LMFDB - Embedded newform 1680.3.l.b.1121.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1680,3,Mod(1121,1680)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1680, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0, 0])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1680.1121"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1680.l (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,8,0,0,0,0,0,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	no
Analytic conductor:	$45.7766844125$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 8 x^{15} + 37 x^{14} - 116 x^{13} + 298 x^{12} - 548 x^{11} + 267 x^{10} + 3264 x^{9} + \cdots + 43046721 $ x^16 - 8x^15 + 37x^14 - 116x^13 + 298x^12 - 548x^11 + 267x^10 + 3264x^9 - 13590x^8 + 29376x^7 + 21627x^6 - 399492x^5 + 1955178x^4 - 6849684x^3 + 19663317x^2 - 38263752*x + 43046721
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{11}\cdot 3^{3} $
Twist minimal:	no (minimal twist has level 420)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1121.6
Root			$-0.895194 + 2.86332i$ of defining polynomial
Character	$\chi$	$=$	1680.1121
Dual form			1680.3.l.b.1121.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.895194 + 2.86332i) q^{3} +2.23607i q^{5} +2.64575 q^{7} +(-7.39726 - 5.12646i) q^{9} -3.00338i q^{11} -24.3241 q^{13} +(-6.40259 - 2.00171i) q^{15} -3.60390i q^{17} +16.8805 q^{19} +(-2.36846 + 7.57565i) q^{21} +12.4015i q^{23} -5.00000 q^{25} +(21.3007 - 16.5916i) q^{27} -13.5807i q^{29} -5.19503 q^{31} +(8.59964 + 2.68860i) q^{33} +5.91608i q^{35} -17.7081 q^{37} +(21.7747 - 69.6477i) q^{39} -19.6909i q^{41} +39.7227 q^{43} +(11.4631 - 16.5408i) q^{45} -72.1833i q^{47} +7.00000 q^{49} +(10.3191 + 3.22619i) q^{51} -14.4407i q^{53} +6.71576 q^{55} +(-15.1113 + 48.3343i) q^{57} -74.4914i q^{59} +66.1551 q^{61} +(-19.5713 - 13.5633i) q^{63} -54.3902i q^{65} +47.8138 q^{67} +(-35.5094 - 11.1017i) q^{69} +46.8183i q^{71} -42.2852 q^{73} +(4.47597 - 14.3166i) q^{75} -7.94619i q^{77} -3.12717 q^{79} +(28.4388 + 75.8435i) q^{81} +98.3606i q^{83} +8.05857 q^{85} +(38.8860 + 12.1574i) q^{87} -154.840i q^{89} -64.3554 q^{91} +(4.65056 - 14.8751i) q^{93} +37.7459i q^{95} -92.3775 q^{97} +(-15.3967 + 22.2168i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 8 q^{3} - 10 q^{9} - 10 q^{15} - 48 q^{19} - 14 q^{21} - 80 q^{25} - 28 q^{27} - 24 q^{31} + 92 q^{33} - 40 q^{37} + 54 q^{39} + 168 q^{43} + 40 q^{45} + 112 q^{49} + 186 q^{51} + 80 q^{55} + 252 q^{57}+ \cdots - 26 q^{99}+O(q^{100}) $ 16 * q + 8 * q^3 - 10 * q^9 - 10 * q^15 - 48 * q^19 - 14 * q^21 - 80 * q^25 - 28 * q^27 - 24 * q^31 + 92 * q^33 - 40 * q^37 + 54 * q^39 + 168 * q^43 + 40 * q^45 + 112 * q^49 + 186 * q^51 + 80 * q^55 + 252 * q^57 - 152 * q^61 + 56 * q^63 - 264 * q^67 - 196 * q^69 + 240 * q^73 - 40 * q^75 - 212 * q^79 - 118 * q^81 - 60 * q^85 - 60 * q^87 - 196 * q^91 - 244 * q^93 - 360 * q^97 - 26 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times$.

$n$	$241$	$337$	$421$	$1121$	$1471$
$\chi(n)$	$1$	$1$	$1$	$-1$	$1$

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
$3$	−0.895194	+	2.86332i	−0.298398	+	0.954442i
$3$	−0.895194	+	2.86332i	−0.298398	+	0.954442i
$4$		0			0
$5$			2.23607i			0.447214i
$5$			2.23607i			0.447214i
$6$		0			0
$7$	2.64575			0.377964
$7$	2.64575			0.377964
$8$		0			0
$9$	−7.39726	−	5.12646i	−0.821917	−	0.569607i
$10$		0			0
$11$		−	3.00338i		−	0.273034i	−0.990638	−	0.136517i	$-0.956409\pi$
$11$		−	3.00338i		−	0.273034i	0.990638	−	0.136517i	$-0.0435909\pi$
$12$		0			0
$13$	−24.3241			−1.87108			−0.935540	−	0.353220i	$-0.885087\pi$
$13$	−24.3241			−1.87108			−0.935540	+	0.353220i	$0.885087\pi$
$14$		0			0
$15$	−6.40259	−	2.00171i	−0.426839	−	0.133448i
$16$		0			0
$17$		−	3.60390i		−	0.211994i	−0.994366	−	0.105997i	$-0.966197\pi$
$17$		−	3.60390i		−	0.211994i	0.994366	−	0.105997i	$-0.0338035\pi$
$18$		0			0
$19$	16.8805			0.888447			0.444223	−	0.895916i	$-0.353480\pi$
$19$	16.8805			0.888447			0.444223	+	0.895916i	$0.353480\pi$
$20$		0			0
$21$	−2.36846	+	7.57565i	−0.112784	+	0.360745i
$22$		0			0
$23$			12.4015i			0.539194i	0.962973	+	0.269597i	$0.0868904\pi$
$23$			12.4015i			0.539194i	−0.962973	+	0.269597i	$0.913110\pi$
$24$		0			0
$25$	−5.00000			−0.200000
$26$		0			0
$27$	21.3007	−	16.5916i	0.788915	−	0.614503i
$28$		0			0
$29$		−	13.5807i		−	0.468301i	−0.972200	−	0.234150i	$-0.924769\pi$
$29$		−	13.5807i		−	0.468301i	0.972200	−	0.234150i	$-0.0752308\pi$
$30$		0			0
$31$	−5.19503			−0.167582			−0.0837908	−	0.996483i	$-0.526703\pi$
$31$	−5.19503			−0.167582			−0.0837908	+	0.996483i	$0.526703\pi$
$32$		0			0
$33$	8.59964	+	2.68860i	0.260595	+	0.0814729i
$34$		0			0
$35$			5.91608i			0.169031i
$36$		0			0
$37$	−17.7081			−0.478598			−0.239299	−	0.970946i	$-0.576918\pi$
$37$	−17.7081			−0.478598			−0.239299	+	0.970946i	$0.576918\pi$
$38$		0			0
$39$	21.7747	−	69.6477i	0.558327	−	1.78584i
$40$		0			0
$41$		−	19.6909i		−	0.480267i	−0.970740	−	0.240133i	$-0.922809\pi$
$41$		−	19.6909i		−	0.480267i	0.970740	−	0.240133i	$-0.0771912\pi$
$42$		0			0
$43$	39.7227			0.923783			0.461891	−	0.886937i	$-0.347171\pi$
$43$	39.7227			0.923783			0.461891	+	0.886937i	$0.347171\pi$
$44$		0			0
$45$	11.4631	−	16.5408i	0.254736	−	0.367573i
$46$		0			0
$47$		−	72.1833i		−	1.53582i	−0.640561	−	0.767908i	$-0.721298\pi$
$47$		−	72.1833i		−	1.53582i	0.640561	−	0.767908i	$-0.278702\pi$
$48$		0			0
$49$	7.00000			0.142857
$50$		0			0
$51$	10.3191	+	3.22619i	0.202336	+	0.0632587i
$52$		0			0
$53$		−	14.4407i		−	0.272467i	−0.990677	−	0.136233i	$-0.956500\pi$
$53$		−	14.4407i		−	0.272467i	0.990677	−	0.136233i	$-0.0434997\pi$
$54$		0			0
$55$	6.71576			0.122105
$56$		0			0
$57$	−15.1113	+	48.3343i	−0.265111	+	0.847971i
$58$		0			0
$59$		−	74.4914i		−	1.26257i	−0.775552	−	0.631283i	$-0.782529\pi$
$59$		−	74.4914i		−	1.26257i	0.775552	−	0.631283i	$-0.217471\pi$
