LMFDB - Embedded newform 600.3.l.g.401.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [600,3,Mod(401,600)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$16.3488158616$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4x^{10} + 30x^{8} - 216x^{6} + 1080x^{4} - 5184x^{2} + 46656 $ x^12 - 4x^10 + 30x^8 - 216x^6 + 1080x^4 - 5184*x^2 + 46656
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{18}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			401.11
Root			$-1.05281 + 2.21170i$ of defining polynomial
Character	$\chi$	$=$	600.401
Dual form			600.3.l.g.401.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.84952 - 0.938195i) q^{3} -6.81219 q^{7} +(7.23958 - 5.34682i) q^{9} +O(q^{10})$	$q+(2.84952 - 0.938195i) q^{3} -6.81219 q^{7} +(7.23958 - 5.34682i) q^{9} -7.52980i q^{11} +16.2362 q^{13} -4.11928i q^{17} -7.86469 q^{19} +(-19.4115 + 6.39117i) q^{21} -19.5246i q^{23} +(15.6130 - 22.0280i) q^{27} -55.8878i q^{29} +43.4375 q^{31} +(-7.06442 - 21.4563i) q^{33} +31.5824 q^{37} +(46.2655 - 15.2328i) q^{39} -51.3487i q^{41} -51.2914 q^{43} +61.7596i q^{47} -2.59407 q^{49} +(-3.86469 - 11.7380i) q^{51} +82.7111i q^{53} +(-22.4106 + 7.37861i) q^{57} -97.6026i q^{59} +4.13531 q^{61} +(-49.3174 + 36.4236i) q^{63} -63.1940 q^{67} +(-18.3179 - 55.6358i) q^{69} +40.3087i q^{71} +78.5111 q^{73} +51.2944i q^{77} +51.0103 q^{79} +(23.8230 - 77.4175i) q^{81} +2.72011i q^{83} +(-52.4337 - 159.254i) q^{87} +70.4279i q^{89} -110.604 q^{91} +(123.776 - 40.7528i) q^{93} -3.44364 q^{97} +(-40.2605 - 54.5126i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 8 q^{9}+O(q^{10}) $ 12 * q - 8 * q^9	$ 12 q - 8 q^{9} + 4 q^{21} - 48 q^{31} + 128 q^{39} + 252 q^{49} + 48 q^{51} + 144 q^{61} - 268 q^{69} - 432 q^{79} - 188 q^{81} + 560 q^{99}+O(q^{100}) $ 12 * q - 8 * q^9 + 4 * q^21 - 48 * q^31 + 128 * q^39 + 252 * q^49 + 48 * q^51 + 144 * q^61 - 268 * q^69 - 432 * q^79 - 188 * q^81 + 560 * q^99

Display 100 coefficients 10 coefficients

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
$3$	2.84952	−	0.938195i	0.949841	−	0.312732i
$3$	2.84952	−	0.938195i	0.949841	−	0.312732i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−6.81219			−0.973170			−0.486585	−	0.873633i	$-0.661758\pi$
$7$	−6.81219			−0.973170			−0.486585	+	0.873633i	$0.661758\pi$
$8$		0			0
$9$	7.23958	−	5.34682i	0.804398	−	0.594091i
$10$		0			0
$11$		−	7.52980i		−	0.684527i	−0.939604	−	0.342263i	$-0.888806\pi$
$11$		−	7.52980i		−	0.684527i	0.939604	−	0.342263i	$-0.111194\pi$
$12$		0			0
$13$	16.2362			1.24894			0.624471	−	0.781048i	$-0.285315\pi$
$13$	16.2362			1.24894			0.624471	+	0.781048i	$0.285315\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	4.11928i		−	0.242311i	−0.992634	−	0.121155i	$-0.961340\pi$
$17$		−	4.11928i		−	0.242311i	0.992634	−	0.121155i	$-0.0386599\pi$
$18$		0			0
$19$	−7.86469			−0.413931			−0.206965	−	0.978348i	$-0.566359\pi$
$19$	−7.86469			−0.413931			−0.206965	+	0.978348i	$0.566359\pi$
$20$		0			0
$21$	−19.4115	+	6.39117i	−0.924357	+	0.304341i
$22$		0			0
$23$		−	19.5246i		−	0.848895i	−0.905453	−	0.424447i	$-0.860468\pi$
$23$		−	19.5246i		−	0.848895i	0.905453	−	0.424447i	$-0.139532\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	15.6130	−	22.0280i	0.578259	−	0.815853i
$28$		0			0
$29$		−	55.8878i		−	1.92717i	−0.267408	−	0.963583i	$-0.586167\pi$
$29$		−	55.8878i		−	1.92717i	0.267408	−	0.963583i	$-0.413833\pi$
$30$		0			0
$31$	43.4375			1.40121			0.700604	−	0.713550i	$-0.252914\pi$
$31$	43.4375			1.40121			0.700604	+	0.713550i	$0.252914\pi$
$32$		0			0
$33$	−7.06442	−	21.4563i	−0.214073	−	0.650192i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	31.5824			0.853579			0.426790	−	0.904351i	$-0.359644\pi$
$37$	31.5824			0.853579			0.426790	+	0.904351i	$0.359644\pi$
$38$		0			0
$39$	46.2655	−	15.2328i	1.18630	−	0.390584i
$40$		0			0
$41$		−	51.3487i		−	1.25241i	−0.779659	−	0.626204i	$-0.784608\pi$
$41$		−	51.3487i		−	1.25241i	0.779659	−	0.626204i	$-0.215392\pi$
$42$		0			0
$43$	−51.2914			−1.19282			−0.596412	−	0.802678i	$-0.703408\pi$
$43$	−51.2914			−1.19282			−0.596412	+	0.802678i	$0.703408\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			61.7596i			1.31403i	0.753875	+	0.657017i	$0.228182\pi$
$47$			61.7596i			1.31403i	−0.753875	+	0.657017i	$0.771818\pi$
$48$		0			0
$49$	−2.59407			−0.0529401
$50$		0			0
$51$	−3.86469	−	11.7380i	−0.0757782	−	0.230157i
$52$		0			0
$53$			82.7111i			1.56059i	0.625414	+	0.780293i	$0.284930\pi$
$53$			82.7111i			1.56059i	−0.625414	+	0.780293i	$0.715070\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−22.4106	+	7.37861i	−0.393169	+	0.129449i
$58$		0			0
$59$		−	97.6026i		−	1.65428i	−0.561994	−	0.827141i	$-0.689966\pi$
$59$		−	97.6026i		−	1.65428i	0.561994	−	0.827141i	$-0.310034\pi$
$60$		0			0
$61$	4.13531			0.0677920			0.0338960	−	0.999425i	$-0.489209\pi$
$61$	4.13531			0.0677920			0.0338960	+	0.999425i	$0.489209\pi$
$62$		0			0
$63$	−49.3174	+	36.4236i	−0.782816	+	0.578152i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−63.1940			−0.943194			−0.471597	−	0.881814i	$-0.656322\pi$
$67$	−63.1940			−0.943194			−0.471597	+	0.881814i	$0.656322\pi$
$68$		0			0
$69$	−18.3179	−	55.6358i	−0.265476	−	0.806316i
$70$		0			0
$71$			40.3087i			0.567729i	0.958864	+	0.283864i	$0.0916165\pi$
$71$			40.3087i			0.567729i	−0.958864	+	0.283864i	$0.908383\pi$
$72$		0			0
$73$	78.5111			1.07549			0.537747	−	0.843106i	$-0.319276\pi$
$73$	78.5111			1.07549			0.537747	+	0.843106i	$0.319276\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			51.2944i			0.666161i
