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Properties

Label 880.2.bo.h
Level $880$
Weight $2$
Character orbit 880.bo
Analytic conductor $7.027$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [880,2,Mod(81,880)]
 
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, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("880.81");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 880 = 2^{4} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 880.bo (of order \(5\), 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: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.13140625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + 5x^{6} - 3x^{5} + 4x^{4} + 3x^{3} + 5x^{2} + 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 55)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{6} + \beta_{5} + \beta_{4} + \cdots + 1) q^{3}+ \cdots + ( - 2 \beta_{7} + 2 \beta_{6} + \cdots - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{6} + \beta_{5} + \beta_{4} + \cdots + 1) q^{3}+ \cdots + (4 \beta_{7} + \beta_{6} + 5 \beta_{5} + \cdots + 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 5 q^{3} + 2 q^{5} + q^{7} - 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 5 q^{3} + 2 q^{5} + q^{7} - 5 q^{9} - 3 q^{11} - 2 q^{13} - 5 q^{15} - 13 q^{17} - 15 q^{19} - 20 q^{21} - 10 q^{23} - 2 q^{25} - 10 q^{27} - 9 q^{29} + 10 q^{31} + 5 q^{33} + 4 q^{35} + 24 q^{37} - 21 q^{39} + 8 q^{41} + 38 q^{43} + q^{49} - q^{51} + 13 q^{53} - 7 q^{55} - 45 q^{57} + 27 q^{59} + 6 q^{61} - 25 q^{63} + 2 q^{65} + 38 q^{67} - q^{69} + 20 q^{71} + 13 q^{73} - 5 q^{75} + 34 q^{77} - 37 q^{79} + 8 q^{81} - 27 q^{83} - 12 q^{85} - 38 q^{87} - 16 q^{89} - 44 q^{91} - 35 q^{93} + 15 q^{95} + 24 q^{97} + 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} + 5x^{6} - 3x^{5} + 4x^{4} + 3x^{3} + 5x^{2} + 3x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} + 2\nu^{6} - 3\nu^{5} - 4\nu^{3} - 7\nu^{2} - 12\nu - 7 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} - 7\nu^{5} + 20\nu^{4} - 16\nu^{3} + 19\nu^{2} + 6\nu + 9 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} + 4\nu^{6} - 9\nu^{5} + 12\nu^{4} - 16\nu^{3} + 13\nu^{2} - 10\nu - 1 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{7} + 10\nu^{6} - 17\nu^{5} + 8\nu^{4} - 4\nu^{3} - 13\nu^{2} - 8\nu - 5 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3\nu^{7} - 12\nu^{6} + 23\nu^{5} - 20\nu^{4} + 16\nu^{3} + \nu^{2} + 6\nu - 1 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -5\nu^{7} + 18\nu^{6} - 35\nu^{5} + 32\nu^{4} - 28\nu^{3} - 11\nu^{2} - 12\nu - 7 ) / 8 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{5} + \beta_{4} - \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + 3\beta_{6} + 2\beta_{5} + \beta_{4} - 3\beta_{2} - 2\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{7} + 4\beta_{6} + \beta_{4} - \beta_{3} - 5\beta_{2} - 4\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{7} - 6\beta_{5} - 4\beta_{3} - 6\beta_{2} - 6\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -16\beta_{6} - 16\beta_{5} - 6\beta_{4} - 6\beta_{3} - 7\beta _1 - 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -16\beta_{7} - 51\beta_{6} - 29\beta_{5} - 23\beta_{4} + 29\beta_{2} - 16 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(111\) \(177\) \(321\) \(661\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{7}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
81.1
0.418926 − 1.28932i
−0.227943 + 0.701538i
−0.386111 − 0.280526i
1.69513 + 1.23158i
0.418926 + 1.28932i
−0.227943 − 0.701538i
−0.386111 + 0.280526i
1.69513 − 1.23158i
0 −0.177837 + 0.547326i 0 0.809017 − 0.587785i 0 1.12773 + 3.47080i 0 2.15911 + 1.56869i 0
81.2 0 0.868820 − 2.67395i 0 0.809017 − 0.587785i 0 −0.318714 − 0.980901i 0 −3.96813 − 2.88301i 0
401.1 0 0.261370 + 0.189896i 0 −0.309017 + 0.951057i 0 2.17239 − 1.57833i 0 −0.894797 − 2.75390i 0
