LMFDB - Embedded newform 4080.2.h.n.3841.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4080,2,Mod(3841,4080)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4080, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4080.3841"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 4080 = 2^{4} \cdot 3 \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4080.h (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,-4,0,0,0,8,0,-4,0,12,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$32.5789640247$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 1020)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3841.1
Root			$0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	4080.3841
Dual form			4080.2.h.n.3841.4

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} -1.00000i q^{5} -3.82843i q^{7} -1.00000 q^{9} +5.82843i q^{11} +4.82843 q^{13} -1.00000 q^{15} +(3.00000 - 2.82843i) q^{17} -3.00000 q^{19} -3.82843 q^{21} -8.48528i q^{23} -1.00000 q^{25} +1.00000i q^{27} -4.17157i q^{29} -6.82843i q^{31} +5.82843 q^{33} -3.82843 q^{35} +9.48528i q^{37} -4.82843i q^{39} +5.82843i q^{41} -8.00000 q^{43} +1.00000i q^{45} +8.65685 q^{47} -7.65685 q^{49} +(-2.82843 - 3.00000i) q^{51} +8.31371 q^{53} +5.82843 q^{55} +3.00000i q^{57} -5.17157 q^{59} -8.48528i q^{61} +3.82843i q^{63} -4.82843i q^{65} +2.48528 q^{67} -8.48528 q^{69} -9.65685i q^{71} -15.8284i q^{73} +1.00000i q^{75} +22.3137 q^{77} +2.00000i q^{79} +1.00000 q^{81} +(-2.82843 - 3.00000i) q^{85} -4.17157 q^{87} +14.1421 q^{89} -18.4853i q^{91} -6.82843 q^{93} +3.00000i q^{95} +14.0000i q^{97} -5.82843i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{9} + 8 q^{13} - 4 q^{15} + 12 q^{17} - 12 q^{19} - 4 q^{21} - 4 q^{25} + 12 q^{33} - 4 q^{35} - 32 q^{43} + 12 q^{47} - 8 q^{49} - 12 q^{53} + 12 q^{55} - 32 q^{59} - 24 q^{67} + 44 q^{77} + 4 q^{81}+ \cdots - 16 q^{93}+O(q^{100}) $ 4 * q - 4 * q^9 + 8 * q^13 - 4 * q^15 + 12 * q^17 - 12 * q^19 - 4 * q^21 - 4 * q^25 + 12 * q^33 - 4 * q^35 - 32 * q^43 + 12 * q^47 - 8 * q^49 - 12 * q^53 + 12 * q^55 - 32 * q^59 - 24 * q^67 + 44 * q^77 + 4 * q^81 - 28 * q^87 - 16 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4080\mathbb{Z}\right)^\times$.

$n$	$241$	$511$	$817$	$1361$	$3061$
$\chi(n)$	$-1$	$1$	$1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		−	1.00000i		−	0.577350i
$3$		−	1.00000i		−	0.577350i
$4$		0			0
$5$		−	1.00000i		−	0.447214i
$5$		−	1.00000i		−	0.447214i
$6$		0			0
$7$		−	3.82843i		−	1.44701i	−0.690319	−	0.723505i	$-0.742530\pi$
$7$		−	3.82843i		−	1.44701i	0.690319	−	0.723505i	$-0.257470\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$			5.82843i			1.75734i	0.477432	+	0.878668i	$0.341568\pi$
$11$			5.82843i			1.75734i	−0.477432	+	0.878668i	$0.658432\pi$
$12$		0			0
$13$	4.82843			1.33916			0.669582	−	0.742738i	$-0.266473\pi$
$13$	4.82843			1.33916			0.669582	+	0.742738i	$0.266473\pi$
$14$		0			0
$15$	−1.00000			−0.258199
$16$		0			0
$17$	3.00000	−	2.82843i	0.727607	−	0.685994i
$17$	3.00000	−	2.82843i	0.727607	−	0.685994i
$18$		0			0
$19$	−3.00000			−0.688247			−0.344124	−	0.938924i	$-0.611824\pi$
$19$	−3.00000			−0.688247			−0.344124	+	0.938924i	$0.611824\pi$
$20$		0			0
$21$	−3.82843			−0.835431
$22$		0			0
$23$		−	8.48528i		−	1.76930i	−0.466252	−	0.884652i	$-0.654396\pi$
$23$		−	8.48528i		−	1.76930i	0.466252	−	0.884652i	$-0.345604\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$		−	4.17157i		−	0.774642i	−0.921945	−	0.387321i	$-0.873401\pi$
$29$		−	4.17157i		−	0.774642i	0.921945	−	0.387321i	$-0.126599\pi$
$30$		0			0
$31$		−	6.82843i		−	1.22642i	−0.789919	−	0.613211i	$-0.789878\pi$
$31$		−	6.82843i		−	1.22642i	0.789919	−	0.613211i	$-0.210122\pi$
$32$		0			0
$33$	5.82843			1.01460
$34$		0			0
$35$	−3.82843			−0.647122
$36$		0			0
$37$			9.48528i			1.55937i	0.626172	+	0.779685i	$0.284621\pi$
$37$			9.48528i			1.55937i	−0.626172	+	0.779685i	$0.715379\pi$
$38$		0			0
$39$		−	4.82843i		−	0.773167i
$40$		0			0
$41$			5.82843i			0.910247i	0.890428	+	0.455124i	$0.150405\pi$
$41$			5.82843i			0.910247i	−0.890428	+	0.455124i	$0.849595\pi$
$42$		0			0
$43$	−8.00000			−1.21999			−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000			−1.21999			−0.609994	+	0.792406i	$0.708828\pi$
$44$		0			0
$45$			1.00000i			0.149071i
$46$		0			0
$47$	8.65685			1.26273			0.631366	−	0.775485i	$-0.282495\pi$
$47$	8.65685			1.26273			0.631366	+	0.775485i	$0.282495\pi$
$48$		0			0
$49$	−7.65685			−1.09384
$50$		0			0
$51$	−2.82843	−	3.00000i	−0.396059	−	0.420084i
$52$		0			0
$53$	8.31371			1.14198			0.570988	−	0.820959i	$-0.306560\pi$
$53$	8.31371			1.14198			0.570988	+	0.820959i	$0.306560\pi$
$54$		0			0
$55$	5.82843			0.785905
$56$		0			0
$57$			3.00000i			0.397360i
$58$		0			0
$59$	−5.17157			−0.673281			−0.336641	−	0.941633i	$-0.609291\pi$
$59$	−5.17157			−0.673281			−0.336641	+	0.941633i	$0.609291\pi$
$60$		0			0
$61$		−	8.48528i		−	1.08643i	−0.839594	−	0.543214i	$-0.817207\pi$
$61$		−	8.48528i		−	1.08643i	0.839594	−	0.543214i	$-0.182793\pi$
$62$		0			0
$63$			3.82843i			0.482336i
$64$		0			0
$65$		−	4.82843i		−	0.598893i
$66$		0			0
$67$	2.48528			0.303625			0.151813	−	0.988409i	$-0.451489\pi$
$67$	2.48528			0.303625			0.151813	+	0.988409i	$0.451489\pi$
$68$		0			0
$69$	−8.48528			−1.02151
$70$		0			0
$71$		−	9.65685i		−	1.14606i	−0.819535	−	0.573029i	$-0.805768\pi$
