Properties

Label 930.2.a.i.1.1
Level $930$
Weight $2$
Character 930.1
Self dual yes
Analytic conductor $7.426$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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,-1] 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: \(0\)
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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -3.00000 q^{11} +1.00000 q^{12} +2.00000 q^{13} +1.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{18} +5.00000 q^{19} +1.00000 q^{20} -1.00000 q^{21} +3.00000 q^{22} +9.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} -2.00000 q^{26} +1.00000 q^{27} -1.00000 q^{28} -1.00000 q^{30} +1.00000 q^{31} -1.00000 q^{32} -3.00000 q^{33} -1.00000 q^{35} +1.00000 q^{36} +8.00000 q^{37} -5.00000 q^{38} +2.00000 q^{39} -1.00000 q^{40} +1.00000 q^{42} +11.0000 q^{43} -3.00000 q^{44} +1.00000 q^{45} -9.00000 q^{46} -6.00000 q^{47} +1.00000 q^{48} -6.00000 q^{49} -1.00000 q^{50} +2.00000 q^{52} -9.00000 q^{53} -1.00000 q^{54} -3.00000 q^{55} +1.00000 q^{56} +5.00000 q^{57} +6.00000 q^{59} +1.00000 q^{60} -10.0000 q^{61} -1.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} +2.00000 q^{65} +3.00000 q^{66} +14.0000 q^{67} +9.00000 q^{69} +1.00000 q^{70} +9.00000 q^{71} -1.00000 q^{72} -1.00000 q^{73} -8.00000 q^{74} +1.00000 q^{75} +5.00000 q^{76} +3.00000 q^{77} -2.00000 q^{78} -1.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} -12.0000 q^{83} -1.00000 q^{84} -11.0000 q^{86} +3.00000 q^{88} -9.00000 q^{89} -1.00000 q^{90} -2.00000 q^{91} +9.00000 q^{92} +1.00000 q^{93} +6.00000 q^{94} +5.00000 q^{95} -1.00000 q^{96} +14.0000 q^{97} +6.00000 q^{98} -3.00000 q^{99} +O(q^{100})\)

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
\(7\) −1.00000 −0.377964 −0.188982 − 0.981981i \(-0.560519\pi\)
−0.188982 + 0.981981i \(0.560519\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) −3.00000 −0.904534 −0.452267 − 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) 1.00000 0.288675
\(13\) 2.00000 0.554700 0.277350 − 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 1.00000 0.267261
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) −1.00000 −0.235702
\(19\) 5.00000 1.14708 0.573539 − 0.819178i \(-0.305570\pi\)
0.573539 + 0.819178i \(0.305570\pi\)
\(20\) 1.00000 0.223607
\(21\) −1.00000 −0.218218
\(22\) 3.00000 0.639602
\(23\) 9.00000 1.87663 0.938315 − 0.345782i \(-0.112386\pi\)
0.938315 + 0.345782i \(0.112386\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) −2.00000 −0.392232
\(27\) 1.00000 0.192450
\(28\) −1.00000 −0.188982
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) −1.00000 −0.182574
\(31\) 1.00000 0.179605
\(32\) −1.00000 −0.176777
\(33\) −3.00000 −0.522233
\(34\) 0 0
\(35\) −1.00000 −0.169031
\(36\) 1.00000 0.166667
\(37\) 8.00000 1.31519 0.657596 − 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) −5.00000 −0.811107
\(39\) 2.00000 0.320256
\(40\) −1.00000 −0.158114
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 1.00000 0.154303
\(43\) 11.0000 1.67748 0.838742 − 0.544529i \(-0.183292\pi\)
0.838742 + 0.544529i \(0.183292\pi\)
\(44\) −3.00000 −0.452267
\(45\) 1.00000 0.149071
\(46\) −9.00000 −1.32698
\(47\) −6.00000 −0.875190 −0.437595 − 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) 1.00000 0.144338
\(49\) −6.00000 −0.857143
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) 2.00000 0.277350
\(53\) −9.00000 −1.23625 −0.618123 − 0.786082i \(-0.712106\pi\)
−0.618123 + 0.786082i \(0.712106\pi\)
\(54\) −1.00000 −0.136083
\(55\) −3.00000 −0.404520
\(56\) 1.00000 0.133631
\(57\) 5.00000 0.662266
\(58\) 0 0
\(59\) 6.00000 0.781133 0.390567 − 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) 1.00000 0.129099
\(61\) −10.0000 −1.28037 −0.640184 − 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) −1.00000 −0.127000
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) 2.00000 0.248069
\(66\) 3.00000 0.369274
\(67\) 14.0000 1.71037 0.855186 − 0.518321i \(-0.173443\pi\)
0.855186 + 0.518321i \(0.173443\pi\)
\(68\) 0 0
\(69\) 9.00000 1.08347
\(70\) 1.00000 0.119523
\(71\) 9.00000 1.06810 0.534052 − 0.845452i \(-0.320669\pi\)
0.534052 + 0.845452i \(0.320669\pi\)
\(72\) −1.00000 −0.117851
\(73\) −1.00000 −0.117041 −0.0585206 − 0.998286i \(-0.518638\pi\)
−0.0585206 + 0.998286i \(0.518638\pi\)
\(74\) −8.00000 −0.929981
\(75\) 1.00000 0.115470
\(76\) 5.00000 0.573539
\(77\) 3.00000 0.341882
\(78\) −2.00000 −0.226455
\(79\) −1.00000 −0.112509 −0.0562544 − 0.998416i \(-0.517916\pi\)
−0.0562544 + 0.998416i \(0.517916\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −12.0000 −1.31717 −0.658586 − 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) −1.00000 −0.109109
\(85\) 0 0
\(86\) −11.0000 −1.18616
\(87\) 0 0
\(88\) 3.00000 0.319801
\(89\) −9.00000 −0.953998 −0.476999 − 0.878904i \(-0.658275\pi\)
−0.476999 + 0.878904i \(0.658275\pi\)
\(90\) −1.00000 −0.105409
\(91\) −2.00000 −0.209657
\(92\) 9.00000 0.938315
\(93\) 1.00000 0.103695
\(94\) 6.00000 0.618853
\(95\) 5.00000 0.512989
\(96\) −1.00000 −0.102062
\(97\) 14.0000 1.42148 0.710742 − 0.703452i \(-0.248359\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) 6.00000 0.606092
\(99\) −3.00000 −0.301511
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.i.1.1 ✓ 1
3.2 odd 2 2790.2.a.r.1.1 1
4.3 odd 2 7440.2.a.k.1.1 1
5.2 odd 4 4650.2.d.d.3349.1 2
5.3 odd 4 4650.2.d.d.3349.2 2
5.4 even 2 4650.2.a.bd.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.a.i.1.1 ✓ 1 1.1 even 1 trivial
2790.2.a.r.1.1 1 3.2 odd 2
4650.2.a.bd.1.1 1 5.4 even 2
4650.2.d.d.3349.1 2 5.2 odd 4
4650.2.d.d.3349.2 2 5.3 odd 4
7440.2.a.k.1.1 1 4.3 odd 2