Properties

Label 525.2.bf.e
Level 525
Weight 2
Character orbit 525.bf
Analytic conductor 4.192
Analytic rank 0
Dimension 16
CM no
Inner twists 16

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Newspace parameters

Level: \( N \) = \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 525.bf (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.19214610612\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{12})\)
Coefficient field: 16.0.63456228123711897600000000.4
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \beta_{5} + \beta_{6} + \beta_{9} ) q^{2} + \beta_{15} q^{3} -3 \beta_{7} q^{4} + ( 2 + \beta_{8} ) q^{6} + ( \beta_{1} + \beta_{9} ) q^{7} + ( \beta_{10} + \beta_{14} ) q^{8} + ( \beta_{2} - \beta_{11} ) q^{9} +O(q^{10})\) \( q + ( \beta_{5} + \beta_{6} + \beta_{9} ) q^{2} + \beta_{15} q^{3} -3 \beta_{7} q^{4} + ( 2 + \beta_{8} ) q^{6} + ( \beta_{1} + \beta_{9} ) q^{7} + ( \beta_{10} + \beta_{14} ) q^{8} + ( \beta_{2} - \beta_{11} ) q^{9} + ( 3 \beta_{6} + 3 \beta_{9} ) q^{12} + ( -2 \beta_{10} + 2 \beta_{14} ) q^{13} + ( -2 \beta_{2} + \beta_{7} + \beta_{11} + 2 \beta_{13} ) q^{14} + \beta_{3} q^{16} + ( -2 \beta_{1} + 2 \beta_{9} ) q^{17} + ( 3 \beta_{12} - 3 \beta_{14} + 2 \beta_{15} ) q^{18} -2 \beta_{11} q^{19} + ( 4 \beta_{3} + \beta_{4} ) q^{21} + ( -3 \beta_{10} - 3 \beta_{12} + 3 \beta_{15} ) q^{23} + ( -2 \beta_{7} + \beta_{13} ) q^{24} + ( -2 \beta_{3} - 4 \beta_{4} ) q^{26} + ( -3 \beta_{1} - 3 \beta_{5} - \beta_{6} ) q^{27} + ( 3 \beta_{10} - 3 \beta_{12} - 3 \beta_{15} ) q^{28} + ( -2 \beta_{2} + \beta_{7} + \beta_{11} + 2 \beta_{13} ) q^{29} + ( -2 - 2 \beta_{3} ) q^{31} + ( 3 \beta_{1} - 3 \beta_{9} ) q^{32} + ( -10 \beta_{7} - 10 \beta_{11} ) q^{34} + ( -3 + 3 \beta_{8} ) q^{36} + ( 4 \beta_{5} - 4 \beta_{6} - 4 \beta_{9} ) q^{37} + ( 2 \beta_{12} - 2 \beta_{14} - 2 \beta_{15} ) q^{38} + ( -8 \beta_{7} - 2 \beta_{13} ) q^{39} + ( 1 - 2 \beta_{8} ) q^{41} + ( 6 \beta_{1} - \beta_{9} ) q^{42} + ( -\beta_{10} + \beta_{14} ) q^{43} + ( 