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Properties

Label 630.4.a.x.1.1
Level $630$
Weight $4$
Character 630.1
Self dual yes
Analytic conductor $37.171$
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] = [630,4,Mod(1,630)] 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("630.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(630, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 630.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,2,0,4,5,0,7,8,0,10,65] 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: \(37.1712033036\)
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 \(\chi\) \(=\) 630.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.00000 q^{2} +4.00000 q^{4} +5.00000 q^{5} +7.00000 q^{7} +8.00000 q^{8} +10.0000 q^{10} +65.0000 q^{11} +13.0000 q^{13} +14.0000 q^{14} +16.0000 q^{16} +73.0000 q^{17} -142.000 q^{19} +20.0000 q^{20} +130.000 q^{22} -130.000 q^{23} +25.0000 q^{25} +26.0000 q^{26} +28.0000 q^{28} -111.000 q^{29} +256.000 q^{31} +32.0000 q^{32} +146.000 q^{34} +35.0000 q^{35} -266.000 q^{37} -284.000 q^{38} +40.0000 q^{40} +424.000 q^{41} +534.000 q^{43} +260.000 q^{44} -260.000 q^{46} +269.000 q^{47} +49.0000 q^{49} +50.0000 q^{50} +52.0000 q^{52} +132.000 q^{53} +325.000 q^{55} +56.0000 q^{56} -222.000 q^{58} +224.000 q^{59} -572.000 q^{61} +512.000 q^{62} +64.0000 q^{64} +65.0000 q^{65} -108.000 q^{67} +292.000 q^{68} +70.0000 q^{70} -560.000 q^{71} +586.000 q^{73} -532.000 q^{74} -568.000 q^{76} +455.000 q^{77} +57.0000 q^{79} +80.0000 q^{80} +848.000 q^{82} -252.000 q^{83} +365.000 q^{85} +1068.00 q^{86} +520.000 q^{88} +184.000 q^{89} +91.0000 q^{91} -520.000 q^{92} +538.000 q^{94} -710.000 q^{95} -605.000 q^{97} +98.0000 q^{98} +O(q^{100})\)

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
\(3\) 0 0
\(4\) 4.00000 0.500000
\(5\) 5.00000 0.447214
\(6\) 0 0
\(7\) 7.00000 0.377964
\(8\) 8.00000 0.353553
\(9\) 0 0
\(10\) 10.0000 0.316228
\(11\) 65.0000 1.78166 0.890829 − 0.454339i \(-0.150124\pi\)
0.890829 + 0.454339i \(0.150124\pi\)
\(12\) 0 0
\(13\) 13.0000 0.277350 0.138675 − 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) 14.0000 0.267261
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) 73.0000 1.04148 0.520738 − 0.853716i \(-0.325657\pi\)
0.520738 + 0.853716i \(0.325657\pi\)
\(18\) 0 0
\(19\) −142.000 −1.71458 −0.857290 − 0.514833i \(-0.827854\pi\)
−0.857290 + 0.514833i \(0.827854\pi\)
\(20\) 20.0000 0.223607
\(21\) 0 0
\(22\) 130.000 1.25982
\(23\) −130.000 −1.17856 −0.589280 − 0.807929i \(-0.700588\pi\)
−0.589280 + 0.807929i \(0.700588\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 26.0000 0.196116
\(27\) 0 0
\(28\) 28.0000 0.188982
\(29\) −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 \(-0.234069\pi\)
0.741596 + 0.670847i \(0.234069\pi\)
