Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [360,2,Mod(109,360)] 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("360.109"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(360, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 360.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.87461447277\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-5})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 4x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 109.3
Root \(1.58114 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 360.109
Dual form 360.2.d.a.109.2

$q$-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)
 
\(f(q)\) \(=\) \(q+1.41421i q^{2} -2.00000 q^{4} -2.23607i q^{5} -2.82843i q^{8} +3.16228 q^{10} -4.47214i q^{11} +6.32456 q^{13} +4.00000 q^{16} +2.82843i q^{17} +4.47214i q^{20} +6.32456 q^{22} -5.65685i q^{23} -5.00000 q^{25} +8.94427i q^{26} -4.47214i q^{29} +2.00000 q^{31} +5.65685i q^{32} -4.00000 q^{34} +6.32456 q^{37} -6.32456 q^{40} -12.6491 q^{43} +8.94427i q^{44} +8.00000 q^{46} +11.3137i q^{47} +7.00000 q^{49} -7.07107i q^{50} -12.6491 q^{52} -10.0000 q^{55} +6.32456 q^{58} -4.47214i q^{59} +2.82843i q^{62} -8.00000 q^{64} -14.1421i q^{65} -12.6491 q^{67} -5.65685i q^{68} +8.94427i q^{74} +14.0000 q^{79} -8.94427i q^{80} +6.32456 q^{85} -17.8885i q^{86} -12.6491 q^{88} +11.3137i q^{92} -16.0000 q^{94} +9.89949i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{4} + 16 q^{16} - 20 q^{25} + 8 q^{31} - 16 q^{34} + 32 q^{46} + 28 q^{49} - 40 q^{55} - 32 q^{64} + 56 q^{79} - 64 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/360\mathbb{Z}\right)^\times\).

