LMFDB - Embedded newform 8046.2.a.s.1.15

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			1.15
Root			$-3.47967$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +3.47967 q^{5} +2.01581 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} +3.47967 q^{5} +2.01581 q^{7} -1.00000 q^{8} -3.47967 q^{10} +5.41891 q^{11} +4.76213 q^{13} -2.01581 q^{14} +1.00000 q^{16} -0.975288 q^{17} +2.00975 q^{19} +3.47967 q^{20} -5.41891 q^{22} -4.93705 q^{23} +7.10807 q^{25} -4.76213 q^{26} +2.01581 q^{28} -4.34340 q^{29} -0.676136 q^{31} -1.00000 q^{32} +0.975288 q^{34} +7.01435 q^{35} -8.00564 q^{37} -2.00975 q^{38} -3.47967 q^{40} +5.01494 q^{41} -2.02437 q^{43} +5.41891 q^{44} +4.93705 q^{46} +9.42387 q^{47} -2.93650 q^{49} -7.10807 q^{50} +4.76213 q^{52} +2.40586 q^{53} +18.8560 q^{55} -2.01581 q^{56} +4.34340 q^{58} -6.80304 q^{59} +14.2856 q^{61} +0.676136 q^{62} +1.00000 q^{64} +16.5706 q^{65} -5.70555 q^{67} -0.975288 q^{68} -7.01435 q^{70} +13.5626 q^{71} +10.9059 q^{73} +8.00564 q^{74} +2.00975 q^{76} +10.9235 q^{77} +0.737648 q^{79} +3.47967 q^{80} -5.01494 q^{82} -12.0263 q^{83} -3.39368 q^{85} +2.02437 q^{86} -5.41891 q^{88} +0.894003 q^{89} +9.59955 q^{91} -4.93705 q^{92} -9.42387 q^{94} +6.99327 q^{95} -6.58391 q^{97} +2.93650 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	3.47967	1.55615	0.778077	−	0.628169i	$-0.216195\pi$
$5$	3.47967	1.55615	0.778077	+	0.628169i	$0.216195\pi$
$6$	0	0
$7$	2.01581	0.761905	0.380953	−	0.924595i	$-0.375596\pi$
$7$	2.01581	0.761905	0.380953	+	0.924595i	$0.375596\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−3.47967	−1.10037
$11$	5.41891	1.63386	0.816931	−	0.576735i	$-0.195674\pi$
$11$	5.41891	1.63386	0.816931	+	0.576735i	$0.195674\pi$
$12$	0	0
$13$	4.76213	1.32078	0.660388	−	0.750924i	$-0.270392\pi$
$13$	4.76213	1.32078	0.660388	+	0.750924i	$0.270392\pi$
$14$	−2.01581	−0.538749
$15$	0	0
$16$	1.00000	0.250000
$17$	−0.975288	−0.236542	−0.118271	−	0.992981i	$-0.537735\pi$
$17$	−0.975288	−0.236542	−0.118271	+	0.992981i	$0.537735\pi$
$18$	0	0
$19$	2.00975	0.461069	0.230534	−	0.973064i	$-0.425953\pi$
$19$	2.00975	0.461069	0.230534	+	0.973064i	$0.425953\pi$
$20$	3.47967	0.778077
$21$	0	0
$22$	−5.41891	−1.15532
$23$	−4.93705	−1.02945	−0.514723	−	0.857356i	$-0.672105\pi$
$23$	−4.93705	−1.02945	−0.514723	+	0.857356i	$0.672105\pi$
$24$	0	0
$25$	7.10807	1.42161
$26$	−4.76213	−0.933930
$27$	0	0
$28$	2.01581	0.380953
$29$	−4.34340	−0.806549	−0.403275	−	0.915079i	$-0.632128\pi$
$29$	−4.34340	−0.806549	−0.403275	+	0.915079i	$0.632128\pi$
$30$	0	0
$31$	−0.676136	−0.121438	−0.0607188	−	0.998155i	$-0.519339\pi$
$31$	−0.676136	−0.121438	−0.0607188	+	0.998155i	$0.519339\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	0.975288	0.167261
$35$	7.01435	1.18564
$36$	0	0
$37$	−8.00564	−1.31612	−0.658060	−	0.752966i	$-0.728623\pi$
$37$	−8.00564	−1.31612	−0.658060	+	0.752966i	$0.728623\pi$
$38$	−2.00975	−0.326025
$39$	0	0
$40$	−3.47967	−0.550183
$41$	5.01494	0.783202	0.391601	−	0.920135i	$-0.371921\pi$
$41$	5.01494	0.783202	0.391601	+	0.920135i	$0.371921\pi$
$42$	0	0
$43$	−2.02437	−0.308713	−0.154357	−	0.988015i	$-0.549330\pi$
$43$	−2.02437	−0.308713	−0.154357	+	0.988015i	$0.549330\pi$
$44$	5.41891	0.816931
$45$	0	0
$46$	4.93705	0.727929
$47$	9.42387	1.37461	0.687306	−	0.726368i	$-0.258793\pi$
$47$	9.42387	1.37461	0.687306	+	0.726368i	$0.258793\pi$
$48$	0	0
$49$	−2.93650	−0.419500
$50$	−7.10807	−1.00523
$51$	0	0
$52$	4.76213	0.660388
$53$	2.40586	0.330470	0.165235	−	0.986254i	$-0.447162\pi$
$53$	2.40586	0.330470	0.165235	+	0.986254i	$0.447162\pi$
$54$	0	0
$55$	18.8560	2.54254
$56$	−2.01581	−0.269374
$57$	0	0
$58$	4.34340	0.570316
$59$	−6.80304	−0.885680	−0.442840	−	0.896601i	$-0.646029\pi$
$59$	−6.80304	−0.885680	−0.442840	+	0.896601i	$0.646029\pi$
$60$	0	0
$61$	14.2856	1.82909	0.914544	−	0.404487i	$-0.132550\pi$
$61$	14.2856	1.82909	0.914544	+	0.404487i	$0.132550\pi$
$62$	0.676136	0.0858694
$63$	0	0
$64$	1.00000	0.125000
$65$	16.5706	2.05533
$66$	0	0
$67$	−5.70555	−0.697043	−0.348522	−	0.937301i	$-0.613316\pi$
$67$	−5.70555	−0.697043	−0.348522	+	0.937301i	$0.613316\pi$
$68$	−0.975288	−0.118271
$69$	0	0
$70$	−7.01435	−0.838376
$71$	13.5626	1.60958	0.804790	−	0.593559i	$-0.202278\pi$
$71$	13.5626	1.60958	0.804790	+	0.593559i	$0.202278\pi$
$72$	0	0
$73$	10.9059	1.27644	0.638222	−	0.769853i	$-0.279670\pi$
$73$	10.9059	1.27644	0.638222	+	0.769853i	$0.279670\pi$
$74$	8.00564	0.930637
$75$	0	0
$76$	2.00975	0.230534
$77$	10.9235	1.24485
$78$	0	0
$79$	0.737648	0.0829919	0.0414959	−	0.999139i	$-0.486788\pi$
$79$	0.737648	0.0829919	0.0414959	+	0.999139i	$0.486788\pi$
$80$	3.47967	0.389038
$81$	0	0
$82$	−5.01494	−0.553807
$83$	−12.0263	−1.32006	−0.660029	−	0.751240i	$-0.729456\pi$
