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Properties

Label 200.3.i.a.93.7
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 93.7
Root \(-0.370982 - 0.364688i\) of defining polynomial
Character \(\chi\) \(=\) 200.93
Dual form 200.3.i.a.157.7

$q$-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+(1.73567 - 0.993706i) q^{2} +(-3.30136 + 3.30136i) q^{3} +(2.02510 - 3.44949i) q^{4} +(-2.44949 + 9.01065i) q^{6} +(-6.68558 + 6.68558i) q^{7} +(0.0871225 - 7.99953i) q^{8} -12.7980i q^{9} +13.9711i q^{11} +(4.70243 + 18.0736i) q^{12} +(-9.57058 + 9.57058i) q^{13} +(-4.96046 + 18.2474i) q^{14} +(-7.79796 - 13.9711i) q^{16} +(-2.45365 + 2.45365i) q^{17} +(-12.7174 - 22.2130i) q^{18} -30.1719 q^{19} -44.1430i q^{21} +(13.8832 + 24.2492i) q^{22} +(15.1494 + 15.1494i) q^{23} +(26.1217 + 26.6969i) q^{24} +(-7.10102 + 26.1217i) q^{26} +(12.5384 + 12.5384i) q^{27} +(9.52288 + 36.6008i) q^{28} +11.7414 q^{29} +47.1918 q^{31} +(-27.4178 - 16.5003i) q^{32} +(-46.1237 - 46.1237i) q^{33} +(-1.82052 + 6.69694i) q^{34} +(-44.1464 - 25.9171i) q^{36} +(7.26973 + 7.26973i) q^{37} +(-52.3684 + 29.9820i) q^{38} -63.1918i q^{39} +15.7980 q^{41} +(-43.8651 - 76.6177i) q^{42} +(34.3139 - 34.3139i) q^{43} +(48.1932 + 28.2929i) q^{44} +(41.3485 + 11.2403i) q^{46} +(-23.6133 + 23.6133i) q^{47} +(71.8675 + 20.3798i) q^{48} -40.3939i q^{49} -16.2008i q^{51} +(13.6323 + 52.3950i) q^{52} +(31.3798 - 31.3798i) q^{53} +(34.2221 + 9.30306i) q^{54} +(52.8990 + 54.0639i) q^{56} +(99.6083 - 99.6083i) q^{57} +(20.3792 - 11.6675i) q^{58} -2.22967 q^{59} +55.8844i q^{61} +(81.9094 - 46.8948i) q^{62} +(85.5617 + 85.5617i) q^{63} +(-63.9848 - 1.39388i) q^{64} +(-125.889 - 34.2221i) q^{66} +(-17.1738 - 17.1738i) q^{67} +(3.49496 + 13.4327i) q^{68} -100.027 q^{69} -78.3837 q^{71} +(-102.378 - 1.11499i) q^{72} +(-31.4017 - 31.4017i) q^{73} +(19.8418 + 5.39388i) q^{74} +(-61.1010 + 104.078i) q^{76} +(-93.4049 - 93.4049i) q^{77} +(-62.7941 - 109.680i) q^{78} +95.1918i q^{79} +32.3939 q^{81} +(27.4200 - 15.6985i) q^{82} +(1.30033 - 1.30033i) q^{83} +(-152.271 - 89.3939i) q^{84} +(25.4597 - 93.6556i) q^{86} +(-38.7627 + 38.7627i) q^{87} +(111.762 + 1.21720i) q^{88} +45.1918i q^{89} -127.970i q^{91} +(82.9369 - 21.5787i) q^{92} +(-155.797 + 155.797i) q^{93} +(-17.5202 + 64.4495i) q^{94} +(144.990 - 36.0426i) q^{96} +(-60.8456 + 60.8456i) q^{97} +(-40.1396 - 70.1104i) q^{98} +178.802 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.73567 − 0.993706i 0.867835 − 0.496853i
\(3\) −3.30136 + 3.30136i −1.10045 + 1.10045i −0.106098 + 0.994356i \(0.533836\pi\)
−0.994356 + 0.106098i \(0.966164\pi\)
\(4\) 2.02510 − 3.44949i 0.506274 − 0.862372i
\(5\) 0 0
\(6\) −2.44949 + 9.01065i −0.408248 + 1.50178i
\(7\) −6.68558 + 6.68558i −0.955082 + 0.955082i −0.999034 − 0.0439513i \(-0.986005\pi\)
0.0439513 + 0.999034i \(0.486005\pi\)
\(8\) 0.0871225 − 7.99953i 0.0108903 − 0.999941i
\(9\) − 12.7980i − 1.42200i
\(10\) 0 0
\(11\) 13.9711i 1.27010i 0.772471 + 0.635050i \(0.219021\pi\)
