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Properties

Label 75.6.l.a
Level $75$
Weight $6$
Character orbit 75.l
Analytic conductor $12.029$
Analytic rank $0$
Dimension $384$
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] = [75,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("75.2");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
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: \(12.0287864860\)
Analytic rank: \(0\)
Dimension: \(384\)
Relative dimension: \(48\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 384 q - 10 q^{3} - 20 q^{4} - 6 q^{6} + 60 q^{7} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 384 q - 10 q^{3} - 20 q^{4} - 6 q^{6} + 60 q^{7} - 10 q^{9} - 1080 q^{10} - 430 q^{12} + 2100 q^{13} + 2990 q^{15} + 21492 q^{16} - 6010 q^{18} - 8800 q^{19} - 6 q^{21} + 6040 q^{22} + 23880 q^{25} + 12950 q^{27} - 7360 q^{28} + 13370 q^{30} - 12 q^{31} - 5770 q^{33} + 38220 q^{34} - 4102 q^{36} - 55740 q^{37} - 7170 q^{39} - 21500 q^{40} - 97890 q^{42} + 23340 q^{43} + 30910 q^{45} - 12 q^{46} + 205780 q^{48} - 16 q^{51} - 203540 q^{52} - 66180 q^{54} - 89140 q^{55} - 124130 q^{57} + 163480 q^{58} - 405970 q^{60} - 12 q^{61} + 283540 q^{63} + 408300 q^{64} - 31110 q^{66} - 21980 q^{67} - 14780 q^{69} - 528960 q^{70} - 278370 q^{72} + 35860 q^{73} - 232170 q^{75} - 8224 q^{76} - 687140 q^{78} - 37660 q^{79} - 58906 q^{81} + 728820 q^{82} + 977270 q^{84} - 163580 q^{85} + 730620 q^{87} - 298680 q^{88} + 77390 q^{90} - 12 q^{91} - 360200 q^{93} + 675060 q^{94} + 37242 q^{96} + 715720 q^{97}+O(q^{100}) \) 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 −1.68409 + 10.6329i 14.4658 − 5.80864i −79.7893 − 25.9251i 46.6296 + 30.8331i 37.4012 + 163.596i 149.114 + 149.114i 253.635 − 497.786i 175.519 − 168.053i −406.374 + 443.884i
2.2 −1.67679 + 10.5869i 7.41769 + 13.7105i −78.8362 − 25.6154i −14.0909 − 54.0966i −157.589 + 55.5404i −11.7073 − 11.7073i 247.659 − 486.058i −132.956 + 203.401i 596.341 − 58.4695i
2.3 −1.61420 + 10.1917i −3.75212 − 15.1302i −70.8310 − 23.0144i −38.6838 + 40.3554i 160.258 − 13.8172i −84.8535 − 84.8535i 198.984 − 390.527i −214.843 + 113.540i −348.846 − 459.395i
2.4 −1.55957 + 9.84676i −13.2493 − 8.21312i −64.0927 − 20.8250i 27.0814 − 48.9040i 101.536 − 117.654i 48.0998 + 48.0998i 160.182 − 314.375i 108.089 + 217.637i 439.311 + 342.933i
2.5 −1.52703 + 9.64126i −10.4038 + 11.6087i −60.1884 − 19.5564i 42.5572 + 36.2475i −96.0355 − 118.032i −81.4490 − 81.4490i 138.646 − 272.108i −26.5228 − 241.548i −414.458 + 354.954i
2.6 −1.42076 + 8.97035i −13.2173 + 8.26464i −48.0148 − 15.6010i −53.1976 − 17.1759i −55.3581 − 130.305i −25.0913 − 25.0913i 76.2207 − 149.592i 106.392 − 218.472i 229.655 − 452.798i
2.7 −1.33610 + 8.43581i 15.4763 + 1.86655i −38.9440 − 12.6537i −48.0252 + 28.6108i −36.4237 + 128.061i −84.4598 − 84.4598i 34.6964 − 68.0955i 236.032 + 57.7744i −177.189 − 443.359i
