LMFDB - Embedded newform 1200.3.l.y.401.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1200,3,Mod(401,1200)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1200, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 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("1200.401"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 1200 = 2^{4} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1200.l (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,0,0,0,0] 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:	$32.6976317232$
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			401.1
Root			$1.05281 - 2.21170i$ of defining polynomial
Character	$\chi$	$=$	1200.401
Dual form			1200.3.l.y.401.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times$.

$n$	$401$	$577$	$751$	$901$
$\chi(n)$	$-1$	$1$	$1$	$1$

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	$-0.338242\pi$
$7$	6.81219			0.973170			0.486585	+	0.873633i	$0.338242\pi$
$8$		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	$0.0386599\pi$
$17$			4.11928i			0.242311i	−0.992634	+	0.121155i	$0.961340\pi$
$18$		0			0
$19$	7.86469			0.413931			0.206965	−	0.978348i	$-0.433641\pi$
$19$	7.86469			0.413931			0.206965	+	0.978348i	$0.433641\pi$
$20$		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	$0.413833\pi$
$29$			55.8878i			1.92717i	−0.267408	+	0.963583i	$0.586167\pi$
$30$		0			0
$31$	−43.4375			−1.40121			−0.700604	−	0.713550i	$-0.747086\pi$
$31$	−43.4375			−1.40121			−0.700604	+	0.713550i	$0.747086\pi$
$32$		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	$0.215392\pi$
$41$			51.3487i			1.25241i	−0.779659	+	0.626204i	$0.784608\pi$
$42$		0			0
$43$	51.2914			1.19282			0.596412	−	0.802678i	$-0.296592\pi$
$43$	51.2914			1.19282			0.596412	+	0.802678i	$0.296592\pi$
$44$		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	$-0.715070\pi$
$53$		−	82.7111i		−	1.56059i	0.625414	−	0.780293i	$-0.284930\pi$
$54$		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	$-0.343678\pi$
$67$	63.1940			0.943194			0.471597	+	0.881814i	$0.343678\pi$
$68$		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	$-0.604641\pi$
$79$	−51.0103			−0.645699			−0.322850	+	0.946450i	$0.604641\pi$
$80$		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	$-0.870515\pi$
$89$		−	70.4279i		−	0.791325i	0.918396	−	0.395662i	$-0.129485\pi$
$90$		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	$-0.921611\pi$
$101$		−	49.2446i		−	0.487570i	0.969829	−	0.243785i	$-0.0783892\pi$
$102$		0			0
$103$	90.5470			0.879097			0.439548	−	0.898219i	$-0.355138\pi$
$103$	90.5470			0.879097			0.439548	+	0.898219i	$0.355138\pi$
$104$		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	$-0.921149\pi$
$113$		−	55.4137i		−	0.490387i	0.969474	−	0.245193i	$-0.0788514\pi$
$114$		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	$-0.595401\pi$
$127$	−74.9923			−0.590491			−0.295245	+	0.955421i	$0.595401\pi$
$128$		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	$-0.863038\pi$
$137$		−	114.292i		−	0.834248i	0.908850	−	0.417124i	$-0.136962\pi$
$138$		0			0
$139$	230.156			1.65580			0.827899	−	0.560878i	$-0.189536\pi$
$139$	230.156			1.65580			0.827899	+	0.560878i	$0.189536\pi$
$140$		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	$-0.990252\pi$
$149$		−	9.12446i		−	0.0612380i	0.999531	−	0.0306190i	$-0.00974785\pi$
$150$		0			0
$151$	163.010			1.07954			0.539769	−	0.841813i	$-0.318512\pi$
$151$	163.010			1.07954			0.539769	+	0.841813i	$0.318512\pi$
$152$		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	$-0.434414\pi$
