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Properties

Label 418.2.n.d
Level $418$
Weight $2$
Character orbit 418.n
Analytic conductor $3.338$
Analytic rank $0$
Dimension $64$
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: \(64\)
Relative dimension: \(8\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 64 q - 8 q^{2} - 6 q^{3} + 8 q^{4} - 7 q^{5} - 4 q^{6} + 22 q^{7} + 16 q^{8} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q - 8 q^{2} - 6 q^{3} + 8 q^{4} - 7 q^{5} - 4 q^{6} + 22 q^{7} + 16 q^{8} + 14 q^{9} - 8 q^{10} - 6 q^{11} - 8 q^{12} + 9 q^{13} + 11 q^{14} + 9 q^{15} + 8 q^{16} - 2 q^{17} - 12 q^{18} + 4 q^{19} + 14 q^{20} - 36 q^{21} + 7 q^{22} + 8 q^{23} - 4 q^{24} + 31 q^{25} - 12 q^{26} + 54 q^{27} + 9 q^{28} + 18 q^{29} + 18 q^{30} + 20 q^{31} + 32 q^{32} + 10 q^{33} + 2 q^{34} - 16 q^{35} - 6 q^{36} + 18 q^{37} - 31 q^{38} + 2 q^{39} - 3 q^{40} + 16 q^{41} + 6 q^{42} + 42 q^{43} - 2 q^{44} - 8 q^{45} - 24 q^{46} - 34 q^{47} - 6 q^{48} - 10 q^{49} - 58 q^{50} - 40 q^{51} - 6 q^{52} + 15 q^{53} - 28 q^{54} + 49 q^{55} + 8 q^{56} + 8 q^{57} + 36 q^{58} - 7 q^{59} + 4 q^{60} - 15 q^{61} - 37 q^{63} - 16 q^{64} - 48 q^{65} - 10 q^{66} - 14 q^{67} + 4 q^{68} - 30 q^{69} - 19 q^{70} - 4 q^{71} - 14 q^{72} + 8 q^{73} + 9 q^{74} - 96 q^{75} - 10 q^{76} - 58 q^{77} + 46 q^{78} + 12 q^{79} + 3 q^{80} - 8 q^{81} + 4 q^{82} - 6 q^{83} - 48 q^{84} + 18 q^{85} + 3 q^{86} - 244 q^{87} + 6 q^{88} - 4 q^{89} - 9 q^{90} - 33 q^{91} + 8 q^{92} + 3 q^{93} + 62 q^{94} - 49 q^{95} - 12 q^{96} - 15 q^{97} - 4 q^{99}+O(q^{100}) \) 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 −2.78374 + 1.23940i 0.913545 + 0.406737i −1.23365 − 1.37011i −2.98059 + 0.633544i 2.74006 − 1.99077i 0.809017 + 0.587785i 4.20569 − 4.67089i −0.921833 − 1.59666i
49.2 0.978148 + 0.207912i −1.41989 + 0.632177i 0.913545 + 0.406737i 1.85966 + 2.06536i −1.52030 + 0.323150i 3.87852 − 2.81791i 0.809017 + 0.587785i −0.390947 + 0.434190i 1.38961 + 2.40687i
49.3 0.978148 + 0.207912i −1.22145 + 0.543824i 0.913545 + 0.406737i −1.54111 − 1.71158i −1.30782 + 0.277987i −1.94898 + 1.41602i 0.809017 + 0.587785i −0.811200 + 0.900929i −1.15158 − 1.99459i
49.4 0.978148 + 0.207912i −0.876283 + 0.390146i 0.913545 + 0.406737i 2.35086 + 2.61090i −0.938250 + 0.199431i −1.45336 + 1.05593i 0.809017 + 0.587785i −1.39173 + 1.54568i 1.75666 + 3.04262i
49.5 0.978148 + 0.207912i −0.690099 + 0.307252i 0.913545 + 0.406737i −2.78499 − 3.09304i −0.738900 + 0.157058i 1.79660 − 1.30531i 0.809017 + 0.587785i −1.62556 + 1.80537i −2.08105 − 3.60448i
49.6 0.978148 + 0.207912i 0.375886 − 0.167355i 0.913545 + 0.406737i 1.03755 + 1.15232i 0.402467 − 0.0855471i −0.764726 + 0.555606i 0.809017 + 0.587785i −1.89411 + 2.10362i 0.775298 + 1.34285i
49.7 0.978148 + 0.207912i 1.44379 − 0.642816i 0.913545 + 0.406737i −0.891991 − 0.990657i 1.54589 − 0.328588i 1.29335 − 0.939674i 0.809017 + 0.587785i −0.336079 + 0.373254i −0.666530 − 1.15446i
49.8 0.978148 + 0.207912i 2.21549 − 0.986399i 0.913545 + 0.406737i 1.36163 + 1.51224i 2.37216 − 0.504218i −1.11442 + 0.809676i 0.809017 + 0.587785i 1.92802 − 2.14128i 1.01746 + 1.76229i
