Properties

Label 552.2.m.a
Level $552$
Weight $2$
Character orbit 552.m
Analytic conductor $4.408$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 552 = 2^{3} \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 552.m (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.40774219157\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.40960000.1
Defining polynomial: \(x^{8} + 7 x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{3} \)
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.

\(f(q)\) \(=\) \( q + ( -1 - \beta_{2} ) q^{3} + \beta_{3} q^{5} + \beta_{5} q^{7} + ( 1 + 2 \beta_{1} + \beta_{2} + \beta_{4} ) q^{9} +O(q^{10})\) \( q + ( -1 - \beta_{2} ) q^{3} + \beta_{3} q^{5} + \beta_{5} q^{7} + ( 1 + 2 \beta_{1} + \beta_{2} + \beta_{4} ) q^{9} + ( -\beta_{3} + 3 \beta_{6} ) q^{11} + ( -2 - \beta_{2} - \beta_{4} ) q^{13} + ( -2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{15} + ( 4 \beta_{3} - \beta_{6} ) q^{17} + ( \beta_{5} + 3 \beta_{7} ) q^{19} + ( \beta_{3} - \beta_{7} ) q^{21} + ( -\beta_{1} - \beta_{2} + \beta_{4} - 3 \beta_{6} ) q^{23} + ( -3 - \beta_{2} - \beta_{4} ) q^{25} + ( -3 - 4 \beta_{1} + \beta_{4} ) q^{27} + ( 4 \beta_{1} + \beta_{2} - \beta_{4} ) q^{29} -6 q^{31} + ( 3 \beta_{3} - \beta_{5} - 4 \beta_{6} + 2 \beta_{7} ) q^{33} + ( \beta_{2} - \beta_{4} ) q^{35} + ( 6 \beta_{5} + \beta_{7} ) q^{37} + ( 4 + 2 \beta_{1} + \beta_{2} + \beta_{4} ) q^{39} + ( 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} ) q^{41} + ( -\beta_{5} + 5 \beta_{7} ) q^{43} + ( -\beta_{3} - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{45} + ( 8 \beta_{1} + 3 \beta_{2} - 3 \beta_{4} ) q^{47} + 5 q^{49} + ( -\beta_{3} - 7 \beta_{5} + 5 \beta_{6} + 3 \beta_{7} ) q^{51} -\beta_{3} q^{53} + ( -2 - 2 \beta_{2} - 2 \beta_{4} ) q^{55} + ( \beta_{3} - 3 \beta_{5} - 3 \beta_{6} - 4 \beta_{7} ) q^{57} -8 \beta_{1} q^{59} + ( 4 \beta_{5} - 3 \beta_{7} ) q^{61} + ( -\beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{63} + 2 \beta_{6} q^{65} + ( -3 \beta_{5} - \beta_{7} ) q^{67} + ( -2 + 3 \beta_{1} + \beta_{2} - 3 \beta_{3} + 3 \beta_{5} + 3 \beta_{6} - 3 \beta_{7} ) q^{69} + ( -6 \beta_{1} - 5 \beta_{2} + 5 \beta_{4} ) q^{71} + ( -6 - \beta_{2} - \beta_{4} ) q^{73} + ( 5 + 2 \beta_{1} + 2 \beta_{2} + \beta_{4} ) q^{75} + ( -6 \beta_{1} - \beta_{2} + \beta_{4} ) q^{77} + ( 5 \beta_{5} - 6 \beta_{7} ) q^{79} + ( 1 + 4 \beta_{1} + 4 \beta_{2} - 4 \beta_{4} ) q^{81} + ( \beta_{3} - 5 \beta_{6} ) q^{83} + ( 8 - 3 \beta_{2} - 3 \beta_{4} ) q^{85} + ( 2 - 6 \beta_{1} - \beta_{2} + 3 \beta_{4} ) q^{87} + ( -2 \beta_{3} - \beta_{6} ) q^{89} -2 \beta_{7} q^{91} + ( 6 + 6 \beta_{2} ) q^{93} + ( 6 \beta_{1} + \beta_{2} - \beta_{4} ) q^{95} + ( 8 \beta_{5} - 6 \beta_{7} ) q^{97} + ( -5 \beta_{3} - 4 \beta_{5} + 5 \beta_{6} - 2 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 4q^{3} + O(q^{10}) \) \( 8q - 4q^{3} - 8q^{13} - 16q^{25} - 28q^{27} - 48q^{31} + 24q^{39} + 40q^{49} - 20q^{69} - 40q^{73} + 28q^{75} + 8q^{81} + 88q^{85} + 8q^{87} + 24q^{93} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 7 x^{4} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{6} + 8 \nu^{2} \)\()/3\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{6} + \nu^{4} + 5 \nu^{2} + 2 \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{7} - 8 \nu^{3} + 3 \nu \)\()/3\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{6} + \nu^{4} - 5 \nu^{2} + 2 \)\()/3\)
\(\beta_{5}\)\(=\)\((\)\( 2 \nu^{7} + \nu^{5} + 13 \nu^{3} + 5 \nu \)\()/3\)
\(\beta_{6}\)\(=\)\((\)\( 2 \nu^{7} - \nu^{5} + 13 \nu^{3} - 5 \nu \)\()/3\)
\(\beta_{7}\)\(=\)\((\)\( 3 \nu^{7} + \nu^{5} + 21 \nu^{3} + 8 \nu \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7} - \beta_{5} + \beta_{3}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{4} - \beta_{2} + 2 \beta_{1}\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(2 \beta_{7} - \beta_{6} - 3 \beta_{5} - 2 \beta_{3}\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(3 \beta_{4} + 3 \beta_{2} - 4\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-5 \beta_{7} - 3 \beta_{6} + 8 \beta_{5} - 5 \beta_{3}\)\()/2\)
\(\nu^{6}\)\(=\)\(-4 \beta_{4} + 4 \beta_{2} - 5 \beta_{1}\)
\(\nu^{7}\)\(=\)\((\)\(-13 \beta_{7} + 8 \beta_{6} + 21 \beta_{5} + 13 \beta_{3}\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/552\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(185\) \(277\) \(415\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(1\)

