LMFDB - Embedded newform 5100.2.k.h.4249.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5100,2,Mod(4249,5100)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5100, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 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("5100.4249"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 5100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5100.k (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,-4,0,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$40.7237050309$
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:	$\Z[a_1, \ldots, a_{11}]$
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			4249.4
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	5100.4249
Dual form			5100.2.k.h.4249.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +3.82843 q^{7} +1.00000 q^{9} +5.82843i q^{11} -4.82843i q^{13} +(-2.82843 + 3.00000i) q^{17} -3.00000 q^{19} -3.82843 q^{21} -8.48528 q^{23} -1.00000 q^{27} -4.17157i q^{29} -6.82843i q^{31} -5.82843i q^{33} +9.48528 q^{37} +4.82843i q^{39} -5.82843i q^{41} -8.00000i q^{43} -8.65685i q^{47} +7.65685 q^{49} +(2.82843 - 3.00000i) q^{51} -8.31371i q^{53} +3.00000 q^{57} -5.17157 q^{59} +8.48528i q^{61} +3.82843 q^{63} -2.48528i q^{67} +8.48528 q^{69} -9.65685i q^{71} +15.8284 q^{73} +22.3137i q^{77} -2.00000i q^{79} +1.00000 q^{81} +4.17157i q^{87} -14.1421 q^{89} -18.4853i q^{91} +6.82843i q^{93} +14.0000 q^{97} +5.82843i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} + 4 q^{7} + 4 q^{9} - 12 q^{19} - 4 q^{21} - 4 q^{27} + 4 q^{37} + 8 q^{49} + 12 q^{57} - 32 q^{59} + 4 q^{63} + 52 q^{73} + 4 q^{81} + 56 q^{97}+O(q^{100}) $ 4 * q - 4 * q^3 + 4 * q^7 + 4 * q^9 - 12 * q^19 - 4 * q^21 - 4 * q^27 + 4 * q^37 + 8 * q^49 + 12 * q^57 - 32 * q^59 + 4 * q^63 + 52 * q^73 + 4 * q^81 + 56 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5100\mathbb{Z}\right)^\times$.

$n$	$2551$	$3301$	$3401$	$3877$
$\chi(n)$	$1$	$-1$	$1$	$-1$

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
$3$	−1.00000			−0.577350
$3$	−1.00000			−0.577350
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	3.82843			1.44701			0.723505	−	0.690319i	$-0.242530\pi$
$7$	3.82843			1.44701			0.723505	+	0.690319i	$0.242530\pi$
$8$		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
$13$		−	4.82843i		−	1.33916i	−0.742738	−	0.669582i	$-0.766473\pi$
$13$		−	4.82843i		−	1.33916i	0.742738	−	0.669582i	$-0.233527\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.82843	+	3.00000i	−0.685994	+	0.727607i
$17$	−2.82843	+	3.00000i	−0.685994	+	0.727607i
$18$		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
$23$	−8.48528			−1.76930			−0.884652	−	0.466252i	$-0.845604\pi$
$23$	−8.48528			−1.76930			−0.884652	+	0.466252i	$0.845604\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−1.00000			−0.192450
$28$		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
$33$		−	5.82843i		−	1.01460i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	9.48528			1.55937			0.779685	−	0.626172i	$-0.215379\pi$
$37$	9.48528			1.55937			0.779685	+	0.626172i	$0.215379\pi$
$38$		0			0
$39$			4.82843i			0.773167i
$40$		0			0
$41$		−	5.82843i		−	0.910247i	−0.890428	−	0.455124i	$-0.849595\pi$
$41$		−	5.82843i		−	0.910247i	0.890428	−	0.455124i	$-0.150405\pi$
$42$		0			0
$43$		−	8.00000i		−	1.21999i	−0.792406	−	0.609994i	$-0.791172\pi$
$43$		−	8.00000i		−	1.21999i	0.792406	−	0.609994i	$-0.208828\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	8.65685i		−	1.26273i	−0.775485	−	0.631366i	$-0.782495\pi$
$47$		−	8.65685i		−	1.26273i	0.775485	−	0.631366i	$-0.217505\pi$
$48$		0			0
$49$	7.65685			1.09384
$50$		0			0
$51$	2.82843	−	3.00000i	0.396059	−	0.420084i
$52$		0			0
$53$		−	8.31371i		−	1.14198i	−0.820959	−	0.570988i	$-0.806560\pi$
$53$		−	8.31371i		−	1.14198i	0.820959	−	0.570988i	$-0.193440\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	3.00000			0.397360
$58$		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	$0.182793\pi$
$61$			8.48528i			1.08643i	−0.839594	+	0.543214i	$0.817207\pi$
$62$		0			0
$63$	3.82843			0.482336
$64$		0			0
$65$		0			0
$66$		0			0
$67$		−	2.48528i		−	0.303625i	−0.988409	−	0.151813i	$-0.951489\pi$
$67$		−	2.48528i		−	0.303625i	0.988409	−	0.151813i	$-0.0485111\pi$
$68$		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
$73$	15.8284			1.85258			0.926289	−	0.376815i	$-0.122981\pi$
$73$	15.8284			1.85258			0.926289	+	0.376815i	$0.122981\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			22.3137i			2.54288i
