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Properties

Label 847.6.a.o
Level $847$
Weight $6$
Character orbit 847.a
Self dual yes
Analytic conductor $135.845$
Analytic rank $0$
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: \(0\)
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 −9.93988 7.80368 66.8013 9.67008 −77.5677 49.0000 −345.921 −182.103 −96.1194
1.2 −9.93957 −25.0148 66.7951 −104.264 248.637 49.0000 −345.849 382.742 1036.34
1.3 −9.63035 −23.2658 60.7436 58.9550 224.058 49.0000 −276.811 298.298 −567.758
1.4 −8.44013 14.1412 39.2357 −33.8338 −119.353 49.0000 −61.0704 −43.0276 285.562
1.5 −8.04735 14.0044 32.7599 61.3336 −112.698 49.0000 −6.11481 −46.8781 −493.573
1.6 −7.28191 −11.0796 21.0263 39.7894 80.6806 49.0000 79.9098 −120.243 −289.743
1.7 −5.15046 21.9629 −5.47279 37.3733 −113.119 49.0000 193.002 239.370 −192.489
1.8 −4.42999 −11.7759 −12.3752 −79.6910 52.1669 49.0000 196.582 −104.329 353.030
1.9 −3.84578 −15.1588 −17.2100 54.8848 58.2974 49.0000 189.251 −13.2108 −211.075
1.10 −1.89449 −12.1732 −28.4109 −74.0437 23.0621 49.0000 114.448 −94.8121 140.275
1.11 −0.937670 27.1643 −31.1208 −6.73074 −25.4711 49.0000 59.1865 494.899 6.31121
1.12 −0.744223 −22.0210 −31.4461 79.5382 16.3886 49.0000 47.2181 241.926 −59.1942
1.13 −0.716090 9.35801 −31.4872 7.43504 −6.70118 49.0000 45.4626 −155.428 −5.32416
1.14 0.958902 6.46272 −31.0805 −36.7648 6.19712 49.0000 −60.4880 −201.233 −35.2538
1.15 1.64681 −27.4349 −29.2880 −24.4594 −45.1801 49.0000 −100.930 509.674 −40.2800
1.16 4.39154 5.70411 −12.7144 97.0946 25.0498 49.0000 −196.365 −210.463 426.394
1.17 5.10979 24.5496 −5.89010 104.032 125.443 49.0000 −193.610 359.681 531.583
1.18 5.23252 17.9604 −4.62078 −74.6363 93.9782 49.0000 −191.619 79.5767 −390.535
1.19 6.01179 −9.11434 4.14161 −4.13530 −54.7935 49.0000 −167.479 −159.929 −24.8606
1.20 6.15390 −1.57723 5.87046 −40.5097 −9.70609 49.0000 −160.799 −240.512 −249.293
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.o yes 26
11.b odd 2 1 847.6.a.n ✓ 26
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
847.6.a.n ✓ 26 11.b odd 2 1
847.6.a.o yes 26 1.a even 1 1 trivial

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