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Properties

Label 731.4.a.a
Level $731$
Weight $4$
Character orbit 731.a
Self dual yes
Analytic conductor $43.130$
Analytic rank $1$
Dimension $35$
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: \(35\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 35 q - 4 q^{2} + 2 q^{3} + 96 q^{4} + 6 q^{5} - 9 q^{6} - 26 q^{7} + 24 q^{8} + 173 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 35 q - 4 q^{2} + 2 q^{3} + 96 q^{4} + 6 q^{5} - 9 q^{6} - 26 q^{7} + 24 q^{8} + 173 q^{9} - 60 q^{10} - 30 q^{11} - 179 q^{12} - 304 q^{13} - 65 q^{14} - 354 q^{15} + 40 q^{16} - 595 q^{17} - 462 q^{18} - 342 q^{19} - 229 q^{20} - 502 q^{21} - 768 q^{22} - 66 q^{23} - 103 q^{24} + 235 q^{25} - 389 q^{26} + 200 q^{27} - 563 q^{28} - 592 q^{29} - 644 q^{30} - 476 q^{31} - 162 q^{32} - 50 q^{33} + 68 q^{34} - 468 q^{35} - 1267 q^{36} - 962 q^{37} - 839 q^{38} - 1062 q^{39} - 1455 q^{40} + 462 q^{41} + 1173 q^{42} + 1505 q^{43} + 520 q^{44} + 230 q^{45} - 1784 q^{46} - 998 q^{47} - 1488 q^{48} - 657 q^{49} - 2198 q^{50} - 34 q^{51} - 1865 q^{52} - 2174 q^{53} - 639 q^{54} - 2576 q^{55} - 1648 q^{56} - 1934 q^{57} + 1475 q^{58} - 1710 q^{59} - 1489 q^{60} - 2778 q^{61} - 577 q^{62} - 3480 q^{63} - 554 q^{64} - 1486 q^{65} - 3327 q^{66} - 4594 q^{67} - 1632 q^{68} - 2130 q^{69} - 429 q^{70} - 1770 q^{71} - 1282 q^{72} - 1162 q^{73} - 2814 q^{74} - 628 q^{75} - 2651 q^{76} - 998 q^{77} + 2120 q^{78} - 4182 q^{79} - 450 q^{80} - 1733 q^{81} - 1385 q^{82} - 660 q^{83} - 3341 q^{84} - 102 q^{85} - 172 q^{86} - 4670 q^{87} - 6848 q^{88} - 1678 q^{89} + 3221 q^{90} - 4250 q^{91} - 1847 q^{92} - 764 q^{93} - 1773 q^{94} - 2338 q^{95} - 6473 q^{96} - 1744 q^{97} - 6794 q^{98} - 2282 q^{99}+O(q^{100}) \) 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 −5.52817 1.55013 22.5607 15.0757 −8.56941 −13.9183 −80.4939 −24.5971 −83.3413
1.2 −4.81597 −8.89762 15.1936 1.32778 42.8507 19.2842 −34.6441 52.1676 −6.39453
1.3 −4.56388 −2.96529 12.8290 −16.1586 13.5332 −13.7615 −22.0389 −18.2070 73.7458
1.4 −4.49307 8.16008 12.1877 −17.6221 −36.6638 −9.61199 −18.8155 39.5869 79.1773
1.5 −4.15327 4.77063 9.24967 7.17683 −19.8137 18.5101 −5.19021 −4.24112 −29.8073
1.6 −4.08798 −2.24400 8.71156 −8.42425 9.17341 36.3023 −2.90886 −21.9645 34.4381
1.7 −3.94862 −6.89549 7.59164 10.6432 27.2277 −14.3173 1.61247 20.5478 −42.0258
1.8 −2.99481 4.16748 0.968882 13.3610 −12.4808 −1.48133 21.0569 −9.63214 −40.0135
1.9 −2.98138 −0.796729 0.888603 8.79315 2.37535 −35.4002 21.2017 −26.3652 −26.2157
1.10 −2.96592 5.35634 0.796660 −7.26138 −15.8865 1.91270 21.3645 1.69036 21.5367
1.11 −2.71383 −7.34651 −0.635147 7.80741 19.9371 22.9255 23.4343 26.9712 −21.1880
1.12 −2.27665 −6.00939 −2.81685 −16.7357 13.6813 −20.2194 24.6262 9.11279 38.1013
1.13 −2.27553 10.0613 −2.82198 11.3516 −22.8948 −28.7635 24.6257 74.2305 −25.8310
1.14 −1.56077 −3.56803 −5.56401 13.4369 5.56886 14.1560 21.1702 −14.2692 −20.9718
1.15 −1.26049 0.642857 −6.41116 −5.92104 −0.810316 6.79726 18.1652 −26.5867 7.46343
1.16 −1.08115 8.18877 −6.83113 −16.3176 −8.85325 −14.4389 16.0346 40.0559 17.6417
1.17 −0.418585 6.79113 −7.82479 −1.63304 −2.84267 26.5347 6.62403 19.1194 0.683565
1.18 −0.116663 5.63234 −7.98639 14.1988 −0.657088 −14.4053 1.86503 4.72324 −1.65648
1.19 0.0743375 −1.66345 −7.99447 −18.3312 −0.123657 16.2165 −1.18899 −24.2329 −1.36269
1.20 0.319709 −7.91160 −7.89779 −7.62726 −2.52941 −20.8168 −5.08266 35.5934 −2.43850
See all 35 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.35
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.a ✓ 35
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
731.4.a.a ✓ 35 1.a even 1 1 trivial