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Properties

Label 448.3.s.d
Level $448$
Weight $3$
Character orbit 448.s
Analytic conductor $12.207$
Analytic rank $0$
Dimension $4$
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] = [448,3,Mod(129,448)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(448, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("448.129");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 448.s (of order \(6\), degree \(2\), 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: \(12.2071158433\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 14)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} + \beta_1 + 2) q^{3} + ( - 2 \beta_{3} + 2 \beta_{2} + \cdots + 1) q^{5}+ \cdots + (2 \beta_{3} + 2 \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + \beta_1 + 2) q^{3} + ( - 2 \beta_{3} + 2 \beta_{2} + \cdots + 1) q^{5}+ \cdots + ( - 36 \beta_{3} + 18 \beta_{2} - 36) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{3} + 6 q^{5} + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 6 q^{3} + 6 q^{5} + 8 q^{7} - 18 q^{11} - 36 q^{15} - 30 q^{17} - 6 q^{19} - 54 q^{21} + 30 q^{23} + 4 q^{25} - 48 q^{29} - 42 q^{31} - 90 q^{33} + 42 q^{35} + 62 q^{37} + 12 q^{39} + 8 q^{43} - 144 q^{45} + 174 q^{47} - 20 q^{49} - 54 q^{51} + 78 q^{53} + 12 q^{57} + 78 q^{59} + 42 q^{61} - 216 q^{63} - 84 q^{65} + 58 q^{67} - 24 q^{71} + 318 q^{73} - 132 q^{75} - 126 q^{77} + 110 q^{79} + 18 q^{81} + 36 q^{85} - 144 q^{87} - 378 q^{89} - 24 q^{91} + 138 q^{93} - 30 q^{95} - 144 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

\(\beta_{1}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 4\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 2\nu ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -4\beta_{3} + 2\beta_{2} ) / 3 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(1\) \(1 + \beta_{1}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
129.1
−0.707107 − 1.22474i
0.707107 + 1.22474i
−0.707107 + 1.22474i
0.707107 − 1.22474i
0 −0.621320 − 0.358719i 0 5.74264 − 3.31552i 0 6.24264 − 3.16693i 0 −4.24264 − 7.34847i 0
129.2 0 3.62132 + 2.09077i 0 −2.74264 + 1.58346i 0 −2.24264 + 6.63103i 0 4.24264 + 7.34847i 0
257.1 0 −0.621320 + 0.358719i 0 5.74264 + 3.31552i 0 6.24264 + 3.16693i 0 −4.24264 + 7.34847i 0
257.2 0 3.62132 − 2.09077i 0 −2.74264 − 1.58346i 0 −2.24264 − 6.63103i 0 4.24264 − 7.34847i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 448.3.s.d 4
4.b odd 2 1 448.3.s.c 4
7.d odd 6 1 inner 448.3.s.d 4
8.b even 2 1 14.3.d.a ✓ 4
8.d odd 2 1 112.3.s.b 4
24.f even 2 1 1008.3.cg.l 4
24.h odd 2 1 126.3.n.c 4
28.f even 6 1 448.3.s.c 4
40.f even 2 1 350.3.k.a 4
40.i odd 4 2 350.3.i.a 8
56.e even 2 1 784.3.s.c 4
56.h odd 2 1 98.3.d.a 4
56.j odd 6 1 14.3.d.a ✓ 4
56.j odd 6 1 98.3.b.b 4
56.k odd 6 1 784.3.c.e 4
56.k odd 6 1 784.3.s.c 4
56.m even 6 1 112.3.s.b 4
56.m even 6 1 784.3.c.e 4
56.p even 6 1 98.3.b.b 4
56.p even 6 1 98.3.d.a 4
168.i even 2 1 882.3.n.b 4
