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Properties

Label 432.6.s.c
Level $432$
Weight $6$
Character orbit 432.s
Analytic conductor $69.286$
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] = [432,6,Mod(143,432)]
 
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([3, 0, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("432.143");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 432 = 2^{4} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 432.s (of order \(6\), 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: \(20\)
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: \( x^{20} - 19 x^{18} - 44 x^{16} - 222597 x^{14} + 1089207 x^{12} + 5219984088 x^{10} + \cdots + 12\!\cdots\!01 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{32}\cdot 3^{38} \)
Twist minimal: no (minimal twist has level 144)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{3} - 6 \beta_1 + 6) q^{5} + (\beta_{9} - \beta_{7}) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} - 6 \beta_1 + 6) q^{5} + (\beta_{9} - \beta_{7}) q^{7} + (\beta_{11} - \beta_{10}) q^{11} + (\beta_{6} + 36 \beta_1 + 36) q^{13} + (2 \beta_{6} + \beta_{5} - \beta_{4} + \cdots + 3) q^{17}+ \cdots + ( - 35 \beta_{14} - 10 \beta_{12} + \cdots - 4530 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 174 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 174 q^{5} + 362 q^{13} + 2944 q^{25} - 9438 q^{29} + 9712 q^{37} - 24426 q^{41} + 36248 q^{49} + 49546 q^{61} + 11418 q^{65} - 137656 q^{73} - 80082 q^{77} + 17700 q^{85} + 41774 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 19 x^{18} - 44 x^{16} - 222597 x^{14} + 1089207 x^{12} + 5219984088 x^{10} + \cdots + 12\!\cdots\!01 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2470379843 \nu^{18} + 329095547674 \nu^{16} - 40326693725770 \nu^{14} + \cdots + 79\!\cdots\!61 ) / 11\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 65498215817 \nu^{18} + 11380527291022 \nu^{16} - 355246482469534 \nu^{14} + \cdots + 87\!\cdots\!63 ) / 44\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1699276160119 \nu^{18} - 106907950452478 \nu^{16} + \cdots - 39\!\cdots\!99 ) / 89\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 19271608 \nu^{18} - 1795087303 \nu^{16} - 11914264793 \nu^{14} + 5728691484405 \nu^{12} + \cdots - 37\!\cdots\!27 ) / 38\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 6777867125929 \nu^{18} - 876606401776910 \nu^{16} + \cdots - 30\!\cdots\!99 ) / 89\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 3795983913565 \nu^{18} - 932467222947230 \nu^{16} + \cdots - 21\!\cdots\!55 ) / 44\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 16481414129111 \nu^{19} + \cdots + 58\!\cdots\!09 \nu ) / 48\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 137911975475 \nu^{19} + 27838326405946 \nu^{17} + \cdots + 66\!\cdots\!73 \nu ) / 34\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 77155849015 \nu^{19} + 12879001354030 \nu^{17} - 267374564048446 \nu^{15} + \cdots + 14\!\cdots\!91 \nu ) / 16\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 112353544651 \nu^{19} - 2035627599812 \nu^{17} + \cdots - 14\!\cdots\!79 \nu ) / 17\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 70771831786237 \nu^{19} + \cdots - 23\!\cdots\!45 \nu ) / 96\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 48399051589847 \nu^{18} + \cdots + 35\!\cdots\!43 ) / 17\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 324478350899 \nu^{19} - 24293565272890 \nu^{17} + \cdots - 59\!\cdots\!77 \nu ) / 34\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 121105870994257 \nu^{18} + \cdots + 15\!\cdots\!09 ) / 35\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 137535627604655 \nu^{18} + \cdots - 12\!\cdots\!41 ) / 35\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 488808904429 \nu^{19} - 25495531334074 \nu^{17} + \cdots + 28\!\cdots\!75 \nu ) / 34\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 279921252181 \nu^{19} + 56635844579050 \nu^{17} - 545133100248346 \nu^{15} + \cdots + 52\!