↑

Properties

Label 3330.2.d.p.1999.1
Level $3330$
Weight $2$
Character 3330.1999
Analytic conductor $26.590$
Analytic rank $0$
Dimension $10$
Inner twists $2$

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] = [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 = [10,0,0,-10,-6,0,0,0,0,2,-6,0,0,-2] 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: \(10\)
Coefficient field: 10.0.12837029094400.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2x^{9} + 2x^{8} - 4x^{7} + 51x^{6} - 124x^{5} + 154x^{4} - 46x^{3} + x^{2} + 4x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(1.24331 + 1.24331i\) of defining polynomial
Character \(\chi\) \(=\) 3330.1999
Dual form 3330.2.d.p.1999.6

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000 q^{4} +(-2.07757 + 0.826871i) q^{5} +4.67211i q^{7} +1.00000i q^{8} +(0.826871 + 2.07757i) q^{10} -0.0451320 q^{11} -5.26514i q^{13} +4.67211 q^{14} +1.00000 q^{16} -3.60861i q^{17} +6.22000 q^{19} +(2.07757 - 0.826871i) q^{20} +0.0451320i q^{22} -2.20164i q^{23} +(3.63257 - 3.43576i) q^{25} -5.26514 q^{26} -4.67211i q^{28} -4.20386 q^{29} -3.01051 q^{31} -1.00000i q^{32} -3.60861 q^{34} +(-3.86323 - 9.70662i) q^{35} +1.00000i q^{37} -6.22000i q^{38} +(-0.826871 - 2.07757i) q^{40} +7.38299 q^{41} +5.54789i q^{43} +0.0451320 q^{44} -2.20164 q^{46} +4.28072i q^{47} -14.8286 q^{49} +(-3.43576 - 3.63257i) q^{50} +5.26514i q^{52} +6.10215i q^{53} +(0.0937647 - 0.0373183i) q^{55} -4.67211 q^{56} +4.20386i q^{58} +13.5275 q^{59} -4.27299 q^{61} +3.01051i q^{62} -1.00000 q^{64} +(4.35359 + 10.9387i) q^{65} +12.3751i q^{67} +3.60861i q^{68} +(-9.70662 + 3.86323i) q^{70} +4.49354 q^{71} -7.03811i q^{73} +1.00000 q^{74} -6.22000 q^{76} -0.210861i q^{77} +8.72499 q^{79} +(-2.07757 + 0.826871i) q^{80} -7.38299i q^{82} -0.880231i q^{83} +(2.98386 + 7.49713i) q^{85} +5.54789 q^{86} -0.0451320i q^{88} +9.97602 q^{89} +24.5993 q^{91} +2.20164i q^{92} +4.28072 q^{94} +(-12.9225 + 5.14314i) q^{95} +0.240408i q^{97} +14.8286i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 10 q^{4} - 6 q^{5} + 2 q^{10} - 6 q^{11} - 2 q^{14} + 10 q^{16} - 8 q^{19} + 6 q^{20} + 4 q^{25} + 12 q^{26} + 22 q^{29} + 46 q^{31} - 18 q^{34} - 32 q^{35} - 2 q^{40} + 14 q^{41} + 6 q^{44} + 12 q^{46}+ \cdots - 12 q^{95}+O(q^{100}) \) 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
\(5\) −2.07757 + 0.826871i −0.929116 + 0.369788i
\(6\) 0 0
\(7\) 4.67211i 1.76589i 0.469475 + 0.882946i \(0.344443\pi\)
−0.469475 + 0.882946i \(0.655557\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) 0.826871 + 2.07757i 0.261480 + 0.656984i
\(11\) −0.0451320 −0.0136078 −0.00680390 − 0.999977i \(-0.502166\pi\)
−0.00680390 + 0.999977i \(0.502166\pi\)
\(12\) 0 0
\(13\) − 5.26514i − 1.46029i −0.683294 − 0.730143i \(-0.739453\pi\)
0.683294 − 0.730143i \(-0.260547\pi\)
\(14\) 4.67211 1.24867
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 3.60861i − 0.875217i −0.899166 − 0.437608i \(-0.855826\pi\)
0.899166 − 0.437608i \(-0.144174\pi\)
\(18\) 0 0
\(19\) 6.22000 1.42697 0.713483 − 0.700672i \(-0.247116\pi\)
0.713483 + 0.700672i \(0.247116\pi\)
\(20\) 2.07757 − 0.826871i 0.464558 − 0.184894i
\(21\) 0 0
\(22\) 0.0451320i 0.00962217i
\(23\) − 2.20164i − 0.459073i −0.973300 − 0.229536i \(-0.926279\pi\)
0.973300 − 0.229536i \(-0.0737210\pi\)
\(24\) 0 0
\(25\) 3.63257 − 3.43576i 0.726514 − 0.687152i
\(26\) −5.26514 −1.03258
