Properties

Label 117.4.a.f
Level $117$
Weight $4$
Character orbit 117.a
Self dual yes
Analytic conductor $6.903$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 117.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.90322347067\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.3144.1
Defining polynomial: \( x^{3} - x^{2} - 16x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 39)
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.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} + (\beta_{2} + 3) q^{4} + (2 \beta_{2} - 2) q^{5} + (6 \beta_1 + 8) q^{7} + ( - 2 \beta_{2} - \beta_1 + 3) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{2} + (\beta_{2} + 3) q^{4} + (2 \beta_{2} - 2) q^{5} + (6 \beta_1 + 8) q^{7} + ( - 2 \beta_{2} - \beta_1 + 3) q^{8} + ( - 4 \beta_{2} + 6 \beta_1 - 2) q^{10} + ( - 6 \beta_{2} - 2 \beta_1 + 8) q^{11} + 13 q^{13} + (6 \beta_{2} + 14 \beta_1 + 52) q^{14} + ( - 5 \beta_{2} - 6 \beta_1 - 33) q^{16} + (8 \beta_1 + 46) q^{17} + (16 \beta_{2} - 6 \beta_1 + 28) q^{19} + ( - 2 \beta_{2} - 12 \beta_1 + 86) q^{20} + (10 \beta_{2} - 18 \beta_1 - 16) q^{22} + (8 \beta_{2} - 32 \beta_1 + 24) q^{23} + ( - 20 \beta_{2} - 24 \beta_1 + 63) q^{25} + (13 \beta_1 - 13) q^{26} + (2 \beta_{2} + 42 \beta_1 + 12) q^{28} + ( - 8 \beta_{2} - 20 \beta_1 + 10) q^{29} + ( - 4 \beta_{2} - 54 \beta_1 + 120) q^{31} + (20 \beta_{2} - 51 \beta_1 - 41) q^{32} + (8 \beta_{2} + 54 \beta_1 + 34) q^{34} + (4 \beta_{2} + 36 \beta_1 - 40) q^{35} + ( - 28 \beta_{2} - 48 \beta_1 + 150) q^{37} + ( - 38 \beta_{2} + 86 \beta_1 - 120) q^{38} + (24 \beta_{2} + 18 \beta_1 - 186) q^{40} + ( - 34 \beta_{2} + 4 \beta_1 - 150) q^{41} + (4 \beta_{2} - 60 \beta_1 - 68) q^{43} + (10 \beta_{2} + 22 \beta_1 - 248) q^{44} + ( - 48 \beta_{2} + 24 \beta_1 - 360) q^{46} + (42 \beta_{2} - 54 \beta_1 + 12) q^{47} + (36 \beta_{2} + 168 \beta_1 + 81) q^{49} + (16 \beta_{2} - 41 \beta_1 - 263) q^{50} + (13 \beta_{2} + 39) q^{52} + (12 \beta_{2} + 108 \beta_1 + 186) q^{53} + (68 \beta_{2} + 60 \beta_1 - 560) q^{55} + ( - 10 \beta_{2} - 50 \beta_1 - 12) q^{56} + ( - 4 \beta_{2} - 42 \beta_1 - 194) q^{58} + ( - 2 \beta_{2} - 82 \beta_1 + 624) q^{59} + ( - 28 \beta_{2} + 24 \beta_1 + 78) q^{61} + ( - 46 \beta_{2} + 50 \beta_1 - 652) q^{62} + ( - 51 \beta_{2} + 36 \beta_1 - 245) q^{64} + (26 \beta_{2} - 26) q^{65} + ( - 76 \beta_{2} + 42 \beta_1 + 36) q^{67} + (38 \beta_{2} + 56 \beta_1 + 122) q^{68} + (28 \beta_{2} + 12 \beta_1 + 392) q^{70} + ( - 14 \beta_{2} + 134 \beta_1 + 276) q^{71} + (12 \beta_{2} - 240 \beta_1 + 2) q^{73} + (8 \beta_{2} - 10 \beta_1 - 574) q^{74} + (34 \beta_{2} - 138 \beta_1 + 832) q^{76} + ( - 24 \beta_{2} - 136 \beta_1 + 16) q^{77} + ( - 24 \beta_{2} + 48 \beta_1 - 16) q^{79} + ( - 14 \beta_{2} + 24 \beta_1 - 370) q^{80} + (72 \beta_{2} - 282 \beta_1 + 258) q^{82} + (10 \beta_{2} - 30 \beta_1 + 272) q^{83} + (76 \beta_{2} + 48 \beta_1 - 124) q^{85} + ( - 68 \beta_{2} - 112 \beta_1 - 540) q^{86} + ( - 78 \beta_{2} - 42 \beta_1 + 576) q^{88} + ( - 30 \beta_{2} - 116 \beta_1 - 430) q^{89} + (78 \beta_1 + 104) q^{91} + (56 \beta_{2} - 272 \beta_1 + 504) q^{92} + ( - 138 \beta_{2} + 126 \beta_1 - 636) q^{94} + ( - 60 \beta_{2} - 228 \beta_1 + 1440) q^{95} + ( - 4 \beta_{2} - 48 \beta_1 + 1098) q^{97} + (96 \beta_{2} + 393 \beta_1 + 1527) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{2} + 10 q^{4} - 4 q^{5} + 30 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 2 q^{2} + 10 q^{4} - 4 q^{5} + 30 q^{7} + 6 q^{8} - 4 q^{10} + 16 q^{11} + 39 q^{13} + 176 q^{14} - 110 q^{16} + 146 q^{17} + 94 q^{19} + 244 q^{20} - 56 q^{22} + 48 q^{23} + 145 q^{25} - 26 q^{26} + 80 q^{28} + 2 q^{29} + 302 q^{31} - 154 q^{32} + 164 q^{34} - 80 q^{35} + 374 q^{37} - 312 q^{38} - 516 q^{40} - 480 q^{41} - 260 q^{43} - 712 