Properties

Label 310.3.o.c
Level $310$
Weight $3$
Character orbit 310.o
Analytic conductor $8.447$
Analytic rank $0$
Dimension $60$
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] = [310,3,Mod(67,310)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(310, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([3, 8]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("310.67");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 310 = 2 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 310.o (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: \(8.44688819517\)
Analytic rank: \(0\)
Dimension: \(60\)
Relative dimension: \(15\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 60 q - 60 q^{2} + 10 q^{3} - 2 q^{5} - 20 q^{6} - 22 q^{7} + 120 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 60 q - 60 q^{2} + 10 q^{3} - 2 q^{5} - 20 q^{6} - 22 q^{7} + 120 q^{8} - 4 q^{10} - 18 q^{11} + 20 q^{12} - 24 q^{13} + 160 q^{15} - 240 q^{16} - 14 q^{17} + 78 q^{18} + 12 q^{20} - 156 q^{21} + 18 q^{22} + 52 q^{23} + 48 q^{26} + 160 q^{27} + 44 q^{28} - 164 q^{30} - 114 q^{31} + 240 q^{32} + 12 q^{33} + 56 q^{35} - 156 q^{36} - 16 q^{37} + 44 q^{38} - 16 q^{40} + 38 q^{41} + 156 q^{42} + 22 q^{43} - 170 q^{45} - 104 q^{46} + 228 q^{47} - 40 q^{48} - 24 q^{50} - 4 q^{51} - 48 q^{52} - 20 q^{53} + 120 q^{55} - 88 q^{56} - 84 q^{58} + 8 q^{60} + 404 q^{61} + 114 q^{62} + 352 q^{63} - 104 q^{65} - 24 q^{66} + 178 q^{67} + 28 q^{68} + 84 q^{70} - 330 q^{71} + 156 q^{72} + 278 q^{73} - 164 q^{75} - 88 q^{76} - 752 q^{77} + 144 q^{78} + 8 q^{80} + 114 q^{81} - 38 q^{82} + 216 q^{83} - 772 q^{85} - 44 q^{86} + 20 q^{87} - 36 q^{88} + 78 q^{90} - 2016 q^{91} + 104 q^{92} + 388 q^{93} + 388 q^{95} + 80 q^{96} + 356 q^{97} - 102 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1 −1.00000 1.00000i −1.44258 5.38379i 2.00000i 3.11675 + 3.90971i −3.94121 + 6.82637i 9.48540 2.54160i 2.00000 2.00000i −19.1099 + 11.0331i 0.792965 7.02646i
67.2 −1.00000 1.00000i −1.30026 4.85263i 2.00000i −4.14890 + 2.79045i −3.55237 + 6.15289i −5.56600 + 1.49140i 2.00000 2.00000i −14.0631 + 8.11934i 6.93935 + 1.35845i
67.3 −1.00000 1.00000i −1.20732 4.50577i 2.00000i 3.15820 3.87631i −3.29845 + 5.71308i −8.21042 + 2.19998i 2.00000 2.00000i −11.0501 + 6.37978i −7.03451 + 0.718103i
67.4 −1.00000 1.00000i −0.748521 2.79352i 2.00000i 3.71811 3.34300i −2.04500 + 3.54204i 3.99794 1.07125i 2.00000 2.00000i 0.550769 0.317987i −7.06111 0.375114i
67.5 −1.00000 1.00000i −0.684780 2.55563i 2.00000i −4.42728 + 2.32362i −1.87085 + 3.24041i 9.25624 2.48020i 2.00000 2.00000i 1.73188 0.999904i 6.75090 + 2.10366i
67.6 −1.00000 1.00000i −0.625156 2.33312i 2.00000i −4.41762 2.34192i −1.70796 + 2.95827i 2.12950 0.570598i 2.00000 2.00000i 2.74162 1.58288i 2.07570 + 6.75955i
67.7 −1.00000 1.00000i −0.587359 2.19206i 2.00000i 3.39124 + 3.67417i −1.60470 + 2.77941i −9.33925 + 2.50244i 2.00000 2.00000i 3.33411 1.92495i 0.282931 7.06541i
