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Properties

Label 616.2.bp.c
Level $616$
Weight $2$
Character orbit 616.bp
Analytic conductor $4.919$
Analytic rank $0$
Dimension $352$
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [616,2,Mod(27,616)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(616, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 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("616.27");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 616 = 2^{3} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 616.bp (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: \(4.91878476451\)
Analytic rank: \(0\)
Dimension: \(352\)
Relative dimension: \(88\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 352 q - 2 q^{2} - 10 q^{4} - 14 q^{8} + 84 q^{9} - 28 q^{11} - 32 q^{14} + 14 q^{16} - 4 q^{18} - 62 q^{22} - 108 q^{25} - 44 q^{28} + 24 q^{30} - 112 q^{32} + 14 q^{35} + 20 q^{36} - 2 q^{42} + 8 q^{43}+ \cdots + 24 q^{99}+O(q^{100}) \) 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 −1.41358 − 0.0421964i −0.639605 + 0.880341i 1.99644 + 0.119296i −1.27214 + 3.91523i 0.941283 − 1.21745i −0.0153490 − 2.64571i −2.81710 − 0.252878i 0.561146 + 1.72703i 1.96348 − 5.48083i
27.2 −1.41358 − 0.0421964i 0.639605 − 0.880341i 1.99644 + 0.119296i 1.27214 − 3.91523i −0.941283 + 1.21745i −2.51147 − 0.832166i −2.81710 − 0.252878i 0.561146 + 1.72703i −1.96348 + 5.48083i
27.3 −1.40439 + 0.166420i −0.157507 + 0.216789i 1.94461 − 0.467435i 1.02719 − 3.16137i 0.185122 − 0.330669i 2.39093 − 1.13291i −2.65319 + 0.980081i 0.904862 + 2.78488i −0.916460 + 4.61073i
27.4 −1.40439 + 0.166420i 0.157507 − 0.216789i 1.94461 − 0.467435i −1.02719 + 3.16137i −0.185122 + 0.330669i −1.81629 + 1.92382i −2.65319 + 0.980081i 0.904862 + 2.78488i 0.916460 − 4.61073i
27.5 −1.37858 − 0.315464i −0.891388 + 1.22689i 1.80096 + 0.869785i −0.0834388 + 0.256798i 1.61589 − 1.41016i 1.62621 − 2.08697i −2.20839 − 1.76721i 0.216364 + 0.665900i 0.196038 − 0.327695i
27.6 −1.37858 − 0.315464i 0.891388 − 1.22689i 1.80096 + 0.869785i 0.0834388 − 0.256798i −1.61589 + 1.41016i −2.48735 + 0.901708i −2.20839 − 1.76721i 0.216364 + 0.665900i −0.196038 + 0.327695i
27.7 −1.37736 − 0.320751i −0.817396 + 1.12505i 1.79424 + 0.883580i 0.552411 − 1.70015i 1.48671 − 1.28742i 1.84190 + 1.89932i −2.18790 − 1.79251i 0.329451 + 1.01395i −1.30619 + 2.16452i
27.8 −1.37736 − 0.320751i 0.817396 − 1.12505i 1.79424 + 0.883580i −0.552411 + 1.70015i −1.48671 + 1.28742i 1.23718 + 2.33867i −2.18790 − 1.79251i 0.329451 + 1.01395i 1.30619 − 2.16452i
27.9 −1.36072 + 0.385294i −1.25998 + 1.73421i 1.70310 − 1.04855i 0.0436016 − 0.134192i 1.04629 − 2.84523i −2.64209 − 0.139117i −1.91343 + 2.08298i −0.492894 − 1.51697i −0.00762601 + 0.199397i
27.10 −1.36072 + 0.385294i 1.25998 − 1.73421i 1.70310 − 1.04855i −0.0436016 + 0.134192i −1.04629 + 2.84523i 0.684143 − 2.55577i −1.91343 + 2.08298i −0.492894 − 1.51697i 0.00762601 − 0.199397i
27.11 −1.35633 + 0.400474i −1.82748 + 2.51531i 1.67924 − 1.08635i −0.705747 + 2.17207i 1.47134 − 4.14344i 1.70685 + 2.02155i −1.84254 + 2.14593i −2.06006 − 6.34020i 0.0873670 − 3.22866i
