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Properties

Label 1296.2.s.j.863.3
Level $1296$
Weight $2$
Character 1296.863
Analytic conductor $10.349$
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] = [1296,2,Mod(431,1296)] 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("1296.431"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1296, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1296 = 2^{4} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1296.s (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.3486121020\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 863.3
Root \(0.965926 - 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 1296.863
Dual form 1296.2.s.j.431.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.776457 + 0.448288i) q^{5} +(1.09808 - 0.633975i) q^{7} +(2.12132 + 3.67423i) q^{11} +(-0.500000 + 0.866025i) q^{13} +0.896575i q^{17} +4.73205i q^{19} +(-3.67423 + 6.36396i) q^{23} +(-2.09808 - 3.63397i) q^{25} +(7.14042 - 4.12252i) q^{29} +(-5.19615 - 3.00000i) q^{31} +1.13681 q^{35} +1.19615 q^{37} +(6.36396 + 3.67423i) q^{41} +(7.09808 - 4.09808i) q^{43} +(5.79555 + 10.0382i) q^{47} +(-2.69615 + 4.66987i) q^{49} -7.34847i q^{53} +3.80385i q^{55} +(-5.79555 + 10.0382i) q^{59} +(1.59808 + 2.76795i) q^{61} +(-0.776457 + 0.448288i) q^{65} +(-9.29423 - 5.36603i) q^{67} +1.13681 q^{71} +9.19615 q^{73} +(4.65874 + 2.68973i) q^{77} +(4.09808 - 2.36603i) q^{79} +(4.24264 + 7.34847i) q^{83} +(-0.401924 + 0.696152i) q^{85} +11.8313i q^{89} +1.26795i q^{91} +(-2.12132 + 3.67423i) q^{95} +(-7.19615 - 12.4641i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 12 q^{7} - 4 q^{13} + 4 q^{25} - 32 q^{37} + 36 q^{43} + 20 q^{49} - 8 q^{61} - 12 q^{67} + 32 q^{73} + 12 q^{79} - 24 q^{85} - 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1296\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1135\) \(1217\)
\(\chi(n)\) \(1\) \(-1\) \(e\left(\frac{5}{6}\right)\)

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 0
\(3\) 0 0
\(4\) 0 0
\(5\) 0.776457 + 0.448288i 0.347242 + 0.200480i 0.663470 − 0.748203i \(-0.269083\pi\)
−0.316228 + 0.948683i \(0.602416\pi\)
\(6\) 0 0
\(7\) 1.09808 − 0.633975i 0.415034 − 0.239620i −0.277916 − 0.960605i \(-0.589644\pi\)
0.692950 + 0.720985i \(0.256311\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.12132 + 3.67423i 0.639602 + 1.10782i 0.985520 + 0.169559i \(0.0542342\pi\)
−0.345918 + 0.938265i \(0.612432\pi\)
\(12\) 0 0
\(13\) −0.500000 + 0.866025i −0.138675 + 0.240192i −0.926995 − 0.375073i \(-0.877618\pi\)
0.788320 + 0.615265i \(0.210951\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.896575i 0.217451i 0.994072 + 0.108726i \(0.0346770\pi\)
−0.994072 + 0.108726i \(0.965323\pi\)
\(18\) 0 0
\(19\) 4.73205i 1.08561i 0.839860 + 0.542803i \(0.182637\pi\)
−0.839860 + 0.542803i \(0.817363\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3.67423 + 6.36396i −0.766131 + 1.32698i 0.173516 + 0.984831i \(0.444487\pi\)
−0.939647 + 0.342147i \(0.888846\pi\)
\(24\) 0 0
\(25\) −2.09808 − 3.63397i −0.419615 − 0.726795i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 7.14042 − 4.12252i 1.32594 − 0.765533i 0.341273 − 0.939964i \(-0.389142\pi\)
0.984669 + 0.174431i \(0.0558086\pi\)
\(30\) 0 0
\(31\) −5.19615 − 3.00000i −0.933257 − 0.538816i −0.0454165 − 0.998968i \(-0.514461\pi\)
−0.887840 + 0.460152i \(0.847795\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.13681 0.192156
\(36\) 0 0
\(37\) 1.19615 0.196646 0.0983231 − 0.995155i \(-0.468652\pi\)
