Properties

Label 2450.4.a.bc.1.1
Level $2450$
Weight $4$
Character 2450.1
Self dual yes
Analytic conductor $144.555$
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] = [2450,4,Mod(1,2450)] 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("2450.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2450, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2450.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,2,-1,4,0,-2,0,8,-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: \(144.554679514\)
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 \(\chi\) \(=\) 2450.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.00000 q^{2} -1.00000 q^{3} +4.00000 q^{4} -2.00000 q^{6} +8.00000 q^{8} -26.0000 q^{9} -65.0000 q^{11} -4.00000 q^{12} +13.0000 q^{13} +16.0000 q^{16} -73.0000 q^{17} -52.0000 q^{18} +142.000 q^{19} -130.000 q^{22} -130.000 q^{23} -8.00000 q^{24} +26.0000 q^{26} +53.0000 q^{27} +111.000 q^{29} -256.000 q^{31} +32.0000 q^{32} +65.0000 q^{33} -146.000 q^{34} -104.000 q^{36} +266.000 q^{37} +284.000 q^{38} -13.0000 q^{39} +424.000 q^{41} -534.000 q^{43} -260.000 q^{44} -260.000 q^{46} -269.000 q^{47} -16.0000 q^{48} +73.0000 q^{51} +52.0000 q^{52} +132.000 q^{53} +106.000 q^{54} -142.000 q^{57} +222.000 q^{58} +224.000 q^{59} +572.000 q^{61} -512.000 q^{62} +64.0000 q^{64} +130.000 q^{66} +108.000 q^{67} -292.000 q^{68} +130.000 q^{69} +560.000 q^{71} -208.000 q^{72} +586.000 q^{73} +532.000 q^{74} +568.000 q^{76} -26.0000 q^{78} +57.0000 q^{79} +649.000 q^{81} +848.000 q^{82} +252.000 q^{83} -1068.00 q^{86} -111.000 q^{87} -520.000 q^{88} +184.000 q^{89} -520.000 q^{92} +256.000 q^{93} -538.000 q^{94} -32.0000 q^{96} -605.000 q^{97} +1690.00 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 0.707107
\(3\) −1.00000 −0.192450 −0.0962250 − 0.995360i \(-0.530677\pi\)
−0.0962250 + 0.995360i \(0.530677\pi\)
\(4\) 4.00000 0.500000
\(5\) 0 0
\(6\) −2.00000 −0.136083
\(7\) 0 0
\(8\) 8.00000 0.353553
\(9\) −26.0000 −0.962963
\(10\) 0 0
\(11\) −65.0000 −1.78166 −0.890829 − 0.454339i \(-0.849876\pi\)
−0.890829 + 0.454339i \(0.849876\pi\)
\(12\) −4.00000 −0.0962250
\(13\) 13.0000 0.277350 0.138675 − 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) −73.0000 −1.04148 −0.520738 − 0.853716i \(-0.674343\pi\)
−0.520738 + 0.853716i \(0.674343\pi\)
\(18\) −52.0000 −0.680918
\(19\) 142.000 1.71458 0.857290 − 0.514833i \(-0.172146\pi\)
0.857290 + 0.514833i \(0.172146\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −130.000 −1.25982
\(23\) −130.000 −1.17856 −0.589280 − 0.807929i \(-0.700588\pi\)
−0.589280 + 0.807929i \(0.700588\pi\)
\(24\) −8.00000 −0.0680414
\(25\) 0 0
\(26\) 26.0000 0.196116
\(27\) 53.0000 0.377772
\(28\) 0 0
\(29\) 111.000 0.710765 0.355382 − 0.934721i \(-0.384351\pi\)
0.355382 + 0.934721i \(0.384351\pi\)
\(30\) 0 0
\(31\) −256.000 −1.48319 −0.741596 − 0.670847i \(-0.765931\pi\)
−0.741596 + 0.670847i \(0.765931\pi\)
\(32\) 32.0000 0.176777
\(33\) 65.0000 0.342880
\(34\) −146.000 −0.736435
\(35\) 0 0
\(36\) −104.000 −0.481481
\(37\) 266.000 1.18190 0.590948 − 0.806710i \(-0.298754\pi\)
0.590948 + 0.806710i \(0.298754\pi\)