$60$		0			0
$61$	66.1551			1.08451			0.542255	−	0.840214i	$-0.317571\pi$
$61$	66.1551			1.08451			0.542255	+	0.840214i	$0.317571\pi$
$62$		0			0
$63$	−19.5713	−	13.5633i	−0.310656	−	0.215291i
$64$		0			0
$65$		−	54.3902i		−	0.836773i
$66$		0			0
$67$	47.8138			0.713639			0.356820	−	0.934173i	$-0.383861\pi$
$67$	47.8138			0.713639			0.356820	+	0.934173i	$0.383861\pi$
$68$		0			0
$69$	−35.5094	−	11.1017i	−0.514629	−	0.160894i
$70$		0			0
$71$			46.8183i			0.659413i	0.944083	+	0.329707i	$0.106950\pi$
$71$			46.8183i			0.659413i	−0.944083	+	0.329707i	$0.893050\pi$
$72$		0			0
$73$	−42.2852			−0.579249			−0.289624	−	0.957140i	$-0.593530\pi$
$73$	−42.2852			−0.579249			−0.289624	+	0.957140i	$0.593530\pi$
$74$		0			0
$75$	4.47597	−	14.3166i	0.0596796	−	0.190888i
$76$		0			0
$77$		−	7.94619i		−	0.103197i
$78$		0			0
$79$	−3.12717			−0.0395844			−0.0197922	−	0.999804i	$-0.506300\pi$
$79$	−3.12717			−0.0395844			−0.0197922	+	0.999804i	$0.506300\pi$
$80$		0			0
$81$	28.4388	+	75.8435i	0.351097	+	0.936339i
$82$		0			0
$83$			98.3606i			1.18507i	0.805545	+	0.592534i	$0.201872\pi$
$83$			98.3606i			1.18507i	−0.805545	+	0.592534i	$0.798128\pi$
$84$		0			0
$85$	8.05857			0.0948068
$86$		0			0
$87$	38.8860	+	12.1574i	0.446966	+	0.139740i
$88$		0			0
$89$		−	154.840i		−	1.73977i	−0.493255	−	0.869885i	$-0.664193\pi$
$89$		−	154.840i		−	1.73977i	0.493255	−	0.869885i	$-0.335807\pi$
$90$		0			0
$91$	−64.3554			−0.707202
$92$		0			0
$93$	4.65056	−	14.8751i	0.0500060	−	0.159947i
$94$		0			0
$95$			37.7459i			0.397325i
$96$		0			0
$97$	−92.3775			−0.952346			−0.476173	−	0.879352i	$-0.657976\pi$
$97$	−92.3775			−0.952346			−0.476173	+	0.879352i	$0.657976\pi$
$98$		0			0
$99$	−15.3967	+	22.2168i	−0.155522	+	0.224412i
$100$		0			0
$101$			94.4792i			0.935438i	0.883877	+	0.467719i	$0.154924\pi$
$101$			94.4792i			0.935438i	−0.883877	+	0.467719i	$0.845076\pi$
$102$		0			0
$103$	−23.8293			−0.231352			−0.115676	−	0.993287i	$-0.536903\pi$
$103$	−23.8293			−0.231352			−0.115676	+	0.993287i	$0.536903\pi$
$104$		0			0
$105$	−16.9397	−	5.29604i	−0.161330	−	0.0504384i
$106$		0			0
$107$			150.037i			1.40221i	0.713056	+	0.701107i	$0.247310\pi$
$107$			150.037i			1.40221i	−0.713056	+	0.701107i	$0.752690\pi$
$108$		0			0
$109$	161.310			1.47991			0.739953	−	0.672658i	$-0.234848\pi$
$109$	161.310			1.47991			0.739953	+	0.672658i	$0.234848\pi$
$110$		0			0
$111$	15.8522	−	50.7042i	0.142813	−	0.456794i
$112$		0			0
$113$		−	200.957i		−	1.77838i	−0.457538	−	0.889190i	$-0.651269\pi$
$113$		−	200.957i		−	1.77838i	0.457538	−	0.889190i	$-0.348731\pi$
$114$		0			0
$115$	−27.7305			−0.241135
$116$		0			0
$117$	179.931	+	124.696i	1.53787	+	1.06578i
$118$		0			0
$119$		−	9.53503i		−	0.0801263i
$120$		0			0
$121$	111.980			0.925452
$122$		0			0
$123$	56.3815	+	17.6272i	0.458387	+	0.143311i
$124$		0			0
$125$		−	11.1803i		−	0.0894427i
$126$		0			0
$127$	−89.2829			−0.703015			−0.351507	−	0.936185i	$-0.614331\pi$
$127$	−89.2829			−0.703015			−0.351507	+	0.936185i	$0.614331\pi$
$128$		0			0
$129$	−35.5595	+	113.739i	−0.275655	+	0.881697i
$130$		0			0
$131$		−	133.670i		−	1.02038i	−0.860061	−	0.510191i	$-0.829575\pi$
$131$		−	133.670i		−	1.02038i	0.860061	−	0.510191i	$-0.170425\pi$
$132$		0			0
$133$	44.6616			0.335801
$134$		0			0
$135$	37.0999	+	47.6298i	0.274814	+	0.352813i
$136$		0			0
$137$		−	51.8272i		−	0.378301i	−0.981948	−	0.189150i	$-0.939427\pi$
$137$		−	51.8272i		−	0.378301i	0.981948	−	0.189150i	$-0.0605733\pi$
$138$		0			0
$139$	190.393			1.36973			0.684866	−	0.728669i	$-0.259861\pi$
$139$	190.393			1.36973			0.684866	+	0.728669i	$0.259861\pi$
$140$		0			0
$141$	206.684	+	64.6180i	1.46585	+	0.458284i
$142$		0			0
$143$			73.0543i			0.510869i
$144$		0			0
$145$	30.3674			0.209431
$146$		0			0
$147$	−6.26636	+	20.0433i	−0.0426283	+	0.136349i
$148$		0			0
$149$			7.21233i			0.0484049i	0.999707	+	0.0242024i	$0.00770463\pi$
$149$			7.21233i			0.0484049i	−0.999707	+	0.0242024i	$0.992295\pi$
$150$		0			0
$151$	113.626			0.752491			0.376245	−	0.926520i	$-0.377215\pi$
$151$	113.626			0.752491			0.376245	+	0.926520i	$0.377215\pi$
$152$		0			0
$153$	−18.4753	+	26.6590i	−0.120753	+	0.174242i
$154$		0			0
$155$		−	11.6164i		−	0.0749448i
$156$		0			0
$157$	204.211			1.30071			0.650355	−	0.759630i	$-0.274620\pi$
$157$	204.211			1.30071			0.650355	+	0.759630i	$0.274620\pi$
$158$		0			0
$159$	41.3485	+	12.9272i	0.260053	+	0.0813034i
$160$		0			0
$161$			32.8112i			0.203796i
$162$		0			0
$163$	−77.2206			−0.473746			−0.236873	−	0.971541i	$-0.576123\pi$
$163$	−77.2206			−0.473746			−0.236873	+	0.971541i	$0.576123\pi$
$164$		0			0
$165$	−6.01190	+	19.2294i	−0.0364358	+	0.116542i
$166$		0			0
$167$		−	284.381i		−	1.70288i	−0.524454	−	0.851439i	$-0.675731\pi$
$167$		−	284.381i		−	1.70288i	0.524454	−	0.851439i	$-0.324269\pi$
$168$		0			0
$169$	422.659			2.50094
$170$		0			0
$171$	−124.869	−	86.5372i	−0.730230	−	0.506065i
$172$		0			0
$173$		−	267.316i		−	1.54518i	−0.634905	−	0.772590i	$-0.718961\pi$
$173$		−	267.316i		−	1.54518i	0.634905	−	0.772590i	$-0.281039\pi$
$174$		0			0
$175$	−13.2288			−0.0755929
$176$		0			0
$177$	213.293	+	66.6842i	1.20505	+	0.376747i
$178$		0			0
$179$			23.3604i			0.130505i	0.997869	+	0.0652525i	$0.0207853\pi$
$179$			23.3604i			0.130505i	−0.997869	+	0.0652525i	$0.979215\pi$
$180$		0			0
$181$	−119.639			−0.660987			−0.330494	−	0.943808i	$-0.607215\pi$
$181$	−119.639			−0.660987			−0.330494	+	0.943808i	$0.607215\pi$
$182$		0			0
$183$	−59.2217	+	189.424i	−0.323616	+	1.03510i
$184$		0			0
$185$		−	39.5966i		−	0.214036i
$186$		0			0
$187$	−10.8239			−0.0578817
$188$		0			0
$189$	56.3563	−	43.8972i	0.298182	−	0.232260i
$190$		0			0
$191$			76.1540i			0.398712i	0.979927	+	0.199356i	$0.0638850\pi$
$191$			76.1540i			0.398712i	−0.979927	+	0.199356i	$0.936115\pi$
$192$		0			0
