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Properties

Label 507.2.q.b
Level $507$
Weight $2$
Character orbit 507.q
Analytic conductor $4.048$
Analytic rank $0$
Dimension $384$
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] = [507,2,Mod(16,507)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(507, base_ring=CyclotomicField(78))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("507.16");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 507.q (of order \(39\), degree \(24\), 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: \(4.04841538248\)
Analytic rank: \(0\)
Dimension: \(384\)
Relative dimension: \(16\) over \(\Q(\zeta_{39})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{39}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 384 q + q^{2} + 16 q^{3} + 17 q^{4} + 6 q^{5} + q^{6} - 3 q^{7} - 18 q^{8} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 384 q + q^{2} + 16 q^{3} + 17 q^{4} + 6 q^{5} + q^{6} - 3 q^{7} - 18 q^{8} + 16 q^{9} + 3 q^{10} - 4 q^{11} - 34 q^{12} - 15 q^{13} + 24 q^{14} - 3 q^{15} + 11 q^{16} - 3 q^{17} - 2 q^{18} + 6 q^{19} + q^{20} + 6 q^{21} - 86 q^{22} - 52 q^{23} + 9 q^{24} - 74 q^{25} - 64 q^{26} - 32 q^{27} - 16 q^{28} - 44 q^{29} + 3 q^{30} - 28 q^{31} + 74 q^{32} - 17 q^{33} - 83 q^{34} + 70 q^{35} + 17 q^{36} + 11 q^{37} - 29 q^{38} + q^{39} - 5 q^{40} + q^{41} + 118 q^{42} - 79 q^{43} - 19 q^{44} - 3 q^{45} + 314 q^{46} + 11 q^{48} - 69 q^{49} - 76 q^{50} + 6 q^{51} - 37 q^{52} - 102 q^{53} + q^{54} - 73 q^{55} - 36 q^{56} + 14 q^{57} - 51 q^{58} + 194 q^{59} + 76 q^{60} + 4 q^{61} - 89 q^{62} - 3 q^{63} - 184 q^{64} - 49 q^{65} - 36 q^{66} - 73 q^{67} + 4 q^{68} - 100 q^{70} + 106 q^{71} + 9 q^{72} - 132 q^{73} - 45 q^{74} + 11 q^{75} + 24 q^{76} - 54 q^{77} - 39 q^{78} - 2 q^{79} + 177 q^{80} + 16 q^{81} - 76 q^{82} - 58 q^{83} - 16 q^{84} - 123 q^{85} - 157 q^{86} - 57 q^{87} + 476 q^{88} + 18 q^{89} - 6 q^{90} + 233 q^{91} - 12 q^{92} + q^{93} - 111 q^{94} + 64 q^{95} - 83 q^{96} - 143 q^{97} - 79 q^{98} - 18 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1 −2.67365 − 0.215839i −0.845190 − 0.534466i 5.12771 + 0.833333i −1.64492 − 1.45728i 2.14438 + 1.61140i 1.09807 − 1.14322i −8.32101 − 2.05094i 0.428693 + 0.903450i 4.08341 + 4.25128i
16.2 −2.48128 − 0.200310i −0.845190 − 0.534466i 4.14255 + 0.673230i 1.36223 + 1.20683i 1.99010 + 1.49546i −0.288513 + 0.300373i −5.30995 − 1.30878i 0.428693 + 0.903450i −3.13834 − 3.26736i
16.3 −2.00855 − 0.162147i −0.845190 − 0.534466i 2.03388 + 0.330538i −1.08961 − 0.965307i 1.61095 + 1.21055i −3.04871 + 3.17404i −0.118504 − 0.0292085i 0.428693 + 0.903450i 2.03201 + 2.11554i
16.4 −1.82353 − 0.147211i −0.845190 − 0.534466i 1.32950 + 0.216064i 1.37382 + 1.21710i 1.46255 + 1.09904i −0.320915 + 0.334108i 1.16003 + 0.285923i 0.428693 + 0.903450i −2.32603 − 2.42166i
16.5 −1.48832 − 0.120150i −0.845190 − 0.534466i 0.226569 + 0.0368210i 1.90213 + 1.68514i 1.19370 + 0.897007i 3.62894 − 3.77813i 2.56677 + 0.632652i 0.428693 + 0.903450i −2.62851 − 2.73657i
16.6 −1.39808 − 0.112865i −0.845190 − 0.534466i −0.0322011 − 0.00523319i −2.59137 − 2.29575i 1.12132 + 0.842620i 2.75817 − 2.87156i 2.76818 + 0.682294i 0.428693 + 0.903450i 3.36384 + 3.50213i
