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Properties

Label 1134.2.m.g.755.4
Level $1134$
Weight $2$
Character 1134.755
Analytic conductor $9.055$
Analytic rank $0$
Dimension $8$
Inner twists $8$

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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(377,1134)] 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("1134.377"); 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, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1134 = 2 \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1134.m (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,4,0,0,4,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.05503558921\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 42)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 755.4
Root \(0.965926 - 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 1134.755
Dual form 1134.2.m.g.377.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.866025 + 0.500000i) q^{2} +(0.500000 + 0.866025i) q^{4} +(1.22474 + 2.12132i) q^{5} +(2.62132 + 0.358719i) q^{7} +1.00000i q^{8} +2.44949i q^{10} +(2.12132 - 1.22474i) q^{13} +(2.09077 + 1.62132i) q^{14} +(-0.500000 + 0.866025i) q^{16} +4.89898 q^{17} -2.44949i q^{19} +(-1.22474 + 2.12132i) q^{20} +(-5.19615 + 3.00000i) q^{23} +(-0.500000 + 0.866025i) q^{25} +2.44949 q^{26} +(1.00000 + 2.44949i) q^{28} +(-5.19615 - 3.00000i) q^{29} +(-0.866025 + 0.500000i) q^{32} +(4.24264 + 2.44949i) q^{34} +(2.44949 + 6.00000i) q^{35} -2.00000 q^{37} +(1.22474 - 2.12132i) q^{38} +(-2.12132 + 1.22474i) q^{40} +(2.44949 + 4.24264i) q^{41} +(-2.00000 + 3.46410i) q^{43} -6.00000 q^{46} +(-2.44949 + 4.24264i) q^{47} +(6.74264 + 1.88064i) q^{49} +(-0.866025 + 0.500000i) q^{50} +(2.12132 + 1.22474i) q^{52} -6.00000i q^{53} +(-0.358719 + 2.62132i) q^{56} +(-3.00000 - 5.19615i) q^{58} +(6.12372 + 10.6066i) q^{59} +(-10.6066 - 6.12372i) q^{61} -1.00000 q^{64} +(5.19615 + 3.00000i) q^{65} +(-4.00000 - 6.92820i) q^{67} +(2.44949 + 4.24264i) q^{68} +(-0.878680 + 6.42090i) q^{70} -9.79796i q^{73} +(-1.73205 - 1.00000i) q^{74} +(2.12132 - 1.22474i) q^{76} +(5.00000 - 8.66025i) q^{79} -2.44949 q^{80} +4.89898i q^{82} +(-1.22474 + 2.12132i) q^{83} +(6.00000 + 10.3923i) q^{85} +(-3.46410 + 2.00000i) q^{86} +(6.00000 - 2.44949i) q^{91} +(-5.19615 - 3.00000i) q^{92} +(-4.24264 + 2.44949i) q^{94} +(5.19615 - 3.00000i) q^{95} +(4.24264 + 2.44949i) q^{97} +(4.89898 + 5.00000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{4} + 4 q^{7} - 4 q^{16} - 4 q^{25} + 8 q^{28} - 16 q^{37} - 16 q^{43} - 48 q^{46} + 20 q^{49} - 24 q^{58} - 8 q^{64} - 32 q^{67} - 24 q^{70} + 40 q^{79} + 48 q^{85} + 48 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(5\) 1.22474 + 2.12132i 0.547723 + 0.948683i 0.998430 + 0.0560116i \(0.0178384\pi\)
−0.450708 + 0.892672i \(0.648828\pi\)
\(6\) 0 0
\(7\) 2.62132 + 0.358719i 0.990766 + 0.135583i
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) 2.44949i 0.774597i
\(11\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(12\) 0 0
\(13\) 2.12132 − 1.22474i 0.588348 − 0.339683i −0.176096 − 0.984373i \(-0.556347\pi\)
0.764444 + 0.644690i \(0.223014\pi\)
\(14\) 2.09077 + 1.62132i 0.558782 + 0.433316i
\(15\) 0 0
\(16\) −0.500000 + 0.866025i −0.125000 + 0.216506i
\(17\) 4.89898 1.18818 0.594089 − 0.804400i \(-0.297513\pi\)
