Properties

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

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 117.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.90322347067\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.5054412.1
Defining polynomial: \( x^{4} + 29x^{2} + 48 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 39)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{3} - 7) q^{4} + \beta_{2} q^{5} - 6 \beta_1 q^{7} + (2 \beta_{2} - 9 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{3} - 7) q^{4} + \beta_{2} q^{5} - 6 \beta_1 q^{7} + (2 \beta_{2} - 9 \beta_1) q^{8} + ( - 2 \beta_{3} + 6) q^{10} + ( - \beta_{2} + 2 \beta_1) q^{11} + (2 \beta_{3} + 3 \beta_{2} - 19) q^{13} + ( - 6 \beta_{3} + 90) q^{14} + ( - 5 \beta_{3} + 91) q^{16} - 54 q^{17} + (6 \beta_{2} - 6 \beta_1) q^{19} + (4 \beta_{2} + 26 \beta_1) q^{20} + (4 \beta_{3} - 36) q^{22} + (12 \beta_{3} - 36) q^{23} + ( - 8 \beta_{3} - 7) q^{25} + ( - 6 \beta_{3} + 4 \beta_{2} - 39 \beta_1 + 18) q^{26} + ( - 12 \beta_{2} + 102 \beta_1) q^{28} + ( - 12 \beta_{3} + 18) q^{29} + (6 \beta_{2} + 18 \beta_1) q^{31} + (6 \beta_{2} + 69 \beta_1) q^{32} - 54 \beta_1 q^{34} + (12 \beta_{3} - 36) q^{35} + ( - 24 \beta_{2} - 48 \beta_1) q^{37} + ( - 18 \beta_{3} + 126) q^{38} + (2 \beta_{3} - 318) q^{40} + ( - 13 \beta_{2} - 4 \beta_1) q^{41} + (12 \beta_{3} + 260) q^{43} - 60 \beta_1 q^{44} + (24 \beta_{2} - 156 \beta_1) q^{46} + ( - 21 \beta_{2} - 122 \beta_1) q^{47} + (36 \beta_{3} - 197) q^{49} + ( - 16 \beta_{2} + 73 \beta_1) q^{50} + ( - 31 \beta_{3} + 12 \beta_{2} + 78 \beta_1 + 457) q^{52} + (36 \beta_{3} + 198) q^{53} + (4 \beta_{3} + 144) q^{55} + (78 \beta_{3} - 882) q^{56} + ( - 24 \beta_{2} + 138 \beta_1) q^{58} + ( - 15 \beta_{2} + 34 \beta_1) q^{59} + ( - 8 \beta_{3} - 566) q^{61} + (6 \beta_{3} - 234) q^{62} + (17 \beta_{3} - 271) q^{64} + ( - 24 \beta_{3} + 3 \beta_{2} + 52 \beta_1 - 396) q^{65} + (12 \beta_{2} - 150 \beta_1) q^{67} + ( - 54 \beta_{3} + 378) q^{68} + (24 \beta_{2} - 156 \beta_1) q^{70} + ( - 23 \beta_{2} + 6 \beta_1) q^{71} - 54 \beta_{2} q^{73} + 576 q^{74} + (12 \beta_{2} + 258 \beta_1) q^{76} + ( - 24 \beta_{3} + 216) q^{77} + ( - 32 \beta_{3} + 88) q^{79} + (36 \beta_{2} - 130 \beta_1) q^{80} + (22 \beta_{3} - 18) q^{82} + (25 \beta_{2} + 138 \beta_1) q^{83} - 54 \beta_{2} q^{85} + (24 \beta_{2} + 140 \beta_1) q^{86} + ( - 28 \beta_{3} + 612) q^{88} + (15 \beta_{2} + 4 \beta_1) q^{89} + (36 \beta_{3} - 24 \beta_{2} + 234 \beta_1 - 108) q^{91} + ( - 108 \beta_{3} + 2196) q^{92} + ( - 80 \beta_{3} + 1704) q^{94} + ( - 36 \beta_{3} - 828) q^{95} + (54 \beta_{2} + 336 \beta_1) q^{97} + (72 \beta_{2} - 557 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 26 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 26 q^{4} + 20 q^{10} - 72 q^{13} + 348 q^{14} + 354 q^{16} - 216 q^{17} - 136 q^{22} - 120 q^{23} - 44 q^{25} + 60 q^{26} + 48 q^{29} - 120 q^{35} + 468 q^{38} - 1268 q^{40} + 1064 q^{43} - 716 q^{49} + 1766 q^{52} + 864 q^{53} + 584 q^{55} - 3372 q^{56} - 2280 q^{61} - 924 q^{62} - 1050 q^{64} - 1632 q^{65} + 1404 q^{68} + 2304 q^{74} + 816 q^{77} + 288 q^{79} - 28 q^{82} + 2392 q^{88} - 360 q^{91} + 8568 q^{92} + 6656 q^{94} - 3384 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 29x^{2} + 48 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 25\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 15 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 15 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{2} - 25\beta_1 \) Copy content Toggle raw display

