Properties

Label 57.4.f.c
Level $57$
Weight $4$
Character orbit 57.f
Analytic conductor $3.363$
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] = [57,4,Mod(8,57)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(57, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("57.8");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 57 = 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 57.f (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: \(3.36310887033\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 88 q^{4} + 25 q^{6} - 56 q^{7} - 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 88 q^{4} + 25 q^{6} - 56 q^{7} - 40 q^{9} - 96 q^{10} - 186 q^{13} - 9 q^{15} - 196 q^{16} + 398 q^{19} + 402 q^{21} + 150 q^{22} + 403 q^{24} + 582 q^{25} - 266 q^{28} + 904 q^{30} + 447 q^{33} - 1032 q^{34} + 237 q^{36} - 2162 q^{39} + 12 q^{40} + 898 q^{42} - 542 q^{43} - 1774 q^{45} - 21 q^{48} - 1632 q^{49} + 1833 q^{51} - 3414 q^{52} + 1520 q^{54} - 352 q^{55} - 901 q^{57} + 3672 q^{58} - 2994 q^{60} - 566 q^{61} - 2614 q^{63} + 10304 q^{64} + 1409 q^{66} + 4800 q^{67} - 7536 q^{70} - 2877 q^{72} - 3836 q^{73} - 1744 q^{76} + 4572 q^{78} + 3882 q^{79} + 1036 q^{81} + 2526 q^{82} - 1378 q^{85} + 1706 q^{87} + 3720 q^{90} + 7764 q^{91} - 790 q^{93} - 17958 q^{96} - 2616 q^{97} + 4069 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1 −2.74302 + 4.75104i 0.850422 5.12609i −11.0483 19.1362i 8.62601 + 4.98023i 22.0215 + 18.1013i 16.5845 77.3340 −25.5536 8.71867i −47.3226 + 27.3217i
8.2 −2.50340 + 4.33601i 3.91185 + 3.42015i −8.53398 14.7813i −10.6454 6.14614i −24.6227 + 8.39985i −23.1478 45.4014 3.60518 + 26.7582i 53.2994 30.7724i
8.3 −2.16509 + 3.75004i −5.01325 + 1.36650i −5.37519 9.31010i −4.00786 2.31394i 5.72967 21.7585i 2.99271 11.9096 23.2653 13.7013i 17.3547 10.0197i
8.4 −1.70071 + 2.94572i 0.694828 + 5.14949i −1.78484 3.09144i 16.5498 + 9.55504i −16.3506 6.71103i 8.54097 −15.0694 −26.0344 + 7.15601i −56.2930 + 32.5008i
8.5 −1.44157 + 2.49688i 5.08799 1.05468i −0.156269 0.270666i 1.32057 + 0.762434i −4.70130 + 14.2245i 22.8542 −22.1641 24.7753 10.7324i −3.80741 + 2.19821i
8.6 −1.31744 + 2.28188i 0.493663 5.17265i 0.528681 + 0.915703i −14.8435 8.56992i 11.1530 + 7.94116i −7.94772 −23.8651 −26.5126 5.10709i 39.1111 22.5808i
8.7 −1.25916 + 2.18093i −3.68522 3.66322i 0.829040 + 1.43594i 13.8138 + 7.97539i 12.6295 3.42461i −28.8532 −24.3221 0.161628 + 26.9995i −34.7875 + 20.0846i
8.8 −0.479152 + 0.829916i −2.14066 + 4.73472i 3.54083 + 6.13289i −8.81261 5.08796i −2.90372 4.04522i −5.02367 −14.4528 −17.8352 20.2708i 8.44517 4.87582i
8.9 0.479152 0.829916i 3.03006 4.22122i 3.54083 + 6.13289i 8.81261 + 5.08796i −2.05140 4.53731i −5.02367 14.4528 −8.63744 25.5811i 8.44517 4.87582i
