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Properties

Label 432.7.q.a
Level $432$
Weight $7$
Character orbit 432.q
Analytic conductor $99.383$
Analytic rank $0$
Dimension $10$
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: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} + 75 x^{8} - 2 x^{7} + 4610 x^{6} - 2412 x^{5} + 66932 x^{4} - 174032 x^{3} + \cdots + 1982464 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{14} \)
Twist minimal: no (minimal twist has level 9)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{7} + \beta_{5} - 15 \beta_{3} + \cdots + 15) q^{5}+ \cdots + (\beta_{9} + 3 \beta_{7} + \cdots - 2 \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{7} + \beta_{5} - 15 \beta_{3} + \cdots + 15) q^{5}+ \cdots + (1011 \beta_{9} + 6183 \beta_{7} + \cdots - 3505 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 219 q^{5} + 121 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 219 q^{5} + 121 q^{7} + 483 q^{11} - 841 q^{13} - 6176 q^{19} + 53565 q^{23} + 8452 q^{25} + 80679 q^{29} + 24601 q^{31} + 12764 q^{37} - 232251 q^{41} + 93271 q^{43} - 142887 q^{47} + 86238 q^{49} + 419982 q^{55} - 995061 q^{59} - 59305 q^{61} - 1642029 q^{65} - 158513 q^{67} + 933896 q^{73} + 2198883 q^{77} - 468707 q^{79} + 3008337 q^{83} - 1189944 q^{85} + 211778 q^{91} - 2562954 q^{95} + 336029 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} + 75 x^{8} - 2 x^{7} + 4610 x^{6} - 2412 x^{5} + 66932 x^{4} - 174032 x^{3} + \cdots + 1982464 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 237483745 \nu^{9} - 2553045007 \nu^{8} + 88893251653 \nu^{7} - 265266893774 \nu^{6} + \cdots - 76\!\cdots\!12 ) / 14\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 4480309 \nu^{9} + 460307201 \nu^{8} - 1146370163 \nu^{7} + 28795204174 \nu^{6} + \cdots + 798806124416192 ) / 8405189131968 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3461507741 \nu^{9} - 4504997485 \nu^{8} + 258824546191 \nu^{7} - 4959213290 \nu^{6} + \cdots - 43\!\cdots\!48 ) / 44\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 616498581 \nu^{9} + 325325413 \nu^{8} - 28321882423 \nu^{7} - 43384613414 \nu^{6} + \cdots - 12\!\cdots\!28 ) / 246552214537728 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 304943719717 \nu^{9} - 403310379317 \nu^{8} + 23041276017575 \nu^{7} + \cdots - 40\!\cdots\!44 ) / 44\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 76279967689 \nu^{9} - 357175151353 \nu^{8} - 5964228234521 \nu^{7} + \cdots - 92\!\cdots\!44 ) / 184914160903296 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 227142042571 \nu^{9} + 578302501715 \nu^{8} - 15147738294065 \nu^{7} + \cdots + 31\!\cdots\!20 ) / 369828321806592 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 949975118761 \nu^{9} - 3915009601193 \nu^{8} + 64497613313027 \nu^{7} + \cdots - 20\!\cdots\!08 ) / 14\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1182537518497 \nu^{9} + 2091940389269 \nu^{8} - 82074515849855 \nu^{7} + \cdots + 14\!\cdots\!32 ) / 11\!\cdots\!76 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} + 3\beta_{3} - \beta _1 + 3 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -4\beta_{5} + 2\beta_{4} + 359\beta_{3} + \beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -12\beta_{8} - 16\beta_{7} + 4\beta_{6} + 163\beta_{4} + 24\beta_{2} + 314\beta _1 - 873 ) / 36 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 20 \beta_{9} + 20 \beta_{8} + 4 \beta_{7} + 8 \beta_{6} + 816 \beta_{5} - 395 \beta_{4} - 54269 \beta_{3} + \cdots - 54269 ) / 36 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -868\beta_{9} + 1468\beta_{7} + 300\beta_{6} + 2136\beta_{5} - 18354\beta_{4} - 120497\beta_{3} - 9611\beta_1 ) / 36 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -356\beta_{8} - 208\beta_{7} - 148\beta_{6} - 10893\beta_{4} - 16920\beta_{2} - 22142\beta _1 + 1047431 ) / 12 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 17692 \beta_{9} + 17692 \beta_{8} - 6292 \beta_{7} - 12584 \beta_{6} - 57672 \beta_{5} + 177813 \beta_{4} + \cdots + 3574691 ) / 12 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 7988 \beta_{9} - 20244 \beta_{7} - 14116 \beta_{6} - 1048072 \beta_{5} + 1694686 \beta_{4} + \cdots + 843349 \beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 3131412 \beta_{8} - 4288144 \beta_{7} + 1156732 \beta_{6} + 36530011 \beta_{4} + 13236504 \beta_{2} + \cdots - 832500609 ) / 36 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(271\) \(325\) \(353\)
\(\chi(n)\) \(1\) \(1\) \(1 + \beta_{3}\)

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
1.07323 − 1.85889i
−2.32209 + 4.02197i
1.22025 − 2.11353i
4.07727 − 7.06203i
−3.54866 + 6.14646i
1.07323 + 1.85889i
−2.32209 − 4.02197i
1.22025 + 2.11353i
4.07727 + 7.06203i
−3.54866 − 6.14646i
0 0 0 −136.563 − 78.8448i 0 256.037 + 443.470i 0 0 0
17.2 0 0 0 −80.3236 − 46.3749i 0 −60.0074 − 103.936i 0 0 0
17.3 0 0 0 64.9866 + 37.5201i 0 −181.066 − 313.616i 0 0 0
17.4 0 0 0 103.839 + 59.9512i 0 128.891 + 223.245i 0 0 0
17.5 0 0 0 157.562 + 90.9682i 0 −83.3541 − 144.373i 0 0 0
305.1 0 0 0 −136.563 + 78.8448i 0 256.037 − 443.470i 0 0 0
305.2 0 0 0 −80.3236 + 46.3749i 0 −60.0074 + 103.936i 0 0 0
305.3 0 0 0 64.9866 − 37.5201i 0 −181.066 + 313.616i 0 0 0
305.4 0 0 0 103.839 − 59.9512i 0 128.891 − 223.245i 0 0 0
305.5 0 0 0 157.562 − 90.9682i 0 −83.3541 + 144.373i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.5
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.a 10
3.b odd 2 1 144.7.q.a 10
4.b odd 2 1 27.7.d.a 10
9.c even 3 1 144.7.q.a 10
9.d odd 6 1 inner 432.7.q.a 10
12.b even 2 1 9.7.d.a ✓ 10
36.f odd 6 1 9.7.d.a ✓ 10
36.f odd 6 1 81.7.b.a 10
36.h even 6 1 27.7.d.a 10
36.h even 6 1 81.7.b.a 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.7.d.a ✓ 10 12.b even 2 1
9.7.d.a ✓ 10 36.f odd 6 1
27.7.d.a 10 4.b odd 2 1
27.7.d.a 10 36.h even 6 1
81.7.b.a 10 36.f odd 6 1
81.7.b.a 10 36.h even 6 1
144.7.q.a 10 3.b odd 2 1
144.7.q.a 10 9.c even 3 1
432.7.q.a 10 1.a even 1 1 trivial
432.7.q.a 10 9.d odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{10} - 219 T_{5}^{9} - 19308 T_{5}^{8} + 7729605 T_{5}^{7} + 501307200 T_{5}^{6} + \cdots + 57\!\cdots\!00 \) acting on \(S_{7}^{\mathrm{new}}(432, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 57\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 91\!\cdots\!96 \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 96\!\cdots\!27 \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 11\!\cdots\!64 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{5} + \cdots + 51\!\cdots\!36)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 17\!\cdots\!92 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 24\!\cdots\!72 \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots + 38\!\cdots\!56 \) Copy content Toggle raw display
$37$ \( (T^{5} + \cdots + 12\!\cdots\!72)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 44\!\cdots\!63 \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 48\!\cdots\!01 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 10\!\cdots\!28 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 41\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 73\!\cdots\!43 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 52\!\cdots\!44 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 73\!\cdots\!41 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{5} + \cdots - 36\!\cdots\!80)^{2} \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 34\!\cdots\!96 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 32\!\cdots\!88 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 14\!\cdots\!25 \) Copy content Toggle raw display
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