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Properties

Label 2700.2.i.d.901.3
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 901.3
Root \(-1.32841 + 0.485097i\) of defining polynomial
Character \(\chi\) \(=\) 2700.901
Dual form 2700.2.i.d.1801.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.340213 + 0.589266i) q^{7} +(-0.840213 - 1.45529i) q^{11} +(2.57251 - 4.45572i) q^{13} -1.31957 q^{17} -0.324642 q^{19} +(-1.89462 + 3.28158i) q^{23} +(-4.32292 - 7.48751i) q^{29} +(2.07251 - 3.58970i) q^{31} -1.35578 q^{37} +(-3.57505 + 6.19216i) q^{41} +(-3.64503 - 6.31337i) q^{43} +(6.48270 + 11.2284i) q^{47} +(3.26851 - 5.66123i) q^{49} -8.83052 q^{53} +(4.40766 - 7.63429i) q^{59} +(-4.98524 - 8.63469i) q^{61} +(-2.08808 + 3.61667i) q^{67} +0.891185 q^{71} +7.82038 q^{73} +(0.571703 - 0.990219i) q^{77} +(-4.82799 - 8.36232i) q^{79} +(-2.42830 - 4.20593i) q^{83} -17.4764 q^{89} +3.50081 q^{91} +(4.46713 + 7.73730i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - q^{7} - 3 q^{11} + 2 q^{13} - 18 q^{17} - 8 q^{19} - 3 q^{23} + 9 q^{29} - 2 q^{31} + 2 q^{37} - 9 q^{41} + 8 q^{43} + 12 q^{47} - 9 q^{49} - 24 q^{53} + 15 q^{59} + q^{61} + 11 q^{67} + 24 q^{71}+ \cdots + 5 q^{97}+O(q^{100}) \) 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
\(7\) 0.340213 + 0.589266i 0.128588 + 0.222722i 0.923130 − 0.384488i \(-0.125622\pi\)
−0.794541 + 0.607210i \(0.792289\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −0.840213 − 1.45529i −0.253334 − 0.438787i 0.711108 − 0.703083i \(-0.248194\pi\)
−0.964442 + 0.264296i \(0.914860\pi\)
\(12\) 0 0
\(13\) 2.57251 − 4.45572i 0.713487 − 1.23580i −0.250054 − 0.968232i \(-0.580448\pi\)
0.963540 − 0.267563i \(-0.0862184\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.31957 −0.320044 −0.160022 − 0.987113i \(-0.551156\pi\)
−0.160022 + 0.987113i \(0.551156\pi\)
\(18\) 0 0
\(19\) −0.324642 −0.0744779 −0.0372390 − 0.999306i \(-0.511856\pi\)
−0.0372390 + 0.999306i \(0.511856\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −1.89462 + 3.28158i −0.395056 + 0.684257i −0.993108 − 0.117200i \(-0.962608\pi\)
0.598053 + 0.801457i \(0.295941\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −4.32292 − 7.48751i −0.802746 − 1.39040i −0.917802 − 0.397037i \(-0.870038\pi\)
0.115057 − 0.993359i \(-0.463295\pi\)
\(30\) 0 0
\(31\) 2.07251 − 3.58970i 0.372234 − 0.644729i −0.617675 − 0.786434i \(-0.711925\pi\)
0.989909 + 0.141705i \(0.0452585\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −1.35578 −0.222890 −0.111445 − 0.993771i \(-0.535548\pi\)
−0.111445 + 0.993771i \(0.535548\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −3.57505 + 6.19216i −0.558328 + 0.967053i 0.439308 + 0.898337i \(0.355224\pi\)
−0.997636 + 0.0687167i \(0.978110\pi\)
\(42\) 0 0
\(43\) −3.64503 − 6.31337i −0.555861 − 0.962780i −0.997836 − 0.0657523i \(-0.979055\pi\)
0.441975 − 0.897027i \(-0.354278\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.48270 + 11.2284i 0.945600 + 1.63783i 0.754546 + 0.656247i \(0.227857\pi\)
0.191054 + 0.981580i \(0.438810\pi\)
\(48\) 0 0
\(49\) 3.26851 − 5.66123i 0.466930 − 0.808747i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −8.83052 −1.21297 −0.606483 − 0.795097i \(-0.707420\pi\)
