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Properties

Label 1134.2.t.c.1025.1
Level $1134$
Weight $2$
Character 1134.1025
Analytic conductor $9.055$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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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 = [4,0,0,2,0,0,2,0,0,6,0,0,0,0] 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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 378)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 1025.1
Root \(0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1134.1025
Dual form 1134.2.t.c.593.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.866025 - 0.500000i) q^{2} +(0.500000 + 0.866025i) q^{4} -1.73205 q^{5} +(0.500000 - 2.59808i) q^{7} -1.00000i q^{8} +(1.50000 + 0.866025i) q^{10} +(-1.73205 + 2.00000i) q^{14} +(-0.500000 + 0.866025i) q^{16} +(-1.73205 + 3.00000i) q^{17} +(6.00000 - 3.46410i) q^{19} +(-0.866025 - 1.50000i) q^{20} -6.00000i q^{23} -2.00000 q^{25} +(2.50000 - 0.866025i) q^{28} +(-7.79423 + 4.50000i) q^{29} +(-3.00000 + 1.73205i) q^{31} +(0.866025 - 0.500000i) q^{32} +(3.00000 - 1.73205i) q^{34} +(-0.866025 + 4.50000i) q^{35} +(-2.00000 - 3.46410i) q^{37} -6.92820 q^{38} +1.73205i q^{40} +(1.73205 - 3.00000i) q^{41} +(4.00000 + 6.92820i) q^{43} +(-3.00000 + 5.19615i) q^{46} +(-1.73205 + 3.00000i) q^{47} +(-6.50000 - 2.59808i) q^{49} +(1.73205 + 1.00000i) q^{50} +(-2.59808 - 1.50000i) q^{53} +(-2.59808 - 0.500000i) q^{56} +9.00000 q^{58} +(-6.06218 - 10.5000i) q^{59} +(-3.00000 - 1.73205i) q^{61} +3.46410 q^{62} -1.00000 q^{64} +(-7.00000 - 12.1244i) q^{67} -3.46410 q^{68} +(3.00000 - 3.46410i) q^{70} +6.00000i q^{71} +(-10.5000 - 6.06218i) q^{73} +4.00000i q^{74} +(6.00000 + 3.46410i) q^{76} +(-5.50000 + 9.52628i) q^{79} +(0.866025 - 1.50000i) q^{80} +(-3.00000 + 1.73205i) q^{82} +(-8.66025 - 15.0000i) q^{83} +(3.00000 - 5.19615i) q^{85} -8.00000i q^{86} +(-5.19615 - 9.00000i) q^{89} +(5.19615 - 3.00000i) q^{92} +(3.00000 - 1.73205i) q^{94} +(-10.3923 + 6.00000i) q^{95} +(6.00000 - 3.46410i) q^{97} +(4.33013 + 5.50000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} + 2 q^{7} + 6 q^{10} - 2 q^{16} + 24 q^{19} - 8 q^{25} + 10 q^{28} - 12 q^{31} + 12 q^{34} - 8 q^{37} + 16 q^{43} - 12 q^{46} - 26 q^{49} + 36 q^{58} - 12 q^{61} - 4 q^{64} - 28 q^{67} + 12 q^{70}+ \cdots + 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(5\) −1.73205 −0.774597 −0.387298 − 0.921954i \(-0.626592\pi\)
−0.387298 + 0.921954i \(0.626592\pi\)
\(6\) 0 0
\(7\) 0.500000 − 2.59808i 0.188982 − 0.981981i
\(8\) − 1.00000i − 0.353553i
\(9\) 0 0
