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Properties

Label 210.4.n.b
Level $210$
Weight $4$
Character orbit 210.n
Analytic conductor $12.390$
Analytic rank $0$
Dimension $28$
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] = [210,4,Mod(79,210)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(210, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("210.79");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 210.n (of order \(6\), degree \(2\), 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: \(12.3904011012\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(14\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 28 q + 56 q^{4} + 6 q^{5} - 168 q^{6} + 126 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q + 56 q^{4} + 6 q^{5} - 168 q^{6} + 126 q^{9} - 20 q^{10} - 18 q^{11} + 60 q^{14} + 60 q^{15} - 224 q^{16} - 166 q^{19} + 48 q^{20} - 54 q^{21} - 336 q^{24} - 132 q^{25} + 36 q^{26} - 408 q^{29} - 36 q^{30} + 208 q^{31} + 320 q^{34} + 276 q^{35} + 1008 q^{36} + 54 q^{39} + 80 q^{40} - 900 q^{41} + 72 q^{44} - 54 q^{45} - 748 q^{46} - 496 q^{49} - 1056 q^{50} - 240 q^{51} - 756 q^{54} + 972 q^{55} + 96 q^{56} + 612 q^{59} + 120 q^{60} + 804 q^{61} - 1792 q^{64} - 948 q^{65} + 108 q^{66} + 2244 q^{69} - 388 q^{70} + 5904 q^{71} + 924 q^{74} + 792 q^{75} - 1328 q^{76} + 2008 q^{79} + 96 q^{80} - 1134 q^{81} - 360 q^{84} - 5936 q^{85} - 2616 q^{86} - 2340 q^{89} - 360 q^{90} + 5968 q^{91} - 1028 q^{94} + 666 q^{95} + 1344 q^{96} - 324 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1 −1.73205 − 1.00000i 2.59808 − 1.50000i 2.00000 + 3.46410i −10.0831 − 4.83028i −6.00000 −8.03751 − 16.6853i − 8.00000i 4.50000 − 7.79423i 12.6341 + 18.4494i
79.2 −1.73205 − 1.00000i 2.59808 − 1.50000i 2.00000 + 3.46410i −8.38725 + 7.39283i −6.00000 −15.8971 + 9.50166i − 8.00000i 4.50000 − 7.79423i 21.9200 − 4.41751i
79.3 −1.73205 − 1.00000i 2.59808 − 1.50000i 2.00000 + 3.46410i −5.13202 + 9.93290i −6.00000 17.2221 + 6.81161i − 8.00000i 4.50000 − 7.79423i 18.8218 − 12.0723i
79.4 −1.73205 − 1.00000i 2.59808 − 1.50000i 2.00000 + 3.46410i 0.878096 − 11.1458i −6.00000 −3.79947 + 18.1263i − 8.00000i 4.50000 − 7.79423i −12.6667 + 18.4270i
79.5 −1.73205 − 1.00000i 2.59808 − 1.50000i 2.00000 + 3.46410i 7.31998 − 8.45091i −6.00000 16.9814 − 7.39130i − 8.00000i 4.50000 − 7.79423i −21.1295 + 7.31744i
79.6 −1.73205 − 1.00000i 2.59808 − 1.50000i 2.00000 + 3.46410i 10.5803 + 3.61355i −6.00000 1.42597 − 18.4653i − 8.00000i 4.50000 − 7.79423i −14.7120 − 16.8391i
79.7 −1.73205 − 1.00000i 2.59808 − 1.50000i 2.00000 + 3.46410i 10.6541 + 3.38964i −6.00000 −14.8236 + 11.1023i − 8.00000i 4.50000 − 7.79423i −15.0638 − 16.5252i
