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Properties

Label 30.4.c.a.19.1
Level $30$
Weight $4$
Character 30.19
Analytic conductor $1.770$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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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] = [30,4,Mod(19,30)] 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("30.19"); S:= CuspForms(chi, 4); 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, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 30 = 2 \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 30.c (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(1.77005730017\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 19.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 30.19
Dual form 30.4.c.a.19.2

$q$-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)
 
\(f(q)\) \(=\) \(q-2.00000i q^{2} -3.00000i q^{3} -4.00000 q^{4} +(2.00000 - 11.0000i) q^{5} -6.00000 q^{6} +2.00000i q^{7} +8.00000i q^{8} -9.00000 q^{9} +(-22.0000 - 4.00000i) q^{10} +70.0000 q^{11} +12.0000i q^{12} +54.0000i q^{13} +4.00000 q^{14} +(-33.0000 - 6.00000i) q^{15} +16.0000 q^{16} +22.0000i q^{17} +18.0000i q^{18} -24.0000 q^{19} +(-8.00000 + 44.0000i) q^{20} +6.00000 q^{21} -140.000i q^{22} -100.000i q^{23} +24.0000 q^{24} +(-117.000 - 44.0000i) q^{25} +108.000 q^{26} +27.0000i q^{27} -8.00000i q^{28} -216.000 q^{29} +(-12.0000 + 66.0000i) q^{30} +208.000 q^{31} -32.0000i q^{32} -210.000i q^{33} +44.0000 q^{34} +(22.0000 + 4.00000i) q^{35} +36.0000 q^{36} +254.000i q^{37} +48.0000i q^{38} +162.000 q^{39} +(88.0000 + 16.0000i) q^{40} -206.000 q^{41} -12.0000i q^{42} +292.000i q^{43} -280.000 q^{44} +(-18.0000 + 99.0000i) q^{45} -200.000 q^{46} +320.000i q^{47} -48.0000i q^{48} +339.000 q^{49} +(-88.0000 + 234.000i) q^{50} +66.0000 q^{51} -216.000i q^{52} -402.000i q^{53} +54.0000 q^{54} +(140.000 - 770.000i) q^{55} -16.0000 q^{56} +72.0000i q^{57} +432.000i q^{58} +370.000 q^{59} +(132.000 + 24.0000i) q^{60} -550.000 q^{61} -416.000i q^{62} -18.0000i q^{63} -64.0000 q^{64} +(594.000 + 108.000i) q^{65} -420.000 q^{66} -728.000i q^{67} -88.0000i q^{68} -300.000 q^{69} +(8.00000 - 44.0000i) q^{70} -540.000 q^{71} -72.0000i q^{72} +604.000i q^{73} +508.000 q^{74} +(-132.000 + 351.000i) q^{75} +96.0000 q^{76} +140.000i q^{77} -324.000i q^{78} -792.000 q^{79} +(32.0000 - 176.000i) q^{80} +81.0000 q^{81} +412.000i q^{82} +404.000i q^{83} -24.0000 q^{84} +(242.000 + 44.0000i) q^{85} +584.000 q^{86} +648.000i q^{87} +560.000i q^{88} +938.000 q^{89} +(198.000 + 36.0000i) q^{90} -108.000 q^{91} +400.000i q^{92} -624.000i q^{93} +640.000 q^{94} +(-48.0000 + 264.000i) q^{95} -96.0000 q^{96} -56.0000i q^{97} -678.000i q^{98} -630.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{4} + 4 q^{5} - 12 q^{6} - 18 q^{9} - 44 q^{10} + 140 q^{11} + 8 q^{14} - 66 q^{15} + 32 q^{16} - 48 q^{19} - 16 q^{20} + 12 q^{21} + 48 q^{24} - 234 q^{25} + 216 q^{26} - 432 q^{29} - 24 q^{30}+ \cdots - 1260 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/30\mathbb{Z}\right)^\times\).

