Properties

Label 1296.3.e.d
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: \( x^{4} + 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3^{3} \)
Twist minimal: no (minimal twist has level 162)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{5} + ( - 2 \beta_{3} - 2) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{5} + ( - 2 \beta_{3} - 2) q^{7} + ( - 2 \beta_{2} - 2 \beta_1) q^{11} + (\beta_{3} - 16) q^{13} + 5 \beta_{2} q^{17} + (2 \beta_{3} - 14) q^{19} + (2 \beta_{2} + 2 \beta_1) q^{23} + (3 \beta_{3} + 7) q^{25} + ( - 7 \beta_{2} + 8 \beta_1) q^{29} - 8 q^{31} + ( - 6 \beta_{2} + 10 \beta_1) q^{35} + (8 \beta_{3} - 19) q^{37} + ( - 3 \beta_{2} - 5 \beta_1) q^{41} + (6 \beta_{3} + 22) q^{43} + ( - 4 \beta_{2} + 4 \beta_1) q^{47} + (8 \beta_{3} + 63) q^{49} + ( - 17 \beta_{2} + 9 \beta_1) q^{53} + ( - 6 \beta_{3} + 54) q^{55} + (20 \beta_{2} - 16 \beta_1) q^{59} - 13 q^{61} + (3 \beta_{2} - 22 \beta_1) q^{65} + (6 \beta_{3} + 10) q^{67} + ( - 2 \beta_{2} + 18 \beta_1) q^{71} + (3 \beta_{3} + 56) q^{73} + (40 \beta_{2} - 32 \beta_1) q^{77} + (14 \beta_{3} - 26) q^{79} + (8 \beta_{2} - 20 \beta_1) q^{83} - 45 q^{85} + (8 \beta_{2} + 21 \beta_1) q^{89} + (30 \beta_{3} - 22) q^{91} + (6 \beta_{2} - 26 \beta_1) q^{95} + (16 \beta_{3} + 8) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{7} - 64 q^{13} - 56 q^{19} + 28 q^{25} - 32 q^{31} - 76 q^{37} + 88 q^{43} + 252 q^{49} + 216 q^{55} - 52 q^{61} + 40 q^{67} + 224 q^{73} - 104 q^{79} - 180 q^{85} - 88 q^{91} + 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 4x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 3\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 3\nu^{3} + 12\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 3\nu^{2} + 6 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 6 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{2} - 4\beta_1 ) / 3 \) 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
1.93185i
0.517638i
0.517638i
1.93185i
0 0 0 5.79555i 0 8.39230 0 0 0
161.2 0 0 0 1.55291i 0 −12.3923 0 0 0
161.3 0 0 0 1.55291i 0 −12.3923 0 0 0
161.4 0 0 0 5.79555i 0 8.39230 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.d 4
3.b odd 2 1 inner 1296.3.e.d 4
4.b odd 2 1 162.3.b.b 4
9.c even 3 2 1296.3.q.o 8
9.d odd 6 2 1296.3.q.o 8
12.b even 2 1 162.3.b.b 4
36.f odd 6 2 162.3.d.c 8
36.h even 6 2 162.3.d.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
162.3.b.b 4 4.b odd 2 1
162.3.b.b 4 12.b even 2 1
162.3.d.c 8 36.f odd 6 2
162.3.d.c 8 36.h even 6 2
1296.3.e.d 4 1.a even 1 1 trivial
1296.3.e.d 4 3.b odd 2 1 inner
1296.3.q.o 8 9.c even 3 2
1296.3.q.o 8 9.d odd 6 2

Hecke kernels

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

\( T_{5}^{4} + 36T_{5}^{2} + 81 \) Copy content Toggle raw display
\( T_{7}^{2} + 4T_{7} - 104 \) 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
$5$ \( T^{4} + 36T^{2} + 81 \) Copy content Toggle raw display
$7$ \( (T^{2} + 4 T - 104)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 216)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 32 T + 229)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 900 T^{2} + 50625 \) Copy content Toggle raw display
$19$ \( (T^{2} + 28 T + 88)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 216)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + 2052 T^{2} + 998001 \) Copy content Toggle raw display
$31$ \( (T + 8)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 38 T - 1367)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 1764 T^{2} + 715716 \) Copy content Toggle raw display
$43$ \( (T^{2} - 44 T - 488)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 288)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 7812 T^{2} + \cdots + 4743684 \) Copy content Toggle raw display
$59$ \( T^{4} + 12096 T^{2} + \cdots + 31539456 \) Copy content Toggle raw display
$61$ \( (T + 13)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} - 20 T - 872)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 10512 T^{2} + \cdots + 2742336 \) Copy content Toggle raw display
$73$ \( (T^{2} - 112 T + 2893)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 52 T - 4616)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 10944 T^{2} + \cdots + 2509056 \) Copy content Toggle raw display
$89$ \( T^{4} + 24228 T^{2} + \cdots + 112211649 \) Copy content Toggle raw display
$97$ \( (T^{2} - 16 T - 6848)^{2} \) Copy content Toggle raw display
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