Properties

Label 525.2.a.j
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
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.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} - 2 \)
\(\beta_{2}\)\(=\)\( -\nu^{2} + 2 \nu + 2 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{1} + 2\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.17009
−1.48119
0.311108
−2.70928 1.00000 5.34017 0 −2.70928 1.00000 −9.04945 1.00000 0
1.2 −0.193937 1.00000 −1.96239 0 −0.193937 1.00000 0.768452 1.00000 0
1.3 1.90321 1.00000 1.62222 0 1.90321 1.00000 −0.719004 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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.j 3
3.b odd 2 1 1575.2.a.x 3
4.b odd 2 1 8400.2.a.dg 3
5.b even 2 1 525.2.a.k 3
5.c odd 4 2 105.2.d.b 6
7.b odd 2 1 3675.2.a.bi 3
15.d odd 2 1 1575.2.a.w 3
15.e even 4 2 315.2.d.e 6
20.d odd 2 1 8400.2.a.dj 3
20.e even 4 2 1680.2.t.k 6
35.c odd 2 1 3675.2.a.bj 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 1.a even 1 1 trivial
525.2.a.k 3 5.b even 2 1
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 15.d odd 2 1
1575.2.a.x 3 3.b 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 7.b odd 2 1
3675.2.a.bj 3 35.c odd 2 1
5040.2.t.v 6 60.l odd 4 2
8400.2.a.dg 3 4.b odd 2 1
8400.2.a.dj 3 20.d odd 2 1

Atkin-Lehner signs

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

Hecke kernels

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

\( T_{2}^{3} + T_{2}^{2} - 5 T_{2} - 1 \)
\( T_{11} - 2 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} + 3 T^{3} + 2 T^{4} + 4 T^{5} + 8 T^{6} \)
$3$ \( ( 1 - T )^{3} \)
$5$ 1
$7$ \( ( 1 - T )^{3} \)
$11$ \( ( 1 - 2 T + 11 T^{2} )^{3} \)
$13$ \( 1 - 6 T + 35 T^{2} - 148 T^{3} + 455 T^{4} - 1014 T^{5} + 2197 T^{6} \)
$17$ \( 1 + 35 T^{2} + 16 T^{3} + 595 T^{4} + 4913 T^{6} \)
$19$ \( 1 - 6 T + 53 T^{2} - 188 T^{3} + 1007 T^{4} - 2166 T^{5} + 6859 T^{6} \)
$23$ \( 1 + 4 T + 61 T^{2} + 168 T^{3} + 1403 T^{4} + 2116 T^{5} + 12167 T^{6} \)
$29$ \( 1 - 2 T + 35 T^{2} - 76 T^{3} + 1015 T^{4} - 1682 T^{5} + 24389 T^{6} \)
$31$ \( 1 - 2 T + 41 T^{2} + 60 T^{3} + 1271 T^{4} - 1922 T^{5} + 29791 T^{6} \)
$37$ \( 1 - 4 T + 31 T^{2} - 232 T^{3} + 1147 T^{4} - 5476 T^{5} + 50653 T^{6} \)
$41$ \( 1 - 2 T + 63 T^{2} + 36 T^{3} + 2583 T^{4} - 3362 T^{5} + 68921 T^{6} \)
$43$ \( 1 + 4 T - 15 T^{2} - 488 T^{3} - 645 T^{4} + 7396 T^{5} + 79507 T^{6} \)
$47$ \( 1 + 8 T + 109 T^{2} + 624 T^{3} + 5123 T^{4} + 17672 T^{5} + 103823 T^{6} \)
$53$ \( 1 + 14 T + 171 T^{2} + 1188 T^{3} + 9063 T^{4} + 39326 T^{5} + 148877 T^{6} \)
$59$ \( 1 - 16 T + 113 T^{2} - 608 T^{3} + 6667 T^{4} - 55696 T^{5} + 205379 T^{6} \)
$61$ \( 1 + 6 T + 131 T^{2} + 484 T^{3} + 7991 T^{4} + 22326 T^{5} + 226981 T^{6} \)
$67$ \( 1 + 8 T + 169 T^{2} + 944 T^{3} + 11323 T^{4} + 35912 T^{5} + 300763 T^{6} \)
$71$ \( ( 1 - 2 T + 71 T^{2} )^{3} \)
$73$ \( 1 - 18 T + 311 T^{2} - 2732 T^{3} + 22703 T^{4} - 95922 T^{5} + 389017 T^{6} \)
$79$ \( 1 - 12 T + 221 T^{2} - 1576 T^{3} + 17459 T^{4} - 74892 T^{5} + 493039 T^{6} \)
$83$ \( 1 + 8 T + 185 T^{2} + 1072 T^{3} + 15355 T^{4} + 55112 T^{5} + 571787 T^{6} \)
$89$ \( 1 - 14 T + 319 T^{2} - 2532 T^{3} + 28391 T^{4} - 110894 T^{5} + 704969 T^{6} \)
$97$ \( 1 - 22 T + 255 T^{2} - 2404 T^{3} + 24735 T^{4} - 206998 T^{5} + 912673 T^{6} \)
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