Properties

Label 3509.2.a.j
Level $3509$
Weight $2$
Character orbit 3509.a
Self dual yes
Analytic conductor $28.020$
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] = [3509,2,Mod(1,3509)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3509, 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("3509.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3509 = 11^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3509.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: \(28.0195060693\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 29)
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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 1) q^{2} + (\beta + 1) q^{3} + (2 \beta + 1) q^{4} - q^{5} + (2 \beta + 3) q^{6} + 2 \beta q^{7} + (\beta + 3) q^{8} + 2 \beta q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 1) q^{2} + (\beta + 1) q^{3} + (2 \beta + 1) q^{4} - q^{5} + (2 \beta + 3) q^{6} + 2 \beta q^{7} + (\beta + 3) q^{8} + 2 \beta q^{9} + ( - \beta - 1) q^{10} + (3 \beta + 5) q^{12} + (2 \beta + 1) q^{13} + (2 \beta + 4) q^{14} + ( - \beta - 1) q^{15} + 3 q^{16} + ( - 2 \beta + 2) q^{17} + (2 \beta + 4) q^{18} - 6 q^{19} + ( - 2 \beta - 1) q^{20} + (2 \beta + 4) q^{21} + (4 \beta - 2) q^{23} + (4 \beta + 5) q^{24} - 4 q^{25} + (3 \beta + 5) q^{26} + ( - \beta + 1) q^{27} + (2 \beta + 8) q^{28} - q^{29} + ( - 2 \beta - 3) q^{30} + (5 \beta + 3) q^{31} + (\beta - 3) q^{32} - 2 q^{34} - 2 \beta q^{35} + (2 \beta + 8) q^{36} - 4 q^{37} + ( - 6 \beta - 6) q^{38} + (3 \beta + 5) q^{39} + ( - \beta - 3) q^{40} + (6 \beta - 4) q^{41} + (6 \beta + 8) q^{42} + (\beta - 5) q^{43} - 2 \beta q^{45} + (2 \beta + 6) q^{46} + ( - 3 \beta + 1) q^{47} + (3 \beta + 3) q^{48} + q^{49} + ( - 4 \beta - 4) q^{50} - 2 q^{51} + (4 \beta + 9) q^{52} + (6 \beta + 1) q^{53} - q^{54} + (6 \beta + 4) q^{56} + ( - 6 \beta - 6) q^{57} + ( - \beta - 1) q^{58} + ( - 4 \beta + 2) q^{59} + ( - 3 \beta - 5) q^{60} + (2 \beta + 2) q^{61} + (8 \beta + 13) q^{62} + 8 q^{63} + ( - 2 \beta - 7) q^{64} + ( - 2 \beta - 1) q^{65} + 4 \beta q^{67} + (2 \beta - 6) q^{68} + (2 \beta + 6) q^{69} + ( - 2 \beta - 4) q^{70} + ( - 2 \beta - 6) q^{71} + (6 \beta + 4) q^{72} - 4 q^{73} + ( - 4 \beta - 4) q^{74} + ( - 4 \beta - 4) q^{75} + ( - 12 \beta - 6) q^{76} + (8 \beta + 11) q^{78} + (\beta + 1) q^{79} - 3 q^{80} + ( - 6 \beta - 1) q^{81} + (2 \beta + 8) q^{82} + ( - 4 \beta - 2) q^{83} + (10 \beta + 12) q^{84} + (2 \beta - 2) q^{85} + ( - 4 \beta - 3) q^{86} + ( - \beta - 1) q^{87} + ( - 6 \beta - 4) q^{89} + ( - 2 \beta - 4) q^{90} + (2 \beta + 8) q^{91} + 14 q^{92} + (8 \beta + 13) q^{93} + ( - 2 \beta - 5) q^{94} + 6 q^{95} + ( - 2 \beta - 1) q^{96} + (6 \beta - 4) q^{97} + (\beta + 1) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} + 6 q^{6} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} + 6 q^{6} + 6 q^{8} - 2 q^{10} + 10 q^{12} + 2 q^{13} + 8 q^{14} - 2 q^{15} + 6 q^{16} + 4 q^{17} + 8 q^{18} - 12 q^{19} - 2 q^{20} + 8 q^{21} - 4 q^{23} + 10 q^{24} - 8 q^{25} + 10 q^{26} + 2 q^{27} + 16 q^{28} - 2 q^{29} - 6 q^{30} + 6 q^{31} - 6 q^{32} - 4 q^{34} + 16 q^{36} - 8 q^{37} - 12 q^{38} + 10 q^{39} - 6 q^{40} - 8 q^{41} + 16 q^{42} - 10 q^{43} + 12 q^{46} + 2 q^{47} + 6 q^{48} + 2 q^{49} - 8 q^{50} - 4 q^{51} + 18 q^{52} + 2 q^{53} - 2 q^{54} + 8 q^{56} - 12 q^{57} - 2 q^{58} + 4 q^{59} - 10 q^{60} + 4 q^{61} + 26 q^{62} + 16 q^{63} - 14 q^{64} - 2 q^{65} - 12 q^{68} + 12 q^{69} - 8 q^{70} - 12 q^{71} + 8 q^{72} - 8 q^{73} - 8 q^{74} - 8 q^{75} - 12 q^{76} + 22 q^{78} + 2 q^{79} - 6 q^{80} - 2 q^{81} + 16 q^{82} - 4 q^{83} + 24 q^{84} - 4 q^{85} - 6 q^{86} - 2 q^{87} - 8 q^{89} - 8 q^{90} + 16 q^{91} + 28 q^{92} + 26 q^{93} - 10 q^{94} + 12 q^{95} - 2 q^{96} - 8 q^{97} + 2 q^{98}+O(q^{100}) \) 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
−1.41421
1.41421
−0.414214 −0.414214 −1.82843 −1.00000 0.171573 −2.82843 1.58579 −2.82843 0.414214
1.2 2.41421 2.41421 3.82843 −1.00000 5.82843 2.82843 4.41421 2.82843 −2.41421
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(11\) \(-1\)
\(29\) \(1\)

