Properties

Label 525.2.d.e
Level 525
Weight 2
Character orbit 525.d
Analytic conductor 4.192
Analytic rank 0
Dimension 4
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 525.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.19214610612\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \beta_{1} - \beta_{3} ) q^{2} + \beta_{3} q^{3} + 3 \beta_{2} q^{4} + ( 1 - \beta_{2} ) q^{6} + \beta_{3} q^{7} + ( -4 \beta_{1} + \beta_{3} ) q^{8} - q^{9} +O(q^{10})\) \( q + ( \beta_{1} - \beta_{3} ) q^{2} + \beta_{3} q^{3} + 3 \beta_{2} q^{4} + ( 1 - \beta_{2} ) q^{6} + \beta_{3} q^{7} + ( -4 \beta_{1} + \beta_{3} ) q^{8} - q^{9} + ( 1 + 4 \beta_{2} ) q^{11} + 3 \beta_{1} q^{12} + ( 2 \beta_{1} + 4 \beta_{3} ) q^{13} + ( 1 - \beta_{2} ) q^{14} + ( 5 - 3 \beta_{2} ) q^{16} + ( -6 \beta_{1} - 2 \beta_{3} ) q^{17} + ( -\beta_{1} + \beta_{3} ) q^{18} -2 \beta_{2} q^{19} - q^{21} + ( -7 \beta_{1} + 3 \beta_{3} ) q^{22} + 5 \beta_{3} q^{23} + ( -1 + 4 \beta_{2} ) q^{24} + 2 q^{26} -\beta_{3} q^{27} + 3 \beta_{1} q^{28} + ( 5 + 6 \beta_{2} ) q^{29} + ( -2 - 4 \beta_{2} ) q^{31} + ( 3 \beta_{1} - 6 \beta_{3} ) q^{32} + ( 4 \beta_{1} + \beta_{3} ) q^{33} + ( 4 - 10 \beta_{2} ) q^{34} -3 \beta_{2} q^{36} + ( 4 \beta_{1} + \beta_{3} ) q^{37} + ( 4 \beta_{1} - 2 \beta_{3} ) q^{38} + ( -4 - 2 \beta_{2} ) q^{39} -8 q^{41} + ( -\beta_{1} + \beta_{3} ) q^{42} + ( -2 \beta_{1} + 5 \beta_{3} ) q^{43} + ( 12 - 9 \beta_{2} ) q^{44} + ( 5 - 5 \beta_{2} ) q^{46} + ( 2 \beta_{1} - 4 \beta_{3} ) q^{47} + ( -3 \beta_{1} + 5 \beta_{3} ) q^{48} - q^{49} + ( 2 + 6 \beta_{2} ) q^{51} + ( 6 \beta_{1} + 6 \beta_{3} ) q^{52} + ( -4 \beta_{1} - 6 \beta_{3} ) q^{53} + ( -1 + \beta_{2} ) q^{54} + ( -1 + 4 \beta_{2} ) q^{56} -2 \beta_{1} q^{57} + ( -7 \beta_{1} + \beta_{3} ) q^{58} + ( 4 + 2 \beta_{2} ) q^{59} + ( 4 + 12 \beta_{2} ) q^{61} + ( 6 \beta_{1} - 2 \beta_{3} ) q^{62} -\beta_{3} q^{63} + ( 1 + 6 \beta_{2} ) q^{64} + ( -3 + 7 \beta_{2} ) q^{66} + ( -6 \beta_{1} - 7 \beta_{3} ) q^{67} + ( 12 \beta_{1} - 18 \beta_{3} ) q^{68} -5 q^{69} + ( -7 - 4 \beta_{2} ) q^{71} + ( 4 \beta_{1} - \beta_{3} ) q^{72} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{73} + ( -3 + 7 \beta_{2} ) q^{74} + ( -6 + 6 \beta_{2} ) q^{76} + ( 4 \beta_{1} + \beta_{3} ) q^{77} + 2 \beta_{3} q^{78} + ( -5 - 2 \beta_{2} ) q^{79} + q^{81} + ( -8 \beta_{1} + 8 \beta_{3} ) q^{82} + ( -4 \beta_{1} + 6 \beta_{3} ) q^{83} -3 \beta_{2} q^{84} + ( 7 - 9 \beta_{2} ) q^{86} + ( 6 \beta_{1} + 5 \beta_{3} ) q^{87} + ( 16 \beta_{1} - 15 \beta_{3} ) q^{88} + ( -4 - 6 \beta_{2} ) q^{89} + ( -4 - 2 \beta_{2} ) q^{91} + 15 \beta_{1} q^{92} + ( -4 \beta_{1} - 2 \beta_{3} ) q^{93} + ( -6 + 8 \beta_{2} ) q^{94} + ( 6 - 3 \beta_{2} ) q^{96} + ( 4 \beta_{1} - 6 \beta_{3} ) q^{97} + ( -\beta_{1} + \beta_{3} ) q^{98} + ( -1 - 4 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 6q^{4} + 6q^{6} - 4q^{9} + O(q^{10}) \) \( 4q - 6q^{4} + 6q^{6} - 4q^{9} - 4q^{11} + 6q^{14} + 26q^{16} + 4q^{19} - 4q^{21} - 12q^{24} + 8q^{26} + 8q^{29} + 36q^{34} + 6q^{36} - 12q^{39} - 32q^{41} + 66q^{44} + 30q^{46} - 4q^{49} - 4q^{51} - 6q^{54} - 12q^{56} + 12q^{59} - 8q^{61} - 8q^{64} - 26q^{66} - 20q^{69} - 20q^{71} - 26q^{74} - 36q^{76} - 16q^{79} + 4q^{81} + 6q^{84} + 46q^{86} - 4q^{89} - 12q^{91} - 40q^{94} + 30q^{96} + 4q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 3 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + 1 \)
\(\beta_{3}\)\(=\)\( \nu^{3} + 2 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 1\)
\(\nu^{3}\)\(=\)\(\beta_{3} - 2 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/525\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(176\) \(451\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
274.1
1.61803i
0.618034i
0.618034i
1.61803i
2.61803i 1.00000i −4.85410 0 2.61803 1.00000i 7.47214i −1.00000 0
274.2 0.381966i 1.00000i 1.85410 0 0.381966 1.00000i 1.47214i −1.00000 0
274.3 0.381966i 1.00000i 1.85410 0 0.381966 1.00000i 1.47214i −1.00000 0
274.4 2.61803i 1.00000i −4.85410 0 2.61803 1.00000i 7.47214i −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 525.2.d.e 4
3.b odd 2 1 1575.2.d.f 4
5.b even 2 1 inner 525.2.d.e 4
5.c odd 4 1 525.2.a.e 2
5.c odd 4 1 525.2.a.i yes 2
15.d odd 2 1 1575.2.d.f 4
15.e even 4 1 1575.2.a.l 2
15.e even 4 1 1575.2.a.v 2
20.e even 4 1 8400.2.a.cy 2
20.e even 4 1 8400.2.a.da 2
35.f even 4 1 3675.2.a.r 2
35.f even 4 1 3675.2.a.bh 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.2.a.e 2 5.c odd 4 1
525.2.a.i yes 2 5.c odd 4 1
525.2.d.e 4 1.a even 1 1 trivial
525.2.d.e 4 5.b even 2 1 inner
1575.2.a.l 2 15.e even 4 1
1575.2.a.v 2 15.e even 4 1
1575.2.d.f 4 3.b odd 2 1
1575.2.d.f 4 15.d odd 2 1
3675.2.a.r 2 35.f even 4 1
3675.2.a.bh 2 35.f even 4 1
8400.2.a.cy 2 20.e even 4 1
8400.2.a.da 2 20.e even 4 1

