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Properties

Label 155.2.h.b
Level $155$
Weight $2$
Character orbit 155.h
Analytic conductor $1.238$
Analytic rank $0$
Dimension $24$
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] = [155,2,Mod(16,155)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(155, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("155.16");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 155 = 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 155.h (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: \(1.23768123133\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(6\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 2 q^{2} - 6 q^{4} - 24 q^{5} - 16 q^{6} - 9 q^{7} + 11 q^{8} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 2 q^{2} - 6 q^{4} - 24 q^{5} - 16 q^{6} - 9 q^{7} + 11 q^{8} - 6 q^{9} - 2 q^{10} + q^{11} + 18 q^{12} - 5 q^{13} + 6 q^{14} - 24 q^{16} + 7 q^{17} - 18 q^{18} + 2 q^{19} + 6 q^{20} - 10 q^{21} + 28 q^{22} - 15 q^{23} - 32 q^{24} + 24 q^{25} + 22 q^{26} + 9 q^{27} + 38 q^{28} + 15 q^{29} + 16 q^{30} + 6 q^{31} + 74 q^{32} + 5 q^{33} - 20 q^{34} + 9 q^{35} - 58 q^{36} - 56 q^{37} - 21 q^{38} - 10 q^{39} - 11 q^{40} - 24 q^{41} - 38 q^{42} + 21 q^{43} + 41 q^{44} + 6 q^{45} + 48 q^{46} - 8 q^{47} - 26 q^{48} - 23 q^{49} + 2 q^{50} + 26 q^{51} - 27 q^{52} + 26 q^{53} + 11 q^{54} - q^{55} - 48 q^{56} + 62 q^{57} + 52 q^{58} + 10 q^{59} - 18 q^{60} - 40 q^{61} - 28 q^{62} + 26 q^{63} + 9 q^{64} + 5 q^{65} - 2 q^{66} - 26 q^{67} - 8 q^{68} + 64 q^{69} - 6 q^{70} - 7 q^{71} - 127 q^{72} - 51 q^{73} - q^{74} + 43 q^{76} - 39 q^{77} - 31 q^{78} + 31 q^{79} + 24 q^{80} + 34 q^{81} + 58 q^{82} + 6 q^{83} + 113 q^{84} - 7 q^{85} - 22 q^{86} + 4 q^{87} - 28 q^{88} + 13 q^{89} + 18 q^{90} + 54 q^{91} - 2 q^{92} - 72 q^{93} - 10 q^{94} - 2 q^{95} + 101 q^{96} - 39 q^{97} - 220 q^{98} - 170 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1 −2.21395 + 1.60853i 2.16266 + 1.57127i 1.69618 − 5.22031i −1.00000 −7.31546 −0.659740 + 2.03047i 2.95046 + 9.08058i 1.28118 + 3.94306i 2.21395 − 1.60853i
16.2 −1.56101 + 1.13414i −0.836830 − 0.607992i 0.532443 − 1.63869i −1.00000 1.99585 −0.394404 + 1.21385i −0.165149 − 0.508276i −0.596422 − 1.83560i 1.56101 − 1.13414i
16.3 0.246271 − 0.178926i −2.37586 − 1.72617i −0.589399 + 1.81398i −1.00000 −0.893963 −0.911543 + 2.80544i 0.367552 + 1.13121i 1.73803 + 5.34911i −0.246271 + 0.178926i
16.4 0.426161 − 0.309624i 1.65764 + 1.20435i −0.532288 + 1.63821i −1.00000 1.07932 0.222489 − 0.684749i 0.605948 + 1.86492i 0.370272 + 1.13958i −0.426161 + 0.309624i
16.5 1.60193 − 1.16387i −1.24021 − 0.901066i 0.593549 − 1.82676i −1.00000 −3.03545 1.38232 − 4.25435i 0.0484821 + 0.149212i −0.200848 − 0.618148i −1.60193 + 1.16387i
16.6 2.00060 − 1.45352i 0.632602 + 0.459612i 1.27165 − 3.91373i −1.00000 1.93364 −1.33011 + 4.09364i −1.61631 − 4.97449i −0.738109 − 2.27167i −2.00060 + 1.45352i
66.1 −0.766659 − 2.35953i −0.478948 + 1.47405i −3.36160 + 2.44235i −1.00000 3.84526 −3.20132 + 2.32590i 4.32573 + 3.14282i 0.483619 + 0.351369i 0.766659 + 2.35953i
