322716	0.161358	+	0.986896i	$0.448413\pi$
$602$	7.39068	0.301222
$603$	0	0
$604$	4.12197	0.167721
$605$	4.15187	0.168797
$606$	0	0
$607$	−25.7295	−1.04433	−0.522165	−	0.852845i	$-0.674875\pi$
$607$	−25.7295	−1.04433	−0.522165	+	0.852845i	$0.674875\pi$
$608$	−3.20948	−0.130162
$609$	0	0
$610$	0.608429	0.0246346
$611$	25.5228	1.03254
$612$	0	0
$613$	−42.8866	−1.73218	−0.866088	−	0.499892i	$-0.833373\pi$
$613$	−42.8866	−1.73218	−0.866088	+	0.499892i	$0.833373\pi$
$614$	1.31921	0.0532390
$615$	0	0
$616$	6.05484	0.243957
$617$	−11.7628	−0.473551	−0.236776	−	0.971564i	$-0.576091\pi$
$617$	−11.7628	−0.473551	−0.236776	+	0.971564i	$0.576091\pi$
$618$	0	0
$619$	47.6826	1.91652	0.958262	−	0.285891i	$-0.0922897\pi$
$619$	47.6826	1.91652	0.958262	+	0.285891i	$0.0922897\pi$
$620$	7.60957	0.305608
$621$	0	0
$622$	8.78774	0.352356
$623$	−17.2321	−0.690391
$624$	0	0
$625$	−0.559793	−0.0223917
$626$	−11.8713	−0.474471
$627$	0	0
$628$	−16.0196	−0.639252
$629$	38.9959	1.55487
$630$	0	0
$631$	−18.6350	−0.741848	−0.370924	−	0.928663i	$-0.620959\pi$
$631$	−18.6350	−0.741848	−0.370924	+	0.928663i	$0.620959\pi$
$632$	−0.914747	−0.0363867
$633$	0	0
$634$	9.89669	0.393048
$635$	20.1642	0.800191
$636$	0	0
$637$	−13.9206	−0.551556
$638$	−0.285228	−0.0112923
$639$	0	0
$640$	−1.40004	−0.0553416
$641$	−39.3534	−1.55436	−0.777182	−	0.629275i	$-0.783352\pi$
$641$	−39.3534	−1.55436	−0.777182	+	0.629275i	$0.783352\pi$
$642$	0	0
$643$	35.5214	1.40083	0.700414	−	0.713737i	$-0.252999\pi$
$643$	35.5214	1.40083	0.700414	+	0.713737i	$0.252999\pi$
$644$	8.48706	0.334437
$645$	0	0
$646$	−18.8490	−0.741602
$647$	−5.96301	−0.234430	−0.117215	−	0.993107i	$-0.537397\pi$
$647$	−5.96301	−0.234430	−0.117215	+	0.993107i	$0.537397\pi$
$648$	0	0
$649$	28.3328	1.11216
$650$	−17.3642	−0.681080
$651$	0	0
$652$	4.47853	0.175393
$653$	2.68049	0.104896	0.0524479	−	0.998624i	$-0.483298\pi$
$653$	2.68049	0.104896	0.0524479	+	0.998624i	$0.483298\pi$
$654$	0	0
$655$	12.8732	0.502996
$656$	−1.02469	−0.0400074
$657$	0	0
$658$	−9.54451	−0.372084
$659$	18.2301	0.710144	0.355072	−	0.934839i	$-0.384456\pi$
$659$	18.2301	0.710144	0.355072	+	0.934839i	$0.384456\pi$
$660$	0	0
$661$	−35.2089	−1.36947	−0.684733	−	0.728794i	$-0.740081\pi$
$661$	−35.2089	−1.36947	−0.684733	+	0.728794i	$0.740081\pi$
$662$	2.86701	0.111429
$663$	0	0
$664$	−9.47992	−0.367892
$665$	−9.59845	−0.372212
$666$	0	0
$667$	−0.399803	−0.0154804
$668$	−15.7335	−0.608746
$669$	0	0
$670$	13.7103	0.529674
$671$	1.23182	0.0475538
$672$	0	0
$673$	−38.3410	−1.47794	−0.738969	−	0.673740i	$-0.764687\pi$
$673$	−38.3410	−1.47794	−0.738969	+	0.673740i	$0.764687\pi$
$674$	−26.9675	−1.03875
$675$	0	0
$676$	19.6286	0.754946
$677$	−27.6779	−1.06375	−0.531875	−	0.846823i	$-0.678512\pi$
