Properties

Label 525.2.r.c
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

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

Character values

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

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

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
−0.866025 + 0.500000i −0.866025 0.500000i −0.500000 + 0.866025i 0 1.00000 −2.59808 + 0.500000i 3.00000i 0.500000 + 0.866025i 0
424.2 0.866025 0.500000i 0.866025 + 0.500000i −0.500000 + 0.866025i 0 1.00000 2.59808 0.500000i 3.00000i 0.500000 + 0.866025i 0
499.1 −0.866025 0.500000i −0.866025 + 0.500000i −0.500000 0.866025i 0 1.00000 −2.59808 0.500000i 3.00000i 0.500000 0.866025i 0
499.2 0.866025 + 0.500000i 0.866025 0.500000i −0.500000 0.866025i 0 1.00000 2.59808 + 0.500000i 3.00000i 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.c 4
5.b even 2 1 inner 525.2.r.c 4
5.c odd 4 1 525.2.i.b 2
5.c odd 4 1 525.2.i.d yes 2
7.c even 3 1 inner 525.2.r.c 4
35.j even 6 1 inner 525.2.r.c 4
35.k even 12 1 3675.2.a.g 1
35.k even 12 1 3675.2.a.k 1
35.l odd 12 1 525.2.i.b 2
35.l odd 12 1 525.2.i.d yes 2
35.l odd 12 1 3675.2.a.e 1
35.l odd 12 1 3675.2.a.m 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.2.i.b 2 5.c odd 4 1
525.2.i.b 2 35.l odd 12 1
525.2.i.d yes 2 5.c odd 4 1
525.2.i.d yes 2 35.l odd 12 1
525.2.r.c 4 1.a even 1 1 trivial
525.2.r.c 4 5.b even 2 1 inner
525.2.r.c 4 7.c even 3 1 inner
525.2.r.c 4 35.j even 6 1 inner
3675.2.a.e 1 35.l odd 12 1
3675.2.a.g 1 35.k even 12 1
3675.2.a.k 1 35.k even 12 1
3675.2.a.m 1 35.l odd 12 1

Hecke kernels

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

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 3 T^{2} + 5 T^{4} + 12 T^{6} + 16 T^{8} \)
$3$ \( 1 - T^{2} + T^{4} \)
$5$ \( \)
$7$ \( 1 - 13 T^{2} + 49 T^{4} \)
$11$ \( ( 1 - 11 T^{2} + 121 T^{4} )^{2} \)
$13$ \( ( 1 - 17 T^{2} + 169 T^{4} )^{2} \)
$17$ \( ( 1 - 8 T + 47 T^{2} - 136 T^{3} + 289 T^{4} )( 1 + 8 T + 47 T^{2} + 136 T^{3} + 289 T^{4} ) \)
$19$ \( ( 1 - 8 T + 19 T^{2} )^{2}( 1 + 7 T + 19 T^{2} )^{2} \)
$23$ \( 1 + 42 T^{2} + 1235 T^{4} + 22218 T^{6} + 279841 T^{8} \)
$29$ \( ( 1 - 8 T + 29 T^{2} )^{4} \)
$31$ \( ( 1 - 8 T + 33 T^{2} - 248 T^{3} + 961 T^{4} )^{2} \)
$37$ \( 1 + 25 T^{2} - 744 T^{4} + 34225 T^{6} + 1874161 T^{8} \)
$41$ \( ( 1 + 41 T^{2} )^{4} \)
$43$ \( ( 1 - 22 T^{2} + 1849 T^{4} )^{2} \)
$47$ \( 1 - 6 T^{2} - 2173 T^{4} - 13254 T^{6} + 4879681 T^{8} \)
$53$ \( ( 1 - 4 T - 37 T^{2} - 212 T^{3} + 2809 T^{4} )( 1 + 4 T - 37 T^{2} + 212 T^{3} + 2809 T^{4} ) \)
$59$ \( ( 1 - 10 T + 41 T^{2} - 590 T^{3} + 3481 T^{4} )^{2} \)
$61$ \( ( 1 + 7 T - 12 T^{2} + 427 T^{3} + 3721 T^{4} )^{2} \)
$67$ \( ( 1 - 13 T^{2} + 4489 T^{4} )( 1 + 122 T^{2} + 4489 T^{4} ) \)
$71$ \( ( 1 + 12 T + 71 T^{2} )^{4} \)
$73$ \( 1 + 25 T^{2} - 4704 T^{4} + 133225 T^{6} + 28398241 T^{8} \)
$79$ \( ( 1 + 7 T - 30 T^{2} + 553 T^{3} + 6241 T^{4} )^{2} \)
$83$ \( ( 1 + 30 T^{2} + 6889 T^{4} )^{2} \)
$89$ \( ( 1 + 6 T - 53 T^{2} + 534 T^{3} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 - 113 T^{2} + 9409 T^{4} )^{2} \)
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