Properties

Label 825.2.o.b
Level $825$
Weight $2$
Character orbit 825.o
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.o (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: \(\Z[a_1, a_2]\)
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.

\(f(q)\) \(=\) \( q + ( \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{2} + \zeta_{10}^{3} q^{3} + ( 1 - \zeta_{10}^{3} ) q^{4} + ( 1 + \zeta_{10} - \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{5} + ( -\zeta_{10}^{2} + \zeta_{10}^{3} ) q^{6} + 3 \zeta_{10}^{3} q^{7} + ( 2 - \zeta_{10} + 2 \zeta_{10}^{2} ) q^{8} -\zeta_{10} q^{9} +O(q^{10})\) \( q + ( \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{2} + \zeta_{10}^{3} q^{3} + ( 1 - \zeta_{10}^{3} ) q^{4} + ( 1 + \zeta_{10} - \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{5} + ( -\zeta_{10}^{2} + \zeta_{10}^{3} ) q^{6} + 3 \zeta_{10}^{3} q^{7} + ( 2 - \zeta_{10} + 2 \zeta_{10}^{2} ) q^{8} -\zeta_{10} q^{9} + ( -1 + 3 \zeta_{10} - \zeta_{10}^{2} ) q^{10} + ( -2 + 4 \zeta_{10} - \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{11} + ( \zeta_{10} + \zeta_{10}^{3} ) q^{12} + ( 2 - 2 \zeta_{10} + 2 \zeta_{10}^{2} ) q^{13} + ( -3 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{14} + ( 2 \zeta_{10} - \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{15} + ( 3 - 3 \zeta_{10} ) q^{16} + ( -4 - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{17} + ( 1 - \zeta_{10} ) q^{18} + ( 1 + 2 \zeta_{10} + \zeta_{10}^{2} ) q^{19} + ( 1 - \zeta_{10} - 3 \zeta_{10}^{3} ) q^{20} -3 \zeta_{10} q^{21} + ( -4 + 3 \zeta_{10} - \zeta_{10}^{2} ) q^{22} + ( -3 + 3 \zeta_{10} + 3 \zeta_{10}^{3} ) q^{23} + ( -1 - \zeta_{10} + \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{24} -5 \zeta_{10}^{3} q^{25} + ( 2 - 2 \zeta_{10} + 2 \zeta_{10}^{3} ) q^{26} + ( 1 - \zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{27} + ( 3 \zeta_{10} + 3 \zeta_{10}^{3} ) q^{28} + ( 6 + \zeta_{10} - \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{29} + ( -2 + 3 \zeta_{10} - 3 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{30} + ( 2 - 2 \zeta_{10} + 7 \zeta_{10}^{3} ) q^{31} + ( 5 + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{32} + ( -3 + 2 \zeta_{10} - 4 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{33} + ( -2 \zeta_{10} - 2 \zeta_{10}^{3} ) q^{34} + ( 6 \zeta_{10} - 3 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{35} + ( -1 - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{36} + ( -4 + 3 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{37} + ( -2 + 2 \zeta_{10} + \zeta_{10}^{3} ) q^{38} + ( -2 \zeta_{10} + 2 \zeta_{10}^{2} ) q^{39} + 5 q^{40} + ( 8 - 3 \zeta_{10} + 3 \zeta_{10}^{2} - 8 \zeta_{10}^{3} ) q^{41} + ( 3 - 3 \zeta_{10} ) q^{42} + 6 q^{43} + ( 1 + 2 \zeta_{10} + 3 \zeta_{10}^{2} ) q^{44} + ( -1 - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{45} -3 q^{46} + ( -5 + 5 \zeta_{10} - 2 \zeta_{10}^{3} ) q^{47} + ( 3 - 3 \zeta_{10} + 3 \zeta_{10}^{2} ) q^{48} -2 \zeta_{10} q^{49} + ( 5 \zeta_{10}^{2} - 5 \zeta_{10}^{3} ) q^{50} + ( 2 - 2 \zeta_{10} - 4 \zeta_{10}^{3} ) q^{51} + 