LMFDB - Embedded newform 600.3.l.c.401.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

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

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

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

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

Newform invariants

comment:select newform

sage:traces = [2,0,2,0,0,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$16.3488158616$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 2 $ x^2 + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			401.2
Root			$-1.41421i$ of defining polynomial
Character	$\chi$	$=$	600.401
Dual form			600.3.l.c.401.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 + 2.82843i) q^{3} -1.00000 q^{7} +(-7.00000 + 5.65685i) q^{9} +8.48528i q^{11} -15.0000 q^{13} -19.7990i q^{17} -23.0000 q^{19} +(-1.00000 - 2.82843i) q^{21} -2.82843i q^{23} +(-23.0000 - 14.1421i) q^{27} +25.4558i q^{29} +33.0000 q^{31} +(-24.0000 + 8.48528i) q^{33} -66.0000 q^{37} +(-15.0000 - 42.4264i) q^{39} +36.7696i q^{41} +7.00000 q^{43} +45.2548i q^{47} -48.0000 q^{49} +(56.0000 - 19.7990i) q^{51} -36.7696i q^{53} +(-23.0000 - 65.0538i) q^{57} -101.823i q^{59} +39.0000 q^{61} +(7.00000 - 5.65685i) q^{63} -113.000 q^{67} +(8.00000 - 2.82843i) q^{69} +25.4558i q^{71} -58.0000 q^{73} -8.48528i q^{77} +70.0000 q^{79} +(17.0000 - 79.1960i) q^{81} -152.735i q^{83} +(-72.0000 + 25.4558i) q^{87} +90.5097i q^{89} +15.0000 q^{91} +(33.0000 + 93.3381i) q^{93} +1.00000 q^{97} +(-48.0000 - 59.3970i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 2 q^{7} - 14 q^{9} - 30 q^{13} - 46 q^{19} - 2 q^{21} - 46 q^{27} + 66 q^{31} - 48 q^{33} - 132 q^{37} - 30 q^{39} + 14 q^{43} - 96 q^{49} + 112 q^{51} - 46 q^{57} + 78 q^{61} + 14 q^{63}+ \cdots - 96 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^7 - 14 * q^9 - 30 * q^13 - 46 * q^19 - 2 * q^21 - 46 * q^27 + 66 * q^31 - 48 * q^33 - 132 * q^37 - 30 * q^39 + 14 * q^43 - 96 * q^49 + 112 * q^51 - 46 * q^57 + 78 * q^61 + 14 * q^63 - 226 * q^67 + 16 * q^69 - 116 * q^73 + 140 * q^79 + 34 * q^81 - 144 * q^87 + 30 * q^91 + 66 * q^93 + 2 * q^97 - 96 * q^99

Character values

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

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	1.00000	+	2.82843i	0.333333	+	0.942809i
$3$	1.00000	+	2.82843i	0.333333	+	0.942809i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−1.00000			−0.142857			−0.0714286	−	0.997446i	$-0.522756\pi$
$7$	−1.00000			−0.142857			−0.0714286	+	0.997446i	$0.522756\pi$
$8$		0			0
$9$	−7.00000	+	5.65685i	−0.777778	+	0.628539i
$10$		0			0
$11$			8.48528i			0.771389i	0.922627	+	0.385695i	$0.126038\pi$
$11$			8.48528i			0.771389i	−0.922627	+	0.385695i	$0.873962\pi$
$12$		0			0
$13$	−15.0000			−1.15385			−0.576923	−	0.816798i	$-0.695747\pi$
$13$	−15.0000			−1.15385			−0.576923	+	0.816798i	$0.695747\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	19.7990i		−	1.16465i	−0.812957	−	0.582323i	$-0.802144\pi$
$17$		−	19.7990i		−	1.16465i	0.812957	−	0.582323i	$-0.197856\pi$
$18$		0			0
$19$	−23.0000			−1.21053			−0.605263	−	0.796025i	$-0.706932\pi$
$19$	−23.0000			−1.21053			−0.605263	+	0.796025i	$0.706932\pi$
$20$		0			0
$21$	−1.00000	−	2.82843i	−0.0476190	−	0.134687i
$22$		0			0
$23$		−	2.82843i		−	0.122975i	−0.998108	−	0.0614875i	$-0.980416\pi$
$23$		−	2.82843i		−	0.122975i	0.998108	−	0.0614875i	$-0.0195844\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−23.0000	−	14.1421i	−0.851852	−	0.523783i
$28$		0			0
$29$			25.4558i			0.877788i	0.898539	+	0.438894i	$0.144630\pi$
$29$			25.4558i			0.877788i	−0.898539	+	0.438894i	$0.855370\pi$
$30$		0			0
$31$	33.0000			1.06452			0.532258	−	0.846582i	$-0.321344\pi$
$31$	33.0000			1.06452			0.532258	+	0.846582i	$0.321344\pi$
$32$		0			0
$33$	−24.0000	+	8.48528i	−0.727273	+	0.257130i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−66.0000			−1.78378			−0.891892	−	0.452249i	$-0.850622\pi$
$37$	−66.0000			−1.78378			−0.891892	+	0.452249i	$0.850622\pi$
$38$		0			0
$39$	−15.0000	−	42.4264i	−0.384615	−	1.08786i
$40$		0			0
$41$			36.7696i			0.896818i	0.893828	+	0.448409i	$0.148009\pi$
$41$			36.7696i			0.896818i	−0.893828	+	0.448409i	$0.851991\pi$
$42$		0			0
$43$	7.00000			0.162791			0.0813953	−	0.996682i	$-0.474062\pi$
$43$	7.00000			0.162791			0.0813953	+	0.996682i	$0.474062\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			45.2548i			0.962869i	0.876482	+	0.481434i	$0.159884\pi$
$47$			45.2548i			0.962869i	−0.876482	+	0.481434i	$0.840116\pi$
$48$		0			0
$49$	−48.0000			−0.979592
$50$		0			0
$51$	56.0000	−	19.7990i	1.09804	−	0.388215i
$52$		0			0
$53$		−	36.7696i		−	0.693765i	−0.937909	−	0.346883i	$-0.887240\pi$
$53$		−	36.7696i		−	0.693765i	0.937909	−	0.346883i	$-0.112760\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−23.0000	−	65.0538i	−0.403509	−	1.14130i
$58$		0			0
$59$		−	101.823i		−	1.72582i	−0.505358	−	0.862910i	$-0.668639\pi$
$59$		−	101.823i		−	1.72582i	0.505358	−	0.862910i	$-0.331361\pi$
$60$		0			0
$61$	39.0000			0.639344			0.319672	−	0.947528i	$-0.396427\pi$
$61$	39.0000			0.639344			0.319672	+	0.947528i	$0.396427\pi$
$62$		0			0
$63$	7.00000	−	5.65685i	0.111111	−	0.0897913i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−113.000			−1.68657			−0.843284	−	0.537469i	$-0.819381\pi$
$67$	−113.000			−1.68657			−0.843284	+	0.537469i	$0.819381\pi$
$68$		0			0
$69$	8.00000	−	2.82843i	0.115942	−	0.0409917i
$70$		0			0
$71$			25.4558i			0.358533i	0.983801	+	0.179267i	$0.0573724\pi$
