Properties

Label 6600.2.d.y
Level $6600$
Weight $2$
Character orbit 6600.d
Analytic conductor $52.701$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 6600 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6600.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(52.7012653340\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1320)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + i q^{3} + 2 i q^{7} - q^{9} +O(q^{10})\) \( q + i q^{3} + 2 i q^{7} - q^{9} + q^{11} + 8 i q^{17} + 8 q^{19} -2 q^{21} + 4 i q^{23} -i q^{27} + 6 q^{29} + i q^{33} -6 i q^{37} -2 q^{41} + 2 i q^{43} + 4 i q^{47} + 3 q^{49} -8 q^{51} -2 i q^{53} + 8 i q^{57} + 12 q^{59} -6 q^{61} -2 i q^{63} -8 i q^{67} -4 q^{69} -8 i q^{73} + 2 i q^{77} -4 q^{79} + q^{81} -6 i q^{83} + 6 i q^{87} + 10 q^{89} + 10 i q^{97} - q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{9} + 2 q^{11} + 16 q^{19} - 4 q^{21} + 12 q^{29} - 4 q^{41} + 6 q^{49} - 16 q^{51} + 24 q^{59} - 12 q^{61} - 8 q^{69} - 8 q^{79} + 2 q^{81} + 20 q^{89} - 2 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6600\mathbb{Z}\right)^\times\).

\(n\) \(1201\) \(2201\) \(2377\) \(3301\) \(4951\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1849.1
1.00000i
1.00000i
0 1.00000i 0 0 0 2.00000i 0 −1.00000 0
1849.2 0 1.00000i 0 0 0 2.00000i 0 −1.00000 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 6600.2.d.y 2
5.b even 2 1 inner 6600.2.d.y 2
5.c odd 4 1 1320.2.a.k 1
5.c odd 4 1 6600.2.a.n 1
15.e even 4 1 3960.2.a.k 1
20.e even 4 1 2640.2.a.c 1
60.l odd 4 1 7920.2.a.bj 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1320.2.a.k 1 5.c odd 4 1
2640.2.a.c 1 20.e even 4 1
3960.2.a.k 1 15.e even 4 1
6600.2.a.n 1 5.c odd 4 1
6600.2.d.y 2 1.a even 1 1 trivial
6600.2.d.y 2 5.b even 2 1 inner
7920.2.a.bj 1 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(6600, [\chi])\):

\( T_{7}^{2} + 4 \)
\( T_{13} \)
\( T_{17}^{2} + 64 \)
\( T_{19} - 8 \)
\( T_{29} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 1 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 4 + T^{2} \)
$11$ \( ( -1 + T )^{2} \)
$13$ \( T^{2} \)
$17$ \( 64 + T^{2} \)
$19$ \( ( -8 + T )^{2} \)
$23$ \( 16 + T^{2} \)
$29$ \( ( -6 + T )^{2} \)
$31$ \( T^{2} \)
$37$ \( 36 + T^{2} \)
$41$ \( ( 2 + T )^{2} \)
$43$ \( 4 + T^{2} \)
$47$ \( 16 + T^{2} \)
$53$ \( 4 + T^{2} \)
$59$ \( ( -12 + T )^{2} \)
$61$ \( ( 6 + T )^{2} \)
$67$ \( 64 + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 64 + T^{2} \)
$79$ \( ( 4 + T )^{2} \)
$83$ \( 36 + T^{2} \)
$89$ \( ( -10 + T )^{2} \)
$97$ \( 100 + T^{2} \)
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