$78$		0			0
$79$		−	2.00000i		−	0.225018i	−0.993651	−	0.112509i	$-0.964111\pi$
$79$		−	2.00000i		−	0.225018i	0.993651	−	0.112509i	$-0.0358886\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$			4.17157i			0.447240i
$88$		0			0
$89$	−14.1421			−1.49906			−0.749532	−	0.661968i	$-0.769721\pi$
$89$	−14.1421			−1.49906			−0.749532	+	0.661968i	$0.769721\pi$
$90$		0			0
$91$		−	18.4853i		−	1.93778i
$92$		0			0
$93$			6.82843i			0.708075i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	14.0000			1.42148			0.710742	−	0.703452i	$-0.248359\pi$
$97$	14.0000			1.42148			0.710742	+	0.703452i	$0.248359\pi$
$98$		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
$103$			6.00000i			0.591198i	0.955312	+	0.295599i	$0.0955191\pi$
$103$			6.00000i			0.591198i	−0.955312	+	0.295599i	$0.904481\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−4.34315			−0.419868			−0.209934	−	0.977716i	$-0.567325\pi$
$107$	−4.34315			−0.419868			−0.209934	+	0.977716i	$0.567325\pi$
$108$		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
$113$	10.8284			1.01865			0.509326	−	0.860573i	$-0.329895\pi$
$113$	10.8284			1.01865			0.509326	+	0.860573i	$0.329895\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		−	4.82843i		−	0.446388i
$118$		0			0
$119$	−10.8284	+	11.4853i	−0.992640	+	1.05285i
$120$		0			0
$121$	−22.9706			−2.08823
$122$		0			0
$123$			5.82843i			0.525532i
$124$		0			0
$125$		0			0
$126$		0			0
$127$			10.1421i			0.899969i	0.893036	+	0.449985i	$0.148570\pi$
$127$			10.1421i			0.899969i	−0.893036	+	0.449985i	$0.851430\pi$
$128$		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
$133$	−11.4853			−0.995900
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	1.34315i		−	0.114753i	−0.998353	−	0.0573763i	$-0.981727\pi$
$137$		−	1.34315i		−	0.114753i	0.998353	−	0.0573763i	$-0.0182735\pi$
$138$		0			0
$139$			6.00000i			0.508913i	0.967084	+	0.254457i	$0.0818966\pi$
$139$			6.00000i			0.508913i	−0.967084	+	0.254457i	$0.918103\pi$
$140$		0			0
$141$			8.65685i			0.729039i
$142$		0			0
$143$	28.1421			2.35336
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−7.65685			−0.631527
$148$		0			0
$149$	13.1716			1.07906			0.539529	−	0.841967i	$-0.318602\pi$
$149$	13.1716			1.07906			0.539529	+	0.841967i	$0.318602\pi$
$150$		0			0
$151$	15.0000			1.22068			0.610341	−	0.792139i	$-0.291032\pi$
$151$	15.0000			1.22068			0.610341	+	0.792139i	$0.291032\pi$
$152$		0			0
$153$	−2.82843	+	3.00000i	−0.228665	+	0.242536i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		−	22.9706i		−	1.83325i	−0.399748	−	0.916625i	$-0.630902\pi$
$157$		−	22.9706i		−	1.83325i	0.399748	−	0.916625i	$-0.369098\pi$
$158$		0			0
$159$			8.31371i			0.659320i
$160$		0			0
$161$	−32.4853			−2.56020
$162$		0			0
$163$	−3.48528			−0.272988			−0.136494	−	0.990641i	$-0.543583\pi$
$163$	−3.48528			−0.272988			−0.136494	+	0.990641i	$0.543583\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	9.65685			0.747270			0.373635	−	0.927576i	$-0.378111\pi$
$167$	9.65685			0.747270			0.373635	+	0.927576i	$0.378111\pi$
$168$		0			0
$169$	−10.3137			−0.793362
$170$		0			0
$171$	−3.00000			−0.229416
$172$		0			0
$173$	20.8284			1.58356			0.791778	−	0.610809i	$-0.209156\pi$
$173$	20.8284			1.58356			0.791778	+	0.610809i	$0.209156\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	5.17157			0.388719
$178$		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	$-0.942569\pi$
$181$		−	4.82843i		−	0.358894i	0.983768	−	0.179447i	$-0.0574309\pi$
$182$		0			0
$183$		−	8.48528i		−	0.627250i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−17.4853	−	16.4853i	−1.27865	−	1.20552i
$188$		0			0
$189$	−3.82843			−0.278477
$190$		0			0
$191$	18.1421			1.31272			0.656359	−	0.754448i	$-0.272096\pi$
$191$	18.1421			1.31272			0.656359	+	0.754448i	$0.272096\pi$
$192$		0			0
$193$	13.3137			0.958342			0.479171	−	0.877722i	$-0.340937\pi$
$193$	13.3137			0.958342			0.479171	+	0.877722i	$0.340937\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−3.65685			−0.260540			−0.130270	−	0.991479i	$-0.541584\pi$
$197$	−3.65685			−0.260540			−0.130270	+	0.991479i	$0.541584\pi$
$198$		0			0
$199$		−	20.9706i		−	1.48656i	−0.668978	−	0.743282i	$-0.733268\pi$
$199$		−	20.9706i		−	1.48656i	0.668978	−	0.743282i	$-0.266732\pi$
$200$		0			0
$201$			2.48528i			0.175298i
$202$		0			0
$203$		−	15.9706i		−	1.12091i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−8.48528			−0.589768
$208$		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
