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Properties

Label 80.11.p.f.17.2
Level $80$
Weight $11$
Character 80.17
Analytic conductor $50.829$
Analytic rank $0$
Dimension $14$
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] = [80,11,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, 11); 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, 11, names="a")
 
Level: \( N \) \(=\) \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 80.p (of order \(4\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [14,0,190] 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: \(50.8285802139\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 57366 x^{11} + 226018348 x^{10} - 639855644 x^{9} + 1645428978 x^{8} - 3566916972676 x^{7} + \cdots + 89\!\cdots\!72 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{47}\cdot 3^{4}\cdot 5^{14} \)
Twist minimal: no (minimal twist has level 40)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 17.2
Root \(-4.46802 + 4.46802i\) of defining polynomial
Character \(\chi\) \(=\) 80.17
Dual form 80.11.p.f.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)
 
\(f(q)\) \(=\) \(q+(-236.074 + 236.074i) q^{3} +(-2741.95 - 1499.11i) q^{5} +(-1877.44 - 1877.44i) q^{7} -52413.3i q^{9} +258101. q^{11} +(-182002. + 182002. i) q^{13} +(1.00121e6 - 293402. i) q^{15} +(824087. + 824087. i) q^{17} +2.00069e6i q^{19} +886431. q^{21} +(162860. - 162860. i) q^{23} +(5.27095e6 + 8.22098e6i) q^{25} +(-1.56653e6 - 1.56653e6i) q^{27} +1.69849e7i q^{29} +4.69161e7 q^{31} +(-6.09311e7 + 6.09311e7i) q^{33} +(2.33335e6 + 7.96234e6i) q^{35} +(-7.70531e7 - 7.70531e7i) q^{37} -8.59321e7i q^{39} +2.92307e6 q^{41} +(1.32675e8 - 1.32675e8i) q^{43} +(-7.85733e7 + 1.43715e8i) q^{45} +(-1.20868e8 - 1.20868e8i) q^{47} -2.75426e8i q^{49} -3.89092e8 q^{51} +(-3.49065e8 + 3.49065e8i) q^{53} +(-7.07701e8 - 3.86923e8i) q^{55} +(-4.72312e8 - 4.72312e8i) q^{57} +9.91185e8i q^{59} -5.60533e8 q^{61} +(-9.84028e7 + 9.84028e7i) q^{63} +(7.71883e8 - 2.26199e8i) q^{65} +(9.67797e8 + 9.67797e8i) q^{67} +7.68942e7i q^{69} -2.10396e9 q^{71} +(-3.40584e8 + 3.40584e8i) q^{73} +(-3.18510e9 - 6.96425e8i) q^{75} +(-4.84570e8 - 4.84570e8i) q^{77} +2.81760e9i q^{79} +3.83458e9 q^{81} +(-2.90486e9 + 2.90486e9i) q^{83} +(-1.02421e9 - 3.49501e9i) q^{85} +(-4.00971e9 - 4.00971e9i) q^{87} -2.05774e9i q^{89} +6.83396e8 q^{91} +(-1.10757e10 + 1.10757e10i) q^{93} +(2.99926e9 - 5.48580e9i) q^{95} +(-1.04120e10 - 1.04120e10i) q^{97} -1.35279e10i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 190 q^{3} - 510 q^{5} + 4646 q^{7} + 57004 q^{11} + 665006 q^{13} - 1478410 q^{15} + 665638 q^{17} - 17091900 q^{21} - 4904618 q^{23} - 10103850 q^{25} - 33966320 q^{27} - 11624148 q^{31} + 2326100 q^{33}+ \cdots + 1599867854 q^{97}+O(q^{100}) \) 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
