LMFDB - Embedded newform 1200.3.l.v.401.3

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 = [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:	$32.6976317232$
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:	no (minimal twist has level 600)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			401.3
Root			$-0.788754 + 0.994180i$ of defining polynomial
Character	$\chi$	$=$	1200.401
Dual form			1200.3.l.v.401.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.78875 - 2.40839i) q^{3} +12.2014 q^{7} +(-2.60072 + 8.61605i) q^{9} -8.79351i q^{11} +6.20144 q^{13} -16.6156i q^{17} +8.93396 q^{19} +(-21.8254 - 29.3859i) q^{21} +40.1210i q^{23} +(25.4029 - 9.14843i) q^{27} +4.94701i q^{29} +50.6043 q^{31} +(-21.1782 + 15.7294i) q^{33} -41.3407 q^{37} +(-11.0928 - 14.9355i) q^{39} -10.8654i q^{41} +39.5422 q^{43} +78.2993i q^{47} +99.8751 q^{49} +(-40.0170 + 29.7213i) q^{51} +69.7316i q^{53} +(-15.9807 - 21.5165i) q^{57} -96.0731i q^{59} +26.7287 q^{61} +(-31.7325 + 105.128i) q^{63} +66.5311 q^{67} +(96.6273 - 71.7667i) q^{69} -34.7444i q^{71} -70.5344 q^{73} -107.293i q^{77} -31.2164 q^{79} +(-67.4725 - 44.8158i) q^{81} -71.5420i q^{83} +(11.9143 - 8.84898i) q^{87} -30.4885i q^{89} +75.6665 q^{91} +(-90.5187 - 121.875i) q^{93} +108.194 q^{97} +(75.7653 + 22.8694i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times$.

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

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
$3$	−1.78875	−	2.40839i	−0.596251	−	0.802798i
$3$	−1.78875	−	2.40839i	−0.596251	−	0.802798i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	12.2014			1.74306			0.871531	−	0.490340i	$-0.163127\pi$
$7$	12.2014			1.74306			0.871531	+	0.490340i	$0.163127\pi$
$8$		0			0
$9$	−2.60072	+	8.61605i	−0.288969	+	0.957339i
$10$		0			0
$11$		−	8.79351i		−	0.799410i	−0.916644	−	0.399705i	$-0.869113\pi$
$11$		−	8.79351i		−	0.799410i	0.916644	−	0.399705i	$-0.130887\pi$
$12$		0			0
$13$	6.20144			0.477034			0.238517	−	0.971138i	$-0.423339\pi$
$13$	6.20144			0.477034			0.238517	+	0.971138i	$0.423339\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	16.6156i		−	0.977391i	−0.872455	−	0.488695i	$-0.837473\pi$
$17$		−	16.6156i		−	0.977391i	0.872455	−	0.488695i	$-0.162527\pi$
$18$		0			0
$19$	8.93396			0.470209			0.235104	−	0.971970i	$-0.424457\pi$
$19$	8.93396			0.470209			0.235104	+	0.971970i	$0.424457\pi$
$20$		0			0
$21$	−21.8254	−	29.3859i	−1.03930	−	1.39933i
$22$		0			0
$23$			40.1210i			1.74439i	0.489156	+	0.872197i	$0.337305\pi$
$23$			40.1210i			1.74439i	−0.489156	+	0.872197i	$0.662695\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	25.4029	−	9.14843i	0.940847	−	0.338831i
$28$		0			0
$29$			4.94701i			0.170587i	0.996356	+	0.0852933i	$0.0271827\pi$
$29$			4.94701i			0.170587i	−0.996356	+	0.0852933i	$0.972817\pi$
$30$		0			0
$31$	50.6043			1.63240			0.816199	−	0.577771i	$-0.196077\pi$
$31$	50.6043			1.63240			0.816199	+	0.577771i	$0.196077\pi$
$32$		0			0
$33$	−21.1782	+	15.7294i	−0.641764	+	0.476649i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−41.3407			−1.11732			−0.558658	−	0.829398i	$-0.688684\pi$
$37$	−41.3407			−1.11732			−0.558658	+	0.829398i	$0.688684\pi$
$38$		0			0
$39$	−11.0928	−	14.9355i	−0.284432	−	0.382962i
$40$		0			0
$41$		−	10.8654i		−	0.265010i	−0.991182	−	0.132505i	$-0.957698\pi$
$41$		−	10.8654i		−	0.265010i	0.991182	−	0.132505i	$-0.0423020\pi$
$42$		0			0
$43$	39.5422			0.919585			0.459792	−	0.888026i	$-0.347924\pi$
$43$	39.5422			0.919585			0.459792	+	0.888026i	$0.347924\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			78.2993i			1.66594i	0.553316	+	0.832972i	$0.313362\pi$
$47$			78.2993i			1.66594i	−0.553316	+	0.832972i	$0.686638\pi$
$48$		0			0
$49$	99.8751			2.03827
$50$		0			0
$51$	−40.0170	+	29.7213i	−0.784647	+	0.582770i
$52$		0			0
$53$			69.7316i			1.31569i	0.753153	+	0.657845i	$0.228532\pi$
$53$			69.7316i			1.31569i	−0.753153	+	0.657845i	$0.771468\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−15.9807	−	21.5165i	−0.280362	−	0.377482i
$58$		0			0
$59$		−	96.0731i		−	1.62836i	−0.580614	−	0.814179i	$-0.697187\pi$
$59$		−	96.0731i		−	1.62836i	0.580614	−	0.814179i	$-0.302813\pi$
$60$		0			0
$61$	26.7287			0.438175			0.219087	−	0.975705i	$-0.429692\pi$
$61$	26.7287			0.438175			0.219087	+	0.975705i	$0.429692\pi$
$62$		0			0
$63$	−31.7325	+	105.128i	−0.503691	+	1.66870i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	66.5311			0.993001			0.496501	−	0.868036i	$-0.334618\pi$
$67$	66.5311			0.993001			0.496501	+	0.868036i	$0.334618\pi$
$68$		0			0
$69$	96.6273	−	71.7667i	1.40040	−	1.04010i
$70$		0			0
$71$		−	34.7444i		−	0.489357i	−0.969604	−	0.244679i	$-0.921318\pi$
$71$		−	34.7444i		−	0.489357i	0.969604	−	0.244679i	$-0.0786824\pi$
$72$		0			0
$73$	−70.5344			−0.966225			−0.483112	−	0.875558i	$-0.660494\pi$
$73$	−70.5344			−0.966225			−0.483112	+	0.875558i	$0.660494\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	107.293i		−	1.39342i
$78$		0			0
$79$	−31.2164			−0.395144			−0.197572	−	0.980288i	$-0.563306\pi$
$79$	−31.2164			−0.395144			−0.197572	+	0.980288i	$0.563306\pi$
$80$		0			0
$81$	−67.4725	−	44.8158i	−0.832994	−	0.553282i
$82$		0			0
$83$		−	71.5420i		−	0.861951i	−0.902364	−	0.430976i	$-0.858169\pi$
$83$		−	71.5420i		−	0.861951i	0.902364	−	0.430976i	$-0.141831\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	11.9143	−	8.84898i	0.136947	−	0.101712i
