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Properties

Label 200.3.i.a.93.4
Level $200$
Weight $3$
Character 200.93
Analytic conductor $5.450$
Analytic rank $0$
Dimension $16$
Inner twists $8$

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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(93,200)] 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("200.93"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(200, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 2, 3])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 200 = 2^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 200.i (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16] 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: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 12x^{14} + 58x^{12} + 132x^{10} - 51x^{8} - 1128x^{6} + 1372x^{4} - 96x^{2} + 100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{22} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 93.4
Root \(1.16721 + 0.201484i\) of defining polynomial
Character \(\chi\) \(=\) 200.93
Dual form 200.3.i.a.157.4

$q$-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+(-0.368691 - 1.96572i) q^{2} +(-1.04930 + 1.04930i) q^{3} +(-3.72813 + 1.44949i) q^{4} +(2.44949 + 1.67576i) q^{6} +(3.91191 - 3.91191i) q^{7} +(4.22383 + 6.79406i) q^{8} +6.79796i q^{9} +10.8078i q^{11} +(2.39097 - 5.43286i) q^{12} +(7.23907 - 7.23907i) q^{13} +(-9.13202 - 6.24745i) q^{14} +(11.7980 - 10.8078i) q^{16} +(14.2120 - 14.2120i) q^{17} +(13.3629 - 2.50635i) q^{18} +19.0173 q^{19} +8.20950i q^{21} +(21.2451 - 3.98473i) q^{22} +(16.6882 + 16.6882i) q^{23} +(-11.5610 - 2.69694i) q^{24} +(-16.8990 - 11.5610i) q^{26} +(-16.5767 - 16.5767i) q^{27} +(-8.91386 + 20.2544i) q^{28} +51.4406 q^{29} -31.1918 q^{31} +(-25.5949 - 19.2068i) q^{32} +(-11.3406 - 11.3406i) q^{33} +(-33.1766 - 22.6969i) q^{34} +(-9.85357 - 25.3437i) q^{36} +(22.8725 + 22.8725i) q^{37} +(-7.01151 - 37.3827i) q^{38} +15.1918i q^{39} -3.79796 q^{41} +(16.1376 - 3.02677i) q^{42} +(-50.7795 + 50.7795i) q^{43} +(-15.6658 - 40.2929i) q^{44} +(26.6515 - 38.9571i) q^{46} +(-37.2882 + 37.2882i) q^{47} +(-1.03899 + 23.7201i) q^{48} +18.3939i q^{49} +29.8251i q^{51} +(-16.4953 + 37.4812i) q^{52} +(61.3784 - 61.3784i) q^{53} +(-26.4736 + 38.6969i) q^{54} +(43.1010 + 10.0545i) q^{56} +(-19.9547 + 19.9547i) q^{57} +(-18.9657 - 101.118i) q^{58} +40.6329 q^{59} +43.2311i q^{61} +(11.5002 + 61.3145i) q^{62} +(26.5930 + 26.5930i) q^{63} +(-28.3186 + 57.3939i) q^{64} +(-18.1112 + 26.4736i) q^{66} +(-26.0204 - 26.0204i) q^{67} +(-32.3840 + 73.5841i) q^{68} -35.0216 q^{69} +78.3837 q^{71} +(-46.1858 + 28.7134i) q^{72} +(-96.6123 - 96.6123i) q^{73} +(36.5281 - 53.3939i) q^{74} +(-70.8990 + 27.5654i) q^{76} +(42.2791 + 42.2791i) q^{77} +(29.8629 - 5.60110i) q^{78} +16.8082i q^{79} -26.3939 q^{81} +(1.40027 + 7.46574i) q^{82} +(-61.2724 + 61.2724i) q^{83} +(-11.8996 - 30.6061i) q^{84} +(118.540 + 81.0964i) q^{86} +(-53.9764 + 53.9764i) q^{87} +(-73.4288 + 45.6502i) q^{88} -33.1918i q^{89} -56.6372i q^{91} +(-86.4050 - 38.0264i) q^{92} +(32.7294 - 32.7294i) q^{93} +(87.0462 + 59.5505i) q^{94} +(47.0102 - 6.70303i) q^{96} +(73.9312 - 73.9312i) q^{97} +(36.1573 - 6.78166i) q^{98} -73.4709 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 32 q^{16} - 192 q^{26} + 128 q^{31} - 432 q^{36} + 96 q^{41} + 544 q^{46} + 768 q^{56} - 1152 q^{66} - 1056 q^{76} + 48 q^{81} + 1152 q^{86} + 1536 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(151\) \(177\)
\(\chi(n)\) \(-1\) \(1\) \(e\left(\frac{3}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.368691 − 1.96572i −0.184346 − 0.982861i
\(3\) −1.04930 + 1.04930i −0.349765 + 0.349765i −0.860022 − 0.510257i \(-0.829550\pi\)
0.510257 + 0.860022i \(0.329550\pi\)
\(4\) −3.72813 + 1.44949i −0.932033 + 0.362372i
\(5\) 0 0
\(6\) 2.44949 + 1.67576i 0.408248 + 0.279293i
\(7\) 3.91191 − 3.91191i 0.558845 − 0.558845i −0.370134 − 0.928978i \(-0.620688\pi\)
0.928978 + 0.370134i \(0.120688\pi\)
\(8\) 4.22383 + 6.79406i 0.527978 + 0.849258i
\(9\) 6.79796i 0.755329i
\(10\) 0 0
\(11\) 10.8078i 0.982526i 0.871011 + 0.491263i \(0.163465\pi\)
−0.871011 + 0.491263i \(0.836535\pi\)
\(12\) 2.39097 − 5.43286i 0.199248 − 0.452738i
\(13\) 7.23907 − 7.23907i 0.556851 − 0.556851i −0.371558 − 0.928410i \(-0.621176\pi\)
0.928410 + 0.371558i \(0.121176\pi\)
\(14\) −9.13202 − 6.24745i −0.652287 − 0.446246i
\(15\) 0 0
\(16\) 11.7980 − 10.8078i 0.737372 − 0.675486i
\(17\) 14.2120 − 14.2120i 0.835997 − 0.835997i −0.152332 − 0.988329i \(-0.548678\pi\)
0.988329 + 0.152332i \(0.0486783\pi\)
\(18\) 13.3629 − 2.50635i 0.742384 − 0.139242i
\(19\) 19.0173 1.00091 0.500455 − 0.865763i \(-0.333166\pi\)
0.500455 + 0.865763i \(0.333166\pi\)
\(20\) 0 0
\(21\) 8.20950i 0.390929i
\(22\) 21.2451 − 3.98473i 0.965687 − 0.181124i
\(23\) 16.6882 + 16.6882i 0.725572 + 0.725572i 0.969734 − 0.244162i \(-0.0785129\pi\)
−0.244162 + 0.969734i \(0.578513\pi\)
\(24\) −11.5610 − 2.69694i −0.481709 − 0.112372i
\(25\) 0 0
\(26\) −16.8990 − 11.5610i −0.649961 − 0.444655i
\(27\) −16.5767 − 16.5767i −0.613953 − 0.613953i
