Properties

Label 30.2.a.a.1.1
Level $30$
Weight $2$
Character 30.1
Self dual yes
Analytic conductor $0.240$
Analytic rank $0$
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] = [30,2,Mod(1,30)] 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("30.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(30, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 30 = 2 \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 30.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(0.239551206064\)
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 \(\chi\) \(=\) 30.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +1.00000 q^{12} +2.00000 q^{13} +4.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} +6.00000 q^{17} -1.00000 q^{18} -4.00000 q^{19} -1.00000 q^{20} -4.00000 q^{21} -1.00000 q^{24} +1.00000 q^{25} -2.00000 q^{26} +1.00000 q^{27} -4.00000 q^{28} -6.00000 q^{29} +1.00000 q^{30} +8.00000 q^{31} -1.00000 q^{32} -6.00000 q^{34} +4.00000 q^{35} +1.00000 q^{36} +2.00000 q^{37} +4.00000 q^{38} +2.00000 q^{39} +1.00000 q^{40} -6.00000 q^{41} +4.00000 q^{42} -4.00000 q^{43} -1.00000 q^{45} +1.00000 q^{48} +9.00000 q^{49} -1.00000 q^{50} +6.00000 q^{51} +2.00000 q^{52} -6.00000 q^{53} -1.00000 q^{54} +4.00000 q^{56} -4.00000 q^{57} +6.00000 q^{58} -1.00000 q^{60} -10.0000 q^{61} -8.00000 q^{62} -4.00000 q^{63} +1.00000 q^{64} -2.00000 q^{65} -4.00000 q^{67} +6.00000 q^{68} -4.00000 q^{70} -1.00000 q^{72} +2.00000 q^{73} -2.00000 q^{74} +1.00000 q^{75} -4.00000 q^{76} -2.00000 q^{78} +8.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} +12.0000 q^{83} -4.00000 q^{84} -6.00000 q^{85} +4.00000 q^{86} -6.00000 q^{87} +18.0000 q^{89} +1.00000 q^{90} -8.00000 q^{91} +8.00000 q^{93} +4.00000 q^{95} -1.00000 q^{96} +2.00000 q^{97} -9.00000 q^{98} +O(q^{100})\)

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
\(7\) −4.00000 −1.51186 −0.755929 − 0.654654i \(-0.772814\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 1.00000 0.288675
\(13\) 2.00000 0.554700 0.277350 − 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 4.00000 1.06904
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) 6.00000 1.45521 0.727607 − 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) −1.00000 −0.235702
\(19\) −4.00000 −0.917663 −0.458831 − 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) −1.00000 −0.223607
\(21\) −4.00000 −0.872872
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) −2.00000 −0.392232
\(27\) 1.00000 0.192450
\(28\) −4.00000 −0.755929
\(29\) −6.00000 −1.11417 −0.557086 − 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 1.00000 0.182574
\(31\) 8.00000 1.43684 0.718421 − 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −6.00000 −1.02899
\(35\) 4.00000 0.676123
\(36\) 1.00000 0.166667
\(37\) 2.00000 0.328798 0.164399 − 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 4.00000 0.648886
\(39\) 2.00000 0.320256
\(40\) 1.00000 0.158114
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 4.00000 0.617213
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 1.00000 0.144338
\(49\) 9.00000 1.28571
\(50\) −1.00000 −0.141421
\(51\) 6.00000 0.840168
\(52\) 2.00000 0.277350
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 4.00000 0.534522
\(57\) −4.00000 −0.529813
\(58\) 6.00000 0.787839
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) −1.00000 −0.129099
\(61\) −10.0000 −1.28037 −0.640184 − 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) −8.00000 −1.01600
\(63\) −4.00000 −0.503953
\(64\) 1.00000 0.125000
\(65\) −2.00000 −0.248069
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 6.00000 0.727607
\(69\) 0 0
\(70\) −4.00000 −0.478091
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) −1.00000 −0.117851
