↑

Properties

Label 618.2.m.d
Level $618$
Weight $2$
Character orbit 618.m
Analytic conductor $4.935$
Analytic rank $0$
Dimension $160$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [618,2,Mod(7,618)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(618, base_ring=CyclotomicField(102))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("618.7");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 618 = 2 \cdot 3 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 618.m (of order \(51\), degree \(32\), 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: \(4.93475484492\)
Analytic rank: \(0\)
Dimension: \(160\)
Relative dimension: \(5\) over \(\Q(\zeta_{51})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{51}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 160 q + 5 q^{2} + 10 q^{3} + 5 q^{4} - 20 q^{5} - 5 q^{6} + 5 q^{7} - 10 q^{8} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 160 q + 5 q^{2} + 10 q^{3} + 5 q^{4} - 20 q^{5} - 5 q^{6} + 5 q^{7} - 10 q^{8} - 10 q^{9} + 6 q^{10} + 33 q^{11} - 5 q^{12} - 11 q^{13} - 10 q^{14} + 3 q^{15} + 5 q^{16} - 2 q^{17} + 5 q^{18} + 20 q^{19} - 3 q^{20} - 5 q^{21} + 19 q^{22} + q^{23} + 10 q^{24} - q^{25} - 37 q^{26} + 10 q^{27} + 5 q^{28} + 10 q^{29} + 11 q^{30} + 52 q^{31} + 5 q^{32} + 18 q^{33} + 4 q^{34} + 5 q^{35} + 5 q^{36} - 80 q^{37} + 20 q^{38} - 23 q^{39} + 31 q^{40} + 35 q^{41} - 7 q^{42} - 22 q^{43} + 33 q^{44} - 3 q^{45} - 60 q^{46} + 91 q^{47} - 5 q^{48} + 20 q^{49} + 16 q^{50} + 53 q^{51} - 3 q^{52} + 36 q^{53} - 5 q^{54} + 3 q^{55} - 29 q^{56} + 48 q^{57} + 44 q^{58} + 18 q^{59} + 3 q^{60} + 20 q^{61} + 8 q^{62} + 5 q^{63} - 10 q^{64} + 143 q^{65} - 2 q^{66} - 57 q^{67} - 2 q^{68} - 35 q^{69} - 80 q^{70} + 167 q^{71} - 10 q^{72} - 42 q^{73} + 23 q^{74} - 16 q^{75} - 6 q^{76} + 4 q^{77} + 3 q^{78} + 104 q^{79} + 6 q^{80} - 10 q^{81} + q^{82} - 3 q^{83} + 12 q^{84} - 71 q^{85} - 73 q^{86} - 44 q^{87} - q^{88} - 63 q^{89} + 6 q^{90} + 44 q^{91} - 26 q^{92} - 52 q^{93} - 29 q^{94} - 108 q^{95} - 5 q^{96} - 29 q^{97} - 150 q^{98} - 35 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 0.650618 − 0.759405i −0.0922684 + 0.995734i −0.153392 − 0.988165i −3.50759 − 0.434333i 0.696134 + 0.717912i 1.71793 + 0.922509i −0.850217 − 0.526432i −0.982973 − 0.183750i −2.61194 + 2.38109i
7.2 0.650618 − 0.759405i −0.0922684 + 0.995734i −0.153392 − 0.988165i −2.08595 − 0.258296i 0.696134 + 0.717912i 0.699862 + 0.375818i −0.850217 − 0.526432i −0.982973 − 0.183750i −1.55331 + 1.41603i
7.3 0.650618 − 0.759405i −0.0922684 + 0.995734i −0.153392 − 0.988165i −0.639654 − 0.0792063i 0.696134 + 0.717912i −0.900101 − 0.483344i −0.850217 − 0.526432i −0.982973 − 0.183750i −0.476320 + 0.434223i
7.4 0.650618 − 0.759405i −0.0922684 + 0.995734i −0.153392 − 0.988165i −0.107446 − 0.0133046i 0.696134 + 0.717912i −4.04110 − 2.17003i −0.850217 − 0.526432i −0.982973 − 0.183750i −0.0800096 + 0.0729384i
7.5 0.650618 − 0.759405i −0.0922684 + 0.995734i −0.153392 − 0.988165i 3.02820 + 0.374972i 0.696134 + 0.717912i 2.73921 + 1.47092i −0.850217 − 0.526432i −0.982973 − 0.183750i 2.25496 − 2.05566i
19.1 −0.0307951 + 0.999526i 0.273663 − 0.961826i −0.998103 − 0.0615609i −3.26162 + 2.62462i 0.952942 + 0.303153i 3.11341 + 1.43240i 0.0922684 − 0.995734i −0.850217 − 0.526432i −2.52293 − 3.34090i