$163$	66.6959			0.409177			0.204589	+	0.978848i	$0.434414\pi$
$164$		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	$0.0319890\pi$
$173$			34.7133i			0.200655i	−0.994954	+	0.100327i	$0.968011\pi$
$174$		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	$-0.798945\pi$
$197$		−	232.643i		−	1.18093i	0.807064	−	0.590464i	$-0.201055\pi$
$198$		0			0
$199$	85.3744			0.429017			0.214509	−	0.976722i	$-0.431185\pi$
$199$	85.3744			0.429017			0.214509	+	0.976722i	$0.431185\pi$
$200$		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	$-0.735532\pi$
$211$	−284.532			−1.34849			−0.674247	+	0.738506i	$0.735532\pi$
$212$		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	$-0.521636\pi$
$223$	−30.2924			−0.135840			−0.0679201	+	0.997691i	$0.521636\pi$
$224$		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	$-0.935443\pi$
$233$		−	93.8635i		−	0.402848i	0.979504	−	0.201424i	$-0.0645569\pi$
$234$		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	$-0.893142\pi$
$257$		−	169.330i		−	0.658870i	0.944178	−	0.329435i	$-0.106858\pi$
$258$		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	$-0.793416\pi$
$269$		−	325.164i		−	1.20879i	0.796686	−	0.604393i	$-0.206584\pi$
$270$		0			0
$271$	132.719			0.489738			0.244869	−	0.969556i	$-0.421255\pi$
$271$	132.719			0.489738			0.244869	+	0.969556i	$0.421255\pi$
$272$		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	$0.0110744\pi$
$281$			19.5488i			0.0695687i	−0.999395	+	0.0347843i	$0.988926\pi$
$282$		0			0
$283$	−109.674			−0.387541			−0.193770	−	0.981047i	$-0.562072\pi$
$283$	−109.674			−0.387541			−0.193770	+	0.981047i	$0.562072\pi$
$284$		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	$0.0984614\pi$
$293$			178.388i			0.608832i	−0.952539	+	0.304416i	$0.901539\pi$
$294$		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	$-0.656020\pi$
$307$	−289.046			−0.941518			−0.470759	+	0.882262i	$0.656020\pi$
$308$		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	$-0.929158\pi$
$317$		−	139.938i		−	0.441445i	0.975337	−	0.220722i	$-0.0708415\pi$
$318$		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	$-0.493022\pi$
$331$	14.5116			0.0438416			0.0219208	+	0.999760i	$0.493022\pi$
$332$		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	$0.364700\pi$
$353$			643.177i			1.82203i	−0.412373	+	0.911015i	$0.635300\pi$
$354$		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	$-0.480596\pi$
$367$	44.7168			0.121844			0.0609221	+	0.998143i	$0.480596\pi$
$368$		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	$-0.732132\pi$
$379$	−505.072			−1.33264			−0.666322	+	0.745664i	$0.732132\pi$
$380$		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	$0.117222\pi$
$389$			280.077i			0.719993i	−0.932954	+	0.359996i	$0.882778\pi$
$390$		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	$-0.888606\pi$
$401$		−	274.969i		−	0.685707i	0.939389	−	0.342854i	$-0.111394\pi$
$402$		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	$-0.448721\pi$
$439$	140.833			0.320804			0.160402	+	0.987052i	$0.448721\pi$
$440$		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	$0.169196\pi$
$449$			455.166i			1.01373i	−0.862025	+	0.506866i	$0.830804\pi$
$450$		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	$-0.868849\pi$
$461$		−	369.227i		−	0.800927i	0.916313	−	0.400463i	$-0.131151\pi$
$462$		0			0
$463$	360.832			0.779336			0.389668	−	0.920955i	$-0.372590\pi$
$463$	360.832			0.779336			0.389668	+	0.920955i	$0.372590\pi$
$464$		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	$-0.488779\pi$
$487$	34.3276			0.0704880			0.0352440	+	0.999379i	$0.488779\pi$
