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Properties

Label 2000.4.a.n.1.2
Level $2000$
Weight $4$
Character 2000.1
Self dual yes
Analytic conductor $118.004$
Analytic rank $0$
Dimension $6$
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] = [2000,4,Mod(1,2000)] 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("2000.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2000, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 2000 = 2^{4} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2000.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,14,0,0,0,67,0,92] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(118.003820011\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.497918125.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 29x^{4} - 6x^{3} + 216x^{2} + 280x + 80 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 125)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-3.45688\) of defining polynomial
Character \(\chi\) \(=\) 2000.1

$q$-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)
 
\(f(q)\) \(=\) \(q-4.57894 q^{3} +18.9048 q^{7} -6.03327 q^{9} +7.94853 q^{11} -43.9341 q^{13} -124.854 q^{17} -93.4871 q^{19} -86.5642 q^{21} -23.6463 q^{23} +151.258 q^{27} -171.051 q^{29} +70.6154 q^{31} -36.3959 q^{33} +275.854 q^{37} +201.172 q^{39} +259.564 q^{41} +433.498 q^{43} -249.622 q^{47} +14.3927 q^{49} +571.698 q^{51} -112.842 q^{53} +428.072 q^{57} -844.191 q^{59} +768.836 q^{61} -114.058 q^{63} -385.978 q^{67} +108.275 q^{69} -350.291 q^{71} -773.075 q^{73} +150.266 q^{77} +497.416 q^{79} -529.701 q^{81} +349.967 q^{83} +783.233 q^{87} -1377.29 q^{89} -830.566 q^{91} -323.344 q^{93} -1205.85 q^{97} -47.9557 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 14 q^{3} + 67 q^{7} + 92 q^{9} - 27 q^{11} - 149 q^{13} - 72 q^{17} + 105 q^{19} - 148 q^{21} + 269 q^{23} + 590 q^{27} - 30 q^{29} + 168 q^{31} + 162 q^{33} - 167 q^{37} - 34 q^{39} + 387 q^{41}+ \cdots + 2511 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −4.57894 −0.881218 −0.440609 − 0.897699i \(-0.645237\pi\)
−0.440609 + 0.897699i \(0.645237\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 18.9048 1.02076 0.510382 − 0.859948i \(-0.329504\pi\)
0.510382 + 0.859948i \(0.329504\pi\)
\(8\) 0 0
\(9\) −6.03327 −0.223455
\(10\) 0 0
\(11\) 7.94853 0.217870 0.108935 − 0.994049i \(-0.465256\pi\)
0.108935 + 0.994049i \(0.465256\pi\)
\(12\) 0 0
\(13\) −43.9341 −0.937317 −0.468658 − 0.883380i \(-0.655262\pi\)
−0.468658 + 0.883380i \(0.655262\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −124.854 −1.78126 −0.890632 − 0.454725i \(-0.849737\pi\)
−0.890632 + 0.454725i \(0.849737\pi\)
\(18\) 0 0
\(19\) −93.4871 −1.12881 −0.564405 − 0.825498i \(-0.690894\pi\)
−0.564405 + 0.825498i \(0.690894\pi\)
\(20\) 0 0
\(21\) −86.5642 −0.899517
\(22\) 0 0
\(23\) −23.6463 −0.214373 −0.107187 − 0.994239i \(-0.534184\pi\)
−0.107187 + 0.994239i \(0.534184\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 151.258 1.07813
\(28\) 0 0
\(29\) −171.051 −1.09529 −0.547644 − 0.836711i \(-0.684475\pi\)