$71$		−	9.65685i		−	1.14606i	0.819535	−	0.573029i	$-0.194232\pi$
$72$		0			0
$73$		−	15.8284i		−	1.85258i	−0.376815	−	0.926289i	$-0.622981\pi$
$73$		−	15.8284i		−	1.85258i	0.376815	−	0.926289i	$-0.377019\pi$
$74$		0			0
$75$			1.00000i			0.115470i
$76$		0			0
$77$	22.3137			2.54288
$78$		0			0
$79$			2.00000i			0.225018i	0.993651	+	0.112509i	$0.0358886\pi$
$79$			2.00000i			0.225018i	−0.993651	+	0.112509i	$0.964111\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$	−2.82843	−	3.00000i	−0.306786	−	0.325396i
$86$		0			0
$87$	−4.17157			−0.447240
$88$		0			0
$89$	14.1421			1.49906			0.749532	−	0.661968i	$-0.230279\pi$
$89$	14.1421			1.49906			0.749532	+	0.661968i	$0.230279\pi$
$90$		0			0
$91$		−	18.4853i		−	1.93778i
$92$		0			0
$93$	−6.82843			−0.708075
$94$		0			0
$95$			3.00000i			0.307794i
$96$		0			0
$97$			14.0000i			1.42148i	0.703452	+	0.710742i	$0.251641\pi$
$97$			14.0000i			1.42148i	−0.703452	+	0.710742i	$0.748359\pi$
$98$		0			0
$99$		−	5.82843i		−	0.585779i
$100$		0			0
$101$	−5.65685			−0.562878			−0.281439	−	0.959579i	$-0.590812\pi$
$101$	−5.65685			−0.562878			−0.281439	+	0.959579i	$0.590812\pi$
$102$		0			0
$103$	6.00000			0.591198			0.295599	−	0.955312i	$-0.404481\pi$
$103$	6.00000			0.591198			0.295599	+	0.955312i	$0.404481\pi$
$104$		0			0
$105$			3.82843i			0.373616i
$106$		0			0
$107$			4.34315i			0.419868i	0.977716	+	0.209934i	$0.0673249\pi$
$107$			4.34315i			0.419868i	−0.977716	+	0.209934i	$0.932675\pi$
$108$		0			0
$109$		−	3.31371i		−	0.317396i	−0.987327	−	0.158698i	$-0.949270\pi$
$109$		−	3.31371i		−	0.317396i	0.987327	−	0.158698i	$-0.0507296\pi$
$110$		0			0
$111$	9.48528			0.900303
$112$		0			0
$113$		−	10.8284i		−	1.01865i	−0.860573	−	0.509326i	$-0.829895\pi$
$113$		−	10.8284i		−	1.01865i	0.860573	−	0.509326i	$-0.170105\pi$
$114$		0			0
$115$	−8.48528			−0.791257
$116$		0			0
$117$	−4.82843			−0.446388
$118$		0			0
$119$	−10.8284	−	11.4853i	−0.992640	−	1.05285i
$120$		0			0
$121$	−22.9706			−2.08823
$122$		0			0
$123$	5.82843			0.525532
$124$		0			0
$125$			1.00000i			0.0894427i
$126$		0			0
$127$	−10.1421			−0.899969			−0.449985	−	0.893036i	$-0.648570\pi$
$127$	−10.1421			−0.899969			−0.449985	+	0.893036i	$0.648570\pi$
$128$		0			0
$129$			8.00000i			0.704361i
$130$		0			0
$131$			7.31371i			0.639002i	0.947586	+	0.319501i	$0.103515\pi$
$131$			7.31371i			0.639002i	−0.947586	+	0.319501i	$0.896485\pi$
$132$		0			0
$133$			11.4853i			0.995900i
$134$		0			0
$135$	1.00000			0.0860663
$136$		0			0
$137$	−1.34315			−0.114753			−0.0573763	−	0.998353i	$-0.518273\pi$
$137$	−1.34315			−0.114753			−0.0573763	+	0.998353i	$0.518273\pi$
$138$		0			0
$139$		−	6.00000i		−	0.508913i	−0.967084	−	0.254457i	$-0.918103\pi$
$139$		−	6.00000i		−	0.508913i	0.967084	−	0.254457i	$-0.0818966\pi$
$140$		0			0
$141$		−	8.65685i		−	0.729039i
$142$		0			0
$143$			28.1421i			2.35336i
$144$		0			0
$145$	−4.17157			−0.346430
$146$		0			0
$147$			7.65685i			0.631527i
$148$		0			0
$149$	−13.1716			−1.07906			−0.539529	−	0.841967i	$-0.681398\pi$
$149$	−13.1716			−1.07906			−0.539529	+	0.841967i	$0.681398\pi$
$150$		0			0
$151$	−15.0000			−1.22068			−0.610341	−	0.792139i	$-0.708968\pi$
$151$	−15.0000			−1.22068			−0.610341	+	0.792139i	$0.708968\pi$
$152$		0			0
$153$	−3.00000	+	2.82843i	−0.242536	+	0.228665i
$154$		0			0
$155$	−6.82843			−0.548472
$156$		0			0
$157$	−22.9706			−1.83325			−0.916625	−	0.399748i	$-0.869098\pi$
$157$	−22.9706			−1.83325			−0.916625	+	0.399748i	$0.869098\pi$
$158$		0			0
$159$		−	8.31371i		−	0.659320i
$160$		0			0
$161$	−32.4853			−2.56020
$162$		0			0
$163$		−	3.48528i		−	0.272988i	−0.990641	−	0.136494i	$-0.956417\pi$
$163$		−	3.48528i		−	0.272988i	0.990641	−	0.136494i	$-0.0435835\pi$
$164$		0			0
$165$		−	5.82843i		−	0.453742i
$166$		0			0
$167$		−	9.65685i		−	0.747270i	−0.927576	−	0.373635i	$-0.878111\pi$
$167$		−	9.65685i		−	0.747270i	0.927576	−	0.373635i	$-0.121889\pi$
$168$		0			0
$169$	10.3137			0.793362
$170$		0			0
$171$	3.00000			0.229416
$172$		0			0
$173$		−	20.8284i		−	1.58356i	−0.610809	−	0.791778i	$-0.709156\pi$
$173$		−	20.8284i		−	1.58356i	0.610809	−	0.791778i	$-0.290844\pi$
$174$		0			0
$175$			3.82843i			0.289402i
$176$		0			0
$177$			5.17157i			0.388719i
$178$		0			0
$179$	1.65685			0.123839			0.0619196	−	0.998081i	$-0.480278\pi$
$179$	1.65685			0.123839			0.0619196	+	0.998081i	$0.480278\pi$
$180$		0			0
$181$			4.82843i			0.358894i	0.983768	+	0.179447i	$0.0574309\pi$
$181$			4.82843i			0.358894i	−0.983768	+	0.179447i	$0.942569\pi$
$182$		0			0
$183$	−8.48528			−0.627250
$184$		0			0
$185$	9.48528			0.697372
$186$		0			0
$187$	16.4853	+	17.4853i	1.20552	+	1.27865i
$188$		0			0
$189$	3.82843			0.278477
$190$		0			0
$191$	−18.1421			−1.31272			−0.656359	−	0.754448i	$-0.727904\pi$
$191$	−18.1421			−1.31272			−0.656359	+	0.754448i	$0.727904\pi$
$192$		0			0
$193$		−	13.3137i		−	0.958342i	−0.877722	−	0.479171i	$-0.840937\pi$
$193$		−	13.3137i		−	0.958342i	0.877722	−	0.479171i	$-0.159063\pi$
$194$		0			0
$195$	−4.82843			−0.345771
$196$		0			0
$197$		−	3.65685i		−	0.260540i	−0.991479	−	0.130270i	$-0.958416\pi$
$197$		−	3.65685i		−	0.260540i	0.991479	−	0.130270i	$-0.0415844\pi$
$198$		0			0
$199$			20.9706i			1.48656i	0.668978	+	0.743282i	$0.266732\pi$
$199$			20.9706i			1.48656i	−0.668978	+	0.743282i	$0.733268\pi$
