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Properties

Label 448.3.r.c
Level $448$
Weight $3$
Character orbit 448.r
Analytic conductor $12.207$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [448,3,Mod(191,448)]
 
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([3, 0, 2]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("448.191");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 448.r (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: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 224)
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 primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 5 \zeta_{12} q^{3} + 9 \zeta_{12}^{2} q^{5} + (3 \zeta_{12}^{3} - 8 \zeta_{12}) q^{7} + 16 \zeta_{12}^{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 5 \zeta_{12} q^{3} + 9 \zeta_{12}^{2} q^{5} + (3 \zeta_{12}^{3} - 8 \zeta_{12}) q^{7} + 16 \zeta_{12}^{2} q^{9} + 3 \zeta_{12} q^{11} - 16 q^{13} + 45 \zeta_{12}^{3} q^{15} + ( - 7 \zeta_{12}^{2} + 7) q^{17} + ( - 11 \zeta_{12}^{3} + 11 \zeta_{12}) q^{19} + ( - 25 \zeta_{12}^{2} - 15) q^{21} + ( - 19 \zeta_{12}^{3} + 19 \zeta_{12}) q^{23} + (56 \zeta_{12}^{2} - 56) q^{25} + 35 \zeta_{12}^{3} q^{27} + 32 q^{29} - 11 \zeta_{12} q^{31} + 15 \zeta_{12}^{2} q^{33} + ( - 45 \zeta_{12}^{3} - 27 \zeta_{12}) q^{35} - \zeta_{12}^{2} q^{37} - 80 \zeta_{12} q^{39} - 40 q^{41} - 40 \zeta_{12}^{3} q^{43} + (144 \zeta_{12}^{2} - 144) q^{45} + ( - 85 \zeta_{12}^{3} + 85 \zeta_{12}) q^{47} + (16 \zeta_{12}^{2} + 39) q^{49} + ( - 35 \zeta_{12}^{3} + 35 \zeta_{12}) q^{51} + ( - 7 \zeta_{12}^{2} + 7) q^{53} + 27 \zeta_{12}^{3} q^{55} + 55 q^{57} + 53 \zeta_{12} q^{59} + 79 \zeta_{12}^{2} q^{61} + ( - 80 \zeta_{12}^{3} - 48 \zeta_{12}) q^{63} - 144 \zeta_{12}^{2} q^{65} - 11 \zeta_{12} q^{67} + 95 q^{69} + 48 \zeta_{12}^{3} q^{71} + (143 \zeta_{12}^{2} - 143) q^{73} + (280 \zeta_{12}^{3} - 280 \zeta_{12}) q^{75} + ( - 15 \zeta_{12}^{2} - 9) q^{77} + ( - 35 \zeta_{12}^{3} + 35 \zeta_{12}) q^{79} + (31 \zeta_{12}^{2} - 31) q^{81} - 8 \zeta_{12}^{3} q^{83} + 63 q^{85} + 160 \zeta_{12} q^{87} + 97 \zeta_{12}^{2} q^{89} + ( - 48 \zeta_{12}^{3} + 128 \zeta_{12}) q^{91} - 55 \zeta_{12}^{2} q^{93} + 99 \zeta_{12} q^{95} - 88 q^{97} + 48 \zeta_{12}^{3} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 18 q^{5} + 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 18 q^{5} + 32 q^{9} - 64 q^{13} + 14 q^{17} - 110 q^{21} - 112 q^{25} + 128 q^{29} + 30 q^{33} - 2 q^{37} - 160 q^{41} - 288 q^{45} + 188 q^{49} + 14 q^{53} + 220 q^{57} + 158 q^{61} - 288 q^{65} + 380 q^{69} - 286 q^{73} - 66 q^{77} - 62 q^{81} + 252 q^{85} + 194 q^{89} - 110 q^{93} - 352 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(-1\) \(-\zeta_{12}^{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
191.1
−0.866025 + 0.500000i
0.866025 − 0.500000i
−0.866025 − 0.500000i
0.866025 + 0.500000i
0 −4.33013 + 2.50000i 0 4.50000 − 7.79423i 0 6.92820 − 1.00000i 0 8.00000 − 13.8564i 0
191.2 0 4.33013 − 2.50000i 0 4.50000 − 7.79423i 0 −6.92820 + 1.00000i 0 8.00000 − 13.8564i 0
319.1 0 −4.33013 − 2.50000i 0 4.50000 + 7.79423i 0 6.92820 + 1.00000i 0 8.00000 + 13.8564i 0
319.2 0 4.33013 + 2.50000i 0 4.50000 + 7.79423i 0 −6.92820 − 1.00000i 0 8.00000 + 13.8564i 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
4.b odd 2 1 inner
7.c even 3 1 inner
28.g 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.r.c 4
4.b odd 2 1 inner 448.3.r.c 4
7.c even 3 1 inner 448.3.r.c 4
8.b even 2 1 224.3.r.a ✓ 4
8.d odd 2 1 224.3.r.a ✓ 4
28.g odd 6 1 inner 448.3.r.c 4
56.j odd 6 1 1568.3.d.a 2
56.k odd 6 1 224.3.r.a ✓ 4
56.k odd 6 1 1568.3.d.e 2
56.m even 6 1 1568.3.d.a 2
56.p even 6 1 224.3.r.a ✓ 4
56.p even 6 1 1568.3.d.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.3.r.a ✓ 4 8.b even 2 1
224.3.r.a ✓ 4 8.d odd 2 1
224.3.r.a ✓ 4 56.k odd 6 1
224.3.r.a ✓ 4 56.p even 6 1
448.3.r.c 4 1.a even 1 1 trivial
448.3.r.c 4 4.b odd 2 1 inner
448.3.r.c 4 7.c even 3 1 inner
448.3.r.c 4 28.g odd 6 1 inner
1568.3.d.a 2 56.j odd 6 1
1568.3.d.a 2 56.m even 6 1
1568.3.d.e 2 56.k odd 6 1
1568.3.d.e 2 56.p even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 25T_{3}^{2} + 625 \) 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
$3$ \( T^{4} - 25T^{2} + 625 \) Copy content Toggle raw display
$5$ \( (T^{2} - 9 T + 81)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} - 94T^{2} + 2401 \) Copy content Toggle raw display
$11$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$13$ \( (T + 16)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} - 7 T + 49)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} - 121 T^{2} + 14641 \) Copy content Toggle raw display
$23$ \( T^{4} - 361 T^{2} + 130321 \) Copy content Toggle raw display
$29$ \( (T - 32)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} - 121 T^{2} + 14641 \) Copy content Toggle raw display
$37$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$41$ \( (T + 40)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 1600)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 7225 T^{2} + 52200625 \) Copy content Toggle raw display
$53$ \( (T^{2} - 7 T + 49)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 2809 T^{2} + 7890481 \) Copy content Toggle raw display
$61$ \( (T^{2} - 79 T + 6241)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 121 T^{2} + 14641 \) Copy content Toggle raw display
$71$ \( (T^{2} + 2304)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 143 T + 20449)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} - 1225 T^{2} + 1500625 \) Copy content Toggle raw display
$83$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 97 T + 9409)^{2} \) Copy content Toggle raw display
$97$ \( (T + 88)^{4} \) Copy content Toggle raw display
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