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Properties

Label 322.4.e.b
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.

\(\operatorname{Tr}(f)(q) = \) \( 22 q - 22 q^{2} + 6 q^{3} - 44 q^{4} + 27 q^{5} - 24 q^{6} + q^{7} + 176 q^{8} - 59 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 22 q - 22 q^{2} + 6 q^{3} - 44 q^{4} + 27 q^{5} - 24 q^{6} + q^{7} + 176 q^{8} - 59 q^{9} + 54 q^{10} - 56 q^{11} + 24 q^{12} - 206 q^{13} - 64 q^{14} + 124 q^{15} - 176 q^{16} + 157 q^{17} - 118 q^{18} + 266 q^{19} - 216 q^{20} - 137 q^{21} + 224 q^{22} + 253 q^{23} + 48 q^{24} - 142 q^{25} + 206 q^{26} - 1062 q^{27} + 124 q^{28} - 390 q^{29} - 124 q^{30} + 316 q^{31} - 352 q^{32} + 682 q^{33} - 628 q^{34} + 539 q^{35} + 472 q^{36} - 347 q^{37} + 532 q^{38} + 99 q^{39} + 216 q^{40} - 1080 q^{41} + 116 q^{42} + 386 q^{43} - 224 q^{44} + 609 q^{45} + 506 q^{46} + 731 q^{47} - 192 q^{48} + 505 q^{49} + 568 q^{50} + 303 q^{51} + 412 q^{52} - 413 q^{53} + 1062 q^{54} - 3618 q^{55} + 8 q^{56} + 686 q^{57} + 390 q^{58} + 1061 q^{59} - 248 q^{60} + 1157 q^{61} - 1264 q^{62} - 557 q^{63} + 1408 q^{64} + 37 q^{65} + 1364 q^{66} + 841 q^{67} + 628 q^{68} + 276 q^{69} - 632 q^{70} + 1738 q^{71} - 472 q^{72} + 409 q^{73} - 694 q^{74} + 2024 q^{75} - 2128 q^{76} + 565 q^{77} - 396 q^{78} - 247 q^{79} + 432 q^{80} - 1019 q^{81} + 1080 q^{82} - 5726 q^{83} + 316 q^{84} + 1570 q^{85} - 386 q^{86} + 670 q^{87} - 448 q^{88} + 1610 q^{89} - 2436 q^{90} - 1766 q^{91} - 2024 q^{92} + 1290 q^{93} + 1462 q^{94} + 209 q^{95} + 192 q^{96} - 3566 q^{97} - 166 q^{98} + 202 q^{99}+O(q^{100}) \) 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 −3.98418 − 6.90081i −2.00000 − 3.46410i 0.156517 − 0.271095i 15.9367 18.4897 + 1.06335i 8.00000 −18.2475 + 31.6055i 0.313034 + 0.542190i
93.2 −1.00000 + 1.73205i −2.93524 − 5.08398i −2.00000 − 3.46410i 2.59907 − 4.50173i 11.7410 −10.4233 − 15.3086i 8.00000 −3.73127 + 6.46274i 5.19815 + 9.00345i
93.3 −1.00000 + 1.73205i −2.42223 − 4.19542i −2.00000 − 3.46410i −3.15820 + 5.47016i 9.68892 −15.0196 + 10.8356i 8.00000 1.76561 − 3.05813i −6.31640 − 10.9403i
93.4 −1.00000 + 1.73205i −2.22951 − 3.86162i −2.00000 − 3.46410i 5.94309 − 10.2937i 8.91803 7.68137 + 16.8522i 8.00000 3.55858 − 6.16365i 11.8862 + 20.5875i
93.5 −1.00000 + 1.73205i −0.188872 − 0.327135i −2.00000 − 3.46410i 9.79257 − 16.9612i 0.755487 −18.4979 + 0.910411i 8.00000 13.4287 − 23.2591i 19.5851 + 33.9224i
93.6 −1.00000 + 1.73205i −0.0271017 − 0.0469415i −2.00000 − 3.46410i −6.54352 + 11.3337i 0.108407 −2.84830 − 18.2999i 8.00000 13.4985 − 23.3801i −13.0870 − 22.6674i
