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Properties

Label 47.18.a.a
Level $47$
Weight $18$
Character orbit 47.a
Self dual yes
Analytic conductor $86.114$
Analytic rank $1$
Dimension $30$
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] = [47,18,Mod(1,47)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(47, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 18, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("47.1");
 
S:= CuspForms(chi, 18);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 47 \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 47.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: \(86.1143810519\)
Analytic rank: \(1\)
Dimension: \(30\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 30 q + 15 q^{2} - 10640 q^{3} + 1742101 q^{4} - 363048 q^{5} - 4186112 q^{6} - 33054460 q^{7} + 17944905 q^{8} + 1195980492 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q + 15 q^{2} - 10640 q^{3} + 1742101 q^{4} - 363048 q^{5} - 4186112 q^{6} - 33054460 q^{7} + 17944905 q^{8} + 1195980492 q^{9} - 278691840 q^{10} - 169088016 q^{11} - 7615730340 q^{12} - 9892163806 q^{13} - 19845806598 q^{14} - 49941733354 q^{15} + 78517013009 q^{16} - 8192485702 q^{17} + 196350519025 q^{18} - 118652670014 q^{19} + 400555462496 q^{20} - 4336107444 q^{21} - 1166477783182 q^{22} - 419455745930 q^{23} - 3808631018808 q^{24} + 208962326022 q^{25} - 8185801854596 q^{26} - 4295082130388 q^{27} - 12236144123248 q^{28} - 5088782848516 q^{29} - 17400147194980 q^{30} + 3900292348348 q^{31} - 7968408700095 q^{32} - 29800025435404 q^{33} - 610768586022 q^{34} - 12289934334988 q^{35} + 142687526511329 q^{36} - 75514477327582 q^{37} + 44204267098266 q^{38} + 8673851284014 q^{39} + 106012964151600 q^{40} + 11225182312722 q^{41} + 196456667039250 q^{42} - 111620108291116 q^{43} + 355959776151498 q^{44} + 47354145018216 q^{45} + 536639103117424 q^{46} - 714338599852830 q^{47} - 273480576965376 q^{48} + 924050852415332 q^{49} + 13\!\cdots\!57 q^{50}+ \cdots - 88\!\cdots\!90 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −698.519 11228.6 356857. −332430. −7.84337e6 7.32360e6 −1.57715e8 −3.05931e6 2.32208e8
1.2 −637.404 −261.371 275212. −655408. 166599. −2.25308e7 −9.18755e7 −1.29072e8 4.17760e8
1.3 −614.457 −17844.6 246486. 654729. 1.09647e7 1.32755e7 −7.09167e7 1.89289e8 −4.02303e8
1.4 −593.911 15808.6 221659. 736380. −9.38889e6 −2.05413e7 −5.38003e7 1.20771e8 −4.37344e8
1.5 −527.116 −6663.51 146780. −403259. 3.51245e6 2.97111e7 −8.27978e6 −8.47378e7 2.12565e8
1.6 −491.662 −21139.3 110659. 237213. 1.03934e7 523219. 1.00362e7 3.17730e8 −1.16628e8
1.7 −428.633 −11521.4 52653.9 −131822. 4.93843e6 −2.70260e7 3.36125e7 3.60153e6 5.65031e7
1.8 −392.867 6574.20 23272.6 1.40322e6 −2.58279e6 −5.44313e6 4.23508e7 −8.59201e7 −5.51278e8
1.9 −320.745 14811.2 −28194.6 353082. −4.75063e6 2.36341e7 5.10840e7 9.02324e7 −1.13249e8
1.10 −302.929 12287.8 −39306.2 −80066.8 −3.72234e6 3.63096e6 5.16124e7 2.18507e7 2.42545e7
1.11 −271.439 −11876.2 −57392.9 1.27904e6 3.22368e6 1.35158e7 5.11567e7 1.19051e7 −3.47181e8
1.12 −223.697 4725.27 −81031.5 −1.50349e6 −1.05703e6 −2.25663e7 4.74470e7 −1.06812e8 3.36327e8
1.13 −170.617 −3920.46 −101962. −1.42936e6 668896. 1.62424e7 3.97595e7 −1.13770e8 2.43873e8
1.14 −8.48237 2.69064 −131000. 454983. −22.8230 −1.16050e7 2.22299e6 −1.29140e8 −3.85933e6
1.15 0.895458 −9298.18 −131071. 166558. −8326.13 −1.43463e7 −234738. −4.26841e7 149146.
1.16 43.4437 18457.2 −129185. −1.47964e6 801851. 2.20266e7 −1.13065e7 2.11529e8 −6.42813e7
1.17 51.0679 17449.1 −128464. −151773. 891092. −405036. −1.32540e7 1.75333e8 −7.75075e6
1.18 102.489 −17606.5 −120568. −959317. −1.80446e6 −1.65642e7 −2.57902e7 1.80848e8 −9.83190e7
1.19 170.539 −8766.11 −101988. 1.32844e6 −1.49496e6 1.20608e7 −3.97459e7 −5.22954e7 2.26551e8
1.20 251.709 −8738.54 −67714.4 −1.19791e6 −2.19957e6 8.79560e6 −5.00364e7 −5.27780e7 −3.01524e8
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.30
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(47\) \(1\)

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 47.18.a.a ✓ 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
47.18.a.a ✓ 30 1.a even 1 1 trivial