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Properties

Label 633.6.a.a
Level $633$
Weight $6$
Character orbit 633.a
Self dual yes
Analytic conductor $101.523$
Analytic rank $1$
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] = [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: \(1\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 38 q - 16 q^{2} + 342 q^{3} + 448 q^{4} - 81 q^{5} - 144 q^{6} - 755 q^{7} - 771 q^{8} + 3078 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 38 q - 16 q^{2} + 342 q^{3} + 448 q^{4} - 81 q^{5} - 144 q^{6} - 755 q^{7} - 771 q^{8} + 3078 q^{9} - 1200 q^{10} - 333 q^{11} + 4032 q^{12} - 2585 q^{13} - 1937 q^{14} - 729 q^{15} + 1944 q^{16} - 2733 q^{17} - 1296 q^{18} - 6278 q^{19} - 5371 q^{20} - 6795 q^{21} - 10614 q^{22} - 16511 q^{23} - 6939 q^{24} + 7875 q^{25} - 1925 q^{26} + 27702 q^{27} - 19830 q^{28} - 10174 q^{29} - 10800 q^{30} - 29890 q^{31} - 18618 q^{32} - 2997 q^{33} - 23109 q^{34} - 25628 q^{35} + 36288 q^{36} - 59337 q^{37} - 27415 q^{38} - 23265 q^{39} - 46066 q^{40} - 3207 q^{41} - 17433 q^{42} - 52399 q^{43} - 17745 q^{44} - 6561 q^{45} - 37693 q^{46} - 64965 q^{47} + 17496 q^{48} + 3649 q^{49} - 68244 q^{50} - 24597 q^{51} - 98428 q^{52} - 124615 q^{53} - 11664 q^{54} - 141189 q^{55} - 198860 q^{56} - 56502 q^{57} - 185669 q^{58} - 115709 q^{59} - 48339 q^{60} - 237132 q^{61} - 175466 q^{62} - 61155 q^{63} - 184797 q^{64} - 200915 q^{65} - 95526 q^{66} - 235458 q^{67} - 331041 q^{68} - 148599 q^{69} - 231911 q^{70} - 219951 q^{71} - 62451 q^{72} - 132528 q^{73} - 125813 q^{74} + 70875 q^{75} - 300627 q^{76} - 248599 q^{77} - 17325 q^{78} - 346545 q^{79} - 220211 q^{80} + 249318 q^{81} - 177411 q^{82} - 327109 q^{83} - 178470 q^{84} - 421323 q^{85} - 56438 q^{86} - 91566 q^{87} - 270585 q^{88} - 4941 q^{89} - 97200 q^{90} - 380913 q^{91} - 126981 q^{92} - 269010 q^{93} - 16032 q^{94} - 93929 q^{95} - 167562 q^{96} - 547281 q^{97} + 335113 q^{98} - 26973 q^{99}+O(q^{100}) \) 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 −10.4017 9.00000 76.1945 −8.51970 −93.6149 141.235 −459.697 81.0000 88.6191
1.2 −10.1090 9.00000 70.1910 54.9734 −90.9806 13.2720 −386.071 81.0000 −555.724
1.3 −10.0030 9.00000 68.0592 66.9836 −90.0266 47.2119 −360.699 81.0000 −670.034
1.4 −9.93720 9.00000 66.7479 −71.8940 −89.4348 −141.887 −345.297 81.0000 714.425
1.5 −9.14008 9.00000 51.5410 −51.9075 −82.2607 −35.1556 −178.606 81.0000 474.438
1.6 −9.08122 9.00000 50.4685 68.9914 −81.7310 −113.463 −167.716 81.0000 −626.526
1.7 −8.17439 9.00000 34.8206 −67.9039 −73.5695 −101.224 −23.0564 81.0000 555.073
1.8 −8.06351 9.00000 33.0202 −44.0907 −72.5716 67.7851 −8.22611 81.0000 355.525
1.9 −7.09051 9.00000 18.2753 −16.7086 −63.8146 155.360 97.3151 81.0000 118.473
1.10 −6.44015 9.00000 9.47547 56.2247 −57.9613 −22.3893 145.061 81.0000 −362.095
1.11 −5.53099 9.00000 −1.40810 108.108 −49.7789 −65.4532 184.780 81.0000 −597.945
1.12 −4.97105 9.00000 −7.28861 −17.3269 −44.7395 −201.425 195.306 81.0000 86.1329
1.13 −3.77253 9.00000 −17.7680 4.41504 −33.9528 230.418 187.751 81.0000 −16.6559
1.14 −3.70554 9.00000 −18.2690 −93.3353 −33.3499 −214.990 186.274 81.0000 345.858
1.15 −3.34184 9.00000 −20.8321 −33.4123 −30.0765 60.0152 176.556 81.0000 111.658
1.16 −3.33343 9.00000 −20.8882 73.5020 −30.0009 −244.443 176.299 81.0000 −245.014
1.17 −3.06899 9.00000 −22.5813 −8.64608 −27.6209 127.032 167.510 81.0000 26.5348
1.18 −1.48146 9.00000 −29.8053 54.4111 −13.3332 0.868198 91.5622 81.0000 −80.6080
1.19 −0.716854 9.00000 −31.4861 51.0870 −6.45168 −147.084 45.5102 81.0000 −36.6219
1.20 0.351933 9.00000 −31.8761 −96.7759 3.16739 139.395 −22.4801 81.0000 −34.0586
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
\(3\) \(-1\)
\(211\) \(-1\)

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