Properties

Label 1296.3.e.g
Level $1296$
Weight $3$
Character orbit 1296.e
Analytic conductor $35.313$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1296 = 2^{4} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1296.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(35.3134422611\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{3} \)
Twist minimal: no (minimal twist has level 18)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{5} + (\beta_{2} + 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{5} + (\beta_{2} + 1) q^{7} + ( - \beta_{3} + \beta_1) q^{11} + ( - 2 \beta_{2} + 5) q^{13} + (2 \beta_{3} + 2 \beta_1) q^{17} + (2 \beta_{2} + 10) q^{19} + (\beta_{3} - \beta_1) q^{23} - 2 q^{25} + (\beta_{3} + 2 \beta_1) q^{29} + ( - 3 \beta_{2} + 19) q^{31} + (\beta_{3} + 9 \beta_1) q^{35} + (2 \beta_{2} + 32) q^{37} + (7 \beta_{3} - 6 \beta_1) q^{41} + ( - 3 \beta_{2} - 23) q^{43} + ( - 3 \beta_{3} - 7 \beta_1) q^{47} + (2 \beta_{2} + 6) q^{49} + (10 \beta_{3} + 10 \beta_1) q^{53} + ( - 3 \beta_{2} + 27) q^{55} + (7 \beta_{3} - 13 \beta_1) q^{59} + (6 \beta_{2} - 31) q^{61} + (5 \beta_{3} - 18 \beta_1) q^{65} + ( - 3 \beta_{2} - 53) q^{67} + (10 \beta_{3} - 8 \beta_1) q^{71} + (6 \beta_{2} - 52) q^{73} + (5 \beta_{3} - 8 \beta_1) q^{77} + (5 \beta_{2} + 7) q^{79} + (21 \beta_{3} + 5 \beta_1) q^{83} + ( - 6 \beta_{2} - 54) q^{85} + ( - 10 \beta_{3} - 22 \beta_1) q^{89} + (3 \beta_{2} - 103) q^{91} + (10 \beta_{3} + 18 \beta_1) q^{95} + ( - 14 \beta_{2} - 7) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{7} + 20 q^{13} + 40 q^{19} - 8 q^{25} + 76 q^{31} + 128 q^{37} - 92 q^{43} + 24 q^{49} + 108 q^{55} - 124 q^{61} - 212 q^{67} - 208 q^{73} + 28 q^{79} - 216 q^{85} - 412 q^{91} - 28 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 3\nu^{3} ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -3\nu^{3} + 12\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 3\nu^{2} - 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 3 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_1 ) / 3 \) Copy content Toggle raw display

Character values

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

\(n\) \(325\) \(1135\) \(1217\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
−1.22474 + 0.707107i
1.22474 0.707107i
−1.22474 0.707107i
1.22474 + 0.707107i
0 0 0 5.19615i 0 −6.34847 0 0 0
161.2 0 0 0 5.19615i 0 8.34847 0 0 0
161.3 0 0 0 5.19615i 0 −6.34847 0 0 0
161.4 0 0 0 5.19615i 0 8.34847 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1296.3.e.g 4
3.b odd 2 1 inner 1296.3.e.g 4
4.b odd 2 1 162.3.b.a 4
9.c even 3 1 144.3.q.c 4
9.c even 3 1 432.3.q.d 4
9.d odd 6 1 144.3.q.c 4
9.d odd 6 1 432.3.q.d 4
12.b even 2 1 162.3.b.a 4
36.f odd 6 1 18.3.d.a 4
36.f odd 6 1 54.3.d.a 4
36.h even 6 1 18.3.d.a 4
36.h even 6 1 54.3.d.a 4
72.j odd 6 1 576.3.q.e 4
72.j odd 6 1 1728.3.q.c 4
72.l even 6 1 576.3.q.f 4
72.l even 6 1 1728.3.q.d 4
72.n even 6 1 576.3.q.e 4
72.n even 6 1 1728.3.q.c 4
72.p odd 6 1 576.3.q.f 4
72.p odd 6 1 1728.3.q.d 4
180.n even 6 1 450.3.i.b 4
180.n even 6 1 1350.3.i.b 4
180.p odd 6 1 450.3.i.b 4
180.p odd 6 1 1350.3.i.b 4
180.v odd 12 2 450.3.k.a 8
180.v odd 12 2 1350.3.k.a 8
180.x even 12 2 450.3.k.a 8
180.x even 12 2 1350.3.k.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.3.d.a 4 36.f odd 6 1
18.3.d.a 4 36.h even 6 1
54.3.d.a 4 36.f odd 6 1
54.3.d.a 4 36.h even 6 1
144.3.q.c 4 9.c even 3 1
144.3.q.c 4 9.d odd 6 1
162.3.b.a 4 4.b odd 2 1
162.3.b.a 4 12.b even 2 1
432.3.q.d 4 9.c even 3 1
432.3.q.d 4 9.d odd 6 1
450.3.i.b 4 180.n even 6 1
450.3.i.b 4 180.p odd 6 1
450.3.k.a 8 180.v odd 12 2
450.3.k.a 8 180.x even 12 2
576.3.q.e 4 72.j odd 6 1
576.3.q.e 4 72.n even 6 1
576.3.q.f 4 72.l even 6 1
576.3.q.f 4 72.p odd 6 1
1296.3.e.g 4 1.a even 1 1 trivial
1296.3.e.g 4 3.b odd 2 1 inner
1350.3.i.b 4 180.n even 6 1
1350.3.i.b 4 180.p odd 6 1
1350.3.k.a 8 180.v odd 12 2
1350.3.k.a 8 180.x even 12 2
1728.3.q.c 4 72.j odd 6 1
1728.3.q.c 4 72.n even 6 1
1728.3.q.d 4 72.l even 6 1
1728.3.q.d 4 72.p odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1296, [\chi])\):

\( T_{5}^{2} + 27 \) Copy content Toggle raw display
\( T_{7}^{2} - 2T_{7} - 53 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + 27)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 2 T - 53)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 90T^{2} + 81 \) Copy content Toggle raw display
$13$ \( (T^{2} - 10 T - 191)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 360T^{2} + 1296 \) Copy content Toggle raw display
$19$ \( (T^{2} - 20 T - 116)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 90T^{2} + 81 \) Copy content Toggle raw display
$29$ \( T^{4} + 198T^{2} + 2025 \) Copy content Toggle raw display
$31$ \( (T^{2} - 38 T - 125)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 64 T + 808)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 3942 T^{2} + 455625 \) Copy content Toggle raw display
$43$ \( (T^{2} + 46 T + 43)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 2250 T^{2} + 408321 \) Copy content Toggle raw display
$53$ \( T^{4} + 9000 T^{2} + 810000 \) Copy content Toggle raw display
$59$ \( T^{4} + 8730 T^{2} + \cdots + 2954961 \) Copy content Toggle raw display
$61$ \( (T^{2} + 62 T - 983)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 106 T + 2323)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 7704 T^{2} + \cdots + 2396304 \) Copy content Toggle raw display
$73$ \( (T^{2} + 104 T + 760)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 14 T - 1301)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 24714 T^{2} + \cdots + 131262849 \) Copy content Toggle raw display
$89$ \( T^{4} + 22824 T^{2} + \cdots + 36144144 \) Copy content Toggle raw display
$97$ \( (T^{2} + 14 T - 10535)^{2} \) Copy content Toggle raw display
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