$193$	−116.189			−0.602015			−0.301007	−	0.953622i	$-0.597323\pi$
$193$	−116.189			−0.602015			−0.301007	+	0.953622i	$0.597323\pi$
$194$		0			0
$195$	155.737	+	48.6898i	0.798651	+	0.249691i
$196$		0			0
$197$			247.485i			1.25627i	0.778105	+	0.628134i	$0.216181\pi$
$197$			247.485i			1.25627i	−0.778105	+	0.628134i	$0.783819\pi$
$198$		0			0
$199$	43.5013			0.218599			0.109300	−	0.994009i	$-0.465139\pi$
$199$	43.5013			0.218599			0.109300	+	0.994009i	$0.465139\pi$
$200$		0			0
$201$	−42.8026	+	136.906i	−0.212948	+	0.681127i
$202$		0			0
$203$		−	35.9312i		−	0.177001i
$204$		0			0
$205$	44.0303			0.214782
$206$		0			0
$207$	63.5756	−	91.7368i	0.307128	−	0.443173i
$208$		0			0
$209$		−	50.6985i		−	0.242576i
$210$		0			0
$211$	283.206			1.34221			0.671104	−	0.741363i	$-0.265820\pi$
$211$	283.206			1.34221			0.671104	+	0.741363i	$0.265820\pi$
$212$		0			0
$213$	−134.056	−	41.9115i	−0.629371	−	0.196767i
$214$		0			0
$215$			88.8226i			0.413128i
$216$		0			0
$217$	−13.7448			−0.0633399
$218$		0			0
$219$	37.8534	−	121.076i	0.172847	−	0.552859i
$220$		0			0
$221$			87.6615i			0.396659i
$222$		0			0
$223$	288.852			1.29530			0.647651	−	0.761937i	$-0.275751\pi$
$223$	288.852			1.29530			0.647651	+	0.761937i	$0.275751\pi$
$224$		0			0
$225$	36.9863	+	25.6323i	0.164383	+	0.113921i
$226$		0			0
$227$			333.944i			1.47112i	0.677460	+	0.735559i	$0.263080\pi$
$227$			333.944i			1.47112i	−0.677460	+	0.735559i	$0.736920\pi$
$228$		0			0
$229$	−11.1825			−0.0488318			−0.0244159	−	0.999702i	$-0.507773\pi$
$229$	−11.1825			−0.0488318			−0.0244159	+	0.999702i	$0.507773\pi$
$230$		0			0
$231$	22.7525	+	7.11338i	0.0984958	+	0.0307938i
$232$		0			0
$233$		−	249.339i		−	1.07012i	−0.844813	−	0.535062i	$-0.820288\pi$
$233$		−	249.339i		−	1.07012i	0.844813	−	0.535062i	$-0.179712\pi$
$234$		0			0
$235$	161.407			0.686837
$236$		0			0
$237$	2.79942	−	8.95410i	0.0118119	−	0.0377810i
$238$		0			0
$239$		−	144.066i		−	0.602786i	−0.953500	−	0.301393i	$-0.902548\pi$
$239$		−	144.066i		−	0.602786i	0.953500	−	0.301393i	$-0.0974516\pi$
$240$		0			0
$241$	−347.745			−1.44292			−0.721462	−	0.692454i	$-0.756530\pi$
$241$	−347.745			−1.44292			−0.721462	+	0.692454i	$0.756530\pi$
$242$		0			0
$243$	−242.623	+	13.5350i	−0.998448	+	0.0556995i
$244$		0			0
$245$			15.6525i			0.0638877i
$246$		0			0
$247$	−410.602			−1.66236
$248$		0			0
$249$	−281.638	−	88.0518i	−1.13108	−	0.353622i
$250$		0			0
$251$		−	453.378i		−	1.80629i	−0.429339	−	0.903143i	$-0.641253\pi$
$251$		−	453.378i		−	1.80629i	0.429339	−	0.903143i	$-0.358747\pi$
$252$		0			0
$253$	37.2463			0.147218
$254$		0			0
$255$	−7.21398	+	23.0743i	−0.0282901	+	0.0904875i
$256$		0			0
$257$		−	422.789i		−	1.64509i	−0.568697	−	0.822547i	$-0.692552\pi$
$257$		−	422.789i		−	1.64509i	0.568697	−	0.822547i	$-0.307448\pi$
$258$		0			0
$259$	−46.8513			−0.180893
$260$		0			0
$261$	−69.6211	+	100.460i	−0.266747	+	0.384905i
$262$		0			0
$263$			375.524i			1.42785i	0.700223	+	0.713924i	$0.253084\pi$
$263$			375.524i			1.42785i	−0.700223	+	0.713924i	$0.746916\pi$
$264$		0			0
$265$	32.2904			0.121851
$266$		0			0
$267$	443.356	+	138.611i	1.66051	+	0.519144i
$268$		0			0
$269$			1.41316i			0.00525338i	0.999997	+	0.00262669i	$0.000836103\pi$
$269$			1.41316i			0.00525338i	−0.999997	+	0.00262669i	$0.999164\pi$
$270$		0			0
$271$	−226.054			−0.834146			−0.417073	−	0.908873i	$-0.636944\pi$
$271$	−226.054			−0.834146			−0.417073	+	0.908873i	$0.636944\pi$
$272$		0			0
$273$	57.6105	−	184.270i	0.211028	−	0.674983i
$274$		0			0
$275$			15.0169i			0.0546069i
$276$		0			0
$277$	−355.341			−1.28282			−0.641410	−	0.767198i	$-0.721650\pi$
$277$	−355.341			−1.28282			−0.641410	+	0.767198i	$0.721650\pi$
$278$		0			0
$279$	38.4290	+	26.6321i	0.137738	+	0.0954556i
$280$		0			0
$281$			195.986i			0.697458i	0.937224	+	0.348729i	$0.113387\pi$
$281$			195.986i			0.697458i	−0.937224	+	0.348729i	$0.886613\pi$
$282$		0			0
$283$	485.087			1.71409			0.857044	−	0.515244i	$-0.172299\pi$
$283$	485.087			1.71409			0.857044	+	0.515244i	$0.172299\pi$
$284$		0			0
$285$	−108.079	−	33.7899i	−0.379224	−	0.118561i
$286$		0			0
$287$		−	52.0973i		−	0.181524i
$288$		0			0
$289$	276.012			0.955058
$290$		0			0
$291$	82.6958	−	264.507i	0.284178	−	0.908958i
$292$		0			0
$293$			378.468i			1.29170i	0.763465	+	0.645849i	$0.223497\pi$
$293$			378.468i			1.29170i	−0.763465	+	0.645849i	$0.776503\pi$
$294$		0			0
$295$	166.568			0.564637
$296$		0			0
$297$	−49.8308	−	63.9740i	−0.167780	−	0.215401i
$298$		0			0
$299$		−	301.654i		−	1.00888i
$300$		0			0
$301$	105.096			0.349157
$302$		0			0
$303$	−270.525	−	84.5772i	−0.892821	−	0.279133i
$304$		0			0
$305$			147.927i			0.485008i
$306$		0			0
$307$	573.684			1.86868			0.934339	−	0.356385i	$-0.115991\pi$
$307$	573.684			1.86868			0.934339	+	0.356385i	$0.115991\pi$
$308$		0			0
$309$	21.3318	−	68.2309i	0.0690349	−	0.220812i
$310$		0			0
$311$		−	441.918i		−	1.42096i	−0.703718	−	0.710479i	$-0.748478\pi$
$311$		−	441.918i		−	1.42096i	0.703718	−	0.710479i	$-0.251522\pi$
$312$		0			0
$313$	−378.973			−1.21078			−0.605388	−	0.795930i	$-0.706982\pi$
$313$	−378.973			−1.21078			−0.605388	+	0.795930i	$0.706982\pi$
$314$		0			0
$315$	30.3285	−	43.7628i	0.0962811	−	0.138929i
$316$		0			0
$317$			588.177i			1.85545i	0.373266	+	0.927724i	$0.378238\pi$
$317$			588.177i			1.85545i	−0.373266	+	0.927724i	$0.621762\pi$
$318$		0			0
$319$	−40.7881			−0.127862
$320$		0			0
$321$	−429.604	−	134.312i	−1.33833	−	0.418418i
$322$		0			0
$323$		−	60.8357i		−	0.188346i
$324$		0			0
$325$	121.620			0.374216
$326$		0			0
$327$	−144.404	+	461.882i	−0.441601	+	1.41248i
$328$		0			0
$329$		−	190.979i		−	0.580484i
$330$		0			0
$331$	−205.546			−0.620984			−0.310492	−	0.950576i	$-0.600494\pi$
$331$	−205.546			−0.620984			−0.310492	+	0.950576i	$0.600494\pi$
$332$		0			0
$333$	130.992	+	90.7801i	0.393368	+	0.272613i
$334$		0			0
