Properties

Label 46.4.a.b.1.1
Level $46$
Weight $4$
Character 46.1
Self dual yes
Analytic conductor $2.714$
Analytic rank $1$
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] = [46,4,Mod(1,46)] 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("46.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(46, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 46 = 2 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 46.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(2.71408786026\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 46.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.00000 q^{2} -9.00000 q^{3} +4.00000 q^{4} -20.0000 q^{5} -18.0000 q^{6} +2.00000 q^{7} +8.00000 q^{8} +54.0000 q^{9} -40.0000 q^{10} -52.0000 q^{11} -36.0000 q^{12} +43.0000 q^{13} +4.00000 q^{14} +180.000 q^{15} +16.0000 q^{16} -50.0000 q^{17} +108.000 q^{18} -74.0000 q^{19} -80.0000 q^{20} -18.0000 q^{21} -104.000 q^{22} -23.0000 q^{23} -72.0000 q^{24} +275.000 q^{25} +86.0000 q^{26} -243.000 q^{27} +8.00000 q^{28} -7.00000 q^{29} +360.000 q^{30} -273.000 q^{31} +32.0000 q^{32} +468.000 q^{33} -100.000 q^{34} -40.0000 q^{35} +216.000 q^{36} -4.00000 q^{37} -148.000 q^{38} -387.000 q^{39} -160.000 q^{40} +123.000 q^{41} -36.0000 q^{42} -152.000 q^{43} -208.000 q^{44} -1080.00 q^{45} -46.0000 q^{46} +75.0000 q^{47} -144.000 q^{48} -339.000 q^{49} +550.000 q^{50} +450.000 q^{51} +172.000 q^{52} +86.0000 q^{53} -486.000 q^{54} +1040.00 q^{55} +16.0000 q^{56} +666.000 q^{57} -14.0000 q^{58} -444.000 q^{59} +720.000 q^{60} +262.000 q^{61} -546.000 q^{62} +108.000 q^{63} +64.0000 q^{64} -860.000 q^{65} +936.000 q^{66} +764.000 q^{67} -200.000 q^{68} +207.000 q^{69} -80.0000 q^{70} -21.0000 q^{71} +432.000 q^{72} +681.000 q^{73} -8.00000 q^{74} -2475.00 q^{75} -296.000 q^{76} -104.000 q^{77} -774.000 q^{78} +426.000 q^{79} -320.000 q^{80} +729.000 q^{81} +246.000 q^{82} +902.000 q^{83} -72.0000 q^{84} +1000.00 q^{85} -304.000 q^{86} +63.0000 q^{87} -416.000 q^{88} -1272.00 q^{89} -2160.00 q^{90} +86.0000 q^{91} -92.0000 q^{92} +2457.00 q^{93} +150.000 q^{94} +1480.00 q^{95} -288.000 q^{96} -342.000 q^{97} -678.000 q^{98} -2808.00 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 0.707107
\(3\) −9.00000 −1.73205 −0.866025 − 0.500000i \(-0.833333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(4\) 4.00000 0.500000
\(5\) −20.0000 −1.78885 −0.894427 − 0.447214i \(-0.852416\pi\)
−0.894427 + 0.447214i \(0.852416\pi\)
\(6\) −18.0000 −1.22474
\(7\) 2.00000 0.107990 0.0539949 − 0.998541i \(-0.482805\pi\)
0.0539949 + 0.998541i \(0.482805\pi\)
\(8\) 8.00000 0.353553
\(9\) 54.0000 2.00000
\(10\) −40.0000 −1.26491
\(11\) −52.0000 −1.42533 −0.712663 − 0.701506i \(-0.752511\pi\)
−0.712663 + 0.701506i \(0.752511\pi\)
\(12\) −36.0000 −0.866025
\(13\) 43.0000 0.917389 0.458694 − 0.888594i \(-0.348317\pi\)
0.458694 + 0.888594i \(0.348317\pi\)
\(14\) 4.00000 0.0763604
\(15\) 180.000 3.09839
\(16\) 16.0000 0.250000
\(17\) −50.0000 −0.713340 −0.356670 − 0.934230i \(-0.616088\pi\)
−0.356670 + 0.934230i \(0.616088\pi\)
\(18\) 108.000 1.41421
\(19\) −74.0000 −0.893514 −0.446757 − 0.894655i \(-0.647421\pi\)
−0.446757 + 0.894655i \(0.647421\pi\)
\(20\) −80.0000 −0.894427
\(21\) −18.0000 −0.187044
\(22\) −104.000 −1.00786
\(23\) −23.0000 −0.208514
\(24\) −72.0000 −0.612372
\(25\) 275.000 2.20000
\(26\) 86.0000 0.648692
\(27\) −243.000 −1.73205
\(28\) 8.00000 0.0539949
\(29\) −7.00000 −0.0448230 −0.0224115 − 0.999749i \(-0.507134\pi\)
−0.0224115 + 0.999749i \(0.507134\pi\)
