LMFDB - Embedded newform 600.3.l.d.401.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [600,3,Mod(401,600)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("600.401"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(600, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 600 = 2^{3} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	600.l (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,-4,0,0,0,10] 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:	$16.3488158616$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.574198272.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 7x^{4} - 34x^{3} + 81x^{2} + 156x + 198 $ x^6 - 2x^5 - 7x^4 - 34x^3 + 81x^2 + 156*x + 198
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4}\cdot 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			401.6
Root			$3.56627 - 0.139571i$ of defining polynomial
Character	$\chi$	$=$	600.401
Dual form			600.3.l.d.401.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.56627 + 1.55378i) q^{3} -1.34301 q^{7} +(4.17150 + 7.97487i) q^{9} -3.66586i q^{11} +7.34301 q^{13} +6.31089i q^{17} +30.7406 q^{19} +(-3.44653 - 2.08675i) q^{21} +26.2426i q^{23} +(-1.68602 + 26.9473i) q^{27} +40.9060i q^{29} -9.97097 q^{31} +(5.69595 - 9.40758i) q^{33} +39.4232 q^{37} +(18.8442 + 11.4095i) q^{39} -68.1695i q^{41} +24.0802 q^{43} +79.7705i q^{47} -47.1963 q^{49} +(-9.80576 + 16.1955i) q^{51} +38.9657i q^{53} +(78.8888 + 47.7643i) q^{57} -39.9648i q^{59} -64.2475 q^{61} +(-5.60237 - 10.7103i) q^{63} +53.9454 q^{67} +(-40.7754 + 67.3457i) q^{69} -136.706i q^{71} -74.6196 q^{73} +4.92328i q^{77} +79.6417 q^{79} +(-46.1971 + 66.5344i) q^{81} +0.0506471i q^{83} +(-63.5592 + 104.976i) q^{87} +22.6492i q^{89} -9.86173 q^{91} +(-25.5882 - 15.4927i) q^{93} +35.0428 q^{97} +(29.2347 - 15.2921i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{3} + 10 q^{7} + 16 q^{9} + 26 q^{13} + 50 q^{19} - 18 q^{21} + 26 q^{27} - 114 q^{31} + 82 q^{33} + 76 q^{37} - 6 q^{39} + 2 q^{43} + 76 q^{49} - 6 q^{51} + 172 q^{57} + 62 q^{61} - 150 q^{63}+ \cdots - 250 q^{99}+O(q^{100}) $ 6 * q - 4 * q^3 + 10 * q^7 + 16 * q^9 + 26 * q^13 + 50 * q^19 - 18 * q^21 + 26 * q^27 - 114 * q^31 + 82 * q^33 + 76 * q^37 - 6 * q^39 + 2 * q^43 + 76 * q^49 - 6 * q^51 + 172 * q^57 + 62 * q^61 - 150 * q^63 + 422 * q^67 + 156 * q^69 + 72 * q^73 - 76 * q^79 - 224 * q^81 - 172 * q^87 - 310 * q^91 + 110 * q^93 - 470 * q^97 - 250 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/600\mathbb{Z}\right)^\times$.

$n$	$151$	$301$	$401$	$577$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$3$	2.56627	+	1.55378i	0.855424	+	0.517928i
$3$	2.56627	+	1.55378i	0.855424	+	0.517928i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−1.34301			−0.191858			−0.0959292	−	0.995388i	$-0.530582\pi$
$7$	−1.34301			−0.191858			−0.0959292	+	0.995388i	$0.530582\pi$
$8$		0			0
$9$	4.17150	+	7.97487i	0.463500	+	0.886097i
$10$		0			0
$11$		−	3.66586i		−	0.333260i	−0.986020	−	0.166630i	$-0.946712\pi$
$11$		−	3.66586i		−	0.333260i	0.986020	−	0.166630i	$-0.0532885\pi$
$12$		0			0
$13$	7.34301			0.564847			0.282423	−	0.959290i	$-0.408862\pi$
$13$	7.34301			0.564847			0.282423	+	0.959290i	$0.408862\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			6.31089i			0.371229i	0.982623	+	0.185614i	$0.0594275\pi$
$17$			6.31089i			0.371229i	−0.982623	+	0.185614i	$0.940572\pi$
$18$		0			0
$19$	30.7406			1.61793			0.808964	−	0.587858i	$-0.200029\pi$
$19$	30.7406			1.61793			0.808964	+	0.587858i	$0.200029\pi$
$20$		0			0
$21$	−3.44653	−	2.08675i	−0.164120	−	0.0993689i
$22$		0			0
$23$			26.2426i			1.14098i	0.821303	+	0.570492i	$0.193247\pi$
$23$			26.2426i			1.14098i	−0.821303	+	0.570492i	$0.806753\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−1.68602	+	26.9473i	−0.0624451	+	0.998048i
$28$		0			0
$29$			40.9060i			1.41055i	0.708932	+	0.705277i	$0.249177\pi$
$29$			40.9060i			1.41055i	−0.708932	+	0.705277i	$0.750823\pi$
$30$		0			0
$31$	−9.97097			−0.321644			−0.160822	−	0.986983i	$-0.551415\pi$
$31$	−9.97097			−0.321644			−0.160822	+	0.986983i	$0.551415\pi$
$32$		0			0
$33$	5.69595	−	9.40758i	0.172605	−	0.285078i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	39.4232			1.06549			0.532746	−	0.846275i	$-0.321160\pi$
$37$	39.4232			1.06549			0.532746	+	0.846275i	$0.321160\pi$
$38$		0			0
$39$	18.8442	+	11.4095i	0.483184	+	0.292550i
$40$		0			0
$41$		−	68.1695i		−	1.66267i	−0.555771	−	0.831335i	$-0.687577\pi$
$41$		−	68.1695i		−	1.66267i	0.555771	−	0.831335i	$-0.312423\pi$
$42$		0			0
$43$	24.0802			0.560005			0.280003	−	0.959999i	$-0.409665\pi$
$43$	24.0802			0.560005			0.280003	+	0.959999i	$0.409665\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			79.7705i			1.69724i	0.529000	+	0.848622i	$0.322567\pi$
$47$			79.7705i			1.69724i	−0.529000	+	0.848622i	$0.677433\pi$
$48$		0			0
$49$	−47.1963			−0.963190
$50$		0			0
$51$	−9.80576	+	16.1955i	−0.192270	+	0.317558i
$52$		0			0
$53$			38.9657i			0.735202i	0.929984	+	0.367601i	$0.119821\pi$
$53$			38.9657i			0.735202i	−0.929984	+	0.367601i	$0.880179\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	78.8888	+	47.7643i	1.38401	+	0.837971i
$58$		0			0
$59$		−	39.9648i		−	0.677369i	−0.940900	−	0.338684i	$-0.890018\pi$
$59$		−	39.9648i		−	0.677369i	0.940900	−	0.338684i	$-0.109982\pi$
$60$		0			0
$61$	−64.2475			−1.05324			−0.526619	−	0.850101i	$-0.676541\pi$
$61$	−64.2475			−1.05324			−0.526619	+	0.850101i	$0.676541\pi$
$62$		0			0
$63$	−5.60237	−	10.7103i	−0.0889265	−	0.170005i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	53.9454			0.805155			0.402577	−	0.915386i	$-0.368114\pi$
