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Properties

Label 210.3.w.a
Level $210$
Weight $3$
Character orbit 210.w
Analytic conductor $5.722$
Analytic rank $0$
Dimension $64$
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] = [210,3,Mod(17,210)]
 
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, 3, 2]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("210.17");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 210.w (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: \(5.72208555157\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 64 q - 32 q^{2} - 6 q^{3} - 12 q^{5} + 4 q^{7} - 128 q^{8} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q - 32 q^{2} - 6 q^{3} - 12 q^{5} + 4 q^{7} - 128 q^{8} - 16 q^{9} + 24 q^{10} + 12 q^{12} - 16 q^{14} - 44 q^{15} + 128 q^{16} - 20 q^{18} + 36 q^{21} + 16 q^{22} - 12 q^{23} - 16 q^{25} + 8 q^{28} - 112 q^{29} + 26 q^{30} + 128 q^{32} + 30 q^{33} + 16 q^{36} - 32 q^{37} + 24 q^{38} + 64 q^{39} - 136 q^{42} + 32 q^{43} - 16 q^{44} - 114 q^{45} - 24 q^{46} - 96 q^{47} + 40 q^{50} - 84 q^{51} + 56 q^{53} - 72 q^{54} - 316 q^{57} + 56 q^{58} + 672 q^{59} + 8 q^{60} + 600 q^{61} - 210 q^{63} + 28 q^{65} + 16 q^{67} + 24 q^{72} - 624 q^{73} - 64 q^{74} + 48 q^{75} + 208 q^{77} - 8 q^{78} - 48 q^{80} - 64 q^{81} - 192 q^{82} + 160 q^{84} - 152 q^{85} + 60 q^{87} - 16 q^{88} + 144 q^{89} - 232 q^{91} + 48 q^{92} - 170 q^{93} + 136 q^{95} - 48 q^{96} + 128 q^{98} + 160 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 −1.36603 − 0.366025i −2.99585 − 0.157742i 1.73205 + 1.00000i 2.05899 + 4.55638i 4.03467 + 1.31204i −6.80611 − 1.63612i −2.00000 − 2.00000i 8.95023 + 0.945145i −1.14488 − 6.97777i
17.2 −1.36603 − 0.366025i −2.72119 − 1.26298i 1.73205 + 1.00000i −2.34013 − 4.41857i 3.25493 + 2.72129i −6.07917 + 3.47039i −2.00000 − 2.00000i 5.80976 + 6.87362i 1.57936 + 6.89243i
17.3 −1.36603 − 0.366025i −2.49614 + 1.66411i 1.73205 + 1.00000i −4.84166 − 1.24831i 4.01890 − 1.35957i −0.400482 − 6.98853i −2.00000 − 2.00000i 3.46146 − 8.30773i 6.15692 + 3.47740i
17.4 −1.36603 − 0.366025i −2.44741 + 1.73499i 1.73205 + 1.00000i 4.38913 + 2.39490i 3.97827 − 1.47423i 5.69479 + 4.07055i −2.00000 − 2.00000i 2.97959 − 8.49247i −5.11906 − 4.87803i
17.5 −1.36603 − 0.366025i −2.25410 − 1.97966i 1.73205 + 1.00000i −3.83488 + 3.20838i 2.35455 + 3.52932i 6.92770 − 1.00349i −2.00000 − 2.00000i 1.16192 + 8.92468i 6.41289 − 2.97907i
17.6 −1.36603 − 0.366025i −1.37229 − 2.66774i 1.73205 + 1.00000i 4.92543 − 0.860317i 0.898127 + 4.14649i 1.68496 + 6.79418i −2.00000 − 2.00000i −5.23363 + 7.32183i −7.04316 − 0.627617i
17.7 −1.36603 − 0.366025i −0.897104 + 2.86273i 1.73205 + 1.00000i 4.28576 − 2.57532i 2.27330 − 3.58219i −1.16075 − 6.90309i −2.00000 − 2.00000i −7.39041 − 5.13633i −6.79709 + 1.94926i
17.8 −1.36603 − 0.366025i −0.197383 + 2.99350i 1.73205 + 1.00000i −1.09236 − 4.87922i 1.36533 − 4.01695i −1.72310 + 6.78461i −2.00000 − 2.00000i −8.92208 − 1.18173i −0.293729 + 7.06496i