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 3509.2.a.j 2
11.b odd 2 1 29.2.a.a 2
33.d even 2 1 261.2.a.d 2
44.c even 2 1 464.2.a.h 2
55.d odd 2 1 725.2.a.b 2
55.e even 4 2 725.2.b.b 4
77.b even 2 1 1421.2.a.j 2
88.b odd 2 1 1856.2.a.r 2
88.g even 2 1 1856.2.a.w 2
132.d odd 2 1 4176.2.a.bq 2
143.d odd 2 1 4901.2.a.g 2
165.d even 2 1 6525.2.a.o 2
187.b odd 2 1 8381.2.a.e 2
319.d odd 2 1 841.2.a.d 2
319.f even 4 2 841.2.b.a 4
319.l odd 14 6 841.2.d.f 12
319.m odd 14 6 841.2.d.j 12
319.q even 28 12 841.2.e.k 24
957.b even 2 1 7569.2.a.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.2.a.a 2 11.b odd 2 1
261.2.a.d 2 33.d even 2 1
464.2.a.h 2 44.c even 2 1
725.2.a.b 2 55.d odd 2 1
725.2.b.b 4 55.e even 4 2
841.2.a.d 2 319.d odd 2 1
841.2.b.a 4 319.f even 4 2
841.2.d.f 12 319.l odd 14 6
841.2.d.j 12 319.m odd 14 6
841.2.e.k 24 319.q even 28 12
1421.2.a.j 2 77.b even 2 1
1856.2.a.r 2 88.b odd 2 1
1856.2.a.w 2 88.g even 2 1
3509.2.a.j 2 1.a even 1 1 trivial
4176.2.a.bq 2 132.d odd 2 1
4901.2.a.g 2 143.d odd 2 1
6525.2.a.o 2 165.d even 2 1
7569.2.a.c 2 957.b even 2 1
8381.2.a.e 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3509))\):

\( T_{2}^{2} - 2T_{2} - 1 \) Copy content Toggle raw display
\( T_{3}^{2} - 2T_{3} - 1 \) Copy content Toggle raw display
\( T_{7}^{2} - 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 2T - 1 \) Copy content Toggle raw display
$3$ \( T^{2} - 2T - 1 \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 8 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 2T - 7 \) Copy content Toggle raw display
$17$ \( T^{2} - 4T - 4 \) Copy content Toggle raw display
$19$ \( (T + 6)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 4T - 28 \) Copy content Toggle raw display
$29$ \( (T + 1)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 6T - 41 \) Copy content Toggle raw display
$37$ \( (T + 4)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 8T - 56 \) Copy content Toggle raw display
$43$ \( T^{2} + 10T + 23 \) Copy content Toggle raw display
$47$ \( T^{2} - 2T - 17 \) Copy content Toggle raw display
$53$ \( T^{2} - 2T - 71 \) Copy content Toggle raw display
$59$ \( T^{2} - 4T - 28 \) Copy content Toggle raw display
$61$ \( T^{2} - 4T - 4 \) Copy content Toggle raw display
$67$ \( T^{2} - 32 \) Copy content Toggle raw display
$71$ \( T^{2} + 12T + 28 \) Copy content Toggle raw display
$73$ \( (T + 4)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 2T - 1 \) Copy content Toggle raw display
$83$ \( T^{2} + 4T - 28 \) Copy content Toggle raw display
$89$ \( T^{2} + 8T - 56 \) Copy content Toggle raw display
$97$ \( T^{2} + 8T - 56 \) Copy content Toggle raw display
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