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Properties

Label 80.9.p.a.17.2
Level $80$
Weight $9$
Character 80.17
Analytic conductor $32.590$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [80,9,Mod(17,80)] 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("80.17"); S:= CuspForms(chi, 9); 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, 0, 1])) N = Newforms(chi, 9, names="a")
 
Level: \( N \) \(=\) \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 80.p (of order \(4\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,-86] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(32.5902888049\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{601})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 301x^{2} + 22500 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 10)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 17.2
Root \(-11.7577i\) of defining polynomial
Character \(\chi\) \(=\) 80.17
Dual form 80.9.p.a.33.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(39.7883 - 39.7883i) q^{3} +(-401.365 + 479.094i) q^{5} +(-144.447 - 144.447i) q^{7} +3394.79i q^{9} -13600.3 q^{11} +(30883.7 - 30883.7i) q^{13} +(3092.72 + 35031.9i) q^{15} +(4949.02 + 4949.02i) q^{17} -176579. i q^{19} -11494.6 q^{21} +(229482. - 229482. i) q^{23} +(-68437.7 - 384583. i) q^{25} +(396123. + 396123. i) q^{27} +144167. i q^{29} -1.50971e6 q^{31} +(-541132. + 541132. i) q^{33} +(127179. - 11227.8i) q^{35} +(-1.60715e6 - 1.60715e6i) q^{37} -2.45761e6i q^{39} +3.62276e6 q^{41} +(575576. - 575576. i) q^{43} +(-1.62642e6 - 1.36255e6i) q^{45} +(-3.11186e6 - 3.11186e6i) q^{47} -5.72307e6i q^{49} +393826. q^{51} +(8.16813e6 - 8.16813e6i) q^{53} +(5.45868e6 - 6.51582e6i) q^{55} +(-7.02575e6 - 7.02575e6i) q^{57} -1.54905e7i q^{59} -1.92688e7 q^{61} +(490366. - 490366. i) q^{63} +(2.40057e6 + 2.71918e7i) q^{65} +(1.00075e7 + 1.00075e7i) q^{67} -1.82614e7i q^{69} -1.84257e7 q^{71} +(2.45702e7 - 2.45702e7i) q^{73} +(-1.80249e7 - 1.25789e7i) q^{75} +(1.96452e6 + 1.96452e6i) q^{77} +2.17799e7i q^{79} +9.24891e6 q^{81} +(-4.00189e6 + 4.00189e6i) q^{83} +(-4.35741e6 + 384685. i) q^{85} +(5.73617e6 + 5.73617e6i) q^{87} +6.14525e7i q^{89} -8.92209e6 q^{91} +(-6.00686e7 + 6.00686e7i) q^{93} +(8.45978e7 + 7.08724e7i) q^{95} +(1.55814e7 + 1.55814e7i) q^{97} -4.61701e7i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 86 q^{3} - 870 q^{5} - 5726 q^{7} + 14732 q^{11} + 45576 q^{13} + 115090 q^{15} + 3616 q^{17} + 877268 q^{21} + 456794 q^{23} - 913600 q^{25} + 889240 q^{27} - 4672348 q^{31} - 4553788 q^{33} - 2956030 q^{35}+ \cdots + 34113396 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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 0
