Properties

Label 650.2.b.b
Level $650$
Weight $2$
Character orbit 650.b
Analytic conductor $5.190$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 650 = 2 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 650.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.19027613138\)
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.

\(f(q)\) \(=\) \( q + i q^{2} + 2 i q^{3} - q^{4} - 2 q^{6} + 5 i q^{7} - i q^{8} - q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + i q^{2} + 2 i q^{3} - q^{4} - 2 q^{6} + 5 i q^{7} - i q^{8} - q^{9} - 3 q^{11} - 2 i q^{12} - i q^{13} - 5 q^{14} + q^{16} + 3 i q^{17} - i q^{18} + 4 q^{19} - 10 q^{21} - 3 i q^{22} - 6 i q^{23} + 2 q^{24} + q^{26} + 4 i q^{27} - 5 i q^{28} - 9 q^{29} + 5 q^{31} + i q^{32} - 6 i q^{33} - 3 q^{34} + q^{36} + 2 i q^{37} + 4 i q^{38} + 2 q^{39} - 10 i q^{42} - 2 i q^{43} + 3 q^{44} + 6 q^{46} - 9 i q^{47} + 2 i q^{48} - 18 q^{49} - 6 q^{51} + i q^{52} + 9 i q^{53} - 4 q^{54} + 5 q^{56} + 8 i q^{57} - 9 i q^{58} + 9 q^{59} - q^{61} + 5 i q^{62} - 5 i q^{63} - q^{64} + 6 q^{66} + 5 i q^{67} - 3 i q^{68} + 12 q^{69} + i q^{72} - 14 i q^{73} - 2 q^{74} - 4 q^{76} - 15 i q^{77} + 2 i q^{78} + 16 q^{79} - 11 q^{81} + 15 i q^{83} + 10 q^{84} + 2 q^{86} - 18 i q^{87} + 3 i q^{88} + 6 q^{89} + 5 q^{91} + 6 i q^{92} + 10 i q^{93} + 9 q^{94} - 2 q^{96} + 8 i q^{97} - 18 i q^{98} + 3 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} - 4 q^{6} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} - 4 q^{6} - 2 q^{9} - 6 q^{11} - 10 q^{14} + 2 q^{16} + 8 q^{19} - 20 q^{21} + 4 q^{24} + 2 q^{26} - 18 q^{29} + 10 q^{31} - 6 q^{34} + 2 q^{36} + 4 q^{39} + 6 q^{44} + 12 q^{46} - 36 q^{49} - 12 q^{51} - 8 q^{54} + 10 q^{56} + 18 q^{59} - 2 q^{61} - 2 q^{64} + 12 q^{66} + 24 q^{69} - 4 q^{74} - 8 q^{76} + 32 q^{79} - 22 q^{81} + 20 q^{84} + 4 q^{86} + 12 q^{89} + 10 q^{91} + 18 q^{94} - 4 q^{96} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/650\mathbb{Z}\right)^\times\).

\(n\) \(27\) \(301\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
599.1
1.00000i
1.00000i
1.00000i 2.00000i −1.00000 0 −2.00000 5.00000i 1.00000i −1.00000 0
599.2 1.00000i 2.00000i −1.00000 0 −2.00000 5.00000i 1.00000i −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
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 650.2.b.b 2
3.b odd 2 1 5850.2.e.ba 2
5.b even 2 1 inner 650.2.b.b 2
5.c odd 4 1 650.2.a.f 1
5.c odd 4 1 650.2.a.h yes 1
15.d odd 2 1 5850.2.e.ba 2
15.e even 4 1 5850.2.a.bb 1
15.e even 4 1 5850.2.a.bc 1
20.e even 4 1 5200.2.a.i 1
20.e even 4 1 5200.2.a.bc 1
65.h odd 4 1 8450.2.a.a 1
65.h odd 4 1 8450.2.a.x 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
650.2.a.f 1 5.c odd 4 1
650.2.a.h yes 1 5.c odd 4 1
650.2.b.b 2 1.a even 1 1 trivial
650.2.b.b 2 5.b even 2 1 inner
5200.2.a.i 1 20.e even 4 1
5200.2.a.bc 1 20.e even 4 1
5850.2.a.bb 1 15.e even 4 1
5850.2.a.bc 1 15.e even 4 1
5850.2.e.ba 2 3.b odd 2 1
5850.2.e.ba 2 15.d odd 2 1
8450.2.a.a 1 65.h odd 4 1
8450.2.a.x 1 65.h odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(650, [\chi])\):

\( T_{3}^{2} + 4 \) Copy content Toggle raw display
\( T_{7}^{2} + 25 \) Copy content Toggle raw display
\( T_{11} + 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{2} + 4 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 25 \) Copy content Toggle raw display
$11$ \( (T + 3)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 1 \) Copy content Toggle raw display
$17$ \( T^{2} + 9 \) Copy content Toggle raw display
$19$ \( (T - 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 36 \) Copy content Toggle raw display
$29$ \( (T + 9)^{2} \) Copy content Toggle raw display
$31$ \( (T - 5)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 4 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 4 \) Copy content Toggle raw display
$47$ \( T^{2} + 81 \) Copy content Toggle raw display
$53$ \( T^{2} + 81 \) Copy content Toggle raw display
$59$ \( (T - 9)^{2} \) Copy content Toggle raw display
$61$ \( (T + 1)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 25 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 196 \) Copy content Toggle raw display
$79$ \( (T - 16)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 225 \) Copy content Toggle raw display
$89$ \( (T - 6)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 64 \) Copy content Toggle raw display
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