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Properties

Label 2000.4.a.t.1.8
Level $2000$
Weight $4$
Character 2000.1
Self dual yes
Analytic conductor $118.004$
Analytic rank $1$
Dimension $8$
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 = [8,0,4,0,0,0,-8,0,60] 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: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 56x^{6} - 44x^{5} + 924x^{4} + 832x^{3} - 5656x^{2} - 3540x + 11255 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 5^{3} \)
Twist minimal: no (minimal twist has level 1000)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(3.52936\) 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)
 
\(f(q)\) \(=\) \(q+9.39735 q^{3} -25.2652 q^{7} +61.3103 q^{9} +25.8725 q^{11} -76.7040 q^{13} +6.74057 q^{17} -65.0844 q^{19} -237.426 q^{21} +141.703 q^{23} +322.426 q^{27} -19.6756 q^{29} -276.763 q^{31} +243.133 q^{33} +298.501 q^{37} -720.815 q^{39} -79.6043 q^{41} -472.042 q^{43} +46.2331 q^{47} +295.332 q^{49} +63.3435 q^{51} -682.053 q^{53} -611.621 q^{57} -95.8988 q^{59} +58.8170 q^{61} -1549.02 q^{63} +911.840 q^{67} +1331.63 q^{69} -1053.69 q^{71} -641.213 q^{73} -653.674 q^{77} -908.871 q^{79} +1374.57 q^{81} -121.056 q^{83} -184.899 q^{87} -622.064 q^{89} +1937.95 q^{91} -2600.84 q^{93} +1122.67 q^{97} +1586.25 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{3} - 8 q^{7} + 60 q^{9} + 16 q^{11} - 40 q^{13} - 112 q^{17} - 16 q^{19} - 256 q^{21} + 312 q^{23} + 268 q^{27} - 188 q^{29} - 464 q^{31} - 144 q^{33} + 72 q^{37} - 672 q^{39} - 380 q^{41} - 44 q^{43}+ \cdots + 832 q^{99}+O(q^{100}) \) 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
\(3\) 9.39735 1.80852 0.904261 − 0.426981i \(-0.140423\pi\)
0.904261 + 0.426981i \(0.140423\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −25.2652 −1.36419 −0.682097 − 0.731261i \(-0.738932\pi\)
−0.682097 + 0.731261i \(0.738932\pi\)
\(8\) 0 0
\(9\) 61.3103 2.27075
\(10\) 0 0
\(11\) 25.8725 0.709167 0.354584 − 0.935024i \(-0.384623\pi\)
0.354584 + 0.935024i \(0.384623\pi\)
\(12\) 0 0
\(13\) −76.7040 −1.63645 −0.818225 − 0.574897i \(-0.805042\pi\)
−0.818225 + 0.574897i \(0.805042\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.74057 0.0961663 0.0480832 − 0.998843i \(-0.484689\pi\)
0.0480832 + 0.998843i \(0.484689\pi\)
\(18\) 0 0
\(19\) −65.0844 −0.785862 −0.392931 − 0.919568i \(-0.628539\pi\)
−0.392931 + 0.919568i \(0.628539\pi\)
\(20\) 0 0
\(21\) −237.426 −2.46718
\(22\) 0 0
\(23\) 141.703 1.28466 0.642329 − 0.766429i \(-0.277968\pi\)
0.642329 + 0.766429i \(0.277968\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 322.426 2.29818
\(28\) 0 0
\(29\) −19.6756 −0.125989 −0.0629943 − 0.998014i \(-0.520065\pi\)
−0.0629943 + 0.998014i \(0.520065\pi\)
\(30\) 0 0