$677$	−27.6779	−1.06375	−0.531875	+	0.846823i	$0.678512\pi$
$678$	0	0
$679$	−0.892220	−0.0342402
$680$	−8.22232	−0.315311
$681$	0	0
$682$	15.4063	0.589936
$683$	24.5302	0.938621	0.469311	−	0.883033i	$-0.344502\pi$
$683$	24.5302	0.938621	0.469311	+	0.883033i	$0.344502\pi$
$684$	0	0
$685$	−10.4848	−0.400603
$686$	20.1585	0.769657
$687$	0	0
$688$	−3.45987	−0.131906
$689$	56.6400	2.15781
$690$	0	0
$691$	27.5033	1.04628	0.523138	−	0.852248i	$-0.324761\pi$
$691$	27.5033	1.04628	0.523138	+	0.852248i	$0.324761\pi$
$692$	15.7737	0.599628
$693$	0	0
$694$	2.86874	0.108896
$695$	7.35823	0.279114
$696$	0	0
$697$	−6.01789	−0.227944
$698$	−19.6662	−0.744378
$699$	0	0
$700$	6.49351	0.245432
$701$	20.6909	0.781484	0.390742	−	0.920500i	$-0.372218\pi$
$701$	20.6909	0.781484	0.390742	+	0.920500i	$0.372218\pi$
$702$	0	0
$703$	−21.3109	−0.803756
$704$	−2.83452	−0.106830
$705$	0	0
$706$	−6.58385	−0.247786
$707$	−6.40492	−0.240882
$708$	0	0
$709$	−10.3179	−0.387498	−0.193749	−	0.981051i	$-0.562065\pi$
$709$	−10.3179	−0.387498	−0.193749	+	0.981051i	$0.562065\pi$
$710$	−16.7892	−0.630089
$711$	0	0
$712$	8.06705	0.302326
$713$	21.5949	0.808735
$714$	0	0
$715$	22.6683	0.847748
$716$	1.28621	0.0480680
$717$	0	0
$718$	−7.90476	−0.295003
$719$	−15.4869	−0.577565	−0.288782	−	0.957395i	$-0.593250\pi$
$719$	−15.4869	−0.577565	−0.288782	+	0.957395i	$0.593250\pi$
$720$	0	0
$721$	−29.1088	−1.08407
$722$	−8.69923	−0.323752
$723$	0	0
$724$	5.65131	0.210029
$725$	−0.305892	−0.0113606
$726$	0	0
$727$	−11.1871	−0.414908	−0.207454	−	0.978245i	$-0.566518\pi$
$727$	−11.1871	−0.414908	−0.207454	+	0.978245i	$0.566518\pi$
$728$	−12.2018	−0.452228
$729$	0	0
$730$	16.0435	0.593797
$731$	−20.3195	−0.751543
$732$	0	0
$733$	−19.9936	−0.738481	−0.369241	−	0.929334i	$-0.620382\pi$
$733$	−19.9936	−0.738481	−0.369241	+	0.929334i	$0.620382\pi$
$734$	8.11501	0.299531
$735$	0	0
$736$	−3.97313	−0.146452
$737$	27.7577	1.02247
$738$	0	0
$739$	−21.6153	−0.795133	−0.397567	−	0.917573i	$-0.630145\pi$
$739$	−21.6153	−0.795133	−0.397567	+	0.917573i	$0.630145\pi$
$740$	−9.29627	−0.341738
$741$	0	0
$742$	−21.1811	−0.777582
$743$	21.5588	0.790916	0.395458	−	0.918484i	$-0.370586\pi$
$743$	21.5588	0.790916	0.395458	+	0.918484i	$0.370586\pi$
$744$	0	0
$745$	−1.40004	−0.0512937
$746$	−4.08805	−0.149674
$747$	0	0
$748$	−16.6468	−0.608668
$749$	29.0327	1.06083
$750$	0	0
$751$	−36.1120	−1.31775	−0.658873	−	0.752254i	$-0.728966\pi$
$751$	−36.1120	−1.31775	−0.658873	+	0.752254i	$0.728966\pi$
$752$	4.46817	0.162937
$753$	0	0
$754$	0.574794	0.0209328
$755$	−5.77095	−0.210026
$756$	0	0
$757$	5.15343	0.187304	0.0936522	−	0.995605i	$-0.470146\pi$
$757$	5.15343	0.187304	0.0936522	+	0.995605i	$0.470146\pi$
$758$	16.2729	0.591059
$759$	0	0
$760$	4.49342	0.162993
