↑

Properties

Label 525.2.g.f
Level $525$
Weight $2$
Character orbit 525.g
Analytic conductor $4.192$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [525,2,Mod(524,525)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(525, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("525.524");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 525.g (of order \(2\), degree \(1\), not 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: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 32x^{14} + 386x^{12} + 2208x^{10} + 6263x^{8} + 8496x^{6} + 4790x^{4} + 704x^{2} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + \beta_{4} q^{3} + ( - \beta_{5} + 1) q^{4} - \beta_{8} q^{6} + (\beta_{9} - \beta_{6} - \beta_{4}) q^{7} + (\beta_{11} + \beta_{2}) q^{8} + ( - \beta_{13} - \beta_{5} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + \beta_{4} q^{3} + ( - \beta_{5} + 1) q^{4} - \beta_{8} q^{6} + (\beta_{9} - \beta_{6} - \beta_{4}) q^{7} + (\beta_{11} + \beta_{2}) q^{8} + ( - \beta_{13} - \beta_{5} - 1) q^{9} + (\beta_{13} + \beta_{10}) q^{11} + (\beta_{15} + \beta_{6} + 2 \beta_{4}) q^{12} + ( - \beta_{6} - \beta_{4}) q^{13} + ( - \beta_{13} + \beta_{8} + \beta_{3}) q^{14} - \beta_{5} q^{16} + ( - \beta_{15} - \beta_{12}) q^{17} + (\beta_{11} + 4 \beta_{9} + \cdots - \beta_1) q^{18}+ \cdots + (4 \beta_{13} - \beta_{10} - 3 \beta_{5} + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 24 q^{4} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 24 q^{4} - 8 q^{9} + 8 q^{16} - 24 q^{21} + 72 q^{36} - 24 q^{39} - 56 q^{46} + 80 q^{49} - 16 q^{64} + 64 q^{79} + 32 q^{81} - 120 q^{84} + 96 q^{91} + 56 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 32x^{14} + 386x^{12} + 2208x^{10} + 6263x^{8} + 8496x^{6} + 4790x^{4} + 704x^{2} + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 1982 \nu^{15} - 58390 \nu^{13} - 615152 \nu^{11} - 2769307 \nu^{9} - 4960972 \nu^{7} + \cdots - 5898695 \nu ) / 730740 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2111 \nu^{14} + 65570 \nu^{12} + 756456 \nu^{10} + 4045936 \nu^{8} + 10451886 \nu^{6} + \cdots + 1240590 ) / 365370 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 8094 \nu^{15} - 3709 \nu^{14} + 263853 \nu^{13} - 106269 \nu^{12} + 3275277 \nu^{11} + \cdots - 2302010 ) / 1461480 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 8537 \nu^{15} + 7764 \nu^{14} - 271988 \nu^{13} + 254549 \nu^{12} - 3258831 \nu^{11} + \cdots + 4385015 ) / 1461480 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 448 \nu^{14} - 14550 \nu^{12} - 179552 \nu^{10} - 1064513 \nu^{8} - 3188560 \nu^{6} + \cdots - 229922 ) / 36537 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 8537 \nu^{15} - 24142 \nu^{14} + 271988 \nu^{13} - 758087 \nu^{12} + 3258831 \nu^{11} + \cdots - 1164895 ) / 1461480 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 32251 \nu^{15} - 11986 \nu^{14} + 1029006 \nu^{13} - 385689 \nu^{12} + 12354197 \nu^{11} + \cdots - 3943235 ) / 1461480 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 2698 \nu^{15} - 9227 \nu^{14} - 87951 \nu^{13} - 292549 \nu^{12} - 1091759 \nu^{11} + \cdots - 3396160 ) / 487160 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 20394 \nu^{15} + 650497 \nu^{13} + 7806514 \nu^{11} + 44273496 \nu^{9} + 123681686 \nu^{7} + \cdots + 7214029 \nu ) / 730740 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 21827 \nu^{15} + 686544 \nu^{13} + 8051137 \nu^{11} + 43834216 \nu^{9} + 113317871 \nu^{7} + \cdots - 4103552 \nu ) / 730740 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 14023 \nu^{14} + 441570 \nu^{12} + 5194823 \nu^{10} + 28523978 \nu^{8} + 75452983 \nu^{6} + \cdots + 2564660 ) / 365370 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 77004 \nu^{15} - 23717 \nu^{14} + 2472421 \nu^{13} - 760867 \nu^{12} + 29962597 \nu^{11} + \cdots - 7159970 ) / 1461480 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 38677 \nu^{15} - 1235424 \nu^{13} - 14856572 \nu^{11} - 84501056 \nu^{9} - 236911486 \nu^{7} + \cdots - 15379688 \nu ) / 730740 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 88 \nu^{15} + 2799 \nu^{13} + 33437 \nu^{11} + 188138 \nu^{9} + 518065 \nu^{7} + 664072 \nu^{5} + \cdots + 27821 \nu ) / 1140 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 94078 \nu^{15} + 23717 \nu^{14} + 3016397 \nu^{13} + 760867 \nu^{12} + 36480259 \nu^{11} + \cdots + 7159970 ) / 1461480 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( -\beta_{15} - \beta_{14} + \beta_{12} + 6\beta_{9} + \beta_{8} - 2\beta_{7} - \beta_{6} + 3\beta_{4} - \beta_{3} ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{15} - \beta_{14} - \beta_{12} + \beta_{8} + 4\beta_{7} + \beta_{6} + 3\beta_{4} - \beta_{3} + 3\beta_{2} - 12 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 4 \beta_{15} + 5 \beta_{14} - 18 \beta_{13} - 10 \beta_{12} - 60 \beta_{9} - 8 \beta_{8} + \cdots + 8 \beta_{3} ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 12 \beta_{15} + 10 \beta_{14} + 12 \beta_{12} - 16 \beta_{8} - 40 \beta_{7} - 6 \beta_{6} - 3 \beta_{5} + \cdots + 93 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 11 \beta_{15} - 36 \beta_{14} + 240 \beta_{13} + 113 \beta_{12} + 576 \beta_{9} + 81 \beta_{8} + \cdots - 30 \beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 131 \beta_{15} - 94 \beta_{14} - 131 \beta_{12} + 3 \beta_{11} + 187 \beta_{8} + 376 \beta_{7} + \cdots - 840 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 91 \beta_{15} + 387 \beta_{14} - 2730 \beta_{13} - 1253 \beta_{12} - 42 \beta_{10} - 5760 \beta_{9} + \cdots + 588 \beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 468 \beta_{15} + 300 \beta_{14} + 468 \beta_{12} - 40 \beta_{11} - 692 \beta_{8} - 1200 \beta_{7} + \cdots + 2694 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 2534 \beta_{15} - 4945 \beta_{14} + 29952 \beta_{13} + 13780 \beta_{12} + 1152 \beta_{10} + \cdots - 8586 \beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 14956 \beta_{15} - 8827 \beta_{14} - 14956 \beta_{12} + 2541 \beta_{11} + 22828 \beta_{8} + 35308 \beta_{7} + \cdots - 80472 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 39881 \beta_{15} + 64502 \beta_{14} - 325512 \beta_{13} - 151333 \beta_{12} - 21054 \beta_{10} + \cdots + 112068 \beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 159138 \beta_{15} + 88336 \beta_{14} + 159138 \beta_{12} - 41796 \beta_{11} - 250798 \beta_{8} + \cdots + 818847 ) / 3 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 538339 \beta_{15} - 822339 \beta_{14} + 3531840 \beta_{13} + 1663073 \beta_{12} + 321360 \beta_{10} + \cdots - 1383252 \beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 1696163 \beta_{15} - 898177 \beta_{14} - 1696163 \beta_{12} + 602826 \beta_{11} + 2757643 \beta_{8} + \cdots - 8459568 ) / 3 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 6782926 \beta_{15} + 10194465 \beta_{14} - 38360154 \beta_{13} - 18295316 \beta_{12} + \cdots + 16535880 \beta_1 ) / 6 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
524.1
− 3.32519i
3.32519i
− 1.32519i
1.32519i
− 3.02470i
3.02470i
− 1.02470i
1.02470i
− 0.378617i
0.378617i
− 2.37862i
2.37862i
1.77038i
− 1.77038i
− 0.229617i
0.229617i
−2.40651 −1.54779 − 0.777403i 3.79129 0 3.72476 + 1.87083i 2.44949 − 1.00000i −4.31075 1.79129 + 2.40651i 0
524.2 −2.40651 −1.54779 + 0.777403i 3.79129 0 3.72476 − 1.87083i 2.44949 + 1.00000i −4.31075 1.79129 − 2.40651i 0
524.3 −2.40651 1.54779 − 0.777403i 3.79129 0 −3.72476 + 1.87083i −2.44949 + 1.00000i −4.31075 1.79129 − 2.40651i 0
524.4 −2.40651 1.54779 + 0.777403i 3.79129 0 −3.72476 − 1.87083i −2.44949 − 1.00000i −4.31075 1.79129 + 2.40651i 0
524.5 −1.09941 −0.323042 − 1.70166i −0.791288 0 0.355157 + 1.87083i −2.44949 − 1.00000i 3.06878 −2.79129 + 1.09941i 0
524.6 −1.09941 −0.323042 + 1.70166i −0.791288 0 0.355157 − 1.87083i −2.44949 + 1.00000i 3.06878 −2.79129 − 1.09941i 0
524.7 −1.09941 0.323042 − 1.70166i −0.791288 0 −0.355157 + 1.87083i 2.44949 + 1.00000i 3.06878 −2.79129 − 1.09941i 0
524.8 −1.09941 0.323042 + 1.70166i −0.791288 0 −0.355157 − 1.87083i 2.44949 − 1.00000i 3.06878 −2.79129 + 1.09941i 0
524.9 1.09941 −0.323042 − 1.70166i −0.791288 0 −0.355157 − 1.87083i −2.44949 + 1.00000i −3.06878 −2.79129 + 1.09941i 0
524.10 1.09941 −0.323042 + 1.70166i −0.791288 0 −0.355157 + 1.87083i −2.44949 − 1.00000i −3.06878 −2.79129 − 1.09941i 0
524.11 1.09941 0.323042 − 1.70166i −0.791288 0 0.355157 − 1.87083i 2.44949 − 1.00000i −3.06878 −2.79129 − 1.09941i 0
524.12 1.09941 0.323042 + 1.70166i −0.791288 0 0.355157 + 1.87083i 2.44949 + 1.00000i −3.06878 −2.79129 + 1.09941i 0
524.13 2.40651 −1.54779 − 0.777403i 3.79129 0 −3.72476 − 1.87083i 2.44949 + 1.00000i 4.31075 1.79129 + 2.40651i 0
524.14 2.40651 −1.54779 + 0.777403i 3.79129 0 −3.72476 + 1.87083i 2.44949 − 1.00000i 4.31075 1.79129 − 2.40651i 0
524.15 2.40651 1.54779 − 0.777403i 3.79129 0 3.72476 − 1.87083i −2.44949 − 1.00000i 4.31075 1.79129 − 2.40651i 0
524.16 2.40651 1.54779 + 0.777403i 3.79129 0 3.72476 + 1.87083i −2.44949 + 1.00000i 4.31075 1.79129 + 2.40651i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 524.16
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
7.b odd 2 1 inner
15.d odd 2 1 inner
21.c even 2 1 inner
35.c odd 2 1 inner
105.g 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.g.f 16
3.b odd 2 1 inner 525.2.g.f 16
5.b even 2 1 inner 525.2.g.f 16
5.c odd 4 1 525.2.b.h ✓ 8
5.c odd 4 1 525.2.b.i yes 8
7.b odd 2 1 inner 525.2.g.f 16
15.d odd 2 1 inner 525.2.g.f 16
15.e even 4 1 525.2.b.h ✓ 8
15.e even 4 1 525.2.b.i yes 8
21.c even 2 1 inner 525.2.g.f 16
35.c odd 2 1 inner 525.2.g.f 16
35.f even 4 1 525.2.b.h ✓ 8
35.f even 4 1 525.2.b.i yes 8
105.g even 2 1 inner 525.2.g.f 16
105.k odd 4 1 525.2.b.h ✓ 8
105.k odd 4 1 525.2.b.i yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.2.b.h ✓ 8 5.c odd 4 1
525.2.b.h ✓ 8 15.e even 4 1
525.2.b.h ✓ 8 35.f even 4 1
525.2.b.h ✓ 8 105.k odd 4 1
525.2.b.i yes 8 5.c odd 4 1
525.2.b.i yes 8 15.e even 4 1
525.2.b.i yes 8 35.f even 4 1
525.2.b.i yes 8 105.k odd 4 1
525.2.g.f 16 1.a even 1 1 trivial
525.2.g.f 16 3.b odd 2 1 inner
525.2.g.f 16 5.b even 2 1 inner
525.2.g.f 16 7.b odd 2 1 inner
525.2.g.f 16 15.d odd 2 1 inner
525.2.g.f 16 21.c even 2 1 inner
525.2.g.f 16 35.c odd 2 1 inner
525.2.g.f 16 105.g even 2 1 inner

