Properties

Label 462.2.j.f
Level $462$
Weight $2$
Character orbit 462.j
Analytic conductor $3.689$
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] = [462,2,Mod(169,462)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(462, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("462.169");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 462 = 2 \cdot 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 462.j (of order \(5\), degree \(4\), 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: \(3.68908857338\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.64000000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{6} + 4x^{4} - 8x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{6} q^{2} + ( - \beta_{6} + \beta_{4} - \beta_{2} + 1) q^{3} - \beta_{2} q^{4} + ( - \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2}) q^{5} - \beta_{4} q^{6} + \beta_{2} q^{7} + (\beta_{6} - \beta_{4} + \beta_{2} - 1) q^{8} - \beta_{6} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{6} q^{2} + ( - \beta_{6} + \beta_{4} - \beta_{2} + 1) q^{3} - \beta_{2} q^{4} + ( - \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2}) q^{5} - \beta_{4} q^{6} + \beta_{2} q^{7} + (\beta_{6} - \beta_{4} + \beta_{2} - 1) q^{8} - \beta_{6} q^{9} + (\beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} - 2 \beta_1) q^{10} + (\beta_{7} - \beta_{6} - \beta_{5} + 2 \beta_{3} + 2) q^{11} - q^{12} + ( - 2 \beta_{6} + 2 \beta_{5} - 2 \beta_{3} - 2 \beta_{2} + \beta_1 + 2) q^{13} + ( - \beta_{6} + \beta_{4} - \beta_{2} + 1) q^{14} + ( - \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 - 1) q^{15} + \beta_{4} q^{16} + ( - \beta_{6} - 2 \beta_{5} - 2 \beta_{3} - \beta_{2}) q^{17} - \beta_{2} q^{18} + (2 \beta_{7} + 3 \beta_{6} - 4 \beta_{4} + \beta_{3} + 4 \beta_{2} - 2 \beta_1 - 3) q^{19} + (\beta_{7} + \beta_{5} + \beta_{2} - 1) q^{20} + q^{21} + ( - 2 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} - \beta_{3} - \beta_{2} + \beta_1) q^{22} + (3 \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} - 2 \beta_1 + 1) q^{23} + \beta_{6} q^{24} + (2 \beta_{6} + 2 \beta_{5} - \beta_{4} + \beta_{2} - 2 \beta_1 - 2) q^{25} + (\beta_{7} - 2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} - 2) q^{26} - \beta_{4} q^{27} - \beta_{4} q^{28} + ( - 3 \beta_{7} + 6 \beta_{5} - 3 \beta_{3} + 2 \beta_{2} + 3 \beta_1) q^{29} + (2 \beta_{7} - \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} - \beta_1) q^{30} + ( - 2 \beta_{7} + 4 \beta_{6} + 4 \beta_{5} - 6 \beta_{3} + 3 \beta_{2} + 3 \beta_1 - 3) q^{31} + q^{32} + ( - 2 \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 2) q^{33} + (2 \beta_{7} + \beta_{6} - 2 \beta_{5} - \beta_{4} + 2 \beta_{3} - 4 \beta_1 - 1) q^{34} + ( - \beta_{7} - \beta_{5} - \beta_{2} + 1) q^{35} + (\beta_{6} - \beta_{4} + \beta_{2} - 1) q^{36} + ( - 3 \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 - 3) q^{37} + (\beta_{7} - \beta_{6} + \beta_{5} + 4 \beta_{4} + \beta_{3} - \beta_{2} + \beta_1) q^{38} + ( - \beta_{7} - 2 \beta_{6} + \beta_{5} + \beta_{3} - 2 \beta_{2} - \beta_1) q^{39} + (\beta_{4} + \beta_{3} - \beta_{2} + \beta_1 + 1) q^{40} + ( - 4 \beta_{7} - 3 \beta_{4} - 2 \beta_{3} + 3 \beta_{2} + 4 \beta_1) q^{41} - \beta_{6} q^{42} + ( - 4 \beta_{7} + 4 \beta_{6} + 3 \beta_{5} - 4 \beta_{4} - 2 \beta_{3} + 6 \beta_1 + 4) q^{43} + (\beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 - 1) q^{44} + (\beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} - 2 \beta_1) q^{45} + (3 \beta_{7} - \beta_{6} - \beta_{5} + 4 \beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{46} + (4 \beta_{7} - 5 \beta_{5} + 2 \beta_{3} + \beta_1) q^{47} + \beta_{2} q^{48} + \beta_{4} q^{49} + (2 \beta_{7} + \beta_{6} + \beta_{4} + \beta_{2} + 2 \beta_1) q^{50} + ( - \beta_{4} - 2 \beta_{3} - 2 \beta_1 - 1) q^{51} + ( - 2 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} - \beta_{3} - 2) q^{52} + ( - 3 \beta_{7} - 3 \beta_{5} + 2 \beta_{2} - 2) q^{53} - q^{54} + ( - 7 \beta_{6} - \beta_{5} + 3 \beta_{4} - 4 \beta_{3} - 2 \beta_{2} + 2) q^{55} - q^{56} + (2 \beta_{7} + 3 \beta_{6} + 2 \beta_{3} - \beta_{2} - \beta_1 + 1) q^{57} + ( - 2 \beta_{6} - 3 \beta_{5} + 2 \beta_{4} - 2 \beta_{2} + 3 \beta_1 + 2) q^{58} + ( - \beta_{7} + 2 \beta_{5} - \beta_{3} + 6 \beta_{2} + \beta_1) q^{59} + (\beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2}) q^{60} + ( - 4 \beta_{6} + \beta_{5} + 8 \beta_{4} + \beta_{3} - 4 \beta_{2}) q^{61} + (3 \beta_{7} - 6 \beta_{5} + 3 \beta_{4} + 4 \beta_{3} + \beta_{2} - 2 \beta_1 + 3) q^{62} + ( - \beta_{6} + \beta_{4} - \beta_{2} + 1) q^{63} - \beta_{6} q^{64} + (3 \beta_{7} + 2 \beta_{6} - 3 \beta_{5} - 2 \beta_{4} + 3 \beta_{3} - 6 \beta_1 + 2) q^{65} + ( - \beta_{7} - \beta_{5} - 2 \beta_{4} + \beta_{3} - 1) q^{66} + ( - 3 \beta_{7} + 3 \beta_{5} - 3 \beta_{3} + 6 \beta_1 - 6) q^{67} + (2 \beta_{7} + \beta_{6} + 2 \beta_{5} + \beta_{2} - 1) q^{68} + (2 \beta_{7} - \beta_{6} + \beta_{5} + \beta_{3} - 3 \beta_1 + 1) q^{69} + ( - \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 - 1) q^{70} + (4 \beta_{6} + 6 \beta_{4} + 4 \beta_{2}) q^{71} + \beta_{4} q^{72} + ( - \beta_{7} + 2 \beta_{5} + \beta_{3} + 3 \beta_1) q^{73} + ( - 2 \beta_{7} + 4 \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 - 4) q^{74} + (2 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} + 4 \beta_{3} + \beta_{2} - 2 \beta_1 - 1) q^{75} + ( - 2 \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + 2 \beta_1 + 3) q^{76} + ( - \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 + 1) q^{77} + (2 \beta_{6} + \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 2 \beta_1 - 2) q^{78} + (\beta_{7} - 4 \beta_{6} - \beta_{5} + 2 \beta_{3} - 8 \beta_{2} - \beta_1 + 8) q^{79} + ( - 2 \beta_{7} + \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + \beta_1) q^{80} - \beta_{2} q^{81} + ( - 2 \beta_{7} - 3 \beta_{6} - 2 \beta_{5} + 3 \beta_{4} - 2 \beta_{3} - 3 \beta_{2} - 2 \beta_1) q^{82} + ( - 4 \beta_{6} - 3 \beta_{5} + 8 \beta_{4} - 3 \beta_{3} - 4 \beta_{2}) q^{83} - \beta_{2} q^{84} + (2 \beta_{7} + 13 \beta_{6} + 2 \beta_{5} - 9 \beta_{4} + \beta_{3} + 9 \beta_{2} + \cdots - 13) q^{85}+ \cdots + ( - 2 