Properties

Label 525.2.b.i
Level 525
Weight 2
Character orbit 525.b
Analytic conductor 4.192
Analytic rank 0
Dimension 8
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 525.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.19214610612\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.624529833984.7
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + \beta_{1} q^{3} + ( -2 + \beta_{4} ) q^{4} -\beta_{5} q^{6} + ( 1 + \beta_{1} - \beta_{2} ) q^{7} + ( -\beta_{3} + \beta_{6} ) q^{8} + ( \beta_{3} + \beta_{4} ) q^{9} +O(q^{10})\) \( q + \beta_{3} q^{2} + \beta_{1} q^{3} + ( -2 + \beta_{4} ) q^{4} -\beta_{5} q^{6} + ( 1 + \beta_{1} - \beta_{2} ) q^{7} + ( -\beta_{3} + \beta_{6} ) q^{8} + ( \beta_{3} + \beta_{4} ) q^{9} + ( \beta_{3} - \beta_{6} ) q^{11} + ( -2 \beta_{1} + \beta_{2} + \beta_{7} ) q^{12} + ( -\beta_{1} + \beta_{2} ) q^{13} + ( \beta_{3} - \beta_{5} + \beta_{7} ) q^{14} + ( 1 - \beta_{4} ) q^{16} + ( \beta_{5} - \beta_{7} ) q^{17} + ( -4 - \beta_{3} + \beta_{4} + \beta_{6} ) q^{18} + ( -2 \beta_{1} + \beta_{5} + \beta_{7} ) q^{19} + ( -2 + \beta_{1} + \beta_{4} - \beta_{6} ) q^{21} + ( -5 + 3 \beta_{4} ) q^{22} + ( \beta_{3} + \beta_{6} ) q^{23} + ( -\beta_{1} + 2 \beta_{2} + \beta_{5} - \beta_{7} ) q^{24} + ( \beta_{5} - \beta_{7} ) q^{26} + ( \beta_{2} - \beta_{5} + \beta_{7} ) q^{27} + ( -2 - 3 \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} + \beta_{7} ) q^{28} + ( \beta_{3} + \beta_{6} ) q^{29} + ( -\beta_{1} - \beta_{2} + \beta_{5} + \beta_{7} ) q^{31} + \beta_{6} q^{32} + ( \beta_{1} - 2 \beta_{2} - \beta_{5} + \beta_{7} ) q^{33} + ( 5 \beta_{1} - 3 \beta_{2} - \beta_{5} - \beta_{7} ) q^{34} + ( 5 - 3 \beta_{3} - \beta_{4} + \beta_{6} ) q^{36} + ( -3 - 2 \beta_{4} ) q^{37} + ( 3 \beta_{1} + \beta_{2} + 3 \beta_{5} - \beta_{7} ) q^{38} + ( 2 - \beta_{4} + \beta_{6} ) q^{39} + ( -3 \beta_{1} - \beta_{2} - \beta_{5} - \beta_{7} ) q^{41} + ( -1 - 3 \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{6} ) q^{42} + ( 1 + 2 \beta_{4} ) q^{43} + ( -6 \beta_{3} + \beta_{6} ) q^{44} + ( -3 - \beta_{4} ) q^{46} + ( \beta_{5} - \beta_{7} ) q^{47} + ( \beta_{1} - \beta_{2} - \beta_{7} ) q^{48} + ( -5 + 2 \beta_{1} - 2 \beta_{2} ) q^{49} + ( 1 + 3 \beta_{3} - 2 \beta_{4} - \beta_{6} ) q^{51} + ( 3 \beta_{1} - \beta_{2} - \beta_{5} - \beta_{7} ) q^{52} + ( 4 \beta_{3} - 2 \beta_{6} ) q^{53} + ( -5 \beta_{1} + 3 \beta_{2} + \beta_{5} ) q^{54} + ( 