Properties

Label 525.2.d.a
Level 525
Weight 2
Character orbit 525.d
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.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

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

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + i q^{2} + i q^{3} + q^{4} - q^{6} + i q^{7} + 3 i q^{8} - q^{9} +O(q^{10})\) \( q + i q^{2} + i q^{3} + q^{4} - q^{6} + i q^{7} + 3 i q^{8} - q^{9} + 4 q^{11} + i q^{12} -2 i q^{13} - q^{14} - q^{16} + 6 i q^{17} -i q^{18} -4 q^{19} - q^{21} + 4 i q^{22} -3 q^{24} + 2 q^{26} -i q^{27} + i q^{28} + 2 q^{29} + 5 i q^{32} + 4 i q^{33} -6 q^{34} - q^{36} -6 i q^{37} -4 i q^{38} + 2 q^{39} + 2 q^{41} -i q^{42} -4 i q^{43} + 4 q^{44} -i q^{48} - q^{49} -6 q^{51} -2 i q^{52} + 6 i q^{53} + q^{54} -3 q^{56} -4 i q^{57} + 2 i q^{58} -12 q^{59} -2 q^{61} -i q^{63} -7 q^{64} -4 q^{66} -4 i q^{67} + 6 i q^{68} -3 i q^{72} -6 i q^{73} + 6 q^{74} -4 q^{76} + 4 i q^{77} + 2 i q^{78} + 16 q^{79} + q^{81} + 2 i q^{82} -12 i q^{83} - q^{84} + 4 q^{86} + 2 i q^{87} + 12 i q^{88} + 14 q^{89} + 2 q^{91} -5 q^{96} -18 i q^{97} -i q^{98} -4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{4} - 2q^{6} - 2q^{9} + O(q^{10}) \) \( 2q + 2q^{4} - 2q^{6} - 2q^{9} + 8q^{11} - 2q^{14} - 2q^{16} - 8q^{19} - 2q^{21} - 6q^{24} + 4q^{26} + 4q^{29} - 12q^{34} - 2q^{36} + 4q^{39} + 4q^{41} + 8q^{44} - 2q^{49} - 12q^{51} + 2q^{54} - 6q^{56} - 24q^{59} - 4q^{61} - 14q^{64} - 8q^{66} + 12q^{74} - 8q^{76} + 32q^{79} + 2q^{81} - 2q^{84} + 8q^{86} + 28q^{89} + 4q^{91} - 10q^{96} - 8q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(127\) \(176\) \(451\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
274.1
1.00000i
1.00000i
1.00000i 1.00000i 1.00000 0 −1.00000 1.00000i 3.00000i −1.00000 0
274.2 1.00000i 1.00000i 1.00000 0 −1.00000 1.00000i 3.00000i −1.00000 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.2.d.a 2
3.b odd 2 1 1575.2.d.a 2
5.b even 2 1 inner 525.2.d.a 2
5.c odd 4 1 21.2.a.a 1
5.c odd 4 1 525.2.a.d 1
15.d odd 2 1 1575.2.d.a 2
15.e even 4 1 63.2.a.a 1
15.e even 4 1 1575.2.a.c 1
20.e even 4 1 336.2.a.a 1
20.e even 4 1 8400.2.a.bn 1
35.f even 4 1 147.2.a.a 1
35.f even 4 1 3675.2.a.n 1
35.k even 12 2 147.2.e.c 2
35.l odd 12 2 147.2.e.b 2
40.i odd 4 1 1344.2.a.g 1
40.k even 4 1 1344.2.a.s 1
45.k odd 12 2 567.2.f.g 2
45.l even 12 2 567.2.f.b 2
55.e even 4 1 2541.2.a.j 1
60.l odd 4 1 1008.2.a.l 1
65.h odd 4 1 3549.2.a.c 1
80.i odd 4 1 5376.2.c.r 2
80.j even 4 1 5376.2.c.l 2
80.s even 4 1 5376.2.c.l 2