$488$		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	$-0.523273\pi$
$499$	−72.9046			−0.146101			−0.0730507	+	0.997328i	$0.523273\pi$
$500$		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	$-0.968941\pi$
$509$		−	99.1735i		−	0.194840i	0.995243	−	0.0974199i	$-0.0310590\pi$
$510$		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	$0.230969\pi$
$521$			691.462i			1.32718i	−0.748095	+	0.663592i	$0.769031\pi$
$522$		0			0
$523$	821.405			1.57056			0.785282	−	0.619138i	$-0.212518\pi$
$523$	821.405			1.57056			0.785282	+	0.619138i	$0.212518\pi$
$524$		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	$-0.646362\pi$
$547$	−485.492			−0.887553			−0.443777	+	0.896137i	$0.646362\pi$
$548$		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	$0.351651\pi$
$557$			995.191i			1.78670i	−0.449363	+	0.893349i	$0.648349\pi$
$558$		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	$-0.775503\pi$
$569$		−	737.704i		−	1.29649i	0.761431	−	0.648246i	$-0.224497\pi$
$570$		0			0
$571$	−805.843			−1.41128			−0.705642	−	0.708569i	$-0.749341\pi$
$571$	−805.843			−1.41128			−0.705642	+	0.708569i	$0.749341\pi$
$572$		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	$-0.942181\pi$
$593$		−	214.249i		−	0.361296i	0.983548	−	0.180648i	$-0.0578195\pi$
$594$		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	$-0.547098\pi$
$607$	−178.972			−0.294847			−0.147423	+	0.989073i	$0.547098\pi$
$608$		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	$0.0514071\pi$
$617$			198.426i			0.321598i	−0.986987	+	0.160799i	$0.948593\pi$
$618$		0			0
$619$	17.1752			0.0277468			0.0138734	−	0.999904i	$-0.495584\pi$
$619$	17.1752			0.0277468			0.0138734	+	0.999904i	$0.495584\pi$
$620$		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	$-0.400293\pi$
$631$	388.876			0.616285			0.308142	+	0.951340i	$0.400293\pi$
$632$		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	$0.0449778\pi$
$641$			180.547i			0.281664i	−0.990034	+	0.140832i	$0.955022\pi$
$642$		0			0
$643$	−701.008			−1.09021			−0.545107	−	0.838367i	$-0.683511\pi$
$643$	−701.008			−1.09021			−0.545107	+	0.838367i	$0.683511\pi$
$644$		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	$-0.940952\pi$
$653$		−	240.882i		−	0.368884i	0.982843	−	0.184442i	$-0.0590479\pi$
$654$		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	$-0.981214\pi$
$677$		−	79.8649i		−	0.117969i	0.998259	−	0.0589844i	$-0.0187862\pi$
$678$		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	$-0.856744\pi$
$691$	−1244.39			−1.80085			−0.900424	+	0.435013i	$0.856744\pi$
$692$		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	$-0.735841\pi$
$701$		−	1034.47i		−	1.47570i	0.674965	−	0.737850i	$-0.264159\pi$
$702$		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	$-0.645180\pi$
$727$	−640.411			−0.880895			−0.440447	+	0.897778i	$0.645180\pi$
$728$		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	$-0.529330\pi$
$739$	−135.992			−0.184022			−0.0920111	+	0.995758i	$0.529330\pi$
$740$		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	$-0.380216\pi$
$751$	551.977			0.734990			0.367495	+	0.930026i	$0.380216\pi$
$752$		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	$-0.993868\pi$
$761$		−	29.3201i		−	0.0385284i	0.999814	−	0.0192642i	$-0.00613236\pi$
$762$		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	$-0.907190\pi$
$773$		−	444.408i		−	0.574913i	0.957794	−	0.287457i	$-0.0928097\pi$
$774$		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	$-0.541930\pi$
$787$	−206.738			−0.262691			−0.131346	+	0.991337i	$0.541930\pi$
