Properties

Label 1296.3.e.c
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: \(x^{4} - 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 72)
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.

\(f(q)\) \(=\) \( q + ( \beta_{1} - \beta_{2} ) q^{5} + ( -3 - \beta_{3} ) q^{7} +O(q^{10})\) \( q + ( \beta_{1} - \beta_{2} ) q^{5} + ( -3 - \beta_{3} ) q^{7} + ( 2 \beta_{1} + 3 \beta_{2} ) q^{11} + ( 7 - \beta_{3} ) q^{13} + ( 2 \beta_{1} - 8 \beta_{2} ) q^{17} + ( -2 - 2 \beta_{3} ) q^{19} + ( -\beta_{1} + 5 \beta_{2} ) q^{23} + ( -10 + 2 \beta_{3} ) q^{25} + ( -3 \beta_{1} + \beta_{2} ) q^{29} + ( 37 - \beta_{3} ) q^{31} -29 \beta_{2} q^{35} + ( -30 + 2 \beta_{3} ) q^{37} + 23 \beta_{2} q^{41} + ( 5 - 4 \beta_{3} ) q^{43} + ( -3 \beta_{1} - 29 \beta_{2} ) q^{47} + ( 56 + 6 \beta_{3} ) q^{49} + ( -8 \beta_{1} - 32 \beta_{2} ) q^{53} + ( -55 - \beta_{3} ) q^{55} + ( 2 \beta_{1} - 3 \beta_{2} ) q^{59} + ( 31 + \beta_{3} ) q^{61} + ( 10 \beta_{1} - 39 \beta_{2} ) q^{65} + ( -11 - 10 \beta_{3} ) q^{67} + ( -2 \beta_{1} - 24 \beta_{2} ) q^{71} + ( 10 + 6 \beta_{3} ) q^{73} + ( -15 \beta_{1} - 73 \beta_{2} ) q^{77} + ( -43 + 7 \beta_{3} ) q^{79} + ( -14 \beta_{1} + 11 \beta_{2} ) q^{83} + ( -88 + 10 \beta_{3} ) q^{85} + ( 6 \beta_{1} - 24 \beta_{2} ) q^{89} + ( 75 - 4 \beta_{3} ) q^{91} + ( 4 \beta_{1} - 62 \beta_{2} ) q^{95} + ( -121 + 2 \beta_{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 12q^{7} + O(q^{10}) \) \( 4q - 12q^{7} + 28q^{13} - 8q^{19} - 40q^{25} + 148q^{31} - 120q^{37} + 20q^{43} + 224q^{49} - 220q^{55} + 124q^{61} - 44q^{67} + 40q^{73} - 172q^{79} - 352q^{85} + 300q^{91} - 484q^{97} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu^{3} \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 1 \)
\(\beta_{3}\)\(=\)\( -2 \nu^{3} + 8 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{1}\)\()/8\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 1\)
\(\nu^{3}\)\(=\)\(\beta_{1}\)\(/2\)

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 7.38891i 0 6.79796 0 0 0
161.2 0 0 0 3.92480i 0 −12.7980 0 0 0
161.3 0 0 0 3.92480i 0 −12.7980 0 0 0
161.4 0 0 0 7.38891i 0 6.79796 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.c 4
3.b odd 2 1 inner 1296.3.e.c 4
4.b odd 2 1 648.3.e.b 4
9.c even 3 1 144.3.q.d 4
9.c even 3 1 432.3.q.c 4
9.d odd 6 1 144.3.q.d 4
9.d odd 6 1 432.3.q.c 4
12.b even 2 1 648.3.e.b 4
36.f odd 6 1 72.3.m.a 4
36.f odd 6 1 216.3.m.a 4
36.h even 6 1 72.3.m.a 4
36.h even 6 1 216.3.m.a 4
72.j odd 6 1 576.3.q.c 4
72.j odd 6 1 1728.3.q.f 4
72.l even 6 1 576.3.q.h 4
72.l even 6 1 1728.3.q.e 4
72.n even 6 1 576.3.q.c 4
72.n even 6 1 1728.3.q.f 4
72.p odd 6 1 576.3.q.h 4
72.p odd 6 1 1728.3.q.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.3.m.a 4 36.f odd 6 1
72.3.m.a 4 36.h even 6 1
144.3.q.d 4 9.c even 3 1
144.3.q.d 4 9.d odd 6 1
216.3.m.a 4 36.f odd 6 1
216.3.m.a 4 36.h even 6 1
432.3.q.c 4 9.c even 3 1
432.3.q.c 4 9.d odd 6 1
576.3.q.c 4 72.j odd 6 1
576.3.q.c 4 72.n even 6 1
576.3.q.h 4 72.l even 6 1
576.3.q.h 4 72.p odd 6 1
648.3.e.b 4 4.b odd 2 1
648.3.e.b 4 12.b even 2 1
1296.3.e.c 4 1.a even 1 1 trivial
1296.3.e.c 4 3.b odd 2 1 inner
1728.3.q.e 4 72.l even 6 1
1728.3.q.e 4 72.p odd 6 1
1728.3.q.f 4 72.j odd 6 1
1728.3.q.f 4 72.n even 6 1

Hecke kernels

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

\( T_{5}^{4} + 70 T_{5}^{2} + 841 \)
\( T_{7}^{2} + 6 T_{7} - 87 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( 841 + 70 T^{2} + T^{4} \)
$7$ \( ( -87 + 6 T + T^{2} )^{2} \)
$11$ \( 10201 + 310 T^{2} + T^{4} \)
$13$ \( ( -47 - 14 T + T^{2} )^{2} \)
$17$ \( 4096 + 640 T^{2} + T^{4} \)
$19$ \( ( -380 + 4 T + T^{2} )^{2} \)
$23$ \( 1849 + 214 T^{2} + T^{4} \)
$29$ \( 81225 + 582 T^{2} + T^{4} \)
$31$ \( ( 1273 - 74 T + T^{2} )^{2} \)
$37$ \( ( 516 + 60 T + T^{2} )^{2} \)
$41$ \( ( 1587 + T^{2} )^{2} \)
$43$ \( ( -1511 - 10 T + T^{2} )^{2} \)
$47$ \( 4995225 + 5622 T^{2} + T^{4} \)
$53$ \( 1048576 + 10240 T^{2} + T^{4} \)
$59$ \( 10201 + 310 T^{2} + T^{4} \)
$61$ \( ( 865 - 62 T + T^{2} )^{2} \)
$67$ \( ( -9479 + 22 T + T^{2} )^{2} \)
$71$ \( 2560000 + 3712 T^{2} + T^{4} \)
$73$ \( ( -3356 - 20 T + T^{2} )^{2} \)
$79$ \( ( -2855 + 86 T + T^{2} )^{2} \)
$83$ \( 34916281 + 13270 T^{2} + T^{4} \)
$89$ \( 331776 + 5760 T^{2} + T^{4} \)
$97$ \( ( 14257 + 242 T + T^{2} )^{2} \)
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