0.594089 + 0.804400i \(0.297513\pi\)
\(18\) 0 0
\(19\) − 2.44949i − 0.561951i −0.959715 − 0.280976i \(-0.909342\pi\)
0.959715 − 0.280976i \(-0.0906580\pi\)
\(20\) −1.22474 + 2.12132i −0.273861 + 0.474342i
\(21\) 0 0
\(22\) 0 0
\(23\) −5.19615 + 3.00000i −1.08347 + 0.625543i −0.931831 − 0.362892i \(-0.881789\pi\)
−0.151642 + 0.988436i \(0.548456\pi\)
\(24\) 0 0
\(25\) −0.500000 + 0.866025i −0.100000 + 0.173205i
\(26\) 2.44949 0.480384
\(27\) 0 0
\(28\) 1.00000 + 2.44949i 0.188982 + 0.462910i
\(29\) −5.19615 − 3.00000i −0.964901 − 0.557086i −0.0672232 − 0.997738i \(-0.521414\pi\)
−0.897678 + 0.440652i \(0.854747\pi\)
\(30\) 0 0
\(31\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(32\) −0.866025 + 0.500000i −0.153093 + 0.0883883i
\(33\) 0 0
\(34\) 4.24264 + 2.44949i 0.727607 + 0.420084i
\(35\) 2.44949 + 6.00000i 0.414039 + 1.01419i
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 − 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 1.22474 − 2.12132i 0.198680 − 0.344124i
\(39\) 0 0
\(40\) −2.12132 + 1.22474i −0.335410 + 0.193649i
\(41\) 2.44949 + 4.24264i 0.382546 + 0.662589i 0.991425 − 0.130674i \(-0.0417140\pi\)
−0.608879 + 0.793263i \(0.708381\pi\)
\(42\) 0 0
\(43\) −2.00000 + 3.46410i −0.304997 + 0.528271i −0.977261 − 0.212041i \(-0.931989\pi\)
0.672264 + 0.740312i \(0.265322\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −6.00000 −0.884652
\(47\) −2.44949 + 4.24264i −0.357295 + 0.618853i −0.987508 − 0.157569i \(-0.949634\pi\)
0.630213 + 0.776422i \(0.282968\pi\)
\(48\) 0 0
\(49\) 6.74264 + 1.88064i 0.963234 + 0.268662i
\(50\) −0.866025 + 0.500000i −0.122474 + 0.0707107i
\(51\) 0 0
\(52\) 2.12132 + 1.22474i 0.294174 + 0.169842i
\(53\) − 6.00000i − 0.824163i −0.911147 − 0.412082i \(-0.864802\pi\)
0.911147 − 0.412082i \(-0.135198\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −0.358719 + 2.62132i −0.0479359 + 0.350289i
\(57\) 0 0
\(58\) −3.00000 − 5.19615i −0.393919 − 0.682288i
\(59\) 6.12372 + 10.6066i 0.797241 + 1.38086i 0.921406 + 0.388600i \(0.127041\pi\)
−0.124165 + 0.992262i \(0.539625\pi\)
\(60\) 0 0
\(61\) −10.6066 − 6.12372i −1.35804 − 0.784063i −0.368677 − 0.929557i \(-0.620189\pi\)
−0.989359 + 0.145495i \(0.953523\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 5.19615 + 3.00000i 0.644503 + 0.372104i
\(66\) 0 0
\(67\) −4.00000 − 6.92820i −0.488678 − 0.846415i 0.511237 − 0.859440i \(-0.329187\pi\)
−0.999915 + 0.0130248i \(0.995854\pi\)
\(68\) 2.44949 + 4.24264i 0.297044 + 0.514496i
\(69\) 0 0
\(70\) −0.878680 + 6.42090i −0.105022 + 0.767444i
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) − 9.79796i − 1.14676i −0.819288 − 0.573382i \(-0.805631\pi\)
0.819288 − 0.573382i \(-0.194369\pi\)
\(74\) −1.73205 − 1.00000i −0.201347 − 0.116248i
\(75\) 0 0
\(76\) 2.12132 − 1.22474i 0.243332 − 0.140488i
\(77\) 0 0
\(78\) 0 0
\(79\) 5.00000 − 8.66025i 0.562544 − 0.974355i −0.434730 − 0.900561i \(-0.643156\pi\)
0.997274 − 0.0737937i \(-0.0235106\pi\)
\(80\) −2.44949 −0.273861
\(81\) 0 0
\(82\) 4.89898i 0.541002i
\(83\) −1.22474 + 2.12132i −0.134433 + 0.232845i −0.925381 − 0.379039i \(-0.876255\pi\)
0.790948 + 0.611884i \(0.209588\pi\)
\(84\) 0 0
\(85\) 6.00000 + 10.3923i 0.650791 + 1.12720i
\(86\) −3.46410 + 2.00000i −0.373544 + 0.215666i
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 6.00000 − 2.44949i 0.628971 − 0.256776i
