Properties

Label 2240.4.a.r.1.1
Level $2240$
Weight $4$
Character 2240.1
Self dual yes
Analytic conductor $132.164$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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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] = [2240,4,Mod(1,2240)] 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("2240.1"); S:= CuspForms(chi, 4); 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, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2240.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,-1,0,5,0,-7,0,-26,0,-65] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(132.164278413\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 70)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2240.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} +5.00000 q^{5} -7.00000 q^{7} -26.0000 q^{9} -65.0000 q^{11} -13.0000 q^{13} -5.00000 q^{15} -73.0000 q^{17} -142.000 q^{19} +7.00000 q^{21} -130.000 q^{23} +25.0000 q^{25} +53.0000 q^{27} -111.000 q^{29} -256.000 q^{31} +65.0000 q^{33} -35.0000 q^{35} +266.000 q^{37} +13.0000 q^{39} -424.000 q^{41} +534.000 q^{43} -130.000 q^{45} +269.000 q^{47} +49.0000 q^{49} +73.0000 q^{51} +132.000 q^{53} -325.000 q^{55} +142.000 q^{57} -224.000 q^{59} +572.000 q^{61} +182.000 q^{63} -65.0000 q^{65} -108.000 q^{67} +130.000 q^{69} -560.000 q^{71} +586.000 q^{73} -25.0000 q^{75} +455.000 q^{77} -57.0000 q^{79} +649.000 q^{81} +252.000 q^{83} -365.000 q^{85} +111.000 q^{87} -184.000 q^{89} +91.0000 q^{91} +256.000 q^{93} -710.000 q^{95} -605.000 q^{97} +1690.00 q^{99} +O(q^{100})\)

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
\(3\) −1.00000 −0.192450 −0.0962250 − 0.995360i \(-0.530677\pi\)
−0.0962250 + 0.995360i \(0.530677\pi\)
\(4\) 0 0
\(5\) 5.00000 0.447214
\(6\) 0 0
\(7\) −7.00000 −0.377964
\(8\) 0 0
\(9\) −26.0000 −0.962963
\(10\) 0 0
\(11\) −65.0000 −1.78166 −0.890829 − 0.454339i \(-0.849876\pi\)
−0.890829 + 0.454339i \(0.849876\pi\)
\(12\) 0 0
\(13\) −13.0000 −0.277350 −0.138675 − 0.990338i \(-0.544284\pi\)
−0.138675 + 0.990338i \(0.544284\pi\)
\(14\) 0 0
\(15\) −5.00000 −0.0860663
\(16\) 0 0
\(17\) −73.0000 −1.04148 −0.520738 − 0.853716i \(-0.674343\pi\)
−0.520738 + 0.853716i \(0.674343\pi\)
\(18\) 0 0
\(19\) −142.000 −1.71458 −0.857290 − 0.514833i \(-0.827854\pi\)
−0.857290 + 0.514833i \(0.827854\pi\)
\(20\) 0 0
\(21\) 7.00000 0.0727393
\(22\) 0 0
\(23\) −130.000 −1.17856 −0.589280 − 0.807929i \(-0.700588\pi\)
−0.589280 + 0.807929i \(0.700588\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) 53.0000 0.377772
\(28\) 0 0
\(29\) −111.000 −0.710765 −0.355382 − 0.934721i \(-0.615649\pi\)
−0.355382 + 0.934721i \(0.615649\pi\)
\(30\) 0 0
\(31\) −256.000 −1.48319 −0.741596 − 0.670847i \(-0.765931\pi\)
−0.741596 + 0.670847i \(0.765931\pi\)
\(32\) 0 0
\(33\) 65.0000 0.342880
\(34\) 0 0
\(35\) −35.0000 −0.169031
\(36\) 0 0
\(37\) 266.000 1.18190 0.590948 − 0.806710i \(-0.298754\pi\)
0.590948 + 0.806710i \(0.298754\pi\)
\(38\) 0 0
