Properties

Label 230.6.a.d.1.3
Level $230$
Weight $6$
Character 230.1
Self dual yes
Analytic conductor $36.888$
Analytic rank $1$
Dimension $3$
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 = [3,12,-34] 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: \(3\)
Coefficient field: 3.3.27980.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 47x - 106 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-2.65230\) of defining polynomial
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)
 
\(f(q)\) \(=\) \(q+4.00000 q^{2} +2.35610 q^{3} +16.0000 q^{4} +25.0000 q^{5} +9.42439 q^{6} -9.44632 q^{7} +64.0000 q^{8} -237.449 q^{9} +100.000 q^{10} -248.872 q^{11} +37.6975 q^{12} -898.786 q^{13} -37.7853 q^{14} +58.9024 q^{15} +256.000 q^{16} -1177.51 q^{17} -949.795 q^{18} +116.462 q^{19} +400.000 q^{20} -22.2564 q^{21} -995.490 q^{22} -529.000 q^{23} +150.790 q^{24} +625.000 q^{25} -3595.14 q^{26} -1131.98 q^{27} -151.141 q^{28} +2766.81 q^{29} +235.610 q^{30} -661.945 q^{31} +1024.00 q^{32} -586.368 q^{33} -4710.03 q^{34} -236.158 q^{35} -3799.18 q^{36} -5965.93 q^{37} +465.848 q^{38} -2117.63 q^{39} +1600.00 q^{40} -7482.54 q^{41} -89.0258 q^{42} +8158.86 q^{43} -3981.96 q^{44} -5936.22 q^{45} -2116.00 q^{46} +14818.9 q^{47} +603.161 q^{48} -16717.8 q^{49} +2500.00 q^{50} -2774.32 q^{51} -14380.6 q^{52} -24766.8 q^{53} -4527.94 q^{54} -6221.81 q^{55} -604.565 q^{56} +274.396 q^{57} +11067.2 q^{58} -7935.48 q^{59} +942.439 q^{60} +5864.34 q^{61} -2647.78 q^{62} +2243.02 q^{63} +4096.00 q^{64} -22469.6 q^{65} -2345.47 q^{66} -5691.52 q^{67} -18840.1 q^{68} -1246.38 q^{69} -944.632 q^{70} -30857.3 q^{71} -15196.7 q^{72} +24861.4 q^{73} -23863.7 q^{74} +1472.56 q^{75} +1863.39 q^{76} +2350.93 q^{77} -8470.51 q^{78} -43677.4 q^{79} +6400.00 q^{80} +55033.0 q^{81} -29930.1 q^{82} -42211.7 q^{83} -356.103 q^{84} -29437.7 q^{85} +32635.4 q^{86} +6518.86 q^{87} -15927.8 q^{88} +92378.1 q^{89} -23744.9 q^{90} +8490.22 q^{91} -8464.00 q^{92} -1559.61 q^{93} +59275.5 q^{94} +2911.55 q^{95} +2412.64 q^{96} -4948.63 q^{97} -66871.1 q^{98} +59094.5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 12 q^{2} - 34 q^{3} + 48 q^{4} + 75 q^{5} - 136 q^{6} - 121 q^{7} + 192 q^{8} + 89 q^{9} + 300 q^{10} - 502 q^{11} - 544 q^{12} + 76 q^{13} - 484 q^{14} - 850 q^{15} + 768 q^{16} - 1167 q^{17} + 356 q^{18}+ \cdots + 344510 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) 2.35610 0.151144 0.0755718 − 0.997140i \(-0.475922\pi\)
0.0755718 + 0.997140i \(0.475922\pi\)
\(4\) 16.0000 0.500000
\(5\) 25.0000 0.447214
\(6\) 9.42439 0.106875
\(7\) −9.44632 −0.0728648 −0.0364324 − 0.999336i \(-0.511599\pi\)
−0.0364324 + 0.999336i \(0.511599\pi\)
\(8\) 64.0000 0.353553
\(9\) −237.449 −0.977156
\(10\) 100.000 0.316228
\(11\) −248.872 −0.620148 −0.310074 − 0.950712i \(-0.600354\pi\)
−0.310074 + 0.950712i \(0.600354\pi\)
\(12\) 37.6975 0.0755718
\(13\) −898.786 −1.47502 −0.737510 − 0.675336i \(-0.763998\pi\)
−0.737510 + 0.675336i \(0.763998\pi\)
\(14\) −37.7853 −0.0515232
\(15\) 58.9024 0.0675935
\(16\) 256.000 0.250000
\(17\) −1177.51 −0.988193 −0.494097 − 0.869407i \(-0.664501\pi\)
−0.494097 + 0.869407i \(0.664501\pi\)
\(18\) −949.795 −0.690953
\(19\) 116.462 0.0740117 0.0370058 − 0.999315i \(-0.488218\pi\)
0.0370058 + 0.999315i \(0.488218\pi\)
\(20\) 400.000 0.223607
\(21\) −22.2564 −0.0110130
