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Properties

Label 475.2.x.a
Level $475$
Weight $2$
Character orbit 475.x
Analytic conductor $3.793$
Analytic rank $0$
Dimension $384$
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] = [475,2,Mod(64,475)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(475, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([9, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("475.64");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 475.x (of order \(30\), 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: \(3.79289409601\)
Analytic rank: \(0\)
Dimension: \(384\)
Relative dimension: \(48\) over \(\Q(\zeta_{30})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 384 q - 5 q^{2} - 5 q^{3} - 49 q^{4} - 5 q^{5} + 5 q^{6} - 50 q^{8} - 47 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 384 q - 5 q^{2} - 5 q^{3} - 49 q^{4} - 5 q^{5} + 5 q^{6} - 50 q^{8} - 47 q^{9} - 3 q^{10} - 6 q^{11} - 20 q^{12} - 5 q^{13} + q^{14} + 19 q^{15} + 35 q^{16} + 12 q^{20} + 26 q^{21} - 75 q^{22} - 35 q^{23} - 20 q^{24} + 5 q^{25} - 4 q^{26} - 50 q^{27} + 15 q^{28} - 21 q^{29} + 8 q^{30} - 36 q^{31} - 15 q^{33} - 3 q^{34} + 9 q^{35} + 19 q^{36} - 20 q^{37} - 75 q^{38} + 24 q^{39} - 40 q^{40} + 13 q^{41} + 25 q^{42} + 47 q^{44} + 132 q^{45} + 28 q^{46} - 75 q^{47} + 15 q^{48} - 300 q^{49} - 50 q^{50} - 70 q^{51} - 15 q^{52} - 25 q^{53} - 10 q^{54} + 70 q^{55} - 20 q^{56} + 290 q^{58} - 51 q^{59} + 26 q^{60} + 11 q^{61} - 30 q^{62} + 45 q^{63} - 2 q^{64} - 96 q^{65} + 13 q^{66} - 25 q^{67} + 24 q^{69} - 32 q^{70} + 38 q^{71} + 50 q^{72} - 85 q^{73} + 2 q^{74} + 116 q^{75} + 36 q^{76} + 160 q^{77} - 80 q^{78} - 19 q^{79} - 45 q^{80} - 65 q^{81} - 90 q^{83} - 44 q^{84} - 60 q^{85} + 35 q^{86} - 230 q^{87} + 100 q^{88} - 21 q^{89} - 143 q^{90} + 34 q^{91} + 5 q^{92} + 72 q^{94} + 58 q^{95} - 190 q^{96} + 10 q^{97} + 60 q^{98} + 28 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
64.1 −1.11698 + 2.50878i −0.451198 − 2.12272i −3.70808 − 4.11824i −0.904987 − 2.04475i 5.82942 + 1.23908i 0.525310i 9.25005 − 3.00552i −1.56172 + 0.695324i 6.14069 + 0.0135314i
64.2 −1.07613 + 2.41703i 0.255014 + 1.19975i −3.34571 − 3.71579i 2.23085 + 0.152621i −3.17426 − 0.674709i 1.90035i 7.54905 − 2.45284i 1.36627 − 0.608304i −2.76958 + 5.22780i
64.3 −1.01622 + 2.28246i −0.167470 − 0.787883i −2.83868 − 3.15267i −0.890888 + 2.05093i 1.96850 + 0.418418i − 4.49119i 5.32821 − 1.73124i 2.14792 − 0.956317i −3.77584 − 4.11761i
64.4 −0.970819 + 2.18049i −0.268308 − 1.26229i −2.47380 − 2.74744i 1.77372 + 1.36159i 3.01289 + 0.640410i 0.521008i 3.85233 − 1.25170i 1.21925 − 0.542845i −4.69089 + 2.54573i
64.5 −0.925844 + 2.07948i 0.675001 + 3.17563i −2.12878 − 2.36426i −0.558369 − 2.16523i −7.22860 − 1.53649i 1.95049i 2.55761 − 0.831018i −6.88837 + 3.06690i 5.01951 + 0.843547i
64.6 −0.912197 + 2.04883i −0.501005 − 2.35704i −2.02733 − 2.25158i −1.65494 + 1.50372i 5.28619 + 1.12362i 1.88351i 2.19652 − 0.713692i −2.56402 + 1.14157i −1.57123 − 4.76238i
