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Properties

Label 18.9.b.a.17.2
Level $18$
Weight $9$
Character 18.17
Analytic conductor $7.333$
Analytic rank $0$
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] = [18,9,Mod(17,18)] 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("18.17"); S:= CuspForms(chi, 9); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(18, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 9, names="a")
 
Level: \( N \) \(=\) \( 18 = 2 \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 18.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,-256,0,0,-7064] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.33281498110\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 17.2
Root \(1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 18.17
Dual form 18.9.b.a.17.1

$q$-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)
 
\(f(q)\) \(=\) \(q+11.3137i q^{2} -128.000 q^{4} +233.345i q^{5} -3532.00 q^{7} -1448.15i q^{8} -2640.00 q^{10} -20178.0i q^{11} -41824.0 q^{13} -39960.0i q^{14} +16384.0 q^{16} +94784.8i q^{17} -36304.0 q^{19} -29868.2i q^{20} +228288. q^{22} +413624. i q^{23} +336175. q^{25} -473185. i q^{26} +452096. q^{28} +269191. i q^{29} -471196. q^{31} +185364. i q^{32} -1.07237e6 q^{34} -824175. i q^{35} -3.00740e6 q^{37} -410733. i q^{38} +337920. q^{40} -1.71534e6i q^{41} +3.62372e6 q^{43} +2.58278e6i q^{44} -4.67962e6 q^{46} -6.01462e6i q^{47} +6.71022e6 q^{49} +3.80339e6i q^{50} +5.35347e6 q^{52} +1.02767e7i q^{53} +4.70844e6 q^{55} +5.11488e6i q^{56} -3.04555e6 q^{58} +2.68810e6i q^{59} -5.44063e6 q^{61} -5.33097e6i q^{62} -2.09715e6 q^{64} -9.75943e6i q^{65} -6.12158e6 q^{67} -1.21325e7i q^{68} +9.32448e6 q^{70} -2.11941e7i q^{71} -4.90312e7 q^{73} -3.40249e7i q^{74} +4.64691e6 q^{76} +7.12687e7i q^{77} +8.35776e6 q^{79} +3.82313e6i q^{80} +1.94068e7 q^{82} -5.13918e7i q^{83} -2.21176e7 q^{85} +4.09977e7i q^{86} -2.92209e7 q^{88} +1.07337e8i q^{89} +1.47722e8 q^{91} -5.29438e7i q^{92} +6.80477e7 q^{94} -8.47137e6i q^{95} +2.04313e7 q^{97} +7.59175e7i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 256 q^{4} - 7064 q^{7} - 5280 q^{10} - 83648 q^{13} + 32768 q^{16} - 72608 q^{19} + 456576 q^{22} + 672350 q^{25} + 904192 q^{28} - 942392 q^{31} - 2144736 q^{34} - 6014804 q^{37} + 675840 q^{40}+ \cdots + 40862656 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/18\mathbb{Z}\right)^\times\).