15 + 15 \beta_{3} ) q^{46} + ( -4 \beta_{5} - 4 \beta_{6} - 4 \beta_{9} ) q^{47} -\beta_{10} q^{48} + 7 \beta_{11} q^{49} + ( -4 \beta_{3} + 2 \beta_{4} ) q^{51} + ( -6 \beta_{1} - 6 \beta_{9} ) q^{52} + ( 2 \beta_{12} - 2 \beta_{14} - 2 \beta_{15} ) q^{53} + ( 2 \beta_{2} - 11 \beta_{11} ) q^{54} + ( 1 + \beta_{3} + 2 \beta_{4} - 2 \beta_{8} ) q^{56} -2 \beta_{6} q^{57} + ( 5 \beta_{10} - 5 \beta_{12} - 5 \beta_{15} ) q^{58} + ( 2 \beta_{7} + 4 \beta_{13} ) q^{59} -11 \beta_{3} q^{61} + ( -2 \beta_{1} - 2 \beta_{5} - 2 \beta_{6} ) q^{62} + ( -4 \beta_{10} - 3 \beta_{14} ) q^{63} + ( 13 \beta_{7} + 13 \beta_{11} ) q^{64} + ( -\beta_{1} - \beta_{9} ) q^{67} + ( 6 \beta_{10} + 6 \beta_{12} - 6 \beta_{15} ) q^{68} + ( 3 \beta_{2} + 6 \beta_{7} + 6 \beta_{11} - 3 \beta_{13} ) q^{69} + ( -2 + 4 \beta_{8} ) q^{71} + ( -3 \beta_{5} + 2 \beta_{6} + 2 \beta_{9} ) q^{72} + ( 2 \beta_{12} - 2 \beta_{14} + 2 \beta_{15} ) q^{73} + ( -4 \beta_{7} - 8 \beta_{13} ) q^{74} -6 q^{76} + ( 2 \beta_{10} + 12 \beta_{14} ) q^{78} -2 \beta_{11} q^{79} + ( -8 - 8 \beta_{3} + \beta_{4} - \beta_{8} ) q^{81} + ( 5 \beta_{5} - 5 \beta_{6} - 5 \beta_{9} ) q^{82} + ( \beta_{10} + \beta_{14} ) q^{83} + ( -3 \beta_{2} + 12 \beta_{7} + 12 \beta_{11} + 3 \beta_{13} ) q^{84} + ( -\beta_{3} - 2 \beta_{4} ) q^{86} + ( 6 \beta_{1} - \beta_{9} ) q^{87} + ( 2 \beta_{2} - \beta_{11} ) q^{89} + ( -14 - 14 \beta_{3} ) q^{91} + ( 9 \beta_{1} + 9 \beta_{5} + 9 \beta_{6} ) q^{92} + ( 2 \beta_{10} - 2 \beta_{15} ) q^{93} + 20 \beta_{7} q^{94} + ( 6 \beta_{3} - 3 \beta_{4} ) q^{96} + ( -4 \beta_{1} - 4 \beta_{5} + 4 \beta_{6} ) q^{97} + ( -7 \beta_{12} + 7 \beta_{14} + 7 \beta_{15} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 40q^{6} + O(q^{10}) \) \( 16q + 40q^{6} - 8q^{16} - 28q^{21} - 16q^{31} - 24q^{36} + 120q^{46} + 40q^{51} + 88q^{61} - 96q^{76} - 68q^{81} - 112q^{91} - 60q^{96} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} + 17 x^{12} + 208 x^{8} + 1377 x^{4} + 6561\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)
