↑

Properties

Label 80.2.s.b.27.7
Level $80$
Weight $2$
Character 80.27
Analytic conductor $0.639$
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] = [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 = [18] 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: \(0\)
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: \( x^{18} + 2 x^{16} - 4 x^{15} - 5 x^{14} - 14 x^{13} - 10 x^{12} + 6 x^{11} + 37 x^{10} + 70 x^{9} + \cdots + 512 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 27.7
Root \(-0.635486 + 1.26339i\) of defining polynomial
Character \(\chi\) \(=\) 80.27
Dual form 80.2.s.b.3.7

$q$-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+(0.828280 - 1.14628i) q^{2} +0.692712 q^{3} +(-0.627905 - 1.89888i) q^{4} +(-0.245325 + 2.22257i) q^{5} +(0.573759 - 0.794040i) q^{6} +(-0.343872 - 0.343872i) q^{7} +(-2.69672 - 0.853049i) q^{8} -2.52015 q^{9} +(2.34448 + 2.12212i) q^{10} +(0.843672 - 0.843672i) q^{11} +(-0.434957 - 1.31538i) q^{12} +3.68390i q^{13} +(-0.678995 + 0.109350i) q^{14} +(-0.169939 + 1.53960i) q^{15} +(-3.21147 + 2.38463i) q^{16} +(0.412137 + 0.412137i) q^{17} +(-2.08739 + 2.88879i) q^{18} +(5.37721 - 5.37721i) q^{19} +(4.37443 - 0.929720i) q^{20} +(-0.238204 - 0.238204i) q^{21} +(-0.268286 - 1.66588i) q^{22} +(-3.08788 + 3.08788i) q^{23} +(-1.86805 - 0.590917i) q^{24} +(-4.87963 - 1.09050i) q^{25} +(4.22278 + 3.05130i) q^{26} -3.82387 q^{27} +(-0.437052 + 0.868890i) q^{28} +(4.22969 + 4.22969i) q^{29} +(1.62405 + 1.47002i) q^{30} -8.75966i q^{31} +(0.0734474 + 5.65638i) q^{32} +(0.584422 - 0.584422i) q^{33} +(0.813788 - 0.131059i) q^{34} +(0.848640 - 0.679919i) q^{35} +(1.58241 + 4.78546i) q^{36} -5.41752i q^{37} +(-1.70994 - 10.6176i) q^{38} +2.55188i q^{39} +(2.55753 - 5.78438i) q^{40} -2.54777i q^{41} +(-0.470348 + 0.0757484i) q^{42} -4.30732i q^{43} +(-2.13178 - 1.07228i) q^{44} +(0.618255 - 5.60121i) q^{45} +(0.981939 + 6.09720i) q^{46} +(-4.56972 + 4.56972i) q^{47} +(-2.22462 + 1.65186i) q^{48} -6.76350i q^{49} +(-5.29172 + 4.69017i) q^{50} +(0.285492 + 0.285492i) q^{51} +(6.99528 - 2.31314i) q^{52} +6.07536 q^{53} +(-3.16724 + 4.38322i) q^{54} +(1.66815 + 2.08209i) q^{55} +(0.633987 + 1.22067i) q^{56} +(3.72486 - 3.72486i) q^{57} +(8.35177 - 1.34503i) q^{58} +(7.33694 + 7.33694i) q^{59} +(3.03022 - 0.644028i) q^{60} +(-4.81576 + 4.81576i) q^{61} +(-10.0410 - 7.25545i) q^{62} +(0.866609 + 0.866609i) q^{63} +(6.54461 + 4.60087i) q^{64} +(-8.18773 - 0.903753i) q^{65} +(-0.185845 - 1.15397i) q^{66} +14.3626i q^{67} +(0.523815 - 1.04138i) q^{68} +(-2.13901 + 2.13901i) q^{69} +(-0.0764647 - 1.53594i) q^{70} -2.97605 q^{71} +(6.79614 + 2.14981i) q^{72} +(6.87152 + 6.87152i) q^{73} +(-6.20998 - 4.48722i) q^{74} +(-3.38018 - 0.755404i) q^{75} +(-13.5870 - 6.83429i) q^{76} -0.580231 q^{77} +(2.92517 + 2.11367i) q^{78} -10.1654 q^{79} +(-4.51215 - 7.72273i) q^{80} +4.91161 q^{81} +(-2.92046 - 2.11027i) q^{82} -7.15276 