$83$	−12.0263	−1.32006	−0.660029	+	0.751240i	$0.729456\pi$
$84$	0	0
$85$	−3.39368	−0.368096
$86$	2.02437	0.218293
$87$	0	0
$88$	−5.41891	−0.577658
$89$	0.894003	0.0947641	0.0473821	−	0.998877i	$-0.484912\pi$
$89$	0.894003	0.0947641	0.0473821	+	0.998877i	$0.484912\pi$
$90$	0	0
$91$	9.59955	1.00631
$92$	−4.93705	−0.514723
$93$	0	0
$94$	−9.42387	−0.971998
$95$	6.99327	0.717494
$96$	0	0
$97$	−6.58391	−0.668494	−0.334247	−	0.942485i	$-0.608482\pi$
$97$	−6.58391	−0.668494	−0.334247	+	0.942485i	$0.608482\pi$
$98$	2.93650	0.296631
$99$	0	0
$100$	7.10807	0.710807
$101$	−8.76010	−0.871663	−0.435831	−	0.900028i	$-0.643546\pi$
$101$	−8.76010	−0.871663	−0.435831	+	0.900028i	$0.643546\pi$
$102$	0	0
$103$	12.8547	1.26661	0.633306	−	0.773902i	$-0.281698\pi$
$103$	12.8547	1.26661	0.633306	+	0.773902i	$0.281698\pi$
$104$	−4.76213	−0.466965
$105$	0	0
$106$	−2.40586	−0.233678
$107$	−12.3420	−1.19315	−0.596573	−	0.802558i	$-0.703472\pi$
$107$	−12.3420	−1.19315	−0.596573	+	0.802558i	$0.703472\pi$
$108$	0	0
$109$	4.08845	0.391603	0.195801	−	0.980644i	$-0.437269\pi$
$109$	4.08845	0.391603	0.195801	+	0.980644i	$0.437269\pi$
$110$	−18.8560	−1.79785
$111$	0	0
$112$	2.01581	0.190476
$113$	5.75758	0.541627	0.270814	−	0.962632i	$-0.412707\pi$
$113$	5.75758	0.541627	0.270814	+	0.962632i	$0.412707\pi$
$114$	0	0
$115$	−17.1793	−1.60198
$116$	−4.34340	−0.403275
$117$	0	0
$118$	6.80304	0.626271
$119$	−1.96600	−0.180223
$120$	0	0
$121$	18.3646	1.66951
$122$	−14.2856	−1.29336
$123$	0	0
$124$	−0.676136	−0.0607188
$125$	7.33539	0.656097
$126$	0	0
$127$	−9.93211	−0.881332	−0.440666	−	0.897671i	$-0.645258\pi$
$127$	−9.93211	−0.881332	−0.440666	+	0.897671i	$0.645258\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−16.5706	−1.45334
$131$	−15.7880	−1.37941	−0.689704	−	0.724092i	$-0.742259\pi$
$131$	−15.7880	−1.37941	−0.689704	+	0.724092i	$0.742259\pi$
$132$	0	0
$133$	4.05128	0.351291
$134$	5.70555	0.492884
$135$	0	0
$136$	0.975288	0.0836303
$137$	−0.857285	−0.0732428	−0.0366214	−	0.999329i	$-0.511660\pi$
$137$	−0.857285	−0.0732428	−0.0366214	+	0.999329i	$0.511660\pi$
$138$	0	0
$139$	13.6243	1.15560	0.577801	−	0.816178i	$-0.303911\pi$
$139$	13.6243	1.15560	0.577801	+	0.816178i	$0.303911\pi$
$140$	7.01435	0.592821
$141$	0	0
$142$	−13.5626	−1.13815
$143$	25.8055	2.15797
$144$	0	0
$145$	−15.1136	−1.25511
$146$	−10.9059	−0.902582
$147$	0	0
$148$	−8.00564	−0.658060
$149$	1.00000	0.0819232
$149$	1.00000	0.0819232
$150$	0	0
$151$	21.1806	1.72365	0.861827	−	0.507202i	$-0.169320\pi$
$151$	21.1806	1.72365	0.861827	+	0.507202i	$0.169320\pi$
$152$	−2.00975	−0.163012
$153$	0	0
$154$	−10.9235	−0.880241
$155$	−2.35273	−0.188976
$156$	0	0
$157$	−5.91355	−0.471952	−0.235976	−	0.971759i	$-0.575829\pi$
$157$	−5.91355	−0.471952	−0.235976	+	0.971759i	$0.575829\pi$
$158$	−0.737648	−0.0586841
$159$	0	0
$160$	−3.47967	−0.275092
$161$	−9.95217	−0.784341
$162$	0	0
$163$	22.3353	1.74943	0.874717	−	0.484634i	$-0.161047\pi$
$163$	22.3353	1.74943	0.874717	+	0.484634i	$0.161047\pi$
$164$	5.01494	0.391601
$165$	0	0
$166$	12.0263	0.933422
$167$	−6.84198	−0.529449	−0.264724	−	0.964324i	$-0.585281\pi$
$167$	−6.84198	−0.529449	−0.264724	+	0.964324i	$0.585281\pi$
$168$	0	0
$169$	9.67785	0.744450
$170$	3.39368	0.260283
$171$	0	0
$172$	−2.02437	−0.154357
$173$	−9.17170	−0.697311	−0.348656	−	0.937251i	$-0.613362\pi$
$173$	−9.17170	−0.697311	−0.348656	+	0.937251i	$0.613362\pi$
$174$	0	0
$175$	14.3285	1.08314
$176$	5.41891	0.408466
$177$	0	0
$178$	−0.894003	−0.0670084
$179$	3.14761	0.235264	0.117632	−	0.993057i	$-0.462470\pi$
$179$	3.14761	0.235264	0.117632	+	0.993057i	$0.462470\pi$
$180$	0	0
$181$	8.54429	0.635092	0.317546	−	0.948243i	$-0.397141\pi$
$181$	8.54429	0.635092	0.317546	+	0.948243i	$0.397141\pi$
$182$	−9.59955	−0.711566
$183$	0	0
$184$	4.93705	0.363964
$185$	−27.8570	−2.04808
$186$	0	0
$187$	−5.28500	−0.386477
$188$	9.42387	0.687306
$189$	0	0
$190$	−6.99327	−0.507345
$191$	−9.91199	−0.717206	−0.358603	−	0.933490i	$-0.616747\pi$
$191$	−9.91199	−0.717206	−0.358603	+	0.933490i	$0.616747\pi$
$192$	0	0
$193$	8.16388	0.587649	0.293824	−	0.955859i	$-0.405072\pi$
$193$	8.16388	0.587649	0.293824	+	0.955859i	$0.405072\pi$
$194$	6.58391	0.472697
$195$	0	0
$196$	−2.93650	−0.209750
$197$	−11.8895	−0.847090	−0.423545	−	0.905875i	$-0.639214\pi$
$197$	−11.8895	−0.847090	−0.423545	+	0.905875i	$0.639214\pi$
$198$	0	0
$199$	12.9235	0.916123	0.458061	−	0.888921i	$-0.348544\pi$
$199$	12.9235	0.916123	0.458061	+	0.888921i	$0.348544\pi$
$200$	−7.10807	−0.502617
$201$	0	0
$202$	8.76010	0.616358
$203$	−8.75548	−0.614514
$204$	0	0
$205$	17.4503	1.21878
$206$	−12.8547	−0.895629
$207$	0	0
$208$	4.76213	0.330194
$209$	10.8907	0.753323
$210$	0	0
$211$	−1.76036	−0.121188	−0.0605942	−	0.998162i	$-0.519300\pi$
$211$	−1.76036	−0.121188	−0.0605942	+	0.998162i	$0.519300\pi$
