Properties

Label 980.6.a.h
Level $980$
Weight $6$
Character orbit 980.a
Self dual yes
Analytic conductor $157.176$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 980.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(157.176143417\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Defining polynomial: \( x^{3} - 499x - 210 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 140)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( - \beta_1 - 2) q^{3} - 25 q^{5} + (\beta_{2} + 5 \beta_1 + 94) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 2) q^{3} - 25 q^{5} + (\beta_{2} + 5 \beta_1 + 94) q^{9} + ( - \beta_{2} + 11 \beta_1 - 5) q^{11} + (2 \beta_{2} - 13 \beta_1 + 2) q^{13} + (25 \beta_1 + 50) q^{15} + (2 \beta_{2} - 31 \beta_1 - 14) q^{17} + ( - 9 \beta_{2} - 6 \beta_1 - 779) q^{19} + ( - \beta_{2} - 70 \beta_1 + 1225) q^{23} + 625 q^{25} + ( - 6 \beta_{2} - 31 \beta_1 - 1244) q^{27} + ( - 15 \beta_{2} - 99 \beta_1 + 1359) q^{29} + (17 \beta_{2} + 248 \beta_1 - 1957) q^{31} + ( - 10 \beta_{2} + 137 \beta_1 - 3776) q^{33} + (\beta_{2} + 64 \beta_1 + 3793) q^{37} + (11 \beta_{2} - 293 \beta_1 + 4571) q^{39} + ( - 31 \beta_{2} - 142 \beta_1 - 3827) q^{41} + ( - 31 \beta_{2} - 322 \beta_1 + 6171) q^{43} + ( - 25 \beta_{2} - 125 \beta_1 - 2350) q^{45} + (16 \beta_{2} - 113 \beta_1 - 7246) q^{47} + (29 \beta_{2} - 223 \beta_1 + 10597) q^{51} + ( - 6 \beta_{2} + 594 \beta_1 + 2496) q^{53} + (25 \beta_{2} - 275 \beta_1 + 125) q^{55} + (15 \beta_{2} + 2282 \beta_1 + 2449) q^{57} + (64 \beta_{2} + 1840 \beta_1 - 4108) q^{59} + (71 \beta_{2} + 110 \beta_1 - 9037) q^{61} + ( - 50 \beta_{2} + 325 \beta_1 - 50) q^{65} + (8 \beta_{2} + 2228 \beta_1 - 2556) q^{67} + (71 \beta_{2} - 850 \beta_1 + 20737) q^{69} + (184 \beta_{2} + 136 \beta_1 + 27392) q^{71} + ( - 250 \beta_{2} + 1772 \beta_1 - 160) q^{73} + ( - 625 \beta_1 - 1250) q^{75} + (95 \beta_{2} - 3919 \beta_1 - 5277) q^{79} + ( - 206 \beta_{2} + 1112 \beta_1 - 10769) q^{81} + (138 \beta_{2} + 960 \beta_1 + 28746) q^{83} + ( - 50 \beta_{2} + 775 \beta_1 + 350) q^{85} + (114 \beta_{2} + 1413 \beta_1 + 28404) q^{87} + ( - 127 \beta_{2} + 2870 \beta_1 + 31861) q^{89} + ( - 265 \beta_{2} - 1592 \beta_1 - 76579) q^{93} + (225 \beta_{2} + 150 \beta_1 + 19475) q^{95} + ( - 322 \beta_{2} - 931 \beta_1 + 58622) q^{97} + (116 \beta_{2} + 2342 \beta_1 - 38084) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 6 q^{3} - 75 q^{5} + 281 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 6 q^{3} - 75 q^{5} + 281 q^{9} - 14 q^{11} + 4 q^{13} + 150 q^{15} - 44 q^{17} - 2328 q^{19} + 3676 q^{23} + 1875 q^{25} - 3726 q^{27} + 4092 q^{29} - 5888 q^{31} - 11318 q^{33} + 11378 q^{37} + 13702 q^{39} - 11450 q^{41} + 18544 q^{43} - 7025 q^{45} - 21754 q^{47} + 31762 q^{51} + 7494 q^{53} + 350 q^{55} + 7332 q^{57} - 12388 q^{59} - 27182 q^{61} - 100 q^{65} - 7676 q^{67} + 62140 q^{69} + 81992 q^{71} - 230 q^{73} - 3750 q^{75} - 15926 q^{79} - 32101 q^{81} + 86100 q^{83} + 1100 q^{85} + 85098 q^{87} + 95710 q^{89} - 229472 q^{93} + 58200 q^{95} + 176188 q^{97} - 114368 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - 499x - 210 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 333 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 333 \) 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
