Properties

Label 1296.3.e.e
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: \(\Q(\sqrt{-3}, \sqrt{-11})\)
Defining polynomial: \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 36)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{5} + (\beta_{2} - 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{5} + (\beta_{2} - 1) q^{7} + ( - \beta_{3} + \beta_1) q^{11} + (\beta_{2} - 3) q^{13} + \beta_{3} q^{17} - \beta_{2} q^{19} + ( - 2 \beta_{3} - \beta_1) q^{23} + (3 \beta_{2} - 8) q^{25} + 7 \beta_1 q^{29} + (3 \beta_{2} - 5) q^{31} + ( - 2 \beta_{3} - 7 \beta_1) q^{35} + ( - 4 \beta_{2} - 14) q^{37} + (\beta_{3} - 7 \beta_1) q^{41} - 23 q^{43} + ( - 2 \beta_{3} + \beta_1) q^{47} + ( - \beta_{2} + 26) q^{49} - 8 \beta_1 q^{53} + ( - 3 \beta_{2} - 21) q^{55} + ( - \beta_{3} - 9 \beta_1) q^{59} + ( - 3 \beta_{2} - 19) q^{61} + ( - 2 \beta_{3} - 9 \beta_1) q^{65} + ( - 6 \beta_{2} + 61) q^{67} + 2 \beta_{3} q^{71} + (3 \beta_{2} + 20) q^{73} + ( - 8 \beta_{3} + 9 \beta_1) q^{77} + (5 \beta_{2} + 39) q^{79} + ( - 2 \beta_{3} + \beta_1) q^{83} + (6 \beta_{2} - 12) q^{85} - 8 \beta_{3} q^{89} + ( - 3 \beta_{2} + 77) q^{91} + (2 \beta_{3} + 6 \beta_1) q^{95} + ( - 2 \beta_{2} + 99) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{7} - 10 q^{13} - 2 q^{19} - 26 q^{25} - 14 q^{31} - 64 q^{37} - 92 q^{43} + 102 q^{49} - 90 q^{55} - 82 q^{61} + 232 q^{67} + 86 q^{73} + 166 q^{79} - 36 q^{85} + 302 q^{91} + 392 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( -\nu^{3} + \nu^{2} - \nu + 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{3} + \nu^{2} + 5\nu + 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 4\nu^{3} + 5\nu^{2} - 5\nu - 21 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - \beta _1 + 1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{3} + 3\beta_{2} + 5\beta _1 + 21 ) / 18 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{3} - 5\beta _1 + 36 ) / 9 \) 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
−1.18614 + 1.26217i
1.68614 + 0.396143i
1.68614 0.396143i
−1.18614 1.26217i
0 0 0 7.57301i 0 −9.11684 0 0 0
161.2 0 0 0 2.37686i 0 8.11684 0 0 0
161.3 0 0 0 2.37686i 0 8.11684 0 0 0
161.4 0 0 0 7.57301i 0 −9.11684 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.e 4
3.b odd 2 1 inner 1296.3.e.e 4
4.b odd 2 1 324.3.c.b 4
9.c even 3 1 144.3.q.b 4
9.c even 3 1 432.3.q.b 4
9.d odd 6 1 144.3.q.b 4
9.d odd 6 1 432.3.q.b 4
12.b even 2 1 324.3.c.b 4
36.f odd 6 1 36.3.g.a 4
36.f odd 6 1 108.3.g.a 4
36.h even 6 1 36.3.g.a 4
36.h even 6 1 108.3.g.a 4
72.j odd 6 1 576.3.q.g 4
72.j odd 6 1 1728.3.q.h 4
72.l even 6 1 576.3.q.d 4
72.l even 6 1 1728.3.q.g 4
72.n even 6 1 576.3.q.g 4
72.n even 6 1 1728.3.q.h 4
72.p odd 6 1 576.3.q.d 4
72.p odd 6 1 1728.3.q.g 4
180.n even 6 1 900.3.p.a 4
180.n even 6 1 2700.3.p.b 4
180.p odd 6 1 900.3.p.a 4
180.p odd 6 1 2700.3.p.b 4
180.v odd 12 2 900.3.u.a 8
180.v odd 12 2 2700.3.u.b 8
180.x even 12 2 900.3.u.a 8
180.x even 12 2 2700.3.u.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.3.g.a 4 36.f odd 6 1
36.3.g.a 4 36.h even 6 1
108.3.g.a 4 36.f odd 6 1
108.3.g.a 4 36.h even 6 1
144.3.q.b 4 9.c even 3 1
144.3.q.b 4 9.d odd 6 1
324.3.c.b 4 4.b odd 2 1
324.3.c.b 4 12.b even 2 1
432.3.q.b 4 9.c even 3 1
432.3.q.b 4 9.d odd 6 1
576.3.q.d 4 72.l even 6 1
576.3.q.d 4 72.p odd 6 1
576.3.q.g 4 72.j odd 6 1
576.3.q.g 4 72.n even 6 1
900.3.p.a 4 180.n even 6 1
900.3.p.a 4 180.p odd 6 1
900.3.u.a 8 180.v odd 12 2
900.3.u.a 8 180.x even 12 2
1296.3.e.e 4 1.a even 1 1 trivial
1296.3.e.e 4 3.b odd 2 1 inner
1728.3.q.g 4 72.l even 6 1
1728.3.q.g 4 72.p odd 6 1
1728.3.q.h 4 72.j odd 6 1
1728.3.q.h 4 72.n even 6 1
2700.3.p.b 4 180.n even 6 1
2700.3.p.b 4 180.p odd 6 1
2700.3.u.b 8 180.v odd 12 2
2700.3.u.b 8 180.x even 12 2

Hecke kernels

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

\( T_{5}^{4} + 63T_{5}^{2} + 324 \) Copy content Toggle raw display
\( T_{7}^{2} + T_{7} - 74 \) 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
$5$ \( T^{4} + 63T^{2} + 324 \) Copy content Toggle raw display
$7$ \( (T^{2} + T - 74)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 414T^{2} + 81 \) Copy content Toggle raw display
$13$ \( (T^{2} + 5 T - 68)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 387 T^{2} + 20736 \) Copy content Toggle raw display
$19$ \( (T^{2} + T - 74)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 1683 T^{2} + 627264 \) Copy content Toggle raw display
$29$ \( T^{4} + 3087 T^{2} + 777924 \) Copy content Toggle raw display
$31$ \( (T^{2} + 7 T - 656)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 32 T - 932)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 3222 T^{2} + \cdots + 2424249 \) Copy content Toggle raw display
$43$ \( (T + 23)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} + 1539 T^{2} + 104976 \) Copy content Toggle raw display
$53$ \( T^{4} + 4032 T^{2} + \cdots + 1327104 \) Copy content Toggle raw display
$59$ \( T^{4} + 5814 T^{2} + 68121 \) Copy content Toggle raw display
$61$ \( (T^{2} + 41 T - 248)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 116 T + 691)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 1548 T^{2} + 331776 \) Copy content Toggle raw display
$73$ \( (T^{2} - 43 T - 206)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 83 T - 134)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 1539 T^{2} + 104976 \) Copy content Toggle raw display
$89$ \( T^{4} + 24768 T^{2} + \cdots + 84934656 \) Copy content Toggle raw display
$97$ \( (T^{2} - 196 T + 9307)^{2} \) Copy content Toggle raw display
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