LMFDB - Embedded newform 8046.2.a.t.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8046,2,Mod(1,8046)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8046, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8046.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$64.2476334663$
Analytic rank:	$0$
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			1.2
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	$-0.783805\pi$
$5$	−3.47967	−1.55615	−0.778077	+	0.628169i	$0.783805\pi$
$6$	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	$-0.804326\pi$
$11$	−5.41891	−1.63386	−0.816931	+	0.576735i	$0.804326\pi$
$12$	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	$-0.462265\pi$
$17$	0.975288	0.236542	0.118271	+	0.992981i	$0.462265\pi$
$18$	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	$-0.327895\pi$
$23$	4.93705	1.02945	0.514723	+	0.857356i	$0.327895\pi$
$24$	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	$-0.367872\pi$
$29$	4.34340	0.806549	0.403275	+	0.915079i	$0.367872\pi$
$30$	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	$-0.628079\pi$
$41$	−5.01494	−0.783202	−0.391601	+	0.920135i	$0.628079\pi$
$42$	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	$-0.741207\pi$
$47$	−9.42387	−1.37461	−0.687306	+	0.726368i	$0.741207\pi$
$48$	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	$-0.552838\pi$
$53$	−2.40586	−0.330470	−0.165235	+	0.986254i	$0.552838\pi$
$54$	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	$-0.353971\pi$
$59$	6.80304	0.885680	0.442840	+	0.896601i	$0.353971\pi$
$60$	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	$-0.797722\pi$
$71$	−13.5626	−1.60958	−0.804790	+	0.593559i	$0.797722\pi$
$72$	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	$-0.270544\pi$
$83$	12.0263	1.32006	0.660029	+	0.751240i	$0.270544\pi$
$84$	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	$-0.515088\pi$
$89$	−0.894003	−0.0947641	−0.0473821	+	0.998877i	$0.515088\pi$
$90$	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	$-0.356454\pi$
$101$	8.76010	0.871663	0.435831	+	0.900028i	$0.356454\pi$
$102$	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	$-0.296528\pi$
$107$	12.3420	1.19315	0.596573	+	0.802558i	$0.296528\pi$
$108$	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	$-0.587293\pi$
$113$	−5.75758	−0.541627	−0.270814	+	0.962632i	$0.587293\pi$
$114$	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	$-0.257741\pi$
$131$	15.7880	1.37941	0.689704	+	0.724092i	$0.257741\pi$
$132$	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	$-0.488340\pi$
$137$	0.857285	0.0732428	0.0366214	+	0.999329i	$0.488340\pi$
$138$	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	$-0.414719\pi$
$167$	6.84198	0.529449	0.264724	+	0.964324i	$0.414719\pi$
$168$	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	$-0.386638\pi$
$173$	9.17170	0.697311	0.348656	+	0.937251i	$0.386638\pi$
$174$	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	$-0.537530\pi$
$179$	−3.14761	−0.235264	−0.117632	+	0.993057i	$0.537530\pi$
$180$	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	$-0.383253\pi$
$191$	9.91199	0.717206	0.358603	+	0.933490i	$0.383253\pi$
$192$	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	$-0.360786\pi$
$197$	11.8895	0.847090	0.423545	+	0.905875i	$0.360786\pi$
$198$	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	$-0.481362\pi$
$227$	1.76335	0.117037	0.0585187	+	0.998286i	$0.481362\pi$
$228$	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	$-0.274491\pi$
$233$	19.8639	1.30132	0.650662	+	0.759367i	$0.274491\pi$
$234$	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	$-0.269774\pi$
$239$	20.4637	1.32369	0.661843	+	0.749642i	$0.269774\pi$
$240$	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	$-0.336254\pi$
$251$	15.5906	0.984068	0.492034	+	0.870576i	$0.336254\pi$
$252$	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	$-0.634351\pi$
$257$	−13.1346	−0.819313	−0.409657	+	0.912240i	$0.634351\pi$
$258$	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	$-0.288455\pi$
