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Properties

Label 8214.2.a.d.1.1
Level $8214$
Weight $2$
Character 8214.1
Self dual yes
Analytic conductor $65.589$
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] = [8214,2,Mod(1,8214)] 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("8214.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8214, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 8214 = 2 \cdot 3 \cdot 37^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8214.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-1,1,1,0,-1,-1] 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: \(65.5891202203\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 222)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 8214.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +3.00000 q^{11} +1.00000 q^{12} +1.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +3.00000 q^{17} -1.00000 q^{18} +7.00000 q^{19} -1.00000 q^{21} -3.00000 q^{22} -3.00000 q^{23} -1.00000 q^{24} -5.00000 q^{25} -1.00000 q^{26} +1.00000 q^{27} -1.00000 q^{28} -2.00000 q^{31} -1.00000 q^{32} +3.00000 q^{33} -3.00000 q^{34} +1.00000 q^{36} -7.00000 q^{38} +1.00000 q^{39} -6.00000 q^{41} +1.00000 q^{42} +4.00000 q^{43} +3.00000 q^{44} +3.00000 q^{46} +6.00000 q^{47} +1.00000 q^{48} -6.00000 q^{49} +5.00000 q^{50} +3.00000 q^{51} +1.00000 q^{52} +9.00000 q^{53} -1.00000 q^{54} +1.00000 q^{56} +7.00000 q^{57} +10.0000 q^{61} +2.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} -3.00000 q^{66} +2.00000 q^{67} +3.00000 q^{68} -3.00000 q^{69} +12.0000 q^{71} -1.00000 q^{72} +5.00000 q^{73} -5.00000 q^{75} +7.00000 q^{76} -3.00000 q^{77} -1.00000 q^{78} -2.00000 q^{79} +1.00000 q^{81} +6.00000 q^{82} +3.00000 q^{83} -1.00000 q^{84} -4.00000 q^{86} -3.00000 q^{88} +3.00000 q^{89} -1.00000 q^{91} -3.00000 q^{92} -2.00000 q^{93} -6.00000 q^{94} -1.00000 q^{96} -2.00000 q^{97} +6.00000 q^{98} +3.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) −1.00000 −0.408248
\(7\) −1.00000 −0.377964 −0.188982 − 0.981981i \(-0.560519\pi\)
−0.188982 + 0.981981i \(0.560519\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 3.00000 0.904534 0.452267 − 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) 1.00000 0.288675
\(13\) 1.00000 0.277350 0.138675 − 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 3.00000 0.727607 0.363803 − 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) −1.00000 −0.235702
\(19\) 7.00000 1.60591 0.802955 − 0.596040i \(-0.203260\pi\)
0.802955 + 0.596040i \(0.203260\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) −3.00000 −0.639602
\(23\) −3.00000 −0.625543 −0.312772 − 0.949828i \(-0.601257\pi\)
−0.312772 + 0.949828i \(0.601257\pi\)
\(24\) −1.00000 −0.204124
\(25\) −5.00000 −1.00000
\(26\) −1.00000 −0.196116
\(27\) 1.00000 0.192450
\(28\) −1.00000 −0.188982
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 − 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) −1.00000 −0.176777
\(33\) 3.00000 0.522233
\(34\) −3.00000 −0.514496
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 0 0
\(38\) −7.00000 −1.13555
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 1.00000 0.154303
\(43\) 4.00000 0.609994 0.304997 − 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 3.00000 0.452267
\(45\) 0 0
\(46\) 3.00000 0.442326
\(47\) 6.00000 0.875190 0.437595 − 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 1.00000 0.144338
\(49\) −6.00000 −0.857143
\(50\) 5.00000 0.707107
\(51\) 3.00000 0.420084
\(52\) 1.00000 0.138675
\(53\) 9.00000 1.23625 0.618123 − 0.786082i \(-0.287894\pi\)
0.618123 + 0.786082i \(0.287894\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 7.00000 0.927173
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 10.0000 1.28037 0.640184 − 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 2.00000 0.254000
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −3.00000 −0.369274
\(67\) 2.00000 0.244339 0.122169 − 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) 3.00000 0.363803
\(69\) −3.00000 −0.361158
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 − 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) −1.00000 −0.117851
\(73\) 5.00000 0.585206 0.292603 − 0.956234i \(-0.405479\pi\)
0.292603 + 0.956234i \(0.405479\pi\)
\(74\) 0 0
\(75\) −5.00000 −0.577350
\(76\) 7.00000 0.802955
\(77\) −3.00000 −0.341882
\(78\) −1.00000 −0.113228
\(79\) −2.00000 −0.225018 −0.112509 − 0.993651i \(-0.535889\pi\)
−0.112509 + 0.993651i \(0.535889\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 6.00000 0.662589
\(83\) 3.00000 0.329293 0.164646 − 0.986353i \(-0.447352\pi\)
0.164646 + 0.986353i \(0.447352\pi\)
\(84\) −1.00000 −0.109109
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) 0 0
\(88\) −3.00000 −0.319801
\(89\) 3.00000 0.317999 0.159000 − 0.987279i \(-0.449173\pi\)
0.159000 + 0.987279i \(0.449173\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) −3.00000 −0.312772
\(93\) −2.00000 −0.207390
\(94\) −6.00000 −0.618853
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) −2.00000 −0.203069 −0.101535 − 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 6.00000 0.606092
\(99\) 3.00000 0.301511
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 8214.2.a.d.1.1 1
37.36 even 2 222.2.a.e.1.1 ✓ 1
111.110 odd 2 666.2.a.a.1.1 1
148.147 odd 2 1776.2.a.c.1.1 1
185.184 even 2 5550.2.a.h.1.1 1
296.147 odd 2 7104.2.a.u.1.1 1
296.221 even 2 7104.2.a.g.1.1 1
444.443 even 2 5328.2.a.l.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
222.2.a.e.1.1 ✓ 1 37.36 even 2
666.2.a.a.1.1 1 111.110 odd 2
1776.2.a.c.1.1 1 148.147 odd 2
5328.2.a.l.1.1 1 444.443 even 2
5550.2.a.h.1.1 1 185.184 even 2
7104.2.a.g.1.1 1 296.221 even 2
7104.2.a.u.1.1 1 296.147 odd 2
8214.2.a.d.1.1 1 1.1 even 1 trivial