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Properties

Label 418.2.n.e
Level $418$
Weight $2$
Character orbit 418.n
Analytic conductor $3.338$
Analytic rank $0$
Dimension $72$
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] = [418,2,Mod(49,418)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(418, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([12, 20]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("418.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 418 = 2 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 418.n (of order \(15\), degree \(8\), 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: \(3.33774680449\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(9\) over \(\Q(\zeta_{15})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 72 q + 9 q^{2} - 2 q^{3} + 9 q^{4} + 8 q^{5} + 3 q^{6} - 20 q^{7} - 18 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q + 9 q^{2} - 2 q^{3} + 9 q^{4} + 8 q^{5} + 3 q^{6} - 20 q^{7} - 18 q^{8} + 9 q^{9} - 22 q^{10} + 10 q^{11} + 4 q^{12} + 15 q^{13} + 10 q^{14} + 9 q^{15} + 9 q^{16} + 6 q^{17} - 28 q^{18} + 7 q^{19} - 16 q^{20} + 48 q^{21} - 15 q^{22} - 20 q^{23} + 3 q^{24} + 7 q^{25} + 30 q^{26} - 56 q^{27} - 10 q^{28} + 35 q^{29} - 18 q^{30} + 52 q^{31} - 36 q^{32} - 12 q^{33} - 4 q^{34} - 9 q^{35} + 14 q^{36} - 52 q^{37} - 2 q^{38} - 42 q^{39} + 3 q^{40} + 41 q^{41} - 2 q^{42} + 14 q^{43} + 116 q^{45} - 50 q^{46} + 19 q^{47} - 2 q^{48} + 46 q^{49} - 44 q^{50} + 33 q^{51} - 15 q^{52} - 15 q^{53} - 2 q^{54} - 55 q^{55} - 33 q^{57} - 70 q^{58} - 13 q^{59} - 6 q^{60} + 8 q^{61} + 19 q^{62} + 40 q^{63} - 18 q^{64} + 120 q^{65} + 23 q^{66} + 2 q^{67} - 12 q^{68} - 194 q^{69} + q^{70} + 52 q^{71} + 9 q^{72} - 48 q^{73} + 26 q^{74} - 158 q^{75} + 20 q^{76} + 130 q^{77} + 46 q^{78} - 48 q^{79} + 3 q^{80} + 48 q^{81} - 14 q^{82} - 62 q^{83} + 44 q^{84} - 27 q^{85} - 16 q^{86} - 164 q^{87} + 10 q^{88} + 20 q^{89} + 52 q^{90} + 4 q^{91} - 15 q^{92} - 39 q^{93} - 8 q^{94} + 69 q^{95} + 4 q^{96} + 2 q^{97} - 48 q^{98} + 9 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1 −0.978148 − 0.207912i −2.93761 + 1.30791i 0.913545 + 0.406737i 1.74891 + 1.94236i 3.14534 − 0.668563i 0.517306 − 0.375844i −0.809017 − 0.587785i 4.91153 − 5.45480i −1.30685 − 2.26353i
49.2 −0.978148 − 0.207912i −2.77573 + 1.23583i 0.913545 + 0.406737i −2.01389 − 2.23665i 2.97202 − 0.631722i −3.06964 + 2.23023i −0.809017 − 0.587785i 4.16999 − 4.63124i 1.50486 + 2.60649i
49.3 −0.978148 − 0.207912i −0.824558 + 0.367117i 0.913545 + 0.406737i 2.20784 + 2.45205i 0.882868 − 0.187659i −3.06526 + 2.22705i −0.809017 − 0.587785i −1.46227 + 1.62402i −1.64978 − 2.85750i
49.4 −0.978148 − 0.207912i −0.619743 + 0.275927i 0.913545 + 0.406737i −0.523627 − 0.581546i 0.663568 − 0.141046i 0.816129 − 0.592952i −0.809017 − 0.587785i −1.69945 + 1.88743i 0.391274 + 0.677706i
49.5 −0.978148 − 0.207912i −0.604189 + 0.269002i 0.913545 + 0.406737i −1.30272 − 1.44681i 0.646915 − 0.137506i −0.622011 + 0.451917i −0.809017 − 0.587785i −1.71471 + 1.90438i 0.973440 + 1.68605i
49.6 −0.978148 − 0.207912i 0.854753 − 0.380561i 0.913545 + 0.406737i 2.66345 + 2.95806i −0.915198 + 0.194531i 3.03005 − 2.20146i −0.809017 − 0.587785i −1.42162 + 1.57886i −1.99023 − 3.44719i
