Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [9610,2,Mod(1,9610)] 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("9610.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9610, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 9610 = 2 \cdot 5 \cdot 31^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9610.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,3,-1,3,-3,-1,4,3,0,-3,-5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(76.7362363425\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.321.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x + 1 \) 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 310)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.239123\) of defining polynomial
Character \(\chi\) \(=\) 9610.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -0.239123 q^{3} +1.00000 q^{4} -1.00000 q^{5} -0.239123 q^{6} +1.23912 q^{7} +1.00000 q^{8} -2.94282 q^{9} -1.00000 q^{10} +4.12476 q^{11} -0.239123 q^{12} -6.12476 q^{13} +1.23912 q^{14} +0.239123 q^{15} +1.00000 q^{16} +3.18194 q^{17} -2.94282 q^{18} -1.94282 q^{19} -1.00000 q^{20} -0.296303 q^{21} +4.12476 q^{22} -7.08126 q^{23} -0.239123 q^{24} +1.00000 q^{25} -6.12476 q^{26} +1.42107 q^{27} +1.23912 q^{28} +8.06758 q^{29} +0.239123 q^{30} +1.00000 q^{32} -0.986327 q^{33} +3.18194 q^{34} -1.23912 q^{35} -2.94282 q^{36} -7.64652 q^{37} -1.94282 q^{38} +1.46457 q^{39} -1.00000 q^{40} +9.54583 q^{41} -0.296303 q^{42} +11.2632 q^{43} +4.12476 q^{44} +2.94282 q^{45} -7.08126 q^{46} +0.942820 q^{47} -0.239123 q^{48} -5.46457 q^{49} +1.00000 q^{50} -0.760877 q^{51} -6.12476 q^{52} -12.5893 q^{53} +1.42107 q^{54} -4.12476 q^{55} +1.23912 q^{56} +0.464574 q^{57} +8.06758 q^{58} +2.64652 q^{59} +0.239123 q^{60} -1.95649 q^{61} -3.64652 q^{63} +1.00000 q^{64} +6.12476 q^{65} -0.986327 q^{66} -8.52175 q^{67} +3.18194 q^{68} +1.69329 q^{69} -1.23912 q^{70} +8.60301 q^{71} -2.94282 q^{72} -11.8993 q^{73} -7.64652 q^{74} -0.239123 q^{75} -1.94282 q^{76} +5.11109 q^{77} +1.46457 q^{78} -5.54583 q^{79} -1.00000 q^{80} +8.48865 q^{81} +9.54583 q^{82} -2.50808 q^{83} -0.296303 q^{84} -3.18194 q^{85} +11.2632 q^{86} -1.92915 q^{87} +4.12476 q^{88} -11.8285 q^{89} +2.94282 q^{90} -7.58934 q^{91} -7.08126 q^{92} +0.942820 q^{94} +1.94282 q^{95} -0.239123 q^{96} -2.29630 q^{97} -5.46457 q^{98} -12.1384 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} - q^{3} + 3 q^{4} - 3 q^{5} - q^{6} + 4 q^{7} + 3 q^{8} - 3 q^{10} - 5 q^{11} - q^{12} - q^{13} + 4 q^{14} + q^{15} + 3 q^{16} + q^{17} + 3 q^{19} - 3 q^{20} - 10 q^{21} - 5 q^{22} - 5 q^{23}+ \cdots - 29 q^{99}+O(q^{100}) \) 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
\(3\) −0.239123 −0.138058 −0.0690289 − 0.997615i \(-0.521990\pi\)
−0.0690289 + 0.997615i \(0.521990\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) −0.239123 −0.0976217
\(7\) 1.23912 0.468345 0.234172 − 0.972195i \(-0.424762\pi\)
0.234172 + 0.972195i \(0.424762\pi\)
\(8\) 1.00000 0.353553
\(9\) −2.94282 −0.980940
\(10\) −1.00000 −0.316228
\(11\) 4.12476 1.24366 0.621831 − 0.783151i \(-0.286389\pi\)
0.621831 + 0.783151i \(0.286389\pi\)
\(12\) −0.239123 −0.0690289
\(13\) −6.12476 −1.69870 −0.849352 − 0.527827i \(-0.823007\pi\)
−0.849352 + 0.527827i \(0.823007\pi\)
\(14\) 1.23912 0.331170
\(15\) 0.239123 0.0617414
\(16\) 1.00000 0.250000
\(17\) 3.18194 0.771735 0.385867 − 0.922554i \(-0.373902\pi\)
0.385867 + 0.922554i \(0.373902\pi\)
\(18\) −2.94282 −0.693629
\(19\) −1.94282 −0.445713 −0.222857 − 0.974851i \(-0.571538\pi\)
−0.222857 + 0.974851i \(0.571538\pi\)
\(20\) −1.00000 −0.223607
\(21\) −0.296303 −0.0646587
\(22\) 4.12476 0.879403
\(23\) −7.08126 −1.47654 −0.738272 − 0.674503i \(-0.764358\pi\)
−0.738272 + 0.674503i \(0.764358\pi\)
\(24\) −0.239123 −0.0488108
