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Properties

Label 1134.2.t.f.1025.1
Level $1134$
Weight $2$
Character 1134.1025
Analytic conductor $9.055$
Analytic rank $0$
Dimension $8$
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 = [8,0,0,4,0,0,4,0,0,0,0,0,-12,0] 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: \(8\)
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 378)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 1025.1
Root \(0.965926 - 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 1134.1025
Dual form 1134.2.t.f.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)
 
\(f(q)\) \(=\) \(q+(-0.866025 - 0.500000i) q^{2} +(0.500000 + 0.866025i) q^{4} -2.44949 q^{5} +(2.62132 - 0.358719i) q^{7} -1.00000i q^{8} +(2.12132 + 1.22474i) q^{10} -4.24264i q^{11} +(0.621320 + 0.358719i) q^{13} +(-2.44949 - 1.00000i) q^{14} +(-0.500000 + 0.866025i) q^{16} +(1.22474 - 2.12132i) q^{17} +(-4.24264 + 2.44949i) q^{19} +(-1.22474 - 2.12132i) q^{20} +(-2.12132 + 3.67423i) q^{22} +6.00000i q^{23} +1.00000 q^{25} +(-0.358719 - 0.621320i) q^{26} +(1.62132 + 2.09077i) q^{28} +(-1.52192 + 0.878680i) q^{29} +(7.86396 - 4.54026i) q^{31} +(0.866025 - 0.500000i) q^{32} +(-2.12132 + 1.22474i) q^{34} +(-6.42090 + 0.878680i) q^{35} +(-2.62132 - 4.54026i) q^{37} +4.89898 q^{38} +2.44949i q^{40} +(-1.22474 + 2.12132i) q^{41} +(-3.50000 - 6.06218i) q^{43} +(3.67423 - 2.12132i) q^{44} +(3.00000 - 5.19615i) q^{46} +(6.42090 - 11.1213i) q^{47} +(6.74264 - 1.88064i) q^{49} +(-0.866025 - 0.500000i) q^{50} +0.717439i q^{52} +(-12.5446 - 7.24264i) q^{53} +10.3923i q^{55} +(-0.358719 - 2.62132i) q^{56} +1.75736 q^{58} +(-1.22474 - 2.12132i) q^{59} +(-3.62132 - 2.09077i) q^{61} -9.08052 q^{62} -1.00000 q^{64} +(-1.52192 - 0.878680i) q^{65} +(-6.74264 - 11.6786i) q^{67} +2.44949 q^{68} +(6.00000 + 2.44949i) q^{70} -12.7279i q^{71} +(-4.75736 - 2.74666i) q^{73} +5.24264i q^{74} +(-4.24264 - 2.44949i) q^{76} +(-1.52192 - 11.1213i) q^{77} +(-0.378680 + 0.655892i) q^{79} +(1.22474 - 2.12132i) q^{80} +(2.12132 - 1.22474i) q^{82} +(7.64564 + 13.2426i) q^{83} +(-3.00000 + 5.19615i) q^{85} +7.00000i q^{86} -4.24264 q^{88} +(1.52192 + 2.63604i) q^{89} +(1.75736 + 0.717439i) q^{91} +(-5.19615 + 3.00000i) q^{92} +(-11.1213 + 6.42090i) q^{94} +(10.3923 - 6.00000i) q^{95} +(2.74264 - 1.58346i) q^{97} +(-6.77962 - 1.74264i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{4} + 4 q^{7} - 12 q^{13} - 4 q^{16} + 8 q^{25} - 4 q^{28} + 12 q^{31} - 4 q^{37} - 28 q^{43} + 24 q^{46} + 20 q^{49} + 48 q^{58} - 12 q^{61} - 8 q^{64} - 20 q^{67} + 48 q^{70} - 72 q^{73} - 20 q^{79}+ \cdots - 12 q^{97}+O(q^{100}) \) 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
\(5\) −2.44949 −1.09545 −0.547723 − 0.836660i \(-0.684505\pi\)
−0.547723 + 0.836660i \(0.684505\pi\)
\(6\) 0 0
\(7\) 2.62132 − 0.358719i 0.990766 − 0.135583i
