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 7.2
Root \(0.500000 + 8.40123i\) of defining polynomial
Character \(\chi\) \(=\) 18.7
Dual form 18.6.c.b.13.2

$q$-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+(-2.00000 + 3.46410i) q^{2} +(-2.43381 - 15.3973i) q^{3} +(-8.00000 - 13.8564i) q^{4} +(-20.8014 - 36.0292i) q^{5} +(58.2054 + 22.3636i) q^{6} +(101.661 - 176.082i) q^{7} +64.0000 q^{8} +(-231.153 + 74.9483i) q^{9} +166.412 q^{10} +(-235.168 + 407.323i) q^{11} +(-193.881 + 156.902i) q^{12} +(-241.506 - 418.300i) q^{13} +(406.643 + 704.326i) q^{14} +(-504.125 + 407.974i) q^{15} +(-128.000 + 221.703i) q^{16} +1259.86 q^{17} +(202.678 - 950.634i) q^{18} +1978.94 q^{19} +(-332.823 + 576.466i) q^{20} +(-2958.60 - 1136.75i) q^{21} +(-940.672 - 1629.29i) q^{22} +(-239.119 - 414.166i) q^{23} +(-155.764 - 985.427i) q^{24} +(697.100 - 1207.41i) q^{25} +1932.04 q^{26} +(1716.58 + 3376.72i) q^{27} -3253.14 q^{28} +(580.249 - 1005.02i) q^{29} +(-405.015 - 2562.29i) q^{30} +(1186.50 + 2055.07i) q^{31} +(-512.000 - 886.810i) q^{32} +(6844.02 + 2629.60i) q^{33} +(-2519.72 + 4364.28i) q^{34} -8458.76 q^{35} +(2887.74 + 2603.37i) q^{36} +8185.10 q^{37} +(-3957.89 + 6855.26i) q^{38} +(-5852.91 + 4736.60i) q^{39} +(-1331.29 - 2305.87i) q^{40} +(-8758.87 - 15170.8i) q^{41} +(9855.02 - 7975.40i) q^{42} +(-11435.0 + 19806.1i) q^{43} +7525.37 q^{44} +(7508.64 + 6769.22i) q^{45} +1912.95 q^{46} +(8685.43 - 15043.6i) q^{47} +(3725.15 + 1431.27i) q^{48} +(-12266.3 - 21245.9i) q^{49} +(2788.40 + 4829.65i) q^{50} +(-3066.26 - 19398.4i) q^{51} +(-3864.09 + 6692.80i) q^{52} -5390.58 q^{53} +(-15130.5 - 807.021i) q^{54} +19567.3 q^{55} +(6506.29 - 11269.2i) q^{56} +(-4816.38 - 30470.4i) q^{57} +(2321.00 + 4020.09i) q^{58} +(22281.8 + 38593.2i) q^{59} +(9686.05 + 3721.56i) q^{60} +(-2084.17 + 3609.88i) q^{61} -9491.97 q^{62} +(-10302.2 + 48321.1i) q^{63} +4096.00 q^{64} +(-10047.3 + 17402.5i) q^{65} +(-22797.3 + 18449.2i) q^{66} +(-1228.46 - 2127.76i) q^{67} +(-10078.9 - 17457.1i) q^{68} +(-5795.07 + 4689.79i) q^{69} +(16917.5 - 29302.0i) q^{70} +2184.37 q^{71} +(-14793.8 + 4796.69i) q^{72} +3037.34 q^{73} +(-16370.2 + 28354.0i) q^{74} +(-20287.5 - 7794.83i) q^{75} +(-15831.5 - 27421.0i) q^{76} +(47814.7 + 82817.5i) q^{77} +(-4702.24 - 29748.3i) q^{78} +(25266.2 - 43762.3i) q^{79} +10650.3 q^{80} +(47814.5 - 34649.1i) q^{81} +70070.9 q^{82} +(25913.9 - 44884.2i) q^{83} +(7917.55 + 50089.6i) q^{84} +(-26206.9 - 45391.7i) q^{85} +(-45740.1 - 79224.3i) q^{86} +(-16886.8 - 6488.23i) q^{87} +(-15050.7 + 26068.7i) q^{88} -20154.7 q^{89} +(-38466.5 + 12472.3i) q^{90} -98206.5 q^{91} +(-3825.91 + 6626.66i) q^{92} +(28754.8 - 23270.5i) q^{93} +(34741.7 + 60174.4i) q^{94} +(-41164.9 - 71299.6i) q^{95} +(-12408.4 + 10041.7i) q^{96} +(-40214.4 + 69653.3i) q^{97} +98130.4 q^{98} +(23831.7 - 111779. i) 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 \(\left(\mathbb{Z}/18\mathbb{Z}\right)^\times\).

