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Properties

Label 1134.2.t.e.593.3
Level $1134$
Weight $2$
Character 1134.593
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,-8,0,0,12,0,0,0,0] 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: \(8\)
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 126)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 593.3
Root \(0.965926 + 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 1134.593
Dual form 1134.2.t.e.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)
 
\(f(q)\) \(=\) \(q+(0.866025 - 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} -0.717439 q^{5} +(-1.00000 + 2.44949i) q^{7} -1.00000i q^{8} +(-0.621320 + 0.358719i) q^{10} +3.00000i q^{11} +(-2.12132 + 1.22474i) q^{13} +(0.358719 + 2.62132i) q^{14} +(-0.500000 - 0.866025i) q^{16} +(2.95680 + 5.12132i) q^{17} +(-5.12132 - 2.95680i) q^{19} +(-0.358719 + 0.621320i) q^{20} +(1.50000 + 2.59808i) q^{22} +4.24264i q^{23} -4.48528 q^{25} +(-1.22474 + 2.12132i) q^{26} +(1.62132 + 2.09077i) q^{28} +(-6.27231 - 3.62132i) q^{29} +(7.86396 + 4.54026i) q^{31} +(-0.866025 - 0.500000i) q^{32} +(5.12132 + 2.95680i) q^{34} +(0.717439 - 1.75736i) q^{35} +(0.121320 - 0.210133i) q^{37} -5.91359 q^{38} +0.717439i q^{40} +(5.91359 + 10.2426i) q^{41} +(0.121320 - 0.210133i) q^{43} +(2.59808 + 1.50000i) q^{44} +(2.12132 + 3.67423i) q^{46} +(2.95680 + 5.12132i) q^{47} +(-5.00000 - 4.89898i) q^{49} +(-3.88437 + 2.24264i) q^{50} +2.44949i q^{52} +(6.27231 - 3.62132i) q^{53} -2.15232i q^{55} +(2.44949 + 1.00000i) q^{56} -7.24264 q^{58} +(-4.03295 + 6.98528i) q^{59} +(-0.878680 + 0.507306i) q^{61} +9.08052 q^{62} -1.00000 q^{64} +(1.52192 - 0.878680i) q^{65} +(5.00000 - 8.66025i) q^{67} +5.91359 q^{68} +(-0.257359 - 1.88064i) q^{70} -1.75736i q^{71} +(-1.24264 + 0.717439i) q^{73} -0.242641i q^{74} +(-5.12132 + 2.95680i) q^{76} +(-7.34847 - 3.00000i) q^{77} +(1.37868 + 2.38794i) q^{79} +(0.358719 + 0.621320i) q^{80} +(10.2426 + 5.91359i) q^{82} +(3.31552 - 5.74264i) q^{83} +(-2.12132 - 3.67423i) q^{85} -0.242641i q^{86} +3.00000 q^{88} +(-5.19615 + 9.00000i) q^{89} +(-0.878680 - 6.42090i) q^{91} +(3.67423 + 2.12132i) q^{92} +(5.12132 + 2.95680i) q^{94} +(3.67423 + 2.12132i) q^{95} +(-11.7426 - 6.77962i) q^{97} +(-6.77962 - 1.74264i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{4} - 8 q^{7} + 12 q^{10} - 4 q^{16} - 24 q^{19} + 12 q^{22} + 32 q^{25} - 4 q^{28} + 12 q^{31} + 24 q^{34} - 16 q^{37} - 16 q^{43} - 40 q^{49} - 24 q^{58} - 24 q^{61} - 8 q^{64} + 40 q^{67} - 36 q^{70}+ \cdots - 60 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(5\) −0.717439 −0.320848 −0.160424 − 0.987048i \(-0.551286\pi\)
