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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1860,2,Mod(929,1860)] 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("1860.929"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1860, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1860 = 2^{2} \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1860.i (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [48] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(14.8521747760\)
Analytic rank: \(0\)
Dimension: \(48\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 929.17
Character \(\chi\) \(=\) 1860.929
Dual form 1860.2.i.b.929.19

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.577293 - 1.63301i) q^{3} +(-2.10556 - 0.752745i) q^{5} -0.552709i q^{7} +(-2.33347 + 1.88545i) q^{9} +5.59856 q^{11} -4.57469 q^{13} +(-0.0137192 + 3.87296i) q^{15} +4.86178i q^{17} +3.76342 q^{19} +(-0.902581 + 0.319075i) q^{21} +6.86328i q^{23} +(3.86675 + 3.16990i) q^{25} +(4.42607 + 2.72212i) q^{27} -8.84843 q^{29} +(0.842462 - 5.50366i) q^{31} +(-3.23201 - 9.14252i) q^{33} +(-0.416049 + 1.16376i) q^{35} +3.00122 q^{37} +(2.64094 + 7.47053i) q^{39} +8.51549i q^{41} -2.35578 q^{43} +(6.33251 - 2.21343i) q^{45} +9.69795 q^{47} +6.69451 q^{49} +(7.93935 - 2.80667i) q^{51} -6.20510i q^{53} +(-11.7881 - 4.21429i) q^{55} +(-2.17260 - 6.14571i) q^{57} -6.68126i q^{59} -0.831471i q^{61} +(1.04211 + 1.28973i) q^{63} +(9.63228 + 3.44358i) q^{65} -2.49238i q^{67} +(11.2078 - 3.96213i) q^{69} +1.93624i q^{71} +3.92767 q^{73} +(2.94424 - 8.14441i) q^{75} -3.09437i q^{77} +13.0652i q^{79} +(1.89012 - 8.79929i) q^{81} +6.73168i q^{83} +(3.65968 - 10.2368i) q^{85} +(5.10814 + 14.4496i) q^{87} +4.67205 q^{89} +2.52847i q^{91} +(-9.47390 + 1.80147i) q^{93} +(-7.92410 - 2.83290i) q^{95} -8.88551i q^{97} +(-13.0640 + 10.5558i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q - 4 q^{9} - 8 q^{19} + 64 q^{25} + 24 q^{31} + 8 q^{39} - 16 q^{45} - 160 q^{49} + 68 q^{51} - 24 q^{69} + 52 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1860\mathbb{Z}\right)^\times\).

\(n\) \(931\) \(1117\) \(1241\) \(1801\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.577293 − 1.63301i −0.333300 − 0.942821i
\(4\) 0 0
\(5\) −2.10556 − 0.752745i −0.941634 − 0.336638i
\(6\) 0 0
\(7\) − 0.552709i − 0.208904i −0.994530 − 0.104452i \(-0.966691\pi\)
0.994530 − 0.104452i \(-0.0333089\pi\)
\(8\) 0 0
\(9\) −2.33347 + 1.88545i −0.777822 + 0.628485i
\(10\) 0 0
\(11\) 5.59856 1.68803 0.844015 − 0.536320i \(-0.180186\pi\)
0.844015 + 0.536320i \(0.180186\pi\)
\(12\) 0 0
\(13\) −4.57469 −1.26879 −0.634396 − 0.773008i \(-0.718751\pi\)
−0.634396 + 0.773008i \(0.718751\pi\)
\(14\) 0 0
\(15\) −0.0137192 + 3.87296i −0.00354227 + 0.999994i
\(16\) 0 0
\(17\) 4.86178i 1.17915i 0.807712 + 0.589577i \(0.200706\pi\)
−0.807712 + 0.589577i \(0.799294\pi\)
\(18\) 0 0
\(19\) 3.76342 0.863388 0.431694 − 0.902020i \(-0.357916\pi\)
0.431694 + 0.902020i \(0.357916\pi\)
\(20\) 0 0
\(21\) −0.902581 + 0.319075i −0.196959 + 0.0696279i
\(22\) 0 0
\(23\) 6.86328i 1.43109i 0.698565 + 0.715547i \(0.253822\pi\)
−0.698565 + 0.715547i \(0.746178\pi\)
\(24\) 0 0
\(25\) 3.86675 + 3.16990i 0.773350 + 0.633980i
\(26\) 0 0
\(27\) 4.42607 + 2.72212i 0.851797 + 0.523872i
\(28\) 0 0
\(29\) −8.84843 −1.64311 −0.821556 − 0.570128i \(-0.806894\pi\)
−0.821556 + 0.570128i \(0.806894\pi\)
\(30\) 0 0
\(31\) 0.842462 − 5.50366i 0.151311 − 0.988486i
\(32\) 0 0
