Properties

Label 2960.2.p.c
Level $2960$
Weight $2$
Character orbit 2960.p
Analytic conductor $23.636$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2960 = 2^{4} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2960.p (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(23.6357189983\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 185)
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.

\(f(q)\) \(=\) \( q - q^{3} + i q^{5} - q^{7} -2 q^{9} +O(q^{10})\) \( q - q^{3} + i q^{5} - q^{7} -2 q^{9} + 3 q^{11} -i q^{15} -6 i q^{17} -6 i q^{19} + q^{21} + 6 i q^{23} - q^{25} + 5 q^{27} + 6 i q^{29} + 6 i q^{31} -3 q^{33} -i q^{35} + ( 1 + 6 i ) q^{37} -3 q^{41} + 6 i q^{43} -2 i q^{45} + 3 q^{47} -6 q^{49} + 6 i q^{51} -9 q^{53} + 3 i q^{55} + 6 i q^{57} -12 i q^{59} -6 i q^{61} + 2 q^{63} -4 q^{67} -6 i q^{69} -9 q^{71} + 7 q^{73} + q^{75} -3 q^{77} -12 i q^{79} + q^{81} -15 q^{83} + 6 q^{85} -6 i q^{87} -6 i q^{93} + 6 q^{95} -6 i q^{97} -6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} - 2q^{7} - 4q^{9} + O(q^{10}) \) \( 2q - 2q^{3} - 2q^{7} - 4q^{9} + 6q^{11} + 2q^{21} - 2q^{25} + 10q^{27} - 6q^{33} + 2q^{37} - 6q^{41} + 6q^{47} - 12q^{49} - 18q^{53} + 4q^{63} - 8q^{67} - 18q^{71} + 14q^{73} + 2q^{75} - 6q^{77} + 2q^{81} - 30q^{83} + 12q^{85} + 12q^{95} - 12q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2960\mathbb{Z}\right)^\times\).

\(n\) \(741\) \(1777\) \(2481\) \(2591\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
961.1
1.00000i
1.00000i
0 −1.00000 0 1.00000i 0 −1.00000 0 −2.00000 0
961.2 0 −1.00000 0 1.00000i 0 −1.00000 0 −2.00000 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
37.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2960.2.p.c 2
4.b odd 2 1 185.2.c.a 2
12.b even 2 1 1665.2.e.b 2
20.d odd 2 1 925.2.c.a 2
20.e even 4 1 925.2.d.b 2
20.e even 4 1 925.2.d.c 2
37.b even 2 1 inner 2960.2.p.c 2
148.b odd 2 1 185.2.c.a 2
148.g even 4 1 6845.2.a.c 1
148.g even 4 1 6845.2.a.d 1
444.g even 2 1 1665.2.e.b 2
740.g odd 2 1 925.2.c.a 2
740.m even 4 1 925.2.d.b 2
740.m even 4 1 925.2.d.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
185.2.c.a 2 4.b odd 2 1
185.2.c.a 2 148.b odd 2 1
925.2.c.a 2 20.d odd 2 1
925.2.c.a 2 740.g odd 2 1
925.2.d.b 2 20.e even 4 1
925.2.d.b 2 740.m even 4 1
925.2.d.c 2 20.e even 4 1
925.2.d.c 2 740.m even 4 1
1665.2.e.b 2 12.b even 2 1
1665.2.e.b 2 444.g even 2 1
2960.2.p.c 2 1.a even 1 1 trivial
2960.2.p.c 2 37.b even 2 1 inner
6845.2.a.c 1 148.g even 4 1
6845.2.a.d 1 148.g even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2960, [\chi])\):

\( T_{3} + 1 \)
\( T_{7} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( 1 + T^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( ( -3 + T )^{2} \)
$13$ \( T^{2} \)
$17$ \( 36 + T^{2} \)
$19$ \( 36 + T^{2} \)
$23$ \( 36 + T^{2} \)
$29$ \( 36 + T^{2} \)
$31$ \( 36 + T^{2} \)
$37$ \( 37 - 2 T + T^{2} \)
$41$ \( ( 3 + T )^{2} \)
$43$ \( 36 + T^{2} \)
$47$ \( ( -3 + T )^{2} \)
$53$ \( ( 9 + T )^{2} \)
$59$ \( 144 + T^{2} \)
$61$ \( 36 + T^{2} \)
$67$ \( ( 4 + T )^{2} \)
$71$ \( ( 9 + T )^{2} \)
$73$ \( ( -7 + T )^{2} \)
$79$ \( 144 + T^{2} \)
$83$ \( ( 15 + T )^{2} \)
$89$ \( T^{2} \)
$97$ \( 36 + T^{2} \)
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