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Properties

Label 45.18.a.e.1.3
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 1.3
Root \(-188.067\) 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)
 
\(f(q)\) \(=\) \(q+670.613 q^{2} +318650. q^{4} +390625. q^{5} -2.70804e7 q^{7} +1.25792e8 q^{8} +2.61958e8 q^{10} -1.01644e9 q^{11} -1.04834e9 q^{13} -1.81605e10 q^{14} +4.25918e10 q^{16} -2.00099e10 q^{17} +1.64417e10 q^{19} +1.24473e11 q^{20} -6.81636e11 q^{22} -5.92721e11 q^{23} +1.52588e11 q^{25} -7.03032e11 q^{26} -8.62917e12 q^{28} -4.83212e11 q^{29} -5.20802e12 q^{31} +1.20748e13 q^{32} -1.34189e13 q^{34} -1.05783e13 q^{35} +3.82367e13 q^{37} +1.10260e13 q^{38} +4.91375e13 q^{40} +6.56402e13 q^{41} -5.83804e13 q^{43} -3.23888e14 q^{44} -3.97486e14 q^{46} -7.43658e12 q^{47} +5.00719e14 q^{49} +1.02327e14 q^{50} -3.34054e14 q^{52} +3.01466e14 q^{53} -3.97046e14 q^{55} -3.40650e15 q^{56} -3.24048e14 q^{58} +3.44389e14 q^{59} -2.43237e15 q^{61} -3.49257e15 q^{62} +2.51491e15 q^{64} -4.09508e14 q^{65} +1.19632e15 q^{67} -6.37614e15 q^{68} -7.09394e15 q^{70} -5.42254e15 q^{71} -5.84732e15 q^{73} +2.56420e16 q^{74} +5.23914e15 q^{76} +2.75256e16 q^{77} +2.03455e15 q^{79} +1.66374e16 q^{80} +4.40192e16 q^{82} +1.18343e16 q^{83} -7.81636e15 q^{85} -3.91507e16 q^{86} -1.27860e17 q^{88} +2.43170e16 q^{89} +2.83895e16 q^{91} -1.88870e17 q^{92} -4.98707e15 q^{94} +6.42253e15 q^{95} -1.06105e17 q^{97} +3.35788e17 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 442 q^{2} + 298148 q^{4} + 1171875 q^{5} + 4962644 q^{7} + 108831912 q^{8} + 172656250 q^{10} - 1049849720 q^{11} - 3091742090 q^{13} - 27586028328 q^{14} + 22392797456 q^{16} + 15119940094 q^{17}+ \cdots + 12\!\cdots\!50 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 670.613 1.85232 0.926162 − 0.377126i \(-0.123088\pi\)
0.926162 + 0.377126i \(0.123088\pi\)
\(3\) 0 0
\(4\) 318650. 2.43110
\(5\) 390625. 0.447214
\(6\) 0 0
\(7\) −2.70804e7 −1.77551 −0.887753 − 0.460320i \(-0.847735\pi\)
−0.887753 + 0.460320i \(0.847735\pi\)
\(8\) 1.25792e8 2.65087
\(9\) 0 0
\(10\) 2.61958e8 0.828385
\(11\) −1.01644e9 −1.42969 −0.714847 − 0.699281i \(-0.753504\pi\)
−0.714847 + 0.699281i \(0.753504\pi\)
\(12\) 0 0
\(13\) −1.04834e9 −0.356438 −0.178219 − 0.983991i \(-0.557034\pi\)
−0.178219 + 0.983991i \(0.557034\pi\)
\(14\) −1.81605e10 −3.28881
\(15\) 0 0
\(16\) 4.25918e10 2.47917
\(17\) −2.00099e10 −0.695711 −0.347855 − 0.937548i \(-0.613090\pi\)
−0.347855 + 0.937548i \(0.613090\pi\)
\(18\) 0 0
\(19\) 1.64417e10 0.222096 0.111048 − 0.993815i \(-0.464579\pi\)
0.111048 + 0.993815i \(0.464579\pi\)
\(20\) 1.24473e11 1.08722
\(21\) 0 0
\(22\) −6.81636e11 −2.64826
\(23\) −5.92721e11 −1.57821 −0.789103 − 0.614261i \(-0.789454\pi\)
−0.789103 + 0.614261i \(0.789454\pi\)
\(24\) 0 0
\(25\) 1.52588e11 0.200000
\(26\) −7.03032e11 −0.660239
\(27\) 0 0
\(28\) −8.62917e12 −4.31644
\(29\) −4.83212e11 −0.179372 −0.0896860 − 0.995970i \(-0.528586\pi\)
−0.0896860 + 0.995970i \(0.528586\pi\)
\(30\) 0 0
