↑

Properties

Label 460.2.i.b
Level $460$
Weight $2$
Character orbit 460.i
Analytic conductor $3.673$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [460,2,Mod(137,460)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(460, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("460.137");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 460 = 2^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 460.i (of order \(4\), degree \(2\), 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: \(3.67311849298\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 18x^{14} + 146x^{12} - 798x^{10} + 3934x^{8} - 19950x^{6} + 91250x^{4} - 281250x^{2} + 390625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{6} q^{3} - \beta_{3} q^{5} + (\beta_{15} - \beta_{3}) q^{7} + (\beta_{12} + 2 \beta_{9} - \beta_{8} + \cdots - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{3} - \beta_{3} q^{5} + (\beta_{15} - \beta_{3}) q^{7} + (\beta_{12} + 2 \beta_{9} - \beta_{8} + \cdots - 1) q^{9}+ \cdots + (\beta_{15} - 3 \beta_{14} + \cdots - 3 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{3}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 4 q^{3} + 12 q^{13} + 12 q^{23} + 36 q^{25} + 52 q^{27} - 8 q^{31} + 16 q^{35} - 48 q^{41} + 4 q^{47} + 24 q^{55} + 8 q^{71} - 52 q^{73} - 56 q^{75} - 64 q^{77} - 152 q^{81} + 28 q^{85} + 28 q^{87} + 84 q^{93} - 68 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 18x^{14} + 146x^{12} - 798x^{10} + 3934x^{8} - 19950x^{6} + 91250x^{4} - 281250x^{2} + 390625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 28 \nu^{14} + 9729 \nu^{12} - 78013 \nu^{10} + 385944 \nu^{8} - 1646452 \nu^{6} + \cdots + 123225000 ) / 11356250 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{15} - 18\nu^{13} + 146\nu^{11} - 798\nu^{9} + 3934\nu^{7} - 19950\nu^{5} + 91250\nu^{3} - 281250\nu ) / 78125 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 6206 \nu^{15} - 145108 \nu^{13} + 1091651 \nu^{11} - 5206913 \nu^{9} + 25292604 \nu^{7} + \cdots - 1562984375 \nu ) / 283906250 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 9539 \nu^{14} + 106427 \nu^{12} - 622119 \nu^{10} + 2938847 \nu^{8} - 15928851 \nu^{6} + \cdots + 434828125 ) / 56781250 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 9679 \nu^{14} + 155072 \nu^{12} - 1012184 \nu^{10} + 4868567 \nu^{8} - 24161111 \nu^{6} + \cdots + 1164515625 ) / 56781250 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1768 \nu^{15} - 26839 \nu^{13} + 159273 \nu^{11} - 818804 \nu^{9} + 3607532 \nu^{7} + \cdots - 119118750 \nu ) / 56781250 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 13152 \nu^{14} - 165036 \nu^{12} + 932717 \nu^{10} - 4530221 \nu^{8} + 23029618 \nu^{6} + \cdots - 766046875 ) / 56781250 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 14766 \nu^{14} + 182013 \nu^{12} - 1107886 \nu^{10} + 5441493 \nu^{8} - 26865744 \nu^{6} + \cdots + 983109375 ) / 56781250 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 37979 \nu^{14} + 527497 \nu^{12} - 3336559 \nu^{10} + 15924867 \nu^{8} - 79211011 \nu^{6} + \cdots + 3273890625 ) / 56781250 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 39617 \nu^{15} + 550556 \nu^{13} - 3408182 \nu^{11} + 16828941 \nu^{9} + \cdots + 3434765625 \nu ) / 283906250 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 10953 \nu^{14} - 151589 \nu^{12} + 959968 \nu^{10} - 4671004 \nu^{8} + 23307357 \nu^{6} + \cdots - 978331250 ) / 11356250 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 62919 \nu^{15} - 763392 \nu^{13} + 4635849 \nu^{11} - 22512212 \nu^{9} + 111486021 \nu^{7} + \cdots - 