\(10\) 1.50000 + 0.866025i 0.474342 + 0.273861i
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(14\) −1.73205 + 2.00000i −0.462910 + 0.534522i
\(15\) 0 0
\(16\) −0.500000 + 0.866025i −0.125000 + 0.216506i
\(17\) −1.73205 + 3.00000i −0.420084 + 0.727607i −0.995947 − 0.0899392i \(-0.971333\pi\)
0.575863 + 0.817546i \(0.304666\pi\)
\(18\) 0 0
\(19\) 6.00000 − 3.46410i 1.37649 − 0.794719i 0.384759 − 0.923017i \(-0.374285\pi\)
0.991736 + 0.128298i \(0.0409513\pi\)
\(20\) −0.866025 − 1.50000i −0.193649 − 0.335410i
\(21\) 0 0
\(22\) 0 0
\(23\) − 6.00000i − 1.25109i −0.780189 − 0.625543i \(-0.784877\pi\)
0.780189 − 0.625543i \(-0.215123\pi\)
\(24\) 0 0
\(25\) −2.00000 −0.400000
\(26\) 0 0
\(27\) 0 0
\(28\) 2.50000 − 0.866025i 0.472456 − 0.163663i
\(29\) −7.79423 + 4.50000i −1.44735 + 0.835629i −0.998323 − 0.0578882i \(-0.981563\pi\)
−0.449029 + 0.893517i \(0.648230\pi\)
\(30\) 0 0
\(31\) −3.00000 + 1.73205i −0.538816 + 0.311086i −0.744599 − 0.667512i \(-0.767359\pi\)
0.205783 + 0.978598i \(0.434026\pi\)
\(32\) 0.866025 − 0.500000i 0.153093 − 0.0883883i
\(33\) 0 0
\(34\) 3.00000 − 1.73205i 0.514496 − 0.297044i
\(35\) −0.866025 + 4.50000i −0.146385 + 0.760639i
\(36\) 0 0
\(37\) −2.00000 − 3.46410i −0.328798 − 0.569495i 0.653476 − 0.756948i \(-0.273310\pi\)
−0.982274 + 0.187453i \(0.939977\pi\)
\(38\) −6.92820 −1.12390
\(39\) 0 0
\(40\) 1.73205i 0.273861i
\(41\) 1.73205 − 3.00000i 0.270501 − 0.468521i −0.698489 − 0.715621i \(-0.746144\pi\)
0.968990 + 0.247099i \(0.0794774\pi\)
\(42\) 0 0
\(43\) 4.00000 + 6.92820i 0.609994 + 1.05654i 0.991241 + 0.132068i \(0.0421616\pi\)
−0.381246 + 0.924473i \(0.624505\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −3.00000 + 5.19615i −0.442326 + 0.766131i
\(47\) −1.73205 + 3.00000i −0.252646 + 0.437595i −0.964253 − 0.264982i \(-0.914634\pi\)
0.711608 + 0.702577i \(0.247967\pi\)
\(48\) 0 0
\(49\) −6.50000 − 2.59808i −0.928571 − 0.371154i
\(50\) 1.73205 + 1.00000i 0.244949 + 0.141421i
\(51\) 0 0
\(52\) 0 0
\(53\) −2.59808 − 1.50000i −0.356873 − 0.206041i 0.310835 − 0.950464i \(-0.399391\pi\)
−0.667708 + 0.744423i \(0.732725\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −2.59808 − 0.500000i −0.347183 − 0.0668153i
\(57\) 0 0
\(58\) 9.00000 1.18176
\(59\) −6.06218 − 10.5000i −0.789228 − 1.36698i −0.926440 − 0.376442i \(-0.877147\pi\)
0.137212 − 0.990542i \(-0.456186\pi\)
\(60\) 0 0
\(61\) −3.00000 − 1.73205i −0.384111 − 0.221766i 0.295495 − 0.955344i \(-0.404516\pi\)
−0.679605 + 0.733578i \(0.737849\pi\)
\(62\) 3.46410 0.439941
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −7.00000 − 12.1244i −0.855186 − 1.48123i −0.876472 − 0.481452i \(-0.840109\pi\)