q^{83} +(-0.302751 + 0.601890i) q^{84} +(-1.01711 + 0.814896i) q^{85} +(-4.93739 - 3.56767i) q^{86} +(2.92996 + 2.92996i) q^{87} +(-2.99484 + 1.55545i) q^{88} -1.10953 q^{89} +(-5.90845 - 5.34806i) q^{90} +(1.26679 - 1.26679i) q^{91} +(7.80240 + 3.92461i) q^{92} -6.06792i q^{93} +(1.45316 + 9.02318i) q^{94} +(10.6321 + 13.2704i) q^{95} +(0.0508779 + 3.91824i) q^{96} +(7.15920 + 7.15920i) q^{97} +(-7.75285 - 5.60207i) q^{98} +(-2.12618 + 2.12618i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 4 q^{4} + 2 q^{5} - 8 q^{6} + 2 q^{7} - 12 q^{8} + 10 q^{9} - 2 q^{11} - 12 q^{14} - 20 q^{15} - 6 q^{17} - 24 q^{18} - 2 q^{19} - 12 q^{20} - 16 q^{21} + 12 q^{22} - 2 q^{23} - 4 q^{24} - 6 q^{25}+ \cdots + 22 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.828280 − 1.14628i 0.585682 − 0.810541i
\(3\) 0.692712 0.399937 0.199969 − 0.979802i \(-0.435916\pi\)
0.199969 + 0.979802i \(0.435916\pi\)
\(4\) −0.627905 − 1.89888i −0.313952 − 0.949439i
\(5\) −0.245325 + 2.22257i −0.109713 + 0.993963i
\(6\) 0.573759 − 0.794040i 0.234236 − 0.324166i
\(7\) −0.343872 − 0.343872i −0.129971 − 0.129971i 0.639129 − 0.769100i \(-0.279295\pi\)
−0.769100 + 0.639129i \(0.779295\pi\)
\(8\) −2.69672 − 0.853049i −0.953435 − 0.301598i
\(9\) −2.52015 −0.840050
\(10\) 2.34448 + 2.12212i 0.741391 + 0.671073i
\(11\) 0.843672 − 0.843672i 0.254377 − 0.254377i −0.568386 − 0.822762i \(-0.692432\pi\)
0.822762 + 0.568386i \(0.192432\pi\)
\(12\) −0.434957 − 1.31538i −0.125561 − 0.379716i
\(13\) 3.68390i 1.02173i 0.859661 + 0.510865i \(0.170675\pi\)
−0.859661 + 0.510865i \(0.829325\pi\)
\(14\) −0.678995 + 0.109350i −0.181469 + 0.0292251i
\(15\) −0.169939 + 1.53960i −0.0438782 + 0.397523i
\(16\) −3.21147 + 2.38463i −0.802868 + 0.596157i
\(17\) 0.412137 + 0.412137i 0.0999579 + 0.0999579i 0.755317 − 0.655359i \(-0.227483\pi\)
−0.655359 + 0.755317i \(0.727483\pi\)
\(18\) −2.08739 + 2.88879i −0.492003 + 0.680895i
\(19\) 5.37721 − 5.37721i 1.23362 − 1.23362i 0.271052 − 0.962565i \(-0.412629\pi\)
0.962565 − 0.271052i \(-0.0873714\pi\)
\(20\) 4.37443 − 0.929720i 0.978152 − 0.207892i
\(21\) −0.238204 − 0.238204i −0.0519804 − 0.0519804i
\(22\) −0.268286 − 1.66588i −0.0571987 − 0.355167i
\(23\) −3.08788 + 3.08788i −0.643868 + 0.643868i −0.951504 − 0.307636i \(-0.900462\pi\)
0.307636 + 0.951504i \(0.400462\pi\)
\(24\) −1.86805 − 0.590917i −0.381314 − 0.120621i
\(25\) −4.87963 − 1.09050i −0.975926 − 0.218101i
\(26\) 4.22278 + 3.05130i 0.828154 + 0.598410i
\(27\) −3.82387 −0.735905
\(28\) −0.437052 + 0.868890i −0.0825951 + 0.164205i
\(29\) 4.22969 + 4.22969i 0.785434 + 0.785434i 0.980742 − 0.195308i \(-0.0625707\pi\)
−0.195308 + 0.980742i \(0.562571\pi\)
\(30\) 1.62405 + 1.47002i 0.296510 + 0.268387i
\(31\) − 8.75966i − 1.57328i −0.617411 − 0.786641i \(-0.711818\pi\)
0.617411 − 0.786641i \(-0.288182\pi\)
\(32\) 0.0734474 + 5.65638i 0.0129838 + 0.999916i
\(33\) 0.584422 − 0.584422i 0.101735 − 0.101735i
