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Properties

Label 102.4.i.b
Level $102$
Weight $4$
Character orbit 102.i
Analytic conductor $6.018$
Analytic rank $0$
Dimension $72$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [102,4,Mod(5,102)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(102, base_ring=CyclotomicField(16))
 
chi = DirichletCharacter(H, H._module([8, 5]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("102.5");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 102 = 2 \cdot 3 \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 102.i (of order \(16\), 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: \(6.01819482059\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(9\) over \(\Q(\zeta_{16})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 72 q + 144 q^{11} - 96 q^{12} + 168 q^{15} + 528 q^{17} + 224 q^{18} + 216 q^{21} - 128 q^{24} - 624 q^{25} + 48 q^{29} + 432 q^{31} - 1040 q^{33} + 336 q^{37} - 720 q^{38} + 1312 q^{39} + 288 q^{41} + 1488 q^{42}+ \cdots - 6440 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 −1.84776 + 0.765367i −5.08778 − 1.05572i 2.82843 − 2.82843i −19.9741 − 3.97309i 10.2090 − 1.94331i 1.21959 + 6.13128i −3.06147 + 7.39104i 24.7709 + 10.7425i 39.9481 − 7.94617i
5.2 −1.84776 + 0.765367i −4.63677 + 2.34528i 2.82843 − 2.82843i 11.0539 + 2.19876i 6.77265 − 7.88234i −3.04002 − 15.2832i −3.06147 + 7.39104i 15.9994 − 21.7490i −22.1079 + 4.39753i
5.3 −1.84776 + 0.765367i −3.71611 − 3.63187i 2.82843 − 2.82843i 8.45257 + 1.68132i 9.64620 + 3.86664i −1.91006 − 9.60254i −3.06147 + 7.39104i 0.618995 + 26.9929i −16.9051 + 3.36264i
5.4 −1.84776 + 0.765367i −2.48392 + 4.56400i 2.82843 − 2.82843i −2.48835 − 0.494963i 1.09655 − 10.3343i 5.44526 + 27.3752i −3.06147 + 7.39104i −14.6603 − 22.6733i 4.97669 − 0.989925i
5.5 −1.84776 + 0.765367i 0.522718 + 5.16979i 2.82843 − 2.82843i −7.93672 − 1.57871i −4.92265 − 9.15246i −5.68524 − 28.5817i −3.06147 + 7.39104i −26.4535 + 5.40469i 15.8734 − 3.15742i
5.6 −1.84776 + 0.765367i 1.99426 − 4.79822i 2.82843 − 2.82843i 9.42218 + 1.87419i −0.0125062 + 10.3923i 5.91243 + 29.7238i −3.06147 + 7.39104i −19.0459 − 19.1378i −18.8444 + 3.74837i
5.7 −1.84776 + 0.765367i 2.36253 − 4.62801i 2.82843 − 2.82843i −9.91973 − 1.97316i −0.823256 + 10.3596i −1.76532 − 8.87488i −3.06147 + 7.39104i −15.8369 − 21.8676i 19.8395 − 3.94631i
5.8 −1.84776 + 0.765367i 4.54774 + 2.51358i 2.82843 − 2.82843i −6.58809 − 1.31045i −10.3269 − 1.16380i 3.74326 + 18.8187i −3.06147 + 7.39104i 14.3638 + 22.8622i 13.1762 − 2.62091i
5.9 −1.84776 + 0.765367i 5.14139 − 0.752393i 2.82843 − 2.82843i 15.6822 + 3.11937i −8.92420 + 5.32529i −3.91988 − 19.7066i −3.06147 + 7.39104i 25.8678 − 7.73669i −31.3643 + 6.23875i
