LMFDB - Embedded newform 8046.2.a.s.1.3

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:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} - 46 x^{14} + 192 x^{13} + 752 x^{12} - 3378 x^{11} - 5277 x^{10} + 27132 x^{9} + \cdots - 4260 $ x^16 - 4x^15 - 46x^14 + 192x^13 + 752x^12 - 3378x^11 - 5277x^10 + 27132x^9 + 15173x^8 - 102466x^7 - 16612x^6 + 186120x^5 + 117x^4 - 156722x^3 + 10545x^2 + 48580*x - 4260
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			1.3
Root			$3.23488$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -3.23488 q^{5} -1.87212 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -3.23488 q^{5} -1.87212 q^{7} -1.00000 q^{8} +3.23488 q^{10} -4.66557 q^{11} +3.77562 q^{13} +1.87212 q^{14} +1.00000 q^{16} -4.68172 q^{17} +2.83250 q^{19} -3.23488 q^{20} +4.66557 q^{22} -4.42544 q^{23} +5.46445 q^{25} -3.77562 q^{26} -1.87212 q^{28} +7.42347 q^{29} +8.38774 q^{31} -1.00000 q^{32} +4.68172 q^{34} +6.05608 q^{35} -5.87734 q^{37} -2.83250 q^{38} +3.23488 q^{40} -7.57269 q^{41} -8.17859 q^{43} -4.66557 q^{44} +4.42544 q^{46} +1.31778 q^{47} -3.49517 q^{49} -5.46445 q^{50} +3.77562 q^{52} -7.98570 q^{53} +15.0926 q^{55} +1.87212 q^{56} -7.42347 q^{58} -0.0111878 q^{59} -13.5308 q^{61} -8.38774 q^{62} +1.00000 q^{64} -12.2137 q^{65} -14.8561 q^{67} -4.68172 q^{68} -6.05608 q^{70} +8.49039 q^{71} -16.8516 q^{73} +5.87734 q^{74} +2.83250 q^{76} +8.73450 q^{77} -4.48279 q^{79} -3.23488 q^{80} +7.57269 q^{82} -11.1948 q^{83} +15.1448 q^{85} +8.17859 q^{86} +4.66557 q^{88} -7.32113 q^{89} -7.06840 q^{91} -4.42544 q^{92} -1.31778 q^{94} -9.16281 q^{95} -12.4460 q^{97} +3.49517 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{2} + 16 q^{4} - 4 q^{5} + 6 q^{7} - 16 q^{8}+O(q^{10}) $ 16 * q - 16 * q^2 + 16 * q^4 - 4 * q^5 + 6 * q^7 - 16 * q^8	$ 16 q - 16 q^{2} + 16 q^{4} - 4 q^{5} + 6 q^{7} - 16 q^{8} + 4 q^{10} - 6 q^{11} + 6 q^{13} - 6 q^{14} + 16 q^{16} - q^{17} + 10 q^{19} - 4 q^{20} + 6 q^{22} - 10 q^{23} + 28 q^{25} - 6 q^{26} + 6 q^{28} - 6 q^{29} + 21 q^{31} - 16 q^{32} + q^{34} - 16 q^{35} + 17 q^{37} - 10 q^{38} + 4 q^{40} + 4 q^{41} + 16 q^{43} - 6 q^{44} + 10 q^{46} - 25 q^{47} + 36 q^{49} - 28 q^{50} + 6 q^{52} - 14 q^{53} + 19 q^{55} - 6 q^{56} + 6 q^{58} - 6 q^{59} + 23 q^{61} - 21 q^{62} + 16 q^{64} - 20 q^{65} + 22 q^{67} - q^{68} + 16 q^{70} - 10 q^{71} + 16 q^{73} - 17 q^{74} + 10 q^{76} + 2 q^{77} + 37 q^{79} - 4 q^{80} - 4 q^{82} - 33 q^{83} + 43 q^{85} - 16 q^{86} + 6 q^{88} + 3 q^{89} + 28 q^{91} - 10 q^{92} + 25 q^{94} - 14 q^{95} - 3 q^{97} - 36 q^{98}+O(q^{100}) $ 16 * q - 16 * q^2 + 16 * q^4 - 4 * q^5 + 6 * q^7 - 16 * q^8 + 4 * q^10 - 6 * q^11 + 6 * q^13 - 6 * q^14 + 16 * q^16 - q^17 + 10 * q^19 - 4 * q^20 + 6 * q^22 - 10 * q^23 + 28 * q^25 - 6 * q^26 + 6 * q^28 - 6 * q^29 + 21 * q^31 - 16 * q^32 + q^34 - 16 * q^35 + 17 * q^37 - 10 * q^38 + 4 * q^40 + 4 * q^41 + 16 * q^43 - 6 * q^44 + 10 * q^46 - 25 * q^47 + 36 * q^49 - 28 * q^50 + 6 * q^52 - 14 * q^53 + 19 * q^55 - 6 * q^56 + 6 * q^58 - 6 * q^59 + 23 * q^61 - 21 * q^62 + 16 * q^64 - 20 * q^65 + 22 * q^67 - q^68 + 16 * q^70 - 10 * q^71 + 16 * q^73 - 17 * q^74 + 10 * q^76 + 2 * q^77 + 37 * q^79 - 4 * q^80 - 4 * q^82 - 33 * q^83 + 43 * q^85 - 16 * q^86 + 6 * q^88 + 3 * q^89 + 28 * q^91 - 10 * q^92 + 25 * q^94 - 14 * q^95 - 3 * q^97 - 36 * 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
$5$	−3.23488	−1.44668	−0.723341	−	0.690491i	$-0.757395\pi$
$5$	−3.23488	−1.44668	−0.723341	+	0.690491i	$0.757395\pi$
$6$	0	0
$7$	−1.87212	−0.707594	−0.353797	−	0.935322i	$-0.615110\pi$
$7$	−1.87212	−0.707594	−0.353797	+	0.935322i	$0.615110\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	3.23488	1.02296
$11$	−4.66557	−1.40672	−0.703361	−	0.710833i	$-0.748318\pi$
$11$	−4.66557	−1.40672	−0.703361	+	0.710833i	$0.748318\pi$
$12$	0	0
$13$	3.77562	1.04717	0.523584	−	0.851974i	$-0.324595\pi$
$13$	3.77562	1.04717	0.523584	+	0.851974i	$0.324595\pi$
$14$	1.87212	0.500345
$15$	0	0
$16$	1.00000	0.250000
$17$	−4.68172	−1.13548	−0.567742	−	0.823207i	$-0.692183\pi$
$17$	−4.68172	−1.13548	−0.567742	+	0.823207i	$0.692183\pi$
$18$	0	0
$19$	2.83250	0.649821	0.324910	−	0.945745i	$-0.394666\pi$
$19$	2.83250	0.649821	0.324910	+	0.945745i	$0.394666\pi$
$20$	−3.23488	−0.723341
$21$	0	0
$22$	4.66557	0.994703
$23$	−4.42544	−0.922767	−0.461384	−	0.887201i	$-0.652647\pi$
$23$	−4.42544	−0.922767	−0.461384	+	0.887201i	$0.652647\pi$
$24$	0	0
$25$	5.46445	1.09289
$26$	−3.77562	−0.740459
$27$	0	0
$28$	−1.87212	−0.353797
$29$	7.42347	1.37850	0.689251	−	0.724522i	$-0.257940\pi$
$29$	7.42347	1.37850	0.689251	+	0.724522i	$0.257940\pi$
$30$	0	0
$31$	8.38774	1.50648	0.753241	−	0.657744i	$-0.228489\pi$
$31$	8.38774	1.50648	0.753241	+	0.657744i	$0.228489\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	4.68172	0.802908
$35$	6.05608	1.02366
$36$	0	0
$37$	−5.87734	−0.966228	−0.483114	−	0.875557i	$-0.660494\pi$
$37$	−5.87734	−0.966228	−0.483114	+	0.875557i	$0.660494\pi$
$38$	−2.83250	−0.459493
$39$	0	0
$40$	3.23488	0.511479
$41$	−7.57269	−1.18266	−0.591328	−	0.806431i	$-0.701396\pi$
$41$	−7.57269	−1.18266	−0.591328	+	0.806431i	$0.701396\pi$
$42$	0	0
$43$	−8.17859	−1.24722	−0.623612	−	0.781734i	$-0.714335\pi$
