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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [160,5,Mod(17,160)] 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("160.17"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(160, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 2, 1])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 160 = 2^{5} \cdot 5 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 160.m (of order \(4\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(16.5391940934\)
Analytic rank: \(0\)
Dimension: \(44\)
Relative dimension: \(22\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 40)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 113.9
Character \(\chi\) \(=\) 160.113
Dual form 160.5.m.a.17.9

$q$-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)
 
\(f(q)\) \(=\) \(q+(-3.10315 + 3.10315i) q^{3} +(-18.7844 + 16.4969i) q^{5} +(52.9781 - 52.9781i) q^{7} +61.7410i q^{9} +95.9857i q^{11} +(23.7878 - 23.7878i) q^{13} +(7.09830 - 109.483i) q^{15} +(-187.913 + 187.913i) q^{17} -312.734 q^{19} +328.798i q^{21} +(-658.173 - 658.173i) q^{23} +(80.7042 - 619.768i) q^{25} +(-442.946 - 442.946i) q^{27} -1301.35 q^{29} +215.144 q^{31} +(-297.858 - 297.858i) q^{33} +(-121.185 + 1869.14i) q^{35} +(886.034 + 886.034i) q^{37} +147.634i q^{39} -1208.71 q^{41} +(-413.820 + 413.820i) q^{43} +(-1018.54 - 1159.76i) q^{45} +(-738.634 + 738.634i) q^{47} -3212.36i q^{49} -1166.24i q^{51} +(-2786.28 + 2786.28i) q^{53} +(-1583.47 - 1803.03i) q^{55} +(970.459 - 970.459i) q^{57} -1389.86 q^{59} -2794.25i q^{61} +(3270.92 + 3270.92i) q^{63} +(-54.4135 + 839.263i) q^{65} +(436.765 + 436.765i) q^{67} +4084.81 q^{69} -1913.45 q^{71} +(5030.71 + 5030.71i) q^{73} +(1672.79 + 2173.67i) q^{75} +(5085.14 + 5085.14i) q^{77} +1410.68i q^{79} -2251.97 q^{81} +(7067.43 - 7067.43i) q^{83} +(429.842 - 6629.81i) q^{85} +(4038.29 - 4038.29i) q^{87} +12805.8i q^{89} -2520.47i q^{91} +(-667.622 + 667.622i) q^{93} +(5874.51 - 5159.14i) q^{95} +(-1957.19 + 1957.19i) q^{97} -5926.25 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 44 q + 4 q^{7} + 4 q^{15} + 236 q^{17} + 4 q^{23} + 332 q^{25} + 264 q^{31} + 320 q^{33} - 8 q^{41} + 5764 q^{47} - 2496 q^{55} - 328 q^{57} - 9924 q^{63} - 3700 q^{65} + 19976 q^{71} + 13196 q^{73} - 13812 q^{81}+ \cdots + 3276 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/160\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(97\) \(101\)
\(\chi(n)\) \(1\) \(e\left(\frac{3}{4}\right)\) \(-1\)

