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Properties

Label 1134.2.t.h.593.3
Level $1134$
Weight $2$
Character 1134.593
Analytic conductor $9.055$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1134,2,Mod(593,1134)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1134.593"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1134, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([1, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1134 = 2 \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1134.t (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0,8,0,0,-4,0,0,0,0,0,12,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.05503558921\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8 x^{15} + 52 x^{14} - 224 x^{13} + 796 x^{12} - 2228 x^{11} + 5254 x^{10} - 10232 x^{9} + \cdots + 225 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 593.3
Root \(0.500000 + 3.05304i\) of defining polynomial
Character \(\chi\) \(=\) 1134.593
Dual form 1134.2.t.h.1025.3

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.866025 + 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} +0.720136 q^{5} +(-2.62682 - 0.315916i) q^{7} +1.00000i q^{8} +(-0.623656 + 0.360068i) q^{10} -4.41031i q^{11} +(-1.08644 + 0.627256i) q^{13} +(2.43285 - 1.03982i) q^{14} +(-0.500000 - 0.866025i) q^{16} +(3.25492 + 5.63769i) q^{17} +(-4.66633 - 2.69411i) q^{19} +(0.360068 - 0.623656i) q^{20} +(2.20516 + 3.81944i) q^{22} +6.87632i q^{23} -4.48140 q^{25} +(0.627256 - 1.08644i) q^{26} +(-1.58700 + 2.11694i) q^{28} +(0.0191795 + 0.0110733i) q^{29} +(-3.21600 - 1.85676i) q^{31} +(0.866025 + 0.500000i) q^{32} +(-5.63769 - 3.25492i) q^{34} +(-1.89167 - 0.227503i) q^{35} +(-1.82735 + 3.16506i) q^{37} +5.38821 q^{38} +0.720136i q^{40} +(0.981388 + 1.69981i) q^{41} +(-4.94223 + 8.56019i) q^{43} +(-3.81944 - 2.20516i) q^{44} +(-3.43816 - 5.95507i) q^{46} +(-4.11735 - 7.13145i) q^{47} +(6.80039 + 1.65971i) q^{49} +(3.88101 - 2.24070i) q^{50} +1.25451i q^{52} +(-2.95555 + 1.70638i) q^{53} -3.17602i q^{55} +(0.315916 - 2.62682i) q^{56} -0.0221465 q^{58} +(2.09076 - 3.62130i) q^{59} +(-7.08448 + 4.09023i) q^{61} +3.71351 q^{62} -1.00000 q^{64} +(-0.782383 + 0.451709i) q^{65} +(-5.60764 + 9.71272i) q^{67} +6.50984 q^{68} +(1.75198 - 0.748811i) q^{70} -2.47961i q^{71} +(8.54555 - 4.93377i) q^{73} -3.65470i q^{74} +(-4.66633 + 2.69411i) q^{76} +(-1.39329 + 11.5851i) q^{77} +(-1.23351 - 2.13650i) q^{79} +(-0.360068 - 0.623656i) q^{80} +(-1.69981 - 0.981388i) q^{82} +(-5.69625 + 9.86619i) q^{83} +(2.34398 + 4.05990i) q^{85} -9.88445i q^{86} +4.41031 q^{88} +(-7.50204 + 12.9939i) q^{89} +(3.05204 - 1.30447i) q^{91} +(5.95507 + 3.43816i) q^{92} +(7.13145 + 4.11735i) q^{94} +(-3.36039 - 1.94012i) q^{95} +(-8.89063 - 5.13301i) q^{97} +(-6.71917 + 1.96284i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{4} - 4 q^{7} + 12 q^{13} + 12 q^{14} - 8 q^{16} + 16 q^{25} + 4 q^{28} + 48 q^{29} - 12 q^{31} - 60 q^{35} + 4 q^{37} - 24 q^{38} - 24 q^{41} + 16 q^{43} - 12 q^{44} + 16 q^{49} - 24 q^{50}+ \cdots - 36 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1134\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(407\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{6}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.866025 + 0.500000i −0.612372 + 0.353553i
