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Properties

Label 105.2.u.a
Level $105$
Weight $2$
Character orbit 105.u
Analytic conductor $0.838$
Analytic rank $0$
Dimension $32$
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] = [105,2,Mod(52,105)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(105, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("105.52");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 105 = 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 105.u (of order \(12\), 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: \(0.838429221223\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(8\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 12 q^{5} + 8 q^{7} - 24 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 12 q^{5} + 8 q^{7} - 24 q^{8} - 12 q^{10} - 8 q^{11} - 8 q^{15} - 8 q^{21} - 8 q^{22} - 8 q^{23} + 12 q^{25} + 24 q^{26} - 24 q^{28} + 8 q^{30} + 24 q^{31} + 24 q^{32} - 36 q^{33} + 44 q^{35} - 32 q^{36} + 4 q^{37} + 12 q^{38} + 12 q^{40} + 16 q^{42} + 40 q^{43} - 40 q^{46} - 60 q^{47} + 72 q^{50} - 8 q^{51} - 108 q^{52} - 24 q^{53} - 48 q^{56} + 16 q^{57} + 4 q^{58} + 20 q^{60} - 24 q^{61} + 4 q^{63} - 4 q^{65} + 72 q^{66} + 8 q^{67} + 132 q^{68} + 4 q^{70} - 16 q^{71} + 12 q^{72} + 36 q^{73} + 48 q^{75} + 60 q^{77} + 80 q^{78} - 12 q^{80} + 16 q^{81} + 12 q^{82} - 72 q^{85} - 16 q^{86} - 24 q^{87} - 32 q^{88} - 24 q^{91} - 56 q^{92} - 24 q^{93} - 12 q^{95} - 72 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
52.1 −0.582519 + 2.17399i −0.965926 + 0.258819i −2.65485 − 1.53278i −0.0591736 + 2.23528i − 2.25068i −2.56205 − 0.660211i 1.69581 − 1.69581i 0.866025 − 0.500000i −4.82502 − 1.43074i
52.2 −0.461174 + 1.72112i 0.965926 − 0.258819i −1.01754 − 0.587476i −1.92237 + 1.14215i 1.78184i 2.04329 + 1.68076i −1.03952 + 1.03952i 0.866025 − 0.500000i −1.07923 − 3.83536i
52.3 −0.401003 + 1.49657i 0.965926 − 0.258819i −0.346853 − 0.200256i 2.14688 − 0.625221i 1.54936i −1.01885 − 2.44171i −1.75234 + 1.75234i 0.866025 − 0.500000i 0.0747765 + 3.46366i
52.4 −0.173702 + 0.648264i −0.965926 + 0.258819i 1.34198 + 0.774791i 1.64858 − 1.51070i − 0.671132i −0.588837 + 2.57939i −1.68450 + 1.68450i 0.866025 − 0.500000i 0.692969 + 1.33112i
52.5 0.105703 − 0.394487i −0.965926 + 0.258819i 1.58760 + 0.916603i −0.699113 + 2.12397i 0.408404i 2.57548 − 0.605712i 1.10697 − 1.10697i 0.866025 − 0.500000i 0.763981 + 0.500300i
52.6 0.259789 − 0.969545i 0.965926 − 0.258819i 0.859523 + 0.496246i −1.40510 − 1.73945i − 1.00375i −1.06195 + 2.42328i 2.12394 − 2.12394i 0.866025 − 0.500000i −2.05151 + 0.910421i
52.7 0.602389 − 2.24814i 0.965926 − 0.258819i −2.95923 − 1.70851i −1.28534 + 1.82973i − 2.32745i 0.519864 − 2.59417i −2.33208 + 2.33208i 0.866025 − 0.500000i 3.33923 + 3.99183i
52.8 0.650518 − 2.42777i −0.965926 + 0.258819i −3.73883 − 2.15861i −1.42436 − 1.72371i 2.51341i 2.09305 + 1.61838i −4.11829 + 4.11829i 0.866025 − 0.500000i −5.11135 + 2.33671i
73.1 −2.42777 − 0.650518i −0.258819 − 0.965926i 3.73883 + 2.15861i 0.780598 + 2.09539i 2.51341i 1.61838 − 2.09305i −4.11829 − 4.11829i −0.866025 + 0.500000i −0.532020 − 5.59492i
73.2 −2.24814 − 0.602389i 0.258819 + 0.965926i 2.95923 + 1.70851i −2.22726 + 0.198269i − 2.32745i −2.59417 − 0.519864i −2.33208 − 2.33208i −0.866025 + 0.500000i 5.12664 + 0.895939i
