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Properties

Label 325.3.w.f
Level $325$
Weight $3$
Character orbit 325.w
Analytic conductor $8.856$
Analytic rank $0$
Dimension $40$
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] = [325,3,Mod(24,325)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(325, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 7]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("325.24");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 325 = 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 325.w (of order \(12\), degree \(4\), not 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: \(8.85560859171\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(10\) over \(\Q(\zeta_{12})\)
Twist minimal: no (minimal twist has level 65)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 12 q^{3} - 12 q^{6} + 44 q^{7} - 36 q^{8} + 72 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q + 12 q^{3} - 12 q^{6} + 44 q^{7} - 36 q^{8} + 72 q^{9} - 12 q^{11} - 120 q^{12} + 36 q^{13} - 48 q^{14} + 128 q^{16} + 32 q^{17} - 136 q^{18} - 68 q^{19} - 48 q^{21} + 72 q^{22} + 28 q^{23} + 56 q^{24} - 84 q^{26} - 76 q^{28} - 28 q^{29} + 128 q^{31} - 84 q^{32} + 68 q^{33} + 28 q^{34} + 300 q^{36} + 260 q^{37} - 8 q^{38} + 88 q^{39} + 68 q^{41} + 480 q^{42} - 32 q^{43} + 240 q^{44} + 260 q^{46} + 152 q^{47} + 480 q^{48} + 132 q^{49} - 488 q^{52} - 152 q^{54} - 288 q^{56} - 252 q^{57} - 468 q^{58} - 492 q^{59} - 100 q^{61} + 8 q^{62} + 356 q^{63} - 456 q^{66} + 56 q^{67} - 12 q^{68} + 576 q^{69} - 132 q^{71} - 396 q^{72} + 424 q^{73} + 160 q^{74} - 992 q^{76} - 344 q^{77} - 12 q^{78} + 248 q^{79} - 248 q^{82} + 112 q^{83} - 1100 q^{84} + 852 q^{86} + 1068 q^{87} - 8 q^{88} - 168 q^{89} - 160 q^{91} + 24 q^{93} - 328 q^{94} + 124 q^{96} - 1020 q^{97} - 1404 q^{98} + 76 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
24.1 −3.73273 − 1.00018i 2.83142 − 1.63472i 9.46878 + 5.46680i 0 −12.2039 + 3.27004i −8.61234 + 2.30767i −18.9464 − 18.9464i 0.844632 − 1.46295i 0
24.2 −3.53665 − 0.947644i −4.05506 + 2.34119i 8.14580 + 4.70298i 0 16.5600 − 4.43723i 10.5506 − 2.82701i −13.9961 − 13.9961i 6.46235 − 11.1931i 0
24.3 −2.20214 − 0.590063i 1.83281 − 1.05817i 1.03716 + 0.598805i 0 −4.66050 + 1.24878i 1.17338 − 0.314408i 4.51768 + 4.51768i −2.26053 + 3.91536i 0
24.4 −1.14195 − 0.305985i 3.98555 − 2.30106i −2.25367 − 1.30116i 0 −5.25539 + 1.40818i 11.4086 − 3.05692i 5.51932 + 5.51932i 6.08971 − 10.5477i 0
24.5 0.346703 + 0.0928988i −0.511202 + 0.295143i −3.35253 − 1.93558i 0 −0.204654 + 0.0548368i 0.327396 − 0.0877254i −1.99774 − 1.99774i −4.32578 + 7.49247i 0
24.6 0.401493 + 0.107580i −3.61165 + 2.08519i −3.31448 − 1.91361i 0 −1.67438 + 0.448649i 10.0361 − 2.68917i −2.30053 − 2.30053i 4.19603 − 7.26773i 0
24.7 1.16577 + 0.312366i 2.06426 − 1.19180i −2.20266 − 1.27171i 0 2.77872 − 0.744556i −7.49531 + 2.00836i −5.58415 − 5.58415i −1.65922 + 2.87385i 0