−0.160424 + 0.987048i \(0.551286\pi\)
\(6\) 0 0
\(7\) −1.00000 + 2.44949i −0.377964 + 0.925820i
\(8\) − 1.00000i − 0.353553i
\(9\) 0 0
\(10\) −0.621320 + 0.358719i −0.196479 + 0.113437i
\(11\) 3.00000i 0.904534i 0.891883 + 0.452267i \(0.149385\pi\)
−0.891883 + 0.452267i \(0.850615\pi\)
\(12\) 0 0
\(13\) −2.12132 + 1.22474i −0.588348 + 0.339683i −0.764444 − 0.644690i \(-0.776986\pi\)
0.176096 + 0.984373i \(0.443653\pi\)
\(14\) 0.358719 + 2.62132i 0.0958718 + 0.700577i
\(15\) 0 0
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) 2.95680 + 5.12132i 0.717128 + 1.24210i 0.962133 + 0.272581i \(0.0878772\pi\)
−0.245005 + 0.969522i \(0.578789\pi\)
\(18\) 0 0
\(19\) −5.12132 − 2.95680i −1.17491 − 0.678335i −0.220080 − 0.975482i \(-0.570632\pi\)
−0.954832 + 0.297146i \(0.903965\pi\)
\(20\) −0.358719 + 0.621320i −0.0802121 + 0.138931i
\(21\) 0 0
\(22\) 1.50000 + 2.59808i 0.319801 + 0.553912i
\(23\) 4.24264i 0.884652i 0.896854 + 0.442326i \(0.145847\pi\)
−0.896854 + 0.442326i \(0.854153\pi\)
\(24\) 0 0
\(25\) −4.48528 −0.897056
\(26\) −1.22474 + 2.12132i −0.240192 + 0.416025i
\(27\) 0 0
\(28\) 1.62132 + 2.09077i 0.306401 + 0.395118i
\(29\) −6.27231 − 3.62132i −1.16474 − 0.672462i −0.212304 − 0.977204i \(-0.568097\pi\)
−0.952435 + 0.304741i \(0.901430\pi\)
\(30\) 0 0
\(31\) 7.86396 + 4.54026i 1.41241 + 0.815455i 0.995615 − 0.0935461i \(-0.0298203\pi\)
0.416794 + 0.909001i \(0.363154\pi\)
\(32\) −0.866025 − 0.500000i −0.153093 − 0.0883883i
\(33\) 0 0
\(34\) 5.12132 + 2.95680i 0.878299 + 0.507086i
\(35\) 0.717439 − 1.75736i 0.121269 − 0.297048i
\(36\) 0 0
\(37\) 0.121320 − 0.210133i 0.0199449 − 0.0345457i −0.855881 − 0.517173i \(-0.826984\pi\)
0.875826 + 0.482628i \(0.160318\pi\)
\(38\) −5.91359 −0.959311
\(39\) 0 0
\(40\) 0.717439i 0.113437i
\(41\) 5.91359 + 10.2426i 0.923548 + 1.59963i 0.793880 + 0.608074i \(0.208058\pi\)
0.129668 + 0.991558i \(0.458609\pi\)
\(42\) 0 0
\(43\) 0.121320 − 0.210133i 0.0185012 − 0.0320450i −0.856627 − 0.515937i \(-0.827444\pi\)
0.875128 + 0.483892i \(0.160777\pi\)
\(44\) 2.59808 + 1.50000i 0.391675 + 0.226134i
\(45\) 0 0
\(46\) 2.12132 + 3.67423i 0.312772 + 0.541736i
\(47\) 2.95680 + 5.12132i 0.431293 + 0.747021i 0.996985 − 0.0775953i \(-0.0247242\pi\)
−0.565692 + 0.824617i \(0.691391\pi\)
\(48\) 0 0
\(49\) −5.00000 − 4.89898i −0.714286 − 0.699854i
