Properties

Label 230.6.a.a.1.1
Level $230$
Weight $6$
Character 230.1
Self dual yes
Analytic conductor $36.888$
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] = [230,6,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, 6); 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, 6, names="a")
 
Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 230.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-4,8] 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: \(36.8882785570\)
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)
 
\(f(q)\) \(=\) \(q-4.00000 q^{2} +8.00000 q^{3} +16.0000 q^{4} -25.0000 q^{5} -32.0000 q^{6} +199.000 q^{7} -64.0000 q^{8} -179.000 q^{9} +100.000 q^{10} +150.000 q^{11} +128.000 q^{12} -1202.00 q^{13} -796.000 q^{14} -200.000 q^{15} +256.000 q^{16} +735.000 q^{17} +716.000 q^{18} -22.0000 q^{19} -400.000 q^{20} +1592.00 q^{21} -600.000 q^{22} -529.000 q^{23} -512.000 q^{24} +625.000 q^{25} +4808.00 q^{26} -3376.00 q^{27} +3184.00 q^{28} -5525.00 q^{29} +800.000 q^{30} -95.0000 q^{31} -1024.00 q^{32} +1200.00 q^{33} -2940.00 q^{34} -4975.00 q^{35} -2864.00 q^{36} -397.000 q^{37} +88.0000 q^{38} -9616.00 q^{39} +1600.00 q^{40} +20633.0 q^{41} -6368.00 q^{42} -11384.0 q^{43} +2400.00 q^{44} +4475.00 q^{45} +2116.00 q^{46} +1992.00 q^{47} +2048.00 q^{48} +22794.0 q^{49} -2500.00 q^{50} +5880.00 q^{51} -19232.0 q^{52} -7349.00 q^{53} +13504.0 q^{54} -3750.00 q^{55} -12736.0 q^{56} -176.000 q^{57} +22100.0 q^{58} -23827.0 q^{59} -3200.00 q^{60} -44016.0 q^{61} +380.000 q^{62} -35621.0 q^{63} +4096.00 q^{64} +30050.0 q^{65} -4800.00 q^{66} -37713.0 q^{67} +11760.0 q^{68} -4232.00 q^{69} +19900.0 q^{70} -50057.0 q^{71} +11456.0 q^{72} -16698.0 q^{73} +1588.00 q^{74} +5000.00 q^{75} -352.000 q^{76} +29850.0 q^{77} +38464.0 q^{78} -31004.0 q^{79} -6400.00 q^{80} +16489.0 q^{81} -82532.0 q^{82} -70077.0 q^{83} +25472.0 q^{84} -18375.0 q^{85} +45536.0 q^{86} -44200.0 q^{87} -9600.00 q^{88} +7676.00 q^{89} -17900.0 q^{90} -239198. q^{91} -8464.00 q^{92} -760.000 q^{93} -7968.00 q^{94} +550.000 q^{95} -8192.00 q^{96} -150094. q^{97} -91176.0 q^{98} -26850.0 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.00000 −0.707107
\(3\) 8.00000 0.513200 0.256600 − 0.966518i \(-0.417398\pi\)
0.256600 + 0.966518i \(0.417398\pi\)
\(4\) 16.0000 0.500000
\(5\) −25.0000 −0.447214
\(6\) −32.0000 −0.362887
\(7\) 199.000 1.53500 0.767499 − 0.641050i \(-0.221501\pi\)
0.767499 + 0.641050i \(0.221501\pi\)
\(8\) −64.0000 −0.353553
\(9\) −179.000 −0.736626
\(10\) 100.000 0.316228
\(11\) 150.000 0.373774 0.186887 − 0.982381i \(-0.440160\pi\)
0.186887 + 0.982381i \(0.440160\pi\)
\(12\) 128.000 0.256600
\(13\) −1202.00 −1.97263 −0.986316 − 0.164866i \(-0.947281\pi\)
−0.986316 + 0.164866i \(0.947281\pi\)
\(14\) −796.000 −1.08541
\(15\) −200.000 −0.229510
\(16\) 256.000 0.250000
\(17\) 735.000 0.616829 0.308415 − 0.951252i \(-0.400202\pi\)
0.308415 + 0.951252i \(0.400202\pi\)
\(18\) 716.000 0.520873
\(19\) −22.0000 −0.0139810 −0.00699051 − 0.999976i \(-0.502225\pi\)
−0.00699051 + 0.999976i \(0.502225\pi\)
\(20\) −400.000 −0.223607
\(21\) 1592.00 0.787762
\(22\) −600.000 −0.264298
\(23\) −529.000 −0.208514
\(24\) −512.000 −0.181444
\(25\) 625.000 0.200000
\(26\) 4808.00 1.39486
