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Properties

Label 612.2.bd.d
Level $612$
Weight $2$
Character orbit 612.bd
Analytic conductor $4.887$
Analytic rank $0$
Dimension $48$
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] = [612,2,Mod(91,612)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(612, base_ring=CyclotomicField(16))
 
chi = DirichletCharacter(H, H._module([8, 0, 15]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("612.91");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 612 = 2^{2} \cdot 3^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 612.bd (of order \(16\), degree \(8\), 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: \(4.88684460370\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(6\) over \(\Q(\zeta_{16})\)
Twist minimal: no (minimal twist has level 68)
Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 8 q^{2} - 8 q^{4} + 16 q^{5} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 8 q^{2} - 8 q^{4} + 16 q^{5} + 8 q^{8} + 16 q^{10} - 16 q^{13} + 8 q^{14} + 16 q^{17} + 24 q^{20} - 8 q^{22} + 16 q^{25} + 16 q^{26} + 40 q^{28} - 32 q^{32} + 56 q^{34} - 16 q^{37} - 32 q^{38} + 56 q^{40} + 48 q^{41} - 24 q^{44} + 8 q^{46} - 16 q^{49} - 16 q^{52} - 48 q^{53} + 48 q^{56} - 64 q^{58} + 16 q^{61} + 64 q^{62} - 56 q^{64} - 96 q^{65} + 32 q^{68} - 80 q^{70} + 64 q^{73} + 16 q^{74} - 64 q^{76} - 16 q^{77} + 24 q^{80} - 40 q^{82} - 80 q^{85} - 64 q^{86} + 56 q^{88} + 16 q^{89} - 104 q^{92} + 88 q^{94} - 16 q^{97} - 72 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
91.1 −1.15431 + 0.817044i 0 0.664880 − 1.88625i 1.26427 − 1.89211i 0 −3.23087 + 2.15880i 0.773668 + 2.72056i 0 0.0865750 + 3.21705i
91.2 −0.368108 + 1.36547i 0 −1.72899 − 1.00528i −0.991762 + 1.48428i 0 1.11612 − 0.745767i 2.00913 − 1.99083i 0 −1.66165 − 1.90059i
91.3 −0.238486 − 1.39396i 0 −1.88625 + 0.664880i 1.26427 − 1.89211i 0 3.23087 − 2.15880i 1.37666 + 2.47079i 0 −2.93903 − 1.31109i
91.4 0.278734 + 1.38647i 0 −1.84461 + 0.772915i 0.561585 − 0.840472i 0 1.73093 − 1.15657i −1.58578 − 2.34207i 0 1.32182 + 0.544354i
91.5 0.705238 − 1.22582i 0 −1.00528 − 1.72899i −0.991762 + 1.48428i 0 −1.11612 + 0.745767i −2.82840 + 0.0129412i 0 1.12003 + 2.26249i
91.6 1.17748 − 0.783289i 0 0.772915 − 1.84461i 0.561585 − 0.840472i 0 −1.73093 + 1.15657i −0.534775 − 2.77741i 0 0.00292235 − 1.42952i
163.1 −1.39534 + 0.230266i 0 1.89395 − 0.642600i 0.803267 − 0.159780i 0 0.704584 − 3.54218i −2.49474 + 1.33276i 0 −1.08404 + 0.407912i
163.2 −0.968540 − 1.03050i 0 −0.123859 + 1.99616i 3.28423 − 0.653273i 0 0.628453 − 3.15945i 2.17701 − 1.80573i 0 −3.85410 − 2.75167i
163.3 −0.121817 − 1.40896i 0 −1.97032 + 0.343271i −1.07382 + 0.213597i 0 −0.127514 + 0.641054i 0.723673 + 2.73428i 0 0.431759 + 1.48695i
163.4 0.823832 + 1.14948i 0 −0.642600 + 1.89395i 0.803267 − 0.159780i 0 −0.704584 + 3.54218i −2.70645 + 0.821646i 0 0.845420 + 0.791706i
163.5 1.08242 − 0.910145i 0 0.343271 − 1.97032i −1.07382 + 0.213597i 0 0.127514 − 0.641054i −1.42172 − 2.44514i 0 −0.967924 + 1.20854i
163.6 1.41353 − 0.0438119i 0 1.99616 − 0.123859i 3.28423 − 0.653273i 0 −0.628453 + 3.15945i 2.81622 − 0.262535i 0 4.61375 − 1.06731i
199.1 −1.39534 − 0.230266i 0 1.89395 + 0.642600i 0.803267 + 0.159780i 0 0.704584 + 3.54218i −2.49474 − 1.33276i 0 −1.08404 − 0.407912i
199.2 −0.968540 + 1.03050i 0 −0.123859 − 1.99616i 3.28423 + 0.653273i 0 0.628453 + 3.15945i 2.17701 + 1.80573i 0 −3.85410 + 2.75167i
199.3 −0.121817 + 1.40896i 0 −1.97032 − 0.343271i −1.07382 − 0.213597i 0 −0.127514 − 0.641054i 0.723673 − 2.73428i 0 0.431759 − 1.48695i
199.4 0.823832 − 1.14948i 0 −0.642600 − 1.89395i 0.803267 + 0.159780i 0 −0.704584 − 3.54218i −2.70645 − 0.821646i 0 0.845420 − 0.791706i
199.5 1.08242 + 0.910145i 0 0.343271 + 1.97032i −1.07382 − 0.213597i 0 0.127514 + 0.641054i −1.42172 + 2.44514i 0 −0.967924 − 1.20854i
199.6 1.41353 + 0.0438119i 0 1.99616 + 0.123859i 3.28423 + 0.653273i 0 −0.628453 − 3.15945i 2.81622 + 0.262535i 0 4.61375 + 1.06731i
235.1 −1.07774 − 0.915678i 0 0.323068 + 1.97373i −0.763429 + 0.510107i 0 0.225807 − 0.337944i 1.45912 − 2.42301i 0 1.28988 + 0.149290i
235.2 −0.114599 + 1.40956i 0 −1.97373 − 0.323068i −0.763429 + 0.510107i 0 −0.225807 + 0.337944i 0.681573 − 2.74508i 0 −0.631540 − 1.13456i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 91.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
17.e odd 16 1 inner
68.i even 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 612.2.bd.d 48
3.b odd 2 1 68.2.i.b ✓ 48
4.b odd 2 1 inner 612.2.bd.d 48
12.b even 2 1 68.2.i.b ✓ 48
17.e odd 16 1 inner 612.2.bd.d 48
51.i even 16 1 68.2.i.b ✓ 48
68.i even 16 1 inner 612.2.bd.d 48
204.t odd 16 1 68.2.i.b ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
68.2.i.b ✓ 48 3.b odd 2 1
68.2.i.b ✓ 48 12.b even 2 1
68.2.i.b ✓ 48 51.i even 16 1
68.2.i.b ✓ 48 204.t odd 16 1
612.2.bd.d 48 1.a even 1 1 trivial
612.2.bd.d 48 4.b odd 2 1 inner
612.2.bd.d 48 17.e odd 16 1 inner
612.2.bd.d 48 68.i even 16 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{24} - 8 T_{5}^{23} + 28 T_{5}^{22} - 56 T_{5}^{21} + 40 T_{5}^{20} + 40 T_{5}^{19} + 472 T_{5}^{18} + \cdots + 2048 \) acting on \(S_{2}^{\mathrm{new}}(612, [\chi])\). Copy content Toggle raw display