\(8\) − 1.00000i − 0.353553i
\(9\) 0 0
\(10\) 2.12132 + 1.22474i 0.670820 + 0.387298i
\(11\) − 4.24264i − 1.27920i −0.768706 − 0.639602i \(-0.779099\pi\)
0.768706 − 0.639602i \(-0.220901\pi\)
\(12\) 0 0
\(13\) 0.621320 + 0.358719i 0.172323 + 0.0994909i 0.583681 − 0.811983i \(-0.301612\pi\)
−0.411358 + 0.911474i \(0.634945\pi\)
\(14\) −2.44949 − 1.00000i −0.654654 − 0.267261i
\(15\) 0 0
\(16\) −0.500000 + 0.866025i −0.125000 + 0.216506i
\(17\) 1.22474 − 2.12132i 0.297044 − 0.514496i −0.678414 − 0.734680i \(-0.737332\pi\)
0.975458 + 0.220184i \(0.0706658\pi\)
\(18\) 0 0
\(19\) −4.24264 + 2.44949i −0.973329 + 0.561951i −0.900249 − 0.435375i \(-0.856616\pi\)
−0.0730792 + 0.997326i \(0.523283\pi\)
\(20\) −1.22474 − 2.12132i −0.273861 − 0.474342i
\(21\) 0 0
\(22\) −2.12132 + 3.67423i −0.452267 + 0.783349i
\(23\) 6.00000i 1.25109i 0.780189 + 0.625543i \(0.215123\pi\)
−0.780189 + 0.625543i \(0.784877\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −0.358719 − 0.621320i −0.0703507 − 0.121851i
\(27\) 0 0
\(28\) 1.62132 + 2.09077i 0.306401 + 0.395118i
\(29\) −1.52192 + 0.878680i −0.282613 + 0.163167i −0.634606 − 0.772836i \(-0.718838\pi\)
0.351993 + 0.936003i \(0.385504\pi\)
\(30\) 0 0
\(31\) 7.86396 − 4.54026i 1.41241 − 0.815455i 0.416794 − 0.909001i \(-0.363154\pi\)
0.995615 + 0.0935461i \(0.0298203\pi\)
\(32\) 0.866025 − 0.500000i 0.153093 − 0.0883883i
\(33\) 0 0
\(34\) −2.12132 + 1.22474i −0.363803 + 0.210042i
\(35\) −6.42090 + 0.878680i −1.08533 + 0.148524i
\(36\) 0 0
\(37\) −2.62132 − 4.54026i −0.430942 − 0.746414i 0.566012 − 0.824397i \(-0.308485\pi\)
−0.996955 + 0.0779826i \(0.975152\pi\)
\(38\) 4.89898 0.794719
\(39\) 0 0
\(40\) 2.44949i 0.387298i
\(41\) −1.22474 + 2.12132i −0.191273 + 0.331295i −0.945672 − 0.325121i \(-0.894595\pi\)
0.754399 + 0.656416i \(0.227928\pi\)
\(42\) 0 0
\(43\) −3.50000 − 6.06218i −0.533745 − 0.924473i −0.999223 − 0.0394140i \(-0.987451\pi\)
0.465478 − 0.885059i \(-0.345882\pi\)
\(44\) 3.67423 − 2.12132i 0.553912 − 0.319801i
\(45\) 0 0
\(46\) 3.00000 − 5.19615i 0.442326 − 0.766131i
\(47\) 6.42090 − 11.1213i 0.936584 − 1.62221i 0.164800 − 0.986327i \(-0.447302\pi\)
0.771784 − 0.635884i \(-0.219364\pi\)
\(48\) 0 0
\(49\) 6.74264 − 1.88064i 0.963234 − 0.268662i
\(50\) −0.866025 − 0.500000i −0.122474 − 0.0707107i
\(51\) 0 0
\(52\) 0.717439i 0.0994909i
\(53\) −12.5446 − 7.24264i −1.72314 − 0.994853i −0.912231 − 0.409675i \(-0.865642\pi\)
−0.810905 − 0.585178i \(-0.801025\pi\)
\(54\) 0 0
\(55\) 10.3923i 1.40130i
\(56\) −0.358719 − 2.62132i −0.0479359 − 0.350289i
\(57\) 0 0
\(58\) 1.75736 0.230753
\(59\) −1.22474 − 2.12132i −0.159448 − 0.276172i 0.775222 − 0.631689i \(-0.217638\pi\)
−0.934670 + 0.355517i \(0.884305\pi\)
\(60\) 0 0
\(61\) −3.62132 − 2.09077i −0.463663 − 0.267696i 0.249920 − 0.968266i \(-0.419596\pi\)
−0.713583 + 0.700571i \(0.752929\pi\)
\(62\) −9.08052 −1.15323
