Properties

Label 368.4.a.e.1.1
Level $368$
Weight $4$
Character 368.1
Self dual yes
Analytic conductor $21.713$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [368,4,Mod(1,368)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("368.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(368, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 368 = 2^{4} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 368.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(21.7127028821\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 46)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 368.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+9.00000 q^{3} -20.0000 q^{5} -2.00000 q^{7} +54.0000 q^{9} +52.0000 q^{11} +43.0000 q^{13} -180.000 q^{15} -50.0000 q^{17} +74.0000 q^{19} -18.0000 q^{21} +23.0000 q^{23} +275.000 q^{25} +243.000 q^{27} -7.00000 q^{29} +273.000 q^{31} +468.000 q^{33} +40.0000 q^{35} -4.00000 q^{37} +387.000 q^{39} +123.000 q^{41} +152.000 q^{43} -1080.00 q^{45} -75.0000 q^{47} -339.000 q^{49} -450.000 q^{51} +86.0000 q^{53} -1040.00 q^{55} +666.000 q^{57} +444.000 q^{59} +262.000 q^{61} -108.000 q^{63} -860.000 q^{65} -764.000 q^{67} +207.000 q^{69} +21.0000 q^{71} +681.000 q^{73} +2475.00 q^{75} -104.000 q^{77} -426.000 q^{79} +729.000 q^{81} -902.000 q^{83} +1000.00 q^{85} -63.0000 q^{87} -1272.00 q^{89} -86.0000 q^{91} +2457.00 q^{93} -1480.00 q^{95} -342.000 q^{97} +2808.00 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 9.00000 1.73205 0.866025 − 0.500000i \(-0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(4\) 0 0
\(5\) −20.0000 −1.78885 −0.894427 − 0.447214i \(-0.852416\pi\)
−0.894427 + 0.447214i \(0.852416\pi\)
\(6\) 0 0
\(7\) −2.00000 −0.107990 −0.0539949 − 0.998541i \(-0.517195\pi\)
−0.0539949 + 0.998541i \(0.517195\pi\)
\(8\) 0 0
\(9\) 54.0000 2.00000
\(10\) 0 0
\(11\) 52.0000 1.42533 0.712663 − 0.701506i \(-0.247489\pi\)
0.712663 + 0.701506i \(0.247489\pi\)
\(12\) 0 0
\(13\) 43.0000 0.917389 0.458694 − 0.888594i \(-0.348317\pi\)
0.458694 + 0.888594i \(0.348317\pi\)
\(14\) 0 0
\(15\) −180.000 −3.09839
\(16\) 0 0
\(17\) −50.0000 −0.713340 −0.356670 − 0.934230i \(-0.616088\pi\)
−0.356670 + 0.934230i \(0.616088\pi\)
\(18\) 0 0
\(19\) 74.0000 0.893514 0.446757 − 0.894655i \(-0.352579\pi\)
0.446757 + 0.894655i \(0.352579\pi\)
\(20\) 0 0
\(21\) −18.0000 −0.187044
\(22\) 0 0
\(23\) 23.0000 0.208514
\(24\) 0 0
\(25\) 275.000 2.20000
\(26\) 0 0
\(27\) 243.000 1.73205
\(28\) 0 0
\(29\) −7.00000 −0.0448230 −0.0224115 − 0.999749i \(-0.507134\pi\)
−0.0224115 + 0.999749i \(0.507134\pi\)
\(30\) 0 0
\(31\) 273.000 1.58169 0.790843 − 0.612019i \(-0.209643\pi\)
0.790843 + 0.612019i \(0.209643\pi\)
\(32\) 0 0
\(33\) 468.000 2.46874
\(34\) 0 0
\(35\) 40.0000 0.193178
\(36\) 0 0
\(37\) −4.00000 −0.0177729 −0.00888643 − 0.999961i \(-0.502829\pi\)
