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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1850,2,Mod(149,1850)] 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("1850.149"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1850, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1850 = 2 \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1850.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,-4,0,-2,0,0,6,0,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(14.7723243739\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) 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 74)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 149.2
Root \(0.618034i\) of defining polynomial
Character \(\chi\) \(=\) 1850.149
Dual form 1850.2.b.j.149.3

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} +0.618034i q^{3} -1.00000 q^{4} +0.618034 q^{6} -1.23607i q^{7} +1.00000i q^{8} +2.61803 q^{9} -3.61803 q^{11} -0.618034i q^{12} +3.85410i q^{13} -1.23607 q^{14} +1.00000 q^{16} -4.47214i q^{17} -2.61803i q^{18} +4.47214 q^{19} +0.763932 q^{21} +3.61803i q^{22} -3.85410i q^{23} -0.618034 q^{24} +3.85410 q^{26} +3.47214i q^{27} +1.23607i q^{28} -6.32624 q^{29} +9.61803 q^{31} -1.00000i q^{32} -2.23607i q^{33} -4.47214 q^{34} -2.61803 q^{36} +1.00000i q^{37} -4.47214i q^{38} -2.38197 q^{39} +7.38197 q^{41} -0.763932i q^{42} -0.763932i q^{43} +3.61803 q^{44} -3.85410 q^{46} -3.23607i q^{47} +0.618034i q^{48} +5.47214 q^{49} +2.76393 q^{51} -3.85410i q^{52} -8.47214i q^{53} +3.47214 q^{54} +1.23607 q^{56} +2.76393i q^{57} +6.32624i q^{58} +9.23607 q^{59} +8.38197 q^{61} -9.61803i q^{62} -3.23607i q^{63} -1.00000 q^{64} -2.23607 q^{66} +10.0902i q^{67} +4.47214i q^{68} +2.38197 q^{69} -14.9443 q^{71} +2.61803i q^{72} -4.09017i q^{73} +1.00000 q^{74} -4.47214 q^{76} +4.47214i q^{77} +2.38197i q^{78} -11.5623 q^{79} +5.70820 q^{81} -7.38197i q^{82} -5.52786i q^{83} -0.763932 q^{84} -0.763932 q^{86} -3.90983i q^{87} -3.61803i q^{88} +10.4721 q^{89} +4.76393 q^{91} +3.85410i q^{92} +5.94427i q^{93} -3.23607 q^{94} +0.618034 q^{96} -8.47214i q^{97} -5.47214i q^{98} -9.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} - 2 q^{6} + 6 q^{9} - 10 q^{11} + 4 q^{14} + 4 q^{16} + 12 q^{21} + 2 q^{24} + 2 q^{26} + 6 q^{29} + 34 q^{31} - 6 q^{36} - 14 q^{39} + 34 q^{41} + 10 q^{44} - 2 q^{46} + 4 q^{49} + 20 q^{51}+ \cdots - 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1850\mathbb{Z}\right)^\times\).

\(n\) \(1001\) \(1777\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.707107i
\(3\) 0.618034i 0.356822i 0.983956 + 0.178411i \(0.0570957\pi\)
−0.983956 + 0.178411i \(0.942904\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 0.618034 0.252311
\(7\) − 1.23607i − 0.467190i −0.972334 − 0.233595i \(-0.924951\pi\)
0.972334 − 0.233595i \(-0.0750489\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 2.61803 0.872678
\(10\) 0 0
\(11\) −3.61803 −1.09088 −0.545439 − 0.838150i \(-0.683637\pi\)
−0.545439 + 0.838150i \(0.683637\pi\)
\(12\) − 0.618034i − 0.178411i
\(13\) 3.85410i 1.06894i 0.845189 + 0.534468i \(0.179488\pi\)
−0.845189 + 0.534468i \(0.820512\pi\)
\(14\) −1.23607 −0.330353
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 4.47214i − 1.08465i −0.840168 − 0.542326i \(-0.817544\pi\)
0.840168 − 0.542326i \(-0.182456\pi\)
\(18\) − 2.61803i − 0.617077i
\(19\) 4.47214 1.02598 0.512989 − 0.858395i \(-0.328538\pi\)
0.512989 + 0.858395i \(0.328538\pi\)
\(20\) 0 0
\(21\) 0.763932 0.166704
\(22\) 3.61803i 0.771367i
\(23\) − 3.85410i − 0.803636i −0.915720 − 0.401818i \(-0.868378\pi\)
0.915720 − 0.401818i \(-0.131622\pi\)
\(24\) −0.618034 −0.126156
\(25\) 0 0
\(26\) 3.85410 0.755852
\(27\) 3.47214i 0.668213i