\(27\) 0 0
\(28\) − 4.67211i − 0.882946i
\(29\) −4.20386 −0.780637 −0.390319 − 0.920680i \(-0.627635\pi\)
−0.390319 + 0.920680i \(0.627635\pi\)
\(30\) 0 0
\(31\) −3.01051 −0.540704 −0.270352 − 0.962762i \(-0.587140\pi\)
−0.270352 + 0.962762i \(0.587140\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 0 0
\(34\) −3.60861 −0.618872
\(35\) −3.86323 − 9.70662i −0.653006 − 1.64072i
\(36\) 0 0
\(37\) 1.00000i 0.164399i
\(38\) − 6.22000i − 1.00902i
\(39\) 0 0
\(40\) −0.826871 − 2.07757i −0.130740 − 0.328492i
\(41\) 7.38299 1.15303 0.576515 − 0.817087i \(-0.304412\pi\)
0.576515 + 0.817087i \(0.304412\pi\)
\(42\) 0 0
\(43\) 5.54789i 0.846046i 0.906119 + 0.423023i \(0.139031\pi\)
−0.906119 + 0.423023i \(0.860969\pi\)
\(44\) 0.0451320 0.00680390
\(45\) 0 0
\(46\) −2.20164 −0.324613
\(47\) 4.28072i 0.624407i 0.950015 + 0.312204i \(0.101067\pi\)
−0.950015 + 0.312204i \(0.898933\pi\)
\(48\) 0 0
\(49\) −14.8286 −2.11837
\(50\) −3.43576 − 3.63257i −0.485890 − 0.513723i
\(51\) 0 0
\(52\) 5.26514i 0.730143i
\(53\) 6.10215i 0.838194i 0.907941 + 0.419097i \(0.137653\pi\)
−0.907941 + 0.419097i \(0.862347\pi\)
\(54\) 0 0
\(55\) 0.0937647 − 0.0373183i 0.0126432 − 0.00503200i
\(56\) −4.67211 −0.624337
\(57\) 0 0
\(58\) 4.20386i 0.551994i
\(59\) 13.5275 1.76113 0.880565 − 0.473926i \(-0.157164\pi\)
0.880565 + 0.473926i \(0.157164\pi\)
\(60\) 0 0
\(61\) −4.27299 −0.547100 −0.273550 − 0.961858i \(-0.588198\pi\)
−0.273550 + 0.961858i \(0.588198\pi\)
\(62\) 3.01051i 0.382335i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 4.35359 + 10.9387i 0.539996 + 1.35678i
\(66\) 0 0
\(67\) 12.3751i 1.51186i 0.654650 + 0.755932i \(0.272816\pi\)
−0.654650 + 0.755932i \(0.727184\pi\)
\(68\) 3.60861i 0.437608i
\(69\) 0 0
\(70\) −9.70662 + 3.86323i −1.16016 + 0.461745i
\(71\) 4.49354 0.533285 0.266642 − 0.963796i \(-0.414086\pi\)
0.266642 + 0.963796i \(0.414086\pi\)
\(72\) 0 0
\(73\) − 7.03811i − 0.823748i −0.911241 − 0.411874i \(-0.864874\pi\)
0.911241 − 0.411874i \(-0.135126\pi\)
\(74\) 1.00000 0.116248
\(75\) 0 0
\(76\) −6.22000 −0.713483
\(77\) − 0.210861i − 0.0240299i
\(78\) 0 0
\(79\) 8.72499 0.981638 0.490819 − 0.871262i \(-0.336698\pi\)
0.490819 + 0.871262i \(0.336698\pi\)
\(80\) −2.07757 + 0.826871i −0.232279 + 0.0924470i
\(81\) 0 0
\(82\) − 7.38299i − 0.815315i
\(83\) − 0.880231i − 0.0966179i −0.998832 − 0.0483090i \(-0.984617\pi\)
0.998832 − 0.0483090i \(-0.0153832\pi\)
\(84\) 0 0
\(85\) 2.98386 + 7.49713i 0.323645 + 0.813178i
\(86\) 5.54789 0.598245
\(87\) 0 0
\(88\) − 0.0451320i − 0.00481108i
\(89\) 9.97602 1.05746 0.528728 − 0.848791i \(-0.322669\pi\)
0.528728 + 0.848791i \(0.322669\pi\)
\(90\) 0 0
\(91\) 24.5993 2.57871
\(92\) 2.20164i 0.229536i
\(93\) 0 0
\(94\) 4.28072 0.441523
\(95\) −12.9225 + 5.14314i −1.32582 + 0.527675i
\(96\) 0 0
\(97\) 0.240408i 0.0244097i 0.999926 + 0.0122049i \(0.00388502\pi\)
−0.999926 + 0.0122049i \(0.996115\pi\)
\(98\) 14.8286i 1.49792i
\(99\) 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.p.1999.1 10
3.2 odd 2 370.2.b.d.149.7 yes 10
5.4 even 2 inner 3330.2.d.p.1999.6 10
15.2 even 4 1850.2.a.bd.1.2 5
15.8 even 4 1850.2.a.be.1.4 5
15.14 odd 2 370.2.b.d.149.4 ✓ 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.b.d.149.4 ✓ 10 15.14 odd 2
370.2.b.d.149.7 yes 10 3.2 odd 2
1850.2.a.bd.1.2 5 15.2 even 4
1850.2.a.be.1.4 5 15.8 even 4
3330.2.d.p.1999.1 10 1.1 even 1 trivial
3330.2.d.p.1999.6 10 5.4 even 2 inner