Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1734,2,Mod(829,1734)] 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("1734.829"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1734, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1734 = 2 \cdot 3 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1734.f (of order \(4\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,-4,0,0,0,0,0,0,0,0,8,0,0,4,0,4,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(21)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(13.8460597105\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) 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 102)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 829.2
Root \(0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 1734.829
Dual form 1734.2.f.e.1483.2

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} +(0.707107 + 0.707107i) q^{3} -1.00000 q^{4} +(1.41421 + 1.41421i) q^{5} +(0.707107 - 0.707107i) q^{6} +1.00000i q^{8} +1.00000i q^{9} +(1.41421 - 1.41421i) q^{10} +(-2.82843 + 2.82843i) q^{11} +(-0.707107 - 0.707107i) q^{12} +2.00000 q^{13} +2.00000i q^{15} +1.00000 q^{16} +1.00000 q^{18} -4.00000i q^{19} +(-1.41421 - 1.41421i) q^{20} +(2.82843 + 2.82843i) q^{22} +(-0.707107 + 0.707107i) q^{24} -1.00000i q^{25} -2.00000i q^{26} +(-0.707107 + 0.707107i) q^{27} +(7.07107 + 7.07107i) q^{29} +2.00000 q^{30} +(5.65685 + 5.65685i) q^{31} -1.00000i q^{32} -4.00000 q^{33} -1.00000i q^{36} +(-1.41421 - 1.41421i) q^{37} -4.00000 q^{38} +(1.41421 + 1.41421i) q^{39} +(-1.41421 + 1.41421i) q^{40} +(-7.07107 + 7.07107i) q^{41} +12.0000i q^{43} +(2.82843 - 2.82843i) q^{44} +(-1.41421 + 1.41421i) q^{45} +(0.707107 + 0.707107i) q^{48} +7.00000i q^{49} -1.00000 q^{50} -2.00000 q^{52} -6.00000i q^{53} +(0.707107 + 0.707107i) q^{54} -8.00000 q^{55} +(2.82843 - 2.82843i) q^{57} +(7.07107 - 7.07107i) q^{58} +12.0000i q^{59} -2.00000i q^{60} +(7.07107 - 7.07107i) q^{61} +(5.65685 - 5.65685i) q^{62} -1.00000 q^{64} +(2.82843 + 2.82843i) q^{65} +4.00000i q^{66} -12.0000 q^{67} -1.00000 q^{72} +(-7.07107 - 7.07107i) q^{73} +(-1.41421 + 1.41421i) q^{74} +(0.707107 - 0.707107i) q^{75} +4.00000i q^{76} +(1.41421 - 1.41421i) q^{78} +(-5.65685 + 5.65685i) q^{79} +(1.41421 + 1.41421i) q^{80} -1.00000 q^{81} +(7.07107 + 7.07107i) q^{82} -4.00000i q^{83} +12.0000 q^{86} +10.0000i q^{87} +(-2.82843 - 2.82843i) q^{88} +6.00000 q^{89} +(1.41421 + 1.41421i) q^{90} +8.00000i q^{93} +(5.65685 - 5.65685i) q^{95} +(0.707107 - 0.707107i) q^{96} +(9.89949 + 9.89949i) q^{97} +7.00000 q^{98} +(-2.82843 - 2.82843i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + 8 q^{13} + 4 q^{16} + 4 q^{18} + 8 q^{30} - 16 q^{33} - 16 q^{38} - 4 q^{50} - 8 q^{52} - 32 q^{55} - 4 q^{64} - 48 q^{67} - 4 q^{72} - 4 q^{81} + 48 q^{86} + 24 q^{89} + 28 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1734\mathbb{Z}\right)^\times\).