80.t odd 4 1 5376.2.c.r 2
85.g odd 4 1 6069.2.a.b 1
95.g even 4 1 7581.2.a.d 1
105.k odd 4 1 441.2.a.f 1
105.w odd 12 2 441.2.e.b 2
105.x even 12 2 441.2.e.a 2
120.q odd 4 1 4032.2.a.k 1
120.w even 4 1 4032.2.a.h 1
140.j odd 4 1 2352.2.a.v 1
140.w even 12 2 2352.2.q.x 2
140.x odd 12 2 2352.2.q.e 2
165.l odd 4 1 7623.2.a.g 1
280.s even 4 1 9408.2.a.bv 1
280.y odd 4 1 9408.2.a.m 1
420.w even 4 1 7056.2.a.p 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.2.a.a 1 5.c odd 4 1
63.2.a.a 1 15.e even 4 1
147.2.a.a 1 35.f even 4 1
147.2.e.b 2 35.l odd 12 2
147.2.e.c 2 35.k even 12 2
336.2.a.a 1 20.e even 4 1
441.2.a.f 1 105.k odd 4 1
441.2.e.a 2 105.x even 12 2
441.2.e.b 2 105.w odd 12 2
525.2.a.d 1 5.c odd 4 1
525.2.d.a 2 1.a even 1 1 trivial
525.2.d.a 2 5.b even 2 1 inner
567.2.f.b 2 45.l even 12 2
567.2.f.g 2 45.k odd 12 2
1008.2.a.l 1 60.l odd 4 1
1344.2.a.g 1 40.i odd 4 1
1344.2.a.s 1 40.k even 4 1
1575.2.a.c 1 15.e even 4 1
1575.2.d.a 2 3.b odd 2 1
1575.2.d.a 2 15.d odd 2 1
2352.2.a.v 1 140.j odd 4 1
2352.2.q.e 2 140.x odd 12 2
2352.2.q.x 2 140.w even 12 2
2541.2.a.j 1 55.e even 4 1
3549.2.a.c 1 65.h odd 4 1
3675.2.a.n 1 35.f even 4 1
4032.2.a.h 1 120.w even 4 1
4032.2.a.k 1 120.q odd 4 1
5376.2.c.l 2 80.j even 4 1
5376.2.c.l 2 80.s even 4 1
5376.2.c.r 2 80.i odd 4 1
5376.2.c.r 2 80.t odd 4 1
6069.2.a.b 1 85.g odd 4 1
7056.2.a.p 1 420.w even 4 1
7581.2.a.d 1 95.g even 4 1
7623.2.a.g 1 165.l odd 4 1
8400.2.a.bn 1 20.e even 4 1
9408.2.a.m 1 280.y odd 4 1
9408.2.a.bv 1 280.s even 4 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 3 T^{2} + 4 T^{4} \)
$3$ \( 1 + T^{2} \)
$5$ 1
$7$ \( 1 + T^{2} \)
$11$ \( ( 1 - 4 T + 11 T^{2} )^{2} \)
$13$ \( 1 - 22 T^{2} + 169 T^{4} \)
$17$ \( 1 + 2 T^{2} + 289 T^{4} \)
$19$ \( ( 1 + 4 T + 19 T^{2} )^{2} \)
$23$ \( ( 1 - 23 T^{2} )^{2} \)
$29$ \( ( 1 - 2 T + 29 T^{2} )^{2} \)
$31$ \( ( 1 + 31 T^{2} )^{2} \)
$37$ \( 1 - 38 T^{2} + 1369 T^{4} \)
$41$ \( ( 1 - 2 T + 41 T^{2} )^{2} \)
$43$ \( 1 - 70 T^{2} + 1849 T^{4} \)
$47$ \( ( 1 - 47 T^{2} )^{2} \)
$53$ \( 1 - 70 T^{2} + 2809 T^{4} \)
$59$ \( ( 1 + 12 T + 59 T^{2} )^{2} \)
$61$ \( ( 1 + 2 T + 61 T^{2} )^{2} \)
$67$ \( 1 - 118 T^{2} + 4489 T^{4} \)
$71$ \( ( 1 + 71 T^{2} )^{2} \)
$73$ \( ( 1 - 16 T + 73 T^{2} )( 1 + 16 T + 73 T^{2} ) \)
$79$ \( ( 1 - 16 T + 79 T^{2} )^{2} \)
$83$ \( 1 - 22 T^{2} + 6889 T^{4} \)
$89$ \( ( 1 - 14 T + 89 T^{2} )^{2} \)
$97$ \( ( 1 - 8 T + 97 T^{2} )( 1 + 8 T + 97 T^{2} ) \)
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