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(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, 7); 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, 7, names="a")
 
Level: \( N \) \(=\) \( 18 = 2 \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 7 \)
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: \(4.14097350516\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 370x^{10} + 51793x^{8} + 3491832x^{6} + 117603792x^{4} + 1832032512x^{2} + 10453017600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 11.2
Root \(-7.20150i\) of defining polynomial
Character \(\chi\) \(=\) 18.11
Dual form 18.7.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+(-4.89898 - 2.82843i) q^{2} +(-11.1983 - 24.5683i) q^{3} +(16.0000 + 27.7128i) q^{4} +(-39.5602 + 22.8401i) q^{5} +(-14.6295 + 152.033i) q^{6} +(-245.097 + 424.521i) q^{7} -181.019i q^{8} +(-478.198 + 550.243i) q^{9} +258.406 q^{10} +(-873.336 - 504.221i) q^{11} +(501.683 - 703.427i) q^{12} +(466.801 + 808.523i) q^{13} +(2401.45 - 1386.48i) q^{14} +(1004.15 + 716.156i) q^{15} +(-512.000 + 886.810i) q^{16} -8090.59i q^{17} +(3899.01 - 1343.08i) q^{18} -7727.36 q^{19} +(-1265.93 - 730.884i) q^{20} +(13174.4 + 1267.72i) q^{21} +(2852.30 + 4940.34i) q^{22} +(-11848.0 + 6840.45i) q^{23} +(-4447.33 + 2027.10i) q^{24} +(-6769.16 + 11724.5i) q^{25} -5281.25i q^{26} +(18873.5 + 5586.73i) q^{27} -15686.2 q^{28} +(-1964.70 - 1134.32i) q^{29} +(-2893.70 - 6348.59i) q^{30} +(-17062.6 - 29553.2i) q^{31} +(5016.55 - 2896.31i) q^{32} +(-2607.99 + 27102.7i) q^{33} +(-22883.6 + 39635.6i) q^{34} -22392.2i q^{35} +(-22900.0 - 4448.33i) q^{36} +92058.0 q^{37} +(37856.2 + 21856.3i) q^{38} +(14636.6 - 20522.5i) q^{39} +(4134.50 + 7161.17i) q^{40} +(-31021.6 + 17910.3i) q^{41} +(-60955.5 - 43473.4i) q^{42} +(34570.9 - 59878.5i) q^{43} -32270.1i q^{44} +(6350.01 - 32689.8i) q^{45} +77390.9 q^{46} +(13211.1 + 7627.46i) q^{47} +(27520.9 + 2648.22i) q^{48} +(-61321.0 - 106211. i) q^{49} +(66323.9 - 38292.1i) q^{50} +(-198772. + 90600.5i) q^{51} +(-14937.6 + 25872.7i) q^{52} +236591. i q^{53} +(-76659.2 - 80751.6i) q^{54} +46065.9 q^{55} +(76846.5 + 44367.4i) q^{56} +(86533.0 + 189848. i) q^{57} +(6416.70 + 11114.0i) q^{58} +(-221890. + 128108. i) q^{59} +(-3780.35 + 39286.2i) q^{60} +(-19919.6 + 34501.8i) q^{61} +193041. i q^{62} +(-116385. - 337868. i) q^{63} -32768.0 q^{64} +(-36933.5 - 21323.6i) q^{65} +(89434.6 - 125399. i) q^{66} +(-160204. - 277482. i) q^{67} +(224213. - 129449. i) q^{68} +(300735. + 214484. i) q^{69} +(-63334.7 + 109699. i) q^{70} -404593. i q^{71} +(99604.7 + 