$213$			9.65685i			0.661677i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		−	26.1421i		−	1.77464i
$218$		0			0
$219$	−15.8284			−1.06959
$220$		0			0
$221$	14.4853	+	13.6569i	0.974385	+	0.918659i
$222$		0			0
$223$		−	13.3137i		−	0.891552i	−0.895145	−	0.445776i	$-0.852928\pi$
$223$		−	13.3137i		−	0.891552i	0.895145	−	0.445776i	$-0.147072\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−4.82843			−0.320474			−0.160237	−	0.987079i	$-0.551226\pi$
$227$	−4.82843			−0.320474			−0.160237	+	0.987079i	$0.551226\pi$
$228$		0			0
$229$	11.0000			0.726900			0.363450	−	0.931614i	$-0.381599\pi$
$229$	11.0000			0.726900			0.363450	+	0.931614i	$0.381599\pi$
$230$		0			0
$231$		−	22.3137i		−	1.46813i
$232$		0			0
$233$	14.4853			0.948962			0.474481	−	0.880266i	$-0.342636\pi$
$233$	14.4853			0.948962			0.474481	+	0.880266i	$0.342636\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$			2.00000i			0.129914i
$238$		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	$-0.984465\pi$
$241$		−	1.51472i		−	0.0975716i	0.998809	−	0.0487858i	$-0.0155352\pi$
$242$		0			0
$243$	−1.00000			−0.0641500
$244$		0			0
$245$		0			0
$246$		0			0
$247$			14.4853i			0.921676i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−21.7990			−1.37594			−0.687970	−	0.725739i	$-0.741498\pi$
$251$	−21.7990			−1.37594			−0.687970	+	0.725739i	$0.741498\pi$
$252$		0			0
$253$		−	49.4558i		−	3.10926i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	22.9706i		−	1.43286i	−0.697657	−	0.716432i	$-0.745774\pi$
$257$		−	22.9706i		−	1.43286i	0.697657	−	0.716432i	$-0.254226\pi$
$258$		0			0
$259$	36.3137			2.25642
$260$		0			0
$261$		−	4.17157i		−	0.258214i
$262$		0			0
$263$		−	0.313708i		−	0.0193441i	−0.999953	−	0.00967205i	$-0.996921\pi$
$263$		−	0.313708i		−	0.0193441i	0.999953	−	0.00967205i	$-0.00307876\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	14.1421			0.865485
$268$		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	$-0.774032\pi$
$271$	−24.9706			−1.51685			−0.758427	+	0.651758i	$0.774032\pi$
$272$		0			0
$273$			18.4853i			1.11878i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	27.6569			1.66174			0.830870	−	0.556467i	$-0.187843\pi$
$277$	27.6569			1.66174			0.830870	+	0.556467i	$0.187843\pi$
$278$		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
$283$	2.51472			0.149485			0.0747423	−	0.997203i	$-0.476187\pi$
$283$	2.51472			0.149485			0.0747423	+	0.997203i	$0.476187\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	22.3137i		−	1.31714i
$288$		0			0
$289$	−1.00000	−	16.9706i	−0.0588235	−	0.998268i
$290$		0			0
$291$	−14.0000			−0.820695
$292$		0			0
$293$			11.3431i			0.662674i	0.943513	+	0.331337i	$0.107500\pi$
$293$			11.3431i			0.662674i	−0.943513	+	0.331337i	$0.892500\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		−	5.82843i		−	0.338200i
$298$		0			0
$299$			40.9706i			2.36939i
$300$		0			0
$301$		−	30.6274i		−	1.76533i
$302$		0			0
$303$	5.65685			0.324978
$304$		0			0
$305$		0			0
$306$		0			0
$307$			19.3137i			1.10229i	0.834409	+	0.551146i	$0.185809\pi$
$307$			19.3137i			1.10229i	−0.834409	+	0.551146i	$0.814191\pi$
$308$		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
$313$	−32.7990			−1.85391			−0.926954	−	0.375174i	$-0.877583\pi$
$313$	−32.7990			−1.85391			−0.926954	+	0.375174i	$0.877583\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−8.14214			−0.457308			−0.228654	−	0.973508i	$-0.573432\pi$
$317$	−8.14214			−0.457308			−0.228654	+	0.973508i	$0.573432\pi$
$318$		0			0
$319$	24.3137			1.36131
$320$		0			0
$321$	4.34315			0.242411
$322$		0			0
$323$	8.48528	−	9.00000i	0.472134	−	0.500773i
$324$		0			0
$325$		0			0
$326$		0			0
$327$			3.31371i			0.183248i
$328$		0			0
$329$		−	33.1421i		−	1.82719i
$330$		0			0
$331$	−27.6274			−1.51854			−0.759270	−	0.650776i	$-0.774444\pi$
$331$	−27.6274			−1.51854			−0.759270	+	0.650776i	$0.774444\pi$
$332$		0			0
$333$	9.48528			0.519790
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−20.7990			−1.13299			−0.566497	−	0.824064i	$-0.691702\pi$
$337$	−20.7990			−1.13299			−0.566497	+	0.824064i	$0.691702\pi$
$338$		0			0
$339$	−10.8284			−0.588119
$340$		0			0
$341$	39.7990			2.15524
$342$		0			0
$343$	2.51472			0.135782
$344$		0			0
$345$		0			0
$346$		0			0
$347$	15.1716			0.814453			0.407226	−	0.913327i	$-0.366496\pi$
$347$	15.1716			0.814453			0.407226	+	0.913327i	$0.366496\pi$
$348$		0			0
$349$	−9.00000			−0.481759			−0.240879	−	0.970555i	$-0.577436\pi$
