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Properties

Label 200.4.d.b.101.9
Level $200$
Weight $4$
Character 200.101
Analytic conductor $11.800$
Analytic rank $0$
Dimension $12$
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] = [200,4,Mod(101,200)] 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("200.101"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(200, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 200 = 2^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 200.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.8003820011\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4x^{11} + 7x^{10} - 12x^{9} + 21x^{8} - 68x^{6} + 336x^{4} - 768x^{3} + 1792x^{2} - 4096x + 4096 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{15}\cdot 5^{4} \)
Twist minimal: no (minimal twist has level 40)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 101.9
Root \(-0.650488 + 1.89126i\) of defining polynomial
Character \(\chi\) \(=\) 200.101
Dual form 200.4.d.b.101.10

$q$-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)
 
\(f(q)\) \(=\) \(q+(2.54175 - 1.24077i) q^{2} +0.888401i q^{3} +(4.92097 - 6.30746i) q^{4} +(1.10230 + 2.25809i) q^{6} -26.6173 q^{7} +(4.68175 - 22.1378i) q^{8} +26.2107 q^{9} -61.7277i q^{11} +(5.60356 + 4.37180i) q^{12} -45.0627i q^{13} +(-67.6545 + 33.0260i) q^{14} +(-15.5681 - 62.0776i) q^{16} +71.8754 q^{17} +(66.6211 - 32.5216i) q^{18} -17.6319i q^{19} -23.6469i q^{21} +(-76.5900 - 156.896i) q^{22} -43.4131 q^{23} +(19.6672 + 4.15927i) q^{24} +(-55.9125 - 114.538i) q^{26} +47.2725i q^{27} +(-130.983 + 167.888i) q^{28} +267.633i q^{29} +50.2133 q^{31} +(-116.594 - 138.469i) q^{32} +54.8390 q^{33} +(182.689 - 89.1810i) q^{34} +(128.982 - 165.323i) q^{36} -75.0720i q^{37} +(-21.8772 - 44.8160i) q^{38} +40.0338 q^{39} -221.685 q^{41} +(-29.3404 - 60.1044i) q^{42} -188.998i q^{43} +(-389.345 - 303.760i) q^{44} +(-110.345 + 53.8658i) q^{46} +384.142 q^{47} +(55.1499 - 13.8307i) q^{48} +365.481 q^{49} +63.8542i q^{51} +(-284.231 - 221.752i) q^{52} +247.445i q^{53} +(58.6544 + 120.155i) q^{54} +(-124.616 + 589.248i) q^{56} +15.6642 q^{57} +(332.072 + 680.256i) q^{58} +518.596i q^{59} -62.0042i q^{61} +(127.630 - 62.3033i) q^{62} -697.660 q^{63} +(-468.162 - 207.287i) q^{64} +(139.387 - 68.0426i) q^{66} +558.476i q^{67} +(353.697 - 453.351i) q^{68} -38.5683i q^{69} +313.194 q^{71} +(122.712 - 580.248i) q^{72} +263.926 q^{73} +(-93.1472 - 190.814i) q^{74} +(-111.213 - 86.7663i) q^{76} +1643.03i q^{77} +(101.756 - 49.6728i) q^{78} -732.940 q^{79} +665.693 q^{81} +(-563.468 + 275.061i) q^{82} -717.705i q^{83} +(-149.152 - 116.365i) q^{84} +(-234.503 - 480.385i) q^{86} -237.766 q^{87} +(-1366.51 - 288.994i) q^{88} +1634.69 q^{89} +1199.45i q^{91} +(-213.635 + 273.827i) q^{92} +44.6096i q^{93} +(976.392 - 476.633i) q^{94} +(123.016 - 103.583i) q^{96} +1367.12 q^{97} +(928.962 - 453.479i) q^{98} -1617.93i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{2} + 16 q^{4} - 36 q^{6} - 28 q^{7} + 40 q^{8} - 108 q^{9} - 188 q^{12} + 68 q^{14} - 56 q^{16} + 206 q^{18} + 164 q^{22} - 604 q^{23} + 360 q^{24} - 308 q^{26} + 436 q^{28} - 264 q^{31} - 72 q^{32}+ \cdots + 7266 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(151\) \(177\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.54175 − 1.24077i 0.898644 − 0.438679i
