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Properties

Label 308.2.t.c
Level $308$
Weight $2$
Character orbit 308.t
Analytic conductor $2.459$
Analytic rank $0$
Dimension $160$
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [308,2,Mod(27,308)]
 
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([5, 5, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("308.27");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 308 = 2^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 308.t (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: \(2.45939238226\)
Analytic rank: \(0\)
Dimension: \(160\)
Relative dimension: \(40\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 160 q - 10 q^{2} - 10 q^{4} + 2 q^{8} - 60 q^{9} - 32 q^{14} - 2 q^{16} + 20 q^{18} - 4 q^{21} - 70 q^{22} + 20 q^{25} - 2 q^{28} - 36 q^{29} - 52 q^{30} + 80 q^{32} - 24 q^{37} + 40 q^{42} - 58 q^{44}+ \cdots - 84 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
27.1 −1.41136 − 0.0897191i −2.41265 − 1.75289i 1.98390 + 0.253253i 0.518254 + 0.168391i 3.24786 + 2.69043i 2.17575 − 1.50536i −2.77729 − 0.535426i 1.82119 + 5.60505i −0.716338 − 0.284158i
27.2 −1.41136 − 0.0897191i 2.41265 + 1.75289i 1.98390 + 0.253253i −0.518254 − 0.168391i −3.24786 − 2.69043i −2.10402 + 1.60408i −2.77729 − 0.535426i 1.82119 + 5.60505i 0.716338 + 0.284158i
27.3 −1.32343 − 0.498531i −1.44459 − 1.04956i 1.50293 + 1.31954i 1.17012 + 0.380196i 1.38858 + 2.10919i −2.18822 + 1.48717i −1.33119 − 2.49558i 0.0582261 + 0.179201i −1.35904 − 1.08650i
27.4 −1.32343 − 0.498531i 1.44459 + 1.04956i 1.50293 + 1.31954i −1.17012 − 0.380196i −1.38858 − 2.10919i 2.09058 − 1.62156i −1.33119 − 2.49558i 0.0582261 + 0.179201i 1.35904 + 1.08650i
27.5 −1.30165 + 0.552901i −0.433766 − 0.315149i 1.38860 − 1.43937i 3.05181 + 0.991593i 0.738858 + 0.170386i 2.57382 + 0.612751i −1.01165 + 2.64132i −0.838217 − 2.57977i −4.52065 + 0.396638i
27.6 −1.30165 + 0.552901i 0.433766 + 0.315149i 1.38860 − 1.43937i −3.05181 − 0.991593i −0.738858 − 0.170386i −0.212593 + 2.63720i −1.01165 + 2.64132i −0.838217 − 2.57977i 4.52065 − 0.396638i
27.7 −1.23496 − 0.689109i −1.21735 − 0.884453i 1.05026 + 1.70205i −3.50785 − 1.13977i 0.893890 + 1.93115i 1.48439 + 2.19011i −0.124133 − 2.82570i −0.227379 − 0.699799i 3.54664 + 3.82486i
27.8 −1.23496 − 0.689109i 1.21735 + 0.884453i 1.05026 + 1.70205i 3.50785 + 1.13977i −0.893890 − 1.93115i 1.62422 + 2.08852i −0.124133 − 2.82570i −0.227379 − 0.699799i −3.54664 − 3.82486i
27.9 −1.13901 + 0.838252i −1.68296 − 1.22274i 0.594668 − 1.90955i −2.50636 − 0.814365i 2.94187 − 0.0180337i −2.25678 − 1.38092i 0.923350 + 2.67347i 0.410207 + 1.26249i 3.53740 − 1.17339i
27.10 −1.13901 + 0.838252i 1.68296 + 1.22274i 0.594668 − 1.90955i 2.50636 + 0.814365i −2.94187 + 0.0180337i −0.615949 − 2.57305i 0.923350 + 2.67347i 0.410207 + 1.26249i −3.53740 + 1.17339i
27.11 −0.854618 + 1.12678i −1.68917 − 1.22726i −0.539257 − 1.92593i −0.865178 − 0.281114i 2.82644 − 0.854488i 2.56910 + 0.632227i 2.63095 + 1.03831i 0.420096 + 1.29292i 1.05615 − 0.734619i
