Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2592,2,Mod(431,2592)] 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("2592.431"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2592, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2592 = 2^{5} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2592.p (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,-12,0,0,0,0,0,12,0,0,0,0,0,16,0,0,0,0,0,-2,0,0,0, 0,0,-12,0,0,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(41)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(20.6972242039\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 72)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 2159.1
Root \(1.22474 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 2592.2159
Dual form 2592.2.p.a.431.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.22474 + 2.12132i) q^{5} +(-3.00000 + 1.73205i) q^{7} +(2.44949 - 1.41421i) q^{11} +(3.00000 + 1.73205i) q^{13} +1.41421i q^{17} +4.00000 q^{19} +(-2.44949 + 4.24264i) q^{23} +(-0.500000 - 0.866025i) q^{25} +(1.22474 + 2.12132i) q^{29} +(-3.00000 - 1.73205i) q^{31} -8.48528i q^{35} +(-1.22474 - 0.707107i) q^{41} +(4.00000 + 6.92820i) q^{43} +(2.44949 + 4.24264i) q^{47} +(2.50000 - 4.33013i) q^{49} -7.34847 q^{53} +6.92820i q^{55} +(-9.79796 - 5.65685i) q^{59} +(12.0000 - 6.92820i) q^{61} +(-7.34847 + 4.24264i) q^{65} +(-2.00000 + 3.46410i) q^{67} -14.6969 q^{71} -4.00000 q^{73} +(-4.89898 + 8.48528i) q^{77} +(-3.00000 + 1.73205i) q^{79} +(-12.2474 + 7.07107i) q^{83} +(-3.00000 - 1.73205i) q^{85} -7.07107i q^{89} -12.0000 q^{91} +(-4.89898 + 8.48528i) q^{95} +(-4.00000 - 6.92820i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{7} + 12 q^{13} + 16 q^{19} - 2 q^{25} - 12 q^{31} + 16 q^{43} + 10 q^{49} + 48 q^{61} - 8 q^{67} - 16 q^{73} - 12 q^{79} - 12 q^{85} - 48 q^{91} - 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2592\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1217\) \(2431\)
\(\chi(n)\) \(-1\) \(e\left(\frac{5}{6}\right)\) \(-1\)

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 0
\(3\) 0 0
\(4\) 0 0
\(5\) −1.22474 + 2.12132i −0.547723 + 0.948683i 0.450708 + 0.892672i \(0.351172\pi\)
−0.998430 + 0.0560116i \(0.982162\pi\)
\(6\) 0 0
\(7\) −3.00000 + 1.73205i −1.13389 + 0.654654i −0.944911 − 0.327327i \(-0.893852\pi\)
−0.188982 + 0.981981i \(0.560519\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.44949 − 1.41421i 0.738549 − 0.426401i −0.0829925 − 0.996550i \(-0.526448\pi\)
0.821541 + 0.570149i \(0.193114\pi\)
\(12\) 0 0
\(13\) 3.00000 + 1.73205i 0.832050 + 0.480384i 0.854554 − 0.519362i \(-0.173830\pi\)
−0.0225039 + 0.999747i \(0.507164\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.41421i 0.342997i 0.985184 + 0.171499i \(0.0548609\pi\)
−0.985184 + 0.171499i \(0.945139\pi\)
\(18\) 0 0
\(19\) 4.00000 0.917663 0.458831 − 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −2.44949 + 4.24264i −0.510754 + 0.884652i 0.489168 + 0.872189i \(0.337300\pi\)
−0.999922 + 0.0124624i \(0.996033\pi\)
\(24\) 0 0
\(25\) −0.500000 − 0.866025i −0.100000 − 0.173205i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.22474 + 2.12132i 0.227429 + 0.393919i 0.957046 − 0.289938i \(-0.0936346\pi\)
