Properties

Label 403.2.k.b
Level $403$
Weight $2$
Character orbit 403.k
Analytic conductor $3.218$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [403,2,Mod(66,403)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(403, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("403.66");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 403 = 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 403.k (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: \(3.21797120146\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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 primitive root of unity \(\zeta_{10}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \zeta_{10} q^{2} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10} - 2) q^{3} - \zeta_{10}^{2} q^{4} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} - 2) q^{5} + 2 q^{6} + (\zeta_{10}^{3} + \zeta_{10}^{2} + \zeta_{10}) q^{7} + 3 \zeta_{10}^{3} q^{8} - \zeta_{10}^{3} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{10} q^{2} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10} - 2) q^{3} - \zeta_{10}^{2} q^{4} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} - 2) q^{5} + 2 q^{6} + (\zeta_{10}^{3} + \zeta_{10}^{2} + \zeta_{10}) q^{7} + 3 \zeta_{10}^{3} q^{8} - \zeta_{10}^{3} q^{9} + (2 \zeta_{10}^{2} + 2) q^{10} + ( - \zeta_{10}^{3} - \zeta_{10}) q^{11} + 2 \zeta_{10} q^{12} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{13} + ( - 2 \zeta_{10}^{3} - \zeta_{10} + 1) q^{14} + ( - 4 \zeta_{10}^{3} + 4) q^{15} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{16} + ( - 4 \zeta_{10}^{3} - \zeta_{10} + 1) q^{17} + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{18} + (2 \zeta_{10}^{3} + 2 \zeta_{10}) q^{20} + ( - 2 \zeta_{10}^{2} - 2 \zeta_{10} - 2) q^{21} + (\zeta_{10}^{3} + \zeta_{10} - 1) q^{22} + ( - 4 \zeta_{10}^{3} - 2 \zeta_{10} + 2) q^{23} - 6 \zeta_{10}^{2} q^{24} + ( - 4 \zeta_{10}^{3} + 4 \zeta_{10}^{2} + 3) q^{25} - q^{26} - 4 \zeta_{10}^{2} q^{27} + ( - 2 \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 2) q^{28} + ( - 5 \zeta_{10}^{2} + 5 \zeta_{10} - 5) q^{29} + (4 \zeta_{10}^{3} - 4 \zeta_{10}^{2} - 4) q^{30} + (6 \zeta_{10}^{2} - 2 \zeta_{10} + 3) q^{31} + 5 q^{32} + (2 \zeta_{10}^{2} + 2) q^{33} + (4 \zeta_{10}^{3} - 3 \zeta_{10}^{2} + 3 \zeta_{10} - 4) q^{34} + ( - 4 \zeta_{10}^{3} - 2 \zeta_{10}^{2} - 4 \zeta_{10}) q^{35} - q^{36} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} + 2) q^{37} + 2 \zeta_{10}^{3} q^{39} + ( - 6 \zeta_{10}^{3} - 6 \zeta_{10} + 6) q^{40} + (2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} + 2 \zeta_{10}) q^{42} + (6 \zeta_{10}^{2} + 6) q^{43} + (\zeta_{10}^{3} - 1) q^{44} + (2 \zeta_{10}^{3} + 2 \zeta_{10} - 2) q^{45} + (4 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10} - 4) q^{46} + ( - 10 \zeta_{10}^{3} + 7 \zeta_{10}^{2} - 7 \zeta_{10} + 10) q^{47} + 2 \zeta_{10}^{3} q^{48} + ( - 2 \zeta_{10}^{3} + 5 \zeta_{10}^{2} - 5 \zeta_{10} + 2) q^{49} + ( - 4 \zeta_{10}^{2} + \zeta_{10} - 4) q^{50} + (2 \zeta_{10}^{3} + 6 \zeta_{10}^{2} + 2 \zeta_{10}) q^{51} - \zeta_{10} q^{52} + 2 \zeta_{10}^{3} q^{53} + 4 \zeta_{10}^{3} q^{54} + (2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} + 2 \zeta_{10}) q^{55} + (3 \zeta_{10}^{3} - 3 \zeta_{10}^{2} - 6) q^{56} + (5 \zeta_{10}^{3} - 5 \zeta_{10}^{2} + 5 \zeta_{10}) q^{58} + (5 \zeta_{10}^{3} - 5) q^{59} + ( - 4 \zeta_{10}^{2} - 4) q^{60} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - 11) q^{61} + ( - 6 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 3 \zeta_{10}) q^{62} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} + 2) q^{63} - 7 \zeta_{10} q^{64} + (2 \zeta_{10}^{3} - 2) q^{65} + ( - 2 \zeta_{10}^{3} - 2 \zeta_{10}) q^{66} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} + 6) q^{67} + (\zeta_{10}^{3} - \zeta_{10}^{2} - 4) q^{68} + (4 \zeta_{10}^{3} + 4 \zeta_{10}^{2} + 4 \zeta_{10}) q^{69} + (6 \zeta_{10}^{3} + 4 \zeta_{10} - 4) q^{70} + ( - \zeta_{10}^{3} - \zeta_{10} + 1) q^{71} + 3 \zeta_{10} q^{72} + ( - 6 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 6 \zeta_{10}) q^{73} + ( - 2 \zeta_{10}^{2} - 2) q^{74} + (6 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} - 6) q^{75} + ( - 3 \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 3) q^{77} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} + 2) q^{78} + ( - 6 \zeta_{10}^{3} - 10 \zeta_{10} + 10) q^{79} + (2 \zeta_{10}^{3} - 2) q^{80} + 11 \zeta_{10} q^{81} + ( - \zeta_{10}^{2} + 4 \zeta_{10} - 1) q^{83} + (4 \zeta_{10}^{3} + 2 \zeta_{10} - 2) q^{84} + (10 \zeta_{10}^{3} + 8 \zeta_{10} - 8) q^{85} + ( - 6 \zeta_{10}^{3} - 6 \zeta_{10}) q^{86} + ( - 10 \zeta_{10}^{3} + 10 \zeta_{10}^{2}) q^{87} + ( - 3 \zeta_{10}^{3} + 3 \zeta_{10}^{2} + 3) q^{88} - 4 \zeta_{10}^{2} q^{89} + ( - 2 \zeta_{10}^{3} + 2) q^{90} + (\zeta_{10}^{2} + \zeta_{10} + 1) q^{91} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} - 4) q^{92} + (6 \zeta_{10}^{3} - 6 \zeta_{10}^{2} - 6 \zeta_{10} - 2) q^{93} + (3 \zeta_{10}^{3} - 3 \zeta_{10}^{2} - 10) q^{94} + (10 \zeta_{10}^{3} - 10 \zeta_{10}^{2} + 10 \zeta_{10} - 10) q^{96} + (10 \zeta_{10}^{3} - 8 \zeta_{10}^{2} + 10 \zeta_{10}) q^{97} + ( - 3 \zeta_{10}^{3} + 3 \zeta_{10}^{2} - 2) q^{98} + (\zeta_{10}^{3} - \zeta_{10}^{2} - 1) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{2} - 2 q^{3} + q^{4} - 4 q^{5} + 8 q^{6} + q^{7} + 3 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{2} - 2 q^{3} + q^{4} - 4 q^{5} + 8 q^{6} + q^{7} + 3 q^{8} - q^{9} + 6 q^{10} - 2 q^{11} + 2 q^{12} + q^{13} + q^{14} + 12 q^{15} + q^{16} - q^{17} - q^{18} + 4 q^{20} - 8 q^{21} - 2 q^{22} + 2 q^{23} + 6 q^{24} + 4 q^{25} - 4 q^{26} + 4 q^{27} + 4 q^{28} - 10 q^{29} - 8 q^{30} + 4 q^{31} + 20 q^{32} + 6 q^{33} - 6 q^{34} - 6 q^{35} - 4 q^{36} + 4 q^{37} + 2 q^{39} + 12 q^{40} + 2 q^{42} + 18 q^{43} - 3 q^{44} - 4 q^{45} - 8 q^{46} + 16 q^{47} + 2 q^{48} - 4 q^{49} - 11 q^{50} - 2 q^{51} - q^{52} + 2 q^{53} + 4 q^{54} + 2 q^{55} - 18 q^{56} + 15 q^{58} - 15 q^{59} - 12 q^{60} - 46 q^{61} - 11 q^{62} + 6 q^{63} - 7 q^{64} - 6 q^{65} - 4 q^{66} + 22 q^{67} - 14 q^{68} + 4 q^{69} - 6 q^{70} + 2 q^{71} + 3 q^{72} - 14 q^{73} - 6 q^{74} - 22 q^{75} + 7 q^{77} + 2 q^{78} + 24 q^{79} - 6 q^{80} + 11 q^{81} + q^{83} - 2 q^{84} - 14 q^{85} - 12 q^{86} - 20 q^{87} + 6 q^{88} + 4 q^{89} + 6 q^{90} + 4 q^{91} - 12 q^{92} - 2 q^{93} - 34 q^{94} - 10 q^{96} + 28 q^{97} - 14 q^{98} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/403\mathbb{Z}\right)^\times\).