$200$		0			0
$201$		−	2.48528i		−	0.175298i
$202$		0			0
$203$	−15.9706			−1.12091
$204$		0			0
$205$	5.82843			0.407075
$206$		0			0
$207$			8.48528i			0.589768i
$208$		0			0
$209$		−	17.4853i		−	1.20948i
$210$		0			0
$211$			13.3137i			0.916553i	0.888810	+	0.458277i	$0.151533\pi$
$211$			13.3137i			0.916553i	−0.888810	+	0.458277i	$0.848467\pi$
$212$		0			0
$213$	−9.65685			−0.661677
$214$		0			0
$215$			8.00000i			0.545595i
$216$		0			0
$217$	−26.1421			−1.77464
$218$		0			0
$219$	−15.8284			−1.06959
$220$		0			0
$221$	14.4853	−	13.6569i	0.974385	−	0.918659i
$222$		0			0
$223$	−13.3137			−0.891552			−0.445776	−	0.895145i	$-0.647072\pi$
$223$	−13.3137			−0.891552			−0.445776	+	0.895145i	$0.647072\pi$
$224$		0			0
$225$	1.00000			0.0666667
$226$		0			0
$227$			4.82843i			0.320474i	0.987079	+	0.160237i	$0.0512258\pi$
$227$			4.82843i			0.320474i	−0.987079	+	0.160237i	$0.948774\pi$
$228$		0			0
$229$	−11.0000			−0.726900			−0.363450	−	0.931614i	$-0.618401\pi$
$229$	−11.0000			−0.726900			−0.363450	+	0.931614i	$0.618401\pi$
$230$		0			0
$231$		−	22.3137i		−	1.46813i
$232$		0			0
$233$		−	14.4853i		−	0.948962i	−0.880266	−	0.474481i	$-0.842636\pi$
$233$		−	14.4853i		−	0.948962i	0.880266	−	0.474481i	$-0.157364\pi$
$234$		0			0
$235$		−	8.65685i		−	0.564711i
$236$		0			0
$237$	2.00000			0.129914
$238$		0			0
$239$	14.4853			0.936975			0.468487	−	0.883470i	$-0.344799\pi$
$239$	14.4853			0.936975			0.468487	+	0.883470i	$0.344799\pi$
$240$		0			0
$241$			1.51472i			0.0975716i	0.998809	+	0.0487858i	$0.0155352\pi$
$241$			1.51472i			0.0975716i	−0.998809	+	0.0487858i	$0.984465\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$			7.65685i			0.489178i
$246$		0			0
$247$	−14.4853			−0.921676
$248$		0			0
$249$		0			0
$250$		0			0
$251$	21.7990			1.37594			0.687970	−	0.725739i	$-0.258502\pi$
$251$	21.7990			1.37594			0.687970	+	0.725739i	$0.258502\pi$
$252$		0			0
$253$	49.4558			3.10926
$254$		0			0
$255$	−3.00000	+	2.82843i	−0.187867	+	0.177123i
$256$		0			0
$257$	−22.9706			−1.43286			−0.716432	−	0.697657i	$-0.754226\pi$
$257$	−22.9706			−1.43286			−0.716432	+	0.697657i	$0.754226\pi$
$258$		0			0
$259$	36.3137			2.25642
$260$		0			0
$261$			4.17157i			0.258214i
$262$		0			0
$263$	−0.313708			−0.0193441			−0.00967205	−	0.999953i	$-0.503079\pi$
$263$	−0.313708			−0.0193441			−0.00967205	+	0.999953i	$0.503079\pi$
$264$		0			0
$265$		−	8.31371i		−	0.510707i
$266$		0			0
$267$		−	14.1421i		−	0.865485i
$268$		0			0
$269$			9.82843i			0.599250i	0.954057	+	0.299625i	$0.0968615\pi$
$269$			9.82843i			0.599250i	−0.954057	+	0.299625i	$0.903138\pi$
$270$		0			0
$271$	24.9706			1.51685			0.758427	−	0.651758i	$-0.225968\pi$
$271$	24.9706			1.51685			0.758427	+	0.651758i	$0.225968\pi$
$272$		0			0
$273$	−18.4853			−1.11878
$274$		0			0
$275$		−	5.82843i		−	0.351467i
$276$		0			0
$277$			27.6569i			1.66174i	0.556467	+	0.830870i	$0.312157\pi$
$277$			27.6569i			1.66174i	−0.556467	+	0.830870i	$0.687843\pi$
$278$		0			0
$279$			6.82843i			0.408807i
$280$		0			0
$281$	−9.31371			−0.555609			−0.277805	−	0.960638i	$-0.589607\pi$
$281$	−9.31371			−0.555609			−0.277805	+	0.960638i	$0.589607\pi$
$282$		0			0
$283$			2.51472i			0.149485i	0.997203	+	0.0747423i	$0.0238134\pi$
$283$			2.51472i			0.149485i	−0.997203	+	0.0747423i	$0.976187\pi$
$284$		0			0
$285$	3.00000			0.177705
$286$		0			0
$287$	22.3137			1.31714
$288$		0			0
$289$	1.00000	−	16.9706i	0.0588235	−	0.998268i
$290$		0			0
$291$	14.0000			0.820695
$292$		0			0
$293$	−11.3431			−0.662674			−0.331337	−	0.943513i	$-0.607500\pi$
$293$	−11.3431			−0.662674			−0.331337	+	0.943513i	$0.607500\pi$
$294$		0			0
$295$			5.17157i			0.301101i
$296$		0			0
$297$	−5.82843			−0.338200
$298$		0			0
$299$		−	40.9706i		−	2.36939i
$300$		0			0
$301$			30.6274i			1.76533i
$302$		0			0
$303$			5.65685i			0.324978i
$304$		0			0
$305$	−8.48528			−0.485866
$306$		0			0
$307$	−19.3137			−1.10229			−0.551146	−	0.834409i	$-0.685809\pi$
$307$	−19.3137			−1.10229			−0.551146	+	0.834409i	$0.685809\pi$
$308$		0			0
$309$		−	6.00000i		−	0.341328i
$310$		0			0
$311$			7.82843i			0.443909i	0.975057	+	0.221955i	$0.0712437\pi$
$311$			7.82843i			0.443909i	−0.975057	+	0.221955i	$0.928756\pi$
$312$		0			0
$313$			32.7990i			1.85391i	0.375174	+	0.926954i	$0.377583\pi$
$313$			32.7990i			1.85391i	−0.375174	+	0.926954i	$0.622417\pi$
$314$		0			0
$315$	3.82843			0.215707
$316$		0			0
$317$		−	8.14214i		−	0.457308i	−0.973508	−	0.228654i	$-0.926568\pi$
$317$		−	8.14214i		−	0.457308i	0.973508	−	0.228654i	$-0.0734324\pi$
$318$		0			0
$319$	24.3137			1.36131
$320$		0			0
$321$	4.34315			0.242411
$322$		0			0
$323$	−9.00000	+	8.48528i	−0.500773	+	0.472134i
$324$		0			0
$325$	−4.82843			−0.267833
$326$		0			0
$327$	−3.31371			−0.183248
$328$		0			0
$329$		−	33.1421i		−	1.82719i
$330$		0			0
$331$	27.6274			1.51854			0.759270	−	0.650776i	$-0.225556\pi$
$331$	27.6274			1.51854			0.759270	+	0.650776i	$0.225556\pi$
$332$		0			0
$333$		−	9.48528i		−	0.519790i
$334$		0			0
$335$		−	2.48528i		−	0.135785i
$336$		0			0
$337$		−	20.7990i		−	1.13299i	−0.824064	−	0.566497i	$-0.808298\pi$
$337$		−	20.7990i		−	1.13299i	0.824064	−	0.566497i	$-0.191702\pi$
$338$		0			0
$339$	−10.8284			−0.588119
$340$		0			0
$341$	39.7990			2.15524
$342$		0			0
$343$			2.51472i			0.135782i
$344$		0			0
$345$			8.48528i			0.456832i
