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Properties

Label 1650.2.c.d.199.1
Level $1650$
Weight $2$
Character 1650.199
Analytic conductor $13.175$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1650,2,Mod(199,1650)] 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("1650.199"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1650, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1650 = 2 \cdot 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1650.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,-2,0,-2,0,0,-2,0,-2,0,0,4,0,2,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(13.1753163335\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 66)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1650.199
Dual form 1650.2.c.d.199.2

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} -1.00000 q^{11} +1.00000i q^{12} +4.00000i q^{13} +2.00000 q^{14} +1.00000 q^{16} -6.00000i q^{17} +1.00000i q^{18} +4.00000 q^{19} +2.00000 q^{21} +1.00000i q^{22} -6.00000i q^{23} +1.00000 q^{24} +4.00000 q^{26} +1.00000i q^{27} -2.00000i q^{28} -6.00000 q^{29} +8.00000 q^{31} -1.00000i q^{32} +1.00000i q^{33} -6.00000 q^{34} +1.00000 q^{36} -10.0000i q^{37} -4.00000i q^{38} +4.00000 q^{39} +6.00000 q^{41} -2.00000i q^{42} -8.00000i q^{43} +1.00000 q^{44} -6.00000 q^{46} -6.00000i q^{47} -1.00000i q^{48} +3.00000 q^{49} -6.00000 q^{51} -4.00000i q^{52} +1.00000 q^{54} -2.00000 q^{56} -4.00000i q^{57} +6.00000i q^{58} +8.00000 q^{61} -8.00000i q^{62} -2.00000i q^{63} -1.00000 q^{64} +1.00000 q^{66} -4.00000i q^{67} +6.00000i q^{68} -6.00000 q^{69} +6.00000 q^{71} -1.00000i q^{72} -2.00000i q^{73} -10.0000 q^{74} -4.00000 q^{76} -2.00000i q^{77} -4.00000i q^{78} -14.0000 q^{79} +1.00000 q^{81} -6.00000i q^{82} +12.0000i q^{83} -2.00000 q^{84} -8.00000 q^{86} +6.00000i q^{87} -1.00000i q^{88} +6.00000 q^{89} -8.00000 q^{91} +6.00000i q^{92} -8.00000i q^{93} -6.00000 q^{94} -1.00000 q^{96} +14.0000i q^{97} -3.00000i q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} - 2 q^{11} + 4 q^{14} + 2 q^{16} + 8 q^{19} + 4 q^{21} + 2 q^{24} + 8 q^{26} - 12 q^{29} + 16 q^{31} - 12 q^{34} + 2 q^{36} + 8 q^{39} + 12 q^{41} + 2 q^{44} - 12 q^{46}+ \cdots + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1650\mathbb{Z}\right)^\times\).

