↑

Properties

Label 16.6.e.a.5.1
Level $16$
Weight $6$
Character 16.5
Analytic conductor $2.566$
Analytic rank $0$
Dimension $18$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [16,6,Mod(5,16)] 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("16.5"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(16, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 16 = 2^{4} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 16.e (of order \(4\), 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: \(2.56614111701\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(9\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 5 x^{16} - 30 x^{15} - 42 x^{14} - 344 x^{13} + 2904 x^{12} + 5344 x^{11} + 16576 x^{10} + \cdots + 68719476736 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{36} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 5.1
Root \(-1.58716 + 3.67164i\) of defining polynomial
Character \(\chi\) \(=\) 16.5
Dual form 16.6.e.a.13.1

$q$-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+(-5.25880 + 2.08448i) q^{2} +(-3.18000 + 3.18000i) q^{3} +(23.3098 - 21.9238i) q^{4} +(-67.3647 - 67.3647i) q^{5} +(10.0943 - 23.3517i) q^{6} -148.379i q^{7} +(-76.8820 + 163.882i) q^{8} +222.775i q^{9} +(494.678 + 213.836i) q^{10} +(-256.205 - 256.205i) q^{11} +(-4.40780 + 143.843i) q^{12} +(-218.586 + 218.586i) q^{13} +(309.293 + 780.294i) q^{14} +428.440 q^{15} +(62.6982 - 1022.08i) q^{16} -463.168 q^{17} +(-464.371 - 1171.53i) q^{18} +(920.791 - 920.791i) q^{19} +(-3047.15 - 93.3741i) q^{20} +(471.845 + 471.845i) q^{21} +(1881.39 + 813.276i) q^{22} -1053.65i q^{23} +(-276.659 - 765.629i) q^{24} +5951.00i q^{25} +(693.860 - 1605.14i) q^{26} +(-1481.17 - 1481.17i) q^{27} +(-3253.02 - 3458.69i) q^{28} +(1290.79 - 1290.79i) q^{29} +(-2253.08 + 893.076i) q^{30} +10036.9 q^{31} +(1800.79 + 5505.60i) q^{32} +1629.47 q^{33} +(2435.71 - 965.466i) q^{34} +(-9995.50 + 9995.50i) q^{35} +(4884.07 + 5192.86i) q^{36} +(-9409.85 - 9409.85i) q^{37} +(-2922.88 + 6761.62i) q^{38} -1390.21i q^{39} +(16219.0 - 5860.70i) q^{40} -368.682i q^{41} +(-3464.89 - 1497.78i) q^{42} +(-9168.57 - 9168.57i) q^{43} +(-11589.1 - 355.126i) q^{44} +(15007.2 - 15007.2i) q^{45} +(2196.32 + 5540.94i) q^{46} -7638.65 q^{47} +(3050.83 + 3449.59i) q^{48} -5209.29 q^{49} +(-12404.8 - 31295.1i) q^{50} +(1472.88 - 1472.88i) q^{51} +(-302.982 + 9887.43i) q^{52} +(-1242.05 - 1242.05i) q^{53} +(10876.6 + 4701.68i) q^{54} +34518.4i q^{55} +(24316.6 + 11407.7i) q^{56} +5856.24i q^{57} +(-4097.36 + 9478.62i) q^{58} +(16255.3 + 16255.3i) q^{59} +(9986.87 - 9393.01i) q^{60} +(-1394.11 + 1394.11i) q^{61} +(-52782.1 + 20921.8i) q^{62} +33055.1 q^{63} +(-20946.3 - 25199.1i) q^{64} +29449.9 q^{65} +(-8569.04 + 3396.60i) q^{66} +(-17400.9 + 17400.9i) q^{67} +(-10796.4 + 