↑

Properties

Label 75.8.g.a
Level $75$
Weight $8$
Character orbit 75.g
Analytic conductor $23.429$
Analytic rank $0$
Dimension $68$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [75,8,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, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("75.16");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 8 \)
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: \(23.4288769113\)
Analytic rank: \(0\)
Dimension: \(68\)
Relative dimension: \(17\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 68 q + 459 q^{3} - 1450 q^{4} - 900 q^{5} + 1676 q^{7} - 1503 q^{8} - 12393 q^{9} + 11705 q^{10} - 20776 q^{11} + 19980 q^{12} + 2054 q^{13} + 16656 q^{14} - 19035 q^{15} - 31506 q^{16} - 63937 q^{17} - 11413 q^{19}+ \cdots + 4894506 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1 −17.9805 − 13.0636i −8.34346 − 25.6785i 113.087 + 348.045i 31.4532 − 277.733i −185.435 + 570.709i −643.929 1634.27 − 5029.78i −589.773 + 428.495i −4193.74 + 4582.89i
16.2 −14.4283 − 10.4828i −8.34346 − 25.6785i 58.7333 + 180.762i −279.474 − 4.38629i −148.800 + 457.960i 1191.57 342.047 − 1052.71i −589.773 + 428.495i 3986.36 + 2992.95i
16.3 −12.6311 − 9.17700i −8.34346 − 25.6785i 35.7721 + 110.095i 278.861 − 19.0184i −130.265 + 400.915i −382.770 −59.0494 + 181.735i −589.773 + 428.495i −3696.84 − 2318.88i
16.4 −12.0199 − 8.73298i −8.34346 − 25.6785i 28.6593 + 88.2043i −161.514 + 228.119i −123.962 + 381.517i −1393.45 −161.870 + 498.184i −589.773 + 428.495i 3933.55 − 1331.46i
16.5 −9.46031 − 6.87332i −8.34346 − 25.6785i 2.70077 + 8.31210i 21.0258 − 278.717i −97.5649 + 300.274i 910.454 −430.948 + 1326.32i −589.773 + 428.495i −2114.62 + 2492.23i
16.6 −7.06256 − 5.13125i −8.34346 − 25.6785i −16.0042 − 49.2558i 185.098 + 209.437i −72.8367 + 224.168i 1475.21 −485.013 + 1492.72i −589.773 + 428.495i −232.593 − 2428.94i
16.7 −5.62885 − 4.08960i −8.34346 − 25.6785i −24.5951 − 75.6958i −116.848 − 253.912i −58.0508 + 178.662i −947.917 −446.327 + 1373.65i −589.773 + 428.495i −380.679 + 1907.10i
16.8 −3.36833 − 2.44723i −8.34346 − 25.6785i −34.1975 − 105.249i −128.688 + 248.122i −34.7378 + 106.912i 152.392 −307.064 + 945.045i −589.773 + 428.495i 1040.68 − 520.825i
16.9 1.29673 + 0.942131i −8.34346 − 25.6785i −38.7603 − 119.292i 212.625 + 181.427i 13.3733 − 41.1588i −1130.50 125.526 − 386.330i −589.773 + 428.495i 104.789 + 435.582i
16.10 2.68477 + 1.95060i −8.34346 − 25.6785i −36.1510 − 111.261i −258.364 − 106.643i 27.6883 − 85.2158i 400.372 251.232 − 773.214i −589.773 + 428.495i −485.631 − 790.279i
16.11 4.53093 + 3.29191i −8.34346 − 25.6785i −29.8616 − 91.9044i 207.683 − 187.064i 46.7278 − 143.813i 1026.06 388.765 − 1196.50i −589.773 + 428.495i 1556.79 − 163.898i
16.12 7.33226 + 5.32720i −8.34346 − 25.6785i −14.1712 − 43.6144i −251.112 − 122.751i 75.6182 − 232.729i −547.317 486.922 − 1498.59i −589.773 + 428.495i −1187.30 − 2237.77i
16.13 8.47550 + 6.15781i −8.34346 − 25.6785i −5.63868 − 17.3541i −122.820 + 251.078i 87.4086 − 269.016i 1115.12 473.454 − 1457.14i −589.773 + 428.495i −2587.05 + 1371.72i
16.14 11.6616 + 8.47268i −8.34346 − 25.6785i 24.6535 + 75.8755i 232.407 − 155.281i 120.267 − 370.145i −1084.01 214.788 − 661.049i −589.773 + 428.495i 4025.89 + 158.281i
16.15 14.3303 + 10.4115i −8.34346 − 25.6785i 57.4019 + 176.665i 199.313 + 195.957i 147.789 − 454.848i 562.195 −316.140 + 972.978i −589.773 + 428.495i 815.997 + 4883.27i
16.16 15.1056 + 10.9749i −8.34346 − 25.6785i 68.1777 + 209.829i −6.56281 − 279.431i 155.786 − 479.459i −243.239 −534.446 + 1644.86i −589.773 + 428.495i 2967.59 − 4293.01i
16.17 17.1621 + 12.4690i −8.34346 − 25.6785i 99.5079 + 306.254i −270.318 + 71.0866i 176.994 − 544.733i 115.292 −1271.83 + 3914.30i −589.773 + 428.495i −5525.60 − 2150.60i
31.1 −6.86094 − 21.1158i 21.8435 − 15.8702i −295.250 + 214.512i −195.105 + 200.148i −484.979 − 352.358i 559.579 4256.13 + 3092.26i 225.273 − 693.320i 5564.88 + 2746.60i
31.2 −5.65098 − 17.3919i 21.8435 − 15.8702i −166.992 + 121.327i −169.570 − 222.196i −399.451 − 290.218i −1508.04 1160.08 + 842.847i 225.273 − 693.320i −2906.18 + 4204.78i
31.3 −5.63831 − 17.3529i 21.8435 − 15.8702i −165.780 + 120.446i 276.141 − 43.2574i −398.555 − 289.567i 225.879 1135.37 + 824.894i 225.273 − 693.320i −2307.61 − 4547.96i
See all 68 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 16.17
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 75.8.g.a ✓ 68
25.d even 5 1 inner 75.8.g.a ✓ 68
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.8.g.a ✓ 68 1.a even 1 1 trivial
75.8.g.a ✓ 68 25.d even 5 1 inner