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Properties

Label 862.6.a.b
Level $862$
Weight $6$
Character orbit 862.a
Self dual yes
Analytic conductor $138.251$
Analytic rank $1$
Dimension $44$
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] = [862,6,Mod(1,862)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(862, 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("862.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 862 = 2 \cdot 431 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 862.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: \(138.250852679\)
Analytic rank: \(1\)
Dimension: \(44\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 44 q + 176 q^{2} - 90 q^{3} + 704 q^{4} - 250 q^{5} - 360 q^{6} - 479 q^{7} + 2816 q^{8} + 3270 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 44 q + 176 q^{2} - 90 q^{3} + 704 q^{4} - 250 q^{5} - 360 q^{6} - 479 q^{7} + 2816 q^{8} + 3270 q^{9} - 1000 q^{10} - 1826 q^{11} - 1440 q^{12} - 2623 q^{13} - 1916 q^{14} - 4443 q^{15} + 11264 q^{16} - 6751 q^{17} + 13080 q^{18} - 7969 q^{19} - 4000 q^{20} - 8220 q^{21} - 7304 q^{22} - 17276 q^{23} - 5760 q^{24} + 21130 q^{25} - 10492 q^{26} - 27963 q^{27} - 7664 q^{28} - 22547 q^{29} - 17772 q^{30} - 34076 q^{31} + 45056 q^{32} - 19225 q^{33} - 27004 q^{34} - 35854 q^{35} + 52320 q^{36} - 26809 q^{37} - 31876 q^{38} - 33759 q^{39} - 16000 q^{40} - 47099 q^{41} - 32880 q^{42} - 44019 q^{43} - 29216 q^{44} - 83014 q^{45} - 69104 q^{46} - 80854 q^{47} - 23040 q^{48} + 98079 q^{49} + 84520 q^{50} - 54896 q^{51} - 41968 q^{52} - 86057 q^{53} - 111852 q^{54} - 111885 q^{55} - 30656 q^{56} - 65903 q^{57} - 90188 q^{58} - 157759 q^{59} - 71088 q^{60} - 163640 q^{61} - 136304 q^{62} - 172844 q^{63} + 180224 q^{64} - 81544 q^{65} - 76900 q^{66} - 106228 q^{67} - 108016 q^{68} - 162750 q^{69} - 143416 q^{70} - 188370 q^{71} + 209280 q^{72} - 135423 q^{73} - 107236 q^{74} - 359459 q^{75} - 127504 q^{76} - 259175 q^{77} - 135036 q^{78} - 194026 q^{79} - 64000 q^{80} + 219444 q^{81} - 188396 q^{82} - 466476 q^{83} - 131520 q^{84} - 92384 q^{85} - 176076 q^{86} - 377355 q^{87} - 116864 q^{88} - 219335 q^{89} - 332056 q^{90} - 483654 q^{91} - 276416 q^{92} - 247383 q^{93} - 323416 q^{94} - 395162 q^{95} - 92160 q^{96} - 290677 q^{97} + 392316 q^{98} - 516636 q^{99}+O(q^{100}) \) 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 4.00000 −30.6776 16.0000 75.1239 −122.710 −140.622 64.0000 698.113 300.495
1.2 4.00000 −29.0374 16.0000 44.8286 −116.150 8.13236 64.0000 600.172 179.315
1.3 4.00000 −28.9632 16.0000 −60.6964 −115.853 −14.5644 64.0000 595.867 −242.786
1.4 4.00000 −28.7748 16.0000 45.3222 −115.099 209.082 64.0000 584.991 181.289
1.5 4.00000 −24.9834 16.0000 −15.3750 −99.9335 −246.831 64.0000 381.169 −61.4999
1.6 4.00000 −24.5160 16.0000 53.1403 −98.0638 39.8923 64.0000 358.032 212.561
1.7 4.00000 −24.2974 16.0000 −106.262 −97.1898 −177.497 64.0000 347.366 −425.050
1.8 4.00000 −20.0207 16.0000 −21.3618 −80.0827 219.361 64.0000 157.828 −85.4474
1.9 4.00000 −20.0132 16.0000 11.9520 −80.0529 235.923 64.0000 157.529 47.8080
1.10 4.00000 −19.0745 16.0000 −107.168 −76.2979 198.696 64.0000 120.836 −428.673
1.11 4.00000 −17.6071 16.0000 −51.8459 −70.4284 8.43190 64.0000 67.0104 −207.384
1.12 4.00000 −17.4835 16.0000 −14.6346 −69.9338 −103.658 64.0000 62.6714 −58.5384
1.13 4.00000 −16.9248 16.0000 28.3452 −67.6991 55.6605 64.0000 43.4477 113.381
1.14 4.00000 −15.8418 16.0000 54.5043 −63.3673 −163.958 64.0000 7.96326 218.017
1.15 4.00000 −12.5783 16.0000 −104.332 −50.3134 −131.070 64.0000 −84.7852 −417.326
1.16 4.00000 −11.7776 16.0000 −3.63672 −47.1103 −124.785 64.0000 −104.289 −14.5469
1.17 4.00000 −9.53887 16.0000 110.044 −38.1555 −152.507 64.0000 −152.010 440.178
1.18 4.00000 −9.22226 16.0000 −86.7297 −36.8890 −111.903 64.0000 −157.950 −346.919
1.19 4.00000 −7.49401 16.0000 32.1951 −29.9760 42.5391 64.0000 −186.840 128.780
1.20 4.00000 −7.28201 16.0000 −42.3458 −29.1280 52.5725 64.0000 −189.972 −169.383
See all 44 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.44
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(431\) \(-1\)

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 862.6.a.b ✓ 44
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
862.6.a.b ✓ 44 1.a even 1 1 trivial