\(92\) −5.19615 − 3.00000i −0.541736 − 0.312772i
\(93\) 0 0
\(94\) −4.24264 + 2.44949i −0.437595 + 0.252646i
\(95\) 5.19615 − 3.00000i 0.533114 − 0.307794i
\(96\) 0 0
\(97\) 4.24264 + 2.44949i 0.430775 + 0.248708i 0.699677 − 0.714460i \(-0.253327\pi\)
−0.268902 + 0.963168i \(0.586661\pi\)
\(98\) 4.89898 + 5.00000i 0.494872 + 0.505076i
\(99\) 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.m.g.755.4 8
3.2 odd 2 inner 1134.2.m.g.755.1 8
7.6 odd 2 inner 1134.2.m.g.755.3 8
9.2 odd 6 42.2.d.a.41.3 yes 4
9.4 even 3 inner 1134.2.m.g.377.2 8
9.5 odd 6 inner 1134.2.m.g.377.3 8
9.7 even 3 42.2.d.a.41.2 yes 4
21.20 even 2 inner 1134.2.m.g.755.2 8
36.7 odd 6 336.2.k.b.209.1 4
36.11 even 6 336.2.k.b.209.3 4
45.2 even 12 1050.2.d.b.1049.4 4
45.7 odd 12 1050.2.d.e.1049.3 4
45.29 odd 6 1050.2.b.b.251.2 4
45.34 even 6 1050.2.b.b.251.3 4
45.38 even 12 1050.2.d.e.1049.1 4
45.43 odd 12 1050.2.d.b.1049.2 4
63.2 odd 6 294.2.f.b.227.3 8
63.11 odd 6 294.2.f.b.215.2 8
63.13 odd 6 inner 1134.2.m.g.377.1 8
63.16 even 3 294.2.f.b.227.1 8
63.20 even 6 42.2.d.a.41.4 yes 4
63.25 even 3 294.2.f.b.215.4 8
63.34 odd 6 42.2.d.a.41.1 ✓ 4
63.38 even 6 294.2.f.b.215.1 8
63.41 even 6 inner 1134.2.m.g.377.4 8
63.47 even 6 294.2.f.b.227.4 8
63.52 odd 6 294.2.f.b.215.3 8
63.61 odd 6 294.2.f.b.227.2 8
72.11 even 6 1344.2.k.d.1217.2 4
72.29 odd 6 1344.2.k.c.1217.3 4
72.43 odd 6 1344.2.k.d.1217.4 4
72.61 even 6 1344.2.k.c.1217.1 4
252.83 odd 6 336.2.k.b.209.2 4
252.223 even 6 336.2.k.b.209.4 4
315.34 odd 6 1050.2.b.b.251.4 4
315.83 odd 12 1050.2.d.e.1049.4 4
315.97 even 12 1050.2.d.e.1049.2 4
315.209 even 6 1050.2.b.b.251.1 4
315.223 even 12 1050.2.d.b.1049.3 4
315.272 odd 12 1050.2.d.b.1049.1 4
504.83 odd 6 1344.2.k.d.1217.3 4
504.349 odd 6 1344.2.k.c.1217.4 4
504.461 even 6 1344.2.k.c.1217.2 4
504.475 even 6 1344.2.k.d.1217.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
42.2.d.a.41.1 ✓ 4 63.34 odd 6
42.2.d.a.41.2 yes 4 9.7 even 3
42.2.d.a.41.3 yes 4 9.2 odd 6
42.2.d.a.41.4 yes 4 63.20 even 6
294.2.f.b.215.1 8 63.38 even 6
294.2.f.b.215.2 8 63.11 odd 6
294.2.f.b.215.3 8 63.52 odd 6
294.2.f.b.215.4 8 63.25 even 3
294.2.f.b.227.1 8 63.16 even 3
294.2.f.b.227.2 8 63.61 odd 6
294.2.f.b.227.3 8 63.2 odd 6
294.2.f.b.227.4 8 63.47 even 6
336.2.k.b.209.1 4 36.7 odd 6
336.2.k.b.209.2 4 252.83 odd 6
336.2.k.b.209.3 4 36.11 even 6
336.2.k.b.209.4 4 252.223 even 6
1050.2.b.b.251.1 4 315.209 even 6
1050.2.b.b.251.2 4 45.29 odd 6
1050.2.b.b.251.3 4 45.34 even 6
1050.2.b.b.251.4 4 315.34 odd 6
1050.2.d.b.1049.1 4 315.272 odd 12
1050.2.d.b.1049.2 4 45.43 odd 12
1050.2.d.b.1049.3 4 315.223 even 12
1050.2.d.b.1049.4 4 45.2 even 12
1050.2.d.e.1049.1 4 45.38 even 12
1050.2.d.e.1049.2 4 315.97 even 12
1050.2.d.e.1049.3 4 45.7 odd 12
1050.2.d.e.1049.4 4 315.83 odd 12
1134.2.m.g.377.1 8 63.13 odd 6 inner
1134.2.m.g.377.2 8 9.4 even 3 inner
1134.2.m.g.377.3 8 9.5 odd 6 inner
1134.2.m.g.377.4 8 63.41 even 6 inner
1134.2.m.g.755.1 8 3.2 odd 2 inner
1134.2.m.g.755.2 8 21.20 even 2 inner
1134.2.m.g.755.3 8 7.6 odd 2 inner
1134.2.m.g.755.4 8 1.1 even 1 trivial
1344.2.k.c.1217.1 4 72.61 even 6
1344.2.k.c.1217.2 4 504.461 even 6
1344.2.k.c.1217.3 4 72.29 odd 6
1344.2.k.c.1217.4 4 504.349 odd 6
1344.2.k.d.1217.1 4 504.475 even 6
1344.2.k.d.1217.2 4 72.11 even 6
1344.2.k.d.1217.3 4 504.83 odd 6
1344.2.k.d.1217.4 4 72.43 odd 6