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Properties

Label 385.2.n.f
Level $385$
Weight $2$
Character orbit 385.n
Analytic conductor $3.074$
Analytic rank $0$
Dimension $36$
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] = [385,2,Mod(36,385)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(385, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("385.36");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 385 = 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 385.n (of order \(5\), degree \(4\), 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: \(3.07424047782\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(9\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q + q^{2} - 17 q^{4} - 9 q^{5} - 13 q^{6} + 9 q^{7} + 3 q^{8} - 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q + q^{2} - 17 q^{4} - 9 q^{5} - 13 q^{6} + 9 q^{7} + 3 q^{8} - 11 q^{9} - 4 q^{10} - 3 q^{11} + 30 q^{12} - q^{13} - q^{14} + 3 q^{16} + 5 q^{17} - 34 q^{18} - 5 q^{19} - 7 q^{20} - 10 q^{21} - 5 q^{22} + 8 q^{23} - 41 q^{24} - 9 q^{25} + 28 q^{26} + 7 q^{28} - 33 q^{29} + 12 q^{30} - q^{31} + 12 q^{32} - 14 q^{33} + 88 q^{34} + 9 q^{35} - 104 q^{36} - 18 q^{37} - 27 q^{38} + 6 q^{39} + 3 q^{40} + 36 q^{41} + 13 q^{42} - 20 q^{43} - 30 q^{44} + 74 q^{45} + 20 q^{46} - 7 q^{47} + 75 q^{48} - 9 q^{49} + q^{50} + 22 q^{51} - 6 q^{52} - 4 q^{53} - 26 q^{54} - 3 q^{55} + 12 q^{56} - 43 q^{57} - 41 q^{58} + 25 q^{59} - 15 q^{60} + 21 q^{61} + 31 q^{62} + 26 q^{63} + 43 q^{64} - 16 q^{65} - 12 q^{66} + 58 q^{67} - 43 q^{68} - 37 q^{69} - q^{70} - 29 q^{71} + 34 q^{72} + 19 q^{73} - 59 q^{74} - 5 q^{75} - 94 q^{76} + 3 q^{77} + 20 q^{78} + 11 q^{79} - 47 q^{80} - 10 q^{81} - 58 q^{82} + 47 q^{83} - 10 q^{85} + 35 q^{86} - 46 q^{87} - 110 q^{88} + 118 q^{89} + 36 q^{90} - 9 q^{91} + 107 q^{92} - 29 q^{93} - 27 q^{94} - 5 q^{95} + 67 q^{96} - 4 q^{98} + 16 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
36.1 −2.14110 + 1.55560i 0.669443 + 2.06033i 1.54638 − 4.75927i −0.809017 − 0.587785i −4.63840 − 3.36999i −0.309017 + 0.951057i 2.45691 + 7.56158i −1.36977 + 0.995196i 2.64654
36.2 −1.58317 + 1.15024i 0.0837431 + 0.257735i 0.565342 − 1.73995i −0.809017 − 0.587785i −0.429037 − 0.311713i −0.309017 + 0.951057i −0.103112 − 0.317347i 2.36764 − 1.72019i 1.95691
36.3 −0.661191 + 0.480383i −0.404309 − 1.24433i −0.411629 + 1.26686i −0.809017 − 0.587785i 0.865083 + 0.628520i −0.309017 + 0.951057i −0.841520 − 2.58993i 1.04215 − 0.757164i 0.817277
36.4 −0.599403 + 0.435491i 1.02764 + 3.16276i −0.448403 + 1.38004i −0.809017 − 0.587785i −1.99333 − 1.44824i −0.309017 + 0.951057i −0.790126 − 2.43176i −6.51996 + 4.73703i 0.740902
36.5 −0.300932 + 0.218640i −0.537710 − 1.65490i −0.575277 + 1.77052i −0.809017 − 0.587785i 0.523642 + 0.380448i −0.309017 + 0.951057i −0.443879 − 1.36612i −0.0225132 + 0.0163568i 0.371973
36.6 0.760972 − 0.552878i 0.264204 + 0.813137i −0.344630 + 1.06066i −0.809017 − 0.587785i 0.650618 + 0.472701i −0.309017 + 0.951057i 0.905495 + 2.78683i 1.83566 − 1.33369i −0.940613
