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Properties

Label 847.6.a.p
Level $847$
Weight $6$
Character orbit 847.a
Self dual yes
Analytic conductor $135.845$
Analytic rank $1$
Dimension $28$
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: \(28\)
Twist minimal: no (minimal twist has level 77)
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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q - 14 q^{2} - 28 q^{3} + 374 q^{4} - 258 q^{5} + 4 q^{6} + 1372 q^{7} - 339 q^{8} + 1660 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q - 14 q^{2} - 28 q^{3} + 374 q^{4} - 258 q^{5} + 4 q^{6} + 1372 q^{7} - 339 q^{8} + 1660 q^{9} - 536 q^{10} - 996 q^{12} + 326 q^{13} - 686 q^{14} - 2561 q^{15} + 5642 q^{16} - 2159 q^{17} - 6642 q^{18} + 2253 q^{19} - 13417 q^{20} - 1372 q^{21} - 6620 q^{23} + 3009 q^{24} + 15576 q^{25} - 4960 q^{26} - 24718 q^{27} + 18326 q^{28} - 15747 q^{29} - 9484 q^{30} - 14455 q^{31} - 13922 q^{32} - 13463 q^{34} - 12642 q^{35} + 27148 q^{36} - 14447 q^{37} - 62211 q^{38} - 28537 q^{39} - 31765 q^{40} - 14604 q^{41} + 196 q^{42} - 6828 q^{43} - 2703 q^{45} - 110251 q^{46} - 51396 q^{47} - 24310 q^{48} + 67228 q^{49} - 17179 q^{50} - 23461 q^{51} + 71461 q^{52} - 134505 q^{53} + 135496 q^{54} - 16611 q^{56} - 45846 q^{57} + 18657 q^{58} - 22255 q^{59} - 176135 q^{60} - 61141 q^{61} + 186610 q^{62} + 81340 q^{63} + 83377 q^{64} - 8616 q^{65} - 34675 q^{67} - 106726 q^{68} - 178024 q^{69} - 26264 q^{70} - 81779 q^{71} - 191763 q^{72} + 91273 q^{73} - 24102 q^{74} + 95115 q^{75} + 578339 q^{76} - 230091 q^{78} - 244684 q^{79} - 296482 q^{80} - 55548 q^{81} - 129920 q^{82} + 89009 q^{83} - 48804 q^{84} + 95335 q^{85} - 32043 q^{86} + 63248 q^{87} - 450187 q^{89} + 408246 q^{90} + 15974 q^{91} - 290507 q^{92} - 411705 q^{93} + 482748 q^{94} - 164289 q^{95} + 149697 q^{96} - 266929 q^{97} - 33614 q^{98}+O(q^{100}) \) 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 −10.8104 −27.2407 84.8641 34.6253 294.482 49.0000 −571.480 499.053 −374.312
1.2 −10.5552 17.8884 79.4132 −90.8756 −188.817 49.0000 −500.457 76.9957 959.214
1.3 −9.73409 −2.05254 62.7525 −29.2303 19.9796 49.0000 −299.348 −238.787 284.530
1.4 −9.31145 11.3708 54.7031 4.67627 −105.879 49.0000 −211.399 −113.705 −43.5428
1.5 −8.48990 −0.443512 40.0785 91.8201 3.76538 49.0000 −68.5855 −242.803 −779.544
1.6 −7.04026 20.0772 17.5653 −10.8600 −141.349 49.0000 101.624 160.095 76.4570
1.7 −6.88812 −29.5999 15.4462 −50.7434 203.888 49.0000 114.025 633.156 349.527
1.8 −6.34511 −17.9633 8.26041 −86.5134 113.979 49.0000 150.630 79.6796 548.937
1.9 −5.64912 26.3683 −0.0874362 0.400475 −148.957 49.0000 181.266 452.285 −2.26233
1.10 −4.04168 −8.44784 −15.6648 −80.4152 34.1435 49.0000 192.646 −171.634 325.013
1.11 −3.49887 −24.3391 −19.7579 27.4633 85.1594 49.0000 181.094 349.394 −96.0904
1.12 −3.24782 1.01621 −21.4517 101.380 −3.30046 49.0000 173.601 −241.967 −329.265
1.13 −2.21821 20.9811 −27.0795 51.8297 −46.5406 49.0000 131.051 197.208 −114.969
1.14 −1.77872 1.24479 −28.8361 −29.8264 −2.21414 49.0000 108.211 −241.450 53.0529
1.15 −0.751775 −9.02588 −31.4348 −0.885950 6.78543 49.0000 47.6887 −161.533 0.666035
1.16 1.24928 22.3033 −30.4393 −97.4016 27.8630 49.0000 −78.0040 254.436 −121.681
1.17 2.20834 16.1100 −27.1232 51.1616 35.5765 49.0000 −130.564 16.5326 112.982
1.18 2.70064 2.75336 −24.7066 −106.708 7.43583 49.0000 −153.144 −235.419 −288.179
1.19 2.81364 −13.2609 −24.0834 69.0036 −37.3115 49.0000 −157.799 −67.1476 194.152
1.20 3.57393 −28.0328 −19.2270 68.0844 −100.187 49.0000 −183.082 542.840 243.329
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.28
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.p 28
11.b odd 2 1 847.6.a.q 28
11.d odd 10 2 77.6.f.a ✓ 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.6.f.a ✓ 56 11.d odd 10 2
847.6.a.p 28 1.a even 1 1 trivial
847.6.a.q 28 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{28} + 14 T_{2}^{27} - 537 T_{2}^{26} - 8021 T_{2}^{25} + 122024 T_{2}^{24} + \cdots - 20\!\cdots\!00 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(847))\). Copy content Toggle raw display