2 q^{52} + ( -10 - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{53} + ( 1 - \zeta_{10} + \zeta_{10}^{2} ) q^{54} + ( 3 \zeta_{10} + 6 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{55} + ( -3 - 3 \zeta_{10} + 3 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{56} + ( -3 + 2 \zeta_{10} - 2 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{57} + ( -1 + 8 \zeta_{10} - \zeta_{10}^{2} ) q^{58} + ( -12 - 4 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{59} + ( 1 + 2 \zeta_{10} + \zeta_{10}^{2} ) q^{60} + ( 1 + 5 \zeta_{10} - 5 \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{61} + ( 2 - 9 \zeta_{10}^{2} + 9 \zeta_{10}^{3} ) q^{62} + ( 3 - 3 \zeta_{10} + 3 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{63} + ( -2 \zeta_{10} + 3 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{64} + ( 4 - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{65} + ( -2 + 3 \zeta_{10} - 3 \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{66} + ( 3 - 3 \zeta_{10} + 3 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{67} + ( -6 + 2 \zeta_{10} - 2 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{68} + ( -3 - 3 \zeta_{10}^{2} ) q^{69} + ( -6 + 9 \zeta_{10} - 9 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{70} + ( -\zeta_{10} - 10 \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{71} + ( -2 \zeta_{10} + \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{72} + ( 9 - 3 \zeta_{10} + 9 \zeta_{10}^{2} ) q^{73} + ( -7 \zeta_{10} + 10 \zeta_{10}^{2} - 7 \zeta_{10}^{3} ) q^{74} + 5 \zeta_{10} q^{75} + ( 4 + 3 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{76} + ( -9 + 6 \zeta_{10} - 12 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{77} + ( 2 - 4 \zeta_{10} + 2 \zeta_{10}^{2} ) q^{78} -5 q^{79} + ( 3 \zeta_{10} - 9 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{80} + \zeta_{10}^{2} q^{81} + ( 3 + 2 \zeta_{10} + 3 \zeta_{10}^{2} ) q^{82} + ( -7 - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{83} + ( -3 - 3 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{84} + ( -8 + 2 \zeta_{10} - 2 \zeta_{10}^{2} + 8 \zeta_{10}^{3} ) q^{85} + ( 6 \zeta_{10} - 6 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{86} + ( 7 \zeta_{10} - \zeta_{10}^{2} + 7 \zeta_{10}^{3} ) q^{87} + ( -4 + 6 \zeta_{10} - 6 \zeta_{10}^{2} + 9 \zeta_{10}^{3} ) q^{88} + ( 2 - 2 \zeta_{10} - \zeta_{10}^{3} ) q^{89} + ( \zeta_{10} - 3 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{90} + ( -6 \zeta_{10} + 6 \zeta_{10}^{2} ) q^{91} + ( 3 \zeta_{10} + 3 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{92} + ( 2 - 9 \zeta_{10} + 2 \zeta_{10}^{2} ) q^{93} + ( -5 + 7 \zeta_{10}^{2} - 7 \zeta_{10}^{3} ) q^{94} + ( 5 + 5 \zeta_{10}^{2} - 5 \zeta_{10}^{3} ) q^{95} + ( -1 + \zeta_{10} + 5 \zeta_{10}^{3} ) q^{96} + ( 9 + 4 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{97} + ( 2 - 2 \zeta_{10} ) q^{98} + ( 2 - 2 \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 3q^{2} + q^{3} + 3q^{4} + 5q^{5} + 2q^{6} + 3q^{7} + 5q^{8} - q^{9} + O(q^{10}) \) \( 4q + 3q^{2} + q^{3} + 3q^{4} + 5q^{5} + 