\(50\) −3.88437 + 2.24264i −0.549333 + 0.317157i
\(51\) 0 0
\(52\) 2.44949i 0.339683i
\(53\) 6.27231 − 3.62132i 0.861568 − 0.497427i −0.00296896 − 0.999996i \(-0.500945\pi\)
0.864537 + 0.502569i \(0.167612\pi\)
\(54\) 0 0
\(55\) − 2.15232i − 0.290218i
\(56\) 2.44949 + 1.00000i 0.327327 + 0.133631i
\(57\) 0 0
\(58\) −7.24264 −0.951005
\(59\) −4.03295 + 6.98528i −0.525046 + 0.909406i 0.474529 + 0.880240i \(0.342619\pi\)
−0.999575 + 0.0291661i \(0.990715\pi\)
\(60\) 0 0
\(61\) −0.878680 + 0.507306i −0.112503 + 0.0649539i −0.555196 − 0.831720i \(-0.687357\pi\)
0.442692 + 0.896674i \(0.354023\pi\)
\(62\) 9.08052 1.15323
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 1.52192 − 0.878680i 0.188771 − 0.108987i
\(66\) 0 0
\(67\) 5.00000 − 8.66025i 0.610847 − 1.05802i −0.380251 − 0.924883i \(-0.624162\pi\)
0.991098 − 0.133135i \(-0.0425044\pi\)
\(68\) 5.91359 0.717128
\(69\) 0 0
\(70\) −0.257359 − 1.88064i −0.0307603 − 0.224779i
\(71\) − 1.75736i − 0.208560i −0.994548 − 0.104280i \(-0.966746\pi\)
0.994548 − 0.104280i \(-0.0332538\pi\)
\(72\) 0 0
\(73\) −1.24264 + 0.717439i −0.145440 + 0.0839699i −0.570954 − 0.820982i \(-0.693427\pi\)
0.425514 + 0.904952i \(0.360093\pi\)
\(74\) − 0.242641i − 0.0282064i
\(75\) 0 0
\(76\) −5.12132 + 2.95680i −0.587456 + 0.339168i
\(77\) −7.34847 − 3.00000i −0.837436 − 0.341882i
\(78\) 0 0
\(79\) 1.37868 + 2.38794i 0.155114 + 0.268665i 0.933100 − 0.359616i \(-0.117092\pi\)
−0.777987 + 0.628281i \(0.783759\pi\)
\(80\) 0.358719 + 0.621320i 0.0401061 + 0.0694657i
\(81\) 0 0
\(82\) 10.2426 + 5.91359i 1.13111 + 0.653047i
\(83\) 3.31552 − 5.74264i 0.363925 − 0.630337i −0.624678 − 0.780882i \(-0.714770\pi\)
0.988603 + 0.150546i \(0.0481031\pi\)
\(84\) 0 0
\(85\) −2.12132 − 3.67423i −0.230089 − 0.398527i
\(86\) − 0.242641i − 0.0261646i
\(87\) 0 0
\(88\) 3.00000 0.319801
\(89\) −5.19615 + 9.00000i −0.550791 + 0.953998i 0.447427 + 0.894321i \(0.352341\pi\)
−0.998218 + 0.0596775i \(0.980993\pi\)
\(90\) 0 0
\(91\) −0.878680 − 6.42090i −0.0921107 − 0.673093i
\(92\) 3.67423 + 2.12132i 0.383065 + 0.221163i
\(93\) 0 0
\(94\) 5.12132 + 2.95680i 0.528224 + 0.304970i
\(95\) 3.67423 + 2.12132i 0.376969 + 0.217643i
\(96\) 0 0
\(97\) −11.7426 − 6.77962i −1.19228 − 0.688366i −0.233460 − 0.972366i \(-0.575005\pi\)
−0.958824 + 0.284001i \(0.908338\pi\)
\(98\) −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.e.593.3 8
3.2 odd 2 inner 1134.2.t.e.593.2 8
7.3 odd 6 1134.2.l.f.269.3 8
9.2 odd 6 126.2.k.a.89.3 yes 8
9.4 even 3 1134.2.l.f.215.4 8
9.5 odd 6 1134.2.l.f.215.1 8
