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 21.8
Root \(1.38652 - 0.278517i\) of defining polynomial
Character \(\chi\) \(=\) 80.21
Dual form 80.2.l.a.61.8

$q$-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+(1.40727 - 0.139945i) q^{2} +(-2.32624 + 2.32624i) q^{3} +(1.96083 - 0.393883i) q^{4} +(0.707107 + 0.707107i) q^{5} +(-2.94811 + 3.59920i) q^{6} -0.982011i q^{7} +(2.70430 - 0.828709i) q^{8} -7.82281i q^{9} +(1.09405 + 0.896135i) q^{10} +(-1.62645 - 1.62645i) q^{11} +(-3.64510 + 5.47764i) q^{12} +(-0.690562 + 0.690562i) q^{13} +(-0.137428 - 1.38196i) q^{14} -3.28980 q^{15} +(3.68971 - 1.54467i) q^{16} -2.19577 q^{17} +(-1.09477 - 11.0088i) q^{18} +(1.92659 - 1.92659i) q^{19} +(1.66503 + 1.10800i) q^{20} +(2.28440 + 2.28440i) q^{21} +(-2.51647 - 2.06124i) q^{22} +2.01442i q^{23} +(-4.36308 + 8.21864i) q^{24} +1.00000i q^{25} +(-0.875168 + 1.06845i) q^{26} +(11.2190 + 11.2190i) q^{27} +(-0.386797 - 1.92556i) q^{28} +(-5.27182 + 5.27182i) q^{29} +(-4.62965 + 0.460393i) q^{30} +0.435286 q^{31} +(4.97626 - 2.69014i) q^{32} +7.56703 q^{33} +(-3.09004 + 0.307288i) q^{34} +(0.694387 - 0.694387i) q^{35} +(-3.08127 - 15.3392i) q^{36} +(-5.79805 - 5.79805i) q^{37} +(2.44162 - 2.98086i) q^{38} -3.21283i q^{39} +(2.49822 + 1.32624i) q^{40} +3.93139i q^{41} +(3.53446 + 2.89508i) q^{42} +(-0.507592 - 0.507592i) q^{43} +(-3.82982 - 2.54856i) q^{44} +(5.53157 - 5.53157i) q^{45} +(0.281909 + 2.83484i) q^{46} -9.21960 q^{47} +(-4.98988 + 12.1765i) q^{48} +6.03565 q^{49} +(0.139945 + 1.40727i) q^{50} +(5.10789 - 5.10789i) q^{51} +(-1.08208 + 1.62608i) q^{52} +(6.29357 + 6.29357i) q^{53} +(17.3583 + 14.2182i) q^{54} -2.30015i q^{55} +(-0.813802 - 2.65565i) q^{56} +8.96345i q^{57} +(-6.68111 + 8.15665i) q^{58} +(-5.67778 - 5.67778i) q^{59} +(-6.45075 + 1.29580i) q^{60} +(-3.60301 + 3.60301i) q^{61} +(0.612566 - 0.0609163i) q^{62} -7.68209 q^{63} +(6.62648 - 4.48216i) q^{64} -0.976603 q^{65} +(10.6489 - 1.05897i) q^{66} +(4.53563 - 4.53563i) q^{67} +(-4.30553 + 0.864875i) q^{68} +(-4.68603 - 4.68603i) q^{69} +(0.880015 - 1.07437i) q^{70} +10.3984i q^{71} +(-6.48284 - 21.1552i) q^{72} -9.24439i q^{73} +(-8.97085 - 7.34803i) q^{74} +(-2.32624 - 2.32624i) q^{75} +(3.01887 - 4.53658i) q^{76} +(-1.59719 + 1.59719i) q^{77} +(-0.449621 - 4.52133i) q^{78} +15.4493 q^{79} +(3.70127 + 1.51677i) q^{80} -28.7280 q^{81} +(0.550180 + 5.53253i) q^{82} +(-0.683244 + 0.683244i) q^{83} +(5.37910 + 3.57953i) q^{84} +(-1.55264 - 1.55264i) q^{85} +(-0.785356 - 0.643285i) q^{86} -24.5271i q^{87} +(-5.74626 - 3.05055i) q^{88} +5.44401i q^{89} +(7.01030 - 8.55854i) q^{90} +(0.678140 + 0.678140i) q^{91} +(0.793445 + 3.94994i) q^{92} +(-1.01258 + 1.01258i) q^{93} +(-12.9745 + 1.29024i) q^{94} +2.72461 q^{95} +(-5.31808 + 17.8339i) q^{96} +5.54540 q^{97} +(8.49381 - 0.844662i) q^{98} +(-12.7234 + 12.7234i) 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 \(\left(\mathbb{Z}/80\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.40727 − 0.139945i 0.995092 − 0.0989564i
