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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2850,2,Mod(799,2850)] 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("2850.799"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2850, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2850.d (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,-2,0,0,0,0,8,0,2,0,0,-2,0,-8,0,0,-2,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(26)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(22.7573645761\)
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 114)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 799.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 2850.799
Dual form 2850.2.d.p.799.2

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +4.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} -1.00000i q^{12} -4.00000i q^{13} +4.00000 q^{14} +1.00000 q^{16} -6.00000i q^{17} +1.00000i q^{18} -1.00000 q^{19} -4.00000 q^{21} -6.00000i q^{23} -1.00000 q^{24} -4.00000 q^{26} -1.00000i q^{27} -4.00000i q^{28} -6.00000 q^{29} +2.00000 q^{31} -1.00000i q^{32} -6.00000 q^{34} +1.00000 q^{36} +4.00000i q^{37} +1.00000i q^{38} +4.00000 q^{39} +6.00000 q^{41} +4.00000i q^{42} -4.00000i q^{43} -6.00000 q^{46} -6.00000i q^{47} +1.00000i q^{48} -9.00000 q^{49} +6.00000 q^{51} +4.00000i q^{52} +6.00000i q^{53} -1.00000 q^{54} -4.00000 q^{56} -1.00000i q^{57} +6.00000i q^{58} +12.0000 q^{59} +14.0000 q^{61} -2.00000i q^{62} -4.00000i q^{63} -1.00000 q^{64} -8.00000i q^{67} +6.00000i q^{68} +6.00000 q^{69} -1.00000i q^{72} +14.0000i q^{73} +4.00000 q^{74} +1.00000 q^{76} -4.00000i q^{78} +10.0000 q^{79} +1.00000 q^{81} -6.00000i q^{82} -12.0000i q^{83} +4.00000 q^{84} -4.00000 q^{86} -6.00000i q^{87} +6.00000 q^{89} +16.0000 q^{91} +6.00000i q^{92} +2.00000i q^{93} -6.00000 q^{94} +1.00000 q^{96} +10.0000i q^{97} +9.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} + 8 q^{14} + 2 q^{16} - 2 q^{19} - 8 q^{21} - 2 q^{24} - 8 q^{26} - 12 q^{29} + 4 q^{31} - 12 q^{34} + 2 q^{36} + 8 q^{39} + 12 q^{41} - 12 q^{46} - 18 q^{49} + 12 q^{51}+ \cdots + 2 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2850\mathbb{Z}\right)^\times\).

\(n\) \(1027\) \(1351\) \(1901\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(7\) 4.00000i 1.51186i 0.654654 + 0.755929i \(0.272814\pi\)
−0.654654 + 0.755929i \(0.727186\pi\)
\(8\) 1.00000i 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) − 1.00000i − 0.288675i
\(13\) − 4.00000i − 1.10940i −0.832050 − 0.554700i \(-0.812833\pi\)
0.832050 − 0.554700i \(-0.187167\pi\)
\(14\) 4.00000 1.06904
\(15\) 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
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −4.00000 −0.872872
\(22\) 0 0
\(23\) − 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
\(28\) − 4.00000i − 0.755929i
\(29\) −6.00000 −1.11417 −0.557086 − 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 − 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 0 0
\(34\) −6.00000 −1.02899
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 4.00000i 0.657596i 0.944400 + 0.328798i \(0.106644\pi\)
−0.944400 + 0.328798i \(0.893356\pi\)
\(38\) 1.00000i 0.162221i
\(39\) 4.00000 0.640513
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 − 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 4.00000i 0.617213i
\(43\) − 4.00000i − 0.609994i −0.952353 − 0.304997i \(-0.901344\pi\)
0.952353 − 0.304997i \(-0.0986555\pi\)
\(44\) 0 0
\(45\) 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
\(49\) −9.00000 −1.28571
\(50\) 0 0
\(51\) 6.00000 0.840168
\(52\) 4.00000i 0.554700i
\(53\) 6.00000i 0.824163i 0.911147 + 0.412082i \(0.135198\pi\)
−0.911147 + 0.412082i \(0.864802\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) −4.00000 −0.534522
\(57\) − 1.00000i − 0.132453i
\(58\) 6.00000i 0.787839i
\(59\) 12.0000 1.56227 0.781133 − 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) 14.0000 1.79252 0.896258 − 0.443533i \(-0.146275\pi\)
0.896258 + 0.443533i \(0.146275\pi\)
\(62\) − 2.00000i − 0.254000i
\(63\) − 4.00000i − 0.503953i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) − 8.00000i − 0.977356i −0.872464 − 0.488678i \(-0.837479\pi\)
0.872464 − 0.488678i \(-0.162521\pi\)
\(68\) 6.00000i 0.727607i
\(69\) 6.00000 0.722315
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) 14.0000i 1.63858i 0.573382 + 0.819288i \(0.305631\pi\)
−0.573382 + 0.819288i \(0.694369\pi\)
\(74\) 4.00000 0.464991
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) 0 0
\(78\) − 4.00000i − 0.452911i
\(79\) 10.0000 1.12509 0.562544 − 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) − 6.00000i − 0.662589i
\(83\) − 12.0000i − 1.31717i −0.752506 − 0.658586i \(-0.771155\pi\)
0.752506 − 0.658586i \(-0.228845\pi\)
\(84\) 4.00000 0.436436
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) − 6.00000i − 0.643268i
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 − 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 16.0000 1.67726
\(92\) 6.00000i 0.625543i
\(93\) 2.00000i 0.207390i
\(94\) −6.00000 −0.618853
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 10.0000i 1.01535i 0.861550 + 0.507673i \(0.169494\pi\)
−0.861550 + 0.507673i \(0.830506\pi\)
\(98\) 9.00000i 0.909137i
\(99\) 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 2850.2.d.p.799.1 2
5.2 odd 4 114.2.a.c.1.1 ✓ 1
5.3 odd 4 2850.2.a.g.1.1 1
5.4 even 2 inner 2850.2.d.p.799.2 2
15.2 even 4 342.2.a.c.1.1 1
15.8 even 4 8550.2.a.bj.1.1 1
20.7 even 4 912.2.a.c.1.1 1
35.27 even 4 5586.2.a.u.1.1 1
40.27 even 4 3648.2.a.bc.1.1 1
40.37 odd 4 3648.2.a.i.1.1 1
60.47 odd 4 2736.2.a.o.1.1 1
95.37 even 4 2166.2.a.a.1.1 1
285.227 odd 4 6498.2.a.t.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
114.2.a.c.1.1 ✓ 1 5.2 odd 4
342.2.a.c.1.1 1 15.2 even 4
912.2.a.c.1.1 1 20.7 even 4
2166.2.a.a.1.1 1 95.37 even 4
2736.2.a.o.1.1 1 60.47 odd 4
2850.2.a.g.1.1 1 5.3 odd 4
2850.2.d.p.799.1 2 1.1 even 1 trivial
2850.2.d.p.799.2 2 5.4 even 2 inner
3648.2.a.i.1.1 1 40.37 odd 4
3648.2.a.bc.1.1 1 40.27 even 4
5586.2.a.u.1.1 1 35.27 even 4
6498.2.a.t.1.1 1 285.227 odd 4
8550.2.a.bj.1.1 1 15.8 even 4