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Properties

Label 2100.2.d.f.1301.4
Level $2100$
Weight $2$
Character 2100.1301
Analytic conductor $16.769$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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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] = [2100,2,Mod(1301,2100)] 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("2100.1301"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2100, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2100.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,-4,0,0,0,0,0,-4,0,0,0,0,0,0,0,-16,0,0,0,8,0,0,0,0,0,20,0, 0,0,0,0,-16,0,0,0,0,0,16,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(43)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(16.7685844245\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{5})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 6x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 420)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1301.4
Root \(-0.874032i\) of defining polynomial
Character \(\chi\) \(=\) 2100.1301
Dual form 2100.2.d.f.1301.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.00000 + 1.41421i) q^{3} +(2.23607 - 1.41421i) q^{7} +(-1.00000 - 2.82843i) q^{9} +2.82843i q^{11} -2.82843i q^{13} -4.00000 q^{17} -6.32456i q^{19} +(-0.236068 + 4.57649i) q^{21} +6.32456i q^{23} +(5.00000 + 1.41421i) q^{27} -5.65685i q^{29} -6.32456i q^{31} +(-4.00000 - 2.82843i) q^{33} -8.94427 q^{37} +(4.00000 + 2.82843i) q^{39} -4.47214 q^{41} -4.47214 q^{43} -6.00000 q^{47} +(3.00000 - 6.32456i) q^{49} +(4.00000 - 5.65685i) q^{51} +6.32456i q^{53} +(8.94427 + 6.32456i) q^{57} -8.94427 q^{59} +(-6.23607 - 4.91034i) q^{63} -4.47214 q^{67} +(-8.94427 - 6.32456i) q^{69} -14.1421i q^{71} +8.48528i q^{73} +(4.00000 + 6.32456i) q^{77} +12.0000 q^{79} +(-7.00000 + 5.65685i) q^{81} +10.0000 q^{83} +(8.00000 + 5.65685i) q^{87} -4.47214 q^{89} +(-4.00000 - 6.32456i) q^{91} +(8.94427 + 6.32456i) q^{93} -8.48528i q^{97} +(8.00000 - 2.82843i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} - 4 q^{9} - 16 q^{17} + 8 q^{21} + 20 q^{27} - 16 q^{33} + 16 q^{39} - 24 q^{47} + 12 q^{49} + 16 q^{51} - 16 q^{63} + 16 q^{77} + 48 q^{79} - 28 q^{81} + 40 q^{83} + 32 q^{87} - 16 q^{91}+ \cdots + 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2100\mathbb{Z}\right)^\times\).

\(n\) \(701\) \(1051\) \(1177\) \(1501\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-1\)