$78$		0			0
$79$	51.0103			0.645699			0.322850	−	0.946450i	$-0.395359\pi$
$79$	51.0103			0.645699			0.322850	+	0.946450i	$0.395359\pi$
$80$		0			0
$81$	23.8230	−	77.4175i	0.294111	−	0.955771i
$82$		0			0
$83$			2.72011i			0.0327725i	0.999866	+	0.0163862i	$0.00521613\pi$
$83$			2.72011i			0.0327725i	−0.999866	+	0.0163862i	$0.994784\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−52.4337	−	159.254i	−0.602686	−	1.83050i
$88$		0			0
$89$			70.4279i			0.791325i	0.918396	+	0.395662i	$0.129485\pi$
$89$			70.4279i			0.791325i	−0.918396	+	0.395662i	$0.870515\pi$
$90$		0			0
$91$	−110.604			−1.21543
$92$		0			0
$93$	123.776	−	40.7528i	1.33093	−	0.438203i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−3.44364			−0.0355015			−0.0177507	−	0.999842i	$-0.505651\pi$
$97$	−3.44364			−0.0355015			−0.0177507	+	0.999842i	$0.505651\pi$
$98$		0			0
$99$	−40.2605	−	54.5126i	−0.406671	−	0.550632i
$100$		0			0
$101$			49.2446i			0.487570i	0.969829	+	0.243785i	$0.0783892\pi$
$101$			49.2446i			0.487570i	−0.969829	+	0.243785i	$0.921611\pi$
$102$		0			0
$103$	−90.5470			−0.879097			−0.439548	−	0.898219i	$-0.644862\pi$
$103$	−90.5470			−0.879097			−0.439548	+	0.898219i	$0.644862\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			65.3836i			0.611062i	0.952182	+	0.305531i	$0.0988339\pi$
$107$			65.3836i			0.611062i	−0.952182	+	0.305531i	$0.901166\pi$
$108$		0			0
$109$	170.469			1.56394			0.781968	−	0.623319i	$-0.214216\pi$
$109$	170.469			1.56394			0.781968	+	0.623319i	$0.214216\pi$
$110$		0			0
$111$	89.9949	−	29.6305i	0.810765	−	0.266941i
$112$		0			0
$113$			55.4137i			0.490387i	0.969474	+	0.245193i	$0.0788514\pi$
$113$			55.4137i			0.490387i	−0.969474	+	0.245193i	$0.921149\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	117.544	−	86.8123i	1.00465	−	0.741985i
$118$		0			0
$119$			28.0613i			0.235809i
$120$		0			0
$121$	64.3022			0.531423
$122$		0			0
$123$	−48.1751	−	146.319i	−0.391668	−	1.18959i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	74.9923			0.590491			0.295245	−	0.955421i	$-0.404599\pi$
$127$	74.9923			0.590491			0.295245	+	0.955421i	$0.404599\pi$
$128$		0			0
$129$	−146.156	+	48.1214i	−1.13299	+	0.373034i
$130$		0			0
$131$		−	101.149i		−	0.772130i	−0.922472	−	0.386065i	$-0.873834\pi$
$131$		−	101.149i		−	0.772130i	0.922472	−	0.386065i	$-0.126166\pi$
$132$		0			0
$133$	53.5758			0.402825
$134$		0			0
$135$		0			0
$136$		0			0
$137$			114.292i			0.834248i	0.908850	+	0.417124i	$0.136962\pi$
$137$			114.292i			0.834248i	−0.908850	+	0.417124i	$0.863038\pi$
$138$		0			0
$139$	−230.156			−1.65580			−0.827899	−	0.560878i	$-0.810464\pi$
$139$	−230.156			−1.65580			−0.827899	+	0.560878i	$0.810464\pi$
$140$		0			0
$141$	57.9426	+	175.986i	0.410940	+	1.24812i
$142$		0			0
$143$		−	122.256i		−	0.854934i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−7.39185	+	2.43374i	−0.0502847	+	0.0165561i
$148$		0			0
$149$			9.12446i			0.0612380i	0.999531	+	0.0306190i	$0.00974785\pi$
$149$			9.12446i			0.0612380i	−0.999531	+	0.0306190i	$0.990252\pi$
$150$		0			0
$151$	−163.010			−1.07954			−0.539769	−	0.841813i	$-0.681488\pi$
$151$	−163.010			−1.07954			−0.539769	+	0.841813i	$0.681488\pi$
$152$		0			0
$153$	−22.0250	−	29.8218i	−0.143955	−	0.194914i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	122.529			0.780442			0.390221	−	0.920721i	$-0.372399\pi$
$157$	122.529			0.780442			0.390221	+	0.920721i	$0.372399\pi$
$158$		0			0
$159$	77.5991	+	235.687i	0.488045	+	1.48231i
$160$		0			0
$161$			133.005i			0.826119i
$162$		0			0
$163$	−66.6959			−0.409177			−0.204589	−	0.978848i	$-0.565586\pi$
$163$	−66.6959			−0.409177			−0.204589	+	0.978848i	$0.565586\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			163.583i			0.979539i	0.871852	+	0.489769i	$0.162919\pi$
$167$			163.583i			0.979539i	−0.871852	+	0.489769i	$0.837081\pi$
$168$		0			0
$169$	94.6153			0.559854
$170$		0			0
$171$	−56.9370	+	42.0511i	−0.332965	+	0.245913i
$172$		0			0
$173$		−	34.7133i		−	0.200655i	−0.994954	−	0.100327i	$-0.968011\pi$
$173$		−	34.7133i		−	0.200655i	0.994954	−	0.100327i	$-0.0319890\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−91.5704	−	278.121i	−0.517347	−	1.57131i
$178$		0			0
$179$			273.448i			1.52764i	0.645429	+	0.763820i	$0.276679\pi$
$179$			273.448i			1.52764i	−0.645429	+	0.763820i	$0.723321\pi$
$180$		0			0
$181$	−40.8749			−0.225828			−0.112914	−	0.993605i	$-0.536019\pi$
$181$	−40.8749			−0.225828			−0.112914	+	0.993605i	$0.536019\pi$
$182$		0			0
$183$	11.7837	−	3.87973i	0.0643916	−	0.0212007i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−31.0173			−0.165868
$188$		0			0
$189$	−106.359	+	150.059i	−0.562744	+	0.793964i
$190$		0			0
$191$		−	40.5934i		−	0.212531i	−0.994338	−	0.106266i	$-0.966111\pi$
$191$		−	40.5934i		−	0.212531i	0.994338	−	0.106266i	$-0.0338894\pi$
$192$		0			0
$193$	141.259			0.731912			0.365956	−	0.930632i	$-0.380742\pi$
$193$	141.259			0.731912			0.365956	+	0.930632i	$0.380742\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			232.643i			1.18093i	0.807064	+	0.590464i	$0.201055\pi$
$197$			232.643i			1.18093i	−0.807064	+	0.590464i	$0.798945\pi$
$198$		0			0
$199$	−85.3744			−0.429017			−0.214509	−	0.976722i	$-0.568815\pi$
$199$	−85.3744			−0.429017			−0.214509	+	0.976722i	$0.568815\pi$
$200$		0			0
$201$	−180.073	+	59.2883i	−0.895885	+	0.294967i
$202$		0			0
$203$			380.719i			1.87546i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−104.394	−	141.350i	−0.504321	−	0.682849i
$208$		0			0
$209$			59.2195i			0.283347i
$210$		0			0
$211$	284.532			1.34849			0.674247	−	0.738506i	$-0.264468\pi$
