↑

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 $1$

Related objects

Downloads

Learn more

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 algebraic \(q\)-expansion of this newform has not been computed, 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