Hecke kernels

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

\( T_{2}^{4} + 7 T_{2}^{2} + 1 \)
\( T_{11}^{2} + 2 T_{11} - 19 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{2} - 3 T^{4} - 4 T^{6} + 16 T^{8} \)
$3$ \( ( 1 + T^{2} )^{2} \)
$5$ 1
$7$ \( ( 1 + T^{2} )^{2} \)
$11$ \( ( 1 + 2 T + 3 T^{2} + 22 T^{3} + 121 T^{4} )^{2} \)
$13$ \( 1 - 24 T^{2} + 302 T^{4} - 4056 T^{6} + 28561 T^{8} \)
$17$ \( 1 + 24 T^{2} + 542 T^{4} + 6936 T^{6} + 83521 T^{8} \)
$19$ \( ( 1 - 2 T + 34 T^{2} - 38 T^{3} + 361 T^{4} )^{2} \)
$23$ \( ( 1 - 21 T^{2} + 529 T^{4} )^{2} \)
$29$ \( ( 1 - 4 T + 17 T^{2} - 116 T^{3} + 841 T^{4} )^{2} \)
$31$ \( ( 1 + 42 T^{2} + 961 T^{4} )^{2} \)
$37$ \( 1 - 106 T^{2} + 5467 T^{4} - 145114 T^{6} + 1874161 T^{8} \)
$41$ \( ( 1 + 8 T + 41 T^{2} )^{4} \)
$43$ \( 1 - 90 T^{2} + 5003 T^{4} - 166410 T^{6} + 3418801 T^{8} \)
$47$ \( 1 - 128 T^{2} + 8014 T^{4} - 282752 T^{6} + 4879681 T^{8} \)
$53$ \( 1 - 140 T^{2} + 9238 T^{4} - 393260 T^{6} + 7890481 T^{8} \)
$59$ \( ( 1 - 6 T + 122 T^{2} - 354 T^{3} + 3481 T^{4} )^{2} \)
$61$ \( ( 1 + 4 T - 54 T^{2} + 244 T^{3} + 3721 T^{4} )^{2} \)
$67$ \( 1 - 146 T^{2} + 11427 T^{4} - 655394 T^{6} + 20151121 T^{8} \)
$71$ \( ( 1 + 10 T + 147 T^{2} + 710 T^{3} + 5041 T^{4} )^{2} \)
$73$ \( 1 - 280 T^{2} + 30238 T^{4} - 1492120 T^{6} + 28398241 T^{8} \)
$79$ \( ( 1 + 8 T + 169 T^{2} + 632 T^{3} + 6241 T^{4} )^{2} \)
$83$ \( 1 - 164 T^{2} + 15382 T^{4} - 1129796 T^{6} + 47458321 T^{8} \)
$89$ \( ( 1 + 2 T + 134 T^{2} + 178 T^{3} + 7921 T^{4} )^{2} \)
$97$ \( 1 - 220 T^{2} + 25798 T^{4} - 2069980 T^{6} + 88529281 T^{8} \)
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