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Properties

Label 432.6.i.e
Level $432$
Weight $6$
Character orbit 432.i
Analytic conductor $69.286$
Analytic rank $0$
Dimension $14$
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] = [432,6,Mod(145,432)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(432, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("432.145");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 432 = 2^{4} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 432.i (of order \(3\), degree \(2\), not 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: \(69.2858101592\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - x^{13} + 129 x^{12} - 54 x^{11} + 11895 x^{10} - 5118 x^{9} + 498525 x^{8} - 176283 x^{7} + \cdots + 1124529156 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{20}\cdot 3^{21} \)
Twist minimal: no (minimal twist has level 72)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{6} - \beta_{3} + 4 \beta_1 - 4) q^{5} + (\beta_{8} - \beta_{7} - 13 \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{6} - \beta_{3} + 4 \beta_1 - 4) q^{5} + (\beta_{8} - \beta_{7} - 13 \beta_1) q^{7} + (\beta_{12} - \beta_{9} + \cdots - 52 \beta_1) q^{11}+ \cdots + ( - 49 \beta_{13} + 250 \beta_{12} + \cdots + 20272 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 25 q^{5} - 93 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 25 q^{5} - 93 q^{7} - 359 q^{11} + 89 q^{13} + 1360 q^{17} - 1768 q^{19} + 2503 q^{23} - 4922 q^{25} - 165 q^{29} - 5143 q^{31} + 22962 q^{35} - 19884 q^{37} + 12273 q^{41} - 10661 q^{43} + 6621 q^{47} - 14972 q^{49} + 36236 q^{53} + 62134 q^{55} - 46241 q^{59} + 52097 q^{61} - 32575 q^{65} - 65899 q^{67} + 39080 q^{71} - 239512 q^{73} - 75579 q^{77} - 52289 q^{79} + 33317 q^{83} + 172990 q^{85} - 34836 q^{89} + 422550 q^{91} + 500 q^{95} + 143167 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - x^{13} + 129 x^{12} - 54 x^{11} + 11895 x^{10} - 5118 x^{9} + 498525 x^{8} - 176283 x^{7} + \cdots + 1124529156 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 55\!\cdots\!41 \nu^{13} + \cdots + 84\!\cdots\!66 ) / 85\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 15\!\cdots\!52 \nu^{13} + \cdots + 43\!\cdots\!73 ) / 21\!\cdots\!55 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 33\!\cdots\!26 \nu^{13} + \cdots + 29\!\cdots\!24 ) / 65\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 35\!\cdots\!06 \nu^{13} + \cdots + 11\!\cdots\!74 ) / 65\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 15\!\cdots\!62 \nu^{13} + \cdots - 10\!\cdots\!22 ) / 19\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 16\!\cdots\!29 \nu^{13} + \cdots + 45\!\cdots\!64 ) / 10\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 39\!\cdots\!58 \nu^{13} + \cdots - 13\!\cdots\!18 ) / 19\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 20\!\cdots\!44 \nu^{13} + \cdots - 35\!\cdots\!76 ) / 45\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 47\!\cdots\!74 \nu^{13} + \cdots - 20\!\cdots\!34 ) / 65\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 40\!\cdots\!54 \nu^{13} + \cdots - 77\!\cdots\!94 ) / 45\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 18\!\cdots\!17 \nu^{13} + \cdots + 16\!\cdots\!68 ) / 15\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 11\!\cdots\!92 \nu^{13} + \cdots - 35\!\cdots\!68 ) / 45\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 48\!\cdots\!02 \nu^{13} + \cdots + 27\!\cdots\!88 ) / 15\!\cdots\!50 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{11} + \beta_{10} - \beta_{6} + \beta_{4} + \beta_{3} - 8\beta _1 + 8 ) / 54 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -9\beta_{13} + 2\beta_{11} + 4\beta_{10} + 11\beta_{6} - 9\beta_{5} - 2\beta_{2} - 3965\beta_1 ) / 108 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 45\beta_{9} - 126\beta_{7} - 27\beta_{5} - 295\beta_{4} + 239\beta_{3} + 193\beta_{2} - 3314 ) / 216 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 1620 \beta_{13} + 243 \beta_{12} - 715 \beta_{11} - 1256 \beta_{10} - 243 \beta_{8} - 1120 \beta_{6} + \cdots - 450931 ) / 216 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 1530 \beta_{13} + 2187 \beta_{12} + 5285 \beta_{11} - 11651 \beta_{10} - 2187 \beta_{9} + \cdots + 209965 \beta_1 ) / 108 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 28953 \beta_{9} + 40536 \beta_{7} + 131355 \beta_{5} + 118253 \beta_{4} - 71887 \beta_{3} + \cdots + 30226570 ) / 216 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 299295 \beta_{13} - 360612 \beta_{12} - 646402 \beta_{11} + 1843285 \beta_{10} + 1307583 \beta_{8} + \cdots + 42871181 ) / 216 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 10325979 \beta_{13} - 2673972 \beta_{12} + 5032606 \beta_{11} + 10121885 \beta_{10} + \cdots - 2186233453 \beta_1 ) / 216 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 28904490 \beta_{9} - 107814447 \beta_{7} - 27585063 \beta_{5} - 145384675 \beta_{4} + \cdots - 4074578723 ) / 216 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 803237472 \beta_{13} + 227091033 \beta_{12} - 380034157 \beta_{11} - 841673858 \beta_{10} + \cdots - 163617588733 ) / 216 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 2457678897 \beta_{13} + 2304120411 \beta_{12} + 2890681699 \beta_{11} - 11444269375 \beta_{10} + \cdots + 373844363210 \beta_1 ) / 216 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 18648017028 \beta_{9} + 36052141077 \beta_{7} + 62285428035 \beta_{5} + 69177119555 \beta_{4} + \cdots + 12442896083221 ) / 216 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 214158583044 \beta_{13} - 183423667887 \beta_{12} - 201401780953 \beta_{11} + 900294602080 \beta_{10} + \cdots + 33481734973691 ) / 216 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/432\mathbb{Z}\right)^\times\).