24.8 2.66986 + 0.715386i −3.65949 + 2.11281i 3.15226 + 1.81996i 0 −11.2818 + 3.02294i −3.52221 + 0.943772i −0.703773 − 0.703773i 4.42790 − 7.66935i 0
24.9 2.72224 + 0.729421i 4.15192 − 2.39711i 3.41442 + 1.97132i 0 13.0510 − 3.49701i 4.75472 − 1.27402i −0.114321 − 0.114321i 6.99232 − 12.1111i 0
24.10 3.30742 + 0.886220i −0.0285599 + 0.0164891i 6.68953 + 3.86220i 0 −0.109072 + 0.0292259i −0.692695 + 0.185607i 9.01754 + 9.01754i −4.49946 + 7.79329i 0
124.1 −1.01171 − 3.77575i 0.585883 + 0.338260i −9.76865 + 5.63993i 0 0.684442 − 2.55437i −1.21531 + 4.53560i 20.1218 + 20.1218i −4.27116 − 7.39787i 0
124.2 −0.683209 − 2.54977i 3.75874 + 2.17011i −2.57045 + 1.48405i 0 2.96528 − 11.0666i 2.17385 − 8.11293i −1.92611 − 1.92611i 4.91876 + 8.51954i 0
124.3 −0.614128 − 2.29196i −1.05591 − 0.609633i −1.41181 + 0.815107i 0 −0.748785 + 2.79450i −1.33475 + 4.98135i −3.97609 − 3.97609i −3.75670 − 6.50679i 0
124.4 −0.598846 − 2.23492i −3.51738 − 2.03076i −1.17216 + 0.676749i 0 −2.43222 + 9.07719i 2.56698 − 9.58012i −4.32988 − 4.32988i 3.74797 + 6.49168i 0
124.5 0.0113525 + 0.0423683i 2.77843 + 1.60413i 3.46244 − 1.99904i 0 −0.0364219 + 0.135928i 2.16799 − 8.09104i 0.248066 + 0.248066i 0.646457 + 1.11970i 0
124.6 0.206966 + 0.772409i −5.13291 − 2.96349i 2.91032 − 1.68027i 0 1.22668 − 4.57805i −0.592215 + 2.21018i 4.16197 + 4.16197i 13.0645 + 22.6284i 0
124.7 0.340684 + 1.27145i 3.38958 + 1.95698i 1.96358 − 1.13368i 0 −1.33342 + 4.97639i −2.05866 + 7.68303i 5.83343 + 5.83343i 3.15951 + 5.47243i 0
124.8 0.624525 + 2.33076i −1.82226 − 1.05208i −1.57831 + 0.911238i 0 1.31411 − 4.90432i −0.565655 + 2.11105i 3.71537 + 3.71537i −2.28624 − 3.95988i 0
124.9 0.749686 + 2.79787i −0.461903 − 0.266680i −3.80193 + 2.19504i 0 0.399852 − 1.49227i 2.69020 − 10.0400i −0.798969 − 0.798969i −4.35776 − 7.54787i 0
124.10 0.974678 + 3.63755i 4.47773 + 2.58522i −8.81765 + 5.09087i 0 −5.03951 + 18.8077i 0.239364 − 0.893318i −16.4612 − 16.4612i 8.86671 + 15.3576i 0
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 24.10
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.s odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 325.3.w.f 40
5.b even 2 1 325.3.w.e 40
5.c odd 4 1 65.3.p.a ✓ 40
5.c odd 4 1 325.3.t.d 40
13.f odd 12 1 325.3.w.e 40
65.o even 12 1 325.3.t.d 40
65.s odd 12 1 inner 325.3.w.f 40
65.t even 12 1 65.3.p.a ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.3.p.a ✓ 40 5.c odd 4 1
65.3.p.a ✓ 40 65.t even 12 1
325.3.t.d 40 5.c odd 4 1
325.3.t.d 40 65.o even 12 1
325.3.w.e 40 5.b even 2 1
325.3.w.e 40 13.f odd 12 1
325.3.w.f 40 1.a even 1 1 trivial
325.3.w.f 40 65.s odd 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{40} + 12 T_{2}^{37} - 336 T_{2}^{36} + 132 T_{2}^{35} + 72 T_{2}^{34} - 5796 T_{2}^{33} + \cdots + 158986881 \) acting on \(S_{3}^{\mathrm{new}}(325, [\chi])\). Copy content Toggle raw display