Properties

Label 1960.2.q.y
Level $1960$
Weight $2$
Character orbit 1960.q
Analytic conductor $15.651$
Analytic rank $0$
Dimension $8$
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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.21913473024.16
Defining polynomial: \(x^{8} - 2 x^{7} + 11 x^{6} - 2 x^{5} + 51 x^{4} + 162 x^{2} + 112 x + 196\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + \beta_{3} q^{5} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + \beta_{3} q^{5} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} ) q^{9} + ( -\beta_{1} - \beta_{6} ) q^{11} + ( -3 + \beta_{2} + \beta_{4} - \beta_{7} ) q^{13} + ( 1 - \beta_{2} ) q^{15} + ( 2 - \beta_{1} - 2 \beta_{3} + 2 \beta_{5} ) q^{17} + ( \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{19} + ( -2 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{23} + ( -1 + \beta_{3} ) q^{25} + ( -3 - \beta_{2} + \beta_{4} + \beta_{7} ) q^{27} + ( -1 + \beta_{2} + 3 \beta_{4} ) q^{29} + ( -2 - 2 \beta_{1} + 2 \beta_{3} + \beta_{5} + 2 \beta_{6} ) q^{31} + ( 1 - \beta_{1} - \beta_{2} + 4 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{33} + ( 2 \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{37} + ( -2 - 3 \beta_{1} + 2 \beta_{3} - 3 \beta_{6} ) q^{39} + ( -2 - 2 \beta_{2} - \beta_{4} - 2 \beta_{7} ) q^{41} + ( -2 + 2 \beta_{4} - 2 \beta_{7} ) q^{43} + ( 1 + \beta_{1} - \beta_{3} - \beta_{6} ) q^{45} + ( -1 + \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{47} + ( -1 + \beta_{1} + \beta_{2} + 7 \beta_{4} + 2 \beta_{5} - 7 \beta_{6} + 2 \beta_{7} ) q^{51} + ( 2 \beta_{1} + \beta_{5} + 2 \beta_{6} ) q^{53} + ( -1 + \beta_{2} - \beta_{4} ) q^{55} + ( -2 + 4 \beta_{4} + 2 \beta_{7} ) q^{57} + ( 2 - 2 \beta_{3} - 5 \beta_{6} ) q^{59} + ( -2 + 2 \beta_{1} + 2 \beta_{2} + 6 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{61} + ( -1 + \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{65} + ( 2 - 2 \beta_{3} - \beta_{5} - 6 \beta_{6} ) q^{67} + ( -6 - 2 \beta_{4} + \beta_{7} ) q^{69} + ( 2 - 2 \beta_{2} + 4 \beta_{7} ) q^{71} + ( 2 + 4 \beta_{1} - 2 \beta_{3} + \beta_{5} + 5 \beta_{6} ) q^{73} + ( 1 - \beta_{1} - \beta_{2} ) q^{75} + ( 1 - \beta_{1} - \beta_{2} - 6 \beta_{3} + 5 \beta_{4} - 5 \beta_{6} ) q^{79} + ( 5 - 5 \beta_{3} - 2 \beta_{5} ) q^{81} + ( -10 + 2 \beta_{2} - \beta_{4} + \beta_{7} ) q^{83} + ( 1 + \beta_{2} - 2 \beta_{7} ) q^{85} + ( -4 - \beta_{1} + 4 \beta_{3} - 3 \beta_{5} + \beta_{6} ) q^{87} + ( 10 \beta_{3} + 3 \beta_{4} - \beta_{5} - 3 \beta_{6} - \beta_{7} ) q^{89} + ( 4 - 4 \beta_{1} - 4 \beta_{2} + 6 \beta_{3} + 2 \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{93} + ( \beta_{5} - \beta_{6} ) q^{95} + ( -7 + \beta_{2} - 2 \beta_{4} - 2 \beta_{7} ) q^{97} + ( 4 - 2 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 2q^{3} + 4q^{5} - 6q^{9} + O(q^{10}) \) \( 8q + 2q^{3} + 4q^{5} - 6q^{9} - 2q^{11} - 20q^{13} + 4q^{15} + 6q^{17} + 4q^{23} - 4q^{25} - 28q^{27} - 4q^{29} - 12q^{31} + 18q^{33} - 14q^{39} - 24q^{41} - 16q^{43} + 6q^{45} - 2q^{47} - 2q^{51} + 4q^{53} - 4q^{55} - 16q^{57} + 8q^{59} + 20q^{61} - 10q^{65} + 8q^{67} - 48q^{69} + 8q^{71} + 16q^{73} + 2q^{75} - 22q^{79} + 20q^{81} - 72q^{83} + 12q^{85} - 18q^{87} + 40q^{89} + 32q^{93} - 52q^{97} + 24q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 2 x^{7} + 11 x^{6} - 2 x^{5} + 51 x^{4} + 162 x^{2} + 112 x + 196\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 329 \nu^{7} + 154 \nu^{6} + 2209 \nu^{5} + 4794 \nu^{4} + 28611 \nu^{3} + 20492 \nu^{2} + 19740 \nu + 169702 \)\()/84134\)
\(\beta_{3}\)\(=\)\((\)\( 3056 \nu^{7} - 8415 \nu^{6} + 32538 \nu^{5} - 21575 \nu^{4} + 122298 \nu^{3} - 200277 \nu^{2} + 351628 \nu + 204092 \)\()/588938\)
\(\beta_{4}\)\(=\)\((\)\( 21 \nu^{7} - 68 \nu^{6} + 141 \nu^{5} + 306 \nu^{4} - 353 \nu^{3} + 1308 \nu^{2} + 1260 \nu + 8108 \)\()/3658\)
\(\beta_{5}\)\(=\)\((\)\( 13 \nu^{7} + 45 \nu^{6} - 174 \nu^{5} + 712 \nu^{4} - 654 \nu^{3} + 1071 \nu^{2} - 2878 \nu + 2058 \)\()/1829\)
\(\beta_{6}\)\(=\)\((\)\( -3270 \nu^{7} + 11895 \nu^{6} - 45994 \nu^{5} + 84562 \nu^{4} - 172874 \nu^{3} + 283101 \nu^{2} - 238267 \nu + 543998 \)\()/294469\)
\(\beta_{7}\)\(=\)\((\)\( -1064 \nu^{7} + 397 \nu^{6} - 7144 \nu^{5} - 15504 \nu^{4} - 25401 \nu^{3} - 66272 \nu^{2} - 63840 \nu - 138894 \)\()/42067\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{6} + \beta_{4} - 4 \beta_{3} + \beta_{2} + \beta_{1} - 1\)
\(\nu^{3}\)\(=\)\(\beta_{7} + \beta_{4} + 5 \beta_{2} - 9\)
\(\nu^{4}\)\(=\)\(9 \beta_{6} - 2 \beta_{5} + 22 \beta_{3} - 9 \beta_{1} - 22\)
\(\nu^{5}\)\(=\)\(-11 \beta_{7} + 17 \beta_{6} - 11 \beta_{5} - 17 \beta_{4} + 40 \beta_{3} - 31 \beta_{2} - 31 \beta_{1} + 31\)
\(\nu^{6}\)\(=\)\(-28 \beta_{7} - 75 \beta_{4} - 71 \beta_{2} + 217\)
\(\nu^{7}\)\(=\)\(-183 \beta_{6} + 103 \beta_{5} - 340 \beta_{3} + 217 \beta_{1} + 340\)

