Properties

Label 414.3.o.a
Level $414$
Weight $3$
Character orbit 414.o
Analytic conductor $11.281$
Analytic rank $0$
Dimension $960$
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] = [414,3,Mod(29,414)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(414, base_ring=CyclotomicField(66))
 
chi = DirichletCharacter(H, H._module([11, 54]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("414.29");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 414 = 2 \cdot 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 414.o (of order \(66\), degree \(20\), 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: \(11.2806829445\)
Analytic rank: \(0\)
Dimension: \(960\)
Relative dimension: \(48\) over \(\Q(\zeta_{66})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{66}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 960 q + 4 q^{3} - 96 q^{4} + 36 q^{5} + 16 q^{6} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 960 q + 4 q^{3} - 96 q^{4} + 36 q^{5} + 16 q^{6} - 4 q^{9} - 8 q^{12} - 72 q^{14} + 300 q^{15} + 192 q^{16} - 160 q^{18} + 72 q^{20} - 158 q^{21} - 18 q^{23} + 16 q^{24} - 228 q^{25} + 310 q^{27} - 36 q^{29} + 328 q^{30} - 60 q^{31} - 688 q^{33} - 16 q^{36} - 168 q^{37} + 156 q^{39} + 288 q^{41} + 128 q^{42} - 116 q^{45} + 24 q^{46} + 72 q^{47} - 32 q^{48} + 120 q^{49} + 288 q^{50} + 56 q^{51} + 420 q^{54} + 264 q^{55} + 648 q^{56} + 662 q^{57} + 120 q^{60} - 96 q^{61} + 310 q^{63} + 768 q^{64} + 1278 q^{65} + 368 q^{66} + 156 q^{67} - 128 q^{69} + 120 q^{70} - 16 q^{72} - 1728 q^{74} - 1062 q^{75} - 540 q^{77} - 716 q^{78} + 44 q^{81} - 2070 q^{83} - 408 q^{84} - 88 q^{87} + 32 q^{90} - 36 q^{92} - 852 q^{93} - 168 q^{94} - 1080 q^{95} - 32 q^{96} + 918 q^{97} - 816 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1 −1.15198 + 0.820324i −2.99334 0.199741i 0.654136 1.89000i −4.08143 + 7.91688i 3.61214 2.22541i 8.43255 + 6.63143i 0.796860 + 2.71386i 8.92021 + 1.19579i −1.79266 12.4682i
29.2 −1.15198 + 0.820324i −2.92929 + 0.647508i 0.654136 1.89000i −0.971524 + 1.88449i 2.84333 3.14889i −7.01983 5.52046i 0.796860 + 2.71386i 8.16147 3.79348i −0.426716 2.96787i
29.3 −1.15198 + 0.820324i −2.84434 0.953805i 0.654136 1.89000i 0.224329 0.435137i 4.05906 1.23451i −1.83852 1.44583i 0.796860 + 2.71386i 7.18051 + 5.42589i 0.0985302 + 0.685293i
29.4 −1.15198 + 0.820324i −2.73798 1.22617i 0.654136 1.89000i 2.56559 4.97655i 4.15996 0.833504i 8.26502 + 6.49968i 0.796860 + 2.71386i 5.99303 + 6.71443i 1.12687 + 7.83753i
29.5 −1.15198 + 0.820324i −2.68304 + 1.34212i 0.654136 1.89000i 3.75061 7.27517i 1.98985 3.74706i 2.16342 + 1.70134i 0.796860 + 2.71386i 5.39744 7.20192i 1.64735 + 11.4576i
29.6 −1.15198 + 0.820324i −2.48619 + 1.67895i 0.654136 1.89000i −2.13326 + 4.13794i 1.48677 3.97360i 1.87456 + 1.47417i 0.796860 + 2.71386i 3.36227 8.34836i −0.936975 6.51680i
29.7 −1.15198 + 0.820324i −1.88417 + 2.33451i 0.654136 1.89000i 0.426441 0.827180i 0.255479 4.23494i 3.49911 + 2.75173i 0.796860 + 2.71386i −1.89983 8.79720i 0.187303 + 1.30272i
