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Properties

Label 16.22.a.f.1.3
Level $16$
Weight $22$
Character 16.1
Self dual yes
Analytic conductor $44.716$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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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] = [16,22,Mod(1,16)] 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("16.1"); S:= CuspForms(chi, 22); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(16, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 22, names="a")
 
Level: \( N \) \(=\) \( 16 = 2^{4} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 16.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,-96764] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(44.7163750859\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4963x + 96223 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{21}\cdot 3\cdot 5\cdot 7 \)
Twist minimal: no (minimal twist has level 8)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-78.2002\) of defining polynomial
Character \(\chi\) \(=\) 16.1

$q$-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)
 
\(f(q)\) \(=\) \(q+97085.2 q^{3} -3.39292e7 q^{5} +5.28731e8 q^{7} -1.03483e9 q^{9} +1.21955e11 q^{11} +4.33605e11 q^{13} -3.29402e12 q^{15} -1.31741e13 q^{17} -2.18808e13 q^{19} +5.13320e13 q^{21} -1.40779e14 q^{23} +6.74355e14 q^{25} -1.11601e15 q^{27} +1.17115e15 q^{29} -9.57156e14 q^{31} +1.18400e16 q^{33} -1.79394e16 q^{35} -3.53720e16 q^{37} +4.20966e16 q^{39} -1.71456e17 q^{41} -1.35679e17 q^{43} +3.51109e16 q^{45} +5.75514e17 q^{47} -2.78989e17 q^{49} -1.27901e18 q^{51} -9.62123e17 q^{53} -4.13782e18 q^{55} -2.12430e18 q^{57} -4.87671e18 q^{59} +4.59195e18 q^{61} -5.47145e17 q^{63} -1.47119e19 q^{65} +1.78611e19 q^{67} -1.36675e19 q^{69} -2.93629e19 q^{71} -8.32585e18 q^{73} +6.54699e19 q^{75} +6.44812e19 q^{77} +7.05718e19 q^{79} -9.75235e19 q^{81} -2.11929e20 q^{83} +4.46988e20 q^{85} +1.13702e20 q^{87} +2.62972e20 q^{89} +2.29260e20 q^{91} -9.29257e19 q^{93} +7.42397e20 q^{95} +3.84293e20 q^{97} -1.26202e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 96764 q^{3} - 24111774 q^{5} - 295988280 q^{7} + 18844697239 q^{9} + 40335108684 q^{11} + 133734425946 q^{13} + 1223136458200 q^{15} + 7797732274422 q^{17} - 35788199781996 q^{19} + 198539224853088 q^{21}+ \cdots + 94\!\cdots\!60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) 97085.2 0.949248 0.474624 − 0.880189i \(-0.342584\pi\)
0.474624 + 0.880189i \(0.342584\pi\)
\(4\) 0 0
\(5\) −3.39292e7 −1.55378 −0.776889 − 0.629638i \(-0.783203\pi\)
−0.776889 + 0.629638i \(0.783203\pi\)
\(6\) 0 0
\(7\) 5.28731e8 0.707466 0.353733 − 0.935346i \(-0.384912\pi\)
0.353733 + 0.935346i \(0.384912\pi\)
\(8\) 0 0
\(9\) −1.03483e9 −0.0989284
\(10\) 0 0
\(11\) 1.21955e11 1.41767 0.708835 − 0.705375i \(-0.249221\pi\)
0.708835 + 0.705375i \(0.249221\pi\)
\(12\) 0 0
\(13\) 4.33605e11 0.872346 0.436173 − 0.899863i \(-0.356334\pi\)
0.436173 + 0.899863i \(0.356334\pi\)
\(14\) 0 0
\(15\) −3.29402e12 −1.47492
\(16\) 0 0
\(17\) −1.31741e13 −1.58492 −0.792461 − 0.609922i \(-0.791201\pi\)
−0.792461 + 0.609922i \(0.791201\pi\)
\(18\) 0 0
\(19\) −2.18808e13 −0.818747 −0.409373 − 0.912367i \(-0.634253\pi\)
