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Properties

Label 847.6.a.n
Level $847$
Weight $6$
Character orbit 847.a
Self dual yes
Analytic conductor $135.845$
Analytic rank $1$
Dimension $26$
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] = [847,6,Mod(1,847)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(847, 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("847.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 847 = 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 847.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: \(135.845095382\)
Analytic rank: \(1\)
Dimension: \(26\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 26 q - 16 q^{2} - 2 q^{3} + 428 q^{4} + 156 q^{5} - 554 q^{6} - 1274 q^{7} - 768 q^{8} + 2368 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 26 q - 16 q^{2} - 2 q^{3} + 428 q^{4} + 156 q^{5} - 554 q^{6} - 1274 q^{7} - 768 q^{8} + 2368 q^{9} - 1132 q^{10} + 658 q^{12} - 1908 q^{13} + 784 q^{14} + 3114 q^{15} + 8772 q^{16} - 4398 q^{17} - 7020 q^{18} - 2922 q^{19} + 6982 q^{20} + 98 q^{21} + 6970 q^{23} - 19924 q^{24} + 10324 q^{25} + 5802 q^{26} - 1346 q^{27} - 20972 q^{28} - 21068 q^{29} + 19072 q^{30} + 1638 q^{31} - 49772 q^{32} + 55400 q^{34} - 7644 q^{35} + 68030 q^{36} + 21362 q^{37} + 20214 q^{38} + 28716 q^{39} - 66894 q^{40} - 30882 q^{41} + 27146 q^{42} - 16068 q^{43} + 32932 q^{45} - 15088 q^{46} - 26000 q^{47} - 10392 q^{48} + 62426 q^{49} - 20744 q^{50} - 41844 q^{51} + 42060 q^{52} + 86508 q^{53} - 160368 q^{54} + 37632 q^{56} - 93764 q^{57} - 52790 q^{58} - 45476 q^{59} - 46376 q^{60} - 101496 q^{61} - 116668 q^{62} - 116032 q^{63} + 121294 q^{64} - 288440 q^{65} + 41942 q^{67} - 255520 q^{68} + 1366 q^{69} + 55468 q^{70} + 86814 q^{71} - 204604 q^{72} - 149254 q^{73} + 161724 q^{74} - 336580 q^{75} + 11930 q^{76} - 617620 q^{78} - 11620 q^{79} + 161660 q^{80} + 195662 q^{81} - 356068 q^{82} - 216994 q^{83} - 32242 q^{84} - 360392 q^{85} + 100438 q^{86} - 49730 q^{87} - 403424 q^{89} - 463182 q^{90} + 93492 q^{91} - 78454 q^{92} + 82690 q^{93} + 467466 q^{94} - 250384 q^{95} - 520006 q^{96} - 360716 q^{97} - 38416 q^{98}+O(q^{100}) \) 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 −11.0476 28.6551 90.0489 −9.68615 −316.569 −49.0000 −641.300 578.114 107.009
1.2 −11.0185 −19.8112 89.4072 97.0419 218.289 −49.0000 −632.541 149.483 −1069.26
1.3 −9.88748 4.47656 65.7623 −59.9620 −44.2619 −49.0000 −333.824 −222.960 592.873
1.4 −9.58852 −2.33610 59.9397 47.7668 22.3998 −49.0000 −267.901 −237.543 −458.013
1.5 −9.27414 26.0706 54.0097 44.3351 −241.782 −49.0000 −204.121 436.675 −411.170
1.6 −6.67645 −29.5505 12.5749 −34.5329 197.293 −49.0000 129.690 630.234 230.557
1.7 −6.15390 −1.57723 5.87046 −40.5097 9.70609 −49.0000 160.799 −240.512 249.293
1.8 −6.01179 −9.11434 4.14161 −4.13530 54.7935 −49.0000 167.479 −159.929 24.8606
1.9 −5.23252 17.9604 −4.62078 −74.6363 −93.9782 −49.0000 191.619 79.5767 390.535
1.10 −5.10979 24.5496 −5.89010 104.032 −125.443 −49.0000 193.610 359.681 −531.583
1.11 −4.39154 5.70411 −12.7144 97.0946 −25.0498 −49.0000 196.365 −210.463 −426.394
1.12 −1.64681 −27.4349 −29.2880 −24.4594 45.1801 −49.0000 100.930 509.674 40.2800
1.13 −0.958902 6.46272 −31.0805 −36.7648 −6.19712 −49.0000 60.4880 −201.233 35.2538
1.14 0.716090 9.35801 −31.4872 7.43504 6.70118 −49.0000 −45.4626 −155.428 5.32416
1.15 0.744223 −22.0210 −31.4461 79.5382 −16.3886 −49.0000 −47.2181 241.926 59.1942
1.16 0.937670 27.1643 −31.1208 −6.73074 25.4711 −49.0000 −59.1865 494.899 −6.31121
1.17 1.89449 −12.1732 −28.4109 −74.0437 −23.0621 −49.0000 −114.448 −94.8121 −140.275
1.18 3.84578 −15.1588 −17.2100 54.8848 −58.2974 −49.0000 −189.251 −13.2108 211.075
1.19 4.42999 −11.7759 −12.3752 −79.6910 −52.1669 −49.0000 −196.582 −104.329 −353.030
1.20 5.15046 21.9629 −5.47279 37.3733 113.119 −49.0000 −193.002 239.370 192.489
See all 26 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.26
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(1\)
\(11\) \(1\)

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 847.6.a.n ✓ 26
11.b odd 2 1 847.6.a.o yes 26
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
847.6.a.n ✓ 26 1.a even 1 1 trivial
847.6.a.o yes 26 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{26} + 16 T_{2}^{25} - 502 T_{2}^{24} - 8800 T_{2}^{23} + 104321 T_{2}^{22} + \cdots - 19\!\cdots\!24 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(847))\). Copy content Toggle raw display