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Properties

Label 3330.2.a.bg.1.2
Level $3330$
Weight $2$
Character 3330.1
Self dual yes
Analytic conductor $26.590$
Analytic rank $1$
Dimension $3$
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] = [3330,2,Mod(1,3330)] 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("3330.1"); 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, 0, 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.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,-3,0,3,3,0,-1,-3,0,-3,-11,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(26.5901838731\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.892.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 8x + 10 \) 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)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.31955\) of defining polynomial
Character \(\chi\) \(=\) 3330.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.31955 q^{7} -1.00000 q^{8} -1.00000 q^{10} +0.258786 q^{11} -5.87847 q^{13} +1.31955 q^{14} +1.00000 q^{16} -4.25879 q^{17} +2.93923 q^{19} +1.00000 q^{20} -0.258786 q^{22} +8.51757 q^{23} +1.00000 q^{25} +5.87847 q^{26} -1.31955 q^{28} +3.61968 q^{29} +3.95865 q^{31} -1.00000 q^{32} +4.25879 q^{34} -1.31955 q^{35} -1.00000 q^{37} -2.93923 q^{38} -1.00000 q^{40} +8.89789 q^{41} -5.61968 q^{43} +0.258786 q^{44} -8.51757 q^{46} -5.57834 q^{47} -5.25879 q^{49} -1.00000 q^{50} -5.87847 q^{52} -12.8979 q^{53} +0.258786 q^{55} +1.31955 q^{56} -3.61968 q^{58} -9.57834 q^{59} +0.380316 q^{61} -3.95865 q^{62} +1.00000 q^{64} -5.87847 q^{65} +5.57834 q^{67} -4.25879 q^{68} +1.31955 q^{70} -11.2394 q^{71} +2.51757 q^{73} +1.00000 q^{74} +2.93923 q^{76} -0.341481 q^{77} -7.69987 q^{79} +1.00000 q^{80} -8.89789 q^{82} +6.17860 q^{83} -4.25879 q^{85} +5.61968 q^{86} -0.258786 q^{88} -6.00000 q^{89} +7.75694 q^{91} +8.51757 q^{92} +5.57834 q^{94} +2.93923 q^{95} -12.8979 q^{97} +5.25879 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} + 3 q^{4} + 3 q^{5} - q^{7} - 3 q^{8} - 3 q^{10} - 11 q^{11} + q^{14} + 3 q^{16} - q^{17} + 3 q^{20} + 11 q^{22} + 2 q^{23} + 3 q^{25} - q^{28} + 5 q^{29} + 3 q^{31} - 3 q^{32} + q^{34}+ \cdots + 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −1.31955 −0.498743 −0.249372 − 0.968408i \(-0.580224\pi\)
−0.249372 + 0.968408i \(0.580224\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) −1.00000 −0.316228
\(11\) 0.258786 0.0780268 0.0390134 − 0.999239i \(-0.487578\pi\)
0.0390134 + 0.999239i \(0.487578\pi\)
\(12\) 0 0
\(13\) −5.87847 −1.63039 −0.815197 − 0.579184i \(-0.803371\pi\)
−0.815197 + 0.579184i \(0.803371\pi\)
\(14\) 1.31955 0.352665
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −4.25879 −1.03291 −0.516454 − 0.856315i \(-0.672748\pi\)
−0.516454 + 0.856315i \(0.672748\pi\)
\(18\) 0 0
\(19\) 2.93923 0.674307 0.337153 − 0.941450i \(-0.390536\pi\)
0.337153 + 0.941450i \(0.390536\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) −0.258786 −0.0551733
\(23\) 8.51757 1.77604 0.888018 − 0.459808i \(-0.152082\pi\)
0.888018 + 0.459808i \(0.152082\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 5.87847 1.15286
