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Properties

Label 1520.2.q.j.961.2
Level $1520$
Weight $2$
Character 1520.961
Analytic conductor $12.137$
Analytic rank $0$
Dimension $6$
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] = [1520,2,Mod(881,1520)] 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("1520.881"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1520, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1520.q (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,1,0,3,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(12.1372611072\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.3518667.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 7x^{4} - 8x^{3} + 43x^{2} - 42x + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 95)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 961.2
Root \(0.610938 - 1.05818i\) of defining polynomial
Character \(\chi\) \(=\) 1520.961
Dual form 1520.2.q.j.881.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.610938 - 1.05818i) q^{3} +(0.500000 - 0.866025i) q^{5} +0.221876 q^{7} +(0.753509 + 1.30512i) q^{9} +0.778124 q^{11} +(2.50000 + 4.33013i) q^{13} +(-0.610938 - 1.05818i) q^{15} +(-3.53865 + 6.12912i) q^{17} +(-1.33281 + 4.15013i) q^{19} +(0.135553 - 0.234784i) q^{21} +(4.03865 + 6.99515i) q^{23} +(-0.500000 - 0.866025i) q^{25} +5.50702 q^{27} +(-0.110938 - 0.192150i) q^{29} -2.50702 q^{31} +(0.475385 - 0.823392i) q^{33} +(0.110938 - 0.192150i) q^{35} -1.90466 q^{37} +6.10938 q^{39} +(3.61796 - 6.26648i) q^{41} +(3.64959 - 6.32128i) q^{43} +1.50702 q^{45} +(-1.39608 - 2.41808i) q^{47} -6.95077 q^{49} +(4.32379 + 7.48903i) q^{51} +(-2.19024 - 3.79361i) q^{53} +(0.389062 - 0.673875i) q^{55} +(3.57730 + 3.94583i) q^{57} +(-1.39608 + 2.41808i) q^{59} +(6.29216 + 10.8983i) q^{61} +(0.167186 + 0.289574i) q^{63} +5.00000 q^{65} +(-5.28514 - 9.15414i) q^{67} +9.86946 q^{69} +(-4.92070 + 8.52289i) q^{71} +(7.03865 - 12.1913i) q^{73} -1.22188 q^{75} +0.172647 q^{77} +(0.792161 - 1.37206i) q^{79} +(1.10392 - 1.91204i) q^{81} +9.52106 q^{83} +(3.53865 + 6.12912i) q^{85} -0.271105 q^{87} +(-1.57028 - 2.71981i) q^{89} +(0.554690 + 0.960752i) q^{91} +(-1.53163 + 2.65287i) q^{93} +(2.92771 + 3.22932i) q^{95} +(3.18122 - 5.51004i) q^{97} +(0.586324 + 1.01554i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + q^{3} + 3 q^{5} - 4 q^{7} - 4 q^{9} + 10 q^{11} + 15 q^{13} - q^{15} - q^{17} + 12 q^{21} + 4 q^{23} - 3 q^{25} + 16 q^{27} + 2 q^{29} + 2 q^{31} - 11 q^{33} - 2 q^{35} - 4 q^{37} + 10 q^{39} + 2 q^{41}+ \cdots - 13 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1520\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(401\) \(1141\) \(1217\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0.610938 − 1.05818i 0.352725 − 0.610938i −0.634001 − 0.773333i \(-0.718588\pi\)
