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Properties

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

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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.

\(\operatorname{Tr}(f)(q) = \) \( 35 q - 5 q^{2} + 3 q^{3} + 49 q^{4} + 16 q^{5} + 10 q^{6} - 35 q^{7} - 3 q^{8} + 56 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 35 q - 5 q^{2} + 3 q^{3} + 49 q^{4} + 16 q^{5} + 10 q^{6} - 35 q^{7} - 3 q^{8} + 56 q^{9} + 18 q^{10} + 35 q^{11} + 20 q^{12} + 26 q^{13} + 5 q^{14} + 18 q^{15} + 73 q^{16} + 23 q^{17} - 32 q^{18} + 4 q^{19} + 21 q^{20} - 3 q^{21} - 5 q^{22} + 10 q^{23} + 28 q^{24} + 61 q^{25} + 19 q^{26} + 36 q^{27} - 49 q^{28} - 10 q^{29} + 8 q^{30} + 39 q^{31} - 51 q^{32} + 3 q^{33} + 36 q^{34} - 16 q^{35} + 111 q^{36} - 3 q^{37} - 8 q^{38} - 6 q^{39} + 39 q^{40} + 17 q^{41} - 10 q^{42} - 35 q^{43} + 49 q^{44} + 60 q^{45} + q^{46} + 32 q^{47} - 3 q^{48} + 35 q^{49} - 36 q^{50} - 6 q^{51} + 62 q^{52} + 38 q^{53} + 47 q^{54} + 16 q^{55} + 3 q^{56} - 22 q^{57} - q^{58} + 61 q^{59} - 67 q^{60} + 30 q^{61} - 58 q^{62} - 56 q^{63} + 107 q^{64} - 17 q^{65} + 10 q^{66} + 26 q^{67} + 31 q^{68} + 58 q^{69} - 18 q^{70} + 23 q^{71} - 77 q^{72} + 17 q^{73} - 21 q^{74} - 19 q^{76} - 35 q^{77} + 4 q^{78} + q^{79} + 5 q^{80} + 115 q^{81} + 46 q^{82} + 23 q^{83} - 20 q^{84} - 10 q^{85} + 5 q^{86} + 15 q^{87} - 3 q^{88} + 82 q^{89} + 95 q^{90} - 26 q^{91} + 45 q^{92} - 16 q^{93} + 87 q^{94} - 11 q^{95} + 57 q^{96} + 88 q^{97} - 5 q^{98} + 56 q^{99}+O(q^{100}) \) 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 −2.82764 −1.86949 5.99555 −3.15724 5.28624 −1.00000 −11.2980 0.494980 8.92753
1.2 −2.74500 −3.21150 5.53501 3.13873 8.81556 −1.00000 −9.70358 7.31373 −8.61580
1.3 −2.67717 3.11549 5.16725 −3.77302 −8.34071 −1.00000 −8.47926 6.70629 10.1010
1.4 −2.64530 0.924459 4.99759 −0.723902 −2.44547 −1.00000 −7.92951 −2.14538 1.91493
1.5 −2.41745 0.932532 3.84404 2.10067 −2.25435 −1.00000 −4.45787 −2.13038 −5.07825
1.6 −2.38493 2.88452 3.68788 4.26222 −6.87938 −1.00000 −4.02548 5.32047 −10.1651
1.7 −1.96914 −3.08264 1.87752 1.46566 6.07015 −1.00000 0.241171 6.50265 −2.88609
1.8 −1.93439 −2.19961 1.74186 −1.95795 4.25490 −1.00000 0.499342 1.83829 3.78744
1.9 −1.86212 3.11103 1.46748 1.03885 −5.79310 −1.00000 0.991615 6.67852 −1.93445
1.10 −1.79276 0.0967668 1.21399 1.66489 −0.173480 −1.00000 1.40912 −2.99064 −2.98475
1.11 −1.74567 −1.30468 1.04735 −1.01064 2.27753 −1.00000 1.66301 −1.29782 1.76425
1.12 −1.23516 0.0333042 −0.474372 3.39916 −0.0411361 −1.00000 3.05625 −2.99889 −4.19852
1.13 −1.11968 2.88075 −0.746322 −2.84200 −3.22551 −1.00000 3.07500 5.29873 3.18213
1.14 −0.941734 −1.26344 −1.11314 −2.36970 1.18982 −1.00000 2.93175 −1.40372 2.23163
1.15 −0.910469 0.785299 −1.17105 −2.46247 −0.714991 −1.00000 2.88714 −2.38331 2.24200
1.16 −0.849355 −0.534669 −1.27860 4.32865 0.454124 −1.00000 2.78469 −2.71413 −3.67657
1.17 −0.708558 −3.09392 −1.49795 −3.81894 2.19222 −1.00000 2.47850 6.57234 2.70594
1.18 −0.144882 2.81659 −1.97901 4.11263 −0.408073 −1.00000 0.576487 4.93316 −0.595847
1.19 −0.0609290 0.272653 −1.99629 −1.19352 −0.0166125 −1.00000 0.243490 −2.92566 0.0727197
1.20 −0.00236574 2.43007 −1.99999 0.788838 −0.00574890 −1.00000 0.00946293 2.90522 −0.00186618
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.i ✓ 35
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3311.2.a.i ✓ 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3311))\):

\( T_{2}^{35} + 5 T_{2}^{34} - 47 T_{2}^{33} - 269 T_{2}^{32} + 929 T_{2}^{31} + 6503 T_{2}^{30} + \cdots - 18 \) Copy content Toggle raw display
\( T_{5}^{35} - 16 T_{5}^{34} + 10 T_{5}^{33} + 1127 T_{5}^{32} - 4880 T_{5}^{31} - 29594 T_{5}^{30} + \cdots + 1483423408 \) Copy content Toggle raw display