$211$	284.532			1.34849			0.674247	+	0.738506i	$0.264468\pi$
$212$		0			0
$213$	37.8175	+	114.861i	0.177547	+	0.539252i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−295.904			−1.36361
$218$		0			0
$219$	223.719	−	73.6587i	1.02155	−	0.336341i
$220$		0			0
$221$		−	66.8816i		−	0.302632i
$222$		0			0
$223$	30.2924			0.135840			0.0679201	−	0.997691i	$-0.478364\pi$
$223$	30.2924			0.135840			0.0679201	+	0.997691i	$0.478364\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	239.380i		−	1.05454i	−0.849698	−	0.527269i	$-0.823216\pi$
$227$		−	239.380i		−	1.05454i	0.849698	−	0.527269i	$-0.176784\pi$
$228$		0			0
$229$	−284.959			−1.24436			−0.622180	−	0.782874i	$-0.713753\pi$
$229$	−284.959			−1.24436			−0.622180	+	0.782874i	$0.713753\pi$
$230$		0			0
$231$	48.1242	+	146.165i	0.208330	+	0.632747i
$232$		0			0
$233$			93.8635i			0.402848i	0.979504	+	0.201424i	$0.0645569\pi$
$233$			93.8635i			0.402848i	−0.979504	+	0.201424i	$0.935443\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	145.355	−	47.8576i	0.613312	−	0.201931i
$238$		0			0
$239$			238.326i			0.997181i	0.866838	+	0.498590i	$0.166149\pi$
$239$			238.326i			0.997181i	−0.866838	+	0.498590i	$0.833851\pi$
$240$		0			0
$241$	44.2919			0.183784			0.0918919	−	0.995769i	$-0.470709\pi$
$241$	44.2919			0.183784			0.0918919	+	0.995769i	$0.470709\pi$
$242$		0			0
$243$	−4.74848	−	242.954i	−0.0195411	−	0.999809i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−127.693			−0.516975
$248$		0			0
$249$	2.55200	+	7.75103i	0.0102490	+	0.0311286i
$250$		0			0
$251$			293.093i			1.16770i	0.811862	+	0.583850i	$0.198454\pi$
$251$			293.093i			1.16770i	−0.811862	+	0.583850i	$0.801546\pi$
$252$		0			0
$253$	−147.016			−0.581092
$254$		0			0
$255$		0			0
$256$		0			0
$257$			169.330i			0.658870i	0.944178	+	0.329435i	$0.106858\pi$
$257$			169.330i			0.658870i	−0.944178	+	0.329435i	$0.893142\pi$
$258$		0			0
$259$	−215.146			−0.830678
$260$		0			0
$261$	−298.822	−	404.604i	−1.14491	−	1.55021i
$262$		0			0
$263$			91.1958i			0.346752i	0.984856	+	0.173376i	$0.0554676\pi$
$263$			91.1958i			0.346752i	−0.984856	+	0.173376i	$0.944532\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	66.0751	+	200.686i	0.247472	+	0.751633i
$268$		0			0
$269$			325.164i			1.20879i	0.796686	+	0.604393i	$0.206584\pi$
$269$			325.164i			1.20879i	−0.796686	+	0.604393i	$0.793416\pi$
$270$		0			0
$271$	−132.719			−0.489738			−0.244869	−	0.969556i	$-0.578745\pi$
$271$	−132.719			−0.489738			−0.244869	+	0.969556i	$0.578745\pi$
$272$		0			0
$273$	−315.170	+	103.768i	−1.15447	+	0.380104i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	254.199			0.917687			0.458844	−	0.888517i	$-0.348264\pi$
$277$	254.199			0.917687			0.458844	+	0.888517i	$0.348264\pi$
$278$		0			0
$279$	314.469	−	232.252i	1.12713	−	0.832446i
$280$		0			0
$281$		−	19.5488i		−	0.0695687i	−0.999395	−	0.0347843i	$-0.988926\pi$
$281$		−	19.5488i		−	0.0695687i	0.999395	−	0.0347843i	$-0.0110744\pi$
$282$		0			0
$283$	109.674			0.387541			0.193770	−	0.981047i	$-0.437928\pi$
$283$	109.674			0.387541			0.193770	+	0.981047i	$0.437928\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			349.797i			1.21881i
$288$		0			0
$289$	272.032			0.941286
$290$		0			0
$291$	−9.81275	+	3.23081i	−0.0337208	+	0.0111024i
$292$		0			0
$293$		−	178.388i		−	0.608832i	−0.952539	−	0.304416i	$-0.901539\pi$
$293$		−	178.388i		−	0.608832i	0.952539	−	0.304416i	$-0.0984614\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−165.867	−	117.563i	−0.558474	−	0.395834i
$298$		0			0
$299$		−	317.006i		−	1.06022i
$300$		0			0
$301$	349.407			1.16082
$302$		0			0
$303$	46.2011	+	140.324i	0.152479	+	0.463115i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	289.046			0.941518			0.470759	−	0.882262i	$-0.343980\pi$
$307$	289.046			0.941518			0.470759	+	0.882262i	$0.343980\pi$
$308$		0			0
$309$	−258.016	+	84.9508i	−0.835003	+	0.274922i
$310$		0			0
$311$			336.061i			1.08058i	0.841478	+	0.540291i	$0.181686\pi$
$311$			336.061i			1.08058i	−0.841478	+	0.540291i	$0.818314\pi$
$312$		0			0
$313$	−563.702			−1.80096			−0.900482	−	0.434894i	$-0.856786\pi$
$313$	−563.702			−1.80096			−0.900482	+	0.434894i	$0.856786\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			139.938i			0.441445i	0.975337	+	0.220722i	$0.0708415\pi$
$317$			139.938i			0.441445i	−0.975337	+	0.220722i	$0.929158\pi$
$318$		0			0
$319$	−420.824			−1.31920
$320$		0			0
$321$	61.3426	+	186.312i	0.191098	+	0.580412i
$322$		0			0
$323$			32.3968i			0.100300i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	485.756	−	159.933i	1.48549	−	0.489092i
$328$		0			0
$329$		−	420.718i		−	1.27878i
$330$		0			0
$331$	−14.5116			−0.0438416			−0.0219208	−	0.999760i	$-0.506978\pi$
$331$	−14.5116			−0.0438416			−0.0219208	+	0.999760i	$0.506978\pi$
$332$		0			0
$333$	228.644	−	168.866i	0.686617	−	0.507104i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−347.492			−1.03113			−0.515566	−	0.856850i	$-0.672418\pi$
$337$	−347.492			−1.03113			−0.515566	+	0.856850i	$0.672418\pi$
$338$		0			0
$339$	51.9889	+	157.903i	0.153359	+	0.465790i
$340$		0			0
$341$		−	327.075i		−	0.959165i
$342$		0			0
$343$	351.469			1.02469
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	273.321i		−	0.787670i	−0.919181	−	0.393835i	$-0.871148\pi$
$347$		−	273.321i		−	0.787670i	0.919181	−	0.393835i	$-0.128852\pi$
$348$		0			0
$349$	550.853			1.57838			0.789188	−	0.614152i	$-0.210502\pi$
$349$	550.853			1.57838			0.789188	+	0.614152i	$0.210502\pi$
$350$		0			0
$351$	253.496	−	357.652i	0.722212	−	1.01895i