$761$	40.7411	1.47686	0.738431	−	0.674329i	$-0.235567\pi$
$761$	40.7411	1.47686	0.738431	+	0.674329i	$0.235567\pi$
$762$	0	0
$763$	11.1810	0.404779
$764$	−19.9838	−0.722990
$765$	0	0
$766$	−3.40986	−0.123203
$767$	−57.0966	−2.06164
$768$	0	0
$769$	−12.4751	−0.449864	−0.224932	−	0.974375i	$-0.572216\pi$
$769$	−12.4751	−0.449864	−0.224932	+	0.974375i	$0.572216\pi$
$770$	−8.47705	−0.305492
$771$	0	0
$772$	−10.7945	−0.388503
$773$	−32.7490	−1.17790	−0.588949	−	0.808170i	$-0.700458\pi$
$773$	−32.7490	−1.17790	−0.588949	+	0.808170i	$0.700458\pi$
$774$	0	0
$775$	16.5224	0.593503
$776$	0.417684	0.0149940
$777$	0	0
$778$	9.04552	0.324298
$779$	3.28872	0.117831
$780$	0	0
$781$	−33.9913	−1.21630
$782$	−23.3338	−0.834415
$783$	0	0
$784$	−2.43702	−0.0870366
$785$	22.4282	0.800495
$786$	0	0
$787$	11.7042	0.417210	0.208605	−	0.978000i	$-0.433108\pi$
$787$	11.7042	0.417210	0.208605	+	0.978000i	$0.433108\pi$
$788$	−5.90747	−0.210445
$789$	0	0
$790$	1.28069	0.0455648
$791$	15.5025	0.551204
$792$	0	0
$793$	−2.48237	−0.0881516
$794$	−24.1560	−0.857264
$795$	0	0
$796$	−10.6822	−0.378622
$797$	−14.0078	−0.496182	−0.248091	−	0.968737i	$-0.579803\pi$
$797$	−14.0078	−0.496182	−0.248091	+	0.968737i	$0.579803\pi$
$798$	0	0
$799$	26.2411	0.928343
$800$	−3.03987	−0.107476
$801$	0	0
$802$	18.6368	0.658089
$803$	32.4815	1.14625
$804$	0	0
$805$	−11.8823	−0.418795
$806$	−31.0468	−1.09358
$807$	0	0
$808$	2.99840	0.105483
$809$	5.47697	0.192560	0.0962800	−	0.995354i	$-0.469306\pi$
$809$	5.47697	0.192560	0.0962800	+	0.995354i	$0.469306\pi$
$810$	0	0
$811$	−51.0670	−1.79321	−0.896603	−	0.442835i	$-0.853973\pi$
$811$	−51.0670	−1.79321	−0.896603	+	0.442835i	$0.853973\pi$
$812$	−0.214950	−0.00754326
$813$	0	0
$814$	−18.8211	−0.659680
$815$	−6.27014	−0.219633
$816$	0	0
$817$	11.1044	0.388494
$818$	25.2949	0.884416
$819$	0	0
$820$	1.43461	0.0500988
$821$	17.7501	0.619483	0.309741	−	0.950821i	$-0.399758\pi$
$821$	17.7501	0.619483	0.309741	+	0.950821i	$0.399758\pi$
$822$	0	0
$823$	−40.3122	−1.40519	−0.702597	−	0.711588i	$-0.747976\pi$
$823$	−40.3122	−1.40519	−0.702597	+	0.711588i	$0.747976\pi$
$824$	13.6270	0.474719
$825$	0	0
$826$	21.3518	0.742926
$827$	−29.3165	−1.01943	−0.509717	−	0.860342i	$-0.670250\pi$
$827$	−29.3165	−1.01943	−0.509717	+	0.860342i	$0.670250\pi$
$828$	0	0
$829$	21.7119	0.754084	0.377042	−	0.926196i	$-0.376941\pi$
$829$	21.7119	0.754084	0.377042	+	0.926196i	$0.376941\pi$
$830$	13.2723	0.460689
$831$	0	0
$832$	5.71214	0.198033
$833$	−14.3124	−0.495895
$834$	0	0
$835$	22.0275	0.762295
$836$	9.09732	0.314638
$837$	0	0
$838$	23.3242	0.805721
$839$	−26.7407	−0.923191	−0.461596	−	0.887090i	$-0.652723\pi$
$839$	−26.7407	−0.923191	−0.461596	+	0.887090i	$0.652723\pi$
$840$	0	0
$841$	−28.9899	−0.999651
$842$	−24.6935	−0.850993
