Properties

Label 525.4.d.f
Level 525
Weight 4
Character orbit 525.d
Analytic conductor 30.976
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: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 525.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(30.9760027530\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 105)
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.

\(f(q)\) \(=\) \( q -3 i q^{3} + 8 q^{4} -7 i q^{7} -9 q^{9} +O(q^{10})\) \( q -3 i q^{3} + 8 q^{4} -7 i q^{7} -9 q^{9} + 42 q^{11} -24 i q^{12} + 20 i q^{13} + 64 q^{16} -66 i q^{17} -38 q^{19} -21 q^{21} + 12 i q^{23} + 27 i q^{27} -56 i q^{28} + 258 q^{29} + 146 q^{31} -126 i q^{33} -72 q^{36} -434 i q^{37} + 60 q^{39} -282 q^{41} + 20 i q^{43} + 336 q^{44} + 72 i q^{47} -192 i q^{48} -49 q^{49} -198 q^{51} + 160 i q^{52} + 336 i q^{53} + 114 i q^{57} + 360 q^{59} -682 q^{61} + 63 i q^{63} + 512 q^{64} -812 i q^{67} -528 i q^{68} + 36 q^{69} + 810 q^{71} -124 i q^{73} -304 q^{76} -294 i q^{77} -1136 q^{79} + 81 q^{81} + 156 i q^{83} -168 q^{84} -774 i q^{87} + 1038 q^{89} + 140 q^{91} + 96 i q^{92} -438 i q^{93} -1208 i q^{97} -378 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 16q^{4} - 18q^{9} + O(q^{10}) \) \( 2q + 16q^{4} - 18q^{9} + 84q^{11} + 128q^{16} - 76q^{19} - 42q^{21} + 516q^{29} + 292q^{31} - 144q^{36} + 120q^{39} - 564q^{41} + 672q^{44} - 98q^{49} - 396q^{51} + 720q^{59} - 1364q^{61} + 1024q^{64} + 72q^{69} + 1620q^{71} - 608q^{76} - 2272q^{79} + 162q^{81} - 336q^{84} + 2076q^{89} + 280q^{91} - 756q^{99} + O(q^{100}) \)

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
0 3.00000i 8.00000 0 0 7.00000i 0 −9.00000 0
274.2 0 3.00000i 8.00000 0 0 7.00000i 0 −9.00000 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 525.4.d.f 2
5.b even 2 1 inner 525.4.d.f 2
5.c odd 4 1 105.4.a.a 1
5.c odd 4 1 525.4.a.e 1
15.e even 4 1 315.4.a.d 1
15.e even 4 1 1575.4.a.f 1
20.e even 4 1 1680.4.a.s 1
35.f even 4 1 735.4.a.c 1
105.k odd 4 1 2205.4.a.o 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.a 1 5.c odd 4 1
315.4.a.d 1 15.e even 4 1
525.4.a.e 1 5.c odd 4 1
525.4.d.f 2 1.a even 1 1 trivial
525.4.d.f 2 5.b even 2 1 inner
735.4.a.c 1 35.f even 4 1
1575.4.a.f 1 15.e even 4 1
1680.4.a.s 1 20.e even 4 1
2205.4.a.o 1 105.k odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(525, [\chi])\):

\( T_{2} \)
\( T_{11} - 42 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 8 T^{2} )^{2} \)
$3$ \( 1 + 9 T^{2} \)
$5$ 1
$7$ \( 1 + 49 T^{2} \)
$11$ \( ( 1 - 42 T + 1331 T^{2} )^{2} \)
$13$ \( 1 - 3994 T^{2} + 4826809 T^{4} \)
$17$ \( 1 - 5470 T^{2} + 24137569 T^{4} \)
$19$ \( ( 1 + 38 T + 6859 T^{2} )^{2} \)
$23$ \( 1 - 24190 T^{2} + 148035889 T^{4} \)
$29$ \( ( 1 - 258 T + 24389 T^{2} )^{2} \)
$31$ \( ( 1 - 146 T + 29791 T^{2} )^{2} \)
$37$ \( 1 + 87050 T^{2} + 2565726409 T^{4} \)
$41$ \( ( 1 + 282 T + 68921 T^{2} )^{2} \)
$43$ \( 1 - 158614 T^{2} + 6321363049 T^{4} \)
$47$ \( 1 - 202462 T^{2} + 10779215329 T^{4} \)
$53$ \( 1 - 184858 T^{2} + 22164361129 T^{4} \)
$59$ \( ( 1 - 360 T + 205379 T^{2} )^{2} \)
$61$ \( ( 1 + 682 T + 226981 T^{2} )^{2} \)
$67$ \( 1 + 57818 T^{2} + 90458382169 T^{4} \)
$71$ \( ( 1 - 810 T + 357911 T^{2} )^{2} \)
$73$ \( 1 - 762658 T^{2} + 151334226289 T^{4} \)
$79$ \( ( 1 + 1136 T + 493039 T^{2} )^{2} \)
$83$ \( 1 - 1119238 T^{2} + 326940373369 T^{4} \)
$89$ \( ( 1 - 1038 T + 704969 T^{2} )^{2} \)
$97$ \( 1 - 366082 T^{2} + 832972004929 T^{4} \)
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