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Properties

Label 210.2.x.a
Level $210$
Weight $2$
Character orbit 210.x
Analytic conductor $1.677$
Analytic rank $0$
Dimension $64$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [210,2,Mod(23,210)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(210, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 9, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("210.23");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 210.x (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: \(1.67685844245\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(16\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 64 q + 8 q^{6} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q + 8 q^{6} + 4 q^{7} + 4 q^{10} - 24 q^{15} + 32 q^{16} - 8 q^{18} + 12 q^{21} - 24 q^{22} - 8 q^{25} - 72 q^{27} + 4 q^{28} - 12 q^{30} + 32 q^{31} - 20 q^{33} - 24 q^{36} - 8 q^{37} - 40 q^{42} - 64 q^{43} - 28 q^{45} + 24 q^{46} - 16 q^{55} + 24 q^{57} - 28 q^{58} + 8 q^{60} + 24 q^{61} - 88 q^{63} - 16 q^{67} - 12 q^{70} + 8 q^{72} - 56 q^{73} + 8 q^{75} - 32 q^{76} - 16 q^{78} + 4 q^{81} + 16 q^{82} + 32 q^{85} + 76 q^{87} + 12 q^{88} + 40 q^{90} - 48 q^{91} + 84 q^{93} + 4 q^{96} + 104 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1 −0.965926 + 0.258819i −1.64884 − 0.530415i 0.866025 − 0.500000i 0.680454 − 2.13002i 1.72993 + 0.0855915i −2.36464 + 1.18680i −0.707107 + 0.707107i 2.43732 + 1.74913i −0.105978 + 2.23356i
23.2 −0.965926 + 0.258819i −1.45883 + 0.933717i 0.866025 − 0.500000i −2.00119 + 0.997619i 1.16745 − 1.27947i −1.20893 − 2.35340i −0.707107 + 0.707107i 1.25635 − 2.72426i 1.67480 − 1.48157i
23.3 −0.965926 + 0.258819i −0.806628 − 1.53276i 0.866025 − 0.500000i 0.480420 + 2.18385i 1.17585 + 1.27176i −0.998710 + 2.45002i −0.707107 + 0.707107i −1.69870 + 2.47273i −1.02927 − 1.98509i
23.4 −0.965926 + 0.258819i −0.755113 + 1.55878i 0.866025 − 0.500000i 1.36035 − 1.77467i 0.325941 − 1.70111i 2.62857 + 0.301012i −0.707107 + 0.707107i −1.85961 − 2.35411i −0.854674 + 2.06628i
23.5 −0.965926 + 0.258819i 0.378582 − 1.69017i 0.866025 − 0.500000i 2.16375 − 0.564071i 0.0717662 + 1.73056i 0.113038 − 2.64334i −0.707107 + 0.707107i −2.71335 − 1.27974i −1.94403 + 1.10487i
23.6 −0.965926 + 0.258819i 0.929705 − 1.46139i 0.866025 − 0.500000i −1.68046 − 1.47514i −0.519792 + 1.65222i 2.42513 + 1.05770i −0.707107 + 0.707107i −1.27130 − 2.71732i 2.00500 + 0.989939i
23.7 −0.965926 + 0.258819i 1.03943 + 1.38549i 0.866025 − 0.500000i −2.22506 − 0.221605i −1.36260 − 1.06926i −0.736347 + 2.54122i −0.707107 + 0.707107i −0.839170 + 2.88024i 2.20660 − 0.361834i
23.8 −0.965926 + 0.258819i 1.35576 + 1.07792i 0.866025 − 0.500000i 0.962923 + 2.01811i −1.58855 − 0.690295i 1.50791 − 2.17399i −0.707107 + 0.707107i 0.676173 + 2.92281i −1.45244 − 1.70012i
23.9 0.965926 − 0.258819i −1.71308 + 0.255627i 0.866025 − 0.500000i −0.962923 − 2.01811i −1.58855 + 0.690295i 1.50791 − 2.17399i 0.707107 − 0.707107i 2.86931 − 0.875819i −1.45244 − 1.70012i
