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Properties

Label 3330.2.a.bg.1.1
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 1.1
Root \(2.59774\) 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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -2.59774 q^{7} -1.00000 q^{8} -1.00000 q^{10} -4.74823 q^{11} +6.69193 q^{13} +2.59774 q^{14} +1.00000 q^{16} +0.748228 q^{17} -3.34596 q^{19} +1.00000 q^{20} +4.74823 q^{22} -1.49646 q^{23} +1.00000 q^{25} -6.69193 q^{26} -2.59774 q^{28} -3.94370 q^{29} +7.79321 q^{31} -1.00000 q^{32} -0.748228 q^{34} -2.59774 q^{35} -1.00000 q^{37} +3.34596 q^{38} -1.00000 q^{40} +6.44724 q^{41} +1.94370 q^{43} -4.74823 q^{44} +1.49646 q^{46} -1.84951 q^{47} -0.251772 q^{49} -1.00000 q^{50} +6.69193 q^{52} -10.4472 q^{53} -4.74823 q^{55} +2.59774 q^{56} +3.94370 q^{58} -5.84951 q^{59} +7.94370 q^{61} -7.79321 q^{62} +1.00000 q^{64} +6.69193 q^{65} +1.84951 q^{67} +0.748228 q^{68} +2.59774 q^{70} +3.88740 q^{71} -7.49646 q^{73} +1.00000 q^{74} -3.34596 q^{76} +12.3346 q^{77} -16.5414 q^{79} +1.00000 q^{80} -6.44724 q^{82} -15.2334 q^{83} +0.748228 q^{85} -1.94370 q^{86} +4.74823 q^{88} -6.00000 q^{89} -17.3839 q^{91} -1.49646 q^{92} +1.84951 q^{94} -3.34596 q^{95} -10.4472 q^{97} +0.251772 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} + 3 q^{4} + 3 q^{5} - q^{7} - 3 q^{8} - 3 q^{10} - 11 q^{11} + q^{14} + 3 q^{16} - q^{17} + 3 q^{20} + 11 q^{22} + 2 q^{23} + 3 q^{25} - q^{28} + 5 q^{29} + 3 q^{31} - 3 q^{32} + q^{34}+ \cdots + 4 q^{98}+O(q^{100}) \) 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
\(7\) −2.59774 −0.981852 −0.490926 − 0.871201i \(-0.663341\pi\)
−0.490926 + 0.871201i \(0.663341\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) −1.00000 −0.316228
\(11\) −4.74823 −1.43164 −0.715822 − 0.698282i \(-0.753948\pi\)
−0.715822 + 0.698282i \(0.753948\pi\)
\(12\) 0 0
\(13\) 6.69193 1.85601 0.928003 − 0.372572i \(-0.121524\pi\)
0.928003 + 0.372572i \(0.121524\pi\)
\(14\) 2.59774 0.694274
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 0.748228 0.181472 0.0907360 − 0.995875i \(-0.471078\pi\)
0.0907360 + 0.995875i \(0.471078\pi\)
\(18\) 0 0
\(19\) −3.34596 −0.767617 −0.383808 − 0.923413i \(-0.625388\pi\)
−0.383808 + 0.923413i \(0.625388\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) 4.74823 1.01233
\(23\) −1.49646 −0.312033 −0.156016 − 0.987754i \(-0.549865\pi\)
−0.156016 + 0.987754i \(0.549865\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −6.69193 −1.31239
\(27\) 0 0
\(28\) −2.59774 −0.490926
\(29\) −3.94370 −0.732326 −0.366163 − 0.930551i \(-0.619329\pi\)
−0.366163 + 0.930551i \(0.619329\pi\)
\(30\) 0 0
\(31\) 7.79321 1.39970 0.699851 − 0.714289i \(-0.253250\pi\)
0.699851 + 0.714289i \(0.253250\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −0.748228 −0.128320