36.7 1.60097 − 1.16317i 0.925528 + 2.84848i 0.592097 − 1.82229i −0.809017 − 0.587785i 4.79501 + 3.48378i −0.309017 + 0.951057i 0.0513277 + 0.157970i −4.83020 + 3.50934i −1.97890
36.8 1.66072 − 1.20659i −1.05519 − 3.24753i 0.684120 − 2.10550i −0.809017 − 0.587785i −5.67081 − 4.12008i −0.309017 + 0.951057i −0.135658 − 0.417514i −7.00601 + 5.09016i −2.05277
36.9 2.07215 − 1.50550i 0.144679 + 0.445277i 1.40922 − 4.33714i −0.809017 − 0.587785i 0.970163 + 0.704865i −0.309017 + 0.951057i −2.02648 − 6.23688i 2.24971 − 1.63451i −2.56132
71.1 −0.858369 − 2.64179i 0.934421 + 0.678897i −4.62421 + 3.35968i 0.309017 − 0.951057i 0.991423 − 3.05129i 0.809017 − 0.587785i 8.35036 + 6.06689i −0.514809 − 1.58442i −2.77774
71.2 −0.781722 − 2.40589i −2.56960 − 1.86692i −3.55920 + 2.58591i 0.309017 − 0.951057i −2.48291 + 7.64160i 0.809017 − 0.587785i 4.91057 + 3.56774i 2.19039 + 6.74133i −2.52971
71.3 −0.429441 − 1.32168i 1.08334 + 0.787091i 0.0556063 − 0.0404004i 0.309017 − 0.951057i 0.575056 − 1.76984i 0.809017 − 0.587785i −2.32586 − 1.68984i −0.372942 − 1.14780i −1.38970
71.4 −0.344095 − 1.05902i −1.84056 − 1.33725i 0.614920 − 0.446766i 0.309017 − 0.951057i −0.782838 + 2.40933i 0.809017 − 0.587785i −2.48643 − 1.80650i 0.672394 + 2.06942i −1.11352
71.5 −0.0738489 − 0.227283i 2.28939 + 1.66334i 1.57183 − 1.14200i 0.309017 − 0.951057i 0.208980 − 0.643175i 0.809017 − 0.587785i −0.762313 − 0.553853i 1.54755 + 4.76288i −0.238980
71.6 0.225377 + 0.693638i −1.37005 − 0.995403i 1.18769 − 0.862910i 0.309017 − 0.951057i 0.381671 − 1.17466i 0.809017 − 0.587785i 2.04631 + 1.48673i −0.0408282 − 0.125656i 0.729335
71.7 0.399135 + 1.22841i 0.377148 + 0.274014i 0.268349 − 0.194967i 0.309017 − 0.951057i −0.186069 + 0.572662i 0.809017 − 0.587785i 2.43650 + 1.77022i −0.859894 − 2.64648i 1.29163
71.8 0.728853 + 2.24318i −2.52872 − 1.83722i −2.88260 + 2.09433i 0.309017 − 0.951057i 2.27816 − 7.01145i 0.809017 − 0.587785i −2.98262 − 2.16700i 2.09199 + 6.43850i 2.35862
71.9 0.825093 + 2.53938i 2.50661 + 1.82116i −4.14962 + 3.01487i 0.309017 − 0.951057i −2.55642 + 7.86786i 0.809017 − 0.587785i −6.75948 − 4.91105i 2.03943 + 6.27673i 2.67006
141.1 −0.858369 + 2.64179i 0.934421 − 0.678897i −4.62421 − 3.35968i 0.309017 + 0.951057i 0.991423 + 3.05129i 0.809017 + 0.587785i 8.35036 − 6.06689i −0.514809 + 1.58442i −2.77774
141.2 −0.781722 + 2.40589i −2.56960 + 1.86692i −3.55920 − 2.58591i 0.309017 + 0.951057i −2.48291 − 7.64160i 0.809017 + 0.587785i 4.91057 − 3.56774i 2.19039 − 6.74133i −2.52971
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 36.9
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 385.2.n.f ✓ 36
11.c even 5 1 inner 385.2.n.f ✓ 36
11.c even 5 1 4235.2.a.bo 18
11.d odd 10 1 4235.2.a.bp 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
385.2.n.f ✓ 36 1.a even 1 1 trivial
385.2.n.f ✓ 36 11.c even 5 1 inner
4235.2.a.bo 18 11.c even 5 1
4235.2.a.bp 18 11.d odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{36} - T_{2}^{35} + 18 T_{2}^{34} - 20 T_{2}^{33} + 192 T_{2}^{32} - 193 T_{2}^{31} + 1571 T_{2}^{30} + \cdots + 30976 \) acting on \(S_{2}^{\mathrm{new}}(385, [\chi])\). Copy content Toggle raw display