$335$			106.915i			0.319149i
$336$		0			0
$337$	456.465			1.35450			0.677248	−	0.735755i	$-0.263172\pi$
$337$	456.465			1.35450			0.677248	+	0.735755i	$0.263172\pi$
$338$		0			0
$339$	575.405	+	179.895i	1.69736	+	0.530665i
$340$		0			0
$341$			15.6026i			0.0457555i
$342$		0			0
$343$	18.5203			0.0539949
$344$		0			0
$345$	24.8242	−	79.4014i	0.0719541	−	0.230149i
$346$		0			0
$347$		−	302.625i		−	0.872118i	−0.899918	−	0.436059i	$-0.856374\pi$
$347$		−	302.625i		−	0.872118i	0.899918	−	0.436059i	$-0.143626\pi$
$348$		0			0
$349$	19.9121			0.0570549			0.0285274	−	0.999593i	$-0.490918\pi$
$349$	19.9121			0.0570549			0.0285274	+	0.999593i	$0.490918\pi$
$350$		0			0
$351$	−518.119	+	403.574i	−1.47612	+	1.14978i
$352$		0			0
$353$			219.478i			0.621750i	0.950451	+	0.310875i	$0.100622\pi$
$353$			219.478i			0.621750i	−0.950451	+	0.310875i	$0.899378\pi$
$354$		0			0
$355$	−104.689			−0.294899
$356$		0			0
$357$	27.3019	+	8.53570i	0.0764759	+	0.0239095i
$358$		0			0
$359$		−	309.375i		−	0.861770i	−0.902407	−	0.430885i	$-0.858201\pi$
$359$		−	309.375i		−	0.861770i	0.902407	−	0.430885i	$-0.141799\pi$
$360$		0			0
$361$	−76.0491			−0.210662
$362$		0			0
$363$	−100.244	+	320.634i	−0.276153	+	0.883290i
$364$		0			0
$365$		−	94.5525i		−	0.259048i
$366$		0			0
$367$	−516.947			−1.40857			−0.704287	−	0.709915i	$-0.748733\pi$
$367$	−516.947			−1.40857			−0.704287	+	0.709915i	$0.748733\pi$
$368$		0			0
$369$	−100.945	+	145.659i	−0.273563	+	0.394740i
$370$		0			0
$371$		−	38.2066i		−	0.102983i
$372$		0			0
$373$	−417.787			−1.12007			−0.560037	−	0.828468i	$-0.689213\pi$
$373$	−417.787			−1.12007			−0.560037	+	0.828468i	$0.689213\pi$
$374$		0			0
$375$	32.0129	+	10.0086i	0.0853678	+	0.0266895i
$376$		0			0
$377$			330.338i			0.876229i
$378$		0			0
$379$	−468.822			−1.23700			−0.618499	−	0.785786i	$-0.712259\pi$
$379$	−468.822			−1.23700			−0.618499	+	0.785786i	$0.712259\pi$
$380$		0			0
$381$	79.9255	−	255.646i	0.209778	−	0.670987i
$382$		0			0
$383$		−	260.516i		−	0.680198i	−0.940390	−	0.340099i	$-0.889539\pi$
$383$		−	260.516i		−	0.680198i	0.940390	−	0.340099i	$-0.110461\pi$
$384$		0			0
$385$	17.7682			0.0461512
$386$		0			0
$387$	−293.839	−	203.637i	−0.759273	−	0.526193i
$388$		0			0
$389$		−	724.894i		−	1.86348i	−0.363125	−	0.931740i	$-0.618290\pi$
$389$		−	724.894i		−	1.86348i	0.363125	−	0.931740i	$-0.381710\pi$
$390$		0			0
$391$	44.6937			0.114306
$392$		0			0
$393$	382.741	+	119.661i	0.973895	+	0.304480i
$394$		0			0
$395$		−	6.99256i		−	0.0177027i
$396$		0			0
$397$	−394.446			−0.993567			−0.496784	−	0.867874i	$-0.665486\pi$
$397$	−394.446			−0.993567			−0.496784	+	0.867874i	$0.665486\pi$
$398$		0			0
$399$	−39.9808	+	127.881i	−0.100202	+	0.320503i
$400$		0			0
$401$		−	340.136i		−	0.848219i	−0.905611	−	0.424110i	$-0.860587\pi$
$401$		−	340.136i		−	0.848219i	0.905611	−	0.424110i	$-0.139413\pi$
$402$		0			0
$403$	126.364			0.313559
$404$		0			0
$405$	−169.591	+	63.5911i	−0.418744	+	0.157015i
$406$		0			0
$407$			53.1842i			0.130674i
$408$		0			0
$409$	−646.306			−1.58021			−0.790105	−	0.612971i	$-0.789974\pi$
$409$	−646.306			−1.58021			−0.790105	+	0.612971i	$0.789974\pi$
$410$		0			0
$411$	148.398	+	46.3954i	0.361066	+	0.112884i
$412$		0			0
$413$		−	197.086i		−	0.477205i
$414$		0			0
$415$	−219.941			−0.529979
$416$		0			0
$417$	−170.438	+	545.157i	−0.408725	+	1.30733i
$418$		0			0
$419$		−	774.244i		−	1.84784i	−0.382590	−	0.923918i	$-0.624968\pi$
$419$		−	774.244i		−	1.84784i	0.382590	−	0.923918i	$-0.375032\pi$
$420$		0			0
$421$	77.6930			0.184544			0.0922719	−	0.995734i	$-0.470587\pi$
$421$	77.6930			0.184544			0.0922719	+	0.995734i	$0.470587\pi$
$422$		0			0
$423$	−370.045	+	533.959i	−0.874811	+	1.26231i
$424$		0			0
$425$			18.0195i			0.0423989i
$426$		0			0
$427$	175.030			0.409906
$428$		0			0
$429$	−209.178	−	65.3977i	−0.487595	−	0.152442i
$430$		0			0
$431$			301.177i			0.698787i	0.936976	+	0.349394i	$0.113612\pi$
$431$			301.177i			0.698787i	−0.936976	+	0.349394i	$0.886388\pi$
$432$		0			0
$433$	−98.3440			−0.227123			−0.113561	−	0.993531i	$-0.536226\pi$
$433$	−98.3440			−0.227123			−0.113561	+	0.993531i	$0.536226\pi$
$434$		0			0
$435$	−27.1847	+	86.9518i	−0.0624936	+	0.199889i
$436$		0			0
$437$			209.343i			0.479045i
$438$		0			0
$439$	−525.993			−1.19816			−0.599081	−	0.800688i	$-0.704467\pi$
$439$	−525.993			−1.19816			−0.599081	+	0.800688i	$0.704467\pi$
$440$		0			0
$441$	−51.7808	−	35.8852i	−0.117417	−	0.0813724i
$442$		0			0
$443$		−	547.933i		−	1.23687i	−0.785837	−	0.618434i	$-0.787767\pi$
$443$		−	547.933i		−	1.23687i	0.785837	−	0.618434i	$-0.212233\pi$
$444$		0			0
$445$	346.232			0.778049
$446$		0			0
$447$	−20.6512	−	6.45643i	−0.0461996	−	0.0144439i
$448$		0			0
$449$		−	302.192i		−	0.673033i	−0.941678	−	0.336517i	$-0.890751\pi$
$449$		−	302.192i		−	0.673033i	0.941678	−	0.336517i	$-0.109249\pi$
$450$		0			0
$451$	−59.1393			−0.131129
$452$		0			0
$453$	−101.717	+	325.348i	−0.224542	+	0.718208i
$454$		0			0
$455$		−	143.903i		−	0.316270i
$456$		0			0
$457$	202.530			0.443172			0.221586	−	0.975141i	$-0.428877\pi$
$457$	202.530			0.443172			0.221586	+	0.975141i	$0.428877\pi$
$458$		0			0
$459$	−59.7944	−	76.7657i	−0.130271	−	0.167245i
$460$		0			0
$461$		−	850.540i		−	1.84499i	−0.386010	−	0.922495i	$-0.626147\pi$
$461$		−	850.540i		−	1.84499i	0.386010	−	0.922495i	$-0.373853\pi$
$462$		0			0
$463$	−371.502			−0.802380			−0.401190	−	0.915995i	$-0.631403\pi$
$463$	−371.502			−0.802380			−0.401190	+	0.915995i	$0.631403\pi$
$464$		0			0
$465$	33.2616	+	10.3990i	0.0715304	+	0.0223634i
$466$		0			0
$467$			571.243i			1.22322i	0.791160	+	0.611609i	$0.209477\pi$
$467$			571.243i			1.22322i	−0.791160	+	0.611609i	$0.790523\pi$
$468$		0			0
$469$	126.503			0.269730
$470$		0			0
$471$	−182.809	+	584.724i	−0.388129	+	1.24145i
$472$		0			0
$473$		−	119.302i		−	0.252224i
