Properties

Label 1216.2.h.d
Level $1216$
Weight $2$
Character orbit 1216.h
Analytic conductor $9.710$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1216.h (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.70980888579\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.14453810176.1
Defining polynomial: \(x^{8} + 3 x^{6} + 6 x^{4} + 12 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 76)
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} + ( 1 + \beta_{1} ) q^{5} -\beta_{7} q^{7} + ( 1 + \beta_{1} ) q^{9} +O(q^{10})\) \( q + \beta_{3} q^{3} + ( 1 + \beta_{1} ) q^{5} -\beta_{7} q^{7} + ( 1 + \beta_{1} ) q^{9} + \beta_{2} q^{11} -\beta_{6} q^{13} + ( 2 \beta_{3} - \beta_{4} ) q^{15} - q^{17} + ( -\beta_{2} + \beta_{3} - \beta_{4} ) q^{19} + ( -\beta_{5} + \beta_{6} ) q^{21} + ( \beta_{2} + \beta_{7} ) q^{23} + \beta_{1} q^{25} + ( -\beta_{3} - \beta_{4} ) q^{27} + ( \beta_{5} + \beta_{6} ) q^{29} -\beta_{4} q^{31} -\beta_{5} q^{33} + ( \beta_{2} - 2 \beta_{7} ) q^{35} -\beta_{5} q^{37} + ( \beta_{2} + 3 \beta_{7} ) q^{39} -\beta_{5} q^{41} + ( 3 \beta_{2} + 2 \beta_{7} ) q^{43} + ( 5 + \beta_{1} ) q^{45} + 3 \beta_{2} q^{47} + ( -4 - 4 \beta_{1} ) q^{49} -\beta_{3} q^{51} + ( \beta_{5} - \beta_{6} ) q^{53} + ( -\beta_{2} - 2 \beta_{7} ) q^{55} + ( 4 + 3 \beta_{1} + \beta_{5} ) q^{57} + ( \beta_{3} - 3 \beta_{4} ) q^{59} + ( -7 - 3 \beta_{1} ) q^{61} + ( \beta_{2} - 2 \beta_{7} ) q^{63} + 2 \beta_{5} q^{65} + ( \beta_{3} + 2 \beta_{4} ) q^{67} -\beta_{6} q^{69} + ( -2 \beta_{3} - 2 \beta_{4} ) q^{71} + ( 3 - 2 \beta_{1} ) q^{73} + ( \beta_{3} - \beta_{4} ) q^{75} + ( -5 - 3 \beta_{1} ) q^{77} + ( 4 \beta_{3} - 3 \beta_{4} ) q^{79} + ( -7 - 2 \beta_{1} ) q^{81} + ( 2 \beta_{2} + 4 \beta_{7} ) q^{83} + ( -1 - \beta_{1} ) q^{85} + ( -3 \beta_{2} - \beta_{7} ) q^{87} + ( \beta_{5} - 2 \beta_{6} ) q^{89} + ( 7 \beta_{3} - \beta_{4} ) q^{91} + 2 \beta_{1} q^{93} + ( \beta_{2} + 4 \beta_{3} + 2 \beta_{7} ) q^{95} + ( 2 \beta_{5} + 2 \beta_{6} ) q^{97} + ( -\beta_{2} - 2 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 4q^{5} + 4q^{9} + O(q^{10}) \) \( 8q + 4q^{5} + 4q^{9} - 8q^{17} - 4q^{25} + 36q^{45} - 16q^{49} + 20q^{57} - 44q^{61} + 32q^{73} - 28q^{77} - 48q^{81} - 4q^{85} - 8q^{93} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 3 x^{6} + 6 x^{4} + 12 x^{2} + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{6} + 3 \nu^{4} + 2 \nu^{2} + 4 \)\()/4\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{6} + \nu^{4} + 2 \nu^{2} \)\()/4\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{7} + \nu^{5} - 2 \nu^{3} - 4 \nu \)\()/8\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{7} + 3 \nu^{5} + 6 \nu^{3} + 4 \nu \)\()/4\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{7} + 3 \nu^{5} + 6 \nu^{3} + 20 \nu \)\()/4\)
\(\beta_{6}\)\(=\)\((\)\( -3 \nu^{7} - 5 \nu^{5} + 2 \nu^{3} - 12 \nu \)\()/8\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{6} + \nu^{4} + 4 \nu^{2} + 6 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{5} - \beta_{4}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{7} + \beta_{2} - \beta_{1} - 2\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{6} + \beta_{4} - \beta_{3}\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(-\beta_{7} + \beta_{2} + 3 \beta_{1}\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-\beta_{6} + \beta_{4} + 5 \beta_{3}\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(\beta_{7} - 5 \beta_{2} + \beta_{1} - 4\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(-3 \beta_{6} - 2 \beta_{5} + \beta_{4} - 9 \beta_{3}\)\()/2\)