−0.772471 + 0.635050i \(0.780979\pi\)
\(12\) 4.70243 + 18.0736i 0.391869 + 1.50613i
\(13\) −9.57058 + 9.57058i −0.736198 + 0.736198i −0.971840 − 0.235642i \(-0.924281\pi\)
0.235642 + 0.971840i \(0.424281\pi\)
\(14\) −4.96046 + 18.2474i −0.354318 + 1.30339i
\(15\) 0 0
\(16\) −7.79796 − 13.9711i −0.487372 − 0.873194i
\(17\) −2.45365 + 2.45365i −0.144332 + 0.144332i −0.775581 − 0.631248i \(-0.782543\pi\)
0.631248 + 0.775581i \(0.282543\pi\)
\(18\) −12.7174 − 22.2130i −0.706523 − 1.23406i
\(19\) −30.1719 −1.58799 −0.793997 − 0.607922i \(-0.792003\pi\)
−0.793997 + 0.607922i \(0.792003\pi\)
\(20\) 0 0
\(21\) − 44.1430i − 2.10205i
\(22\) 13.8832 + 24.2492i 0.631053 + 1.10224i
\(23\) 15.1494 + 15.1494i 0.658671 + 0.658671i 0.955066 − 0.296395i \(-0.0957844\pi\)
−0.296395 + 0.955066i \(0.595784\pi\)
\(24\) 26.1217 + 26.6969i 1.08840 + 1.11237i
\(25\) 0 0
\(26\) −7.10102 + 26.1217i −0.273116 + 1.00468i
\(27\) 12.5384 + 12.5384i 0.464386 + 0.464386i
\(28\) 9.52288 + 36.6008i 0.340103 + 1.30717i
\(29\) 11.7414 0.404877 0.202439 − 0.979295i \(-0.435113\pi\)
0.202439 + 0.979295i \(0.435113\pi\)
\(30\) 0 0
\(31\) 47.1918 1.52232 0.761159 − 0.648566i \(-0.224631\pi\)
0.761159 + 0.648566i \(0.224631\pi\)
\(32\) −27.4178 − 16.5003i −0.856808 − 0.515636i
\(33\) −46.1237 − 46.1237i −1.39769 − 1.39769i
\(34\) −1.82052 + 6.69694i −0.0535447 + 0.196969i
\(35\) 0 0
\(36\) −44.1464 − 25.9171i −1.22629 − 0.719920i
\(37\) 7.26973 + 7.26973i 0.196479 + 0.196479i 0.798489 − 0.602010i \(-0.205633\pi\)
−0.602010 + 0.798489i \(0.705633\pi\)
\(38\) −52.3684 + 29.9820i −1.37812 + 0.788999i
\(39\) − 63.1918i − 1.62030i
\(40\) 0 0
\(41\) 15.7980 0.385316 0.192658 − 0.981266i \(-0.438289\pi\)
0.192658 + 0.981266i \(0.438289\pi\)
\(42\) −43.8651 − 76.6177i −1.04441 − 1.82423i
\(43\) 34.3139 − 34.3139i 0.797998 − 0.797998i −0.184781 − 0.982780i \(-0.559158\pi\)
0.982780 + 0.184781i \(0.0591577\pi\)
\(44\) 48.1932 + 28.2929i 1.09530 + 0.643019i
\(45\) 0 0
\(46\) 41.3485 + 11.2403i 0.898880 + 0.244355i
\(47\) −23.6133 + 23.6133i −0.502410 + 0.502410i −0.912186 − 0.409776i \(-0.865607\pi\)
0.409776 + 0.912186i \(0.365607\pi\)
\(48\) 71.8675 + 20.3798i 1.49724 + 0.424579i
\(49\) − 40.3939i − 0.824365i
\(50\) 0 0
\(51\) − 16.2008i − 0.317662i
\(52\) 13.6323 + 52.3950i 0.262159 + 1.00760i
\(53\) 31.3798 − 31.3798i 0.592071 − 0.592071i −0.346119 − 0.938190i \(-0.612501\pi\)
0.938190 + 0.346119i \(0.112501\pi\)
\(54\) 34.2221 + 9.30306i 0.633742 + 0.172279i
\(55\) 0 0
\(56\) 52.8990 + 54.0639i 0.944625 + 0.965427i
\(57\) 99.6083 − 99.6083i 1.74751 − 1.74751i
\(58\) 20.3792 − 11.6675i 0.351366 − 0.201164i
\(59\) −2.22967 −0.0377911 −0.0188955 − 0.999821i \(-0.506015\pi\)
−0.0188955 + 0.999821i \(0.506015\pi\)
\(60\) 0 0
\(61\) 55.8844i 0.916138i 0.888917 + 0.458069i \(0.151459\pi\)
−0.888917 + 0.458069i \(0.848541\pi\)
\(62\) 81.9094 − 46.8948i 1.32112 − 0.756368i
\(63\) 85.5617 + 85.5617i 1.35812 + 1.35812i
\(64\) −63.9848 − 1.39388i −0.999763 − 0.0217793i
\(65\) 0 0
