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Properties

Label 3022.2.a.b
Level $3022$
Weight $2$
Character orbit 3022.a
Self dual yes
Analytic conductor $24.131$
Analytic rank $1$
Dimension $28$
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] = [3022,2,Mod(1,3022)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3022, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3022.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3022 = 2 \cdot 1511 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3022.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: \(24.1307914908\)
Analytic rank: \(1\)
Dimension: \(28\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q + 28 q^{2} - 9 q^{3} + 28 q^{4} - 16 q^{5} - 9 q^{6} - 23 q^{7} + 28 q^{8} + 15 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q + 28 q^{2} - 9 q^{3} + 28 q^{4} - 16 q^{5} - 9 q^{6} - 23 q^{7} + 28 q^{8} + 15 q^{9} - 16 q^{10} - 32 q^{11} - 9 q^{12} - 21 q^{13} - 23 q^{14} - 34 q^{15} + 28 q^{16} - 21 q^{17} + 15 q^{18} - 34 q^{19} - 16 q^{20} - 18 q^{21} - 32 q^{22} - 44 q^{23} - 9 q^{24} + 16 q^{25} - 21 q^{26} - 21 q^{27} - 23 q^{28} - 33 q^{29} - 34 q^{30} - 19 q^{31} + 28 q^{32} - 20 q^{33} - 21 q^{34} - 40 q^{35} + 15 q^{36} - 30 q^{37} - 34 q^{38} - 34 q^{39} - 16 q^{40} - 9 q^{41} - 18 q^{42} - 53 q^{43} - 32 q^{44} - 28 q^{45} - 44 q^{46} - 53 q^{47} - 9 q^{48} + 23 q^{49} + 16 q^{50} - 43 q^{51} - 21 q^{52} - 82 q^{53} - 21 q^{54} - 20 q^{55} - 23 q^{56} - 15 q^{57} - 33 q^{58} - 45 q^{59} - 34 q^{60} - 14 q^{61} - 19 q^{62} - 66 q^{63} + 28 q^{64} - 18 q^{65} - 20 q^{66} - 66 q^{67} - 21 q^{68} - 7 q^{69} - 40 q^{70} - 52 q^{71} + 15 q^{72} + 17 q^{73} - 30 q^{74} + 24 q^{75} - 34 q^{76} - 14 q^{77} - 34 q^{78} - 57 q^{79} - 16 q^{80} + 24 q^{81} - 9 q^{82} - 35 q^{83} - 18 q^{84} - 11 q^{85} - 53 q^{86} + 5 q^{87} - 32 q^{88} - 18 q^{89} - 28 q^{90} - 17 q^{91} - 44 q^{92} - 40 q^{93} - 53 q^{94} - 10 q^{95} - 9 q^{96} + 18 q^{97} + 23 q^{98} - 44 q^{99}+O(q^{100}) \) 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 1.00000 −3.27621 1.00000 1.93107 −3.27621 0.738442 1.00000 7.73354 1.93107
1.2 1.00000 −2.99734 1.00000 2.60589 −2.99734 −4.84606 1.00000 5.98405 2.60589
1.3 1.00000 −2.88235 1.00000 −1.77651 −2.88235 −3.07827 1.00000 5.30794 −1.77651
1.4 1.00000 −2.82324 1.00000 1.54794 −2.82324 −0.801132 1.00000 4.97066 1.54794
1.5 1.00000 −2.22103 1.00000 −0.811878 −2.22103 −3.61511 1.00000 1.93296 −0.811878
1.6 1.00000 −2.22048 1.00000 −4.31592 −2.22048 2.07760 1.00000 1.93054 −4.31592
1.7 1.00000 −2.13531 1.00000 −0.668042 −2.13531 1.55222 1.00000 1.55955 −0.668042
1.8 1.00000 −2.05587 1.00000 −3.42459 −2.05587 0.486010 1.00000 1.22659 −3.42459
1.9 1.00000 −1.89771 1.00000 −0.173021 −1.89771 5.26672 1.00000 0.601286 −0.173021
1.10 1.00000 −1.60470 1.00000 2.31688 −1.60470 −0.0835122 1.00000 −0.424924 2.31688
1.11 1.00000 −0.775430 1.00000 1.37036 −0.775430 −1.97918 1.00000 −2.39871 1.37036
1.12 1.00000 −0.667463 1.00000 4.16695 −0.667463 −4.44454 1.00000 −2.55449 4.16695
1.13 1.00000 −0.528500 1.00000 0.915709 −0.528500 1.27781 1.00000 −2.72069 0.915709
1.14 1.00000 −0.401192 1.00000 −1.34190 −0.401192 0.895522 1.00000 −2.83905 −1.34190
1.15 1.00000 −0.230581 1.00000 −2.65439 −0.230581 −1.00000 1.00000 −2.94683 −2.65439
1.16 1.00000 −0.0987369 1.00000 3.39357 −0.0987369 −2.37243 1.00000 −2.99025 3.39357
1.17 1.00000 −0.0562230 1.00000 −1.22175 −0.0562230 −1.45836 1.00000 −2.99684 −1.22175
1.18 1.00000 −0.0366391 1.00000 −2.33800 −0.0366391 3.34955 1.00000 −2.99866 −2.33800
1.19 1.00000 1.02973 1.00000 −0.278191 1.02973 0.0953259 1.00000 −1.93965 −0.278191
1.20 1.00000 1.13770 1.00000 1.03605 1.13770 −0.688437 1.00000 −1.70564 1.03605
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.28
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(1511\) \(-1\)

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 3022.2.a.b ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3022.2.a.b ✓ 28 1.a even 1 1 trivial