\cdots\!21 \nu ) / 16\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 217713152653 \nu^{19} - 25094875980346 \nu^{17} - 363157384900502 \nu^{15} + \cdots - 10\!\cdots\!77 \nu ) / 11\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 62426499834935 \nu^{19} + \cdots + 12\!\cdots\!65 \nu ) / 16\!\cdots\!56 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{18} + 2\beta_{16} + \beta_{13} ) / 81 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{15} - \beta_{12} - \beta_{5} - \beta_{3} + 2\beta_{2} + 253\beta _1 + 178 ) / 27 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 9 \beta_{19} - 3 \beta_{18} + 18 \beta_{17} + 6 \beta_{16} + 7 \beta_{13} + 81 \beta_{11} + \cdots + 207 \beta_{7} ) / 81 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 13 \beta_{15} + 16 \beta_{14} - 35 \beta_{12} - 27 \beta_{6} + 14 \beta_{5} + 216 \beta_{4} + \cdots - 3884 ) / 27 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 720 \beta_{19} + 298 \beta_{18} + 3087 \beta_{17} - 659 \beta_{16} + 1656 \beta_{13} + \cdots + 8946 \beta_{7} ) / 81 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 2967 \beta_{15} - 929 \beta_{14} - 1240 \beta_{12} + 7074 \beta_{6} - 2352 \beta_{5} + \cdots + 2756598 ) / 27 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 65916 \beta_{19} + 130450 \beta_{18} + 143199 \beta_{17} + 644860 \beta_{16} + \cdots - 1906182 \beta_{7} ) / 81 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 15002 \beta_{15} - 510237 \beta_{14} + 220969 \beta_{12} - 864297 \beta_{6} - 146465 \beta_{5} + \cdots - 175208803 ) / 27 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 4914585 \beta_{19} + 9615504 \beta_{18} + 4593249 \beta_{17} - 9248001 \beta_{16} + \cdots + 79839711 \beta_{7} ) / 81 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 8079286 \beta_{15} + 3864503 \beta_{14} - 2557837 \beta_{12} + 25787916 \beta_{6} + \cdots - 71347794205 ) / 27 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 114947028 \beta_{19} + 4399371908 \beta_{18} + 445447746 \beta_{17} - 3918956344 \beta_{16} + \cdots - 2437493832 \beta_{7} ) / 81 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 2810959734 \beta_{15} + 822675638 \beta_{14} + 1868292160 \beta_{12} - 4488694812 \beta_{6} + \cdots - 3483950976447 ) / 27 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 58241048148 \beta_{19} + 160956845987 \beta_{18} - 39820409586 \beta_{17} + \cdots - 449195796792 \beta_{7} ) / 81 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 69122119445 \beta_{15} - 40038005322 \beta_{14} + 298626198443 \beta_{12} + 183289539852 \beta_{6} + \cdots + 131119387795960 ) / 27 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 5816296300599 \beta_{19} + 682041482241 \beta_{18} - 8456263490196 \beta_{17} + \cdots - 54450172229265 \beta_{7} ) / 81 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 17206221820279 \beta_{15} + 7121057955910 \beta_{14} + 6605756749009 \beta_{12} - 34392105405279 \beta_{6} + \cdots - 27\!\cdots\!82 ) / 27 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 561716664003036 \beta_{19} + 356112143233678 \beta_{18} - 459457451886735 \beta_{17} + \cdots + 54\!\cdots\!42 \beta_{7} ) / 81 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( - 213147693082797 \beta_{15} + \cdots + 72\!\cdots\!68 ) / 27 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( 43\!\cdots\!88 \beta_{19} + \cdots - 31\!\cdots\!66 \beta_{7} ) / 81 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
143.1
8.44629 + 3.10810i
−8.44629 − 3.10810i
−5.31078 + 7.26606i
5.31078 − 7.26606i
1.18210 + 8.92203i
−1.18210 − 8.92203i
−5.37530 − 7.21846i
5.37530 + 7.21846i
8.79856 − 1.89349i
−8.79856 + 1.89349i
8.44629 − 3.10810i
−8.44629 + 3.10810i
−5.31078 − 7.26606i
5.31078 + 7.26606i
1.18210 − 8.92203i
−1.18210 + 8.92203i
−5.37530 + 7.21846i
5.37530 − 7.21846i
8.79856 + 1.89349i
−8.79856 − 1.89349i