$43$	−8.17859	−1.24722	−0.623612	+	0.781734i	$0.714335\pi$
$44$	−4.66557	−0.703361
$45$	0	0
$46$	4.42544	0.652495
$47$	1.31778	0.192218	0.0961088	−	0.995371i	$-0.469360\pi$
$47$	1.31778	0.192218	0.0961088	+	0.995371i	$0.469360\pi$
$48$	0	0
$49$	−3.49517	−0.499310
$50$	−5.46445	−0.772790
$51$	0	0
$52$	3.77562	0.523584
$53$	−7.98570	−1.09692	−0.548460	−	0.836177i	$-0.684786\pi$
$53$	−7.98570	−1.09692	−0.548460	+	0.836177i	$0.684786\pi$
$54$	0	0
$55$	15.0926	2.03508
$56$	1.87212	0.250172
$57$	0	0
$58$	−7.42347	−0.974749
$59$	−0.0111878	−0.00145653	−0.000728263	−	1.00000i	$-0.500232\pi$
$59$	−0.0111878	−0.00145653	−0.000728263		1.00000i	$0.500232\pi$
$60$	0	0
$61$	−13.5308	−1.73244	−0.866221	−	0.499660i	$-0.833458\pi$
$61$	−13.5308	−1.73244	−0.866221	+	0.499660i	$0.833458\pi$
$62$	−8.38774	−1.06524
$63$	0	0
$64$	1.00000	0.125000
$65$	−12.2137	−1.51492
$66$	0	0
$67$	−14.8561	−1.81496	−0.907481	−	0.420093i	$-0.861997\pi$
$67$	−14.8561	−1.81496	−0.907481	+	0.420093i	$0.861997\pi$
$68$	−4.68172	−0.567742
$69$	0	0
$70$	−6.05608	−0.723840
$71$	8.49039	1.00762	0.503812	−	0.863814i	$-0.331931\pi$
$71$	8.49039	1.00762	0.503812	+	0.863814i	$0.331931\pi$
$72$	0	0
$73$	−16.8516	−1.97233	−0.986167	−	0.165753i	$-0.946995\pi$
$73$	−16.8516	−1.97233	−0.986167	+	0.165753i	$0.946995\pi$
$74$	5.87734	0.683227
$75$	0	0
$76$	2.83250	0.324910
$77$	8.73450	0.995389
$78$	0	0
$79$	−4.48279	−0.504354	−0.252177	−	0.967681i	$-0.581146\pi$
$79$	−4.48279	−0.504354	−0.252177	+	0.967681i	$0.581146\pi$
$80$	−3.23488	−0.361671
$81$	0	0
$82$	7.57269	0.836264
$83$	−11.1948	−1.22879	−0.614394	−	0.789000i	$-0.710599\pi$
$83$	−11.1948	−1.22879	−0.614394	+	0.789000i	$0.710599\pi$
$84$	0	0
$85$	15.1448	1.64268
$86$	8.17859	0.881920
$87$	0	0
$88$	4.66557	0.497351
$89$	−7.32113	−0.776039	−0.388019	−	0.921651i	$-0.626841\pi$
$89$	−7.32113	−0.776039	−0.388019	+	0.921651i	$0.626841\pi$
$90$	0	0
$91$	−7.06840	−0.740970
$92$	−4.42544	−0.461384
$93$	0	0
$94$	−1.31778	−0.135918
$95$	−9.16281	−0.940085
$96$	0	0
$97$	−12.4460	−1.26370	−0.631850	−	0.775091i	$-0.717704\pi$
$97$	−12.4460	−1.26370	−0.631850	+	0.775091i	$0.717704\pi$
$98$	3.49517	0.353066
$99$	0	0
$100$	5.46445	0.546445
$101$	3.51868	0.350121	0.175061	−	0.984558i	$-0.443988\pi$
$101$	3.51868	0.350121	0.175061	+	0.984558i	$0.443988\pi$
$102$	0	0
$103$	6.14904	0.605883	0.302941	−	0.953009i	$-0.402031\pi$
$103$	6.14904	0.605883	0.302941	+	0.953009i	$0.402031\pi$
$104$	−3.77562	−0.370230
$105$	0	0
$106$	7.98570	0.775640
$107$	−0.862074	−0.0833398	−0.0416699	−	0.999131i	$-0.513268\pi$
$107$	−0.862074	−0.0833398	−0.0416699	+	0.999131i	$0.513268\pi$
$108$	0	0
$109$	11.0784	1.06112	0.530560	−	0.847647i	$-0.321982\pi$
$109$	11.0784	1.06112	0.530560	+	0.847647i	$0.321982\pi$
$110$	−15.0926	−1.43902
$111$	0	0
$112$	−1.87212	−0.176899
$113$	16.0867	1.51331	0.756654	−	0.653815i	$-0.226833\pi$
$113$	16.0867	1.51331	0.756654	+	0.653815i	$0.226833\pi$
$114$	0	0
$115$	14.3158	1.33495
$116$	7.42347	0.689251
$117$	0	0
$118$	0.0111878	0.00102992
$119$	8.76473	0.803462
$120$	0	0
$121$	10.7675	0.978868
$122$	13.5308	1.22502
$123$	0	0
$124$	8.38774	0.753241
$125$	−1.50243	−0.134382
$126$	0	0
$127$	5.16232	0.458082	0.229041	−	0.973417i	$-0.426441\pi$
$127$	5.16232	0.458082	0.229041	+	0.973417i	$0.426441\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	12.2137	1.07121
$131$	−3.81504	−0.333322	−0.166661	−	0.986014i	$-0.553299\pi$
$131$	−3.81504	−0.333322	−0.166661	+	0.986014i	$0.553299\pi$
$132$	0	0
$133$	−5.30278	−0.459810
$134$	14.8561	1.28337
$135$	0	0
$136$	4.68172	0.401454
$137$	−8.93528	−0.763392	−0.381696	−	0.924288i	$-0.624660\pi$
$137$	−8.93528	−0.763392	−0.381696	+	0.924288i	$0.624660\pi$
$138$	0	0
$139$	11.9035	1.00964	0.504819	−	0.863225i	$-0.331559\pi$
$139$	11.9035	1.00964	0.504819	+	0.863225i	$0.331559\pi$
$140$	6.05608	0.511832
$141$	0	0
$142$	−8.49039	−0.712497
$143$	−17.6154	−1.47307
$144$	0	0
$145$	−24.0140	−1.99426
$146$	16.8516	1.39465
$147$	0	0
$148$	−5.87734	−0.483114
$149$	1.00000	0.0819232
$149$	1.00000	0.0819232
$150$	0	0
$151$	−19.5730	−1.59283	−0.796413	−	0.604752i	$-0.793272\pi$
$151$	−19.5730	−1.59283	−0.796413	+	0.604752i	$0.793272\pi$
$152$	−2.83250	−0.229746
$153$	0	0
$154$	−8.73450	−0.703846
$155$	−27.1333	−2.17940
$156$	0	0
$157$	8.16484	0.651625	0.325812	−	0.945434i	$-0.394362\pi$
$157$	8.16484	0.651625	0.325812	+	0.945434i	$0.394362\pi$
$158$	4.48279	0.356632
$159$	0	0
$160$	3.23488	0.255740
$161$	8.28494	0.652945
$162$	0	0
$163$	−14.7237	−1.15325	−0.576623	−	0.817010i	$-0.695630\pi$
$163$	−14.7237	−1.15325	−0.576623	+	0.817010i	$0.695630\pi$
$164$	−7.57269	−0.591328
$165$	0	0
$166$	11.1948	0.868884
$167$	−3.40133	−0.263203	−0.131601	−	0.991303i	$-0.542012\pi$
$167$	−3.40133	−0.263203	−0.131601	+	0.991303i	$0.542012\pi$
$168$	0	0
$169$	1.25528	0.0965597
$170$	−15.1448	−1.16155
$171$	0	0
$172$	−8.17859	−0.623612
$173$	12.3853	0.941635	0.470817	−	0.882231i	$-0.343959\pi$
$173$	12.3853	0.941635	0.470817	+	0.882231i	$0.343959\pi$
$174$	0	0
$175$	−10.2301	−0.773322
$176$	−4.66557	−0.351681
$177$	0	0
$178$	7.32113	0.548742
$179$	−4.42416	−0.330677	−0.165339	−	0.986237i	$-0.552872\pi$
$179$	−4.42416	−0.330677	−0.165339	+	0.986237i	$0.552872\pi$
$180$	0	0