\(28\) 1.23607i 0.233595i
\(29\) −6.32624 −1.17475 −0.587376 − 0.809314i \(-0.699839\pi\)
−0.587376 + 0.809314i \(0.699839\pi\)
\(30\) 0 0
\(31\) 9.61803 1.72745 0.863725 − 0.503964i \(-0.168125\pi\)
0.863725 + 0.503964i \(0.168125\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) − 2.23607i − 0.389249i
\(34\) −4.47214 −0.766965
\(35\) 0 0
\(36\) −2.61803 −0.436339
\(37\) 1.00000i 0.164399i
\(38\) − 4.47214i − 0.725476i
\(39\) −2.38197 −0.381420
\(40\) 0 0
\(41\) 7.38197 1.15287 0.576435 − 0.817143i \(-0.304444\pi\)
0.576435 + 0.817143i \(0.304444\pi\)
\(42\) − 0.763932i − 0.117877i
\(43\) − 0.763932i − 0.116499i −0.998302 − 0.0582493i \(-0.981448\pi\)
0.998302 − 0.0582493i \(-0.0185518\pi\)
\(44\) 3.61803 0.545439
\(45\) 0 0
\(46\) −3.85410 −0.568256
\(47\) − 3.23607i − 0.472029i −0.971750 − 0.236015i \(-0.924159\pi\)
0.971750 − 0.236015i \(-0.0758413\pi\)
\(48\) 0.618034i 0.0892055i
\(49\) 5.47214 0.781734
\(50\) 0 0
\(51\) 2.76393 0.387028
\(52\) − 3.85410i − 0.534468i
\(53\) − 8.47214i − 1.16374i −0.813283 − 0.581869i \(-0.802322\pi\)
0.813283 − 0.581869i \(-0.197678\pi\)
\(54\) 3.47214 0.472498
\(55\) 0 0
\(56\) 1.23607 0.165177
\(57\) 2.76393i 0.366092i
\(58\) 6.32624i 0.830676i
\(59\) 9.23607 1.20243 0.601217 − 0.799086i \(-0.294683\pi\)
0.601217 + 0.799086i \(0.294683\pi\)
\(60\) 0 0
\(61\) 8.38197 1.07320 0.536600 − 0.843836i \(-0.319708\pi\)
0.536600 + 0.843836i \(0.319708\pi\)
\(62\) − 9.61803i − 1.22149i
\(63\) − 3.23607i − 0.407706i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −2.23607 −0.275241
\(67\) 10.0902i 1.23271i 0.787468 + 0.616355i \(0.211391\pi\)
−0.787468 + 0.616355i \(0.788609\pi\)
\(68\) 4.47214i 0.542326i
\(69\) 2.38197 0.286755
\(70\) 0 0
\(71\) −14.9443 −1.77356 −0.886779 − 0.462193i \(-0.847063\pi\)
−0.886779 + 0.462193i \(0.847063\pi\)
\(72\) 2.61803i 0.308538i
\(73\) − 4.09017i − 0.478718i −0.970931 − 0.239359i \(-0.923063\pi\)
0.970931 − 0.239359i \(-0.0769373\pi\)
\(74\) 1.00000 0.116248
\(75\) 0 0
\(76\) −4.47214 −0.512989
\(77\) 4.47214i 0.509647i
\(78\) 2.38197i 0.269705i
\(79\) −11.5623 −1.30086 −0.650431 − 0.759566i \(-0.725411\pi\)
−0.650431 + 0.759566i \(0.725411\pi\)
\(80\) 0 0
\(81\) 5.70820 0.634245
\(82\) − 7.38197i − 0.815202i
\(83\) − 5.52786i − 0.606762i −0.952869 − 0.303381i \(-0.901885\pi\)
0.952869 − 0.303381i \(-0.0981155\pi\)
\(84\) −0.763932 −0.0833518
\(85\) 0 0
\(86\) −0.763932 −0.0823769
\(87\) − 3.90983i − 0.419178i
\(88\) − 3.61803i − 0.385684i
\(89\) 10.4721 1.11004 0.555022 − 0.831836i \(-0.312710\pi\)
0.555022 + 0.831836i \(0.312710\pi\)
\(90\) 0 0
\(91\) 4.76393 0.499396
\(92\) 3.85410i 0.401818i
\(93\) 5.94427i 0.616392i
\(94\) −3.23607 −0.333775
\(95\) 0 0
\(96\) 0.618034 0.0630778
\(97\) − 8.47214i − 0.860215i −0.902778 − 0.430108i \(-0.858476\pi\)
0.902778 − 0.430108i \(-0.141524\pi\)
\(98\) − 5.47214i − 0.552769i
\(99\) −9.47214 −0.951985
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 1850.2.b.j.149.2 4
5.2 odd 4 74.2.a.b.1.2 ✓ 2
5.3 odd 4 1850.2.a.t.1.1 2
5.4 even 2 inner 1850.2.b.j.149.3 4
15.2 even 4 666.2.a.i.1.2 2
20.7 even 4 592.2.a.g.1.1 2
35.27 even 4 3626.2.a.s.1.1 2
40.27 even 4 2368.2.a.u.1.2 2
40.37 odd 4 2368.2.a.y.1.1 2
55.32 even 4 8954.2.a.j.1.2 2
60.47 odd 4 5328.2.a.bc.1.2 2
185.147 odd 4 2738.2.a.g.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.a.b.1.2 ✓ 2 5.2 odd 4
592.2.a.g.1.1 2 20.7 even 4
666.2.a.i.1.2 2 15.2 even 4
1850.2.a.t.1.1 2 5.3 odd 4
1850.2.b.j.149.2 4 1.1 even 1 trivial
1850.2.b.j.149.3 4 5.4 even 2 inner
2368.2.a.u.1.2 2 40.27 even 4
2368.2.a.y.1.1 2 40.37 odd 4
2738.2.a.g.1.2 2 185.147 odd 4
3626.2.a.s.1.1 2 35.27 even 4
5328.2.a.bc.1.2 2 60.47 odd 4
8954.2.a.j.1.2 2 55.32 even 4