\(n\) \(271\) \(325\) \(353\)
\(\chi(n)\) \(1\) \(1\) \(-1 + \beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
145.1
1.96567 − 3.40465i
−4.24874 + 7.35903i
0.363092 − 0.628893i
3.03456 − 5.25602i
4.46537 − 7.73425i
−2.26483 + 3.92281i
−2.81513 + 4.87594i
1.96567 + 3.40465i
−4.24874 − 7.35903i
0.363092 + 0.628893i
3.03456 + 5.25602i
4.46537 + 7.73425i
−2.26483 − 3.92281i
−2.81513 − 4.87594i
0 0 0 −49.9293 + 86.4801i 0 −123.572 − 214.034i 0 0 0
145.2 0 0 0 −27.1257 + 46.9830i 0 41.4529 + 71.7986i 0 0 0
145.3 0 0 0 −24.8287 + 43.0045i 0 20.5752 + 35.6373i 0 0 0
145.4 0 0 0 9.09373 − 15.7508i 0 112.185 + 194.310i 0 0 0
145.5 0 0 0 10.1127 − 17.5157i 0 −29.6263 − 51.3143i 0 0 0
145.6 0 0 0 24.9232 − 43.1683i 0 −32.9730 − 57.1109i 0 0 0
145.7 0 0 0 45.2540 − 78.3822i 0 −34.5411 − 59.8269i 0 0 0
289.1 0 0 0 −49.9293 − 86.4801i 0 −123.572 + 214.034i 0 0 0
289.2 0 0 0 −27.1257 − 46.9830i 0 41.4529 − 71.7986i 0 0 0
289.3 0 0 0 −24.8287 − 43.0045i 0 20.5752 − 35.6373i 0 0 0
289.4 0 0 0 9.09373 + 15.7508i 0 112.185 − 194.310i 0 0 0
289.5 0 0 0 10.1127 + 17.5157i 0 −29.6263 + 51.3143i 0 0 0
289.6 0 0 0 24.9232 + 43.1683i 0 −32.9730 + 57.1109i 0 0 0
289.7 0 0 0 45.2540 + 78.3822i 0 −34.5411 + 59.8269i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 145.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 432.6.i.e 14
3.b odd 2 1 144.6.i.e 14
4.b odd 2 1 216.6.i.a 14
9.c even 3 1 inner 432.6.i.e 14
9.d odd 6 1 144.6.i.e 14
12.b even 2 1 72.6.i.a ✓ 14
36.f odd 6 1 216.6.i.a 14
36.f odd 6 1 648.6.a.f 7
36.h even 6 1 72.6.i.a ✓ 14
36.h even 6 1 648.6.a.e 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.6.i.a ✓ 14 12.b even 2 1
72.6.i.a ✓ 14 36.h even 6 1
144.6.i.e 14 3.b odd 2 1
144.6.i.e 14 9.d odd 6 1
216.6.i.a 14 4.b odd 2 1
216.6.i.a 14 36.f odd 6 1
432.6.i.e 14 1.a even 1 1 trivial
432.6.i.e 14 9.c even 3 1 inner
648.6.a.e 7 36.h even 6 1
648.6.a.f 7 36.f odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{14} + 25 T_{5}^{13} + 13711 T_{5}^{12} + 72014 T_{5}^{11} + 134257517 T_{5}^{10} + \cdots + 19\!\cdots\!84 \) acting on \(S_{6}^{\mathrm{new}}(432, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( T^{14} \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots + 19\!\cdots\!84 \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots + 26\!\cdots\!76 \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 94\!\cdots\!21 \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots + 71\!\cdots\!76 \) Copy content Toggle raw display
$17$ \( (T^{7} + \cdots + 11\!\cdots\!24)^{2} \) Copy content Toggle raw display
$19$ \( (T^{7} + \cdots - 12\!\cdots\!16)^{2} \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 32\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 15\!\cdots\!36 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 29\!\cdots\!24 \) Copy content Toggle raw display
$37$ \( (T^{7} + \cdots - 27\!\cdots\!16)^{2} \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 68\!\cdots\!41 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 25\!\cdots\!49 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 13\!\cdots\!64 \) Copy content Toggle raw display
$53$ \( (T^{7} + \cdots + 85\!\cdots\!68)^{2} \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 11\!\cdots\!01 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 13\!\cdots\!36 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 39\!\cdots\!29 \) Copy content Toggle raw display
$71$ \( (T^{7} + \cdots + 41\!\cdots\!16)^{2} \) Copy content Toggle raw display
$73$ \( (T^{7} + \cdots - 27\!\cdots\!76)^{2} \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 53\!\cdots\!44 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 21\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( (T^{7} + \cdots - 10\!\cdots\!36)^{2} \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 20\!\cdots\!69 \) Copy content Toggle raw display
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