49.7 −0.978148 − 0.207912i 1.25549 − 0.558979i 0.913545 + 0.406737i −0.0680196 − 0.0755435i −1.34427 + 0.285733i −3.31089 + 2.40550i −0.809017 − 0.587785i −0.743602 + 0.825853i 0.0508269 + 0.0880347i
49.8 −0.978148 − 0.207912i 2.37757 − 1.05856i 0.913545 + 0.406737i −1.31390 − 1.45923i −2.54571 + 0.541107i 3.94781 − 2.86825i −0.809017 − 0.587785i 2.52491 − 2.80420i 0.981796 + 1.70052i
49.9 −0.978148 − 0.207912i 2.70941 − 1.20631i 0.913545 + 0.406737i 1.53406 + 1.70375i −2.90101 + 0.616628i −1.86152 + 1.35247i −0.809017 − 0.587785i 3.87833 − 4.30732i −1.14631 − 1.98547i
125.1 0.913545 − 0.406737i −2.00528 − 2.22709i 0.669131 − 0.743145i −0.275672 − 2.62284i −2.73776 − 1.21893i 0.0729011 + 0.224366i 0.309017 − 0.951057i −0.625195 + 5.94833i −1.31864 − 2.28396i
125.2 0.913545 − 0.406737i −1.53847 − 1.70865i 0.669131 − 0.743145i −0.0583470 − 0.555134i −2.10043 − 0.935173i −1.47197 − 4.53025i 0.309017 − 0.951057i −0.238990 + 2.27384i −0.279096 − 0.483408i
125.3 0.913545 − 0.406737i −0.547677 − 0.608257i 0.669131 − 0.743145i −0.129363 − 1.23080i −0.747728 − 0.332910i 0.847071 + 2.60701i 0.309017 − 0.951057i 0.243559 − 2.31731i −0.618791 − 1.07178i
125.4 0.913545 − 0.406737i −0.470759 − 0.522831i 0.669131 − 0.743145i 0.330325 + 3.14283i −0.642715 − 0.286155i 0.775272 + 2.38604i 0.309017 − 0.951057i 0.261847 − 2.49131i 1.58007 + 2.73676i
125.5 0.913545 − 0.406737i −0.0737020 − 0.0818544i 0.669131 − 0.743145i −0.396813 − 3.77542i −0.100623 − 0.0448004i 0.602017 + 1.85282i 0.309017 − 0.951057i 0.312317 − 2.97150i −1.89811 − 3.28762i
125.6 0.913545 − 0.406737i 0.667839 + 0.741710i 0.669131 − 0.743145i −0.142258 − 1.35350i 0.911782 + 0.405951i −1.16508 − 3.58574i 0.309017 − 0.951057i 0.209460 − 1.99288i −0.680475 − 1.17862i
125.7 0.913545 − 0.406737i 0.927739 + 1.03036i 0.669131 − 0.743145i 0.231918 + 2.20655i 1.26662 + 0.563934i −0.102451 − 0.315310i 0.309017 − 0.951057i 0.112646 − 1.07176i 1.10935 + 1.92145i
125.8 0.913545 − 0.406737i 1.97217 + 2.19032i 0.669131 − 0.743145i 0.204659 + 1.94720i 2.69255 + 1.19880i −0.518534 − 1.59588i 0.309017 − 0.951057i −0.594450 + 5.65582i 0.978965 + 1.69562i
125.9 0.913545 − 0.406737i 2.15082 + 2.38873i 0.669131 − 0.743145i −0.456223 − 4.34067i 2.93645 + 1.30739i −0.421199 − 1.29632i 0.309017 − 0.951057i −0.766406 + 7.29186i −2.18229 − 3.77983i
159.1 0.669131 − 0.743145i −0.310012 + 2.94957i −0.104528 − 0.994522i −2.24252 + 0.476663i 1.98452 + 2.20403i −1.86152 + 1.35247i −0.809017 − 0.587785i −5.66942 − 1.20507i −1.14631 + 1.98547i
159.2 0.669131 − 0.743145i −0.272044 + 2.58832i −0.104528 − 0.994522i 1.92068 − 0.408254i 1.74147 + 1.93409i 3.94781 − 2.86825i −0.809017 − 0.587785i −3.69096 − 0.784537i 0.981796 − 1.70052i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.9
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner
19.c even 3 1 inner
209.n even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 418.2.n.e ✓ 72
11.c even 5 1 inner 418.2.n.e ✓ 72
19.c even 3 1 inner 418.2.n.e ✓ 72
209.n even 15 1 inner 418.2.n.e ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.n.e ✓ 72 1.a even 1 1 trivial
418.2.n.e ✓ 72 11.c even 5 1 inner
418.2.n.e ✓ 72 19.c even 3 1 inner
418.2.n.e ✓ 72 209.n even 15 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{72} + 2 T_{3}^{71} - 16 T_{3}^{70} - 4 T_{3}^{69} + 104 T_{3}^{68} - 352 T_{3}^{67} + \cdots + 2562890625 \) acting on \(S_{2}^{\mathrm{new}}(418, [\chi])\). Copy content Toggle raw display