$474$		0			0
$475$	−84.4024			−0.177689
$476$		0			0
$477$	−74.0298	+	106.822i	−0.155199	+	0.223945i
$478$		0			0
$479$		−	433.086i		−	0.904145i	−0.891981	−	0.452073i	$-0.850685\pi$
$479$		−	433.086i		−	0.904145i	0.891981	−	0.452073i	$-0.149315\pi$
$480$		0			0
$481$	430.734			0.895496
$482$		0			0
$483$	−93.9490	−	29.3723i	−0.194511	−	0.0608123i
$484$		0			0
$485$		−	206.562i		−	0.425902i
$486$		0			0
$487$	319.390			0.655831			0.327916	−	0.944707i	$-0.393654\pi$
$487$	319.390			0.655831			0.327916	+	0.944707i	$0.393654\pi$
$488$		0			0
$489$	69.1274	−	221.108i	0.141365	−	0.452163i
$490$		0			0
$491$			106.542i			0.216989i	0.994097	+	0.108495i	$0.0346031\pi$
$491$			106.542i			0.216989i	−0.994097	+	0.108495i	$0.965397\pi$
$492$		0			0
$493$	−48.9436			−0.0992772
$494$		0			0
$495$	−49.6782	−	34.4281i	−0.100360	−	0.0695516i
$496$		0			0
$497$			123.870i			0.249235i
$498$		0			0
$499$	−189.829			−0.380420			−0.190210	−	0.981743i	$-0.560917\pi$
$499$	−189.829			−0.380420			−0.190210	+	0.981743i	$0.560917\pi$
$500$		0			0
$501$	814.274	+	254.576i	1.62530	+	0.508135i
$502$		0			0
$503$		−	18.3298i		−	0.0364410i	−0.999834	−	0.0182205i	$-0.994200\pi$
$503$		−	18.3298i		−	0.0364410i	0.999834	−	0.0182205i	$-0.00580008\pi$
$504$		0			0
$505$	−211.262			−0.418340
$506$		0			0
$507$	−378.362	+	1210.21i	−0.746276	+	2.38700i
$508$		0			0
$509$		−	102.736i		−	0.201839i	−0.994895	−	0.100920i	$-0.967821\pi$
$509$		−	102.736i		−	0.201839i	0.994895	−	0.100920i	$-0.0321785\pi$
$510$		0			0
$511$	−111.876			−0.218936
$512$		0			0
$513$	359.566	−	280.074i	0.700909	−	0.545953i
$514$		0			0
$515$		−	53.2838i		−	0.103464i
$516$		0			0
$517$	−216.794			−0.419330
$518$		0			0
$519$	765.413	+	239.300i	1.47478	+	0.461079i
$520$		0			0
$521$			341.510i			0.655489i	0.944766	+	0.327744i	$0.106289\pi$
$521$			341.510i			0.655489i	−0.944766	+	0.327744i	$0.893711\pi$
$522$		0			0
$523$	−400.488			−0.765751			−0.382875	−	0.923800i	$-0.625066\pi$
$523$	−400.488			−0.765751			−0.382875	+	0.923800i	$0.625066\pi$
$524$		0			0
$525$	11.8423	−	37.8782i	0.0225568	−	0.0721490i
$526$		0			0
$527$			18.7224i			0.0355264i
$528$		0			0
$529$	375.204			0.709270
$530$		0			0
$531$	−381.877	+	551.032i	−0.719166	+	1.03773i
$532$		0			0
$533$			478.963i			0.898618i
$534$		0			0
$535$	−335.493			−0.627089
$536$		0			0
$537$	−66.8884	−	20.9121i	−0.124559	−	0.0389424i
$538$		0			0
$539$		−	21.0236i		−	0.0390049i
$540$		0			0
$541$	−516.907			−0.955465			−0.477733	−	0.878505i	$-0.658541\pi$
$541$	−516.907			−0.955465			−0.477733	+	0.878505i	$0.658541\pi$
$542$		0			0
$543$	107.100	−	342.564i	0.197237	−	0.630874i
$544$		0			0
$545$			360.700i			0.661834i
$546$		0			0
$547$	725.077			1.32555			0.662776	−	0.748818i	$-0.269378\pi$
$547$	725.077			1.32555			0.662776	+	0.748818i	$0.269378\pi$
$548$		0			0
$549$	−489.367	−	339.142i	−0.891378	−	0.617744i
$550$		0			0
$551$		−	229.249i		−	0.416060i
$552$		0			0
$553$	−8.27371			−0.0149615
$554$		0			0
$555$	113.378	+	35.4466i	0.204285	+	0.0638678i
$556$		0			0
$557$		−	741.121i		−	1.33056i	−0.746595	−	0.665279i	$-0.768313\pi$
$557$		−	741.121i		−	1.33056i	0.746595	−	0.665279i	$-0.231687\pi$
$558$		0			0
$559$	−966.216			−1.72847
$560$		0			0
$561$	9.68947	−	30.9923i	0.0172718	−	0.0552447i
$562$		0			0
$563$		−	926.643i		−	1.64590i	−0.568112	−	0.822951i	$-0.692326\pi$
$563$		−	926.643i		−	1.64590i	0.568112	−	0.822951i	$-0.307674\pi$
$564$		0			0
$565$	449.353			0.795316
$566$		0			0
$567$	75.2420	+	200.663i	0.132702	+	0.353903i
$568$		0			0
$569$			116.739i			0.205166i	0.994724	+	0.102583i	$0.0327107\pi$
$569$			116.739i			0.205166i	−0.994724	+	0.102583i	$0.967289\pi$
$570$		0			0
$571$	202.431			0.354520			0.177260	−	0.984164i	$-0.443277\pi$
$571$	202.431			0.354520			0.177260	+	0.984164i	$0.443277\pi$
$572$		0			0
$573$	−218.054	−	68.1726i	−0.380547	−	0.118975i
$574$		0			0
$575$		−	62.0073i		−	0.107839i
$576$		0			0
$577$	−82.6440			−0.143230			−0.0716152	−	0.997432i	$-0.522815\pi$
$577$	−82.6440			−0.143230			−0.0716152	+	0.997432i	$0.522815\pi$
$578$		0			0
$579$	104.012	−	332.686i	0.179640	−	0.574588i
$580$		0			0
$581$			260.238i			0.447914i
$582$		0			0
$583$	−43.3710			−0.0743927
$584$		0			0
$585$	−278.829	+	402.339i	−0.476631	+	0.687758i
$586$		0			0
$587$		−	108.578i		−	0.184970i	−0.995714	−	0.0924851i	$-0.970519\pi$
$587$		−	108.578i		−	0.184970i	0.995714	−	0.0924851i	$-0.0294811\pi$
$588$		0			0
$589$	−87.6947			−0.148887
$590$		0			0
$591$	−708.630	−	221.547i	−1.19903	−	0.374868i
$592$		0			0
$593$		−	395.830i		−	0.667504i	−0.942661	−	0.333752i	$-0.891685\pi$
$593$		−	395.830i		−	0.667504i	0.942661	−	0.333752i	$-0.108315\pi$
$594$		0			0
$595$	21.3210			0.0358336
$596$		0			0
$597$	−38.9420	+	124.558i	−0.0652296	+	0.208640i
$598$		0			0
$599$			59.2409i			0.0988996i	0.998777	+	0.0494498i	$0.0157468\pi$
$599$			59.2409i			0.0988996i	−0.998777	+	0.0494498i	$0.984253\pi$
$600$		0			0
$601$	−350.659			−0.583460			−0.291730	−	0.956501i	$-0.594231\pi$
$601$	−350.659			−0.583460			−0.291730	+	0.956501i	$0.594231\pi$
$602$		0			0
$603$	−353.691	−	245.116i	−0.586552	−	0.406494i
$604$		0			0
$605$			250.394i			0.413875i
$606$		0			0
$607$	−319.382			−0.526164			−0.263082	−	0.964773i	$-0.584739\pi$
$607$	−319.382			−0.526164			−0.263082	+	0.964773i	$0.584739\pi$
$608$		0			0
$609$	102.883	+	32.1654i	0.168937	+	0.0528168i
$610$		0			0
$611$			1755.79i			2.87363i
$612$		0			0
$613$	423.614			0.691051			0.345525	−	0.938409i	$-0.387701\pi$
$613$	423.614			0.691051			0.345525	+	0.938409i	$0.387701\pi$
$614$		0			0
$615$	−39.4156	+	126.073i	−0.0640904	+	0.204997i
$616$		0			0
$617$			986.842i			1.59942i	0.600386	+	0.799710i	$0.295014\pi$
$617$			986.842i			1.59942i	−0.600386	+	0.799710i	$0.704986\pi$
$618$		0			0
$619$	−857.133			−1.38471			−0.692353	−	0.721559i	$-0.743426\pi$
