Properties

Label 825.2.c.c
Level $825$
Weight $2$
Character orbit 825.c
Analytic conductor $6.588$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 825.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.58765816676\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 165)
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.

\(f(q)\) \(=\) \( q + ( -1 + 2 \zeta_{12}^{2} ) q^{2} + \zeta_{12}^{3} q^{3} - q^{4} + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{6} -2 \zeta_{12}^{3} q^{7} + ( -1 + 2 \zeta_{12}^{2} ) q^{8} - q^{9} +O(q^{10})\) \( q + ( -1 + 2 \zeta_{12}^{2} ) q^{2} + \zeta_{12}^{3} q^{3} - q^{4} + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{6} -2 \zeta_{12}^{3} q^{7} + ( -1 + 2 \zeta_{12}^{2} ) q^{8} - q^{9} - q^{11} -\zeta_{12}^{3} q^{12} + ( -2 + 4 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{13} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{14} -5 q^{16} + ( 1 - 2 \zeta_{12}^{2} ) q^{18} + ( -2 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{19} + 2 q^{21} + ( 1 - 2 \zeta_{12}^{2} ) q^{22} + ( -4 + 8 \zeta_{12}^{2} ) q^{23} + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{24} + ( -6 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{26} -\zeta_{12}^{3} q^{27} + 2 \zeta_{12}^{3} q^{28} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{29} + ( -4 - 8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{31} + ( 3 - 6 \zeta_{12}^{2} ) q^{32} -\zeta_{12}^{3} q^{33} + q^{36} + ( -4 + 8 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{37} + ( 2 - 4 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{38} + ( -2 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{39} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{41} + ( -2 + 4 \zeta_{12}^{2} ) q^{42} + ( 4 - 8 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{43} + q^{44} -12 q^{46} + ( -4 + 8 \zeta_{12}^{2} ) q^{47} -5 \zeta_{12}^{3} q^{48} + 3 q^{49} + ( 2 - 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{52} + ( -4 + 8 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{53} + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{54} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{56} + ( 2 - 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{57} + 6 \zeta_{12}^{3} q^{58} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{59} + 2 q^{61} + ( 4 - 8 \zeta_{12}^{2} - 12 \zeta_{12}^{3} ) q^{62} + 2 \zeta_{12}^{3} q^{63} - q^{64} + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{66} -8 \zeta_{12}^{3} q^{67} + ( -8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{69} + ( 16 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{71} + ( 1 - 2 \zeta_{12}^{2} ) q^{72} + ( 6 - 12 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{73} + ( -12 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{74} + ( 2 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{76} + 2 \zeta_{12}^{3} q^{77} + ( 2 - 4 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{78} + ( 10 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{79} + q^{81} + 6 \zeta_{12}^{3} q^{82} + ( 2 - 4 \zeta_{12}^{2} + 12 \zeta_{12}^{3} ) q^{83} -2 q^{84} + ( 12 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{86} + ( -2 + 4 \zeta_{12}^{2} ) q^{87} + ( 1 - 2 \zeta_{12}^{2} ) q^{88} + ( 6 - 8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{89} + ( 4 + 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{91} + ( 4 - 8 \zeta_{12}^{2} ) q^{92} + ( 4 - 8 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{93} -12 q^{94} + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{96} + 10 \zeta_{12}^{3} q^{97} + ( -3 + 6 \zeta_{12}^{2} ) q^{98} + q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{4} - 4q^{9} + O(q^{10}) \) \( 4q - 4q^{4} - 4q^{9} - 4q^{11} - 20q^{16} - 8q^{19} + 8q^{21} - 24q^{26} - 16q^{31} + 4q^{36} - 8q^{39} + 4q^{44} - 48q^{46} + 12q^{49} + 8q^{61} - 4q^{64} - 48q^{74} + 8q^{76} + 40q^{79} + 4q^{81} - 8q^{84} + 48q^{86} + 24q^{89} + 16q^{91} - 48q^{94} + 4q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
0.866025 0.500000i
−0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
1.73205i 1.00000i −1.00000 0 −1.73205 2.00000i 1.73205i −1.00000 0
199.2 1.73205i 1.00000i −1.00000 0 1.73205 2.00000i 1.73205i −1.00000 0
199.3 1.73205i 1.00000i −1.00000 0 1.73205 2.00000i 1.73205i −1.00000 0
199.4 1.73205i 1.00000i −1.00000 0 −1.73205 2.00000i 1.73205i −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 825.2.c.c 4
3.b odd 2 1 2475.2.c.n 4
5.b even 2 1 inner 825.2.c.c 4
5.c odd 4 1 165.2.a.b 2
5.c odd 4 1 825.2.a.e 2
15.d odd 2 1 2475.2.c.n 4
15.e even 4 1 495.2.a.c 2
15.e even 4 1 2475.2.a.r 2
20.e even 4 1 2640.2.a.x 2
35.f even 4 1 8085.2.a.bd 2
55.e even 4 1 1815.2.a.i 2
55.e even 4 1 9075.2.a.bh 2
60.l odd 4 1 7920.2.a.bz 2
165.l odd 4 1 5445.2.a.s 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.a.b 2 5.c odd 4 1
495.2.a.c 2 15.e even 4 1
825.2.a.e 2 5.c odd 4 1
825.2.c.c 4 1.a even 1 1 trivial
825.2.c.c 4 5.b even 2 1 inner
1815.2.a.i 2 55.e even 4 1
2475.2.a.r 2 15.e even 4 1
2475.2.c.n 4 3.b odd 2 1
2475.2.c.n 4 15.d odd 2 1
2640.2.a.x 2 20.e even 4 1
5445.2.a.s 2 165.l odd 4 1
7920.2.a.bz 2 60.l odd 4 1
8085.2.a.bd 2 35.f even 4 1
9075.2.a.bh 2 55.e even 4 1

Hecke kernels

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

\( T_{2}^{2} + 3 \)
\( T_{7}^{2} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 3 + T^{2} )^{2} \)
$3$ \( ( 1 + T^{2} )^{2} \)
$5$ \( T^{4} \)
$7$ \( ( 4 + T^{2} )^{2} \)
$11$ \( ( 1 + T )^{4} \)
$13$ \( 64 + 32 T^{2} + T^{4} \)
$17$ \( T^{4} \)
$19$ \( ( -8 + 4 T + T^{2} )^{2} \)
$23$ \( ( 48 + T^{2} )^{2} \)
$29$ \( ( -12 + T^{2} )^{2} \)
$31$ \( ( -32 + 8 T + T^{2} )^{2} \)
$37$ \( 1936 + 104 T^{2} + T^{4} \)
$41$ \( ( -12 + T^{2} )^{2} \)
$43$ \( 1936 + 104 T^{2} + T^{4} \)
$47$ \( ( 48 + T^{2} )^{2} \)
$53$ \( 144 + 168 T^{2} + T^{4} \)
$59$ \( ( -48 + T^{2} )^{2} \)
$61$ \( ( -2 + T )^{4} \)
$67$ \( ( 64 + T^{2} )^{2} \)
$71$ \( ( -192 + T^{2} )^{2} \)
$73$ \( 10816 + 224 T^{2} + T^{4} \)
$79$ \( ( 88 - 20 T + T^{2} )^{2} \)
$83$ \( 17424 + 312 T^{2} + T^{4} \)
$89$ \( ( -12 - 12 T + T^{2} )^{2} \)
$97$ \( ( 100 + T^{2} )^{2} \)
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