Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [80,2,Mod(21,80)] 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("80.21"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(80, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 80.l (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: \(0.638803216170\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} + 4 x^{14} + 7 x^{12} - 8 x^{11} - 28 x^{10} + 28 x^{9} + 17 x^{8} + 56 x^{7} + \cdots + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 61.1
Root \(1.26868 + 0.624862i\) of defining polynomial
Character \(\chi\) \(=\) 80.61
Dual form 80.2.l.a.21.1

$q$-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+(-1.37735 - 0.320793i) q^{2} +(-0.720673 - 0.720673i) q^{3} +(1.79418 + 0.883688i) q^{4} +(0.707107 - 0.707107i) q^{5} +(0.761432 + 1.22381i) q^{6} -4.02840i q^{7} +(-2.18774 - 1.79271i) q^{8} -1.96126i q^{9} +(-1.20077 + 0.747098i) q^{10} +(-0.646837 + 0.646837i) q^{11} +(-0.656170 - 1.92987i) q^{12} +(4.91492 + 4.91492i) q^{13} +(-1.29228 + 5.54851i) q^{14} -1.01919 q^{15} +(2.43819 + 3.17100i) q^{16} -2.70862 q^{17} +(-0.629159 + 2.70134i) q^{18} +(-0.438397 - 0.438397i) q^{19} +(1.89354 - 0.643818i) q^{20} +(-2.90316 + 2.90316i) q^{21} +(1.09842 - 0.683420i) q^{22} +3.60080i q^{23} +(0.284686 + 2.86860i) q^{24} -1.00000i q^{25} +(-5.19289 - 8.34623i) q^{26} +(-3.57545 + 3.57545i) q^{27} +(3.55985 - 7.22769i) q^{28} +(2.00921 + 2.00921i) q^{29} +(1.40377 + 0.326948i) q^{30} +4.30994 q^{31} +(-2.34101 - 5.14973i) q^{32} +0.932316 q^{33} +(3.73072 + 0.868908i) q^{34} +(-2.84851 - 2.84851i) q^{35} +(1.73314 - 3.51886i) q^{36} +(-0.743961 + 0.743961i) q^{37} +(0.463191 + 0.744461i) q^{38} -7.08410i q^{39} +(-2.81460 + 0.279327i) q^{40} -0.603979i q^{41} +(4.92998 - 3.06735i) q^{42} +(-5.03010 + 5.03010i) q^{43} +(-1.73215 + 0.588942i) q^{44} +(-1.38682 - 1.38682i) q^{45} +(1.15511 - 4.95956i) q^{46} +10.8177 q^{47} +(0.528115 - 4.04239i) q^{48} -9.22800 q^{49} +(-0.320793 + 1.37735i) q^{50} +(1.95203 + 1.95203i) q^{51} +(4.47501 + 13.1615i) q^{52} +(4.07420 - 4.07420i) q^{53} +(6.07162 - 3.77766i) q^{54} +0.914766i q^{55} +(-7.22175 + 8.81308i) q^{56} +0.631882i q^{57} +(-2.12285 - 3.41193i) q^{58} +(1.22845 - 1.22845i) q^{59} +(-1.82861 - 0.900642i) q^{60} +(-6.98912 - 6.98912i) q^{61} +(-5.93629 - 1.38260i) q^{62} -7.90074 q^{63} +(1.57239 + 7.84395i) q^{64} +6.95074 q^{65} +(-1.28413 - 0.299081i) q^{66} +(5.24219 + 5.24219i) q^{67} +(-4.85977 - 2.39358i) q^{68} +(2.59500 - 2.59500i) q^{69} +(3.00961 + 4.83717i) q^{70} +13.7940i q^{71} +(-3.51597 + 4.29072i) q^{72} -1.30876i q^{73} +(1.26335 - 0.786036i) q^{74} +(-0.720673 + 0.720673i) q^{75} +(-0.399159 - 1.17397i) q^{76} +(2.60572 + 2.60572i) q^{77} +(-2.27253 + 9.75728i) q^{78} +0.611127 q^{79} +(3.96630 + 0.518173i) q^{80} -0.730326 q^{81} +(-0.193752 + 0.831890i) q^{82} +(1.29471 + 1.29471i) q^{83} +(-7.77429 + 