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Properties

Label 16.18.a.d.1.2
Level $16$
Weight $18$
Character 16.1
Self dual yes
Analytic conductor $29.316$
Analytic rank $1$
Dimension $2$
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] = [16,18,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, 18); 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, 18, names="a")
 
Level: \( N \) \(=\) \( 16 = 2^{4} \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 16.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,952] 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: \(29.3155339751\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2146}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2146 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 8)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(46.3249\) 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)
 
\(f(q)\) \(=\) \(q+12335.2 q^{3} -738361. q^{5} +1.17649e7 q^{7} +2.30166e7 q^{9} -1.10233e9 q^{11} -2.57536e9 q^{13} -9.10782e9 q^{15} +4.96650e10 q^{17} -7.58630e10 q^{19} +1.45122e11 q^{21} -4.80343e10 q^{23} -2.17762e11 q^{25} -1.30905e12 q^{27} -4.46967e12 q^{29} +2.89743e12 q^{31} -1.35974e13 q^{33} -8.68672e12 q^{35} +1.98048e13 q^{37} -3.17675e13 q^{39} -5.72233e13 q^{41} -1.17723e14 q^{43} -1.69946e13 q^{45} -1.99303e14 q^{47} -9.42184e13 q^{49} +6.12627e14 q^{51} +3.19512e14 q^{53} +8.13917e14 q^{55} -9.35784e14 q^{57} -3.92786e14 q^{59} +2.54361e15 q^{61} +2.70787e14 q^{63} +1.90154e15 q^{65} -2.36089e14 q^{67} -5.92511e14 q^{69} +7.35004e14 q^{71} +7.70307e15 q^{73} -2.68614e15 q^{75} -1.29688e16 q^{77} +8.20252e15 q^{79} -1.91198e16 q^{81} +2.50479e15 q^{83} -3.66707e16 q^{85} -5.51342e16 q^{87} +1.99635e16 q^{89} -3.02987e16 q^{91} +3.57403e16 q^{93} +5.60143e16 q^{95} +5.80049e16 q^{97} -2.53719e16 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 952 q^{3} - 53620 q^{5} + 333168 q^{7} + 23453338 q^{9} - 430974680 q^{11} + 2667521948 q^{13} - 16902353840 q^{15} + 60673503268 q^{17} - 178629960040 q^{19} + 275250928704 q^{21} - 528756594608 q^{23}+ \cdots - 25\!\cdots\!64 q^{99}+O(q^{100}) \) 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
\(3\) 12335.2 1.08546 0.542731 − 0.839906i \(-0.317390\pi\)
0.542731 + 0.839906i \(0.317390\pi\)
\(4\) 0 0
\(5\) −738361. −0.845325 −0.422663 − 0.906287i \(-0.638904\pi\)
−0.422663 + 0.906287i \(0.638904\pi\)
\(6\) 0 0
\(7\) 1.17649e7 0.771354 0.385677 − 0.922634i \(-0.373968\pi\)
0.385677 + 0.922634i \(0.373968\pi\)
\(8\) 0 0
\(9\) 2.30166e7 0.178230
\(10\) 0 0
\(11\) −1.10233e9 −1.55051 −0.775253 − 0.631651i \(-0.782378\pi\)
−0.775253 + 0.631651i \(0.782378\pi\)
\(12\) 0 0
\(13\) −2.57536e9 −0.875627 −0.437814 − 0.899066i \(-0.644247\pi\)
−0.437814 + 0.899066i \(0.644247\pi\)
\(14\) 0 0
\(15\) −9.10782e9 −0.917569
\(16\) 0 0
\(17\) 4.96650e10 1.72677 0.863386 − 0.504544i \(-0.168339\pi\)
0.863386 + 0.504544i \(0.168339\pi\)
\(18\) 0 0
\(19\) −7.58630e10 −1.02477 −0.512383 − 0.858757i \(-0.671237\pi\)
−0.512383 + 0.858757i \(0.671237\pi\)
\(20\) 0 0
\(21\) 1.45122e11 0.837276
\(22\) 0 0
\(23\) −4.80343e10 −0.127898 −0.0639491 − 0.997953i \(-0.520370\pi\)