27.12 −1.35633 + 0.400474i 1.82748 − 2.51531i 1.67924 − 1.08635i 0.705747 − 2.17207i −1.47134 + 4.14344i 1.39516 + 2.24800i −1.84254 + 2.14593i −2.06006 − 6.34020i −0.0873670 + 3.22866i
27.13 −1.30368 − 0.548105i −1.45558 + 2.00343i 1.39916 + 1.42911i 0.401910 − 1.23695i 2.99570 − 1.81402i −1.21393 + 2.35083i −1.04076 − 2.62999i −0.967981 − 2.97914i −1.20194 + 1.39230i
27.14 −1.30368 − 0.548105i 1.45558 − 2.00343i 1.39916 + 1.42911i −0.401910 + 1.23695i −2.99570 + 1.81402i 2.61089 − 0.428069i −1.04076 − 2.62999i −0.967981 − 2.97914i 1.20194 − 1.39230i
27.15 −1.19374 + 0.758283i −0.793216 + 1.09177i 0.850015 − 1.81038i −0.358256 + 1.10260i 0.119022 − 1.90477i 2.45449 + 0.987654i 0.358085 + 2.80567i 0.364285 + 1.12115i −0.408418 − 1.58787i
27.16 −1.19374 + 0.758283i 0.793216 − 1.09177i 0.850015 − 1.81038i 0.358256 − 1.10260i −0.119022 + 1.90477i 0.180835 + 2.63956i 0.358085 + 2.80567i 0.364285 + 1.12115i 0.408418 + 1.58787i
27.17 −1.05400 − 0.942910i −1.34043 + 1.84494i 0.221842 + 1.98766i 1.27821 − 3.93392i 3.15243 − 0.680670i −1.37149 − 2.26252i 1.64036 − 2.30417i −0.680011 − 2.09286i −5.05656 + 2.94112i
27.18 −1.05400 − 0.942910i 1.34043 − 1.84494i 0.221842 + 1.98766i −1.27821 + 3.93392i −3.15243 + 0.680670i −1.72797 − 2.00352i 1.64036 − 2.30417i −0.680011 − 2.09286i 5.05656 − 2.94112i
27.19 −1.04343 + 0.954597i −1.29235 + 1.77877i 0.177487 − 1.99211i 1.23186 − 3.79128i −0.349532 − 3.08970i −1.69416 + 2.03220i 1.71647 + 2.24805i −0.566803 − 1.74444i 2.33379 + 5.13186i
27.20 −1.04343 + 0.954597i 1.29235 − 1.77877i 0.177487 − 1.99211i −1.23186 + 3.79128i 0.349532 + 3.08970i 2.45626 − 0.983259i 1.71647 + 2.24805i −0.566803 − 1.74444i −2.33379 − 5.13186i
See next 80 embeddings (of 352 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 27.88
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
8.d odd 2 1 inner
11.c even 5 1 inner
56.e even 2 1 inner
77.j odd 10 1 inner
88.l odd 10 1 inner
616.bp even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 616.2.bp.c ✓ 352
7.b odd 2 1 inner 616.2.bp.c ✓ 352
8.d odd 2 1 inner 616.2.bp.c ✓ 352
11.c even 5 1 inner 616.2.bp.c ✓ 352
56.e even 2 1 inner 616.2.bp.c ✓ 352
77.j odd 10 1 inner 616.2.bp.c ✓ 352
88.l odd 10 1 inner 616.2.bp.c ✓ 352
616.bp even 10 1 inner 616.2.bp.c ✓ 352
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
616.2.bp.c ✓ 352 1.a even 1 1 trivial
616.2.bp.c ✓ 352 7.b odd 2 1 inner
616.2.bp.c ✓ 352 8.d odd 2 1 inner
616.2.bp.c ✓ 352 11.c even 5 1 inner
616.2.bp.c ✓ 352 56.e even 2 1 inner
616.2.bp.c ✓ 352 77.j odd 10 1 inner
616.2.bp.c ✓ 352 88.l odd 10 1 inner
616.2.bp.c ✓ 352 616.bp 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 \(S_{2}^{\mathrm{new}}(616, [\chi])\):

\( T_{3}^{176} - 87 T_{3}^{174} + 4107 T_{3}^{172} - 139603 T_{3}^{170} + 3825410 T_{3}^{168} + \cdots + 71\!\cdots\!36 \) Copy content Toggle raw display
\( T_{29}^{176} - 495 T_{29}^{174} + 141387 T_{29}^{172} - 30356430 T_{29}^{170} + 5443913335 T_{29}^{168} + \cdots + 75\!\cdots\!00 \) Copy content Toggle raw display