Properties

Label 1350.2.f.d
Level 1350
Weight 2
Character orbit 1350.f
Analytic conductor 10.780
Analytic rank 0
Dimension 8
CM no
Inner twists 8

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Newspace parameters

Level: \( N \) = \( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1350.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.7798042729\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{24})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
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 \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\zeta_{24}^{3} q^{2} + \zeta_{24}^{6} q^{4} + ( -\zeta_{24}^{3} + 2 \zeta_{24}^{7} ) q^{7} + ( \zeta_{24} - \zeta_{24}^{5} ) q^{8} +O(q^{10})\) \( q -\zeta_{24}^{3} q^{2} + \zeta_{24}^{6} q^{4} + ( -\zeta_{24}^{3} + 2 \zeta_{24}^{7} ) q^{7} + ( \zeta_{24} - \zeta_{24}^{5} ) q^{8} + ( 2 \zeta_{24}^{2} - \zeta_{24}^{6} ) q^{14} - q^{16} -6 \zeta_{24}^{3} q^{17} + \zeta_{24}^{6} q^{19} + ( 6 \zeta_{24} - 6 \zeta_{24}^{5} ) q^{23} + ( -\zeta_{24} - \zeta_{24}^{5} ) q^{28} + ( 12 \zeta_{24}^{2} - 6 \zeta_{24}^{6} ) q^{29} + 7 q^{31} + \zeta_{24}^{3} q^{32} + 6 \zeta_{24}^{6} q^{34} + ( -5 \zeta_{24}^{3} + 10 \zeta_{24}^{7} ) q^{37} + ( \zeta_{24} - \zeta_{24}^{5} ) q^{38} + ( \zeta_{24} + \zeta_{24}^{5} ) q^{43} -6 q^{46} + 4 \zeta_{24}^{6} q^{49} + ( 12 \zeta_{24} - 12 \zeta_{24}^{5} ) q^{53} + ( -1 + 2 \zeta_{24}^{4} ) q^{56} + ( -6 \zeta_{24} - 6 \zeta_{24}^{5} ) q^{58} + ( 12 \zeta_{24}^{2} - 6 \zeta_{24}^{6} ) q^{59} + 5 q^{61} -7 \zeta_{24}^{3} q^{62} -\zeta_{24}^{6} q^{64} + ( 6 \zeta_{24}^{3} - 12 \zeta_{24}^{7} ) q^{67} + ( 6 \zeta_{24} - 6 \zeta_{24}^{5} ) q^{68} + ( -6 + 12 \zeta_{24}^{4} ) q^{71} + ( 7 \zeta_{24} + 7 \zeta_{24}^{5} ) q^{73} + ( 10 \zeta_{24}^{2} - 5 \zeta_{24}^{6} ) q^{74} - q^{76} -13 \zeta_{24}^{6} q^{79} + ( 6 \zeta_{24} - 6 \zeta_{24}^{5} ) q^{83} + ( 1 - 2 \zeta_{24}^{4} ) q^{86} + ( -12 \zeta_{24}^{2} + 6 \zeta_{24}^{6} ) q^{89} + 6 \zeta_{24}^{3} q^{92} + ( -5 \zeta_{24}^{3} + 10 \zeta_{24}^{7} ) q^{97} + ( 4 \zeta_{24} - 4 \zeta_{24}^{5} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q - 8q^{16} + 56q^{31} - 48q^{46} + 40q^{61} - 8q^{76} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1350\mathbb{Z}\right)^\times\).