$67$	53.9454			0.805155			0.402577	+	0.915386i	$0.368114\pi$
$68$		0			0
$69$	−40.7754	+	67.3457i	−0.590948	+	0.976025i
$70$		0			0
$71$		−	136.706i		−	1.92543i	−0.270511	−	0.962717i	$-0.587193\pi$
$71$		−	136.706i		−	1.92543i	0.270511	−	0.962717i	$-0.412807\pi$
$72$		0			0
$73$	−74.6196			−1.02219			−0.511093	−	0.859526i	$-0.670759\pi$
$73$	−74.6196			−1.02219			−0.511093	+	0.859526i	$0.670759\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			4.92328i			0.0639387i
$78$		0			0
$79$	79.6417			1.00812			0.504062	−	0.863668i	$-0.331838\pi$
$79$	79.6417			1.00812			0.504062	+	0.863668i	$0.331838\pi$
$80$		0			0
$81$	−46.1971	+	66.5344i	−0.570335	+	0.821412i
$82$		0			0
$83$			0.0506471i			0.000610206i	1.00000		0.000305103i	$9.71174e-5\pi$
$83$			0.0506471i			0.000610206i	−1.00000		0.000305103i	$0.999903\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−63.5592	+	104.976i	−0.730566	+	1.20662i
$88$		0			0
$89$			22.6492i			0.254485i	0.991872	+	0.127243i	$0.0406126\pi$
$89$			22.6492i			0.254485i	−0.991872	+	0.127243i	$0.959387\pi$
$90$		0			0
$91$	−9.86173			−0.108371
$92$		0			0
$93$	−25.5882	−	15.4927i	−0.275142	−	0.166589i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	35.0428			0.361266			0.180633	−	0.983551i	$-0.442185\pi$
$97$	35.0428			0.361266			0.180633	+	0.983551i	$0.442185\pi$
$98$		0			0
$99$	29.2347	−	15.2921i	0.295300	−	0.154466i
$100$		0			0
$101$		−	10.2711i		−	0.101694i	−0.998706	−	0.0508472i	$-0.983808\pi$
$101$		−	10.2711i		−	0.101694i	0.998706	−	0.0508472i	$-0.0161921\pi$
$102$		0			0
$103$	108.963			1.05789			0.528944	−	0.848656i	$-0.322588\pi$
$103$	108.963			1.05789			0.528944	+	0.848656i	$0.322588\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	113.251i		−	1.05842i	−0.848492	−	0.529209i	$-0.822489\pi$
$107$		−	113.251i		−	1.05842i	0.848492	−	0.529209i	$-0.177511\pi$
$108$		0			0
$109$	−105.722			−0.969926			−0.484963	−	0.874535i	$-0.661167\pi$
$109$	−105.722			−0.969926			−0.484963	+	0.874535i	$0.661167\pi$
$110$		0			0
$111$	101.171	+	61.2552i	0.911448	+	0.551849i
$112$		0			0
$113$			168.318i			1.48954i	0.667321	+	0.744770i	$0.267441\pi$
$113$			168.318i			1.48954i	−0.667321	+	0.744770i	$0.732559\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	30.6314	+	58.5595i	0.261807	+	0.500509i
$118$		0			0
$119$		−	8.47558i		−	0.0712234i
$120$		0			0
$121$	107.562			0.888938
$122$		0			0
$123$	105.921	−	174.941i	0.861144	−	1.42229i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−97.8159			−0.770204			−0.385102	−	0.922874i	$-0.625834\pi$
$127$	−97.8159			−0.770204			−0.385102	+	0.922874i	$0.625834\pi$
$128$		0			0
$129$	61.7964	+	37.4155i	0.479042	+	0.290043i
$130$		0			0
$131$		−	88.9361i		−	0.678902i	−0.940624	−	0.339451i	$-0.889759\pi$
$131$		−	88.9361i		−	0.678902i	0.940624	−	0.339451i	$-0.110241\pi$
$132$		0			0
$133$	−41.2850			−0.310413
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	241.635i		−	1.76376i	−0.471474	−	0.881880i	$-0.656278\pi$
$137$		−	241.635i		−	1.76376i	0.471474	−	0.881880i	$-0.343722\pi$
$138$		0			0
$139$	215.214			1.54830			0.774149	−	0.633003i	$-0.218178\pi$
$139$	215.214			1.54830			0.774149	+	0.633003i	$0.218178\pi$
$140$		0			0
$141$	−123.946	+	204.713i	−0.879051	+	1.45186i
$142$		0			0
$143$		−	26.9184i		−	0.188241i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−121.119	−	73.3329i	−0.823936	−	0.498864i
$148$		0			0
$149$		−	104.338i		−	0.700257i	−0.936702	−	0.350128i	$-0.886138\pi$
$149$		−	104.338i		−	0.700257i	0.936702	−	0.350128i	$-0.113862\pi$
$150$		0			0
$151$	92.1895			0.610526			0.305263	−	0.952268i	$-0.401256\pi$
$151$	92.1895			0.610526			0.305263	+	0.952268i	$0.401256\pi$
$152$		0			0
$153$	−50.3285	+	26.3259i	−0.328945	+	0.172065i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−250.985			−1.59863			−0.799314	−	0.600913i	$-0.794804\pi$
$157$	−250.985			−1.59863			−0.799314	+	0.600913i	$0.794804\pi$
$158$		0			0
$159$	−60.5443	+	99.9966i	−0.380782	+	0.628909i
$160$		0			0
$161$		−	35.2441i		−	0.218907i
$162$		0			0
$163$	87.6008			0.537428			0.268714	−	0.963220i	$-0.413401\pi$
$163$	87.6008			0.537428			0.268714	+	0.963220i	$0.413401\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	116.164i		−	0.695590i	−0.937571	−	0.347795i	$-0.886931\pi$
$167$		−	116.164i		−	0.695590i	0.937571	−	0.347795i	$-0.113069\pi$
$168$		0			0
$169$	−115.080			−0.680948
$170$		0			0
$171$	128.235	+	245.153i	0.749911	+	1.43364i
$172$		0			0
$173$			76.7440i			0.443607i	0.975091	+	0.221804i	$0.0711944\pi$
$173$			76.7440i			0.443607i	−0.975091	+	0.221804i	$0.928806\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	62.0967	−	102.560i	0.350829	−	0.579438i
$178$		0			0
$179$		−	281.899i		−	1.57486i	−0.616406	−	0.787428i	$-0.711412\pi$
$179$		−	281.899i		−	1.57486i	0.616406	−	0.787428i	$-0.288588\pi$
$180$		0			0
$181$	−37.8892			−0.209333			−0.104666	−	0.994507i	$-0.533377\pi$
$181$	−37.8892			−0.209333			−0.104666	+	0.994507i	$0.533377\pi$
$182$		0			0
$183$	−164.877	−	99.8268i	−0.900965	−	0.545502i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	23.1348			0.123716
$188$		0			0
$189$	2.26434	−	36.1905i	0.0119806	−	0.191484i
$190$		0			0
$191$		−	62.1819i		−	0.325560i	−0.986662	−	0.162780i	$-0.947954\pi$
$191$		−	62.1819i		−	0.325560i	0.986662	−	0.162780i	$-0.0520460\pi$
$192$		0			0
$193$	−263.150			−1.36347			−0.681737	−	0.731597i	$-0.738775\pi$
$193$	−263.150			−1.36347			−0.681737	+	0.731597i	$0.738775\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	180.658i		−	0.917044i	−0.888683	−	0.458522i	$-0.848379\pi$
