Properties

Label 525.2.r.b
Level 525
Weight 2
Character orbit 525.r
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.r (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.19214610612\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 105)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{12} q^{3} + ( -2 + 2 \zeta_{12}^{2} ) q^{4} + ( -\zeta_{12} + 3 \zeta_{12}^{3} ) q^{7} + \zeta_{12}^{2} q^{9} +O(q^{10})\) \( q + \zeta_{12} q^{3} + ( -2 + 2 \zeta_{12}^{2} ) q^{4} + ( -\zeta_{12} + 3 \zeta_{12}^{3} ) q^{7} + \zeta_{12}^{2} q^{9} + ( -2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{12} + \zeta_{12}^{3} q^{13} -4 \zeta_{12}^{2} q^{16} -6 \zeta_{12} q^{17} + 5 \zeta_{12}^{2} q^{19} + ( -3 + 2 \zeta_{12}^{2} ) q^{21} + ( -6 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{23} + \zeta_{12}^{3} q^{27} + ( -4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{28} + 6 q^{29} + ( -5 + 5 \zeta_{12}^{2} ) q^{31} -2 q^{36} + ( -7 \zeta_{12} + 7 \zeta_{12}^{3} ) q^{37} + ( -1 + \zeta_{12}^{2} ) q^{39} + 12 q^{41} + \zeta_{12}^{3} q^{43} + ( 6 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{47} -4 \zeta_{12}^{3} q^{48} + ( -3 - 5 \zeta_{12}^{2} ) q^{49} -6 \zeta_{12}^{2} q^{51} -2 \zeta_{12} q^{52} + 5 \zeta_{12}^{3} q^{57} + ( -6 + 6 \zeta_{12}^{2} ) q^{59} -2 \zeta_{12}^{2} q^{61} + ( -3 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{63} + 8 q^{64} + 7 \zeta_{12} q^{67} + ( 12 \zeta_{12} - 12 \zeta_{12}^{3} ) q^{68} -6 q^{69} + 12 q^{71} + 11 \zeta_{12} q^{73} -10 q^{76} -13 \zeta_{12}^{2} q^{79} + ( -1 + \zeta_{12}^{2} ) q^{81} + 12 \zeta_{12}^{3} q^{83} + ( 2 - 6 \zeta_{12}^{2} ) q^{84} + 6 \zeta_{12} q^{87} + 6 \zeta_{12}^{2} q^{89} + ( -2 - \zeta_{12}^{2} ) q^{91} -12 \zeta_{12}^{3} q^{92} + ( -5 \zeta_{12} + 5 \zeta_{12}^{3} ) q^{93} -10 \zeta_{12}^{3} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{4} + 2q^{9} + O(q^{10}) \) \( 4q - 4q^{4} + 2q^{9} - 8q^{16} + 10q^{19} - 8q^{21} + 24q^{29} - 10q^{31} - 8q^{36} - 2q^{39} + 48q^{41} - 22q^{49} - 12q^{51} - 12q^{59} - 4q^{61} + 32q^{64} - 24q^{69} + 48q^{71} - 40q^{76} - 26q^{79} - 2q^{81} - 4q^{84} + 12q^{89} - 10q^{91} + O(q^{100}) \)

