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Properties

Label 51.4.h.a
Level $51$
Weight $4$
Character orbit 51.h
Analytic conductor $3.009$
Analytic rank $0$
Dimension $40$
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] = [51,4,Mod(19,51)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(51, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 7]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("51.19");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 51 = 3 \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 51.h (of order \(8\), 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: \(3.00909741029\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(10\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 32 q^{5} - 24 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q + 32 q^{5} - 24 q^{6} + 128 q^{10} + 112 q^{11} + 256 q^{14} - 1024 q^{16} - 112 q^{17} - 32 q^{19} - 640 q^{20} + 728 q^{22} + 208 q^{23} + 456 q^{24} + 296 q^{25} + 1472 q^{26} - 328 q^{28} - 1272 q^{29} - 192 q^{31} - 960 q^{32} - 912 q^{33} + 24 q^{34} - 992 q^{35} - 216 q^{36} - 1120 q^{37} - 48 q^{39} + 2984 q^{40} - 1320 q^{41} + 1032 q^{42} + 672 q^{43} + 3744 q^{44} + 504 q^{45} - 384 q^{46} - 2096 q^{49} - 448 q^{50} + 960 q^{52} + 2120 q^{53} - 216 q^{54} + 5680 q^{56} + 768 q^{57} + 1000 q^{58} - 832 q^{59} - 1320 q^{60} - 208 q^{61} - 4432 q^{62} - 2376 q^{65} - 1248 q^{66} - 2272 q^{67} - 4560 q^{68} + 3456 q^{69} - 4792 q^{70} + 256 q^{71} - 1416 q^{73} - 2704 q^{74} + 96 q^{75} - 264 q^{76} + 6048 q^{77} - 4872 q^{78} + 1920 q^{79} + 1312 q^{80} + 8328 q^{82} + 4960 q^{83} - 3888 q^{84} + 3192 q^{85} - 15616 q^{86} - 1872 q^{87} - 248 q^{88} - 1152 q^{90} + 1232 q^{91} + 944 q^{92} + 2544 q^{93} - 10000 q^{94} + 4528 q^{95} + 4416 q^{96} + 1008 q^{97} + 1008 q^{99}+O(q^{100}) \) 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 −3.79946 − 3.79946i 2.77164 − 1.14805i 20.8719i 4.62215 + 11.1589i −14.8927 − 6.16877i −11.9634 + 28.8822i 48.9062 − 48.9062i 6.36396 − 6.36396i 24.8360 − 59.9594i
19.2 −2.85730 − 2.85730i −2.77164 + 1.14805i 8.32833i 7.19045 + 17.3593i 11.1997 + 4.63908i 10.1833 − 24.5847i 0.938143 − 0.938143i 6.36396 − 6.36396i 29.0554 − 70.1460i
19.3 −2.07008 − 2.07008i 2.77164 − 1.14805i 0.570424i −1.15618 − 2.79128i −8.11405 − 3.36095i 7.11409 − 17.1749i −15.3798 + 15.3798i 6.36396 − 6.36396i −3.38476 + 8.17154i
19.4 −1.89364 − 1.89364i −2.77164 + 1.14805i − 0.828264i −1.56537 − 3.77915i 7.42248 + 3.07449i −11.5719 + 27.9371i −16.7175 + 16.7175i 6.36396 − 6.36396i −4.19208 + 10.1206i
19.5 0.320127 + 0.320127i −2.77164 + 1.14805i − 7.79504i −2.81499 − 6.79598i −1.25480 − 0.519754i 6.62120 − 15.9850i 5.05641 − 5.05641i 6.36396 − 6.36396i 1.27442 − 3.07673i
19.6 0.511853 + 0.511853i 2.77164 − 1.14805i − 7.47601i −6.64023 − 16.0309i 2.00630 + 0.831038i −5.63793 + 13.6112i 7.92144 − 7.92144i 6.36396 − 6.36396i 4.80665 − 11.6043i
