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Properties

Label 618.2.m.c
Level $618$
Weight $2$
Character orbit 618.m
Analytic conductor $4.935$
Analytic rank $0$
Dimension $160$
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] = [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.

\(\operatorname{Tr}(f)(q) = \) \( 160 q - 5 q^{2} - 10 q^{3} + 5 q^{4} - 16 q^{5} - 5 q^{6} - 3 q^{7} + 10 q^{8} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 160 q - 5 q^{2} - 10 q^{3} + 5 q^{4} - 16 q^{5} - 5 q^{6} - 3 q^{7} + 10 q^{8} - 10 q^{9} + 2 q^{10} - 31 q^{11} + 5 q^{12} - 11 q^{13} - 6 q^{14} + q^{15} + 5 q^{16} - 5 q^{18} + 6 q^{19} + q^{20} - 3 q^{21} - 11 q^{22} + 21 q^{23} + 10 q^{24} - 9 q^{25} - 31 q^{26} - 10 q^{27} - 3 q^{28} - 12 q^{29} + 19 q^{30} - 72 q^{31} - 5 q^{32} + 54 q^{33} + 13 q^{35} + 5 q^{36} + 60 q^{37} - 6 q^{38} - 11 q^{39} + 33 q^{40} - 27 q^{41} + 11 q^{42} - 30 q^{43} - 31 q^{44} + q^{45} + 2 q^{46} - 39 q^{47} + 5 q^{48} + 76 q^{49} - 8 q^{50} - 51 q^{51} - 3 q^{52} + 8 q^{53} - 5 q^{54} - 17 q^{55} + 3 q^{56} - 28 q^{57} + 46 q^{58} + 38 q^{59} + q^{60} - 36 q^{61} - 2 q^{62} - 3 q^{63} - 10 q^{64} + 103 q^{65} + 6 q^{66} - 59 q^{67} + 21 q^{69} + 72 q^{70} - 73 q^{71} + 10 q^{72} - 34 q^{73} - 21 q^{74} + 8 q^{75} + 22 q^{76} + 106 q^{77} + 3 q^{78} + 48 q^{79} - 2 q^{80} - 10 q^{81} - 7 q^{82} + 17 q^{83} + 14 q^{84} + 69 q^{85} + 13 q^{86} - 46 q^{87} - 3 q^{88} - 95 q^{89} + 2 q^{90} - 20 q^{91} - 36 q^{92} - 72 q^{93} - 27 q^{94} - 36 q^{95} - 5 q^{96} + 211 q^{97} - 110 q^{98} - 31 q^{99}+O(q^{100}) \) 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 −3.22817 − 0.399734i 0.696134 + 0.717912i 2.72475 + 1.46316i 0.850217 + 0.526432i −0.982973 − 0.183750i 2.40386 − 2.19141i
7.2 −0.650618 + 0.759405i 0.0922684 − 0.995734i −0.153392 − 0.988165i −1.39741 − 0.173037i 0.696134 + 0.717912i −1.60261 − 0.860583i 0.850217 + 0.526432i −0.982973 − 0.183750i 1.04059 − 0.948622i
7.3 −0.650618 + 0.759405i 0.0922684 − 0.995734i −0.153392 − 0.988165i 0.348107 + 0.0431050i 0.696134 + 0.717912i −0.0985250 − 0.0529068i 0.850217 + 0.526432i −0.982973 − 0.183750i −0.259219 + 0.236309i
7.4 −0.650618 + 0.759405i 0.0922684 − 0.995734i −0.153392 − 0.988165i 2.67351 + 0.331052i 0.696134 + 0.717912i 4.36586 + 2.34442i 0.850217 + 0.526432i −0.982973 − 0.183750i −1.99084 + 1.81489i
7.5 −0.650618 + 0.759405i 0.0922684 − 0.995734i −0.153392 − 0.988165i 3.36907 + 0.417181i 0.696134 + 0.717912i −3.84325 − 2.06378i 0.850217 + 0.526432i −0.982973 − 0.183750i −2.50879 + 2.28706i
19.1 0.0307951 − 0.999526i −0.273663 + 0.961826i −0.998103 − 0.0615609i −3.37844 + 2.71862i 0.952942 + 0.303153i −2.35250 − 1.08232i −0.0922684 + 0.995734i −0.850217 − 0.526432i 2.61329 + 3.46056i
19.2 0.0307951 − 0.999526i −0.273663 + 0.961826i −0.998103 − 0.0615609i −1.32194 + 1.06376i 0.952942 + 0.303153i 2.92540 + 1.34590i −0.0922684 + 0.995734i −0.850217 − 0.526432i 1.02255 + 1.35407i
