↑

Properties

Label 775.2.k.d
Level $775$
Weight $2$
Character orbit 775.k
Analytic conductor $6.188$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [775,2,Mod(101,775)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(775, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("775.101");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 775 = 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 775.k (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: \(6.18840615665\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(6\) over \(\Q(\zeta_{5})\)
Twist minimal: no (minimal twist has level 155)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 2 q^{2} - 6 q^{4} - 16 q^{6} + 9 q^{7} - 11 q^{8} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 2 q^{2} - 6 q^{4} - 16 q^{6} + 9 q^{7} - 11 q^{8} - 6 q^{9} + q^{11} - 18 q^{12} + 5 q^{13} + 6 q^{14} - 24 q^{16} - 7 q^{17} + 18 q^{18} + 2 q^{19} - 10 q^{21} - 28 q^{22} + 15 q^{23} - 32 q^{24} + 22 q^{26} - 9 q^{27} - 38 q^{28} + 15 q^{29} + 6 q^{31} - 74 q^{32} - 5 q^{33} - 20 q^{34} - 58 q^{36} + 56 q^{37} + 21 q^{38} - 10 q^{39} - 24 q^{41} + 38 q^{42} - 21 q^{43} + 41 q^{44} + 48 q^{46} + 8 q^{47} + 26 q^{48} - 23 q^{49} + 26 q^{51} + 27 q^{52} - 26 q^{53} + 11 q^{54} - 48 q^{56} - 62 q^{57} - 52 q^{58} + 10 q^{59} - 40 q^{61} + 28 q^{62} - 26 q^{63} + 9 q^{64} - 2 q^{66} + 26 q^{67} + 8 q^{68} + 64 q^{69} - 7 q^{71} + 127 q^{72} + 51 q^{73} - q^{74} + 43 q^{76} + 39 q^{77} + 31 q^{78} + 31 q^{79} + 34 q^{81} - 58 q^{82} - 6 q^{83} + 113 q^{84} - 22 q^{86} - 4 q^{87} + 28 q^{88} + 13 q^{89} + 54 q^{91} + 2 q^{92} + 72 q^{93} - 10 q^{94} + 101 q^{96} + 39 q^{97} + 220 q^{98} - 170 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1 −0.810275 + 2.49377i 0.0815674 + 0.251039i −3.94431 − 2.86571i 0 −0.692125 0.126491 + 0.0919008i 6.09973 − 4.43171i 2.37068 − 1.72240i 0
101.2 −0.420591 + 1.29445i 0.973089 + 2.99486i 0.119342 + 0.0867070i 0 −4.28595 0.741309 + 0.538593i −2.36467 + 1.71804i −5.59523 + 4.06517i 0
101.3 −0.352241 + 1.08409i −0.0978947 − 0.301289i 0.566864 + 0.411851i 0 0.361106 −2.11961 − 1.53999i −2.49051 + 1.80946i 2.34586 − 1.70437i 0
101.4 −0.199171 + 0.612984i −0.903105 − 2.77947i 1.28195 + 0.931393i 0 1.88364 2.42006 + 1.75828i −1.86913 + 1.35800i −4.48281 + 3.25695i 0
101.5 0.515618 − 1.58691i −0.532605 − 1.63919i −0.634386 − 0.460908i 0 −2.87586 −1.56056 − 1.13381i 1.64129 − 1.19247i 0.0237792 − 0.0172766i 0
101.6 0.766659 − 2.35953i 0.478948 + 1.47405i −3.36160 − 2.44235i 0 3.84526 3.20132 + 2.32590i −4.32573 + 3.14282i 0.483619 − 0.351369i 0
