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Properties

Label 930.2.bo.b
Level $930$
Weight $2$
Character orbit 930.bo
Analytic conductor $7.426$
Analytic rank $0$
Dimension $256$
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] = [930,2,Mod(179,930)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(930, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([15, 15, 13]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("930.179");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 930.bo (of order \(30\), 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: \(7.42608738798\)
Analytic rank: \(0\)
Dimension: \(256\)
Relative dimension: \(32\) over \(\Q(\zeta_{30})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 256 q + 64 q^{2} - 64 q^{4} + 2 q^{5} + 64 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 256 q + 64 q^{2} - 64 q^{4} + 2 q^{5} + 64 q^{8} + 4 q^{9} - 2 q^{10} - 10 q^{15} - 64 q^{16} + 6 q^{17} + 6 q^{18} - 4 q^{19} - 3 q^{20} - 20 q^{23} - 2 q^{25} + 42 q^{31} - 256 q^{32} - 8 q^{33} + 14 q^{34} + 16 q^{35} + 4 q^{36} - 36 q^{38} + 8 q^{39} + 3 q^{40} + 55 q^{45} - 10 q^{46} + 6 q^{47} - 40 q^{49} + 7 q^{50} + 68 q^{51} + 34 q^{53} + 6 q^{57} + 10 q^{60} - 2 q^{62} + 72 q^{63} - 64 q^{64} + 8 q^{66} + 6 q^{68} + 10 q^{69} - 16 q^{70} + 6 q^{72} - 80 q^{75} - 24 q^{76} - 100 q^{77} - 8 q^{78} + 40 q^{79} + 2 q^{80} + 12 q^{81} + 26 q^{83} - 30 q^{85} - 16 q^{87} - 25 q^{90} - 20 q^{91} + 22 q^{93} + 4 q^{94} - 56 q^{95} - 130 q^{98} + 102 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
179.1 −0.309017 − 0.951057i −1.72309 + 0.175914i −0.809017 + 0.587785i −2.22856 + 0.183115i 0.699770 + 1.58440i 0.318388 − 0.715111i 0.809017 + 0.587785i 2.93811 − 0.606234i 0.862815 + 2.06290i
179.2 −0.309017 − 0.951057i −1.70041 − 0.329548i −0.809017 + 0.587785i 1.50750 − 1.65150i 0.212037 + 1.71902i 0.732525 − 1.64528i 0.809017 + 0.587785i 2.78280 + 1.12073i −2.03651 − 0.923374i
179.3 −0.309017 − 0.951057i −1.69055 − 0.376882i −0.809017 + 0.587785i 1.53101 + 1.62973i 0.163972 + 1.72427i 0.325961 − 0.732121i 0.809017 + 0.587785i 2.71592 + 1.27428i 1.07685 − 1.95969i
179.4 −0.309017 − 0.951057i −1.56532 + 0.741477i −0.809017 + 0.587785i −0.493029 − 2.18104i 1.18890 + 1.25957i −0.863747 + 1.94001i 0.809017 + 0.587785i 1.90042 − 2.32129i −1.92193 + 1.14288i
179.5 −0.309017 − 0.951057i −1.51134 − 0.846071i −0.809017 + 0.587785i 1.97712 − 1.04451i −0.337630 + 1.69882i −1.83029 + 4.11091i 0.809017 + 0.587785i 1.56833 + 2.55741i −1.60435 − 1.55758i
179.6 −0.309017 − 0.951057i −1.41182 + 1.00338i −0.809017 + 0.587785i 1.92997 + 1.12925i 1.39055 + 1.03266i −0.934663 + 2.09929i 0.809017 + 0.587785i 0.986461 − 2.83318i 0.477584 − 2.18447i
179.7 −0.309017 − 0.951057i −1.32355 + 1.11724i −0.809017 + 0.587785i −0.575623 + 2.16071i 1.47156 + 0.913521i −1.66082 + 3.73026i 0.809017 + 0.587785i 0.503549 − 2.95744i 2.23283 − 0.120245i
179.8 −0.309017 − 0.951057i −1.30748 − 1.13599i −0.809017 + 0.587785i −0.719639 − 2.11710i −0.676360 + 1.59453i 1.13296 − 2.54466i 0.809017 + 0.587785i 0.419032 + 2.97059i −1.79110 + 1.33864i
179.9 −0.309017 − 0.951057i −1.28150 − 1.16523i −0.809017 + 0.587785i −1.79380 + 1.33503i −0.712193 + 1.57885i 0.308507 − 0.692917i 0.809017 + 0.587785i 0.284484 + 2.98648i 1.82400 + 1.29346i
