Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [495,6,Mod(1,495)] 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("495.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(495, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 495.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,5,0,61] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(79.3899908074\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 33x^{2} - 8x + 116 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: no (minimal twist has level 55)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-3.95665\) of defining polynomial
Character \(\chi\) \(=\) 495.1

$q$-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)
 
\(f(q)\) \(=\) \(q-7.82466 q^{2} +29.2253 q^{4} +25.0000 q^{5} -125.436 q^{7} +21.7111 q^{8} -195.616 q^{10} +121.000 q^{11} +532.300 q^{13} +981.491 q^{14} -1105.09 q^{16} +1373.09 q^{17} -554.639 q^{19} +730.632 q^{20} -946.784 q^{22} -4250.72 q^{23} +625.000 q^{25} -4165.06 q^{26} -3665.89 q^{28} +6973.40 q^{29} +3130.03 q^{31} +7952.21 q^{32} -10744.0 q^{34} -3135.89 q^{35} +1384.70 q^{37} +4339.86 q^{38} +542.778 q^{40} -679.385 q^{41} -1721.06 q^{43} +3536.26 q^{44} +33260.4 q^{46} +15143.3 q^{47} -1072.91 q^{49} -4890.41 q^{50} +15556.6 q^{52} +9544.44 q^{53} +3025.00 q^{55} -2723.35 q^{56} -54564.5 q^{58} -27582.7 q^{59} -40527.5 q^{61} -24491.5 q^{62} -26860.4 q^{64} +13307.5 q^{65} -58726.5 q^{67} +40129.0 q^{68} +24537.3 q^{70} +42527.8 q^{71} +23753.0 q^{73} -10834.8 q^{74} -16209.5 q^{76} -15177.7 q^{77} -78690.1 q^{79} -27627.3 q^{80} +5315.96 q^{82} -52252.1 q^{83} +34327.3 q^{85} +13466.7 q^{86} +2627.05 q^{88} -8156.09 q^{89} -66769.3 q^{91} -124228. q^{92} -118491. q^{94} -13866.0 q^{95} +79010.1 q^{97} +8395.17 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 5 q^{2} + 61 q^{4} + 100 q^{5} - 90 q^{7} + 135 q^{8} + 125 q^{10} + 484 q^{11} + 820 q^{13} + 1687 q^{14} - 2671 q^{16} + 3800 q^{17} - 3394 q^{19} + 1525 q^{20} + 605 q^{22} + 3020 q^{23} + 2500 q^{25}+ \cdots + 393590 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −7.82466 −1.38322 −0.691609 − 0.722272i \(-0.743098\pi\)
−0.691609 + 0.722272i \(0.743098\pi\)
\(3\) 0 0
\(4\) 29.2253 0.913290
\(5\) 25.0000 0.447214
\(6\) 0 0
\(7\) −125.436 −0.967555 −0.483778 − 0.875191i \(-0.660736\pi\)
−0.483778 + 0.875191i \(0.660736\pi\)
\(8\) 21.7111 0.119938
\(9\) 0 0
\(10\) −195.616 −0.618594
\(11\) 121.000 0.301511
\(12\) 0 0
\(13\) 532.300 0.873570 0.436785 − 0.899566i \(-0.356117\pi\)
0.436785 + 0.899566i \(0.356117\pi\)
\(14\) 981.491 1.33834
\(15\) 0 0
\(16\) −1105.09 −1.07919
\(17\) 1373.09 1.15233 0.576166 − 0.817333i \(-0.304548\pi\)
0.576166 + 0.817333i \(0.304548\pi\)
\(18\) 0 0
\(19\) −554.639 −0.352474 −0.176237 − 0.984348i \(-0.556392\pi\)
−0.176237 + 0.984348i \(0.556392\pi\)
\(20\) 730.632 0.408436
\(21\) 0 0
\(22\) −946.784 −0.417056
\(23\) −4250.72 −1.67549 −0.837746 − 0.546060i \(-0.816127\pi\)
−0.837746 + 0.546060i \(0.816127\pi\)
\(24\) 0 0
\(25\) 625.000 0.200000
\(26\) −4165.06 −1.20834
\(27\) 0 0
\(28\) −3665.89 −0.883659
\(29\) 6973.40 1.53975 0.769874 − 0.638196i \(-0.220319\pi\)
