Properties

Label 230.4.a.e.1.1
Level $230$
Weight $4$
Character 230.1
Self dual yes
Analytic conductor $13.570$
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] = [230,4,Mod(1,230)] 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("230.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(230, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 230.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: \(13.5704393013\)
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 \(\chi\) \(=\) 230.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.00000 q^{2} +1.00000 q^{3} +4.00000 q^{4} -5.00000 q^{5} +2.00000 q^{6} -18.0000 q^{7} +8.00000 q^{8} -26.0000 q^{9} -10.0000 q^{10} -32.0000 q^{11} +4.00000 q^{12} -47.0000 q^{13} -36.0000 q^{14} -5.00000 q^{15} +16.0000 q^{16} +20.0000 q^{17} -52.0000 q^{18} +36.0000 q^{19} -20.0000 q^{20} -18.0000 q^{21} -64.0000 q^{22} -23.0000 q^{23} +8.00000 q^{24} +25.0000 q^{25} -94.0000 q^{26} -53.0000 q^{27} -72.0000 q^{28} -27.0000 q^{29} -10.0000 q^{30} -33.0000 q^{31} +32.0000 q^{32} -32.0000 q^{33} +40.0000 q^{34} +90.0000 q^{35} -104.000 q^{36} +56.0000 q^{37} +72.0000 q^{38} -47.0000 q^{39} -40.0000 q^{40} -157.000 q^{41} -36.0000 q^{42} +18.0000 q^{43} -128.000 q^{44} +130.000 q^{45} -46.0000 q^{46} +65.0000 q^{47} +16.0000 q^{48} -19.0000 q^{49} +50.0000 q^{50} +20.0000 q^{51} -188.000 q^{52} -14.0000 q^{53} -106.000 q^{54} +160.000 q^{55} -144.000 q^{56} +36.0000 q^{57} -54.0000 q^{58} -744.000 q^{59} -20.0000 q^{60} +552.000 q^{61} -66.0000 q^{62} +468.000 q^{63} +64.0000 q^{64} +235.000 q^{65} -64.0000 q^{66} -156.000 q^{67} +80.0000 q^{68} -23.0000 q^{69} +180.000 q^{70} +699.000 q^{71} -208.000 q^{72} -609.000 q^{73} +112.000 q^{74} +25.0000 q^{75} +144.000 q^{76} +576.000 q^{77} -94.0000 q^{78} -644.000 q^{79} -80.0000 q^{80} +649.000 q^{81} -314.000 q^{82} +512.000 q^{83} -72.0000 q^{84} -100.000 q^{85} +36.0000 q^{86} -27.0000 q^{87} -256.000 q^{88} -102.000 q^{89} +260.000 q^{90} +846.000 q^{91} -92.0000 q^{92} -33.0000 q^{93} +130.000 q^{94} -180.000 q^{95} +32.0000 q^{96} +578.000 q^{97} -38.0000 q^{98} +832.000 q^{99} +O(q^{100})\)

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 \(-0.469323\pi\)
0.0962250 + 0.995360i \(0.469323\pi\)
\(4\) 4.00000 0.500000
\(5\) −5.00000 −0.447214
\(6\) 2.00000 0.136083
\(7\) −18.0000 −0.971909 −0.485954 − 0.873984i \(-0.661528\pi\)
−0.485954 + 0.873984i \(0.661528\pi\)
\(8\) 8.00000 0.353553
\(9\) −26.0000 −0.962963
\(10\) −10.0000 −0.316228
\(11\) −32.0000 −0.877124 −0.438562 − 0.898701i \(-0.644512\pi\)
−0.438562 + 0.898701i \(0.644512\pi\)
\(12\) 4.00000 0.0962250
\(13\) −47.0000 −1.00273 −0.501364 − 0.865237i \(-0.667168\pi\)
−0.501364 + 0.865237i \(0.667168\pi\)
\(14\) −36.0000 −0.687243
\(15\) −5.00000 −0.0860663
\(16\) 16.0000 0.250000
\(17\) 20.0000 0.285336 0.142668 − 0.989771i \(-0.454432\pi\)
0.142668 + 0.989771i \(0.454432\pi\)
\(18\) −52.0000 −0.680918
\(19\) 36.0000 0.434682 0.217341 − 0.976096i \(-0.430262\pi\)
0.217341 + 0.976096i \(0.430262\pi\)
\(20\) −20.0000 −0.223607
\(21\) −18.0000 −0.187044
\(22\) −64.0000 −0.620220
\(23\) −23.0000 −0.208514
\(24\) 8.00000 0.0680414
\(25\) 25.0000 0.200000
\(26\) −94.0000 −0.709035
\(27\) −53.0000 −0.377772
\(28\) −72.0000 −0.485954
\(29\) −27.0000 −0.172889 −0.0864444 − 0.996257i \(-0.527550\pi\)
−0.0864444 + 0.996257i \(0.527550\pi\)