66.2 −0.515618 − 1.58691i 0.532605 − 1.63919i −0.634386 + 0.460908i −1.00000 −2.87586 1.56056 − 1.13381i −1.64129 − 1.19247i 0.0237792 + 0.0172766i 0.515618 + 1.58691i
66.3 0.199171 + 0.612984i 0.903105 − 2.77947i 1.28195 − 0.931393i −1.00000 1.88364 −2.42006 + 1.75828i 1.86913 + 1.35800i −4.48281 − 3.25695i −0.199171 − 0.612984i
66.4 0.352241 + 1.08409i 0.0978947 − 0.301289i 0.566864 − 0.411851i −1.00000 0.361106 2.11961 − 1.53999i 2.49051 + 1.80946i 2.34586 + 1.70437i −0.352241 − 1.08409i
66.5 0.420591 + 1.29445i −0.973089 + 2.99486i 0.119342 − 0.0867070i −1.00000 −4.28595 −0.741309 + 0.538593i 2.36467 + 1.71804i −5.59523 − 4.06517i −0.420591 − 1.29445i
66.6 0.810275 + 2.49377i −0.0815674 + 0.251039i −3.94431 + 2.86571i −1.00000 −0.692125 −0.126491 + 0.0919008i −6.09973 − 4.43171i 2.37068 + 1.72240i −0.810275 − 2.49377i
101.1 −0.766659 + 2.35953i −0.478948 − 1.47405i −3.36160 − 2.44235i −1.00000 3.84526 −3.20132 − 2.32590i 4.32573 − 3.14282i 0.483619 − 0.351369i 0.766659 − 2.35953i
101.2 −0.515618 + 1.58691i 0.532605 + 1.63919i −0.634386 − 0.460908i −1.00000 −2.87586 1.56056 + 1.13381i −1.64129 + 1.19247i 0.0237792 − 0.0172766i 0.515618 − 1.58691i
101.3 0.199171 − 0.612984i 0.903105 + 2.77947i 1.28195 + 0.931393i −1.00000 1.88364 −2.42006 − 1.75828i 1.86913 − 1.35800i −4.48281 + 3.25695i −0.199171 + 0.612984i
101.4 0.352241 − 1.08409i 0.0978947 + 0.301289i 0.566864 + 0.411851i −1.00000 0.361106 2.11961 + 1.53999i 2.49051 − 1.80946i 2.34586 − 1.70437i −0.352241 + 1.08409i
101.5 0.420591 − 1.29445i −0.973089 − 2.99486i 0.119342 + 0.0867070i −1.00000 −4.28595 −0.741309 − 0.538593i 2.36467 − 1.71804i −5.59523 + 4.06517i −0.420591 + 1.29445i
101.6 0.810275 − 2.49377i −0.0815674 − 0.251039i −3.94431 − 2.86571i −1.00000 −0.692125 −0.126491 − 0.0919008i −6.09973 + 4.43171i 2.37068 − 1.72240i −0.810275 + 2.49377i
126.1 −2.21395 − 1.60853i 2.16266 − 1.57127i 1.69618 + 5.22031i −1.00000 −7.31546 −0.659740 − 2.03047i 2.95046 − 9.08058i 1.28118 − 3.94306i 2.21395 + 1.60853i
126.2 −1.56101 − 1.13414i −0.836830 + 0.607992i 0.532443 + 1.63869i −1.00000 1.99585 −0.394404 − 1.21385i −0.165149 + 0.508276i −0.596422 + 1.83560i 1.56101 + 1.13414i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 16.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 155.2.h.b ✓ 24
5.b even 2 1 775.2.k.d 24
5.c odd 4 2 775.2.bf.c 48
31.d even 5 1 inner 155.2.h.b ✓ 24
31.d even 5 1 4805.2.a.u 12
31.f odd 10 1 4805.2.a.v 12
155.n even 10 1 775.2.k.d 24
155.s odd 20 2 775.2.bf.c 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
155.2.h.b ✓ 24 1.a even 1 1 trivial
155.2.h.b ✓ 24 31.d even 5 1 inner
775.2.k.d 24 5.b even 2 1
775.2.k.d 24 155.n even 10 1
775.2.bf.c 48 5.c odd 4 2
775.2.bf.c 48 155.s odd 20 2
4805.2.a.u 12 31.d even 5 1
4805.2.a.v 12 31.f odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} - 2 T_{2}^{23} + 11 T_{2}^{22} - 19 T_{2}^{21} + 93 T_{2}^{20} - 191 T_{2}^{19} + 791 T_{2}^{18} + \cdots + 2025 \) acting on \(S_{2}^{\mathrm{new}}(155, [\chi])\). Copy content Toggle raw display