$346$		0			0
$347$		−	15.1716i		−	0.814453i	−0.913327	−	0.407226i	$-0.866496\pi$
$347$		−	15.1716i		−	0.814453i	0.913327	−	0.407226i	$-0.133504\pi$
$348$		0			0
$349$	9.00000			0.481759			0.240879	−	0.970555i	$-0.422564\pi$
$349$	9.00000			0.481759			0.240879	+	0.970555i	$0.422564\pi$
$350$		0			0
$351$			4.82843i			0.257722i
$352$		0			0
$353$	15.6274			0.831763			0.415882	−	0.909419i	$-0.363473\pi$
$353$	15.6274			0.831763			0.415882	+	0.909419i	$0.363473\pi$
$354$		0			0
$355$	−9.65685			−0.512533
$356$		0			0
$357$	−11.4853	+	10.8284i	−0.607866	+	0.573101i
$358$		0			0
$359$	−3.17157			−0.167389			−0.0836946	−	0.996491i	$-0.526672\pi$
$359$	−3.17157			−0.167389			−0.0836946	+	0.996491i	$0.526672\pi$
$360$		0			0
$361$	−10.0000			−0.526316
$362$		0			0
$363$			22.9706i			1.20564i
$364$		0			0
$365$	−15.8284			−0.828498
$366$		0			0
$367$		−	8.00000i		−	0.417597i	−0.977959	−	0.208798i	$-0.933045\pi$
$367$		−	8.00000i		−	0.417597i	0.977959	−	0.208798i	$-0.0669552\pi$
$368$		0			0
$369$		−	5.82843i		−	0.303416i
$370$		0			0
$371$		−	31.8284i		−	1.65245i
$372$		0			0
$373$	2.68629			0.139091			0.0695455	−	0.997579i	$-0.477845\pi$
$373$	2.68629			0.139091			0.0695455	+	0.997579i	$0.477845\pi$
$374$		0			0
$375$	1.00000			0.0516398
$376$		0			0
$377$		−	20.1421i		−	1.03737i
$378$		0			0
$379$			10.0000i			0.513665i	0.966456	+	0.256833i	$0.0826790\pi$
$379$			10.0000i			0.513665i	−0.966456	+	0.256833i	$0.917321\pi$
$380$		0			0
$381$			10.1421i			0.519597i
$382$		0			0
$383$	−7.34315			−0.375217			−0.187609	−	0.982244i	$-0.560074\pi$
$383$	−7.34315			−0.375217			−0.187609	+	0.982244i	$0.560074\pi$
$384$		0			0
$385$		−	22.3137i		−	1.13721i
$386$		0			0
$387$	8.00000			0.406663
$388$		0			0
$389$	1.02944			0.0521945			0.0260973	−	0.999659i	$-0.491692\pi$
$389$	1.02944			0.0521945			0.0260973	+	0.999659i	$0.491692\pi$
$390$		0			0
$391$	−24.0000	−	25.4558i	−1.21373	−	1.28736i
$392$		0			0
$393$	7.31371			0.368928
$394$		0			0
$395$	2.00000			0.100631
$396$		0			0
$397$		−	37.4853i		−	1.88133i	−0.339333	−	0.940666i	$-0.610201\pi$
$397$		−	37.4853i		−	1.88133i	0.339333	−	0.940666i	$-0.389799\pi$
$398$		0			0
$399$	11.4853			0.574983
$400$		0			0
$401$			7.82843i			0.390933i	0.980710	+	0.195466i	$0.0626221\pi$
$401$			7.82843i			0.390933i	−0.980710	+	0.195466i	$0.937378\pi$
$402$		0			0
$403$		−	32.9706i		−	1.64238i
$404$		0			0
$405$		−	1.00000i		−	0.0496904i
$406$		0			0
$407$	−55.2843			−2.74034
$408$		0			0
$409$	−17.3431			−0.857563			−0.428782	−	0.903408i	$-0.641057\pi$
$409$	−17.3431			−0.857563			−0.428782	+	0.903408i	$0.641057\pi$
$410$		0			0
$411$			1.34315i			0.0662525i
$412$		0			0
$413$			19.7990i			0.974245i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−6.00000			−0.293821
$418$		0			0
$419$		−	21.1421i		−	1.03286i	−0.856329	−	0.516430i	$-0.827260\pi$
$419$		−	21.1421i		−	1.03286i	0.856329	−	0.516430i	$-0.172740\pi$
$420$		0			0
$421$	6.65685			0.324435			0.162218	−	0.986755i	$-0.448135\pi$
$421$	6.65685			0.324435			0.162218	+	0.986755i	$0.448135\pi$
$422$		0			0
$423$	−8.65685			−0.420911
$424$		0			0
$425$	−3.00000	+	2.82843i	−0.145521	+	0.137199i
$426$		0			0
$427$	−32.4853			−1.57207
$428$		0			0
$429$	28.1421			1.35872
$430$		0			0
$431$		−	16.1716i		−	0.778957i	−0.921036	−	0.389479i	$-0.872655\pi$
$431$		−	16.1716i		−	0.778957i	0.921036	−	0.389479i	$-0.127345\pi$
$432$		0			0
$433$	24.6274			1.18352			0.591759	−	0.806115i	$-0.298434\pi$
$433$	24.6274			1.18352			0.591759	+	0.806115i	$0.298434\pi$
$434$		0			0
$435$			4.17157i			0.200012i
$436$		0			0
$437$			25.4558i			1.21772i
$438$		0			0
$439$			8.48528i			0.404980i	0.979284	+	0.202490i	$0.0649034\pi$
$439$			8.48528i			0.404980i	−0.979284	+	0.202490i	$0.935097\pi$
$440$		0			0
$441$	7.65685			0.364612
$442$		0			0
$443$	34.6274			1.64520			0.822599	−	0.568622i	$-0.192523\pi$
$443$	34.6274			1.64520			0.822599	+	0.568622i	$0.192523\pi$
$444$		0			0
$445$		−	14.1421i		−	0.670402i
$446$		0			0
$447$			13.1716i			0.622994i
$448$		0			0
$449$			5.31371i			0.250769i	0.992108	+	0.125385i	$0.0400165\pi$
$449$			5.31371i			0.250769i	−0.992108	+	0.125385i	$0.959983\pi$
$450$		0			0
$451$	−33.9706			−1.59961
$452$		0			0
$453$			15.0000i			0.704761i
$454$		0			0
$455$	−18.4853			−0.866603
$456$		0			0
$457$	11.5147			0.538636			0.269318	−	0.963051i	$-0.413202\pi$
$457$	11.5147			0.538636			0.269318	+	0.963051i	$0.413202\pi$
$458$		0			0
$459$	2.82843	+	3.00000i	0.132020	+	0.140028i
$460$		0			0
$461$	17.1716			0.799760			0.399880	−	0.916568i	$-0.369052\pi$
$461$	17.1716			0.799760			0.399880	+	0.916568i	$0.369052\pi$
$462$		0			0
$463$	20.9706			0.974585			0.487292	−	0.873239i	$-0.337985\pi$
$463$	20.9706			0.974585			0.487292	+	0.873239i	$0.337985\pi$
$464$		0			0
$465$			6.82843i			0.316661i
$466$		0			0
$467$	27.0000			1.24941			0.624705	−	0.780860i	$-0.285219\pi$
$467$	27.0000			1.24941			0.624705	+	0.780860i	$0.285219\pi$
$468$		0			0
$469$		−	9.51472i		−	0.439349i
$470$		0			0
$471$			22.9706i			1.05843i
$472$		0			0
$473$		−	46.6274i		−	2.14393i
$474$		0			0
$475$	3.00000			0.137649
$476$		0			0
$477$	−8.31371			−0.380659
$478$		0			0
$479$			13.6569i			0.623998i	0.950082	+	0.311999i	$0.100998\pi$
$479$			13.6569i			0.623998i	−0.950082	+	0.311999i	$0.899002\pi$
$480$		0			0
$481$			45.7990i			2.08825i
