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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1682,2,Mod(1,1682)] 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("1682.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1682, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1682 = 2 \cdot 29^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1682.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,1,1,1,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(13.4308376200\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 58)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1682.1

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

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 0.288675 − 0.957427i \(-0.406785\pi\)
0.288675 + 0.957427i \(0.406785\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214 −0.223607 − 0.974679i \(-0.571783\pi\)
−0.223607 + 0.974679i \(0.571783\pi\)
\(6\) 1.00000 0.408248
\(7\) −2.00000 −0.755929 −0.377964 − 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 1.00000 0.353553
\(9\) −2.00000 −0.666667
\(10\) −1.00000 −0.316228
\(11\) −5.00000 −1.50756 −0.753778 − 0.657129i \(-0.771771\pi\)
−0.753778 + 0.657129i \(0.771771\pi\)
\(12\) 1.00000 0.288675
\(13\) 1.00000 0.277350 0.138675 − 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) −2.00000 −0.534522
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) −2.00000 −0.485071 −0.242536 − 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) −2.00000 −0.471405
\(19\) 4.00000 0.917663 0.458831 − 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) −1.00000 −0.223607
\(21\) −2.00000 −0.436436
\(22\) −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
\(26\) 1.00000 0.196116
\(27\) −5.00000 −0.962250
\(28\) −2.00000 −0.377964
\(29\) 0 0
\(30\) −1.00000 −0.182574
\(31\) 5.00000 0.898027 0.449013 − 0.893525i \(-0.351776\pi\)
0.449013 + 0.893525i \(0.351776\pi\)
\(32\) 1.00000 0.176777
\(33\) −5.00000 −0.870388
\(34\) −2.00000 −0.342997
\(35\) 2.00000 0.338062
\(36\) −2.00000 −0.333333
\(37\) −8.00000 −1.31519 −0.657596 − 0.753371i \(-0.728427\pi\)
−0.657596 + 0.753371i \(0.728427\pi\)
\(38\) 4.00000 0.648886
\(39\) 1.00000 0.160128
\(40\) −1.00000 −0.158114
\(41\) 10.0000 1.56174 0.780869 − 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) −2.00000 −0.308607
\(43\) −9.00000 −1.37249 −0.686244 − 0.727372i \(-0.740742\pi\)
−0.686244 + 0.727372i \(0.740742\pi\)
\(44\) −5.00000 −0.753778
\(45\) 2.00000 0.298142
\(46\) −6.00000 −0.884652
\(47\) −3.00000 −0.437595 −0.218797 − 0.975770i \(-0.570213\pi\)
−0.218797 + 0.975770i \(0.570213\pi\)
\(48\) 1.00000 0.144338
\(49\) −3.00000 −0.428571
\(50\) −4.00000 −0.565685
\(51\) −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
\(55\) 5.00000 0.674200
\(56\) −2.00000 −0.267261
\(57\) 4.00000 0.529813
\(58\) 0 0
\(59\) 10.0000 1.30189 0.650945 − 0.759125i \(-0.274373\pi\)
0.650945 + 0.759125i \(0.274373\pi\)
\(60\) −1.00000 −0.129099
\(61\) −10.0000 −1.28037 −0.640184 − 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 5.00000 0.635001
\(63\) 4.00000 0.503953
\(64\) 1.00000 0.125000
\(65\) −1.00000 −0.124035
\(66\) −5.00000 −0.615457
\(67\) −8.00000 −0.977356 −0.488678 − 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) −2.00000 −0.242536
\(69\) −6.00000 −0.722315
\(70\) 2.00000 0.239046
\(71\) 8.00000 0.949425 0.474713 − 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) −2.00000 −0.235702
\(73\) −16.0000 −1.87266 −0.936329 − 0.351123i \(-0.885800\pi\)
−0.936329 + 0.351123i \(0.885800\pi\)
\(74\) −8.00000 −0.929981
\(75\) −4.00000 −0.461880
\(76\) 4.00000 0.458831
\(77\) 10.0000 1.13961
\(78\) 1.00000 0.113228
\(79\) −1.00000 −0.112509 −0.0562544 − 0.998416i \(-0.517916\pi\)
−0.0562544 + 0.998416i \(0.517916\pi\)
\(80\) −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 \(0.221078\pi\)
\(84\) −2.00000 −0.218218
\(85\) 2.00000 0.216930
\(86\) −9.00000 −0.970495
\(87\) 0 0
\(88\) −5.00000 −0.533002
\(89\) 14.0000 1.48400 0.741999 − 0.670402i \(-0.233878\pi\)
0.741999 + 0.670402i \(0.233878\pi\)
\(90\) 2.00000 0.210819
\(91\) −2.00000 −0.209657
\(92\) −6.00000 −0.625543
\(93\) 5.00000 0.518476
\(94\) −3.00000 −0.309426
\(95\) −4.00000 −0.410391
\(96\) 1.00000 0.102062
\(97\) 2.00000 0.203069 0.101535 − 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) −3.00000 −0.303046
\(99\) 10.0000 1.00504
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 1682.2.a.g.1.1 1
29.12 odd 4 58.2.b.a.57.2 yes 2
29.17 odd 4 58.2.b.a.57.1 ✓ 2
29.28 even 2 1682.2.a.c.1.1 1
87.17 even 4 522.2.d.a.289.2 2
87.41 even 4 522.2.d.a.289.1 2
116.75 even 4 464.2.e.c.289.2 2
116.99 even 4 464.2.e.c.289.1 2
145.12 even 4 1450.2.d.b.1449.1 2
145.17 even 4 1450.2.d.c.1449.1 2
145.99 odd 4 1450.2.c.a.1101.1 2
145.104 odd 4 1450.2.c.a.1101.2 2
145.128 even 4 1450.2.d.c.1449.2 2
145.133 even 4 1450.2.d.b.1449.2 2
232.75 even 4 1856.2.e.d.1217.1 2
232.99 even 4 1856.2.e.d.1217.2 2
232.133 odd 4 1856.2.e.b.1217.2 2
232.157 odd 4 1856.2.e.b.1217.1 2
348.191 odd 4 4176.2.o.d.289.2 2
348.215 odd 4 4176.2.o.d.289.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
58.2.b.a.57.1 ✓ 2 29.17 odd 4
58.2.b.a.57.2 yes 2 29.12 odd 4
464.2.e.c.289.1 2 116.99 even 4
464.2.e.c.289.2 2 116.75 even 4
522.2.d.a.289.1 2 87.41 even 4
522.2.d.a.289.2 2 87.17 even 4
1450.2.c.a.1101.1 2 145.99 odd 4
1450.2.c.a.1101.2 2 145.104 odd 4
1450.2.d.b.1449.1 2 145.12 even 4
1450.2.d.b.1449.2 2 145.133 even 4
1450.2.d.c.1449.1 2 145.17 even 4
1450.2.d.c.1449.2 2 145.128 even 4
1682.2.a.c.1.1 1 29.28 even 2
1682.2.a.g.1.1 1 1.1 even 1 trivial
1856.2.e.b.1217.1 2 232.157 odd 4
1856.2.e.b.1217.2 2 232.133 odd 4
1856.2.e.d.1217.1 2 232.75 even 4
1856.2.e.d.1217.2 2 232.99 even 4
4176.2.o.d.289.1 2 348.215 odd 4
4176.2.o.d.289.2 2 348.191 odd 4