−0.0639491 + 0.997953i \(0.520370\pi\)
\(24\) 0 0
\(25\) −2.17762e11 −0.285426
\(26\) 0 0
\(27\) −1.30905e12 −0.892001
\(28\) 0 0
\(29\) −4.46967e12 −1.65918 −0.829589 − 0.558375i \(-0.811425\pi\)
−0.829589 + 0.558375i \(0.811425\pi\)
\(30\) 0 0
\(31\) 2.89743e12 0.610152 0.305076 − 0.952328i \(-0.401318\pi\)
0.305076 + 0.952328i \(0.401318\pi\)
\(32\) 0 0
\(33\) −1.35974e13 −1.68302
\(34\) 0 0
\(35\) −8.68672e12 −0.652045
\(36\) 0 0
\(37\) 1.98048e13 0.926950 0.463475 − 0.886110i \(-0.346602\pi\)
0.463475 + 0.886110i \(0.346602\pi\)
\(38\) 0 0
\(39\) −3.17675e13 −0.950461
\(40\) 0 0
\(41\) −5.72233e13 −1.11921 −0.559604 − 0.828760i \(-0.689047\pi\)
−0.559604 + 0.828760i \(0.689047\pi\)
\(42\) 0 0
\(43\) −1.17723e14 −1.53596 −0.767978 − 0.640476i \(-0.778737\pi\)
−0.767978 + 0.640476i \(0.778737\pi\)
\(44\) 0 0
\(45\) −1.69946e13 −0.150662
\(46\) 0 0
\(47\) −1.99303e14 −1.22090 −0.610452 − 0.792053i \(-0.709012\pi\)
−0.610452 + 0.792053i \(0.709012\pi\)
\(48\) 0 0
\(49\) −9.42184e13 −0.405013
\(50\) 0 0
\(51\) 6.12627e14 1.87435
\(52\) 0 0
\(53\) 3.19512e14 0.704924 0.352462 − 0.935826i \(-0.385345\pi\)
0.352462 + 0.935826i \(0.385345\pi\)
\(54\) 0 0
\(55\) 8.13917e14 1.31068
\(56\) 0 0
\(57\) −9.35784e14 −1.11234
\(58\) 0 0
\(59\) −3.92786e14 −0.348268 −0.174134 − 0.984722i \(-0.555713\pi\)
−0.174134 + 0.984722i \(0.555713\pi\)
\(60\) 0 0
\(61\) 2.54361e15 1.69882 0.849409 − 0.527736i \(-0.176959\pi\)
0.849409 + 0.527736i \(0.176959\pi\)
\(62\) 0 0
\(63\) 2.70787e14 0.137478
\(64\) 0 0
\(65\) 1.90154e15 0.740189
\(66\) 0 0
\(67\) −2.36089e14 −0.0710296 −0.0355148 − 0.999369i \(-0.511307\pi\)
−0.0355148 + 0.999369i \(0.511307\pi\)
\(68\) 0 0
\(69\) −5.92511e14 −0.138829
\(70\) 0 0
\(71\) 7.35004e14 0.135081 0.0675404 − 0.997717i \(-0.478485\pi\)
0.0675404 + 0.997717i \(0.478485\pi\)
\(72\) 0 0
\(73\) 7.70307e15 1.11794 0.558971 − 0.829187i \(-0.311196\pi\)
0.558971 + 0.829187i \(0.311196\pi\)
\(74\) 0 0
\(75\) −2.68614e15 −0.309819
\(76\) 0 0
\(77\) −1.29688e16 −1.19599
\(78\) 0 0
\(79\) 8.20252e15 0.608299 0.304149 − 0.952624i \(-0.401628\pi\)
0.304149 + 0.952624i \(0.401628\pi\)
\(80\) 0 0
\(81\) −1.91198e16 −1.14646
\(82\) 0 0
\(83\) 2.50479e15 0.122070 0.0610348 − 0.998136i \(-0.480560\pi\)
0.0610348 + 0.998136i \(0.480560\pi\)
\(84\) 0 0
\(85\) −3.66707e16 −1.45968
\(86\) 0 0
\(87\) −5.51342e16 −1.80098
\(88\) 0 0
\(89\) 1.99635e16 0.537554 0.268777 − 0.963202i \(-0.413381\pi\)
0.268777 + 0.963202i \(0.413381\pi\)
\(90\) 0 0
\(91\) −3.02987e16 −0.675418
\(92\) 0 0
\(93\) 3.57403e16 0.662297
\(94\) 0 0
\(95\) 5.60143e16 0.866260
\(96\) 0 0
\(97\) 5.80049e16 0.751458 0.375729 − 0.926730i \(-0.377392\pi\)
0.375729 + 0.926730i \(0.377392\pi\)
\(98\) 0 0
\(99\) −2.53719e16 −0.276346
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 16.18.a.d.1.2 2
4.3 odd 2 8.18.a.a.1.1 ✓ 2
8.3 odd 2 64.18.a.k.1.2 2
8.5 even 2 64.18.a.h.1.1 2
12.11 even 2 72.18.a.c.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8.18.a.a.1.1 ✓ 2 4.3 odd 2
16.18.a.d.1.2 2 1.1 even 1 trivial
64.18.a.h.1.1 2 8.5 even 2
64.18.a.k.1.2 2 8.3 odd 2
72.18.a.c.1.2 2 12.11 even 2