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

Related objects

Downloads

Learn more about

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: \(\Q(\sqrt{-3}, \sqrt{-11})\)
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.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( -\nu^{3} + \nu^{2} - \nu + 3 \)
\(\beta_{2}\)\(=\)\( -\nu^{3} + \nu^{2} + 5 \nu + 2 \)
\(\beta_{3}\)\(=\)\( 4 \nu^{3} + 5 \nu^{2} - 5 \nu - 21 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} - \beta_{1} + 1\)\()/6\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{3} + 3 \beta_{2} + 5 \beta_{1} + 21\)\()/18\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{3} - 5 \beta_{1} + 36\)\()/9\)

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 \(S_{3}^{\mathrm{new}}(1296, [\chi])\):

\( T_{5}^{4} + 63 T_{5}^{2} + 324 \)
\( T_{7}^{2} + T_{7} - 74 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 - 37 T^{2} + 924 T^{4} - 23125 T^{6} + 390625 T^{8} \)
$7$ \( ( 1 + T + 24 T^{2} + 49 T^{3} + 2401 T^{4} )^{2} \)
$11$ \( 1 - 70 T^{2} - 12261 T^{4} - 1024870 T^{6} + 214358881 T^{8} \)
$13$ \( ( 1 + 5 T + 270 T^{2} + 845 T^{3} + 28561 T^{4} )^{2} \)
$17$ \( 1 - 769 T^{2} + 298176 T^{4} - 64227649 T^{6} + 6975757441 T^{8} \)
$19$ \( ( 1 + T + 648 T^{2} + 361 T^{3} + 130321 T^{4} )^{2} \)
$23$ \( 1 - 433 T^{2} + 525696 T^{4} - 121171153 T^{6} + 78310985281 T^{8} \)
$29$ \( 1 - 277 T^{2} - 170724 T^{4} - 195916837 T^{6} + 500246412961 T^{8} \)
$31$ \( ( 1 + 7 T + 1266 T^{2} + 6727 T^{3} + 923521 T^{4} )^{2} \)
$37$ \( ( 1 + 32 T + 1806 T^{2} + 43808 T^{3} + 1874161 T^{4} )^{2} \)
$41$ \( 1 - 3502 T^{2} + 8546451 T^{4} - 9895815022 T^{6} + 7984925229121 T^{8} \)
$43$ \( ( 1 + 23 T + 1849 T^{2} )^{4} \)
$47$ \( 1 - 7297 T^{2} + 22583760 T^{4} - 35607032257 T^{6} + 23811286661761 T^{8} \)
$53$ \( 1 - 7204 T^{2} + 26018214 T^{4} - 56843025124 T^{6} + 62259690411361 T^{8} \)
$59$ \( 1 - 8110 T^{2} + 32295219 T^{4} - 98271797710 T^{6} + 146830437604321 T^{8} \)
$61$ \( ( 1 + 41 T + 7194 T^{2} + 152561 T^{3} + 13845841 T^{4} )^{2} \)
$67$ \( ( 1 - 116 T + 9669 T^{2} - 520724 T^{3} + 20151121 T^{4} )^{2} \)
$71$ \( 1 - 18616 T^{2} + 137194926 T^{4} - 473063853496 T^{6} + 645753531245761 T^{8} \)
$73$ \( ( 1 - 43 T + 10452 T^{2} - 229147 T^{3} + 28398241 T^{4} )^{2} \)
$79$ \( ( 1 - 83 T + 12348 T^{2} - 518003 T^{3} + 38950081 T^{4} )^{2} \)
$83$ \( 1 - 26017 T^{2} + 263650560 T^{4} - 1234723137457 T^{6} + 2252292232139041 T^{8} \)
$89$ \( 1 - 6916 T^{2} + 69013446 T^{4} - 433925338756 T^{6} + 3936588805702081 T^{8} \)
$97$ \( ( 1 - 196 T + 28125 T^{2} - 1844164 T^{3} + 88529281 T^{4} )^{2} \)
show more
show less