\(34\) 0.813788 − 0.131059i 0.139564 − 0.0224764i
\(35\) 0.848640 − 0.679919i 0.143446 − 0.114927i
\(36\) 1.58241 + 4.78546i 0.263736 + 0.797576i
\(37\) − 5.41752i − 0.890634i −0.895373 − 0.445317i \(-0.853091\pi\)
0.895373 − 0.445317i \(-0.146909\pi\)
\(38\) −1.70994 − 10.6176i −0.277389 − 1.72240i
\(39\) 2.55188i 0.408628i
\(40\) 2.55753 − 5.78438i 0.404382 − 0.914590i
\(41\) − 2.54777i − 0.397895i −0.980010 − 0.198948i \(-0.936248\pi\)
0.980010 − 0.198948i \(-0.0637524\pi\)
\(42\) −0.470348 + 0.0757484i −0.0725763 + 0.0116882i
\(43\) − 4.30732i − 0.656861i −0.944528 − 0.328430i \(-0.893480\pi\)
0.944528 − 0.328430i \(-0.106520\pi\)
\(44\) −2.13178 − 1.07228i −0.321377 − 0.161653i
\(45\) 0.618255 − 5.60121i 0.0921641 − 0.834979i
\(46\) 0.981939 + 6.09720i 0.144779 + 0.898983i
\(47\) −4.56972 + 4.56972i −0.666562 + 0.666562i −0.956919 − 0.290356i \(-0.906226\pi\)
0.290356 + 0.956919i \(0.406226\pi\)
\(48\) −2.22462 + 1.65186i −0.321097 + 0.238425i
\(49\) − 6.76350i − 0.966215i
\(50\) −5.29172 + 4.69017i −0.748362 + 0.663290i
\(51\) 0.285492 + 0.285492i 0.0399769 + 0.0399769i
\(52\) 6.99528 − 2.31314i 0.970071 − 0.320775i
\(53\) 6.07536 0.834515 0.417257 − 0.908788i \(-0.362991\pi\)
0.417257 + 0.908788i \(0.362991\pi\)
\(54\) −3.16724 + 4.38322i −0.431007 + 0.596481i
\(55\) 1.66815 + 2.08209i 0.224933 + 0.280749i
\(56\) 0.633987 + 1.22067i 0.0847201 + 0.163118i
\(57\) 3.72486 − 3.72486i 0.493369 − 0.493369i
\(58\) 8.35177 − 1.34503i 1.09664 − 0.176611i
\(59\) 7.33694 + 7.33694i 0.955189 + 0.955189i 0.999038 − 0.0438495i \(-0.0139622\pi\)
−0.0438495 + 0.999038i \(0.513962\pi\)
\(60\) 3.03022 − 0.644028i 0.391200 − 0.0831437i
\(61\) −4.81576 + 4.81576i −0.616595 + 0.616595i −0.944656 − 0.328062i \(-0.893605\pi\)
0.328062 + 0.944656i \(0.393605\pi\)
\(62\) −10.0410 − 7.25545i −1.27521 − 0.921444i
\(63\) 0.866609 + 0.866609i 0.109183 + 0.109183i
\(64\) 6.54461 + 4.60087i 0.818077 + 0.575109i
\(65\) −8.18773 − 0.903753i −1.01556 − 0.112097i
\(66\) −0.185845 − 1.15397i −0.0228759 − 0.142044i
\(67\) 14.3626i 1.75467i 0.479880 + 0.877334i \(0.340680\pi\)
−0.479880 + 0.877334i \(0.659320\pi\)
\(68\) 0.523815 − 1.04138i 0.0635219 − 0.126286i
\(69\) −2.13901 + 2.13901i −0.257507 + 0.257507i
\(70\) −0.0764647 − 1.53594i −0.00913928 − 0.183580i
\(71\) −2.97605 −0.353193 −0.176596 − 0.984283i \(-0.556509\pi\)
−0.176596 + 0.984283i \(0.556509\pi\)
\(72\) 6.79614 + 2.14981i 0.800933 + 0.253358i
\(73\) 6.87152 + 6.87152i 0.804250 + 0.804250i 0.983757 − 0.179507i \(-0.0574501\pi\)
−0.179507 + 0.983757i \(0.557450\pi\)
\(74\) −6.20998 − 4.48722i −0.721895 − 0.521629i
\(75\) −3.38018 − 0.755404i −0.390309 − 0.0872266i
\(76\) −13.5870 − 6.83429i −1.55854 − 0.783947i
\(77\) −0.580231 −0.0661234
\(78\) 2.92517 + 2.11367i 0.331210 + 0.239326i
\(79\) −10.1654 −1.14369 −0.571847 − 0.820360i \(-0.693773\pi\)
−0.571847 + 0.820360i \(0.693773\pi\)
\(80\) −4.51215 − 7.72273i −0.504473 − 0.863427i
