Properties

Label 2960.2.p.b
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 1480)
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.

\(f(q)\) \(=\) \( q -2 q^{3} + i q^{5} -4 q^{7} + q^{9} +O(q^{10})\) \( q -2 q^{3} + i q^{5} -4 q^{7} + q^{9} -4 q^{11} -2 i q^{13} -2 i q^{15} + 6 i q^{17} + 8 i q^{19} + 8 q^{21} + 4 i q^{23} - q^{25} + 4 q^{27} + 6 i q^{29} -2 i q^{31} + 8 q^{33} -4 i q^{35} + ( -6 + i ) q^{37} + 4 i q^{39} -2 q^{41} + 4 i q^{43} + i q^{45} -12 q^{47} + 9 q^{49} -12 i q^{51} -12 q^{53} -4 i q^{55} -16 i q^{57} -8 i q^{59} -2 i q^{61} -4 q^{63} + 2 q^{65} + 2 q^{67} -8 i q^{69} + 8 q^{71} + 6 q^{73} + 2 q^{75} + 16 q^{77} -6 i q^{79} -11 q^{81} -14 q^{83} -6 q^{85} -12 i q^{87} -4 i q^{89} + 8 i q^{91} + 4 i q^{93} -8 q^{95} + 2 i q^{97} -4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{3} - 8q^{7} + 2q^{9} + O(q^{10}) \) \( 2q - 4q^{3} - 8q^{7} + 2q^{9} - 8q^{11} + 16q^{21} - 2q^{25} + 8q^{27} + 16q^{33} - 12q^{37} - 4q^{41} - 24q^{47} + 18q^{49} - 24q^{53} - 8q^{63} + 4q^{65} + 4q^{67} + 16q^{71} + 12q^{73} + 4q^{75} + 32q^{77} - 22q^{81} - 28q^{83} - 12q^{85} - 16q^{95} - 8q^{99} + O(q^{100}) \)

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 −2.00000 0 1.00000i 0 −4.00000 0 1.00000 0
961.2 0 −2.00000 0 1.00000i 0 −4.00000 0 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
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.b 2
4.b odd 2 1 1480.2.p.a 2
37.b even 2 1 inner 2960.2.p.b 2
148.b odd 2 1 1480.2.p.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1480.2.p.a 2 4.b odd 2 1
1480.2.p.a 2 148.b odd 2 1
2960.2.p.b 2 1.a even 1 1 trivial
2960.2.p.b 2 37.b even 2 1 inner

Hecke kernels

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

\( T_{3} + 2 \)
\( T_{7} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( 2 + T )^{2} \)
$5$ \( 1 + T^{2} \)
$7$ \( ( 4 + T )^{2} \)
$11$ \( ( 4 + T )^{2} \)
$13$ \( 4 + T^{2} \)
$17$ \( 36 + T^{2} \)
$19$ \( 64 + T^{2} \)
$23$ \( 16 + T^{2} \)
$29$ \( 36 + T^{2} \)
$31$ \( 4 + T^{2} \)
$37$ \( 37 + 12 T + T^{2} \)
$41$ \( ( 2 + T )^{2} \)
$43$ \( 16 + T^{2} \)
$47$ \( ( 12 + T )^{2} \)
$53$ \( ( 12 + T )^{2} \)
$59$ \( 64 + T^{2} \)
$61$ \( 4 + T^{2} \)
$67$ \( ( -2 + T )^{2} \)
$71$ \( ( -8 + T )^{2} \)
$73$ \( ( -6 + T )^{2} \)
$79$ \( 36 + T^{2} \)
$83$ \( ( 14 + T )^{2} \)
$89$ \( 16 + T^{2} \)
$97$ \( 4 + T^{2} \)
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