Properties

Label 930.2.a.h.1.1
Level $930$
Weight $2$
Character 930.1
Self dual yes
Analytic conductor $7.426$
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] = [930,2,Mod(1,930)] 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("930.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(930, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 930.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-1,1,1,1,-1,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(7.42608738798\)
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 \(\chi\) \(=\) 930.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -4.00000 q^{11} +1.00000 q^{12} -4.00000 q^{13} +2.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} +2.00000 q^{17} -1.00000 q^{18} -8.00000 q^{19} +1.00000 q^{20} -2.00000 q^{21} +4.00000 q^{22} -8.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} +4.00000 q^{26} +1.00000 q^{27} -2.00000 q^{28} +4.00000 q^{29} -1.00000 q^{30} -1.00000 q^{31} -1.00000 q^{32} -4.00000 q^{33} -2.00000 q^{34} -2.00000 q^{35} +1.00000 q^{36} -12.0000 q^{37} +8.00000 q^{38} -4.00000 q^{39} -1.00000 q^{40} +10.0000 q^{41} +2.00000 q^{42} +8.00000 q^{43} -4.00000 q^{44} +1.00000 q^{45} +8.00000 q^{46} -4.00000 q^{47} +1.00000 q^{48} -3.00000 q^{49} -1.00000 q^{50} +2.00000 q^{51} -4.00000 q^{52} +6.00000 q^{53} -1.00000 q^{54} -4.00000 q^{55} +2.00000 q^{56} -8.00000 q^{57} -4.00000 q^{58} +2.00000 q^{59} +1.00000 q^{60} +10.0000 q^{61} +1.00000 q^{62} -2.00000 q^{63} +1.00000 q^{64} -4.00000 q^{65} +4.00000 q^{66} -6.00000 q^{67} +2.00000 q^{68} -8.00000 q^{69} +2.00000 q^{70} +6.00000 q^{71} -1.00000 q^{72} -4.00000 q^{73} +12.0000 q^{74} +1.00000 q^{75} -8.00000 q^{76} +8.00000 q^{77} +4.00000 q^{78} -8.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} -10.0000 q^{82} +4.00000 q^{83} -2.00000 q^{84} +2.00000 q^{85} -8.00000 q^{86} +4.00000 q^{87} +4.00000 q^{88} -1.00000 q^{90} +8.00000 q^{91} -8.00000 q^{92} -1.00000 q^{93} +4.00000 q^{94} -8.00000 q^{95} -1.00000 q^{96} -18.0000 q^{97} +3.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) −1.00000 −0.408248
\(7\) −2.00000 −0.755929 −0.377964 − 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) −4.00000 −1.20605 −0.603023 − 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 1.00000 0.288675
\(13\) −4.00000 −1.10940 −0.554700 − 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 2.00000 0.534522
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) 2.00000 0.485071 0.242536 − 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) −1.00000 −0.235702
\(19\) −8.00000 −1.83533 −0.917663 − 0.397360i \(-0.869927\pi\)
−0.917663 + 0.397360i \(0.869927\pi\)
\(20\) 1.00000 0.223607
\(21\) −2.00000 −0.436436
\(22\) 4.00000 0.852803
\(23\) −8.00000 −1.66812 −0.834058 − 0.551677i \(-0.813988\pi\)
−0.834058 + 0.551677i \(0.813988\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) 4.00000 0.784465
\(27\) 1.00000 0.192450
\(28\) −2.00000 −0.377964
\(29\) 4.00000 0.742781 0.371391 − 0.928477i \(-0.378881\pi\)