126.1 −2.00060 − 1.45352i −0.632602 + 0.459612i 1.27165 + 3.91373i 0 1.93364 1.33011 + 4.09364i 1.61631 − 4.97449i −0.738109 + 2.27167i 0
126.2 −1.60193 − 1.16387i 1.24021 − 0.901066i 0.593549 + 1.82676i 0 −3.03545 −1.38232 − 4.25435i −0.0484821 + 0.149212i −0.200848 + 0.618148i 0
126.3 −0.426161 − 0.309624i −1.65764 + 1.20435i −0.532288 − 1.63821i 0 1.07932 −0.222489 − 0.684749i −0.605948 + 1.86492i 0.370272 − 1.13958i 0
126.4 −0.246271 − 0.178926i 2.37586 − 1.72617i −0.589399 − 1.81398i 0 −0.893963 0.911543 + 2.80544i −0.367552 + 1.13121i 1.73803 − 5.34911i 0
126.5 1.56101 + 1.13414i 0.836830 − 0.607992i 0.532443 + 1.63869i 0 1.99585 0.394404 + 1.21385i 0.165149 − 0.508276i −0.596422 + 1.83560i 0
126.6 2.21395 + 1.60853i −2.16266 + 1.57127i 1.69618 + 5.22031i 0 −7.31546 0.659740 + 2.03047i −2.95046 + 9.08058i 1.28118 − 3.94306i 0
326.1 −2.00060 + 1.45352i −0.632602 − 0.459612i 1.27165 − 3.91373i 0 1.93364 1.33011 − 4.09364i 1.61631 + 4.97449i −0.738109 − 2.27167i 0
326.2 −1.60193 + 1.16387i 1.24021 + 0.901066i 0.593549 − 1.82676i 0 −3.03545 −1.38232 + 4.25435i −0.0484821 − 0.149212i −0.200848 − 0.618148i 0
326.3 −0.426161 + 0.309624i −1.65764 − 1.20435i −0.532288 + 1.63821i 0 1.07932 −0.222489 + 0.684749i −0.605948 − 1.86492i 0.370272 + 1.13958i 0
326.4 −0.246271 + 0.178926i 2.37586 + 1.72617i −0.589399 + 1.81398i 0 −0.893963 0.911543 − 2.80544i −0.367552 − 1.13121i 1.73803 + 5.34911i 0
326.5 1.56101 − 1.13414i 0.836830 + 0.607992i 0.532443 − 1.63869i 0 1.99585 0.394404 − 1.21385i 0.165149 + 0.508276i −0.596422 − 1.83560i 0
326.6 2.21395 − 1.60853i −2.16266 − 1.57127i 1.69618 − 5.22031i 0 −7.31546 0.659740 − 2.03047i −2.95046 − 9.08058i 1.28118 + 3.94306i 0
376.1 −0.810275 − 2.49377i 0.0815674 − 0.251039i −3.94431 + 2.86571i 0 −0.692125 0.126491 − 0.0919008i 6.09973 + 4.43171i 2.37068 + 1.72240i 0
376.2 −0.420591 − 1.29445i 0.973089 − 2.99486i 0.119342 − 0.0867070i 0 −4.28595 0.741309 − 0.538593i −2.36467 − 1.71804i −5.59523 − 4.06517i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 101.6
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 775.2.k.d 24
5.b even 2 1 155.2.h.b ✓ 24
5.c odd 4 2 775.2.bf.c 48
31.d even 5 1 inner 775.2.k.d 24
155.m odd 10 1 4805.2.a.v 12
155.n even 10 1 155.2.h.b ✓ 24
155.n even 10 1 4805.2.a.u 12
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 5.b even 2 1
155.2.h.b ✓ 24 155.n even 10 1
775.2.k.d 24 1.a even 1 1 trivial
775.2.k.d 24 31.d even 5 1 inner
775.2.bf.c 48 5.c odd 4 2
775.2.bf.c 48 155.s odd 20 2
4805.2.a.u 12 155.n even 10 1
4805.2.a.v 12 155.m 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 \(S_{2}^{\mathrm{new}}(775, [\chi])\). Copy content Toggle raw display