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Properties

Label 91.11.t.a
Level $91$
Weight $11$
Character orbit 91.t
Analytic conductor $57.818$
Analytic rank $0$
Dimension $184$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [91,11,Mod(62,91)]
 
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([3, 5]))
 
N = Newforms(chi, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("91.62");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 91.t (of order \(6\), 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: \(57.8175099933\)
Analytic rank: \(0\)
Dimension: \(184\)
Relative dimension: \(92\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 184 q - 6 q^{2} + 47102 q^{4} + 25536 q^{7} + 1811178 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 184 q - 6 q^{2} + 47102 q^{4} + 25536 q^{7} + 1811178 q^{9} - 48126 q^{11} + 1283876 q^{14} - 1385550 q^{15} - 25576682 q^{16} - 8133630 q^{22} - 5480968 q^{23} + 342975860 q^{25} + 108259200 q^{28} - 4968710 q^{29} - 76434328 q^{30} + 6291450 q^{32} + 66195156 q^{35} - 854321884 q^{36} + 86623056 q^{37} - 285817068 q^{39} - 52409430 q^{42} + 254978266 q^{43} + 211133496 q^{46} - 144319414 q^{49} + 1904662866 q^{50} + 2153691004 q^{51} + 886096864 q^{53} + 2199144978 q^{56} + 126043374 q^{58} + 5893009974 q^{63} - 27152051968 q^{64} - 884979422 q^{65} + 8098632972 q^{67} - 4201602180 q^{71} + 4275296250 q^{72} + 6045727536 q^{74} + 4990323640 q^{77} + 13850997130 q^{78} + 6539724272 q^{79} - 26894778780 q^{81} + 1234521894 q^{84} - 27714259314 q^{85} + 8842027002 q^{88} - 12218486364 q^{91} + 70727933144 q^{92} + 29083152426 q^{93} - 32499914946 q^{95} + 30930836538 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
62.1 −54.5984 + 31.5224i −90.8612 + 52.4587i 1475.32 − 2555.34i −4854.56 3307.25 − 5728.33i −16259.2 + 4255.86i 121465.i −24020.7 + 41605.0i 265051. − 153027.i
62.2 −54.5984 + 31.5224i 90.8612 − 52.4587i 1475.32 − 2555.34i 4854.56 −3307.25 + 5728.33i −4443.94 − 16208.8i 121465.i −24020.7 + 41605.0i −265051. + 153027.i
62.3 −52.3451 + 30.2214i −252.766 + 145.934i 1314.67 − 2277.08i 3058.66 8820.69 − 15277.9i 6983.98 + 15287.2i 97031.7i 13069.2 − 22636.4i −160106. + 92437.1i
62.4 −52.3451 + 30.2214i 252.766 − 145.934i 1314.67 − 2277.08i −3058.66 −8820.69 + 15277.9i 16731.1 − 1595.30i 97031.7i 13069.2 − 22636.4i 160106. − 92437.1i
62.5 −50.5935 + 29.2102i −383.200 + 221.241i 1194.47 − 2068.88i 1674.45 12925.0 − 22386.7i 3095.86 − 16519.4i 79739.8i 68370.5 − 118421.i −84716.4 + 48911.0i
62.6 −50.5935 + 29.2102i 383.200 − 221.241i 1194.47 − 2068.88i −1674.45 −12925.0 + 22386.7i −12758.3 + 10940.8i 79739.8i 68370.5 − 118421.i 84716.4 − 48911.0i
62.7 −48.2644 + 27.8654i −77.7522 + 44.8903i 1040.97 − 1803.00i −1966.87 2501.77 − 4333.20i 16427.9 − 3549.49i 58959.4i −25494.2 + 44157.3i 94929.6 − 54807.6i