64.7 −0.902639 + 2.02736i 0.292883 + 1.37791i −1.95717 − 2.17366i 1.60858 − 1.55321i −3.05788 − 0.649973i − 3.15833i 1.95220 − 0.634308i 0.927786 − 0.413077i 1.69695 + 4.66317i
64.8 −0.802128 + 1.80161i −0.628840 − 2.95846i −1.26412 − 1.40395i 2.17715 − 0.509932i 5.83440 + 1.24014i 2.88993i −0.207810 + 0.0675216i −5.61641 + 2.50058i −0.827652 + 4.33140i
64.9 −0.750712 + 1.68613i 0.417381 + 1.96363i −0.941196 − 1.04530i 0.00110333 + 2.23607i −3.62426 − 0.770359i 1.76901i −1.04164 + 0.338449i −0.940980 + 0.418951i −3.77112 − 1.67678i
64.10 −0.738156 + 1.65792i −0.0112865 − 0.0530989i −0.865580 − 0.961324i −0.113934 − 2.23316i 0.0963651 + 0.0204830i 4.56089i −1.21927 + 0.396164i 2.73794 − 1.21901i 3.78652 + 1.45953i
64.11 −0.702502 + 1.57784i −0.0448160 − 0.210843i −0.657824 − 0.730587i −2.00025 + 0.999496i 0.364160 + 0.0774047i 2.33475i −1.67039 + 0.542743i 2.69819 − 1.20131i −0.171868 − 3.85823i
64.12 −0.700986 + 1.57444i −0.324542 − 1.52685i −0.649219 − 0.721031i 0.889074 − 2.05172i 2.63144 + 0.559329i − 2.92955i −1.68786 + 0.548420i 0.514686 − 0.229153i 2.60708 + 2.83802i
64.13 −0.667147 + 1.49844i 0.526408 + 2.47655i −0.461969 − 0.513068i −2.19694 + 0.416479i −4.06215 − 0.863437i − 3.81386i −2.04293 + 0.663787i −3.11557 + 1.38714i 0.841616 − 3.56983i
64.14 −0.515607 + 1.15807i −0.0880499 − 0.414242i 0.262980 + 0.292069i −1.94463 − 1.10382i 0.525122 + 0.111618i − 1.35326i −2.88508 + 0.937419i 2.57679 − 1.14726i 2.28097 − 1.68289i
64.15 −0.459516 + 1.03209i −0.692119 − 3.25617i 0.484208 + 0.537768i 0.0316692 + 2.23584i 3.67869 + 0.781930i − 4.13155i −2.92646 + 0.950865i −7.38295 + 3.28710i −2.32214 − 0.994719i
64.16 −0.447967 + 1.00615i 0.513234 + 2.41458i 0.526596 + 0.584844i 1.99783 + 1.00433i −2.65934 − 0.565261i − 1.93939i −2.91927 + 0.948528i −2.82614 + 1.25828i −1.90547 + 1.56021i
64.17 −0.442045 + 0.992849i 0.000780969 0.00367417i 0.547916 + 0.608523i 1.40448 + 1.73995i −0.00399312 0.000848763i − 3.64114i −2.91361 + 0.946688i 2.74062 − 1.22020i −2.34835 + 0.625296i
64.18 −0.298050 + 0.669431i 0.109759 + 0.516378i 0.978957 + 1.08724i 2.22395 − 0.232484i −0.378393 − 0.0804300i 4.06457i −2.41345 + 0.784177i 2.48604 − 1.10686i −0.507216 + 1.55807i
64.19 −0.238228 + 0.535068i 0.278295 + 1.30928i 1.10872 + 1.23135i −0.605568 − 2.15251i −0.766849 − 0.162999i − 0.191694i −2.03706 + 0.661881i 1.10388 − 0.491480i 1.29600 + 0.188766i
64.20 −0.231243 + 0.519380i −0.365614 − 1.72008i 1.12198 + 1.24608i −2.18350 + 0.482007i 0.977921 + 0.207863i 0.163044i −1.98805 + 0.645958i −0.0843645 + 0.0375615i 0.254574 − 1.24553i
See next 80 embeddings (of 384 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 64.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner
25.e even 10 1 inner
475.x even 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 475.2.x.a ✓ 384
19.c even 3 1 inner 475.2.x.a ✓ 384
25.e even 10 1 inner 475.2.x.a ✓ 384
475.x even 30 1 inner 475.2.x.a ✓ 384
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
475.2.x.a ✓ 384 1.a even 1 1 trivial
475.2.x.a ✓ 384 19.c even 3 1 inner
475.2.x.a ✓ 384 25.e even 10 1 inner
475.2.x.a ✓ 384 475.x even 30 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(475, [\chi])\).