10154.4i) q^{68} +(3350.62 + 3350.62i) q^{69} +(31728.8 - 73399.7i) q^{70} -67414.6i q^{71} +(-36508.7 - 17127.4i) q^{72} +19543.2i q^{73} +(69099.2 + 29869.8i) q^{74} +(-18924.2 - 18924.2i) q^{75} +(1276.31 - 41650.7i) q^{76} +(-38015.5 + 38015.5i) q^{77} +(2897.87 + 7310.82i) q^{78} -43996.5 q^{79} +(-73075.7 + 64628.4i) q^{80} -44714.1 q^{81} +(768.512 + 1938.82i) q^{82} +(73286.1 - 73286.1i) q^{83} +(21343.3 + 654.024i) q^{84} +(31201.2 + 31201.2i) q^{85} +(67327.4 + 29103.9i) q^{86} +8209.42i q^{87} +(61684.9 - 22289.8i) q^{88} -88671.1i q^{89} +(-47637.4 + 110202. i) q^{90} +(32433.5 + 32433.5i) q^{91} +(-23100.0 - 24560.5i) q^{92} +(-31917.5 + 31917.5i) q^{93} +(40170.1 - 15922.6i) q^{94} -124058. q^{95} +(-23234.3 - 11781.3i) q^{96} -73610.3 q^{97} +(27394.6 - 10858.7i) q^{98} +(57076.2 - 57076.2i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 2 q^{2} - 2 q^{3} - 24 q^{4} - 2 q^{5} + 112 q^{6} + 244 q^{8} - 436 q^{10} - 606 q^{11} + 4 q^{12} - 2 q^{13} - 100 q^{14} + 1796 q^{15} - 872 q^{16} - 4 q^{17} - 3138 q^{18} - 2362 q^{19} + 2972 q^{20}+ \cdots - 296770 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/16\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(15\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.25880 + 2.08448i −0.929632 + 0.368488i
\(3\) −3.18000 + 3.18000i −0.203997 + 0.203997i −0.801710 − 0.597713i \(-0.796076\pi\)
0.597713 + 0.801710i \(0.296076\pi\)
\(4\) 23.3098 − 21.9238i 0.728433 − 0.685117i
\(5\) −67.3647 − 67.3647i −1.20506 − 1.20506i −0.972609 − 0.232447i \(-0.925327\pi\)
−0.232447 − 0.972609i \(-0.574673\pi\)
\(6\) 10.0943 − 23.3517i 0.114472 − 0.264813i
\(7\) − 148.379i − 1.14453i −0.820069 − 0.572265i \(-0.806065\pi\)
0.820069 − 0.572265i \(-0.193935\pi\)
\(8\) −76.8820 + 163.882i −0.424717 + 0.905326i
\(9\) 222.775i 0.916770i
\(10\) 494.678 + 213.836i 1.56431 + 0.676210i
\(11\) −256.205 − 256.205i −0.638420 − 0.638420i 0.311745 − 0.950166i \(-0.399086\pi\)
−0.950166 + 0.311745i \(0.899086\pi\)
\(12\) −4.40780 + 143.843i −0.00883626 + 0.288360i
\(13\) −218.586 + 218.586i −0.358727 + 0.358727i −0.863343 − 0.504617i \(-0.831634\pi\)
0.504617 + 0.863343i \(0.331634\pi\)
\(14\) 309.293 + 780.294i 0.421746 + 1.06399i
\(15\) 428.440 0.491657
\(16\) 62.6982 − 1022.08i 0.0612287 − 0.998124i
\(17\) −463.168 −0.388701 −0.194351 − 0.980932i \(-0.562260\pi\)
−0.194351 + 0.980932i \(0.562260\pi\)
\(18\) −464.371 − 1171.53i −0.337819 − 0.852259i
\(19\) 920.791 − 920.791i 0.585163 − 0.585163i −0.351154 − 0.936318i \(-0.614211\pi\)
0.936318 + 0.351154i \(0.114211\pi\)
\(20\) −3047.15 − 93.3741i −1.70341 − 0.0521977i
\(21\) 471.845 + 471.845i 0.233481 + 0.233481i
\(22\) 1881.39 + 813.276i 0.828746 + 0.358246i
\(23\) − 1053.65i − 0.415315i −0.978202 − 0.207657i \(-0.933416\pi\)