$197$		−	180.658i		−	0.917044i	0.888683	−	0.458522i	$-0.151621\pi$
$198$		0			0
$199$	−136.985			−0.688365			−0.344183	−	0.938903i	$-0.611844\pi$
$199$	−136.985			−0.688365			−0.344183	+	0.938903i	$0.611844\pi$
$200$		0			0
$201$	138.439	+	83.8195i	0.688749	+	0.417013i
$202$		0			0
$203$		−	54.9372i		−	0.270627i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−209.282	+	109.471i	−1.01102	+	0.528847i
$208$		0			0
$209$		−	112.691i		−	0.539190i
$210$		0			0
$211$	−139.434			−0.660822			−0.330411	−	0.943837i	$-0.607187\pi$
$211$	−139.434			−0.660822			−0.330411	+	0.943837i	$0.607187\pi$
$212$		0			0
$213$	212.411	−	350.824i	0.997237	−	1.64706i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	13.3911			0.0617102
$218$		0			0
$219$	−191.494	−	115.943i	−0.874402	−	0.529419i
$220$		0			0
$221$			46.3409i			0.209687i
$222$		0			0
$223$	−4.94040			−0.0221543			−0.0110771	−	0.999939i	$-0.503526\pi$
$223$	−4.94040			−0.0221543			−0.0110771	+	0.999939i	$0.503526\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	334.372i		−	1.47301i	−0.676434	−	0.736503i	$-0.736476\pi$
$227$		−	334.372i		−	1.47301i	0.676434	−	0.736503i	$-0.263524\pi$
$228$		0			0
$229$	−337.101			−1.47206			−0.736028	−	0.676951i	$-0.763301\pi$
$229$	−337.101			−1.47206			−0.736028	+	0.676951i	$0.763301\pi$
$230$		0			0
$231$	−7.64971	+	12.6345i	−0.0331156	+	0.0546947i
$232$		0			0
$233$		−	113.099i		−	0.485402i	−0.970101	−	0.242701i	$-0.921967\pi$
$233$		−	113.099i		−	0.485402i	0.970101	−	0.242701i	$-0.0780334\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	204.382	+	123.746i	0.862373	+	0.522135i
$238$		0			0
$239$		−	130.740i		−	0.547029i	−0.961868	−	0.273514i	$-0.911814\pi$
$239$		−	130.740i		−	0.547029i	0.961868	−	0.273514i	$-0.0881862\pi$
$240$		0			0
$241$	336.727			1.39721			0.698604	−	0.715508i	$-0.253805\pi$
$241$	336.727			1.39721			0.698604	+	0.715508i	$0.253805\pi$
$242$		0			0
$243$	−221.934	+	98.9650i	−0.913311	+	0.407264i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	225.729			0.913882
$248$		0			0
$249$	−0.0786948	+	0.129974i	−0.000316043	+	0.000521985i
$250$		0			0
$251$			135.453i			0.539655i	0.962909	+	0.269828i	$0.0869667\pi$
$251$			135.453i			0.539655i	−0.962909	+	0.269828i	$0.913033\pi$
$252$		0			0
$253$	96.2017			0.380244
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	104.440i		−	0.406379i	−0.979139	−	0.203190i	$-0.934869\pi$
$257$		−	104.440i		−	0.406379i	0.979139	−	0.203190i	$-0.0651308\pi$
$258$		0			0
$259$	−52.9457			−0.204424
$260$		0			0
$261$	−326.220	+	170.640i	−1.24989	+	0.653792i
$262$		0			0
$263$			326.365i			1.24093i	0.784234	+	0.620466i	$0.213056\pi$
$263$			326.365i			1.24093i	−0.784234	+	0.620466i	$0.786944\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−35.1919	+	58.1239i	−0.131805	+	0.217693i
$268$		0			0
$269$		−	56.9353i		−	0.211656i	−0.994384	−	0.105828i	$-0.966251\pi$
$269$		−	56.9353i		−	0.211656i	0.994384	−	0.105828i	$-0.0337492\pi$
$270$		0			0
$271$	−110.853			−0.409053			−0.204526	−	0.978861i	$-0.565565\pi$
$271$	−110.853			−0.409053			−0.204526	+	0.978861i	$0.565565\pi$
$272$		0			0
$273$	−25.3079	−	15.3230i	−0.0927028	−	0.0561282i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−492.842			−1.77921			−0.889606	−	0.456728i	$-0.849021\pi$
$277$	−492.842			−1.77921			−0.889606	+	0.456728i	$0.849021\pi$
$278$		0			0
$279$	−41.5940	−	79.5172i	−0.149082	−	0.285008i
$280$		0			0
$281$		−	428.705i		−	1.52564i	−0.646611	−	0.762820i	$-0.723814\pi$
$281$		−	428.705i		−	1.52564i	0.646611	−	0.762820i	$-0.276186\pi$
$282$		0			0
$283$	−289.871			−1.02428			−0.512139	−	0.858903i	$-0.671147\pi$
$283$	−289.871			−1.02428			−0.512139	+	0.858903i	$0.671147\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			91.5522i			0.318997i
$288$		0			0
$289$	249.173			0.862189
$290$		0			0
$291$	89.9293	+	54.4490i	0.309035	+	0.187110i
$292$		0			0
$293$		−	407.033i		−	1.38919i	−0.719400	−	0.694596i	$-0.755583\pi$
$293$		−	407.033i		−	1.38919i	0.719400	−	0.694596i	$-0.244417\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	98.7849	+	6.18070i	0.332609	+	0.0208104i
$298$		0			0
$299$			192.700i			0.644481i
$300$		0			0
$301$	−32.3400			−0.107442
$302$		0			0
$303$	15.9591	−	26.3585i	0.0526704	−	0.0869918i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	284.594			0.927016			0.463508	−	0.886093i	$-0.346590\pi$
$307$	284.594			0.927016			0.463508	+	0.886093i	$0.346590\pi$
$308$		0			0
$309$	279.628	+	169.304i	0.904944	+	0.547911i
$310$		0			0
$311$			32.0008i			0.102896i	0.998676	+	0.0514482i	$0.0163837\pi$
$311$			32.0008i			0.102896i	−0.998676	+	0.0514482i	$0.983616\pi$
$312$		0			0
$313$	142.954			0.456722			0.228361	−	0.973576i	$-0.426663\pi$
$313$	142.954			0.456722			0.228361	+	0.973576i	$0.426663\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			248.052i			0.782500i	0.920284	+	0.391250i	$0.127957\pi$
$317$			248.052i			0.782500i	−0.920284	+	0.391250i	$0.872043\pi$
$318$		0			0
$319$	149.956			0.470080
$320$		0			0
$321$	175.967	−	290.632i	0.548184	−	0.905395i
$322$		0			0
$323$			194.001i			0.600622i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−271.311	−	164.269i	−0.829698	−	0.502352i
$328$		0			0
$329$		−	107.132i		−	0.325630i
$330$		0			0
$331$	−377.660			−1.14097			−0.570484	−	0.821309i	$-0.693244\pi$
$331$	−377.660			−1.14097			−0.570484	+	0.821309i	$0.693244\pi$
$332$		0			0
$333$	164.454	+	314.395i	0.493856	+	0.944129i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	30.2254			0.0896896			0.0448448	−	0.998994i	$-0.485721\pi$
$337$	30.2254			0.0896896			0.0448448	+	0.998994i	$0.485721\pi$