$71$			25.4558i			0.358533i	−0.983801	+	0.179267i	$0.942628\pi$
$72$		0			0
$73$	−58.0000			−0.794521			−0.397260	−	0.917706i	$-0.630039\pi$
$73$	−58.0000			−0.794521			−0.397260	+	0.917706i	$0.630039\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	8.48528i		−	0.110198i
$78$		0			0
$79$	70.0000			0.886076			0.443038	−	0.896503i	$-0.353901\pi$
$79$	70.0000			0.886076			0.443038	+	0.896503i	$0.353901\pi$
$80$		0			0
$81$	17.0000	−	79.1960i	0.209877	−	0.977728i
$82$		0			0
$83$		−	152.735i		−	1.84018i	−0.391705	−	0.920091i	$-0.628115\pi$
$83$		−	152.735i		−	1.84018i	0.391705	−	0.920091i	$-0.371885\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−72.0000	+	25.4558i	−0.827586	+	0.292596i
$88$		0			0
$89$			90.5097i			1.01696i	0.861073	+	0.508481i	$0.169793\pi$
$89$			90.5097i			1.01696i	−0.861073	+	0.508481i	$0.830207\pi$
$90$		0			0
$91$	15.0000			0.164835
$92$		0			0
$93$	33.0000	+	93.3381i	0.354839	+	1.00364i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	1.00000			0.0103093			0.00515464	−	0.999987i	$-0.498359\pi$
$97$	1.00000			0.0103093			0.00515464	+	0.999987i	$0.498359\pi$
$98$		0			0
$99$	−48.0000	−	59.3970i	−0.484848	−	0.599969i
$100$		0			0
$101$			189.505i			1.87628i	0.346252	+	0.938142i	$0.387454\pi$
$101$			189.505i			1.87628i	−0.346252	+	0.938142i	$0.612546\pi$
$102$		0			0
$103$	26.0000			0.252427			0.126214	−	0.992003i	$-0.459718\pi$
$103$	26.0000			0.252427			0.126214	+	0.992003i	$0.459718\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			48.0833i			0.449376i	0.974431	+	0.224688i	$0.0721363\pi$
$107$			48.0833i			0.449376i	−0.974431	+	0.224688i	$0.927864\pi$
$108$		0			0
$109$	−145.000			−1.33028			−0.665138	−	0.746721i	$-0.731627\pi$
$109$	−145.000			−1.33028			−0.665138	+	0.746721i	$0.731627\pi$
$110$		0			0
$111$	−66.0000	−	186.676i	−0.594595	−	1.68177i
$112$		0			0
$113$			152.735i			1.35164i	0.737068	+	0.675819i	$0.236210\pi$
$113$			152.735i			1.35164i	−0.737068	+	0.675819i	$0.763790\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	105.000	−	84.8528i	0.897436	−	0.725238i
$118$		0			0
$119$			19.7990i			0.166378i
$120$		0			0
$121$	49.0000			0.404959
$122$		0			0
$123$	−104.000	+	36.7696i	−0.845528	+	0.298939i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	226.000			1.77953			0.889764	−	0.456421i	$-0.150869\pi$
$127$	226.000			1.77953			0.889764	+	0.456421i	$0.150869\pi$
$128$		0			0
$129$	7.00000	+	19.7990i	0.0542636	+	0.153481i
$130$		0			0
$131$			104.652i			0.798869i	0.916762	+	0.399434i	$0.130793\pi$
$131$			104.652i			0.798869i	−0.916762	+	0.399434i	$0.869207\pi$
$132$		0			0
$133$	23.0000			0.172932
$134$		0			0
$135$		0			0
$136$		0			0
$137$			16.9706i			0.123873i	0.998080	+	0.0619364i	$0.0197276\pi$
$137$			16.9706i			0.123873i	−0.998080	+	0.0619364i	$0.980272\pi$
$138$		0			0
$139$	102.000			0.733813			0.366906	−	0.930258i	$-0.380417\pi$
$139$	102.000			0.733813			0.366906	+	0.930258i	$0.380417\pi$
$140$		0			0
$141$	−128.000	+	45.2548i	−0.907801	+	0.320956i
$142$		0			0
$143$		−	127.279i		−	0.890064i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−48.0000	−	135.765i	−0.326531	−	0.923568i
$148$		0			0
$149$			288.500i			1.93624i	0.250489	+	0.968119i	$0.419408\pi$
$149$			288.500i			1.93624i	−0.250489	+	0.968119i	$0.580592\pi$
$150$		0			0
$151$	−207.000			−1.37086			−0.685430	−	0.728138i	$-0.740386\pi$
$151$	−207.000			−1.37086			−0.685430	+	0.728138i	$0.740386\pi$
$152$		0			0
$153$	112.000	+	138.593i	0.732026	+	0.905836i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	89.0000			0.566879			0.283439	−	0.958990i	$-0.408524\pi$
$157$	89.0000			0.566879			0.283439	+	0.958990i	$0.408524\pi$
$158$		0			0
$159$	104.000	−	36.7696i	0.654088	−	0.231255i
$160$		0			0
$161$			2.82843i			0.0175679i
$162$		0			0
$163$	−17.0000			−0.104294			−0.0521472	−	0.998639i	$-0.516607\pi$
$163$	−17.0000			−0.104294			−0.0521472	+	0.998639i	$0.516607\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			268.701i			1.60899i	0.593962	+	0.804493i	$0.297563\pi$
$167$			268.701i			1.60899i	−0.593962	+	0.804493i	$0.702437\pi$
$168$		0			0
$169$	56.0000			0.331361
$170$		0			0
$171$	161.000	−	130.108i	0.941520	−	0.760863i
$172$		0			0
$173$		−	110.309i		−	0.637622i	−0.947818	−	0.318811i	$-0.896716\pi$
$173$		−	110.309i		−	0.637622i	0.947818	−	0.318811i	$-0.103284\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	288.000	−	101.823i	1.62712	−	0.575273i
$178$		0			0
$179$			73.5391i			0.410833i	0.978675	+	0.205416i	$0.0658549\pi$
$179$			73.5391i			0.410833i	−0.978675	+	0.205416i	$0.934145\pi$
$180$		0			0
$181$	−145.000			−0.801105			−0.400552	−	0.916274i	$-0.631182\pi$
$181$	−145.000			−0.801105			−0.400552	+	0.916274i	$0.631182\pi$
$182$		0			0
$183$	39.0000	+	110.309i	0.213115	+	0.602780i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	168.000			0.898396
$188$		0			0
$189$	23.0000	+	14.1421i	0.121693	+	0.0748261i
$190$		0			0
$191$		−	53.7401i		−	0.281362i	−0.990055	−	0.140681i	$-0.955071\pi$
$191$		−	53.7401i		−	0.281362i	0.990055	−	0.140681i	$-0.0449292\pi$
$192$		0			0
$193$	57.0000			0.295337			0.147668	−	0.989037i	$-0.452823\pi$
$193$	57.0000			0.295337			0.147668	+	0.989037i	$0.452823\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			226.274i			1.14860i	0.818645	+	0.574300i	$0.194726\pi$
$197$			226.274i			1.14860i	−0.818645	+	0.574300i	$0.805274\pi$
$198$		0			0
$199$	225.000			1.13065			0.565327	−	0.824867i	$-0.308750\pi$
$199$	225.000			1.13065			0.565327	+	0.824867i	$0.308750\pi$
$200$		0			0