$88$		0			0
$89$		−	30.4885i		−	0.342568i	−0.985222	−	0.171284i	$-0.945208\pi$
$89$		−	30.4885i		−	0.342568i	0.985222	−	0.171284i	$-0.0547916\pi$
$90$		0			0
$91$	75.6665			0.831500
$92$		0			0
$93$	−90.5187	−	121.875i	−0.973319	−	1.31049i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	108.194			1.11540			0.557699	−	0.830043i	$-0.311684\pi$
$97$	108.194			1.11540			0.557699	+	0.830043i	$0.311684\pi$
$98$		0			0
$99$	75.7653	+	22.8694i	0.765306	+	0.231005i
$100$		0			0
$101$		−	30.7501i		−	0.304456i	−0.988345	−	0.152228i	$-0.951355\pi$
$101$		−	30.7501i		−	0.304456i	0.988345	−	0.152228i	$-0.0486448\pi$
$102$		0			0
$103$	−49.7358			−0.482872			−0.241436	−	0.970417i	$-0.577618\pi$
$103$	−49.7358			−0.482872			−0.241436	+	0.970417i	$0.577618\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	188.098i		−	1.75792i	−0.476894	−	0.878961i	$-0.658238\pi$
$107$		−	188.098i		−	1.75792i	0.476894	−	0.878961i	$-0.341762\pi$
$108$		0			0
$109$	−72.7585			−0.667509			−0.333755	−	0.942660i	$-0.608316\pi$
$109$	−72.7585			−0.667509			−0.333755	+	0.942660i	$0.608316\pi$
$110$		0			0
$111$	73.9484	+	99.5647i	0.666201	+	0.896979i
$112$		0			0
$113$			79.0278i			0.699361i	0.936869	+	0.349681i	$0.113710\pi$
$113$			79.0278i			0.699361i	−0.936869	+	0.349681i	$0.886290\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−16.1282	+	53.4319i	−0.137848	+	0.456683i
$118$		0			0
$119$		−	202.735i		−	1.70365i
$120$		0			0
$121$	43.6742			0.360944
$122$		0			0
$123$	−26.1681	+	19.4355i	−0.212749	+	0.158012i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	194.410			1.53078			0.765392	−	0.643565i	$-0.222545\pi$
$127$	194.410			1.53078			0.765392	+	0.643565i	$0.222545\pi$
$128$		0			0
$129$	−70.7312	−	95.2331i	−0.548304	−	0.738241i
$130$		0			0
$131$		−	221.663i		−	1.69209i	−0.533114	−	0.846044i	$-0.678978\pi$
$131$		−	221.663i		−	1.69209i	0.533114	−	0.846044i	$-0.321022\pi$
$132$		0			0
$133$	109.007			0.819603
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	254.898i		−	1.86057i	−0.366839	−	0.930285i	$-0.619560\pi$
$137$		−	254.898i		−	1.86057i	0.366839	−	0.930285i	$-0.380440\pi$
$138$		0			0
$139$	103.142			0.742029			0.371015	−	0.928627i	$-0.379010\pi$
$139$	103.142			0.742029			0.371015	+	0.928627i	$0.379010\pi$
$140$		0			0
$141$	188.576	−	140.058i	1.33742	−	0.993321i
$142$		0			0
$143$		−	54.5324i		−	0.381345i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−178.652	−	240.539i	−1.21532	−	1.63632i
$148$		0			0
$149$		−	130.351i		−	0.874836i	−0.899258	−	0.437418i	$-0.855893\pi$
$149$		−	130.351i		−	0.874836i	0.899258	−	0.437418i	$-0.144107\pi$
$150$		0			0
$151$	−82.4800			−0.546225			−0.273113	−	0.961982i	$-0.588053\pi$
$151$	−82.4800			−0.546225			−0.273113	+	0.961982i	$0.588053\pi$
$152$		0			0
$153$	143.161	+	43.2126i	0.935694	+	0.282435i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	189.015			1.20392			0.601958	−	0.798527i	$-0.294387\pi$
$157$	189.015			1.20392			0.601958	+	0.798527i	$0.294387\pi$
$158$		0			0
$159$	167.941	−	124.733i	1.05623	−	0.784482i
$160$		0			0
$161$			489.534i			3.04059i
$162$		0			0
$163$	−222.963			−1.36787			−0.683935	−	0.729543i	$-0.739733\pi$
$163$	−222.963			−1.36787			−0.683935	+	0.729543i	$0.739733\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			96.2445i			0.576314i	0.957583	+	0.288157i	$0.0930426\pi$
$167$			96.2445i			0.576314i	−0.957583	+	0.288157i	$0.906957\pi$
$168$		0			0
$169$	−130.542			−0.772439
$170$		0			0
$171$	−23.2347	+	76.9754i	−0.135876	+	0.450149i
$172$		0			0
$173$		−	61.4068i		−	0.354952i	−0.984125	−	0.177476i	$-0.943207\pi$
$173$		−	61.4068i		−	0.354952i	0.984125	−	0.177476i	$-0.0567933\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−231.382	+	171.851i	−1.30724	+	0.970911i
$178$		0			0
$179$			172.095i			0.961425i	0.876878	+	0.480713i	$0.159622\pi$
$179$			172.095i			0.961425i	−0.876878	+	0.480713i	$0.840378\pi$
$180$		0			0
$181$	101.512			0.560841			0.280421	−	0.959877i	$-0.409526\pi$
$181$	101.512			0.560841			0.280421	+	0.959877i	$0.409526\pi$
$182$		0			0
$183$	−47.8110	−	64.3731i	−0.261262	−	0.351766i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−146.110			−0.781336
$188$		0			0
$189$	309.952	−	111.624i	1.63996	−	0.590603i
$190$		0			0
$191$		−	46.5031i		−	0.243472i	−0.992563	−	0.121736i	$-0.961154\pi$
$191$		−	46.5031i		−	0.243472i	0.992563	−	0.121736i	$-0.0388461\pi$
$192$		0			0
$193$	−241.951			−1.25363			−0.626816	−	0.779167i	$-0.715642\pi$
$193$	−241.951			−1.25363			−0.626816	+	0.779167i	$0.715642\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			375.594i			1.90657i	0.302074	+	0.953285i	$0.402321\pi$
$197$			375.594i			1.90657i	−0.302074	+	0.953285i	$0.597679\pi$
$198$		0			0
$199$	75.0149			0.376959			0.188480	−	0.982077i	$-0.439644\pi$
$199$	75.0149			0.376959			0.188480	+	0.982077i	$0.439644\pi$
$200$		0			0
$201$	−119.008	−	160.233i	−0.592078	−	0.797179i
$202$		0			0
$203$			60.3606i			0.297343i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−345.685	−	104.344i	−1.66997	−	0.504075i
$208$		0			0
$209$		−	78.5609i		−	0.375889i
$210$		0			0
$211$	−64.6919			−0.306597			−0.153298	−	0.988180i	$-0.548990\pi$
$211$	−64.6919			−0.306597			−0.153298	+	0.988180i	$0.548990\pi$
$212$		0			0
$213$	−83.6781	+	62.1491i	−0.392855	+	0.291780i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	617.446			2.84537
$218$		0			0