$352$		0			0
$353$		−	643.177i		−	1.82203i	−0.412373	−	0.911015i	$-0.635300\pi$
$353$		−	643.177i		−	1.82203i	0.412373	−	0.911015i	$-0.364700\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	26.3270	+	79.9614i	0.0737451	+	0.223982i
$358$		0			0
$359$		−	465.873i		−	1.29770i	−0.760918	−	0.648849i	$-0.775251\pi$
$359$		−	465.873i		−	1.29770i	0.760918	−	0.648849i	$-0.224749\pi$
$360$		0			0
$361$	−299.147			−0.828661
$362$		0			0
$363$	183.231	−	60.3280i	0.504767	−	0.166193i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−44.7168			−0.121844			−0.0609221	−	0.998143i	$-0.519404\pi$
$367$	−44.7168			−0.121844			−0.0609221	+	0.998143i	$0.519404\pi$
$368$		0			0
$369$	−274.552	−	371.743i	−0.744044	−	1.00743i
$370$		0			0
$371$		−	563.443i		−	1.51872i
$372$		0			0
$373$	275.158			0.737688			0.368844	−	0.929491i	$-0.379754\pi$
$373$	275.158			0.737688			0.368844	+	0.929491i	$0.379754\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	907.408i		−	2.40692i
$378$		0			0
$379$	505.072			1.33264			0.666322	−	0.745664i	$-0.267868\pi$
$379$	505.072			1.33264			0.666322	+	0.745664i	$0.267868\pi$
$380$		0			0
$381$	213.692	−	70.3575i	0.560873	−	0.184665i
$382$		0			0
$383$		−	193.577i		−	0.505422i	−0.967542	−	0.252711i	$-0.918678\pi$
$383$		−	193.577i		−	0.505422i	0.967542	−	0.252711i	$-0.0813222\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−371.328	+	274.246i	−0.959505	+	0.708646i
$388$		0			0
$389$		−	280.077i		−	0.719993i	−0.932954	−	0.359996i	$-0.882778\pi$
$389$		−	280.077i		−	0.719993i	0.932954	−	0.359996i	$-0.117222\pi$
$390$		0			0
$391$	−80.4272			−0.205696
$392$		0			0
$393$	−94.8975	−	288.226i	−0.241469	−	0.733401i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	468.421			1.17990			0.589951	−	0.807439i	$-0.299147\pi$
$397$	468.421			1.17990			0.589951	+	0.807439i	$0.299147\pi$
$398$		0			0
$399$	152.665	−	50.2645i	0.382620	−	0.125976i
$400$		0			0
$401$			274.969i			0.685707i	0.939389	+	0.342854i	$0.111394\pi$
$401$			274.969i			0.685707i	−0.939389	+	0.342854i	$0.888606\pi$
$402$		0			0
$403$	705.261			1.75003
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	237.809i		−	0.584298i
$408$		0			0
$409$	202.697			0.495592			0.247796	−	0.968812i	$-0.420294\pi$
$409$	202.697			0.495592			0.247796	+	0.968812i	$0.420294\pi$
$410$		0			0
$411$	107.228	+	325.678i	0.260896	+	0.792403i
$412$		0			0
$413$			664.888i			1.60990i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−655.835	+	215.931i	−1.57274	+	0.517820i
$418$		0			0
$419$			76.6340i			0.182897i	0.995810	+	0.0914487i	$0.0291497\pi$
$419$			76.6340i			0.182897i	−0.995810	+	0.0914487i	$0.970850\pi$
$420$		0			0
$421$	−183.991			−0.437033			−0.218516	−	0.975833i	$-0.570122\pi$
$421$	−183.991			−0.437033			−0.218516	+	0.975833i	$0.570122\pi$
$422$		0			0
$423$	330.218	+	447.114i	0.780657	+	1.05701i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−28.1705			−0.0659731
$428$		0			0
$429$	−114.700	−	348.370i	−0.267365	−	0.812052i
$430$		0			0
$431$		−	671.408i		−	1.55779i	−0.627153	−	0.778896i	$-0.715780\pi$
$431$		−	671.408i		−	1.55779i	0.627153	−	0.778896i	$-0.284220\pi$
$432$		0			0
$433$	−399.351			−0.922288			−0.461144	−	0.887325i	$-0.652561\pi$
$433$	−399.351			−0.922288			−0.461144	+	0.887325i	$0.652561\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			153.555i			0.351384i
$438$		0			0
$439$	−140.833			−0.320804			−0.160402	−	0.987052i	$-0.551279\pi$
$439$	−140.833			−0.320804			−0.160402	+	0.987052i	$0.551279\pi$
$440$		0			0
$441$	−18.7799	+	13.8700i	−0.0425849	+	0.0314513i
$442$		0			0
$443$		−	211.416i		−	0.477237i	−0.971113	−	0.238618i	$-0.923306\pi$
$443$		−	211.416i		−	0.477237i	0.971113	−	0.238618i	$-0.0766945\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	8.56052	+	26.0004i	0.0191511	+	0.0581664i
$448$		0			0
$449$		−	455.166i		−	1.01373i	−0.862025	−	0.506866i	$-0.830804\pi$
$449$		−	455.166i		−	1.01373i	0.862025	−	0.506866i	$-0.169196\pi$
$450$		0			0
$451$	−386.645			−0.857307
$452$		0			0
$453$	−464.502	+	152.935i	−1.02539	+	0.337606i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−555.776			−1.21614			−0.608070	−	0.793883i	$-0.708056\pi$
$457$	−555.776			−1.21614			−0.608070	+	0.793883i	$0.708056\pi$
$458$		0			0
$459$	−90.7396	−	64.3143i	−0.197690	−	0.140118i
$460$		0			0
$461$			369.227i			0.800927i	0.916313	+	0.400463i	$0.131151\pi$
$461$			369.227i			0.800927i	−0.916313	+	0.400463i	$0.868849\pi$
$462$		0			0
$463$	−360.832			−0.779336			−0.389668	−	0.920955i	$-0.627410\pi$
$463$	−360.832			−0.779336			−0.389668	+	0.920955i	$0.627410\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			496.306i			1.06275i	0.847135	+	0.531377i	$0.178325\pi$
$467$			496.306i			1.06275i	−0.847135	+	0.531377i	$0.821675\pi$
$468$		0			0
$469$	430.490			0.917888
$470$		0			0
$471$	349.151	−	114.957i	0.741296	−	0.244069i
$472$		0			0
$473$			386.214i			0.816520i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	442.241	+	598.793i	0.927131	+	1.25533i
$478$		0			0
$479$		−	286.032i		−	0.597145i	−0.954387	−	0.298572i	$-0.903490\pi$
$479$		−	286.032i		−	0.597145i	0.954387	−	0.298572i	$-0.0965105\pi$
$480$		0			0
$481$	512.780			1.06607
$482$		0			0
$483$	124.785	+	379.002i	0.258354	+	0.784682i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−34.3276			−0.0704880			−0.0352440	−	0.999379i	$-0.511221\pi$
$487$	−34.3276			−0.0704880			−0.0352440	+	0.999379i	$0.511221\pi$
$488$		0			0
$489$	−190.052	+	62.5738i	−0.388653	+	0.127963i
$490$		0			0
$491$		−	26.4204i		−	0.0538094i	−0.999638	−	0.0269047i	$-0.991435\pi$