\(n\) \(11\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 11.3137i 0.707107i
\(3\) 0 0
\(4\) −128.000 −0.500000
\(5\) 233.345i 0.373352i 0.982421 + 0.186676i \(0.0597715\pi\)
−0.982421 + 0.186676i \(0.940228\pi\)
\(6\) 0 0
\(7\) −3532.00 −1.47105 −0.735527 − 0.677496i \(-0.763065\pi\)
−0.735527 + 0.677496i \(0.763065\pi\)
\(8\) − 1448.15i − 0.353553i
\(9\) 0 0
\(10\) −2640.00 −0.264000
\(11\) − 20178.0i − 1.37818i −0.724674 − 0.689092i \(-0.758009\pi\)
0.724674 − 0.689092i \(-0.241991\pi\)
\(12\) 0 0
\(13\) −41824.0 −1.46437 −0.732187 − 0.681103i \(-0.761500\pi\)
−0.732187 + 0.681103i \(0.761500\pi\)
\(14\) − 39960.0i − 1.04019i
\(15\) 0 0
\(16\) 16384.0 0.250000
\(17\) 94784.8i 1.13486i 0.823421 + 0.567431i \(0.192063\pi\)
−0.823421 + 0.567431i \(0.807937\pi\)
\(18\) 0 0
\(19\) −36304.0 −0.278574 −0.139287 − 0.990252i \(-0.544481\pi\)
−0.139287 + 0.990252i \(0.544481\pi\)
\(20\) − 29868.2i − 0.186676i
\(21\) 0 0
\(22\) 228288. 0.974524
\(23\) 413624.i 1.47807i 0.673669 + 0.739033i \(0.264717\pi\)
−0.673669 + 0.739033i \(0.735283\pi\)
\(24\) 0 0
\(25\) 336175. 0.860608
\(26\) − 473185.i − 1.03547i
\(27\) 0 0
\(28\) 452096. 0.735527
\(29\) 269191.i 0.380600i 0.981726 + 0.190300i \(0.0609461\pi\)
−0.981726 + 0.190300i \(0.939054\pi\)
\(30\) 0 0
\(31\) −471196. −0.510217 −0.255108 − 0.966912i \(-0.582111\pi\)
−0.255108 + 0.966912i \(0.582111\pi\)
\(32\) 185364.i 0.176777i
\(33\) 0 0
\(34\) −1.07237e6 −0.802469
\(35\) − 824175.i − 0.549221i
\(36\) 0 0
\(37\) −3.00740e6 −1.60467 −0.802333 − 0.596877i \(-0.796408\pi\)
−0.802333 + 0.596877i \(0.796408\pi\)
\(38\) − 410733.i − 0.196981i
\(39\) 0 0
\(40\) 337920. 0.132000
\(41\) − 1.71534e6i − 0.607036i −0.952826 − 0.303518i \(-0.901839\pi\)
0.952826 − 0.303518i \(-0.0981612\pi\)
\(42\) 0 0
\(43\) 3.62372e6 1.05994 0.529969 − 0.848017i \(-0.322203\pi\)
0.529969 + 0.848017i \(0.322203\pi\)
\(44\) 2.58278e6i 0.689092i
\(45\) 0 0
\(46\) −4.67962e6 −1.04515
\(47\) − 6.01462e6i − 1.23259i −0.787517 − 0.616293i \(-0.788634\pi\)
0.787517 − 0.616293i \(-0.211366\pi\)
\(48\) 0 0
\(49\) 6.71022e6 1.16400
\(50\) 3.80339e6i 0.608542i
\(51\) 0 0
\(52\) 5.35347e6 0.732187
\(53\) 1.02767e7i 1.30241i 0.758901 + 0.651206i \(0.225737\pi\)
−0.758901 + 0.651206i \(0.774263\pi\)
\(54\) 0 0
\(55\) 4.70844e6 0.514548
\(56\) 5.11488e6i 0.520096i
\(57\) 0 0
\(58\) −3.04555e6 −0.269125
\(59\) 2.68810e6i 0.221839i 0.993829 + 0.110919i \(0.0353796\pi\)
−0.993829 + 0.110919i \(0.964620\pi\)
\(60\) 0 0
\(61\) −5.44063e6 −0.392943 −0.196472 − 0.980510i \(-0.562948\pi\)
−0.196472 + 0.980510i \(0.562948\pi\)
\(62\) − 5.33097e6i − 0.360778i
\(63\) 0 0
\(64\) −2.09715e6 −0.125000
\(65\) − 9.75943e6i − 0.546728i
\(66\) 0 0
\(67\) −6.12158e6 −0.303783 −0.151892 − 0.988397i \(-0.548536\pi\)
−0.151892 + 0.988397i \(0.548536\pi\)
\(68\) − 1.21325e7i − 0.567431i
\(69\) 0 0
\(70\) 9.32448e6 0.388358
\(71\) − 2.11941e7i − 0.834029i −0.908900 − 0.417014i \(-0.863076\pi\)
0.908900 − 0.417014i \(-0.136924\pi\)
\(72\) 0 0
\(73\) −4.90312e7 −1.72656 −0.863278 − 0.504729i \(-0.831593\pi\)
−0.863278 + 0.504729i \(0.831593\pi\)
\(74\) − 3.40249e7i − 1.13467i
\(75\) 0 0
\(76\) 4.64691e6 0.139287
\(77\) 7.12687e7i 2.02738i
\(78\) 0 0
\(79\) 8.35776e6 0.214576 0.107288 − 0.994228i \(-0.465783\pi\)
0.107288 + 0.994228i \(0.465783\pi\)
\(80\) 3.82313e6i 0.0933381i
\(81\) 0 0
\(82\) 1.94068e7 0.429239
\(83\) − 5.13918e7i − 1.08288i −0.840738 − 0.541441i \(-0.817879\pi\)
0.840738 − 0.541441i \(-0.182121\pi\)
\(84\) 0 0
\(85\) −2.21176e7 −0.423704
\(86\) 4.09977e7i 0.749490i
\(87\) 0 0
\(88\) −2.92209e7 −0.487262
\(89\) 1.07337e8i 1.71076i 0.517997 + 0.855382i \(0.326678\pi\)
−0.517997 + 0.855382i \(0.673322\pi\)
\(90\) 0 0
\(91\) 1.47722e8 2.15417
\(92\) − 5.29438e7i − 0.739033i
\(93\) 0 0
\(94\) 6.80477e7 0.871569
\(95\) − 8.47137e6i − 0.104006i
\(96\) 0 0
\(97\) 2.04313e7 0.230786 0.115393 − 0.993320i \(-0.463187\pi\)
0.115393 + 0.993320i \(0.463187\pi\)
\(98\) 7.59175e7i 0.823072i
\(99\) 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 18.9.b.a.17.2 yes 2
3.2 odd 2 inner 18.9.b.a.17.1 ✓ 2
4.3 odd 2 144.9.e.d.17.2 2
5.2 odd 4 450.9.b.a.449.1 4
5.3 odd 4 450.9.b.a.449.4 4
5.4 even 2 450.9.d.b.251.1 2
9.2 odd 6 162.9.d.d.53.1 4
9.4 even 3 162.9.d.d.107.1 4
9.5 odd 6 162.9.d.d.107.2 4
9.7 even 3 162.9.d.d.53.2 4
12.11 even 2 144.9.e.d.17.1 2
15.2 even 4 450.9.b.a.449.3 4
15.8 even 4 450.9.b.a.449.2 4
15.14 odd 2 450.9.d.b.251.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
18.9.b.a.17.1 ✓ 2 3.2 odd 2 inner
18.9.b.a.17.2 yes 2 1.1 even 1 trivial
144.9.e.d.17.1 2 12.11 even 2
144.9.e.d.17.2 2 4.3 odd 2
162.9.d.d.53.1 4 9.2 odd 6
162.9.d.d.53.2 4 9.7 even 3
162.9.d.d.107.1 4 9.4 even 3
162.9.d.d.107.2 4 9.5 odd 6
450.9.b.a.449.1 4 5.2 odd 4
450.9.b.a.449.2 4 15.8 even 4
450.9.b.a.449.3 4 15.2 even 4
450.9.b.a.449.4 4 5.3 odd 4
450.9.d.b.251.1 2 5.4 even 2
450.9.d.b.251.2 2 15.14 odd 2