$619$	−857.133			−1.38471			−0.692353	+	0.721559i	$0.743426\pi$
$620$		0			0
$621$	205.760	+	264.160i	0.331336	+	0.425378i
$622$		0			0
$623$		−	409.667i		−	0.657571i
$624$		0			0
$625$	25.0000			0.0400000
$626$		0			0
$627$	145.166	+	45.3850i	0.231525	+	0.0723843i
$628$		0			0
$629$			63.8184i			0.101460i
$630$		0			0
$631$	320.357			0.507697			0.253848	−	0.967244i	$-0.418304\pi$
$631$	320.357			0.507697			0.253848	+	0.967244i	$0.418304\pi$
$632$		0			0
$633$	−253.524	+	810.911i	−0.400512	+	1.28106i
$634$		0			0
$635$		−	199.643i		−	0.314398i
$636$		0			0
$637$	−170.268			−0.267297
$638$		0			0
$639$	240.012	−	346.327i	0.375606	−	0.541983i
$640$		0			0
$641$		−	280.940i		−	0.438284i	−0.975693	−	0.219142i	$-0.929674\pi$
$641$		−	280.940i		−	0.438284i	0.975693	−	0.219142i	$-0.0703259\pi$
$642$		0			0
$643$	−206.903			−0.321777			−0.160889	−	0.986973i	$-0.551436\pi$
$643$	−206.903			−0.321777			−0.160889	+	0.986973i	$0.551436\pi$
$644$		0			0
$645$	−254.328	−	79.5134i	−0.394307	−	0.123277i
$646$		0			0
$647$			907.218i			1.40219i	0.713067	+	0.701096i	$0.247306\pi$
$647$			907.218i			1.40219i	−0.713067	+	0.701096i	$0.752694\pi$
$648$		0			0
$649$	−223.726			−0.344724
$650$		0			0
$651$	12.3042	−	39.3557i	0.0189005	−	0.0604542i
$652$		0			0
$653$			93.6270i			0.143380i	0.997427	+	0.0716899i	$0.0228392\pi$
$653$			93.6270i			0.143380i	−0.997427	+	0.0716899i	$0.977161\pi$
$654$		0			0
$655$	298.895			0.456329
$656$		0			0
$657$	312.794	+	216.773i	0.476095	+	0.329944i
$658$		0			0
$659$			223.298i			0.338844i	0.985544	+	0.169422i	$0.0541900\pi$
$659$			223.298i			0.338844i	−0.985544	+	0.169422i	$0.945810\pi$
$660$		0			0
$661$	−409.255			−0.619145			−0.309572	−	0.950876i	$-0.600186\pi$
$661$	−409.255			−0.619145			−0.309572	+	0.950876i	$0.600186\pi$
$662$		0			0
$663$	−251.003	−	78.4741i	−0.378587	−	0.118362i
$664$		0			0
$665$			99.8663i			0.150175i
$666$		0			0
$667$	168.421			0.252505
$668$		0			0
$669$	−258.579	+	827.078i	−0.386515	+	1.23629i
$670$		0			0
$671$		−	198.689i		−	0.296109i
$672$		0			0
$673$	−256.438			−0.381037			−0.190519	−	0.981684i	$-0.561017\pi$
$673$	−256.438			−0.381037			−0.190519	+	0.981684i	$0.561017\pi$
$674$		0			0
$675$	−106.503	+	82.9579i	−0.157783	+	0.122901i
$676$		0			0
$677$			186.843i			0.275986i	0.990433	+	0.137993i	$0.0440652\pi$
$677$			186.843i			0.275986i	−0.990433	+	0.137993i	$0.955935\pi$
$678$		0			0
$679$	−244.408			−0.359953
$680$		0			0
$681$	−956.190	−	298.944i	−1.40410	−	0.438979i
$682$		0			0
$683$		−	639.899i		−	0.936895i	−0.883491	−	0.468447i	$-0.844814\pi$
$683$		−	639.899i		−	0.936895i	0.883491	−	0.468447i	$-0.155186\pi$
$684$		0			0
$685$	115.889			0.169181
$686$		0			0
$687$	10.0105	−	32.0191i	0.0145713	−	0.0466071i
$688$		0			0
$689$			351.257i			0.509807i
$690$		0			0
$691$	784.721			1.13563			0.567815	−	0.823156i	$-0.307789\pi$
$691$	784.721			1.13563			0.567815	+	0.823156i	$0.307789\pi$
$692$		0			0
$693$	−40.7358	+	58.7800i	−0.0587819	+	0.0848196i
$694$		0			0
$695$			425.731i			0.612563i
$696$		0			0
$697$	−70.9642			−0.101814
$698$		0			0
$699$	713.938	+	223.206i	1.02137	+	0.319322i
$700$		0			0
$701$		−	743.171i		−	1.06016i	−0.847948	−	0.530079i	$-0.822162\pi$
$701$		−	743.171i		−	1.06016i	0.847948	−	0.530079i	$-0.177838\pi$
$702$		0			0
$703$	−298.922			−0.425209
$704$		0			0
$705$	−144.490	+	462.160i	−0.204951	+	0.655546i
$706$		0			0
$707$			249.968i			0.353562i
$708$		0			0
$709$	−742.471			−1.04721			−0.523605	−	0.851961i	$-0.675413\pi$
$709$	−742.471			−1.04721			−0.523605	+	0.851961i	$0.675413\pi$
$710$		0			0
$711$	23.1325	+	16.0313i	0.0325351	+	0.0225475i
$712$		0			0
$713$		−	64.4259i		−	0.0903590i
$714$		0			0
$715$	−163.354			−0.228468
$716$		0			0
$717$	412.507	+	128.967i	0.575324	+	0.179870i
$718$		0			0
$719$			1075.69i			1.49609i	0.663648	+	0.748045i	$0.269007\pi$
$719$			1075.69i			1.49609i	−0.663648	+	0.748045i	$0.730993\pi$
$720$		0			0
$721$	−63.0463			−0.0874428
$722$		0			0
$723$	311.299	−	995.706i	0.430565	−	1.37719i
$724$		0			0
$725$			67.9036i			0.0936602i
$726$		0			0
$727$	−276.007			−0.379652			−0.189826	−	0.981818i	$-0.560792\pi$
$727$	−276.007			−0.379652			−0.189826	+	0.981818i	$0.560792\pi$
$728$		0			0
$729$	178.439	−	706.824i	0.244773	−	0.969580i
$730$		0			0
$731$		−	143.157i		−	0.195837i
$732$		0			0
$733$	−1323.23			−1.80523			−0.902616	−	0.430448i	$-0.858356\pi$
$733$	−1323.23			−1.80523			−0.902616	+	0.430448i	$0.858356\pi$
$734$		0			0
$735$	−44.8181	−	14.0120i	−0.0609770	−	0.0190639i
$736$		0			0
$737$		−	143.603i		−	0.194848i
$738$		0			0
$739$	−11.1722			−0.0151181			−0.00755903	−	0.999971i	$-0.502406\pi$
$739$	−11.1722			−0.0151181			−0.00755903	+	0.999971i	$0.502406\pi$
$740$		0			0
$741$	367.568	−	1175.69i	0.496043	−	1.58662i
$742$		0			0
$743$		−	532.660i		−	0.716904i	−0.933548	−	0.358452i	$-0.883305\pi$
$743$		−	532.660i		−	0.716904i	0.933548	−	0.358452i	$-0.116695\pi$
$744$		0			0
$745$	−16.1273			−0.0216473
$746$		0			0
$747$	504.242	−	727.599i	0.675023	−	0.974028i
$748$		0			0
$749$			396.960i			0.529987i
$750$		0			0
$751$	843.951			1.12377			0.561885	−	0.827215i	$-0.310076\pi$
$751$	843.951			1.12377			0.561885	+	0.827215i	$0.310076\pi$
$752$		0			0
$753$	1298.17	+	405.861i	1.72400	+	0.538992i
$754$		0			0
$755$			254.076i			0.336524i
$756$		0			0
$757$	−303.221			−0.400556			−0.200278	−	0.979739i	$-0.564185\pi$
$757$	−303.221			−0.400556			−0.200278	+	0.979739i	$0.564185\pi$
$758$		0			0
$759$	−33.3426	+	106.648i	−0.0439297	+	0.140511i
$760$		0			0
$761$			115.096i			0.151243i	0.997137	+	0.0756215i	$0.0240941\pi$
$761$			115.096i			0.151243i	−0.997137	+	0.0756215i	$0.975906\pi$
$762$		0			0
$763$	426.786			0.559352
$764$		0			0
$765$	−59.6113	−	41.3120i	−0.0779233	−	0.0540026i
$766$		0			0
$767$			1811.93i			2.36236i
$768$		0			0
$769$	801.344			1.04206			0.521030	−	0.853538i	$-0.325548\pi$