$212$	2.40586	0.165235
$213$	0	0
$214$	12.3420	0.843682
$215$	−7.04412	−0.480405
$216$	0	0
$217$	−1.36296	−0.0925240
$218$	−4.08845	−0.276905
$219$	0	0
$220$	18.8560	1.27127
$221$	−4.64445	−0.312419
$222$	0	0
$223$	14.0022	0.937658	0.468829	−	0.883289i	$-0.344676\pi$
$223$	14.0022	0.937658	0.468829	+	0.883289i	$0.344676\pi$
$224$	−2.01581	−0.134687
$225$	0	0
$226$	−5.75758	−0.382988
$227$	−1.76335	−0.117037	−0.0585187	−	0.998286i	$-0.518638\pi$
$227$	−1.76335	−0.117037	−0.0585187	+	0.998286i	$0.518638\pi$
$228$	0	0
$229$	−24.6220	−1.62706	−0.813532	−	0.581520i	$-0.802458\pi$
$229$	−24.6220	−1.62706	−0.813532	+	0.581520i	$0.802458\pi$
$230$	17.1793	1.13277
$231$	0	0
$232$	4.34340	0.285158
$233$	−19.8639	−1.30132	−0.650662	−	0.759367i	$-0.725509\pi$
$233$	−19.8639	−1.30132	−0.650662	+	0.759367i	$0.725509\pi$
$234$	0	0
$235$	32.7919	2.13911
$236$	−6.80304	−0.442840
$237$	0	0
$238$	1.96600	0.127437
$239$	−20.4637	−1.32369	−0.661843	−	0.749642i	$-0.730226\pi$
$239$	−20.4637	−1.32369	−0.661843	+	0.749642i	$0.730226\pi$
$240$	0	0
$241$	9.65057	0.621648	0.310824	−	0.950468i	$-0.399395\pi$
$241$	9.65057	0.621648	0.310824	+	0.950468i	$0.399395\pi$
$242$	−18.3646	−1.18052
$243$	0	0
$244$	14.2856	0.914544
$245$	−10.2180	−0.652807
$246$	0	0
$247$	9.57070	0.608969
$248$	0.676136	0.0429347
$249$	0	0
$250$	−7.33539	−0.463931
$251$	−15.5906	−0.984068	−0.492034	−	0.870576i	$-0.663746\pi$
$251$	−15.5906	−0.984068	−0.492034	+	0.870576i	$0.663746\pi$
$252$	0	0
$253$	−26.7534	−1.68197
$254$	9.93211	0.623196
$255$	0	0
$256$	1.00000	0.0625000
$257$	13.1346	0.819313	0.409657	−	0.912240i	$-0.365649\pi$
$257$	13.1346	0.819313	0.409657	+	0.912240i	$0.365649\pi$
$258$	0	0
$259$	−16.1379	−1.00276
$260$	16.5706	1.02767
$261$	0	0
$262$	15.7880	0.975388
$263$	−20.0035	−1.23347	−0.616735	−	0.787171i	$-0.711545\pi$
$263$	−20.0035	−1.23347	−0.616735	+	0.787171i	$0.711545\pi$
$264$	0	0
$265$	8.37159	0.514263
$266$	−4.05128	−0.248400
$267$	0	0
$268$	−5.70555	−0.348522
$269$	15.6684	0.955317	0.477659	−	0.878546i	$-0.341486\pi$
$269$	15.6684	0.955317	0.477659	+	0.878546i	$0.341486\pi$
$270$	0	0
$271$	−18.0835	−1.09849	−0.549247	−	0.835660i	$-0.685086\pi$
$271$	−18.0835	−1.09849	−0.549247	+	0.835660i	$0.685086\pi$
$272$	−0.975288	−0.0591355
$273$	0	0
$274$	0.857285	0.0517905
$275$	38.5180	2.32272
$276$	0	0
$277$	19.4875	1.17089	0.585444	−	0.810713i	$-0.300920\pi$
$277$	19.4875	1.17089	0.585444	+	0.810713i	$0.300920\pi$
$278$	−13.6243	−0.817134
$279$	0	0
$280$	−7.01435	−0.419188
$281$	−3.45732	−0.206246	−0.103123	−	0.994669i	$-0.532884\pi$
$281$	−3.45732	−0.206246	−0.103123	+	0.994669i	$0.532884\pi$
$282$	0	0
$283$	−13.4408	−0.798973	−0.399487	−	0.916739i	$-0.630812\pi$
$283$	−13.4408	−0.798973	−0.399487	+	0.916739i	$0.630812\pi$
$284$	13.5626	0.804790
$285$	0	0
$286$	−25.8055	−1.52591
$287$	10.1092	0.596726
$288$	0	0
$289$	−16.0488	−0.944048
$290$	15.1136	0.887500
$291$	0	0
$292$	10.9059	0.638222
$293$	31.3727	1.83281	0.916407	−	0.400247i	$-0.131076\pi$
$293$	31.3727	1.83281	0.916407	+	0.400247i	$0.131076\pi$
$294$	0	0
$295$	−23.6723	−1.37826
$296$	8.00564	0.465318
$297$	0	0
$298$	−1.00000	−0.0579284
$299$	−23.5109	−1.35967
$300$	0	0
$301$	−4.08075	−0.235210
$302$	−21.1806	−1.21881
$303$	0	0
$304$	2.00975	0.115267
$305$	49.7092	2.84634
$306$	0	0
$307$	−28.8531	−1.64673	−0.823365	−	0.567512i	$-0.807906\pi$
$307$	−28.8531	−1.64673	−0.823365	+	0.567512i	$0.807906\pi$
$308$	10.9235	0.622424
$309$	0	0
$310$	2.35273	0.133626
$311$	−2.34657	−0.133062	−0.0665308	−	0.997784i	$-0.521193\pi$
$311$	−2.34657	−0.133062	−0.0665308	+	0.997784i	$0.521193\pi$
$312$	0	0
$313$	18.3626	1.03792	0.518958	−	0.854800i	$-0.326320\pi$
$313$	18.3626	1.03792	0.518958	+	0.854800i	$0.326320\pi$
$314$	5.91355	0.333721
$315$	0	0
$316$	0.737648	0.0414959
$317$	−9.52950	−0.535230	−0.267615	−	0.963526i	$-0.586236\pi$
$317$	−9.52950	−0.535230	−0.267615	+	0.963526i	$0.586236\pi$
$318$	0	0
$319$	−23.5365	−1.31779
$320$	3.47967	0.194519
$321$	0	0
$322$	9.95217	0.554613
$323$	−1.96009	−0.109062
$324$	0	0
$325$	33.8495	1.87763
$326$	−22.3353	−1.23704
$327$	0	0
$328$	−5.01494	−0.276904
$329$	18.9967	1.04732
$330$	0	0
$331$	−12.6765	−0.696763	−0.348382	−	0.937353i	$-0.613269\pi$
$331$	−12.6765	−0.696763	−0.348382	+	0.937353i	$0.613269\pi$
$332$	−12.0263	−0.660029
$333$	0	0
$334$	6.84198	0.374377
$335$	−19.8534	−1.08471
$336$	0	0
$337$	5.00190	0.272471	0.136236	−	0.990676i	$-0.456500\pi$
$337$	5.00190	0.272471	0.136236	+	0.990676i	$0.456500\pi$
$338$	−9.67785	−0.526406
$339$	0	0
$340$	−3.39368	−0.184048
$341$	−3.66392	−0.198412
$342$	0	0
$343$	−20.0301	−1.08152
$344$	2.02437	0.109147
$345$	0	0
$346$	9.17170	0.493074
$347$	−15.9520	−0.856350	−0.428175	−	0.903696i	$-0.640843\pi$