0.986726 + 0.162394i \(0.0519217\pi\)
\(4\) 0 0
\(5\) 0.500000 − 0.866025i 0.223607 − 0.387298i
\(6\) 0 0
\(7\) 0.221876 0.0838613 0.0419307 − 0.999121i \(-0.486649\pi\)
0.0419307 + 0.999121i \(0.486649\pi\)
\(8\) 0 0
\(9\) 0.753509 + 1.30512i 0.251170 + 0.435039i
\(10\) 0 0
\(11\) 0.778124 0.234613 0.117307 − 0.993096i \(-0.462574\pi\)
0.117307 + 0.993096i \(0.462574\pi\)
\(12\) 0 0
\(13\) 2.50000 + 4.33013i 0.693375 + 1.20096i 0.970725 + 0.240192i \(0.0772105\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) −0.610938 − 1.05818i −0.157744 − 0.273220i
\(16\) 0 0
\(17\) −3.53865 + 6.12912i −0.858249 + 1.48653i 0.0153485 + 0.999882i \(0.495114\pi\)
−0.873598 + 0.486649i \(0.838219\pi\)
\(18\) 0 0
\(19\) −1.33281 + 4.15013i −0.305769 + 0.952106i
\(20\) 0 0
\(21\) 0.135553 − 0.234784i 0.0295800 − 0.0512341i
\(22\) 0 0
\(23\) 4.03865 + 6.99515i 0.842117 + 1.45859i 0.888101 + 0.459647i \(0.152024\pi\)
−0.0459843 + 0.998942i \(0.514642\pi\)
\(24\) 0 0
\(25\) −0.500000 − 0.866025i −0.100000 − 0.173205i
\(26\) 0 0
\(27\) 5.50702 1.05983
\(28\) 0 0
\(29\) −0.110938 − 0.192150i −0.0206007 − 0.0356814i 0.855541 − 0.517735i \(-0.173225\pi\)
−0.876142 + 0.482053i \(0.839891\pi\)
\(30\) 0 0
\(31\) −2.50702 −0.450274 −0.225137 − 0.974327i \(-0.572283\pi\)
−0.225137 + 0.974327i \(0.572283\pi\)
\(32\) 0 0
\(33\) 0.475385 − 0.823392i 0.0827540 − 0.143334i
\(34\) 0 0
\(35\) 0.110938 − 0.192150i 0.0187520 − 0.0324793i
\(36\) 0 0
\(37\) −1.90466 −0.313124 −0.156562 − 0.987668i \(-0.550041\pi\)
−0.156562 + 0.987668i \(0.550041\pi\)
\(38\) 0 0
\(39\) 6.10938 0.978284
\(40\) 0 0
\(41\) 3.61796 − 6.26648i 0.565030 − 0.978661i −0.432017 − 0.901865i \(-0.642198\pi\)
0.997047 − 0.0767950i \(-0.0244687\pi\)
\(42\) 0 0
\(43\) 3.64959 − 6.32128i 0.556557 − 0.963985i −0.441223 − 0.897397i \(-0.645455\pi\)
0.997781 − 0.0665881i \(-0.0212113\pi\)
\(44\) 0 0
\(45\) 1.50702 0.224653
\(46\) 0 0
\(47\) −1.39608 − 2.41808i −0.203639 − 0.352714i 0.746059 − 0.665880i \(-0.231944\pi\)
−0.949698 + 0.313166i \(0.898610\pi\)
\(48\) 0 0
\(49\) −6.95077 −0.992967
\(50\) 0 0
\(51\) 4.32379 + 7.48903i 0.605452 + 1.04867i
\(52\) 0 0
\(53\) −2.19024 − 3.79361i −0.300853 − 0.521093i 0.675476 − 0.737382i \(-0.263938\pi\)
−0.976329 + 0.216289i \(0.930605\pi\)
\(54\) 0 0
\(55\) 0.389062 − 0.673875i 0.0524611 − 0.0908653i
\(56\) 0 0
\(57\) 3.57730 + 3.94583i 0.473825 + 0.522637i
\(58\) 0 0
\(59\) −1.39608 + 2.41808i −0.181754 + 0.314808i −0.942478 − 0.334268i \(-0.891511\pi\)
0.760724 + 0.649076i \(0.224844\pi\)
\(60\) 0 0
\(61\) 6.29216 + 10.8983i 0.805629 + 1.39539i 0.915866 + 0.401484i \(0.131506\pi\)
