Show commands: Magma / Pari/GP / SageMath

Newspace parameters

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

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,8] 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: \(8.26959704671\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(i)\)
Coefficient field: 6.0.313431616.3
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 27x^{4} + 145x^{2} + 144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{5}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 40)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 17.3
Root \(-2.35597i\) of defining polynomial
Character \(\chi\) \(=\) 80.17
Dual form 80.5.p.g.33.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(11.3304 - 11.3304i) q^{3} +(9.33042 - 23.1936i) q^{5} +(42.8544 + 42.8544i) q^{7} -175.757i q^{9} +20.3872 q^{11} +(-151.697 + 151.697i) q^{13} +(-157.076 - 368.511i) q^{15} +(-150.269 - 150.269i) q^{17} +277.202i q^{19} +971.118 q^{21} +(363.891 - 363.891i) q^{23} +(-450.886 - 432.812i) q^{25} +(-1073.64 - 1073.64i) q^{27} +413.744i q^{29} +659.526 q^{31} +(230.996 - 230.996i) q^{33} +(1393.80 - 594.099i) q^{35} +(1058.50 + 1058.50i) q^{37} +3437.59i q^{39} -801.206 q^{41} +(-2008.78 + 2008.78i) q^{43} +(-4076.44 - 1639.89i) q^{45} +(1471.26 + 1471.26i) q^{47} +1272.01i q^{49} -3405.23 q^{51} +(-1321.08 + 1321.08i) q^{53} +(190.221 - 472.853i) q^{55} +(3140.82 + 3140.82i) q^{57} +937.237i q^{59} +2543.20 q^{61} +(7531.97 - 7531.97i) q^{63} +(2103.01 + 4933.80i) q^{65} +(-140.942 - 140.942i) q^{67} -8246.08i q^{69} -2532.44 q^{71} +(5900.02 - 5900.02i) q^{73} +(-10012.7 + 204.788i) q^{75} +(873.682 + 873.682i) q^{77} +196.937i q^{79} -10093.2 q^{81} +(9130.34 - 9130.34i) q^{83} +(-4887.36 + 2083.21i) q^{85} +(4687.89 + 4687.89i) q^{87} +10678.4i q^{89} -13001.8 q^{91} +(7472.70 - 7472.70i) q^{93} +(6429.32 + 2586.41i) q^{95} +(-11052.2 - 11052.2i) q^{97} -3583.19i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 8 q^{3} - 4 q^{5} + 40 q^{7} - 72 q^{11} - 62 q^{13} + 280 q^{15} + 418 q^{17} + 1736 q^{21} + 1760 q^{23} - 2974 q^{25} - 2248 q^{27} - 2864 q^{31} - 8 q^{33} + 4224 q^{35} + 5366 q^{37} + 1656 q^{41}+ \cdots - 9418 q^{97}+O(q^{100}) \) 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
\(3\) 11.3304 − 11.3304i 1.25894 − 1.25894i 0.307334 − 0.951602i \(-0.400563\pi\)
0.951602 − 0.307334i \(-0.0994369\pi\)
\(4\) 0 0
\(5\) 9.33042 − 23.1936i 0.373217 − 0.927744i
\(6\) 0 0