\(33\) −3.23201 − 9.14252i −0.562621 − 1.59151i
\(34\) 0 0
\(35\) −0.416049 + 1.16376i −0.0703251 + 0.196711i
\(36\) 0 0
\(37\) 3.00122 0.493398 0.246699 − 0.969092i \(-0.420654\pi\)
0.246699 + 0.969092i \(0.420654\pi\)
\(38\) 0 0
\(39\) 2.64094 + 7.47053i 0.422889 + 1.19624i
\(40\) 0 0
\(41\) 8.51549i 1.32990i 0.746890 + 0.664948i \(0.231546\pi\)
−0.746890 + 0.664948i \(0.768454\pi\)
\(42\) 0 0
\(43\) −2.35578 −0.359253 −0.179626 − 0.983735i \(-0.557489\pi\)
−0.179626 + 0.983735i \(0.557489\pi\)
\(44\) 0 0
\(45\) 6.33251 − 2.21343i 0.943995 − 0.329958i
\(46\) 0 0
\(47\) 9.69795 1.41459 0.707296 − 0.706918i \(-0.249915\pi\)
0.707296 + 0.706918i \(0.249915\pi\)
\(48\) 0 0
\(49\) 6.69451 0.956359
\(50\) 0 0
\(51\) 7.93935 − 2.80667i 1.11173 − 0.393012i
\(52\) 0 0
\(53\) − 6.20510i − 0.852336i −0.904644 − 0.426168i \(-0.859863\pi\)
0.904644 − 0.426168i \(-0.140137\pi\)
\(54\) 0 0
\(55\) −11.7881 − 4.21429i −1.58951 − 0.568255i
\(56\) 0 0
\(57\) −2.17260 − 6.14571i −0.287767 − 0.814020i
\(58\) 0 0
\(59\) − 6.68126i − 0.869826i −0.900472 − 0.434913i \(-0.856779\pi\)
0.900472 − 0.434913i \(-0.143221\pi\)
\(60\) 0 0
\(61\) − 0.831471i − 0.106459i −0.998582 − 0.0532295i \(-0.983049\pi\)
0.998582 − 0.0532295i \(-0.0169515\pi\)
\(62\) 0 0
\(63\) 1.04211 + 1.28973i 0.131293 + 0.162490i
\(64\) 0 0
\(65\) 9.63228 + 3.44358i 1.19474 + 0.427123i
\(66\) 0 0
\(67\) − 2.49238i − 0.304493i −0.988343 − 0.152246i \(-0.951349\pi\)
0.988343 − 0.152246i \(-0.0486507\pi\)
\(68\) 0 0
\(69\) 11.2078 − 3.96213i 1.34926 − 0.476984i
\(70\) 0 0
\(71\) 1.93624i 0.229789i 0.993378 + 0.114895i \(0.0366530\pi\)
−0.993378 + 0.114895i \(0.963347\pi\)
\(72\) 0 0
\(73\) 3.92767 0.459699 0.229849 − 0.973226i \(-0.426177\pi\)
0.229849 + 0.973226i \(0.426177\pi\)
\(74\) 0 0
\(75\) 2.94424 − 8.14441i 0.339971 − 0.940436i
\(76\) 0 0
\(77\) − 3.09437i − 0.352637i
\(78\) 0 0
\(79\) 13.0652i 1.46995i 0.678094 + 0.734975i \(0.262806\pi\)
−0.678094 + 0.734975i \(0.737194\pi\)
\(80\) 0 0
\(81\) 1.89012 − 8.79929i 0.210014 − 0.977698i
\(82\) 0 0
\(83\) 6.73168i 0.738898i 0.929251 + 0.369449i \(0.120454\pi\)
−0.929251 + 0.369449i \(0.879546\pi\)
\(84\) 0 0
\(85\) 3.65968 − 10.2368i 0.396948 − 1.11033i
\(86\) 0 0
\(87\) 5.10814 + 14.4496i 0.547650 + 1.54916i
\(88\) 0 0
\(89\) 4.67205 0.495236 0.247618 − 0.968858i \(-0.420352\pi\)
0.247618 + 0.968858i \(0.420352\pi\)
\(90\) 0 0
\(91\) 2.52847i 0.265056i
\(92\) 0 0
\(93\) −9.47390 + 1.80147i −0.982397 + 0.186804i
\(94\) 0 0
\(95\) −7.92410 − 2.83290i −0.812995 − 0.290649i
\(96\) 0 0
\(97\) − 8.88551i − 0.902186i −0.892477 − 0.451093i \(-0.851034\pi\)
0.892477 − 0.451093i \(-0.148966\pi\)
\(98\) 0 0
\(99\) −13.0640 + 10.5558i −1.31299 + 1.06090i
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 1860.2.i.b.929.17 ✓ 48
3.2 odd 2 inner 1860.2.i.b.929.20 yes 48
5.4 even 2 inner 1860.2.i.b.929.32 yes 48
15.14 odd 2 inner 1860.2.i.b.929.29 yes 48
31.30 odd 2 inner 1860.2.i.b.929.31 yes 48
93.92 even 2 inner 1860.2.i.b.929.30 yes 48
155.154 odd 2 inner 1860.2.i.b.929.18 yes 48
465.464 even 2 inner 1860.2.i.b.929.19 yes 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1860.2.i.b.929.17 ✓ 48 1.1 even 1 trivial
1860.2.i.b.929.18 yes 48 155.154 odd 2 inner
1860.2.i.b.929.19 yes 48 465.464 even 2 inner
1860.2.i.b.929.20 yes 48 3.2 odd 2 inner
1860.2.i.b.929.29 yes 48 15.14 odd 2 inner
1860.2.i.b.929.30 yes 48 93.92 even 2 inner
1860.2.i.b.929.31 yes 48 31.30 odd 2 inner
1860.2.i.b.929.32 yes 48 5.4 even 2 inner