179.10 −0.309017 − 0.951057i −1.18086 + 1.26711i −0.809017 + 0.587785i 2.09420 − 0.783790i 1.57000 + 0.731511i 1.79370 − 4.02872i 0.809017 + 0.587785i −0.211119 − 2.99256i −1.39257 − 1.74950i
179.11 −0.309017 − 0.951057i −0.955557 + 1.44461i −0.809017 + 0.587785i −1.43804 − 1.71232i 1.66919 + 0.462379i 0.429946 − 0.965674i 0.809017 + 0.587785i −1.17382 − 2.76082i −1.18414 + 1.89679i
179.12 −0.309017 − 0.951057i −0.624185 − 1.61567i −0.809017 + 0.587785i 2.09670 + 0.777090i −1.34371 + 1.09291i 1.84852 − 4.15185i 0.809017 + 0.587785i −2.22078 + 2.01696i 0.0911422 − 2.23421i
179.13 −0.309017 − 0.951057i −0.608421 + 1.62167i −0.809017 + 0.587785i −1.34040 + 1.78978i 1.73032 + 0.0775177i 1.34591 − 3.02296i 0.809017 + 0.587785i −2.25965 − 1.97332i 2.11639 + 0.721724i
179.14 −0.309017 − 0.951057i −0.518984 − 1.65247i −0.809017 + 0.587785i −2.18990 + 0.452039i −1.41122 + 1.00422i 0.478693 − 1.07516i 0.809017 + 0.587785i −2.46131 + 1.71521i 1.10663 + 1.94303i
179.15 −0.309017 − 0.951057i −0.198005 − 1.72070i −0.809017 + 0.587785i 0.703473 + 2.12253i −1.57529 + 0.720038i −0.478693 + 1.07516i 0.809017 + 0.587785i −2.92159 + 0.681412i 1.80126 − 1.32494i
179.16 −0.309017 − 0.951057i −0.145674 + 1.72591i −0.809017 + 0.587785i −2.23544 − 0.0527903i 1.68646 − 0.394792i −0.915394 + 2.05601i 0.809017 + 0.587785i −2.95756 − 0.502843i 0.640584 + 2.14235i
179.17 −0.309017 − 0.951057i −0.0869307 − 1.72987i −0.809017 + 0.587785i −1.72133 − 1.42725i −1.61834 + 0.617235i −1.84852 + 4.15185i 0.809017 + 0.587785i −2.98489 + 0.300757i −0.825472 + 2.07812i
179.18 −0.309017 − 0.951057i 0.128921 + 1.72725i −0.809017 + 0.587785i 2.12123 − 0.707392i 1.60287 − 0.656359i −1.37162 + 3.08072i 0.809017 + 0.587785i −2.96676 + 0.445355i −1.32826 − 1.79881i
179.19 −0.309017 − 0.951057i 0.584760 + 1.63035i −0.809017 + 0.587785i −0.447993 − 2.19073i 1.36986 − 1.05995i 1.37162 − 3.08072i 0.809017 + 0.587785i −2.31611 + 1.90673i −1.94507 + 1.10304i
179.20 −0.309017 − 0.951057i 0.696767 − 1.58572i −0.809017 + 0.587785i −0.259267 + 2.22099i −1.72342 − 0.172650i −0.308507 + 0.692917i 0.809017 + 0.587785i −2.02903 − 2.20976i 2.19240 − 0.439745i
See next 80 embeddings (of 256 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 179.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 inner
31.h odd 30 1 inner
465.bm even 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.2.bo.b yes 256
3.b odd 2 1 930.2.bo.a ✓ 256
5.b even 2 1 930.2.bo.a ✓ 256
15.d odd 2 1 inner 930.2.bo.b yes 256
31.h odd 30 1 inner 930.2.bo.b yes 256
93.p even 30 1 930.2.bo.a ✓ 256
155.v odd 30 1 930.2.bo.a ✓ 256
465.bm even 30 1 inner 930.2.bo.b yes 256
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.bo.a ✓ 256 3.b odd 2 1
930.2.bo.a ✓ 256 5.b even 2 1
930.2.bo.a ✓ 256 93.p even 30 1
930.2.bo.a ✓ 256 155.v odd 30 1
930.2.bo.b yes 256 1.a even 1 1 trivial
930.2.bo.b yes 256 15.d odd 2 1 inner
930.2.bo.b yes 256 31.h odd 30 1 inner
930.2.bo.b yes 256 465.bm even 30 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{17}^{128} - 3 T_{17}^{127} + 248 T_{17}^{126} - 420 T_{17}^{125} + 28137 T_{17}^{124} + \cdots + 12\!\cdots\!36 \) acting on \(S_{2}^{\mathrm{new}}(930, [\chi])\). Copy content Toggle raw display