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Properties

Label 198.6.e.b
Level $198$
Weight $6$
Character orbit 198.e
Analytic conductor $31.756$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [198,6,Mod(67,198)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(198, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("198.67");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 198 = 2 \cdot 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 198.e (of order \(3\), degree \(2\), minimal)

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: no
Analytic conductor: \(31.7559963230\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 48 q^{2} + 21 q^{3} - 192 q^{4} - 7 q^{5} + 168 q^{6} + 24 q^{7} - 1536 q^{8} - 543 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 48 q^{2} + 21 q^{3} - 192 q^{4} - 7 q^{5} + 168 q^{6} + 24 q^{7} - 1536 q^{8} - 543 q^{9} - 56 q^{10} - 1452 q^{11} + 336 q^{12} - 688 q^{13} - 96 q^{14} - 1500 q^{15} - 3072 q^{16} + 3512 q^{17} + 1164 q^{18} - 3836 q^{19} - 112 q^{20} + 1842 q^{21} + 5808 q^{22} - 680 q^{23} - 1344 q^{24} - 7759 q^{25} - 5504 q^{26} + 3024 q^{27} - 768 q^{28} - 3210 q^{29} - 4596 q^{30} + 857 q^{31} + 12288 q^{32} - 5082 q^{33} + 7024 q^{34} + 29824 q^{35} + 13344 q^{36} + 7166 q^{37} - 7672 q^{38} - 23076 q^{39} + 448 q^{40} - 27726 q^{41} - 10680 q^{42} - 15092 q^{43} + 46464 q^{44} + 38667 q^{45} - 5440 q^{46} - 28938 q^{47} - 10752 q^{48} + 17322 q^{49} + 31036 q^{50} + 37032 q^{51} - 11008 q^{52} - 12652 q^{53} + 59832 q^{54} + 1694 q^{55} - 1536 q^{56} - 168660 q^{57} + 12840 q^{58} - 33123 q^{59} + 5616 q^{60} + 36262 q^{61} + 6856 q^{62} + 47496 q^{63} + 98304 q^{64} - 31470 q^{65} - 10164 q^{66} + 35563 q^{67} - 28096 q^{68} + 123459 q^{69} + 59648 q^{70} + 295146 q^{71} + 34752 q^{72} + 12056 q^{73} + 14332 q^{74} - 244221 q^{75} + 30688 q^{76} + 2904 q^{77} - 64200 q^{78} + 128058 q^{79} + 3584 q^{80} + 90621 q^{81} - 221808 q^{82} - 209888 q^{83} - 72192 q^{84} + 228182 q^{85} + 60368 q^{86} - 89802 q^{87} + 92928 q^{88} - 46724 q^{89} + 96972 q^{90} - 305924 q^{91} - 10880 q^{92} - 187767 q^{93} + 115752 q^{94} - 118660 q^{95} - 21504 q^{96} + 240799 q^{97} + 138576 q^{98} - 35211 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1 2.00000 + 3.46410i −15.1654 + 3.60721i −8.00000 + 13.8564i 23.5761 − 40.8350i −42.8265 − 45.3199i 19.3963 + 33.5953i −64.0000 216.976 − 109.409i 188.609
67.2 2.00000 + 3.46410i −14.2828 − 6.24519i −8.00000 + 13.8564i −2.34189 + 4.05628i −6.93156 − 61.9674i −87.2715 − 151.159i −64.0000 164.995 + 178.397i −18.7352
67.3 2.00000 + 3.46410i −8.80127 − 12.8661i −8.00000 + 13.8564i 25.5446 − 44.2445i 26.9671 − 56.2208i 79.0911 + 136.990i −64.0000 −88.0752 + 226.477i 204.357
67.4 2.00000 + 3.46410i −7.29840 + 13.7744i −8.00000 + 13.8564i −6.92025 + 11.9862i −62.3126 + 2.26633i 62.0895 + 107.542i −64.0000 −136.467 − 201.062i −55.3620
