Properties

Label 414.4.a.b.1.1
Level $414$
Weight $4$
Character 414.1
Self dual yes
Analytic conductor $24.427$
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] = [414,4,Mod(1,414)] 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("414.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(414, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 414 = 2 \cdot 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 414.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-2,0,4,20,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(24.4267907424\)
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 \(\chi\) \(=\) 414.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} +4.00000 q^{4} +20.0000 q^{5} +2.00000 q^{7} -8.00000 q^{8} -40.0000 q^{10} +52.0000 q^{11} +43.0000 q^{13} -4.00000 q^{14} +16.0000 q^{16} +50.0000 q^{17} -74.0000 q^{19} +80.0000 q^{20} -104.000 q^{22} +23.0000 q^{23} +275.000 q^{25} -86.0000 q^{26} +8.00000 q^{28} +7.00000 q^{29} -273.000 q^{31} -32.0000 q^{32} -100.000 q^{34} +40.0000 q^{35} -4.00000 q^{37} +148.000 q^{38} -160.000 q^{40} -123.000 q^{41} -152.000 q^{43} +208.000 q^{44} -46.0000 q^{46} -75.0000 q^{47} -339.000 q^{49} -550.000 q^{50} +172.000 q^{52} -86.0000 q^{53} +1040.00 q^{55} -16.0000 q^{56} -14.0000 q^{58} +444.000 q^{59} +262.000 q^{61} +546.000 q^{62} +64.0000 q^{64} +860.000 q^{65} +764.000 q^{67} +200.000 q^{68} -80.0000 q^{70} +21.0000 q^{71} +681.000 q^{73} +8.00000 q^{74} -296.000 q^{76} +104.000 q^{77} +426.000 q^{79} +320.000 q^{80} +246.000 q^{82} -902.000 q^{83} +1000.00 q^{85} +304.000 q^{86} -416.000 q^{88} +1272.00 q^{89} +86.0000 q^{91} +92.0000 q^{92} +150.000 q^{94} -1480.00 q^{95} -342.000 q^{97} +678.000 q^{98} +O(q^{100})\)

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
\(3\) 0 0
\(4\) 4.00000 0.500000
\(5\) 20.0000 1.78885 0.894427 − 0.447214i \(-0.147584\pi\)
0.894427 + 0.447214i \(0.147584\pi\)
\(6\) 0 0
\(7\) 2.00000 0.107990 0.0539949 − 0.998541i \(-0.482805\pi\)
0.0539949 + 0.998541i \(0.482805\pi\)
\(8\) −8.00000 −0.353553
\(9\) 0 0
\(10\) −40.0000 −1.26491
\(11\) 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 \(0.348317\pi\)
\(14\) −4.00000 −0.0763604
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) 50.0000 0.713340 0.356670 − 0.934230i \(-0.383912\pi\)
0.356670 + 0.934230i \(0.383912\pi\)
\(18\) 0 0
\(19\) −74.0000 −0.893514 −0.446757 − 0.894655i \(-0.647421\pi\)
−0.446757 + 0.894655i \(0.647421\pi\)
\(20\) 80.0000 0.894427
\(21\) 0 0
\(22\) −104.000 −1.00786
\(23\) 23.0000 0.208514
\(24\) 0 0
\(25\) 275.000 2.20000
\(26\) −86.0000 −0.648692
\(27\) 0 0
\(28\) 8.00000 0.0539949
\(29\) 7.00000 0.0448230 0.0224115 − 0.999749i \(-0.492866\pi\)
0.0224115 + 0.999749i \(0.492866\pi\)
\(30\) 0 0