$201$	−113.000	−	319.612i	−0.562189	−	1.59011i
$202$		0			0
$203$		−	25.4558i		−	0.125398i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	16.0000	+	19.7990i	0.0772947	+	0.0956473i
$208$		0			0
$209$		−	195.161i		−	0.933787i
$210$		0			0
$211$	−231.000			−1.09479			−0.547393	−	0.836875i	$-0.684380\pi$
$211$	−231.000			−1.09479			−0.547393	+	0.836875i	$0.684380\pi$
$212$		0			0
$213$	−72.0000	+	25.4558i	−0.338028	+	0.119511i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−33.0000			−0.152074
$218$		0			0
$219$	−58.0000	−	164.049i	−0.264840	−	0.749081i
$220$		0			0
$221$			296.985i			1.34382i
$222$		0			0
$223$	327.000			1.46637			0.733184	−	0.680030i	$-0.238033\pi$
$223$	327.000			1.46637			0.733184	+	0.680030i	$0.238033\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			76.3675i			0.336421i	0.985751	+	0.168210i	$0.0537988\pi$
$227$			76.3675i			0.336421i	−0.985751	+	0.168210i	$0.946201\pi$
$228$		0			0
$229$	183.000			0.799127			0.399563	−	0.916706i	$-0.369162\pi$
$229$	183.000			0.799127			0.399563	+	0.916706i	$0.369162\pi$
$230$		0			0
$231$	24.0000	−	8.48528i	0.103896	−	0.0367328i
$232$		0			0
$233$		−	98.9949i		−	0.424871i	−0.977175	−	0.212436i	$-0.931860\pi$
$233$		−	98.9949i		−	0.424871i	0.977175	−	0.212436i	$-0.0681395\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	70.0000	+	197.990i	0.295359	+	0.835400i
$238$		0			0
$239$			8.48528i			0.0355033i	0.999842	+	0.0177516i	$0.00565082\pi$
$239$			8.48528i			0.0355033i	−0.999842	+	0.0177516i	$0.994349\pi$
$240$		0			0
$241$	215.000			0.892116			0.446058	−	0.895004i	$-0.352827\pi$
$241$	215.000			0.892116			0.446058	+	0.895004i	$0.352827\pi$
$242$		0			0
$243$	241.000	−	31.1127i	0.991770	−	0.128036i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	345.000			1.39676
$248$		0			0
$249$	432.000	−	152.735i	1.73494	−	0.613394i
$250$		0			0
$251$			164.049i			0.653581i	0.945097	+	0.326790i	$0.105967\pi$
$251$			164.049i			0.653581i	−0.945097	+	0.326790i	$0.894033\pi$
$252$		0			0
$253$	24.0000			0.0948617
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	152.735i		−	0.594300i	−0.954831	−	0.297150i	$-0.903964\pi$
$257$		−	152.735i		−	0.594300i	0.954831	−	0.297150i	$-0.0960361\pi$
$258$		0			0
$259$	66.0000			0.254826
$260$		0			0
$261$	−144.000	−	178.191i	−0.551724	−	0.682724i
$262$		0			0
$263$			59.3970i			0.225844i	0.993604	+	0.112922i	$0.0360210\pi$
$263$			59.3970i			0.225844i	−0.993604	+	0.112922i	$0.963979\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−256.000	+	90.5097i	−0.958801	+	0.338988i
$268$		0			0
$269$			8.48528i			0.0315438i	0.999876	+	0.0157719i	$0.00502056\pi$
$269$			8.48528i			0.0315438i	−0.999876	+	0.0157719i	$0.994979\pi$
$270$		0			0
$271$	86.0000			0.317343			0.158672	−	0.987331i	$-0.449279\pi$
$271$	86.0000			0.317343			0.158672	+	0.987331i	$0.449279\pi$
$272$		0			0
$273$	15.0000	+	42.4264i	0.0549451	+	0.155408i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−319.000			−1.15162			−0.575812	−	0.817582i	$-0.695314\pi$
$277$	−319.000			−1.15162			−0.575812	+	0.817582i	$0.695314\pi$
$278$		0			0
$279$	−231.000	+	186.676i	−0.827957	+	0.669090i
$280$		0			0
$281$		−	427.092i		−	1.51990i	−0.649980	−	0.759951i	$-0.725223\pi$
$281$		−	427.092i		−	1.51990i	0.649980	−	0.759951i	$-0.274777\pi$
$282$		0			0
$283$	−1.00000			−0.00353357			−0.00176678	−	0.999998i	$-0.500562\pi$
$283$	−1.00000			−0.00353357			−0.00176678	+	0.999998i	$0.500562\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	36.7696i		−	0.128117i
$288$		0			0
$289$	−103.000			−0.356401
$290$		0			0
$291$	1.00000	+	2.82843i	0.00343643	+	0.00971968i
$292$		0			0
$293$		−	486.489i		−	1.66037i	−0.557485	−	0.830187i	$-0.688234\pi$
$293$		−	486.489i		−	1.66037i	0.557485	−	0.830187i	$-0.311766\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	120.000	−	195.161i	0.404040	−	0.657109i
$298$		0			0
$299$			42.4264i			0.141894i
$300$		0			0
$301$	−7.00000			−0.0232558
$302$		0			0
$303$	−536.000	+	189.505i	−1.76898	+	0.625428i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	215.000			0.700326			0.350163	−	0.936689i	$-0.386126\pi$
$307$	215.000			0.700326			0.350163	+	0.936689i	$0.386126\pi$
$308$		0			0
$309$	26.0000	+	73.5391i	0.0841424	+	0.237991i
$310$		0			0
$311$			214.960i			0.691191i	0.938384	+	0.345596i	$0.112323\pi$
$311$			214.960i			0.691191i	−0.938384	+	0.345596i	$0.887677\pi$
$312$		0			0
$313$	−535.000			−1.70927			−0.854633	−	0.519233i	$-0.826218\pi$
$313$	−535.000			−1.70927			−0.854633	+	0.519233i	$0.826218\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			469.519i			1.48113i	0.671984	+	0.740566i	$0.265443\pi$
$317$			469.519i			1.48113i	−0.671984	+	0.740566i	$0.734557\pi$
$318$		0			0
$319$	−216.000			−0.677116
$320$		0			0
$321$	−136.000	+	48.0833i	−0.423676	+	0.149792i
$322$		0			0
$323$			455.377i			1.40984i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−145.000	−	410.122i	−0.443425	−	1.25420i
$328$		0			0
$329$		−	45.2548i		−	0.137553i
$330$		0			0
$331$	94.0000			0.283988			0.141994	−	0.989868i	$-0.454649\pi$
$331$	94.0000			0.283988			0.141994	+	0.989868i	$0.454649\pi$
$332$		0			0
$333$	462.000	−	373.352i	1.38739	−	1.12118i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−295.000			−0.875371			−0.437685	−	0.899128i	$-0.644202\pi$
$337$	−295.000			−0.875371			−0.437685	+	0.899128i	$0.644202\pi$
$338$		0			0
$339$	−432.000	+	152.735i	−1.27434	+	0.450546i
$340$		0			0
$341$			280.014i			0.821156i
$342$		0			0
$343$	97.0000			0.282799