$347$	−15.9520	−0.856350	−0.428175	+	0.903696i	$0.640843\pi$
$348$	0	0
$349$	−1.23296	−0.0659990	−0.0329995	−	0.999455i	$-0.510506\pi$
$349$	−1.23296	−0.0659990	−0.0329995	+	0.999455i	$0.510506\pi$
$350$	−14.3285	−0.765893
$351$	0	0
$352$	−5.41891	−0.288829
$353$	−34.9387	−1.85960	−0.929799	−	0.368068i	$-0.880020\pi$
$353$	−34.9387	−1.85960	−0.929799	+	0.368068i	$0.880020\pi$
$354$	0	0
$355$	47.1932	2.50476
$356$	0.894003	0.0473821
$357$	0	0
$358$	−3.14761	−0.166356
$359$	29.9852	1.58256	0.791279	−	0.611455i	$-0.209415\pi$
$359$	29.9852	1.58256	0.791279	+	0.611455i	$0.209415\pi$
$360$	0	0
$361$	−14.9609	−0.787415
$362$	−8.54429	−0.449078
$363$	0	0
$364$	9.59955	0.503153
$365$	37.9490	1.98634
$366$	0	0
$367$	−1.51456	−0.0790595	−0.0395298	−	0.999218i	$-0.512586\pi$
$367$	−1.51456	−0.0790595	−0.0395298	+	0.999218i	$0.512586\pi$
$368$	−4.93705	−0.257362
$369$	0	0
$370$	27.8570	1.44821
$371$	4.84976	0.251787
$372$	0	0
$373$	−2.43005	−0.125823	−0.0629117	−	0.998019i	$-0.520039\pi$
$373$	−2.43005	−0.125823	−0.0629117	+	0.998019i	$0.520039\pi$
$374$	5.28500	0.273281
$375$	0	0
$376$	−9.42387	−0.485999
$377$	−20.6838	−1.06527
$378$	0	0
$379$	0.467938	0.0240363	0.0120182	−	0.999928i	$-0.496174\pi$
$379$	0.467938	0.0240363	0.0120182	+	0.999928i	$0.496174\pi$
$380$	6.99327	0.358747
$381$	0	0
$382$	9.91199	0.507141
$383$	29.1832	1.49119	0.745596	−	0.666398i	$-0.232165\pi$
$383$	29.1832	1.49119	0.745596	+	0.666398i	$0.232165\pi$
$384$	0	0
$385$	38.0101	1.93718
$386$	−8.16388	−0.415530
$387$	0	0
$388$	−6.58391	−0.334247
$389$	−22.9274	−1.16247	−0.581234	−	0.813737i	$-0.697430\pi$
$389$	−22.9274	−1.16247	−0.581234	+	0.813737i	$0.697430\pi$
$390$	0	0
$391$	4.81505	0.243508
$392$	2.93650	0.148316
$393$	0	0
$394$	11.8895	0.598983
$395$	2.56677	0.129148
$396$	0	0
$397$	−22.2395	−1.11617	−0.558085	−	0.829784i	$-0.688464\pi$
$397$	−22.2395	−1.11617	−0.558085	+	0.829784i	$0.688464\pi$
$398$	−12.9235	−0.647797
$399$	0	0
$400$	7.10807	0.355404
$401$	37.2988	1.86261	0.931306	−	0.364237i	$-0.118670\pi$
$401$	37.2988	1.86261	0.931306	+	0.364237i	$0.118670\pi$
$402$	0	0
$403$	−3.21985	−0.160392
$404$	−8.76010	−0.435831
$405$	0	0
$406$	8.75548	0.434527
$407$	−43.3818	−2.15036
$408$	0	0
$409$	−1.76177	−0.0871137	−0.0435569	−	0.999051i	$-0.513869\pi$
$409$	−1.76177	−0.0871137	−0.0435569	+	0.999051i	$0.513869\pi$
$410$	−17.4503	−0.861809
$411$	0	0
$412$	12.8547	0.633306
$413$	−13.7137	−0.674805
$414$	0	0
$415$	−41.8475	−2.05421
$416$	−4.76213	−0.233482
$417$	0	0
$418$	−10.8907	−0.532680
$419$	11.6337	0.568345	0.284172	−	0.958773i	$-0.408281\pi$
$419$	11.6337	0.568345	0.284172	+	0.958773i	$0.408281\pi$
$420$	0	0
$421$	17.1886	0.837723	0.418861	−	0.908050i	$-0.362429\pi$
$421$	17.1886	0.837723	0.418861	+	0.908050i	$0.362429\pi$
$422$	1.76036	0.0856932
$423$	0	0
$424$	−2.40586	−0.116839
$425$	−6.93242	−0.336272
$426$	0	0
$427$	28.7972	1.39359
$428$	−12.3420	−0.596573
$429$	0	0
$430$	7.04412	0.339698
$431$	−13.7670	−0.663134	−0.331567	−	0.943432i	$-0.607577\pi$
$431$	−13.7670	−0.663134	−0.331567	+	0.943432i	$0.607577\pi$
$432$	0	0
$433$	30.8957	1.48475	0.742376	−	0.669984i	$-0.233699\pi$
$433$	30.8957	1.48475	0.742376	+	0.669984i	$0.233699\pi$
$434$	1.36296	0.0654244
$435$	0	0
$436$	4.08845	0.195801
$437$	−9.92226	−0.474646
$438$	0	0
$439$	−23.5069	−1.12193	−0.560963	−	0.827841i	$-0.689569\pi$
$439$	−23.5069	−1.12193	−0.560963	+	0.827841i	$0.689569\pi$
$440$	−18.8560	−0.898924
$441$	0	0
$442$	4.64445	0.220914
$443$	−35.5813	−1.69052	−0.845260	−	0.534355i	$-0.820555\pi$
$443$	−35.5813	−1.69052	−0.845260	+	0.534355i	$0.820555\pi$
$444$	0	0
$445$	3.11083	0.147468
$446$	−14.0022	−0.663025
$447$	0	0
$448$	2.01581	0.0952382
$449$	6.38040	0.301110	0.150555	−	0.988602i	$-0.451894\pi$
$449$	6.38040	0.301110	0.150555	+	0.988602i	$0.451894\pi$
$450$	0	0
$451$	27.1755	1.27964
$452$	5.75758	0.270814
$453$	0	0
$454$	1.76335	0.0827579
$455$	33.4032	1.56597
$456$	0	0
$457$	−2.21340	−0.103538	−0.0517692	−	0.998659i	$-0.516486\pi$
$457$	−2.21340	−0.103538	−0.0517692	+	0.998659i	$0.516486\pi$
$458$	24.6220	1.15051
$459$	0	0
$460$	−17.1793	−0.800989
$461$	−12.8846	−0.600094	−0.300047	−	0.953924i	$-0.597002\pi$
$461$	−12.8846	−0.600094	−0.300047	+	0.953924i	$0.597002\pi$
$462$	0	0
$463$	−22.6870	−1.05435	−0.527177	−	0.849755i	$-0.676750\pi$
$463$	−22.6870	−1.05435	−0.527177	+	0.849755i	$0.676750\pi$
$464$	−4.34340	−0.201637
$465$	0	0
$466$	19.8639	0.920175
$467$	31.4227	1.45407	0.727035	−	0.686600i	$-0.240898\pi$
$467$	31.4227	1.45407	0.727035	+	0.686600i	$0.240898\pi$
$468$	0	0
$469$	−11.5013	−0.531081
$470$	−32.7919	−1.51258
$471$	0	0
$472$	6.80304	0.313135
$473$	−10.9699	−0.504395
$474$	0	0
$475$	14.2855	0.655462
$476$	−1.96600	−0.0901114
$477$	0	0
$478$	20.4637	0.935988
$479$	−3.55649	−0.162500	−0.0812501	−	0.996694i	$-0.525891\pi$
