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Properties

Label 230.2.a.a.1.1
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: \(\Q(\sqrt{21}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 5 \) 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 \(2.79129\) 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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -2.79129 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.79129 q^{6} -1.79129 q^{7} -1.00000 q^{8} +4.79129 q^{9} +1.00000 q^{10} -0.791288 q^{11} -2.79129 q^{12} +5.79129 q^{13} +1.79129 q^{14} +2.79129 q^{15} +1.00000 q^{16} +0.791288 q^{17} -4.79129 q^{18} +5.79129 q^{19} -1.00000 q^{20} +5.00000 q^{21} +0.791288 q^{22} +1.00000 q^{23} +2.79129 q^{24} +1.00000 q^{25} -5.79129 q^{26} -5.00000 q^{27} -1.79129 q^{28} +7.58258 q^{29} -2.79129 q^{30} -3.37386 q^{31} -1.00000 q^{32} +2.20871 q^{33} -0.791288 q^{34} +1.79129 q^{35} +4.79129 q^{36} -4.00000 q^{37} -5.79129 q^{38} -16.1652 q^{39} +1.00000 q^{40} -6.79129 q^{41} -5.00000 q^{42} +11.1652 q^{43} -0.791288 q^{44} -4.79129 q^{45} -1.00000 q^{46} -4.41742 q^{47} -2.79129 q^{48} -3.79129 q^{49} -1.00000 q^{50} -2.20871 q^{51} +5.79129 q^{52} +6.00000 q^{53} +5.00000 q^{54} +0.791288 q^{55} +1.79129 q^{56} -16.1652 q^{57} -7.58258 q^{58} -13.5826 q^{59} +2.79129 q^{60} +10.3739 q^{61} +3.37386 q^{62} -8.58258 q^{63} +1.00000 q^{64} -5.79129 q^{65} -2.20871 q^{66} +11.1652 q^{67} +0.791288 q^{68} -2.79129 q^{69} -1.79129 q^{70} +8.37386 q^{71} -4.79129 q^{72} +12.7477 q^{73} +4.00000 q^{74} -2.79129 q^{75} +5.79129 q^{76} +1.41742 q^{77} +16.1652 q^{78} +8.00000 q^{79} -1.00000 q^{80} -0.417424 q^{81} +6.79129 q^{82} -6.00000 q^{83} +5.00000 q^{84} -0.791288 q^{85} -11.1652 q^{86} -21.1652 q^{87} +0.791288 q^{88} +15.1652 q^{89} +4.79129 q^{90} -10.3739 q^{91} +1.00000 q^{92} +9.41742 q^{93} +4.41742 q^{94} -5.79129 q^{95} +2.79129 q^{96} -7.95644 q^{97} +3.79129 q^{98} -3.79129 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - q^{3} + 2 q^{4} - 2 q^{5} + q^{6} + q^{7} - 2 q^{8} + 5 q^{9} + 2 q^{10} + 3 q^{11} - q^{12} + 7 q^{13} - q^{14} + q^{15} + 2 q^{16} - 3 q^{17} - 5 q^{18} + 7 q^{19} - 2 q^{20} + 10 q^{21}+ \cdots - 3 q^{99}+O(q^{100}) \) 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
\(3\) −2.79129 −1.61155 −0.805775 − 0.592221i \(-0.798251\pi\)
−0.805775 + 0.592221i \(0.798251\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 2.79129 1.13954
\(7\) −1.79129 −0.677043 −0.338522 − 0.940959i \(-0.609927\pi\)
−0.338522 + 0.940959i \(0.609927\pi\)
\(8\) −1.00000 −0.353553
\(9\) 4.79129 1.59710
\(10\) 1.00000 0.316228
\(11\) −0.791288 −0.238582 −0.119291 − 0.992859i \(-0.538062\pi\)
−0.119291 + 0.992859i \(0.538062\pi\)
\(12\) −2.79129 −0.805775
\(13\) 5.79129 1.60621 0.803107 − 0.595835i \(-0.203179\pi\)
0.803107 + 0.595835i \(0.203179\pi\)
\(14\) 1.79129 0.478742
\(15\) 2.79129 0.720707
\(16\) 1.00000 0.250000
\(17\) 0.791288 0.191915 0.0959577 − 0.995385i \(-0.469409\pi\)
0.0959577 + 0.995385i \(0.469409\pi\)
\(18\) −4.79129 −1.12932
\(19\) 5.79129 1.32861 0.664306 − 0.747460i \(-0.268727\pi\)
0.664306 + 0.747460i \(0.268727\pi\)
\(20\) −1.00000 −0.223607
\(21\) 5.00000 1.09109
\(22\) 0.791288 0.168703
\(23\) 1.00000 0.208514
\(24\) 2.79129 0.569769
\(25\) 1.00000 0.200000
\(26\) −5.79129 −1.13576
\(27\) −5.00000 −0.962250
\(28\) −1.79129 −0.338522
\(29\) 7.58258 1.40805 0.704024 − 0.710176i \(-0.251385\pi\)
0.704024 + 0.710176i \(0.251385\pi\)
