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Properties

Label 731.4.a.b
Level $731$
Weight $4$
Character orbit 731.a
Self dual yes
Analytic conductor $43.130$
Analytic rank $1$
Dimension $37$
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] = [731,4,Mod(1,731)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(731, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("731.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 731 = 17 \cdot 43 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 731.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: \(43.1303962142\)
Analytic rank: \(1\)
Dimension: \(37\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 37 q - 4 q^{2} - 16 q^{3} + 120 q^{4} - 64 q^{5} - 21 q^{6} - 74 q^{7} - 84 q^{8} + 227 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 37 q - 4 q^{2} - 16 q^{3} + 120 q^{4} - 64 q^{5} - 21 q^{6} - 74 q^{7} - 84 q^{8} + 227 q^{9} - 158 q^{10} - 52 q^{11} - 51 q^{12} - 216 q^{13} - 177 q^{14} - 18 q^{15} + 248 q^{16} + 629 q^{17} - 12 q^{18} - 334 q^{19} - 283 q^{20} - 654 q^{21} - 62 q^{22} - 338 q^{23} - 9 q^{24} + 761 q^{25} - 267 q^{26} - 748 q^{27} - 955 q^{28} - 1086 q^{29} - 628 q^{30} - 384 q^{31} - 616 q^{32} - 1030 q^{33} - 68 q^{34} - 60 q^{35} + 521 q^{36} - 1680 q^{37} + 313 q^{38} - 1034 q^{39} - 1595 q^{40} - 906 q^{41} - 803 q^{42} - 1591 q^{43} - 1414 q^{44} - 1660 q^{45} - 1828 q^{46} - 670 q^{47} - 752 q^{48} + 281 q^{49} - 1154 q^{50} - 272 q^{51} - 1377 q^{52} - 762 q^{53} - 931 q^{54} + 76 q^{55} - 834 q^{56} + 202 q^{57} - 729 q^{58} - 1366 q^{59} - 1517 q^{60} - 4548 q^{61} - 3069 q^{62} - 176 q^{63} - 1906 q^{64} - 2214 q^{65} - 1699 q^{66} + 470 q^{67} + 2040 q^{68} - 2634 q^{69} - 465 q^{70} - 30 q^{71} - 3012 q^{72} - 2878 q^{73} - 1228 q^{74} - 1294 q^{75} - 5097 q^{76} - 2826 q^{77} - 4720 q^{78} - 3334 q^{79} - 4778 q^{80} - 1015 q^{81} - 1223 q^{82} + 232 q^{83} - 5937 q^{84} - 1088 q^{85} + 172 q^{86} + 298 q^{87} - 4424 q^{88} - 3246 q^{89} - 8473 q^{90} - 3330 q^{91} - 7009 q^{92} - 5768 q^{93} - 4291 q^{94} - 2678 q^{95} - 1083 q^{96} - 1416 q^{97} + 2844 q^{98} - 924 q^{99}+O(q^{100}) \) 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 −5.30505 3.34990 20.1436 −3.54575 −17.7714 16.0851 −64.4224 −15.7782 18.8104
1.2 −5.20183 −4.41466 19.0590 −7.41657 22.9643 −31.5086 −57.5271 −7.51076 38.5797
1.3 −5.10172 −4.93980 18.0275 7.95773 25.2014 6.82384 −51.1575 −2.59841 −40.5981
1.4 −4.94224 7.24458 16.4258 16.7459 −35.8045 −19.2709 −41.6421 25.4840 −82.7624
1.5 −4.42561 −9.41763 11.5861 −11.2063 41.6788 6.09512 −15.8705 61.6918 49.5947
1.6 −4.26913 2.50867 10.2254 7.08171 −10.7098 −11.7339 −9.50069 −20.7066 −30.2327
1.7 −3.96569 9.32086 7.72666 −11.1203 −36.9636 5.12681 1.08396 59.8784 44.0998
1.8 −3.63273 −3.03123 5.19671 −7.06829 11.0116 12.8241 10.1836 −17.8116 25.6772
1.9 −3.56841 2.90448 4.73352 −20.8039 −10.3644 1.69242 11.6561 −18.5640 74.2368
1.10 −3.36871 −9.05833 3.34821 20.1944 30.5149 26.1440 15.6705 55.0533 −68.0292
1.11 −2.55430 0.389981 −1.47556 8.43087 −0.996127 30.1169 24.2034 −26.8479 −21.5350
1.12 −2.50065 2.28546 −1.74673 18.7401 −5.71516 −20.4219 24.3732 −21.7767 −46.8624
1.13 −2.41123 −0.646145 −2.18595 −18.8140 1.55801 −27.0297 24.5607 −26.5825 45.3650
1.14 −1.32526 7.18446 −6.24369 4.80719 −9.52126 4.75998 18.8766 24.6165 −6.37077
1.15 −1.25085 6.64508 −6.43537 −1.59238 −8.31201 −22.2515 18.0565 17.1571 1.99183
1.16 −1.14747 −6.36366 −6.68332 11.1956 7.30210 −19.5471 16.8486 13.4962 −12.8466
1.17 −0.686786 −4.75390 −7.52832 −8.64111 3.26491 −19.3151 10.6646 −4.40046 5.93460
1.18 −0.518920 −5.52741 −7.73072 −13.9011 2.86828 29.0294 8.16298 3.55231 7.21357
1.19 −0.202223 3.70747 −7.95911 −0.838774 −0.749735 15.7752 3.22730 −13.2547 0.169619
1.20 0.149313 −3.11026 −7.97771 −12.8807 −0.464402 0.941422 −2.38567 −17.3263 −1.92325
See all 37 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.37
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(17\) \(-1\)
\(43\) \(1\)

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 731.4.a.b ✓ 37
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
731.4.a.b ✓ 37 1.a even 1 1 trivial