Properties

Label 402.2.a.f
Level $402$
Weight $2$
Character orbit 402.a
Self dual yes
Analytic conductor $3.210$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [402,2,Mod(1,402)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(402, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("402.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 402 = 2 \cdot 3 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 402.a (trivial)

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: yes
Analytic conductor: \(3.20998616126\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{41}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = \frac{1}{2}(1 + \sqrt{41})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} + \beta q^{5} - q^{6} - \beta q^{7} + q^{8} + q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - q^{3} + q^{4} + \beta q^{5} - q^{6} - \beta q^{7} + q^{8} + q^{9} + \beta q^{10} + 4 q^{11} - q^{12} + 4 q^{13} - \beta q^{14} - \beta q^{15} + q^{16} - 2 q^{17} + q^{18} - 2 \beta q^{19} + \beta q^{20} + \beta q^{21} + 4 q^{22} + (\beta + 4) q^{23} - q^{24} + (\beta + 5) q^{25} + 4 q^{26} - q^{27} - \beta q^{28} - 4 q^{29} - \beta q^{30} + (\beta - 4) q^{31} + q^{32} - 4 q^{33} - 2 q^{34} + ( - \beta - 10) q^{35} + q^{36} + ( - \beta + 8) q^{37} - 2 \beta q^{38} - 4 q^{39} + \beta q^{40} - \beta q^{41} + \beta q^{42} + (\beta - 6) q^{43} + 4 q^{44} + \beta q^{45} + (\beta + 4) q^{46} + (2 \beta - 6) q^{47} - q^{48} + (\beta + 3) q^{49} + (\beta + 5) q^{50} + 2 q^{51} + 4 q^{52} - 3 \beta q^{53} - q^{54} + 4 \beta q^{55} - \beta q^{56} + 2 \beta q^{57} - 4 q^{58} + ( - 3 \beta - 2) q^{59} - \beta q^{60} + (2 \beta - 8) q^{61} + (\beta - 4) q^{62} - \beta q^{63} + q^{64} + 4 \beta q^{65} - 4 q^{66} - q^{67} - 2 q^{68} + ( - \beta - 4) q^{69} + ( - \beta - 10) q^{70} + ( - 2 \beta - 6) q^{71} + q^{72} + ( - 3 \beta + 8) q^{73} + ( - \beta + 8) q^{74} + ( - \beta - 5) q^{75} - 2 \beta q^{76} - 4 \beta q^{77} - 4 q^{78} + ( - 2 \beta - 2) q^{79} + \beta q^{80} + q^{81} - \beta q^{82} + (3 \beta + 6) q^{83} + \beta q^{84} - 2 \beta q^{85} + (\beta - 6) q^{86} + 4 q^{87} + 4 q^{88} + (2 \beta - 10) q^{89} + \beta q^{90} - 4 \beta q^{91} + (\beta + 4) q^{92} + ( - \beta + 4) q^{93} + (2 \beta - 6) q^{94} + ( - 2 \beta - 20) q^{95} - q^{96} + ( - 2 \beta - 6) q^{97} + (\beta + 3) q^{98} + 4 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} + q^{5} - 2 q^{6} - q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} + q^{5} - 2 q^{6} - q^{7} + 2 q^{8} + 2 q^{9} + q^{10} + 8 q^{11} - 2 q^{12} + 8 q^{13} - q^{14} - q^{15} + 2 q^{16} - 4 q^{17} + 2 q^{18} - 2 q^{19} + q^{20} + q^{21} + 8 q^{22} + 9 q^{23} - 2 q^{24} + 11 q^{25} + 8 q^{26} - 2 q^{27} - q^{28} - 8 q^{29} - q^{30} - 7 q^{31} + 2 q^{32} - 8 q^{33} - 4 q^{34} - 21 q^{35} + 2 q^{36} + 15 q^{37} - 2 q^{38} - 8 q^{39} + q^{40} - q^{41} + q^{42} - 11 q^{43} + 8 q^{44} + q^{45} + 9 q^{46} - 10 q^{47} - 2 q^{48} + 7 q^{49} + 11 q^{50} + 4 q^{51} + 8 q^{52} - 3 q^{53} - 2 q^{54} + 4 q^{55} - q^{56} + 2 q^{57} - 8 q^{58} - 7 q^{59} - q^{60} - 14 q^{61} - 7 q^{62} - q^{63} + 2 q^{64} + 4 q^{65} - 8 q^{66} - 2 q^{67} - 4 q^{68} - 9 q^{69} - 21 q^{70} - 14 q^{71} + 2 q^{72} + 13 q^{73} + 15 q^{74} - 11 q^{75} - 2 q^{76} - 4 q^{77} - 8 q^{78} - 6 q^{79} + q^{80} + 2 q^{81} - q^{82} + 15 q^{83} + q^{84} - 2 q^{85} - 11 q^{86} + 8 q^{87} + 8 q^{88} - 18 q^{89} + q^{90} - 4 q^{91} + 9 q^{92} + 7 q^{93} - 10 q^{94} - 42 q^{95} - 2 q^{96} - 14 q^{97} + 7 q^{98} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.70156
3.70156
1.00000 −1.00000 1.00000 −2.70156 −1.00000 2.70156 1.00000 1.00000 −2.70156
1.2 1.00000 −1.00000 1.00000 3.70156 −1.00000 −3.70156 1.00000 1.00000 3.70156
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(67\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 402.2.a.f 2
3.b odd 2 1 1206.2.a.h 2
4.b odd 2 1 3216.2.a.p 2
12.b even 2 1 9648.2.a.z 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
402.2.a.f 2 1.a even 1 1 trivial
1206.2.a.h 2 3.b odd 2 1
3216.2.a.p 2 4.b odd 2 1
9648.2.a.z 2 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(402))\):

\( T_{5}^{2} - T_{5} - 10 \) Copy content Toggle raw display
\( T_{7}^{2} + T_{7} - 10 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{2} \) Copy content Toggle raw display
$3$ \( (T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - T - 10 \) Copy content Toggle raw display
$7$ \( T^{2} + T - 10 \) Copy content Toggle raw display
$11$ \( (T - 4)^{2} \) Copy content Toggle raw display
$13$ \( (T - 4)^{2} \) Copy content Toggle raw display
$17$ \( (T + 2)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 2T - 40 \) Copy content Toggle raw display
$23$ \( T^{2} - 9T + 10 \) Copy content Toggle raw display
$29$ \( (T + 4)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 7T + 2 \) Copy content Toggle raw display
$37$ \( T^{2} - 15T + 46 \) Copy content Toggle raw display
$41$ \( T^{2} + T - 10 \) Copy content Toggle raw display
$43$ \( T^{2} + 11T + 20 \) Copy content Toggle raw display
$47$ \( T^{2} + 10T - 16 \) Copy content Toggle raw display
$53$ \( T^{2} + 3T - 90 \) Copy content Toggle raw display
$59$ \( T^{2} + 7T - 80 \) Copy content Toggle raw display
$61$ \( T^{2} + 14T + 8 \) Copy content Toggle raw display
$67$ \( (T + 1)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 14T + 8 \) Copy content Toggle raw display
$73$ \( T^{2} - 13T - 50 \) Copy content Toggle raw display
$79$ \( T^{2} + 6T - 32 \) Copy content Toggle raw display
$83$ \( T^{2} - 15T - 36 \) Copy content Toggle raw display
$89$ \( T^{2} + 18T + 40 \) Copy content Toggle raw display
$97$ \( T^{2} + 14T + 8 \) Copy content Toggle raw display
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