\(30\) −2.79129 −0.509617
\(31\) −3.37386 −0.605964 −0.302982 − 0.952996i \(-0.597982\pi\)
−0.302982 + 0.952996i \(0.597982\pi\)
\(32\) −1.00000 −0.176777
\(33\) 2.20871 0.384487
\(34\) −0.791288 −0.135705
\(35\) 1.79129 0.302783
\(36\) 4.79129 0.798548
\(37\) −4.00000 −0.657596 −0.328798 − 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) −5.79129 −0.939471
\(39\) −16.1652 −2.58850
\(40\) 1.00000 0.158114
\(41\) −6.79129 −1.06062 −0.530310 − 0.847804i \(-0.677925\pi\)
−0.530310 + 0.847804i \(0.677925\pi\)
\(42\) −5.00000 −0.771517
\(43\) 11.1652 1.70267 0.851335 − 0.524623i \(-0.175794\pi\)
0.851335 + 0.524623i \(0.175794\pi\)
\(44\) −0.791288 −0.119291
\(45\) −4.79129 −0.714243
\(46\) −1.00000 −0.147442
\(47\) −4.41742 −0.644348 −0.322174 − 0.946681i \(-0.604414\pi\)
−0.322174 + 0.946681i \(0.604414\pi\)
\(48\) −2.79129 −0.402888
\(49\) −3.79129 −0.541613
\(50\) −1.00000 −0.141421
\(51\) −2.20871 −0.309282
\(52\) 5.79129 0.803107
\(53\) 6.00000 0.824163 0.412082 − 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 5.00000 0.680414
\(55\) 0.791288 0.106697
\(56\) 1.79129 0.239371
\(57\) −16.1652 −2.14113
\(58\) −7.58258 −0.995641
\(59\) −13.5826 −1.76830 −0.884150 − 0.467202i \(-0.845262\pi\)
−0.884150 + 0.467202i \(0.845262\pi\)
\(60\) 2.79129 0.360354
\(61\) 10.3739 1.32824 0.664119 − 0.747627i \(-0.268807\pi\)
0.664119 + 0.747627i \(0.268807\pi\)
\(62\) 3.37386 0.428481
\(63\) −8.58258 −1.08130
\(64\) 1.00000 0.125000
\(65\) −5.79129 −0.718321
\(66\) −2.20871 −0.271874
\(67\) 11.1652 1.36404 0.682020 − 0.731333i \(-0.261102\pi\)
0.682020 + 0.731333i \(0.261102\pi\)
\(68\) 0.791288 0.0959577
\(69\) −2.79129 −0.336032
\(70\) −1.79129 −0.214100
\(71\) 8.37386 0.993795 0.496897 − 0.867809i \(-0.334473\pi\)
0.496897 + 0.867809i \(0.334473\pi\)
\(72\) −4.79129 −0.564659
\(73\) 12.7477 1.49201 0.746004 − 0.665941i \(-0.231970\pi\)
0.746004 + 0.665941i \(0.231970\pi\)
\(74\) 4.00000 0.464991
\(75\) −2.79129 −0.322310
\(76\) 5.79129 0.664306
\(77\) 1.41742 0.161530
\(78\) 16.1652 1.83034
\(79\) 8.00000 0.900070 0.450035 − 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) −1.00000 −0.111803
\(81\) −0.417424 −0.0463805
\(82\) 6.79129 0.749972
\(83\) −6.00000 −0.658586 −0.329293 − 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 5.00000 0.545545
\(85\) −0.791288 −0.0858272
\(86\) −11.1652 −1.20397
\(87\) −21.1652 −2.26914
\(88\) 0.791288 0.0843516
\(89\) 15.1652 1.60750 0.803751 − 0.594965i \(-0.202834\pi\)
0.803751 + 0.594965i \(0.202834\pi\)
\(90\) 4.79129 0.505046
\(91\) −10.3739 −1.08748
\(92\) 1.00000 0.104257
\(93\) 9.41742 0.976541
\(94\) 4.41742 0.455623
\(95\) −5.79129 −0.594174
\(96\) 2.79129 0.284885
\(97\) −7.95644 −0.807854 −0.403927 − 0.914791i \(-0.632355\pi\)
−0.403927 + 0.914791i \(0.632355\pi\)
\(98\) 3.79129 0.382978
\(99\) −3.79129 −0.381039
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.a.1.1 ✓ 2
3.2 odd 2 2070.2.a.x.1.1 2
4.3 odd 2 1840.2.a.n.1.2 2
5.2 odd 4 1150.2.b.g.599.2 4
5.3 odd 4 1150.2.b.g.599.3 4
5.4 even 2 1150.2.a.o.1.2 2
8.3 odd 2 7360.2.a.bk.1.1 2
8.5 even 2 7360.2.a.bq.1.2 2
20.19 odd 2 9200.2.a.bs.1.1 2
23.22 odd 2 5290.2.a.e.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.a.1.1 ✓ 2 1.1 even 1 trivial
1150.2.a.o.1.2 2 5.4 even 2
1150.2.b.g.599.2 4 5.2 odd 4
1150.2.b.g.599.3 4 5.3 odd 4
1840.2.a.n.1.2 2 4.3 odd 2
2070.2.a.x.1.1 2 3.2 odd 2
5290.2.a.e.1.1 2 23.22 odd 2
7360.2.a.bk.1.1 2 8.3 odd 2
7360.2.a.bq.1.2 2 8.5 even 2
9200.2.a.bs.1.1 2 20.19 odd 2