$181$	5.56556	0.413685	0.206842	−	0.978374i	$-0.433681\pi$
$181$	5.56556	0.413685	0.206842	+	0.978374i	$0.433681\pi$
$182$	7.06840	0.523945
$183$	0	0
$184$	4.42544	0.326247
$185$	19.0125	1.39783
$186$	0	0
$187$	21.8429	1.59731
$188$	1.31778	0.0961088
$189$	0	0
$190$	9.16281	0.664740
$191$	19.4431	1.40685	0.703427	−	0.710768i	$-0.251652\pi$
$191$	19.4431	1.40685	0.703427	+	0.710768i	$0.251652\pi$
$192$	0	0
$193$	18.1823	1.30879	0.654397	−	0.756151i	$-0.272922\pi$
$193$	18.1823	1.30879	0.654397	+	0.756151i	$0.272922\pi$
$194$	12.4460	0.893571
$195$	0	0
$196$	−3.49517	−0.249655
$197$	−3.60218	−0.256645	−0.128323	−	0.991732i	$-0.540959\pi$
$197$	−3.60218	−0.256645	−0.128323	+	0.991732i	$0.540959\pi$
$198$	0	0
$199$	10.7345	0.760948	0.380474	−	0.924792i	$-0.375761\pi$
$199$	10.7345	0.760948	0.380474	+	0.924792i	$0.375761\pi$
$200$	−5.46445	−0.386395
$201$	0	0
$202$	−3.51868	−0.247573
$203$	−13.8976	−0.975421
$204$	0	0
$205$	24.4967	1.71093
$206$	−6.14904	−0.428424
$207$	0	0
$208$	3.77562	0.261792
$209$	−13.2152	−0.914118
$210$	0	0
$211$	19.4045	1.33586	0.667929	−	0.744225i	$-0.267181\pi$
$211$	19.4045	1.33586	0.667929	+	0.744225i	$0.267181\pi$
$212$	−7.98570	−0.548460
$213$	0	0
$214$	0.862074	0.0589302
$215$	26.4568	1.80434
$216$	0	0
$217$	−15.7028	−1.06598
$218$	−11.0784	−0.750325
$219$	0	0
$220$	15.0926	1.01754
$221$	−17.6764	−1.18904
$222$	0	0
$223$	0.0723012	0.00484164	0.00242082	−	0.999997i	$-0.499229\pi$
$223$	0.0723012	0.00484164	0.00242082	+	0.999997i	$0.499229\pi$
$224$	1.87212	0.125086
$225$	0	0
$226$	−16.0867	−1.07007
$227$	11.1563	0.740471	0.370236	−	0.928938i	$-0.379277\pi$
$227$	11.1563	0.740471	0.370236	+	0.928938i	$0.379277\pi$
$228$	0	0
$229$	18.4325	1.21806	0.609028	−	0.793149i	$-0.291560\pi$
$229$	18.4325	1.21806	0.609028	+	0.793149i	$0.291560\pi$
$230$	−14.3158	−0.943953
$231$	0	0
$232$	−7.42347	−0.487374
$233$	−15.3706	−1.00696	−0.503481	−	0.864006i	$-0.667948\pi$
$233$	−15.3706	−1.00696	−0.503481	+	0.864006i	$0.667948\pi$
$234$	0	0
$235$	−4.26285	−0.278078
$236$	−0.0111878	−0.000728263 0
$237$	0	0
$238$	−8.76473	−0.568133
$239$	10.1084	0.653858	0.326929	−	0.945049i	$-0.393986\pi$
$239$	10.1084	0.653858	0.326929	+	0.945049i	$0.393986\pi$
$240$	0	0
$241$	10.3305	0.665447	0.332724	−	0.943024i	$-0.392032\pi$
$241$	10.3305	0.665447	0.332724	+	0.943024i	$0.392032\pi$
$242$	−10.7675	−0.692164
$243$	0	0
$244$	−13.5308	−0.866221
$245$	11.3065	0.722343
$246$	0	0
$247$	10.6944	0.680471
$248$	−8.38774	−0.532622
$249$	0	0
$250$	1.50243	0.0950222
$251$	−29.1816	−1.84193	−0.920964	−	0.389647i	$-0.872597\pi$
$251$	−29.1816	−1.84193	−0.920964	+	0.389647i	$0.872597\pi$
$252$	0	0
$253$	20.6472	1.29808
$254$	−5.16232	−0.323913
$255$	0	0
$256$	1.00000	0.0625000
$257$	16.6147	1.03639	0.518197	−	0.855261i	$-0.326603\pi$
$257$	16.6147	1.03639	0.518197	+	0.855261i	$0.326603\pi$
$258$	0	0
$259$	11.0031	0.683698
$260$	−12.2137	−0.757459
$261$	0	0
$262$	3.81504	0.235694
$263$	−24.8311	−1.53115	−0.765575	−	0.643346i	$-0.777546\pi$
$263$	−24.8311	−1.53115	−0.765575	+	0.643346i	$0.777546\pi$
$264$	0	0
$265$	25.8328	1.58690
$266$	5.30278	0.325134
$267$	0	0
$268$	−14.8561	−0.907481
$269$	−7.21483	−0.439896	−0.219948	−	0.975512i	$-0.570589\pi$
$269$	−7.21483	−0.439896	−0.219948	+	0.975512i	$0.570589\pi$
$270$	0	0
$271$	−16.0375	−0.974207	−0.487103	−	0.873344i	$-0.661946\pi$
$271$	−16.0375	−0.974207	−0.487103	+	0.873344i	$0.661946\pi$
$272$	−4.68172	−0.283871
$273$	0	0
$274$	8.93528	0.539800
$275$	−25.4948	−1.53739
$276$	0	0
$277$	−20.3652	−1.22363	−0.611813	−	0.791003i	$-0.709559\pi$
$277$	−20.3652	−1.22363	−0.611813	+	0.791003i	$0.709559\pi$
$278$	−11.9035	−0.713922
$279$	0	0
$280$	−6.05608	−0.361920
$281$	32.1042	1.91518	0.957588	−	0.288139i	$-0.0930367\pi$
$281$	32.1042	1.91518	0.957588	+	0.288139i	$0.0930367\pi$
$282$	0	0
$283$	−9.53278	−0.566665	−0.283332	−	0.959022i	$-0.591440\pi$
$283$	−9.53278	−0.566665	−0.283332	+	0.959022i	$0.591440\pi$
$284$	8.49039	0.503812
$285$	0	0
$286$	17.6154	1.04162
$287$	14.1770	0.836840
$288$	0	0
$289$	4.91849	0.289323
$290$	24.0140	1.41015
$291$	0	0
$292$	−16.8516	−0.986167
$293$	19.7079	1.15135	0.575673	−	0.817680i	$-0.304740\pi$
$293$	19.7079	1.15135	0.575673	+	0.817680i	$0.304740\pi$
$294$	0	0
$295$	0.0361912	0.00210713
$296$	5.87734	0.341613
$297$	0	0
$298$	−1.00000	−0.0579284
$299$	−16.7087	−0.966292
$300$	0	0
$301$	15.3113	0.882528
$302$	19.5730	1.12630
$303$	0	0
$304$	2.83250	0.162455
$305$	43.7706	2.50629
$306$	0	0
$307$	32.3057	1.84379	0.921893	−	0.387444i	$-0.126642\pi$
$307$	32.3057	1.84379	0.921893	+	0.387444i	$0.126642\pi$
$308$	8.73450	0.497694
$309$	0	0
$310$	27.1333	1.54107
$311$	−10.6783	−0.605514	−0.302757	−	0.953068i	$-0.597907\pi$
$311$	−10.6783	−0.605514	−0.302757	+	0.953068i	$0.597907\pi$
$312$	0	0
$313$	−1.11853	−0.0632229	−0.0316115	−	0.999500i	$-0.510064\pi$
$313$	−1.11853	−0.0632229	−0.0316115	+	0.999500i	$0.510064\pi$
$314$	−8.16484	−0.460768
$315$	0	0
$316$	−4.48279	−0.252177
$317$	16.0206	0.899808	0.449904	−	0.893077i	$-0.351458\pi$
$317$	16.0206	0.899808	0.449904	+	0.893077i	$0.351458\pi$
$318$	0	0
$319$	−34.6347	−1.93917
$320$	−3.23488	−0.180835
$321$	0	0
$322$	−8.28494	−0.461702
$323$	−13.2610	−0.737861
$324$	0	0
$325$	20.6317	1.14444
