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Properties

Label 2000.4.a.q.1.6
Level $2000$
Weight $4$
Character 2000.1
Self dual yes
Analytic conductor $118.004$
Analytic rank $1$
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] = [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 = [8,0,0,0,0,0,0,0,96] 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: \( x^{8} - 26x^{6} + 201x^{4} - 416x^{2} + 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6}\cdot 5^{4} \)
Twist minimal: no (minimal twist has level 125)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(-3.12336\) 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)
 
\(f(q)\) \(=\) \(q+4.33288 q^{3} +25.2127 q^{7} -8.22615 q^{9} +48.7634 q^{11} +10.6911 q^{13} -49.3465 q^{17} -132.448 q^{19} +109.244 q^{21} -135.001 q^{23} -152.631 q^{27} -175.367 q^{29} -207.098 q^{31} +211.286 q^{33} -195.008 q^{37} +46.3231 q^{39} -102.077 q^{41} -413.419 q^{43} -159.545 q^{47} +292.681 q^{49} -213.813 q^{51} -477.605 q^{53} -573.881 q^{57} +249.456 q^{59} +771.392 q^{61} -207.404 q^{63} -8.51845 q^{67} -584.944 q^{69} -254.666 q^{71} +207.364 q^{73} +1229.46 q^{77} -299.523 q^{79} -439.224 q^{81} +939.409 q^{83} -759.845 q^{87} +316.866 q^{89} +269.551 q^{91} -897.333 q^{93} +172.399 q^{97} -401.135 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 96 q^{9} - 76 q^{11} - 280 q^{19} + 236 q^{21} + 300 q^{29} - 996 q^{31} - 352 q^{39} - 764 q^{41} + 1404 q^{49} - 436 q^{51} - 660 q^{59} + 756 q^{61} - 3688 q^{69} - 1896 q^{71} - 940 q^{79} + 1588 q^{81}+ \cdots - 9912 q^{99}+O(q^{100}) \) 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
\(3\) 4.33288 0.833863 0.416932 − 0.908938i \(-0.363105\pi\)
0.416932 + 0.908938i \(0.363105\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 25.2127 1.36136 0.680679 − 0.732581i \(-0.261685\pi\)
0.680679 + 0.732581i \(0.261685\pi\)
\(8\) 0 0
\(9\) −8.22615 −0.304672
\(10\) 0 0
\(11\) 48.7634 1.33661 0.668305 − 0.743887i \(-0.267020\pi\)
0.668305 + 0.743887i \(0.267020\pi\)
\(12\) 0 0
\(13\) 10.6911 0.228090 0.114045 − 0.993476i \(-0.463619\pi\)
0.114045 + 0.993476i \(0.463619\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −49.3465 −0.704017 −0.352008 − 0.935997i \(-0.614501\pi\)
−0.352008 + 0.935997i \(0.614501\pi\)
\(18\) 0 0
\(19\) −132.448 −1.59924 −0.799622 − 0.600503i \(-0.794967\pi\)
−0.799622 + 0.600503i \(0.794967\pi\)
\(20\) 0 0
\(21\) 109.244 1.13519
\(22\) 0 0
\(23\) −135.001 −1.22390 −0.611950 − 0.790897i \(-0.709614\pi\)
−0.611950 + 0.790897i \(0.709614\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −152.631 −1.08792
\(28\) 0 0
\(29\) −175.367 −1.12293 −0.561463 − 0.827502i \(-0.689761\pi\)
−0.561463 + 0.827502i \(0.689761\pi\)
\(30\) 0 0
\(31\) −207.098 −1.19987 −0.599935 − 0.800049i \(-0.704807\pi\)
−0.599935 + 0.800049i \(0.704807\pi\)