\(22\) −995.490 −0.438511
\(23\) −529.000 −0.208514
\(24\) 150.790 0.0534374
\(25\) 625.000 0.200000
\(26\) −3595.14 −1.04300
\(27\) −1131.98 −0.298835
\(28\) −151.141 −0.0364324
\(29\) 2766.81 0.610919 0.305459 − 0.952205i \(-0.401190\pi\)
0.305459 + 0.952205i \(0.401190\pi\)
\(30\) 235.610 0.0477958
\(31\) −661.945 −0.123714 −0.0618568 − 0.998085i \(-0.519702\pi\)
−0.0618568 + 0.998085i \(0.519702\pi\)
\(32\) 1024.00 0.176777
\(33\) −586.368 −0.0937314
\(34\) −4710.03 −0.698758
\(35\) −236.158 −0.0325861
\(36\) −3799.18 −0.488578
\(37\) −5965.93 −0.716430 −0.358215 − 0.933639i \(-0.616614\pi\)
−0.358215 + 0.933639i \(0.616614\pi\)
\(38\) 465.848 0.0523342
\(39\) −2117.63 −0.222940
\(40\) 1600.00 0.158114
\(41\) −7482.54 −0.695167 −0.347584 − 0.937649i \(-0.612998\pi\)
−0.347584 + 0.937649i \(0.612998\pi\)
\(42\) −89.0258 −0.00778740
\(43\) 8158.86 0.672912 0.336456 − 0.941699i \(-0.390772\pi\)
0.336456 + 0.941699i \(0.390772\pi\)
\(44\) −3981.96 −0.310074
\(45\) −5936.22 −0.436997
\(46\) −2116.00 −0.147442
\(47\) 14818.9 0.978522 0.489261 − 0.872137i \(-0.337267\pi\)
0.489261 + 0.872137i \(0.337267\pi\)
\(48\) 603.161 0.0377859
\(49\) −16717.8 −0.994691
\(50\) 2500.00 0.141421
\(51\) −2774.32 −0.149359
\(52\) −14380.6 −0.737510
\(53\) −24766.8 −1.21110 −0.605551 − 0.795807i \(-0.707047\pi\)
−0.605551 + 0.795807i \(0.707047\pi\)
\(54\) −4527.94 −0.211308
\(55\) −6221.81 −0.277339
\(56\) −604.565 −0.0257616
\(57\) 274.396 0.0111864
\(58\) 11067.2 0.431985
\(59\) −7935.48 −0.296786 −0.148393 − 0.988928i \(-0.547410\pi\)
−0.148393 + 0.988928i \(0.547410\pi\)
\(60\) 942.439 0.0337968
\(61\) 5864.34 0.201788 0.100894 − 0.994897i \(-0.467830\pi\)
0.100894 + 0.994897i \(0.467830\pi\)
\(62\) −2647.78 −0.0874787
\(63\) 2243.02 0.0712002
\(64\) 4096.00 0.125000
\(65\) −22469.6 −0.659649
\(66\) −2345.47 −0.0662781
\(67\) −5691.52 −0.154896 −0.0774482 − 0.996996i \(-0.524677\pi\)
−0.0774482 + 0.996996i \(0.524677\pi\)
\(68\) −18840.1 −0.494097
\(69\) −1246.38 −0.0315156
\(70\) −944.632 −0.0230419
\(71\) −30857.3 −0.726460 −0.363230 − 0.931699i \(-0.618326\pi\)
−0.363230 + 0.931699i \(0.618326\pi\)
\(72\) −15196.7 −0.345477
\(73\) 24861.4 0.546032 0.273016 − 0.962009i \(-0.411979\pi\)
0.273016 + 0.962009i \(0.411979\pi\)
\(74\) −23863.7 −0.506592
\(75\) 1472.56 0.0302287
\(76\) 1863.39 0.0370058
\(77\) 2350.93 0.0451869
\(78\) −8470.51 −0.157642
\(79\) −43677.4 −0.787389 −0.393694 − 0.919241i \(-0.628803\pi\)
−0.393694 + 0.919241i \(0.628803\pi\)
\(80\) 6400.00 0.111803
\(81\) 55033.0 0.931989
\(82\) −29930.1 −0.491557
\(83\) −42211.7 −0.672570 −0.336285 − 0.941760i \(-0.609170\pi\)
−0.336285 + 0.941760i \(0.609170\pi\)
\(84\) −356.103 −0.00550652
\(85\) −29437.7 −0.441933
\(86\) 32635.4 0.475821
\(87\) 6518.86 0.0923365
\(88\) −15927.8 −0.219255
\(89\) 92378.1 1.23621 0.618107 − 0.786094i \(-0.287900\pi\)
0.618107 + 0.786094i \(0.287900\pi\)
\(90\) −23744.9 −0.309004
\(91\) 8490.22 0.107477
\(92\) −8464.00 −0.104257
\(93\) −1559.61 −0.0186985
\(94\) 59275.5 0.691920
\(95\) 2911.55 0.0330990
\(96\) 2412.64 0.0267187
\(97\) −4948.63 −0.0534018 −0.0267009 − 0.999643i \(-0.508500\pi\)
−0.0267009 + 0.999643i \(0.508500\pi\)
\(98\) −66871.1 −0.703353
\(99\) 59094.5 0.605981
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.d.1.3 ✓ 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.6.a.d.1.3 ✓ 3 1.1 even 1 trivial