16.7 −0.529051 − 0.0427094i −0.845190 − 0.534466i −1.69603 − 0.275632i 0.694311 + 0.615105i 0.424322 + 0.318857i 0.272511 − 0.283714i 1.91621 + 0.472304i 0.428693 + 0.903450i −0.341055 − 0.355076i
16.8 −0.488381 − 0.0394262i −0.845190 − 0.534466i −1.73714 − 0.282313i −1.47540 − 1.30709i 0.391703 + 0.294346i −2.51230 + 2.61558i 1.78872 + 0.440880i 0.428693 + 0.903450i 0.669022 + 0.696526i
16.9 0.0277086 + 0.00223687i −0.845190 − 0.534466i −1.97334 − 0.320699i 0.676752 + 0.599550i −0.0222235 − 0.0166999i −0.402242 + 0.418779i −0.107943 − 0.0266056i 0.428693 + 0.903450i 0.0174108 + 0.0181265i
16.10 0.655764 + 0.0529387i −0.845190 − 0.534466i −1.54688 − 0.251392i −1.61643 − 1.43203i −0.525951 − 0.395227i 1.45558 − 1.51542i −2.27864 − 0.561634i 0.428693 + 0.903450i −0.984187 − 1.02465i
16.11 0.910598 + 0.0735111i −0.845190 − 0.534466i −1.15032 − 0.186944i 3.23408 + 2.86515i −0.730339 − 0.548814i −1.27940 + 1.33200i −2.80776 − 0.692051i 0.428693 + 0.903450i 2.73433 + 2.84674i
16.12 1.45946 + 0.117820i −0.845190 − 0.534466i 0.142046 + 0.0230847i −0.575893 − 0.510197i −1.17055 − 0.879613i −2.84784 + 2.96491i −2.63873 − 0.650390i 0.428693 + 0.903450i −0.780383 − 0.812465i
16.13 1.80906 + 0.146043i −0.845190 − 0.534466i 1.27728 + 0.207579i −0.0943821 − 0.0836152i −1.45095 − 1.09032i 1.37395 − 1.43043i −1.24405 − 0.306631i 0.428693 + 0.903450i −0.158532 − 0.165049i
16.14 2.08178 + 0.168058i −0.845190 − 0.534466i 2.33145 + 0.378897i −3.16521 − 2.80413i −1.66968 − 1.25468i 1.01537 − 1.05711i 0.734161 + 0.180955i 0.428693 + 0.903450i −6.11801 − 6.36952i
16.15 2.27058 + 0.183300i −0.845190 − 0.534466i 3.14784 + 0.511574i 2.05738 + 1.82268i −1.82111 − 1.36847i 0.613173 − 0.638380i 2.63011 + 0.648264i 0.428693 + 0.903450i 4.33734 + 4.51565i
16.16 2.67914 + 0.216283i −0.845190 − 0.534466i 5.15693 + 0.838083i 1.04622 + 0.926874i −2.14879 − 1.61471i −2.79839 + 2.91343i 8.41538 + 2.07420i 0.428693 + 0.903450i 2.60252 + 2.70951i
55.1 −1.73567 + 2.12581i −0.996757 − 0.0804666i −1.10646 − 5.41982i 0.355841 + 2.93061i 1.90110 − 1.97925i 0.0396979 − 0.0251034i 8.58190 + 4.50413i 0.987050 + 0.160411i −6.84755 − 4.33013i
55.2 −1.55963 + 1.91020i −0.996757 − 0.0804666i −0.816373 − 3.99886i −0.349402 − 2.87759i 1.70828 − 1.77851i −1.82814 + 1.15605i 4.54475 + 2.38527i 0.987050 + 0.160411i 6.04172 + 3.82055i
55.3 −1.21429 + 1.48724i −0.996757 − 0.0804666i −0.337321 − 1.65231i −0.0596193 − 0.491009i 1.33003 − 1.38471i 0.351566 − 0.222317i −0.533160 − 0.279824i 0.987050 + 0.160411i 0.802643 + 0.507561i
55.4 −1.17775 + 1.44248i −0.996757 − 0.0804666i −0.293610 − 1.43820i 0.180273 + 1.48468i 1.29000 − 1.34304i 3.26448 − 2.06433i −0.877451 − 0.460522i 0.987050 + 0.160411i −2.35394 − 1.48854i
See next 80 embeddings (of 384 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 16.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
169.i even 39 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.2.q.b ✓ 384
169.i even 39 1 inner 507.2.q.b ✓ 384
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
507.2.q.b ✓ 384 1.a even 1 1 trivial
507.2.q.b ✓ 384 169.i even 39 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{384} - T_{2}^{383} - 24 T_{2}^{382} + 33 T_{2}^{381} + 219 T_{2}^{380} - 477 T_{2}^{379} + \cdots + 17\!\cdots\!36 \) acting on \(S_{2}^{\mathrm{new}}(507, [\chi])\). Copy content Toggle raw display