$349$	−9.00000			−0.481759			−0.240879	+	0.970555i	$0.577436\pi$
$350$		0			0
$351$			4.82843i			0.257722i
$352$		0			0
$353$		−	15.6274i		−	0.831763i	−0.909419	−	0.415882i	$-0.863473\pi$
$353$		−	15.6274i		−	0.831763i	0.909419	−	0.415882i	$-0.136527\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	10.8284	−	11.4853i	0.573101	−	0.607866i
$358$		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
$363$	22.9706			1.20564
$364$		0			0
$365$		0			0
$366$		0			0
$367$	8.00000			0.417597			0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000			0.417597			0.208798	+	0.977959i	$0.433045\pi$
$368$		0			0
$369$		−	5.82843i		−	0.303416i
$370$		0			0
$371$		−	31.8284i		−	1.65245i
$372$		0			0
$373$		−	2.68629i		−	0.139091i	−0.997579	−	0.0695455i	$-0.977845\pi$
$373$		−	2.68629i		−	0.139091i	0.997579	−	0.0695455i	$-0.0221549\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−20.1421			−1.03737
$378$		0			0
$379$		−	10.0000i		−	0.513665i	−0.966456	−	0.256833i	$-0.917321\pi$
$379$		−	10.0000i		−	0.513665i	0.966456	−	0.256833i	$-0.0826790\pi$
$380$		0			0
$381$		−	10.1421i		−	0.519597i
$382$		0			0
$383$		−	7.34315i		−	0.375217i	−0.982244	−	0.187609i	$-0.939926\pi$
$383$		−	7.34315i		−	0.375217i	0.982244	−	0.187609i	$-0.0600737\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		−	8.00000i		−	0.406663i
$388$		0			0
$389$	−1.02944			−0.0521945			−0.0260973	−	0.999659i	$-0.508308\pi$
$389$	−1.02944			−0.0521945			−0.0260973	+	0.999659i	$0.508308\pi$
$390$		0			0
$391$	24.0000	−	25.4558i	1.21373	−	1.28736i
$392$		0			0
$393$		−	7.31371i		−	0.368928i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−37.4853			−1.88133			−0.940666	−	0.339333i	$-0.889799\pi$
$397$	−37.4853			−1.88133			−0.940666	+	0.339333i	$0.889799\pi$
$398$		0			0
$399$	11.4853			0.574983
$400$		0			0
$401$		−	7.82843i		−	0.390933i	−0.980710	−	0.195466i	$-0.937378\pi$
$401$		−	7.82843i		−	0.390933i	0.980710	−	0.195466i	$-0.0626221\pi$
$402$		0			0
$403$	−32.9706			−1.64238
$404$		0			0
$405$		0			0
$406$		0			0
$407$			55.2843i			2.74034i
$408$		0			0
$409$	17.3431			0.857563			0.428782	−	0.903408i	$-0.358943\pi$
$409$	17.3431			0.857563			0.428782	+	0.903408i	$0.358943\pi$
$410$		0			0
$411$			1.34315i			0.0662525i
$412$		0			0
$413$	−19.7990			−0.974245
$414$		0			0
$415$		0			0
$416$		0			0
$417$		−	6.00000i		−	0.293821i
$418$		0			0
$419$			21.1421i			1.03286i	0.856329	+	0.516430i	$0.172740\pi$
$419$			21.1421i			1.03286i	−0.856329	+	0.516430i	$0.827260\pi$
$420$		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
$423$		−	8.65685i		−	0.420911i
$424$		0			0
$425$		0			0
$426$		0			0
$427$			32.4853i			1.57207i
$428$		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
$433$		−	24.6274i		−	1.18352i	−0.806115	−	0.591759i	$-0.798434\pi$
$433$		−	24.6274i		−	1.18352i	0.806115	−	0.591759i	$-0.201566\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	25.4558			1.21772
$438$		0			0
$439$		−	8.48528i		−	0.404980i	−0.979284	−	0.202490i	$-0.935097\pi$
$439$		−	8.48528i		−	0.404980i	0.979284	−	0.202490i	$-0.0649034\pi$
$440$		0			0
$441$	7.65685			0.364612
$442$		0			0
$443$			34.6274i			1.64520i	0.568622	+	0.822599i	$0.307477\pi$
$443$			34.6274i			1.64520i	−0.568622	+	0.822599i	$0.692523\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−13.1716			−0.622994
$448$		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
$453$	−15.0000			−0.704761
$454$		0			0
$455$		0			0
$456$		0			0
$457$			11.5147i			0.538636i	0.963051	+	0.269318i	$0.0867982\pi$
$457$			11.5147i			0.538636i	−0.963051	+	0.269318i	$0.913202\pi$
$458$		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
$463$			20.9706i			0.974585i	0.873239	+	0.487292i	$0.162015\pi$
$463$			20.9706i			0.974585i	−0.873239	+	0.487292i	$0.837985\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	27.0000i		−	1.24941i	−0.780860	−	0.624705i	$-0.785219\pi$
$467$		−	27.0000i		−	1.24941i	0.780860	−	0.624705i	$-0.214781\pi$
$468$		0			0
$469$		−	9.51472i		−	0.439349i
$470$		0			0
$471$			22.9706i			1.05843i
$472$		0			0
$473$	46.6274			2.14393
$474$		0			0
$475$		0			0
$476$		0			0
$477$		−	8.31371i		−	0.380659i
$478$		0			0
$479$		−	13.6569i		−	0.623998i	−0.950082	−	0.311999i	$-0.899002\pi$
$479$		−	13.6569i		−	0.623998i	0.950082	−	0.311999i	$-0.100998\pi$
$480$		0			0
$481$		−	45.7990i		−	2.08825i
$482$		0			0
$483$	32.4853			1.47813
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−10.3431			−0.468693			−0.234346	−	0.972153i	$-0.575295\pi$
