Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [18,6,Mod(7,18)] 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("18.7"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(18, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 18 = 2 \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 18.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.88690875663\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.47347183152.3
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} + 118x^{4} - 231x^{3} + 3700x^{2} - 3585x + 32331 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 13.1
Root \(0.500000 + 4.03013i\) of defining polynomial
Character \(\chi\) \(=\) 18.13
Dual form 18.6.c.b.7.1

$q$-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+(-2.00000 - 3.46410i) q^{2} +(-8.58066 - 13.0143i) q^{3} +(-8.00000 + 13.8564i) q^{4} +(-39.2420 + 67.9691i) q^{5} +(-27.9216 + 55.7529i) q^{6} +(-110.566 - 191.505i) q^{7} +64.0000 q^{8} +(-95.7446 + 223.343i) q^{9} +313.936 q^{10} +(-115.144 - 199.436i) q^{11} +(248.977 - 14.7826i) q^{12} +(385.793 - 668.213i) q^{13} +(-442.262 + 766.020i) q^{14} +(1221.29 - 72.5123i) q^{15} +(-128.000 - 221.703i) q^{16} -769.880 q^{17} +(965.171 - 115.016i) q^{18} -383.367 q^{19} +(-627.872 - 1087.51i) q^{20} +(-1543.58 + 3082.17i) q^{21} +(-460.577 + 797.743i) q^{22} +(-193.178 + 334.594i) q^{23} +(-549.162 - 832.916i) q^{24} +(-1517.37 - 2628.15i) q^{25} -3086.34 q^{26} +(3728.20 - 670.377i) q^{27} +3538.10 q^{28} +(394.883 + 683.958i) q^{29} +(-2693.78 - 4085.66i) q^{30} +(-1609.97 + 2788.54i) q^{31} +(-512.000 + 886.810i) q^{32} +(-1607.51 + 3209.82i) q^{33} +(1539.76 + 2666.94i) q^{34} +17355.2 q^{35} +(-2328.77 - 3113.42i) q^{36} +2465.33 q^{37} +(766.733 + 1328.02i) q^{38} +(-12006.7 + 712.877i) q^{39} +(-2511.49 + 4350.02i) q^{40} +(4621.73 - 8005.07i) q^{41} +(13764.1 - 817.223i) q^{42} +(-5315.76 - 9207.16i) q^{43} +3684.62 q^{44} +(-11423.2 - 15272.1i) q^{45} +1545.42 q^{46} +(-976.032 - 1690.54i) q^{47} +(-1786.98 + 3568.19i) q^{48} +(-16046.0 + 27792.4i) q^{49} +(-6069.46 + 10512.6i) q^{50} +(6606.08 + 10019.5i) q^{51} +(6172.68 + 10691.4i) q^{52} -32589.2 q^{53} +(-9778.66 - 11574.1i) q^{54} +18074.0 q^{55} +(-7076.19 - 12256.3i) q^{56} +(3289.54 + 4989.25i) q^{57} +(1579.53 - 2735.83i) q^{58} +(11916.1 - 20639.2i) q^{59} +(-8765.58 + 17502.8i) q^{60} +(-18804.4 - 32570.2i) q^{61} +12879.7 q^{62} +(53357.3 - 6358.42i) q^{63} +4096.00 q^{64} +(30278.5 + 52444.0i) q^{65} +(14334.1 - 851.066i) q^{66} +(11525.6 - 19962.9i) q^{67} +(6159.04 - 10667.8i) q^{68} +(6012.10 - 356.959i) q^{69} +(-34710.5 - 