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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [80,2,Mod(3,80)] 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("80.3"); 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([2, 3, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 80.s (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2] 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: \(0.638803216170\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 27.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 80.27
Dual form 80.2.s.a.3.1

$q$-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+(-1.00000 + 1.00000i) q^{2} -2.00000 q^{3} -2.00000i q^{4} +(-2.00000 + 1.00000i) q^{5} +(2.00000 - 2.00000i) q^{6} +(-3.00000 - 3.00000i) q^{7} +(2.00000 + 2.00000i) q^{8} +1.00000 q^{9} +(1.00000 - 3.00000i) q^{10} +(-1.00000 + 1.00000i) q^{11} +4.00000i q^{12} +2.00000i q^{13} +6.00000 q^{14} +(4.00000 - 2.00000i) q^{15} -4.00000 q^{16} +(1.00000 + 1.00000i) q^{17} +(-1.00000 + 1.00000i) q^{18} +(-3.00000 + 3.00000i) q^{19} +(2.00000 + 4.00000i) q^{20} +(6.00000 + 6.00000i) q^{21} -2.00000i q^{22} +(-1.00000 + 1.00000i) q^{23} +(-4.00000 - 4.00000i) q^{24} +(3.00000 - 4.00000i) q^{25} +(-2.00000 - 2.00000i) q^{26} +4.00000 q^{27} +(-6.00000 + 6.00000i) q^{28} +(-7.00000 - 7.00000i) q^{29} +(-2.00000 + 6.00000i) q^{30} -2.00000i q^{31} +(4.00000 - 4.00000i) q^{32} +(2.00000 - 2.00000i) q^{33} -2.00000 q^{34} +(9.00000 + 3.00000i) q^{35} -2.00000i q^{36} -6.00000i q^{37} -6.00000i q^{38} -4.00000i q^{39} +(-6.00000 - 2.00000i) q^{40} +4.00000i q^{41} -12.0000 q^{42} -4.00000i q^{43} +(2.00000 + 2.00000i) q^{44} +(-2.00000 + 1.00000i) q^{45} -2.00000i q^{46} +(-7.00000 + 7.00000i) q^{47} +8.00000 q^{48} +11.0000i q^{49} +(1.00000 + 7.00000i) q^{50} +(-2.00000 - 2.00000i) q^{51} +4.00000 q^{52} -8.00000 q^{53} +(-4.00000 + 4.00000i) q^{54} +(1.00000 - 3.00000i) q^{55} -12.0000i q^{56} +(6.00000 - 6.00000i) q^{57} +14.0000 q^{58} +(3.00000 + 3.00000i) q^{59} +(-4.00000 - 8.00000i) q^{60} +(-1.00000 + 1.00000i) q^{61} +(2.00000 + 2.00000i) q^{62} +(-3.00000 - 3.00000i) q^{63} +8.00000i q^{64} +(-2.00000 - 4.00000i) q^{65} +4.00000i q^{66} +4.00000i q^{67} +(2.00000 - 2.00000i) q^{68} +(2.00000 - 2.00000i) q^{69} +(-12.0000 + 6.00000i) q^{70} +(2.00000 + 2.00000i) q^{72} +(3.00000 + 3.00000i) q^{73} +(6.00000 + 6.00000i) q^{74} +(-6.00000 + 8.00000i) q^{75} +(6.00000 + 6.00000i) q^{76} +6.00000 q^{77} +(4.00000 + 4.00000i) q^{78} +8.00000 q^{79} +(8.00000 - 4.00000i) q^{80} -11.0000 q^{81} +(-4.00000 - 4.00000i) q^{82} -2.00000 q^{83} +(12.0000 - 12.0000i) q^{84} +(-3.00000 - 1.00000i) q^{85} +(4.00000 + 4.00000i) q^{86} +(14.0000 + 14.0000i) q^{87} -4.00000 q^{88} -6.00000 q^{89} +(1.00000 - 3.00000i) q^{90} +(6.00000 - 6.00000i) q^{91} +(2.00000 + 2.00000i) q^{92} +4.00000i q^{93} -14.0000i q^{94} +(3.00000 - 9.00000i) q^{95} +(-8.00000 + 8.00000i) q^{96} +(-11.0000 - 11.0000i) q^{97} +(-11.0000 - 11.0000i) q^{98} +(-1.00000 + 1.00000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 4 q^{3} - 4 q^{5} + 4 q^{6} - 6 q^{7} + 4 q^{8} + 2 q^{9} + 2 q^{10} - 2 q^{11} + 12 q^{14} + 8 q^{15} - 8 q^{16} + 2 q^{17} - 2 q^{18} - 6 q^{19} + 4 q^{20} + 12 q^{21} - 2 q^{23} - 8 q^{24}+ \cdots - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 + 1.00000i −0.707107 + 0.707107i
\(3\) −2.00000 −1.15470 −0.577350 − 0.816497i \(-0.695913\pi\)
−0.577350 + 0.816497i \(0.695913\pi\)
\(4\) − 2.00000i − 1.00000i
\(5\) −2.00000 + 1.00000i −0.894427 + 0.447214i
\(6\) 2.00000 − 2.00000i 0.816497 − 0.816497i
\(7\) −3.00000 − 3.00000i −1.13389 − 1.13389i −0.989524 − 0.144370i \(-0.953885\pi\)
−0.144370 − 0.989524i \(-0.546115\pi\)
\(8\) 2.00000 + 2.00000i 0.707107 + 0.707107i
\(9\) 1.00000 0.333333
\(10\) 1.00000 − 3.00000i 0.316228 − 0.948683i
\(11\) −1.00000 + 1.00000i −0.301511 + 0.301511i −0.841605 − 0.540094i \(-0.818389\pi\)
0.540094 + 0.841605i \(0.318389\pi\)
\(12\) 4.00000i 1.15470i
\(13\) 2.00000i 0.554700i 0.960769 + 0.277350i \(0.0894562\pi\)
−0.960769 + 0.277350i \(0.910544\pi\)
\(14\) 6.00000 1.60357
\(15\) 4.00000 − 2.00000i 1.03280 − 0.516398i
\(16\) −4.00000 −1.00000
\(17\) 1.00000 + 1.00000i 0.242536 + 0.242536i 0.817898 − 0.575363i \(-0.195139\pi\)
−0.575363 + 0.817898i \(0.695139\pi\)
\(18\) −1.00000 + 1.00000i −0.235702 + 0.235702i
\(19\) −3.00000 + 3.00000i −0.688247 + 0.688247i −0.961844 − 0.273597i \(-0.911786\pi\)
0.273597 + 0.961844i \(0.411786\pi\)
\(20\) 2.00000 + 4.00000i 0.447214 + 0.894427i
\(21\) 6.00000 + 6.00000i 1.30931 + 1.30931i
\(22\) − 2.00000i − 0.426401i
\(23\) −1.00000 + 1.00000i −0.208514 + 0.208514i −0.803636 − 0.595121i \(-0.797104\pi\)
0.595121 + 0.803636i \(0.297104\pi\)
\(24\) −4.00000 − 4.00000i −0.816497 − 0.816497i
\(25\) 3.00000 − 4.00000i 0.600000 − 0.800000i
\(26\) −2.00000 − 2.00000i −0.392232 − 0.392232i
\(27\) 4.00000 0.769800
\(28\) −6.00000 + 6.00000i −1.13389 + 1.13389i
\(29\) −7.00000 − 7.00000i −1.29987 − 1.29987i −0.928477 − 0.371391i \(-0.878881\pi\)
−0.371391 − 0.928477i \(-0.621119\pi\)
\(30\) −2.00000 + 6.00000i −0.365148 + 1.09545i
\(31\) − 2.00000i − 0.359211i −0.983739 − 0.179605i \(-0.942518\pi\)
0.983739 − 0.179605i \(-0.0574821\pi\)
\(32\) 4.00000 − 4.00000i 0.707107 − 0.707107i
\(33\) 2.00000 − 2.00000i 0.348155 − 0.348155i
\(34\) −2.00000 −0.342997
