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Properties

Label 731.6.a.c
Level $731$
Weight $6$
Character orbit 731.a
Self dual yes
Analytic conductor $117.241$
Analytic rank $0$
Dimension $75$
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: \(0\)
Dimension: \(75\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 75 q + 8 q^{2} + 53 q^{3} + 1312 q^{4} + 329 q^{5} + 159 q^{6} + 547 q^{7} + 804 q^{8} + 6854 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 75 q + 8 q^{2} + 53 q^{3} + 1312 q^{4} + 329 q^{5} + 159 q^{6} + 547 q^{7} + 804 q^{8} + 6854 q^{9} + 2362 q^{10} + 716 q^{11} + 7647 q^{12} + 3114 q^{13} + 3087 q^{14} + 2559 q^{15} + 26744 q^{16} + 21675 q^{17} - 598 q^{18} + 7204 q^{19} + 6527 q^{20} + 7616 q^{21} - 524 q^{22} + 3387 q^{23} + 10607 q^{24} + 52144 q^{25} + 3527 q^{26} + 14108 q^{27} + 29197 q^{28} + 33077 q^{29} + 38008 q^{30} + 9272 q^{31} + 31086 q^{32} + 35242 q^{33} + 2312 q^{34} - 7797 q^{35} + 135185 q^{36} + 56926 q^{37} - 7303 q^{38} + 23775 q^{39} + 105461 q^{40} + 38142 q^{41} - 90687 q^{42} - 138675 q^{43} - 42310 q^{44} + 76448 q^{45} + 46324 q^{46} + 6023 q^{47} + 269320 q^{48} + 272518 q^{49} + 24450 q^{50} + 15317 q^{51} + 18671 q^{52} + 162961 q^{53} - 17803 q^{54} + 111398 q^{55} + 95186 q^{56} + 192841 q^{57} + 3963 q^{58} - 18436 q^{59} + 41755 q^{60} + 293756 q^{61} + 108983 q^{62} + 125905 q^{63} + 577886 q^{64} + 245982 q^{65} + 200965 q^{66} + 40128 q^{67} + 379168 q^{68} + 210891 q^{69} + 216091 q^{70} + 102165 q^{71} + 37850 q^{72} + 229105 q^{73} + 172840 q^{74} + 118535 q^{75} + 252969 q^{76} + 183543 q^{77} - 254128 q^{78} + 208488 q^{79} + 89656 q^{80} + 837859 q^{81} + 202697 q^{82} - 206648 q^{83} + 584551 q^{84} + 95081 q^{85} - 14792 q^{86} - 158480 q^{87} + 207664 q^{88} + 163975 q^{89} + 851219 q^{90} - 154796 q^{91} + 170279 q^{92} + 195119 q^{93} + 248591 q^{94} + 233919 q^{95} + 309613 q^{96} - 50172 q^{97} + 470358 q^{98} - 221219 q^{99}+O(q^{100}) \) 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 −11.1162 30.6335 91.5695 −76.5050 −340.528 163.488 −662.186 695.411 850.443
1.2 −11.0105 16.0849 89.2319 54.3271 −177.103 −79.6239 −630.154 15.7234 −598.171
1.3 −10.7978 10.0780 84.5934 14.7519 −108.821 14.7222 −567.896 −141.434 −159.288
1.4 −10.7037 −6.26576 82.5694 −46.5727 67.0669 −166.964 −541.280 −203.740 498.501
1.5 −9.87298 −22.5065 65.4757 −79.5798 222.206 28.3588 −330.505 263.543 785.689
1.6 −9.84623 −28.6057 64.9482 59.3316 281.658 −130.460 −324.415 575.284 −584.193
1.7 −9.73393 4.18682 62.7493 39.0426 −40.7541 −224.321 −299.311 −225.471 −380.038
1.8 −9.66280 2.52996 61.3697 −101.590 −24.4465 196.694 −283.794 −236.599 981.640
1.9 −9.55192 −17.0073 59.2392 26.0526 162.452 186.405 −260.187 46.2484 −248.852
1.10 −8.54125 7.82500 40.9530 10.0722 −66.8353 −58.6423 −76.4701 −181.769 −86.0288
1.11 −8.31320 21.3523 37.1094 −69.6982 −177.506 −95.0190 −42.4751 212.919 579.416
1.12 −8.25206 11.7091 36.0964 108.863 −96.6243 186.538 −33.8041 −105.897 −898.341
1.13 −8.18042 26.7726 34.9192 58.4462 −219.011 −57.0892 −23.8806 473.770 −478.114
1.14 −8.05869 −12.1491 32.9424 27.5436 97.9059 209.880 −7.59468 −95.3989 −221.965
1.15 −7.82084 −15.8461 29.1656 −37.1968 123.930 −122.225 22.1675 8.09833 290.910
1.16 −7.67167 25.5431 26.8546 45.5711 −195.958 250.593 39.4742 409.449 −349.606
1.17 −7.09996 −26.1211 18.4095 63.1352 185.459 96.7285 96.4921 439.311 −448.257
1.18 −6.89494 26.7733 15.5402 −68.8943 −184.600 93.2799 113.489 473.809 475.022
1.19 −6.86346 −17.6215 15.1070 84.7289 120.945 −73.8704 115.944 67.5183 −581.533
1.20 −6.70343 −16.1740 12.9360 −87.4587 108.421 −112.117 127.794 18.5972 586.273
See all 75 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.75
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.c ✓ 75
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
731.6.a.c ✓ 75 1.a even 1 1 trivial