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 = [8,0,0,-8,0,0,4,0,0,12,0] 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: \(8\)
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: \(\Z[a_1, \ldots, a_{11}]\)
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 269.1
Root \(0.258819 - 0.965926i\) of defining polynomial
Character \(\chi\) \(=\) 1134.269
Dual form 1134.2.l.f.215.3

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000 q^{4} +(-2.09077 + 3.62132i) q^{5} +(2.62132 - 0.358719i) q^{7} +1.00000i q^{8} +(3.62132 + 2.09077i) q^{10} +(-2.59808 + 1.50000i) q^{11} +(-2.12132 + 1.22474i) q^{13} +(-0.358719 - 2.62132i) q^{14} +1.00000 q^{16} +(0.507306 - 0.878680i) q^{17} +(-0.878680 + 0.507306i) q^{19} +(2.09077 - 3.62132i) q^{20} +(1.50000 + 2.59808i) q^{22} +(-3.67423 - 2.12132i) q^{23} +(-6.24264 - 10.8126i) q^{25} +(1.22474 + 2.12132i) q^{26} +(-2.62132 + 0.358719i) q^{28} +(-1.07616 - 0.621320i) q^{29} -5.61642i q^{31} -1.00000i q^{32} +(-0.878680 - 0.507306i) q^{34} +(-4.18154 + 10.2426i) q^{35} +(-4.12132 - 7.13834i) q^{37} +(0.507306 + 0.878680i) q^{38} +(-3.62132 - 2.09077i) q^{40} +(1.01461 + 1.75736i) q^{41} +(-4.12132 + 7.13834i) q^{43} +(2.59808 - 1.50000i) q^{44} +(-2.12132 + 3.67423i) q^{46} -1.01461 q^{47} +(6.74264 - 1.88064i) q^{49} +(-10.8126 + 6.24264i) q^{50} +(2.12132 - 1.22474i) q^{52} +(-1.07616 - 0.621320i) q^{53} -12.5446i q^{55} +(0.358719 + 2.62132i) q^{56} +(-0.621320 + 1.07616i) q^{58} -11.5300 q^{59} +5.91359i q^{61} -5.61642 q^{62} -1.00000 q^{64} -10.2426i q^{65} -10.0000 q^{67} +(-0.507306 + 0.878680i) q^{68} +(10.2426 + 4.18154i) q^{70} +10.2426i q^{71} +(7.24264 + 4.18154i) q^{73} +(-7.13834 + 4.12132i) q^{74} +(0.878680 - 0.507306i) q^{76} +(-6.27231 + 4.86396i) q^{77} -11.2426 q^{79} +(-2.09077 + 3.62132i) q^{80} +(1.75736 - 1.01461i) q^{82} +(-1.58346 + 2.74264i) q^{83} +(2.12132 + 3.67423i) q^{85} +(7.13834 + 4.12132i) q^{86} +(-1.50000 - 2.59808i) q^{88} +(-5.19615 - 9.00000i) q^{89} +(-5.12132 + 3.97141i) q^{91} +(3.67423 + 2.12132i) q^{92} +1.01461i q^{94} -4.24264i q^{95} +(3.25736 + 1.88064i) q^{97} +(-1.88064 - 6.74264i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{4} + 4 q^{7} + 12 q^{10} + 8 q^{16} - 24 q^{19} + 12 q^{22} - 16 q^{25} - 4 q^{28} - 24 q^{34} - 16 q^{37} - 12 q^{40} - 16 q^{43} + 20 q^{49} + 12 q^{58} - 8 q^{64} - 80 q^{67} + 48 q^{70} + 24 q^{73}+ \cdots + 60 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.707107i