$479$	−3.55649	−0.162500	−0.0812501	+	0.996694i	$0.525891\pi$
$480$	0	0
$481$	−38.1239	−1.73830
$482$	−9.65057	−0.439571
$483$	0	0
$484$	18.3646	0.834753
$485$	−22.9098	−1.04028
$486$	0	0
$487$	−32.2388	−1.46088	−0.730439	−	0.682978i	$-0.760685\pi$
$487$	−32.2388	−1.46088	−0.730439	+	0.682978i	$0.760685\pi$
$488$	−14.2856	−0.646680
$489$	0	0
$490$	10.2180	0.461604
$491$	−19.5620	−0.882819	−0.441409	−	0.897306i	$-0.645521\pi$
$491$	−19.5620	−0.882819	−0.441409	+	0.897306i	$0.645521\pi$
$492$	0	0
$493$	4.23607	0.190783
$494$	−9.57070	−0.430606
$495$	0	0
$496$	−0.676136	−0.0303594
$497$	27.3396	1.22635
$498$	0	0
$499$	21.1874	0.948477	0.474238	−	0.880397i	$-0.342724\pi$
$499$	21.1874	0.948477	0.474238	+	0.880397i	$0.342724\pi$
$500$	7.33539	0.328049
$501$	0	0
$502$	15.5906	0.695841
$503$	22.5342	1.00475	0.502376	−	0.864649i	$-0.332459\pi$
$503$	22.5342	1.00475	0.502376	+	0.864649i	$0.332459\pi$
$504$	0	0
$505$	−30.4822	−1.35644
$506$	26.7534	1.18934
$507$	0	0
$508$	−9.93211	−0.440666
$509$	−2.14538	−0.0950924	−0.0475462	−	0.998869i	$-0.515140\pi$
$509$	−2.14538	−0.0950924	−0.0475462	+	0.998869i	$0.515140\pi$
$510$	0	0
$511$	21.9843	0.972529
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−13.1346	−0.579342
$515$	44.7301	1.97104
$516$	0	0
$517$	51.0671	2.24593
$518$	16.1379	0.709057
$519$	0	0
$520$	−16.5706	−0.726669
$521$	−16.9257	−0.741530	−0.370765	−	0.928727i	$-0.620905\pi$
$521$	−16.9257	−0.741530	−0.370765	+	0.928727i	$0.620905\pi$
$522$	0	0
$523$	2.37984	0.104063	0.0520316	−	0.998645i	$-0.483430\pi$
$523$	2.37984	0.104063	0.0520316	+	0.998645i	$0.483430\pi$
$524$	−15.7880	−0.689704
$525$	0	0
$526$	20.0035	0.872195
$527$	0.659428	0.0287251
$528$	0	0
$529$	1.37450	0.0597607
$530$	−8.37159	−0.363639
$531$	0	0
$532$	4.05128	0.175645
$533$	23.8818	1.03443
$534$	0	0
$535$	−42.9461	−1.85672
$536$	5.70555	0.246442
$537$	0	0
$538$	−15.6684	−0.675511
$539$	−15.9126	−0.685405
$540$	0	0
$541$	−12.2640	−0.527271	−0.263636	−	0.964622i	$-0.584922\pi$
$541$	−12.2640	−0.527271	−0.263636	+	0.964622i	$0.584922\pi$
$542$	18.0835	0.776753
$543$	0	0
$544$	0.975288	0.0418151
$545$	14.2264	0.609394
$546$	0	0
$547$	3.75872	0.160711	0.0803557	−	0.996766i	$-0.474394\pi$
$547$	3.75872	0.160711	0.0803557	+	0.996766i	$0.474394\pi$
$548$	−0.857285	−0.0366214
$549$	0	0
$550$	−38.5180	−1.64241
$551$	−8.72916	−0.371875
$552$	0	0
$553$	1.48696	0.0632320
$554$	−19.4875	−0.827943
$555$	0	0
$556$	13.6243	0.577801
$557$	−13.6461	−0.578205	−0.289102	−	0.957298i	$-0.593357\pi$
$557$	−13.6461	−0.578205	−0.289102	+	0.957298i	$0.593357\pi$
$558$	0	0
$559$	−9.64030	−0.407741
$560$	7.01435	0.296411
$561$	0	0
$562$	3.45732	0.145838
$563$	13.0353	0.549370	0.274685	−	0.961534i	$-0.411426\pi$
$563$	13.0353	0.549370	0.274685	+	0.961534i	$0.411426\pi$
$564$	0	0
$565$	20.0344	0.842855
$566$	13.4408	0.564959
$567$	0	0
$568$	−13.5626	−0.569073
$569$	19.5853	0.821057	0.410528	−	0.911848i	$-0.365344\pi$
$569$	19.5853	0.821057	0.410528	+	0.911848i	$0.365344\pi$
$570$	0	0
$571$	−14.5084	−0.607160	−0.303580	−	0.952806i	$-0.598182\pi$
$571$	−14.5084	−0.607160	−0.303580	+	0.952806i	$0.598182\pi$
$572$	25.8055	1.07898
$573$	0	0
$574$	−10.1092	−0.421949
$575$	−35.0929	−1.46348
$576$	0	0
$577$	−25.0069	−1.04105	−0.520526	−	0.853846i	$-0.674264\pi$
$577$	−25.0069	−1.04105	−0.520526	+	0.853846i	$0.674264\pi$
$578$	16.0488	0.667543
$579$	0	0
$580$	−15.1136	−0.627557
$581$	−24.2428	−1.00576
$582$	0	0
$583$	13.0371	0.539943
$584$	−10.9059	−0.451291
$585$	0	0
$586$	−31.3727	−1.29600
$587$	5.25911	0.217067	0.108533	−	0.994093i	$-0.465385\pi$
$587$	5.25911	0.217067	0.108533	+	0.994093i	$0.465385\pi$
$588$	0	0
$589$	−1.35887	−0.0559911
$590$	23.6723	0.974573
$591$	0	0
$592$	−8.00564	−0.329030
$593$	25.6632	1.05386	0.526930	−	0.849909i	$-0.323343\pi$
$593$	25.6632	1.05386	0.526930	+	0.849909i	$0.323343\pi$
$594$	0	0
$595$	−6.84102	−0.280454
$596$	1.00000	0.0409616
$597$	0	0
$598$	23.5109	0.961431
$599$	30.6881	1.25388	0.626940	−	0.779067i	$-0.284307\pi$
$599$	30.6881	1.25388	0.626940	+	0.779067i	$0.284307\pi$
$600$	0	0
$601$	30.2941	1.23572	0.617861	−	0.786288i	$-0.288000\pi$
$601$	30.2941	1.23572	0.617861	+	0.786288i	$0.288000\pi$
$602$	4.08075	0.166319
$603$	0	0
$604$	21.1806	0.861827
$605$	63.9026	2.59801
$606$	0	0
$607$	26.5235	1.07656	0.538278	−	0.842767i	$-0.319075\pi$
$607$	26.5235	1.07656	0.538278	+	0.842767i	$0.319075\pi$
$608$	−2.00975	−0.0815062
$609$	0	0
$610$	−49.7092	−2.01267
$611$	44.8776	1.81556
$612$	0	0
$613$	18.7897	0.758907	0.379453	−	0.925211i	$-0.376112\pi$
$613$	18.7897	0.758907	0.379453	+	0.925211i	$0.376112\pi$
$614$	28.8531	1.16441
$615$	0	0
$616$	−10.9235	−0.440120
$617$	−21.6050	−0.869784	−0.434892	−	0.900483i	$-0.643214\pi$