q^{44} - 1104 q^{46} + 24 q^{47} + 447 q^{49} - 814 q^{50} + 130 q^{52} + 678 q^{53} - 1552 q^{55} - 96 q^{56} - 628 q^{58} + 1788 q^{59} + 230 q^{61} - 1952 q^{62} - 750 q^{64} - 52 q^{65} + 74 q^{67} + 460 q^{68} + 1216 q^{70} + 948 q^{71} - 222 q^{73} - 1724 q^{74} + 2392 q^{76} - 112 q^{77} - 24 q^{79} - 1100 q^{80} + 564 q^{82} + 796 q^{83} - 248 q^{85} - 1800 q^{86} + 1608 q^{88} - 1436 q^{89} + 390 q^{91} + 1296 q^{92} - 1920 q^{94} + 4032 q^{95} + 3242 q^{97} + 5070 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 16x - 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2\nu - 10 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2\beta _1 + 10 \) 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
−3.20905
−0.526440
4.73549
−4.20905 0 9.71610 11.4322 0 −11.2543 −7.22315 0 −48.1187
1.2 −1.52644 0 −5.66998 −19.3400 0 4.84136 20.8664 0 29.5213
1.3 3.73549 0 5.95388 3.90776 0 36.4129 −7.64325 0 14.5974
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(13\) \(-1\)

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 117.4.a.f 3
3.b odd 2 1 39.4.a.c 3
4.b odd 2 1 1872.4.a.bk 3
12.b even 2 1 624.4.a.t 3
13.b even 2 1 1521.4.a.u 3
15.d odd 2 1 975.4.a.l 3
21.c even 2 1 1911.4.a.k 3
24.f even 2 1 2496.4.a.bp 3
24.h odd 2 1 2496.4.a.bl 3
39.d odd 2 1 507.4.a.h 3
39.f even 4 2 507.4.b.g 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.a.c 3 3.b odd 2 1
117.4.a.f 3 1.a even 1 1 trivial
507.4.a.h 3 39.d odd 2 1
507.4.b.g 6 39.f even 4 2
624.4.a.t 3 12.b even 2 1
975.4.a.l 3 15.d odd 2 1
1521.4.a.u 3 13.b even 2 1
1872.4.a.bk 3 4.b odd 2 1
1911.4.a.k 3 21.c even 2 1
2496.4.a.bl 3 24.h odd 2 1
2496.4.a.bp 3 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} + 2T_{2}^{2} - 15T_{2} - 24 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(117))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} + 2 T^{2} - 15 T - 24 \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( T^{3} + 4 T^{2} - 252 T + 864 \) Copy content Toggle raw display
$7$ \( T^{3} - 30 T^{2} - 288 T + 1984 \) Copy content Toggle raw display
$11$ \( T^{3} - 16 T^{2} - 2256 T - 30336 \) Copy content Toggle raw display
$13$ \( (T - 13)^{3} \) Copy content Toggle raw display
$17$ \( T^{3} - 146 T^{2} + 6060 T - 71256 \) Copy content Toggle raw display
$19$ \( T^{3} - 94 T^{2} - 14432 T + 779616 \) Copy content Toggle raw display
$23$ \( T^{3} - 48 T^{2} - 20928 T - 534528 \) Copy content Toggle raw display
$29$ \( T^{3} - 2 T^{2} - 10116 T + 199176 \) Copy content Toggle raw display
$31$ \( T^{3} - 302 T^{2} - 17536 T + 7197248 \) Copy content Toggle raw display
$37$ \( T^{3} - 374 T^{2} - 36964 T + 7758104 \) Copy content Toggle raw display
$41$ \( T^{3} + 480 T^{2} + \cdots - 12919824 \) Copy content Toggle raw display
$43$ \( T^{3} + 260 T^{2} - 38096 T - 3663168 \) Copy content Toggle raw display
$47$ \( T^{3} - 24 T^{2} - 168480 T - 18102528 \) Copy content Toggle raw display
$53$ \( T^{3} - 678 T^{2} - 42228 T + 1471608 \) Copy content Toggle raw display
$59$ \( T^{3} - 1788 T^{2} + \cdots - 137423808 \) Copy content Toggle raw display
$61$ \( T^{3} - 230 T^{2} - 44452 T + 6279512 \) Copy content Toggle raw display
$67$ \( T^{3} - 74 T^{2} - 409216 T + 4260896 \) Copy content Toggle raw display
$71$ \( T^{3} - 948 T^{2} + \cdots + 70464384 \) Copy content Toggle raw display
$73$ \( T^{3} + 222 T^{2} + \cdots + 22780552 \) Copy content Toggle raw display
$79$ \( T^{3} + 24 T^{2} - 78336 T + 7757824 \) Copy content Toggle raw display
$83$ \( T^{3} - 796 T^{2} + \cdots - 13963968 \) Copy content Toggle raw display
$89$ \( T^{3} + 1436 T^{2} + \cdots + 30129888 \) Copy content Toggle raw display
$97$ \( T^{3} - 3242 T^{2} + \cdots - 1218481048 \) Copy content Toggle raw display
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