Properties

Label 117.10.a.e
Level $117$
Weight $10$
Character orbit 117.a
Self dual yes
Analytic conductor $60.259$
Analytic rank $1$
Dimension $5$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [117,10,Mod(1,117)]
 
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, 0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("117.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 117.a (trivial)

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: yes
Analytic conductor: \(60.2591928312\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 1438x^{3} - 4164x^{2} + 396957x - 59580 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: no (minimal twist has level 13)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 3) q^{2} + (\beta_{4} + 10 \beta_1 + 72) q^{4} + (3 \beta_{4} - \beta_{3} - 29 \beta_1 - 361) q^{5} + (4 \beta_{4} + 14 \beta_{3} + \cdots + 2020) q^{7}+ \cdots + ( - 25 \beta_{4} - 4 \beta_{3} + \cdots - 4630) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 3) q^{2} + (\beta_{4} + 10 \beta_1 + 72) q^{4} + (3 \beta_{4} - \beta_{3} - 29 \beta_1 - 361) q^{5} + (4 \beta_{4} + 14 \beta_{3} + \cdots + 2020) q^{7}+ \cdots + (293988 \beta_{4} + 572997 \beta_{3} + \cdots - 314218242) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 15 q^{2} + 361 q^{4} - 1803 q^{5} + 10099 q^{7} - 23151 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 15 q^{2} + 361 q^{4} - 1803 q^{5} + 10099 q^{7} - 23151 q^{8} + 84505 q^{10} - 121746 q^{11} + 142805 q^{13} - 8475 q^{14} - 322463 q^{16} + 495669 q^{17} - 840738 q^{19} + 1595607 q^{20} - 2023594 q^{22} + 592152 q^{23} + 1670362 q^{25} - 428415 q^{26} + 2587955 q^{28} - 10678182 q^{29} + 12885296 q^{31} - 3282927 q^{32} - 9934079 q^{34} - 8380731 q^{35} + 7171823 q^{37} + 25568814 q^{38} - 54359445 q^{40} - 9294012 q^{41} + 12831975 q^{43} + 41479074 q^{44} - 59319696 q^{46} - 43354215 q^{47} + 25249488 q^{49} + 16270770 q^{50} + 10310521 q^{52} - 93231780 q^{53} + 99448846 q^{55} - 199599225 q^{56} + 151020970 q^{58} - 246496182 q^{59} - 132232612 q^{61} - 158135724 q^{62} + 91019105 q^{64} - 51495483 q^{65} - 369388534 q^{67} - 238172073 q^{68} - 144857425 q^{70} - 212150457 q^{71} - 252729806 q^{73} - 192105957 q^{74} - 953775990 q^{76} - 449666118 q^{77} - 1247271728 q^{79} - 900649725 q^{80} + 169559388 q^{82} - 1696894296 q^{83} - 775363765 q^{85} - 3291621459 q^{86} - 220227222 q^{88} + 753854382 q^{89} + 288437539 q^{91} - 13876128 q^{92} + 272071215 q^{94} - 1442632962 q^{95} + 3824606 q^{97} - 1570614816 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 1438x^{3} - 4164x^{2} + 396957x - 59580 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{4} + 55\nu^{3} + 317\nu^{2} - 39383\nu + 189604 ) / 1088 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 7\nu^{4} - 113\nu^{3} - 6571\nu^{2} + 54545\nu + 495172 ) / 1088 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{2} - 4\nu - 575 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + 4\beta _1 + 575 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 16\beta_{4} + 4\beta_{3} + 28\beta_{2} + 877\beta _1 + 2500 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 1197\beta_{4} + 220\beta_{3} + 452\beta_{2} + 10120\beta _1 + 509379 \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
35.1685
16.7176
0.150341
−24.3176
−27.7188
−38.1685 0 944.833 109.762 0 5947.44 −16520.6 0 −4189.45
1.2 −19.7176 0 −123.217 −1555.58 0 −8329.39 12524.9 0 30672.3
1.3 −3.15034 0 −502.075 −2554.62 0 9399.91 3194.68 0 8047.92
1.4 21.3176 0 −57.5590 1277.14 0 −2277.98 −12141.6 0 27225.6
1.5 24.7188 0 99.0182 920.299 0 5359.02 −10208.4 0 22748.7
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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 117.10.a.e 5
3.b odd 2 1 13.10.a.b 5
12.b even 2 1 208.10.a.h 5
15.d odd 2 1 325.10.a.b 5
39.d odd 2 1 169.10.a.b 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.10.a.b 5 3.b odd 2 1
117.10.a.e 5 1.a even 1 1 trivial
169.10.a.b 5 39.d odd 2 1
208.10.a.h 5 12.b even 2 1
325.10.a.b 5 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{5} + 15T_{2}^{4} - 1348T_{2}^{3} - 8508T_{2}^{2} + 383520T_{2} + 1249344 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(117))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} + 15 T^{4} + \cdots + 1249344 \) Copy content Toggle raw display
$3$ \( T^{5} \) Copy content Toggle raw display
$5$ \( T^{5} + \cdots - 512670311383500 \) Copy content Toggle raw display
$7$ \( T^{5} + \cdots - 56\!\cdots\!88 \) Copy content Toggle raw display
$11$ \( T^{5} + \cdots + 19\!\cdots\!96 \) Copy content Toggle raw display
$13$ \( (T - 28561)^{5} \) Copy content Toggle raw display
$17$ \( T^{5} + \cdots + 39\!\cdots\!92 \) Copy content Toggle raw display
$19$ \( T^{5} + \cdots - 29\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{5} + \cdots + 42\!\cdots\!52 \) Copy content Toggle raw display
$29$ \( T^{5} + \cdots + 44\!\cdots\!20 \) Copy content Toggle raw display
$31$ \( T^{5} + \cdots + 48\!\cdots\!64 \) Copy content Toggle raw display
$37$ \( T^{5} + \cdots - 51\!\cdots\!48 \) Copy content Toggle raw display
$41$ \( T^{5} + \cdots - 49\!\cdots\!12 \) Copy content Toggle raw display
$43$ \( T^{5} + \cdots + 16\!\cdots\!08 \) Copy content Toggle raw display
$47$ \( T^{5} + \cdots + 10\!\cdots\!64 \) Copy content Toggle raw display
$53$ \( T^{5} + \cdots + 36\!\cdots\!12 \) Copy content Toggle raw display
$59$ \( T^{5} + \cdots + 43\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{5} + \cdots + 57\!\cdots\!08 \) Copy content Toggle raw display
$67$ \( T^{5} + \cdots - 85\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( T^{5} + \cdots + 41\!\cdots\!92 \) Copy content Toggle raw display
$73$ \( T^{5} + \cdots + 16\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( T^{5} + \cdots - 41\!\cdots\!40 \) Copy content Toggle raw display
$83$ \( T^{5} + \cdots + 52\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( T^{5} + \cdots + 19\!\cdots\!60 \) Copy content Toggle raw display
$97$ \( T^{5} + \cdots - 14\!\cdots\!32 \) Copy content Toggle raw display
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