$617$	−21.6050	−0.869784	−0.434892	+	0.900483i	$0.643214\pi$
$618$	0	0
$619$	−5.22268	−0.209917	−0.104959	−	0.994477i	$-0.533471\pi$
$619$	−5.22268	−0.209917	−0.104959	+	0.994477i	$0.533471\pi$
$620$	−2.35273	−0.0944878
$621$	0	0
$622$	2.34657	0.0940888
$623$	1.80214	0.0722013
$624$	0	0
$625$	−10.0157	−0.400627
$626$	−18.3626	−0.733918
$627$	0	0
$628$	−5.91355	−0.235976
$629$	7.80781	0.311318
$630$	0	0
$631$	−14.9786	−0.596289	−0.298144	−	0.954521i	$-0.596368\pi$
$631$	−14.9786	−0.596289	−0.298144	+	0.954521i	$0.596368\pi$
$632$	−0.737648	−0.0293421
$633$	0	0
$634$	9.52950	0.378465
$635$	−34.5604	−1.37149
$636$	0	0
$637$	−13.9840	−0.554066
$638$	23.5365	0.931818
$639$	0	0
$640$	−3.47967	−0.137546
$641$	−17.8757	−0.706046	−0.353023	−	0.935615i	$-0.614846\pi$
$641$	−17.8757	−0.706046	−0.353023	+	0.935615i	$0.614846\pi$
$642$	0	0
$643$	25.3424	0.999408	0.499704	−	0.866196i	$-0.333442\pi$
$643$	25.3424	0.999408	0.499704	+	0.866196i	$0.333442\pi$
$644$	−9.95217	−0.392171
$645$	0	0
$646$	1.96009	0.0771186
$647$	−24.5412	−0.964815	−0.482408	−	0.875947i	$-0.660238\pi$
$647$	−24.5412	−0.964815	−0.482408	+	0.875947i	$0.660238\pi$
$648$	0	0
$649$	−36.8651	−1.44708
$650$	−33.8495	−1.32769
$651$	0	0
$652$	22.3353	0.874717
$653$	−33.7995	−1.32268	−0.661339	−	0.750087i	$-0.730011\pi$
$653$	−33.7995	−1.32268	−0.661339	+	0.750087i	$0.730011\pi$
$654$	0	0
$655$	−54.9371	−2.14657
$656$	5.01494	0.195800
$657$	0	0
$658$	−18.9967	−0.740570
$659$	−24.4532	−0.952563	−0.476281	−	0.879293i	$-0.658016\pi$
$659$	−24.4532	−0.952563	−0.476281	+	0.879293i	$0.658016\pi$
$660$	0	0
$661$	49.4040	1.92159	0.960797	−	0.277253i	$-0.0894239\pi$
$661$	49.4040	1.92159	0.960797	+	0.277253i	$0.0894239\pi$
$662$	12.6765	0.492686
$663$	0	0
$664$	12.0263	0.466711
$665$	14.0971	0.546663
$666$	0	0
$667$	21.4436	0.830299
$668$	−6.84198	−0.264724
$669$	0	0
$670$	19.8534	0.767004
$671$	77.4125	2.98848
$672$	0	0
$673$	24.9585	0.962078	0.481039	−	0.876699i	$-0.340260\pi$
$673$	24.9585	0.962078	0.481039	+	0.876699i	$0.340260\pi$
$674$	−5.00190	−0.192666
$675$	0	0
$676$	9.67785	0.372225
$677$	9.53517	0.366466	0.183233	−	0.983069i	$-0.441344\pi$
$677$	9.53517	0.366466	0.183233	+	0.983069i	$0.441344\pi$
$678$	0	0
$679$	−13.2719	−0.509330
$680$	3.39368	0.130142
$681$	0	0
$682$	3.66392	0.140299
$683$	−24.2009	−0.926023	−0.463011	−	0.886352i	$-0.653231\pi$
$683$	−24.2009	−0.926023	−0.463011	+	0.886352i	$0.653231\pi$
$684$	0	0
$685$	−2.98306	−0.113977
$686$	20.0301	0.764754
$687$	0	0
$688$	−2.02437	−0.0771783
$689$	11.4570	0.436477
$690$	0	0
$691$	21.5469	0.819681	0.409840	−	0.912157i	$-0.365584\pi$
$691$	21.5469	0.819681	0.409840	+	0.912157i	$0.365584\pi$
$692$	−9.17170	−0.348656
$693$	0	0
$694$	15.9520	0.605531
$695$	47.4081	1.79829
$696$	0	0
$697$	−4.89101	−0.185260
$698$	1.23296	0.0466684
$699$	0	0
$700$	14.3285	0.541568
$701$	32.7007	1.23509	0.617543	−	0.786537i	$-0.288128\pi$
$701$	32.7007	1.23509	0.617543	+	0.786537i	$0.288128\pi$
$702$	0	0
$703$	−16.0894	−0.606822
$704$	5.41891	0.204233
$705$	0	0
$706$	34.9387	1.31493
$707$	−17.6587	−0.664124
$708$	0	0
$709$	33.5648	1.26055	0.630276	−	0.776371i	$-0.282942\pi$
$709$	33.5648	1.26055	0.630276	+	0.776371i	$0.282942\pi$
$710$	−47.1932	−1.77113
$711$	0	0
$712$	−0.894003	−0.0335042
$713$	3.33812	0.125014
$714$	0	0
$715$	89.7946	3.35813
$716$	3.14761	0.117632
$717$	0	0
$718$	−29.9852	−1.11904
$719$	−35.0203	−1.30604	−0.653018	−	0.757342i	$-0.726498\pi$
$719$	−35.0203	−1.30604	−0.653018	+	0.757342i	$0.726498\pi$
$720$	0	0
$721$	25.9127	0.965038
$722$	14.9609	0.556787
$723$	0	0
$724$	8.54429	0.317546
$725$	−30.8732	−1.14660
$726$	0	0
$727$	−30.3502	−1.12563	−0.562814	−	0.826584i	$-0.690281\pi$
$727$	−30.3502	−1.12563	−0.562814	+	0.826584i	$0.690281\pi$
$728$	−9.59955	−0.355783
$729$	0	0
$730$	−37.9490	−1.40456
$731$	1.97434	0.0730237
$732$	0	0
$733$	−3.14151	−0.116034	−0.0580171	−	0.998316i	$-0.518478\pi$
$733$	−3.14151	−0.116034	−0.0580171	+	0.998316i	$0.518478\pi$
$734$	1.51456	0.0559035
$735$	0	0
$736$	4.93705	0.181982
$737$	−30.9178	−1.13887
$738$	0	0
$739$	−2.77796	−0.102189	−0.0510944	−	0.998694i	$-0.516271\pi$
$739$	−2.77796	−0.102189	−0.0510944	+	0.998694i	$0.516271\pi$
$740$	−27.8570	−1.02404
$741$	0	0
$742$	−4.84976	−0.178040
$743$	−13.9626	−0.512239	−0.256119	−	0.966645i	$-0.582444\pi$
$743$	−13.9626	−0.512239	−0.256119	+	0.966645i	$0.582444\pi$
$744$	0	0
$745$	3.47967	0.127485
$746$	2.43005	0.0889706
$747$	0	0
$748$	−5.28500	−0.193239
$749$	−24.8792	−0.909065
$750$	0	0
$751$	8.35496	0.304877	0.152438	−	0.988313i	$-0.451287\pi$
$751$	8.35496	0.304877	0.152438	+	0.988313i	$0.451287\pi$
$752$	9.42387	0.343653
$753$	0	0
$754$	20.6838	0.753260
$755$	73.7015	2.68227
$756$	0	0
$757$	−41.2693	−1.49996	−0.749979	−	0.661462i	$-0.769936\pi$
