Properties

Label 6525.2.a.o
Level $6525$
Weight $2$
Character orbit 6525.a
Self dual yes
Analytic conductor $52.102$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6525 = 3^{2} \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6525.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(52.1023873189\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 29)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta - 1) q^{2} + ( - 2 \beta + 1) q^{4} - 2 \beta q^{7} + (\beta - 3) q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + (\beta - 1) q^{2} + ( - 2 \beta + 1) q^{4} - 2 \beta q^{7} + (\beta - 3) q^{8} + ( - \beta - 1) q^{11} + ( - 2 \beta + 1) q^{13} + (2 \beta - 4) q^{14} + 3 q^{16} + ( - 2 \beta - 2) q^{17} + 6 q^{19} - q^{22} + ( - 4 \beta - 2) q^{23} + (3 \beta - 5) q^{26} + ( - 2 \beta + 8) q^{28} - q^{29} + ( - 5 \beta + 3) q^{31} + (\beta + 3) q^{32} - 2 q^{34} + 4 q^{37} + (6 \beta - 6) q^{38} + ( - 6 \beta - 4) q^{41} + ( - \beta - 5) q^{43} + (\beta + 3) q^{44} + (2 \beta - 6) q^{46} + (3 \beta + 1) q^{47} + q^{49} + ( - 4 \beta + 9) q^{52} + ( - 6 \beta + 1) q^{53} + (6 \beta - 4) q^{56} + ( - \beta + 1) q^{58} + ( - 4 \beta - 2) q^{59} + (2 \beta - 2) q^{61} + (8 \beta - 13) q^{62} + (2 \beta - 7) q^{64} + 4 \beta q^{67} + (2 \beta + 6) q^{68} + ( - 2 \beta + 6) q^{71} - 4 q^{73} + (4 \beta - 4) q^{74} + ( - 12 \beta + 6) q^{76} + (2 \beta + 4) q^{77} + (\beta - 1) q^{79} + (2 \beta - 8) q^{82} + ( - 4 \beta + 2) q^{83} + ( - 4 \beta + 3) q^{86} + (2 \beta + 1) q^{88} + ( - 6 \beta + 4) q^{89} + ( - 2 \beta + 8) q^{91} + 14 q^{92} + ( - 2 \beta + 5) q^{94} + (6 \beta + 4) q^{97} + (\beta - 1) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} - 6 q^{8} - 2 q^{11} + 2 q^{13} - 8 q^{14} + 6 q^{16} - 4 q^{17} + 12 q^{19} - 2 q^{22} - 4 q^{23} - 10 q^{26} + 16 q^{28} - 2 q^{29} + 6 q^{31} + 6 q^{32} - 4 q^{34} + 8 q^{37} - 12 q^{38} - 8 q^{41} - 10 q^{43} + 6 q^{44} - 12 q^{46} + 2 q^{47} + 2 q^{49} + 18 q^{52} + 2 q^{53} - 8 q^{56} + 2 q^{58} - 4 q^{59} - 4 q^{61} - 26 q^{62} - 14 q^{64} + 12 q^{68} + 12 q^{71} - 8 q^{73} - 8 q^{74} + 12 q^{76} + 8 q^{77} - 2 q^{79} - 16 q^{82} + 4 q^{83} + 6 q^{86} + 2 q^{88} + 8 q^{89} + 16 q^{91} + 28 q^{92} + 10 q^{94} + 8 q^{97} - 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.41421
1.41421
−2.41421 0 3.82843 0 0 2.82843 −4.41421 0 0
1.2 0.414214 0 −1.82843 0 0 −2.82843 −1.58579 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)
\(29\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6525.2.a.o 2
3.b odd 2 1 725.2.a.b 2
5.b even 2 1 261.2.a.d 2
15.d odd 2 1 29.2.a.a 2
15.e even 4 2 725.2.b.b 4
20.d odd 2 1 4176.2.a.bq 2
60.h even 2 1 464.2.a.h 2
105.g even 2 1 1421.2.a.j 2
120.i odd 2 1 1856.2.a.r 2
120.m even 2 1 1856.2.a.w 2
145.d even 2 1 7569.2.a.c 2
165.d even 2 1 3509.2.a.j 2
195.e odd 2 1 4901.2.a.g 2
255.h odd 2 1 8381.2.a.e 2
435.b odd 2 1 841.2.a.d 2
435.l even 4 2 841.2.b.a 4
435.w odd 14 6 841.2.d.j 12
435.bb odd 14 6 841.2.d.f 12
435.bk even 28 12 841.2.e.k 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.2.a.a 2 15.d odd 2 1
261.2.a.d 2 5.b even 2 1
464.2.a.h 2 60.h even 2 1
725.2.a.b 2 3.b odd 2 1
725.2.b.b 4 15.e even 4 2
841.2.a.d 2 435.b odd 2 1
841.2.b.a 4 435.l even 4 2
841.2.d.f 12 435.bb odd 14 6
841.2.d.j 12 435.w odd 14 6
841.2.e.k 24 435.bk even 28 12
1421.2.a.j 2 105.g even 2 1
1856.2.a.r 2 120.i odd 2 1
1856.2.a.w 2 120.m even 2 1
3509.2.a.j 2 165.d even 2 1
4176.2.a.bq 2 20.d odd 2 1
4901.2.a.g 2 195.e odd 2 1
6525.2.a.o 2 1.a even 1 1 trivial
7569.2.a.c 2 145.d even 2 1
8381.2.a.e 2 255.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6525))\):

\( T_{2}^{2} + 2T_{2} - 1 \) Copy content Toggle raw display
\( T_{7}^{2} - 8 \) Copy content Toggle raw display
\( T_{11}^{2} + 2T_{11} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2T - 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 8 \) Copy content Toggle raw display
$11$ \( T^{2} + 2T - 1 \) Copy content Toggle raw display
$13$ \( T^{2} - 2T - 7 \) Copy content Toggle raw display
$17$ \( T^{2} + 4T - 4 \) Copy content Toggle raw display
$19$ \( (T - 6)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 4T - 28 \) Copy content Toggle raw display
$29$ \( (T + 1)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 6T - 41 \) Copy content Toggle raw display
$37$ \( (T - 4)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 8T - 56 \) Copy content Toggle raw display
$43$ \( T^{2} + 10T + 23 \) Copy content Toggle raw display
$47$ \( T^{2} - 2T - 17 \) Copy content Toggle raw display
$53$ \( T^{2} - 2T - 71 \) Copy content Toggle raw display
$59$ \( T^{2} + 4T - 28 \) Copy content Toggle raw display
$61$ \( T^{2} + 4T - 4 \) Copy content Toggle raw display
$67$ \( T^{2} - 32 \) Copy content Toggle raw display
$71$ \( T^{2} - 12T + 28 \) Copy content Toggle raw display
$73$ \( (T + 4)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 2T - 1 \) Copy content Toggle raw display
$83$ \( T^{2} - 4T - 28 \) Copy content Toggle raw display
$89$ \( T^{2} - 8T - 56 \) Copy content Toggle raw display
$97$ \( T^{2} - 8T - 56 \) Copy content Toggle raw display
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