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Properties

Label 312.2.bt.c
Level $312$
Weight $2$
Character orbit 312.bt
Analytic conductor $2.491$
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] = [312,2,Mod(19,312)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(312, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 6, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("312.19");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 312 = 2^{3} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 312.bt (of order \(12\), 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: \(2.49133254306\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(12\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 2 q^{2} - 24 q^{3} + 2 q^{6} - 4 q^{8} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 2 q^{2} - 24 q^{3} + 2 q^{6} - 4 q^{8} - 24 q^{9} - 6 q^{10} + 20 q^{11} - 28 q^{14} + 4 q^{16} + 12 q^{17} - 4 q^{18} + 20 q^{19} + 4 q^{20} + 14 q^{22} + 2 q^{24} - 12 q^{26} + 48 q^{27} - 2 q^{28} + 6 q^{30} - 18 q^{32} - 28 q^{33} + 12 q^{35} + 52 q^{40} - 36 q^{41} + 14 q^{42} + 12 q^{43} + 12 q^{44} - 14 q^{46} + 4 q^{48} + 24 q^{49} - 32 q^{50} + 24 q^{52} + 2 q^{54} - 54 q^{56} - 40 q^{57} - 22 q^{58} + 96 q^{59} - 56 q^{60} - 108 q^{62} - 52 q^{65} - 28 q^{66} - 12 q^{67} - 6 q^{68} - 52 q^{70} + 2 q^{72} - 48 q^{73} + 114 q^{76} + 36 q^{78} - 16 q^{80} - 24 q^{81} - 60 q^{82} + 48 q^{83} + 10 q^{84} + 24 q^{86} + 102 q^{88} + 80 q^{89} - 152 q^{91} - 100 q^{92} - 10 q^{94} - 24 q^{96} - 64 q^{97} - 6 q^{98} + 8 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −1.39432 − 0.236374i −0.500000 + 0.866025i 1.88825 + 0.659163i 1.86647 + 1.86647i 0.901866 − 1.08933i 4.71214 + 1.26261i −2.47702 − 1.36542i −0.500000 − 0.866025i −2.16128 − 3.04365i
19.2 −1.34885 + 0.424973i −0.500000 + 0.866025i 1.63880 − 1.14645i −0.728760 − 0.728760i 0.306387 − 1.38063i −0.0829661 − 0.0222307i −1.72328 + 2.24284i −0.500000 − 0.866025i 1.29269 + 0.673285i
19.3 −0.839367 − 1.13818i −0.500000 + 0.866025i −0.590927 + 1.91071i 2.14006 + 2.14006i 1.40538 − 0.157821i −1.94933 − 0.522322i 2.67074 − 0.931201i −0.500000 − 0.866025i 0.639489 − 4.23209i
19.4 −0.785789 − 1.17581i −0.500000 + 0.866025i −0.765072 + 1.84788i −0.835211 − 0.835211i 1.41118 − 0.0926066i −0.794398 − 0.212858i 2.77395 − 0.552463i −0.500000 − 0.866025i −0.325753 + 1.63835i
19.5 −0.614612 + 1.27368i −0.500000 + 0.866025i −1.24450 − 1.56563i −2.47909 − 2.47909i −0.795730 − 1.16911i 3.75755 + 1.00683i 2.75900 − 0.622837i −0.500000 − 0.866025i 4.68125 − 1.63388i
19.6 0.0926066 − 1.41118i −0.500000 + 0.866025i −1.98285 − 0.261369i 0.835211 + 0.835211i 1.17581 + 0.785789i 0.794398 + 0.212858i −0.552463 + 2.77395i −0.500000 − 0.866025i 1.25598 − 1.10129i
19.7 0.157821 − 1.40538i −0.500000 + 0.866025i −1.95019 − 0.443596i −2.14006 − 2.14006i 1.13818 + 0.839367i 1.94933 + 0.522322i −0.931201 + 2.67074i −0.500000 − 0.866025i −3.34535 + 2.66986i
19.8 0.210345 + 1.39848i −0.500000 + 0.866025i −1.91151 + 0.588328i −1.60057 − 1.60057i −1.31629 − 0.517077i −3.02018 − 0.809254i −1.22484 − 2.54946i −0.500000 − 0.866025i 1.90170 − 2.57504i
