Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1776,2,Mod(961,1776)] 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("1776.961"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1776, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1776 = 2^{4} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1776.h (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,4,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(14.1814313990\)
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: \( x^{4} + 6x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 444)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 961.2
Root \(0.874032i\) of defining polynomial
Character \(\chi\) \(=\) 1776.961
Dual form 1776.2.h.f.961.3

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -0.540182i q^{5} +1.23607 q^{7} +1.00000 q^{9} -2.47214 q^{11} -4.57649i q^{13} -0.540182i q^{15} -3.36861i q^{17} -1.74806i q^{19} +1.23607 q^{21} +8.61280i q^{23} +4.70820 q^{25} +1.00000 q^{27} -7.94510i q^{29} -6.32456i q^{31} -2.47214 q^{33} -0.667701i q^{35} +(-2.23607 - 5.65685i) q^{37} -4.57649i q^{39} +2.00000 q^{41} -2.82843i q^{43} -0.540182i q^{45} -4.00000 q^{47} -5.47214 q^{49} -3.36861i q^{51} +10.9443 q^{53} +1.33540i q^{55} -1.74806i q^{57} +2.28825i q^{59} +5.65685i q^{61} +1.23607 q^{63} -2.47214 q^{65} +7.70820 q^{67} +8.61280i q^{69} +8.94427 q^{71} +3.23607 q^{73} +4.70820 q^{75} -3.05573 q^{77} -9.56564i q^{79} +1.00000 q^{81} -1.52786 q^{83} -1.81966 q^{85} -7.94510i q^{87} +8.61280i q^{89} -5.65685i q^{91} -6.32456i q^{93} -0.944272 q^{95} +13.7295i q^{97} -2.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} - 4 q^{7} + 4 q^{9} + 8 q^{11} - 4 q^{21} - 8 q^{25} + 4 q^{27} + 8 q^{33} + 8 q^{41} - 16 q^{47} - 4 q^{49} + 8 q^{53} - 4 q^{63} + 8 q^{65} + 4 q^{67} + 4 q^{73} - 8 q^{75} - 48 q^{77} + 4 q^{81}+ \cdots + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1776\mathbb{Z}\right)^\times\).

\(n\) \(223\) \(593\) \(1297\) \(1333\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) − 0.540182i − 0.241577i −0.992678 − 0.120788i \(-0.961458\pi\)
0.992678 − 0.120788i \(-0.0385422\pi\)
\(6\) 0 0
\(7\) 1.23607 0.467190 0.233595 − 0.972334i \(-0.424951\pi\)
0.233595 + 0.972334i \(0.424951\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.47214 −0.745377 −0.372689 − 0.927957i \(-0.621564\pi\)
−0.372689 + 0.927957i \(0.621564\pi\)
\(12\) 0 0
\(13\) − 4.57649i − 1.26929i −0.772804 − 0.634645i \(-0.781146\pi\)
0.772804 − 0.634645i \(-0.218854\pi\)
\(14\) 0 0
\(15\) − 0.540182i − 0.139474i
\(16\) 0 0
\(17\) − 3.36861i − 0.817008i −0.912757 − 0.408504i \(-0.866051\pi\)
0.912757 − 0.408504i \(-0.133949\pi\)
\(18\) 0 0
\(19\) − 1.74806i − 0.401033i −0.979690 − 0.200517i \(-0.935738\pi\)
0.979690 − 0.200517i \(-0.0642621\pi\)
\(20\) 0 0
\(21\) 1.23607 0.269732
\(22\) 0 0
\(23\) 8.61280i 1.79589i 0.440104 + 0.897947i \(0.354941\pi\)
−0.440104 + 0.897947i \(0.645059\pi\)
\(24\) 0 0
\(25\) 4.70820 0.941641
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) − 7.94510i − 1.47537i −0.675146 − 0.737684i \(-0.735919\pi\)