\(n\) \(1001\) \(1027\)
\(\chi(n)\) \(-1\) \(-\zeta_{24}^{3}\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1
0.965926 0.258819i
−0.258819 + 0.965926i
0.258819 0.965926i
−0.965926 + 0.258819i
0.965926 + 0.258819i
−0.258819 0.965926i
0.258819 + 0.965926i
−0.965926 0.258819i
−0.707107 + 0.707107i 0 1.00000i 0 0 −1.22474 1.22474i 0.707107 + 0.707107i 0 0
107.2 −0.707107 + 0.707107i 0 1.00000i 0 0 1.22474 + 1.22474i 0.707107 + 0.707107i 0 0
107.3 0.707107 0.707107i 0 1.00000i 0 0 −1.22474 1.22474i −0.707107 0.707107i 0 0
107.4 0.707107 0.707107i 0 1.00000i 0 0 1.22474 + 1.22474i −0.707107 0.707107i 0 0
593.1 −0.707107 0.707107i 0 1.00000i 0 0 −1.22474 + 1.22474i 0.707107 0.707107i 0 0
593.2 −0.707107 0.707107i 0 1.00000i 0 0 1.22474 1.22474i 0.707107 0.707107i 0 0
593.3 0.707107 + 0.707107i 0 1.00000i 0 0 −1.22474 + 1.22474i −0.707107 + 0.707107i 0 0
593.4 0.707107 + 0.707107i 0 1.00000i 0 0 1.22474 1.22474i −0.707107 + 0.707107i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 593.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
5.c odd 4 2 inner
15.d odd 2 1 inner
15.e even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1350.2.f.d 8
3.b odd 2 1 inner 1350.2.f.d 8
5.b even 2 1 inner 1350.2.f.d 8
5.c odd 4 2 inner 1350.2.f.d 8
15.d odd 2 1 inner 1350.2.f.d 8
15.e even 4 2 inner 1350.2.f.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1350.2.f.d 8 1.a even 1 1 trivial
1350.2.f.d 8 3.b odd 2 1 inner
1350.2.f.d 8 5.b even 2 1 inner
1350.2.f.d 8 5.c odd 4 2 inner
1350.2.f.d 8 15.d odd 2 1 inner
1350.2.f.d 8 15.e even 4 2 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1350, [\chi])\):

\( T_{7}^{4} + 9 \)
\( T_{29}^{2} - 108 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{4} )^{2} \)
$3$ \( \)
$5$ \( \)
$7$ \( ( 1 + 23 T^{4} + 2401 T^{8} )^{2} \)
$11$ \( ( 1 - 11 T^{2} )^{8} \)
$13$ \( ( 1 + 169 T^{4} )^{4} \)
$17$ \( ( 1 - 8 T + 32 T^{2} - 136 T^{3} + 289 T^{4} )^{2}( 1 + 8 T + 32 T^{2} + 136 T^{3} + 289 T^{4} )^{2} \)
$19$ \( ( 1 - 37 T^{2} + 361 T^{4} )^{4} \)
$23$ \( ( 1 - 958 T^{4} + 279841 T^{8} )^{2} \)
$29$ \( ( 1 - 50 T^{2} + 841 T^{4} )^{4} \)
$31$ \( ( 1 - 7 T + 31 T^{2} )^{8} \)
$37$ \( ( 1 - 2737 T^{4} + 1874161 T^{8} )^{2} \)
$41$ \( ( 1 - 41 T^{2} )^{8} \)
$43$ \( ( 1 + 3191 T^{4} + 3418801 T^{8} )^{2} \)
$47$ \( ( 1 + 2209 T^{4} )^{4} \)
$53$ \( ( 1 - 4174 T^{4} + 7890481 T^{8} )^{2} \)
$59$ \( ( 1 + 10 T^{2} + 3481 T^{4} )^{4} \)
$61$ \( ( 1 - 5 T + 61 T^{2} )^{8} \)
$67$ \( ( 1 - 8302 T^{4} + 20151121 T^{8} )^{2} \)
$71$ \( ( 1 - 34 T^{2} + 5041 T^{4} )^{4} \)
$73$ \( ( 1 - 10657 T^{4} + 28398241 T^{8} )^{2} \)
$79$ \( ( 1 + 11 T^{2} + 6241 T^{4} )^{4} \)
$83$ \( ( 1 + 3122 T^{4} + 47458321 T^{8} )^{2} \)
$89$ \( ( 1 + 70 T^{2} + 7921 T^{4} )^{4} \)
$97$ \( ( 1 - 4657 T^{4} + 88529281 T^{8} )^{2} \)
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