67.5 2.00000 + 3.46410i −2.92981 − 15.3107i −8.00000 + 13.8564i −33.1336 + 57.3891i 47.1780 − 40.7705i −58.2968 − 100.973i −64.0000 −225.832 + 89.7147i −265.069
67.6 2.00000 + 3.46410i −2.38562 + 15.4048i −8.00000 + 13.8564i −45.8911 + 79.4857i −58.1351 + 22.5456i −48.6889 − 84.3317i −64.0000 −231.618 − 73.5002i −367.129
67.7 2.00000 + 3.46410i 3.31846 + 15.2311i −8.00000 + 13.8564i 25.8529 − 44.7785i −46.1253 + 41.9578i −53.8618 − 93.2913i −64.0000 −220.976 + 101.088i 206.823
67.8 2.00000 + 3.46410i 6.46136 − 14.1863i −8.00000 + 13.8564i −15.4896 + 26.8288i 62.0655 − 5.98979i 29.1617 + 50.5095i −64.0000 −159.502 − 183.325i −123.917
67.9 2.00000 + 3.46410i 8.66870 − 12.9558i −8.00000 + 13.8564i 51.5279 − 89.2490i 62.2177 + 4.11761i 17.1308 + 29.6714i −64.0000 −92.7072 − 224.621i 412.223
67.10 2.00000 + 3.46410i 13.5237 − 7.75299i −8.00000 + 13.8564i 2.38434 − 4.12980i 53.9046 + 31.3416i −89.1267 − 154.372i −64.0000 122.782 − 209.699i 19.0747
67.11 2.00000 + 3.46410i 14.1729 + 6.49073i −8.00000 + 13.8564i 23.5822 − 40.8457i 5.86120 + 62.0777i 78.0548 + 135.195i −64.0000 158.741 + 183.985i 188.658
67.12 2.00000 + 3.46410i 15.2181 − 3.37772i −8.00000 + 13.8564i −52.1915 + 90.3984i 42.1370 + 45.9617i 64.3216 + 111.408i −64.0000 220.182 − 102.805i −417.532
133.1 2.00000 − 3.46410i −15.1654 − 3.60721i −8.00000 − 13.8564i 23.5761 + 40.8350i −42.8265 + 45.3199i 19.3963 − 33.5953i −64.0000 216.976 + 109.409i 188.609
133.2 2.00000 − 3.46410i −14.2828 + 6.24519i −8.00000 − 13.8564i −2.34189 − 4.05628i −6.93156 + 61.9674i −87.2715 + 151.159i −64.0000 164.995 − 178.397i −18.7352
133.3 2.00000 − 3.46410i −8.80127 + 12.8661i −8.00000 − 13.8564i 25.5446 + 44.2445i 26.9671 + 56.2208i 79.0911 − 136.990i −64.0000 −88.0752 − 226.477i 204.357
133.4 2.00000 − 3.46410i −7.29840 − 13.7744i −8.00000 − 13.8564i −6.92025 − 11.9862i −62.3126 − 2.26633i 62.0895 − 107.542i −64.0000 −136.467 + 201.062i −55.3620
133.5 2.00000 − 3.46410i −2.92981 + 15.3107i −8.00000 − 13.8564i −33.1336 − 57.3891i 47.1780 + 40.7705i −58.2968 + 100.973i −64.0000 −225.832 − 89.7147i −265.069
133.6 2.00000 − 3.46410i −2.38562 − 15.4048i −8.00000 − 13.8564i −45.8911 − 79.4857i −58.1351 − 22.5456i −48.6889 + 84.3317i −64.0000 −231.618 + 73.5002i −367.129
133.7 2.00000 − 3.46410i 3.31846 − 15.2311i −8.00000 − 13.8564i 25.8529 + 44.7785i −46.1253 − 41.9578i −53.8618 + 93.2913i −64.0000 −220.976 − 101.088i 206.823
133.8 2.00000 − 3.46410i 6.46136 + 14.1863i −8.00000 − 13.8564i −15.4896 − 26.8288i 62.0655 + 5.98979i 29.1617 − 50.5095i −64.0000 −159.502 + 183.325i −123.917
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 67.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 198.6.e.b ✓ 24
3.b odd 2 1 594.6.e.a 24
9.c even 3 1 inner 198.6.e.b ✓ 24
9.d odd 6 1 594.6.e.a 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
198.6.e.b ✓ 24 1.a even 1 1 trivial
198.6.e.b ✓ 24 9.c even 3 1 inner
594.6.e.a 24 3.b odd 2 1
594.6.e.a 24 9.d odd 6 1