4064531250 \nu ) / 283906250 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 14766 \nu^{15} - 182013 \nu^{13} + 1107886 \nu^{11} - 5441493 \nu^{9} + 26865744 \nu^{7} + \cdots - 983109375 \nu ) / 56781250 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 99216 \nu^{15} - 1413363 \nu^{13} + 9018211 \nu^{11} - 43634468 \nu^{9} + 219432044 \nu^{7} + \cdots - 9530000000 \nu ) / 283906250 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{6} + \beta_{5} + \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{15} - 3\beta_{14} + 4\beta_{13} - 4\beta_{11} - \beta_{7} + 3\beta_{4} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{12} + \beta_{10} - 5\beta_{9} - \beta_{8} - 9\beta_{6} + 2\beta_{2} + 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -8\beta_{15} + 3\beta_{14} + \beta_{13} - 13\beta_{11} - 9\beta_{7} + 16\beta_{4} - 7\beta_{3} - \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 11\beta_{12} + 18\beta_{10} + 28\beta_{9} - \beta_{8} - 24\beta_{6} - 29\beta_{5} + 8\beta_{2} + 39 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 7\beta_{15} + 22\beta_{14} - 49\beta_{13} - 45\beta_{11} - 89\beta_{7} - 35\beta_{4} - 4\beta_{3} + 65\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -6\beta_{12} - 109\beta_{10} + 294\beta_{9} + 6\beta_{8} - 38\beta_{6} - 11\beta_{5} + 154\beta_{2} - 49 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 63\beta_{15} - 426\beta_{14} + 389\beta_{13} - 129\beta_{11} - 381\beta_{7} + 3\beta_{4} - 285\beta_{3} + 60\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -360\beta_{12} - 867\beta_{10} + 164\beta_{9} - 744\beta_{8} - 3\beta_{6} + 102\beta_{5} + 441\beta_{2} + 594 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 471 \beta_{15} - 809 \beta_{14} + 1593 \beta_{13} + 2235 \beta_{11} - 612 \beta_{7} + 1380 \beta_{4} + \cdots + 2058 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1042 \beta_{12} - 512 \beta_{10} - 1083 \beta_{9} - 5507 \beta_{8} + 1173 \beta_{6} + 875 \beta_{5} + \cdots + 4014 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 437 \beta_{15} - 3337 \beta_{14} + 8644 \beta_{13} + 6996 \beta_{11} - 4706 \beta_{7} + \cdots + 19038 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 1021 \beta_{12} - 4853 \beta_{10} - 18728 \beta_{9} - 17355 \beta_{8} - 2656 \beta_{6} + 14916 \beta_{5} + \cdots - 35425 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 30996 \beta_{15} - 34848 \beta_{14} + 81152 \beta_{13} - 20644 \beta_{11} - 37568 \beta_{7} + \cdots + 4436 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/460\mathbb{Z}\right)^\times\).

\(n\) \(231\) \(277\) \(281\)
\(\chi(n)\) \(1\) \(-\beta_{9}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
137.1
−2.19448 + 0.429243i
2.19448 − 0.429243i
−2.23266 − 0.123447i
2.23266 + 0.123447i
−1.88618 − 1.20098i
1.88618 + 1.20098i
−1.06856 + 1.96422i
1.06856 − 1.96422i
−2.19448 − 0.429243i
2.19448 + 0.429243i
−2.23266 + 0.123447i
2.23266 − 0.123447i
−1.88618 + 1.20098i
1.88618 − 1.20098i
−1.06856 − 1.96422i
1.06856 + 1.96422i
0 −2.36693 + 2.36693i 0 −2.19448 − 0.429243i 0 −2.93008 + 2.93008i 0 − 8.20472i 0
137.2 0 −2.36693 + 2.36693i 0 2.19448 + 0.429243i 0 2.93008 − 2.93008i 0 − 8.20472i 0
137.3 0 0.237125 − 0.237125i 0 −2.23266 + 0.123447i 0 1.69421 − 1.69421i 0 2.88754i 0
137.4 0 0.237125 − 0.237125i 0 2.23266 − 0.123447i 0 −1.69421 + 1.69421i 0 2.88754i 0
137.5 0 1.09396 − 1.09396i 0 −1.88618 + 1.20098i 0 −2.67082 + 2.67082i 0 0.606488i 0
137.6 0 1.09396 − 1.09396i 0 1.88618 − 1.20098i 0 2.67082 − 2.67082i 0 0.606488i 0
137.7 0 2.03584 − 2.03584i 0 −1.06856 − 1.96422i 0 0.641104 − 0.641104i 0 − 5.28931i 0
