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Properties

Label 97.8.h.a
Level $97$
Weight $8$
Character orbit 97.h
Analytic conductor $30.301$
Analytic rank $0$
Dimension $440$
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] = [97,8,Mod(8,97)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(97, base_ring=CyclotomicField(16)) chi = DirichletCharacter(H, H._module([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("97.8"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Level: \( N \) \(=\) \( 97 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 97.h (of order \(16\), degree \(8\), 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: \(30.3013474720\)
Analytic rank: \(0\)
Dimension: \(440\)
Relative dimension: \(55\) over \(\Q(\zeta_{16})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 440 q - 8 q^{2} + 96 q^{3} - 8 q^{4} - 8 q^{5} - 8 q^{7} - 8 q^{8} - 13424 q^{9} - 1672 q^{10} - 8 q^{11} - 18952 q^{12} - 8 q^{13} - 35264 q^{14} - 99880 q^{15} - 7504 q^{17} + 103208 q^{18} - 157440 q^{19}+ \cdots - 243024 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1 −20.2712 − 8.39660i −24.3454 − 58.7751i 249.908 + 249.908i 417.025 + 82.9514i 1395.86i −436.387 + 86.8028i −1892.79 − 4569.60i −1315.37 + 1315.37i −7757.08 − 5183.11i
8.2 −20.1509 − 8.34677i 22.2046 + 53.6066i 245.880 + 245.880i −194.096 − 38.6080i − 1265.56i 893.707 − 177.769i −1834.01 − 4427.68i −834.178 + 834.178i 3588.95 + 2398.06i
8.3 −19.6534 − 8.14069i −14.8615 − 35.8787i 229.474 + 229.474i −167.126 − 33.2435i 826.120i 179.469 − 35.6986i −1599.86 − 3862.39i 480.023 − 480.023i 3013.97 + 2013.87i
8.4 −18.3208 − 7.58872i 16.3453 + 39.4610i 187.553 + 187.553i 28.0467 + 5.57883i − 846.996i −910.476 + 181.105i −1041.48 − 2514.35i 256.442 − 256.442i −471.501 − 315.047i
8.5 −17.2851 − 7.15970i 6.15294 + 14.8545i 157.002 + 157.002i 350.463 + 69.7114i − 300.814i 524.297 − 104.289i −673.259 − 1625.39i 1363.64 − 1363.64i −5558.66 − 3714.18i
8.6 −16.9940 − 7.03913i −10.1290 − 24.4535i 148.736 + 148.736i −499.378 − 99.3324i 486.862i 208.091 − 41.3920i −579.633 − 1399.36i 1051.06 − 1051.06i 7787.20 + 5203.24i
8.7 −16.6711 − 6.90540i 1.04022 + 2.51132i 139.732 + 139.732i −71.3894 − 14.2002i − 49.0496i −1704.25 + 338.997i −480.689 − 1160.49i 1541.22 − 1541.22i 1092.08 + 729.706i
8.8 −16.1903 − 6.70626i 34.6581 + 83.6721i 126.644 + 126.644i 481.745 + 95.8250i − 1587.11i −159.733 + 31.7728i −342.697 − 827.345i −4253.40 + 4253.40i −7156.99 − 4782.15i
8.9 −15.4197 − 6.38703i −34.4893 − 83.2646i 106.462 + 106.462i −285.686 − 56.8266i 1504.20i −910.212 + 181.052i −144.093 − 347.872i −4197.04 + 4197.04i 4042.24 + 2700.94i
8.10 −14.8540 − 6.15271i −26.3081 − 63.5135i 92.2744 + 92.2744i 133.110 + 26.4773i 1105.29i 671.524 − 133.574i −15.3550 − 37.0702i −1795.40 + 1795.40i −1814.31 − 1212.28i
8.11 −14.8378 − 6.14601i 29.4785 + 71.1674i 91.8765 + 91.8765i −331.728 − 65.9848i − 1237.14i −681.501 + 135.559i −11.8803 − 28.6815i −2649.38 + 2649.38i 4516.56 + 3017.87i
8.12 −14.3189 − 5.93107i −13.0172 − 31.4263i 79.3427 + 79.3427i −56.0713 − 11.1533i 527.195i 1335.12 − 265.571i 93.6661 + 226.130i 728.279 − 728.279i 736.726 + 492.265i
8.13 −12.7349 − 5.27495i 20.7134 + 50.0066i 43.8417 + 43.8417i −95.6555 − 19.0271i − 746.089i 1710.21 − 340.182i 348.138 + 840.480i −525.171 + 525.171i 1117.79 + 746.885i
8.14 −11.9808 − 4.96263i −19.1529 − 46.2391i 28.4033 + 28.4033i 338.783 + 67.3881i 649.032i −1451.58 + 288.737i 435.876 + 1052.30i −224.782 + 224.782i −3724.48 − 2488.62i
8.15 −11.0476 − 4.57606i −2.12866 − 5.13904i 10.5992 + 10.5992i 299.878 + 59.6495i 66.5149i −474.287 + 94.3415i 517.143 + 1248.49i 1524.56 − 1524.56i −3039.97 − 2031.24i
8.16 −10.9086 − 4.51848i 8.54779 + 20.6362i 8.07072 + 8.07072i −399.200 − 79.4059i − 263.734i −102.762 + 20.4405i 526.793 + 1271.79i 1193.66 − 1193.66i 3995.91 + 2669.98i
8.17 −8.78157 − 3.63744i 18.9318 + 45.7053i −26.6248 − 26.6248i 180.794 + 35.9622i − 470.228i −381.684 + 75.9216i 602.554 + 1454.69i −184.123 + 184.123i −1456.84 − 973.433i
8.18 −8.18508 − 3.39037i 23.4510 + 56.6158i −35.0088 − 35.0088i 138.946 + 27.6380i − 542.913i 41.0820 − 8.17173i 601.824 + 1452.93i −1108.96 + 1108.96i −1043.58 − 697.297i
8.19 −8.12427 − 3.36518i −5.89910 − 14.2417i −35.8303 − 35.8303i −190.256 − 37.8443i 135.555i 170.751 − 33.9644i 601.263 + 1451.58i 1378.42 − 1378.42i 1418.34 + 947.705i
8.20 −7.09290 − 2.93797i −28.3325 − 68.4006i −48.8322 − 48.8322i 351.179 + 69.8538i 568.398i 765.341 − 152.236i 578.955 + 1397.72i −2329.47 + 2329.47i −2285.65 − 1527.22i
See next 80 embeddings (of 440 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 8.55
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
97.h even 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 97.8.h.a ✓ 440
97.h even 16 1 inner 97.8.h.a ✓ 440
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
97.8.h.a ✓ 440 1.a even 1 1 trivial
97.8.h.a ✓ 440 97.h even 16 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(97, [\chi])\).