Properties

Label 1296.3.q.o
Level $1296$
Weight $3$
Character orbit 1296.q
Analytic conductor $35.313$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1296 = 2^{4} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1296.q (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(35.3134422611\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 3^{8} \)
Twist minimal: no (minimal twist has level 162)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{6} q^{5} + (2 \beta_{3} + 2 \beta_{2}) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{5} + (2 \beta_{3} + 2 \beta_{2}) q^{7} + (2 \beta_{7} - 2 \beta_{6} - 4 \beta_{5} + 2 \beta_1) q^{11} + ( - \beta_{4} + \beta_{3} - 16 \beta_{2} + 16) q^{13} + (5 \beta_{7} - 5 \beta_{6} - 5 \beta_{5}) q^{17} + (2 \beta_{4} - 14) q^{19} + (2 \beta_{6} + 2 \beta_1) q^{23} + ( - 3 \beta_{3} - 7 \beta_{2}) q^{25} + (7 \beta_{7} - 7 \beta_{6} + \beta_{5} + 7 \beta_1) q^{29} + ( - 8 \beta_{2} + 8) q^{31} + ( - 6 \beta_{7} - 4 \beta_{6} - 4 \beta_{5}) q^{35} + (8 \beta_{4} - 19) q^{37} + ( - 5 \beta_{6} - 3 \beta_1) q^{41} + ( - 6 \beta_{3} - 22 \beta_{2}) q^{43} + (4 \beta_{7} - 4 \beta_{6} + 4 \beta_1) q^{47} + ( - 8 \beta_{4} + 8 \beta_{3} + 63 \beta_{2} - 63) q^{49} + ( - 17 \beta_{7} + 8 \beta_{6} + 8 \beta_{5}) q^{53} + ( - 6 \beta_{4} + 54) q^{55} + ( - 16 \beta_{6} + 20 \beta_1) q^{59} + 13 \beta_{2} q^{61} + ( - 3 \beta_{7} + 3 \beta_{6} - 19 \beta_{5} - 3 \beta_1) q^{65} + ( - 6 \beta_{4} + 6 \beta_{3} + 10 \beta_{2} - 10) q^{67} + ( - 2 \beta_{7} - 16 \beta_{6} - 16 \beta_{5}) q^{71} + (3 \beta_{4} + 56) q^{73} + ( - 32 \beta_{6} + 40 \beta_1) q^{77} + ( - 14 \beta_{3} + 26 \beta_{2}) q^{79} + ( - 8 \beta_{7} + 8 \beta_{6} - 12 \beta_{5} - 8 \beta_1) q^{83} + ( - 45 \beta_{2} + 45) q^{85} + (8 \beta_{7} - 29 \beta_{6} - 29 \beta_{5}) q^{89} + (30 \beta_{4} - 22) q^{91} + ( - 26 \beta_{6} + 6 \beta_1) q^{95} + ( - 16 \beta_{3} - 8 \beta_{2}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{7} + 64 q^{13} - 112 q^{19} - 28 q^{25} + 32 q^{31} - 152 q^{37} - 88 q^{43} - 252 q^{49} + 432 q^{55} + 52 q^{61} - 40 q^{67} + 448 q^{73} + 104 q^{79} + 180 q^{85} - 176 q^{91} - 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 3\zeta_{24}^{3} + 3\zeta_{24} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{24}^{4} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 3\zeta_{24}^{6} + 3\zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -3\zeta_{24}^{6} + 6\zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( 3\zeta_{24}^{5} - 3\zeta_{24}^{3} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( 3\zeta_{24}^{7} - 3\zeta_{24}^{5} + 3\zeta_{24} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -3\zeta_{24}^{5} - 3\zeta_{24}^{3} + 3\zeta_{24} \) Copy content Toggle raw display
\(\zeta_{24}\)\(=\) \( ( \beta_{7} + \beta_{5} + 2\beta_1 ) / 9 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( \beta_{4} + \beta_{3} ) / 9 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( -\beta_{7} - \beta_{5} + \beta_1 ) / 9 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( -\beta_{7} + 2\beta_{5} + \beta_1 ) / 9 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( ( -\beta_{4} + 2\beta_{3} ) / 9 \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( -2\beta_{7} + 3\beta_{6} + \beta_{5} - \beta_1 ) / 9 \) Copy content Toggle raw display