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	441.235i		−	1.27157i	−0.771866	−	0.635785i	$-0.780677\pi$
$347$		−	441.235i		−	1.27157i	0.771866	−	0.635785i	$-0.219323\pi$
$348$		0			0
$349$	−374.000			−1.07163			−0.535817	−	0.844334i	$-0.679996\pi$
$349$	−374.000			−1.07163			−0.535817	+	0.844334i	$0.679996\pi$
$350$		0			0
$351$	345.000	+	212.132i	0.982906	+	0.604365i
$352$		0			0
$353$			280.014i			0.793242i	0.917983	+	0.396621i	$0.129817\pi$
$353$			280.014i			0.793242i	−0.917983	+	0.396621i	$0.870183\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−56.0000	+	19.7990i	−0.156863	+	0.0554594i
$358$		0			0
$359$			370.524i			1.03210i	0.856558	+	0.516050i	$0.172598\pi$
$359$			370.524i			1.03210i	−0.856558	+	0.516050i	$0.827402\pi$
$360$		0			0
$361$	168.000			0.465374
$362$		0			0
$363$	49.0000	+	138.593i	0.134986	+	0.381799i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−241.000			−0.656676			−0.328338	−	0.944560i	$-0.606488\pi$
$367$	−241.000			−0.656676			−0.328338	+	0.944560i	$0.606488\pi$
$368$		0			0
$369$	−208.000	−	257.387i	−0.563686	−	0.697525i
$370$		0			0
$371$			36.7696i			0.0991093i
$372$		0			0
$373$	−47.0000			−0.126005			−0.0630027	−	0.998013i	$-0.520068\pi$
$373$	−47.0000			−0.126005			−0.0630027	+	0.998013i	$0.520068\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	381.838i		−	1.01283i
$378$		0			0
$379$	73.0000			0.192612			0.0963061	−	0.995352i	$-0.469297\pi$
$379$	73.0000			0.192612			0.0963061	+	0.995352i	$0.469297\pi$
$380$		0			0
$381$	226.000	+	639.225i	0.593176	+	1.67775i
$382$		0			0
$383$			113.137i			0.295397i	0.989032	+	0.147699i	$0.0471865\pi$
$383$			113.137i			0.295397i	−0.989032	+	0.147699i	$0.952813\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−49.0000	+	39.5980i	−0.126615	+	0.102320i
$388$		0			0
$389$		−	534.573i		−	1.37422i	−0.726552	−	0.687111i	$-0.758878\pi$
$389$		−	534.573i		−	1.37422i	0.726552	−	0.687111i	$-0.241122\pi$
$390$		0			0
$391$	−56.0000			−0.143223
$392$		0			0
$393$	−296.000	+	104.652i	−0.753181	+	0.266290i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−695.000			−1.75063			−0.875315	−	0.483553i	$-0.839346\pi$
$397$	−695.000			−1.75063			−0.875315	+	0.483553i	$0.839346\pi$
$398$		0			0
$399$	23.0000	+	65.0538i	0.0576441	+	0.163042i
$400$		0			0
$401$		−	721.249i		−	1.79863i	−0.437306	−	0.899313i	$-0.644067\pi$
$401$		−	721.249i		−	1.79863i	0.437306	−	0.899313i	$-0.355933\pi$
$402$		0			0
$403$	−495.000			−1.22829
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	560.029i		−	1.37599i
$408$		0			0
$409$	−425.000			−1.03912			−0.519560	−	0.854434i	$-0.673904\pi$
$409$	−425.000			−1.03912			−0.519560	+	0.854434i	$0.673904\pi$
$410$		0			0
$411$	−48.0000	+	16.9706i	−0.116788	+	0.0412909i
$412$		0			0
$413$			101.823i			0.246546i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	102.000	+	288.500i	0.244604	+	0.691845i
$418$		0			0
$419$			659.024i			1.57285i	0.617687	+	0.786424i	$0.288070\pi$
$419$			659.024i			1.57285i	−0.617687	+	0.786424i	$0.711930\pi$
$420$		0			0
$421$	306.000			0.726841			0.363420	−	0.931625i	$-0.381609\pi$
$421$	306.000			0.726841			0.363420	+	0.931625i	$0.381609\pi$
$422$		0			0
$423$	−256.000	−	316.784i	−0.605201	−	0.748898i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−39.0000			−0.0913349
$428$		0			0
$429$	360.000	−	127.279i	0.839161	−	0.296688i
$430$		0			0
$431$			104.652i			0.242812i	0.992603	+	0.121406i	$0.0387402\pi$
$431$			104.652i			0.242812i	−0.992603	+	0.121406i	$0.961260\pi$
$432$		0			0
$433$	281.000			0.648961			0.324480	−	0.945892i	$-0.394811\pi$
$433$	281.000			0.648961			0.324480	+	0.945892i	$0.394811\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			65.0538i			0.148865i
$438$		0			0
$439$	−671.000			−1.52847			−0.764237	−	0.644936i	$-0.776884\pi$
$439$	−671.000			−1.52847			−0.764237	+	0.644936i	$0.776884\pi$
$440$		0			0
$441$	336.000	−	271.529i	0.761905	−	0.615712i
$442$		0			0
$443$		−	424.264i		−	0.957707i	−0.877895	−	0.478853i	$-0.841053\pi$
$443$		−	424.264i		−	0.957707i	0.877895	−	0.478853i	$-0.158947\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−816.000	+	288.500i	−1.82550	+	0.645413i
$448$		0			0
$449$		−	605.283i		−	1.34807i	−0.738699	−	0.674035i	$-0.764560\pi$
$449$		−	605.283i		−	1.34807i	0.738699	−	0.674035i	$-0.235440\pi$
$450$		0			0
$451$	−312.000			−0.691796
$452$		0			0
$453$	−207.000	−	585.484i	−0.456954	−	1.29246i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−202.000			−0.442013			−0.221007	−	0.975272i	$-0.570934\pi$
$457$	−202.000			−0.442013			−0.221007	+	0.975272i	$0.570934\pi$
$458$		0			0
$459$	−280.000	+	455.377i	−0.610022	+	0.992106i
$460$		0			0
$461$			336.583i			0.730115i	0.930985	+	0.365057i	$0.118951\pi$
$461$			336.583i			0.730115i	−0.930985	+	0.365057i	$0.881049\pi$
$462$		0			0
$463$	−166.000			−0.358531			−0.179266	−	0.983801i	$-0.557372\pi$
$463$	−166.000			−0.358531			−0.179266	+	0.983801i	$0.557372\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	121.622i		−	0.260433i	−0.991486	−	0.130217i	$-0.958433\pi$
$467$		−	121.622i		−	0.260433i	0.991486	−	0.130217i	$-0.0415673\pi$
$468$		0			0
$469$	113.000			0.240938
$470$		0			0
$471$	89.0000	+	251.730i	0.188960	+	0.534459i
$472$		0			0
$473$			59.3970i			0.125575i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	208.000	+	257.387i	0.436059	+	0.539595i
$478$		0			0
$479$		−	797.616i		−	1.66517i	−0.553897	−	0.832585i	$-0.686860\pi$
$479$		−	797.616i		−	1.66517i	0.553897	−	0.832585i	$-0.313140\pi$
$480$		0			0
$481$	990.000			2.05821
$482$		0			0
