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Properties

Label 616.2.bi.c
Level $616$
Weight $2$
Character orbit 616.bi
Analytic conductor $4.919$
Analytic rank $0$
Dimension $352$
CM no
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(13,616)]
 
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([0, 5, 5, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("616.13");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 616 = 2^{3} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 616.bi (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 dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 352 q - 10 q^{2} - 10 q^{4} - 10 q^{7} - 10 q^{8} - 108 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 352 q - 10 q^{2} - 10 q^{4} - 10 q^{7} - 10 q^{8} - 108 q^{9} + 10 q^{14} + 12 q^{15} + 14 q^{16} - 20 q^{18} + 54 q^{22} - 72 q^{23} - 108 q^{25} + 30 q^{28} + 20 q^{30} - 20 q^{39} - 26 q^{42} + 26 q^{44} - 50 q^{46} - 34 q^{49} + 40 q^{50} + 44 q^{56} - 20 q^{57} + 78 q^{58} + 88 q^{60} + 20 q^{63} - 34 q^{64} - 72 q^{70} - 152 q^{71} + 90 q^{72} - 150 q^{74} + 116 q^{78} - 200 q^{79} - 12 q^{81} - 50 q^{84} + 50 q^{86} - 38 q^{88} + 16 q^{92} - 20 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 −1.41421 − 0.00244584i −2.62406 + 1.90649i 1.99999 + 0.00691788i 0.0836925 + 0.257579i 3.71563 − 2.68976i −2.26578 − 1.36610i −2.82839 − 0.0146750i 2.32392 − 7.15229i −0.117729 − 0.364476i
13.2 −1.41421 − 0.00244584i 2.62406 − 1.90649i 1.99999 + 0.00691788i −0.0836925 − 0.257579i −3.71563 + 2.68976i 1.99940 + 1.73274i −2.82839 − 0.0146750i 2.32392 − 7.15229i 0.117729 + 0.364476i
13.3 −1.41415 − 0.0129456i −1.66625 + 1.21060i 1.99966 + 0.0366142i 0.191944 + 0.590744i 2.37200 − 1.69040i 2.23539 − 1.41529i −2.82736 − 0.0776650i 0.383777 − 1.18114i −0.263791 − 0.837888i
13.4 −1.41415 − 0.0129456i 1.66625 − 1.21060i 1.99966 + 0.0366142i −0.191944 − 0.590744i −2.37200 + 1.69040i 0.655244 − 2.56333i −2.82736 − 0.0776650i 0.383777 − 1.18114i 0.263791 + 0.837888i
13.5 −1.41337 + 0.0487330i −1.21258 + 0.880988i 1.99525 − 0.137756i −1.22746 − 3.77772i 1.67089 − 1.30426i 2.64103 + 0.157973i −2.81332 + 0.291935i −0.232850 + 0.716639i 1.91895 + 5.27952i
13.6 −1.41337 + 0.0487330i 1.21258 − 0.880988i 1.99525 − 0.137756i 1.22746 + 3.77772i −1.67089 + 1.30426i −0.966365 − 2.46295i −2.81332 + 0.291935i −0.232850 + 0.716639i −1.91895 − 5.27952i
13.7 −1.39240 − 0.247420i −0.531724 + 0.386320i 1.87757 + 0.689016i 0.488477 + 1.50338i 0.835957 − 0.406354i −2.64550 − 0.0363803i −2.44385 − 1.42394i −0.793564 + 2.44234i −0.308190 − 2.21416i
13.8 −1.39240 − 0.247420i 0.531724 − 0.386320i 1.87757 + 0.689016i −0.488477 − 1.50338i −0.835957 + 0.406354i 0.852105 + 2.50478i −2.44385 − 1.42394i −0.793564 + 2.44234i 0.308190 + 2.21416i
