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Properties

Label 841.6.a.h
Level $841$
Weight $6$
Character orbit 841.a
Self dual yes
Analytic conductor $134.883$
Analytic rank $1$
Dimension $33$
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] = [841,6,Mod(1,841)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(841, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("841.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 841 = 29^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 841.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: \(134.882792463\)
Analytic rank: \(1\)
Dimension: \(33\)
Twist minimal: no (minimal twist has level 29)
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.

\(\operatorname{Tr}(f)(q) = \) \( 33 q - 12 q^{2} - q^{3} + 422 q^{4} - 157 q^{5} - 166 q^{6} - 331 q^{7} - 453 q^{8} + 1622 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 33 q - 12 q^{2} - q^{3} + 422 q^{4} - 157 q^{5} - 166 q^{6} - 331 q^{7} - 453 q^{8} + 1622 q^{9} + 291 q^{10} + 491 q^{11} + 124 q^{12} - 1157 q^{13} - 620 q^{14} + 3960 q^{15} + 4566 q^{16} - 338 q^{17} - 6865 q^{18} + 451 q^{19} - 6636 q^{20} + 97 q^{21} - 11899 q^{22} - 8247 q^{23} - 13519 q^{24} + 7262 q^{25} + 11758 q^{26} - 9886 q^{27} + 3754 q^{28} - 20009 q^{30} + 11231 q^{31} - 17863 q^{32} - 22142 q^{33} + 7463 q^{34} - 26372 q^{35} - 516 q^{36} - 7149 q^{37} - 13318 q^{38} - 25700 q^{39} + 18611 q^{40} + 9360 q^{41} + 41603 q^{42} + 55165 q^{43} + 84442 q^{44} + 10619 q^{45} - 7924 q^{46} - 46957 q^{47} + 69535 q^{48} - 16038 q^{49} - 42663 q^{50} - 125432 q^{51} + 7049 q^{52} - 54523 q^{53} - 15016 q^{54} - 118472 q^{55} - 50285 q^{56} - 20081 q^{57} - 166756 q^{59} + 165453 q^{60} - 87433 q^{61} - 103656 q^{62} - 67869 q^{63} - 31945 q^{64} - 27116 q^{65} + 62548 q^{66} - 100267 q^{67} - 92443 q^{68} + 208538 q^{69} + 255839 q^{70} - 117725 q^{71} + 29044 q^{72} + 96829 q^{73} - 147314 q^{74} - 269104 q^{75} + 85334 q^{76} + 237412 q^{77} - 168156 q^{78} - 364713 q^{79} - 629809 q^{80} + 93985 q^{81} - 238717 q^{82} - 203241 q^{83} - 552428 q^{84} + 19206 q^{85} - 174200 q^{86} - 492819 q^{88} + 309027 q^{89} - 252630 q^{90} - 369702 q^{91} - 194235 q^{92} + 86514 q^{93} - 632929 q^{94} + 272214 q^{95} - 361521 q^{96} - 186709 q^{97} - 651857 q^{98} - 219304 q^{99}+O(q^{100}) \) 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 −10.8533 0.765894 85.7949 −85.9391 −8.31251 175.866 −583.854 −242.413 932.726
1.2 −10.3192 27.1100 74.4856 32.5269 −279.753 −202.054 −438.417 491.950 −335.651
1.3 −10.0113 −12.6180 68.2261 −68.7741 126.323 90.8898 −362.671 −83.7858 688.519
1.4 −9.84368 5.83838 64.8981 −17.8351 −57.4712 −26.7610 −323.839 −208.913 175.564
1.5 −8.50819 16.2520 40.3893 88.9618 −138.275 −26.5712 −71.3774 21.1288 −756.903
1.6 −7.77901 −23.1465 28.5130 18.6120 180.057 −99.8084 27.1251 292.760 −144.783
1.7 −7.47341 −28.5348 23.8518 −46.4428 213.252 83.4382 60.8947 571.237 347.086
1.8 −6.80751 −2.82559 14.3423 43.1294 19.2353 1.57354 120.205 −235.016 −293.604
1.9 −6.34408 −10.7257 8.24734 92.8824 68.0449 −145.389 150.689 −127.959 −589.253
1.10 −5.86143 26.8668 2.35639 21.0392 −157.478 107.078 173.754 478.824 −123.320
1.11 −4.98230 14.8253 −7.17669 −66.4528 −73.8639 −156.220 195.190 −23.2117 331.088
1.12 −3.76979 21.5399 −17.7887 −23.8546 −81.2008 230.214 187.693 220.966 89.9267
1.13 −3.41635 −30.2746 −20.3286 −17.5410 103.428 −63.8846 178.773 673.548 59.9263
1.14 −2.67324 −7.64195 −24.8538 8.47169 20.4287 148.100 151.984 −184.601 −22.6468
1.15 −2.62243 −1.15253 −25.1229 −101.892 3.02244 −74.2998 149.801 −241.672 267.203
1.16 −1.52850 7.14829 −29.6637 −79.7929 −10.9262 −245.424 94.2529 −191.902 121.963
1.17 −1.31634 −2.46928 −30.2673 31.3078 3.25041 191.440 81.9648 −236.903 −41.2117
1.18 1.71154 18.0299 −29.0706 16.6425 30.8588 −240.415 −104.525 82.0757 28.4844
1.19 1.80700 24.5303 −28.7348 24.8498 44.3261 86.3727 −109.748 358.734 44.9036
1.20 1.88537 −8.46545 −28.4454 77.5774 −15.9605 17.2386 −113.962 −171.336 146.262
See all 33 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.33
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(29\) \(1\)

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 841.6.a.h 33
29.b even 2 1 841.6.a.i 33
29.e even 14 2 29.6.d.a ✓ 66
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.6.d.a ✓ 66 29.e even 14 2
841.6.a.h 33 1.a even 1 1 trivial
841.6.a.i 33 29.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{33} + 12 T_{2}^{32} - 667 T_{2}^{31} - 8173 T_{2}^{30} + 197435 T_{2}^{29} + \cdots + 11\!\cdots\!84 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(841))\). Copy content Toggle raw display