Properties

Label 912.3.h.d
Level $912$
Weight $3$
Character orbit 912.h
Analytic conductor $24.850$
Analytic rank $0$
Dimension $36$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [912,3,Mod(305,912)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(912, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("912.305");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 912.h (of order \(2\), degree \(1\), not 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: \(24.8502001097\)
Analytic rank: \(0\)
Dimension: \(36\)
Twist minimal: no (minimal twist has level 456)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 8 q^{7} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q - 8 q^{7} + 16 q^{9} - 32 q^{13} + 48 q^{15} + 24 q^{21} - 140 q^{25} - 72 q^{27} + 24 q^{31} - 64 q^{33} - 48 q^{37} + 108 q^{39} - 16 q^{43} - 96 q^{45} + 276 q^{49} - 312 q^{51} - 32 q^{55} + 416 q^{61} - 28 q^{63} + 48 q^{67} + 160 q^{69} - 264 q^{73} + 400 q^{75} + 168 q^{79} + 216 q^{81} - 464 q^{85} + 68 q^{87} + 48 q^{91} - 200 q^{93} + 200 q^{97} - 104 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
305.1 0 −2.98372 0.312096i 0 2.00477i 0 9.74789 0 8.80519 + 1.86242i 0
305.2 0 −2.98372 + 0.312096i 0 2.00477i 0 9.74789 0 8.80519 1.86242i 0
305.3 0 −2.96520 0.455642i 0 8.19836i 0 −10.4735 0 8.58478 + 2.70214i 0
305.4 0 −2.96520 + 0.455642i 0 8.19836i 0 −10.4735 0 8.58478 2.70214i 0
305.5 0 −2.76466 1.16475i 0 7.59780i 0 −5.98667 0 6.28671 + 6.44028i 0
305.6 0 −2.76466 + 1.16475i 0 7.59780i 0 −5.98667 0 6.28671 6.44028i 0
305.7 0 −2.72635 1.25180i 0 0.528472i 0 −1.87136 0 5.86601 + 6.82568i 0
305.8 0 −2.72635 + 1.25180i 0 0.528472i 0 −1.87136 0 5.86601 6.82568i 0
305.9 0 −2.23280 2.00365i 0 8.74486i 0 7.20322 0 0.970748 + 8.94749i 0
305.10 0 −2.23280 + 2.00365i 0 8.74486i 0 7.20322 0 0.970748 8.94749i 0
305.11 0 −1.89465 2.32600i 0 1.91924i 0 −7.15920 0 −1.82058 + 8.81394i 0
305.12 0 −1.89465 + 2.32600i 0 1.91924i 0 −7.15920 0 −1.82058 8.81394i 0
305.13 0 −1.27146 2.71724i 0 4.57275i 0 6.79364 0 −5.76676 + 6.90974i 0
305.14 0 −1.27146 + 2.71724i 0 4.57275i 0 6.79364 0 −5.76676 6.90974i 0
305.15 0 −0.922699 2.85458i 0 8.24976i 0 6.69140 0 −7.29725 + 5.26784i 0
305.16 0 −0.922699 + 2.85458i 0 8.24976i 0 6.69140 0 −7.29725 5.26784i 0
305.17 0 0.316327 2.98328i 0 4.15317i 0 −12.8970 0 −8.79987 1.88738i 0
305.18 0 0.316327 + 2.98328i 0 4.15317i 0 −12.8970 0 −8.79987 + 1.88738i 0
305.19 0 0.423217 2.97000i 0 0.567002i 0 −4.38953 0 −8.64177 2.51391i 0
305.20 0 0.423217 + 2.97000i 0 0.567002i 0 −4.38953 0 −8.64177 + 2.51391i 0
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 305.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 912.3.h.d 36
3.b odd 2 1 inner 912.3.h.d 36
4.b odd 2 1 456.3.h.a 36
12.b even 2 1 456.3.h.a 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
456.3.h.a 36 4.b odd 2 1
456.3.h.a 36 12.b even 2 1
912.3.h.d 36 1.a even 1 1 trivial
912.3.h.d 36 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{36} + 520 T_{5}^{34} + 122068 T_{5}^{32} + 17126088 T_{5}^{30} + 1602774310 T_{5}^{28} + 105771193736 T_{5}^{26} + 5074684791836 T_{5}^{24} + 179915051470136 T_{5}^{22} + \cdots + 16\!\cdots\!00 \) acting on \(S_{3}^{\mathrm{new}}(912, [\chi])\). Copy content Toggle raw display