Hecke kernels

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

\( T_{2}^{4} - 7T_{2}^{2} + 7 \) Copy content Toggle raw display
\( T_{41}^{4} - 126T_{41}^{2} + 2268 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 7 T^{2} + 7)^{4} \) Copy content Toggle raw display
$3$ \( (T^{8} + 2 T^{6} - 2 T^{4} + \cdots + 81)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{4} - 10 T^{2} + 49)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} + 28 T^{2} + 175)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} - 6)^{8} \) Copy content Toggle raw display
$17$ \( (T^{4} + 42 T^{2} + 252)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} + 66 T^{2} + 900)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} - 28 T^{2} + 7)^{4} \) Copy content Toggle raw display
$29$ \( (T^{4} + 28 T^{2} + 7)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 66 T^{2} + 900)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + 74 T^{2} + 25)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} - 126 T^{2} + 2268)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 50 T^{2} + 289)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} + 42 T^{2} + 252)^{4} \) Copy content Toggle raw display
$53$ \( (T^{4} - 196 T^{2} + 2800)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} - 168 T^{2} + 6300)^{4} \) Copy content Toggle raw display
$61$ \( (T^{4} + 66 T^{2} + 900)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + 218 T^{2} + 3481)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} + 28 T^{2} + 175)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} - 300 T^{2} + 10404)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - 8 T - 5)^{8} \) Copy content Toggle raw display
$83$ \( (T^{4} + 294 T^{2} + 12348)^{4} \) Copy content Toggle raw display
$89$ \( (T^{4} - 168 T^{2} + 6300)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} - 138 T^{2} + 36)^{4} \) Copy content Toggle raw display
show more
show less