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Properties

Label 630.6.a.n.1.1
Level $630$
Weight $6$
Character 630.1
Self dual yes
Analytic conductor $101.042$
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] = [630,6,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, 6); 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, 6, names="a")
 
Level: \( N \) \(=\) \( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 630.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,4,0,16,25,0,49,64,0,100,-405] 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: \(101.041806482\)
Analytic rank: \(1\)
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)
 
\(f(q)\) \(=\) \(q+4.00000 q^{2} +16.0000 q^{4} +25.0000 q^{5} +49.0000 q^{7} +64.0000 q^{8} +100.000 q^{10} -405.000 q^{11} -391.000 q^{13} +196.000 q^{14} +256.000 q^{16} -999.000 q^{17} +2342.00 q^{19} +400.000 q^{20} -1620.00 q^{22} -2430.00 q^{23} +625.000 q^{25} -1564.00 q^{26} +784.000 q^{28} -8259.00 q^{29} +4016.00 q^{31} +1024.00 q^{32} -3996.00 q^{34} +1225.00 q^{35} -7042.00 q^{37} +9368.00 q^{38} +1600.00 q^{40} -3336.00 q^{41} -23518.0 q^{43} -6480.00 q^{44} -9720.00 q^{46} -10317.0 q^{47} +2401.00 q^{49} +2500.00 q^{50} -6256.00 q^{52} -3084.00 q^{53} -10125.0 q^{55} +3136.00 q^{56} -33036.0 q^{58} +18816.0 q^{59} +21668.0 q^{61} +16064.0 q^{62} +4096.00 q^{64} -9775.00 q^{65} +52124.0 q^{67} -15984.0 q^{68} +4900.00 q^{70} +28560.0 q^{71} -70342.0 q^{73} -28168.0 q^{74} +37472.0 q^{76} -19845.0 q^{77} +58823.0 q^{79} +6400.00 q^{80} -13344.0 q^{82} -756.000 q^{83} -24975.0 q^{85} -94072.0 q^{86} -25920.0 q^{88} -135384. q^{89} -19159.0 q^{91} -38880.0 q^{92} -41268.0 q^{94} +58550.0 q^{95} +110435. q^{97} +9604.00 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.00000 0.707107
\(3\) 0 0
\(4\) 16.0000 0.500000
\(5\) 25.0000 0.447214
\(6\) 0 0
\(7\) 49.0000 0.377964
\(8\) 64.0000 0.353553
\(9\) 0 0
\(10\) 100.000 0.316228
\(11\) −405.000 −1.00919 −0.504595 − 0.863356i \(-0.668358\pi\)
−0.504595 + 0.863356i \(0.668358\pi\)
\(12\) 0 0
\(13\) −391.000 −0.641680 −0.320840 − 0.947133i \(-0.603965\pi\)
−0.320840 + 0.947133i \(0.603965\pi\)
\(14\) 196.000 0.267261
\(15\) 0 0
\(16\) 256.000 0.250000
\(17\) −999.000 −0.838384 −0.419192 − 0.907898i \(-0.637687\pi\)
−0.419192 + 0.907898i \(0.637687\pi\)
\(18\) 0 0
\(19\) 2342.00 1.48834 0.744171 − 0.667989i \(-0.232845\pi\)
0.744171 + 0.667989i \(0.232845\pi\)
\(20\) 400.000 0.223607
\(21\) 0 0
\(22\) −1620.00 −0.713606
\(23\) −2430.00 −0.957826 −0.478913 − 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(24\) 0 0
\(25\) 625.000 0.200000
\(26\) −1564.00 −0.453736
\(27\) 0 0
\(28\) 784.000 0.188982
\(29\) −8259.00 −1.82361 −0.911806 − 0.410621i \(-0.865312\pi\)
−0.911806 + 0.410621i \(0.865312\pi\)
\(30\) 0 0
\(31\) 4016.00 0.750567 0.375284 − 0.926910i \(-0.377545\pi\)
0.375284 + 0.926910i \(0.377545\pi\)
\(32\) 1024.00 0.176777
