Properties

Label 464.2.u.e
Level $464$
Weight $2$
Character orbit 464.u
Analytic conductor $3.705$
Analytic rank $0$
Dimension $6$
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] = [464,2,Mod(49,464)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(464, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([0, 0, 12]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("464.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 464 = 2^{4} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 464.u (of order \(7\), degree \(6\), 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: \(3.70505865379\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{14})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 232)
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-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 \(\zeta_{14}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{14}^{5} + \zeta_{14}^{4} - \zeta_{14} + 1) q^{3} + (\zeta_{14}^{5} + \zeta_{14}^{3}) q^{5} + ( - 3 \zeta_{14}^{5} + 3 \zeta_{14}^{4} - 3 \zeta_{14}^{3} + 3 \zeta_{14}^{2} - 3 \zeta_{14} + 3) q^{7} + (\zeta_{14}^{5} + \zeta_{14}^{3} + \zeta_{14}^{2} - \zeta_{14} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{14}^{5} + \zeta_{14}^{4} - \zeta_{14} + 1) q^{3} + (\zeta_{14}^{5} + \zeta_{14}^{3}) q^{5} + ( - 3 \zeta_{14}^{5} + 3 \zeta_{14}^{4} - 3 \zeta_{14}^{3} + 3 \zeta_{14}^{2} - 3 \zeta_{14} + 3) q^{7} + (\zeta_{14}^{5} + \zeta_{14}^{3} + \zeta_{14}^{2} - \zeta_{14} - 1) q^{9} + (\zeta_{14}^{4} - \zeta_{14}^{3} + \zeta_{14}^{2} - \zeta_{14} + 1) q^{11} + (\zeta_{14}^{4} - \zeta_{14}^{3} + 4 \zeta_{14}^{2} - \zeta_{14} + 1) q^{13} + \zeta_{14}^{3} q^{15} + (\zeta_{14}^{5} - 3 \zeta_{14}^{4} + 3 \zeta_{14}^{3} - \zeta_{14}^{2} + 1) q^{17} + ( - 2 \zeta_{14}^{5} + 2 \zeta_{14}^{4} + \zeta_{14}^{2} + \zeta_{14} + 1) q^{19} + ( - 3 \zeta_{14}^{5} + 3 \zeta_{14}^{2} - 3 \zeta_{14}) q^{21} + ( - 2 \zeta_{14}^{5} - \zeta_{14}^{4} - \zeta_{14}^{3} - \zeta_{14}^{2} - 2 \zeta_{14}) q^{23} + (\zeta_{14}^{5} - \zeta_{14}^{4} - \zeta_{14}^{2} + 4 \zeta_{14} - 1) q^{25} + (2 \zeta_{14}^{5} + 2 \zeta_{14}^{3} - \zeta_{14} + 1) q^{27} + (\zeta_{14}^{5} + 3 \zeta_{14}^{4} + 3 \zeta_{14}^{3} - 3 \zeta_{14}^{2} + \zeta_{14} - 1) q^{29} + (3 \zeta_{14}^{5} - 5 \zeta_{14}^{4} + 3 \zeta_{14}^{3} + 3 \zeta_{14} - 3) q^{31} + ( - \zeta_{14}^{5} + \zeta_{14}^{4} + \zeta_{14}^{2} - 2 \zeta_{14} + 1) q^{33} + (3 \zeta_{14}^{4} + 3 \zeta_{14}^{2}) q^{35} + ( - 6 \zeta_{14}^{5} - 3 \zeta_{14}^{3} + 2 \zeta_{14}^{2} - 2 \zeta_{14} + 3) q^{37} + (2 \zeta_{14}^{5} - 2 \zeta_{14}^{4} + \zeta_{14}^{2} + \zeta_{14} + 1) q^{39} + (5 \zeta_{14}^{4} - 5 \zeta_{14}^{3} + 5) q^{41} + (\zeta_{14}^{5} - \zeta_{14}^{4} + 9 \zeta_{14}^{3} - \zeta_{14}^{2} + \zeta_{14}) q^{43} + ( - \zeta_{14}^{4} - 2 \zeta_{14}^{3} - 2 \zeta_{14} - 1) q^{45} + ( - 6 \zeta_{14}^{4} - 3 \zeta_{14}^{3} + 3 \zeta_{14}^{2} - 3 \zeta_{14} - 6) q^{47} - 2 \zeta_{14}^{5} q^{49} + (\zeta_{14}^{5} - 3 \zeta_{14}^{4} + 3 \zeta_{14}^{3} - 3 \zeta_{14}^{2} + 3 \zeta_{14} - 1) q^{51} + ( - 6 \zeta_{14}^{5} + 3 \zeta_{14}^{4} - 6 \zeta_{14}^{3} - 3 \zeta_{14} + 3) q^{53} + (\zeta_{14}^{4} - \zeta_{14}^{3} + \zeta_{14}^{2} - \zeta_{14}) q^{55} + ( - 2 \zeta_{14}^{5} + \zeta_{14}^{4} - \zeta_{14}^{3} + 2 \zeta_{14}^{2}) q^{57} + (3 \zeta_{14}^{5} - \zeta_{14}^{4} + \zeta_{14}^{3} - 3 \zeta_{14}^{2} + 3) q^{59} + (\zeta_{14}^{5} + 3 \zeta_{14}^{3} - 3 \zeta_{14}^{2} - 1) q^{61} + (3 \zeta_{14}^{5} + 3 \zeta_{14}^{3} + 6 \zeta_{14} - 6) q^{63} + (3 \zeta_{14}^{5} + \zeta_{14}^{4} - \zeta_{14}^{3} + \zeta_{14}^{2} - \zeta_{14} - 3) q^{65} + (3 \zeta_{14}^{5} - 8 \zeta_{14}^{2} + 8 \zeta_{14}) q^{67} + ( - 3 \zeta_{14}^{4} + \zeta_{14}^{3} - \zeta_{14}^{2} + \zeta_{14} - 3) q^{69} + ( - 3 \zeta_{14}^{4} - 6 \zeta_{14}^{3} - 6 \zeta_{14} - 3) q^{71} + (5 \zeta_{14}^{5} - 4 \zeta_{14}^{4} - \zeta_{14}^{3} - 4 \zeta_{14}^{2} + 5 \zeta_{14}) q^{73} + (\zeta_{14}^{5} + 4 \zeta_{14}^{4} - 4 \zeta_{14}^{3} - \zeta_{14}^{2} + 4) q^{75} + ( - 3 \zeta_{14}^{5} + 3 \zeta_{14}^{4}) q^{77} + (4 \zeta_{14}^{5} - 8 \zeta_{14}^{3} + 2 \zeta_{14}^{2} - 2 \zeta_{14} + 8) q^{79} + ( - 7 \zeta_{14}^{5} + 3 \zeta_{14}^{4} - 3 \zeta_{14}^{3} + 3 \zeta_{14}^{2} - 7 \zeta_{14}) q^{81} + ( - \zeta_{14}^{5} + \zeta_{14}^{4} - 3 \zeta_{14}^{2} - 4 \zeta_{14} - 3) q^{83} + (3 \zeta_{14}^{5} - 3 \zeta_{14}^{4} + 3 \zeta_{14}^{3} - \zeta_{14} + 1) q^{85} + ( - 5 \zeta_{14}^{5} + 4 \zeta_{14}^{4} + 2 \zeta_{14}^{3} + 3 \zeta_{14}^{2} - 3 \zeta_{14} - 2) q^{87} + (2 \zeta_{14}^{5} + 9 \zeta_{14}^{4} + 2 \zeta_{14}^{3} + 2 \zeta_{14} - 2) q^{89} + ( - 3 \zeta_{14}^{5} + 3 \zeta_{14}^{4} + 9 \zeta_{14}) q^{91} + (8 \zeta_{14}^{5} - 5 \zeta_{14}^{4} - 5 \zeta_{14}^{2} + 8 \zeta_{14}) q^{93} + (3 \zeta_{14}^{5} + 4 \zeta_{14}^{3} - 3 \zeta_{14}^{2} + 3 \zeta_{14} - 4) q^{95} + (8 \zeta_{14}^{5} - 8 \zeta_{14}^{4} - 10 \zeta_{14}^{2} + 6 \zeta_{14} - 10) q^{97} + ( - \zeta_{14}^{5} + \zeta_{14}^{4} - \zeta_{14}^{3} + \zeta_{14}^{2} - 2) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{3} + 2 q^{5} + 3 q^{7} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 3 q^{3} + 2 q^{5} + 3 q^{7} - 6 q^{9} + 2 q^{11} - q^{13} + q^{15} + 14 q^{17} + 2 q^{19} - 9 q^{21} - 3 q^{23} + q^{25} + 9 q^{27} - q^{29} - 4 q^{31} + q^{33} - 6 q^{35} + 5 q^{37} + 10 q^{39} + 20 q^{41} + 13 q^{43} - 9 q^{45} - 39 q^{47} - 2 q^{49} + 7 q^{51} - 4 q^{55} - 6 q^{57} + 26 q^{59} + q^{61} - 24 q^{63} - 19 q^{65} + 19 q^{67} - 12 q^{69} - 27 q^{71} + 17 q^{73} + 18 q^{75} - 6 q^{77} + 40 q^{79} - 23 q^{81} - 21 q^{83} + 14 q^{85} - 25 q^{87} - 15 q^{89} + 3 q^{91} + 26 q^{93} - 11 q^{95} - 28 q^{97} - 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/464\mathbb{Z}\right)^\times\).

