↑

Properties

Label 97.7.j.a
Level $97$
Weight $7$
Character orbit 97.j
Analytic conductor $22.315$
Analytic rank $0$
Dimension $768$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [97,7,Mod(19,97)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(97, base_ring=CyclotomicField(32)) chi = DirichletCharacter(H, H._module([27])) N = Newforms(chi, 7, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("97.19"); S:= CuspForms(chi, 7); N := Newforms(S);
 
Level: \( N \) \(=\) \( 97 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 97.j (of order \(32\), degree \(16\), 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: \(22.3152461111\)
Analytic rank: \(0\)
Dimension: \(768\)
Relative dimension: \(48\) over \(\Q(\zeta_{32})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{32}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 768 q - 16 q^{2} - 16 q^{3} - 16 q^{4} - 16 q^{5} - 16 q^{6} - 16 q^{7} - 16 q^{8} - 11056 q^{9} - 16 q^{10} - 16 q^{11} - 15376 q^{12} - 16 q^{13} - 16 q^{14} + 7408 q^{15} - 16 q^{16} + 16624 q^{17} - 16 q^{18}+ \cdots - 6220816 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −8.79004 + 13.1552i 1.06910 + 0.212658i −71.3034 − 172.142i −131.771 + 70.4331i −12.1950 + 12.1950i −340.019 − 181.744i 1898.19 + 377.574i −672.410 − 278.522i 231.710 − 2352.59i
19.2 −8.08643 + 12.1022i 19.7519 + 3.92890i −56.5811 − 136.599i 102.565 − 54.8223i −207.271 + 207.271i −28.4535 − 15.2087i 1197.05 + 238.108i −298.806 − 123.769i −165.917 + 1684.58i
19.3 −7.93371 + 11.8736i −38.8820 − 7.73411i −53.5477 − 129.276i −50.1015 + 26.7798i 400.310 − 400.310i 20.2645 + 10.8316i 1063.43 + 211.529i 778.485 + 322.459i 79.5171 − 807.350i
19.4 −7.69492 + 11.5163i 50.9813 + 10.1408i −48.9208 − 118.105i −42.9030 + 22.9321i −509.081 + 509.081i 117.577 + 62.8462i 867.173 + 172.491i 1822.75 + 755.006i 66.0427 − 670.543i
19.5 −7.63247 + 11.4228i −29.3932 − 5.84667i −47.7340 − 115.240i 177.811 − 95.0420i 291.128 − 291.128i 220.494 + 117.856i 818.348 + 162.780i 156.269 + 64.7286i −271.492 + 2756.50i
19.6 −7.44558 + 11.1431i 17.2512 + 3.43147i −44.2402 − 106.805i −89.5143 + 47.8464i −166.682 + 166.682i 592.462 + 316.678i 678.307 + 134.924i −387.681 − 160.583i 133.329 − 1353.71i
19.7 −6.78987 + 10.1618i 15.2804 + 3.03947i −32.6673 − 78.8658i 107.652 − 57.5412i −134.639 + 134.639i −172.387 − 92.1430i 256.078 + 50.9372i −449.254 − 186.087i −146.223 + 1484.63i
19.8 −6.14859 + 9.20201i −8.41134 − 1.67312i −22.3801 − 54.0304i −46.9219 + 25.0803i 67.1139 − 67.1139i 521.829 + 278.924i −59.8941 − 11.9137i −605.557 − 250.830i 57.7144 − 585.984i
19.9 −5.92118 + 8.86167i −21.8054 − 4.33737i −18.9771 − 45.8147i −143.084 + 76.4799i 167.550 − 167.550i 15.6889 + 8.38587i −150.634 − 29.9631i −216.844 − 89.8196i 169.485 − 1720.81i
19.10 −5.89540 + 8.82309i 20.5481 + 4.08728i −18.5994 − 44.9029i −208.842 + 111.628i −157.202 + 157.202i −245.674 − 131.315i −160.250 − 31.8756i −267.988 − 111.004i 246.299 − 2500.72i
19.11 −5.88431 + 8.80650i −20.5586 − 4.08936i −18.4375 − 44.5120i 61.0885 − 32.6525i 156.986 − 156.986i −528.374 − 282.422i −164.343 − 32.6898i −267.576 − 110.833i −71.9098 + 730.113i
19.12 −5.23530 + 7.83517i 37.4064 + 7.44060i −9.48989 − 22.9106i −26.3162 + 14.0663i −254.132 + 254.132i −345.967 − 184.923i −362.311 − 72.0681i 670.370 + 277.676i 27.5611 − 279.833i
19.13 −4.52976 + 6.77927i 38.8146 + 7.72070i −0.947976 − 2.28862i 220.298 − 117.752i −228.162 + 228.162i 356.239 + 190.414i −491.979 − 97.8607i 773.455 + 320.376i −199.627 + 2026.84i
19.14 −4.46871 + 6.68790i −49.0270 − 9.75207i −0.266874 − 0.644290i −104.105 + 55.6455i 284.308 − 284.308i 92.2469 + 49.3070i −499.389 − 99.3347i 1635.03 + 677.252i 93.0655 − 944.910i
19.15 −3.82791 + 5.72887i 21.1096 + 4.19895i 6.32464 + 15.2690i −6.96191 + 3.72122i −104.861 + 104.861i 109.780 + 58.6788i −544.175 − 108.243i −245.526 − 101.700i 5.33118 − 54.1284i
19.16 −3.74297 + 5.60175i −47.6605 − 9.48025i 7.12199 + 17.1940i 153.939 − 82.2822i 231.497 − 231.497i −50.7344 − 27.1181i −545.867 − 108.580i 1508.14 + 624.690i −115.265 + 1170.31i
19.17 −3.60786 + 5.39954i −5.65428 − 1.12471i 8.35334 + 20.1667i 75.8939 − 40.5661i 26.4728 − 26.4728i 182.189 + 97.3818i −546.657 − 108.737i −642.802 − 266.257i −54.7759 + 556.149i
19.18 −2.48629 + 3.72100i −25.2506 − 5.02265i 16.8276 + 40.6253i 100.797 − 53.8771i 81.4696 − 81.4696i 262.607 + 140.366i −473.915 − 94.2675i −61.1431 − 25.3263i −50.1341 + 509.020i
19.19 −1.81050 + 2.70961i 41.4098 + 8.23692i 20.4277 + 49.3168i −131.101 + 70.0748i −97.2914 + 97.2914i 372.276 + 198.986i −375.171 − 74.6261i 973.414 + 403.201i 47.4830 − 482.103i
19.20 −1.66408 + 2.49047i −1.43974 − 0.286382i 21.0585 + 50.8396i −142.655 + 76.2507i 3.10907 − 3.10907i −77.1178 − 41.2203i −349.670 − 69.5537i −671.517 − 278.152i 47.4890 − 482.164i
See next 80 embeddings (of 768 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
97.j odd 32 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 97.7.j.a ✓ 768
97.j odd 32 1 inner 97.7.j.a ✓ 768
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
97.7.j.a ✓ 768 1.a even 1 1 trivial
97.7.j.a ✓ 768 97.j odd 32 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(97, [\chi])\).