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Properties

Label 1296.2.s.l.431.4
Level $1296$
Weight $2$
Character 1296.431
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 431.4
Root \(0.258819 - 0.965926i\) of defining polynomial
Character \(\chi\) \(=\) 1296.431
Dual form 1296.2.s.l.863.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(2.89778 - 1.67303i) q^{5} +(4.09808 + 2.36603i) q^{7} +(2.12132 - 3.67423i) q^{11} +(-0.500000 - 0.866025i) q^{13} -3.34607i q^{17} -1.26795i q^{19} +(3.67423 + 6.36396i) q^{23} +(3.09808 - 5.36603i) q^{25} +(-3.46618 - 2.00120i) q^{29} +(-5.19615 + 3.00000i) q^{31} +15.8338 q^{35} -9.19615 q^{37} +(-6.36396 + 3.67423i) q^{41} +(-1.90192 - 1.09808i) q^{43} +(-1.55291 + 2.68973i) q^{47} +(7.69615 + 13.3301i) q^{49} -7.34847i q^{53} -14.1962i q^{55} +(1.55291 + 2.68973i) q^{59} +(-3.59808 + 6.23205i) q^{61} +(-2.89778 - 1.67303i) q^{65} +(-6.29423 + 3.63397i) q^{67} +15.8338 q^{71} -1.19615 q^{73} +(17.3867 - 10.0382i) q^{77} +(1.09808 + 0.633975i) q^{79} +(4.24264 - 7.34847i) q^{83} +(-5.59808 - 9.69615i) q^{85} -9.38186i q^{89} -4.73205i q^{91} +(-2.12132 - 3.67423i) q^{95} +(3.19615 - 5.53590i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 12 q^{7} - 4 q^{13} + 4 q^{25} - 32 q^{37} - 36 q^{43} + 20 q^{49} - 8 q^{61} + 12 q^{67} + 32 q^{73} - 12 q^{79} - 24 q^{85} - 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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 0
\(3\) 0 0
\(4\) 0 0
\(5\) 2.89778 − 1.67303i 1.29593 − 0.748203i 0.316228 − 0.948683i \(-0.397584\pi\)
0.979698 + 0.200480i \(0.0642503\pi\)
\(6\) 0 0
\(7\) 4.09808 + 2.36603i 1.54893 + 0.894274i 0.998224 + 0.0595724i \(0.0189737\pi\)
0.550703 + 0.834701i \(0.314360\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.12132 − 3.67423i 0.639602 − 1.10782i −0.345918 − 0.938265i \(-0.612432\pi\)
0.985520 − 0.169559i \(-0.0542342\pi\)
\(12\) 0 0
\(13\) −0.500000 − 0.866025i −0.138675 − 0.240192i 0.788320 − 0.615265i \(-0.210951\pi\)
−0.926995 + 0.375073i \(0.877618\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 3.34607i − 0.811540i −0.913975 − 0.405770i \(-0.867003\pi\)
0.913975 − 0.405770i \(-0.132997\pi\)
\(18\) 0 0
\(19\) − 1.26795i − 0.290887i −0.989367 − 0.145444i \(-0.953539\pi\)
0.989367 − 0.145444i \(-0.0464610\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.67423 + 6.36396i 0.766131 + 1.32698i 0.939647 + 0.342147i \(0.111154\pi\)
−0.173516 + 0.984831i \(0.555513\pi\)
\(24\) 0 0
\(25\) 3.09808 − 5.36603i 0.619615 − 1.07321i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −3.46618 − 2.00120i −0.643654 − 0.371614i 0.142367 − 0.989814i \(-0.454529\pi\)
−0.786021 + 0.618200i \(0.787862\pi\)
\(30\) 0 0
\(31\) −5.19615 + 3.00000i −0.933257 + 0.538816i −0.887840 − 0.460152i \(-0.847795\pi\)
−0.0454165 + 0.998968i \(0.514461\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 15.8338 2.67639
\(36\) 0 0
\(37\) −9.19615 −1.51184 −0.755919 − 0.654665i \(-0.772810\pi\)
−0.755919 + 0.654665i \(0.772810\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −6.36396 + 3.67423i −0.993884 + 0.573819i −0.906433 − 0.422350i \(-0.861205\pi\)
−0.0874508 + 0.996169i \(0.527872\pi\)
\(42\) 0 0
