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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [58,2,Mod(57,58)] 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("58.57"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(58, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 58 = 2 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 58.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.463132331723\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 57.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 58.57
Dual form 58.2.b.a.57.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -2.00000 q^{7} -1.00000i q^{8} +2.00000 q^{9} +1.00000i q^{10} -5.00000i q^{11} -1.00000i q^{12} -1.00000 q^{13} -2.00000i q^{14} +1.00000i q^{15} +1.00000 q^{16} -2.00000i q^{17} +2.00000i q^{18} +4.00000i q^{19} -1.00000 q^{20} -2.00000i q^{21} +5.00000 q^{22} -6.00000 q^{23} +1.00000 q^{24} -4.00000 q^{25} -1.00000i q^{26} +5.00000i q^{27} +2.00000 q^{28} +(5.00000 + 2.00000i) q^{29} -1.00000 q^{30} +5.00000i q^{31} +1.00000i q^{32} +5.00000 q^{33} +2.00000 q^{34} -2.00000 q^{35} -2.00000 q^{36} +8.00000i q^{37} -4.00000 q^{38} -1.00000i q^{39} -1.00000i q^{40} -10.0000i q^{41} +2.00000 q^{42} -9.00000i q^{43} +5.00000i q^{44} +2.00000 q^{45} -6.00000i q^{46} +3.00000i q^{47} +1.00000i q^{48} -3.00000 q^{49} -4.00000i q^{50} +2.00000 q^{51} +1.00000 q^{52} -1.00000 q^{53} -5.00000 q^{54} -5.00000i q^{55} +2.00000i q^{56} -4.00000 q^{57} +(-2.00000 + 5.00000i) q^{58} +10.0000 q^{59} -1.00000i q^{60} -10.0000i q^{61} -5.00000 q^{62} -4.00000 q^{63} -1.00000 q^{64} -1.00000 q^{65} +5.00000i q^{66} +8.00000 q^{67} +2.00000i q^{68} -6.00000i q^{69} -2.00000i q^{70} -8.00000 q^{71} -2.00000i q^{72} +16.0000i q^{73} -8.00000 q^{74} -4.00000i q^{75} -4.00000i q^{76} +10.0000i q^{77} +1.00000 q^{78} -1.00000i q^{79} +1.00000 q^{80} +1.00000 q^{81} +10.0000 q^{82} +14.0000 q^{83} +2.00000i q^{84} -2.00000i q^{85} +9.00000 q^{86} +(-2.00000 + 5.00000i) q^{87} -5.00000 q^{88} +14.0000i q^{89} +2.00000i q^{90} +2.00000 q^{91} +6.00000 q^{92} -5.00000 q^{93} -3.00000 q^{94} +4.00000i q^{95} -1.00000 q^{96} -2.00000i q^{97} -3.00000i q^{98} -10.0000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} + 2 q^{5} - 2 q^{6} - 4 q^{7} + 4 q^{9} - 2 q^{13} + 2 q^{16} - 2 q^{20} + 10 q^{22} - 12 q^{23} + 2 q^{24} - 8 q^{25} + 4 q^{28} + 10 q^{29} - 2 q^{30} + 10 q^{33} + 4 q^{34} - 4 q^{35} - 4 q^{36}+ \cdots - 2 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/58\mathbb{Z}\right)^\times\).