$483$	−8.00000	+	2.82843i	−0.0165631	+	0.00585596i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−369.000			−0.757700			−0.378850	−	0.925458i	$-0.623680\pi$
$487$	−369.000			−0.757700			−0.378850	+	0.925458i	$0.623680\pi$
$488$		0			0
$489$	−17.0000	−	48.0833i	−0.0347648	−	0.0983298i
$490$		0			0
$491$		−	588.313i		−	1.19819i	−0.800677	−	0.599097i	$-0.795527\pi$
$491$		−	588.313i		−	1.19819i	0.800677	−	0.599097i	$-0.204473\pi$
$492$		0			0
$493$	504.000			1.02231
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	25.4558i		−	0.0512190i
$498$		0			0
$499$	785.000			1.57315			0.786573	−	0.617497i	$-0.211853\pi$
$499$	785.000			1.57315			0.786573	+	0.617497i	$0.211853\pi$
$500$		0			0
$501$	−760.000	+	268.701i	−1.51697	+	0.536328i
$502$		0			0
$503$			280.014i			0.556688i	0.960481	+	0.278344i	$0.0897856\pi$
$503$			280.014i			0.556688i	−0.960481	+	0.278344i	$0.910214\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	56.0000	+	158.392i	0.110454	+	0.312410i
$508$		0			0
$509$		−	220.617i		−	0.433433i	−0.976235	−	0.216716i	$-0.930465\pi$
$509$		−	220.617i		−	0.433433i	0.976235	−	0.216716i	$-0.0695347\pi$
$510$		0			0
$511$	58.0000			0.113503
$512$		0			0
$513$	529.000	+	325.269i	1.03119	+	0.634053i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−384.000			−0.742747
$518$		0			0
$519$	312.000	−	110.309i	0.601156	−	0.212541i
$520$		0			0
$521$		−	659.024i		−	1.26492i	−0.774593	−	0.632460i	$-0.782045\pi$
$521$		−	659.024i		−	1.26492i	0.774593	−	0.632460i	$-0.217955\pi$
$522$		0			0
$523$	−129.000			−0.246654			−0.123327	−	0.992366i	$-0.539356\pi$
$523$	−129.000			−0.246654			−0.123327	+	0.992366i	$0.539356\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	653.367i		−	1.23978i
$528$		0			0
$529$	521.000			0.984877
$530$		0			0
$531$	576.000	+	712.764i	1.08475	+	1.34230i
$532$		0			0
$533$		−	551.543i		−	1.03479i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−208.000	+	73.5391i	−0.387337	+	0.136944i
$538$		0			0
$539$		−	407.294i		−	0.755647i
$540$		0			0
$541$	119.000			0.219963			0.109982	−	0.993934i	$-0.464921\pi$
$541$	119.000			0.219963			0.109982	+	0.993934i	$0.464921\pi$
$542$		0			0
$543$	−145.000	−	410.122i	−0.267035	−	0.755289i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	650.000			1.18830			0.594150	−	0.804354i	$-0.297489\pi$
$547$	650.000			1.18830			0.594150	+	0.804354i	$0.297489\pi$
$548$		0			0
$549$	−273.000	+	220.617i	−0.497268	+	0.401853i
$550$		0			0
$551$		−	585.484i		−	1.06259i
$552$		0			0
$553$	−70.0000			−0.126582
$554$		0			0
$555$		0			0
$556$		0			0
$557$			107.480i			0.192963i	0.995335	+	0.0964814i	$0.0307588\pi$
$557$			107.480i			0.192963i	−0.995335	+	0.0964814i	$0.969241\pi$
$558$		0			0
$559$	−105.000			−0.187835
$560$		0			0
$561$	168.000	+	475.176i	0.299465	+	0.847016i
$562$		0			0
$563$			789.131i			1.40165i	0.713331	+	0.700827i	$0.247185\pi$
$563$			789.131i			1.40165i	−0.713331	+	0.700827i	$0.752815\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−17.0000	+	79.1960i	−0.0299824	+	0.139675i
$568$		0			0
$569$		−	523.259i		−	0.919612i	−0.888019	−	0.459806i	$-0.847919\pi$
$569$		−	523.259i		−	0.919612i	0.888019	−	0.459806i	$-0.152081\pi$
$570$		0			0
$571$	−503.000			−0.880911			−0.440455	−	0.897775i	$-0.645183\pi$
$571$	−503.000			−0.880911			−0.440455	+	0.897775i	$0.645183\pi$
$572$		0			0
$573$	152.000	−	53.7401i	0.265271	−	0.0937873i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	353.000			0.611785			0.305893	−	0.952066i	$-0.401045\pi$
$577$	353.000			0.611785			0.305893	+	0.952066i	$0.401045\pi$
$578$		0			0
$579$	57.0000	+	161.220i	0.0984456	+	0.278446i
$580$		0			0
$581$			152.735i			0.262883i
$582$		0			0
$583$	312.000			0.535163
$584$		0			0
$585$		0			0
$586$		0			0
$587$			1077.63i			1.83583i	0.396780	+	0.917914i	$0.370128\pi$
$587$			1077.63i			1.83583i	−0.396780	+	0.917914i	$0.629872\pi$
$588$		0			0
$589$	−759.000			−1.28862
$590$		0			0
$591$	−640.000	+	226.274i	−1.08291	+	0.382867i
$592$		0			0
$593$			726.906i			1.22581i	0.790156	+	0.612905i	$0.209999\pi$
$593$			726.906i			1.22581i	−0.790156	+	0.612905i	$0.790001\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	225.000	+	636.396i	0.376884	+	1.06599i
$598$		0			0
$599$			871.156i			1.45435i	0.686452	+	0.727175i	$0.259167\pi$
$599$			871.156i			1.45435i	−0.686452	+	0.727175i	$0.740833\pi$
$600$		0			0
$601$	−1033.00			−1.71880			−0.859401	−	0.511302i	$-0.829163\pi$
$601$	−1033.00			−1.71880			−0.859401	+	0.511302i	$0.829163\pi$
$602$		0			0
$603$	791.000	−	639.225i	1.31177	−	1.06007i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	330.000			0.543657			0.271829	−	0.962346i	$-0.412372\pi$
$607$	330.000			0.543657			0.271829	+	0.962346i	$0.412372\pi$
$608$		0			0
$609$	72.0000	−	25.4558i	0.118227	−	0.0417994i
$610$		0			0
$611$		−	678.823i		−	1.11100i
$612$		0			0
$613$	−770.000			−1.25612			−0.628059	−	0.778166i	$-0.716150\pi$
$613$	−770.000			−1.25612			−0.628059	+	0.778166i	$0.716150\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	630.739i		−	1.02227i	−0.859501	−	0.511134i	$-0.829226\pi$
$617$		−	630.739i		−	1.02227i	0.859501	−	0.511134i	$-0.170774\pi$
$618$		0			0
$619$	401.000			0.647819			0.323910	−	0.946088i	$-0.395003\pi$
$619$	401.000			0.647819			0.323910	+	0.946088i	$0.395003\pi$
$620$		0			0
$621$	−40.0000	+	65.0538i	−0.0644122	+	0.104757i
$622$		0			0
$623$		−	90.5097i		−	0.145280i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	552.000	−	195.161i	0.880383	−	0.311262i
$628$		0			0
$629$			1306.73i			2.07748i
