Properties

Label 578.2.b.a
Level $578$
Weight $2$
Character orbit 578.b
Analytic conductor $4.615$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 578 = 2 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 578.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.61535323683\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 34)
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.

\(f(q)\) \(=\) \( q - q^{2} + 2 i q^{3} + q^{4} -2 i q^{6} -4 i q^{7} - q^{8} - q^{9} +O(q^{10})\) \( q - q^{2} + 2 i q^{3} + q^{4} -2 i q^{6} -4 i q^{7} - q^{8} - q^{9} + 6 i q^{11} + 2 i q^{12} + 2 q^{13} + 4 i q^{14} + q^{16} + q^{18} + 4 q^{19} + 8 q^{21} -6 i q^{22} -2 i q^{24} + 5 q^{25} -2 q^{26} + 4 i q^{27} -4 i q^{28} + 4 i q^{31} - q^{32} -12 q^{33} - q^{36} + 4 i q^{37} -4 q^{38} + 4 i q^{39} + 6 i q^{41} -8 q^{42} -8 q^{43} + 6 i q^{44} + 2 i q^{48} -9 q^{49} -5 q^{50} + 2 q^{52} + 6 q^{53} -4 i q^{54} + 4 i q^{56} + 8 i q^{57} -4 i q^{61} -4 i q^{62} + 4 i q^{63} + q^{64} + 12 q^{66} + 8 q^{67} + q^{72} -2 i q^{73} -4 i q^{74} + 10 i q^{75} + 4 q^{76} + 24 q^{77} -4 i q^{78} + 8 i q^{79} -11 q^{81} -6 i q^{82} + 8 q^{84} + 8 q^{86} -6 i q^{88} -6 q^{89} -8 i q^{91} -8 q^{93} -2 i q^{96} -14 i q^{97} + 9 q^{98} -6 i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{4} - 2q^{8} - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{4} - 2q^{8} - 2q^{9} + 4q^{13} + 2q^{16} + 2q^{18} + 8q^{19} + 16q^{21} + 10q^{25} - 4q^{26} - 2q^{32} - 24q^{33} - 2q^{36} - 8q^{38} - 16q^{42} - 16q^{43} - 18q^{49} - 10q^{50} + 4q^{52} + 12q^{53} + 2q^{64} + 24q^{66} + 16q^{67} + 2q^{72} + 8q^{76} + 48q^{77} - 22q^{81} + 16q^{84} + 16q^{86} - 12q^{89} - 16q^{93} + 18q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/578\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
577.1
1.00000i
1.00000i
−1.00000 2.00000i 1.00000 0 2.00000i 4.00000i −1.00000 −1.00000 0
577.2 −1.00000 2.00000i 1.00000 0 2.00000i 4.00000i −1.00000 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 578.2.b.a 2
17.b even 2 1 inner 578.2.b.a 2
17.c even 4 1 34.2.a.a 1
17.c even 4 1 578.2.a.a 1
17.d even 8 4 578.2.c.e 4
17.e odd 16 8 578.2.d.e 8
51.f odd 4 1 306.2.a.a 1
51.f odd 4 1 5202.2.a.d 1
68.f odd 4 1 272.2.a.d 1
68.f odd 4 1 4624.2.a.a 1
85.f odd 4 1 850.2.c.b 2
85.i odd 4 1 850.2.c.b 2
85.j even 4 1 850.2.a.e 1
119.f odd 4 1 1666.2.a.m 1
136.i even 4 1 1088.2.a.l 1
136.j odd 4 1 1088.2.a.d 1
187.f odd 4 1 4114.2.a.a 1
204.l even 4 1 2448.2.a.k 1
221.k even 4 1 5746.2.a.b 1
255.i odd 4 1 7650.2.a.ci 1
340.n odd 4 1 6800.2.a.b 1
408.q even 4 1 9792.2.a.bj 1
408.t odd 4 1 9792.2.a.y 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
34.2.a.a 1 17.c even 4 1
272.2.a.d 1 68.f odd 4 1
306.2.a.a 1 51.f odd 4 1
578.2.a.a 1 17.c even 4 1
578.2.b.a 2 1.a even 1 1 trivial
578.2.b.a 2 17.b even 2 1 inner
578.2.c.e 4 17.d even 8 4
578.2.d.e 8 17.e odd 16 8
850.2.a.e 1 85.j even 4 1
850.2.c.b 2 85.f odd 4 1
850.2.c.b 2 85.i odd 4 1
1088.2.a.d 1 136.j odd 4 1
1088.2.a.l 1 136.i even 4 1
1666.2.a.m 1 119.f odd 4 1
2448.2.a.k 1 204.l even 4 1
4114.2.a.a 1 187.f odd 4 1
4624.2.a.a 1 68.f odd 4 1
5202.2.a.d 1 51.f odd 4 1
5746.2.a.b 1 221.k even 4 1
6800.2.a.b 1 340.n odd 4 1
7650.2.a.ci 1 255.i odd 4 1
9792.2.a.y 1 408.t odd 4 1
9792.2.a.bj 1 408.q even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(578, [\chi])\).

Hecke characteristic polynomials

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