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Properties

Label 775.2.df.b
Level $775$
Weight $2$
Character orbit 775.df
Analytic conductor $6.188$
Analytic rank $0$
Dimension $224$
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] = [775,2,Mod(43,775)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(775, base_ring=CyclotomicField(60))
 
chi = DirichletCharacter(H, H._module([45, 38]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("775.43");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 775 = 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 775.df (of order \(60\), degree \(16\), 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: \(6.18840615665\)
Analytic rank: \(0\)
Dimension: \(224\)
Relative dimension: \(14\) over \(\Q(\zeta_{60})\)
Twist minimal: no (minimal twist has level 155)
Sato-Tate group: $\mathrm{SU}(2)[C_{60}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 224 q + 12 q^{2} + 14 q^{3} - 36 q^{6} - 6 q^{7} + 40 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 224 q + 12 q^{2} + 14 q^{3} - 36 q^{6} - 6 q^{7} + 40 q^{8} - 28 q^{11} + 40 q^{12} + 14 q^{13} + 24 q^{16} + 14 q^{17} + 40 q^{18} - 44 q^{21} - 30 q^{22} + 30 q^{23} - 48 q^{26} - 100 q^{27} + 48 q^{28} - 44 q^{31} - 28 q^{32} - 20 q^{33} + 16 q^{36} - 30 q^{37} - 38 q^{38} - 20 q^{41} + 130 q^{42} - 22 q^{43} + 28 q^{47} + 156 q^{48} - 36 q^{51} + 10 q^{52} - 26 q^{53} - 48 q^{56} + 20 q^{58} - 30 q^{62} - 152 q^{63} - 152 q^{66} - 38 q^{67} + 126 q^{68} - 128 q^{71} + 156 q^{72} + 30 q^{73} + 128 q^{76} - 30 q^{78} - 64 q^{81} - 8 q^{82} + 82 q^{83} - 28 q^{86} + 24 q^{87} - 252 q^{88} - 194 q^{93} + 60 q^{96} + 38 q^{97} - 174 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1 −2.49026 + 0.394419i 0.213619 − 0.0820006i 4.14372 − 1.34638i 0 −0.499624 + 0.288458i −0.0529094 − 1.00957i −5.29491 + 2.69789i −2.19053 + 1.97236i 0
43.2 −2.32070 + 0.367563i 2.10158 − 0.806721i 3.34844 − 1.08797i 0 −4.58062 + 2.64462i 0.0583537 + 1.11346i −3.18375 + 1.62220i 1.53640 − 1.38338i 0
43.3 −1.60231 + 0.253782i −1.85797 + 0.713206i 0.600894 − 0.195242i 0 2.79605 − 1.61430i 0.210014 + 4.00730i 1.97767 − 1.00767i 0.713938 − 0.642832i 0
43.4 −1.45779 + 0.230891i −1.01398 + 0.389230i 0.169727 − 0.0551478i 0 1.38830 − 0.801535i −0.0319091 − 0.608862i 2.39549 − 1.22056i −1.35278 + 1.21805i 0
43.5 −0.953273 + 0.150984i 1.76032 − 0.675724i −1.01618 + 0.330177i 0 −1.57604 + 0.909928i −0.0750900 − 1.43280i 2.63876 − 1.34452i 0.412691 − 0.371589i 0
43.6 −0.183400 + 0.0290477i −2.31655 + 0.889239i −1.86932 + 0.607379i 0 0.399024 − 0.230376i −0.0762747 − 1.45541i 0.656085 − 0.334292i 2.34621 − 2.11253i 0
43.7 0.121149 − 0.0191881i 0.312498 − 0.119957i −1.88780 + 0.613385i 0 0.0355570 − 0.0205289i −0.183445 − 3.50033i −0.435516 + 0.221907i −2.14617 + 1.93242i 0
43.8 0.574598 − 0.0910074i 2.98200 − 1.14468i −1.58023 + 0.513449i 0 1.60928 − 0.929116i −0.0473284 − 0.903079i −1.89798 + 0.967067i 5.35258 − 4.81948i 0