\(3\) −2.32624 + 2.32624i −1.34306 + 1.34306i −0.450058 + 0.893000i \(0.648597\pi\)
−0.893000 + 0.450058i \(0.851403\pi\)
\(4\) 1.96083 − 0.393883i 0.980415 − 0.196941i
\(5\) 0.707107 + 0.707107i 0.316228 + 0.316228i
\(6\) −2.94811 + 3.59920i −1.20356 + 1.46937i
\(7\) − 0.982011i − 0.371165i −0.982629 − 0.185583i \(-0.940583\pi\)
0.982629 − 0.185583i \(-0.0594172\pi\)
\(8\) 2.70430 − 0.828709i 0.956115 − 0.292993i
\(9\) − 7.82281i − 2.60760i
\(10\) 1.09405 + 0.896135i 0.345968 + 0.283383i
\(11\) −1.62645 − 1.62645i −0.490393 − 0.490393i 0.418037 − 0.908430i \(-0.362718\pi\)
−0.908430 + 0.418037i \(0.862718\pi\)
\(12\) −3.64510 + 5.47764i −1.05225 + 1.58126i
\(13\) −0.690562 + 0.690562i −0.191528 + 0.191528i −0.796356 − 0.604828i \(-0.793242\pi\)
0.604828 + 0.796356i \(0.293242\pi\)
\(14\) −0.137428 − 1.38196i −0.0367292 − 0.369344i
\(15\) −3.28980 −0.849424
\(16\) 3.68971 − 1.54467i 0.922428 − 0.386169i
\(17\) −2.19577 −0.532552 −0.266276 − 0.963897i \(-0.585793\pi\)
−0.266276 + 0.963897i \(0.585793\pi\)
\(18\) −1.09477 − 11.0088i −0.258039 − 2.59481i
\(19\) 1.92659 − 1.92659i 0.441991 − 0.441991i −0.450690 − 0.892681i \(-0.648822\pi\)
0.892681 + 0.450690i \(0.148822\pi\)
\(20\) 1.66503 + 1.10800i 0.372313 + 0.247756i
\(21\) 2.28440 + 2.28440i 0.498496 + 0.498496i
\(22\) −2.51647 − 2.06124i −0.536513 − 0.439458i
\(23\) 2.01442i 0.420035i 0.977698 + 0.210018i \(0.0673522\pi\)
−0.977698 + 0.210018i \(0.932648\pi\)
\(24\) −4.36308 + 8.21864i −0.890610 + 1.67762i
\(25\) 1.00000i 0.200000i
\(26\) −0.875168 + 1.06845i −0.171635 + 0.209540i
\(27\) 11.2190 + 11.2190i 2.15911 + 2.15911i
\(28\) −0.386797 − 1.92556i −0.0730978 − 0.363896i
\(29\) −5.27182 + 5.27182i −0.978952 + 0.978952i −0.999783 − 0.0208314i \(-0.993369\pi\)
0.0208314 + 0.999783i \(0.493369\pi\)
\(30\) −4.62965 + 0.460393i −0.845255 + 0.0840559i
\(31\) 0.435286 0.0781797 0.0390898 − 0.999236i \(-0.487554\pi\)
0.0390898 + 0.999236i \(0.487554\pi\)
\(32\) 4.97626 − 2.69014i 0.879687 − 0.475553i
\(33\) 7.56703 1.31725
\(34\) −3.09004 + 0.307288i −0.529938 + 0.0526994i
\(35\) 0.694387 − 0.694387i 0.117373 − 0.117373i
\(36\) −3.08127 − 15.3392i −0.513545 − 2.55654i
\(37\) −5.79805 − 5.79805i −0.953194 − 0.953194i 0.0457583 − 0.998953i \(-0.485430\pi\)
−0.998953 + 0.0457583i \(0.985430\pi\)
\(38\) 2.44162 − 2.98086i 0.396084 − 0.483559i
\(39\) − 3.21283i − 0.514465i
