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Properties

Label 369.2.u.a
Level $369$
Weight $2$
Character orbit 369.u
Analytic conductor $2.946$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [369,2,Mod(46,369)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(369, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([0, 11]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("369.46");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 369 = 3^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 369.u (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: \(2.94647983459\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(3\) over \(\Q(\zeta_{20})\)
Twist minimal: no (minimal twist has level 41)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 10 q^{2} + 10 q^{5} - 8 q^{7} + 10 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 10 q^{2} + 10 q^{5} - 8 q^{7} + 10 q^{8} + 6 q^{10} + 16 q^{11} - 14 q^{14} - 20 q^{16} - 8 q^{17} + 16 q^{19} - 20 q^{20} + 6 q^{22} - 12 q^{23} - 8 q^{25} + 28 q^{26} + 18 q^{28} - 40 q^{29} - 12 q^{31} - 16 q^{34} + 36 q^{35} - 46 q^{38} - 44 q^{40} + 4 q^{41} + 48 q^{44} + 70 q^{46} + 12 q^{47} - 30 q^{49} + 20 q^{52} + 26 q^{53} + 20 q^{55} - 106 q^{56} - 20 q^{58} - 6 q^{59} + 30 q^{61} + 10 q^{62} + 70 q^{64} - 68 q^{65} - 22 q^{67} + 20 q^{68} - 20 q^{70} - 4 q^{71} - 10 q^{74} - 128 q^{76} + 20 q^{77} - 2 q^{79} + 70 q^{80} - 90 q^{82} - 80 q^{83} - 56 q^{85} + 46 q^{86} + 10 q^{88} + 72 q^{89} - 18 q^{94} + 40 q^{95} - 22 q^{97} - 6 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
46.1 −1.42655 − 1.96347i 0 −1.20216 + 3.69986i −0.110420 − 0.0358775i 0 −0.422339 − 2.66654i 4.36309 − 1.41766i 0 0.0870742 + 0.267987i
46.2 0.522120 + 0.718637i 0 0.374204 − 1.15168i −1.02071 − 0.331648i 0 0.625712 + 3.95059i 2.71264 − 0.881390i 0 −0.294598 − 0.906679i
46.3 1.00762 + 1.38687i 0 −0.290083 + 0.892782i 2.57687 + 0.837277i 0 −0.252316 − 1.59306i 1.73027 − 0.562198i 0 1.43532 + 4.41746i
118.1 −0.746800 − 1.02788i 0 0.119203 − 0.366868i 3.27974 + 1.06565i 0 1.70635 − 0.270260i −2.88281 + 0.936683i 0 −1.35394 − 4.16701i
118.2 0.415383 + 0.571726i 0 0.463706 − 1.42714i −2.26179 − 0.734900i 0 −4.85225 + 0.768522i 2.35276 − 0.764458i 0 −0.519348 − 1.59839i
118.3 1.61018 + 2.21623i 0 −1.70094 + 5.23496i 1.15434 + 0.375067i 0 1.19485 − 0.189245i −9.13005 + 2.96653i 0 1.02746 + 3.16221i
172.1 −0.746800 + 1.02788i 0 0.119203 + 0.366868i 3.27974 − 1.06565i 0 1.70635 + 0.270260i −2.88281 − 0.936683i 0 −1.35394 + 4.16701i
172.2 0.415383 − 0.571726i 0 0.463706 + 1.42714i −2.26179 + 0.734900i 0 −4.85225 − 0.768522i 2.35276 + 0.764458i 0 −0.519348 + 1.59839i
172.3 1.61018 − 2.21623i 0 −1.70094 − 5.23496i 1.15434 − 0.375067i 0 1.19485 + 0.189245i −9.13005 − 2.96653i 0 1.02746 − 3.16221i
226.1 −0.698642 − 0.227002i 0 −1.18146 − 0.858384i 0.422124 − 0.581004i 0 −1.42228 + 2.79137i 1.49413 + 2.05650i 0 −0.426803 + 0.310091i
226.2 1.40718 + 0.457221i 0 0.153075 + 0.111216i 0.455161 − 0.626475i 0 2.12336 − 4.16732i −1.57482 − 2.16755i 0 0.926931 − 0.673455i
226.3 2.05153 + 0.666584i 0 2.14642 + 1.55947i −1.72514 + 2.37446i 0 −1.11329 + 2.18496i 0.828108 + 1.13979i 0 −5.12196 + 3.72132i
244.1 −1.14785 + 0.372958i 0 −0.439578 + 0.319372i −1.49595 − 2.05900i 0 −1.01547 + 0.517405i 1.80427 − 2.48337i 0 2.48504 + 1.80549i
244.2 −0.290902 + 0.0945196i 0 −1.54234 + 1.12058i 2.03721 + 2.80398i 0 1.29765 − 0.661185i 0.702328 − 0.966671i 0 −0.857659 − 0.623126i
244.3 2.29671 − 0.746246i 0 3.09996 − 2.25225i 1.68856 + 2.32411i 0 −1.86997 + 0.952797i 2.60008 − 3.57870i 0 5.61249 + 4.07771i
289.1 −0.698642 + 0.227002i 0 −1.18146 + 0.858384i 0.422124 + 0.581004i 0 −1.42228 − 2.79137i 1.49413 − 2.05650i 0 −0.426803 − 0.310091i
289.2 1.40718 − 0.457221i 0 0.153075 − 0.111216i 0.455161 + 0.626475i 0 2.12336 + 4.16732i −1.57482 + 2.16755i 0 0.926931 + 0.673455i
289.3 2.05153 − 0.666584i 0 2.14642 − 1.55947i −1.72514 − 2.37446i 0 −1.11329 − 2.18496i 0.828108 − 1.13979i 0 −5.12196 − 3.72132i
307.1 −1.14785 − 0.372958i 0 −0.439578 − 0.319372i −1.49595 + 2.05900i 0 −1.01547 − 0.517405i 1.80427 + 2.48337i 0 2.48504 − 1.80549i
307.2 −0.290902 − 0.0945196i 0 −1.54234 − 1.12058i 2.03721 − 2.80398i 0 1.29765 + 0.661185i 0.702328 + 0.966671i 0 −0.857659 + 0.623126i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 46.3
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 369.2.u.a 24
3.b odd 2 1 41.2.g.a ✓ 24
12.b even 2 1 656.2.bs.d 24
41.g even 20 1 inner 369.2.u.a 24
123.m odd 20 1 41.2.g.a ✓ 24
123.o even 40 2 1681.2.a.m 24
492.y even 20 1 656.2.bs.d 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
41.2.g.a ✓ 24 3.b odd 2 1
41.2.g.a ✓ 24 123.m odd 20 1
369.2.u.a 24 1.a even 1 1 trivial
369.2.u.a 24 41.g even 20 1 inner
656.2.bs.d 24 12.b even 2 1
656.2.bs.d 24 492.y even 20 1
1681.2.a.m 24 123.o even 40 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} - 10 T_{2}^{23} + 44 T_{2}^{22} - 110 T_{2}^{21} + 200 T_{2}^{20} - 480 T_{2}^{19} + \cdots + 361 \) acting on \(S_{2}^{\mathrm{new}}(369, [\chi])\). Copy content Toggle raw display