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Properties

Label 75.8.i.a
Level $75$
Weight $8$
Character orbit 75.i
Analytic conductor $23.429$
Analytic rank $0$
Dimension $144$
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(4,75)] 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, 1])) 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("75.4"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 75.i (of order \(10\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(23.4288769113\)
Analytic rank: \(0\)
Dimension: \(144\)
Relative dimension: \(36\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 144 q + 2432 q^{4} + 444 q^{5} - 432 q^{6} + 5010 q^{8} + 26244 q^{9} + 6686 q^{10} + 11586 q^{11} + 17628 q^{14} + 17496 q^{15} - 171180 q^{16} - 63830 q^{17} + 76118 q^{19} + 251756 q^{20} - 37044 q^{21}+ \cdots + 5630796 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −20.7551 − 6.74374i −15.8702 + 21.8435i 281.742 + 204.698i 264.211 + 91.2007i 476.695 − 346.339i − 646.924i −2825.26 − 3888.64i −225.273 − 693.320i −4868.69 − 3674.65i
4.2 −20.4935 − 6.65874i 15.8702 − 21.8435i 272.090 + 197.685i 182.194 − 211.968i −470.686 + 341.973i 657.484i −2638.54 − 3631.64i −225.273 − 693.320i −5145.23 + 3130.78i
4.3 −19.2349 − 6.24980i −15.8702 + 21.8435i 227.367 + 165.192i −267.057 + 82.4957i 441.779 − 320.971i 279.921i −1819.33 − 2504.10i −225.273 − 693.320i 5652.40 + 82.2569i
4.4 −18.5349 − 6.02234i 15.8702 − 21.8435i 203.719 + 148.010i 10.9269 + 279.295i −425.701 + 309.290i − 1019.16i −1418.27 − 1952.07i −225.273 − 693.320i 1479.48 − 5242.50i
4.5 −16.4262 − 5.33719i 15.8702 − 21.8435i 137.779 + 100.103i −252.155 − 120.593i −377.269 + 274.102i − 989.987i −429.476 − 591.123i −225.273 − 693.320i 3498.32 + 3326.68i
4.6 −15.9978 − 5.19799i −15.8702 + 21.8435i 125.355 + 91.0761i −164.963 − 225.638i 367.430 − 266.954i − 761.019i −266.438 − 366.721i −225.273 − 693.320i 1466.17 + 4467.17i
4.7 −15.1895 − 4.93537i −15.8702 + 21.8435i 102.809 + 74.6952i 137.774 − 243.194i 348.866 − 253.466i 1729.43i 8.64518 + 11.8991i −225.273 − 693.320i −3292.98 + 3014.03i
4.8 −14.2990 − 4.64601i 15.8702 − 21.8435i 79.3205 + 57.6297i 276.954 + 37.7019i −328.412 + 238.605i 671.369i 264.715 + 364.349i −225.273 − 693.320i −3784.99 − 1825.83i
4.9 −11.9387 − 3.87913i 15.8702 − 21.8435i 23.9313 + 17.3871i −68.3062 − 271.034i −274.203 + 199.220i 342.329i 726.190 + 999.515i −225.273 − 693.320i −235.885 + 3500.76i
4.10 −11.8447 − 3.84857i −15.8702 + 21.8435i 21.9310 + 15.9338i −108.372 + 257.644i 272.044 − 197.651i − 783.153i 738.570 + 1016.55i −225.273 − 693.320i 2275.20 − 2634.63i
4.11 −11.3253 − 3.67981i 15.8702 − 21.8435i 11.1669 + 8.11322i −216.143 + 177.221i −260.114 + 188.984i 1198.17i 799.311 + 1100.16i −225.273 − 693.320i 3100.02 − 1211.71i
4.12 −9.85069 − 3.20068i −15.8702 + 21.8435i −16.7624 − 12.1786i 279.506 + 1.22238i 226.246 − 164.378i − 633.339i 905.414 + 1246.19i −225.273 − 693.320i −2749.41 − 906.651i
4.13 −8.41215 − 2.73327i 15.8702 − 21.8435i −40.2607 − 29.2511i 235.359 − 150.768i −193.207 + 140.373i − 1792.05i 924.199 + 1272.05i −225.273 − 693.320i −2391.97 + 624.985i
4.14 −5.67378 − 1.84352i −15.8702 + 21.8435i −74.7610 − 54.3171i −268.202 − 78.6934i 130.313 − 94.6778i 1072.72i 772.886 + 1063.79i −225.273 − 693.320i 1376.65 + 940.925i
4.15 −3.41271 − 1.10886i −15.8702 + 21.8435i −93.1372 − 67.6681i 114.335 + 255.054i 78.3816 − 56.9475i 1230.60i 512.789 + 705.794i −225.273 − 693.320i −107.373 − 997.205i
4.16 −2.76941 − 0.899835i 15.8702 − 21.8435i −96.6943 − 70.2525i 214.025 + 179.773i −63.6066 + 46.2129i 166.889i 423.654 + 583.109i −225.273 − 693.320i −430.957 − 690.452i
4.17 −2.63463 − 0.856043i −15.8702 + 21.8435i −97.3457 − 70.7258i 2.73943 − 279.495i 60.5111 − 43.9639i − 280.140i 404.347 + 556.536i −225.273 − 693.320i −246.477 + 734.021i
4.18 −1.85809 − 0.603729i 15.8702 − 21.8435i −100.466 − 72.9929i −39.7417 + 276.669i −42.6758 + 31.0058i − 16.6474i 289.597 + 398.597i −225.273 − 693.320i 240.877 − 490.082i
4.19 0.257467 + 0.0836561i 15.8702 − 21.8435i −103.495 − 75.1934i −269.409 − 74.4581i 5.91339 − 4.29633i − 942.224i −40.7239 − 56.0517i −225.273 − 693.320i −63.1350 − 41.7082i
4.20 1.88540 + 0.612605i −15.8702 + 21.8435i −100.375 − 72.9265i 14.8566 − 279.113i −43.3032 + 31.4616i − 1231.97i −293.723 − 404.275i −225.273 − 693.320i 198.997 − 517.140i
See next 80 embeddings (of 144 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.36
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.e even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 75.8.i.a ✓ 144
25.e even 10 1 inner 75.8.i.a ✓ 144
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.8.i.a ✓ 144 1.a even 1 1 trivial
75.8.i.a ✓ 144 25.e even 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(75, [\chi])\).