73.3 −0.969545 − 0.259789i 0.258819 + 0.965926i −0.859523 − 0.496246i 0.803857 + 2.08658i − 1.00375i 2.42328 + 1.06195i 2.12394 + 2.12394i −0.866025 + 0.500000i −0.237305 − 2.23187i
73.4 −0.394487 − 0.105703i −0.258819 − 0.965926i −1.58760 − 0.916603i −2.18897 − 0.456535i 0.408404i −0.605712 − 2.57548i 1.10697 + 1.10697i −0.866025 + 0.500000i 0.815263 + 0.411477i
73.5 0.648264 + 0.173702i −0.258819 − 0.965926i −1.34198 − 0.774791i 2.13259 − 0.672361i − 0.671132i 2.57939 + 0.588837i −1.68450 − 1.68450i −0.866025 + 0.500000i 1.49927 − 0.0654328i
73.6 1.49657 + 0.401003i 0.258819 + 0.965926i 0.346853 + 0.200256i 1.61490 − 1.54664i 1.54936i −2.44171 + 1.01885i −1.75234 − 1.75234i −0.866025 + 0.500000i 3.03701 − 1.66707i
73.7 1.72112 + 0.461174i 0.258819 + 0.965926i 1.01754 + 0.587476i −1.95031 + 1.09375i 1.78184i 1.68076 − 2.04329i −1.03952 − 1.03952i −0.866025 + 0.500000i −3.86114 + 0.983040i
73.8 2.17399 + 0.582519i −0.258819 − 0.965926i 2.65485 + 1.53278i −1.96540 − 1.06640i − 2.25068i −0.660211 + 2.56205i 1.69581 + 1.69581i −0.866025 + 0.500000i −3.65156 − 3.46322i
82.1 −2.42777 + 0.650518i −0.258819 + 0.965926i 3.73883 − 2.15861i 0.780598 − 2.09539i − 2.51341i 1.61838 + 2.09305i −4.11829 + 4.11829i −0.866025 − 0.500000i −0.532020 + 5.59492i
82.2 −2.24814 + 0.602389i 0.258819 − 0.965926i 2.95923 − 1.70851i −2.22726 − 0.198269i 2.32745i −2.59417 + 0.519864i −2.33208 + 2.33208i −0.866025 − 0.500000i 5.12664 − 0.895939i
82.3 −0.969545 + 0.259789i 0.258819 − 0.965926i −0.859523 + 0.496246i 0.803857 − 2.08658i 1.00375i 2.42328 − 1.06195i 2.12394 − 2.12394i −0.866025 − 0.500000i −0.237305 + 2.23187i
82.4 −0.394487 + 0.105703i −0.258819 + 0.965926i −1.58760 + 0.916603i −2.18897 + 0.456535i − 0.408404i −0.605712 + 2.57548i 1.10697 − 1.10697i −0.866025 − 0.500000i 0.815263 − 0.411477i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 52.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
7.d odd 6 1 inner
35.k even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 105.2.u.a ✓ 32
3.b odd 2 1 315.2.bz.d 32
5.b even 2 1 525.2.bc.e 32
5.c odd 4 1 inner 105.2.u.a ✓ 32
5.c odd 4 1 525.2.bc.e 32
7.b odd 2 1 735.2.v.b 32
7.c even 3 1 735.2.m.c 32
7.c even 3 1 735.2.v.b 32
7.d odd 6 1 inner 105.2.u.a ✓ 32
7.d odd 6 1 735.2.m.c 32
15.e even 4 1 315.2.bz.d 32
21.g even 6 1 315.2.bz.d 32
35.f even 4 1 735.2.v.b 32
35.i odd 6 1 525.2.bc.e 32
35.k even 12 1 inner 105.2.u.a ✓ 32
35.k even 12 1 525.2.bc.e 32
35.k even 12 1 735.2.m.c 32
35.l odd 12 1 735.2.m.c 32
35.l odd 12 1 735.2.v.b 32
105.w odd 12 1 315.2.bz.d 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.u.a ✓ 32 1.a even 1 1 trivial
105.2.u.a ✓ 32 5.c odd 4 1 inner
105.2.u.a ✓ 32 7.d odd 6 1 inner
105.2.u.a ✓ 32 35.k even 12 1 inner
315.2.bz.d 32 3.b odd 2 1
315.2.bz.d 32 15.e even 4 1
315.2.bz.d 32 21.g even 6 1
315.2.bz.d 32 105.w odd 12 1
525.2.bc.e 32 5.b even 2 1
525.2.bc.e 32 5.c odd 4 1
525.2.bc.e 32 35.i odd 6 1
525.2.bc.e 32 35.k even 12 1
735.2.m.c 32 7.c even 3 1
735.2.m.c 32 7.d odd 6 1
735.2.m.c 32 35.k even 12 1
735.2.m.c 32 35.l odd 12 1
735.2.v.b 32 7.b odd 2 1
735.2.v.b 32 7.c even 3 1
735.2.v.b 32 35.f even 4 1
735.2.v.b 32 35.l odd 12 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(105, [\chi])\).