↑

Properties

Label 41.2.g.a
Level $41$
Weight $2$
Character orbit 41.g
Analytic conductor $0.327$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [41,2,Mod(2,41)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(41, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([13]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("41.2");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 41.g (of order \(20\), degree \(8\), 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: \(0.327386648287\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(3\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 10 q^{2} - 6 q^{3} - 10 q^{5} - 2 q^{6} - 8 q^{7} - 10 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 10 q^{2} - 6 q^{3} - 10 q^{5} - 2 q^{6} - 8 q^{7} - 10 q^{8} + 6 q^{10} - 16 q^{11} + 2 q^{12} + 14 q^{14} + 8 q^{15} - 20 q^{16} + 8 q^{17} + 16 q^{19} + 20 q^{20} - 10 q^{21} + 6 q^{22} + 12 q^{23} + 68 q^{24} - 8 q^{25} - 28 q^{26} - 6 q^{27} + 18 q^{28} + 40 q^{29} - 36 q^{30} - 12 q^{31} + 10 q^{33} - 16 q^{34} - 36 q^{35} - 40 q^{36} + 46 q^{38} - 50 q^{39} - 44 q^{40} - 4 q^{41} - 40 q^{42} - 48 q^{44} + 16 q^{45} + 70 q^{46} - 12 q^{47} - 50 q^{48} - 30 q^{49} - 24 q^{51} + 20 q^{52} - 26 q^{53} + 68 q^{54} + 20 q^{55} + 106 q^{56} + 10 q^{57} - 20 q^{58} + 6 q^{59} + 76 q^{60} + 30 q^{61} - 10 q^{62} + 92 q^{63} + 70 q^{64} + 68 q^{65} + 34 q^{66} - 22 q^{67} - 20 q^{68} - 38 q^{69} - 20 q^{70} + 4 q^{71} - 74 q^{72} + 10 q^{74} + 4 q^{75} - 128 q^{76} - 20 q^{77} - 10 q^{78} - 2 q^{79} - 70 q^{80} + 28 q^{81} - 90 q^{82} + 80 q^{83} - 30 q^{84} - 56 q^{85} - 46 q^{86} - 10 q^{87} + 10 q^{88} - 72 q^{89} - 70 q^{90} - 6 q^{93} - 18 q^{94} - 40 q^{95} + 66 q^{96} - 22 q^{97} + 6 q^{98} + 14 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −2.05153 + 0.666584i 1.49921 − 1.49921i 2.14642 − 1.55947i 1.72514 + 2.37446i −2.07633 + 4.07502i −1.11329 − 2.18496i −0.828108 + 1.13979i − 1.49525i −5.12196 − 3.72132i
2.2 −1.40718 + 0.457221i −1.90046 + 1.90046i 0.153075 − 0.111216i −0.455161 − 0.626475i 1.80536 − 3.54322i 2.12336 + 4.16732i 1.57482 − 2.16755i − 4.22349i 0.926931 + 0.673455i
2.3 0.698642 − 0.227002i 0.0432913 − 0.0432913i −1.18146 + 0.858384i −0.422124 − 0.581004i 0.0204179 − 0.0400723i −1.42228 − 2.79137i −1.49413 + 2.05650i 2.99625i −0.426803 − 0.310091i
5.1 −1.00762 − 1.38687i 2.22848 + 2.22848i −0.290083 + 0.892782i −2.57687 − 0.837277i 0.845152 − 5.33608i −0.252316 − 1.59306i −1.73027 + 0.562198i 6.93221i 1.43532 + 4.41746i
5.2 −0.522120 − 0.718637i −0.983583 − 0.983583i 0.374204 − 1.15168i 1.02071 + 0.331648i −0.193291 + 1.22039i 0.625712 + 3.95059i −2.71264 + 0.881390i − 1.06513i −0.294598 − 0.906679i
5.3 1.42655 + 1.96347i −2.02366 − 2.02366i −1.20216 + 3.69986i 0.110420 + 0.0358775i 1.08656 − 6.86025i −0.422339 − 2.66654i −4.36309 + 1.41766i 5.19040i 0.0870742 + 0.267987i
8.1 −1.61018 + 2.21623i 1.24382 + 1.24382i −1.70094 − 5.23496i −1.15434 + 0.375067i −4.75938 + 0.753811i 1.19485 + 0.189245i 9.13005 + 2.96653i 0.0941838i 1.02746 − 3.16221i