\(31\) −276.763 −1.60349 −0.801744 − 0.597667i \(-0.796094\pi\)
−0.801744 + 0.597667i \(0.796094\pi\)
\(32\) 0 0
\(33\) 243.133 1.28254
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 298.501 1.32630 0.663151 − 0.748486i \(-0.269219\pi\)
0.663151 + 0.748486i \(0.269219\pi\)
\(38\) 0 0
\(39\) −720.815 −2.95956
\(40\) 0 0
\(41\) −79.6043 −0.303222 −0.151611 − 0.988440i \(-0.548446\pi\)
−0.151611 + 0.988440i \(0.548446\pi\)
\(42\) 0 0
\(43\) −472.042 −1.67409 −0.837044 − 0.547136i \(-0.815718\pi\)
−0.837044 + 0.547136i \(0.815718\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 46.2331 0.143485 0.0717425 − 0.997423i \(-0.477144\pi\)
0.0717425 + 0.997423i \(0.477144\pi\)
\(48\) 0 0
\(49\) 295.332 0.861027
\(50\) 0 0
\(51\) 63.3435 0.173919
\(52\) 0 0
\(53\) −682.053 −1.76768 −0.883842 − 0.467786i \(-0.845052\pi\)
−0.883842 + 0.467786i \(0.845052\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −611.621 −1.42125
\(58\) 0 0
\(59\) −95.8988 −0.211609 −0.105805 − 0.994387i \(-0.533742\pi\)
−0.105805 + 0.994387i \(0.533742\pi\)
\(60\) 0 0
\(61\) 58.8170 0.123455 0.0617274 − 0.998093i \(-0.480339\pi\)
0.0617274 + 0.998093i \(0.480339\pi\)
\(62\) 0 0
\(63\) −1549.02 −3.09775
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 911.840 1.66267 0.831336 − 0.555770i \(-0.187576\pi\)
0.831336 + 0.555770i \(0.187576\pi\)
\(68\) 0 0
\(69\) 1331.63 2.32333
\(70\) 0 0
\(71\) −1053.69 −1.76127 −0.880635 − 0.473795i \(-0.842884\pi\)
−0.880635 + 0.473795i \(0.842884\pi\)
\(72\) 0 0
\(73\) −641.213 −1.02806 −0.514030 − 0.857772i \(-0.671848\pi\)
−0.514030 + 0.857772i \(0.671848\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −653.674 −0.967442
\(78\) 0 0
\(79\) −908.871 −1.29438 −0.647190 − 0.762329i \(-0.724056\pi\)
−0.647190 + 0.762329i \(0.724056\pi\)
\(80\) 0 0
\(81\) 1374.57 1.88556
\(82\) 0 0
\(83\) −121.056 −0.160092 −0.0800458 − 0.996791i \(-0.525507\pi\)
−0.0800458 + 0.996791i \(0.525507\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −184.899 −0.227853
\(88\) 0 0
\(89\) −622.064 −0.740884 −0.370442 − 0.928856i \(-0.620794\pi\)
−0.370442 + 0.928856i \(0.620794\pi\)
\(90\) 0 0
\(91\) 1937.95 2.23244
\(92\) 0 0
\(93\) −2600.84 −2.89994
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1122.67 1.17516 0.587579 − 0.809167i \(-0.300081\pi\)
0.587579 + 0.809167i \(0.300081\pi\)
\(98\) 0 0
\(99\) 1586.25 1.61034
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.t.1.8 8
4.3 odd 2 1000.4.a.a.1.1 ✓ 8
5.4 even 2 2000.4.a.o.1.1 8
20.3 even 4 1000.4.c.a.249.1 16
20.7 even 4 1000.4.c.a.249.16 16
20.19 odd 2 1000.4.a.d.1.8 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1000.4.a.a.1.1 ✓ 8 4.3 odd 2
1000.4.a.d.1.8 yes 8 20.19 odd 2
1000.4.c.a.249.1 16 20.3 even 4
1000.4.c.a.249.16 16 20.7 even 4
2000.4.a.o.1.1 8 5.4 even 2
2000.4.a.t.1.8 8 1.1 even 1 trivial