$326$	14.7237	0.815468
$327$	0	0
$328$	7.57269	0.418132
$329$	−2.46703	−0.136012
$330$	0	0
$331$	−0.490745	−0.0269738	−0.0134869	−	0.999909i	$-0.504293\pi$
$331$	−0.490745	−0.0269738	−0.0134869	+	0.999909i	$0.504293\pi$
$332$	−11.1948	−0.614394
$333$	0	0
$334$	3.40133	0.186113
$335$	48.0577	2.62567
$336$	0	0
$337$	10.6204	0.578532	0.289266	−	0.957249i	$-0.406589\pi$
$337$	10.6204	0.578532	0.289266	+	0.957249i	$0.406589\pi$
$338$	−1.25528	−0.0682780
$339$	0	0
$340$	15.1448	0.821342
$341$	−39.1336	−2.11920
$342$	0	0
$343$	19.6482	1.06090
$344$	8.17859	0.440960
$345$	0	0
$346$	−12.3853	−0.665836
$347$	18.6395	1.00062	0.500312	−	0.865845i	$-0.333219\pi$
$347$	18.6395	1.00062	0.500312	+	0.865845i	$0.333219\pi$
$348$	0	0
$349$	25.5174	1.36591	0.682957	−	0.730459i	$-0.260694\pi$
$349$	25.5174	1.36591	0.682957	+	0.730459i	$0.260694\pi$
$350$	10.2301	0.546822
$351$	0	0
$352$	4.66557	0.248676
$353$	−4.63419	−0.246653	−0.123327	−	0.992366i	$-0.539356\pi$
$353$	−4.63419	−0.246653	−0.123327	+	0.992366i	$0.539356\pi$
$354$	0	0
$355$	−27.4654	−1.45771
$356$	−7.32113	−0.388019
$357$	0	0
$358$	4.42416	0.233824
$359$	−4.43020	−0.233817	−0.116908	−	0.993143i	$-0.537298\pi$
$359$	−4.43020	−0.233817	−0.116908	+	0.993143i	$0.537298\pi$
$360$	0	0
$361$	−10.9769	−0.577733
$362$	−5.56556	−0.292519
$363$	0	0
$364$	−7.06840	−0.370485
$365$	54.5130	2.85334
$366$	0	0
$367$	−2.04147	−0.106564	−0.0532821	−	0.998580i	$-0.516968\pi$
$367$	−2.04147	−0.106564	−0.0532821	+	0.998580i	$0.516968\pi$
$368$	−4.42544	−0.230692
$369$	0	0
$370$	−19.0125	−0.988412
$371$	14.9502	0.776175
$372$	0	0
$373$	−7.59725	−0.393371	−0.196685	−	0.980467i	$-0.563018\pi$
$373$	−7.59725	−0.393371	−0.196685	+	0.980467i	$0.563018\pi$
$374$	−21.8429	−1.12947
$375$	0	0
$376$	−1.31778	−0.0679592
$377$	28.0282	1.44352
$378$	0	0
$379$	−15.5108	−0.796735	−0.398368	−	0.917226i	$-0.630423\pi$
$379$	−15.5108	−0.796735	−0.398368	+	0.917226i	$0.630423\pi$
$380$	−9.16281	−0.470042
$381$	0	0
$382$	−19.4431	−0.994796
$383$	−18.4202	−0.941227	−0.470614	−	0.882339i	$-0.655967\pi$
$383$	−18.4202	−0.941227	−0.470614	+	0.882339i	$0.655967\pi$
$384$	0	0
$385$	−28.2551	−1.44001
$386$	−18.1823	−0.925457
$387$	0	0
$388$	−12.4460	−0.631850
$389$	1.57147	0.0796766	0.0398383	−	0.999206i	$-0.487316\pi$
$389$	1.57147	0.0796766	0.0398383	+	0.999206i	$0.487316\pi$
$390$	0	0
$391$	20.7186	1.04779
$392$	3.49517	0.176533
$393$	0	0
$394$	3.60218	0.181475
$395$	14.5013	0.729640
$396$	0	0
$397$	−14.4991	−0.727689	−0.363844	−	0.931460i	$-0.618536\pi$
$397$	−14.4991	−0.727689	−0.363844	+	0.931460i	$0.618536\pi$
$398$	−10.7345	−0.538072
$399$	0	0
$400$	5.46445	0.273222
$401$	30.5872	1.52745	0.763726	−	0.645540i	$-0.223368\pi$
$401$	30.5872	1.52745	0.763726	+	0.645540i	$0.223368\pi$
$402$	0	0
$403$	31.6689	1.57754
$404$	3.51868	0.175061
$405$	0	0
$406$	13.8976	0.689727
$407$	27.4211	1.35922
$408$	0	0
$409$	−7.58789	−0.375197	−0.187599	−	0.982246i	$-0.560070\pi$
$409$	−7.58789	−0.375197	−0.187599	+	0.982246i	$0.560070\pi$
$410$	−24.4967	−1.20981
$411$	0	0
$412$	6.14904	0.302941
$413$	0.0209449	0.00103063
$414$	0	0
$415$	36.2138	1.77767
$416$	−3.77562	−0.185115
$417$	0	0
$418$	13.2152	0.646379
$419$	−34.4523	−1.68310	−0.841551	−	0.540177i	$-0.818357\pi$
$419$	−34.4523	−1.68310	−0.841551	+	0.540177i	$0.818357\pi$
$420$	0	0
$421$	−26.9298	−1.31248	−0.656240	−	0.754552i	$-0.727854\pi$
$421$	−26.9298	−1.31248	−0.656240	+	0.754552i	$0.727854\pi$
$422$	−19.4045	−0.944594
$423$	0	0
$424$	7.98570	0.387820
$425$	−25.5830	−1.24096
$426$	0	0
$427$	25.3313	1.22587
$428$	−0.862074	−0.0416699
$429$	0	0
$430$	−26.4568	−1.27586
$431$	−11.5963	−0.558576	−0.279288	−	0.960207i	$-0.590098\pi$
$431$	−11.5963	−0.558576	−0.279288	+	0.960207i	$0.590098\pi$
$432$	0	0
$433$	39.6098	1.90352	0.951762	−	0.306837i	$-0.0992706\pi$
$433$	39.6098	1.90352	0.951762	+	0.306837i	$0.0992706\pi$
$434$	15.7028	0.753761
$435$	0	0
$436$	11.0784	0.530560
$437$	−12.5351	−0.599633
$438$	0	0
$439$	20.9278	0.998828	0.499414	−	0.866363i	$-0.333549\pi$
$439$	20.9278	0.998828	0.499414	+	0.866363i	$0.333549\pi$
$440$	−15.0926	−0.719509
$441$	0	0
$442$	17.6764	0.840779
$443$	−6.11587	−0.290574	−0.145287	−	0.989390i	$-0.546411\pi$
$443$	−6.11587	−0.290574	−0.145287	+	0.989390i	$0.546411\pi$
$444$	0	0
$445$	23.6830	1.12268
$446$	−0.0723012	−0.00342356
$447$	0	0
$448$	−1.87212	−0.0884493
$449$	15.0463	0.710079	0.355040	−	0.934851i	$-0.384467\pi$
$449$	15.0463	0.710079	0.355040	+	0.934851i	$0.384467\pi$
$450$	0	0
$451$	35.3309	1.66367
$452$	16.0867	0.756654
$453$	0	0
$454$	−11.1563	−0.523592
$455$	22.8654	1.07195
$456$	0	0
$457$	−20.2281	−0.946229	−0.473114	−	0.881001i	$-0.656870\pi$
$457$	−20.2281	−0.946229	−0.473114	+	0.881001i	$0.656870\pi$
$458$	−18.4325	−0.861295
$459$	0	0
$460$	14.3158	0.667475
$461$	5.16604	0.240606	0.120303	−	0.992737i	$-0.461613\pi$
$461$	5.16604	0.240606	0.120303	+	0.992737i	$0.461613\pi$
$462$	0	0
$463$	−24.7033	−1.14806	−0.574030	−	0.818834i	$-0.694621\pi$
$463$	−24.7033	−1.14806	−0.574030	+	0.818834i	$0.694621\pi$
$464$	7.42347	0.344626
$465$	0	0
$466$	15.3706	0.712029
$467$	41.0322	1.89875	0.949373	−	0.314151i	$-0.101720\pi$
$467$	41.0322	1.89875	0.949373	+	0.314151i	$0.101720\pi$
$468$	0	0
