↑

Properties

Label 880.2.bf.g
Level $880$
Weight $2$
Character orbit 880.bf
Analytic conductor $7.027$
Analytic rank $0$
Dimension $20$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [880,2,Mod(287,880)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(880, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("880.287");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 880 = 2^{4} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 880.bf (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: \(7.02683537787\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 2 x^{19} + 2 x^{18} + 2 x^{17} + 147 x^{16} - 300 x^{15} + 308 x^{14} + 436 x^{13} + 3579 x^{12} + \cdots + 144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{11} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} - \beta_{14} q^{5} - \beta_{19} q^{7} + ( - \beta_{19} - \beta_{18} + \cdots + \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{3} - \beta_{14} q^{5} - \beta_{19} q^{7} + ( - \beta_{19} - \beta_{18} + \cdots + \beta_{2}) q^{9}+ \cdots + ( - \beta_{19} + \beta_{17} + \cdots - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 2 q^{3}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 2 q^{3} + 4 q^{13} - 8 q^{15} - 8 q^{17} + 8 q^{19} + 32 q^{21} - 18 q^{23} + 10 q^{25} - 2 q^{27} + 2 q^{33} + 20 q^{35} - 14 q^{37} + 32 q^{39} + 8 q^{41} - 16 q^{43} - 18 q^{45} + 40 q^{47} + 12 q^{53} - 6 q^{55} + 64 q^{57} + 60 q^{59} - 8 q^{61} - 52 q^{63} + 8 q^{65} - 2 q^{67} - 20 q^{73} - 24 q^{75} + 16 q^{79} - 72 q^{81} - 32 q^{83} + 44 q^{85} + 112 q^{87} + 34 q^{93} - 20 q^{95} + 2 q^{97} - 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 2 x^{19} + 2 x^{18} + 2 x^{17} + 147 x^{16} - 300 x^{15} + 308 x^{14} + 436 x^{13} + 3579 x^{12} + \cdots + 144 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 11\!\cdots\!57 \nu^{19} + \cdots + 31\!\cdots\!88 ) / 74\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 15\!\cdots\!19 \nu^{19} + \cdots + 13\!\cdots\!88 ) / 37\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 28\!\cdots\!93 \nu^{19} + \cdots + 64\!\cdots\!76 ) / 14\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 76\!\cdots\!00 \nu^{19} + \cdots - 42\!\cdots\!76 ) / 37\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 90\!\cdots\!77 \nu^{19} + \cdots - 66\!\cdots\!36 ) / 37\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 19\!\cdots\!97 \nu^{19} + \cdots - 63\!\cdots\!76 ) / 49\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 39\!\cdots\!55 \nu^{19} + \cdots + 19\!\cdots\!80 ) / 74\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 43\!\cdots\!84 \nu^{19} + \cdots - 13\!\cdots\!72 ) / 74\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 75\!\cdots\!89 \nu^{19} + \cdots - 79\!\cdots\!96 ) / 10\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 11\!\cdots\!27 \nu^{19} + \cdots + 11\!\cdots\!60 ) / 14\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 11\!\cdots\!11 \nu^{19} + \cdots + 66\!\cdots\!76 ) / 14\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 13\!\cdots\!17 \nu^{19} + \cdots + 59\!\cdots\!92 ) / 14\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 17\!\cdots\!95 \nu^{19} + \cdots - 41\!\cdots\!36 ) / 14\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 20\!\cdots\!75 \nu^{19} + \cdots - 16\!\cdots\!64 ) / 14\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 26\!\cdots\!33 \nu^{19} + \cdots + 18\!\cdots\!08 ) / 14\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 31\!\cdots\!49 \nu^{19} + \cdots - 18\!\cdots\!64 ) / 14\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 31\!\cdots\!95 \nu^{19} + \cdots - 22\!\cdots\!44 ) / 14\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 13\!\cdots\!00 \nu^{19} + \cdots + 43\!\cdots\!60 ) / 49\!