29.8 −1.15198 + 0.820324i −1.57511 2.55324i 0.654136 1.89000i −3.77717 + 7.32668i 3.90899 + 1.64919i −7.64250 6.01013i 0.796860 + 2.71386i −4.03806 + 8.04326i −1.65902 11.5387i
29.9 −1.15198 + 0.820324i −1.39829 2.65420i 0.654136 1.89000i 1.56148 3.02884i 3.78812 + 1.91054i 0.462983 + 0.364094i 0.796860 + 2.71386i −5.08956 + 7.42269i 0.685837 + 4.77010i
29.10 −1.15198 + 0.820324i −0.995486 2.83002i 0.654136 1.89000i 2.41189 4.67841i 3.46832 + 2.44352i −4.90570 3.85789i 0.796860 + 2.71386i −7.01802 + 5.63449i 1.05936 + 7.36799i
29.11 −1.15198 + 0.820324i −0.858721 + 2.87447i 0.654136 1.89000i 1.27146 2.46629i −1.36877 4.01578i −9.21164 7.24412i 0.796860 + 2.71386i −7.52520 4.93674i 0.558454 + 3.88413i
29.12 −1.15198 + 0.820324i −0.211823 + 2.99251i 0.654136 1.89000i −2.84452 + 5.51760i −2.21081 3.62109i 1.88035 + 1.47873i 0.796860 + 2.71386i −8.91026 1.26777i −1.24938 8.68961i
29.13 −1.15198 + 0.820324i −0.00945685 2.99999i 0.654136 1.89000i −2.36088 + 4.57947i 2.47185 + 3.44818i 7.63159 + 6.00155i 0.796860 + 2.71386i −8.99982 + 0.0567408i −1.03695 7.21217i
29.14 −1.15198 + 0.820324i 0.355720 + 2.97884i 0.654136 1.89000i 1.78394 3.46036i −2.85340 3.13977i 10.3427 + 8.13360i 0.796860 + 2.71386i −8.74693 + 2.11926i 0.783547 + 5.44969i
29.15 −1.15198 + 0.820324i 1.32781 + 2.69015i 0.654136 1.89000i 3.22219 6.25018i −3.73641 2.00978i −4.61888 3.63233i 0.796860 + 2.71386i −5.47384 + 7.14402i 1.41526 + 9.84335i
29.16 −1.15198 + 0.820324i 1.35985 2.67410i 0.654136 1.89000i −2.17040 + 4.20998i 0.627101 + 4.19604i −1.31557 1.03458i 0.796860 + 2.71386i −5.30161 7.27275i −0.953288 6.63026i
29.17 −1.15198 + 0.820324i 1.38285 2.66228i 0.654136 1.89000i 1.47947 2.86978i 0.590910 + 4.20129i −8.32828 6.54943i 0.796860 + 2.71386i −5.17545 7.36306i 0.649819 + 4.51959i
29.18 −1.15198 + 0.820324i 1.68468 + 2.48231i 0.654136 1.89000i −3.43268 + 6.65847i −3.97702 1.47760i 0.376015 + 0.295701i 0.796860 + 2.71386i −3.32371 + 8.36379i −1.50771 10.4864i
29.19 −1.15198 + 0.820324i 1.93664 2.29116i 0.654136 1.89000i 0.257284 0.499061i −0.351483 + 4.22806i 2.18749 + 1.72026i 0.796860 + 2.71386i −1.49885 8.87431i 0.113005 + 0.785967i
29.20 −1.15198 + 0.820324i 2.48787 + 1.67646i 0.654136 1.89000i 0.670722 1.30102i −4.24122 + 0.109603i 1.82494 + 1.43515i 0.796860 + 2.71386i 3.37897 + 8.34162i 0.294596 + 2.04896i
See next 80 embeddings (of 960 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner
23.c even 11 1 inner
207.n odd 66 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 414.3.o.a 960
9.d odd 6 1 inner 414.3.o.a 960
23.c even 11 1 inner 414.3.o.a 960
207.n odd 66 1 inner 414.3.o.a 960
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
414.3.o.a 960 1.a even 1 1 trivial
414.3.o.a 960 9.d odd 6 1 inner
414.3.o.a 960 23.c even 11 1 inner
414.3.o.a 960 207.n odd 66 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(414, [\chi])\).