Character values

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

\(n\) \(981\) \(1081\) \(1177\) \(1471\)
\(\chi(n)\) \(1\) \(-\beta_{3}\) \(1\) \(1\)

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
−0.939980 + 1.62809i
−0.591990 + 1.02536i
1.09199 1.89138i
1.43998 2.49412i
−0.939980 1.62809i
−0.591990 1.02536i
1.09199 + 1.89138i
1.43998 + 2.49412i
0 −0.939980 + 1.62809i 0 0.500000 + 0.866025i 0 0 0 −0.267126 0.462676i 0
361.2 0 −0.591990 + 1.02536i 0 0.500000 + 0.866025i 0 0 0 0.799096 + 1.38408i 0
361.3 0 1.09199 1.89138i 0 0.500000 + 0.866025i 0 0 0 −0.884883 1.53266i 0
361.4 0 1.43998 2.49412i 0 0.500000 + 0.866025i 0 0 0 −2.64709 4.58489i 0
961.1 0 −0.939980 1.62809i 0 0.500000 0.866025i 0 0 0 −0.267126 + 0.462676i 0
961.2 0 −0.591990 1.02536i 0 0.500000 0.866025i 0 0 0 0.799096 1.38408i 0
961.3 0 1.09199 + 1.89138i 0 0.500000 0.866025i 0 0 0 −0.884883 + 1.53266i 0
961.4 0 1.43998 + 2.49412i 0 0.500000 0.866025i 0 0 0 −2.64709 + 4.58489i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 961.4
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.y 8
7.b odd 2 1 1960.2.q.x 8
7.c even 3 1 1960.2.a.x 4
7.c even 3 1 inner 1960.2.q.y 8
7.d odd 6 1 1960.2.a.y yes 4
7.d odd 6 1 1960.2.q.x 8
28.f even 6 1 3920.2.a.cd 4
28.g odd 6 1 3920.2.a.ce 4
35.i odd 6 1 9800.2.a.cl 4
35.j even 6 1 9800.2.a.cs 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1960.2.a.x 4 7.c even 3 1
1960.2.a.y yes 4 7.d odd 6 1
1960.2.q.x 8 7.b odd 2 1
1960.2.q.x 8 7.d odd 6 1
1960.2.q.y 8 1.a even 1 1 trivial
1960.2.q.y 8 7.c even 3 1 inner
3920.2.a.cd 4 28.f even 6 1
3920.2.a.ce 4 28.g odd 6 1
9800.2.a.cl 4 35.i odd 6 1
9800.2.a.cs 4 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 \(S_{2}^{\mathrm{new}}(1960, [\chi])\):