401.2 0 1.54765 + 1.12443i 0 −0.309017 + 0.951057i 0 −2.48141 + 1.80285i 0 0.203814 + 0.627276i 0
641.1 0 −0.177837 − 0.547326i 0 0.809017 + 0.587785i 0 1.12773 − 3.47080i 0 2.15911 − 1.56869i 0
641.2 0 0.868820 + 2.67395i 0 0.809017 + 0.587785i 0 −0.318714 + 0.980901i 0 −3.96813 + 2.88301i 0
801.1 0 0.261370 − 0.189896i 0 −0.309017 − 0.951057i 0 2.17239 + 1.57833i 0 −0.894797 + 2.75390i 0
801.2 0 1.54765 − 1.12443i 0 −0.309017 − 0.951057i 0 −2.48141 − 1.80285i 0 0.203814 − 0.627276i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 81.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 880.2.bo.h 8
4.b odd 2 1 55.2.g.b ✓ 8
11.c even 5 1 inner 880.2.bo.h 8
11.c even 5 1 9680.2.a.cn 4
11.d odd 10 1 9680.2.a.cm 4
12.b even 2 1 495.2.n.e 8
20.d odd 2 1 275.2.h.a 8
20.e even 4 2 275.2.z.a 16
44.c even 2 1 605.2.g.k 8
44.g even 10 1 605.2.a.k 4
44.g even 10 2 605.2.g.e 8
44.g even 10 1 605.2.g.k 8
44.h odd 10 1 55.2.g.b ✓ 8
44.h odd 10 1 605.2.a.j 4
44.h odd 10 2 605.2.g.m 8
132.n odd 10 1 5445.2.a.bi 4
132.o even 10 1 495.2.n.e 8
132.o even 10 1 5445.2.a.bp 4
220.n odd 10 1 275.2.h.a 8
220.n odd 10 1 3025.2.a.bd 4
220.o even 10 1 3025.2.a.w 4
220.v even 20 2 275.2.z.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.g.b ✓ 8 4.b odd 2 1
55.2.g.b ✓ 8 44.h odd 10 1
275.2.h.a 8 20.d odd 2 1
275.2.h.a 8 220.n odd 10 1
275.2.z.a 16 20.e even 4 2
275.2.z.a 16 220.v even 20 2
495.2.n.e 8 12.b even 2 1
495.2.n.e 8 132.o even 10 1
605.2.a.j 4 44.h odd 10 1
605.2.a.k 4 44.g even 10 1
605.2.g.e 8 44.g even 10 2
605.2.g.k 8 44.c even 2 1
605.2.g.k 8 44.g even 10 1
605.2.g.m 8 44.h odd 10 2
880.2.bo.h 8 1.a even 1 1 trivial
880.2.bo.h 8 11.c even 5 1 inner
3025.2.a.w 4 220.o even 10 1
3025.2.a.bd 4 220.n odd 10 1
5445.2.a.bi 4 132.n odd 10 1
5445.2.a.bp 4 132.o even 10 1
9680.2.a.cm 4 11.d odd 10 1
9680.2.a.cn 4 11.c even 5 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 5T_{3}^{7} + 18T_{3}^{6} - 35T_{3}^{5} + 39T_{3}^{4} - 15T_{3}^{3} + 12T_{3}^{2} - 5T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(880, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 5 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} - T^{7} + \cdots + 961 \) Copy content Toggle raw display
$11$ \( T^{8} + 3 T^{7} + \cdots + 14641 \) Copy content Toggle raw display
$13$ \( T^{8} + 2 T^{7} + \cdots + 19321 \) Copy content Toggle raw display
$17$ \( T^{8} + 13 T^{7} + \cdots + 361 \) Copy content Toggle raw display
$19$ \( T^{8} + 15 T^{7} + \cdots + 625 \) Copy content Toggle raw display
$23$ \( (T^{4} + 5 T^{3} + 4 T^{2} + \cdots - 11)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + 9 T^{7} + \cdots + 203401 \) Copy content Toggle raw display
$31$ \( T^{8} - 10 T^{7} + \cdots + 390625 \) Copy content Toggle raw display
$37$ \( T^{8} - 24 T^{7} + \cdots + 1324801 \) Copy content Toggle raw display
$41$ \( T^{8} - 8 T^{7} + \cdots + 101761 \) Copy content Toggle raw display
$43$ \( (T^{4} - 19 T^{3} + \cdots + 211)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 23 T^{6} + \cdots + 28561 \) Copy content Toggle raw display
$53$ \( T^{8} - 13 T^{7} + \cdots + 885481 \) Copy content Toggle raw display
$59$ \( T^{8} - 27 T^{7} + \cdots + 687241 \) Copy content Toggle raw display
$61$ \( T^{8} - 6 T^{7} + \cdots + 28561 \) Copy content Toggle raw display
$67$ \( (T^{4} - 19 T^{3} + \cdots - 4079)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} - 20 T^{7} + \cdots + 17161 \) Copy content Toggle raw display
$73$ \( T^{8} - 13 T^{7} + \cdots + 121 \) Copy content Toggle raw display
$79$ \( T^{8} + 37 T^{7} + \cdots + 45954841 \) Copy content Toggle raw display
$83$ \( T^{8} + 27 T^{7} + \cdots + 2886601 \) Copy content Toggle raw display
$89$ \( (T^{4} + 8 T^{3} + \cdots + 1861)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} - 24 T^{7} + \cdots + 9066121 \) Copy content Toggle raw display
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