$219$	126.169	+	169.875i	0.576113	+	0.775683i
$220$		0			0
$221$		−	103.041i		−	0.466248i
$222$		0			0
$223$	250.487			1.12326			0.561629	−	0.827389i	$-0.310175\pi$
$223$	250.487			1.12326			0.561629	+	0.827389i	$0.310175\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			191.981i			0.845733i	0.906192	+	0.422866i	$0.138976\pi$
$227$			191.981i			0.845733i	−0.906192	+	0.422866i	$0.861024\pi$
$228$		0			0
$229$	−112.598			−0.491693			−0.245846	−	0.969309i	$-0.579066\pi$
$229$	−112.598			−0.491693			−0.245846	+	0.969309i	$0.579066\pi$
$230$		0			0
$231$	−258.405	+	191.922i	−1.11864	+	0.830829i
$232$		0			0
$233$			402.723i			1.72843i	0.503125	+	0.864213i	$0.332183\pi$
$233$			402.723i			1.72843i	−0.503125	+	0.864213i	$0.667817\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	55.8384	+	75.1813i	0.235605	+	0.317221i
$238$		0			0
$239$		−	99.9740i		−	0.418301i	−0.977883	−	0.209151i	$-0.932930\pi$
$239$		−	99.9740i		−	0.418301i	0.977883	−	0.209151i	$-0.0670699\pi$
$240$		0			0
$241$	1.82898			0.00758914			0.00379457	−	0.999993i	$-0.498792\pi$
$241$	1.82898			0.00758914			0.00379457	+	0.999993i	$0.498792\pi$
$242$		0			0
$243$	12.7575	+	242.665i	0.0525001	+	0.998621i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	55.4034			0.224305
$248$		0			0
$249$	−172.301	+	127.971i	−0.691973	+	0.513940i
$250$		0			0
$251$		−	349.307i		−	1.39166i	−0.718206	−	0.695831i	$-0.755036\pi$
$251$		−	349.307i		−	1.39166i	0.718206	−	0.695831i	$-0.244964\pi$
$252$		0			0
$253$	352.805			1.39448
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	12.7334i		−	0.0495463i	−0.999693	−	0.0247731i	$-0.992114\pi$
$257$		−	12.7334i		−	0.0495463i	0.999693	−	0.0247731i	$-0.00788634\pi$
$258$		0			0
$259$	−504.416			−1.94755
$260$		0			0
$261$	−42.6237	−	12.8658i	−0.163309	−	0.0492942i
$262$		0			0
$263$		−	188.806i		−	0.717892i	−0.933358	−	0.358946i	$-0.883136\pi$
$263$		−	188.806i		−	0.717892i	0.933358	−	0.358946i	$-0.116864\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−73.4284	+	54.5365i	−0.275013	+	0.204257i
$268$		0			0
$269$			113.044i			0.420237i	0.977676	+	0.210118i	$0.0673849\pi$
$269$			113.044i			0.420237i	−0.977676	+	0.210118i	$0.932615\pi$
$270$		0			0
$271$	−22.6737			−0.0836667			−0.0418334	−	0.999125i	$-0.513320\pi$
$271$	−22.6737			−0.0836667			−0.0418334	+	0.999125i	$0.513320\pi$
$272$		0			0
$273$	−135.349	−	182.235i	−0.495783	−	0.667526i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−341.284			−1.23207			−0.616035	−	0.787719i	$-0.711262\pi$
$277$	−341.284			−1.23207			−0.616035	+	0.787719i	$0.711262\pi$
$278$		0			0
$279$	−131.608	+	436.009i	−0.471712	+	1.56276i
$280$		0			0
$281$			5.77214i			0.0205414i	0.999947	+	0.0102707i	$0.00326933\pi$
$281$			5.77214i			0.0205414i	−0.999947	+	0.0102707i	$0.996731\pi$
$282$		0			0
$283$	14.9406			0.0527937			0.0263968	−	0.999652i	$-0.491597\pi$
$283$	14.9406			0.0527937			0.0263968	+	0.999652i	$0.491597\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	132.573i		−	0.461928i
$288$		0			0
$289$	12.9205			0.0447075
$290$		0			0
$291$	−193.532	−	260.573i	−0.665058	−	0.895440i
$292$		0			0
$293$			6.23602i			0.0212833i	0.999943	+	0.0106417i	$0.00338741\pi$
$293$			6.23602i			0.0212833i	−0.999943	+	0.0106417i	$0.996613\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−80.4468	−	223.380i	−0.270865	−	0.752123i
$298$		0			0
$299$			248.808i			0.832134i
$300$		0			0
$301$	482.471			1.60289
$302$		0			0
$303$	−74.0583	+	55.0043i	−0.244417	+	0.181532i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	283.513			0.923496			0.461748	−	0.887011i	$-0.347222\pi$
$307$	283.513			0.923496			0.461748	+	0.887011i	$0.347222\pi$
$308$		0			0
$309$	88.9652	+	119.784i	0.287913	+	0.387649i
$310$		0			0
$311$			108.620i			0.349260i	0.984634	+	0.174630i	$0.0558729\pi$
$311$			108.620i			0.349260i	−0.984634	+	0.174630i	$0.944127\pi$
$312$		0			0
$313$	387.197			1.23705			0.618525	−	0.785765i	$-0.287730\pi$
$313$	387.197			1.23705			0.618525	+	0.785765i	$0.287730\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	118.772i		−	0.374675i	−0.982296	−	0.187338i	$-0.940014\pi$
$317$		−	118.772i		−	0.374675i	0.982296	−	0.187338i	$-0.0599858\pi$
$318$		0			0
$319$	43.5016			0.136369
$320$		0			0
$321$	−453.013	+	336.460i	−1.41126	+	1.04816i
$322$		0			0
$323$		−	148.444i		−	0.459577i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	130.147	+	175.231i	0.398003	+	0.535875i
$328$		0			0
$329$			955.365i			2.90384i
$330$		0			0
$331$	494.645			1.49439			0.747197	−	0.664602i	$-0.231399\pi$
$331$	494.645			1.49439			0.747197	+	0.664602i	$0.231399\pi$
$332$		0			0
$333$	107.516	−	356.193i	0.322870	−	1.06965i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−14.6414			−0.0434463			−0.0217231	−	0.999764i	$-0.506915\pi$
$337$	−14.6414			−0.0434463			−0.0217231	+	0.999764i	$0.506915\pi$
$338$		0			0
$339$	190.330	−	141.361i	0.561446	−	0.416995i
$340$		0			0
$341$		−	444.989i		−	1.30495i
$342$		0			0
$343$	620.750			1.80977
$344$		0			0
$345$		0			0
$346$		0			0
$347$			146.105i			0.421053i	0.977588	+	0.210526i	$0.0675178\pi$
$347$			146.105i			0.421053i	−0.977588	+	0.210526i	$0.932482\pi$
$348$		0			0
$349$	−594.189			−1.70255			−0.851273	−	0.524723i	$-0.824169\pi$
$349$	−594.189			−1.70255			−0.851273	+	0.524723i	$0.824169\pi$
$350$		0			0
$351$	157.534	−	56.7334i	0.448816	−	0.161634i
$352$		0			0