$757$	−41.2693	−1.49996	−0.749979	+	0.661462i	$0.769936\pi$
$758$	−0.467938	−0.0169963
$759$	0	0
$760$	−6.99327	−0.253672
$761$	0.412554	0.0149551	0.00747753	−	0.999972i	$-0.497620\pi$
$761$	0.412554	0.0149551	0.00747753	+	0.999972i	$0.497620\pi$
$762$	0	0
$763$	8.24155	0.298364
$764$	−9.91199	−0.358603
$765$	0	0
$766$	−29.1832	−1.05443
$767$	−32.3969	−1.16979
$768$	0	0
$769$	8.78229	0.316698	0.158349	−	0.987383i	$-0.449383\pi$
$769$	8.78229	0.316698	0.158349	+	0.987383i	$0.449383\pi$
$770$	−38.0101	−1.36979
$771$	0	0
$772$	8.16388	0.293824
$773$	39.4326	1.41829	0.709145	−	0.705062i	$-0.249081\pi$
$773$	39.4326	1.41829	0.709145	+	0.705062i	$0.249081\pi$
$774$	0	0
$775$	−4.80603	−0.172638
$776$	6.58391	0.236348
$777$	0	0
$778$	22.9274	0.821989
$779$	10.0788	0.361110
$780$	0	0
$781$	73.4943	2.62983
$782$	−4.81505	−0.172186
$783$	0	0
$784$	−2.93650	−0.104875
$785$	−20.5772	−0.734430
$786$	0	0
$787$	−0.528869	−0.0188521	−0.00942606	−	0.999956i	$-0.503000\pi$
$787$	−0.528869	−0.0188521	−0.00942606	+	0.999956i	$0.503000\pi$
$788$	−11.8895	−0.423545
$789$	0	0
$790$	−2.56677	−0.0913215
$791$	11.6062	0.412669
$792$	0	0
$793$	68.0300	2.41582
$794$	22.2395	0.789251
$795$	0	0
$796$	12.9235	0.458061
$797$	40.8195	1.44590	0.722950	−	0.690900i	$-0.242785\pi$
$797$	40.8195	1.44590	0.722950	+	0.690900i	$0.242785\pi$
$798$	0	0
$799$	−9.19099	−0.325154
$800$	−7.10807	−0.251308
$801$	0	0
$802$	−37.2988	−1.31707
$803$	59.0983	2.08553
$804$	0	0
$805$	−34.6302	−1.22056
$806$	3.21985	0.113414
$807$	0	0
$808$	8.76010	0.308179
$809$	2.09921	0.0738044	0.0369022	−	0.999319i	$-0.488251\pi$
$809$	2.09921	0.0738044	0.0369022	+	0.999319i	$0.488251\pi$
$810$	0	0
$811$	16.7659	0.588731	0.294366	−	0.955693i	$-0.404892\pi$
$811$	16.7659	0.588731	0.294366	+	0.955693i	$0.404892\pi$
$812$	−8.75548	−0.307257
$813$	0	0
$814$	43.3818	1.52053
$815$	77.7193	2.72239
$816$	0	0
$817$	−4.06848	−0.142338
$818$	1.76177	0.0615987
$819$	0	0
$820$	17.4503	0.609391
$821$	12.2985	0.429219	0.214610	−	0.976700i	$-0.431152\pi$
$821$	12.2985	0.429219	0.214610	+	0.976700i	$0.431152\pi$
$822$	0	0
$823$	−2.21029	−0.0770458	−0.0385229	−	0.999258i	$-0.512265\pi$
$823$	−2.21029	−0.0770458	−0.0385229	+	0.999258i	$0.512265\pi$
$824$	−12.8547	−0.447815
$825$	0	0
$826$	13.7137	0.477159
$827$	−33.7571	−1.17385	−0.586925	−	0.809641i	$-0.699662\pi$
$827$	−33.7571	−1.17385	−0.586925	+	0.809641i	$0.699662\pi$
$828$	0	0
$829$	34.4237	1.19559	0.597793	−	0.801651i	$-0.296045\pi$
$829$	34.4237	1.19559	0.597793	+	0.801651i	$0.296045\pi$
$830$	41.8475	1.45255
$831$	0	0
$832$	4.76213	0.165097
$833$	2.86393	0.0992294
$834$	0	0
$835$	−23.8078	−0.823903
$836$	10.8907	0.376662
$837$	0	0
$838$	−11.6337	−0.401881
$839$	22.5617	0.778917	0.389458	−	0.921044i	$-0.372662\pi$
$839$	22.5617	0.778917	0.389458	+	0.921044i	$0.372662\pi$
$840$	0	0
$841$	−10.1349	−0.349478
$842$	−17.1886	−0.592360
$843$	0	0
$844$	−1.76036	−0.0605942
$845$	33.6757	1.15848
$846$	0	0
$847$	37.0195	1.27201
$848$	2.40586	0.0826176
$849$	0	0
$850$	6.93242	0.237780
$851$	39.5243	1.35487
$852$	0	0
$853$	1.88445	0.0645225	0.0322612	−	0.999479i	$-0.489729\pi$
$853$	1.88445	0.0645225	0.0322612	+	0.999479i	$0.489729\pi$
$854$	−28.7972	−0.985418
$855$	0	0
$856$	12.3420	0.421841
$857$	6.91499	0.236212	0.118106	−	0.993001i	$-0.462318\pi$
$857$	6.91499	0.236212	0.118106	+	0.993001i	$0.462318\pi$
$858$	0	0
$859$	−32.9567	−1.12447	−0.562235	−	0.826978i	$-0.690058\pi$
$859$	−32.9567	−1.12447	−0.562235	+	0.826978i	$0.690058\pi$
$860$	−7.04412	−0.240203
$861$	0	0
$862$	13.7670	0.468907
$863$	−47.0948	−1.60312	−0.801562	−	0.597911i	$-0.795998\pi$
$863$	−47.0948	−1.60312	−0.801562	+	0.597911i	$0.795998\pi$
$864$	0	0
$865$	−31.9144	−1.08512
$866$	−30.8957	−1.04988
$867$	0	0
$868$	−1.36296	−0.0462620
$869$	3.99725	0.135597
$870$	0	0
$871$	−27.1705	−0.920638
$872$	−4.08845	−0.138452
$873$	0	0
$874$	9.92226	0.335625
$875$	14.7868	0.499884
$876$	0	0
$877$	4.18726	0.141394	0.0706969	−	0.997498i	$-0.477478\pi$
$877$	4.18726	0.141394	0.0706969	+	0.997498i	$0.477478\pi$
$878$	23.5069	0.793321
$879$	0	0
$880$	18.8560	0.635635
$881$	35.4781	1.19529	0.597644	−	0.801762i	$-0.296104\pi$
$881$	35.4781	1.19529	0.597644	+	0.801762i	$0.296104\pi$
$882$	0	0
$883$	54.2771	1.82657	0.913284	−	0.407323i	$-0.133538\pi$
$883$	54.2771	1.82657	0.913284	+	0.407323i	$0.133538\pi$
$884$	−4.64445	−0.156210
$885$	0	0
$886$	35.5813	1.19538
$887$	−3.77073	−0.126609	−0.0633043	−	0.997994i	$-0.520164\pi$
$887$	−3.77073	−0.126609	−0.0633043	+	0.997994i	$0.520164\pi$
$888$	0	0
$889$	−20.0213	−0.671492
$890$	−3.11083	−0.104275
$891$	0	0
$892$	14.0022	0.468829
$893$	18.9396	0.633791
$894$	0	0
$895$	10.9526	0.366106
$896$	−2.01581	−0.0673436
$897$	0	0
$898$	−6.38040	−0.212917
$899$	2.93673	0.0979455
$900$	0	0
