Properties

Label 9.18.a.d.1.1
Level $9$
Weight $18$
Character 9.1
Self dual yes
Analytic conductor $16.490$
Analytic rank $1$
Dimension $2$
Inner twists $2$

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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] = [9,18,Mod(1,9)] 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("9.1"); S:= CuspForms(chi, 18); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 18, names="a")
 
Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0] 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: \(16.4899878610\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{910}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 910 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-30.1662\) of defining polynomial
Character \(\chi\) \(=\) 9.1

$q$-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)
 
\(f(q)\) \(=\) \(q-361.994 q^{2} -32.0000 q^{4} +101358. q^{5} -196420. q^{7} +4.74589e7 q^{8} -3.66912e7 q^{10} +1.10678e9 q^{11} -1.30863e9 q^{13} +7.11030e7 q^{14} -1.71757e10 q^{16} -3.49921e10 q^{17} -1.11323e11 q^{19} -3.24347e6 q^{20} -4.00647e11 q^{22} +3.45621e11 q^{23} -7.52666e11 q^{25} +4.73718e11 q^{26} +6.28544e6 q^{28} -3.38248e12 q^{29} +4.07372e12 q^{31} -3.03700e9 q^{32} +1.26670e13 q^{34} -1.99088e10 q^{35} +1.71554e13 q^{37} +4.02981e13 q^{38} +4.81036e12 q^{40} -4.84420e13 q^{41} -6.03415e13 q^{43} -3.54168e10 q^{44} -1.25113e14 q^{46} -1.95023e14 q^{47} -2.32592e14 q^{49} +2.72461e14 q^{50} +4.18762e10 q^{52} +3.02369e14 q^{53} +1.12181e14 q^{55} -9.32188e12 q^{56} +1.22444e15 q^{58} -2.96910e14 q^{59} +7.02560e14 q^{61} -1.47466e15 q^{62} +2.25235e15 q^{64} -1.32641e14 q^{65} -2.18203e15 q^{67} +1.11975e12 q^{68} +7.20689e12 q^{70} +9.52807e15 q^{71} -1.09475e16 q^{73} -6.21018e15 q^{74} +3.56232e12 q^{76} -2.17393e14 q^{77} -1.60620e16 q^{79} -1.74090e15 q^{80} +1.75357e16 q^{82} -3.08532e16 q^{83} -3.54675e15 q^{85} +2.18433e16 q^{86} +5.25264e16 q^{88} +3.54252e16 q^{89} +2.57041e14 q^{91} -1.10599e13 q^{92} +7.05973e16 q^{94} -1.12835e16 q^{95} -5.46564e16 q^{97} +8.41970e16 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 64 q^{4} - 392840 q^{7} - 73382400 q^{10} - 2617263740 q^{13} - 34351347712 q^{16} - 222645082352 q^{19} - 801293875200 q^{22} - 1505331834250 q^{25} + 12570880 q^{28} + 8147437309816 q^{31} + 25333923363840 q^{34}+ \cdots - 10\!\cdots\!40 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −361.994 −0.999878 −0.499939 − 0.866061i \(-0.666644\pi\)
−0.499939 + 0.866061i \(0.666644\pi\)
\(3\) 0 0
\(4\) −32.0000 −0.000244141 0
\(5\) 101358. 0.116042 0.0580210 − 0.998315i \(-0.481521\pi\)
0.0580210 + 0.998315i \(0.481521\pi\)
\(6\) 0 0
\(7\) −196420. −0.0128781 −0.00643906 − 0.999979i \(-0.502050\pi\)
−0.00643906 + 0.999979i \(0.502050\pi\)
\(8\) 4.74589e7 1.00012
\(9\) 0 0
\(10\) −3.66912e7 −0.116028
\(11\) 1.10678e9 1.55676 0.778381 − 0.627793i \(-0.216041\pi\)
0.778381 + 0.627793i \(0.216041\pi\)
\(12\) 0 0
\(13\) −1.30863e9 −0.444937 −0.222469 − 0.974940i \(-0.571412\pi\)
−0.222469 + 0.974940i \(0.571412\pi\)
\(14\) 7.11030e7 0.0128765
\(15\) 0 0
\(16\) −1.71757e10 −0.999756
\(17\) −3.49921e10 −1.21662 −0.608310 − 0.793700i \(-0.708152\pi\)
−0.608310 + 0.793700i \(0.708152\pi\)
\(18\) 0 0
\(19\) −1.11323e11 −1.50376 −0.751878 − 0.659302i \(-0.770852\pi\)
−0.751878 + 0.659302i \(0.770852\pi\)
