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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1950,2,Mod(649,1950)] 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("1950.649"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1950, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1950.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,2,0,2,0,0,-4,2,-2,0,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(15.5708283941\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 78)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 649.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1950.649
Dual form 1950.2.f.g.649.2

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000i q^{3} +1.00000 q^{4} -1.00000i q^{6} -2.00000 q^{7} +1.00000 q^{8} -1.00000 q^{9} -1.00000i q^{12} +(2.00000 + 3.00000i) q^{13} -2.00000 q^{14} +1.00000 q^{16} -2.00000i q^{17} -1.00000 q^{18} -6.00000i q^{19} +2.00000i q^{21} -4.00000i q^{23} -1.00000i q^{24} +(2.00000 + 3.00000i) q^{26} +1.00000i q^{27} -2.00000 q^{28} +10.0000 q^{29} -10.0000i q^{31} +1.00000 q^{32} -2.00000i q^{34} -1.00000 q^{36} +8.00000 q^{37} -6.00000i q^{38} +(3.00000 - 2.00000i) q^{39} -10.0000i q^{41} +2.00000i q^{42} -4.00000i q^{43} -4.00000i q^{46} -12.0000 q^{47} -1.00000i q^{48} -3.00000 q^{49} -2.00000 q^{51} +(2.00000 + 3.00000i) q^{52} +6.00000i q^{53} +1.00000i q^{54} -2.00000 q^{56} -6.00000 q^{57} +10.0000 q^{58} +4.00000i q^{59} +2.00000 q^{61} -10.0000i q^{62} +2.00000 q^{63} +1.00000 q^{64} -2.00000 q^{67} -2.00000i q^{68} -4.00000 q^{69} -1.00000 q^{72} +4.00000 q^{73} +8.00000 q^{74} -6.00000i q^{76} +(3.00000 - 2.00000i) q^{78} +1.00000 q^{81} -10.0000i q^{82} +4.00000 q^{83} +2.00000i q^{84} -4.00000i q^{86} -10.0000i q^{87} -6.00000i q^{89} +(-4.00000 - 6.00000i) q^{91} -4.00000i q^{92} -10.0000 q^{93} -12.0000 q^{94} -1.00000i q^{96} -12.0000 q^{97} -3.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} - 4 q^{7} + 2 q^{8} - 2 q^{9} + 4 q^{13} - 4 q^{14} + 2 q^{16} - 2 q^{18} + 4 q^{26} - 4 q^{28} + 20 q^{29} + 2 q^{32} - 2 q^{36} + 16 q^{37} + 6 q^{39} - 24 q^{47} - 6 q^{49} - 4 q^{51}+ \cdots - 6 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1950\mathbb{Z}\right)^\times\).

\(n\) \(301\) \(1301\) \(1327\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) − 1.00000i − 0.577350i
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) − 1.00000i − 0.408248i
\(7\) −2.00000 −0.755929 −0.377964 − 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 1.00000 0.353553
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) − 1.00000i − 0.288675i
\(13\) 2.00000 + 3.00000i 0.554700 + 0.832050i
\(14\) −2.00000 −0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 2.00000i − 0.485071i −0.970143 − 0.242536i \(-0.922021\pi\)
0.970143 − 0.242536i \(-0.0779791\pi\)
\(18\) −1.00000 −0.235702
\(19\) − 6.00000i − 1.37649i −0.725476 − 0.688247i \(-0.758380\pi\)
0.725476 − 0.688247i \(-0.241620\pi\)
\(20\) 0 0
\(21\) 2.00000i 0.436436i
\(22\) 0 0
\(23\) − 4.00000i − 0.834058i −0.908893 − 0.417029i \(-0.863071\pi\)
0.908893 − 0.417029i \(-0.136929\pi\)
\(24\) − 1.00000i − 0.204124i
\(25\) 0 0
\(26\) 2.00000 + 3.00000i 0.392232 + 0.588348i
\(27\) 1.00000i 0.192450i
\(28\) −2.00000 −0.377964
\(29\) 10.0000 1.85695 0.928477 − 0.371391i \(-0.121119\pi\)
0.928477 + 0.371391i \(0.121119\pi\)
\(30\) 0 0
\(31\) − 10.0000i − 1.79605i −0.439941 − 0.898027i \(-0.645001\pi\)
0.439941 − 0.898027i \(-0.354999\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) − 2.00000i − 0.342997i
\(35\) 0 0
\(36\) −1.00000 −0.166667
\(37\) 8.00000 1.31519 0.657596 − 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) − 6.00000i − 0.973329i
\(39\) 3.00000 − 2.00000i 0.480384 − 0.320256i
\(40\) 0 0
\(41\) − 10.0000i − 1.56174i −0.624695 − 0.780869i \(-0.714777\pi\)
0.624695 − 0.780869i \(-0.285223\pi\)
\(42\) 2.00000i 0.308607i
\(43\) − 4.00000i − 0.609994i −0.952353 − 0.304997i \(-0.901344\pi\)
0.952353 − 0.304997i \(-0.0986555\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) − 4.00000i − 0.589768i