\(n\) \(31\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) 1.00000i 0.577350i 0.957427 + 0.288675i \(0.0932147\pi\)
−0.957427 + 0.288675i \(0.906785\pi\)
\(4\) −1.00000 −0.500000
\(5\) 1.00000 0.447214 0.223607 − 0.974679i \(-0.428217\pi\)
0.223607 + 0.974679i \(0.428217\pi\)
\(6\) −1.00000 −0.408248
\(7\) −2.00000 −0.755929 −0.377964 − 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) 2.00000 0.666667
\(10\) 1.00000i 0.316228i
\(11\) − 5.00000i − 1.50756i −0.657129 − 0.753778i \(-0.728229\pi\)
0.657129 − 0.753778i \(-0.271771\pi\)
\(12\) − 1.00000i − 0.288675i
\(13\) −1.00000 −0.277350 −0.138675 − 0.990338i \(-0.544284\pi\)
−0.138675 + 0.990338i \(0.544284\pi\)
\(14\) − 2.00000i − 0.534522i
\(15\) 1.00000i 0.258199i
\(16\) 1.00000 0.250000
\(17\) − 2.00000i − 0.485071i −0.970143 − 0.242536i \(-0.922021\pi\)
0.970143 − 0.242536i \(-0.0779791\pi\)
\(18\) 2.00000i 0.471405i
\(19\) 4.00000i 0.917663i 0.888523 + 0.458831i \(0.151732\pi\)
−0.888523 + 0.458831i \(0.848268\pi\)
\(20\) −1.00000 −0.223607
\(21\) − 2.00000i − 0.436436i
\(22\) 5.00000 1.06600
\(23\) −6.00000 −1.25109 −0.625543 − 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 1.00000 0.204124
\(25\) −4.00000 −0.800000
\(26\) − 1.00000i − 0.196116i
\(27\) 5.00000i 0.962250i
\(28\) 2.00000 0.377964
\(29\) 5.00000 + 2.00000i 0.928477 + 0.371391i
\(30\) −1.00000 −0.182574
\(31\) 5.00000i 0.898027i 0.893525 + 0.449013i \(0.148224\pi\)
−0.893525 + 0.449013i \(0.851776\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 5.00000 0.870388
\(34\) 2.00000 0.342997
\(35\) −2.00000 −0.338062
\(36\) −2.00000 −0.333333
\(37\) 8.00000i 1.31519i 0.753371 + 0.657596i \(0.228427\pi\)
−0.753371 + 0.657596i \(0.771573\pi\)
\(38\) −4.00000 −0.648886
\(39\) − 1.00000i − 0.160128i
\(40\) − 1.00000i − 0.158114i
\(41\) − 10.0000i − 1.56174i −0.624695 − 0.780869i \(-0.714777\pi\)
0.624695 − 0.780869i \(-0.285223\pi\)
\(42\) 2.00000 0.308607
\(43\) − 9.00000i − 1.37249i −0.727372 − 0.686244i \(-0.759258\pi\)
0.727372 − 0.686244i \(-0.240742\pi\)
\(44\) 5.00000i 0.753778i
\(45\) 2.00000 0.298142
\(46\) − 6.00000i − 0.884652i
\(47\) 3.00000i 0.437595i 0.975770 + 0.218797i \(0.0702134\pi\)
−0.975770 + 0.218797i \(0.929787\pi\)
\(48\) 1.00000i 0.144338i
\(49\) −3.00000 −0.428571
\(50\) − 4.00000i − 0.565685i
\(51\) 2.00000 0.280056
\(52\) 1.00000 0.138675
\(53\) −1.00000 −0.137361 −0.0686803 − 0.997639i \(-0.521879\pi\)
−0.0686803 + 0.997639i \(0.521879\pi\)
\(54\) −5.00000 −0.680414
\(55\) − 5.00000i − 0.674200i
\(56\) 2.00000i 0.267261i
\(57\) −4.00000 −0.529813
\(58\) −2.00000 + 5.00000i −0.262613 + 0.656532i
\(59\) 10.0000 1.30189 0.650945 − 0.759125i \(-0.274373\pi\)
0.650945 + 0.759125i \(0.274373\pi\)
\(60\) − 1.00000i − 0.129099i
\(61\) − 10.0000i − 1.28037i −0.768221 − 0.640184i \(-0.778858\pi\)
0.768221 − 0.640184i \(-0.221142\pi\)
\(62\) −5.00000 −0.635001
