Properties

Label 825.2.k.f
Level $825$
Weight $2$
Character orbit 825.k
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.k (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.58765816676\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 165)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(376\) \(551\) \(727\)
\(\chi(n)\) \(1\) \(-1\) \(\zeta_{8}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
518.1
−0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 0.707107i
0.707107 + 0.707107i
0.292893 + 0.292893i 1.00000 1.41421i 1.82843i 0 0.707107 0.121320i 3.41421 3.41421i 1.12132 1.12132i −1.00000 2.82843i 0
518.2 1.70711 + 1.70711i 1.00000 + 1.41421i 3.82843i 0 −0.707107 + 4.12132i 0.585786 0.585786i −3.12132 + 3.12132i −1.00000 + 2.82843i 0
782.1 0.292893 0.292893i 1.00000 + 1.41421i 1.82843i 0 0.707107 + 0.121320i 3.41421 + 3.41421i 1.12132 + 1.12132i −1.00000 + 2.82843i 0
782.2 1.70711 1.70711i 1.00000 1.41421i 3.82843i 0 −0.707107 4.12132i 0.585786 + 0.585786i −3.12132 3.12132i −1.00000 2.82843i 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
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 825.2.k.f 4
3.b odd 2 1 825.2.k.c 4
5.b even 2 1 165.2.k.a 4
5.c odd 4 1 165.2.k.b yes 4
5.c odd 4 1 825.2.k.c 4
15.d odd 2 1 165.2.k.b yes 4
15.e even 4 1 165.2.k.a 4
15.e even 4 1 inner 825.2.k.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.k.a 4 5.b even 2 1
165.2.k.a 4 15.e even 4 1
165.2.k.b yes 4 5.c odd 4 1
165.2.k.b yes 4 15.d odd 2 1
825.2.k.c 4 3.b odd 2 1
825.2.k.c 4 5.c odd 4 1
825.2.k.f 4 1.a even 1 1 trivial
825.2.k.f 4 15.e even 4 1 inner

Hecke kernels

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

\( T_{2}^{4} - 4 T_{2}^{3} + 8 T_{2}^{2} - 4 T_{2} + 1 \)
\( T_{7}^{4} - 8 T_{7}^{3} + 32 T_{7}^{2} - 32 T_{7} + 16 \)
\( T_{13}^{2} - 4 T_{13} + 8 \)
\( T_{29}^{2} + 4 T_{29} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 4 T + 8 T^{2} - 4 T^{3} + T^{4} \)
$3$ \( ( 3 - 2 T + T^{2} )^{2} \)
$5$ \( T^{4} \)
$7$ \( 16 - 32 T + 32 T^{2} - 8 T^{3} + T^{4} \)
$11$ \( ( 1 + T^{2} )^{2} \)
$13$ \( ( 8 - 4 T + T^{2} )^{2} \)
$17$ \( 256 + T^{4} \)
$19$ \( ( 8 + T^{2} )^{2} \)
$23$ \( 1156 + 136 T + 8 T^{2} - 4 T^{3} + T^{4} \)
$29$ \( ( -4 + 4 T + T^{2} )^{2} \)
$31$ \( ( 28 + 12 T + T^{2} )^{2} \)
$37$ \( 4 - 24 T + 72 T^{2} - 12 T^{3} + T^{4} \)
$41$ \( 784 + 72 T^{2} + T^{4} \)
$43$ \( 16 + 64 T + 128 T^{2} - 16 T^{3} + T^{4} \)
$47$ \( 324 + 216 T + 72 T^{2} - 12 T^{3} + T^{4} \)
$53$ \( ( 2 + 2 T + T^{2} )^{2} \)
$59$ \( ( 4 + T )^{4} \)
$61$ \( ( -68 - 4 T + T^{2} )^{2} \)
$67$ \( 2116 - 920 T + 200 T^{2} - 20 T^{3} + T^{4} \)
$71$ \( 1296 + 216 T^{2} + T^{4} \)
$73$ \( T^{4} \)
$79$ \( 3136 + 144 T^{2} + T^{4} \)
$83$ \( 8464 + 736 T + 32 T^{2} - 8 T^{3} + T^{4} \)
$89$ \( ( -16 + 8 T + T^{2} )^{2} \)
$97$ \( 196 - 280 T + 200 T^{2} + 20 T^{3} + T^{4} \)
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