Properties

Label 1296.3.g.j
Level $1296$
Weight $3$
Character orbit 1296.g
Analytic conductor $35.313$
Analytic rank $0$
Dimension $8$
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.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(35.3134422611\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.856615824.2
Defining polynomial: \(x^{8} + 11 x^{6} + 36 x^{4} + 32 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{6} \)
Twist minimal: no (minimal twist has level 144)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( -1 - \beta_{3} ) q^{5} + \beta_{6} q^{7} +O(q^{10})\) \( q + ( -1 - \beta_{3} ) q^{5} + \beta_{6} q^{7} + ( -\beta_{1} + \beta_{2} + \beta_{6} ) q^{11} + ( -1 + \beta_{3} + \beta_{4} ) q^{13} + ( 1 - \beta_{4} - \beta_{7} ) q^{17} + ( -\beta_{2} + 3 \beta_{5} + \beta_{6} ) q^{19} + ( \beta_{1} + 3 \beta_{2} + \beta_{5} ) q^{23} + ( 7 + 4 \beta_{3} - 2 \beta_{4} - \beta_{7} ) q^{25} + ( -17 + \beta_{3} + 3 \beta_{4} ) q^{29} + ( -3 \beta_{1} - \beta_{2} + 3 \beta_{5} + 2 \beta_{6} ) q^{31} + ( 4 \beta_{1} + 7 \beta_{2} - 4 \beta_{5} + \beta_{6} ) q^{35} + ( -4 - 4 \beta_{3} - \beta_{4} + 2 \beta_{7} ) q^{37} + ( -15 - 4 \beta_{3} - \beta_{4} + 2 \beta_{7} ) q^{41} + ( 6 \beta_{1} - 2 \beta_{2} - 3 \beta_{5} - 2 \beta_{6} ) q^{43} + ( -2 \beta_{1} + 10 \beta_{2} + 2 \beta_{5} - 5 \beta_{6} ) q^{47} + ( -9 + 6 \beta_{3} + 3 \beta_{4} + \beta_{7} ) q^{49} + ( -30 + 6 \beta_{3} - 3 \beta_{4} ) q^{53} + ( 3 \beta_{1} + 3 \beta_{2} - 3 \beta_{5} ) q^{55} + ( -4 \beta_{1} + 14 \beta_{2} + 7 \beta_{5} + 2 \beta_{6} ) q^{59} + ( -3 - 7 \beta_{3} - \beta_{4} - 2 \beta_{7} ) q^{61} + ( -22 + 6 \beta_{3} + \beta_{4} + \beta_{7} ) q^{65} + ( -3 \beta_{1} + \beta_{2} - 6 \beta_{5} - 3 \beta_{6} ) q^{67} + ( -5 \beta_{1} + 19 \beta_{2} - \beta_{5} - 5 \beta_{6} ) q^{71} + ( 9 - 2 \beta_{3} + \beta_{4} - \beta_{7} ) q^{73} + ( -51 + 3 \beta_{3} ) q^{77} + ( -9 \beta_{1} + 5 \beta_{2} - 3 \beta_{5} - 2 \beta_{6} ) q^{79} + ( -2 \beta_{1} + 23 \beta_{2} - 6 \beta_{5} + 5 \beta_{6} ) q^{83} + ( 4 - 16 \beta_{3} - \beta_{4} - 2 \beta_{7} ) q^{85} + ( -22 - 4 \beta_{3} + 3 \beta_{4} ) q^{89} + ( -9 \beta_{1} + 3 \beta_{5} - 8 \beta_{6} ) q^{91} + ( 10 \beta_{1} + 26 \beta_{2} - 18 \beta_{5} - 10 \beta_{6} ) q^{95} + ( 3 + 6 \beta_{3} + 3 \beta_{4} + 4 \beta_{7} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 6q^{5} + O(q^{10}) \) \( 8q - 6q^{5} - 10q^{13} + 6q^{17} + 46q^{25} - 138q^{29} - 20q^{37} - 108q^{41} - 82q^{49} - 252q^{53} - 14q^{61} - 186q^{65} + 74q^{73} - 414q^{77} + 60q^{85} - 168q^{89} + 20q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 11 x^{6} + 36 x^{4} + 32 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 3 \nu^{3} + 12 \nu \)