$353$		−	81.0080i		−	0.229484i	−0.993395	−	0.114742i	$-0.963396\pi$
$353$		−	81.0080i		−	0.229484i	0.993395	−	0.114742i	$-0.0366042\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−488.265	+	362.643i	−1.36769	+	1.01581i
$358$		0			0
$359$			290.075i			0.808009i	0.914757	+	0.404005i	$0.132382\pi$
$359$			290.075i			0.808009i	−0.914757	+	0.404005i	$0.867618\pi$
$360$		0			0
$361$	−281.184			−0.778904
$362$		0			0
$363$	−78.1224	−	105.185i	−0.215213	−	0.289765i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	168.056			0.457918			0.228959	−	0.973436i	$-0.426468\pi$
$367$	168.056			0.457918			0.228959	+	0.973436i	$0.426468\pi$
$368$		0			0
$369$	93.6167	+	28.2578i	0.253704	+	0.0765795i
$370$		0			0
$371$			850.826i			2.29333i
$372$		0			0
$373$	237.510			0.636756			0.318378	−	0.947964i	$-0.396862\pi$
$373$	237.510			0.636756			0.318378	+	0.947964i	$0.396862\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			30.6786i			0.0813756i
$378$		0			0
$379$	−136.385			−0.359854			−0.179927	−	0.983680i	$-0.557586\pi$
$379$	−136.385			−0.359854			−0.179927	+	0.983680i	$0.557586\pi$
$380$		0			0
$381$	−347.751	−	468.215i	−0.912732	−	1.22891i
$382$		0			0
$383$			202.859i			0.529658i	0.964295	+	0.264829i	$0.0853155\pi$
$383$			202.859i			0.529658i	−0.964295	+	0.264829i	$0.914685\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−102.838	+	340.697i	−0.265731	+	0.880354i
$388$		0			0
$389$		−	60.7715i		−	0.156225i	−0.996945	−	0.0781125i	$-0.975111\pi$
$389$		−	60.7715i		−	0.156225i	0.996945	−	0.0781125i	$-0.0248893\pi$
$390$		0			0
$391$	666.637			1.70495
$392$		0			0
$393$	−533.853	+	396.501i	−1.35840	+	1.00891i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−456.360			−1.14952			−0.574761	−	0.818321i	$-0.694905\pi$
$397$	−456.360			−1.14952			−0.574761	+	0.818321i	$0.694905\pi$
$398$		0			0
$399$	−194.987	−	262.532i	−0.488689	−	0.657976i
$400$		0			0
$401$			479.363i			1.19542i	0.801712	+	0.597710i	$0.203923\pi$
$401$			479.363i			1.19542i	−0.801712	+	0.597710i	$0.796077\pi$
$402$		0			0
$403$	313.820			0.778709
$404$		0			0
$405$		0			0
$406$		0			0
$407$			363.530i			0.893194i
$408$		0			0
$409$	−45.2033			−0.110522			−0.0552608	−	0.998472i	$-0.517599\pi$
$409$	−45.2033			−0.110522			−0.0552608	+	0.998472i	$0.517599\pi$
$410$		0			0
$411$	−613.895	+	455.950i	−1.49366	+	1.10937i
$412$		0			0
$413$		−	1172.23i		−	2.83833i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−184.496	−	248.407i	−0.442436	−	0.595699i
$418$		0			0
$419$		−	199.218i		−	0.475461i	−0.971331	−	0.237730i	$-0.923597\pi$
$419$		−	199.218i		−	0.475461i	0.971331	−	0.237730i	$-0.0764034\pi$
$420$		0			0
$421$	−73.7967			−0.175289			−0.0876445	−	0.996152i	$-0.527934\pi$
$421$	−73.7967			−0.175289			−0.0876445	+	0.996152i	$0.527934\pi$
$422$		0			0
$423$	−674.631	−	203.635i	−1.59487	−	0.481406i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	326.128			0.763766
$428$		0			0
$429$	−131.335	+	97.5450i	−0.306143	+	0.227378i
$430$		0			0
$431$		−	42.2198i		−	0.0979577i	−0.998800	−	0.0489789i	$-0.984403\pi$
$431$		−	42.2198i		−	0.0979577i	0.998800	−	0.0489789i	$-0.0155967\pi$
$432$		0			0
$433$	−434.287			−1.00297			−0.501486	−	0.865166i	$-0.667213\pi$
$433$	−434.287			−1.00297			−0.501486	+	0.865166i	$0.667213\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			358.440i			0.820228i
$438$		0			0
$439$	−739.972			−1.68558			−0.842792	−	0.538239i	$-0.819090\pi$
$439$	−739.972			−1.68558			−0.842792	+	0.538239i	$0.819090\pi$
$440$		0			0
$441$	−259.747	+	860.529i	−0.588996	+	1.95131i
$442$		0			0
$443$		−	127.665i		−	0.288184i	−0.989564	−	0.144092i	$-0.953974\pi$
$443$		−	127.665i		−	0.288184i	0.989564	−	0.144092i	$-0.0460261\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−313.935	+	233.165i	−0.702316	+	0.521622i
$448$		0			0
$449$		−	200.120i		−	0.445701i	−0.974853	−	0.222850i	$-0.928464\pi$
$449$		−	200.120i		−	0.445701i	0.974853	−	0.222850i	$-0.0715361\pi$
$450$		0			0
$451$	−95.5449			−0.211851
$452$		0			0
$453$	147.536	+	198.644i	0.325687	+	0.438508i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−364.615			−0.797845			−0.398923	−	0.916985i	$-0.630616\pi$
$457$	−364.615			−0.797845			−0.398923	+	0.916985i	$0.630616\pi$
$458$		0			0
$459$	−152.007	−	422.085i	−0.331170	−	0.919575i
$460$		0			0
$461$		−	39.0715i		−	0.0847538i	−0.999102	−	0.0423769i	$-0.986507\pi$
$461$		−	39.0715i		−	0.0847538i	0.999102	−	0.0423769i	$-0.0134930\pi$
$462$		0			0
$463$	−403.683			−0.871887			−0.435943	−	0.899974i	$-0.643585\pi$
$463$	−403.683			−0.871887			−0.435943	+	0.899974i	$0.643585\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			535.689i			1.14709i	0.819175	+	0.573543i	$0.194432\pi$
$467$			535.689i			1.14709i	−0.819175	+	0.573543i	$0.805568\pi$
$468$		0			0
$469$	811.775			1.73086
$470$		0			0
$471$	−338.101	−	455.222i	−0.717837	−	0.966502i
$472$		0			0
$473$		−	347.714i		−	0.735125i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−600.811	−	181.352i	−1.25956	−	0.380194i
$478$		0			0
$479$			768.476i			1.60433i	0.597099	+	0.802167i	$0.296320\pi$
$479$			768.476i			1.60433i	−0.597099	+	0.802167i	$0.703680\pi$
$480$		0			0
$481$	−256.372			−0.532998
$482$		0			0
$483$	1178.99	−	875.657i	2.44098	−	1.81295i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−130.785			−0.268553			−0.134276	−	0.990944i	$-0.542871\pi$
$487$	−130.785			−0.268553			−0.134276	+	0.990944i	$0.542871\pi$