\(30\) −10.0000 −0.0608581
\(31\) −33.0000 −0.191193 −0.0955964 − 0.995420i \(-0.530476\pi\)
−0.0955964 + 0.995420i \(0.530476\pi\)
\(32\) 32.0000 0.176777
\(33\) −32.0000 −0.168803
\(34\) 40.0000 0.201763
\(35\) 90.0000 0.434651
\(36\) −104.000 −0.481481
\(37\) 56.0000 0.248820 0.124410 − 0.992231i \(-0.460296\pi\)
0.124410 + 0.992231i \(0.460296\pi\)
\(38\) 72.0000 0.307367
\(39\) −47.0000 −0.192975
\(40\) −40.0000 −0.158114
\(41\) −157.000 −0.598031 −0.299016 − 0.954248i \(-0.596658\pi\)
−0.299016 + 0.954248i \(0.596658\pi\)
\(42\) −36.0000 −0.132260
\(43\) 18.0000 0.0638366 0.0319183 − 0.999490i \(-0.489838\pi\)
0.0319183 + 0.999490i \(0.489838\pi\)
\(44\) −128.000 −0.438562
\(45\) 130.000 0.430650
\(46\) −46.0000 −0.147442
\(47\) 65.0000 0.201728 0.100864 − 0.994900i \(-0.467839\pi\)
0.100864 + 0.994900i \(0.467839\pi\)
\(48\) 16.0000 0.0481125
\(49\) −19.0000 −0.0553936
\(50\) 50.0000 0.141421
\(51\) 20.0000 0.0549129
\(52\) −188.000 −0.501364
\(53\) −14.0000 −0.0362839 −0.0181420 − 0.999835i \(-0.505775\pi\)
−0.0181420 + 0.999835i \(0.505775\pi\)
\(54\) −106.000 −0.267125
\(55\) 160.000 0.392262
\(56\) −144.000 −0.343622
\(57\) 36.0000 0.0836547
\(58\) −54.0000 −0.122251
\(59\) −744.000 −1.64170 −0.820852 − 0.571141i \(-0.806501\pi\)
−0.820852 + 0.571141i \(0.806501\pi\)
\(60\) −20.0000 −0.0430331
\(61\) 552.000 1.15863 0.579314 − 0.815104i \(-0.303320\pi\)
0.579314 + 0.815104i \(0.303320\pi\)
\(62\) −66.0000 −0.135194
\(63\) 468.000 0.935912
\(64\) 64.0000 0.125000
\(65\) 235.000 0.448433
\(66\) −64.0000 −0.119361
\(67\) −156.000 −0.284454 −0.142227 − 0.989834i \(-0.545426\pi\)
−0.142227 + 0.989834i \(0.545426\pi\)
\(68\) 80.0000 0.142668
\(69\) −23.0000 −0.0401286
\(70\) 180.000 0.307344
\(71\) 699.000 1.16839 0.584197 − 0.811612i \(-0.301409\pi\)
0.584197 + 0.811612i \(0.301409\pi\)
\(72\) −208.000 −0.340459
\(73\) −609.000 −0.976412 −0.488206 − 0.872728i \(-0.662348\pi\)
−0.488206 + 0.872728i \(0.662348\pi\)
\(74\) 112.000 0.175942
\(75\) 25.0000 0.0384900
\(76\) 144.000 0.217341
\(77\) 576.000 0.852484
\(78\) −94.0000 −0.136454
\(79\) −644.000 −0.917160 −0.458580 − 0.888653i \(-0.651642\pi\)
−0.458580 + 0.888653i \(0.651642\pi\)
\(80\) −80.0000 −0.111803
\(81\) 649.000 0.890261
\(82\) −314.000 −0.422872
\(83\) 512.000 0.677100 0.338550 − 0.940948i \(-0.390064\pi\)
0.338550 + 0.940948i \(0.390064\pi\)
\(84\) −72.0000 −0.0935220
\(85\) −100.000 −0.127606
\(86\) 36.0000 0.0451393
\(87\) −27.0000 −0.0332725
\(88\) −256.000 −0.310110
\(89\) −102.000 −0.121483 −0.0607415 − 0.998154i \(-0.519347\pi\)
−0.0607415 + 0.998154i \(0.519347\pi\)
\(90\) 260.000 0.304516
\(91\) 846.000 0.974559
\(92\) −92.0000 −0.104257
\(93\) −33.0000 −0.0367951
\(94\) 130.000 0.142643
\(95\) −180.000 −0.194396
\(96\) 32.0000 0.0340207
\(97\) 578.000 0.605021 0.302510 − 0.953146i \(-0.402175\pi\)
0.302510 + 0.953146i \(0.402175\pi\)
\(98\) −38.0000 −0.0391692
\(99\) 832.000 0.844638
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 230.4.a.e.1.1 ✓ 1
3.2 odd 2 2070.4.a.e.1.1 1
4.3 odd 2 1840.4.a.d.1.1 1
5.2 odd 4 1150.4.b.f.599.2 2
5.3 odd 4 1150.4.b.f.599.1 2
5.4 even 2 1150.4.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.4.a.e.1.1 ✓ 1 1.1 even 1 trivial
1150.4.a.b.1.1 1 5.4 even 2
1150.4.b.f.599.1 2 5.3 odd 4
1150.4.b.f.599.2 2 5.2 odd 4
1840.4.a.d.1.1 1 4.3 odd 2
2070.4.a.e.1.1 1 3.2 odd 2