\(n\) \(117\) \(175\) \(321\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{14}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
−0.623490 + 0.781831i
0.900969 0.433884i
0.222521 + 0.974928i
−0.623490 0.781831i
0.900969 + 0.433884i
0.222521 0.974928i
0 0.500000 + 0.626980i 0 1.12349 0.541044i 0 −1.87047 2.34549i 0 0.524459 2.29780i 0
65.1 0 0.500000 + 0.240787i 0 −0.400969 1.75676i 0 2.70291 + 1.30165i 0 −1.67845 2.10471i 0
81.1 0 0.500000 2.19064i 0 0.277479 0.347948i 0 0.667563 2.92478i 0 −1.84601 0.888992i 0
161.1 0 0.500000 0.626980i 0 1.12349 + 0.541044i 0 −1.87047 + 2.34549i 0 0.524459 + 2.29780i 0
257.1 0 0.500000 0.240787i 0 −0.400969 + 1.75676i 0 2.70291 1.30165i 0 −1.67845 + 2.10471i 0
401.1 0 0.500000 + 2.19064i 0 0.277479 + 0.347948i 0 0.667563 + 2.92478i 0 −1.84601 + 0.888992i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.1
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 464.2.u.e 6
4.b odd 2 1 232.2.m.a 6
29.d even 7 1 inner 464.2.u.e 6
116.h odd 14 1 6728.2.a.i 3
116.j odd 14 1 232.2.m.a 6
116.j odd 14 1 6728.2.a.q 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
232.2.m.a 6 4.b odd 2 1
232.2.m.a 6 116.j odd 14 1
464.2.u.e 6 1.a even 1 1 trivial
464.2.u.e 6 29.d even 7 1 inner
6728.2.a.i 3 116.h odd 14 1
6728.2.a.q 3 116.j odd 14 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} - 3T_{3}^{5} + 9T_{3}^{4} - 13T_{3}^{3} + 11T_{3}^{2} - 5T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(464, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - 3 T^{5} + 9 T^{4} - 13 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{6} - 2 T^{5} + 4 T^{4} - 8 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{6} - 3 T^{5} + 9 T^{4} - 27 T^{3} + \cdots + 729 \) Copy content Toggle raw display
$11$ \( T^{6} - 2 T^{5} + 4 T^{4} - 8 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{6} + T^{5} + 22 T^{4} + T^{3} + \cdots + 841 \) Copy content Toggle raw display
$17$ \( (T^{3} - 7 T^{2} + 7)^{2} \) Copy content Toggle raw display
$19$ \( T^{6} - 2 T^{5} - 3 T^{4} - 22 T^{3} + \cdots + 169 \) Copy content Toggle raw display
$23$ \( T^{6} + 3 T^{5} + 30 T^{4} + 13 T^{3} + \cdots + 169 \) Copy content Toggle raw display
$29$ \( T^{6} + T^{5} + 57 T^{4} + 15 T^{3} + \cdots + 24389 \) Copy content Toggle raw display
$31$ \( T^{6} + 4 T^{5} + 16 T^{4} + \cdots + 1849 \) Copy content Toggle raw display
$37$ \( T^{6} - 5 T^{5} + 25 T^{4} + 337 T^{3} + \cdots + 841 \) Copy content Toggle raw display
$41$ \( (T^{3} - 10 T^{2} - 25 T + 125)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} - 13 T^{5} + 99 T^{4} + \cdots + 312481 \) Copy content Toggle raw display
$47$ \( T^{6} + 39 T^{5} + 765 T^{4} + \cdots + 1347921 \) Copy content Toggle raw display
$53$ \( T^{6} + 63 T^{4} + 378 T^{3} + \cdots + 35721 \) Copy content Toggle raw display
$59$ \( (T^{3} - 13 T^{2} + 40 T + 13)^{2} \) Copy content Toggle raw display
$61$ \( T^{6} - T^{5} + 8 T^{4} + 55 T^{3} + \cdots + 1849 \) Copy content Toggle raw display
$67$ \( T^{6} - 19 T^{5} + 193 T^{4} + \cdots + 6889 \) Copy content Toggle raw display
$71$ \( T^{6} + 27 T^{5} + 477 T^{4} + \cdots + 123201 \) Copy content Toggle raw display
$73$ \( T^{6} - 17 T^{5} + 177 T^{4} + \cdots + 1681 \) Copy content Toggle raw display
$79$ \( T^{6} - 40 T^{5} + 760 T^{4} + \cdots + 817216 \) Copy content Toggle raw display
$83$ \( T^{6} + 21 T^{5} + 210 T^{4} + \cdots + 8281 \) Copy content Toggle raw display
$89$ \( T^{6} + 15 T^{5} + 225 T^{4} + \cdots + 1018081 \) Copy content Toggle raw display
$97$ \( T^{6} + 28 T^{5} + 364 T^{4} + \cdots + 529984 \) Copy content Toggle raw display
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