$469$	27.8124	1.28426
$470$	4.26285	0.196631
$471$	0	0
$472$	0.0111878	0.000514960 0
$473$	38.1578	1.75450
$474$	0	0
$475$	15.4781	0.710183
$476$	8.76473	0.401731
$477$	0	0
$478$	−10.1084	−0.462347
$479$	−9.10686	−0.416103	−0.208052	−	0.978118i	$-0.566712\pi$
$479$	−9.10686	−0.416103	−0.208052	+	0.978118i	$0.566712\pi$
$480$	0	0
$481$	−22.1906	−1.01180
$482$	−10.3305	−0.470542
$483$	0	0
$484$	10.7675	0.489434
$485$	40.2613	1.82817
$486$	0	0
$487$	18.5260	0.839495	0.419747	−	0.907641i	$-0.362119\pi$
$487$	18.5260	0.839495	0.419747	+	0.907641i	$0.362119\pi$
$488$	13.5308	0.612511
$489$	0	0
$490$	−11.3065	−0.510774
$491$	31.1935	1.40774	0.703871	−	0.710328i	$-0.251453\pi$
$491$	31.1935	1.40774	0.703871	+	0.710328i	$0.251453\pi$
$492$	0	0
$493$	−34.7546	−1.56527
$494$	−10.6944	−0.481166
$495$	0	0
$496$	8.38774	0.376621
$497$	−15.8950	−0.712989
$498$	0	0
$499$	−3.12704	−0.139985	−0.0699927	−	0.997548i	$-0.522298\pi$
$499$	−3.12704	−0.139985	−0.0699927	+	0.997548i	$0.522298\pi$
$500$	−1.50243	−0.0671909
$501$	0	0
$502$	29.1816	1.30244
$503$	−40.7902	−1.81874	−0.909372	−	0.415984i	$-0.863437\pi$
$503$	−40.7902	−1.81874	−0.909372	+	0.415984i	$0.863437\pi$
$504$	0	0
$505$	−11.3825	−0.506514
$506$	−20.6472	−0.917879
$507$	0	0
$508$	5.16232	0.229041
$509$	−27.4998	−1.21891	−0.609455	−	0.792821i	$-0.708612\pi$
$509$	−27.4998	−1.21891	−0.609455	+	0.792821i	$0.708612\pi$
$510$	0	0
$511$	31.5483	1.39561
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−16.6147	−0.732841
$515$	−19.8914	−0.876520
$516$	0	0
$517$	−6.14818	−0.270397
$518$	−11.0031	−0.483447
$519$	0	0
$520$	12.2137	0.535605
$521$	23.3356	1.02235	0.511175	−	0.859476i	$-0.329210\pi$
$521$	23.3356	1.02235	0.511175	+	0.859476i	$0.329210\pi$
$522$	0	0
$523$	−27.0766	−1.18398	−0.591989	−	0.805946i	$-0.701657\pi$
$523$	−27.0766	−1.18398	−0.591989	+	0.805946i	$0.701657\pi$
$524$	−3.81504	−0.166661
$525$	0	0
$526$	24.8311	1.08269
$527$	−39.2690	−1.71059
$528$	0	0
$529$	−3.41552	−0.148501
$530$	−25.8328	−1.12210
$531$	0	0
$532$	−5.30278	−0.229905
$533$	−28.5916	−1.23844
$534$	0	0
$535$	2.78871	0.120566
$536$	14.8561	0.641686
$537$	0	0
$538$	7.21483	0.311054
$539$	16.3070	0.702391
$540$	0	0
$541$	−32.3614	−1.39132	−0.695662	−	0.718369i	$-0.744889\pi$
$541$	−32.3614	−1.39132	−0.695662	+	0.718369i	$0.744889\pi$
$542$	16.0375	0.688868
$543$	0	0
$544$	4.68172	0.200727
$545$	−35.8374	−1.53510
$546$	0	0
$547$	−31.0227	−1.32644	−0.663218	−	0.748426i	$-0.730810\pi$
$547$	−31.0227	−1.32644	−0.663218	+	0.748426i	$0.730810\pi$
$548$	−8.93528	−0.381696
$549$	0	0
$550$	25.4948	1.08710
$551$	21.0270	0.895780
$552$	0	0
$553$	8.39232	0.356878
$554$	20.3652	0.865234
$555$	0	0
$556$	11.9035	0.504819
$557$	−3.24463	−0.137479	−0.0687397	−	0.997635i	$-0.521898\pi$
$557$	−3.24463	−0.137479	−0.0687397	+	0.997635i	$0.521898\pi$
$558$	0	0
$559$	−30.8792	−1.30605
$560$	6.05608	0.255916
$561$	0	0
$562$	−32.1042	−1.35423
$563$	24.0563	1.01385	0.506925	−	0.861990i	$-0.330782\pi$
$563$	24.0563	1.01385	0.506925	+	0.861990i	$0.330782\pi$
$564$	0	0
$565$	−52.0385	−2.18928
$566$	9.53278	0.400693
$567$	0	0
$568$	−8.49039	−0.356249
$569$	22.6225	0.948384	0.474192	−	0.880421i	$-0.342740\pi$
$569$	22.6225	0.948384	0.474192	+	0.880421i	$0.342740\pi$
$570$	0	0
$571$	−10.0927	−0.422367	−0.211184	−	0.977446i	$-0.567732\pi$
$571$	−10.0927	−0.422367	−0.211184	+	0.977446i	$0.567732\pi$
$572$	−17.6154	−0.736537
$573$	0	0
$574$	−14.1770	−0.591736
$575$	−24.1826	−1.00848
$576$	0	0
$577$	−33.9585	−1.41371	−0.706856	−	0.707358i	$-0.749887\pi$
$577$	−33.9585	−1.41371	−0.706856	+	0.707358i	$0.749887\pi$
$578$	−4.91849	−0.204582
$579$	0	0
$580$	−24.0140	−0.997128
$581$	20.9580	0.869483
$582$	0	0
$583$	37.2578	1.54306
$584$	16.8516	0.697326
$585$	0	0
$586$	−19.7079	−0.814124
$587$	−12.2053	−0.503767	−0.251883	−	0.967758i	$-0.581050\pi$
$587$	−12.2053	−0.503767	−0.251883	+	0.967758i	$0.581050\pi$
$588$	0	0
$589$	23.7583	0.978944
$590$	−0.0361912	−0.00148997
$591$	0	0
$592$	−5.87734	−0.241557
$593$	−17.2868	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$593$	−17.2868	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$594$	0	0
$595$	−28.3529	−1.16235
$596$	1.00000	0.0409616
$597$	0	0
$598$	16.7087	0.683271
$599$	−32.9099	−1.34466	−0.672331	−	0.740251i	$-0.734707\pi$
$599$	−32.9099	−1.34466	−0.672331	+	0.740251i	$0.734707\pi$
$600$	0	0
$601$	37.5213	1.53053	0.765263	−	0.643718i	$-0.222609\pi$
$601$	37.5213	1.53053	0.765263	+	0.643718i	$0.222609\pi$
$602$	−15.3113	−0.624042
$603$	0	0
$604$	−19.5730	−0.796413
$605$	−34.8317	−1.41611
$606$	0	0
$607$	34.3048	1.39239	0.696194	−	0.717854i	$-0.254875\pi$
$607$	34.3048	1.39239	0.696194	+	0.717854i	$0.254875\pi$
$608$	−2.83250	−0.114873
$609$	0	0
$610$	−43.7706	−1.77222
$611$	4.97542	0.201284
$612$	0	0
$613$	−46.8061	−1.89048	−0.945241	−	0.326373i	$-0.894174\pi$
$613$	−46.8061	−1.89048	−0.945241	+	0.326373i	$0.894174\pi$
$614$	−32.3057	−1.30375
$615$	0	0
$616$	−8.73450	−0.351923
$617$	28.3845	1.14272	0.571359	−	0.820700i	$-0.306417\pi$
$617$	28.3845	1.14272	0.571359	+	0.820700i	$0.306417\pi$
$618$	0	0
$619$	18.0578	0.725803	0.362901	−	0.931828i	$-0.381786\pi$
$619$	18.0578	0.725803	0.362901	+	0.931828i	$0.381786\pi$
$620$	−27.1333	−1.08970