\cdots\!08 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{19} + \beta_{18} + \beta_{13} - \beta_{11} - \beta_{10} + \beta_{8} - \beta_{7} + \cdots - \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{16} - \beta_{14} - \beta_{13} + \beta_{12} + \beta_{10} + \beta_{6} - 7\beta_{3} - \beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 13 \beta_{19} + 11 \beta_{17} - 10 \beta_{15} - 2 \beta_{14} + 13 \beta_{13} - 2 \beta_{12} + \cdots - 19 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 13 \beta_{16} - 16 \beta_{15} + 2 \beta_{14} - 2 \beta_{13} + 13 \beta_{12} - 3 \beta_{11} - \beta_{10} + \cdots + 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 147 \beta_{19} - 115 \beta_{18} + 32 \beta_{16} - 14 \beta_{15} - 26 \beta_{14} - 85 \beta_{13} + \cdots + 24 \beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 4 \beta_{18} + 4 \beta_{17} - 149 \beta_{16} - 20 \beta_{15} + 147 \beta_{14} + 155 \beta_{13} + \cdots - 80 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1605 \beta_{19} - 1211 \beta_{17} + 1038 \beta_{15} + 390 \beta_{14} - 1545 \beta_{13} + 406 \beta_{12} + \cdots + 1511 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 340 \beta_{19} - 120 \beta_{18} + 120 \beta_{17} + 1655 \beta_{16} + 2108 \beta_{15} - 488 \beta_{14} + \cdots - 1132 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 17299 \beta_{19} + 12839 \beta_{18} - 4800 \beta_{16} + 1566 \beta_{15} + 3434 \beta_{14} + \cdots - 4926 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 2296 \beta_{18} - 2296 \beta_{17} + 18187 \beta_{16} + 4216 \beta_{15} - 17141 \beta_{14} + \cdots + 14252 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 185521 \beta_{19} + 136663 \beta_{17} - 112450 \beta_{15} - 46562 \beta_{14} + 179817 \beta_{13} + \cdots - 160923 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 99760 \beta_{19} + 36280 \beta_{18} - 36280 \beta_{17} - 199121 \beta_{16} - 226816 \beta_{15} + \cdots + 170796 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 1986555 \beta_{19} - 1458411 \beta_{18} + 627904 \beta_{16} - 137454 \beta_{15} - 436474 \beta_{14} + \cdots + 754184 \beta_1 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 517388 \beta_{18} + 517388 \beta_{17} - 2177925 \beta_{16} - 680940 \beta_{15} + 1942603 \beta_{14} + \cdots - 1998840 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 21273925 \beta_{19} - 15593867 \beta_{17} + 12351750 \beta_{15} + 5162670 \beta_{14} - 21010273 \beta_{13} + \cdots + 18520871 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 18965364 \beta_{19} - 6933168 \beta_{18} + 6933168 \beta_{17} + 23823439 \beta_{16} + 23358884 \beta_{15} + \cdots - 23096820 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 228016499 \beta_{19} + 167018887 \beta_{18} - 79962976 \beta_{16} + 9992918 \beta_{15} + \cdots - 106467286 \beta_1 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 89173728 \beta_{18} - 89173728 \beta_{17} + 260723971 \beta_{16} + 99583792 \beta_{15} + \cdots + 264859268 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/880\mathbb{Z}\right)^\times\).

\(n\) \(111\) \(177\) \(321\) \(661\)
\(\chi(n)\) \(-1\) \(-\beta_{6}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
287.1
2.25896 − 2.25896i
1.62599 − 1.62599i
1.59344 − 1.59344i
0.636383 − 0.636383i
0.261641 − 0.261641i
0.155966 − 0.155966i
−0.802942 + 0.802942i
−0.862365 + 0.862365i
−1.51471 + 1.51471i
−2.35237 + 2.35237i
2.25896 + 2.25896i
1.62599 + 1.62599i
1.59344 + 1.59344i
0.636383 + 0.636383i
0.261641 + 0.261641i
0.155966 + 0.155966i
−0.802942 − 0.802942i
−0.862365 − 0.862365i
−1.51471 − 1.51471i
−2.35237 − 2.35237i
0 −2.25896 − 2.25896i 0 −1.92419 − 1.13907i 0 −3.06397 + 3.06397i 0 7.20584i 0
287.2 0 −1.62599 − 1.62599i 0 −0.262111 + 2.22065i 0 1.47454 − 1.47454i 0 2.28770i 0
287.3 0 −1.59344 − 1.59344i 0 1.95624 − 1.08310i 0 −1.39615 + 1.39615i 0 2.07813i 0
287.4 0 −0.636383 − 0.636383i 0 0.986043 − 2.00692i 0 0.890190 − 0.890190i 0 − 2.19003i 0
287.5 0 −0.261641 − 0.261641i 0 1.79637 + 1.33156i 0 −2.21442 + 2.21442i 0 − 2.86309i 0
