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Properties

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

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [75,8,Mod(16,75)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage: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")
 
Copy content magma://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

Copy content comment:select newform
 
Copy content sage:traces = [68,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content 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.

\(\operatorname{Tr}(f)(q) = \) \( 68 q + 16 q^{2} - 459 q^{3} - 598 q^{4} + 510 q^{5} + 432 q^{6} - 3812 q^{7} + 1569 q^{8} - 12393 q^{9} - 19295 q^{10} + 9190 q^{11} - 35316 q^{12} + 19630 q^{13} + 972 q^{14} + 22815 q^{15} - 84034 q^{16}+ \cdots + 736290 q^{99}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1 −15.4173 − 11.2013i 8.34346 + 25.6785i 72.6689 + 223.652i 279.489 − 3.31800i 159.000 − 489.350i 1675.76 631.060 − 1942.20i −589.773 + 428.495i −4346.12 − 3079.48i
16.2 −15.4060 − 11.1931i 8.34346 + 25.6785i 72.5042 + 223.145i 241.416 + 140.867i 158.883 − 488.991i −1629.23 627.462 − 1931.13i −589.773 + 428.495i −2142.51 − 4872.37i
16.3 −15.2028 − 11.0455i 8.34346 + 25.6785i 69.5683 + 214.109i −48.0211 − 275.352i 156.788 − 482.543i −430.761 564.017 − 1735.86i −589.773 + 428.495i −2311.34 + 4716.54i
16.4 −10.1660 − 7.38600i 8.34346 + 25.6785i 9.23955 + 28.4364i −269.898 − 72.6623i 104.842 − 322.672i 520.686 −380.928 + 1172.38i −589.773 + 428.495i 2207.09 + 2732.15i
16.5 −9.71791 − 7.06047i 8.34346 + 25.6785i 5.03327 + 15.4908i −32.7326 + 277.585i 100.222 − 308.450i 287.687 −414.665 + 1276.21i −589.773 + 428.495i 2277.98 − 2466.44i
16.6 −7.32176 − 5.31957i 8.34346 + 25.6785i −14.2438 − 43.8379i 144.102 − 239.498i 75.5100 − 232.396i −555.608 −486.882 + 1498.47i −589.773 + 428.495i −2329.11 + 986.987i
16.7 −2.12025 − 1.54045i 8.34346 + 25.6785i −37.4317 − 115.203i −277.872 + 30.2012i 21.8663 − 67.2976i −1578.19 −201.763 + 620.961i −589.773 + 428.495i 635.682 + 364.014i
16.8 −2.11488 − 1.53655i 8.34346 + 25.6785i −37.4424 − 115.236i 278.610 + 22.3962i 21.8110 − 67.1273i 1261.72 −201.280 + 619.476i −589.773 + 428.495i −554.815 − 475.464i
16.9 1.53106 + 1.11238i 8.34346 + 25.6785i −38.4474 − 118.329i 150.682 + 235.415i −15.7900 + 48.5966i −601.639 147.618 − 454.321i −589.773 + 428.495i −31.1682 + 528.051i
16.10 2.88988 + 2.09962i 8.34346 + 25.6785i −35.6112 − 109.600i −10.8682 − 279.297i −29.8036 + 91.7259i 55.7761 268.497 − 826.349i −589.773 + 428.495i 555.010 − 829.954i
16.11 5.18878 + 3.76987i 8.34346 + 25.6785i −26.8426 − 82.6132i −172.703 + 219.769i −53.5123 + 164.694i 1259.26 425.848 − 1310.63i −589.773 + 428.495i −1724.62 + 489.266i
16.12 8.84199 + 6.42408i 8.34346 + 25.6785i −2.64224 − 8.13198i 262.176 − 96.8955i −91.1881 + 280.648i −855.431 461.177 − 1419.36i −589.773 + 428.495i 2940.62 + 827.490i
16.13 10.1219 + 7.35396i 8.34346 + 25.6785i 8.81706 + 27.1361i −261.464 − 98.8006i −104.388 + 321.272i 202.841 384.561 − 1183.56i −589.773 + 428.495i −1919.93 − 2922.84i
16.14 12.7016 + 9.22823i 8.34346 + 25.6785i 36.6154 + 112.691i 54.9726 + 274.049i −130.992 + 403.153i −1037.66 46.1390 − 142.001i −589.773 + 428.495i −1830.75 + 3988.15i
16.15 15.6042 + 11.3371i 8.34346 + 25.6785i 75.4060 + 232.076i 152.824 − 234.029i −160.927 + 495.282i 1185.40 −691.502 + 2128.23i −589.773 + 428.495i 5037.90 − 1919.25i
16.16 16.5608 + 12.0321i 8.34346 + 25.6785i 89.9338 + 276.788i −129.637 − 247.627i −170.793 + 525.646i −1547.60 −1031.28 + 3173.97i −589.773 + 428.495i 832.597 − 5660.71i
16.17 16.9710 + 12.3301i 8.34346 + 25.6785i 96.4279 + 296.775i −15.5581 + 279.075i −175.023 + 538.666i 990.513 −1193.06 + 3671.85i −589.773 + 428.495i −3705.08 + 4544.35i
31.1 −6.59147 − 20.2865i −21.8435 + 15.8702i −264.539 + 192.199i −238.214 − 146.216i 465.931 + 338.519i −896.383 3433.88 + 2494.86i 225.273 − 693.320i −1396.02 + 5796.30i
31.2 −5.67748 − 17.4735i −21.8435 + 15.8702i −169.535 + 123.174i 33.8509 + 277.451i 401.324 + 291.579i 642.984 1212.24 + 880.747i 225.273 − 693.320i 4655.85 − 2166.72i
31.3 −5.54320 − 17.0602i −21.8435 + 15.8702i −156.770 + 113.900i 98.0640 − 261.741i 391.832 + 284.682i 509.951 954.587 + 693.548i 225.273 − 693.320i −5008.95 − 222.109i
See all 68 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 16.17
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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.b ✓ 68
25.d even 5 1 inner 75.8.g.b ✓ 68
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.8.g.b ✓ 68 1.a even 1 1 trivial
75.8.g.b ✓ 68 25.d even 5 1 inner