0.769874 + 0.638196i \(0.220319\pi\)
\(30\) 0 0
\(31\) 3130.03 0.584985 0.292493 − 0.956268i \(-0.405515\pi\)
0.292493 + 0.956268i \(0.405515\pi\)
\(32\) 7952.21 1.37282
\(33\) 0 0
\(34\) −10744.0 −1.59392
\(35\) −3135.89 −0.432704
\(36\) 0 0
\(37\) 1384.70 0.166284 0.0831422 − 0.996538i \(-0.473504\pi\)
0.0831422 + 0.996538i \(0.473504\pi\)
\(38\) 4339.86 0.487548
\(39\) 0 0
\(40\) 542.778 0.0536380
\(41\) −679.385 −0.0631185 −0.0315592 − 0.999502i \(-0.510047\pi\)
−0.0315592 + 0.999502i \(0.510047\pi\)
\(42\) 0 0
\(43\) −1721.06 −0.141946 −0.0709732 − 0.997478i \(-0.522610\pi\)
−0.0709732 + 0.997478i \(0.522610\pi\)
\(44\) 3536.26 0.275367
\(45\) 0 0
\(46\) 33260.4 2.31757
\(47\) 15143.3 0.999945 0.499972 − 0.866041i \(-0.333344\pi\)
0.499972 + 0.866041i \(0.333344\pi\)
\(48\) 0 0
\(49\) −1072.91 −0.0638372
\(50\) −4890.41 −0.276643
\(51\) 0 0
\(52\) 15556.6 0.797823
\(53\) 9544.44 0.466724 0.233362 − 0.972390i \(-0.425027\pi\)
0.233362 + 0.972390i \(0.425027\pi\)
\(54\) 0 0
\(55\) 3025.00 0.134840
\(56\) −2723.35 −0.116047
\(57\) 0 0
\(58\) −54564.5 −2.12981
\(59\) −27582.7 −1.03159 −0.515794 − 0.856713i \(-0.672503\pi\)
−0.515794 + 0.856713i \(0.672503\pi\)
\(60\) 0 0
\(61\) −40527.5 −1.39452 −0.697261 − 0.716817i \(-0.745598\pi\)
−0.697261 + 0.716817i \(0.745598\pi\)
\(62\) −24491.5 −0.809162
\(63\) 0 0
\(64\) −26860.4 −0.819714
\(65\) 13307.5 0.390672
\(66\) 0 0
\(67\) −58726.5 −1.59826 −0.799130 − 0.601159i \(-0.794706\pi\)
−0.799130 + 0.601159i \(0.794706\pi\)
\(68\) 40129.0 1.05241
\(69\) 0 0
\(70\) 24537.3 0.598523
\(71\) 42527.8 1.00121 0.500607 − 0.865675i \(-0.333110\pi\)
0.500607 + 0.865675i \(0.333110\pi\)
\(72\) 0 0
\(73\) 23753.0 0.521690 0.260845 − 0.965381i \(-0.415999\pi\)
0.260845 + 0.965381i \(0.415999\pi\)
\(74\) −10834.8 −0.230007
\(75\) 0 0
\(76\) −16209.5 −0.321911
\(77\) −15177.7 −0.291729
\(78\) 0 0
\(79\) −78690.1 −1.41857 −0.709287 − 0.704919i \(-0.750983\pi\)
−0.709287 + 0.704919i \(0.750983\pi\)
\(80\) −27627.3 −0.482629
\(81\) 0 0
\(82\) 5315.96 0.0873066
\(83\) −52252.1 −0.832546 −0.416273 − 0.909240i \(-0.636664\pi\)
−0.416273 + 0.909240i \(0.636664\pi\)
\(84\) 0 0
\(85\) 34327.3 0.515338
\(86\) 13466.7 0.196343
\(87\) 0 0
\(88\) 2627.05 0.0361627
\(89\) −8156.09 −0.109146 −0.0545729 − 0.998510i \(-0.517380\pi\)
−0.0545729 + 0.998510i \(0.517380\pi\)
\(90\) 0 0
\(91\) −66769.3 −0.845227
\(92\) −124228. −1.53021
\(93\) 0 0
\(94\) −118491. −1.38314
\(95\) −13866.0 −0.157631
\(96\) 0 0
\(97\) 79010.1 0.852615 0.426308 − 0.904578i \(-0.359814\pi\)
0.426308 + 0.904578i \(0.359814\pi\)
\(98\) 8395.17 0.0883007
\(99\) 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 495.6.a.g.1.1 4
3.2 odd 2 55.6.a.b.1.4 ✓ 4
12.11 even 2 880.6.a.n.1.4 4
15.2 even 4 275.6.b.d.199.7 8
15.8 even 4 275.6.b.d.199.2 8
15.14 odd 2 275.6.a.d.1.1 4
33.32 even 2 605.6.a.c.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
55.6.a.b.1.4 ✓ 4 3.2 odd 2
275.6.a.d.1.1 4 15.14 odd 2
275.6.b.d.199.2 8 15.8 even 4
275.6.b.d.199.7 8 15.2 even 4
495.6.a.g.1.1 4 1.1 even 1 trivial
605.6.a.c.1.1 4 33.32 even 2
880.6.a.n.1.4 4 12.11 even 2