\(25\) 1.00000 0.200000
\(26\) −6.12476 −1.20116
\(27\) 1.42107 0.273484
\(28\) 1.23912 0.234172
\(29\) 8.06758 1.49811 0.749056 − 0.662506i \(-0.230507\pi\)
0.749056 + 0.662506i \(0.230507\pi\)
\(30\) 0.239123 0.0436577
\(31\) 0 0
\(32\) 1.00000 0.176777
\(33\) −0.986327 −0.171697
\(34\) 3.18194 0.545699
\(35\) −1.23912 −0.209450
\(36\) −2.94282 −0.490470
\(37\) −7.64652 −1.25708 −0.628540 − 0.777777i \(-0.716347\pi\)
−0.628540 + 0.777777i \(0.716347\pi\)
\(38\) −1.94282 −0.315167
\(39\) 1.46457 0.234519
\(40\) −1.00000 −0.158114
\(41\) 9.54583 1.49081 0.745404 − 0.666613i \(-0.232257\pi\)
0.745404 + 0.666613i \(0.232257\pi\)
\(42\) −0.296303 −0.0457206
\(43\) 11.2632 1.71762 0.858811 − 0.512293i \(-0.171204\pi\)
0.858811 + 0.512293i \(0.171204\pi\)
\(44\) 4.12476 0.621831
\(45\) 2.94282 0.438690
\(46\) −7.08126 −1.04407
\(47\) 0.942820 0.137524 0.0687622 − 0.997633i \(-0.478095\pi\)
0.0687622 + 0.997633i \(0.478095\pi\)
\(48\) −0.239123 −0.0345145
\(49\) −5.46457 −0.780653
\(50\) 1.00000 0.141421
\(51\) −0.760877 −0.106544
\(52\) −6.12476 −0.849352
\(53\) −12.5893 −1.72928 −0.864639 − 0.502393i \(-0.832453\pi\)
−0.864639 + 0.502393i \(0.832453\pi\)
\(54\) 1.42107 0.193383
\(55\) −4.12476 −0.556183
\(56\) 1.23912 0.165585
\(57\) 0.464574 0.0615343
\(58\) 8.06758 1.05933
\(59\) 2.64652 0.344547 0.172274 − 0.985049i \(-0.444889\pi\)
0.172274 + 0.985049i \(0.444889\pi\)
\(60\) 0.239123 0.0308707
\(61\) −1.95649 −0.250503 −0.125252 − 0.992125i \(-0.539974\pi\)
−0.125252 + 0.992125i \(0.539974\pi\)
\(62\) 0 0
\(63\) −3.64652 −0.459418
\(64\) 1.00000 0.125000
\(65\) 6.12476 0.759683
\(66\) −0.986327 −0.121408
\(67\) −8.52175 −1.04110 −0.520549 − 0.853832i \(-0.674273\pi\)
−0.520549 + 0.853832i \(0.674273\pi\)
\(68\) 3.18194 0.385867
\(69\) 1.69329 0.203849
\(70\) −1.23912 −0.148104
\(71\) 8.60301 1.02099 0.510495 − 0.859881i \(-0.329462\pi\)
0.510495 + 0.859881i \(0.329462\pi\)
\(72\) −2.94282 −0.346815
\(73\) −11.8993 −1.39271 −0.696355 − 0.717698i \(-0.745196\pi\)
−0.696355 + 0.717698i \(0.745196\pi\)
\(74\) −7.64652 −0.888890
\(75\) −0.239123 −0.0276116
\(76\) −1.94282 −0.222857
\(77\) 5.11109 0.582463
\(78\) 1.46457 0.165830
\(79\) −5.54583 −0.623955 −0.311977 − 0.950090i \(-0.600991\pi\)
−0.311977 + 0.950090i \(0.600991\pi\)
\(80\) −1.00000 −0.111803
\(81\) 8.48865 0.943183
\(82\) 9.54583 1.05416
\(83\) −2.50808 −0.275298 −0.137649 − 0.990481i \(-0.543955\pi\)
−0.137649 + 0.990481i \(0.543955\pi\)
\(84\) −0.296303 −0.0323293
\(85\) −3.18194 −0.345130
\(86\) 11.2632 1.21454
\(87\) −1.92915 −0.206826
\(88\) 4.12476 0.439701
\(89\) −11.8285 −1.25381 −0.626907 − 0.779094i \(-0.715679\pi\)
−0.626907 + 0.779094i \(0.715679\pi\)
\(90\) 2.94282 0.310200
\(91\) −7.58934 −0.795579
\(92\) −7.08126 −0.738272
\(93\) 0 0
\(94\) 0.942820 0.0972445
\(95\) 1.94282 0.199329
\(96\) −0.239123 −0.0244054
\(97\) −2.29630 −0.233154 −0.116577 − 0.993182i \(-0.537192\pi\)
−0.116577 + 0.993182i \(0.537192\pi\)
\(98\) −5.46457 −0.552005
\(99\) −12.1384 −1.21996
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 9610.2.a.v.1.2 3
31.6 odd 6 310.2.e.c.191.2 ✓ 6
31.26 odd 6 310.2.e.c.211.2 yes 6
31.30 odd 2 9610.2.a.x.1.2 3
155.37 even 12 1550.2.p.g.749.5 12
155.57 even 12 1550.2.p.g.149.5 12
155.68 even 12 1550.2.p.g.749.2 12
155.88 even 12 1550.2.p.g.149.2 12
155.99 odd 6 1550.2.e.l.501.2 6
155.119 odd 6 1550.2.e.l.1451.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
310.2.e.c.191.2 ✓ 6 31.6 odd 6
310.2.e.c.211.2 yes 6 31.26 odd 6
1550.2.e.l.501.2 6 155.99 odd 6
1550.2.e.l.1451.2 6 155.119 odd 6
1550.2.p.g.149.2 12 155.88 even 12
1550.2.p.g.149.5 12 155.57 even 12
1550.2.p.g.749.2 12 155.68 even 12
1550.2.p.g.749.5 12 155.37 even 12
9610.2.a.v.1.2 3 1.1 even 1 trivial
9610.2.a.x.1.2 3 31.30 odd 2