$630$		0			0
$631$	889.000			1.40887			0.704437	−	0.709766i	$-0.251199\pi$
$631$	889.000			1.40887			0.704437	+	0.709766i	$0.251199\pi$
$632$		0			0
$633$	−231.000	−	653.367i	−0.364929	−	1.03217i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	720.000			1.13030
$638$		0			0
$639$	−144.000	−	178.191i	−0.225352	−	0.278859i
$640$		0			0
$641$		−	678.823i		−	1.05901i	−0.848308	−	0.529503i	$-0.822379\pi$
$641$		−	678.823i		−	1.05901i	0.848308	−	0.529503i	$-0.177621\pi$
$642$		0			0
$643$	42.0000			0.0653188			0.0326594	−	0.999467i	$-0.489602\pi$
$643$	42.0000			0.0653188			0.0326594	+	0.999467i	$0.489602\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	458.205i		−	0.708200i	−0.935208	−	0.354100i	$-0.884787\pi$
$647$		−	458.205i		−	0.708200i	0.935208	−	0.354100i	$-0.115213\pi$
$648$		0			0
$649$	864.000			1.33128
$650$		0			0
$651$	−33.0000	−	93.3381i	−0.0506912	−	0.143376i
$652$		0			0
$653$			1001.26i			1.53333i	0.642049	+	0.766664i	$0.278085\pi$
$653$			1001.26i			1.53333i	−0.642049	+	0.766664i	$0.721915\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	406.000	−	328.098i	0.617960	−	0.499387i
$658$		0			0
$659$		−	376.181i		−	0.570836i	−0.958403	−	0.285418i	$-0.907868\pi$
$659$		−	376.181i		−	0.570836i	0.958403	−	0.285418i	$-0.0921324\pi$
$660$		0			0
$661$	−974.000			−1.47352			−0.736762	−	0.676152i	$-0.763646\pi$
$661$	−974.000			−1.47352			−0.736762	+	0.676152i	$0.763646\pi$
$662$		0			0
$663$	−840.000	+	296.985i	−1.26697	+	0.447941i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	72.0000			0.107946
$668$		0			0
$669$	327.000	+	924.896i	0.488789	+	1.38250i
$670$		0			0
$671$			330.926i			0.493183i
$672$		0			0
$673$	−26.0000			−0.0386330			−0.0193165	−	0.999813i	$-0.506149\pi$
$673$	−26.0000			−0.0386330			−0.0193165	+	0.999813i	$0.506149\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	93.3381i		−	0.137870i	−0.997621	−	0.0689351i	$-0.978040\pi$
$677$		−	93.3381i		−	0.137870i	0.997621	−	0.0689351i	$-0.0219601\pi$
$678$		0			0
$679$	−1.00000			−0.00147275
$680$		0			0
$681$	−216.000	+	76.3675i	−0.317181	+	0.112140i
$682$		0			0
$683$			16.9706i			0.0248471i	0.999923	+	0.0124235i	$0.00395464\pi$
$683$			16.9706i			0.0248471i	−0.999923	+	0.0124235i	$0.996045\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	183.000	+	517.602i	0.266376	+	0.753424i
$688$		0			0
$689$			551.543i			0.800498i
$690$		0			0
$691$	−722.000			−1.04486			−0.522431	−	0.852681i	$-0.674975\pi$
$691$	−722.000			−1.04486			−0.522431	+	0.852681i	$0.674975\pi$
$692$		0			0
$693$	48.0000	+	59.3970i	0.0692641	+	0.0857099i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	728.000			1.04448
$698$		0			0
$699$	280.000	−	98.9949i	0.400572	−	0.141624i
$700$		0			0
$701$		−	65.0538i		−	0.0928015i	−0.998923	−	0.0464007i	$-0.985225\pi$
$701$		−	65.0538i		−	0.0928015i	0.998923	−	0.0464007i	$-0.0147751\pi$
$702$		0			0
$703$	1518.00			2.15932
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	189.505i		−	0.268040i
$708$		0			0
$709$	103.000			0.145275			0.0726375	−	0.997358i	$-0.476858\pi$
$709$	103.000			0.145275			0.0726375	+	0.997358i	$0.476858\pi$
$710$		0			0
$711$	−490.000	+	395.980i	−0.689170	+	0.556934i
$712$		0			0
$713$		−	93.3381i		−	0.130909i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−24.0000	+	8.48528i	−0.0334728	+	0.0118344i
$718$		0			0
$719$		−	670.337i		−	0.932319i	−0.884701	−	0.466159i	$-0.845637\pi$
$719$		−	670.337i		−	0.932319i	0.884701	−	0.466159i	$-0.154363\pi$
$720$		0			0
$721$	−26.0000			−0.0360610
$722$		0			0
$723$	215.000	+	608.112i	0.297372	+	0.841095i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	895.000			1.23109			0.615543	−	0.788103i	$-0.288937\pi$
$727$	895.000			1.23109			0.615543	+	0.788103i	$0.288937\pi$
$728$		0			0
$729$	329.000	+	650.538i	0.451303	+	0.892371i
$730$		0			0
$731$		−	138.593i		−	0.189594i
$732$		0			0
$733$	406.000			0.553888			0.276944	−	0.960886i	$-0.410678\pi$
$733$	406.000			0.553888			0.276944	+	0.960886i	$0.410678\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	958.837i		−	1.30100i
$738$		0			0
$739$	−578.000			−0.782138			−0.391069	−	0.920361i	$-0.627895\pi$
$739$	−578.000			−0.782138			−0.391069	+	0.920361i	$0.627895\pi$
$740$		0			0
$741$	345.000	+	975.807i	0.465587	+	1.31688i
$742$		0			0
$743$			492.146i			0.662377i	0.943565	+	0.331189i	$0.107450\pi$
$743$			492.146i			0.662377i	−0.943565	+	0.331189i	$0.892550\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	864.000	+	1069.15i	1.15663	+	1.43125i
$748$		0			0
$749$		−	48.0833i		−	0.0641966i
$750$		0			0
$751$	30.0000			0.0399467			0.0199734	−	0.999801i	$-0.493642\pi$
$751$	30.0000			0.0399467			0.0199734	+	0.999801i	$0.493642\pi$
$752$		0			0
$753$	−464.000	+	164.049i	−0.616202	+	0.217860i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	897.000			1.18494			0.592470	−	0.805592i	$-0.298153\pi$
$757$	897.000			1.18494			0.592470	+	0.805592i	$0.298153\pi$
$758$		0			0
$759$	24.0000	+	67.8823i	0.0316206	+	0.0894364i
$760$		0			0
$761$		−	164.049i		−	0.215570i	−0.994174	−	0.107785i	$-0.965624\pi$
$761$		−	164.049i		−	0.215570i	0.994174	−	0.107785i	$-0.0343758\pi$
$762$		0			0
$763$	145.000			0.190039
$764$		0			0
$765$		0			0
$766$		0			0
$767$			1527.35i			1.99133i
$768$		0			0
$769$	−465.000			−0.604681			−0.302341	−	0.953200i	$-0.597768\pi$
$769$	−465.000			−0.604681			−0.302341	+	0.953200i	$0.597768\pi$
$770$		0			0
$771$	432.000	−	152.735i	0.560311	−	0.198100i
$772$		0			0
$773$		−	172.534i		−	0.223201i	−0.993753	−	0.111600i	$-0.964402\pi$
