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Properties

Label 95.4.k.b
Level $95$
Weight $4$
Character orbit 95.k
Analytic conductor $5.605$
Analytic rank $0$
Dimension $60$
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] = [95,4,Mod(6,95)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(95, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 14]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("95.6");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 95 = 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 95.k (of order \(9\), degree \(6\), 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: \(5.60518145055\)
Analytic rank: \(0\)
Dimension: \(60\)
Relative dimension: \(10\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 60 q + 9 q^{2} - 3 q^{3} + 9 q^{4} - 39 q^{6} + 36 q^{8} + 93 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 60 q + 9 q^{2} - 3 q^{3} + 9 q^{4} - 39 q^{6} + 36 q^{8} + 93 q^{9} - 45 q^{10} + 72 q^{12} + 150 q^{13} - 150 q^{14} + 15 q^{15} - 375 q^{16} - 162 q^{17} + 216 q^{18} - 192 q^{19} - 1200 q^{20} - 426 q^{21} - 18 q^{22} - 33 q^{23} + 162 q^{24} + 90 q^{26} + 417 q^{27} + 2493 q^{28} + 408 q^{29} + 60 q^{30} - 714 q^{31} + 645 q^{32} - 951 q^{33} + 114 q^{34} - 585 q^{35} - 588 q^{36} + 1836 q^{37} + 984 q^{38} - 3456 q^{39} - 360 q^{40} + 606 q^{41} - 123 q^{42} + 72 q^{43} + 441 q^{44} + 1350 q^{45} + 129 q^{46} + 2814 q^{47} + 3321 q^{48} - 1356 q^{49} + 2049 q^{51} - 3843 q^{52} - 2928 q^{53} - 4221 q^{54} + 585 q^{55} + 3456 q^{56} + 876 q^{57} - 2472 q^{58} - 3993 q^{59} + 1170 q^{60} + 3084 q^{61} - 3354 q^{62} + 2205 q^{63} + 3564 q^{64} + 705 q^{65} + 5946 q^{66} + 2079 q^{67} + 432 q^{68} - 1380 q^{69} - 1590 q^{70} - 3378 q^{71} - 1410 q^{72} - 4110 q^{73} + 3348 q^{74} - 3372 q^{76} - 3906 q^{77} - 9366 q^{78} + 204 q^{79} - 1185 q^{80} - 8649 q^{81} - 1917 q^{82} + 1470 q^{83} + 2136 q^{84} + 810 q^{85} + 5193 q^{86} - 1788 q^{87} - 4521 q^{88} + 2550 q^{89} + 3495 q^{90} - 1008 q^{91} - 660 q^{92} + 17484 q^{93} + 13896 q^{94} + 1200 q^{95} - 414 q^{96} - 939 q^{97} + 2031 q^{98} - 114 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
6.1 −4.11693 + 3.45451i −4.83870 − 1.76114i 3.62625 − 20.5655i −0.868241 − 4.92404i 26.0045 − 9.46486i −0.675317 + 1.16968i 34.6177 + 59.9597i −0.371804 − 0.311981i 20.5846 + 17.2726i
6.2 −3.16499 + 2.65575i 3.89076 + 1.41612i 1.57502 − 8.93238i −0.868241 − 4.92404i −16.0751 + 5.85086i 9.34491 − 16.1859i 2.21078 + 3.82918i −7.55056 − 6.33568i 15.8250 + 13.2787i
6.3 −2.84678 + 2.38873i 6.40608 + 2.33162i 1.00893 − 5.72194i −0.868241 − 4.92404i −23.8063 + 8.66480i −13.6371 + 23.6201i −4.06886 − 7.04748i 14.9182 + 12.5179i 14.2339 + 11.9437i
6.4 −1.97601 + 1.65807i −3.68433 − 1.34099i −0.233758 + 1.32571i −0.868241 − 4.92404i 9.50376 − 3.45908i −0.488995 + 0.846964i −12.0542 − 20.8785i −8.90713 − 7.47397i 9.88007 + 8.29037i
6.5 −0.218503 + 0.183346i −1.99430 − 0.725867i −1.37506 + 7.79834i −0.868241 − 4.92404i 0.568845 − 0.207043i −1.41581 + 2.45225i −2.27028 − 3.93223i −17.2328 − 14.4601i 1.09251 + 0.916728i
6.6 0.478693 − 0.401671i 8.62374 + 3.13878i −1.32138 + 7.49391i −0.868241 − 4.92404i 5.38888 − 1.96139i 7.60280 − 13.1684i 4.87711 + 8.44740i 43.8337 + 36.7808i −2.39347 − 2.00836i