\(47\) −12.0000 −1.75038 −0.875190 − 0.483779i \(-0.839264\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) − 1.00000i − 0.144338i
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) −2.00000 −0.280056
\(52\) 2.00000 + 3.00000i 0.277350 + 0.416025i
\(53\) 6.00000i 0.824163i 0.911147 + 0.412082i \(0.135198\pi\)
−0.911147 + 0.412082i \(0.864802\pi\)
\(54\) 1.00000i 0.136083i
\(55\) 0 0
\(56\) −2.00000 −0.267261
\(57\) −6.00000 −0.794719
\(58\) 10.0000 1.31306
\(59\) 4.00000i 0.520756i 0.965507 + 0.260378i \(0.0838471\pi\)
−0.965507 + 0.260378i \(0.916153\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 − 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) − 10.0000i − 1.27000i
\(63\) 2.00000 0.251976
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −2.00000 −0.244339 −0.122169 − 0.992509i \(-0.538985\pi\)
−0.122169 + 0.992509i \(0.538985\pi\)
\(68\) − 2.00000i − 0.242536i
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) −1.00000 −0.117851
\(73\) 4.00000 0.468165 0.234082 − 0.972217i \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) 8.00000 0.929981
\(75\) 0 0
\(76\) − 6.00000i − 0.688247i
\(77\) 0 0
\(78\) 3.00000 − 2.00000i 0.339683 − 0.226455i
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) − 10.0000i − 1.10432i
\(83\) 4.00000 0.439057 0.219529 − 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 2.00000i 0.218218i
\(85\) 0 0
\(86\) − 4.00000i − 0.431331i
\(87\) − 10.0000i − 1.07211i
\(88\) 0 0
\(89\) − 6.00000i − 0.635999i −0.948091 − 0.317999i \(-0.896989\pi\)
0.948091 − 0.317999i \(-0.103011\pi\)
\(90\) 0 0
\(91\) −4.00000 − 6.00000i −0.419314 − 0.628971i
\(92\) − 4.00000i − 0.417029i
\(93\) −10.0000 −1.03695
\(94\) −12.0000 −1.23771
\(95\) 0 0
\(96\) − 1.00000i − 0.102062i
\(97\) −12.0000 −1.21842 −0.609208 − 0.793011i \(-0.708512\pi\)
−0.609208 + 0.793011i \(0.708512\pi\)
\(98\) −3.00000 −0.303046
\(99\) 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 1950.2.f.g.649.1 2
5.2 odd 4 1950.2.b.c.1351.2 2
5.3 odd 4 78.2.b.a.25.1 ✓ 2
5.4 even 2 1950.2.f.d.649.2 2
13.12 even 2 1950.2.f.d.649.1 2
15.8 even 4 234.2.b.a.181.2 2
20.3 even 4 624.2.c.a.337.1 2
35.13 even 4 3822.2.c.d.883.1 2
40.3 even 4 2496.2.c.m.961.2 2
40.13 odd 4 2496.2.c.f.961.2 2
60.23 odd 4 1872.2.c.b.1585.2 2
65.3 odd 12 1014.2.i.c.823.2 4
65.8 even 4 1014.2.a.g.1.1 1
65.12 odd 4 1950.2.b.c.1351.1 2
65.18 even 4 1014.2.a.b.1.1 1
65.23 odd 12 1014.2.i.c.823.1 4
65.28 even 12 1014.2.e.e.529.1 2
65.33 even 12 1014.2.e.b.991.1 2
65.38 odd 4 78.2.b.a.25.2 yes 2
65.43 odd 12 1014.2.i.c.361.2 4
65.48 odd 12 1014.2.i.c.361.1 4
65.58 even 12 1014.2.e.e.991.1 2
65.63 even 12 1014.2.e.b.529.1 2
65.64 even 2 inner 1950.2.f.g.649.2 2
195.8 odd 4 3042.2.a.c.1.1 1
195.38 even 4 234.2.b.a.181.1 2
195.83 odd 4 3042.2.a.n.1.1 1
260.83 odd 4 8112.2.a.g.1.1 1
260.103 even 4 624.2.c.a.337.2 2
260.203 odd 4 8112.2.a.j.1.1 1
455.363 even 4 3822.2.c.d.883.2 2
520.363 even 4 2496.2.c.m.961.1 2
520.493 odd 4 2496.2.c.f.961.1 2
780.623 odd 4 1872.2.c.b.1585.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
78.2.b.a.25.1 ✓ 2 5.3 odd 4
78.2.b.a.25.2 yes 2 65.38 odd 4
234.2.b.a.181.1 2 195.38 even 4
234.2.b.a.181.2 2 15.8 even 4
624.2.c.a.337.1 2 20.3 even 4
624.2.c.a.337.2 2 260.103 even 4
1014.2.a.b.1.1 1 65.18 even 4
1014.2.a.g.1.1 1 65.8 even 4
1014.2.e.b.529.1 2 65.63 even 12
1014.2.e.b.991.1 2 65.33 even 12
1014.2.e.e.529.1 2 65.28 even 12
1014.2.e.e.991.1 2 65.58 even 12
1014.2.i.c.361.1 4 65.48 odd 12
1014.2.i.c.361.2 4 65.43 odd 12
1014.2.i.c.823.1 4 65.23 odd 12
1014.2.i.c.823.2 4 65.3 odd 12
1872.2.c.b.1585.1 2 780.623 odd 4
1872.2.c.b.1585.2 2 60.23 odd 4
1950.2.b.c.1351.1 2 65.12 odd 4
1950.2.b.c.1351.2 2 5.2 odd 4
1950.2.f.d.649.1 2 13.12 even 2
1950.2.f.d.649.2 2 5.4 even 2
1950.2.f.g.649.1 2 1.1 even 1 trivial
1950.2.f.g.649.2 2 65.64 even 2 inner
2496.2.c.f.961.1 2 520.493 odd 4
2496.2.c.f.961.2 2 40.13 odd 4
2496.2.c.m.961.1 2 520.363 even 4
2496.2.c.m.961.2 2 40.3 even 4
3042.2.a.c.1.1 1 195.8 odd 4
3042.2.a.n.1.1 1 195.83 odd 4
3822.2.c.d.883.1 2 35.13 even 4
3822.2.c.d.883.2 2 455.363 even 4
8112.2.a.g.1.1 1 260.83 odd 4
8112.2.a.j.1.1 1 260.203 odd 4