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Properties

Label 891.6.a.f
Level $891$
Weight $6$
Character orbit 891.a
Self dual yes
Analytic conductor $142.902$
Analytic rank $1$
Dimension $23$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [891,6,Mod(1,891)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(891, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("891.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 891 = 3^{4} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 891.a (trivial)

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: yes
Analytic conductor: \(142.901983453\)
Analytic rank: \(1\)
Dimension: \(23\)
Twist minimal: no (minimal twist has level 99)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 23 q + 320 q^{4} + 36 q^{5} - 167 q^{7} + 213 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 23 q + 320 q^{4} + 36 q^{5} - 167 q^{7} + 213 q^{8} - 600 q^{10} - 2783 q^{11} - 1871 q^{13} + 1329 q^{14} + 3584 q^{16} + 267 q^{17} - 3641 q^{19} + 1917 q^{20} + 8292 q^{23} + 10049 q^{25} + 9570 q^{26} + 3793 q^{28} + 5970 q^{29} - 9542 q^{31} + 3831 q^{32} - 2982 q^{34} + 3240 q^{35} - 16007 q^{37} - 1221 q^{38} - 40635 q^{40} - 12030 q^{41} - 25943 q^{43} - 38720 q^{44} - 77004 q^{46} - 9756 q^{47} + 6990 q^{49} - 101805 q^{50} - 144446 q^{52} + 53919 q^{53} - 4356 q^{55} + 16602 q^{56} - 95367 q^{58} + 20310 q^{59} - 100247 q^{61} - 15297 q^{62} - 84577 q^{64} - 20931 q^{65} - 84956 q^{67} - 168471 q^{68} - 212292 q^{70} + 36093 q^{71} - 173444 q^{73} - 86619 q^{74} - 340334 q^{76} + 20207 q^{77} - 123113 q^{79} - 15123 q^{80} - 199983 q^{82} - 30672 q^{83} - 268335 q^{85} + 211260 q^{86} - 25773 q^{88} + 32514 q^{89} - 328021 q^{91} + 196731 q^{92} - 230262 q^{94} + 325926 q^{95} - 357002 q^{97} - 214464 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −10.2329 0 72.7122 −40.3523 0 −223.015 −416.604 0 412.921
1.2 −9.97631 0 67.5268 106.587 0 112.524 −354.426 0 −1063.34
1.3 −9.45471 0 57.3915 26.8297 0 −2.53640 −240.069 0 −253.667
1.4 −8.43505 0 39.1501 −9.84195 0 100.946 −60.3116 0 83.0173
1.5 −6.68259 0 12.6571 −79.9903 0 156.773 129.261 0 534.543
1.6 −6.63730 0 12.0538 −35.4703 0 −101.899 132.389 0 235.427
1.7 −5.85920 0 2.33021 53.6372 0 −25.4659 173.841 0 −314.271
1.8 −3.87267 0 −17.0024 −87.2369 0 −140.451 189.770 0 337.840
1.9 −2.59705 0 −25.2554 1.42265 0 160.104 148.695 0 −3.69468
1.10 −2.40104 0 −26.2350 105.603 0 −208.805 139.825 0 −253.557
1.11 −2.38380 0 −26.3175 81.0633 0 5.57865 139.017 0 −193.239
1.12 −0.204234 0 −31.9583 −18.4559 0 −133.899 13.0625 0 3.76934
1.13 1.53916 0 −29.6310 17.6583 0 165.539 −94.8598 0 27.1789
1.14 1.81826 0 −28.6939 −96.6452 0 −6.20964 −110.357 0 −175.726
1.15 3.39140 0 −20.4984 −25.0381 0 −249.837 −178.043 0 −84.9142
1.16 4.33614 0 −13.1979 −59.3925 0 60.8126 −195.984 0 −257.534
1.17 5.15043 0 −5.47303 72.2685 0 93.7551 −193.002 0 372.214
1.18 6.18511 0 6.25560 52.8876 0 22.4230 −159.232 0 327.116
1.19 7.48876 0 24.0815 −5.07543 0 170.013 −59.2996 0 −38.0087
1.20 9.17326 0 52.1486 52.6232 0 −114.930 184.828 0 482.726
See all 23 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.23
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(11\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 891.6.a.f 23
3.b odd 2 1 891.6.a.e 23
9.c even 3 2 99.6.e.a ✓ 46
9.d odd 6 2 297.6.e.a 46
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.6.e.a ✓ 46 9.c even 3 2
297.6.e.a 46 9.d odd 6 2
891.6.a.e 23 3.b odd 2 1
891.6.a.f 23 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{23} - 528 T_{2}^{21} - 71 T_{2}^{20} + 119040 T_{2}^{19} + 31269 T_{2}^{18} + \cdots - 21\!\cdots\!40 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(891))\). Copy content Toggle raw display