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Properties

Label 405.3.w.a
Level $405$
Weight $3$
Character orbit 405.w
Analytic conductor $11.035$
Analytic rank $0$
Dimension $3816$
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] = [405,3,Mod(7,405)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(405, base_ring=CyclotomicField(108))
 
chi = DirichletCharacter(H, H._module([32, 27]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("405.7");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 405.w (of order \(108\), degree \(36\), 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: \(11.0354507066\)
Analytic rank: \(0\)
Dimension: \(3816\)
Relative dimension: \(106\) over \(\Q(\zeta_{108})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{108}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 3816 q - 36 q^{2} - 36 q^{3} - 36 q^{5} - 72 q^{6} - 36 q^{7} - 36 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 3816 q - 36 q^{2} - 36 q^{3} - 36 q^{5} - 72 q^{6} - 36 q^{7} - 36 q^{8} - 36 q^{10} - 72 q^{11} - 36 q^{12} - 36 q^{13} - 36 q^{15} - 72 q^{16} - 36 q^{17} - 36 q^{18} + 396 q^{20} - 72 q^{21} - 36 q^{22} - 36 q^{23} - 36 q^{25} - 36 q^{26} - 36 q^{27} - 18 q^{28} - 468 q^{30} - 72 q^{31} - 36 q^{32} - 36 q^{33} - 522 q^{35} - 72 q^{36} - 36 q^{37} - 36 q^{38} - 36 q^{40} + 360 q^{41} - 36 q^{42} - 36 q^{43} - 36 q^{45} - 72 q^{46} - 36 q^{47} - 36 q^{48} - 36 q^{50} + 360 q^{51} + 36 q^{52} - 18 q^{53} - 18 q^{55} - 828 q^{56} - 36 q^{57} - 36 q^{58} - 36 q^{60} - 72 q^{61} - 36 q^{62} - 36 q^{63} - 36 q^{65} - 1944 q^{66} + 666 q^{67} - 36 q^{68} - 36 q^{70} - 72 q^{71} - 3492 q^{72} - 36 q^{73} - 36 q^{75} - 72 q^{76} - 900 q^{77} + 126 q^{78} - 72 q^{80} - 72 q^{81} - 72 q^{82} - 36 q^{83} - 36 q^{85} - 72 q^{86} + 1980 q^{87} - 36 q^{88} - 36 q^{90} - 72 q^{91} + 5148 q^{92} + 2484 q^{93} - 36 q^{95} - 72 q^{96} - 954 q^{97} + 3042 q^{98}+O(q^{100}) \) 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 −0.561706 + 3.83473i −1.38259 + 2.66241i −10.5577 − 3.16077i 4.80872 − 1.36974i −9.43303 − 6.79737i −11.1743 + 0.325140i 11.4994 − 24.6605i −5.17687 − 7.36206i 2.55150 + 19.2096i
7.2 −0.556532 + 3.79941i 2.48035 − 1.68756i −10.2939 − 3.08178i −4.50811 + 2.16262i 5.03135 + 10.3631i 10.3325 − 0.300644i 10.9465 − 23.4748i 3.30428 − 8.37148i −5.70779 − 18.3317i
7.3 −0.548708 + 3.74600i −2.90639 + 0.743584i −9.89944 − 2.96370i −0.933045 + 4.91217i −1.19071 − 11.2953i 5.11735 − 0.148900i 10.1338 − 21.7321i 7.89417 − 4.32228i −17.8890 − 6.19053i
7.4 −0.543003 + 3.70705i 1.19626 + 2.75118i −9.61544 − 2.87867i −4.19944 − 2.71379i −10.8483 + 2.94069i −0.715076 + 0.0208066i 9.55907 − 20.4995i −6.13794 + 6.58223i 12.3405 − 14.0940i
7.5 −0.541343 + 3.69572i −1.87923 − 2.33848i −9.53335 − 2.85410i 4.11428 + 2.84125i 9.65968 − 5.67920i 1.99695 − 0.0581054i 9.39458 − 20.1467i −1.93697 + 8.78909i −12.7277 + 13.6671i
7.6 −0.516736 + 3.52773i −2.89503 − 0.786623i −8.34589 − 2.49860i −0.625219 − 4.96076i 4.27096 − 9.80642i 1.93587 − 0.0563279i 7.09983 − 15.2256i 7.76245 + 4.55460i 17.8233 + 0.357800i