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
\(3\) −1.00000 + 1.41421i −0.577350 + 0.816497i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 2.23607 − 1.41421i 0.845154 − 0.534522i
\(8\) 0 0
\(9\) −1.00000 − 2.82843i −0.333333 − 0.942809i
\(10\) 0 0
\(11\) 2.82843i 0.852803i 0.904534 + 0.426401i \(0.140219\pi\)
−0.904534 + 0.426401i \(0.859781\pi\)
\(12\) 0 0
\(13\) − 2.82843i − 0.784465i −0.919866 − 0.392232i \(-0.871703\pi\)
0.919866 − 0.392232i \(-0.128297\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.00000 −0.970143 −0.485071 − 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) 0 0
\(19\) − 6.32456i − 1.45095i −0.688247 − 0.725476i \(-0.741620\pi\)
0.688247 − 0.725476i \(-0.258380\pi\)
\(20\) 0 0
\(21\) −0.236068 + 4.57649i −0.0515143 + 0.998672i
\(22\) 0 0
\(23\) 6.32456i 1.31876i 0.751809 + 0.659380i \(0.229181\pi\)
−0.751809 + 0.659380i \(0.770819\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 5.00000 + 1.41421i 0.962250 + 0.272166i
\(28\) 0 0
\(29\) − 5.65685i − 1.05045i −0.850963 − 0.525226i \(-0.823981\pi\)
0.850963 − 0.525226i \(-0.176019\pi\)
\(30\) 0 0
\(31\) − 6.32456i − 1.13592i −0.823055 − 0.567962i \(-0.807732\pi\)
0.823055 − 0.567962i \(-0.192268\pi\)
\(32\) 0 0
\(33\) −4.00000 − 2.82843i −0.696311 − 0.492366i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −8.94427 −1.47043 −0.735215 − 0.677834i \(-0.762919\pi\)
−0.735215 + 0.677834i \(0.762919\pi\)
\(38\) 0 0
\(39\) 4.00000 + 2.82843i 0.640513 + 0.452911i
\(40\) 0 0
\(41\) −4.47214 −0.698430 −0.349215 − 0.937043i \(-0.613552\pi\)
−0.349215 + 0.937043i \(0.613552\pi\)
\(42\) 0 0
\(43\) −4.47214 −0.681994 −0.340997 − 0.940064i \(-0.610765\pi\)
−0.340997 + 0.940064i \(0.610765\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −6.00000 −0.875190 −0.437595 − 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) 0 0
\(49\) 3.00000 − 6.32456i 0.428571 − 0.903508i
\(50\) 0 0
\(51\) 4.00000 − 5.65685i 0.560112 − 0.792118i
\(52\) 0 0
\(53\) 6.32456i 0.868744i 0.900733 + 0.434372i \(0.143030\pi\)
−0.900733 + 0.434372i \(0.856970\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 8.94427 + 6.32456i 1.18470 + 0.837708i
\(58\) 0 0
\(59\) −8.94427 −1.16445 −0.582223 − 0.813029i \(-0.697817\pi\)
−0.582223 + 0.813029i \(0.697817\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) −6.23607 − 4.91034i −0.785671 − 0.618645i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −4.47214 −0.546358 −0.273179 − 0.961963i \(-0.588075\pi\)
−0.273179 + 0.961963i \(0.588075\pi\)
\(68\) 0 0
\(69\) −8.94427 − 6.32456i −1.07676 − 0.761387i
\(70\) 0 0
\(71\) − 14.1421i − 1.67836i −0.543852 − 0.839181i \(-0.683035\pi\)
0.543852 − 0.839181i \(-0.316965\pi\)
\(72\) 0 0
\(73\) 8.48528i 0.993127i 0.868000 + 0.496564i \(0.165405\pi\)
−0.868000 + 0.496564i \(0.834595\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 4.00000 + 6.32456i 0.455842 + 0.720750i
\(78\) 0 0
\(79\) 12.0000 1.35011 0.675053 − 0.737769i \(-0.264121\pi\)
0.675053 + 0.737769i \(0.264121\pi\)
\(80\) 0 0
\(81\) −7.00000 + 5.65685i −0.777778 + 0.628539i
\(82\) 0 0
\(83\) 10.0000 1.09764 0.548821 − 0.835940i \(-0.315077\pi\)
0.548821 + 0.835940i \(0.315077\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 8.00000 + 5.65685i 0.857690 + 0.606478i
\(88\) 0 0
\(89\) −4.47214 −0.474045 −0.237023 − 0.971504i \(-0.576172\pi\)
−0.237023 + 0.971504i \(0.576172\pi\)
\(90\) 0 0
\(91\) −4.00000 − 6.32456i −0.419314 − 0.662994i
\(92\) 0 0
\(93\) 8.94427 + 6.32456i 0.927478 + 0.655826i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 8.48528i − 0.861550i −0.902459 − 0.430775i \(-0.858240\pi\)
0.902459 − 0.430775i \(-0.141760\pi\)
\(98\) 0 0
\(99\) 8.00000 − 2.82843i 0.804030 − 0.284268i
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 2100.2.d.f.1301.4 4
3.2 odd 2 2100.2.d.i.1301.4 4
5.2 odd 4 420.2.f.b.209.7 yes 8
5.3 odd 4 420.2.f.b.209.1 ✓ 8
5.4 even 2 2100.2.d.i.1301.1 4
7.6 odd 2 2100.2.d.i.1301.2 4
15.2 even 4 420.2.f.b.209.6 yes 8
15.8 even 4 420.2.f.b.209.4 yes 8
15.14 odd 2 inner 2100.2.d.f.1301.1 4
20.3 even 4 1680.2.k.g.209.7 8
20.7 even 4 1680.2.k.g.209.1 8
21.20 even 2 inner 2100.2.d.f.1301.2 4
35.13 even 4 420.2.f.b.209.8 yes 8
35.27 even 4 420.2.f.b.209.2 yes 8
35.34 odd 2 inner 2100.2.d.f.1301.3 4
60.23 odd 4 1680.2.k.g.209.6 8
60.47 odd 4 1680.2.k.g.209.4 8
105.62 odd 4 420.2.f.b.209.3 yes 8
105.83 odd 4 420.2.f.b.209.5 yes 8
105.104 even 2 2100.2.d.i.1301.3 4
140.27 odd 4 1680.2.k.g.209.8 8
140.83 odd 4 1680.2.k.g.209.2 8
420.83 even 4 1680.2.k.g.209.3 8
420.167 even 4 1680.2.k.g.209.5 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
420.2.f.b.209.1 ✓ 8 5.3 odd 4
420.2.f.b.209.2 yes 8 35.27 even 4
420.2.f.b.209.3 yes 8 105.62 odd 4
420.2.f.b.209.4 yes 8 15.8 even 4
420.2.f.b.209.5 yes 8 105.83 odd 4
420.2.f.b.209.6 yes 8 15.2 even 4
420.2.f.b.209.7 yes 8 5.2 odd 4
420.2.f.b.209.8 yes 8 35.13 even 4
1680.2.k.g.209.1 8 20.7 even 4
1680.2.k.g.209.2 8 140.83 odd 4
1680.2.k.g.209.3 8 420.83 even 4
1680.2.k.g.209.4 8 60.47 odd 4
1680.2.k.g.209.5 8 420.167 even 4
1680.2.k.g.209.6 8 60.23 odd 4
1680.2.k.g.209.7 8 20.3 even 4
1680.2.k.g.209.8 8 140.27 odd 4
2100.2.d.f.1301.1 4 15.14 odd 2 inner
2100.2.d.f.1301.2 4 21.20 even 2 inner
2100.2.d.f.1301.3 4 35.34 odd 2 inner
2100.2.d.f.1301.4 4 1.1 even 1 trivial
2100.2.d.i.1301.1 4 5.4 even 2
2100.2.d.i.1301.2 4 7.6 odd 2
2100.2.d.i.1301.3 4 105.104 even 2
2100.2.d.i.1301.4 4 3.2 odd 2