$769$	801.344			1.04206			0.521030	+	0.853538i	$0.325548\pi$
$770$		0			0
$771$	1210.58	+	378.478i	1.57015	+	0.490893i
$772$		0			0
$773$			94.0269i			0.121639i	0.998149	+	0.0608194i	$0.0193714\pi$
$773$			94.0269i			0.121639i	−0.998149	+	0.0608194i	$0.980629\pi$
$774$		0			0
$775$	25.9752			0.0335163
$776$		0			0
$777$	41.9410	−	134.151i	0.0539781	−	0.172652i
$778$		0			0
$779$		−	332.393i		−	0.426691i
$780$		0			0
$781$	140.613			0.180042
$782$		0			0
$783$	−225.326	−	289.279i	−0.287772	−	0.369450i
$784$		0			0
$785$			456.631i			0.581695i
$786$		0			0
$787$	114.020			0.144880			0.0724398	−	0.997373i	$-0.476921\pi$
$787$	114.020			0.144880			0.0724398	+	0.997373i	$0.476921\pi$
$788$		0			0
$789$	−1075.25	−	336.167i	−1.36280	−	0.426067i
$790$		0			0
$791$		−	531.682i		−	0.672165i
$792$		0			0
$793$	−1609.16			−2.02921
$794$		0			0
$795$	−28.9062	+	92.4580i	−0.0363600	+	0.116299i
$796$		0			0
$797$			879.789i			1.10388i	0.833885	+	0.551938i	$0.186111\pi$
$797$			879.789i			1.10388i	−0.833885	+	0.551938i	$0.813889\pi$
$798$		0			0
$799$	−260.142			−0.325584
$800$		0			0
$801$	−793.779	+	1145.39i	−0.990984	+	1.42995i
$802$		0			0
$803$			126.998i			0.158155i
$804$		0			0
$805$	−73.3680			−0.0911404
$806$		0			0
$807$	−4.04634	−	1.26505i	−0.00501405	−	0.00156760i
$808$		0			0
$809$		−	1429.94i		−	1.76753i	−0.467927	−	0.883767i	$-0.654999\pi$
$809$		−	1429.94i		−	1.76753i	0.467927	−	0.883767i	$-0.345001\pi$
$810$		0			0
$811$	1369.69			1.68889			0.844445	−	0.535643i	$-0.179931\pi$
$811$	1369.69			1.68889			0.844445	+	0.535643i	$0.179931\pi$
$812$		0			0
$813$	202.362	−	647.265i	0.248907	−	0.796144i
$814$		0			0
$815$		−	172.670i		−	0.211866i
$816$		0			0
$817$	670.538			0.820732
$818$		0			0
$819$	476.053	+	329.915i	0.581262	+	0.402827i
$820$		0			0
$821$		−	1221.52i		−	1.48785i	−0.668265	−	0.743923i	$-0.732963\pi$
$821$		−	1221.52i		−	1.48785i	0.668265	−	0.743923i	$-0.267037\pi$
$822$		0			0
$823$	223.454			0.271512			0.135756	−	0.990742i	$-0.456654\pi$
$823$	223.454			0.271512			0.135756	+	0.990742i	$0.456654\pi$
$824$		0			0
$825$	−42.9982	−	13.4430i	−0.0521191	−	0.0162946i
$826$		0			0
$827$		−	150.303i		−	0.181744i	−0.995863	−	0.0908722i	$-0.971035\pi$
$827$		−	150.303i		−	0.181744i	0.995863	−	0.0908722i	$-0.0289655\pi$
$828$		0			0
$829$	−994.512			−1.19965			−0.599826	−	0.800130i	$-0.704764\pi$
$829$	−994.512			−1.19965			−0.599826	+	0.800130i	$0.704764\pi$
$830$		0			0
$831$	318.099	−	1017.46i	0.382791	−	1.22438i
$832$		0			0
$833$		−	25.2273i		−	0.0302849i
$834$		0			0
$835$	635.894			0.761550
$836$		0			0
$837$	−110.658	+	86.1937i	−0.132208	+	0.102979i
$838$		0			0
$839$			582.305i			0.694047i	0.937857	+	0.347023i	$0.112808\pi$
$839$			582.305i			0.694047i	−0.937857	+	0.347023i	$0.887192\pi$
$840$		0			0
$841$	656.564			0.780694
$842$		0			0
$843$	−561.171	−	175.445i	−0.665683	−	0.208120i
$844$		0			0
$845$			945.095i			1.11846i
$846$		0			0
$847$	296.271			0.349788
$848$		0			0
$849$	−434.246	+	1388.96i	−0.511480	+	1.63600i
$850$		0			0
$851$		−	219.607i		−	0.258057i
$852$		0			0
$853$	1513.46			1.77428			0.887139	−	0.461503i	$-0.152690\pi$
$853$	1513.46			1.77428			0.887139	+	0.461503i	$0.152690\pi$
$854$		0			0
$855$	193.503	−	279.216i	0.226319	−	0.326569i
$856$		0			0
$857$		−	1275.06i		−	1.48782i	−0.668278	−	0.743911i	$-0.732969\pi$
$857$		−	1275.06i		−	1.48782i	0.668278	−	0.743911i	$-0.267031\pi$
$858$		0			0
$859$	330.107			0.384292			0.192146	−	0.981366i	$-0.438455\pi$
$859$	330.107			0.384292			0.192146	+	0.981366i	$0.438455\pi$
$860$		0			0
$861$	149.172	+	46.6372i	0.173254	+	0.0541663i
$862$		0			0
$863$		−	51.0607i		−	0.0591665i	−0.999562	−	0.0295832i	$-0.990582\pi$
$863$		−	51.0607i		−	0.0591665i	0.999562	−	0.0295832i	$-0.00941801\pi$
$864$		0			0
$865$	597.737			0.691026
$866$		0			0
$867$	−247.084	+	790.312i	−0.284987	+	0.911547i
$868$		0			0
$869$			9.39207i			0.0108079i
$870$		0			0
$871$	−1163.03			−1.33528
$872$		0			0
$873$	683.340	+	473.570i	0.782750	+	0.542463i
$874$		0			0
$875$		−	29.5804i		−	0.0338062i
$876$		0			0
$877$	19.8915			0.0226813			0.0113406	−	0.999936i	$-0.496390\pi$
$877$	19.8915			0.0226813			0.0113406	+	0.999936i	$0.496390\pi$
$878$		0			0
$879$	−1083.68	−	338.802i	−1.23285	−	0.385440i
$880$		0			0
$881$		−	946.321i		−	1.07414i	−0.843536	−	0.537072i	$-0.819530\pi$
$881$		−	946.321i		−	1.07414i	0.843536	−	0.537072i	$-0.180470\pi$
$882$		0			0
$883$	1500.11			1.69888			0.849441	−	0.527684i	$-0.176940\pi$
$883$	1500.11			1.69888			0.849441	+	0.527684i	$0.176940\pi$
$884$		0			0
$885$	−149.110	+	476.938i	−0.168486	+	0.538913i
$886$		0			0
$887$			515.237i			0.580876i	0.956894	+	0.290438i	$0.0938010\pi$
$887$			515.237i			0.580876i	−0.956894	+	0.290438i	$0.906199\pi$
$888$		0			0
$889$	−236.220			−0.265715
$890$		0			0
$891$	227.787	−	85.4125i	0.255653	−	0.0958614i
$892$		0			0
$893$		−	1218.49i		−	1.36449i
$894$		0			0
$895$	−52.2355			−0.0583637
$896$		0			0
$897$	863.732	+	270.038i	0.962912	+	0.301046i
$898$		0			0
$899$			70.5523i			0.0784786i
$900$		0			0
$901$	−52.0430			−0.0577614
$902$		0			0
$903$	−94.0815	+	300.925i	−0.104188	+	0.333250i
$904$		0			0
$905$		−	267.520i		−	0.295602i
$906$		0			0
$907$	−841.383			−0.927655			−0.463827	−	0.885926i	$-0.653524\pi$
$907$	−841.383			−0.927655			−0.463827	+	0.885926i	$0.653524\pi$
$908$		0			0
$909$	484.344	−	698.887i	0.532831	−	0.768852i
$910$		0			0
$911$		−	564.445i		−	0.619589i	−0.950804	−	0.309794i	$-0.899740\pi$
$911$		−	564.445i		−	0.619589i	0.950804	−	0.309794i	$-0.100260\pi$
$912$		0			0
$913$	295.414			0.323564
$914$		0			0
$915$	−423.564	−	132.424i	−0.462912	−	0.144725i
$916$		0			0
$917$		−	353.658i		−	0.385668i
$918$		0			0
$919$	−1171.81			−1.27509			−0.637547	−	0.770411i	$-0.720051\pi$
$919$	−1171.81			−1.27509			−0.637547	+	0.770411i	$0.720051\pi$