8.10 1.25916 2.18093i −5.01505 1.35988i 0.829040 + 1.43594i −13.8138 7.97539i −9.28054 + 9.22515i −28.8532 24.3221 23.3015 + 13.6397i −34.7875 + 20.0846i
8.11 1.31744 2.28188i −4.23281 + 3.01385i 0.528681 + 0.915703i 14.8435 + 8.56992i 1.30075 + 13.6294i −7.94772 23.8651 8.83343 25.5141i 39.1111 22.5808i
8.12 1.44157 2.49688i 1.63061 + 4.93367i −0.156269 0.270666i −1.32057 0.762434i 14.6694 + 3.04080i 22.8542 22.1641 −21.6822 + 16.0898i −3.80741 + 2.19821i
8.13 1.70071 2.94572i 4.80700 1.97300i −1.78484 3.09144i −16.5498 9.55504i 2.36340 17.5156i 8.54097 15.0694 19.2145 18.9685i −56.2930 + 32.5008i
8.14 2.16509 3.75004i −1.32320 5.02485i −5.37519 9.31010i 4.00786 + 2.31394i −21.7082 5.91720i 2.99271 −11.9096 −23.4983 + 13.2977i 17.3547 10.0197i
8.15 2.50340 4.33601i 4.91786 + 1.67769i −8.53398 14.7813i 10.6454 + 6.14614i 19.5858 17.1240i −23.1478 −45.4014 21.3707 + 16.5013i 53.2994 30.7724i
8.16 2.74302 4.75104i −4.01411 + 3.29953i −11.0483 19.1362i −8.62601 4.98023i 4.66544 + 28.1219i 16.5845 −77.3340 5.22619 26.4894i −47.3226 + 27.3217i
50.1 −2.74302 4.75104i 0.850422 + 5.12609i −11.0483 + 19.1362i 8.62601 4.98023i 22.0215 18.1013i 16.5845 77.3340 −25.5536 + 8.71867i −47.3226 27.3217i
50.2 −2.50340 4.33601i 3.91185 3.42015i −8.53398 + 14.7813i −10.6454 + 6.14614i −24.6227 8.39985i −23.1478 45.4014 3.60518 26.7582i 53.2994 + 30.7724i
50.3 −2.16509 3.75004i −5.01325 1.36650i −5.37519 + 9.31010i −4.00786 + 2.31394i 5.72967 + 21.7585i 2.99271 11.9096 23.2653 + 13.7013i 17.3547 + 10.0197i
50.4 −1.70071 2.94572i 0.694828 5.14949i −1.78484 + 3.09144i 16.5498 9.55504i −16.3506 + 6.71103i 8.54097 −15.0694 −26.0344 7.15601i −56.2930 32.5008i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 8.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
19.d odd 6 1 inner
57.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 57.4.f.c 32
3.b odd 2 1 inner 57.4.f.c 32
19.d odd 6 1 inner 57.4.f.c 32
57.f even 6 1 inner 57.4.f.c 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.4.f.c 32 1.a even 1 1 trivial
57.4.f.c 32 3.b odd 2 1 inner
57.4.f.c 32 19.d odd 6 1 inner
57.4.f.c 32 57.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 \(S_{4}^{\mathrm{new}}(57, [\chi])\):

\( T_{2}^{32} + 108 T_{2}^{30} + 6921 T_{2}^{28} + 292684 T_{2}^{26} + 9187074 T_{2}^{24} + 218025072 T_{2}^{22} + 4053581529 T_{2}^{20} + 59091427476 T_{2}^{18} + 685653653682 T_{2}^{16} + \cdots + 30\!\cdots\!36 \) Copy content Toggle raw display
\( T_{7}^{8} + 14 T_{7}^{7} - 1070 T_{7}^{6} - 9260 T_{7}^{5} + 327467 T_{7}^{4} + 1064026 T_{7}^{3} - 21464462 T_{7}^{2} - 39409468 T_{7} + 258350560 \) Copy content Toggle raw display