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Properties

Label 507.2.m.a
Level $507$
Weight $2$
Character orbit 507.m
Analytic conductor $4.048$
Analytic rank $0$
Dimension $180$
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(40,507)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(507, base_ring=CyclotomicField(26))
 
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("507.40");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 507.m (of order \(13\), degree \(12\), 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: \(180\)
Relative dimension: \(15\) over \(\Q(\zeta_{13})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{13}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 180 q - q^{2} + 15 q^{3} - 15 q^{4} - 2 q^{5} + q^{6} + 4 q^{7} + 3 q^{8} - 15 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 180 q - q^{2} + 15 q^{3} - 15 q^{4} - 2 q^{5} + q^{6} + 4 q^{7} + 3 q^{8} - 15 q^{9} - 2 q^{10} - 4 q^{11} + 15 q^{12} - 14 q^{13} + 6 q^{14} + 2 q^{15} - 15 q^{16} - 2 q^{17} - q^{18} + 2 q^{20} - 4 q^{21} - 28 q^{22} - 52 q^{23} - 3 q^{24} - 67 q^{25} - 40 q^{26} + 15 q^{27} - 4 q^{28} - 27 q^{29} + 2 q^{30} + 22 q^{31} - 5 q^{32} - 9 q^{33} + 63 q^{34} - 31 q^{35} - 15 q^{36} + 2 q^{37} + 65 q^{38} + q^{39} + 45 q^{40} - 6 q^{41} + 59 q^{42} - 60 q^{43} - 35 q^{44} - 2 q^{45} - 156 q^{46} + 15 q^{48} + 59 q^{49} - 51 q^{50} + 2 q^{51} + 66 q^{52} + 50 q^{53} + q^{54} + 55 q^{55} - 14 q^{56} - 13 q^{57} + 36 q^{58} + 92 q^{59} - 15 q^{60} + 6 q^{61} + 61 q^{62} + 4 q^{63} - 203 q^{64} - 54 q^{65} + 54 q^{66} + 86 q^{67} + 32 q^{68} + 112 q^{70} + 39 q^{71} + 3 q^{72} - 158 q^{73} - 80 q^{74} + 15 q^{75} + 130 q^{76} - 64 q^{77} + 66 q^{78} - 10 q^{79} - 310 q^{80} - 15 q^{81} + 59 q^{82} - 82 q^{83} + 4 q^{84} + 22 q^{85} - q^{86} + 40 q^{87} + 10 q^{88} + 2 q^{89} - 2 q^{90} - 100 q^{91} - 54 q^{92} + 43 q^{93} + 65 q^{94} + 58 q^{95} - 60 q^{96} + 16 q^{97} - 113 q^{98} - 30 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
40.1 −2.58082 − 0.636114i −0.120537 + 0.992709i 4.48506 + 2.35394i 0.134864 − 0.195384i 0.942559 − 2.48533i 1.30603 + 1.15704i −6.09859 − 5.40288i −0.970942 − 0.239316i −0.472345 + 0.418461i
40.2 −2.31903 − 0.571589i −0.120537 + 0.992709i 3.28027 + 1.72162i −2.03486 + 2.94800i 0.846950 − 2.23322i −0.933245 − 0.826783i −3.04745 − 2.69981i −0.970942 − 0.239316i 6.40394 − 5.67340i
40.3 −1.76615 − 0.435317i −0.120537 + 0.992709i 1.15887 + 0.608224i 1.67811 − 2.43117i 0.645029 − 1.70080i 3.62124 + 3.20814i 0.941117 + 0.833757i −0.970942 − 0.239316i −4.02213 + 3.56330i
40.4 −1.73397 − 0.427384i −0.120537 + 0.992709i 1.05307 + 0.552694i 0.457873 − 0.663343i 0.633275 − 1.66981i −0.852902 − 0.755605i 1.08370 + 0.960071i −0.970942 − 0.239316i −1.07744 + 0.954527i
40.5 −1.56270 − 0.385172i −0.120537 + 0.992709i 0.522773 + 0.274373i −0.878504 + 1.27273i 0.570727 − 1.50488i 0.273931 + 0.242682i 1.69816 + 1.50443i −0.970942 − 0.239316i 1.86306 − 1.65053i
40.6 −0.926138 − 0.228272i −0.120537 + 0.992709i −0.965289 − 0.506623i −0.278592 + 0.403610i 0.338242 − 0.891870i −3.10155 − 2.74774i 2.20628 + 1.95460i −0.970942 − 0.239316i 0.350148 − 0.310204i
