Properties

Label 825.2.r.a
Level $825$
Weight $2$
Character orbit 825.r
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.r (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.

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

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
−0.309017 0.951057i
0.809017 + 0.587785i
0.809017 0.587785i
−0.309017 + 0.951057i
2.23607 0.309017 0.951057i 3.00000 1.80902 + 1.31433i 0.690983 2.12663i −0.309017 0.224514i 2.23607 −0.809017 0.587785i 4.04508 + 2.93893i
91.1 −2.23607 −0.809017 + 0.587785i 3.00000 0.690983 2.12663i 1.80902 1.31433i 0.809017 2.48990i −2.23607 0.309017 0.951057i −1.54508 + 4.75528i
136.1 −2.23607 −0.809017 0.587785i 3.00000 0.690983 + 2.12663i 1.80902 + 1.31433i 0.809017 + 2.48990i −2.23607 0.309017 + 0.951057i −1.54508 4.75528i
346.1 2.23607 0.309017 + 0.951057i 3.00000 1.80902 1.31433i 0.690983 + 2.12663i −0.309017 + 0.224514i 2.23607 −0.809017 + 0.587785i 4.04508 2.93893i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
275.l 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.r.a yes 4
11.c even 5 1 825.2.p.a 4
25.d even 5 1 825.2.p.a 4
275.l even 5 1 inner 825.2.r.a yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.2.p.a 4 11.c even 5 1
825.2.p.a 4 25.d even 5 1
825.2.r.a yes 4 1.a even 1 1 trivial
825.2.r.a yes 4 275.l even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 5 \) acting on \(S_{2}^{\mathrm{new}}(825, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -5 + T^{2} )^{2} \)
$3$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
$5$ \( 25 - 25 T + 15 T^{2} - 5 T^{3} + T^{4} \)
$7$ \( 1 + 4 T + 6 T^{2} - T^{3} + T^{4} \)
$11$ \( 121 + 99 T + 41 T^{2} + 9 T^{3} + T^{4} \)
$13$ \( 1296 - 216 T + 36 T^{2} - 6 T^{3} + T^{4} \)
$17$ \( 121 - 11 T + 31 T^{2} + 9 T^{3} + T^{4} \)
$19$ \( ( 29 - 11 T + T^{2} )^{2} \)
$23$ \( 361 + 171 T + 61 T^{2} + 11 T^{3} + T^{4} \)
$29$ \( ( -29 - 3 T + T^{2} )^{2} \)
$31$ \( 121 - 143 T + 69 T^{2} - 7 T^{3} + T^{4} \)
$37$ \( 81 - 27 T + 54 T^{2} + 12 T^{3} + T^{4} \)
$41$ \( 121 + 99 T + 31 T^{2} - T^{3} + T^{4} \)
$43$ \( ( -79 + 2 T + T^{2} )^{2} \)
$47$ \( 625 + 250 T + 100 T^{2} + 15 T^{3} + T^{4} \)
$53$ \( 1 - T + T^{2} - T^{3} + T^{4} \)
$59$ \( 400 - 400 T + 160 T^{2} - 10 T^{3} + T^{4} \)
$61$ \( 3025 + 825 T + 190 T^{2} + 20 T^{3} + T^{4} \)
$67$ \( 25 + 50 T + 40 T^{2} + 5 T^{3} + T^{4} \)
$71$ \( 121 - 143 T + 69 T^{2} - 7 T^{3} + T^{4} \)
$73$ \( 16 - 16 T + 16 T^{2} - 6 T^{3} + T^{4} \)
$79$ \( 4096 - 1024 T + 256 T^{2} - 24 T^{3} + T^{4} \)
$83$ \( 9801 - 2376 T + 306 T^{2} - 21 T^{3} + T^{4} \)
$89$ \( 3481 + 1711 T + 346 T^{2} + 16 T^{3} + T^{4} \)
$97$ \( 841 + 493 T + 109 T^{2} - 3 T^{3} + T^{4} \)
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