$488$		0			0
$489$	398.825	+	536.982i	0.815594	+	1.09812i
$490$		0			0
$491$			576.106i			1.17333i	0.809829	+	0.586666i	$0.199560\pi$
$491$			576.106i			1.17333i	−0.809829	+	0.586666i	$0.800440\pi$
$492$		0			0
$493$	82.1977			0.166730
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	423.931i		−	0.852980i
$498$		0			0
$499$	−569.803			−1.14189			−0.570945	−	0.820988i	$-0.693423\pi$
$499$	−569.803			−1.14189			−0.570945	+	0.820988i	$0.693423\pi$
$500$		0			0
$501$	231.795	−	172.158i	0.462664	−	0.343628i
$502$		0			0
$503$			167.894i			0.333784i	0.985975	+	0.166892i	$0.0533732\pi$
$503$			167.894i			0.333784i	−0.985975	+	0.166892i	$0.946627\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	233.508	+	314.397i	0.460568	+	0.620112i
$508$		0			0
$509$		−	422.910i		−	0.830865i	−0.909624	−	0.415433i	$-0.863630\pi$
$509$		−	422.910i		−	0.830865i	0.909624	−	0.415433i	$-0.136370\pi$
$510$		0			0
$511$	−860.621			−1.68419
$512$		0			0
$513$	226.948	−	81.7317i	0.442394	−	0.159321i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	688.526			1.33177
$518$		0			0
$519$	−147.892	+	109.842i	−0.284955	+	0.211641i
$520$		0			0
$521$		−	412.630i		−	0.791995i	−0.918251	−	0.395998i	$-0.870399\pi$
$521$		−	412.630i		−	0.791995i	0.918251	−	0.395998i	$-0.129601\pi$
$522$		0			0
$523$	38.1647			0.0729727			0.0364864	−	0.999334i	$-0.488383\pi$
$523$	38.1647			0.0729727			0.0364864	+	0.999334i	$0.488383\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	840.823i		−	1.59549i
$528$		0			0
$529$	−1080.70			−2.04291
$530$		0			0
$531$	827.771	+	249.859i	1.55889	+	0.470545i
$532$		0			0
$533$		−	67.3811i		−	0.126419i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	414.473	−	307.836i	0.771830	−	0.573251i
$538$		0			0
$539$		−	878.253i		−	1.62941i
$540$		0			0
$541$	671.986			1.24212			0.621059	−	0.783764i	$-0.286703\pi$
$541$	671.986			1.24212			0.621059	+	0.783764i	$0.286703\pi$
$542$		0			0
$543$	−181.580	−	244.482i	−0.334402	−	0.450242i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−56.8126			−0.103862			−0.0519311	−	0.998651i	$-0.516538\pi$
$547$	−56.8126			−0.103862			−0.0519311	+	0.998651i	$0.516538\pi$
$548$		0			0
$549$	−69.5137	+	230.295i	−0.126619	+	0.419481i
$550$		0			0
$551$			44.1964i			0.0802113i
$552$		0			0
$553$	−380.885			−0.688761
$554$		0			0
$555$		0			0
$556$		0			0
$557$			240.349i			0.431507i	0.976448	+	0.215753i	$0.0692207\pi$
$557$			240.349i			0.431507i	−0.976448	+	0.215753i	$0.930779\pi$
$558$		0			0
$559$	245.218			0.438673
$560$		0			0
$561$	261.354	+	351.890i	0.465872	+	0.627255i
$562$		0			0
$563$			156.807i			0.278521i	0.990256	+	0.139261i	$0.0444725\pi$
$563$			156.807i			0.278521i	−0.990256	+	0.139261i	$0.955527\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−823.262	−	546.818i	−1.45196	−	0.964405i
$568$		0			0
$569$		−	457.623i		−	0.804258i	−0.915583	−	0.402129i	$-0.868270\pi$
$569$		−	457.623i		−	0.804258i	0.915583	−	0.402129i	$-0.131730\pi$
$570$		0			0
$571$	609.680			1.06774			0.533870	−	0.845566i	$-0.320737\pi$
$571$	609.680			1.06774			0.533870	+	0.845566i	$0.320737\pi$
$572$		0			0
$573$	−111.998	+	83.1827i	−0.195459	+	0.145170i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	565.696			0.980409			0.490204	−	0.871608i	$-0.336922\pi$
$577$	565.696			0.980409			0.490204	+	0.871608i	$0.336922\pi$
$578$		0			0
$579$	432.791	+	582.714i	0.747480	+	1.00641i
$580$		0			0
$581$		−	872.915i		−	1.50244i
$582$		0			0
$583$	613.185			1.05178
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	555.622i		−	0.946545i	−0.880916	−	0.473272i	$-0.843073\pi$
$587$		−	555.622i		−	0.946545i	0.880916	−	0.473272i	$-0.156927\pi$
$588$		0			0
$589$	452.097			0.767567
$590$		0			0
$591$	904.578	−	671.845i	1.53059	−	1.13679i
$592$		0			0
$593$			166.238i			0.280333i	0.990128	+	0.140167i	$0.0447638\pi$
$593$			166.238i			0.280333i	−0.990128	+	0.140167i	$0.955236\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−134.183	−	180.665i	−0.224763	−	0.302622i
$598$		0			0
$599$			629.119i			1.05028i	0.851015	+	0.525141i	$0.175987\pi$
$599$			629.119i			1.05028i	−0.851015	+	0.525141i	$0.824013\pi$
$600$		0			0
$601$	−616.381			−1.02559			−0.512796	−	0.858511i	$-0.671390\pi$
$601$	−616.381			−1.02559			−0.512796	+	0.858511i	$0.671390\pi$
$602$		0			0
$603$	−173.029	+	573.235i	−0.286946	+	0.950638i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	835.637			1.37667			0.688334	−	0.725394i	$-0.258343\pi$
$607$	835.637			1.37667			0.688334	+	0.725394i	$0.258343\pi$
$608$		0			0
$609$	145.372	−	107.970i	0.238706	−	0.177291i
$610$		0			0
$611$			485.569i			0.794711i
$612$		0			0
$613$	111.889			0.182527			0.0912634	−	0.995827i	$-0.470909\pi$
$613$	111.889			0.182527			0.0912634	+	0.995827i	$0.470909\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			749.519i			1.21478i	0.794404	+	0.607390i	$0.207783\pi$
$617$			749.519i			1.21478i	−0.794404	+	0.607390i	$0.792217\pi$
$618$		0			0
$619$	195.232			0.315398			0.157699	−	0.987487i	$-0.449592\pi$
$619$	195.232			0.315398			0.157699	+	0.987487i	$0.449592\pi$
$620$		0			0
$621$	367.045	+	1019.19i	0.591054	+	1.64121i
$622$		0			0
$623$		−	372.004i		−	0.597117i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−189.205	+	140.526i	−0.301763	+	0.224124i
$628$		0			0
$629$			686.902i			1.09205i
$630$		0			0
$631$	−273.937			−0.434132			−0.217066	−	0.976157i	$-0.569649\pi$