67.8 −1.00000 1.00000i −0.133096 0.496721i 2.00000i −1.38599 4.80406i −0.363625 + 0.629817i 3.18158 0.852501i 2.00000 2.00000i 7.56521 4.36778i −3.41807 + 6.19006i
67.9 −1.00000 1.00000i −0.00691904 0.0258222i 2.00000i −1.45893 + 4.78242i −0.0189032 + 0.0327412i −4.08615 + 1.09488i 2.00000 2.00000i 7.79361 4.49964i 6.24135 3.32349i
67.10 −1.00000 1.00000i 0.495067 + 1.84762i 2.00000i −1.47284 4.77815i 1.35255 2.34268i −8.53581 + 2.28716i 2.00000 2.00000i 4.62564 2.67061i −3.30532 + 6.25099i
67.11 −1.00000 1.00000i 0.547772 + 2.04431i 2.00000i 2.08818 + 4.54307i 1.49654 2.59209i 5.63559 1.51005i 2.00000 2.00000i 3.91506 2.26036i 2.45489 6.63125i
67.12 −1.00000 1.00000i 0.596728 + 2.22702i 2.00000i 4.61146 1.93247i 1.63029 2.82375i 7.04237 1.88700i 2.00000 2.00000i 3.19070 1.84215i −6.54393 2.67900i
67.13 −1.00000 1.00000i 0.758485 + 2.83070i 2.00000i −4.88288 + 1.07588i 2.07222 3.58919i −7.55379 + 2.02403i 2.00000 2.00000i 0.356638 0.205905i 5.95876 + 3.80699i
67.14 −1.00000 1.00000i 1.14761 + 4.28293i 2.00000i 4.96940 + 0.552337i 3.13532 5.43054i −10.4628 + 2.80349i 2.00000 2.00000i −9.23228 + 5.33026i −4.41706 5.52174i
67.15 −1.00000 1.00000i 1.36020 + 5.07634i 2.00000i −0.760826 4.94178i 3.71614 6.43654i −2.00070 + 0.536087i 2.00000 2.00000i −16.1248 + 9.30968i −4.18095 + 5.70260i
87.1 −1.00000 1.00000i −5.07634 1.36020i 2.00000i −3.89929 + 3.12978i 3.71614 + 6.43654i 0.536087 2.00070i 2.00000 2.00000i 16.1248 + 9.30968i 7.02907 + 0.769507i
87.2 −1.00000 1.00000i −4.28293 1.14761i 2.00000i −2.00636 4.57979i 3.13532 + 5.43054i 2.80349 10.4628i 2.00000 2.00000i 9.23228 + 5.33026i −2.57343 + 6.58616i
87.3 −1.00000 1.00000i −2.83070 0.758485i 2.00000i 3.37318 + 3.69075i 2.07222 + 3.58919i 2.02403 7.55379i 2.00000 2.00000i −0.356638 0.205905i 0.317571 7.06393i
87.4 −1.00000 1.00000i −2.22702 0.596728i 2.00000i −3.97929 3.02741i 1.63029 + 2.82375i −1.88700 + 7.04237i 2.00000 2.00000i −3.19070 1.84215i 0.951885 + 7.00671i
87.5 −1.00000 1.00000i −2.04431 0.547772i 2.00000i 2.89033 4.07995i 1.49654 + 2.59209i −1.51005 + 5.63559i 2.00000 2.00000i −3.91506 2.26036i −6.97028 + 1.18963i
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 67.15
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
31.c even 3 1 inner
155.o odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 310.3.o.c 60
5.c odd 4 1 inner 310.3.o.c 60
31.c even 3 1 inner 310.3.o.c 60
155.o odd 12 1 inner 310.3.o.c 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
310.3.o.c 60 1.a even 1 1 trivial
310.3.o.c 60 5.c odd 4 1 inner
310.3.o.c 60 31.c even 3 1 inner
310.3.o.c 60 155.o odd 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{60} - 10 T_{3}^{59} + 50 T_{3}^{58} - 340 T_{3}^{57} + 380 T_{3}^{56} + 9668 T_{3}^{55} - 57880 T_{3}^{54} + 419978 T_{3}^{53} - 955032 T_{3}^{52} - 7784772 T_{3}^{51} + 51536312 T_{3}^{50} + \cdots + 53\!\cdots\!00 \) acting on \(S_{3}^{\mathrm{new}}(310, [\chi])\). Copy content Toggle raw display