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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1053,2,Mod(649,1053)] 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("1053.649"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1053, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1053 = 3^{4} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1053.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,0,0,-12,0,0,0,0,0,-8,0,0,4,18,0,-4,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.40824733284\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 16x^{8} + 91x^{6} + 222x^{4} + 228x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{3} \)
Twist minimal: no (minimal twist has level 117)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 649.7
Root \(1.07384i\) of defining polynomial
Character \(\chi\) \(=\) 1053.649
Dual form 1053.2.b.j.649.4

$q$-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)
 
\(f(q)\) \(=\) \(q+1.07384i q^{2} +0.846877 q^{4} -1.27644i q^{5} +1.02875i q^{7} +3.05708i q^{8} +1.37069 q^{10} +4.66263i q^{11} +(1.25783 + 3.37903i) q^{13} -1.10471 q^{14} -1.58904 q^{16} -0.476187 q^{17} -6.69096i q^{19} -1.08099i q^{20} -5.00690 q^{22} +0.959735 q^{23} +3.37069 q^{25} +(-3.62852 + 1.35071i) q^{26} +0.871227i q^{28} -9.37759 q^{29} +1.92751i q^{31} +4.40778i q^{32} -0.511346i q^{34} +1.31314 q^{35} +4.94666i q^{37} +7.18499 q^{38} +3.90219 q^{40} +1.52414i q^{41} +2.62852 q^{43} +3.94868i q^{44} +1.03060i q^{46} +6.83948i q^{47} +5.94167 q^{49} +3.61957i q^{50} +(1.06523 + 2.86163i) q^{52} +0.582145 q^{53} +5.95159 q^{55} -3.14498 q^{56} -10.0700i q^{58} +4.21233i q^{59} +9.43290 q^{61} -2.06983 q^{62} -7.91132 q^{64} +(4.31314 - 1.60555i) q^{65} +2.32275i q^{67} -0.403272 q^{68} +1.41010i q^{70} +1.35071i q^{71} +12.8687i q^{73} -5.31190 q^{74} -5.66642i q^{76} -4.79669 q^{77} -12.9083 q^{79} +2.02833i q^{80} -1.63667 q^{82} -10.2328i q^{83} +0.607825i q^{85} +2.82260i q^{86} -14.2540 q^{88} -6.85985i q^{89} +(-3.47619 + 1.29400i) q^{91} +0.812778 q^{92} -7.34448 q^{94} -8.54063 q^{95} -17.2784i q^{97} +6.38037i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 12 q^{4} - 8 q^{10} + 4 q^{13} + 18 q^{14} - 4 q^{16} - 6 q^{17} + 10 q^{22} - 24 q^{23} + 12 q^{25} - 6 q^{26} - 12 q^{29} - 6 q^{35} - 12 q^{38} + 8 q^{40} - 4 q^{43} + 10 q^{49} + 54 q^{53} + 10 q^{55}+ \cdots - 24 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1053\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(730\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.07384i 0.759316i 0.925127 + 0.379658i \(0.123958\pi\)
−0.925127 + 0.379658i \(0.876042\pi\)
\(3\) 0 0
\(4\) 0.846877 0.423439
\(5\) − 1.27644i − 0.570843i −0.958402 − 0.285422i \(-0.907866\pi\)
0.958402 − 0.285422i \(-0.0921336\pi\)
\(6\) 0 0
\(7\) 1.02875i 0.388832i 0.980919 + 0.194416i \(0.0622811\pi\)
−0.980919 + 0.194416i \(0.937719\pi\)
\(8\) 3.05708i 1.08084i
\(9\) 0 0
\(10\) 1.37069 0.433450
\(11\) 4.66263i 1.40584i 0.711271 + 0.702918i \(0.248120\pi\)
−0.711271 + 0.702918i \(0.751880\pi\)
\(12\) 0 0
\(13\) 1.25783 + 3.37903i 0.348860 + 0.937175i
\(14\) −1.10471 −0.295246
\(15\) 0 0
\(16\) −1.58904 −0.397261
\(17\) −0.476187 −0.115492 −0.0577461 − 0.998331i \(-0.518391\pi\)
−0.0577461 + 0.998331i \(0.518391\pi\)
\(18\) 0 0
\(19\) − 6.69096i − 1.53501i −0.641042 − 0.767505i \(-0.721498\pi\)
0.641042 − 0.767505i \(-0.278502\pi\)
\(20\) − 1.08099i − 0.241717i
\(21\) 0 0
\(22\) −5.00690 −1.06747
\(23\) 0.959735 0.200119 0.100059 − 0.994981i \(-0.468097\pi\)
0.100059 + 0.994981i \(0.468097\pi\)
\(24\) 0 0
\(25\) 3.37069 0.674138
\(26\) −3.62852 + 1.35071i −0.711612 + 0.264895i
\(27\) 0 0
\(28\) 0.871227i 0.164646i