−0.606483 + 0.795097i \(0.707420\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.40766 − 7.63429i 0.573828 − 0.993900i −0.422340 − 0.906438i \(-0.638791\pi\)
0.996168 − 0.0874619i \(-0.0278756\pi\)
\(60\) 0 0
\(61\) −4.98524 − 8.63469i −0.638294 − 1.10556i −0.985807 − 0.167883i \(-0.946307\pi\)
0.347513 − 0.937675i \(-0.387026\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −2.08808 + 3.61667i −0.255100 + 0.441846i −0.964923 − 0.262534i \(-0.915442\pi\)
0.709823 + 0.704380i \(0.248775\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0.891185 0.105764 0.0528821 − 0.998601i \(-0.483159\pi\)
0.0528821 + 0.998601i \(0.483159\pi\)
\(72\) 0 0
\(73\) 7.82038 0.915307 0.457653 − 0.889131i \(-0.348690\pi\)
0.457653 + 0.889131i \(0.348690\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0.571703 − 0.990219i 0.0651516 − 0.112846i
\(78\) 0 0
\(79\) −4.82799 − 8.36232i −0.543191 − 0.940834i −0.998718 − 0.0506124i \(-0.983883\pi\)
0.455528 − 0.890222i \(-0.349451\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −2.42830 − 4.20593i −0.266540 − 0.461661i 0.701426 − 0.712743i \(-0.252547\pi\)
−0.967966 + 0.251081i \(0.919214\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −17.4764 −1.85249 −0.926245 − 0.376922i \(-0.876982\pi\)
−0.926245 + 0.376922i \(0.876982\pi\)
\(90\) 0 0
\(91\) 3.50081 0.366985
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 4.46713 + 7.73730i 0.453569 + 0.785604i 0.998605 − 0.0528087i \(-0.0168173\pi\)
−0.545036 + 0.838413i \(0.683484\pi\)
\(98\) 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.901.3 8
3.2 odd 2 900.2.i.e.301.2 yes 8
5.2 odd 4 2700.2.s.d.1549.3 16
5.3 odd 4 2700.2.s.d.1549.6 16
5.4 even 2 2700.2.i.e.901.2 8
9.2 odd 6 900.2.i.e.601.2 yes 8
9.4 even 3 8100.2.a.ba.1.2 4
9.5 odd 6 8100.2.a.z.1.2 4
9.7 even 3 inner 2700.2.i.d.1801.3 8
15.2 even 4 900.2.s.d.49.1 16
15.8 even 4 900.2.s.d.49.8 16
15.14 odd 2 900.2.i.d.301.3 ✓ 8
45.2 even 12 900.2.s.d.349.8 16
45.4 even 6 8100.2.a.y.1.3 4
45.7 odd 12 2700.2.s.d.2449.6 16
45.13 odd 12 8100.2.d.s.649.6 8
45.14 odd 6 8100.2.a.x.1.3 4
45.22 odd 12 8100.2.d.s.649.3 8
45.23 even 12 8100.2.d.q.649.6 8
45.29 odd 6 900.2.i.d.601.3 yes 8
45.32 even 12 8100.2.d.q.649.3 8
45.34 even 6 2700.2.i.e.1801.2 8
45.38 even 12 900.2.s.d.349.1 16
45.43 odd 12 2700.2.s.d.2449.3 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
900.2.i.d.301.3 ✓ 8 15.14 odd 2
900.2.i.d.601.3 yes 8 45.29 odd 6
900.2.i.e.301.2 yes 8 3.2 odd 2
900.2.i.e.601.2 yes 8 9.2 odd 6
900.2.s.d.49.1 16 15.2 even 4
900.2.s.d.49.8 16 15.8 even 4
900.2.s.d.349.1 16 45.38 even 12
900.2.s.d.349.8 16 45.2 even 12
2700.2.i.d.901.3 8 1.1 even 1 trivial
2700.2.i.d.1801.3 8 9.7 even 3 inner
2700.2.i.e.901.2 8 5.4 even 2
2700.2.i.e.1801.2 8 45.34 even 6
2700.2.s.d.1549.3 16 5.2 odd 4
2700.2.s.d.1549.6 16 5.3 odd 4
2700.2.s.d.2449.3 16 45.43 odd 12
2700.2.s.d.2449.6 16 45.7 odd 12
8100.2.a.x.1.3 4 45.14 odd 6
8100.2.a.y.1.3 4 45.4 even 6
8100.2.a.z.1.2 4 9.5 odd 6
8100.2.a.ba.1.2 4 9.4 even 3
8100.2.d.q.649.3 8 45.32 even 12
8100.2.d.q.649.6 8 45.23 even 12
8100.2.d.s.649.3 8 45.22 odd 12
8100.2.d.s.649.6 8 45.13 odd 12