# 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

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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

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