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Properties

Label 76.8.i.a
Level $76$
Weight $8$
Character orbit 76.i
Analytic conductor $23.741$
Analytic rank $0$
Dimension $72$
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] = [76,8,Mod(5,76)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(76, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 16]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("76.5");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 76.i (of order \(9\), degree \(6\), 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: \(23.7412619368\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(12\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 72 q - 39 q^{3} + 591 q^{7} - 1677 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q - 39 q^{3} + 591 q^{7} - 1677 q^{9} - 5793 q^{11} + 22083 q^{13} - 70707 q^{15} + 29409 q^{17} + 12756 q^{19} - 40419 q^{21} - 116796 q^{23} + 26934 q^{25} + 439347 q^{27} - 148005 q^{29} + 141366 q^{31} + 1206654 q^{33} - 531651 q^{35} - 1508004 q^{37} - 2140644 q^{39} + 494157 q^{41} + 923511 q^{43} + 1790028 q^{45} - 662955 q^{47} - 4191369 q^{49} + 3753363 q^{51} - 246513 q^{53} + 2650185 q^{55} + 5831286 q^{57} - 2244432 q^{59} - 10302306 q^{61} - 12102117 q^{63} + 3985635 q^{65} + 11688288 q^{67} + 12632535 q^{69} + 10541736 q^{71} + 560730 q^{73} - 17718750 q^{75} - 22266192 q^{77} + 9424767 q^{79} + 16585437 q^{81} + 2359839 q^{83} - 20859762 q^{85} + 2348574 q^{87} - 4397130 q^{89} - 12653112 q^{91} + 52259766 q^{93} + 46248669 q^{95} - 34112766 q^{97} - 93757926 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 0 −14.6146 + 82.8834i 0 −59.8786 − 50.2441i 0 603.312 + 1044.97i 0 −4600.96 − 1674.61i 0
5.2 0 −14.4533 + 81.9690i 0 267.986 + 224.867i 0 −728.602 − 1261.98i 0 −4454.90 − 1621.45i 0
5.3 0 −8.77591 + 49.7706i 0 −247.952 − 208.056i 0 12.6981 + 21.9937i 0 −344.992 − 125.567i 0
5.4 0 −7.07386 + 40.1179i 0 −209.079 − 175.438i 0 −296.174 − 512.988i 0 495.705 + 180.422i 0
5.5 0 −6.04978 + 34.3100i 0 311.028 + 260.984i 0 682.442 + 1182.02i 0 914.532 + 332.862i 0
5.6 0 −5.17297 + 29.3374i 0 80.9108 + 67.8923i 0 −461.043 − 798.550i 0 1221.19 + 444.476i 0
5.7 0 1.71959 − 9.75230i 0 −215.225 − 180.596i 0 406.860 + 704.703i 0 1962.96 + 714.458i 0
5.8 0 3.98192 − 22.5826i 0 279.342 + 234.395i 0 −171.377 − 296.833i 0 1560.99 + 568.154i 0
5.9 0 5.48485 − 31.1062i 0 84.7947 + 71.1512i 0 −132.049 − 228.716i 0 1117.60 + 406.773i 0
5.10 0 10.1867 − 57.7718i 0 −349.972 − 293.661i 0 −759.486 − 1315.47i 0 −1178.70 − 429.013i 0
5.11 0 11.5723 − 65.6297i 0 −139.868 − 117.363i 0 803.700 + 1392.05i 0 −2118.24 − 770.975i 0
5.12 0 14.2640 − 80.8950i 0 197.147 + 165.426i 0 −125.442 − 217.272i 0 −4285.43 − 1559.77i 0
9.1 0 −62.6763 + 52.5916i 0 458.247 + 166.788i 0 128.407 − 222.408i 0 782.666 − 4438.72i 0
9.2 0 −60.5402 + 50.7993i 0 −295.109 − 107.411i 0 156.140 − 270.443i 0 704.783 − 3997.02i 0
9.3 0 −39.9118 + 33.4899i 0 −63.3488 − 23.0571i 0 546.634 − 946.798i 0 91.6039 − 519.511i 0
9.4 0 −39.6110 + 33.2376i 0 72.0424 + 26.2213i 0 −610.937 + 1058.17i 0 84.5259 − 479.370i 0
9.5 0 −10.5751 + 8.87354i 0 −438.581 − 159.630i 0 −408.557 + 707.641i 0 −346.676 + 1966.10i 0
9.6 0 −5.61353 + 4.71031i 0 101.799 + 37.0519i 0 618.350 − 1071.01i 0 −370.444 + 2100.89i 0
9.7 0 3.99587 − 3.35293i 0 24.8731 + 9.05307i 0 24.2221 − 41.9538i 0 −375.044 + 2126.98i 0
9.8 0 6.91272 − 5.80046i 0 511.939 + 186.330i 0 −283.678 + 491.345i 0 −365.628 + 2073.58i 0
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 76.8.i.a ✓ 72
19.e even 9 1 inner 76.8.i.a ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.8.i.a ✓ 72 1.a even 1 1 trivial
76.8.i.a ✓ 72 19.e even 9 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(76, [\chi])\).