$482$		0			0
$483$			32.4853i			1.47813i
$484$		0			0
$485$	14.0000			0.635707
$486$		0			0
$487$			10.3431i			0.468693i	0.972153	+	0.234346i	$0.0752950\pi$
$487$			10.3431i			0.468693i	−0.972153	+	0.234346i	$0.924705\pi$
$488$		0			0
$489$	−3.48528			−0.157610
$490$		0			0
$491$	26.6274			1.20168			0.600839	−	0.799370i	$-0.294833\pi$
$491$	26.6274			1.20168			0.600839	+	0.799370i	$0.294833\pi$
$492$		0			0
$493$	−11.7990	−	12.5147i	−0.531400	−	0.563635i
$494$		0			0
$495$	−5.82843			−0.261968
$496$		0			0
$497$	−36.9706			−1.65836
$498$		0			0
$499$			9.79899i			0.438663i	0.975650	+	0.219332i	$0.0703876\pi$
$499$			9.79899i			0.438663i	−0.975650	+	0.219332i	$0.929612\pi$
$500$		0			0
$501$	−9.65685			−0.431436
$502$		0			0
$503$			1.31371i			0.0585754i	0.999571	+	0.0292877i	$0.00932389\pi$
$503$			1.31371i			0.0585754i	−0.999571	+	0.0292877i	$0.990676\pi$
$504$		0			0
$505$			5.65685i			0.251727i
$506$		0			0
$507$		−	10.3137i		−	0.458048i
$508$		0			0
$509$	32.1421			1.42468			0.712338	−	0.701837i	$-0.247637\pi$
$509$	32.1421			1.42468			0.712338	+	0.701837i	$0.247637\pi$
$510$		0			0
$511$	−60.5980			−2.68070
$512$		0			0
$513$		−	3.00000i		−	0.132453i
$514$		0			0
$515$		−	6.00000i		−	0.264392i
$516$		0			0
$517$			50.4558i			2.21905i
$518$		0			0
$519$	−20.8284			−0.914266
$520$		0			0
$521$			32.7990i			1.43695i	0.695553	+	0.718475i	$0.255159\pi$
$521$			32.7990i			1.43695i	−0.695553	+	0.718475i	$0.744841\pi$
$522$		0			0
$523$	−16.9706			−0.742071			−0.371035	−	0.928619i	$-0.620997\pi$
$523$	−16.9706			−0.742071			−0.371035	+	0.928619i	$0.620997\pi$
$524$		0			0
$525$	3.82843			0.167086
$526$		0			0
$527$	−19.3137	−	20.4853i	−0.841318	−	0.892353i
$528$		0			0
$529$	−49.0000			−2.13043
$530$		0			0
$531$	5.17157			0.224427
$532$		0			0
$533$			28.1421i			1.21897i
$534$		0			0
$535$	4.34315			0.187771
$536$		0			0
$537$		−	1.65685i		−	0.0714985i
$538$		0			0
$539$		−	44.6274i		−	1.92224i
$540$		0			0
$541$		−	11.5147i		−	0.495056i	−0.968881	−	0.247528i	$-0.920382\pi$
$541$		−	11.5147i		−	0.495056i	0.968881	−	0.247528i	$-0.0796183\pi$
$542$		0			0
$543$	4.82843			0.207208
$544$		0			0
$545$	−3.31371			−0.141944
$546$		0			0
$547$			10.8579i			0.464249i	0.972686	+	0.232124i	$0.0745676\pi$
$547$			10.8579i			0.464249i	−0.972686	+	0.232124i	$0.925432\pi$
$548$		0			0
$549$			8.48528i			0.362143i
$550$		0			0
$551$			12.5147i			0.533145i
$552$		0			0
$553$	7.65685			0.325603
$554$		0			0
$555$		−	9.48528i		−	0.402628i
$556$		0			0
$557$	−11.6569			−0.493917			−0.246958	−	0.969026i	$-0.579431\pi$
$557$	−11.6569			−0.493917			−0.246958	+	0.969026i	$0.579431\pi$
$558$		0			0
$559$	−38.6274			−1.63377
$560$		0			0
$561$	17.4853	−	16.4853i	0.738229	−	0.696009i
$562$		0			0
$563$	21.2843			0.897025			0.448513	−	0.893777i	$-0.351954\pi$
$563$	21.2843			0.897025			0.448513	+	0.893777i	$0.351954\pi$
$564$		0			0
$565$	−10.8284			−0.455555
$566$		0			0
$567$		−	3.82843i		−	0.160779i
$568$		0			0
$569$	7.31371			0.306607			0.153303	−	0.988179i	$-0.451009\pi$
$569$	7.31371			0.306607			0.153303	+	0.988179i	$0.451009\pi$
$570$		0			0
$571$			26.4853i			1.10837i	0.832392	+	0.554187i	$0.186971\pi$
$571$			26.4853i			1.10837i	−0.832392	+	0.554187i	$0.813029\pi$
$572$		0			0
$573$			18.1421i			0.757899i
$574$		0			0
$575$			8.48528i			0.353861i
$576$		0			0
$577$	31.6569			1.31789			0.658946	−	0.752190i	$-0.271003\pi$
$577$	31.6569			1.31789			0.658946	+	0.752190i	$0.271003\pi$
$578$		0			0
$579$	−13.3137			−0.553299
$580$		0			0
$581$		0			0
$582$		0			0
$583$			48.4558i			2.00684i
$584$		0			0
$585$			4.82843i			0.199631i
$586$		0			0
$587$	1.62742			0.0671707			0.0335853	−	0.999436i	$-0.489307\pi$
$587$	1.62742			0.0671707			0.0335853	+	0.999436i	$0.489307\pi$
$588$		0			0
$589$			20.4853i			0.844081i
$590$		0			0
$591$	−3.65685			−0.150423
$592$		0			0
$593$	45.6274			1.87369			0.936847	−	0.349740i	$-0.113730\pi$
$593$	45.6274			1.87369			0.936847	+	0.349740i	$0.113730\pi$
$594$		0			0
$595$	−11.4853	+	10.8284i	−0.470851	+	0.443922i
$596$		0			0
$597$	20.9706			0.858268
$598$		0			0
$599$	39.9411			1.63195			0.815975	−	0.578087i	$-0.196201\pi$
$599$	39.9411			1.63195			0.815975	+	0.578087i	$0.196201\pi$
$600$		0			0
$601$		−	4.48528i		−	0.182958i	−0.995807	−	0.0914792i	$-0.970840\pi$
$601$		−	4.48528i		−	0.182958i	0.995807	−	0.0914792i	$-0.0291595\pi$
$602$		0			0
$603$	−2.48528			−0.101208
$604$		0			0
$605$			22.9706i			0.933886i
$606$		0			0
$607$			20.1127i			0.816349i	0.912904	+	0.408175i	$0.133835\pi$
$607$			20.1127i			0.816349i	−0.912904	+	0.408175i	$0.866165\pi$
$608$		0			0
$609$			15.9706i			0.647160i
$610$		0			0
$611$	41.7990			1.69101
$612$		0			0
$613$	−18.6274			−0.752354			−0.376177	−	0.926548i	$-0.622762\pi$
$613$	−18.6274			−0.752354			−0.376177	+	0.926548i	$0.622762\pi$
$614$		0			0
$615$		−	5.82843i		−	0.235025i
$616$		0			0
$617$			5.17157i			0.208200i	0.994567	+	0.104100i	$0.0331962\pi$
$617$			5.17157i			0.208200i	−0.994567	+	0.104100i	$0.966804\pi$
$618$		0			0
$619$		−	34.0000i		−	1.36658i	−0.730149	−	0.683288i	$-0.760549\pi$
$619$		−	34.0000i		−	1.36658i	0.730149	−	0.683288i	$-0.239451\pi$
$620$		0			0
$621$	8.48528			0.340503
$622$		0			0
$623$		−	54.1421i		−	2.16916i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	−17.4853			−0.698295
$628$		0			0