\(n\) \(11\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\)

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
\(3\) −2.43381 − 15.3973i −0.156129 − 0.987737i
\(4\) −8.00000 − 13.8564i −0.250000 − 0.433013i
\(5\) −20.8014 − 36.0292i −0.372108 − 0.644509i 0.617782 − 0.786349i \(-0.288031\pi\)
−0.989890 + 0.141840i \(0.954698\pi\)
\(6\) 58.2054 + 22.3636i 0.660063 + 0.253608i
\(7\) 101.661 − 176.082i 0.784166 − 1.35822i −0.145330 − 0.989383i \(-0.546424\pi\)
0.929496 − 0.368832i \(-0.120242\pi\)
\(8\) 64.0000 0.353553
\(9\) −231.153 + 74.9483i −0.951247 + 0.308429i
\(10\) 166.412 0.526239
\(11\) −235.168 + 407.323i −0.585998 + 1.01498i 0.408752 + 0.912646i \(0.365964\pi\)
−0.994750 + 0.102333i \(0.967369\pi\)
\(12\) −193.881 + 156.902i −0.388670 + 0.314540i
\(13\) −241.506 − 418.300i −0.396341 − 0.686482i 0.596931 − 0.802293i \(-0.296387\pi\)
−0.993271 + 0.115811i \(0.963053\pi\)
\(14\) 406.643 + 704.326i 0.554489 + 0.960403i
\(15\) −504.125 + 407.974i −0.578508 + 0.468171i
\(16\) −128.000 + 221.703i −0.125000 + 0.216506i
\(17\) 1259.86 1.05730 0.528652 − 0.848839i \(-0.322698\pi\)
0.528652 + 0.848839i \(0.322698\pi\)
\(18\) 202.678 − 950.634i 0.147443 − 0.691564i
\(19\) 1978.94 1.25762 0.628810 − 0.777559i \(-0.283542\pi\)
0.628810 + 0.777559i \(0.283542\pi\)
\(20\) −332.823 + 576.466i −0.186054 + 0.322255i
\(21\) −2958.60 − 1136.75i −1.46399 − 0.562492i
\(22\) −940.672 − 1629.29i −0.414363 − 0.717698i
\(23\) −239.119 − 414.166i −0.0942529 − 0.163251i 0.815044 − 0.579400i \(-0.196713\pi\)
−0.909297 + 0.416149i \(0.863380\pi\)
\(24\) −155.764 − 985.427i −0.0552000 − 0.349218i
\(25\) 697.100 − 1207.41i 0.223072 − 0.386372i
\(26\) 1932.04 0.560511
\(27\) 1716.58 + 3376.72i 0.453164 + 0.891427i
\(28\) −3253.14 −0.784166
\(29\) 580.249 − 1005.02i 0.128121 − 0.221912i −0.794828 − 0.606835i \(-0.792439\pi\)
0.922949 + 0.384923i \(0.125772\pi\)
\(30\) −405.015 − 2562.29i −0.0821614 − 0.519786i
\(31\) 1186.50 + 2055.07i 0.221749 + 0.384081i 0.955339 − 0.295511i \(-0.0954901\pi\)
−0.733590 + 0.679592i \(0.762157\pi\)
\(32\) −512.000 − 886.810i −0.0883883 − 0.153093i
\(33\) 6844.02 + 2629.60i 1.09402 + 0.420344i
\(34\) −2519.72 + 4364.28i −0.373813 + 0.647464i
\(35\) −8458.76 −1.16718
