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Properties

Label 45.18.a.e.1.1
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 1.1
Root \(-9.44141\) 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)
 
\(f(q)\) \(=\) \(q-442.567 q^{2} +64793.8 q^{4} +390625. q^{5} +2.47993e7 q^{7} +2.93326e7 q^{8} -1.72878e8 q^{10} +7.66684e6 q^{11} -3.40086e9 q^{13} -1.09754e10 q^{14} -2.14743e10 q^{16} +5.35306e10 q^{17} -1.29760e11 q^{19} +2.53101e10 q^{20} -3.39309e9 q^{22} -2.11422e11 q^{23} +1.52588e11 q^{25} +1.50511e12 q^{26} +1.60684e12 q^{28} -4.13975e12 q^{29} -5.31377e12 q^{31} +5.65914e12 q^{32} -2.36909e13 q^{34} +9.68724e12 q^{35} +1.85634e13 q^{37} +5.74274e13 q^{38} +1.14580e13 q^{40} +5.57404e13 q^{41} -7.31648e13 q^{43} +4.96764e11 q^{44} +9.35684e13 q^{46} +1.22507e14 q^{47} +3.82376e14 q^{49} -6.75304e13 q^{50} -2.20355e14 q^{52} -2.74251e14 q^{53} +2.99486e12 q^{55} +7.27428e14 q^{56} +1.83212e15 q^{58} -6.31977e14 q^{59} -1.14361e15 q^{61} +2.35170e15 q^{62} +3.10129e14 q^{64} -1.32846e15 q^{65} +6.08292e14 q^{67} +3.46845e15 q^{68} -4.28726e15 q^{70} -1.25518e15 q^{71} +7.71376e15 q^{73} -8.21553e15 q^{74} -8.40763e15 q^{76} +1.90133e14 q^{77} -2.63076e16 q^{79} -8.38839e15 q^{80} -2.46689e16 q^{82} -1.45466e16 q^{83} +2.09104e16 q^{85} +3.23803e16 q^{86} +2.24888e14 q^{88} -2.61450e16 q^{89} -8.43391e16 q^{91} -1.36988e16 q^{92} -5.42175e16 q^{94} -5.06874e16 q^{95} -9.64972e15 q^{97} -1.69227e17 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 442 q^{2} + 298148 q^{4} + 1171875 q^{5} + 4962644 q^{7} + 108831912 q^{8} + 172656250 q^{10} - 1049849720 q^{11} - 3091742090 q^{13} - 27586028328 q^{14} + 22392797456 q^{16} + 15119940094 q^{17}+ \cdots + 12\!\cdots\!50 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −442.567 −1.22243 −0.611215 − 0.791464i \(-0.709319\pi\)
−0.611215 + 0.791464i \(0.709319\pi\)
\(3\) 0 0
\(4\) 64793.8 0.494337
\(5\) 390625. 0.447214
\(6\) 0 0
\(7\) 2.47993e7 1.62595 0.812974 − 0.582300i \(-0.197847\pi\)
0.812974 + 0.582300i \(0.197847\pi\)
\(8\) 2.93326e7 0.618138
\(9\) 0 0
\(10\) −1.72878e8 −0.546688
\(11\) 7.66684e6 0.0107840 0.00539199 − 0.999985i \(-0.498284\pi\)
0.00539199 + 0.999985i \(0.498284\pi\)
\(12\) 0 0
\(13\) −3.40086e9 −1.15630 −0.578150 − 0.815931i \(-0.696225\pi\)
−0.578150 + 0.815931i \(0.696225\pi\)
\(14\) −1.09754e10 −1.98761
\(15\) 0 0
\(16\) −2.14743e10 −1.24997
\(17\) 5.35306e10 1.86117 0.930586 − 0.366073i \(-0.119298\pi\)
0.930586 + 0.366073i \(0.119298\pi\)
\(18\) 0 0
\(19\) −1.29760e11 −1.75281 −0.876404 − 0.481576i \(-0.840065\pi\)
−0.876404 + 0.481576i \(0.840065\pi\)
\(20\) 2.53101e10 0.221074
\(21\) 0 0
\(22\) −3.39309e9 −0.0131827
\(23\) −2.11422e11 −0.562941 −0.281471 − 0.959570i \(-0.590822\pi\)
−0.281471 + 0.959570i \(0.590822\pi\)
\(24\) 0 0
\(25\) 1.52588e11 0.200000
\(26\) 1.50511e12 1.41350
\(27\) 0 0
\(28\) 1.60684e12 0.803767
\(29\) −4.13975e12 −1.53671 −0.768354 − 0.640026i \(-0.778924\pi\)
−0.768354 + 0.640026i \(0.778924\pi\)
\(30\) 0 0
