Properties

Label 525.4.d.d
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: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + 3 i q^{2} -3 i q^{3} - q^{4} + 9 q^{6} -7 i q^{7} + 21 i q^{8} -9 q^{9} +O(q^{10})\) \( q + 3 i q^{2} -3 i q^{3} - q^{4} + 9 q^{6} -7 i q^{7} + 21 i q^{8} -9 q^{9} -6 q^{11} + 3 i q^{12} + 41 i q^{13} + 21 q^{14} -71 q^{16} -27 i q^{17} -27 i q^{18} + 4 q^{19} -21 q^{21} -18 i q^{22} + 75 i q^{23} + 63 q^{24} -123 q^{26} + 27 i q^{27} + 7 i q^{28} + 123 q^{29} -205 q^{31} -45 i q^{32} + 18 i q^{33} + 81 q^{34} + 9 q^{36} + 262 i q^{37} + 12 i q^{38} + 123 q^{39} + 57 q^{41} -63 i q^{42} + 407 i q^{43} + 6 q^{44} -225 q^{46} + 60 i q^{47} + 213 i q^{48} -49 q^{49} -81 q^{51} -41 i q^{52} + 327 i q^{53} -81 q^{54} + 147 q^{56} -12 i q^{57} + 369 i q^{58} -33 q^{59} -427 q^{61} -615 i q^{62} + 63 i q^{63} -433 q^{64} -54 q^{66} + 628 i q^{67} + 27 i q^{68} + 225 q^{69} + 300 q^{71} -189 i q^{72} + 98 i q^{73} -786 q^{74} -4 q^{76} + 42 i q^{77} + 369 i q^{78} -686 q^{79} + 81 q^{81} + 171 i q^{82} + 1401 i q^{83} + 21 q^{84} -1221 q^{86} -369 i q^{87} -126 i q^{88} -714 q^{89} + 287 q^{91} -75 i q^{92} + 615 i q^{93} -180 q^{94} -135 q^{96} -494 i q^{97} -147 i q^{98} + 54 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + 18q^{6} - 18q^{9} + O(q^{10}) \) \( 2q - 2q^{4} + 18q^{6} - 18q^{9} - 12q^{11} + 42q^{14} - 142q^{16} + 8q^{19} - 42q^{21} + 126q^{24} - 246q^{26} + 246q^{29} - 410q^{31} + 162q^{34} + 18q^{36} + 246q^{39} + 114q^{41} + 12q^{44} - 450q^{46} - 98q^{49} - 162q^{51} - 162q^{54} + 294q^{56} - 66q^{59} - 854q^{61} - 866q^{64} - 108q^{66} + 450q^{69} + 600q^{71} - 1572q^{74} - 8q^{76} - 1372q^{79} + 162q^{81} + 42q^{84} - 2442q^{86} - 1428q^{89} + 574q^{91} - 360q^{94} - 270q^{96} + 108q^{99} + O(q^{100}) \)

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
3.00000i 3.00000i −1.00000 0 9.00000 7.00000i 21.0000i −9.00000 0
274.2 3.00000i 3.00000i −1.00000 0 9.00000 7.00000i 21.0000i −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.d 2
5.b even 2 1 inner 525.4.d.d 2
5.c odd 4 1 525.4.a.c 1
5.c odd 4 1 525.4.a.h yes 1
15.e even 4 1 1575.4.a.a 1
15.e even 4 1 1575.4.a.j 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.4.a.c 1 5.c odd 4 1
525.4.a.h yes 1 5.c odd 4 1
525.4.d.d 2 1.a even 1 1 trivial
525.4.d.d 2 5.b even 2 1 inner
1575.4.a.a 1 15.e even 4 1
1575.4.a.j 1 15.e even 4 1

Hecke kernels

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

\( T_{2}^{2} + 9 \)
\( T_{11} + 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 7 T^{2} + 64 T^{4} \)
$3$ \( 1 + 9 T^{2} \)
$5$ 1
$7$ \( 1 + 49 T^{2} \)
$11$ \( ( 1 + 6 T + 1331 T^{2} )^{2} \)
$13$ \( 1 - 2713 T^{2} + 4826809 T^{4} \)
$17$ \( 1 - 9097 T^{2} + 24137569 T^{4} \)
$19$ \( ( 1 - 4 T + 6859 T^{2} )^{2} \)
$23$ \( 1 - 18709 T^{2} + 148035889 T^{4} \)
$29$ \( ( 1 - 123 T + 24389 T^{2} )^{2} \)
$31$ \( ( 1 + 205 T + 29791 T^{2} )^{2} \)
$37$ \( 1 - 32662 T^{2} + 2565726409 T^{4} \)
$41$ \( ( 1 - 57 T + 68921 T^{2} )^{2} \)
$43$ \( 1 + 6635 T^{2} + 6321363049 T^{4} \)
$47$ \( 1 - 204046 T^{2} + 10779215329 T^{4} \)
$53$ \( 1 - 190825 T^{2} + 22164361129 T^{4} \)
$59$ \( ( 1 + 33 T + 205379 T^{2} )^{2} \)
$61$ \( ( 1 + 427 T + 226981 T^{2} )^{2} \)
$67$ \( 1 - 207142 T^{2} + 90458382169 T^{4} \)
$71$ \( ( 1 - 300 T + 357911 T^{2} )^{2} \)
$73$ \( 1 - 768430 T^{2} + 151334226289 T^{4} \)
$79$ \( ( 1 + 686 T + 493039 T^{2} )^{2} \)
$83$ \( 1 + 819227 T^{2} + 326940373369 T^{4} \)
$89$ \( ( 1 + 714 T + 704969 T^{2} )^{2} \)
$97$ \( 1 - 1581310 T^{2} + 832972004929 T^{4} \)
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