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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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [525,2,Mod(32,525)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(525, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 3, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("525.32");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
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

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
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
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 17x^{12} + 208x^{8} + 1377x^{4} + 6561 \) Copy content Toggle raw display
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

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

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.

\(f(q)\) \(=\) \( q + (\beta_{9} + \beta_{6} + \beta_{5}) q^{2} + \beta_{15} q^{3} - 3 \beta_{7} q^{4} + (\beta_{8} + 2) q^{6} + (\beta_{9} + \beta_1) q^{7} + (\beta_{14} + \beta_{10}) q^{8} + ( - \beta_{11} + \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{9} + \beta_{6} + \beta_{5}) q^{2} + \beta_{15} q^{3} - 3 \beta_{7} q^{4} + (\beta_{8} + 2) q^{6} + (\beta_{9} + \beta_1) q^{7} + (\beta_{14} + \beta_{10}) q^{8} + ( - \beta_{11} + \beta_{2}) q^{9} + (3 \beta_{9} + 3 \beta_{6}) q^{12} + (2 \beta_{14} - 2 \beta_{10}) q^{13} + (2 \beta_{13} + \beta_{11} + \beta_{7} - 2 \beta_{2}) q^{14} + \beta_{3} q^{16} + (2 \beta_{9} - 2 \beta_1) q^{17} + (2 \beta_{15} - 3 \beta_{14} + 3 \beta_{12}) q^{18} - 2 \beta_{11} q^{19} + (\beta_{4} + 4 \beta_{3}) q^{21} + (3 \beta_{15} - 3 \beta_{12} - 3 \beta_{10}) q^{23} + (\beta_{13} - 2 \beta_{7}) q^{24} + ( - 4 \beta_{4} - 2 \beta_{3}) q^{26} + ( - \beta_{6} - 3 \beta_{5} - 3 \beta_1) q^{27} + ( - 3 \beta_{15} - 3 \beta_{12} + 3 \beta_{10}) q^{28} + (2 \beta_{13} + \beta_{11} + \beta_{7} - 2 \beta_{2}) q^{29} + ( - 2 \beta_{3} - 2) q^{31} + ( - 3 \beta_{9} + 3 \beta_1) q^{32} + ( - 10 \beta_{11} - 10 \beta_{7}) q^{34} + (3 \beta_{8} - 3) q^{36} + ( - 4 \beta_{9} - 4 \beta_{6} + 4 \beta_{5}) q^{37} + ( - 2 \beta_{15} - 2 \beta_{14} + 2 \beta_{12}) q^{38} + ( - 2 \beta_{13} - 8 \beta_{7}) q^{39} + ( - 2 \beta_{8} + 1) q^{41} + ( - \beta_{9} + 6 \beta_1) q^{42} + (\beta_{14} - \beta_{10}) q^{43} + (15 \beta_{3} + 15) q^{46} + ( - 4 \beta_{9} - 4 \beta_{6} - 4 \beta_{5}) q^{47} - \beta_{10} q^{48} + 7 \beta_{11} q^{49} + (2 \beta_{4} - 4 \beta_{3}) q^{51} + ( - 6 \beta_{9} - 6 \beta_1) q^{52} + ( - 2 \beta_{15} - 2 \beta_{14} + 2 \beta_{12}) q^{53} + ( - 11 \beta_{11} + 2 \beta_{2}) q^{54} + ( - 2 \beta_{8} + 2 \beta_{4} + \beta_{3} + 1) q^{56} - 2 \beta_{6} q^{57} + ( - 5 \beta_{15} - 5 \beta_{12} + 5 \beta_{10}) q^{58} + (4 \beta_{13} + 2 \beta_{7}) q^{59} - 11 \beta_{3} q^{61} + ( - 2 \beta_{6} - 2 \beta_{5} - 2 \beta_1) q^{62} + ( - 3 \beta_{14} - 4 \beta_{10}) q^{63} + (13 \beta_{11} + 13 \beta_{7}) q^{64} + ( - \beta_{9} - \beta_1) q^{67} + ( - 6 \beta_{15} + 6 \beta_{12} + 6 \beta_{10}) q^{68} + ( - 3 \beta_{13} + 6 \beta_{11} + 6 \beta_{7} + 3 \beta_{2}) q^{69} + (4 \beta_{8} - 2) q^{71} + (2 \beta_{9} + 2 \beta_{6} - 3 \beta_{5}) q^{72} + (2 \beta_{15} - 2 \beta_{14} + 2 \beta_{12}) q^{73} + ( - 8 \beta_{13} - 4 \beta_{7}) q^{74} - 6 q^{76} + (12 \beta_{14} + 2 \beta_{10}) q^{78} - 2 \beta_{11} q^{79} + ( - \beta_{8} + \beta_{4} - 8 \beta_{3} - 8) q^{81} + ( - 5 \beta_{9} - 5 \beta_{6} + 5 \beta_{5}) q^{82} + (\beta_{14} + \beta_{10}) q^{83} + (3 \beta_{13} + 12 \beta_{11} + 12 \beta_{7} - 3 \beta_{2}) q^{84} + ( - 2 \beta_{4} - \beta_{3}) q^{86} + ( - \beta_{9} + 6 \beta_1) q^{87} + ( - \beta_{11} + 2 \beta_{2}) q^{89} + ( - 14 \beta_{3} - 14) q^{91} + (9 \beta_{6} + 9 \beta_{5} + 9 \beta_1) q^{92} + ( - 2 \beta_{15} + 2 \beta_{10}) q^{93} + 20 \beta_{7} q^{94} + ( - 3 \beta_{4} + 6 \beta_{3}) q^{96} + (4 \beta_{6} - 4 \beta_{5} - 4 \beta_1) q^{97} + (7 \beta_{15} + 7 \beta_{14} - 7 \beta_{12}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 40 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 40 q^{6} - 8 q^{16} - 28 q^{21} - 16 q^{31} - 24 q^{36} + 120 q^{46} + 40 q^{51} + 88 q^{61} - 96 q^{76} - 68 q^{81} - 112 q^{91} - 60 q^{96}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -17\nu^{12} - 208\nu^{8} - 3536\nu^{4} - 23409 ) / 16848 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{12} + 26\nu^{8} + 208\nu^{4} + 1377 ) / 234 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 17\nu^{13} + 208\nu^{9} + 3536\nu^{5} + 6561\nu ) / 16848 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 55\nu^{13} + 1664\nu^{9} + 11440\nu^{5} + 75735\nu ) / 50544 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 55\nu^{14} + 1664\nu^{10} + 11440\nu^{6} + 75735\nu^{2} ) / 151632 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -\nu^{12} + 495 ) / 208 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{13} - 287\nu ) / 624 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( \nu^{15} - 2159\nu^{3} ) / 5616 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( \nu^{14} - 287\nu^{2} ) / 1872 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 55\nu^{15} + 1664\nu^{11} + 11440\nu^{7} + 75735\nu^{3} ) / 151632 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 17\nu^{14} + 208\nu^{10} + 3536\nu^{6} + 23409\nu^{2} ) / 16848 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( -\nu^{15} + 287\nu^{3} ) / 1872 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 289\nu^{15} + 3536\nu^{11} + 43264\nu^{7} + 111537\nu^{3} ) / 454896 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{14} - 3\beta_{10} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{8} - \beta_{4} - 9\beta_{3} - 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -3\beta_{9} - 3\beta_{6} + 8\beta_{5} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 8\beta_{13} - 9\beta_{11} - 9\beta_{7} - 8\beta_{2} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 24\beta_{15} + 17\beta_{14} - 17\beta_{12} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 17\beta_{4} + 72\beta_{3} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 51\beta_{6} - 55\beta_{5} - 55\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -55\beta_{13} + 153\beta_{7} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -165\beta_{15} + 208\beta_{12} + 165\beta_{10} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -208\beta_{8} + 495 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 624\beta_{9} + 287\beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 1872\beta_{11} + 287\beta_{2} \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( -2159\beta_{14} - 861\beta_{10} \) Copy content Toggle raw display

