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Properties

Label 880.2.cd.b
Level $880$
Weight $2$
Character orbit 880.cd
Analytic conductor $7.027$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [880,2,Mod(49,880)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(880, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 5, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("880.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 880 = 2^{4} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 880.cd (of order \(10\), degree \(4\), 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: \(7.02683537787\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} + 8 x^{14} + 10 x^{13} - 109 x^{12} + 280 x^{11} - 198 x^{10} - 1168 x^{9} + \cdots + 390625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 110)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{12} + \beta_{10} + \beta_{6}) q^{3} - \beta_{8} q^{5} + ( - \beta_{13} + \beta_{12} + \beta_{11} + \cdots + 1) q^{7}+ \cdots + (2 \beta_{9} + \beta_{4} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{12} + \beta_{10} + \beta_{6}) q^{3} - \beta_{8} q^{5} + ( - \beta_{13} + \beta_{12} + \beta_{11} + \cdots + 1) q^{7}+ \cdots + (2 \beta_{15} - \beta_{14} + \beta_{13} + \cdots + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 6 q^{5} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 6 q^{5} + 4 q^{9} + 20 q^{11} - 2 q^{15} + 16 q^{19} - 16 q^{21} + 16 q^{29} + 4 q^{31} + 48 q^{35} + 8 q^{39} + 40 q^{41} - 4 q^{45} + 84 q^{49} + 4 q^{51} - 32 q^{55} + 20 q^{61} + 72 q^{65} + 56 q^{71} - 32 q^{75} - 36 q^{79} - 56 q^{81} - 54 q^{85} - 8 q^{89} + 40 q^{91} + 50 q^{95} + 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} + 8 x^{14} + 10 x^{13} - 109 x^{12} + 280 x^{11} - 198 x^{10} - 1168 x^{9} + \cdots + 390625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 173441 \nu^{15} - 13142014 \nu^{14} + 62145978 \nu^{13} - 45934140 \nu^{12} + \cdots - 1109367578125 ) / 130247390625 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 21238 \nu^{15} + 1031322 \nu^{14} - 5312684 \nu^{13} + 3514030 \nu^{12} + \cdots + 99028437500 ) / 11840671875 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 58612 \nu^{15} - 640840 \nu^{14} + 1540234 \nu^{13} + 633079 \nu^{12} - 16605168 \nu^{11} + \cdots - 8784171875 ) / 11840671875 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 735828 \nu^{15} + 1774345 \nu^{14} + 3100799 \nu^{13} - 21315936 \nu^{12} + \cdots - 271245109375 ) / 130247390625 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 6436462 \nu^{15} + 6651203 \nu^{14} + 28931559 \nu^{13} - 104135105 \nu^{12} + \cdots - 738595781250 ) / 651236953125 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1813699 \nu^{15} + 16036663 \nu^{14} - 30900230 \nu^{13} - 47371799 \nu^{12} + \cdots + 253811312500 ) / 130247390625 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 9454026 \nu^{15} + 69998414 \nu^{14} - 108888223 \nu^{13} - 239198055 \nu^{12} + \cdots + 2335828906250 ) / 651236953125 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 969684 \nu^{15} + 7259596 \nu^{14} - 19918312 \nu^{13} - 11443860 \nu^{12} + \cdots + 366095625000 ) / 59203359375 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 21389 \nu^{15} + 15849 \nu^{14} - 192528 \nu^{13} + 559035 \nu^{12} + 343314 \nu^{11} + \cdots + 5258500000 ) / 1058921875 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 1267564 \nu^{15} - 4964066 \nu^{14} + 4983902 \nu^{13} + 39239060 \nu^{12} + \cdots - 108288437500 ) / 59203359375 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 2839981 \nu^{15} + 11186483 \nu^{14} - 9577834 \nu^{13} - 90545788 \nu^{12} + \cdots + 180205890625 ) / 130247390625 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 3163593 \nu^{15} + 21134817 \nu^{14} - 40809799 \nu^{13} - 59283170 \nu^{12} + \cdots + 1200271796875 ) / 130247390625 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 616596 \nu^{15} - 2891274 \nu^{14} + 2637478 \nu^{13} + 14416640 \nu^{12} - 75328514 \nu^{11} + \cdots + 12354453125 ) / 11840671875 