Properties

Label 117.4.g.d
Level $117$
Weight $4$
Character orbit 117.g
Analytic conductor $6.903$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 117.g (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.90322347067\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{17})\)
Defining polynomial: \( x^{4} - x^{3} + 5x^{2} + 4x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 13)
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.

\(f(q)\) \(=\) \( q + (\beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{2} - 5 \beta_1 q^{4} + (5 \beta_{3} + 10) q^{5} + ( - 7 \beta_{2} - \beta_1) q^{7} + ( - 7 \beta_{3} + 4) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{2} - 5 \beta_1 q^{4} + (5 \beta_{3} + 10) q^{5} + ( - 7 \beta_{2} - \beta_1) q^{7} + ( - 7 \beta_{3} + 4) q^{8} + ( - 5 \beta_{3} - 5 \beta_1) q^{10} + ( - 15 \beta_{3} - \beta_{2} - 15 \beta_1 + 1) q^{11} + ( - 2 \beta_{3} - 40 \beta_{2} + 5 \beta_1 + 49) q^{13} + ( - 10 \beta_{3} + 18) q^{14} + ( - 15 \beta_{3} + 36 \beta_{2} - 15 \beta_1 - 36) q^{16} + (43 \beta_{2} - 16 \beta_1) q^{17} + (73 \beta_{2} - 5 \beta_1) q^{19} + (100 \beta_{2} - 25 \beta_1) q^{20} + (62 \beta_{2} + 46 \beta_1) q^{22} + (33 \beta_{3} - 89 \beta_{2} + 33 \beta_1 + 89) q^{23} + (75 \beta_{3} + 75) q^{25} + (30 \beta_{3} + 106 \beta_{2} + 55 \beta_1 - 46) q^{26} + (40 \beta_{3} + 20 \beta_{2} + 40 \beta_1 - 20) q^{28} + ( - 60 \beta_{3} + 13 \beta_{2} - 60 \beta_1 - 13) q^{29} + ( - 100 \beta_{3} - 120) q^{31} + (20 \beta_{2} + 65 \beta_1) q^{32} + ( - 5 \beta_{3} - 22) q^{34} + ( - 50 \beta_{2} + 30 \beta_1) q^{35} + ( - 44 \beta_{3} - 73 \beta_{2} - 44 \beta_1 + 73) q^{37} + (58 \beta_{3} - 126) q^{38} + ( - 15 \beta_{3} - 100) q^{40} + ( - 20 \beta_{3} - 259 \beta_{2} - 20 \beta_1 + 259) q^{41} + ( - 179 \beta_{2} - 97 \beta_1) q^{43} + (80 \beta_{3} - 300) q^{44} + (46 \beta_{2} - 10 \beta_1) q^{46} + ( - 140 \beta_{3} - 100) q^{47} + (15 \beta_{3} - 290 \beta_{2} + 15 \beta_1 + 290) q^{49} + ( - 150 \beta_{3} - 150 \beta_{2} - 150 \beta_1 + 150) q^{50} + (175 \beta_{3} - 140 \beta_{2} - 80 \beta_1 + 100) q^{52} + (165 \beta_{3} - 190) q^{53} + ( - 70 \beta_{3} + 290 \beta_{2} - 70 \beta_1 - 290) q^{55} + ( - 56 \beta_{2} - 60 \beta_1) q^{56} + (214 \beta_{2} + 167 \beta_1) q^{58} + ( - 377 \beta_{2} - 55 \beta_1) q^{59} + ( - 351 \beta_{2} + 200 \beta_1) q^{61} + (180 \beta_{3} + 160 \beta_{2} + 180 \beta_1 - 160) q^{62} + (95 \beta_{3} - 588) q^{64} + (235 \beta_{3} - 500 \beta_{2} + 225 \beta_1 + 450) q^{65} + (91 \beta_{3} - 283 \beta_{2} + 91 \beta_1 + 283) q^{67} + ( - 135 \beta_{3} + 320 \beta_{2} - 135 \beta_1 - 320) q^{68} + (40 \beta_{3} - 20) q^{70} + (11 \beta_{2} + 105 \beta_1) q^{71} + ( - 85 \beta_{3} + 250) q^{73} + (322 \beta_{2} + 205 \beta_1) q^{74} + ( - 340 \beta_{3} + 100 \beta_{2} - 340 \beta_1 - 100) q^{76} + (121 \beta_{3} - 67) q^{77} + (40 \beta_{3} + 140) q^{79} + ( - 255 \beta_{3} + 660 \beta_{2} - 255 \beta_1 - 660) q^{80} + (598 \beta_{2} + 319 \beta_1) q^{82} + ( - 100 \beta_{3} - 180) q^{83} + (750 \beta_{2} - 295 \beta_1) q^{85} + ( - 470 \beta_{3} + 746) q^{86} + ( - 172 \beta_{3} - 424 \beta_{2} - 172 \beta_1 + 424) q^{88} + (125 \beta_{3} - 523 \beta_{2} + 125 \beta_1 + 523) q^{89} + ( - 91 \beta_{2} - 65 \beta_1 - 260) q^{91} + (280 \beta_{3} + 660) q^{92} + (320 \beta_{3} + 360 \beta_{2} + 320 \beta_1 - 360) q^{94} + (830 \beta_{2} - 390 \beta_1) q^{95} + ( - 27 \beta_{2} + 469 \beta_1) q^{97} + (520 \beta_{2} + 245 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 5 q^{2} - 5 q^{4} + 30 q^{5} - 15 q^{7} + 30 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 5 q^{2} - 5 q^{4} + 30 q^{5} - 15 q^{7} + 30 q^{8} + 5 q^{10} + 17 q^{11} + 125 q^{13} + 92 q^{14} - 57 q^{16} + 70 q^{17} + 141 q^{19} + 175 q^{20} + 170 q^{22} + 145 q^{23} + 150 q^{25} + 23 q^{26} - 80 q^{28} + 34 q^{29} - 280 q^{31} + 105 q^{32} - 78 q^{34} - 70 q^{35} + 190 q^{37} - 620 q^{38} - 370 q^{40} + 538 q^{41} - 455 q^{43} - 1360 q^{44} + 82 q^{46} - 120 q^{47} + 565 q^{49} + 450 q^{50} - 310 q^{52} - 1090 q^{53} - 510 q^{55} - 172 q^{56} + 595 q^{58} - 809 q^{59} - 502 q^{61} - 500 q^{62} - 2542 q^{64} + 555 q^{65} + 475 q^{67} - 505 q^{68} - 160 q^{70} + 127 q^{71} + 1170 q^{73} + 849 q^{74} + 140 q^{76} - 510 q^{77} + 480 q^{79} - 1065 q^{80} + 1515 q^{82} - 520 q^{83} + 1205 q^{85} + 3924 q^{86} + 1020 q^{88} + 921 q^{89} - 1287 q^{91} + 2080 q^{92} - 1040 q^{94} + 1270 q^{95} + 415 q^{97} + 1285 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 5\nu^{2} - 5\nu + 16 ) / 20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 4 ) / 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 4\beta_{2} + \beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 5\beta_{3} - 4 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/117\mathbb{Z}\right)^\times\).