−0.409373 + 0.912367i \(0.634253\pi\)
\(20\) 0 0
\(21\) 5.13320e13 0.671561
\(22\) 0 0
\(23\) −1.40779e14 −0.708589 −0.354294 − 0.935134i \(-0.615279\pi\)
−0.354294 + 0.935134i \(0.615279\pi\)
\(24\) 0 0
\(25\) 6.74355e14 1.41423
\(26\) 0 0
\(27\) −1.11601e15 −1.04316
\(28\) 0 0
\(29\) 1.17115e15 0.516934 0.258467 − 0.966020i \(-0.416783\pi\)
0.258467 + 0.966020i \(0.416783\pi\)
\(30\) 0 0
\(31\) −9.57156e14 −0.209742 −0.104871 − 0.994486i \(-0.533443\pi\)
−0.104871 + 0.994486i \(0.533443\pi\)
\(32\) 0 0
\(33\) 1.18400e16 1.34572
\(34\) 0 0
\(35\) −1.79394e16 −1.09925
\(36\) 0 0
\(37\) −3.53720e16 −1.20932 −0.604661 − 0.796483i \(-0.706691\pi\)
−0.604661 + 0.796483i \(0.706691\pi\)
\(38\) 0 0
\(39\) 4.20966e16 0.828073
\(40\) 0 0
\(41\) −1.71456e17 −1.99490 −0.997451 − 0.0713504i \(-0.977269\pi\)
−0.997451 + 0.0713504i \(0.977269\pi\)
\(42\) 0 0
\(43\) −1.35679e17 −0.957398 −0.478699 − 0.877979i \(-0.658892\pi\)
−0.478699 + 0.877979i \(0.658892\pi\)
\(44\) 0 0
\(45\) 3.51109e16 0.153713
\(46\) 0 0
\(47\) 5.75514e17 1.59598 0.797991 − 0.602669i \(-0.205896\pi\)
0.797991 + 0.602669i \(0.205896\pi\)
\(48\) 0 0
\(49\) −2.78989e17 −0.499492
\(50\) 0 0
\(51\) −1.27901e18 −1.50448
\(52\) 0 0
\(53\) −9.62123e17 −0.755673 −0.377837 − 0.925872i \(-0.623332\pi\)
−0.377837 + 0.925872i \(0.623332\pi\)
\(54\) 0 0
\(55\) −4.13782e18 −2.20274
\(56\) 0 0
\(57\) −2.12430e18 −0.777194
\(58\) 0 0
\(59\) −4.87671e18 −1.24217 −0.621085 − 0.783743i \(-0.713308\pi\)
−0.621085 + 0.783743i \(0.713308\pi\)
\(60\) 0 0
\(61\) 4.59195e18 0.824202 0.412101 − 0.911138i \(-0.364795\pi\)
0.412101 + 0.911138i \(0.364795\pi\)
\(62\) 0 0
\(63\) −5.47145e17 −0.0699885
\(64\) 0 0
\(65\) −1.47119e19 −1.35543
\(66\) 0 0
\(67\) 1.78611e19 1.19708 0.598539 − 0.801094i \(-0.295748\pi\)
0.598539 + 0.801094i \(0.295748\pi\)
\(68\) 0 0
\(69\) −1.36675e19 −0.672626
\(70\) 0 0
\(71\) −2.93629e19 −1.07050 −0.535250 − 0.844694i \(-0.679783\pi\)
−0.535250 + 0.844694i \(0.679783\pi\)
\(72\) 0 0
\(73\) −8.32585e18 −0.226745 −0.113373 − 0.993553i \(-0.536165\pi\)
−0.113373 + 0.993553i \(0.536165\pi\)
\(74\) 0 0
\(75\) 6.54699e19 1.34245
\(76\) 0 0
\(77\) 6.44812e19 1.00295
\(78\) 0 0
\(79\) 7.05718e19 0.838584 0.419292 − 0.907851i \(-0.362278\pi\)
0.419292 + 0.907851i \(0.362278\pi\)
\(80\) 0 0
\(81\) −9.75235e19 −0.891285
\(82\) 0 0
\(83\) −2.11929e20 −1.49924 −0.749620 − 0.661869i \(-0.769764\pi\)
−0.749620 + 0.661869i \(0.769764\pi\)
\(84\) 0 0
\(85\) 4.46988e20 2.46262
\(86\) 0 0
\(87\) 1.13702e20 0.490698
\(88\) 0 0
\(89\) 2.62972e20 0.893952 0.446976 − 0.894546i \(-0.352501\pi\)
0.446976 + 0.894546i \(0.352501\pi\)
\(90\) 0 0
\(91\) 2.29260e20 0.617155
\(92\) 0 0
\(93\) −9.29257e19 −0.199097
\(94\) 0 0
\(95\) 7.42397e20 1.27215
\(96\) 0 0
\(97\) 3.84293e20 0.529126 0.264563 − 0.964368i \(-0.414772\pi\)
0.264563 + 0.964368i \(0.414772\pi\)
\(98\) 0 0
\(99\) −1.26202e20 −0.140248
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 16.22.a.f.1.3 3
4.3 odd 2 8.22.a.b.1.1 ✓ 3
8.3 odd 2 64.22.a.l.1.3 3
8.5 even 2 64.22.a.m.1.1 3
12.11 even 2 72.22.a.f.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8.22.a.b.1.1 ✓ 3 4.3 odd 2
16.22.a.f.1.3 3 1.1 even 1 trivial
64.22.a.l.1.3 3 8.3 odd 2
64.22.a.m.1.1 3 8.5 even 2
72.22.a.f.1.3 3 12.11 even 2