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Properties

Label 220.3.c.a
Level $220$
Weight $3$
Character orbit 220.c
Analytic conductor $5.995$
Analytic rank $0$
Dimension $40$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [220,3,Mod(111,220)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(220, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("220.111");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 220 = 2^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 220.c (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(5.99456581593\)
Analytic rank: \(0\)
Dimension: \(40\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 4 q^{4} + 36 q^{8} - 120 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 4 q^{4} + 36 q^{8} - 120 q^{9} + 20 q^{10} - 80 q^{12} + 32 q^{13} - 56 q^{14} + 40 q^{16} + 16 q^{18} - 20 q^{20} - 80 q^{21} + 104 q^{24} + 200 q^{25} + 100 q^{26} + 60 q^{28} - 48 q^{29} - 280 q^{32} + 8 q^{34} + 160 q^{36} + 160 q^{37} - 200 q^{38} - 80 q^{40} - 120 q^{42} + 44 q^{44} + 80 q^{45} + 120 q^{46} - 16 q^{48} - 376 q^{49} - 36 q^{52} + 296 q^{54} - 40 q^{56} + 288 q^{57} + 96 q^{58} + 60 q^{60} - 160 q^{61} - 16 q^{62} + 500 q^{64} - 220 q^{66} + 116 q^{68} - 304 q^{69} - 60 q^{70} - 380 q^{72} - 480 q^{73} - 320 q^{74} - 480 q^{76} + 744 q^{78} - 80 q^{80} + 280 q^{81} - 480 q^{82} + 16 q^{84} + 320 q^{85} - 460 q^{86} + 224 q^{89} - 60 q^{90} + 24 q^{92} + 480 q^{93} + 432 q^{94} + 424 q^{96} + 720 q^{97} + 680 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
111.1 −1.99979 − 0.0291589i − 1.72793i 3.99830 + 0.116623i 2.23607 −0.0503843 + 3.45548i − 8.42008i −7.99235 − 0.349807i 6.01427 −4.47166 − 0.0652012i
111.2 −1.99979 + 0.0291589i 1.72793i 3.99830 − 0.116623i 2.23607 −0.0503843 − 3.45548i 8.42008i −7.99235 + 0.349807i 6.01427 −4.47166 + 0.0652012i
111.3 −1.99818 − 0.0853244i 4.89415i 3.98544 + 0.340987i −2.23607 0.417590 − 9.77938i − 7.01759i −7.93453 − 1.02141i −14.9527 4.46806 + 0.190791i
111.4 −1.99818 + 0.0853244i − 4.89415i 3.98544 − 0.340987i −2.23607 0.417590 + 9.77938i 7.01759i −7.93453 + 1.02141i −14.9527 4.46806 − 0.190791i
111.5 −1.89682 − 0.634098i 1.40441i 3.19584 + 2.40554i −2.23607 0.890535 − 2.66392i − 1.67390i −4.53658 − 6.58934i 7.02763 4.24141 + 1.41789i
111.6 −1.89682 + 0.634098i − 1.40441i 3.19584 − 2.40554i −2.23607 0.890535 + 2.66392i 1.67390i −4.53658 + 6.58934i 7.02763 4.24141 − 1.41789i
111.7 −1.50888 − 1.31274i 3.61410i 0.553451 + 3.96153i 2.23607 4.74436 − 5.45326i − 9.37232i 4.36535 − 6.70401i −4.06174 −3.37396 − 2.93537i
111.8 −1.50888 + 1.31274i − 3.61410i 0.553451 − 3.96153i 2.23607 4.74436 + 5.45326i 9.37232i 4.36535 + 6.70401i −4.06174 −3.37396 + 2.93537i
111.9 −1.24010 − 1.56913i 5.26184i −0.924308 + 3.89174i −2.23607 8.25649 − 6.52521i 11.5241i 7.25286 − 3.37579i −18.6870 2.77295 + 3.50867i
111.10 −1.24010 + 1.56913i − 5.26184i −0.924308 − 3.89174i −2.23607 8.25649 + 6.52521i − 11.5241i 7.25286 + 3.37579i −18.6870 2.77295 − 3.50867i
111.11 −1.12341 − 1.65468i − 0.776243i −1.47591 + 3.71775i −2.23607 −1.28443 + 0.872037i − 8.96218i 7.80973 − 1.73439i 8.39745 2.51201 + 3.69997i
111.12 −1.12341 + 1.65468i 0.776243i −1.47591 − 3.71775i −2.23607 −1.28443 − 0.872037i 8.96218i 7.80973 + 1.73439i 8.39745 2.51201 − 3.69997i
111.13 −0.966908 − 1.75074i − 1.35561i −2.13018 + 3.38561i −2.23607 −2.37333 + 1.31075i 6.68479i 7.98700 + 0.455814i 7.16231 2.16207 + 3.91477i
111.14 −0.966908 + 1.75074i 1.35561i −2.13018 − 3.38561i −2.23607 −2.37333 − 1.31075i − 6.68479i 7.98700 − 0.455814i 7.16231 2.16207 − 3.91477i
111.15 −0.679211 − 1.88114i − 5.18603i −3.07735 + 2.55538i 2.23607 −9.75563 + 3.52241i − 11.9764i 6.89717 + 4.05327i −17.8949 −1.51876 − 4.20635i
111.16 −0.679211 + 1.88114i 5.18603i −3.07735 − 2.55538i 2.23607 −9.75563 − 3.52241i 11.9764i 6.89717 − 4.05327i −17.8949 −1.51876 + 4.20635i
111.17 −0.536785 − 1.92662i − 0.401733i −3.42372 + 2.06836i 2.23607 −0.773986 + 0.215644i 4.24999i 5.82275 + 5.48594i 8.83861 −1.20029 − 4.30805i
111.18 −0.536785 + 1.92662i 0.401733i −3.42372 − 2.06836i 2.23607 −0.773986 − 0.215644i − 4.24999i 5.82275 − 5.48594i 8.83861 −1.20029 + 4.30805i
111.19 −0.274639 − 1.98105i 4.52775i −3.84915 + 1.08815i 2.23607 8.96972 − 1.24350i 1.58481i 3.21281 + 7.32652i −11.5005 −0.614111 − 4.42977i
111.20 −0.274639 + 1.98105i − 4.52775i −3.84915 − 1.08815i 2.23607 8.96972 + 1.24350i − 1.58481i 3.21281 − 7.32652i −11.5005 −0.614111 + 4.42977i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 111.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 220.3.c.a ✓ 40
4.b odd 2 1 inner 220.3.c.a ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
220.3.c.a ✓ 40 1.a even 1 1 trivial
220.3.c.a ✓ 40 4.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(220, [\chi])\).