−0.729616 + 0.683857i \(0.760301\pi\)
\(30\) 0 0
\(31\) −3.00000 − 1.73205i −0.538816 − 0.311086i 0.205783 − 0.978598i \(-0.434026\pi\)
−0.744599 + 0.667512i \(0.767359\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 8.48528i − 1.43427i
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −1.22474 − 0.707107i −0.191273 − 0.110432i 0.401305 − 0.915944i \(-0.368557\pi\)
−0.592578 + 0.805513i \(0.701890\pi\)
\(42\) 0 0
\(43\) 4.00000 + 6.92820i 0.609994 + 1.05654i 0.991241 + 0.132068i \(0.0421616\pi\)
−0.381246 + 0.924473i \(0.624505\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.44949 + 4.24264i 0.357295 + 0.618853i 0.987508 − 0.157569i \(-0.0503658\pi\)
−0.630213 + 0.776422i \(0.717032\pi\)
\(48\) 0 0
\(49\) 2.50000 − 4.33013i 0.357143 − 0.618590i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −7.34847 −1.00939 −0.504695 − 0.863298i \(-0.668395\pi\)
−0.504695 + 0.863298i \(0.668395\pi\)
\(54\) 0 0
\(55\) 6.92820i 0.934199i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −9.79796 − 5.65685i −1.27559 − 0.736460i −0.299552 − 0.954080i \(-0.596837\pi\)
−0.976034 + 0.217620i \(0.930171\pi\)
\(60\) 0 0
\(61\) 12.0000 − 6.92820i 1.53644 − 0.887066i 0.537400 − 0.843328i \(-0.319407\pi\)
0.999043 − 0.0437377i \(-0.0139266\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −7.34847 + 4.24264i −0.911465 + 0.526235i
\(66\) 0 0
\(67\) −2.00000 + 3.46410i −0.244339 + 0.423207i −0.961946 − 0.273241i \(-0.911904\pi\)
0.717607 + 0.696449i \(0.245238\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −14.6969 −1.74421 −0.872103 − 0.489323i \(-0.837244\pi\)
−0.872103 + 0.489323i \(0.837244\pi\)
\(72\) 0 0
\(73\) −4.00000 −0.468165 −0.234082 − 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −4.89898 + 8.48528i −0.558291 + 0.966988i
\(78\) 0 0
\(79\) −3.00000 + 1.73205i −0.337526 + 0.194871i −0.659178 − 0.751987i \(-0.729095\pi\)
0.321651 + 0.946858i \(0.395762\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −12.2474 + 7.07107i −1.34433 + 0.776151i −0.987440 − 0.157995i \(-0.949497\pi\)
−0.356892 + 0.934146i \(0.616164\pi\)
\(84\) 0 0
\(85\) −3.00000 − 1.73205i −0.325396 − 0.187867i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 7.07107i − 0.749532i −0.927119 − 0.374766i \(-0.877723\pi\)
0.927119 − 0.374766i \(-0.122277\pi\)
\(90\) 0 0
\(91\) −12.0000 −1.25794
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −4.89898 + 8.48528i −0.502625 + 0.870572i
\(96\) 0 0
\(97\) −4.00000 − 6.92820i −0.406138 − 0.703452i 0.588315 − 0.808632i \(-0.299792\pi\)
−0.994453 + 0.105180i \(0.966458\pi\)
\(98\) 0 0
\(99\) 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 2592.2.p.a.2159.1 4
3.2 odd 2 inner 2592.2.p.a.2159.2 4
4.3 odd 2 648.2.l.c.539.2 4
8.3 odd 2 2592.2.p.c.2159.2 4
8.5 even 2 648.2.l.a.539.1 4
9.2 odd 6 2592.2.p.c.431.2 4
9.4 even 3 288.2.f.a.143.3 4
9.5 odd 6 288.2.f.a.143.1 4
9.7 even 3 2592.2.p.c.431.1 4
12.11 even 2 648.2.l.c.539.1 4
24.5 odd 2 648.2.l.a.539.2 4
24.11 even 2 2592.2.p.c.2159.1 4
36.7 odd 6 648.2.l.a.107.1 4
36.11 even 6 648.2.l.a.107.2 4
36.23 even 6 72.2.f.a.35.3 yes 4
36.31 odd 6 72.2.f.a.35.2 yes 4
45.4 even 6 7200.2.b.c.4751.4 4
45.13 odd 12 7200.2.m.c.3599.4 8