2.8 −1.26293 + 7.97384i 1.88797 + 15.4737i −31.5534 − 10.2523i −26.5989 + 49.1680i −125.769 − 4.48788i 164.867 + 164.867i 4.31459 − 8.46787i −235.871 + 58.4278i −358.466 − 274.192i
2.9 −1.22902 + 7.75970i 5.37355 − 14.6330i −28.2686 − 9.18502i 55.1156 − 9.34170i 106.944 + 59.6813i −124.351 − 124.351i −8.12019 + 15.9368i −185.250 − 157.262i 4.75079 + 439.162i
2.10 −1.16560 + 7.35928i 2.90460 − 15.3155i −22.3666 − 7.26734i −48.7964 − 27.2747i 109.325 + 39.2274i 155.205 + 155.205i −28.6933 + 56.3137i −226.127 − 88.9707i 257.599 − 327.315i
2.11 −1.14011 + 7.19840i 14.7306 − 5.09980i −20.0832 − 6.52544i −3.67438 − 55.7808i 19.9158 + 111.851i −20.4935 − 20.4935i −36.0098 + 70.6733i 190.984 − 150.247i 405.722 + 37.1468i
2.12 −1.06958 + 6.75307i 8.91357 + 12.7886i −14.0262 − 4.55738i 54.8174 − 10.9568i −95.8961 + 46.5155i −36.2336 − 36.2336i −53.5510 + 105.100i −84.0967 + 227.984i 15.3606 + 381.905i
2.13 −1.01682 + 6.41997i −13.9314 − 6.99399i −9.74831 − 3.16742i 20.0490 + 52.1827i 59.0670 − 82.3276i 27.8128 + 27.8128i −64.1829 + 125.966i 145.168 + 194.872i −355.398 + 75.6534i
2.14 −0.845592 + 5.33886i −12.5432 + 9.25569i 2.64544 + 0.859556i 42.9017 − 35.8391i −38.8083 − 74.7930i 152.334 + 152.334i −85.3541 + 167.517i 71.6645 − 232.192i 155.063 + 259.351i
2.15 −0.769527 + 4.85860i −15.5108 − 1.55378i 7.41996 + 2.41089i −55.3704 − 7.68875i 19.4852 − 74.1653i −36.3024 − 36.3024i −88.8876 + 174.452i 238.172 + 48.2007i 79.9656 − 263.106i
2.16 −0.645548 + 4.07583i −4.73185 + 14.8529i 14.2381 + 4.62625i −0.352288 − 55.9006i −57.4834 − 28.8745i −48.0778 − 48.0778i −87.9978 + 172.705i −198.219 − 140.564i 228.069 + 34.6507i
2.17 −0.575257 + 3.63203i 7.40071 − 13.7197i 17.5731 + 5.70984i −3.44517 + 55.7954i 45.5730 + 34.7719i 23.1563 + 23.1563i −84.2701 + 165.389i −133.459 − 203.071i −200.669 − 44.6097i
2.18 −0.564970 + 3.56708i −8.02108 − 13.3665i 18.0289 + 5.85795i −11.6928 − 54.6652i 52.2109 − 21.0602i −123.814 − 123.814i −83.5491 + 163.974i −114.325 + 214.427i 201.601 − 10.8249i
2.19 −0.546989 + 3.45355i 15.3119 + 2.92339i 18.8060 + 6.11044i 34.8257 + 43.7283i −18.4715 + 51.2813i 14.4941 + 14.4941i −82.1869 + 161.301i 225.908 + 89.5253i −170.067 + 96.3536i
2.20 −0.506700 + 3.19918i −0.903480 + 15.5623i 20.4558 + 6.64650i −28.4220 + 48.1372i −49.3286 − 10.7758i −167.086 − 167.086i −78.6843 + 154.427i −241.367 − 28.1204i −139.598 − 115.318i
See next 80 embeddings (of 384 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.48
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 75.6.l.a ✓ 384
3.b odd 2 1 inner 75.6.l.a ✓ 384
25.f odd 20 1 inner 75.6.l.a ✓ 384
75.l even 20 1 inner 75.6.l.a ✓ 384
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.6.l.a ✓ 384 1.a even 1 1 trivial
75.6.l.a ✓ 384 3.b odd 2 1 inner
75.6.l.a ✓ 384 25.f odd 20 1 inner
75.6.l.a ✓ 384 75.l even 20 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(75, [\chi])\).