\(\beta_{2}\)\(=\)\( \nu^{5} + 7 \nu^{3} + 10 \nu \)
\(\beta_{3}\)\(=\)\( \nu^{6} + 8 \nu^{4} + 14 \nu^{2} \)
\(\beta_{4}\)\(=\)\( -\nu^{6} - 5 \nu^{4} - 2 \nu^{2} - 4 \)
\(\beta_{5}\)\(=\)\((\)\( 3 \nu^{7} + 30 \nu^{5} + 87 \nu^{3} + 66 \nu \)\()/2\)
\(\beta_{6}\)\(=\)\((\)\( -3 \nu^{7} - 32 \nu^{5} - 95 \nu^{3} - 50 \nu \)\()/2\)
\(\beta_{7}\)\(=\)\( 3 \nu^{6} + 30 \nu^{4} + 84 \nu^{2} + 43 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{6} + \beta_{5} + \beta_{2} - \beta_{1}\)\()/6\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{7} - 2 \beta_{4} - 5 \beta_{3} - 51\)\()/18\)
\(\nu^{3}\)\(=\)\((\)\(-2 \beta_{6} - 2 \beta_{5} - 2 \beta_{2} + 3 \beta_{1}\)\()/3\)
\(\nu^{4}\)\(=\)\((\)\(-2 \beta_{7} + 7 \beta_{4} + 13 \beta_{3} + 114\)\()/9\)
\(\nu^{5}\)\(=\)\((\)\(9 \beta_{6} + 9 \beta_{5} + 12 \beta_{2} - 16 \beta_{1}\)\()/3\)
\(\nu^{6}\)\(=\)\((\)\(3 \beta_{7} - 14 \beta_{4} - 20 \beta_{3} - 185\)\()/3\)
\(\nu^{7}\)\(=\)\((\)\(-43 \beta_{6} - 41 \beta_{5} - 73 \beta_{2} + 84 \beta_{1}\)\()/3\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1135.1
2.06288i
2.06288i
2.33086i
2.33086i
0.385731i
0.385731i
1.07834i
1.07834i
0 0 0 −9.23321 0 6.15562i 0 0 0
1135.2 0 0 0 −9.23321 0 6.15562i 0 0 0
1135.3 0 0 0 −0.710609 0 3.12324i 0 0 0
1135.4 0 0 0 −0.710609 0 3.12324i 0 0 0
1135.5 0 0 0 0.909226 0 7.05186i 0 0 0
1135.6 0 0 0 0.909226 0 7.05186i 0 0 0
1135.7 0 0 0 6.03459 0 11.8163i 0 0 0
1135.8 0 0 0 6.03459 0 11.8163i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1135.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.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.g.j 8
3.b odd 2 1 1296.3.g.k 8
4.b odd 2 1 inner 1296.3.g.j 8
9.c even 3 1 144.3.o.a 8
9.c even 3 1 144.3.o.c yes 8
9.d odd 6 1 432.3.o.a 8
9.d odd 6 1 432.3.o.b 8
12.b even 2 1 1296.3.g.k 8
36.f odd 6 1 144.3.o.a 8
36.f odd 6 1 144.3.o.c yes 8
36.h even 6 1 432.3.o.a 8
36.h even 6 1 432.3.o.b 8
72.j odd 6 1 1728.3.o.e 8
72.j odd 6 1 1728.3.o.f 8
72.l even 6 1 1728.3.o.e 8
72.l even 6 1 1728.3.o.f 8
72.n even 6 1 576.3.o.d 8
72.n even 6 1 576.3.o.f 8
72.p odd 6 1 576.3.o.d 8
72.p odd 6 1 576.3.o.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.3.o.a 8 9.c even 3 1
144.3.o.a 8 36.f odd 6 1