\(35\) 9.00000 + 3.00000i 1.52128 + 0.507093i
\(36\) − 2.00000i − 0.333333i
\(37\) − 6.00000i − 0.986394i −0.869918 − 0.493197i \(-0.835828\pi\)
0.869918 − 0.493197i \(-0.164172\pi\)
\(38\) − 6.00000i − 0.973329i
\(39\) − 4.00000i − 0.640513i
\(40\) −6.00000 − 2.00000i −0.948683 − 0.316228i
\(41\) 4.00000i 0.624695i 0.949968 + 0.312348i \(0.101115\pi\)
−0.949968 + 0.312348i \(0.898885\pi\)
\(42\) −12.0000 −1.85164
\(43\) − 4.00000i − 0.609994i −0.952353 − 0.304997i \(-0.901344\pi\)
0.952353 − 0.304997i \(-0.0986555\pi\)
\(44\) 2.00000 + 2.00000i 0.301511 + 0.301511i
\(45\) −2.00000 + 1.00000i −0.298142 + 0.149071i
\(46\) − 2.00000i − 0.294884i
\(47\) −7.00000 + 7.00000i −1.02105 + 1.02105i −0.0212814 + 0.999774i \(0.506775\pi\)
−0.999774 + 0.0212814i \(0.993225\pi\)
\(48\) 8.00000 1.15470
\(49\) 11.0000i 1.57143i
\(50\) 1.00000 + 7.00000i 0.141421 + 0.989949i
\(51\) −2.00000 − 2.00000i −0.280056 − 0.280056i
\(52\) 4.00000 0.554700
\(53\) −8.00000 −1.09888 −0.549442 − 0.835532i \(-0.685160\pi\)
−0.549442 + 0.835532i \(0.685160\pi\)
\(54\) −4.00000 + 4.00000i −0.544331 + 0.544331i
\(55\) 1.00000 − 3.00000i 0.134840 − 0.404520i
\(56\) − 12.0000i − 1.60357i
\(57\) 6.00000 − 6.00000i 0.794719 − 0.794719i
\(58\) 14.0000 1.83829
\(59\) 3.00000 + 3.00000i 0.390567 + 0.390567i 0.874889 − 0.484323i \(-0.160934\pi\)
−0.484323 + 0.874889i \(0.660934\pi\)
\(60\) −4.00000 − 8.00000i −0.516398 − 1.03280i
\(61\) −1.00000 + 1.00000i −0.128037 + 0.128037i −0.768221 − 0.640184i \(-0.778858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 2.00000 + 2.00000i 0.254000 + 0.254000i
\(63\) −3.00000 − 3.00000i −0.377964 − 0.377964i
\(64\) 8.00000i 1.00000i
\(65\) −2.00000 − 4.00000i −0.248069 − 0.496139i
\(66\) 4.00000i 0.492366i
\(67\) 4.00000i 0.488678i 0.969690 + 0.244339i \(0.0785709\pi\)
−0.969690 + 0.244339i \(0.921429\pi\)
\(68\) 2.00000 − 2.00000i 0.242536 − 0.242536i
\(69\) 2.00000 − 2.00000i 0.240772 − 0.240772i
\(70\) −12.0000 + 6.00000i −1.43427 + 0.717137i
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 2.00000 + 2.00000i 0.235702 + 0.235702i
\(73\) 3.00000 + 3.00000i 0.351123 + 0.351123i 0.860527 − 0.509404i \(-0.170134\pi\)
−0.509404 + 0.860527i \(0.670134\pi\)
\(74\) 6.00000 + 6.00000i 0.697486 + 0.697486i
\(75\) −6.00000 + 8.00000i −0.692820 + 0.923760i
\(76\) 6.00000 + 6.00000i 0.688247 + 0.688247i
\(77\) 6.00000 0.683763
\(78\) 4.00000 + 4.00000i 0.452911 + 0.452911i
\(79\) 8.00000 0.900070 0.450035 − 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 8.00000 − 4.00000i 0.894427 − 0.447214i
\(81\) −11.0000 −1.22222
\(82\) −4.00000 − 4.00000i −0.441726 − 0.441726i
\(83\) −2.00000 −0.219529 −0.109764 − 0.993958i \(-0.535010\pi\)
