Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3468,2,Mod(829,3468)] 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("3468.829"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3468, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3468 = 2^{2} \cdot 3 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3468.j (of order \(4\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,0,0,0,0,20,0,0,0,0,0,0,0,0] 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: \(27.6921194210\)
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 204)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 3217.1
Root \(0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 3468.3217
Dual form 3468.2.j.f.829.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.707107 + 0.707107i) q^{3} +(0.707107 - 0.707107i) q^{5} -1.00000i q^{9} +(-3.53553 - 3.53553i) q^{11} +5.00000 q^{13} +1.00000i q^{15} +1.00000i q^{19} +(2.12132 + 2.12132i) q^{23} +4.00000i q^{25} +(0.707107 + 0.707107i) q^{27} +(1.41421 - 1.41421i) q^{29} +(-1.41421 + 1.41421i) q^{31} +5.00000 q^{33} +(5.65685 - 5.65685i) q^{37} +(-3.53553 + 3.53553i) q^{39} +(-3.53553 - 3.53553i) q^{41} +9.00000i q^{43} +(-0.707107 - 0.707107i) q^{45} -6.00000 q^{47} -7.00000i q^{49} -6.00000i q^{53} -5.00000 q^{55} +(-0.707107 - 0.707107i) q^{57} -6.00000i q^{59} +(-2.82843 - 2.82843i) q^{61} +(3.53553 - 3.53553i) q^{65} +12.0000 q^{67} -3.00000 q^{69} +(8.48528 - 8.48528i) q^{71} +(-1.41421 + 1.41421i) q^{73} +(-2.82843 - 2.82843i) q^{75} +(-7.07107 - 7.07107i) q^{79} -1.00000 q^{81} -2.00000i q^{83} +2.00000i q^{87} -12.0000 q^{89} -2.00000i q^{93} +(0.707107 + 0.707107i) q^{95} +(11.3137 - 11.3137i) q^{97} +(-3.53553 + 3.53553i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 20 q^{13} + 20 q^{33} - 24 q^{47} - 20 q^{55} + 48 q^{67} - 12 q^{69} - 4 q^{81} - 48 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3468\mathbb{Z}\right)^\times\).

\(n\) \(1157\) \(1735\) \(2893\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{3}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.707107 + 0.707107i −0.408248 + 0.408248i
\(4\) 0 0
\(5\) 0.707107 − 0.707107i 0.316228 − 0.316228i −0.531089 − 0.847316i \(-0.678217\pi\)
0.847316 + 0.531089i \(0.178217\pi\)
\(6\) 0 0
\(7\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(8\) 0 0
\(9\) − 1.00000i − 0.333333i
\(10\) 0 0
\(11\) −3.53553 − 3.53553i −1.06600 − 1.06600i −0.997662 − 0.0683416i \(-0.978229\pi\)
−0.0683416 − 0.997662i \(-0.521771\pi\)
\(12\) 0 0
\(13\) 5.00000 1.38675 0.693375 − 0.720577i \(-0.256123\pi\)
0.693375 + 0.720577i \(0.256123\pi\)
\(14\) 0 0
\(15\) 1.00000i 0.258199i
\(16\) 0 0
\(17\) 0 0
\(18\) 0 0
\(19\) 1.00000i 0.229416i 0.993399 + 0.114708i \(0.0365932\pi\)
−0.993399 + 0.114708i \(0.963407\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.12132 + 2.12132i 0.442326 + 0.442326i 0.892793 − 0.450467i \(-0.148743\pi\)
−0.450467 + 0.892793i \(0.648743\pi\)
\(24\) 0 0
\(25\) 4.00000i 0.800000i
\(26\) 0 0
\(27\) 0.707107 + 0.707107i 0.136083 + 0.136083i
\(28\) 0 0
\(29\) 1.41421 − 1.41421i 0.262613 − 0.262613i −0.563502 − 0.826115i \(-0.690546\pi\)
0.826115 + 0.563502i \(0.190546\pi\)
\(30\) 0 0
\(31\) −1.41421 + 1.41421i −0.254000 + 0.254000i −0.822608 − 0.568608i \(-0.807482\pi\)
0.568608 + 0.822608i \(0.307482\pi\)
\(32\) 0 0
\(33\) 5.00000 0.870388
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 5.65685 − 5.65685i 0.929981 − 0.929981i −0.0677230 − 0.997704i \(-0.521573\pi\)
0.997704 + 0.0677230i \(0.0215734\pi\)
\(38\) 0 0
\(39\) −3.53553 + 3.53553i −0.566139 + 0.566139i
\(40\) 0 0
