Properties

Label 6600.2.d.n
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 264)
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.

\(f(q)\) \(=\) \( q -i q^{3} + 4 i q^{7} - q^{9} +O(q^{10})\) \( q -i q^{3} + 4 i q^{7} - q^{9} - q^{11} -6 i q^{13} + 6 i q^{17} + 8 q^{19} + 4 q^{21} + i q^{27} + 6 q^{29} + i q^{33} + 6 i q^{37} -6 q^{39} -10 q^{41} + 8 i q^{43} -9 q^{49} + 6 q^{51} -6 i q^{53} -8 i q^{57} -4 q^{59} -2 q^{61} -4 i q^{63} -12 i q^{67} -8 q^{71} -2 i q^{73} -4 i q^{77} + 4 q^{79} + q^{81} + 12 i q^{83} -6 i q^{87} + 6 q^{89} + 24 q^{91} + 2 i q^{97} + q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{9} - 2q^{11} + 16q^{19} + 8q^{21} + 12q^{29} - 12q^{39} - 20q^{41} - 18q^{49} + 12q^{51} - 8q^{59} - 4q^{61} - 16q^{71} + 8q^{79} + 2q^{81} + 12q^{89} + 48q^{91} + 2q^{99} + O(q^{100}) \)

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 4.00000i 0 −1.00000 0
1849.2 0 1.00000i 0 0 0 4.00000i 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.n 2
5.b even 2 1 inner 6600.2.d.n 2
5.c odd 4 1 264.2.a.b 1
5.c odd 4 1 6600.2.a.a 1
15.e even 4 1 792.2.a.f 1
20.e even 4 1 528.2.a.b 1
40.i odd 4 1 2112.2.a.m 1
40.k even 4 1 2112.2.a.y 1
55.e even 4 1 2904.2.a.i 1
60.l odd 4 1 1584.2.a.n 1
120.q odd 4 1 6336.2.a.o 1
120.w even 4 1 6336.2.a.v 1
165.l odd 4 1 8712.2.a.r 1
220.i odd 4 1 5808.2.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
264.2.a.b 1 5.c odd 4 1
528.2.a.b 1 20.e even 4 1
792.2.a.f 1 15.e even 4 1
1584.2.a.n 1 60.l odd 4 1
2112.2.a.m 1 40.i odd 4 1
2112.2.a.y 1 40.k even 4 1
2904.2.a.i 1 55.e even 4 1
5808.2.a.f 1 220.i odd 4 1
6336.2.a.o 1 120.q odd 4 1
6336.2.a.v 1 120.w even 4 1
6600.2.a.a 1 5.c odd 4 1
6600.2.d.n 2 1.a even 1 1 trivial
6600.2.d.n 2 5.b even 2 1 inner
8712.2.a.r 1 165.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 \(S_{2}^{\mathrm{new}}(6600, [\chi])\):

\( T_{7}^{2} + 16 \)
\( T_{13}^{2} + 36 \)
\( T_{17}^{2} + 36 \)
\( T_{19} - 8 \)
\( T_{29} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + T^{2} \)
$5$ 1
$7$ \( 1 + 2 T^{2} + 49 T^{4} \)
$11$ \( ( 1 + T )^{2} \)
$13$ \( ( 1 - 4 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} ) \)
$17$ \( 1 + 2 T^{2} + 289 T^{4} \)
$19$ \( ( 1 - 8 T + 19 T^{2} )^{2} \)
$23$ \( ( 1 - 23 T^{2} )^{2} \)
$29$ \( ( 1 - 6 T + 29 T^{2} )^{2} \)
$31$ \( ( 1 + 31 T^{2} )^{2} \)
$37$ \( 1 - 38 T^{2} + 1369 T^{4} \)
$41$ \( ( 1 + 10 T + 41 T^{2} )^{2} \)
$43$ \( 1 - 22 T^{2} + 1849 T^{4} \)
$47$ \( ( 1 - 47 T^{2} )^{2} \)
$53$ \( 1 - 70 T^{2} + 2809 T^{4} \)
$59$ \( ( 1 + 4 T + 59 T^{2} )^{2} \)
$61$ \( ( 1 + 2 T + 61 T^{2} )^{2} \)
$67$ \( 1 + 10 T^{2} + 4489 T^{4} \)
$71$ \( ( 1 + 8 T + 71 T^{2} )^{2} \)
$73$ \( 1 - 142 T^{2} + 5329 T^{4} \)
$79$ \( ( 1 - 4 T + 79 T^{2} )^{2} \)
$83$ \( 1 - 22 T^{2} + 6889 T^{4} \)
$89$ \( ( 1 - 6 T + 89 T^{2} )^{2} \)
$97$ \( 1 - 190 T^{2} + 9409 T^{4} \)
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