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Properties

Label 75.8.l.a
Level $75$
Weight $8$
Character orbit 75.l
Analytic conductor $23.429$
Analytic rank $0$
Dimension $544$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [75,8,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, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("75.2");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 8 \)
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: \(23.4288769113\)
Analytic rank: \(0\)
Dimension: \(544\)
Relative dimension: \(68\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 544 q - 34 q^{3} - 20 q^{4} - 6 q^{6} - 1364 q^{7} - 10 q^{9} + 3160 q^{10} + 2018 q^{12} - 16868 q^{13} - 37570 q^{15} + 507892 q^{16} + 106550 q^{18} + 70800 q^{19} - 6 q^{21} - 195160 q^{22} - 444200 q^{25}+ \cdots - 22089384 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −3.52237 + 22.2394i 2.63401 + 46.6911i −360.447 − 117.116i −181.072 + 212.927i −1047.66 − 105.885i −180.295 − 180.295i 2565.76 − 5035.59i −2173.12 + 245.970i −4097.56 − 4776.93i
2.2 −3.30156 + 20.8452i −46.7631 − 0.456621i −301.887 − 98.0892i 263.042 + 94.5194i 163.910 − 973.280i 47.5755 + 47.5755i 1814.96 − 3562.05i 2186.58 + 42.7061i −2838.73 + 5171.10i
2.3 −3.27990 + 20.7085i 45.8746 + 9.08428i −296.348 − 96.2892i 264.930 − 89.0893i −338.585 + 920.196i −826.593 − 826.593i 1747.61 − 3429.87i 2021.95 + 833.475i 975.957 + 5778.51i
2.4 −3.19418 + 20.1673i 38.5888 − 26.4179i −274.781 − 89.2816i −278.663 + 21.7191i 409.517 + 862.614i 365.838 + 365.838i 1491.72 − 2927.67i 791.191 − 2038.87i 452.087 − 5689.25i
2.5 −3.09412 + 19.5355i 4.16744 − 46.5793i −250.326 − 81.3359i 138.197 − 242.954i 897.055 + 225.535i 3.00418 + 3.00418i 1214.10 − 2382.81i −2152.26 − 388.233i 4318.62 + 3451.48i
2.6 −3.08634 + 19.4864i −41.4224 − 21.7068i −248.459 − 80.7292i −249.237 − 126.515i 550.831 − 740.179i −994.647 − 994.647i 1193.47 − 2342.31i 1244.63 + 1798.30i 3234.55 − 4466.26i
2.7 −3.04456 + 19.2226i −36.2819 + 29.5063i −238.504 − 77.4946i −121.311 − 251.811i −456.725 − 787.267i 962.826 + 962.826i 1084.82 − 2129.08i 445.759 − 2141.09i 5209.79 − 1565.27i
2.8 −3.00217 + 18.9549i −13.4723 − 44.7828i −228.542 − 74.2578i 56.3184 + 273.776i 889.301 − 120.922i 507.559 + 507.559i 978.456 − 1920.33i −1823.99 + 1206.66i −5358.49 + 245.591i
2.9 −2.78472 + 17.5820i 42.5275 + 19.4528i −179.638 − 58.3678i −75.2428 − 269.190i −460.446 + 693.549i 758.406 + 758.406i 492.023 − 965.649i 1430.18 + 1654.55i 4942.44 − 573.301i
2.10 −2.68688 + 16.9643i 2.70038 + 46.6873i −158.832 − 51.6076i 277.208 + 35.7878i −799.272 − 79.6331i 577.729 + 577.729i 304.152 − 596.931i −2172.42 + 252.147i −1351.94 + 4606.47i
2.11 −2.55265 + 16.1168i 36.2961 − 29.4889i −131.500 − 42.7270i 40.5144 + 276.557i 382.616 + 660.251i −970.273 − 970.273i 76.0619 − 149.280i 447.807 − 2140.66i −4560.63 − 52.9906i
2.12 −2.55129 + 16.1082i −12.9716 + 44.9304i −131.229 − 42.6390i 79.1005 − 268.082i −690.652 − 323.579i −1066.81 − 1066.81i 73.9138 − 145.064i −1850.47 − 1165.64i 4116.51 + 1958.12i
2.13 −2.53512 + 16.0061i 41.4008 + 21.7479i −128.035 − 41.6010i 67.8172 + 271.156i −453.056 + 607.535i 364.928 + 364.928i 48.7316 − 95.6411i 1241.06 + 1800.76i −4512.10 + 398.077i
2.14 −2.30743 + 14.5685i −38.7959 + 26.1128i −85.1830 − 27.6776i −102.175 + 260.164i −290.906 − 625.453i −331.771 − 331.771i −257.366 + 505.109i 823.247 − 2026.14i −3554.45 − 2088.85i
2.15 −2.27542 + 14.3665i −40.9482 − 22.5886i −79.4821 − 25.8253i −249.935 + 125.130i 417.693 − 536.882i 933.969 + 933.969i −293.379 + 575.788i 1166.51 + 1849.93i −1228.96 − 3875.40i
2.16 −2.19123 + 13.8349i 26.4684 + 38.5542i −64.8671 − 21.0766i −278.769 − 20.3187i −591.390 + 281.706i −491.802 − 491.802i −380.246 + 746.275i −785.845 + 2040.94i 891.953 − 3812.21i
2.17 −1.89065 + 11.9371i 5.16139 − 46.4797i −17.1845 − 5.58357i −260.391 − 101.596i 545.074 + 149.489i −382.688 − 382.688i −603.179 + 1183.81i −2133.72 − 479.799i 1705.07 − 2916.22i
2.18 −1.88257 + 11.8861i 33.2863 − 32.8484i −15.9995 − 5.19856i 279.504 − 1.66925i 327.775 + 457.484i 759.321 + 759.321i −607.408 + 1192.11i 28.9621 − 2186.81i −506.344 + 3325.34i
2.19 −1.85418 + 11.7069i −46.5389 + 4.59638i −11.8772 − 3.85915i 250.787 − 123.414i 32.4826 − 553.347i −2.39286 − 2.39286i −621.573 + 1219.91i 2144.75 − 427.821i 979.788 + 3164.76i
2.20 −1.79599 + 11.3395i −28.9982 − 36.6893i −3.62242 − 1.17700i 245.145 + 134.273i 468.117 − 262.930i −1082.77 − 1082.77i −647.306 + 1270.41i −505.206 + 2127.85i −1962.86 + 2538.65i
See next 80 embeddings (of 544 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.68
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 75.8.l.a ✓ 544
3.b odd 2 1 inner 75.8.l.a ✓ 544
25.f odd 20 1 inner 75.8.l.a ✓ 544
75.l even 20 1 inner 75.8.l.a ✓ 544
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.8.l.a ✓ 544 1.a even 1 1 trivial
75.8.l.a ✓ 544 3.b odd 2 1 inner
75.8.l.a ✓ 544 25.f odd 20 1 inner
75.8.l.a ✓ 544 75.l even 20 1 inner

Hecke kernels

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