2q^{6} + 3q^{7} + 5q^{8} - q^{9} - q^{11} + 2q^{12} + 4q^{13} + 6q^{14} + 5q^{15} + 9q^{16} - 12q^{17} + 3q^{18} + 5q^{19} - 3q^{21} - 12q^{22} - 6q^{23} - 5q^{24} - 5q^{25} + 8q^{26} + q^{27} + 6q^{28} + 20q^{29} + 13q^{31} + 18q^{32} - 4q^{33} - 4q^{34} + 15q^{35} - 2q^{36} - 22q^{37} - 5q^{38} - 4q^{39} + 20q^{40} + 18q^{41} + 9q^{42} + 24q^{43} + 3q^{44} - 12q^{46} - 17q^{47} + 6q^{48} - 2q^{49} - 10q^{50} + 2q^{51} + 8q^{52} - 36q^{53} + 2q^{54} - 5q^{55} - 15q^{56} - 5q^{57} + 5q^{58} - 40q^{59} + 5q^{60} + 13q^{61} + 26q^{62} + 3q^{63} - 7q^{64} + 20q^{65} - 3q^{66} + 3q^{67} - 14q^{68} - 9q^{69} + 8q^{71} - 5q^{72} + 24q^{73} - 24q^{74} + 5q^{75} + 10q^{76} - 12q^{77} + 2q^{78} - 20q^{79} + 15q^{80} - q^{81} + 11q^{82} - 26q^{83} - 6q^{84} - 20q^{85} + 18q^{86} + 15q^{87} + 5q^{88} + 5q^{89} + 5q^{90} - 12q^{91} + 3q^{92} - 3q^{93} - 34q^{94} + 10q^{95} + 2q^{96} + 28q^{97} + 6q^{98} + 9q^{99} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
421.1
0.809017 + 0.587785i
−0.309017 + 0.951057i
0.809017 0.587785i
−0.309017 0.951057i
0.190983 + 0.587785i −0.309017 + 0.951057i 1.30902 0.951057i 1.80902 1.31433i −0.618034 −0.927051 + 2.85317i 1.80902 + 1.31433i −0.809017 0.587785i 1.11803 + 0.812299i
511.1 1.30902 + 0.951057i 0.809017 0.587785i 0.190983 + 0.587785i 0.690983 + 2.12663i 1.61803 2.42705 1.76336i 0.690983 2.12663i 0.309017 0.951057i −1.11803 + 3.44095i
631.1 0.190983 0.587785i −0.309017 0.951057i 1.30902 + 0.951057i 1.80902 + 1.31433i −0.618034 −0.927051 2.85317i 1.80902 1.31433i −0.809017 + 0.587785i 1.11803 0.812299i
691.1 1.30902 0.951057i 0.809017 + 0.587785i 0.190983 0.587785i 0.690983 2.12663i 1.61803 2.42705 + 1.76336i 0.690983 + 2.12663i 0.309017 + 0.951057i −1.11803 3.44095i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 3 T_{2}^{3} + 4 T_{2}^{2} - 2 T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(825, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T + 4 T^{2} - 3 T^{3} + T^{4} \)
$3$ \( 1 - T + T^{2} - T^{3} + T^{4} \)
$5$ \( 25 - 25 T + 15 T^{2} - 5 T^{3} + T^{4} \)
$7$ \( 81 - 27 T + 9 T^{2} - 3 T^{3} + T^{4} \)
$11$ \( 121 + 11 T + 21 T^{2} + T^{3} + T^{4} \)
$13$ \( 16 - 24 T + 16 T^{2} - 4 T^{3} + T^{4} \)
$17$ \( ( 4 + 6 T + T^{2} )^{2} \)
$19$ \( 25 + 10 T^{2} - 5 T^{3} + T^{4} \)
$23$ \( 81 + 81 T + 36 T^{2} + 6 T^{3} + T^{4} \)
$29$ \( 3025 - 825 T + 190 T^{2} - 20 T^{3} + T^{4} \)
$31$ \( 3481 - 177 T + 79 T^{2} - 13 T^{3} + T^{4} \)
$37$ \( ( 19 + 11 T + T^{2} )^{2} \)
$41$ \( 1 - 7 T + 124 T^{2} - 18 T^{3} + T^{4} \)
$43$ \( ( -6 + T )^{4} \)
$47$ \( 121 + 88 T + 114 T^{2} + 17 T^{3} + T^{4} \)
$53$ \( ( 76 + 18 T + T^{2} )^{2} \)
$59$ \( ( 80 + 20 T + T^{2} )^{2} \)
$61$ \( 1681 - 697 T + 139 T^{2} - 13 T^{3} + T^{4} \)
$67$ \( 81 - 27 T + 9 T^{2} - 3 T^{3} + T^{4} \)
$71$ \( 11881 - 1417 T + 114 T^{2} - 8 T^{3} + T^{4} \)
$73$ \( 9801 - 2079 T + 306 T^{2} - 24 T^{3} + T^{4} \)
$79$ \( ( 5 + T )^{4} \)
$83$ \( ( 41 + 13 T + T^{2} )^{2} \)
$89$ \( 25 - 25 T + 15 T^{2} - 5 T^{3} + T^{4} \)
$97$ \( ( 29 - 14 T + T^{2} )^{2} \)
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