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Properties

Label 448.6.e.b
Level $448$
Weight $6$
Character orbit 448.e
Analytic conductor $71.852$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [448,6,Mod(223,448)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(448, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("448.223");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 448.e (of order \(2\), degree \(1\), 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: \(71.8519512762\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 8714990x^{12} + 32218285101715x^{8} + 35910809316510003054x^{4} + 12224517313371131727050625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{40}\cdot 3^{4} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{8} q^{3} + \beta_{12} q^{5} + (\beta_{9} - 10 \beta_{4}) q^{7} + ( - \beta_1 + 13) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{8} q^{3} + \beta_{12} q^{5} + (\beta_{9} - 10 \beta_{4}) q^{7} + ( - \beta_1 + 13) q^{9} - \beta_{6} q^{11} + (4 \beta_{12} + 11 \beta_{11}) q^{13} + (3 \beta_{5} - 108 \beta_{4}) q^{15} + \beta_{13} q^{17} + (3 \beta_{15} - \beta_{8}) q^{19} + (2 \beta_{12} - 5 \beta_{11} + \beta_{2}) q^{21} + (6 \beta_{5} - 123 \beta_{4}) q^{23} + ( - 5 \beta_1 - 679) q^{25} + (3 \beta_{15} + 32 \beta_{8}) q^{27} + ( - 3 \beta_{3} + 2 \beta_{2}) q^{29} + ( - \beta_{14} - 7 \beta_{9} + \cdots - 3 \beta_{4}) q^{31}+ \cdots + (10 \beta_{7} - 227 \beta_{6}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 208 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 208 q^{9} - 10864 q^{25} + 108752 q^{49} + 9056 q^{57} + 282912 q^{65} - 61840 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 8714990x^{12} + 32218285101715x^{8} + 35910809316510003054x^{4} + 12224517313371131727050625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 563\nu^{12} + 2910598266\nu^{8} + 3477758381935686\nu^{4} - 11034018228592008651131 ) / 53786412350774068966 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 27866871589 \nu^{12} + \cdots - 41\!\cdots\!67 ) / 19\!\cdots\!67 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1785918165145 \nu^{12} + \cdots + 28\!\cdots\!65 ) / 11\!\cdots\!02 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 5506477996 \nu^{14} + \cdots - 32\!\cdots\!34 \nu^{2} ) / 43\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 141272 \nu^{14} - 2575937036905 \nu^{10} + \cdots - 23\!\cdots\!13 \nu^{2} ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 15482048578562 \nu^{14} + \cdots + 24\!\cdots\!83 \nu^{2} ) / 17\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 104595593466202 \nu^{14} + \cdots - 16\!\cdots\!83 \nu^{2} ) / 86\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 11\!\cdots\!03 \nu^{15} + \cdots - 16\!\cdots\!75 \nu ) / 25\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 27\!\cdots\!23 \nu^{15} + \cdots + 67\!\cdots\!75 \nu ) / 25\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 91\!\cdots\!41 \nu^{15} + \cdots + 22\!\cdots\!25 \nu ) / 21\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 99\!\cdots\!74 \nu^{15} + \cdots + 13\!\cdots\!75 \nu ) / 58\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 14\!\cdots\!98 \nu^{15} + \cdots + 19\!\cdots\!75 \nu ) / 76\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 94\!\cdots\!24 \nu^{15} + \cdots + 28\!\cdots\!75 \nu ) / 47\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 94\!\cdots\!23 \nu^{15} + \cdots - 29\!\cdots\!75 \nu ) / 25\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 27\!\cdots\!56 \nu^{15} + \cdots - 37\!\cdots\!25 \nu ) / 38\!\cdots\!50 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{12} + \beta_{11} + \beta_{10} + 4\beta_{9} + 4\beta_{8} + 2\beta_{5} + 2\beta_{4} ) / 16 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + \beta_{6} + 40\beta_{5} - 2893\beta_{4} ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 93 \beta_{15} - 63 \beta_{14} - 126 \beta_{13} - 5947 \beta_{12} + 6319 \beta_{11} + \cdots - 3156 \beta_{4} ) / 16 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -210\beta_{3} - 4013\beta_{2} - 24772\beta _1 - 8714990 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 433605 \beta_{15} - 145103 \beta_{14} + 290206 \beta_{13} + 19302514 \beta_{12} + \cdots - 4765778 \beta_{4} ) / 16 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -5819532\beta_{7} - 7554804\beta_{6} - 81673568\beta_{5} + 2997188447\beta_{4} ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 1295662836 \beta_{15} + 203516404 \beta_{14} + 407032808 \beta_{13} + 50903523835 \beta_{12} + \cdots + 5873512414 \beta_{4} ) / 16 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 2420047266\beta_{3} + 28724593867\beta_{2} + 40244915128\beta _1 + 11514480496670 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 3157348122867 \beta_{15} + 78222201561 \beta_{14} - 156444403122 \beta_{13} + \cdots + 2913564973932 \beta_{4} ) / 16 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 63010237692071\beta_{7} + 85346341156823\beta_{6} - 134232328929800\beta_{5} + 2795617519045307\beta_{4} ) / 8 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 67\!