\(3\) −236.074 + 236.074i −0.971500 + 0.971500i −0.999605 − 0.0281053i \(-0.991053\pi\)
0.0281053 + 0.999605i \(0.491053\pi\)
\(4\) 0 0
\(5\) −2741.95 − 1499.11i −0.877424 − 0.479716i
\(6\) 0 0
\(7\) −1877.44 − 1877.44i −0.111706 − 0.111706i 0.649045 − 0.760750i \(-0.275169\pi\)
−0.760750 + 0.649045i \(0.775169\pi\)
\(8\) 0 0
\(9\) − 52413.3i − 0.887623i
\(10\) 0 0
\(11\) 258101. 1.60261 0.801303 − 0.598258i \(-0.204140\pi\)
0.801303 + 0.598258i \(0.204140\pi\)
\(12\) 0 0
\(13\) −182002. + 182002.i −0.490185 + 0.490185i −0.908364 − 0.418180i \(-0.862668\pi\)
0.418180 + 0.908364i \(0.362668\pi\)
\(14\) 0 0
\(15\) 1.00121e6 − 293402.i 1.31846 − 0.386374i
\(16\) 0 0
\(17\) 824087. + 824087.i 0.580402 + 0.580402i 0.935014 − 0.354612i \(-0.115387\pi\)
−0.354612 + 0.935014i \(0.615387\pi\)
\(18\) 0 0
\(19\) 2.00069e6i 0.808002i 0.914759 + 0.404001i \(0.132381\pi\)
−0.914759 + 0.404001i \(0.867619\pi\)
\(20\) 0 0
\(21\) 886431. 0.217044
\(22\) 0 0
\(23\) 162860. − 162860.i 0.0253032 − 0.0253032i −0.694342 − 0.719645i \(-0.744304\pi\)
0.719645 + 0.694342i \(0.244304\pi\)
\(24\) 0 0
\(25\) 5.27095e6 + 8.22098e6i 0.539746 + 0.841828i
\(26\) 0 0
\(27\) −1.56653e6 − 1.56653e6i −0.109174 − 0.109174i
\(28\) 0 0
\(29\) 1.69849e7i 0.828083i 0.910258 + 0.414041i \(0.135883\pi\)
−0.910258 + 0.414041i \(0.864117\pi\)
\(30\) 0 0
\(31\) 4.69161e7 1.63875 0.819376 − 0.573257i \(-0.194320\pi\)
0.819376 + 0.573257i \(0.194320\pi\)
\(32\) 0 0
\(33\) −6.09311e7 + 6.09311e7i −1.55693 + 1.55693i
\(34\) 0 0
\(35\) 2.33335e6 + 7.96234e6i 0.0444263 + 0.151600i
\(36\) 0 0
\(37\) −7.70531e7 − 7.70531e7i −1.11117 − 1.11117i −0.992992 − 0.118180i \(-0.962294\pi\)
−0.118180 − 0.992992i \(-0.537706\pi\)
\(38\) 0 0
\(39\) − 8.59321e7i − 0.952429i
\(40\) 0 0
\(41\) 2.92307e6 0.0252301 0.0126151 − 0.999920i \(-0.495984\pi\)
0.0126151 + 0.999920i \(0.495984\pi\)
\(42\) 0 0
\(43\) 1.32675e8 − 1.32675e8i 0.902496 − 0.902496i −0.0931552 − 0.995652i \(-0.529695\pi\)
0.995652 + 0.0931552i \(0.0296953\pi\)
\(44\) 0 0
\(45\) −7.85733e7 + 1.43715e8i −0.425807 + 0.778822i
\(46\) 0 0
\(47\) −1.20868e8 − 1.20868e8i −0.527015 − 0.527015i 0.392666 − 0.919681i \(-0.371553\pi\)
−0.919681 + 0.392666i \(0.871553\pi\)
\(48\) 0 0
\(49\) − 2.75426e8i − 0.975044i
\(50\) 0 0
\(51\) −3.89092e8 −1.12772
