Properties

Label 8100.2.d.l
Level $8100$
Weight $2$
Character orbit 8100.d
Analytic conductor $64.679$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 8100 = 2^{2} \cdot 3^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8100.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(64.6788256372\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 1620)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{7} +O(q^{10})\) \( q + ( 1 - 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{7} + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{11} + ( 2 - 4 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{13} + ( -1 + 2 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{17} + ( 1 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{19} + ( -2 + 4 \zeta_{12}^{2} ) q^{23} + ( -6 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{29} + ( -1 - 8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{31} + ( 3 - 6 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{37} + ( 6 - 6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{41} + ( -1 + 2 \zeta_{12}^{2} + 5 \zeta_{12}^{3} ) q^{43} + ( 1 - 2 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{47} + ( 3 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{49} + ( -1 + 2 \zeta_{12}^{2} + 9 \zeta_{12}^{3} ) q^{53} + ( -6 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{59} -4 q^{61} + ( -6 + 12 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{67} + ( 6 - 6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{71} + ( -3 + 6 \zeta_{12}^{2} + 5 \zeta_{12}^{3} ) q^{73} + ( -1 + 2 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{77} + ( 4 - 12 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{79} + ( 7 - 14 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{83} + ( -6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{89} + ( -8 + 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{91} + ( -1 + 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + O(q^{10}) \) \( 4 q + 4 q^{19} - 24 q^{29} - 4 q^{31} + 24 q^{41} + 12 q^{49} - 24 q^{59} - 16 q^{61} + 24 q^{71} + 16 q^{79} - 32 q^{91} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/8100\mathbb{Z}\right)^\times\).

\(n\) \(4051\) \(6401\) \(7777\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
649.1
−0.866025 0.500000i
0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
0 0 0 0 0 2.73205i 0 0 0
649.2 0 0 0 0 0 0.732051i 0 0 0
649.3 0 0 0 0 0 0.732051i 0 0 0
649.4 0 0 0 0 0 2.73205i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8100.2.d.l 4
3.b odd 2 1 8100.2.d.m 4
5.b even 2 1 inner 8100.2.d.l 4
5.c odd 4 1 1620.2.a.h yes 2
5.c odd 4 1 8100.2.a.s 2
15.d odd 2 1 8100.2.d.m 4
15.e even 4 1 1620.2.a.g 2
15.e even 4 1 8100.2.a.t 2
20.e even 4 1 6480.2.a.bp 2
45.k odd 12 2 1620.2.i.m 4
45.l even 12 2 1620.2.i.n 4
60.l odd 4 1 6480.2.a.bh 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1620.2.a.g 2 15.e even 4 1
1620.2.a.h yes 2 5.c odd 4 1
1620.2.i.m 4 45.k odd 12 2
1620.2.i.n 4 45.l even 12 2
6480.2.a.bh 2 60.l odd 4 1
6480.2.a.bp 2 20.e even 4 1
8100.2.a.s 2 5.c odd 4 1
8100.2.a.t 2 15.e even 4 1
8100.2.d.l 4 1.a even 1 1 trivial
8100.2.d.l 4 5.b even 2 1 inner
8100.2.d.m 4 3.b odd 2 1
8100.2.d.m 4 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(8100, [\chi])\):

\( T_{7}^{4} + 8 T_{7}^{2} + 4 \)
\( T_{11}^{2} - 3 \)
\( T_{29}^{2} + 12 T_{29} + 33 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( T^{4} \)
$7$ \( 4 + 8 T^{2} + T^{4} \)
$11$ \( ( -3 + T^{2} )^{2} \)
$13$ \( 64 + 32 T^{2} + T^{4} \)
$17$ \( 36 + 24 T^{2} + T^{4} \)
$19$ \( ( -11 - 2 T + T^{2} )^{2} \)
$23$ \( ( 12 + T^{2} )^{2} \)
$29$ \( ( 33 + 12 T + T^{2} )^{2} \)
$31$ \( ( -47 + 2 T + T^{2} )^{2} \)
$37$ \( 676 + 56 T^{2} + T^{4} \)
$41$ \( ( 9 - 12 T + T^{2} )^{2} \)
$43$ \( 484 + 56 T^{2} + T^{4} \)
$47$ \( 36 + 24 T^{2} + T^{4} \)
$53$ \( 6084 + 168 T^{2} + T^{4} \)
$59$ \( ( 33 + 12 T + T^{2} )^{2} \)
$61$ \( ( 4 + T )^{4} \)
$67$ \( 8464 + 248 T^{2} + T^{4} \)
$71$ \( ( 9 - 12 T + T^{2} )^{2} \)
$73$ \( 4 + 104 T^{2} + T^{4} \)
$79$ \( ( -92 - 8 T + T^{2} )^{2} \)
$83$ \( 19044 + 312 T^{2} + T^{4} \)
$89$ \( ( -27 + T^{2} )^{2} \)
$97$ \( 4 + 8 T^{2} + T^{4} \)
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