13.9 −1.35856 + 0.392821i −1.28776 + 0.935610i 1.69138 − 1.06734i 0.995233 + 3.06301i 1.38197 − 1.77694i −0.792771 + 2.52419i −1.87857 + 2.11447i −0.144099 + 0.443493i −2.55530 − 3.77034i
13.10 −1.35856 + 0.392821i 1.28776 − 0.935610i 1.69138 − 1.06734i −0.995233 − 3.06301i −1.38197 + 1.77694i −2.15566 + 1.53399i −1.87857 + 2.11447i −0.144099 + 0.443493i 2.55530 + 3.77034i
13.11 −1.27191 − 0.618266i −0.531724 + 0.386320i 1.23549 + 1.57275i 0.488477 + 1.50338i 0.915153 − 0.162616i 0.852105 + 2.50478i −0.599051 − 2.76426i −0.793564 + 2.44234i 0.308190 − 2.21416i
13.12 −1.27191 − 0.618266i 0.531724 − 0.386320i 1.23549 + 1.57275i −0.488477 − 1.50338i −0.915153 + 0.162616i −2.64550 − 0.0363803i −0.599051 − 2.76426i −0.793564 + 2.44234i −0.308190 + 2.21416i
13.13 −1.26963 + 0.622922i −2.02333 + 1.47003i 1.22394 − 1.58177i −0.495683 − 1.52556i 1.65317 − 3.12678i −0.564533 + 2.58482i −0.568632 + 2.77068i 1.00581 − 3.09555i 1.57964 + 1.62812i
13.14 −1.26963 + 0.622922i 2.02333 − 1.47003i 1.22394 − 1.58177i 0.495683 + 1.52556i −1.65317 + 3.12678i −2.28386 + 1.33566i −0.568632 + 2.77068i 1.00581 − 3.09555i −1.57964 − 1.62812i
13.15 −1.26637 + 0.629539i −0.250134 + 0.181733i 1.20736 − 1.59445i −0.645803 − 1.98758i 0.202353 − 0.387610i −1.50618 − 2.17518i −0.525193 + 2.77924i −0.897511 + 2.76225i 2.06908 + 2.11044i
13.16 −1.26637 + 0.629539i 0.250134 − 0.181733i 1.20736 − 1.59445i 0.645803 + 1.98758i −0.202353 + 0.387610i 2.53416 + 0.760297i −0.525193 + 2.77924i −0.897511 + 2.76225i −2.06908 − 2.11044i
13.17 −1.15168 − 0.820746i −1.66625 + 1.21060i 0.652753 + 1.89048i 0.191944 + 0.590744i 2.91258 − 0.0266627i 0.655244 − 2.56333i 0.799839 − 2.71298i 0.383777 − 1.18114i 0.263791 − 0.837888i
13.18 −1.15168 − 0.820746i 1.66625 − 1.21060i 0.652753 + 1.89048i −0.191944 − 0.590744i −2.91258 + 0.0266627i 2.23539 − 1.41529i 0.799839 − 2.71298i 0.383777 − 1.18114i −0.263791 + 0.837888i
13.19 −1.14556 − 0.829274i −2.62406 + 1.90649i 0.624610 + 1.89996i 0.0836925 + 0.257579i 4.58701 − 0.00793312i 1.99940 + 1.73274i 0.860064 − 2.69449i 2.32392 − 7.15229i 0.117729 − 0.364476i
13.20 −1.14556 − 0.829274i 2.62406 − 1.90649i 0.624610 + 1.89996i −0.0836925 − 0.257579i −4.58701 + 0.00793312i −2.26578 − 1.36610i 0.860064 − 2.69449i 2.32392 − 7.15229i −0.117729 + 0.364476i
See next 80 embeddings (of 352 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.88
Significant digits:
Format:

Inner twists

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

\( T_{3}^{176} + 93 T_{3}^{174} + 4623 T_{3}^{172} + 163413 T_{3}^{170} + 4612806 T_{3}^{168} + \cdots + 11\!\cdots\!56 \) Copy content Toggle raw display
\( T_{29}^{176} + 593 T_{29}^{174} + 190271 T_{29}^{172} + 43892602 T_{29}^{170} + 8179551559 T_{29}^{168} + \cdots + 24\!\cdots\!96 \) Copy content Toggle raw display