\(20\) −3.24347e6 −2.83305e−5 0
\(21\) 0 0
\(22\) −4.00647e11 −1.55657
\(23\) 3.45621e11 0.920266 0.460133 − 0.887850i \(-0.347802\pi\)
0.460133 + 0.887850i \(0.347802\pi\)
\(24\) 0 0
\(25\) −7.52666e11 −0.986534
\(26\) 4.73718e11 0.444883
\(27\) 0 0
\(28\) 6.28544e6 3.14407e−6 0
\(29\) −3.38248e12 −1.25560 −0.627801 − 0.778374i \(-0.716045\pi\)
−0.627801 + 0.778374i \(0.716045\pi\)
\(30\) 0 0
\(31\) 4.07372e12 0.857860 0.428930 − 0.903338i \(-0.358891\pi\)
0.428930 + 0.903338i \(0.358891\pi\)
\(32\) −3.03700e9 −0.000488281 0
\(33\) 0 0
\(34\) 1.26670e13 1.21647
\(35\) −1.99088e10 −0.00149440
\(36\) 0 0
\(37\) 1.71554e13 0.802948 0.401474 − 0.915870i \(-0.368498\pi\)
0.401474 + 0.915870i \(0.368498\pi\)
\(38\) 4.02981e13 1.50357
\(39\) 0 0
\(40\) 4.81036e12 0.116056
\(41\) −4.84420e13 −0.947456 −0.473728 − 0.880671i \(-0.657092\pi\)
−0.473728 + 0.880671i \(0.657092\pi\)
\(42\) 0 0
\(43\) −6.03415e13 −0.787289 −0.393644 − 0.919263i \(-0.628786\pi\)
−0.393644 + 0.919263i \(0.628786\pi\)
\(44\) −3.54168e10 −0.000380069 0
\(45\) 0 0
\(46\) −1.25113e14 −0.920153
\(47\) −1.95023e14 −1.19469 −0.597343 − 0.801986i \(-0.703777\pi\)
−0.597343 + 0.801986i \(0.703777\pi\)
\(48\) 0 0
\(49\) −2.32592e14 −0.999834
\(50\) 2.72461e14 0.986414
\(51\) 0 0
\(52\) 4.18762e10 0.000108627 0
\(53\) 3.02369e14 0.667102 0.333551 − 0.942732i \(-0.391753\pi\)
0.333551 + 0.942732i \(0.391753\pi\)
\(54\) 0 0
\(55\) 1.12181e14 0.180650
\(56\) −9.32188e12 −0.0128797
\(57\) 0 0
\(58\) 1.22444e15 1.25545
\(59\) −2.96910e14 −0.263259 −0.131629 − 0.991299i \(-0.542021\pi\)
−0.131629 + 0.991299i \(0.542021\pi\)
\(60\) 0 0
\(61\) 7.02560e14 0.469224 0.234612 − 0.972089i \(-0.424618\pi\)
0.234612 + 0.972089i \(0.424618\pi\)
\(62\) −1.47466e15 −0.857756
\(63\) 0 0
\(64\) 2.25235e15 1.00024
\(65\) −1.32641e14 −0.0516314
\(66\) 0 0
\(67\) −2.18203e15 −0.656485 −0.328242 − 0.944594i \(-0.606456\pi\)
−0.328242 + 0.944594i \(0.606456\pi\)
\(68\) 1.11975e12 0.000297026 0
\(69\) 0 0
\(70\) 7.20689e12 0.00149422
\(71\) 9.52807e15 1.75109 0.875546 − 0.483134i \(-0.160502\pi\)
0.875546 + 0.483134i \(0.160502\pi\)
\(72\) 0 0
\(73\) −1.09475e16 −1.58881 −0.794403 − 0.607391i \(-0.792216\pi\)
−0.794403 + 0.607391i \(0.792216\pi\)
\(74\) −6.21018e15 −0.802850
\(75\) 0 0
\(76\) 3.56232e12 0.000367128 0
\(77\) −2.17393e14 −0.0200482
\(78\) 0 0
\(79\) −1.60620e16 −1.19116 −0.595579 − 0.803297i \(-0.703077\pi\)
−0.595579 + 0.803297i \(0.703077\pi\)
\(80\) −1.74090e15 −0.116014
\(81\) 0 0
\(82\) 1.75357e16 0.947340
\(83\) −3.08532e16 −1.50361 −0.751807 − 0.659383i \(-0.770817\pi\)
−0.751807 + 0.659383i \(0.770817\pi\)
\(84\) 0 0
\(85\) −3.54675e15 −0.141179
\(86\) 2.18433e16 0.787193
\(87\) 0 0
\(88\) 5.25264e16 1.55695
\(89\) 3.54252e16 0.953888 0.476944 − 0.878934i \(-0.341745\pi\)
0.476944 + 0.878934i \(0.341745\pi\)
\(90\) 0 0
\(91\) 2.57041e14 0.00572996
\(92\) −1.10599e13 −0.000224674 0
\(93\) 0 0
\(94\) 7.05973e16 1.19454
\(95\) −1.12835e16 −0.174499
\(96\) 0 0
\(97\) −5.46564e16 −0.708079 −0.354039 − 0.935231i \(-0.615192\pi\)
−0.354039 + 0.935231i \(0.615192\pi\)
\(98\) 8.41970e16 0.999712
\(99\) 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 9.18.a.d.1.1 ✓ 2
3.2 odd 2 inner 9.18.a.d.1.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9.18.a.d.1.1 ✓ 2 1.1 even 1 trivial
9.18.a.d.1.2 yes 2 3.2 odd 2 inner