Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [18,7,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, 7); 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, 7, names="a")
 
Level: \( N \) \(=\) \( 18 = 2 \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 18.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.14097350516\)
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 18.7.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)
 
\(f(q)\) \(=\) \(q+5.65685i q^{2} -32.0000 q^{4} +173.948i q^{5} -484.000 q^{7} -181.019i q^{8} -984.000 q^{10} +1340.67i q^{11} +3368.00 q^{13} -2737.92i q^{14} +1024.00 q^{16} +12.7279i q^{17} +5744.00 q^{19} -5566.34i q^{20} -7584.00 q^{22} -3377.14i q^{23} -14633.0 q^{25} +19052.3i q^{26} +15488.0 q^{28} +29354.8i q^{29} -39796.0 q^{31} +5792.62i q^{32} -72.0000 q^{34} -84191.0i q^{35} +52526.0 q^{37} +32493.0i q^{38} +31488.0 q^{40} -37042.5i q^{41} +3800.00 q^{43} -42901.6i q^{44} +19104.0 q^{46} +76791.8i q^{47} +116607. q^{49} -82776.7i q^{50} -107776. q^{52} +238738. i q^{53} -233208. q^{55} +87613.4i q^{56} -166056. q^{58} -249841. i q^{59} +13250.0 q^{61} -225120. i q^{62} -32768.0 q^{64} +585858. i q^{65} +168968. q^{67} -407.294i q^{68} +476256. q^{70} -531467. i q^{71} +236144. q^{73} +297132. i q^{74} -183808. q^{76} -648886. i q^{77} -35116.0 q^{79} +178123. i q^{80} +209544. q^{82} -10980.0i q^{83} -2214.00 q^{85} +21496.0i q^{86} +242688. q^{88} -129328. i q^{89} -1.63011e6 q^{91} +108069. i q^{92} -434400. q^{94} +999159. i q^{95} -321424. q^{97} +659629. i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 64 q^{4} - 968 q^{7} - 1968 q^{10} + 6736 q^{13} + 2048 q^{16} + 11488 q^{19} - 15168 q^{22} - 29266 q^{25} + 30976 q^{28} - 79592 q^{31} - 144 q^{34} + 105052 q^{37} + 62976 q^{40} + 7600 q^{43}+ \cdots - 642848 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.65685i 0.707107i
\(3\) 0 0
\(4\) −32.0000 −0.500000
\(5\) 173.948i 1.39159i 0.718242 + 0.695793i \(0.244947\pi\)
−0.718242 + 0.695793i \(0.755053\pi\)
\(6\) 0 0
\(7\) −484.000 −1.41108 −0.705539 − 0.708671i \(-0.749295\pi\)
−0.705539 + 0.708671i \(0.749295\pi\)
\(8\) − 181.019i − 0.353553i
\(9\) 0 0
\(10\) −984.000 −0.984000
\(11\) 1340.67i 1.00727i 0.863917 + 0.503634i \(0.168004\pi\)
−0.863917 + 0.503634i \(0.831996\pi\)
\(12\) 0 0
\(13\) 3368.00 1.53300 0.766500 − 0.642245i \(-0.221997\pi\)
0.766500 + 0.642245i \(0.221997\pi\)
\(14\) − 2737.92i − 0.997783i
\(15\) 0 0
\(16\) 1024.00 0.250000
\(17\) 12.7279i 0.00259066i 0.999999 + 0.00129533i \(0.000412317\pi\)
−0.999999 + 0.00129533i \(0.999588\pi\)
\(18\) 0 0
\(19\) 5744.00 0.837440 0.418720 − 0.908115i \(-0.362479\pi\)
0.418720 + 0.908115i \(0.362479\pi\)
\(20\) − 5566.34i − 0.695793i
\(21\) 0 0
\(22\) −7584.00 −0.712246
\(23\) − 3377.14i − 0.277566i −0.990323 − 0.138783i \(-0.955681\pi\)
0.990323 − 0.138783i \(-0.0443190\pi\)
\(24\) 0 0
\(25\) −14633.0 −0.936512
\(26\) 19052.3i 1.08399i
\(27\) 0 0
\(28\) 15488.0 0.705539
\(29\) 29354.8i 1.20361i 0.798643 + 0.601805i \(0.205551\pi\)
−0.798643 + 0.601805i \(0.794449\pi\)
\(30\) 0 0
\(31\) −39796.0 −1.33584 −0.667920 − 0.744233i \(-0.732815\pi\)
−0.667920 + 0.744233i \(0.732815\pi\)
\(32\) 5792.62i 0.176777i
\(33\) 0 0
\(34\) −72.0000 −0.00183187
\(35\) − 84191.0i − 1.96364i
\(36\) 0 0
\(37\) 52526.0 1.03698 0.518489 − 0.855085i \(-0.326495\pi\)
0.518489 + 0.855085i \(0.326495\pi\)
\(38\) 32493.0i 0.592159i
\(39\) 0 0
