Properties

Label 276.2.g.a
Level $276$
Weight $2$
Character orbit 276.g
Analytic conductor $2.204$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 276 = 2^{2} \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 276.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.20387109579\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.14453810176.7
Defining polynomial: \(x^{8} + 9 x^{6} + 15 x^{4} - 30 x^{2} + 64\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + \beta_{4} q^{3} + \beta_{5} q^{5} -\beta_{7} q^{7} + ( -1 - \beta_{3} ) q^{9} +O(q^{10})\) \( q + \beta_{4} q^{3} + \beta_{5} q^{5} -\beta_{7} q^{7} + ( -1 - \beta_{3} ) q^{9} + ( \beta_{5} + \beta_{6} ) q^{11} + ( -\beta_{2} + \beta_{4} ) q^{13} + ( -\beta_{1} + \beta_{6} ) q^{15} + ( -\beta_{5} - 2 \beta_{6} ) q^{17} + ( -2 \beta_{1} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{19} + ( \beta_{1} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{21} + ( \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} ) q^{23} + ( 1 + 2 \beta_{2} - 2 \beta_{4} ) q^{25} + ( 2 + 3 \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{27} + ( -\beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{29} + ( -4 - 3 \beta_{2} + 3 \beta_{4} ) q^{31} + ( -\beta_{1} + \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{33} + ( 2 \beta_{3} + 2 \beta_{4} ) q^{35} + ( 2 \beta_{1} - \beta_{5} - \beta_{6} ) q^{37} + ( 2 - \beta_{3} ) q^{39} + ( -3 \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{41} + ( 2 \beta_{1} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{43} + ( \beta_{1} - 2 \beta_{5} + \beta_{7} ) q^{45} + ( -\beta_{2} - 2 \beta_{3} - 3 \beta_{4} ) q^{47} + ( -3 + 2 \beta_{2} - 2 \beta_{4} ) q^{49} + ( \beta_{1} - 2 \beta_{5} - 3 \beta_{6} - 2 \beta_{7} ) q^{51} + ( 2 \beta_{5} - 3 \beta_{6} ) q^{53} + 4 q^{55} + ( -4 \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{57} + ( -2 \beta_{2} - 2 \beta_{4} ) q^{59} + 2 \beta_{7} q^{61} + ( \beta_{1} + 3 \beta_{5} + 2 \beta_{6} ) q^{63} + ( -\beta_{5} + \beta_{6} ) q^{65} + 3 \beta_{7} q^{67} + ( -1 - \beta_{1} + 3 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} ) q^{69} + ( \beta_{2} + \beta_{4} ) q^{71} + ( 2 + 3 \beta_{2} - 3 \beta_{4} ) q^{73} + ( -4 + 2 \beta_{3} + \beta_{4} ) q^{75} + ( -4 \beta_{2} + 4 \beta_{3} ) q^{77} + ( -2 \beta_{1} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{79} + ( -5 + 3 \beta_{2} + \beta_{3} + \beta_{4} ) q^{81} + ( -3 \beta_{5} + \beta_{6} ) q^{83} + ( -2 + 2 \beta_{2} - 2 \beta_{4} ) q^{85} + ( -2 - 6 \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{87} + ( -2 \beta_{5} + \beta_{6} ) q^{89} + ( 2 \beta_{1} - \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{91} + ( 6 - 3 \beta_{3} - 4 \beta_{4} ) q^{93} + ( 4 \beta_{2} + 2 \beta_{3} + 6 \beta_{4} ) q^{95} + ( -2 \beta_{1} + \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{97} + ( -\beta_{1} + \beta_{6} + 3 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 2q^{3} - 6q^{9} + O(q^{10}) \) \( 8q + 2q^{3} - 6q^{9} + 4q^{13} + 8q^{27} - 20q^{31} + 18q^{39} - 32q^{49} + 32q^{55} - 14q^{69} + 4q^{73} - 34q^{75} - 46q^{81} - 24q^{85} - 2q^{87} + 46q^{93} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 9 x^{6} + 15 x^{4} - 30 x^{2} + 64\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu \)
\(\beta_{2}\)\(=\)\((\)\( -3 \nu^{6} - 20 \nu^{4} - 67 \nu^{2} - 97 \)\()/103\)
\(\beta_{3}\)\(=\)\((\)\( 4 \nu^{6} + 61 \nu^{4} + 158 \nu^{2} - 214 \)\()/103\)
\(\beta_{4}\)\(=\)\((\)\( -6 \nu^{6} - 40 \nu^{4} - 31 \nu^{2} + 115 \)\()/103\)
