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Properties

Label 80.11.h.c.79.4
Level $80$
Weight $11$
Character 80.79
Analytic conductor $50.829$
Analytic rank $0$
Dimension $20$
Inner twists $4$

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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] = [80,11,Mod(79,80)] 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("80.79"); S:= CuspForms(chi, 11); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(80, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1])) N = Newforms(chi, 11, names="a")
 
Level: \( N \) \(=\) \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 80.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [20] 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: \(50.8285802139\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 10 x^{19} - 214065 x^{18} + 1926870 x^{17} + 18968501725 x^{16} - 151791690812 x^{15} + \cdots + 19\!\cdots\!07 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{139}\cdot 3^{14}\cdot 5^{12}\cdot 7^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 79.4
Root \(197.935 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 80.79
Dual form 80.11.h.c.79.3

$q$-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)
 
\(f(q)\) \(=\) \(q-394.871 q^{3} +(540.369 + 3077.93i) q^{5} +29913.4 q^{7} +96874.0 q^{9} +105843. i q^{11} +356330. i q^{13} +(-213376. - 1.21538e6i) q^{15} -1.69563e6i q^{17} -3.65226e6i q^{19} -1.18119e7 q^{21} +1.00168e7 q^{23} +(-9.18163e6 + 3.32643e6i) q^{25} -1.49360e7 q^{27} -6.03662e6 q^{29} -1.91238e7i q^{31} -4.17942e7i q^{33} +(1.61643e7 + 9.20713e7i) q^{35} +5.62702e7i q^{37} -1.40704e8i q^{39} +2.78531e6 q^{41} +1.65546e8 q^{43} +(5.23477e7 + 2.98171e8i) q^{45} +1.89866e8 q^{47} +6.12337e8 q^{49} +6.69555e8i q^{51} -6.83741e8i q^{53} +(-3.25776e8 + 5.71941e7i) q^{55} +1.44217e9i q^{57} +2.98791e8i q^{59} -4.22796e7 q^{61} +2.89783e9 q^{63} +(-1.09676e9 + 1.92549e8i) q^{65} +7.22294e8 q^{67} -3.95533e9 q^{69} -3.22537e9i q^{71} +4.43270e8i q^{73} +(3.62556e9 - 1.31351e9i) q^{75} +3.16612e9i q^{77} +2.31113e9i q^{79} +1.77476e8 q^{81} +2.74762e9 q^{83} +(5.21902e9 - 9.16265e8i) q^{85} +2.38369e9 q^{87} +3.11807e9 q^{89} +1.06590e10i q^{91} +7.55141e9i q^{93} +(1.12414e10 - 1.97357e9i) q^{95} +1.58875e10i q^{97} +1.02534e10i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 8644 q^{5} + 531980 q^{9} - 9918720 q^{21} - 27949132 q^{25} + 72610760 q^{29} - 22549160 q^{41} - 376019996 q^{45} + 1955095660 q^{49} + 1108414520 q^{61} + 1858963008 q^{65} - 3380864640 q^{69}+ \cdots - 5262787480 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(17\) \(21\) \(31\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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
\(3\) −394.871 −1.62498 −0.812492 − 0.582973i \(-0.801889\pi\)
−0.812492 + 0.582973i \(0.801889\pi\)
\(4\) 0 0
\(5\) 540.369 + 3077.93i 0.172918 + 0.984936i
\(6\) 0 0
\(7\) 29913.4 1.77982 0.889909 − 0.456137i \(-0.150767\pi\)
0.889909 + 0.456137i \(0.150767\pi\)
\(8\) 0 0
\(9\) 96874.0 1.64057
\(10\) 0 0
\(11\) 105843.i 0.657200i 0.944469 + 0.328600i \(0.106577\pi\)
−0.944469 + 0.328600i \(0.893423\pi\)
\(12\) 0 0
\(13\) 356330.i 0.959700i 0.877351 + 0.479850i \(0.159309\pi\)
−0.877351 + 0.479850i \(0.840691\pi\)
\(14\) 0 0
\(15\) −213376. − 1.21538e6i −0.280989 − 1.60050i
\(16\) 0 0
\(17\) − 1.69563e6i − 1.19423i −0.802157 − 0.597113i \(-0.796314\pi\)
0.802157 − 0.597113i \(-0.203686\pi\)
\(18\) 0 0
