Properties

Label 3850.2.a.bl
Level $3850$
Weight $2$
Character orbit 3850.a
Self dual yes
Analytic conductor $30.742$
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] = [3850,2,Mod(1,3850)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3850, 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("3850.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3850 = 2 \cdot 5^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3850.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: \(30.7424047782\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) 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 \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + (\beta - 1) q^{3} + q^{4} + (\beta - 1) q^{6} - q^{7} + q^{8} + ( - 2 \beta + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + (\beta - 1) q^{3} + q^{4} + (\beta - 1) q^{6} - q^{7} + q^{8} + ( - 2 \beta + 1) q^{9} + q^{11} + (\beta - 1) q^{12} + (\beta - 2) q^{13} - q^{14} + q^{16} + ( - \beta - 3) q^{17} + ( - 2 \beta + 1) q^{18} + ( - 2 \beta - 1) q^{19} + ( - \beta + 1) q^{21} + q^{22} - \beta q^{23} + (\beta - 1) q^{24} + (\beta - 2) q^{26} - 4 q^{27} - q^{28} + ( - 2 \beta + 3) q^{29} + (2 \beta - 1) q^{31} + q^{32} + (\beta - 1) q^{33} + ( - \beta - 3) q^{34} + ( - 2 \beta + 1) q^{36} + (2 \beta - 2) q^{37} + ( - 2 \beta - 1) q^{38} + ( - 3 \beta + 5) q^{39} + ( - 2 \beta - 6) q^{41} + ( - \beta + 1) q^{42} + ( - \beta + 4) q^{43} + q^{44} - \beta q^{46} - 6 q^{47} + (\beta - 1) q^{48} + q^{49} - 2 \beta q^{51} + (\beta - 2) q^{52} + ( - \beta - 3) q^{53} - 4 q^{54} - q^{56} + (\beta - 5) q^{57} + ( - 2 \beta + 3) q^{58} - 4 \beta q^{59} + (6 \beta - 4) q^{61} + (2 \beta - 1) q^{62} + (2 \beta - 1) q^{63} + q^{64} + (\beta - 1) q^{66} + ( - \beta - 5) q^{67} + ( - \beta - 3) q^{68} + (\beta - 3) q^{69} - 5 \beta q^{71} + ( - 2 \beta + 1) q^{72} + ( - 4 \beta + 4) q^{73} + (2 \beta - 2) q^{74} + ( - 2 \beta - 1) q^{76} - q^{77} + ( - 3 \beta + 5) q^{78} + (5 \beta - 1) q^{79} + (2 \beta + 1) q^{81} + ( - 2 \beta - 6) q^{82} - 3 q^{83} + ( - \beta + 1) q^{84} + ( - \beta + 4) q^{86} + (5 \beta - 9) q^{87} + q^{88} - \beta q^{89} + ( - \beta + 2) q^{91} - \beta q^{92} + ( - 3 \beta + 7) q^{93} - 6 q^{94} + (\beta - 1) q^{96} + ( - 3 \beta - 2) q^{97} + q^{98} + ( - 2 \beta + 1) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{6} - 2 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{6} - 2 q^{7} + 2 q^{8} + 2 q^{9} + 2 q^{11} - 2 q^{12} - 4 q^{13} - 2 q^{14} + 2 q^{16} - 6 q^{17} + 2 q^{18} - 2 q^{19} + 2 q^{21} + 2 q^{22} - 2 q^{24} - 4 q^{26} - 8 q^{27} - 2 q^{28} + 6 q^{29} - 2 q^{31} + 2 q^{32} - 2 q^{33} - 6 q^{34} + 2 q^{36} - 4 q^{37} - 2 q^{38} + 10 q^{39} - 12 q^{41} + 2 q^{42} + 8 q^{43} + 2 q^{44} - 12 q^{47} - 2 q^{48} + 2 q^{49} - 4 q^{52} - 6 q^{53} - 8 q^{54} - 2 q^{56} - 10 q^{57} + 6 q^{58} - 8 q^{61} - 2 q^{62} - 2 q^{63} + 2 q^{64} - 2 q^{66} - 10 q^{67} - 6 q^{68} - 6 q^{69} + 2 q^{72} + 8 q^{73} - 4 q^{74} - 2 q^{76} - 2 q^{77} + 10 q^{78} - 2 q^{79} + 2 q^{81} - 12 q^{82} - 6 q^{83} + 2 q^{84} + 8 q^{86} - 18 q^{87} + 2 q^{88} + 4 q^{91} + 14 q^{93} - 12 q^{94} - 2 q^{96} - 4 q^{97} + 2 q^{98} + 2 q^{99}+O(q^{100}) \) 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
−1.73205
1.73205
1.00000 −2.73205 1.00000 0 −2.73205 −1.00000 1.00000 4.46410 0
1.2 1.00000 0.732051 1.00000 0 0.732051 −1.00000 1.00000 −2.46410 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)
\(7\) \(1\)
\(11\) \(-1\)

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 3850.2.a.bl yes 2
5.b even 2 1 3850.2.a.bk 2
5.c odd 4 2 3850.2.c.r 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3850.2.a.bk 2 5.b even 2 1
3850.2.a.bl yes 2 1.a even 1 1 trivial
3850.2.c.r 4 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3850))\):

\( T_{3}^{2} + 2T_{3} - 2 \) Copy content Toggle raw display
\( T_{13}^{2} + 4T_{13} + 1 \) Copy content Toggle raw display
\( T_{17}^{2} + 6T_{17} + 6 \) Copy content Toggle raw display
\( T_{19}^{2} + 2T_{19} - 11 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 2T - 2 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T + 1)^{2} \) Copy content Toggle raw display
$11$ \( (T - 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 4T + 1 \) Copy content Toggle raw display
$17$ \( T^{2} + 6T + 6 \) Copy content Toggle raw display
$19$ \( T^{2} + 2T - 11 \) Copy content Toggle raw display
$23$ \( T^{2} - 3 \) Copy content Toggle raw display
$29$ \( T^{2} - 6T - 3 \) Copy content Toggle raw display
$31$ \( T^{2} + 2T - 11 \) Copy content Toggle raw display
$37$ \( T^{2} + 4T - 8 \) Copy content Toggle raw display
$41$ \( T^{2} + 12T + 24 \) Copy content Toggle raw display
$43$ \( T^{2} - 8T + 13 \) Copy content Toggle raw display
$47$ \( (T + 6)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 6T + 6 \) Copy content Toggle raw display
$59$ \( T^{2} - 48 \) Copy content Toggle raw display
$61$ \( T^{2} + 8T - 92 \) Copy content Toggle raw display
$67$ \( T^{2} + 10T + 22 \) Copy content Toggle raw display
$71$ \( T^{2} - 75 \) Copy content Toggle raw display
$73$ \( T^{2} - 8T - 32 \) Copy content Toggle raw display
$79$ \( T^{2} + 2T - 74 \) Copy content Toggle raw display
$83$ \( (T + 3)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 3 \) Copy content Toggle raw display
$97$ \( T^{2} + 4T - 23 \) Copy content Toggle raw display
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