\(40\) 2.49822 + 1.32624i 0.395003 + 0.209697i
\(41\) 3.93139i 0.613980i 0.951713 + 0.306990i \(0.0993218\pi\)
−0.951713 + 0.306990i \(0.900678\pi\)
\(42\) 3.53446 + 2.89508i 0.545379 + 0.446720i
\(43\) −0.507592 − 0.507592i −0.0774071 − 0.0774071i 0.667343 − 0.744750i \(-0.267431\pi\)
−0.744750 + 0.667343i \(0.767431\pi\)
\(44\) −3.82982 − 2.54856i −0.577367 − 0.384210i
\(45\) 5.53157 − 5.53157i 0.824597 − 0.824597i
\(46\) 0.281909 + 2.83484i 0.0415652 + 0.417974i
\(47\) −9.21960 −1.34482 −0.672409 − 0.740180i \(-0.734740\pi\)
−0.672409 + 0.740180i \(0.734740\pi\)
\(48\) −4.98988 + 12.1765i −0.720227 + 1.75752i
\(49\) 6.03565 0.862236
\(50\) 0.139945 + 1.40727i 0.0197913 + 0.199018i
\(51\) 5.10789 − 5.10789i 0.715248 − 0.715248i
\(52\) −1.08208 + 1.62608i −0.150057 + 0.225496i
\(53\) 6.29357 + 6.29357i 0.864488 + 0.864488i 0.991856 − 0.127367i \(-0.0406527\pi\)
−0.127367 + 0.991856i \(0.540653\pi\)
\(54\) 17.3583 + 14.2182i 2.36216 + 1.93485i
\(55\) − 2.30015i − 0.310152i
\(56\) −0.813802 − 2.65565i −0.108749 − 0.354877i
\(57\) 8.96345i 1.18724i
\(58\) −6.68111 + 8.15665i −0.877273 + 1.07102i
\(59\) −5.67778 − 5.67778i −0.739183 − 0.739183i 0.233237 − 0.972420i \(-0.425068\pi\)
−0.972420 + 0.233237i \(0.925068\pi\)
\(60\) −6.45075 + 1.29580i −0.832788 + 0.167287i
\(61\) −3.60301 + 3.60301i −0.461318 + 0.461318i −0.899087 − 0.437770i \(-0.855769\pi\)
0.437770 + 0.899087i \(0.355769\pi\)
\(62\) 0.612566 − 0.0609163i 0.0777959 − 0.00773637i
\(63\) −7.68209 −0.967852
\(64\) 6.62648 − 4.48216i 0.828310 − 0.560270i
\(65\) −0.976603 −0.121133
\(66\) 10.6489 − 1.05897i 1.31079 − 0.130350i
\(67\) 4.53563 − 4.53563i 0.554116 − 0.554116i −0.373510 − 0.927626i \(-0.621846\pi\)
0.927626 + 0.373510i \(0.121846\pi\)
\(68\) −4.30553 + 0.864875i −0.522122 + 0.104882i
\(69\) −4.68603 − 4.68603i −0.564132 − 0.564132i
\(70\) 0.880015 − 1.07437i 0.105182 − 0.128411i
\(71\) 10.3984i 1.23407i 0.786937 + 0.617033i \(0.211665\pi\)
−0.786937 + 0.617033i \(0.788335\pi\)
\(72\) −6.48284 − 21.1552i −0.764010 − 2.49317i
\(73\) − 9.24439i − 1.08197i −0.841031 − 0.540987i \(-0.818051\pi\)
0.841031 − 0.540987i \(-0.181949\pi\)
\(74\) −8.97085 − 7.34803i −1.04284 − 0.854191i
\(75\) −2.32624 − 2.32624i −0.268611 − 0.268611i
\(76\) 3.01887 − 4.53658i 0.346288 − 0.520381i
\(77\) −1.59719 + 1.59719i −0.182017 + 0.182017i
\(78\) −0.449621 − 4.52133i −0.0509096 − 0.511940i
\(79\) 15.4493 1.73818 0.869091 − 0.494653i \(-0.164705\pi\)
0.869091 + 0.494653i \(0.164705\pi\)
\(80\) 3.70127 + 1.51677i 0.413815 + 0.169580i
\(81\) −28.7280 −3.19200
\(82\) 0.550180 + 5.53253i 0.0607572 + 0.610966i
\(83\) −0.683244 + 0.683244i −0.0749957 + 0.0749957i −0.743610 − 0.668614i \(-0.766888\pi\)
