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Properties

Label 475.2.l.e
Level $475$
Weight $2$
Character orbit 475.l
Analytic conductor $3.793$
Analytic rank $0$
Dimension $42$
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] = [475,2,Mod(101,475)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(475, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 14]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("475.101");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 475.l (of order \(9\), degree \(6\), 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: \(3.79289409601\)
Analytic rank: \(0\)
Dimension: \(42\)
Relative dimension: \(7\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 42 q + 3 q^{3} - 6 q^{4} - 3 q^{6} - 18 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 42 q + 3 q^{3} - 6 q^{4} - 3 q^{6} - 18 q^{8} + 9 q^{9} - 24 q^{12} + 27 q^{13} + 12 q^{14} + 18 q^{16} + 36 q^{18} + 12 q^{19} + 15 q^{21} - 12 q^{22} + 12 q^{23} + 33 q^{24} - 18 q^{26} - 30 q^{27} + 42 q^{28} - 3 q^{29} - 6 q^{31} + 60 q^{32} - 36 q^{33} - 42 q^{34} - 63 q^{36} + 48 q^{37} + 24 q^{38} - 60 q^{39} - 9 q^{41} + 165 q^{42} - 33 q^{44} - 15 q^{46} + 9 q^{47} - 84 q^{48} - 33 q^{49} - 36 q^{51} - 105 q^{52} - 6 q^{53} - 18 q^{54} + 36 q^{56} - 21 q^{57} - 24 q^{58} - 45 q^{59} + 9 q^{61} - 57 q^{62} - 51 q^{63} - 48 q^{66} - 27 q^{67} - 78 q^{68} + 15 q^{69} - 6 q^{71} + 24 q^{72} - 81 q^{73} + 72 q^{74} - 114 q^{76} + 18 q^{77} - 66 q^{78} + 27 q^{79} - 21 q^{81} - 9 q^{82} + 36 q^{84} - 60 q^{86} - 6 q^{87} - 60 q^{88} + 84 q^{89} - 33 q^{91} + 18 q^{92} - 21 q^{93} - 36 q^{94} + 222 q^{96} - 51 q^{97} + 180 q^{98} - 18 q^{99}+O(q^{100}) \) 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 −1.71612 + 1.44000i −1.18870 − 0.432650i 0.524184 − 2.97280i 0 2.66296 − 0.969238i −0.687421 + 1.19065i 1.14102 + 1.97630i −1.07232 − 0.899784i 0
101.2 −0.726194 + 0.609349i 2.21591 + 0.806526i −0.191245 + 1.08460i 0 −2.10064 + 0.764570i −1.22454 + 2.12097i −1.47000 − 2.54612i 1.96165 + 1.64602i 0
101.3 −0.473166 + 0.397033i 0.0218424 + 0.00794998i −0.281046 + 1.59389i 0 −0.0134915 + 0.00491050i 2.33559 − 4.04536i −1.11752 − 1.93560i −2.29772 − 1.92802i 0
101.4 0.0337662 − 0.0283332i −3.14687 − 1.14537i −0.346959 + 1.96770i 0 −0.138710 + 0.0504863i −0.741448 + 1.28423i 0.0881145 + 0.152619i 6.29282 + 5.28030i 0
101.5 0.976672 − 0.819525i −0.184051 − 0.0669891i −0.0650294 + 0.368800i 0 −0.234657 + 0.0854081i −0.699275 + 1.21118i 1.51368 + 2.62178i −2.26875 − 1.90370i 0
101.6 1.44827 − 1.21525i 2.45373 + 0.893086i 0.273377 − 1.55040i 0 4.63900 − 1.68846i 0.487743 − 0.844795i 0.402396 + 0.696970i 2.92507 + 2.45443i 0
101.7 1.98886 − 1.66885i −1.55125 − 0.564610i 0.823199 − 4.66860i 0 −4.02747 + 1.46588i −1.00274 + 1.73679i −3.55769 − 6.16210i −0.210533 − 0.176658i 0
176.1 −2.58100 + 0.939407i 0.510652 − 2.89605i 4.24698 − 3.56364i 0 1.40258 + 7.95442i −0.689697 − 1.19459i −4.86711 + 8.43008i −5.30728 − 1.93169i 0