Character values

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

\(n\) \(127\) \(176\) \(451\)
\(\chi(n)\) \(-1\) \(1\) \(-\zeta_{12}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
424.1
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
0 −0.866025 0.500000i −1.00000 + 1.73205i 0 0 0.866025 2.50000i 0 0.500000 + 0.866025i 0
424.2 0 0.866025 + 0.500000i −1.00000 + 1.73205i 0 0 −0.866025 + 2.50000i 0 0.500000 + 0.866025i 0
499.1 0 −0.866025 + 0.500000i −1.00000 1.73205i 0 0 0.866025 + 2.50000i 0 0.500000 0.866025i 0
499.2 0 0.866025 0.500000i −1.00000 1.73205i 0 0 −0.866025 2.50000i 0 0.500000 0.866025i 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
7.c even 3 1 inner
35.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.2.r.b 4
5.b even 2 1 inner 525.2.r.b 4
5.c odd 4 1 105.2.i.a 2
5.c odd 4 1 525.2.i.c 2
7.c even 3 1 inner 525.2.r.b 4
15.e even 4 1 315.2.j.b 2
20.e even 4 1 1680.2.bg.m 2
35.f even 4 1 735.2.i.c 2
35.j even 6 1 inner 525.2.r.b 4
35.k even 12 1 735.2.a.d 1
35.k even 12 1 735.2.i.c 2
35.k even 12 1 3675.2.a.i 1
35.l odd 12 1 105.2.i.a 2
35.l odd 12 1 525.2.i.c 2
35.l odd 12 1 735.2.a.e 1
35.l odd 12 1 3675.2.a.h 1
105.w odd 12 1 2205.2.a.d 1
105.x even 12 1 315.2.j.b 2
105.x even 12 1 2205.2.a.f 1
140.w even 12 1 1680.2.bg.m 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.i.a 2 5.c odd 4 1
105.2.i.a 2 35.l odd 12 1
315.2.j.b 2 15.e even 4 1
315.2.j.b 2 105.x even 12 1
525.2.i.c 2 5.c odd 4 1
525.2.i.c 2 35.l odd 12 1
525.2.r.b 4 1.a even 1 1 trivial
525.2.r.b 4 5.b even 2 1 inner
525.2.r.b 4 7.c even 3 1 inner
525.2.r.b 4 35.j even 6 1 inner
735.2.a.d 1 35.k even 12 1
735.2.a.e 1 35.l odd 12 1
735.2.i.c 2 35.f even 4 1
735.2.i.c 2 35.k even 12 1
1680.2.bg.m 2 20.e even 4 1
1680.2.bg.m 2 140.w even 12 1
2205.2.a.d 1 105.w odd 12 1
2205.2.a.f 1 105.x even 12 1
3675.2.a.h 1 35.l odd 12 1
3675.2.a.i 1 35.k even 12 1

Hecke kernels

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

\( T_{2} \)
\( T_{11} \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 2 T^{2} + 4 T^{4} )^{2} \)
$3$ \( 1 - T^{2} + T^{4} \)
$5$ \( \)
$7$ \( 1 + 11 T^{2} + 49 T^{4} \)
$11$ \( ( 1 - 11 T^{2} + 121 T^{4} )^{2} \)
$13$ \( ( 1 - 25 T^{2} + 169 T^{4} )^{2} \)
$17$ \( 1 - 2 T^{2} - 285 T^{4} - 578 T^{6} + 83521 T^{8} \)
$19$ \( ( 1 - 5 T + 6 T^{2} - 95 T^{3} + 361 T^{4} )^{2} \)
$23$ \( 1 + 10 T^{2} - 429 T^{4} + 5290 T^{6} + 279841 T^{8} \)
$29$ \( ( 1 - 6 T + 29 T^{2} )^{4} \)
$31$ \( ( 1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4} )^{2} \)
$37$ \( 1 + 25 T^{2} - 744 T^{4} + 34225 T^{6} + 1874161 T^{8} \)
$41$ \( ( 1 - 12 T + 41 T^{2} )^{4} \)
$43$ \( ( 1 - 85 T^{2} + 1849 T^{4} )^{2} \)
$47$ \( 1 + 58 T^{2} + 1155 T^{4} + 128122 T^{6} + 4879681 T^{8} \)
$53$ \( ( 1 + 53 T^{2} + 2809 T^{4} )^{2} \)
$59$ \( ( 1 + 6 T - 23 T^{2} + 354 T^{3} + 3481 T^{4} )^{2} \)
$61$ \( ( 1 + 2 T - 57 T^{2} + 122 T^{3} + 3721 T^{4} )^{2} \)
$67$ \( 1 + 85 T^{2} + 2736 T^{4} + 381565 T^{6} + 20151121 T^{8} \)
$71$ \( ( 1 - 12 T + 71 T^{2} )^{4} \)
$73$ \( 1 + 25 T^{2} - 4704 T^{4} + 133225 T^{6} + 28398241 T^{8} \)
$79$ \( ( 1 - 4 T + 79 T^{2} )^{2}( 1 + 17 T + 79 T^{2} )^{2} \)
$83$ \( ( 1 - 22 T^{2} + 6889 T^{4} )^{2} \)
$89$ \( ( 1 - 6 T - 53 T^{2} - 534 T^{3} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 - 94 T^{2} + 9409 T^{4} )^{2} \)
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