Properties

Label 825.2.k.h
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: yes
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.

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

Character values

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

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

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 1.70711 0.292893i 1.82843i 0 0.585786 + 0.414214i 0.585786 0.585786i 1.12132 1.12132i 2.82843 1.00000i 0
518.2 1.70711 + 1.70711i 0.292893 1.70711i 3.82843i 0 3.41421 2.41421i 3.41421 3.41421i −3.12132 + 3.12132i −2.82843 1.00000i 0
782.1 0.292893 0.292893i 1.70711 + 0.292893i 1.82843i 0 0.585786 0.414214i 0.585786 + 0.585786i 1.12132 + 1.12132i 2.82843 + 1.00000i 0
782.2 1.70711 1.70711i 0.292893 + 1.70711i 3.82843i 0 3.41421 + 2.41421i 3.41421 + 3.41421i −3.12132 3.12132i −2.82843 + 1.00000i 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.h yes 4
3.b odd 2 1 825.2.k.b yes 4
5.b even 2 1 825.2.k.a 4
5.c odd 4 1 825.2.k.b yes 4
5.c odd 4 1 825.2.k.g yes 4
15.d odd 2 1 825.2.k.g yes 4
15.e even 4 1 825.2.k.a 4
15.e even 4 1 inner 825.2.k.h yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.2.k.a 4 5.b even 2 1
825.2.k.a 4 15.e even 4 1
825.2.k.b yes 4 3.b odd 2 1
825.2.k.b yes 4 5.c odd 4 1
825.2.k.g yes 4 5.c odd 4 1
825.2.k.g yes 4 15.d odd 2 1
825.2.k.h yes 4 1.a even 1 1 trivial
825.2.k.h yes 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 \(S_{2}^{\mathrm{new}}(825, [\chi])\):

\( T_{2}^{4} - 4 T_{2}^{3} + 8 T_{2}^{2} - 4 T_{2} + 1 \)
\( T_{7}^{4} - 8 T_{7}^{3} + 32 T_{7}^{2} - 32 T_{7} + 16 \)
\( T_{13}^{4} - 8 T_{13}^{3} + 32 T_{13}^{2} - 32 T_{13} + 16 \)
\( T_{29}^{2} + 12 T_{29} + 28 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 4 T + 8 T^{2} - 4 T^{3} + T^{4} \)
$3$ \( 9 - 12 T + 8 T^{2} - 4 T^{3} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( 16 - 32 T + 32 T^{2} - 8 T^{3} + T^{4} \)
$11$ \( ( 1 + T^{2} )^{2} \)
$13$ \( 16 - 32 T + 32 T^{2} - 8 T^{3} + T^{4} \)
$17$ \( ( 8 + 4 T + T^{2} )^{2} \)
$19$ \( 16 + 24 T^{2} + T^{4} \)
$23$ \( 64 + 64 T + 32 T^{2} - 8 T^{3} + T^{4} \)
$29$ \( ( 28 + 12 T + T^{2} )^{2} \)
$31$ \( ( -4 + T )^{4} \)
$37$ \( 4096 + T^{4} \)
$41$ \( 16 + 24 T^{2} + T^{4} \)
$43$ \( 4624 - 1632 T + 288 T^{2} - 24 T^{3} + T^{4} \)
$47$ \( 64 - 64 T + 32 T^{2} + 8 T^{3} + T^{4} \)
$53$ \( 20736 + T^{4} \)
$59$ \( ( 32 + 16 T + T^{2} )^{2} \)
$61$ \( ( 6 + T )^{4} \)
$67$ \( 784 + 448 T + 128 T^{2} + 16 T^{3} + T^{4} \)
$71$ \( 256 + 96 T^{2} + T^{4} \)
$73$ \( 4624 - 1632 T + 288 T^{2} - 24 T^{3} + T^{4} \)
$79$ \( 16 + 24 T^{2} + T^{4} \)
$83$ \( 1296 + T^{4} \)
$89$ \( ( -112 + 8 T + T^{2} )^{2} \)
$97$ \( 1024 + 512 T + 128 T^{2} - 16 T^{3} + T^{4} \)
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