Character values

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

\(n\) \(325\) \(1135\) \(1217\)
\(\chi(n)\) \(1\) \(1\) \(1 - \beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
593.1
0.258819 + 0.965926i
−0.965926 + 0.258819i
0.965926 0.258819i
−0.258819 0.965926i
0.258819 0.965926i
−0.965926 0.258819i
0.965926 + 0.258819i
−0.258819 + 0.965926i
0 0 0 −5.01910 + 2.89778i 0 −4.19615 + 7.26795i 0 0 0
593.2 0 0 0 −1.34486 + 0.776457i 0 6.19615 10.7321i 0 0 0
593.3 0 0 0 1.34486 0.776457i 0 6.19615 10.7321i 0 0 0
593.4 0 0 0 5.01910 2.89778i 0 −4.19615 + 7.26795i 0 0 0
1025.1 0 0 0 −5.01910 2.89778i 0 −4.19615 7.26795i 0 0 0
1025.2 0 0 0 −1.34486 0.776457i 0 6.19615 + 10.7321i 0 0 0
1025.3 0 0 0 1.34486 + 0.776457i 0 6.19615 + 10.7321i 0 0 0
1025.4 0 0 0 5.01910 + 2.89778i 0 −4.19615 7.26795i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1025.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1296.3.q.o 8
3.b odd 2 1 inner 1296.3.q.o 8
4.b odd 2 1 162.3.d.c 8
9.c even 3 1 1296.3.e.d 4
9.c even 3 1 inner 1296.3.q.o 8
9.d odd 6 1 1296.3.e.d 4
9.d odd 6 1 inner 1296.3.q.o 8
12.b even 2 1 162.3.d.c 8
36.f odd 6 1 162.3.b.b 4
36.f odd 6 1 162.3.d.c 8
36.h even 6 1 162.3.b.b 4
36.h even 6 1 162.3.d.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
162.3.b.b 4 36.f odd 6 1
162.3.b.b 4 36.h even 6 1
162.3.d.c 8 4.b odd 2 1
162.3.d.c 8 12.b even 2 1
162.3.d.c 8 36.f odd 6 1
162.3.d.c 8 36.h even 6 1
1296.3.e.d 4 9.c even 3 1
1296.3.e.d 4 9.d odd 6 1
1296.3.q.o 8 1.a even 1 1 trivial
1296.3.q.o 8 3.b odd 2 1 inner
1296.3.q.o 8 9.c even 3 1 inner
1296.3.q.o 8 9.d odd 6 1 inner

Hecke kernels

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

\( T_{5}^{8} - 36T_{5}^{6} + 1215T_{5}^{4} - 2916T_{5}^{2} + 6561 \) Copy content Toggle raw display
\( T_{7}^{4} - 4T_{7}^{3} + 120T_{7}^{2} + 416T_{7} + 10816 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 36 T^{6} + 1215 T^{4} + \cdots + 6561 \) Copy content Toggle raw display
$7$ \( (T^{4} - 4 T^{3} + 120 T^{2} + 416 T + 10816)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 216 T^{2} + 46656)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 32 T^{3} + 795 T^{2} + \cdots + 52441)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 900 T^{2} + 50625)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 28 T + 88)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} - 216 T^{2} + 46656)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} - 2052 T^{6} + \cdots + 996005996001 \) Copy content Toggle raw display
$31$ \( (T^{2} - 8 T + 64)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 38 T - 1367)^{4} \) Copy content Toggle raw display
$41$ \( T^{8} - 1764 T^{6} + \cdots + 512249392656 \) Copy content Toggle raw display
$43$ \( (T^{4} + 44 T^{3} + 2424 T^{2} + \cdots + 238144)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 288 T^{2} + 82944)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 7812 T^{2} + 4743684)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 994737284775936 \) Copy content Toggle raw display
$61$ \( (T^{2} - 13 T + 169)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + 20 T^{3} + 1272 T^{2} + \cdots + 760384)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 10512 T^{2} + 2742336)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 112 T + 2893)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} - 52 T^{3} + 7320 T^{2} + \cdots + 21307456)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} - 10944 T^{6} + \cdots + 6295362011136 \) Copy content Toggle raw display
$89$ \( (T^{4} + 24228 T^{2} + \cdots + 112211649)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 16 T^{3} + 7104 T^{2} + \cdots + 46895104)^{2} \) Copy content Toggle raw display
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