Properties

Label 3850.2.a.bq
Level $3850$
Weight $2$
Character orbit 3850.a
Self dual yes
Analytic conductor $30.742$
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] = [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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{10}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 10 \) 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 \(\beta = \sqrt{10}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + \beta q^{3} + q^{4} + \beta q^{6} + q^{7} + q^{8} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + \beta q^{3} + q^{4} + \beta q^{6} + q^{7} + q^{8} + 7 q^{9} - q^{11} + \beta q^{12} + ( - \beta + 3) q^{13} + q^{14} + q^{16} + ( - \beta + 4) q^{17} + 7 q^{18} - q^{19} + \beta q^{21} - q^{22} + ( - \beta + 1) q^{23} + \beta q^{24} + ( - \beta + 3) q^{26} + 4 \beta q^{27} + q^{28} + 3 q^{29} - 7 q^{31} + q^{32} - \beta q^{33} + ( - \beta + 4) q^{34} + 7 q^{36} + 2 \beta q^{37} - q^{38} + (3 \beta - 10) q^{39} + (2 \beta + 4) q^{41} + \beta q^{42} + ( - \beta + 3) q^{43} - q^{44} + ( - \beta + 1) q^{46} + (2 \beta - 2) q^{47} + \beta q^{48} + q^{49} + (4 \beta - 10) q^{51} + ( - \beta + 3) q^{52} + ( - \beta + 4) q^{53} + 4 \beta q^{54} + q^{56} - \beta q^{57} + 3 q^{58} - 12 q^{59} + (2 \beta + 6) q^{61} - 7 q^{62} + 7 q^{63} + q^{64} - \beta q^{66} + ( - 3 \beta + 2) q^{67} + ( - \beta + 4) q^{68} + (\beta - 10) q^{69} + ( - \beta + 7) q^{71} + 7 q^{72} - 4 q^{73} + 2 \beta q^{74} - q^{76} - q^{77} + (3 \beta - 10) q^{78} - \beta q^{79} + 19 q^{81} + (2 \beta + 4) q^{82} + ( - 4 \beta + 1) q^{83} + \beta q^{84} + ( - \beta + 3) q^{86} + 3 \beta q^{87} - q^{88} + (\beta - 1) q^{89} + ( - \beta + 3) q^{91} + ( - \beta + 1) q^{92} - 7 \beta q^{93} + (2 \beta - 2) q^{94} + \beta q^{96} + ( - \beta + 9) q^{97} + q^{98} - 7 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{7} + 2 q^{8} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{7} + 2 q^{8} + 14 q^{9} - 2 q^{11} + 6 q^{13} + 2 q^{14} + 2 q^{16} + 8 q^{17} + 14 q^{18} - 2 q^{19} - 2 q^{22} + 2 q^{23} + 6 q^{26} + 2 q^{28} + 6 q^{29} - 14 q^{31} + 2 q^{32} + 8 q^{34} + 14 q^{36} - 2 q^{38} - 20 q^{39} + 8 q^{41} + 6 q^{43} - 2 q^{44} + 2 q^{46} - 4 q^{47} + 2 q^{49} - 20 q^{51} + 6 q^{52} + 8 q^{53} + 2 q^{56} + 6 q^{58} - 24 q^{59} + 12 q^{61} - 14 q^{62} + 14 q^{63} + 2 q^{64} + 4 q^{67} + 8 q^{68} - 20 q^{69} + 14 q^{71} + 14 q^{72} - 8 q^{73} - 2 q^{76} - 2 q^{77} - 20 q^{78} + 38 q^{81} + 8 q^{82} + 2 q^{83} + 6 q^{86} - 2 q^{88} - 2 q^{89} + 6 q^{91} + 2 q^{92} - 4 q^{94} + 18 q^{97} + 2 q^{98} - 14 q^{99}+O(q^{100}) \) 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
−3.16228
3.16228
1.00000 −3.16228 1.00000 0 −3.16228 1.00000 1.00000 7.00000 0
1.2 1.00000 3.16228 1.00000 0 3.16228 1.00000 1.00000 7.00000 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.bq yes 2
5.b even 2 1 3850.2.a.bf 2
5.c odd 4 2 3850.2.c.t 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3850.2.a.bf 2 5.b even 2 1
3850.2.a.bq yes 2 1.a even 1 1 trivial
3850.2.c.t 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3850))\):

\( T_{3}^{2} - 10 \) Copy content Toggle raw display
\( T_{13}^{2} - 6T_{13} - 1 \) Copy content Toggle raw display
\( T_{17}^{2} - 8T_{17} + 6 \) Copy content Toggle raw display
\( T_{19} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 10 \) 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
$13$ \( T^{2} - 6T - 1 \) Copy content Toggle raw display
$17$ \( T^{2} - 8T + 6 \) Copy content Toggle raw display
$19$ \( (T + 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 2T - 9 \) Copy content Toggle raw display
$29$ \( (T - 3)^{2} \) Copy content Toggle raw display
$31$ \( (T + 7)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 40 \) Copy content Toggle raw display
$41$ \( T^{2} - 8T - 24 \) Copy content Toggle raw display
$43$ \( T^{2} - 6T - 1 \) Copy content Toggle raw display
$47$ \( T^{2} + 4T - 36 \) Copy content Toggle raw display
$53$ \( T^{2} - 8T + 6 \) Copy content Toggle raw display
$59$ \( (T + 12)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 12T - 4 \) Copy content Toggle raw display
$67$ \( T^{2} - 4T - 86 \) Copy content Toggle raw display
$71$ \( T^{2} - 14T + 39 \) Copy content Toggle raw display
$73$ \( (T + 4)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 10 \) Copy content Toggle raw display
$83$ \( T^{2} - 2T - 159 \) Copy content Toggle raw display
$89$ \( T^{2} + 2T - 9 \) Copy content Toggle raw display
$97$ \( T^{2} - 18T + 71 \) Copy content Toggle raw display
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