0.978202 − 0.207657i \(-0.0665839\pi\)
\(24\) −276.659 − 765.629i −0.0980430 − 0.271325i
\(25\) 5951.00i 1.90432i
\(26\) 693.860 − 1605.14i 0.201297 − 0.465671i
\(27\) −1481.17 − 1481.17i −0.391016 − 0.391016i
\(28\) −3253.02 − 3458.69i −0.784137 − 0.833713i
\(29\) 1290.79 − 1290.79i 0.285010 − 0.285010i −0.550093 − 0.835103i \(-0.685408\pi\)
0.835103 + 0.550093i \(0.185408\pi\)
\(30\) −2253.08 + 893.076i −0.457060 + 0.181170i
\(31\) 10036.9 1.87584 0.937921 − 0.346848i \(-0.112748\pi\)
0.937921 + 0.346848i \(0.112748\pi\)
\(32\) 1800.79 + 5505.60i 0.310877 + 0.950450i
\(33\) 1629.47 0.260472
\(34\) 2435.71 − 965.466i 0.361349 − 0.143232i
\(35\) −9995.50 + 9995.50i −1.37922 + 1.37922i
\(36\) 4884.07 + 5192.86i 0.628095 + 0.667805i
\(37\) −9409.85 − 9409.85i −1.13000 − 1.13000i −0.990176 − 0.139824i \(-0.955346\pi\)
−0.139824 − 0.990176i \(-0.544654\pi\)
\(38\) −2922.88 + 6761.62i −0.328361 + 0.759612i
\(39\) − 1390.21i − 0.146359i
\(40\) 16219.0 − 5860.70i 1.60278 − 0.579161i
\(41\) − 368.682i − 0.0342525i −0.999853 − 0.0171263i \(-0.994548\pi\)
0.999853 − 0.0171263i \(-0.00545173\pi\)
\(42\) −3464.89 − 1497.78i −0.303086 − 0.131016i
\(43\) −9168.57 − 9168.57i −0.756189 − 0.756189i 0.219438 − 0.975627i \(-0.429578\pi\)
−0.975627 + 0.219438i \(0.929578\pi\)
\(44\) −11589.1 − 355.126i −0.902439 − 0.0276535i
\(45\) 15007.2 − 15007.2i 1.10476 − 1.10476i
\(46\) 2196.32 + 5540.94i 0.153039 + 0.386090i
\(47\) −7638.65 −0.504396 −0.252198 − 0.967676i \(-0.581153\pi\)
−0.252198 + 0.967676i \(0.581153\pi\)
\(48\) 3050.83 + 3449.59i 0.191124 + 0.216105i
\(49\) −5209.29 −0.309948
\(50\) −12404.8 − 31295.1i −0.701720 − 1.77032i
\(51\) 1472.88 − 1472.88i 0.0792941 − 0.0792941i
\(52\) −302.982 + 9887.43i −0.0155385 + 0.507078i
\(53\) −1242.05 − 1242.05i −0.0607363 − 0.0607363i 0.676086 − 0.736823i \(-0.263675\pi\)
−0.736823 + 0.676086i \(0.763675\pi\)
\(54\) 10876.6 + 4701.68i 0.507586 + 0.219416i
\(55\) 34518.4i 1.53866i
\(56\) 24316.6 + 11407.7i 1.03617 + 0.486101i
\(57\) 5856.24i 0.238743i
\(58\) −4097.36 + 9478.62i −0.159932 + 0.369977i
\(59\) 16255.3 + 16255.3i 0.607945 + 0.607945i 0.942409 − 0.334464i \(-0.108555\pi\)
−0.334464 + 0.942409i \(0.608555\pi\)
\(60\) 9986.87 − 9393.01i 0.358139 − 0.336842i
\(61\) −1394.11 + 1394.11i −0.0479704 + 0.0479704i −0.730685 − 0.682715i \(-0.760799\pi\)
0.682715 + 0.730685i \(0.260799\pi\)
\(62\) −52782.1 + 20921.8i −1.74384 + 0.691226i
\(63\) 33055.1 1.04927
\(64\) −20946.3 − 25199.1i −0.639231 − 0.769015i
\(65\) 29449.9 0.864572
\(66\) −8569.04 + 3396.60i −0.242143 + 0.0959809i
\(67\) −17400.9 + 17400.9i −0.473569 + 0.473569i −0.903068 − 0.429498i \(-0.858690\pi\)
0.429498 + 0.903068i \(0.358690\pi\)
\(68\) −10796.4 + 10154.4i −0.283143 + 0.266306i
\(69\) 3350.62 + 3350.62i 0.0847231 + 0.0847231i