$631$	−273.937			−0.434132			−0.217066	+	0.976157i	$0.569649\pi$
$632$		0			0
$633$	115.718	+	155.804i	0.182809	+	0.246135i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	619.369			0.972323
$638$		0			0
$639$	299.359	+	90.3603i	0.468480	+	0.141409i
$640$		0			0
$641$			22.1793i			0.0346010i	0.999850	+	0.0173005i	$0.00550720\pi$
$641$			22.1793i			0.0346010i	−0.999850	+	0.0173005i	$0.994493\pi$
$642$		0			0
$643$	−1188.17			−1.84785			−0.923926	−	0.382572i	$-0.875039\pi$
$643$	−1188.17			−1.84785			−0.923926	+	0.382572i	$0.875039\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	333.016i		−	0.514708i	−0.966317	−	0.257354i	$-0.917149\pi$
$647$		−	333.016i		−	0.514708i	0.966317	−	0.257354i	$-0.0828507\pi$
$648$		0			0
$649$	−844.820			−1.30173
$650$		0			0
$651$	−1104.46	−	1487.05i	−1.69656	−	2.28426i
$652$		0			0
$653$		−	71.5403i		−	0.109556i	−0.998499	−	0.0547781i	$-0.982555\pi$
$653$		−	71.5403i		−	0.109556i	0.998499	−	0.0547781i	$-0.0174452\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	183.440	−	607.728i	0.279209	−	0.925004i
$658$		0			0
$659$			724.319i			1.09912i	0.835455	+	0.549559i	$0.185204\pi$
$659$			724.319i			1.09912i	−0.835455	+	0.549559i	$0.814796\pi$
$660$		0			0
$661$	545.047			0.824579			0.412289	−	0.911053i	$-0.364729\pi$
$661$	545.047			0.824579			0.412289	+	0.911053i	$0.364729\pi$
$662$		0			0
$663$	−248.163	+	184.315i	−0.374303	+	0.278001i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−198.479			−0.297570
$668$		0			0
$669$	−448.059	−	603.270i	−0.669744	−	0.901749i
$670$		0			0
$671$		−	235.039i		−	0.350281i
$672$		0			0
$673$	1106.67			1.64439			0.822193	−	0.569208i	$-0.192750\pi$
$673$	1106.67			1.64439			0.822193	+	0.569208i	$0.192750\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			1011.02i			1.49338i	0.665172	+	0.746690i	$0.268358\pi$
$677$			1011.02i			1.49338i	−0.665172	+	0.746690i	$0.731642\pi$
$678$		0			0
$679$	1320.12			1.94421
$680$		0			0
$681$	462.367	−	343.407i	0.678953	−	0.504269i
$682$		0			0
$683$			1073.43i			1.57163i	0.618460	+	0.785816i	$0.287757\pi$
$683$			1073.43i			1.57163i	−0.618460	+	0.785816i	$0.712243\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	201.410	+	271.180i	0.293172	+	0.394730i
$688$		0			0
$689$			432.436i			0.627629i
$690$		0			0
$691$	−685.207			−0.991617			−0.495809	−	0.868432i	$-0.665128\pi$
$691$	−685.207			−0.991617			−0.495809	+	0.868432i	$0.665128\pi$
$692$		0			0
$693$	924.445	+	279.040i	1.33398	+	0.402655i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−180.535			−0.259018
$698$		0			0
$699$	969.917	−	720.373i	1.38758	−	1.03058i
$700$		0			0
$701$		−	475.846i		−	0.678810i	−0.940640	−	0.339405i	$-0.889774\pi$
$701$		−	475.846i		−	0.678810i	0.940640	−	0.339405i	$-0.110226\pi$
$702$		0			0
$703$	−369.336			−0.525372
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	375.195i		−	0.530686i
$708$		0			0
$709$	296.954			0.418835			0.209418	−	0.977826i	$-0.432843\pi$
$709$	296.954			0.418835			0.209418	+	0.977826i	$0.432843\pi$
$710$		0			0
$711$	81.1850	−	268.962i	0.114184	−	0.378287i
$712$		0			0
$713$			2030.30i			2.84754i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−240.777	+	178.829i	−0.335811	+	0.249413i
$718$		0			0
$719$		−	1335.46i		−	1.85739i	−0.370844	−	0.928695i	$-0.620932\pi$
$719$		−	1335.46i		−	1.85739i	0.370844	−	0.928695i	$-0.379068\pi$
$720$		0			0
$721$	−606.849			−0.841677
$722$		0			0
$723$	−3.27160	−	4.40491i	−0.00452503	−	0.00609254i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−941.614			−1.29521			−0.647603	−	0.761978i	$-0.724228\pi$
$727$	−941.614			−1.29521			−0.647603	+	0.761978i	$0.724228\pi$
$728$		0			0
$729$	561.612	−	464.793i	0.770387	−	0.637576i
$730$		0			0
$731$		−	657.018i		−	0.898794i
$732$		0			0
$733$	−1073.15			−1.46405			−0.732025	−	0.681278i	$-0.761425\pi$
$733$	−1073.15			−1.46405			−0.732025	+	0.681278i	$0.761425\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	585.042i		−	0.793815i
$738$		0			0
$739$	76.4461			0.103445			0.0517226	−	0.998661i	$-0.483529\pi$
$739$	76.4461			0.103445			0.0517226	+	0.998661i	$0.483529\pi$
$740$		0			0
$741$	−99.1031	−	133.433i	−0.133742	−	0.180072i
$742$		0			0
$743$			78.2278i			0.105286i	0.998613	+	0.0526432i	$0.0167646\pi$
$743$			78.2278i			0.105286i	−0.998613	+	0.0526432i	$0.983235\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	616.409	+	186.061i	0.825179	+	0.249077i
$748$		0			0
$749$		−	2295.06i		−	3.06417i
$750$		0			0
$751$	−1150.75			−1.53229			−0.766143	−	0.642670i	$-0.777827\pi$
$751$	−1150.75			−1.53229			−0.766143	+	0.642670i	$0.777827\pi$
$752$		0			0
$753$	−841.269	+	624.824i	−1.11722	+	0.829780i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	257.571			0.340252			0.170126	−	0.985422i	$-0.445582\pi$
$757$	257.571			0.340252			0.170126	+	0.985422i	$0.445582\pi$
$758$		0			0
$759$	−631.081	−	849.693i	−0.831463	−	1.11949i
$760$		0			0
$761$			1001.67i			1.31626i	0.752906	+	0.658128i	$0.228652\pi$
$761$			1001.67i			1.31626i	−0.752906	+	0.658128i	$0.771348\pi$
$762$		0			0
$763$	−887.759			−1.16351
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	595.792i		−	0.776782i
$768$		0			0
$769$	−132.122			−0.171810			−0.0859050	−	0.996303i	$-0.527378\pi$
$769$	−132.122			−0.171810			−0.0859050	+	0.996303i	$0.527378\pi$
$770$		0			0
$771$	−30.6670	+	22.7769i	−0.0397757	+	0.0295420i
$772$		0			0
$773$			254.447i			0.329168i	0.986363	+	0.164584i	$0.0526281\pi$
