Properties

Label 552.2.f.a
Level $552$
Weight $2$
Character orbit 552.f
Analytic conductor $4.408$
Analytic rank $0$
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.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.40774219157\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + i ) q^{2} + i q^{3} + 2 i q^{4} + 2 i q^{5} + ( -1 + i ) q^{6} + 4 q^{7} + ( -2 + 2 i ) q^{8} - q^{9} +O(q^{10})\) \( q + ( 1 + i ) q^{2} + i q^{3} + 2 i q^{4} + 2 i q^{5} + ( -1 + i ) q^{6} + 4 q^{7} + ( -2 + 2 i ) q^{8} - q^{9} + ( -2 + 2 i ) q^{10} + 2 i q^{11} -2 q^{12} -4 i q^{13} + ( 4 + 4 i ) q^{14} -2 q^{15} -4 q^{16} + ( -1 - i ) q^{18} -4 i q^{19} -4 q^{20} + 4 i q^{21} + ( -2 + 2 i ) q^{22} + q^{23} + ( -2 - 2 i ) q^{24} + q^{25} + ( 4 - 4 i ) q^{26} -i q^{27} + 8 i q^{28} -2 i q^{29} + ( -2 - 2 i ) q^{30} -6 q^{31} + ( -4 - 4 i ) q^{32} -2 q^{33} + 8 i q^{35} -2 i q^{36} + 6 i q^{37} + ( 4 - 4 i ) q^{38} + 4 q^{39} + ( -4 - 4 i ) q^{40} + 2 q^{41} + ( -4 + 4 i ) q^{42} -8 i q^{43} -4 q^{44} -2 i q^{45} + ( 1 + i ) q^{46} -8 q^{47} -4 i q^{48} + 9 q^{49} + ( 1 + i ) q^{50} + 8 q^{52} + 6 i q^{53} + ( 1 - i ) q^{54} -4 q^{55} + ( -8 + 8 i ) q^{56} + 4 q^{57} + ( 2 - 2 i ) q^{58} -4 i q^{59} -4 i q^{60} + 6 i q^{61} + ( -6 - 6 i ) q^{62} -4 q^{63} -8 i q^{64} + 8 q^{65} + ( -2 - 2 i ) q^{66} -4 i q^{67} + i q^{69} + ( -8 + 8 i ) q^{70} + 12 q^{71} + ( 2 - 2 i ) q^{72} + 14 q^{73} + ( -6 + 6 i ) q^{74} + i q^{75} + 8 q^{76} + 8 i q^{77} + ( 4 + 4 i ) q^{78} + 8 q^{79} -8 i q^{80} + q^{81} + ( 2 + 2 i ) q^{82} -14 i q^{83} -8 q^{84} + ( 8 - 8 i ) q^{86} + 2 q^{87} + ( -4 - 4 i ) q^{88} + 12 q^{89} + ( 2 - 2 i ) q^{90} -16 i q^{91} + 2 i q^{92} -6 i q^{93} + ( -8 - 8 i ) q^{94} + 8 q^{95} + ( 4 - 4 i ) q^{96} -14 q^{97} + ( 9 + 9 i ) q^{98} -2 i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} - 2q^{6} + 8q^{7} - 4q^{8} - 2q^{9} + O(q^{10}) \) \( 2q + 2q^{2} - 2q^{6} + 8q^{7} - 4q^{8} - 2q^{9} - 4q^{10} - 4q^{12} + 8q^{14} - 4q^{15} - 8q^{16} - 2q^{18} - 8q^{20} - 4q^{22} + 2q^{23} - 4q^{24} + 2q^{25} + 8q^{26} - 4q^{30} - 12q^{31} - 8q^{32} - 4q^{33} + 8q^{38} + 8q^{39} - 8q^{40} + 4q^{41} - 8q^{42} - 8q^{44} + 2q^{46} - 16q^{47} + 18q^{49} + 2q^{50} + 16q^{52} + 2q^{54} - 8q^{55} - 16q^{56} + 8q^{57} + 4q^{58} - 12q^{62} - 8q^{63} + 16q^{65} - 4q^{66} - 16q^{70} + 24q^{71} + 4q^{72} + 28q^{73} - 12q^{74} + 16q^{76} + 8q^{78} + 16q^{79} + 2q^{81} + 4q^{82} - 16q^{84} + 16q^{86} + 4q^{87} - 8q^{88} + 24q^{89} + 4q^{90} - 16q^{94} + 16q^{95} + 8q^{96} - 28q^{97} + 18q^{98} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
277.1
1.00000i
1.00000i
1.00000 1.00000i 1.00000i 2.00000i 2.00000i −1.00000 1.00000i 4.00000 −2.00000 2.00000i −1.00000 −2.00000 2.00000i
277.2 1.00000 + 1.00000i 1.00000i 2.00000i 2.00000i −1.00000 + 1.00000i 4.00000 −2.00000 + 2.00000i −1.00000 −2.00000 + 2.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 2 - 2 T + T^{2} \)
$3$ \( 1 + T^{2} \)
$5$ \( 4 + T^{2} \)
$7$ \( ( -4 + T )^{2} \)
$11$ \( 4 + T^{2} \)
$13$ \( 16 + T^{2} \)
$17$ \( T^{2} \)
$19$ \( 16 + T^{2} \)
$23$ \( ( -1 + T )^{2} \)
$29$ \( 4 + T^{2} \)
$31$ \( ( 6 + T )^{2} \)
$37$ \( 36 + T^{2} \)
$41$ \( ( -2 + T )^{2} \)
$43$ \( 64 + T^{2} \)
$47$ \( ( 8 + T )^{2} \)
$53$ \( 36 + T^{2} \)
$59$ \( 16 + T^{2} \)
$61$ \( 36 + T^{2} \)
$67$ \( 16 + T^{2} \)
$71$ \( ( -12 + T )^{2} \)
$73$ \( ( -14 + T )^{2} \)
$79$ \( ( -8 + T )^{2} \)
$83$ \( 196 + T^{2} \)
$89$ \( ( -12 + T )^{2} \)
$97$ \( ( 14 + T )^{2} \)
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