↑

Properties

Label 2880.3.c.j
Level $2880$
Weight $3$
Character orbit 2880.c
Analytic conductor $78.474$
Analytic rank $0$
Dimension $12$
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] = [2880,3,Mod(449,2880)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2880, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2880.449");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2880.c (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: \(78.4743161358\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4 x^{11} + 8 x^{10} - 76 x^{9} + 33 x^{8} - 112 x^{7} + 3072 x^{6} - 2032 x^{5} - 17864 x^{4} + \cdots + 810000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{16}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 360)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{10} q^{5} - \beta_{9} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{10} q^{5} - \beta_{9} q^{7} + \beta_{7} q^{11} - \beta_{3} q^{13} + (\beta_{11} + \beta_{10} + \beta_{5}) q^{17} - \beta_1 q^{19} + (\beta_{11} - 2 \beta_{10} + \cdots - 2 \beta_{5}) q^{23}+ \cdots + (3 \beta_{9} - \beta_{3} - 11 \beta_{2}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 24 q^{25} + 96 q^{31} + 156 q^{49} + 144 q^{55} + 120 q^{61} - 384 q^{79} + 300 q^{85} + 48 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4 x^{11} + 8 x^{10} - 76 x^{9} + 33 x^{8} - 112 x^{7} + 3072 x^{6} - 2032 x^{5} - 17864 x^{4} + \cdots + 810000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 4604738 \nu^{11} + 42942375 \nu^{10} + 543121266 \nu^{9} - 5843704058 \nu^{8} + \cdots + 69968084290488 ) / 3976236626976 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 113285587 \nu^{11} + 3069782188 \nu^{10} + 2178876244 \nu^{9} + 42918256132 \nu^{8} + \cdots - 549698871276000 ) / 41419131531000 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 7382523711 \nu^{11} - 144520107739 \nu^{10} + 212434458368 \nu^{9} - 1655196755846 \nu^{8} + \cdots + 10\!\cdots\!00 ) / 19\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 47522 \nu^{11} + 116577 \nu^{10} - 517534 \nu^{9} + 1409098 \nu^{8} - 22164714 \nu^{7} + \cdots + 12807693000 ) / 6106896000 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 13221193685 \nu^{11} - 129727996867 \nu^{10} + 280123571668 \nu^{9} - 1625987889646 \nu^{8} + \cdots + 64\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 24472339721 \nu^{11} + 191771366979 \nu^{10} - 369082790748 \nu^{9} + \cdots - 11\!\cdots\!00 ) / 19\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 108553180306 \nu^{11} + 763025410179 \nu^{10} - 6018051805618 \nu^{9} + \cdots + 10\!\cdots\!00 ) / 59\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 791793 \nu^{11} + 4298897 \nu^{10} - 9872944 \nu^{9} + 61102618 \nu^{8} + \cdots - 166191129000 ) / 35159724000 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 9156724511 \nu^{11} - 17097991039 \nu^{10} + 92694072368 \nu^{9} - 553018767246 \nu^{8} + \cdots + 866139378003000 ) / 397623662697600 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 137819890001 \nu^{11} - 555839083319 \nu^{10} + 1935291284268 \nu^{9} + \cdots + 11\!\cdots\!00 ) / 59\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 98500310901 \nu^{11} - 503427082304 \nu^{10} + 1154556251908 \nu^{9} + \cdots + 17\!