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 0
\(3\) −3.10315 + 3.10315i −0.344794 + 0.344794i −0.858166 − 0.513372i \(-0.828396\pi\)
0.513372 + 0.858166i \(0.328396\pi\)
\(4\) 0 0
\(5\) −18.7844 + 16.4969i −0.751374 + 0.659876i
\(6\) 0 0
\(7\) 52.9781 − 52.9781i 1.08119 − 1.08119i 0.0847872 − 0.996399i \(-0.472979\pi\)
0.996399 − 0.0847872i \(-0.0270211\pi\)
\(8\) 0 0
\(9\) 61.7410i 0.762234i
\(10\) 0 0
\(11\) 95.9857i 0.793271i 0.917976 + 0.396635i \(0.129822\pi\)
−0.917976 + 0.396635i \(0.870178\pi\)
\(12\) 0 0
\(13\) 23.7878 − 23.7878i 0.140756 − 0.140756i −0.633218 − 0.773974i \(-0.718266\pi\)
0.773974 + 0.633218i \(0.218266\pi\)
\(14\) 0 0
\(15\) 7.09830 − 109.483i 0.0315480 − 0.486591i
\(16\) 0 0
\(17\) −187.913 + 187.913i −0.650218 + 0.650218i −0.953045 − 0.302827i \(-0.902069\pi\)
0.302827 + 0.953045i \(0.402069\pi\)
\(18\) 0 0
\(19\) −312.734 −0.866299 −0.433150 − 0.901322i \(-0.642598\pi\)
−0.433150 + 0.901322i \(0.642598\pi\)
\(20\) 0 0
\(21\) 328.798i 0.745573i
\(22\) 0 0
\(23\) −658.173 − 658.173i −1.24418 − 1.24418i −0.958249 − 0.285934i \(-0.907696\pi\)
−0.285934 − 0.958249i \(-0.592304\pi\)
\(24\) 0 0
\(25\) 80.7042 − 619.768i 0.129127 − 0.991628i
\(26\) 0 0
\(27\) −442.946 − 442.946i −0.607608 − 0.607608i
\(28\) 0 0
\(29\) −1301.35 −1.54739 −0.773695 − 0.633559i \(-0.781594\pi\)
−0.773695 + 0.633559i \(0.781594\pi\)
\(30\) 0 0
\(31\) 215.144 0.223875 0.111937 − 0.993715i \(-0.464294\pi\)
0.111937 + 0.993715i \(0.464294\pi\)
\(32\) 0 0
\(33\) −297.858 − 297.858i −0.273515 − 0.273515i
\(34\) 0 0
\(35\) −121.185 + 1869.14i −0.0989265 + 1.52582i
\(36\) 0 0
\(37\) 886.034 + 886.034i 0.647213 + 0.647213i 0.952318 − 0.305106i \(-0.0986919\pi\)
−0.305106 + 0.952318i \(0.598692\pi\)
\(38\) 0 0
\(39\) 147.634i 0.0970637i
\(40\) 0 0
\(41\) −1208.71 −0.719040 −0.359520 − 0.933137i \(-0.617060\pi\)
−0.359520 + 0.933137i \(0.617060\pi\)
\(42\) 0 0
\(43\) −413.820 + 413.820i −0.223807 + 0.223807i −0.810100 − 0.586292i \(-0.800587\pi\)
0.586292 + 0.810100i \(0.300587\pi\)
\(44\) 0 0
\(45\) −1018.54 − 1159.76i −0.502980 − 0.572723i
\(46\) 0 0
\(47\) −738.634 + 738.634i −0.334375 + 0.334375i −0.854245 − 0.519870i \(-0.825980\pi\)
0.519870 + 0.854245i \(0.325980\pi\)
\(48\) 0 0
\(49\) − 3212.36i − 1.33793i
\(50\) 0 0
\(51\) − 1166.24i − 0.448383i
\(52\) 0 0
\(53\) −2786.28 + 2786.28i −0.991913 + 0.991913i −0.999968 − 0.00805505i \(-0.997436\pi\)
0.00805505 + 0.999968i \(0.497436\pi\)
\(54\) 0 0
\(55\) −1583.47 − 1803.03i −0.523460 − 0.596043i
\(56\) 0 0
\(57\) 970.459 − 970.459i 0.298695 − 0.298695i
\(58\) 0 0
\(59\) −1389.86 −0.399271 −0.199636 − 0.979870i \(-0.563976\pi\)
−0.199636 + 0.979870i \(0.563976\pi\)
\(60\) 0 0
\(61\) − 2794.25i − 0.750940i −0.926835 − 0.375470i \(-0.877481\pi\)
0.926835 − 0.375470i \(-0.122519\pi\)
\(62\) 0 0
\(63\) 3270.92 + 3270.92i 0.824117 + 0.824117i
\(64\) 0 0
\(65\) −54.4135 + 839.263i −0.0128789 + 0.198642i
\(66\) 0 0
\(67\) 436.765 + 436.765i 0.0972968 + 0.0972968i 0.754080 − 0.656783i \(-0.228083\pi\)
−0.656783 + 0.754080i \(0.728083\pi\)
\(68\) 0 0
\(69\) 4084.81 0.857974
\(70\) 0 0
\(71\) −1913.45 −0.379578 −0.189789 − 0.981825i \(-0.560780\pi\)
−0.189789 + 0.981825i \(0.560780\pi\)
\(72\) 0 0
\(73\) 5030.71 + 5030.71i 0.944025 + 0.944025i 0.998514 − 0.0544892i \(-0.0173530\pi\)
−0.0544892 + 0.998514i \(0.517353\pi\)
\(74\) 0 0
\(75\) 1672.79 + 2173.67i 0.297385 + 0.386429i
\(76\) 0 0
\(77\) 5085.14 + 5085.14i 0.857673 + 0.857673i
\(78\) 0 0
\(79\) 1410.68i 0.226034i 0.993593 + 0.113017i \(0.0360514\pi\)
−0.993593 + 0.113017i \(0.963949\pi\)
\(80\) 0 0
\(81\) −2251.97 −0.343235
\(82\) 0 0
\(83\) 7067.43 − 7067.43i 1.02590 − 1.02590i 0.0262455 − 0.999656i \(-0.491645\pi\)
0.999656 − 0.0262455i \(-0.00835517\pi\)
\(84\) 0 0
\(85\) 429.842 − 6629.81i 0.0594937 − 0.917621i
\(86\) 0 0
\(87\) 4038.29 − 4038.29i 0.533530 − 0.533530i
\(88\) 0 0
\(89\) 12805.8i 1.61669i 0.588710 + 0.808345i \(0.299636\pi\)
−0.588710 + 0.808345i \(0.700364\pi\)
\(90\) 0 0
\(91\) − 2520.47i − 0.304367i
\(92\) 0 0
\(93\) −667.622 + 667.622i −0.0771907 + 0.0771907i
\(94\) 0 0
\(95\) 5874.51 − 5159.14i 0.650915 − 0.571650i
\(96\) 0 0
\(97\) −1957.19 + 1957.19i −0.208013 + 0.208013i −0.803422 − 0.595409i \(-0.796990\pi\)
0.595409 + 0.803422i \(0.296990\pi\)
\(98\) 0 0
\(99\) −5926.25 −0.604658
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 160.5.m.a.113.9 44
4.3 odd 2 40.5.i.a.13.11 yes 44
5.2 odd 4 inner 160.5.m.a.17.14 44
8.3 odd 2 40.5.i.a.13.2 ✓ 44
8.5 even 2 inner 160.5.m.a.113.14 44
20.7 even 4 40.5.i.a.37.2 yes 44
40.27 even 4 40.5.i.a.37.11 yes 44
40.37 odd 4 inner 160.5.m.a.17.9 44
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.5.i.a.13.2 ✓ 44 8.3 odd 2
40.5.i.a.13.11 yes 44 4.3 odd 2
40.5.i.a.37.2 yes 44 20.7 even 4
40.5.i.a.37.11 yes 44 40.27 even 4
160.5.m.a.17.9 44 40.37 odd 4 inner
160.5.m.a.17.14 44 5.2 odd 4 inner
160.5.m.a.113.9 44 1.1 even 1 trivial
160.5.m.a.113.14 44 8.5 even 2 inner