\(43\) −1.90192 − 1.09808i −0.290041 − 0.167455i 0.347920 − 0.937524i \(-0.386888\pi\)
−0.637960 + 0.770069i \(0.720222\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −1.55291 + 2.68973i −0.226516 + 0.392337i −0.956773 − 0.290835i \(-0.906067\pi\)
0.730257 + 0.683172i \(0.239400\pi\)
\(48\) 0 0
\(49\) 7.69615 + 13.3301i 1.09945 + 1.90430i
\(50\) 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
\(55\) − 14.1962i − 1.91421i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 1.55291 + 2.68973i 0.202172 + 0.350173i 0.949228 − 0.314589i \(-0.101867\pi\)
−0.747056 + 0.664761i \(0.768533\pi\)
\(60\) 0 0
\(61\) −3.59808 + 6.23205i −0.460686 + 0.797932i −0.998995 − 0.0448153i \(-0.985730\pi\)
0.538309 + 0.842748i \(0.319063\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −2.89778 − 1.67303i −0.359425 − 0.207514i
\(66\) 0 0
\(67\) −6.29423 + 3.63397i −0.768962 + 0.443961i −0.832504 − 0.554019i \(-0.813094\pi\)
0.0635419 + 0.997979i \(0.479760\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 15.8338 1.87912 0.939560 − 0.342384i \(-0.111234\pi\)
0.939560 + 0.342384i \(0.111234\pi\)
\(72\) 0 0
\(73\) −1.19615 −0.139999 −0.0699995 − 0.997547i \(-0.522300\pi\)
−0.0699995 + 0.997547i \(0.522300\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 17.3867 − 10.0382i 1.98139 − 1.14396i
\(78\) 0 0
\(79\) 1.09808 + 0.633975i 0.123543 + 0.0713277i 0.560498 − 0.828156i \(-0.310610\pi\)
−0.436955 + 0.899483i \(0.643943\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 4.24264 − 7.34847i 0.465690 − 0.806599i −0.533542 − 0.845774i \(-0.679139\pi\)
0.999232 + 0.0391742i \(0.0124727\pi\)
\(84\) 0 0
\(85\) −5.59808 − 9.69615i −0.607197 − 1.05170i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 9.38186i − 0.994475i −0.867615 − 0.497237i \(-0.834348\pi\)
0.867615 − 0.497237i \(-0.165652\pi\)
\(90\) 0 0
\(91\) − 4.73205i − 0.496054i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −2.12132 − 3.67423i −0.217643 − 0.376969i
\(96\) 0 0
\(97\) 3.19615 − 5.53590i 0.324520 − 0.562085i −0.656895 − 0.753982i \(-0.728131\pi\)
0.981415 + 0.191897i \(0.0614639\pi\)
\(98\) 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.l.431.4 8
3.2 odd 2 inner 1296.2.s.l.431.1 8
4.3 odd 2 1296.2.s.j.431.4 8
9.2 odd 6 1296.2.c.g.1295.1 ✓ 8
9.4 even 3 1296.2.s.j.863.1 8
9.5 odd 6 1296.2.s.j.863.4 8
9.7 even 3 1296.2.c.g.1295.7 yes 8
12.11 even 2 1296.2.s.j.431.1 8
36.7 odd 6 1296.2.c.g.1295.8 yes 8
36.11 even 6 1296.2.c.g.1295.2 yes 8
36.23 even 6 inner 1296.2.s.l.863.4 8
36.31 odd 6 inner 1296.2.s.l.863.1 8
72.11 even 6 5184.2.c.h.5183.8 8
72.29 odd 6 5184.2.c.h.5183.7 8
72.43 odd 6 5184.2.c.h.5183.2 8
72.61 even 6 5184.2.c.h.5183.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1296.2.c.g.1295.1 ✓ 8 9.2 odd 6
1296.2.c.g.1295.2 yes 8 36.11 even 6
1296.2.c.g.1295.7 yes 8 9.7 even 3
1296.2.c.g.1295.8 yes 8 36.7 odd 6
1296.2.s.j.431.1 8 12.11 even 2
1296.2.s.j.431.4 8 4.3 odd 2
1296.2.s.j.863.1 8 9.4 even 3
1296.2.s.j.863.4 8 9.5 odd 6
1296.2.s.l.431.1 8 3.2 odd 2 inner
1296.2.s.l.431.4 8 1.1 even 1 trivial
1296.2.s.l.863.1 8 36.31 odd 6 inner
1296.2.s.l.863.4 8 36.23 even 6 inner
5184.2.c.h.5183.1 8 72.61 even 6
5184.2.c.h.5183.2 8 72.43 odd 6
5184.2.c.h.5183.7 8 72.29 odd 6
5184.2.c.h.5183.8 8 72.11 even 6