\(40\) 31488.0 0.492000
\(41\) − 37042.5i − 0.537463i −0.963215 − 0.268732i \(-0.913396\pi\)
0.963215 − 0.268732i \(-0.0866045\pi\)
\(42\) 0 0
\(43\) 3800.00 0.0477945 0.0238973 − 0.999714i \(-0.492393\pi\)
0.0238973 + 0.999714i \(0.492393\pi\)
\(44\) − 42901.6i − 0.503634i
\(45\) 0 0
\(46\) 19104.0 0.196269
\(47\) 76791.8i 0.739641i 0.929103 + 0.369821i \(0.120581\pi\)
−0.929103 + 0.369821i \(0.879419\pi\)
\(48\) 0 0
\(49\) 116607. 0.991143
\(50\) − 82776.7i − 0.662214i
\(51\) 0 0
\(52\) −107776. −0.766500
\(53\) 238738.i 1.60359i 0.597599 + 0.801795i \(0.296121\pi\)
−0.597599 + 0.801795i \(0.703879\pi\)
\(54\) 0 0
\(55\) −233208. −1.40170
\(56\) 87613.4i 0.498892i
\(57\) 0 0
\(58\) −166056. −0.851080
\(59\) − 249841.i − 1.21649i −0.793751 − 0.608243i \(-0.791875\pi\)
0.793751 − 0.608243i \(-0.208125\pi\)
\(60\) 0 0
\(61\) 13250.0 0.0583749 0.0291875 − 0.999574i \(-0.490708\pi\)
0.0291875 + 0.999574i \(0.490708\pi\)
\(62\) − 225120.i − 0.944581i
\(63\) 0 0
\(64\) −32768.0 −0.125000
\(65\) 585858.i 2.13330i
\(66\) 0 0
\(67\) 168968. 0.561798 0.280899 − 0.959737i \(-0.409367\pi\)
0.280899 + 0.959737i \(0.409367\pi\)
\(68\) − 407.294i − 0.00129533i
\(69\) 0 0
\(70\) 476256. 1.38850
\(71\) − 531467.i − 1.48491i −0.669894 − 0.742457i \(-0.733660\pi\)
0.669894 − 0.742457i \(-0.266340\pi\)
\(72\) 0 0
\(73\) 236144. 0.607027 0.303514 − 0.952827i \(-0.401840\pi\)
0.303514 + 0.952827i \(0.401840\pi\)
\(74\) 297132.i 0.733254i
\(75\) 0 0
\(76\) −183808. −0.418720
\(77\) − 648886.i − 1.42134i
\(78\) 0 0
\(79\) −35116.0 −0.0712236 −0.0356118 − 0.999366i \(-0.511338\pi\)
−0.0356118 + 0.999366i \(0.511338\pi\)
\(80\) 178123.i 0.347897i
\(81\) 0 0
\(82\) 209544. 0.380044
\(83\) − 10980.0i − 0.0192029i −0.999954 − 0.00960144i \(-0.996944\pi\)
0.999954 − 0.00960144i \(-0.00305628\pi\)
\(84\) 0 0
\(85\) −2214.00 −0.00360513
\(86\) 21496.0i 0.0337958i
\(87\) 0 0
\(88\) 242688. 0.356123
\(89\) − 129328.i − 0.183453i −0.995784 − 0.0917263i \(-0.970762\pi\)
0.995784 − 0.0917263i \(-0.0292385\pi\)
\(90\) 0 0
\(91\) −1.63011e6 −2.16318
\(92\) 108069.i 0.138783i
\(93\) 0 0
\(94\) −434400. −0.523005
\(95\) 999159.i 1.16537i
\(96\) 0 0
\(97\) −321424. −0.352179 −0.176089 − 0.984374i \(-0.556345\pi\)
−0.176089 + 0.984374i \(0.556345\pi\)
\(98\) 659629.i 0.700844i
\(99\) 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 18.7.b.a.17.2 yes 2
3.2 odd 2 inner 18.7.b.a.17.1 ✓ 2
4.3 odd 2 144.7.e.d.17.2 2
5.2 odd 4 450.7.b.a.449.1 4
5.3 odd 4 450.7.b.a.449.4 4
5.4 even 2 450.7.d.a.251.1 2
8.3 odd 2 576.7.e.k.449.1 2
8.5 even 2 576.7.e.b.449.1 2
9.2 odd 6 162.7.d.d.53.1 4
9.4 even 3 162.7.d.d.107.1 4
9.5 odd 6 162.7.d.d.107.2 4
9.7 even 3 162.7.d.d.53.2 4
12.11 even 2 144.7.e.d.17.1 2
15.2 even 4 450.7.b.a.449.3 4
15.8 even 4 450.7.b.a.449.2 4
15.14 odd 2 450.7.d.a.251.2 2
24.5 odd 2 576.7.e.b.449.2 2
24.11 even 2 576.7.e.k.449.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
18.7.b.a.17.1 ✓ 2 3.2 odd 2 inner
18.7.b.a.17.2 yes 2 1.1 even 1 trivial
144.7.e.d.17.1 2 12.11 even 2
144.7.e.d.17.2 2 4.3 odd 2
162.7.d.d.53.1 4 9.2 odd 6
162.7.d.d.53.2 4 9.7 even 3
162.7.d.d.107.1 4 9.4 even 3
162.7.d.d.107.2 4 9.5 odd 6
450.7.b.a.449.1 4 5.2 odd 4
450.7.b.a.449.2 4 15.8 even 4
450.7.b.a.449.3 4 15.2 even 4
450.7.b.a.449.4 4 5.3 odd 4
450.7.d.a.251.1 2 5.4 even 2
450.7.d.a.251.2 2 15.14 odd 2
576.7.e.b.449.1 2 8.5 even 2
576.7.e.b.449.2 2 24.5 odd 2
576.7.e.k.449.1 2 8.3 odd 2
576.7.e.k.449.2 2 24.11 even 2