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Properties

Label 230.2.a.d.1.1
Level $230$
Weight $2$
Character 230.1
Self dual yes
Analytic conductor $1.837$
Analytic rank $0$
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] = [230,2,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, 2); 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, 2, names="a")
 
Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 230.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,3,1] 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: \(1.83655924649\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.1101.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 9x + 12 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-3.11903\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -3.11903 q^{3} +1.00000 q^{4} -1.00000 q^{5} -3.11903 q^{6} +4.50973 q^{7} +1.00000 q^{8} +6.72833 q^{9} -1.00000 q^{10} +4.33763 q^{11} -3.11903 q^{12} -3.72833 q^{13} +4.50973 q^{14} +3.11903 q^{15} +1.00000 q^{16} +1.11903 q^{17} +6.72833 q^{18} +4.50973 q^{19} -1.00000 q^{20} -14.0660 q^{21} +4.33763 q^{22} -1.00000 q^{23} -3.11903 q^{24} +1.00000 q^{25} -3.72833 q^{26} -11.6288 q^{27} +4.50973 q^{28} -8.23805 q^{29} +3.11903 q^{30} +1.72833 q^{31} +1.00000 q^{32} -13.5292 q^{33} +1.11903 q^{34} -4.50973 q^{35} +6.72833 q^{36} -0.781399 q^{37} +4.50973 q^{38} +11.6288 q^{39} -1.00000 q^{40} +3.90043 q^{41} -14.0660 q^{42} +8.00000 q^{43} +4.33763 q^{44} -6.72833 q^{45} -1.00000 q^{46} -11.4567 q^{47} -3.11903 q^{48} +13.3376 q^{49} +1.00000 q^{50} -3.49027 q^{51} -3.72833 q^{52} -6.00000 q^{53} -11.6288 q^{54} -4.33763 q^{55} +4.50973 q^{56} -14.0660 q^{57} -8.23805 q^{58} -2.23805 q^{59} +3.11903 q^{60} +3.55623 q^{61} +1.72833 q^{62} +30.3429 q^{63} +1.00000 q^{64} +3.72833 q^{65} -13.5292 q^{66} +2.43720 q^{67} +1.11903 q^{68} +3.11903 q^{69} -4.50973 q^{70} +7.11903 q^{71} +6.72833 q^{72} -9.45665 q^{73} -0.781399 q^{74} -3.11903 q^{75} +4.50973 q^{76} +19.5615 q^{77} +11.6288 q^{78} -14.9133 q^{79} -1.00000 q^{80} +16.0854 q^{81} +3.90043 q^{82} +2.78140 q^{83} -14.0660 q^{84} -1.11903 q^{85} +8.00000 q^{86} +25.6947 q^{87} +4.33763 q^{88} -7.69471 q^{89} -6.72833 q^{90} -16.8137 q^{91} -1.00000 q^{92} -5.39070 q^{93} -11.4567 q^{94} -4.50973 q^{95} -3.11903 q^{96} -0.642920 q^{97} +13.3376 q^{98} +29.1850 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + q^{3} + 3 q^{4} - 3 q^{5} + q^{6} + 3 q^{7} + 3 q^{8} + 10 q^{9} - 3 q^{10} + 3 q^{11} + q^{12} - q^{13} + 3 q^{14} - q^{15} + 3 q^{16} - 7 q^{17} + 10 q^{18} + 3 q^{19} - 3 q^{20}+ \cdots + 57 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −3.11903 −1.80077 −0.900385 − 0.435093i \(-0.856715\pi\)
−0.900385 + 0.435093i \(0.856715\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) −3.11903 −1.27334
\(7\) 4.50973 1.70452 0.852258 − 0.523122i \(-0.175233\pi\)
0.852258 + 0.523122i \(0.175233\pi\)
\(8\) 1.00000 0.353553
\(9\) 6.72833 2.24278
\(10\) −1.00000 −0.316228
\(11\) 4.33763 1.30784 0.653922 − 0.756562i \(-0.273122\pi\)
0.653922 + 0.756562i \(0.273122\pi\)
\(12\) −3.11903 −0.900385
\(13\) −3.72833 −1.03405 −0.517026 − 0.855970i \(-0.672961\pi\)
−0.517026 + 0.855970i \(0.672961\pi\)
\(14\) 4.50973 1.20527
\(15\) 3.11903 0.805329
\(16\) 1.00000 0.250000
\(17\) 1.11903 0.271404 0.135702 − 0.990750i \(-0.456671\pi\)
0.135702 + 0.990750i \(0.456671\pi\)
\(18\) 6.72833 1.58588
\(19\) 4.50973 1.03460 0.517301 − 0.855803i \(-0.326937\pi\)
0.517301 + 0.855803i \(0.326937\pi\)
\(20\) −1.00000 −0.223607
\(21\) −14.0660 −3.06944
\(22\) 4.33763 0.924785
\(23\) −1.00000 −0.208514
\(24\) −3.11903 −0.636669
\(25\) 1.00000 0.200000
