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Properties

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

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [117,4,Mod(64,117)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(117, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("117.64");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 117.b (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.90322347067\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.1362828.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 23x^{2} + 48 \) 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

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{3} - 4) q^{4} + ( - \beta_{2} + 2 \beta_1) q^{5} + ( - 2 \beta_{2} + 2 \beta_1) q^{7} + (2 \beta_{2} - 3 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{3} - 4) q^{4} + ( - \beta_{2} + 2 \beta_1) q^{5} + ( - 2 \beta_{2} + 2 \beta_1) q^{7} + (2 \beta_{2} - 3 \beta_1) q^{8} + (4 \beta_{3} - 24) q^{10} + ( - 3 \beta_{2} - 8 \beta_1) q^{11} + ( - 4 \beta_{3} - \beta_{2} - 8 \beta_1 - 1) q^{13} + (6 \beta_{3} - 24) q^{14} + (\beta_{3} + 4) q^{16} + (12 \beta_{3} - 30) q^{17} + (8 \beta_{2} + 10 \beta_1) q^{19} - 36 \beta_1 q^{20} + ( - 2 \beta_{3} + 96) q^{22} + 72 q^{23} + (4 \beta_{3} - 7) q^{25} + ( - 6 \beta_{3} - 8 \beta_{2} + 27 \beta_1 + 96) q^{26} + ( - 4 \beta_{2} - 50 \beta_1) q^{28} + ( - 12 \beta_{3} + 102) q^{29} + ( - 20 \beta_{2} + 38 \beta_1) q^{31} + (18 \beta_{2} - 27 \beta_1) q^{32} + (24 \beta_{2} - 114 \beta_1) q^{34} - 216 q^{35} + (4 \beta_{2} + 68 \beta_1) q^{37} + ( - 6 \beta_{3} - 120) q^{38} + ( - 4 \beta_{3} + 240) q^{40} + (13 \beta_{2} + 66 \beta_1) q^{41} + (12 \beta_{3} - 328) q^{43} + ( - 28 \beta_{2} + 46 \beta_1) q^{44} + 72 \beta_1 q^{46} + (\beta_{2} + 36 \beta_1) q^{47} + ( - 12 \beta_{3} - 41) q^{49} + (8 \beta_{2} - 35 \beta_1) q^{50} + (11 \beta_{3} - 20 \beta_{2} + 74 \beta_1 - 332) q^{52} + ( - 48 \beta_{3} - 222) q^{53} + ( - 44 \beta_{3} - 60) q^{55} + (6 \beta_{3} + 408) q^{56} + ( - 24 \beta_{2} + 186 \beta_1) q^{58} + ( - \beta_{2} - 52 \beta_1) q^{59} + (28 \beta_{3} + 58) q^{61} + (78 \beta_{3} - 456) q^{62} + ( - 55 \beta_{3} + 356) q^{64} + ( - 36 \beta_{3} + 17 \beta_{2} + 110 \beta_1 + 108) q^{65} + (54 \beta_{2} + 54 \beta_1) q^{67} + ( - 66 \beta_{3} + 1128) q^{68} - 216 \beta_1 q^{70} + ( - 41 \beta_{2} + 168 \beta_1) q^{71} + ( - 42 \beta_{2} - 156 \beta_1) q^{73} + (60 \beta_{3} - 816) q^{74} + (52 \beta_{2} + 2 \beta_1) q^{76} + ( - 84 \beta_{3} - 312) q^{77} + (16 \beta_{3} + 1072) q^{79} + ( - 8 \beta_{2} - 20 \beta_1) q^{80} + (40 \beta_{3} - 792) q^{82} + (63 \beta_{2} - 188 \beta_1) q^{83} + ( - 18 \beta_{2} - 396 \beta_1) q^{85} + (24 \beta_{2} - 412 \beta_1) q^{86} + (86 \beta_{3} + 216) q^{88} + (25 \beta_{2} + 138 \beta_1) q^{89} + ( - 60 \beta_{3} + 50 \beta_{2} + 166 \beta_1 + 24) q^{91} + (72 \beta_{3} - 288) q^{92} + (34 \beta_{3} - 432) q^{94} + (72 \beta_{3} + 432) q^{95} + ( - 14 \beta_{2} + 68 \beta_1) q^{97} + ( - 24 \beta_{2} + 43 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 14 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 14 q^{4} - 88 q^{10} - 12 q^{13} - 84 q^{14} + 18 q^{16} - 96 q^{17} + 380 q^{22} + 288 q^{23} - 20 q^{25} + 372 q^{26} + 384 q^{29} - 864 q^{35} - 492 q^{38} + 952 q^{40} - 1288 q^{43} - 188 q^{49} - 1306 q^{52} - 984 q^{53} - 328 q^{55} + 1644 q^{56} + 288 q^{61} - 1668 q^{62} + 1314 q^{64} + 360 q^{65} + 4380 q^{68} - 3144 q^{74} - 1416 q^{77} + 4320 q^{79} - 3088 q^{82} + 1036 q^{88} - 24 q^{91} - 1008 q^{92} - 1660 q^{94} + 1872 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 23x^{2} + 48 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 19\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 12 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{2} - 19\beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