$629$	26.8284	+	28.4558i	1.06972	+	1.13461i
$630$		0			0
$631$	1.00000			0.0398094			0.0199047	−	0.999802i	$-0.493664\pi$
$631$	1.00000			0.0398094			0.0199047	+	0.999802i	$0.493664\pi$
$632$		0			0
$633$	13.3137			0.529172
$634$		0			0
$635$			10.1421i			0.402478i
$636$		0			0
$637$	−36.9706			−1.46483
$638$		0			0
$639$			9.65685i			0.382019i
$640$		0			0
$641$		−	35.1421i		−	1.38803i	−0.719960	−	0.694015i	$-0.755840\pi$
$641$		−	35.1421i		−	1.38803i	0.719960	−	0.694015i	$-0.244160\pi$
$642$		0			0
$643$		−	36.4558i		−	1.43768i	−0.695177	−	0.718839i	$-0.744674\pi$
$643$		−	36.4558i		−	1.43768i	0.695177	−	0.718839i	$-0.255326\pi$
$644$		0			0
$645$	8.00000			0.315000
$646$		0			0
$647$	44.2843			1.74099			0.870497	−	0.492173i	$-0.163797\pi$
$647$	44.2843			1.74099			0.870497	+	0.492173i	$0.163797\pi$
$648$		0			0
$649$		−	30.1421i		−	1.18318i
$650$		0			0
$651$			26.1421i			1.02459i
$652$		0			0
$653$			17.6569i			0.690966i	0.938425	+	0.345483i	$0.112285\pi$
$653$			17.6569i			0.690966i	−0.938425	+	0.345483i	$0.887715\pi$
$654$		0			0
$655$	7.31371			0.285770
$656$		0			0
$657$			15.8284i			0.617526i
$658$		0			0
$659$	−6.68629			−0.260461			−0.130230	−	0.991484i	$-0.541572\pi$
$659$	−6.68629			−0.260461			−0.130230	+	0.991484i	$0.541572\pi$
$660$		0			0
$661$	−24.9411			−0.970097			−0.485049	−	0.874487i	$-0.661198\pi$
$661$	−24.9411			−0.970097			−0.485049	+	0.874487i	$0.661198\pi$
$662$		0			0
$663$	−13.6569	−	14.4853i	−0.530388	−	0.562562i
$664$		0			0
$665$	11.4853			0.445380
$666$		0			0
$667$	−35.3970			−1.37058
$668$		0			0
$669$			13.3137i			0.514738i
$670$		0			0
$671$	49.4558			1.90922
$672$		0			0
$673$			27.9411i			1.07705i	0.842609	+	0.538526i	$0.181018\pi$
$673$			27.9411i			1.07705i	−0.842609	+	0.538526i	$0.818982\pi$
$674$		0			0
$675$		−	1.00000i		−	0.0384900i
$676$		0			0
$677$		−	2.34315i		−	0.0900544i	−0.998986	−	0.0450272i	$-0.985663\pi$
$677$		−	2.34315i		−	0.0900544i	0.998986	−	0.0450272i	$-0.0143375\pi$
$678$		0			0
$679$	53.5980			2.05690
$680$		0			0
$681$	4.82843			0.185026
$682$		0			0
$683$		−	7.02944i		−	0.268974i	−0.990915	−	0.134487i	$-0.957061\pi$
$683$		−	7.02944i		−	0.268974i	0.990915	−	0.134487i	$-0.0429386\pi$
$684$		0			0
$685$			1.34315i			0.0513190i
$686$		0			0
$687$			11.0000i			0.419676i
$688$		0			0
$689$	40.1421			1.52929
$690$		0			0
$691$			40.3431i			1.53473i	0.641213	+	0.767363i	$0.278431\pi$
$691$			40.3431i			1.53473i	−0.641213	+	0.767363i	$0.721569\pi$
$692$		0			0
$693$	−22.3137			−0.847628
$694$		0			0
$695$	−6.00000			−0.227593
$696$		0			0
$697$	16.4853	+	17.4853i	0.624425	+	0.662302i
$698$		0			0
$699$	−14.4853			−0.547884
$700$		0			0
$701$	−3.51472			−0.132749			−0.0663745	−	0.997795i	$-0.521143\pi$
$701$	−3.51472			−0.132749			−0.0663745	+	0.997795i	$0.521143\pi$
$702$		0			0
$703$		−	28.4558i		−	1.07323i
$704$		0			0
$705$	−8.65685			−0.326036
$706$		0			0
$707$			21.6569i			0.814490i
$708$		0			0
$709$		−	34.9706i		−	1.31335i	−0.754175	−	0.656674i	$-0.771963\pi$
$709$		−	34.9706i		−	1.31335i	0.754175	−	0.656674i	$-0.228037\pi$
$710$		0			0
$711$		−	2.00000i		−	0.0750059i
$712$		0			0
$713$	−57.9411			−2.16991
$714$		0			0
$715$	28.1421			1.05246
$716$		0			0
$717$		−	14.4853i		−	0.540963i
$718$		0			0
$719$			22.1127i			0.824664i	0.911034	+	0.412332i	$0.135286\pi$
$719$			22.1127i			0.824664i	−0.911034	+	0.412332i	$0.864714\pi$
$720$		0			0
$721$		−	22.9706i		−	0.855468i
$722$		0			0
$723$	1.51472			0.0563330
$724$		0			0
$725$			4.17157i			0.154928i
$726$		0			0
$727$	38.9706			1.44534			0.722669	−	0.691194i	$-0.242915\pi$
$727$	38.9706			1.44534			0.722669	+	0.691194i	$0.242915\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	−24.0000	+	22.6274i	−0.887672	+	0.836905i
$732$		0			0
$733$	40.4853			1.49536			0.747679	−	0.664060i	$-0.231168\pi$
$733$	40.4853			1.49536			0.747679	+	0.664060i	$0.231168\pi$
$734$		0			0
$735$	7.65685			0.282427
$736$		0			0
$737$			14.4853i			0.533572i
$738$		0			0
$739$	−8.28427			−0.304742			−0.152371	−	0.988323i	$-0.548691\pi$
$739$	−8.28427			−0.304742			−0.152371	+	0.988323i	$0.548691\pi$
$740$		0			0
$741$			14.4853i			0.532130i
$742$		0			0
$743$			39.9411i			1.46530i	0.680607	+	0.732649i	$0.261716\pi$
$743$			39.9411i			1.46530i	−0.680607	+	0.732649i	$0.738284\pi$
$744$		0			0
$745$			13.1716i			0.482569i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	16.6274			0.607553
$750$		0			0
$751$		−	19.8579i		−	0.724624i	−0.932057	−	0.362312i	$-0.881988\pi$
$751$		−	19.8579i		−	0.724624i	0.932057	−	0.362312i	$-0.118012\pi$
$752$		0			0
$753$		−	21.7990i		−	0.794399i
$754$		0			0
$755$			15.0000i			0.545906i
$756$		0			0
$757$	27.6569			1.00521			0.502603	−	0.864517i	$-0.332376\pi$
$757$	27.6569			1.00521			0.502603	+	0.864517i	$0.332376\pi$
$758$		0			0
$759$		−	49.4558i		−	1.79513i
$760$		0			0
$761$	−21.5147			−0.779908			−0.389954	−	0.920834i	$-0.627509\pi$
$761$	−21.5147			−0.779908			−0.389954	+	0.920834i	$0.627509\pi$
$762$		0			0
$763$	−12.6863			−0.459275
$764$		0			0
$765$	2.82843	+	3.00000i	0.102262	+	0.108465i
$766$		0			0
$767$	−24.9706			−0.901635
$768$		0			0
$769$	24.6569			0.889149			0.444574	−	0.895742i	$-0.353355\pi$
$769$	24.6569			0.889149			0.444574	+	0.895742i	$0.353355\pi$
$770$		0			0
$771$			22.9706i			0.827265i
$772$		0			0
