Properties

Label 525.2.i.b
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q -\zeta_{6} q^{2} + ( -1 + \zeta_{6} ) q^{3} + ( 1 - \zeta_{6} ) q^{4} + q^{6} + ( 1 - 3 \zeta_{6} ) q^{7} -3 q^{8} -\zeta_{6} q^{9} +O(q^{10})\) \( q -\zeta_{6} q^{2} + ( -1 + \zeta_{6} ) q^{3} + ( 1 - \zeta_{6} ) q^{4} + q^{6} + ( 1 - 3 \zeta_{6} ) q^{7} -3 q^{8} -\zeta_{6} q^{9} + \zeta_{6} q^{12} -3 q^{13} + ( -3 + 2 \zeta_{6} ) q^{14} + \zeta_{6} q^{16} + ( -2 + 2 \zeta_{6} ) q^{17} + ( -1 + \zeta_{6} ) q^{18} -\zeta_{6} q^{19} + ( 2 + \zeta_{6} ) q^{21} -2 \zeta_{6} q^{23} + ( 3 - 3 \zeta_{6} ) q^{24} + 3 \zeta_{6} q^{26} + q^{27} + ( -2 - \zeta_{6} ) q^{28} -8 q^{29} + ( 8 - 8 \zeta_{6} ) q^{31} + ( -5 + 5 \zeta_{6} ) q^{32} + 2 q^{34} - q^{36} -7 \zeta_{6} q^{37} + ( -1 + \zeta_{6} ) q^{38} + ( 3 - 3 \zeta_{6} ) q^{39} + ( 1 - 3 \zeta_{6} ) q^{42} -8 q^{43} + ( -2 + 2 \zeta_{6} ) q^{46} + 10 \zeta_{6} q^{47} - q^{48} + ( -8 + 3 \zeta_{6} ) q^{49} -2 \zeta_{6} q^{51} + ( -3 + 3 \zeta_{6} ) q^{52} + ( 14 - 14 \zeta_{6} ) q^{53} -\zeta_{6} q^{54} + ( -3 + 9 \zeta_{6} ) q^{56} + q^{57} + 8 \zeta_{6} q^{58} + ( -10 + 10 \zeta_{6} ) q^{59} -7 \zeta_{6} q^{61} -8 q^{62} + ( -3 + 2 \zeta_{6} ) q^{63} + 7 q^{64} + ( 5 - 5 \zeta_{6} ) q^{67} + 2 \zeta_{6} q^{68} + 2 q^{69} -12 q^{71} + 3 \zeta_{6} q^{72} + ( 11 - 11 \zeta_{6} ) q^{73} + ( -7 + 7 \zeta_{6} ) q^{74} - q^{76} -3 q^{78} + 7 \zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{81} + 14 q^{83} + ( 3 - 2 \zeta_{6} ) q^{84} + 8 \zeta_{6} q^{86} + ( 8 - 8 \zeta_{6} ) q^{87} + 6 \zeta_{6} q^{89} + ( -3 + 9 \zeta_{6} ) q^{91} -2 q^{92} + 8 \zeta_{6} q^{93} + ( 10 - 10 \zeta_{6} ) q^{94} -5 \zeta_{6} q^{96} + 9 q^{97} + ( 3 + 5 \zeta_{6} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} - q^{3} + q^{4} + 2q^{6} - q^{7} - 6q^{8} - q^{9} + O(q^{10}) \) \( 2q - q^{2} - q^{3} + q^{4} + 2q^{6} - q^{7} - 6q^{8} - q^{9} + q^{12} - 6q^{13} - 4q^{14} + q^{16} - 2q^{17} - q^{18} - q^{19} + 5q^{21} - 2q^{23} + 3q^{24} + 3q^{26} + 2q^{27} - 5q^{28} - 16q^{29} + 8q^{31} - 5q^{32} + 4q^{34} - 2q^{36} - 7q^{37} - q^{38} + 3q^{39} - q^{42} - 16q^{43} - 2q^{46} + 10q^{47} - 2q^{48} - 13q^{49} - 2q^{51} - 3q^{52} + 14q^{53} - q^{54} + 3q^{56} + 2q^{57} + 8q^{58} - 10q^{59} - 7q^{61} - 16q^{62} - 4q^{63} + 14q^{64} + 5q^{67} + 2q^{68} + 4q^{69} - 24q^{71} + 3q^{72} + 11q^{73} - 7q^{74} - 2q^{76} - 6q^{78} + 7q^{79} - q^{81} + 28q^{83} + 4q^{84} + 8q^{86} + 8q^{87} + 6q^{89} + 3q^{91} - 4q^{92} + 8q^{93} + 10q^{94} - 5q^{96} + 18q^{97} + 11q^{98} + O(q^{100}) \)