168.s odd 6 1 882.3.c.f 4
168.s odd 6 1 882.3.n.b 4
168.ba even 6 1 126.3.n.c 4
168.ba even 6 1 882.3.c.f 4
168.be odd 6 1 1008.3.cg.l 4
280.bk odd 6 1 350.3.k.a 4
280.bv even 12 2 350.3.i.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.3.d.a ✓ 4 8.b even 2 1
14.3.d.a ✓ 4 56.j odd 6 1
98.3.b.b 4 56.j odd 6 1
98.3.b.b 4 56.p even 6 1
98.3.d.a 4 56.h odd 2 1
98.3.d.a 4 56.p even 6 1
112.3.s.b 4 8.d odd 2 1
112.3.s.b 4 56.m even 6 1
126.3.n.c 4 24.h odd 2 1
126.3.n.c 4 168.ba even 6 1
350.3.i.a 8 40.i odd 4 2
350.3.i.a 8 280.bv even 12 2
350.3.k.a 4 40.f even 2 1
350.3.k.a 4 280.bk odd 6 1
448.3.s.c 4 4.b odd 2 1
448.3.s.c 4 28.f even 6 1
448.3.s.d 4 1.a even 1 1 trivial
448.3.s.d 4 7.d odd 6 1 inner
784.3.c.e 4 56.k odd 6 1
784.3.c.e 4 56.m even 6 1
784.3.s.c 4 56.e even 2 1
784.3.s.c 4 56.k odd 6 1
882.3.c.f 4 168.s odd 6 1
882.3.c.f 4 168.ba even 6 1
882.3.n.b 4 168.i even 2 1
882.3.n.b 4 168.s odd 6 1
1008.3.cg.l 4 24.f even 2 1
1008.3.cg.l 4 168.be odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 6T_{3}^{3} + 9T_{3}^{2} + 18T_{3} + 9 \) acting on \(S_{3}^{\mathrm{new}}(448, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 6 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$5$ \( T^{4} - 6 T^{3} + \cdots + 441 \) Copy content Toggle raw display
$7$ \( T^{4} - 8 T^{3} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( T^{4} + 18 T^{3} + \cdots + 3969 \) Copy content Toggle raw display
$13$ \( T^{4} + 264T^{2} + 7056 \) Copy content Toggle raw display
$17$ \( T^{4} + 30 T^{3} + \cdots + 2601 \) Copy content Toggle raw display
$19$ \( T^{4} + 6 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$23$ \( T^{4} - 30 T^{3} + \cdots + 3969 \) Copy content Toggle raw display
$29$ \( (T^{2} + 24 T + 72)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 42 T^{3} + \cdots + 1447209 \) Copy content Toggle raw display
$37$ \( T^{4} - 62 T^{3} + \cdots + 36481 \) Copy content Toggle raw display
$41$ \( T^{4} + 1224 T^{2} + 345744 \) Copy content Toggle raw display
$43$ \( (T^{2} - 4 T - 68)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 174 T^{3} + \cdots + 6335289 \) Copy content Toggle raw display
$53$ \( T^{4} - 78 T^{3} + \cdots + 1520289 \) Copy content Toggle raw display
$59$ \( T^{4} - 78 T^{3} + \cdots + 10517049 \) Copy content Toggle raw display
$61$ \( T^{4} - 42 T^{3} + \cdots + 35964009 \) Copy content Toggle raw display
$67$ \( T^{4} - 58 T^{3} + \cdots + 10297681 \) Copy content Toggle raw display
$71$ \( (T^{2} + 12 T - 1764)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 318 T^{3} + \cdots + 47485881 \) Copy content Toggle raw display
$79$ \( T^{4} - 110 T^{3} + \cdots + 6630625 \) Copy content Toggle raw display
$83$ \( T^{4} + 27936 T^{2} + 189778176 \) Copy content Toggle raw display
$89$ \( T^{4} + 378 T^{3} + \cdots + 71419401 \) Copy content Toggle raw display
$97$ \( T^{4} + 11016 T^{2} + 6780816 \) Copy content Toggle raw display
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