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 593.1
Root \(0.258819 - 0.965926i\) of defining polynomial
Character \(\chi\) \(=\) 1134.593
Dual form 1134.2.t.e.1025.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.866025 + 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} -4.18154 q^{5} +(-1.00000 - 2.44949i) q^{7} +1.00000i q^{8} +(3.62132 - 2.09077i) q^{10} -3.00000i q^{11} +(2.12132 - 1.22474i) q^{13} +(2.09077 + 1.62132i) q^{14} +(-0.500000 - 0.866025i) q^{16} +(-0.507306 - 0.878680i) q^{17} +(-0.878680 - 0.507306i) q^{19} +(-2.09077 + 3.62132i) q^{20} +(1.50000 + 2.59808i) q^{22} +4.24264i q^{23} +12.4853 q^{25} +(-1.22474 + 2.12132i) q^{26} +(-2.62132 - 0.358719i) q^{28} +(-1.07616 - 0.621320i) q^{29} +(-4.86396 - 2.80821i) q^{31} +(0.866025 + 0.500000i) q^{32} +(0.878680 + 0.507306i) q^{34} +(4.18154 + 10.2426i) q^{35} +(-4.12132 + 7.13834i) q^{37} +1.01461 q^{38} -4.18154i q^{40} +(-1.01461 - 1.75736i) q^{41} +(-4.12132 + 7.13834i) q^{43} +(-2.59808 - 1.50000i) q^{44} +(-2.12132 - 3.67423i) q^{46} +(-0.507306 - 0.878680i) q^{47} +(-5.00000 + 4.89898i) q^{49} +(-10.8126 + 6.24264i) q^{50} -2.44949i q^{52} +(1.07616 - 0.621320i) q^{53} +12.5446i q^{55} +(2.44949 - 1.00000i) q^{56} +1.24264 q^{58} +(-5.76500 + 9.98528i) q^{59} +(-5.12132 + 2.95680i) q^{61} +5.61642 q^{62} -1.00000 q^{64} +(-8.87039 + 5.12132i) q^{65} +(5.00000 - 8.66025i) q^{67} -1.01461 q^{68} +(-8.74264 - 6.77962i) q^{70} +10.2426i q^{71} +(7.24264 - 4.18154i) q^{73} -8.24264i q^{74} +(-0.878680 + 0.507306i) q^{76} +(-7.34847 + 3.00000i) q^{77} +(5.62132 + 9.73641i) q^{79} +(2.09077 + 3.62132i) q^{80} +(1.75736 + 1.01461i) q^{82} +(1.58346 - 2.74264i) q^{83} +(2.12132 + 3.67423i) q^{85} -8.24264i q^{86} +3.00000 q^{88} +(5.19615 - 9.00000i) q^{89} +(-5.12132 - 3.97141i) q^{91} +(3.67423 + 2.12132i) q^{92} +(0.878680 + 0.507306i) q^{94} +(3.67423 + 2.12132i) q^{95} +(-3.25736 - 1.88064i) q^{97} +(1.88064 - 6.74264i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{4} - 8 q^{7} + 12 q^{10} - 4 q^{16} - 24 q^{19} + 12 q^{22} + 32 q^{25} - 4 q^{28} + 12 q^{31} + 24 q^{34} - 16 q^{37} - 16 q^{43} - 40 q^{49} - 24 q^{58} - 24 q^{61} - 8 q^{64} + 40 q^{67} - 36 q^{70}+ \cdots - 60 q^{97}+O(q^{100}) \) 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
\(5\) −4.18154 −1.87004 −0.935021 − 0.354593i \(-0.884620\pi\)
−0.935021 + 0.354593i \(0.884620\pi\)
\(6\) 0 0
\(7\) −1.00000 − 2.44949i −0.377964 − 0.925820i
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) 3.62132 − 2.09077i 1.14516 − 0.661160i
\(11\) − 3.00000i − 0.904534i −0.891883 − 0.452267i \(-0.850615\pi\)
0.891883 − 0.452267i \(-0.149385\pi\)
\(12\) 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
\(17\) −0.507306 − 0.878680i −0.123040 − 0.213111i 0.797925 − 0.602756i \(-0.205931\pi\)
−0.920965 + 0.389645i \(0.872598\pi\)
\(18\) 0 0