\(32\) 0 0
\(33\) 211.286 1.11455
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −195.008 −0.866462 −0.433231 − 0.901283i \(-0.642627\pi\)
−0.433231 + 0.901283i \(0.642627\pi\)
\(38\) 0 0
\(39\) 46.3231 0.190196
\(40\) 0 0
\(41\) −102.077 −0.388824 −0.194412 − 0.980920i \(-0.562280\pi\)
−0.194412 + 0.980920i \(0.562280\pi\)
\(42\) 0 0
\(43\) −413.419 −1.46618 −0.733091 − 0.680131i \(-0.761923\pi\)
−0.733091 + 0.680131i \(0.761923\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −159.545 −0.495149 −0.247574 − 0.968869i \(-0.579633\pi\)
−0.247574 + 0.968869i \(0.579633\pi\)
\(48\) 0 0
\(49\) 292.681 0.853297
\(50\) 0 0
\(51\) −213.813 −0.587054
\(52\) 0 0
\(53\) −477.605 −1.23781 −0.618907 − 0.785465i \(-0.712424\pi\)
−0.618907 + 0.785465i \(0.712424\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −573.881 −1.33355
\(58\) 0 0
\(59\) 249.456 0.550447 0.275223 − 0.961380i \(-0.411248\pi\)
0.275223 + 0.961380i \(0.411248\pi\)
\(60\) 0 0
\(61\) 771.392 1.61913 0.809563 − 0.587034i \(-0.199704\pi\)
0.809563 + 0.587034i \(0.199704\pi\)
\(62\) 0 0
\(63\) −207.404 −0.414768
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −8.51845 −0.0155328 −0.00776638 − 0.999970i \(-0.502472\pi\)
−0.00776638 + 0.999970i \(0.502472\pi\)
\(68\) 0 0
\(69\) −584.944 −1.02056
\(70\) 0 0
\(71\) −254.666 −0.425679 −0.212840 − 0.977087i \(-0.568271\pi\)
−0.212840 + 0.977087i \(0.568271\pi\)
\(72\) 0 0
\(73\) 207.364 0.332468 0.166234 − 0.986086i \(-0.446839\pi\)
0.166234 + 0.986086i \(0.446839\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1229.46 1.81961
\(78\) 0 0
\(79\) −299.523 −0.426569 −0.213284 − 0.976990i \(-0.568416\pi\)
−0.213284 + 0.976990i \(0.568416\pi\)
\(80\) 0 0
\(81\) −439.224 −0.602502
\(82\) 0 0
\(83\) 939.409 1.24233 0.621166 − 0.783679i \(-0.286659\pi\)
0.621166 + 0.783679i \(0.286659\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −759.845 −0.936367
\(88\) 0 0
\(89\) 316.866 0.377390 0.188695 − 0.982036i \(-0.439574\pi\)
0.188695 + 0.982036i \(0.439574\pi\)
\(90\) 0 0
\(91\) 269.551 0.310512
\(92\) 0 0
\(93\) −897.333 −1.00053
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 172.399 0.180459 0.0902294 − 0.995921i \(-0.471240\pi\)
0.0902294 + 0.995921i \(0.471240\pi\)
\(98\) 0 0
\(99\) −401.135 −0.407228
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.q.1.6 8
4.3 odd 2 125.4.a.d.1.2 ✓ 8
5.4 even 2 inner 2000.4.a.q.1.3 8
12.11 even 2 1125.4.a.o.1.7 8
20.3 even 4 125.4.b.b.124.7 8
20.7 even 4 125.4.b.b.124.2 8
20.19 odd 2 125.4.a.d.1.7 yes 8
60.59 even 2 1125.4.a.o.1.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
125.4.a.d.1.2 ✓ 8 4.3 odd 2
125.4.a.d.1.7 yes 8 20.19 odd 2
125.4.b.b.124.2 8 20.7 even 4
125.4.b.b.124.7 8 20.3 even 4
1125.4.a.o.1.2 8 60.59 even 2
1125.4.a.o.1.7 8 12.11 even 2
2000.4.a.q.1.3 8 5.4 even 2 inner
2000.4.a.q.1.6 8 1.1 even 1 trivial