Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [50,11,Mod(7,50)] 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("50.7"); S:= CuspForms(chi, 11); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(50, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 11, names="a")
 
Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 50.c (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,96,-128] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(31.7678626337\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 1148x^{3} + 68121x^{2} - 299628x + 658952 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{9}\cdot 5^{8} \)
Twist minimal: no (minimal twist has level 10)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 43.2
Root \(10.1043 + 10.1043i\) of defining polynomial
Character \(\chi\) \(=\) 50.43
Dual form 50.11.c.e.7.2

$q$-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+(16.0000 - 16.0000i) q^{2} +(-4.29207 - 4.29207i) q^{3} -512.000i q^{4} -137.346 q^{6} +(-21284.7 + 21284.7i) q^{7} +(-8192.00 - 8192.00i) q^{8} -59012.2i q^{9} +155649. q^{11} +(-2197.54 + 2197.54i) q^{12} +(358614. + 358614. i) q^{13} +681110. i q^{14} -262144. q^{16} +(609397. - 609397. i) q^{17} +(-944194. - 944194. i) q^{18} -335226. i q^{19} +182711. q^{21} +(2.49038e6 - 2.49038e6i) q^{22} +(5.50132e6 + 5.50132e6i) q^{23} +70321.3i q^{24} +1.14757e7 q^{26} +(-506727. + 506727. i) q^{27} +(1.08978e7 + 1.08978e7i) q^{28} -1.33815e6i q^{29} +2.59306e7 q^{31} +(-4.19430e6 + 4.19430e6i) q^{32} +(-668055. - 668055. i) q^{33} -1.95007e7i q^{34} -3.02142e7 q^{36} +(5.50890e7 - 5.50890e7i) q^{37} +(-5.36362e6 - 5.36362e6i) q^{38} -3.07840e6i q^{39} +1.37064e8 q^{41} +(2.92337e6 - 2.92337e6i) q^{42} +(9.34290e7 + 9.34290e7i) q^{43} -7.96921e7i q^{44} +1.76042e8 q^{46} +(-1.14979e8 + 1.14979e8i) q^{47} +(1.12514e6 + 1.12514e6i) q^{48} -6.23600e8i q^{49} -5.23115e6 q^{51} +(1.83611e8 - 1.83611e8i) q^{52} +(2.29978e7 + 2.29978e7i) q^{53} +1.62153e7i q^{54} +3.48728e8 q^{56} +(-1.43882e6 + 1.43882e6i) q^{57} +(-2.14104e7 - 2.14104e7i) q^{58} +8.73535e8i q^{59} +5.87905e8 q^{61} +(4.14890e8 - 4.14890e8i) q^{62} +(1.25605e9 + 1.25605e9i) q^{63} +1.34218e8i q^{64} -2.13778e7 q^{66} +(6.75197e8 - 6.75197e8i) q^{67} +(-3.12011e8 - 3.12011e8i) q^{68} -4.72241e7i q^{69} -5.54597e8 q^{71} +(-4.83428e8 + 4.83428e8i) q^{72} +(-8.91747e8 - 8.91747e8i) q^{73} -1.76285e9i q^{74} -1.71636e8 q^{76} +(-3.31293e9 + 3.31293e9i) q^{77} +(-4.92544e7 - 4.92544e7i) q^{78} +1.69149e9i q^{79} -3.48026e9 q^{81} +(2.19302e9 - 2.19302e9i) q^{82} +(-1.96163e9 - 1.96163e9i) q^{83} -9.35479e7i q^{84} +2.98973e9 q^{86} +(-5.74345e6 + 5.74345e6i) q^{87} +(-1.27507e9 - 1.27507e9i) q^{88} +7.73241e9i q^{89} -1.52660e10 q^{91} +(2.81667e9 - 2.81667e9i) q^{92} +(-1.11296e8 - 1.11296e8i) q^{93} +3.67931e9i q^{94} +3.60045e7 q^{96} +(-1.16280e10 + 1.16280e10i) q^{97} +(-9.97760e9 - 9.97760e9i) q^{98} -9.18516e9i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 96 q^{2} - 128 q^{3} - 4096 q^{6} - 13512 q^{7} - 49152 q^{8} + 647832 q^{11} - 65536 q^{12} + 742902 q^{13} - 1572864 q^{16} + 755118 q^{17} + 5683744 q^{18} + 12277112 q^{21} + 10365312 q^{22} + 15052992 q^{23}+ \cdots - 12874047264 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/50\mathbb{Z}\right)^\times\).