\(31\) −273.000 −1.58169 −0.790843 − 0.612019i \(-0.790357\pi\)
−0.790843 + 0.612019i \(0.790357\pi\)
\(32\) −32.0000 −0.176777
\(33\) 0 0
\(34\) −100.000 −0.504408
\(35\) 40.0000 0.193178
\(36\) 0 0
\(37\) −4.00000 −0.0177729 −0.00888643 − 0.999961i \(-0.502829\pi\)
−0.00888643 + 0.999961i \(0.502829\pi\)
\(38\) 148.000 0.631810
\(39\) 0 0
\(40\) −160.000 −0.632456
\(41\) −123.000 −0.468521 −0.234261 − 0.972174i \(-0.575267\pi\)
−0.234261 + 0.972174i \(0.575267\pi\)
\(42\) 0 0
\(43\) −152.000 −0.539065 −0.269532 − 0.962991i \(-0.586869\pi\)
−0.269532 + 0.962991i \(0.586869\pi\)
\(44\) 208.000 0.712663
\(45\) 0 0
\(46\) −46.0000 −0.147442
\(47\) −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
\(50\) −550.000 −1.55563
\(51\) 0 0
\(52\) 172.000 0.458694
\(53\) −86.0000 −0.222887 −0.111443 − 0.993771i \(-0.535547\pi\)
−0.111443 + 0.993771i \(0.535547\pi\)
\(54\) 0 0
\(55\) 1040.00 2.54970
\(56\) −16.0000 −0.0381802
\(57\) 0 0
\(58\) −14.0000 −0.0316947
\(59\) 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 \(0.411334\pi\)
\(62\) 546.000 1.11842
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) 860.000 1.64107
\(66\) 0 0
\(67\) 764.000 1.39310 0.696548 − 0.717510i \(-0.254718\pi\)
0.696548 + 0.717510i \(0.254718\pi\)
\(68\) 200.000 0.356670
\(69\) 0 0
\(70\) −80.0000 −0.136598
\(71\) 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 \(0.316178\pi\)
\(74\) 8.00000 0.0125673
\(75\) 0 0
\(76\) −296.000 −0.446757
\(77\) 104.000 0.153921
\(78\) 0 0
\(79\) 426.000 0.606693 0.303346 − 0.952880i \(-0.401896\pi\)
0.303346 + 0.952880i \(0.401896\pi\)
\(80\) 320.000 0.447214
\(81\) 0 0
\(82\) 246.000 0.331295
\(83\) −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
\(86\) 304.000 0.381176
\(87\) 0 0
\(88\) −416.000 −0.503929
\(89\) 1272.00 1.51496 0.757482 − 0.652856i \(-0.226430\pi\)
0.757482 + 0.652856i \(0.226430\pi\)
\(90\) 0 0
\(91\) 86.0000 0.0990687
\(92\) 92.0000 0.104257
\(93\) 0 0
\(94\) 150.000 0.164588
\(95\) −1480.00 −1.59837
\(96\) 0 0
\(97\) −342.000 −0.357988 −0.178994 − 0.983850i \(-0.557284\pi\)
−0.178994 + 0.983850i \(0.557284\pi\)
\(98\) 678.000 0.698861
\(99\) 0 0
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 414.4.a.b.1.1 1
3.2 odd 2 46.4.a.b.1.1 ✓ 1
12.11 even 2 368.4.a.e.1.1 1
15.2 even 4 1150.4.b.a.599.2 2
15.8 even 4 1150.4.b.a.599.1 2
15.14 odd 2 1150.4.a.d.1.1 1
21.20 even 2 2254.4.a.b.1.1 1
24.5 odd 2 1472.4.a.j.1.1 1
24.11 even 2 1472.4.a.a.1.1 1
69.68 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 3.2 odd 2
368.4.a.e.1.1 1 12.11 even 2
414.4.a.b.1.1 1 1.1 even 1 trivial
1058.4.a.b.1.1 1 69.68 even 2
1150.4.a.d.1.1 1 15.14 odd 2
1150.4.b.a.599.1 2 15.8 even 4
1150.4.b.a.599.2 2 15.2 even 4
1472.4.a.a.1.1 1 24.11 even 2
1472.4.a.j.1.1 1 24.5 odd 2
2254.4.a.b.1.1 1 21.20 even 2