Properties

Label 418.2.a.d
Level $418$
Weight $2$
Character orbit 418.a
Self dual yes
Analytic conductor $3.338$
Analytic rank $1$
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] = [418,2,Mod(1,418)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(418, 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("418.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 418 = 2 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 418.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: \(3.33774680449\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{13})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + ( - \beta - 1) q^{3} + q^{4} + (\beta - 1) q^{5} + (\beta + 1) q^{6} - \beta q^{7} - q^{8} + (3 \beta + 1) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + ( - \beta - 1) q^{3} + q^{4} + (\beta - 1) q^{5} + (\beta + 1) q^{6} - \beta q^{7} - q^{8} + (3 \beta + 1) q^{9} + ( - \beta + 1) q^{10} + q^{11} + ( - \beta - 1) q^{12} + (\beta - 2) q^{13} + \beta q^{14} + ( - \beta - 2) q^{15} + q^{16} + (2 \beta - 2) q^{17} + ( - 3 \beta - 1) q^{18} + q^{19} + (\beta - 1) q^{20} + (2 \beta + 3) q^{21} - q^{22} + ( - 2 \beta - 4) q^{23} + (\beta + 1) q^{24} + ( - \beta - 1) q^{25} + ( - \beta + 2) q^{26} + ( - 4 \beta - 7) q^{27} - \beta q^{28} + (\beta - 7) q^{29} + (\beta + 2) q^{30} + (\beta - 2) q^{31} - q^{32} + ( - \beta - 1) q^{33} + ( - 2 \beta + 2) q^{34} - 3 q^{35} + (3 \beta + 1) q^{36} - 4 \beta q^{37} - q^{38} - q^{39} + ( - \beta + 1) q^{40} - 3 \beta q^{41} + ( - 2 \beta - 3) q^{42} + (3 \beta + 5) q^{43} + q^{44} + (\beta + 8) q^{45} + (2 \beta + 4) q^{46} - 6 q^{47} + ( - \beta - 1) q^{48} + (\beta - 4) q^{49} + (\beta + 1) q^{50} + ( - 2 \beta - 4) q^{51} + (\beta - 2) q^{52} + (2 \beta - 8) q^{53} + (4 \beta + 7) q^{54} + (\beta - 1) q^{55} + \beta q^{56} + ( - \beta - 1) q^{57} + ( - \beta + 7) q^{58} + ( - \beta - 2) q^{60} + ( - 4 \beta + 6) q^{61} + ( - \beta + 2) q^{62} + ( - 4 \beta - 9) q^{63} + q^{64} + ( - 2 \beta + 5) q^{65} + (\beta + 1) q^{66} + 5 \beta q^{67} + (2 \beta - 2) q^{68} + (8 \beta + 10) q^{69} + 3 q^{70} + ( - \beta - 11) q^{71} + ( - 3 \beta - 1) q^{72} + 2 \beta q^{73} + 4 \beta q^{74} + (3 \beta + 4) q^{75} + q^{76} - \beta q^{77} + q^{78} + ( - 6 \beta + 8) q^{79} + (\beta - 1) q^{80} + (6 \beta + 16) q^{81} + 3 \beta q^{82} + ( - \beta + 13) q^{83} + (2 \beta + 3) q^{84} + ( - 2 \beta + 8) q^{85} + ( - 3 \beta - 5) q^{86} + (5 \beta + 4) q^{87} - q^{88} + ( - 2 \beta + 2) q^{89} + ( - \beta - 8) q^{90} + (\beta - 3) q^{91} + ( - 2 \beta - 4) q^{92} - q^{93} + 6 q^{94} + (\beta - 1) q^{95} + (\beta + 1) q^{96} + 8 q^{97} + ( - \beta + 4) q^{98} + (3 \beta + 1) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 3 q^{3} + 2 q^{4} - q^{5} + 3 q^{6} - q^{7} - 2 q^{8} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 3 q^{3} + 2 q^{4} - q^{5} + 3 q^{6} - q^{7} - 2 q^{8} + 5 q^{9} + q^{10} + 2 q^{11} - 3 q^{12} - 3 q^{13} + q^{14} - 5 q^{15} + 2 q^{16} - 2 q^{17} - 5 q^{18} + 2 q^{19} - q^{20} + 8 q^{21} - 2 q^{22} - 10 q^{23} + 3 