\(n\) \(1157\) \(1159\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{4}\right)\)

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
\(3\) 0.707107 + 0.707107i 0.408248 + 0.408248i
\(4\) −1.00000 −0.500000
\(5\) 1.41421 + 1.41421i 0.632456 + 0.632456i 0.948683 − 0.316228i \(-0.102416\pi\)
−0.316228 + 0.948683i \(0.602416\pi\)
\(6\) 0.707107 − 0.707107i 0.288675 − 0.288675i
\(7\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 1.00000i 0.333333i
\(10\) 1.41421 − 1.41421i 0.447214 − 0.447214i
\(11\) −2.82843 + 2.82843i −0.852803 + 0.852803i −0.990478 − 0.137675i \(-0.956037\pi\)
0.137675 + 0.990478i \(0.456037\pi\)
\(12\) −0.707107 − 0.707107i −0.204124 − 0.204124i
\(13\) 2.00000 0.554700 0.277350 − 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) 2.00000i 0.516398i
\(16\) 1.00000 0.250000
\(17\) 0 0
\(18\) 1.00000 0.235702
\(19\) − 4.00000i − 0.917663i −0.888523 − 0.458831i \(-0.848268\pi\)
0.888523 − 0.458831i \(-0.151732\pi\)
\(20\) −1.41421 − 1.41421i −0.316228 − 0.316228i
\(21\) 0 0
\(22\) 2.82843 + 2.82843i 0.603023 + 0.603023i
\(23\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(24\) −0.707107 + 0.707107i −0.144338 + 0.144338i
\(25\) − 1.00000i − 0.200000i
\(26\) − 2.00000i − 0.392232i
\(27\) −0.707107 + 0.707107i −0.136083 + 0.136083i
\(28\) 0 0
\(29\) 7.07107 + 7.07107i 1.31306 + 1.31306i 0.919145 + 0.393919i \(0.128881\pi\)
0.393919 + 0.919145i \(0.371119\pi\)
\(30\) 2.00000 0.365148
\(31\) 5.65685 + 5.65685i 1.01600 + 1.01600i 0.999870 + 0.0161311i \(0.00513492\pi\)
0.0161311 + 0.999870i \(0.494865\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) −4.00000 −0.696311
\(34\) 0 0
\(35\) 0 0
\(36\) − 1.00000i − 0.166667i
\(37\) −1.41421 − 1.41421i −0.232495 − 0.232495i 0.581238 − 0.813733i \(-0.302568\pi\)
−0.813733 + 0.581238i \(0.802568\pi\)
\(38\) −4.00000 −0.648886
\(39\) 1.41421 + 1.41421i 0.226455 + 0.226455i
\(40\) −1.41421 + 1.41421i −0.223607 + 0.223607i
\(41\) −7.07107 + 7.07107i −1.10432 + 1.10432i −0.110432 + 0.993884i \(0.535223\pi\)
−0.993884 + 0.110432i \(0.964777\pi\)
\(42\) 0 0
\(43\) 12.0000i 1.82998i 0.403473 + 0.914991i \(0.367803\pi\)
−0.403473 + 0.914991i \(0.632197\pi\)
\(44\) 2.82843 − 2.82843i 0.426401 − 0.426401i
\(45\) −1.41421 + 1.41421i −0.210819 + 0.210819i
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0.707107 + 0.707107i 0.102062 + 0.102062i
\(49\) 7.00000i 1.00000i
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) −2.00000 −0.277350
\(53\) − 6.00000i − 0.824163i −0.911147 − 0.412082i \(-0.864802\pi\)
0.911147 − 0.412082i \(-0.135198\pi\)
\(54\) 0.707107 + 0.707107i 0.0962250 + 0.0962250i
\(55\) −8.00000 −1.07872
\(56\) 0 0
\(57\) 2.82843 − 2.82843i 0.374634 − 0.374634i
\(58\) 7.07107 − 7.07107i 0.928477 − 0.928477i
\(59\) 12.0000i 1.56227i 0.624364 + 0.781133i \(0.285358\pi\)
−0.624364 + 0.781133i \(0.714642\pi\)
\(60\) − 2.00000i − 0.258199i
\(61\) 7.07107 − 7.07107i 0.905357 − 0.905357i −0.0905357 − 0.995893i \(-0.528858\pi\)
0.995893 + 0.0905357i \(0.0288579\pi\)
