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Properties

Label 117.5.j.a
Level $117$
Weight $5$
Character orbit 117.j
Analytic conductor $12.094$
Analytic rank $0$
Dimension $6$
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] = [117,5,Mod(73,117)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(117, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("117.73");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 117.j (of order \(4\), degree \(2\), 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: \(12.0942856808\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(i)\)
Coefficient field: 6.0.53039932416.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} - 12x^{3} + 529x^{2} - 1334x + 1682 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 13)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + ( - \beta_{5} - \beta_{3} + \cdots + \beta_1) q^{4}+ \cdots + ( - \beta_{5} + \beta_{4} - 12 \beta_{2} + \cdots + 12) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} + ( - \beta_{5} - \beta_{3} + \cdots + \beta_1) q^{4}+ \cdots + (431 \beta_{5} - 431 \beta_{4} + \cdots + 7008) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{2} + 14 q^{5} + 48 q^{7} + 96 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 2 q^{2} + 14 q^{5} + 48 q^{7} + 96 q^{8} + 32 q^{11} + 244 q^{14} + 1044 q^{16} + 732 q^{19} - 428 q^{20} - 1632 q^{22} - 910 q^{26} + 1884 q^{28} - 4184 q^{29} - 3468 q^{31} - 2092 q^{32} - 5304 q^{34} + 4204 q^{35} - 1758 q^{37} - 708 q^{40} - 4750 q^{41} + 3956 q^{44} + 516 q^{46} + 6872 q^{47} + 322 q^{50} + 3900 q^{52} - 2108 q^{53} + 6408 q^{55} + 6516 q^{58} - 4372 q^{59} + 5988 q^{61} + 5018 q^{65} + 72 q^{67} + 10572 q^{68} + 7368 q^{70} + 14672 q^{71} + 5874 q^{73} - 1544 q^{74} + 3576 q^{76} + 2616 q^{79} + 12080 q^{80} - 19264 q^{83} + 4164 q^{85} - 29376 q^{86} + 986 q^{89} - 30888 q^{91} - 5304 q^{92} - 36156 q^{94} - 23154 q^{97} + 41426 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} + 2x^{4} - 12x^{3} + 529x^{2} - 1334x + 1682 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -581\nu^{5} + 437\nu^{4} + 114\nu^{3} - 11153\nu^{2} - 302999\nu + 392573 ) / 424531 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -25\nu^{5} + 44\nu^{4} - 625\nu^{3} + 150\nu^{2} - 13189\nu + 33698 ) / 14639 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 31\nu^{5} + 531\nu^{4} + 775\nu^{3} - 186\nu^{2} - 1798\nu + 173115 ) / 14639 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -9152\nu^{5} + 6153\nu^{4} + 20063\nu^{3} + 230580\nu^{2} - 4343971\nu + 5696499 ) / 424531 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{3} - 17\beta_{2} - \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{4} - 21\beta_{3} + 12\beta_{2} + 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 25\beta_{4} + 31\beta_{3} + 31\beta _1 - 367 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -19\beta_{5} + 19\beta_{4} - 402\beta_{2} - 479\beta _1 + 402 \) Copy content Toggle raw display

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

Character values

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

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

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
73.1
−3.48832 + 3.48832i
1.30633 − 1.30633i
3.18200 − 3.18200i
−3.48832 − 3.48832i
1.30633 + 1.30633i
3.18200 + 3.18200i
−3.48832 − 3.48832i 0 8.33680i 6.84848 + 6.84848i 0 15.2891 − 15.2891i −26.7317 + 26.7317i 0 − 47.7794i
73.2 1.30633 + 1.30633i 0 − 12.5870i −9.28070 − 9.28070i 0 −46.1782 + 46.1782i 37.3440 − 37.3440i 0 − 24.2472i
73.3 3.18200 + 3.18200i 0 4.25023i 9.43223 + 9.43223i 0 54.8891 − 54.8891i 37.3878 − 37.3878i 0 60.0267i
109.1 −3.48832 + 3.48832i 0 − 8.33680i 6.84848 − 6.84848i 0 15.2891 + 15.2891i −26.7317 − 26.7317i 0 47.7794i
109.2 1.30633 − 1.30633i 0 12.5870i −9.28070 + 9.28070i 0 −46.1782 − 46.1782i 37.3440 + 37.3440i 0 24.2472i
109.3 3.18200 − 3.18200i 0 − 4.25023i 9.43223 − 9.43223i 0 54.8891 + 54.8891i 37.3878 + 37.3878i 0 − 60.0267i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 73.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.d odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 117.5.j.a 6
3.b odd 2 1 13.5.d.a ✓ 6
12.b even 2 1 208.5.t.c 6
13.d odd 4 1 inner 117.5.j.a 6
39.d odd 2 1 169.5.d.a 6
39.f even 4 1 13.5.d.a ✓ 6
39.f even 4 1 169.5.d.a 6
156.l odd 4 1 208.5.t.c 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.5.d.a ✓ 6 3.b odd 2 1
13.5.d.a ✓ 6 39.f even 4 1
117.5.j.a 6 1.a even 1 1 trivial
117.5.j.a 6 13.d odd 4 1 inner
169.5.d.a 6 39.d odd 2 1
169.5.d.a 6 39.f even 4 1
208.5.t.c 6 12.b even 2 1
208.5.t.c 6 156.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - 2T_{2}^{5} + 2T_{2}^{4} - 12T_{2}^{3} + 529T_{2}^{2} - 1334T_{2} + 1682 \) acting on \(S_{5}^{\mathrm{new}}(117, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - 2 T^{5} + \cdots + 1682 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} - 14 T^{5} + \cdots + 2875202 \) Copy content Toggle raw display
$7$ \( T^{6} + \cdots + 12014360072 \) Copy content Toggle raw display
$11$ \( T^{6} + \cdots + 88136171552 \) Copy content Toggle raw display
$13$ \( T^{6} + \cdots + 23298085122481 \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots + 11\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( T^{6} + \cdots + 15\!\cdots\!68 \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots + 26482633392384 \) Copy content Toggle raw display
$29$ \( (T^{3} + 2092 T^{2} + \cdots - 695715376)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + \cdots + 15\!\cdots\!48 \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots + 11\!\cdots\!18 \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots + 10\!\cdots\!92 \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 19\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 16\!\cdots\!92 \) Copy content Toggle raw display
$53$ \( (T^{3} + 1054 T^{2} + \cdots + 7634592356)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 10\!\cdots\!48 \) Copy content Toggle raw display
$61$ \( (T^{3} - 2994 T^{2} + \cdots + 1197889048)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 55\!\cdots\!68 \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots + 10\!\cdots\!68 \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 22\!\cdots\!08 \) Copy content Toggle raw display
$79$ \( (T^{3} - 1308 T^{2} + \cdots - 7469664296)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 75\!\cdots\!52 \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots + 14\!\cdots\!12 \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots + 20\!\cdots\!88 \) Copy content Toggle raw display
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