Properties

Label 140.6.a.d
Level $140$
Weight $6$
Character orbit 140.a
Self dual yes
Analytic conductor $22.454$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 140 = 2^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 140.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(22.4537347738\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Defining polynomial: \(x^{3} - 499 x - 210\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + ( 2 + \beta_{1} ) q^{3} + 25 q^{5} + 49 q^{7} + ( 94 + 5 \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})\) \( q + ( 2 + \beta_{1} ) q^{3} + 25 q^{5} + 49 q^{7} + ( 94 + 5 \beta_{1} + \beta_{2} ) q^{9} + ( -5 + 11 \beta_{1} - \beta_{2} ) q^{11} + ( -2 + 13 \beta_{1} - 2 \beta_{2} ) q^{13} + ( 50 + 25 \beta_{1} ) q^{15} + ( 14 + 31 \beta_{1} - 2 \beta_{2} ) q^{17} + ( 779 + 6 \beta_{1} + 9 \beta_{2} ) q^{19} + ( 98 + 49 \beta_{1} ) q^{21} + ( 1225 - 70 \beta_{1} - \beta_{2} ) q^{23} + 625 q^{25} + ( 1244 + 31 \beta_{1} + 6 \beta_{2} ) q^{27} + ( 1359 - 99 \beta_{1} - 15 \beta_{2} ) q^{29} + ( 1957 - 248 \beta_{1} - 17 \beta_{2} ) q^{31} + ( 3776 - 137 \beta_{1} + 10 \beta_{2} ) q^{33} + 1225 q^{35} + ( 3793 + 64 \beta_{1} + \beta_{2} ) q^{37} + ( 4571 - 293 \beta_{1} + 11 \beta_{2} ) q^{39} + ( 3827 + 142 \beta_{1} + 31 \beta_{2} ) q^{41} + ( 6171 - 322 \beta_{1} - 31 \beta_{2} ) q^{43} + ( 2350 + 125 \beta_{1} + 25 \beta_{2} ) q^{45} + ( 7246 + 113 \beta_{1} - 16 \beta_{2} ) q^{47} + 2401 q^{49} + ( 10597 - 223 \beta_{1} + 29 \beta_{2} ) q^{51} + ( 2496 + 594 \beta_{1} - 6 \beta_{2} ) q^{53} + ( -125 + 275 \beta_{1} - 25 \beta_{2} ) q^{55} + ( 2449 + 2282 \beta_{1} + 15 \beta_{2} ) q^{57} + ( 4108 - 1840 \beta_{1} - 64 \beta_{2} ) q^{59} + ( 9037 - 110 \beta_{1} - 71 \beta_{2} ) q^{61} + ( 4606 + 245 \beta_{1} + 49 \beta_{2} ) q^{63} + ( -50 + 325 \beta_{1} - 50 \beta_{2} ) q^{65} + ( -2556 + 2228 \beta_{1} + 8 \beta_{2} ) q^{67} + ( -20737 + 850 \beta_{1} - 71 \beta_{2} ) q^{69} + ( 27392 + 136 \beta_{1} + 184 \beta_{2} ) q^{71} + ( 160 - 1772 \beta_{1} + 250 \beta_{2} ) q^{73} + ( 1250 + 625 \beta_{1} ) q^{75} + ( -245 + 539 \beta_{1} - 49 \beta_{2} ) q^{77} + ( -5277 - 3919 \beta_{1} + 95 \beta_{2} ) q^{79} + ( -10769 + 1112 \beta_{1} - 206 \beta_{2} ) q^{81} + ( -28746 - 960 \beta_{1} - 138 \beta_{2} ) q^{83} + ( 350 + 775 \beta_{1} - 50 \beta_{2} ) q^{85} + ( -28404 - 1413 \beta_{1} - 114 \beta_{2} ) q^{87} + ( -31861 - 2870 \beta_{1} + 127 \beta_{2} ) q^{89} + ( -98 + 637 \beta_{1} - 98 \beta_{2} ) q^{91} + ( -76579 - 1592 \beta_{1} - 265 \beta_{2} ) q^{93} + ( 19475 + 150 \beta_{1} + 225 \beta_{2} ) q^{95} + ( -58622 + 931 \beta_{1} + 322 \beta_{2} ) q^{97} + ( -38084 + 2342 \beta_{1} + 116 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 6 q^{3} + 75 q^{5} + 147 q^{7} + 281 q^{9} + O(q^{10}) \) \( 3 q + 6 q^{3} + 75 q^{5} + 147 q^{7} + 281 q^{9} - 14 q^{11} - 