\(39\) 13.0000 0.0533761
\(40\) 0 0
\(41\) −424.000 −1.61507 −0.807533 − 0.589823i \(-0.799198\pi\)
−0.807533 + 0.589823i \(0.799198\pi\)
\(42\) 0 0
\(43\) 534.000 1.89382 0.946910 − 0.321500i \(-0.104187\pi\)
0.946910 + 0.321500i \(0.104187\pi\)
\(44\) 0 0
\(45\) −130.000 −0.430650
\(46\) 0 0
\(47\) 269.000 0.834844 0.417422 − 0.908713i \(-0.362934\pi\)
0.417422 + 0.908713i \(0.362934\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) 73.0000 0.200432
\(52\) 0 0
\(53\) 132.000 0.342106 0.171053 − 0.985262i \(-0.445283\pi\)
0.171053 + 0.985262i \(0.445283\pi\)
\(54\) 0 0
\(55\) −325.000 −0.796782
\(56\) 0 0
\(57\) 142.000 0.329971
\(58\) 0 0
\(59\) −224.000 −0.494277 −0.247138 − 0.968980i \(-0.579490\pi\)
−0.247138 + 0.968980i \(0.579490\pi\)
\(60\) 0 0
\(61\) 572.000 1.20061 0.600304 − 0.799772i \(-0.295046\pi\)
0.600304 + 0.799772i \(0.295046\pi\)
\(62\) 0 0
\(63\) 182.000 0.363966
\(64\) 0 0
\(65\) −65.0000 −0.124035
\(66\) 0 0
\(67\) −108.000 −0.196930 −0.0984649 − 0.995141i \(-0.531393\pi\)
−0.0984649 + 0.995141i \(0.531393\pi\)
\(68\) 0 0
\(69\) 130.000 0.226814
\(70\) 0 0
\(71\) −560.000 −0.936053 −0.468027 − 0.883714i \(-0.655035\pi\)
−0.468027 + 0.883714i \(0.655035\pi\)
\(72\) 0 0
\(73\) 586.000 0.939536 0.469768 − 0.882790i \(-0.344338\pi\)
0.469768 + 0.882790i \(0.344338\pi\)
\(74\) 0 0
\(75\) −25.0000 −0.0384900
\(76\) 0 0
\(77\) 455.000 0.673403
\(78\) 0 0
\(79\) −57.0000 −0.0811772 −0.0405886 − 0.999176i \(-0.512923\pi\)
−0.0405886 + 0.999176i \(0.512923\pi\)
\(80\) 0 0
\(81\) 649.000 0.890261
\(82\) 0 0
\(83\) 252.000 0.333260 0.166630 − 0.986019i \(-0.446711\pi\)
0.166630 + 0.986019i \(0.446711\pi\)
\(84\) 0 0
\(85\) −365.000 −0.465762
\(86\) 0 0
\(87\) 111.000 0.136787
\(88\) 0 0
\(89\) −184.000 −0.219146 −0.109573 − 0.993979i \(-0.534948\pi\)
−0.109573 + 0.993979i \(0.534948\pi\)
\(90\) 0 0
\(91\) 91.0000 0.104828
\(92\) 0 0
\(93\) 256.000 0.285440
\(94\) 0 0
\(95\) −710.000 −0.766784
\(96\) 0 0
\(97\) −605.000 −0.633283 −0.316641 − 0.948545i \(-0.602555\pi\)
−0.316641 + 0.948545i \(0.602555\pi\)
\(98\) 0 0
\(99\) 1690.00 1.71567
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 2240.4.a.r.1.1 1
4.3 odd 2 2240.4.a.v.1.1 1
8.3 odd 2 70.4.a.c.1.1 ✓ 1
8.5 even 2 560.4.a.i.1.1 1
24.11 even 2 630.4.a.x.1.1 1
40.3 even 4 350.4.c.h.99.2 2
40.19 odd 2 350.4.a.r.1.1 1
40.27 even 4 350.4.c.h.99.1 2
56.3 even 6 490.4.e.n.471.1 2
56.11 odd 6 490.4.e.o.471.1 2
56.19 even 6 490.4.e.n.361.1 2
56.27 even 2 490.4.a.d.1.1 1
56.51 odd 6 490.4.e.o.361.1 2
280.139 even 2 2450.4.a.bc.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
70.4.a.c.1.1 ✓ 1 8.3 odd 2
350.4.a.r.1.1 1 40.19 odd 2
350.4.c.h.99.1 2 40.27 even 4
350.4.c.h.99.2 2 40.3 even 4
490.4.a.d.1.1 1 56.27 even 2
490.4.e.n.361.1 2 56.19 even 6
490.4.e.n.471.1 2 56.3 even 6
490.4.e.o.361.1 2 56.51 odd 6
490.4.e.o.471.1 2 56.11 odd 6
560.4.a.i.1.1 1 8.5 even 2
630.4.a.x.1.1 1 24.11 even 2
2240.4.a.r.1.1 1 1.1 even 1 trivial
2240.4.a.v.1.1 1 4.3 odd 2
2450.4.a.bc.1.1 1 280.139 even 2