$338$		0			0
$339$	−261.530	+	431.950i	−0.771475	+	1.27419i
$340$		0			0
$341$			36.5521i			0.107191i
$342$		0			0
$343$	129.193			0.376655
$344$		0			0
$345$		0			0
$346$		0			0
$347$			424.938i			1.22460i	0.790624	+	0.612302i	$0.209756\pi$
$347$			424.938i			1.22460i	−0.790624	+	0.612302i	$0.790244\pi$
$348$		0			0
$349$	263.069			0.753779			0.376889	−	0.926258i	$-0.376994\pi$
$349$	263.069			0.753779			0.376889	+	0.926258i	$0.376994\pi$
$350$		0			0
$351$	−12.3804	+	197.874i	−0.0352719	+	0.563744i
$352$		0			0
$353$			352.772i			0.999353i	0.866212	+	0.499677i	$0.166548\pi$
$353$			352.772i			0.999353i	−0.866212	+	0.499677i	$0.833452\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	13.1692	−	21.7506i	0.0368886	−	0.0609262i
$358$		0			0
$359$			421.619i			1.17443i	0.809432	+	0.587214i	$0.199775\pi$
$359$			421.619i			1.17443i	−0.809432	+	0.587214i	$0.800225\pi$
$360$		0			0
$361$	583.987			1.61769
$362$		0			0
$363$	276.032	+	167.127i	0.760419	+	0.460406i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	605.435			1.64969			0.824844	−	0.565361i	$-0.191263\pi$
$367$	605.435			1.64969			0.824844	+	0.565361i	$0.191263\pi$
$368$		0			0
$369$	543.643	−	284.369i	1.47329	−	0.770649i
$370$		0			0
$371$		−	52.3313i		−	0.141055i
$372$		0			0
$373$	691.549			1.85402			0.927009	−	0.375040i	$-0.122371\pi$
$373$	691.549			1.85402			0.927009	+	0.375040i	$0.122371\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			300.373i			0.796747i
$378$		0			0
$379$	179.894			0.474655			0.237327	−	0.971430i	$-0.423729\pi$
$379$	179.894			0.474655			0.237327	+	0.971430i	$0.423729\pi$
$380$		0			0
$381$	−251.022	−	151.985i	−0.658851	−	0.398910i
$382$		0			0
$383$		−	137.087i		−	0.357929i	−0.983856	−	0.178965i	$-0.942725\pi$
$383$		−	137.087i		−	0.357929i	0.983856	−	0.178965i	$-0.0572747\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	100.451	+	192.037i	0.259563	+	0.496219i
$388$		0			0
$389$			682.795i			1.75526i	0.479341	+	0.877629i	$0.340876\pi$
$389$			682.795i			1.75526i	−0.479341	+	0.877629i	$0.659124\pi$
$390$		0			0
$391$	−165.614			−0.423566
$392$		0			0
$393$	138.188	−	228.234i	0.351622	−	0.580749i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−538.381			−1.35612			−0.678062	−	0.735005i	$-0.737180\pi$
$397$	−538.381			−1.35612			−0.678062	+	0.735005i	$0.737180\pi$
$398$		0			0
$399$	−105.948	−	64.1479i	−0.265535	−	0.160772i
$400$		0			0
$401$		−	113.772i		−	0.283720i	−0.989887	−	0.141860i	$-0.954692\pi$
$401$		−	113.772i		−	0.283720i	0.989887	−	0.141860i	$-0.0453083\pi$
$402$		0			0
$403$	−73.2169			−0.181680
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	144.520i		−	0.355086i
$408$		0			0
$409$	438.806			1.07288			0.536438	−	0.843940i	$-0.319770\pi$
$409$	438.806			1.07288			0.536438	+	0.843940i	$0.319770\pi$
$410$		0			0
$411$	375.449	−	620.101i	0.913501	−	1.50876i
$412$		0			0
$413$			53.6730i			0.129959i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	552.296	+	334.396i	1.32445	+	0.801908i
$418$		0			0
$419$			544.982i			1.30067i	0.759646	+	0.650337i	$0.225372\pi$
$419$			544.982i			1.30067i	−0.759646	+	0.650337i	$0.774628\pi$
$420$		0			0
$421$	−557.806			−1.32495			−0.662477	−	0.749082i	$-0.730495\pi$
$421$	−557.806			−1.32495			−0.662477	+	0.749082i	$0.730495\pi$
$422$		0			0
$423$	−636.159	+	332.763i	−1.50392	+	0.786673i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	86.2850			0.202073
$428$		0			0
$429$	41.8254	−	69.0800i	0.0974952	−	0.161026i
$430$		0			0
$431$		−	41.0366i		−	0.0952124i	−0.998866	−	0.0476062i	$-0.984841\pi$
$431$		−	41.0366i		−	0.0952124i	0.998866	−	0.0476062i	$-0.0151593\pi$
$432$		0			0
$433$	−589.338			−1.36106			−0.680529	−	0.732721i	$-0.738250\pi$
$433$	−589.338			−1.36106			−0.680529	+	0.732721i	$0.738250\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			806.715i			1.84603i
$438$		0			0
$439$	−363.125			−0.827165			−0.413582	−	0.910467i	$-0.635723\pi$
$439$	−363.125			−0.827165			−0.413582	+	0.910467i	$0.635723\pi$
$440$		0			0
$441$	−196.880	−	376.385i	−0.446439	−	0.853480i
$442$		0			0
$443$			142.457i			0.321573i	0.986989	+	0.160786i	$0.0514030\pi$
$443$			142.457i			0.321573i	−0.986989	+	0.160786i	$0.948597\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	162.119	−	267.760i	0.362683	−	0.599016i
$448$		0			0
$449$		−	613.747i		−	1.36692i	−0.729988	−	0.683460i	$-0.760474\pi$
$449$		−	613.747i		−	1.36692i	0.729988	−	0.683460i	$-0.239526\pi$
$450$		0			0
$451$	−249.900			−0.554101
$452$		0			0
$453$	236.583	+	143.243i	0.522259	+	0.316209i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	66.7594			0.146082			0.0730409	−	0.997329i	$-0.476730\pi$
$457$	66.7594			0.146082			0.0730409	+	0.997329i	$0.476730\pi$
$458$		0			0
$459$	−170.061	−	10.6403i	−0.370504	−	0.0231814i
$460$		0			0
$461$			891.544i			1.93393i	0.254900	+	0.966967i	$0.417957\pi$
$461$			891.544i			1.93393i	−0.254900	+	0.966967i	$0.582043\pi$
$462$		0			0
$463$	381.895			0.824826			0.412413	−	0.910997i	$-0.364686\pi$
$463$	381.895			0.824826			0.412413	+	0.910997i	$0.364686\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			546.640i			1.17054i	0.810840	+	0.585268i	$0.199011\pi$
$467$			546.640i			1.17054i	−0.810840	+	0.585268i	$0.800989\pi$
$468$		0			0
$469$	−72.4491			−0.154476
$470$		0			0
$471$	−644.095	−	389.976i	−1.36751	−	0.827975i
$472$		0			0
$473$		−	88.2746i		−	0.186627i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−310.746	+	162.546i	−0.651460	+	0.340766i
$478$		0			0
$479$			227.312i			0.474555i	0.971442	+	0.237277i	$0.0762550\pi$
$479$			227.312i			0.474555i	−0.971442	+	0.237277i	$0.923745\pi$
