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Properties

Label 117.3.p.a
Level $117$
Weight $3$
Character orbit 117.p
Analytic conductor $3.188$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [117,3,Mod(35,117)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(117, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 4]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("117.35");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 117.p (of order \(6\), 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: \(3.18801909302\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 32 x^{18} + 690 x^{16} - 7984 x^{14} + 66147 x^{12} - 315440 x^{10} + 1074610 x^{8} + \cdots + 1327104 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{5} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{5} q^{2} + ( - \beta_{4} + 2 \beta_{3}) q^{4} + ( - \beta_{15} + \beta_{8}) q^{5} + (\beta_{6} - \beta_{3}) q^{7} + ( - \beta_{19} - \beta_{18} + \cdots - 3 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{2} + ( - \beta_{4} + 2 \beta_{3}) q^{4} + ( - \beta_{15} + \beta_{8}) q^{5} + (\beta_{6} - \beta_{3}) q^{7} + ( - \beta_{19} - \beta_{18} + \cdots - 3 \beta_1) q^{8}+ \cdots + ( - 6 \beta_{19} + \beta_{18} + \cdots - 2 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 24 q^{4} - 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 24 q^{4} - 6 q^{7} + 12 q^{10} - 2 q^{13} - 104 q^{16} - 92 q^{19} + 44 q^{22} - 116 q^{25} + 76 q^{28} - 156 q^{31} + 80 q^{34} + 148 q^{37} + 328 q^{40} + 186 q^{43} + 164 q^{46} + 8 q^{49} + 392 q^{52} - 208 q^{55} - 236 q^{58} + 210 q^{61} - 1568 q^{64} - 158 q^{67} - 1048 q^{70} + 156 q^{73} + 764 q^{76} - 132 q^{79} + 648 q^{82} + 44 q^{85} + 372 q^{88} + 834 q^{91} - 220 q^{94} - 14 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 32 x^{18} + 690 x^{16} - 7984 x^{14} + 66147 x^{12} - 315440 x^{10} + 1074610 x^{8} + \cdots + 1327104 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 51179873616221 \nu^{18} + \cdots + 29\!\cdots\!46 ) / 60\!\cdots\!61 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 20\!\cdots\!95 \nu^{18} + \cdots + 39\!\cdots\!00 ) / 23\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 49576277103901 \nu^{18} + \cdots + 10\!\cdots\!20 ) / 11\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 20\!\cdots\!95 \nu^{19} + \cdots - 39\!\cdots\!00 \nu ) / 23\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 23\!\cdots\!19 \nu^{18} + \cdots + 25\!\cdots\!36 ) / 58\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 22\!\cdots\!85 \nu^{18} + \cdots - 21\!\cdots\!36 ) / 38\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 24\!\cdots\!47 \nu^{19} + \cdots - 16\!\cdots\!96 \nu ) / 18\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 23\!\cdots\!99 \nu^{18} + \cdots - 29\!\cdots\!28 ) / 23\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 15\!\cdots\!33 \nu^{18} + \cdots + 26\!\cdots\!72 ) / 11\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 45\!\cdots\!93 \nu^{19} + \cdots + 47\!\cdots\!44 \nu ) / 18\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 93\!\cdots\!71 \nu^{19} + \cdots + 11\!\cdots\!56 \nu ) / 30\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 27\!\cdots\!49 \nu^{19} + \cdots + 28\!\cdots\!96 \nu ) / 71\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 27\!\cdots\!83 \nu^{18} + \cdots - 23\!\cdots\!88 ) / 58\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 11\!\cdots\!74 \nu^{19} + \cdots - 18\!\cdots\!19 \nu ) / 14\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 22\!\cdots\!45 \nu^{18} + \cdots + 19\!\cdots\!56 ) / 38\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 35\!\cdots\!92 \nu^{19} + \cdots - 34\!\cdots\!73 \nu ) / 14\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 27\!\cdots\!06 \nu^{19} + \cdots + 20\!\cdots\!89 \nu ) / 72\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 20\!\cdots\!66 \nu^{19} + \cdots - 21\!\cdots\!77 \nu ) / 55\!