\(7\) 42.8544 + 42.8544i 0.874581 + 0.874581i 0.992968 − 0.118387i \(-0.0377723\pi\)
−0.118387 + 0.992968i \(0.537772\pi\)
\(8\) 0 0
\(9\) − 175.757i − 2.16984i
\(10\) 0 0
\(11\) 20.3872 0.168489 0.0842447 − 0.996445i \(-0.473152\pi\)
0.0842447 + 0.996445i \(0.473152\pi\)
\(12\) 0 0
\(13\) −151.697 + 151.697i −0.897617 + 0.897617i −0.995225 − 0.0976082i \(-0.968881\pi\)
0.0976082 + 0.995225i \(0.468881\pi\)
\(14\) 0 0
\(15\) −157.076 − 368.511i −0.698114 − 1.63783i
\(16\) 0 0
\(17\) −150.269 − 150.269i −0.519963 − 0.519963i 0.397597 − 0.917560i \(-0.369844\pi\)
−0.917560 + 0.397597i \(0.869844\pi\)
\(18\) 0 0
\(19\) 277.202i 0.767873i 0.923359 + 0.383937i \(0.125432\pi\)
−0.923359 + 0.383937i \(0.874568\pi\)
\(20\) 0 0
\(21\) 971.118 2.20208
\(22\) 0 0
\(23\) 363.891 − 363.891i 0.687885 − 0.687885i −0.273879 − 0.961764i \(-0.588307\pi\)
0.961764 + 0.273879i \(0.0883069\pi\)
\(24\) 0 0
\(25\) −450.886 − 432.812i −0.721418 − 0.692500i
\(26\) 0 0
\(27\) −1073.64 − 1073.64i −1.47275 − 1.47275i
\(28\) 0 0
\(29\) 413.744i 0.491967i 0.969274 + 0.245983i \(0.0791108\pi\)
−0.969274 + 0.245983i \(0.920889\pi\)
\(30\) 0 0
\(31\) 659.526 0.686291 0.343145 − 0.939282i \(-0.388508\pi\)
0.343145 + 0.939282i \(0.388508\pi\)
\(32\) 0 0
\(33\) 230.996 − 230.996i 0.212117 − 0.212117i
\(34\) 0 0
\(35\) 1393.80 − 594.099i 1.13780 − 0.484979i
\(36\) 0 0
\(37\) 1058.50 + 1058.50i 0.773193 + 0.773193i 0.978663 − 0.205471i \(-0.0658725\pi\)
−0.205471 + 0.978663i \(0.565873\pi\)
\(38\) 0 0
\(39\) 3437.59i 2.26008i
\(40\) 0 0
\(41\) −801.206 −0.476624 −0.238312 − 0.971189i \(-0.576594\pi\)
−0.238312 + 0.971189i \(0.576594\pi\)
\(42\) 0 0
\(43\) −2008.78 + 2008.78i −1.08642 + 1.08642i −0.0905217 + 0.995894i \(0.528853\pi\)
−0.995894 + 0.0905217i \(0.971147\pi\)
\(44\) 0 0
\(45\) −4076.44 − 1639.89i −2.01306 − 0.809821i
\(46\) 0 0
\(47\) 1471.26 + 1471.26i 0.666030 + 0.666030i 0.956795 − 0.290765i \(-0.0939097\pi\)
−0.290765 + 0.956795i \(0.593910\pi\)
\(48\) 0 0
\(49\) 1272.01i 0.529782i
\(50\) 0 0
\(51\) −3405.23 −1.30920
\(52\) 0 0
\(53\) −1321.08 + 1321.08i −0.470304 + 0.470304i −0.902013 − 0.431709i \(-0.857911\pi\)
0.431709 + 0.902013i \(0.357911\pi\)
\(54\) 0 0
\(55\) 190.221 − 472.853i 0.0628831 − 0.156315i
\(56\) 0 0
\(57\) 3140.82 + 3140.82i 0.966703 + 0.966703i
\(58\) 0 0
\(59\) 937.237i 0.269243i 0.990897 + 0.134622i \(0.0429819\pi\)
−0.990897 + 0.134622i \(0.957018\pi\)
\(60\) 0 0
\(61\) 2543.20 0.683473 0.341736 − 0.939796i \(-0.388985\pi\)