2.64331i) q^{84} +(-1.91529 + 1.91529i) q^{85} +(8.54182 - 5.31458i) q^{86} -2.89597i q^{87} +(2.57470 - 0.255519i) q^{88} -10.9236i q^{89} +(1.46525 + 2.35502i) q^{90} +(19.7993 - 19.7993i) q^{91} +(-3.18199 + 6.46050i) q^{92} +(-3.10606 - 3.10606i) q^{93} +(-14.8998 - 3.47025i) q^{94} -0.619987 q^{95} +(-2.02417 + 5.39837i) q^{96} -12.7571 q^{97} +(12.7102 + 2.96028i) q^{98} +(1.26862 + 1.26862i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{4} - 12 q^{6} + 4 q^{10} - 8 q^{11} - 12 q^{12} + 4 q^{14} - 8 q^{15} + 16 q^{16} - 8 q^{19} + 8 q^{20} - 20 q^{22} + 8 q^{24} - 16 q^{26} + 24 q^{27} - 4 q^{28} - 16 q^{29} + 16 q^{34} - 4 q^{36}+ \cdots - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/80\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(21\) \(31\)
\(\chi(n)\) \(1\) \(e\left(\frac{3}{4}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.37735 − 0.320793i −0.973933 − 0.226835i
\(3\) −0.720673 − 0.720673i −0.416081 − 0.416081i 0.467770 − 0.883850i \(-0.345058\pi\)
−0.883850 + 0.467770i \(0.845058\pi\)
\(4\) 1.79418 + 0.883688i 0.897092 + 0.441844i
\(5\) 0.707107 − 0.707107i 0.316228 − 0.316228i
\(6\) 0.761432 + 1.22381i 0.310853 + 0.499617i
\(7\) − 4.02840i − 1.52259i −0.648405 − 0.761296i \(-0.724563\pi\)
0.648405 − 0.761296i \(-0.275437\pi\)
\(8\) −2.18774 − 1.79271i −0.773482 − 0.633818i
\(9\) − 1.96126i − 0.653754i
\(10\) −1.20077 + 0.747098i −0.379716 + 0.236253i
\(11\) −0.646837 + 0.646837i −0.195029 + 0.195029i −0.797865 − 0.602836i \(-0.794037\pi\)
0.602836 + 0.797865i \(0.294037\pi\)
\(12\) −0.656170 − 1.92987i −0.189420 − 0.557106i
\(13\) 4.91492 + 4.91492i 1.36315 + 1.36315i 0.869867 + 0.493286i \(0.164204\pi\)
0.493286 + 0.869867i \(0.335796\pi\)
\(14\) −1.29228 + 5.54851i −0.345377 + 1.48290i
\(15\) −1.01919 −0.263153
\(16\) 2.43819 + 3.17100i 0.609548 + 0.792749i
\(17\) −2.70862 −0.656938 −0.328469 − 0.944515i \(-0.606533\pi\)
−0.328469 + 0.944515i \(0.606533\pi\)
\(18\) −0.629159 + 2.70134i −0.148294 + 0.636712i
\(19\) −0.438397 − 0.438397i −0.100575 − 0.100575i 0.655029 − 0.755604i \(-0.272656\pi\)
−0.755604 + 0.655029i \(0.772656\pi\)
\(20\) 1.89354 − 0.643818i 0.423409 − 0.143962i
\(21\) −2.90316 + 2.90316i −0.633521 + 0.633521i
\(22\) 1.09842 − 0.683420i 0.234184 − 0.145706i
\(23\) 3.60080i 0.750819i 0.926859 + 0.375410i \(0.122498\pi\)
−0.926859 + 0.375410i \(0.877502\pi\)
\(24\) 0.284686 + 2.86860i 0.0581113 + 0.585551i
\(25\) − 1.00000i − 0.200000i
\(26\) −5.19289 − 8.34623i −1.01841 − 1.63683i
\(27\) −3.57545 + 3.57545i −0.688095 + 0.688095i
\(28\) 3.55985 − 7.22769i 0.672748 − 1.36590i
\(29\) 2.00921 + 2.00921i 0.373102 + 0.373102i 0.868606 − 0.495504i \(-0.165017\pi\)
−0.495504 + 0.868606i \(0.665017\pi\)
\(30\) 1.40377 + 0.326948i 0.256293 + 0.0596922i
\(31\) 4.30994 0.774087 0.387044 − 0.922061i \(-0.373496\pi\)
0.387044 + 0.922061i \(0.373496\pi\)
\(32\) −2.34101 − 5.14973i −0.413835 − 0.910352i