\cdots\!64 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} - 6\beta_{3} - \beta_{2} + 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{19} + \beta_{18} + \beta_{13} - \beta_{12} + 11\beta_{5} + 11\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{10} + \beta_{9} + \beta_{7} + 2\beta_{6} + 15\beta_{4} - 67\beta_{3} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 22\beta_{13} - 16\beta_{12} - 8\beta_{11} + 4\beta_{8} + 143\beta_{5} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 20\beta_{16} + 48\beta_{14} + 30\beta_{10} + 60\beta_{9} + 20\beta_{7} + 48\beta_{6} + 213\beta_{2} - 912 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -399\beta_{19} - 233\beta_{18} + 228\beta_{17} + 88\beta_{15} - 1997\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 321 \beta_{16} + 886 \beta_{14} + 1254 \beta_{10} + 627 \beta_{9} - 3095 \beta_{4} + 13241 \beta_{3} + \cdots - 13241 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 6748 \beta_{19} - 3416 \beta_{18} + 4648 \beta_{17} + 1528 \beta_{15} - 6748 \beta_{13} + \cdots - 29037 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 11396\beta_{10} - 11396\beta_{9} - 4944\beta_{7} - 15024\beta_{6} - 46033\beta_{4} + 197970\beta_{3} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -110269\beta_{13} + 50977\beta_{12} + 83400\beta_{11} - 24912\beta_{8} - 433079\beta_{5} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 75889 \beta_{16} - 245450 \beta_{14} - 193669 \beta_{10} - 387338 \beta_{9} - 75889 \beta_{7} + \cdots + 3008767 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 1768186\beta_{19} + 772168\beta_{18} - 1407464\beta_{17} - 397228\beta_{15} + 6566051\beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 1169396 \beta_{16} - 3933600 \beta_{14} - 6351300 \beta_{10} - 3175650 \beta_{9} + 10650741 \beta_{4} + \cdots + 46181340 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 28044891 \beta_{19} + 11820137 \beta_{18} - 22987500 \beta_{17} - 6272392 \beta_{15} + \cdots + 100604441 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 51032391 \beta_{10} + 51032391 \beta_{9} + 18092529 \beta_{7} + 62362174 \beta_{6} + \cdots - 713178821 \beta_{3} \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 441931264\beta_{13} - 182202272\beta_{12} - 368556520\beta_{11} + 98547232\beta_{8} + 1551820065\beta_{5} \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 280749504 \beta_{16} + 982409760 \beta_{14} + 810487784 \beta_{10} + 1620975568 \beta_{9} + \cdots - 11055615678 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 6936641017 \beta_{19} - 2821107649 \beta_{18} + 5845336464 \beta_{17} + 1543908768 \beta_{15} - 24038155907 \beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(28\) \(92\)
\(\chi(n)\) \(-\beta_{3}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
35.1
−3.42041 + 1.97477i
−2.68869 + 1.55231i
−1.73011 + 0.998881i
−1.02559 + 0.592124i
−1.01324 + 0.584997i
1.01324 − 0.584997i
1.02559 − 0.592124i
1.73011 − 0.998881i
2.68869 − 1.55231i
3.42041 − 1.97477i
−3.42041 − 1.97477i
−2.68869 − 1.55231i
−1.73011 − 0.998881i
−1.02559 − 0.592124i
−1.01324 − 0.584997i
1.01324 + 0.584997i
1.02559 + 0.592124i
1.73011 + 0.998881i
2.68869 + 1.55231i
3.42041 + 1.97477i
−3.42041 − 1.97477i 0 5.79946 + 10.0450i 3.48286i 0 −3.91511 − 6.78116i − 30.0123i 0 6.87786 − 11.9128i
35.2 −2.68869 − 1.55231i 0 2.81935 + 4.88326i − 1.11552i 0 3.02043 + 5.23153i − 5.08757i 0 −1.73164 + 2.99928i
35.3 −1.73011 − 0.998881i 0 −0.00447433 − 0.00774977i − 8.68283i 0 3.03342 + 5.25404i 8.00892i 0 −8.67311 + 15.0223i
35.4 −1.02559 − 0.592124i 0 −1.29878 − 2.24955i 4.21775i 0 −4.97984 − 8.62533i 7.81314i 0 2.49743 − 4.32568i
35.5 −1.01324 − 0.584997i 0 −1.31556 − 2.27861i 6.88799i 0 1.34109 + 2.32284i 7.75836i 0 4.02945 − 6.97921i
35.6 1.01324 + 0.584997i 0 −1.31556 − 2.27861i − 6.88799i 0 1.34109 + 2.32284i − 7.75836i 0 4.02945 − 6.97921i
35.7 1.02559 + 0.592124i 0 −1.29878 − 2.24955i − 4.21775i 0 −4.97984 − 8.62533i − 7.81314i 0 2.49743 − 4.32568i