$487$	−10.3431			−0.468693			−0.234346	+	0.972153i	$0.575295\pi$
$488$		0			0
$489$	3.48528			0.157610
$490$		0			0
$491$	−26.6274			−1.20168			−0.600839	−	0.799370i	$-0.705167\pi$
$491$	−26.6274			−1.20168			−0.600839	+	0.799370i	$0.705167\pi$
$492$		0			0
$493$	12.5147	+	11.7990i	0.563635	+	0.531400i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	36.9706i		−	1.65836i
$498$		0			0
$499$		−	9.79899i		−	0.438663i	−0.975650	−	0.219332i	$-0.929612\pi$
$499$		−	9.79899i		−	0.438663i	0.975650	−	0.219332i	$-0.0703876\pi$
$500$		0			0
$501$	−9.65685			−0.431436
$502$		0			0
$503$	1.31371			0.0585754			0.0292877	−	0.999571i	$-0.490676\pi$
$503$	1.31371			0.0585754			0.0292877	+	0.999571i	$0.490676\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	10.3137			0.458048
$508$		0			0
$509$	−32.1421			−1.42468			−0.712338	−	0.701837i	$-0.752363\pi$
$509$	−32.1421			−1.42468			−0.712338	+	0.701837i	$0.752363\pi$
$510$		0			0
$511$	60.5980			2.68070
$512$		0			0
$513$	3.00000			0.132453
$514$		0			0
$515$		0			0
$516$		0			0
$517$	50.4558			2.21905
$518$		0			0
$519$	−20.8284			−0.914266
$520$		0			0
$521$		−	32.7990i		−	1.43695i	−0.695553	−	0.718475i	$-0.744841\pi$
$521$		−	32.7990i		−	1.43695i	0.695553	−	0.718475i	$-0.255159\pi$
$522$		0			0
$523$		−	16.9706i		−	0.742071i	−0.928619	−	0.371035i	$-0.879003\pi$
$523$		−	16.9706i		−	0.742071i	0.928619	−	0.371035i	$-0.120997\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	20.4853	+	19.3137i	0.892353	+	0.841318i
$528$		0			0
$529$	49.0000			2.13043
$530$		0			0
$531$	−5.17157			−0.224427
$532$		0			0
$533$	−28.1421			−1.21897
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−1.65685			−0.0714985
$538$		0			0
$539$			44.6274i			1.92224i
$540$		0			0
$541$			11.5147i			0.495056i	0.968881	+	0.247528i	$0.0796183\pi$
$541$			11.5147i			0.495056i	−0.968881	+	0.247528i	$0.920382\pi$
$542$		0			0
$543$			4.82843i			0.207208i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−10.8579			−0.464249			−0.232124	−	0.972686i	$-0.574568\pi$
$547$	−10.8579			−0.464249			−0.232124	+	0.972686i	$0.574568\pi$
$548$		0			0
$549$			8.48528i			0.362143i
$550$		0			0
$551$			12.5147i			0.533145i
$552$		0			0
$553$		−	7.65685i		−	0.325603i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	11.6569i		−	0.493917i	−0.969026	−	0.246958i	$-0.920569\pi$
$557$		−	11.6569i		−	0.493917i	0.969026	−	0.246958i	$-0.0794311\pi$
$558$		0			0
$559$	−38.6274			−1.63377
$560$		0			0
$561$	17.4853	+	16.4853i	0.738229	+	0.696009i
$562$		0			0
$563$			21.2843i			0.897025i	0.893777	+	0.448513i	$0.148046\pi$
$563$			21.2843i			0.897025i	−0.893777	+	0.448513i	$0.851954\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	3.82843			0.160779
$568$		0			0
$569$	−7.31371			−0.306607			−0.153303	−	0.988179i	$-0.548991\pi$
$569$	−7.31371			−0.306607			−0.153303	+	0.988179i	$0.548991\pi$
$570$		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
$573$	−18.1421			−0.757899
$574$		0			0
$575$		0			0
$576$		0			0
$577$			31.6569i			1.31789i	0.752190	+	0.658946i	$0.228997\pi$
$577$			31.6569i			1.31789i	−0.752190	+	0.658946i	$0.771003\pi$
$578$		0			0
$579$	−13.3137			−0.553299
$580$		0			0
$581$		0			0
$582$		0			0
$583$	48.4558			2.00684
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	1.62742i		−	0.0671707i	−0.999436	−	0.0335853i	$-0.989307\pi$
$587$		−	1.62742i		−	0.0671707i	0.999436	−	0.0335853i	$-0.0106926\pi$
$588$		0			0
$589$			20.4853i			0.844081i
$590$		0			0
$591$	3.65685			0.150423
$592$		0			0
$593$		−	45.6274i		−	1.87369i	−0.349740	−	0.936847i	$-0.613730\pi$
$593$		−	45.6274i		−	1.87369i	0.349740	−	0.936847i	$-0.386270\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$			20.9706i			0.858268i
$598$		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	$0.0291595\pi$
$601$			4.48528i			0.182958i	−0.995807	+	0.0914792i	$0.970840\pi$
$602$		0			0
$603$		−	2.48528i		−	0.101208i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−20.1127			−0.816349			−0.408175	−	0.912904i	$-0.633835\pi$
$607$	−20.1127			−0.816349			−0.408175	+	0.912904i	$0.633835\pi$
$608$		0			0
$609$			15.9706i			0.647160i
$610$		0			0
$611$	−41.7990			−1.69101
$612$		0			0
$613$			18.6274i			0.752354i	0.926548	+	0.376177i	$0.122762\pi$
$613$			18.6274i			0.752354i	−0.926548	+	0.376177i	$0.877238\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	5.17157			0.208200			0.104100	−	0.994567i	$-0.466804\pi$
$617$	5.17157			0.208200			0.104100	+	0.994567i	$0.466804\pi$
$618$		0			0