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} - \beta_{3} - \beta_{2} + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{2} + 2 q^{3} - 2 q^{4} - 6 q^{5} + 2 q^{6} + 2 q^{7} - 2 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{2} + 2 q^{3} - 2 q^{4} - 6 q^{5} + 2 q^{6} + 2 q^{7} - 2 q^{8} - 2 q^{9} + 4 q^{10} + 14 q^{11} - 8 q^{12} + 8 q^{13} + 2 q^{14} - 4 q^{15} - 2 q^{16} - 4 q^{17} - 2 q^{18} - 2 q^{19} - 6 q^{20} + 8 q^{21} - 6 q^{22} + 4 q^{23} + 2 q^{24} - 8 q^{25} - 12 q^{26} + 2 q^{27} + 2 q^{28} + 4 q^{29} - 4 q^{30} - 10 q^{31} + 8 q^{32} + 6 q^{33} - 4 q^{34} + 6 q^{35} - 2 q^{36} - 20 q^{37} - 12 q^{38} - 8 q^{39} + 4 q^{40} + 12 q^{41} - 2 q^{42} + 48 q^{43} - 6 q^{44} + 4 q^{45} + 4 q^{46} + 2 q^{48} - 2 q^{49} + 2 q^{50} - 6 q^{51} - 12 q^{52} - 12 q^{53} - 8 q^{54} - 8 q^{55} - 8 q^{56} + 12 q^{57} + 4 q^{58} + 12 q^{59} + 6 q^{60} - 32 q^{61} + 20 q^{62} + 2 q^{63} - 2 q^{64} + 24 q^{65} - 4 q^{66} - 48 q^{67} - 4 q^{68} + 6 q^{69} - 4 q^{70} + 4 q^{71} - 2 q^{72} - 20 q^{74} - 2 q^{75} + 28 q^{76} + 6 q^{77} - 8 q^{78} + 40 q^{79} + 4 q^{80} - 2 q^{81} - 18 q^{82} - 32 q^{83} - 2 q^{84} - 42 q^{85} - 32 q^{86} + 16 q^{87} - 6 q^{88} + 12 q^{89} - 6 q^{90} + 12 q^{91} - 6 q^{92} + 10 q^{93} + 54 q^{95} + 2 q^{96} + 8 q^{97} + 8 q^{98} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{6} + 4x^{4} - 8x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} ) / 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{4} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{5} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 8\beta_{6} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 8\beta_{7} \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/462\mathbb{Z}\right)^\times\).

\(n\) \(155\) \(199\) \(211\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
169.1
−0.831254 + 1.14412i
0.831254 1.14412i
1.34500 + 0.437016i
−1.34500 0.437016i
1.34500 0.437016i
−1.34500 + 0.437016i
−0.831254 1.14412i
0.831254 + 1.14412i
−0.809017 0.587785i −0.309017 + 0.951057i 0.309017 + 0.951057i −2.65401 + 1.92825i 0.809017 0.587785i −0.309017 0.951057i 0.309017 0.951057i −0.809017 0.587785i 3.28054
169.2 −0.809017 0.587785i −0.309017 + 0.951057i 0.309017 + 0.951057i 0.0359800 0.0261410i 0.809017 0.587785i −0.309017 0.951057i 0.309017 0.951057i −0.809017 0.587785i −0.0444738
295.1 0.309017 0.951057i 0.809017 0.587785i −0.809017 0.587785i −1.02224 3.14612i −0.309017 0.951057i 0.809017 + 0.587785i −0.809017 + 0.587785i 0.309017 0.951057i −3.30803
295.2 0.309017 0.951057i 0.809017 0.587785i −0.809017 0.587785i 0.640271 + 1.97055i −0.309017 0.951057i 0.809017 + 0.587785i −0.809017 + 0.587785i 0.309017 0.951057i 2.07196
379.1 0.309017 + 0.951057i 0.809017 + 0.587785i −0.809017 + 0.587785i −1.02224 + 3.14612i −0.309017 + 0.951057i 0.809017 0.587785i −0.809017 0.587785i 0.309017 + 0.951057i −3.30803
379.2 0.309017 + 0.951057i 0.809017 + 0.587785i −0.809017 + 0.587785i 0.640271 1.97055i −0.309017 + 0.951057i 0.809017 0.587785i −0.809017 0.587785i 0.309017 + 0.951057i 2.07196
421.1 −0.809017 + 0.587785i −0.309017 0.951057i 0.309017 0.951057i −2.65401 1.92825i 0.809017 + 0.587785i −0.309017 + 0.951057i 0.309017 + 0.951057i −0.809017 + 0.587785i 3.28054