\(70\) 31728.8 − 73399.7i 0.773943 − 1.79040i
\(71\) − 67414.6i − 1.58711i −0.608495 − 0.793557i \(-0.708227\pi\)
0.608495 − 0.793557i \(-0.291773\pi\)
\(72\) −36508.7 − 17127.4i −0.829976 − 0.389368i
\(73\) 19543.2i 0.429228i 0.976699 + 0.214614i \(0.0688494\pi\)
−0.976699 + 0.214614i \(0.931151\pi\)
\(74\) 69099.2 + 29869.8i 1.46688 + 0.634093i
\(75\) −18924.2 − 18924.2i −0.388476 − 0.388476i
\(76\) 1276.31 − 41650.7i 0.0253467 − 0.827157i
\(77\) −38015.5 + 38015.5i −0.730691 + 0.730691i
\(78\) 2897.87 + 7310.82i 0.0539314 + 0.136060i
\(79\) −43996.5 −0.793140 −0.396570 − 0.918004i \(-0.629800\pi\)
−0.396570 + 0.918004i \(0.629800\pi\)
\(80\) −73075.7 + 64628.4i −1.27658 + 1.12901i
\(81\) −44714.1 −0.757238
\(82\) 768.512 + 1938.82i 0.0126217 + 0.0318423i
\(83\) 73286.1 − 73286.1i 1.16769 − 1.16769i 0.184937 − 0.982750i \(-0.440792\pi\)
0.982750 − 0.184937i \(-0.0592080\pi\)
\(84\) 21343.3 + 654.024i 0.330037 + 0.0101134i
\(85\) 31201.2 + 31201.2i 0.468407 + 0.468407i
\(86\) 67327.4 + 29103.9i 0.981625 + 0.424331i
\(87\) 8209.42i 0.116283i
\(88\) 61684.9 − 22289.8i 0.849126 − 0.306831i
\(89\) − 88671.1i − 1.18661i −0.804979 − 0.593304i \(-0.797823\pi\)
0.804979 − 0.593304i \(-0.202177\pi\)
\(90\) −47637.4 + 110202.i −0.619929 + 1.43411i
\(91\) 32433.5 + 32433.5i 0.410573 + 0.410573i
\(92\) −23100.0 − 24560.5i −0.284539 − 0.302529i
\(93\) −31917.5 + 31917.5i −0.382667 + 0.382667i
\(94\) 40170.1 − 15922.6i 0.468903 − 0.185864i
\(95\) −124058. −1.41031
\(96\) −23234.3 − 11781.3i −0.257307 − 0.130471i
\(97\) −73610.3 −0.794345 −0.397173 − 0.917744i \(-0.630009\pi\)
−0.397173 + 0.917744i \(0.630009\pi\)
\(98\) 27394.6 − 10858.7i 0.288137 − 0.114212i
\(99\) 57076.2 − 57076.2i 0.585285 − 0.585285i
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 16.6.e.a.5.1 ✓ 18
3.2 odd 2 144.6.k.a.37.9 18
4.3 odd 2 64.6.e.a.49.5 18
8.3 odd 2 128.6.e.a.97.5 18
8.5 even 2 128.6.e.b.97.5 18
12.11 even 2 576.6.k.a.433.9 18
16.3 odd 4 64.6.e.a.17.5 18
16.5 even 4 128.6.e.b.33.5 18
16.11 odd 4 128.6.e.a.33.5 18
16.13 even 4 inner 16.6.e.a.13.1 yes 18
32.3 odd 8 1024.6.a.l.1.10 18
32.13 even 8 1024.6.a.k.1.10 18
32.19 odd 8 1024.6.a.l.1.9 18
32.29 even 8 1024.6.a.k.1.9 18
48.29 odd 4 144.6.k.a.109.9 18
48.35 even 4 576.6.k.a.145.9 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
16.6.e.a.5.1 ✓ 18 1.1 even 1 trivial
16.6.e.a.13.1 yes 18 16.13 even 4 inner
64.6.e.a.17.5 18 16.3 odd 4
64.6.e.a.49.5 18 4.3 odd 2
128.6.e.a.33.5 18 16.11 odd 4
128.6.e.a.97.5 18 8.3 odd 2
128.6.e.b.33.5 18 16.5 even 4
128.6.e.b.97.5 18 8.5 even 2
144.6.k.a.37.9 18 3.2 odd 2
144.6.k.a.109.9 18 48.29 odd 4
576.6.k.a.145.9 18 48.35 even 4
576.6.k.a.433.9 18 12.11 even 2
1024.6.a.k.1.9 18 32.29 even 8
1024.6.a.k.1.10 18 32.13 even 8
1024.6.a.l.1.9 18 32.19 odd 8
1024.6.a.l.1.10 18 32.3 odd 8