Properties

Label 3675.2.a.y
Level $3675$
Weight $2$
Character orbit 3675.a
Self dual yes
Analytic conductor $29.345$
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] = [3675,2,Mod(1,3675)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3675, 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("3675.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3675 = 3 \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3675.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: \(29.3450227428\)
Analytic rank: \(0\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 105)
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 \(\beta = \sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{2} - q^{3} + 3 q^{4} + \beta q^{6} - \beta q^{8} + q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{2} - q^{3} + 3 q^{4} + \beta q^{6} - \beta q^{8} + q^{9} + ( - 2 \beta + 2) q^{11} - 3 q^{12} - 2 \beta q^{13} - q^{16} - 2 q^{17} - \beta q^{18} + ( - 2 \beta - 2) q^{19} + ( - 2 \beta + 10) q^{22} - 4 q^{23} + \beta q^{24} + 10 q^{26} - q^{27} - 2 q^{29} + ( - 2 \beta - 6) q^{31} + 3 \beta q^{32} + (2 \beta - 2) q^{33} + 2 \beta q^{34} + 3 q^{36} + ( - 4 \beta - 2) q^{37} + (2 \beta + 10) q^{38} + 2 \beta q^{39} + 2 q^{41} + 4 \beta q^{43} + ( - 6 \beta + 6) q^{44} + 4 \beta q^{46} + ( - 4 \beta + 4) q^{47} + q^{48} + 2 q^{51} - 6 \beta q^{52} + (2 \beta + 8) q^{53} + \beta q^{54} + (2 \beta + 2) q^{57} + 2 \beta q^{58} - 4 \beta q^{59} + 2 q^{61} + (6 \beta + 10) q^{62} - 13 q^{64} + (2 \beta - 10) q^{66} + 4 q^{67} - 6 q^{68} + 4 q^{69} + (2 \beta + 10) q^{71} - \beta q^{72} + (2 \beta - 8) q^{73} + (2 \beta + 20) q^{74} + ( - 6 \beta - 6) q^{76} - 10 q^{78} + ( - 4 \beta + 4) q^{79} + q^{81} - 2 \beta q^{82} + (4 \beta - 8) q^{83} - 20 q^{86} + 2 q^{87} + ( - 2 \beta + 10) q^{88} + 2 q^{89} - 12 q^{92} + (2 \beta + 6) q^{93} + ( - 4 \beta + 20) q^{94} - 3 \beta q^{96} + ( - 2 \beta + 4) q^{97} + ( - 2 \beta + 2) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 6 q^{4} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + 6 q^{4} + 2 q^{9} + 4 q^{11} - 6 q^{12} - 2 q^{16} - 4 q^{17} - 4 q^{19} + 20 q^{22} - 8 q^{23} + 20 q^{26} - 2 q^{27} - 4 q^{29} - 12 q^{31} - 4 q^{33} + 6 q^{36} - 4 q^{37} + 20 q^{38} + 4 q^{41} + 12 q^{44} + 8 q^{47} + 2 q^{48} + 4 q^{51} + 16 q^{53} + 4 q^{57} + 4 q^{61} + 20 q^{62} - 26 q^{64} - 20 q^{66} + 8 q^{67} - 12 q^{68} + 8 q^{69} + 20 q^{71} - 16 q^{73} + 40 q^{74} - 12 q^{76} - 20 q^{78} + 8 q^{79} + 2 q^{81} - 16 q^{83} - 40 q^{86} + 4 q^{87} + 20 q^{88} + 4 q^{89} - 24 q^{92} + 12 q^{93} + 40 q^{94} + 8 q^{97} + 4 q^{99}+O(q^{100}) \) 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
1.61803
−0.618034
−2.23607 −1.00000 3.00000 0 2.23607 0 −2.23607 1.00000 0
1.2 2.23607 −1.00000 3.00000 0 −2.23607 0 2.23607 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(1\)
\(7\) \(-1\)

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 3675.2.a.y 2
5.b even 2 1 735.2.a.k 2
7.b odd 2 1 525.2.a.g 2
15.d odd 2 1 2205.2.a.w 2
21.c even 2 1 1575.2.a.r 2
28.d even 2 1 8400.2.a.cx 2
35.c odd 2 1 105.2.a.b 2
35.f even 4 2 525.2.d.c 4
35.i odd 6 2 735.2.i.k 4
35.j even 6 2 735.2.i.i 4
105.g even 2 1 315.2.a.d 2
105.k odd 4 2 1575.2.d.d 4
140.c even 2 1 1680.2.a.v 2
280.c odd 2 1 6720.2.a.cx 2
280.n even 2 1 6720.2.a.cs 2
420.o odd 2 1 5040.2.a.bw 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.a.b 2 35.c odd 2 1
315.2.a.d 2 105.g even 2 1
525.2.a.g 2 7.b odd 2 1
525.2.d.c 4 35.f even 4 2
735.2.a.k 2 5.b even 2 1
735.2.i.i 4 35.j even 6 2
735.2.i.k 4 35.i odd 6 2
1575.2.a.r 2 21.c even 2 1
1575.2.d.d 4 105.k odd 4 2
1680.2.a.v 2 140.c even 2 1
2205.2.a.w 2 15.d odd 2 1
3675.2.a.y 2 1.a even 1 1 trivial
5040.2.a.bw 2 420.o odd 2 1
6720.2.a.cs 2 280.n even 2 1
6720.2.a.cx 2 280.c odd 2 1
8400.2.a.cx 2 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3675))\):

\( T_{2}^{2} - 5 \) Copy content Toggle raw display
\( T_{11}^{2} - 4T_{11} - 16 \) Copy content Toggle raw display
\( T_{13}^{2} - 20 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 5 \) Copy content Toggle raw display
$3$ \( (T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 4T - 16 \) Copy content Toggle raw display
$13$ \( T^{2} - 20 \) Copy content Toggle raw display
$17$ \( (T + 2)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 4T - 16 \) Copy content Toggle raw display
$23$ \( (T + 4)^{2} \) Copy content Toggle raw display
$29$ \( (T + 2)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 12T + 16 \) Copy content Toggle raw display
$37$ \( T^{2} + 4T - 76 \) Copy content Toggle raw display
$41$ \( (T - 2)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 80 \) Copy content Toggle raw display
$47$ \( T^{2} - 8T - 64 \) Copy content Toggle raw display
$53$ \( T^{2} - 16T + 44 \) Copy content Toggle raw display
$59$ \( T^{2} - 80 \) Copy content Toggle raw display
$61$ \( (T - 2)^{2} \) Copy content Toggle raw display
$67$ \( (T - 4)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} - 20T + 80 \) Copy content Toggle raw display
$73$ \( T^{2} + 16T + 44 \) Copy content Toggle raw display
$79$ \( T^{2} - 8T - 64 \) Copy content Toggle raw display
$83$ \( T^{2} + 16T - 16 \) Copy content Toggle raw display
$89$ \( (T - 2)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 8T - 4 \) Copy content Toggle raw display
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