6.7 1.29856 − 1.08962i −8.85206 − 3.22189i −0.890203 + 5.04859i −0.868241 − 4.92404i −15.0055 + 5.46157i 3.60688 − 6.24730i 11.1257 + 19.2702i 47.2952 + 39.6854i −6.49279 − 5.44810i
6.8 2.39352 − 2.00841i −0.310724 − 0.113094i 0.306081 − 1.73587i −0.868241 − 4.92404i −0.970865 + 0.353366i 18.0464 − 31.2572i 9.74437 + 16.8777i −20.5994 − 17.2850i −11.9676 − 10.0420i
6.9 3.58404 − 3.00736i 5.73918 + 2.08889i 2.41190 − 13.6786i −0.868241 − 4.92404i 26.8515 − 9.77314i −4.62023 + 8.00248i −13.7775 − 23.8634i 7.89151 + 6.62176i −17.9202 − 15.0368i
6.10 3.77028 − 3.16364i −6.41934 − 2.33645i 2.81719 − 15.9771i −0.868241 − 4.92404i −31.5943 + 11.4994i −7.43684 + 12.8810i −20.2372 − 35.0518i 15.0657 + 12.6416i −18.8514 − 15.8182i
16.1 −4.11693 − 3.45451i −4.83870 + 1.76114i 3.62625 + 20.5655i −0.868241 + 4.92404i 26.0045 + 9.46486i −0.675317 − 1.16968i 34.6177 − 59.9597i −0.371804 + 0.311981i 20.5846 − 17.2726i
16.2 −3.16499 − 2.65575i 3.89076 − 1.41612i 1.57502 + 8.93238i −0.868241 + 4.92404i −16.0751 − 5.85086i 9.34491 + 16.1859i 2.21078 − 3.82918i −7.55056 + 6.33568i 15.8250 − 13.2787i
16.3 −2.84678 − 2.38873i 6.40608 − 2.33162i 1.00893 + 5.72194i −0.868241 + 4.92404i −23.8063 − 8.66480i −13.6371 − 23.6201i −4.06886 + 7.04748i 14.9182 − 12.5179i 14.2339 − 11.9437i
16.4 −1.97601 − 1.65807i −3.68433 + 1.34099i −0.233758 − 1.32571i −0.868241 + 4.92404i 9.50376 + 3.45908i −0.488995 − 0.846964i −12.0542 + 20.8785i −8.90713 + 7.47397i 9.88007 − 8.29037i
16.5 −0.218503 − 0.183346i −1.99430 + 0.725867i −1.37506 − 7.79834i −0.868241 + 4.92404i 0.568845 + 0.207043i −1.41581 − 2.45225i −2.27028 + 3.93223i −17.2328 + 14.4601i 1.09251 − 0.916728i
16.6 0.478693 + 0.401671i 8.62374 − 3.13878i −1.32138 − 7.49391i −0.868241 + 4.92404i 5.38888 + 1.96139i 7.60280 + 13.1684i 4.87711 − 8.44740i 43.8337 − 36.7808i −2.39347 + 2.00836i
16.7 1.29856 + 1.08962i −8.85206 + 3.22189i −0.890203 − 5.04859i −0.868241 + 4.92404i −15.0055 − 5.46157i 3.60688 + 6.24730i 11.1257 − 19.2702i 47.2952 − 39.6854i −6.49279 + 5.44810i
16.8 2.39352 + 2.00841i −0.310724 + 0.113094i 0.306081 + 1.73587i −0.868241 + 4.92404i −0.970865 − 0.353366i 18.0464 + 31.2572i 9.74437 − 16.8777i −20.5994 + 17.2850i −11.9676 + 10.0420i
16.9 3.58404 + 3.00736i 5.73918 − 2.08889i 2.41190 + 13.6786i −0.868241 + 4.92404i 26.8515 + 9.77314i −4.62023 − 8.00248i −13.7775 + 23.8634i 7.89151 − 6.62176i −17.9202 + 15.0368i
16.10 3.77028 + 3.16364i −6.41934 + 2.33645i 2.81719 + 15.9771i −0.868241 + 4.92404i −31.5943 − 11.4994i −7.43684 − 12.8810i −20.2372 + 35.0518i 15.0657 − 12.6416i −18.8514 + 15.8182i
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 6.10
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 95.4.k.b ✓ 60
19.e even 9 1 inner 95.4.k.b ✓ 60
19.e even 9 1 1805.4.a.x 30
19.f odd 18 1 1805.4.a.y 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.4.k.b ✓ 60 1.a even 1 1 trivial
95.4.k.b ✓ 60 19.e even 9 1 inner
1805.4.a.x 30 19.e even 9 1
1805.4.a.y 30 19.f odd 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{60} - 9 T_{2}^{59} + 36 T_{2}^{58} - 93 T_{2}^{57} + 411 T_{2}^{56} - 3450 T_{2}^{55} + \cdots + 11\!\cdots\!04 \) acting on \(S_{4}^{\mathrm{new}}(95, [\chi])\). Copy content Toggle raw display