Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1445,4,Mod(1,1445)] 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("1445.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1445, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1445 = 5 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1445.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [5,2,1,34,-25] 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: \(85.2577599583\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 20x^{3} + 38x^{2} + 69x - 126 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 85)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-2.08822\) of defining polynomial
Character \(\chi\) \(=\) 1445.1

$q$-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)
 
\(f(q)\) \(=\) \(q+5.46013 q^{2} +0.820806 q^{3} +21.8130 q^{4} -5.00000 q^{5} +4.48171 q^{6} +22.0083 q^{7} +75.4209 q^{8} -26.3263 q^{9} -27.3007 q^{10} -8.31663 q^{11} +17.9043 q^{12} -24.7022 q^{13} +120.168 q^{14} -4.10403 q^{15} +237.304 q^{16} -143.745 q^{18} +79.8929 q^{19} -109.065 q^{20} +18.0646 q^{21} -45.4099 q^{22} +157.610 q^{23} +61.9060 q^{24} +25.0000 q^{25} -134.877 q^{26} -43.7706 q^{27} +480.068 q^{28} +226.519 q^{29} -22.4086 q^{30} -30.3326 q^{31} +692.342 q^{32} -6.82635 q^{33} -110.042 q^{35} -574.256 q^{36} +232.860 q^{37} +436.225 q^{38} -20.2757 q^{39} -377.105 q^{40} -384.878 q^{41} +98.6348 q^{42} +422.692 q^{43} -181.411 q^{44} +131.631 q^{45} +860.570 q^{46} +596.896 q^{47} +194.780 q^{48} +141.365 q^{49} +136.503 q^{50} -538.829 q^{52} -402.902 q^{53} -238.993 q^{54} +41.5832 q^{55} +1659.89 q^{56} +65.5766 q^{57} +1236.82 q^{58} -299.311 q^{59} -89.5213 q^{60} -527.970 q^{61} -165.620 q^{62} -579.397 q^{63} +1881.85 q^{64} +123.511 q^{65} -37.2727 q^{66} +462.088 q^{67} +129.367 q^{69} -600.841 q^{70} -335.298 q^{71} -1985.55 q^{72} +171.122 q^{73} +1271.45 q^{74} +20.5202 q^{75} +1742.70 q^{76} -183.035 q^{77} -110.708 q^{78} -414.795 q^{79} -1186.52 q^{80} +674.882 q^{81} -2101.48 q^{82} -1313.29 q^{83} +394.043 q^{84} +2307.95 q^{86} +185.928 q^{87} -627.248 q^{88} +844.369 q^{89} +718.725 q^{90} -543.653 q^{91} +3437.95 q^{92} -24.8972 q^{93} +3259.13 q^{94} -399.464 q^{95} +568.279 q^{96} +1201.83 q^{97} +771.873 q^{98} +218.946 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 2 q^{2} + q^{3} + 34 q^{4} - 25 q^{5} + 5 q^{6} - 10 q^{7} + 30 q^{8} - 30 q^{9} - 10 q^{10} - 126 q^{11} - 15 q^{12} + 83 q^{13} - 90 q^{14} - 5 q^{15} + 322 q^{16} - 97 q^{18} + 55 q^{19} - 170 q^{20}+ \cdots + 858 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.46013 1.93045 0.965224 − 0.261425i \(-0.0841924\pi\)
0.965224 + 0.261425i \(0.0841924\pi\)
\(3\) 0.820806 0.157964 0.0789821 − 0.996876i \(-0.474833\pi\)
0.0789821 + 0.996876i \(0.474833\pi\)
\(4\) 21.8130 2.72663
\(5\) −5.00000 −0.447214
\(6\) 4.48171 0.304942
\(7\) 22.0083 1.18834 0.594168 − 0.804341i \(-0.297481\pi\)
0.594168 + 0.804341i \(0.297481\pi\)
\(8\) 75.4209 3.33316
\(9\) −26.3263 −0.975047
\(10\) −27.3007 −0.863322
\(11\) −8.31663 −0.227960 −0.113980 − 0.993483i \(-0.536360\pi\)
−0.113980 + 0.993483i \(0.536360\pi\)
\(12\) 17.9043 0.430710
\(13\) −24.7022 −0.527011 −0.263506 − 0.964658i \(-0.584879\pi\)
−0.263506 + 0.964658i \(0.584879\pi\)
\(14\) 120.168 2.29402
\(15\) −4.10403 −0.0706438
\(16\) 237.304 3.70787
\(17\) 0 0
\(18\) −143.745 −1.88228
\(19\) 79.8929 0.964667 0.482334 − 0.875988i \(-0.339789\pi\)
0.482334 + 0.875988i \(0.339789\pi\)
\(20\) −109.065 −1.21939
\(21\) 18.0646 0.187715
\(22\) −45.4099 −0.440065
\(23\) 157.610 1.42887 0.714433 − 0.699704i \(-0.246685\pi\)
