Properties

Label 130.2.d.a
Level $130$
Weight $2$
Character orbit 130.d
Analytic conductor $1.038$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 130 = 2 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 130.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.03805522628\)
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.

\(f(q)\) \(=\) \( q + i q^{2} + 2 q^{3} - q^{4} + i q^{5} + 2 i q^{6} -i q^{8} + q^{9} +O(q^{10})\) \( q + i q^{2} + 2 q^{3} - q^{4} + i q^{5} + 2 i q^{6} -i q^{8} + q^{9} - q^{10} -2 q^{12} + ( 2 - 3 i ) q^{13} + 2 i q^{15} + q^{16} -6 q^{17} + i q^{18} -i q^{20} -2 i q^{24} - q^{25} + ( 3 + 2 i ) q^{26} -4 q^{27} + 6 q^{29} -2 q^{30} -6 i q^{31} + i q^{32} -6 i q^{34} - q^{36} -6 i q^{37} + ( 4 - 6 i ) q^{39} + q^{40} -10 q^{43} + i q^{45} + 12 i q^{47} + 2 q^{48} + 7 q^{49} -i q^{50} -12 q^{51} + ( -2 + 3 i ) q^{52} -4 i q^{54} + 6 i q^{58} + 12 i q^{59} -2 i q^{60} + 10 q^{61} + 6 q^{62} - q^{64} + ( 3 + 2 i ) q^{65} + 12 i q^{67} + 6 q^{68} + 6 i q^{71} -i q^{72} + 6 i q^{73} + 6 q^{74} -2 q^{75} + ( 6 + 4 i ) q^{78} -8 q^{79} + i q^{80} -11 q^{81} -12 i q^{83} -6 i q^{85} -10 i q^{86} + 12 q^{87} + 12 i q^{89} - q^{90} -12 i q^{93} -12 q^{94} + 2 i q^{96} -18 i q^{97} + 7 i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{3} - 2q^{4} + 2q^{9} + O(q^{10}) \) \( 2q + 4q^{3} - 2q^{4} + 2q^{9} - 2q^{10} - 4q^{12} + 4q^{13} + 2q^{16} - 12q^{17} - 2q^{25} + 6q^{26} - 8q^{27} + 12q^{29} - 4q^{30} - 2q^{36} + 8q^{39} + 2q^{40} - 20q^{43} + 4q^{48} + 14q^{49} - 24q^{51} - 4q^{52} + 20q^{61} + 12q^{62} - 2q^{64} + 6q^{65} + 12q^{68} + 12q^{74} - 4q^{75} + 12q^{78} - 16q^{79} - 22q^{81} + 24q^{87} - 2q^{90} - 24q^{94} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/130\mathbb{Z}\right)^\times\).

\(n\) \(27\) \(41\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
51.1
1.00000i
1.00000i
1.00000i 2.00000 −1.00000 1.00000i 2.00000i 0 1.00000i 1.00000 −1.00000
51.2 1.00000i 2.00000 −1.00000 1.00000i 2.00000i 0 1.00000i 1.00000 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 130.2.d.a 2
3.b odd 2 1 1170.2.b.a 2
4.b odd 2 1 1040.2.k.a 2
5.b even 2 1 650.2.d.a 2
5.c odd 4 1 650.2.c.b 2
5.c odd 4 1 650.2.c.c 2
13.b even 2 1 inner 130.2.d.a 2
13.c even 3 2 1690.2.l.b 4
13.d odd 4 1 1690.2.a.d 1
13.d odd 4 1 1690.2.a.i 1
13.e even 6 2 1690.2.l.b 4
13.f odd 12 2 1690.2.e.b 2
13.f odd 12 2 1690.2.e.f 2
39.d odd 2 1 1170.2.b.a 2
52.b odd 2 1 1040.2.k.a 2
65.d even 2 1 650.2.d.a 2
65.g odd 4 1 8450.2.a.b 1
65.g odd 4 1 8450.2.a.o 1
65.h odd 4 1 650.2.c.b 2
65.h odd 4 1 650.2.c.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.2.d.a 2 1.a even 1 1 trivial
130.2.d.a 2 13.b even 2 1 inner
650.2.c.b 2 5.c odd 4 1
650.2.c.b 2 65.h odd 4 1
650.2.c.c 2 5.c odd 4 1
650.2.c.c 2 65.h odd 4 1
650.2.d.a 2 5.b even 2 1
650.2.d.a 2 65.d even 2 1
1040.2.k.a 2 4.b odd 2 1
1040.2.k.a 2 52.b odd 2 1
1170.2.b.a 2 3.b odd 2 1
1170.2.b.a 2 39.d odd 2 1
1690.2.a.d 1 13.d odd 4 1
1690.2.a.i 1 13.d odd 4 1
1690.2.e.b 2 13.f odd 12 2
1690.2.e.f 2 13.f odd 12 2
1690.2.l.b 4 13.c even 3 2
1690.2.l.b 4 13.e even 6 2
8450.2.a.b 1 65.g odd 4 1
8450.2.a.o 1 65.g odd 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(130, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( ( -2 + T )^{2} \)
$5$ \( 1 + T^{2} \)
$7$ \( T^{2} \)
$11$ \( T^{2} \)
$13$ \( 13 - 4 T + T^{2} \)
$17$ \( ( 6 + T )^{2} \)
$19$ \( T^{2} \)
$23$ \( T^{2} \)
$29$ \( ( -6 + T )^{2} \)
$31$ \( 36 + T^{2} \)
$37$ \( 36 + T^{2} \)
$41$ \( T^{2} \)
$43$ \( ( 10 + T )^{2} \)
$47$ \( 144 + T^{2} \)
$53$ \( T^{2} \)
$59$ \( 144 + T^{2} \)
$61$ \( ( -10 + T )^{2} \)
$67$ \( 144 + T^{2} \)
$71$ \( 36 + T^{2} \)
$73$ \( 36 + T^{2} \)
$79$ \( ( 8 + T )^{2} \)
$83$ \( 144 + T^{2} \)
$89$ \( 144 + T^{2} \)
$97$ \( 324 + T^{2} \)
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