0.371391 + 0.928477i \(0.378881\pi\)
\(30\) −1.00000 −0.182574
\(31\) −1.00000 −0.179605
\(32\) −1.00000 −0.176777
\(33\) −4.00000 −0.696311
\(34\) −2.00000 −0.342997
\(35\) −2.00000 −0.338062
\(36\) 1.00000 0.166667
\(37\) −12.0000 −1.97279 −0.986394 − 0.164399i \(-0.947432\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) 8.00000 1.29777
\(39\) −4.00000 −0.640513
\(40\) −1.00000 −0.158114
\(41\) 10.0000 1.56174 0.780869 − 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) 2.00000 0.308607
\(43\) 8.00000 1.21999 0.609994 − 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) −4.00000 −0.603023
\(45\) 1.00000 0.149071
\(46\) 8.00000 1.17954
\(47\) −4.00000 −0.583460 −0.291730 − 0.956501i \(-0.594231\pi\)
−0.291730 + 0.956501i \(0.594231\pi\)
\(48\) 1.00000 0.144338
\(49\) −3.00000 −0.428571
\(50\) −1.00000 −0.141421
\(51\) 2.00000 0.280056
\(52\) −4.00000 −0.554700
\(53\) 6.00000 0.824163 0.412082 − 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) −1.00000 −0.136083
\(55\) −4.00000 −0.539360
\(56\) 2.00000 0.267261
\(57\) −8.00000 −1.05963
\(58\) −4.00000 −0.525226
\(59\) 2.00000 0.260378 0.130189 − 0.991489i \(-0.458442\pi\)
0.130189 + 0.991489i \(0.458442\pi\)
\(60\) 1.00000 0.129099
\(61\) 10.0000 1.28037 0.640184 − 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 1.00000 0.127000
\(63\) −2.00000 −0.251976
\(64\) 1.00000 0.125000
\(65\) −4.00000 −0.496139
\(66\) 4.00000 0.492366
\(67\) −6.00000 −0.733017 −0.366508 − 0.930415i \(-0.619447\pi\)
−0.366508 + 0.930415i \(0.619447\pi\)
\(68\) 2.00000 0.242536
\(69\) −8.00000 −0.963087
\(70\) 2.00000 0.239046
\(71\) 6.00000 0.712069 0.356034 − 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) −1.00000 −0.117851
\(73\) −4.00000 −0.468165 −0.234082 − 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) 12.0000 1.39497
\(75\) 1.00000 0.115470
\(76\) −8.00000 −0.917663
\(77\) 8.00000 0.911685
\(78\) 4.00000 0.452911
\(79\) −8.00000 −0.900070 −0.450035 − 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) −10.0000 −1.10432
\(83\) 4.00000 0.439057 0.219529 − 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) −2.00000 −0.218218
\(85\) 2.00000 0.216930
\(86\) −8.00000 −0.862662
\(87\) 4.00000 0.428845
\(88\) 4.00000 0.426401
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) −1.00000 −0.105409
\(91\) 8.00000 0.838628
\(92\) −8.00000 −0.834058
\(93\) −1.00000 −0.103695
\(94\) 4.00000 0.412568
\(95\) −8.00000 −0.820783
\(96\) −1.00000 −0.102062
\(97\) −18.0000 −1.82762 −0.913812 − 0.406138i \(-0.866875\pi\)
−0.913812 + 0.406138i \(0.866875\pi\)
\(98\) 3.00000 0.303046
\(99\) −4.00000 −0.402015
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 930.2.a.h.1.1 ✓ 1
3.2 odd 2 2790.2.a.p.1.1 1
4.3 odd 2 7440.2.a.m.1.1 1
5.2 odd 4 4650.2.d.b.3349.1 2
5.3 odd 4 4650.2.d.b.3349.2 2
5.4 even 2 4650.2.a.bg.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.a.h.1.1 ✓ 1 1.1 even 1 trivial
2790.2.a.p.1.1 1 3.2 odd 2
4650.2.a.bg.1.1 1 5.4 even 2
4650.2.d.b.3349.1 2 5.2 odd 4
4650.2.d.b.3349.2 2 5.3 odd 4
7440.2.a.m.1.1 1 4.3 odd 2