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Properties

Label 351.4.t.a
Level $351$
Weight $4$
Character orbit 351.t
Analytic conductor $20.710$
Analytic rank $0$
Dimension $80$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [351,4,Mod(64,351)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(351, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 3]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("351.64");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 351 = 3^{3} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 351.t (of order \(6\), degree \(2\), 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: \(20.7096704120\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(40\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 117)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 80 q + 150 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 80 q + 150 q^{4} - 40 q^{10} - 13 q^{13} - 78 q^{14} - 530 q^{16} - 264 q^{17} - 34 q^{22} + 174 q^{23} + 798 q^{25} + 1032 q^{26} - 642 q^{29} + 2136 q^{35} + 708 q^{38} + 88 q^{40} + 166 q^{43} + 1610 q^{49} + 786 q^{52} - 1296 q^{53} - 508 q^{55} + 888 q^{56} + 838 q^{61} - 3540 q^{62} - 3652 q^{64} + 201 q^{65} - 612 q^{68} - 4458 q^{74} + 2166 q^{77} + 514 q^{79} - 5188 q^{82} + 2338 q^{88} - 1086 q^{91} - 516 q^{92} + 488 q^{94} - 2136 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
64.1 −4.76791 + 2.75276i 0 11.1553 − 19.3216i −10.5269 − 6.07771i 0 −15.9136 + 9.18770i 78.7874i 0 66.9218
64.2 −4.51411 + 2.60622i 0 9.58480 − 16.6014i 14.3282 + 8.27241i 0 26.6627 − 15.3937i 58.2209i 0 −86.2390
64.3 −4.45290 + 2.57088i 0 9.21885 − 15.9675i 4.08224 + 2.35688i 0 −7.28951 + 4.20860i 53.6682i 0 −24.2370
64.4 −4.04161 + 2.33342i 0 6.88974 − 11.9334i 9.89367 + 5.71211i 0 6.90773 − 3.98818i 26.9720i 0 −53.3152
64.5 −3.99826 + 2.30840i 0 6.65738 − 11.5309i −5.26914 − 3.04214i 0 −14.5470 + 8.39873i 24.5372i 0 28.0898
64.6 −3.79588 + 2.19155i 0 5.60582 − 9.70957i −14.9801 − 8.64875i 0 3.77018 − 2.17672i 14.0770i 0 75.8168
64.7 −3.57012 + 2.06121i 0 4.49718 − 7.78935i −11.2072 − 6.47045i 0 30.5463 − 17.6359i 4.09918i 0 53.3479
64.8 −3.17659 + 1.83400i 0 2.72714 − 4.72354i 1.60561 + 0.926997i 0 12.1556 − 7.01804i − 9.33774i 0 −6.80046
64.9 −3.10562 + 1.79303i 0 2.42992 − 4.20874i 6.91588 + 3.99289i 0 −30.1595 + 17.4126i − 11.2608i 0 −28.6375
64.10 −2.81478 + 1.62512i 0 1.28200 − 2.22049i 8.08964 + 4.67056i 0 −5.60310 + 3.23495i − 17.6683i 0 −30.3608
64.11 −2.63673 + 1.52232i 0 0.634892 − 1.09966i −4.80701 − 2.77533i 0 8.73788 − 5.04482i − 20.4910i 0 16.8997
64.12 −2.61982 + 1.51255i 0 0.575620 − 0.997004i 18.8187 + 10.8650i 0 −0.934009 + 0.539251i − 20.7182i 0 −65.7353
64.13 −1.88513 + 1.08838i 0 −1.63085 + 2.82471i −12.6071 − 7.27874i 0 −9.24346 + 5.33672i − 24.5141i 0 31.6882
64.14 −1.73586 + 1.00220i 0 −1.99119 + 3.44883i −16.4185 − 9.47923i 0 −21.8241 + 12.6002i − 24.0175i 0 38.0004
64.15 −1.55219 + 0.896159i 0 −2.39380 + 4.14618i 1.81395 + 1.04728i 0 22.1814 − 12.8064i − 22.9194i 0 −3.75413
64.16 −1.12540 + 0.649749i 0 −3.15565 + 5.46575i 4.00874 + 2.31444i 0 −7.19572 + 4.15445i − 18.5975i 0 −6.01524
64.17 −1.00243 + 0.578751i 0 −3.33009 + 5.76789i 9.98169 + 5.76293i 0 −23.3870 + 13.5025i − 16.9692i 0 −13.3412
64.18 −0.916837 + 0.529336i 0 −3.43961 + 5.95757i −2.58576 − 1.49289i 0 19.0506 − 10.9989i − 15.7522i 0 3.16096
64.19 −0.457077 + 0.263894i 0 −3.86072 + 6.68696i 14.6710 + 8.47032i 0 18.0982 − 10.4490i − 8.29758i 0 −8.94106
64.20 −0.254683 + 0.147041i 0 −3.95676 + 6.85331i −10.1633 − 5.86777i 0 3.99110 − 2.30426i − 4.67989i 0 3.45122
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 64.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner
13.b even 2 1 inner
117.t even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 351.4.t.a 80
3.b odd 2 1 117.4.t.a ✓ 80
9.c even 3 1 inner 351.4.t.a 80
9.d odd 6 1 117.4.t.a ✓ 80
13.b even 2 1 inner 351.4.t.a 80
39.d odd 2 1 117.4.t.a ✓ 80
117.n odd 6 1 117.4.t.a ✓ 80
117.t even 6 1 inner 351.4.t.a 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
117.4.t.a ✓ 80 3.b odd 2 1
117.4.t.a ✓ 80 9.d odd 6 1
117.4.t.a ✓ 80 39.d odd 2 1
117.4.t.a ✓ 80 117.n odd 6 1
351.4.t.a 80 1.a even 1 1 trivial
351.4.t.a 80 9.c even 3 1 inner
351.4.t.a 80 13.b even 2 1 inner
351.4.t.a 80 117.t even 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(351, [\chi])\).