\(3\) 0.888401i 0.170973i 0.996339 + 0.0854865i \(0.0272444\pi\)
−0.996339 + 0.0854865i \(0.972756\pi\)
\(4\) 4.92097 − 6.30746i 0.615121 − 0.788433i
\(5\) 0 0
\(6\) 1.10230 + 2.25809i 0.0750023 + 0.153644i
\(7\) −26.6173 −1.43720 −0.718600 − 0.695424i \(-0.755217\pi\)
−0.718600 + 0.695424i \(0.755217\pi\)
\(8\) 4.68175 − 22.1378i 0.206906 − 0.978361i
\(9\) 26.2107 0.970768
\(10\) 0 0
\(11\) − 61.7277i − 1.69196i −0.533212 − 0.845982i \(-0.679015\pi\)
0.533212 − 0.845982i \(-0.320985\pi\)
\(12\) 5.60356 + 4.37180i 0.134801 + 0.105169i
\(13\) − 45.0627i − 0.961396i −0.876886 − 0.480698i \(-0.840383\pi\)
0.876886 − 0.480698i \(-0.159617\pi\)
\(14\) −67.6545 + 33.0260i −1.29153 + 0.630470i
\(15\) 0 0
\(16\) −15.5681 − 62.0776i −0.243252 − 0.969963i
\(17\) 71.8754 1.02543 0.512716 − 0.858558i \(-0.328639\pi\)
0.512716 + 0.858558i \(0.328639\pi\)
\(18\) 66.6211 − 32.5216i 0.872375 − 0.425856i
\(19\) − 17.6319i − 0.212897i −0.994318 − 0.106449i \(-0.966052\pi\)
0.994318 − 0.106449i \(-0.0339480\pi\)
\(20\) 0 0
\(21\) − 23.6469i − 0.245722i
\(22\) −76.5900 − 156.896i −0.742229 − 1.52047i
\(23\) −43.4131 −0.393577 −0.196788 − 0.980446i \(-0.563051\pi\)
−0.196788 + 0.980446i \(0.563051\pi\)
\(24\) 19.6672 + 4.15927i 0.167273 + 0.0353753i
\(25\) 0 0
\(26\) −55.9125 − 114.538i −0.421744 − 0.863952i
\(27\) 47.2725i 0.336948i
\(28\) −130.983 + 167.888i −0.884052 + 1.13314i
\(29\) 267.633i 1.71373i 0.515540 + 0.856866i \(0.327591\pi\)
−0.515540 + 0.856866i \(0.672409\pi\)
\(30\) 0 0
\(31\) 50.2133 0.290922 0.145461 − 0.989364i \(-0.453533\pi\)
0.145461 + 0.989364i \(0.453533\pi\)
\(32\) −116.594 − 138.469i −0.644099 − 0.764942i
\(33\) 54.8390 0.289280
\(34\) 182.689 − 89.1810i 0.921498 − 0.449836i
\(35\) 0 0
\(36\) 128.982 − 165.323i 0.597140 − 0.765385i
\(37\) − 75.0720i − 0.333561i −0.985994 − 0.166781i \(-0.946663\pi\)
0.985994 − 0.166781i \(-0.0533372\pi\)
\(38\) −21.8772 − 44.8160i −0.0933935 − 0.191319i
\(39\) 40.0338 0.164373
\(40\) 0 0
\(41\) −221.685 −0.844425 −0.422213 − 0.906497i \(-0.638746\pi\)
−0.422213 + 0.906497i \(0.638746\pi\)
\(42\) −29.3404 − 60.1044i −0.107793 − 0.220817i
\(43\) − 188.998i − 0.670277i −0.942169 − 0.335139i \(-0.891217\pi\)
0.942169 − 0.335139i \(-0.108783\pi\)
\(44\) −389.345 − 303.760i −1.33400 − 1.04076i
\(45\) 0 0
\(46\) −110.345 + 53.8658i −0.353685 + 0.172654i
\(47\) 384.142 1.19219 0.596094 − 0.802914i \(-0.296718\pi\)
0.596094 + 0.802914i \(0.296718\pi\)
\(48\) 55.1499 − 13.8307i 0.165837 − 0.0415894i
\(49\) 365.481 1.06554
\(50\) 0 0
\(51\) 63.8542i 0.175321i
\(52\) −284.231 − 221.752i −0.757996 − 0.591375i
\(53\) 247.445i 0.641306i 0.947197 + 0.320653i \(0.103902\pi\)
−0.947197 + 0.320653i \(0.896098\pi\)
\(54\) 58.6544 + 120.155i 0.147812 + 0.302796i
\(55\) 0 0
\(56\) −124.616 + 589.248i −0.297365 + 1.40610i
\(57\) 15.6642 0.0363997
\(58\) 332.072 + 680.256i 0.751778 + 1.54003i
\(59\) 518.596i 1.14433i 0.820139 + 0.572165i \(0.193896\pi\)
−0.820139 + 0.572165i \(0.806104\pi\)
\(60\) 0 0
\(61\) − 62.0042i − 0.130145i −0.997881 − 0.0650723i \(-0.979272\pi\)
0.997881 − 0.0650723i \(-0.0207278\pi\)
\(62\) 127.630 − 62.3033i 0.261435 − 0.127621i
\(63\) −697.660 −1.39519
