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Properties

Label 799.6.a.b
Level $799$
Weight $6$
Character orbit 799.a
Self dual yes
Analytic conductor $128.147$
Analytic rank $1$
Dimension $74$
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] = [799,6,Mod(1,799)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(799, 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("799.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 799 = 17 \cdot 47 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 799.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: \(128.146672031\)
Analytic rank: \(1\)
Dimension: \(74\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 74 q - 11 q^{2} - 9 q^{3} + 1179 q^{4} - 241 q^{5} - 121 q^{6} + 285 q^{7} - 633 q^{8} + 4941 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 74 q - 11 q^{2} - 9 q^{3} + 1179 q^{4} - 241 q^{5} - 121 q^{6} + 285 q^{7} - 633 q^{8} + 4941 q^{9} + 248 q^{10} - 1270 q^{11} - 576 q^{12} - 1013 q^{13} - 3283 q^{14} - 1747 q^{15} + 17235 q^{16} - 21386 q^{17} - 4590 q^{18} - 3667 q^{19} - 14481 q^{20} - 2734 q^{21} - 3479 q^{22} - 5392 q^{23} - 13278 q^{24} + 40025 q^{25} - 33429 q^{26} - 8130 q^{27} - 7995 q^{28} - 23366 q^{29} - 39813 q^{30} - 15026 q^{31} - 25377 q^{32} - 53461 q^{33} + 3179 q^{34} - 38407 q^{35} + 18760 q^{36} + 19268 q^{37} - 74683 q^{38} - 28763 q^{39} - 6093 q^{40} - 107517 q^{41} - 37599 q^{42} - 18147 q^{43} - 116048 q^{44} - 110691 q^{45} + 3222 q^{46} - 163466 q^{47} + 7448 q^{48} + 72417 q^{49} - 23365 q^{50} + 2601 q^{51} + 32676 q^{52} - 57465 q^{53} - 38029 q^{54} - 52591 q^{55} - 209326 q^{56} + 28852 q^{57} + 38283 q^{58} - 142921 q^{59} - 211643 q^{60} - 77283 q^{61} - 149161 q^{62} + 38431 q^{63} + 214505 q^{64} - 173008 q^{65} - 77702 q^{66} - 121434 q^{67} - 340731 q^{68} - 175237 q^{69} - 72845 q^{70} - 170497 q^{71} - 241272 q^{72} + 16640 q^{73} - 194853 q^{74} - 77098 q^{75} - 267523 q^{76} - 121556 q^{77} - 408640 q^{78} + 160210 q^{79} - 407078 q^{80} + 26306 q^{81} - 178373 q^{82} - 145869 q^{83} - 311284 q^{84} + 69649 q^{85} - 427978 q^{86} - 307073 q^{87} - 88793 q^{88} - 962939 q^{89} + 16472 q^{90} - 257603 q^{91} - 512134 q^{92} - 503761 q^{93} + 24299 q^{94} - 311968 q^{95} - 423682 q^{96} - 40515 q^{97} - 757696 q^{98} - 145721 q^{99}+O(q^{100}) \) 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 −11.1014 −10.3842 91.2409 67.7694 115.279 179.312 −657.656 −135.169 −752.335
1.2 −10.7367 13.8742 83.2775 −94.8177 −148.964 113.747 −550.553 −50.5066 1018.03
1.3 −10.7074 −20.4261 82.6494 −66.0410 218.711 50.3511 −542.326 174.224 707.130
1.4 −10.3028 22.3559 74.1485 −76.3154 −230.329 −199.417 −434.249 256.784 786.266
1.5 −10.0271 −4.64974 68.5423 −61.7111 46.6233 −39.3286 −366.413 −221.380 618.783
1.6 −10.0196 −1.98157 68.3928 22.8753 19.8546 13.5142 −364.642 −239.073 −229.201
1.7 −9.96766 24.2138 67.3543 4.95867 −241.355 238.321 −352.399 343.306 −49.4263
1.8 −9.82804 −20.6965 64.5903 81.0981 203.406 −82.4706 −320.299 185.347 −797.035
1.9 −9.78308 20.4067 63.7086 66.3117 −199.640 4.39508 −310.208 173.432 −648.733
1.10 −9.08695 25.0101 50.5727 17.9801 −227.266 −171.333 −168.769 382.506 −163.384
1.11 −8.77046 −10.4486 44.9210 −13.2973 91.6392 −226.801 −113.323 −133.826 116.623
1.12 −8.52184 −26.4127 40.6218 −26.6415 225.085 −190.678 −73.4735 454.630 227.035
1.13 −8.48444 −0.186766 39.9857 99.2091 1.58460 −58.9798 −67.7541 −242.965 −841.734
1.14 −8.19067 0.871511 35.0871 −30.3780 −7.13827 137.640 −25.2857 −242.240 248.816
1.15 −8.17672 13.6134 34.8587 −6.72221 −111.313 154.915 −23.3750 −57.6755 54.9656
1.16 −7.69864 −29.9868 27.2690 −76.3317 230.858 137.618 36.4220 656.208 587.650
1.17 −6.75108 −11.9164 13.5771 −82.3496 80.4487 174.766 124.374 −100.999 555.949
1.18 −6.67783 10.3916 12.5934 −46.9210 −69.3933 −138.103 129.594 −135.015 313.331
1.19 −6.36337 −5.65267 8.49252 −22.2886 35.9701 −36.8776 149.587 −211.047 141.831
1.20 −6.05362 −21.7774 4.64627 22.5178 131.832 −0.379842 165.589 231.256 −136.314
See all 74 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.74
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(17\) \(1\)
\(47\) \(1\)

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 799.6.a.b ✓ 74
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
799.6.a.b ✓ 74 1.a even 1 1 trivial