62.8 −48.2644 + 27.8654i 77.7522 − 44.8903i 1040.97 − 1803.00i 1966.87 −2501.77 + 4333.20i 5140.01 + 16001.7i 58959.4i −25494.2 + 44157.3i −94929.6 + 54807.6i
62.9 −46.2649 + 26.7111i −280.010 + 161.664i 914.961 − 1584.76i −650.485 8636.43 − 14958.7i −15336.0 − 6876.13i 43054.1i 22746.0 − 39397.2i 30094.6 − 17375.1i
62.10 −46.2649 + 26.7111i 280.010 − 161.664i 914.961 − 1584.76i 650.485 −8636.43 + 14958.7i −13622.9 − 9843.33i 43054.1i 22746.0 − 39397.2i −30094.6 + 17375.1i
62.11 −45.5235 + 26.2830i −324.247 + 187.204i 869.594 − 1506.18i −2587.82 9840.57 − 17044.4i 3752.25 + 16382.8i 37594.6i 40566.1 − 70262.5i 117807. − 68015.7i
62.12 −45.5235 + 26.2830i 324.247 − 187.204i 869.594 − 1506.18i 2587.82 −9840.57 + 17044.4i 16064.0 − 4941.85i 37594.6i 40566.1 − 70262.5i −117807. + 68015.7i
62.13 −43.3845 + 25.0480i −8.06004 + 4.65346i 742.808 − 1286.58i 3448.47 233.120 − 403.776i −14646.9 + 8242.81i 23125.2i −29481.2 + 51062.9i −149610. + 86377.4i
62.14 −43.3845 + 25.0480i 8.06004 − 4.65346i 742.808 − 1286.58i −3448.47 −233.120 + 403.776i −184.956 − 16806.0i 23125.2i −29481.2 + 51062.9i 149610. − 86377.4i
62.15 −39.8414 + 23.0024i −135.543 + 78.2557i 546.222 − 946.085i 3067.35 3600.14 − 6235.63i 10053.0 − 13468.9i 3148.78i −17276.6 + 29923.9i −122207. + 70556.4i
62.16 −39.8414 + 23.0024i 135.543 − 78.2557i 546.222 − 946.085i −3067.35 −3600.14 + 6235.63i −6637.90 + 15440.6i 3148.78i −17276.6 + 29923.9i 122207. − 70556.4i
62.17 −38.4412 + 22.1940i −322.678 + 186.298i 473.150 − 819.521i −5575.61 8269.42 − 14323.1i 16290.3 − 4135.34i − 3448.91i 39889.6 − 69090.9i 214333. − 123745.i
62.18 −38.4412 + 22.1940i 322.678 − 186.298i 473.150 − 819.521i 5575.61 −8269.42 + 14323.1i 4563.85 + 16175.5i − 3448.91i 39889.6 − 69090.9i −214333. + 123745.i
62.19 −36.4610 + 21.0508i −237.637 + 137.200i 374.269 − 648.252i 2917.77 5776.32 − 10004.9i −16807.0 + 31.3034i − 11597.4i 8123.12 − 14069.7i −106385. + 61421.3i
62.20 −36.4610 + 21.0508i 237.637 − 137.200i 374.269 − 648.252i −2917.77 −5776.32 + 10004.9i −8376.38 − 14570.9i − 11597.4i 8123.12 − 14069.7i 106385. − 61421.3i
See next 80 embeddings (of 184 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 62.92
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
13.e even 6 1 inner
91.t odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 91.11.t.a ✓ 184
7.b odd 2 1 inner 91.11.t.a ✓ 184
13.e even 6 1 inner 91.11.t.a ✓ 184
91.t odd 6 1 inner 91.11.t.a ✓ 184
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.11.t.a ✓ 184 1.a even 1 1 trivial
91.11.t.a ✓ 184 7.b odd 2 1 inner
91.11.t.a ✓ 184 13.e even 6 1 inner
91.11.t.a ✓ 184 91.t odd 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{11}^{\mathrm{new}}(91, [\chi])\).