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Properties

Label 1001.6.a.e
Level $1001$
Weight $6$
Character orbit 1001.a
Self dual yes
Analytic conductor $160.544$
Analytic rank $0$
Dimension $38$
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] = [1001,6,Mod(1,1001)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1001, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1001.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1001 = 7 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 1001.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: \(160.544203633\)
Analytic rank: \(0\)
Dimension: \(38\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 38 q + 12 q^{3} + 650 q^{4} - 51 q^{5} - 151 q^{6} + 1862 q^{7} + 237 q^{8} + 3132 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 38 q + 12 q^{3} + 650 q^{4} - 51 q^{5} - 151 q^{6} + 1862 q^{7} + 237 q^{8} + 3132 q^{9} + 1495 q^{10} - 4598 q^{11} + 967 q^{12} - 6422 q^{13} + 702 q^{15} + 12798 q^{16} + 3183 q^{17} + 1872 q^{18} + 2838 q^{19} + 1668 q^{20} + 588 q^{21} - 585 q^{23} - 20077 q^{24} + 21341 q^{25} + 13644 q^{27} + 31850 q^{28} - 1886 q^{29} + 12100 q^{30} + 10566 q^{31} - 5936 q^{32} - 1452 q^{33} + 30843 q^{34} - 2499 q^{35} + 84153 q^{36} + 7055 q^{37} - 7976 q^{38} - 2028 q^{39} + 139076 q^{40} + 30336 q^{41} - 7399 q^{42} + 46404 q^{43} - 78650 q^{44} + 3078 q^{45} + 60486 q^{46} + 75880 q^{47} + 103582 q^{48} + 91238 q^{49} + 65317 q^{50} + 87407 q^{51} - 109850 q^{52} - 15493 q^{53} + 100228 q^{54} + 6171 q^{55} + 11613 q^{56} + 4166 q^{57} + 65236 q^{58} - 57578 q^{59} + 137515 q^{60} + 160738 q^{61} + 235004 q^{62} + 153468 q^{63} + 322681 q^{64} + 8619 q^{65} + 18271 q^{66} + 17621 q^{67} + 97918 q^{68} - 24251 q^{69} + 73255 q^{70} - 2845 q^{71} - 50187 q^{72} + 335493 q^{73} - 137541 q^{74} + 242726 q^{75} + 444822 q^{76} - 225302 q^{77} + 25519 q^{78} + 303044 q^{79} + 141730 q^{80} + 322798 q^{81} - 161012 q^{82} - 32423 q^{83} + 47383 q^{84} - 188205 q^{85} + 275102 q^{86} + 335315 q^{87} - 28677 q^{88} + 2989 q^{89} + 567334 q^{90} - 314678 q^{91} - 54146 q^{92} - 177572 q^{93} + 565112 q^{94} - 119994 q^{95} - 885020 q^{96} + 430504 q^{97} - 378972 q^{99}+O(q^{100}) \) 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 −11.2762 0.905398 95.1517 −16.7874 −10.2094 49.0000 −712.109 −242.180 189.298
1.2 −10.7062 28.6870 82.6225 −83.5752 −307.128 49.0000 −541.974 579.944 894.772
1.3 −9.86138 −24.5564 65.2468 −46.6399 242.160 49.0000 −327.859 360.015 459.934
1.4 −9.79990 15.3650 64.0380 41.4089 −150.575 49.0000 −313.969 −6.91709 −405.803
1.5 −8.93598 20.2993 47.8518 92.9324 −181.394 49.0000 −141.651 169.060 −830.442
1.6 −8.91966 −10.6790 47.5603 −94.8982 95.2529 49.0000 −138.793 −128.959 846.459
1.7 −8.44202 8.35465 39.2677 −27.9347 −70.5301 49.0000 −61.3536 −173.200 235.825
1.8 −7.79809 −24.8308 28.8102 42.9134 193.633 49.0000 24.8747 373.569 −334.643
1.9 −6.69377 −5.49043 12.8065 49.7445 36.7517 49.0000 128.477 −212.855 −332.978
1.10 −5.74719 −20.1351 1.03021 −12.5963 115.720 49.0000 177.989 162.421 72.3935
1.11 −5.69286 3.50356 0.408641 −12.1064 −19.9453 49.0000 179.845 −230.725 68.9199
1.12 −5.43467 22.7836 −2.46436 −21.9421 −123.821 49.0000 187.302 276.093 119.248
1.13 −4.99511 −22.3049 −7.04891 76.4412 111.415 49.0000 195.053 254.508 −381.832
1.14 −3.33040 5.72369 −20.9084 85.3969 −19.0622 49.0000 176.206 −210.239 −284.406
1.15 −2.25912 −12.1897 −26.8964 −70.5821 27.5380 49.0000 133.054 −94.4103 159.453
1.16 −2.24058 9.97364 −26.9798 −71.9451 −22.3467 49.0000 132.149 −143.526 161.199
1.17 −1.98133 28.0842 −28.0743 18.9520 −55.6441 49.0000 119.027 545.722 −37.5501
1.18 −1.54105 18.0561 −29.6251 −60.2076 −27.8254 49.0000 94.9677 83.0228 92.7832
1.19 −1.39001 −24.5539 −30.0679 −82.9237 34.1301 49.0000 86.2749 359.893 115.265
1.20 −1.13686 −7.78015 −30.7076 6.77272 8.84492 49.0000 71.2895 −182.469 −7.69961
See all 38 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.38
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(-1\)
\(11\) \(1\)
\(13\) \(1\)

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 1001.6.a.e ✓ 38
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1001.6.a.e ✓ 38 1.a even 1 1 trivial