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Properties

Label 1344.4.b.e
Level $1344$
Weight $4$
Character orbit 1344.b
Analytic conductor $79.299$
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] = [1344,4,Mod(895,1344)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1344, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1344.895");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1344.b (of order \(2\), degree \(1\), 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: \(79.2985670477\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 158x^{6} + 8461x^{4} + 180672x^{2} + 1232100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{15}\cdot 3^{3} \)
Twist minimal: no (minimal twist has level 336)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - 3 q^{3} + \beta_{5} q^{5} - \beta_{3} q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 q^{3} + \beta_{5} q^{5} - \beta_{3} q^{7} + 9 q^{9} + ( - \beta_{5} + \beta_{3} + \cdots + \beta_1) q^{11}+ \cdots + ( - 9 \beta_{5} + 9 \beta_{3} + \cdots + 9 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 24 q^{3} + 4 q^{7} + 72 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 24 q^{3} + 4 q^{7} + 72 q^{9} - 56 q^{19} - 12 q^{21} - 656 q^{25} - 216 q^{27} - 240 q^{29} - 320 q^{31} + 600 q^{35} - 392 q^{37} + 816 q^{47} - 16 q^{49} - 288 q^{53} + 456 q^{55} + 168 q^{57} - 1824 q^{59} + 36 q^{63} - 816 q^{65} + 1968 q^{75} + 2064 q^{77} + 648 q^{81} + 1680 q^{83} - 2568 q^{85} + 720 q^{87} - 864 q^{91} + 960 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 158x^{6} + 8461x^{4} + 180672x^{2} + 1232100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -2\nu^{7} - 316\nu^{5} - 14702\nu^{3} - 185964\nu ) / 11655 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 169 \nu^{7} + 740 \nu^{6} - 20597 \nu^{5} + 88060 \nu^{4} - 690094 \nu^{3} + 2786840 \nu^{2} + \cdots + 21107760 ) / 93240 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 169 \nu^{7} - 740 \nu^{6} - 20597 \nu^{5} - 88060 \nu^{4} - 690094 \nu^{3} - 2786840 \nu^{2} + \cdots - 21107760 ) / 93240 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 251\nu^{7} + 31333\nu^{5} + 1117496\nu^{3} + 11580252\nu ) / 93240 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -257\nu^{7} - 31171\nu^{5} - 1027292\nu^{3} - 8481804\nu ) / 93240 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} + 119\nu^{4} + 3892\nu^{2} + 33438 ) / 21 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5\nu^{6} + 613\nu^{4} + 20630\nu^{2} + 177216 ) / 63 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -6\beta_{5} + 2\beta_{4} + 6\beta_{3} + 6\beta_{2} + \beta_1 ) / 24 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} + 3\beta_{3} - 3\beta_{2} - 234 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 396\beta_{5} - 44\beta_{4} - 336\beta_{3} - 336\beta_{2} + 47\beta_1 ) / 24 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 21\beta_{7} - 100\beta_{6} - 195\beta_{3} + 195\beta_{2} + 11868 ) / 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -26136\beta_{5} + 752\beta_{4} + 20892\beta_{3} + 20892\beta_{2} - 9737\beta_1 ) / 24 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -2499\beta_{7} + 8134\beta_{6} + 11529\beta_{3} - 11529\beta_{2} - 702192 ) / 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1776384\beta_{5} + 18664\beta_{4} - 1388892\beta_{3} - 1388892\beta_{2} + 960107\beta_1 ) / 24 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(449\) \(577\) \(1093\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
895.1
6.15149i
− 5.92762i
8.49618i
− 3.58293i
3.58293i
− 8.49618i
5.92762i
− 6.15149i
0 −3.00000 0 − 20.9291i 0 17.6950 + 5.46693i 0 9.00000 0
895.2 0 −3.00000 0 − 17.7376i 0 −2.09706 + 18.4011i 0 9.00000 0
895.3 0 −3.00000 0 − 7.52281i 0 4.86375 − 17.8702i 0 9.00000 0
895.4 0 −3.00000 0 − 4.33129i 0 −18.4617 − 1.47188i 0 9.00000 0
895.5 0 −3.00000 0 4.33129i 0 −18.4617 + 1.47188i 0 9.00000 0
895.6 0 −3.00000 0 7.52281i 0 4.86375 + 17.8702i 0 9.00000 0
895.7 0 −3.00000 0 17.7376i 0 −2.09706 − 18.4011i 0 9.00000 0
895.8 0 −3.00000 0 20.9291i 0 17.6950 − 5.46693i 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 895.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1344.4.b.e 8
4.b odd 2 1 1344.4.b.f 8
7.b odd 2 1 1344.4.b.f 8
8.b even 2 1 336.4.b.f yes 8
8.d odd 2 1 336.4.b.e ✓ 8
24.f even 2 1 1008.4.b.i 8
24.h odd 2 1 1008.4.b.k 8
28.d even 2 1 inner 1344.4.b.e 8
56.e even 2 1 336.4.b.f yes 8
56.h odd 2 1 336.4.b.e ✓ 8
168.e odd 2 1 1008.4.b.k 8
168.i even 2 1 1008.4.b.i 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.4.b.e ✓ 8 8.d odd 2 1
336.4.b.e ✓ 8 56.h odd 2 1
336.4.b.f yes 8 8.b even 2 1
336.4.b.f yes 8 56.e even 2 1
1008.4.b.i 8 24.f even 2 1
1008.4.b.i 8 168.i even 2 1
1008.4.b.k 8 24.h odd 2 1
1008.4.b.k 8 168.e odd 2 1
1344.4.b.e 8 1.a even 1 1 trivial
1344.4.b.e 8 28.d even 2 1 inner
1344.4.b.f 8 4.b odd 2 1
1344.4.b.f 8 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(1344, [\chi])\):

\( T_{5}^{8} + 828T_{5}^{6} + 195588T_{5}^{4} + 11183616T_{5}^{2} + 146313216 \) Copy content Toggle raw display
\( T_{19}^{4} + 28T_{19}^{3} - 10380T_{19}^{2} - 500320T_{19} - 5935552 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T + 3)^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 828 T^{6} + \cdots + 146313216 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 13841287201 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 238089347136 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 624529833984 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 104135983727616 \) Copy content Toggle raw display
$19$ \( (T^{4} + 28 T^{3} + \cdots - 5935552)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 58\!\cdots\!44 \) Copy content Toggle raw display
$29$ \( (T^{4} + 120 T^{3} + \cdots + 242576208)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 160 T^{3} + \cdots - 1109014528)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 196 T^{3} + \cdots + 300652096)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 11\!\cdots\!96 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 24\!\cdots\!04 \) Copy content Toggle raw display
$47$ \( (T^{4} - 408 T^{3} + \cdots - 1755758592)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 144 T^{3} + \cdots - 460467504)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 912 T^{3} + \cdots - 4942861056)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 999247734374400 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 89\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 72\!\cdots\!04 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 86\!\cdots\!36 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 38\!\cdots\!04 \) Copy content Toggle raw display
$83$ \( (T^{4} - 840 T^{3} + \cdots - 57542400000)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 38\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 20\!\cdots\!56 \) Copy content Toggle raw display
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