$773$			254.447i			0.329168i	−0.986363	+	0.164584i	$0.947372\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	902.276	+	1214.83i	1.16123	+	1.56349i
$778$		0			0
$779$		−	97.0710i		−	0.124610i
$780$		0			0
$781$	−305.525			−0.391197
$782$		0			0
$783$	45.2574	+	125.668i	0.0578000	+	0.160496i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−1278.73			−1.62481			−0.812407	−	0.583091i	$-0.801843\pi$
$787$	−1278.73			−1.62481			−0.812407	+	0.583091i	$0.801843\pi$
$788$		0			0
$789$	−454.718	+	337.727i	−0.576323	+	0.428044i
$790$		0			0
$791$			964.253i			1.21903i
$792$		0			0
$793$	165.756			0.209024
$794$		0			0
$795$		0			0
$796$		0			0
$797$			747.744i			0.938199i	0.883145	+	0.469099i	$0.155421\pi$
$797$			747.744i			0.938199i	−0.883145	+	0.469099i	$0.844579\pi$
$798$		0			0
$799$	1300.99			1.62828
$800$		0			0
$801$	262.691	+	79.2922i	0.327954	+	0.0989915i
$802$		0			0
$803$			620.245i			0.772410i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	272.254	−	202.207i	0.337365	−	0.250567i
$808$		0			0
$809$		−	340.146i		−	0.420453i	−0.977653	−	0.210226i	$-0.932580\pi$
$809$		−	340.146i		−	0.420453i	0.977653	−	0.210226i	$-0.0674201\pi$
$810$		0			0
$811$	1112.71			1.37202			0.686011	−	0.727591i	$-0.259360\pi$
$811$	1112.71			1.37202			0.686011	+	0.727591i	$0.259360\pi$
$812$		0			0
$813$	40.5576	+	54.6071i	0.0498864	+	0.0671675i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	353.268			0.432397
$818$		0			0
$819$	−196.787	+	651.946i	−0.240278	+	0.796027i
$820$		0			0
$821$		−	1283.05i		−	1.56279i	−0.624036	−	0.781395i	$-0.714508\pi$
$821$		−	1283.05i		−	1.56279i	0.624036	−	0.781395i	$-0.285492\pi$
$822$		0			0
$823$	1430.32			1.73794			0.868968	−	0.494867i	$-0.164783\pi$
$823$	1430.32			1.73794			0.868968	+	0.494867i	$0.164783\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	144.113i		−	0.174260i	−0.996197	−	0.0871300i	$-0.972230\pi$
$827$		−	144.113i		−	0.174260i	0.996197	−	0.0871300i	$-0.0277695\pi$
$828$		0			0
$829$	1116.06			1.34627			0.673137	−	0.739518i	$-0.264946\pi$
$829$	1116.06			1.34627			0.673137	+	0.739518i	$0.264946\pi$
$830$		0			0
$831$	610.472	+	821.945i	0.734624	+	0.989104i
$832$		0			0
$833$		−	1659.49i		−	1.99218i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	1285.50	−	462.950i	1.53584	−	0.553106i
$838$		0			0
$839$			949.194i			1.13134i	0.824632	+	0.565670i	$0.191382\pi$
$839$			949.194i			1.13134i	−0.824632	+	0.565670i	$0.808618\pi$
$840$		0			0
$841$	816.527			0.970900
$842$		0			0
$843$	13.9016	−	10.3249i	0.0164906	−	0.0122479i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	532.888			0.629148
$848$		0			0
$849$	−26.7251	−	35.9829i	−0.0314783	−	0.0423827i
$850$		0			0
$851$		−	1658.63i		−	1.94904i
$852$		0			0
$853$	−666.717			−0.781614			−0.390807	−	0.920473i	$-0.627804\pi$
$853$	−666.717			−0.781614			−0.390807	+	0.920473i	$0.627804\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	788.585i		−	0.920169i	−0.887875	−	0.460085i	$-0.847819\pi$
$857$		−	788.585i		−	0.920169i	0.887875	−	0.460085i	$-0.152181\pi$
$858$		0			0
$859$	−406.487			−0.473209			−0.236605	−	0.971606i	$-0.576035\pi$
$859$	−406.487			−0.473209			−0.236605	+	0.971606i	$0.576035\pi$
$860$		0			0
$861$	−319.289	+	237.141i	−0.370835	+	0.275425i
$862$		0			0
$863$			1031.38i			1.19512i	0.801826	+	0.597558i	$0.203862\pi$
$863$			1031.38i			1.19512i	−0.801826	+	0.597558i	$0.796138\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−23.1116	−	31.1176i	−0.0266569	−	0.0358911i
$868$		0			0
$869$			274.501i			0.315882i
$870$		0			0
$871$	412.588			0.473695
$872$		0			0
$873$	−281.381	+	932.202i	−0.322316	+	1.06781i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−725.461			−0.827208			−0.413604	−	0.910457i	$-0.635730\pi$
$877$	−725.461			−0.827208			−0.413604	+	0.910457i	$0.635730\pi$
$878$		0			0
$879$	15.0188	−	11.1547i	0.0170862	−	0.0126902i
$880$		0			0
$881$		−	282.083i		−	0.320185i	−0.987102	−	0.160092i	$-0.948821\pi$
$881$		−	282.083i		−	0.320185i	0.987102	−	0.160092i	$-0.0511792\pi$
$882$		0			0
$883$	−454.628			−0.514868			−0.257434	−	0.966296i	$-0.582877\pi$
$883$	−454.628			−0.514868			−0.257434	+	0.966296i	$0.582877\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	309.991i		−	0.349482i	−0.984614	−	0.174741i	$-0.944091\pi$
$887$		−	309.991i		−	0.349482i	0.984614	−	0.174741i	$-0.0559088\pi$
$888$		0			0
$889$	2372.08			2.66825
$890$		0			0
$891$	−394.088	+	593.320i	−0.442299	+	0.665904i
$892$		0			0
$893$			699.523i			0.783341i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	599.228	−	445.057i	0.668036	−	0.496161i
$898$		0			0
$899$			250.340i			0.278465i
$900$		0			0
$901$	1158.64			1.28594
$902$		0			0
$903$	−863.022	−	1161.98i	−0.955728	−	1.28680i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	87.5518			0.0965290			0.0482645	−	0.998835i	$-0.484631\pi$
$907$	87.5518			0.0965290			0.0482645	+	0.998835i	$0.484631\pi$
$908$		0			0
$909$	264.944	+	79.9723i	0.291468	+	0.0879784i
$910$		0			0
$911$		−	685.361i		−	0.752317i	−0.926555	−	0.376158i	$-0.877245\pi$
$911$		−	685.361i		−	0.752317i	0.926555	−	0.376158i	$-0.122755\pi$
$912$		0			0
$913$	−629.105			−0.689052
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	2704.61i		−	2.94941i
$918$		0			0
$919$	285.772			0.310960			0.155480	−	0.987839i	$-0.450308\pi$
$919$	285.772			0.310960			0.155480	+	0.987839i	$0.450308\pi$