\(28\) −8.91386 + 20.2544i −0.318352 + 0.723372i
\(29\) 51.4406 1.77381 0.886907 − 0.461947i \(-0.152849\pi\)
0.886907 + 0.461947i \(0.152849\pi\)
\(30\) 0 0
\(31\) −31.1918 −1.00619 −0.503094 − 0.864232i \(-0.667805\pi\)
−0.503094 + 0.864232i \(0.667805\pi\)
\(32\) −25.5949 − 19.2068i −0.799841 − 0.600212i
\(33\) −11.3406 − 11.3406i −0.343653 − 0.343653i
\(34\) −33.1766 − 22.6969i −0.975782 − 0.667557i
\(35\) 0 0
\(36\) −9.85357 − 25.3437i −0.273710 − 0.703992i
\(37\) 22.8725 + 22.8725i 0.618176 + 0.618176i 0.945063 − 0.326888i \(-0.106000\pi\)
−0.326888 + 0.945063i \(0.606000\pi\)
\(38\) −7.01151 − 37.3827i −0.184513 − 0.983756i
\(39\) 15.1918i 0.389534i
\(40\) 0 0
\(41\) −3.79796 −0.0926331 −0.0463166 − 0.998927i \(-0.514748\pi\)
−0.0463166 + 0.998927i \(0.514748\pi\)
\(42\) 16.1376 − 3.02677i 0.384229 − 0.0720660i
\(43\) −50.7795 + 50.7795i −1.18092 + 1.18092i −0.201411 + 0.979507i \(0.564553\pi\)
−0.979507 + 0.201411i \(0.935447\pi\)
\(44\) −15.6658 − 40.2929i −0.356040 − 0.915747i
\(45\) 0 0
\(46\) 26.6515 − 38.9571i 0.579381 − 0.846893i
\(47\) −37.2882 + 37.2882i −0.793367 + 0.793367i −0.982040 − 0.188673i \(-0.939581\pi\)
0.188673 + 0.982040i \(0.439581\pi\)
\(48\) −1.03899 + 23.7201i −0.0216456 + 0.494169i
\(49\) 18.3939i 0.375385i
\(50\) 0 0
\(51\) 29.8251i 0.584805i
\(52\) −16.4953 + 37.4812i −0.317216 + 0.720792i
\(53\) 61.3784 − 61.3784i 1.15808 − 1.15808i 0.173196 − 0.984887i \(-0.444591\pi\)
0.984887 − 0.173196i \(-0.0554094\pi\)
\(54\) −26.4736 + 38.6969i −0.490251 + 0.716610i
\(55\) 0 0
\(56\) 43.1010 + 10.0545i 0.769661 + 0.179545i
\(57\) −19.9547 + 19.9547i −0.350083 + 0.350083i
\(58\) −18.9657 − 101.118i −0.326995 − 1.74341i
\(59\) 40.6329 0.688692 0.344346 − 0.938843i \(-0.388101\pi\)
0.344346 + 0.938843i \(0.388101\pi\)
\(60\) 0 0
\(61\) 43.2311i 0.708707i 0.935112 + 0.354354i \(0.115299\pi\)
−0.935112 + 0.354354i \(0.884701\pi\)
\(62\) 11.5002 + 61.3145i 0.185486 + 0.988944i
\(63\) 26.5930 + 26.5930i 0.422111 + 0.422111i
\(64\) −28.3186 + 57.3939i −0.442478 + 0.896779i
\(65\) 0 0
\(66\) −18.1112 + 26.4736i −0.274413 + 0.401114i
\(67\) −26.0204 − 26.0204i −0.388364 − 0.388364i 0.485740 − 0.874104i \(-0.338550\pi\)
−0.874104 + 0.485740i \(0.838550\pi\)
\(68\) −32.3840 + 73.5841i −0.476235 + 1.08212i
\(69\) −35.0216 −0.507560
\(70\) 0 0
\(71\) 78.3837 1.10400 0.551998 − 0.833846i \(-0.313866\pi\)
0.551998 + 0.833846i \(0.313866\pi\)
\(72\) −46.1858 + 28.7134i −0.641469 + 0.398797i
\(73\) −96.6123 − 96.6123i −1.32346 − 1.32346i −0.910959 − 0.412497i \(-0.864657\pi\)
