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(1849,1850)] 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("1850.1849"); 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, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1850 = 2 \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1850.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,-4,0,4,0,0,0,-4,-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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 74)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1849.2
Root \(-1.79129i\) of defining polynomial
Character \(\chi\) \(=\) 1850.1849
Dual form 1850.2.c.g.1849.3

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.79129i q^{3} +1.00000 q^{4} +1.79129i q^{6} -2.00000i q^{7} -1.00000 q^{8} -0.208712 q^{9} -0.791288 q^{11} -1.79129i q^{12} +3.79129 q^{13} +2.00000i q^{14} +1.00000 q^{16} +7.58258 q^{17} +0.208712 q^{18} +1.58258i q^{19} -3.58258 q^{21} +0.791288 q^{22} +3.79129 q^{23} +1.79129i q^{24} -3.79129 q^{26} -5.00000i q^{27} -2.00000i q^{28} +3.79129i q^{29} +8.37386i q^{31} -1.00000 q^{32} +1.41742i q^{33} -7.58258 q^{34} -0.208712 q^{36} +(-4.58258 + 4.00000i) q^{37} -1.58258i q^{38} -6.79129i q^{39} +9.79129 q^{41} +3.58258 q^{42} +6.00000 q^{43} -0.791288 q^{44} -3.79129 q^{46} -7.58258i q^{47} -1.79129i q^{48} +3.00000 q^{49} -13.5826i q^{51} +3.79129 q^{52} +1.58258i q^{53} +5.00000i q^{54} +2.00000i q^{56} +2.83485 q^{57} -3.79129i q^{58} +1.58258i q^{59} -12.7913i q^{61} -8.37386i q^{62} +0.417424i q^{63} +1.00000 q^{64} -1.41742i q^{66} +6.37386i q^{67} +7.58258 q^{68} -6.79129i q^{69} -9.16515 q^{71} +0.208712 q^{72} +4.37386i q^{73} +(4.58258 - 4.00000i) q^{74} +1.58258i q^{76} +1.58258i q^{77} +6.79129i q^{78} +8.20871i q^{79} -9.58258 q^{81} -9.79129 q^{82} -15.1652i q^{83} -3.58258 q^{84} -6.00000 q^{86} +6.79129 q^{87} +0.791288 q^{88} +6.00000i q^{89} -7.58258i q^{91} +3.79129 q^{92} +15.0000 q^{93} +7.58258i q^{94} +1.79129i q^{96} -13.5826 q^{97} -3.00000 q^{98} +0.165151 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8} - 10 q^{9} + 6 q^{11} + 6 q^{13} + 4 q^{16} + 12 q^{17} + 10 q^{18} + 4 q^{21} - 6 q^{22} + 6 q^{23} - 6 q^{26} - 4 q^{32} - 12 q^{34} - 10 q^{36} + 30 q^{41} - 4 q^{42}+ \cdots - 36 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) − 1.79129i − 1.03420i −0.855925 − 0.517100i \(-0.827011\pi\)
0.855925 − 0.517100i \(-0.172989\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 1.79129i 0.731290i
\(7\) − 2.00000i − 0.755929i −0.925820 − 0.377964i \(-0.876624\pi\)
0.925820 − 0.377964i \(-0.123376\pi\)
\(8\) −1.00000 −0.353553
\(9\) −0.208712 −0.0695707
\(10\) 0 0
\(11\) −0.791288 −0.238582 −0.119291 − 0.992859i \(-0.538062\pi\)
−0.119291 + 0.992859i \(0.538062\pi\)
\(12\) − 1.79129i − 0.517100i
\(13\) 3.79129 1.05151 0.525757 − 0.850635i \(-0.323782\pi\)
0.525757 + 0.850635i \(0.323782\pi\)
\(14\) 2.00000i 0.534522i
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 7.58258 1.83904 0.919522 − 0.393038i \(-0.128576\pi\)
0.919522 + 0.393038i \(0.128576\pi\)
\(18\) 0.208712 0.0491939
\(19\) 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
\(22\) 0.791288 0.168703
\(23\) 3.79129 0.790538 0.395269 − 0.918565i \(-0.370651\pi\)
0.395269 + 0.918565i \(0.370651\pi\)
\(24\) 1.79129i 0.365645i
\(25\) 0 0
\(26\) −3.79129 −0.743533
\(27\) − 5.00000i − 0.962250i
\(28\) − 2.00000i − 0.377964i
\(29\) 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 \(0.270908\pi\)
−0.659169 + 0.751995i \(0.729092\pi\)
\(32\) −1.00000 −0.176777
\(33\) 1.41742i 0.246742i
\(34\) −7.58258 −1.30040
\(35\) 0 0
\(36\) −0.208712 −0.0347854
