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Properties

Label 45.18.a.d.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 = [3,253] 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: \( x^{3} - x^{2} - 182396x + 3921120 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{2}\cdot 3\cdot 5 \)
Twist minimal: no (minimal twist has level 15)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(21.5502\) 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)
 
\(f(q)\) \(=\) \(q+105.550 q^{2} -119931. q^{4} +390625. q^{5} -2.39385e7 q^{7} -2.64934e7 q^{8} +4.12305e7 q^{10} +5.09412e8 q^{11} +4.71682e9 q^{13} -2.52671e9 q^{14} +1.29232e10 q^{16} +4.44993e10 q^{17} -4.25101e10 q^{19} -4.68481e10 q^{20} +5.37686e10 q^{22} -4.70013e11 q^{23} +1.52588e11 q^{25} +4.97862e11 q^{26} +2.87097e12 q^{28} -3.15724e12 q^{29} +3.46790e12 q^{31} +4.83660e12 q^{32} +4.69691e12 q^{34} -9.35098e12 q^{35} +3.18740e13 q^{37} -4.48695e12 q^{38} -1.03490e13 q^{40} -8.13537e13 q^{41} -1.24496e14 q^{43} -6.10944e13 q^{44} -4.96099e13 q^{46} +7.05130e13 q^{47} +3.40421e14 q^{49} +1.61057e13 q^{50} -5.65694e14 q^{52} -2.51778e14 q^{53} +1.98989e14 q^{55} +6.34213e14 q^{56} -3.33247e14 q^{58} +6.39615e14 q^{59} -4.96148e13 q^{61} +3.66037e14 q^{62} -1.18337e15 q^{64} +1.84251e15 q^{65} -2.75976e15 q^{67} -5.33686e15 q^{68} -9.86997e14 q^{70} -2.50612e15 q^{71} +8.85474e14 q^{73} +3.36430e15 q^{74} +5.09829e15 q^{76} -1.21946e16 q^{77} -6.85544e13 q^{79} +5.04814e15 q^{80} -8.58690e15 q^{82} -3.38359e16 q^{83} +1.73826e16 q^{85} -1.31405e16 q^{86} -1.34961e16 q^{88} +3.37871e16 q^{89} -1.12914e17 q^{91} +5.63692e16 q^{92} +7.44266e15 q^{94} -1.66055e16 q^{95} +6.48330e16 q^{97} +3.59315e16 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 253 q^{2} - 7087 q^{4} + 1171875 q^{5} - 4332484 q^{7} + 16188513 q^{8} + 98828125 q^{10} - 943563680 q^{11} + 4257013150 q^{13} - 1847483988 q^{14} - 21943871359 q^{16} - 3063169514 q^{17} - 78122492996 q^{19}+ \cdots + 896231490630245 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 105.550 0.291544 0.145772 − 0.989318i \(-0.453433\pi\)
0.145772 + 0.989318i \(0.453433\pi\)
\(3\) 0 0
\(4\) −119931. −0.915002
\(5\) 390625. 0.447214
\(6\) 0 0
\(7\) −2.39385e7 −1.56951 −0.784754 − 0.619807i \(-0.787211\pi\)
−0.784754 + 0.619807i \(0.787211\pi\)
\(8\) −2.64934e7 −0.558307
\(9\) 0 0
\(10\) 4.12305e7 0.130382
\(11\) 5.09412e8 0.716526 0.358263 − 0.933621i \(-0.383369\pi\)
0.358263 + 0.933621i \(0.383369\pi\)
\(12\) 0 0
\(13\) 4.71682e9 1.60373 0.801865 − 0.597506i \(-0.203841\pi\)
0.801865 + 0.597506i \(0.203841\pi\)
\(14\) −2.52671e9 −0.457580
\(15\) 0 0
\(16\) 1.29232e10 0.752231
\(17\) 4.44993e10 1.54717 0.773584 − 0.633693i \(-0.218462\pi\)
0.773584 + 0.633693i \(0.218462\pi\)
\(18\) 0 0
\(19\) −4.25101e10 −0.574231 −0.287116 − 0.957896i \(-0.592696\pi\)
−0.287116 + 0.957896i \(0.592696\pi\)
\(20\) −4.68481e10 −0.409201
\(21\) 0 0
\(22\) 5.37686e10 0.208899
\(23\) −4.70013e11 −1.25148 −0.625739 − 0.780033i \(-0.715202\pi\)
−0.625739 + 0.780033i \(0.715202\pi\)
\(24\) 0 0
\(25\) 1.52588e11 0.200000
\(26\) 4.97862e11 0.467558
\(27\) 0 0
\(28\) 2.87097e12 1.43610
