Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [18,3,Mod(5,18)] 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("18.5"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(18, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([5])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 18 = 2 \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 18.d (of order \(6\), degree \(2\), 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: \(0.490464475849\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 11.2
Root \(1.22474 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 18.11
Dual form 18.3.d.a.5.2

$q$-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+(1.22474 + 0.707107i) q^{2} +(-2.44949 - 1.73205i) q^{3} +(1.00000 + 1.73205i) q^{4} +(-4.50000 + 2.59808i) q^{5} +(-1.77526 - 3.85337i) q^{6} +(4.17423 - 7.22999i) q^{7} +2.82843i q^{8} +(3.00000 + 8.48528i) q^{9} -7.34847 q^{10} +(0.825765 + 0.476756i) q^{11} +(0.550510 - 5.97469i) q^{12} +(4.84847 + 8.39780i) q^{13} +(10.2247 - 5.90326i) q^{14} +(15.5227 + 1.43027i) q^{15} +(-2.00000 + 3.46410i) q^{16} -18.8776i q^{17} +(-2.32577 + 12.5136i) q^{18} -24.6969 q^{19} +(-9.00000 - 5.19615i) q^{20} +(-22.7474 + 10.4798i) q^{21} +(0.674235 + 1.16781i) q^{22} +(0.825765 - 0.476756i) q^{23} +(4.89898 - 6.92820i) q^{24} +(1.00000 - 1.73205i) q^{25} +13.7135i q^{26} +(7.34847 - 25.9808i) q^{27} +16.6969 q^{28} +(11.8485 + 6.84072i) q^{29} +(18.0000 + 12.7279i) q^{30} +(-1.52270 - 2.63740i) q^{31} +(-4.89898 + 2.82843i) q^{32} +(-1.19694 - 2.59808i) q^{33} +(13.3485 - 23.1202i) q^{34} +43.3799i q^{35} +(-11.6969 + 13.6814i) q^{36} +46.6969 q^{37} +(-30.2474 - 17.4634i) q^{38} +(2.66913 - 28.9681i) q^{39} +(-7.34847 - 12.7279i) q^{40} +(-9.45459 + 5.45861i) q^{41} +(-35.2702 - 3.24980i) q^{42} +(-22.5227 + 39.0105i) q^{43} +1.90702i q^{44} +(-35.5454 - 30.3895i) q^{45} +1.34847 q^{46} +(39.2196 + 22.6435i) q^{47} +(10.8990 - 5.02118i) q^{48} +(-10.3485 - 17.9241i) q^{49} +(2.44949 - 1.41421i) q^{50} +(-32.6969 + 46.2405i) q^{51} +(-9.69694 + 16.7956i) q^{52} -94.3879i q^{53} +(27.3712 - 26.6237i) q^{54} -4.95459 q^{55} +(20.4495 + 11.8065i) q^{56} +(60.4949 + 42.7764i) q^{57} +(9.67423 + 16.7563i) q^{58} +(-16.2650 + 9.39063i) q^{59} +(13.0454 + 28.3164i) q^{60} +(-6.54541 + 11.3370i) q^{61} -4.30686i q^{62} +(73.8712 + 13.7296i) q^{63} -8.00000 q^{64} +(-43.6362 - 25.1934i) q^{65} +(0.371173 - 4.02834i) q^{66} +(-37.5227 - 64.9912i) q^{67} +(32.6969 - 18.8776i) q^{68} +(-2.84847 - 0.262459i) q^{69} +(-30.6742 + 53.1293i) q^{70} +18.0204i q^{71} +(-24.0000 + 8.48528i) q^{72} -7.90918 q^{73} +(57.1918 + 33.0197i) q^{74} +(-5.44949 + 2.51059i) q^{75} +(-24.6969 - 42.7764i) q^{76} +(6.89388 - 3.98018i) q^{77} +(23.7526 - 33.5912i) q^{78} +(21.8712 - 37.8820i) q^{79} -20.7846i q^{80} +(-63.0000 + 50.9117i) q^{81} -15.4393 q^{82} +(-112.871 - 65.1662i) q^{83} +(-40.8990 - 28.9199i) q^{84} +(49.0454 + 84.9491i) q^{85} +(-55.1691 + 31.8519i) q^{86} +(-17.1742 - 37.2784i) q^{87} +(-1.34847 + 2.33562i) q^{88} +145.300i q^{89} +(-22.0454 - 62.3538i) q^{90} +80.9546 q^{91} +(1.65153 + 0.953512i) q^{92} +(-0.838264 + 9.09769i) q^{93} +(32.0227 + 55.4650i) q^{94} +(111.136 - 64.1645i) q^{95} +(16.8990 + 1.55708i) q^{96} +(54.9393 - 95.1576i) q^{97} -29.2699i q^{98} +(-1.56811 + 8.43712i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{4} - 18 q^{5} - 12 q^{6} + 2 q^{7} + 12 q^{9} + 18 q^{11} + 12 q^{12} - 10 q^{13} + 36 q^{14} + 18 q^{15} - 8 q^{16} - 24 q^{18} - 40 q^{19} - 36 q^{20} - 42 q^{21} - 12 q^{22} + 18 q^{23} + 4 q^{25}+ \cdots + 126 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(11\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.22474 + 0.707107i 0.612372 + 0.353553i
\(3\) −2.44949 − 1.73205i −0.816497 − 0.577350i
\(4\) 1.00000 + 1.73205i 0.250000 + 0.433013i
\(5\) −4.50000 + 2.59808i −0.900000 + 0.519615i −0.877200 − 0.480125i \(-0.840591\pi\)
−0.0227998 + 0.999740i \(0.507258\pi\)
\(6\) −1.77526 − 3.85337i −0.295876 − 0.642229i
\(7\) 4.17423 − 7.22999i 0.596319 − 1.03286i −0.397040 − 0.917801i \(-0.629963\pi\)
0.993359 − 0.115054i \(-0.0367041\pi\)
\(8\) 2.82843i 0.353553i
\(9\) 3.00000 + 8.48528i 0.333333 + 0.942809i
\(10\) −7.34847 −0.734847
\(11\) 0.825765 + 0.476756i 0.0750696 + 0.0433414i 0.537065 − 0.843541i \(-0.319533\pi\)
−0.461995 + 0.886882i \(0.652866\pi\)
\(12\) 0.550510 − 5.97469i 0.0458759 − 0.497891i
\(13\) 4.84847 + 8.39780i 0.372959 + 0.645984i 0.990019 − 0.140932i \(-0.0450098\pi\)
−0.617060 + 0.786916i \(0.711676\pi\)
\(14\) 10.2247 − 5.90326i 0.730339 − 0.421661i
\(15\) 15.5227 + 1.43027i 1.03485 + 0.0953512i
\(16\) −2.00000 + 3.46410i −0.125000 + 0.216506i
\(17\) − 18.8776i − 1.11045i −0.831701 − 0.555223i \(-0.812633\pi\)
0.831701 − 0.555223i \(-0.187367\pi\)
\(18\) −2.32577 + 12.5136i −0.129209 + 0.695201i
\(19\) −24.6969 −1.29984 −0.649919 − 0.760003i \(-0.725197\pi\)
−0.649919 + 0.760003i \(0.725197\pi\)
\(20\) −9.00000 − 5.19615i −0.450000 − 0.259808i
\(21\) −22.7474 + 10.4798i −1.08321 + 0.499038i
\(22\) 0.674235 + 1.16781i 0.0306470 + 0.0530822i
\(23\) 0.825765 − 0.476756i 0.0359028 − 0.0207285i −0.481941 − 0.876204i \(-0.660068\pi\)
0.517844 + 0.855475i \(0.326735\pi\)
\(24\) 4.89898 − 6.92820i 0.204124 − 0.288675i
\(25\) 1.00000 − 1.73205i 0.0400000 − 0.0692820i
\(26\) 13.7135i 0.527444i
\(27\) 7.34847 − 25.9808i 0.272166 − 0.962250i
\(28\) 16.6969 0.596319
\(29\) 11.8485 + 6.84072i 0.408568 + 0.235887i 0.690174 − 0.723643i \(-0.257534\pi\)
−0.281606 + 0.959530i \(0.590867\pi\)
\(30\) 18.0000 + 12.7279i 0.600000 + 0.424264i
\(31\) −1.52270 − 2.63740i −0.0491195 − 0.0850774i 0.840420 − 0.541935i \(-0.182308\pi\)
−0.889540 + 0.456858i \(0.848975\pi\)
\(32\) −4.89898 + 2.82843i −0.153093 + 0.0883883i
\(33\) −1.19694 − 2.59808i −0.0362709 − 0.0787296i
\(34\) 13.3485 − 23.1202i 0.392602 − 0.680007i
\(35\) 43.3799i 1.23943i
\(36\) −11.6969 + 13.6814i −0.324915 + 0.380040i
