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(1701,1850)] 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("1850.1701"); 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([0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1850 = 2 \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1850.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,2,-4,0,0,8,0,10,0,6] 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: \(\Q(i, \sqrt{21})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 11x^{2} + 25 \) 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 1701.3
Root \(-2.79129i\) of defining polynomial
Character \(\chi\) \(=\) 1850.1701
Dual form 1850.2.d.e.1701.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.79129 q^{3} -1.00000 q^{4} -1.79129i q^{6} +2.00000 q^{7} -1.00000i q^{8} +0.208712 q^{9} -0.791288 q^{11} +1.79129 q^{12} +3.79129i q^{13} +2.00000i q^{14} +1.00000 q^{16} -7.58258i q^{17} +0.208712i q^{18} +1.58258i q^{19} -3.58258 q^{21} -0.791288i q^{22} +3.79129i q^{23} +1.79129i q^{24} -3.79129 q^{26} +5.00000 q^{27} -2.00000 q^{28} +3.79129i q^{29} -8.37386i q^{31} +1.00000i q^{32} +1.41742 q^{33} +7.58258 q^{34} -0.208712 q^{36} +(-4.00000 + 4.58258i) q^{37} -1.58258 q^{38} -6.79129i q^{39} +9.79129 q^{41} -3.58258i q^{42} +6.00000i q^{43} +0.791288 q^{44} -3.79129 q^{46} +7.58258 q^{47} -1.79129 q^{48} -3.00000 q^{49} +13.5826i q^{51} -3.79129i q^{52} +1.58258 q^{53} +5.00000i q^{54} -2.00000i q^{56} -2.83485i q^{57} -3.79129 q^{58} +1.58258i q^{59} +12.7913i q^{61} +8.37386 q^{62} +0.417424 q^{63} -1.00000 q^{64} +1.41742i q^{66} -6.37386 q^{67} +7.58258i q^{68} -6.79129i q^{69} -9.16515 q^{71} -0.208712i q^{72} +4.37386 q^{73} +(-4.58258 - 4.00000i) q^{74} -1.58258i q^{76} -1.58258 q^{77} +6.79129 q^{78} +8.20871i q^{79} -9.58258 q^{81} +9.79129i q^{82} -15.1652 q^{83} +3.58258 q^{84} -6.00000 q^{86} -6.79129i q^{87} +0.791288i q^{88} +6.00000i q^{89} +7.58258i q^{91} -3.79129i q^{92} +15.0000i q^{93} +7.58258i q^{94} -1.79129i q^{96} +13.5826i q^{97} -3.00000i q^{98} -0.165151 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} - 4 q^{4} + 8 q^{7} + 10 q^{9} + 6 q^{11} - 2 q^{12} + 4 q^{16} + 4 q^{21} - 6 q^{26} + 20 q^{27} - 8 q^{28} + 24 q^{33} + 12 q^{34} - 10 q^{36} - 16 q^{37} + 12 q^{38} + 30 q^{41} - 6 q^{44}+ \cdots + 36 q^{99}+O(q^{100}) \) 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
\(3\) −1.79129 −1.03420 −0.517100 − 0.855925i \(-0.672989\pi\)
−0.517100 + 0.855925i \(0.672989\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) − 1.79129i − 0.731290i
