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Properties

Label 230.2.a.c.1.2
Level $230$
Weight $2$
Character 230.1
Self dual yes
Analytic conductor $1.837$
Analytic rank $0$
Dimension $2$
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 = [2,2,1] 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: \(1.83655924649\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) 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 1.2
Root \(1.61803\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.61803 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.61803 q^{6} -0.618034 q^{7} +1.00000 q^{8} -0.381966 q^{9} +1.00000 q^{10} -2.85410 q^{11} +1.61803 q^{12} -7.09017 q^{13} -0.618034 q^{14} +1.61803 q^{15} +1.00000 q^{16} +6.09017 q^{17} -0.381966 q^{18} +1.85410 q^{19} +1.00000 q^{20} -1.00000 q^{21} -2.85410 q^{22} +1.00000 q^{23} +1.61803 q^{24} +1.00000 q^{25} -7.09017 q^{26} -5.47214 q^{27} -0.618034 q^{28} -9.23607 q^{29} +1.61803 q^{30} +9.09017 q^{31} +1.00000 q^{32} -4.61803 q^{33} +6.09017 q^{34} -0.618034 q^{35} -0.381966 q^{36} +6.47214 q^{37} +1.85410 q^{38} -11.4721 q^{39} +1.00000 q^{40} +3.32624 q^{41} -1.00000 q^{42} -2.85410 q^{44} -0.381966 q^{45} +1.00000 q^{46} -3.70820 q^{47} +1.61803 q^{48} -6.61803 q^{49} +1.00000 q^{50} +9.85410 q^{51} -7.09017 q^{52} +0.472136 q^{53} -5.47214 q^{54} -2.85410 q^{55} -0.618034 q^{56} +3.00000 q^{57} -9.23607 q^{58} +1.70820 q^{59} +1.61803 q^{60} -9.32624 q^{61} +9.09017 q^{62} +0.236068 q^{63} +1.00000 q^{64} -7.09017 q^{65} -4.61803 q^{66} +14.4721 q^{67} +6.09017 q^{68} +1.61803 q^{69} -0.618034 q^{70} -4.09017 q^{71} -0.381966 q^{72} +3.23607 q^{73} +6.47214 q^{74} +1.61803 q^{75} +1.85410 q^{76} +1.76393 q^{77} -11.4721 q^{78} +1.52786 q^{79} +1.00000 q^{80} -7.70820 q^{81} +3.32624 q^{82} -6.94427 q^{83} -1.00000 q^{84} +6.09017 q^{85} -14.9443 q^{87} -2.85410 q^{88} -10.4721 q^{89} -0.381966 q^{90} +4.38197 q^{91} +1.00000 q^{92} +14.7082 q^{93} -3.70820 q^{94} +1.85410 q^{95} +1.61803 q^{96} +12.3820 q^{97} -6.61803 q^{98} +1.09017 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + q^{3} + 2 q^{4} + 2 q^{5} + q^{6} + q^{7} + 2 q^{8} - 3 q^{9} + 2 q^{10} + q^{11} + q^{12} - 3 q^{13} + q^{14} + q^{15} + 2 q^{16} + q^{17} - 3 q^{18} - 3 q^{19} + 2 q^{20} - 2 q^{21}+ \cdots - 9 q^{99}+O(q^{100}) \) 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
\(3\) 1.61803 0.934172 0.467086 − 0.884212i \(-0.345304\pi\)
0.467086 + 0.884212i \(0.345304\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 1.61803 0.660560
\(7\) −0.618034 −0.233595 −0.116797 − 0.993156i \(-0.537263\pi\)
−0.116797 + 0.993156i \(0.537263\pi\)
\(8\) 1.00000 0.353553
\(9\) −0.381966 −0.127322
\(10\) 1.00000 0.316228
\(11\) −2.85410 −0.860544 −0.430272 − 0.902699i \(-0.641582\pi\)
−0.430272 + 0.902699i \(0.641582\pi\)
\(12\) 1.61803 0.467086
\(13\) −7.09017 −1.96646 −0.983230 − 0.182372i \(-0.941623\pi\)
−0.983230 + 0.182372i \(0.941623\pi\)
\(14\) −0.618034 −0.165177
\(15\) 1.61803 0.417775
\(16\) 1.00000 0.250000
\(17\) 6.09017 1.47708 0.738542 − 0.674208i \(-0.235515\pi\)
0.738542 + 0.674208i \(0.235515\pi\)
\(18\) −0.381966 −0.0900303
\(19\) 1.85410 0.425360 0.212680 − 0.977122i \(-0.431781\pi\)
0.212680 + 0.977122i \(0.431781\pi\)
\(20\) 1.00000 0.223607
\(21\) −1.00000 −0.218218
\(22\) −2.85410 −0.608497
\(23\) 1.00000 0.208514
\(24\) 1.61803 0.330280
\(25\) 1.00000 0.200000
\(26\) −7.09017 −1.39050
\(27\) −5.47214 −1.05311
\(28\) −0.618034 −0.116797
\(29\) −9.23607 −1.71509 −0.857547 − 0.514405i \(-0.828013\pi\)