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) −2.09077 + 3.62132i −0.935021 + 1.61950i −0.160424 + 0.987048i \(0.551286\pi\)
−0.774597 + 0.632456i \(0.782047\pi\)
\(6\) 0 0
\(7\) 2.62132 − 0.358719i 0.990766 − 0.135583i
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) 3.62132 + 2.09077i 1.14516 + 0.661160i
\(11\) −2.59808 + 1.50000i −0.783349 + 0.452267i −0.837616 − 0.546259i \(-0.816051\pi\)
0.0542666 + 0.998526i \(0.482718\pi\)
\(12\) 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
\(16\) 1.00000 0.250000
\(17\) 0.507306 − 0.878680i 0.123040 − 0.213111i −0.797925 − 0.602756i \(-0.794069\pi\)
0.920965 + 0.389645i \(0.127402\pi\)
\(18\) 0 0
\(19\) −0.878680 + 0.507306i −0.201583 + 0.116384i −0.597394 − 0.801948i \(-0.703797\pi\)
0.395811 + 0.918332i \(0.370464\pi\)
\(20\) 2.09077 − 3.62132i 0.467510 − 0.809752i
\(21\) 0 0
\(22\) 1.50000 + 2.59808i 0.319801 + 0.553912i
\(23\) −3.67423 − 2.12132i −0.766131 − 0.442326i 0.0653618 − 0.997862i \(-0.479180\pi\)
−0.831493 + 0.555536i \(0.812513\pi\)
\(24\) 0 0
\(25\) −6.24264 − 10.8126i −1.24853 − 2.16251i
\(26\) 1.22474 + 2.12132i 0.240192 + 0.416025i
\(27\) 0 0
\(28\) −2.62132 + 0.358719i −0.495383 + 0.0677916i
\(29\) −1.07616 − 0.621320i −0.199838 − 0.115376i 0.396742 − 0.917930i \(-0.370141\pi\)
−0.596580 + 0.802554i \(0.703474\pi\)
\(30\) 0 0
\(31\) − 5.61642i − 1.00874i −0.863488 − 0.504369i \(-0.831725\pi\)
0.863488 − 0.504369i \(-0.168275\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 0 0
\(34\) −0.878680 − 0.507306i −0.150692 − 0.0870023i
\(35\) −4.18154 + 10.2426i −0.706809 + 1.73132i
\(36\) 0 0
\(37\) −4.12132 − 7.13834i −0.677541 − 1.17354i −0.975719 − 0.219025i \(-0.929712\pi\)
0.298178 − 0.954510i \(-0.403621\pi\)
\(38\) 0.507306 + 0.878680i 0.0822959 + 0.142541i
\(39\) 0 0
\(40\) −3.62132 − 2.09077i −0.572581 − 0.330580i
\(41\) 1.01461 + 1.75736i 0.158456 + 0.274453i 0.934312 − 0.356456i \(-0.116015\pi\)
−0.775856 + 0.630910i \(0.782682\pi\)
\(42\) 0 0
\(43\) −4.12132 + 7.13834i −0.628495 + 1.08859i 0.359358 + 0.933200i \(0.382996\pi\)
−0.987854 + 0.155386i \(0.950338\pi\)
\(44\) 2.59808 − 1.50000i 0.391675 − 0.226134i
\(45\) 0 0
\(46\) −2.12132 + 3.67423i −0.312772 + 0.541736i
\(47\) −1.01461 −0.147996 −0.0739982 − 0.997258i \(-0.523576\pi\)
−0.0739982 + 0.997258i \(0.523576\pi\)
\(48\) 0 0
\(49\) 6.74264 − 1.88064i 0.963234 − 0.268662i
\(50\) −10.8126 + 6.24264i −1.52913 + 0.882843i
\(51\) 0 0