\(73\) 2.00000 0.234082 0.117041 − 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) −2.00000 −0.232495
\(75\) 1.00000 0.115470
\(76\) −4.00000 −0.458831
\(77\) 0 0
\(78\) −2.00000 −0.226455
\(79\) 8.00000 0.900070 0.450035 − 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) 6.00000 0.662589
\(83\) 12.0000 1.31717 0.658586 − 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) −4.00000 −0.436436
\(85\) −6.00000 −0.650791
\(86\) 4.00000 0.431331
\(87\) −6.00000 −0.643268
\(88\) 0 0
\(89\) 18.0000 1.90800 0.953998 − 0.299813i \(-0.0969242\pi\)
0.953998 + 0.299813i \(0.0969242\pi\)
\(90\) 1.00000 0.105409
\(91\) −8.00000 −0.838628
\(92\) 0 0
\(93\) 8.00000 0.829561
\(94\) 0 0
\(95\) 4.00000 0.410391
\(96\) −1.00000 −0.102062
\(97\) 2.00000 0.203069 0.101535 − 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) −9.00000 −0.909137
\(99\) 0 0
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 30.2.a.a.1.1 ✓ 1
3.2 odd 2 90.2.a.c.1.1 1
4.3 odd 2 240.2.a.b.1.1 1
5.2 odd 4 150.2.c.a.49.1 2
5.3 odd 4 150.2.c.a.49.2 2
5.4 even 2 150.2.a.b.1.1 1
7.2 even 3 1470.2.i.o.361.1 2
7.3 odd 6 1470.2.i.q.961.1 2
7.4 even 3 1470.2.i.o.961.1 2
7.5 odd 6 1470.2.i.q.361.1 2
7.6 odd 2 1470.2.a.d.1.1 1
8.3 odd 2 960.2.a.p.1.1 1
8.5 even 2 960.2.a.e.1.1 1
9.2 odd 6 810.2.e.b.271.1 2
9.4 even 3 810.2.e.l.541.1 2
9.5 odd 6 810.2.e.b.541.1 2
9.7 even 3 810.2.e.l.271.1 2
11.10 odd 2 3630.2.a.w.1.1 1
12.11 even 2 720.2.a.j.1.1 1
13.5 odd 4 5070.2.b.k.1351.2 2
13.8 odd 4 5070.2.b.k.1351.1 2
13.12 even 2 5070.2.a.w.1.1 1
15.2 even 4 450.2.c.b.199.2 2
15.8 even 4 450.2.c.b.199.1 2
15.14 odd 2 450.2.a.d.1.1 1
16.3 odd 4 3840.2.k.f.1921.1 2
16.5 even 4 3840.2.k.y.1921.1 2
16.11 odd 4 3840.2.k.f.1921.2 2
16.13 even 4 3840.2.k.y.1921.2 2
17.16 even 2 8670.2.a.g.1.1 1
20.3 even 4 1200.2.f.e.49.1 2
20.7 even 4 1200.2.f.e.49.2 2
20.19 odd 2 1200.2.a.k.1.1 1
21.20 even 2 4410.2.a.z.1.1 1
24.5 odd 2 2880.2.a.a.1.1 1
24.11 even 2 2880.2.a.q.1.1 1
35.34 odd 2 7350.2.a.ct.1.1 1
40.3 even 4 4800.2.f.w.3649.2 2
40.13 odd 4 4800.2.f.p.3649.1 2
40.19 odd 2 4800.2.a.d.1.1 1
40.27 even 4 4800.2.f.w.3649.1 2
40.29 even 2 4800.2.a.cq.1.1 1
40.37 odd 4 4800.2.f.p.3649.2 2
60.23 odd 4 3600.2.f.i.2449.1 2
60.47 odd 4 3600.2.f.i.2449.2 2
60.59 even 2 3600.2.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
30.2.a.a.1.1 ✓ 1 1.1 even 1 trivial
90.2.a.c.1.1 1 3.2 odd 2
150.2.a.b.1.1 1 5.4 even 2
150.2.c.a.49.1 2 5.2 odd 4
150.2.c.a.49.2 2 5.3 odd 4
240.2.a.b.1.1 1 4.3 odd 2
450.2.a.d.1.1 1 15.14 odd 2
450.2.c.b.199.1 2 15.8 even 4
450.2.c.b.199.2 2 15.2 even 4
720.2.a.j.1.1 1 12.11 even 2
810.2.e.b.271.1 2 9.2 odd 6
810.2.e.b.541.1 2 9.5 odd 6
810.2.e.l.271.1 2 9.7 even 3
810.2.e.l.541.1 2 9.4 even 3
960.2.a.e.1.1 1 8.5 even 2
960.2.a.p.1.1 1 8.3 odd 2
1200.2.a.k.1.1 1 20.19 odd 2
1200.2.f.e.49.1 2 20.3 even 4
1200.2.f.e.49.2 2 20.7 even 4
1470.2.a.d.1.1 1 7.6 odd 2
1470.2.i.o.361.1 2 7.2 even 3
1470.2.i.o.961.1 2 7.4 even 3
1470.2.i.q.361.1 2 7.5 odd 6
1470.2.i.q.961.1 2 7.3 odd 6
2880.2.a.a.1.1 1 24.5 odd 2
2880.2.a.q.1.1 1 24.11 even 2
3600.2.a.f.1.1 1 60.59 even 2
3600.2.f.i.2449.1 2 60.23 odd 4
3600.2.f.i.2449.2 2 60.47 odd 4
3630.2.a.w.1.1 1 11.10 odd 2
3840.2.k.f.1921.1 2 16.3 odd 4
3840.2.k.f.1921.2 2 16.11 odd 4
3840.2.k.y.1921.1 2 16.5 even 4
3840.2.k.y.1921.2 2 16.13 even 4
4410.2.a.z.1.1 1 21.20 even 2
4800.2.a.d.1.1 1 40.19 odd 2
4800.2.a.cq.1.1 1 40.29 even 2
4800.2.f.p.3649.1 2 40.13 odd 4
4800.2.f.p.3649.2 2 40.37 odd 4
4800.2.f.w.3649.1 2 40.27 even 4
4800.2.f.w.3649.2 2 40.3 even 4
5070.2.a.w.1.1 1 13.12 even 2
5070.2.b.k.1351.1 2 13.8 odd 4
5070.2.b.k.1351.2 2 13.5 odd 4
7350.2.a.ct.1.1 1 35.34 odd 2
8670.2.a.g.1.1 1 17.16 even 2