$491$		−	26.4204i		−	0.0538094i	0.999638	−	0.0269047i	$-0.00856507\pi$
$492$		0			0
$493$	−230.218			−0.466973
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	274.591i		−	0.552496i
$498$		0			0
$499$	72.9046			0.146101			0.0730507	−	0.997328i	$-0.476727\pi$
$499$	72.9046			0.146101			0.0730507	+	0.997328i	$0.476727\pi$
$500$		0			0
$501$	153.473	+	466.134i	0.306333	+	0.930407i
$502$		0			0
$503$		−	228.404i		−	0.454083i	−0.973885	−	0.227041i	$-0.927095\pi$
$503$		−	228.404i		−	0.454083i	0.973885	−	0.227041i	$-0.0729052\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	269.609	−	88.7677i	0.531773	−	0.175084i
$508$		0			0
$509$			99.1735i			0.194840i	0.995243	+	0.0974199i	$0.0310590\pi$
$509$			99.1735i			0.194840i	−0.995243	+	0.0974199i	$0.968941\pi$
$510$		0			0
$511$	−534.832			−1.04664
$512$		0			0
$513$	−122.791	+	173.244i	−0.239359	+	0.337707i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	465.037			0.899492
$518$		0			0
$519$	−32.5678	−	98.9163i	−0.0627511	−	0.190590i
$520$		0			0
$521$		−	691.462i		−	1.32718i	−0.748095	−	0.663592i	$-0.769031\pi$
$521$		−	691.462i		−	1.32718i	0.748095	−	0.663592i	$-0.230969\pi$
$522$		0			0
$523$	−821.405			−1.57056			−0.785282	−	0.619138i	$-0.787482\pi$
$523$	−821.405			−1.57056			−0.785282	+	0.619138i	$0.787482\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	178.931i		−	0.339528i
$528$		0			0
$529$	147.791			0.279377
$530$		0			0
$531$	−521.864	−	706.602i	−0.982795	−	1.33070i
$532$		0			0
$533$		−	833.710i		−	1.56418i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	256.547	+	779.196i	0.477742	+	1.45102i
$538$		0			0
$539$			19.5328i			0.0362389i
$540$		0			0
$541$	446.354			0.825054			0.412527	−	0.910945i	$-0.364646\pi$
$541$	446.354			0.825054			0.412527	+	0.910945i	$0.364646\pi$
$542$		0			0
$543$	−116.474	+	38.3487i	−0.214501	+	0.0706237i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	485.492			0.887553			0.443777	−	0.896137i	$-0.353638\pi$
$547$	485.492			0.887553			0.443777	+	0.896137i	$0.353638\pi$
$548$		0			0
$549$	29.9379	−	22.1108i	0.0545317	−	0.0402746i
$550$		0			0
$551$			439.540i			0.797714i
$552$		0			0
$553$	−347.492			−0.628375
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	995.191i		−	1.78670i	−0.449363	−	0.893349i	$-0.648349\pi$
$557$		−	995.191i		−	1.78670i	0.449363	−	0.893349i	$-0.351651\pi$
$558$		0			0
$559$	−832.780			−1.48977
$560$		0			0
$561$	−88.3847	+	29.1003i	−0.157548	+	0.0518722i
$562$		0			0
$563$			486.571i			0.864248i	0.901814	+	0.432124i	$0.142236\pi$
$563$			486.571i			0.864248i	−0.901814	+	0.432124i	$0.857764\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−162.287	+	527.383i	−0.286220	+	0.930128i
$568$		0			0
$569$			737.704i			1.29649i	0.761431	+	0.648246i	$0.224497\pi$
$569$			737.704i			1.29649i	−0.761431	+	0.648246i	$0.775503\pi$
$570$		0			0
$571$	805.843			1.41128			0.705642	−	0.708569i	$-0.250659\pi$
$571$	805.843			1.41128			0.705642	+	0.708569i	$0.250659\pi$
$572$		0			0
$573$	−38.0846	−	115.672i	−0.0664652	−	0.201871i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−690.512			−1.19673			−0.598364	−	0.801225i	$-0.704182\pi$
$577$	−690.512			−1.19673			−0.598364	+	0.801225i	$0.704182\pi$
$578$		0			0
$579$	402.521	−	132.529i	0.695201	−	0.228892i
$580$		0			0
$581$		−	18.5299i		−	0.0318932i
$582$		0			0
$583$	622.798			1.06826
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	670.611i		−	1.14244i	−0.820798	−	0.571219i	$-0.806471\pi$
$587$		−	670.611i		−	1.14244i	0.820798	−	0.571219i	$-0.193529\pi$
$588$		0			0
$589$	−341.622			−0.580004
$590$		0			0
$591$	218.265	+	662.922i	0.369314	+	1.12170i
$592$		0			0
$593$			214.249i			0.361296i	0.983548	+	0.180648i	$0.0578195\pi$
$593$			214.249i			0.361296i	−0.983548	+	0.180648i	$0.942181\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−243.276	+	80.0979i	−0.407498	+	0.134167i
$598$		0			0
$599$		−	303.315i		−	0.506368i	−0.967418	−	0.253184i	$-0.918522\pi$
$599$		−	303.315i		−	0.506368i	0.967418	−	0.253184i	$-0.0814779\pi$
$600$		0			0
$601$	66.0555			0.109909			0.0549546	−	0.998489i	$-0.482499\pi$
$601$	66.0555			0.109909			0.0549546	+	0.998489i	$0.482499\pi$
$602$		0			0
$603$	−457.498	+	337.887i	−0.758703	+	0.560343i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	178.972			0.294847			0.147423	−	0.989073i	$-0.452902\pi$
$607$	178.972			0.294847			0.147423	+	0.989073i	$0.452902\pi$
$608$		0			0
$609$	357.188	+	1084.87i	0.586516	+	1.78139i
$610$		0			0
$611$			1002.74i			1.64115i
$612$		0			0
$613$	728.201			1.18793			0.593965	−	0.804491i	$-0.297562\pi$
$613$	728.201			1.18793			0.593965	+	0.804491i	$0.297562\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	198.426i		−	0.321598i	−0.986987	−	0.160799i	$-0.948593\pi$
$617$		−	198.426i		−	0.321598i	0.986987	−	0.160799i	$-0.0514071\pi$
$618$		0			0
$619$	−17.1752			−0.0277468			−0.0138734	−	0.999904i	$-0.504416\pi$
$619$	−17.1752			−0.0277468			−0.0138734	+	0.999904i	$0.504416\pi$
$620$		0			0
$621$	−430.088	−	304.837i	−0.692574	−	0.490881i
$622$		0			0
$623$		−	479.768i		−	0.770094i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	55.5595	+	168.747i	0.0886116	+	0.269135i
$628$		0			0
$629$		−	130.097i		−	0.206831i
$630$		0			0
$631$	−388.876			−0.616285			−0.308142	−	0.951340i	$-0.599707\pi$
$631$	−388.876			−0.616285			−0.308142	+	0.951340i	$0.599707\pi$
$632$		0			0
$633$	810.781	−	266.947i	1.28085	−	0.421717i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−42.1179			−0.0661191
$638$		0			0
$639$	215.524	+	291.818i	0.337283	+	0.456680i
