↑

Properties

Label 75.6.g.b
Level $75$
Weight $6$
Character orbit 75.g
Analytic conductor $12.029$
Analytic rank $0$
Dimension $52$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [75,6,Mod(16,75)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(75, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("75.16");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 75.g (of order \(5\), degree \(4\), 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: \(52\)
Relative dimension: \(13\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 52 q + 8 q^{2} + 117 q^{3} - 158 q^{4} + 30 q^{5} - 72 q^{6} - 236 q^{7} + 753 q^{8} - 1053 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 52 q + 8 q^{2} + 117 q^{3} - 158 q^{4} + 30 q^{5} - 72 q^{6} - 236 q^{7} + 753 q^{8} - 1053 q^{9} - 215 q^{10} + 860 q^{11} + 2412 q^{12} + 430 q^{13} + 3132 q^{14} + 585 q^{15} - 9906 q^{16} - 99 q^{17} - 2592 q^{18} - 2249 q^{19} - 4625 q^{20} - 1431 q^{21} - 3353 q^{22} - 1883 q^{23} + 16038 q^{24} - 5120 q^{25} + 34434 q^{26} + 9477 q^{27} + 1538 q^{28} - 16533 q^{29} - 13185 q^{30} - 4199 q^{31} - 72836 q^{32} + 4545 q^{33} - 49116 q^{34} + 28520 q^{35} - 21708 q^{36} + 22894 q^{37} + 30886 q^{38} - 3870 q^{39} + 36340 q^{40} - 10414 q^{41} + 24507 q^{42} - 98524 q^{43} + 1003 q^{44} + 2430 q^{45} - 77714 q^{46} + 85977 q^{47} + 89154 q^{48} + 330088 q^{49} + 21955 q^{50} + 61506 q^{51} - 66754 q^{52} + 43262 q^{53} - 5832 q^{54} - 135335 q^{55} - 116640 q^{56} - 64494 q^{57} - 45423 q^{58} - 31396 q^{59} + 14265 q^{60} - 23492 q^{61} + 68613 q^{62} - 3321 q^{63} - 177441 q^{64} + 221020 q^{65} - 44028 q^{66} - 15694 q^{67} - 344874 q^{68} - 43038 q^{69} + 458620 q^{70} - 106571 q^{71} + 60993 q^{72} + 211232 q^{73} + 341662 q^{74} - 33120 q^{75} + 132144 q^{76} - 237267 q^{77} - 38826 q^{78} - 248710 q^{79} - 344450 q^{80} - 85293 q^{81} + 67248 q^{82} - 15026 q^{83} + 8478 q^{84} - 467385 q^{85} - 140350 q^{86} - 291438 q^{87} - 16337 q^{88} - 319689 q^{89} - 110160 q^{90} + 436761 q^{91} - 30361 q^{92} - 152874 q^{93} + 606525 q^{94} + 729675 q^{95} - 170541 q^{96} + 554006 q^{97} + 1222905 q^{98} - 57510 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1 −8.07213 − 5.86475i −2.78115 − 8.55951i 20.8755 + 64.2482i −50.9230 − 23.0618i −27.7495 + 85.4042i −110.265 109.625 − 337.390i −65.5304 + 47.6106i 275.806 + 484.808i
16.2 −8.00931 − 5.81911i −2.78115 − 8.55951i 20.3985 + 62.7803i 3.94263 + 55.7625i −27.5336 + 84.7396i 68.0562 104.050 − 320.232i −65.5304 + 47.6106i 292.910 − 469.562i
16.3 −6.21312 − 4.51410i −2.78115 − 8.55951i 8.33725 + 25.6594i 49.9920 − 25.0159i −21.3588 + 65.7356i −86.3247 −11.9137 + 36.6666i −65.5304 + 47.6106i −423.531 − 70.2417i
16.4 −3.25710 − 2.36642i −2.78115 − 8.55951i −4.87979 − 15.0184i −52.7813 + 18.4157i −11.1969 + 34.4606i −30.1836 −59.4573 + 182.991i −65.5304 + 47.6106i 215.493 + 64.9212i
16.5 −3.06820 − 2.22917i −2.78115 − 8.55951i −5.44394 − 16.7547i 49.2880 + 26.3760i −10.5475 + 32.4619i 166.973 −58.1484 + 178.962i −65.5304 + 47.6106i −92.4285 − 190.798i
