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Properties

Label 3330.2.d.f.1999.1
Level $3330$
Weight $2$
Character 3330.1999
Analytic conductor $26.590$
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] = [3330,2,Mod(1999,3330)] 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("3330.1999"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3330, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3330.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,-2,4,0,0,0,0,2,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(26.5901838731\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 370)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1999.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 3330.1999
Dual form 3330.2.d.f.1999.2

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000 q^{4} +(2.00000 + 1.00000i) q^{5} -2.00000i q^{7} +1.00000i q^{8} +(1.00000 - 2.00000i) q^{10} -2.00000i q^{13} -2.00000 q^{14} +1.00000 q^{16} +6.00000i q^{17} +6.00000 q^{19} +(-2.00000 - 1.00000i) q^{20} +4.00000i q^{23} +(3.00000 + 4.00000i) q^{25} -2.00000 q^{26} +2.00000i q^{28} -4.00000 q^{31} -1.00000i q^{32} +6.00000 q^{34} +(2.00000 - 4.00000i) q^{35} -1.00000i q^{37} -6.00000i q^{38} +(-1.00000 + 2.00000i) q^{40} +10.0000 q^{41} +4.00000i q^{43} +4.00000 q^{46} +2.00000i q^{47} +3.00000 q^{49} +(4.00000 - 3.00000i) q^{50} +2.00000i q^{52} +2.00000i q^{53} +2.00000 q^{56} -6.00000 q^{59} +4.00000i q^{62} -1.00000 q^{64} +(2.00000 - 4.00000i) q^{65} +8.00000i q^{67} -6.00000i q^{68} +(-4.00000 - 2.00000i) q^{70} -8.00000i q^{73} -1.00000 q^{74} -6.00000 q^{76} -4.00000 q^{79} +(2.00000 + 1.00000i) q^{80} -10.0000i q^{82} -12.0000i q^{83} +(-6.00000 + 12.0000i) q^{85} +4.00000 q^{86} +6.00000 q^{89} -4.00000 q^{91} -4.00000i q^{92} +2.00000 q^{94} +(12.0000 + 6.00000i) q^{95} -10.0000i q^{97} -3.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} + 4 q^{5} + 2 q^{10} - 4 q^{14} + 2 q^{16} + 12 q^{19} - 4 q^{20} + 6 q^{25} - 4 q^{26} - 8 q^{31} + 12 q^{34} + 4 q^{35} - 2 q^{40} + 20 q^{41} + 8 q^{46} + 6 q^{49} + 8 q^{50} + 4 q^{56}+ \cdots + 24 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3330\mathbb{Z}\right)^\times\).