\(31\) −5.20802e12 −1.09673 −0.548363 − 0.836240i \(-0.684749\pi\)
−0.548363 + 0.836240i \(0.684749\pi\)
\(32\) 1.20748e13 1.94135
\(33\) 0 0
\(34\) −1.34189e13 −1.28868
\(35\) −1.05783e13 −0.794030
\(36\) 0 0
\(37\) 3.82367e13 1.78964 0.894820 − 0.446427i \(-0.147304\pi\)
0.894820 + 0.446427i \(0.147304\pi\)
\(38\) 1.10260e13 0.411393
\(39\) 0 0
\(40\) 4.91375e13 1.18551
\(41\) 6.56402e13 1.28383 0.641915 − 0.766776i \(-0.278140\pi\)
0.641915 + 0.766776i \(0.278140\pi\)
\(42\) 0 0
\(43\) −5.83804e13 −0.761703 −0.380851 − 0.924636i \(-0.624369\pi\)
−0.380851 + 0.924636i \(0.624369\pi\)
\(44\) −3.23888e14 −3.47573
\(45\) 0 0
\(46\) −3.97486e14 −2.92335
\(47\) −7.43658e12 −0.0455556 −0.0227778 − 0.999741i \(-0.507251\pi\)
−0.0227778 + 0.999741i \(0.507251\pi\)
\(48\) 0 0
\(49\) 5.00719e14 2.15242
\(50\) 1.02327e14 0.370465
\(51\) 0 0
\(52\) −3.34054e14 −0.866539
\(53\) 3.01466e14 0.665109 0.332555 − 0.943084i \(-0.392089\pi\)
0.332555 + 0.943084i \(0.392089\pi\)
\(54\) 0 0
\(55\) −3.97046e14 −0.639378
\(56\) −3.40650e15 −4.70664
\(57\) 0 0
\(58\) −3.24048e14 −0.332255
\(59\) 3.44389e14 0.305357 0.152678 − 0.988276i \(-0.451210\pi\)
0.152678 + 0.988276i \(0.451210\pi\)
\(60\) 0 0
\(61\) −2.43237e15 −1.62452 −0.812261 − 0.583294i \(-0.801763\pi\)
−0.812261 + 0.583294i \(0.801763\pi\)
\(62\) −3.49257e15 −2.03149
\(63\) 0 0
\(64\) 2.51491e15 1.11684
\(65\) −4.09508e14 −0.159404
\(66\) 0 0
\(67\) 1.19632e15 0.359924 0.179962 − 0.983674i \(-0.442403\pi\)
0.179962 + 0.983674i \(0.442403\pi\)
\(68\) −6.37614e15 −1.69135
\(69\) 0 0
\(70\) −7.09394e15 −1.47080
\(71\) −5.42254e15 −0.996567 −0.498283 − 0.867014i \(-0.666036\pi\)
−0.498283 + 0.867014i \(0.666036\pi\)
\(72\) 0 0
\(73\) −5.84732e15 −0.848619 −0.424309 − 0.905517i \(-0.639483\pi\)
−0.424309 + 0.905517i \(0.639483\pi\)
\(74\) 2.56420e16 3.31499
\(75\) 0 0
\(76\) 5.23914e15 0.539938
\(77\) 2.75256e16 2.53843
\(78\) 0 0
\(79\) 2.03455e15 0.150882 0.0754411 − 0.997150i \(-0.475964\pi\)
0.0754411 + 0.997150i \(0.475964\pi\)
\(80\) 1.66374e16 1.10872
\(81\) 0 0
\(82\) 4.40192e16 2.37807
\(83\) 1.18343e16 0.576740 0.288370 − 0.957519i \(-0.406887\pi\)
0.288370 + 0.957519i \(0.406887\pi\)
\(84\) 0 0
\(85\) −7.81636e15 −0.311131
\(86\) −3.91507e16 −1.41092
\(87\) 0 0
\(88\) −1.27860e17 −3.78993
\(89\) 2.43170e16 0.654780 0.327390 − 0.944889i \(-0.393831\pi\)
0.327390 + 0.944889i \(0.393831\pi\)
\(90\) 0 0
\(91\) 2.83895e16 0.632858
\(92\) −1.88870e17 −3.83678
\(93\) 0 0
\(94\) −4.98707e15 −0.0843837
\(95\) 6.42253e15 0.0993243
\(96\) 0 0
\(97\) −1.06105e17 −1.37459 −0.687296 − 0.726377i \(-0.741203\pi\)
−0.687296 + 0.726377i \(0.741203\pi\)
\(98\) 3.35788e17 3.98698
\(99\) 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.1.3 3
3.2 odd 2 15.18.a.b.1.1 ✓ 3
15.2 even 4 75.18.b.d.49.1 6
15.8 even 4 75.18.b.d.49.6 6
15.14 odd 2 75.18.a.e.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.18.a.b.1.1 ✓ 3 3.2 odd 2
45.18.a.e.1.3 3 1.1 even 1 trivial
75.18.a.e.1.3 3 15.14 odd 2
75.18.b.d.49.1 6 15.2 even 4
75.18.b.d.49.6 6 15.8 even 4