19.3 0.0307951 − 0.999526i −0.273663 + 0.961826i −0.998103 − 0.0615609i 0.538231 − 0.433113i 0.952942 + 0.303153i 0.298054 + 0.137126i −0.0922684 + 0.995734i −0.850217 − 0.526432i −0.416333 − 0.551314i
19.4 0.0307951 − 0.999526i −0.273663 + 0.961826i −0.998103 − 0.0615609i 2.01662 − 1.62277i 0.952942 + 0.303153i −3.63421 − 1.67200i −0.0922684 + 0.995734i −0.850217 − 0.526432i −1.55990 − 2.06564i
19.5 0.0307951 − 0.999526i −0.273663 + 0.961826i −0.998103 − 0.0615609i 2.46613 − 1.98448i 0.952942 + 0.303153i 1.87732 + 0.863705i −0.0922684 + 0.995734i −0.850217 − 0.526432i −1.90760 − 2.52607i
25.1 0.908465 − 0.417960i 0.739009 − 0.673696i 0.650618 − 0.759405i −4.20592 − 0.259412i 0.389786 − 0.920906i 2.33983 + 0.588491i 0.273663 − 0.961826i 0.0922684 − 0.995734i −3.92936 + 1.52224i
25.2 0.908465 − 0.417960i 0.739009 − 0.673696i 0.650618 − 0.759405i −1.54681 − 0.0954042i 0.389786 − 0.920906i −4.27888 − 1.07618i 0.273663 − 0.961826i 0.0922684 − 0.995734i −1.44510 + 0.559835i
25.3 0.908465 − 0.417960i 0.739009 − 0.673696i 0.650618 − 0.759405i −0.215178 − 0.0132717i 0.389786 − 0.920906i 3.40503 + 0.856399i 0.273663 − 0.961826i 0.0922684 − 0.995734i −0.201029 + 0.0778789i
25.4 0.908465 − 0.417960i 0.739009 − 0.673696i 0.650618 − 0.759405i 2.15809 + 0.133107i 0.389786 − 0.920906i 1.72022 + 0.432652i 0.273663 − 0.961826i 0.0922684 − 0.995734i 2.01618 − 0.781074i
25.5 0.908465 − 0.417960i 0.739009 − 0.673696i 0.650618 − 0.759405i 2.37253 + 0.146333i 0.389786 − 0.920906i −2.69469 − 0.677741i 0.273663 − 0.961826i 0.0922684 − 0.995734i 2.21652 − 0.858686i
49.1 0.153392 + 0.988165i −0.982973 − 0.183750i −0.952942 + 0.303153i −3.14494 − 0.790985i 0.0307951 − 0.999526i 1.55283 + 2.34345i −0.445738 − 0.895163i 0.932472 + 0.361242i 0.299216 − 3.22905i
49.2 0.153392 + 0.988165i −0.982973 − 0.183750i −0.952942 + 0.303153i −2.71799 − 0.683602i 0.0307951 − 0.999526i 0.0272750 + 0.0411621i −0.445738 − 0.895163i 0.932472 + 0.361242i 0.258595 − 2.79068i
49.3 0.153392 + 0.988165i −0.982973 − 0.183750i −0.952942 + 0.303153i −0.373340 − 0.0938988i 0.0307951 − 0.999526i −0.115163 − 0.173798i −0.445738 − 0.895163i 0.932472 + 0.361242i 0.0355203 − 0.383325i
49.4 0.153392 + 0.988165i −0.982973 − 0.183750i −0.952942 + 0.303153i 1.85855 + 0.467443i 0.0307951 − 0.999526i −1.70652 − 2.57540i −0.445738 − 0.895163i 0.932472 + 0.361242i −0.176826 + 1.90825i
49.5 0.153392 + 0.988165i −0.982973 − 0.183750i −0.952942 + 0.303153i 3.58236 + 0.901000i 0.0307951 − 0.999526i 2.30079 + 3.47223i −0.445738 − 0.895163i 0.932472 + 0.361242i −0.340833 + 3.67817i
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.c ✓ 160
103.g even 51 1 inner 618.2.m.c ✓ 160
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
618.2.m.c ✓ 160 1.a even 1 1 trivial
618.2.m.c ✓ 160 103.g even 51 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{160} + 16 T_{5}^{159} + 120 T_{5}^{158} + 483 T_{5}^{157} + 502 T_{5}^{156} + \cdots + 13\!\cdots\!89 \) acting on \(S_{2}^{\mathrm{new}}(618, [\chi])\). Copy content Toggle raw display