\(31\) −5.31377e12 −1.11900 −0.559498 − 0.828832i \(-0.689006\pi\)
−0.559498 + 0.828832i \(0.689006\pi\)
\(32\) 5.65914e12 0.909862
\(33\) 0 0
\(34\) −2.36909e13 −2.27515
\(35\) 9.68724e12 0.727146
\(36\) 0 0
\(37\) 1.85634e13 0.868844 0.434422 − 0.900709i \(-0.356953\pi\)
0.434422 + 0.900709i \(0.356953\pi\)
\(38\) 5.74274e13 2.14269
\(39\) 0 0
\(40\) 1.14580e13 0.276440
\(41\) 5.57404e13 1.09020 0.545101 − 0.838370i \(-0.316491\pi\)
0.545101 + 0.838370i \(0.316491\pi\)
\(42\) 0 0
\(43\) −7.31648e13 −0.954597 −0.477299 − 0.878741i \(-0.658384\pi\)
−0.477299 + 0.878741i \(0.658384\pi\)
\(44\) 4.96764e11 0.00533092
\(45\) 0 0
\(46\) 9.35684e13 0.688157
\(47\) 1.22507e14 0.750461 0.375231 − 0.926931i \(-0.377564\pi\)
0.375231 + 0.926931i \(0.377564\pi\)
\(48\) 0 0
\(49\) 3.82376e14 1.64371
\(50\) −6.75304e13 −0.244486
\(51\) 0 0
\(52\) −2.20355e14 −0.571602
\(53\) −2.74251e14 −0.605068 −0.302534 − 0.953139i \(-0.597832\pi\)
−0.302534 + 0.953139i \(0.597832\pi\)
\(54\) 0 0
\(55\) 2.99486e12 0.00482274
\(56\) 7.27428e14 1.00506
\(57\) 0 0
\(58\) 1.83212e15 1.87852
\(59\) −6.31977e14 −0.560350 −0.280175 − 0.959949i \(-0.590392\pi\)
−0.280175 + 0.959949i \(0.590392\pi\)
\(60\) 0 0
\(61\) −1.14361e15 −0.763791 −0.381895 − 0.924206i \(-0.624729\pi\)
−0.381895 + 0.924206i \(0.624729\pi\)
\(62\) 2.35170e15 1.36790
\(63\) 0 0
\(64\) 3.10129e14 0.137725
\(65\) −1.32846e15 −0.517113
\(66\) 0 0
\(67\) 6.08292e14 0.183011 0.0915053 − 0.995805i \(-0.470832\pi\)
0.0915053 + 0.995805i \(0.470832\pi\)
\(68\) 3.46845e15 0.920047
\(69\) 0 0
\(70\) −4.28726e15 −0.888886
\(71\) −1.25518e15 −0.230681 −0.115340 − 0.993326i \(-0.536796\pi\)
−0.115340 + 0.993326i \(0.536796\pi\)
\(72\) 0 0
\(73\) 7.71376e15 1.11949 0.559747 − 0.828664i \(-0.310898\pi\)
0.559747 + 0.828664i \(0.310898\pi\)
\(74\) −8.21553e15 −1.06210
\(75\) 0 0
\(76\) −8.40763e15 −0.866479
\(77\) 1.90133e14 0.0175342
\(78\) 0 0
\(79\) −2.63076e16 −1.95097 −0.975486 − 0.220061i \(-0.929374\pi\)
−0.975486 + 0.220061i \(0.929374\pi\)
\(80\) −8.38839e15 −0.559003
\(81\) 0 0
\(82\) −2.46689e16 −1.33270
\(83\) −1.45466e16 −0.708922 −0.354461 − 0.935071i \(-0.615336\pi\)
−0.354461 + 0.935071i \(0.615336\pi\)
\(84\) 0 0
\(85\) 2.09104e16 0.832342
\(86\) 3.23803e16 1.16693
\(87\) 0 0
\(88\) 2.24888e14 0.00666598
\(89\) −2.61450e16 −0.704001 −0.352000 − 0.936000i \(-0.614498\pi\)
−0.352000 + 0.936000i \(0.614498\pi\)
\(90\) 0 0
\(91\) −8.43391e16 −1.88008
\(92\) −1.36988e16 −0.278283
\(93\) 0 0
\(94\) −5.42175e16 −0.917387
\(95\) −5.06874e16 −0.783880
\(96\) 0 0
\(97\) −9.64972e15 −0.125013 −0.0625064 − 0.998045i \(-0.519909\pi\)
−0.0625064 + 0.998045i \(0.519909\pi\)
\(98\) −1.69227e17 −2.00932
\(99\) 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.1.1 3
3.2 odd 2 15.18.a.b.1.3 ✓ 3
15.2 even 4 75.18.b.d.49.5 6
15.8 even 4 75.18.b.d.49.2 6
15.14 odd 2 75.18.a.e.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.18.a.b.1.3 ✓ 3 3.2 odd 2
45.18.a.e.1.1 3 1.1 even 1 trivial
75.18.a.e.1.1 3 15.14 odd 2
75.18.b.d.49.2 6 15.8 even 4
75.18.b.d.49.5 6 15.2 even 4