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Properties

Label 322.4.e.a
Level $322$
Weight $4$
Character orbit 322.e
Analytic conductor $18.999$
Analytic rank $0$
Dimension $22$
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] = [322,4,Mod(93,322)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(322, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("322.93");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 322 = 2 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 322.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: \(18.9986150218\)
Analytic rank: \(0\)
Dimension: \(22\)
Relative dimension: \(11\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 22 q - 22 q^{2} - 18 q^{3} - 44 q^{4} - 33 q^{5} + 72 q^{6} + 33 q^{7} + 176 q^{8} - 171 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 22 q - 22 q^{2} - 18 q^{3} - 44 q^{4} - 33 q^{5} + 72 q^{6} + 33 q^{7} + 176 q^{8} - 171 q^{9} - 66 q^{10} - 8 q^{11} - 72 q^{12} + 370 q^{13} - 372 q^{15} - 176 q^{16} - 107 q^{17} - 342 q^{18} - 114 q^{19} + 264 q^{20} + 141 q^{21} + 32 q^{22} - 253 q^{23} - 144 q^{24} - 494 q^{25} - 370 q^{26} + 1122 q^{27} - 132 q^{28} - 158 q^{29} + 372 q^{30} - 488 q^{31} - 352 q^{32} - 304 q^{33} + 428 q^{34} - 117 q^{35} + 1368 q^{36} + 41 q^{37} - 228 q^{38} - 425 q^{39} - 264 q^{40} + 1624 q^{41} - 828 q^{42} + 438 q^{43} - 32 q^{44} - 619 q^{45} - 506 q^{46} - 1141 q^{47} + 576 q^{48} + 121 q^{49} + 1976 q^{50} - 83 q^{51} - 740 q^{52} - 579 q^{53} - 1122 q^{54} + 1902 q^{55} + 264 q^{56} + 766 q^{57} + 158 q^{58} - 1695 q^{59} + 744 q^{60} - 1105 q^{61} + 1952 q^{62} + 583 q^{63} + 1408 q^{64} + 347 q^{65} - 608 q^{66} - 695 q^{67} - 428 q^{68} + 828 q^{69} - 1104 q^{70} + 130 q^{71} - 1368 q^{72} - 1935 q^{73} + 82 q^{74} - 1712 q^{75} + 912 q^{76} + 1045 q^{77} + 1700 q^{78} - 447 q^{79} - 528 q^{80} - 2203 q^{81} - 1624 q^{82} + 2778 q^{83} + 1092 q^{84} + 306 q^{85} - 438 q^{86} - 3354 q^{87} - 64 q^{88} - 2698 q^{89} + 2476 q^{90} + 2052 q^{91} + 2024 q^{92} + 686 q^{93} - 2282 q^{94} - 827 q^{95} - 576 q^{96} + 3178 q^{97} + 458 q^{98} - 3458 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
93.1 −1.00000 + 1.73205i −5.14226 − 8.90665i −2.00000 − 3.46410i 8.15435 − 14.1237i 20.5690 −18.0548 + 4.12587i 8.00000 −39.3856 + 68.2179i 16.3087 + 28.2475i
93.2 −1.00000 + 1.73205i −4.79988 − 8.31364i −2.00000 − 3.46410i −6.15920 + 10.6680i 19.1995 9.32988 − 15.9985i 8.00000 −32.5777 + 56.4262i −12.3184 − 21.3361i
93.3 −1.00000 + 1.73205i −3.65227 − 6.32591i −2.00000 − 3.46410i 0.623993 − 1.08079i 14.6091 3.16968 + 18.2470i 8.00000 −13.1781 + 22.8252i 1.24799 + 2.16157i
93.4 −1.00000 + 1.73205i −2.84399 − 4.92594i −2.00000 − 3.46410i −7.70651 + 13.3481i 11.3760 −17.7872 − 5.15912i 8.00000 −2.67661 + 4.63603i −15.4130 − 26.6961i