\(33\) 0.932316 0.162295
\(34\) 3.73072 + 0.868908i 0.639814 + 0.149016i
\(35\) −2.84851 − 2.84851i −0.481486 − 0.481486i
\(36\) 1.73314 − 3.51886i 0.288857 − 0.586477i
\(37\) −0.743961 + 0.743961i −0.122306 + 0.122306i −0.765611 − 0.643304i \(-0.777563\pi\)
0.643304 + 0.765611i \(0.277563\pi\)
\(38\) 0.463191 + 0.744461i 0.0751395 + 0.120767i
\(39\) − 7.08410i − 1.13436i
\(40\) −2.81460 + 0.279327i −0.445027 + 0.0441655i
\(41\) − 0.603979i − 0.0943256i −0.998887 − 0.0471628i \(-0.984982\pi\)
0.998887 − 0.0471628i \(-0.0150180\pi\)
\(42\) 4.92998 − 3.06735i 0.760712 − 0.473303i
\(43\) −5.03010 + 5.03010i −0.767083 + 0.767083i −0.977592 − 0.210509i \(-0.932488\pi\)
0.210509 + 0.977592i \(0.432488\pi\)
\(44\) −1.73215 + 0.588942i −0.261131 + 0.0887864i
\(45\) −1.38682 − 1.38682i −0.206735 − 0.206735i
\(46\) 1.15511 − 4.95956i 0.170312 − 0.731248i
\(47\) 10.8177 1.57793 0.788963 − 0.614440i \(-0.210618\pi\)
0.788963 + 0.614440i \(0.210618\pi\)
\(48\) 0.528115 − 4.04239i 0.0762268 − 0.583469i
\(49\) −9.22800 −1.31829
\(50\) −0.320793 + 1.37735i −0.0453670 + 0.194787i
\(51\) 1.95203 + 1.95203i 0.273339 + 0.273339i
\(52\) 4.47501 + 13.1615i 0.620572 + 1.82517i
\(53\) 4.07420 − 4.07420i 0.559634 − 0.559634i −0.369569 − 0.929203i \(-0.620495\pi\)
0.929203 + 0.369569i \(0.120495\pi\)
\(54\) 6.07162 − 3.77766i 0.826243 − 0.514075i
\(55\) 0.914766i 0.123347i
\(56\) −7.22175 + 8.81308i −0.965047 + 1.17770i
\(57\) 0.631882i 0.0836948i
\(58\) −2.12285 − 3.41193i −0.278744 − 0.448009i
\(59\) 1.22845 − 1.22845i 0.159931 − 0.159931i −0.622605 − 0.782536i \(-0.713926\pi\)
0.782536 + 0.622605i \(0.213926\pi\)
\(60\) −1.82861 − 0.900642i −0.236072 − 0.116272i
\(61\) −6.98912 − 6.98912i −0.894865 − 0.894865i 0.100112 − 0.994976i \(-0.468080\pi\)
−0.994976 + 0.100112i \(0.968080\pi\)
\(62\) −5.93629 − 1.38260i −0.753909 − 0.175590i
\(63\) −7.90074 −0.995400
\(64\) 1.57239 + 7.84395i 0.196548 + 0.980494i
\(65\) 6.95074 0.862134
\(66\) −1.28413 − 0.299081i −0.158065 − 0.0368143i
\(67\) 5.24219 + 5.24219i 0.640435 + 0.640435i 0.950662 − 0.310227i \(-0.100405\pi\)
−0.310227 + 0.950662i \(0.600405\pi\)
\(68\) −4.85977 − 2.39358i −0.589334 − 0.290264i
\(69\) 2.59500 − 2.59500i 0.312401 − 0.312401i
\(70\) 3.00961 + 4.83717i 0.359717 + 0.578153i
\(71\) 13.7940i 1.63704i 0.574475 + 0.818522i \(0.305206\pi\)
−0.574475 + 0.818522i \(0.694794\pi\)
\(72\) −3.51597 + 4.29072i −0.414361 + 0.505667i
\(73\) − 1.30876i − 0.153179i −0.997063 − 0.0765895i \(-0.975597\pi\)
0.997063 − 0.0765895i \(-0.0244031\pi\)
\(74\) 1.26335 − 0.786036i 0.146862 − 0.0913749i
\(75\) −0.720673 + 0.720673i −0.0832162 + 0.0832162i
\(76\) −0.399159 − 1.17397i −0.0457866 − 0.134664i
\(77\) 2.60572 + 2.60572i 0.296949 + 0.296949i
\(78\) −2.27253 + 9.75728i −0.257313 + 1.10479i
\(79\) 0.611127 0.0687571 0.0343786 − 0.999409i \(-0.489055\pi\)
0.0343786 + 0.999409i \(0.489055\pi\)
\(80\) 3.96630 + 0.518173i 0.443445 + 0.0579335i