$263$	20.0035	1.23347	0.616735	+	0.787171i	$0.288455\pi$
$264$	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	$-0.658514\pi$
$269$	−15.6684	−0.955317	−0.477659	+	0.878546i	$0.658514\pi$
$270$	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	$-0.467116\pi$
$281$	3.45732	0.206246	0.103123	+	0.994669i	$0.467116\pi$
$282$	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	$-0.868924\pi$
$293$	−31.3727	−1.83281	−0.916407	+	0.400247i	$0.868924\pi$
$294$	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	$-0.478807\pi$
$311$	2.34657	0.133062	0.0665308	+	0.997784i	$0.478807\pi$
$312$	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	$-0.413764\pi$
$317$	9.52950	0.535230	0.267615	+	0.963526i	$0.413764\pi$
$318$	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	$-0.359157\pi$
$347$	15.9520	0.856350	0.428175	+	0.903696i	$0.359157\pi$
$348$	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	$-0.119980\pi$
$353$	34.9387	1.85960	0.929799	+	0.368068i	$0.119980\pi$
$354$	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	$-0.790585\pi$
$359$	−29.9852	−1.58256	−0.791279	+	0.611455i	$0.790585\pi$
$360$	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	$-0.767835\pi$
$383$	−29.1832	−1.49119	−0.745596	+	0.666398i	$0.767835\pi$
$384$	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	$-0.302570\pi$
$389$	22.9274	1.16247	0.581234	+	0.813737i	$0.302570\pi$
$390$	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	$-0.881330\pi$
$401$	−37.2988	−1.86261	−0.931306	+	0.364237i	$0.881330\pi$
$402$	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	$-0.591719\pi$
$419$	−11.6337	−0.568345	−0.284172	+	0.958773i	$0.591719\pi$
$420$	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	$-0.392423\pi$
$431$	13.7670	0.663134	0.331567	+	0.943432i	$0.392423\pi$
$432$	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	$-0.179445\pi$
$443$	35.5813	1.69052	0.845260	+	0.534355i	$0.179445\pi$
$444$	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	$-0.548106\pi$
$449$	−6.38040	−0.301110	−0.150555	+	0.988602i	$0.548106\pi$
$450$	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	$-0.402998\pi$
$461$	12.8846	0.600094	0.300047	+	0.953924i	$0.402998\pi$
$462$	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	$-0.759102\pi$
$467$	−31.4227	−1.45407	−0.727035	+	0.686600i	$0.759102\pi$
$468$	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	$-0.474109\pi$
$479$	3.55649	0.162500	0.0812501	+	0.996694i	$0.474109\pi$
$480$	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	$-0.354479\pi$
$491$	19.5620	0.882819	0.441409	+	0.897306i	$0.354479\pi$
$492$	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	$-0.667541\pi$
$503$	−22.5342	−1.00475	−0.502376	+	0.864649i	$0.667541\pi$
$504$	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	$-0.484860\pi$
$509$	2.14538	0.0950924	0.0475462	+	0.998869i	$0.484860\pi$
$510$	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	$-0.379095\pi$
$521$	16.9257	0.741530	0.370765	+	0.928727i	$0.379095\pi$
$522$	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	$-0.406643\pi$
$557$	13.6461	0.578205	0.289102	+	0.957298i	$0.406643\pi$
$558$	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	$-0.588574\pi$
$563$	−13.0353	−0.549370	−0.274685	+	0.961534i	$0.588574\pi$
$564$	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	$-0.634656\pi$
$569$	−19.5853	−0.821057	−0.410528	+	0.911848i	$0.634656\pi$
$570$	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	$-0.534615\pi$
$587$	−5.25911	−0.217067	−0.108533	+	0.994093i	$0.534615\pi$
$588$	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	$-0.676657\pi$
$593$	−25.6632	−1.05386	−0.526930	+	0.849909i	$0.676657\pi$
$594$	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	$-0.715693\pi$
$599$	−30.6881	−1.25388	−0.626940	+	0.779067i	$0.715693\pi$
$600$	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	$-0.356786\pi$
$617$	21.6050	0.869784	0.434892	+	0.900483i	$0.356786\pi$
$618$	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	$-0.385154\pi$
