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Properties

Label 310.3.l.a
Level $310$
Weight $3$
Character orbit 310.l
Analytic conductor $8.447$
Analytic rank $0$
Dimension $128$
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(29,310)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(310, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("310.29");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 310 = 2 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 310.l (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: \(8.44688819517\)
Analytic rank: \(0\)
Dimension: \(128\)
Relative dimension: \(32\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 128 q + 64 q^{4} - 12 q^{5} - 56 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 128 q + 64 q^{4} - 12 q^{5} - 56 q^{9} - 16 q^{10} - 70 q^{15} - 128 q^{16} + 16 q^{19} + 24 q^{20} + 120 q^{21} - 52 q^{25} + 20 q^{29} - 252 q^{31} - 80 q^{34} - 156 q^{35} - 848 q^{36} - 160 q^{39} - 48 q^{40} + 228 q^{41} - 144 q^{45} + 160 q^{46} + 388 q^{49} - 112 q^{50} + 104 q^{51} - 120 q^{55} + 520 q^{59} - 140 q^{60} + 256 q^{64} + 830 q^{65} + 176 q^{66} - 392 q^{69} + 376 q^{70} - 364 q^{71} + 160 q^{75} + 48 q^{76} + 440 q^{79} + 72 q^{80} - 876 q^{81} + 240 q^{85} + 600 q^{86} + 300 q^{89} + 8 q^{90} - 220 q^{91} + 176 q^{94} + 574 q^{95}+O(q^{100}) \) 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 −0.831254 + 1.14412i −4.43391 + 3.22143i −0.618034 − 1.90211i −4.79498 − 1.41711i − 7.75076i −8.13247 + 2.64240i 2.68999 + 0.874032i 6.50084 − 20.0075i 5.60719 − 4.30806i
29.2 −0.831254 + 1.14412i −3.86333 + 2.80688i −0.618034 − 1.90211i −3.94507 + 3.07187i − 6.75336i 10.7711 − 3.49975i 2.68999 + 0.874032i 4.26564 − 13.1283i −0.235243 − 7.06715i
29.3 −0.831254 + 1.14412i −3.83890 + 2.78913i −0.618034 − 1.90211i 4.98030 + 0.443448i − 6.71065i −1.98165 + 0.643877i 2.68999 + 0.874032i 4.17679 − 12.8548i −4.64725 + 5.32945i
29.4 −0.831254 + 1.14412i −3.08902 + 2.24430i −0.618034 − 1.90211i 1.51996 + 4.76337i − 5.39980i −0.275791 + 0.0896098i 2.68999 + 0.874032i 1.72399 − 5.30588i −6.71336 − 2.22055i
29.5 −0.831254 + 1.14412i −2.57728 + 1.87250i −0.618034 − 1.90211i 3.32988 − 3.72987i − 4.50525i −0.0684838 + 0.0222517i 2.68999 + 0.874032i 0.354944 − 1.09241i 1.49945 + 6.91026i
29.6 −0.831254 + 1.14412i −1.53813 + 1.11752i −0.618034 − 1.90211i −2.38982 − 4.39190i − 2.68875i 7.91423 − 2.57149i 2.68999 + 0.874032i −1.66415 + 5.12174i 7.01142 + 0.916540i
29.7 −0.831254 + 1.14412i −0.809292 + 0.587985i −0.618034 − 1.90211i −0.0956107 + 4.99909i − 1.41469i −11.6840 + 3.79635i 2.68999 + 0.874032i −2.47193 + 7.60781i −5.64009 − 4.26490i
29.8 −0.831254 + 1.14412i −0.401640 + 0.291808i −0.618034 − 1.90211i −4.98216 + 0.421995i − 0.702092i 3.16199 − 1.02739i 2.68999 + 0.874032i −2.70499 + 8.32510i 3.65863 − 6.05099i
