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Properties

Label 91.10.g.a
Level $91$
Weight $10$
Character orbit 91.g
Analytic conductor $46.868$
Analytic rank $0$
Dimension $164$
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] = [91,10,Mod(9,91)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(91, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 4]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("91.9");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 91.g (of order \(3\), degree \(2\), 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: \(46.8682610909\)
Analytic rank: \(0\)
Dimension: \(164\)
Relative dimension: \(82\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 164 q + q^{2} - 326 q^{3} - 20479 q^{4} - 2 q^{5} - 1026 q^{6} + 113 q^{7} - 1032 q^{8} + 1025210 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 164 q + q^{2} - 326 q^{3} - 20479 q^{4} - 2 q^{5} - 1026 q^{6} + 113 q^{7} - 1032 q^{8} + 1025210 q^{9} - 77954 q^{10} + 118570 q^{11} + 44088 q^{12} + 232454 q^{13} + 17108 q^{14} - 407531 q^{15} - 5248827 q^{16} + 697543 q^{17} - 76575 q^{18} - 2415090 q^{19} + 713618 q^{20} + 1790137 q^{21} + 199216 q^{22} - 1074031 q^{23} - 6871664 q^{24} - 29660780 q^{25} + 1438932 q^{26} - 9655538 q^{27} + 1986426 q^{28} + 996812 q^{29} - 9821348 q^{30} + 7678612 q^{31} + 6792183 q^{32} + 6164830 q^{33} + 32705872 q^{34} - 25367458 q^{35} - 134627249 q^{36} + 12618565 q^{37} + 25519874 q^{38} + 26474816 q^{39} + 45727182 q^{40} - 13667582 q^{41} + 182796702 q^{42} + 26867814 q^{43} - 75673614 q^{44} - 14796775 q^{45} + 35446678 q^{46} - 23074040 q^{47} + 29166348 q^{48} + 106344145 q^{49} + 77715962 q^{50} - 94065066 q^{51} - 528916 q^{52} + 46529776 q^{53} + 7237275 q^{54} + 218800681 q^{55} + 104202771 q^{56} - 143710238 q^{57} - 2285402 q^{58} - 219565615 q^{59} - 557690411 q^{60} - 1017389486 q^{61} + 307731320 q^{62} + 10604096 q^{63} + 1713678376 q^{64} + 312379252 q^{65} + 278425259 q^{66} - 74837910 q^{67} + 645242415 q^{68} + 290468024 q^{69} - 182370322 q^{70} - 415622020 q^{71} - 1441664786 q^{72} + 514120432 q^{73} - 64494881 q^{74} - 120674222 q^{75} + 985961470 q^{76} - 1189530907 q^{77} + 2167273673 q^{78} - 376448286 q^{79} + 964789424 q^{80} + 6262613948 q^{81} + 1943122144 q^{82} + 971157612 q^{83} - 4584540912 q^{84} + 538436691 q^{85} + 1470174876 q^{86} + 456274843 q^{87} + 1259346774 q^{88} - 349701971 q^{89} - 8083448244 q^{90} - 1357541371 q^{91} + 2309185554 q^{92} - 410352447 q^{93} - 838713772 q^{94} + 2641623 q^{95} - 4707829686 q^{96} + 1339771120 q^{97} + 2454784069 q^{98} + 1482300132 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1 −22.4007 − 38.7992i 112.724 −747.586 + 1294.86i −713.691 + 1236.15i −2525.11 − 4373.62i 5091.63 − 3798.54i 44047.5 −6976.19 63948.8
9.2 −21.2434 − 36.7947i −258.040 −646.568 + 1119.89i 1072.79 − 1858.13i 5481.67 + 9494.52i −865.905 − 6293.16i 33188.0 46901.8 −91159.0