$773$		−	172.534i		−	0.223201i	0.993753	−	0.111600i	$-0.0355976\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	66.0000	+	186.676i	0.0849421	+	0.240252i
$778$		0			0
$779$		−	845.700i		−	1.08562i
$780$		0			0
$781$	−216.000			−0.276569
$782$		0			0
$783$	360.000	−	585.484i	0.459770	−	0.747745i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−801.000			−1.01779			−0.508895	−	0.860829i	$-0.669946\pi$
$787$	−801.000			−1.01779			−0.508895	+	0.860829i	$0.669946\pi$
$788$		0			0
$789$	−168.000	+	59.3970i	−0.212928	+	0.0752813i
$790$		0			0
$791$		−	152.735i		−	0.193091i
$792$		0			0
$793$	−585.000			−0.737705
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	197.990i		−	0.248419i	−0.992256	−	0.124209i	$-0.960361\pi$
$797$		−	197.990i		−	0.248419i	0.992256	−	0.124209i	$-0.0396395\pi$
$798$		0			0
$799$	896.000			1.12140
$800$		0			0
$801$	−512.000	−	633.568i	−0.639201	−	0.790971i
$802$		0			0
$803$		−	492.146i		−	0.612885i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−24.0000	+	8.48528i	−0.0297398	+	0.0105146i
$808$		0			0
$809$			670.337i			0.828600i	0.910140	+	0.414300i	$0.135974\pi$
$809$			670.337i			0.828600i	−0.910140	+	0.414300i	$0.864026\pi$
$810$		0			0
$811$	−879.000			−1.08385			−0.541924	−	0.840428i	$-0.682304\pi$
$811$	−879.000			−1.08385			−0.541924	+	0.840428i	$0.682304\pi$
$812$		0			0
$813$	86.0000	+	243.245i	0.105781	+	0.299194i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−161.000			−0.197062
$818$		0			0
$819$	−105.000	+	84.8528i	−0.128205	+	0.103605i
$820$		0			0
$821$		−	842.871i		−	1.02664i	−0.858197	−	0.513320i	$-0.828415\pi$
$821$		−	842.871i		−	1.02664i	0.858197	−	0.513320i	$-0.171585\pi$
$822$		0			0
$823$	−521.000			−0.633050			−0.316525	−	0.948584i	$-0.602516\pi$
$823$	−521.000			−0.633050			−0.316525	+	0.948584i	$0.602516\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	172.534i		−	0.208626i	−0.994544	−	0.104313i	$-0.966736\pi$
$827$		−	172.534i		−	0.208626i	0.994544	−	0.104313i	$-0.0332644\pi$
$828$		0			0
$829$	818.000			0.986731			0.493366	−	0.869822i	$-0.335767\pi$
$829$	818.000			0.986731			0.493366	+	0.869822i	$0.335767\pi$
$830$		0			0
$831$	−319.000	−	902.268i	−0.383875	−	1.08576i
$832$		0			0
$833$			950.352i			1.14088i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−759.000	−	466.690i	−0.906810	−	0.557575i
$838$		0			0
$839$		−	1309.56i		−	1.56086i	−0.625243	−	0.780430i	$-0.715000\pi$
$839$		−	1309.56i		−	1.56086i	0.625243	−	0.780430i	$-0.285000\pi$
$840$		0			0
$841$	193.000			0.229489
$842$		0			0
$843$	1208.00	−	427.092i	1.43298	−	0.506634i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−49.0000			−0.0578512
$848$		0			0
$849$	−1.00000	−	2.82843i	−0.00117786	−	0.00333148i
$850$		0			0
$851$			186.676i			0.219361i
$852$		0			0
$853$	233.000			0.273154			0.136577	−	0.990629i	$-0.456390\pi$
$853$	233.000			0.273154			0.136577	+	0.990629i	$0.456390\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			808.930i			0.943909i	0.881623	+	0.471955i	$0.156451\pi$
$857$			808.930i			0.943909i	−0.881623	+	0.471955i	$0.843549\pi$
$858$		0			0
$859$	1230.00			1.43190			0.715949	−	0.698153i	$-0.245994\pi$
$859$	1230.00			1.43190			0.715949	+	0.698153i	$0.245994\pi$
$860$		0			0
$861$	104.000	−	36.7696i	0.120790	−	0.0427056i
$862$		0			0
$863$		−	1284.11i		−	1.48796i	−0.668204	−	0.743978i	$-0.732937\pi$
$863$		−	1284.11i		−	1.48796i	0.668204	−	0.743978i	$-0.267063\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−103.000	−	291.328i	−0.118800	−	0.336018i
$868$		0			0
$869$			593.970i			0.683509i
$870$		0			0
$871$	1695.00			1.94604
$872$		0			0
$873$	−7.00000	+	5.65685i	−0.00801833	+	0.00647979i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	113.000			0.128848			0.0644242	−	0.997923i	$-0.479479\pi$
$877$	113.000			0.128848			0.0644242	+	0.997923i	$0.479479\pi$
$878$		0			0
$879$	1376.00	−	486.489i	1.56542	−	0.553458i
$880$		0			0
$881$			1295.42i			1.47040i	0.677852	+	0.735198i	$0.262911\pi$
$881$			1295.42i			1.47040i	−0.677852	+	0.735198i	$0.737089\pi$
$882$		0			0
$883$	−793.000			−0.898075			−0.449037	−	0.893513i	$-0.648233\pi$
$883$	−793.000			−0.898075			−0.449037	+	0.893513i	$0.648233\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			67.8823i			0.0765302i	0.999268	+	0.0382651i	$0.0121831\pi$
$887$			67.8823i			0.0765302i	−0.999268	+	0.0382651i	$0.987817\pi$
$888$		0			0
$889$	−226.000			−0.254218
$890$		0			0
$891$	672.000	+	144.250i	0.754209	+	0.161897i
$892$		0			0
$893$		−	1040.86i		−	1.16558i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−120.000	+	42.4264i	−0.133779	+	0.0472981i
$898$		0			0
$899$			840.043i			0.934419i
$900$		0			0
$901$	−728.000			−0.807991
$902$		0			0
$903$	−7.00000	−	19.7990i	−0.00775194	−	0.0219258i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−1678.00			−1.85006			−0.925028	−	0.379900i	$-0.875958\pi$
$907$	−1678.00			−1.85006			−0.925028	+	0.379900i	$0.875958\pi$
$908$		0			0
$909$	−1072.00	−	1326.53i	−1.17932	−	1.45933i
$910$		0			0
$911$		−	1702.71i		−	1.86906i	−0.355885	−	0.934530i	$-0.615821\pi$
$911$		−	1702.71i		−	1.86906i	0.355885	−	0.934530i	$-0.384179\pi$
$912$		0			0
$913$	1296.00			1.41950
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	104.652i		−	0.114124i
$918$		0			0
$919$	−1079.00			−1.17410			−0.587051	−	0.809550i	$-0.699711\pi$
$919$	−1079.00			−1.17410			−0.587051	+	0.809550i	$0.699711\pi$
$920$		0			0
$921$	215.000	+	608.112i	0.233442	+	0.660273i
$922$		0			0
