Properties

Label 525.2.a.k
Level $525$
Weight $2$
Character orbit 525.a
Self dual yes
Analytic conductor $4.192$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 525.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.19214610612\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 105)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - q^{3} + (\beta_{2} + \beta_1 + 1) q^{4} - \beta_1 q^{6} - q^{7} + (\beta_{2} + 2 \beta_1 + 2) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - q^{3} + (\beta_{2} + \beta_1 + 1) q^{4} - \beta_1 q^{6} - q^{7} + (\beta_{2} + 2 \beta_1 + 2) q^{8} + q^{9} + 2 q^{11} + ( - \beta_{2} - \beta_1 - 1) q^{12} + ( - \beta_{2} + \beta_1 - 2) q^{13} - \beta_1 q^{14} + (4 \beta_1 + 3) q^{16} + (\beta_{2} - \beta_1) q^{17} + \beta_1 q^{18} + ( - \beta_{2} + \beta_1 + 2) q^{19} + q^{21} + 2 \beta_1 q^{22} + ( - \beta_{2} - \beta_1 + 2) q^{23} + ( - \beta_{2} - 2 \beta_1 - 2) q^{24} + (\beta_{2} - 3 \beta_1 + 4) q^{26} - q^{27} + ( - \beta_{2} - \beta_1 - 1) q^{28} + ( - 2 \beta_{2} - 2 \beta_1 + 2) q^{29} + ( - \beta_{2} - 3 \beta_1 + 2) q^{31} + (2 \beta_{2} + 3 \beta_1 + 8) q^{32} - 2 q^{33} + ( - \beta_{2} + \beta_1 - 4) q^{34} + (\beta_{2} + \beta_1 + 1) q^{36} - 4 \beta_1 q^{37} + (\beta_{2} + \beta_1 + 4) q^{38} + (\beta_{2} - \beta_1 + 2) q^{39} + ( - \beta_{2} + 3 \beta_1) q^{41} + \beta_1 q^{42} + 4 \beta_{2} q^{43} + (2 \beta_{2} + 2 \beta_1 + 2) q^{44} + ( - \beta_{2} - \beta_1 - 2) q^{46} + ( - 2 \beta_{2} - 2 \beta_1 + 4) q^{47} + ( - 4 \beta_1 - 3) q^{48} + q^{49} + ( - \beta_{2} + \beta_1) q^{51} + ( - \beta_{2} + \beta_1 - 6) q^{52} + ( - \beta_{2} - 3 \beta_1 + 6) q^{53} - \beta_1 q^{54} + ( - \beta_{2} - 2 \beta_1 - 2) q^{56} + (\beta_{2} - \beta_1 - 2) q^{57} + ( - 2 \beta_{2} - 4 \beta_1 - 4) q^{58} + (4 \beta_{2} + 4) q^{59} + (2 \beta_{2} - 2 \beta_1 - 2) q^{61} + ( - 3 \beta_{2} - 3 \beta_1 - 8) q^{62} - q^{63} + (3 \beta_{2} + 7 \beta_1 + 1) q^{64} - 2 \beta_1 q^{66} + ( - 2 \beta_{2} - 2 \beta_1 + 4) q^{67} + ( - \beta_{2} - 3 \beta_1 + 4) q^{68} + (\beta_{2} + \beta_1 - 2) q^{69} + 2 q^{71} + (\beta_{2} + 2 \beta_1 + 2) q^{72} + (\beta_{2} - \beta_1 - 6) q^{73} + ( - 4 \beta_{2} - 4 \beta_1 - 12) q^{74} + (3 \beta_{2} + 5 \beta_1 - 2) q^{76} - 2 q^{77} + ( - \beta_{2} + 3 \beta_1 - 4) q^{78} + ( - 2 \beta_{2} + 2 \beta_1 + 4) q^{79} + q^{81} + (3 \beta_{2} + \beta_1 + 10) q^{82} + ( - 4 \beta_1 + 4) q^{83} + (\beta_{2} + \beta_1 + 1) q^{84} + (8 \beta_1 - 4) q^{86} + (2 \beta_{2} + 2 \beta_1 - 2) q^{87} + (2 \beta_{2} + 4 \beta_1 + 4) q^{88} + (\beta_{2} + \beta_1 + 4) q^{89} + (\beta_{2} - \beta_1 + 2) q^{91} + (\beta_{2} - 3 \beta_1 - 6) q^{92} + (\beta_{2} + 3 \beta_1 - 2) q^{93} + ( - 2 \beta_{2} - 2 \beta_1 - 4) q^{94} + ( - 2 \beta_{2} - 3 \beta_1 - 8) q^{96} + (3 \beta_{2} + 5 \beta_1 - 10) q^{97} + \beta_1 q^{98} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{2} - 3 q^{3} + 5 q^{4} - q^{6} - 3 q^{7} + 9 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + q^{2} - 3 q^{3} + 5 q^{4} - q^{6} - 3 q^{7} + 9 q^{8} + 3 q^{9} + 6 q^{11} - 5 q^{12} - 6 q^{13} - q^{14} + 13 q^{16} + q^{18} + 6 q^{19} + 3 q^{21} + 2 q^{22} + 4 q^{23} - 9 q^{24} + 10 q^{26} - 3 q^{27} - 5 q^{28} + 2 q^{29} + 2 q^{31} + 29 q^{32} - 6 q^{33} - 12 q^{34} + 5 q^{36} - 4 q^{37} + 14 q^{38} + 6 q^{39} + 2 q^{41} + q^{42} + 4 q^{43} + 10 q^{44} - 8 q^{46} + 8 q^{47} - 13 q^{48} + 3 q^{49} - 18 q^{52} + 14 q^{53} - q^{54} - 9 q^{56} - 6 q^{57} - 18 q^{58} + 16 q^{59} - 6 q^{61} - 30 q^{62} - 3 q^{63} + 13 q^{64} - 2 q^{66} + 8 q^{67} + 8 q^{68} - 4 q^{69} + 6 q^{71} + 9 q^{72} - 18 q^{73} - 44 q^{74} + 2 q^{76} - 6 q^{77} - 10 q^{78} + 12 q^{79} + 3 q^{81} + 34 q^{82} + 8 q^{83} + 5 q^{84} - 4 q^{86} - 2 q^{87} + 18 q^{88} + 14 q^{89} + 6 q^{91} - 20 q^{92} - 2 q^{93} - 16 q^{94} - 29 q^{96} - 22 q^{97} + q^{98} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 3x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{2} + 2\nu + 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta _1 + 2 \) 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
0.311108
−1.48119
2.17009
−1.90321 −1.00000 1.62222 0 1.90321 −1.00000 0.719004 1.00000 0
1.2 0.193937 −1.00000 −1.96239 0 −0.193937 −1.00000 −0.768452 1.00000 0
1.3 2.70928 −1.00000 5.34017 0 −2.70928 −1.00000 9.04945 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(-1\)
\(7\) \(1\)

