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Properties

Label 2240.2.g.a.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,-4,0,0,0,4,0,-2,0,0,0,-2,0,0,0,8] 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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 280)
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.a.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)
 
\(f(q)\) \(=\) \(q-1.00000i q^{3} +(-2.00000 - 1.00000i) q^{5} +1.00000i q^{7} +2.00000 q^{9} -1.00000 q^{11} -1.00000i q^{13} +(-1.00000 + 2.00000i) q^{15} -3.00000i q^{17} +4.00000 q^{19} +1.00000 q^{21} +2.00000i q^{23} +(3.00000 + 4.00000i) q^{25} -5.00000i q^{27} -1.00000 q^{29} +6.00000 q^{31} +1.00000i q^{33} +(1.00000 - 2.00000i) q^{35} -2.00000i q^{37} -1.00000 q^{39} -10.0000 q^{41} +(-4.00000 - 2.00000i) q^{45} -9.00000i q^{47} -1.00000 q^{49} -3.00000 q^{51} -14.0000i q^{53} +(2.00000 + 1.00000i) q^{55} -4.00000i q^{57} -6.00000 q^{59} +4.00000 q^{61} +2.00000i q^{63} +(-1.00000 + 2.00000i) q^{65} +10.0000i q^{67} +2.00000 q^{69} +16.0000 q^{71} -10.0000i q^{73} +(4.00000 - 3.00000i) q^{75} -1.00000i q^{77} -11.0000 q^{79} +1.00000 q^{81} -4.00000i q^{83} +(-3.00000 + 6.00000i) q^{85} +1.00000i q^{87} -12.0000 q^{89} +1.00000 q^{91} -6.00000i q^{93} +(-8.00000 - 4.00000i) q^{95} -19.0000i q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{5} + 4 q^{9} - 2 q^{11} - 2 q^{15} + 8 q^{19} + 2 q^{21} + 6 q^{25} - 2 q^{29} + 12 q^{31} + 2 q^{35} - 2 q^{39} - 20 q^{41} - 8 q^{45} - 2 q^{49} - 6 q^{51} + 4 q^{55} - 12 q^{59} + 8 q^{61}+ \cdots - 4 q^{99}+O(q^{100}) \) 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
\(3\) − 1.00000i − 0.577350i −0.957427 − 0.288675i \(-0.906785\pi\)
0.957427 − 0.288675i \(-0.0932147\pi\)
\(4\) 0 0
\(5\) −2.00000 − 1.00000i −0.894427 − 0.447214i
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) 2.00000 0.666667
\(10\) 0 0
\(11\) −1.00000 −0.301511 −0.150756 − 0.988571i \(-0.548171\pi\)
−0.150756 + 0.988571i \(0.548171\pi\)
\(12\) 0 0
\(13\) − 1.00000i − 0.277350i −0.990338 − 0.138675i \(-0.955716\pi\)
0.990338 − 0.138675i \(-0.0442844\pi\)
\(14\) 0 0
\(15\) −1.00000 + 2.00000i −0.258199 + 0.516398i
\(16\) 0 0
\(17\) − 3.00000i − 0.727607i −0.931476 − 0.363803i \(-0.881478\pi\)
0.931476 − 0.363803i \(-0.118522\pi\)
\(18\) 0 0
\(19\) 4.00000 0.917663 0.458831 − 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) 2.00000i 0.417029i 0.978019 + 0.208514i \(0.0668628\pi\)
−0.978019 + 0.208514i \(0.933137\pi\)
\(24\) 0 0
\(25\) 3.00000 + 4.00000i 0.600000 + 0.800000i
\(26\) 0 0
\(27\) − 5.00000i − 0.962250i
\(28\) 0 0
\(29\) −1.00000 −0.185695 −0.0928477 − 0.995680i \(-0.529597\pi\)
−0.0928477 + 0.995680i \(0.529597\pi\)
\(30\) 0 0
\(31\) 6.00000 1.07763 0.538816 − 0.842424i \(-0.318872\pi\)
0.538816 + 0.842424i \(0.318872\pi\)
\(32\) 0 0
\(33\) 1.00000i 0.174078i
\(34\) 0 0
\(35\) 1.00000 − 2.00000i 0.169031 − 0.338062i
\(36\) 0 0
\(37\) − 2.00000i − 0.328798i −0.986394 − 0.164399i \(-0.947432\pi\)
0.986394 − 0.164399i \(-0.0525685\pi\)
\(38\) 0 0
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) −10.0000 −1.56174 −0.780869 − 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) −4.00000 − 2.00000i −0.596285 − 0.298142i
\(46\) 0 0