\(63\) −4.00000 −0.503953
\(64\) −1.00000 −0.125000
\(65\) −1.00000 −0.124035
\(66\) 5.00000i 0.615457i
\(67\) 8.00000 0.977356 0.488678 − 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) 2.00000i 0.242536i
\(69\) − 6.00000i − 0.722315i
\(70\) − 2.00000i − 0.239046i
\(71\) −8.00000 −0.949425 −0.474713 − 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) − 2.00000i − 0.235702i
\(73\) 16.0000i 1.87266i 0.351123 + 0.936329i \(0.385800\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) −8.00000 −0.929981
\(75\) − 4.00000i − 0.461880i
\(76\) − 4.00000i − 0.458831i
\(77\) 10.0000i 1.13961i
\(78\) 1.00000 0.113228
\(79\) − 1.00000i − 0.112509i −0.998416 − 0.0562544i \(-0.982084\pi\)
0.998416 − 0.0562544i \(-0.0179158\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) 10.0000 1.10432
\(83\) 14.0000 1.53670 0.768350 − 0.640030i \(-0.221078\pi\)
0.768350 + 0.640030i \(0.221078\pi\)
\(84\) 2.00000i 0.218218i
\(85\) − 2.00000i − 0.216930i
\(86\) 9.00000 0.970495
\(87\) −2.00000 + 5.00000i −0.214423 + 0.536056i
\(88\) −5.00000 −0.533002
\(89\) 14.0000i 1.48400i 0.670402 + 0.741999i \(0.266122\pi\)
−0.670402 + 0.741999i \(0.733878\pi\)
\(90\) 2.00000i 0.210819i
\(91\) 2.00000 0.209657
\(92\) 6.00000 0.625543
\(93\) −5.00000 −0.518476
\(94\) −3.00000 −0.309426
\(95\) 4.00000i 0.410391i
\(96\) −1.00000 −0.102062
\(97\) − 2.00000i − 0.203069i −0.994832 − 0.101535i \(-0.967625\pi\)
0.994832 − 0.101535i \(-0.0323753\pi\)
\(98\) − 3.00000i − 0.303046i
\(99\) − 10.0000i − 1.00504i
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 58.2.b.a.57.2 yes 2
3.2 odd 2 522.2.d.a.289.1 2
4.3 odd 2 464.2.e.c.289.1 2
5.2 odd 4 1450.2.d.b.1449.1 2
5.3 odd 4 1450.2.d.c.1449.2 2
5.4 even 2 1450.2.c.a.1101.1 2
8.3 odd 2 1856.2.e.d.1217.2 2
8.5 even 2 1856.2.e.b.1217.1 2
12.11 even 2 4176.2.o.d.289.1 2
29.12 odd 4 1682.2.a.c.1.1 1
29.17 odd 4 1682.2.a.g.1.1 1
29.28 even 2 inner 58.2.b.a.57.1 ✓ 2
87.86 odd 2 522.2.d.a.289.2 2
116.115 odd 2 464.2.e.c.289.2 2
145.28 odd 4 1450.2.d.b.1449.2 2
145.57 odd 4 1450.2.d.c.1449.1 2
145.144 even 2 1450.2.c.a.1101.2 2
232.115 odd 2 1856.2.e.d.1217.1 2
232.173 even 2 1856.2.e.b.1217.2 2
348.347 even 2 4176.2.o.d.289.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
58.2.b.a.57.1 ✓ 2 29.28 even 2 inner
58.2.b.a.57.2 yes 2 1.1 even 1 trivial
464.2.e.c.289.1 2 4.3 odd 2
464.2.e.c.289.2 2 116.115 odd 2
522.2.d.a.289.1 2 3.2 odd 2
522.2.d.a.289.2 2 87.86 odd 2
1450.2.c.a.1101.1 2 5.4 even 2
1450.2.c.a.1101.2 2 145.144 even 2
1450.2.d.b.1449.1 2 5.2 odd 4
1450.2.d.b.1449.2 2 145.28 odd 4
1450.2.d.c.1449.1 2 145.57 odd 4
1450.2.d.c.1449.2 2 5.3 odd 4
1682.2.a.c.1.1 1 29.12 odd 4
1682.2.a.g.1.1 1 29.17 odd 4
1856.2.e.b.1217.1 2 8.5 even 2
1856.2.e.b.1217.2 2 232.173 even 2
1856.2.e.d.1217.1 2 232.115 odd 2
1856.2.e.d.1217.2 2 8.3 odd 2
4176.2.o.d.289.1 2 12.11 even 2
4176.2.o.d.289.2 2 348.347 even 2