$621$	0	0
$622$	10.6783	0.428163
$623$	13.7060	0.549120
$624$	0	0
$625$	−22.4620	−0.898482
$626$	1.11853	0.0447054
$627$	0	0
$628$	8.16484	0.325812
$629$	27.5160	1.09714
$630$	0	0
$631$	−38.5705	−1.53547	−0.767734	−	0.640769i	$-0.778616\pi$
$631$	−38.5705	−1.53547	−0.767734	+	0.640769i	$0.778616\pi$
$632$	4.48279	0.178316
$633$	0	0
$634$	−16.0206	−0.636261
$635$	−16.6995	−0.662699
$636$	0	0
$637$	−13.1964	−0.522862
$638$	34.6347	1.37120
$639$	0	0
$640$	3.23488	0.127870
$641$	32.2745	1.27477	0.637383	−	0.770547i	$-0.280017\pi$
$641$	32.2745	1.27477	0.637383	+	0.770547i	$0.280017\pi$
$642$	0	0
$643$	12.0476	0.475111	0.237556	−	0.971374i	$-0.423654\pi$
$643$	12.0476	0.475111	0.237556	+	0.971374i	$0.423654\pi$
$644$	8.28494	0.326472
$645$	0	0
$646$	13.2610	0.521747
$647$	3.15117	0.123885	0.0619427	−	0.998080i	$-0.480270\pi$
$647$	3.15117	0.123885	0.0619427	+	0.998080i	$0.480270\pi$
$648$	0	0
$649$	0.0521974	0.00204893
$650$	−20.6317	−0.809240
$651$	0	0
$652$	−14.7237	−0.576623
$653$	−31.4558	−1.23096	−0.615480	−	0.788152i	$-0.711038\pi$
$653$	−31.4558	−1.23096	−0.615480	+	0.788152i	$0.711038\pi$
$654$	0	0
$655$	12.3412	0.482211
$656$	−7.57269	−0.295664
$657$	0	0
$658$	2.46703	0.0961750
$659$	−29.5585	−1.15144	−0.575718	−	0.817648i	$-0.695277\pi$
$659$	−29.5585	−1.15144	−0.575718	+	0.817648i	$0.695277\pi$
$660$	0	0
$661$	4.95740	0.192820	0.0964102	−	0.995342i	$-0.469264\pi$
$661$	4.95740	0.192820	0.0964102	+	0.995342i	$0.469264\pi$
$662$	0.490745	0.0190734
$663$	0	0
$664$	11.1948	0.434442
$665$	17.1539	0.665198
$666$	0	0
$667$	−32.8521	−1.27204
$668$	−3.40133	−0.131601
$669$	0	0
$670$	−48.0577	−1.85663
$671$	63.1289	2.43707
$672$	0	0
$673$	40.5689	1.56381	0.781907	−	0.623395i	$-0.214247\pi$
$673$	40.5689	1.56381	0.781907	+	0.623395i	$0.214247\pi$
$674$	−10.6204	−0.409084
$675$	0	0
$676$	1.25528	0.0482799
$677$	3.47343	0.133495	0.0667473	−	0.997770i	$-0.478738\pi$
$677$	3.47343	0.133495	0.0667473	+	0.997770i	$0.478738\pi$
$678$	0	0
$679$	23.3004	0.894187
$680$	−15.1448	−0.580777
$681$	0	0
$682$	39.1336	1.49850
$683$	25.7397	0.984904	0.492452	−	0.870340i	$-0.336101\pi$
$683$	25.7397	0.984904	0.492452	+	0.870340i	$0.336101\pi$
$684$	0	0
$685$	28.9046	1.10439
$686$	−19.6482	−0.750172
$687$	0	0
$688$	−8.17859	−0.311806
$689$	−30.1509	−1.14866
$690$	0	0
$691$	19.3628	0.736595	0.368298	−	0.929708i	$-0.379941\pi$
$691$	19.3628	0.736595	0.368298	+	0.929708i	$0.379941\pi$
$692$	12.3853	0.470817
$693$	0	0
$694$	−18.6395	−0.707547
$695$	−38.5063	−1.46063
$696$	0	0
$697$	35.4532	1.34289
$698$	−25.5174	−0.965846
$699$	0	0
$700$	−10.2301	−0.386661
$701$	−21.6490	−0.817673	−0.408837	−	0.912608i	$-0.634065\pi$
$701$	−21.6490	−0.817673	−0.408837	+	0.912608i	$0.634065\pi$
$702$	0	0
$703$	−16.6476	−0.627876
$704$	−4.66557	−0.175840
$705$	0	0
$706$	4.63419	0.174410
$707$	−6.58738	−0.247744
$708$	0	0
$709$	37.2314	1.39825	0.699127	−	0.714998i	$-0.253572\pi$
$709$	37.2314	1.39825	0.699127	+	0.714998i	$0.253572\pi$
$710$	27.4654	1.03076
$711$	0	0
$712$	7.32113	0.274371
$713$	−37.1194	−1.39013
$714$	0	0
$715$	56.9837	2.13107
$716$	−4.42416	−0.165339
$717$	0	0
$718$	4.43020	0.165333
$719$	1.16406	0.0434119	0.0217060	−	0.999764i	$-0.493090\pi$
$719$	1.16406	0.0434119	0.0217060	+	0.999764i	$0.493090\pi$
$720$	0	0
$721$	−11.5117	−0.428719
$722$	10.9769	0.408519
$723$	0	0
$724$	5.56556	0.206842
$725$	40.5651	1.50655
$726$	0	0
$727$	−10.1336	−0.375836	−0.187918	−	0.982185i	$-0.560174\pi$
$727$	−10.1336	−0.375836	−0.187918	+	0.982185i	$0.560174\pi$
$728$	7.06840	0.261972
$729$	0	0
$730$	−54.5130	−2.01762
$731$	38.2899	1.41620
$732$	0	0
$733$	53.0744	1.96035	0.980173	−	0.198144i	$-0.0634912\pi$
$733$	53.0744	1.96035	0.980173	+	0.198144i	$0.0634912\pi$
$734$	2.04147	0.0753522
$735$	0	0
$736$	4.42544	0.163124
$737$	69.3122	2.55315
$738$	0	0
$739$	2.42744	0.0892947	0.0446473	−	0.999003i	$-0.485784\pi$
$739$	2.42744	0.0892947	0.0446473	+	0.999003i	$0.485784\pi$
$740$	19.0125	0.698913
$741$	0	0
$742$	−14.9502	−0.548838
$743$	26.3368	0.966205	0.483103	−	0.875564i	$-0.339510\pi$
$743$	26.3368	0.966205	0.483103	+	0.875564i	$0.339510\pi$
$744$	0	0
$745$	−3.23488	−0.118517
$746$	7.59725	0.278155
$747$	0	0
$748$	21.8429	0.798655
$749$	1.61390	0.0589708
$750$	0	0
$751$	22.9669	0.838074	0.419037	−	0.907969i	$-0.362368\pi$
$751$	22.9669	0.838074	0.419037	+	0.907969i	$0.362368\pi$
$752$	1.31778	0.0480544
$753$	0	0
$754$	−28.0282	−1.02073
$755$	63.3163	2.30431
$756$	0	0
$757$	13.6173	0.494930	0.247465	−	0.968897i	$-0.420402\pi$
$757$	13.6173	0.494930	0.247465	+	0.968897i	$0.420402\pi$
$758$	15.5108	0.563377
$759$	0	0
$760$	9.16281	0.332370
$761$	9.75835	0.353740	0.176870	−	0.984234i	$-0.443403\pi$
$761$	9.75835	0.353740	0.176870	+	0.984234i	$0.443403\pi$
$762$	0	0
$763$	−20.7401	−0.750842
$764$	19.4431	0.703427
$765$	0	0
$766$	18.4202	0.665548
$767$	−0.0422408	−0.00152523
$768$	0	0
$769$	−17.0007	−0.613061	−0.306531	−	0.951861i	$-0.599168\pi$
$769$	−17.0007	−0.613061	−0.306531	+	0.951861i	$0.599168\pi$
$770$	28.2551	1.01824
$771$	0	0
$772$	18.1823	0.654397
$773$	3.43559	0.123570	0.0617849	−	0.998089i	$-0.480321\pi$
$773$	3.43559	0.123570	0.0617849	+	0.998089i	$0.480321\pi$
$774$	0	0
$775$	45.8344	1.64642