\cdots\!00 ) / 14\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - 4 \beta_{11} - 8 \beta_{10} + 10 \beta_{9} + \beta_{8} - 4 \beta_{6} + 16 \beta_{5} + 4 \beta_{4} + \cdots + 16 ) / 48 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -10\beta_{11} + 4\beta_{10} - 6\beta_{9} - 32\beta_{8} + 4\beta_{5} - 18\beta_{3} - 3\beta_{2} ) / 24 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 32 \beta_{11} - 70 \beta_{10} - 146 \beta_{9} - 211 \beta_{8} - 66 \beta_{7} - 52 \beta_{6} + \cdots + 784 ) / 48 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 39 \beta_{10} - 158 \beta_{9} - 9 \beta_{7} - 316 \beta_{6} + 39 \beta_{5} + 145 \beta_{4} + \cdots + 1912 ) / 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 608 \beta_{11} - 1942 \beta_{10} + 1958 \beta_{9} - 3337 \beta_{8} + 726 \beta_{7} - 1316 \beta_{6} + \cdots + 10736 ) / 48 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 3338 \beta_{11} - 2140 \beta_{10} + 618 \beta_{9} - 12304 \beta_{8} - 2140 \beta_{5} + \cdots + 873 \beta_{2} ) / 24 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 17752 \beta_{11} + 46762 \beta_{10} - 53662 \beta_{9} - 44765 \beta_{8} - 8082 \beta_{7} + \cdots + 186608 ) / 48 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 24939 \beta_{10} - 44386 \beta_{9} + 1365 \beta_{7} - 88772 \beta_{6} - 24939 \beta_{5} + \cdots + 1064072 ) / 24 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 275008 \beta_{11} - 698810 \beta_{10} + 740506 \beta_{9} - 768119 \beta_{8} + 148794 \beta_{7} + \cdots + 4994128 ) / 48 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 1270918 \beta_{11} - 242324 \beta_{10} + 354054 \beta_{9} - 3785852 \beta_{8} + \cdots + 58167 \beta_{2} ) / 24 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 5774648 \beta_{11} + 10774478 \beta_{10} - 17917922 \beta_{9} - 19598515 \beta_{8} - 4108038 \beta_{7} + \cdots + 75801808 ) / 48 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2880\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(641\) \(901\) \(2431\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
−0.856820 + 1.92442i
−0.856820 − 1.92442i
−2.11143 + 2.52283i
−2.11143 − 2.52283i
0.0410965 + 4.32874i
0.0410965 − 4.32874i
4.32874 − 0.0410965i
4.32874 + 0.0410965i
2.52283 + 2.11143i
2.52283 − 2.11143i
−1.92442 − 0.856820i
−1.92442 + 0.856820i
0 0 0 −4.98699 − 0.360490i 0 − 5.56248i 0 0 0
449.2 0 0 0 −4.98699 + 0.360490i 0 5.56248i 0 0 0
449.3 0 0 0 −3.09475 − 3.92715i 0 − 0.822805i 0 0 0
449.4 0 0 0 −3.09475 + 3.92715i 0 0.822805i 0 0 0
449.5 0 0 0 −0.229082 − 4.99475i 0 − 8.73967i 0 0 0
449.6 0 0 0 −0.229082 + 4.99475i 0 8.73967i 0 0 0
449.7 0 0 0 0.229082 − 4.99475i 0 8.73967i 0 0 0
449.8 0 0 0 0.229082 + 4.99475i 0 − 8.73967i 0 0 0
449.9 0 0 0 3.09475 − 3.92715i 0 0.822805i 0 0 0
449.10 0 0 0 3.09475 + 3.92715i 0 − 0.822805i 0 0 0
449.11 0 0 0 4.98699 − 0.360490i 0 5.56248i 0 0 0
449.12 0 0 0 4.98699 + 0.360490i 0 − 5.56248i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 449.12
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
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2880.3.c.j 12
3.b odd 2 1 inner 2880.3.c.j 12
4.b odd 2 1 2880.3.c.i 12
5.b even 2 1 inner 2880.3.c.j 12
8.b even 2 1 360.3.c.a ✓ 12
8.d odd 2 1 720.3.c.d 12
12.b even 2 1 2880.3.c.i 12
15.d odd 2 1 inner 2880.3.c.j 12
20.d odd 2 1 2880.3.c.i 12
24.f even 2 1 720.3.c.d 12
24.h odd 2 1 360.3.c.a ✓ 12
40.e odd 2 1 720.3.c.d 12
