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Properties

Label 75.4.l.a
Level $75$
Weight $4$
Character orbit 75.l
Analytic conductor $4.425$
Analytic rank $0$
Dimension $224$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [75,4,Mod(2,75)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(75, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([10, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("75.2");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 75.l (of order \(20\), 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: \(4.42514325043\)
Analytic rank: \(0\)
Dimension: \(224\)
Relative dimension: \(28\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 224 q - 4 q^{3} - 20 q^{4} - 6 q^{6} - 4 q^{7} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 224 q - 4 q^{3} - 20 q^{4} - 6 q^{6} - 4 q^{7} - 10 q^{9} + 80 q^{10} - 142 q^{12} - 88 q^{13} - 100 q^{15} + 692 q^{16} + 230 q^{18} + 100 q^{19} - 6 q^{21} - 240 q^{22} - 1240 q^{25} - 712 q^{27} - 1248 q^{28} - 670 q^{30} - 12 q^{31} + 230 q^{33} + 2620 q^{34} - 262 q^{36} + 1136 q^{37} + 2430 q^{39} + 580 q^{40} + 150 q^{42} - 568 q^{43} - 2510 q^{45} - 12 q^{46} - 5084 q^{48} - 16 q^{51} - 84 q^{52} - 3780 q^{54} - 480 q^{55} - 134 q^{57} - 80 q^{58} + 110 q^{60} - 12 q^{61} + 982 q^{63} - 2900 q^{64} - 870 q^{66} + 296 q^{67} + 3070 q^{69} + 6520 q^{70} + 11310 q^{72} + 5372 q^{73} + 8820 q^{75} - 544 q^{76} + 6820 q^{78} + 2140 q^{79} - 646 q^{81} + 6900 q^{82} - 1930 q^{84} + 6920 q^{85} - 2430 q^{87} - 7160 q^{88} - 10570 q^{90} - 12 q^{91} - 6098 q^{93} - 19340 q^{94} + 1242 q^{96} - 14944 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −0.839064 + 5.29764i 3.18492 − 4.10564i −19.7525 − 6.41799i −5.18161 − 9.90711i 19.0778 + 20.3175i −11.1652 − 11.1652i 31.0934 − 61.0242i −6.71252 − 26.1523i 56.8321 − 19.1376i
2.2 −0.836586 + 5.28200i −5.19205 + 0.206332i −19.5912 − 6.36556i −7.32393 + 8.44749i 3.25376 − 27.5970i 17.6352 + 17.6352i 30.5896 − 60.0355i 26.9149 − 2.14257i −38.4925 − 45.7520i
2.3 −0.770659 + 4.86575i 3.65029 + 3.69802i −15.4732 − 5.02753i 4.29906 + 10.3208i −20.8068 + 14.9115i −14.4144 − 14.4144i 18.4949 − 36.2984i −0.350737 + 26.9977i −53.5314 + 12.9643i
2.4 −0.637815 + 4.02700i −1.27842 − 5.03643i −8.20150 − 2.66483i 10.8407 + 2.73475i 21.0971 − 1.93588i 11.0103 + 11.0103i 1.15422 − 2.26529i −23.7313 + 12.8773i −17.9272 + 41.9113i
2.5 −0.620102 + 3.91517i −5.19087 + 0.234278i −7.33558 − 2.38347i 6.18957 − 9.31071i 2.30163 − 20.4684i −21.2563 − 21.2563i −0.516339 + 1.01337i 26.8902 − 2.43221i 32.6148 + 30.0068i
2.6 −0.592624 + 3.74168i −0.463682 + 5.17542i −6.04051 − 1.96268i −9.85935 − 5.27193i −19.0900 − 4.80203i −0.782697 − 0.782697i −2.83542 + 5.56482i −26.5700 − 4.79951i 25.5688 − 33.7663i
2.7 −0.577725 + 3.64761i 4.88611 + 1.76803i −5.36285 − 1.74250i 6.99657 − 8.72055i −9.27191 + 16.8012i 19.7780 + 19.7780i −3.95878 + 7.76955i 20.7481 + 17.2776i 27.7671 + 30.5589i