−0.547644 + 0.836711i \(0.684475\pi\)
\(30\) 0 0
\(31\) 70.6154 0.409126 0.204563 − 0.978853i \(-0.434423\pi\)
0.204563 + 0.978853i \(0.434423\pi\)
\(32\) 0 0
\(33\) −36.3959 −0.191991
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 275.854 1.22568 0.612840 − 0.790207i \(-0.290027\pi\)
0.612840 + 0.790207i \(0.290027\pi\)
\(38\) 0 0
\(39\) 201.172 0.825980
\(40\) 0 0
\(41\) 259.564 0.988710 0.494355 − 0.869260i \(-0.335404\pi\)
0.494355 + 0.869260i \(0.335404\pi\)
\(42\) 0 0
\(43\) 433.498 1.53739 0.768695 − 0.639616i \(-0.220907\pi\)
0.768695 + 0.639616i \(0.220907\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −249.622 −0.774706 −0.387353 − 0.921932i \(-0.626610\pi\)
−0.387353 + 0.921932i \(0.626610\pi\)
\(48\) 0 0
\(49\) 14.3927 0.0419612
\(50\) 0 0
\(51\) 571.698 1.56968
\(52\) 0 0
\(53\) −112.842 −0.292452 −0.146226 − 0.989251i \(-0.546713\pi\)
−0.146226 + 0.989251i \(0.546713\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 428.072 0.994728
\(58\) 0 0
\(59\) −844.191 −1.86279 −0.931393 − 0.364016i \(-0.881405\pi\)
−0.931393 + 0.364016i \(0.881405\pi\)
\(60\) 0 0
\(61\) 768.836 1.61376 0.806880 − 0.590715i \(-0.201154\pi\)
0.806880 + 0.590715i \(0.201154\pi\)
\(62\) 0 0
\(63\) −114.058 −0.228095
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −385.978 −0.703802 −0.351901 − 0.936037i \(-0.614465\pi\)
−0.351901 + 0.936037i \(0.614465\pi\)
\(68\) 0 0
\(69\) 108.275 0.188910
\(70\) 0 0
\(71\) −350.291 −0.585520 −0.292760 − 0.956186i \(-0.594574\pi\)
−0.292760 + 0.956186i \(0.594574\pi\)
\(72\) 0 0
\(73\) −773.075 −1.23947 −0.619737 − 0.784810i \(-0.712761\pi\)
−0.619737 + 0.784810i \(0.712761\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 150.266 0.222394
\(78\) 0 0
\(79\) 497.416 0.708401 0.354201 − 0.935169i \(-0.384753\pi\)
0.354201 + 0.935169i \(0.384753\pi\)
\(80\) 0 0
\(81\) −529.701 −0.726613
\(82\) 0 0
\(83\) 349.967 0.462818 0.231409 − 0.972857i \(-0.425666\pi\)
0.231409 + 0.972857i \(0.425666\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 783.233 0.965188
\(88\) 0 0
\(89\) −1377.29 −1.64036 −0.820181 − 0.572104i \(-0.806127\pi\)
−0.820181 + 0.572104i \(0.806127\pi\)
\(90\) 0 0
\(91\) −830.566 −0.956780
\(92\) 0 0
\(93\) −323.344 −0.360529
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1205.85 −1.26222 −0.631111 − 0.775693i \(-0.717401\pi\)
−0.631111 + 0.775693i \(0.717401\pi\)
\(98\) 0 0
\(99\) −47.9557 −0.0486841
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 2000.4.a.n.1.2 6
4.3 odd 2 125.4.a.b.1.4 ✓ 6
5.4 even 2 2000.4.a.k.1.5 6
12.11 even 2 1125.4.a.k.1.3 6
20.3 even 4 125.4.b.c.124.7 12
20.7 even 4 125.4.b.c.124.6 12
20.19 odd 2 125.4.a.c.1.3 yes 6
60.59 even 2 1125.4.a.f.1.4 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
125.4.a.b.1.4 ✓ 6 4.3 odd 2
125.4.a.c.1.3 yes 6 20.19 odd 2
125.4.b.c.124.6 12 20.7 even 4
125.4.b.c.124.7 12 20.3 even 4
1125.4.a.f.1.4 6 60.59 even 2
1125.4.a.k.1.3 6 12.11 even 2
2000.4.a.k.1.5 6 5.4 even 2
2000.4.a.n.1.2 6 1.1 even 1 trivial