\(37\) 46.6969 1.26208 0.631040 − 0.775751i \(-0.282628\pi\)
0.631040 + 0.775751i \(0.282628\pi\)
\(38\) −30.2474 − 17.4634i −0.795985 − 0.459562i
\(39\) 2.66913 − 28.9681i 0.0684393 − 0.742772i
\(40\) −7.34847 − 12.7279i −0.183712 − 0.318198i
\(41\) −9.45459 + 5.45861i −0.230600 + 0.133137i −0.610849 − 0.791747i \(-0.709172\pi\)
0.380249 + 0.924884i \(0.375838\pi\)
\(42\) −35.2702 − 3.24980i −0.839766 − 0.0773763i
\(43\) −22.5227 + 39.0105i −0.523784 + 0.907220i 0.475833 + 0.879536i \(0.342147\pi\)
−0.999617 + 0.0276845i \(0.991187\pi\)
\(44\) 1.90702i 0.0433414i
\(45\) −35.5454 − 30.3895i −0.789898 − 0.675323i
\(46\) 1.34847 0.0293145
\(47\) 39.2196 + 22.6435i 0.834460 + 0.481776i 0.855377 − 0.518005i \(-0.173325\pi\)
−0.0209170 + 0.999781i \(0.506659\pi\)
\(48\) 10.8990 − 5.02118i 0.227062 − 0.104608i
\(49\) −10.3485 − 17.9241i −0.211193 − 0.365797i
\(50\) 2.44949 − 1.41421i 0.0489898 − 0.0282843i
\(51\) −32.6969 + 46.2405i −0.641116 + 0.906676i
\(52\) −9.69694 + 16.7956i −0.186480 + 0.322992i
\(53\) − 94.3879i − 1.78090i −0.455077 − 0.890452i \(-0.650388\pi\)
0.455077 − 0.890452i \(-0.349612\pi\)
\(54\) 27.3712 − 26.6237i 0.506874 − 0.493031i
\(55\) −4.95459 −0.0900835
\(56\) 20.4495 + 11.8065i 0.365169 + 0.210831i
\(57\) 60.4949 + 42.7764i 1.06131 + 0.750462i
\(58\) 9.67423 + 16.7563i 0.166797 + 0.288901i
\(59\) −16.2650 + 9.39063i −0.275679 + 0.159163i −0.631466 − 0.775404i \(-0.717546\pi\)
0.355787 + 0.934567i \(0.384213\pi\)
\(60\) 13.0454 + 28.3164i 0.217423 + 0.471940i
\(61\) −6.54541 + 11.3370i −0.107302 + 0.185852i −0.914676 − 0.404187i \(-0.867554\pi\)
0.807375 + 0.590039i \(0.200888\pi\)
\(62\) − 4.30686i − 0.0694654i
\(63\) 73.8712 + 13.7296i 1.17256 + 0.217930i
\(64\) −8.00000 −0.125000
\(65\) −43.6362 − 25.1934i −0.671327 − 0.387591i
\(66\) 0.371173 − 4.02834i 0.00562383 − 0.0610355i
\(67\) −37.5227 − 64.9912i −0.560040 − 0.970018i −0.997492 − 0.0707765i \(-0.977452\pi\)
0.437452 − 0.899242i \(-0.355881\pi\)
\(68\) 32.6969 − 18.8776i 0.480837 − 0.277612i
\(69\) −2.84847 − 0.262459i −0.0412822 − 0.00380375i
\(70\) −30.6742 + 53.1293i −0.438203 + 0.758990i
\(71\) 18.0204i 0.253808i 0.991915 + 0.126904i \(0.0405041\pi\)
−0.991915 + 0.126904i \(0.959496\pi\)
\(72\) −24.0000 + 8.48528i −0.333333 + 0.117851i
\(73\) −7.90918 −0.108345 −0.0541725 − 0.998532i \(-0.517252\pi\)
−0.0541725 + 0.998532i \(0.517252\pi\)
\(74\) 57.1918 + 33.0197i 0.772863 + 0.446212i
\(75\) −5.44949 + 2.51059i −0.0726599 + 0.0334745i
\(76\) −24.6969 − 42.7764i −0.324960 − 0.562847i
\(77\) 6.89388 − 3.98018i 0.0895309 − 0.0516907i
\(78\) 23.7526 − 33.5912i 0.304520 − 0.430656i
\(79\) 21.8712 − 37.8820i 0.276850 − 0.479519i −0.693750 − 0.720216i \(-0.744043\pi\)
0.970600 + 0.240697i \(0.0773761\pi\)
\(80\) − 20.7846i − 0.259808i
\(81\) −63.0000 + 50.9117i −0.777778 + 0.628539i
\(82\) −15.4393 −0.188284
\(83\) −112.871 − 65.1662i −1.35989 − 0.785135i −0.370284 − 0.928918i \(-0.620740\pi\)