2.8 −0.448710 + 2.83304i 4.85329 − 1.85623i −0.216326 − 0.0702887i −10.0886 + 4.81869i 3.08107 + 14.5825i 9.76442 + 9.76442i −10.1214 + 19.8645i 20.1088 − 18.0177i −9.12467 − 30.7437i
2.9 −0.411963 + 2.60103i −2.27616 − 4.67109i 1.01281 + 0.329081i −9.32049 + 6.17482i 13.0873 − 3.99606i −12.7719 − 12.7719i −10.8377 + 21.2702i −16.6382 + 21.2643i −12.2212 − 26.7867i
2.10 −0.350641 + 2.21386i −3.18020 + 4.10930i 2.83022 + 0.919594i 7.20797 + 8.54664i −7.98231 − 8.48141i 3.19492 + 3.19492i −11.1690 + 21.9205i −6.77267 − 26.1368i −21.4485 + 12.9606i
2.11 −0.201927 + 1.27492i −5.10839 − 0.950963i 6.02381 + 1.95726i −4.85392 − 10.0717i 2.24392 − 6.32075i 22.2801 + 22.2801i −8.39983 + 16.4856i 25.1913 + 9.71578i 13.8207 − 4.15459i
2.12 −0.197834 + 1.24908i 4.84546 − 1.87656i 6.08740 + 1.97792i 10.8606 + 2.65460i 1.38537 + 6.42360i −20.0447 − 20.0447i −8.26796 + 16.2268i 19.9570 − 18.1856i −5.46440 + 13.0406i
2.13 −0.108958 + 0.687933i 1.21934 − 5.05106i 7.14707 + 2.32222i −1.50552 − 11.0785i 3.34194 + 1.38917i −1.62076 − 1.62076i −4.90593 + 9.62843i −24.0264 − 12.3179i 7.78531 + 0.171393i
2.14 −0.00143654 + 0.00906996i 1.15875 + 5.06530i 7.60837 + 2.47211i 7.27358 − 8.49088i −0.0476067 + 0.00323326i −0.680990 − 0.680990i −0.0667037 + 0.130913i −24.3146 + 11.7388i 0.0665631 + 0.0781686i
2.15 0.00143654 − 0.00906996i 3.41682 + 3.91476i 7.60837 + 2.47211i −7.27358 + 8.49088i 0.0404151 − 0.0253667i −0.680990 − 0.680990i 0.0667037 − 0.130913i −3.65064 + 26.7521i 0.0665631 + 0.0781686i
2.16 0.108958 − 0.687933i −4.80310 − 1.98248i 7.14707 + 2.32222i 1.50552 + 11.0785i −1.88715 + 3.08821i −1.62076 − 1.62076i 4.90593 − 9.62843i 19.1396 + 19.0441i 7.78531 + 0.171393i
2.17 0.197834 − 1.24908i −4.36626 + 2.81705i 6.08740 + 1.97792i −10.8606 − 2.65460i 2.65491 + 6.01110i −20.0447 − 20.0447i 8.26796 − 16.2268i 11.1285 − 24.5999i −5.46440 + 13.0406i
2.18 0.201927 − 1.27492i 2.23329 − 4.69174i 6.02381 + 1.95726i 4.85392 + 10.0717i −5.53061 − 3.79465i 22.2801 + 22.2801i 8.39983 − 16.4856i −17.0248 − 20.9560i 13.8207 − 4.15459i
2.19 0.350641 − 2.21386i 5.19377 − 0.157450i 2.83022 + 0.919594i −7.20797 − 8.54664i 1.47258 − 11.5535i 3.19492 + 3.19492i 11.1690 − 21.9205i 26.9504 − 1.63551i −21.4485 + 12.9606i
2.20 0.411963 − 2.60103i −2.44110 − 4.58705i 1.01281 + 0.329081i 9.32049 − 6.17482i −12.9367 + 4.45967i −12.7719 − 12.7719i 10.8377 − 21.2702i −15.0821 + 22.3949i −12.2212 − 26.7867i
See next 80 embeddings (of 224 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
25.f odd 20 1 inner
75.l even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 75.4.l.a ✓ 224
3.b odd 2 1 inner 75.4.l.a ✓ 224
25.f odd 20 1 inner 75.4.l.a ✓ 224
75.l even 20 1 inner 75.4.l.a ✓ 224
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.4.l.a ✓ 224 1.a even 1 1 trivial
75.4.l.a ✓ 224 3.b odd 2 1 inner
75.4.l.a ✓ 224 25.f odd 20 1 inner
75.4.l.a ✓ 224 75.l even 20 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(75, [\chi])\).