\(81\) 4.91161 0.545734
\(82\) −2.92046 − 2.11027i −0.322510 − 0.233040i
\(83\) −7.15276 −0.785118 −0.392559 − 0.919727i \(-0.628410\pi\)
−0.392559 + 0.919727i \(0.628410\pi\)
\(84\) −0.302751 + 0.601890i −0.0330329 + 0.0656716i
\(85\) −1.01711 + 0.814896i −0.110321 + 0.0883878i
\(86\) −4.93739 − 3.56767i −0.532412 − 0.384712i
\(87\) 2.92996 + 2.92996i 0.314124 + 0.314124i
\(88\) −2.99484 + 1.55545i −0.319251 + 0.165812i
\(89\) −1.10953 −0.117610 −0.0588050 − 0.998269i \(-0.518729\pi\)
−0.0588050 + 0.998269i \(0.518729\pi\)
\(90\) −5.90845 − 5.34806i −0.622806 − 0.563735i
\(91\) 1.26679 − 1.26679i 0.132796 − 0.132796i
\(92\) 7.80240 + 3.92461i 0.813457 + 0.409169i
\(93\) − 6.06792i − 0.629214i
\(94\) 1.45316 + 9.02318i 0.149882 + 0.930670i
\(95\) 10.6321 + 13.2704i 1.09083 + 1.36151i
\(96\) 0.0508779 + 3.91824i 0.00519270 + 0.399904i
\(97\) 7.15920 + 7.15920i 0.726906 + 0.726906i 0.970002 − 0.243096i \(-0.0781630\pi\)
−0.243096 + 0.970002i \(0.578163\pi\)
\(98\) −7.75285 − 5.60207i −0.783156 − 0.565895i
\(99\) −2.12618 + 2.12618i −0.213689 + 0.213689i
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.b.27.7 yes 18
3.2 odd 2 720.2.z.g.667.3 18
4.3 odd 2 320.2.s.b.207.4 18
5.2 odd 4 400.2.j.d.43.7 18
5.3 odd 4 80.2.j.b.43.3 ✓ 18
5.4 even 2 400.2.s.d.107.3 18
8.3 odd 2 640.2.s.c.287.6 18
8.5 even 2 640.2.s.d.287.4 18
15.8 even 4 720.2.bd.g.523.7 18
16.3 odd 4 80.2.j.b.67.3 yes 18
16.5 even 4 640.2.j.c.607.4 18
16.11 odd 4 640.2.j.d.607.6 18
16.13 even 4 320.2.j.b.47.6 18
20.3 even 4 320.2.j.b.143.4 18
20.7 even 4 1600.2.j.d.143.6 18
20.19 odd 2 1600.2.s.d.207.6 18
40.3 even 4 640.2.j.c.543.6 18
40.13 odd 4 640.2.j.d.543.4 18
48.35 even 4 720.2.bd.g.307.7 18
80.3 even 4 inner 80.2.s.b.3.7 yes 18
80.13 odd 4 320.2.s.b.303.4 18
80.19 odd 4 400.2.j.d.307.7 18
80.29 even 4 1600.2.j.d.1007.4 18
80.43 even 4 640.2.s.d.223.4 18
80.53 odd 4 640.2.s.c.223.6 18
80.67 even 4 400.2.s.d.243.3 18
80.77 odd 4 1600.2.s.d.943.6 18
240.83 odd 4 720.2.z.g.163.3 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.2.j.b.43.3 ✓ 18 5.3 odd 4
80.2.j.b.67.3 yes 18 16.3 odd 4
80.2.s.b.3.7 yes 18 80.3 even 4 inner
80.2.s.b.27.7 yes 18 1.1 even 1 trivial
320.2.j.b.47.6 18 16.13 even 4
320.2.j.b.143.4 18 20.3 even 4
320.2.s.b.207.4 18 4.3 odd 2
320.2.s.b.303.4 18 80.13 odd 4
400.2.j.d.43.7 18 5.2 odd 4
400.2.j.d.307.7 18 80.19 odd 4
400.2.s.d.107.3 18 5.4 even 2
400.2.s.d.243.3 18 80.67 even 4
640.2.j.c.543.6 18 40.3 even 4
640.2.j.c.607.4 18 16.5 even 4
640.2.j.d.543.4 18 40.13 odd 4
640.2.j.d.607.6 18 16.11 odd 4
640.2.s.c.223.6 18 80.53 odd 4
640.2.s.c.287.6 18 8.3 odd 2
640.2.s.d.223.4 18 80.43 even 4
640.2.s.d.287.4 18 8.5 even 2
720.2.z.g.163.3 18 240.83 odd 4
720.2.z.g.667.3 18 3.2 odd 2
720.2.bd.g.307.7 18 48.35 even 4
720.2.bd.g.523.7 18 15.8 even 4
1600.2.j.d.143.6 18 20.7 even 4
1600.2.j.d.1007.4 18 80.29 even 4
1600.2.s.d.207.6 18 20.19 odd 2
1600.2.s.d.943.6 18 80.77 odd 4