\(\beta_{3}\)\(=\)\((\)\( -17 \nu^{12} - 208 \nu^{8} - 3536 \nu^{4} - 23409 \)\()/16848\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{12} + 26 \nu^{8} + 208 \nu^{4} + 1377 \)\()/234\)
\(\beta_{5}\)\(=\)\((\)\( 17 \nu^{13} + 208 \nu^{9} + 3536 \nu^{5} + 6561 \nu \)\()/16848\)
\(\beta_{6}\)\(=\)\((\)\( 55 \nu^{13} + 1664 \nu^{9} + 11440 \nu^{5} + 75735 \nu \)\()/50544\)
\(\beta_{7}\)\(=\)\((\)\( 55 \nu^{14} + 1664 \nu^{10} + 11440 \nu^{6} + 75735 \nu^{2} \)\()/151632\)
\(\beta_{8}\)\(=\)\((\)\( -\nu^{12} + 495 \)\()/208\)
\(\beta_{9}\)\(=\)\((\)\( \nu^{13} - 287 \nu \)\()/624\)
\(\beta_{10}\)\(=\)\((\)\( \nu^{15} - 2159 \nu^{3} \)\()/5616\)
\(\beta_{11}\)\(=\)\((\)\( \nu^{14} - 287 \nu^{2} \)\()/1872\)
\(\beta_{12}\)\(=\)\((\)\( 55 \nu^{15} + 1664 \nu^{11} + 11440 \nu^{7} + 75735 \nu^{3} \)\()/151632\)
\(\beta_{13}\)\(=\)\((\)\( 17 \nu^{14} + 208 \nu^{10} + 3536 \nu^{6} + 23409 \nu^{2} \)\()/16848\)
\(\beta_{14}\)\(=\)\((\)\( -\nu^{15} + 287 \nu^{3} \)\()/1872\)
\(\beta_{15}\)\(=\)\((\)\( 289 \nu^{15} + 3536 \nu^{11} + 43264 \nu^{7} + 111537 \nu^{3} \)\()/454896\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\)
\(\nu^{3}\)\(=\)\(-\beta_{14} - 3 \beta_{10}\)
\(\nu^{4}\)\(=\)\(\beta_{8} - \beta_{4} - 9 \beta_{3} - 9\)
\(\nu^{5}\)\(=\)\(-3 \beta_{9} - 3 \beta_{6} + 8 \beta_{5}\)
\(\nu^{6}\)\(=\)\(8 \beta_{13} - 9 \beta_{11} - 9 \beta_{7} - 8 \beta_{2}\)
\(\nu^{7}\)\(=\)\(24 \beta_{15} + 17 \beta_{14} - 17 \beta_{12}\)
\(\nu^{8}\)\(=\)\(17 \beta_{4} + 72 \beta_{3}\)
\(\nu^{9}\)\(=\)\(51 \beta_{6} - 55 \beta_{5} - 55 \beta_{1}\)
\(\nu^{10}\)\(=\)\(-55 \beta_{13} + 153 \beta_{7}\)
\(\nu^{11}\)\(=\)\(-165 \beta_{15} + 208 \beta_{12} + 165 \beta_{10}\)
\(\nu^{12}\)\(=\)\(-208 \beta_{8} + 495\)
\(\nu^{13}\)\(=\)\(624 \beta_{9} + 287 \beta_{1}\)
\(\nu^{14}\)\(=\)\(1872 \beta_{11} + 287 \beta_{2}\)
\(\nu^{15}\)\(=\)\(-2159 \beta_{14} - 861 \beta_{10}\)

