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Properties

Label 200.3.u.b.33.5
Level $200$
Weight $3$
Character 200.33
Analytic conductor $5.450$
Analytic rank $0$
Dimension $64$
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] = [200,3,Mod(17,200)] 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("200.17"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(200, base_ring=CyclotomicField(20)) chi = DirichletCharacter(H, H._module([0, 0, 13])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 200 = 2^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 200.u (of order \(20\), degree \(8\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [64] 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: \(5.44960528721\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(8\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label 33.5
Character \(\chi\) \(=\) 200.33
Dual form 200.3.u.b.97.5

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.535050 + 0.0847436i) q^{3} +(2.05203 - 4.55951i) q^{5} +(4.47022 - 4.47022i) q^{7} +(-8.28041 - 2.69047i) q^{9} +(-1.20728 - 3.71563i) q^{11} +(-8.27903 + 4.21838i) q^{13} +(1.48433 - 2.26567i) q^{15} +(11.2207 - 1.77718i) q^{17} +(20.6896 - 28.4767i) q^{19} +(2.77061 - 2.01297i) q^{21} +(4.19198 - 8.22722i) q^{23} +(-16.5783 - 18.7125i) q^{25} +(-8.54651 - 4.35467i) q^{27} +(23.1727 + 31.8945i) q^{29} +(16.1948 + 11.7662i) q^{31} +(-0.331080 - 2.09036i) q^{33} +(-11.2090 - 29.5550i) q^{35} +(-10.4140 - 20.4385i) q^{37} +(-4.78718 + 1.55545i) q^{39} +(-16.0730 + 49.4677i) q^{41} +(17.2999 + 17.2999i) q^{43} +(-29.2589 + 32.2337i) q^{45} +(2.99450 - 18.9065i) q^{47} +9.03431i q^{49} +6.15422 q^{51} +(-75.9092 - 12.0228i) q^{53} +(-19.4189 - 2.11998i) q^{55} +(13.4832 - 13.4832i) q^{57} +(101.199 + 32.8817i) q^{59} +(10.5593 + 32.4981i) q^{61} +(-49.0422 + 24.9883i) q^{63} +(2.24491 + 46.4046i) q^{65} +(58.9216 - 9.33226i) q^{67} +(2.94012 - 4.04673i) q^{69} +(-88.8498 + 64.5532i) q^{71} +(-27.0876 + 53.1624i) q^{73} +(-7.28447 - 11.4170i) q^{75} +(-22.0065 - 11.2129i) q^{77} +(-14.1699 - 19.5033i) q^{79} +(59.1899 + 43.0040i) q^{81} +(17.1948 + 108.564i) q^{83} +(14.9221 - 54.8076i) q^{85} +(9.69572 + 19.0289i) q^{87} +(-3.07504 + 0.999140i) q^{89} +(-18.1520 + 55.8661i) q^{91} +(7.66789 + 7.66789i) q^{93} +(-87.3844 - 152.769i) q^{95} +(24.7036 - 155.972i) q^{97} +34.0151i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 64 q + 6 q^{5} - 4 q^{7} - 40 q^{9} + 16 q^{11} + 24 q^{13} + 82 q^{15} - 8 q^{17} - 50 q^{19} - 100 q^{21} - 48 q^{23} - 200 q^{25} + 90 q^{27} + 108 q^{31} + 260 q^{33} + 2 q^{35} - 94 q^{37} + 320 q^{39}+ \cdots - 544 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/200\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(151\) \(177\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{3}{20}\right)\)