\(37\) −4.58258 + 4.00000i −0.753371 + 0.657596i
\(38\) − 1.58258i − 0.256728i
\(39\) − 6.79129i − 1.08748i
\(40\) 0 0
\(41\) 9.79129 1.52914 0.764571 − 0.644539i \(-0.222951\pi\)
0.764571 + 0.644539i \(0.222951\pi\)
\(42\) 3.58258 0.552803
\(43\) 6.00000 0.914991 0.457496 − 0.889212i \(-0.348747\pi\)
0.457496 + 0.889212i \(0.348747\pi\)
\(44\) −0.791288 −0.119291
\(45\) 0 0
\(46\) −3.79129 −0.558995
\(47\) − 7.58258i − 1.10603i −0.833171 − 0.553016i \(-0.813477\pi\)
0.833171 − 0.553016i \(-0.186523\pi\)
\(48\) − 1.79129i − 0.258550i
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) − 13.5826i − 1.90194i
\(52\) 3.79129 0.525757
\(53\) 1.58258i 0.217383i 0.994076 + 0.108692i \(0.0346661\pi\)
−0.994076 + 0.108692i \(0.965334\pi\)
\(54\) 5.00000i 0.680414i
\(55\) 0 0
\(56\) 2.00000i 0.267261i
\(57\) 2.83485 0.375485
\(58\) − 3.79129i − 0.497820i
\(59\) 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 \(-0.694596\pi\)
0.573967 − 0.818878i \(-0.305404\pi\)
\(62\) − 8.37386i − 1.06348i
\(63\) 0.417424i 0.0525905i
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) − 1.41742i − 0.174473i
\(67\) 6.37386i 0.778691i 0.921092 + 0.389346i \(0.127299\pi\)
−0.921092 + 0.389346i \(0.872701\pi\)
\(68\) 7.58258 0.919522
\(69\) − 6.79129i − 0.817575i
\(70\) 0 0
\(71\) −9.16515 −1.08770 −0.543852 − 0.839181i \(-0.683035\pi\)
−0.543852 + 0.839181i \(0.683035\pi\)
\(72\) 0.208712 0.0245970
\(73\) 4.37386i 0.511922i 0.966687 + 0.255961i \(0.0823919\pi\)
−0.966687 + 0.255961i \(0.917608\pi\)
\(74\) 4.58258 − 4.00000i 0.532714 − 0.464991i
\(75\) 0 0
\(76\) 1.58258i 0.181534i
\(77\) 1.58258i 0.180351i
\(78\) 6.79129i 0.768962i
\(79\) 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
\(82\) −9.79129 −1.08127
\(83\) − 15.1652i − 1.66459i −0.554332 − 0.832296i \(-0.687026\pi\)
0.554332 − 0.832296i \(-0.312974\pi\)
\(84\) −3.58258 −0.390891
\(85\) 0 0
\(86\) −6.00000 −0.646997
\(87\) 6.79129 0.728102
\(88\) 0.791288 0.0843516
\(89\) 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
\(92\) 3.79129 0.395269
\(93\) 15.0000 1.55543
\(94\) 7.58258i 0.782083i
\(95\) 0 0
\(96\) 1.79129i 0.182823i
\(97\) −13.5826 −1.37910 −0.689551 − 0.724237i \(-0.742192\pi\)
−0.689551 + 0.724237i \(0.742192\pi\)
\(98\) −3.00000 −0.303046
\(99\) 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.c.g.1849.2 4
5.2 odd 4 1850.2.d.e.1701.1 4
5.3 odd 4 74.2.b.a.73.4 yes 4
5.4 even 2 1850.2.c.h.1849.3 4
15.8 even 4 666.2.c.b.73.2 4
20.3 even 4 592.2.g.c.369.1 4
37.36 even 2 1850.2.c.h.1849.2 4
40.3 even 4 2368.2.g.h.961.4 4
40.13 odd 4 2368.2.g.j.961.2 4
60.23 odd 4 5328.2.h.m.2737.3 4
185.43 even 4 2738.2.a.k.1.1 2
185.68 even 4 2738.2.a.h.1.1 2
185.73 odd 4 74.2.b.a.73.2 ✓ 4
185.147 odd 4 1850.2.d.e.1701.3 4
185.184 even 2 inner 1850.2.c.g.1849.3 4
555.443 even 4 666.2.c.b.73.3 4
740.443 even 4 592.2.g.c.369.2 4
1480.443 even 4 2368.2.g.h.961.3 4
1480.813 odd 4 2368.2.g.j.961.1 4
2220.443 odd 4 5328.2.h.m.2737.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.b.a.73.2 ✓ 4 185.73 odd 4
74.2.b.a.73.4 yes 4 5.3 odd 4
592.2.g.c.369.1 4 20.3 even 4
592.2.g.c.369.2 4 740.443 even 4
666.2.c.b.73.2 4 15.8 even 4
666.2.c.b.73.3 4 555.443 even 4
1850.2.c.g.1849.2 4 1.1 even 1 trivial
1850.2.c.g.1849.3 4 185.184 even 2 inner
1850.2.c.h.1849.2 4 37.36 even 2
1850.2.c.h.1849.3 4 5.4 even 2
1850.2.d.e.1701.1 4 5.2 odd 4
1850.2.d.e.1701.3 4 185.147 odd 4
2368.2.g.h.961.3 4 1480.443 even 4
2368.2.g.h.961.4 4 40.3 even 4
2368.2.g.j.961.1 4 1480.813 odd 4
2368.2.g.j.961.2 4 40.13 odd 4
2738.2.a.h.1.1 2 185.68 even 4
2738.2.a.k.1.1 2 185.43 even 4
5328.2.h.m.2737.2 4 2220.443 odd 4
5328.2.h.m.2737.3 4 60.23 odd 4