Properties

Label 418.2.a.g.1.1
Level $418$
Weight $2$
Character 418.1
Self dual yes
Analytic conductor $3.338$
Analytic rank $1$
Dimension $3$
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] = [418,2,Mod(1,418)] 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("418.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(418, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 418 = 2 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 418.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,-3,0] 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: \(3.33774680449\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.621.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 6x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.14510\) of defining polynomial
Character \(\chi\) \(=\) 418.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -2.14510 q^{3} +1.00000 q^{4} -1.60147 q^{5} +2.14510 q^{6} +2.89167 q^{7} -1.00000 q^{8} +1.60147 q^{9} +1.60147 q^{10} -1.00000 q^{11} -2.14510 q^{12} +4.89167 q^{13} -2.89167 q^{14} +3.43531 q^{15} +1.00000 q^{16} -5.74657 q^{17} -1.60147 q^{18} -1.00000 q^{19} -1.60147 q^{20} -6.20293 q^{21} +1.00000 q^{22} -7.74657 q^{23} +2.14510 q^{24} -2.43531 q^{25} -4.89167 q^{26} +3.00000 q^{27} +2.89167 q^{28} +5.34803 q^{29} -3.43531 q^{30} -7.43531 q^{31} -1.00000 q^{32} +2.14510 q^{33} +5.74657 q^{34} -4.63091 q^{35} +1.60147 q^{36} -7.49314 q^{37} +1.00000 q^{38} -10.4931 q^{39} +1.60147 q^{40} -2.05783 q^{41} +6.20293 q^{42} -10.6887 q^{43} -1.00000 q^{44} -2.56469 q^{45} +7.74657 q^{46} -7.08727 q^{47} -2.14510 q^{48} +1.36176 q^{49} +2.43531 q^{50} +12.3270 q^{51} +4.89167 q^{52} +8.32698 q^{53} -3.00000 q^{54} +1.60147 q^{55} -2.89167 q^{56} +2.14510 q^{57} -5.34803 q^{58} -2.54364 q^{59} +3.43531 q^{60} +13.4931 q^{61} +7.43531 q^{62} +4.63091 q^{63} +1.00000 q^{64} -7.83384 q^{65} -2.14510 q^{66} -10.3848 q^{67} -5.74657 q^{68} +16.6172 q^{69} +4.63091 q^{70} +14.9789 q^{71} -1.60147 q^{72} -3.74657 q^{73} +7.49314 q^{74} +5.22399 q^{75} -1.00000 q^{76} -2.89167 q^{77} +10.4931 q^{78} -8.69607 q^{79} -1.60147 q^{80} -11.2397 q^{81} +2.05783 q^{82} -6.10833 q^{83} -6.20293 q^{84} +9.20293 q^{85} +10.6887 q^{86} -11.4721 q^{87} +1.00000 q^{88} +3.20293 q^{89} +2.56469 q^{90} +14.1451 q^{91} -7.74657 q^{92} +15.9495 q^{93} +7.08727 q^{94} +1.60147 q^{95} +2.14510 q^{96} -6.98627 q^{97} -1.36176 q^{98} -1.60147 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} + 3 q^{4} - 3 q^{5} - 6 q^{7} - 3 q^{8} + 3 q^{9} + 3 q^{10} - 3 q^{11} + 6 q^{14} - 9 q^{15} + 3 q^{16} - 9 q^{17} - 3 q^{18} - 3 q^{19} - 3 q^{20} - 15 q^{21} + 3 q^{22} - 15 q^{23} + 12 q^{25}+ \cdots - 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −2.14510 −1.23848 −0.619238 − 0.785204i \(-0.712558\pi\)
−0.619238 + 0.785204i \(0.712558\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.60147 −0.716197 −0.358099 − 0.933684i \(-0.616575\pi\)
−0.358099 + 0.933684i \(0.616575\pi\)
\(6\) 2.14510 0.875735
\(7\) 2.89167 1.09295 0.546474 − 0.837476i \(-0.315970\pi\)
0.546474 + 0.837476i \(0.315970\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.60147 0.533822
\(10\) 1.60147 0.506428
\(11\) −1.00000 −0.301511
\(12\) −2.14510 −0.619238
\(13\) 4.89167 1.35671 0.678353 − 0.734736i \(-0.262694\pi\)
0.678353 + 0.734736i \(0.262694\pi\)
\(14\) −2.89167 −0.772832
\(15\) 3.43531 0.886993
\(16\) 1.00000 0.250000
\(17\) −5.74657 −1.39375 −0.696874 − 0.717194i \(-0.745426\pi\)
−0.696874 + 0.717194i \(0.745426\pi\)
\(18\) −1.60147 −0.377469
\(19\) −1.00000 −0.229416
\(20\) −1.60147 −0.358099
\(21\) −6.20293 −1.35359
