Properties

Label 525.2.i.e
Level 525
Weight 2
Character orbit 525.i
Analytic conductor 4.192
Analytic rank 0
Dimension 2
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.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.19214610612\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(127\) \(176\) \(451\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
151.1
0.500000 + 0.866025i
0.500000 0.866025i
1.00000 + 1.73205i 0.500000 0.866025i −1.00000 + 1.73205i 0 2.00000 2.50000 + 0.866025i 0 −0.500000 0.866025i 0
226.1 1.00000 1.73205i 0.500000 + 0.866025i −1.00000 1.73205i 0 2.00000 2.50000 0.866025i 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
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.2.i.e 2
5.b even 2 1 21.2.e.a 2
5.c odd 4 2 525.2.r.e 4
7.c even 3 1 inner 525.2.i.e 2
7.c even 3 1 3675.2.a.a 1
7.d odd 6 1 3675.2.a.c 1
15.d odd 2 1 63.2.e.b 2
20.d odd 2 1 336.2.q.f 2
35.c odd 2 1 147.2.e.a 2
35.i odd 6 1 147.2.a.b 1
35.i odd 6 1 147.2.e.a 2
35.j even 6 1 21.2.e.a 2
35.j even 6 1 147.2.a.c 1
35.l odd 12 2 525.2.r.e 4
40.e odd 2 1 1344.2.q.c 2
40.f even 2 1 1344.2.q.m 2
45.h odd 6 1 567.2.g.f 2
45.h odd 6 1 567.2.h.a 2
45.j even 6 1 567.2.g.a 2
45.j even 6 1 567.2.h.f 2
60.h even 2 1 1008.2.s.d 2
105.g even 2 1 441.2.e.e 2
105.o odd 6 1 63.2.e.b 2
105.o odd 6 1 441.2.a.b 1
105.p even 6 1 441.2.a.a 1
105.p even 6 1 441.2.e.e 2
140.c even 2 1 2352.2.q.c 2
140.p odd 6 1 336.2.q.f 2
140.p odd 6 1 2352.2.a.d 1
140.s even 6 1 2352.2.a.w 1
140.s even 6 1 2352.2.q.c 2
280.ba even 6 1 9408.2.a.k 1
280.bf even 6 1 1344.2.q.m 2
280.bf even 6 1 9408.2.a.bg 1
280.bi odd 6 1 1344.2.q.c 2
280.bi odd 6 1 9408.2.a.cv 1
280.bk odd 6 1 9408.2.a.bz 1
315.r even 6 1 567.2.g.a 2
315.v odd 6 1 567.2.h.a 2
315.bo even 6 1 567.2.h.f 2
315.br odd 6 1 567.2.g.f 2
420.ba even 6 1 1008.2.s.d 2
420.ba even 6 1 7056.2.a.bp 1
420.be odd 6 1 7056.2.a.m 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.2.e.a 2 5.b even 2 1
21.2.e.a 2 35.j even 6 1
63.2.e.b 2 15.d odd 2 1
63.2.e.b 2 105.o odd 6 1
147.2.a.b 1 35.i odd 6 1
147.2.a.c 1 35.j even 6 1
147.2.e.a 2 35.c odd 2 1
147.2.e.a 2 35.i odd 6 1
336.2.q.f 2 20.d odd 2 1
336.2.q.f 2 140.p odd 6 1
441.2.a.a 1 105.p even 6 1
441.2.a.b 1 105.o odd 6 1
441.2.e.e 2 105.g even 2 1
441.2.e.e 2 105.p even 6 1
525.2.i.e 2 1.a even 1 1 trivial
525.2.i.e 2 7.c even 3 1 inner
525.2.r.e 4 5.c odd 4 2
525.2.r.e 4 35.l odd 12 2
567.2.g.a 2 45.j even 6 1
567.2.g.a 2 315.r even 6 1
567.2.g.f 2 45.h odd 6 1
567.2.g.f 2 315.br odd 6 1
567.2.h.a 2 45.h odd 6 1
567.2.h.a 2 315.v odd 6 1
567.2.h.f 2 45.j even 6 1
567.2.h.f 2 315.bo even 6 1
1008.2.s.d 2 60.h even 2 1
1008.2.s.d 2 420.ba even 6 1
1344.2.q.c 2 40.e odd 2 1
1344.2.q.c 2 280.bi odd 6 1
1344.2.q.m 2 40.f even 2 1
1344.2.q.m 2 280.bf even 6 1
2352.2.a.d 1 140.p odd 6 1
2352.2.a.w 1 140.s even 6 1
2352.2.q.c 2 140.c even 2 1
2352.2.q.c 2 140.s even 6 1
3675.2.a.a 1 7.c even 3 1
3675.2.a.c 1 7.d odd 6 1
7056.2.a.m 1 420.be odd 6 1
7056.2.a.bp 1 420.ba even 6 1
9408.2.a.k 1 280.ba even 6 1
9408.2.a.bg 1 280.bf even 6 1
9408.2.a.bz 1 280.bk odd 6 1
9408.2.a.cv 1 280.bi odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 2 T_{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(525, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T + 2 T^{2} - 4 T^{3} + 4 T^{4} \)
$3$ \( 1 - T + T^{2} \)
$5$ \( \)
$7$ \( 1 - 5 T + 7 T^{2} \)
$11$ \( 1 - 2 T - 7 T^{2} - 22 T^{3} + 121 T^{4} \)
$13$ \( ( 1 + T + 13 T^{2} )^{2} \)
$17$ \( 1 - 17 T^{2} + 289 T^{4} \)
$19$ \( ( 1 - 7 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \)
$23$ \( 1 - 23 T^{2} + 529 T^{4} \)
$29$ \( ( 1 - 4 T + 29 T^{2} )^{2} \)
$31$ \( 1 + 9 T + 50 T^{2} + 279 T^{3} + 961 T^{4} \)
$37$ \( 1 - 3 T - 28 T^{2} - 111 T^{3} + 1369 T^{4} \)
$41$ \( ( 1 + 10 T + 41 T^{2} )^{2} \)
$43$ \( ( 1 + 5 T + 43 T^{2} )^{2} \)
$47$ \( 1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4} \)
$53$ \( 1 - 12 T + 91 T^{2} - 636 T^{3} + 2809 T^{4} \)
$59$ \( 1 - 12 T + 85 T^{2} - 708 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 10 T + 39 T^{2} + 610 T^{3} + 3721 T^{4} \)
$67$ \( ( 1 - 11 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \)
$71$ \( ( 1 + 6 T + 71 T^{2} )^{2} \)
$73$ \( 1 + 3 T - 64 T^{2} + 219 T^{3} + 5329 T^{4} \)
$79$ \( 1 - T - 78 T^{2} - 79 T^{3} + 6241 T^{4} \)
$83$ \( ( 1 + 6 T + 83 T^{2} )^{2} \)
$89$ \( 1 + 16 T + 167 T^{2} + 1424 T^{3} + 7921 T^{4} \)
$97$ \( ( 1 - 6 T + 97 T^{2} )^{2} \)
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