Properties

Label 3630.2.a.bi
Level $3630$
Weight $2$
Character orbit 3630.a
Self dual yes
Analytic conductor $28.986$
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] = [3630,2,Mod(1,3630)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3630, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("3630.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3630 = 2 \cdot 3 \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3630.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: \(28.9856959337\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 330)
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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} + ( - \beta - 2) q^{7} + q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} + ( - \beta - 2) q^{7} + q^{8} + q^{9} - q^{10} - q^{12} + ( - \beta + 1) q^{13} + ( - \beta - 2) q^{14} + q^{15} + q^{16} + 4 \beta q^{17} + q^{18} + (\beta - 1) q^{19} - q^{20} + (\beta + 2) q^{21} + 3 \beta q^{23} - q^{24} + q^{25} + ( - \beta + 1) q^{26} - q^{27} + ( - \beta - 2) q^{28} + ( - 6 \beta + 6) q^{29} + q^{30} + ( - 2 \beta - 6) q^{31} + q^{32} + 4 \beta q^{34} + (\beta + 2) q^{35} + q^{36} + ( - \beta + 1) q^{37} + (\beta - 1) q^{38} + (\beta - 1) q^{39} - q^{40} + (7 \beta - 2) q^{41} + (\beta + 2) q^{42} + (2 \beta - 4) q^{43} - q^{45} + 3 \beta q^{46} + ( - \beta - 1) q^{47} - q^{48} + (5 \beta - 2) q^{49} + q^{50} - 4 \beta q^{51} + ( - \beta + 1) q^{52} + ( - 5 \beta + 2) q^{53} - q^{54} + ( - \beta - 2) q^{56} + ( - \beta + 1) q^{57} + ( - 6 \beta + 6) q^{58} + ( - \beta - 13) q^{59} + q^{60} + ( - 2 \beta - 8) q^{61} + ( - 2 \beta - 6) q^{62} + ( - \beta - 2) q^{63} + q^{64} + (\beta - 1) q^{65} + ( - 10 \beta + 8) q^{67} + 4 \beta q^{68} - 3 \beta q^{69} + (\beta + 2) q^{70} + (6 \beta - 10) q^{71} + q^{72} + ( - 8 \beta + 2) q^{73} + ( - \beta + 1) q^{74} - q^{75} + (\beta - 1) q^{76} + (\beta - 1) q^{78} + (4 \beta - 2) q^{79} - q^{80} + q^{81} + (7 \beta - 2) q^{82} + (8 \beta - 8) q^{83} + (\beta + 2) q^{84} - 4 \beta q^{85} + (2 \beta - 4) q^{86} + (6 \beta - 6) q^{87} + (3 \beta - 10) q^{89} - q^{90} + (2 \beta - 1) q^{91} + 3 \beta q^{92} + (2 \beta + 6) q^{93} + ( - \beta - 1) q^{94} + ( - \beta + 1) q^{95} - q^{96} + (8 \beta + 2) q^{97} + (5 \beta - 2) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} - 5 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} - 5 q^{7} + 2 q^{8} + 2 q^{9} - 2 q^{10} - 2 q^{12} + q^{13} - 5 q^{14} + 2 q^{15} + 2 q^{16} + 4 q^{17} + 2 q^{18} - q^{19} - 2 q^{20} + 5 q^{21} + 3 q^{23} - 2 q^{24} + 2 q^{25} + q^{26} - 2 q^{27} - 5 q^{28} + 6 q^{29} + 2 q^{30} - 14 q^{31} + 2 q^{32} + 4 q^{34} + 5 q^{35} + 2 q^{36} + q^{37} - q^{38} - q^{39} - 2 q^{40} + 3 q^{41} + 5 q^{42} - 6 q^{43} - 2 q^{45} + 3 q^{46} - 3 q^{47} - 2 q^{48} + q^{49} + 2 q^{50} - 4 q^{51} + q^{52} - q^{53} - 2 q^{54} - 5 q^{56} + q^{57} + 6 q^{58} - 27 q^{59} + 2 q^{60} - 18 q^{61} - 14 q^{62} - 5 q^{63} + 2 q^{64} - q^{65} + 6 q^{67} + 4 q^{68} - 3 q^{69} + 5 q^{70} - 14 q^{71} + 2 q^{72} - 4 q^{73} + q^{74} - 2 q^{75} - q^{76} - q^{78} - 2 q^{80} + 2 q^{81} + 3 q^{82} - 8 q^{83} + 5 q^{84} - 4 q^{85} - 6 q^{86} - 6 q^{87} - 17 q^{89} - 2 q^{90} + 3 q^{92} + 14 q^{93} - 3 q^{94} + q^{95} - 2 q^{96} + 12 q^{97} + q^{98}+O(q^{100}) \) 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
1.61803
−0.618034
1.00000 −1.00000 1.00000 −1.00000 −1.00000 −3.61803 1.00000 1.00000 −1.00000
1.2 1.00000 −1.00000 1.00000 −1.00000 −1.00000 −1.38197 1.00000 1.00000 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(5\) \(1\)
\(11\) \(-1\)

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 3630.2.a.bi 2
11.b odd 2 1 3630.2.a.bc 2
11.d odd 10 2 330.2.m.d 4
33.f even 10 2 990.2.n.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
330.2.m.d 4 11.d odd 10 2
990.2.n.a 4 33.f even 10 2
3630.2.a.bc 2 11.b odd 2 1
3630.2.a.bi 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3630))\):

\( T_{7}^{2} + 5T_{7} + 5 \) Copy content Toggle raw display
\( T_{13}^{2} - T_{13} - 1 \) 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
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 5T + 5 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - T - 1 \) Copy content Toggle raw display
$17$ \( T^{2} - 4T - 16 \) Copy content Toggle raw display
$19$ \( T^{2} + T - 1 \) Copy content Toggle raw display
$23$ \( T^{2} - 3T - 9 \) Copy content Toggle raw display
$29$ \( T^{2} - 6T - 36 \) Copy content Toggle raw display
$31$ \( T^{2} + 14T + 44 \) Copy content Toggle raw display
$37$ \( T^{2} - T - 1 \) Copy content Toggle raw display
$41$ \( T^{2} - 3T - 59 \) Copy content Toggle raw display
$43$ \( T^{2} + 6T + 4 \) Copy content Toggle raw display
$47$ \( T^{2} + 3T + 1 \) Copy content Toggle raw display
$53$ \( T^{2} + T - 31 \) Copy content Toggle raw display
$59$ \( T^{2} + 27T + 181 \) Copy content Toggle raw display
$61$ \( T^{2} + 18T + 76 \) Copy content Toggle raw display
$67$ \( T^{2} - 6T - 116 \) Copy content Toggle raw display
$71$ \( T^{2} + 14T + 4 \) Copy content Toggle raw display
$73$ \( T^{2} + 4T - 76 \) Copy content Toggle raw display
$79$ \( T^{2} - 20 \) Copy content Toggle raw display
$83$ \( T^{2} + 8T - 64 \) Copy content Toggle raw display
$89$ \( T^{2} + 17T + 61 \) Copy content Toggle raw display
$97$ \( T^{2} - 12T - 44 \) Copy content Toggle raw display
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