60120.3i) q^{70} -66050.4 q^{71} +(-6127.65 + 14293.9i) q^{72} +65130.0 q^{73} +(-4930.66 - 8540.16i) q^{74} +(-21183.6 + 42298.8i) q^{75} +(3066.93 - 5312.08i) q^{76} +(-25462.0 + 44101.5i) q^{77} +(26482.8 + 40166.6i) q^{78} +(35433.7 + 61373.0i) q^{79} +20091.9 q^{80} +(-40714.9 - 42767.7i) q^{81} -36973.8 q^{82} +(-27643.5 - 47880.0i) q^{83} +(-30359.2 - 46045.9i) q^{84} +(30211.6 - 52328.0i) q^{85} +(-21263.0 + 36828.6i) q^{86} +(5512.88 - 11007.9i) q^{87} +(-7369.24 - 12763.9i) q^{88} +10598.6 q^{89} +(-30057.7 + 70115.3i) q^{90} -170622. q^{91} +(-3090.85 - 5353.50i) q^{92} +(50105.6 - 2974.94i) q^{93} +(-3904.13 + 6762.15i) q^{94} +(15044.1 - 26057.1i) q^{95} +(15934.5 - 946.086i) q^{96} +(41409.9 + 71724.1i) q^{97} +128368. q^{98} +(55567.0 - 6621.74i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 12 q^{2} + 9 q^{3} - 48 q^{4} - 54 q^{5} - 12 q^{6} - 132 q^{7} + 384 q^{8} - 177 q^{9} + 432 q^{10} - 315 q^{11} - 96 q^{12} - 744 q^{13} - 528 q^{14} + 2286 q^{15} - 768 q^{16} + 2898 q^{17} + 1056 q^{18}+ \cdots + 282168 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(11\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 − 3.46410i −0.353553 − 0.612372i
\(3\) −8.58066 − 13.0143i −0.550449 − 0.834868i
\(4\) −8.00000 + 13.8564i −0.250000 + 0.433013i
\(5\) −39.2420 + 67.9691i −0.701982 + 1.21587i 0.265788 + 0.964032i \(0.414368\pi\)
−0.967770 + 0.251837i \(0.918965\pi\)
\(6\) −27.9216 + 55.7529i −0.316637 + 0.632251i
\(7\) −110.566 − 191.505i −0.852854 − 1.47719i −0.878622 − 0.477518i \(-0.841536\pi\)
0.0257678 − 0.999668i \(-0.491797\pi\)
\(8\) 64.0000 0.353553
\(9\) −95.7446 + 223.343i −0.394011 + 0.919106i
\(10\) 313.936 0.992752
\(11\) −115.144 − 199.436i −0.286920 − 0.496960i 0.686153 − 0.727457i \(-0.259298\pi\)
−0.973073 + 0.230497i \(0.925965\pi\)
\(12\) 248.977 − 14.7826i 0.499121 − 0.0296345i
\(13\) 385.793 − 668.213i 0.633134 − 1.09662i −0.353773 − 0.935331i \(-0.615101\pi\)
0.986907 − 0.161289i \(-0.0515652\pi\)
\(14\) −442.262 + 766.020i −0.603059 + 1.04453i
\(15\) 1221.29 − 72.5123i 1.40150 − 0.0832115i
\(16\) −128.000 − 221.703i −0.125000 − 0.216506i
\(17\) −769.880 −0.646101 −0.323051 − 0.946382i \(-0.604708\pi\)
−0.323051 + 0.946382i \(0.604708\pi\)
\(18\) 965.171 − 115.016i 0.702139 − 0.0836717i
\(19\) −383.367 −0.243630 −0.121815 − 0.992553i \(-0.538871\pi\)
−0.121815 + 0.992553i \(0.538871\pi\)
\(20\) −627.872 − 1087.51i −0.350991 − 0.607934i
\(21\) −1543.58 + 3082.17i −0.763803 + 1.52514i
\(22\) −460.577 + 797.743i −0.202883 + 0.351404i
\(23\) −193.178 + 334.594i −0.0761444 + 0.131886i −0.901583 − 0.432605i \(-0.857594\pi\)
