Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [6300,2,Mod(1,6300)] 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("6300.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6300, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 6300 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6300.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,0,0,0,0,-1,0,0,0,4,0,6,0,0,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(50.3057532734\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 420)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 6300.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{7} +4.00000 q^{11} +6.00000 q^{13} +2.00000 q^{17} +6.00000 q^{19} +2.00000 q^{23} -6.00000 q^{29} -2.00000 q^{31} +4.00000 q^{37} -8.00000 q^{41} +4.00000 q^{43} +4.00000 q^{47} +1.00000 q^{49} +6.00000 q^{53} -4.00000 q^{59} +14.0000 q^{61} -4.00000 q^{67} +10.0000 q^{73} -4.00000 q^{77} -16.0000 q^{83} -8.00000 q^{89} -6.00000 q^{91} -10.0000 q^{97} +O(q^{100})\)

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 0
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 4.00000 1.20605 0.603023 − 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 0 0
\(13\) 6.00000 1.66410 0.832050 − 0.554700i \(-0.187167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.00000 0.485071 0.242536 − 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) 6.00000 1.37649 0.688247 − 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.00000 0.417029 0.208514 − 0.978019i \(-0.433137\pi\)
0.208514 + 0.978019i \(0.433137\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 − 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 − 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 4.00000 0.657596 0.328798 − 0.944400i \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −8.00000 −1.24939 −0.624695 − 0.780869i \(-0.714777\pi\)
−0.624695 + 0.780869i \(0.714777\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 − 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.00000 0.583460 0.291730 − 0.956501i \(-0.405769\pi\)
0.291730 + 0.956501i \(0.405769\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 − 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −4.00000 −0.520756 −0.260378 − 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) 14.0000 1.79252 0.896258 − 0.443533i \(-0.146275\pi\)
0.896258 + 0.443533i \(0.146275\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 10.0000 1.17041 0.585206 − 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −4.00000 −0.455842
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −16.0000 −1.75623 −0.878114 − 0.478451i \(-0.841198\pi\)
−0.878114 + 0.478451i \(0.841198\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −8.00000 −0.847998 −0.423999 − 0.905663i \(-0.639374\pi\)
−0.423999 + 0.905663i \(0.639374\pi\)
\(90\) 0 0
\(91\) −6.00000 −0.628971
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −10.0000 −1.01535 −0.507673 − 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 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 6300.2.a.n.1.1 1
3.2 odd 2 2100.2.a.j.1.1 1
5.2 odd 4 1260.2.k.d.1009.1 2
5.3 odd 4 1260.2.k.d.1009.2 2
5.4 even 2 6300.2.a.bc.1.1 1
12.11 even 2 8400.2.a.bh.1.1 1
15.2 even 4 420.2.k.a.169.1 ✓ 2
15.8 even 4 420.2.k.a.169.2 yes 2
15.14 odd 2 2100.2.a.e.1.1 1
20.3 even 4 5040.2.t.o.1009.2 2
20.7 even 4 5040.2.t.o.1009.1 2
60.23 odd 4 1680.2.t.a.1009.1 2
60.47 odd 4 1680.2.t.a.1009.2 2
60.59 even 2 8400.2.a.cd.1.1 1
105.2 even 12 2940.2.bb.h.949.2 4
105.17 odd 12 2940.2.bb.c.1549.2 4
105.23 even 12 2940.2.bb.h.949.1 4
105.32 even 12 2940.2.bb.h.1549.1 4
105.38 odd 12 2940.2.bb.c.1549.1 4
105.47 odd 12 2940.2.bb.c.949.1 4
105.53 even 12 2940.2.bb.h.1549.2 4
105.62 odd 4 2940.2.k.d.589.2 2
105.68 odd 12 2940.2.bb.c.949.2 4
105.83 odd 4 2940.2.k.d.589.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
420.2.k.a.169.1 ✓ 2 15.2 even 4
420.2.k.a.169.2 yes 2 15.8 even 4
1260.2.k.d.1009.1 2 5.2 odd 4
1260.2.k.d.1009.2 2 5.3 odd 4
1680.2.t.a.1009.1 2 60.23 odd 4
1680.2.t.a.1009.2 2 60.47 odd 4
2100.2.a.e.1.1 1 15.14 odd 2
2100.2.a.j.1.1 1 3.2 odd 2
2940.2.k.d.589.1 2 105.83 odd 4
2940.2.k.d.589.2 2 105.62 odd 4
2940.2.bb.c.949.1 4 105.47 odd 12
2940.2.bb.c.949.2 4 105.68 odd 12
2940.2.bb.c.1549.1 4 105.38 odd 12
2940.2.bb.c.1549.2 4 105.17 odd 12
2940.2.bb.h.949.1 4 105.23 even 12
2940.2.bb.h.949.2 4 105.2 even 12
2940.2.bb.h.1549.1 4 105.32 even 12
2940.2.bb.h.1549.2 4 105.53 even 12
5040.2.t.o.1009.1 2 20.7 even 4
5040.2.t.o.1009.2 2 20.3 even 4
6300.2.a.n.1.1 1 1.1 even 1 trivial
6300.2.a.bc.1.1 1 5.4 even 2
8400.2.a.bh.1.1 1 12.11 even 2
8400.2.a.cd.1.1 1 60.59 even 2