$640$		0			0
$641$		−	180.547i		−	0.281664i	−0.990034	−	0.140832i	$-0.955022\pi$
$641$		−	180.547i		−	0.281664i	0.990034	−	0.140832i	$-0.0449778\pi$
$642$		0			0
$643$	701.008			1.09021			0.545107	−	0.838367i	$-0.316489\pi$
$643$	701.008			1.09021			0.545107	+	0.838367i	$0.316489\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	114.266i		−	0.176609i	−0.996094	−	0.0883046i	$-0.971855\pi$
$647$		−	114.266i		−	0.176609i	0.996094	−	0.0883046i	$-0.0281449\pi$
$648$		0			0
$649$	−734.928			−1.13240
$650$		0			0
$651$	−843.187	+	277.616i	−1.29522	+	0.426446i
$652$		0			0
$653$			240.882i			0.368884i	0.982843	+	0.184442i	$0.0590479\pi$
$653$			240.882i			0.368884i	−0.982843	+	0.184442i	$0.940952\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	568.387	−	419.785i	0.865125	−	0.638942i
$658$		0			0
$659$			1218.15i			1.84848i	0.381807	+	0.924242i	$0.375302\pi$
$659$			1218.15i			1.84848i	−0.381807	+	0.924242i	$0.624698\pi$
$660$		0			0
$661$	−1075.32			−1.62681			−0.813404	−	0.581699i	$-0.802388\pi$
$661$	−1075.32			−1.62681			−0.813404	+	0.581699i	$0.802388\pi$
$662$		0			0
$663$	−62.7480	−	190.581i	−0.0946425	−	0.287452i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−1091.19			−1.63596
$668$		0			0
$669$	86.3189	−	28.4202i	0.129027	−	0.0424816i
$670$		0			0
$671$		−	31.1381i		−	0.0464054i
$672$		0			0
$673$	356.367			0.529521			0.264760	−	0.964314i	$-0.414707\pi$
$673$	356.367			0.529521			0.264760	+	0.964314i	$0.414707\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			79.8649i			0.117969i	0.998259	+	0.0589844i	$0.0187862\pi$
$677$			79.8649i			0.117969i	−0.998259	+	0.0589844i	$0.981214\pi$
$678$		0			0
$679$	23.4588			0.0345490
$680$		0			0
$681$	−224.585	−	682.120i	−0.329788	−	1.00164i
$682$		0			0
$683$		−	1270.41i		−	1.86004i	−0.367507	−	0.930021i	$-0.619789\pi$
$683$		−	1270.41i		−	1.86004i	0.367507	−	0.930021i	$-0.380211\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−811.996	+	267.347i	−1.18195	+	0.389151i
$688$		0			0
$689$			1342.92i			1.94908i
$690$		0			0
$691$	1244.39			1.80085			0.900424	−	0.435013i	$-0.143256\pi$
$691$	1244.39			1.80085			0.900424	+	0.435013i	$0.143256\pi$
$692$		0			0
$693$	274.262	+	371.350i	0.395760	+	0.535858i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−211.520			−0.303472
$698$		0			0
$699$	88.0623	+	267.466i	0.125983	+	0.382641i
$700$		0			0
$701$			1034.47i			1.47570i	0.674965	+	0.737850i	$0.264159\pi$
$701$			1034.47i			1.47570i	−0.674965	+	0.737850i	$0.735841\pi$
$702$		0			0
$703$	−248.386			−0.353323
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	335.464i		−	0.474489i
$708$		0			0
$709$	−659.081			−0.929592			−0.464796	−	0.885418i	$-0.653872\pi$
$709$	−659.081			−0.929592			−0.464796	+	0.885418i	$0.653872\pi$
$710$		0			0
$711$	369.293	−	272.743i	0.519399	−	0.383604i
$712$		0			0
$713$		−	848.099i		−	1.18948i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	223.597	+	679.116i	0.311850	+	0.947164i
$718$		0			0
$719$			471.213i			0.655373i	0.944787	+	0.327686i	$0.106269\pi$
$719$			471.213i			0.655373i	−0.944787	+	0.327686i	$0.893731\pi$
$720$		0			0
$721$	616.823			0.855511
$722$		0			0
$723$	126.211	−	41.5545i	0.174566	−	0.0574750i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	640.411			0.880895			0.440447	−	0.897778i	$-0.354820\pi$
$727$	640.411			0.880895			0.440447	+	0.897778i	$0.354820\pi$
$728$		0			0
$729$	−241.469	−	687.847i	−0.331233	−	0.943549i
$730$		0			0
$731$			211.284i			0.289034i
$732$		0			0
$733$	619.831			0.845608			0.422804	−	0.906221i	$-0.361046\pi$
$733$	619.831			0.845608			0.422804	+	0.906221i	$0.361046\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			475.838i			0.645642i
$738$		0			0
$739$	135.992			0.184022			0.0920111	−	0.995758i	$-0.470670\pi$
$739$	135.992			0.184022			0.0920111	+	0.995758i	$0.470670\pi$
$740$		0			0
$741$	−363.864	+	119.801i	−0.491045	+	0.161675i
$742$		0			0
$743$			986.468i			1.32768i	0.747874	+	0.663841i	$0.231075\pi$
$743$			986.468i			1.32768i	−0.747874	+	0.663841i	$0.768925\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	14.5440	+	19.6925i	0.0194698	+	0.0263621i
$748$		0			0
$749$		−	445.405i		−	0.594667i
$750$		0			0
$751$	−551.977			−0.734990			−0.367495	−	0.930026i	$-0.619784\pi$
$751$	−551.977			−0.734990			−0.367495	+	0.930026i	$0.619784\pi$
$752$		0			0
$753$	274.978	+	835.175i	0.365177	+	1.10913i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	1143.91			1.51111			0.755556	−	0.655084i	$-0.227367\pi$
$757$	1143.91			1.51111			0.755556	+	0.655084i	$0.227367\pi$
$758$		0			0
$759$	−418.926	+	137.930i	−0.551945	+	0.181726i
$760$		0			0
$761$			29.3201i			0.0385284i	0.999814	+	0.0192642i	$0.00613236\pi$
$761$			29.3201i			0.0385284i	−0.999814	+	0.0192642i	$0.993868\pi$
$762$		0			0
$763$	−1161.27			−1.52198
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	1584.70i		−	2.06610i
$768$		0			0
$769$	61.7841			0.0803434			0.0401717	−	0.999193i	$-0.487210\pi$
$769$	61.7841			0.0803434			0.0401717	+	0.999193i	$0.487210\pi$
$770$		0			0
$771$	158.864	+	482.509i	0.206050	+	0.625822i
$772$		0			0
$773$			444.408i			0.574913i	0.957794	+	0.287457i	$0.0928097\pi$
$773$			444.408i			0.574913i	−0.957794	+	0.287457i	$0.907190\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−613.063	+	201.849i	−0.789012	+	0.259779i
$778$		0			0
$779$			403.842i			0.518410i
$780$		0			0
$781$	303.517			0.388626
$782$		0			0
$783$	−1231.10	−	872.576i	−1.57229	−	1.11440i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	206.738			0.262691			0.131346	−	0.991337i	$-0.458070\pi$