0.825439 + 0.564491i \(0.190928\pi\)
\(24\) −549.162 − 832.916i −0.194613 − 0.295171i
\(25\) −1517.37 − 2628.15i −0.485557 − 0.841009i
\(26\) −3086.34 −0.895387
\(27\) 3728.20 − 670.377i 0.984215 − 0.176974i
\(28\) 3538.10 0.852854
\(29\) 394.883 + 683.958i 0.0871914 + 0.151020i 0.906323 − 0.422586i \(-0.138877\pi\)
−0.819131 + 0.573606i \(0.805544\pi\)
\(30\) −2693.78 − 4085.66i −0.546460 − 0.828817i
\(31\) −1609.97 + 2788.54i −0.300893 + 0.521163i −0.976339 − 0.216247i \(-0.930618\pi\)
0.675445 + 0.737410i \(0.263952\pi\)
\(32\) −512.000 + 886.810i −0.0883883 + 0.153093i
\(33\) −1607.51 + 3209.82i −0.256961 + 0.513092i
\(34\) 1539.76 + 2666.94i 0.228431 + 0.395655i
\(35\) 17355.2 2.39475
\(36\) −2328.77 − 3113.42i −0.299482 − 0.400388i
\(37\) 2465.33 0.296054 0.148027 − 0.988983i \(-0.452708\pi\)
0.148027 + 0.988983i \(0.452708\pi\)
\(38\) 766.733 + 1328.02i 0.0861361 + 0.149192i
\(39\) −12006.7 + 712.877i −1.26404 + 0.0750505i
\(40\) −2511.49 + 4350.02i −0.248188 + 0.429874i
\(41\) 4621.73 − 8005.07i 0.429383 − 0.743713i −0.567436 − 0.823418i \(-0.692064\pi\)
0.996819 + 0.0797047i \(0.0253977\pi\)
\(42\) 13764.1 − 817.223i 1.20400 − 0.0714854i
\(43\) −5315.76 − 9207.16i −0.438424 − 0.759372i 0.559145 − 0.829070i \(-0.311130\pi\)
−0.997568 + 0.0696983i \(0.977796\pi\)
\(44\) 3684.62 0.286920
\(45\) −11423.2 − 15272.1i −0.840923 − 1.12426i
\(46\) 1545.42 0.107684
\(47\) −976.032 − 1690.54i −0.0644495 − 0.111630i 0.832000 − 0.554775i \(-0.187196\pi\)
−0.896450 + 0.443146i \(0.853862\pi\)
\(48\) −1786.98 + 3568.19i −0.111948 + 0.223534i
\(49\) −16046.0 + 27792.4i −0.954720 + 1.65362i
\(50\) −6069.46 + 10512.6i −0.343341 + 0.594683i
\(51\) 6606.08 + 10019.5i 0.355646 + 0.539410i
\(52\) 6172.68 + 10691.4i 0.316567 + 0.548310i
\(53\) −32589.2 −1.59362 −0.796809 − 0.604231i \(-0.793480\pi\)
−0.796809 + 0.604231i \(0.793480\pi\)
\(54\) −9778.66 − 11574.1i −0.456347 − 0.540137i
\(55\) 18074.0 0.805651
\(56\) −7076.19 − 12256.3i −0.301529 − 0.522264i
\(57\) 3289.54 + 4989.25i 0.134106 + 0.203399i
\(58\) 1579.53 − 2735.83i 0.0616537 − 0.106787i
\(59\) 11916.1 − 20639.2i 0.445659 − 0.771903i −0.552439 − 0.833553i \(-0.686303\pi\)
0.998098 + 0.0616498i \(0.0196362\pi\)
\(60\) −8765.58 + 17502.8i −0.314342 + 0.627668i
\(61\) −18804.4 − 32570.2i −0.647046 − 1.12072i −0.983825 − 0.179133i \(-0.942671\pi\)
0.336779 − 0.941584i \(-0.390663\pi\)
\(62\) 12879.7 0.425528
\(63\) 53357.3 − 6358.42i 1.69372 − 0.201836i
\(64\) 4096.00 0.125000
\(65\) 30278.5 + 52444.0i 0.888897 + 1.53962i
\(66\) 14334.1 − 851.066i 0.405053 − 0.0240494i