$480$		0			0
$481$	289.485			0.601840
$482$		0			0
$483$	54.7617	−	90.4459i	0.113378	−	0.187259i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	896.798			1.84147			0.920737	−	0.390185i	$-0.127589\pi$
$487$	896.798			1.84147			0.920737	+	0.390185i	$0.127589\pi$
$488$		0			0
$489$	224.808	+	136.113i	0.459729	+	0.278349i
$490$		0			0
$491$		−	368.073i		−	0.749639i	−0.927098	−	0.374819i	$-0.877705\pi$
$491$		−	368.073i		−	0.749639i	0.927098	−	0.374819i	$-0.122295\pi$
$492$		0			0
$493$	−258.154			−0.523638
$494$		0			0
$495$		0			0
$496$		0			0
$497$			183.597i			0.369411i
$498$		0			0
$499$	−471.446			−0.944782			−0.472391	−	0.881389i	$-0.656609\pi$
$499$	−471.446			−0.944782			−0.472391	+	0.881389i	$0.656609\pi$
$500$		0			0
$501$	180.493	−	298.107i	0.360266	−	0.595024i
$502$		0			0
$503$		−	601.673i		−	1.19617i	−0.801433	−	0.598085i	$-0.795929\pi$
$503$		−	601.673i		−	1.19617i	0.801433	−	0.598085i	$-0.204071\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−295.327	−	178.810i	−0.582499	−	0.352682i
$508$		0			0
$509$		−	471.175i		−	0.925687i	−0.886440	−	0.462844i	$-0.846829\pi$
$509$		−	471.175i		−	0.925687i	0.886440	−	0.462844i	$-0.153171\pi$
$510$		0			0
$511$	100.215			0.196115
$512$		0			0
$513$	−51.8293	+	828.377i	−0.101032	+	1.61477i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	292.427			0.565623
$518$		0			0
$519$	−119.244	+	196.946i	−0.229757	+	0.379472i
$520$		0			0
$521$		−	324.027i		−	0.621932i	−0.950421	−	0.310966i	$-0.899347\pi$
$521$		−	324.027i		−	0.621932i	0.950421	−	0.310966i	$-0.100653\pi$
$522$		0			0
$523$	−88.5290			−0.169272			−0.0846358	−	0.996412i	$-0.526973\pi$
$523$	−88.5290			−0.169272			−0.0846358	+	0.996412i	$0.526973\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	62.9257i		−	0.119404i
$528$		0			0
$529$	−159.675			−0.301844
$530$		0			0
$531$	318.714	−	166.713i	0.600214	−	0.313961i
$532$		0			0
$533$		−	500.569i		−	0.939154i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	438.011	−	723.431i	0.815663	−	1.34717i
$538$		0			0
$539$			173.015i			0.320992i
$540$		0			0
$541$	−566.555			−1.04724			−0.523619	−	0.851953i	$-0.675418\pi$
$541$	−566.555			−1.04724			−0.523619	+	0.851953i	$0.675418\pi$
$542$		0			0
$543$	−97.2341	−	58.8717i	−0.179068	−	0.108419i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	8.15550			0.0149095			0.00745475	−	0.999972i	$-0.497627\pi$
$547$	8.15550			0.0149095			0.00745475	+	0.999972i	$0.497627\pi$
$548$		0			0
$549$	−268.009	−	512.366i	−0.488176	−	0.933271i
$550$		0			0
$551$			1257.48i			2.28217i
$552$		0			0
$553$	−106.960			−0.193417
$554$		0			0
$555$		0			0
$556$		0			0
$557$			403.842i			0.725031i	0.931978	+	0.362516i	$0.118082\pi$
$557$			403.842i			0.725031i	−0.931978	+	0.362516i	$0.881918\pi$
$558$		0			0
$559$	176.821			0.316317
$560$		0			0
$561$	59.3702	+	35.9465i	0.105829	+	0.0640758i
$562$		0			0
$563$		−	319.709i		−	0.567867i	−0.958844	−	0.283933i	$-0.908361\pi$
$563$		−	319.709i		−	0.567867i	0.958844	−	0.283933i	$-0.0916395\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	62.0431	−	89.3563i	0.109423	−	0.157595i
$568$		0			0
$569$			705.574i			1.24003i	0.784592	+	0.620013i	$0.212873\pi$
$569$			705.574i			1.24003i	−0.784592	+	0.620013i	$0.787127\pi$
$570$		0			0
$571$	−13.6020			−0.0238214			−0.0119107	−	0.999929i	$-0.503791\pi$
$571$	−13.6020			−0.0238214			−0.0119107	+	0.999929i	$0.503791\pi$
$572$		0			0
$573$	96.6173	−	159.576i	0.168617	−	0.278492i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−426.416			−0.739023			−0.369512	−	0.929226i	$-0.620475\pi$
$577$	−426.416			−0.739023			−0.369512	+	0.929226i	$0.620475\pi$
$578$		0			0
$579$	−675.316	−	408.879i	−1.16635	−	0.706182i
$580$		0			0
$581$		−	0.0680196i		−	0.000117073i
$582$		0			0
$583$	142.843			0.245013
$584$		0			0
$585$		0			0
$586$		0			0
$587$			178.246i			0.303656i	0.988407	+	0.151828i	$0.0485160\pi$
$587$			178.246i			0.303656i	−0.988407	+	0.151828i	$0.951484\pi$
$588$		0			0
$589$	−306.514			−0.520397
$590$		0			0
$591$	280.703	−	463.617i	0.474963	−	0.784461i
$592$		0			0
$593$			122.209i			0.206086i	0.994677	+	0.103043i	$0.0328579\pi$
$593$			122.209i			0.206086i	−0.994677	+	0.103043i	$0.967142\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−351.540	−	212.845i	−0.588844	−	0.356524i
$598$		0			0
$599$		−	874.960i		−	1.46070i	−0.683072	−	0.730351i	$-0.739357\pi$
$599$		−	874.960i		−	1.46070i	0.683072	−	0.730351i	$-0.260643\pi$
$600$		0			0
$601$	976.356			1.62455			0.812276	−	0.583273i	$-0.198228\pi$
$601$	976.356			1.62455			0.812276	+	0.583273i	$0.198228\pi$
$602$		0			0
$603$	225.033	+	430.207i	0.373190	+	0.713445i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−728.093			−1.19949			−0.599747	−	0.800189i	$-0.704732\pi$
$607$	−728.093			−1.19949			−0.599747	+	0.800189i	$0.704732\pi$
$608$		0			0
$609$	85.3606	−	140.984i	0.140165	−	0.231500i
$610$		0			0
$611$			585.755i			0.958683i
$612$		0			0
$613$	368.041			0.600394			0.300197	−	0.953877i	$-0.402948\pi$
$613$	368.041			0.600394			0.300197	+	0.953877i	$0.402948\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	550.637i		−	0.892442i	−0.894923	−	0.446221i	$-0.852769\pi$
$617$		−	550.637i		−	0.892442i	0.894923	−	0.446221i	$-0.147231\pi$
$618$		0			0
$619$	757.081			1.22307			0.611536	−	0.791217i	$-0.290552\pi$
$619$	757.081			1.22307			0.611536	+	0.791217i	$0.290552\pi$
$620$		0			0
$621$	−707.168	−	44.2455i	−1.13876	−	0.0712488i
$622$		0			0
$623$		−	30.4180i		−	0.0488251i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	175.097	−	289.195i	0.279262	−	0.461236i
$628$		0			0
