↑

Properties

Label 3311.2.a.j
Level $3311$
Weight $2$
Character orbit 3311.a
Self dual yes
Analytic conductor $26.438$
Analytic rank $0$
Dimension $35$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3311,2,Mod(1,3311)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3311, 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("3311.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3311 = 7 \cdot 11 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3311.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: \(26.4384681092\)
Analytic rank: \(0\)
Dimension: \(35\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 35 q + 2 q^{2} - 4 q^{3} + 46 q^{4} - q^{5} + 12 q^{6} + 35 q^{7} + 6 q^{8} + 53 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 35 q + 2 q^{2} - 4 q^{3} + 46 q^{4} - q^{5} + 12 q^{6} + 35 q^{7} + 6 q^{8} + 53 q^{9} + 20 q^{10} - 35 q^{11} - 2 q^{12} + 22 q^{13} + 2 q^{14} + 19 q^{15} + 60 q^{16} + 17 q^{17} + 15 q^{18} + 14 q^{19} - 3 q^{20} - 4 q^{21} - 2 q^{22} + 19 q^{23} + 28 q^{24} + 80 q^{25} - 9 q^{26} - 25 q^{27} + 46 q^{28} + 20 q^{29} - 20 q^{30} + 20 q^{31} + 4 q^{32} + 4 q^{33} + 34 q^{34} - q^{35} + 84 q^{36} + 42 q^{37} - 26 q^{38} + 50 q^{39} + 63 q^{40} + 29 q^{41} + 12 q^{42} - 35 q^{43} - 46 q^{44} - 2 q^{45} + 53 q^{46} - 14 q^{47} - 39 q^{48} + 35 q^{49} - 11 q^{50} + 4 q^{51} + 16 q^{52} + 28 q^{53} + 53 q^{54} + q^{55} + 6 q^{56} + 62 q^{57} - q^{58} - 34 q^{59} + 11 q^{60} + 48 q^{61} - 6 q^{62} + 53 q^{63} + 114 q^{64} + 39 q^{65} - 12 q^{66} + 41 q^{67} + 43 q^{68} + 13 q^{69} + 20 q^{70} - 2 q^{71} + 14 q^{72} + 29 q^{73} + 19 q^{74} - 96 q^{75} + 21 q^{76} - 35 q^{77} - 12 q^{78} + 81 q^{79} - 91 q^{80} + 87 q^{81} + 2 q^{82} + 19 q^{83} - 2 q^{84} + 54 q^{85} - 2 q^{86} + q^{87} - 6 q^{88} - 3 q^{89} + 29 q^{90} + 22 q^{91} + 3 q^{92} - q^{93} + 29 q^{94} + 7 q^{95} + 77 q^{96} + 27 q^{97} + 2 q^{98} - 53 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −2.74415 −3.17936 5.53035 −3.71681 8.72463 1.00000 −9.68781 7.10832 10.1995
1.2 −2.72953 −1.06550 5.45035 0.613146 2.90831 1.00000 −9.41786 −1.86471 −1.67360
1.3 −2.67150 2.31501 5.13690 −1.94291 −6.18454 1.00000 −8.38023 2.35926 5.19047
1.4 −2.35716 0.186474 3.55620 2.89875 −0.439550 1.00000 −3.66821 −2.96523 −6.83281
1.5 −2.27113 −1.17182 3.15804 −4.10127 2.66135 1.00000 −2.63005 −1.62685 9.31451
1.6 −2.07971 1.00156 2.32520 −2.81435 −2.08295 1.00000 −0.676314 −1.99688 5.85303
1.7 −2.05838 −3.12473 2.23693 3.28521 6.43187 1.00000 −0.487689 6.76391 −6.76222
1.8 −1.98140 2.75303 1.92593 2.22951 −5.45485 1.00000 0.146765 4.57919 −4.41754
1.9 −1.62585 −1.90962 0.643402 2.23807 3.10476 1.00000 2.20563 0.646631 −3.63878
1.10 −1.45686 2.23102 0.122431 0.271793 −3.25027 1.00000 2.73535 1.97743 −0.395963
1.11 −1.31599 0.887058 −0.268178 −3.96165 −1.16736 1.00000 2.98489 −2.21313 5.21349
1.12 −1.07412 −2.05007 −0.846274 1.06589 2.20201 1.00000 3.05723 1.20278 −1.14489
1.13 −0.939540 0.701398 −1.11726 0.584863 −0.658992 1.00000 2.92879 −2.50804 −0.549502
1.14 −0.861294 2.50306 −1.25817 3.88239 −2.15587 1.00000 2.80624 3.26532 −3.34388
1.15 −0.608414 −2.02343 −1.62983 −1.75387 1.23108 1.00000 2.20844 1.09428 1.06708
1.16 −0.446883 −3.35469 −1.80030 −4.25894 1.49916 1.00000 1.69829 8.25397 1.90325
1.17 −0.397869 −2.48783 −1.84170 −0.0624101 0.989831 1.00000 1.52849 3.18929 0.0248311
1.18 0.394006 0.893470 −1.84476 −1.83223 0.352033 1.00000 −1.51486 −2.20171 −0.721909
1.19 0.422033 3.22122 −1.82189 −1.93868 1.35946 1.00000 −1.61296 7.37623 −0.818185
1.20 0.528315 1.67249 −1.72088 3.83464 0.883601 1.00000 −1.96580 −0.202778 2.02590
See all 35 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.35
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(-1\)
\(11\) \(1\)
\(43\) \(1\)

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 3311.2.a.j ✓ 35
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3311.2.a.j ✓ 35 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3311))\):

\( T_{2}^{35} - 2 T_{2}^{34} - 56 T_{2}^{33} + 110 T_{2}^{32} + 1422 T_{2}^{31} - 2736 T_{2}^{30} + \cdots + 88946 \) Copy content Toggle raw display
\( T_{5}^{35} + T_{5}^{34} - 127 T_{5}^{33} - 112 T_{5}^{32} + 7300 T_{5}^{31} + 5618 T_{5}^{30} + \cdots - 1303283824 \) Copy content Toggle raw display