Properties

Label 3751.2.a.b
Level $3751$
Weight $2$
Character orbit 3751.a
Self dual yes
Analytic conductor $29.952$
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] = [3751,2,Mod(1,3751)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3751, 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("3751.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3751 = 11^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3751.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: \(29.9518857982\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 31)
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.

\(f(q)\) \(=\) \( q - \beta q^{2} - 2 \beta q^{3} + (\beta - 1) q^{4} + q^{5} + (2 \beta + 2) q^{6} + ( - 2 \beta + 3) q^{7} + (2 \beta - 1) q^{8} + (4 \beta + 1) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{2} - 2 \beta q^{3} + (\beta - 1) q^{4} + q^{5} + (2 \beta + 2) q^{6} + ( - 2 \beta + 3) q^{7} + (2 \beta - 1) q^{8} + (4 \beta + 1) q^{9} - \beta q^{10} - 2 q^{12} + 2 \beta q^{13} + ( - \beta + 2) q^{14} - 2 \beta q^{15} - 3 \beta q^{16} + (2 \beta - 4) q^{17} + ( - 5 \beta - 4) q^{18} + (2 \beta - 1) q^{19} + (\beta - 1) q^{20} + ( - 2 \beta + 4) q^{21} + (6 \beta - 4) q^{23} + ( - 2 \beta - 4) q^{24} - 4 q^{25} + ( - 2 \beta - 2) q^{26} + ( - 4 \beta - 8) q^{27} + (3 \beta - 5) q^{28} + (2 \beta - 6) q^{29} + (2 \beta + 2) q^{30} + q^{31} + ( - \beta + 5) q^{32} + (2 \beta - 2) q^{34} + ( - 2 \beta + 3) q^{35} + (\beta + 3) q^{36} - 2 q^{37} + ( - \beta - 2) q^{38} + ( - 4 \beta - 4) q^{39} + (2 \beta - 1) q^{40} - 7 q^{41} + ( - 2 \beta + 2) q^{42} + ( - 2 \beta + 2) q^{43} + (4 \beta + 1) q^{45} + ( - 2 \beta - 6) q^{46} + (4 \beta - 4) q^{47} + (6 \beta + 6) q^{48} + ( - 8 \beta + 6) q^{49} + 4 \beta q^{50} + (4 \beta - 4) q^{51} + 2 q^{52} + ( - 4 \beta - 4) q^{53} + (12 \beta + 4) q^{54} + (4 \beta - 7) q^{56} + ( - 2 \beta - 4) q^{57} + (4 \beta - 2) q^{58} + (2 \beta - 1) q^{59} - 2 q^{60} + ( - 10 \beta + 8) q^{61} - \beta q^{62} + (2 \beta - 5) q^{63} + (2 \beta + 1) q^{64} + 2 \beta q^{65} + 8 q^{67} + ( - 4 \beta + 6) q^{68} + ( - 4 \beta - 12) q^{69} + ( - \beta + 2) q^{70} + ( - 10 \beta + 7) q^{71} + (6 \beta + 7) q^{72} + ( - 4 \beta - 2) q^{73} + 2 \beta q^{74} + 8 \beta q^{75} + ( - \beta + 3) q^{76} + (8 \beta + 4) q^{78} + (6 \beta + 2) q^{79} - 3 \beta q^{80} + (12 \beta + 5) q^{81} + 7 \beta q^{82} + (8 \beta + 2) q^{83} + (4 \beta - 6) q^{84} + (2 \beta - 4) q^{85} + 2 q^{86} + (8 \beta - 4) q^{87} + (6 \beta + 2) q^{89} + ( - 5 \beta - 4) q^{90} + (2 \beta - 4) q^{91} + ( - 4 \beta + 10) q^{92} - 2 \beta q^{93} - 4 q^{94} + (2 \beta - 1) q^{95} + ( - 8 \beta + 2) q^{96} + ( - 8 \beta - 3) q^{97} + (2 \beta + 8) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - 2 q^{3} - q^{4} + 2 q^{5} + 6 q^{6} + 4 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} - 2 q^{3} - q^{4} + 2 q^{5} + 6 q^{6} + 4 q^{7} + 6 q^{9} - q^{10} - 4 q^{12} + 2 q^{13} + 3 q^{14} - 2 q^{15} - 3 q^{16} - 6 q^{17} - 13 q^{18} - q^{20} + 6 q^{21} - 2 q^{23} - 10 q^{24} - 8 q^{25} - 6 q^{26} - 20 q^{27} - 7 q^{28} - 10 q^{29} + 6 q^{30} + 2 q^{31} + 9 q^{32} - 2 q^{34} + 4 q^{35} + 7 q^{36} - 4 q^{37} - 5 q^{38} - 12 q^{39} - 14 q^{41} + 2 q^{42} + 2 q^{43} + 6 q^{45} - 14 q^{46} - 4 q^{47} + 18 q^{48} + 4 q^{49} + 4 q^{50} - 4 q^{51} + 4 q^{52} - 12 q^{53} + 20 q^{54} - 10 q^{56} - 10 q^{57} - 4 q^{60} + 6 q^{61} - q^{62} - 8 q^{63} + 4 q^{64} + 2 q^{65} + 16 q^{67} + 8 q^{68} - 28 q^{69} + 3 q^{70} + 4 q^{71} + 20 q^{72} - 8 q^{73} + 2 q^{74} + 8 q^{75} + 5 q^{76} + 16 q^{78} + 10 q^{79} - 3 q^{80} + 22 q^{81} + 7 q^{82} + 12 q^{83} - 8 q^{84} - 6 q^{85} + 4 q^{86} + 10 q^{89} - 13 q^{90} - 6 q^{91} + 16 q^{92} - 2 q^{93} - 8 q^{94} - 4 q^{96} - 14 q^{97} + 18 q^{98}+O(q^{100}) \) 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
−1.61803 −3.23607 0.618034 1.00000 5.23607 −0.236068 2.23607 7.47214 −1.61803
1.2 0.618034 1.23607 −1.61803 1.00000 0.763932 4.23607 −2.23607 −1.47214 0.618034
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(11\) \(-1\)
\(31\) \(-1\)