0.0212861 − 0.999773i \(-0.493224\pi\)
\(68\) −3.46410 −0.420084
\(69\) 0 0
\(70\) 3.00000 − 3.46410i 0.358569 − 0.414039i
\(71\) 6.00000i 0.712069i 0.934473 + 0.356034i \(0.115871\pi\)
−0.934473 + 0.356034i \(0.884129\pi\)
\(72\) 0 0
\(73\) −10.5000 − 6.06218i −1.22893 − 0.709524i −0.262126 − 0.965034i \(-0.584423\pi\)
−0.966807 + 0.255510i \(0.917757\pi\)
\(74\) 4.00000i 0.464991i
\(75\) 0 0
\(76\) 6.00000 + 3.46410i 0.688247 + 0.397360i
\(77\) 0 0
\(78\) 0 0
\(79\) −5.50000 + 9.52628i −0.618798 + 1.07179i 0.370907 + 0.928670i \(0.379047\pi\)
−0.989705 + 0.143120i \(0.954286\pi\)
\(80\) 0.866025 − 1.50000i 0.0968246 − 0.167705i
\(81\) 0 0
\(82\) −3.00000 + 1.73205i −0.331295 + 0.191273i
\(83\) −8.66025 − 15.0000i −0.950586 − 1.64646i −0.744160 − 0.668002i \(-0.767150\pi\)
−0.206427 − 0.978462i \(-0.566184\pi\)
\(84\) 0 0
\(85\) 3.00000 − 5.19615i 0.325396 − 0.563602i
\(86\) − 8.00000i − 0.862662i
\(87\) 0 0
\(88\) 0 0
\(89\) −5.19615 − 9.00000i −0.550791 − 0.953998i −0.998218 − 0.0596775i \(-0.980993\pi\)
0.447427 − 0.894321i \(-0.352341\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 5.19615 − 3.00000i 0.541736 − 0.312772i
\(93\) 0 0
\(94\) 3.00000 − 1.73205i 0.309426 − 0.178647i
\(95\) −10.3923 + 6.00000i −1.06623 + 0.615587i
\(96\) 0 0
\(97\) 6.00000 − 3.46410i 0.609208 − 0.351726i −0.163448 − 0.986552i \(-0.552261\pi\)
0.772655 + 0.634826i \(0.218928\pi\)
\(98\) 4.33013 + 5.50000i 0.437409 + 0.555584i
\(99\) 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.c.1025.1 4
3.2 odd 2 inner 1134.2.t.c.1025.2 4
7.5 odd 6 1134.2.l.b.215.2 4
9.2 odd 6 1134.2.l.b.269.1 4
9.4 even 3 378.2.k.c.269.2 yes 4
9.5 odd 6 378.2.k.c.269.1 yes 4
9.7 even 3 1134.2.l.b.269.2 4
21.5 even 6 1134.2.l.b.215.1 4
63.4 even 3 2646.2.d.a.2645.3 4
63.5 even 6 378.2.k.c.215.2 yes 4
63.31 odd 6 2646.2.d.a.2645.4 4
63.32 odd 6 2646.2.d.a.2645.2 4
63.40 odd 6 378.2.k.c.215.1 ✓ 4
63.47 even 6 inner 1134.2.t.c.593.1 4
63.59 even 6 2646.2.d.a.2645.1 4
63.61 odd 6 inner 1134.2.t.c.593.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
378.2.k.c.215.1 ✓ 4 63.40 odd 6
378.2.k.c.215.2 yes 4 63.5 even 6
378.2.k.c.269.1 yes 4 9.5 odd 6
378.2.k.c.269.2 yes 4 9.4 even 3
1134.2.l.b.215.1 4 21.5 even 6
1134.2.l.b.215.2 4 7.5 odd 6
1134.2.l.b.269.1 4 9.2 odd 6
1134.2.l.b.269.2 4 9.7 even 3
1134.2.t.c.593.1 4 63.47 even 6 inner
1134.2.t.c.593.2 4 63.61 odd 6 inner
1134.2.t.c.1025.1 4 1.1 even 1 trivial
1134.2.t.c.1025.2 4 3.2 odd 2 inner
2646.2.d.a.2645.1 4 63.59 even 6
2646.2.d.a.2645.2 4 63.32 odd 6
2646.2.d.a.2645.3 4 63.4 even 3
2646.2.d.a.2645.4 4 63.31 odd 6