4 q^{13} + 150 q^{15} + 44 q^{17} + 2328 q^{19} + 294 q^{21} + 3676 q^{23} + 1875 q^{25} + 3726 q^{27} + 4092 q^{29} + 5888 q^{31} + 11318 q^{33} + 3675 q^{35} + 11378 q^{37} + 13702 q^{39} + 11450 q^{41} + 18544 q^{43} + 7025 q^{45} + 21754 q^{47} + 7203 q^{49} + 31762 q^{51} + 7494 q^{53} - 350 q^{55} + 7332 q^{57} + 12388 q^{59} + 27182 q^{61} + 13769 q^{63} - 100 q^{65} - 7676 q^{67} - 62140 q^{69} + 81992 q^{71} + 230 q^{73} + 3750 q^{75} - 686 q^{77} - 15926 q^{79} - 32101 q^{81} - 86100 q^{83} + 1100 q^{85} - 85098 q^{87} - 95710 q^{89} - 196 q^{91} - 229472 q^{93} + 58200 q^{95} - 176188 q^{97} - 114368 q^{99} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 333 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 333\)

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
−22.1248
−0.420991
22.5458
0 −20.1248 0 25.0000 0 49.0000 0 162.009 0
1.2 0 1.57901 0 25.0000 0 49.0000 0 −240.507 0
1.3 0 24.5458 0 25.0000 0 49.0000 0 359.498 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 140.6.a.d 3
4.b odd 2 1 560.6.a.t 3
5.b even 2 1 700.6.a.i 3
5.c odd 4 2 700.6.e.g 6
7.b odd 2 1 980.6.a.h 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.6.a.d 3 1.a even 1 1 trivial
560.6.a.t 3 4.b odd 2 1
700.6.a.i 3 5.b even 2 1
700.6.e.g 6 5.c odd 4 2
980.6.a.h 3 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{3} - 6 T_{3}^{2} - 487 T_{3} + 780 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(140))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( 780 - 487 T - 6 T^{2} + T^{3} \)
$5$ \( ( -25 + T )^{3} \)
$7$ \( ( -49 + T )^{3} \)
$11$ \( 12436740 - 147231 T + 14 T^{2} + T^{3} \)
$13$ \( -6140734 - 425371 T + 4 T^{2} + T^{3} \)
$17$ \( 279119070 - 833187 T - 44 T^{2} + T^{3} \)
$19$ \( 11429521264 - 4903980 T - 2328 T^{2} + T^{3} \)
$23$ \( 2083660704 + 2002116 T - 3676 T^{2} + T^{3} \)
$29$ \( 17580868722 - 17408331 T - 4092 T^{2} + T^{3} \)
$31$ \( 211802104832 - 41532064 T - 5888 T^{2} + T^{3} \)
$37$ \( -47286923800 + 41049740 T - 11378 T^{2} + T^{3} \)
$41$ \( 478579953600 - 44378496 T - 11450 T^{2} + T^{3} \)
$43$ \( 750561676176 - 13075724 T - 18544 T^{2} + T^{3} \)
$47$ \( -173657376144 + 129493881 T - 21754 T^{2} + T^{3} \)
$53$ \( 743911257600 - 161640000 T - 7494 T^{2} + T^{3} \)
$59$ \( 41176040028480 - 1934357328 T - 12388 T^{2} + T^{3} \)
$61$ \( -169955356480 - 174621088 T - 27182 T^{2} + T^{3} \)
$67$ \( -15170685707520 - 2456132096 T + 7676 T^{2} + T^{3} \)
$71$ \( 113425504819200 - 564748800 T - 81992 T^{2} + T^{3} \)
$73$ \( -11043630664360 - 6909191716 T - 230 T^{2} + T^{3} \)
$79$ \( -274092525845520 - 8464280087 T + 15926 T^{2} + T^{3} \)
$83$ \( -40289422939200 + 481780656 T + 86100 T^{2} + T^{3} \)
$89$ \( -305457269205600 - 2527476864 T + 95710 T^{2} + T^{3} \)
$97$ \( -41652594334882 + 1432985861 T + 176188 T^{2} + T^{3} \)
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