\(47\) − 9.00000i − 1.31278i −0.754420 − 0.656392i \(-0.772082\pi\)
0.754420 − 0.656392i \(-0.227918\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) −3.00000 −0.420084
\(52\) 0 0
\(53\) − 14.0000i − 1.92305i −0.274721 − 0.961524i \(-0.588586\pi\)
0.274721 − 0.961524i \(-0.411414\pi\)
\(54\) 0 0
\(55\) 2.00000 + 1.00000i 0.269680 + 0.134840i
\(56\) 0 0
\(57\) − 4.00000i − 0.529813i
\(58\) 0 0
\(59\) −6.00000 −0.781133 −0.390567 − 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 0 0
\(61\) 4.00000 0.512148 0.256074 − 0.966657i \(-0.417571\pi\)
0.256074 + 0.966657i \(0.417571\pi\)
\(62\) 0 0
\(63\) 2.00000i 0.251976i
\(64\) 0 0
\(65\) −1.00000 + 2.00000i −0.124035 + 0.248069i
\(66\) 0 0
\(67\) 10.0000i 1.22169i 0.791748 + 0.610847i \(0.209171\pi\)
−0.791748 + 0.610847i \(0.790829\pi\)
\(68\) 0 0
\(69\) 2.00000 0.240772
\(70\) 0 0
\(71\) 16.0000 1.89885 0.949425 − 0.313993i \(-0.101667\pi\)
0.949425 + 0.313993i \(0.101667\pi\)
\(72\) 0 0
\(73\) − 10.0000i − 1.17041i −0.810885 − 0.585206i \(-0.801014\pi\)
0.810885 − 0.585206i \(-0.198986\pi\)
\(74\) 0 0
\(75\) 4.00000 − 3.00000i 0.461880 − 0.346410i
\(76\) 0 0
\(77\) − 1.00000i − 0.113961i
\(78\) 0 0
\(79\) −11.0000 −1.23760 −0.618798 − 0.785550i \(-0.712380\pi\)
−0.618798 + 0.785550i \(0.712380\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) − 4.00000i − 0.439057i −0.975606 − 0.219529i \(-0.929548\pi\)
0.975606 − 0.219529i \(-0.0704519\pi\)
\(84\) 0 0
\(85\) −3.00000 + 6.00000i −0.325396 + 0.650791i
\(86\) 0 0
\(87\) 1.00000i 0.107211i
\(88\) 0 0
\(89\) −12.0000 −1.27200 −0.635999 − 0.771690i \(-0.719412\pi\)
−0.635999 + 0.771690i \(0.719412\pi\)
\(90\) 0 0
\(91\) 1.00000 0.104828
\(92\) 0 0
\(93\) − 6.00000i − 0.622171i
\(94\) 0 0
\(95\) −8.00000 − 4.00000i −0.820783 − 0.410391i
\(96\) 0 0
\(97\) − 19.0000i − 1.92916i −0.263795 − 0.964579i \(-0.584974\pi\)
0.263795 − 0.964579i \(-0.415026\pi\)
\(98\) 0 0
\(99\) −2.00000 −0.201008
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.a.449.1 2
4.3 odd 2 2240.2.g.b.449.2 2
5.4 even 2 inner 2240.2.g.a.449.2 2
8.3 odd 2 280.2.g.a.169.1 ✓ 2
8.5 even 2 560.2.g.d.449.2 2
20.19 odd 2 2240.2.g.b.449.1 2
24.5 odd 2 5040.2.t.a.1009.1 2
24.11 even 2 2520.2.t.a.1009.1 2
40.3 even 4 1400.2.a.j.1.1 1
40.13 odd 4 2800.2.a.k.1.1 1
40.19 odd 2 280.2.g.a.169.2 yes 2
40.27 even 4 1400.2.a.d.1.1 1
40.29 even 2 560.2.g.d.449.1 2
40.37 odd 4 2800.2.a.u.1.1 1
56.27 even 2 1960.2.g.a.1569.2 2
120.29 odd 2 5040.2.t.a.1009.2 2
120.59 even 2 2520.2.t.a.1009.2 2
280.27 odd 4 9800.2.a.bb.1.1 1
280.83 odd 4 9800.2.a.p.1.1 1
280.139 even 2 1960.2.g.a.1569.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
280.2.g.a.169.1 ✓ 2 8.3 odd 2
280.2.g.a.169.2 yes 2 40.19 odd 2
560.2.g.d.449.1 2 40.29 even 2
560.2.g.d.449.2 2 8.5 even 2
1400.2.a.d.1.1 1 40.27 even 4
1400.2.a.j.1.1 1 40.3 even 4
1960.2.g.a.1569.1 2 280.139 even 2
1960.2.g.a.1569.2 2 56.27 even 2
2240.2.g.a.449.1 2 1.1 even 1 trivial
2240.2.g.a.449.2 2 5.4 even 2 inner
2240.2.g.b.449.1 2 20.19 odd 2
2240.2.g.b.449.2 2 4.3 odd 2
2520.2.t.a.1009.1 2 24.11 even 2
2520.2.t.a.1009.2 2 120.59 even 2
2800.2.a.k.1.1 1 40.13 odd 4
2800.2.a.u.1.1 1 40.37 odd 4
5040.2.t.a.1009.1 2 24.5 odd 2
5040.2.t.a.1009.2 2 120.29 odd 2
9800.2.a.p.1.1 1 280.83 odd 4
9800.2.a.bb.1.1 1 280.27 odd 4