Properties

Label 432.2.k.b
Level 432
Weight 2
Character orbit 432.k
Analytic conductor 3.450
Analytic rank 0
Dimension 4
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 432 = 2^{4} \cdot 3^{3} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 432.k (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.44953736732\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \zeta_{8} + \zeta_{8}^{3} ) q^{2} -2 q^{4} -\zeta_{8} q^{5} -3 \zeta_{8}^{2} q^{7} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{8} +O(q^{10})\) \( q + ( \zeta_{8} + \zeta_{8}^{3} ) q^{2} -2 q^{4} -\zeta_{8} q^{5} -3 \zeta_{8}^{2} q^{7} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{8} + ( 1 - \zeta_{8}^{2} ) q^{10} + 5 \zeta_{8} q^{11} + ( 1 - \zeta_{8}^{2} ) q^{13} + ( 3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{14} + 4 q^{16} + ( 3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{17} + ( 4 - 4 \zeta_{8}^{2} ) q^{19} + 2 \zeta_{8} q^{20} + ( -5 + 5 \zeta_{8}^{2} ) q^{22} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{23} -4 \zeta_{8}^{2} q^{25} + 2 \zeta_{8} q^{26} + 6 \zeta_{8}^{2} q^{28} -4 \zeta_{8}^{3} q^{29} -7 q^{31} + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{32} + 6 \zeta_{8}^{2} q^{34} + 3 \zeta_{8}^{3} q^{35} + ( -2 - 2 \zeta_{8}^{2} ) q^{37} + 8 \zeta_{8} q^{38} + ( -2 + 2 \zeta_{8}^{2} ) q^{40} + ( -4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{41} + ( 7 + 7 \zeta_{8}^{2} ) q^{43} -10 \zeta_{8} q^{44} -4 q^{46} + ( -3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{47} -2 q^{49} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{50} + ( -2 + 2 \zeta_{8}^{2} ) q^{52} -\zeta_{8} q^{53} -5 \zeta_{8}^{2} q^{55} + ( -6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{56} + ( 4 + 4 \zeta_{8}^{2} ) q^{58} -10 \zeta_{8} q^{59} + ( 10 - 10 \zeta_{8}^{2} ) q^{61} + ( -7 \zeta_{8} - 7 \zeta_{8}^{3} ) q^{62} -8 q^{64} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{65} + ( 1 - \zeta_{8}^{2} ) q^{67} + ( -6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{68} + ( -3 - 3 \zeta_{8}^{2} ) q^{70} + ( 11 \zeta_{8} + 11 \zeta_{8}^{3} ) q^{71} + 15 \zeta_{8}^{2} q^{73} -4 \zeta_{8}^{3} q^{74} + ( -8 + 8 \zeta_{8}^{2} ) q^{76} -15 \zeta_{8}^{3} q^{77} + 2 q^{79} -4 \zeta_{8} q^{80} + 8 q^{82} + 11 \zeta_{8}^{3} q^{83} + ( -3 - 3 \zeta_{8}^{2} ) q^{85} + 14 \zeta_{8}^{3} q^{86} + ( 10 - 10 \zeta_{8}^{2} ) q^{88} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{89} + ( -3 - 3 \zeta_{8}^{2} ) q^{91} + ( -4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{92} -6 \zeta_{8}^{2} q^{94} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{95} -7 q^{97} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 8q^{4} + O(q^{10}) \) \( 4q - 8q^{4} + 4q^{10} + 4q^{13} + 16q^{16} + 16q^{19} - 20q^{22} - 28q^{31} - 8q^{37} - 8q^{40} + 28q^{43} - 16q^{46} - 8q^{49} - 8q^{52} + 16q^{58} + 40q^{61} - 32q^{64} + 4q^{67} - 12q^{70} - 32q^{76} + 8q^{79} + 32q^{82} - 12q^{85} + 40q^{88} - 12q^{91} - 28q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(271\) \(325\) \(353\)