9.7 even 3 126.2.k.a.89.2 yes 8
21.17 even 6 1134.2.l.f.269.2 8
36.7 odd 6 1008.2.bt.c.593.3 8
36.11 even 6 1008.2.bt.c.593.2 8
45.2 even 12 3150.2.bp.b.1349.3 8
45.7 odd 12 3150.2.bp.e.1349.3 8
45.29 odd 6 3150.2.bf.a.1601.1 8
45.34 even 6 3150.2.bf.a.1601.3 8
45.38 even 12 3150.2.bp.e.1349.2 8
45.43 odd 12 3150.2.bp.b.1349.2 8
63.2 odd 6 882.2.d.a.881.3 8
63.11 odd 6 882.2.k.a.521.1 8
63.16 even 3 882.2.d.a.881.6 8
63.20 even 6 882.2.k.a.215.4 8
63.25 even 3 882.2.k.a.521.4 8
63.31 odd 6 inner 1134.2.t.e.1025.2 8
63.34 odd 6 882.2.k.a.215.1 8
63.38 even 6 126.2.k.a.17.2 ✓ 8
63.47 even 6 882.2.d.a.881.2 8
63.52 odd 6 126.2.k.a.17.3 yes 8
63.59 even 6 inner 1134.2.t.e.1025.3 8
63.61 odd 6 882.2.d.a.881.7 8
252.47 odd 6 7056.2.k.f.881.4 8
252.79 odd 6 7056.2.k.f.881.3 8
252.115 even 6 1008.2.bt.c.17.2 8
252.187 even 6 7056.2.k.f.881.5 8
252.191 even 6 7056.2.k.f.881.6 8
252.227 odd 6 1008.2.bt.c.17.3 8
315.38 odd 12 3150.2.bp.e.899.3 8
315.52 even 12 3150.2.bp.e.899.2 8
315.164 even 6 3150.2.bf.a.1151.3 8
315.178 even 12 3150.2.bp.b.899.3 8
315.227 odd 12 3150.2.bp.b.899.2 8
315.304 odd 6 3150.2.bf.a.1151.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
126.2.k.a.17.2 ✓ 8 63.38 even 6
126.2.k.a.17.3 yes 8 63.52 odd 6
126.2.k.a.89.2 yes 8 9.7 even 3
126.2.k.a.89.3 yes 8 9.2 odd 6
882.2.d.a.881.2 8 63.47 even 6
882.2.d.a.881.3 8 63.2 odd 6
882.2.d.a.881.6 8 63.16 even 3
882.2.d.a.881.7 8 63.61 odd 6
882.2.k.a.215.1 8 63.34 odd 6
882.2.k.a.215.4 8 63.20 even 6
882.2.k.a.521.1 8 63.11 odd 6
882.2.k.a.521.4 8 63.25 even 3
1008.2.bt.c.17.2 8 252.115 even 6
1008.2.bt.c.17.3 8 252.227 odd 6
1008.2.bt.c.593.2 8 36.11 even 6
1008.2.bt.c.593.3 8 36.7 odd 6
1134.2.l.f.215.1 8 9.5 odd 6
1134.2.l.f.215.4 8 9.4 even 3
1134.2.l.f.269.2 8 21.17 even 6
1134.2.l.f.269.3 8 7.3 odd 6
1134.2.t.e.593.2 8 3.2 odd 2 inner
1134.2.t.e.593.3 8 1.1 even 1 trivial
1134.2.t.e.1025.2 8 63.31 odd 6 inner
1134.2.t.e.1025.3 8 63.59 even 6 inner
3150.2.bf.a.1151.1 8 315.304 odd 6
3150.2.bf.a.1151.3 8 315.164 even 6
3150.2.bf.a.1601.1 8 45.29 odd 6
3150.2.bf.a.1601.3 8 45.34 even 6
3150.2.bp.b.899.2 8 315.227 odd 12
3150.2.bp.b.899.3 8 315.178 even 12
3150.2.bp.b.1349.2 8 45.43 odd 12
3150.2.bp.b.1349.3 8 45.2 even 12
3150.2.bp.e.899.2 8 315.52 even 12
3150.2.bp.e.899.3 8 315.38 odd 12
3150.2.bp.e.1349.2 8 45.38 even 12
3150.2.bp.e.1349.3 8 45.7 odd 12
7056.2.k.f.881.3 8 252.79 odd 6
7056.2.k.f.881.4 8 252.47 odd 6
7056.2.k.f.881.5 8 252.187 even 6
7056.2.k.f.881.6 8 252.191 even 6