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 525.2.a.k 3
3.b odd 2 1 1575.2.a.w 3
4.b odd 2 1 8400.2.a.dj 3
5.b even 2 1 525.2.a.j 3
5.c odd 4 2 105.2.d.b 6
7.b odd 2 1 3675.2.a.bj 3
15.d odd 2 1 1575.2.a.x 3
15.e even 4 2 315.2.d.e 6
20.d odd 2 1 8400.2.a.dg 3
20.e even 4 2 1680.2.t.k 6
35.c odd 2 1 3675.2.a.bi 3
35.f even 4 2 735.2.d.b 6
35.k even 12 4 735.2.q.f 12
35.l odd 12 4 735.2.q.e 12
60.l odd 4 2 5040.2.t.v 6
105.k odd 4 2 2205.2.d.l 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.d.b 6 5.c odd 4 2
315.2.d.e 6 15.e even 4 2
525.2.a.j 3 5.b even 2 1
525.2.a.k 3 1.a even 1 1 trivial
735.2.d.b 6 35.f even 4 2
735.2.q.e 12 35.l odd 12 4
735.2.q.f 12 35.k even 12 4
1575.2.a.w 3 3.b odd 2 1
1575.2.a.x 3 15.d odd 2 1
1680.2.t.k 6 20.e even 4 2
2205.2.d.l 6 105.k odd 4 2
3675.2.a.bi 3 35.c odd 2 1
3675.2.a.bj 3 7.b odd 2 1
5040.2.t.v 6 60.l odd 4 2
8400.2.a.dg 3 20.d odd 2 1
8400.2.a.dj 3 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(525))\):

\( T_{2}^{3} - T_{2}^{2} - 5T_{2} + 1 \) Copy content Toggle raw display
\( T_{11} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - T^{2} - 5T + 1 \) Copy content Toggle raw display
$3$ \( (T + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( (T + 1)^{3} \) Copy content Toggle raw display
$11$ \( (T - 2)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} + 6 T^{2} - 4 T - 8 \) Copy content Toggle raw display
$17$ \( T^{3} - 16T - 16 \) Copy content Toggle raw display
$19$ \( T^{3} - 6 T^{2} - 4 T + 40 \) Copy content Toggle raw display
$23$ \( T^{3} - 4 T^{2} - 8 T + 16 \) Copy content Toggle raw display
$29$ \( T^{3} - 2 T^{2} - 52 T + 40 \) Copy content Toggle raw display
$31$ \( T^{3} - 2 T^{2} - 52 T + 184 \) Copy content Toggle raw display
$37$ \( T^{3} + 4 T^{2} - 80 T - 64 \) Copy content Toggle raw display
$41$ \( T^{3} - 2 T^{2} - 60 T + 200 \) Copy content Toggle raw display
$43$ \( T^{3} - 4 T^{2} - 144 T + 832 \) Copy content Toggle raw display
$47$ \( T^{3} - 8 T^{2} - 32 T + 128 \) Copy content Toggle raw display
$53$ \( T^{3} - 14 T^{2} + 12 T + 296 \) Copy content Toggle raw display
$59$ \( T^{3} - 16 T^{2} - 64 T + 1280 \) Copy content Toggle raw display
$61$ \( T^{3} + 6 T^{2} - 52 T - 248 \) Copy content Toggle raw display
$67$ \( T^{3} - 8 T^{2} - 32 T + 128 \) Copy content Toggle raw display
$71$ \( (T - 2)^{3} \) Copy content Toggle raw display
$73$ \( T^{3} + 18 T^{2} + 92 T + 104 \) Copy content Toggle raw display
$79$ \( T^{3} - 12 T^{2} - 16 T + 320 \) Copy content Toggle raw display
$83$ \( T^{3} - 8 T^{2} - 64 T + 256 \) Copy content Toggle raw display
$89$ \( T^{3} - 14 T^{2} + 52 T - 40 \) Copy content Toggle raw display
$97$ \( T^{3} + 22 T^{2} - 36 T - 1864 \) Copy content Toggle raw display
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