7.7 −0.516390 + 3.52537i 2.92098 + 0.683995i −8.32960 − 2.49372i 4.34213 − 2.47910i −3.91970 + 9.94434i −1.44263 + 0.0419761i 7.06947 − 15.1605i 8.06430 + 3.99588i 6.49749 + 16.5878i
7.8 −0.511146 + 3.48956i 1.62053 + 2.52466i −8.08383 − 2.42014i 3.57903 + 3.49150i −9.63828 + 4.36447i 11.5853 − 0.337097i 6.61530 − 14.1865i −3.74777 + 8.18255i −14.0132 + 10.7046i
7.9 −0.508103 + 3.46879i −1.07891 − 2.79928i −7.94237 − 2.37779i −4.90325 + 0.978836i 10.2583 − 2.32018i −6.80760 + 0.198081i 6.35713 − 13.6329i −6.67192 + 6.04032i −0.904019 − 17.5057i
7.10 −0.498828 + 3.40547i 1.85275 − 2.35952i −7.51643 − 2.25027i 2.09319 + 4.54077i 7.11106 + 7.48647i −9.24023 + 0.268863i 5.59435 − 11.9971i −2.13464 − 8.74319i −16.5076 + 4.86323i
7.11 −0.494813 + 3.37806i 2.95688 − 0.506836i −7.33449 − 2.19580i −0.479780 − 4.97693i 0.249021 + 10.2393i −1.81061 + 0.0526835i 5.27530 − 11.3129i 8.48624 − 2.99730i 17.0498 + 0.841923i
7.12 −0.475852 + 3.24861i 1.40237 − 2.65205i −6.49509 − 1.94450i 4.80051 − 1.39825i 7.94816 + 5.81773i 5.79559 − 0.168634i 3.85734 − 8.27208i −5.06672 − 7.43830i 2.25804 + 16.2604i
7.13 −0.475507 + 3.24626i 2.69056 + 1.32698i −6.48014 − 1.94003i −2.43971 + 4.36438i −5.58710 + 8.10328i −8.15667 + 0.237335i 3.83291 − 8.21971i 5.47825 + 7.14064i −13.0078 − 9.99523i
7.14 −0.465467 + 3.17771i −2.70153 + 1.30450i −6.04925 − 1.81103i −4.99400 − 0.244784i −2.88787 − 9.19190i −11.4596 + 0.333439i 3.14148 − 6.73693i 5.59654 − 7.04831i 3.10239 − 15.7556i
7.15 −0.459599 + 3.13766i −1.20767 + 2.74619i −5.80170 − 1.73691i 0.676995 − 4.95396i −8.06155 − 5.05139i 10.8995 − 0.317144i 2.75557 − 5.90933i −6.08308 − 6.63296i 15.2327 + 4.40101i
7.16 −0.451248 + 3.08065i −1.09389 + 2.79346i −5.45481 − 1.63306i −0.550304 + 4.96962i −8.11205 − 4.63043i 0.414689 − 0.0120662i 2.22904 − 4.78019i −6.60682 − 6.11146i −15.0613 − 3.93783i
7.17 −0.425828 + 2.90710i −0.0373859 − 2.99977i −4.43797 − 1.32864i −3.18375 − 3.85536i 8.73656 + 1.16870i 9.02310 − 0.262546i 0.785488 − 1.68448i −8.99720 + 0.224298i 12.5637 − 7.61376i
7.18 −0.393062 + 2.68341i −2.57288 + 1.54281i −3.21425 − 0.962282i 4.73342 − 1.61081i −3.12870 − 7.51053i 4.52616 − 0.131698i −0.739044 + 1.58488i 4.23946 − 7.93895i 2.46192 + 13.3349i
7.19 −0.392014 + 2.67626i −2.29035 − 1.93760i −3.17673 − 0.951050i 4.58645 − 1.99109i 6.08337 − 5.37001i −8.37087 + 0.243568i −0.781842 + 1.67667i 1.49142 + 8.87557i 3.53072 + 13.0551i
7.20 −0.368389 + 2.51497i −2.86810 − 0.879768i −2.35741 − 0.705762i −4.10404 + 2.85603i 3.26917 − 6.88909i 8.50000 − 0.247325i −1.65345 + 3.54582i 7.45202 + 5.04653i −5.67094 − 11.3737i
See next 80 embeddings (of 3816 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.106
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
81.g even 27 1 inner
405.w odd 108 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 405.3.w.a ✓ 3816
5.c odd 4 1 inner 405.3.w.a ✓ 3816
81.g even 27 1 inner 405.3.w.a ✓ 3816
405.w odd 108 1 inner 405.3.w.a ✓ 3816
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
405.3.w.a ✓ 3816 1.a even 1 1 trivial
405.3.w.a ✓ 3816 5.c odd 4 1 inner
405.3.w.a ✓ 3816 81.g even 27 1 inner
405.3.w.a ✓ 3816 405.w odd 108 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(405, [\chi])\).