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Properties

Label 1134.2.l.d.215.2
Level $1134$
Weight $2$
Character 1134.215
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(215,1134)] 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("1134.215"); 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([5, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1134 = 2 \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1134.l (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,-4,0,0,8,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == 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: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 378)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 215.2
Root \(0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1134.215
Dual form 1134.2.l.d.269.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000 q^{4} +(2.00000 - 1.73205i) q^{7} -1.00000i q^{8} +(2.59808 + 1.50000i) q^{11} +(-3.00000 - 1.73205i) q^{13} +(1.73205 + 2.00000i) q^{14} +1.00000 q^{16} +(-1.50000 + 2.59808i) q^{22} +(2.50000 - 4.33013i) q^{25} +(1.73205 - 3.00000i) q^{26} +(-2.00000 + 1.73205i) q^{28} +(7.79423 - 4.50000i) q^{29} -1.73205i q^{31} +1.00000i q^{32} +(4.00000 - 6.92820i) q^{37} +(-5.19615 + 9.00000i) q^{41} +(-2.00000 - 3.46410i) q^{43} +(-2.59808 - 1.50000i) q^{44} +10.3923 q^{47} +(1.00000 - 6.92820i) q^{49} +(4.33013 + 2.50000i) q^{50} +(3.00000 + 1.73205i) q^{52} +(5.19615 - 3.00000i) q^{53} +(-1.73205 - 2.00000i) q^{56} +(4.50000 + 7.79423i) q^{58} +5.19615 q^{59} +13.8564i q^{61} +1.73205 q^{62} -1.00000 q^{64} +2.00000 q^{67} +12.0000i q^{71} +(4.50000 - 2.59808i) q^{73} +(6.92820 + 4.00000i) q^{74} +(7.79423 - 1.50000i) q^{77} -13.0000 q^{79} +(-9.00000 - 5.19615i) q^{82} +(2.59808 + 4.50000i) q^{83} +(3.46410 - 2.00000i) q^{86} +(1.50000 - 2.59808i) q^{88} +(5.19615 - 9.00000i) q^{89} +(-9.00000 + 1.73205i) q^{91} +10.3923i q^{94} +(-7.50000 + 4.33013i) q^{97} +(6.92820 + 1.00000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + 8 q^{7} - 12 q^{13} + 4 q^{16} - 6 q^{22} + 10 q^{25} - 8 q^{28} + 16 q^{37} - 8 q^{43} + 4 q^{49} + 12 q^{52} + 18 q^{58} - 4 q^{64} + 8 q^{67} + 18 q^{73} - 52 q^{79} - 36 q^{82} + 6 q^{88}+ \cdots - 30 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(6\) 0 0
\(7\) 2.00000 − 1.73205i 0.755929 − 0.654654i
\(8\) − 1.00000i − 0.353553i
\(9\) 0 0
\(10\) 0 0
\(11\) 2.59808 + 1.50000i 0.783349 + 0.452267i 0.837616 − 0.546259i \(-0.183949\pi\)
−0.0542666 + 0.998526i \(0.517282\pi\)
\(12\) 0 0
\(13\) −3.00000 − 1.73205i −0.832050 − 0.480384i 0.0225039 − 0.999747i \(-0.492836\pi\)
−0.854554 + 0.519362i \(0.826170\pi\)
\(14\) 1.73205 + 2.00000i 0.462910 + 0.534522i
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(18\) 0 0
\(19\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −1.50000 + 2.59808i −0.319801 + 0.553912i
\(23\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(24\) 0 0
\(25\) 2.50000 − 4.33013i 0.500000 − 0.866025i
\(26\) 1.73205 − 3.00000i 0.339683 − 0.588348i
\(27\) 0 0
\(28\) −2.00000 + 1.73205i −0.377964 + 0.327327i
\(29\) 7.79423 − 4.50000i 1.44735 − 0.835629i 0.449029 − 0.893517i \(-0.351770\pi\)
0.998323 + 0.0578882i \(0.0184367\pi\)
\(30\) 0 0
\(31\) − 1.73205i − 0.311086i −0.987829 − 0.155543i \(-0.950287\pi\)
0.987829 − 0.155543i \(-0.0497126\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 4.00000 − 6.92820i 0.657596 − 1.13899i −0.323640 − 0.946180i \(-0.604907\pi\)
0.981236 − 0.192809i \(-0.0617599\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −5.19615 + 9.00000i −0.811503 + 1.40556i 0.100309 + 0.994956i \(0.468017\pi\)