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 3495548 \nu^{15} - 16936892 \nu^{14} + 32507734 \nu^{13} + 71121680 \nu^{12} + \cdots - 710165312500 ) / 59203359375 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{14} - \beta_{13} - 5\beta_{5} + \beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 2 \beta_{15} - 3 \beta_{12} - 2 \beta_{11} - 2 \beta_{9} - 3 \beta_{7} - 3 \beta_{6} - 5 \beta_{5} + \cdots - 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 2 \beta_{14} - 4 \beta_{13} + 6 \beta_{12} + 3 \beta_{11} + 6 \beta_{10} + 12 \beta_{9} - 6 \beta_{8} + \cdots + 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 18 \beta_{14} - 12 \beta_{11} + 10 \beta_{10} + 18 \beta_{9} + 30 \beta_{6} + 30 \beta_{5} - 30 \beta_{4} + \cdots - 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 28 \beta_{15} + 15 \beta_{13} + 62 \beta_{12} + 28 \beta_{11} + 13 \beta_{9} - 15 \beta_{8} + 15 \beta_{7} + \cdots + 15 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 15 \beta_{15} + 79 \beta_{13} - 90 \beta_{12} - 129 \beta_{11} - 75 \beta_{10} - 204 \beta_{9} + \cdots - 129 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 4 \beta_{15} - 204 \beta_{14} + 200 \beta_{12} + 4 \beta_{11} + 204 \beta_{10} - 275 \beta_{9} + \cdots + 116 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 67 \beta_{14} - 944 \beta_{12} - 1011 \beta_{11} + 76 \beta_{10} - 87 \beta_{9} - 96 \beta_{8} + \cdots + 9 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 9 \beta_{15} + 9 \beta_{14} + 9 \beta_{13} - 9 \beta_{12} + 9 \beta_{11} + 335 \beta_{10} + \cdots + 9 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 353 \beta_{15} + 2310 \beta_{13} - 4928 \beta_{12} - 2002 \beta_{11} - 4620 \beta_{10} + \cdots - 2310 \beta_{4} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 2310 \beta_{15} - 2002 \beta_{14} - 308 \beta_{13} + 4620 \beta_{12} + 8091 \beta_{11} + 2310 \beta_{10} + \cdots + 8091 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 5769 \beta_{14} - 4241 \beta_{12} + 5769 \beta_{10} + 5781 \beta_{9} - 5769 \beta_{8} + 1540 \beta_{6} + \cdots + 10010 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 616 \beta_{14} - 17023 \beta_{13} + 15578 \beta_{12} + 44423 \beta_{10} + 16407 \beta_{9} + \cdots - 16407 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 26784 \beta_{15} + 26784 \beta_{13} - 26784 \beta_{12} - 82035 \beta_{11} + 3080 \beta_{10} + \cdots - 111899 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(111\) \(177\) \(321\) \(661\)
\(\chi(n)\) \(1\) \(-1\) \(\beta_{4}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
0.774688 − 2.09758i
0.0885831 + 2.23431i
−2.09758 − 0.774688i
2.23431 − 0.0885831i
1.89162 + 1.19237i
0.147217 − 2.23122i
−2.23122 − 0.147217i
1.19237 − 1.89162i
0.774688 + 2.09758i
0.0885831 − 2.23431i
−2.09758 + 0.774688i
2.23431 + 0.0885831i
1.89162 − 1.19237i
0.147217 + 2.23122i
−2.23122 + 0.147217i
1.19237 + 1.89162i
0 −0.363271 + 0.500000i 0 −1.38496 − 1.75553i 0 −0.175555 − 0.241631i 0 0.809017 + 2.48990i 0
49.2 0 −0.363271 + 0.500000i 0 0.606193 + 2.15233i 0 2.07767 + 2.85966i 0 0.809017 + 2.48990i 0
49.3 0 0.363271 − 0.500000i 0 −1.75553 + 1.38496i 0 −2.07767 − 2.85966i 0 0.809017 + 2.48990i 0
49.4 0 0.363271 − 0.500000i 0 2.15233 − 0.606193i 0 0.175555 + 0.241631i 0 0.809017 + 2.48990i 0
289.1 0 −1.53884 + 0.500000i 0 −2.07652 − 0.829496i 0 −3.59321 − 1.16751i 0 −0.309017 + 0.224514i 0
289.2 0 −1.53884 + 0.500000i 0 1.71856 − 1.43058i 0 4.76878 + 1.54947i 0 −0.309017 + 0.224514i 0
289.3 0 1.53884 − 0.500000i 0 −1.43058 − 1.71856i 0 3.59321 + 1.16751i 0 −0.309017 + 0.224514i 0
289.4 0 1.53884 − 0.500000i 0 −0.829496 + 2.07652i 0 −4.76878 − 1.54947i 0 −0.309017 + 0.224514i 0
449.1 0 −0.363271 − 0.500000i 0 −1.38496 + 1.75553i 0 −0.175555 + 0.241631i 0 0.809017 − 2.48990i 0
449.2 0 −0.363271 − 0.500000i 0 0.606193 − 2.15233i 0 2.07767 − 2.85966i 0 0.809017 − 2.48990i 0
449.3 0 0.363271 + 0.500000i 0 −1.75553 − 1.38496i 0 −2.07767 + 2.85966i 0 0.809017 − 2.48990i 0
449.4 0 0.363271 + 0.500000i 0 2.15233 + 0.606193i 0 0.175555 − 0.241631i 0 0.809017 − 2.48990i 0