q^{24} - 3 q^{25} + 3 q^{26} - 18 q^{27} - q^{28} - 13 q^{29} + 5 q^{30} - 3 q^{31} - 2 q^{32} - 3 q^{33} + 2 q^{34} - 6 q^{35} + 5 q^{36} - 4 q^{37} - 2 q^{38} - 2 q^{39} + q^{40} - 3 q^{41} - 8 q^{42} + 13 q^{43} + 2 q^{44} + 17 q^{45} + 10 q^{46} - 12 q^{47} - 3 q^{48} - 7 q^{49} + 3 q^{50} - 10 q^{51} - 3 q^{52} - 14 q^{53} + 18 q^{54} - q^{55} + q^{56} - 3 q^{57} + 13 q^{58} - 5 q^{60} + 8 q^{61} + 3 q^{62} - 22 q^{63} + 2 q^{64} + 8 q^{65} + 3 q^{66} + 5 q^{67} - 2 q^{68} + 28 q^{69} + 6 q^{70} - 23 q^{71} - 5 q^{72} + 2 q^{73} + 4 q^{74} + 11 q^{75} + 2 q^{76} - q^{77} + 2 q^{78} + 10 q^{79} - q^{80} + 38 q^{81} + 3 q^{82} + 25 q^{83} + 8 q^{84} + 14 q^{85} - 13 q^{86} + 13 q^{87} - 2 q^{88} + 2 q^{89} - 17 q^{90} - 5 q^{91} - 10 q^{92} - 2 q^{93} + 12 q^{94} - q^{95} + 3 q^{96} + 16 q^{97} + 7 q^{98} + 5 q^{99}+O(q^{100}) \) 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
2.30278
−1.30278
−1.00000 −3.30278 1.00000 1.30278 3.30278 −2.30278 −1.00000 7.90833 −1.30278
1.2 −1.00000 0.302776 1.00000 −2.30278 −0.302776 1.30278 −1.00000 −2.90833 2.30278
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(11\) \( -1 \)
\(19\) \( -1 \)

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 418.2.a.d 2
3.b odd 2 1 3762.2.a.bb 2
4.b odd 2 1 3344.2.a.o 2
11.b odd 2 1 4598.2.a.bc 2
19.b odd 2 1 7942.2.a.bb 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.a.d 2 1.a even 1 1 trivial
3344.2.a.o 2 4.b odd 2 1
3762.2.a.bb 2 3.b odd 2 1
4598.2.a.bc 2 11.b odd 2 1
7942.2.a.bb 2 19.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 3T_{3} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(418))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 3T - 1 \) Copy content Toggle raw display
$5$ \( T^{2} + T - 3 \) Copy content Toggle raw display
$7$ \( T^{2} + T - 3 \) Copy content Toggle raw display
$11$ \( (T - 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 3T - 1 \) Copy content Toggle raw display
$17$ \( T^{2} + 2T - 12 \) Copy content Toggle raw display
$19$ \( (T - 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 10T + 12 \) Copy content Toggle raw display
$29$ \( T^{2} + 13T + 39 \) Copy content Toggle raw display
$31$ \( T^{2} + 3T - 1 \) Copy content Toggle raw display
$37$ \( T^{2} + 4T - 48 \) Copy content Toggle raw display
$41$ \( T^{2} + 3T - 27 \) Copy content Toggle raw display
$43$ \( T^{2} - 13T + 13 \) Copy content Toggle raw display
$47$ \( (T + 6)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 14T + 36 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 8T - 36 \) Copy content Toggle raw display
$67$ \( T^{2} - 5T - 75 \) Copy content Toggle raw display
$71$ \( T^{2} + 23T + 129 \) Copy content Toggle raw display
$73$ \( T^{2} - 2T - 12 \) Copy content Toggle raw display
$79$ \( T^{2} - 10T - 92 \) Copy content Toggle raw display
$83$ \( T^{2} - 25T + 153 \) Copy content Toggle raw display
$89$ \( T^{2} - 2T - 12 \) Copy content Toggle raw display
$97$ \( (T - 8)^{2} \) Copy content Toggle raw display
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