$629$			248.796i			0.395541i
$630$		0			0
$631$	101.694			0.161164			0.0805820	−	0.996748i	$-0.474322\pi$
$631$	101.694			0.161164			0.0805820	+	0.996748i	$0.474322\pi$
$632$		0			0
$633$	−357.824	−	216.650i	−0.565283	−	0.342259i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−346.563			−0.544055
$638$		0			0
$639$	1090.21	−	570.269i	1.70612	−	0.892439i
$640$		0			0
$641$			1008.28i			1.57298i	0.617601	+	0.786491i	$0.288105\pi$
$641$			1008.28i			1.57298i	−0.617601	+	0.786491i	$0.711895\pi$
$642$		0			0
$643$	−1126.32			−1.75167			−0.875833	−	0.482615i	$-0.839687\pi$
$643$	−1126.32			−1.75167			−0.875833	+	0.482615i	$0.839687\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	410.296i		−	0.634152i	−0.948400	−	0.317076i	$-0.897299\pi$
$647$		−	410.296i		−	0.634152i	0.948400	−	0.317076i	$-0.102701\pi$
$648$		0			0
$649$	−146.505			−0.225740
$650$		0			0
$651$	34.3652	+	20.8069i	0.0527884	+	0.0319614i
$652$		0			0
$653$		−	393.103i		−	0.601996i	−0.953625	−	0.300998i	$-0.902680\pi$
$653$		−	393.103i		−	0.601996i	0.953625	−	0.300998i	$-0.0973197\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−311.276	−	595.081i	−0.473784	−	0.905755i
$658$		0			0
$659$			882.684i			1.33943i	0.742618	+	0.669715i	$0.233584\pi$
$659$			882.684i			1.33943i	−0.742618	+	0.669715i	$0.766416\pi$
$660$		0			0
$661$	260.491			0.394087			0.197043	−	0.980395i	$-0.436866\pi$
$661$	260.491			0.394087			0.197043	+	0.980395i	$0.436866\pi$
$662$		0			0
$663$	−72.0038	+	118.923i	−0.108603	+	0.179372i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−1073.48			−1.60942
$668$		0			0
$669$	−12.6784	−	7.67632i	−0.0189513	−	0.0114743i
$670$		0			0
$671$			235.522i			0.351002i
$672$		0			0
$673$	−69.6242			−0.103454			−0.0517268	−	0.998661i	$-0.516472\pi$
$673$	−69.6242			−0.103454			−0.0517268	+	0.998661i	$0.516472\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	1148.22i		−	1.69604i	−0.529967	−	0.848019i	$-0.677796\pi$
$677$		−	1148.22i		−	1.69604i	0.529967	−	0.848019i	$-0.322204\pi$
$678$		0			0
$679$	−47.0628			−0.0693119
$680$		0			0
$681$	519.543	−	858.091i	0.762912	−	1.26004i
$682$		0			0
$683$		−	205.570i		−	0.300981i	−0.988611	−	0.150490i	$-0.951915\pi$
$683$		−	205.570i		−	0.300981i	0.988611	−	0.150490i	$-0.0480852\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−865.092	−	523.782i	−1.25923	−	0.762420i
$688$		0			0
$689$			286.125i			0.415276i
$690$		0			0
$691$	−45.2341			−0.0654618			−0.0327309	−	0.999464i	$-0.510420\pi$
$691$	−45.2341			−0.0654618			−0.0327309	+	0.999464i	$0.510420\pi$
$692$		0			0
$693$	−39.2625	+	20.5375i	−0.0566558	+	0.0296356i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	430.210			0.617231
$698$		0			0
$699$	175.731	−	290.242i	0.251404	−	0.415225i
$700$		0			0
$701$			743.766i			1.06101i	0.847683	+	0.530504i	$0.177997\pi$
$701$			743.766i			1.06101i	−0.847683	+	0.530504i	$0.822003\pi$
$702$		0			0
$703$	1211.90			1.72389
$704$		0			0
$705$		0			0
$706$		0			0
$707$			13.7942i			0.0195109i
$708$		0			0
$709$	−283.784			−0.400259			−0.200130	−	0.979769i	$-0.564136\pi$
$709$	−283.784			−0.400259			−0.200130	+	0.979769i	$0.564136\pi$
$710$		0			0
$711$	332.226	+	635.132i	0.467266	+	0.893294i
$712$		0			0
$713$		−	261.665i		−	0.366991i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	203.142	−	335.514i	0.283322	−	0.467942i
$718$		0			0
$719$		−	1088.15i		−	1.51343i	−0.653747	−	0.756713i	$-0.726804\pi$
$719$		−	1088.15i		−	1.51343i	0.653747	−	0.756713i	$-0.273196\pi$
$720$		0			0
$721$	−146.338			−0.202965
$722$		0			0
$723$	864.134	+	523.202i	1.19521	+	0.723654i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−248.930			−0.342407			−0.171203	−	0.985236i	$-0.554766\pi$
$727$	−248.930			−0.342407			−0.171203	+	0.985236i	$0.554766\pi$
$728$		0			0
$729$	−723.315	−	90.8673i	−0.992201	−	0.124646i
$730$		0			0
$731$			151.968i			0.207890i
$732$		0			0
$733$	235.310			0.321023			0.160512	−	0.987034i	$-0.448686\pi$
$733$	235.310			0.321023			0.160512	+	0.987034i	$0.448686\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	197.756i		−	0.268326i
$738$		0			0
$739$	356.976			0.483053			0.241527	−	0.970394i	$-0.422352\pi$
$739$	356.976			0.483053			0.241527	+	0.970394i	$0.422352\pi$
$740$		0			0
$741$	579.282	+	350.734i	0.781756	+	0.473325i
$742$		0			0
$743$		−	230.874i		−	0.310732i	−0.987857	−	0.155366i	$-0.950344\pi$
$743$		−	230.874i		−	0.310732i	0.987857	−	0.155366i	$-0.0496557\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−0.403904	+	0.211275i	−0.000540702	+	0.000282831i
$748$		0			0
$749$			152.097i			0.203066i
$750$		0			0
$751$	620.963			0.826848			0.413424	−	0.910539i	$-0.364333\pi$
$751$	620.963			0.826848			0.413424	+	0.910539i	$0.364333\pi$
$752$		0			0
$753$	−210.466	+	347.610i	−0.279503	+	0.461634i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	28.7800			0.0380184			0.0190092	−	0.999819i	$-0.493949\pi$
$757$	28.7800			0.0380184			0.0190092	+	0.999819i	$0.493949\pi$
$758$		0			0
$759$	246.880	+	149.477i	0.325270	+	0.196939i
$760$		0			0
$761$			439.052i			0.576940i	0.957489	+	0.288470i	$0.0931466\pi$
$761$			439.052i			0.576940i	−0.957489	+	0.288470i	$0.906853\pi$
$762$		0			0
$763$	141.986			0.186088
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	293.462i		−	0.382610i
$768$		0			0
$769$	−354.830			−0.461417			−0.230709	−	0.973023i	$-0.574104\pi$
$769$	−354.830			−0.461417			−0.230709	+	0.973023i	$0.574104\pi$
$770$		0			0
$771$	162.277	−	268.020i	0.210475	−	0.347627i
$772$		0			0
$773$		−	954.595i		−	1.23492i	−0.786601	−	0.617461i	$-0.788161\pi$