$776$	12.4460	0.446785
$777$	0	0
$778$	−1.57147	−0.0563398
$779$	−21.4497	−0.768515
$780$	0	0
$781$	−39.6125	−1.41745
$782$	−20.7186	−0.740897
$783$	0	0
$784$	−3.49517	−0.124828
$785$	−26.4123	−0.942694
$786$	0	0
$787$	42.5941	1.51832	0.759158	−	0.650906i	$-0.225611\pi$
$787$	42.5941	1.51832	0.759158	+	0.650906i	$0.225611\pi$
$788$	−3.60218	−0.128323
$789$	0	0
$790$	−14.5013	−0.515933
$791$	−30.1162	−1.07081
$792$	0	0
$793$	−51.0871	−1.81416
$794$	14.4991	0.514553
$795$	0	0
$796$	10.7345	0.380474
$797$	21.1891	0.750558	0.375279	−	0.926912i	$-0.377547\pi$
$797$	21.1891	0.750558	0.375279	+	0.926912i	$0.377547\pi$
$798$	0	0
$799$	−6.16946	−0.218260
$800$	−5.46445	−0.193197
$801$	0	0
$802$	−30.5872	−1.08007
$803$	78.6225	2.77453
$804$	0	0
$805$	−26.8008	−0.944604
$806$	−31.6689	−1.11549
$807$	0	0
$808$	−3.51868	−0.123787
$809$	−30.0685	−1.05715	−0.528577	−	0.848885i	$-0.677274\pi$
$809$	−30.0685	−1.05715	−0.528577	+	0.848885i	$0.677274\pi$
$810$	0	0
$811$	23.7888	0.835337	0.417669	−	0.908599i	$-0.362847\pi$
$811$	23.7888	0.835337	0.417669	+	0.908599i	$0.362847\pi$
$812$	−13.8976	−0.487710
$813$	0	0
$814$	−27.4211	−0.961110
$815$	47.6293	1.66838
$816$	0	0
$817$	−23.1659	−0.810472
$818$	7.58789	0.265304
$819$	0	0
$820$	24.4967	0.855464
$821$	−14.5095	−0.506385	−0.253193	−	0.967416i	$-0.581481\pi$
$821$	−14.5095	−0.506385	−0.253193	+	0.967416i	$0.581481\pi$
$822$	0	0
$823$	−0.563994	−0.0196596	−0.00982980	−	0.999952i	$-0.503129\pi$
$823$	−0.563994	−0.0196596	−0.00982980	+	0.999952i	$0.503129\pi$
$824$	−6.14904	−0.214212
$825$	0	0
$826$	−0.0209449	−0.000728765 0
$827$	−7.59775	−0.264200	−0.132100	−	0.991236i	$-0.542172\pi$
$827$	−7.59775	−0.264200	−0.132100	+	0.991236i	$0.542172\pi$
$828$	0	0
$829$	−17.7016	−0.614803	−0.307401	−	0.951580i	$-0.599459\pi$
$829$	−17.7016	−0.614803	−0.307401	+	0.951580i	$0.599459\pi$
$830$	−36.2138	−1.25700
$831$	0	0
$832$	3.77562	0.130896
$833$	16.3634	0.566959
$834$	0	0
$835$	11.0029	0.380771
$836$	−13.2152	−0.457059
$837$	0	0
$838$	34.4523	1.19013
$839$	−52.2507	−1.80389	−0.901947	−	0.431846i	$-0.857862\pi$
$839$	−52.2507	−1.80389	−0.901947	+	0.431846i	$0.857862\pi$
$840$	0	0
$841$	26.1078	0.900270
$842$	26.9298	0.928063
$843$	0	0
$844$	19.4045	0.667929
$845$	−4.06067	−0.139691
$846$	0	0
$847$	−20.1581	−0.692641
$848$	−7.98570	−0.274230
$849$	0	0
$850$	25.5830	0.877490
$851$	26.0098	0.891604
$852$	0	0
$853$	−50.6035	−1.73263	−0.866315	−	0.499497i	$-0.833518\pi$
$853$	−50.6035	−1.73263	−0.866315	+	0.499497i	$0.833518\pi$
$854$	−25.3313	−0.866819
$855$	0	0
$856$	0.862074	0.0294651
$857$	21.6346	0.739023	0.369511	−	0.929226i	$-0.379525\pi$
$857$	21.6346	0.739023	0.369511	+	0.929226i	$0.379525\pi$
$858$	0	0
$859$	−25.1183	−0.857025	−0.428513	−	0.903536i	$-0.640962\pi$
$859$	−25.1183	−0.857025	−0.428513	+	0.903536i	$0.640962\pi$
$860$	26.4568	0.902168
$861$	0	0
$862$	11.5963	0.394973
$863$	−16.4271	−0.559185	−0.279593	−	0.960119i	$-0.590199\pi$
$863$	−16.4271	−0.559185	−0.279593	+	0.960119i	$0.590199\pi$
$864$	0	0
$865$	−40.0649	−1.36225
$866$	−39.6098	−1.34599
$867$	0	0
$868$	−15.7028	−0.532989
$869$	20.9148	0.709486
$870$	0	0
$871$	−56.0910	−1.90057
$872$	−11.0784	−0.375163
$873$	0	0
$874$	12.5351	0.424005
$875$	2.81273	0.0950877
$876$	0	0
$877$	−32.2537	−1.08913	−0.544565	−	0.838719i	$-0.683305\pi$
$877$	−32.2537	−1.08913	−0.544565	+	0.838719i	$0.683305\pi$
$878$	−20.9278	−0.706278
$879$	0	0
$880$	15.0926	0.508770
$881$	12.5362	0.422355	0.211177	−	0.977448i	$-0.432270\pi$
$881$	12.5362	0.422355	0.211177	+	0.977448i	$0.432270\pi$
$882$	0	0
$883$	53.2761	1.79288	0.896441	−	0.443162i	$-0.146143\pi$
$883$	53.2761	1.79288	0.896441	+	0.443162i	$0.146143\pi$
$884$	−17.6764	−0.594521
$885$	0	0
$886$	6.11587	0.205467
$887$	28.0539	0.941957	0.470979	−	0.882145i	$-0.343901\pi$
$887$	28.0539	0.941957	0.470979	+	0.882145i	$0.343901\pi$
$888$	0	0
$889$	−9.66447	−0.324136
$890$	−23.6830	−0.793855
$891$	0	0
$892$	0.0723012	0.00242082
$893$	3.73261	0.124907
$894$	0	0
$895$	14.3116	0.478385
$896$	1.87212	0.0625431
$897$	0	0
$898$	−15.0463	−0.502102
$899$	62.2661	2.07669
$900$	0	0
$901$	37.3868	1.24554
$902$	−35.3309	−1.17639
$903$	0	0
$904$	−16.0867	−0.535035
$905$	−18.0039	−0.598471
$906$	0	0
$907$	46.8565	1.55585	0.777923	−	0.628360i	$-0.216273\pi$
$907$	46.8565	1.55585	0.777923	+	0.628360i	$0.216273\pi$
$908$	11.1563	0.370236
$909$	0	0
$910$	−22.8654	−0.757981
$911$	−33.7262	−1.11740	−0.558699	−	0.829371i	$-0.688699\pi$
$911$	−33.7262	−1.11740	−0.558699	+	0.829371i	$0.688699\pi$
$912$	0	0
$913$	52.2301	1.72856
$914$	20.2281	0.669085
$915$	0	0
$916$	18.4325	0.609028
$917$	7.14221	0.235857
$918$	0	0
$919$	12.8406	0.423572	0.211786	−	0.977316i	$-0.432072\pi$
$919$	12.8406	0.423572	0.211786	+	0.977316i	$0.432072\pi$
$920$	−14.3158	−0.471976
$921$	0	0
$922$	−5.16604	−0.170134
$923$	32.0564	1.05515
$924$	0	0
$925$	−32.1164	−1.05598
$926$	24.7033	0.811801
$927$	0	0
$928$	−7.42347	−0.243687
$929$	52.1975	1.71254	0.856272	−	0.516526i	$-0.172775\pi$
$929$	52.1975	1.71254	0.856272	+	0.516526i	$0.172775\pi$
$930$	0	0
$931$	−9.90009	−0.324462
$932$	−15.3706	−0.503481
$933$	0	0
$934$	−41.0322	−1.34262
$935$	−70.6591	−2.31080
$936$	0	0
$937$	8.26863	0.270124	0.135062	−	0.990837i	$-0.456877\pi$
