Show commands: Magma / Pari/GP / SageMath

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 149.1
Root \(-1.61803i\) of defining polynomial
Character \(\chi\) \(=\) 1850.149
Dual form 1850.2.b.j.149.4

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.61803i q^{3} -1.00000 q^{4} -1.61803 q^{6} +3.23607i q^{7} +1.00000i q^{8} +0.381966 q^{9} -1.38197 q^{11} +1.61803i q^{12} -2.85410i q^{13} +3.23607 q^{14} +1.00000 q^{16} +4.47214i q^{17} -0.381966i q^{18} -4.47214 q^{19} +5.23607 q^{21} +1.38197i q^{22} +2.85410i q^{23} +1.61803 q^{24} -2.85410 q^{26} -5.47214i q^{27} -3.23607i q^{28} +9.32624 q^{29} +7.38197 q^{31} -1.00000i q^{32} +2.23607i q^{33} +4.47214 q^{34} -0.381966 q^{36} +1.00000i q^{37} +4.47214i q^{38} -4.61803 q^{39} +9.61803 q^{41} -5.23607i q^{42} -5.23607i q^{43} +1.38197 q^{44} +2.85410 q^{46} +1.23607i q^{47} -1.61803i q^{48} -3.47214 q^{49} +7.23607 q^{51} +2.85410i q^{52} +0.472136i q^{53} -5.47214 q^{54} -3.23607 q^{56} +7.23607i q^{57} -9.32624i q^{58} +4.76393 q^{59} +10.6180 q^{61} -7.38197i q^{62} +1.23607i q^{63} -1.00000 q^{64} +2.23607 q^{66} -1.09017i q^{67} -4.47214i q^{68} +4.61803 q^{69} +2.94427 q^{71} +0.381966i q^{72} +7.09017i q^{73} +1.00000 q^{74} +4.47214 q^{76} -4.47214i q^{77} +4.61803i q^{78} +8.56231 q^{79} -7.70820 q^{81} -9.61803i q^{82} -14.4721i q^{83} -5.23607 q^{84} -5.23607 q^{86} -15.0902i q^{87} -1.38197i q^{88} +1.52786 q^{89} +9.23607 q^{91} -2.85410i q^{92} -11.9443i q^{93} +1.23607 q^{94} -1.61803 q^{96} +0.472136i q^{97} +3.47214i q^{98} -0.527864 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} - 2 q^{6} + 6 q^{9} - 10 q^{11} + 4 q^{14} + 4 q^{16} + 12 q^{21} + 2 q^{24} + 2 q^{26} + 6 q^{29} + 34 q^{31} - 6 q^{36} - 14 q^{39} + 34 q^{41} + 10 q^{44} - 2 q^{46} + 4 q^{49} + 20 q^{51}+ \cdots - 20 q^{99}+O(q^{100}) \) 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
\(3\) − 1.61803i − 0.934172i −0.884212 − 0.467086i \(-0.845304\pi\)
0.884212 − 0.467086i \(-0.154696\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) −1.61803 −0.660560
\(7\) 3.23607i 1.22312i 0.791199 + 0.611559i \(0.209457\pi\)
−0.791199 + 0.611559i \(0.790543\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 0.381966 0.127322
\(10\) 0 0
\(11\) −1.38197 −0.416678 −0.208339 − 0.978057i \(-0.566806\pi\)
−0.208339 + 0.978057i \(0.566806\pi\)
\(12\) 1.61803i 0.467086i
\(13\) − 2.85410i − 0.791585i −0.918340 − 0.395793i \(-0.870470\pi\)
0.918340 − 0.395793i \(-0.129530\pi\)
\(14\) 3.23607 0.864876
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 4.47214i 1.08465i 0.840168 + 0.542326i \(0.182456\pi\)
−0.840168 + 0.542326i \(0.817544\pi\)
\(18\) − 0.381966i − 0.0900303i
\(19\) −4.47214 −1.02598 −0.512989 − 0.858395i \(-0.671462\pi\)
−0.512989 + 0.858395i \(0.671462\pi\)
\(20\) 0 0
\(21\) 5.23607 1.14260
\(22\) 1.38197i 0.294636i
\(23\) 2.85410i 0.595121i 0.954703 + 0.297561i \(0.0961731\pi\)
−0.954703 + 0.297561i \(0.903827\pi\)
\(24\) 1.61803 0.330280
\(25\) 0 0
\(26\) −2.85410 −0.559735
\(27\) − 5.47214i − 1.05311i
\(28\) − 3.23607i − 0.611559i
\(29\) 9.32624 1.73184 0.865919 − 0.500183i \(-0.166734\pi\)
0.865919 + 0.500183i \(0.166734\pi\)
\(30\) 0 0
\(31\) 7.38197 1.32584 0.662920 − 0.748690i \(-0.269317\pi\)
0.662920 + 0.748690i \(0.269317\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 2.23607i 0.389249i