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
\(3\) 0.535050 + 0.0847436i 0.178350 + 0.0282479i 0.244971 − 0.969531i \(-0.421222\pi\)
−0.0666206 + 0.997778i \(0.521222\pi\)
\(4\) 0 0
\(5\) 2.05203 − 4.55951i 0.410406 − 0.911903i
\(6\) 0 0
\(7\) 4.47022 − 4.47022i 0.638602 − 0.638602i −0.311608 − 0.950211i \(-0.600868\pi\)
0.950211 + 0.311608i \(0.100868\pi\)
\(8\) 0 0
\(9\) −8.28041 − 2.69047i −0.920046 − 0.298941i
\(10\) 0 0
\(11\) −1.20728 − 3.71563i −0.109753 − 0.337785i 0.881064 − 0.472998i \(-0.156828\pi\)
−0.990817 + 0.135213i \(0.956828\pi\)
\(12\) 0 0
\(13\) −8.27903 + 4.21838i −0.636848 + 0.324491i −0.742433 − 0.669920i \(-0.766328\pi\)
0.105585 + 0.994410i \(0.466328\pi\)
\(14\) 0 0
\(15\) 1.48433 − 2.26567i 0.0989553 − 0.151045i
\(16\) 0 0
\(17\) 11.2207 − 1.77718i 0.660040 − 0.104540i 0.182576 − 0.983192i \(-0.441556\pi\)
0.477463 + 0.878652i \(0.341556\pi\)
\(18\) 0 0
\(19\) 20.6896 − 28.4767i 1.08892 − 1.49878i 0.239623 − 0.970866i \(-0.422976\pi\)
0.849301 − 0.527909i \(-0.177024\pi\)
\(20\) 0 0
\(21\) 2.77061 − 2.01297i 0.131934 − 0.0958556i
\(22\) 0 0
\(23\) 4.19198 − 8.22722i 0.182260 − 0.357705i −0.781741 − 0.623603i \(-0.785668\pi\)
0.964001 + 0.265897i \(0.0856683\pi\)
\(24\) 0 0
\(25\) −16.5783 − 18.7125i −0.663133 − 0.748501i
\(26\) 0 0
\(27\) −8.54651 − 4.35467i −0.316538 − 0.161284i
\(28\) 0 0
\(29\) 23.1727 + 31.8945i 0.799060 + 1.09981i 0.992920 + 0.118783i \(0.0378992\pi\)
−0.193860 + 0.981029i \(0.562101\pi\)
\(30\) 0 0
\(31\) 16.1948 + 11.7662i 0.522411 + 0.379554i 0.817512 − 0.575912i \(-0.195353\pi\)
−0.295100 + 0.955466i \(0.595353\pi\)
\(32\) 0 0
\(33\) −0.331080 − 2.09036i −0.0100327 − 0.0633442i
\(34\) 0 0
\(35\) −11.2090 − 29.5550i −0.320257 − 0.844430i
\(36\) 0 0
\(37\) −10.4140 − 20.4385i −0.281458 − 0.552393i 0.706388 − 0.707824i \(-0.250323\pi\)
−0.987847 + 0.155432i \(0.950323\pi\)
\(38\) 0 0
\(39\) −4.78718 + 1.55545i −0.122748 + 0.0398833i
\(40\) 0 0
\(41\) −16.0730 + 49.4677i −0.392026 + 1.20653i 0.539228 + 0.842160i \(0.318716\pi\)
−0.931254 + 0.364371i \(0.881284\pi\)
\(42\) 0 0
\(43\) 17.2999 + 17.2999i 0.402323 + 0.402323i 0.879051 − 0.476728i \(-0.158177\pi\)
−0.476728 + 0.879051i \(0.658177\pi\)
\(44\) 0 0
\(45\) −29.2589 + 32.2337i −0.650198 + 0.716305i
\(46\) 0 0
\(47\) 2.99450 − 18.9065i 0.0637127 − 0.402266i −0.935136 − 0.354290i \(-0.884723\pi\)
0.998848 − 0.0479767i \(-0.0152773\pi\)