\(3\) 39.7883 − 39.7883i 0.491213 − 0.491213i −0.417475 − 0.908688i \(-0.637085\pi\)
0.908688 + 0.417475i \(0.137085\pi\)
\(4\) 0 0
\(5\) −401.365 + 479.094i −0.642184 + 0.766551i
\(6\) 0 0
\(7\) −144.447 − 144.447i −0.0601610 − 0.0601610i 0.676386 − 0.736547i \(-0.263545\pi\)
−0.736547 + 0.676386i \(0.763545\pi\)
\(8\) 0 0
\(9\) 3394.79i 0.517420i
\(10\) 0 0
\(11\) −13600.3 −0.928918 −0.464459 − 0.885595i \(-0.653751\pi\)
−0.464459 + 0.885595i \(0.653751\pi\)
\(12\) 0 0
\(13\) 30883.7 − 30883.7i 1.08132 − 1.08132i 0.0849366 − 0.996386i \(-0.472931\pi\)
0.996386 − 0.0849366i \(-0.0270688\pi\)
\(14\) 0 0
\(15\) 3092.72 + 35031.9i 0.0610908 + 0.691989i
\(16\) 0 0
\(17\) 4949.02 + 4949.02i 0.0592549 + 0.0592549i 0.736113 − 0.676858i \(-0.236659\pi\)
−0.676858 + 0.736113i \(0.736659\pi\)
\(18\) 0 0
\(19\) − 176579.i − 1.35495i −0.735546 − 0.677475i \(-0.763074\pi\)
0.735546 − 0.677475i \(-0.236926\pi\)
\(20\) 0 0
\(21\) −11494.6 −0.0591038
\(22\) 0 0
\(23\) 229482. − 229482.i 0.820043 − 0.820043i −0.166071 − 0.986114i \(-0.553108\pi\)
0.986114 + 0.166071i \(0.0531080\pi\)
\(24\) 0 0
\(25\) −68437.7 − 384583.i −0.175200 − 0.984533i
\(26\) 0 0
\(27\) 396123. + 396123.i 0.745376 + 0.745376i
\(28\) 0 0
\(29\) 144167.i 0.203833i 0.994793 + 0.101917i \(0.0324975\pi\)
−0.994793 + 0.101917i \(0.967503\pi\)
\(30\) 0 0
\(31\) −1.50971e6 −1.63473 −0.817365 − 0.576120i \(-0.804566\pi\)
−0.817365 + 0.576120i \(0.804566\pi\)
\(32\) 0 0
\(33\) −541132. + 541132.i −0.456297 + 0.456297i
\(34\) 0 0
\(35\) 127179. − 11227.8i 0.0847509 − 0.00748206i
\(36\) 0 0
\(37\) −1.60715e6 − 1.60715e6i −0.857532 − 0.857532i 0.133515 − 0.991047i \(-0.457374\pi\)
−0.991047 + 0.133515i \(0.957374\pi\)
\(38\) 0 0
\(39\) − 2.45761e6i − 1.06232i
\(40\) 0 0
\(41\) 3.62276e6 1.28205 0.641023 − 0.767522i \(-0.278510\pi\)
0.641023 + 0.767522i \(0.278510\pi\)
\(42\) 0 0
\(43\) 575576. − 575576.i 0.168356 − 0.168356i −0.617900 − 0.786256i \(-0.712017\pi\)
0.786256 + 0.617900i \(0.212017\pi\)
\(44\) 0 0
\(45\) −1.62642e6 − 1.36255e6i −0.396628 − 0.332278i
\(46\) 0 0
\(47\) −3.11186e6 − 3.11186e6i −0.637718 − 0.637718i 0.312274 − 0.949992i \(-0.398909\pi\)
−0.949992 + 0.312274i \(0.898909\pi\)
\(48\) 0 0
\(49\) − 5.72307e6i − 0.992761i
\(50\) 0 0
\(51\) 393826. 0.0582135
\(52\) 0 0
\(53\) 8.16813e6 − 8.16813e6i 1.03519 − 1.03519i 0.0358294 − 0.999358i \(-0.488593\pi\)