\(66\) −125.889 − 34.2221i −1.90741 − 0.518516i
\(67\) −17.1738 − 17.1738i −0.256326 − 0.256326i 0.567232 − 0.823558i \(-0.308014\pi\)
−0.823558 + 0.567232i \(0.808014\pi\)
\(68\) 3.49496 + 13.4327i 0.0513965 + 0.197540i
\(69\) −100.027 −1.44967
\(70\) 0 0
\(71\) −78.3837 −1.10400 −0.551998 − 0.833846i \(-0.686134\pi\)
−0.551998 + 0.833846i \(0.686134\pi\)
\(72\) −102.378 − 1.11499i −1.42191 − 0.0154860i
\(73\) −31.4017 − 31.4017i −0.430161 − 0.430161i 0.458522 − 0.888683i \(-0.348379\pi\)
−0.888683 + 0.458522i \(0.848379\pi\)
\(74\) 19.8418 + 5.39388i 0.268133 + 0.0728902i
\(75\) 0 0
\(76\) −61.1010 + 104.078i −0.803961 + 1.36944i
\(77\) −93.4049 − 93.4049i −1.21305 − 1.21305i
\(78\) −62.7941 − 109.680i −0.805052 − 1.40616i
\(79\) 95.1918i 1.20496i 0.798134 + 0.602480i \(0.205821\pi\)
−0.798134 + 0.602480i \(0.794179\pi\)
\(80\) 0 0
\(81\) 32.3939 0.399924
\(82\) 27.4200 − 15.6985i 0.334391 − 0.191445i
\(83\) 1.30033 − 1.30033i 0.0156666 − 0.0156666i −0.699230 − 0.714897i \(-0.746474\pi\)
0.714897 + 0.699230i \(0.246474\pi\)
\(84\) −152.271 − 89.3939i −1.81275 − 1.06421i
\(85\) 0 0
\(86\) 25.4597 − 93.6556i 0.296043 − 1.08902i
\(87\) −38.7627 + 38.7627i −0.445548 + 0.445548i
\(88\) 111.762 + 1.21720i 1.27003 + 0.0138318i
\(89\) 45.1918i 0.507773i 0.967234 + 0.253887i \(0.0817090\pi\)
−0.967234 + 0.253887i \(0.918291\pi\)
\(90\) 0 0
\(91\) − 127.970i − 1.40626i
\(92\) 82.9369 − 21.5787i 0.901488 − 0.234551i
\(93\) −155.797 + 155.797i −1.67524 + 1.67524i
\(94\) −17.5202 + 64.4495i −0.186385 + 0.685633i
\(95\) 0 0
\(96\) 144.990 − 36.0426i 1.51031 − 0.375444i
\(97\) −60.8456 + 60.8456i −0.627274 + 0.627274i −0.947381 − 0.320107i \(-0.896281\pi\)
0.320107 + 0.947381i \(0.396281\pi\)
\(98\) −40.1396 − 70.1104i −0.409588 − 0.715412i
\(99\) 178.802 1.80608
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.93.7 yes 16
4.3 odd 2 800.3.m.a.593.8 16
5.2 odd 4 inner 200.3.i.a.157.6 yes 16
5.3 odd 4 inner 200.3.i.a.157.3 yes 16
5.4 even 2 inner 200.3.i.a.93.2 ✓ 16
8.3 odd 2 800.3.m.a.593.2 16
8.5 even 2 inner 200.3.i.a.93.6 yes 16
20.3 even 4 800.3.m.a.657.7 16
20.7 even 4 800.3.m.a.657.2 16
20.19 odd 2 800.3.m.a.593.1 16
40.3 even 4 800.3.m.a.657.1 16
40.13 odd 4 inner 200.3.i.a.157.2 yes 16
40.19 odd 2 800.3.m.a.593.7 16
40.27 even 4 800.3.m.a.657.8 16
40.29 even 2 inner 200.3.i.a.93.3 yes 16
40.37 odd 4 inner 200.3.i.a.157.7 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
200.3.i.a.93.2 ✓ 16 5.4 even 2 inner
200.3.i.a.93.3 yes 16 40.29 even 2 inner
200.3.i.a.93.6 yes 16 8.5 even 2 inner
200.3.i.a.93.7 yes 16 1.1 even 1 trivial
200.3.i.a.157.2 yes 16 40.13 odd 4 inner
200.3.i.a.157.3 yes 16 5.3 odd 4 inner
200.3.i.a.157.6 yes 16 5.2 odd 4 inner
200.3.i.a.157.7 yes 16 40.37 odd 4 inner
800.3.m.a.593.1 16 20.19 odd 2
800.3.m.a.593.2 16 8.3 odd 2
800.3.m.a.593.7 16 40.19 odd 2
800.3.m.a.593.8 16 4.3 odd 2
800.3.m.a.657.1 16 40.3 even 4
800.3.m.a.657.2 16 20.7 even 4
800.3.m.a.657.7 16 20.3 even 4
800.3.m.a.657.8 16 40.27 even 4