Properties

Label 525.2.a.i
Level 525
Weight 2
Character orbit 525.a
Self dual yes
Analytic conductor 4.192
Analytic rank 0
Dimension 2
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: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

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

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
−0.618034
1.61803
0.381966 1.00000 −1.85410 0 0.381966 −1.00000 −1.47214 1.00000 0
1.2 2.61803 1.00000 4.85410 0 2.61803 −1.00000 7.47214 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.i yes 2
3.b odd 2 1 1575.2.a.l 2
4.b odd 2 1 8400.2.a.cy 2
5.b even 2 1 525.2.a.e 2
5.c odd 4 2 525.2.d.e 4
7.b odd 2 1 3675.2.a.bh 2
15.d odd 2 1 1575.2.a.v 2
15.e even 4 2 1575.2.d.f 4
20.d odd 2 1 8400.2.a.da 2
35.c odd 2 1 3675.2.a.r 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.2.a.e 2 5.b even 2 1
525.2.a.i yes 2 1.a even 1 1 trivial
525.2.d.e 4 5.c odd 4 2
1575.2.a.l 2 3.b odd 2 1
1575.2.a.v 2 15.d odd 2 1
1575.2.d.f 4 15.e even 4 2
3675.2.a.r 2 35.c odd 2 1
3675.2.a.bh 2 7.b odd 2 1
8400.2.a.cy 2 4.b odd 2 1
8400.2.a.da 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(525))\):

\( T_{2}^{2} - 3 T_{2} + 1 \)
\( T_{11}^{2} + 2 T_{11} - 19 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 3 T + 5 T^{2} - 6 T^{3} + 4 T^{4} \)
$3$ \( ( 1 - T )^{2} \)
$5$ \( \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( 1 + 2 T + 3 T^{2} + 22 T^{3} + 121 T^{4} \)
$13$ \( 1 - 6 T + 30 T^{2} - 78 T^{3} + 169 T^{4} \)
$17$ \( 1 + 2 T - 10 T^{2} + 34 T^{3} + 289 T^{4} \)
$19$ \( 1 + 2 T + 34 T^{2} + 38 T^{3} + 361 T^{4} \)
$23$ \( ( 1 - 5 T + 23 T^{2} )^{2} \)
$29$ \( 1 + 4 T + 17 T^{2} + 116 T^{3} + 841 T^{4} \)
$31$ \( 1 + 42 T^{2} + 961 T^{4} \)
$37$ \( 1 - 2 T + 55 T^{2} - 74 T^{3} + 1369 T^{4} \)
$41$ \( ( 1 + 8 T + 41 T^{2} )^{2} \)
$43$ \( 1 - 12 T + 117 T^{2} - 516 T^{3} + 1849 T^{4} \)
$47$ \( 1 - 10 T + 114 T^{2} - 470 T^{3} + 2209 T^{4} \)
$53$ \( 1 + 8 T + 102 T^{2} + 424 T^{3} + 2809 T^{4} \)
$59$ \( 1 + 6 T + 122 T^{2} + 354 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 4 T - 54 T^{2} + 244 T^{3} + 3721 T^{4} \)
$67$ \( 1 - 8 T + 105 T^{2} - 536 T^{3} + 4489 T^{4} \)
$71$ \( 1 + 10 T + 147 T^{2} + 710 T^{3} + 5041 T^{4} \)
$73$ \( 1 + 2 T + 142 T^{2} + 146 T^{3} + 5329 T^{4} \)
$79$ \( 1 - 8 T + 169 T^{2} - 632 T^{3} + 6241 T^{4} \)
$83$ \( 1 - 16 T + 210 T^{2} - 1328 T^{3} + 6889 T^{4} \)
$89$ \( 1 - 2 T + 134 T^{2} - 178 T^{3} + 7921 T^{4} \)
$97$ \( 1 - 16 T + 238 T^{2} - 1552 T^{3} + 9409 T^{4} \)
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