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Properties

Label 775.2.bl.e
Level $775$
Weight $2$
Character orbit 775.bl
Analytic conductor $6.188$
Analytic rank $0$
Dimension $88$
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] = [775,2,Mod(51,775)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(775, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([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("775.51");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 775 = 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 775.bl (of order \(15\), 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: \(6.18840615665\)
Analytic rank: \(0\)
Dimension: \(88\)
Relative dimension: \(11\) over \(\Q(\zeta_{15})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 88 q + 3 q^{3} - 20 q^{4} + q^{6} - 2 q^{7} + 9 q^{8} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 88 q + 3 q^{3} - 20 q^{4} + q^{6} - 2 q^{7} + 9 q^{8} + 12 q^{9} + 6 q^{11} - 10 q^{12} + 16 q^{13} + 20 q^{14} - 28 q^{16} + 12 q^{17} - 5 q^{18} - 26 q^{19} + 10 q^{21} + 6 q^{22} - 28 q^{23} + 46 q^{24} - 2 q^{26} - 54 q^{27} - 54 q^{28} + 8 q^{29} - 12 q^{31} + 28 q^{32} - 11 q^{33} + 31 q^{34} - 69 q^{36} - 27 q^{37} - 20 q^{38} - 21 q^{39} - 18 q^{41} + 66 q^{42} + 17 q^{43} + 18 q^{44} - 32 q^{46} - 24 q^{47} - 7 q^{48} - 25 q^{49} + 22 q^{51} + 18 q^{52} + 17 q^{53} - 56 q^{54} + 53 q^{56} - 62 q^{57} - 80 q^{58} - 24 q^{59} + 42 q^{61} - 102 q^{62} - 42 q^{63} - 77 q^{64} - 58 q^{66} + 3 q^{67} - 143 q^{68} + 14 q^{69} - 38 q^{71} - 134 q^{72} - 11 q^{73} - 57 q^{74} - 65 q^{76} + 56 q^{77} - 60 q^{78} - 45 q^{79} + 39 q^{81} + 110 q^{82} + 57 q^{83} - 71 q^{84} - 91 q^{86} - 13 q^{87} - 96 q^{88} - 34 q^{91} + 238 q^{92} - 8 q^{93} - 30 q^{94} - 130 q^{96} + 64 q^{97} + 102 q^{98} - 47 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
51.1 −0.815637 − 2.51027i −0.0352620 − 0.0391624i −4.01817 + 2.91937i 0 −0.0695474 + 0.120460i −0.206303 − 1.96284i 6.33506 + 4.60269i 0.313295 − 2.98080i 0
51.2 −0.670122 − 2.06242i 2.21494 + 2.45994i −2.18650 + 1.58858i 0 3.58917 − 6.21662i 0.321728 + 3.06104i 1.23275 + 0.895646i −0.831765 + 7.91371i 0
51.3 −0.542143 − 1.66854i −0.426864 − 0.474080i −0.872089 + 0.633610i 0 −0.559603 + 0.969261i 0.389029 + 3.70137i −1.30870 − 0.950823i 0.271046 − 2.57883i 0
51.4 −0.315467 − 0.970908i 0.470915 + 0.523004i 0.774891 − 0.562991i 0 0.359231 − 0.622206i −0.224881 − 2.13960i −2.44287 − 1.77485i 0.261813 − 2.49098i 0
51.5 −0.0170192 − 0.0523797i 1.21604 + 1.35055i 1.61558 − 1.17379i 0 0.0500455 − 0.0866813i 0.227901 + 2.16834i −0.178092 − 0.129392i −0.0316464 + 0.301095i 0
51.6 0.0136821 + 0.0421092i −1.46288 − 1.62470i 1.61645 − 1.17442i 0 0.0483994 − 0.0838302i −0.378239 − 3.59871i 0.143211 + 0.104049i −0.186026 + 1.76992i 0
51.7 0.111255 + 0.342407i −1.77458 − 1.97087i 1.51317 − 1.09938i 0 0.477408 − 0.826895i 0.492083 + 4.68185i 1.12732 + 0.819045i −0.421610 + 4.01135i 0