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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 32.4
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 \(S_{2}^{\mathrm{new}}(525, [\chi])\):

\( T_{2}^{8} - 25T_{2}^{4} + 625 \) Copy content Toggle raw display
\( T_{13}^{4} + 784 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - 25 T^{4} + 625)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} + 17 T^{12} + 208 T^{8} + \cdots + 6561 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{8} - 49 T^{4} + 2401)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} \) Copy content Toggle raw display
$13$ \( (T^{4} + 784)^{4} \) Copy content Toggle raw display
$17$ \( (T^{8} - 400 T^{4} + 160000)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 4 T^{2} + 16)^{4} \) Copy content Toggle raw display
$23$ \( (T^{8} - 2025 T^{4} + 4100625)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 35)^{8} \) Copy content Toggle raw display
$31$ \( (T^{2} + 2 T + 4)^{8} \) Copy content Toggle raw display
$37$ \( (T^{8} - 12544 T^{4} + \cdots + 157351936)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 35)^{8} \) Copy content Toggle raw display
$43$ \( (T^{4} + 49)^{4} \) Copy content Toggle raw display
$47$ \( (T^{8} - 6400 T^{4} + 40960000)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} - 400 T^{4} + 160000)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 140 T^{2} + 19600)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} - 11 T + 121)^{8} \) Copy content Toggle raw display
$67$ \( (T^{8} - 49 T^{4} + 2401)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 140)^{8} \) Copy content Toggle raw display
$73$ \( (T^{8} - 784 T^{4} + 614656)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 4 T^{2} + 16)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 25)^{4} \) Copy content Toggle raw display
$89$ \( (T^{4} + 35 T^{2} + 1225)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 12544)^{4} \) Copy content Toggle raw display
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