Properties

Label 96.9.m.a
Level $96$
Weight $9$
Character orbit 96.m
Analytic conductor $39.108$
Analytic rank $0$
Dimension $256$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [96,9,Mod(19,96)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(96, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([4, 7, 0]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("96.19");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 96 = 2^{5} \cdot 3 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 96.m (of order \(8\), degree \(4\), 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: \(39.1083465659\)
Analytic rank: \(0\)
Dimension: \(256\)
Relative dimension: \(64\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 256 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 256 q - 35000 q^{10} + 45360 q^{12} - 291168 q^{14} + 111352 q^{16} + 52488 q^{18} - 1380000 q^{20} - 540184 q^{22} - 1691136 q^{23} + 149688 q^{24} - 3364200 q^{26} + 3615240 q^{28} - 4840920 q^{32} - 8276520 q^{34} + 4831488 q^{35} + 16322040 q^{38} + 14395976 q^{40} + 3719424 q^{43} + 22652136 q^{44} - 11790688 q^{46} + 5145192 q^{50} + 27724032 q^{51} + 33503192 q^{52} + 10717440 q^{53} - 9920232 q^{54} - 46326784 q^{55} - 56874888 q^{56} + 11797280 q^{58} + 89877504 q^{59} + 81854064 q^{60} - 48952064 q^{61} + 43931664 q^{62} - 5886312 q^{64} - 113657904 q^{66} - 37352192 q^{67} + 1507800 q^{68} + 17273088 q^{69} + 288032952 q^{70} + 159664128 q^{71} - 56030304 q^{74} + 25837056 q^{75} - 264189016 q^{76} + 189928704 q^{77} - 183989880 q^{78} - 144406528 q^{79} + 360917832 q^{80} + 428364200 q^{82} + 18570048 q^{86} - 463482160 q^{88} - 350400768 q^{91} + 152934384 q^{92} + 354090144 q^{94} - 343391400 q^{96} - 239975856 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −15.9958 0.366123i 43.2056 + 17.8963i 255.732 + 11.7129i 268.312 111.139i −684.556 302.085i −1378.29 1378.29i −4086.35 280.986i 1546.44 + 1546.44i −4332.56 + 1679.52i
19.2 −15.8347 + 2.29379i −43.2056 17.8963i 245.477 72.6430i −747.761 + 309.733i 725.199 + 184.279i −28.3179 28.3179i −3720.43 + 1713.36i 1546.44 + 1546.44i 11130.1 6619.74i
19.3 −15.6447 3.35309i −43.2056 17.8963i 233.514 + 104.916i 801.566 332.020i 615.930 + 424.855i 191.054 + 191.054i −3301.46 2424.37i 1546.44 + 1546.44i −13653.6 + 2506.63i
19.4 −15.6067 + 3.52573i 43.2056 + 17.8963i 231.138 110.050i −1121.83 + 464.676i −737.394 126.972i 2627.04 + 2627.04i −3219.30 + 2532.45i 1546.44 + 1546.44i 15869.7 11207.3i
19.5 −15.5621 3.71751i 43.2056 + 17.8963i 228.360 + 115.705i 776.323 321.564i −605.841 439.122i 1757.58 + 1757.58i −3123.64 2649.54i 1546.44 + 1546.44i −13276.7 + 2118.23i
19.6 −15.2618 + 4.80392i −43.2056 17.8963i 209.845 146.633i 316.822 131.232i 745.367 + 65.5739i −1732.33 1732.33i −2498.19 + 3245.96i 1546.44 + 1546.44i −4204.84 + 3524.82i
19.7 −15.1654 + 5.10003i 43.2056 + 17.8963i 203.979 154.688i 570.142 236.161i −746.502 51.0555i 3017.40 + 3017.40i −2304.52 + 3386.21i 1546.44 + 1546.44i −7442.02 + 6489.22i
19.8 −14.9989 5.57072i −43.2056 17.8963i 193.934 + 167.109i −717.074 + 297.022i 548.341 + 509.112i −2002.68 2002.68i −1977.88 3586.81i 1546.44 + 1546.44i 12409.9 460.378i
19.9 −14.5447 6.66724i 43.2056 + 17.8963i 167.096 + 193.946i −348.304 + 144.272i −509.092 548.359i 801.130 + 801.130i −1137.27 3934.95i 1546.44 + 1546.44i 6027.88 + 223.833i
19.10 −13.7949 8.10556i −43.2056 17.8963i 124.600 + 223.631i −476.820 + 197.505i 450.958 + 597.084i 2814.95 + 2814.95i 93.8068 4094.93i 1546.44 + 1546.44i 8178.59 + 1140.32i
19.11 −13.7485 + 8.18409i 43.2056 + 17.8963i 122.041 225.038i −193.504 + 80.1521i −740.476 + 107.551i −431.718 431.718i 163.849 + 4092.72i 1546.44 + 1546.44i 2004.42 2685.63i
19.12 −12.9519 + 9.39411i 43.2056 + 17.8963i 79.5015 243.342i 125.666 52.0526i −727.713 + 174.087i −1487.78 1487.78i 1256.29 + 3898.58i 1546.44 + 1546.44i −1138.62 + 1854.70i
19.13 −12.6198 + 9.83572i −43.2056 17.8963i 62.5174 248.249i 826.983 342.548i 721.268 199.110i 2310.06 + 2310.06i 1652.75 + 3747.75i 1546.44 + 1546.44i −7067.14 + 12456.8i
19.14 −12.5530 + 9.92074i −43.2056 17.8963i 59.1576 249.071i −504.310 + 208.892i 719.906 203.978i 1931.13 + 1931.13i 1728.36 + 3713.49i 1546.44 + 1546.44i 4258.26 7625.36i
19.15 −12.2933 + 10.2409i −43.2056 17.8963i 46.2485 251.788i 329.017 136.283i 714.412 222.459i −2609.12 2609.12i 2009.98 + 3568.92i 1546.44 + 1546.44i −2649.03 + 5044.79i
19.16 −12.0641 10.5098i 43.2056 + 17.8963i 35.0864 + 253.584i −790.406 + 327.397i −333.150 669.987i −895.524 895.524i 2241.84 3428.02i 1546.44 + 1546.44i 12976.5 + 4357.29i
19.17 −11.3532 11.2741i 43.2056 + 17.8963i 1.79111 + 255.994i 673.833 279.111i −288.758 690.283i −2327.03 2327.03i 2865.75 2926.55i 1546.44 + 1546.44i −10796.9 4428.03i
19.18 −10.9060 11.7072i −43.2056 17.8963i −18.1177 + 255.358i 549.015 227.410i 261.684 + 700.994i 728.365 + 728.365i 3187.12 2572.83i 1546.44 + 1546.44i −8649.90 3947.31i
19.19 −10.3475 12.2036i −43.2056 17.8963i −41.8574 + 252.555i 0.0489834 0.0202896i 228.670 + 712.448i −1476.87 1476.87i 3515.21 2102.51i 1546.44 + 1546.44i −0.754464 0.387828i
19.20 −8.18068 + 13.7505i 43.2056 + 17.8963i −122.153 224.977i −830.774 + 344.118i −599.535 + 447.694i 409.454 + 409.454i 4092.84 + 160.803i 1546.44 + 1546.44i 2064.51 14238.7i
See next 80 embeddings (of 256 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.64
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
32.h odd 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 96.9.m.a 256
32.h odd 8 1 inner 96.9.m.a 256
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
96.9.m.a 256 1.a even 1 1 trivial
96.9.m.a 256 32.h odd 8 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(96, [\chi])\).