LMFDB - Embedded newform 8046.2.a.e.1.7

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			1.7
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	$-0.398646\pi$
$5$	1.40004	0.626119	0.313060	+	0.949733i	$0.398646\pi$
$6$	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	$-0.359458\pi$
$11$	2.83452	0.854638	0.427319	+	0.904101i	$0.359458\pi$
$12$	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	$-0.752298\pi$
$17$	−5.87290	−1.42439	−0.712193	+	0.701983i	$0.752298\pi$
$18$	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	$-0.364052\pi$
$23$	3.97313	0.828455	0.414228	+	0.910173i	$0.364052\pi$
$24$	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	$-0.502974\pi$
$29$	−0.100627	−0.0186859	−0.00934295	+	0.999956i	$0.502974\pi$
$30$	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	$-0.474503\pi$
$41$	1.02469	0.160030	0.0800148	+	0.996794i	$0.474503\pi$
$42$	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	$-0.605659\pi$
$47$	−4.46817	−0.651749	−0.325875	+	0.945413i	$0.605659\pi$
$48$	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	$-0.738461\pi$
$53$	−9.91571	−1.36203	−0.681014	+	0.732270i	$0.738461\pi$
$54$	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	$-0.274492\pi$
$59$	9.99566	1.30132	0.650662	+	0.759368i	$0.274492\pi$
$60$	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	$-0.752025\pi$
$71$	−11.9919	−1.42318	−0.711590	+	0.702595i	$0.752025\pi$
$72$	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	$-0.325828\pi$
$83$	9.47992	1.04056	0.520278	+	0.853997i	$0.325828\pi$
$84$	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	$-0.640624\pi$
$89$	−8.06705	−0.855106	−0.427553	+	0.903990i	$0.640624\pi$
$90$	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	$-0.547662\pi$
$101$	−2.99840	−0.298352	−0.149176	+	0.988811i	$0.547662\pi$
$102$	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	$-0.271841\pi$
$107$	13.5914	1.31393	0.656963	+	0.753923i	$0.271841\pi$
$108$	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	$-0.389114\pi$
$113$	7.25732	0.682712	0.341356	+	0.939934i	$0.389114\pi$
$114$	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	$-0.368427\pi$
$131$	9.19483	0.803356	0.401678	+	0.915781i	$0.368427\pi$
$132$	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	$-0.603652\pi$
$137$	−7.48889	−0.639819	−0.319909	+	0.947448i	$0.603652\pi$
$138$	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	$-0.291673\pi$
$167$	15.7335	1.21749	0.608746	+	0.793365i	$0.291673\pi$
$168$	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	$-0.704685\pi$
$173$	−15.7737	−1.19926	−0.599628	+	0.800279i	$0.704685\pi$
$174$	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	$-0.515306\pi$
$179$	−1.28621	−0.0961361	−0.0480680	+	0.998844i	$0.515306\pi$
$180$	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	$-0.242767\pi$
$191$	19.9838	1.44598	0.722990	+	0.690858i	$0.242767\pi$
$192$	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	$-0.432509\pi$
$197$	5.90747	0.420890	0.210445	+	0.977606i	$0.432509\pi$
$198$	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	$-0.254324\pi$
$227$	21.0159	1.39487	0.697437	+	0.716646i	$0.254324\pi$
$228$	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	$-0.238632\pi$
$233$	22.3440	1.46381	0.731904	+	0.681408i	$0.238632\pi$
$234$	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	$-0.603738\pi$
$239$	−9.89922	−0.640327	−0.320164	+	0.947362i	$0.603738\pi$
$240$	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	$-0.650071\pi$
$251$	−14.3914	−0.908377	−0.454188	+	0.890906i	$0.650071\pi$
$252$	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	$-0.474173\pi$
$257$	2.59865	0.162099	0.0810495	+	0.996710i	$0.474173\pi$
$258$	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	$-0.480197\pi$
$263$	2.01653	0.124345	0.0621724	+	0.998065i	$0.480197\pi$
$264$	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	$-0.237730\pi$