\(n\) \(28\) \(92\)
\(\chi(n)\) \(-\beta_{2}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1
1.28078 2.21837i
−0.780776 + 1.35234i
1.28078 + 2.21837i
−0.780776 1.35234i
−2.28078 3.95042i 0 −6.40388 + 11.0918i −2.80776 0 −4.78078 + 8.28055i 21.9309 0 6.40388 + 11.0918i
55.2 −0.219224 0.379706i 0 3.90388 6.76172i 17.8078 0 −2.71922 + 4.70983i −6.93087 0 −3.90388 6.76172i
100.1 −2.28078 + 3.95042i 0 −6.40388 11.0918i −2.80776 0 −4.78078 8.28055i 21.9309 0 6.40388 11.0918i
100.2 −0.219224 + 0.379706i 0 3.90388 + 6.76172i 17.8078 0 −2.71922 4.70983i −6.93087 0 −3.90388 + 6.76172i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 117.4.g.d 4
3.b odd 2 1 13.4.c.b 4
12.b even 2 1 208.4.i.e 4
13.c even 3 1 inner 117.4.g.d 4
13.c even 3 1 1521.4.a.t 2
13.e even 6 1 1521.4.a.l 2
39.d odd 2 1 169.4.c.f 4
39.f even 4 2 169.4.e.g 8
39.h odd 6 1 169.4.a.j 2
39.h odd 6 1 169.4.c.f 4
39.i odd 6 1 13.4.c.b 4
39.i odd 6 1 169.4.a.f 2
39.k even 12 2 169.4.b.e 4
39.k even 12 2 169.4.e.g 8
156.p even 6 1 208.4.i.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.c.b 4 3.b odd 2 1
13.4.c.b 4 39.i odd 6 1
117.4.g.d 4 1.a even 1 1 trivial
117.4.g.d 4 13.c even 3 1 inner
169.4.a.f 2 39.i odd 6 1
169.4.a.j 2 39.h odd 6 1
169.4.b.e 4 39.k even 12 2
169.4.c.f 4 39.d odd 2 1
169.4.c.f 4 39.h odd 6 1
169.4.e.g 8 39.f even 4 2
169.4.e.g 8 39.k even 12 2
208.4.i.e 4 12.b even 2 1
208.4.i.e 4 156.p even 6 1
1521.4.a.l 2 13.e even 6 1
1521.4.a.t 2 13.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 5T_{2}^{3} + 23T_{2}^{2} + 10T_{2} + 4 \) acting on \(S_{4}^{\mathrm{new}}(117, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 5 T^{3} + 23 T^{2} + 10 T + 4 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} - 15 T - 50)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 15 T^{3} + 173 T^{2} + \cdots + 2704 \) Copy content Toggle raw display
$11$ \( T^{4} - 17 T^{3} + 1173 T^{2} + \cdots + 781456 \) Copy content Toggle raw display
$13$ \( T^{4} - 125 T^{3} + 7956 T^{2} + \cdots + 4826809 \) Copy content Toggle raw display
$17$ \( T^{4} - 70 T^{3} + 4763 T^{2} + \cdots + 18769 \) Copy content Toggle raw display
$19$ \( T^{4} - 141 T^{3} + \cdots + 23658496 \) Copy content Toggle raw display
$23$ \( T^{4} - 145 T^{3} + 20397 T^{2} + \cdots + 394384 \) Copy content Toggle raw display
$29$ \( T^{4} - 34 T^{3} + \cdots + 225330121 \) Copy content Toggle raw display
$31$ \( (T^{2} + 140 T - 37600)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - 190 T^{3} + 35303 T^{2} + \cdots + 635209 \) Copy content Toggle raw display
$41$ \( T^{4} - 538 T^{3} + \cdots + 4992976921 \) Copy content Toggle raw display
$43$ \( T^{4} + 455 T^{3} + \cdots + 138485824 \) Copy content Toggle raw display
$47$ \( (T^{2} + 60 T - 82400)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 545 T - 41450)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 809 T^{3} + \cdots + 22729783696 \) Copy content Toggle raw display
$61$ \( T^{4} + 502 T^{3} + \cdots + 11448786001 \) Copy content Toggle raw display
$67$ \( T^{4} - 475 T^{3} + \cdots + 449948944 \) Copy content Toggle raw display
$71$ \( T^{4} - 127 T^{3} + \cdots + 1833894976 \) Copy content Toggle raw display
$73$ \( (T^{2} - 585 T + 54850)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 240 T + 7600)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 260 T - 25600)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 921 T^{3} + \cdots + 21215087716 \) Copy content Toggle raw display
$97$ \( T^{4} - 415 T^{3} + \cdots + 795268001284 \) Copy content Toggle raw display
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