609.1 0 −1.53884 − 0.500000i 0 −2.07652 + 0.829496i 0 −3.59321 + 1.16751i 0 −0.309017 − 0.224514i 0
609.2 0 −1.53884 − 0.500000i 0 1.71856 + 1.43058i 0 4.76878 − 1.54947i 0 −0.309017 − 0.224514i 0
609.3 0 1.53884 + 0.500000i 0 −1.43058 + 1.71856i 0 3.59321 − 1.16751i 0 −0.309017 − 0.224514i 0
609.4 0 1.53884 + 0.500000i 0 −0.829496 − 2.07652i 0 −4.76878 + 1.54947i 0 −0.309017 − 0.224514i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
11.c even 5 1 inner
55.j even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 880.2.cd.b 16
4.b odd 2 1 110.2.j.b ✓ 16
5.b even 2 1 inner 880.2.cd.b 16
11.c even 5 1 inner 880.2.cd.b 16
12.b even 2 1 990.2.ba.h 16
20.d odd 2 1 110.2.j.b ✓ 16
20.e even 4 1 550.2.h.j 8
20.e even 4 1 550.2.h.n 8
44.g even 10 1 1210.2.b.l 8
44.h odd 10 1 110.2.j.b ✓ 16
44.h odd 10 1 1210.2.b.k 8
55.j even 10 1 inner 880.2.cd.b 16
60.h even 2 1 990.2.ba.h 16
132.o even 10 1 990.2.ba.h 16
220.n odd 10 1 110.2.j.b ✓ 16
220.n odd 10 1 1210.2.b.k 8
220.o even 10 1 1210.2.b.l 8
220.v even 20 1 550.2.h.j 8
220.v even 20 1 550.2.h.n 8
220.v even 20 1 6050.2.a.dd 4
220.v even 20 1 6050.2.a.di 4
220.w odd 20 1 6050.2.a.da 4
220.w odd 20 1 6050.2.a.dl 4
660.bd even 10 1 990.2.ba.h 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
110.2.j.b ✓ 16 4.b odd 2 1
110.2.j.b ✓ 16 20.d odd 2 1
110.2.j.b ✓ 16 44.h odd 10 1
110.2.j.b ✓ 16 220.n odd 10 1
550.2.h.j 8 20.e even 4 1
550.2.h.j 8 220.v even 20 1
550.2.h.n 8 20.e even 4 1
550.2.h.n 8 220.v even 20 1
880.2.cd.b 16 1.a even 1 1 trivial
880.2.cd.b 16 5.b even 2 1 inner
880.2.cd.b 16 11.c even 5 1 inner
880.2.cd.b 16 55.j even 10 1 inner
990.2.ba.h 16 12.b even 2 1
990.2.ba.h 16 60.h even 2 1
990.2.ba.h 16 132.o even 10 1
990.2.ba.h 16 660.bd even 10 1
1210.2.b.k 8 44.h odd 10 1
1210.2.b.k 8 220.n odd 10 1
1210.2.b.l 8 44.g even 10 1
1210.2.b.l 8 220.o even 10 1
6050.2.a.da 4 220.w odd 20 1
6050.2.a.dd 4 220.v even 20 1
6050.2.a.di 4 220.v even 20 1
6050.2.a.dl 4 220.w odd 20 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 4T_{3}^{6} + 6T_{3}^{4} + T_{3}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(880, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{8} - 4 T^{6} + 6 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} + 6 T^{15} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( T^{16} - 56 T^{14} + \cdots + 160000 \) Copy content Toggle raw display
$11$ \( (T^{8} - 10 T^{7} + \cdots + 14641)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} - 44 T^{14} + \cdots + 160000 \) Copy content Toggle raw display
$17$ \( T^{16} - 76 T^{14} + \cdots + 625 \) Copy content Toggle raw display
$19$ \( (T^{8} - 8 T^{7} + \cdots + 2401)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 76 T^{6} + \cdots + 1936)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} - 8 T^{7} + \cdots + 144400)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} - 2 T^{7} + \cdots + 3168400)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 401469235456 \) Copy content Toggle raw display
$41$ \( (T^{8} - 20 T^{7} + \cdots + 14641)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 15 T^{2} + 25)^{4} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 10038758560000 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 20851360000 \) Copy content Toggle raw display
$59$ \( (T^{8} + 9 T^{6} + \cdots + 57121)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} - 10 T^{7} + \cdots + 1210000)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 294 T^{6} + \cdots + 5285401)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} - 28 T^{7} + \cdots + 28944400)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} - 60 T^{14} + \cdots + 5764801 \) Copy content Toggle raw display
$79$ \( (T^{8} + 18 T^{7} + \cdots + 99856)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} - 86 T^{14} + \cdots + 9150625 \) Copy content Toggle raw display
$89$ \( (T^{2} + T - 31)^{8} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 10\!\cdots\!21 \) Copy content Toggle raw display
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