3 \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{5} + \beta_{6} ) q^{56} + ( 7 - 3 \beta_{3} - 2 \beta_{4} - \beta_{6} ) q^{57} + ( -3 - \beta_{4} ) q^{58} + ( 3 \beta_{1} + \beta_{2} + 2 \beta_{5} ) q^{59} + ( \beta_{1} + \beta_{2} - \beta_{5} - \beta_{7} ) q^{61} + ( 3 \beta_{1} + \beta_{2} + 2 \beta_{5} ) q^{62} + ( -\beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{7} ) q^{63} + ( 3 - 4 \beta_{4} ) q^{64} + ( -5 \beta_{1} + 3 \beta_{2} + 3 \beta_{7} ) q^{66} + ( 7 - 4 \beta_{4} ) q^{67} + ( -3 \beta_{1} - \beta_{2} - 4 \beta_{5} + 2 \beta_{7} ) q^{68} + ( -\beta_{1} + 2 \beta_{2} - \beta_{5} - \beta_{7} ) q^{69} + ( \beta_{3} - \beta_{6} ) q^{71} + ( 5 + 4 \beta_{3} - 3 \beta_{4} + \beta_{6} ) q^{72} + ( \beta_{1} + 3 \beta_{2} - 2 \beta_{5} - 2 \beta_{7} ) q^{73} + ( -\beta_{3} - 2 \beta_{6} ) q^{74} + ( 9 \beta_{1} - 5 \beta_{2} - 2 \beta_{5} - 2 \beta_{7} ) q^{76} + ( -3 \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{5} - \beta_{6} ) q^{77} + ( 1 + 3 \beta_{3} - 2 \beta_{4} - \beta_{6} ) q^{78} + ( -5 + 2 \beta_{4} ) q^{79} + ( 1 - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{6} ) q^{81} + ( -3 \beta_{1} - \beta_{2} + 2 \beta_{5} + 2 \beta_{7} ) q^{82} + ( -3 \beta_{1} - \beta_{2} + \beta_{5} - 3 \beta_{7} ) q^{83} + ( 9 - 2 \beta_{1} + \beta_{2} - 3 \beta_{3} - 3 \beta_{4} + \beta_{7} ) q^{84} + ( -\beta_{3} + 2 \beta_{6} ) q^{86} + ( -\beta_{1} + 2 \beta_{2} - \beta_{5} - \beta_{7} ) q^{87} + ( 15 - 2 \beta_{4} ) q^{88} + ( -3 \beta_{1} - \beta_{2} - 2 \beta_{5} ) q^{89} + ( 6 - \beta_{1} + \beta_{2} ) q^{91} + \beta_{6} q^{92} + ( 5 - 3 \beta_{3} - \beta_{4} - 2 \beta_{6} ) q^{93} + ( 5 \beta_{1} - 3 \beta_{2} - \beta_{5} - \beta_{7} ) q^{94} + ( -\beta_{1} + 2 \beta_{2} - \beta_{7} ) q^{96} + ( \beta_{1} - 3 \beta_{2} + \beta_{5} + \beta_{7} ) q^{97} + ( -5 \beta_{3} - 2 \beta_{5} + 2 \beta_{7} ) q^{98} + ( -5 - 4 \beta_{3} + 3 \beta_{4} - \beta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 12q^{4} + 8q^{7} + 4q^{9} + O(q^{10}) \) \( 8q - 12q^{4} + 8q^{7} + 4q^{9} + 4q^{16} - 28q^{18} - 12q^{21} - 28q^{22} - 12q^{28} + 36q^{36} - 32q^{37} + 12q^{39} + 16q^{43} - 28q^{46} - 40q^{49} + 48q^{57} - 28q^{58} + 4q^{63} + 8q^{64} + 40q^{67} + 28q^{72} - 32q^{79} + 16q^{81} + 60q^{84} + 112q^{88} + 48q^{91} + 36q^{93} - 28q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 2 x^{6} - 2 x^{4} - 18 x^{2} + 