11.1 0.765367 − 1.84776i −4.89130 − 1.75362i −2.82843 − 2.82843i −3.44244 + 5.15198i −6.98391 + 7.69578i −3.12286 + 2.08663i −7.39104 + 3.06147i 20.8496 + 17.1550i 6.88488 + 10.3040i
11.2 0.765367 − 1.84776i −3.85766 + 3.48115i −2.82843 − 2.82843i −2.26019 + 3.38262i 3.47980 + 9.79239i 23.3608 − 15.6092i −7.39104 + 3.06147i 2.76315 − 26.8582i 4.52039 + 6.76524i
11.3 0.765367 − 1.84776i −2.62362 − 4.48516i −2.82843 − 2.82843i 8.13727 − 12.1783i −10.2955 + 1.41502i 6.46051 − 4.31678i −7.39104 + 3.06147i −13.2333 + 23.5347i −16.2745 − 24.3566i
11.4 0.765367 − 1.84776i −1.05214 + 5.08852i −2.82843 − 2.82843i −0.312864 + 0.468234i 8.59708 + 5.83868i −19.0832 + 12.7510i −7.39104 + 3.06147i −24.7860 − 10.7076i 0.625728 + 0.936468i
11.5 0.765367 − 1.84776i 0.943618 − 5.10975i −2.82843 − 2.82843i −9.01808 + 13.4965i −8.71938 − 5.65442i −29.1873 + 19.5024i −7.39104 + 3.06147i −25.2192 − 9.64332i 18.0362 + 26.9930i
11.6 0.765367 − 1.84776i 3.26855 − 4.03937i −2.82843 − 2.82843i 6.25043 − 9.35443i −4.96215 − 9.13111i −0.311950 + 0.208438i −7.39104 + 3.06147i −5.63310 − 26.4058i −12.5009 − 18.7089i
11.7 0.765367 − 1.84776i 3.55328 + 3.79133i −2.82843 − 2.82843i 8.81029 − 13.1855i 9.72503 − 3.66385i 3.44910 − 2.30462i −7.39104 + 3.06147i −1.74838 + 26.9433i −17.6206 − 26.3711i
11.8 0.765367 − 1.84776i 3.78694 + 3.55796i −2.82843 − 2.82843i −11.1669 + 16.7125i 9.47265 − 4.27421i 3.40122 − 2.27262i −7.39104 + 3.06147i 1.68182 + 26.9476i 22.3338 + 33.4249i
11.9 0.765367 − 1.84776i 4.84140 − 1.88701i −2.82843 − 2.82843i −2.54078 + 3.80255i 0.218711 − 10.3900i 15.0337 − 10.0452i −7.39104 + 3.06147i 19.8784 − 18.2716i 5.08156 + 7.60510i
23.1 −0.765367 + 1.84776i −4.91913 + 1.67397i −2.82843 − 2.82843i −10.4404 − 6.97608i 0.671841 − 10.3706i 7.64039 + 11.4346i 7.39104 − 3.06147i 21.3956 − 16.4690i 20.8809 − 13.9522i
23.2 −0.765367 + 1.84776i −4.35908 − 2.82816i −2.82843 − 2.82843i 0.344220 + 0.230000i 8.56204 − 5.88994i −2.37864 − 3.55988i 7.39104 − 3.06147i 11.0031 + 24.6563i −0.688440 + 0.460001i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.9
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
51.i even 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 102.4.i.b yes 72
3.b odd 2 1 102.4.i.a ✓ 72
17.e odd 16 1 102.4.i.a ✓ 72
51.i even 16 1 inner 102.4.i.b yes 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
102.4.i.a ✓ 72 3.b odd 2 1
102.4.i.a ✓ 72 17.e odd 16 1
102.4.i.b yes 72 1.a even 1 1 trivial
102.4.i.b yes 72 51.i even 16 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{72} + 312 T_{5}^{70} + 48 T_{5}^{69} + 41988 T_{5}^{68} + 3015360 T_{5}^{67} + \cdots + 65\!\cdots\!72 \) acting on \(S_{4}^{\mathrm{new}}(102, [\chi])\). Copy content Toggle raw display