\(35\) −2.59774 −0.439097
\(36\) 0 0
\(37\) −1.00000 −0.164399
\(38\) 3.34596 0.542787
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 6.44724 1.00689 0.503445 − 0.864027i \(-0.332066\pi\)
0.503445 + 0.864027i \(0.332066\pi\)
\(42\) 0 0
\(43\) 1.94370 0.296411 0.148206 − 0.988957i \(-0.452650\pi\)
0.148206 + 0.988957i \(0.452650\pi\)
\(44\) −4.74823 −0.715822
\(45\) 0 0
\(46\) 1.49646 0.220640
\(47\) −1.84951 −0.269778 −0.134889 − 0.990861i \(-0.543068\pi\)
−0.134889 + 0.990861i \(0.543068\pi\)
\(48\) 0 0
\(49\) −0.251772 −0.0359674
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) 6.69193 0.928003
\(53\) −10.4472 −1.43504 −0.717520 − 0.696538i \(-0.754723\pi\)
−0.717520 + 0.696538i \(0.754723\pi\)
\(54\) 0 0
\(55\) −4.74823 −0.640251
\(56\) 2.59774 0.347137
\(57\) 0 0
\(58\) 3.94370 0.517833
\(59\) −5.84951 −0.761541 −0.380770 − 0.924670i \(-0.624341\pi\)
−0.380770 + 0.924670i \(0.624341\pi\)
\(60\) 0 0
\(61\) 7.94370 1.01709 0.508543 − 0.861036i \(-0.330184\pi\)
0.508543 + 0.861036i \(0.330184\pi\)
\(62\) −7.79321 −0.989738
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 6.69193 0.830031
\(66\) 0 0
\(67\) 1.84951 0.225953 0.112977 − 0.993598i \(-0.463961\pi\)
0.112977 + 0.993598i \(0.463961\pi\)
\(68\) 0.748228 0.0907360
\(69\) 0 0
\(70\) 2.59774 0.310489
\(71\) 3.88740 0.461349 0.230675 − 0.973031i \(-0.425907\pi\)
0.230675 + 0.973031i \(0.425907\pi\)
\(72\) 0 0
\(73\) −7.49646 −0.877394 −0.438697 − 0.898635i \(-0.644560\pi\)
−0.438697 + 0.898635i \(0.644560\pi\)
\(74\) 1.00000 0.116248
\(75\) 0 0
\(76\) −3.34596 −0.383808
\(77\) 12.3346 1.40566
\(78\) 0 0
\(79\) −16.5414 −1.86106 −0.930528 − 0.366220i \(-0.880652\pi\)
−0.930528 + 0.366220i \(0.880652\pi\)
\(80\) 1.00000 0.111803
\(81\) 0 0
\(82\) −6.44724 −0.711979
\(83\) −15.2334 −1.67208 −0.836039 − 0.548670i \(-0.815134\pi\)
−0.836039 + 0.548670i \(0.815134\pi\)
\(84\) 0 0
\(85\) 0.748228 0.0811567
\(86\) −1.94370 −0.209594
\(87\) 0 0
\(88\) 4.74823 0.506163
\(89\) −6.00000 −0.635999 −0.317999 − 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) −17.3839 −1.82232
\(92\) −1.49646 −0.156016
\(93\) 0 0
\(94\) 1.84951 0.190762
\(95\) −3.34596 −0.343289
\(96\) 0 0
\(97\) −10.4472 −1.06076 −0.530378 − 0.847761i \(-0.677950\pi\)
−0.530378 + 0.847761i \(0.677950\pi\)
\(98\) 0.251772 0.0254328
\(99\) 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.1.1 3
3.2 odd 2 370.2.a.g.1.1 ✓ 3
12.11 even 2 2960.2.a.u.1.3 3
15.2 even 4 1850.2.b.o.149.6 6
15.8 even 4 1850.2.b.o.149.1 6
15.14 odd 2 1850.2.a.z.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.a.g.1.1 ✓ 3 3.2 odd 2
1850.2.a.z.1.3 3 15.14 odd 2
1850.2.b.o.149.1 6 15.8 even 4
1850.2.b.o.149.6 6 15.2 even 4
2960.2.a.u.1.3 3 12.11 even 2
3330.2.a.bg.1.1 3 1.1 even 1 trivial