Properties

Label 1960.2.a.s
Level $1960$
Weight $2$
Character orbit 1960.a
Self dual yes
Analytic conductor $15.651$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1960 = 2^{3} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1960.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(15.6506787962\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Defining polynomial: \( x^{2} - x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 280)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{33})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} + q^{5} + (\beta + 5) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{3} + q^{5} + (\beta + 5) q^{9} + ( - \beta + 4) q^{11} + (\beta - 2) q^{13} + \beta q^{15} + ( - \beta - 2) q^{17} - 2 \beta q^{19} + 2 \beta q^{23} + q^{25} + (3 \beta + 8) q^{27} + (\beta - 2) q^{29} + 8 q^{31} + (3 \beta - 8) q^{33} - 2 q^{37} + ( - \beta + 8) q^{39} + (2 \beta - 2) q^{41} + (2 \beta - 4) q^{43} + (\beta + 5) q^{45} - 3 \beta q^{47} + ( - 3 \beta - 8) q^{51} + ( - 2 \beta + 6) q^{53} + ( - \beta + 4) q^{55} + ( - 2 \beta - 16) q^{57} - 8 q^{59} + ( - 2 \beta - 2) q^{61} + (\beta - 2) q^{65} - 4 q^{67} + (2 \beta + 16) q^{69} + 8 q^{71} + 6 q^{73} + \beta q^{75} + ( - 3 \beta + 8) q^{79} + (8 \beta + 9) q^{81} - 4 \beta q^{83} + ( - \beta - 2) q^{85} + ( - \beta + 8) q^{87} + (2 \beta - 10) q^{89} + 8 \beta q^{93} - 2 \beta q^{95} + ( - 5 \beta - 2) q^{97} + ( - 2 \beta + 12) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} + 2 q^{5} + 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{3} + 2 q^{5} + 11 q^{9} + 7 q^{11} - 3 q^{13} + q^{15} - 5 q^{17} - 2 q^{19} + 2 q^{23} + 2 q^{25} + 19 q^{27} - 3 q^{29} + 16 q^{31} - 13 q^{33} - 4 q^{37} + 15 q^{39} - 2 q^{41} - 6 q^{43} + 11 q^{45} - 3 q^{47} - 19 q^{51} + 10 q^{53} + 7 q^{55} - 34 q^{57} - 16 q^{59} - 6 q^{61} - 3 q^{65} - 8 q^{67} + 34 q^{69} + 16 q^{71} + 12 q^{73} + q^{75} + 13 q^{79} + 26 q^{81} - 4 q^{83} - 5 q^{85} + 15 q^{87} - 18 q^{89} + 8 q^{93} - 2 q^{95} - 9 q^{97} + 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.37228
3.37228
0 −2.37228 0 1.00000 0 0 0 2.62772 0
1.2 0 3.37228 0 1.00000 0 0 0 8.37228 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)
\(7\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1960.2.a.s 2
4.b odd 2 1 3920.2.a.bt 2
5.b even 2 1 9800.2.a.bu 2
7.b odd 2 1 280.2.a.c 2
7.c even 3 2 1960.2.q.r 4
7.d odd 6 2 1960.2.q.t 4
21.c even 2 1 2520.2.a.x 2
28.d even 2 1 560.2.a.h 2
35.c odd 2 1 1400.2.a.r 2
35.f even 4 2 1400.2.g.i 4
56.e even 2 1 2240.2.a.bg 2
56.h odd 2 1 2240.2.a.bk 2
84.h odd 2 1 5040.2.a.by 2
140.c even 2 1 2800.2.a.bk 2
140.j odd 4 2 2800.2.g.r 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.a.c 2 7.b odd 2 1
560.2.a.h 2 28.d even 2 1
1400.2.a.r 2 35.c odd 2 1
1400.2.g.i 4 35.f even 4 2
1960.2.a.s 2 1.a even 1 1 trivial
1960.2.q.r 4 7.c even 3 2
1960.2.q.t 4 7.d odd 6 2
2240.2.a.bg 2 56.e even 2 1
2240.2.a.bk 2 56.h odd 2 1
2520.2.a.x 2 21.c even 2 1
2800.2.a.bk 2 140.c even 2 1
2800.2.g.r 4 140.j odd 4 2
3920.2.a.bt 2 4.b odd 2 1
5040.2.a.by 2 84.h odd 2 1
9800.2.a.bu 2 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1960))\):

\( T_{3}^{2} - T_{3} - 8 \) Copy content Toggle raw display
\( T_{11}^{2} - 7T_{11} + 4 \) Copy content Toggle raw display
\( T_{13}^{2} + 3T_{13} - 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - T - 8 \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 7T + 4 \) Copy content Toggle raw display
$13$ \( T^{2} + 3T - 6 \) Copy content Toggle raw display
$17$ \( T^{2} + 5T - 2 \) Copy content Toggle raw display
$19$ \( T^{2} + 2T - 32 \) Copy content Toggle raw display
$23$ \( T^{2} - 2T - 32 \) Copy content Toggle raw display
$29$ \( T^{2} + 3T - 6 \) Copy content Toggle raw display
$31$ \( (T - 8)^{2} \) Copy content Toggle raw display
$37$ \( (T + 2)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 2T - 32 \) Copy content Toggle raw display
$43$ \( T^{2} + 6T - 24 \) Copy content Toggle raw display
$47$ \( T^{2} + 3T - 72 \) Copy content Toggle raw display
$53$ \( T^{2} - 10T - 8 \) Copy content Toggle raw display
$59$ \( (T + 8)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 6T - 24 \) Copy content Toggle raw display
$67$ \( (T + 4)^{2} \) Copy content Toggle raw display
$71$ \( (T - 8)^{2} \) Copy content Toggle raw display
$73$ \( (T - 6)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 13T - 32 \) Copy content Toggle raw display
$83$ \( T^{2} + 4T - 128 \) Copy content Toggle raw display
$89$ \( T^{2} + 18T + 48 \) Copy content Toggle raw display
$97$ \( T^{2} + 9T - 186 \) Copy content Toggle raw display
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