0.714433 + 0.699704i \(0.246685\pi\)
\(24\) 61.9060 0.526521
\(25\) 25.0000 0.200000
\(26\) −134.877 −1.01737
\(27\) −43.7706 −0.311987
\(28\) 480.068 3.24015
\(29\) 226.519 1.45046 0.725232 − 0.688505i \(-0.241732\pi\)
0.725232 + 0.688505i \(0.241732\pi\)
\(30\) −22.4086 −0.136374
\(31\) −30.3326 −0.175739 −0.0878694 − 0.996132i \(-0.528006\pi\)
−0.0878694 + 0.996132i \(0.528006\pi\)
\(32\) 692.342 3.82469
\(33\) −6.82635 −0.0360095
\(34\) 0 0
\(35\) −110.042 −0.531440
\(36\) −574.256 −2.65859
\(37\) 232.860 1.03465 0.517324 − 0.855789i \(-0.326928\pi\)
0.517324 + 0.855789i \(0.326928\pi\)
\(38\) 436.225 1.86224
\(39\) −20.2757 −0.0832490
\(40\) −377.105 −1.49064
\(41\) −384.878 −1.46604 −0.733022 − 0.680205i \(-0.761891\pi\)
−0.733022 + 0.680205i \(0.761891\pi\)
\(42\) 98.6348 0.362373
\(43\) 422.692 1.49907 0.749534 − 0.661965i \(-0.230277\pi\)
0.749534 + 0.661965i \(0.230277\pi\)
\(44\) −181.411 −0.621562
\(45\) 131.631 0.436054
\(46\) 860.570 2.75835
\(47\) 596.896 1.85247 0.926237 − 0.376942i \(-0.123024\pi\)
0.926237 + 0.376942i \(0.123024\pi\)
\(48\) 194.780 0.585711
\(49\) 141.365 0.412144
\(50\) 136.503 0.386090
\(51\) 0 0
\(52\) −538.829 −1.43696
\(53\) −402.902 −1.04421 −0.522103 − 0.852883i \(-0.674852\pi\)
−0.522103 + 0.852883i \(0.674852\pi\)
\(54\) −238.993 −0.602274
\(55\) 41.5832 0.101947
\(56\) 1659.89 3.96092
\(57\) 65.5766 0.152383
\(58\) 1236.82 2.80004
\(59\) −299.311 −0.660457 −0.330228 − 0.943901i \(-0.607126\pi\)
−0.330228 + 0.943901i \(0.607126\pi\)
\(60\) −89.5213 −0.192619
\(61\) −527.970 −1.10819 −0.554096 − 0.832453i \(-0.686936\pi\)
−0.554096 + 0.832453i \(0.686936\pi\)
\(62\) −165.620 −0.339255
\(63\) −579.397 −1.15868
\(64\) 1881.85 3.67549
\(65\) 123.511 0.235687
\(66\) −37.2727 −0.0695145
\(67\) 462.088 0.842583 0.421291 − 0.906925i \(-0.361577\pi\)
0.421291 + 0.906925i \(0.361577\pi\)
\(68\) 0 0
\(69\) 129.367 0.225710
\(70\) −600.841 −1.02592
\(71\) −335.298 −0.560458 −0.280229 − 0.959933i \(-0.590410\pi\)
−0.280229 + 0.959933i \(0.590410\pi\)
\(72\) −1985.55 −3.24999
\(73\) 171.122 0.274361 0.137180 − 0.990546i \(-0.456196\pi\)
0.137180 + 0.990546i \(0.456196\pi\)
\(74\) 1271.45 1.99734
\(75\) 20.5202 0.0315929
\(76\) 1742.70 2.63029
\(77\) −183.035 −0.270893
\(78\) −110.708 −0.160708
\(79\) −414.795 −0.590736 −0.295368 − 0.955384i \(-0.595442\pi\)
−0.295368 + 0.955384i \(0.595442\pi\)
\(80\) −1186.52 −1.65821
\(81\) 674.882 0.925765
\(82\) −2101.48 −2.83012
\(83\) −1313.29 −1.73677 −0.868386 − 0.495889i \(-0.834842\pi\)
−0.868386 + 0.495889i \(0.834842\pi\)
\(84\) 394.043 0.511828
\(85\) 0 0
\(86\) 2307.95 2.89387
\(87\) 185.928 0.229122
\(88\) −627.248 −0.759828
\(89\) 844.369 1.00565 0.502826 − 0.864388i \(-0.332294\pi\)
0.502826 + 0.864388i \(0.332294\pi\)
\(90\) 718.725 0.841780
\(91\) −543.653 −0.626267
\(92\) 3437.95 3.89599
\(93\) −24.8972 −0.0277605
\(94\) 3259.13 3.57610
\(95\) −399.464 −0.431412
\(96\) 568.279 0.604164
\(97\) 1201.83 1.25802 0.629008 − 0.777399i \(-0.283461\pi\)
0.629008 + 0.777399i \(0.283461\pi\)
\(98\) 771.873 0.795622
\(99\) 218.946 0.222272
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 1445.4.a.l.1.5 5
17.16 even 2 85.4.a.g.1.5 ✓ 5
51.50 odd 2 765.4.a.m.1.1 5
68.67 odd 2 1360.4.a.w.1.3 5
85.33 odd 4 425.4.b.i.324.1 10
85.67 odd 4 425.4.b.i.324.10 10
85.84 even 2 425.4.a.i.1.1 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.4.a.g.1.5 ✓ 5 17.16 even 2
425.4.a.i.1.1 5 85.84 even 2
425.4.b.i.324.1 10 85.33 odd 4
425.4.b.i.324.10 10 85.67 odd 4
765.4.a.m.1.1 5 51.50 odd 2
1360.4.a.w.1.3 5 68.67 odd 2
1445.4.a.l.1.5 5 1.1 even 1 trivial