Properties

Label 525.2.t.a
Level 525
Weight 2
Character orbit 525.t
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.t (of order \(6\), 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: no (minimal twist has level 105)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -2 + \zeta_{6} ) q^{2} + ( -1 + 2 \zeta_{6} ) q^{3} + ( 1 - \zeta_{6} ) q^{4} -3 \zeta_{6} q^{6} + ( 3 - \zeta_{6} ) q^{7} + ( 1 - 2 \zeta_{6} ) q^{8} -3 q^{9} +O(q^{10})\) \( q + ( -2 + \zeta_{6} ) q^{2} + ( -1 + 2 \zeta_{6} ) q^{3} + ( 1 - \zeta_{6} ) q^{4} -3 \zeta_{6} q^{6} + ( 3 - \zeta_{6} ) q^{7} + ( 1 - 2 \zeta_{6} ) q^{8} -3 q^{9} + ( 2 + 2 \zeta_{6} ) q^{11} + ( 1 + \zeta_{6} ) q^{12} + ( -2 + 4 \zeta_{6} ) q^{13} + ( -5 + 4 \zeta_{6} ) q^{14} + 5 \zeta_{6} q^{16} + ( -6 + 6 \zeta_{6} ) q^{17} + ( 6 - 3 \zeta_{6} ) q^{18} + ( -8 + 4 \zeta_{6} ) q^{19} + ( -1 + 5 \zeta_{6} ) q^{21} -6 q^{22} + ( 2 - \zeta_{6} ) q^{23} + 3 q^{24} -6 \zeta_{6} q^{26} + ( 3 - 6 \zeta_{6} ) q^{27} + ( 2 - 3 \zeta_{6} ) q^{28} + ( 1 - 2 \zeta_{6} ) q^{29} + ( -2 - 2 \zeta_{6} ) q^{31} + ( -3 - 3 \zeta_{6} ) q^{32} + ( -6 + 6 \zeta_{6} ) q^{33} + ( 6 - 12 \zeta_{6} ) q^{34} + ( -3 + 3 \zeta_{6} ) q^{36} + 4 \zeta_{6} q^{37} + ( 12 - 12 \zeta_{6} ) q^{38} -6 q^{39} + 3 q^{41} + ( -3 - 6 \zeta_{6} ) q^{42} - q^{43} + ( 4 - 2 \zeta_{6} ) q^{44} + ( -3 + 3 \zeta_{6} ) q^{46} + ( -10 + 5 \zeta_{6} ) q^{48} + ( 8 - 5 \zeta_{6} ) q^{49} + ( -6 - 6 \zeta_{6} ) q^{51} + ( 2 + 2 \zeta_{6} ) q^{52} + 9 \zeta_{6} q^{54} + ( 1 - 5 \zeta_{6} ) q^{56} -12 \zeta_{6} q^{57} + 3 \zeta_{6} q^{58} + ( -6 + 3 \zeta_{6} ) q^{61} + 6 q^{62} + ( -9 + 3 \zeta_{6} ) q^{63} - q^{64} + ( 6 - 12 \zeta_{6} ) q^{66} + ( -13 + 13 \zeta_{6} ) q^{67} + 6 \zeta_{6} q^{68} + 3 \zeta_{6} q^{69} + ( 4 - 8 \zeta_{6} ) q^{71} + ( -3 + 6 \zeta_{6} ) q^{72} + ( -2 - 2 \zeta_{6} ) q^{73} + ( -4 - 4 \zeta_{6} ) q^{74} + ( -4 + 8 \zeta_{6} ) q^{76} + ( 8 + 2 \zeta_{6} ) q^{77} + ( 12 - 6 \zeta_{6} ) q^{78} + 16 \zeta_{6} q^{79} + 9 q^{81} + ( -6 + 3 \zeta_{6} ) q^{82} -9 q^{83} + ( 4 + \zeta_{6} ) q^{84} + ( 2 - \zeta_{6} ) q^{86} + 3 q^{87} + ( 6 - 6 \zeta_{6} ) q^{88} + 3 \zeta_{6} q^{89} + ( -2 + 10 \zeta_{6} ) q^{91} + ( 1 - 2 \zeta_{6} ) q^{92} + ( 6 - 6 \zeta_{6} ) q^{93} + ( 9 - 9 \zeta_{6} ) q^{96} + ( 6 - 12 \zeta_{6} ) q^{97} + ( -11 + 13 \zeta_{6} ) q^{98} + ( -6 - 6 \zeta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 3q^{2} + q^{4} - 3q^{6} + 5q^{7} - 6q^{9} + O(q^{10}) \) \( 2q - 3q^{2} + q^{4} - 3q^{6} + 5q^{7} - 6q^{9} + 6q^{11} + 3q^{12} - 6q^{14} + 5q^{16} - 6q^{17} + 9q^{18} - 12q^{19} + 3q^{21} - 12q^{22} + 3q^{23} + 6q^{24} - 6q^{26} + q^{28} - 6q^{31} - 9q^{32} - 6q^{33} - 3q^{36} + 4q^{37} + 12q^{38} - 12q^{39} + 6q^{41} - 12q^{42} - 2q^{43} + 6q^{44} - 3q^{46} - 15q^{48} + 11q^{49} - 18q^{51} + 6q^{52} + 9q^{54} - 3q^{56} - 12q^{57} + 3q^{58} - 9q^{61} + 12q^{62} - 15q^{63} - 2q^{64} - 13q^{67} + 6q^{68} + 3q^{69} - 6q^{73} - 12q^{74} + 18q^{77} + 18q^{78} + 16q^{79} + 18q^{81} - 9q^{82} - 18q^{83} + 9q^{84} + 3q^{86} + 6q^{87} + 6q^{88} + 3q^{89} + 6q^{91} + 6q^{93} + 9q^{96} - 9q^{98} - 18q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(127\) \(176\) \(451\)
\(\chi(n)\) \(1\) \(-1\) \(\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1
0.500000 0.866025i
0.500000 + 0.866025i
−1.50000 0.866025i 1.73205i 0.500000 + 0.866025i 0 −1.50000 + 2.59808i 2.50000 + 0.866025i 1.73205i −3.00000 0
101.1 −1.50000 + 0.866025i 1.73205i 0.500000 0.866025i 0 −1.50000 2.59808i 2.50000 0.866025i 1.73205i −3.00000 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
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.2.t.a 2
3.b odd 2 1 525.2.t.e 2
5.b even 2 1 105.2.s.b yes 2
5.c odd 4 2 525.2.q.a 4
7.d odd 6 1 525.2.t.e 2
15.d odd 2 1 105.2.s.a 2
15.e even 4 2 525.2.q.b 4
21.g even 6 1 inner 525.2.t.a 2
35.c odd 2 1 735.2.s.e 2
35.i odd 6 1 105.2.s.a 2
35.i odd 6 1 735.2.b.b 2
35.j even 6 1 735.2.b.a 2
35.j even 6 1 735.2.s.c 2
35.k even 12 2 525.2.q.b 4
105.g even 2 1 735.2.s.c 2
105.o odd 6 1 735.2.b.b 2
105.o odd 6 1 735.2.s.e 2
105.p even 6 1 105.2.s.b yes 2
105.p even 6 1 735.2.b.a 2
105.w odd 12 2 525.2.q.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.s.a 2 15.d odd 2 1
105.2.s.a 2 35.i odd 6 1
105.2.s.b yes 2 5.b even 2 1
105.2.s.b yes 2 105.p even 6 1
525.2.q.a 4 5.c odd 4 2
525.2.q.a 4 105.w odd 12 2
525.2.q.b 4 15.e even 4 2
525.2.q.b 4 35.k even 12 2
525.2.t.a 2 1.a even 1 1 trivial
525.2.t.a 2 21.g even 6 1 inner
525.2.t.e 2 3.b odd 2 1
525.2.t.e 2 7.d odd 6 1
735.2.b.a 2 35.j even 6 1
735.2.b.a 2 105.p even 6 1
735.2.b.b 2 35.i odd 6 1
735.2.b.b 2 105.o odd 6 1
735.2.s.c 2 35.j even 6 1
735.2.s.c 2 105.g even 2 1
735.2.s.e 2 35.c odd 2 1
735.2.s.e 2 105.o odd 6 1

