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Properties

Label 374.2.o.a
Level $374$
Weight $2$
Character orbit 374.o
Analytic conductor $2.986$
Analytic rank $0$
Dimension $64$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [374,2,Mod(47,374)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(374, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([16, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("374.47");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 374 = 2 \cdot 11 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 374.o (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: \(2.98640503560\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(8\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 64 q + 16 q^{4} - 4 q^{5} - 10 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q + 16 q^{4} - 4 q^{5} - 10 q^{7} + 4 q^{10} - 2 q^{11} + 16 q^{13} - 10 q^{14} - 16 q^{16} + 6 q^{17} - 12 q^{18} + 4 q^{20} - 32 q^{21} + 8 q^{22} - 12 q^{23} - 30 q^{27} - 10 q^{28} + 6 q^{29} + 4 q^{31} + 36 q^{33} - 20 q^{34} - 24 q^{37} + 16 q^{38} - 20 q^{39} + 6 q^{40} + 30 q^{41} - 8 q^{44} + 20 q^{45} - 32 q^{46} - 16 q^{47} - 4 q^{50} + 4 q^{52} + 16 q^{55} + 12 q^{57} - 6 q^{58} + 4 q^{61} - 4 q^{62} - 70 q^{63} + 16 q^{64} - 32 q^{65} - 88 q^{67} - 6 q^{68} + 28 q^{69} - 52 q^{71} - 48 q^{72} - 16 q^{73} + 24 q^{74} + 90 q^{75} - 70 q^{79} + 6 q^{80} - 76 q^{81} + 10 q^{82} + 12 q^{84} + 24 q^{85} + 80 q^{86} - 18 q^{88} + 32 q^{89} + 20 q^{90} + 42 q^{91} - 38 q^{92} + 62 q^{95} - 18 q^{97} - 8 q^{98} + 118 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1 −0.587785 − 0.809017i −1.40398 + 2.75546i −0.309017 + 0.951057i −0.591255 − 3.73303i 3.05445 − 0.483777i 1.14262 + 2.24251i 0.951057 − 0.309017i −3.85804 − 5.31014i −2.67256 + 2.67256i
47.2 −0.587785 − 0.809017i −0.995860 + 1.95449i −0.309017 + 0.951057i 0.448337 + 2.83069i 2.16656 − 0.343150i 0.712141 + 1.39766i 0.951057 − 0.309017i −1.06492 − 1.46574i 2.02655 − 2.02655i
47.3 −0.587785 − 0.809017i −0.947790 + 1.86014i −0.309017 + 0.951057i 0.0756605 + 0.477702i 2.06198 − 0.326586i −0.00745527 − 0.0146318i 0.951057 − 0.309017i −0.798467 − 1.09900i 0.341997 − 0.341997i
47.4 −0.587785 − 0.809017i −0.197352 + 0.387325i −0.309017 + 0.951057i 0.152558 + 0.963212i 0.429353 − 0.0680029i −1.75945 − 3.45311i 0.951057 − 0.309017i 1.65228 + 2.27417i 0.689584 − 0.689584i
47.5 −0.587785 − 0.809017i 0.0669364 − 0.131370i −0.309017 + 0.951057i −0.306165 − 1.93305i −0.145625 + 0.0230647i 1.30353 + 2.55833i 0.951057 − 0.309017i 1.75058 + 2.40946i −1.38391 + 1.38391i
47.6 −0.587785 − 0.809017i 0.769378 − 1.50999i −0.309017 + 0.951057i −0.324210 − 2.04698i −1.67384 + 0.265109i −0.570816 − 1.12029i 0.951057 − 0.309017i 0.0752325 + 0.103549i −1.46547 + 1.46547i
47.7 −0.587785 − 0.809017i 1.25645 − 2.46591i −0.309017 + 0.951057i 0.464706 + 2.93404i −2.73349 + 0.432942i 1.67728 + 3.29185i 0.951057 − 0.309017i −2.73872 − 3.76952i 2.10054 − 2.10054i
47.8 −0.587785 − 0.809017i 1.45222 − 2.85014i −0.309017 + 0.951057i −0.0563617 − 0.355854i −3.15940 + 0.500400i −1.06221 − 2.08471i 0.951057 − 0.309017i −4.25099 − 5.85099i −0.254763 + 0.254763i
