Properties

Label 70.4.a.c.1.1
Level $70$
Weight $4$
Character 70.1
Self dual yes
Analytic conductor $4.130$
Analytic rank $1$
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] = [70,4,Mod(1,70)] 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("70.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(70, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 70 = 2 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 70.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-2,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(4.13013370040\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 70.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} -1.00000 q^{3} +4.00000 q^{4} -5.00000 q^{5} +2.00000 q^{6} +7.00000 q^{7} -8.00000 q^{8} -26.0000 q^{9} +10.0000 q^{10} -65.0000 q^{11} -4.00000 q^{12} +13.0000 q^{13} -14.0000 q^{14} +5.00000 q^{15} +16.0000 q^{16} -73.0000 q^{17} +52.0000 q^{18} -142.000 q^{19} -20.0000 q^{20} -7.00000 q^{21} +130.000 q^{22} +130.000 q^{23} +8.00000 q^{24} +25.0000 q^{25} -26.0000 q^{26} +53.0000 q^{27} +28.0000 q^{28} +111.000 q^{29} -10.0000 q^{30} +256.000 q^{31} -32.0000 q^{32} +65.0000 q^{33} +146.000 q^{34} -35.0000 q^{35} -104.000 q^{36} -266.000 q^{37} +284.000 q^{38} -13.0000 q^{39} +40.0000 q^{40} -424.000 q^{41} +14.0000 q^{42} +534.000 q^{43} -260.000 q^{44} +130.000 q^{45} -260.000 q^{46} -269.000 q^{47} -16.0000 q^{48} +49.0000 q^{49} -50.0000 q^{50} +73.0000 q^{51} +52.0000 q^{52} -132.000 q^{53} -106.000 q^{54} +325.000 q^{55} -56.0000 q^{56} +142.000 q^{57} -222.000 q^{58} -224.000 q^{59} +20.0000 q^{60} -572.000 q^{61} -512.000 q^{62} -182.000 q^{63} +64.0000 q^{64} -65.0000 q^{65} -130.000 q^{66} -108.000 q^{67} -292.000 q^{68} -130.000 q^{69} +70.0000 q^{70} +560.000 q^{71} +208.000 q^{72} +586.000 q^{73} +532.000 q^{74} -25.0000 q^{75} -568.000 q^{76} -455.000 q^{77} +26.0000 q^{78} +57.0000 q^{79} -80.0000 q^{80} +649.000 q^{81} +848.000 q^{82} +252.000 q^{83} -28.0000 q^{84} +365.000 q^{85} -1068.00 q^{86} -111.000 q^{87} +520.000 q^{88} -184.000 q^{89} -260.000 q^{90} +91.0000 q^{91} +520.000 q^{92} -256.000 q^{93} +538.000 q^{94} +710.000 q^{95} +32.0000 q^{96} -605.000 q^{97} -98.0000 q^{98} +1690.00 q^{99} +O(q^{100})\)

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
\(5\) −5.00000 −0.447214
\(6\) 2.00000 0.136083
\(7\) 7.00000 0.377964
\(8\) −8.00000 −0.353553
\(9\) −26.0000 −0.962963
\(10\) 10.0000 0.316228
\(11\) −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 \(0.455716\pi\)
\(14\) −14.0000 −0.267261
\(15\) 5.00000 0.0860663
\(16\) 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 \(-0.827854\pi\)
−0.857290 + 0.514833i \(0.827854\pi\)
\(20\) −20.0000 −0.223607
\(21\) −7.00000 −0.0727393
\(22\) 130.000 1.25982
\(23\) 130.000 1.17856 0.589280 − 0.807929i \(-0.299412\pi\)
0.589280 + 0.807929i \(0.299412\pi\)
\(24\) 8.00000 0.0680414
\(25\) 25.0000 0.200000
\(26\) −26.0000 −0.196116
\(27\) 53.0000 0.377772
\(28\) 28.0000 0.188982
\(29\) 111.000 0.710765 0.355382 − 0.934721i \(-0.384351\pi\)
0.355382 + 0.934721i \(0.384351\pi\)
\(30\) −10.0000 −0.0608581
\(31\) 256.000 1.48319 0.741596 − 0.670847i \(-0.234069\pi\)
0.741596 + 0.670847i \(0.234069\pi\)
\(32\) −32.0000 −0.176777