\(81\) −0.730326 −0.0811473
\(82\) −0.193752 + 0.831890i −0.0213963 + 0.0918669i
\(83\) 1.29471 + 1.29471i 0.142113 + 0.142113i 0.774584 − 0.632471i \(-0.217959\pi\)
−0.632471 + 0.774584i \(0.717959\pi\)
\(84\) −7.77429 + 2.64331i −0.848244 + 0.288409i
\(85\) −1.91529 + 1.91529i −0.207742 + 0.207742i
\(86\) 8.54182 − 5.31458i 0.921089 − 0.573086i
\(87\) − 2.89597i − 0.310481i
\(88\) 2.57470 − 0.255519i 0.274464 − 0.0272384i
\(89\) − 10.9236i − 1.15790i −0.815363 − 0.578950i \(-0.803463\pi\)
0.815363 − 0.578950i \(-0.196537\pi\)
\(90\) 1.46525 + 2.35502i 0.154451 + 0.248241i
\(91\) 19.7993 − 19.7993i 2.07553 − 2.07553i
\(92\) −3.18199 + 6.46050i −0.331745 + 0.673554i
\(93\) −3.10606 − 3.10606i −0.322083 − 0.322083i
\(94\) −14.8998 − 3.47025i −1.53680 − 0.357929i
\(95\) −0.619987 −0.0636093
\(96\) −2.02417 + 5.39837i −0.206591 + 0.550969i
\(97\) −12.7571 −1.29528 −0.647642 − 0.761945i \(-0.724245\pi\)
−0.647642 + 0.761945i \(0.724245\pi\)
\(98\) 12.7102 + 2.96028i 1.28392 + 0.299033i
\(99\) 1.26862 + 1.26862i 0.127501 + 0.127501i
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 80.2.l.a.61.1 yes 16
3.2 odd 2 720.2.t.c.541.8 16
4.3 odd 2 320.2.l.a.81.6 16
5.2 odd 4 400.2.q.h.349.4 16
5.3 odd 4 400.2.q.g.349.5 16
5.4 even 2 400.2.l.h.301.8 16
8.3 odd 2 640.2.l.a.161.3 16
8.5 even 2 640.2.l.b.161.6 16
12.11 even 2 2880.2.t.c.721.4 16
16.3 odd 4 640.2.l.a.481.3 16
16.5 even 4 inner 80.2.l.a.21.1 ✓ 16
16.11 odd 4 320.2.l.a.241.6 16
16.13 even 4 640.2.l.b.481.6 16
20.3 even 4 1600.2.q.h.849.3 16
20.7 even 4 1600.2.q.g.849.6 16
20.19 odd 2 1600.2.l.i.401.3 16
32.5 even 8 5120.2.a.v.1.5 8
32.11 odd 8 5120.2.a.u.1.5 8
32.21 even 8 5120.2.a.s.1.4 8
32.27 odd 8 5120.2.a.t.1.4 8
48.5 odd 4 720.2.t.c.181.8 16
48.11 even 4 2880.2.t.c.2161.1 16
80.27 even 4 1600.2.q.h.49.3 16
80.37 odd 4 400.2.q.g.149.5 16
80.43 even 4 1600.2.q.g.49.6 16
80.53 odd 4 400.2.q.h.149.4 16
80.59 odd 4 1600.2.l.i.1201.3 16
80.69 even 4 400.2.l.h.101.8 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.2.l.a.21.1 ✓ 16 16.5 even 4 inner
80.2.l.a.61.1 yes 16 1.1 even 1 trivial
320.2.l.a.81.6 16 4.3 odd 2
320.2.l.a.241.6 16 16.11 odd 4
400.2.l.h.101.8 16 80.69 even 4
400.2.l.h.301.8 16 5.4 even 2
400.2.q.g.149.5 16 80.37 odd 4
400.2.q.g.349.5 16 5.3 odd 4
400.2.q.h.149.4 16 80.53 odd 4
400.2.q.h.349.4 16 5.2 odd 4
640.2.l.a.161.3 16 8.3 odd 2
640.2.l.a.481.3 16 16.3 odd 4
640.2.l.b.161.6 16 8.5 even 2
640.2.l.b.481.6 16 16.13 even 4
720.2.t.c.181.8 16 48.5 odd 4
720.2.t.c.541.8 16 3.2 odd 2
1600.2.l.i.401.3 16 20.19 odd 2
1600.2.l.i.1201.3 16 80.59 odd 4
1600.2.q.g.49.6 16 80.43 even 4
1600.2.q.g.849.6 16 20.7 even 4
1600.2.q.h.49.3 16 80.27 even 4
1600.2.q.h.849.3 16 20.3 even 4
2880.2.t.c.721.4 16 12.11 even 2
2880.2.t.c.2161.1 16 48.11 even 4
5120.2.a.s.1.4 8 32.21 even 8
5120.2.a.t.1.4 8 32.27 odd 8
5120.2.a.u.1.5 8 32.11 odd 8
5120.2.a.v.1.5 8 32.5 even 8