$901$	−2.34641	−0.0781701
$902$	−27.1755	−0.904845
$903$	0	0
$904$	−5.75758	−0.191494
$905$	29.7313	0.988301
$906$	0	0
$907$	2.17099	0.0720864	0.0360432	−	0.999350i	$-0.488525\pi$
$907$	2.17099	0.0720864	0.0360432	+	0.999350i	$0.488525\pi$
$908$	−1.76335	−0.0585187
$909$	0	0
$910$	−33.4032	−1.10731
$911$	12.8909	0.427094	0.213547	−	0.976933i	$-0.431498\pi$
$911$	12.8909	0.427094	0.213547	+	0.976933i	$0.431498\pi$
$912$	0	0
$913$	−65.1694	−2.15679
$914$	2.21340	0.0732128
$915$	0	0
$916$	−24.6220	−0.813532
$917$	−31.8257	−1.05098
$918$	0	0
$919$	−27.6889	−0.913374	−0.456687	−	0.889627i	$-0.650964\pi$
$919$	−27.6889	−0.913374	−0.456687	+	0.889627i	$0.650964\pi$
$920$	17.1793	0.566385
$921$	0	0
$922$	12.8846	0.424331
$923$	64.5867	2.12590
$924$	0	0
$925$	−56.9047	−1.87101
$926$	22.6870	0.745541
$927$	0	0
$928$	4.34340	0.142579
$929$	42.2570	1.38641	0.693203	−	0.720743i	$-0.256199\pi$
$929$	42.2570	1.38641	0.693203	+	0.720743i	$0.256199\pi$
$930$	0	0
$931$	−5.90164	−0.193418
$932$	−19.8639	−0.650662
$933$	0	0
$934$	−31.4227	−1.02818
$935$	−18.3900	−0.601418
$936$	0	0
$937$	50.4698	1.64878	0.824388	−	0.566025i	$-0.191520\pi$
$937$	50.4698	1.64878	0.824388	+	0.566025i	$0.191520\pi$
$938$	11.5013	0.375531
$939$	0	0
$940$	32.7919	1.06955
$941$	−47.7943	−1.55805	−0.779025	−	0.626992i	$-0.784286\pi$
$941$	−47.7943	−1.55805	−0.779025	+	0.626992i	$0.784286\pi$
$942$	0	0
$943$	−24.7590	−0.806264
$944$	−6.80304	−0.221420
$945$	0	0
$946$	10.9699	0.356661
$947$	−30.7873	−1.00045	−0.500227	−	0.865894i	$-0.666750\pi$
$947$	−30.7873	−1.00045	−0.500227	+	0.865894i	$0.666750\pi$
$948$	0	0
$949$	51.9355	1.68590
$950$	−14.2855	−0.463482
$951$	0	0
$952$	1.96600	0.0637184
$953$	−42.3743	−1.37264	−0.686319	−	0.727301i	$-0.740775\pi$
$953$	−42.3743	−1.37264	−0.686319	+	0.727301i	$0.740775\pi$
$954$	0	0
$955$	−34.4904	−1.11608
$956$	−20.4637	−0.661843
$957$	0	0
$958$	3.55649	0.114905
$959$	−1.72812	−0.0558041
$960$	0	0
$961$	−30.5428	−0.985253
$962$	38.1239	1.22916
$963$	0	0
$964$	9.65057	0.310824
$965$	28.4076	0.914472
$966$	0	0
$967$	38.0152	1.22249	0.611243	−	0.791443i	$-0.290670\pi$
$967$	38.0152	1.22249	0.611243	+	0.791443i	$0.290670\pi$
$968$	−18.3646	−0.590260
$969$	0	0
$970$	22.9098	0.735589
$971$	14.4352	0.463248	0.231624	−	0.972805i	$-0.425596\pi$
$971$	14.4352	0.463248	0.231624	+	0.972805i	$0.425596\pi$
$972$	0	0
$973$	27.4641	0.880459
$974$	32.2388	1.03300
$975$	0	0
$976$	14.2856	0.457272
$977$	27.8670	0.891545	0.445773	−	0.895146i	$-0.352929\pi$
$977$	27.8670	0.891545	0.445773	+	0.895146i	$0.352929\pi$
$978$	0	0
$979$	4.84452	0.154832
$980$	−10.2180	−0.326403
$981$	0	0
$982$	19.5620	0.624247
$983$	−49.2649	−1.57131	−0.785653	−	0.618667i	$-0.787673\pi$
$983$	−49.2649	−1.57131	−0.785653	+	0.618667i	$0.787673\pi$
$984$	0	0
$985$	−41.3714	−1.31820
$986$	−4.23607	−0.134904
$987$	0	0
$988$	9.57070	0.304484
$989$	9.99441	0.317804
$990$	0	0
$991$	−39.6323	−1.25896	−0.629481	−	0.777016i	$-0.716733\pi$
$991$	−39.6323	−1.25896	−0.629481	+	0.777016i	$0.716733\pi$
$992$	0.676136	0.0214674
$993$	0	0
$994$	−27.3396	−0.867159
$995$	44.9695	1.42563
$996$	0	0
$997$	−48.5956	−1.53904	−0.769519	−	0.638624i	$-0.779504\pi$
$997$	−48.5956	−1.53904	−0.769519	+	0.638624i	$0.779504\pi$
$998$	−21.1874	−0.670674
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.s.1.15	✓	16
3.2	odd	2		8046.2.a.t.1.2	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.s.1.15	✓	16	1.1	even	1	trivial
8046.2.a.t.1.2	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +3.47967 q^{5} +2.01581 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +3.47967 q^{5} +2.01581 q^{7} -1.00000 q^{8} -3.47967 q^{10} +5.41891 q^{11} +4.76213 q^{13} -2.01581 q^{14} +1.00000 q^{16} -0.975288 q^{17} +2.00975 q^{19} +3.47967 q^{20} -5.41891 q^{22} -4.93705 q^{23} +7.10807 q^{25} -4.76213 q^{26} +2.01581 q^{28} -4.34340 q^{29} -0.676136 q^{31} -1.00000 q^{32} +0.975288 q^{34} +7.01435 q^{35} -8.00564 q^{37} -2.00975 q^{38} -3.47967 q^{40} +5.01494 q^{41} -2.02437 q^{43} +5.41891 q^{44} +4.93705 q^{46} +9.42387 q^{47} -2.93650 q^{49} -7.10807 q^{50} +4.76213 q^{52} +2.40586 q^{53} +18.8560 q^{55} -2.01581 q^{56} +4.34340 q^{58} -6.80304 q^{59} +14.2856 q^{61} +0.676136 q^{62} +1.00000 q^{64} +16.5706 q^{65} -5.70555 q^{67} -0.975288 q^{68} -7.01435 q^{70} +13.5626 q^{71} +10.9059 q^{73} +8.00564 q^{74} +2.00975 q^{76} +10.9235 q^{77} +0.737648 q^{79} +3.47967 q^{80} -5.01494 q^{82} -12.0263 q^{83} -3.39368 q^{85} +2.02437 q^{86} -5.41891 q^{88} +0.894003 q^{89} +9.59955 q^{91} -4.93705 q^{92} -9.42387 q^{94} +6.99327 q^{95} -6.58391 q^{97} +2.93650 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{2} + 16 q^{4} - 4 q^{5} + 6 q^{7} - 16 q^{8}+O(q^{10}) \) 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