40.7 −0.584736 − 0.144124i −0.120537 + 0.992709i −1.44977 − 0.760897i −2.25739 + 3.27039i 0.213556 − 0.563100i 2.17116 + 1.92348i 1.63963 + 1.45258i −0.970942 − 0.239316i 1.79132 − 1.58697i
40.8 −0.0301922 − 0.00744170i −0.120537 + 0.992709i −1.77006 − 0.928997i 1.67342 − 2.42437i 0.0110267 − 0.0290750i −1.02495 − 0.908024i 0.0930795 + 0.0824612i −0.970942 − 0.239316i −0.0685655 + 0.0607437i
40.9 −0.00839886 − 0.00207013i −0.120537 + 0.992709i −1.77085 − 0.929412i 0.646404 − 0.936478i 0.00306741 − 0.00808810i 2.26833 + 2.00956i 0.0258987 + 0.0229442i −0.970942 − 0.239316i −0.00736769 + 0.00652721i
40.10 0.869211 + 0.214241i −0.120537 + 0.992709i −1.06128 − 0.557005i −1.13901 + 1.65015i −0.317451 + 0.837049i −1.32969 − 1.17800i −2.14332 − 1.89881i −0.970942 − 0.239316i −1.34357 + 1.19030i
40.11 1.36051 + 0.335337i −0.120537 + 0.992709i −0.0323644 − 0.0169861i 1.40752 − 2.03915i −0.496883 + 1.31017i −3.59937 − 3.18876i −2.13601 − 1.89234i −0.970942 − 0.239316i 2.59876 − 2.30230i
40.12 1.56583 + 0.385942i −0.120537 + 0.992709i 0.531960 + 0.279194i 1.72727 − 2.50239i −0.571868 + 1.50789i 1.75934 + 1.55864i −1.68903 − 1.49635i −0.970942 − 0.239316i 3.67039 − 3.25168i
40.13 1.74042 + 0.428975i −0.120537 + 0.992709i 1.07413 + 0.563748i −1.82535 + 2.64447i −0.635632 + 1.67602i 2.27352 + 2.01416i −1.05581 − 0.935370i −0.970942 − 0.239316i −4.31129 + 3.81947i
40.14 2.46984 + 0.608762i −0.120537 + 0.992709i 3.95863 + 2.07765i 0.951102 − 1.37791i −0.902030 + 2.37846i 0.576934 + 0.511119i 4.70433 + 4.16767i −0.970942 − 0.239316i 3.18789 − 2.82423i
40.15 2.53537 + 0.624913i −0.120537 + 0.992709i 4.26668 + 2.23932i −2.06622 + 2.99344i −0.925962 + 2.44156i −3.04788 − 2.70019i 5.50913 + 4.88066i −0.970942 − 0.239316i −7.10928 + 6.29828i
79.1 −2.26213 − 1.18726i 0.970942 + 0.239316i 2.57152 + 3.72549i −1.11131 − 2.93028i −1.91227 − 1.69412i 0.198155 + 1.63195i −0.778110 − 6.40831i 0.885456 + 0.464723i −0.965071 + 7.94807i
79.2 −2.25517 − 1.18361i 0.970942 + 0.239316i 2.54876 + 3.69251i 1.29246 + 3.40793i −1.90639 − 1.68891i 0.394148 + 3.24610i −0.763417 − 6.28731i 0.885456 + 0.464723i 1.11893 − 9.21525i
79.3 −1.85232 − 0.972175i 0.970942 + 0.239316i 1.34985 + 1.95560i 0.359355 + 0.947542i −1.56584 − 1.38722i −0.194000 − 1.59773i −0.0948689 − 0.781316i 0.885456 + 0.464723i 0.255534 − 2.10451i
79.4 −1.32045 − 0.693027i 0.970942 + 0.239316i 0.127180 + 0.184253i −0.0244689 − 0.0645191i −1.11623 − 0.988894i 0.0316924 + 0.261010i 0.319262 + 2.62936i 0.885456 + 0.464723i −0.0124035 + 0.102152i
79.5 −0.890994 − 0.467630i 0.970942 + 0.239316i −0.560937 − 0.812658i −1.54798 − 4.08168i −0.753192 − 0.667270i −0.324090 − 2.66912i 0.362350 + 2.98422i 0.885456 + 0.464723i −0.529477 + 4.36063i
See next 80 embeddings (of 180 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 40.15
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
169.g even 13 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.2.m.a ✓ 180
169.g even 13 1 inner 507.2.m.a ✓ 180
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
507.2.m.a ✓ 180 1.a even 1 1 trivial
507.2.m.a ✓ 180 169.g even 13 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{180} + T_{2}^{179} + 23 T_{2}^{178} + 23 T_{2}^{177} + 321 T_{2}^{176} + 321 T_{2}^{175} + \cdots + 28561 \) acting on \(S_{2}^{\mathrm{new}}(507, [\chi])\). Copy content Toggle raw display