\(7\) 2.00000 0.755929 0.377964 − 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) 0.208712 0.0695707
\(10\) 0 0
\(11\) −0.791288 −0.238582 −0.119291 − 0.992859i \(-0.538062\pi\)
−0.119291 + 0.992859i \(0.538062\pi\)
\(12\) 1.79129 0.517100
\(13\) 3.79129i 1.05151i 0.850635 + 0.525757i \(0.176218\pi\)
−0.850635 + 0.525757i \(0.823782\pi\)
\(14\) 2.00000i 0.534522i
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 7.58258i − 1.83904i −0.393038 − 0.919522i \(-0.628576\pi\)
0.393038 − 0.919522i \(-0.371424\pi\)
\(18\) 0.208712i 0.0491939i
\(19\) 1.58258i 0.363068i 0.983385 + 0.181534i \(0.0581062\pi\)
−0.983385 + 0.181534i \(0.941894\pi\)
\(20\) 0 0
\(21\) −3.58258 −0.781782
\(22\) − 0.791288i − 0.168703i
\(23\) 3.79129i 0.790538i 0.918565 + 0.395269i \(0.129349\pi\)
−0.918565 + 0.395269i \(0.870651\pi\)
\(24\) 1.79129i 0.365645i
\(25\) 0 0
\(26\) −3.79129 −0.743533
\(27\) 5.00000 0.962250
\(28\) −2.00000 −0.377964
\(29\) 3.79129i 0.704024i 0.935995 + 0.352012i \(0.114502\pi\)
−0.935995 + 0.352012i \(0.885498\pi\)
\(30\) 0 0
\(31\) − 8.37386i − 1.50399i −0.659169 − 0.751995i \(-0.729092\pi\)
0.659169 − 0.751995i \(-0.270908\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 1.41742 0.246742
\(34\) 7.58258 1.30040
\(35\) 0 0
\(36\) −0.208712 −0.0347854
\(37\) −4.00000 + 4.58258i −0.657596 + 0.753371i
\(38\) −1.58258 −0.256728
\(39\) − 6.79129i − 1.08748i
\(40\) 0 0
\(41\) 9.79129 1.52914 0.764571 − 0.644539i \(-0.222951\pi\)
0.764571 + 0.644539i \(0.222951\pi\)
\(42\) − 3.58258i − 0.552803i
\(43\) 6.00000i 0.914991i 0.889212 + 0.457496i \(0.151253\pi\)
−0.889212 + 0.457496i \(0.848747\pi\)
\(44\) 0.791288 0.119291
\(45\) 0 0
\(46\) −3.79129 −0.558995
\(47\) 7.58258 1.10603 0.553016 − 0.833171i \(-0.313477\pi\)
0.553016 + 0.833171i \(0.313477\pi\)
\(48\) −1.79129 −0.258550
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 13.5826i 1.90194i
\(52\) − 3.79129i − 0.525757i
\(53\) 1.58258 0.217383 0.108692 − 0.994076i \(-0.465334\pi\)
0.108692 + 0.994076i \(0.465334\pi\)
\(54\) 5.00000i 0.680414i
\(55\) 0 0
\(56\) − 2.00000i − 0.267261i
\(57\) − 2.83485i − 0.375485i
\(58\) −3.79129 −0.497820
\(59\) 1.58258i 0.206034i 0.994680 + 0.103017i \(0.0328496\pi\)
−0.994680 + 0.103017i \(0.967150\pi\)
\(60\) 0 0
\(61\) 12.7913i 1.63776i 0.573967 + 0.818878i \(0.305404\pi\)
−0.573967 + 0.818878i \(0.694596\pi\)
\(62\) 8.37386 1.06348
\(63\) 0.417424 0.0525905
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 1.41742i 0.174473i
\(67\) −6.37386 −0.778691 −0.389346 − 0.921092i \(-0.627299\pi\)
−0.389346 + 0.921092i \(0.627299\pi\)
\(68\) 7.58258i 0.919522i