\(19\) − 3.65226e6i − 1.47501i −0.675344 − 0.737503i \(-0.736005\pi\)
0.675344 − 0.737503i \(-0.263995\pi\)
\(20\) 0 0
\(21\) −1.18119e7 −2.89218
\(22\) 0 0
\(23\) 1.00168e7 1.55628 0.778142 − 0.628089i \(-0.216163\pi\)
0.778142 + 0.628089i \(0.216163\pi\)
\(24\) 0 0
\(25\) −9.18163e6 + 3.32643e6i −0.940199 + 0.340626i
\(26\) 0 0
\(27\) −1.49360e7 −1.04092
\(28\) 0 0
\(29\) −6.03662e6 −0.294309 −0.147155 − 0.989113i \(-0.547012\pi\)
−0.147155 + 0.989113i \(0.547012\pi\)
\(30\) 0 0
\(31\) − 1.91238e7i − 0.667982i −0.942576 − 0.333991i \(-0.891605\pi\)
0.942576 − 0.333991i \(-0.108395\pi\)
\(32\) 0 0
\(33\) − 4.17942e7i − 1.06794i
\(34\) 0 0
\(35\) 1.61643e7 + 9.20713e7i 0.307763 + 1.75301i
\(36\) 0 0
\(37\) 5.62702e7i 0.811465i 0.913992 + 0.405732i \(0.132984\pi\)
−0.913992 + 0.405732i \(0.867016\pi\)
\(38\) 0 0
\(39\) − 1.40704e8i − 1.55950i
\(40\) 0 0
\(41\) 2.78531e6 0.0240411 0.0120205 − 0.999928i \(-0.496174\pi\)
0.0120205 + 0.999928i \(0.496174\pi\)
\(42\) 0 0
\(43\) 1.65546e8 1.12610 0.563050 − 0.826423i \(-0.309628\pi\)
0.563050 + 0.826423i \(0.309628\pi\)
\(44\) 0 0
\(45\) 5.23477e7 + 2.98171e8i 0.283684 + 1.61586i
\(46\) 0 0
\(47\) 1.89866e8 0.827860 0.413930 − 0.910309i \(-0.364156\pi\)
0.413930 + 0.910309i \(0.364156\pi\)
\(48\) 0 0
\(49\) 6.12337e8 2.16775
\(50\) 0 0
\(51\) 6.69555e8i 1.94060i
\(52\) 0 0
\(53\) − 6.83741e8i − 1.63498i −0.575943 − 0.817490i \(-0.695365\pi\)
0.575943 − 0.817490i \(-0.304635\pi\)
\(54\) 0 0
\(55\) −3.25776e8 + 5.71941e7i −0.647300 + 0.113642i
\(56\) 0 0
\(57\) 1.44217e9i 2.39686i
\(58\) 0 0
\(59\) 2.98791e8i 0.417934i 0.977923 + 0.208967i \(0.0670101\pi\)
−0.977923 + 0.208967i \(0.932990\pi\)
\(60\) 0 0
\(61\) −4.22796e7 −0.0500589 −0.0250295 − 0.999687i \(-0.507968\pi\)
−0.0250295 + 0.999687i \(0.507968\pi\)
\(62\) 0 0
\(63\) 2.89783e9 2.91992
\(64\) 0 0
\(65\) −1.09676e9 + 1.92549e8i −0.945243 + 0.165949i
\(66\) 0 0
\(67\) 7.22294e8 0.534983 0.267491 − 0.963560i \(-0.413805\pi\)
0.267491 + 0.963560i \(0.413805\pi\)
\(68\) 0 0
\(69\) −3.95533e9 −2.52893
\(70\) 0 0
\(71\) − 3.22537e9i − 1.78767i −0.448394 − 0.893836i \(-0.648004\pi\)
0.448394 − 0.893836i \(-0.351996\pi\)
\(72\) 0 0
\(73\) 4.43270e8i 0.213823i 0.994269 + 0.106911i \(0.0340961\pi\)
−0.994269 + 0.106911i \(0.965904\pi\)
\(74\) 0 0
\(75\) 3.62556e9 − 1.31351e9i 1.52781 − 0.553512i
\(76\) 0 0
\(77\) 3.16612e9i 1.16970i
\(78\) 0 0
\(79\) 2.31113e9i 0.751085i 0.926805 + 0.375542i \(0.122544\pi\)
−0.926805 + 0.375542i \(0.877456\pi\)
\(80\) 0 0
\(81\) 1.77476e8 0.0508995
\(82\) 0 0
\(83\) 2.74762e9 0.697536 0.348768 − 0.937209i \(-0.386600\pi\)
0.348768 + 0.937209i \(0.386600\pi\)
\(84\) 0 0
\(85\) 5.21902e9 − 9.16265e8i 1.17624 − 0.206503i
\(86\) 0 0
\(87\) 2.38369e9 0.478248
\(88\) 0 0
\(89\) 3.11807e9 0.558387 0.279194 − 0.960235i \(-0.409933\pi\)
0.279194 + 0.960235i \(0.409933\pi\)
\(90\) 0 0
\(91\) 1.06590e10i 1.70809i
\(92\) 0 0
\(93\) 7.55141e9i 1.08546i
\(94\) 0 0
\(95\) 1.12414e10 − 1.97357e9i 1.45279 − 0.255055i
\(96\) 0 0
\(97\) 1.58875e10i 1.85011i 0.379830 + 0.925056i \(0.375983\pi\)
−0.379830 + 0.925056i \(0.624017\pi\)
\(98\) 0 0
\(99\) 1.02534e10i 1.07818i
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 80.11.h.c.79.4 yes 20
4.3 odd 2 inner 80.11.h.c.79.18 yes 20
5.4 even 2 inner 80.11.h.c.79.17 yes 20
20.19 odd 2 inner 80.11.h.c.79.3 ✓ 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.11.h.c.79.3 ✓ 20 20.19 odd 2 inner
80.11.h.c.79.4 yes 20 1.1 even 1 trivial
80.11.h.c.79.17 yes 20 5.4 even 2 inner
80.11.h.c.79.18 yes 20 4.3 odd 2 inner