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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [40,5,Mod(17,40)] 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("40.17"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(40, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 40 = 2^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 40.l (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6] 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: \(4.13479852335\)
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: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 17.2
Root \(-1.13432i\) of defining polynomial
Character \(\chi\) \(=\) 40.17
Dual form 40.5.l.c.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)
 
\(f(q)\) \(=\) \(q+(-0.958314 + 0.958314i) q^{3} +(-1.04169 + 24.9783i) q^{5} +(26.0617 + 26.0617i) q^{7} +79.1633i q^{9} +75.9566 q^{11} +(-99.1428 + 99.1428i) q^{13} +(-22.9388 - 24.9353i) q^{15} +(77.9171 + 77.9171i) q^{17} -640.445i q^{19} -49.9505 q^{21} +(117.134 - 117.134i) q^{23} +(-622.830 - 52.0390i) q^{25} +(-153.487 - 153.487i) q^{27} -752.534i q^{29} +708.205 q^{31} +(-72.7903 + 72.7903i) q^{33} +(-678.124 + 623.828i) q^{35} +(1170.51 + 1170.51i) q^{37} -190.020i q^{39} +1822.21 q^{41} +(-2372.12 + 2372.12i) q^{43} +(-1977.36 - 82.4632i) q^{45} +(2458.79 + 2458.79i) q^{47} -1042.58i q^{49} -149.338 q^{51} +(2118.26 - 2118.26i) q^{53} +(-79.1229 + 1897.27i) q^{55} +(613.748 + 613.748i) q^{57} -5251.96i q^{59} -1312.89 q^{61} +(-2063.13 + 2063.13i) q^{63} +(-2373.14 - 2579.69i) q^{65} +(-5569.35 - 5569.35i) q^{67} +224.503i q^{69} +7554.66 q^{71} +(1206.35 - 1206.35i) q^{73} +(646.737 - 546.997i) q^{75} +(1979.55 + 1979.55i) q^{77} -3397.97i q^{79} -6118.05 q^{81} +(928.421 - 928.421i) q^{83} +(-2027.40 + 1865.07i) q^{85} +(721.164 + 721.164i) q^{87} +11981.9i q^{89} -5167.65 q^{91} +(-678.683 + 678.683i) q^{93} +(15997.2 + 667.143i) q^{95} +(4683.76 + 4683.76i) q^{97} +6012.97i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 8 q^{3} - 4 q^{5} - 40 q^{7} + 72 q^{11} - 62 q^{13} - 280 q^{15} + 418 q^{17} + 1736 q^{21} - 1760 q^{23} - 2974 q^{25} + 2248 q^{27} + 2864 q^{31} - 8 q^{33} - 4224 q^{35} + 5366 q^{37} + 1656 q^{41}+ \cdots - 9418 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/40\mathbb{Z}\right)^\times\).

