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Properties

Label 500.4.g.a.301.3
Level $500$
Weight $4$
Character 500.301
Analytic conductor $29.501$
Analytic rank $0$
Dimension $28$
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] = [500,4,Mod(101,500)] 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("500.101"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(500, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 6])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 500 = 2^{2} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 500.g (of order \(5\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [28] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(29.5009550029\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(7\) over \(\Q(\zeta_{5})\)
Twist minimal: no (minimal twist has level 100)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label 301.3
Character \(\chi\) \(=\) 500.301
Dual form 500.4.g.a.201.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.696421 + 2.14336i) q^{3} +29.6874 q^{7} +(17.7345 + 12.8848i) q^{9} +(-32.9943 + 23.9718i) q^{11} +(8.23451 + 5.98272i) q^{13} +(-4.14936 - 12.7704i) q^{17} +(19.1682 + 58.9938i) q^{19} +(-20.6749 + 63.6309i) q^{21} +(133.629 - 97.0873i) q^{23} +(-89.1954 + 64.8043i) q^{27} +(2.90164 - 8.93033i) q^{29} +(-64.0758 - 197.205i) q^{31} +(-28.4023 - 87.4132i) q^{33} +(-101.178 - 73.5098i) q^{37} +(-18.5578 + 13.4831i) q^{39} +(349.755 + 254.112i) q^{41} +163.372 q^{43} +(-143.860 + 442.755i) q^{47} +538.341 q^{49} +30.2614 q^{51} +(-147.434 + 453.756i) q^{53} -139.794 q^{57} +(338.366 + 245.838i) q^{59} +(-746.693 + 542.504i) q^{61} +(526.490 + 382.517i) q^{63} +(35.4642 + 109.148i) q^{67} +(115.031 + 354.030i) q^{69} +(-263.913 + 812.242i) q^{71} +(699.672 - 508.342i) q^{73} +(-979.514 + 711.659i) q^{77} +(61.9704 - 190.725i) q^{79} +(106.115 + 326.589i) q^{81} +(-49.9514 - 153.735i) q^{83} +(17.1202 + 12.4385i) q^{87} +(-296.211 + 215.210i) q^{89} +(244.461 + 177.611i) q^{91} +467.306 q^{93} +(324.116 - 997.528i) q^{97} -894.008 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 28 q + 4 q^{3} - 16 q^{7} - 13 q^{9} - 20 q^{11} + 86 q^{13} + 178 q^{17} + 2 q^{19} + 108 q^{21} - 102 q^{23} - 92 q^{27} + 192 q^{29} + 378 q^{31} - 560 q^{33} + 399 q^{37} + 592 q^{39} + 298 q^{41} + 180 q^{43}+ \cdots - 2100 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/500\mathbb{Z}\right)^\times\).

\(n\) \(251\) \(377\)
\(\chi(n)\) \(1\) \(e\left(\frac{4}{5}\right)\)

