Properties

Label 1960.2.q.u
Level $1960$
Weight $2$
Character orbit 1960.q
Analytic conductor $15.651$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1960 = 2^{3} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1960.q (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(15.6506787962\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{17})\)
Defining polynomial: \(x^{4} - x^{3} + 5 x^{2} + 4 x + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 280)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + ( 1 - \beta_{2} ) q^{5} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + ( 1 - \beta_{2} ) q^{5} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{9} + \beta_{1} q^{11} + ( -2 - 3 \beta_{3} ) q^{13} -\beta_{3} q^{15} + ( -\beta_{1} + 6 \beta_{2} ) q^{17} + ( -4 - 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{19} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{23} -\beta_{2} q^{25} + ( -4 - \beta_{3} ) q^{27} + ( 2 - \beta_{3} ) q^{29} -4 \beta_{1} q^{31} + ( -4 + \beta_{1} + 4 \beta_{2} + \beta_{3} ) q^{33} + ( 2 + 4 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} ) q^{37} + ( \beta_{1} + 12 \beta_{2} ) q^{39} + ( -2 + 2 \beta_{3} ) q^{41} + ( -4 - 2 \beta_{3} ) q^{43} + ( \beta_{1} + \beta_{2} ) q^{45} + ( 4 - \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{47} + ( 4 + 5 \beta_{1} - 4 \beta_{2} + 5 \beta_{3} ) q^{51} + ( -2 \beta_{1} + 10 \beta_{2} ) q^{53} -\beta_{3} q^{55} + ( 8 + 2 \beta_{3} ) q^{57} -4 \beta_{2} q^{59} + ( -10 + 2 \beta_{1} + 10 \beta_{2} + 2 \beta_{3} ) q^{61} + ( -2 - 3 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} ) q^{65} + ( 4 \beta_{1} + 4 \beta_{2} ) q^{67} + ( 8 - 2 \beta_{3} ) q^{69} + ( 4 \beta_{1} + 2 \beta_{2} ) q^{73} + ( -\beta_{1} - \beta_{3} ) q^{75} + ( 4 - 3 \beta_{1} - 4 \beta_{2} - 3 \beta_{3} ) q^{79} + 7 \beta_{2} q^{81} -12 q^{83} + ( 6 + \beta_{3} ) q^{85} + ( 3 \beta_{1} + 4 \beta_{2} ) q^{87} + ( 2 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{89} + ( 16 - 4 \beta_{1} - 16 \beta_{2} - 4 \beta_{3} ) q^{93} + ( -2 \beta_{1} + 4 \beta_{2} ) q^{95} + ( -6 + 3 \beta_{3} ) q^{97} + ( -4 + 2 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + q^{3} + 2q^{5} - 3q^{9} + O(q^{10}) \) \( 4q + q^{3} + 2q^{5} - 3q^{9} + q^{11} - 2q^{13} + 2q^{15} + 11q^{17} - 6q^{19} + 2q^{23} - 2q^{25} - 14q^{27} + 10q^{29} - 4q^{31} - 9q^{33} + 25q^{39} - 12q^{41} - 12q^{43} + 3q^{45} + 9q^{47} + 3q^{51} + 18q^{53} + 2q^{55} + 28q^{57} - 8q^{59} - 22q^{61} - q^{65} + 12q^{67} + 36q^{69} + 8q^{73} + q^{75} + 11q^{79} + 14q^{81} - 48q^{83} + 22q^{85} + 11q^{87} + 2q^{89} + 36q^{93} + 6q^{95} - 30q^{97} - 20q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} + 5 x^{2} + 4 x + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + 5 \nu^{2} - 5 \nu + 16 \)\()/20\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} + 4 \)\()/5\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 4 \beta_{2} + \beta_{1} - 4\)
\(\nu^{3}\)\(=\)\(5 \beta_{3} - 4\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1960\mathbb{Z}\right)^\times\).