$937$	8.26863	0.270124	0.135062	+	0.990837i	$0.456877\pi$
$938$	−27.8124	−0.908107
$939$	0	0
$940$	−4.26285	−0.139039
$941$	−3.85862	−0.125787	−0.0628937	−	0.998020i	$-0.520033\pi$
$941$	−3.85862	−0.125787	−0.0628937	+	0.998020i	$0.520033\pi$
$942$	0	0
$943$	33.5125	1.09132
$944$	−0.0111878	−0.000364132 0
$945$	0	0
$946$	−38.1578	−1.24062
$947$	−48.1525	−1.56475	−0.782373	−	0.622810i	$-0.785991\pi$
$947$	−48.1525	−1.56475	−0.782373	+	0.622810i	$0.785991\pi$
$948$	0	0
$949$	−63.6253	−2.06536
$950$	−15.4781	−0.502175
$951$	0	0
$952$	−8.76473	−0.284067
$953$	37.1054	1.20196	0.600981	−	0.799263i	$-0.294777\pi$
$953$	37.1054	1.20196	0.600981	+	0.799263i	$0.294777\pi$
$954$	0	0
$955$	−62.8961	−2.03527
$956$	10.1084	0.326929
$957$	0	0
$958$	9.10686	0.294229
$959$	16.7279	0.540172
$960$	0	0
$961$	39.3542	1.26949
$962$	22.1906	0.715453
$963$	0	0
$964$	10.3305	0.332724
$965$	−58.8177	−1.89341
$966$	0	0
$967$	3.85309	0.123907	0.0619536	−	0.998079i	$-0.480267\pi$
$967$	3.85309	0.123907	0.0619536	+	0.998079i	$0.480267\pi$
$968$	−10.7675	−0.346082
$969$	0	0
$970$	−40.2613	−1.29271
$971$	18.2980	0.587211	0.293605	−	0.955927i	$-0.405145\pi$
$971$	18.2980	0.587211	0.293605	+	0.955927i	$0.405145\pi$
$972$	0	0
$973$	−22.2847	−0.714415
$974$	−18.5260	−0.593613
$975$	0	0
$976$	−13.5308	−0.433111
$977$	−45.3303	−1.45024	−0.725122	−	0.688621i	$-0.758217\pi$
$977$	−45.3303	−1.45024	−0.725122	+	0.688621i	$0.758217\pi$
$978$	0	0
$979$	34.1573	1.09167
$980$	11.3065	0.361172
$981$	0	0
$982$	−31.1935	−0.995424
$983$	−59.1676	−1.88715	−0.943577	−	0.331154i	$-0.892562\pi$
$983$	−59.1676	−1.88715	−0.943577	+	0.331154i	$0.892562\pi$
$984$	0	0
$985$	11.6526	0.371284
$986$	34.7546	1.10681
$987$	0	0
$988$	10.6944	0.340236
$989$	36.1938	1.15090
$990$	0	0
$991$	12.4592	0.395780	0.197890	−	0.980224i	$-0.436591\pi$
$991$	12.4592	0.395780	0.197890	+	0.980224i	$0.436591\pi$
$992$	−8.38774	−0.266311
$993$	0	0
$994$	15.8950	0.504159
$995$	−34.7248	−1.10085
$996$	0	0
$997$	12.2829	0.389003	0.194502	−	0.980902i	$-0.437691\pi$
$997$	12.2829	0.389003	0.194502	+	0.980902i	$0.437691\pi$
$998$	3.12704	0.0989847
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.s.1.3	✓	16
3.2	odd	2		8046.2.a.t.1.14	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.s.1.3	✓	16	1.1	even	1	trivial
8046.2.a.t.1.14	yes	16	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -3.23488 q^{5} -1.87212 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -3.23488 q^{5} -1.87212 q^{7} -1.00000 q^{8} +3.23488 q^{10} -4.66557 q^{11} +3.77562 q^{13} +1.87212 q^{14} +1.00000 q^{16} -4.68172 q^{17} +2.83250 q^{19} -3.23488 q^{20} +4.66557 q^{22} -4.42544 q^{23} +5.46445 q^{25} -3.77562 q^{26} -1.87212 q^{28} +7.42347 q^{29} +8.38774 q^{31} -1.00000 q^{32} +4.68172 q^{34} +6.05608 q^{35} -5.87734 q^{37} -2.83250 q^{38} +3.23488 q^{40} -7.57269 q^{41} -8.17859 q^{43} -4.66557 q^{44} +4.42544 q^{46} +1.31778 q^{47} -3.49517 q^{49} -5.46445 q^{50} +3.77562 q^{52} -7.98570 q^{53} +15.0926 q^{55} +1.87212 q^{56} -7.42347 q^{58} -0.0111878 q^{59} -13.5308 q^{61} -8.38774 q^{62} +1.00000 q^{64} -12.2137 q^{65} -14.8561 q^{67} -4.68172 q^{68} -6.05608 q^{70} +8.49039 q^{71} -16.8516 q^{73} +5.87734 q^{74} +2.83250 q^{76} +8.73450 q^{77} -4.48279 q^{79} -3.23488 q^{80} +7.57269 q^{82} -11.1948 q^{83} +15.1448 q^{85} +8.17859 q^{86} +4.66557 q^{88} -7.32113 q^{89} -7.06840 q^{91} -4.42544 q^{92} -1.31778 q^{94} -9.16281 q^{95} -12.4460 q^{97} +3.49517 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{2} + 16 q^{4} - 4 q^{5} + 6 q^{7} - 16 q^{8}+O(q^{10}) \) 16 * q - 16 * q^2 + 16 * q^4 - 4 * q^5 + 6 * q^7 - 16 * q^8	\( 16 q - 16 q^{2} + 16 q^{4} - 4 q^{5} + 6 q^{7} - 16 q^{8} + 4 q^{10} - 6 q^{11} + 6 q^{13} - 6 q^{14} + 16 q^{16} - q^{17} + 10 q^{19} - 4 q^{20} + 6 q^{22} - 10 q^{23} + 28 q^{25} - 6 q^{26} + 6 q^{28} - 6 q^{29} + 21 q^{31} - 16 q^{32} + q^{34} - 16 q^{35} + 17 q^{37} - 10 q^{38} + 4 q^{40} + 4 q^{41} + 16 q^{43} - 6 q^{44} + 10 q^{46} - 25 q^{47} + 36 q^{49} - 28 q^{50} + 6 q^{52} - 14 q^{53} + 19 q^{55} - 6 q^{56} + 6 q^{58} - 6 q^{59} + 23 q^{61} - 21 q^{62} + 16 q^{64} - 20 q^{65} + 22 q^{67} - q^{68} + 16 q^{70} - 10 q^{71} + 16 q^{73} - 17 q^{74} + 10 q^{76} + 2 q^{77} + 37 q^{79} - 4 q^{80} - 4 q^{82} - 33 q^{83} + 43 q^{85} - 16 q^{86} + 6 q^{88} + 3 q^{89} + 28 q^{91} - 10 q^{92} + 25 q^{94} - 14 q^{95} - 3 q^{97} - 36 q^{98}+O(q^{100}) \) 16 * q - 16 * q^2 + 16 * q^4 - 4 * q^5 + 6 * q^7 - 16 * q^8 + 4 * q^10 - 6 * q^11 + 6 * q^13 - 6 * q^14 + 16 * q^16 - q^17 + 10 * q^19 - 4 * q^20 + 6 * q^22 - 10 * q^23 + 28 * q^25 - 6 * q^26 + 6 * q^28 - 6 * q^29 + 21 * q^31 - 16 * q^32 + q^34 - 16 * q^35 + 17 * q^37 - 10 * q^38 + 4 * q^40 + 4 * q^41 + 16 * q^43 - 6 * q^44 + 10 * q^46 - 25 * q^47 + 36 * q^49 - 28 * q^50 + 6 * q^52 - 14 * q^53 + 19 * q^55 - 6 * q^56 + 6 * q^58 - 6 * q^59 + 23 * q^61 - 21 * q^62 + 16 * q^64 - 20 * q^65 + 22 * q^67 - q^68 + 16 * q^70 - 10 * q^71 + 16 * q^73 - 17 * q^74 + 10 * q^76 + 2 * q^77 + 37 * q^79 - 4 * q^80 - 4 * q^82 - 33 * q^83 + 43 * q^85 - 16 * q^86 + 6 * q^88 + 3 * q^89 + 28 * q^91 - 10 * q^92 + 25 * q^94 - 14 * q^95 - 3 * q^97 - 36 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more