\(\beta_{5}\)\(=\)\((\)\( -23 \nu^{7} - 119 \nu^{5} - 33 \nu^{3} + 458 \nu \)\()/824\)
\(\beta_{6}\)\(=\)\((\)\( -27 \nu^{7} - 283 \nu^{5} - 397 \nu^{3} + 2114 \nu \)\()/824\)
\(\beta_{7}\)\(=\)\((\)\( -29 \nu^{7} - 365 \nu^{5} - 1403 \nu^{3} - 354 \nu \)\()/824\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/2\)
\(\nu^{2}\)\(=\)\(\beta_{4} - 2 \beta_{2} - 3\)
\(\nu^{3}\)\(=\)\((\)\(-2 \beta_{7} + 3 \beta_{6} - \beta_{5} - 4 \beta_{1}\)\()/2\)
\(\nu^{4}\)\(=\)\(-2 \beta_{4} + 3 \beta_{3} + 8 \beta_{2} + 16\)
\(\nu^{5}\)\(=\)\((\)\(5 \beta_{7} - 19 \beta_{6} + 16 \beta_{5} + 21 \beta_{1}\)\()/2\)
\(\nu^{6}\)\(=\)\(-9 \beta_{4} - 20 \beta_{3} - 43 \beta_{2} - 72\)
\(\nu^{7}\)\(=\)\((\)\(-23 \beta_{7} + 94 \beta_{6} - 153 \beta_{5} - 83 \beta_{1}\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/276\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(139\) \(185\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
137.1
−0.331077 2.33494i
0.331077 + 2.33494i
−0.331077 + 2.33494i
0.331077 2.33494i
−1.06789 0.545948i
1.06789 + 0.545948i
−1.06789 + 0.545948i
1.06789 0.545948i
0 −0.780776 1.54609i 0 −3.02045 0 2.62238i 0 −1.78078 + 2.41430i 0
137.2 0 −0.780776 1.54609i 0 3.02045 0 2.62238i 0 −1.78078 + 2.41430i 0
137.3 0 −0.780776 + 1.54609i 0 −3.02045 0 2.62238i 0 −1.78078 2.41430i 0
137.4 0 −0.780776 + 1.54609i 0 3.02045 0 2.62238i 0 −1.78078 2.41430i 0
137.5 0 1.28078 1.16602i 0 −0.936426 0 3.88884i 0 0.280776 2.98683i 0
137.6 0 1.28078 1.16602i 0 0.936426 0 3.88884i 0 0.280776 2.98683i 0
137.7 0 1.28078 + 1.16602i 0 −0.936426 0 3.88884i 0 0.280776 + 2.98683i 0
137.8 0 1.28078 + 1.16602i 0 0.936426 0 3.88884i 0 0.280776 + 2.98683i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 137.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
23.b odd 2 1 inner
69.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 276.2.g.a 8
3.b odd 2 1 inner 276.2.g.a 8
4.b odd 2 1 1104.2.m.b 8
12.b even 2 1 1104.2.m.b 8
23.b odd 2 1 inner 276.2.g.a 8
69.c even 2 1 inner 276.2.g.a 8
92.b even 2 1 1104.2.m.b 8
276.h odd 2 1 1104.2.m.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
276.2.g.a 8 1.a even 1 1 trivial
276.2.g.a 8 3.b odd 2 1 inner
276.2.g.a 8 23.b odd 2 1 inner
276.2.g.a 8 69.c even 2 1 inner
1104.2.m.b 8 4.b odd 2 1
1104.2.m.b 8 12.b even 2 1
1104.2.m.b 8 92.b even 2 1
1104.2.m.b 8 276.h odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(276, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( 9 - 3 T + 2 T^{2} - T^{3} + T^{4} )^{2} \)
$5$ \( ( 8 - 10 T^{2} + T^{4} )^{2} \)
$7$ \( ( 104 + 22 T^{2} + T^{4} )^{2} \)
$11$ \( ( 32 - 20 T^{2} + T^{4} )^{2} \)
$13$ \( ( -4 - T + T^{2} )^{4} \)
$17$ \( ( 8 - 58 T^{2} + T^{4} )^{2} \)
$19$ \( ( 1664 + 82 T^{2} + T^{4} )^{2} \)
$23$ \( 279841 - 10580 T^{2} + 70 T^{4} - 20 T^{6} + T^{8} \)
$29$ \( ( 832 + 59 T^{2} + T^{4} )^{2} \)
$31$ \( ( -32 + 5 T + T^{2} )^{4} \)
$37$ \( ( 416 + 92 T^{2} + T^{4} )^{2} \)
$41$ \( ( 208 + 115 T^{2} + T^{4} )^{2} \)
$43$ \( ( 1664 + 82 T^{2} + T^{4} )^{2} \)
$47$ \( ( 208 + 123 T^{2} + T^{4} )^{2} \)
$53$ \( ( 8192 - 190 T^{2} + T^{4} )^{2} \)
$59$ \( ( 832 + 60 T^{2} + T^{4} )^{2} \)
$61$ \( ( 1664 + 88 T^{2} + T^{4} )^{2} \)
$67$ \( ( 8424 + 198 T^{2} + T^{4} )^{2} \)
$71$ \( ( 52 + 15 T^{2} + T^{4} )^{2} \)
$73$ \( ( -38 - T + T^{2} )^{4} \)
$79$ \( ( 416 + 146 T^{2} + T^{4} )^{2} \)
$83$ \( ( 32 - 116 T^{2} + T^{4} )^{2} \)
$89$ \( ( 128 - 62 T^{2} + T^{4} )^{2} \)
$97$ \( ( 1664 + 116 T^{2} + T^{4} )^{2} \)
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