\(3\) 0 0
\(4\) 0.500000 − 0.866025i 0.250000 − 0.433013i
\(5\) 0.720136 0.322055 0.161027 − 0.986950i \(-0.448519\pi\)
0.161027 + 0.986950i \(0.448519\pi\)
\(6\) 0 0
\(7\) −2.62682 − 0.315916i −0.992846 − 0.119405i
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) −0.623656 + 0.360068i −0.197217 + 0.113863i
\(11\) − 4.41031i − 1.32976i −0.746950 − 0.664880i \(-0.768483\pi\)
0.746950 − 0.664880i \(-0.231517\pi\)
\(12\) 0 0
\(13\) −1.08644 + 0.627256i −0.301324 + 0.173969i −0.643037 − 0.765835i \(-0.722326\pi\)
0.341714 + 0.939804i \(0.388993\pi\)
\(14\) 2.43285 − 1.03982i 0.650207 − 0.277903i
\(15\) 0 0
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) 3.25492 + 5.63769i 0.789434 + 1.36734i 0.926314 + 0.376752i \(0.122959\pi\)
−0.136880 + 0.990588i \(0.543708\pi\)
\(18\) 0 0
\(19\) −4.66633 − 2.69411i −1.07053 − 0.618070i −0.142203 − 0.989838i \(-0.545419\pi\)
−0.928326 + 0.371767i \(0.878752\pi\)
\(20\) 0.360068 − 0.623656i 0.0805136 − 0.139454i
\(21\) 0 0
\(22\) 2.20516 + 3.81944i 0.470141 + 0.814308i
\(23\) 6.87632i 1.43381i 0.697170 + 0.716906i \(0.254442\pi\)
−0.697170 + 0.716906i \(0.745558\pi\)
\(24\) 0 0
\(25\) −4.48140 −0.896281
\(26\) 0.627256 − 1.08644i 0.123015 − 0.213068i
\(27\) 0 0
\(28\) −1.58700 + 2.11694i −0.299915 + 0.400063i
\(29\) 0.0191795 + 0.0110733i 0.00356154 + 0.00205625i 0.501780 − 0.864995i \(-0.332679\pi\)
−0.498218 + 0.867052i \(0.666012\pi\)
\(30\) 0 0
\(31\) −3.21600 − 1.85676i −0.577610 − 0.333483i 0.182573 − 0.983192i \(-0.441557\pi\)
−0.760183 + 0.649709i \(0.774891\pi\)
\(32\) 0.866025 + 0.500000i 0.153093 + 0.0883883i
\(33\) 0 0
\(34\) −5.63769 − 3.25492i −0.966855 − 0.558214i
\(35\) −1.89167 − 0.227503i −0.319750 − 0.0384550i
\(36\) 0 0
\(37\) −1.82735 + 3.16506i −0.300414 + 0.520333i −0.976230 − 0.216738i \(-0.930458\pi\)
0.675815 + 0.737071i \(0.263792\pi\)
\(38\) 5.38821 0.874083
\(39\) 0 0
\(40\) 0.720136i 0.113863i
\(41\) 0.981388 + 1.69981i 0.153267 + 0.265466i 0.932427 − 0.361359i \(-0.117687\pi\)
−0.779160 + 0.626826i \(0.784354\pi\)
\(42\) 0 0
\(43\) −4.94223 + 8.56019i −0.753682 + 1.30542i 0.192344 + 0.981328i \(0.438391\pi\)
−0.946027 + 0.324089i \(0.894942\pi\)
\(44\) −3.81944 − 2.20516i −0.575803 − 0.332440i
\(45\) 0 0
\(46\) −3.43816 − 5.95507i −0.506929 − 0.878027i
\(47\) −4.11735 − 7.13145i −0.600577 − 1.04023i −0.992734 − 0.120331i \(-0.961604\pi\)
0.392157 − 0.919898i \(-0.371729\pi\)
\(48\) 0 0
\(49\) 6.80039 + 1.65971i 0.971485 + 0.237102i
\(50\) 3.88101 − 2.24070i 0.548858 − 0.316883i
\(51\) 0 0
\(52\) 1.25451i 0.173969i
\(53\) −2.95555 + 1.70638i −0.405975 + 0.234390i −0.689059 − 0.724705i \(-0.741976\pi\)
0.283084 + 0.959095i \(0.408643\pi\)
\(54\) 0 0
\(55\) − 3.17602i − 0.428255i
