Properties

Label 8619.2.a.q
Level $8619$
Weight $2$
Character orbit 8619.a
Self dual yes
Analytic conductor $68.823$
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] = [8619,2,Mod(1,8619)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8619, 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("8619.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8619 = 3 \cdot 13^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8619.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: \(68.8230615021\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 51)
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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} - q^{3} + (\beta + 2) q^{4} + ( - \beta - 1) q^{5} - \beta q^{6} + (\beta + 4) q^{8} + q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} - q^{3} + (\beta + 2) q^{4} + ( - \beta - 1) q^{5} - \beta q^{6} + (\beta + 4) q^{8} + q^{9} + ( - 2 \beta - 4) q^{10} + ( - \beta + 1) q^{11} + ( - \beta - 2) q^{12} + (\beta + 1) q^{15} + 3 \beta q^{16} + q^{17} + \beta q^{18} + (3 \beta - 3) q^{19} + ( - 4 \beta - 6) q^{20} - 4 q^{22} + (\beta - 5) q^{23} + ( - \beta - 4) q^{24} + 3 \beta q^{25} - q^{27} + ( - 4 \beta + 2) q^{29} + (2 \beta + 4) q^{30} + ( - 2 \beta + 2) q^{31} + (\beta + 4) q^{32} + (\beta - 1) q^{33} + \beta q^{34} + (\beta + 2) q^{36} + 2 \beta q^{37} + 12 q^{38} + ( - 6 \beta - 8) q^{40} + (\beta + 1) q^{41} + (3 \beta - 3) q^{43} + ( - 2 \beta - 2) q^{44} + ( - \beta - 1) q^{45} + ( - 4 \beta + 4) q^{46} + (2 \beta + 6) q^{47} - 3 \beta q^{48} - 7 q^{49} + (3 \beta + 12) q^{50} - q^{51} + (4 \beta + 2) q^{53} - \beta q^{54} + (\beta + 3) q^{55} + ( - 3 \beta + 3) q^{57} + ( - 2 \beta - 16) q^{58} + ( - 2 \beta - 2) q^{59} + (4 \beta + 6) q^{60} + (2 \beta + 4) q^{61} - 8 q^{62} + ( - \beta + 4) q^{64} + 4 q^{66} - 4 q^{67} + (\beta + 2) q^{68} + ( - \beta + 5) q^{69} + (4 \beta - 4) q^{71} + (\beta + 4) q^{72} + (4 \beta + 2) q^{73} + (2 \beta + 8) q^{74} - 3 \beta q^{75} + (6 \beta + 6) q^{76} + ( - 6 \beta + 6) q^{79} + ( - 6 \beta - 12) q^{80} + q^{81} + (2 \beta + 4) q^{82} + ( - 2 \beta + 6) q^{83} + ( - \beta - 1) q^{85} + 12 q^{86} + (4 \beta - 2) q^{87} - 4 \beta q^{88} + (2 \beta - 4) q^{89} + ( - 2 \beta - 4) q^{90} + ( - 2 \beta - 6) q^{92} + (2 \beta - 2) q^{93} + (8 \beta + 8) q^{94} + ( - 3 \beta - 9) q^{95} + ( - \beta - 4) q^{96} + ( - 2 \beta + 8) q^{97} - 7 \beta q^{98} + ( - \beta + 1) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - 2 q^{3} + 5 q^{4} - 3 q^{5} - q^{6} + 9 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - 2 q^{3} + 5 q^{4} - 3 q^{5} - q^{6} + 9 q^{8} + 2 q^{9} - 10 q^{10} + q^{11} - 5 q^{12} + 3 q^{15} + 3 q^{16} + 2 q^{17} + q^{18} - 3 q^{19} - 16 q^{20} - 8 q^{22} - 9 q^{23} - 9 q^{24} + 3 q^{25} - 2 q^{27} + 10 q^{30} + 2 q^{31} + 9 q^{32} - q^{33} + q^{34} + 5 q^{36} + 2 q^{37} + 24 q^{38} - 22 q^{40} + 3 q^{41} - 3 q^{43} - 6 q^{44} - 3 q^{45} + 4 q^{46} + 14 q^{47} - 3 q^{48} - 14 q^{49} + 27 q^{50} - 2 q^{51} + 8 q^{53} - q^{54} + 7 q^{55} + 3 q^{57} - 34 q^{58} - 6 q^{59} + 16 q^{60} + 10 q^{61} - 16 q^{62} + 7 q^{64} + 8 q^{66} - 8 q^{67} + 5 q^{68} + 9 q^{69} - 4 q^{71} + 9 q^{72} + 8 q^{73} + 18 q^{74} - 3 q^{75} + 18 q^{76} + 6 q^{79} - 30 q^{80} + 2 q^{81} + 10 q^{82} + 10 q^{83} - 3 q^{85} + 24 q^{86} - 4 q^{88} - 6 q^{89} - 10 q^{90} - 14 q^{92} - 2 q^{93} + 24 q^{94} - 21 q^{95} - 9 q^{96} + 14 q^{97} - 7 q^{98} + q^{99}+O(q^{100}) \) 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
−1.56155
2.56155
−1.56155 −1.00000 0.438447 0.561553 1.56155 0 2.43845 1.00000 −0.876894
1.2 2.56155 −1.00000 4.56155 −3.56155 −2.56155 0 6.56155 1.00000 −9.12311
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(13\) \(1\)
\(17\) \(-1\)