\cdots\!39 \beta_{15} + 681562265337835 \beta_{14} + \cdots + 14\!\cdots\!34 \beta_{4} ) / 16 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -2803488506150598\beta_{3} - 30927845787804852\beta_{2} + 81776154464268704\beta _1 + 18175255536347778847 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 12\!\cdots\!00 \beta_{15} + \cdots + 69\!\cdots\!42 \beta_{4} ) / 16 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 21\!\cdots\!09 \beta_{7} + \cdots - 13\!\cdots\!43 \beta_{4} ) / 8 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 19\!\cdots\!17 \beta_{15} + \cdots - 21\!\cdots\!96 \beta_{4} ) / 16 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/448\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
223.1
−26.8462 − 37.5434i
37.5434 + 26.8462i
−37.5434 + 26.8462i
26.8462 − 37.5434i
−20.5686 − 21.3238i
21.3238 + 20.5686i
−21.3238 + 20.5686i
20.5686 − 21.3238i
−20.5686 + 21.3238i
21.3238 − 20.5686i
−21.3238 − 20.5686i
20.5686 + 21.3238i
−26.8462 + 37.5434i
37.5434 − 26.8462i
−37.5434 − 26.8462i
26.8462 + 37.5434i
0 − 21.3944i 0 −36.1580 0 −128.779 − 14.9298i 0 −214.719 0
223.2 0 − 21.3944i 0 −36.1580 0 128.779 − 14.9298i 0 −214.719 0
223.3 0 − 21.3944i 0 36.1580 0 −128.779 + 14.9298i 0 −214.719 0
223.4 0 − 21.3944i 0 36.1580 0 128.779 + 14.9298i 0 −214.719 0
223.5 0 − 1.51026i 0 −59.8715 0 −83.7848 + 98.9298i 0 240.719 0
223.6 0 − 1.51026i 0 −59.8715 0 83.7848 + 98.9298i 0 240.719 0
223.7 0 − 1.51026i 0 59.8715 0 −83.7848 − 98.9298i 0 240.719 0
223.8 0 − 1.51026i 0 59.8715 0 83.7848 − 98.9298i 0 240.719 0
223.9 0 1.51026i 0 −59.8715 0 −83.7848 − 98.9298i 0 240.719 0
223.10 0 1.51026i 0 −59.8715 0 83.7848 − 98.9298i 0 240.719 0
223.11 0 1.51026i 0 59.8715 0 −83.7848 + 98.9298i 0 240.719 0
223.12 0 1.51026i 0 59.8715 0 83.7848 + 98.9298i 0 240.719 0
223.13 0 21.3944i 0 −36.1580 0 −128.779 + 14.9298i 0 −214.719 0
223.14 0 21.3944i 0 −36.1580 0 128.779 + 14.9298i 0 −214.719 0
223.15 0 21.3944i 0 36.1580 0 −128.779 − 14.9298i 0 −214.719 0
223.16 0 21.3944i 0 36.1580 0 128.779 − 14.9298i 0 −214.719 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 223.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
28.d even 2 1 inner
56.e even 2 1 inner
56.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 448.6.e.b ✓ 16
4.b odd 2 1 inner 448.6.e.b ✓ 16
7.b odd 2 1 inner 448.6.e.b ✓ 16
8.b even 2 1 inner 448.6.e.b ✓ 16
8.d odd 2 1 inner 448.6.e.b ✓ 16
28.d even 2 1 inner 448.6.e.b ✓ 16
56.e even 2 1 inner 448.6.e.b ✓ 16
56.h odd 2 1 inner 448.6.e.b ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
448.6.e.b ✓ 16 1.a even 1 1 trivial
448.6.e.b ✓ 16 4.b odd 2 1 inner
448.6.e.b ✓ 16 7.b odd 2 1 inner
448.6.e.b ✓ 16 8.b even 2 1 inner
448.6.e.b ✓ 16 8.d odd 2 1 inner
448.6.e.b ✓ 16 28.d even 2 1 inner
448.6.e.b ✓ 16 56.e even 2 1 inner
448.6.e.b ✓ 16 56.h odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 460T_{3}^{2} + 1044 \) acting on \(S_{6}^{\mathrm{new}}(448, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{4} + 460 T^{2} + 1044)^{4} \) Copy content Toggle raw display
$5$ \( (T^{4} - 4892 T^{2} + 4686516)^{4} \) Copy content Toggle raw display
$7$ \( (T^{8} + \cdots + 79\!\cdots\!01)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 477912 T^{2} + 6890086800)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} - 850780 T^{2} + 110092264884)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} + \cdots + 4683396324672)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} + 473548 T^{2} + 4723190676)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} + 1417536 T^{2} + 50463129600)^{4} \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots + 429253510053888)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots + 121665475835712)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots + 22\!\cdots\!92)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots + 30\!\cdots\!00)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots + 35\!\cdots\!88)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} + \cdots + 17\!\cdots\!88)^{4} \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 21\!\cdots\!48)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} + \cdots + 85\!\cdots\!16)^{4} \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 20\!\cdots\!00)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots + 17\!\cdots\!08)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 46\!\cdots\!00)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots + 24\!\cdots\!52)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 10\!\cdots\!00)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + \cdots + 174087562925844)^{4} \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 19\!\cdots\!68)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 58\!\cdots\!00)^{4} \) Copy content Toggle raw display
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