$641$	17.8757	0.706046	0.353023	+	0.935615i	$0.385154\pi$
$642$	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	$-0.339762\pi$
$647$	24.5412	0.964815	0.482408	+	0.875947i	$0.339762\pi$
$648$	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	$-0.269989\pi$
$653$	33.7995	1.32268	0.661339	+	0.750087i	$0.269989\pi$
$654$	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	$-0.341984\pi$
$659$	24.4532	0.952563	0.476281	+	0.879293i	$0.341984\pi$
$660$	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	$-0.558656\pi$
$677$	−9.53517	−0.366466	−0.183233	+	0.983069i	$0.558656\pi$
$678$	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	$-0.346769\pi$
$683$	24.2009	0.926023	0.463011	+	0.886352i	$0.346769\pi$
$684$	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	$-0.711872\pi$
$701$	−32.7007	−1.23509	−0.617543	+	0.786537i	$0.711872\pi$
$702$	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	$-0.273502\pi$
$719$	35.0203	1.30604	0.653018	+	0.757342i	$0.273502\pi$
$720$	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	$-0.417556\pi$
$743$	13.9626	0.512239	0.256119	+	0.966645i	$0.417556\pi$
$744$	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	$-0.502380\pi$
$761$	−0.412554	−0.0149551	−0.00747753	+	0.999972i	$0.502380\pi$
$762$	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	$-0.750919\pi$
$773$	−39.4326	−1.41829	−0.709145	+	0.705062i	$0.750919\pi$
$774$	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	$-0.757215\pi$
$797$	−40.8195	−1.44590	−0.722950	+	0.690900i	$0.757215\pi$
$798$	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	$-0.511749\pi$
$809$	−2.09921	−0.0738044	−0.0369022	+	0.999319i	$0.511749\pi$
$810$	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	$-0.568848\pi$
$821$	−12.2985	−0.429219	−0.214610	+	0.976700i	$0.568848\pi$
$822$	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	$-0.300338\pi$
$827$	33.7571	1.17385	0.586925	+	0.809641i	$0.300338\pi$
$828$	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	$-0.627338\pi$
$839$	−22.5617	−0.778917	−0.389458	+	0.921044i	$0.627338\pi$
$840$	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	$-0.537682\pi$
$857$	−6.91499	−0.236212	−0.118106	+	0.993001i	$0.537682\pi$
$858$	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	$-0.204002\pi$
$863$	47.0948	1.60312	0.801562	+	0.597911i	$0.204002\pi$
$864$	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	$-0.703896\pi$
$881$	−35.4781	−1.19529	−0.597644	+	0.801762i	$0.703896\pi$
$882$	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	$-0.479836\pi$
$887$	3.77073	0.126609	0.0633043	+	0.997994i	$0.479836\pi$
$888$	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	$-0.568502\pi$
$911$	−12.8909	−0.427094	−0.213547	+	0.976933i	$0.568502\pi$
$912$	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	$-0.743801\pi$
$929$	−42.2570	−1.38641	−0.693203	+	0.720743i	$0.743801\pi$
$930$	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	$-0.215714\pi$
$941$	47.7943	1.55805	0.779025	+	0.626992i	$0.215714\pi$
$942$	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	$-0.333250\pi$
$947$	30.7873	1.00045	0.500227	+	0.865894i	$0.333250\pi$
$948$	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	$-0.259225\pi$
$953$	42.3743	1.37264	0.686319	+	0.727301i	$0.259225\pi$
$954$	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	$-0.574404\pi$
$971$	−14.4352	−0.463248	−0.231624	+	0.972805i	$0.574404\pi$
$972$	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	$-0.647071\pi$
$977$	−27.8670	−0.891545	−0.445773	+	0.895146i	$0.647071\pi$
$978$	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	$-0.212327\pi$
$983$	49.2649	1.57131	0.785653	+	0.618667i	$0.212327\pi$
$984$	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.t.1.2	yes	16
3.2	odd	2		8046.2.a.s.1.15	✓	16

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

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

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Database

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