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 8619.2.a.q 2
13.b even 2 1 51.2.a.b 2
39.d odd 2 1 153.2.a.e 2
52.b odd 2 1 816.2.a.m 2
65.d even 2 1 1275.2.a.n 2
65.h odd 4 2 1275.2.b.d 4
91.b odd 2 1 2499.2.a.o 2
104.e even 2 1 3264.2.a.bl 2
104.h odd 2 1 3264.2.a.bg 2
143.d odd 2 1 6171.2.a.p 2
156.h even 2 1 2448.2.a.v 2
195.e odd 2 1 3825.2.a.s 2
221.b even 2 1 867.2.a.f 2
221.k even 4 2 867.2.d.c 4
221.p even 8 4 867.2.e.f 8
221.y odd 16 8 867.2.h.j 16
273.g even 2 1 7497.2.a.v 2
312.b odd 2 1 9792.2.a.cy 2
312.h even 2 1 9792.2.a.cz 2
663.g odd 2 1 2601.2.a.t 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
51.2.a.b 2 13.b even 2 1
153.2.a.e 2 39.d odd 2 1
816.2.a.m 2 52.b odd 2 1
867.2.a.f 2 221.b even 2 1
867.2.d.c 4 221.k even 4 2
867.2.e.f 8 221.p even 8 4
867.2.h.j 16 221.y odd 16 8
1275.2.a.n 2 65.d even 2 1
1275.2.b.d 4 65.h odd 4 2
2448.2.a.v 2 156.h even 2 1
2499.2.a.o 2 91.b odd 2 1
2601.2.a.t 2 663.g odd 2 1
3264.2.a.bg 2 104.h odd 2 1
3264.2.a.bl 2 104.e even 2 1
3825.2.a.s 2 195.e odd 2 1
6171.2.a.p 2 143.d odd 2 1
7497.2.a.v 2 273.g even 2 1
8619.2.a.q 2 1.a even 1 1 trivial
9792.2.a.cy 2 312.b odd 2 1
9792.2.a.cz 2 312.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8619))\):

\( T_{2}^{2} - T_{2} - 4 \) Copy content Toggle raw display
\( T_{5}^{2} + 3T_{5} - 2 \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T - 4 \) Copy content Toggle raw display
$3$ \( (T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 3T - 2 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - T - 4 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( (T - 1)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 3T - 36 \) Copy content Toggle raw display
$23$ \( T^{2} + 9T + 16 \) Copy content Toggle raw display
$29$ \( T^{2} - 68 \) Copy content Toggle raw display
$31$ \( T^{2} - 2T - 16 \) Copy content Toggle raw display
$37$ \( T^{2} - 2T - 16 \) Copy content Toggle raw display
$41$ \( T^{2} - 3T - 2 \) Copy content Toggle raw display
$43$ \( T^{2} + 3T - 36 \) Copy content Toggle raw display
$47$ \( T^{2} - 14T + 32 \) Copy content Toggle raw display
$53$ \( T^{2} - 8T - 52 \) Copy content Toggle raw display
$59$ \( T^{2} + 6T - 8 \) Copy content Toggle raw display
$61$ \( T^{2} - 10T + 8 \) Copy content Toggle raw display
$67$ \( (T + 4)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 4T - 64 \) Copy content Toggle raw display
$73$ \( T^{2} - 8T - 52 \) Copy content Toggle raw display
$79$ \( T^{2} - 6T - 144 \) Copy content Toggle raw display
$83$ \( T^{2} - 10T + 8 \) Copy content Toggle raw display
$89$ \( T^{2} + 6T - 8 \) Copy content Toggle raw display
$97$ \( T^{2} - 14T + 32 \) Copy content Toggle raw display
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