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) −1.52192 − 0.878680i −0.188771 − 0.108987i
\(66\) 0 0
\(67\) −6.74264 − 11.6786i −0.823745 − 1.42677i −0.902875 − 0.429903i \(-0.858548\pi\)
0.0791303 − 0.996864i \(-0.474786\pi\)
\(68\) 2.44949 0.297044
\(69\) 0 0
\(70\) 6.00000 + 2.44949i 0.717137 + 0.292770i
\(71\) − 12.7279i − 1.51053i −0.655422 − 0.755263i \(-0.727509\pi\)
0.655422 − 0.755263i \(-0.272491\pi\)
\(72\) 0 0
\(73\) −4.75736 − 2.74666i −0.556807 − 0.321473i 0.195056 − 0.980792i \(-0.437511\pi\)
−0.751863 + 0.659320i \(0.770844\pi\)
\(74\) 5.24264i 0.609445i
\(75\) 0 0
\(76\) −4.24264 − 2.44949i −0.486664 − 0.280976i
\(77\) −1.52192 − 11.1213i −0.173439 − 1.26739i
\(78\) 0 0
\(79\) −0.378680 + 0.655892i −0.0426048 + 0.0737937i −0.886541 − 0.462649i \(-0.846899\pi\)
0.843937 + 0.536443i \(0.180232\pi\)
\(80\) 1.22474 − 2.12132i 0.136931 − 0.237171i
\(81\) 0 0
\(82\) 2.12132 − 1.22474i 0.234261 − 0.135250i
\(83\) 7.64564 + 13.2426i 0.839218 + 1.45357i 0.890549 + 0.454887i \(0.150320\pi\)
−0.0513309 + 0.998682i \(0.516346\pi\)
\(84\) 0 0
\(85\) −3.00000 + 5.19615i −0.325396 + 0.563602i
\(86\) 7.00000i 0.754829i
\(87\) 0 0
\(88\) −4.24264 −0.452267
\(89\) 1.52192 + 2.63604i 0.161323 + 0.279420i 0.935343 − 0.353741i \(-0.115091\pi\)
−0.774020 + 0.633161i \(0.781757\pi\)
\(90\) 0 0
\(91\) 1.75736 + 0.717439i 0.184221 + 0.0752080i
\(92\) −5.19615 + 3.00000i −0.541736 + 0.312772i
\(93\) 0 0
\(94\) −11.1213 + 6.42090i −1.14708 + 0.662265i
\(95\) 10.3923 − 6.00000i 1.06623 − 0.615587i
\(96\) 0 0
\(97\) 2.74264 − 1.58346i 0.278473 − 0.160776i −0.354259 − 0.935147i \(-0.615267\pi\)
0.632732 + 0.774371i \(0.281934\pi\)
\(98\) −6.77962 − 1.74264i −0.684845 − 0.176033i
\(99\) 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.f.1025.1 8
3.2 odd 2 inner 1134.2.t.f.1025.4 8
7.5 odd 6 1134.2.l.e.215.3 8
9.2 odd 6 1134.2.l.e.269.1 8
9.4 even 3 378.2.k.d.269.4 yes 8
9.5 odd 6 378.2.k.d.269.1 yes 8
9.7 even 3 1134.2.l.e.269.4 8
21.5 even 6 1134.2.l.e.215.2 8
63.4 even 3 2646.2.d.d.2645.5 8
63.5 even 6 378.2.k.d.215.4 yes 8
63.31 odd 6 2646.2.d.d.2645.7 8
63.32 odd 6 2646.2.d.d.2645.4 8
63.40 odd 6 378.2.k.d.215.1 ✓ 8
63.47 even 6 inner 1134.2.t.f.593.1 8
63.59 even 6 2646.2.d.d.2645.2 8
63.61 odd 6 inner 1134.2.t.f.593.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
378.2.k.d.215.1 ✓ 8 63.40 odd 6
378.2.k.d.215.4 yes 8 63.5 even 6
378.2.k.d.269.1 yes 8 9.5 odd 6
378.2.k.d.269.4 yes 8 9.4 even 3
1134.2.l.e.215.2 8 21.5 even 6
1134.2.l.e.215.3 8 7.5 odd 6
1134.2.l.e.269.1 8 9.2 odd 6
1134.2.l.e.269.4 8 9.7 even 3
1134.2.t.f.593.1 8 63.47 even 6 inner
1134.2.t.f.593.4 8 63.61 odd 6 inner
1134.2.t.f.1025.1 8 1.1 even 1 trivial
1134.2.t.f.1025.4 8 3.2 odd 2 inner
2646.2.d.d.2645.2 8 63.59 even 6
2646.2.d.d.2645.4 8 63.32 odd 6
2646.2.d.d.2645.5 8 63.4 even 3
2646.2.d.d.2645.7 8 63.31 odd 6