\(19\) −0.878680 − 0.507306i −0.201583 − 0.116384i 0.395811 − 0.918332i \(-0.370464\pi\)
−0.597394 + 0.801948i \(0.703797\pi\)
\(20\) −2.09077 + 3.62132i −0.467510 + 0.809752i
\(21\) 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
\(25\) 12.4853 2.49706
\(26\) −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
\(31\) −4.86396 − 2.80821i −0.873593 − 0.504369i −0.00505256 − 0.999987i \(-0.501608\pi\)
−0.868541 + 0.495618i \(0.834942\pi\)
\(32\) 0.866025 + 0.500000i 0.153093 + 0.0883883i
\(33\) 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 0.298178 + 0.954510i \(0.403621\pi\)
−0.975719 + 0.219025i \(0.929712\pi\)
\(38\) 1.01461 0.164592
\(39\) 0 0
\(40\) − 4.18154i − 0.661160i
\(41\) −1.01461 − 1.75736i −0.158456 − 0.274453i 0.775856 − 0.630910i \(-0.217318\pi\)
−0.934312 + 0.356456i \(0.883985\pi\)
\(42\) 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
\(47\) −0.507306 − 0.878680i −0.0739982 − 0.128169i 0.826652 − 0.562713i \(-0.190243\pi\)
−0.900650 + 0.434545i \(0.856909\pi\)
\(48\) 0 0
\(49\) −5.00000 + 4.89898i −0.714286 + 0.699854i
\(50\) −10.8126 + 6.24264i −1.52913 + 0.882843i
\(51\) 0 0
\(52\) − 2.44949i − 0.339683i
\(53\) 1.07616 − 0.621320i 0.147822 − 0.0853449i −0.424265 − 0.905538i \(-0.639467\pi\)
0.572087 + 0.820193i \(0.306134\pi\)
\(54\) 0 0
\(55\) 12.5446i 1.69152i
\(56\) 2.44949 − 1.00000i 0.327327 − 0.133631i
\(57\) 0 0
\(58\) 1.24264 0.163167
\(59\) −5.76500 + 9.98528i −0.750540 + 1.29997i 0.197022 + 0.980399i \(0.436873\pi\)
−0.947561 + 0.319574i \(0.896460\pi\)
\(60\) 0 0
\(61\) −5.12132 + 2.95680i −0.655718 + 0.378579i −0.790643 − 0.612277i \(-0.790254\pi\)
0.134926 + 0.990856i \(0.456920\pi\)
\(62\) 5.61642 0.713286
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) −8.87039 + 5.12132i −1.10024 + 0.635222i
\(66\) 0 0
\(67\) 5.00000 − 8.66025i 0.610847 − 1.05802i −0.380251 − 0.924883i \(-0.624162\pi\)
0.991098 − 0.133135i \(-0.0425044\pi\)
\(68\) −1.01461 −0.123040
\(69\) 0 0
\(70\) −8.74264 − 6.77962i −1.04495 − 0.810319i
\(71\) 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 −0.0121828 − 0.999926i \(-0.503878\pi\)
0.859870 + 0.510513i \(0.170545\pi\)
\(74\) − 8.24264i − 0.958188i
\(75\) 0 0
\(76\) −0.878680 + 0.507306i −0.100791 + 0.0581920i
\(77\) −7.34847 + 3.00000i −0.837436 + 0.341882i
\(78\) 0 0
\(79\) 5.62132 + 9.73641i 0.632448 + 1.09543i 0.987050 + 0.160415i \(0.0512831\pi\)
−0.354602 + 0.935017i \(0.615384\pi\)
\(80\) 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 −0.765940 − 0.642912i \(-0.777726\pi\)
0.939748 + 0.341868i \(0.111060\pi\)
\(84\) 0 0
\(85\) 2.12132 + 3.67423i 0.230089 + 0.398527i
\(86\) − 8.24264i − 0.888827i
\(87\) 0 0
\(88\) 3.00000 0.319801
\(89\) 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
\(91\) −5.12132 − 3.97141i −0.536860 − 0.416317i
\(92\) 3.67423 + 2.12132i 0.383065 + 0.221163i
\(93\) 0 0
\(94\) 0.878680 + 0.507306i 0.0906289 + 0.0523246i