\(n\) \(249\) \(313\)
\(\chi(n)\) \(1\) \(-\zeta_{10}^{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
66.1
−0.309017 0.951057i
0.809017 + 0.587785i
−0.309017 + 0.951057i
0.809017 0.587785i
0.309017 + 0.951057i 0.618034 1.90211i 0.809017 0.587785i 1.23607 2.00000 −0.309017 + 0.224514i 2.42705 + 1.76336i −0.809017 0.587785i 0.381966 + 1.17557i
157.1 −0.809017 0.587785i −1.61803 + 1.17557i −0.309017 0.951057i −3.23607 2.00000 0.809017 + 2.48990i −0.927051 + 2.85317i 0.309017 0.951057i 2.61803 + 1.90211i
287.1 0.309017 0.951057i 0.618034 + 1.90211i 0.809017 + 0.587785i 1.23607 2.00000 −0.309017 0.224514i 2.42705 1.76336i −0.809017 + 0.587785i 0.381966 1.17557i
326.1 −0.809017 + 0.587785i −1.61803 1.17557i −0.309017 + 0.951057i −3.23607 2.00000 0.809017 2.48990i −0.927051 2.85317i 0.309017 + 0.951057i 2.61803 1.90211i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 403.2.k.b 4
31.d even 5 1 inner 403.2.k.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
403.2.k.b 4 1.a even 1 1 trivial
403.2.k.b 4 31.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + T_{2}^{3} + T_{2}^{2} + T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(403, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + T^{3} + T^{2} + T + 1 \) Copy content Toggle raw display
$3$ \( T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16 \) Copy content Toggle raw display
$5$ \( (T^{2} + 2 T - 4)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} - T^{3} + 6 T^{2} + 4 T + 1 \) Copy content Toggle raw display
$11$ \( T^{4} + 2 T^{3} + 4 T^{2} + 3 T + 1 \) Copy content Toggle raw display
$13$ \( T^{4} - T^{3} + T^{2} - T + 1 \) Copy content Toggle raw display
$17$ \( T^{4} + T^{3} + 16 T^{2} + 66 T + 121 \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} - 2 T^{3} + 24 T^{2} + 32 T + 16 \) Copy content Toggle raw display
$29$ \( T^{4} + 10 T^{3} + 100 T^{2} + \cdots + 625 \) Copy content Toggle raw display
$31$ \( T^{4} - 4 T^{3} + 46 T^{2} - 124 T + 961 \) Copy content Toggle raw display
$37$ \( (T^{2} - 2 T - 4)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( T^{4} - 18 T^{3} + 144 T^{2} + \cdots + 1296 \) Copy content Toggle raw display
$47$ \( T^{4} - 16 T^{3} + 106 T^{2} + \cdots + 3721 \) Copy content Toggle raw display
$53$ \( T^{4} - 2 T^{3} + 4 T^{2} - 8 T + 16 \) Copy content Toggle raw display
$59$ \( T^{4} + 15 T^{3} + 100 T^{2} + \cdots + 625 \) Copy content Toggle raw display
$61$ \( (T^{2} + 23 T + 131)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 11 T + 29)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} - 2 T^{3} + 4 T^{2} - 3 T + 1 \) Copy content Toggle raw display
$73$ \( T^{4} + 14 T^{3} + 136 T^{2} + \cdots + 1936 \) Copy content Toggle raw display
$79$ \( T^{4} - 24 T^{3} + 376 T^{2} + \cdots + 15376 \) Copy content Toggle raw display
$83$ \( T^{4} - T^{3} + 16 T^{2} - 66 T + 121 \) Copy content Toggle raw display
$89$ \( T^{4} - 4 T^{3} + 16 T^{2} - 64 T + 256 \) Copy content Toggle raw display
$97$ \( T^{4} - 28 T^{3} + 384 T^{2} + \cdots + 13456 \) Copy content Toggle raw display
show more
show less