176.2 −1.83299 + 0.667154i −0.167298 + 0.948795i 1.38267 − 1.16020i 0 −0.326336 − 1.85074i 2.62481 + 4.54630i 0.190241 − 0.329507i 1.94685 + 0.708597i 0
176.3 −1.46987 + 0.534990i −0.0502129 + 0.284771i 0.342220 − 0.287157i 0 −0.0785432 − 0.445441i −2.00998 − 3.48138i 1.21481 − 2.10411i 2.74050 + 0.997462i 0
176.4 −0.347463 + 0.126466i 0.422373 − 2.39540i −1.42735 + 1.19769i 0 0.156178 + 0.885729i 1.49845 + 2.59540i 0.714247 − 1.23711i −2.74046 − 0.997445i 0
176.5 0.524915 − 0.191054i 0.116387 − 0.660064i −1.29305 + 1.08500i 0 −0.0650142 − 0.368714i −1.27745 − 2.21261i −1.03005 + 1.78411i 2.39694 + 0.872414i 0
176.6 1.65028 − 0.600653i −0.347636 + 1.97154i 0.830552 − 0.696916i 0 0.610515 + 3.46240i 0.238393 + 0.412909i −0.804153 + 1.39283i −0.947041 − 0.344695i 0
176.7 2.17674 − 0.792270i 0.363030 − 2.05885i 2.57843 − 2.16356i 0 −0.840938 − 4.76919i 1.49485 + 2.58916i 1.58201 − 2.74013i −1.28798 − 0.468785i 0
226.1 −0.397029 + 2.25166i −0.775093 − 0.650380i −3.03298 − 1.10391i 0 1.77217 − 1.48703i 0.160076 + 0.277260i 1.40342 − 2.43080i −0.343170 − 1.94621i 0
226.2 −0.373691 + 2.11931i 2.49205 + 2.09108i −2.47243 − 0.899890i 0 −5.36289 + 4.50000i −1.31120 − 2.27106i 0.679065 − 1.17618i 1.31676 + 7.46771i 0
226.3 −0.0818155 + 0.463999i 0.820655 + 0.688611i 1.67078 + 0.608116i 0 −0.386657 + 0.324444i 1.37205 + 2.37647i −0.890018 + 1.54156i −0.321656 − 1.82420i 0
226.4 0.101945 − 0.578157i −2.01873 − 1.69392i 1.55551 + 0.566160i 0 −1.18515 + 0.994459i 0.996491 + 1.72597i 1.07298 − 1.85846i 0.684982 + 3.88473i 0
226.5 0.204768 − 1.16130i 1.05573 + 0.885861i 0.572702 + 0.208447i 0 1.24493 − 1.04462i −2.10998 − 3.65460i 1.53855 − 2.66485i −0.191133 − 1.08397i 0
226.6 0.428489 − 2.43008i 2.17135 + 1.82198i −3.84230 − 1.39848i 0 5.35796 − 4.49586i 2.33368 + 4.04204i −2.57724 + 4.46391i 0.874210 + 4.95789i 0
See all 42 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 101.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 475.2.l.e yes 42
5.b even 2 1 475.2.l.d ✓ 42
5.c odd 4 2 475.2.u.d 84
19.e even 9 1 inner 475.2.l.e yes 42
19.e even 9 1 9025.2.a.cs 21
19.f odd 18 1 9025.2.a.cp 21
95.o odd 18 1 9025.2.a.cr 21
95.p even 18 1 475.2.l.d ✓ 42
95.p even 18 1 9025.2.a.cq 21
95.q odd 36 2 475.2.u.d 84
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
475.2.l.d ✓ 42 5.b even 2 1
475.2.l.d ✓ 42 95.p even 18 1
475.2.l.e yes 42 1.a even 1 1 trivial
475.2.l.e yes 42 19.e even 9 1 inner
475.2.u.d 84 5.c odd 4 2
475.2.u.d 84 95.q odd 36 2
9025.2.a.cp 21 19.f odd 18 1
9025.2.a.cq 21 95.p even 18 1
9025.2.a.cr 21 95.o odd 18 1
9025.2.a.cs 21 19.e even 9 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{42} + 3 T_{2}^{40} + 14 T_{2}^{39} - 18 T_{2}^{38} + 30 T_{2}^{37} + 369 T_{2}^{36} - 246 T_{2}^{35} + \cdots + 729 \) acting on \(S_{2}^{\mathrm{new}}(475, [\chi])\). Copy content Toggle raw display