137.8 0 2.03584 − 2.03584i 0 1.06856 + 1.96422i 0 −0.641104 + 0.641104i 0 − 5.28931i 0
413.1 0 −2.36693 − 2.36693i 0 −2.19448 + 0.429243i 0 −2.93008 − 2.93008i 0 8.20472i 0
413.2 0 −2.36693 − 2.36693i 0 2.19448 − 0.429243i 0 2.93008 + 2.93008i 0 8.20472i 0
413.3 0 0.237125 + 0.237125i 0 −2.23266 − 0.123447i 0 1.69421 + 1.69421i 0 − 2.88754i 0
413.4 0 0.237125 + 0.237125i 0 2.23266 + 0.123447i 0 −1.69421 − 1.69421i 0 − 2.88754i 0
413.5 0 1.09396 + 1.09396i 0 −1.88618 − 1.20098i 0 −2.67082 − 2.67082i 0 − 0.606488i 0
413.6 0 1.09396 + 1.09396i 0 1.88618 + 1.20098i 0 2.67082 + 2.67082i 0 − 0.606488i 0
413.7 0 2.03584 + 2.03584i 0 −1.06856 + 1.96422i 0 0.641104 + 0.641104i 0 5.28931i 0
413.8 0 2.03584 + 2.03584i 0 1.06856 − 1.96422i 0 −0.641104 − 0.641104i 0 5.28931i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 137.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
23.b odd 2 1 inner
115.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 460.2.i.b ✓ 16
5.b even 2 1 2300.2.i.d 16
5.c odd 4 1 inner 460.2.i.b ✓ 16
5.c odd 4 1 2300.2.i.d 16
23.b odd 2 1 inner 460.2.i.b ✓ 16
115.c odd 2 1 2300.2.i.d 16
115.e even 4 1 inner 460.2.i.b ✓ 16
115.e even 4 1 2300.2.i.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
460.2.i.b ✓ 16 1.a even 1 1 trivial
460.2.i.b ✓ 16 5.c odd 4 1 inner
460.2.i.b ✓ 16 23.b odd 2 1 inner
460.2.i.b ✓ 16 115.e even 4 1 inner
2300.2.i.d 16 5.b even 2 1
2300.2.i.d 16 5.c odd 4 1
2300.2.i.d 16 115.c odd 2 1
2300.2.i.d 16 115.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 2T_{3}^{7} + 2T_{3}^{6} - 6T_{3}^{5} + 110T_{3}^{4} - 270T_{3}^{3} + 338T_{3}^{2} - 130T_{3} + 25 \) acting on \(S_{2}^{\mathrm{new}}(460, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{8} - 2 T^{7} + 2 T^{6} + \cdots + 25)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} - 18 T^{14} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( T^{16} + 532 T^{12} + \cdots + 1336336 \) Copy content Toggle raw display
$11$ \( (T^{8} + 50 T^{6} + \cdots + 2500)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} - 6 T^{7} + \cdots + 13225)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + 880 T^{12} + \cdots + 16 \) Copy content Toggle raw display
$19$ \( (T^{8} - 36 T^{6} + \cdots + 100)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 78310985281 \) Copy content Toggle raw display
$29$ \( (T^{8} + 88 T^{6} + \cdots + 28561)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 2 T^{3} - 20 T^{2} + \cdots + 11)^{4} \) Copy content Toggle raw display
$37$ \( T^{16} + 1348 T^{12} + \cdots + 456976 \) Copy content Toggle raw display
$41$ \( (T^{4} + 12 T^{3} + \cdots - 107)^{4} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 127880620816 \) Copy content Toggle raw display
$47$ \( (T^{8} - 2 T^{7} + \cdots + 24025)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 533794816 \) Copy content Toggle raw display
$59$ \( (T^{8} + 168 T^{6} + \cdots + 110224)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 224 T^{6} + \cdots + 100)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 36\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( (T^{4} - 2 T^{3} + \cdots + 169)^{4} \) Copy content Toggle raw display
$73$ \( (T^{8} + 26 T^{7} + \cdots + 25)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} - 424 T^{6} + \cdots + 115562500)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 17799252215056 \) Copy content Toggle raw display
$89$ \( (T^{8} - 260 T^{6} + \cdots + 1144900)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 583506543376 \) Copy content Toggle raw display
show more
show less