0.341736 + 0.939796i \(0.388985\pi\)
\(62\) 0 0
\(63\) 7531.97 − 7531.97i 1.89770 − 1.89770i
\(64\) 0 0
\(65\) 2103.01 + 4933.80i 0.497753 + 1.16776i
\(66\) 0 0
\(67\) −140.942 − 140.942i −0.0313971 − 0.0313971i 0.691234 − 0.722631i \(-0.257067\pi\)
−0.722631 + 0.691234i \(0.757067\pi\)
\(68\) 0 0
\(69\) − 8246.08i − 1.73200i
\(70\) 0 0
\(71\) −2532.44 −0.502368 −0.251184 − 0.967939i \(-0.580820\pi\)
−0.251184 + 0.967939i \(0.580820\pi\)
\(72\) 0 0
\(73\) 5900.02 − 5900.02i 1.10715 − 1.10715i 0.113630 − 0.993523i \(-0.463752\pi\)
0.993523 − 0.113630i \(-0.0362479\pi\)
\(74\) 0 0
\(75\) −10012.7 + 204.788i −1.78003 + 0.0364068i
\(76\) 0 0
\(77\) 873.682 + 873.682i 0.147357 + 0.147357i
\(78\) 0 0
\(79\) 196.937i 0.0315554i 0.999876 + 0.0157777i \(0.00502240\pi\)
−0.999876 + 0.0157777i \(0.994978\pi\)
\(80\) 0 0
\(81\) −10093.2 −1.53836
\(82\) 0 0
\(83\) 9130.34 − 9130.34i 1.32535 − 1.32535i 0.415973 − 0.909377i \(-0.363441\pi\)
0.909377 − 0.415973i \(-0.136559\pi\)
\(84\) 0 0
\(85\) −4887.36 + 2083.21i −0.676452 + 0.288334i
\(86\) 0 0
\(87\) 4687.89 + 4687.89i 0.619354 + 0.619354i
\(88\) 0 0
\(89\) 10678.4i 1.34811i 0.738680 + 0.674056i \(0.235449\pi\)
−0.738680 + 0.674056i \(0.764551\pi\)
\(90\) 0 0
\(91\) −13001.8 −1.57008
\(92\) 0 0
\(93\) 7472.70 − 7472.70i 0.863996 − 0.863996i
\(94\) 0 0
\(95\) 6429.32 + 2586.41i 0.712390 + 0.286583i
\(96\) 0 0
\(97\) −11052.2 − 11052.2i −1.17464 − 1.17464i −0.981090 − 0.193550i \(-0.938000\pi\)
−0.193550 − 0.981090i \(-0.562000\pi\)
\(98\) 0 0
\(99\) − 3583.19i − 0.365595i
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 80.5.p.g.17.3 6
4.3 odd 2 40.5.l.c.17.1 ✓ 6
5.2 odd 4 400.5.p.o.193.1 6
5.3 odd 4 inner 80.5.p.g.33.3 6
5.4 even 2 400.5.p.o.257.1 6
8.3 odd 2 320.5.p.p.257.3 6
8.5 even 2 320.5.p.o.257.1 6
12.11 even 2 360.5.v.c.217.1 6
20.3 even 4 40.5.l.c.33.1 yes 6
20.7 even 4 200.5.l.e.193.3 6
20.19 odd 2 200.5.l.e.57.3 6
40.3 even 4 320.5.p.p.193.3 6
40.13 odd 4 320.5.p.o.193.1 6
60.23 odd 4 360.5.v.c.73.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.5.l.c.17.1 ✓ 6 4.3 odd 2
40.5.l.c.33.1 yes 6 20.3 even 4
80.5.p.g.17.3 6 1.1 even 1 trivial
80.5.p.g.33.3 6 5.3 odd 4 inner
200.5.l.e.57.3 6 20.19 odd 2
200.5.l.e.193.3 6 20.7 even 4
320.5.p.o.193.1 6 40.13 odd 4
320.5.p.o.257.1 6 8.5 even 2
320.5.p.p.193.3 6 40.3 even 4
320.5.p.p.257.3 6 8.3 odd 2
360.5.v.c.73.1 6 60.23 odd 4
360.5.v.c.217.1 6 12.11 even 2
400.5.p.o.193.1 6 5.2 odd 4
400.5.p.o.257.1 6 5.4 even 2