81.1 0.587785 − 0.809017i −2.85014 + 1.45222i −0.309017 − 0.951057i −0.355854 − 0.0563617i −0.500400 + 3.15940i 2.08471 + 1.06221i −0.951057 − 0.309017i 4.25099 − 5.85099i −0.254763 + 0.254763i
81.2 0.587785 − 0.809017i −2.46591 + 1.25645i −0.309017 − 0.951057i 2.93404 + 0.464706i −0.432942 + 2.73349i −3.29185 − 1.67728i −0.951057 − 0.309017i 2.73872 − 3.76952i 2.10054 − 2.10054i
81.3 0.587785 − 0.809017i −1.50999 + 0.769378i −0.309017 − 0.951057i −2.04698 − 0.324210i −0.265109 + 1.67384i 1.12029 + 0.570816i −0.951057 − 0.309017i −0.0752325 + 0.103549i −1.46547 + 1.46547i
81.4 0.587785 − 0.809017i −0.131370 + 0.0669364i −0.309017 − 0.951057i −1.93305 − 0.306165i −0.0230647 + 0.145625i −2.55833 − 1.30353i −0.951057 − 0.309017i −1.75058 + 2.40946i −1.38391 + 1.38391i
81.5 0.587785 − 0.809017i 0.387325 − 0.197352i −0.309017 − 0.951057i 0.963212 + 0.152558i 0.0680029 − 0.429353i 3.45311 + 1.75945i −0.951057 − 0.309017i −1.65228 + 2.27417i 0.689584 − 0.689584i
81.6 0.587785 − 0.809017i 1.86014 − 0.947790i −0.309017 − 0.951057i 0.477702 + 0.0756605i 0.326586 − 2.06198i 0.0146318 + 0.00745527i −0.951057 − 0.309017i 0.798467 − 1.09900i 0.341997 − 0.341997i
81.7 0.587785 − 0.809017i 1.95449 − 0.995860i −0.309017 − 0.951057i 2.83069 + 0.448337i 0.343150 − 2.16656i −1.39766 − 0.712141i −0.951057 − 0.309017i 1.06492 − 1.46574i 2.02655 − 2.02655i
81.8 0.587785 − 0.809017i 2.75546 − 1.40398i −0.309017 − 0.951057i −3.73303 − 0.591255i 0.483777 − 3.05445i −2.24251 − 1.14262i −0.951057 − 0.309017i 3.85804 − 5.31014i −2.67256 + 2.67256i
115.1 −0.951057 + 0.309017i −0.462064 − 2.91736i 0.809017 − 0.587785i −0.864653 + 0.440563i 1.34096 + 2.63179i −0.385216 + 2.43216i −0.587785 + 0.809017i −5.44431 + 1.76896i 0.686192 − 0.686192i
115.2 −0.951057 + 0.309017i −0.309426 − 1.95364i 0.809017 − 0.587785i −0.0286893 + 0.0146179i 0.897989 + 1.76240i 0.347561 − 2.19441i −0.587785 + 0.809017i −0.867792 + 0.281963i 0.0227679 − 0.0227679i
115.3 −0.951057 + 0.309017i −0.174229 − 1.10004i 0.809017 − 0.587785i 1.93936 − 0.988152i 0.505631 + 0.992357i −0.591632 + 3.73542i −0.587785 + 0.809017i 1.67345 − 0.543735i −1.53908 + 1.53908i
115.4 −0.951057 + 0.309017i −0.0634921 − 0.400874i 0.809017 − 0.587785i −3.06569 + 1.56205i 0.184261 + 0.361633i 0.391331 − 2.47077i −0.587785 + 0.809017i 2.69650 − 0.876146i 2.43294 − 2.43294i
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner
17.c even 4 1 inner
187.p even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 374.2.o.a ✓ 64
11.c even 5 1 inner 374.2.o.a ✓ 64
17.c even 4 1 inner 374.2.o.a ✓ 64
187.p even 20 1 inner 374.2.o.a ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
374.2.o.a ✓ 64 1.a even 1 1 trivial
374.2.o.a ✓ 64 11.c even 5 1 inner
374.2.o.a ✓ 64 17.c even 4 1 inner
374.2.o.a ✓ 64 187.p even 20 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{64} + 10 T_{3}^{61} - 26 T_{3}^{60} - 4 T_{3}^{59} + 50 T_{3}^{58} - 954 T_{3}^{57} + \cdots + 577200625 \) acting on \(S_{2}^{\mathrm{new}}(374, [\chi])\). Copy content Toggle raw display