−0.110237 + 0.993905i \(0.535161\pi\)
\(62\) 0 0
\(63\) 0.167186 + 0.289574i 0.0210634 + 0.0364829i
\(64\) 0 0
\(65\) 5.00000 0.620174
\(66\) 0 0
\(67\) −5.28514 − 9.15414i −0.645683 − 1.11836i −0.984143 − 0.177375i \(-0.943239\pi\)
0.338460 − 0.940981i \(-0.390094\pi\)
\(68\) 0 0
\(69\) 9.86946 1.18814
\(70\) 0 0
\(71\) −4.92070 + 8.52289i −0.583979 + 1.01148i 0.411023 + 0.911625i \(0.365172\pi\)
−0.995002 + 0.0998563i \(0.968162\pi\)
\(72\) 0 0
\(73\) 7.03865 − 12.1913i 0.823812 − 1.42688i −0.0790121 − 0.996874i \(-0.525177\pi\)
0.902824 − 0.430010i \(-0.141490\pi\)
\(74\) 0 0
\(75\) −1.22188 −0.141090
\(76\) 0 0
\(77\) 0.172647 0.0196750
\(78\) 0 0
\(79\) 0.792161 − 1.37206i 0.0891251 − 0.154369i −0.818016 − 0.575195i \(-0.804926\pi\)
0.907142 + 0.420826i \(0.138260\pi\)
\(80\) 0 0
\(81\) 1.10392 − 1.91204i 0.122658 − 0.212449i
\(82\) 0 0
\(83\) 9.52106 1.04507 0.522536 − 0.852617i \(-0.324986\pi\)
0.522536 + 0.852617i \(0.324986\pi\)
\(84\) 0 0
\(85\) 3.53865 + 6.12912i 0.383821 + 0.664797i
\(86\) 0 0
\(87\) −0.271105 −0.0290655
\(88\) 0 0
\(89\) −1.57028 − 2.71981i −0.166450 − 0.288300i 0.770719 − 0.637175i \(-0.219897\pi\)
−0.937169 + 0.348875i \(0.886564\pi\)
\(90\) 0 0
\(91\) 0.554690 + 0.960752i 0.0581474 + 0.100714i
\(92\) 0 0
\(93\) −1.53163 + 2.65287i −0.158823 + 0.275089i
\(94\) 0 0
\(95\) 2.92771 + 3.22932i 0.300377 + 0.331321i
\(96\) 0 0
\(97\) 3.18122 − 5.51004i 0.323004 − 0.559460i −0.658102 − 0.752929i \(-0.728641\pi\)
0.981106 + 0.193469i \(0.0619738\pi\)
\(98\) 0 0
\(99\) 0.586324 + 1.01554i 0.0589277 + 0.102066i
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 1520.2.q.j.961.2 6
4.3 odd 2 95.2.e.b.11.3 ✓ 6
12.11 even 2 855.2.k.g.676.1 6
19.7 even 3 inner 1520.2.q.j.881.2 6
20.3 even 4 475.2.j.b.49.1 12
20.7 even 4 475.2.j.b.49.6 12
20.19 odd 2 475.2.e.d.201.1 6
76.7 odd 6 95.2.e.b.26.3 yes 6
76.11 odd 6 1805.2.a.h.1.1 3
76.27 even 6 1805.2.a.g.1.3 3
228.83 even 6 855.2.k.g.406.1 6
380.7 even 12 475.2.j.b.349.1 12
380.83 even 12 475.2.j.b.349.6 12
380.159 odd 6 475.2.e.d.26.1 6
380.179 even 6 9025.2.a.ba.1.1 3
380.239 odd 6 9025.2.a.z.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.2.e.b.11.3 ✓ 6 4.3 odd 2
95.2.e.b.26.3 yes 6 76.7 odd 6
475.2.e.d.26.1 6 380.159 odd 6
475.2.e.d.201.1 6 20.19 odd 2
475.2.j.b.49.1 12 20.3 even 4
475.2.j.b.49.6 12 20.7 even 4
475.2.j.b.349.1 12 380.7 even 12
475.2.j.b.349.6 12 380.83 even 12
855.2.k.g.406.1 6 228.83 even 6
855.2.k.g.676.1 6 12.11 even 2
1520.2.q.j.881.2 6 19.7 even 3 inner
1520.2.q.j.961.2 6 1.1 even 1 trivial
1805.2.a.g.1.3 3 76.27 even 6
1805.2.a.h.1.1 3 76.11 odd 6
9025.2.a.z.1.3 3 380.239 odd 6
9025.2.a.ba.1.1 3 380.179 even 6