0 0 0 −63.8385 + 36.8572i 0 −20.9224 − 12.0795i 0 0 0
143.2 0 0 0 −63.8385 + 36.8572i 0 20.9224 + 12.0795i 0 0 0
143.3 0 0 0 −25.2081 + 14.5539i 0 −5.79097 − 3.34342i 0 0 0
143.4 0 0 0 −25.2081 + 14.5539i 0 5.79097 + 3.34342i 0 0 0
143.5 0 0 0 14.3487 − 8.28421i 0 −149.607 − 86.3756i 0 0 0
143.6 0 0 0 14.3487 − 8.28421i 0 149.607 + 86.3756i 0 0 0
143.7 0 0 0 37.6433 − 21.7334i 0 −213.857 − 123.470i 0 0 0
143.8 0 0 0 37.6433 − 21.7334i 0 213.857 + 123.470i 0 0 0
143.9 0 0 0 80.5546 − 46.5082i 0 −89.6151 − 51.7393i 0 0 0
143.10 0 0 0 80.5546 − 46.5082i 0 89.6151 + 51.7393i 0 0 0
287.1 0 0 0 −63.8385 − 36.8572i 0 −20.9224 + 12.0795i 0 0 0
287.2 0 0 0 −63.8385 − 36.8572i 0 20.9224 − 12.0795i 0 0 0
287.3 0 0 0 −25.2081 − 14.5539i 0 −5.79097 + 3.34342i 0 0 0
287.4 0 0 0 −25.2081 − 14.5539i 0 5.79097 − 3.34342i 0 0 0
287.5 0 0 0 14.3487 + 8.28421i 0 −149.607 + 86.3756i 0 0 0
287.6 0 0 0 14.3487 + 8.28421i 0 149.607 − 86.3756i 0 0 0
287.7 0 0 0 37.6433 + 21.7334i 0 −213.857 + 123.470i 0 0 0
287.8 0 0 0 37.6433 + 21.7334i 0 213.857 − 123.470i 0 0 0
287.9 0 0 0 80.5546 + 46.5082i 0 −89.6151 + 51.7393i 0 0 0
287.10 0 0 0 80.5546 + 46.5082i 0 89.6151 − 51.7393i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 143.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
9.d odd 6 1 inner
36.h even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 432.6.s.c 20
3.b odd 2 1 144.6.s.b ✓ 20
4.b odd 2 1 inner 432.6.s.c 20
9.c even 3 1 144.6.s.b ✓ 20
9.d odd 6 1 inner 432.6.s.c 20
12.b even 2 1 144.6.s.b ✓ 20
36.f odd 6 1 144.6.s.b ✓ 20
36.h even 6 1 inner 432.6.s.c 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.6.s.b ✓ 20 3.b odd 2 1
144.6.s.b ✓ 20 9.c even 3 1
144.6.s.b ✓ 20 12.b even 2 1
144.6.s.b ✓ 20 36.f odd 6 1
432.6.s.c 20 1.a even 1 1 trivial
432.6.s.c 20 4.b odd 2 1 inner
432.6.s.c 20 9.d odd 6 1 inner
432.6.s.c 20 36.h even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(432, [\chi])\):

\( T_{5}^{10} - 87 T_{5}^{9} - 4764 T_{5}^{8} + 633969 T_{5}^{7} + 33760512 T_{5}^{6} + \cdots + 20\!\cdots\!48 \) Copy content Toggle raw display
\( T_{7}^{20} - 102159 T_{7}^{18} + 7580300922 T_{7}^{16} - 249295276800999 T_{7}^{14} + \cdots + 25\!\cdots\!76 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( (T^{10} + \cdots + 20\!\cdots\!48)^{2} \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 25\!\cdots\!76 \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 77\!\cdots\!29 \) Copy content Toggle raw display
$13$ \( (T^{10} + \cdots + 18\!\cdots\!16)^{2} \) Copy content Toggle raw display
$17$ \( (T^{10} + \cdots + 48\!\cdots\!92)^{2} \) Copy content Toggle raw display
$19$ \( (T^{10} + \cdots + 11\!\cdots\!04)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 18\!\cdots\!04 \) Copy content Toggle raw display
$29$ \( (T^{10} + \cdots + 96\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 10\!\cdots\!76 \) Copy content Toggle raw display
$37$ \( (T^{5} + \cdots + 21\!\cdots\!84)^{4} \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots + 69\!\cdots\!47)^{2} \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 84\!\cdots\!61 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 15\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots + 65\!\cdots\!28)^{2} \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 51\!\cdots\!49 \) Copy content Toggle raw display
$61$ \( (T^{10} + \cdots + 21\!\cdots\!64)^{2} \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 14\!\cdots\!41 \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots - 10\!\cdots\!92)^{2} \) Copy content Toggle raw display
$73$ \( (T^{5} + \cdots + 17\!\cdots\!16)^{4} \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 81\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 24\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots + 12\!\cdots\!72)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots + 13\!\cdots\!61)^{2} \) Copy content Toggle raw display
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