81\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{7} + 7 \nu^{5} + 7 \nu^{3} - 9 \nu \)\()/54\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{6} - 2 \nu^{4} + 7 \nu^{2} - 18 \)\()/18\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{6} + 2 \nu^{4} + 11 \nu^{2} + 18 \)\()/18\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{7} + 2 \nu^{5} - 7 \nu^{3} + 18 \nu \)\()/18\)
\(\beta_{6}\)\(=\)\((\)\( 2 \nu^{6} + 5 \nu^{4} - 4 \nu^{2} - 45 \)\()/18\)
\(\beta_{7}\)\(=\)\((\)\( -4 \nu^{7} - \nu^{5} + 26 \nu^{3} + 63 \nu \)\()/54\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{4} + \beta_{3}\)
\(\nu^{3}\)\(=\)\(\beta_{7} - \beta_{5} + \beta_{2}\)
\(\nu^{4}\)\(=\)\(2 \beta_{6} + 2 \beta_{4} - 2 \beta_{3} + 1\)
\(\nu^{5}\)\(=\)\(2 \beta_{5} + 6 \beta_{2} - \beta_{1}\)
\(\nu^{6}\)\(=\)\(4 \beta_{6} - 3 \beta_{4} + 7 \beta_{3} + 20\)
\(\nu^{7}\)\(=\)\(-7 \beta_{7} - 7 \beta_{5} + 5 \beta_{2} + 16 \beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
251.1
−0.777403 + 1.54779i
0.777403 1.54779i
−1.70166 + 0.323042i
1.70166 0.323042i
−1.70166 0.323042i
1.70166 + 0.323042i
−0.777403 1.54779i
0.777403 + 1.54779i
2.40651i −0.777403 + 1.54779i −3.79129 0 3.72476 + 1.87083i 1.00000 + 2.44949i 4.31075i −1.79129 2.40651i 0
251.2 2.40651i 0.777403 1.54779i −3.79129 0 −3.72476 1.87083i 1.00000 2.44949i 4.31075i −1.79129 2.40651i 0
251.3 1.09941i −1.70166 + 0.323042i 0.791288 0 0.355157 + 1.87083i 1.00000 2.44949i 3.06878i 2.79129 1.09941i 0
251.4 1.09941i 1.70166 0.323042i 0.791288 0 −0.355157 1.87083i 1.00000 + 2.44949i 3.06878i 2.79129 1.09941i 0
251.5 1.09941i −1.70166 0.323042i 0.791288 0 0.355157 1.87083i 1.00000 + 2.44949i 3.06878i 2.79129 + 1.09941i 0
251.6 1.09941i 1.70166 + 0.323042i 0.791288 0 −0.355157 + 1.87083i 1.00000 2.44949i 3.06878i 2.79129 + 1.09941i 0
251.7 2.40651i −0.777403 1.54779i −3.79129 0 3.72476 1.87083i 1.00000 2.44949i 4.31075i −1.79129 + 2.40651i 0
251.8 2.40651i 0.777403 + 1.54779i −3.79129 0 −3.72476 + 1.87083i 1.00000 + 2.44949i 4.31075i −1.79129 + 2.40651i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 251.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.2.b.i yes 8
3.b odd 2 1 inner 525.2.b.i yes 8
5.b even 2 1 525.2.b.h 8
5.c odd 4 2 525.2.g.f 16
7.b odd 2 1 inner 525.2.b.i yes 8
15.d odd 2 1 525.2.b.h 8
15.e even 4 2 525.2.g.f 16
21.c even 2 1 inner 525.2.b.i yes 8
35.c odd 2 1 525.2.b.h 8
35.f even 4 2 525.2.g.f 16
