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Properties

Label 308.3.r.a
Level $308$
Weight $3$
Character orbit 308.r
Analytic conductor $8.392$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [308,3,Mod(29,308)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(308, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 7]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("308.29");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 308 = 2^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 308.r (of order \(10\), 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: \(8.39239214230\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(12\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 10 q^{3} + 6 q^{5} - 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 10 q^{3} + 6 q^{5} - 40 q^{9} - 10 q^{11} + 30 q^{13} + 24 q^{15} + 60 q^{19} - 132 q^{23} - 186 q^{25} - 110 q^{27} - 90 q^{29} - 26 q^{31} + 46 q^{33} + 82 q^{37} + 290 q^{39} - 336 q^{45} + 84 q^{47} + 84 q^{49} - 20 q^{51} + 58 q^{53} + 370 q^{55} - 20 q^{57} + 436 q^{59} + 160 q^{61} + 276 q^{67} - 118 q^{69} - 150 q^{71} - 320 q^{73} - 692 q^{75} + 28 q^{77} - 560 q^{79} + 122 q^{81} - 630 q^{83} + 220 q^{85} - 444 q^{89} - 126 q^{91} + 500 q^{93} + 440 q^{95} - 80 q^{97} + 1034 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1 0 −1.77709 + 5.46932i 0 1.19908 − 0.871185i 0 −2.51626 + 0.817582i 0 −19.4742 − 14.1489i 0
29.2 0 −1.08666 + 3.34439i 0 4.48598 − 3.25926i 0 2.51626 − 0.817582i 0 −2.72296 − 1.97835i 0
29.3 0 −0.859178 + 2.64428i 0 1.72124 − 1.25056i 0 −2.51626 + 0.817582i 0 1.02714 + 0.746261i 0
29.4 0 −0.629070 + 1.93608i 0 −1.14060 + 0.828696i 0 2.51626 − 0.817582i 0 3.92849 + 2.85421i 0
29.5 0 −0.367779 + 1.13191i 0 5.33735 − 3.87781i 0 −2.51626 + 0.817582i 0 6.13520 + 4.45748i 0
29.6 0 −0.0336547 + 0.103579i 0 −4.89286 + 3.55487i 0 −2.51626 + 0.817582i 0 7.27156 + 5.28310i 0
29.7 0 0.450106 − 1.38528i 0 6.79782 − 4.93891i 0 2.51626 − 0.817582i 0 5.56474 + 4.04302i 0
29.8 0 0.476956 − 1.46792i 0 −4.13612 + 3.00507i 0 2.51626 − 0.817582i 0 5.35385 + 3.88980i 0
29.9 0 0.715503 − 2.20209i 0 −3.95398 + 2.87274i 0 −2.51626 + 0.817582i 0 2.94389 + 2.13886i 0
29.10 0 1.34580 − 4.14196i 0 2.74800 − 1.99654i 0 2.51626 − 0.817582i 0 −8.06346 − 5.85845i 0
29.11 0 1.45805 − 4.48741i 0 3.01622 − 2.19141i 0 −2.51626 + 0.817582i 0 −10.7298 − 7.79564i 0
29.12 0 1.68898 − 5.19814i 0 −6.32803 + 4.59758i 0 2.51626 − 0.817582i 0 −16.8869 − 12.2690i 0
57.1 0 −3.27779 + 2.38145i 0 0.110332 + 0.339566i 0 −1.55513 + 2.14046i 0 2.29142 − 7.05226i 0
57.2 0 −3.23386 + 2.34954i 0 −2.54199 − 7.82345i 0 1.55513 − 2.14046i 0 2.15638 − 6.63666i 0
57.3 0 −2.66310 + 1.93486i 0 −0.225454 − 0.693877i 0 1.55513 − 2.14046i 0 0.567294 − 1.74595i 0
57.4 0 −1.70156 + 1.23625i 0 2.48745 + 7.65558i 0 1.55513 − 2.14046i 0 −1.41418 + 4.35239i 0
57.5 0 −0.376205 + 0.273329i 0 −1.04910 − 3.22879i 0 −1.55513 + 2.14046i 0 −2.71433 + 8.35385i 0
57.6 0 −0.0173009 + 0.0125698i 0 −1.51166 − 4.65241i 0 −1.55513 + 2.14046i 0 −2.78101 + 8.55907i 0
57.7 0 0.251197 − 0.182505i 0 2.69700 + 8.30052i 0 −1.55513 + 2.14046i 0 −2.75136 + 8.46782i 0
57.8 0 1.16003 − 0.842810i 0 1.03518 + 3.18595i 0 1.55513 − 2.14046i 0 −2.14582 + 6.60414i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.d odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 308.3.r.a ✓ 48
11.d odd 10 1 inner 308.3.r.a ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
308.3.r.a ✓ 48 1.a even 1 1 trivial
308.3.r.a ✓ 48 11.d odd 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(308, [\chi])\).