Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1305,2,Mod(784,1305)] 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("1305.784"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1305, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1305.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,0,-4,12] 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: no
Analytic conductor: \(10.4204774638\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{3})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 145)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 784.2
Root \(-0.517638i\) of defining polynomial
Character \(\chi\) \(=\) 1305.784
Dual form 1305.2.c.f.784.3

$q$-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)
 
\(f(q)\) \(=\) \(q-0.517638i q^{2} +1.73205 q^{4} +(1.73205 + 1.41421i) q^{5} -2.44949i q^{7} -1.93185i q^{8} +(0.732051 - 0.896575i) q^{10} +1.26795 q^{11} +1.79315i q^{13} -1.26795 q^{14} +2.46410 q^{16} -1.41421i q^{17} +3.26795 q^{19} +(3.00000 + 2.44949i) q^{20} -0.656339i q^{22} -6.31319i q^{23} +(1.00000 + 4.89898i) q^{25} +0.928203 q^{26} -4.24264i q^{28} +1.00000 q^{29} -8.73205 q^{31} -5.13922i q^{32} -0.732051 q^{34} +(3.46410 - 4.24264i) q^{35} +9.14162i q^{37} -1.69161i q^{38} +(2.73205 - 3.34607i) q^{40} +6.92820 q^{41} -9.14162i q^{43} +2.19615 q^{44} -3.26795 q^{46} -1.41421i q^{47} +1.00000 q^{49} +(2.53590 - 0.517638i) q^{50} +3.10583i q^{52} +5.93426i q^{53} +(2.19615 + 1.79315i) q^{55} -4.73205 q^{56} -0.517638i q^{58} -10.3923 q^{59} +2.92820 q^{61} +4.52004i q^{62} +2.26795 q^{64} +(-2.53590 + 3.10583i) q^{65} +4.24264i q^{67} -2.44949i q^{68} +(-2.19615 - 1.79315i) q^{70} -3.46410 q^{71} +7.34847i q^{73} +4.73205 q^{74} +5.66025 q^{76} -3.10583i q^{77} +4.19615 q^{79} +(4.26795 + 3.48477i) q^{80} -3.58630i q^{82} +10.1769i q^{83} +(2.00000 - 2.44949i) q^{85} -4.73205 q^{86} -2.44949i q^{88} +10.3923 q^{89} +4.39230 q^{91} -10.9348i q^{92} -0.732051 q^{94} +(5.66025 + 4.62158i) q^{95} -10.9348i q^{97} -0.517638i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{10} + 12 q^{11} - 12 q^{14} - 4 q^{16} + 20 q^{19} + 12 q^{20} + 4 q^{25} - 24 q^{26} + 4 q^{29} - 28 q^{31} + 4 q^{34} + 4 q^{40} - 12 q^{44} - 20 q^{46} + 4 q^{49} + 24 q^{50} - 12 q^{55} - 12 q^{56}+ \cdots - 12 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1305\mathbb{Z}\right)^\times\).

