Properties

Label 88.5.l.a
Level $88$
Weight $5$
Character orbit 88.l
Analytic conductor $9.097$
Analytic rank $0$
Dimension $8$
CM discriminant -8
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [88,5,Mod(3,88)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(88, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 5, 8]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("88.3");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 88 = 2^{3} \cdot 11 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 88.l (of order \(10\), degree \(4\), 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: \(9.09655675138\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: 8.0.64000000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{6} + 4x^{4} - 8x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{10}]$

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

\(f(q)\) \(=\) \( q + 4 \beta_{4} q^{2} + (7 \beta_{4} - \beta_{3} - 7 \beta_{2} - \beta_1 + 7) q^{3} + (16 \beta_{6} - 16 \beta_{4} + 16 \beta_{2} - 16) q^{4} + ( - 4 \beta_{7} - 4 \beta_{5} + 28 \beta_{2} - 28) q^{6} - 64 \beta_{2} q^{8} + ( - 14 \beta_{7} - 17 \beta_{6} + 81 \beta_{4} - 17 \beta_{2} - 14 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 4 \beta_{4} q^{2} + (7 \beta_{4} - \beta_{3} - 7 \beta_{2} - \beta_1 + 7) q^{3} + (16 \beta_{6} - 16 \beta_{4} + 16 \beta_{2} - 16) q^{4} + ( - 4 \beta_{7} - 4 \beta_{5} + 28 \beta_{2} - 28) q^{6} - 64 \beta_{2} q^{8} + ( - 14 \beta_{7} - 17 \beta_{6} + 81 \beta_{4} - 17 \beta_{2} - 14 \beta_1) q^{9} + ( - 21 \beta_{7} + 23 \beta_{2}) q^{11} + ( - 16 \beta_{7} + 112 \beta_{6} + 16 \beta_{5} - 112 \beta_{4} + \cdots + 32 \beta_1) q^{12}+ \cdots + (35 \beta_{7} + 10880 \beta_{6} - 35 \beta_{5} - 18816 \beta_{4} + \cdots - 9017) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{2} + 28 q^{3} - 32 q^{4} - 168 q^{6} - 128 q^{8} - 230 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{2} + 28 q^{3} - 32 q^{4} - 168 q^{6} - 128 q^{8} - 230 q^{9} + 46 q^{11} + 448 q^{12} - 512 q^{16} + 1148 q^{17} - 240 q^{18} + 1302 q^{19} + 184 q^{22} - 2688 q^{24} - 1250 q^{25} - 4718 q^{27} + 8192 q^{32} - 5754 q^{33} + 4592 q^{34} - 960 q^{36} - 3472 q^{38} + 2492 q^{41} + 7004 q^{43} - 2944 q^{44} + 7168 q^{48} - 4802 q^{49} - 5000 q^{50} + 2098 q^{51} + 3808 q^{54} + 13282 q^{57} - 714 q^{59} - 8192 q^{64} - 2576 q^{66} + 10268 q^{67} + 18368 q^{68} - 14720 q^{72} - 19012 q^{73} - 26250 q^{75} - 13888 q^{76} + 19072 q^{81} - 14952 q^{82} + 33558 q^{83} - 42024 q^{86} + 2944 q^{88} - 10948 q^{89} + 28672 q^{96} - 29946 q^{97} + 76832 q^{98} + 5290 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{6} + 4x^{4} - 8x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 4\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{3} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{4} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 8\beta_{6} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2\beta_{7} \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/88\mathbb{Z}\right)^\times\).

\(n\) \(23\) \(45\) \(57\)
\(\chi(n)\) \(-1\) \(-1\) \(-1 + \beta_{2} - \beta_{4} + \beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
−0.831254 + 1.14412i
0.831254 1.14412i
1.34500 + 0.437016i
−1.34500 0.437016i
−0.831254 1.14412i
0.831254 + 1.14412i
1.34500 0.437016i
−1.34500 + 0.437016i
−3.23607 + 2.35114i 1.44503 + 4.44734i 4.94427 15.2169i 0 −15.1325 10.9944i 0 19.7771 + 60.8676i 47.8397 34.7576i 0
3.2 −3.23607 + 2.35114i 5.55497 + 17.0964i 4.94427 15.2169i 0 −58.1724 42.2647i 0 19.7771 + 60.8676i −195.900 + 142.330i 0
27.1 1.23607 + 3.80423i −5.20500 3.78166i −12.9443 + 9.40456i 0 7.95254 24.4754i 0 −51.7771 37.6183i −12.2392 37.6685i 0
27.2 1.23607 + 3.80423i 12.2050 + 8.86745i −12.9443 + 9.40456i 0 −18.6476 + 57.3914i 0 −51.7771 37.6183i 45.3000 + 139.419i 0
59.1 −3.23607 2.35114i 1.44503 4.44734i 4.94427 + 15.2169i 0 −15.1325 + 10.9944i 0 19.7771 60.8676i 47.8397 + 34.7576i 0
59.2 −3.23607 2.35114i 5.55497 17.0964i 4.94427 + 15.2169i 0 −58.1724 + 42.2647i 0 19.7771 60.8676i −195.900 142.330i 0
75.1 1.23607 3.80423i −5.20500 + 3.78166i −12.9443 9.40456i 0 7.95254 + 24.4754i 0 −51.7771 + 37.6183i −12.2392 + 37.6685i 0
75.2 1.23607 3.80423i 12.2050 8.86745i −12.9443 9.40456i 0 −18.6476 57.3914i 0 −51.7771 + 37.6183i 45.3000 139.419i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
11.c even 5 1 inner
88.l odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 88.5.l.a 8
8.d odd 2 1 CM 88.5.l.a 8
11.c even 5 1 inner 88.5.l.a 8
88.l odd 10 1 inner 88.5.l.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
88.5.l.a 8 1.a even 1 1 trivial
88.5.l.a 8 8.d odd 2 1 CM
88.5.l.a 8 11.c even 5 1 inner
88.5.l.a 8 88.l odd 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 28 T_{3}^{7} + 588 T_{3}^{6} - 5306 T_{3}^{5} + 19550 T_{3}^{4} + 264124 T_{3}^{3} + 2058753 T_{3}^{2} - 1484938 T_{3} + 66569281 \) acting on \(S_{5}^{\mathrm{new}}(88, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 4 T^{3} + 16 T^{2} + 64 T + 256)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} - 28 T^{7} + 588 T^{6} + \cdots + 66569281 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} - 46 T^{7} + \cdots + 45\!\cdots\!61 \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( T^{8} - 1148 T^{7} + \cdots + 34\!\cdots\!01 \) Copy content Toggle raw display
$19$ \( T^{8} - 1302 T^{7} + \cdots + 54\!\cdots\!21 \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( T^{8} - 2492 T^{7} + \cdots + 41\!\cdots\!61 \) Copy content Toggle raw display
$43$ \( (T^{4} - 3502 T^{3} + \cdots - 794150195999)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( T^{8} + 714 T^{7} + \cdots + 53\!\cdots\!41 \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( (T^{4} - 5134 T^{3} + \cdots + 69368428937761)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( T^{8} + 19012 T^{7} + \cdots + 40\!\cdots\!41 \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( T^{8} - 33558 T^{7} + \cdots + 76\!\cdots\!81 \) Copy content Toggle raw display
$89$ \( (T^{4} + 5474 T^{3} + \cdots + 11\!\cdots\!01)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 29946 T^{7} + \cdots + 25\!\cdots\!21 \) Copy content Toggle raw display
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