Properties

Label 552.2.j.a
Level $552$
Weight $2$
Character orbit 552.j
Analytic conductor $4.408$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(4.40774219157\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \(x^{2} + 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + ( -1 + \beta ) q^{3} -2 q^{4} -4 q^{5} + ( -2 - \beta ) q^{6} + 2 \beta q^{7} -2 \beta q^{8} + ( -1 - 2 \beta ) q^{9} +O(q^{10})\) \( q + \beta q^{2} + ( -1 + \beta ) q^{3} -2 q^{4} -4 q^{5} + ( -2 - \beta ) q^{6} + 2 \beta q^{7} -2 \beta q^{8} + ( -1 - 2 \beta ) q^{9} -4 \beta q^{10} + 4 \beta q^{11} + ( 2 - 2 \beta ) q^{12} -4 \beta q^{13} -4 q^{14} + ( 4 - 4 \beta ) q^{15} + 4 q^{16} + 2 \beta q^{17} + ( 4 - \beta ) q^{18} -2 q^{19} + 8 q^{20} + ( -4 - 2 \beta ) q^{21} -8 q^{22} + q^{23} + ( 4 + 2 \beta ) q^{24} + 11 q^{25} + 8 q^{26} + ( 5 + \beta ) q^{27} -4 \beta q^{28} + ( 8 + 4 \beta ) q^{30} -4 \beta q^{31} + 4 \beta q^{32} + ( -8 - 4 \beta ) q^{33} -4 q^{34} -8 \beta q^{35} + ( 2 + 4 \beta ) q^{36} + 2 \beta q^{37} -2 \beta q^{38} + ( 8 + 4 \beta ) q^{39} + 8 \beta q^{40} -4 \beta q^{41} + ( 4 - 4 \beta ) q^{42} -6 q^{43} -8 \beta q^{44} + ( 4 + 8 \beta ) q^{45} + \beta q^{46} + ( -4 + 4 \beta ) q^{48} - q^{49} + 11 \beta q^{50} + ( -4 - 2 \beta ) q^{51} + 8 \beta q^{52} -12 q^{53} + ( -2 + 5 \beta ) q^{54} -16 \beta q^{55} + 8 q^{56} + ( 2 - 2 \beta ) q^{57} -2 \beta q^{59} + ( -8 + 8 \beta ) q^{60} -2 \beta q^{61} + 8 q^{62} + ( 8 - 2 \beta ) q^{63} -8 q^{64} + 16 \beta q^{65} + ( 8 - 8 \beta ) q^{66} + 2 q^{67} -4 \beta q^{68} + ( -1 + \beta ) q^{69} + 16 q^{70} -8 q^{71} + ( -8 + 2 \beta ) q^{72} -6 q^{73} -4 q^{74} + ( -11 + 11 \beta ) q^{75} + 4 q^{76} -16 q^{77} + ( -8 + 8 \beta ) q^{78} + 2 \beta q^{79} -16 q^{80} + ( -7 + 4 \beta ) q^{81} + 8 q^{82} -4 \beta q^{83} + ( 8 + 4 \beta ) q^{84} -8 \beta q^{85} -6 \beta q^{86} + 16 q^{88} -6 \beta q^{89} + ( -16 + 4 \beta ) q^{90} + 16 q^{91} -2 q^{92} + ( 8 + 4 \beta ) q^{93} + 8 q^{95} + ( -8 - 4 \beta ) q^{96} -2 q^{97} -\beta q^{98} + ( 16 - 4 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} - 4q^{4} - 8q^{5} - 4q^{6} - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} - 4q^{4} - 8q^{5} - 4q^{6} - 2q^{9} + 4q^{12} - 8q^{14} + 8q^{15} + 8q^{16} + 8q^{18} - 4q^{19} + 16q^{20} - 8q^{21} - 16q^{22} + 2q^{23} + 8q^{24} + 22q^{25} + 16q^{26} + 10q^{27} + 16q^{30} - 16q^{33} - 8q^{34} + 4q^{36} + 16q^{39} + 8q^{42} - 12q^{43} + 8q^{45} - 8q^{48} - 2q^{49} - 8q^{51} - 24q^{53} - 4q^{54} + 16q^{56} + 4q^{57} - 16q^{60} + 16q^{62} + 16q^{63} - 16q^{64} + 16q^{66} + 4q^{67} - 2q^{69} + 32q^{70} - 16q^{71} - 16q^{72} - 12q^{73} - 8q^{74} - 22q^{75} + 8q^{76} - 32q^{77} - 16q^{78} - 32q^{80} - 14q^{81} + 16q^{82} + 16q^{84} + 32q^{88} - 32q^{90} + 32q^{91} - 4q^{92} + 16q^{93} + 16q^{95} - 16q^{96} - 4q^{97} + 32q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
323.1
1.41421i
1.41421i
1.41421i −1.00000 1.41421i −2.00000 −4.00000 −2.00000 + 1.41421i 2.82843i 2.82843i −1.00000 + 2.82843i 5.65685i
323.2 1.41421i −1.00000 + 1.41421i −2.00000 −4.00000 −2.00000 1.41421i 2.82843i 2.82843i −1.00000 2.82843i 5.65685i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.f 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.j.a 2
3.b odd 2 1 552.2.j.b yes 2
4.b odd 2 1 2208.2.j.a 2
8.b even 2 1 2208.2.j.b 2
8.d odd 2 1 552.2.j.b yes 2
12.b even 2 1 2208.2.j.b 2
24.f even 2 1 inner 552.2.j.a 2
24.h odd 2 1 2208.2.j.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
552.2.j.a 2 1.a even 1 1 trivial
552.2.j.a 2 24.f even 2 1 inner
552.2.j.b yes 2 3.b odd 2 1
552.2.j.b yes 2 8.d odd 2 1
2208.2.j.a 2 4.b odd 2 1
2208.2.j.a 2 24.h odd 2 1
2208.2.j.b 2 8.b even 2 1
2208.2.j.b 2 12.b even 2 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 2 + T^{2} \)
$3$ \( 3 + 2 T + T^{2} \)
$5$ \( ( 4 + T )^{2} \)
$7$ \( 8 + T^{2} \)
$11$ \( 32 + T^{2} \)
$13$ \( 32 + T^{2} \)
$17$ \( 8 + T^{2} \)
$19$ \( ( 2 + T )^{2} \)
$23$ \( ( -1 + T )^{2} \)
$29$ \( T^{2} \)
$31$ \( 32 + T^{2} \)
$37$ \( 8 + T^{2} \)
$41$ \( 32 + T^{2} \)
$43$ \( ( 6 + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( ( 12 + T )^{2} \)
$59$ \( 8 + T^{2} \)
$61$ \( 8 + T^{2} \)
$67$ \( ( -2 + T )^{2} \)
$71$ \( ( 8 + T )^{2} \)
$73$ \( ( 6 + T )^{2} \)
$79$ \( 8 + T^{2} \)
$83$ \( 32 + T^{2} \)
$89$ \( 72 + T^{2} \)
$97$ \( ( 2 + T )^{2} \)
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