\(29\) −3.15724e12 −1.17199 −0.585996 − 0.810314i \(-0.699296\pi\)
−0.585996 + 0.810314i \(0.699296\pi\)
\(30\) 0 0
\(31\) 3.46790e12 0.730285 0.365142 − 0.930952i \(-0.381020\pi\)
0.365142 + 0.930952i \(0.381020\pi\)
\(32\) 4.83660e12 0.777616
\(33\) 0 0
\(34\) 4.69691e12 0.451068
\(35\) −9.35098e12 −0.701905
\(36\) 0 0
\(37\) 3.18740e13 1.49184 0.745919 − 0.666037i \(-0.232011\pi\)
0.745919 + 0.666037i \(0.232011\pi\)
\(38\) −4.48695e12 −0.167414
\(39\) 0 0
\(40\) −1.03490e13 −0.249683
\(41\) −8.13537e13 −1.59116 −0.795581 − 0.605847i \(-0.792834\pi\)
−0.795581 + 0.605847i \(0.792834\pi\)
\(42\) 0 0
\(43\) −1.24496e14 −1.62432 −0.812161 − 0.583433i \(-0.801709\pi\)
−0.812161 + 0.583433i \(0.801709\pi\)
\(44\) −6.10944e13 −0.655623
\(45\) 0 0
\(46\) −4.96099e13 −0.364861
\(47\) 7.05130e13 0.431954 0.215977 − 0.976398i \(-0.430706\pi\)
0.215977 + 0.976398i \(0.430706\pi\)
\(48\) 0 0
\(49\) 3.40421e14 1.46336
\(50\) 1.61057e13 0.0583088
\(51\) 0 0
\(52\) −5.65694e14 −1.46742
\(53\) −2.51778e14 −0.555485 −0.277743 − 0.960656i \(-0.589586\pi\)
−0.277743 + 0.960656i \(0.589586\pi\)
\(54\) 0 0
\(55\) 1.98989e14 0.320440
\(56\) 6.34213e14 0.876268
\(57\) 0 0
\(58\) −3.33247e14 −0.341687
\(59\) 6.39615e14 0.567122 0.283561 − 0.958954i \(-0.408484\pi\)
0.283561 + 0.958954i \(0.408484\pi\)
\(60\) 0 0
\(61\) −4.96148e13 −0.0331366 −0.0165683 − 0.999863i \(-0.505274\pi\)
−0.0165683 + 0.999863i \(0.505274\pi\)
\(62\) 3.66037e14 0.212910
\(63\) 0 0
\(64\) −1.18337e15 −0.525522
\(65\) 1.84251e15 0.717210
\(66\) 0 0
\(67\) −2.75976e15 −0.830300 −0.415150 − 0.909753i \(-0.636271\pi\)
−0.415150 + 0.909753i \(0.636271\pi\)
\(68\) −5.33686e15 −1.41566
\(69\) 0 0
\(70\) −9.86997e14 −0.204636
\(71\) −2.50612e15 −0.460582 −0.230291 − 0.973122i \(-0.573968\pi\)
−0.230291 + 0.973122i \(0.573968\pi\)
\(72\) 0 0
\(73\) 8.85474e14 0.128508 0.0642542 − 0.997934i \(-0.479533\pi\)
0.0642542 + 0.997934i \(0.479533\pi\)
\(74\) 3.36430e15 0.434936
\(75\) 0 0
\(76\) 5.09829e15 0.525423
\(77\) −1.21946e16 −1.12459
\(78\) 0 0
\(79\) −6.85544e13 −0.00508400 −0.00254200 − 0.999997i \(-0.500809\pi\)
−0.00254200 + 0.999997i \(0.500809\pi\)
\(80\) 5.04814e15 0.336408
\(81\) 0 0
\(82\) −8.58690e15 −0.463894
\(83\) −3.38359e16 −1.64897 −0.824487 − 0.565881i \(-0.808536\pi\)
−0.824487 + 0.565881i \(0.808536\pi\)
\(84\) 0 0
\(85\) 1.73826e16 0.691915
\(86\) −1.31405e16 −0.473561
\(87\) 0 0
\(88\) −1.34961e16 −0.400041
\(89\) 3.37871e16 0.909778 0.454889 − 0.890548i \(-0.349679\pi\)
0.454889 + 0.890548i \(0.349679\pi\)
\(90\) 0 0
\(91\) −1.12914e17 −2.51707
\(92\) 5.63692e16 1.14510
\(93\) 0 0
\(94\) 7.44266e15 0.125934
\(95\) −1.66055e16 −0.256804
\(96\) 0 0
\(97\) 6.48330e16 0.839917 0.419958 − 0.907543i \(-0.362045\pi\)
0.419958 + 0.907543i \(0.362045\pi\)
\(98\) 3.59315e16 0.426632
\(99\) 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.d.1.2 3
3.2 odd 2 15.18.a.c.1.2 ✓ 3
15.2 even 4 75.18.b.e.49.3 6
15.8 even 4 75.18.b.e.49.4 6
15.14 odd 2 75.18.a.d.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.18.a.c.1.2 ✓ 3 3.2 odd 2
45.18.a.d.1.2 3 1.1 even 1 trivial
75.18.a.d.1.2 3 15.14 odd 2
75.18.b.e.49.3 6 15.2 even 4
75.18.b.e.49.4 6 15.8 even 4