−0.857547 + 0.514405i \(0.828013\pi\)
\(30\) 1.61803 0.295411
\(31\) 9.09017 1.63264 0.816321 − 0.577598i \(-0.196010\pi\)
0.816321 + 0.577598i \(0.196010\pi\)
\(32\) 1.00000 0.176777
\(33\) −4.61803 −0.803897
\(34\) 6.09017 1.04446
\(35\) −0.618034 −0.104467
\(36\) −0.381966 −0.0636610
\(37\) 6.47214 1.06401 0.532006 − 0.846740i \(-0.321438\pi\)
0.532006 + 0.846740i \(0.321438\pi\)
\(38\) 1.85410 0.300775
\(39\) −11.4721 −1.83701
\(40\) 1.00000 0.158114
\(41\) 3.32624 0.519471 0.259736 − 0.965680i \(-0.416365\pi\)
0.259736 + 0.965680i \(0.416365\pi\)
\(42\) −1.00000 −0.154303
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) −2.85410 −0.430272
\(45\) −0.381966 −0.0569401
\(46\) 1.00000 0.147442
\(47\) −3.70820 −0.540897 −0.270449 − 0.962734i \(-0.587172\pi\)
−0.270449 + 0.962734i \(0.587172\pi\)
\(48\) 1.61803 0.233543
\(49\) −6.61803 −0.945433
\(50\) 1.00000 0.141421
\(51\) 9.85410 1.37985
\(52\) −7.09017 −0.983230
\(53\) 0.472136 0.0648529 0.0324264 − 0.999474i \(-0.489677\pi\)
0.0324264 + 0.999474i \(0.489677\pi\)
\(54\) −5.47214 −0.744663
\(55\) −2.85410 −0.384847
\(56\) −0.618034 −0.0825883
\(57\) 3.00000 0.397360
\(58\) −9.23607 −1.21276
\(59\) 1.70820 0.222389 0.111195 − 0.993799i \(-0.464532\pi\)
0.111195 + 0.993799i \(0.464532\pi\)
\(60\) 1.61803 0.208887
\(61\) −9.32624 −1.19410 −0.597051 − 0.802203i \(-0.703661\pi\)
−0.597051 + 0.802203i \(0.703661\pi\)
\(62\) 9.09017 1.15445
\(63\) 0.236068 0.0297418
\(64\) 1.00000 0.125000
\(65\) −7.09017 −0.879427
\(66\) −4.61803 −0.568441
\(67\) 14.4721 1.76805 0.884026 − 0.467437i \(-0.154823\pi\)
0.884026 + 0.467437i \(0.154823\pi\)
\(68\) 6.09017 0.738542
\(69\) 1.61803 0.194788
\(70\) −0.618034 −0.0738692
\(71\) −4.09017 −0.485414 −0.242707 − 0.970100i \(-0.578035\pi\)
−0.242707 + 0.970100i \(0.578035\pi\)
\(72\) −0.381966 −0.0450151
\(73\) 3.23607 0.378753 0.189377 − 0.981905i \(-0.439353\pi\)
0.189377 + 0.981905i \(0.439353\pi\)
\(74\) 6.47214 0.752371
\(75\) 1.61803 0.186834
\(76\) 1.85410 0.212680
\(77\) 1.76393 0.201019
\(78\) −11.4721 −1.29896
\(79\) 1.52786 0.171898 0.0859491 − 0.996300i \(-0.472608\pi\)
0.0859491 + 0.996300i \(0.472608\pi\)
\(80\) 1.00000 0.111803
\(81\) −7.70820 −0.856467
\(82\) 3.32624 0.367322
\(83\) −6.94427 −0.762233 −0.381116 − 0.924527i \(-0.624460\pi\)
−0.381116 + 0.924527i \(0.624460\pi\)
\(84\) −1.00000 −0.109109
\(85\) 6.09017 0.660572
\(86\) 0 0
\(87\) −14.9443 −1.60219
\(88\) −2.85410 −0.304248
\(89\) −10.4721 −1.11004 −0.555022 − 0.831836i \(-0.687290\pi\)
−0.555022 + 0.831836i \(0.687290\pi\)
\(90\) −0.381966 −0.0402628
\(91\) 4.38197 0.459355
\(92\) 1.00000 0.104257
\(93\) 14.7082 1.52517
\(94\) −3.70820 −0.382472
\(95\) 1.85410 0.190227
\(96\) 1.61803 0.165140
\(97\) 12.3820 1.25720 0.628599 − 0.777730i \(-0.283629\pi\)
0.628599 + 0.777730i \(0.283629\pi\)
\(98\) −6.61803 −0.668522
\(99\) 1.09017 0.109566
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.c.1.2 ✓ 2
3.2 odd 2 2070.2.a.u.1.1 2
4.3 odd 2 1840.2.a.l.1.1 2
5.2 odd 4 1150.2.b.i.599.3 4
5.3 odd 4 1150.2.b.i.599.2 4
5.4 even 2 1150.2.a.j.1.1 2
8.3 odd 2 7360.2.a.bn.1.2 2
8.5 even 2 7360.2.a.bh.1.1 2
20.19 odd 2 9200.2.a.bu.1.2 2
23.22 odd 2 5290.2.a.o.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.c.1.2 ✓ 2 1.1 even 1 trivial
1150.2.a.j.1.1 2 5.4 even 2
1150.2.b.i.599.2 4 5.3 odd 4
1150.2.b.i.599.3 4 5.2 odd 4
1840.2.a.l.1.1 2 4.3 odd 2
2070.2.a.u.1.1 2 3.2 odd 2
5290.2.a.o.1.2 2 23.22 odd 2
7360.2.a.bh.1.1 2 8.5 even 2
7360.2.a.bn.1.2 2 8.3 odd 2
9200.2.a.bu.1.2 2 20.19 odd 2