Properties

Label 825.2.m.b
Level $825$
Weight $2$
Character orbit 825.m
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.m (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.58765816676\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Defining polynomial: \(x^{4} - x^{3} + x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{10}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\zeta_{10} q^{3} -2 \zeta_{10}^{2} q^{4} + ( 2 - \zeta_{10} + 2 \zeta_{10}^{2} ) q^{5} + ( 1 - \zeta_{10} + 3 \zeta_{10}^{3} ) q^{7} + \zeta_{10}^{2} q^{9} +O(q^{10})\) \( q -\zeta_{10} q^{3} -2 \zeta_{10}^{2} q^{4} + ( 2 - \zeta_{10} + 2 \zeta_{10}^{2} ) q^{5} + ( 1 - \zeta_{10} + 3 \zeta_{10}^{3} ) q^{7} + \zeta_{10}^{2} q^{9} + ( 2 + 2 \zeta_{10} - \zeta_{10}^{2} ) q^{11} + 2 \zeta_{10}^{3} q^{12} + ( \zeta_{10} - 4 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{13} + ( -2 \zeta_{10} + \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{15} + ( -4 + 4 \zeta_{10} - 4 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{16} + ( 2 \zeta_{10} + 3 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{17} + ( -4 + 4 \zeta_{10} - \zeta_{10}^{3} ) q^{19} + ( 4 - 4 \zeta_{10} - 2 \zeta_{10}^{3} ) q^{20} + ( 3 - 4 \zeta_{10} + 4 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{21} + ( \zeta_{10} + 3 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{23} + 5 \zeta_{10}^{2} q^{25} -\zeta_{10}^{3} q^{27} + ( 6 - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{28} + ( -6 \zeta_{10} + 2 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{29} -2 q^{31} + ( -2 \zeta_{10} - 2 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{33} + ( -1 - 6 \zeta_{10} + 6 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{35} + ( 2 - 2 \zeta_{10} + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{36} + ( 2 + 7 \zeta_{10} + 2 \zeta_{10}^{2} ) q^{37} + ( 1 - \zeta_{10} + 3 \zeta_{10}^{3} ) q^{39} + ( 5 + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{41} + ( 5 + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{43} + ( -2 + 2 \zeta_{10} - 6 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{44} + ( -2 + 2 \zeta_{10} + \zeta_{10}^{3} ) q^{45} + ( 1 - 3 \zeta_{10} + \zeta_{10}^{2} ) q^{47} + 4 q^{48} + ( 7 - 10 \zeta_{10} + 7 \zeta_{10}^{2} ) q^{49} + ( 2 - 2 \zeta_{10} - 5 \zeta_{10}^{3} ) q^{51} + ( -6 + 8 \zeta_{10} - 8 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{52} -9 \zeta_{10}^{3} q^{53} + ( 6 + 2 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{55} + ( -1 + 5 \zeta_{10} - 5 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{57} + ( 2 + 9 \zeta_{10} + 2 \zeta_{10}^{2} ) q^{59} + ( -2 - 2 \zeta_{10} + 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{60} + ( 8 - 8 \zeta_{10} - 9 \zeta_{10}^{3} ) q^{61} + ( -3 + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{63} + 8 \zeta_{10} q^{64} + ( 7 - 7 \zeta_{10} - \zeta_{10}^{3} ) q^{65} + ( -2 + \zeta_{10} - 2 \zeta_{10}^{2} ) q^{67} + ( 10 - 6 \zeta_{10} + 6 \zeta_{10}^{2} - 10 \zeta_{10}^{3} ) q^{68} + ( 1 - \zeta_{10} - 4 \zeta_{10}^{3} ) q^{69} + ( -9 - 4 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{71} + ( -4 - 6 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{73} -5 \zeta_{10}^{3} q^{75} + ( -2 + 8 \zeta_{10}^{2} - 8 \zeta_{10}^{3} ) q^{76} + ( -1 + 6 \zeta_{10} - 9 \zeta_{10}^{2} + 13 \zeta_{10}^{3} ) q^{77} + ( 5 - 5 \zeta_{10} + 4 \zeta_{10}^{3} ) q^{79} + ( -4 - 8 \zeta_{10}^{2} + 8 \zeta_{10}^{3} ) q^{80} + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{81} + ( -5 \zeta_{10} + 9 \zeta_{10}^{2} - 5 \zeta_{10}^{3} ) q^{83} + ( 2 - 8 \zeta_{10} + 2 \zeta_{10}^{2} ) q^{84} + ( -8 + 8 \zeta_{10} + 9 \zeta_{10}^{3} ) q^{85} + ( -6 + 6 \zeta_{10} + 4 \zeta_{10}^{3} ) q^{87} + ( -6 \zeta_{10} + 9 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{89} + ( 10 - 7 \zeta_{10}^{2} + 7 \zeta_{10}^{3} ) q^{91} + ( 8 - 6 \zeta_{10} + 6 \zeta_{10}^{2} - 8 \zeta_{10}^{3} ) q^{92} + 2 \zeta_{10} q^{93} + ( -7 + 13 \zeta_{10} - 13 \zeta_{10}^{2} + 7 \zeta_{10}^{3} ) q^{95} + ( 1 - \zeta_{10} + 5 \zeta_{10}^{3} ) q^{97} + ( 1 - \zeta_{10} + 3 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{3} + 2 q^{4} + 5 q^{5} + 6 q^{7} - q^{9} + O(q^{10}) \) \( 4 q - q^{3} + 2 q^{4} + 5 q^{5} + 6 q^{7} - q^{9} + 11 q^{11} + 2 q^{12} + 6 q^{13} - 5 q^{15} - 4 q^{16} + q^{17} - 13 q^{19} + 10 q^{20} + q^{21} - q^{23} - 5 q^{25} - q^{27} + 28 q^{28} - 14 q^{29} - 8 q^{31} + q^{33} - 15 q^{35} + 2 q^{36} + 13 q^{37} + 6 q^{39} + 16 q^{41} + 16 q^{43} - 2 q^{44} - 5 q^{45} + 16 q^{48} + 11 q^{49} + q^{51} - 2 q^{52} - 9 q^{53} + 25 q^{55} + 7 q^{57} + 15 q^{59} - 10 q^{60} + 15 q^{61} - 14 q^{63} + 8 q^{64} + 20 q^{65} - 5 q^{67} + 18 q^{68} - q^{69} - 28 q^{71} - 4 q^{73} - 5 q^{75} - 24 q^{76} + 24 q^{77} + 19 q^{79} - q^{81} - 19 q^{83} - 2 q^{84} - 15 q^{85} - 14 q^{87} - 21 q^{89} + 54 q^{91} + 12 q^{92} + 2 q^{93} + 5 q^{95} + 8 q^{97} + q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(376\) \(551\) \(727\)
\(\chi(n)\) \(-\zeta_{10}^{3}\) \(1\) \(-1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1
−0.309017 + 0.951057i
0.809017 + 0.587785i
−0.309017 0.951057i
0.809017 0.587785i
0 0.309017 0.951057i 1.61803 + 1.17557i 0.690983 2.12663i 0 3.73607 2.71441i 0 −0.809017 0.587785i 0
256.1 0 −0.809017 0.587785i −0.618034 1.90211i 1.80902 + 1.31433i 0 −0.736068 + 2.26538i 0 0.309017 + 0.951057i 0
361.1 0 0.309017 + 0.951057i 1.61803 1.17557i 0.690983 + 2.12663i 0 3.73607 + 2.71441i 0 −0.809017 + 0.587785i 0
796.1 0 −0.809017 + 0.587785i −0.618034 + 1.90211i 1.80902 1.31433i 0 −0.736068 2.26538i 0 0.309017 0.951057i 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
275.g even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 825.2.m.b 4
11.c even 5 1 825.2.o.a yes 4
25.d even 5 1 825.2.o.a yes 4
275.g even 5 1 inner 825.2.m.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.2.m.b 4 1.a even 1 1 trivial
825.2.m.b 4 275.g even 5 1 inner
825.2.o.a yes 4 11.c even 5 1
825.2.o.a yes 4 25.d even 5 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{2}^{\mathrm{new}}(825, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
$5$ \( 25 - 25 T + 15 T^{2} - 5 T^{3} + T^{4} \)
$7$ \( 121 - 11 T + 16 T^{2} - 6 T^{3} + T^{4} \)
$11$ \( 121 - 121 T + 51 T^{2} - 11 T^{3} + T^{4} \)
$13$ \( 121 - 11 T + 16 T^{2} - 6 T^{3} + T^{4} \)
$17$ \( 121 + 99 T + 31 T^{2} - T^{3} + T^{4} \)
$19$ \( 121 + 77 T + 69 T^{2} + 13 T^{3} + T^{4} \)
$23$ \( 121 + 66 T + 16 T^{2} + T^{3} + T^{4} \)
$29$ \( 1936 + 704 T + 136 T^{2} + 14 T^{3} + T^{4} \)
$31$ \( ( 2 + T )^{4} \)
$37$ \( 3481 - 177 T + 79 T^{2} - 13 T^{3} + T^{4} \)
$41$ \( ( 11 - 8 T + T^{2} )^{2} \)
$43$ \( ( 11 - 8 T + T^{2} )^{2} \)
$47$ \( 25 + 25 T + 10 T^{2} + T^{4} \)
$53$ \( 6561 + 729 T + 81 T^{2} + 9 T^{3} + T^{4} \)
$59$ \( 9025 - 475 T + 115 T^{2} - 15 T^{3} + T^{4} \)
$61$ \( 3025 - 1375 T + 265 T^{2} - 15 T^{3} + T^{4} \)
$67$ \( 25 + 25 T + 15 T^{2} + 5 T^{3} + T^{4} \)
$71$ \( ( 29 + 14 T + T^{2} )^{2} \)
$73$ \( ( -44 + 2 T + T^{2} )^{2} \)
$79$ \( 121 + 66 T + 136 T^{2} - 19 T^{3} + T^{4} \)
$83$ \( 121 - 66 T + 136 T^{2} + 19 T^{3} + T^{4} \)
$89$ \( 81 + 81 T + 171 T^{2} + 21 T^{3} + T^{4} \)
$97$ \( 841 - 87 T + 34 T^{2} - 8 T^{3} + T^{4} \)
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