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Properties

Label 731.4.a.c
Level $731$
Weight $4$
Character orbit 731.a
Self dual yes
Analytic conductor $43.130$
Analytic rank $0$
Dimension $47$
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: \(0\)
Dimension: \(47\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 47 q + 4 q^{2} + 2 q^{3} + 216 q^{4} - 14 q^{5} + 39 q^{6} + 58 q^{7} + 120 q^{8} + 497 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 47 q + 4 q^{2} + 2 q^{3} + 216 q^{4} - 14 q^{5} + 39 q^{6} + 58 q^{7} + 120 q^{8} + 497 q^{9} + 100 q^{10} + 58 q^{11} + 255 q^{12} + 216 q^{13} - 9 q^{14} + 114 q^{15} + 1096 q^{16} - 799 q^{17} + 548 q^{18} + 266 q^{19} + 117 q^{20} + 938 q^{21} + 730 q^{22} - 158 q^{23} + 473 q^{24} + 1879 q^{25} + 401 q^{26} + 392 q^{27} + 445 q^{28} + 592 q^{29} + 436 q^{30} + 20 q^{31} + 844 q^{32} + 742 q^{33} - 68 q^{34} + 360 q^{35} + 1973 q^{36} + 1058 q^{37} - 151 q^{38} - 6 q^{39} + 465 q^{40} + 790 q^{41} + 2097 q^{42} - 2021 q^{43} + 1520 q^{44} + 590 q^{45} + 2564 q^{46} - 434 q^{47} + 1984 q^{48} + 4335 q^{49} + 178 q^{50} - 34 q^{51} + 1791 q^{52} + 2222 q^{53} + 1229 q^{54} + 3596 q^{55} - 1548 q^{56} + 2650 q^{57} + 3331 q^{58} - 550 q^{59} + 3439 q^{60} + 3278 q^{61} - 1321 q^{62} + 3748 q^{63} + 8278 q^{64} + 2306 q^{65} + 1945 q^{66} + 874 q^{67} - 3672 q^{68} + 2334 q^{69} + 7827 q^{70} + 1682 q^{71} + 5396 q^{72} + 230 q^{73} + 2106 q^{74} + 272 q^{75} + 2071 q^{76} + 3086 q^{77} - 1456 q^{78} + 3718 q^{79} + 320 q^{80} + 7015 q^{81} + 1367 q^{82} + 4536 q^{83} + 10291 q^{84} + 238 q^{85} - 172 q^{86} + 826 q^{87} + 5080 q^{88} + 1682 q^{89} + 257 q^{90} + 9106 q^{91} - 2951 q^{92} + 3416 q^{93} + 1073 q^{94} - 1858 q^{95} - 1097 q^{96} + 1688 q^{97} - 4 q^{98} + 8122 q^{99}+O(q^{100}) \) 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 −5.34906 2.69317 20.6124 −19.7228 −14.4059 9.07046 −67.4646 −19.7468 105.498
1.2 −5.32700 9.06484 20.3769 10.6670 −48.2884 25.7817 −65.9319 55.1713 −56.8229
1.3 −5.31092 −3.67223 20.2059 −4.01671 19.5029 25.1576 −64.8243 −13.5147 21.3324
1.4 −5.17936 −6.64126 18.8257 2.89649 34.3974 −17.4021 −56.0704 17.1063 −15.0019
1.5 −4.57828 1.92351 12.9607 −12.0211 −8.80636 −1.20710 −22.7114 −23.3001 55.0361
1.6 −4.46587 6.31763 11.9440 4.44083 −28.2137 −31.8334 −17.6134 12.9124 −19.8322
1.7 −4.46368 −1.66954 11.9244 18.6424 7.45230 25.0182 −17.5174 −24.2126 −83.2138
1.8 −4.29641 −8.44212 10.4592 −11.3947 36.2709 −11.3763 −10.5656 44.2694 48.9564
1.9 −4.28557 −3.94953 10.3661 21.5115 16.9260 −21.3995 −10.1401 −11.4012 −92.1888
1.10 −3.68242 0.237304 5.56024 1.72295 −0.873852 3.59163 8.98423 −26.9437 −6.34463
1.11 −3.41219 6.64518 3.64307 −14.4420 −22.6746 −35.7030 14.8667 17.1584 49.2790
1.12 −2.74559 6.60458 −0.461742 18.3894 −18.1335 1.89520 23.2325 16.6205 −50.4898
1.13 −2.70204 4.36125 −0.698991 −16.9120 −11.7843 27.4083 23.5050 −7.97950 45.6968
1.14 −2.53468 −1.83762 −1.57540 −3.88604 4.65778 −6.64757 24.2706 −23.6232 9.84986
1.15 −2.38890 −1.99706 −2.29317 −2.92267 4.77077 −8.78012 24.5893 −23.0117 6.98196
1.16 −2.28992 −9.14488 −2.75625 −11.6015 20.9411 10.2654 24.6310 56.6288 26.5665
1.17 −2.23215 −7.54937 −3.01752 −21.1497 16.8513 29.1903 24.5927 29.9930 47.2091
1.18 −2.01765 9.89595 −3.92909 −8.38047 −19.9665 32.5851 24.0687 70.9297 16.9088
1.19 −1.20661 −9.51690 −6.54409 6.44305 11.4832 −16.0902 17.5491 63.5715 −7.77426
1.20 −1.02914 −8.14141 −6.94088 17.5538 8.37862 −10.4412 15.3762 39.2825 −18.0653
See all 47 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.47
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.c ✓ 47
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
731.4.a.c ✓ 47 1.a even 1 1 trivial