86563.1i) q^{72} +393719. q^{73} +(-450990. - 260379. i) q^{74} +(363854. + 35012.2i) q^{75} +(-123638. - 214147. i) q^{76} +(428105. - 247167. i) q^{77} +(-129751. + 59140.8i) q^{78} +(-449184. + 778010. i) q^{79} -46776.5i q^{80} +(-74094.1 - 526251. i) q^{81} +202632. q^{82} +(154916. + 89441.0i) q^{83} +(175658. + 385383. i) q^{84} +(184790. + 320066. i) q^{85} +(-338724. + 195562. i) q^{86} +(-5867.06 + 60971.8i) q^{87} +(-91273.8 + 158091. i) q^{88} -826458. i q^{89} +(-123569. + 142186. i) q^{90} -457647. q^{91} +(-379136. - 218894. i) q^{92} +(-535000. + 750142. i) q^{93} +(-43147.4 - 74733.5i) q^{94} +(305696. - 176494. i) q^{95} +(-127334. - 90814.4i) q^{96} +(-317981. + 550760. i) q^{97} +693768. i q^{98} +(695072. - 239430. i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 42 q^{3} + 192 q^{4} + 432 q^{5} - 144 q^{6} + 240 q^{7} + 2190 q^{9} + 378 q^{11} + 384 q^{12} + 1680 q^{13} - 4752 q^{14} - 10872 q^{15} - 6144 q^{16} - 2976 q^{18} - 2820 q^{19} + 13824 q^{20}+ \cdots + 4398804 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.89898 − 2.82843i −0.612372 − 0.353553i
\(3\) −11.1983 − 24.5683i −0.414750 − 0.909935i
\(4\) 16.0000 + 27.7128i 0.250000 + 0.433013i
\(5\) −39.5602 + 22.8401i −0.316482 + 0.182721i −0.649823 − 0.760085i \(-0.725157\pi\)
0.333341 + 0.942806i \(0.391824\pi\)
\(6\) −14.6295 + 152.033i −0.0677291 + 0.703856i
\(7\) −245.097 + 424.521i −0.714570 + 1.23767i 0.248556 + 0.968618i \(0.420044\pi\)
−0.963125 + 0.269053i \(0.913289\pi\)
\(8\) − 181.019i − 0.353553i
\(9\) −478.198 + 550.243i −0.655965 + 0.754792i
\(10\) 258.406 0.258406
\(11\) −873.336 − 504.221i −0.656151 − 0.378829i 0.134658 − 0.990892i \(-0.457006\pi\)
−0.790809 + 0.612063i \(0.790340\pi\)
\(12\) 501.683 − 703.427i 0.290326 − 0.407076i
\(13\) 466.801 + 808.523i 0.212472 + 0.368012i 0.952488 − 0.304577i \(-0.0985152\pi\)
−0.740016 + 0.672590i \(0.765182\pi\)
\(14\) 2401.45 − 1386.48i 0.875166 − 0.505277i
\(15\) 1004.15 + 716.156i 0.297525 + 0.212194i
\(16\) −512.000 + 886.810i −0.125000 + 0.216506i
\(17\) − 8090.59i − 1.64677i −0.567482 − 0.823386i \(-0.692082\pi\)
0.567482 − 0.823386i \(-0.307918\pi\)
\(18\) 3899.01 − 1343.08i 0.668554 − 0.230295i
\(19\) −7727.36 −1.12660 −0.563301 − 0.826252i \(-0.690469\pi\)
−0.563301 + 0.826252i \(0.690469\pi\)
\(20\) −1265.93 − 730.884i −0.158241 − 0.0913604i
\(21\) 13174.4 + 1267.72i 1.42257 + 0.136888i
\(22\) 2852.30 + 4940.34i 0.267872 + 0.463969i
\(23\) −11848.0 + 6840.45i −0.973782 + 0.562213i −0.900387 − 0.435090i \(-0.856717\pi\)
−0.0733950 + 0.997303i \(0.523383\pi\)
\(24\) −4447.33 + 2027.10i −0.321711 + 0.146636i
