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Properties

Label 538.2.d.b
Level $538$
Weight $2$
Character orbit 538.d
Analytic conductor $4.296$
Analytic rank $0$
Dimension $792$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [538,2,Mod(5,538)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(538, base_ring=CyclotomicField(134))
 
chi = DirichletCharacter(H, H._module([104]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("538.5");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 538 = 2 \cdot 269 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 538.d (of order \(67\), degree \(66\), 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: \(4.29595162874\)
Analytic rank: \(0\)
Dimension: \(792\)
Relative dimension: \(12\) over \(\Q(\zeta_{67})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{67}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 792 q - 12 q^{2} - 5 q^{3} - 12 q^{4} - 9 q^{5} - 5 q^{6} - 8 q^{7} - 12 q^{8} - 23 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 792 q - 12 q^{2} - 5 q^{3} - 12 q^{4} - 9 q^{5} - 5 q^{6} - 8 q^{7} - 12 q^{8} - 23 q^{9} - 9 q^{10} - 11 q^{11} - 5 q^{12} - 13 q^{13} - 8 q^{14} - 26 q^{15} - 12 q^{16} - 18 q^{17} - 23 q^{18} - 17 q^{19} - 9 q^{20} + 179 q^{21} - 11 q^{22} - 26 q^{23} - 5 q^{24} - 55 q^{25} - 13 q^{26} + 169 q^{27} - 8 q^{28} - 33 q^{29} + 41 q^{30} - 40 q^{31} - 12 q^{32} - 54 q^{33} - 18 q^{34} - 58 q^{35} - 23 q^{36} + 154 q^{37} - 17 q^{38} - 44 q^{39} - 9 q^{40} + 88 q^{41} - 22 q^{42} - 23 q^{43} + 56 q^{44} - 85 q^{45} - 26 q^{46} + 167 q^{47} - 5 q^{48} - 64 q^{49} - 55 q^{50} - 38 q^{51} - 13 q^{52} + 61 q^{53} - 32 q^{54} - 40 q^{55} - 8 q^{56} - 68 q^{57} - 33 q^{58} - 53 q^{59} + 108 q^{60} + 12 q^{61} - 40 q^{62} - 88 q^{63} - 12 q^{64} - 84 q^{65} - 54 q^{66} - 61 q^{67} - 18 q^{68} - 60 q^{69} - 58 q^{70} - 100 q^{71} - 23 q^{72} + 15 q^{73} - 47 q^{74} - 81 q^{75} - 17 q^{76} - 98 q^{77} - 44 q^{78} - 3 q^{79} - 9 q^{80} + 148 q^{81} + 222 q^{82} + 120 q^{83} + 246 q^{84} + 26 q^{85} - 23 q^{86} - 146 q^{87} - 11 q^{88} - 102 q^{89} - 85 q^{90} - 80 q^{91} - 26 q^{92} + 12 q^{93} - 34 q^{94} - 78 q^{95} - 5 q^{96} - 90 q^{97} - 64 q^{98} - 123 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −0.762685 + 0.646770i −1.02751 + 3.26626i 0.163377 − 0.986564i −0.677917 − 3.16502i −1.32885 − 3.15569i 0.132038 − 0.135170i 0.513474 + 0.858105i −7.15300 − 4.99471i 2.56407 + 1.97545i
5.2 −0.762685 + 0.646770i −0.676007 + 2.14890i 0.163377 − 0.986564i 0.281122 + 1.31248i −0.874262 − 2.07615i 2.08367 − 2.13311i 0.513474 + 0.858105i −1.70108 − 1.18781i −1.06328 − 0.819191i
5.3 −0.762685 + 0.646770i −0.557254 + 1.77140i 0.163377 − 0.986564i 0.533389 + 2.49025i −0.720682 − 1.71144i −3.49366 + 3.57654i 0.513474 + 0.858105i −0.367646 − 0.256715i −2.01743 − 1.55430i
5.4 −0.762685 + 0.646770i −0.538053 + 1.71037i 0.163377 − 0.986564i −0.140340 − 0.655208i −0.695850 − 1.65247i 1.24564 − 1.27519i 0.513474 + 0.858105i −0.176166 − 0.123011i 0.530804 + 0.408950i
5.5 −0.762685 + 0.646770i −0.312794 + 0.994315i 0.163377 − 0.986564i −0.660524 − 3.08381i −0.404529 − 0.960655i −2.22093 + 2.27362i 0.513474 + 0.858105i 1.56887 + 1.09549i 2.49829 + 1.92477i
5.6 −0.762685 + 0.646770i 0.000979586 − 0.00311392i 0.163377 − 0.986564i 0.832147 + 3.88507i 0.00126687 + 0.00300851i 1.48520 − 1.52043i 0.513474 + 0.858105i 2.45969 + 1.71752i −3.14741 − 2.42488i