Character values

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

\(n\) \(191\) \(705\) \(837\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1215.1
−0.331077 1.37491i
−0.331077 + 1.37491i
1.06789 + 0.927153i
1.06789 0.927153i
−1.06789 0.927153i
−1.06789 + 0.927153i
0.331077 + 1.37491i
0.331077 1.37491i
0 −2.35829 0 2.56155 0 4.15286i 0 2.56155 0
1215.2 0 −2.35829 0 2.56155 0 4.15286i 0 2.56155 0
1215.3 0 −1.19935 0 −1.56155 0 0.868210i 0 −1.56155 0
1215.4 0 −1.19935 0 −1.56155 0 0.868210i 0 −1.56155 0
1215.5 0 1.19935 0 −1.56155 0 0.868210i 0 −1.56155 0
1215.6 0 1.19935 0 −1.56155 0 0.868210i 0 −1.56155 0
1215.7 0 2.35829 0 2.56155 0 4.15286i 0 2.56155 0
1215.8 0 2.35829 0 2.56155 0 4.15286i 0 2.56155 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1215.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
19.b odd 2 1 inner
76.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1216.2.h.d 8
4.b odd 2 1 inner 1216.2.h.d 8
8.b even 2 1 76.2.d.a 8
8.d odd 2 1 76.2.d.a 8
19.b odd 2 1 inner 1216.2.h.d 8
24.f even 2 1 684.2.f.b 8
24.h odd 2 1 684.2.f.b 8
76.d even 2 1 inner 1216.2.h.d 8
152.b even 2 1 76.2.d.a 8
152.g odd 2 1 76.2.d.a 8
456.l odd 2 1 684.2.f.b 8
456.p even 2 1 684.2.f.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.2.d.a 8 8.b even 2 1
76.2.d.a 8 8.d odd 2 1
76.2.d.a 8 152.b even 2 1
76.2.d.a 8 152.g odd 2 1
684.2.f.b 8 24.f even 2 1
684.2.f.b 8 24.h odd 2 1
684.2.f.b 8 456.l odd 2 1
684.2.f.b 8 456.p even 2 1
1216.2.h.d 8 1.a even 1 1 trivial
1216.2.h.d 8 4.b odd 2 1 inner
1216.2.h.d 8 19.b odd 2 1 inner
1216.2.h.d 8 76.d even 2 1 inner

Hecke kernels

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

\( T_{3}^{4} - 7 T_{3}^{2} + 8 \)
\( T_{5}^{2} - T_{5} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( 8 - 7 T^{2} + T^{4} )^{2} \)
$5$ \( ( -4 - T + T^{2} )^{4} \)
$7$ \( ( 13 + 18 T^{2} + T^{4} )^{2} \)
$11$ \( ( 52 + 15 T^{2} + T^{4} )^{2} \)
$13$ \( ( 416 + 41 T^{2} + T^{4} )^{2} \)
$17$ \( ( 1 + T )^{8} \)
$19$ \( 130321 - 5776 T^{2} + 718 T^{4} - 16 T^{6} + T^{8} \)
$23$ \( ( 52 + 19 T^{2} + T^{4} )^{2} \)
$29$ \( ( 104 + 73 T^{2} + T^{4} )^{2} \)
$31$ \( ( 32 - 20 T^{2} + T^{4} )^{2} \)
$37$ \( ( 416 + 44 T^{2} + T^{4} )^{2} \)
$41$ \( ( 416 + 44 T^{2} + T^{4} )^{2} \)
$43$ \( ( 208 + 123 T^{2} + T^{4} )^{2} \)
$47$ \( ( 4212 + 135 T^{2} + T^{4} )^{2} \)
$53$ \( ( 104 + 97 T^{2} + T^{4} )^{2} \)
$59$ \( ( 5408 - 175 T^{2} + T^{4} )^{2} \)
$61$ \( ( -8 + 11 T + T^{2} )^{4} \)
$67$ \( ( 8 - 95 T^{2} + T^{4} )^{2} \)
$71$ \( ( 512 - 124 T^{2} + T^{4} )^{2} \)
$73$ \( ( -1 - 8 T + T^{2} )^{4} \)
$79$ \( ( 11552 - 244 T^{2} + T^{4} )^{2} \)
$83$ \( ( 13312 + 236 T^{2} + T^{4} )^{2} \)
$89$ \( ( 6656 + 232 T^{2} + T^{4} )^{2} \)
$97$ \( ( 1664 + 292 T^{2} + T^{4} )^{2} \)
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