Properties

Label 633.6.a.b
Level $633$
Weight $6$
Character orbit 633.a
Self dual yes
Analytic conductor $101.523$
Analytic rank $0$
Dimension $43$
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] = [633,6,Mod(1,633)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(633, 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("633.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 633 = 3 \cdot 211 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 633.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: \(101.522957942\)
Analytic rank: \(0\)
Dimension: \(43\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 43 q + 20 q^{2} - 387 q^{3} + 688 q^{4} + 75 q^{5} - 180 q^{6} + 519 q^{7} + 963 q^{8} + 3483 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 43 q + 20 q^{2} - 387 q^{3} + 688 q^{4} + 75 q^{5} - 180 q^{6} + 519 q^{7} + 963 q^{8} + 3483 q^{9} - 264 q^{10} + 313 q^{11} - 6192 q^{12} - 1177 q^{13} + 521 q^{14} - 675 q^{15} + 11872 q^{16} + 2329 q^{17} + 1620 q^{18} - 786 q^{19} + 4513 q^{20} - 4671 q^{21} + 3310 q^{22} + 19965 q^{23} - 8667 q^{24} + 30338 q^{25} + 3609 q^{26} - 31347 q^{27} + 13446 q^{28} + 7750 q^{29} + 2376 q^{30} - 4476 q^{31} + 46086 q^{32} - 2817 q^{33} + 2835 q^{34} + 17204 q^{35} + 55728 q^{36} + 3783 q^{37} + 22991 q^{38} + 10593 q^{39} - 17030 q^{40} + 16757 q^{41} - 4689 q^{42} + 21113 q^{43} + 21971 q^{44} + 6075 q^{45} + 11063 q^{46} + 51299 q^{47} - 106848 q^{48} + 73196 q^{49} + 75226 q^{50} - 20961 q^{51} - 38180 q^{52} + 23427 q^{53} - 14580 q^{54} - 3525 q^{55} - 86476 q^{56} + 7074 q^{57} - 115921 q^{58} - 18255 q^{59} - 40617 q^{60} - 131786 q^{61} - 131770 q^{62} + 42039 q^{63} + 177619 q^{64} - 32323 q^{65} - 29790 q^{66} + 83936 q^{67} - 135317 q^{68} - 179685 q^{69} - 188931 q^{70} + 258561 q^{71} + 78003 q^{72} - 50400 q^{73} - 63427 q^{74} - 273042 q^{75} - 92719 q^{76} + 68287 q^{77} - 32481 q^{78} + 78587 q^{79} + 363843 q^{80} + 282123 q^{81} + 54685 q^{82} + 199043 q^{83} - 121014 q^{84} + 5265 q^{85} + 193706 q^{86} - 69750 q^{87} + 336235 q^{88} + 259389 q^{89} - 21384 q^{90} + 173275 q^{91} + 1017819 q^{92} + 40284 q^{93} + 282828 q^{94} + 550281 q^{95} - 414774 q^{96} - 18215 q^{97} + 905479 q^{98} + 25353 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 −10.4485 −9.00000 77.1701 32.3344 94.0361 106.301 −471.958 81.0000 −337.844
1.2 −10.3217 −9.00000 74.5376 22.9849 92.8954 124.903 −439.061 81.0000 −237.243
1.3 −9.95524 −9.00000 67.1069 −51.5368 89.5972 −153.923 −349.497 81.0000 513.061
1.4 −9.63790 −9.00000 60.8892 22.2688 86.7411 −180.808 −278.431 81.0000 −214.625
1.5 −8.70572 −9.00000 43.7895 93.3937 78.3515 196.820 −102.636 81.0000 −813.059
1.6 −8.60921 −9.00000 42.1186 −87.1953 77.4829 95.7003 −87.1129 81.0000 750.683
1.7 −8.40613 −9.00000 38.6630 −14.5903 75.6552 −42.4616 −56.0100 81.0000 122.648
1.8 −8.37304 −9.00000 38.1079 8.08613 75.3574 183.646 −51.1415 81.0000 −67.7055
1.9 −7.93873 −9.00000 31.0235 76.9177 71.4486 9.33725 7.75222 81.0000 −610.629
1.10 −6.27027 −9.00000 7.31630 −72.9967 56.4324 −132.363 154.773 81.0000 457.709
1.11 −5.12190 −9.00000 −5.76617 82.2225 46.0971 121.113 193.434 81.0000 −421.135
1.12 −4.79000 −9.00000 −9.05586 −94.8244 43.1100 −26.7586 196.658 81.0000 454.209
1.13 −4.69088 −9.00000 −9.99560 97.6143 42.2180 −138.970 196.997 81.0000 −457.897
1.14 −4.50212 −9.00000 −11.7309 −35.2568 40.5191 −74.8838 196.882 81.0000 158.730
1.15 −3.78564 −9.00000 −17.6689 −94.6688 34.0707 65.0006 188.029 81.0000 358.382
1.16 −3.18591 −9.00000 −21.8500 29.8535 28.6732 −163.113 171.561 81.0000 −95.1108
1.17 −2.08669 −9.00000 −27.6457 60.2817 18.7802 −112.841 124.462 81.0000 −125.789
1.18 −1.34138 −9.00000 −30.2007 −20.1300 12.0724 −159.138 83.4346 81.0000 27.0020
1.19 −1.05971 −9.00000 −30.8770 −9.18692 9.53741 152.911 66.6315 81.0000 9.73549
1.20 −0.834509 −9.00000 −31.3036 −46.8133 7.51058 118.944 52.8274 81.0000 39.0662
See all 43 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.43
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(211\) \(-1\)

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 633.6.a.b 43
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
633.6.a.b 43 1.a even 1 1 trivial