Character values

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

\(n\) \(127\) \(176\) \(451\)
\(\chi(n)\) \(-\beta_{7} - \beta_{11}\) \(-1\) \(\beta_{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
32.1
1.56717 + 0.737552i
−0.988431 + 1.42232i
0.988431 1.42232i
−1.56717 0.737552i
1.42232 0.988431i
0.737552 + 1.56717i
−0.737552 1.56717i
−1.42232 + 0.988431i
1.42232 + 0.988431i
0.737552 1.56717i
−0.737552 + 1.56717i
−1.42232 0.988431i
1.56717 0.737552i
−0.988431 1.42232i
0.988431 + 1.42232i
−1.56717 + 0.737552i
−2.15988 0.578737i −1.42232 0.988431i 2.59808 + 1.50000i 0 2.50000 + 2.95804i 2.55560 0.684771i −1.58114 1.58114i 1.04601 + 2.81174i 0
32.2 −2.15988 0.578737i −0.737552 + 1.56717i 2.59808 + 1.50000i 0 2.50000 2.95804i −2.55560 + 0.684771i −1.58114 1.58114i −1.91203 2.31174i 0
32.3 2.15988 + 0.578737i 0.737552 1.56717i 2.59808 + 1.50000i 0 2.50000 2.95804i 2.55560 0.684771i 1.58114 + 1.58114i −1.91203 2.31174i 0
32.4 2.15988 + 0.578737i 1.42232 + 0.988431i 2.59808 + 1.50000i 0 2.50000 + 2.95804i −2.55560 + 0.684771i 1.58114 + 1.58114i 1.04601 + 2.81174i 0
107.1 −0.578737 2.15988i −1.56717 + 0.737552i −2.59808 + 1.50000i 0 2.50000 + 2.95804i 0.684771 2.55560i 1.58114 + 1.58114i 1.91203 2.31174i 0
107.2 −0.578737 2.15988i 0.988431 + 1.42232i −2.59808 + 1.50000i 0 2.50000 2.95804i −0.684771 + 2.55560i 1.58114 + 1.58114i −1.04601 + 2.81174i 0
107.3 0.578737 + 2.15988i −0.988431 1.42232i −2.59808 + 1.50000i 0 2.50000 2.95804i 0.684771 2.55560i −1.58114 1.58114i −1.04601 + 2.81174i 0
107.4 0.578737 + 2.15988i 1.56717 0.737552i −2.59808 + 1.50000i 0 2.50000 + 2.95804i −0.684771 + 2.55560i −1.58114 1.58114i 1.91203 2.31174i 0
368.1 −0.578737 + 2.15988i −1.56717 0.737552i −2.59808 1.50000i 0 2.50000 2.95804i 0.684771 + 2.55560i 1.58114 1.58114i 1.91203 + 2.31174i 0
368.2 −0.578737 + 2.15988i 0.988431 1.42232i −2.59808 1.50000i 0 2.50000 + 2.95804i −0.684771 2.55560i 1.58114 1.58114i −1.04601 2.81174i 0
368.3 0.578737 2.15988i −0.988431 + 1.42232i −2.59808 1.50000i 0 2.50000 + 2.95804i 0.684771 + 2.55560i −1.58114 + 1.58114i −1.04601 2.81174i 0
368.4 0.578737 2.15988i 1.56717 + 0.737552i −2.59808 1.50000i 0 2.50000 2.95804i −0.684771 2.55560i −1.58114 + 1.58114i 1.91203 + 2.31174i 0
443.1 −2.15988 + 0.578737i −1.42232 + 0.988431i 2.59808 1.50000i 0 2.50000 2.95804i 2.55560 + 0.684771i −1.58114 + 1.58114i 1.04601 2.81174i 0
443.2 −2.15988 + 0.578737i −0.737552 1.56717i 2.59808 1.50000i 0 2.50000 + 2.95804i −2.55560 0.684771i −1.58114 + 1.58114i −1.91203 + 2.31174i 0
443.3 2.15988 0.578737i 0.737552 + 1.56717i 2.59808 1.50000i 0 2.50000 + 2.95804i 2.55560 + 0.684771i 1.58114 1.58114i −1.91203 + 2.31174i 0
443.4 2.15988 0.578737i 1.42232 0.988431i 2.59808 1.50000i 0 2.50000 2.95804i −2.55560 0.684771i 1.58114 1.58114i 1.04601 2.81174i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 443.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
5.c odd 4 2 inner
7.c even 3 1 inner
15.d odd 2 1 inner
15.e even 4 2 inner
21.h odd 6 1 inner
35.j even 6 1 inner
35.l odd 12 2 inner
105.o odd 6 1 inner
105.x even 12 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.2.bf.e 16
3.b odd 2 1 inner 525.2.bf.e 16
5.b even 2 1 inner 525.2.bf.e 16
5.c odd 4 2 inner 525.2.bf.e 16
7.c even 3 1 inner 525.2.bf.e 16
15.d odd 2 1 inner 525.2.bf.e 16
15.e even 4 2 inner 525.2.bf.e 16
21.h odd 6 1 inner 525.2.bf.e 16
35.j even 6 1 inner 525.2.bf.e 16
35.l odd 12 2 inner 525.2.bf.e 16
105.o odd 6 1 inner 525.2.bf.e 16
105.x even 12 2 inner 525.2.bf.e 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.2.bf.e 16 1.a even 1 1 trivial
525.2.bf.e 16 3.b odd 2 1 inner
525.2.bf.e 16 5.b even 2 1 inner
525.2.bf.e 16 5.c odd 4 2 inner
525.2.bf.e 16 7.c even 3 1 inner
525.2.bf.e 16 15.d odd 2 1 inner
525.2.bf.e 16 15.e even 4 2 inner
525.2.bf.e 16 21.h odd 6 1 inner
525.2.bf.e 16 35.j even 6 1 inner
525.2.bf.e 16 35.l odd 12 2 inner
525.2.bf.e 16 105.o odd 6 1 inner
525.2.bf.e 16 105.x even 12 2 inner