\(n\) \(7\) \(11\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.00000i − 0.707107i
\(3\) − 3.00000i − 0.577350i
\(4\) −4.00000 −0.500000
\(5\) 2.00000 − 11.0000i 0.178885 − 0.983870i
\(6\) −6.00000 −0.408248
\(7\) 2.00000i 0.107990i 0.998541 + 0.0539949i \(0.0171955\pi\)
−0.998541 + 0.0539949i \(0.982805\pi\)
\(8\) 8.00000i 0.353553i
\(9\) −9.00000 −0.333333
\(10\) −22.0000 − 4.00000i −0.695701 − 0.126491i
\(11\) 70.0000 1.91871 0.959354 − 0.282204i \(-0.0910657\pi\)
0.959354 + 0.282204i \(0.0910657\pi\)
\(12\) 12.0000i 0.288675i
\(13\) 54.0000i 1.15207i 0.817425 + 0.576035i \(0.195401\pi\)
−0.817425 + 0.576035i \(0.804599\pi\)
\(14\) 4.00000 0.0763604
\(15\) −33.0000 − 6.00000i −0.568038 − 0.103280i
\(16\) 16.0000 0.250000
\(17\) 22.0000i 0.313870i 0.987609 + 0.156935i \(0.0501613\pi\)
−0.987609 + 0.156935i \(0.949839\pi\)
\(18\) 18.0000i 0.235702i
\(19\) −24.0000 −0.289788 −0.144894 − 0.989447i \(-0.546284\pi\)
−0.144894 + 0.989447i \(0.546284\pi\)
\(20\) −8.00000 + 44.0000i −0.0894427 + 0.491935i
\(21\) 6.00000 0.0623480
\(22\) − 140.000i − 1.35673i
\(23\) − 100.000i − 0.906584i −0.891362 − 0.453292i \(-0.850249\pi\)
0.891362 − 0.453292i \(-0.149751\pi\)
\(24\) 24.0000 0.204124
\(25\) −117.000 − 44.0000i −0.936000 − 0.352000i
\(26\) 108.000 0.814636
\(27\) 27.0000i 0.192450i
\(28\) − 8.00000i − 0.0539949i
\(29\) −216.000 −1.38311 −0.691555 − 0.722324i \(-0.743074\pi\)
−0.691555 + 0.722324i \(0.743074\pi\)
\(30\) −12.0000 + 66.0000i −0.0730297 + 0.401663i
\(31\) 208.000 1.20509 0.602547 − 0.798084i \(-0.294153\pi\)
0.602547 + 0.798084i \(0.294153\pi\)
\(32\) − 32.0000i − 0.176777i
\(33\) − 210.000i − 1.10777i
\(34\) 44.0000 0.221939
\(35\) 22.0000 + 4.00000i 0.106248 + 0.0193178i
\(36\) 36.0000 0.166667
\(37\) 254.000i 1.12858i 0.825578 + 0.564288i \(0.190849\pi\)
−0.825578 + 0.564288i \(0.809151\pi\)
\(38\) 48.0000i 0.204911i
\(39\) 162.000 0.665148
\(40\) 88.0000 + 16.0000i 0.347851 + 0.0632456i
\(41\) −206.000 −0.784678 −0.392339 − 0.919821i \(-0.628334\pi\)
−0.392339 + 0.919821i \(0.628334\pi\)
\(42\) − 12.0000i − 0.0440867i
\(43\) 292.000i 1.03557i 0.855510 + 0.517786i \(0.173244\pi\)
−0.855510 + 0.517786i \(0.826756\pi\)
\(44\) −280.000 −0.959354
\(45\) −18.0000 + 99.0000i −0.0596285 + 0.327957i
\(46\) −200.000 −0.641052
\(47\) 320.000i 0.993123i 0.868001 + 0.496562i \(0.165404\pi\)
−0.868001 + 0.496562i \(0.834596\pi\)
\(48\) − 48.0000i − 0.144338i
\(49\) 339.000 0.988338
\(50\) −88.0000 + 234.000i −0.248902 + 0.661852i
\(51\) 66.0000 0.181213
\(52\) − 216.000i − 0.576035i
\(53\) − 402.000i − 1.04187i −0.853597 − 0.520933i \(-0.825584\pi\)
0.853597 − 0.520933i \(-0.174416\pi\)
\(54\) 54.0000 0.136083
\(55\) 140.000 − 770.000i 0.343229 − 1.88776i
\(56\) −16.0000 −0.0381802
\(57\) 72.0000i 0.167309i
\(58\) 432.000i 0.978007i
\(59\) 370.000 0.816439 0.408219 − 0.912884i \(-0.366150\pi\)
0.408219 + 0.912884i \(0.366150\pi\)
\(60\) 132.000 + 24.0000i 0.284019 + 0.0516398i
