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Properties

Label 2000.4.a.v.1.2
Level $2000$
Weight $4$
Character 2000.1
Self dual yes
Analytic conductor $118.004$
Analytic rank $0$
Dimension $10$
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 = [10,0,-1,0,0,0,48,0,131] 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: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 3 x^{9} - 115 x^{8} + 152 x^{7} + 4978 x^{6} + 1245 x^{5} - 90069 x^{4} - 138850 x^{3} + \cdots + 873521 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8}\cdot 5^{5} \)
Twist minimal: no (minimal twist has level 1000)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.91692\) 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)
 
\(f(q)\) \(=\) \(q-7.07684 q^{3} +19.7613 q^{7} +23.0817 q^{9} -63.8400 q^{11} -82.1391 q^{13} -59.2047 q^{17} -98.7379 q^{19} -139.848 q^{21} -210.868 q^{23} +27.7291 q^{27} +115.576 q^{29} -207.743 q^{31} +451.786 q^{33} +124.192 q^{37} +581.286 q^{39} -50.4293 q^{41} -416.166 q^{43} +455.394 q^{47} +47.5089 q^{49} +418.983 q^{51} +172.230 q^{53} +698.753 q^{57} -711.678 q^{59} -93.8934 q^{61} +456.125 q^{63} -887.110 q^{67} +1492.28 q^{69} -594.121 q^{71} +344.656 q^{73} -1261.56 q^{77} -682.398 q^{79} -819.441 q^{81} +404.203 q^{83} -817.911 q^{87} +317.633 q^{89} -1623.18 q^{91} +1470.17 q^{93} +806.662 q^{97} -1473.54 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - q^{3} + 48 q^{7} + 131 q^{9} - 92 q^{11} + 13 q^{13} + 16 q^{17} - 83 q^{19} + 279 q^{21} + 138 q^{23} + 338 q^{27} + 196 q^{29} - 113 q^{31} + 151 q^{33} + 56 q^{37} - 147 q^{39} + 188 q^{41} + 224 q^{43}+ \cdots - 8897 q^{99}+O(q^{100}) \) 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
\(3\) −7.07684 −1.36194 −0.680970 − 0.732312i \(-0.738441\pi\)
−0.680970 + 0.732312i \(0.738441\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 19.7613 1.06701 0.533505 − 0.845797i \(-0.320875\pi\)
0.533505 + 0.845797i \(0.320875\pi\)
\(8\) 0 0
\(9\) 23.0817 0.854878
\(10\) 0 0
\(11\) −63.8400 −1.74986 −0.874932 − 0.484247i \(-0.839094\pi\)
−0.874932 + 0.484247i \(0.839094\pi\)
\(12\) 0 0
\(13\) −82.1391 −1.75241 −0.876204 − 0.481941i \(-0.839932\pi\)
−0.876204 + 0.481941i \(0.839932\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −59.2047 −0.844662 −0.422331 − 0.906442i \(-0.638788\pi\)
−0.422331 + 0.906442i \(0.638788\pi\)
\(18\) 0 0
\(19\) −98.7379 −1.19221 −0.596106 − 0.802906i \(-0.703286\pi\)
−0.596106 + 0.802906i \(0.703286\pi\)
\(20\) 0 0
\(21\) −139.848 −1.45320
\(22\) 0 0
\(23\) −210.868 −1.91169 −0.955847 − 0.293864i \(-0.905059\pi\)
−0.955847 + 0.293864i \(0.905059\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 27.7291 0.197647
\(28\) 0 0
\(29\) 115.576 0.740064 0.370032 − 0.929019i \(-0.379347\pi\)
0.370032 + 0.929019i \(0.379347\pi\)
\(30\) 0 0
