↑

Properties

Label 432.6.c.g
Level $432$
Weight $6$
Character orbit 432.c
Analytic conductor $69.286$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [432,6,Mod(431,432)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(432, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("432.431");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 432 = 2^{4} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 432.c (of order \(2\), degree \(1\), 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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} + 43 x^{10} - 802 x^{9} + 2077 x^{8} - 26672 x^{7} + 276788 x^{6} - 792632 x^{5} + \cdots + 2123735056 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{34}\cdot 3^{24} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{5} + ( - \beta_{8} + 3 \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{5} + ( - \beta_{8} + 3 \beta_1) q^{7} + (\beta_{5} - \beta_{3}) q^{11} + ( - \beta_{7} - 170) q^{13} + (\beta_{10} - \beta_{9} + 3 \beta_{2}) q^{17} + ( - 3 \beta_{8} - \beta_{4} - 10 \beta_1) q^{19} + ( - \beta_{6} + 5 \beta_{5} - 3 \beta_{3}) q^{23} + (\beta_{11} - 7 \beta_{7} - 652) q^{25} + (3 \beta_{10} - 4 \beta_{9} - 21 \beta_{2}) q^{29} + (18 \beta_{8} - 5 \beta_{4} - 142 \beta_1) q^{31} + ( - 2 \beta_{6} - 10 \beta_{5} + 21 \beta_{3}) q^{35} + (4 \beta_{11} - 5 \beta_{7} + 1520) q^{37} + ( - 17 \beta_{10} + \cdots + 15 \beta_{2}) q^{41}+ \cdots + (8 \beta_{11} - 108 \beta_{7} - 23525) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 2040 q^{13} - 7824 q^{25} + 18240 q^{37} - 69936 q^{49} + 105936 q^{61} + 200172 q^{73} - 126072 q^{85} - 282300 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} + 43 x^{10} - 802 x^{9} + 2077 x^{8} - 26672 x^{7} + 276788 x^{6} - 792632 x^{5} + \cdots + 2123735056 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 943919106134176 \nu^{11} + \cdots + 12\!\cdots\!92 ) / 59\!\cdots\!43 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 43\!\cdots\!07 \nu^{11} + \cdots + 55\!\cdots\!80 ) / 98\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 5118456906 \nu^{11} + 47335697433 \nu^{10} - 879319276038 \nu^{9} + \cdots + 73\!\cdots\!52 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 63\!\cdots\!78 \nu^{11} + \cdots - 57\!\cdots\!24 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 18\!\cdots\!28 \nu^{11} + \cdots + 38\!\cdots\!76 ) / 33\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 17\!\cdots\!70 \nu^{11} + \cdots + 64\!\cdots\!32 ) / 26\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 219755188696 \nu^{11} + 3415968145959 \nu^{10} - 11429138097064 \nu^{9} + \cdots + 12\!\cdots\!60 ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 10\!\cdots\!46 \nu^{11} + \cdots - 13\!\cdots\!32 ) / 59\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 97\!\cdots\!43 \nu^{11} + \cdots - 24\!\cdots\!80 ) / 25\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 16\!\cdots\!09 \nu^{11} + \cdots + 56\!\cdots\!60 ) / 32\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 254490330311 \nu^{11} + 3156510140169 \nu^{10} + 3688242809026 \nu^{9} + \cdots + 18\!\cdots\!60 ) / 37\!\cdots\!25 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 3 \beta_{11} + 12 \beta_{10} - 5 \beta_{9} + 48 \beta_{8} - 24 \beta_{7} + 9 \beta_{6} + 36 \beta_{5} + \cdots + 864 ) / 5184 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3 \beta_{11} - 32 \beta_{10} + \beta_{9} - 36 \beta_{8} - 36 \beta_{7} - 3 \beta_{6} + 96 \beta_{5} + \cdots - 11808 ) / 1728 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -105\beta_{11} - 336\beta_{10} + 325\beta_{9} + 156\beta_{7} + 4898\beta_{2} + 465696 ) / 2592 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 35 \beta_{11} + 40 \beta_{10} - 115 \beta_{9} + 140 \beta_{8} - 700 \beta_{7} + 615 \beta_{6} + \cdots + 54432 ) / 576 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 1713 \beta_{11} - 16008 \beta_{10} - 18545 \beta_{9} - 33348 \beta_{8} - 7764 \beta_{7} + \cdots - 11663136 ) / 5184 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -6573\beta_{11} - 31768\beta_{10} + 31069\beta_{9} + 1476\beta_{7} + 922298\beta_{2} + 13628448 ) / 864 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 103575 \beta_{11} - 280824 \beta_{10} - 688945 \beta_{9} - 3227916 \beta_{8} - 742116 \beta_{7} + \cdots + 336390624 ) / 5184 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 8695 \beta_{11} - 232120 \beta_{10} - 745255 \beta_{9} - 941980 \beta_{8} + 733300 \beta_{7} + \cdots - 99121248 ) / 576 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 4317927 \beta_{11} - 20971272 \beta_{10} + 22293185 \beta_{9} - 33704796 \beta_{7} + \cdots - 16859146464 ) / 2592 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 24434907 \beta_{11} - 40477832 \beta_{10} - 80601189 \beta_{9} - 506729484 \beta_{8} + \cdots + 65409350112 ) / 1728 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 245864055 \beta_{11} - 59794584 \beta_{10} - 1251809665 \beta_{9} + 1514532396 \beta_{8} + \cdots + 742912664544 ) / 5184 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(271\) \(325\) \(353\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
431.1
−2.05548 + 6.65289i
−4.73383 − 5.10655i
6.16887 − 1.14932i
−2.08910 + 5.91706i
4.64459 + 1.60185i
−0.935047 − 4.82326i
−0.935047 + 4.82326i
4.64459 − 1.60185i
−2.08910 − 5.91706i
6.16887 + 1.14932i
−4.73383 + 5.10655i
−2.05548 − 6.65289i
0 0 0 − 96.4718i 0 − 43.9240i 0 0 0
431.2 0 0 0 − 96.4718i 0 43.9240i 0 0 0
431.3 0 0 0 − 33.9573i 0 − 80.4860i 0 0 0
431.4 0 0 0 − 33.9573i 0 80.4860i 0 0 0
431.5 0 0 0 − 29.5145i 0 − 243.921i 0 0 0
431.6 0 0 0 − 29.5145i 0 243.921i 0 0 0
431.7 0 0 0 29.5145i 0 − 243.921i 0 0 0
431.8 0 0 0 29.5145i 0 243.921i 0 0 0
431.9 0 0 0 33.9573i 0 − 80.4860i 0 0 0
431.10 0 0 0 33.9573i 0 80.4860i 0 0 0
431.11 0 0 0 96.4718i 0 − 43.9240i 0 0 0
431.12 0 0 0 96.4718i 0 43.9240i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 431.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 432.6.c.g ✓ 12
3.b odd 2 1 inner 432.6.c.g ✓ 12
4.b odd 2 1 inner 432.6.c.g ✓ 12
12.b even 2 1 inner 432.6.c.g ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
432.6.c.g ✓ 12 1.a even 1 1 trivial
432.6.c.g ✓ 12 3.b odd 2 1 inner
432.6.c.g ✓ 12 4.b odd 2 1 inner
432.6.c.g ✓ 12 12.b even 2 1 inner