8.2 −0.415383 + 0.571726i −0.242613 − 0.242613i 0.463706 + 1.42714i 2.26179 − 0.734900i 0.239486 − 0.0379308i −4.85225 − 0.768522i −2.35276 − 0.764458i − 2.88228i −0.519348 + 1.59839i
8.3 0.746800 − 1.02788i −0.604406 − 0.604406i 0.119203 + 0.366868i −3.27974 + 1.06565i −1.07263 + 0.169888i 1.70635 + 0.270260i 2.88281 + 0.936683i − 2.26939i −1.35394 + 4.16701i
20.1 −2.29671 − 0.746246i −1.67347 + 1.67347i 3.09996 + 2.25225i −1.68856 + 2.32411i 5.09230 − 2.59466i −1.86997 − 0.952797i −2.60008 − 3.57870i − 2.60102i 5.61249 − 4.07771i
20.2 0.290902 + 0.0945196i 0.964912 − 0.964912i −1.54234 − 1.12058i −2.03721 + 2.80398i 0.371897 − 0.189491i 1.29765 + 0.661185i −0.702328 − 0.966671i 1.13789i −0.857659 + 0.623126i
20.3 1.14785 + 0.372958i −1.55151 + 1.55151i −0.439578 − 0.319372i 1.49595 − 2.05900i −2.35955 + 1.20225i −1.01547 − 0.517405i −1.80427 − 2.48337i − 1.81438i 2.48504 − 1.80549i
21.1 −2.05153 − 0.666584i 1.49921 + 1.49921i 2.14642 + 1.55947i 1.72514 − 2.37446i −2.07633 − 4.07502i −1.11329 + 2.18496i −0.828108 − 1.13979i 1.49525i −5.12196 + 3.72132i
21.2 −1.40718 − 0.457221i −1.90046 − 1.90046i 0.153075 + 0.111216i −0.455161 + 0.626475i 1.80536 + 3.54322i 2.12336 − 4.16732i 1.57482 + 2.16755i 4.22349i 0.926931 − 0.673455i
21.3 0.698642 + 0.227002i 0.0432913 + 0.0432913i −1.18146 − 0.858384i −0.422124 + 0.581004i 0.0204179 + 0.0400723i −1.42228 + 2.79137i −1.49413 − 2.05650i − 2.99625i −0.426803 + 0.310091i
33.1 −1.00762 + 1.38687i 2.22848 − 2.22848i −0.290083 − 0.892782i −2.57687 + 0.837277i 0.845152 + 5.33608i −0.252316 + 1.59306i −1.73027 − 0.562198i − 6.93221i 1.43532 − 4.41746i
33.2 −0.522120 + 0.718637i −0.983583 + 0.983583i 0.374204 + 1.15168i 1.02071 − 0.331648i −0.193291 − 1.22039i 0.625712 − 3.95059i −2.71264 − 0.881390i 1.06513i −0.294598 + 0.906679i
33.3 1.42655 − 1.96347i −2.02366 + 2.02366i −1.20216 − 3.69986i 0.110420 − 0.0358775i 1.08656 + 6.86025i −0.422339 + 2.66654i −4.36309 − 1.41766i − 5.19040i 0.0870742 − 0.267987i
36.1 −1.61018 − 2.21623i 1.24382 − 1.24382i −1.70094 + 5.23496i −1.15434 − 0.375067i −4.75938 − 0.753811i 1.19485 − 0.189245i 9.13005 − 2.96653i − 0.0941838i 1.02746 + 3.16221i
36.2 −0.415383 − 0.571726i −0.242613 + 0.242613i 0.463706 − 1.42714i 2.26179 + 0.734900i 0.239486 + 0.0379308i −4.85225 + 0.768522i −2.35276 + 0.764458i 2.88228i −0.519348 − 1.59839i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
41.g even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 41.2.g.a ✓ 24
3.b odd 2 1 369.2.u.a 24
4.b odd 2 1 656.2.bs.d 24
41.g even 20 1 inner 41.2.g.a ✓ 24
41.h odd 40 2 1681.2.a.m 24
123.m odd 20 1 369.2.u.a 24
164.n odd 20 1 656.2.bs.d 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
41.2.g.a ✓ 24 1.a even 1 1 trivial
41.2.g.a ✓ 24 41.g even 20 1 inner
369.2.u.a 24 3.b odd 2 1
369.2.u.a 24 123.m odd 20 1
656.2.bs.d 24 4.b odd 2 1
656.2.bs.d 24 164.n odd 20 1
1681.2.a.m 24 41.h odd 40 2

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(41, [\chi])\).