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Properties

Label 756.2.bf.c
Level $756$
Weight $2$
Character orbit 756.bf
Analytic conductor $6.037$
Analytic rank $0$
Dimension $32$
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] = [756,2,Mod(271,756)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(756, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 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("756.271");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 756.bf (of order \(6\), degree \(2\), 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: \(6.03669039281\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 6 q^{11} + 17 q^{14} - 4 q^{16} - 8 q^{20} + 2 q^{22} + 14 q^{25} - 15 q^{26} - 13 q^{28} - 15 q^{32} - 6 q^{35} + 4 q^{37} + q^{38} - 15 q^{40} + 42 q^{44} - 9 q^{46} + 4 q^{47} + 14 q^{49} - 9 q^{52} - 45 q^{56} + 10 q^{58} + 16 q^{59} - 42 q^{64} + 49 q^{68} - 33 q^{70} + 36 q^{73} + 54 q^{74} + 15 q^{80} - 51 q^{82} - 20 q^{83} + 16 q^{85} - 78 q^{86} - 2 q^{88} - 27 q^{94} - 24 q^{95} + 46 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
271.1 −1.38270 + 0.296888i 0 1.82372 − 0.821014i −0.421763 − 0.243505i 0 0.250520 + 2.63386i −2.27790 + 1.67665i 0 0.655465 + 0.211478i
271.2 −1.31567 − 0.518653i 0 1.46200 + 1.36476i −2.03190 − 1.17312i 0 2.03407 − 1.69191i −1.21568 − 2.55385i 0 2.06488 + 2.59729i
271.3 −1.29094 + 0.577482i 0 1.33303 − 1.49098i 3.03704 + 1.75344i 0 −0.151085 − 2.64143i −0.859840 + 2.69456i 0 −4.93321 − 0.509738i
271.4 −1.10700 − 0.880081i 0 0.450916 + 1.94851i 2.03190 + 1.17312i 0 −2.03407 + 1.69191i 1.21568 − 2.55385i 0 −1.21688 − 3.08688i
271.5 −0.996999 + 1.00299i 0 −0.0119854 − 1.99996i −3.53919 − 2.04335i 0 −2.14499 − 1.54888i 2.01790 + 1.98194i 0 5.57803 − 1.51256i
271.6 −0.434237 − 1.34590i 0 −1.62288 + 1.16888i 0.421763 + 0.243505i 0 −0.250520 − 2.63386i 2.27790 + 1.67665i 0 0.144587 − 0.673389i
271.7 −0.428236 + 1.34782i 0 −1.63323 − 1.15437i 0.703801 + 0.406340i 0 2.60258 − 0.476010i 2.25528 − 1.70695i 0 −0.849065 + 0.774587i
271.8 −0.145354 − 1.40672i 0 −1.95774 + 0.408946i −3.03704 − 1.75344i 0 0.151085 + 2.64143i 0.859840 + 2.69456i 0 −2.02516 + 4.52715i
271.9 −0.0660073 + 1.41267i 0 −1.99129 − 0.186493i 1.13024 + 0.652543i 0 −2.50492 − 0.851694i 0.394894 − 2.80072i 0 −0.996433 + 1.55358i
271.10 0.370117 − 1.36492i 0 −1.72603 − 1.01036i 3.53919 + 2.04335i 0 2.14499 + 1.54888i −2.01790 + 1.98194i 0 4.09893 − 4.07444i
271.11 0.870799 + 1.11432i 0 −0.483419 + 1.94070i −1.71130 − 0.988019i 0 2.27191 − 1.35588i −2.58352 + 1.15127i 0 −0.389228 − 2.76730i
271.12 0.953127 − 1.04477i 0 −0.183097 − 1.99160i −0.703801 − 0.406340i 0 −2.60258 + 0.476010i −2.25528 − 1.70695i 0 −1.09534 + 0.348018i
271.13 1.02662 + 0.972657i 0 0.107877 + 1.99709i 2.11960 + 1.22375i 0 1.64814 + 2.06969i −1.83173 + 2.15517i 0 0.985722 + 3.31796i
271.14 1.19041 − 0.763500i 0 0.834135 − 1.81775i −1.13024 − 0.652543i 0 2.50492 + 0.851694i −0.394894 − 2.80072i 0 −1.84366 + 0.0861452i
271.15 1.35565 + 0.402746i 0 1.67559 + 1.09197i −2.11960 − 1.22375i 0 −1.64814 − 2.06969i 1.83173 + 2.15517i 0 −2.38058 − 2.51264i
271.16 1.40043 + 0.196974i 0 1.92240 + 0.551696i 1.71130 + 0.988019i 0 −2.27191 + 1.35588i 2.58352 + 1.15127i 0 2.20194 + 1.72073i
703.1 −1.38270 − 0.296888i 0 1.82372 + 0.821014i −0.421763 + 0.243505i 0 0.250520 − 2.63386i −2.27790 − 1.67665i 0 0.655465 − 0.211478i
703.2 −1.31567 + 0.518653i 0 1.46200 − 1.36476i −2.03190 + 1.17312i 0 2.03407 + 1.69191i −1.21568 + 2.55385i 0 2.06488 − 2.59729i
703.3 −1.29094 − 0.577482i 0 1.33303 + 1.49098i 3.03704 − 1.75344i 0 −0.151085 + 2.64143i −0.859840 − 2.69456i 0 −4.93321 + 0.509738i
703.4 −1.10700 + 0.880081i 0 0.450916 − 1.94851i 2.03190 − 1.17312i 0 −2.03407 − 1.69191i 1.21568 + 2.55385i 0 −1.21688 + 3.08688i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 271.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
12.b even 2 1 inner
21.g even 6 1 inner
28.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 756.2.bf.c yes 32
3.b odd 2 1 756.2.bf.b ✓ 32
4.b odd 2 1 756.2.bf.b ✓ 32
7.d odd 6 1 756.2.bf.b ✓ 32
12.b even 2 1 inner 756.2.bf.c yes 32
21.g even 6 1 inner 756.2.bf.c yes 32
28.f even 6 1 inner 756.2.bf.c yes 32
84.j odd 6 1 756.2.bf.b ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.2.bf.b ✓ 32 3.b odd 2 1
756.2.bf.b ✓ 32 4.b odd 2 1
756.2.bf.b ✓ 32 7.d odd 6 1
756.2.bf.b ✓ 32 84.j odd 6 1
756.2.bf.c yes 32 1.a even 1 1 trivial
756.2.bf.c yes 32 12.b even 2 1 inner
756.2.bf.c yes 32 21.g even 6 1 inner
756.2.bf.c yes 32 28.f even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(756, [\chi])\):

\( T_{5}^{32} - 47 T_{5}^{30} + 1362 T_{5}^{28} - 24763 T_{5}^{26} + 328426 T_{5}^{24} - 3131493 T_{5}^{22} + \cdots + 49787136 \) Copy content Toggle raw display
\( T_{11}^{16} - 3 T_{11}^{15} - 46 T_{11}^{14} + 147 T_{11}^{13} + 1632 T_{11}^{12} - 2943 T_{11}^{11} + \cdots + 47251876 \) Copy content Toggle raw display
\( T_{19}^{32} + 202 T_{19}^{30} + 24749 T_{19}^{28} + 1963254 T_{19}^{26} + 115014031 T_{19}^{24} + \cdots + 5742272860416 \) Copy content Toggle raw display