27.12 −0.854618 + 1.12678i 1.68917 + 1.22726i −0.539257 − 1.92593i 0.865178 + 0.281114i −2.82644 + 0.854488i −0.192612 + 2.63873i 2.63095 + 1.03831i 0.420096 + 1.29292i −1.05615 + 0.734619i
27.13 −0.765067 − 1.18940i −0.171309 − 0.124464i −0.829346 + 1.81994i −1.47093 − 0.477935i −0.0169739 + 0.298978i 1.30415 − 2.30200i 2.79914 − 0.405951i −0.913195 − 2.81053i 0.556905 + 2.11518i
27.14 −0.765067 − 1.18940i 0.171309 + 0.124464i −0.829346 + 1.81994i 1.47093 + 0.477935i 0.0169739 − 0.298978i −2.59233 + 0.528968i 2.79914 − 0.405951i −0.913195 − 2.81053i −0.556905 − 2.11518i
27.15 −0.549899 − 1.30292i −2.49350 − 1.81164i −1.39522 + 1.43295i −2.98770 − 0.970763i −0.989248 + 4.24506i −2.10593 − 1.60158i 2.63426 + 1.02989i 2.00848 + 6.18146i 0.378104 + 4.42657i
27.16 −0.549899 − 1.30292i 2.49350 + 1.81164i −1.39522 + 1.43295i 2.98770 + 0.970763i 0.989248 − 4.24506i −0.872428 − 2.49777i 2.63426 + 1.02989i 2.00848 + 6.18146i −0.378104 − 4.42657i
27.17 −0.320962 − 1.37731i −2.49350 − 1.81164i −1.79397 + 0.884128i 2.98770 + 0.970763i −1.69487 + 4.01579i 0.872428 + 2.49777i 1.79351 + 2.18708i 2.00848 + 6.18146i 0.378104 − 4.42657i
27.18 −0.320962 − 1.37731i 2.49350 + 1.81164i −1.79397 + 0.884128i −2.98770 − 0.970763i 1.69487 − 4.01579i 2.10593 + 1.60158i 1.79351 + 2.18708i 2.00848 + 6.18146i −0.378104 + 4.42657i
27.19 −0.0801601 − 1.41194i −0.171309 − 0.124464i −1.98715 + 0.226362i 1.47093 + 0.477935i −0.162003 + 0.251856i 2.59233 − 0.528968i 0.478900 + 2.78759i −0.913195 − 2.81053i 0.556905 − 2.11518i
27.20 −0.0801601 − 1.41194i 0.171309 + 0.124464i −1.98715 + 0.226362i −1.47093 − 0.477935i 0.162003 − 0.251856i −1.30415 + 2.30200i 0.478900 + 2.78759i −0.913195 − 2.81053i −0.556905 + 2.11518i
See next 80 embeddings (of 160 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 27.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.b odd 2 1 inner
11.c even 5 1 inner
28.d even 2 1 inner
44.h odd 10 1 inner
77.j odd 10 1 inner
308.t even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 308.2.t.c ✓ 160
4.b odd 2 1 inner 308.2.t.c ✓ 160
7.b odd 2 1 inner 308.2.t.c ✓ 160
11.c even 5 1 inner 308.2.t.c ✓ 160
28.d even 2 1 inner 308.2.t.c ✓ 160
44.h odd 10 1 inner 308.2.t.c ✓ 160
77.j odd 10 1 inner 308.2.t.c ✓ 160
308.t even 10 1 inner 308.2.t.c ✓ 160
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
308.2.t.c ✓ 160 1.a even 1 1 trivial
308.2.t.c ✓ 160 4.b odd 2 1 inner
308.2.t.c ✓ 160 7.b odd 2 1 inner
308.2.t.c ✓ 160 11.c even 5 1 inner
308.2.t.c ✓ 160 28.d even 2 1 inner
308.2.t.c ✓ 160 44.h odd 10 1 inner
308.2.t.c ✓ 160 77.j odd 10 1 inner
308.2.t.c ✓ 160 308.t even 10 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(308, [\chi])\):

\( T_{3}^{80} + 45 T_{3}^{78} + 1143 T_{3}^{76} + 21711 T_{3}^{74} + 346846 T_{3}^{72} + \cdots + 168283265539216 \) Copy content Toggle raw display
\( T_{71}^{80} - 1062 T_{71}^{78} + 577846 T_{71}^{76} - 214984652 T_{71}^{74} + 63284314539 T_{71}^{72} + \cdots + 22\!\cdots\!16 \) Copy content Toggle raw display