\(\chi(n)\) \(1\) \(\zeta_{8}^{2}\) \(1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
0.707107 0.707107i
−0.707107 + 0.707107i
−0.707107 0.707107i
0.707107 + 0.707107i
1.41421i 0 −2.00000 −0.707107 + 0.707107i 0 3.00000i 2.82843i 0 1.00000 + 1.00000i
109.2 1.41421i 0 −2.00000 0.707107 0.707107i 0 3.00000i 2.82843i 0 1.00000 + 1.00000i
325.1 1.41421i 0 −2.00000 0.707107 + 0.707107i 0 3.00000i 2.82843i 0 1.00000 1.00000i
325.2 1.41421i 0 −2.00000 −0.707107 0.707107i 0 3.00000i 2.82843i 0 1.00000 1.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
16.e even 4 1 inner
48.i odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 432.2.k.b 4
3.b odd 2 1 inner 432.2.k.b 4
4.b odd 2 1 1728.2.k.b 4
12.b even 2 1 1728.2.k.b 4
16.e even 4 1 inner 432.2.k.b 4
16.f odd 4 1 1728.2.k.b 4
48.i odd 4 1 inner 432.2.k.b 4
48.k even 4 1 1728.2.k.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
432.2.k.b 4 1.a even 1 1 trivial
432.2.k.b 4 3.b odd 2 1 inner
432.2.k.b 4 16.e even 4 1 inner
432.2.k.b 4 48.i odd 4 1 inner
1728.2.k.b 4 4.b odd 2 1
1728.2.k.b 4 12.b even 2 1
1728.2.k.b 4 16.f odd 4 1
1728.2.k.b 4 48.k even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(432, [\chi])\):

\( T_{5}^{4} + 1 \)
\( T_{11}^{4} + 625 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 2 T^{2} )^{2} \)
$3$ \( \)
$5$ \( 1 + 31 T^{4} + 625 T^{8} \)
$7$ \( ( 1 - 5 T^{2} + 49 T^{4} )^{2} \)
$11$ \( 1 - 233 T^{4} + 14641 T^{8} \)
$13$ \( ( 1 - 6 T + 13 T^{2} )^{2}( 1 + 4 T + 13 T^{2} )^{2} \)
$17$ \( ( 1 + 16 T^{2} + 289 T^{4} )^{2} \)
$19$ \( ( 1 - 8 T + 32 T^{2} - 152 T^{3} + 361 T^{4} )^{2} \)
$23$ \( ( 1 - 38 T^{2} + 529 T^{4} )^{2} \)
$29$ \( ( 1 - 40 T^{2} + 841 T^{4} )( 1 + 40 T^{2} + 841 T^{4} ) \)
$31$ \( ( 1 + 7 T + 31 T^{2} )^{4} \)
$37$ \( ( 1 + 4 T + 8 T^{2} + 148 T^{3} + 1369 T^{4} )^{2} \)
$41$ \( ( 1 - 50 T^{2} + 1681 T^{4} )^{2} \)
$43$ \( ( 1 - 14 T + 98 T^{2} - 602 T^{3} + 1849 T^{4} )^{2} \)
$47$ \( ( 1 + 76 T^{2} + 2209 T^{4} )^{2} \)
$53$ \( 1 + 5407 T^{4} + 7890481 T^{8} \)
$59$ \( 1 - 6638 T^{4} + 12117361 T^{8} \)
$61$ \( ( 1 - 20 T + 200 T^{2} - 1220 T^{3} + 3721 T^{4} )^{2} \)
$67$ \( ( 1 - 2 T + 2 T^{2} - 134 T^{3} + 4489 T^{4} )^{2} \)
$71$ \( ( 1 + 100 T^{2} + 5041 T^{4} )^{2} \)
$73$ \( ( 1 + 79 T^{2} + 5329 T^{4} )^{2} \)
$79$ \( ( 1 - 2 T + 79 T^{2} )^{4} \)
$83$ \( 1 - 11753 T^{4} + 47458321 T^{8} \)
$89$ \( ( 1 - 176 T^{2} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 + 7 T + 97 T^{2} )^{4} \)
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