\(67\) 11525.6 − 19962.9i 0.313672 − 0.543295i −0.665483 − 0.746413i \(-0.731774\pi\)
0.979154 + 0.203118i \(0.0651075\pi\)
\(68\) 6159.04 − 10667.8i 0.161525 − 0.279770i
\(69\) 6012.10 − 356.959i 0.152021 − 0.00902600i
\(70\) −34710.5 − 60120.3i −0.846673 − 1.46648i
\(71\) −66050.4 −1.55500 −0.777498 − 0.628885i \(-0.783511\pi\)
−0.777498 + 0.628885i \(0.783511\pi\)
\(72\) −6127.65 + 14293.9i −0.139304 + 0.324953i
\(73\) 65130.0 1.43045 0.715227 − 0.698893i \(-0.246324\pi\)
0.715227 + 0.698893i \(0.246324\pi\)
\(74\) −4930.66 − 8540.16i −0.104671 − 0.181295i
\(75\) −21183.6 + 42298.8i −0.434858 + 0.868309i
\(76\) 3066.93 − 5312.08i 0.0609074 − 0.105495i
\(77\) −25462.0 + 44101.5i −0.489402 + 0.847669i
\(78\) 26482.8 + 40166.6i 0.492865 + 0.747530i
\(79\) 35433.7 + 61373.0i 0.638776 + 1.10639i 0.985702 + 0.168500i \(0.0538924\pi\)
−0.346925 + 0.937893i \(0.612774\pi\)
\(80\) 20091.9 0.350991
\(81\) −40714.9 − 42767.7i −0.689511 − 0.724275i
\(82\) −36973.8 −0.607239
\(83\) −27643.5 − 47880.0i −0.440451 − 0.762884i 0.557272 − 0.830330i \(-0.311848\pi\)
−0.997723 + 0.0674462i \(0.978515\pi\)
\(84\) −30359.2 − 46045.9i −0.469453 − 0.712021i
\(85\) 30211.6 − 52328.0i 0.453551 − 0.785574i
\(86\) −21263.0 + 36828.6i −0.310012 + 0.536957i
\(87\) 5512.88 − 11007.9i 0.0780873 − 0.155922i
\(88\) −7369.24 − 12763.9i −0.101442 − 0.175702i
\(89\) 10598.6 0.141831 0.0709156 − 0.997482i \(-0.477408\pi\)
0.0709156 + 0.997482i \(0.477408\pi\)
\(90\) −30057.7 + 70115.3i −0.391155 + 0.912444i
\(91\) −170622. −2.15988
\(92\) −3090.85 − 5353.50i −0.0380722 − 0.0659430i
\(93\) 50105.6 − 2974.94i 0.600729 − 0.0356673i
\(94\) −3904.13 + 6762.15i −0.0455727 + 0.0789342i
\(95\) 15044.1 − 26057.1i 0.171024 − 0.296222i
\(96\) 15934.5 − 946.086i 0.176466 − 0.0104774i
\(97\) 41409.9 + 71724.1i 0.446864 + 0.773991i 0.998180 − 0.0603057i \(-0.0192076\pi\)
−0.551316 + 0.834296i \(0.685874\pi\)
\(98\) 128368. 1.35018
\(99\) 55567.0 − 6621.74i 0.569809 − 0.0679023i
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 18.6.c.b.13.1 yes 6
3.2 odd 2 54.6.c.b.37.3 6
4.3 odd 2 144.6.i.b.49.3 6
9.2 odd 6 54.6.c.b.19.3 6
9.4 even 3 162.6.a.j.1.3 3
9.5 odd 6 162.6.a.i.1.1 3
9.7 even 3 inner 18.6.c.b.7.1 ✓ 6
12.11 even 2 432.6.i.b.145.3 6
36.7 odd 6 144.6.i.b.97.3 6
36.11 even 6 432.6.i.b.289.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
18.6.c.b.7.1 ✓ 6 9.7 even 3 inner
18.6.c.b.13.1 yes 6 1.1 even 1 trivial
54.6.c.b.19.3 6 9.2 odd 6
54.6.c.b.37.3 6 3.2 odd 2
144.6.i.b.49.3 6 4.3 odd 2
144.6.i.b.97.3 6 36.7 odd 6
162.6.a.i.1.1 3 9.5 odd 6
162.6.a.j.1.3 3 9.4 even 3
432.6.i.b.145.3 6 12.11 even 2
432.6.i.b.289.3 6 36.11 even 6