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Properties

Label 432.7.q.b
Level $432$
Weight $7$
Character orbit 432.q
Analytic conductor $99.383$
Analytic rank $0$
Dimension $12$
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] = [432,7,Mod(17,432)]
 
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, 5]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("432.17");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 432 = 2^{4} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 432.q (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: \(99.3833641238\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 370x^{10} + 51793x^{8} + 3491832x^{6} + 117603792x^{4} + 1832032512x^{2} + 10453017600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{20} \)
Twist minimal: no (minimal twist has level 18)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{7} + 24 \beta_{2} - 48) q^{5} + (\beta_{10} + \beta_{8} + 2 \beta_{7} + \cdots - 40) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{7} + 24 \beta_{2} - 48) q^{5} + (\beta_{10} + \beta_{8} + 2 \beta_{7} + \cdots - 40) q^{7}+ \cdots + ( - 430 \beta_{11} + 28 \beta_{10} + \cdots - 6479) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 432 q^{5} - 240 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 432 q^{5} - 240 q^{7} + 378 q^{11} + 1680 q^{13} + 2820 q^{19} - 76248 q^{23} + 8094 q^{25} - 97092 q^{29} - 21480 q^{31} - 25536 q^{37} + 410562 q^{41} - 71430 q^{43} + 347652 q^{47} - 135954 q^{49} - 580392 q^{55} + 369738 q^{59} + 135744 q^{61} + 753840 q^{65} + 289938 q^{67} - 977700 q^{73} + 159192 q^{77} + 764796 q^{79} + 396900 q^{83} + 1619568 q^{85} - 355584 q^{91} - 2089260 q^{95} - 38874 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 370x^{10} + 51793x^{8} + 3491832x^{6} + 117603792x^{4} + 1832032512x^{2} + 10453017600 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 545 \nu^{10} + 914786 \nu^{8} + 213017969 \nu^{6} + 15650944776 \nu^{4} + 284531955984 \nu^{2} - 2422892805120 ) / 12347432640 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 119 \nu^{11} + 59366 \nu^{9} + 10447223 \nu^{7} + 794976432 \nu^{5} + 25420007664 \nu^{3} + \cdots + 25705589760 ) / 51411179520 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 79977 \nu^{11} - 234158 \nu^{10} + 22976562 \nu^{9} - 160687484 \nu^{8} + \cdots - 10\!\cdots\!40 ) / 1753335434880 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 79977 \nu^{11} + 234158 \nu^{10} + 22976562 \nu^{9} + 160687484 \nu^{8} + \cdots + 10\!\cdots\!40 ) / 1753335434880 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 763583 \nu^{11} - 175389312 \nu^{10} - 57134314 \nu^{9} - 56467506432 \nu^{8} + \cdots - 50\!\cdots\!00 ) / 14026683479040 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 763583 \nu^{11} - 175389312 \nu^{10} + 57134314 \nu^{9} - 56467506432 \nu^{8} + \cdots - 50\!\cdots\!00 ) / 14026683479040 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2804005 \nu^{11} - 2835456 \nu^{10} + 937140106 \nu^{9} + 243106272 \nu^{8} + \cdots + 78\!\cdots\!80 ) / 4675561159680 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 2804005 \nu^{11} + 2835456 \nu^{10} + 937140106 \nu^{9} - 243106272 \nu^{8} + \cdots - 78\!\cdots\!80 ) / 4675561159680 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 8217881 \nu^{11} + 18176568 \nu^{10} - 2832073634 \nu^{9} + 5016715728 \nu^{8} + \cdots - 43\!\cdots\!80 ) / 7013341739520 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 8217881 \nu^{11} + 18176568 \nu^{10} + 2832073634 \nu^{9} + 5016715728 \nu^{8} + \cdots - 43\!\cdots\!80 ) / 7013341739520 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 24178287 \nu^{11} + 4407964 \nu^{10} + 8235349626 \nu^{9} - 104860184 \nu^{8} + \cdots - 12\!