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

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
\(3\) −0.958314 + 0.958314i −0.106479 + 0.106479i −0.758339 − 0.651860i \(-0.773989\pi\)
0.651860 + 0.758339i \(0.273989\pi\)
\(4\) 0 0
\(5\) −1.04169 + 24.9783i −0.0416674 + 0.999132i
\(6\) 0 0
\(7\) 26.0617 + 26.0617i 0.531871 + 0.531871i 0.921129 − 0.389258i \(-0.127269\pi\)
−0.389258 + 0.921129i \(0.627269\pi\)
\(8\) 0 0
\(9\) 79.1633i 0.977324i
\(10\) 0 0
\(11\) 75.9566 0.627740 0.313870 − 0.949466i \(-0.398374\pi\)
0.313870 + 0.949466i \(0.398374\pi\)
\(12\) 0 0
\(13\) −99.1428 + 99.1428i −0.586644 + 0.586644i −0.936721 − 0.350077i \(-0.886155\pi\)
0.350077 + 0.936721i \(0.386155\pi\)
\(14\) 0 0
\(15\) −22.9388 − 24.9353i −0.101950 − 0.110824i
\(16\) 0 0
\(17\) 77.9171 + 77.9171i 0.269609 + 0.269609i 0.828943 − 0.559333i \(-0.188943\pi\)
−0.559333 + 0.828943i \(0.688943\pi\)
\(18\) 0 0
\(19\) − 640.445i − 1.77409i −0.461686 − 0.887044i \(-0.652755\pi\)
0.461686 − 0.887044i \(-0.347245\pi\)
\(20\) 0 0
\(21\) −49.9505 −0.113267
\(22\) 0 0
\(23\) 117.134 − 117.134i 0.221426 − 0.221426i −0.587673 − 0.809099i \(-0.699956\pi\)
0.809099 + 0.587673i \(0.199956\pi\)
\(24\) 0 0
\(25\) −622.830 − 52.0390i −0.996528 − 0.0832625i
\(26\) 0 0
\(27\) −153.487 − 153.487i −0.210544 − 0.210544i
\(28\) 0 0
\(29\) − 752.534i − 0.894809i −0.894332 − 0.447404i \(-0.852348\pi\)
0.894332 − 0.447404i \(-0.147652\pi\)
\(30\) 0 0
\(31\) 708.205 0.736946 0.368473 − 0.929638i \(-0.379881\pi\)
0.368473 + 0.929638i \(0.379881\pi\)
\(32\) 0 0
\(33\) −72.7903 + 72.7903i −0.0668414 + 0.0668414i
\(34\) 0 0
\(35\) −678.124 + 623.828i −0.553570 + 0.509247i
\(36\) 0 0
\(37\) 1170.51 + 1170.51i 0.855013 + 0.855013i 0.990745 − 0.135733i \(-0.0433389\pi\)
−0.135733 + 0.990745i \(0.543339\pi\)
\(38\) 0 0
\(39\) − 190.020i − 0.124931i
\(40\) 0 0
\(41\) 1822.21 1.08400 0.542001 − 0.840378i \(-0.317667\pi\)
0.542001 + 0.840378i \(0.317667\pi\)
\(42\) 0 0
\(43\) −2372.12 + 2372.12i −1.28292 + 1.28292i −0.343921 + 0.938998i \(0.611755\pi\)
−0.938998 + 0.343921i \(0.888245\pi\)
\(44\) 0 0
\(45\) −1977.36 − 82.4632i −0.976476 − 0.0407226i
\(46\) 0 0
\(47\) 2458.79 + 2458.79i 1.11308 + 1.11308i 0.992732 + 0.120344i \(0.0383997\pi\)
0.120344 + 0.992732i \(0.461600\pi\)
\(48\) 0 0
\(49\) − 1042.58i − 0.434227i
\(50\) 0 0
\(51\) −149.338 −0.0574157
\(52\) 0 0
\(53\) 2118.26 − 2118.26i 0.754097 − 0.754097i −0.221144 − 0.975241i \(-0.570979\pi\)
0.975241 + 0.221144i \(0.0709792\pi\)
\(54\) 0 0
\(55\) −79.1229 + 1897.27i −0.0261563 + 0.627195i
\(56\) 0 0
\(57\) 613.748 + 613.748i 0.188904 + 0.188904i
\(58\) 0 0