45.14 odd 6 7200.2.b.c.4751.3 4
45.22 odd 12 7200.2.m.c.3599.7 8
45.23 even 12 7200.2.m.c.3599.1 8
45.32 even 12 7200.2.m.c.3599.6 8
72.5 odd 6 72.2.f.a.35.1 ✓ 4
72.11 even 6 inner 2592.2.p.a.431.1 4
72.13 even 6 72.2.f.a.35.4 yes 4
72.29 odd 6 648.2.l.c.107.2 4
72.43 odd 6 inner 2592.2.p.a.431.2 4
72.59 even 6 288.2.f.a.143.4 4
72.61 even 6 648.2.l.c.107.1 4
72.67 odd 6 288.2.f.a.143.2 4
144.5 odd 12 2304.2.c.i.2303.3 8
144.13 even 12 2304.2.c.i.2303.4 8
144.59 even 12 2304.2.c.i.2303.1 8
144.67 odd 12 2304.2.c.i.2303.2 8
144.77 odd 12 2304.2.c.i.2303.8 8
144.85 even 12 2304.2.c.i.2303.7 8
144.131 even 12 2304.2.c.i.2303.6 8
144.139 odd 12 2304.2.c.i.2303.5 8
180.23 odd 12 1800.2.m.c.899.2 8
180.59 even 6 1800.2.b.c.251.2 4
180.67 even 12 1800.2.m.c.899.1 8
180.103 even 12 1800.2.m.c.899.8 8
180.139 odd 6 1800.2.b.c.251.3 4
180.167 odd 12 1800.2.m.c.899.7 8
360.13 odd 12 1800.2.m.c.899.5 8
360.59 even 6 7200.2.b.c.4751.1 4
360.67 even 12 7200.2.m.c.3599.3 8
360.77 even 12 1800.2.m.c.899.6 8
360.139 odd 6 7200.2.b.c.4751.2 4
360.149 odd 6 1800.2.b.c.251.4 4
360.157 odd 12 1800.2.m.c.899.4 8
360.203 odd 12 7200.2.m.c.3599.5 8
360.229 even 6 1800.2.b.c.251.1 4
360.283 even 12 7200.2.m.c.3599.8 8
360.293 even 12 1800.2.m.c.899.3 8
360.347 odd 12 7200.2.m.c.3599.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
72.2.f.a.35.1 ✓ 4 72.5 odd 6
72.2.f.a.35.2 yes 4 36.31 odd 6
72.2.f.a.35.3 yes 4 36.23 even 6
72.2.f.a.35.4 yes 4 72.13 even 6
288.2.f.a.143.1 4 9.5 odd 6
288.2.f.a.143.2 4 72.67 odd 6
288.2.f.a.143.3 4 9.4 even 3
288.2.f.a.143.4 4 72.59 even 6
648.2.l.a.107.1 4 36.7 odd 6
648.2.l.a.107.2 4 36.11 even 6
648.2.l.a.539.1 4 8.5 even 2
648.2.l.a.539.2 4 24.5 odd 2
648.2.l.c.107.1 4 72.61 even 6
648.2.l.c.107.2 4 72.29 odd 6
648.2.l.c.539.1 4 12.11 even 2
648.2.l.c.539.2 4 4.3 odd 2
1800.2.b.c.251.1 4 360.229 even 6
1800.2.b.c.251.2 4 180.59 even 6
1800.2.b.c.251.3 4 180.139 odd 6
1800.2.b.c.251.4 4 360.149 odd 6
1800.2.m.c.899.1 8 180.67 even 12
1800.2.m.c.899.2 8 180.23 odd 12
1800.2.m.c.899.3 8 360.293 even 12
1800.2.m.c.899.4 8 360.157 odd 12
1800.2.m.c.899.5 8 360.13 odd 12
1800.2.m.c.899.6 8 360.77 even 12
1800.2.m.c.899.7 8 180.167 odd 12
1800.2.m.c.899.8 8 180.103 even 12
2304.2.c.i.2303.1 8 144.59 even 12
2304.2.c.i.2303.2 8 144.67 odd 12
2304.2.c.i.2303.3 8 144.5 odd 12
2304.2.c.i.2303.4 8 144.13 even 12
2304.2.c.i.2303.5 8 144.139 odd 12
2304.2.c.i.2303.6 8 144.131 even 12
2304.2.c.i.2303.7 8 144.85 even 12
2304.2.c.i.2303.8 8 144.77 odd 12
2592.2.p.a.431.1 4 72.11 even 6 inner
2592.2.p.a.431.2 4 72.43 odd 6 inner
2592.2.p.a.2159.1 4 1.1 even 1 trivial
2592.2.p.a.2159.2 4 3.2 odd 2 inner
2592.2.p.c.431.1 4 9.7 even 3
2592.2.p.c.431.2 4 9.2 odd 6
2592.2.p.c.2159.1 4 24.11 even 2
2592.2.p.c.2159.2 4 8.3 odd 2
7200.2.b.c.4751.1 4 360.59 even 6
7200.2.b.c.4751.2 4 360.139 odd 6
7200.2.b.c.4751.3 4 45.14 odd 6
7200.2.b.c.4751.4 4 45.4 even 6
7200.2.m.c.3599.1 8 45.23 even 12
7200.2.m.c.3599.2 8 360.347 odd 12
7200.2.m.c.3599.3 8 360.67 even 12
7200.2.m.c.3599.4 8 45.13 odd 12
7200.2.m.c.3599.5 8 360.203 odd 12
7200.2.m.c.3599.6 8 45.32 even 12
7200.2.m.c.3599.7 8 45.22 odd 12
7200.2.m.c.3599.8 8 360.283 even 12