$920$		0			0
$921$	−507.135	−	682.811i	−0.550635	−	0.741380i
$922$		0			0
$923$		−	215.465i		−	0.233440i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	129.349	−	428.526i	0.139535	−	0.462272i
$928$		0			0
$929$		−	1319.46i		−	1.42031i	−0.704048	−	0.710153i	$-0.748626\pi$
$929$		−	1319.46i		−	1.42031i	0.704048	−	0.710153i	$-0.251374\pi$
$930$		0			0
$931$	892.281			0.958411
$932$		0			0
$933$	261.599	−	194.294i	0.280385	−	0.208246i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−820.446			−0.875609			−0.437805	−	0.899070i	$-0.644244\pi$
$937$	−820.446			−0.875609			−0.437805	+	0.899070i	$0.644244\pi$
$938$		0			0
$939$	−692.600	−	932.522i	−0.737593	−	0.993102i
$940$		0			0
$941$			1134.15i			1.20526i	0.798020	+	0.602631i	$0.205881\pi$
$941$			1134.15i			1.20526i	−0.798020	+	0.602631i	$0.794119\pi$
$942$		0			0
$943$	435.931			0.462281
$944$		0			0
$945$		0			0
$946$		0			0
$947$			513.273i			0.541999i	0.962580	+	0.270999i	$0.0873541\pi$
$947$			513.273i			0.541999i	−0.962580	+	0.270999i	$0.912646\pi$
$948$		0			0
$949$	−437.415			−0.460922
$950$		0			0
$951$	−286.050	+	212.454i	−0.300788	+	0.223401i
$952$		0			0
$953$		−	742.919i		−	0.779559i	−0.920908	−	0.389779i	$-0.872551\pi$
$953$		−	742.919i		−	0.779559i	0.920908	−	0.389779i	$-0.127449\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−77.8136	−	104.769i	−0.0813099	−	0.109476i
$958$		0			0
$959$		−	3110.12i		−	3.24309i
$960$		0			0
$961$	1599.80			1.66472
$962$		0			0
$963$	1620.66	+	489.189i	1.68293	+	0.507985i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	152.272			0.157468			0.0787340	−	0.996896i	$-0.474912\pi$
$967$	152.272			0.157468			0.0787340	+	0.996896i	$0.474912\pi$
$968$		0			0
$969$	−357.510	+	265.529i	−0.368948	+	0.274024i
$970$		0			0
$971$			52.6817i			0.0542551i	0.999632	+	0.0271275i	$0.00863602\pi$
$971$			52.6817i			0.0542551i	−0.999632	+	0.0271275i	$0.991364\pi$
$972$		0			0
$973$	1258.48			1.29340
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	386.394i		−	0.395490i	−0.980253	−	0.197745i	$-0.936638\pi$
$977$		−	386.394i		−	0.395490i	0.980253	−	0.197745i	$-0.0633618\pi$
$978$		0			0
$979$	−268.101			−0.273852
$980$		0			0
$981$	189.225	−	626.891i	0.192889	−	0.639032i
$982$		0			0
$983$		−	1706.21i		−	1.73572i	−0.496809	−	0.867860i	$-0.665495\pi$
$983$		−	1706.21i		−	1.73572i	0.496809	−	0.867860i	$-0.334505\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	2300.89	−	1708.91i	2.33120	−	1.73142i
$988$		0			0
$989$			1586.47i			1.60412i
$990$		0			0
$991$	−217.282			−0.219255			−0.109627	−	0.993973i	$-0.534966\pi$
$991$	−217.282			−0.219255			−0.109627	+	0.993973i	$0.534966\pi$
$992$		0			0
$993$	−884.798	−	1191.30i	−0.891035	−	1.19970i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−730.791			−0.732990			−0.366495	−	0.930420i	$-0.619442\pi$
$997$	−730.791			−0.732990			−0.366495	+	0.930420i	$0.619442\pi$
$998$		0			0
$999$	−1050.17	+	378.203i	−1.05122	+	0.378581i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1200.3.l.v.401.3		6
3.2	odd	2	inner	1200.3.l.v.401.4		6
4.3	odd	2		600.3.l.e.401.4	yes	6
5.2	odd	4		1200.3.c.l.449.2		12
5.3	odd	4		1200.3.c.l.449.11		12
5.4	even	2		1200.3.l.w.401.4		6
12.11	even	2		600.3.l.e.401.3	yes	6
15.2	even	4		1200.3.c.l.449.12		12
15.8	even	4		1200.3.c.l.449.1		12
15.14	odd	2		1200.3.l.w.401.3		6
20.3	even	4		600.3.c.c.449.2		12
20.7	even	4		600.3.c.c.449.11		12
20.19	odd	2		600.3.l.d.401.3	✓	6
60.23	odd	4		600.3.c.c.449.12		12
60.47	odd	4		600.3.c.c.449.1		12
60.59	even	2		600.3.l.d.401.4	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
600.3.c.c.449.1		12	60.47	odd	4
600.3.c.c.449.2		12	20.3	even	4
600.3.c.c.449.11		12	20.7	even	4
600.3.c.c.449.12		12	60.23	odd	4
600.3.l.d.401.3	✓	6	20.19	odd	2
600.3.l.d.401.4	yes	6	60.59	even	2
600.3.l.e.401.3	yes	6	12.11	even	2
600.3.l.e.401.4	yes	6	4.3	odd	2
1200.3.c.l.449.1		12	15.8	even	4
1200.3.c.l.449.2		12	5.2	odd	4
1200.3.c.l.449.11		12	5.3	odd	4
1200.3.c.l.449.12		12	15.2	even	4
1200.3.l.v.401.3		6	1.1	even	1	trivial
1200.3.l.v.401.4		6	3.2	odd	2	inner
1200.3.l.w.401.3		6	15.14	odd	2
1200.3.l.w.401.4		6	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+(-1.78875 - 2.40839i) q^{3} +12.2014 q^{7} +(-2.60072 + 8.61605i) q^{9} -8.79351i q^{11} +6.20144 q^{13} -16.6156i q^{17} +8.93396 q^{19} +(-21.8254 - 29.3859i) q^{21} +40.1210i q^{23} +(25.4029 - 9.14843i) q^{27} +4.94701i q^{29} +50.6043 q^{31} +(-21.1782 + 15.7294i) q^{33} -41.3407 q^{37} +(-11.0928 - 14.9355i) q^{39} -10.8654i q^{41} +39.5422 q^{43} +78.2993i q^{47} +99.8751 q^{49} +(-40.0170 + 29.7213i) q^{51} +69.7316i q^{53} +(-15.9807 - 21.5165i) q^{57} -96.0731i q^{59} +26.7287 q^{61} +(-31.7325 + 105.128i) q^{63} +66.5311 q^{67} +(96.6273 - 71.7667i) q^{69} -34.7444i q^{71} -70.5344 q^{73} -107.293i q^{77} -31.2164 q^{79} +(-67.4725 - 44.8158i) q^{81} -71.5420i q^{83} +(11.9143 - 8.84898i) q^{87} -30.4885i q^{89} +75.6665 q^{91} +(-90.5187 - 121.875i) q^{93} +108.194 q^{97} +(75.7653 + 22.8694i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{3} + 10 q^{7} + 16 q^{9} - 26 q^{13} - 50 q^{19} - 18 q^{21} + 26 q^{27} + 114 q^{31} - 82 q^{33} - 76 q^{37} + 6 q^{39} + 2 q^{43} + 76 q^{49} + 6 q^{51} - 172 q^{57} + 62 q^{61} - 150 q^{63}+ \cdots + 250 q^{99}+O(q^{100}) \) 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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