Properties

Label 5290.2.a.j
Level $5290$
Weight $2$
Character orbit 5290.a
Self dual yes
Analytic conductor $42.241$
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] = [5290,2,Mod(1,5290)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5290, 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("5290.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5290 = 2 \cdot 5 \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5290.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: \(42.2408626693\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 230)
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 \(\beta = \frac{1}{2}(1 + \sqrt{13})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + (\beta + 1) q^{3} + q^{4} - q^{5} + ( - \beta - 1) q^{6} + (\beta - 2) q^{7} - q^{8} + (3 \beta + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + (\beta + 1) q^{3} + q^{4} - q^{5} + ( - \beta - 1) q^{6} + (\beta - 2) q^{7} - q^{8} + (3 \beta + 1) q^{9} + q^{10} + (\beta + 3) q^{11} + (\beta + 1) q^{12} + ( - \beta + 2) q^{13} + ( - \beta + 2) q^{14} + ( - \beta - 1) q^{15} + q^{16} + (3 \beta - 3) q^{17} + ( - 3 \beta - 1) q^{18} + (3 \beta - 2) q^{19} - q^{20} + q^{21} + ( - \beta - 3) q^{22} + ( - \beta - 1) q^{24} + q^{25} + (\beta - 2) q^{26} + (4 \beta + 7) q^{27} + (\beta - 2) q^{28} + 2 \beta q^{29} + (\beta + 1) q^{30} + (3 \beta - 4) q^{31} - q^{32} + (5 \beta + 6) q^{33} + ( - 3 \beta + 3) q^{34} + ( - \beta + 2) q^{35} + (3 \beta + 1) q^{36} - 8 q^{37} + ( - 3 \beta + 2) q^{38} - q^{39} + q^{40} + ( - 3 \beta - 3) q^{41} - q^{42} + ( - 4 \beta + 4) q^{43} + (\beta + 3) q^{44} + ( - 3 \beta - 1) q^{45} + 2 \beta q^{47} + (\beta + 1) q^{48} - 3 \beta q^{49} - q^{50} + (3 \beta + 6) q^{51} + ( - \beta + 2) q^{52} + ( - 4 \beta + 6) q^{53} + ( - 4 \beta - 7) q^{54} + ( - \beta - 3) q^{55} + ( - \beta + 2) q^{56} + (4 \beta + 7) q^{57} - 2 \beta q^{58} + ( - 2 \beta - 6) q^{59} + ( - \beta - 1) q^{60} + (5 \beta - 5) q^{61} + ( - 3 \beta + 4) q^{62} + ( - 2 \beta + 7) q^{63} + q^{64} + (\beta - 2) q^{65} + ( - 5 \beta - 6) q^{66} + 4 q^{67} + (3 \beta - 3) q^{68} + (\beta - 2) q^{70} + (\beta - 15) q^{71} + ( - 3 \beta - 1) q^{72} + (6 \beta + 2) q^{73} + 8 q^{74} + (\beta + 1) q^{75} + (3 \beta - 2) q^{76} + (2 \beta - 3) q^{77} + q^{78} + ( - 8 \beta + 4) q^{79} - q^{80} + (6 \beta + 16) q^{81} + (3 \beta + 3) q^{82} + (4 \beta - 6) q^{83} + q^{84} + ( - 3 \beta + 3) q^{85} + (4 \beta - 4) q^{86} + (4 \beta + 6) q^{87} + ( - \beta - 3) q^{88} + (3 \beta + 1) q^{90} + (3 \beta - 7) q^{91} + (2 \beta + 5) q^{93} - 2 \beta q^{94} + ( - 3 \beta + 2) q^{95} + ( - \beta - 1) q^{96} + (\beta - 5) q^{97} + 3 \beta q^{98} + (13 \beta + 12) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 3 q^{3} + 2 q^{4} - 2 q^{5} - 3 q^{6} - 3 q^{7} - 2 q^{8} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 3 q^{3} + 2 q^{4} - 2 q^{5} - 3 q^{6} - 3 q^{7} - 2 q^{8} + 5 q^{9} + 2 q^{10} + 7 q^{11} + 3 q^{12} + 3 q^{13} + 3 q^{14} - 3 q^{15} + 2 q^{16} - 3 q^{17} - 5 q^{18} - q^{19} - 2 q^{20} + 2 q^{21} - 7 q^{22} - 3 q^{24} + 2 q^{25} - 3 q^{26} + 18 q^{27} - 3 q^{28} + 2 q^{29} + 3 q^{30} - 5 q^{31} - 2 q^{32} + 17 q^{33} + 3 q^{34} + 3 q^{35} + 5 q^{36} - 16 q^{37} + q^{38} - 2 q^{39} + 2 q^{40} - 9 q^{41} - 2 q^{42} + 4 q^{43} + 7 q^{44} - 5 q^{45} + 2 q^{47} + 3 q^{48} - 3 q^{49} - 2 q^{50} + 15 q^{51} + 3 q^{52} + 8 q^{53} - 18 q^{54} - 7 q^{55} + 3 q^{56} + 18 q^{57} - 2 q^{58} - 14 q^{59} - 3 q^{60} - 5 q^{61} + 5 q^{62} + 12 q^{63} + 2 q^{64} - 3 q^{65} - 17 q^{66} + 8 q^{67} - 3 q^{68} - 3 q^{70} - 29 q^{71} - 5 q^{72} + 10 q^{73} + 16 q^{74} + 3 q^{75} - q^{76} - 4 q^{77} + 2 q^{78} - 2 q^{80} + 38 q^{81} + 9 q^{82} - 8 q^{83} + 2 q^{84} + 3 q^{85} - 4 q^{86} + 16 q^{87} - 7 q^{88} + 5 q^{90} - 11 q^{91} + 12 q^{93} - 2 q^{94} + q^{95} - 3 q^{96} - 9 q^{97} + 3 q^{98} + 37 q^{99}+O(q^{100}) \) 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
−1.30278
2.30278
−1.00000 −0.302776 1.00000 −1.00000 0.302776 −3.30278 −1.00000 −2.90833 1.00000
1.2 −1.00000 3.30278 1.00000 −1.00000 −3.30278 0.302776 −1.00000 7.90833 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(5\) \( +1 \)
\(23\) \( -1 \)

