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Properties

Label 2700.2.i.d.901.1
Level $2700$
Weight $2$
Character 2700.901
Analytic conductor $21.560$
Analytic rank $0$
Dimension $8$
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] = [2700,2,Mod(901,2700)] 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("2700.901"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2700, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2700 = 2^{2} \cdot 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2700.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0,0,-1,0,0,0,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(21.5596085457\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.142635249.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + 3x^{6} + 3x^{5} - 11x^{4} + 6x^{3} + 12x^{2} - 24x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3^{5} \)
Twist minimal: no (minimal twist has level 900)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 901.1
Root \(0.818235 - 1.15347i\) of defining polynomial
Character \(\chi\) \(=\) 2700.901
Dual form 2700.2.i.d.1801.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-2.49787 - 4.32643i) q^{7} +(1.99787 + 3.46041i) q^{11} +(0.771582 - 1.33642i) q^{13} -6.99574 q^{17} -2.25667 q^{19} +(-3.89778 + 6.75116i) q^{23} +(3.08304 + 5.33998i) q^{29} +(0.271582 - 0.470394i) q^{31} +6.25240 q^{37} +(0.0979532 - 0.169660i) q^{41} +(-0.0431636 - 0.0747616i) q^{43} +(1.91483 + 3.31658i) q^{47} +(-8.97869 + 15.5515i) q^{49} +4.19164 q^{53} +(3.51278 - 6.08432i) q^{59} +(1.45470 + 2.51962i) q^{61} +(4.48295 - 7.76470i) q^{67} -8.79130 q^{71} +2.28650 q^{73} +(9.98082 - 17.2873i) q^{77} +(6.32211 + 10.9502i) q^{79} +(6.98082 + 12.0911i) q^{83} +10.3577 q^{89} -7.70924 q^{91} +(4.66936 + 8.08758i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - q^{7} - 3 q^{11} + 2 q^{13} - 18 q^{17} - 8 q^{19} - 3 q^{23} + 9 q^{29} - 2 q^{31} + 2 q^{37} - 9 q^{41} + 8 q^{43} + 12 q^{47} - 9 q^{49} - 24 q^{53} + 15 q^{59} + q^{61} + 11 q^{67} + 24 q^{71}+ \cdots + 5 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2700\mathbb{Z}\right)^\times\).

\(n\) \(1001\) \(1351\) \(2377\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(1\)