Character values

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

\(n\) \(28\) \(92\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
64.1
5.21898i
1.32750i
1.32750i
5.21898i
5.21898i 0 −19.2377 5.83936i 0 31.3139i 58.6495i 0 30.4755
64.2 1.32750i 0 6.23774 15.4241i 0 7.96501i 18.9006i 0 −20.4755
64.3 1.32750i 0 6.23774 15.4241i 0 7.96501i 18.9006i 0 −20.4755
64.4 5.21898i 0 −19.2377 5.83936i 0 31.3139i 58.6495i 0 30.4755
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 117.4.b.d 4
3.b odd 2 1 39.4.b.a 4
12.b even 2 1 624.4.c.e 4
13.b even 2 1 inner 117.4.b.d 4
13.d odd 4 2 1521.4.a.x 4
39.d odd 2 1 39.4.b.a 4
39.f even 4 2 507.4.a.j 4
156.h even 2 1 624.4.c.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.b.a 4 3.b odd 2 1
39.4.b.a 4 39.d odd 2 1
117.4.b.d 4 1.a even 1 1 trivial
117.4.b.d 4 13.b even 2 1 inner
507.4.a.j 4 39.f even 4 2
624.4.c.e 4 12.b even 2 1
624.4.c.e 4 156.h even 2 1
1521.4.a.x 4 13.d odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 29T_{2}^{2} + 48 \) acting on \(S_{4}^{\mathrm{new}}(117, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 29T^{2} + 48 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 272T^{2} + 8112 \) Copy content Toggle raw display
$7$ \( T^{4} + 1044 T^{2} + 62208 \) Copy content Toggle raw display
$11$ \( T^{4} + 428 T^{2} + 43200 \) Copy content Toggle raw display
$13$ \( T^{4} + 72 T^{3} + 3094 T^{2} + \cdots + 4826809 \) Copy content Toggle raw display
$17$ \( (T + 54)^{4} \) Copy content Toggle raw display
$19$ \( T^{4} + 11556 T^{2} + \cdots + 31492800 \) Copy content Toggle raw display
$23$ \( (T^{2} + 60 T - 22464)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 24 T - 23220)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 17028 T^{2} + \cdots + 47044800 \) Copy content Toggle raw display
$37$ \( T^{4} + 200448 T^{2} + \cdots + 2293235712 \) Copy content Toggle raw display
$41$ \( T^{4} + 45392 T^{2} + \cdots + 128314800 \) Copy content Toggle raw display
$43$ \( (T^{2} - 532 T + 47392)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 500348 T^{2} + \cdots + 62387841792 \) Copy content Toggle raw display
$53$ \( (T^{2} - 432 T - 163620)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 104924 T^{2} + \cdots + 2436066048 \) Copy content Toggle raw display
$61$ \( (T^{2} + 1140 T + 314516)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 727668 T^{2} + \cdots + 143327232 \) Copy content Toggle raw display
$71$ \( T^{4} + 147692 T^{2} + \cdots + 3298756800 \) Copy content Toggle raw display
$73$ \( T^{4} + 793152 T^{2} + \cdots + 68976790272 \) Copy content Toggle raw display
$79$ \( (T^{2} - 144 T - 160960)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 653276 T^{2} + \cdots + 106682723328 \) Copy content Toggle raw display
$89$ \( T^{4} + 60464 T^{2} + \cdots + 249304368 \) Copy content Toggle raw display
$97$ \( T^{4} + 3704256 T^{2} + \cdots + 3383532000000 \) Copy content Toggle raw display
show more
show less