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Properties

Label 667.6.a.d
Level $667$
Weight $6$
Character orbit 667.a
Self dual yes
Analytic conductor $106.976$
Analytic rank $0$
Dimension $68$
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] = [667,6,Mod(1,667)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(667, 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("667.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 667 = 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 667.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: \(106.976007815\)
Analytic rank: \(0\)
Dimension: \(68\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 68 q + 12 q^{2} + 65 q^{3} + 1200 q^{4} + 400 q^{5} + 248 q^{6} + 149 q^{7} + 1080 q^{8} + 6513 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 68 q + 12 q^{2} + 65 q^{3} + 1200 q^{4} + 400 q^{5} + 248 q^{6} + 149 q^{7} + 1080 q^{8} + 6513 q^{9} + 410 q^{10} + 836 q^{11} + 2539 q^{12} + 3383 q^{13} - 32 q^{14} + 1082 q^{15} + 24632 q^{16} + 4848 q^{17} + 3633 q^{18} - 3188 q^{19} + 15117 q^{20} + 4645 q^{21} - 2803 q^{22} + 35972 q^{23} + 24918 q^{24} + 50070 q^{25} - 7441 q^{26} + 18644 q^{27} + 11429 q^{28} - 57188 q^{29} + 38179 q^{30} + 12708 q^{31} + 44632 q^{32} + 32111 q^{33} + 28583 q^{34} + 55901 q^{35} + 145830 q^{36} + 3619 q^{37} + 53094 q^{38} + 13362 q^{39} - 10720 q^{40} + 52589 q^{41} + 45856 q^{42} + 18811 q^{43} + 67325 q^{44} + 109746 q^{45} + 6348 q^{46} + 83125 q^{47} + 63139 q^{48} + 222017 q^{49} + 117173 q^{50} + 13050 q^{51} + 110606 q^{52} + 149661 q^{53} + 28435 q^{54} + 105421 q^{55} - 87907 q^{56} + 92850 q^{57} - 10092 q^{58} + 107189 q^{59} + 55455 q^{60} - 85242 q^{61} + 155043 q^{62} + 268989 q^{63} + 503178 q^{64} - 37001 q^{65} + 278263 q^{66} + 51109 q^{67} + 52407 q^{68} + 34385 q^{69} - 218616 q^{70} + 324575 q^{71} + 311755 q^{72} + 203910 q^{73} + 265799 q^{74} + 310165 q^{75} - 228724 q^{76} + 325643 q^{77} - 73814 q^{78} + 62611 q^{79} + 521110 q^{80} + 614756 q^{81} + 178657 q^{82} + 275824 q^{83} + 317338 q^{84} + 208202 q^{85} + 191461 q^{86} - 54665 q^{87} - 12040 q^{88} + 254454 q^{89} + 267435 q^{90} + 368455 q^{91} + 634800 q^{92} + 78666 q^{93} + 254027 q^{94} + 472148 q^{95} + 1110064 q^{96} + 85014 q^{97} + 551534 q^{98} + 80458 q^{99}+O(q^{100}) \) 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 −11.2364 6.55083 94.2569 9.74485 −73.6078 −171.199 −699.544 −200.087 −109.497
1.2 −10.4932 −1.56766 78.1063 −11.6506 16.4497 254.086 −483.801 −240.542 122.252
1.3 −10.3976 −11.3529 76.1108 −46.3121 118.044 −157.043 −458.648 −114.111 481.537
1.4 −10.2717 −26.7942 73.5072 108.223 275.221 176.943 −426.349 474.928 −1111.63
1.5 −10.2251 17.6191 72.5529 11.3572 −180.157 −26.7961 −414.657 67.4315 −116.129
1.6 −9.90822 −28.2205 66.1728 7.89735 279.615 −39.8665 −338.592 553.396 −78.2486
1.7 −9.83161 30.3154 64.6605 58.1724 −298.049 243.149 −321.105 676.023 −571.928
1.8 −9.45813 22.9194 57.4563 −73.1853 −216.774 −52.3331 −240.769 282.297 692.196
1.9 −9.10016 −16.8638 50.8130 −64.0221 153.463 −9.75921 −171.201 41.3882 582.611
1.10 −8.57671 15.2862 41.5599 102.744 −131.106 −247.432 −81.9923 −9.33071 −881.208
1.11 −8.41048 −18.6264 38.7362 58.1900 156.657 144.954 −56.6550 103.942 −489.406
1.12 −7.68146 6.63937 27.0048 0.0116236 −51.0001 100.274 38.3705 −198.919 −0.0892865
1.13 −7.66603 12.5215 26.7680 −89.8875 −95.9905 99.2645 40.1085 −86.2110 689.080
1.14 −7.58596 −16.3824 25.5468 53.9495 124.276 9.22636 48.9539 25.3821 −409.259
1.15 −7.55155 2.61534 25.0259 63.7730 −19.7499 139.983 52.6651 −236.160 −481.585
1.16 −7.03612 −1.98489 17.5069 −59.1202 13.9659 −124.750 101.975 −239.060 415.977
1.17 −6.36379 −19.2048 8.49786 46.3482 122.216 −221.400 149.563 125.825 −294.950
1.18 −6.10974 27.7733 5.32897 82.2850 −169.688 59.2986 162.953 528.358 −502.740
1.19 −5.85338 19.6616 2.26207 17.4729 −115.087 −181.049 174.067 143.577 −102.275
1.20 −5.18878 24.7919 −5.07655 −60.4685 −128.640 24.7678 192.382 371.640 313.758
See all 68 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.68
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(23\) \(-1\)
\(29\) \(1\)

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 667.6.a.d ✓ 68
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
667.6.a.d ✓ 68 1.a even 1 1 trivial