\(41\) −3.53553 − 3.53553i −0.552158 − 0.552158i 0.374905 − 0.927063i \(-0.377675\pi\)
−0.927063 + 0.374905i \(0.877675\pi\)
\(42\) 0 0
\(43\) 9.00000i 1.37249i 0.727372 + 0.686244i \(0.240742\pi\)
−0.727372 + 0.686244i \(0.759258\pi\)
\(44\) 0 0
\(45\) −0.707107 − 0.707107i −0.105409 − 0.105409i
\(46\) 0 0
\(47\) −6.00000 −0.875190 −0.437595 − 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) 0 0
\(49\) − 7.00000i − 1.00000i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 6.00000i − 0.824163i −0.911147 − 0.412082i \(-0.864802\pi\)
0.911147 − 0.412082i \(-0.135198\pi\)
\(54\) 0 0
\(55\) −5.00000 −0.674200
\(56\) 0 0
\(57\) −0.707107 − 0.707107i −0.0936586 − 0.0936586i
\(58\) 0 0
\(59\) − 6.00000i − 0.781133i −0.920575 − 0.390567i \(-0.872279\pi\)
0.920575 − 0.390567i \(-0.127721\pi\)
\(60\) 0 0
\(61\) −2.82843 − 2.82843i −0.362143 − 0.362143i 0.502458 − 0.864601i \(-0.332429\pi\)
−0.864601 + 0.502458i \(0.832429\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 3.53553 − 3.53553i 0.438529 − 0.438529i
\(66\) 0 0
\(67\) 12.0000 1.46603 0.733017 − 0.680211i \(-0.238112\pi\)
0.733017 + 0.680211i \(0.238112\pi\)
\(68\) 0 0
\(69\) −3.00000 −0.361158
\(70\) 0 0
\(71\) 8.48528 − 8.48528i 1.00702 − 1.00702i 0.00704243 − 0.999975i \(-0.497758\pi\)
0.999975 − 0.00704243i \(-0.00224169\pi\)
\(72\) 0 0
\(73\) −1.41421 + 1.41421i −0.165521 + 0.165521i −0.785007 − 0.619486i \(-0.787341\pi\)
0.619486 + 0.785007i \(0.287341\pi\)
\(74\) 0 0
\(75\) −2.82843 − 2.82843i −0.326599 − 0.326599i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −7.07107 − 7.07107i −0.795557 − 0.795557i 0.186834 − 0.982391i \(-0.440177\pi\)
−0.982391 + 0.186834i \(0.940177\pi\)
\(80\) 0 0
\(81\) −1.00000 −0.111111
\(82\) 0 0
\(83\) − 2.00000i − 0.219529i −0.993958 − 0.109764i \(-0.964990\pi\)
0.993958 − 0.109764i \(-0.0350096\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 2.00000i 0.214423i
\(88\) 0 0
\(89\) −12.0000 −1.27200 −0.635999 − 0.771690i \(-0.719412\pi\)
−0.635999 + 0.771690i \(0.719412\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) − 2.00000i − 0.207390i
\(94\) 0 0
\(95\) 0.707107 + 0.707107i 0.0725476 + 0.0725476i
\(96\) 0 0
\(97\) 11.3137 − 11.3137i 1.14873 − 1.14873i 0.161931 − 0.986802i \(-0.448228\pi\)
0.986802 − 0.161931i \(-0.0517722\pi\)
\(98\) 0 0
\(99\) −3.53553 + 3.53553i −0.355335 + 0.355335i
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 3468.2.j.f.3217.1 4
17.2 even 8 3468.2.b.a.577.2 2
17.4 even 4 inner 3468.2.j.f.829.2 4
17.8 even 8 204.2.a.b.1.1 ✓ 1
17.9 even 8 3468.2.a.a.1.1 1
17.13 even 4 inner 3468.2.j.f.829.1 4
17.15 even 8 3468.2.b.a.577.1 2
17.16 even 2 inner 3468.2.j.f.3217.2 4
51.8 odd 8 612.2.a.b.1.1 1
68.59 odd 8 816.2.a.e.1.1 1
85.8 odd 8 5100.2.g.l.2449.2 2
85.42 odd 8 5100.2.g.l.2449.1 2
85.59 even 8 5100.2.a.f.1.1 1
119.76 odd 8 9996.2.a.b.1.1 1
136.59 odd 8 3264.2.a.u.1.1 1
136.93 even 8 3264.2.a.f.1.1 1
204.59 even 8 2448.2.a.e.1.1 1
408.59 even 8 9792.2.a.bn.1.1 1
408.365 odd 8 9792.2.a.bo.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
204.2.a.b.1.1 ✓ 1 17.8 even 8
612.2.a.b.1.1 1 51.8 odd 8
816.2.a.e.1.1 1 68.59 odd 8
2448.2.a.e.1.1 1 204.59 even 8
3264.2.a.f.1.1 1 136.93 even 8
3264.2.a.u.1.1 1 136.59 odd 8
3468.2.a.a.1.1 1 17.9 even 8
3468.2.b.a.577.1 2 17.15 even 8
3468.2.b.a.577.2 2 17.2 even 8
3468.2.j.f.829.1 4 17.13 even 4 inner
3468.2.j.f.829.2 4 17.4 even 4 inner
3468.2.j.f.3217.1 4 1.1 even 1 trivial
3468.2.j.f.3217.2 4 17.16 even 2 inner
5100.2.a.f.1.1 1 85.59 even 8
5100.2.g.l.2449.1 2 85.42 odd 8
5100.2.g.l.2449.2 2 85.8 odd 8
9792.2.a.bn.1.1 1 408.59 even 8
9792.2.a.bo.1.1 1 408.365 odd 8
9996.2.a.b.1.1 1 119.76 odd 8