\(38\) 284.000 1.21239
\(39\) −13.0000 −0.0533761
\(40\) 0 0
\(41\) 424.000 1.61507 0.807533 − 0.589823i \(-0.200802\pi\)
0.807533 + 0.589823i \(0.200802\pi\)
\(42\) 0 0
\(43\) −534.000 −1.89382 −0.946910 − 0.321500i \(-0.895813\pi\)
−0.946910 + 0.321500i \(0.895813\pi\)
\(44\) −260.000 −0.890829
\(45\) 0 0
\(46\) −260.000 −0.833368
\(47\) −269.000 −0.834844 −0.417422 − 0.908713i \(-0.637066\pi\)
−0.417422 + 0.908713i \(0.637066\pi\)
\(48\) −16.0000 −0.0481125
\(49\) 0 0
\(50\) 0 0
\(51\) 73.0000 0.200432
\(52\) 52.0000 0.138675
\(53\) 132.000 0.342106 0.171053 − 0.985262i \(-0.445283\pi\)
0.171053 + 0.985262i \(0.445283\pi\)
\(54\) 106.000 0.267125
\(55\) 0 0
\(56\) 0 0
\(57\) −142.000 −0.329971
\(58\) 222.000 0.502587
\(59\) 224.000 0.494277 0.247138 − 0.968980i \(-0.420510\pi\)
0.247138 + 0.968980i \(0.420510\pi\)
\(60\) 0 0
\(61\) 572.000 1.20061 0.600304 − 0.799772i \(-0.295046\pi\)
0.600304 + 0.799772i \(0.295046\pi\)
\(62\) −512.000 −1.04878
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) 0 0
\(66\) 130.000 0.242453
\(67\) 108.000 0.196930 0.0984649 − 0.995141i \(-0.468607\pi\)
0.0984649 + 0.995141i \(0.468607\pi\)
\(68\) −292.000 −0.520738
\(69\) 130.000 0.226814
\(70\) 0 0
\(71\) 560.000 0.936053 0.468027 − 0.883714i \(-0.344965\pi\)
0.468027 + 0.883714i \(0.344965\pi\)
\(72\) −208.000 −0.340459
\(73\) 586.000 0.939536 0.469768 − 0.882790i \(-0.344338\pi\)
0.469768 + 0.882790i \(0.344338\pi\)
\(74\) 532.000 0.835726
\(75\) 0 0
\(76\) 568.000 0.857290
\(77\) 0 0
\(78\) −26.0000 −0.0377426
\(79\) 57.0000 0.0811772 0.0405886 − 0.999176i \(-0.487077\pi\)
0.0405886 + 0.999176i \(0.487077\pi\)
\(80\) 0 0
\(81\) 649.000 0.890261
\(82\) 848.000 1.14202
\(83\) 252.000 0.333260 0.166630 − 0.986019i \(-0.446711\pi\)
0.166630 + 0.986019i \(0.446711\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −1068.00 −1.33913
\(87\) −111.000 −0.136787
\(88\) −520.000 −0.629911
\(89\) 184.000 0.219146 0.109573 − 0.993979i \(-0.465052\pi\)
0.109573 + 0.993979i \(0.465052\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −520.000 −0.589280
\(93\) 256.000 0.285440
\(94\) −538.000 −0.590324
\(95\) 0 0
\(96\) −32.0000 −0.0340207
\(97\) −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 2450.4.a.bc.1.1 1
5.4 even 2 490.4.a.d.1.1 1
7.6 odd 2 350.4.a.r.1.1 1
35.4 even 6 490.4.e.n.471.1 2
35.9 even 6 490.4.e.n.361.1 2
35.13 even 4 350.4.c.h.99.1 2
35.19 odd 6 490.4.e.o.361.1 2
35.24 odd 6 490.4.e.o.471.1 2
35.27 even 4 350.4.c.h.99.2 2
35.34 odd 2 70.4.a.c.1.1 ✓ 1
105.104 even 2 630.4.a.x.1.1 1
140.139 even 2 560.4.a.i.1.1 1
280.69 odd 2 2240.4.a.v.1.1 1
280.139 even 2 2240.4.a.r.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
70.4.a.c.1.1 ✓ 1 35.34 odd 2
350.4.a.r.1.1 1 7.6 odd 2
350.4.c.h.99.1 2 35.13 even 4
350.4.c.h.99.2 2 35.27 even 4
490.4.a.d.1.1 1 5.4 even 2
490.4.e.n.361.1 2 35.9 even 6
490.4.e.n.471.1 2 35.4 even 6
490.4.e.o.361.1 2 35.19 odd 6
490.4.e.o.471.1 2 35.24 odd 6
560.4.a.i.1.1 1 140.139 even 2
630.4.a.x.1.1 1 105.104 even 2
2240.4.a.r.1.1 1 280.139 even 2
2240.4.a.v.1.1 1 280.69 odd 2
2450.4.a.bc.1.1 1 1.1 even 1 trivial