\(n\) \(981\) \(1081\) \(1177\) \(1471\)
\(\chi(n)\) \(1\) \(-1 + \beta_{2}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
−0.780776 + 1.35234i
1.28078 2.21837i
−0.780776 1.35234i
1.28078 + 2.21837i
0 −0.780776 + 1.35234i 0 0.500000 + 0.866025i 0 0 0 0.280776 + 0.486319i 0
361.2 0 1.28078 2.21837i 0 0.500000 + 0.866025i 0 0 0 −1.78078 3.08440i 0
961.1 0 −0.780776 1.35234i 0 0.500000 0.866025i 0 0 0 0.280776 0.486319i 0
961.2 0 1.28078 + 2.21837i 0 0.500000 0.866025i 0 0 0 −1.78078 + 3.08440i 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
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1960.2.q.u 4
7.b odd 2 1 1960.2.q.s 4
7.c even 3 1 1960.2.a.r 2
7.c even 3 1 inner 1960.2.q.u 4
7.d odd 6 1 280.2.a.d 2
7.d odd 6 1 1960.2.q.s 4
21.g even 6 1 2520.2.a.w 2
28.f even 6 1 560.2.a.g 2
28.g odd 6 1 3920.2.a.bu 2
35.i odd 6 1 1400.2.a.p 2
35.j even 6 1 9800.2.a.by 2
35.k even 12 2 1400.2.g.k 4
56.j odd 6 1 2240.2.a.be 2
56.m even 6 1 2240.2.a.bi 2
84.j odd 6 1 5040.2.a.bq 2
140.s even 6 1 2800.2.a.bn 2
140.x odd 12 2 2800.2.g.u 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.a.d 2 7.d odd 6 1
560.2.a.g 2 28.f even 6 1
1400.2.a.p 2 35.i odd 6 1
1400.2.g.k 4 35.k even 12 2
1960.2.a.r 2 7.c even 3 1
1960.2.q.s 4 7.b odd 2 1
1960.2.q.s 4 7.d odd 6 1
1960.2.q.u 4 1.a even 1 1 trivial
1960.2.q.u 4 7.c even 3 1 inner
2240.2.a.be 2 56.j odd 6 1
2240.2.a.bi 2 56.m even 6 1
2520.2.a.w 2 21.g even 6 1
2800.2.a.bn 2 140.s even 6 1
2800.2.g.u 4 140.x odd 12 2
3920.2.a.bu 2 28.g odd 6 1
5040.2.a.bq 2 84.j odd 6 1
9800.2.a.by 2 35.j even 6 1

Hecke kernels

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

\( T_{3}^{4} - T_{3}^{3} + 5 T_{3}^{2} + 4 T_{3} + 16 \)
\( T_{11}^{4} - T_{11}^{3} + 5 T_{11}^{2} + 4 T_{11} + 16 \)
\( T_{13}^{2} + T_{13} - 38 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 16 + 4 T + 5 T^{2} - T^{3} + T^{4} \)
$5$ \( ( 1 - T + T^{2} )^{2} \)
$7$ \( T^{4} \)
$11$ \( 16 + 4 T + 5 T^{2} - T^{3} + T^{4} \)
$13$ \( ( -38 + T + T^{2} )^{2} \)
$17$ \( 676 - 286 T + 95 T^{2} - 11 T^{3} + T^{4} \)
$19$ \( 64 - 48 T + 44 T^{2} + 6 T^{3} + T^{4} \)
$23$ \( 256 + 32 T + 20 T^{2} - 2 T^{3} + T^{4} \)
$29$ \( ( 2 - 5 T + T^{2} )^{2} \)
$31$ \( 4096 - 256 T + 80 T^{2} + 4 T^{3} + T^{4} \)
$37$ \( 4624 + 68 T^{2} + T^{4} \)
$41$ \( ( -8 + 6 T + T^{2} )^{2} \)
$43$ \( ( -8 + 6 T + T^{2} )^{2} \)
$47$ \( 256 - 144 T + 65 T^{2} - 9 T^{3} + T^{4} \)
$53$ \( 4096 - 1152 T + 260 T^{2} - 18 T^{3} + T^{4} \)
$59$ \( ( 16 + 4 T + T^{2} )^{2} \)
$61$ \( 10816 + 2288 T + 380 T^{2} + 22 T^{3} + T^{4} \)
$67$ \( 1024 + 384 T + 176 T^{2} - 12 T^{3} + T^{4} \)
$71$ \( T^{4} \)
$73$ \( 2704 + 416 T + 116 T^{2} - 8 T^{3} + T^{4} \)
$79$ \( 64 + 88 T + 129 T^{2} - 11 T^{3} + T^{4} \)
$83$ \( ( 12 + T )^{4} \)
$89$ \( 256 + 32 T + 20 T^{2} - 2 T^{3} + T^{4} \)
$97$ \( ( 18 + 15 T + T^{2} )^{2} \)
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