$619$			34.0000i			1.36658i	0.730149	+	0.683288i	$0.239451\pi$
$619$			34.0000i			1.36658i	−0.730149	+	0.683288i	$0.760549\pi$
$620$		0			0
$621$	8.48528			0.340503
$622$		0			0
$623$	−54.1421			−2.16916
$624$		0			0
$625$		0			0
$626$		0			0
$627$			17.4853i			0.698295i
$628$		0			0
$629$	−26.8284	+	28.4558i	−1.06972	+	1.13461i
$630$		0			0
$631$	−1.00000			−0.0398094			−0.0199047	−	0.999802i	$-0.506336\pi$
$631$	−1.00000			−0.0398094			−0.0199047	+	0.999802i	$0.506336\pi$
$632$		0			0
$633$		−	13.3137i		−	0.529172i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		−	36.9706i		−	1.46483i
$638$		0			0
$639$		−	9.65685i		−	0.382019i
$640$		0			0
$641$			35.1421i			1.38803i	0.719960	+	0.694015i	$0.244160\pi$
$641$			35.1421i			1.38803i	−0.719960	+	0.694015i	$0.755840\pi$
$642$		0			0
$643$	−36.4558			−1.43768			−0.718839	−	0.695177i	$-0.755326\pi$
$643$	−36.4558			−1.43768			−0.718839	+	0.695177i	$0.755326\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	44.2843i		−	1.74099i	−0.492173	−	0.870497i	$-0.663797\pi$
$647$		−	44.2843i		−	1.74099i	0.492173	−	0.870497i	$-0.336203\pi$
$648$		0			0
$649$		−	30.1421i		−	1.18318i
$650$		0			0
$651$			26.1421i			1.02459i
$652$		0			0
$653$	−17.6569			−0.690966			−0.345483	−	0.938425i	$-0.612285\pi$
$653$	−17.6569			−0.690966			−0.345483	+	0.938425i	$0.612285\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	15.8284			0.617526
$658$		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
$663$	−14.4853	−	13.6569i	−0.562562	−	0.530388i
$664$		0			0
$665$		0			0
$666$		0			0
$667$			35.3970i			1.37058i
$668$		0			0
$669$			13.3137i			0.514738i
$670$		0			0
$671$	−49.4558			−1.90922
$672$		0			0
$673$	−27.9411			−1.07705			−0.538526	−	0.842609i	$-0.681018\pi$
$673$	−27.9411			−1.07705			−0.538526	+	0.842609i	$0.681018\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−2.34315			−0.0900544			−0.0450272	−	0.998986i	$-0.514337\pi$
$677$	−2.34315			−0.0900544			−0.0450272	+	0.998986i	$0.514337\pi$
$678$		0			0
$679$	53.5980			2.05690
$680$		0			0
$681$	4.82843			0.185026
$682$		0			0
$683$	−7.02944			−0.268974			−0.134487	−	0.990915i	$-0.542939\pi$
$683$	−7.02944			−0.268974			−0.134487	+	0.990915i	$0.542939\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−11.0000			−0.419676
$688$		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
$693$			22.3137i			0.847628i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	17.4853	+	16.4853i	0.662302	+	0.624425i
$698$		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
$703$	−28.4558			−1.07323
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−21.6569			−0.814490
$708$		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
$713$			57.9411i			2.16991i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−14.4853			−0.540963
$718$		0			0
$719$		−	22.1127i		−	0.824664i	−0.911034	−	0.412332i	$-0.864714\pi$
$719$		−	22.1127i		−	0.824664i	0.911034	−	0.412332i	$-0.135286\pi$
$720$		0			0
$721$			22.9706i			0.855468i
$722$		0			0
$723$			1.51472i			0.0563330i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	38.9706i		−	1.44534i	−0.691194	−	0.722669i	$-0.742915\pi$
$727$		−	38.9706i		−	1.44534i	0.691194	−	0.722669i	$-0.257085\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	24.0000	+	22.6274i	0.887672	+	0.836905i
$732$		0			0
$733$		−	40.4853i		−	1.49536i	−0.664060	−	0.747679i	$-0.731168\pi$
$733$		−	40.4853i		−	1.49536i	0.664060	−	0.747679i	$-0.268832\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	14.4853			0.533572
$738$		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
$743$	39.9411			1.46530			0.732649	−	0.680607i	$-0.238284\pi$
$743$	39.9411			1.46530			0.732649	+	0.680607i	$0.238284\pi$
$744$		0			0
$745$		0			0
$746$		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
$753$	21.7990			0.794399
$754$		0			0
$755$		0			0
$756$		0			0
$757$			27.6569i			1.00521i	0.864517	+	0.502603i	$0.167624\pi$
$757$			27.6569i			1.00521i	−0.864517	+	0.502603i	$0.832376\pi$
$758$		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
$763$		−	12.6863i		−	0.459275i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			24.9706i			0.901635i
$768$		0			0
$769$	−24.6569			−0.889149			−0.444574	−	0.895742i	$-0.646645\pi$
$769$	−24.6569			−0.889149			−0.444574	+	0.895742i	$0.646645\pi$
$770$		0			0
$771$			22.9706i			0.827265i
$772$		0			0
$773$		−	19.6863i		−	0.708067i	−0.935233	−	0.354033i	$-0.884810\pi$
$773$		−	19.6863i		−	0.708067i	0.935233	−	0.354033i	$-0.115190\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−36.3137			−1.30275
