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Properties

Label 45.18.a.e.1.2
Level $45$
Weight $18$
Character 45.1
Self dual yes
Analytic conductor $82.450$
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] = [45,18,Mod(1,45)] 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("45.1"); S:= CuspForms(chi, 18); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(45, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 18, names="a")
 
Level: \( N \) \(=\) \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 45.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,442] 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: yes
Analytic conductor: \(82.4499393051\)
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: \( x^{3} - 37234x - 350700 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 3\cdot 5 \)
Twist minimal: no (minimal twist has level 15)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(197.509\) of defining polynomial
Character \(\chi\) \(=\) 45.1

$q$-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)
 
\(f(q)\) \(=\) \(q+213.954 q^{2} -85295.6 q^{4} +390625. q^{5} +7.24373e6 q^{7} -4.62928e7 q^{8} +8.35759e7 q^{10} -4.10790e7 q^{11} +1.35746e9 q^{13} +1.54983e9 q^{14} +1.27533e9 q^{16} -1.84008e10 q^{17} +1.42200e9 q^{19} -3.33186e10 q^{20} -8.78904e9 q^{22} +1.96071e10 q^{23} +1.52588e11 q^{25} +2.90434e11 q^{26} -6.17858e11 q^{28} +3.22761e12 q^{29} -1.93640e12 q^{31} +6.34055e12 q^{32} -3.93694e12 q^{34} +2.82958e12 q^{35} -2.47063e13 q^{37} +3.04244e11 q^{38} -1.80831e13 q^{40} -1.91037e13 q^{41} -9.80229e12 q^{43} +3.50386e12 q^{44} +4.19503e12 q^{46} -6.08880e13 q^{47} -1.80159e14 q^{49} +3.26468e13 q^{50} -1.15785e14 q^{52} -3.72760e14 q^{53} -1.60465e13 q^{55} -3.35332e14 q^{56} +6.90561e14 q^{58} -1.78795e15 q^{59} +1.98485e15 q^{61} -4.14302e14 q^{62} +1.18943e15 q^{64} +5.30258e14 q^{65} -4.22235e15 q^{67} +1.56951e15 q^{68} +6.05401e14 q^{70} +1.65823e15 q^{71} -1.26706e16 q^{73} -5.28602e15 q^{74} -1.21291e14 q^{76} -2.97565e14 q^{77} -1.63691e16 q^{79} +4.98175e14 q^{80} -4.08732e15 q^{82} +2.45355e16 q^{83} -7.18782e15 q^{85} -2.09724e15 q^{86} +1.90166e15 q^{88} +4.52816e15 q^{89} +9.83307e15 q^{91} -1.67240e15 q^{92} -1.30272e16 q^{94} +5.55471e14 q^{95} +3.59936e16 q^{97} -3.85458e16 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 442 q^{2} + 298148 q^{4} + 1171875 q^{5} + 4962644 q^{7} + 108831912 q^{8} + 172656250 q^{10} - 1049849720 q^{11} - 3091742090 q^{13} - 27586028328 q^{14} + 22392797456 q^{16} + 15119940094 q^{17}+ \cdots + 12\!\cdots\!50 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 213.954 0.590971 0.295485 − 0.955347i \(-0.404519\pi\)
0.295485 + 0.955347i \(0.404519\pi\)
\(3\) 0 0
\(4\) −85295.6 −0.650754
\(5\) 390625. 0.447214
\(6\) 0 0
\(7\) 7.24373e6 0.474929 0.237465 − 0.971396i \(-0.423684\pi\)
0.237465 + 0.971396i \(0.423684\pi\)
\(8\) −4.62928e7 −0.975547
\(9\) 0 0
\(10\) 8.35759e7 0.264290
\(11\) −4.10790e7 −0.0577807 −0.0288903 − 0.999583i \(-0.509197\pi\)
−0.0288903 + 0.999583i \(0.509197\pi\)
\(12\) 0 0
\(13\) 1.35746e9 0.461539 0.230769 − 0.973008i \(-0.425876\pi\)
0.230769 + 0.973008i \(0.425876\pi\)
\(14\) 1.54983e9 0.280669
\(15\) 0 0
\(16\) 1.27533e9 0.0742338
\(17\) −1.84008e10 −0.639766 −0.319883 − 0.947457i \(-0.603644\pi\)
−0.319883 + 0.947457i \(0.603644\pi\)
\(18\) 0 0
\(19\) 1.42200e9 0.0192086 0.00960429 − 0.999954i \(-0.496943\pi\)
0.00960429 + 0.999954i \(0.496943\pi\)
