Properties

Label 1216.2.t.b
Level $1216$
Weight $2$
Character orbit 1216.t
Analytic conductor $9.710$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1216.t (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.70980888579\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.3317760000.2
Defining polynomial: \(x^{8} - 25 x^{4} + 625\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + ( 3 \beta_{1} - 3 \beta_{3} ) q^{3} + \beta_{5} q^{5} -\beta_{7} q^{7} + ( 6 - 6 \beta_{2} ) q^{9} +O(q^{10})\) \( q + ( 3 \beta_{1} - 3 \beta_{3} ) q^{3} + \beta_{5} q^{5} -\beta_{7} q^{7} + ( 6 - 6 \beta_{2} ) q^{9} -3 \beta_{3} q^{11} + 2 \beta_{6} q^{13} + ( -3 \beta_{4} + 3 \beta_{7} ) q^{15} -4 \beta_{2} q^{17} + ( 3 \beta_{1} + 2 \beta_{3} ) q^{19} -3 \beta_{5} q^{21} + ( 3 \beta_{4} - 3 \beta_{7} ) q^{23} + ( 5 - 5 \beta_{2} ) q^{25} -9 \beta_{3} q^{27} -\beta_{6} q^{29} -\beta_{7} q^{31} -9 \beta_{2} q^{33} + ( -10 \beta_{1} + 10 \beta_{3} ) q^{35} + ( \beta_{5} - \beta_{6} ) q^{37} + 6 \beta_{7} q^{39} + 3 \beta_{2} q^{41} + ( 10 \beta_{1} - 10 \beta_{3} ) q^{43} + ( 6 \beta_{5} - 6 \beta_{6} ) q^{45} + ( \beta_{4} - \beta_{7} ) q^{47} + 3 q^{49} -12 \beta_{1} q^{51} -3 \beta_{4} q^{55} + ( 9 + 6 \beta_{2} ) q^{57} + ( -7 \beta_{1} + 7 \beta_{3} ) q^{59} -\beta_{6} q^{61} + ( 6 \beta_{4} - 6 \beta_{7} ) q^{63} + 20 q^{65} -5 \beta_{1} q^{67} + ( -9 \beta_{5} + 9 \beta_{6} ) q^{69} + 7 \beta_{2} q^{73} -15 \beta_{3} q^{75} + ( -3 \beta_{5} + 3 \beta_{6} ) q^{77} -2 \beta_{4} q^{79} -9 \beta_{2} q^{81} + 7 \beta_{3} q^{83} -4 \beta_{6} q^{85} -3 \beta_{7} q^{87} + ( -8 + 8 \beta_{2} ) q^{89} -20 \beta_{1} q^{91} -3 \beta_{5} q^{93} + ( 2 \beta_{4} + 3 \beta_{7} ) q^{95} + 9 \beta_{2} q^{97} -18 \beta_{1} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 24q^{9} + O(q^{10}) \) \( 8q + 24q^{9} - 16q^{17} + 20q^{25} - 36q^{33} + 12q^{41} + 24q^{49} + 96q^{57} + 160q^{65} + 28q^{73} - 36q^{81} - 32q^{89} + 36q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 25 x^{4} + 625\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} \)\(/5\)
\(\beta_{2}\)\(=\)\( \nu^{4} \)\(/25\)
\(\beta_{3}\)\(=\)\( \nu^{6} \)\(/125\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{7} + 125 \nu \)\()/125\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{7} + 125 \nu \)\()/125\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{7} + 5 \nu^{5} + 25 \nu^{3} \)\()/125\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{5} + 5 \nu^{3} + 25 \nu \)\()/25\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{5} + \beta_{4}\)\()/2\)
\(\nu^{2}\)\(=\)\(5 \beta_{1}\)
\(\nu^{3}\)\(=\)\((\)\(5 \beta_{7} + 5 \beta_{6} - 5 \beta_{5}\)\()/2\)
\(\nu^{4}\)\(=\)\(25 \beta_{2}\)
\(\nu^{5}\)\(=\)\((\)\(-25 \beta_{7} + 25 \beta_{6} + 25 \beta_{4}\)\()/2\)
\(\nu^{6}\)\(=\)\(125 \beta_{3}\)
