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Properties

Label 310.3.o.d
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.

\(\operatorname{Tr}(f)(q) = \) \( 60 q + 60 q^{2} - 2 q^{3} - 4 q^{6} - 12 q^{7} - 120 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 60 q + 60 q^{2} - 2 q^{3} - 4 q^{6} - 12 q^{7} - 120 q^{8} + 20 q^{10} - 6 q^{11} - 4 q^{12} - 10 q^{13} - 104 q^{15} - 240 q^{16} - 8 q^{17} - 78 q^{18} + 40 q^{20} + 48 q^{21} - 6 q^{22} - 56 q^{23} - 20 q^{25} - 20 q^{26} + 160 q^{27} + 24 q^{28} - 148 q^{30} + 22 q^{31} - 240 q^{32} - 260 q^{33} + 12 q^{35} - 156 q^{36} - 208 q^{37} + 96 q^{38} + 40 q^{40} - 10 q^{41} + 48 q^{42} + 136 q^{43} - 104 q^{45} - 112 q^{46} + 208 q^{47} + 8 q^{48} - 76 q^{50} - 48 q^{51} - 20 q^{52} + 150 q^{53} + 246 q^{55} + 48 q^{56} + 364 q^{57} - 140 q^{58} - 88 q^{60} - 108 q^{61} + 22 q^{62} + 364 q^{63} + 70 q^{65} - 520 q^{66} + 60 q^{67} + 16 q^{68} + 108 q^{70} + 146 q^{71} - 156 q^{72} - 240 q^{73} - 92 q^{75} + 192 q^{76} + 180 q^{77} - 40 q^{78} + 306 q^{81} - 10 q^{82} + 296 q^{83} + 808 q^{85} + 272 q^{86} + 392 q^{87} + 12 q^{88} - 74 q^{90} + 704 q^{91} - 112 q^{92} - 660 q^{93} + 556 q^{95} + 16 q^{96} - 172 q^{97} + 474 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1 1.00000 + 1.00000i −1.33685 − 4.98921i 2.00000i 2.76825 − 4.16374i 3.65236 − 6.32607i 11.6300 − 3.11624i −2.00000 + 2.00000i −15.3108 + 8.83970i 6.93200 − 1.39549i
67.2 1.00000 + 1.00000i −1.30700 − 4.87778i 2.00000i 3.86665 + 3.17002i 3.57078 − 6.18477i −12.5375 + 3.35941i −2.00000 + 2.00000i −14.2902 + 8.25047i 0.696626 + 7.03667i
67.3 1.00000 + 1.00000i −1.24558 − 4.64857i 2.00000i −4.33045 − 2.49944i 3.40299 − 5.89415i −4.63461 + 1.24184i −2.00000 + 2.00000i −12.2635 + 7.08033i −1.83101 − 6.82989i
67.4 1.00000 + 1.00000i −0.697803 − 2.60424i 2.00000i −4.48272 + 2.21477i 1.90643 − 3.30204i 1.20013 − 0.321573i −2.00000 + 2.00000i 1.49911 − 0.865510i −6.69749 − 2.26795i
67.5 1.00000 + 1.00000i −0.445605 − 1.66302i 2.00000i 1.04221 − 4.89017i 1.21742 − 2.10863i −1.49319 + 0.400098i −2.00000 + 2.00000i 5.22715 − 3.01790i 5.93238 − 3.84796i
67.6 1.00000 + 1.00000i −0.351384 − 1.31138i 2.00000i 4.42335 − 2.33109i 0.959998 − 1.66277i −9.01063 + 2.41439i −2.00000 + 2.00000i 6.19798 − 3.57840i 6.75444 + 2.09226i
67.7 1.00000 + 1.00000i 0.0979777 + 0.365658i 2.00000i 4.99996 − 0.0197110i −0.267680 + 0.463635i 2.61029 − 0.699424i −2.00000 + 2.00000i 7.67012 − 4.42835i 5.01967 + 4.98025i
67.8 1.00000 + 1.00000i 0.103194 + 0.385125i 2.00000i −4.04832 − 2.93447i −0.281931 + 0.488319i 6.92905 − 1.85663i −2.00000 + 2.00000i 7.65656 − 4.42051i −1.11385 − 6.98279i