\(34\) 4.47214 0.766965
\(35\) 0 0
\(36\) −0.381966 −0.0636610
\(37\) 1.00000i 0.164399i
\(38\) 4.47214i 0.725476i
\(39\) −4.61803 −0.739477
\(40\) 0 0
\(41\) 9.61803 1.50208 0.751042 − 0.660254i \(-0.229551\pi\)
0.751042 + 0.660254i \(0.229551\pi\)
\(42\) − 5.23607i − 0.807943i
\(43\) − 5.23607i − 0.798493i −0.916844 − 0.399246i \(-0.869272\pi\)
0.916844 − 0.399246i \(-0.130728\pi\)
\(44\) 1.38197 0.208339
\(45\) 0 0
\(46\) 2.85410 0.420814
\(47\) 1.23607i 0.180299i 0.995928 + 0.0901495i \(0.0287345\pi\)
−0.995928 + 0.0901495i \(0.971266\pi\)
\(48\) − 1.61803i − 0.233543i
\(49\) −3.47214 −0.496019
\(50\) 0 0
\(51\) 7.23607 1.01325
\(52\) 2.85410i 0.395793i
\(53\) 0.472136i 0.0648529i 0.999474 + 0.0324264i \(0.0103235\pi\)
−0.999474 + 0.0324264i \(0.989677\pi\)
\(54\) −5.47214 −0.744663
\(55\) 0 0
\(56\) −3.23607 −0.432438
\(57\) 7.23607i 0.958441i
\(58\) − 9.32624i − 1.22460i
\(59\) 4.76393 0.620211 0.310106 − 0.950702i \(-0.399636\pi\)
0.310106 + 0.950702i \(0.399636\pi\)
\(60\) 0 0
\(61\) 10.6180 1.35950 0.679750 − 0.733444i \(-0.262088\pi\)
0.679750 + 0.733444i \(0.262088\pi\)
\(62\) − 7.38197i − 0.937511i
\(63\) 1.23607i 0.155730i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 2.23607 0.275241
\(67\) − 1.09017i − 0.133185i −0.997780 − 0.0665927i \(-0.978787\pi\)
0.997780 − 0.0665927i \(-0.0212128\pi\)
\(68\) − 4.47214i − 0.542326i
\(69\) 4.61803 0.555946
\(70\) 0 0
\(71\) 2.94427 0.349421 0.174710 − 0.984620i \(-0.444101\pi\)
0.174710 + 0.984620i \(0.444101\pi\)
\(72\) 0.381966i 0.0450151i
\(73\) 7.09017i 0.829842i 0.909858 + 0.414921i \(0.136191\pi\)
−0.909858 + 0.414921i \(0.863809\pi\)
\(74\) 1.00000 0.116248
\(75\) 0 0
\(76\) 4.47214 0.512989
\(77\) − 4.47214i − 0.509647i
\(78\) 4.61803i 0.522889i
\(79\) 8.56231 0.963335 0.481667 − 0.876354i \(-0.340031\pi\)
0.481667 + 0.876354i \(0.340031\pi\)
\(80\) 0 0
\(81\) −7.70820 −0.856467
\(82\) − 9.61803i − 1.06213i
\(83\) − 14.4721i − 1.58852i −0.607576 − 0.794262i \(-0.707858\pi\)
0.607576 − 0.794262i \(-0.292142\pi\)
\(84\) −5.23607 −0.571302
\(85\) 0 0
\(86\) −5.23607 −0.564620
\(87\) − 15.0902i − 1.61784i
\(88\) − 1.38197i − 0.147318i
\(89\) 1.52786 0.161953 0.0809766 − 0.996716i \(-0.474196\pi\)
0.0809766 + 0.996716i \(0.474196\pi\)
\(90\) 0 0
\(91\) 9.23607 0.968203
\(92\) − 2.85410i − 0.297561i
\(93\) − 11.9443i − 1.23856i
\(94\) 1.23607 0.127491
\(95\) 0 0
\(96\) −1.61803 −0.165140
\(97\) 0.472136i 0.0479381i 0.999713 + 0.0239691i \(0.00763032\pi\)
−0.999713 + 0.0239691i \(0.992370\pi\)
\(98\) 3.47214i 0.350739i
\(99\) −0.527864 −0.0530523
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.149.1 4
5.2 odd 4 74.2.a.b.1.1 ✓ 2
5.3 odd 4 1850.2.a.t.1.2 2
5.4 even 2 inner 1850.2.b.j.149.4 4
15.2 even 4 666.2.a.i.1.1 2
20.7 even 4 592.2.a.g.1.2 2
35.27 even 4 3626.2.a.s.1.2 2
40.27 even 4 2368.2.a.u.1.1 2
40.37 odd 4 2368.2.a.y.1.2 2
55.32 even 4 8954.2.a.j.1.1 2
60.47 odd 4 5328.2.a.bc.1.1 2
185.147 odd 4 2738.2.a.g.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.a.b.1.1 ✓ 2 5.2 odd 4
592.2.a.g.1.2 2 20.7 even 4
666.2.a.i.1.1 2 15.2 even 4
1850.2.a.t.1.2 2 5.3 odd 4
1850.2.b.j.149.1 4 1.1 even 1 trivial
1850.2.b.j.149.4 4 5.4 even 2 inner
2368.2.a.u.1.1 2 40.27 even 4
2368.2.a.y.1.2 2 40.37 odd 4
2738.2.a.g.1.1 2 185.147 odd 4
3626.2.a.s.1.2 2 35.27 even 4
5328.2.a.bc.1.1 2 60.47 odd 4
8954.2.a.j.1.1 2 55.32 even 4