35.8 1.73011 + 0.998881i 0 −0.00447433 − 0.00774977i 8.68283i 0 3.03342 + 5.25404i − 8.00892i 0 −8.67311 + 15.0223i
35.9 2.68869 + 1.55231i 0 2.81935 + 4.88326i 1.11552i 0 3.02043 + 5.23153i 5.08757i 0 −1.73164 + 2.99928i
35.10 3.42041 + 1.97477i 0 5.79946 + 10.0450i − 3.48286i 0 −3.91511 − 6.78116i 30.0123i 0 6.87786 − 11.9128i
107.1 −3.42041 + 1.97477i 0 5.79946 − 10.0450i − 3.48286i 0 −3.91511 + 6.78116i 30.0123i 0 6.87786 + 11.9128i
107.2 −2.68869 + 1.55231i 0 2.81935 − 4.88326i 1.11552i 0 3.02043 − 5.23153i 5.08757i 0 −1.73164 − 2.99928i
107.3 −1.73011 + 0.998881i 0 −0.00447433 + 0.00774977i 8.68283i 0 3.03342 − 5.25404i − 8.00892i 0 −8.67311 − 15.0223i
107.4 −1.02559 + 0.592124i 0 −1.29878 + 2.24955i − 4.21775i 0 −4.97984 + 8.62533i − 7.81314i 0 2.49743 + 4.32568i
107.5 −1.01324 + 0.584997i 0 −1.31556 + 2.27861i − 6.88799i 0 1.34109 − 2.32284i − 7.75836i 0 4.02945 + 6.97921i
107.6 1.01324 − 0.584997i 0 −1.31556 + 2.27861i 6.88799i 0 1.34109 − 2.32284i 7.75836i 0 4.02945 + 6.97921i
107.7 1.02559 − 0.592124i 0 −1.29878 + 2.24955i 4.21775i 0 −4.97984 + 8.62533i 7.81314i 0 2.49743 + 4.32568i
107.8 1.73011 − 0.998881i 0 −0.00447433 + 0.00774977i − 8.68283i 0 3.03342 − 5.25404i 8.00892i 0 −8.67311 − 15.0223i
107.9 2.68869 − 1.55231i 0 2.81935 − 4.88326i − 1.11552i 0 3.02043 − 5.23153i − 5.08757i 0 −1.73164 − 2.99928i
107.10 3.42041 − 1.97477i 0 5.79946 − 10.0450i 3.48286i 0 −3.91511 + 6.78116i − 30.0123i 0 6.87786 + 11.9128i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 35.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
13.c even 3 1 inner
39.i odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 117.3.p.a ✓ 20
3.b odd 2 1 inner 117.3.p.a ✓ 20
13.c even 3 1 inner 117.3.p.a ✓ 20
39.i odd 6 1 inner 117.3.p.a ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
117.3.p.a ✓ 20 1.a even 1 1 trivial
117.3.p.a ✓ 20 3.b odd 2 1 inner
117.3.p.a ✓ 20 13.c even 3 1 inner
117.3.p.a ✓ 20 39.i odd 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(117, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} - 32 T^{18} + \cdots + 1327104 \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( (T^{10} + 154 T^{8} + \cdots + 960498)^{2} \) Copy content Toggle raw display
$7$ \( (T^{10} + 3 T^{9} + \cdots + 58767556)^{2} \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 15\!\cdots\!84 \) Copy content Toggle raw display
$13$ \( (T^{10} + \cdots + 137858491849)^{2} \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 1773456850944 \) Copy content Toggle raw display
$19$ \( (T^{10} + \cdots + 17480694104064)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 25\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 89\!\cdots\!44 \) Copy content Toggle raw display
$31$ \( (T^{5} + 39 T^{4} + \cdots - 1474928)^{4} \) Copy content Toggle raw display
$37$ \( (T^{10} + \cdots + 30\!\cdots\!36)^{2} \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 21\!\cdots\!64 \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots + 8559030038724)^{2} \) Copy content Toggle raw display
$47$ \( (T^{10} + \cdots + 10\!\cdots\!08)^{2} \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots + 312107323433472)^{2} \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 34\!\cdots\!44 \) Copy content Toggle raw display
$61$ \( (T^{10} + \cdots + 11\!\cdots\!29)^{2} \) Copy content Toggle raw display
$67$ \( (T^{10} + \cdots + 16\!\cdots\!84)^{2} \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 69\!\cdots\!64 \) Copy content Toggle raw display
$73$ \( (T^{5} - 39 T^{4} + \cdots - 6060767)^{4} \) Copy content Toggle raw display
$79$ \( (T^{5} + 33 T^{4} + \cdots + 1476171424)^{4} \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots + 26\!\cdots\!72)^{2} \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 15\!\cdots\!44 \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots + 98\!\cdots\!84)^{2} \) Copy content Toggle raw display
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