17.9 −1.36603 − 0.366025i 0.587638 − 2.94188i 1.73205 + 1.00000i −3.75994 − 3.29588i −1.87953 + 3.80360i 5.65896 − 4.12022i −2.00000 − 2.00000i −8.30936 − 3.45752i 3.92980 + 5.87849i
17.10 −1.36603 − 0.366025i 0.941591 − 2.84840i 1.73205 + 1.00000i 2.72423 − 4.19268i −2.32883 + 3.54635i −5.35730 − 4.50548i −2.00000 − 2.00000i −7.22681 − 5.36406i −5.25600 + 4.73017i
17.11 −1.36603 − 0.366025i 1.05588 + 2.80804i 1.73205 + 1.00000i −1.47500 + 4.77749i −0.414547 − 4.22234i 6.99874 + 0.132847i −2.00000 − 2.00000i −6.77022 + 5.92993i 3.76357 − 5.98628i
17.12 −1.36603 − 0.366025i 1.06031 − 2.80638i 1.73205 + 1.00000i −0.695196 + 4.95143i −2.47561 + 3.44548i −2.48720 + 6.54323i −2.00000 − 2.00000i −6.75151 − 5.95123i 2.76201 − 6.50933i
17.13 −1.36603 − 0.366025i 2.42903 + 1.76063i 1.73205 + 1.00000i −4.98956 + 0.322873i −2.67368 − 3.29415i −4.85698 − 5.04081i −2.00000 − 2.00000i 2.80035 + 8.55325i 6.93405 + 1.38525i
17.14 −1.36603 − 0.366025i 2.82505 − 1.00950i 1.73205 + 1.00000i 2.52260 + 4.31700i −4.22859 + 0.344962i 0.692615 − 6.96565i −2.00000 − 2.00000i 6.96182 − 5.70377i −1.86581 − 6.82047i
17.15 −1.36603 − 0.366025i 2.86958 − 0.874940i 1.73205 + 1.00000i −4.99957 + 0.0656422i −4.24017 + 0.144852i −2.70652 + 6.45560i −2.00000 − 2.00000i 7.46896 − 5.02142i 6.85357 + 1.74030i
17.16 −1.36603 − 0.366025i 2.97842 + 0.359186i 1.73205 + 1.00000i 2.39011 − 4.39174i −3.93713 − 1.58084i 6.65191 + 2.17994i −2.00000 − 2.00000i 8.74197 + 2.13962i −4.87243 + 5.12439i
47.1 0.366025 + 1.36603i −2.99647 − 0.145428i −1.73205 + 1.00000i 3.20777 + 3.83539i −0.898127 − 4.14649i 6.79418 + 1.68496i −2.00000 − 2.00000i 8.95770 + 0.871542i −4.06511 + 5.78575i
47.2 0.366025 + 1.36603i −2.84148 + 0.962278i −1.73205 + 1.00000i −4.69598 − 1.71691i −2.35455 − 3.52932i −1.00349 + 6.92770i −2.00000 − 2.00000i 7.14804 − 5.46859i 0.626491 − 7.04326i
47.3 0.366025 + 1.36603i −2.45437 + 1.72513i −1.73205 + 1.00000i 2.65653 − 4.23590i −3.25493 − 2.72129i 3.47039 − 6.07917i −2.00000 − 2.00000i 3.04785 − 8.46821i 6.75870 + 2.07845i
47.4 0.366025 + 1.36603i −2.25393 − 1.97985i −1.73205 + 1.00000i 0.974348 − 4.90415i 1.87953 − 3.80360i −4.12022 + 5.65896i −2.00000 − 2.00000i 1.16038 + 8.92488i 7.05582 − 0.464058i
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
15.e even 4 1 inner
105.w odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 210.3.w.a ✓ 64
3.b odd 2 1 210.3.w.b yes 64
5.c odd 4 1 210.3.w.b yes 64
7.d odd 6 1 inner 210.3.w.a ✓ 64
15.e even 4 1 inner 210.3.w.a ✓ 64
21.g even 6 1 210.3.w.b yes 64
35.k even 12 1 210.3.w.b yes 64
105.w odd 12 1 inner 210.3.w.a ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.3.w.a ✓ 64 1.a even 1 1 trivial
210.3.w.a ✓ 64 7.d odd 6 1 inner
210.3.w.a ✓ 64 15.e even 4 1 inner
210.3.w.a ✓ 64 105.w odd 12 1 inner
210.3.w.b yes 64 3.b odd 2 1
210.3.w.b yes 64 5.c odd 4 1
210.3.w.b yes 64 21.g even 6 1
210.3.w.b yes 64 35.k even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{17}^{64} - 1232998 T_{17}^{60} - 1668936 T_{17}^{59} + 749871912 T_{17}^{57} + 986760210559 T_{17}^{56} + \cdots + 47\!\cdots\!00 \) acting on \(S_{3}^{\mathrm{new}}(210, [\chi])\). Copy content Toggle raw display