\(29\) −9.37759 −1.74137 −0.870687 − 0.491837i \(-0.836326\pi\)
−0.870687 + 0.491837i \(0.836326\pi\)
\(30\) 0 0
\(31\) 1.92751i 0.346191i 0.984905 + 0.173095i \(0.0553769\pi\)
−0.984905 + 0.173095i \(0.944623\pi\)
\(32\) 4.40778i 0.779193i
\(33\) 0 0
\(34\) − 0.511346i − 0.0876951i
\(35\) 1.31314 0.221962
\(36\) 0 0
\(37\) 4.94666i 0.813226i 0.913601 + 0.406613i \(0.133290\pi\)
−0.913601 + 0.406613i \(0.866710\pi\)
\(38\) 7.18499 1.16556
\(39\) 0 0
\(40\) 3.90219 0.616990
\(41\) 1.52414i 0.238030i 0.992892 + 0.119015i \(0.0379737\pi\)
−0.992892 + 0.119015i \(0.962026\pi\)
\(42\) 0 0
\(43\) 2.62852 0.400846 0.200423 − 0.979709i \(-0.435768\pi\)
0.200423 + 0.979709i \(0.435768\pi\)
\(44\) 3.94868i 0.595285i
\(45\) 0 0
\(46\) 1.03060i 0.151953i
\(47\) 6.83948i 0.997641i 0.866705 + 0.498820i \(0.166233\pi\)
−0.866705 + 0.498820i \(0.833767\pi\)
\(48\) 0 0
\(49\) 5.94167 0.848810
\(50\) 3.61957i 0.511884i
\(51\) 0 0
\(52\) 1.06523 + 2.86163i 0.147721 + 0.396836i
\(53\) 0.582145 0.0799637 0.0399819 − 0.999200i \(-0.487270\pi\)
0.0399819 + 0.999200i \(0.487270\pi\)
\(54\) 0 0
\(55\) 5.95159 0.802512
\(56\) −3.14498 −0.420265
\(57\) 0 0
\(58\) − 10.0700i − 1.32225i
\(59\) 4.21233i 0.548399i 0.961673 + 0.274199i \(0.0884128\pi\)
−0.961673 + 0.274199i \(0.911587\pi\)
\(60\) 0 0
\(61\) 9.43290 1.20776 0.603880 − 0.797076i \(-0.293621\pi\)
0.603880 + 0.797076i \(0.293621\pi\)
\(62\) −2.06983 −0.262868
\(63\) 0 0
\(64\) −7.91132 −0.988915
\(65\) 4.31314 − 1.60555i 0.534980 − 0.199144i
\(66\) 0 0
\(67\) 2.32275i 0.283769i 0.989883 + 0.141885i \(0.0453162\pi\)
−0.989883 + 0.141885i \(0.954684\pi\)
\(68\) −0.403272 −0.0489039
\(69\) 0 0
\(70\) 1.41010i 0.168539i
\(71\) 1.35071i 0.160299i 0.996783 + 0.0801497i \(0.0255398\pi\)
−0.996783 + 0.0801497i \(0.974460\pi\)
\(72\) 0 0
\(73\) 12.8687i 1.50617i 0.657923 + 0.753085i \(0.271435\pi\)
−0.657923 + 0.753085i \(0.728565\pi\)
\(74\) −5.31190 −0.617496
\(75\) 0 0
\(76\) − 5.66642i − 0.649983i
\(77\) −4.79669 −0.546634
\(78\) 0 0
\(79\) −12.9083 −1.45230 −0.726149 − 0.687538i \(-0.758692\pi\)
−0.726149 + 0.687538i \(0.758692\pi\)
\(80\) 2.02833i 0.226774i
\(81\) 0 0
\(82\) −1.63667 −0.180740
\(83\) − 10.2328i − 1.12320i −0.827409 − 0.561599i \(-0.810186\pi\)
0.827409 − 0.561599i \(-0.189814\pi\)
\(84\) 0 0
\(85\) 0.607825i 0.0659279i
\(86\) 2.82260i 0.304369i
\(87\) 0 0
\(88\) −14.2540 −1.51948
\(89\) − 6.85985i − 0.727143i −0.931566 − 0.363572i \(-0.881557\pi\)
0.931566 − 0.363572i \(-0.118443\pi\)
\(90\) 0 0
\(91\) −3.47619 + 1.29400i −0.364403 + 0.135648i
\(92\) 0.812778 0.0847379
\(93\) 0 0
\(94\) −7.34448 −0.757525
\(95\) −8.54063 −0.876250
\(96\) 0 0
\(97\) − 17.2784i − 1.75435i −0.480170 − 0.877175i \(-0.659425\pi\)
0.480170 − 0.877175i \(-0.340575\pi\)
\(98\) 6.38037i 0.644515i
\(99\) 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 1053.2.b.j.649.7 10
3.2 odd 2 1053.2.b.i.649.4 10
9.2 odd 6 351.2.t.c.64.4 20
9.4 even 3 117.2.t.c.25.4 ✓ 20
9.5 odd 6 351.2.t.c.181.7 20
9.7 even 3 117.2.t.c.103.7 yes 20
13.12 even 2 inner 1053.2.b.j.649.4 10
39.38 odd 2 1053.2.b.i.649.7 10
117.25 even 6 117.2.t.c.103.4 yes 20
117.38 odd 6 351.2.t.c.64.7 20
117.77 odd 6 351.2.t.c.181.4 20
117.103 even 6 117.2.t.c.25.7 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
117.2.t.c.25.4 ✓ 20 9.4 even 3
117.2.t.c.25.7 yes 20 117.103 even 6
117.2.t.c.103.4 yes 20 117.25 even 6
117.2.t.c.103.7 yes 20 9.7 even 3
351.2.t.c.64.4 20 9.2 odd 6
351.2.t.c.64.7 20 117.38 odd 6
351.2.t.c.181.4 20 117.77 odd 6
351.2.t.c.181.7 20 9.5 odd 6
1053.2.b.i.649.4 10 3.2 odd 2
1053.2.b.i.649.7 10 39.38 odd 2
1053.2.b.j.649.4 10 13.12 even 2 inner
1053.2.b.j.649.7 10 1.1 even 1 trivial