−0.911812 + 0.410608i \(0.865317\pi\)
\(42\) 0 0
\(43\) −2.00000 − 3.46410i −0.304997 − 0.528271i 0.672264 − 0.740312i \(-0.265322\pi\)
−0.977261 + 0.212041i \(0.931989\pi\)
\(44\) −2.59808 − 1.50000i −0.391675 − 0.226134i
\(45\) 0 0
\(46\) 0 0
\(47\) 10.3923 1.51587 0.757937 − 0.652328i \(-0.226208\pi\)
0.757937 + 0.652328i \(0.226208\pi\)
\(48\) 0 0
\(49\) 1.00000 − 6.92820i 0.142857 − 0.989743i
\(50\) 4.33013 + 2.50000i 0.612372 + 0.353553i
\(51\) 0 0
\(52\) 3.00000 + 1.73205i 0.416025 + 0.240192i
\(53\) 5.19615 − 3.00000i 0.713746 − 0.412082i −0.0987002 − 0.995117i \(-0.531468\pi\)
0.812447 + 0.583036i \(0.198135\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −1.73205 − 2.00000i −0.231455 − 0.267261i
\(57\) 0 0
\(58\) 4.50000 + 7.79423i 0.590879 + 1.02343i
\(59\) 5.19615 0.676481 0.338241 − 0.941060i \(-0.390168\pi\)
0.338241 + 0.941060i \(0.390168\pi\)
\(60\) 0 0
\(61\) 13.8564i 1.77413i 0.461644 + 0.887066i \(0.347260\pi\)
−0.461644 + 0.887066i \(0.652740\pi\)
\(62\) 1.73205 0.219971
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 2.00000 0.244339 0.122169 − 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 12.0000i 1.42414i 0.702109 + 0.712069i \(0.252242\pi\)
−0.702109 + 0.712069i \(0.747758\pi\)
\(72\) 0 0
\(73\) 4.50000 − 2.59808i 0.526685 − 0.304082i −0.212980 − 0.977056i \(-0.568317\pi\)
0.739666 + 0.672975i \(0.234984\pi\)
\(74\) 6.92820 + 4.00000i 0.805387 + 0.464991i
\(75\) 0 0
\(76\) 0 0
\(77\) 7.79423 − 1.50000i 0.888235 − 0.170941i
\(78\) 0 0
\(79\) −13.0000 −1.46261 −0.731307 − 0.682048i \(-0.761089\pi\)
−0.731307 + 0.682048i \(0.761089\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −9.00000 − 5.19615i −0.993884 − 0.573819i
\(83\) 2.59808 + 4.50000i 0.285176 + 0.493939i 0.972652 − 0.232268i \(-0.0746146\pi\)
−0.687476 + 0.726207i \(0.741281\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 3.46410 − 2.00000i 0.373544 − 0.215666i
\(87\) 0 0
\(88\) 1.50000 − 2.59808i 0.159901 − 0.276956i
\(89\) 5.19615 − 9.00000i 0.550791 − 0.953998i −0.447427 − 0.894321i \(-0.647659\pi\)
0.998218 − 0.0596775i \(-0.0190072\pi\)
\(90\) 0 0
\(91\) −9.00000 + 1.73205i −0.943456 + 0.181568i
\(92\) 0 0
\(93\) 0 0
\(94\) 10.3923i 1.07188i
\(95\) 0 0
\(96\) 0 0
\(97\) −7.50000 + 4.33013i −0.761510 + 0.439658i −0.829837 − 0.558005i \(-0.811567\pi\)
0.0683279 + 0.997663i \(0.478234\pi\)
\(98\) 6.92820 + 1.00000i 0.699854 + 0.101015i
\(99\) 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.l.d.215.2 4
3.2 odd 2 inner 1134.2.l.d.215.1 4
7.3 odd 6 1134.2.t.a.1025.1 4
9.2 odd 6 1134.2.t.a.593.1 4
9.4 even 3 378.2.k.a.215.1 ✓ 4
9.5 odd 6 378.2.k.a.215.2 yes 4
9.7 even 3 1134.2.t.a.593.2 4
21.17 even 6 1134.2.t.a.1025.2 4
63.5 even 6 2646.2.d.c.2645.1 4
63.23 odd 6 2646.2.d.c.2645.2 4
63.31 odd 6 378.2.k.a.269.2 yes 4
63.38 even 6 inner 1134.2.l.d.269.1 4
63.40 odd 6 2646.2.d.c.2645.3 4
63.52 odd 6 inner 1134.2.l.d.269.2 4
63.58 even 3 2646.2.d.c.2645.4 4
63.59 even 6 378.2.k.a.269.1 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
378.2.k.a.215.1 ✓ 4 9.4 even 3
378.2.k.a.215.2 yes 4 9.5 odd 6
378.2.k.a.269.1 yes 4 63.59 even 6
378.2.k.a.269.2 yes 4 63.31 odd 6
1134.2.l.d.215.1 4 3.2 odd 2 inner
1134.2.l.d.215.2 4 1.1 even 1 trivial
1134.2.l.d.269.1 4 63.38 even 6 inner
1134.2.l.d.269.2 4 63.52 odd 6 inner
1134.2.t.a.593.1 4 9.2 odd 6
1134.2.t.a.593.2 4 9.7 even 3
1134.2.t.a.1025.1 4 7.3 odd 6
1134.2.t.a.1025.2 4 21.17 even 6
2646.2.d.c.2645.1 4 63.5 even 6
2646.2.d.c.2645.2 4 63.23 odd 6
2646.2.d.c.2645.3 4 63.40 odd 6
2646.2.d.c.2645.4 4 63.58 even 3