\(n\) \(27\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 16.0000 − 16.0000i 0.500000 − 0.500000i
\(3\) −4.29207 − 4.29207i −0.0176629 − 0.0176629i 0.698220 − 0.715883i \(-0.253976\pi\)
−0.715883 + 0.698220i \(0.753976\pi\)
\(4\) − 512.000i − 0.500000i
\(5\) 0 0
\(6\) −137.346 −0.0176629
\(7\) −21284.7 + 21284.7i −1.26642 + 1.26642i −0.318492 + 0.947926i \(0.603176\pi\)
−0.947926 + 0.318492i \(0.896824\pi\)
\(8\) −8192.00 − 8192.00i −0.250000 − 0.250000i
\(9\) − 59012.2i − 0.999376i
\(10\) 0 0
\(11\) 155649. 0.966456 0.483228 − 0.875495i \(-0.339464\pi\)
0.483228 + 0.875495i \(0.339464\pi\)
\(12\) −2197.54 + 2197.54i −0.00883143 + 0.00883143i
\(13\) 358614. + 358614.i 0.965853 + 0.965853i 0.999436 − 0.0335834i \(-0.0106919\pi\)
−0.0335834 + 0.999436i \(0.510692\pi\)
\(14\) 681110.i 1.26642i
\(15\) 0 0
\(16\) −262144. −0.250000
\(17\) 609397. − 609397.i 0.429196 − 0.429196i −0.459158 − 0.888354i \(-0.651849\pi\)
0.888354 + 0.459158i \(0.151849\pi\)
\(18\) −944194. − 944194.i −0.499688 − 0.499688i
\(19\) − 335226.i − 0.135385i −0.997706 − 0.0676924i \(-0.978436\pi\)
0.997706 − 0.0676924i \(-0.0215637\pi\)
\(20\) 0 0
\(21\) 182711. 0.0447371
\(22\) 2.49038e6 − 2.49038e6i 0.483228 − 0.483228i
\(23\) 5.50132e6 + 5.50132e6i 0.854727 + 0.854727i 0.990711 − 0.135984i \(-0.0434196\pi\)
−0.135984 + 0.990711i \(0.543420\pi\)
\(24\) 70321.3i 0.00883143i
\(25\) 0 0
\(26\) 1.14757e7 0.965853
\(27\) −506727. + 506727.i −0.0353147 + 0.0353147i
\(28\) 1.08978e7 + 1.08978e7i 0.633209 + 0.633209i
\(29\) − 1.33815e6i − 0.0652402i −0.999468 − 0.0326201i \(-0.989615\pi\)
0.999468 − 0.0326201i \(-0.0103851\pi\)
\(30\) 0 0
\(31\) 2.59306e7 0.905743 0.452871 − 0.891576i \(-0.350400\pi\)
0.452871 + 0.891576i \(0.350400\pi\)
\(32\) −4.19430e6 + 4.19430e6i −0.125000 + 0.125000i
\(33\) −668055. − 668055.i −0.0170704 − 0.0170704i
\(34\) − 1.95007e7i − 0.429196i
\(35\) 0 0
\(36\) −3.02142e7 −0.499688
\(37\) 5.50890e7 − 5.50890e7i 0.794431 − 0.794431i −0.187780 − 0.982211i \(-0.560129\pi\)
0.982211 + 0.187780i \(0.0601292\pi\)
\(38\) −5.36362e6 − 5.36362e6i −0.0676924 − 0.0676924i
\(39\) − 3.07840e6i − 0.0341194i
\(40\) 0 0
\(41\) 1.37064e8 1.18305 0.591525 − 0.806287i \(-0.298526\pi\)
0.591525 + 0.806287i \(0.298526\pi\)
\(42\) 2.92337e6 − 2.92337e6i 0.0223685 − 0.0223685i
\(43\) 9.34290e7 + 9.34290e7i 0.635535 + 0.635535i 0.949451 − 0.313916i \(-0.101641\pi\)
−0.313916 + 0.949451i \(0.601641\pi\)
\(44\) − 7.96921e7i − 0.483228i
\(45\) 0 0
\(46\) 1.76042e8 0.854727
\(47\) −1.14979e8 + 1.14979e8i −0.501334 + 0.501334i −0.911852 − 0.410518i \(-0.865348\pi\)
0.410518 + 0.911852i \(0.365348\pi\)
\(48\) 1.12514e6 + 1.12514e6i 0.00441571 + 0.00441571i
\(49\) − 6.23600e8i − 2.20763i
\(50\) 0 0
\(51\) −5.23115e6 −0.0151616
\(52\) 1.83611e8 − 1.83611e8i 0.482926 − 0.482926i
\(53\) 2.29978e7 + 2.29978e7i 0.0549930 + 0.0549930i 0.734068 − 0.679075i \(-0.237619\pi\)