\(25\) −6769.16 + 11724.5i −0.433226 + 0.750370i
\(26\) − 5281.25i − 0.300481i
\(27\) 18873.5 + 5586.73i 0.958873 + 0.283835i
\(28\) −15686.2 −0.714570
\(29\) −1964.70 − 1134.32i −0.0805570 − 0.0465096i 0.459180 − 0.888343i \(-0.348143\pi\)
−0.539737 + 0.841833i \(0.681476\pi\)
\(30\) −2893.70 − 6348.59i −0.107174 − 0.235133i
\(31\) −17062.6 − 29553.2i −0.572742 − 0.992019i −0.996283 − 0.0861417i \(-0.972546\pi\)
0.423541 − 0.905877i \(-0.360787\pi\)
\(32\) 5016.55 − 2896.31i 0.153093 − 0.0883883i
\(33\) −2607.99 + 27102.7i −0.0725710 + 0.754174i
\(34\) −22883.6 + 39635.6i −0.582222 + 1.00844i
\(35\) − 22392.2i − 0.522267i
\(36\) −22900.0 − 4448.33i −0.490826 − 0.0953430i
\(37\) 92058.0 1.81743 0.908713 − 0.417422i \(-0.137066\pi\)
0.908713 + 0.417422i \(0.137066\pi\)
\(38\) 37856.2 + 21856.3i 0.689900 + 0.398314i
\(39\) 14636.6 − 20522.5i 0.246745 − 0.345969i
\(40\) 4134.50 + 7161.17i 0.0646016 + 0.111893i
\(41\) −31021.6 + 17910.3i −0.450103 + 0.259867i −0.707874 − 0.706339i \(-0.750345\pi\)
0.257771 + 0.966206i \(0.417012\pi\)
\(42\) −60955.5 − 43473.4i −0.822745 − 0.586780i
\(43\) 34570.9 − 59878.5i 0.434816 − 0.753123i −0.562465 − 0.826821i \(-0.690147\pi\)
0.997281 + 0.0736985i \(0.0234803\pi\)
\(44\) − 32270.1i − 0.378829i
\(45\) 6350.01 − 32689.8i 0.0696847 − 0.358736i
\(46\) 77390.9 0.795090
\(47\) 13211.1 + 7627.46i 0.127247 + 0.0734659i 0.562272 − 0.826952i \(-0.309927\pi\)
−0.435025 + 0.900418i \(0.643261\pi\)
\(48\) 27520.9 + 2648.22i 0.248851 + 0.0239459i
\(49\) −61321.0 − 106211.i −0.521220 − 0.902779i
\(50\) 66323.9 − 38292.1i 0.530592 − 0.306337i
\(51\) −198772. + 90600.5i −1.49846 + 0.682999i
\(52\) −14937.6 + 25872.7i −0.106236 + 0.184006i
\(53\) 236591.i 1.58917i 0.607153 + 0.794585i \(0.292312\pi\)
−0.607153 + 0.794585i \(0.707688\pi\)
\(54\) −76659.2 − 80751.6i −0.486837 − 0.512826i
\(55\) 46065.9 0.276880
\(56\) 76846.5 + 44367.4i 0.437583 + 0.252639i
\(57\) 86533.0 + 189848.i 0.467258 + 1.02513i
\(58\) 6416.70 + 11114.0i 0.0328873 + 0.0569624i
\(59\) −221890. + 128108.i −1.08039 + 0.623766i −0.931003 − 0.365012i \(-0.881065\pi\)
−0.149392 + 0.988778i \(0.547732\pi\)
\(60\) −3780.35 + 39286.2i −0.0175016 + 0.181881i
\(61\) −19919.6 + 34501.8i −0.0877589 + 0.152003i −0.906563 − 0.422069i \(-0.861304\pi\)
0.818805 + 0.574072i \(0.194637\pi\)
\(62\) 193041.i 0.809980i
\(63\) −116385. − 337868.i −0.465451 − 1.35122i
\(64\) −32768.0 −0.125000
\(65\) −36933.5 − 21323.6i −0.134487 − 0.0776462i
\(66\) 89434.6 − 125399.i 0.311081 − 0.436178i
\(67\) −160204. − 277482.i −0.532660 − 0.922593i −0.999273 − 0.0381319i \(-0.987859\pi\)