$787$	206.738			0.262691			0.131346	+	0.991337i	$0.458070\pi$
$788$		0			0
$789$	85.5595	+	259.865i	0.108440	+	0.329360i
$790$		0			0
$791$		−	377.489i		−	0.477230i
$792$		0			0
$793$	67.1419			0.0846682
$794$		0			0
$795$		0			0
$796$		0			0
$797$			235.281i			0.295208i	0.989047	+	0.147604i	$0.0471562\pi$
$797$			235.281i			0.295208i	−0.989047	+	0.147604i	$0.952844\pi$
$798$		0			0
$799$	254.405			0.318404
$800$		0			0
$801$	376.565	+	509.868i	0.470119	+	0.636540i
$802$		0			0
$803$		−	591.172i		−	0.736205i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	305.067	+	926.561i	0.378026	+	1.14816i
$808$		0			0
$809$		−	56.7919i		−	0.0702001i	−0.999384	−	0.0351000i	$-0.988825\pi$
$809$		−	56.7919i		−	0.0702001i	0.999384	−	0.0351000i	$-0.0111750\pi$
$810$		0			0
$811$	743.889			0.917249			0.458625	−	0.888630i	$-0.348342\pi$
$811$	743.889			0.917249			0.458625	+	0.888630i	$0.348342\pi$
$812$		0			0
$813$	−378.186	+	124.516i	−0.465174	+	0.153157i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	403.391			0.493747
$818$		0			0
$819$	−800.729	+	591.382i	−0.977691	+	0.722078i
$820$		0			0
$821$		−	314.095i		−	0.382576i	−0.981534	−	0.191288i	$-0.938733\pi$
$821$		−	314.095i		−	0.382576i	0.981534	−	0.191288i	$-0.0612665\pi$
$822$		0			0
$823$	−1059.68			−1.28758			−0.643789	−	0.765203i	$-0.722639\pi$
$823$	−1059.68			−1.28758			−0.643789	+	0.765203i	$0.722639\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			834.408i			1.00896i	0.863424	+	0.504479i	$0.168315\pi$
$827$			834.408i			1.00896i	−0.863424	+	0.504479i	$0.831685\pi$
$828$		0			0
$829$	−19.3660			−0.0233607			−0.0116803	−	0.999932i	$-0.503718\pi$
$829$	−19.3660			−0.0233607			−0.0116803	+	0.999932i	$0.503718\pi$
$830$		0			0
$831$	724.347	−	238.489i	0.871658	−	0.286990i
$832$		0			0
$833$			10.6857i			0.0128279i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	678.189	−	956.842i	0.810262	−	1.14318i
$838$		0			0
$839$			1565.62i			1.86605i	0.359808	+	0.933026i	$0.382842\pi$
$839$			1565.62i			1.86605i	−0.359808	+	0.933026i	$0.617158\pi$
$840$		0			0
$841$	−2282.45			−2.71397
$842$		0			0
$843$	−18.3406	−	55.7048i	−0.0217563	−	0.0660792i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−438.039			−0.517165
$848$		0			0
$849$	312.519	−	102.896i	0.368102	−	0.121196i
$850$		0			0
$851$		−	616.634i		−	0.724599i
$852$		0			0
$853$	20.9266			0.0245329			0.0122665	−	0.999925i	$-0.496095\pi$
$853$	20.9266			0.0245329			0.0122665	+	0.999925i	$0.496095\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	236.110i		−	0.275508i	−0.990466	−	0.137754i	$-0.956012\pi$
$857$		−	236.110i		−	0.275508i	0.990466	−	0.137754i	$-0.0439884\pi$
$858$		0			0
$859$	997.970			1.16178			0.580891	−	0.813982i	$-0.302704\pi$
$859$	997.970			1.16178			0.580891	+	0.813982i	$0.302704\pi$
$860$		0			0
$861$	328.178	+	996.756i	0.381159	+	1.15767i
$862$		0			0
$863$			1147.63i			1.32982i	0.746924	+	0.664909i	$0.231530\pi$
$863$			1147.63i			1.32982i	−0.746924	+	0.664909i	$0.768470\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	775.161	−	255.219i	0.894072	−	0.294370i
$868$		0			0
$869$		−	384.097i		−	0.441999i
$870$		0			0
$871$	−1026.03			−1.17799
$872$		0			0
$873$	−24.9305	+	18.4125i	−0.0285573	+	0.0210911i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−166.283			−0.189604			−0.0948022	−	0.995496i	$-0.530222\pi$
$877$	−166.283			−0.189604			−0.0948022	+	0.995496i	$0.530222\pi$
$878$		0			0
$879$	−167.363	−	508.321i	−0.190401	−	0.578294i
$880$		0			0
$881$			275.137i			0.312301i	0.987733	+	0.156150i	$0.0499085\pi$
$881$			275.137i			0.312301i	−0.987733	+	0.156150i	$0.950092\pi$
$882$		0			0
$883$	−262.830			−0.297656			−0.148828	−	0.988863i	$-0.547550\pi$
$883$	−262.830			−0.297656			−0.148828	+	0.988863i	$0.547550\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			406.164i			0.457908i	0.973437	+	0.228954i	$0.0735306\pi$
$887$			406.164i			0.457908i	−0.973437	+	0.228954i	$0.926469\pi$
$888$		0			0
$889$	−510.862			−0.574648
$890$		0			0
$891$	−582.938	−	179.382i	−0.654251	−	0.201327i
$892$		0			0
$893$		−	485.720i		−	0.543920i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−297.413	−	903.316i	−0.331564	−	1.00704i
$898$		0			0
$899$		−	2427.63i		−	2.70036i
$900$		0			0
$901$	340.710			0.378146
$902$		0			0
$903$	995.644	−	327.812i	1.10260	−	0.363026i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−1284.19			−1.41587			−0.707934	−	0.706278i	$-0.750373\pi$
$907$	−1284.19			−1.41587			−0.707934	+	0.706278i	$0.750373\pi$
$908$		0			0
$909$	263.302	+	356.510i	0.289661	+	0.392201i
$910$		0			0
$911$			353.054i			0.387546i	0.981046	+	0.193773i	$0.0620725\pi$
$911$			353.054i			0.387546i	−0.981046	+	0.193773i	$0.937928\pi$
$912$		0			0
$913$	20.4819			0.0224336
$914$		0			0
$915$		0			0
$916$		0			0
$917$			689.046i			0.751413i
$918$		0			0
$919$	−210.284			−0.228818			−0.114409	−	0.993434i	$-0.536497\pi$
$919$	−210.284			−0.228818			−0.114409	+	0.993434i	$0.536497\pi$
$920$		0			0
$921$	823.644	−	271.182i	0.894293	−	0.294443i
$922$		0			0
$923$			654.462i			0.709060i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−655.522	+	484.139i	−0.707143	+	0.522264i
$928$		0			0
$929$		−	1095.27i		−	1.17898i	−0.807777	−	0.589488i	$-0.799330\pi$
$929$		−	1095.27i		−	1.17898i	0.807777	−	0.589488i	$-0.200670\pi$
$930$		0			0
$931$	20.4015			0.0219136
$932$		0			0
$933$	315.291	+	957.614i	0.337932	+	1.02638i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	567.824			0.606002			0.303001	−	0.952990i	$-0.402011\pi$