19.2 −0.0307951 + 0.999526i 0.273663 − 0.961826i −0.998103 − 0.0615609i −1.82552 + 1.46899i 0.952942 + 0.303153i −2.07873 − 0.956370i 0.0922684 − 0.995734i −0.850217 − 0.526432i −1.41208 − 1.86989i
19.3 −0.0307951 + 0.999526i 0.273663 − 0.961826i −0.998103 − 0.0615609i 0.230870 − 0.185780i 0.952942 + 0.303153i −2.97379 − 1.36816i 0.0922684 − 0.995734i −0.850217 − 0.526432i 0.178582 + 0.236481i
19.4 −0.0307951 + 0.999526i 0.273663 − 0.961826i −0.998103 − 0.0615609i 1.63080 − 1.31230i 0.952942 + 0.303153i −1.79432 − 0.825518i 0.0922684 − 0.995734i −0.850217 − 0.526432i 1.26146 + 1.67044i
19.5 −0.0307951 + 0.999526i 0.273663 − 0.961826i −0.998103 − 0.0615609i 3.38289 − 2.72220i 0.952942 + 0.303153i 2.80243 + 1.28932i 0.0922684 − 0.995734i −0.850217 − 0.526432i 2.61673 + 3.46511i
25.1 −0.908465 + 0.417960i −0.739009 + 0.673696i 0.650618 − 0.759405i −4.05752 − 0.250259i 0.389786 − 0.920906i −2.80746 − 0.706104i −0.273663 + 0.961826i 0.0922684 − 0.995734i 3.79071 − 1.46853i
25.2 −0.908465 + 0.417960i −0.739009 + 0.673696i 0.650618 − 0.759405i −2.10356 − 0.129743i 0.389786 − 0.920906i 4.19174 + 1.05427i −0.273663 + 0.961826i 0.0922684 − 0.995734i 1.96524 − 0.761336i
25.3 −0.908465 + 0.417960i −0.739009 + 0.673696i 0.650618 − 0.759405i 1.00583 + 0.0620372i 0.389786 − 0.920906i −2.47376 − 0.622176i −0.273663 + 0.961826i 0.0922684 − 0.995734i −0.939687 + 0.364037i
25.4 −0.908465 + 0.417960i −0.739009 + 0.673696i 0.650618 − 0.759405i 1.08837 + 0.0671282i 0.389786 − 0.920906i −1.64265 − 0.413143i −0.273663 + 0.961826i 0.0922684 − 0.995734i −1.01680 + 0.393911i
25.5 −0.908465 + 0.417960i −0.739009 + 0.673696i 0.650618 − 0.759405i 3.29890 + 0.203469i 0.389786 − 0.920906i 4.18020 + 1.05136i −0.273663 + 0.961826i 0.0922684 − 0.995734i −3.08198 + 1.19396i
49.1 −0.153392 − 0.988165i 0.982973 + 0.183750i −0.952942 + 0.303153i −2.71902 − 0.683861i 0.0307951 − 0.999526i 1.65769 + 2.50171i 0.445738 + 0.895163i 0.932472 + 0.361242i −0.258693 + 2.79174i
49.2 −0.153392 − 0.988165i 0.982973 + 0.183750i −0.952942 + 0.303153i −2.28409 − 0.574471i 0.0307951 − 0.999526i −1.78076 − 2.68743i 0.445738 + 0.895163i 0.932472 + 0.361242i −0.217313 + 2.34518i
49.3 −0.153392 − 0.988165i 0.982973 + 0.183750i −0.952942 + 0.303153i −2.05535 − 0.516940i 0.0307951 − 0.999526i −0.362529 − 0.547111i 0.445738 + 0.895163i 0.932472 + 0.361242i −0.195550 + 2.11032i
49.4 −0.153392 − 0.988165i 0.982973 + 0.183750i −0.952942 + 0.303153i 1.70128 + 0.427888i 0.0307951 − 0.999526i 1.09564 + 1.65349i 0.445738 + 0.895163i 0.932472 + 0.361242i 0.161863 − 1.74678i
49.5 −0.153392 − 0.988165i 0.982973 + 0.183750i −0.952942 + 0.303153i 3.45211 + 0.868240i 0.0307951 − 0.999526i −1.56452 − 2.36110i 0.445738 + 0.895163i 0.932472 + 0.361242i 0.328440 − 3.54443i
See next 80 embeddings (of 160 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
103.g even 51 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 618.2.m.d ✓ 160
103.g even 51 1 inner 618.2.m.d ✓ 160
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
618.2.m.d ✓ 160 1.a even 1 1 trivial
618.2.m.d ✓ 160 103.g even 51 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{160} + 20 T_{5}^{159} + 188 T_{5}^{158} + 1155 T_{5}^{157} + 5558 T_{5}^{156} + \cdots + 20\!\cdots\!09 \) acting on \(S_{2}^{\mathrm{new}}(618, [\chi])\). Copy content Toggle raw display