19.7 0.865877 + 0.865877i 2.77164 − 1.14805i − 6.50051i 6.89196 + 16.6387i 3.39397 + 1.40583i 6.38012 − 15.4030i 12.5557 − 12.5557i 6.36396 − 6.36396i −8.43944 + 20.3746i
19.8 1.98341 + 1.98341i −2.77164 + 1.14805i − 0.132152i 6.98831 + 16.8713i −7.77436 − 3.22024i −9.61579 + 23.2146i 16.1294 − 16.1294i 6.36396 − 6.36396i −19.6020 + 47.3234i
19.9 3.18525 + 3.18525i 2.77164 − 1.14805i 12.2916i −0.657844 − 1.58818i 12.4852 + 5.17153i −2.66469 + 6.43312i −13.6698 + 13.6698i 6.36396 − 6.36396i 2.96334 − 7.15413i
19.10 3.75396 + 3.75396i −2.77164 + 1.14805i 20.1845i −0.615614 − 1.48622i −14.7144 − 6.09489i 8.32655 − 20.1021i −45.7401 + 45.7401i 6.36396 − 6.36396i 3.26824 − 7.89022i
25.1 −3.59414 + 3.59414i −1.14805 + 2.77164i − 17.8356i −10.5477 − 4.36900i −5.83540 − 14.0879i 6.74598 − 2.79428i 35.3506 + 35.3506i −6.36396 − 6.36396i 53.6127 − 22.2071i
25.2 −3.45749 + 3.45749i 1.14805 − 2.77164i − 15.9085i 13.3151 + 5.51528i 5.61354 + 13.5523i 7.75774 − 3.21336i 27.3435 + 27.3435i −6.36396 − 6.36396i −65.1057 + 26.9677i
25.3 −2.21747 + 2.21747i 1.14805 − 2.77164i − 1.83439i −12.9753 − 5.37456i 3.60027 + 8.69181i −7.27652 + 3.01403i −13.6721 − 13.6721i −6.36396 − 6.36396i 40.6904 − 16.8545i
25.4 −2.15816 + 2.15816i −1.14805 + 2.77164i − 1.31528i 13.0882 + 5.42131i −3.50396 − 8.45930i −26.4168 + 10.9422i −14.4267 − 14.4267i −6.36396 − 6.36396i −39.9464 + 16.5464i
25.5 −0.0147863 + 0.0147863i −1.14805 + 2.77164i 7.99956i −18.9786 − 7.86121i −0.0240068 − 0.0579575i −5.95742 + 2.46764i −0.236574 − 0.236574i −6.36396 − 6.36396i 0.396861 − 0.164385i
25.6 0.378339 − 0.378339i 1.14805 − 2.77164i 7.71372i 7.33680 + 3.03900i −0.614267 − 1.48297i 24.6775 − 10.2218i 5.94511 + 5.94511i −6.36396 − 6.36396i 3.92557 − 1.62602i
25.7 1.41071 − 1.41071i −1.14805 + 2.77164i 4.01979i 9.58757 + 3.97130i 2.29041 + 5.52955i 5.26522 − 2.18092i 16.9564 + 16.9564i −6.36396 − 6.36396i 19.1277 − 7.92293i
25.8 2.73634 − 2.73634i 1.14805 − 2.77164i − 6.97513i −12.5970 − 5.21785i −4.44269 − 10.7256i 19.5561 − 8.10040i 2.80440 + 2.80440i −6.36396 − 6.36396i −48.7476 + 20.1919i
25.9 3.10148 − 3.10148i 1.14805 − 2.77164i − 11.2384i 14.1902 + 5.87778i −5.03553 − 12.1568i −30.3663 + 12.5781i −10.0437 − 10.0437i −6.36396 − 6.36396i 62.2405 − 25.7809i
25.10 3.81517 − 3.81517i −1.14805 + 2.77164i − 21.1111i 1.33820 + 0.554300i 6.19427 + 14.9543i 8.84293 − 3.66286i −50.0210 − 50.0210i −6.36396 − 6.36396i 7.22021 − 2.99071i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.d even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 51.4.h.a ✓ 40
3.b odd 2 1 153.4.l.c 40
17.d even 8 1 inner 51.4.h.a ✓ 40
17.e odd 16 1 867.4.a.v 20
17.e odd 16 1 867.4.a.w 20
51.g odd 8 1 153.4.l.c 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
51.4.h.a ✓ 40 1.a even 1 1 trivial
51.4.h.a ✓ 40 17.d even 8 1 inner
153.4.l.c 40 3.b odd 2 1
153.4.l.c 40 51.g odd 8 1
867.4.a.v 20 17.e odd 16 1
867.4.a.w 20 17.e odd 16 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(51, [\chi])\).