Properties

Label 825.2.f.a
Level 825
Weight 2
Character orbit 825.f
Analytic conductor 6.588
Analytic rank 0
Dimension 2
CM discriminant -11
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 825.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.58765816676\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-11}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{-11})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta q^{3} -2 q^{4} + ( -3 + \beta ) q^{9} +O(q^{10})\) \( q -\beta q^{3} -2 q^{4} + ( -3 + \beta ) q^{9} + ( -1 + 2 \beta ) q^{11} + 2 \beta q^{12} + 4 q^{16} + ( -1 + 2 \beta ) q^{23} + ( 3 + 2 \beta ) q^{27} + 5 q^{31} + ( 6 - \beta ) q^{33} + ( 6 - 2 \beta ) q^{36} + 7 q^{37} + ( 2 - 4 \beta ) q^{44} + ( 2 - 4 \beta ) q^{47} -4 \beta q^{48} + 7 q^{49} + ( -4 + 8 \beta ) q^{53} + ( 1 - 2 \beta ) q^{59} -8 q^{64} + 13 q^{67} + ( 6 - \beta ) q^{69} + ( -5 + 10 \beta ) q^{71} + ( 6 - 5 \beta ) q^{81} + ( -5 + 10 \beta ) q^{89} + ( 2 - 4 \beta ) q^{92} -5 \beta q^{93} -17 q^{97} + ( -3 - 5 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{3} - 4q^{4} - 5q^{9} + O(q^{10}) \) \( 2q - q^{3} - 4q^{4} - 5q^{9} + 2q^{12} + 8q^{16} + 8q^{27} + 10q^{31} + 11q^{33} + 10q^{36} + 14q^{37} - 4q^{48} + 14q^{49} - 16q^{64} + 26q^{67} + 11q^{69} + 7q^{81} - 5q^{93} - 34q^{97} - 11q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/825\mathbb{Z}\right)^\times\).

\(n\) \(376\) \(551\) \(727\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
626.1
0.500000 + 1.65831i
0.500000 1.65831i
0 −0.500000 1.65831i −2.00000 0 0 0 0 −2.50000 + 1.65831i 0
626.2 0 −0.500000 + 1.65831i −2.00000 0 0 0 0 −2.50000 1.65831i 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
11.b odd 2 1 CM by \(\Q(\sqrt{-11}) \)
3.b odd 2 1 inner
33.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 825.2.f.a 2
3.b odd 2 1 inner 825.2.f.a 2
5.b even 2 1 33.2.d.a 2
5.c odd 4 2 825.2.d.a 4
11.b odd 2 1 CM 825.2.f.a 2
15.d odd 2 1 33.2.d.a 2
15.e even 4 2 825.2.d.a 4
20.d odd 2 1 528.2.b.a 2
33.d even 2 1 inner 825.2.f.a 2
40.e odd 2 1 2112.2.b.f 2
40.f even 2 1 2112.2.b.e 2
45.h odd 6 2 891.2.g.a 4
45.j even 6 2 891.2.g.a 4
55.d odd 2 1 33.2.d.a 2
55.e even 4 2 825.2.d.a 4
55.h odd 10 4 363.2.f.c 8
55.j even 10 4 363.2.f.c 8
60.h even 2 1 528.2.b.a 2
120.i odd 2 1 2112.2.b.e 2
120.m even 2 1 2112.2.b.f 2
165.d even 2 1 33.2.d.a 2
165.l odd 4 2 825.2.d.a 4
165.o odd 10 4 363.2.f.c 8
165.r even 10 4 363.2.f.c 8
220.g even 2 1 528.2.b.a 2
440.c even 2 1 2112.2.b.f 2
440.o odd 2 1 2112.2.b.e 2
495.o odd 6 2 891.2.g.a 4
495.r even 6 2 891.2.g.a 4
660.g odd 2 1 528.2.b.a 2
1320.b odd 2 1 2112.2.b.f 2
1320.u even 2 1 2112.2.b.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.2.d.a 2 5.b even 2 1
33.2.d.a 2 15.d odd 2 1
33.2.d.a 2 55.d odd 2 1
33.2.d.a 2 165.d even 2 1
363.2.f.c 8 55.h odd 10 4
363.2.f.c 8 55.j even 10 4
363.2.f.c 8 165.o odd 10 4
363.2.f.c 8 165.r even 10 4
528.2.b.a 2 20.d odd 2 1
528.2.b.a 2 60.h even 2 1
528.2.b.a 2 220.g even 2 1
528.2.b.a 2 660.g odd 2 1
825.2.d.a 4 5.c odd 4 2
825.2.d.a 4 15.e even 4 2
825.2.d.a 4 55.e even 4 2
825.2.d.a 4 165.l odd 4 2
825.2.f.a 2 1.a even 1 1 trivial
825.2.f.a 2 3.b odd 2 1 inner
825.2.f.a 2 11.b odd 2 1 CM
825.2.f.a 2 33.d even 2 1 inner
891.2.g.a 4 45.h odd 6 2
891.2.g.a 4 45.j even 6 2
891.2.g.a 4 495.o odd 6 2
891.2.g.a 4 495.r even 6 2
2112.2.b.e 2 40.f even 2 1
2112.2.b.e 2 120.i odd 2 1
2112.2.b.e 2 440.o odd 2 1
2112.2.b.e 2 1320.u even 2 1
2112.2.b.f 2 40.e odd 2 1
2112.2.b.f 2 120.m even 2 1
2112.2.b.f 2 440.c even 2 1
2112.2.b.f 2 1320.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(825, [\chi])\):

\( T_{2} \)
\( T_{23}^{2} + 11 \)
\( T_{37} - 7 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 2 T^{2} )^{2} \)
$3$ \( 1 + T + 3 T^{2} \)
$5$ \( \)
$7$ \( ( 1 - 7 T^{2} )^{2} \)
$11$ \( 1 + 11 T^{2} \)
$13$ \( ( 1 - 13 T^{2} )^{2} \)
$17$ \( ( 1 + 17 T^{2} )^{2} \)
$19$ \( ( 1 - 19 T^{2} )^{2} \)
$23$ \( ( 1 - 9 T + 23 T^{2} )( 1 + 9 T + 23 T^{2} ) \)
$29$ \( ( 1 + 29 T^{2} )^{2} \)
$31$ \( ( 1 - 5 T + 31 T^{2} )^{2} \)
$37$ \( ( 1 - 7 T + 37 T^{2} )^{2} \)
$41$ \( ( 1 + 41 T^{2} )^{2} \)
$43$ \( ( 1 - 43 T^{2} )^{2} \)
$47$ \( ( 1 - 12 T + 47 T^{2} )( 1 + 12 T + 47 T^{2} ) \)
$53$ \( ( 1 - 6 T + 53 T^{2} )( 1 + 6 T + 53 T^{2} ) \)
$59$ \( ( 1 - 15 T + 59 T^{2} )( 1 + 15 T + 59 T^{2} ) \)
$61$ \( ( 1 - 61 T^{2} )^{2} \)
$67$ \( ( 1 - 13 T + 67 T^{2} )^{2} \)
$71$ \( ( 1 - 3 T + 71 T^{2} )( 1 + 3 T + 71 T^{2} ) \)
$73$ \( ( 1 - 73 T^{2} )^{2} \)
$79$ \( ( 1 - 79 T^{2} )^{2} \)
$83$ \( ( 1 + 83 T^{2} )^{2} \)
$89$ \( ( 1 - 9 T + 89 T^{2} )( 1 + 9 T + 89 T^{2} ) \)
$97$ \( ( 1 + 17 T + 97 T^{2} )^{2} \)
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