\cdots\!00 ) / 7013341739520 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2 \beta_{10} - 2 \beta_{9} - 4 \beta_{8} - 4 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} - 3 \beta_{4} + \cdots - 36 ) / 108 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 5 \beta_{10} + 5 \beta_{9} - 2 \beta_{8} + 2 \beta_{7} + \beta_{6} + \beta_{5} + 7 \beta_{4} + \cdots - 3330 ) / 54 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 32 \beta_{11} - 95 \beta_{10} + 95 \beta_{9} + 130 \beta_{8} + 162 \beta_{7} + 107 \beta_{6} + \cdots + 5670 ) / 54 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 977 \beta_{10} - 977 \beta_{9} + 266 \beta_{8} - 266 \beta_{7} - 205 \beta_{6} - 205 \beta_{5} + \cdots + 299826 ) / 54 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 5000 \beta_{11} + 8731 \beta_{10} - 8731 \beta_{9} - 9266 \beta_{8} - 14266 \beta_{7} + \cdots - 986022 ) / 54 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 151881 \beta_{10} + 151881 \beta_{9} - 29178 \beta_{8} + 29178 \beta_{7} + 32913 \beta_{6} + \cdots - 32744394 ) / 54 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 731688 \beta_{11} - 902903 \beta_{10} + 902903 \beta_{9} + 643402 \beta_{8} + 1375090 \beta_{7} + \cdots + 150200550 ) / 54 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 22070137 \beta_{10} - 22070137 \beta_{9} + 3058234 \beta_{8} - 3058234 \beta_{7} - 4884461 \beta_{6} + \cdots + 3980601810 ) / 54 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 105940840 \beta_{11} + 105108331 \beta_{10} - 105108331 \beta_{9} - 40008722 \beta_{8} + \cdots - 21849040278 ) / 54 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 3127309105 \beta_{10} + 3127309105 \beta_{9} - 323447242 \beta_{8} + 323447242 \beta_{7} + \cdots - 514655128170 ) / 54 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 15181664872 \beta_{11} - 13272884855 \beta_{10} + 13272884855 \beta_{9} + 1746634090 \beta_{8} + \cdots + 3115054097910 ) / 54 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(271\) \(325\) \(353\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
8.88570i
4.28281i
− 3.87527i
− 11.8022i
7.20150i
− 8.15670i
− 8.88570i
− 4.28281i
3.87527i
11.8022i
− 7.20150i
8.15670i
0 0 0 −202.253 − 116.771i 0 −95.5752 − 165.541i 0 0 0
17.2 0 0 0 −156.951 − 90.6160i 0 −104.306 − 180.663i 0 0 0
17.3 0 0 0 −1.59771 − 0.922438i 0 −6.34411 − 10.9883i 0 0 0
17.4 0 0 0 9.39126 + 5.42205i 0 −322.041 − 557.792i 0 0 0
17.5 0 0 0 39.5602 + 22.8401i 0 245.097 + 424.521i 0 0 0
17.6 0 0 0 95.8504 + 55.3393i 0 163.169 + 282.617i 0 0 0
305.1 0 0 0 −202.253 + 116.771i 0 −95.5752 + 165.541i 0 0 0
305.2 0 0 0 −156.951 + 90.6160i 0 −104.306 + 180.663i 0 0 0
305.3 0 0 0 −1.59771 + 0.922438i 0 −6.34411 + 10.9883i 0 0 0
305.4 0 0 0 9.39126 − 5.42205i 0 −322.041 + 557.792i 0 0 0
305.5 0 0 0 39.5602 − 22.8401i 0 245.097 − 424.521i 0 0 0
305.6 0 0 0 95.8504 − 55.3393i 0 163.169 − 282.617i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 432.7.q.b 12
3.b odd 2 1 144.7.q.c 12
4.b odd 2 1 54.7.d.a 12
9.c even 3 1 144.7.q.c 12
9.d odd 6 1 inner 432.7.q.b 12
12.b even 2 1 18.7.d.a ✓ 12
36.f odd 6 1 18.7.d.a ✓ 12
36.f odd 6 1 162.7.b.c 12
36.h even 6 1 54.7.d.a 12
36.h even 6 1 162.7.b.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.7.d.a ✓ 12 12.b even 2 1
18.7.d.a ✓ 12 36.f odd 6 1
54.7.d.a 12 4.b odd 2 1
54.7.d.a 12 36.h even 6 1
144.7.q.c 12 3.b odd 2 1
144.7.q.c 12 9.c even 3 1
162.7.b.c 12 36.f odd 6 1
162.7.b.c 12 36.h even 6 1
432.7.q.b 12 1.a even 1 1 trivial
432.7.q.b 12 9.d odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} + 432 T_{5}^{11} + 42390 T_{5}^{10} - 8561376 T_{5}^{9} - 951920181 T_{5}^{8} + \cdots + 18\!\cdots\!00 \) acting on \(S_{7}^{\mathrm{new}}(432, [\chi])\). 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
$5$ \( T^{12} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 83\!\cdots\!09 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 98\!\cdots\!16 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 64\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{6} + \cdots - 71\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 18\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 50\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots + 54\!\cdots\!48)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 10\!\cdots\!25 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 19\!\cdots\!25 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 11\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 10\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 10\!\cdots\!04 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 50\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 58\!\cdots\!76 \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots - 49\!\cdots\!60)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 42\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 74\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 16\!\cdots\!25 \) Copy content Toggle raw display
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