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
−0.437016 0.437016i
0.437016 + 0.437016i
−0.437016 + 0.437016i
0.437016 0.437016i
−1.14412 + 1.14412i
1.14412 1.14412i
−1.14412 1.14412i
1.14412 + 1.14412i
0 −1.61803 0.618034i 0 −0.874032 0 1.41421i 0 2.23607 + 2.00000i 0
137.2 0 −1.61803 0.618034i 0 0.874032 0 1.41421i 0 2.23607 + 2.00000i 0
137.3 0 −1.61803 + 0.618034i 0 −0.874032 0 1.41421i 0 2.23607 2.00000i 0
137.4 0 −1.61803 + 0.618034i 0 0.874032 0 1.41421i 0 2.23607 2.00000i 0
137.5 0 0.618034 1.61803i 0 −2.28825 0 1.41421i 0 −2.23607 2.00000i 0
137.6 0 0.618034 1.61803i 0 2.28825 0 1.41421i 0 −2.23607 2.00000i 0
137.7 0 0.618034 + 1.61803i 0 −2.28825 0 1.41421i 0 −2.23607 + 2.00000i 0
137.8 0 0.618034 + 1.61803i 0 2.28825 0 1.41421i 0 −2.23607 + 2.00000i 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 552.2.m.a 8
3.b odd 2 1 inner 552.2.m.a 8
4.b odd 2 1 1104.2.m.c 8
12.b even 2 1 1104.2.m.c 8
23.b odd 2 1 inner 552.2.m.a 8
69.c even 2 1 inner 552.2.m.a 8
92.b even 2 1 1104.2.m.c 8
276.h odd 2 1 1104.2.m.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
552.2.m.a 8 1.a even 1 1 trivial
552.2.m.a 8 3.b odd 2 1 inner
552.2.m.a 8 23.b odd 2 1 inner
552.2.m.a 8 69.c even 2 1 inner
1104.2.m.c 8 4.b odd 2 1
1104.2.m.c 8 12.b even 2 1
1104.2.m.c 8 92.b even 2 1
1104.2.m.c 8 276.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 6 T_{5}^{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(552, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( 9 + 6 T + 2 T^{2} + 2 T^{3} + T^{4} )^{2} \)
$5$ \( ( 4 - 6 T^{2} + T^{4} )^{2} \)
$7$ \( ( 2 + T^{2} )^{4} \)
$11$ \( ( 100 - 30 T^{2} + T^{4} )^{2} \)
$13$ \( ( -4 + 2 T + T^{2} )^{4} \)
$17$ \( ( 1444 - 84 T^{2} + T^{4} )^{2} \)
$19$ \( ( 100 + 70 T^{2} + T^{4} )^{2} \)
$23$ \( ( 529 - 26 T^{2} + T^{4} )^{2} \)
$29$ \( ( 16 + 28 T^{2} + T^{4} )^{2} \)
$31$ \( ( 6 + T )^{8} \)
$37$ \( ( 6724 + 174 T^{2} + T^{4} )^{2} \)
$41$ \( ( 16 + 72 T^{2} + T^{4} )^{2} \)
$43$ \( ( 3364 + 134 T^{2} + T^{4} )^{2} \)
$47$ \( ( 400 + 140 T^{2} + T^{4} )^{2} \)
$53$ \( ( 4 - 6 T^{2} + T^{4} )^{2} \)
$59$ \( ( 64 + T^{2} )^{4} \)
$61$ \( ( 100 + 70 T^{2} + T^{4} )^{2} \)
$67$ \( ( 484 + 54 T^{2} + T^{4} )^{2} \)
$71$ \( ( 15376 + 252 T^{2} + T^{4} )^{2} \)
$73$ \( ( 20 + 10 T + T^{2} )^{4} \)
$79$ \( ( 6724 + 196 T^{2} + T^{4} )^{2} \)
$83$ \( ( 1444 - 86 T^{2} + T^{4} )^{2} \)
$89$ \( ( 4 - 36 T^{2} + T^{4} )^{2} \)
$97$ \( ( 1600 + 280 T^{2} + T^{4} )^{2} \)
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