Properties

Label 825.2.c.a
Level $825$
Weight $2$
Character orbit 825.c
Analytic conductor $6.588$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

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

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
1.00000i
1.00000i
1.00000i 1.00000i 1.00000 0 −1.00000 4.00000i 3.00000i −1.00000 0
199.2 1.00000i 1.00000i 1.00000 0 −1.00000 4.00000i 3.00000i −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.a 2
3.b odd 2 1 2475.2.c.d 2
5.b even 2 1 inner 825.2.c.a 2
5.c odd 4 1 33.2.a.a 1
5.c odd 4 1 825.2.a.a 1
15.d odd 2 1 2475.2.c.d 2
15.e even 4 1 99.2.a.b 1
15.e even 4 1 2475.2.a.g 1
20.e even 4 1 528.2.a.g 1
35.f even 4 1 1617.2.a.j 1
40.i odd 4 1 2112.2.a.bb 1
40.k even 4 1 2112.2.a.j 1
45.k odd 12 2 891.2.e.e 2
45.l even 12 2 891.2.e.g 2
55.e even 4 1 363.2.a.b 1
55.e even 4 1 9075.2.a.q 1
55.k odd 20 4 363.2.e.e 4
55.l even 20 4 363.2.e.g 4
60.l odd 4 1 1584.2.a.o 1
65.h odd 4 1 5577.2.a.a 1
85.g odd 4 1 9537.2.a.m 1
105.k odd 4 1 4851.2.a.b 1
120.q odd 4 1 6336.2.a.n 1
120.w even 4 1 6336.2.a.x 1
165.l odd 4 1 1089.2.a.j 1
220.i odd 4 1 5808.2.a.t 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.2.a.a 1 5.c odd 4 1
99.2.a.b 1 15.e even 4 1
363.2.a.b 1 55.e even 4 1
363.2.e.e 4 55.k odd 20 4
363.2.e.g 4 55.l even 20 4
528.2.a.g 1 20.e even 4 1
825.2.a.a 1 5.c odd 4 1
825.2.c.a 2 1.a even 1 1 trivial
825.2.c.a 2 5.b even 2 1 inner
891.2.e.e 2 45.k odd 12 2
891.2.e.g 2 45.l even 12 2
1089.2.a.j 1 165.l odd 4 1
1584.2.a.o 1 60.l odd 4 1
1617.2.a.j 1 35.f even 4 1
2112.2.a.j 1 40.k even 4 1
2112.2.a.bb 1 40.i odd 4 1
2475.2.a.g 1 15.e even 4 1
2475.2.c.d 2 3.b odd 2 1
2475.2.c.d 2 15.d odd 2 1
4851.2.a.b 1 105.k odd 4 1
5577.2.a.a 1 65.h odd 4 1
5808.2.a.t 1 220.i odd 4 1
6336.2.a.n 1 120.q odd 4 1
6336.2.a.x 1 120.w even 4 1
9075.2.a.q 1 55.e even 4 1
9537.2.a.m 1 85.g odd 4 1

Hecke kernels

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

\( T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{2} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 16 \) Copy content Toggle raw display
$11$ \( (T - 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 4 \) Copy content Toggle raw display
$17$ \( T^{2} + 4 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 64 \) Copy content Toggle raw display
$29$ \( (T - 6)^{2} \) Copy content Toggle raw display
$31$ \( (T + 8)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 36 \) Copy content Toggle raw display
$41$ \( (T + 2)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 64 \) Copy content Toggle raw display
$53$ \( T^{2} + 36 \) Copy content Toggle raw display
$59$ \( (T - 4)^{2} \) Copy content Toggle raw display
$61$ \( (T - 6)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 16 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 196 \) Copy content Toggle raw display
$79$ \( (T - 4)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 144 \) Copy content Toggle raw display
$89$ \( (T - 6)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 4 \) Copy content Toggle raw display
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