421.2 −0.809017 + 0.587785i −0.309017 0.951057i 0.309017 0.951057i 0.0359800 + 0.0261410i 0.809017 + 0.587785i −0.309017 + 0.951057i 0.309017 + 0.951057i −0.809017 + 0.587785i −0.0444738
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 169.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 462.2.j.f 8
11.c even 5 1 inner 462.2.j.f 8
11.c even 5 1 5082.2.a.cc 4
11.d odd 10 1 5082.2.a.bx 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.j.f 8 1.a even 1 1 trivial
462.2.j.f 8 11.c even 5 1 inner
5082.2.a.bx 4 11.d odd 10 1
5082.2.a.cc 4 11.c even 5 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 6T_{5}^{7} + 27T_{5}^{6} + 68T_{5}^{5} + 150T_{5}^{4} + 182T_{5}^{3} + 492T_{5}^{2} - 36T_{5} + 1 \) acting on \(S_{2}^{\mathrm{new}}(462, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + T^{3} + T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{4} - T^{3} + T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} + 6 T^{7} + 27 T^{6} + 68 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( (T^{4} - T^{3} + T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} - 14 T^{7} + 105 T^{6} + \cdots + 14641 \) Copy content Toggle raw display
$13$ \( T^{8} - 8 T^{7} + 38 T^{6} + \cdots + 26896 \) Copy content Toggle raw display
$17$ \( T^{8} + 4 T^{7} + 52 T^{6} + \cdots + 78961 \) Copy content Toggle raw display
$19$ \( T^{8} + 2 T^{7} + 23 T^{6} + \cdots + 72361 \) Copy content Toggle raw display
$23$ \( (T^{4} - 2 T^{3} - 51 T^{2} + 2 T + 451)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} - 4 T^{7} + 102 T^{6} + \cdots + 1628176 \) Copy content Toggle raw display
$31$ \( T^{8} + 10 T^{7} + 55 T^{6} + \cdots + 7425625 \) Copy content Toggle raw display
$37$ \( T^{8} + 20 T^{7} + 180 T^{6} + \cdots + 3025 \) Copy content Toggle raw display
$41$ \( T^{8} - 12 T^{7} + 68 T^{6} + \cdots + 1957201 \) Copy content Toggle raw display
$43$ \( (T^{4} - 24 T^{3} + 76 T^{2} + 1616 T - 9644)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 210 T^{6} + 16840 T^{4} + \cdots + 400 \) Copy content Toggle raw display
$53$ \( T^{8} + 12 T^{7} + 158 T^{6} + \cdots + 839056 \) Copy content Toggle raw display
$59$ \( T^{8} - 12 T^{7} + 118 T^{6} + \cdots + 913936 \) Copy content Toggle raw display
$61$ \( T^{8} + 32 T^{7} + 458 T^{6} + \cdots + 274576 \) Copy content Toggle raw display
$67$ \( (T^{4} + 24 T^{3} + 126 T^{2} - 216 T - 324)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 2 T^{3} + 124 T^{2} + 792 T + 1936)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} - 30 T^{6} + 2440 T^{4} + \cdots + 400 \) Copy content Toggle raw display
$79$ \( T^{8} - 40 T^{7} + 880 T^{6} + \cdots + 31584400 \) Copy content Toggle raw display
$83$ \( T^{8} + 32 T^{7} + 538 T^{6} + \cdots + 28344976 \) Copy content Toggle raw display
$89$ \( (T^{4} - 6 T^{3} - 29 T^{2} + 74 T + 241)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} - 8 T^{7} - 22 T^{6} + \cdots + 633616 \) Copy content Toggle raw display
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