$788$		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	$-0.952844\pi$
$797$		−	235.281i		−	0.295208i	0.989047	−	0.147604i	$-0.0471562\pi$
$798$		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	$0.0111750\pi$
$809$			56.7919i			0.0702001i	−0.999384	+	0.0351000i	$0.988825\pi$
$810$		0			0
$811$	−743.889			−0.917249			−0.458625	−	0.888630i	$-0.651658\pi$
$811$	−743.889			−0.917249			−0.458625	+	0.888630i	$0.651658\pi$
$812$		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	$0.0612665\pi$
$821$			314.095i			0.382576i	−0.981534	+	0.191288i	$0.938733\pi$
$822$		0			0
$823$	1059.68			1.28758			0.643789	−	0.765203i	$-0.277361\pi$
$823$	1059.68			1.28758			0.643789	+	0.765203i	$0.277361\pi$
$824$		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	$0.0439884\pi$
$857$			236.110i			0.275508i	−0.990466	+	0.137754i	$0.956012\pi$
$858$		0			0
$859$	−997.970			−1.16178			−0.580891	−	0.813982i	$-0.697296\pi$
$859$	−997.970			−1.16178			−0.580891	+	0.813982i	$0.697296\pi$
$860$		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	$-0.950092\pi$
$881$		−	275.137i		−	0.312301i	0.987733	−	0.156150i	$-0.0499085\pi$
$882$		0			0
$883$	262.830			0.297656			0.148828	−	0.988863i	$-0.452450\pi$
$883$	262.830			0.297656			0.148828	+	0.988863i	$0.452450\pi$
$884$		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	$-0.249627\pi$
$907$	1284.19			1.41587			0.707934	+	0.706278i	$0.249627\pi$
$908$		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	$-0.463503\pi$
$919$	210.284			0.228818			0.114409	+	0.993434i	$0.463503\pi$
$920$		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	$0.200670\pi$
$929$			1095.27i			1.17898i	−0.807777	+	0.589488i	$0.799330\pi$
$930$		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	$0.126070\pi$
$941$			726.052i			0.771575i	−0.922588	+	0.385787i	$0.873930\pi$
$942$		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	$-0.800401\pi$
$953$		−	1118.37i		−	1.17353i	0.809758	−	0.586765i	$-0.199599\pi$
$954$		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	$-0.371489\pi$
$967$	759.775			0.785704			0.392852	+	0.919602i	$0.371489\pi$
$968$		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	$-0.848181\pi$
$977$		−	897.030i		−	0.918147i	0.888398	−	0.459074i	$-0.151819\pi$
$978$		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	$-0.463913\pi$
$991$	224.222			0.226258			0.113129	+	0.993580i	$0.463913\pi$
$992$		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	1200.3.l.y.401.1		12
3.2	odd	2	inner	1200.3.l.y.401.2		12
4.3	odd	2		600.3.l.g.401.12		12
5.2	odd	4		240.3.c.e.209.6		12
5.3	odd	4		240.3.c.e.209.7		12
5.4	even	2	inner	1200.3.l.y.401.12		12
12.11	even	2		600.3.l.g.401.11		12
15.2	even	4		240.3.c.e.209.8		12
15.8	even	4		240.3.c.e.209.5		12
15.14	odd	2	inner	1200.3.l.y.401.11		12
20.3	even	4		120.3.c.a.89.6	yes	12
20.7	even	4		120.3.c.a.89.7	yes	12
20.19	odd	2		600.3.l.g.401.1		12
40.3	even	4		960.3.c.k.449.7		12
40.13	odd	4		960.3.c.j.449.6		12
40.27	even	4		960.3.c.k.449.6		12
40.37	odd	4		960.3.c.j.449.7		12
60.23	odd	4		120.3.c.a.89.8	yes	12
60.47	odd	4		120.3.c.a.89.5	✓	12
60.59	even	2		600.3.l.g.401.2		12
120.53	even	4		960.3.c.j.449.8		12
120.77	even	4		960.3.c.j.449.5		12
120.83	odd	4		960.3.c.k.449.5		12
120.107	odd	4		960.3.c.k.449.8		12

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

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:	\( N \)	\(=\)	\( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1200.l (of order \(2\), degree \(1\), not minimal)

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