0.999358 − 0.0358294i \(-0.0114073\pi\)
\(54\) 0 0
\(55\) 5.45868e6 − 6.51582e6i 0.596536 − 0.712063i
\(56\) 0 0
\(57\) −7.02575e6 − 7.02575e6i −0.665569 − 0.665569i
\(58\) 0 0
\(59\) − 1.54905e7i − 1.27837i −0.769051 − 0.639187i \(-0.779271\pi\)
0.769051 − 0.639187i \(-0.220729\pi\)
\(60\) 0 0
\(61\) −1.92688e7 −1.39167 −0.695833 − 0.718204i \(-0.744965\pi\)
−0.695833 + 0.718204i \(0.744965\pi\)
\(62\) 0 0
\(63\) 490366. − 490366.i 0.0311285 − 0.0311285i
\(64\) 0 0
\(65\) 2.40057e6 + 2.71918e7i 0.134481 + 1.52330i
\(66\) 0 0
\(67\) 1.00075e7 + 1.00075e7i 0.496624 + 0.496624i 0.910385 − 0.413761i \(-0.135785\pi\)
−0.413761 + 0.910385i \(0.635785\pi\)
\(68\) 0 0
\(69\) − 1.82614e7i − 0.805632i
\(70\) 0 0
\(71\) −1.84257e7 −0.725087 −0.362543 − 0.931967i \(-0.618092\pi\)
−0.362543 + 0.931967i \(0.618092\pi\)
\(72\) 0 0
\(73\) 2.45702e7 − 2.45702e7i 0.865203 − 0.865203i −0.126734 − 0.991937i \(-0.540449\pi\)
0.991937 + 0.126734i \(0.0404494\pi\)
\(74\) 0 0
\(75\) −1.80249e7 − 1.25789e7i −0.569676 − 0.397555i
\(76\) 0 0
\(77\) 1.96452e6 + 1.96452e6i 0.0558847 + 0.0558847i
\(78\) 0 0
\(79\) 2.17799e7i 0.559176i 0.960120 + 0.279588i \(0.0901978\pi\)
−0.960120 + 0.279588i \(0.909802\pi\)
\(80\) 0 0
\(81\) 9.24891e6 0.214857
\(82\) 0 0
\(83\) −4.00189e6 + 4.00189e6i −0.0843242 + 0.0843242i −0.748011 − 0.663687i \(-0.768991\pi\)
0.663687 + 0.748011i \(0.268991\pi\)
\(84\) 0 0
\(85\) −4.35741e6 + 384685.i −0.0834744 + 0.00736936i
\(86\) 0 0
\(87\) 5.73617e6 + 5.73617e6i 0.100126 + 0.100126i
\(88\) 0 0
\(89\) 6.14525e7i 0.979444i 0.871879 + 0.489722i \(0.162902\pi\)
−0.871879 + 0.489722i \(0.837098\pi\)
\(90\) 0 0
\(91\) −8.92209e6 −0.130107
\(92\) 0 0
\(93\) −6.00686e7 + 6.00686e7i −0.803001 + 0.803001i
\(94\) 0 0
\(95\) 8.45978e7 + 7.08724e7i 1.03864 + 0.870127i
\(96\) 0 0
\(97\) 1.55814e7 + 1.55814e7i 0.176003 + 0.176003i 0.789611 − 0.613608i \(-0.210282\pi\)
−0.613608 + 0.789611i \(0.710282\pi\)
\(98\) 0 0
\(99\) − 4.61701e7i − 0.480640i
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 80.9.p.a.17.2 4
4.3 odd 2 10.9.c.b.7.1 yes 4
5.3 odd 4 inner 80.9.p.a.33.2 4
12.11 even 2 90.9.g.a.37.2 4
20.3 even 4 10.9.c.b.3.1 ✓ 4
20.7 even 4 50.9.c.b.43.2 4
20.19 odd 2 50.9.c.b.7.2 4
60.23 odd 4 90.9.g.a.73.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.9.c.b.3.1 ✓ 4 20.3 even 4
10.9.c.b.7.1 yes 4 4.3 odd 2
50.9.c.b.7.2 4 20.19 odd 2
50.9.c.b.43.2 4 20.7 even 4
80.9.p.a.17.2 4 1.1 even 1 trivial
80.9.p.a.33.2 4 5.3 odd 4 inner
90.9.g.a.37.2 4 12.11 even 2
90.9.g.a.73.2 4 60.23 odd 4