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Properties

Label 432.4.s.d
Level $432$
Weight $4$
Character orbit 432.s
Analytic conductor $25.489$
Analytic rank $0$
Dimension $10$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [432,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("432.143");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 432 = 2^{4} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 4 \)
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: \(25.4888251225\)
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: \( x^{10} - 2 x^{9} + 15 x^{8} + 26 x^{7} + 101 x^{6} + 396 x^{5} + 1292 x^{4} + 2864 x^{3} + 7860 x^{2} + \cdots + 26368 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{10} \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{7} - \beta_1) q^{5} + (\beta_{9} - \beta_{7} - \beta_{5} + \cdots + 6) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{7} - \beta_1) q^{5} + (\beta_{9} - \beta_{7} - \beta_{5} + \cdots + 6) q^{7}+ \cdots + (19 \beta_{9} + 19 \beta_{7} + \cdots + 9) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 9 q^{5} + 33 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 9 q^{5} + 33 q^{7} + 27 q^{11} + 55 q^{13} - 153 q^{23} + 56 q^{25} + 99 q^{29} + 351 q^{31} + 990 q^{35} + 1100 q^{37} - 783 q^{41} - 333 q^{43} - 603 q^{47} + 442 q^{49} - 423 q^{59} - 325 q^{61} + 3735 q^{65} + 753 q^{67} + 5040 q^{71} + 736 q^{73} - 2565 q^{77} + 237 q^{79} - 1323 q^{83} + 1452 q^{85} - 2322 q^{95} + 553 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2 x^{9} + 15 x^{8} + 26 x^{7} + 101 x^{6} + 396 x^{5} + 1292 x^{4} + 2864 x^{3} + 7860 x^{2} + \cdots + 26368 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 29 \nu^{9} + 272 \nu^{8} - 487 \nu^{7} + 4348 \nu^{6} - 3117 \nu^{5} + 37158 \nu^{4} + \cdots + 979808 ) / 1049760 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 61 \nu^{9} - 88 \nu^{8} + 10493 \nu^{7} - 29852 \nu^{6} + 118923 \nu^{5} - 40062 \nu^{4} + \cdots - 8318272 ) / 524880 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 229 \nu^{9} - 3308 \nu^{8} + 17773 \nu^{7} - 64672 \nu^{6} + 109623 \nu^{5} - 243582 \nu^{4} + \cdots + 2356768 ) / 1049760 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 47 \nu^{9} - 884 \nu^{8} + 3169 \nu^{7} - 8056 \nu^{6} - 8925 \nu^{5} + 75378 \nu^{4} + \cdots + 2113024 ) / 209952 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 49 \nu^{9} - 832 \nu^{8} + 1091 \nu^{7} - 7868 \nu^{6} - 21279 \nu^{5} - 59790 \nu^{4} + \cdots - 377920 ) / 209952 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 179 \nu^{9} + 118 \nu^{8} - 7133 \nu^{7} + 21362 \nu^{6} - 114243 \nu^{5} + 46632 \nu^{4} + \cdots - 2787968 ) / 524880 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 71 \nu^{9} - 98 \nu^{8} + 913 \nu^{7} + 2138 \nu^{6} + 6123 \nu^{5} + 18768 \nu^{4} + 37924 \nu^{3} + \cdots + 136888 ) / 87480 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 1691 \nu^{9} + 4688 \nu^{8} - 44653 \nu^{7} + 24052 \nu^{6} - 322023 \nu^{5} - 542838 \nu^{4} + \cdots - 23005888 ) / 1049760 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 529 \nu^{9} + 2176 \nu^{8} - 12671 \nu^{7} + 4652 \nu^{6} - 55149 \nu^{5} - 216114 \nu^{4} + \cdots - 7425824 ) / 209952 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{6} - \beta_{5} + \beta_{3} - 4\beta _1 + 5 ) / 9 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{9} - 3\beta_{8} + 3\beta_{7} - \beta_{5} + 3\beta_{4} + \beta_{3} - 2\beta_{2} - 11\beta _1 - 20 ) / 9 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{9} - 8\beta_{8} - 6\beta_{7} - 4\beta_{6} + 5\beta_{5} - 4\beta_{3} - 5\beta_{2} + 20\beta _1 - 161 ) / 9 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 18 \beta_{9} + 6 \beta_{8} - 45 \beta_{7} - 12 \beta_{6} + \beta_{5} - 6 \beta_{4} - 28 \beta_{3} + \cdots - 352 ) / 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 99 \beta_{9} + 112 \beta_{8} - 63 \beta_{7} - 61 \beta_{6} - 30 \beta_{5} - 75 \beta_{4} - 50 \beta_{3} + \cdots + 396 ) / 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 210 \beta_{9} + 422 \beta_{8} + 330 \beta_{7} - 82 \beta_{6} + 85 \beta_{5} - 204 \beta_{4} + \cdots + 2955 ) / 9 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 372 \beta_{9} + 154 \beta_{8} + 1587 \beta_{7} + 539 \beta_{6} + 451 \beta_{5} + 213 \beta_{4} + \cdots + 12953 ) / 9 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 4104 \beta_{9} - 4690 \beta_{8} + 2079 \beta_{7} + 3914 \beta_{6} - 900 \beta_{5} + 2580 \beta_{4} + \cdots + 31867 ) / 9 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 11355 \beta_{9} - 20590 \beta_{8} - 3396 \beta_{7} + 9456 \beta_{6} - 5381 \beta_{5} + 7476 \beta_{4} + \cdots - 44003 ) / 9 \) 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
