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Properties

Label 633.6.a.d
Level $633$
Weight $6$
Character orbit 633.a
Self dual yes
Analytic conductor $101.523$
Analytic rank $0$
Dimension $49$
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: \(49\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 49 q + 16 q^{2} + 441 q^{3} + 928 q^{4} + 119 q^{5} + 144 q^{6} + 813 q^{7} + 765 q^{8} + 3969 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 49 q + 16 q^{2} + 441 q^{3} + 928 q^{4} + 119 q^{5} + 144 q^{6} + 813 q^{7} + 765 q^{8} + 3969 q^{9} + 1400 q^{10} + 1119 q^{11} + 8352 q^{12} + 2823 q^{13} + 807 q^{14} + 1071 q^{15} + 19352 q^{16} + 1891 q^{17} + 1296 q^{18} + 5996 q^{19} + 4229 q^{20} + 7317 q^{21} + 9714 q^{22} + 15229 q^{23} + 6885 q^{24} + 48500 q^{25} + 8891 q^{26} + 35721 q^{27} + 36618 q^{28} + 6646 q^{29} + 12600 q^{30} + 23926 q^{31} + 38726 q^{32} + 10071 q^{33} + 34691 q^{34} + 13572 q^{35} + 75168 q^{36} + 44707 q^{37} + 24569 q^{38} + 25407 q^{39} + 78734 q^{40} + 23689 q^{41} + 7263 q^{42} + 69635 q^{43} + 40335 q^{44} + 9639 q^{45} + 55411 q^{46} + 63157 q^{47} + 174168 q^{48} + 217338 q^{49} + 31756 q^{50} + 17019 q^{51} + 107076 q^{52} + 85268 q^{53} + 11664 q^{54} + 151104 q^{55} + 177032 q^{56} + 53964 q^{57} + 264091 q^{58} + 169432 q^{59} + 38061 q^{60} + 237394 q^{61} + 268138 q^{62} + 65853 q^{63} + 405027 q^{64} + 156412 q^{65} + 87426 q^{66} + 257256 q^{67} + 440001 q^{68} + 137061 q^{69} + 368749 q^{70} + 247151 q^{71} + 61965 q^{72} + 198135 q^{73} + 235383 q^{74} + 436500 q^{75} + 303609 q^{76} + 56711 q^{77} + 80019 q^{78} + 352045 q^{79} + 156173 q^{80} + 321489 q^{81} + 333841 q^{82} - 6138 q^{83} + 329562 q^{84} + 136161 q^{85} + 106974 q^{86} + 59814 q^{87} + 505697 q^{88} + 123891 q^{89} + 113400 q^{90} + 380939 q^{91} + 332093 q^{92} + 215334 q^{93} + 265276 q^{94} + 371100 q^{95} + 348534 q^{96} + 165051 q^{97} - 503229 q^{98} + 90639 q^{99}+O(q^{100}) \) 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 −11.1856 9.00000 93.1179 4.30937 −100.671 −107.087 −683.641 81.0000 −48.2029
1.2 −10.8272 9.00000 85.2283 −105.445 −97.4448 173.921 −576.313 81.0000 1141.67
1.3 −10.4881 9.00000 78.0006 −24.8385 −94.3931 89.8129 −482.460 81.0000 260.509
1.4 −10.0331 9.00000 68.6633 76.2471 −90.2980 −221.851 −367.847 81.0000 −764.996
1.5 −9.25942 9.00000 53.7368 86.4594 −83.3348 179.108 −201.270 81.0000 −800.564
1.6 −9.07137 9.00000 50.2897 −53.2375 −81.6423 −240.273 −165.912 81.0000 482.937
1.7 −9.06783 9.00000 50.2256 44.2978 −81.6105 220.734 −165.266 81.0000 −401.685
1.8 −8.27090 9.00000 36.4078 51.0812 −74.4381 −105.431 −36.4568 81.0000 −422.488
1.9 −8.19847 9.00000 35.2149 −45.6386 −73.7862 220.606 −26.3569 81.0000 374.166
1.10 −7.48942 9.00000 24.0914 −78.2626 −67.4048 120.657 59.2311 81.0000 586.141
1.11 −6.88987 9.00000 15.4703 33.7384 −62.0089 −83.7986 113.887 81.0000 −232.453
1.12 −6.63548 9.00000 12.0296 53.8111 −59.7193 160.843 132.513 81.0000 −357.062
1.13 −6.61244 9.00000 11.7244 −96.9704 −59.5120 60.0218 134.071 81.0000 641.211
1.14 −6.39288 9.00000 8.86887 14.4380 −57.5359 −145.227 147.874 81.0000 −92.3005
1.15 −5.34871 9.00000 −3.39129 −51.1571 −48.1384 −113.881 189.298 81.0000 273.624
1.16 −5.17582 9.00000 −5.21091 107.711 −46.5824 82.1489 192.597 81.0000 −557.493
1.17 −3.41938 9.00000 −20.3079 −90.5460 −30.7744 13.3868 178.860 81.0000 309.611
1.18 −2.98190 9.00000 −23.1083 −60.0447 −26.8371 112.410 164.327 81.0000 179.047
1.19 −2.72605 9.00000 −24.5687 −8.11409 −24.5344 36.1086 154.209 81.0000 22.1194
1.20 −2.00974 9.00000 −27.9609 51.4090 −18.0877 −80.5111 120.506 81.0000 −103.319
See all 49 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.49
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.d ✓ 49
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
633.6.a.d ✓ 49 1.a even 1 1 trivial