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
\(3\) −0.696421 + 2.14336i −0.134026 + 0.412491i −0.995437 − 0.0954180i \(-0.969581\pi\)
0.861411 + 0.507909i \(0.169581\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 29.6874 1.60297 0.801484 − 0.598016i \(-0.204044\pi\)
0.801484 + 0.598016i \(0.204044\pi\)
\(8\) 0 0
\(9\) 17.7345 + 12.8848i 0.656832 + 0.477216i
\(10\) 0 0
\(11\) −32.9943 + 23.9718i −0.904378 + 0.657069i −0.939587 − 0.342311i \(-0.888790\pi\)
0.0352091 + 0.999380i \(0.488790\pi\)
\(12\) 0 0
\(13\) 8.23451 + 5.98272i 0.175680 + 0.127639i 0.672150 − 0.740415i \(-0.265371\pi\)
−0.496470 + 0.868054i \(0.665371\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.14936 − 12.7704i −0.0591981 − 0.182193i 0.917085 − 0.398693i \(-0.130536\pi\)
−0.976283 + 0.216500i \(0.930536\pi\)
\(18\) 0 0
\(19\) 19.1682 + 58.9938i 0.231447 + 0.712321i 0.997573 + 0.0696305i \(0.0221820\pi\)
−0.766126 + 0.642691i \(0.777818\pi\)
\(20\) 0 0
\(21\) −20.6749 + 63.6309i −0.214840 + 0.661209i
\(22\) 0 0
\(23\) 133.629 − 97.0873i 1.21146 − 0.880178i 0.216099 − 0.976371i \(-0.430667\pi\)
0.995363 + 0.0961931i \(0.0306666\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −89.1954 + 64.8043i −0.635765 + 0.461911i
\(28\) 0 0
\(29\) 2.90164 − 8.93033i 0.0185800 − 0.0571835i −0.941337 − 0.337469i \(-0.890429\pi\)
0.959917 + 0.280286i \(0.0904292\pi\)
\(30\) 0 0
\(31\) −64.0758 − 197.205i −0.371237 − 1.14255i −0.945982 − 0.324218i \(-0.894899\pi\)
0.574745 − 0.818332i \(-0.305101\pi\)
\(32\) 0 0
\(33\) −28.4023 − 87.4132i −0.149824 − 0.461112i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −101.178 − 73.5098i −0.449554 − 0.326620i 0.339866 − 0.940474i \(-0.389618\pi\)
−0.789420 + 0.613854i \(0.789618\pi\)
\(38\) 0 0
\(39\) −18.5578 + 13.4831i −0.0761957 + 0.0553594i
\(40\) 0 0
\(41\) 349.755 + 254.112i 1.33226 + 0.967941i 0.999691 + 0.0248605i \(0.00791416\pi\)
0.332565 + 0.943080i \(0.392086\pi\)
\(42\) 0 0
\(43\) 163.372 0.579395 0.289698 − 0.957118i \(-0.406445\pi\)
0.289698 + 0.957118i \(0.406445\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −143.860 + 442.755i −0.446471 + 1.37410i 0.434392 + 0.900724i \(0.356963\pi\)
−0.880863 + 0.473372i \(0.843037\pi\)
\(48\) 0 0
\(49\) 538.341 1.56951
\(50\) 0 0
\(51\) 30.2614 0.0830871
\(52\) 0 0
\(53\) −147.434 + 453.756i −0.382107 + 1.17600i 0.556450 + 0.830881i \(0.312163\pi\)
−0.938557 + 0.345123i \(0.887837\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −139.794 −0.324846
\(58\) 0 0
\(59\) 338.366 + 245.838i 0.746637 + 0.542463i 0.894783 − 0.446502i \(-0.147330\pi\)
−0.148146 + 0.988966i \(0.547330\pi\)
\(60\) 0 0
\(61\) −746.693 + 542.504i −1.56728 + 1.13870i −0.637582 + 0.770382i \(0.720065\pi\)
−0.929701 + 0.368315i \(0.879935\pi\)
\(62\) 0 0
\(63\) 526.490 + 382.517i 1.05288 + 0.764962i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 35.4642 + 109.148i 0.0646663 + 0.199022i 0.978169 − 0.207810i \(-0.0666335\pi\)
−0.913503 + 0.406832i \(0.866633\pi\)
\(68\) 0 0
\(69\) 115.031 + 354.030i 0.200698 + 0.617684i
\(70\) 0 0
\(71\) −263.913 + 812.242i −0.441137 + 1.35768i 0.445527 + 0.895268i \(0.353016\pi\)
−0.886665 + 0.462413i \(0.846984\pi\)
\(72\) 0 0
\(73\) 699.672 − 508.342i 1.12179 − 0.815026i 0.137308 − 0.990528i \(-0.456155\pi\)
0.984479 + 0.175502i \(0.0561550\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −979.514 + 711.659i −1.44969 + 1.05326i
\(78\) 0 0
\(79\) 61.9704 − 190.725i 0.0882559 − 0.271624i −0.897182 − 0.441662i \(-0.854389\pi\)
0.985437 + 0.170038i \(0.0543891\pi\)
\(80\) 0 0
\(81\) 106.115 + 326.589i 0.145563 + 0.447996i
\(82\) 0 0
\(83\) −49.9514 − 153.735i −0.0660588 − 0.203308i 0.912579 − 0.408901i \(-0.134088\pi\)
−0.978638 + 0.205593i \(0.934088\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 17.1202 + 12.4385i 0.0210974 + 0.0153282i
\(88\) 0 0
\(89\) −296.211 + 215.210i −0.352790 + 0.256317i −0.750038 − 0.661394i \(-0.769965\pi\)
0.397249 + 0.917711i \(0.369965\pi\)
\(90\) 0 0
\(91\) 244.461 + 177.611i 0.281610 + 0.204602i
\(92\) 0 0
\(93\) 467.306 0.521047
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 324.116 − 997.528i 0.339268 − 1.04416i −0.625313 − 0.780374i \(-0.715029\pi\)
0.964581 − 0.263787i \(-0.0849715\pi\)
\(98\) 0 0
\(99\) −894.008 −0.907588
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 500.4.g.a.301.3 28
5.2 odd 4 500.4.i.b.449.9 56
5.3 odd 4 500.4.i.b.449.6 56
5.4 even 2 100.4.g.a.61.5 yes 28
25.4 even 10 2500.4.a.c.1.9 14
25.9 even 10 100.4.g.a.41.5 ✓ 28
25.12 odd 20 500.4.i.b.49.6 56
25.13 odd 20 500.4.i.b.49.9 56
25.16 even 5 inner 500.4.g.a.201.3 28
25.21 even 5 2500.4.a.d.1.6 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
100.4.g.a.41.5 ✓ 28 25.9 even 10
100.4.g.a.61.5 yes 28 5.4 even 2
500.4.g.a.201.3 28 25.16 even 5 inner
500.4.g.a.301.3 28 1.1 even 1 trivial
500.4.i.b.49.6 56 25.12 odd 20
500.4.i.b.49.9 56 25.13 odd 20
500.4.i.b.449.6 56 5.3 odd 4
500.4.i.b.449.9 56 5.2 odd 4
2500.4.a.c.1.9 14 25.4 even 10
2500.4.a.d.1.6 14 25.21 even 5