\(33\) 0 0
\(34\) −3996.00 −0.592827
\(35\) 1225.00 0.169031
\(36\) 0 0
\(37\) −7042.00 −0.845652 −0.422826 − 0.906211i \(-0.638962\pi\)
−0.422826 + 0.906211i \(0.638962\pi\)
\(38\) 9368.00 1.05242
\(39\) 0 0
\(40\) 1600.00 0.158114
\(41\) −3336.00 −0.309932 −0.154966 − 0.987920i \(-0.549527\pi\)
−0.154966 + 0.987920i \(0.549527\pi\)
\(42\) 0 0
\(43\) −23518.0 −1.93968 −0.969838 − 0.243750i \(-0.921622\pi\)
−0.969838 + 0.243750i \(0.921622\pi\)
\(44\) −6480.00 −0.504595
\(45\) 0 0
\(46\) −9720.00 −0.677285
\(47\) −10317.0 −0.681254 −0.340627 − 0.940199i \(-0.610639\pi\)
−0.340627 + 0.940199i \(0.610639\pi\)
\(48\) 0 0
\(49\) 2401.00 0.142857
\(50\) 2500.00 0.141421
\(51\) 0 0
\(52\) −6256.00 −0.320840
\(53\) −3084.00 −0.150808 −0.0754041 − 0.997153i \(-0.524025\pi\)
−0.0754041 + 0.997153i \(0.524025\pi\)
\(54\) 0 0
\(55\) −10125.0 −0.451324
\(56\) 3136.00 0.133631
\(57\) 0 0
\(58\) −33036.0 −1.28949
\(59\) 18816.0 0.703716 0.351858 − 0.936053i \(-0.385550\pi\)
0.351858 + 0.936053i \(0.385550\pi\)
\(60\) 0 0
\(61\) 21668.0 0.745580 0.372790 − 0.927916i \(-0.378401\pi\)
0.372790 + 0.927916i \(0.378401\pi\)
\(62\) 16064.0 0.530731
\(63\) 0 0
\(64\) 4096.00 0.125000
\(65\) −9775.00 −0.286968
\(66\) 0 0
\(67\) 52124.0 1.41857 0.709285 − 0.704922i \(-0.249018\pi\)
0.709285 + 0.704922i \(0.249018\pi\)
\(68\) −15984.0 −0.419192
\(69\) 0 0
\(70\) 4900.00 0.119523
\(71\) 28560.0 0.672376 0.336188 − 0.941795i \(-0.390862\pi\)
0.336188 + 0.941795i \(0.390862\pi\)
\(72\) 0 0
\(73\) −70342.0 −1.54493 −0.772463 − 0.635060i \(-0.780975\pi\)
−0.772463 + 0.635060i \(0.780975\pi\)
\(74\) −28168.0 −0.597966
\(75\) 0 0
\(76\) 37472.0 0.744171
\(77\) −19845.0 −0.381438
\(78\) 0 0
\(79\) 58823.0 1.06042 0.530212 − 0.847865i \(-0.322112\pi\)
0.530212 + 0.847865i \(0.322112\pi\)
\(80\) 6400.00 0.111803
\(81\) 0 0
\(82\) −13344.0 −0.219155
\(83\) −756.000 −0.0120455 −0.00602277 − 0.999982i \(-0.501917\pi\)
−0.00602277 + 0.999982i \(0.501917\pi\)
\(84\) 0 0
\(85\) −24975.0 −0.374937
\(86\) −94072.0 −1.37156
\(87\) 0 0
\(88\) −25920.0 −0.356803
\(89\) −135384. −1.81173 −0.905863 − 0.423572i \(-0.860776\pi\)
−0.905863 + 0.423572i \(0.860776\pi\)
\(90\) 0 0
\(91\) −19159.0 −0.242532
\(92\) −38880.0 −0.478913
\(93\) 0 0
\(94\) −41268.0 −0.481719
\(95\) 58550.0 0.665607
\(96\) 0 0
\(97\) 110435. 1.19173 0.595864 − 0.803085i \(-0.296810\pi\)
0.595864 + 0.803085i \(0.296810\pi\)
\(98\) 9604.00 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 630.6.a.n.1.1 1
3.2 odd 2 70.6.a.c.1.1 ✓ 1
12.11 even 2 560.6.a.d.1.1 1
15.2 even 4 350.6.c.e.99.1 2
15.8 even 4 350.6.c.e.99.2 2
15.14 odd 2 350.6.a.k.1.1 1
21.20 even 2 490.6.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
70.6.a.c.1.1 ✓ 1 3.2 odd 2
350.6.a.k.1.1 1 15.14 odd 2
350.6.c.e.99.1 2 15.2 even 4
350.6.c.e.99.2 2 15.8 even 4
490.6.a.e.1.1 1 21.20 even 2
560.6.a.d.1.1 1 12.11 even 2
630.6.a.n.1.1 1 1.1 even 1 trivial