19.9 0.517077 + 1.31629i −0.500000 + 0.866025i −1.46526 + 1.36125i 1.60057 + 1.60057i −1.39848 − 0.210345i 3.02018 + 0.809254i −2.54946 − 1.22484i −0.500000 − 0.866025i −1.27920 + 2.93444i
19.10 1.08933 − 0.901866i −0.500000 + 0.866025i 0.373276 − 1.96486i −1.86647 − 1.86647i 0.236374 + 1.39432i −4.71214 − 1.26261i −1.36542 − 2.47702i −0.500000 − 0.866025i −3.71651 − 0.349895i
19.11 1.16911 + 0.795730i −0.500000 + 0.866025i 0.733627 + 1.86059i 2.47909 + 2.47909i −1.27368 + 0.614612i −3.75755 − 1.00683i −0.622837 + 2.75900i −0.500000 − 0.866025i 0.925639 + 4.87102i
19.12 1.38063 − 0.306387i −0.500000 + 0.866025i 1.81225 − 0.846013i 0.728760 + 0.728760i −0.424973 + 1.34885i 0.0829661 + 0.0222307i 2.24284 − 1.72328i −0.500000 − 0.866025i 1.22943 + 0.782862i
67.1 −1.39746 − 0.217051i −0.500000 − 0.866025i 1.90578 + 0.606638i 0.706529 + 0.706529i 0.510758 + 1.31876i −0.914686 − 3.41366i −2.53157 − 1.26140i −0.500000 + 0.866025i −0.833992 − 1.14070i
67.2 −1.31876 − 0.510758i −0.500000 − 0.866025i 1.47825 + 1.34713i −0.706529 − 0.706529i 0.217051 + 1.39746i 0.914686 + 3.41366i −1.26140 − 2.53157i −0.500000 + 0.866025i 0.570877 + 1.29261i
67.3 −1.22892 + 0.699821i −0.500000 − 0.866025i 1.02050 − 1.72005i −0.957986 − 0.957986i 1.22052 + 0.714368i 0.656748 + 2.45102i −0.0503920 + 2.82798i −0.500000 + 0.866025i 1.84771 + 0.506873i
67.4 −0.714368 − 1.22052i −0.500000 − 0.866025i −0.979356 + 1.74381i 0.957986 + 0.957986i −0.699821 + 1.22892i −0.656748 − 2.45102i 2.82798 − 0.0503920i −0.500000 + 0.866025i 0.484890 − 1.85360i
67.5 −0.620496 + 1.27082i −0.500000 − 0.866025i −1.22997 − 1.57708i 2.72895 + 2.72895i 1.41081 − 0.0980454i −0.628900 − 2.34709i 2.76737 − 0.584503i −0.500000 + 0.866025i −5.16131 + 1.77470i
67.6 0.0980454 − 1.41081i −0.500000 − 0.866025i −1.98077 − 0.276647i −2.72895 − 2.72895i −1.27082 + 0.620496i 0.628900 + 2.34709i −0.584503 + 2.76737i −0.500000 + 0.866025i −4.11759 + 3.58247i
67.7 0.610305 + 1.27575i −0.500000 − 0.866025i −1.25506 + 1.55719i 1.07676 + 1.07676i 0.799676 − 1.16641i 0.276971 + 1.03367i −2.75254 − 0.650774i −0.500000 + 0.866025i −0.716521 + 2.03083i
67.8 0.818904 + 1.15299i −0.500000 − 0.866025i −0.658792 + 1.88838i −2.80467 − 2.80467i 0.589070 − 1.28569i −0.527394 − 1.96826i −2.71678 + 0.786822i −0.500000 + 0.866025i 0.937012 − 5.53052i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
13.f odd 12 1 inner
104.u even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 312.2.bt.c ✓ 48
3.b odd 2 1 936.2.ed.c 48
8.d odd 2 1 inner 312.2.bt.c ✓ 48
13.f odd 12 1 inner 312.2.bt.c ✓ 48
24.f even 2 1 936.2.ed.c 48
39.k even 12 1 936.2.ed.c 48
104.u even 12 1 inner 312.2.bt.c ✓ 48
312.bq odd 12 1 936.2.ed.c 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
312.2.bt.c ✓ 48 1.a even 1 1 trivial
312.2.bt.c ✓ 48 8.d odd 2 1 inner
312.2.bt.c ✓ 48 13.f odd 12 1 inner
312.2.bt.c ✓ 48 104.u even 12 1 inner
936.2.ed.c 48 3.b odd 2 1
936.2.ed.c 48 24.f even 2 1
936.2.ed.c 48 39.k even 12 1
936.2.ed.c 48 312.bq odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{48} + 792 T_{5}^{44} + 241916 T_{5}^{40} + 36083752 T_{5}^{36} + 2772003622 T_{5}^{32} + \cdots + 1712789917696 \) acting on \(S_{2}^{\mathrm{new}}(312, [\chi])\). Copy content Toggle raw display