\(64\) −468.162 − 207.287i −0.914380 − 0.404858i
\(65\) 0 0
\(66\) 139.387 − 68.0426i 0.259960 − 0.126901i
\(67\) 558.476i 1.01834i 0.860666 + 0.509169i \(0.170047\pi\)
−0.860666 + 0.509169i \(0.829953\pi\)
\(68\) 353.697 − 453.351i 0.630765 − 0.808484i
\(69\) − 38.5683i − 0.0672910i
\(70\) 0 0
\(71\) 313.194 0.523512 0.261756 − 0.965134i \(-0.415699\pi\)
0.261756 + 0.965134i \(0.415699\pi\)
\(72\) 122.712 − 580.248i 0.200858 − 0.949762i
\(73\) 263.926 0.423153 0.211576 − 0.977361i \(-0.432140\pi\)
0.211576 + 0.977361i \(0.432140\pi\)
\(74\) −93.1472 − 190.814i −0.146326 − 0.299753i
\(75\) 0 0
\(76\) −111.213 − 86.7663i −0.167855 − 0.130958i
\(77\) 1643.03i 2.43169i
\(78\) 101.756 − 49.6728i 0.147712 − 0.0721069i
\(79\) −732.940 −1.04383 −0.521913 − 0.852999i \(-0.674781\pi\)
−0.521913 + 0.852999i \(0.674781\pi\)
\(80\) 0 0
\(81\) 665.693 0.913159
\(82\) −563.468 + 275.061i −0.758837 + 0.370432i
\(83\) − 717.705i − 0.949137i −0.880218 − 0.474569i \(-0.842604\pi\)
0.880218 − 0.474569i \(-0.157396\pi\)
\(84\) −149.152 − 116.365i −0.193735 − 0.151149i
\(85\) 0 0
\(86\) −234.503 − 480.385i −0.294037 − 0.602341i
\(87\) −237.766 −0.293002
\(88\) −1366.51 − 288.994i −1.65535 − 0.350077i
\(89\) 1634.69 1.94693 0.973463 − 0.228845i \(-0.0734949\pi\)
0.973463 + 0.228845i \(0.0734949\pi\)
\(90\) 0 0
\(91\) 1199.45i 1.38172i
\(92\) −213.635 + 273.827i −0.242097 + 0.310309i
\(93\) 44.6096i 0.0497398i
\(94\) 976.392 − 476.633i 1.07135 − 0.522988i
\(95\) 0 0
\(96\) 123.016 − 103.583i 0.130784 − 0.110124i
\(97\) 1367.12 1.43103 0.715516 − 0.698596i \(-0.246191\pi\)
0.715516 + 0.698596i \(0.246191\pi\)
\(98\) 928.962 − 453.479i 0.957544 − 0.467432i
\(99\) − 1617.93i − 1.64250i
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 200.4.d.b.101.9 12
4.3 odd 2 800.4.d.d.401.6 12
5.2 odd 4 200.4.f.c.149.8 12
5.3 odd 4 200.4.f.b.149.5 12
5.4 even 2 40.4.d.a.21.4 yes 12
8.3 odd 2 800.4.d.d.401.7 12
8.5 even 2 inner 200.4.d.b.101.10 12
15.14 odd 2 360.4.k.c.181.9 12
20.3 even 4 800.4.f.c.49.5 12
20.7 even 4 800.4.f.b.49.8 12
20.19 odd 2 160.4.d.a.81.7 12
40.3 even 4 800.4.f.b.49.7 12
40.13 odd 4 200.4.f.c.149.7 12
40.19 odd 2 160.4.d.a.81.6 12
40.27 even 4 800.4.f.c.49.6 12
40.29 even 2 40.4.d.a.21.3 ✓ 12
40.37 odd 4 200.4.f.b.149.6 12
60.59 even 2 1440.4.k.c.721.7 12
80.19 odd 4 1280.4.a.ba.1.4 6
80.29 even 4 1280.4.a.bc.1.3 6
80.59 odd 4 1280.4.a.bd.1.3 6
80.69 even 4 1280.4.a.bb.1.4 6
120.29 odd 2 360.4.k.c.181.10 12
120.59 even 2 1440.4.k.c.721.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.4.d.a.21.3 ✓ 12 40.29 even 2
40.4.d.a.21.4 yes 12 5.4 even 2
160.4.d.a.81.6 12 40.19 odd 2
160.4.d.a.81.7 12 20.19 odd 2
200.4.d.b.101.9 12 1.1 even 1 trivial
200.4.d.b.101.10 12 8.5 even 2 inner
200.4.f.b.149.5 12 5.3 odd 4
200.4.f.b.149.6 12 40.37 odd 4
200.4.f.c.149.7 12 40.13 odd 4
200.4.f.c.149.8 12 5.2 odd 4
360.4.k.c.181.9 12 15.14 odd 2
360.4.k.c.181.10 12 120.29 odd 2
800.4.d.d.401.6 12 4.3 odd 2
800.4.d.d.401.7 12 8.3 odd 2
800.4.f.b.49.7 12 40.3 even 4
800.4.f.b.49.8 12 20.7 even 4
800.4.f.c.49.5 12 20.3 even 4
800.4.f.c.49.6 12 40.27 even 4
1280.4.a.ba.1.4 6 80.19 odd 4
1280.4.a.bb.1.4 6 80.69 even 4
1280.4.a.bc.1.3 6 80.29 even 4
1280.4.a.bd.1.3 6 80.59 odd 4
1440.4.k.c.721.1 12 120.59 even 2
1440.4.k.c.721.7 12 60.59 even 2