29.9 −0.831254 + 1.14412i 0.902094 − 0.655410i −0.618034 − 1.90211i −0.346560 + 4.98798i 1.57692i 5.45906 − 1.77376i 2.68999 + 0.874032i −2.39694 + 7.37703i −5.41878 − 4.54278i
29.10 −0.831254 + 1.14412i 1.23740 − 0.899026i −0.618034 − 1.90211i 3.69658 − 3.36679i 2.16306i 4.37080 − 1.42016i 2.68999 + 0.874032i −2.05823 + 6.33459i 0.779224 + 7.02800i
29.11 −0.831254 + 1.14412i 1.44587 − 1.05048i −0.618034 − 1.90211i −4.76702 − 1.50849i 2.52747i −2.45402 + 0.797360i 2.68999 + 0.874032i −1.79414 + 5.52178i 5.68850 − 4.20012i
29.12 −0.831254 + 1.14412i 1.46683 − 1.06571i −0.618034 − 1.90211i 4.97359 + 0.513202i 2.56411i −9.66347 + 3.13985i 2.68999 + 0.874032i −1.76531 + 5.43308i −4.72148 + 5.26380i
29.13 −0.831254 + 1.14412i 3.32133 − 2.41309i −0.618034 − 1.90211i −1.91795 − 4.61752i 5.80590i −10.8265 + 3.51773i 2.68999 + 0.874032i 2.42710 − 7.46983i 6.87731 + 1.64395i
29.14 −0.831254 + 1.14412i 3.47472 − 2.52453i −0.618034 − 1.90211i 4.20395 + 2.70681i 6.07404i 4.21719 − 1.37025i 2.68999 + 0.874032i 2.91927 − 8.98460i −6.59147 + 2.55979i
29.15 −0.831254 + 1.14412i 4.32119 − 3.13952i −0.618034 − 1.90211i −3.29017 + 3.76494i 7.55371i −2.86214 + 0.929964i 2.68999 + 0.874032i 6.03487 − 18.5734i −1.57258 − 6.89398i
29.16 −0.831254 + 1.14412i 4.38207 − 3.18376i −0.618034 − 1.90211i −1.02902 − 4.89297i 7.66014i 12.0541 − 3.91660i 2.68999 + 0.874032i 6.28506 − 19.3434i 6.45353 + 2.88997i
29.17 0.831254 − 1.14412i −4.38207 + 3.18376i −0.618034 − 1.90211i −1.02902 + 4.89297i 7.66014i −12.0541 + 3.91660i −2.68999 − 0.874032i 6.28506 − 19.3434i 4.74278 + 5.24462i
29.18 0.831254 − 1.14412i −4.32119 + 3.13952i −0.618034 − 1.90211i −3.29017 − 3.76494i 7.55371i 2.86214 − 0.929964i −2.68999 − 0.874032i 6.03487 − 18.5734i −7.04252 + 0.634740i
29.19 0.831254 − 1.14412i −3.47472 + 2.52453i −0.618034 − 1.90211i 4.20395 − 2.70681i 6.07404i −4.21719 + 1.37025i −2.68999 − 0.874032i 2.91927 − 8.98460i 0.397626 − 7.05988i
29.20 0.831254 − 1.14412i −3.32133 + 2.41309i −0.618034 − 1.90211i −1.91795 + 4.61752i 5.80590i 10.8265 − 3.51773i −2.68999 − 0.874032i 2.42710 − 7.46983i 3.68870 + 6.03270i
See next 80 embeddings (of 128 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
31.f odd 10 1 inner
155.m odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 310.3.l.a ✓ 128
5.b even 2 1 inner 310.3.l.a ✓ 128
31.f odd 10 1 inner 310.3.l.a ✓ 128
155.m odd 10 1 inner 310.3.l.a ✓ 128
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
310.3.l.a ✓ 128 1.a even 1 1 trivial
310.3.l.a ✓ 128 5.b even 2 1 inner
310.3.l.a ✓ 128 31.f odd 10 1 inner
310.3.l.a ✓ 128 155.m odd 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(310, [\chi])\).