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Properties

Label 731.6.a.a
Level $731$
Weight $6$
Character orbit 731.a
Self dual yes
Analytic conductor $117.241$
Analytic rank $1$
Dimension $63$
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] = [731,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("731.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 731 = 17 \cdot 43 \)
Weight: \( k \) \(=\) \( 6 \)
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: \(117.240572283\)
Analytic rank: \(1\)
Dimension: \(63\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 63 q - 8 q^{2} - 55 q^{3} + 832 q^{4} - 271 q^{5} - 417 q^{6} - 433 q^{7} + 36 q^{8} + 3938 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 63 q - 8 q^{2} - 55 q^{3} + 832 q^{4} - 271 q^{5} - 417 q^{6} - 433 q^{7} + 36 q^{8} + 3938 q^{9} - 438 q^{10} - 1704 q^{11} - 3767 q^{12} - 3646 q^{13} - 2401 q^{14} - 2775 q^{15} + 9848 q^{16} + 18207 q^{17} - 3204 q^{18} - 4348 q^{19} - 5391 q^{20} - 12052 q^{21} - 610 q^{22} - 13541 q^{23} - 17041 q^{24} + 20652 q^{25} + 10949 q^{26} - 10024 q^{27} - 17843 q^{28} - 30421 q^{29} + 5608 q^{30} - 25324 q^{31} + 14964 q^{32} + 9106 q^{33} - 2312 q^{34} - 9241 q^{35} + 18545 q^{36} - 49618 q^{37} - 4663 q^{38} + 12765 q^{39} - 28939 q^{40} - 55994 q^{41} + 25653 q^{42} + 116487 q^{43} - 52574 q^{44} - 109852 q^{45} - 139176 q^{46} - 20485 q^{47} - 123576 q^{48} + 54504 q^{49} - 95822 q^{50} - 15895 q^{51} - 76969 q^{52} - 54883 q^{53} - 4743 q^{54} - 117426 q^{55} - 69722 q^{56} - 97199 q^{57} - 170965 q^{58} - 65484 q^{59} - 44341 q^{60} - 220168 q^{61} - 106281 q^{62} - 128619 q^{63} + 12638 q^{64} - 96322 q^{65} + 140685 q^{66} - 145412 q^{67} + 240448 q^{68} - 82215 q^{69} + 59419 q^{70} - 120265 q^{71} + 71768 q^{72} - 85083 q^{73} - 137048 q^{74} - 286465 q^{75} - 131565 q^{76} - 47913 q^{77} - 339352 q^{78} - 540432 q^{79} - 179886 q^{80} + 129271 q^{81} - 131575 q^{82} - 143204 q^{83} + 116071 q^{84} - 78319 q^{85} - 14792 q^{86} - 91808 q^{87} - 37320 q^{88} - 7351 q^{89} + 292871 q^{90} - 186128 q^{91} - 642265 q^{92} - 281849 q^{93} - 178771 q^{94} - 388261 q^{95} - 722579 q^{96} - 494156 q^{97} - 436884 q^{98} - 241379 q^{99}+O(q^{100}) \) 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 −10.8080 −10.3524 84.8127 −65.2147 111.888 −35.5897 −570.800 −135.828 704.840
1.2 −10.7869 −12.5617 84.3562 94.3410 135.502 28.6534 −564.759 −85.2028 −1017.64
1.3 −10.1622 21.3560 71.2710 −27.2072 −217.025 139.749 −399.081 213.078 276.486
1.4 −10.0035 0.116794 68.0694 −6.83209 −1.16835 113.916 −360.819 −242.986 68.3446
1.5 −9.69602 −25.8846 62.0127 −22.3964 250.978 −6.30778 −291.004 427.014 217.155
1.6 −9.56139 17.9145 59.4202 −98.7887 −171.288 −159.273 −262.175 77.9306 944.557
1.7 −9.30545 16.7253 54.5914 26.5781 −155.637 97.8267 −210.223 36.7365 −247.321
1.8 −8.95179 26.0717 48.1346 −14.5303 −233.389 −172.849 −144.433 436.735 130.072
1.9 −8.92930 −3.83049 47.7323 84.8527 34.2036 −230.285 −140.479 −228.327 −757.675
1.10 −8.58437 −21.4349 41.6914 7.80142 184.005 −178.201 −83.1945 216.456 −66.9702
1.11 −8.06224 4.67688 32.9997 95.3328 −37.7062 45.8191 −8.06002 −221.127 −768.596
1.12 −7.01653 −22.9074 17.2316 −47.4552 160.731 14.2227 103.623 281.750 332.971
1.13 −6.99904 −0.830897 16.9865 −57.5126 5.81548 93.3702 105.080 −242.310 402.533
1.14 −6.88149 −28.7676 15.3549 −83.7666 197.964 252.065 114.543 584.575 576.439
1.15 −6.60115 19.6233 11.5752 27.1235 −129.536 −161.768 134.827 142.073 −179.046
1.16 −6.55781 −3.63421 11.0049 −102.249 23.8324 −20.5612 137.682 −229.793 670.532
1.17 −5.88125 13.0725 2.58908 82.3049 −76.8827 33.9233 172.973 −72.1092 −484.055
1.18 −5.53460 −0.00584061 −1.36824 12.7663 0.0323254 189.734 184.680 −243.000 −70.6563
1.19 −5.36281 −8.88397 −3.24026 60.1336 47.6431 −70.4141 188.987 −164.075 −322.485
1.20 −4.97476 28.2654 −7.25178 6.37305 −140.613 56.2401 195.268 555.932 −31.7044
See all 63 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.63
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 731.6.a.a ✓ 63
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
731.6.a.a ✓ 63 1.a even 1 1 trivial