0.675146 − 0.737684i \(-0.264081\pi\)
\(30\) 0 0
\(31\) − 6.32456i − 1.13592i −0.823055 − 0.567962i \(-0.807732\pi\)
0.823055 − 0.567962i \(-0.192268\pi\)
\(32\) 0 0
\(33\) −2.47214 −0.430344
\(34\) 0 0
\(35\) − 0.667701i − 0.112862i
\(36\) 0 0
\(37\) −2.23607 − 5.65685i −0.367607 − 0.929981i
\(38\) 0 0
\(39\) − 4.57649i − 0.732825i
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 − 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) − 2.82843i − 0.431331i −0.976467 − 0.215666i \(-0.930808\pi\)
0.976467 − 0.215666i \(-0.0691921\pi\)
\(44\) 0 0
\(45\) − 0.540182i − 0.0805255i
\(46\) 0 0
\(47\) −4.00000 −0.583460 −0.291730 − 0.956501i \(-0.594231\pi\)
−0.291730 + 0.956501i \(0.594231\pi\)
\(48\) 0 0
\(49\) −5.47214 −0.781734
\(50\) 0 0
\(51\) − 3.36861i − 0.471700i
\(52\) 0 0
\(53\) 10.9443 1.50331 0.751656 − 0.659556i \(-0.229256\pi\)
0.751656 + 0.659556i \(0.229256\pi\)
\(54\) 0 0
\(55\) 1.33540i 0.180066i
\(56\) 0 0
\(57\) − 1.74806i − 0.231537i
\(58\) 0 0
\(59\) 2.28825i 0.297904i 0.988844 + 0.148952i \(0.0475900\pi\)
−0.988844 + 0.148952i \(0.952410\pi\)
\(60\) 0 0
\(61\) 5.65685i 0.724286i 0.932123 + 0.362143i \(0.117955\pi\)
−0.932123 + 0.362143i \(0.882045\pi\)
\(62\) 0 0
\(63\) 1.23607 0.155730
\(64\) 0 0
\(65\) −2.47214 −0.306631
\(66\) 0 0
\(67\) 7.70820 0.941707 0.470853 − 0.882211i \(-0.343946\pi\)
0.470853 + 0.882211i \(0.343946\pi\)
\(68\) 0 0
\(69\) 8.61280i 1.03686i
\(70\) 0 0
\(71\) 8.94427 1.06149 0.530745 − 0.847532i \(-0.321912\pi\)
0.530745 + 0.847532i \(0.321912\pi\)
\(72\) 0 0
\(73\) 3.23607 0.378753 0.189377 − 0.981905i \(-0.439353\pi\)
0.189377 + 0.981905i \(0.439353\pi\)
\(74\) 0 0
\(75\) 4.70820 0.543657
\(76\) 0 0
\(77\) −3.05573 −0.348233
\(78\) 0 0
\(79\) − 9.56564i − 1.07622i −0.842875 − 0.538110i \(-0.819139\pi\)
0.842875 − 0.538110i \(-0.180861\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −1.52786 −0.167705 −0.0838524 − 0.996478i \(-0.526722\pi\)
−0.0838524 + 0.996478i \(0.526722\pi\)
\(84\) 0 0
\(85\) −1.81966 −0.197370
\(86\) 0 0
\(87\) − 7.94510i − 0.851804i
\(88\) 0 0
\(89\) 8.61280i 0.912955i 0.889735 + 0.456478i \(0.150889\pi\)
−0.889735 + 0.456478i \(0.849111\pi\)
\(90\) 0 0
\(91\) − 5.65685i − 0.592999i
\(92\) 0 0
\(93\) − 6.32456i − 0.655826i
\(94\) 0 0
\(95\) −0.944272 −0.0968803
\(96\) 0 0
\(97\) 13.7295i 1.39402i 0.717063 + 0.697008i \(0.245486\pi\)
−0.717063 + 0.697008i \(0.754514\pi\)
\(98\) 0 0
\(99\) −2.47214 −0.248459
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 1776.2.h.f.961.2 4
3.2 odd 2 5328.2.h.g.2737.3 4
4.3 odd 2 444.2.e.a.73.2 ✓ 4
12.11 even 2 1332.2.e.e.73.3 4
37.36 even 2 inner 1776.2.h.f.961.3 4
111.110 odd 2 5328.2.h.g.2737.2 4
148.147 odd 2 444.2.e.a.73.3 yes 4
444.443 even 2 1332.2.e.e.73.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
444.2.e.a.73.2 ✓ 4 4.3 odd 2
444.2.e.a.73.3 yes 4 148.147 odd 2
1332.2.e.e.73.2 4 444.443 even 2
1332.2.e.e.73.3 4 12.11 even 2
1776.2.h.f.961.2 4 1.1 even 1 trivial
1776.2.h.f.961.3 4 37.36 even 2 inner
5328.2.h.g.2737.2 4 111.110 odd 2
5328.2.h.g.2737.3 4 3.2 odd 2