Properties

Label 525.2.r.e
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: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
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.

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

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
−1.73205 + 1.00000i −0.866025 0.500000i 1.00000 1.73205i 0 2.00000 −0.866025 + 2.50000i 0 0.500000 + 0.866025i 0
424.2 1.73205 1.00000i 0.866025 + 0.500000i 1.00000 1.73205i 0 2.00000 0.866025 2.50000i 0 0.500000 + 0.866025i 0
499.1 −1.73205 1.00000i −0.866025 + 0.500000i 1.00000 + 1.73205i 0 2.00000 −0.866025 2.50000i 0 0.500000 0.866025i 0
499.2 1.73205 + 1.00000i 0.866025 0.500000i 1.00000 + 1.73205i 0 2.00000 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.e 4
5.b even 2 1 inner 525.2.r.e 4
5.c odd 4 1 21.2.e.a 2
5.c odd 4 1 525.2.i.e 2
7.c even 3 1 inner 525.2.r.e 4
15.e even 4 1 63.2.e.b 2
20.e even 4 1 336.2.q.f 2
35.f even 4 1 147.2.e.a 2
35.j even 6 1 inner 525.2.r.e 4
35.k even 12 1 147.2.a.b 1
35.k even 12 1 147.2.e.a 2
35.k even 12 1 3675.2.a.c 1
35.l odd 12 1 21.2.e.a 2
35.l odd 12 1 147.2.a.c 1
35.l odd 12 1 525.2.i.e 2
35.l odd 12 1 3675.2.a.a 1
40.i odd 4 1 1344.2.q.m 2
40.k even 4 1 1344.2.q.c 2
45.k odd 12 1 567.2.g.a 2
45.k odd 12 1 567.2.h.f 2
45.l even 12 1 567.2.g.f 2
45.l even 12 1 567.2.h.a 2
60.l odd 4 1 1008.2.s.d 2
105.k odd 4 1 441.2.e.e 2
105.w odd 12 1 441.2.a.a 1
105.w odd 12 1 441.2.e.e 2
105.x even 12 1 63.2.e.b 2
105.x even 12 1 441.2.a.b 1
140.j odd 4 1 2352.2.q.c 2
140.w even 12 1 336.2.q.f 2
140.w even 12 1 2352.2.a.d 1
140.x odd 12 1 2352.2.a.w 1
140.x odd 12 1 2352.2.q.c 2
280.bp odd 12 1 9408.2.a.k 1
280.br even 12 1 1344.2.q.c 2
280.br even 12 1 9408.2.a.cv 1
280.bt odd 12 1 1344.2.q.m 2
280.bt odd 12 1 9408.2.a.bg 1
280.bv even 12 1 9408.2.a.bz 1
315.bt odd 12 1 567.2.g.a 2
315.bv even 12 1 567.2.g.f 2
315.bx even 12 1 567.2.h.a 2
315.ch odd 12 1 567.2.h.f 2
420.bp odd 12 1 1008.2.s.d 2
420.bp odd 12 1 7056.2.a.bp 1
420.br even 12 1 7056.2.a.m 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.2.e.a 2 5.c odd 4 1
21.2.e.a 2 35.l odd 12 1
63.2.e.b 2 15.e even 4 1
63.2.e.b 2 105.x even 12 1
147.2.a.b 1 35.k even 12 1
147.2.a.c 1 35.l odd 12 1
147.2.e.a 2 35.f even 4 1
147.2.e.a 2 35.k even 12 1
336.2.q.f 2 20.e even 4 1
336.2.q.f 2 140.w even 12 1
441.2.a.a 1 105.w odd 12 1
441.2.a.b 1 105.x even 12 1
441.2.e.e 2 105.k odd 4 1
441.2.e.e 2 105.w odd 12 1
525.2.i.e 2 5.c odd 4 1
525.2.i.e 2 35.l odd 12 1
525.2.r.e 4 1.a even 1 1 trivial
525.2.r.e 4 5.b even 2 1 inner
525.2.r.e 4 7.c even 3 1 inner
525.2.r.e 4 35.j even 6 1 inner
567.2.g.a 2 45.k odd 12 1
567.2.g.a 2 315.bt odd 12 1
567.2.g.f 2 45.l even 12 1
567.2.g.f 2 315.bv even 12 1
567.2.h.a 2 45.l even 12 1
567.2.h.a 2 315.bx even 12 1
567.2.h.f 2 45.k odd 12 1
567.2.h.f 2 315.ch odd 12 1
1008.2.s.d 2 60.l odd 4 1
1008.2.s.d 2 420.bp odd 12 1
1344.2.q.c 2 40.k even 4 1
1344.2.q.c 2 280.br even 12 1
1344.2.q.m 2 40.i odd 4 1
1344.2.q.m 2 280.bt odd 12 1
2352.2.a.d 1 140.w even 12 1
2352.2.a.w 1 140.x odd 12 1
2352.2.q.c 2 140.j odd 4 1
2352.2.q.c 2 140.x odd 12 1
3675.2.a.a 1 35.l odd 12 1
3675.2.a.c 1 35.k even 12 1
7056.2.a.m 1 420.br even 12 1
7056.2.a.bp 1 420.bp odd 12 1
9408.2.a.k 1 280.bp odd 12 1
9408.2.a.bg 1 280.bt odd 12 1
9408.2.a.bz 1 280.bv even 12 1
9408.2.a.cv 1 280.br even 12 1

