Properties

Label 525.2.b.e
Level 525
Weight 2
Character orbit 525.b
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.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.19214610612\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-11})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 105)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 2 x^{2} - 3 x + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 2 \nu^{2} + 4 \nu - 9 \)\()/6\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} - 2 \nu^{2} + 2 \nu + 6 \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + 2 \nu + 3 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + 2 \beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{3} - 3 \beta_{2} + 3\)\()/2\)
\(\nu^{3}\)\(=\)\(-2 \beta_{3} + \beta_{2} + 2 \beta_{1} + 4\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
251.1
−1.18614 + 1.26217i
1.68614 + 0.396143i
1.68614 0.396143i
−1.18614 1.26217i
2.52434i −1.68614 0.396143i −4.37228 0 −1.00000 + 4.25639i 2.00000 + 1.73205i 5.98844i 2.68614 + 1.33591i 0
251.2 0.792287i 1.18614 1.26217i 1.37228 0 −1.00000 0.939764i 2.00000 1.73205i 2.67181i −0.186141 2.99422i 0
251.3 0.792287i 1.18614 + 1.26217i 1.37228 0 −1.00000 + 0.939764i 2.00000 + 1.73205i 2.67181i −0.186141 + 2.99422i 0
251.4 2.52434i −1.68614 + 0.396143i −4.37228 0 −1.00000 4.25639i 2.00000 1.73205i 5.98844i 2.68614 1.33591i 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
21.c 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.b.e 4
3.b odd 2 1 525.2.b.g 4
5.b even 2 1 105.2.b.d yes 4
5.c odd 4 2 525.2.g.d 8
7.b odd 2 1 525.2.b.g 4
15.d odd 2 1 105.2.b.c 4
15.e even 4 2 525.2.g.e 8
20.d odd 2 1 1680.2.f.g 4
21.c even 2 1 inner 525.2.b.e 4
35.c odd 2 1 105.2.b.c 4
35.f even 4 2 525.2.g.e 8
35.i odd 6 1 735.2.s.h 4
35.i odd 6 1 735.2.s.i 4
35.j even 6 1 735.2.s.g 4
35.j even 6 1 735.2.s.j 4
60.h even 2 1 1680.2.f.h 4
105.g even 2 1 105.2.b.d yes 4
105.k odd 4 2 525.2.g.d 8
105.o odd 6 1 735.2.s.h 4
105.o odd 6 1 735.2.s.i 4
105.p even 6 1 735.2.s.g 4
105.p even 6 1 735.2.s.j 4
140.c even 2 1 1680.2.f.h 4
420.o odd 2 1 1680.2.f.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.b.c 4 15.d odd 2 1
105.2.b.c 4 35.c odd 2 1
105.2.b.d yes 4 5.b even 2 1
105.2.b.d yes 4 105.g even 2 1
525.2.b.e 4 1.a even 1 1 trivial
525.2.b.e 4 21.c even 2 1 inner
525.2.b.g 4 3.b odd 2 1
525.2.b.g 4 7.b odd 2 1
525.2.g.d 8 5.c odd 4 2
525.2.g.d 8 105.k odd 4 2
525.2.g.e 8 15.e even 4 2
525.2.g.e 8 35.f even 4 2
735.2.s.g 4 35.j even 6 1
735.2.s.g 4 105.p even 6 1
735.2.s.h 4 35.i odd 6 1
735.2.s.h 4 105.o odd 6 1
735.2.s.i 4 35.i odd 6 1
735.2.s.i 4 105.o odd 6 1
735.2.s.j 4 35.j even 6 1
735.2.s.j 4 105.p even 6 1
1680.2.f.g 4 20.d odd 2 1
1680.2.f.g 4 420.o odd 2 1
1680.2.f.h 4 60.h even 2 1
1680.2.f.h 4 140.c even 2 1

Hecke kernels

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

\( T_{2}^{4} + 7 T_{2}^{2} + 4 \)
\( T_{11}^{4} + 7 T_{11}^{2} + 4 \)
\( T_{17}^{2} + 3 T_{17} - 6 \)
\( T_{37}^{2} - 2 T_{37} - 32 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{2} - 4 T^{6} + 16 T^{8} \)
$3$ \( 1 + T - 2 T^{2} + 3 T^{3} + 9 T^{4} \)
$5$ \( \)
$7$ \( ( 1 - 4 T + 7 T^{2} )^{2} \)
$11$ \( 1 - 37 T^{2} + 576 T^{4} - 4477 T^{6} + 14641 T^{8} \)
$13$ \( 1 - T^{2} + 264 T^{4} - 169 T^{6} + 28561 T^{8} \)
$17$ \( ( 1 + 3 T + 28 T^{2} + 51 T^{3} + 289 T^{4} )^{2} \)
$19$ \( ( 1 - 8 T + 19 T^{2} )^{2}( 1 + 8 T + 19 T^{2} )^{2} \)
$23$ \( 1 - 16 T^{2} - 66 T^{4} - 8464 T^{6} + 279841 T^{8} \)
$29$ \( 1 - 97 T^{2} + 3960 T^{4} - 81577 T^{6} + 707281 T^{8} \)
$31$ \( ( 1 - 50 T^{2} + 961 T^{4} )^{2} \)
$37$ \( ( 1 - 2 T + 42 T^{2} - 74 T^{3} + 1369 T^{4} )^{2} \)
$41$ \( ( 1 + 6 T + 41 T^{2} )^{4} \)
$43$ \( ( 1 - 2 T + 54 T^{2} - 86 T^{3} + 1849 T^{4} )^{2} \)
$47$ \( ( 1 - 9 T + 106 T^{2} - 423 T^{3} + 2209 T^{4} )^{2} \)
$53$ \( 1 - 136 T^{2} + 9054 T^{4} - 382024 T^{6} + 7890481 T^{8} \)
$59$ \( ( 1 + 6 T + 94 T^{2} + 354 T^{3} + 3481 T^{4} )^{2} \)
$61$ \( ( 1 - 14 T + 61 T^{2} )^{2}( 1 + 14 T + 61 T^{2} )^{2} \)
$67$ \( ( 1 - 2 T + 102 T^{2} - 134 T^{3} + 4489 T^{4} )^{2} \)
$71$ \( 1 - 100 T^{2} + 4134 T^{4} - 504100 T^{6} + 25411681 T^{8} \)
$73$ \( ( 1 - 98 T^{2} + 5329 T^{4} )^{2} \)
$79$ \( ( 1 - T + 150 T^{2} - 79 T^{3} + 6241 T^{4} )^{2} \)
$83$ \( ( 1 - 12 T + 70 T^{2} - 996 T^{3} + 6889 T^{4} )^{2} \)
$89$ \( ( 1 - 18 T + 226 T^{2} - 1602 T^{3} + 7921 T^{4} )^{2} \)
$97$ \( 1 - 265 T^{2} + 32736 T^{4} - 2493385 T^{6} + 88529281 T^{8} \)
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