Properties

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

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(4.19214610612\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
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 primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
524.1
0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
−0.866025 0.500000i
−1.73205 −1.73205 1.00000 0 3.00000 −1.73205 2.00000i 1.73205 3.00000 0
524.2 −1.73205 −1.73205 1.00000 0 3.00000 −1.73205 + 2.00000i 1.73205 3.00000 0
524.3 1.73205 1.73205 1.00000 0 3.00000 1.73205 2.00000i −1.73205 3.00000 0
524.4 1.73205 1.73205 1.00000 0 3.00000 1.73205 + 2.00000i −1.73205 3.00000 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
21.c even 2 1 inner
105.g 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.g.c 4
3.b odd 2 1 525.2.g.b 4
5.b even 2 1 inner 525.2.g.c 4
5.c odd 4 1 105.2.b.b yes 2
5.c odd 4 1 525.2.b.b 2
7.b odd 2 1 525.2.g.b 4
15.d odd 2 1 525.2.g.b 4
15.e even 4 1 105.2.b.a 2
15.e even 4 1 525.2.b.a 2
20.e even 4 1 1680.2.f.c 2
21.c even 2 1 inner 525.2.g.c 4
35.c odd 2 1 525.2.g.b 4
35.f even 4 1 105.2.b.a 2
35.f even 4 1 525.2.b.a 2
35.k even 12 1 735.2.s.b 2
35.k even 12 1 735.2.s.d 2
35.l odd 12 1 735.2.s.a 2
35.l odd 12 1 735.2.s.f 2
60.l odd 4 1 1680.2.f.b 2
105.g even 2 1 inner 525.2.g.c 4
105.k odd 4 1 105.2.b.b yes 2
105.k odd 4 1 525.2.b.b 2
105.w odd 12 1 735.2.s.a 2
105.w odd 12 1 735.2.s.f 2
105.x even 12 1 735.2.s.b 2
105.x even 12 1 735.2.s.d 2
140.j odd 4 1 1680.2.f.b 2
420.w even 4 1 1680.2.f.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.b.a 2 15.e even 4 1
105.2.b.a 2 35.f even 4 1
105.2.b.b yes 2 5.c odd 4 1
105.2.b.b yes 2 105.k odd 4 1
525.2.b.a 2 15.e even 4 1
525.2.b.a 2 35.f even 4 1
525.2.b.b 2 5.c odd 4 1
525.2.b.b 2 105.k odd 4 1
525.2.g.b 4 3.b odd 2 1
525.2.g.b 4 7.b odd 2 1
525.2.g.b 4 15.d odd 2 1
525.2.g.b 4 35.c odd 2 1
525.2.g.c 4 1.a even 1 1 trivial
525.2.g.c 4 5.b even 2 1 inner
525.2.g.c 4 21.c even 2 1 inner
525.2.g.c 4 105.g even 2 1 inner
735.2.s.a 2 35.l odd 12 1
735.2.s.a 2 105.w odd 12 1
735.2.s.b 2 35.k even 12 1
735.2.s.b 2 105.x even 12 1
735.2.s.d 2 35.k even 12 1
735.2.s.d 2 105.x even 12 1
735.2.s.f 2 35.l odd 12 1
735.2.s.f 2 105.w odd 12 1
1680.2.f.b 2 60.l odd 4 1
1680.2.f.b 2 140.j odd 4 1
1680.2.f.c 2 20.e even 4 1
1680.2.f.c 2 420.w even 4 1

Hecke kernels

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

\( T_{2}^{2} - 3 \)
\( T_{41} + 6 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} + 4 T^{4} )^{2} \)
$3$ \( ( 1 - 3 T^{2} )^{2} \)
$5$ 1
$7$ \( 1 + 2 T^{2} + 49 T^{4} \)
$11$ \( ( 1 - 10 T^{2} + 121 T^{4} )^{2} \)
$13$ \( ( 1 + 13 T^{2} )^{4} \)
$17$ \( ( 1 + 2 T^{2} + 289 T^{4} )^{2} \)
$19$ \( ( 1 - 8 T + 19 T^{2} )^{2}( 1 + 8 T + 19 T^{2} )^{2} \)
$23$ \( ( 1 + 34 T^{2} + 529 T^{4} )^{2} \)
$29$ \( ( 1 - 10 T^{2} + 841 T^{4} )^{2} \)
$31$ \( ( 1 - 50 T^{2} + 961 T^{4} )^{2} \)
$37$ \( ( 1 - 12 T + 37 T^{2} )^{2}( 1 + 12 T + 37 T^{2} )^{2} \)
$41$ \( ( 1 + 6 T + 41 T^{2} )^{4} \)
$43$ \( ( 1 - 22 T^{2} + 1849 T^{4} )^{2} \)
$47$ \( ( 1 + 50 T^{2} + 2209 T^{4} )^{2} \)
$53$ \( ( 1 + 53 T^{2} )^{4} \)
$59$ \( ( 1 + 12 T + 59 T^{2} )^{4} \)
$61$ \( ( 1 - 14 T + 61 T^{2} )^{2}( 1 + 14 T + 61 T^{2} )^{2} \)
$67$ \( ( 1 - 70 T^{2} + 4489 T^{4} )^{2} \)
$71$ \( ( 1 - 130 T^{2} + 5041 T^{4} )^{2} \)
$73$ \( ( 1 + 98 T^{2} + 5329 T^{4} )^{2} \)
$79$ \( ( 1 + 8 T + 79 T^{2} )^{4} \)
$83$ \( ( 1 - 83 T^{2} )^{4} \)
$89$ \( ( 1 - 6 T + 89 T^{2} )^{4} \)
$97$ \( ( 1 + 146 T^{2} + 9409 T^{4} )^{2} \)
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