$843$	0	0
$844$	−9.02578	−0.310680
$845$	−27.4809	−0.945372
$846$	0	0
$847$	6.33469	0.217663
$848$	9.91571	0.340507
$849$	0	0
$850$	−17.8529	−0.612348
$851$	−26.3815	−0.904347
$852$	0	0
$853$	−56.0203	−1.91810	−0.959049	−	0.283240i	$-0.908591\pi$
$853$	−56.0203	−1.91810	−0.959049	+	0.283240i	$0.908591\pi$
$854$	0.928307	0.0317660
$855$	0	0
$856$	−13.5914	−0.464543
$857$	4.97876	0.170071	0.0850355	−	0.996378i	$-0.472900\pi$
$857$	4.97876	0.170071	0.0850355	+	0.996378i	$0.472900\pi$
$858$	0	0
$859$	35.7104	1.21842	0.609211	−	0.793008i	$-0.291486\pi$
$859$	35.7104	1.21842	0.609211	+	0.793008i	$0.291486\pi$
$860$	4.84398	0.165178
$861$	0	0
$862$	15.8410	0.539548
$863$	8.22800	0.280085	0.140042	−	0.990146i	$-0.455276\pi$
$863$	8.22800	0.280085	0.140042	+	0.990146i	$0.455276\pi$
$864$	0	0
$865$	−22.0839	−0.750877
$866$	29.0445	0.986972
$867$	0	0
$868$	11.6103	0.394078
$869$	2.59286	0.0879568
$870$	0	0
$871$	−55.9376	−1.89537
$872$	−5.23427	−0.177255
$873$	0	0
$874$	12.7517	0.431332
$875$	−24.0445	−0.812852
$876$	0	0
$877$	24.8923	0.840553	0.420277	−	0.907396i	$-0.361933\pi$
$877$	24.8923	0.840553	0.420277	+	0.907396i	$0.361933\pi$
$878$	−17.2149	−0.580975
$879$	0	0
$880$	3.96845	0.133776
$881$	−18.5260	−0.624158	−0.312079	−	0.950056i	$-0.601025\pi$
$881$	−18.5260	−0.624158	−0.312079	+	0.950056i	$0.601025\pi$
$882$	0	0
$883$	−24.7335	−0.832350	−0.416175	−	0.909285i	$-0.636630\pi$
$883$	−24.7335	−0.832350	−0.416175	+	0.909285i	$0.636630\pi$
$884$	33.5468	1.12830
$885$	0	0
$886$	−1.17623	−0.0395161
$887$	38.1760	1.28182	0.640912	−	0.767614i	$-0.278556\pi$
$887$	38.1760	1.28182	0.640912	+	0.767614i	$0.278556\pi$
$888$	0	0
$889$	30.7654	1.03184
$890$	−11.2942	−0.378584
$891$	0	0
$892$	−24.8499	−0.832037
$893$	−14.3405	−0.479887
$894$	0	0
$895$	−1.80076	−0.0601926
$896$	−2.13611	−0.0713625
$897$	0	0
$898$	1.59665	0.0532808
$899$	−0.546929	−0.0182411
$900$	0	0
$901$	58.2340	1.94005
$902$	2.90450	0.0967092
$903$	0	0
$904$	−7.25732	−0.241375
$905$	−7.91209	−0.263007
$906$	0	0
$907$	53.3879	1.77271	0.886357	−	0.463002i	$-0.153228\pi$
$907$	53.3879	1.77271	0.886357	+	0.463002i	$0.153228\pi$
$908$	−21.0159	−0.697437
$909$	0	0
$910$	17.0830	0.566297
$911$	25.5963	0.848045	0.424022	−	0.905652i	$-0.360618\pi$
$911$	25.5963	0.848045	0.424022	+	0.905652i	$0.360618\pi$
$912$	0	0
$913$	26.8710	0.889299
$914$	11.8057	0.390498
$915$	0	0
$916$	−6.46347	−0.213559
$917$	19.6412	0.648609
$918$	0	0
$919$	12.1548	0.400950	0.200475	−	0.979699i	$-0.435751\pi$
$919$	12.1548	0.400950	0.200475	+	0.979699i	$0.435751\pi$
$920$	5.56256	0.183392
$921$	0	0
$922$	32.8342	1.08134
$923$	68.4996	2.25469
$924$	0	0
$925$	−20.1847	−0.663669
$926$	33.9261	1.11488
$927$	0	0
$928$	0.100627	0.00330323