\(52\) 0 0
\(53\) −3.49065e8 + 3.49065e8i −0.834692 + 0.834692i −0.988155 − 0.153462i \(-0.950958\pi\)
0.153462 + 0.988155i \(0.450958\pi\)
\(54\) 0 0
\(55\) −7.07701e8 − 3.86923e8i −1.40617 − 0.768795i
\(56\) 0 0
\(57\) −4.72312e8 − 4.72312e8i −0.784973 − 0.784973i
\(58\) 0 0
\(59\) 9.91185e8i 1.38642i 0.720736 + 0.693210i \(0.243804\pi\)
−0.720736 + 0.693210i \(0.756196\pi\)
\(60\) 0 0
\(61\) −5.60533e8 −0.663670 −0.331835 − 0.943338i \(-0.607668\pi\)
−0.331835 + 0.943338i \(0.607668\pi\)
\(62\) 0 0
\(63\) −9.84028e7 + 9.84028e7i −0.0991527 + 0.0991527i
\(64\) 0 0
\(65\) 7.71883e8 − 2.26199e8i 0.665249 − 0.194951i
\(66\) 0 0
\(67\) 9.67797e8 + 9.67797e8i 0.716820 + 0.716820i 0.967953 − 0.251133i \(-0.0808030\pi\)
−0.251133 + 0.967953i \(0.580803\pi\)
\(68\) 0 0
\(69\) 7.68942e7i 0.0491641i
\(70\) 0 0
\(71\) −2.10396e9 −1.16613 −0.583063 − 0.812427i \(-0.698146\pi\)
−0.583063 + 0.812427i \(0.698146\pi\)
\(72\) 0 0
\(73\) −3.40584e8 + 3.40584e8i −0.164290 + 0.164290i −0.784464 − 0.620174i \(-0.787062\pi\)
0.620174 + 0.784464i \(0.287062\pi\)
\(74\) 0 0
\(75\) −3.18510e9 − 6.96425e8i −1.34220 − 0.293473i
\(76\) 0 0
\(77\) −4.84570e8 − 4.84570e8i −0.179020 − 0.179020i
\(78\) 0 0
\(79\) 2.81760e9i 0.915680i 0.889035 + 0.457840i \(0.151377\pi\)
−0.889035 + 0.457840i \(0.848623\pi\)
\(80\) 0 0
\(81\) 3.83458e9 1.09975
\(82\) 0 0
\(83\) −2.90486e9 + 2.90486e9i −0.737454 + 0.737454i −0.972085 − 0.234631i \(-0.924612\pi\)
0.234631 + 0.972085i \(0.424612\pi\)
\(84\) 0 0
\(85\) −1.02421e9 − 3.49501e9i −0.230831 − 0.787686i
\(86\) 0 0
\(87\) −4.00971e9 − 4.00971e9i −0.804482 − 0.804482i
\(88\) 0 0
\(89\) − 2.05774e9i − 0.368502i −0.982879 − 0.184251i \(-0.941014\pi\)
0.982879 − 0.184251i \(-0.0589859\pi\)
\(90\) 0 0
\(91\) 6.83396e8 0.109513
\(92\) 0 0
\(93\) −1.10757e10 + 1.10757e10i −1.59205 + 1.59205i
\(94\) 0 0
\(95\) 2.99926e9 − 5.48580e9i 0.387611 − 0.708960i
\(96\) 0 0
\(97\) −1.04120e10 − 1.04120e10i −1.21248 − 1.21248i −0.970209 − 0.242270i \(-0.922108\pi\)
−0.242270 − 0.970209i \(-0.577892\pi\)
\(98\) 0 0
\(99\) − 1.35279e10i − 1.42251i
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.p.f.17.2 14
4.3 odd 2 40.11.l.a.17.6 ✓ 14
5.3 odd 4 inner 80.11.p.f.33.2 14
20.3 even 4 40.11.l.a.33.6 yes 14
20.7 even 4 200.11.l.c.193.2 14
20.19 odd 2 200.11.l.c.57.2 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.11.l.a.17.6 ✓ 14 4.3 odd 2
40.11.l.a.33.6 yes 14 20.3 even 4
80.11.p.f.17.2 14 1.1 even 1 trivial
80.11.p.f.33.2 14 5.3 odd 4 inner
200.11.l.c.57.2 14 20.19 odd 2
200.11.l.c.193.2 14 20.7 even 4