22.5458
−0.420991
−22.1248
0 −24.5458 0 −25.0000 0 0 0 359.498 0
1.2 0 −1.57901 0 −25.0000 0 0 0 −240.507 0
1.3 0 20.1248 0 −25.0000 0 0 0 162.009 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 980.6.a.h 3
7.b odd 2 1 140.6.a.d 3
28.d even 2 1 560.6.a.t 3
35.c odd 2 1 700.6.a.i 3
35.f even 4 2 700.6.e.g 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.6.a.d 3 7.b odd 2 1
560.6.a.t 3 28.d even 2 1
700.6.a.i 3 35.c odd 2 1
700.6.e.g 6 35.f even 4 2
980.6.a.h 3 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{3} + 6T_{3}^{2} - 487T_{3} - 780 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(980))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} + 6 T^{2} - 487 T - 780 \) Copy content Toggle raw display
$5$ \( (T + 25)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} \) Copy content Toggle raw display
$11$ \( T^{3} + 14 T^{2} - 147231 T + 12436740 \) Copy content Toggle raw display
$13$ \( T^{3} - 4 T^{2} - 425371 T + 6140734 \) Copy content Toggle raw display
$17$ \( T^{3} + 44 T^{2} + \cdots - 279119070 \) Copy content Toggle raw display
$19$ \( T^{3} + 2328 T^{2} + \cdots - 11429521264 \) Copy content Toggle raw display
$23$ \( T^{3} - 3676 T^{2} + \cdots + 2083660704 \) Copy content Toggle raw display
$29$ \( T^{3} - 4092 T^{2} + \cdots + 17580868722 \) Copy content Toggle raw display
$31$ \( T^{3} + 5888 T^{2} + \cdots - 211802104832 \) Copy content Toggle raw display
$37$ \( T^{3} - 11378 T^{2} + \cdots - 47286923800 \) Copy content Toggle raw display
$41$ \( T^{3} + 11450 T^{2} + \cdots - 478579953600 \) Copy content Toggle raw display
$43$ \( T^{3} - 18544 T^{2} + \cdots + 750561676176 \) Copy content Toggle raw display
$47$ \( T^{3} + 21754 T^{2} + \cdots + 173657376144 \) Copy content Toggle raw display
$53$ \( T^{3} - 7494 T^{2} + \cdots + 743911257600 \) Copy content Toggle raw display
$59$ \( T^{3} + 12388 T^{2} + \cdots - 41176040028480 \) Copy content Toggle raw display
$61$ \( T^{3} + 27182 T^{2} + \cdots + 169955356480 \) Copy content Toggle raw display
$67$ \( T^{3} + 7676 T^{2} + \cdots - 15170685707520 \) Copy content Toggle raw display
$71$ \( T^{3} + \cdots + 113425504819200 \) Copy content Toggle raw display
$73$ \( T^{3} + 230 T^{2} + \cdots + 11043630664360 \) Copy content Toggle raw display
$79$ \( T^{3} + \cdots - 274092525845520 \) Copy content Toggle raw display
$83$ \( T^{3} - 86100 T^{2} + \cdots + 40289422939200 \) Copy content Toggle raw display
$89$ \( T^{3} + \cdots + 305457269205600 \) Copy content Toggle raw display
$97$ \( T^{3} - 176188 T^{2} + \cdots + 41652594334882 \) Copy content Toggle raw display
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