$269$	24.0715	1.46766	0.733832	+	0.679331i	$0.237730\pi$
$270$	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	$-0.537621\pi$
$281$	−3.95321	−0.235829	−0.117914	+	0.993024i	$0.537621\pi$
$282$	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	$-0.302809\pi$
$293$	19.8773	1.16125	0.580623	+	0.814172i	$0.302809\pi$
$294$	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	$-0.580152\pi$
$311$	−8.78774	−0.498307	−0.249153	+	0.968464i	$0.580152\pi$
$312$	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	$-0.589647\pi$
$317$	−9.89669	−0.555853	−0.277927	+	0.960602i	$0.589647\pi$
$318$	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	$-0.524534\pi$
$347$	−2.86874	−0.154002	−0.0770010	+	0.997031i	$0.524534\pi$
$348$	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	$-0.443939\pi$
$353$	6.58385	0.350423	0.175211	+	0.984531i	$0.443939\pi$
$354$	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	$-0.433110\pi$
$359$	7.90476	0.417197	0.208599	+	0.978001i	$0.433110\pi$
$360$	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	$-0.472234\pi$
$383$	3.40986	0.174236	0.0871179	+	0.996198i	$0.472234\pi$
$384$	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	$-0.573648\pi$
$389$	−9.04552	−0.458626	−0.229313	+	0.973353i	$0.573648\pi$
$390$	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	$-0.654068\pi$
$401$	−18.6368	−0.930679	−0.465340	+	0.885132i	$0.654068\pi$
$402$	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	$-0.692953\pi$
$419$	−23.3242	−1.13946	−0.569731	+	0.821831i	$0.692953\pi$
$420$	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	$-0.624599\pi$
$431$	−15.8410	−0.763036	−0.381518	+	0.924361i	$0.624599\pi$
$432$	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	$-0.491105\pi$
$443$	1.17623	0.0558843	0.0279421	+	0.999610i	$0.491105\pi$
$444$	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	$-0.511995\pi$
$449$	−1.59665	−0.0753504	−0.0376752	+	0.999290i	$0.511995\pi$
$450$	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	$-0.777074\pi$
$461$	−32.8342	−1.52924	−0.764621	+	0.644480i	$0.777074\pi$
$462$	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	$-0.578170\pi$
$467$	−10.5076	−0.486232	−0.243116	+	0.969997i	$0.578170\pi$
$468$	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	$-0.592344\pi$
$479$	−12.5213	−0.572111	−0.286056	+	0.958213i	$0.592344\pi$
$480$	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	$-0.641501\pi$
$491$	−19.0581	−0.860080	−0.430040	+	0.902810i	$0.641501\pi$
$492$	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	$-0.408871\pi$
$503$	12.6670	0.564792	0.282396	+	0.959298i	$0.408871\pi$
$504$	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	$-0.480764\pi$
$509$	2.72517	0.120791	0.0603954	+	0.998175i	$0.480764\pi$
$510$	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	$-0.554256\pi$
$521$	−7.74356	−0.339252	−0.169626	+	0.985509i	$0.554256\pi$
$522$	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	$-0.755103\pi$
$557$	−33.9075	−1.43670	−0.718352	+	0.695679i	$0.755103\pi$
$558$	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	$-0.729907\pi$
$563$	−31.3723	−1.32219	−0.661093	+	0.750304i	$0.729907\pi$
$564$	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	$-0.342013\pi$
$569$	22.7184	0.952405	0.476202	+	0.879336i	$0.342013\pi$
$570$	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	$-0.630447\pi$
$587$	−19.3067	−0.796874	−0.398437	+	0.917196i	$0.630447\pi$
$588$	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	$-0.649139\pi$
$593$	−21.9933	−0.903155	−0.451578	+	0.892232i	$0.649139\pi$
$594$	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	$-0.468716\pi$
$599$	4.80300	0.196245	0.0981227	+	0.995174i	$0.468716\pi$
$600$	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	$-0.423909\pi$
$617$	11.7628	0.473551	0.236776	+	0.971564i	$0.423909\pi$
$618$	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	$-0.216648\pi$