\(\nu^{7}\)\(=\)\((\)\(-125 \beta_{5} + 125 \beta_{4}\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(705\) \(837\)
\(\chi(n)\) \(1\) \(-1 + \beta_{2}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
353.1
−0.578737 2.15988i
0.578737 + 2.15988i
−2.15988 + 0.578737i
2.15988 0.578737i
−0.578737 + 2.15988i
0.578737 2.15988i
−2.15988 0.578737i
2.15988 + 0.578737i
0 −2.59808 1.50000i 0 −2.73861 1.58114i 0 −3.16228 0 3.00000 + 5.19615i 0
353.2 0 −2.59808 1.50000i 0 2.73861 + 1.58114i 0 3.16228 0 3.00000 + 5.19615i 0
353.3 0 2.59808 + 1.50000i 0 −2.73861 1.58114i 0 3.16228 0 3.00000 + 5.19615i 0
353.4 0 2.59808 + 1.50000i 0 2.73861 + 1.58114i 0 −3.16228 0 3.00000 + 5.19615i 0
1185.1 0 −2.59808 + 1.50000i 0 −2.73861 + 1.58114i 0 −3.16228 0 3.00000 5.19615i 0
1185.2 0 −2.59808 + 1.50000i 0 2.73861 1.58114i 0 3.16228 0 3.00000 5.19615i 0
1185.3 0 2.59808 1.50000i 0 −2.73861 + 1.58114i 0 3.16228 0 3.00000 5.19615i 0
1185.4 0 2.59808 1.50000i 0 2.73861 1.58114i 0 −3.16228 0 3.00000 5.19615i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1185.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
19.c even 3 1 inner
76.g odd 6 1 inner
152.k odd 6 1 inner
152.p even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1216.2.t.b 8
4.b odd 2 1 inner 1216.2.t.b 8
8.b even 2 1 inner 1216.2.t.b 8
8.d odd 2 1 inner 1216.2.t.b 8
19.c even 3 1 inner 1216.2.t.b 8
76.g odd 6 1 inner 1216.2.t.b 8
152.k odd 6 1 inner 1216.2.t.b 8
152.p even 6 1 inner 1216.2.t.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1216.2.t.b 8 1.a even 1 1 trivial
1216.2.t.b 8 4.b odd 2 1 inner
1216.2.t.b 8 8.b even 2 1 inner
1216.2.t.b 8 8.d odd 2 1 inner
1216.2.t.b 8 19.c even 3 1 inner
1216.2.t.b 8 76.g odd 6 1 inner
1216.2.t.b 8 152.k odd 6 1 inner
1216.2.t.b 8 152.p even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1216, [\chi])\):

\( T_{3}^{4} - 9 T_{3}^{2} + 81 \)
\( T_{5}^{4} - 10 T_{5}^{2} + 100 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( 81 - 9 T^{2} + T^{4} )^{2} \)
$5$ \( ( 100 - 10 T^{2} + T^{4} )^{2} \)
$7$ \( ( -10 + T^{2} )^{4} \)
$11$ \( ( 9 + T^{2} )^{4} \)
$13$ \( ( 1600 - 40 T^{2} + T^{4} )^{2} \)
$17$ \( ( 16 + 4 T + T^{2} )^{4} \)
$19$ \( ( 361 + 11 T^{2} + T^{4} )^{2} \)
$23$ \( ( 8100 + 90 T^{2} + T^{4} )^{2} \)
$29$ \( ( 100 - 10 T^{2} + T^{4} )^{2} \)
$31$ \( ( -10 + T^{2} )^{4} \)
$37$ \( ( 10 + T^{2} )^{4} \)
$41$ \( ( 9 - 3 T + T^{2} )^{4} \)
$43$ \( ( 10000 - 100 T^{2} + T^{4} )^{2} \)
$47$ \( ( 100 + 10 T^{2} + T^{4} )^{2} \)
$53$ \( T^{8} \)
$59$ \( ( 2401 - 49 T^{2} + T^{4} )^{2} \)
$61$ \( ( 100 - 10 T^{2} + T^{4} )^{2} \)
$67$ \( ( 625 - 25 T^{2} + T^{4} )^{2} \)
$71$ \( T^{8} \)
$73$ \( ( 49 - 7 T + T^{2} )^{4} \)
$79$ \( ( 1600 + 40 T^{2} + T^{4} )^{2} \)
$83$ \( ( 49 + T^{2} )^{4} \)
$89$ \( ( 64 + 8 T + T^{2} )^{4} \)
$97$ \( ( 81 - 9 T + T^{2} )^{4} \)
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