\(n\) \(551\) \(727\) \(1201\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.707107i
\(3\) − 1.00000i − 0.577350i
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) 2.00000i 0.755929i 0.925820 + 0.377964i \(0.123376\pi\)
−0.925820 + 0.377964i \(0.876624\pi\)
\(8\) 1.00000i 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 1.00000i 0.288675i
\(13\) 4.00000i 1.10940i 0.832050 + 0.554700i \(0.187167\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 6.00000i − 1.45521i −0.685994 − 0.727607i \(-0.740633\pi\)
0.685994 − 0.727607i \(-0.259367\pi\)
\(18\) 1.00000i 0.235702i
\(19\) 4.00000 0.917663 0.458831 − 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 1.00000i 0.213201i
\(23\) − 6.00000i − 1.25109i −0.780189 − 0.625543i \(-0.784877\pi\)
0.780189 − 0.625543i \(-0.215123\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) 4.00000 0.784465
\(27\) 1.00000i 0.192450i
\(28\) − 2.00000i − 0.377964i
\(29\) −6.00000 −1.11417 −0.557086 − 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 − 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 1.00000i 0.174078i
\(34\) −6.00000 −1.02899
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) − 10.0000i − 1.64399i −0.569495 − 0.821995i \(-0.692861\pi\)
0.569495 − 0.821995i \(-0.307139\pi\)
\(38\) − 4.00000i − 0.648886i
\(39\) 4.00000 0.640513
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 − 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) − 2.00000i − 0.308607i
\(43\) − 8.00000i − 1.21999i −0.792406 − 0.609994i \(-0.791172\pi\)
0.792406 − 0.609994i \(-0.208828\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) −6.00000 −0.884652
\(47\) − 6.00000i − 0.875190i −0.899172 − 0.437595i \(-0.855830\pi\)
0.899172 − 0.437595i \(-0.144170\pi\)
\(48\) − 1.00000i − 0.144338i
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) −6.00000 −0.840168
\(52\) − 4.00000i − 0.554700i
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) −2.00000 −0.267261
\(57\) − 4.00000i − 0.529813i
\(58\) 6.00000i 0.787839i
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 8.00000 1.02430 0.512148 − 0.858898i \(-0.328850\pi\)
0.512148 + 0.858898i \(0.328850\pi\)
\(62\) − 8.00000i − 1.01600i
\(63\) − 2.00000i − 0.251976i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 1.00000 0.123091
\(67\) − 4.00000i − 0.488678i −0.969690 − 0.244339i \(-0.921429\pi\)
0.969690 − 0.244339i \(-0.0785709\pi\)
\(68\) 6.00000i 0.727607i
\(69\) −6.00000 −0.722315
\(70\) 0 0
\(71\) 6.00000 0.712069 0.356034 − 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) − 2.00000i − 0.234082i −0.993127 − 0.117041i \(-0.962659\pi\)
0.993127 − 0.117041i \(-0.0373409\pi\)
\(74\) −10.0000 −1.16248
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) − 2.00000i − 0.227921i
\(78\) − 4.00000i − 0.452911i
\(79\) −14.0000 −1.57512 −0.787562 − 0.616236i \(-0.788657\pi\)
−0.787562 + 0.616236i \(0.788657\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) − 6.00000i − 0.662589i
\(83\) 12.0000i 1.31717i 0.752506 + 0.658586i \(0.228845\pi\)
−0.752506 + 0.658586i \(0.771155\pi\)
\(84\) −2.00000 −0.218218
\(85\) 0 0
\(86\) −8.00000 −0.862662
\(87\) 6.00000i 0.643268i
\(88\) − 1.00000i − 0.106600i
\(89\) 6.00000 0.635999 0.317999 − 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) −8.00000 −0.838628
\(92\) 6.00000i 0.625543i
\(93\) − 8.00000i − 0.829561i
\(94\) −6.00000 −0.618853
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) 14.0000i 1.42148i 0.703452 + 0.710742i \(0.251641\pi\)
−0.703452 + 0.710742i \(0.748359\pi\)
\(98\) − 3.00000i − 0.303046i
\(99\) 1.00000 0.100504
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 1650.2.c.d.199.1 2
3.2 odd 2 4950.2.c.r.199.2 2
5.2 odd 4 1650.2.a.m.1.1 1
5.3 odd 4 66.2.a.a.1.1 ✓ 1
5.4 even 2 inner 1650.2.c.d.199.2 2
15.2 even 4 4950.2.a.g.1.1 1
15.8 even 4 198.2.a.e.1.1 1
15.14 odd 2 4950.2.c.r.199.1 2
20.3 even 4 528.2.a.d.1.1 1
35.13 even 4 3234.2.a.d.1.1 1
40.3 even 4 2112.2.a.v.1.1 1
40.13 odd 4 2112.2.a.i.1.1 1
45.13 odd 12 1782.2.e.s.1189.1 2
45.23 even 12 1782.2.e.f.1189.1 2
45.38 even 12 1782.2.e.f.595.1 2
45.43 odd 12 1782.2.e.s.595.1 2
55.3 odd 20 726.2.e.k.493.1 4
55.8 even 20 726.2.e.b.493.1 4
55.13 even 20 726.2.e.b.565.1 4
55.18 even 20 726.2.e.b.511.1 4
55.28 even 20 726.2.e.b.487.1 4
55.38 odd 20 726.2.e.k.487.1 4
55.43 even 4 726.2.a.i.1.1 1
55.48 odd 20 726.2.e.k.511.1 4
55.53 odd 20 726.2.e.k.565.1 4
60.23 odd 4 1584.2.a.h.1.1 1
105.83 odd 4 9702.2.a.bu.1.1 1
120.53 even 4 6336.2.a.bj.1.1 1
120.83 odd 4 6336.2.a.bf.1.1 1
165.98 odd 4 2178.2.a.b.1.1 1
220.43 odd 4 5808.2.a.l.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
66.2.a.a.1.1 ✓ 1 5.3 odd 4
198.2.a.e.1.1 1 15.8 even 4
528.2.a.d.1.1 1 20.3 even 4
726.2.a.i.1.1 1 55.43 even 4
726.2.e.b.487.1 4 55.28 even 20
726.2.e.b.493.1 4 55.8 even 20
726.2.e.b.511.1 4 55.18 even 20
726.2.e.b.565.1 4 55.13 even 20
726.2.e.k.487.1 4 55.38 odd 20
726.2.e.k.493.1 4 55.3 odd 20
726.2.e.k.511.1 4 55.48 odd 20
726.2.e.k.565.1 4 55.53 odd 20
1584.2.a.h.1.1 1 60.23 odd 4
1650.2.a.m.1.1 1 5.2 odd 4
1650.2.c.d.199.1 2 1.1 even 1 trivial
1650.2.c.d.199.2 2 5.4 even 2 inner
1782.2.e.f.595.1 2 45.38 even 12
1782.2.e.f.1189.1 2 45.23 even 12
1782.2.e.s.595.1 2 45.43 odd 12
1782.2.e.s.1189.1 2 45.13 odd 12
2112.2.a.i.1.1 1 40.13 odd 4
2112.2.a.v.1.1 1 40.3 even 4
2178.2.a.b.1.1 1 165.98 odd 4
3234.2.a.d.1.1 1 35.13 even 4
4950.2.a.g.1.1 1 15.2 even 4
4950.2.c.r.199.1 2 15.14 odd 2
4950.2.c.r.199.2 2 3.2 odd 2
5808.2.a.l.1.1 1 220.43 odd 4
6336.2.a.bf.1.1 1 120.83 odd 4
6336.2.a.bj.1.1 1 120.53 even 4
9702.2.a.bu.1.1 1 105.83 odd 4