\(31\) −207.743 −1.20361 −0.601803 − 0.798644i \(-0.705551\pi\)
−0.601803 + 0.798644i \(0.705551\pi\)
\(32\) 0 0
\(33\) 451.786 2.38321
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 124.192 0.551811 0.275906 − 0.961185i \(-0.411022\pi\)
0.275906 + 0.961185i \(0.411022\pi\)
\(38\) 0 0
\(39\) 581.286 2.38667
\(40\) 0 0
\(41\) −50.4293 −0.192091 −0.0960455 − 0.995377i \(-0.530619\pi\)
−0.0960455 + 0.995377i \(0.530619\pi\)
\(42\) 0 0
\(43\) −416.166 −1.47592 −0.737962 − 0.674842i \(-0.764212\pi\)
−0.737962 + 0.674842i \(0.764212\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 455.394 1.41332 0.706660 − 0.707553i \(-0.250201\pi\)
0.706660 + 0.707553i \(0.250201\pi\)
\(48\) 0 0
\(49\) 47.5089 0.138510
\(50\) 0 0
\(51\) 418.983 1.15038
\(52\) 0 0
\(53\) 172.230 0.446371 0.223185 − 0.974776i \(-0.428355\pi\)
0.223185 + 0.974776i \(0.428355\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 698.753 1.62372
\(58\) 0 0
\(59\) −711.678 −1.57038 −0.785191 − 0.619254i \(-0.787435\pi\)
−0.785191 + 0.619254i \(0.787435\pi\)
\(60\) 0 0
\(61\) −93.8934 −0.197079 −0.0985395 − 0.995133i \(-0.531417\pi\)
−0.0985395 + 0.995133i \(0.531417\pi\)
\(62\) 0 0
\(63\) 456.125 0.912163
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −887.110 −1.61758 −0.808789 − 0.588099i \(-0.799877\pi\)
−0.808789 + 0.588099i \(0.799877\pi\)
\(68\) 0 0
\(69\) 1492.28 2.60361
\(70\) 0 0
\(71\) −594.121 −0.993088 −0.496544 − 0.868012i \(-0.665398\pi\)
−0.496544 + 0.868012i \(0.665398\pi\)
\(72\) 0 0
\(73\) 344.656 0.552587 0.276294 − 0.961073i \(-0.410894\pi\)
0.276294 + 0.961073i \(0.410894\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1261.56 −1.86712
\(78\) 0 0
\(79\) −682.398 −0.971845 −0.485922 − 0.874002i \(-0.661516\pi\)
−0.485922 + 0.874002i \(0.661516\pi\)
\(80\) 0 0
\(81\) −819.441 −1.12406
\(82\) 0 0
\(83\) 404.203 0.534542 0.267271 − 0.963621i \(-0.413878\pi\)
0.267271 + 0.963621i \(0.413878\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −817.911 −1.00792
\(88\) 0 0
\(89\) 317.633 0.378304 0.189152 − 0.981948i \(-0.439426\pi\)
0.189152 + 0.981948i \(0.439426\pi\)
\(90\) 0 0
\(91\) −1623.18 −1.86984
\(92\) 0 0
\(93\) 1470.17 1.63924
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 806.662 0.844372 0.422186 − 0.906509i \(-0.361263\pi\)
0.422186 + 0.906509i \(0.361263\pi\)
\(98\) 0 0
\(99\) −1473.54 −1.49592
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.v.1.2 10
4.3 odd 2 1000.4.a.g.1.9 yes 10
5.4 even 2 2000.4.a.w.1.9 10
20.3 even 4 1000.4.c.c.249.17 20
20.7 even 4 1000.4.c.c.249.4 20
20.19 odd 2 1000.4.a.f.1.2 ✓ 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1000.4.a.f.1.2 ✓ 10 20.19 odd 2
1000.4.a.g.1.9 yes 10 4.3 odd 2
1000.4.c.c.249.4 20 20.7 even 4
1000.4.c.c.249.17 20 20.3 even 4
2000.4.a.v.1.2 10 1.1 even 1 trivial
2000.4.a.w.1.9 10 5.4 even 2