Hecke kernels

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

\( T_{2}^{8} - 25 T_{2}^{4} + 625 \)
\( T_{13}^{4} + 784 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 7 T^{4} + 33 T^{8} + 112 T^{12} + 256 T^{16} )^{2} \)
$3$ \( 1 + 17 T^{4} + 208 T^{8} + 1377 T^{12} + 6561 T^{16} \)
$5$ \( \)
$7$ \( ( 1 - 49 T^{4} + 2401 T^{8} )^{2} \)
$11$ \( ( 1 + 11 T^{2} + 121 T^{4} )^{8} \)
$13$ \( ( 1 - 334 T^{4} + 28561 T^{8} )^{4} \)
$17$ \( ( 1 + 382 T^{4} + 62403 T^{8} + 31905022 T^{12} + 6975757441 T^{16} )^{2} \)
$19$ \( ( 1 + 34 T^{2} + 795 T^{4} + 12274 T^{6} + 130321 T^{8} )^{4} \)
$23$ \( ( 1 + 1057 T^{4} + 837408 T^{8} + 295791937 T^{12} + 78310985281 T^{16} )^{2} \)
$29$ \( ( 1 + 23 T^{2} + 841 T^{4} )^{8} \)
$31$ \( ( 1 + 2 T - 27 T^{2} + 62 T^{3} + 961 T^{4} )^{8} \)
$37$ \( ( 1 + 1294 T^{4} - 199725 T^{8} + 2425164334 T^{12} + 3512479453921 T^{16} )^{2} \)
$41$ \( ( 1 - 47 T^{2} + 1681 T^{4} )^{8} \)
$43$ \( ( 1 + 2543 T^{4} + 3418801 T^{8} )^{4} \)
$47$ \( ( 1 + 4222 T^{4} + 12945603 T^{8} + 20602013182 T^{12} + 23811286661761 T^{16} )^{2} \)
$53$ \( ( 1 - 1778 T^{4} - 4729197 T^{8} - 14029275218 T^{12} + 62259690411361 T^{16} )^{2} \)
$59$ \( ( 1 + 22 T^{2} - 2997 T^{4} + 76582 T^{6} + 12117361 T^{8} )^{4} \)
$61$ \( ( 1 - 11 T + 60 T^{2} - 671 T^{3} + 3721 T^{4} )^{8} \)
$67$ \( ( 1 - 7151 T^{4} + 30985680 T^{8} - 144100666271 T^{12} + 406067677556641 T^{16} )^{2} \)
$71$ \( ( 1 - 12 T + 71 T^{2} )^{8}( 1 + 12 T + 71 T^{2} )^{8} \)
$73$ \( ( 1 - 3266 T^{4} - 17731485 T^{8} - 92748655106 T^{12} + 806460091894081 T^{16} )^{2} \)
$79$ \( ( 1 + 154 T^{2} + 17475 T^{4} + 961114 T^{6} + 38950081 T^{8} )^{4} \)
$83$ \( ( 1 + 12143 T^{4} + 47458321 T^{8} )^{4} \)
$89$ \( ( 1 - 143 T^{2} + 12528 T^{4} - 1132703 T^{6} + 62742241 T^{8} )^{4} \)
$97$ \( ( 1 - 12094 T^{4} + 88529281 T^{8} )^{4} \)
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