\(22\) 1.00000 0.213201
\(23\) −7.74657 −1.61527 −0.807636 − 0.589682i \(-0.799253\pi\)
−0.807636 + 0.589682i \(0.799253\pi\)
\(24\) 2.14510 0.437867
\(25\) −2.43531 −0.487062
\(26\) −4.89167 −0.959336
\(27\) 3.00000 0.577350
\(28\) 2.89167 0.546474
\(29\) 5.34803 0.993105 0.496552 − 0.868007i \(-0.334599\pi\)
0.496552 + 0.868007i \(0.334599\pi\)
\(30\) −3.43531 −0.627199
\(31\) −7.43531 −1.33542 −0.667710 − 0.744421i \(-0.732726\pi\)
−0.667710 + 0.744421i \(0.732726\pi\)
\(32\) −1.00000 −0.176777
\(33\) 2.14510 0.373414
\(34\) 5.74657 0.985528
\(35\) −4.63091 −0.782767
\(36\) 1.60147 0.266911
\(37\) −7.49314 −1.23186 −0.615932 − 0.787799i \(-0.711220\pi\)
−0.615932 + 0.787799i \(0.711220\pi\)
\(38\) 1.00000 0.162221
\(39\) −10.4931 −1.68025
\(40\) 1.60147 0.253214
\(41\) −2.05783 −0.321379 −0.160689 − 0.987005i \(-0.551372\pi\)
−0.160689 + 0.987005i \(0.551372\pi\)
\(42\) 6.20293 0.957133
\(43\) −10.6887 −1.63002 −0.815009 − 0.579449i \(-0.803268\pi\)
−0.815009 + 0.579449i \(0.803268\pi\)
\(44\) −1.00000 −0.150756
\(45\) −2.56469 −0.382322
\(46\) 7.74657 1.14217
\(47\) −7.08727 −1.03379 −0.516893 − 0.856050i \(-0.672911\pi\)
−0.516893 + 0.856050i \(0.672911\pi\)
\(48\) −2.14510 −0.309619
\(49\) 1.36176 0.194537
\(50\) 2.43531 0.344405
\(51\) 12.3270 1.72612
\(52\) 4.89167 0.678353
\(53\) 8.32698 1.14380 0.571899 − 0.820324i \(-0.306207\pi\)
0.571899 + 0.820324i \(0.306207\pi\)
\(54\) −3.00000 −0.408248
\(55\) 1.60147 0.215942
\(56\) −2.89167 −0.386416
\(57\) 2.14510 0.284126
\(58\) −5.34803 −0.702231
\(59\) −2.54364 −0.331153 −0.165577 − 0.986197i \(-0.552949\pi\)
−0.165577 + 0.986197i \(0.552949\pi\)
\(60\) 3.43531 0.443496
\(61\) 13.4931 1.72762 0.863810 − 0.503818i \(-0.168072\pi\)
0.863810 + 0.503818i \(0.168072\pi\)
\(62\) 7.43531 0.944285
\(63\) 4.63091 0.583440
\(64\) 1.00000 0.125000
\(65\) −7.83384 −0.971669
\(66\) −2.14510 −0.264044
\(67\) −10.3848 −1.26871 −0.634353 − 0.773043i \(-0.718733\pi\)
−0.634353 + 0.773043i \(0.718733\pi\)
\(68\) −5.74657 −0.696874
\(69\) 16.6172 2.00047
\(70\) 4.63091 0.553500
\(71\) 14.9789 1.77767 0.888837 − 0.458224i \(-0.151514\pi\)
0.888837 + 0.458224i \(0.151514\pi\)
\(72\) −1.60147 −0.188735
\(73\) −3.74657 −0.438503 −0.219251 − 0.975668i \(-0.570361\pi\)
−0.219251 + 0.975668i \(0.570361\pi\)
\(74\) 7.49314 0.871059
\(75\) 5.22399 0.603214
\(76\) −1.00000 −0.114708
\(77\) −2.89167 −0.329536
\(78\) 10.4931 1.18811
\(79\) −8.69607 −0.978384 −0.489192 − 0.872176i \(-0.662708\pi\)
−0.489192 + 0.872176i \(0.662708\pi\)
\(80\) −1.60147 −0.179049
\(81\) −11.2397 −1.24886
\(82\) 2.05783 0.227249
\(83\) −6.10833 −0.670476 −0.335238 − 0.942133i \(-0.608817\pi\)
−0.335238 + 0.942133i \(0.608817\pi\)
\(84\) −6.20293 −0.676795
\(85\) 9.20293 0.998198
\(86\) 10.6887 1.15260
\(87\) −11.4721 −1.22994
\(88\) 1.00000 0.106600
\(89\) 3.20293 0.339510 0.169755 − 0.985486i \(-0.445702\pi\)
0.169755 + 0.985486i \(0.445702\pi\)
\(90\) 2.56469 0.270342
\(91\) 14.1451 1.48281
\(92\) −7.74657 −0.807636
\(93\) 15.9495 1.65389
\(94\) 7.08727 0.730997
\(95\) 1.60147 0.164307
\(96\) 2.14510 0.218934
\(97\) −6.98627 −0.709349 −0.354674 − 0.934990i \(-0.615408\pi\)
−0.354674 + 0.934990i \(0.615408\pi\)
\(98\) −1.36176 −0.137559
\(99\) −1.60147 −0.160953
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 418.2.a.g.1.1 ✓ 3
3.2 odd 2 3762.2.a.bg.1.2 3
4.3 odd 2 3344.2.a.q.1.3 3
11.10 odd 2 4598.2.a.bo.1.1 3
19.18 odd 2 7942.2.a.bi.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
418.2.a.g.1.1 ✓ 3 1.1 even 1 trivial
3344.2.a.q.1.3 3 4.3 odd 2
3762.2.a.bg.1.2 3 3.2 odd 2
4598.2.a.bo.1.1 3 11.10 odd 2
7942.2.a.bi.1.3 3 19.18 odd 2