$641$	39.3534	1.55436	0.777182	+	0.629275i	$0.216648\pi$
$642$	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	$-0.462603\pi$
$647$	5.96301	0.234430	0.117215	+	0.993107i	$0.462603\pi$
$648$	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	$-0.516702\pi$
$653$	−2.68049	−0.104896	−0.0524479	+	0.998624i	$0.516702\pi$
$654$	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	$-0.615544\pi$
$659$	−18.2301	−0.710144	−0.355072	+	0.934839i	$0.615544\pi$
$660$	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	$-0.321488\pi$
$677$	27.6779	1.06375	0.531875	+	0.846823i	$0.321488\pi$
$678$	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	$-0.655498\pi$
$683$	−24.5302	−0.938621	−0.469311	+	0.883033i	$0.655498\pi$
$684$	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	$-0.627782\pi$
$701$	−20.6909	−0.781484	−0.390742	+	0.920500i	$0.627782\pi$
$702$	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	$-0.406750\pi$
$719$	15.4869	0.577565	0.288782	+	0.957395i	$0.406750\pi$
$720$	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	$-0.629414\pi$
$743$	−21.5588	−0.790916	−0.395458	+	0.918484i	$0.629414\pi$
$744$	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	$-0.764433\pi$
$761$	−40.7411	−1.47686	−0.738431	+	0.674329i	$0.764433\pi$
$762$	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	$-0.299542\pi$
$773$	32.7490	1.17790	0.588949	+	0.808170i	$0.299542\pi$
$774$	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	$-0.420197\pi$
$797$	14.0078	0.496182	0.248091	+	0.968737i	$0.420197\pi$
$798$	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	$-0.530694\pi$
$809$	−5.47697	−0.192560	−0.0962800	+	0.995354i	$0.530694\pi$
$810$	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	$-0.600242\pi$
$821$	−17.7501	−0.619483	−0.309741	+	0.950821i	$0.600242\pi$
$822$	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	$-0.329750\pi$
$827$	29.3165	1.01943	0.509717	+	0.860342i	$0.329750\pi$
$828$	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	$-0.347277\pi$
$839$	26.7407	0.923191	0.461596	+	0.887090i	$0.347277\pi$
$840$	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	$-0.527100\pi$
$857$	−4.97876	−0.170071	−0.0850355	+	0.996378i	$0.527100\pi$
$858$	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	$-0.544724\pi$
$863$	−8.22800	−0.280085	−0.140042	+	0.990146i	$0.544724\pi$
$864$	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	$-0.398975\pi$
$881$	18.5260	0.624158	0.312079	+	0.950056i	$0.398975\pi$
$882$	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	$-0.721444\pi$
$887$	−38.1760	−1.28182	−0.640912	+	0.767614i	$0.721444\pi$
$888$	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	$-0.639382\pi$
$911$	−25.5963	−0.848045	−0.424022	+	0.905652i	$0.639382\pi$
$912$	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	$-0.498536\pi$
$929$	0.280436	0.00920081	0.00460040	+	0.999989i	$0.498536\pi$
$930$	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	$-0.700880\pi$
$941$	−36.1986	−1.18004	−0.590020	+	0.807388i	$0.700880\pi$
$942$	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	$-0.654533\pi$
$947$	−28.7197	−0.933265	−0.466633	+	0.884451i	$0.654533\pi$
$948$	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	$-0.724472\pi$
$953$	−40.0200	−1.29637	−0.648187	+	0.761481i	$0.724472\pi$
$954$	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	$-0.755423\pi$
$971$	−44.8124	−1.43810	−0.719050	+	0.694959i	$0.755423\pi$
$972$	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	$-0.700222\pi$
$977$	−36.7800	−1.17670	−0.588349	+	0.808607i	$0.700222\pi$
$978$	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	$-0.571947\pi$
$983$	−14.0529	−0.448219	−0.224110	+	0.974564i	$0.571947\pi$
$984$	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.e.1.7	✓	8
3.2	odd	2		8046.2.a.f.1.2	yes	8

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

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