$920$		0			0
$921$	−513.558	+	1642.64i	−0.557610	+	1.78354i
$922$		0			0
$923$		−	1138.81i		−	1.23382i
$924$		0			0
$925$	88.5407			0.0957197
$926$		0			0
$927$	176.271	+	122.160i	0.190152	+	0.131780i
$928$		0			0
$929$		−	812.845i		−	0.874967i	−0.899226	−	0.437484i	$-0.855870\pi$
$929$		−	812.845i		−	0.874967i	0.899226	−	0.437484i	$-0.144130\pi$
$930$		0			0
$931$	118.163			0.126921
$932$		0			0
$933$	1265.36	+	395.602i	1.35622	+	0.424011i
$934$		0			0
$935$		−	24.2029i		−	0.0258855i
$936$		0			0
$937$	1673.57			1.78610			0.893048	−	0.449962i	$-0.148563\pi$
$937$	1673.57			1.78610			0.893048	+	0.449962i	$0.148563\pi$
$938$		0			0
$939$	339.254	−	1085.12i	0.361293	−	1.15562i
$940$		0			0
$941$			974.517i			1.03562i	0.855496	+	0.517809i	$0.173252\pi$
$941$			974.517i			1.03562i	−0.855496	+	0.517809i	$0.826748\pi$
$942$		0			0
$943$	244.196			0.258957
$944$		0			0
$945$	98.1571	+	126.017i	0.103870	+	0.133351i
$946$		0			0
$947$		−	876.037i		−	0.925066i	−0.886602	−	0.462533i	$-0.846941\pi$
$947$		−	876.037i		−	0.925066i	0.886602	−	0.462533i	$-0.153059\pi$
$948$		0			0
$949$	1028.55			1.08382
$950$		0			0
$951$	−1684.14	−	526.533i	−1.77092	−	0.553662i
$952$		0			0
$953$			6.64325i			0.00697088i	0.999994	+	0.00348544i	$0.00110945\pi$
$953$			6.64325i			0.00697088i	−0.999994	+	0.00348544i	$0.998891\pi$
$954$		0			0
$955$	−170.286			−0.178309
$956$		0			0
$957$	36.5132	−	116.789i	0.0381538	−	0.122037i
$958$		0			0
$959$		−	137.122i		−	0.142984i
$960$		0			0
$961$	−934.012			−0.971916
$962$		0			0
$963$	769.158	−	1109.86i	0.798710	−	1.15250i
$964$		0			0
$965$		−	259.806i		−	0.269229i
$966$		0			0
$967$	1147.22			1.18637			0.593184	−	0.805067i	$-0.297871\pi$
$967$	1147.22			1.18637			0.593184	+	0.805067i	$0.297871\pi$
$968$		0			0
$969$	174.192	+	54.4597i	0.179765	+	0.0562020i
$970$		0			0
$971$			48.8617i			0.0503210i	0.999683	+	0.0251605i	$0.00800968\pi$
$971$			48.8617i			0.0503210i	−0.999683	+	0.0251605i	$0.991990\pi$
$972$		0			0
$973$	503.732			0.517710
$974$		0			0
$975$	−108.874	+	348.238i	−0.111665	+	0.357167i
$976$		0			0
$977$		−	454.682i		−	0.465386i	−0.972550	−	0.232693i	$-0.925246\pi$
$977$		−	454.682i		−	0.465386i	0.972550	−	0.232693i	$-0.0747537\pi$
$978$		0			0
$979$	−465.042			−0.475017
$980$		0			0
$981$	−1193.25	−	826.948i	−1.21636	−	0.842965i
$982$		0			0
$983$			493.543i			0.502078i	0.967977	+	0.251039i	$0.0807722\pi$
$983$			493.543i			0.502078i	−0.967977	+	0.251039i	$0.919228\pi$
$984$		0			0
$985$	−553.393			−0.561820
$986$		0			0
$987$	546.835	+	170.963i	0.554038	+	0.173215i
$988$		0			0
$989$			492.619i			0.498098i
$990$		0			0
$991$	−321.693			−0.324614			−0.162307	−	0.986740i	$-0.551893\pi$
$991$	−321.693			−0.324614			−0.162307	+	0.986740i	$0.551893\pi$
$992$		0			0
$993$	184.003	−	588.544i	0.185300	−	0.592693i
$994$		0			0
$995$			97.2718i			0.0977606i
$996$		0			0
$997$	−1254.19			−1.25796			−0.628980	−	0.777421i	$-0.716527\pi$
$997$	−1254.19			−1.25796			−0.628980	+	0.777421i	$0.716527\pi$
$998$		0			0
$999$	−377.196	+	293.806i	−0.377573	+	0.294100i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1680.3.l.b.1121.6		16
3.2	odd	2	inner	1680.3.l.b.1121.5		16
4.3	odd	2		420.3.g.a.281.11	✓	16
12.11	even	2		420.3.g.a.281.12	yes	16
20.3	even	4		2100.3.e.c.449.30		32
20.7	even	4		2100.3.e.c.449.3		32
20.19	odd	2		2100.3.g.d.701.6		16
60.23	odd	4		2100.3.e.c.449.4		32
60.47	odd	4		2100.3.e.c.449.29		32
60.59	even	2		2100.3.g.d.701.5		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
420.3.g.a.281.11	✓	16	4.3	odd	2
420.3.g.a.281.12	yes	16	12.11	even	2
1680.3.l.b.1121.5		16	3.2	odd	2	inner
1680.3.l.b.1121.6		16	1.1	even	1	trivial
2100.3.e.c.449.3		32	20.7	even	4
2100.3.e.c.449.4		32	60.23	odd	4
2100.3.e.c.449.29		32	60.47	odd	4
2100.3.e.c.449.30		32	20.3	even	4
2100.3.g.d.701.5		16	60.59	even	2
2100.3.g.d.701.6		16	20.19	odd	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:	\( N \)	\(=\)	\( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1680.l (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.895194 + 2.86332i) q^{3} +2.23607i q^{5} +2.64575 q^{7} +(-7.39726 - 5.12646i) q^{9} -3.00338i q^{11} -24.3241 q^{13} +(-6.40259 - 2.00171i) q^{15} -3.60390i q^{17} +16.8805 q^{19} +(-2.36846 + 7.57565i) q^{21} +12.4015i q^{23} -5.00000 q^{25} +(21.3007 - 16.5916i) q^{27} -13.5807i q^{29} -5.19503 q^{31} +(8.59964 + 2.68860i) q^{33} +5.91608i q^{35} -17.7081 q^{37} +(21.7747 - 69.6477i) q^{39} -19.6909i q^{41} +39.7227 q^{43} +(11.4631 - 16.5408i) q^{45} -72.1833i q^{47} +7.00000 q^{49} +(10.3191 + 3.22619i) q^{51} -14.4407i q^{53} +6.71576 q^{55} +(-15.1113 + 48.3343i) q^{57} -74.4914i q^{59} +66.1551 q^{61} +(-19.5713 - 13.5633i) q^{63} -54.3902i q^{65} +47.8138 q^{67} +(-35.5094 - 11.1017i) q^{69} +46.8183i q^{71} -42.2852 q^{73} +(4.47597 - 14.3166i) q^{75} -7.94619i q^{77} -3.12717 q^{79} +(28.4388 + 75.8435i) q^{81} +98.3606i q^{83} +8.05857 q^{85} +(38.8860 + 12.1574i) q^{87} -154.840i q^{89} -64.3554 q^{91} +(4.65056 - 14.8751i) q^{93} +37.7459i q^{95} -92.3775 q^{97} +(-15.3967 + 22.2168i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 8 q^{3} - 10 q^{9} - 10 q^{15} - 48 q^{19} - 14 q^{21} - 80 q^{25} - 28 q^{27} - 24 q^{31} + 92 q^{33} - 40 q^{37} + 54 q^{39} + 168 q^{43} + 40 q^{45} + 112 q^{49} + 186 q^{51} + 80 q^{55} + 252 q^{57}+ \cdots - 26 q^{99}+O(q^{100}) \) 16 * q + 8 * q^3 - 10 * q^9 - 10 * q^15 - 48 * q^19 - 14 * q^21 - 80 * q^25 - 28 * q^27 - 24 * q^31 + 92 * q^33 - 40 * q^37 + 54 * q^39 + 168 * q^43 + 40 * q^45 + 112 * q^49 + 186 * q^51 + 80 * q^55 + 252 * q^57 - 152 * q^61 + 56 * q^63 - 264 * q^67 - 196 * q^69 + 240 * q^73 - 40 * q^75 - 212 * q^79 - 118 * q^81 - 60 * q^85 - 60 * q^87 - 196 * q^91 - 244 * q^93 - 360 * q^97 - 26 * q^99

Introduction

L-functions

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