Hecke kernels

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

\( T_{2}^{4} - 4 T_{2}^{2} + 16 \)
\( T_{11}^{2} - 2 T_{11} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 2 T + 2 T^{2} - 4 T^{3} + 4 T^{4} )( 1 + 2 T + 2 T^{2} + 4 T^{3} + 4 T^{4} ) \)
$3$ \( 1 - T^{2} + T^{4} \)
$5$ 1
$7$ \( 1 + 11 T^{2} + 49 T^{4} \)
$11$ \( ( 1 - 2 T - 7 T^{2} - 22 T^{3} + 121 T^{4} )^{2} \)
$13$ \( ( 1 - 25 T^{2} + 169 T^{4} )^{2} \)
$17$ \( ( 1 + 17 T^{2} + 289 T^{4} )^{2} \)
$19$ \( ( 1 - 8 T + 19 T^{2} )^{2}( 1 + 7 T + 19 T^{2} )^{2} \)
$23$ \( ( 1 + 23 T^{2} + 529 T^{4} )^{2} \)
$29$ \( ( 1 + 4 T + 29 T^{2} )^{4} \)
$31$ \( ( 1 + 9 T + 50 T^{2} + 279 T^{3} + 961 T^{4} )^{2} \)
$37$ \( 1 + 65 T^{2} + 2856 T^{4} + 88985 T^{6} + 1874161 T^{8} \)
$41$ \( ( 1 + 10 T + 41 T^{2} )^{4} \)
$43$ \( ( 1 - 61 T^{2} + 1849 T^{4} )^{2} \)
$47$ \( 1 + 58 T^{2} + 1155 T^{4} + 128122 T^{6} + 4879681 T^{8} \)
$53$ \( 1 - 38 T^{2} - 1365 T^{4} - 106742 T^{6} + 7890481 T^{8} \)
$59$ \( ( 1 + 12 T + 85 T^{2} + 708 T^{3} + 3481 T^{4} )^{2} \)
$61$ \( ( 1 + 10 T + 39 T^{2} + 610 T^{3} + 3721 T^{4} )^{2} \)
$67$ \( ( 1 - 13 T^{2} + 4489 T^{4} )( 1 + 122 T^{2} + 4489 T^{4} ) \)
$71$ \( ( 1 + 6 T + 71 T^{2} )^{4} \)
$73$ \( 1 + 137 T^{2} + 13440 T^{4} + 730073 T^{6} + 28398241 T^{8} \)
$79$ \( ( 1 + T - 78 T^{2} + 79 T^{3} + 6241 T^{4} )^{2} \)
$83$ \( ( 1 - 130 T^{2} + 6889 T^{4} )^{2} \)
$89$ \( ( 1 - 16 T + 167 T^{2} - 1424 T^{3} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 - 158 T^{2} + 9409 T^{4} )^{2} \)
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