\(30\) 360.000 2.19089
\(31\) −273.000 −1.58169 −0.790843 − 0.612019i \(-0.790357\pi\)
−0.790843 + 0.612019i \(0.790357\pi\)
\(32\) 32.0000 0.176777
\(33\) 468.000 2.46874
\(34\) −100.000 −0.504408
\(35\) −40.0000 −0.193178
\(36\) 216.000 1.00000
\(37\) −4.00000 −0.0177729 −0.00888643 − 0.999961i \(-0.502829\pi\)
−0.00888643 + 0.999961i \(0.502829\pi\)
\(38\) −148.000 −0.631810
\(39\) −387.000 −1.58896
\(40\) −160.000 −0.632456
\(41\) 123.000 0.468521 0.234261 − 0.972174i \(-0.424733\pi\)
0.234261 + 0.972174i \(0.424733\pi\)
\(42\) −36.0000 −0.132260
\(43\) −152.000 −0.539065 −0.269532 − 0.962991i \(-0.586869\pi\)
−0.269532 + 0.962991i \(0.586869\pi\)
\(44\) −208.000 −0.712663
\(45\) −1080.00 −3.57771
\(46\) −46.0000 −0.147442
\(47\) 75.0000 0.232763 0.116382 − 0.993205i \(-0.462870\pi\)
0.116382 + 0.993205i \(0.462870\pi\)
\(48\) −144.000 −0.433013
\(49\) −339.000 −0.988338
\(50\) 550.000 1.55563
\(51\) 450.000 1.23554
\(52\) 172.000 0.458694
\(53\) 86.0000 0.222887 0.111443 − 0.993771i \(-0.464453\pi\)
0.111443 + 0.993771i \(0.464453\pi\)
\(54\) −486.000 −1.22474
\(55\) 1040.00 2.54970
\(56\) 16.0000 0.0381802
\(57\) 666.000 1.54761
\(58\) −14.0000 −0.0316947
\(59\) −444.000 −0.979727 −0.489863 − 0.871799i \(-0.662953\pi\)
−0.489863 + 0.871799i \(0.662953\pi\)
\(60\) 720.000 1.54919
\(61\) 262.000 0.549929 0.274964 − 0.961454i \(-0.411334\pi\)
0.274964 + 0.961454i \(0.411334\pi\)
\(62\) −546.000 −1.11842
\(63\) 108.000 0.215980
\(64\) 64.0000 0.125000
\(65\) −860.000 −1.64107
\(66\) 936.000 1.74566
\(67\) 764.000 1.39310 0.696548 − 0.717510i \(-0.254718\pi\)
0.696548 + 0.717510i \(0.254718\pi\)
\(68\) −200.000 −0.356670
\(69\) 207.000 0.361158
\(70\) −80.0000 −0.136598
\(71\) −21.0000 −0.0351020 −0.0175510 − 0.999846i \(-0.505587\pi\)
−0.0175510 + 0.999846i \(0.505587\pi\)
\(72\) 432.000 0.707107
\(73\) 681.000 1.09185 0.545925 − 0.837834i \(-0.316178\pi\)
0.545925 + 0.837834i \(0.316178\pi\)
\(74\) −8.00000 −0.0125673
\(75\) −2475.00 −3.81051
\(76\) −296.000 −0.446757
\(77\) −104.000 −0.153921
\(78\) −774.000 −1.12357
\(79\) 426.000 0.606693 0.303346 − 0.952880i \(-0.401896\pi\)
0.303346 + 0.952880i \(0.401896\pi\)
\(80\) −320.000 −0.447214
\(81\) 729.000 1.00000
\(82\) 246.000 0.331295
\(83\) 902.000 1.19286 0.596430 − 0.802665i \(-0.296585\pi\)
0.596430 + 0.802665i \(0.296585\pi\)
\(84\) −72.0000 −0.0935220
\(85\) 1000.00 1.27606
\(86\) −304.000 −0.381176
\(87\) 63.0000 0.0776357
\(88\) −416.000 −0.503929
\(89\) −1272.00 −1.51496 −0.757482 − 0.652856i \(-0.773570\pi\)
−0.757482 + 0.652856i \(0.773570\pi\)
\(90\) −2160.00 −2.52982
\(91\) 86.0000 0.0990687
\(92\) −92.0000 −0.104257
\(93\) 2457.00 2.73956
\(94\) 150.000 0.164588
\(95\) 1480.00 1.59837
\(96\) −288.000 −0.306186
\(97\) −342.000 −0.357988 −0.178994 − 0.983850i \(-0.557284\pi\)
−0.178994 + 0.983850i \(0.557284\pi\)
\(98\) −678.000 −0.698861
\(99\) −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 46.4.a.b.1.1 ✓ 1
3.2 odd 2 414.4.a.b.1.1 1
4.3 odd 2 368.4.a.e.1.1 1
5.2 odd 4 1150.4.b.a.599.2 2
5.3 odd 4 1150.4.b.a.599.1 2
5.4 even 2 1150.4.a.d.1.1 1
7.6 odd 2 2254.4.a.b.1.1 1
8.3 odd 2 1472.4.a.a.1.1 1
8.5 even 2 1472.4.a.j.1.1 1
23.22 odd 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 1.1 even 1 trivial
368.4.a.e.1.1 1 4.3 odd 2
414.4.a.b.1.1 1 3.2 odd 2
1058.4.a.b.1.1 1 23.22 odd 2
1150.4.a.d.1.1 1 5.4 even 2
1150.4.b.a.599.1 2 5.3 odd 4
1150.4.b.a.599.2 2 5.2 odd 4
1472.4.a.a.1.1 1 8.3 odd 2
1472.4.a.j.1.1 1 8.5 even 2
2254.4.a.b.1.1 1 7.6 odd 2