−0.109764 + 0.993958i \(0.535010\pi\)
\(84\) 12.0000 − 12.0000i 1.30931 − 1.30931i
\(85\) −3.00000 − 1.00000i −0.325396 − 0.108465i
\(86\) 4.00000 + 4.00000i 0.431331 + 0.431331i
\(87\) 14.0000 + 14.0000i 1.50096 + 1.50096i
\(88\) −4.00000 −0.426401
\(89\) −6.00000 −0.635999 −0.317999 − 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 1.00000 − 3.00000i 0.105409 − 0.316228i
\(91\) 6.00000 − 6.00000i 0.628971 − 0.628971i
\(92\) 2.00000 + 2.00000i 0.208514 + 0.208514i
\(93\) 4.00000i 0.414781i
\(94\) − 14.0000i − 1.44399i
\(95\) 3.00000 − 9.00000i 0.307794 − 0.923381i
\(96\) −8.00000 + 8.00000i −0.816497 + 0.816497i
\(97\) −11.0000 − 11.0000i −1.11688 − 1.11688i −0.992196 − 0.124684i \(-0.960208\pi\)
−0.124684 − 0.992196i \(-0.539792\pi\)
\(98\) −11.0000 − 11.0000i −1.11117 − 1.11117i
\(99\) −1.00000 + 1.00000i −0.100504 + 0.100504i
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.s.a.27.1 yes 2
3.2 odd 2 720.2.z.d.667.1 2
4.3 odd 2 320.2.s.a.207.1 2
5.2 odd 4 400.2.j.a.43.1 2
5.3 odd 4 80.2.j.a.43.1 ✓ 2
5.4 even 2 400.2.s.a.107.1 2
8.3 odd 2 640.2.s.a.287.1 2
8.5 even 2 640.2.s.b.287.1 2
15.8 even 4 720.2.bd.a.523.1 2
16.3 odd 4 80.2.j.a.67.1 yes 2
16.5 even 4 640.2.j.b.607.1 2
16.11 odd 4 640.2.j.a.607.1 2
16.13 even 4 320.2.j.a.47.1 2
20.3 even 4 320.2.j.a.143.1 2
20.7 even 4 1600.2.j.a.143.1 2
20.19 odd 2 1600.2.s.a.207.1 2
40.3 even 4 640.2.j.b.543.1 2
40.13 odd 4 640.2.j.a.543.1 2
48.35 even 4 720.2.bd.a.307.1 2
80.3 even 4 inner 80.2.s.a.3.1 yes 2
80.13 odd 4 320.2.s.a.303.1 2
80.19 odd 4 400.2.j.a.307.1 2
80.29 even 4 1600.2.j.a.1007.1 2
80.43 even 4 640.2.s.b.223.1 2
80.53 odd 4 640.2.s.a.223.1 2
80.67 even 4 400.2.s.a.243.1 2
80.77 odd 4 1600.2.s.a.943.1 2
240.83 odd 4 720.2.z.d.163.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.2.j.a.43.1 ✓ 2 5.3 odd 4
80.2.j.a.67.1 yes 2 16.3 odd 4
80.2.s.a.3.1 yes 2 80.3 even 4 inner
80.2.s.a.27.1 yes 2 1.1 even 1 trivial
320.2.j.a.47.1 2 16.13 even 4
320.2.j.a.143.1 2 20.3 even 4
320.2.s.a.207.1 2 4.3 odd 2
320.2.s.a.303.1 2 80.13 odd 4
400.2.j.a.43.1 2 5.2 odd 4
400.2.j.a.307.1 2 80.19 odd 4
400.2.s.a.107.1 2 5.4 even 2
400.2.s.a.243.1 2 80.67 even 4
640.2.j.a.543.1 2 40.13 odd 4
640.2.j.a.607.1 2 16.11 odd 4
640.2.j.b.543.1 2 40.3 even 4
640.2.j.b.607.1 2 16.5 even 4
640.2.s.a.223.1 2 80.53 odd 4
640.2.s.a.287.1 2 8.3 odd 2
640.2.s.b.223.1 2 80.43 even 4
640.2.s.b.287.1 2 8.5 even 2
720.2.z.d.163.1 2 240.83 odd 4
720.2.z.d.667.1 2 3.2 odd 2
720.2.bd.a.307.1 2 48.35 even 4
720.2.bd.a.523.1 2 15.8 even 4
1600.2.j.a.143.1 2 20.7 even 4
1600.2.j.a.1007.1 2 80.29 even 4
1600.2.s.a.207.1 2 20.19 odd 2
1600.2.s.a.943.1 2 80.77 odd 4