\(26\) −3.72833 −0.731185
\(27\) −11.6288 −2.23795
\(28\) 4.50973 0.852258
\(29\) −8.23805 −1.52977 −0.764884 − 0.644168i \(-0.777204\pi\)
−0.764884 + 0.644168i \(0.777204\pi\)
\(30\) 3.11903 0.569454
\(31\) 1.72833 0.310417 0.155208 − 0.987882i \(-0.450395\pi\)
0.155208 + 0.987882i \(0.450395\pi\)
\(32\) 1.00000 0.176777
\(33\) −13.5292 −2.35513
\(34\) 1.11903 0.191911
\(35\) −4.50973 −0.762283
\(36\) 6.72833 1.12139
\(37\) −0.781399 −0.128461 −0.0642306 − 0.997935i \(-0.520459\pi\)
−0.0642306 + 0.997935i \(0.520459\pi\)
\(38\) 4.50973 0.731574
\(39\) 11.6288 1.86209
\(40\) −1.00000 −0.158114
\(41\) 3.90043 0.609144 0.304572 − 0.952489i \(-0.401487\pi\)
0.304572 + 0.952489i \(0.401487\pi\)
\(42\) −14.0660 −2.17042
\(43\) 8.00000 1.21999 0.609994 − 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 4.33763 0.653922
\(45\) −6.72833 −1.00300
\(46\) −1.00000 −0.147442
\(47\) −11.4567 −1.67112 −0.835562 − 0.549396i \(-0.814858\pi\)
−0.835562 + 0.549396i \(0.814858\pi\)
\(48\) −3.11903 −0.450193
\(49\) 13.3376 1.90538
\(50\) 1.00000 0.141421
\(51\) −3.49027 −0.488736
\(52\) −3.72833 −0.517026
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) −11.6288 −1.58247
\(55\) −4.33763 −0.584886
\(56\) 4.50973 0.602637
\(57\) −14.0660 −1.86308
\(58\) −8.23805 −1.08171
\(59\) −2.23805 −0.291370 −0.145685 − 0.989331i \(-0.546539\pi\)
−0.145685 + 0.989331i \(0.546539\pi\)
\(60\) 3.11903 0.402665
\(61\) 3.55623 0.455329 0.227664 − 0.973740i \(-0.426891\pi\)
0.227664 + 0.973740i \(0.426891\pi\)
\(62\) 1.72833 0.219498
\(63\) 30.3429 3.82285
\(64\) 1.00000 0.125000
\(65\) 3.72833 0.462442
\(66\) −13.5292 −1.66533
\(67\) 2.43720 0.297752 0.148876 − 0.988856i \(-0.452435\pi\)
0.148876 + 0.988856i \(0.452435\pi\)
\(68\) 1.11903 0.135702
\(69\) 3.11903 0.375487
\(70\) −4.50973 −0.539015
\(71\) 7.11903 0.844873 0.422437 − 0.906393i \(-0.361175\pi\)
0.422437 + 0.906393i \(0.361175\pi\)
\(72\) 6.72833 0.792941
\(73\) −9.45665 −1.10682 −0.553409 − 0.832910i \(-0.686673\pi\)
−0.553409 + 0.832910i \(0.686673\pi\)
\(74\) −0.781399 −0.0908357
\(75\) −3.11903 −0.360154
\(76\) 4.50973 0.517301
\(77\) 19.5615 2.22924
\(78\) 11.6288 1.31670
\(79\) −14.9133 −1.67788 −0.838939 − 0.544225i \(-0.816824\pi\)
−0.838939 + 0.544225i \(0.816824\pi\)
\(80\) −1.00000 −0.111803
\(81\) 16.0854 1.78727
\(82\) 3.90043 0.430730
\(83\) 2.78140 0.305298 0.152649 − 0.988280i \(-0.451220\pi\)
0.152649 + 0.988280i \(0.451220\pi\)
\(84\) −14.0660 −1.53472
\(85\) −1.11903 −0.121375
\(86\) 8.00000 0.862662
\(87\) 25.6947 2.75476
\(88\) 4.33763 0.462393
\(89\) −7.69471 −0.815637 −0.407819 − 0.913063i \(-0.633710\pi\)
−0.407819 + 0.913063i \(0.633710\pi\)
\(90\) −6.72833 −0.709228
\(91\) −16.8137 −1.76256
\(92\) −1.00000 −0.104257
\(93\) −5.39070 −0.558989
\(94\) −11.4567 −1.18166
\(95\) −4.50973 −0.462688
\(96\) −3.11903 −0.318334
\(97\) −0.642920 −0.0652786 −0.0326393 − 0.999467i \(-0.510391\pi\)
−0.0326393 + 0.999467i \(0.510391\pi\)
\(98\) 13.3376 1.34730
\(99\) 29.1850 2.93320
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 230.2.a.d.1.1 ✓ 3
3.2 odd 2 2070.2.a.z.1.3 3
4.3 odd 2 1840.2.a.r.1.3 3
5.2 odd 4 1150.2.b.j.599.6 6
5.3 odd 4 1150.2.b.j.599.1 6
5.4 even 2 1150.2.a.q.1.3 3
8.3 odd 2 7360.2.a.ce.1.1 3
8.5 even 2 7360.2.a.bz.1.3 3
20.19 odd 2 9200.2.a.cf.1.1 3
23.22 odd 2 5290.2.a.r.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.d.1.1 ✓ 3 1.1 even 1 trivial
1150.2.a.q.1.3 3 5.4 even 2
1150.2.b.j.599.1 6 5.3 odd 4
1150.2.b.j.599.6 6 5.2 odd 4
1840.2.a.r.1.3 3 4.3 odd 2
2070.2.a.z.1.3 3 3.2 odd 2
5290.2.a.r.1.1 3 23.22 odd 2
7360.2.a.bz.1.3 3 8.5 even 2
7360.2.a.ce.1.1 3 8.3 odd 2
9200.2.a.cf.1.1 3 20.19 odd 2