2.22708 + 2.73757i
−1.92524 + 0.665541i
1.90731 − 2.85951i
−1.46035 − 1.71659i
0.251202 + 2.90505i
2.22708 − 2.73757i
−1.92524 − 0.665541i
1.90731 + 2.85951i
−1.46035 + 1.71659i
0.251202 − 2.90505i
0 0 0 −13.2594 + 7.65534i 0 4.22536 + 2.43952i 0 0 0
143.2 0 0 0 −5.73939 + 3.31364i 0 −25.3681 − 14.6463i 0 0 0
143.3 0 0 0 −5.32688 + 3.07547i 0 −1.70936 − 0.986900i 0 0 0
143.4 0 0 0 3.70432 − 2.13869i 0 29.1403 + 16.8241i 0 0 0
143.5 0 0 0 16.1214 − 9.30769i 0 10.2118 + 5.89578i 0 0 0
287.1 0 0 0 −13.2594 − 7.65534i 0 4.22536 − 2.43952i 0 0 0
287.2 0 0 0 −5.73939 − 3.31364i 0 −25.3681 + 14.6463i 0 0 0
287.3 0 0 0 −5.32688 − 3.07547i 0 −1.70936 + 0.986900i 0 0 0
287.4 0 0 0 3.70432 + 2.13869i 0 29.1403 − 16.8241i 0 0 0
287.5 0 0 0 16.1214 + 9.30769i 0 10.2118 − 5.89578i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 143.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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 432.4.s.d 10
3.b odd 2 1 144.4.s.d yes 10
4.b odd 2 1 432.4.s.c 10
9.c even 3 1 144.4.s.c ✓ 10
9.c even 3 1 1296.4.c.e 10
9.d odd 6 1 432.4.s.c 10
9.d odd 6 1 1296.4.c.f 10
12.b even 2 1 144.4.s.c ✓ 10
36.f odd 6 1 144.4.s.d yes 10
36.f odd 6 1 1296.4.c.f 10
36.h even 6 1 inner 432.4.s.d 10
36.h even 6 1 1296.4.c.e 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.4.s.c ✓ 10 9.c even 3 1
144.4.s.c ✓ 10 12.b even 2 1
144.4.s.d yes 10 3.b odd 2 1
144.4.s.d yes 10 36.f odd 6 1
432.4.s.c 10 4.b odd 2 1
432.4.s.c 10 9.d odd 6 1
432.4.s.d 10 1.a even 1 1 trivial
432.4.s.d 10 36.h even 6 1 inner
1296.4.c.e 10 9.c even 3 1
1296.4.c.e 10 36.h even 6 1
1296.4.c.f 10 9.d odd 6 1
1296.4.c.f 10 36.f odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(432, [\chi])\):

\( T_{5}^{10} + 9 T_{5}^{9} - 300 T_{5}^{8} - 2943 T_{5}^{7} + 89136 T_{5}^{6} + 1358937 T_{5}^{5} + \cdots + 2469692592 \) Copy content Toggle raw display
\( T_{7}^{10} - 33 T_{7}^{9} - 534 T_{7}^{8} + 29601 T_{7}^{7} + 553266 T_{7}^{6} - 22321737 T_{7}^{5} + \cdots + 12527233200 \) 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
$5$ \( T^{10} + \cdots + 2469692592 \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 12527233200 \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 22979185782921 \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 18\!\cdots\!84 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 71\!\cdots\!28 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 68\!\cdots\!72 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 83\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots + 49\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{5} - 550 T^{4} + \cdots - 2882353280)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 13\!\cdots\!03 \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 46\!\cdots\!87 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 48\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 16\!\cdots\!68 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 16\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 64\!\cdots\!83 \) Copy content Toggle raw display
$71$ \( (T^{5} + \cdots + 77392428682176)^{2} \) Copy content Toggle raw display
$73$ \( (T^{5} + \cdots + 54363390126368)^{2} \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 46\!\cdots\!52 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 17\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 19\!\cdots\!72 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 47\!\cdots\!01 \) Copy content Toggle raw display
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