\(61\) −550.000 −1.15443 −0.577215 − 0.816592i \(-0.695861\pi\)
−0.577215 + 0.816592i \(0.695861\pi\)
\(62\) − 416.000i − 0.852130i
\(63\) − 18.0000i − 0.0359966i
\(64\) −64.0000 −0.125000
\(65\) 594.000 + 108.000i 1.13349 + 0.206088i
\(66\) −420.000 −0.783309
\(67\) − 728.000i − 1.32745i −0.747975 − 0.663727i \(-0.768974\pi\)
0.747975 − 0.663727i \(-0.231026\pi\)
\(68\) − 88.0000i − 0.156935i
\(69\) −300.000 −0.523417
\(70\) 8.00000 − 44.0000i 0.0136598 − 0.0751287i
\(71\) −540.000 −0.902623 −0.451311 − 0.892367i \(-0.649044\pi\)
−0.451311 + 0.892367i \(0.649044\pi\)
\(72\) − 72.0000i − 0.117851i
\(73\) 604.000i 0.968395i 0.874959 + 0.484198i \(0.160888\pi\)
−0.874959 + 0.484198i \(0.839112\pi\)
\(74\) 508.000 0.798024
\(75\) −132.000 + 351.000i −0.203227 + 0.540400i
\(76\) 96.0000 0.144894
\(77\) 140.000i 0.207201i
\(78\) − 324.000i − 0.470330i
\(79\) −792.000 −1.12794 −0.563968 − 0.825797i \(-0.690726\pi\)
−0.563968 + 0.825797i \(0.690726\pi\)
\(80\) 32.0000 − 176.000i 0.0447214 − 0.245967i
\(81\) 81.0000 0.111111
\(82\) 412.000i 0.554851i
\(83\) 404.000i 0.534274i 0.963659 + 0.267137i \(0.0860777\pi\)
−0.963659 + 0.267137i \(0.913922\pi\)
\(84\) −24.0000 −0.0311740
\(85\) 242.000 + 44.0000i 0.308807 + 0.0561467i
\(86\) 584.000 0.732260
\(87\) 648.000i 0.798539i
\(88\) 560.000i 0.678366i
\(89\) 938.000 1.11717 0.558583 − 0.829449i \(-0.311345\pi\)
0.558583 + 0.829449i \(0.311345\pi\)
\(90\) 198.000 + 36.0000i 0.231900 + 0.0421637i
\(91\) −108.000 −0.124412
\(92\) 400.000i 0.453292i
\(93\) − 624.000i − 0.695761i
\(94\) 640.000 0.702244
\(95\) −48.0000 + 264.000i −0.0518389 + 0.285114i
\(96\) −96.0000 −0.102062
\(97\) − 56.0000i − 0.0586179i −0.999570 − 0.0293090i \(-0.990669\pi\)
0.999570 − 0.0293090i \(-0.00933067\pi\)
\(98\) − 678.000i − 0.698861i
\(99\) −630.000 −0.639570
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 30.4.c.a.19.1 ✓ 2
3.2 odd 2 90.4.c.a.19.2 2
4.3 odd 2 240.4.f.d.49.2 2
5.2 odd 4 150.4.a.f.1.1 1
5.3 odd 4 150.4.a.d.1.1 1
5.4 even 2 inner 30.4.c.a.19.2 yes 2
8.3 odd 2 960.4.f.d.769.1 2
8.5 even 2 960.4.f.c.769.2 2
12.11 even 2 720.4.f.c.289.2 2
15.2 even 4 450.4.a.e.1.1 1
15.8 even 4 450.4.a.p.1.1 1
15.14 odd 2 90.4.c.a.19.1 2
20.3 even 4 1200.4.a.h.1.1 1
20.7 even 4 1200.4.a.bc.1.1 1
20.19 odd 2 240.4.f.d.49.1 2
40.19 odd 2 960.4.f.d.769.2 2
40.29 even 2 960.4.f.c.769.1 2
60.59 even 2 720.4.f.c.289.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
30.4.c.a.19.1 ✓ 2 1.1 even 1 trivial
30.4.c.a.19.2 yes 2 5.4 even 2 inner
90.4.c.a.19.1 2 15.14 odd 2
90.4.c.a.19.2 2 3.2 odd 2
150.4.a.d.1.1 1 5.3 odd 4
150.4.a.f.1.1 1 5.2 odd 4
240.4.f.d.49.1 2 20.19 odd 2
240.4.f.d.49.2 2 4.3 odd 2
450.4.a.e.1.1 1 15.2 even 4
450.4.a.p.1.1 1 15.8 even 4
720.4.f.c.289.1 2 60.59 even 2
720.4.f.c.289.2 2 12.11 even 2
960.4.f.c.769.1 2 40.29 even 2
960.4.f.c.769.2 2 8.5 even 2
960.4.f.d.769.1 2 8.3 odd 2
960.4.f.d.769.2 2 40.19 odd 2
1200.4.a.h.1.1 1 20.3 even 4
1200.4.a.bc.1.1 1 20.7 even 4