↑

Properties

Label 6192.2.l.g.2321.3
Level $6192$
Weight $2$
Character 6192.2321
Analytic conductor $49.443$
Analytic rank $0$
Dimension $6$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [6192,2,Mod(2321,6192)] 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("6192.2321"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6192, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 6192 = 2^{4} \cdot 3^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6192.l (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0, -12,0,0,0,0,0,0,0,0,0,0,0,-6,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 36] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(65)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(49.4433689316\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.30233088.3
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 6x^{3} + 27 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 774)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2321.3
Root \(-0.352860 - 1.69573i\) of defining polynomial
Character \(\chi\) \(=\) 6192.2321
Dual form 6192.2.l.g.2321.4

$q$-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)
 
\(f(q)\) \(=\) \(q-0.705720 q^{5} -3.39145i q^{7} +1.97724i q^{11} +1.29428 q^{13} +1.41421i q^{17} -4.38949i q^{19} -0.416175i q^{23} -4.50196 q^{25} -8.88676 q^{29} +1.38480 q^{31} +2.39342i q^{35} +1.99608i q^{37} +7.19908i q^{41} +(-6.50196 - 0.851187i) q^{43} +0.979202i q^{47} -4.50196 q^{49} +9.19516i q^{53} -1.39538i q^{55} -9.19516i q^{59} +7.48724i q^{61} -0.913399 q^{65} +2.79624 q^{67} +4.79624 q^{71} -5.78487i q^{73} +6.70572 q^{77} +9.00392 q^{79} -0.0188362i q^{83} -0.998038i q^{85} -5.00392 q^{89} -4.38949i q^{91} +3.09775i q^{95} -10.9134 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 12 q^{13} + 6 q^{25} - 12 q^{31} - 6 q^{43} + 6 q^{49} + 36 q^{65} - 12 q^{67} + 36 q^{77} - 12 q^{79} + 36 q^{89} - 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6192\mathbb{Z}\right)^\times\).

\(n\) \(433\) \(3871\) \(4645\) \(4817\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-1\)

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 0
\(4\) 0 0
\(5\) −0.705720 −0.315607 −0.157804 − 0.987471i \(-0.550441\pi\)
−0.157804 + 0.987471i \(0.550441\pi\)
\(6\) 0 0
\(7\) − 3.39145i − 1.28185i −0.767604 − 0.640925i \(-0.778551\pi\)
0.767604 − 0.640925i \(-0.221449\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.97724i 0.596160i 0.954541 + 0.298080i \(0.0963463\pi\)
−0.954541 + 0.298080i \(0.903654\pi\)
\(12\) 0 0
\(13\) 1.29428 0.358969 0.179484 − 0.983761i \(-0.442557\pi\)
0.179484 + 0.983761i \(0.442557\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.41421i 0.342997i 0.985184 + 0.171499i \(0.0548609\pi\)
−0.985184 + 0.171499i \(0.945139\pi\)
\(18\) 0 0
\(19\) − 4.38949i − 1.00702i −0.863990 − 0.503509i \(-0.832042\pi\)
0.863990 − 0.503509i \(-0.167958\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 0.416175i − 0.0867785i −0.999058 − 0.0433893i \(-0.986184\pi\)
0.999058 − 0.0433893i \(-0.0138156\pi\)
\(24\) 0 0
\(25\) −4.50196 −0.900392
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −8.88676 −1.65023 −0.825115 − 0.564965i \(-0.808890\pi\)
−0.825115 + 0.564965i \(0.808890\pi\)
\(30\) 0 0
\(31\) 1.38480 0.248718 0.124359 − 0.992237i \(-0.460313\pi\)
0.124359 + 0.992237i \(0.460313\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.39342i 0.404561i
\(36\) 0 0
\(37\) 1.99608i 0.328153i 0.986448 + 0.164076i \(0.0524644\pi\)
−0.986448 + 0.164076i \(0.947536\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 7.19908i 1.12431i 0.827033 + 0.562154i \(0.190027\pi\)
−0.827033 + 0.562154i \(0.809973\pi\)
\(42\) 0 0
\(43\) −6.50196 − 0.851187i −0.991540 − 0.129805i
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0.979202i 0.142831i 0.997447 + 0.0714157i \(0.0227517\pi\)
−0.997447 + 0.0714157i \(0.977248\pi\)
\(48\) 0 0
\(49\) −4.50196 −0.643137
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 9.19516i 1.26305i 0.775355 + 0.631526i \(0.217571\pi\)
−0.775355 + 0.631526i \(0.782429\pi\)
\(54\) 0 0
\(55\) − 1.39538i − 0.188153i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 9.19516i − 1.19711i −0.801083 − 0.598554i \(-0.795742\pi\)
0.801083 − 0.598554i \(-0.204258\pi\)
\(60\) 0 0
\(61\) 7.48724i 0.958643i 0.877639 + 0.479322i \(0.159117\pi\)
−0.877639 + 0.479322i \(0.840883\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −0.913399 −0.113293
\(66\) 0 0
\(67\) 2.79624 0.341615 0.170808 − 0.985304i \(-0.445362\pi\)
0.170808 + 0.985304i \(0.445362\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 4.79624 0.569209 0.284604 − 0.958645i \(-0.408138\pi\)
0.284604 + 0.958645i \(0.408138\pi\)
\(72\) 0 0
\(73\) − 5.78487i − 0.677068i −0.940954 − 0.338534i \(-0.890069\pi\)
0.940954 − 0.338534i \(-0.109931\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 6.70572 0.764188
\(78\) 0 0
\(79\) 9.00392 1.01302 0.506510 − 0.862234i \(-0.330935\pi\)
0.506510 + 0.862234i \(0.330935\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 0.0188362i − 0.00206754i −0.999999 − 0.00103377i \(-0.999671\pi\)
0.999999 − 0.00103377i \(-0.000329060\pi\)
\(84\) 0 0
\(85\) − 0.998038i − 0.108252i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −5.00392 −0.530414 −0.265207 − 0.964191i \(-0.585440\pi\)
−0.265207 + 0.964191i \(0.585440\pi\)
\(90\) 0 0
\(91\) − 4.38949i − 0.460144i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 3.09775i 0.317823i
\(96\) 0 0
\(97\) −10.9134 −1.10809 −0.554044 − 0.832487i \(-0.686916\pi\)
−0.554044 + 0.832487i \(0.686916\pi\)
\(98\) 0 0
\(99\) 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 6192.2.l.g.2321.3 6
3.2 odd 2 6192.2.l.f.2321.3 6
4.3 odd 2 774.2.d.a.773.4 yes 6
12.11 even 2 774.2.d.b.773.4 yes 6
43.42 odd 2 6192.2.l.f.2321.4 6
129.128 even 2 inner 6192.2.l.g.2321.4 6
172.171 even 2 774.2.d.b.773.3 yes 6
516.515 odd 2 774.2.d.a.773.3 ✓ 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
774.2.d.a.773.3 ✓ 6 516.515 odd 2
774.2.d.a.773.4 yes 6 4.3 odd 2
774.2.d.b.773.3 yes 6 172.171 even 2
774.2.d.b.773.4 yes 6 12.11 even 2
6192.2.l.f.2321.3 6 3.2 odd 2
6192.2.l.f.2321.4 6 43.42 odd 2
6192.2.l.g.2321.3 6 1.1 even 1 trivial
6192.2.l.g.2321.4 6 129.128 even 2 inner