64.1
− 4.54739i
− 1.52356i
1.52356i
4.54739i
− 4.54739i 0 −12.6788 − 12.9118i 0 − 16.7289i 21.2762i 0 −58.7151
64.2 − 1.52356i 0 5.67878 9.65841i 0 22.3639i − 20.8404i 0 14.7151
64.3 1.52356i 0 5.67878 − 9.65841i 0 − 22.3639i 20.8404i 0 14.7151
64.4 4.54739i 0 −12.6788 12.9118i 0 16.7289i − 21.2762i 0 −58.7151
\(n\): 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.e 4
3.b odd 2 1 39.4.b.b ✓ 4
12.b even 2 1 624.4.c.c 4
13.b even 2 1 inner 117.4.b.e 4
13.d odd 4 2 1521.4.a.w 4
39.d odd 2 1 39.4.b.b ✓ 4
39.f even 4 2 507.4.a.l 4
156.h even 2 1 624.4.c.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.b.b ✓ 4 3.b odd 2 1
39.4.b.b ✓ 4 39.d odd 2 1
117.4.b.e 4 1.a even 1 1 trivial
117.4.b.e 4 13.b even 2 1 inner
507.4.a.l 4 39.f even 4 2
624.4.c.c 4 12.b even 2 1
624.4.c.c 4 156.h even 2 1
1521.4.a.w 4 13.d odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 23T_{2}^{2} + 48 \) acting on \(S_{4}^{\mathrm{new}}(117, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 23T^{2} + 48 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 260 T^{2} + 15552 \) Copy content Toggle raw display
$7$ \( T^{4} + 780 T^{2} + 139968 \) Copy content Toggle raw display
$11$ \( T^{4} + 3152 T^{2} + \cdots + 1572528 \) Copy content Toggle raw display
$13$ \( T^{4} + 12 T^{3} - 962 T^{2} + \cdots + 4826809 \) Copy content Toggle raw display
$17$ \( (T^{2} + 48 T - 11556)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 13884 T^{2} + \cdots + 3048192 \) Copy content Toggle raw display
$23$ \( (T - 72)^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} - 192 T - 2916)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 100572 T^{2} + \cdots + 2389782528 \) Copy content Toggle raw display
$37$ \( T^{4} + 110256 T^{2} + \cdots + 2060577792 \) Copy content Toggle raw display
$41$ \( T^{4} + 133364 T^{2} + \cdots + 4431055872 \) Copy content Toggle raw display
$43$ \( (T^{2} + 644 T + 91552)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 30128 T^{2} + \cdots + 116663088 \) Copy content Toggle raw display
$53$ \( (T^{2} + 492 T - 133596)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 62576 T^{2} + \cdots + 457419312 \) Copy content Toggle raw display
$61$ \( (T^{2} - 144 T - 60868)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 591948 T^{2} + \cdots + 918330048 \) Copy content Toggle raw display
$71$ \( T^{4} + 917456 T^{2} + \cdots + 59484058032 \) Copy content Toggle raw display
$73$ \( T^{4} + 896400 T^{2} + \cdots + 179358354432 \) Copy content Toggle raw display
$79$ \( (T^{2} - 2160 T + 1144832)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 1464080 T^{2} + \cdots + 317023217328 \) Copy content Toggle raw display
$89$ \( T^{4} + 561812 T^{2} + \cdots + 78903164928 \) Copy content Toggle raw display
$97$ \( T^{4} + 137040 T^{2} + \cdots + 725594112 \) Copy content Toggle raw display
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