−0.00888643 + 0.999961i \(0.502829\pi\)
\(38\) 0 0
\(39\) 387.000 1.58896
\(40\) 0 0
\(41\) 123.000 0.468521 0.234261 − 0.972174i \(-0.424733\pi\)
0.234261 + 0.972174i \(0.424733\pi\)
\(42\) 0 0
\(43\) 152.000 0.539065 0.269532 − 0.962991i \(-0.413131\pi\)
0.269532 + 0.962991i \(0.413131\pi\)
\(44\) 0 0
\(45\) −1080.00 −3.57771
\(46\) 0 0
\(47\) −75.0000 −0.232763 −0.116382 − 0.993205i \(-0.537130\pi\)
−0.116382 + 0.993205i \(0.537130\pi\)
\(48\) 0 0
\(49\) −339.000 −0.988338
\(50\) 0 0
\(51\) −450.000 −1.23554
\(52\) 0 0
\(53\) 86.0000 0.222887 0.111443 − 0.993771i \(-0.464453\pi\)
0.111443 + 0.993771i \(0.464453\pi\)
\(54\) 0 0
\(55\) −1040.00 −2.54970
\(56\) 0 0
\(57\) 666.000 1.54761
\(58\) 0 0
\(59\) 444.000 0.979727 0.489863 − 0.871799i \(-0.337047\pi\)
0.489863 + 0.871799i \(0.337047\pi\)
\(60\) 0 0
\(61\) 262.000 0.549929 0.274964 − 0.961454i \(-0.411334\pi\)
0.274964 + 0.961454i \(0.411334\pi\)
\(62\) 0 0
\(63\) −108.000 −0.215980
\(64\) 0 0
\(65\) −860.000 −1.64107
\(66\) 0 0
\(67\) −764.000 −1.39310 −0.696548 − 0.717510i \(-0.745282\pi\)
−0.696548 + 0.717510i \(0.745282\pi\)
\(68\) 0 0
\(69\) 207.000 0.361158
\(70\) 0 0
\(71\) 21.0000 0.0351020 0.0175510 − 0.999846i \(-0.494413\pi\)
0.0175510 + 0.999846i \(0.494413\pi\)
\(72\) 0 0
\(73\) 681.000 1.09185 0.545925 − 0.837834i \(-0.316178\pi\)
0.545925 + 0.837834i \(0.316178\pi\)
\(74\) 0 0
\(75\) 2475.00 3.81051
\(76\) 0 0
\(77\) −104.000 −0.153921
\(78\) 0 0
\(79\) −426.000 −0.606693 −0.303346 − 0.952880i \(-0.598104\pi\)
−0.303346 + 0.952880i \(0.598104\pi\)
\(80\) 0 0
\(81\) 729.000 1.00000
\(82\) 0 0
\(83\) −902.000 −1.19286 −0.596430 − 0.802665i \(-0.703415\pi\)
−0.596430 + 0.802665i \(0.703415\pi\)
\(84\) 0 0
\(85\) 1000.00 1.27606
\(86\) 0 0
\(87\) −63.0000 −0.0776357
\(88\) 0 0
\(89\) −1272.00 −1.51496 −0.757482 − 0.652856i \(-0.773570\pi\)
−0.757482 + 0.652856i \(0.773570\pi\)
\(90\) 0 0
\(91\) −86.0000 −0.0990687
\(92\) 0 0
\(93\) 2457.00 2.73956
\(94\) 0 0
\(95\) −1480.00 −1.59837
\(96\) 0 0
\(97\) −342.000 −0.357988 −0.178994 − 0.983850i \(-0.557284\pi\)
−0.178994 + 0.983850i \(0.557284\pi\)
\(98\) 0 0
\(99\) 2808.00 2.85065
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 368.4.a.e.1.1 1
4.3 odd 2 46.4.a.b.1.1 ✓ 1
8.3 odd 2 1472.4.a.j.1.1 1
8.5 even 2 1472.4.a.a.1.1 1
12.11 even 2 414.4.a.b.1.1 1
20.3 even 4 1150.4.b.a.599.1 2
20.7 even 4 1150.4.b.a.599.2 2
20.19 odd 2 1150.4.a.d.1.1 1
28.27 even 2 2254.4.a.b.1.1 1
92.91 even 2 1058.4.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
46.4.a.b.1.1 ✓ 1 4.3 odd 2
368.4.a.e.1.1 1 1.1 even 1 trivial
414.4.a.b.1.1 1 12.11 even 2
1058.4.a.b.1.1 1 92.91 even 2
1150.4.a.d.1.1 1 20.19 odd 2
1150.4.b.a.599.1 2 20.3 even 4
1150.4.b.a.599.2 2 20.7 even 4
1472.4.a.a.1.1 1 8.5 even 2
1472.4.a.j.1.1 1 8.3 odd 2
2254.4.a.b.1.1 1 28.27 even 2