−0.679075 + 0.734068i \(0.737619\pi\)
\(54\) 1.62153e7i 0.0353147i
\(55\) 0 0
\(56\) 3.48728e8 0.633209
\(57\) −1.43882e6 + 1.43882e6i −0.00239128 + 0.00239128i
\(58\) −2.14104e7 − 2.14104e7i −0.0326201 − 0.0326201i
\(59\) 8.73535e8i 1.22186i 0.791686 + 0.610929i \(0.209204\pi\)
−0.791686 + 0.610929i \(0.790796\pi\)
\(60\) 0 0
\(61\) 5.87905e8 0.696079 0.348039 − 0.937480i \(-0.386848\pi\)
0.348039 + 0.937480i \(0.386848\pi\)
\(62\) 4.14890e8 − 4.14890e8i 0.452871 − 0.452871i
\(63\) 1.25605e9 + 1.25605e9i 1.26563 + 1.26563i
\(64\) 1.34218e8i 0.125000i
\(65\) 0 0
\(66\) −2.13778e7 −0.0170704
\(67\) 6.75197e8 − 6.75197e8i 0.500100 − 0.500100i −0.411369 − 0.911469i \(-0.634949\pi\)
0.911469 + 0.411369i \(0.134949\pi\)
\(68\) −3.12011e8 − 3.12011e8i −0.214598 − 0.214598i
\(69\) − 4.72241e7i − 0.0301938i
\(70\) 0 0
\(71\) −5.54597e8 −0.307387 −0.153694 − 0.988119i \(-0.549117\pi\)
−0.153694 + 0.988119i \(0.549117\pi\)
\(72\) −4.83428e8 + 4.83428e8i −0.249844 + 0.249844i
\(73\) −8.91747e8 − 8.91747e8i −0.430158 − 0.430158i 0.458524 − 0.888682i \(-0.348378\pi\)
−0.888682 + 0.458524i \(0.848378\pi\)
\(74\) − 1.76285e9i − 0.794431i
\(75\) 0 0
\(76\) −1.71636e8 −0.0676924
\(77\) −3.31293e9 + 3.31293e9i −1.22394 + 1.22394i
\(78\) −4.92544e7 − 4.92544e7i −0.0170597 − 0.0170597i
\(79\) 1.69149e9i 0.549711i 0.961486 + 0.274856i \(0.0886300\pi\)
−0.961486 + 0.274856i \(0.911370\pi\)
\(80\) 0 0
\(81\) −3.48026e9 −0.998129
\(82\) 2.19302e9 − 2.19302e9i 0.591525 − 0.591525i
\(83\) −1.96163e9 − 1.96163e9i −0.497996 − 0.497996i 0.412818 − 0.910814i \(-0.364545\pi\)
−0.910814 + 0.412818i \(0.864545\pi\)
\(84\) − 9.35479e7i − 0.0223685i
\(85\) 0 0
\(86\) 2.98973e9 0.635535
\(87\) −5.74345e6 + 5.74345e6i −0.00115233 + 0.00115233i
\(88\) −1.27507e9 − 1.27507e9i −0.241614 − 0.241614i
\(89\) 7.73241e9i 1.38473i 0.721548 + 0.692364i \(0.243431\pi\)
−0.721548 + 0.692364i \(0.756569\pi\)
\(90\) 0 0
\(91\) −1.52660e10 −2.44635
\(92\) 2.81667e9 − 2.81667e9i 0.427363 − 0.427363i
\(93\) −1.11296e8 − 1.11296e8i −0.0159980 − 0.0159980i
\(94\) 3.67931e9i 0.501334i
\(95\) 0 0
\(96\) 3.60045e7 0.00441571
\(97\) −1.16280e10 + 1.16280e10i −1.35408 + 1.35408i −0.473039 + 0.881041i \(0.656843\pi\)
−0.881041 + 0.473039i \(0.843157\pi\)
\(98\) −9.97760e9 − 9.97760e9i −1.10381 − 1.10381i
\(99\) − 9.18516e9i − 0.965853i
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 50.11.c.e.43.2 6
5.2 odd 4 inner 50.11.c.e.7.2 6
5.3 odd 4 10.11.c.c.7.2 yes 6
5.4 even 2 10.11.c.c.3.2 ✓ 6
15.8 even 4 90.11.g.c.37.3 6
15.14 odd 2 90.11.g.c.73.3 6
20.3 even 4 80.11.p.c.17.2 6
20.19 odd 2 80.11.p.c.33.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.11.c.c.3.2 ✓ 6 5.4 even 2
10.11.c.c.7.2 yes 6 5.3 odd 4
50.11.c.e.7.2 6 5.2 odd 4 inner
50.11.c.e.43.2 6 1.1 even 1 trivial
80.11.p.c.17.2 6 20.3 even 4
80.11.p.c.33.2 6 20.19 odd 2
90.11.g.c.37.3 6 15.8 even 4
90.11.g.c.73.3 6 15.14 odd 2