Properties

Label 825.2.c.d
Level $825$
Weight $2$
Character orbit 825.c
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.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.58765816676\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

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

Character values

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

\(n\) \(376\) \(551\) \(727\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
0.707107 0.707107i
−0.707107 0.707107i
−0.707107 + 0.707107i
0.707107 + 0.707107i
2.41421i 1.00000i −3.82843 0 2.41421 0.414214i 4.41421i −1.00000 0
199.2 0.414214i 1.00000i 1.82843 0 −0.414214 2.41421i 1.58579i −1.00000 0
199.3 0.414214i 1.00000i 1.82843 0 −0.414214 2.41421i 1.58579i −1.00000 0
199.4 2.41421i 1.00000i −3.82843 0 2.41421 0.414214i 4.41421i −1.00000 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
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 825.2.c.d 4
3.b odd 2 1 2475.2.c.o 4
5.b even 2 1 inner 825.2.c.d 4
5.c odd 4 1 825.2.a.d 2
5.c odd 4 1 825.2.a.f yes 2
15.d odd 2 1 2475.2.c.o 4
15.e even 4 1 2475.2.a.l 2
15.e even 4 1 2475.2.a.w 2
55.e even 4 1 9075.2.a.w 2
55.e even 4 1 9075.2.a.ca 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.2.a.d 2 5.c odd 4 1
825.2.a.f yes 2 5.c odd 4 1
825.2.c.d 4 1.a even 1 1 trivial
825.2.c.d 4 5.b even 2 1 inner
2475.2.a.l 2 15.e even 4 1
2475.2.a.w 2 15.e even 4 1
2475.2.c.o 4 3.b odd 2 1
2475.2.c.o 4 15.d odd 2 1
9075.2.a.w 2 55.e even 4 1
9075.2.a.ca 2 55.e even 4 1

Hecke kernels

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

\( T_{2}^{4} + 6 T_{2}^{2} + 1 \)
\( T_{7}^{4} + 6 T_{7}^{2} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 6 T^{2} + T^{4} \)
$3$ \( ( 1 + T^{2} )^{2} \)
$5$ \( T^{4} \)
$7$ \( 1 + 6 T^{2} + T^{4} \)
$11$ \( ( 1 + T )^{4} \)
$13$ \( ( 8 + T^{2} )^{2} \)
$17$ \( 1 + 6 T^{2} + T^{4} \)
$19$ \( ( 23 + 10 T + T^{2} )^{2} \)
$23$ \( ( 1 + T^{2} )^{2} \)
$29$ \( ( 8 + 8 T + T^{2} )^{2} \)
$31$ \( ( -72 + T^{2} )^{2} \)
$37$ \( 1 + 34 T^{2} + T^{4} \)
$41$ \( ( -97 + 2 T + T^{2} )^{2} \)
$43$ \( 16 + 136 T^{2} + T^{4} \)
$47$ \( 5041 + 146 T^{2} + T^{4} \)
$53$ \( 784 + 72 T^{2} + T^{4} \)
$59$ \( ( 11 + T )^{4} \)
$61$ \( ( 28 - 12 T + T^{2} )^{2} \)
$67$ \( 16 + 136 T^{2} + T^{4} \)
$71$ \( ( 17 - 10 T + T^{2} )^{2} \)
$73$ \( 784 + 88 T^{2} + T^{4} \)
$79$ \( ( 63 + 18 T + T^{2} )^{2} \)
$83$ \( 3136 + 176 T^{2} + T^{4} \)
$89$ \( ( -28 - 4 T + T^{2} )^{2} \)
$97$ \( 1 + 34 T^{2} + T^{4} \)
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