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Properties

Label 2240.2.g.c.449.1
Level $2240$
Weight $2$
Character 2240.449
Analytic conductor $17.886$
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] = [2240,2,Mod(449,2240)] 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("2240.449"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2240, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2240.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,-2,0,0,0,6,0,0,0,0,0,0,0,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(17.8864900528\)
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 449.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 2240.449
Dual form 2240.2.g.c.449.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.00000 - 2.00000i) q^{5} -1.00000i q^{7} +3.00000 q^{9} +4.00000i q^{13} +4.00000i q^{17} -4.00000 q^{19} +8.00000i q^{23} +(-3.00000 + 4.00000i) q^{25} +2.00000 q^{29} +8.00000 q^{31} +(-2.00000 + 1.00000i) q^{35} +8.00000i q^{37} +6.00000 q^{41} -8.00000i q^{43} +(-3.00000 - 6.00000i) q^{45} -8.00000i q^{47} -1.00000 q^{49} +4.00000 q^{59} +6.00000 q^{61} -3.00000i q^{63} +(8.00000 - 4.00000i) q^{65} +8.00000i q^{67} -12.0000 q^{71} +4.00000i q^{73} -4.00000 q^{79} +9.00000 q^{81} +(8.00000 - 4.00000i) q^{85} +10.0000 q^{89} +4.00000 q^{91} +(4.00000 + 8.00000i) q^{95} -12.0000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} + 6 q^{9} - 8 q^{19} - 6 q^{25} + 4 q^{29} + 16 q^{31} - 4 q^{35} + 12 q^{41} - 6 q^{45} - 2 q^{49} + 8 q^{59} + 12 q^{61} + 16 q^{65} - 24 q^{71} - 8 q^{79} + 18 q^{81} + 16 q^{85} + 20 q^{89}+ \cdots + 8 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2240\mathbb{Z}\right)^\times\).

\(n\) \(897\) \(1471\) \(1541\) \(1921\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) 0 0
\(5\) −1.00000 − 2.00000i −0.447214 − 0.894427i
\(6\) 0 0
\(7\) − 1.00000i − 0.377964i
\(8\) 0 0
\(9\) 3.00000 1.00000
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 4.00000i 1.10940i 0.832050 + 0.554700i \(0.187167\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.00000i 0.970143i 0.874475 + 0.485071i \(0.161206\pi\)
−0.874475 + 0.485071i \(0.838794\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 − 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 8.00000i 1.66812i 0.551677 + 0.834058i \(0.313988\pi\)
−0.551677 + 0.834058i \(0.686012\pi\)
\(24\) 0 0
\(25\) −3.00000 + 4.00000i −0.600000 + 0.800000i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 2.00000 0.371391 0.185695 − 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 − 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.00000 + 1.00000i −0.338062 + 0.169031i
\(36\) 0 0
\(37\) 8.00000i 1.31519i 0.753371 + 0.657596i \(0.228427\pi\)
−0.753371 + 0.657596i \(0.771573\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 − 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) − 8.00000i − 1.21999i −0.792406 − 0.609994i \(-0.791172\pi\)
0.792406 − 0.609994i \(-0.208828\pi\)
\(44\) 0 0
\(45\) −3.00000 − 6.00000i −0.447214 − 0.894427i
\(46\) 0 0