0.0983231 + 0.995155i \(0.468652\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.36396 + 3.67423i 0.993884 + 0.573819i 0.906433 − 0.422350i \(-0.138795\pi\)
0.0874508 + 0.996169i \(0.472128\pi\)
\(42\) 0 0
\(43\) 7.09808 − 4.09808i 1.08245 − 0.624951i 0.150891 − 0.988550i \(-0.451786\pi\)
0.931555 + 0.363600i \(0.118452\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 5.79555 + 10.0382i 0.845369 + 1.46422i 0.885301 + 0.465019i \(0.153952\pi\)
−0.0399322 + 0.999202i \(0.512714\pi\)
\(48\) 0 0
\(49\) −2.69615 + 4.66987i −0.385165 + 0.667125i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 7.34847i − 1.00939i −0.863298 − 0.504695i \(-0.831605\pi\)
0.863298 − 0.504695i \(-0.168395\pi\)
\(54\) 0 0
\(55\) 3.80385i 0.512911i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −5.79555 + 10.0382i −0.754517 + 1.30686i 0.191097 + 0.981571i \(0.438795\pi\)
−0.945614 + 0.325291i \(0.894538\pi\)
\(60\) 0 0
\(61\) 1.59808 + 2.76795i 0.204613 + 0.354400i 0.950009 − 0.312222i \(-0.101073\pi\)
−0.745397 + 0.666621i \(0.767740\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −0.776457 + 0.448288i −0.0963077 + 0.0556033i
\(66\) 0 0
\(67\) −9.29423 − 5.36603i −1.13547 − 0.655564i −0.190166 − 0.981752i \(-0.560903\pi\)
−0.945305 + 0.326188i \(0.894236\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 1.13681 0.134915 0.0674574 − 0.997722i \(-0.478511\pi\)
0.0674574 + 0.997722i \(0.478511\pi\)
\(72\) 0 0
\(73\) 9.19615 1.07633 0.538164 − 0.842840i \(-0.319118\pi\)
0.538164 + 0.842840i \(0.319118\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 4.65874 + 2.68973i 0.530913 + 0.306523i
\(78\) 0 0
\(79\) 4.09808 − 2.36603i 0.461070 − 0.266199i −0.251424 − 0.967877i \(-0.580899\pi\)
0.712494 + 0.701678i \(0.247566\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 4.24264 + 7.34847i 0.465690 + 0.806599i 0.999232 − 0.0391742i \(-0.0124727\pi\)
−0.533542 + 0.845774i \(0.679139\pi\)
\(84\) 0 0
\(85\) −0.401924 + 0.696152i −0.0435948 + 0.0755083i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 11.8313i 1.25412i 0.778971 + 0.627060i \(0.215742\pi\)
−0.778971 + 0.627060i \(0.784258\pi\)
\(90\) 0 0
\(91\) 1.26795i 0.132917i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −2.12132 + 3.67423i −0.217643 + 0.376969i
\(96\) 0 0
\(97\) −7.19615 − 12.4641i −0.730659 − 1.26554i −0.956602 − 0.291397i \(-0.905880\pi\)
0.225944 − 0.974140i \(-0.427454\pi\)
\(98\) 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 1296.2.s.j.863.3 8
3.2 odd 2 inner 1296.2.s.j.863.2 8
4.3 odd 2 1296.2.s.l.863.3 8
9.2 odd 6 1296.2.s.l.431.3 8
9.4 even 3 1296.2.c.g.1295.4 yes 8
9.5 odd 6 1296.2.c.g.1295.6 yes 8
9.7 even 3 1296.2.s.l.431.2 8
12.11 even 2 1296.2.s.l.863.2 8
36.7 odd 6 inner 1296.2.s.j.431.2 8
36.11 even 6 inner 1296.2.s.j.431.3 8
36.23 even 6 1296.2.c.g.1295.5 yes 8
36.31 odd 6 1296.2.c.g.1295.3 ✓ 8
72.5 odd 6 5184.2.c.h.5183.4 8
72.13 even 6 5184.2.c.h.5183.6 8
72.59 even 6 5184.2.c.h.5183.3 8
72.67 odd 6 5184.2.c.h.5183.5 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1296.2.c.g.1295.3 ✓ 8 36.31 odd 6
1296.2.c.g.1295.4 yes 8 9.4 even 3
1296.2.c.g.1295.5 yes 8 36.23 even 6
1296.2.c.g.1295.6 yes 8 9.5 odd 6
1296.2.s.j.431.2 8 36.7 odd 6 inner
1296.2.s.j.431.3 8 36.11 even 6 inner
1296.2.s.j.863.2 8 3.2 odd 2 inner
1296.2.s.j.863.3 8 1.1 even 1 trivial
1296.2.s.l.431.2 8 9.7 even 3
1296.2.s.l.431.3 8 9.2 odd 6
1296.2.s.l.863.2 8 12.11 even 2
1296.2.s.l.863.3 8 4.3 odd 2
5184.2.c.h.5183.3 8 72.59 even 6
5184.2.c.h.5183.4 8 72.5 odd 6
5184.2.c.h.5183.5 8 72.67 odd 6
5184.2.c.h.5183.6 8 72.13 even 6