$778$		0			0
$779$			17.4853i			0.626475i
$780$		0			0
$781$	56.2843			2.01401
$782$		0			0
$783$			4.17157i			0.149080i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−5.48528			−0.195529			−0.0977646	−	0.995210i	$-0.531169\pi$
$787$	−5.48528			−0.195529			−0.0977646	+	0.995210i	$0.531169\pi$
$788$		0			0
$789$			0.313708i			0.0111683i
$790$		0			0
$791$	41.4558			1.47400
$792$		0			0
$793$	40.9706			1.45491
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	1.97056i		−	0.0698009i	−0.999391	−	0.0349005i	$-0.988889\pi$
$797$		−	1.97056i		−	0.0698009i	0.999391	−	0.0349005i	$-0.0111114\pi$
$798$		0			0
$799$	25.9706	+	24.4853i	0.918772	+	0.866227i
$800$		0			0
$801$	−14.1421			−0.499688
$802$		0			0
$803$			92.2548i			3.25560i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		−	9.82843i		−	0.345977i
$808$		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
$813$	24.9706			0.875756
$814$		0			0
$815$		0			0
$816$		0			0
$817$			24.0000i			0.839654i
$818$		0			0
$819$		−	18.4853i		−	0.645928i
$820$		0			0
$821$			29.3137i			1.02306i	0.859267	+	0.511528i	$0.170920\pi$
$821$			29.3137i			1.02306i	−0.859267	+	0.511528i	$0.829080\pi$
$822$		0			0
$823$	−49.8284			−1.73691			−0.868455	−	0.495768i	$-0.834887\pi$
$823$	−49.8284			−1.73691			−0.868455	+	0.495768i	$0.834887\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	38.1421			1.32633			0.663166	−	0.748472i	$-0.269212\pi$
$827$	38.1421			1.32633			0.663166	+	0.748472i	$0.269212\pi$
$828$		0			0
$829$	22.0294			0.765114			0.382557	−	0.923932i	$-0.375044\pi$
$829$	22.0294			0.765114			0.382557	+	0.923932i	$0.375044\pi$
$830$		0			0
$831$	−27.6569			−0.959406
$832$		0			0
$833$	−21.6569	+	22.9706i	−0.750366	+	0.795883i
$834$		0			0
$835$		0			0
$836$		0			0
$837$			6.82843i			0.236025i
$838$		0			0
$839$		−	6.17157i		−	0.213066i	−0.994309	−	0.106533i	$-0.966025\pi$
$839$		−	6.17157i		−	0.213066i	0.994309	−	0.106533i	$-0.0339750\pi$
$840$		0			0
$841$	11.5980			0.399930
$842$		0			0
$843$	9.31371			0.320781
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−87.9411			−3.02169
$848$		0			0
$849$	−2.51472			−0.0863049
$850$		0			0
$851$	−80.4853			−2.75900
$852$		0			0
$853$	1.31371			0.0449805			0.0224903	−	0.999747i	$-0.492841\pi$
$853$	1.31371			0.0449805			0.0224903	+	0.999747i	$0.492841\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	31.3137			1.06966			0.534828	−	0.844961i	$-0.320376\pi$
$857$	31.3137			1.06966			0.534828	+	0.844961i	$0.320376\pi$
$858$		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
$863$			27.3431i			0.930772i	0.885108	+	0.465386i	$0.154084\pi$
$863$			27.3431i			0.930772i	−0.885108	+	0.465386i	$0.845916\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	1.00000	+	16.9706i	0.0339618	+	0.576351i
$868$		0			0
$869$	11.6569			0.395432
$870$		0			0
$871$	−12.0000			−0.406604
$872$		0			0
$873$	14.0000			0.473828
$874$		0			0
$875$		0			0
$876$		0			0
$877$	8.45584			0.285534			0.142767	−	0.989756i	$-0.454400\pi$
$877$	8.45584			0.285534			0.142767	+	0.989756i	$0.454400\pi$
$878$		0			0
$879$		−	11.3431i		−	0.382595i
$880$		0			0
$881$		−	10.1716i		−	0.342689i	−0.985211	−	0.171344i	$-0.945189\pi$
$881$		−	10.1716i		−	0.342689i	0.985211	−	0.171344i	$-0.0548111\pi$
$882$		0			0
$883$		−	18.4853i		−	0.622079i	−0.950397	−	0.311040i	$-0.899323\pi$
$883$		−	18.4853i		−	0.622079i	0.950397	−	0.311040i	$-0.100677\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−16.3431			−0.548749			−0.274375	−	0.961623i	$-0.588471\pi$
$887$	−16.3431			−0.548749			−0.274375	+	0.961623i	$0.588471\pi$
$888$		0			0
$889$			38.8284i			1.30226i
$890$		0			0
$891$			5.82843i			0.195260i
$892$		0			0
$893$			25.9706i			0.869072i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		−	40.9706i		−	1.36797i
$898$		0			0
$899$	−28.4853			−0.950037
$900$		0			0
$901$	24.9411	+	23.5147i	0.830909	+	0.783389i
$902$		0			0
$903$			30.6274i			1.01922i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−2.51472			−0.0834999			−0.0417499	−	0.999128i	$-0.513293\pi$
$907$	−2.51472			−0.0834999			−0.0417499	+	0.999128i	$0.513293\pi$
$908$		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
$915$		0			0
$916$		0			0
$917$			28.0000i			0.924641i
$918$		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
$923$	−46.6274			−1.53476
$924$		0			0
$925$		0			0
$926$		0			0
$927$			6.00000i			0.197066i
$928$		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
