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Properties

Label 6300.2.k.p.6049.2
Level $6300$
Weight $2$
Character 6300.6049
Analytic conductor $50.306$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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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(6049,6300)] 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("6300.6049"); 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, 1, 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.k (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,-12,0,0,0,0,0,0,0,0,0,-18, 0,-8,0,0,0,0,0,0,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(41)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(50.3057532734\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 140)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 6049.2
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 6300.6049
Dual form 6300.2.k.p.6049.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000i q^{7} +5.00000 q^{11} -3.00000i q^{13} -1.00000i q^{17} -6.00000 q^{19} -6.00000i q^{23} -9.00000 q^{29} -4.00000 q^{31} -2.00000i q^{37} +4.00000 q^{41} +10.0000i q^{43} -1.00000i q^{47} -1.00000 q^{49} -4.00000i q^{53} -8.00000 q^{59} -8.00000 q^{61} -12.0000i q^{67} -8.00000 q^{71} +2.00000i q^{73} +5.00000i q^{77} -13.0000 q^{79} +4.00000i q^{83} +4.00000 q^{89} +3.00000 q^{91} +13.0000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 10 q^{11} - 12 q^{19} - 18 q^{29} - 8 q^{31} + 8 q^{41} - 2 q^{49} - 16 q^{59} - 16 q^{61} - 16 q^{71} - 26 q^{79} + 8 q^{89} + 6 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6300\mathbb{Z}\right)^\times\).

\(n\) \(2801\) \(3151\) \(3277\) \(3601\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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 0
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 5.00000 1.50756 0.753778 − 0.657129i \(-0.228229\pi\)
0.753778 + 0.657129i \(0.228229\pi\)
\(12\) 0 0
\(13\) − 3.00000i − 0.832050i −0.909353 − 0.416025i \(-0.863423\pi\)
0.909353 − 0.416025i \(-0.136577\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 1.00000i − 0.242536i −0.992620 − 0.121268i \(-0.961304\pi\)
0.992620 − 0.121268i \(-0.0386960\pi\)
\(18\) 0 0
\(19\) −6.00000 −1.37649 −0.688247 − 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 6.00000i − 1.25109i −0.780189 − 0.625543i \(-0.784877\pi\)
0.780189 − 0.625543i \(-0.215123\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −9.00000 −1.67126 −0.835629 − 0.549294i \(-0.814897\pi\)
−0.835629 + 0.549294i \(0.814897\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 − 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 2.00000i − 0.328798i −0.986394 − 0.164399i \(-0.947432\pi\)
0.986394 − 0.164399i \(-0.0525685\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 4.00000 0.624695 0.312348 − 0.949968i \(-0.398885\pi\)
0.312348 + 0.949968i \(0.398885\pi\)
\(42\) 0 0
\(43\) 10.0000i 1.52499i 0.646997 + 0.762493i \(0.276025\pi\)
−0.646997 + 0.762493i \(0.723975\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 1.00000i − 0.145865i −0.997337 − 0.0729325i \(-0.976764\pi\)
0.997337 − 0.0729325i \(-0.0232358\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 4.00000i − 0.549442i −0.961524 − 0.274721i \(-0.911414\pi\)
0.961524 − 0.274721i \(-0.0885855\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −8.00000 −1.04151 −0.520756 − 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 0 0
\(61\) −8.00000 −1.02430 −0.512148 − 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 12.0000i − 1.46603i −0.680211 − 0.733017i \(-0.738112\pi\)
0.680211 − 0.733017i \(-0.261888\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 − 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) 2.00000i 0.234082i 0.993127 + 0.117041i \(0.0373409\pi\)
−0.993127 + 0.117041i \(0.962659\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 5.00000i 0.569803i
\(78\) 0 0
\(79\) −13.0000 −1.46261 −0.731307 − 0.682048i \(-0.761089\pi\)
−0.731307 + 0.682048i \(0.761089\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 4.00000i 0.439057i 0.975606 + 0.219529i \(0.0704519\pi\)
−0.975606 + 0.219529i \(0.929548\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 4.00000 0.423999 0.212000 − 0.977270i \(-0.432002\pi\)
0.212000 + 0.977270i \(0.432002\pi\)
\(90\) 0 0
\(91\) 3.00000 0.314485
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 13.0000i 1.31995i 0.751288 + 0.659975i \(0.229433\pi\)
−0.751288 + 0.659975i \(0.770567\pi\)
\(98\) 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.k.p.6049.2 2
3.2 odd 2 700.2.e.a.449.2 2
5.2 odd 4 1260.2.a.h.1.1 1
5.3 odd 4 6300.2.a.bf.1.1 1
5.4 even 2 inner 6300.2.k.p.6049.1 2
12.11 even 2 2800.2.g.c.449.1 2
15.2 even 4 140.2.a.b.1.1 ✓ 1
15.8 even 4 700.2.a.b.1.1 1
15.14 odd 2 700.2.e.a.449.1 2
20.7 even 4 5040.2.a.bd.1.1 1
21.20 even 2 4900.2.e.a.2549.1 2
35.27 even 4 8820.2.a.n.1.1 1
60.23 odd 4 2800.2.a.be.1.1 1
60.47 odd 4 560.2.a.a.1.1 1
60.59 even 2 2800.2.g.c.449.2 2
105.2 even 12 980.2.i.b.361.1 2
105.17 odd 12 980.2.i.j.961.1 2
105.32 even 12 980.2.i.b.961.1 2
105.47 odd 12 980.2.i.j.361.1 2
105.62 odd 4 980.2.a.b.1.1 1
105.83 odd 4 4900.2.a.u.1.1 1
105.104 even 2 4900.2.e.a.2549.2 2
120.77 even 4 2240.2.a.c.1.1 1
120.107 odd 4 2240.2.a.bb.1.1 1
420.167 even 4 3920.2.a.bl.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
140.2.a.b.1.1 ✓ 1 15.2 even 4
560.2.a.a.1.1 1 60.47 odd 4
700.2.a.b.1.1 1 15.8 even 4
700.2.e.a.449.1 2 15.14 odd 2
700.2.e.a.449.2 2 3.2 odd 2
980.2.a.b.1.1 1 105.62 odd 4
980.2.i.b.361.1 2 105.2 even 12
980.2.i.b.961.1 2 105.32 even 12
980.2.i.j.361.1 2 105.47 odd 12
980.2.i.j.961.1 2 105.17 odd 12
1260.2.a.h.1.1 1 5.2 odd 4
2240.2.a.c.1.1 1 120.77 even 4
2240.2.a.bb.1.1 1 120.107 odd 4
2800.2.a.be.1.1 1 60.23 odd 4
2800.2.g.c.449.1 2 12.11 even 2
2800.2.g.c.449.2 2 60.59 even 2
3920.2.a.bl.1.1 1 420.167 even 4
4900.2.a.u.1.1 1 105.83 odd 4
4900.2.e.a.2549.1 2 21.20 even 2
4900.2.e.a.2549.2 2 105.104 even 2
5040.2.a.bd.1.1 1 20.7 even 4
6300.2.a.bf.1.1 1 5.3 odd 4
6300.2.k.p.6049.1 2 5.4 even 2 inner
6300.2.k.p.6049.2 2 1.1 even 1 trivial
8820.2.a.n.1.1 1 35.27 even 4