\(47\) − 8.00000i − 1.16692i −0.812142 − 0.583460i \(-0.801699\pi\)
0.812142 − 0.583460i \(-0.198301\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.00000 0.520756 0.260378 − 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 − 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 0 0
\(63\) − 3.00000i − 0.377964i
\(64\) 0 0
\(65\) 8.00000 − 4.00000i 0.992278 − 0.496139i
\(66\) 0 0
\(67\) 8.00000i 0.977356i 0.872464 + 0.488678i \(0.162521\pi\)
−0.872464 + 0.488678i \(0.837479\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 − 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 0 0
\(73\) 4.00000i 0.468165i 0.972217 + 0.234082i \(0.0752085\pi\)
−0.972217 + 0.234082i \(0.924791\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 − 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 8.00000 − 4.00000i 0.867722 − 0.433861i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 10.0000 1.06000 0.529999 − 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 4.00000 + 8.00000i 0.410391 + 0.820783i
\(96\) 0 0
\(97\) − 12.0000i − 1.21842i −0.793011 − 0.609208i \(-0.791488\pi\)
0.793011 − 0.609208i \(-0.208512\pi\)
\(98\) 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 2240.2.g.c.449.1 2
4.3 odd 2 2240.2.g.d.449.1 2
5.4 even 2 inner 2240.2.g.c.449.2 2
8.3 odd 2 140.2.e.b.29.2 yes 2
8.5 even 2 560.2.g.c.449.2 2
20.19 odd 2 2240.2.g.d.449.2 2
24.5 odd 2 5040.2.t.g.1009.1 2
24.11 even 2 1260.2.k.b.1009.1 2
40.3 even 4 700.2.a.h.1.1 1
40.13 odd 4 2800.2.a.o.1.1 1
40.19 odd 2 140.2.e.b.29.1 ✓ 2
40.27 even 4 700.2.a.f.1.1 1
40.29 even 2 560.2.g.c.449.1 2
40.37 odd 4 2800.2.a.s.1.1 1
56.3 even 6 980.2.q.e.569.2 4
56.11 odd 6 980.2.q.d.569.1 4
56.19 even 6 980.2.q.e.949.1 4
56.27 even 2 980.2.e.a.589.1 2
56.51 odd 6 980.2.q.d.949.2 4
120.29 odd 2 5040.2.t.g.1009.2 2
120.59 even 2 1260.2.k.b.1009.2 2
120.83 odd 4 6300.2.a.y.1.1 1
120.107 odd 4 6300.2.a.g.1.1 1
280.19 even 6 980.2.q.e.949.2 4
280.27 odd 4 4900.2.a.m.1.1 1
280.59 even 6 980.2.q.e.569.1 4
280.83 odd 4 4900.2.a.l.1.1 1
280.139 even 2 980.2.e.a.589.2 2
280.179 odd 6 980.2.q.d.569.2 4
280.219 odd 6 980.2.q.d.949.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
140.2.e.b.29.1 ✓ 2 40.19 odd 2
140.2.e.b.29.2 yes 2 8.3 odd 2
560.2.g.c.449.1 2 40.29 even 2
560.2.g.c.449.2 2 8.5 even 2
700.2.a.f.1.1 1 40.27 even 4
700.2.a.h.1.1 1 40.3 even 4
980.2.e.a.589.1 2 56.27 even 2
980.2.e.a.589.2 2 280.139 even 2
980.2.q.d.569.1 4 56.11 odd 6
980.2.q.d.569.2 4 280.179 odd 6
980.2.q.d.949.1 4 280.219 odd 6
980.2.q.d.949.2 4 56.51 odd 6
980.2.q.e.569.1 4 280.59 even 6
980.2.q.e.569.2 4 56.3 even 6
980.2.q.e.949.1 4 56.19 even 6
980.2.q.e.949.2 4 280.19 even 6
1260.2.k.b.1009.1 2 24.11 even 2
1260.2.k.b.1009.2 2 120.59 even 2
2240.2.g.c.449.1 2 1.1 even 1 trivial
2240.2.g.c.449.2 2 5.4 even 2 inner
2240.2.g.d.449.1 2 4.3 odd 2
2240.2.g.d.449.2 2 20.19 odd 2
2800.2.a.o.1.1 1 40.13 odd 4
2800.2.a.s.1.1 1 40.37 odd 4
4900.2.a.l.1.1 1 280.83 odd 4
4900.2.a.m.1.1 1 280.27 odd 4
5040.2.t.g.1009.1 2 24.5 odd 2
5040.2.t.g.1009.2 2 120.29 odd 2
6300.2.a.g.1.1 1 120.107 odd 4
6300.2.a.y.1.1 1 120.83 odd 4