144.3.o.c yes 8 9.c even 3 1
144.3.o.c yes 8 36.f odd 6 1
432.3.o.a 8 9.d odd 6 1
432.3.o.a 8 36.h even 6 1
432.3.o.b 8 9.d odd 6 1
432.3.o.b 8 36.h even 6 1
576.3.o.d 8 72.n even 6 1
576.3.o.d 8 72.p odd 6 1
576.3.o.f 8 72.n even 6 1
576.3.o.f 8 72.p odd 6 1
1296.3.g.j 8 1.a even 1 1 trivial
1296.3.g.j 8 4.b odd 2 1 inner
1296.3.g.k 8 3.b odd 2 1
1296.3.g.k 8 12.b even 2 1
1728.3.o.e 8 72.j odd 6 1
1728.3.o.e 8 72.l even 6 1
1728.3.o.f 8 72.j odd 6 1
1728.3.o.f 8 72.l even 6 1

Hecke kernels

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

\( T_{5}^{4} + 3 T_{5}^{3} - 57 T_{5}^{2} + 9 T_{5} + 36 \)
\( T_{17}^{4} - 3 T_{17}^{3} - 822 T_{17}^{2} + 1908 T_{17} + 84168 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( ( 36 + 9 T - 57 T^{2} + 3 T^{3} + T^{4} )^{2} \)
$7$ \( 2566404 + 400815 T^{2} + 16335 T^{4} + 237 T^{6} + T^{8} \)
$11$ \( 12131289 + 1221804 T^{2} + 37422 T^{4} + 396 T^{6} + T^{8} \)
$13$ \( ( -3194 - 2353 T - 291 T^{2} + 5 T^{3} + T^{4} )^{2} \)
$17$ \( ( 84168 + 1908 T - 822 T^{2} - 3 T^{3} + T^{4} )^{2} \)
$19$ \( 2931572736 + 85791744 T^{2} + 739584 T^{4} + 1731 T^{6} + T^{8} \)
$23$ \( 19131876 + 5137263 T^{2} + 273375 T^{4} + 1053 T^{6} + T^{8} \)
$29$ \( ( -2011626 - 119457 T - 579 T^{2} + 69 T^{3} + T^{4} )^{2} \)
$31$ \( 944784 + 49200939 T^{2} + 1357155 T^{4} + 2673 T^{6} + T^{8} \)
$37$ \( ( -613568 - 117320 T - 3756 T^{2} + 10 T^{3} + T^{4} )^{2} \)
$41$ \( ( -2330613 - 190998 T - 2700 T^{2} + 54 T^{3} + T^{4} )^{2} \)
$43$ \( 29016737649 + 3679120116 T^{2} + 9809910 T^{4} + 6372 T^{6} + T^{8} \)
$47$ \( 28643839776036 + 59032500543 T^{2} + 41801103 T^{4} + 11709 T^{6} + T^{8} \)
$53$ \( ( -6508512 - 274104 T + 972 T^{2} + 126 T^{3} + T^{4} )^{2} \)
$59$ \( 48359409452649 + 114842866284 T^{2} + 77288094 T^{4} + 15948 T^{6} + T^{8} \)
$61$ \( ( -556736 - 222251 T - 6357 T^{2} + 7 T^{3} + T^{4} )^{2} \)
$67$ \( 68036119056801 + 102586425588 T^{2} + 54676998 T^{4} + 12324 T^{6} + T^{8} \)
$71$ \( 726110197530624 + 765915906816 T^{2} + 231242688 T^{4} + 26208 T^{6} + T^{8} \)
$73$ \( ( 416536 + 25628 T - 1002 T^{2} - 37 T^{3} + T^{4} )^{2} \)
$79$ \( 240627852449856 + 354653956251 T^{2} + 162185139 T^{4} + 23169 T^{6} + T^{8} \)
$83$ \( 1517530356962064 + 2504009122227 T^{2} + 545735475 T^{4} + 40761 T^{6} + T^{8} \)
$89$ \( ( -1161936 - 109152 T - 984 T^{2} + 84 T^{3} + T^{4} )^{2} \)
$97$ \( ( -5516309 + 583178 T - 14988 T^{2} - 10 T^{3} + T^{4} )^{2} \)
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