51.8 0.390313 + 1.20126i −0.0873606 − 0.0970238i 0.327350 − 0.237834i 0 0.0824529 − 0.142813i −0.175285 − 1.66773i 2.45718 + 1.78524i 0.311804 − 2.96661i 0
51.9 0.406696 + 1.25168i 2.10309 + 2.33571i 0.216726 − 0.157461i 0 −2.06825 + 3.58232i −0.446441 − 4.24760i 2.41472 + 1.75440i −0.719001 + 6.84084i 0
51.10 0.710118 + 2.18552i −1.44998 − 1.61037i −2.65419 + 1.92838i 0 2.48983 − 4.31251i −0.00344974 − 0.0328221i −2.38107 − 1.72995i −0.177253 + 1.68645i 0
51.11 0.728325 + 2.24155i 1.23933 + 1.37642i −2.87607 + 2.08958i 0 −2.18268 + 3.78052i 0.412834 + 3.92786i −2.96507 − 2.15425i −0.0449980 + 0.428127i 0
76.1 −0.815637 + 2.51027i −0.0352620 + 0.0391624i −4.01817 − 2.91937i 0 −0.0695474 − 0.120460i −0.206303 + 1.96284i 6.33506 − 4.60269i 0.313295 + 2.98080i 0
76.2 −0.670122 + 2.06242i 2.21494 − 2.45994i −2.18650 − 1.58858i 0 3.58917 + 6.21662i 0.321728 − 3.06104i 1.23275 − 0.895646i −0.831765 − 7.91371i 0
76.3 −0.542143 + 1.66854i −0.426864 + 0.474080i −0.872089 − 0.633610i 0 −0.559603 − 0.969261i 0.389029 − 3.70137i −1.30870 + 0.950823i 0.271046 + 2.57883i 0
76.4 −0.315467 + 0.970908i 0.470915 − 0.523004i 0.774891 + 0.562991i 0 0.359231 + 0.622206i −0.224881 + 2.13960i −2.44287 + 1.77485i 0.261813 + 2.49098i 0
76.5 −0.0170192 + 0.0523797i 1.21604 − 1.35055i 1.61558 + 1.17379i 0 0.0500455 + 0.0866813i 0.227901 − 2.16834i −0.178092 + 0.129392i −0.0316464 − 0.301095i 0
76.6 0.0136821 − 0.0421092i −1.46288 + 1.62470i 1.61645 + 1.17442i 0 0.0483994 + 0.0838302i −0.378239 + 3.59871i 0.143211 − 0.104049i −0.186026 − 1.76992i 0
76.7 0.111255 − 0.342407i −1.77458 + 1.97087i 1.51317 + 1.09938i 0 0.477408 + 0.826895i 0.492083 − 4.68185i 1.12732 − 0.819045i −0.421610 − 4.01135i 0
76.8 0.390313 − 1.20126i −0.0873606 + 0.0970238i 0.327350 + 0.237834i 0 0.0824529 + 0.142813i −0.175285 + 1.66773i 2.45718 − 1.78524i 0.311804 + 2.96661i 0
76.9 0.406696 − 1.25168i 2.10309 − 2.33571i 0.216726 + 0.157461i 0 −2.06825 − 3.58232i −0.446441 + 4.24760i 2.41472 − 1.75440i −0.719001 − 6.84084i 0
See all 88 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 51.11
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.g even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 775.2.bl.e yes 88
5.b even 2 1 775.2.bl.d ✓ 88
5.c odd 4 2 775.2.ck.d 176
31.g even 15 1 inner 775.2.bl.e yes 88
155.u even 30 1 775.2.bl.d ✓ 88
155.w odd 60 2 775.2.ck.d 176
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
775.2.bl.d ✓ 88 5.b even 2 1
775.2.bl.d ✓ 88 155.u even 30 1
775.2.bl.e yes 88 1.a even 1 1 trivial
775.2.bl.e yes 88 31.g even 15 1 inner
775.2.ck.d 176 5.c odd 4 2
775.2.ck.d 176 155.w odd 60 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{88} + 32 T_{2}^{86} - 3 T_{2}^{85} + 608 T_{2}^{84} - 92 T_{2}^{83} + 8924 T_{2}^{82} + \cdots + 6561 \) acting on \(S_{2}^{\mathrm{new}}(775, [\chi])\). Copy content Toggle raw display