\(52\) 2.12132 − 1.22474i 0.294174 − 0.169842i
\(53\) −1.07616 − 0.621320i −0.147822 − 0.0853449i 0.424265 − 0.905538i \(-0.360533\pi\)
−0.572087 + 0.820193i \(0.693866\pi\)
\(54\) 0 0
\(55\) − 12.5446i − 1.69152i
\(56\) 0.358719 + 2.62132i 0.0479359 + 0.350289i
\(57\) 0 0
\(58\) −0.621320 + 1.07616i −0.0815834 + 0.141307i
\(59\) −11.5300 −1.50108 −0.750540 − 0.660825i \(-0.770206\pi\)
−0.750540 + 0.660825i \(0.770206\pi\)
\(60\) 0 0
\(61\) 5.91359i 0.757158i 0.925569 + 0.378579i \(0.123587\pi\)
−0.925569 + 0.378579i \(0.876413\pi\)
\(62\) −5.61642 −0.713286
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) − 10.2426i − 1.27044i
\(66\) 0 0
\(67\) −10.0000 −1.22169 −0.610847 − 0.791748i \(-0.709171\pi\)
−0.610847 + 0.791748i \(0.709171\pi\)
\(68\) −0.507306 + 0.878680i −0.0615199 + 0.106556i
\(69\) 0 0
\(70\) 10.2426 + 4.18154i 1.22423 + 0.499790i
\(71\) 10.2426i 1.21558i 0.794099 + 0.607789i \(0.207943\pi\)
−0.794099 + 0.607789i \(0.792057\pi\)
\(72\) 0 0
\(73\) 7.24264 + 4.18154i 0.847687 + 0.489412i 0.859870 − 0.510513i \(-0.170545\pi\)
−0.0121828 + 0.999926i \(0.503878\pi\)
\(74\) −7.13834 + 4.12132i −0.829815 + 0.479094i
\(75\) 0 0
\(76\) 0.878680 − 0.507306i 0.100791 − 0.0581920i
\(77\) −6.27231 + 4.86396i −0.714796 + 0.554300i
\(78\) 0 0
\(79\) −11.2426 −1.26490 −0.632448 − 0.774603i \(-0.717950\pi\)
−0.632448 + 0.774603i \(0.717950\pi\)
\(80\) −2.09077 + 3.62132i −0.233755 + 0.404876i
\(81\) 0 0
\(82\) 1.75736 − 1.01461i 0.194068 − 0.112045i
\(83\) −1.58346 + 2.74264i −0.173808 + 0.301044i −0.939748 − 0.341868i \(-0.888940\pi\)
0.765940 + 0.642912i \(0.222274\pi\)
\(84\) 0 0
\(85\) 2.12132 + 3.67423i 0.230089 + 0.398527i
\(86\) 7.13834 + 4.12132i 0.769747 + 0.444413i
\(87\) 0 0
\(88\) −1.50000 − 2.59808i −0.159901 − 0.276956i
\(89\) −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
\(91\) −5.12132 + 3.97141i −0.536860 + 0.416317i
\(92\) 3.67423 + 2.12132i 0.383065 + 0.221163i
\(93\) 0 0
\(94\) 1.01461i 0.104649i
\(95\) − 4.24264i − 0.435286i
\(96\) 0 0
\(97\) 3.25736 + 1.88064i 0.330735 + 0.190950i 0.656167 − 0.754615i \(-0.272177\pi\)
−0.325433 + 0.945565i \(0.605510\pi\)
\(98\) −1.88064 − 6.74264i −0.189973 − 0.681110i
\(99\) 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.f.269.1 8
3.2 odd 2 inner 1134.2.l.f.269.4 8
7.5 odd 6 1134.2.t.e.593.1 8
9.2 odd 6 126.2.k.a.17.4 yes 8
9.4 even 3 1134.2.t.e.1025.4 8
9.5 odd 6 1134.2.t.e.1025.1 8
9.7 even 3 126.2.k.a.17.1 ✓ 8
21.5 even 6 1134.2.t.e.593.4 8