79.8 1.73205 + 1.00000i −2.59808 + 1.50000i 2.00000 + 3.46410i −8.41956 − 7.35601i −6.00000 −1.42597 + 18.4653i 8.00000i 4.50000 − 7.79423i −7.22710 − 21.1606i
79.9 1.73205 + 1.00000i −2.59808 + 1.50000i 2.00000 + 3.46410i −8.26258 − 7.53192i −6.00000 14.8236 − 11.1023i 8.00000i 4.50000 − 7.79423i −6.77928 − 21.3082i
79.10 1.73205 + 1.00000i −2.59808 + 1.50000i 2.00000 + 3.46410i −6.03613 + 9.41090i −6.00000 −17.2221 − 6.81161i 8.00000i 4.50000 − 7.79423i −19.8658 + 10.2640i
79.11 1.73205 + 1.00000i −2.59808 + 1.50000i 2.00000 + 3.46410i −2.20876 + 10.9600i −6.00000 15.8971 − 9.50166i 8.00000i 4.50000 − 7.79423i −14.7857 + 16.7745i
79.12 1.73205 + 1.00000i −2.59808 + 1.50000i 2.00000 + 3.46410i 3.65872 − 10.5647i −6.00000 −16.9814 + 7.39130i 8.00000i 4.50000 − 7.79423i 16.9018 − 14.6400i
79.13 1.73205 + 1.00000i −2.59808 + 1.50000i 2.00000 + 3.46410i 9.21350 − 6.33336i −6.00000 3.79947 − 18.1263i 8.00000i 4.50000 − 7.79423i 22.2916 − 1.75619i
79.14 1.73205 + 1.00000i −2.59808 + 1.50000i 2.00000 + 3.46410i 9.22468 + 6.31706i −6.00000 8.03751 + 16.6853i 8.00000i 4.50000 − 7.79423i 9.66056 + 20.1662i
109.1 −1.73205 + 1.00000i 2.59808 + 1.50000i 2.00000 − 3.46410i −10.0831 + 4.83028i −6.00000 −8.03751 + 16.6853i 8.00000i 4.50000 + 7.79423i 12.6341 − 18.4494i
109.2 −1.73205 + 1.00000i 2.59808 + 1.50000i 2.00000 − 3.46410i −8.38725 − 7.39283i −6.00000 −15.8971 − 9.50166i 8.00000i 4.50000 + 7.79423i 21.9200 + 4.41751i
109.3 −1.73205 + 1.00000i 2.59808 + 1.50000i 2.00000 − 3.46410i −5.13202 − 9.93290i −6.00000 17.2221 − 6.81161i 8.00000i 4.50000 + 7.79423i 18.8218 + 12.0723i
109.4 −1.73205 + 1.00000i 2.59808 + 1.50000i 2.00000 − 3.46410i 0.878096 + 11.1458i −6.00000 −3.79947 − 18.1263i 8.00000i 4.50000 + 7.79423i −12.6667 − 18.4270i
109.5 −1.73205 + 1.00000i 2.59808 + 1.50000i 2.00000 − 3.46410i 7.31998 + 8.45091i −6.00000 16.9814 + 7.39130i 8.00000i 4.50000 + 7.79423i −21.1295 − 7.31744i
109.6 −1.73205 + 1.00000i 2.59808 + 1.50000i 2.00000 − 3.46410i 10.5803 − 3.61355i −6.00000 1.42597 + 18.4653i 8.00000i 4.50000 + 7.79423i −14.7120 + 16.8391i
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 79.14
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 210.4.n.b ✓ 28
5.b even 2 1 inner 210.4.n.b ✓ 28
7.c even 3 1 inner 210.4.n.b ✓ 28
35.j even 6 1 inner 210.4.n.b ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.4.n.b ✓ 28 1.a even 1 1 trivial
210.4.n.b ✓ 28 5.b even 2 1 inner
210.4.n.b ✓ 28 7.c even 3 1 inner
210.4.n.b ✓ 28 35.j even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{14} + 9 T_{11}^{13} + 6234 T_{11}^{12} + 78589 T_{11}^{11} + 28676706 T_{11}^{10} + \cdots + 11\!\cdots\!00 \) acting on \(S_{4}^{\mathrm{new}}(210, [\chi])\). Copy content Toggle raw display