\(56\) 0.315916 − 2.62682i 0.0422161 − 0.351024i
\(57\) 0 0
\(58\) −0.0221465 −0.00290798
\(59\) 2.09076 − 3.62130i 0.272193 − 0.471452i −0.697230 − 0.716848i \(-0.745584\pi\)
0.969423 + 0.245395i \(0.0789177\pi\)
\(60\) 0 0
\(61\) −7.08448 + 4.09023i −0.907075 + 0.523700i −0.879489 − 0.475920i \(-0.842115\pi\)
−0.0275859 + 0.999619i \(0.508782\pi\)
\(62\) 3.71351 0.471616
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) −0.782383 + 0.451709i −0.0970427 + 0.0560276i
\(66\) 0 0
\(67\) −5.60764 + 9.71272i −0.685083 + 1.18660i 0.288328 + 0.957532i \(0.406901\pi\)
−0.973411 + 0.229066i \(0.926433\pi\)
\(68\) 6.50984 0.789434
\(69\) 0 0
\(70\) 1.75198 − 0.748811i 0.209402 − 0.0895001i
\(71\) − 2.47961i − 0.294275i −0.989116 − 0.147138i \(-0.952994\pi\)
0.989116 − 0.147138i \(-0.0470060\pi\)
\(72\) 0 0
\(73\) 8.54555 − 4.93377i 1.00018 − 0.577454i 0.0918788 − 0.995770i \(-0.470713\pi\)
0.908302 + 0.418316i \(0.137379\pi\)
\(74\) − 3.65470i − 0.424850i
\(75\) 0 0
\(76\) −4.66633 + 2.69411i −0.535265 + 0.309035i
\(77\) −1.39329 + 11.5851i −0.158780 + 1.32025i
\(78\) 0 0
\(79\) −1.23351 − 2.13650i −0.138781 − 0.240375i 0.788255 − 0.615349i \(-0.210985\pi\)
−0.927035 + 0.374974i \(0.877652\pi\)
\(80\) −0.360068 − 0.623656i −0.0402568 − 0.0697269i
\(81\) 0 0
\(82\) −1.69981 − 0.981388i −0.187713 − 0.108376i
\(83\) −5.69625 + 9.86619i −0.625244 + 1.08296i 0.363249 + 0.931692i \(0.381667\pi\)
−0.988494 + 0.151263i \(0.951666\pi\)
\(84\) 0 0
\(85\) 2.34398 + 4.05990i 0.254241 + 0.440358i
\(86\) − 9.88445i − 1.06587i
\(87\) 0 0
\(88\) 4.41031 0.470141
\(89\) −7.50204 + 12.9939i −0.795214 + 1.37735i 0.127489 + 0.991840i \(0.459308\pi\)
−0.922703 + 0.385512i \(0.874025\pi\)
\(90\) 0 0
\(91\) 3.05204 − 1.30447i 0.319941 − 0.136745i
\(92\) 5.95507 + 3.43816i 0.620859 + 0.358453i
\(93\) 0 0
\(94\) 7.13145 + 4.11735i 0.735553 + 0.424672i
\(95\) −3.36039 − 1.94012i −0.344769 − 0.199052i
\(96\) 0 0
\(97\) −8.89063 − 5.13301i −0.902707 − 0.521178i −0.0246295 − 0.999697i \(-0.507841\pi\)
−0.878077 + 0.478519i \(0.841174\pi\)
\(98\) −6.71917 + 1.96284i −0.678739 + 0.198277i
\(99\) 0 0
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1134.2.t.h.593.3 16
3.2 odd 2 1134.2.t.g.593.6 16
7.3 odd 6 1134.2.l.g.269.3 16
9.2 odd 6 1134.2.k.d.971.3 yes 16
9.4 even 3 1134.2.l.h.215.2 16
9.5 odd 6 1134.2.l.g.215.7 16
9.7 even 3 1134.2.k.c.971.6 yes 16
21.17 even 6 1134.2.l.h.269.6 16
63.31 odd 6 1134.2.t.g.1025.6 16
63.38 even 6 1134.2.k.c.647.6 ✓ 16
63.52 odd 6 1134.2.k.d.647.3 yes 16
63.59 even 6 inner 1134.2.t.h.1025.3 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1134.2.k.c.647.6 ✓ 16 63.38 even 6
1134.2.k.c.971.6 yes 16 9.7 even 3
1134.2.k.d.647.3 yes 16 63.52 odd 6
1134.2.k.d.971.3 yes 16 9.2 odd 6
1134.2.l.g.215.7 16 9.5 odd 6
1134.2.l.g.269.3 16 7.3 odd 6
1134.2.l.h.215.2 16 9.4 even 3
1134.2.l.h.269.6 16 21.17 even 6
1134.2.t.g.593.6 16 3.2 odd 2
1134.2.t.g.1025.6 16 63.31 odd 6
1134.2.t.h.593.3 16 1.1 even 1 trivial
1134.2.t.h.1025.3 16 63.59 even 6 inner