Properties

Label 525.2.i.c
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 105)
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.

\(f(q)\) \(=\) \( q + ( 1 - \zeta_{6} ) q^{3} + ( 2 - 2 \zeta_{6} ) q^{4} + ( -2 - \zeta_{6} ) q^{7} -\zeta_{6} q^{9} +O(q^{10})\) \( q + ( 1 - \zeta_{6} ) q^{3} + ( 2 - 2 \zeta_{6} ) q^{4} + ( -2 - \zeta_{6} ) q^{7} -\zeta_{6} q^{9} -2 \zeta_{6} q^{12} + q^{13} -4 \zeta_{6} q^{16} + ( 6 - 6 \zeta_{6} ) q^{17} -5 \zeta_{6} q^{19} + ( -3 + 2 \zeta_{6} ) q^{21} + 6 \zeta_{6} q^{23} - q^{27} + ( -6 + 4 \zeta_{6} ) q^{28} -6 q^{29} + ( -5 + 5 \zeta_{6} ) q^{31} -2 q^{36} -7 \zeta_{6} q^{37} + ( 1 - \zeta_{6} ) q^{39} + 12 q^{41} + q^{43} + 6 \zeta_{6} q^{47} -4 q^{48} + ( 3 + 5 \zeta_{6} ) q^{49} -6 \zeta_{6} q^{51} + ( 2 - 2 \zeta_{6} ) q^{52} -5 q^{57} + ( 6 - 6 \zeta_{6} ) q^{59} -2 \zeta_{6} q^{61} + ( -1 + 3 \zeta_{6} ) q^{63} -8 q^{64} + ( -7 + 7 \zeta_{6} ) q^{67} -12 \zeta_{6} q^{68} + 6 q^{69} + 12 q^{71} + ( 11 - 11 \zeta_{6} ) q^{73} -10 q^{76} + 13 \zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{81} + 12 q^{83} + ( -2 + 6 \zeta_{6} ) q^{84} + ( -6 + 6 \zeta_{6} ) q^{87} -6 \zeta_{6} q^{89} + ( -2 - \zeta_{6} ) q^{91} + 12 q^{92} + 5 \zeta_{6} q^{93} + 10 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{3} + 2q^{4} - 5q^{7} - q^{9} + O(q^{10}) \) \( 2q + q^{3} + 2q^{4} - 5q^{7} - q^{9} - 2q^{12} + 2q^{13} - 4q^{16} + 6q^{17} - 5q^{19} - 4q^{21} + 6q^{23} - 2q^{27} - 8q^{28} - 12q^{29} - 5q^{31} - 4q^{36} - 7q^{37} + q^{39} + 24q^{41} + 2q^{43} + 6q^{47} - 8q^{48} + 11q^{49} - 6q^{51} + 2q^{52} - 10q^{57} + 6q^{59} - 2q^{61} + q^{63} - 16q^{64} - 7q^{67} - 12q^{68} + 12q^{69} + 24q^{71} + 11q^{73} - 20q^{76} + 13q^{79} - q^{81} + 24q^{83} + 2q^{84} - 6q^{87} - 6q^{89} - 5q^{91} + 24q^{92} + 5q^{93} + 20q^{97} + O(q^{100}) \)

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
0 0.500000 0.866025i 1.00000 1.73205i 0 0 −2.50000 0.866025i 0 −0.500000 0.866025i 0
226.1 0 0.500000 + 0.866025i 1.00000 + 1.73205i 0 0 −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.c 2
5.b even 2 1 105.2.i.a 2
5.c odd 4 2 525.2.r.b 4
7.c even 3 1 inner 525.2.i.c 2
7.c even 3 1 3675.2.a.h 1
7.d odd 6 1 3675.2.a.i 1
15.d odd 2 1 315.2.j.b 2
20.d odd 2 1 1680.2.bg.m 2
35.c odd 2 1 735.2.i.c 2
35.i odd 6 1 735.2.a.d 1
35.i odd 6 1 735.2.i.c 2
35.j even 6 1 105.2.i.a 2
35.j even 6 1 735.2.a.e 1
35.l odd 12 2 525.2.r.b 4
105.o odd 6 1 315.2.j.b 2
105.o odd 6 1 2205.2.a.f 1
105.p even 6 1 2205.2.a.d 1
140.p odd 6 1 1680.2.bg.m 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.i.a 2 5.b even 2 1
105.2.i.a 2 35.j even 6 1
315.2.j.b 2 15.d odd 2 1
315.2.j.b 2 105.o odd 6 1
525.2.i.c 2 1.a even 1 1 trivial
525.2.i.c 2 7.c even 3 1 inner
525.2.r.b 4 5.c odd 4 2
525.2.r.b 4 35.l odd 12 2
735.2.a.d 1 35.i odd 6 1
735.2.a.e 1 35.j even 6 1
735.2.i.c 2 35.c odd 2 1
735.2.i.c 2 35.i odd 6 1
1680.2.bg.m 2 20.d odd 2 1
1680.2.bg.m 2 140.p odd 6 1
2205.2.a.d 1 105.p even 6 1
2205.2.a.f 1 105.o odd 6 1
3675.2.a.h 1 7.c even 3 1
3675.2.a.i 1 7.d odd 6 1

Hecke kernels

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T^{2} + 4 T^{4} \)
$3$ \( 1 - T + T^{2} \)
$5$ 1
$7$ \( 1 + 5 T + 7 T^{2} \)
$11$ \( 1 - 11 T^{2} + 121 T^{4} \)
$13$ \( ( 1 - T + 13 T^{2} )^{2} \)
$17$ \( 1 - 6 T + 19 T^{2} - 102 T^{3} + 289 T^{4} \)
$19$ \( 1 + 5 T + 6 T^{2} + 95 T^{3} + 361 T^{4} \)
$23$ \( 1 - 6 T + 13 T^{2} - 138 T^{3} + 529 T^{4} \)
$29$ \( ( 1 + 6 T + 29 T^{2} )^{2} \)
$31$ \( 1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4} \)
$37$ \( 1 + 7 T + 12 T^{2} + 259 T^{3} + 1369 T^{4} \)
$41$ \( ( 1 - 12 T + 41 T^{2} )^{2} \)
$43$ \( ( 1 - T + 43 T^{2} )^{2} \)
$47$ \( 1 - 6 T - 11 T^{2} - 282 T^{3} + 2209 T^{4} \)
$53$ \( 1 - 53 T^{2} + 2809 T^{4} \)
$59$ \( 1 - 6 T - 23 T^{2} - 354 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 2 T - 57 T^{2} + 122 T^{3} + 3721 T^{4} \)
$67$ \( 1 + 7 T - 18 T^{2} + 469 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 - 12 T + 71 T^{2} )^{2} \)
$73$ \( 1 - 11 T + 48 T^{2} - 803 T^{3} + 5329 T^{4} \)
$79$ \( ( 1 - 17 T + 79 T^{2} )( 1 + 4 T + 79 T^{2} ) \)
$83$ \( ( 1 - 12 T + 83 T^{2} )^{2} \)
$89$ \( 1 + 6 T - 53 T^{2} + 534 T^{3} + 7921 T^{4} \)
$97$ \( ( 1 - 10 T + 97 T^{2} )^{2} \)
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