287.6 0 −0.155966 − 0.155966i 0 −1.37957 + 1.75977i 0 3.01657 − 3.01657i 0 − 2.95135i 0
287.7 0 0.802942 + 0.802942i 0 −2.20978 + 0.341894i 0 1.03403 − 1.03403i 0 − 1.71057i 0
287.8 0 0.862365 + 0.862365i 0 −1.91625 − 1.15239i 0 −1.17526 + 1.17526i 0 − 1.51265i 0
287.9 0 1.51471 + 1.51471i 0 2.14128 + 0.644153i 0 1.76926 − 1.76926i 0 1.58871i 0
287.10 0 2.35237 + 2.35237i 0 0.811961 + 2.08344i 0 −0.334794 + 0.334794i 0 8.06731i 0
463.1 0 −2.25896 + 2.25896i 0 −1.92419 + 1.13907i 0 −3.06397 − 3.06397i 0 − 7.20584i 0
463.2 0 −1.62599 + 1.62599i 0 −0.262111 − 2.22065i 0 1.47454 + 1.47454i 0 − 2.28770i 0
463.3 0 −1.59344 + 1.59344i 0 1.95624 + 1.08310i 0 −1.39615 − 1.39615i 0 − 2.07813i 0
463.4 0 −0.636383 + 0.636383i 0 0.986043 + 2.00692i 0 0.890190 + 0.890190i 0 2.19003i 0
463.5 0 −0.261641 + 0.261641i 0 1.79637 − 1.33156i 0 −2.21442 − 2.21442i 0 2.86309i 0
463.6 0 −0.155966 + 0.155966i 0 −1.37957 − 1.75977i 0 3.01657 + 3.01657i 0 2.95135i 0
463.7 0 0.802942 − 0.802942i 0 −2.20978 − 0.341894i 0 1.03403 + 1.03403i 0 1.71057i 0
463.8 0 0.862365 − 0.862365i 0 −1.91625 + 1.15239i 0 −1.17526 − 1.17526i 0 1.51265i 0
463.9 0 1.51471 − 1.51471i 0 2.14128 − 0.644153i 0 1.76926 + 1.76926i 0 − 1.58871i 0
463.10 0 2.35237 − 2.35237i 0 0.811961 − 2.08344i 0 −0.334794 − 0.334794i 0 − 8.06731i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 287.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 880.2.bf.g ✓ 20
4.b odd 2 1 880.2.bf.h yes 20
5.c odd 4 1 880.2.bf.h yes 20
20.e even 4 1 inner 880.2.bf.g ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
880.2.bf.g ✓ 20 1.a even 1 1 trivial
880.2.bf.g ✓ 20 20.e even 4 1 inner
880.2.bf.h yes 20 4.b odd 2 1
880.2.bf.h yes 20 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(880, [\chi])\):

\( T_{3}^{20} + 2 T_{3}^{19} + 2 T_{3}^{18} - 2 T_{3}^{17} + 147 T_{3}^{16} + 300 T_{3}^{15} + 308 T_{3}^{14} + \cdots + 144 \) Copy content Toggle raw display
\( T_{7}^{20} - 8 T_{7}^{17} + 434 T_{7}^{16} - 88 T_{7}^{15} + 32 T_{7}^{14} - 2408 T_{7}^{13} + \cdots + 746496 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} + 2 T^{19} + \cdots + 144 \) Copy content Toggle raw display
$5$ \( T^{20} - 5 T^{18} + \cdots + 9765625 \) Copy content Toggle raw display
$7$ \( T^{20} - 8 T^{17} + \cdots + 746496 \) Copy content Toggle raw display
$11$ \( (T^{2} + 1)^{10} \) Copy content Toggle raw display
$13$ \( T^{20} - 4 T^{19} + \cdots + 5308416 \) Copy content Toggle raw display
$17$ \( T^{20} + 8 T^{19} + \cdots + 32809984 \) Copy content Toggle raw display
$19$ \( (T^{10} - 4 T^{9} + \cdots - 6912)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} + 18 T^{19} + \cdots + 2304 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 7439539912704 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 2689809924096 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 6511254544656 \) Copy content Toggle raw display
$41$ \( (T^{10} - 4 T^{9} + \cdots - 5313024)^{2} \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 253576833417216 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 2256340460544 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 315454257705024 \) Copy content Toggle raw display
$59$ \( (T^{10} - 30 T^{9} + \cdots - 5859072)^{2} \) Copy content Toggle raw display
$61$ \( (T^{10} + 4 T^{9} + \cdots + 2380032)^{2} \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 18\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 32886829967616 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 274877906944 \) Copy content Toggle raw display
$79$ \( (T^{10} - 8 T^{9} + \cdots + 6690816)^{2} \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 21\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 46\!\cdots\!16 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 38\!\cdots\!36 \) Copy content Toggle raw display
show more
show less