$933$		−	7.82843i		−	0.256291i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	15.4558i		−	0.504920i	−0.967607	−	0.252460i	$-0.918760\pi$
$937$		−	15.4558i		−	0.504920i	0.967607	−	0.252460i	$-0.0812397\pi$
$938$		0			0
$939$	32.7990			1.07035
$940$		0			0
$941$		−	30.0000i		−	0.977972i	−0.872292	−	0.488986i	$-0.837367\pi$
$941$		−	30.0000i		−	0.977972i	0.872292	−	0.488986i	$-0.162633\pi$
$942$		0			0
$943$			49.4558i			1.61050i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−12.6274			−0.410336			−0.205168	−	0.978727i	$-0.565774\pi$
$947$	−12.6274			−0.410336			−0.205168	+	0.978727i	$0.565774\pi$
$948$		0			0
$949$		−	76.4264i		−	2.48091i
$950$		0			0
$951$	8.14214			0.264027
$952$		0			0
$953$			30.3137i			0.981957i	0.871172	+	0.490979i	$0.163361\pi$
$953$			30.3137i			0.981957i	−0.871172	+	0.490979i	$0.836639\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−24.3137			−0.785951
$958$		0			0
$959$		−	5.14214i		−	0.166048i
$960$		0			0
$961$	−15.6274			−0.504110
$962$		0			0
$963$	−4.34315			−0.139956
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	31.1716i		−	1.00241i	−0.865329	−	0.501205i	$-0.832890\pi$
$967$		−	31.1716i		−	1.00241i	0.865329	−	0.501205i	$-0.167110\pi$
$968$		0			0
$969$	−8.48528	+	9.00000i	−0.272587	+	0.289122i
$970$		0			0
$971$	12.1421			0.389660			0.194830	−	0.980837i	$-0.437585\pi$
$971$	12.1421			0.389660			0.194830	+	0.980837i	$0.437585\pi$
$972$		0			0
$973$			22.9706i			0.736402i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	18.6863i		−	0.597827i	−0.954280	−	0.298914i	$-0.903376\pi$
$977$		−	18.6863i		−	0.597827i	0.954280	−	0.298914i	$-0.0966243\pi$
$978$		0			0
$979$		−	82.4264i		−	2.63436i
$980$		0			0
$981$		−	3.31371i		−	0.105799i
$982$		0			0
$983$	−39.7990			−1.26939			−0.634695	−	0.772762i	$-0.718874\pi$
$983$	−39.7990			−1.26939			−0.634695	+	0.772762i	$0.718874\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$			33.1421i			1.05493i
$988$		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
$993$	27.6274			0.876730
$994$		0			0
$995$		0			0
$996$		0			0
$997$	27.8284			0.881335			0.440668	−	0.897670i	$-0.354742\pi$
$997$	27.8284			0.881335			0.440668	+	0.897670i	$0.354742\pi$
$998$		0			0
$999$	−9.48528			−0.300101

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5100.2.k.h.4249.4		4
5.2	odd	4		5100.2.e.f.1801.4		4
5.3	odd	4		1020.2.e.d.781.1	✓	4
5.4	even	2		5100.2.k.i.4249.2		4
15.8	even	4		3060.2.e.g.1801.1		4
17.16	even	2		5100.2.k.i.4249.1		4
20.3	even	4		4080.2.h.n.3841.4		4
85.33	odd	4		1020.2.e.d.781.4	yes	4
85.67	odd	4		5100.2.e.f.1801.1		4
85.84	even	2	inner	5100.2.k.h.4249.3		4
255.203	even	4		3060.2.e.g.1801.4		4
340.203	even	4		4080.2.h.n.3841.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1020.2.e.d.781.1	✓	4	5.3	odd	4
1020.2.e.d.781.4	yes	4	85.33	odd	4
3060.2.e.g.1801.1		4	15.8	even	4
3060.2.e.g.1801.4		4	255.203	even	4
4080.2.h.n.3841.1		4	340.203	even	4
4080.2.h.n.3841.4		4	20.3	even	4
5100.2.e.f.1801.1		4	85.67	odd	4
5100.2.e.f.1801.4		4	5.2	odd	4
5100.2.k.h.4249.3		4	85.84	even	2	inner
5100.2.k.h.4249.4		4	1.1	even	1	trivial
5100.2.k.i.4249.1		4	17.16	even	2
5100.2.k.i.4249.2		4	5.4	even	2

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:	\( N \)	\(=\)	\( 5100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5100.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +3.82843 q^{7} +1.00000 q^{9} +5.82843i q^{11} -4.82843i q^{13} +(-2.82843 + 3.00000i) q^{17} -3.00000 q^{19} -3.82843 q^{21} -8.48528 q^{23} -1.00000 q^{27} -4.17157i q^{29} -6.82843i q^{31} -5.82843i q^{33} +9.48528 q^{37} +4.82843i q^{39} -5.82843i q^{41} -8.00000i q^{43} -8.65685i q^{47} +7.65685 q^{49} +(2.82843 - 3.00000i) q^{51} -8.31371i q^{53} +3.00000 q^{57} -5.17157 q^{59} +8.48528i q^{61} +3.82843 q^{63} -2.48528i q^{67} +8.48528 q^{69} -9.65685i q^{71} +15.8284 q^{73} +22.3137i q^{77} -2.00000i q^{79} +1.00000 q^{81} +4.17157i q^{87} -14.1421 q^{89} -18.4853i q^{91} +6.82843i q^{93} +14.0000 q^{97} +5.82843i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} + 4 q^{7} + 4 q^{9} - 12 q^{19} - 4 q^{21} - 4 q^{27} + 4 q^{37} + 8 q^{49} + 12 q^{57} - 32 q^{59} + 4 q^{63} + 52 q^{73} + 4 q^{81} + 56 q^{97}+O(q^{100}) \) 4 * q - 4 * q^3 + 4 * q^7 + 4 * q^9 - 12 * q^19 - 4 * q^21 - 4 * q^27 + 4 * q^37 + 8 * q^49 + 12 * q^57 - 32 * q^59 + 4 * q^63 + 52 * q^73 + 4 * q^81 + 56 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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