5.7 −0.762685 + 0.646770i 0.223877 − 0.711662i 0.163377 − 0.986564i −0.137385 − 0.641417i 0.289534 + 0.687571i −1.23253 + 1.26177i 0.513474 + 0.858105i 2.00335 + 1.39888i 0.519631 + 0.400342i
5.8 −0.762685 + 0.646770i 0.401412 − 1.27601i 0.163377 − 0.986564i −0.385553 − 1.80004i 0.519136 + 1.23282i 2.46099 − 2.51937i 0.513474 + 0.858105i 0.992618 + 0.693113i 1.45827 + 1.12350i
5.9 −0.762685 + 0.646770i 0.542648 − 1.72497i 0.163377 − 0.986564i 0.554118 + 2.58703i 0.701792 + 1.66658i −1.07852 + 1.10411i 0.513474 + 0.858105i −0.221376 − 0.154580i −2.09583 − 1.61470i
5.10 −0.762685 + 0.646770i 0.575510 − 1.82944i 0.163377 − 0.986564i −0.587428 − 2.74255i 0.744292 + 1.76751i 2.19300 − 2.24502i 0.513474 + 0.858105i −0.555931 − 0.388189i 2.22182 + 1.71177i
5.11 −0.762685 + 0.646770i 0.879426 − 2.79553i 0.163377 − 0.986564i 0.432277 + 2.01819i 1.13734 + 2.70089i 0.389133 − 0.398365i 0.513474 + 0.858105i −4.58189 − 3.19939i −1.63499 − 1.25966i
5.12 −0.762685 + 0.646770i 0.984397 − 3.12921i 0.163377 − 0.986564i −0.644536 − 3.00917i 1.27310 + 3.02328i −3.15935 + 3.23430i 0.513474 + 0.858105i −6.36323 − 4.44324i 2.43782 + 1.87818i
21.1 0.0702762 + 0.997528i −3.02870 − 0.574812i −0.990123 + 0.140205i 1.66099 + 3.09656i 0.360546 − 3.06161i 1.09100 + 0.260573i −0.209440 − 0.977821i 6.05122 + 2.38273i −2.97218 + 1.87449i
21.2 0.0702762 + 0.997528i −2.80197 − 0.531782i −0.990123 + 0.140205i −1.30130 − 2.42600i 0.333555 − 2.83242i −3.55510 − 0.849099i −0.209440 − 0.977821i 4.77686 + 1.88094i 2.32855 − 1.46857i
21.3 0.0702762 + 0.997528i −1.99844 − 0.379280i −0.990123 + 0.140205i 0.415351 + 0.774336i 0.237900 − 2.02015i 0.244678 + 0.0584386i −0.209440 − 0.977821i 1.05850 + 0.416796i −0.743232 + 0.468742i
21.4 0.0702762 + 0.997528i −1.37341 − 0.260657i −0.990123 + 0.140205i −0.374139 − 0.697505i 0.163495 − 1.38833i 2.33207 + 0.556990i −0.209440 − 0.977821i −0.973088 − 0.383163i 0.669487 − 0.422232i
21.5 0.0702762 + 0.997528i −1.04025 − 0.197428i −0.990123 + 0.140205i 0.306230 + 0.570902i 0.123835 − 1.05155i −2.22357 − 0.531076i −0.209440 − 0.977821i −1.74825 − 0.688392i −0.547970 + 0.345594i
21.6 0.0702762 + 0.997528i −0.217891 − 0.0413533i −0.990123 + 0.140205i −1.27898 − 2.38440i 0.0259385 − 0.220259i 2.69946 + 0.644736i −0.209440 − 0.977821i −2.74563 − 1.08112i 2.28862 − 1.44339i
21.7 0.0702762 + 0.997528i −0.0186022 − 0.00353047i −0.990123 + 0.140205i 1.95454 + 3.64383i 0.00221446 − 0.0188043i −3.49574 − 0.834920i −0.209440 − 0.977821i −2.79106 − 1.09901i −3.49746 + 2.20578i
21.8 0.0702762 + 0.997528i 1.38143 + 0.262179i −0.990123 + 0.140205i 1.40783 + 2.62461i −0.164449 + 1.39644i 3.20066 + 0.764442i −0.209440 − 0.977821i −0.951795 − 0.374779i −2.51919 + 1.58880i
See next 80 embeddings (of 792 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
269.d even 67 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 538.2.d.b ✓ 792
269.d even 67 1 inner 538.2.d.b ✓ 792
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
538.2.d.b ✓ 792 1.a even 1 1 trivial
538.2.d.b ✓ 792 269.d even 67 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{792} + 5 T_{3}^{791} + 42 T_{3}^{790} + 122 T_{3}^{789} + 667 T_{3}^{788} + 1335 T_{3}^{787} + \cdots + 22\!\cdots\!56 \) acting on \(S_{2}^{\mathrm{new}}(538, [\chi])\). Copy content Toggle raw display