$773$		−	954.595i		−	1.23492i	0.786601	−	0.617461i	$-0.211839\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−135.873	−	82.2663i	−0.174869	−	0.105877i
$778$		0			0
$779$		−	2095.57i		−	2.69008i
$780$		0			0
$781$	−501.144			−0.641669
$782$		0			0
$783$	−1102.31	−	68.9683i	−1.40780	−	0.0880822i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−679.060			−0.862846			−0.431423	−	0.902150i	$-0.641988\pi$
$787$	−679.060			−0.862846			−0.431423	+	0.902150i	$0.641988\pi$
$788$		0			0
$789$	−507.101	+	837.541i	−0.642713	+	1.06152i
$790$		0			0
$791$		−	226.053i		−	0.285781i
$792$		0			0
$793$	−471.770			−0.594918
$794$		0			0
$795$		0			0
$796$		0			0
$797$			284.382i			0.356815i	0.983957	+	0.178408i	$0.0570946\pi$
$797$			284.382i			0.356815i	−0.983957	+	0.178408i	$0.942905\pi$
$798$		0			0
$799$	−503.422			−0.630066
$800$		0			0
$801$	−180.624	+	94.4811i	−0.225498	+	0.117954i
$802$		0			0
$803$			273.545i			0.340653i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	88.4653	−	146.112i	0.109622	−	0.181055i
$808$		0			0
$809$		−	33.2847i		−	0.0411431i	−0.999788	−	0.0205715i	$-0.993451\pi$
$809$		−	33.2847i		−	0.0411431i	0.999788	−	0.0205715i	$-0.00654859\pi$
$810$		0			0
$811$	930.829			1.14775			0.573877	−	0.818941i	$-0.305439\pi$
$811$	930.829			1.14775			0.573877	+	0.818941i	$0.305439\pi$
$812$		0			0
$813$	−284.480	−	172.242i	−0.349914	−	0.211860i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	740.241			0.906048
$818$		0			0
$819$	−41.1382	−	78.6460i	−0.0502298	−	0.0960268i
$820$		0			0
$821$		−	192.628i		−	0.234626i	−0.993095	−	0.117313i	$-0.962572\pi$
$821$		−	192.628i		−	0.234626i	0.993095	−	0.117313i	$-0.0374281\pi$
$822$		0			0
$823$	−326.743			−0.397015			−0.198507	−	0.980099i	$-0.563609\pi$
$823$	−326.743			−0.397015			−0.198507	+	0.980099i	$0.563609\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			225.268i			0.272392i	0.990682	+	0.136196i	$0.0434876\pi$
$827$			225.268i			0.272392i	−0.990682	+	0.136196i	$0.956512\pi$
$828$		0			0
$829$	−875.339			−1.05590			−0.527949	−	0.849276i	$-0.677039\pi$
$829$	−875.339			−1.05590			−0.527949	+	0.849276i	$0.677039\pi$
$830$		0			0
$831$	−1264.77	−	765.770i	−1.52198	−	0.921505i
$832$		0			0
$833$		−	297.851i		−	0.357564i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	16.8112	−	268.691i	0.0200851	−	0.321017i
$838$		0			0
$839$			1614.70i			1.92456i	0.272064	+	0.962279i	$0.412294\pi$
$839$			1614.70i			1.92456i	−0.272064	+	0.962279i	$0.587706\pi$
$840$		0			0
$841$	−832.305			−0.989661
$842$		0			0
$843$	666.115	−	1100.17i	0.790172	−	1.30507i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−144.456			−0.170550
$848$		0			0
$849$	−743.887	−	450.396i	−0.876191	−	0.530502i
$850$		0			0
$851$			1034.57i			1.21571i
$852$		0			0
$853$	−1133.27			−1.32856			−0.664282	−	0.747482i	$-0.731263\pi$
$853$	−1133.27			−1.32856			−0.664282	+	0.747482i	$0.731263\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	800.665i		−	0.934264i	−0.884187	−	0.467132i	$-0.845287\pi$
$857$		−	800.665i		−	0.934264i	0.884187	−	0.467132i	$-0.154713\pi$
$858$		0			0
$859$	1156.84			1.34672			0.673362	−	0.739313i	$-0.264850\pi$
$859$	1156.84			1.34672			0.673362	+	0.739313i	$0.264850\pi$
$860$		0			0
$861$	−142.252	+	234.948i	−0.165218	+	0.272878i
$862$		0			0
$863$		−	178.321i		−	0.206629i	−0.994649	−	0.103315i	$-0.967055\pi$
$863$		−	178.321i		−	0.206629i	0.994649	−	0.103315i	$-0.0329448\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	639.445	+	387.161i	0.737537	+	0.446552i
$868$		0			0
$869$		−	291.955i		−	0.335967i
$870$		0			0
$871$	396.121			0.454789
$872$		0			0
$873$	146.181	+	279.462i	0.167447	+	0.320116i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	354.008			0.403658			0.201829	−	0.979421i	$-0.435311\pi$
$877$	354.008			0.403658			0.201829	+	0.979421i	$0.435311\pi$
$878$		0			0
$879$	632.442	−	1044.56i	0.719502	−	1.18835i
$880$		0			0
$881$		−	828.055i		−	0.939903i	−0.882692	−	0.469951i	$-0.844271\pi$
$881$		−	828.055i		−	0.939903i	0.882692	−	0.469951i	$-0.155729\pi$
$882$		0			0
$883$	−943.554			−1.06858			−0.534289	−	0.845302i	$-0.679420\pi$
$883$	−943.554			−1.06858			−0.534289	+	0.845302i	$0.679420\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			1503.35i			1.69487i	0.530896	+	0.847437i	$0.321856\pi$
$887$			1503.35i			1.69487i	−0.530896	+	0.847437i	$0.678144\pi$
$888$		0			0
$889$	131.368			0.147770
$890$		0			0
$891$	243.906	+	169.352i	0.273744	+	0.190069i
$892$		0			0
$893$			2452.19i			2.74602i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−299.414	+	494.520i	−0.333795	+	0.551305i
$898$		0			0
$899$		−	407.873i		−	0.453696i
$900$		0			0
$901$	−245.908			−0.272928
$902$		0			0
$903$	−82.9931	−	50.2493i	−0.0919082	−	0.0556471i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−268.683			−0.296233			−0.148116	−	0.988970i	$-0.547321\pi$
$907$	−268.683			−0.296233			−0.148116	+	0.988970i	$0.547321\pi$
$908$		0			0
$909$	81.9110	−	42.8461i	0.0901111	−	0.0471354i
$910$		0			0
$911$		−	807.241i		−	0.886104i	−0.896496	−	0.443052i	$-0.853896\pi$
$911$		−	807.241i		−	0.886104i	0.896496	−	0.443052i	$-0.146104\pi$
$912$		0			0
$913$	0.185665			0.000203357
$914$		0			0
$915$		0			0
$916$		0			0
$917$			119.442i			0.130253i
$918$		0			0
$919$	−1359.86			−1.47972			−0.739860	−	0.672761i	$-0.765108\pi$
$919$	−1359.86			−1.47972			−0.739860	+	0.672761i	$0.765108\pi$
$920$		0			0
$921$	730.346	+	442.198i	0.792992	+	0.480128i
$922$		0			0
$923$		−	1003.83i		−	1.08758i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	454.538	+	868.962i	0.490332	+	0.937392i
$928$		0			0
$929$			294.426i			0.316928i	0.987365	+	0.158464i	$0.0506542\pi$