36.7 odd 6 1008.2.bt.c.17.1 8
36.11 even 6 1008.2.bt.c.17.4 8
45.2 even 12 3150.2.bp.e.899.4 8
45.7 odd 12 3150.2.bp.b.899.4 8
45.29 odd 6 3150.2.bf.a.1151.2 8
45.34 even 6 3150.2.bf.a.1151.4 8
45.38 even 12 3150.2.bp.b.899.1 8
45.43 odd 12 3150.2.bp.e.899.1 8
63.2 odd 6 882.2.k.a.215.2 8
63.5 even 6 inner 1134.2.l.f.215.3 8
63.11 odd 6 882.2.d.a.881.5 8
63.16 even 3 882.2.k.a.215.3 8
63.20 even 6 882.2.k.a.521.3 8
63.25 even 3 882.2.d.a.881.4 8
63.34 odd 6 882.2.k.a.521.2 8
63.38 even 6 882.2.d.a.881.8 8
63.40 odd 6 inner 1134.2.l.f.215.2 8
63.47 even 6 126.2.k.a.89.1 yes 8
63.52 odd 6 882.2.d.a.881.1 8
63.61 odd 6 126.2.k.a.89.4 yes 8
252.11 even 6 7056.2.k.f.881.1 8
252.47 odd 6 1008.2.bt.c.593.1 8
252.115 even 6 7056.2.k.f.881.2 8
252.151 odd 6 7056.2.k.f.881.8 8
252.187 even 6 1008.2.bt.c.593.4 8
252.227 odd 6 7056.2.k.f.881.7 8
315.47 odd 12 3150.2.bp.e.1349.1 8
315.124 odd 6 3150.2.bf.a.1601.2 8
315.173 odd 12 3150.2.bp.b.1349.4 8
315.187 even 12 3150.2.bp.b.1349.1 8
315.299 even 6 3150.2.bf.a.1601.4 8
315.313 even 12 3150.2.bp.e.1349.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
126.2.k.a.17.1 ✓ 8 9.7 even 3
126.2.k.a.17.4 yes 8 9.2 odd 6
126.2.k.a.89.1 yes 8 63.47 even 6
126.2.k.a.89.4 yes 8 63.61 odd 6
882.2.d.a.881.1 8 63.52 odd 6
882.2.d.a.881.4 8 63.25 even 3
882.2.d.a.881.5 8 63.11 odd 6
882.2.d.a.881.8 8 63.38 even 6
882.2.k.a.215.2 8 63.2 odd 6
882.2.k.a.215.3 8 63.16 even 3
882.2.k.a.521.2 8 63.34 odd 6
882.2.k.a.521.3 8 63.20 even 6
1008.2.bt.c.17.1 8 36.7 odd 6
1008.2.bt.c.17.4 8 36.11 even 6
1008.2.bt.c.593.1 8 252.47 odd 6
1008.2.bt.c.593.4 8 252.187 even 6
1134.2.l.f.215.2 8 63.40 odd 6 inner
1134.2.l.f.215.3 8 63.5 even 6 inner
1134.2.l.f.269.1 8 1.1 even 1 trivial
1134.2.l.f.269.4 8 3.2 odd 2 inner
1134.2.t.e.593.1 8 7.5 odd 6
1134.2.t.e.593.4 8 21.5 even 6
1134.2.t.e.1025.1 8 9.5 odd 6
1134.2.t.e.1025.4 8 9.4 even 3
3150.2.bf.a.1151.2 8 45.29 odd 6
3150.2.bf.a.1151.4 8 45.34 even 6
3150.2.bf.a.1601.2 8 315.124 odd 6
3150.2.bf.a.1601.4 8 315.299 even 6
3150.2.bp.b.899.1 8 45.38 even 12
3150.2.bp.b.899.4 8 45.7 odd 12
3150.2.bp.b.1349.1 8 315.187 even 12
3150.2.bp.b.1349.4 8 315.173 odd 12
3150.2.bp.e.899.1 8 45.43 odd 12
3150.2.bp.e.899.4 8 45.2 even 12
3150.2.bp.e.1349.1 8 315.47 odd 12
3150.2.bp.e.1349.4 8 315.313 even 12
7056.2.k.f.881.1 8 252.11 even 6
7056.2.k.f.881.2 8 252.115 even 6
7056.2.k.f.881.7 8 252.227 odd 6
7056.2.k.f.881.8 8 252.151 odd 6