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 0
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −2.49787 − 4.32643i −0.944105 − 1.63524i −0.757533 − 0.652797i \(-0.773595\pi\)
−0.186573 − 0.982441i \(-0.559738\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.99787 + 3.46041i 0.602380 + 1.04335i 0.992460 + 0.122571i \(0.0391141\pi\)
−0.390080 + 0.920781i \(0.627553\pi\)
\(12\) 0 0
\(13\) 0.771582 − 1.33642i 0.213998 − 0.370656i −0.738964 − 0.673745i \(-0.764685\pi\)
0.952962 + 0.303089i \(0.0980180\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −6.99574 −1.69672 −0.848358 − 0.529424i \(-0.822408\pi\)
−0.848358 + 0.529424i \(0.822408\pi\)
\(18\) 0 0
\(19\) −2.25667 −0.517715 −0.258857 − 0.965916i \(-0.583346\pi\)
−0.258857 + 0.965916i \(0.583346\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3.89778 + 6.75116i −0.812744 + 1.40771i 0.0981929 + 0.995167i \(0.468694\pi\)
−0.910937 + 0.412546i \(0.864640\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 3.08304 + 5.33998i 0.572506 + 0.991609i 0.996308 + 0.0858540i \(0.0273619\pi\)
−0.423802 + 0.905755i \(0.639305\pi\)
\(30\) 0 0
\(31\) 0.271582 − 0.470394i 0.0487775 − 0.0844852i −0.840606 − 0.541647i \(-0.817801\pi\)
0.889383 + 0.457162i \(0.151134\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 6.25240 1.02789 0.513944 − 0.857824i \(-0.328184\pi\)
0.513944 + 0.857824i \(0.328184\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0.0979532 − 0.169660i 0.0152977 − 0.0264964i −0.858275 − 0.513190i \(-0.828464\pi\)
0.873573 + 0.486693i \(0.161797\pi\)
\(42\) 0 0
\(43\) −0.0431636 − 0.0747616i −0.00658239 − 0.0114010i 0.862715 − 0.505690i \(-0.168762\pi\)
−0.869298 + 0.494289i \(0.835429\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.91483 + 3.31658i 0.279307 + 0.483774i 0.971213 − 0.238214i \(-0.0765620\pi\)
−0.691906 + 0.721988i \(0.743229\pi\)
\(48\) 0 0
\(49\) −8.97869 + 15.5515i −1.28267 + 2.22165i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 4.19164 0.575766 0.287883 − 0.957666i \(-0.407049\pi\)
0.287883 + 0.957666i \(0.407049\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 3.51278 − 6.08432i 0.457326 − 0.792111i −0.541493 − 0.840705i \(-0.682141\pi\)
0.998819 + 0.0485942i \(0.0154741\pi\)
\(60\) 0 0
\(61\) 1.45470 + 2.51962i 0.186256 + 0.322604i 0.943999 − 0.329948i \(-0.107031\pi\)
−0.757743 + 0.652553i \(0.773698\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 4.48295 − 7.76470i 0.547680 − 0.948609i −0.450753 − 0.892649i \(-0.648845\pi\)
0.998433 − 0.0559605i \(-0.0178221\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −8.79130 −1.04334 −0.521668 − 0.853149i \(-0.674690\pi\)
−0.521668 + 0.853149i \(0.674690\pi\)
\(72\) 0 0
\(73\) 2.28650 0.267614 0.133807 − 0.991007i \(-0.457280\pi\)
0.133807 + 0.991007i \(0.457280\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 9.98082 − 17.2873i 1.13742 − 1.97007i
\(78\) 0 0
\(79\) 6.32211 + 10.9502i 0.711293 + 1.23199i 0.964372 + 0.264549i \(0.0852232\pi\)
−0.253080 + 0.967445i \(0.581443\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 6.98082 + 12.0911i 0.766244 + 1.32717i 0.939586 + 0.342313i \(0.111210\pi\)
−0.173342 + 0.984862i \(0.555456\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 10.3577 1.09792 0.548958 − 0.835850i \(-0.315025\pi\)
0.548958 + 0.835850i \(0.315025\pi\)
\(90\) 0 0
\(91\) −7.70924 −0.808148
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 4.66936 + 8.08758i 0.474102 + 0.821169i 0.999560 − 0.0296505i \(-0.00943942\pi\)
−0.525458 + 0.850819i \(0.676106\pi\)
\(98\) 0 0
\(99\) 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 2700.2.i.d.901.1 8
3.2 odd 2 900.2.i.e.301.3 yes 8
5.2 odd 4 2700.2.s.d.1549.8 16
5.3 odd 4 2700.2.s.d.1549.1 16
5.4 even 2 2700.2.i.e.901.4 8
9.2 odd 6 900.2.i.e.601.3 yes 8
9.4 even 3 8100.2.a.ba.1.4 4
9.5 odd 6 8100.2.a.z.1.4 4
9.7 even 3 inner 2700.2.i.d.1801.1 8
15.2 even 4 900.2.s.d.49.7 16
15.8 even 4 900.2.s.d.49.2 16
15.14 odd 2 900.2.i.d.301.2 ✓ 8
45.2 even 12 900.2.s.d.349.2 16
45.4 even 6 8100.2.a.y.1.1 4
45.7 odd 12 2700.2.s.d.2449.1 16
45.13 odd 12 8100.2.d.s.649.1 8
45.14 odd 6 8100.2.a.x.1.1 4
45.22 odd 12 8100.2.d.s.649.8 8
45.23 even 12 8100.2.d.q.649.1 8
45.29 odd 6 900.2.i.d.601.2 yes 8
45.32 even 12 8100.2.d.q.649.8 8
45.34 even 6 2700.2.i.e.1801.4 8
45.38 even 12 900.2.s.d.349.7 16
45.43 odd 12 2700.2.s.d.2449.8 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
900.2.i.d.301.2 ✓ 8 15.14 odd 2
900.2.i.d.601.2 yes 8 45.29 odd 6
900.2.i.e.301.3 yes 8 3.2 odd 2
900.2.i.e.601.3 yes 8 9.2 odd 6
900.2.s.d.49.2 16 15.8 even 4
900.2.s.d.49.7 16 15.2 even 4
900.2.s.d.349.2 16 45.2 even 12
900.2.s.d.349.7 16 45.38 even 12
2700.2.i.d.901.1 8 1.1 even 1 trivial
2700.2.i.d.1801.1 8 9.7 even 3 inner
2700.2.i.e.901.4 8 5.4 even 2
2700.2.i.e.1801.4 8 45.34 even 6
2700.2.s.d.1549.1 16 5.3 odd 4
2700.2.s.d.1549.8 16 5.2 odd 4
2700.2.s.d.2449.1 16 45.7 odd 12
2700.2.s.d.2449.8 16 45.43 odd 12
8100.2.a.x.1.1 4 45.14 odd 6
8100.2.a.y.1.1 4 45.4 even 6
8100.2.a.z.1.4 4 9.5 odd 6
8100.2.a.ba.1.4 4 9.4 even 3
8100.2.d.q.649.1 8 45.23 even 12
8100.2.d.q.649.8 8 45.32 even 12
8100.2.d.s.649.1 8 45.13 odd 12
8100.2.d.s.649.8 8 45.22 odd 12