67.9 1.00000 + 1.00000i 0.133984 + 0.500036i 2.00000i 0.824848 + 4.93149i −0.366051 + 0.634020i −5.70790 + 1.52943i −2.00000 + 2.00000i 7.56214 − 4.36601i −4.10664 + 5.75634i
67.10 1.00000 + 1.00000i 0.176233 + 0.657711i 2.00000i 3.78920 + 3.26220i −0.481478 + 0.833944i 11.6244 − 3.11476i −2.00000 + 2.00000i 7.39270 − 4.26818i 0.527005 + 7.05140i
67.11 1.00000 + 1.00000i 0.661779 + 2.46979i 2.00000i −3.10942 − 3.91554i −1.80801 + 3.13157i −12.7293 + 3.41080i −2.00000 + 2.00000i 2.13231 − 1.23109i 0.806120 − 7.02497i
67.12 1.00000 + 1.00000i 1.02832 + 3.83776i 2.00000i 1.72916 − 4.69148i −2.80944 + 4.86608i 7.57712 − 2.03028i −2.00000 + 2.00000i −5.87672 + 3.39293i 6.42064 − 2.96232i
67.13 1.00000 + 1.00000i 1.05621 + 3.94184i 2.00000i −4.52312 + 2.13106i −2.88563 + 4.99806i −1.21778 + 0.326302i −2.00000 + 2.00000i −6.62831 + 3.82686i −6.65418 − 2.39206i
67.14 1.00000 + 1.00000i 1.16472 + 4.34681i 2.00000i 0.715169 + 4.94859i −3.18208 + 5.51153i 4.33487 − 1.16152i −2.00000 + 2.00000i −9.74393 + 5.62566i −4.23342 + 5.66376i
67.15 1.00000 + 1.00000i 1.32782 + 4.95549i 2.00000i 4.99548 − 0.212475i −3.62767 + 6.28331i −6.77114 + 1.81432i −2.00000 + 2.00000i −14.9995 + 8.65999i 5.20796 + 4.78301i
87.1 1.00000 + 1.00000i −4.95549 − 1.32782i 2.00000i −2.68175 − 4.21998i −3.62767 − 6.28331i 1.81432 − 6.77114i −2.00000 + 2.00000i 14.9995 + 8.65999i 1.53823 − 6.90173i
87.2 1.00000 + 1.00000i −4.34681 − 1.16472i 2.00000i 3.92802 − 3.09365i −3.18208 − 5.51153i −1.16152 + 4.33487i −2.00000 + 2.00000i 9.74393 + 5.62566i 7.02167 + 0.834371i
87.3 1.00000 + 1.00000i −3.94184 − 1.05621i 2.00000i 4.10711 + 2.85161i −2.88563 − 4.99806i 0.326302 − 1.21778i −2.00000 + 2.00000i 6.62831 + 3.82686i 1.25550 + 6.95871i
87.4 1.00000 + 1.00000i −3.83776 − 1.02832i 2.00000i −4.92752 + 0.848245i −2.80944 − 4.86608i −2.03028 + 7.57712i −2.00000 + 2.00000i 5.87672 + 3.39293i −5.77577 − 4.07928i
87.5 1.00000 + 1.00000i −2.46979 − 0.661779i 2.00000i −1.83625 + 4.65061i −1.80801 − 3.13157i 3.41080 − 12.7293i −2.00000 + 2.00000i −2.13231 − 1.23109i −6.48686 + 2.81436i
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.d ✓ 60
5.c odd 4 1 inner 310.3.o.d ✓ 60
31.c even 3 1 inner 310.3.o.d ✓ 60
155.o odd 12 1 inner 310.3.o.d ✓ 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
310.3.o.d ✓ 60 1.a even 1 1 trivial
310.3.o.d ✓ 60 5.c odd 4 1 inner
310.3.o.d ✓ 60 31.c even 3 1 inner
310.3.o.d ✓ 60 155.o odd 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{60} + 2 T_{3}^{59} + 2 T_{3}^{58} - 28 T_{3}^{57} - 1876 T_{3}^{56} - 3108 T_{3}^{55} + \cdots + 26\!\cdots\!00 \) acting on \(S_{3}^{\mathrm{new}}(310, [\chi])\). Copy content Toggle raw display