93.7 −1.00000 + 1.73205i 1.12707 + 1.95214i −2.00000 − 3.46410i −2.97911 + 5.15996i −4.50828 18.5180 − 0.291445i 8.00000 10.9594 − 18.9823i −5.95821 − 10.3199i
93.8 −1.00000 + 1.73205i 1.87478 + 3.24721i −2.00000 − 3.46410i −7.08923 + 12.2789i −7.49911 −5.26649 + 17.7557i 8.00000 6.47043 − 11.2071i −14.1785 − 24.5578i
93.9 −1.00000 + 1.73205i 2.90702 + 5.03510i −2.00000 − 3.46410i 11.0010 − 19.0544i −11.6281 14.5875 + 11.4108i 8.00000 −3.40150 + 5.89157i 22.0021 + 38.1087i
93.10 −1.00000 + 1.73205i 3.73605 + 6.47103i −2.00000 − 3.46410i 2.65469 − 4.59806i −14.9442 9.31488 − 16.0073i 8.00000 −14.4162 + 24.9695i 5.30939 + 9.19613i
93.11 −1.00000 + 1.73205i 5.14222 + 8.90659i −2.00000 − 3.46410i 1.12307 − 1.94522i −20.5689 −16.0357 + 9.26579i 8.00000 −39.3849 + 68.2166i 2.24614 + 3.89043i
277.1 −1.00000 − 1.73205i −3.98418 + 6.90081i −2.00000 + 3.46410i 0.156517 + 0.271095i 15.9367 18.4897 − 1.06335i 8.00000 −18.2475 − 31.6055i 0.313034 − 0.542190i
277.2 −1.00000 − 1.73205i −2.93524 + 5.08398i −2.00000 + 3.46410i 2.59907 + 4.50173i 11.7410 −10.4233 + 15.3086i 8.00000 −3.73127 − 6.46274i 5.19815 − 9.00345i
277.3 −1.00000 − 1.73205i −2.42223 + 4.19542i −2.00000 + 3.46410i −3.15820 − 5.47016i 9.68892 −15.0196 − 10.8356i 8.00000 1.76561 + 3.05813i −6.31640 + 10.9403i
277.4 −1.00000 − 1.73205i −2.22951 + 3.86162i −2.00000 + 3.46410i 5.94309 + 10.2937i 8.91803 7.68137 − 16.8522i 8.00000 3.55858 + 6.16365i 11.8862 − 20.5875i
277.5 −1.00000 − 1.73205i −0.188872 + 0.327135i −2.00000 + 3.46410i 9.79257 + 16.9612i 0.755487 −18.4979 − 0.910411i 8.00000 13.4287 + 23.2591i 19.5851 − 33.9224i
277.6 −1.00000 − 1.73205i −0.0271017 + 0.0469415i −2.00000 + 3.46410i −6.54352 − 11.3337i 0.108407 −2.84830 + 18.2999i 8.00000 13.4985 + 23.3801i −13.0870 + 22.6674i
277.7 −1.00000 − 1.73205i 1.12707 − 1.95214i −2.00000 + 3.46410i −2.97911 − 5.15996i −4.50828 18.5180 + 0.291445i 8.00000 10.9594 + 18.9823i −5.95821 + 10.3199i
277.8 −1.00000 − 1.73205i 1.87478 − 3.24721i −2.00000 + 3.46410i −7.08923 − 12.2789i −7.49911 −5.26649 − 17.7557i 8.00000 6.47043 + 11.2071i −14.1785 + 24.5578i
277.9 −1.00000 − 1.73205i 2.90702 − 5.03510i −2.00000 + 3.46410i 11.0010 + 19.0544i −11.6281 14.5875 − 11.4108i 8.00000 −3.40150 − 5.89157i 22.0021 − 38.1087i
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.b ✓ 22
7.c even 3 1 inner 322.4.e.b ✓ 22
7.c even 3 1 2254.4.a.w 11
7.d odd 6 1 2254.4.a.x 11
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
322.4.e.b ✓ 22 1.a even 1 1 trivial
322.4.e.b ✓ 22 7.c even 3 1 inner
2254.4.a.w 11 7.c even 3 1
2254.4.a.x 11 7.d odd 6 1