Properties

Label 41.2.b.a
Level $41$
Weight $2$
Character orbit 41.b
Analytic conductor $0.327$
Analytic rank $0$
Dimension $2$
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] = [41,2,Mod(40,41)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(41, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("41.40");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 41.b (of order \(2\), degree \(1\), 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: \(0.327386648287\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(\beta = 2\sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + \beta q^{3} - q^{4} + 2 q^{5} - \beta q^{6} - \beta q^{7} + 3 q^{8} - 5 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + \beta q^{3} - q^{4} + 2 q^{5} - \beta q^{6} - \beta q^{7} + 3 q^{8} - 5 q^{9} - 2 q^{10} + \beta q^{11} - \beta q^{12} - 2 \beta q^{13} + \beta q^{14} + 2 \beta q^{15} - q^{16} + 5 q^{18} + \beta q^{19} - 2 q^{20} + 8 q^{21} - \beta q^{22} + 3 \beta q^{24} - q^{25} + 2 \beta q^{26} - 2 \beta q^{27} + \beta q^{28} - 2 \beta q^{29} - 2 \beta q^{30} - 8 q^{31} - 5 q^{32} - 8 q^{33} - 2 \beta q^{35} + 5 q^{36} + 2 q^{37} - \beta q^{38} + 16 q^{39} + 6 q^{40} + (2 \beta - 3) q^{41} - 8 q^{42} + 4 q^{43} - \beta q^{44} - 10 q^{45} - \beta q^{47} - \beta q^{48} - q^{49} + q^{50} + 2 \beta q^{52} + 2 \beta q^{53} + 2 \beta q^{54} + 2 \beta q^{55} - 3 \beta q^{56} - 8 q^{57} + 2 \beta q^{58} + 4 q^{59} - 2 \beta q^{60} - 2 q^{61} + 8 q^{62} + 5 \beta q^{63} + 7 q^{64} - 4 \beta q^{65} + 8 q^{66} - 3 \beta q^{67} + 2 \beta q^{70} + 3 \beta q^{71} - 15 q^{72} + 14 q^{73} - 2 q^{74} - \beta q^{75} - \beta q^{76} + 8 q^{77} - 16 q^{78} - \beta q^{79} - 2 q^{80} + q^{81} + ( - 2 \beta + 3) q^{82} - 12 q^{83} - 8 q^{84} - 4 q^{86} + 16 q^{87} + 3 \beta q^{88} + 4 \beta q^{89} + 10 q^{90} - 16 q^{91} - 8 \beta q^{93} + \beta q^{94} + 2 \beta q^{95} - 5 \beta q^{96} + q^{98} - 5 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 2 q^{4} + 4 q^{5} + 6 q^{8} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 2 q^{4} + 4 q^{5} + 6 q^{8} - 10 q^{9} - 4 q^{10} - 2 q^{16} + 10 q^{18} - 4 q^{20} + 16 q^{21} - 2 q^{25} - 16 q^{31} - 10 q^{32} - 16 q^{33} + 10 q^{36} + 4 q^{37} + 32 q^{39} + 12 q^{40} - 6 q^{41} - 16 q^{42} + 8 q^{43} - 20 q^{45} - 2 q^{49} + 2 q^{50} - 16 q^{57} + 8 q^{59} - 4 q^{61} + 16 q^{62} + 14 q^{64} + 16 q^{66} - 30 q^{72} + 28 q^{73} - 4 q^{74} + 16 q^{77} - 32 q^{78} - 4 q^{80} + 2 q^{81} + 6 q^{82} - 24 q^{83} - 16 q^{84} - 8 q^{86} + 32 q^{87} + 20 q^{90} - 32 q^{91} + 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/41\mathbb{Z}\right)^\times\).

\(n\) \(6\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
40.1
1.41421i
1.41421i
−1.00000 2.82843i −1.00000 2.00000 2.82843i 2.82843i 3.00000 −5.00000 −2.00000
40.2 −1.00000 2.82843i −1.00000 2.00000 2.82843i 2.82843i 3.00000 −5.00000 −2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
41.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 41.2.b.a 2
3.b odd 2 1 369.2.d.c 2
4.b odd 2 1 656.2.d.c 2
5.b even 2 1 1025.2.d.a 2
5.c odd 4 2 1025.2.c.b 4
8.b even 2 1 2624.2.d.a 2
8.d odd 2 1 2624.2.d.b 2
41.b even 2 1 inner 41.2.b.a 2
41.c even 4 2 1681.2.a.a 2
123.b odd 2 1 369.2.d.c 2
164.d odd 2 1 656.2.d.c 2
205.c even 2 1 1025.2.d.a 2
205.g odd 4 2 1025.2.c.b 4
328.c odd 2 1 2624.2.d.b 2
328.g even 2 1 2624.2.d.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
41.2.b.a 2 1.a even 1 1 trivial
41.2.b.a 2 41.b even 2 1 inner
369.2.d.c 2 3.b odd 2 1
369.2.d.c 2 123.b odd 2 1
656.2.d.c 2 4.b odd 2 1
656.2.d.c 2 164.d odd 2 1
1025.2.c.b 4 5.c odd 4 2
1025.2.c.b 4 205.g odd 4 2
1025.2.d.a 2 5.b even 2 1
1025.2.d.a 2 205.c even 2 1
1681.2.a.a 2 41.c even 4 2
2624.2.d.a 2 8.b even 2 1
2624.2.d.a 2 328.g even 2 1
2624.2.d.b 2 8.d odd 2 1
2624.2.d.b 2 328.c odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(41, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 8 \) Copy content Toggle raw display
$5$ \( (T - 2)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 8 \) Copy content Toggle raw display
$11$ \( T^{2} + 8 \) Copy content Toggle raw display
$13$ \( T^{2} + 32 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 8 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 32 \) Copy content Toggle raw display
$31$ \( (T + 8)^{2} \) Copy content Toggle raw display
$37$ \( (T - 2)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 6T + 41 \) Copy content Toggle raw display
$43$ \( (T - 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 8 \) Copy content Toggle raw display
$53$ \( T^{2} + 32 \) Copy content Toggle raw display
$59$ \( (T - 4)^{2} \) Copy content Toggle raw display
$61$ \( (T + 2)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 72 \) Copy content Toggle raw display
$71$ \( T^{2} + 72 \) Copy content Toggle raw display
$73$ \( (T - 14)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 8 \) Copy content Toggle raw display
$83$ \( (T + 12)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 128 \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
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