\(27\) −3376.00 −0.891237
\(28\) 3184.00 0.767499
\(29\) −5525.00 −1.21994 −0.609968 − 0.792426i \(-0.708818\pi\)
−0.609968 + 0.792426i \(0.708818\pi\)
\(30\) 800.000 0.162288
\(31\) −95.0000 −0.0177549 −0.00887747 − 0.999961i \(-0.502826\pi\)
−0.00887747 + 0.999961i \(0.502826\pi\)
\(32\) −1024.00 −0.176777
\(33\) 1200.00 0.191821
\(34\) −2940.00 −0.436164
\(35\) −4975.00 −0.686472
\(36\) −2864.00 −0.368313
\(37\) −397.000 −0.0476745 −0.0238373 − 0.999716i \(-0.507588\pi\)
−0.0238373 + 0.999716i \(0.507588\pi\)
\(38\) 88.0000 0.00988607
\(39\) −9616.00 −1.01236
\(40\) 1600.00 0.158114
\(41\) 20633.0 1.91691 0.958457 − 0.285236i \(-0.0920721\pi\)
0.958457 + 0.285236i \(0.0920721\pi\)
\(42\) −6368.00 −0.557032
\(43\) −11384.0 −0.938910 −0.469455 − 0.882957i \(-0.655550\pi\)
−0.469455 + 0.882957i \(0.655550\pi\)
\(44\) 2400.00 0.186887
\(45\) 4475.00 0.329429
\(46\) 2116.00 0.147442
\(47\) 1992.00 0.131536 0.0657680 − 0.997835i \(-0.479050\pi\)
0.0657680 + 0.997835i \(0.479050\pi\)
\(48\) 2048.00 0.128300
\(49\) 22794.0 1.35622
\(50\) −2500.00 −0.141421
\(51\) 5880.00 0.316557
\(52\) −19232.0 −0.986316
\(53\) −7349.00 −0.359367 −0.179684 − 0.983724i \(-0.557507\pi\)
−0.179684 + 0.983724i \(0.557507\pi\)
\(54\) 13504.0 0.630199
\(55\) −3750.00 −0.167157
\(56\) −12736.0 −0.542704
\(57\) −176.000 −0.00717506
\(58\) 22100.0 0.862625
\(59\) −23827.0 −0.891126 −0.445563 − 0.895250i \(-0.646997\pi\)
−0.445563 + 0.895250i \(0.646997\pi\)
\(60\) −3200.00 −0.114755
\(61\) −44016.0 −1.51456 −0.757279 − 0.653091i \(-0.773472\pi\)
−0.757279 + 0.653091i \(0.773472\pi\)
\(62\) 380.000 0.0125546
\(63\) −35621.0 −1.13072
\(64\) 4096.00 0.125000
\(65\) 30050.0 0.882188
\(66\) −4800.00 −0.135638
\(67\) −37713.0 −1.02637 −0.513185 − 0.858278i \(-0.671535\pi\)
−0.513185 + 0.858278i \(0.671535\pi\)
\(68\) 11760.0 0.308415
\(69\) −4232.00 −0.107010
\(70\) 19900.0 0.485409
\(71\) −50057.0 −1.17847 −0.589236 − 0.807961i \(-0.700571\pi\)
−0.589236 + 0.807961i \(0.700571\pi\)
\(72\) 11456.0 0.260436
\(73\) −16698.0 −0.366739 −0.183370 − 0.983044i \(-0.558701\pi\)
−0.183370 + 0.983044i \(0.558701\pi\)
\(74\) 1588.00 0.0337110
\(75\) 5000.00 0.102640
\(76\) −352.000 −0.00699051
\(77\) 29850.0 0.573743
\(78\) 38464.0 0.715843
\(79\) −31004.0 −0.558920 −0.279460 − 0.960157i \(-0.590156\pi\)
−0.279460 + 0.960157i \(0.590156\pi\)
\(80\) −6400.00 −0.111803
\(81\) 16489.0 0.279243
\(82\) −82532.0 −1.35546
\(83\) −70077.0 −1.11656 −0.558278 − 0.829654i \(-0.688538\pi\)
−0.558278 + 0.829654i \(0.688538\pi\)
\(84\) 25472.0 0.393881
\(85\) −18375.0 −0.275854
\(86\) 45536.0 0.663909
\(87\) −44200.0 −0.626072
\(88\) −9600.00 −0.132149
\(89\) 7676.00 0.102721 0.0513606 − 0.998680i \(-0.483644\pi\)
0.0513606 + 0.998680i \(0.483644\pi\)
\(90\) −17900.0 −0.232941
\(91\) −239198. −3.02799
\(92\) −8464.00 −0.104257
\(93\) −760.000 −0.00911184
\(94\) −7968.00 −0.0930100
\(95\) 550.000 0.00625250
\(96\) −8192.00 −0.0907218
\(97\) −150094. −1.61970 −0.809849 − 0.586639i \(-0.800451\pi\)
−0.809849 + 0.586639i \(0.800451\pi\)
\(98\) −91176.0 −0.958993
\(99\) −26850.0 −0.275332
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 230.6.a.a.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.6.a.a.1.1 ✓ 1 1.1 even 1 trivial