\(48\) 0 0
\(49\) 9.03431i 0.184374i
\(50\) 0 0
\(51\) 6.15422 0.120671
\(52\) 0 0
\(53\) −75.9092 − 12.0228i −1.43225 − 0.226846i −0.608389 − 0.793639i \(-0.708184\pi\)
−0.823860 + 0.566793i \(0.808184\pi\)
\(54\) 0 0
\(55\) −19.4189 − 2.11998i −0.353070 − 0.0385450i
\(56\) 0 0
\(57\) 13.4832 − 13.4832i 0.236547 − 0.236547i
\(58\) 0 0
\(59\) 101.199 + 32.8817i 1.71524 + 0.557317i 0.991192 − 0.132430i \(-0.0422778\pi\)
0.724051 + 0.689746i \(0.242278\pi\)
\(60\) 0 0
\(61\) 10.5593 + 32.4981i 0.173103 + 0.532755i 0.999542 − 0.0302710i \(-0.00963702\pi\)
−0.826439 + 0.563026i \(0.809637\pi\)
\(62\) 0 0
\(63\) −49.0422 + 24.9883i −0.778448 + 0.396639i
\(64\) 0 0
\(65\) 2.24491 + 46.4046i 0.0345371 + 0.713917i
\(66\) 0 0
\(67\) 58.9216 − 9.33226i 0.879426 − 0.139287i 0.299636 − 0.954053i \(-0.403135\pi\)
0.579790 + 0.814766i \(0.303135\pi\)
\(68\) 0 0
\(69\) 2.94012 − 4.04673i 0.0426105 − 0.0586483i
\(70\) 0 0
\(71\) −88.8498 + 64.5532i −1.25141 + 0.909200i −0.998303 − 0.0582393i \(-0.981451\pi\)
−0.253104 + 0.967439i \(0.581451\pi\)
\(72\) 0 0
\(73\) −27.0876 + 53.1624i −0.371063 + 0.728252i −0.998738 − 0.0502232i \(-0.984007\pi\)
0.627675 + 0.778476i \(0.284007\pi\)
\(74\) 0 0
\(75\) −7.28447 − 11.4170i −0.0971262 − 0.152227i
\(76\) 0 0
\(77\) −22.0065 − 11.2129i −0.285799 − 0.145622i
\(78\) 0 0
\(79\) −14.1699 − 19.5033i −0.179366 − 0.246877i 0.709861 − 0.704341i \(-0.248758\pi\)
−0.889228 + 0.457465i \(0.848758\pi\)
\(80\) 0 0
\(81\) 59.1899 + 43.0040i 0.730739 + 0.530913i
\(82\) 0 0
\(83\) 17.1948 + 108.564i 0.207167 + 1.30800i 0.843728 + 0.536772i \(0.180356\pi\)
−0.636561 + 0.771226i \(0.719644\pi\)
\(84\) 0 0
\(85\) 14.9221 − 54.8076i 0.175554 − 0.644796i
\(86\) 0 0
\(87\) 9.69572 + 19.0289i 0.111445 + 0.218723i
\(88\) 0 0
\(89\) −3.07504 + 0.999140i −0.0345510 + 0.0112263i −0.326241 − 0.945286i \(-0.605782\pi\)
0.291691 + 0.956513i \(0.405782\pi\)
\(90\) 0 0
\(91\) −18.1520 + 55.8661i −0.199473 + 0.613913i
\(92\) 0 0
\(93\) 7.66789 + 7.66789i 0.0824505 + 0.0824505i
\(94\) 0 0
\(95\) −87.3844 − 152.769i −0.919836 − 1.60810i
\(96\) 0 0
\(97\) 24.7036 − 155.972i 0.254676 − 1.60796i −0.446371 − 0.894848i \(-0.647284\pi\)
0.701047 − 0.713115i \(-0.252716\pi\)
\(98\) 0 0
\(99\) 34.0151i 0.343587i
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 200.3.u.b.33.5 ✓ 64
4.3 odd 2 400.3.bg.f.33.4 64
25.22 odd 20 inner 200.3.u.b.97.5 yes 64
100.47 even 20 400.3.bg.f.97.4 64
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
200.3.u.b.33.5 ✓ 64 1.1 even 1 trivial
200.3.u.b.97.5 yes 64 25.22 odd 20 inner
400.3.bg.f.33.4 64 4.3 odd 2
400.3.bg.f.97.4 64 100.47 even 20