125.1 −0.913545 + 0.406737i −1.96426 − 2.18153i 0.669131 − 0.743145i −0.0453791 − 0.431753i 2.68175 + 1.19399i 0.873693 + 2.68895i −0.309017 + 0.951057i −0.587180 + 5.58664i 0.217066 + 0.375969i
125.2 −0.913545 + 0.406737i −1.84910 − 2.05364i 0.669131 − 0.743145i 0.213696 + 2.03318i 2.52453 + 1.12399i −0.473002 − 1.45575i −0.309017 + 0.951057i −0.484658 + 4.61121i −1.02219 − 1.77049i
125.3 −0.913545 + 0.406737i −0.311198 − 0.345621i 0.669131 − 0.743145i 0.187144 + 1.78056i 0.424871 + 0.189165i −0.928913 − 2.85890i −0.309017 + 0.951057i 0.290976 − 2.76845i −0.895184 − 1.55050i
125.4 −0.913545 + 0.406737i 0.0875155 + 0.0971958i 0.669131 − 0.743145i 0.00825402 + 0.0785317i −0.119482 − 0.0531970i 1.37067 + 4.21848i −0.309017 + 0.951057i 0.311797 − 2.96655i −0.0394821 − 0.0683851i
125.5 −0.913545 + 0.406737i 0.344284 + 0.382366i 0.669131 − 0.743145i 0.0740341 + 0.704387i −0.470041 − 0.209276i −0.424117 − 1.30530i −0.309017 + 0.951057i 0.285913 − 2.72028i −0.354134 − 0.613377i
125.6 −0.913545 + 0.406737i 1.27149 + 1.41213i 0.669131 − 0.743145i −0.245336 − 2.33422i −1.73593 − 0.772884i −0.645619 − 1.98701i −0.309017 + 0.951057i −0.0638446 + 0.607441i 1.17354 + 2.03263i
125.7 −0.913545 + 0.406737i 1.35422 + 1.50401i 0.669131 − 0.743145i 0.371716 + 3.53664i −1.84887 − 0.823172i 1.09191 + 3.36056i −0.309017 + 0.951057i −0.114558 + 1.08994i −1.77806 − 3.07970i
125.8 −0.913545 + 0.406737i 1.89415 + 2.10367i 0.669131 − 0.743145i −0.121340 − 1.15447i −2.58603 − 1.15138i 0.208330 + 0.641173i −0.309017 + 0.951057i −0.524026 + 4.98578i 0.580415 + 1.00531i
159.1 −0.669131 + 0.743145i −0.253498 + 2.41187i −0.104528 − 0.994522i −1.99045 + 0.423084i −1.62275 − 1.80224i −1.11442 + 0.809676i 0.809017 + 0.587785i −2.81841 − 0.599072i 1.01746 − 1.76229i
159.2 −0.669131 + 0.743145i −0.165199 + 1.57177i −0.104528 − 0.994522i 1.30393 − 0.277159i −1.05751 − 1.17448i 1.29335 − 0.939674i 0.809017 + 0.587785i 0.491287 + 0.104426i −0.666530 + 1.15446i
159.3 −0.669131 + 0.743145i −0.0430092 + 0.409205i −0.104528 − 0.994522i −1.51671 + 0.322387i −0.275320 − 0.305774i −0.764726 + 0.555606i 0.809017 + 0.587785i 2.76884 + 0.588536i 0.775298 − 1.34285i
159.4 −0.669131 + 0.743145i 0.0789616 − 0.751269i −0.104528 − 0.994522i 4.07114 − 0.865348i 0.505466 + 0.561377i 1.79660 − 1.30531i 0.809017 + 0.587785i 2.37627 + 0.505092i −2.08105 + 3.60448i
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.8
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.d ✓ 64
11.c even 5 1 inner 418.2.n.d ✓ 64
19.c even 3 1 inner 418.2.n.d ✓ 64
209.n even 15 1 inner 418.2.n.d ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.n.d ✓ 64 1.a even 1 1 trivial
418.2.n.d ✓ 64 11.c even 5 1 inner
418.2.n.d ✓ 64 19.c even 3 1 inner
418.2.n.d ✓ 64 209.n even 15 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{64} + 6 T_{3}^{63} - T_{3}^{62} - 80 T_{3}^{61} - 140 T_{3}^{60} + 214 T_{3}^{59} + 758 T_{3}^{58} + \cdots + 160000 \) acting on \(S_{2}^{\mathrm{new}}(418, [\chi])\). Copy content Toggle raw display