$937$	567.824			0.606002			0.303001	+	0.952990i	$0.402011\pi$
$938$		0			0
$939$	−1606.28	+	528.862i	−1.71063	+	0.563219i
$940$		0			0
$941$		−	726.052i		−	0.771575i	−0.922588	−	0.385787i	$-0.873930\pi$
$941$		−	726.052i		−	0.771575i	0.922588	−	0.385787i	$-0.126070\pi$
$942$		0			0
$943$	−1002.56			−1.06316
$944$		0			0
$945$		0			0
$946$		0			0
$947$			568.400i			0.600211i	0.953906	+	0.300105i	$0.0970219\pi$
$947$			568.400i			0.600211i	−0.953906	+	0.300105i	$0.902978\pi$
$948$		0			0
$949$	1274.72			1.34323
$950$		0			0
$951$	131.289	+	398.757i	0.138054	+	0.419303i
$952$		0			0
$953$			1118.37i			1.17353i	0.809758	+	0.586765i	$0.199599\pi$
$953$			1118.37i			1.17353i	−0.809758	+	0.586765i	$0.800401\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−1199.15	+	394.815i	−1.25303	+	0.412555i
$958$		0			0
$959$		−	778.578i		−	0.811865i
$960$		0			0
$961$	925.814			0.963386
$962$		0			0
$963$	349.594	+	473.350i	0.363026	+	0.491537i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−759.775			−0.785704			−0.392852	−	0.919602i	$-0.628511\pi$
$967$	−759.775			−0.785704			−0.392852	+	0.919602i	$0.628511\pi$
$968$		0			0
$969$	30.3946	+	92.3156i	0.0313669	+	0.0952689i
$970$		0			0
$971$		−	1342.50i		−	1.38259i	−0.722570	−	0.691297i	$-0.757040\pi$
$971$		−	1342.50i		−	1.38259i	0.722570	−	0.691297i	$-0.242960\pi$
$972$		0			0
$973$	1567.87			1.61137
$974$		0			0
$975$		0			0
$976$		0			0
$977$			897.030i			0.918147i	0.888398	+	0.459074i	$0.151819\pi$
$977$			897.030i			0.918147i	−0.888398	+	0.459074i	$0.848181\pi$
$978$		0			0
$979$	530.308			0.541683
$980$		0			0
$981$	1234.12	−	911.467i	1.25803	−	0.929121i
$982$		0			0
$983$		−	377.542i		−	0.384071i	−0.981388	−	0.192035i	$-0.938491\pi$
$983$		−	377.542i		−	0.384071i	0.981388	−	0.192035i	$-0.0615088\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−394.716	−	1198.85i	−0.399915	−	1.21464i
$988$		0			0
$989$			1001.44i			1.01258i
$990$		0			0
$991$	−224.222			−0.226258			−0.113129	−	0.993580i	$-0.536087\pi$
$991$	−224.222			−0.226258			−0.113129	+	0.993580i	$0.536087\pi$
$992$		0			0
$993$	−41.3511	+	13.6147i	−0.0416426	+	0.0137107i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−1018.21			−1.02127			−0.510635	−	0.859797i	$-0.670590\pi$
$997$	−1018.21			−1.02127			−0.510635	+	0.859797i	$0.670590\pi$
$998$		0			0
$999$	493.096	−	695.699i	0.493590	−	0.696396i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	600.3.l.g.401.11		12
3.2	odd	2	inner	600.3.l.g.401.12		12
4.3	odd	2		1200.3.l.y.401.2		12
5.2	odd	4		120.3.c.a.89.5	✓	12
5.3	odd	4		120.3.c.a.89.8	yes	12
5.4	even	2	inner	600.3.l.g.401.2		12
12.11	even	2		1200.3.l.y.401.1		12
15.2	even	4		120.3.c.a.89.7	yes	12
15.8	even	4		120.3.c.a.89.6	yes	12
15.14	odd	2	inner	600.3.l.g.401.1		12
20.3	even	4		240.3.c.e.209.5		12
20.7	even	4		240.3.c.e.209.8		12
20.19	odd	2		1200.3.l.y.401.11		12
40.3	even	4		960.3.c.j.449.8		12
40.13	odd	4		960.3.c.k.449.5		12
40.27	even	4		960.3.c.j.449.5		12
40.37	odd	4		960.3.c.k.449.8		12
60.23	odd	4		240.3.c.e.209.7		12
60.47	odd	4		240.3.c.e.209.6		12
60.59	even	2		1200.3.l.y.401.12		12
120.53	even	4		960.3.c.k.449.7		12
120.77	even	4		960.3.c.k.449.6		12
120.83	odd	4		960.3.c.j.449.6		12
120.107	odd	4		960.3.c.j.449.7		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.3.c.a.89.5	✓	12	5.2	odd	4
120.3.c.a.89.6	yes	12	15.8	even	4
120.3.c.a.89.7	yes	12	15.2	even	4
120.3.c.a.89.8	yes	12	5.3	odd	4
240.3.c.e.209.5		12	20.3	even	4
240.3.c.e.209.6		12	60.47	odd	4
240.3.c.e.209.7		12	60.23	odd	4
240.3.c.e.209.8		12	20.7	even	4
600.3.l.g.401.1		12	15.14	odd	2	inner
600.3.l.g.401.2		12	5.4	even	2	inner
600.3.l.g.401.11		12	1.1	even	1	trivial
600.3.l.g.401.12		12	3.2	odd	2	inner
960.3.c.j.449.5		12	40.27	even	4
960.3.c.j.449.6		12	120.83	odd	4
960.3.c.j.449.7		12	120.107	odd	4
960.3.c.j.449.8		12	40.3	even	4
960.3.c.k.449.5		12	40.13	odd	4
960.3.c.k.449.6		12	120.77	even	4
960.3.c.k.449.7		12	120.53	even	4
960.3.c.k.449.8		12	40.37	odd	4
1200.3.l.y.401.1		12	12.11	even	2
1200.3.l.y.401.2		12	4.3	odd	2
1200.3.l.y.401.11		12	20.19	odd	2
1200.3.l.y.401.12		12	60.59	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+(2.84952 - 0.938195i) q^{3} -6.81219 q^{7} +(7.23958 - 5.34682i) q^{9} +O(q^{10})\)	\(q+(2.84952 - 0.938195i) q^{3} -6.81219 q^{7} +(7.23958 - 5.34682i) q^{9} -7.52980i q^{11} +16.2362 q^{13} -4.11928i q^{17} -7.86469 q^{19} +(-19.4115 + 6.39117i) q^{21} -19.5246i q^{23} +(15.6130 - 22.0280i) q^{27} -55.8878i q^{29} +43.4375 q^{31} +(-7.06442 - 21.4563i) q^{33} +31.5824 q^{37} +(46.2655 - 15.2328i) q^{39} -51.3487i q^{41} -51.2914 q^{43} +61.7596i q^{47} -2.59407 q^{49} +(-3.86469 - 11.7380i) q^{51} +82.7111i q^{53} +(-22.4106 + 7.37861i) q^{57} -97.6026i q^{59} +4.13531 q^{61} +(-49.3174 + 36.4236i) q^{63} -63.1940 q^{67} +(-18.3179 - 55.6358i) q^{69} +40.3087i q^{71} +78.5111 q^{73} +51.2944i q^{77} +51.0103 q^{79} +(23.8230 - 77.4175i) q^{81} +2.72011i q^{83} +(-52.4337 - 159.254i) q^{87} +70.4279i q^{89} -110.604 q^{91} +(123.776 - 40.7528i) q^{93} -3.44364 q^{97} +(-40.2605 - 54.5126i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 8 q^{9}+O(q^{10}) \) 12 * q - 8 * q^9	\( 12 q - 8 q^{9} + 4 q^{21} - 48 q^{31} + 128 q^{39} + 252 q^{49} + 48 q^{51} + 144 q^{61} - 268 q^{69} - 432 q^{79} - 188 q^{81} + 560 q^{99}+O(q^{100}) \) 12 * q - 8 * q^9 + 4 * q^21 - 48 * q^31 + 128 * q^39 + 252 * q^49 + 48 * q^51 + 144 * q^61 - 268 * q^69 - 432 * q^79 - 188 * q^81 + 560 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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