105.g even 2 1 525.2.b.h 8
105.k odd 4 2 525.2.g.f 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.2.b.h 8 5.b even 2 1
525.2.b.h 8 15.d odd 2 1
525.2.b.h 8 35.c odd 2 1
525.2.b.h 8 105.g even 2 1
525.2.b.i yes 8 1.a even 1 1 trivial
525.2.b.i yes 8 3.b odd 2 1 inner
525.2.b.i yes 8 7.b odd 2 1 inner
525.2.b.i yes 8 21.c even 2 1 inner
525.2.g.f 16 5.c odd 4 2
525.2.g.f 16 15.e even 4 2
525.2.g.f 16 35.f even 4 2
525.2.g.f 16 105.k odd 4 2

Hecke kernels

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

\( T_{2}^{4} + 7 T_{2}^{2} + 7 \)
\( T_{11}^{4} + 28 T_{11}^{2} + 175 \)
\( T_{17}^{4} - 42 T_{17}^{2} + 252 \)
\( T_{37}^{2} + 8 T_{37} - 5 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T^{2} + 3 T^{4} - 4 T^{6} + 16 T^{8} )^{2} \)
$3$ \( 1 - 2 T^{2} - 2 T^{4} - 18 T^{6} + 81 T^{8} \)
$5$ \( \)
$7$ \( ( 1 - 2 T + 7 T^{2} )^{4} \)
$11$ \( ( 1 - 16 T^{2} + 285 T^{4} - 1936 T^{6} + 14641 T^{8} )^{2} \)
$13$ \( ( 1 - 20 T^{2} + 169 T^{4} )^{4} \)
$17$ \( ( 1 + 26 T^{2} + 558 T^{4} + 7514 T^{6} + 83521 T^{8} )^{2} \)
$19$ \( ( 1 - 10 T^{2} + 558 T^{4} - 3610 T^{6} + 130321 T^{8} )^{2} \)
$23$ \( ( 1 - 64 T^{2} + 1893 T^{4} - 33856 T^{6} + 279841 T^{8} )^{2} \)
$29$ \( ( 1 - 88 T^{2} + 3429 T^{4} - 74008 T^{6} + 707281 T^{8} )^{2} \)
$31$ \( ( 1 - 58 T^{2} + 2574 T^{4} - 55738 T^{6} + 923521 T^{8} )^{2} \)
$37$ \( ( 1 + 8 T + 69 T^{2} + 296 T^{3} + 1369 T^{4} )^{4} \)
$41$ \( ( 1 + 38 T^{2} + 2022 T^{4} + 63878 T^{6} + 2825761 T^{8} )^{2} \)
$43$ \( ( 1 - 4 T + 69 T^{2} - 172 T^{3} + 1849 T^{4} )^{4} \)
$47$ \( ( 1 + 146 T^{2} + 9558 T^{4} + 322514 T^{6} + 4879681 T^{8} )^{2} \)
$53$ \( ( 1 - 16 T^{2} - 1122 T^{4} - 44944 T^{6} + 7890481 T^{8} )^{2} \)
$59$ \( ( 1 + 68 T^{2} + 7362 T^{4} + 236708 T^{6} + 12117361 T^{8} )^{2} \)
$61$ \( ( 1 - 178 T^{2} + 15174 T^{4} - 662338 T^{6} + 13845841 T^{8} )^{2} \)
$67$ \( ( 1 - 10 T + 75 T^{2} - 670 T^{3} + 4489 T^{4} )^{4} \)
$71$ \( ( 1 - 256 T^{2} + 26445 T^{4} - 1290496 T^{6} + 25411681 T^{8} )^{2} \)
$73$ \( ( 1 + 8 T^{2} - 1422 T^{4} + 42632 T^{6} + 28398241 T^{8} )^{2} \)
$79$ \( ( 1 + 8 T + 153 T^{2} + 632 T^{3} + 6241 T^{4} )^{4} \)
$83$ \( ( 1 + 38 T^{2} + 4878 T^{4} + 261782 T^{6} + 47458321 T^{8} )^{2} \)
$89$ \( ( 1 + 188 T^{2} + 23922 T^{4} + 1489148 T^{6} + 62742241 T^{8} )^{2} \)
$97$ \( ( 1 - 250 T^{2} + 29718 T^{4} - 2352250 T^{6} + 88529281 T^{8} )^{2} \)
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