\(95\) 3.67423 + 2.12132i 0.376969 + 0.217643i
\(96\) 0 0
\(97\) −3.25736 − 1.88064i −0.330735 − 0.190950i 0.325433 − 0.945565i \(-0.394490\pi\)
−0.656167 + 0.754615i \(0.727823\pi\)
\(98\) 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.t.e.593.1 8
3.2 odd 2 inner 1134.2.t.e.593.4 8
7.3 odd 6 1134.2.l.f.269.1 8
9.2 odd 6 126.2.k.a.89.1 yes 8
9.4 even 3 1134.2.l.f.215.2 8
9.5 odd 6 1134.2.l.f.215.3 8
9.7 even 3 126.2.k.a.89.4 yes 8
21.17 even 6 1134.2.l.f.269.4 8
36.7 odd 6 1008.2.bt.c.593.4 8
36.11 even 6 1008.2.bt.c.593.1 8
45.2 even 12 3150.2.bp.e.1349.1 8
45.7 odd 12 3150.2.bp.b.1349.1 8
45.29 odd 6 3150.2.bf.a.1601.4 8
45.34 even 6 3150.2.bf.a.1601.2 8
45.38 even 12 3150.2.bp.b.1349.4 8
45.43 odd 12 3150.2.bp.e.1349.4 8
63.2 odd 6 882.2.d.a.881.8 8
63.11 odd 6 882.2.k.a.521.3 8
63.16 even 3 882.2.d.a.881.1 8
63.20 even 6 882.2.k.a.215.2 8
63.25 even 3 882.2.k.a.521.2 8
63.31 odd 6 inner 1134.2.t.e.1025.4 8
63.34 odd 6 882.2.k.a.215.3 8
63.38 even 6 126.2.k.a.17.4 yes 8
63.47 even 6 882.2.d.a.881.5 8
63.52 odd 6 126.2.k.a.17.1 ✓ 8
63.59 even 6 inner 1134.2.t.e.1025.1 8
63.61 odd 6 882.2.d.a.881.4 8
252.47 odd 6 7056.2.k.f.881.1 8
252.79 odd 6 7056.2.k.f.881.2 8
252.115 even 6 1008.2.bt.c.17.1 8
252.187 even 6 7056.2.k.f.881.8 8
252.191 even 6 7056.2.k.f.881.7 8
252.227 odd 6 1008.2.bt.c.17.4 8
315.38 odd 12 3150.2.bp.b.899.1 8
315.52 even 12 3150.2.bp.b.899.4 8
315.164 even 6 3150.2.bf.a.1151.2 8
315.178 even 12 3150.2.bp.e.899.1 8
315.227 odd 12 3150.2.bp.e.899.4 8
315.304 odd 6 3150.2.bf.a.1151.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
126.2.k.a.17.1 ✓ 8 63.52 odd 6
126.2.k.a.17.4 yes 8 63.38 even 6
126.2.k.a.89.1 yes 8 9.2 odd 6
126.2.k.a.89.4 yes 8 9.7 even 3
882.2.d.a.881.1 8 63.16 even 3
882.2.d.a.881.4 8 63.61 odd 6
882.2.d.a.881.5 8 63.47 even 6
882.2.d.a.881.8 8 63.2 odd 6
882.2.k.a.215.2 8 63.20 even 6
882.2.k.a.215.3 8 63.34 odd 6
882.2.k.a.521.2 8 63.25 even 3
882.2.k.a.521.3 8 63.11 odd 6
1008.2.bt.c.17.1 8 252.115 even 6
1008.2.bt.c.17.4 8 252.227 odd 6
1008.2.bt.c.593.1 8 36.11 even 6
1008.2.bt.c.593.4 8 36.7 odd 6
1134.2.l.f.215.2 8 9.4 even 3
1134.2.l.f.215.3 8 9.5 odd 6
1134.2.l.f.269.1 8 7.3 odd 6
1134.2.l.f.269.4 8 21.17 even 6
1134.2.t.e.593.1 8 1.1 even 1 trivial
1134.2.t.e.593.4 8 3.2 odd 2 inner
1134.2.t.e.1025.1 8 63.59 even 6 inner
1134.2.t.e.1025.4 8 63.31 odd 6 inner
3150.2.bf.a.1151.2 8 315.164 even 6
3150.2.bf.a.1151.4 8 315.304 odd 6
3150.2.bf.a.1601.2 8 45.34 even 6
3150.2.bf.a.1601.4 8 45.29 odd 6
3150.2.bp.b.899.1 8 315.38 odd 12
3150.2.bp.b.899.4 8 315.52 even 12
3150.2.bp.b.1349.1 8 45.7 odd 12
3150.2.bp.b.1349.4 8 45.38 even 12
3150.2.bp.e.899.1 8 315.178 even 12
3150.2.bp.e.899.4 8 315.227 odd 12
3150.2.bp.e.1349.1 8 45.2 even 12
3150.2.bp.e.1349.4 8 45.43 odd 12
7056.2.k.f.881.1 8 252.47 odd 6
7056.2.k.f.881.2 8 252.79 odd 6
7056.2.k.f.881.7 8 252.191 even 6
7056.2.k.f.881.8 8 252.187 even 6