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 5290.2.a.j 2
23.b odd 2 1 230.2.a.b 2
69.c even 2 1 2070.2.a.w 2
92.b even 2 1 1840.2.a.j 2
115.c odd 2 1 1150.2.a.m 2
115.e even 4 2 1150.2.b.f 4
184.e odd 2 1 7360.2.a.bc 2
184.h even 2 1 7360.2.a.bu 2
460.g even 2 1 9200.2.a.ca 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.2.a.b 2 23.b odd 2 1
1150.2.a.m 2 115.c odd 2 1
1150.2.b.f 4 115.e even 4 2
1840.2.a.j 2 92.b even 2 1
2070.2.a.w 2 69.c even 2 1
5290.2.a.j 2 1.a even 1 1 trivial
7360.2.a.bc 2 184.e odd 2 1
7360.2.a.bu 2 184.h even 2 1
9200.2.a.ca 2 460.g even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5290))\):

\( T_{3}^{2} - 3T_{3} - 1 \) Copy content Toggle raw display
\( T_{7}^{2} + 3T_{7} - 1 \) Copy content Toggle raw display
\( T_{11}^{2} - 7T_{11} + 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 3T - 1 \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 3T - 1 \) Copy content Toggle raw display
$11$ \( T^{2} - 7T + 9 \) Copy content Toggle raw display
$13$ \( T^{2} - 3T - 1 \) Copy content Toggle raw display
$17$ \( T^{2} + 3T - 27 \) Copy content Toggle raw display
$19$ \( T^{2} + T - 29 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 2T - 12 \) Copy content Toggle raw display
$31$ \( T^{2} + 5T - 23 \) Copy content Toggle raw display
$37$ \( (T + 8)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 9T - 9 \) Copy content Toggle raw display
$43$ \( T^{2} - 4T - 48 \) Copy content Toggle raw display
$47$ \( T^{2} - 2T - 12 \) Copy content Toggle raw display
$53$ \( T^{2} - 8T - 36 \) Copy content Toggle raw display
$59$ \( T^{2} + 14T + 36 \) Copy content Toggle raw display
$61$ \( T^{2} + 5T - 75 \) Copy content Toggle raw display
$67$ \( (T - 4)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 29T + 207 \) Copy content Toggle raw display
$73$ \( T^{2} - 10T - 92 \) Copy content Toggle raw display
$79$ \( T^{2} - 208 \) Copy content Toggle raw display
$83$ \( T^{2} + 8T - 36 \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 9T + 17 \) Copy content Toggle raw display
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