40.f even 2 1 360.3.c.a ✓ 12
40.i odd 4 1 1800.3.l.h 6
40.i odd 4 1 1800.3.l.i 6
40.k even 4 1 3600.3.l.w 6
40.k even 4 1 3600.3.l.x 6
60.h even 2 1 2880.3.c.i 12
120.i odd 2 1 360.3.c.a ✓ 12
120.m even 2 1 720.3.c.d 12
120.q odd 4 1 3600.3.l.w 6
120.q odd 4 1 3600.3.l.x 6
120.w even 4 1 1800.3.l.h 6
120.w even 4 1 1800.3.l.i 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.3.c.a ✓ 12 8.b even 2 1
360.3.c.a ✓ 12 24.h odd 2 1
360.3.c.a ✓ 12 40.f even 2 1
360.3.c.a ✓ 12 120.i odd 2 1
720.3.c.d 12 8.d odd 2 1
720.3.c.d 12 24.f even 2 1
720.3.c.d 12 40.e odd 2 1
720.3.c.d 12 120.m even 2 1
1800.3.l.h 6 40.i odd 4 1
1800.3.l.h 6 120.w even 4 1
1800.3.l.i 6 40.i odd 4 1
1800.3.l.i 6 120.w even 4 1
2880.3.c.i 12 4.b odd 2 1
2880.3.c.i 12 12.b even 2 1
2880.3.c.i 12 20.d odd 2 1
2880.3.c.i 12 60.h even 2 1
2880.3.c.j 12 1.a even 1 1 trivial
2880.3.c.j 12 3.b odd 2 1 inner
2880.3.c.j 12 5.b even 2 1 inner
2880.3.c.j 12 15.d odd 2 1 inner
3600.3.l.w 6 40.k even 4 1
3600.3.l.w 6 120.q odd 4 1
3600.3.l.x 6 40.k even 4 1
3600.3.l.x 6 120.q odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(2880, [\chi])\):

\( T_{7}^{6} + 108T_{7}^{4} + 2436T_{7}^{2} + 1600 \) Copy content Toggle raw display
\( T_{17}^{6} - 1506T_{17}^{4} + 684864T_{17}^{2} - 83980800 \) Copy content Toggle raw display
\( T_{19}^{3} - 528T_{19} + 1408 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 244140625 \) Copy content Toggle raw display
$7$ \( (T^{6} + 108 T^{4} + \cdots + 1600)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + 516 T^{4} + \cdots + 1438208)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + 192 T^{4} + \cdots + 85264)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} - 1506 T^{4} + \cdots - 83980800)^{2} \) Copy content Toggle raw display
$19$ \( (T^{3} - 528 T + 1408)^{4} \) Copy content Toggle raw display
$23$ \( (T^{6} - 1944 T^{4} + \cdots - 90747392)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + 3042 T^{4} + \cdots + 446168192)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 24 T^{2} + \cdots + 3888)^{4} \) Copy content Toggle raw display
$37$ \( (T^{6} + 6876 T^{4} + \cdots + 1233414400)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + 3462 T^{4} + \cdots + 483605000)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + 4464 T^{4} + \cdots + 967458816)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} - 10848 T^{4} + \cdots - 16564912128)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} - 2802 T^{4} + \cdots - 327680000)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + 3396 T^{4} + \cdots + 17428608)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 30 T^{2} + \cdots - 82312)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} + 19584 T^{4} + \cdots + 18828230656)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + 7392 T^{4} + \cdots + 99123200)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 14004 T^{4} + \cdots + 93733945600)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + 96 T^{2} + \cdots - 858320)^{4} \) Copy content Toggle raw display
$83$ \( (T^{6} - 9984 T^{4} + \cdots - 8506253312)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + 25830 T^{4} + \cdots + 418887045000)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 4190700294400)^{2} \) Copy content Toggle raw display
show more
show less