$923$		−	381.838i		−	0.413692i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−182.000	+	147.078i	−0.196332	+	0.158660i
$928$		0			0
$929$		−	144.250i		−	0.155274i	−0.996982	−	0.0776371i	$-0.975262\pi$
$929$		−	144.250i		−	0.155274i	0.996982	−	0.0776371i	$-0.0247376\pi$
$930$		0			0
$931$	1104.00			1.18582
$932$		0			0
$933$	−608.000	+	214.960i	−0.651661	+	0.230397i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	73.0000			0.0779082			0.0389541	−	0.999241i	$-0.487597\pi$
$937$	73.0000			0.0779082			0.0389541	+	0.999241i	$0.487597\pi$
$938$		0			0
$939$	−535.000	−	1513.21i	−0.569755	−	1.61151i
$940$		0			0
$941$			1368.96i			1.45479i	0.686218	+	0.727396i	$0.259269\pi$
$941$			1368.96i			1.45479i	−0.686218	+	0.727396i	$0.740731\pi$
$942$		0			0
$943$	104.000			0.110286
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	596.798i		−	0.630199i	−0.949059	−	0.315099i	$-0.897962\pi$
$947$		−	596.798i		−	0.630199i	0.949059	−	0.315099i	$-0.102038\pi$
$948$		0			0
$949$	870.000			0.916754
$950$		0			0
$951$	−1328.00	+	469.519i	−1.39642	+	0.493711i
$952$		0			0
$953$			763.675i			0.801338i	0.916223	+	0.400669i	$0.131222\pi$
$953$			763.675i			0.801338i	−0.916223	+	0.400669i	$0.868778\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−216.000	−	610.940i	−0.225705	−	0.638391i
$958$		0			0
$959$		−	16.9706i		−	0.0176961i
$960$		0			0
$961$	128.000			0.133195
$962$		0			0
$963$	−272.000	−	336.583i	−0.282451	−	0.349515i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	10.0000			0.0103413			0.00517063	−	0.999987i	$-0.498354\pi$
$967$	10.0000			0.0103413			0.00517063	+	0.999987i	$0.498354\pi$
$968$		0			0
$969$	−1288.00	+	455.377i	−1.32921	+	0.469945i
$970$		0			0
$971$			486.489i			0.501019i	0.968114	+	0.250510i	$0.0805981\pi$
$971$			486.489i			0.501019i	−0.968114	+	0.250510i	$0.919402\pi$
$972$		0			0
$973$	−102.000			−0.104830
$974$		0			0
$975$		0			0
$976$		0			0
$977$			820.244i			0.839554i	0.907627	+	0.419777i	$0.137892\pi$
$977$			820.244i			0.839554i	−0.907627	+	0.419777i	$0.862108\pi$
$978$		0			0
$979$	−768.000			−0.784474
$980$		0			0
$981$	1015.00	−	820.244i	1.03466	−	0.836130i
$982$		0			0
$983$			135.765i			0.138112i	0.997613	+	0.0690562i	$0.0219988\pi$
$983$			135.765i			0.138112i	−0.997613	+	0.0690562i	$0.978001\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	128.000	−	45.2548i	0.129686	−	0.0458509i
$988$		0			0
$989$		−	19.7990i		−	0.0200192i
$990$		0			0
$991$	−79.0000			−0.0797175			−0.0398587	−	0.999205i	$-0.512691\pi$
$991$	−79.0000			−0.0797175			−0.0398587	+	0.999205i	$0.512691\pi$
$992$		0			0
$993$	94.0000	+	265.872i	0.0946626	+	0.267746i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−378.000			−0.379137			−0.189569	−	0.981867i	$-0.560709\pi$
$997$	−378.000			−0.379137			−0.189569	+	0.981867i	$0.560709\pi$
$998$		0			0
$999$	1518.00	+	933.381i	1.51952	+	0.934315i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	600.3.l.c.401.2	yes	2
3.2	odd	2	inner	600.3.l.c.401.1	yes	2
4.3	odd	2		1200.3.l.j.401.1		2
5.2	odd	4		600.3.c.b.449.3		4
5.3	odd	4		600.3.c.b.449.2		4
5.4	even	2		600.3.l.a.401.1	✓	2
12.11	even	2		1200.3.l.j.401.2		2
15.2	even	4		600.3.c.b.449.1		4
15.8	even	4		600.3.c.b.449.4		4
15.14	odd	2		600.3.l.a.401.2	yes	2
20.3	even	4		1200.3.c.g.449.3		4
20.7	even	4		1200.3.c.g.449.2		4
20.19	odd	2		1200.3.l.o.401.2		2
60.23	odd	4		1200.3.c.g.449.1		4
60.47	odd	4		1200.3.c.g.449.4		4
60.59	even	2		1200.3.l.o.401.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
600.3.c.b.449.1		4	15.2	even	4
600.3.c.b.449.2		4	5.3	odd	4
600.3.c.b.449.3		4	5.2	odd	4
600.3.c.b.449.4		4	15.8	even	4
600.3.l.a.401.1	✓	2	5.4	even	2
600.3.l.a.401.2	yes	2	15.14	odd	2
600.3.l.c.401.1	yes	2	3.2	odd	2	inner
600.3.l.c.401.2	yes	2	1.1	even	1	trivial
1200.3.c.g.449.1		4	60.23	odd	4
1200.3.c.g.449.2		4	20.7	even	4
1200.3.c.g.449.3		4	20.3	even	4
1200.3.c.g.449.4		4	60.47	odd	4
1200.3.l.j.401.1		2	4.3	odd	2
1200.3.l.j.401.2		2	12.11	even	2
1200.3.l.o.401.1		2	60.59	even	2
1200.3.l.o.401.2		2	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(1.00000 + 2.82843i) q^{3} -1.00000 q^{7} +(-7.00000 + 5.65685i) q^{9} +8.48528i q^{11} -15.0000 q^{13} -19.7990i q^{17} -23.0000 q^{19} +(-1.00000 - 2.82843i) q^{21} -2.82843i q^{23} +(-23.0000 - 14.1421i) q^{27} +25.4558i q^{29} +33.0000 q^{31} +(-24.0000 + 8.48528i) q^{33} -66.0000 q^{37} +(-15.0000 - 42.4264i) q^{39} +36.7696i q^{41} +7.00000 q^{43} +45.2548i q^{47} -48.0000 q^{49} +(56.0000 - 19.7990i) q^{51} -36.7696i q^{53} +(-23.0000 - 65.0538i) q^{57} -101.823i q^{59} +39.0000 q^{61} +(7.00000 - 5.65685i) q^{63} -113.000 q^{67} +(8.00000 - 2.82843i) q^{69} +25.4558i q^{71} -58.0000 q^{73} -8.48528i q^{77} +70.0000 q^{79} +(17.0000 - 79.1960i) q^{81} -152.735i q^{83} +(-72.0000 + 25.4558i) q^{87} +90.5097i q^{89} +15.0000 q^{91} +(33.0000 + 93.3381i) q^{93} +1.00000 q^{97} +(-48.0000 - 59.3970i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{7} - 14 q^{9} - 30 q^{13} - 46 q^{19} - 2 q^{21} - 46 q^{27} + 66 q^{31} - 48 q^{33} - 132 q^{37} - 30 q^{39} + 14 q^{43} - 96 q^{49} + 112 q^{51} - 46 q^{57} + 78 q^{61} + 14 q^{63}+ \cdots - 96 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^7 - 14 * q^9 - 30 * q^13 - 46 * q^19 - 2 * q^21 - 46 * q^27 + 66 * q^31 - 48 * q^33 - 132 * q^37 - 30 * q^39 + 14 * q^43 - 96 * q^49 + 112 * q^51 - 46 * q^57 + 78 * q^61 + 14 * q^63 - 226 * q^67 + 16 * q^69 - 116 * q^73 + 140 * q^79 + 34 * q^81 - 144 * q^87 + 30 * q^91 + 66 * q^93 + 2 * q^97 - 96 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more