9.3 −21.2245 − 36.7619i 212.909 −644.956 + 1117.10i 466.430 − 807.881i −4518.89 − 7826.95i −6349.41 − 196.356i 33021.6 25647.4 −39598.9
9.4 −21.1158 − 36.5737i −87.9870 −635.758 + 1101.16i −364.214 + 630.837i 1857.92 + 3218.01i −6070.85 + 1870.39i 32075.6 −11941.3 30762.7
9.5 −20.9348 − 36.2601i −184.365 −620.528 + 1074.79i −486.133 + 842.007i 3859.64 + 6685.09i 4600.30 + 4380.74i 30525.3 14307.4 40708.3
9.6 −20.1675 − 34.9312i 69.8993 −557.458 + 965.545i 1239.05 − 2146.10i −1409.70 − 2441.66i 6207.57 − 1348.94i 24318.6 −14797.1 −99954.3
9.7 −19.5414 − 33.8467i −75.1493 −507.733 + 879.419i 963.563 − 1668.94i 1468.52 + 2543.56i −2401.96 + 5880.83i 19676.9 −14035.6 −75317.5
9.8 −19.1647 − 33.1942i −54.0849 −478.571 + 828.910i −125.252 + 216.944i 1036.52 + 1795.31i −580.546 − 6325.87i 17062.1 −16757.8 9601.70
9.9 −18.8614 − 32.6689i 130.928 −455.506 + 788.960i 100.333 − 173.782i −2469.49 − 4277.28i −502.173 + 6332.57i 15051.9 −2540.88 −7569.68
9.10 −18.4481 − 31.9531i 262.246 −424.666 + 735.543i −282.496 + 489.297i −4837.94 − 8379.56i 3924.01 + 4995.57i 12446.3 49089.8 20846.1
9.11 −17.5926 − 30.4713i 95.9820 −363.000 + 628.734i −1292.25 + 2238.24i −1688.57 − 2924.70i −6340.28 − 393.011i 7529.64 −10470.5 90936.0
9.12 −16.9313 − 29.3259i −230.014 −317.338 + 549.646i −1103.55 + 1911.40i 3894.43 + 6745.35i 2711.31 − 5744.77i 4154.14 33223.3 74738.1
9.13 −16.8358 − 29.1605i −79.4436 −310.891 + 538.480i 30.2571 − 52.4068i 1337.50 + 2316.62i −3465.48 − 5323.91i 3696.57 −13371.7 −2037.61
9.14 −16.4825 − 28.5485i 174.350 −287.345 + 497.696i 707.045 − 1224.64i −2873.72 − 4977.43i −2983.54 − 5608.22i 2066.58 10714.9 −46615.4
9.15 −16.0520 − 27.8030i −24.7851 −259.336 + 449.183i 133.374 − 231.010i 397.851 + 689.098i 6284.64 − 925.725i 214.212 −19068.7 −8563.70
9.16 −15.7462 − 27.2732i 54.6994 −239.883 + 415.490i −1015.07 + 1758.16i −861.306 − 1491.83i 4165.71 + 4795.88i −1015.11 −16691.0 63934.0
9.17 −14.6376 − 25.3531i 219.288 −172.519 + 298.812i −669.989 + 1160.45i −3209.85 − 5559.63i 2761.73 − 5720.70i −4887.85 28404.3 39228.1
9.18 −14.4815 − 25.0827i −201.094 −163.428 + 283.066i −943.098 + 1633.49i 2912.14 + 5043.97i −4745.27 + 4223.27i −5362.32 20755.6 54629.9
9.19 −13.8425 − 23.9759i −145.671 −127.230 + 220.368i 517.524 − 896.377i 2016.46 + 3492.60i 6211.06 − 1332.81i −7130.01 1537.12 −28655.3
9.20 −13.4961 − 23.3759i −231.386 −108.290 + 187.563i 693.566 − 1201.29i 3122.82 + 5408.88i 2818.33 + 5693.04i −7974.05 33856.7 −37441.8
See next 80 embeddings (of 164 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.82
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.g even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 91.10.g.a ✓ 164
7.c even 3 1 91.10.h.a yes 164
13.c even 3 1 91.10.h.a yes 164
91.g even 3 1 inner 91.10.g.a ✓ 164
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.10.g.a ✓ 164 1.a even 1 1 trivial
91.10.g.a ✓ 164 91.g even 3 1 inner
91.10.h.a yes 164 7.c even 3 1
91.10.h.a yes 164 13.c even 3 1

Hecke kernels

This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(91, [\chi])\).