\(n\) \(181\) \(217\) \(271\) \(281\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.41421i 1.00000i
\(3\) 0 0
\(4\) −2.00000 −1.00000
\(5\) − 2.23607i − 1.00000i
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) − 2.82843i − 1.00000i
\(9\) 0 0
\(10\) 3.16228 1.00000
\(11\) − 4.47214i − 1.34840i −0.738549 − 0.674200i \(-0.764489\pi\)
0.738549 − 0.674200i \(-0.235511\pi\)
\(12\) 0 0
\(13\) 6.32456 1.75412 0.877058 − 0.480384i \(-0.159503\pi\)
0.877058 + 0.480384i \(0.159503\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 4.00000 1.00000
\(17\) 2.82843i 0.685994i 0.939336 + 0.342997i \(0.111442\pi\)
−0.939336 + 0.342997i \(0.888558\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 4.47214i 1.00000i
\(21\) 0 0
\(22\) 6.32456 1.34840
\(23\) − 5.65685i − 1.17954i −0.807573 − 0.589768i \(-0.799219\pi\)
0.807573 − 0.589768i \(-0.200781\pi\)
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) 8.94427i 1.75412i
\(27\) 0 0
\(28\) 0 0
\(29\) − 4.47214i − 0.830455i −0.909718 − 0.415227i \(-0.863702\pi\)
0.909718 − 0.415227i \(-0.136298\pi\)
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 − 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) 5.65685i 1.00000i
\(33\) 0 0
\(34\) −4.00000 −0.685994
\(35\) 0 0
\(36\) 0 0
\(37\) 6.32456 1.03975 0.519875 − 0.854242i \(-0.325978\pi\)
0.519875 + 0.854242i \(0.325978\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −6.32456 −1.00000
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) −12.6491 −1.92897 −0.964486 − 0.264135i \(-0.914913\pi\)
−0.964486 + 0.264135i \(0.914913\pi\)
\(44\) 8.94427i 1.34840i
\(45\) 0 0
\(46\) 8.00000 1.17954
\(47\) 11.3137i 1.65027i 0.564933 + 0.825137i \(0.308902\pi\)
−0.564933 + 0.825137i \(0.691098\pi\)
\(48\) 0 0
\(49\) 7.00000 1.00000
\(50\) − 7.07107i − 1.00000i
\(51\) 0 0
\(52\) −12.6491 −1.75412
\(53\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) −10.0000 −1.34840
\(56\) 0 0
\(57\) 0 0
\(58\) 6.32456 0.830455
\(59\) − 4.47214i − 0.582223i −0.956689 − 0.291111i \(-0.905975\pi\)
0.956689 − 0.291111i \(-0.0940250\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 2.82843i 0.359211i
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) − 14.1421i − 1.75412i
\(66\) 0 0
\(67\) −12.6491 −1.54533 −0.772667 − 0.634811i \(-0.781078\pi\)
−0.772667 + 0.634811i \(0.781078\pi\)
\(68\) − 5.65685i − 0.685994i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 8.94427i 1.03975i
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 14.0000 1.57512 0.787562 − 0.616236i \(-0.211343\pi\)
0.787562 + 0.616236i \(0.211343\pi\)
\(80\) − 8.94427i − 1.00000i
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 6.32456 0.685994
\(86\) − 17.8885i − 1.92897i
\(87\) 0 0
\(88\) −12.6491 −1.34840
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 11.3137i 1.17954i
\(93\) 0 0
\(94\) −16.0000 −1.65027
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 9.89949i 1.00000i
\(99\) 0 0
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 360.2.d.a.109.3 yes 4
3.2 odd 2 inner 360.2.d.a.109.2 yes 4
4.3 odd 2 1440.2.d.a.1009.2 4
5.2 odd 4 1800.2.k.o.901.1 4
5.3 odd 4 1800.2.k.o.901.3 4
5.4 even 2 inner 360.2.d.a.109.1 ✓ 4
8.3 odd 2 1440.2.d.a.1009.3 4
8.5 even 2 inner 360.2.d.a.109.4 yes 4
12.11 even 2 1440.2.d.a.1009.4 4
15.2 even 4 1800.2.k.o.901.4 4
15.8 even 4 1800.2.k.o.901.2 4
15.14 odd 2 inner 360.2.d.a.109.4 yes 4
20.3 even 4 7200.2.k.m.3601.4 4
20.7 even 4 7200.2.k.m.3601.3 4
20.19 odd 2 1440.2.d.a.1009.1 4
24.5 odd 2 inner 360.2.d.a.109.1 ✓ 4
24.11 even 2 1440.2.d.a.1009.1 4
40.3 even 4 7200.2.k.m.3601.1 4
40.13 odd 4 1800.2.k.o.901.4 4
40.19 odd 2 1440.2.d.a.1009.4 4
40.27 even 4 7200.2.k.m.3601.2 4
40.29 even 2 inner 360.2.d.a.109.2 yes 4
40.37 odd 4 1800.2.k.o.901.2 4
60.23 odd 4 7200.2.k.m.3601.2 4
60.47 odd 4 7200.2.k.m.3601.1 4
60.59 even 2 1440.2.d.a.1009.3 4
120.29 odd 2 CM 360.2.d.a.109.3 yes 4
120.53 even 4 1800.2.k.o.901.1 4
120.59 even 2 1440.2.d.a.1009.2 4
120.77 even 4 1800.2.k.o.901.3 4
120.83 odd 4 7200.2.k.m.3601.3 4
120.107 odd 4 7200.2.k.m.3601.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
360.2.d.a.109.1 ✓ 4 5.4 even 2 inner
360.2.d.a.109.1 ✓ 4 24.5 odd 2 inner
360.2.d.a.109.2 yes 4 3.2 odd 2 inner
360.2.d.a.109.2 yes 4 40.29 even 2 inner
360.2.d.a.109.3 yes 4 1.1 even 1 trivial
360.2.d.a.109.3 yes 4 120.29 odd 2 CM
360.2.d.a.109.4 yes 4 8.5 even 2 inner
360.2.d.a.109.4 yes 4 15.14 odd 2 inner
1440.2.d.a.1009.1 4 20.19 odd 2
1440.2.d.a.1009.1 4 24.11 even 2
1440.2.d.a.1009.2 4 4.3 odd 2
1440.2.d.a.1009.2 4 120.59 even 2
1440.2.d.a.1009.3 4 8.3 odd 2
1440.2.d.a.1009.3 4 60.59 even 2
1440.2.d.a.1009.4 4 12.11 even 2
1440.2.d.a.1009.4 4 40.19 odd 2
1800.2.k.o.901.1 4 5.2 odd 4
1800.2.k.o.901.1 4 120.53 even 4
1800.2.k.o.901.2 4 15.8 even 4
1800.2.k.o.901.2 4 40.37 odd 4
1800.2.k.o.901.3 4 5.3 odd 4
1800.2.k.o.901.3 4 120.77 even 4
1800.2.k.o.901.4 4 15.2 even 4
1800.2.k.o.901.4 4 40.13 odd 4
7200.2.k.m.3601.1 4 40.3 even 4
7200.2.k.m.3601.1 4 60.47 odd 4
7200.2.k.m.3601.2 4 40.27 even 4
7200.2.k.m.3601.2 4 60.23 odd 4
7200.2.k.m.3601.3 4 20.7 even 4
7200.2.k.m.3601.3 4 120.83 odd 4
7200.2.k.m.3601.4 4 20.3 even 4
7200.2.k.m.3601.4 4 120.107 odd 4