93.5 −1.00000 + 1.73205i −1.91677 − 3.31994i −2.00000 − 3.46410i −8.85109 + 15.3305i 7.66708 14.3167 + 11.7487i 8.00000 6.15199 − 10.6556i −17.7022 − 30.6611i
93.6 −1.00000 + 1.73205i −1.77936 − 3.08193i −2.00000 − 3.46410i 8.99920 − 15.5871i 7.11742 7.71568 − 16.8365i 8.00000 7.16779 − 12.4150i 17.9984 + 31.1742i
93.7 −1.00000 + 1.73205i −0.435789 − 0.754809i −2.00000 − 3.46410i 3.07860 − 5.33229i 1.74316 14.8584 − 11.0557i 8.00000 13.1202 − 22.7248i 6.15720 + 10.6646i
93.8 −1.00000 + 1.73205i 0.926564 + 1.60486i −2.00000 − 3.46410i −1.40500 + 2.43353i −3.70626 −18.3052 + 2.81391i 8.00000 11.7830 − 20.4087i −2.81000 − 4.86706i
93.9 −1.00000 + 1.73205i 3.05297 + 5.28791i −2.00000 − 3.46410i 2.60213 − 4.50702i −12.2119 4.58385 + 17.9440i 8.00000 −5.14131 + 8.90500i 5.20425 + 9.01403i
93.10 −1.00000 + 1.73205i 3.60587 + 6.24555i −2.00000 − 3.46410i −8.13490 + 14.0901i −14.4235 −1.58655 − 18.4522i 8.00000 −12.5046 + 21.6586i −16.2698 − 28.1801i
93.11 −1.00000 + 1.73205i 3.98491 + 6.90206i −2.00000 − 3.46410i −7.70158 + 13.3395i −15.9396 18.2596 + 3.09617i 8.00000 −18.2590 + 31.6255i −15.4032 − 26.6790i
277.1 −1.00000 − 1.73205i −5.14226 + 8.90665i −2.00000 + 3.46410i 8.15435 + 14.1237i 20.5690 −18.0548 − 4.12587i 8.00000 −39.3856 − 68.2179i 16.3087 − 28.2475i
277.2 −1.00000 − 1.73205i −4.79988 + 8.31364i −2.00000 + 3.46410i −6.15920 − 10.6680i 19.1995 9.32988 + 15.9985i 8.00000 −32.5777 − 56.4262i −12.3184 + 21.3361i
277.3 −1.00000 − 1.73205i −3.65227 + 6.32591i −2.00000 + 3.46410i 0.623993 + 1.08079i 14.6091 3.16968 − 18.2470i 8.00000 −13.1781 − 22.8252i 1.24799 − 2.16157i
277.4 −1.00000 − 1.73205i −2.84399 + 4.92594i −2.00000 + 3.46410i −7.70651 − 13.3481i 11.3760 −17.7872 + 5.15912i 8.00000 −2.67661 − 4.63603i −15.4130 + 26.6961i
277.5 −1.00000 − 1.73205i −1.91677 + 3.31994i −2.00000 + 3.46410i −8.85109 − 15.3305i 7.66708 14.3167 − 11.7487i 8.00000 6.15199 + 10.6556i −17.7022 + 30.6611i
277.6 −1.00000 − 1.73205i −1.77936 + 3.08193i −2.00000 + 3.46410i 8.99920 + 15.5871i 7.11742 7.71568 + 16.8365i 8.00000 7.16779 + 12.4150i 17.9984 − 31.1742i
277.7 −1.00000 − 1.73205i −0.435789 + 0.754809i −2.00000 + 3.46410i 3.07860 + 5.33229i 1.74316 14.8584 + 11.0557i 8.00000 13.1202 + 22.7248i 6.15720 − 10.6646i
277.8 −1.00000 − 1.73205i 0.926564 − 1.60486i −2.00000 + 3.46410i −1.40500 − 2.43353i −3.70626 −18.3052 − 2.81391i 8.00000 11.7830 + 20.4087i −2.81000 + 4.86706i
277.9 −1.00000 − 1.73205i 3.05297 − 5.28791i −2.00000 + 3.46410i 2.60213 + 4.50702i −12.2119 4.58385 − 17.9440i 8.00000 −5.14131 − 8.90500i 5.20425 − 9.01403i
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 93.11
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 322.4.e.a ✓ 22
7.c even 3 1 inner 322.4.e.a ✓ 22
7.c even 3 1 2254.4.a.y 11
7.d odd 6 1 2254.4.a.v 11
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
322.4.e.a ✓ 22 1.a even 1 1 trivial
322.4.e.a ✓ 22 7.c even 3 1 inner
2254.4.a.v 11 7.d odd 6 1
2254.4.a.y 11 7.c even 3 1