\(62\) 5.65685 − 5.65685i 0.718421 − 0.718421i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 2.82843 + 2.82843i 0.350823 + 0.350823i
\(66\) 4.00000i 0.492366i
\(67\) −12.0000 −1.46603 −0.733017 − 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(72\) −1.00000 −0.117851
\(73\) −7.07107 − 7.07107i −0.827606 − 0.827606i 0.159579 − 0.987185i \(-0.448986\pi\)
−0.987185 + 0.159579i \(0.948986\pi\)
\(74\) −1.41421 + 1.41421i −0.164399 + 0.164399i
\(75\) 0.707107 − 0.707107i 0.0816497 − 0.0816497i
\(76\) 4.00000i 0.458831i
\(77\) 0 0
\(78\) 1.41421 − 1.41421i 0.160128 − 0.160128i
\(79\) −5.65685 + 5.65685i −0.636446 + 0.636446i −0.949677 − 0.313231i \(-0.898589\pi\)
0.313231 + 0.949677i \(0.398589\pi\)
\(80\) 1.41421 + 1.41421i 0.158114 + 0.158114i
\(81\) −1.00000 −0.111111
\(82\) 7.07107 + 7.07107i 0.780869 + 0.780869i
\(83\) − 4.00000i − 0.439057i −0.975606 − 0.219529i \(-0.929548\pi\)
0.975606 − 0.219529i \(-0.0704519\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 12.0000 1.29399
\(87\) 10.0000i 1.07211i
\(88\) −2.82843 − 2.82843i −0.301511 − 0.301511i
\(89\) 6.00000 0.635999 0.317999 − 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 1.41421 + 1.41421i 0.149071 + 0.149071i
\(91\) 0 0
\(92\) 0 0
\(93\) 8.00000i 0.829561i
\(94\) 0 0
\(95\) 5.65685 − 5.65685i 0.580381 − 0.580381i
\(96\) 0.707107 − 0.707107i 0.0721688 − 0.0721688i
\(97\) 9.89949 + 9.89949i 1.00514 + 1.00514i 0.999987 + 0.00515471i \(0.00164080\pi\)
0.00515471 + 0.999987i \(0.498359\pi\)
\(98\) 7.00000 0.707107
\(99\) −2.82843 − 2.82843i −0.284268 − 0.284268i
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 1734.2.f.e.829.2 4
17.2 even 8 102.2.a.c.1.1 ✓ 1
17.4 even 4 inner 1734.2.f.e.1483.2 4
17.8 even 8 1734.2.b.b.577.1 2
17.9 even 8 1734.2.b.b.577.2 2
17.13 even 4 inner 1734.2.f.e.1483.1 4
17.15 even 8 1734.2.a.j.1.1 1
17.16 even 2 inner 1734.2.f.e.829.1 4
51.2 odd 8 306.2.a.b.1.1 1
51.32 odd 8 5202.2.a.c.1.1 1
68.19 odd 8 816.2.a.b.1.1 1
85.2 odd 8 2550.2.d.m.2449.2 2
85.19 even 8 2550.2.a.c.1.1 1
85.53 odd 8 2550.2.d.m.2449.1 2
119.104 odd 8 4998.2.a.be.1.1 1
136.19 odd 8 3264.2.a.bc.1.1 1
136.53 even 8 3264.2.a.m.1.1 1
204.155 even 8 2448.2.a.p.1.1 1
255.104 odd 8 7650.2.a.ca.1.1 1
408.53 odd 8 9792.2.a.k.1.1 1
408.155 even 8 9792.2.a.l.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
102.2.a.c.1.1 ✓ 1 17.2 even 8
306.2.a.b.1.1 1 51.2 odd 8
816.2.a.b.1.1 1 68.19 odd 8
1734.2.a.j.1.1 1 17.15 even 8
1734.2.b.b.577.1 2 17.8 even 8
1734.2.b.b.577.2 2 17.9 even 8
1734.2.f.e.829.1 4 17.16 even 2 inner
1734.2.f.e.829.2 4 1.1 even 1 trivial
1734.2.f.e.1483.1 4 17.13 even 4 inner
1734.2.f.e.1483.2 4 17.4 even 4 inner
2448.2.a.p.1.1 1 204.155 even 8
2550.2.a.c.1.1 1 85.19 even 8
2550.2.d.m.2449.1 2 85.53 odd 8
2550.2.d.m.2449.2 2 85.2 odd 8
3264.2.a.m.1.1 1 136.53 even 8
3264.2.a.bc.1.1 1 136.19 odd 8
4998.2.a.be.1.1 1 119.104 odd 8
5202.2.a.c.1.1 1 51.32 odd 8
7650.2.a.ca.1.1 1 255.104 odd 8
9792.2.a.k.1.1 1 408.53 odd 8
9792.2.a.l.1.1 1 408.155 even 8