\(20\) −3.33186e10 −0.291026
\(21\) 0 0
\(22\) −8.78904e9 −0.0341467
\(23\) 1.96071e10 0.0522068 0.0261034 − 0.999659i \(-0.491690\pi\)
0.0261034 + 0.999659i \(0.491690\pi\)
\(24\) 0 0
\(25\) 1.52588e11 0.200000
\(26\) 2.90434e11 0.272756
\(27\) 0 0
\(28\) −6.17858e11 −0.309062
\(29\) 3.22761e12 1.19811 0.599057 − 0.800707i \(-0.295542\pi\)
0.599057 + 0.800707i \(0.295542\pi\)
\(30\) 0 0
\(31\) −1.93640e12 −0.407776 −0.203888 − 0.978994i \(-0.565358\pi\)
−0.203888 + 0.978994i \(0.565358\pi\)
\(32\) 6.34055e12 1.01942
\(33\) 0 0
\(34\) −3.93694e12 −0.378083
\(35\) 2.82958e12 0.212395
\(36\) 0 0
\(37\) −2.47063e13 −1.15636 −0.578180 − 0.815909i \(-0.696237\pi\)
−0.578180 + 0.815909i \(0.696237\pi\)
\(38\) 3.04244e11 0.0113517
\(39\) 0 0
\(40\) −1.80831e13 −0.436278
\(41\) −1.91037e13 −0.373641 −0.186821 − 0.982394i \(-0.559818\pi\)
−0.186821 + 0.982394i \(0.559818\pi\)
\(42\) 0 0
\(43\) −9.80229e12 −0.127893 −0.0639463 − 0.997953i \(-0.520369\pi\)
−0.0639463 + 0.997953i \(0.520369\pi\)
\(44\) 3.50386e12 0.0376010
\(45\) 0 0
\(46\) 4.19503e12 0.0308527
\(47\) −6.08880e13 −0.372992 −0.186496 − 0.982456i \(-0.559713\pi\)
−0.186496 + 0.982456i \(0.559713\pi\)
\(48\) 0 0
\(49\) −1.80159e14 −0.774442
\(50\) 3.26468e13 0.118194
\(51\) 0 0
\(52\) −1.15785e14 −0.300348
\(53\) −3.72760e14 −0.822402 −0.411201 − 0.911545i \(-0.634891\pi\)
−0.411201 + 0.911545i \(0.634891\pi\)
\(54\) 0 0
\(55\) −1.60465e13 −0.0258403
\(56\) −3.35332e14 −0.463316
\(57\) 0 0
\(58\) 6.90561e14 0.708050
\(59\) −1.78795e15 −1.58530 −0.792652 − 0.609675i \(-0.791300\pi\)
−0.792652 + 0.609675i \(0.791300\pi\)
\(60\) 0 0
\(61\) 1.98485e15 1.32563 0.662817 − 0.748781i \(-0.269361\pi\)
0.662817 + 0.748781i \(0.269361\pi\)
\(62\) −4.14302e14 −0.240983
\(63\) 0 0
\(64\) 1.18943e15 0.528212
\(65\) 5.30258e14 0.206406
\(66\) 0 0
\(67\) −4.22235e15 −1.27033 −0.635167 − 0.772375i \(-0.719069\pi\)
−0.635167 + 0.772375i \(0.719069\pi\)
\(68\) 1.56951e15 0.416330
\(69\) 0 0
\(70\) 6.05401e14 0.125519
\(71\) 1.65823e15 0.304754 0.152377 − 0.988322i \(-0.451307\pi\)
0.152377 + 0.988322i \(0.451307\pi\)
\(72\) 0 0
\(73\) −1.26706e16 −1.83888 −0.919439 − 0.393233i \(-0.871357\pi\)
−0.919439 + 0.393233i \(0.871357\pi\)
\(74\) −5.28602e15 −0.683375
\(75\) 0 0
\(76\) −1.21291e14 −0.0125001
\(77\) −2.97565e14 −0.0274417
\(78\) 0 0
\(79\) −1.63691e16 −1.21393 −0.606965 − 0.794728i \(-0.707613\pi\)
−0.606965 + 0.794728i \(0.707613\pi\)
\(80\) 4.98175e14 0.0331984
\(81\) 0 0
\(82\) −4.08732e15 −0.220811
\(83\) 2.45355e16 1.19573 0.597863 − 0.801598i \(-0.296017\pi\)
0.597863 + 0.801598i \(0.296017\pi\)
\(84\) 0 0
\(85\) −7.18782e15 −0.286112
\(86\) −2.09724e15 −0.0755808
\(87\) 0 0
\(88\) 1.90166e15 0.0563678
\(89\) 4.52816e15 0.121929 0.0609645 − 0.998140i \(-0.480582\pi\)
0.0609645 + 0.998140i \(0.480582\pi\)
\(90\) 0 0
\(91\) 9.83307e15 0.219198
\(92\) −1.67240e15 −0.0339738
\(93\) 0 0
\(94\) −1.30272e16 −0.220427
\(95\) 5.55471e14 0.00859034
\(96\) 0 0
\(97\) 3.59936e16 0.466301 0.233150 − 0.972441i \(-0.425097\pi\)
0.233150 + 0.972441i \(0.425097\pi\)
\(98\) −3.85458e16 −0.457673
\(99\) 0 0
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 45.18.a.e.1.2 3
3.2 odd 2 15.18.a.b.1.2 ✓ 3
15.2 even 4 75.18.b.d.49.3 6
15.8 even 4 75.18.b.d.49.4 6
15.14 odd 2 75.18.a.e.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.18.a.b.1.2 ✓ 3 3.2 odd 2
45.18.a.e.1.2 3 1.1 even 1 trivial
75.18.a.e.1.2 3 15.14 odd 2
75.18.b.d.49.3 6 15.2 even 4
75.18.b.d.49.4 6 15.8 even 4