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
−0.500000 0.866025i −0.500000 + 0.866025i 0.500000 0.866025i 0 1.00000 −0.500000 2.59808i −3.00000 −0.500000 0.866025i 0
226.1 −0.500000 + 0.866025i −0.500000 0.866025i 0.500000 + 0.866025i 0 1.00000 −0.500000 + 2.59808i −3.00000 −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.b 2
5.b even 2 1 525.2.i.d yes 2
5.c odd 4 2 525.2.r.c 4
7.c even 3 1 inner 525.2.i.b 2
7.c even 3 1 3675.2.a.m 1
7.d odd 6 1 3675.2.a.k 1
35.i odd 6 1 3675.2.a.g 1
35.j even 6 1 525.2.i.d yes 2
35.j even 6 1 3675.2.a.e 1
35.l odd 12 2 525.2.r.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.2.i.b 2 1.a even 1 1 trivial
525.2.i.b 2 7.c even 3 1 inner
525.2.i.d yes 2 5.b even 2 1
525.2.i.d yes 2 35.j even 6 1
525.2.r.c 4 5.c odd 4 2
525.2.r.c 4 35.l odd 12 2
3675.2.a.e 1 35.j even 6 1
3675.2.a.g 1 35.i odd 6 1
3675.2.a.k 1 7.d odd 6 1
3675.2.a.m 1 7.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(525, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + T - T^{2} + 2 T^{3} + 4 T^{4} \)
$3$ \( 1 + T + T^{2} \)
$5$ \( \)
$7$ \( 1 + T + 7 T^{2} \)
$11$ \( 1 - 11 T^{2} + 121 T^{4} \)
$13$ \( ( 1 + 3 T + 13 T^{2} )^{2} \)
$17$ \( 1 + 2 T - 13 T^{2} + 34 T^{3} + 289 T^{4} \)
$19$ \( ( 1 - 7 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \)
$23$ \( 1 + 2 T - 19 T^{2} + 46 T^{3} + 529 T^{4} \)
$29$ \( ( 1 + 8 T + 29 T^{2} )^{2} \)
$31$ \( 1 - 8 T + 33 T^{2} - 248 T^{3} + 961 T^{4} \)
$37$ \( 1 + 7 T + 12 T^{2} + 259 T^{3} + 1369 T^{4} \)
$41$ \( ( 1 + 41 T^{2} )^{2} \)
$43$ \( ( 1 + 8 T + 43 T^{2} )^{2} \)
$47$ \( 1 - 10 T + 53 T^{2} - 470 T^{3} + 2209 T^{4} \)
$53$ \( 1 - 14 T + 143 T^{2} - 742 T^{3} + 2809 T^{4} \)
$59$ \( 1 + 10 T + 41 T^{2} + 590 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 7 T - 12 T^{2} + 427 T^{3} + 3721 T^{4} \)
$67$ \( ( 1 - 16 T + 67 T^{2} )( 1 + 11 T + 67 T^{2} ) \)
$71$ \( ( 1 + 12 T + 71 T^{2} )^{2} \)
$73$ \( 1 - 11 T + 48 T^{2} - 803 T^{3} + 5329 T^{4} \)
$79$ \( 1 - 7 T - 30 T^{2} - 553 T^{3} + 6241 T^{4} \)
$83$ \( ( 1 - 14 T + 83 T^{2} )^{2} \)
$89$ \( 1 - 6 T - 53 T^{2} - 534 T^{3} + 7921 T^{4} \)
$97$ \( ( 1 - 9 T + 97 T^{2} )^{2} \)
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