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Properties

Label 896.2.i.e
Level $896$
Weight $2$
Character orbit 896.i
Analytic conductor $7.155$
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] = [896,2,Mod(513,896)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(896, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([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("896.513");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 896 = 2^{7} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 896.i (of order \(3\), 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: \(7.15459602111\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 8x^{6} + 21x^{4} - 4x^{3} + 28x^{2} + 12x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{6} + \beta_{4}) q^{3} + (\beta_{7} + \beta_{5} + \beta_{4} - 1) q^{5} + (\beta_{7} + \beta_{5} + \beta_{3}) q^{7} + ( - \beta_{6} + \beta_{4} + \beta_{3} + \cdots - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{6} + \beta_{4}) q^{3} + (\beta_{7} + \beta_{5} + \beta_{4} - 1) q^{5} + (\beta_{7} + \beta_{5} + \beta_{3}) q^{7} + ( - \beta_{6} + \beta_{4} + \beta_{3} + \cdots - 1) q^{9}+ \cdots + (5 \beta_{7} + 2 \beta_{2} + 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{3} - 2 q^{5} - 2 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{3} - 2 q^{5} - 2 q^{7} - 4 q^{9} + 16 q^{13} - 4 q^{15} + 4 q^{17} + 4 q^{19} + 2 q^{21} + 4 q^{23} - 16 q^{27} - 16 q^{29} + 10 q^{31} + 8 q^{33} - 20 q^{35} + 2 q^{37} + 22 q^{39} - 24 q^{41} - 8 q^{43} + 14 q^{47} - 4 q^{49} - 8 q^{51} - 10 q^{53} - 48 q^{55} + 24 q^{57} + 6 q^{59} - 6 q^{61} + 14 q^{63} - 8 q^{65} + 22 q^{67} + 68 q^{69} + 48 q^{71} + 16 q^{73} + 32 q^{75} - 26 q^{77} - 8 q^{79} + 24 q^{81} - 32 q^{83} + 12 q^{85} - 18 q^{87} + 8 q^{89} + 20 q^{91} - 2 q^{93} + 16 q^{95} - 32 q^{97} + 60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 8x^{6} + 21x^{4} - 4x^{3} + 28x^{2} + 12x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 68\nu^{7} - 215\nu^{6} + 357\nu^{5} + 646\nu^{4} - 1444\nu^{3} + 1156\nu^{2} + 561\nu + 5468 ) / 4243 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 84\nu^{7} - 16\nu^{6} + 441\nu^{5} + 798\nu^{4} + 3208\nu^{3} + 1428\nu^{2} + 693\nu + 2262 ) / 4243 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -754\nu^{7} + 1760\nu^{6} - 6080\nu^{5} + 1323\nu^{4} - 13440\nu^{3} + 12640\nu^{2} - 16828\nu + 5760 ) / 12729 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -815\nu^{7} + 3388\nu^{6} - 11704\nu^{5} + 15594\nu^{4} - 25872\nu^{3} + 24332\nu^{2} - 56579\nu + 11088 ) / 12729 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1052\nu^{7} - 2827\nu^{6} + 9766\nu^{5} - 6978\nu^{4} + 21588\nu^{3} - 20303\nu^{2} + 17165\nu - 9252 ) / 12729 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -556\nu^{7} + 510\nu^{6} - 2919\nu^{5} - 5282\nu^{4} - 8909\nu^{3} - 9452\nu^{2} - 4587\nu - 13760 ) / 4243 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + 2\beta_{4} + \beta_{3} + \beta_{2} + \beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + 5\beta_{3} + 2\beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -8\beta_{6} - 2\beta_{5} - 9\beta_{4} - 10\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -8\beta_{7} - 20\beta_{6} - 8\beta_{5} - 18\beta_{4} - 33\beta_{3} - 20\beta_{2} - 33\beta _1 + 18 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -20\beta_{7} - 83\beta_{3} - 61\beta_{2} + 58 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 164\beta_{6} + 61\beta_{5} + 146\beta_{4} + 243\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/896\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(645\)
\(\chi(n)\) \(1\) \(-1 + \beta_{4}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
513.1
1.39083 + 2.40898i
−0.758290 − 1.31340i
−0.276205 − 0.478401i
0.643668 + 1.11487i
1.39083 − 2.40898i
−0.758290 + 1.31340i
−0.276205 + 0.478401i
0.643668 − 1.11487i
0 −0.977971 − 1.69390i 0 −0.351579 + 0.608953i 0 −2.63323 − 0.257073i 0 −0.412855 + 0.715087i 0