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 3751.2.a.b 2
11.b odd 2 1 31.2.a.a 2
33.d even 2 1 279.2.a.a 2
44.c even 2 1 496.2.a.i 2
55.d odd 2 1 775.2.a.d 2
55.e even 4 2 775.2.b.d 4
77.b even 2 1 1519.2.a.a 2
88.b odd 2 1 1984.2.a.r 2
88.g even 2 1 1984.2.a.n 2
132.d odd 2 1 4464.2.a.bf 2
143.d odd 2 1 5239.2.a.f 2
165.d even 2 1 6975.2.a.y 2
187.b odd 2 1 8959.2.a.b 2
341.b even 2 1 961.2.a.f 2
341.l odd 6 2 961.2.c.e 4
341.m even 6 2 961.2.c.c 4
341.z odd 10 2 961.2.d.c 4
341.z odd 10 2 961.2.d.d 4
341.bd even 10 2 961.2.d.a 4
341.bd even 10 2 961.2.d.g 4
341.bu even 30 4 961.2.g.d 8
341.bu even 30 4 961.2.g.e 8
341.by odd 30 4 961.2.g.a 8
341.by odd 30 4 961.2.g.h 8
1023.g odd 2 1 8649.2.a.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
31.2.a.a 2 11.b odd 2 1
279.2.a.a 2 33.d even 2 1
496.2.a.i 2 44.c even 2 1
775.2.a.d 2 55.d odd 2 1
775.2.b.d 4 55.e even 4 2
961.2.a.f 2 341.b even 2 1
961.2.c.c 4 341.m even 6 2
961.2.c.e 4 341.l odd 6 2
961.2.d.a 4 341.bd even 10 2
961.2.d.c 4 341.z odd 10 2
961.2.d.d 4 341.z odd 10 2
961.2.d.g 4 341.bd even 10 2
961.2.g.a 8 341.by odd 30 4
961.2.g.d 8 341.bu even 30 4
961.2.g.e 8 341.bu even 30 4
961.2.g.h 8 341.by odd 30 4
1519.2.a.a 2 77.b even 2 1
1984.2.a.n 2 88.g even 2 1
1984.2.a.r 2 88.b odd 2 1
3751.2.a.b 2 1.a even 1 1 trivial
4464.2.a.bf 2 132.d odd 2 1
5239.2.a.f 2 143.d odd 2 1
6975.2.a.y 2 165.d even 2 1
8649.2.a.c 2 1023.g odd 2 1
8959.2.a.b 2 187.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3751))\):

\( T_{2}^{2} + T_{2} - 1 \) Copy content Toggle raw display
\( T_{3}^{2} + 2T_{3} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T - 1 \) Copy content Toggle raw display
$3$ \( T^{2} + 2T - 4 \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 4T - 1 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 2T - 4 \) Copy content Toggle raw display
$17$ \( T^{2} + 6T + 4 \) Copy content Toggle raw display
$19$ \( T^{2} - 5 \) Copy content Toggle raw display
$23$ \( T^{2} + 2T - 44 \) Copy content Toggle raw display
$29$ \( T^{2} + 10T + 20 \) Copy content Toggle raw display
$31$ \( (T - 1)^{2} \) Copy content Toggle raw display
$37$ \( (T + 2)^{2} \) Copy content Toggle raw display
$41$ \( (T + 7)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 2T - 4 \) Copy content Toggle raw display
$47$ \( T^{2} + 4T - 16 \) Copy content Toggle raw display
$53$ \( T^{2} + 12T + 16 \) Copy content Toggle raw display
$59$ \( T^{2} - 5 \) Copy content Toggle raw display
$61$ \( T^{2} - 6T - 116 \) Copy content Toggle raw display
$67$ \( (T - 8)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} - 4T - 121 \) Copy content Toggle raw display
$73$ \( T^{2} + 8T - 4 \) Copy content Toggle raw display
$79$ \( T^{2} - 10T - 20 \) Copy content Toggle raw display
$83$ \( T^{2} - 12T - 44 \) Copy content Toggle raw display
$89$ \( T^{2} - 10T - 20 \) Copy content Toggle raw display
$97$ \( T^{2} + 14T - 31 \) Copy content Toggle raw display
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