Hecke kernels

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

\( T_{2}^{2} + 3 T_{2} + 3 \)
\( T_{13}^{2} + 12 \)
\( T_{37}^{2} - 4 T_{37} + 16 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 3 T + 5 T^{2} + 6 T^{3} + 4 T^{4} \)
$3$ \( 1 + 3 T^{2} \)
$5$ \( \)
$7$ \( 1 - 5 T + 7 T^{2} \)
$11$ \( 1 - 6 T + 23 T^{2} - 66 T^{3} + 121 T^{4} \)
$13$ \( 1 - 14 T^{2} + 169 T^{4} \)
$17$ \( 1 + 6 T + 19 T^{2} + 102 T^{3} + 289 T^{4} \)
$19$ \( 1 + 12 T + 67 T^{2} + 228 T^{3} + 361 T^{4} \)
$23$ \( 1 - 3 T + 26 T^{2} - 69 T^{3} + 529 T^{4} \)
$29$ \( 1 - 55 T^{2} + 841 T^{4} \)
$31$ \( 1 + 6 T + 43 T^{2} + 186 T^{3} + 961 T^{4} \)
$37$ \( 1 - 4 T - 21 T^{2} - 148 T^{3} + 1369 T^{4} \)
$41$ \( ( 1 - 3 T + 41 T^{2} )^{2} \)
$43$ \( ( 1 + T + 43 T^{2} )^{2} \)
$47$ \( 1 - 47 T^{2} + 2209 T^{4} \)
$53$ \( 1 + 53 T^{2} + 2809 T^{4} \)
$59$ \( 1 - 59 T^{2} + 3481 T^{4} \)
$61$ \( 1 + 9 T + 88 T^{2} + 549 T^{3} + 3721 T^{4} \)
$67$ \( 1 + 13 T + 102 T^{2} + 871 T^{3} + 4489 T^{4} \)
$71$ \( 1 - 94 T^{2} + 5041 T^{4} \)
$73$ \( 1 + 6 T + 85 T^{2} + 438 T^{3} + 5329 T^{4} \)
$79$ \( 1 - 16 T + 177 T^{2} - 1264 T^{3} + 6241 T^{4} \)
$83$ \( ( 1 + 9 T + 83 T^{2} )^{2} \)
$89$ \( 1 - 3 T - 80 T^{2} - 267 T^{3} + 7921 T^{4} \)
$97$ \( 1 - 86 T^{2} + 9409 T^{4} \)
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