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Properties

Label 475.2.l.d
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 −1.98886 + 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
101.2 −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
101.3 −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
101.4 −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 0.473166 − 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
101.6 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
101.7 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
176.1 −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
176.2 −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
176.3 −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
176.4 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 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
176.6 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
176.7 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
226.1 −0.464630 + 2.63505i 1.71387 + 1.43810i −4.84822 − 1.76461i 0 −4.58579 + 3.84794i 1.78841 + 3.09762i 4.22676 − 7.32097i 0.348248 + 1.97501i 0
226.2 −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
226.3 −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
226.4 −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 0.0818155 − 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
226.6 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
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.d ✓ 42
5.b even 2 1 475.2.l.e yes 42
5.c odd 4 2 475.2.u.d 84
19.e even 9 1 inner 475.2.l.d ✓ 42
19.e even 9 1 9025.2.a.cq 21
19.f odd 18 1 9025.2.a.cr 21
95.o odd 18 1 9025.2.a.cp 21
95.p even 18 1 475.2.l.e yes 42
95.p even 18 1 9025.2.a.cs 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 1.a even 1 1 trivial
475.2.l.d ✓ 42 19.e even 9 1 inner
475.2.l.e yes 42 5.b even 2 1
475.2.l.e yes 42 95.p even 18 1
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 95.o odd 18 1
9025.2.a.cq 21 19.e even 9 1
9025.2.a.cr 21 19.f odd 18 1
9025.2.a.cs 21 95.p even 18 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