\( T_{3}^{8} - 2 T_{3}^{7} + 11 T_{3}^{6} - 2 T_{3}^{5} + 51 T_{3}^{4} + 162 T_{3}^{2} + 112 T_{3} + 196 \)
\(T_{11}^{8} + \cdots\)
\( T_{13}^{4} + 10 T_{13}^{3} + 19 T_{13}^{2} - 52 T_{13} - 124 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( 196 + 112 T + 162 T^{2} + 51 T^{4} - 2 T^{5} + 11 T^{6} - 2 T^{7} + T^{8} \)
$5$ \( ( 1 - T + T^{2} )^{4} \)
$7$ \( T^{8} \)
$11$ \( 16 + 80 T + 356 T^{2} + 236 T^{3} + 165 T^{4} + 18 T^{5} + 15 T^{6} + 2 T^{7} + T^{8} \)
$13$ \( ( -124 - 52 T + 19 T^{2} + 10 T^{3} + T^{4} )^{2} \)
$17$ \( 188356 + 105896 T + 81670 T^{2} - 7236 T^{3} + 3631 T^{4} - 182 T^{5} + 87 T^{6} - 6 T^{7} + T^{8} \)
$19$ \( 64 + 192 T + 784 T^{2} - 624 T^{3} + 668 T^{4} - 48 T^{5} + 26 T^{6} + T^{8} \)
$23$ \( 256 - 768 T + 2912 T^{2} + 1952 T^{3} + 1236 T^{4} + 248 T^{5} + 54 T^{6} - 4 T^{7} + T^{8} \)
$29$ \( ( 188 - 20 T - 43 T^{2} + 2 T^{3} + T^{4} )^{2} \)
$31$ \( 614656 + 225792 T + 90784 T^{2} + 15936 T^{3} + 4340 T^{4} + 696 T^{5} + 134 T^{6} + 12 T^{7} + T^{8} \)
$37$ \( 64 + 320 T + 1936 T^{2} - 1680 T^{3} + 1756 T^{4} - 80 T^{5} + 42 T^{6} + T^{8} \)
$41$ \( ( -4228 - 1528 T - 96 T^{2} + 12 T^{3} + T^{4} )^{2} \)
$43$ \( ( 752 - 192 T - 48 T^{2} + 8 T^{3} + T^{4} )^{2} \)
$47$ \( 3136 - 5376 T + 6920 T^{2} - 3712 T^{3} + 1545 T^{4} - 274 T^{5} + 45 T^{6} + 2 T^{7} + T^{8} \)
$53$ \( 322624 - 240832 T + 140016 T^{2} - 34224 T^{3} + 7164 T^{4} - 568 T^{5} + 86 T^{6} - 4 T^{7} + T^{8} \)
$59$ \( ( 2116 + 184 T + 62 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$61$ \( 28558336 - 5814272 T + 1418880 T^{2} - 165888 T^{3} + 29040 T^{4} - 3056 T^{5} + 356 T^{6} - 20 T^{7} + T^{8} \)
$67$ \( 5456896 + 1009152 T + 452928 T^{2} - 11872 T^{3} + 14116 T^{4} + 48 T^{5} + 178 T^{6} - 8 T^{7} + T^{8} \)
$71$ \( ( 10976 - 252 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$73$ \( 201412864 - 53134848 T + 11604896 T^{2} - 1090624 T^{3} + 102996 T^{4} - 4768 T^{5} + 426 T^{6} - 16 T^{7} + T^{8} \)
$79$ \( 3136 + 27328 T + 242232 T^{2} - 33160 T^{3} + 16121 T^{4} + 2582 T^{5} + 411 T^{6} + 22 T^{7} + T^{8} \)
$83$ \( ( 256 + 1600 T + 418 T^{2} + 36 T^{3} + T^{4} )^{2} \)
$89$ \( 29246464 - 16872960 T + 6716736 T^{2} - 1308320 T^{3} + 181156 T^{4} - 16080 T^{5} + 1042 T^{6} - 40 T^{7} + T^{8} \)
$97$ \( ( -1022 + 36 T + 157 T^{2} + 26 T^{3} + T^{4} )^{2} \)
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