16.6 −0.374585 − 0.272152i −2.78115 − 8.55951i −9.82230 − 30.2299i 17.8544 − 52.9738i −1.28771 + 3.96316i −250.965 −9.12636 + 28.0881i −65.5304 + 47.6106i −21.1049 + 14.9841i
16.7 0.391279 + 0.284281i −2.78115 − 8.55951i −9.81626 − 30.2113i −6.85789 − 55.4795i 1.34510 − 4.13978i 147.533 9.53017 − 29.3308i −65.5304 + 47.6106i 13.0884 − 23.6575i
16.8 1.89994 + 1.38039i −2.78115 − 8.55951i −8.18425 − 25.1885i −11.0011 + 54.8085i 6.53140 − 20.1016i −12.9596 42.4431 − 130.626i −65.5304 + 47.6106i −96.5584 + 88.9470i
16.9 3.90568 + 2.83764i −2.78115 − 8.55951i −2.68643 − 8.26798i 51.6888 + 21.2901i 13.4265 − 41.3226i −21.5163 60.7080 − 186.840i −65.5304 + 47.6106i 141.466 + 229.827i
16.10 5.77150 + 4.19324i −2.78115 − 8.55951i 5.83843 + 17.9688i −54.6824 + 11.6117i 19.8407 − 61.0633i 250.575 28.8934 − 88.9247i −65.5304 + 47.6106i −364.290 − 162.280i
16.11 6.42660 + 4.66920i −2.78115 − 8.55951i 9.61125 + 29.5804i −36.9775 − 41.9245i 22.0927 − 67.9943i −121.974 2.20276 − 6.77939i −65.5304 + 47.6106i −41.8858 − 442.088i
16.12 8.11007 + 5.89231i −2.78115 − 8.55951i 21.1654 + 65.1403i −10.6481 + 54.8782i 27.8799 − 85.8056i −200.169 −113.046 + 347.919i −65.5304 + 47.6106i −409.716 + 382.325i
16.13 8.96151 + 6.51092i −2.78115 − 8.55951i 28.0280 + 86.2614i 41.8350 − 37.0788i 30.8069 − 94.8140i 186.941 −200.932 + 618.405i −65.5304 + 47.6106i 616.321 − 59.8977i
31.1 −3.24904 − 9.99951i 7.28115 − 5.29007i −63.5454 + 46.1684i 38.5958 − 40.4397i −76.5548 − 55.6203i −222.740 395.928 + 287.658i 25.0304 − 77.0356i −529.776 − 254.549i
31.2 −2.93726 − 9.03997i 7.28115 − 5.29007i −47.2049 + 34.2964i 41.1893 + 37.7947i −69.2087 − 50.2831i 216.193 202.616 + 147.209i 25.0304 − 77.0356i 220.679 − 483.363i
31.3 −2.72816 − 8.39643i 7.28115 − 5.29007i −37.1686 + 27.0046i −47.8713 + 28.8675i −64.2819 − 46.7035i −28.9752 99.5861 + 72.3535i 25.0304 − 77.0356i 372.985 + 323.193i
31.4 −1.99814 − 6.14964i 7.28115 − 5.29007i −7.93702 + 5.76658i −16.2718 − 53.4811i −47.0808 − 34.2062i 97.8556 −116.077 − 84.3346i 25.0304 − 77.0356i −296.377 + 206.928i
31.5 −1.47508 − 4.53983i 7.28115 − 5.29007i 7.45434 − 5.41589i 24.4143 + 50.2886i −34.7563 − 25.2519i −198.437 −159.161 − 115.637i 25.0304 − 77.0356i 192.289 − 185.017i
31.6 −0.583439 − 1.79564i 7.28115 − 5.29007i 23.0046 − 16.7138i 55.8709 + 1.85435i −13.7472 − 9.98791i 129.317 −92.3127 − 67.0691i 25.0304 − 77.0356i −29.2676 − 101.406i
31.7 −0.500337 − 1.53988i 7.28115 − 5.29007i 23.7676 − 17.2682i −29.9286 − 47.2153i −11.7891 − 8.56529i −132.446 −80.3996 − 58.4138i 25.0304 − 77.0356i −57.7315 + 69.7100i
See all 52 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 16.13
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 75.6.g.b ✓ 52
25.d even 5 1 inner 75.6.g.b ✓ 52
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.6.g.b ✓ 52 1.a even 1 1 trivial
75.6.g.b ✓ 52 25.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{52} - 8 T_{2}^{51} + 319 T_{2}^{50} - 2803 T_{2}^{49} + 69477 T_{2}^{48} - 520313 T_{2}^{47} + \cdots + 88\!\cdots\!36 \) acting on \(S_{6}^{\mathrm{new}}(75, [\chi])\). Copy content Toggle raw display