Hecke kernels

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

\( T_{5}^{6} + 11331T_{5}^{4} + 19843299T_{5}^{2} + 9348375969 \) Copy content Toggle raw display
\( T_{7}^{6} + 67905T_{7}^{4} + 512713611T_{7}^{2} + 743608674675 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T^{6} + 11331 T^{4} + \cdots + 9348375969)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} + 67905 T^{4} + \cdots + 743608674675)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + \cdots - 38636410731075)^{2} \) Copy content Toggle raw display
$13$ \( (T^{3} + 510 T^{2} + \cdots - 55890136)^{4} \) Copy content Toggle raw display
$17$ \( (T^{6} + \cdots + 885827597276736)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + \cdots + 21\!\cdots\!88)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots - 33\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots + 29\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots + 19\!\cdots\!67)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} - 4560 T^{2} + \cdots - 167667833600)^{4} \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots + 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots - 27\!\cdots\!08)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots + 15\!\cdots\!69)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots - 79\!\cdots\!72)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + \cdots + 14293424185408)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 72\!\cdots\!28)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots - 18\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} + \cdots + 41059756302695)^{4} \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 25\!\cdots\!32)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots - 76\!\cdots\!03)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots + 79\!\cdots\!16)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} + 70575 T^{2} + \cdots + 887388032285)^{4} \) Copy content Toggle raw display
show more
show less