0.466613 − 0.884461i \(-0.345474\pi\)
\(68\) 224213. − 129449.i 0.713073 − 0.411693i
\(69\) 300735. + 214484.i 0.915454 + 0.652901i
\(70\) −63334.7 + 109699.i −0.184649 + 0.319822i
\(71\) − 404593.i − 1.13043i −0.824944 − 0.565215i \(-0.808793\pi\)
0.824944 − 0.565215i \(-0.191207\pi\)
\(72\) 99604.7 + 86563.1i 0.266859 + 0.231918i
\(73\) 393719. 1.01209 0.506044 − 0.862508i \(-0.331107\pi\)
0.506044 + 0.862508i \(0.331107\pi\)
\(74\) −450990. − 260379.i −1.11294 − 0.642557i
\(75\) 363854. + 35012.2i 0.862469 + 0.0829918i
\(76\) −123638. − 214147.i −0.281650 − 0.487833i
\(77\) 428105. − 247167.i 0.937731 − 0.541399i
\(78\) −129751. + 59140.8i −0.273418 + 0.124624i
\(79\) −449184. + 778010.i −0.911052 + 1.57799i −0.0984709 + 0.995140i \(0.531395\pi\)
−0.812581 + 0.582848i \(0.801938\pi\)
\(80\) − 46776.5i − 0.0913604i
\(81\) −74094.1 − 526251.i −0.139421 − 0.990233i
\(82\) 202632. 0.367508
\(83\) 154916. + 89441.0i 0.270934 + 0.156424i 0.629312 − 0.777153i \(-0.283337\pi\)
−0.358378 + 0.933576i \(0.616670\pi\)
\(84\) 175658. + 385383.i 0.296368 + 0.650212i
\(85\) 184790. + 320066.i 0.300900 + 0.521173i
\(86\) −338724. + 195562.i −0.532538 + 0.307461i
\(87\) −5867.06 + 60971.8i −0.00890970 + 0.0925915i
\(88\) −91273.8 + 158091.i −0.133936 + 0.231984i
\(89\) − 826458.i − 1.17233i −0.810191 − 0.586166i \(-0.800636\pi\)
0.810191 − 0.586166i \(-0.199364\pi\)
\(90\) −123569. + 142186.i −0.169505 + 0.195043i
\(91\) −457647. −0.607304
\(92\) −379136. − 218894.i −0.486891 − 0.281107i
\(93\) −535000. + 750142.i −0.665128 + 0.932599i
\(94\) −43147.4 − 74733.5i −0.0519483 − 0.0899770i
\(95\) 305696. − 176494.i 0.356549 − 0.205854i
\(96\) −127334. − 90814.4i −0.143923 − 0.102646i
\(97\) −317981. + 550760.i −0.348406 + 0.603458i −0.985967 − 0.166943i \(-0.946610\pi\)
0.637560 + 0.770401i \(0.279944\pi\)
\(98\) 693768.i 0.737116i
\(99\) 695072. − 239430.i 0.716348 − 0.246759i
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.d.a.11.2 yes 12
3.2 odd 2 54.7.d.a.35.6 12
4.3 odd 2 144.7.q.c.65.3 12
9.2 odd 6 162.7.b.c.161.5 12
9.4 even 3 54.7.d.a.17.6 12
9.5 odd 6 inner 18.7.d.a.5.2 ✓ 12
9.7 even 3 162.7.b.c.161.8 12
12.11 even 2 432.7.q.b.305.5 12
36.23 even 6 144.7.q.c.113.3 12
36.31 odd 6 432.7.q.b.17.5 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
18.7.d.a.5.2 ✓ 12 9.5 odd 6 inner
18.7.d.a.11.2 yes 12 1.1 even 1 trivial
54.7.d.a.17.6 12 9.4 even 3
54.7.d.a.35.6 12 3.2 odd 2
144.7.q.c.65.3 12 4.3 odd 2
144.7.q.c.113.3 12 36.23 even 6
162.7.b.c.161.5 12 9.2 odd 6
162.7.b.c.161.8 12 9.7 even 3
432.7.q.b.17.5 12 36.31 odd 6
432.7.q.b.305.5 12 12.11 even 2