−0.989609 + 0.143783i \(0.954073\pi\)
\(84\) −40.8990 − 28.9199i −0.486893 − 0.344285i
\(85\) 49.0454 + 84.9491i 0.577005 + 0.999402i
\(86\) −55.1691 + 31.8519i −0.641502 + 0.370371i
\(87\) −17.1742 − 37.2784i −0.197405 − 0.428488i
\(88\) −1.34847 + 2.33562i −0.0153235 + 0.0265411i
\(89\) 145.300i 1.63258i 0.577642 + 0.816290i \(0.303973\pi\)
−0.577642 + 0.816290i \(0.696027\pi\)
\(90\) −22.0454 − 62.3538i −0.244949 − 0.692820i
\(91\) 80.9546 0.889611
\(92\) 1.65153 + 0.953512i 0.0179514 + 0.0103643i
\(93\) −0.838264 + 9.09769i −0.00901359 + 0.0978246i
\(94\) 32.0227 + 55.4650i 0.340667 + 0.590053i
\(95\) 111.136 − 64.1645i 1.16985 − 0.675416i
\(96\) 16.8990 + 1.55708i 0.176031 + 0.0162196i
\(97\) 54.9393 − 95.1576i 0.566384 − 0.981007i −0.430535 − 0.902574i \(-0.641675\pi\)
0.996919 − 0.0784327i \(-0.0249916\pi\)
\(98\) − 29.2699i − 0.298672i
\(99\) −1.56811 + 8.43712i −0.0158395 + 0.0852234i
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 18.3.d.a.11.2 yes 4
3.2 odd 2 54.3.d.a.35.1 4
4.3 odd 2 144.3.q.c.65.2 4
5.2 odd 4 450.3.k.a.299.1 8
5.3 odd 4 450.3.k.a.299.4 8
5.4 even 2 450.3.i.b.101.1 4
8.3 odd 2 576.3.q.e.65.1 4
8.5 even 2 576.3.q.f.65.2 4
9.2 odd 6 162.3.b.a.161.4 4
9.4 even 3 54.3.d.a.17.1 4
9.5 odd 6 inner 18.3.d.a.5.2 ✓ 4
9.7 even 3 162.3.b.a.161.1 4
12.11 even 2 432.3.q.d.305.1 4
15.2 even 4 1350.3.k.a.899.4 8
15.8 even 4 1350.3.k.a.899.1 8
15.14 odd 2 1350.3.i.b.251.2 4
24.5 odd 2 1728.3.q.d.1601.2 4
24.11 even 2 1728.3.q.c.1601.1 4
36.7 odd 6 1296.3.e.g.161.2 4
36.11 even 6 1296.3.e.g.161.4 4
36.23 even 6 144.3.q.c.113.2 4
36.31 odd 6 432.3.q.d.17.1 4
45.4 even 6 1350.3.i.b.1151.2 4
45.13 odd 12 1350.3.k.a.449.4 8
45.14 odd 6 450.3.i.b.401.1 4
45.22 odd 12 1350.3.k.a.449.1 8
45.23 even 12 450.3.k.a.149.1 8
45.32 even 12 450.3.k.a.149.4 8
72.5 odd 6 576.3.q.f.257.2 4
72.13 even 6 1728.3.q.d.449.2 4
72.59 even 6 576.3.q.e.257.1 4
72.67 odd 6 1728.3.q.c.449.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
18.3.d.a.5.2 ✓ 4 9.5 odd 6 inner
18.3.d.a.11.2 yes 4 1.1 even 1 trivial
54.3.d.a.17.1 4 9.4 even 3
54.3.d.a.35.1 4 3.2 odd 2
144.3.q.c.65.2 4 4.3 odd 2
144.3.q.c.113.2 4 36.23 even 6
162.3.b.a.161.1 4 9.7 even 3
162.3.b.a.161.4 4 9.2 odd 6
432.3.q.d.17.1 4 36.31 odd 6
432.3.q.d.305.1 4 12.11 even 2
450.3.i.b.101.1 4 5.4 even 2
450.3.i.b.401.1 4 45.14 odd 6
450.3.k.a.149.1 8 45.23 even 12
450.3.k.a.149.4 8 45.32 even 12
450.3.k.a.299.1 8 5.2 odd 4
450.3.k.a.299.4 8 5.3 odd 4
576.3.q.e.65.1 4 8.3 odd 2
576.3.q.e.257.1 4 72.59 even 6
576.3.q.f.65.2 4 8.5 even 2
576.3.q.f.257.2 4 72.5 odd 6
1296.3.e.g.161.2 4 36.7 odd 6
1296.3.e.g.161.4 4 36.11 even 6
1350.3.i.b.251.2 4 15.14 odd 2
1350.3.i.b.1151.2 4 45.4 even 6
1350.3.k.a.449.1 8 45.22 odd 12
1350.3.k.a.449.4 8 45.13 odd 12
1350.3.k.a.899.1 8 15.8 even 4
1350.3.k.a.899.4 8 15.2 even 4
1728.3.q.c.449.1 4 72.67 odd 6
1728.3.q.c.1601.1 4 24.11 even 2
1728.3.q.d.449.2 4 72.13 even 6
1728.3.q.d.1601.2 4 24.5 odd 2