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Properties

Label 368.2.m.f
Level $368$
Weight $2$
Character orbit 368.m
Analytic conductor $2.938$
Analytic rank $0$
Dimension $30$
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] = [368,2,Mod(49,368)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(368, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([0, 0, 16]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("368.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 368 = 2^{4} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 368.m (of order \(11\), degree \(10\), not 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: \(2.93849479438\)
Analytic rank: \(0\)
Dimension: \(30\)
Relative dimension: \(3\) over \(\Q(\zeta_{11})\)
Twist minimal: no (minimal twist has level 184)
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 30 q + 2 q^{3} + 2 q^{5} - 13 q^{7} - 23 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q + 2 q^{3} + 2 q^{5} - 13 q^{7} - 23 q^{9} + 4 q^{11} + 2 q^{15} + 26 q^{17} + 5 q^{19} + 2 q^{21} + q^{23} - 11 q^{25} - 19 q^{27} - 19 q^{29} + 6 q^{31} + 20 q^{33} + 63 q^{35} + 36 q^{37} + 22 q^{39} - 11 q^{41} - 13 q^{43} - 92 q^{45} - 24 q^{47} - 48 q^{49} + 41 q^{51} + 27 q^{53} + 59 q^{55} - 2 q^{57} + 11 q^{59} - 31 q^{61} - 82 q^{63} - 30 q^{65} - 21 q^{67} + 18 q^{69} + 52 q^{71} + 32 q^{73} - 3 q^{75} - 21 q^{77} - 54 q^{79} + 3 q^{81} - 70 q^{83} - 8 q^{85} - 33 q^{87} + 77 q^{89} - 26 q^{91} + 82 q^{93} - 21 q^{95} + 31 q^{97} - 201 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1 0 −1.14478 + 2.50671i 0 2.05971 − 2.37704i 0 2.73308 − 1.75644i 0 −3.00852 − 3.47201i 0
49.2 0 −0.0870522 + 0.190618i 0 0.0866818 − 0.100036i 0 −2.06143 + 1.32480i 0 1.93583 + 2.23406i 0
49.3 0 1.35007 − 2.95624i 0 −0.836673 + 0.965572i 0 −0.644005 + 0.413877i 0 −4.95209 − 5.71501i 0
81.1 0 −0.278454 + 1.93669i 0 1.46929 − 0.431422i 0 0.616482 + 0.711458i 0 −0.794747 − 0.233359i 0
81.2 0 0.0579878 − 0.403314i 0 −2.25586 + 0.662379i 0 −2.18085 − 2.51684i 0 2.71918 + 0.798423i 0
81.3 0 0.459912 − 3.19876i 0 2.70555 − 0.794422i 0 −1.70484 − 1.96749i 0 −7.14204 − 2.09709i 0
177.1 0 −1.97449 + 1.26892i 0 −0.417997 − 0.915286i 0 −0.323060 + 0.0948589i 0 1.04218 − 2.28205i 0
177.2 0 0.741585 − 0.476588i 0 1.02542 + 2.24535i 0 1.34070 − 0.393666i 0 −0.923433 + 2.02204i 0
177.3 0 2.33471 − 1.50043i 0 −1.43825 − 3.14933i 0 −3.52121 + 1.03392i 0 1.95334 − 4.27722i 0
193.1 0 −1.83472 − 2.11738i 0 0.104983 + 0.730173i 0 −1.59958 + 3.50260i 0 −0.690154 + 4.80013i 0
193.2 0 0.211214 + 0.243754i 0 −0.354040 − 2.46240i 0 −0.114164 + 0.249984i 0 0.412140 − 2.86650i 0
193.3 0 0.366836 + 0.423352i 0 0.533687 + 3.71187i 0 1.40226 − 3.07052i 0 0.382287 − 2.65886i 0
209.1 0 −0.278454 − 1.93669i 0 1.46929 + 0.431422i 0 0.616482 − 0.711458i 0 −0.794747 + 0.233359i 0
209.2 0 0.0579878 + 0.403314i 0 −2.25586 − 0.662379i 0 −2.18085 + 2.51684i 0 2.71918 − 0.798423i 0
209.3 0 0.459912 + 3.19876i 0 2.70555 + 0.794422i 0 −1.70484 + 1.96749i 0 −7.14204 + 2.09709i 0
225.1 0 −1.83472 + 2.11738i 0 0.104983 − 0.730173i 0 −1.59958 − 3.50260i 0 −0.690154 − 4.80013i 0
225.2 0 0.211214 − 0.243754i 0 −0.354040 + 2.46240i 0 −0.114164 − 0.249984i 0 0.412140 + 2.86650i 0
225.3 0 0.366836 − 0.423352i 0 0.533687 − 3.71187i 0 1.40226 + 3.07052i 0 0.382287 + 2.65886i 0
257.1 0 −1.37739 + 0.404437i 0 0.696214 + 0.447429i 0 −0.722941 − 5.02816i 0 −0.790138 + 0.507791i 0
257.2 0 0.262516 − 0.0770816i 0 −3.42011 − 2.19797i 0 0.433778 + 3.01699i 0 −2.46079 + 1.58145i 0
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.c even 11 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 368.2.m.f 30
4.b odd 2 1 184.2.i.a ✓ 30
23.c even 11 1 inner 368.2.m.f 30
23.c even 11 1 8464.2.a.ci 15
23.d odd 22 1 8464.2.a.cj 15
92.g odd 22 1 184.2.i.a ✓ 30
92.g odd 22 1 4232.2.a.y 15
92.h even 22 1 4232.2.a.z 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
184.2.i.a ✓ 30 4.b odd 2 1
184.2.i.a ✓ 30 92.g odd 22 1
368.2.m.f 30 1.a even 1 1 trivial
368.2.m.f 30 23.c even 11 1 inner
4232.2.a.y 15 92.g odd 22 1
4232.2.a.z 15 92.h even 22 1
8464.2.a.ci 15 23.c even 11 1
8464.2.a.cj 15 23.d odd 22 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{30} - 2 T_{3}^{29} + 18 T_{3}^{28} - 31 T_{3}^{27} + 171 T_{3}^{26} - 413 T_{3}^{25} + 1383 T_{3}^{24} + \cdots + 121 \) acting on \(S_{2}^{\mathrm{new}}(368, [\chi])\). Copy content Toggle raw display