43.9 0.992172 − 0.157145i −2.87341 + 1.10300i −0.942403 + 0.306205i 0 −2.67759 + 1.54591i 0.141383 + 2.69775i −2.67701 + 1.36400i 4.81045 − 4.33135i 0
43.10 1.09991 − 0.174208i 0.311667 − 0.119638i −0.722666 + 0.234808i 0 0.321963 − 0.185885i 0.00304009 + 0.0580084i −2.73844 + 1.39531i −2.14661 + 1.93282i 0
43.11 1.41300 − 0.223797i 0.590210 − 0.226560i 0.0443669 − 0.0144157i 0 0.783262 − 0.452217i 0.220513 + 4.20765i −2.48990 + 1.26867i −1.93242 + 1.73996i 0
43.12 2.03530 − 0.322360i −1.93556 + 0.742990i 2.13642 − 0.694166i 0 −3.69993 + 2.13615i −0.205418 − 3.91961i 0.452349 − 0.230483i 0.964905 − 0.868805i 0
43.13 2.35109 − 0.372377i 1.93777 − 0.743841i 3.48687 − 1.13295i 0 4.27890 − 2.47042i 0.0525542 + 1.00279i 3.53416 − 1.80075i 0.972230 − 0.875400i 0
43.14 2.68059 − 0.424564i −1.64952 + 0.633191i 5.10320 − 1.65813i 0 −4.15286 + 2.39765i 0.177053 + 3.37838i 8.13923 − 4.14714i 0.0905470 − 0.0815288i 0
168.1 −0.394419 + 2.49026i −0.0820006 + 0.213619i −4.14372 − 1.34638i 0 −0.499624 − 0.288458i 1.00957 + 0.0529094i 2.69789 − 5.29491i 2.19053 + 1.97236i 0
168.2 −0.367563 + 2.32070i −0.806721 + 2.10158i −3.34844 − 1.08797i 0 −4.58062 − 2.64462i −1.11346 − 0.0583537i 1.62220 − 3.18375i −1.53640 − 1.38338i 0
168.3 −0.253782 + 1.60231i 0.713206 − 1.85797i −0.600894 − 0.195242i 0 2.79605 + 1.61430i −4.00730 − 0.210014i −1.00767 + 1.97767i −0.713938 − 0.642832i 0
168.4 −0.230891 + 1.45779i 0.389230 − 1.01398i −0.169727 − 0.0551478i 0 1.38830 + 0.801535i 0.608862 + 0.0319091i −1.22056 + 2.39549i 1.35278 + 1.21805i 0
168.5 −0.150984 + 0.953273i −0.675724 + 1.76032i 1.01618 + 0.330177i 0 −1.57604 − 0.909928i 1.43280 + 0.0750900i −1.34452 + 2.63876i −0.412691 − 0.371589i 0
168.6 −0.0290477 + 0.183400i 0.889239 − 2.31655i 1.86932 + 0.607379i 0 0.399024 + 0.230376i 1.45541 + 0.0762747i −0.334292 + 0.656085i −2.34621 − 2.11253i 0
See next 80 embeddings (of 224 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
31.h odd 30 1 inner
155.x even 60 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 775.2.df.b 224
5.b even 2 1 155.2.x.a ✓ 224
5.c odd 4 1 155.2.x.a ✓ 224
5.c odd 4 1 inner 775.2.df.b 224
31.h odd 30 1 inner 775.2.df.b 224
155.v odd 30 1 155.2.x.a ✓ 224
155.x even 60 1 155.2.x.a ✓ 224
155.x even 60 1 inner 775.2.df.b 224
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
155.2.x.a ✓ 224 5.b even 2 1
155.2.x.a ✓ 224 5.c odd 4 1
155.2.x.a ✓ 224 155.v odd 30 1
155.2.x.a ✓ 224 155.x even 60 1
775.2.df.b 224 1.a even 1 1 trivial
775.2.df.b 224 5.c odd 4 1 inner
775.2.df.b 224 31.h odd 30 1 inner
775.2.df.b 224 155.x even 60 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{224} - 12 T_{2}^{223} + 72 T_{2}^{222} - 312 T_{2}^{221} + 944 T_{2}^{220} + \cdots + 91048256370241 \) acting on \(S_{2}^{\mathrm{new}}(775, [\chi])\). Copy content Toggle raw display