0.668614 + 0.743610i \(0.266888\pi\)
\(84\) 5.37910 + 3.57953i 0.586908 + 0.390559i
\(85\) −1.55264 − 1.55264i −0.168408 − 0.168408i
\(86\) −0.785356 − 0.643285i −0.0846871 − 0.0693672i
\(87\) − 24.5271i − 2.62958i
\(88\) −5.74626 − 3.05055i −0.612553 − 0.325190i
\(89\) 5.44401i 0.577064i 0.957470 + 0.288532i \(0.0931672\pi\)
−0.957470 + 0.288532i \(0.906833\pi\)
\(90\) 7.01030 − 8.55854i 0.738951 − 0.902149i
\(91\) 0.678140 + 0.678140i 0.0710884 + 0.0710884i
\(92\) 0.793445 + 3.94994i 0.0827223 + 0.411809i
\(93\) −1.01258 + 1.01258i −0.105000 + 0.105000i
\(94\) −12.9745 + 1.29024i −1.33822 + 0.133078i
\(95\) 2.72461 0.279540
\(96\) −5.31808 + 17.8339i −0.542775 + 1.82016i
\(97\) 5.54540 0.563050 0.281525 − 0.959554i \(-0.409160\pi\)
0.281525 + 0.959554i \(0.409160\pi\)
\(98\) 8.49381 − 0.844662i 0.858004 − 0.0853238i
\(99\) −12.7234 + 12.7234i −1.27875 + 1.27875i
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.21.8 ✓ 16
3.2 odd 2 720.2.t.c.181.1 16
4.3 odd 2 320.2.l.a.241.8 16
5.2 odd 4 400.2.q.g.149.6 16
5.3 odd 4 400.2.q.h.149.3 16
5.4 even 2 400.2.l.h.101.1 16
8.3 odd 2 640.2.l.a.481.1 16
8.5 even 2 640.2.l.b.481.8 16
12.11 even 2 2880.2.t.c.2161.3 16
16.3 odd 4 320.2.l.a.81.8 16
16.5 even 4 640.2.l.b.161.8 16
16.11 odd 4 640.2.l.a.161.1 16
16.13 even 4 inner 80.2.l.a.61.8 yes 16
20.3 even 4 1600.2.q.g.49.8 16
20.7 even 4 1600.2.q.h.49.1 16
20.19 odd 2 1600.2.l.i.1201.1 16
32.3 odd 8 5120.2.a.u.1.8 8
32.13 even 8 5120.2.a.v.1.8 8
32.19 odd 8 5120.2.a.t.1.1 8
32.29 even 8 5120.2.a.s.1.1 8
48.29 odd 4 720.2.t.c.541.1 16
48.35 even 4 2880.2.t.c.721.2 16
80.3 even 4 1600.2.q.h.849.1 16
80.13 odd 4 400.2.q.g.349.6 16
80.19 odd 4 1600.2.l.i.401.1 16
80.29 even 4 400.2.l.h.301.1 16
80.67 even 4 1600.2.q.g.849.8 16
80.77 odd 4 400.2.q.h.349.3 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.2.l.a.21.8 ✓ 16 1.1 even 1 trivial
80.2.l.a.61.8 yes 16 16.13 even 4 inner
320.2.l.a.81.8 16 16.3 odd 4
320.2.l.a.241.8 16 4.3 odd 2
400.2.l.h.101.1 16 5.4 even 2
400.2.l.h.301.1 16 80.29 even 4
400.2.q.g.149.6 16 5.2 odd 4
400.2.q.g.349.6 16 80.13 odd 4
400.2.q.h.149.3 16 5.3 odd 4
400.2.q.h.349.3 16 80.77 odd 4
640.2.l.a.161.1 16 16.11 odd 4
640.2.l.a.481.1 16 8.3 odd 2
640.2.l.b.161.8 16 16.5 even 4
640.2.l.b.481.8 16 8.5 even 2
720.2.t.c.181.1 16 3.2 odd 2
720.2.t.c.541.1 16 48.29 odd 4
1600.2.l.i.401.1 16 80.19 odd 4
1600.2.l.i.1201.1 16 20.19 odd 2
1600.2.q.g.49.8 16 20.3 even 4
1600.2.q.g.849.8 16 80.67 even 4
1600.2.q.h.49.1 16 20.7 even 4
1600.2.q.h.849.1 16 80.3 even 4
2880.2.t.c.721.2 16 48.35 even 4
2880.2.t.c.2161.3 16 12.11 even 2
5120.2.a.s.1.1 8 32.29 even 8
5120.2.a.t.1.1 8 32.19 odd 8
5120.2.a.u.1.8 8 32.3 odd 8
5120.2.a.v.1.8 8 32.13 even 8