Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [4598,2,Mod(1,4598)] 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("4598.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4598, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4598 = 2 \cdot 11^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4598.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,3,0,3,-3,0,6,3,3,-3,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(36.7152148494\)
Analytic rank: \(0\)
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: no (minimal twist has level 418)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.14510\) of defining polynomial
Character \(\chi\) \(=\) 4598.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -2.14510 q^{3} +1.00000 q^{4} -1.60147 q^{5} -2.14510 q^{6} -2.89167 q^{7} +1.00000 q^{8} +1.60147 q^{9} -1.60147 q^{10} -2.14510 q^{12} -4.89167 q^{13} -2.89167 q^{14} +3.43531 q^{15} +1.00000 q^{16} +5.74657 q^{17} +1.60147 q^{18} +1.00000 q^{19} -1.60147 q^{20} +6.20293 q^{21} -7.74657 q^{23} -2.14510 q^{24} -2.43531 q^{25} -4.89167 q^{26} +3.00000 q^{27} -2.89167 q^{28} -5.34803 q^{29} +3.43531 q^{30} -7.43531 q^{31} +1.00000 q^{32} +5.74657 q^{34} +4.63091 q^{35} +1.60147 q^{36} -7.49314 q^{37} +1.00000 q^{38} +10.4931 q^{39} -1.60147 q^{40} +2.05783 q^{41} +6.20293 q^{42} +10.6887 q^{43} -2.56469 q^{45} -7.74657 q^{46} -7.08727 q^{47} -2.14510 q^{48} +1.36176 q^{49} -2.43531 q^{50} -12.3270 q^{51} -4.89167 q^{52} +8.32698 q^{53} +3.00000 q^{54} -2.89167 q^{56} -2.14510 q^{57} -5.34803 q^{58} -2.54364 q^{59} +3.43531 q^{60} -13.4931 q^{61} -7.43531 q^{62} -4.63091 q^{63} +1.00000 q^{64} +7.83384 q^{65} -10.3848 q^{67} +5.74657 q^{68} +16.6172 q^{69} +4.63091 q^{70} +14.9789 q^{71} +1.60147 q^{72} +3.74657 q^{73} -7.49314 q^{74} +5.22399 q^{75} +1.00000 q^{76} +10.4931 q^{78} +8.69607 q^{79} -1.60147 q^{80} -11.2397 q^{81} +2.05783 q^{82} +6.10833 q^{83} +6.20293 q^{84} -9.20293 q^{85} +10.6887 q^{86} +11.4721 q^{87} +3.20293 q^{89} -2.56469 q^{90} +14.1451 q^{91} -7.74657 q^{92} +15.9495 q^{93} -7.08727 q^{94} -1.60147 q^{95} -2.14510 q^{96} -6.98627 q^{97} +1.36176 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 3 q^{4} - 3 q^{5} + 6 q^{7} + 3 q^{8} + 3 q^{9} - 3 q^{10} + 6 q^{14} - 9 q^{15} + 3 q^{16} + 9 q^{17} + 3 q^{18} + 3 q^{19} - 3 q^{20} + 15 q^{21} - 15 q^{23} + 12 q^{25} + 9 q^{27} + 6 q^{28}+ \cdots + 27 q^{98}+O(q^{100}) \) 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 \(-0.684030\pi\)
−0.546474 + 0.837476i \(0.684030\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.60147 0.533822
\(10\) −1.60147 −0.506428
\(11\) 0 0
\(12\) −2.14510 −0.619238
\(13\) −4.89167 −1.35671 −0.678353 − 0.734736i \(-0.737306\pi\)
−0.678353 + 0.734736i \(0.737306\pi\)
\(14\) −2.89167 −0.772832
\(15\) 3.43531 0.886993
\(16\) 1.00000 0.250000
\(17\) 5.74657 1.39375 0.696874 − 0.717194i \(-0.254574\pi\)
0.696874 + 0.717194i \(0.254574\pi\)
\(18\) 1.60147 0.377469
\(19\) 1.00000 0.229416
\(20\) −1.60147 −0.358099
\(21\) 6.20293 1.35359
\(22\) 0 0
\(23\) −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 \(-0.665401\pi\)
−0.496552 + 0.868007i \(0.665401\pi\)
\(30\) 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
\(33\) 0 0
\(34\) 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 \(-0.448628\pi\)
0.160689 + 0.987005i \(0.448628\pi\)
\(42\) 6.20293 0.957133
\(43\) 10.6887 1.63002 0.815009 − 0.579449i \(-0.196732\pi\)
0.815009 + 0.579449i \(0.196732\pi\)
\(44\) 0 0
\(45\) −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
\(55\) 0 0
\(56\) −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 \(-0.831928\pi\)
−0.863810 + 0.503818i \(0.831928\pi\)
\(62\) −7.43531 −0.944285
\(63\) −4.63091 −0.583440
\(64\) 1.00000 0.125000
\(65\) 7.83384 0.971669
\(66\) 0 0
\(67\) −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 \(-0.429639\pi\)
0.219251 + 0.975668i \(0.429639\pi\)
\(74\) −7.49314 −0.871059
\(75\) 5.22399 0.603214
\(76\) 1.00000 0.114708
\(77\) 0 0
\(78\) 10.4931 1.18811
\(79\) 8.69607 0.978384 0.489192 − 0.872176i \(-0.337292\pi\)
0.489192 + 0.872176i \(0.337292\pi\)
\(80\) −1.60147 −0.179049
\(81\) −11.2397 −1.24886
\(82\) 2.05783 0.227249
\(83\) 6.10833 0.670476 0.335238 − 0.942133i \(-0.391183\pi\)
0.335238 + 0.942133i \(0.391183\pi\)
\(84\) 6.20293 0.676795
\(85\) −9.20293 −0.998198
\(86\) 10.6887 1.15260
\(87\) 11.4721 1.22994
\(88\) 0 0
\(89\) 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
\(99\) 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 4598.2.a.bo.1.1 3
11.10 odd 2 418.2.a.g.1.1 ✓ 3
33.32 even 2 3762.2.a.bg.1.2 3
44.43 even 2 3344.2.a.q.1.3 3
209.208 even 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 11.10 odd 2
3344.2.a.q.1.3 3 44.43 even 2
3762.2.a.bg.1.2 3 33.32 even 2
4598.2.a.bo.1.1 3 1.1 even 1 trivial
7942.2.a.bi.1.3 3 209.208 even 2