\(32\) 32.0000 0.176777
\(33\) 0 0
\(34\) 146.000 0.736435
\(35\) 35.0000 0.169031
\(36\) 0 0
\(37\) −266.000 −1.18190 −0.590948 − 0.806710i \(-0.701246\pi\)
−0.590948 + 0.806710i \(0.701246\pi\)
\(38\) −284.000 −1.21239
\(39\) 0 0
\(40\) 40.0000 0.158114
\(41\) 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 \(-0.104187\pi\)
0.946910 + 0.321500i \(0.104187\pi\)
\(44\) 260.000 0.890829
\(45\) 0 0
\(46\) −260.000 −0.833368
\(47\) 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
\(50\) 50.0000 0.141421
\(51\) 0 0
\(52\) 52.0000 0.138675
\(53\) 132.000 0.342106 0.171053 − 0.985262i \(-0.445283\pi\)
0.171053 + 0.985262i \(0.445283\pi\)
\(54\) 0 0
\(55\) 325.000 0.796782
\(56\) 56.0000 0.133631
\(57\) 0 0
\(58\) −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 \(-0.704954\pi\)
−0.600304 + 0.799772i \(0.704954\pi\)
\(62\) 512.000 1.04878
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) 65.0000 0.124035
\(66\) 0 0
\(67\) −108.000 −0.196930 −0.0984649 − 0.995141i \(-0.531393\pi\)
−0.0984649 + 0.995141i \(0.531393\pi\)
\(68\) 292.000 0.520738
\(69\) 0 0
\(70\) 70.0000 0.119523
\(71\) −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 \(0.344338\pi\)
\(74\) −532.000 −0.835726
\(75\) 0 0
\(76\) −568.000 −0.857290
\(77\) 455.000 0.673403
\(78\) 0 0
\(79\) 57.0000 0.0811772 0.0405886 − 0.999176i \(-0.487077\pi\)
0.0405886 + 0.999176i \(0.487077\pi\)
\(80\) 80.0000 0.111803
\(81\) 0 0
\(82\) 848.000 1.14202
\(83\) −252.000 −0.333260 −0.166630 − 0.986019i \(-0.553289\pi\)
−0.166630 + 0.986019i \(0.553289\pi\)
\(84\) 0 0
\(85\) 365.000 0.465762
\(86\) 1068.00 1.33913
\(87\) 0 0
\(88\) 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
\(91\) 91.0000 0.104828
\(92\) −520.000 −0.589280
\(93\) 0 0
\(94\) 538.000 0.590324
\(95\) −710.000 −0.766784
\(96\) 0 0
\(97\) −605.000 −0.633283 −0.316641 − 0.948545i \(-0.602555\pi\)
−0.316641 + 0.948545i \(0.602555\pi\)
\(98\) 98.0000 0.101015
\(99\) 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 630.4.a.x.1.1 1
3.2 odd 2 70.4.a.c.1.1 ✓ 1
12.11 even 2 560.4.a.i.1.1 1
15.2 even 4 350.4.c.h.99.1 2
15.8 even 4 350.4.c.h.99.2 2
15.14 odd 2 350.4.a.r.1.1 1
21.2 odd 6 490.4.e.o.361.1 2
21.5 even 6 490.4.e.n.361.1 2
21.11 odd 6 490.4.e.o.471.1 2
21.17 even 6 490.4.e.n.471.1 2
21.20 even 2 490.4.a.d.1.1 1
24.5 odd 2 2240.4.a.v.1.1 1
24.11 even 2 2240.4.a.r.1.1 1
105.104 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 3.2 odd 2
350.4.a.r.1.1 1 15.14 odd 2
350.4.c.h.99.1 2 15.2 even 4
350.4.c.h.99.2 2 15.8 even 4
490.4.a.d.1.1 1 21.20 even 2
490.4.e.n.361.1 2 21.5 even 6
490.4.e.n.471.1 2 21.17 even 6
490.4.e.o.361.1 2 21.2 odd 6
490.4.e.o.471.1 2 21.11 odd 6
560.4.a.i.1.1 1 12.11 even 2
630.4.a.x.1.1 1 1.1 even 1 trivial
2240.4.a.r.1.1 1 24.11 even 2
2240.4.a.v.1.1 1 24.5 odd 2
2450.4.a.bc.1.1 1 105.104 even 2