\(59\) − 5251.96i − 1.50875i −0.656443 − 0.754376i \(-0.727940\pi\)
0.656443 − 0.754376i \(-0.272060\pi\)
\(60\) 0 0
\(61\) −1312.89 −0.352831 −0.176416 − 0.984316i \(-0.556450\pi\)
−0.176416 + 0.984316i \(0.556450\pi\)
\(62\) 0 0
\(63\) −2063.13 + 2063.13i −0.519810 + 0.519810i
\(64\) 0 0
\(65\) −2373.14 − 2579.69i −0.561691 − 0.610578i
\(66\) 0 0
\(67\) −5569.35 − 5569.35i −1.24067 − 1.24067i −0.959725 − 0.280940i \(-0.909354\pi\)
−0.280940 − 0.959725i \(-0.590646\pi\)
\(68\) 0 0
\(69\) 224.503i 0.0471545i
\(70\) 0 0
\(71\) 7554.66 1.49864 0.749322 − 0.662206i \(-0.230380\pi\)
0.749322 + 0.662206i \(0.230380\pi\)
\(72\) 0 0
\(73\) 1206.35 − 1206.35i 0.226374 − 0.226374i −0.584802 − 0.811176i \(-0.698828\pi\)
0.811176 + 0.584802i \(0.198828\pi\)
\(74\) 0 0
\(75\) 646.737 − 546.997i 0.114975 − 0.0972439i
\(76\) 0 0
\(77\) 1979.55 + 1979.55i 0.333877 + 0.333877i
\(78\) 0 0
\(79\) − 3397.97i − 0.544460i −0.962232 − 0.272230i \(-0.912239\pi\)
0.962232 − 0.272230i \(-0.0877612\pi\)
\(80\) 0 0
\(81\) −6118.05 −0.932487
\(82\) 0 0
\(83\) 928.421 − 928.421i 0.134769 − 0.134769i −0.636504 − 0.771273i \(-0.719620\pi\)
0.771273 + 0.636504i \(0.219620\pi\)
\(84\) 0 0
\(85\) −2027.40 + 1865.07i −0.280609 + 0.258141i
\(86\) 0 0
\(87\) 721.164 + 721.164i 0.0952787 + 0.0952787i
\(88\) 0 0
\(89\) 11981.9i 1.51268i 0.654180 + 0.756339i \(0.273014\pi\)
−0.654180 + 0.756339i \(0.726986\pi\)
\(90\) 0 0
\(91\) −5167.65 −0.624037
\(92\) 0 0
\(93\) −678.683 + 678.683i −0.0784695 + 0.0784695i
\(94\) 0 0
\(95\) 15997.2 + 667.143i 1.77255 + 0.0739216i
\(96\) 0 0
\(97\) 4683.76 + 4683.76i 0.497796 + 0.497796i 0.910751 − 0.412955i \(-0.135503\pi\)
−0.412955 + 0.910751i \(0.635503\pi\)
\(98\) 0 0
\(99\) 6012.97i 0.613506i
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 40.5.l.c.17.2 ✓ 6
3.2 odd 2 360.5.v.c.217.2 6
4.3 odd 2 80.5.p.g.17.2 6
5.2 odd 4 200.5.l.e.193.2 6
5.3 odd 4 inner 40.5.l.c.33.2 yes 6
5.4 even 2 200.5.l.e.57.2 6
8.3 odd 2 320.5.p.o.257.2 6
8.5 even 2 320.5.p.p.257.2 6
15.8 even 4 360.5.v.c.73.2 6
20.3 even 4 80.5.p.g.33.2 6
20.7 even 4 400.5.p.o.193.2 6
20.19 odd 2 400.5.p.o.257.2 6
40.3 even 4 320.5.p.o.193.2 6
40.13 odd 4 320.5.p.p.193.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.5.l.c.17.2 ✓ 6 1.1 even 1 trivial
40.5.l.c.33.2 yes 6 5.3 odd 4 inner
80.5.p.g.17.2 6 4.3 odd 2
80.5.p.g.33.2 6 20.3 even 4
200.5.l.e.57.2 6 5.4 even 2
200.5.l.e.193.2 6 5.2 odd 4
320.5.p.o.193.2 6 40.3 even 4
320.5.p.o.257.2 6 8.3 odd 2
320.5.p.p.193.2 6 40.13 odd 4
320.5.p.p.257.2 6 8.5 even 2
360.5.v.c.73.2 6 15.8 even 4
360.5.v.c.217.2 6 3.2 odd 2
400.5.p.o.193.2 6 20.7 even 4
400.5.p.o.257.2 6 20.19 odd 2