\(36\) 2887.74 + 2603.37i 0.371366 + 0.334795i
\(37\) 8185.10 0.982923 0.491462 − 0.870899i \(-0.336463\pi\)
0.491462 + 0.870899i \(0.336463\pi\)
\(38\) −3957.89 + 6855.26i −0.444636 + 0.770132i
\(39\) −5852.91 + 4736.60i −0.616183 + 0.498660i
\(40\) −1331.29 − 2305.87i −0.131560 − 0.227868i
\(41\) −8758.87 − 15170.8i −0.813745 − 1.40945i −0.910226 − 0.414113i \(-0.864092\pi\)
0.0964809 − 0.995335i \(-0.469241\pi\)
\(42\) 9855.02 − 7975.40i 0.862054 − 0.697636i
\(43\) −11435.0 + 19806.1i −0.943119 + 1.63353i −0.183644 + 0.982993i \(0.558789\pi\)
−0.759475 + 0.650537i \(0.774544\pi\)
\(44\) 7525.37 0.585998
\(45\) 7508.64 + 6769.22i 0.552752 + 0.498319i
\(46\) 1912.95 0.133294
\(47\) 8685.43 − 15043.6i 0.573518 − 0.993362i −0.422683 − 0.906277i \(-0.638912\pi\)
0.996201 − 0.0870843i \(-0.0277550\pi\)
\(48\) 3725.15 + 1431.27i 0.233367 + 0.0896641i
\(49\) −12266.3 − 21245.9i −0.729833 − 1.26411i
\(50\) 2788.40 + 4829.65i 0.157736 + 0.273206i
\(51\) −3066.26 − 19398.4i −0.165076 − 1.04434i
\(52\) −3864.09 + 6692.80i −0.198170 + 0.343241i
\(53\) −5390.58 −0.263600 −0.131800 − 0.991276i \(-0.542076\pi\)
−0.131800 + 0.991276i \(0.542076\pi\)
\(54\) −15130.5 − 807.021i −0.706103 − 0.0376617i
\(55\) 19567.3 0.872218
\(56\) 6506.29 − 11269.2i 0.277245 − 0.480202i
\(57\) −4816.38 − 30470.4i −0.196351 − 1.24220i
\(58\) 2321.00 + 4020.09i 0.0905951 + 0.156915i
\(59\) 22281.8 + 38593.2i 0.833335 + 1.44338i 0.895379 + 0.445306i \(0.146905\pi\)
−0.0620431 + 0.998073i \(0.519762\pi\)
\(60\) 9686.05 + 3721.56i 0.347351 + 0.133459i
\(61\) −2084.17 + 3609.88i −0.0717147 + 0.124213i −0.899653 − 0.436606i \(-0.856180\pi\)
0.827938 + 0.560819i \(0.189514\pi\)
\(62\) −9491.97 −0.313601
\(63\) −10302.2 + 48321.1i −0.327023 + 1.53386i
\(64\) 4096.00 0.125000
\(65\) −10047.3 + 17402.5i −0.294963 + 0.510891i
\(66\) −22797.3 + 18449.2i −0.644203 + 0.521336i
\(67\) −1228.46 − 2127.76i −0.0334330 − 0.0579076i 0.848825 − 0.528674i \(-0.177311\pi\)
−0.882258 + 0.470767i \(0.843977\pi\)
\(68\) −10078.9 − 17457.1i −0.264326 − 0.457826i
\(69\) −5795.07 + 4689.79i −0.146533 + 0.118585i
\(70\) 16917.5 − 29302.0i 0.412659 − 0.714747i
\(71\) 2184.37 0.0514256 0.0257128 − 0.999669i \(-0.491814\pi\)
0.0257128 + 0.999669i \(0.491814\pi\)