$773$	19.6863			0.708067			0.354033	−	0.935233i	$-0.384810\pi$
$773$	19.6863			0.708067			0.354033	+	0.935233i	$0.384810\pi$
$774$		0			0
$775$			6.82843i			0.245284i
$776$		0			0
$777$		−	36.3137i		−	1.30275i
$778$		0			0
$779$		−	17.4853i		−	0.626475i
$780$		0			0
$781$	56.2843			2.01401
$782$		0			0
$783$	4.17157			0.149080
$784$		0			0
$785$			22.9706i			0.819855i
$786$		0			0
$787$			5.48528i			0.195529i	0.995210	+	0.0977646i	$0.0311692\pi$
$787$			5.48528i			0.195529i	−0.995210	+	0.0977646i	$0.968831\pi$
$788$		0			0
$789$			0.313708i			0.0111683i
$790$		0			0
$791$	−41.4558			−1.47400
$792$		0			0
$793$		−	40.9706i		−	1.45491i
$794$		0			0
$795$	−8.31371			−0.294857
$796$		0			0
$797$	−1.97056			−0.0698009			−0.0349005	−	0.999391i	$-0.511111\pi$
$797$	−1.97056			−0.0698009			−0.0349005	+	0.999391i	$0.511111\pi$
$798$		0			0
$799$	25.9706	−	24.4853i	0.918772	−	0.866227i
$800$		0			0
$801$	−14.1421			−0.499688
$802$		0			0
$803$	92.2548			3.25560
$804$		0			0
$805$			32.4853i			1.14496i
$806$		0			0
$807$	9.82843			0.345977
$808$		0			0
$809$		−	46.2843i		−	1.62727i	−0.581377	−	0.813634i	$-0.697486\pi$
$809$		−	46.2843i		−	1.62727i	0.581377	−	0.813634i	$-0.302514\pi$
$810$		0			0
$811$		−	13.4558i		−	0.472499i	−0.971692	−	0.236249i	$-0.924082\pi$
$811$		−	13.4558i		−	0.472499i	0.971692	−	0.236249i	$-0.0759182\pi$
$812$		0			0
$813$		−	24.9706i		−	0.875756i
$814$		0			0
$815$	−3.48528			−0.122084
$816$		0			0
$817$	24.0000			0.839654
$818$		0			0
$819$			18.4853i			0.645928i
$820$		0			0
$821$		−	29.3137i		−	1.02306i	−0.859267	−	0.511528i	$-0.829080\pi$
$821$		−	29.3137i		−	1.02306i	0.859267	−	0.511528i	$-0.170920\pi$
$822$		0			0
$823$		−	49.8284i		−	1.73691i	−0.495768	−	0.868455i	$-0.665113\pi$
$823$		−	49.8284i		−	1.73691i	0.495768	−	0.868455i	$-0.334887\pi$
$824$		0			0
$825$	−5.82843			−0.202920
$826$		0			0
$827$		−	38.1421i		−	1.32633i	−0.748472	−	0.663166i	$-0.769212\pi$
$827$		−	38.1421i		−	1.32633i	0.748472	−	0.663166i	$-0.230788\pi$
$828$		0			0
$829$	−22.0294			−0.765114			−0.382557	−	0.923932i	$-0.624956\pi$
$829$	−22.0294			−0.765114			−0.382557	+	0.923932i	$0.624956\pi$
$830$		0			0
$831$	27.6569			0.959406
$832$		0			0
$833$	−22.9706	+	21.6569i	−0.795883	+	0.750366i
$834$		0			0
$835$	−9.65685			−0.334189
$836$		0			0
$837$	6.82843			0.236025
$838$		0			0
$839$			6.17157i			0.213066i	0.994309	+	0.106533i	$0.0339750\pi$
$839$			6.17157i			0.213066i	−0.994309	+	0.106533i	$0.966025\pi$
$840$		0			0
$841$	11.5980			0.399930
$842$		0			0
$843$			9.31371i			0.320781i
$844$		0			0
$845$		−	10.3137i		−	0.354802i
$846$		0			0
$847$			87.9411i			3.02169i
$848$		0			0
$849$	2.51472			0.0863049
$850$		0			0
$851$	80.4853			2.75900
$852$		0			0
$853$		−	1.31371i		−	0.0449805i	−0.999747	−	0.0224903i	$-0.992841\pi$
$853$		−	1.31371i		−	0.0449805i	0.999747	−	0.0224903i	$-0.00715948\pi$
$854$		0			0
$855$		−	3.00000i		−	0.102598i
$856$		0			0
$857$			31.3137i			1.06966i	0.844961	+	0.534828i	$0.179624\pi$
$857$			31.3137i			1.06966i	−0.844961	+	0.534828i	$0.820376\pi$
$858$		0			0
$859$	−41.2843			−1.40860			−0.704301	−	0.709902i	$-0.748739\pi$
$859$	−41.2843			−1.40860			−0.704301	+	0.709902i	$0.748739\pi$
$860$		0			0
$861$		−	22.3137i		−	0.760449i
$862$		0			0
$863$	27.3431			0.930772			0.465386	−	0.885108i	$-0.345916\pi$
$863$	27.3431			0.930772			0.465386	+	0.885108i	$0.345916\pi$
$864$		0			0
$865$	−20.8284			−0.708188
$866$		0			0
$867$	−16.9706	−	1.00000i	−0.576351	−	0.0339618i
$868$		0			0
$869$	−11.6569			−0.395432
$870$		0			0
$871$	12.0000			0.406604
$872$		0			0
$873$		−	14.0000i		−	0.473828i
$874$		0			0
$875$	3.82843			0.129424
$876$		0			0
$877$			8.45584i			0.285534i	0.989756	+	0.142767i	$0.0455999\pi$
$877$			8.45584i			0.285534i	−0.989756	+	0.142767i	$0.954400\pi$
$878$		0			0
$879$			11.3431i			0.382595i
$880$		0			0
$881$			10.1716i			0.342689i	0.985211	+	0.171344i	$0.0548111\pi$
$881$			10.1716i			0.342689i	−0.985211	+	0.171344i	$0.945189\pi$
$882$		0			0
$883$	−18.4853			−0.622079			−0.311040	−	0.950397i	$-0.600677\pi$
$883$	−18.4853			−0.622079			−0.311040	+	0.950397i	$0.600677\pi$
$884$		0			0
$885$	5.17157			0.173841
$886$		0			0
$887$			16.3431i			0.548749i	0.961623	+	0.274375i	$0.0884708\pi$
$887$			16.3431i			0.548749i	−0.961623	+	0.274375i	$0.911529\pi$
$888$		0			0
$889$			38.8284i			1.30226i
$890$		0			0
$891$			5.82843i			0.195260i
$892$		0			0
$893$	−25.9706			−0.869072
$894$		0			0
$895$		−	1.65685i		−	0.0553825i
$896$		0			0
$897$	−40.9706			−1.36797
$898$		0			0
$899$	−28.4853			−0.950037
$900$		0			0
$901$	24.9411	−	23.5147i	0.830909	−	0.783389i
$902$		0			0
$903$	30.6274			1.01922
$904$		0			0
$905$	4.82843			0.160502
$906$		0			0
$907$			2.51472i			0.0834999i	0.999128	+	0.0417499i	$0.0132933\pi$
$907$			2.51472i			0.0834999i	−0.999128	+	0.0417499i	$0.986707\pi$
$908$		0			0
$909$	5.65685			0.187626
$910$		0			0
$911$		−	11.8284i		−	0.391893i	−0.980615	−	0.195947i	$-0.937222\pi$
$911$		−	11.8284i		−	0.391893i	0.980615	−	0.195947i	$-0.0627779\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$			8.48528i			0.280515i
$916$		0			0
$917$	28.0000			0.924641
$918$		0			0
$919$	39.0000			1.28649			0.643246	−	0.765660i	$-0.277587\pi$
$919$	39.0000			1.28649			0.643246	+	0.765660i	$0.277587\pi$
$920$		0			0
$921$			19.3137i			0.636408i
$922$		0			0
$923$		−	46.6274i		−	1.53476i