\(n\) \(146\) \(262\) \(901\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 0.517638i − 0.366025i −0.983111 − 0.183013i \(-0.941415\pi\)
0.983111 − 0.183013i \(-0.0585849\pi\)
\(3\) 0 0
\(4\) 1.73205 0.866025
\(5\) 1.73205 + 1.41421i 0.774597 + 0.632456i
\(6\) 0 0
\(7\) − 2.44949i − 0.925820i −0.886405 − 0.462910i \(-0.846805\pi\)
0.886405 − 0.462910i \(-0.153195\pi\)
\(8\) − 1.93185i − 0.683013i
\(9\) 0 0
\(10\) 0.732051 − 0.896575i 0.231495 − 0.283522i
\(11\) 1.26795 0.382301 0.191151 − 0.981561i \(-0.438778\pi\)
0.191151 + 0.981561i \(0.438778\pi\)
\(12\) 0 0
\(13\) 1.79315i 0.497331i 0.968589 + 0.248665i \(0.0799919\pi\)
−0.968589 + 0.248665i \(0.920008\pi\)
\(14\) −1.26795 −0.338874
\(15\) 0 0
\(16\) 2.46410 0.616025
\(17\) − 1.41421i − 0.342997i −0.985184 − 0.171499i \(-0.945139\pi\)
0.985184 − 0.171499i \(-0.0548609\pi\)
\(18\) 0 0
\(19\) 3.26795 0.749719 0.374859 − 0.927082i \(-0.377691\pi\)
0.374859 + 0.927082i \(0.377691\pi\)
\(20\) 3.00000 + 2.44949i 0.670820 + 0.547723i
\(21\) 0 0
\(22\) − 0.656339i − 0.139932i
\(23\) − 6.31319i − 1.31639i −0.752847 − 0.658196i \(-0.771320\pi\)
0.752847 − 0.658196i \(-0.228680\pi\)
\(24\) 0 0
\(25\) 1.00000 + 4.89898i 0.200000 + 0.979796i
\(26\) 0.928203 0.182036
\(27\) 0 0
\(28\) − 4.24264i − 0.801784i
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) −8.73205 −1.56832 −0.784161 − 0.620557i \(-0.786907\pi\)
−0.784161 + 0.620557i \(0.786907\pi\)
\(32\) − 5.13922i − 0.908494i
\(33\) 0 0
\(34\) −0.732051 −0.125546
\(35\) 3.46410 − 4.24264i 0.585540 − 0.717137i
\(36\) 0 0
\(37\) 9.14162i 1.50287i 0.659805 + 0.751437i \(0.270639\pi\)
−0.659805 + 0.751437i \(0.729361\pi\)
\(38\) − 1.69161i − 0.274416i
\(39\) 0 0
\(40\) 2.73205 − 3.34607i 0.431975 − 0.529059i
\(41\) 6.92820 1.08200 0.541002 − 0.841021i \(-0.318045\pi\)
0.541002 + 0.841021i \(0.318045\pi\)
\(42\) 0 0
\(43\) − 9.14162i − 1.39408i −0.717030 − 0.697042i \(-0.754499\pi\)
0.717030 − 0.697042i \(-0.245501\pi\)
\(44\) 2.19615 0.331082
\(45\) 0 0
\(46\) −3.26795 −0.481833
\(47\) − 1.41421i − 0.206284i −0.994667 − 0.103142i \(-0.967110\pi\)
0.994667 − 0.103142i \(-0.0328896\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 2.53590 − 0.517638i 0.358630 − 0.0732051i
\(51\) 0 0
\(52\) 3.10583i 0.430701i
\(53\) 5.93426i 0.815133i 0.913176 + 0.407566i \(0.133622\pi\)
−0.913176 + 0.407566i \(0.866378\pi\)
\(54\) 0 0
\(55\) 2.19615 + 1.79315i 0.296129 + 0.241788i
\(56\) −4.73205 −0.632347
\(57\) 0 0
\(58\) − 0.517638i − 0.0679692i
\(59\) −10.3923 −1.35296 −0.676481 − 0.736460i \(-0.736496\pi\)
−0.676481 + 0.736460i \(0.736496\pi\)
\(60\) 0 0
\(61\) 2.92820 0.374918 0.187459 − 0.982272i \(-0.439975\pi\)
0.187459 + 0.982272i \(0.439975\pi\)
\(62\) 4.52004i 0.574046i
\(63\) 0 0
\(64\) 2.26795 0.283494
\(65\) −2.53590 + 3.10583i −0.314539 + 0.385231i
\(66\) 0 0
\(67\) 4.24264i 0.518321i 0.965834 + 0.259161i \(0.0834459\pi\)
−0.965834 + 0.259161i \(0.916554\pi\)
\(68\) − 2.44949i − 0.297044i
\(69\) 0 0
\(70\) −2.19615 − 1.79315i −0.262490 − 0.214323i
\(71\) −3.46410 −0.411113 −0.205557 − 0.978645i \(-0.565900\pi\)
−0.205557 + 0.978645i \(0.565900\pi\)
\(72\) 0 0
\(73\) 7.34847i 0.860073i 0.902811 + 0.430037i \(0.141499\pi\)
−0.902811 + 0.430037i \(0.858501\pi\)
\(74\) 4.73205 0.550090
\(75\) 0 0
\(76\) 5.66025 0.649276
\(77\) − 3.10583i − 0.353942i
\(78\) 0 0
\(79\) 4.19615 0.472104 0.236052 − 0.971740i \(-0.424146\pi\)
0.236052 + 0.971740i \(0.424146\pi\)
\(80\) 4.26795 + 3.48477i 0.477171 + 0.389609i
\(81\) 0 0
\(82\) − 3.58630i − 0.396041i
\(83\) 10.1769i 1.11706i 0.829484 + 0.558530i \(0.188634\pi\)
−0.829484 + 0.558530i \(0.811366\pi\)
\(84\) 0 0
\(85\) 2.00000 − 2.44949i 0.216930 − 0.265684i
\(86\) −4.73205 −0.510270
\(87\) 0 0
\(88\) − 2.44949i − 0.261116i
\(89\) 10.3923 1.10158 0.550791 − 0.834643i \(-0.314326\pi\)
0.550791 + 0.834643i \(0.314326\pi\)
\(90\) 0 0
\(91\) 4.39230 0.460439
\(92\) − 10.9348i − 1.14003i
\(93\) 0 0
\(94\) −0.732051 −0.0755053
\(95\) 5.66025 + 4.62158i 0.580730 + 0.474164i
\(96\) 0 0
\(97\) − 10.9348i − 1.11026i −0.831764 − 0.555129i \(-0.812669\pi\)
0.831764 − 0.555129i \(-0.187331\pi\)
\(98\) − 0.517638i − 0.0522893i
\(99\) 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 1305.2.c.f.784.2 4
3.2 odd 2 145.2.b.b.59.3 yes 4
5.2 odd 4 6525.2.a.bj.1.3 4
5.3 odd 4 6525.2.a.bj.1.2 4
5.4 even 2 inner 1305.2.c.f.784.3 4
12.11 even 2 2320.2.d.f.929.3 4
15.2 even 4 725.2.a.f.1.2 4
15.8 even 4 725.2.a.f.1.3 4
15.14 odd 2 145.2.b.b.59.2 ✓ 4
60.59 even 2 2320.2.d.f.929.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
145.2.b.b.59.2 ✓ 4 15.14 odd 2
145.2.b.b.59.3 yes 4 3.2 odd 2
725.2.a.f.1.2 4 15.2 even 4
725.2.a.f.1.3 4 15.8 even 4
1305.2.c.f.784.2 4 1.1 even 1 trivial
1305.2.c.f.784.3 4 5.4 even 2 inner
2320.2.d.f.929.1 4 60.59 even 2
2320.2.d.f.929.3 4 12.11 even 2
6525.2.a.bj.1.2 4 5.3 odd 4
6525.2.a.bj.1.3 4 5.2 odd 4