$929$	−0.280436	−0.00920081	−0.00460040	−	0.999989i	$-0.501464\pi$
$929$	−0.280436	−0.00920081	−0.00460040	+	0.999989i	$0.501464\pi$
$930$	0	0
$931$	7.82158	0.256342
$932$	−22.3440	−0.731904
$933$	0	0
$934$	10.5076	0.343818
$935$	23.3063	0.762197
$936$	0	0
$937$	38.0650	1.24353	0.621765	−	0.783204i	$-0.286416\pi$
$937$	38.0650	1.24353	0.621765	+	0.783204i	$0.286416\pi$
$938$	20.9184	0.683010
$939$	0	0
$940$	−6.25564	−0.204036
$941$	36.1986	1.18004	0.590020	−	0.807388i	$-0.299120\pi$
$941$	36.1986	1.18004	0.590020	+	0.807388i	$0.299120\pi$
$942$	0	0
$943$	4.07123	0.132577
$944$	−9.99566	−0.325331
$945$	0	0
$946$	9.80706	0.318855
$947$	28.7197	0.933265	0.466633	−	0.884451i	$-0.345467\pi$
$947$	28.7197	0.933265	0.466633	+	0.884451i	$0.345467\pi$
$948$	0	0
$949$	−65.4570	−2.12483
$950$	9.75642	0.316540
$951$	0	0
$952$	−12.5452	−0.406591
$953$	40.0200	1.29637	0.648187	−	0.761481i	$-0.275528\pi$
$953$	40.0200	1.29637	0.648187	+	0.761481i	$0.275528\pi$
$954$	0	0
$955$	27.9783	0.905356
$956$	9.89922	0.320164
$957$	0	0
$958$	12.5213	0.404544
$959$	−15.9971	−0.516573
$960$	0	0
$961$	−1.45823	−0.0470397
$962$	37.9285	1.22287
$963$	0	0
$964$	−16.2922	−0.524735
$965$	15.1128	0.486499
$966$	0	0
$967$	14.4863	0.465848	0.232924	−	0.972495i	$-0.425171\pi$
$967$	14.4863	0.465848	0.232924	+	0.972495i	$0.425171\pi$
$968$	−2.96552	−0.0953156
$969$	0	0
$970$	−0.584776	−0.0187760
$971$	44.8124	1.43810	0.719050	−	0.694959i	$-0.244577\pi$
$971$	44.8124	1.43810	0.719050	+	0.694959i	$0.244577\pi$
$972$	0	0
$973$	11.2268	0.359914
$974$	25.7221	0.824190
$975$	0	0
$976$	−0.434578	−0.0139105
$977$	36.7800	1.17670	0.588349	−	0.808607i	$-0.299778\pi$
$977$	36.7800	1.17670	0.588349	+	0.808607i	$0.299778\pi$
$978$	0	0
$979$	−22.8662	−0.730806
$980$	3.41194	0.108991
$981$	0	0
$982$	19.0581	0.608168
$983$	14.0529	0.448219	0.224110	−	0.974564i	$-0.428053\pi$
$983$	14.0529	0.448219	0.224110	+	0.974564i	$0.428053\pi$
$984$	0	0
$985$	8.27073	0.263527
$986$	0.590970	0.0188203
$987$	0	0
$988$	−18.3330	−0.583251
$989$	13.7465	0.437114
$990$	0	0
$991$	−41.7742	−1.32700	−0.663501	−	0.748176i	$-0.730930\pi$
$991$	−41.7742	−1.32700	−0.663501	+	0.748176i	$0.730930\pi$
$992$	−5.43523	−0.172569
$993$	0	0
$994$	−25.6161	−0.812493
$995$	14.9556	0.474125
$996$	0	0
$997$	−59.3478	−1.87956	−0.939782	−	0.341775i	$-0.888972\pi$
$997$	−59.3478	−1.87956	−0.939782	+	0.341775i	$0.888972\pi$
$998$	32.8816	1.04085
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.f.1.2	yes	8
3.2	odd	2		8046.2.a.e.1.7	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.e.1.7	✓	8	3.2	odd	2