\(27\) 0 0
\(28\) −1.31955 −0.249372
\(29\) 3.61968 0.672158 0.336079 − 0.941834i \(-0.390899\pi\)
0.336079 + 0.941834i \(0.390899\pi\)
\(30\) 0 0
\(31\) 3.95865 0.710995 0.355497 − 0.934677i \(-0.384311\pi\)
0.355497 + 0.934677i \(0.384311\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 4.25879 0.730376
\(35\) −1.31955 −0.223045
\(36\) 0 0
\(37\) −1.00000 −0.164399
\(38\) −2.93923 −0.476807
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 8.89789 1.38962 0.694808 − 0.719195i \(-0.255489\pi\)
0.694808 + 0.719195i \(0.255489\pi\)
\(42\) 0 0
\(43\) −5.61968 −0.856994 −0.428497 − 0.903543i \(-0.640957\pi\)
−0.428497 + 0.903543i \(0.640957\pi\)
\(44\) 0.258786 0.0390134
\(45\) 0 0
\(46\) −8.51757 −1.25585
\(47\) −5.57834 −0.813684 −0.406842 − 0.913499i \(-0.633370\pi\)
−0.406842 + 0.913499i \(0.633370\pi\)
\(48\) 0 0
\(49\) −5.25879 −0.751255
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) −5.87847 −0.815197
\(53\) −12.8979 −1.77166 −0.885831 − 0.464009i \(-0.846411\pi\)
−0.885831 + 0.464009i \(0.846411\pi\)
\(54\) 0 0
\(55\) 0.258786 0.0348947
\(56\) 1.31955 0.176332
\(57\) 0 0
\(58\) −3.61968 −0.475288
\(59\) −9.57834 −1.24699 −0.623497 − 0.781826i \(-0.714288\pi\)
−0.623497 + 0.781826i \(0.714288\pi\)
\(60\) 0 0
\(61\) 0.380316 0.0486945 0.0243472 − 0.999704i \(-0.492249\pi\)
0.0243472 + 0.999704i \(0.492249\pi\)
\(62\) −3.95865 −0.502749
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −5.87847 −0.729134
\(66\) 0 0
\(67\) 5.57834 0.681502 0.340751 − 0.940154i \(-0.389319\pi\)
0.340751 + 0.940154i \(0.389319\pi\)
\(68\) −4.25879 −0.516454
\(69\) 0 0
\(70\) 1.31955 0.157716
\(71\) −11.2394 −1.33387 −0.666934 − 0.745117i \(-0.732394\pi\)
−0.666934 + 0.745117i \(0.732394\pi\)
\(72\) 0 0
\(73\) 2.51757 0.294659 0.147330 − 0.989087i \(-0.452932\pi\)
0.147330 + 0.989087i \(0.452932\pi\)
\(74\) 1.00000 0.116248
\(75\) 0 0
\(76\) 2.93923 0.337153
\(77\) −0.341481 −0.0389154
\(78\) 0 0
\(79\) −7.69987 −0.866303 −0.433151 − 0.901321i \(-0.642598\pi\)
−0.433151 + 0.901321i \(0.642598\pi\)
\(80\) 1.00000 0.111803
\(81\) 0 0
\(82\) −8.89789 −0.982607
\(83\) 6.17860 0.678190 0.339095 − 0.940752i \(-0.389879\pi\)
0.339095 + 0.940752i \(0.389879\pi\)
\(84\) 0 0
\(85\) −4.25879 −0.461930
\(86\) 5.61968 0.605986
\(87\) 0 0
\(88\) −0.258786 −0.0275866
\(89\) −6.00000 −0.635999 −0.317999 − 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 7.75694 0.813148
\(92\) 8.51757 0.888018
\(93\) 0 0
\(94\) 5.57834 0.575361
\(95\) 2.93923 0.301559
\(96\) 0 0
\(97\) −12.8979 −1.30958 −0.654791 − 0.755810i \(-0.727243\pi\)
−0.654791 + 0.755810i \(0.727243\pi\)
\(98\) 5.25879 0.531218
\(99\) 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.a.bg.1.2 3
3.2 odd 2 370.2.a.g.1.3 ✓ 3
12.11 even 2 2960.2.a.u.1.1 3
15.2 even 4 1850.2.b.o.149.4 6
15.8 even 4 1850.2.b.o.149.3 6
15.14 odd 2 1850.2.a.z.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.a.g.1.3 ✓ 3 3.2 odd 2
1850.2.a.z.1.1 3 15.14 odd 2
1850.2.b.o.149.3 6 15.8 even 4
1850.2.b.o.149.4 6 15.2 even 4
2960.2.a.u.1.1 3 12.11 even 2
3330.2.a.bg.1.2 3 1.1 even 1 trivial