−0.412497 − 0.910959i \(-0.635343\pi\)
\(74\) 36.5281 − 53.3939i 0.493623 − 0.721539i
\(75\) 0 0
\(76\) −70.8990 + 27.5654i −0.932881 + 0.362702i
\(77\) 42.2791 + 42.2791i 0.549079 + 0.549079i
\(78\) 29.8629 − 5.60110i 0.382858 − 0.0718089i
\(79\) 16.8082i 0.212762i 0.994325 + 0.106381i \(0.0339263\pi\)
−0.994325 + 0.106381i \(0.966074\pi\)
\(80\) 0 0
\(81\) −26.3939 −0.325850
\(82\) 1.40027 + 7.46574i 0.0170765 + 0.0910456i
\(83\) −61.2724 + 61.2724i −0.738222 + 0.738222i −0.972234 − 0.234012i \(-0.924815\pi\)
0.234012 + 0.972234i \(0.424815\pi\)
\(84\) −11.8996 − 30.6061i −0.141662 − 0.364359i
\(85\) 0 0
\(86\) 118.540 + 81.0964i 1.37838 + 0.942982i
\(87\) −53.9764 + 53.9764i −0.620419 + 0.620419i
\(88\) −73.4288 + 45.6502i −0.834418 + 0.518752i
\(89\) − 33.1918i − 0.372942i −0.982460 − 0.186471i \(-0.940295\pi\)
0.982460 − 0.186471i \(-0.0597050\pi\)
\(90\) 0 0
\(91\) − 56.6372i − 0.622387i
\(92\) −86.4050 − 38.0264i −0.939185 − 0.413330i
\(93\) 32.7294 − 32.7294i 0.351930 − 0.351930i
\(94\) 87.0462 + 59.5505i 0.926024 + 0.633516i
\(95\) 0 0
\(96\) 47.0102 − 6.70303i 0.489690 − 0.0698232i
\(97\) 73.9312 − 73.9312i 0.762177 − 0.762177i −0.214539 − 0.976716i \(-0.568825\pi\)
0.976716 + 0.214539i \(0.0688248\pi\)
\(98\) 36.1573 − 6.78166i 0.368952 − 0.0692006i
\(99\) −73.4709 −0.742130
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.i.a.93.4 yes 16
4.3 odd 2 800.3.m.a.593.5 16
5.2 odd 4 inner 200.3.i.a.157.8 yes 16
5.3 odd 4 inner 200.3.i.a.157.1 yes 16
5.4 even 2 inner 200.3.i.a.93.5 yes 16
8.3 odd 2 800.3.m.a.593.3 16
8.5 even 2 inner 200.3.i.a.93.8 yes 16
20.3 even 4 800.3.m.a.657.6 16
20.7 even 4 800.3.m.a.657.3 16
20.19 odd 2 800.3.m.a.593.4 16
40.3 even 4 800.3.m.a.657.4 16
40.13 odd 4 inner 200.3.i.a.157.5 yes 16
40.19 odd 2 800.3.m.a.593.6 16
40.27 even 4 800.3.m.a.657.5 16
40.29 even 2 inner 200.3.i.a.93.1 ✓ 16
40.37 odd 4 inner 200.3.i.a.157.4 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
200.3.i.a.93.1 ✓ 16 40.29 even 2 inner
200.3.i.a.93.4 yes 16 1.1 even 1 trivial
200.3.i.a.93.5 yes 16 5.4 even 2 inner
200.3.i.a.93.8 yes 16 8.5 even 2 inner
200.3.i.a.157.1 yes 16 5.3 odd 4 inner
200.3.i.a.157.4 yes 16 40.37 odd 4 inner
200.3.i.a.157.5 yes 16 40.13 odd 4 inner
200.3.i.a.157.8 yes 16 5.2 odd 4 inner
800.3.m.a.593.3 16 8.3 odd 2
800.3.m.a.593.4 16 20.19 odd 2
800.3.m.a.593.5 16 4.3 odd 2
800.3.m.a.593.6 16 40.19 odd 2
800.3.m.a.657.3 16 20.7 even 4
800.3.m.a.657.4 16 40.3 even 4
800.3.m.a.657.5 16 40.27 even 4
800.3.m.a.657.6 16 20.3 even 4