Properties

Label 525.6.a.b
Level $525$
Weight $6$
Character orbit 525.a
Self dual yes
Analytic conductor $84.202$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 525.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(84.2015054018\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 5 q^{2} - 9 q^{3} - 7 q^{4} + 45 q^{6} + 49 q^{7} + 195 q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 5 q^{2} - 9 q^{3} - 7 q^{4} + 45 q^{6} + 49 q^{7} + 195 q^{8} + 81 q^{9} + 52 q^{11} + 63 q^{12} + 770 q^{13} - 245 q^{14} - 751 q^{16} + 2022 q^{17} - 405 q^{18} + 1732 q^{19} - 441 q^{21} - 260 q^{22} + 576 q^{23} - 1755 q^{24} - 3850 q^{26} - 729 q^{27} - 343 q^{28} + 5518 q^{29} + 6336 q^{31} - 2485 q^{32} - 468 q^{33} - 10110 q^{34} - 567 q^{36} + 7338 q^{37} - 8660 q^{38} - 6930 q^{39} - 3262 q^{41} + 2205 q^{42} - 5420 q^{43} - 364 q^{44} - 2880 q^{46} - 864 q^{47} + 6759 q^{48} + 2401 q^{49} - 18198 q^{51} - 5390 q^{52} - 4182 q^{53} + 3645 q^{54} + 9555 q^{56} - 15588 q^{57} - 27590 q^{58} - 11220 q^{59} - 45602 q^{61} - 31680 q^{62} + 3969 q^{63} + 36457 q^{64} + 2340 q^{66} - 1396 q^{67} - 14154 q^{68} - 5184 q^{69} + 18720 q^{71} + 15795 q^{72} - 46362 q^{73} - 36690 q^{74} - 12124 q^{76} + 2548 q^{77} + 34650 q^{78} + 97424 q^{79} + 6561 q^{81} + 16310 q^{82} + 81228 q^{83} + 3087 q^{84} + 27100 q^{86} - 49662 q^{87} + 10140 q^{88} - 3182 q^{89} + 37730 q^{91} - 4032 q^{92} - 57024 q^{93} + 4320 q^{94} + 22365 q^{96} - 4914 q^{97} - 12005 q^{98} + 4212 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
−5.00000 −9.00000 −7.00000 0 45.0000 49.0000 195.000 81.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(1\)
\(7\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.6.a.b 1
5.b even 2 1 21.6.a.c 1
5.c odd 4 2 525.6.d.c 2
15.d odd 2 1 63.6.a.b 1
20.d odd 2 1 336.6.a.i 1
35.c odd 2 1 147.6.a.f 1
35.i odd 6 2 147.6.e.d 2
35.j even 6 2 147.6.e.c 2
60.h even 2 1 1008.6.a.a 1
105.g even 2 1 441.6.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.a.c 1 5.b even 2 1
63.6.a.b 1 15.d odd 2 1
147.6.a.f 1 35.c odd 2 1
147.6.e.c 2 35.j even 6 2
147.6.e.d 2 35.i odd 6 2
336.6.a.i 1 20.d odd 2 1
441.6.a.c 1 105.g even 2 1
525.6.a.b 1 1.a even 1 1 trivial
525.6.d.c 2 5.c odd 4 2
1008.6.a.a 1 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 5 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(525))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 5 \) Copy content Toggle raw display
$3$ \( T + 9 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T - 49 \) Copy content Toggle raw display
$11$ \( T - 52 \) Copy content Toggle raw display
$13$ \( T - 770 \) Copy content Toggle raw display
$17$ \( T - 2022 \) Copy content Toggle raw display
$19$ \( T - 1732 \) Copy content Toggle raw display
$23$ \( T - 576 \) Copy content Toggle raw display
$29$ \( T - 5518 \) Copy content Toggle raw display
$31$ \( T - 6336 \) Copy content Toggle raw display
$37$ \( T - 7338 \) Copy content Toggle raw display
$41$ \( T + 3262 \) Copy content Toggle raw display
$43$ \( T + 5420 \) Copy content Toggle raw display
$47$ \( T + 864 \) Copy content Toggle raw display
$53$ \( T + 4182 \) Copy content Toggle raw display
$59$ \( T + 11220 \) Copy content Toggle raw display
$61$ \( T + 45602 \) Copy content Toggle raw display
$67$ \( T + 1396 \) Copy content Toggle raw display
$71$ \( T - 18720 \) Copy content Toggle raw display
$73$ \( T + 46362 \) Copy content Toggle raw display
$79$ \( T - 97424 \) Copy content Toggle raw display
$83$ \( T - 81228 \) Copy content Toggle raw display
$89$ \( T + 3182 \) Copy content Toggle raw display
$97$ \( T + 4914 \) Copy content Toggle raw display
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