23.10 0.965926 − 0.258819i −1.59292 + 0.680155i 0.866025 − 0.500000i 2.22506 + 0.221605i −1.36260 + 1.06926i −0.736347 + 2.54122i 0.707107 − 0.707107i 2.07478 − 2.16686i 2.20660 − 0.361834i
23.11 0.965926 − 0.258819i −0.125444 + 1.72750i 0.866025 − 0.500000i −1.36035 + 1.77467i 0.325941 + 1.70111i 2.62857 + 0.301012i 0.707107 − 0.707107i −2.96853 − 0.433411i −0.854674 + 2.06628i
23.12 0.965926 − 0.258819i −0.0744552 − 1.73045i 0.866025 − 0.500000i 1.68046 + 1.47514i −0.519792 − 1.65222i 2.42513 + 1.05770i 0.707107 − 0.707107i −2.98891 + 0.257682i 2.00500 + 0.989939i
23.13 0.965926 − 0.258819i 0.517224 − 1.65302i 0.866025 − 0.500000i −2.16375 + 0.564071i 0.0717662 − 1.73056i 0.113038 − 2.64334i 0.707107 − 0.707107i −2.46496 − 1.70996i −1.94403 + 1.10487i
23.14 0.965926 − 0.258819i 0.796522 + 1.53804i 0.866025 − 0.500000i 2.00119 − 0.997619i 1.16745 + 1.27947i −1.20893 − 2.35340i 0.707107 − 0.707107i −1.73111 + 2.45016i 1.67480 − 1.48157i
23.15 0.965926 − 0.258819i 1.46494 − 0.924094i 0.866025 − 0.500000i −0.480420 − 2.18385i 1.17585 − 1.27176i −0.998710 + 2.45002i 0.707107 − 0.707107i 1.29210 − 2.70749i −1.02927 − 1.98509i
23.16 0.965926 − 0.258819i 1.69314 + 0.365065i 0.866025 − 0.500000i −0.680454 + 2.13002i 1.72993 − 0.0855915i −2.36464 + 1.18680i 0.707107 − 0.707107i 2.73346 + 1.23621i −0.105978 + 2.23356i
53.1 −0.258819 + 0.965926i −1.69017 + 0.378582i −0.866025 − 0.500000i 0.593376 − 2.15590i 0.0717662 − 1.73056i 2.64334 − 0.113038i 0.707107 − 0.707107i 2.71335 − 1.27974i 1.92886 + 1.13115i
53.2 −0.258819 + 0.965926i −1.53276 − 0.806628i −0.866025 − 0.500000i 2.13148 + 0.675868i 1.17585 − 1.27176i −2.45002 + 0.998710i 0.707107 − 0.707107i 1.69870 + 2.47273i −1.20451 + 1.88392i
53.3 −0.258819 + 0.965926i −1.46139 + 0.929705i −0.866025 − 0.500000i −2.11774 + 0.717755i −0.519792 − 1.65222i −1.05770 − 2.42513i 0.707107 − 0.707107i 1.27130 − 2.71732i −0.145187 − 2.23135i
53.4 −0.258819 + 0.965926i −0.530415 − 1.64884i −0.866025 − 0.500000i −1.50442 − 1.65430i 1.72993 − 0.0855915i −1.18680 + 2.36464i 0.707107 − 0.707107i −2.43732 + 1.74913i 1.98730 − 1.02500i
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 23.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
7.c even 3 1 inner
15.e even 4 1 inner
21.h odd 6 1 inner
35.l odd 12 1 inner
105.x even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 210.2.x.a ✓ 64
3.b odd 2 1 inner 210.2.x.a ✓ 64
5.c odd 4 1 inner 210.2.x.a ✓ 64
7.c even 3 1 inner 210.2.x.a ✓ 64
15.e even 4 1 inner 210.2.x.a ✓ 64
21.h odd 6 1 inner 210.2.x.a ✓ 64
35.l odd 12 1 inner 210.2.x.a ✓ 64
105.x even 12 1 inner 210.2.x.a ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.x.a ✓ 64 1.a even 1 1 trivial
210.2.x.a ✓ 64 3.b odd 2 1 inner
210.2.x.a ✓ 64 5.c odd 4 1 inner
210.2.x.a ✓ 64 7.c even 3 1 inner
210.2.x.a ✓ 64 15.e even 4 1 inner
210.2.x.a ✓ 64 21.h odd 6 1 inner
210.2.x.a ✓ 64 35.l odd 12 1 inner
210.2.x.a ✓ 64 105.x even 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(210, [\chi])\).