8046.2.a.f.1.2	yes	8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8046 = 2 \cdot 3^{3} \cdot 149 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8046.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.40004 q^{5} -2.13611 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.40004 q^{5} -2.13611 q^{7} +1.00000 q^{8} -1.40004 q^{10} -2.83452 q^{11} +5.71214 q^{13} -2.13611 q^{14} +1.00000 q^{16} +5.87290 q^{17} -3.20948 q^{19} -1.40004 q^{20} -2.83452 q^{22} -3.97313 q^{23} -3.03987 q^{25} +5.71214 q^{26} -2.13611 q^{28} +0.100627 q^{29} -5.43523 q^{31} +1.00000 q^{32} +5.87290 q^{34} +2.99065 q^{35} +6.63998 q^{37} -3.20948 q^{38} -1.40004 q^{40} -1.02469 q^{41} -3.45987 q^{43} -2.83452 q^{44} -3.97313 q^{46} +4.46817 q^{47} -2.43702 q^{49} -3.03987 q^{50} +5.71214 q^{52} +9.91571 q^{53} +3.96845 q^{55} -2.13611 q^{56} +0.100627 q^{58} -9.99566 q^{59} -0.434578 q^{61} -5.43523 q^{62} +1.00000 q^{64} -7.99726 q^{65} -9.79274 q^{67} +5.87290 q^{68} +2.99065 q^{70} +11.9919 q^{71} -11.4593 q^{73} +6.63998 q^{74} -3.20948 q^{76} +6.05484 q^{77} -0.914747 q^{79} -1.40004 q^{80} -1.02469 q^{82} -9.47992 q^{83} -8.22232 q^{85} -3.45987 q^{86} -2.83452 q^{88} +8.06705 q^{89} -12.2018 q^{91} -3.97313 q^{92} +4.46817 q^{94} +4.49342 q^{95} +0.417684 q^{97} -2.43702 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} + 8 q^{4} - 5 q^{7} + 8 q^{8}+O(q^{10}) \) 8 * q + 8 * q^2 + 8 * q^4 - 5 * q^7 + 8 * q^8	\( 8 q + 8 q^{2} + 8 q^{4} - 5 q^{7} + 8 q^{8} - 6 q^{11} - 8 q^{13} - 5 q^{14} + 8 q^{16} + 5 q^{17} - 14 q^{19} - 6 q^{22} - 21 q^{23} - 6 q^{25} - 8 q^{26} - 5 q^{28} - 3 q^{29} - 4 q^{31} + 8 q^{32} + 5 q^{34} + 2 q^{35} - 3 q^{37} - 14 q^{38} - 7 q^{41} - 12 q^{43} - 6 q^{44} - 21 q^{46} - 25 q^{47} - 7 q^{49} - 6 q^{50} - 8 q^{52} - 3 q^{53} - 9 q^{55} - 5 q^{56} - 3 q^{58} - 2 q^{59} - 17 q^{61} - 4 q^{62} + 8 q^{64} - 32 q^{65} - 14 q^{67} + 5 q^{68} + 2 q^{70} - 7 q^{71} - 10 q^{73} - 3 q^{74} - 14 q^{76} - 12 q^{77} - 33 q^{79} - 7 q^{82} - 13 q^{83} - 33 q^{85} - 12 q^{86} - 6 q^{88} + 22 q^{89} - 22 q^{91} - 21 q^{92} - 25 q^{94} + 14 q^{95} - 11 q^{97} - 7 q^{98}+O(q^{100}) \) 8 * q + 8 * q^2 + 8 * q^4 - 5 * q^7 + 8 * q^8 - 6 * q^11 - 8 * q^13 - 5 * q^14 + 8 * q^16 + 5 * q^17 - 14 * q^19 - 6 * q^22 - 21 * q^23 - 6 * q^25 - 8 * q^26 - 5 * q^28 - 3 * q^29 - 4 * q^31 + 8 * q^32 + 5 * q^34 + 2 * q^35 - 3 * q^37 - 14 * q^38 - 7 * q^41 - 12 * q^43 - 6 * q^44 - 21 * q^46 - 25 * q^47 - 7 * q^49 - 6 * q^50 - 8 * q^52 - 3 * q^53 - 9 * q^55 - 5 * q^56 - 3 * q^58 - 2 * q^59 - 17 * q^61 - 4 * q^62 + 8 * q^64 - 32 * q^65 - 14 * q^67 + 5 * q^68 + 2 * q^70 - 7 * q^71 - 10 * q^73 - 3 * q^74 - 14 * q^76 - 12 * q^77 - 33 * q^79 - 7 * q^82 - 13 * q^83 - 33 * q^85 - 12 * q^86 - 6 * q^88 + 22 * q^89 - 22 * q^91 - 21 * q^92 - 25 * q^94 + 14 * q^95 - 11 * q^97 - 7 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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