\(n\) \(371\) \(631\) \(667\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.707107i
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) 2.00000 + 1.00000i 0.894427 + 0.447214i
\(6\) 0 0
\(7\) − 2.00000i − 0.755929i −0.925820 − 0.377964i \(-0.876624\pi\)
0.925820 − 0.377964i \(-0.123376\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) 1.00000 − 2.00000i 0.316228 − 0.632456i
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) − 2.00000i − 0.554700i −0.960769 − 0.277350i \(-0.910544\pi\)
0.960769 − 0.277350i \(-0.0894562\pi\)
\(14\) −2.00000 −0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 6.00000i 1.45521i 0.685994 + 0.727607i \(0.259367\pi\)
−0.685994 + 0.727607i \(0.740633\pi\)
\(18\) 0 0
\(19\) 6.00000 1.37649 0.688247 − 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) −2.00000 − 1.00000i −0.447214 − 0.223607i
\(21\) 0 0
\(22\) 0 0
\(23\) 4.00000i 0.834058i 0.908893 + 0.417029i \(0.136929\pi\)
−0.908893 + 0.417029i \(0.863071\pi\)
\(24\) 0 0
\(25\) 3.00000 + 4.00000i 0.600000 + 0.800000i
\(26\) −2.00000 −0.392232
\(27\) 0 0
\(28\) 2.00000i 0.377964i
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 − 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 0 0
\(34\) 6.00000 1.02899
\(35\) 2.00000 − 4.00000i 0.338062 − 0.676123i
\(36\) 0 0
\(37\) − 1.00000i − 0.164399i
\(38\) − 6.00000i − 0.973329i
\(39\) 0 0
\(40\) −1.00000 + 2.00000i −0.158114 + 0.316228i
\(41\) 10.0000 1.56174 0.780869 − 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) 0 0
\(43\) 4.00000i 0.609994i 0.952353 + 0.304997i \(0.0986555\pi\)
−0.952353 + 0.304997i \(0.901344\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 4.00000 0.589768
\(47\) 2.00000i 0.291730i 0.989305 + 0.145865i \(0.0465965\pi\)
−0.989305 + 0.145865i \(0.953403\pi\)
\(48\) 0 0
\(49\) 3.00000 0.428571
\(50\) 4.00000 − 3.00000i 0.565685 − 0.424264i
\(51\) 0 0
\(52\) 2.00000i 0.277350i
\(53\) 2.00000i 0.274721i 0.990521 + 0.137361i \(0.0438619\pi\)
−0.990521 + 0.137361i \(0.956138\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 2.00000 0.267261
\(57\) 0 0
\(58\) 0 0
\(59\) −6.00000 −0.781133 −0.390567 − 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 0 0
\(61\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) 4.00000i 0.508001i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 2.00000 − 4.00000i 0.248069 − 0.496139i
\(66\) 0 0
\(67\) 8.00000i 0.977356i 0.872464 + 0.488678i \(0.162521\pi\)
−0.872464 + 0.488678i \(0.837479\pi\)
\(68\) − 6.00000i − 0.727607i
\(69\) 0 0
\(70\) −4.00000 − 2.00000i −0.478091 − 0.239046i
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) − 8.00000i − 0.936329i −0.883641 − 0.468165i \(-0.844915\pi\)
0.883641 − 0.468165i \(-0.155085\pi\)
\(74\) −1.00000 −0.116248
\(75\) 0 0
\(76\) −6.00000 −0.688247
\(77\) 0 0
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 − 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 2.00000 + 1.00000i 0.223607 + 0.111803i
\(81\) 0 0
\(82\) − 10.0000i − 1.10432i
\(83\) − 12.0000i − 1.31717i −0.752506 − 0.658586i \(-0.771155\pi\)
0.752506 − 0.658586i \(-0.228845\pi\)
\(84\) 0 0
\(85\) −6.00000 + 12.0000i −0.650791 + 1.30158i
\(86\) 4.00000 0.431331
\(87\) 0 0
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 − 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) −4.00000 −0.419314
\(92\) − 4.00000i − 0.417029i
\(93\) 0 0
\(94\) 2.00000 0.206284
\(95\) 12.0000 + 6.00000i 1.23117 + 0.615587i
\(96\) 0 0
\(97\) − 10.0000i − 1.01535i −0.861550 − 0.507673i \(-0.830506\pi\)
0.861550 − 0.507673i \(-0.169494\pi\)
\(98\) − 3.00000i − 0.303046i
\(99\) 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 3330.2.d.f.1999.1 2
3.2 odd 2 370.2.b.a.149.2 yes 2
5.4 even 2 inner 3330.2.d.f.1999.2 2
15.2 even 4 1850.2.a.e.1.1 1
15.8 even 4 1850.2.a.j.1.1 1
15.14 odd 2 370.2.b.a.149.1 ✓ 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.b.a.149.1 ✓ 2 15.14 odd 2
370.2.b.a.149.2 yes 2 3.2 odd 2
1850.2.a.e.1.1 1 15.2 even 4
1850.2.a.j.1.1 1 15.8 even 4
3330.2.d.f.1999.1 2 1.1 even 1 trivial
3330.2.d.f.1999.2 2 5.4 even 2 inner