513.2 0 −0.408297 − 0.707191i 0 0.269222 − 0.466307i 0 2.28580 − 1.33233i 0 1.16659 − 2.02059i 0
513.3 0 1.07122 + 1.85540i 0 −1.93917 + 3.35875i 0 −0.886763 + 2.49272i 0 −0.795012 + 1.37700i 0
513.4 0 1.31505 + 2.27774i 0 1.02153 − 1.76934i 0 0.234193 − 2.63537i 0 −1.95872 + 3.39260i 0
641.1 0 −0.977971 + 1.69390i 0 −0.351579 − 0.608953i 0 −2.63323 + 0.257073i 0 −0.412855 − 0.715087i 0
641.2 0 −0.408297 + 0.707191i 0 0.269222 + 0.466307i 0 2.28580 + 1.33233i 0 1.16659 + 2.02059i 0
641.3 0 1.07122 − 1.85540i 0 −1.93917 − 3.35875i 0 −0.886763 − 2.49272i 0 −0.795012 − 1.37700i 0
641.4 0 1.31505 − 2.27774i 0 1.02153 + 1.76934i 0 0.234193 + 2.63537i 0 −1.95872 − 3.39260i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 513.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 896.2.i.e yes 8
4.b odd 2 1 896.2.i.b ✓ 8
7.c even 3 1 inner 896.2.i.e yes 8
7.c even 3 1 6272.2.a.bd 4
7.d odd 6 1 6272.2.a.bq 4
8.b even 2 1 896.2.i.d yes 8
8.d odd 2 1 896.2.i.g yes 8
28.f even 6 1 6272.2.a.ba 4
28.g odd 6 1 896.2.i.b ✓ 8
28.g odd 6 1 6272.2.a.bt 4
56.j odd 6 1 6272.2.a.be 4
56.k odd 6 1 896.2.i.g yes 8
56.k odd 6 1 6272.2.a.z 4
56.m even 6 1 6272.2.a.bu 4
56.p even 6 1 896.2.i.d yes 8
56.p even 6 1 6272.2.a.bp 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
896.2.i.b ✓ 8 4.b odd 2 1
896.2.i.b ✓ 8 28.g odd 6 1
896.2.i.d yes 8 8.b even 2 1
896.2.i.d yes 8 56.p even 6 1
896.2.i.e yes 8 1.a even 1 1 trivial
896.2.i.e yes 8 7.c even 3 1 inner
896.2.i.g yes 8 8.d odd 2 1
896.2.i.g yes 8 56.k odd 6 1
6272.2.a.z 4 56.k odd 6 1
6272.2.a.ba 4 28.f even 6 1
6272.2.a.bd 4 7.c even 3 1
6272.2.a.be 4 56.j odd 6 1
6272.2.a.bp 4 56.p even 6 1
6272.2.a.bq 4 7.d odd 6 1
6272.2.a.bt 4 28.g odd 6 1
6272.2.a.bu 4 56.m even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(896, [\chi])\):

\( T_{3}^{8} - 2T_{3}^{7} + 10T_{3}^{6} - 4T_{3}^{5} + 43T_{3}^{4} - 12T_{3}^{3} + 118T_{3}^{2} + 72T_{3} + 81 \) Copy content Toggle raw display
\( T_{5}^{8} + 2T_{5}^{7} + 12T_{5}^{6} - 12T_{5}^{5} + 65T_{5}^{4} + 4T_{5}^{3} + 28T_{5}^{2} - 6T_{5} + 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 2 T^{7} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( T^{8} + 2 T^{7} + \cdots + 9 \) Copy content Toggle raw display
$7$ \( T^{8} + 2 T^{7} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( T^{8} + 38 T^{6} + \cdots + 38809 \) Copy content Toggle raw display
$13$ \( (T^{4} - 8 T^{3} + \cdots - 116)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} - 4 T^{7} + \cdots + 9409 \) Copy content Toggle raw display
$19$ \( T^{8} - 4 T^{7} + \cdots + 22801 \) Copy content Toggle raw display
$23$ \( T^{8} - 4 T^{7} + \cdots + 15625 \) Copy content Toggle raw display
$29$ \( (T^{4} + 8 T^{3} - 20 T^{2} + \cdots + 12)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} - 10 T^{7} + \cdots + 121 \) Copy content Toggle raw display
$37$ \( T^{8} - 2 T^{7} + \cdots + 63001 \) Copy content Toggle raw display
$41$ \( (T^{4} + 12 T^{3} + \cdots - 44)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 4 T^{3} - 32 T^{2} + \cdots - 64)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 14 T^{7} + \cdots + 40401 \) Copy content Toggle raw display
$53$ \( T^{8} + 10 T^{7} + \cdots + 366025 \) Copy content Toggle raw display
$59$ \( T^{8} - 6 T^{7} + \cdots + 9409 \) Copy content Toggle raw display
$61$ \( T^{8} + 6 T^{7} + \cdots + 12652249 \) Copy content Toggle raw display
$67$ \( T^{8} - 22 T^{7} + \cdots + 121 \) Copy content Toggle raw display
$71$ \( (T - 6)^{8} \) Copy content Toggle raw display
$73$ \( T^{8} - 16 T^{7} + \cdots + 30946969 \) Copy content Toggle raw display
$79$ \( T^{8} + 8 T^{7} + \cdots + 10029889 \) Copy content Toggle raw display
$83$ \( (T^{4} + 16 T^{3} + \cdots - 14640)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} - 8 T^{7} + \cdots + 1505529 \) Copy content Toggle raw display
$97$ \( (T^{4} + 16 T^{3} + \cdots - 6828)^{2} \) Copy content Toggle raw display
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