\(72\) −14793.8 + 4796.69i −0.336317 + 0.109046i
\(73\) 3037.34 0.0667093 0.0333546 − 0.999444i \(-0.489381\pi\)
0.0333546 + 0.999444i \(0.489381\pi\)
\(74\) −16370.2 + 28354.0i −0.347516 + 0.601915i
\(75\) −20287.5 − 7794.83i −0.416462 − 0.160012i
\(76\) −15831.5 − 27421.0i −0.314405 − 0.544565i
\(77\) 47814.7 + 82817.5i 0.919040 + 1.59182i
\(78\) −4702.24 − 29748.3i −0.0875121 − 0.553637i
\(79\) 25266.2 − 43762.3i 0.455482 − 0.788918i −0.543234 − 0.839582i \(-0.682800\pi\)
0.998716 + 0.0506633i \(0.0161335\pi\)
\(80\) 10650.3 0.186054
\(81\) 47814.5 − 34649.1i 0.809743 − 0.586785i
\(82\) 70070.9 1.15081
\(83\) 25913.9 − 44884.2i 0.412893 − 0.715152i −0.582311 − 0.812966i \(-0.697852\pi\)
0.995205 + 0.0978135i \(0.0311849\pi\)
\(84\) 7917.55 + 50089.6i 0.122431 + 0.774550i
\(85\) −26206.9 − 45391.7i −0.393431 − 0.681442i
\(86\) −45740.1 − 79224.3i −0.666886 − 1.15508i
\(87\) −16886.8 − 6488.23i −0.239194 − 0.0919027i
\(88\) −15050.7 + 26068.7i −0.207182 + 0.358849i
\(89\) −20154.7 −0.269713 −0.134857 − 0.990865i \(-0.543057\pi\)
−0.134857 + 0.990865i \(0.543057\pi\)
\(90\) −38466.5 + 12472.3i −0.500584 + 0.162308i
\(91\) −98206.5 −1.24319
\(92\) −3825.91 + 6626.66i −0.0471264 + 0.0816254i
\(93\) 28754.8 − 23270.5i 0.344749 − 0.278996i
\(94\) 34741.7 + 60174.4i 0.405538 + 0.702413i
\(95\) −41164.9 − 71299.6i −0.467970 − 0.810547i
\(96\) −12408.4 + 10041.7i −0.137416 + 0.111207i
\(97\) −40214.4 + 69653.3i −0.433962 + 0.751644i −0.997210 − 0.0746433i \(-0.976218\pi\)
0.563248 + 0.826288i \(0.309552\pi\)
\(98\) 98130.4 1.03214
\(99\) 23831.7 − 111779.i 0.244380 − 1.14623i
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.7.2 ✓ 6
3.2 odd 2 54.6.c.b.19.2 6
4.3 odd 2 144.6.i.b.97.2 6
9.2 odd 6 162.6.a.i.1.2 3
9.4 even 3 inner 18.6.c.b.13.2 yes 6
9.5 odd 6 54.6.c.b.37.2 6
9.7 even 3 162.6.a.j.1.2 3
12.11 even 2 432.6.i.b.289.2 6
36.23 even 6 432.6.i.b.145.2 6
36.31 odd 6 144.6.i.b.49.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
18.6.c.b.7.2 ✓ 6 1.1 even 1 trivial
18.6.c.b.13.2 yes 6 9.4 even 3 inner
54.6.c.b.19.2 6 3.2 odd 2
54.6.c.b.37.2 6 9.5 odd 6
144.6.i.b.49.2 6 36.31 odd 6
144.6.i.b.97.2 6 4.3 odd 2
162.6.a.i.1.2 3 9.2 odd 6
162.6.a.j.1.2 3 9.7 even 3
432.6.i.b.145.2 6 36.23 even 6
432.6.i.b.289.2 6 12.11 even 2