Properties

Label 575.2.b.b
Level $575$
Weight $2$
Character orbit 575.b
Analytic conductor $4.591$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 575 = 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 575.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.59139811622\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + i q^{2} + q^{4} - i q^{7} + 3 i q^{8} + 3 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + i q^{2} + q^{4} - i q^{7} + 3 i q^{8} + 3 q^{9} - q^{11} + i q^{13} + q^{14} - q^{16} + 3 i q^{18} + 5 q^{19} - i q^{22} + i q^{23} - q^{26} - i q^{28} + 5 q^{29} - 2 q^{31} + 5 i q^{32} + 3 q^{36} + 4 i q^{37} + 5 i q^{38} - 5 q^{41} - 9 i q^{43} - q^{44} - q^{46} + 6 i q^{47} + 6 q^{49} + i q^{52} + 2 i q^{53} + 3 q^{56} + 5 i q^{58} - 8 q^{59} - 8 q^{61} - 2 i q^{62} - 3 i q^{63} - 7 q^{64} - 8 i q^{67} - 10 q^{71} + 9 i q^{72} - 3 i q^{73} - 4 q^{74} + 5 q^{76} + i q^{77} + 3 q^{79} + 9 q^{81} - 5 i q^{82} + 3 i q^{83} + 9 q^{86} - 3 i q^{88} - 10 q^{89} + q^{91} + i q^{92} - 6 q^{94} + 2 i q^{97} + 6 i q^{98} - 3 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{4} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{4} + 6 q^{9} - 2 q^{11} + 2 q^{14} - 2 q^{16} + 10 q^{19} - 2 q^{26} + 10 q^{29} - 4 q^{31} + 6 q^{36} - 10 q^{41} - 2 q^{44} - 2 q^{46} + 12 q^{49} + 6 q^{56} - 16 q^{59} - 16 q^{61} - 14 q^{64} - 20 q^{71} - 8 q^{74} + 10 q^{76} + 6 q^{79} + 18 q^{81} + 18 q^{86} - 20 q^{89} + 2 q^{91} - 12 q^{94} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/575\mathbb{Z}\right)^\times\).

\(n\) \(51\) \(277\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
24.1
1.00000i
1.00000i
1.00000i 0 1.00000 0 0 1.00000i 3.00000i 3.00000 0
24.2 1.00000i 0 1.00000 0 0 1.00000i 3.00000i 3.00000 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
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 575.2.b.b 2
5.b even 2 1 inner 575.2.b.b 2
5.c odd 4 1 575.2.a.c 1
5.c odd 4 1 575.2.a.d yes 1
15.e even 4 1 5175.2.a.e 1
15.e even 4 1 5175.2.a.u 1
20.e even 4 1 9200.2.a.r 1
20.e even 4 1 9200.2.a.u 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
575.2.a.c 1 5.c odd 4 1
575.2.a.d yes 1 5.c odd 4 1
575.2.b.b 2 1.a even 1 1 trivial
575.2.b.b 2 5.b even 2 1 inner
5175.2.a.e 1 15.e even 4 1
5175.2.a.u 1 15.e even 4 1
9200.2.a.r 1 20.e even 4 1
9200.2.a.u 1 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(575, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 1 \) Copy content Toggle raw display
$11$ \( (T + 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 1 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( (T - 5)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 1 \) Copy content Toggle raw display
$29$ \( (T - 5)^{2} \) Copy content Toggle raw display
$31$ \( (T + 2)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 16 \) Copy content Toggle raw display
$41$ \( (T + 5)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 81 \) Copy content Toggle raw display
$47$ \( T^{2} + 36 \) Copy content Toggle raw display
$53$ \( T^{2} + 4 \) Copy content Toggle raw display
$59$ \( (T + 8)^{2} \) Copy content Toggle raw display
$61$ \( (T + 8)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 64 \) Copy content Toggle raw display
$71$ \( (T + 10)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 9 \) Copy content Toggle raw display
$79$ \( (T - 3)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 9 \) Copy content Toggle raw display
$89$ \( (T + 10)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 4 \) Copy content Toggle raw display
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