\(33\) 65.0000 0.342880
\(34\) 146.000 0.736435
\(35\) −35.0000 −0.169031
\(36\) −104.000 −0.481481
\(37\) −266.000 −1.18190 −0.590948 − 0.806710i \(-0.701246\pi\)
−0.590948 + 0.806710i \(0.701246\pi\)
\(38\) 284.000 1.21239
\(39\) −13.0000 −0.0533761
\(40\) 40.0000 0.158114
\(41\) −424.000 −1.61507 −0.807533 − 0.589823i \(-0.799198\pi\)
−0.807533 + 0.589823i \(0.799198\pi\)
\(42\) 14.0000 0.0514344
\(43\) 534.000 1.89382 0.946910 − 0.321500i \(-0.104187\pi\)
0.946910 + 0.321500i \(0.104187\pi\)
\(44\) −260.000 −0.890829
\(45\) 130.000 0.430650
\(46\) −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
\(49\) 49.0000 0.142857
\(50\) −50.0000 −0.141421
\(51\) 73.0000 0.200432
\(52\) 52.0000 0.138675
\(53\) −132.000 −0.342106 −0.171053 − 0.985262i \(-0.554717\pi\)
−0.171053 + 0.985262i \(0.554717\pi\)
\(54\) −106.000 −0.267125
\(55\) 325.000 0.796782
\(56\) −56.0000 −0.133631
\(57\) 142.000 0.329971
\(58\) −222.000 −0.502587
\(59\) −224.000 −0.494277 −0.247138 − 0.968980i \(-0.579490\pi\)
−0.247138 + 0.968980i \(0.579490\pi\)
\(60\) 20.0000 0.0430331
\(61\) −572.000 −1.20061 −0.600304 − 0.799772i \(-0.704954\pi\)
−0.600304 + 0.799772i \(0.704954\pi\)
\(62\) −512.000 −1.04878
\(63\) −182.000 −0.363966
\(64\) 64.0000 0.125000
\(65\) −65.0000 −0.124035
\(66\) −130.000 −0.242453
\(67\) −108.000 −0.196930 −0.0984649 − 0.995141i \(-0.531393\pi\)
−0.0984649 + 0.995141i \(0.531393\pi\)
\(68\) −292.000 −0.520738
\(69\) −130.000 −0.226814
\(70\) 70.0000 0.119523
\(71\) 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
\(75\) −25.0000 −0.0384900
\(76\) −568.000 −0.857290
\(77\) −455.000 −0.673403
\(78\) 26.0000 0.0377426
\(79\) 57.0000 0.0811772 0.0405886 − 0.999176i \(-0.487077\pi\)
0.0405886 + 0.999176i \(0.487077\pi\)
\(80\) −80.0000 −0.111803
\(81\) 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 \(0.446711\pi\)
\(84\) −28.0000 −0.0363696
\(85\) 365.000 0.465762
\(86\) −1068.00 −1.33913
\(87\) −111.000 −0.136787
\(88\) 520.000 0.629911
\(89\) −184.000 −0.219146 −0.109573 − 0.993979i \(-0.534948\pi\)
−0.109573 + 0.993979i \(0.534948\pi\)
\(90\) −260.000 −0.304516
\(91\) 91.0000 0.104828
\(92\) 520.000 0.589280
\(93\) −256.000 −0.285440
\(94\) 538.000 0.590324
\(95\) 710.000 0.766784
\(96\) 32.0000 0.0340207
\(97\) −605.000 −0.633283 −0.316641 − 0.948545i \(-0.602555\pi\)
−0.316641 + 0.948545i \(0.602555\pi\)
\(98\) −98.0000 −0.101015
\(99\) 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 70.4.a.c.1.1 ✓ 1
3.2 odd 2 630.4.a.x.1.1 1
4.3 odd 2 560.4.a.i.1.1 1
5.2 odd 4 350.4.c.h.99.1 2
5.3 odd 4 350.4.c.h.99.2 2
5.4 even 2 350.4.a.r.1.1 1
7.2 even 3 490.4.e.o.361.1 2
7.3 odd 6 490.4.e.n.471.1 2
7.4 even 3 490.4.e.o.471.1 2
7.5 odd 6 490.4.e.n.361.1 2
7.6 odd 2 490.4.a.d.1.1 1
8.3 odd 2 2240.4.a.r.1.1 1
8.5 even 2 2240.4.a.v.1.1 1
35.34 odd 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 1.1 even 1 trivial
350.4.a.r.1.1 1 5.4 even 2
350.4.c.h.99.1 2 5.2 odd 4
350.4.c.h.99.2 2 5.3 odd 4
490.4.a.d.1.1 1 7.6 odd 2
490.4.e.n.361.1 2 7.5 odd 6
490.4.e.n.471.1 2 7.3 odd 6
490.4.e.o.361.1 2 7.2 even 3
490.4.e.o.471.1 2 7.4 even 3
560.4.a.i.1.1 1 4.3 odd 2
630.4.a.x.1.1 1 3.2 odd 2
2240.4.a.r.1.1 1 8.3 odd 2
2240.4.a.v.1.1 1 8.5 even 2
2450.4.a.bc.1.1 1 35.34 odd 2