\(69\) − 6.79129i − 0.817575i
\(70\) 0 0
\(71\) −9.16515 −1.08770 −0.543852 − 0.839181i \(-0.683035\pi\)
−0.543852 + 0.839181i \(0.683035\pi\)
\(72\) − 0.208712i − 0.0245970i
\(73\) 4.37386 0.511922 0.255961 − 0.966687i \(-0.417608\pi\)
0.255961 + 0.966687i \(0.417608\pi\)
\(74\) −4.58258 − 4.00000i −0.532714 − 0.464991i
\(75\) 0 0
\(76\) − 1.58258i − 0.181534i
\(77\) −1.58258 −0.180351
\(78\) 6.79129 0.768962
\(79\) 8.20871i 0.923552i 0.886997 + 0.461776i \(0.152788\pi\)
−0.886997 + 0.461776i \(0.847212\pi\)
\(80\) 0 0
\(81\) −9.58258 −1.06473
\(82\) 9.79129i 1.08127i
\(83\) −15.1652 −1.66459 −0.832296 − 0.554332i \(-0.812974\pi\)
−0.832296 + 0.554332i \(0.812974\pi\)
\(84\) 3.58258 0.390891
\(85\) 0 0
\(86\) −6.00000 −0.646997
\(87\) − 6.79129i − 0.728102i
\(88\) 0.791288i 0.0843516i
\(89\) 6.00000i 0.635999i 0.948091 + 0.317999i \(0.103011\pi\)
−0.948091 + 0.317999i \(0.896989\pi\)
\(90\) 0 0
\(91\) 7.58258i 0.794870i
\(92\) − 3.79129i − 0.395269i
\(93\) 15.0000i 1.55543i
\(94\) 7.58258i 0.782083i
\(95\) 0 0
\(96\) − 1.79129i − 0.182823i
\(97\) 13.5826i 1.37910i 0.724237 + 0.689551i \(0.242192\pi\)
−0.724237 + 0.689551i \(0.757808\pi\)
\(98\) − 3.00000i − 0.303046i
\(99\) −0.165151 −0.0165983
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.d.e.1701.3 4
5.2 odd 4 1850.2.c.g.1849.3 4
5.3 odd 4 1850.2.c.h.1849.2 4
5.4 even 2 74.2.b.a.73.2 ✓ 4
15.14 odd 2 666.2.c.b.73.3 4
20.19 odd 2 592.2.g.c.369.2 4
37.36 even 2 inner 1850.2.d.e.1701.1 4
40.19 odd 2 2368.2.g.h.961.3 4
40.29 even 2 2368.2.g.j.961.1 4
60.59 even 2 5328.2.h.m.2737.2 4
185.73 odd 4 1850.2.c.g.1849.2 4
185.147 odd 4 1850.2.c.h.1849.3 4
185.154 odd 4 2738.2.a.h.1.1 2
185.179 odd 4 2738.2.a.k.1.1 2
185.184 even 2 74.2.b.a.73.4 yes 4
555.554 odd 2 666.2.c.b.73.2 4
740.739 odd 2 592.2.g.c.369.1 4
1480.739 odd 2 2368.2.g.h.961.4 4
1480.1109 even 2 2368.2.g.j.961.2 4
2220.2219 even 2 5328.2.h.m.2737.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.b.a.73.2 ✓ 4 5.4 even 2
74.2.b.a.73.4 yes 4 185.184 even 2
592.2.g.c.369.1 4 740.739 odd 2
592.2.g.c.369.2 4 20.19 odd 2
666.2.c.b.73.2 4 555.554 odd 2
666.2.c.b.73.3 4 15.14 odd 2
1850.2.c.g.1849.2 4 185.73 odd 4
1850.2.c.g.1849.3 4 5.2 odd 4
1850.2.c.h.1849.2 4 5.3 odd 4
1850.2.c.h.1849.3 4 185.147 odd 4
1850.2.d.e.1701.1 4 37.36 even 2 inner
1850.2.d.e.1701.3 4 1.1 even 1 trivial
2368.2.g.h.961.3 4 40.19 odd 2
2368.2.g.h.961.4 4 1480.739 odd 2
2368.2.g.j.961.1 4 40.29 even 2
2368.2.g.j.961.2 4 1480.1109 even 2
2738.2.a.h.1.1 2 185.154 odd 4
2738.2.a.k.1.1 2 185.179 odd 4
5328.2.h.m.2737.2 4 60.59 even 2
5328.2.h.m.2737.3 4 2220.2219 even 2