$924$		0			0
$925$		−	9.48528i		−	0.311874i
$926$		0			0
$927$	−6.00000			−0.197066
$928$		0			0
$929$		−	39.4264i		−	1.29354i	−0.762686	−	0.646769i	$-0.776120\pi$
$929$		−	39.4264i		−	1.29354i	0.762686	−	0.646769i	$-0.223880\pi$
$930$		0			0
$931$	22.9706			0.752830
$932$		0			0
$933$	7.82843			0.256291
$934$		0			0
$935$	17.4853	−	16.4853i	0.571830	−	0.539126i
$936$		0			0
$937$	−15.4558			−0.504920			−0.252460	−	0.967607i	$-0.581240\pi$
$937$	−15.4558			−0.504920			−0.252460	+	0.967607i	$0.581240\pi$
$938$		0			0
$939$	32.7990			1.07035
$940$		0			0
$941$			30.0000i			0.977972i	0.872292	+	0.488986i	$0.162633\pi$
$941$			30.0000i			0.977972i	−0.872292	+	0.488986i	$0.837367\pi$
$942$		0			0
$943$	49.4558			1.61050
$944$		0			0
$945$		−	3.82843i		−	0.124539i
$946$		0			0
$947$			12.6274i			0.410336i	0.978727	+	0.205168i	$0.0657741\pi$
$947$			12.6274i			0.410336i	−0.978727	+	0.205168i	$0.934226\pi$
$948$		0			0
$949$		−	76.4264i		−	2.48091i
$950$		0			0
$951$	−8.14214			−0.264027
$952$		0			0
$953$	−30.3137			−0.981957			−0.490979	−	0.871172i	$-0.663361\pi$
$953$	−30.3137			−0.981957			−0.490979	+	0.871172i	$0.663361\pi$
$954$		0			0
$955$			18.1421i			0.587066i
$956$		0			0
$957$		−	24.3137i		−	0.785951i
$958$		0			0
$959$			5.14214i			0.166048i
$960$		0			0
$961$	−15.6274			−0.504110
$962$		0			0
$963$		−	4.34315i		−	0.139956i
$964$		0			0
$965$	−13.3137			−0.428583
$966$		0			0
$967$	31.1716			1.00241			0.501205	−	0.865329i	$-0.332890\pi$
$967$	31.1716			1.00241			0.501205	+	0.865329i	$0.332890\pi$
$968$		0			0
$969$	8.48528	+	9.00000i	0.272587	+	0.289122i
$970$		0			0
$971$	−12.1421			−0.389660			−0.194830	−	0.980837i	$-0.562415\pi$
$971$	−12.1421			−0.389660			−0.194830	+	0.980837i	$0.562415\pi$
$972$		0			0
$973$	−22.9706			−0.736402
$974$		0			0
$975$			4.82843i			0.154633i
$976$		0			0
$977$	−18.6863			−0.597827			−0.298914	−	0.954280i	$-0.596624\pi$
$977$	−18.6863			−0.597827			−0.298914	+	0.954280i	$0.596624\pi$
$978$		0			0
$979$			82.4264i			2.63436i
$980$		0			0
$981$			3.31371i			0.105799i
$982$		0			0
$983$		−	39.7990i		−	1.26939i	−0.772762	−	0.634695i	$-0.781126\pi$
$983$		−	39.7990i		−	1.26939i	0.772762	−	0.634695i	$-0.218874\pi$
$984$		0			0
$985$	−3.65685			−0.116517
$986$		0			0
$987$	−33.1421			−1.05493
$988$		0			0
$989$			67.8823i			2.15853i
$990$		0			0
$991$		−	52.1421i		−	1.65635i	−0.560470	−	0.828175i	$-0.689379\pi$
$991$		−	52.1421i		−	1.65635i	0.560470	−	0.828175i	$-0.310621\pi$
$992$		0			0
$993$		−	27.6274i		−	0.876730i
$994$		0			0
$995$	20.9706			0.664812
$996$		0			0
$997$			27.8284i			0.881335i	0.897670	+	0.440668i	$0.145258\pi$
$997$			27.8284i			0.881335i	−0.897670	+	0.440668i	$0.854742\pi$
$998$		0			0
$999$	−9.48528			−0.300101

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4080.2.h.n.3841.1		4
4.3	odd	2		1020.2.e.d.781.4	yes	4
12.11	even	2		3060.2.e.g.1801.4		4
17.16	even	2	inner	4080.2.h.n.3841.4		4
20.3	even	4		5100.2.k.h.4249.3		4
20.7	even	4		5100.2.k.i.4249.1		4
20.19	odd	2		5100.2.e.f.1801.1		4
68.67	odd	2		1020.2.e.d.781.1	✓	4
204.203	even	2		3060.2.e.g.1801.1		4
340.67	even	4		5100.2.k.h.4249.4		4
340.203	even	4		5100.2.k.i.4249.2		4
340.339	odd	2		5100.2.e.f.1801.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1020.2.e.d.781.1	✓	4	68.67	odd	2
1020.2.e.d.781.4	yes	4	4.3	odd	2
3060.2.e.g.1801.1		4	204.203	even	2
3060.2.e.g.1801.4		4	12.11	even	2
4080.2.h.n.3841.1		4	1.1	even	1	trivial
4080.2.h.n.3841.4		4	17.16	even	2	inner
5100.2.e.f.1801.1		4	20.19	odd	2
5100.2.e.f.1801.4		4	340.339	odd	2
5100.2.k.h.4249.3		4	20.3	even	4
5100.2.k.h.4249.4		4	340.67	even	4
5100.2.k.i.4249.1		4	20.7	even	4
5100.2.k.i.4249.2		4	340.203	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4080 = 2^{4} \cdot 3 \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4080.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} -1.00000i q^{5} -3.82843i q^{7} -1.00000 q^{9} +5.82843i q^{11} +4.82843 q^{13} -1.00000 q^{15} +(3.00000 - 2.82843i) q^{17} -3.00000 q^{19} -3.82843 q^{21} -8.48528i q^{23} -1.00000 q^{25} +1.00000i q^{27} -4.17157i q^{29} -6.82843i q^{31} +5.82843 q^{33} -3.82843 q^{35} +9.48528i q^{37} -4.82843i q^{39} +5.82843i q^{41} -8.00000 q^{43} +1.00000i q^{45} +8.65685 q^{47} -7.65685 q^{49} +(-2.82843 - 3.00000i) q^{51} +8.31371 q^{53} +5.82843 q^{55} +3.00000i q^{57} -5.17157 q^{59} -8.48528i q^{61} +3.82843i q^{63} -4.82843i q^{65} +2.48528 q^{67} -8.48528 q^{69} -9.65685i q^{71} -15.8284i q^{73} +1.00000i q^{75} +22.3137 q^{77} +2.00000i q^{79} +1.00000 q^{81} +(-2.82843 - 3.00000i) q^{85} -4.17157 q^{87} +14.1421 q^{89} -18.4853i q^{91} -6.82843 q^{93} +3.00000i q^{95} +14.0000i q^{97} -5.82843i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{9} + 8 q^{13} - 4 q^{15} + 12 q^{17} - 12 q^{19} - 4 q^{21} - 4 q^{25} + 12 q^{33} - 4 q^{35} - 32 q^{43} + 12 q^{47} - 8 q^{49} - 12 q^{53} + 12 q^{55} - 32 q^{59} - 24 q^{67} + 44 q^{77} + 4 q^{81}+ \cdots - 16 q^{93}+O(q^{100}) \) 4 * q - 4 * q^9 + 8 * q^13 - 4 * q^15 + 12 * q^17 - 12 * q^19 - 4 * q^21 - 4 * q^25 + 12 * q^33 - 4 * q^35 - 32 * q^43 + 12 * q^47 - 8 * q^49 - 12 * q^53 + 12 * q^55 - 32 * q^59 - 24 * q^67 + 44 * q^77 + 4 * q^81 - 28 * q^87 - 16 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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