$929$			294.426i			0.316928i	−0.987365	+	0.158464i	$0.949346\pi$
$930$		0			0
$931$	−1450.85			−1.55837
$932$		0			0
$933$	−49.7223	+	82.1227i	−0.0532929	+	0.0880200i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	1449.73			1.54721			0.773603	−	0.633671i	$-0.218453\pi$
$937$	1449.73			1.54721			0.773603	+	0.633671i	$0.218453\pi$
$938$		0			0
$939$	366.859	+	222.120i	0.390691	+	0.236549i
$940$		0			0
$941$		−	386.698i		−	0.410944i	−0.978663	−	0.205472i	$-0.934127\pi$
$941$		−	386.698i		−	0.410944i	0.978663	−	0.205472i	$-0.0658729\pi$
$942$		0			0
$943$	1788.95			1.89708
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	1187.76i		−	1.25423i	−0.778927	−	0.627115i	$-0.784236\pi$
$947$		−	1187.76i		−	1.25423i	0.778927	−	0.627115i	$-0.215764\pi$
$948$		0			0
$949$	−547.932			−0.577378
$950$		0			0
$951$	−385.420	+	636.570i	−0.405279	+	0.669369i
$952$		0			0
$953$		−	127.552i		−	0.133842i	−0.997758	−	0.0669211i	$-0.978682\pi$
$953$		−	127.552i		−	0.133842i	0.997758	−	0.0669211i	$-0.0213176\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	384.827	+	232.999i	0.402118	+	0.243468i
$958$		0			0
$959$			324.518i			0.338392i
$960$		0			0
$961$	−861.580			−0.896545
$962$		0			0
$963$	903.159	−	472.426i	0.937860	−	0.490577i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−68.0305			−0.0703521			−0.0351760	−	0.999381i	$-0.511199\pi$
$967$	−68.0305			−0.0703521			−0.0351760	+	0.999381i	$0.511199\pi$
$968$		0			0
$969$	−301.435	+	497.859i	−0.311079	+	0.513786i
$970$		0			0
$971$		−	1507.80i		−	1.55283i	−0.630220	−	0.776417i	$-0.717035\pi$
$971$		−	1507.80i		−	1.55283i	0.630220	−	0.776417i	$-0.282965\pi$
$972$		0			0
$973$	−289.034			−0.297054
$974$		0			0
$975$		0			0
$976$		0			0
$977$			1349.37i			1.38113i	0.723269	+	0.690566i	$0.242639\pi$
$977$			1349.37i			1.38113i	−0.723269	+	0.690566i	$0.757361\pi$
$978$		0			0
$979$	83.0286			0.0848096
$980$		0			0
$981$	−441.020	−	843.119i	−0.449561	−	0.859448i
$982$		0			0
$983$			710.166i			0.722448i	0.932479	+	0.361224i	$0.117641\pi$
$983$			710.166i			0.722448i	−0.932479	+	0.361224i	$0.882359\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	166.461	−	274.931i	0.168653	−	0.278552i
$988$		0			0
$989$			631.928i			0.638957i
$990$		0			0
$991$	1736.30			1.75207			0.876034	−	0.482249i	$-0.160180\pi$
$991$	1736.30			1.75207			0.876034	+	0.482249i	$0.160180\pi$
$992$		0			0
$993$	−969.179	−	586.803i	−0.976012	−	0.590940i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	812.058			0.814502			0.407251	−	0.913316i	$-0.366488\pi$
$997$	812.058			0.814502			0.407251	+	0.913316i	$0.366488\pi$
$998$		0			0
$999$	−66.4683	+	1062.35i	−0.0665348	+	1.06341i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	600.3.l.d.401.6	yes	6
3.2	odd	2	inner	600.3.l.d.401.5	✓	6
4.3	odd	2		1200.3.l.w.401.1		6
5.2	odd	4		600.3.c.c.449.9		12
5.3	odd	4		600.3.c.c.449.4		12
5.4	even	2		600.3.l.e.401.1	yes	6
12.11	even	2		1200.3.l.w.401.2		6
15.2	even	4		600.3.c.c.449.3		12
15.8	even	4		600.3.c.c.449.10		12
15.14	odd	2		600.3.l.e.401.2	yes	6
20.3	even	4		1200.3.c.l.449.9		12
20.7	even	4		1200.3.c.l.449.4		12
20.19	odd	2		1200.3.l.v.401.6		6
60.23	odd	4		1200.3.c.l.449.3		12
60.47	odd	4		1200.3.c.l.449.10		12
60.59	even	2		1200.3.l.v.401.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
600.3.c.c.449.3		12	15.2	even	4
600.3.c.c.449.4		12	5.3	odd	4
600.3.c.c.449.9		12	5.2	odd	4
600.3.c.c.449.10		12	15.8	even	4
600.3.l.d.401.5	✓	6	3.2	odd	2	inner
600.3.l.d.401.6	yes	6	1.1	even	1	trivial
600.3.l.e.401.1	yes	6	5.4	even	2
600.3.l.e.401.2	yes	6	15.14	odd	2
1200.3.c.l.449.3		12	60.23	odd	4
1200.3.c.l.449.4		12	20.7	even	4
1200.3.c.l.449.9		12	20.3	even	4
1200.3.c.l.449.10		12	60.47	odd	4
1200.3.l.v.401.5		6	60.59	even	2
1200.3.l.v.401.6		6	20.19	odd	2
1200.3.l.w.401.1		6	4.3	odd	2
1200.3.l.w.401.2		6	12.11	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:	\( N \)	\(=\)	\( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	600.l (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(2.56627 + 1.55378i) q^{3} -1.34301 q^{7} +(4.17150 + 7.97487i) q^{9} -3.66586i q^{11} +7.34301 q^{13} +6.31089i q^{17} +30.7406 q^{19} +(-3.44653 - 2.08675i) q^{21} +26.2426i q^{23} +(-1.68602 + 26.9473i) q^{27} +40.9060i q^{29} -9.97097 q^{31} +(5.69595 - 9.40758i) q^{33} +39.4232 q^{37} +(18.8442 + 11.4095i) q^{39} -68.1695i q^{41} +24.0802 q^{43} +79.7705i q^{47} -47.1963 q^{49} +(-9.80576 + 16.1955i) q^{51} +38.9657i q^{53} +(78.8888 + 47.7643i) q^{57} -39.9648i q^{59} -64.2475 q^{61} +(-5.60237 - 10.7103i) q^{63} +53.9454 q^{67} +(-40.7754 + 67.3457i) q^{69} -136.706i q^{71} -74.6196 q^{73} +4.92328i q^{77} +79.6417 q^{79} +(-46.1971 + 66.5344i) q^{81} +0.0506471i q^{83} +(-63.5592 + 104.976i) q^{87} +22.6492i q^{89} -9.86173 q^{91} +(-25.5882 - 15.4927i) q^{93} +35.0428 q^{97} +(29.2347 - 15.2921i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{3} + 10 q^{7} + 16 q^{9} + 26 q^{13} + 50 q^{19} - 18 q^{21} + 26 q^{27} - 114 q^{31} + 82 q^{33} + 76 q^{37} - 6 q^{39} + 2 q^{43} + 76 q^{49} - 6 q^{51} + 172 q^{57} + 62 q^{61} - 150 q^{63}+ \cdots - 250 q^{99}+O(q^{100}) \) 6 * q - 4 * q^3 + 10 * q^7 + 16 * q^9 + 26 * q^13 + 50 * q^19 - 18 * q^21 + 26 * q^27 - 114 * q^31 + 82 * q^33 + 76 * q^37 - 6 * q^39 + 2 * q^43 + 76 * q^49 - 6 * q^51 + 172 * q^57 + 62 * q^61 - 150 * q^63 + 422 * q^67 + 156 * q^69 + 72 * q^73 - 76 * q^79 - 224 * q^81 - 172 * q^87 - 310 * q^91 + 110 * q^93 - 470 * q^97 - 250 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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