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Properties

Label 799.6.a.c
Level $799$
Weight $6$
Character orbit 799.a
Self dual yes
Analytic conductor $128.147$
Analytic rank $0$
Dimension $80$
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: \(0\)
Dimension: \(80\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 80 q + 13 q^{2} + 27 q^{3} + 1339 q^{4} + 341 q^{5} + 193 q^{6} + 187 q^{7} + 519 q^{8} + 7695 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 80 q + 13 q^{2} + 27 q^{3} + 1339 q^{4} + 341 q^{5} + 193 q^{6} + 187 q^{7} + 519 q^{8} + 7695 q^{9} + 1248 q^{10} + 2600 q^{11} + 1440 q^{12} + 1691 q^{13} + 2205 q^{14} + 1853 q^{15} + 24011 q^{16} + 23120 q^{17} + 3382 q^{18} + 393 q^{19} + 15897 q^{20} + 242 q^{21} + 7643 q^{22} + 9010 q^{23} + 17040 q^{24} + 55931 q^{25} + 1771 q^{26} + 3534 q^{27} + 3661 q^{28} + 29386 q^{29} - 9045 q^{30} + 25482 q^{31} + 24011 q^{32} + 44071 q^{33} + 3757 q^{34} + 31513 q^{35} + 99112 q^{36} + 28856 q^{37} - 3451 q^{38} - 233 q^{39} - 20355 q^{40} + 137899 q^{41} + 44609 q^{42} + 4701 q^{43} + 89572 q^{44} + 122417 q^{45} + 62470 q^{46} - 176720 q^{47} + 78534 q^{48} + 294863 q^{49} + 55535 q^{50} + 7803 q^{51} + 67064 q^{52} + 154579 q^{53} + 204233 q^{54} - 64691 q^{55} + 38960 q^{56} + 192850 q^{57} + 79801 q^{58} + 286363 q^{59} - 38843 q^{60} + 34347 q^{61} + 48895 q^{62} + 63491 q^{63} + 486049 q^{64} + 311002 q^{65} + 194966 q^{66} + 100774 q^{67} + 386971 q^{68} + 123347 q^{69} + 59407 q^{70} + 81419 q^{71} - 67296 q^{72} + 380310 q^{73} + 79199 q^{74} - 191698 q^{75} - 52227 q^{76} + 50064 q^{77} + 115452 q^{78} + 102756 q^{79} + 416790 q^{80} + 1055588 q^{81} + 414653 q^{82} + 129691 q^{83} + 31220 q^{84} + 98549 q^{85} + 217030 q^{86} - 28461 q^{87} + 188571 q^{88} + 883673 q^{89} + 117552 q^{90} + 137071 q^{91} + 443062 q^{92} - 208277 q^{93} - 28717 q^{94} - 405434 q^{95} + 675518 q^{96} + 612251 q^{97} + 294096 q^{98} + 688085 q^{99}+O(q^{100}) \) 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 −10.8996 −10.2833 86.8023 −24.7133 112.084 −106.692 −597.326 −137.254 269.366
1.2 −10.8555 6.97278 85.8411 71.0229 −75.6927 59.8647 −584.470 −194.380 −770.986
1.3 −10.8080 −30.2705 84.8118 58.2420 327.162 147.081 −570.788 673.303 −629.477
1.4 −10.6415 16.9234 81.2422 77.4397 −180.091 −254.573 −524.012 43.4007 −824.077
1.5 −10.5838 2.49594 80.0165 −50.4168 −26.4165 −111.262 −508.196 −236.770 533.601
1.6 −10.2821 18.0545 73.7212 −11.0807 −185.638 234.041 −428.981 82.9644 113.933
1.7 −9.54894 22.8262 59.1823 50.5509 −217.966 −1.87136 −259.562 278.037 −482.708
1.8 −9.53812 −25.1473 58.9757 −109.777 239.858 −75.6522 −257.297 389.388 1047.06
1.9 −9.33762 −8.32526 55.1911 −15.0517 77.7381 103.565 −216.549 −173.690 140.547
1.10 −9.27886 24.8750 54.0973 −75.0547 −230.811 −11.9692 −205.038 375.764 696.422
1.11 −8.42986 7.08647 39.0625 −4.75724 −59.7379 −53.6199 −59.5362 −192.782 40.1029
1.12 −8.26531 −23.4606 36.3154 55.2488 193.909 −212.016 −35.6678 307.399 −456.649
1.13 −7.94368 4.61361 31.1020 85.0725 −36.6490 −97.2026 7.13330 −221.715 −675.788
1.14 −7.75487 −21.4396 28.1380 −16.5209 166.261 201.080 29.9492 216.655 128.117
1.15 −7.74644 4.23082 28.0073 −65.7260 −32.7738 195.696 30.9291 −225.100 509.142
1.16 −7.49404 −15.7543 24.1607 3.29606 118.064 −223.060 58.7484 5.19888 −24.7008
1.17 −7.36112 27.7425 22.1861 70.4053 −204.216 153.958 72.2410 526.644 −518.262
1.18 −7.19686 −26.1312 19.7947 108.568 188.063 194.349 87.8395 439.841 −781.350
1.19 −7.01696 5.00283 17.2377 −94.4612 −35.1046 −14.1915 103.587 −217.972 662.830
1.20 −6.91980 −15.0237 15.8836 52.3366 103.961 113.538 111.522 −17.2893 −362.158
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.80
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.c ✓ 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
799.6.a.c ✓ 80 1.a even 1 1 trivial