LMFDB - Embedded newform 1200.3.c.i.449.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1200,3,Mod(449,1200)] 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("1200.449"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1200, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 3, names="a")

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

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$32.6976317232$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 24)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.1
Root			$0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1200.449
Dual form			1200.3.c.i.449.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.82843 - 1.00000i) q^{3} -6.00000i q^{7} +(7.00000 + 5.65685i) q^{9} -5.65685i q^{11} +10.0000i q^{13} -22.6274 q^{17} +2.00000 q^{19} +(-6.00000 + 16.9706i) q^{21} -11.3137 q^{23} +(-14.1421 - 23.0000i) q^{27} -16.9706i q^{29} +22.0000 q^{31} +(-5.65685 + 16.0000i) q^{33} +6.00000i q^{37} +(10.0000 - 28.2843i) q^{39} +33.9411i q^{41} -82.0000i q^{43} -67.8823 q^{47} +13.0000 q^{49} +(64.0000 + 22.6274i) q^{51} +62.2254 q^{53} +(-5.65685 - 2.00000i) q^{57} +73.5391i q^{59} -86.0000 q^{61} +(33.9411 - 42.0000i) q^{63} +2.00000i q^{67} +(32.0000 + 11.3137i) q^{69} +124.451i q^{71} +82.0000i q^{73} -33.9411 q^{77} +10.0000 q^{79} +(17.0000 + 79.1960i) q^{81} -73.5391 q^{83} +(-16.9706 + 48.0000i) q^{87} +33.9411i q^{89} +60.0000 q^{91} +(-62.2254 - 22.0000i) q^{93} +94.0000i q^{97} +(32.0000 - 39.5980i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 28 q^{9} + 8 q^{19} - 24 q^{21} + 88 q^{31} + 40 q^{39} + 52 q^{49} + 256 q^{51} - 344 q^{61} + 128 q^{69} + 40 q^{79} + 68 q^{81} + 240 q^{91} + 128 q^{99}+O(q^{100}) $ 4 * q + 28 * q^9 + 8 * q^19 - 24 * q^21 + 88 * q^31 + 40 * q^39 + 52 * q^49 + 256 * q^51 - 344 * q^61 + 128 * q^69 + 40 * q^79 + 68 * q^81 + 240 * q^91 + 128 * 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
$3$	−2.82843	−	1.00000i	−0.942809	−	0.333333i
$3$	−2.82843	−	1.00000i	−0.942809	−	0.333333i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$		−	6.00000i		−	0.857143i	−0.903508	−	0.428571i	$-0.859017\pi$
$7$		−	6.00000i		−	0.857143i	0.903508	−	0.428571i	$-0.140983\pi$
$8$		0			0
$9$	7.00000	+	5.65685i	0.777778	+	0.628539i
$10$		0			0
$11$		−	5.65685i		−	0.514259i	−0.966377	−	0.257130i	$-0.917223\pi$
$11$		−	5.65685i		−	0.514259i	0.966377	−	0.257130i	$-0.0827768\pi$
$12$		0			0
$13$			10.0000i			0.769231i	0.923077	+	0.384615i	$0.125666\pi$
$13$			10.0000i			0.769231i	−0.923077	+	0.384615i	$0.874334\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−22.6274			−1.33102			−0.665512	−	0.746387i	$-0.731787\pi$
$17$	−22.6274			−1.33102			−0.665512	+	0.746387i	$0.731787\pi$
$18$		0			0
$19$	2.00000			0.105263			0.0526316	−	0.998614i	$-0.483239\pi$
$19$	2.00000			0.105263			0.0526316	+	0.998614i	$0.483239\pi$
$20$		0			0
$21$	−6.00000	+	16.9706i	−0.285714	+	0.808122i
$22$		0			0
$23$	−11.3137			−0.491900			−0.245950	−	0.969282i	$-0.579100\pi$
$23$	−11.3137			−0.491900			−0.245950	+	0.969282i	$0.579100\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−14.1421	−	23.0000i	−0.523783	−	0.851852i
$28$		0			0
$29$		−	16.9706i		−	0.585192i	−0.956236	−	0.292596i	$-0.905481\pi$
$29$		−	16.9706i		−	0.585192i	0.956236	−	0.292596i	$-0.0945191\pi$
$30$		0			0
$31$	22.0000			0.709677			0.354839	−	0.934928i	$-0.384536\pi$
$31$	22.0000			0.709677			0.354839	+	0.934928i	$0.384536\pi$
$32$		0			0
$33$	−5.65685	+	16.0000i	−0.171420	+	0.484848i
$34$		0			0
$35$		0			0
$36$		0			0
$37$			6.00000i			0.162162i	0.996708	+	0.0810811i	$0.0258373\pi$
$37$			6.00000i			0.162162i	−0.996708	+	0.0810811i	$0.974163\pi$
$38$		0			0
$39$	10.0000	−	28.2843i	0.256410	−	0.725238i
$40$		0			0
$41$			33.9411i			0.827832i	0.910315	+	0.413916i	$0.135839\pi$
$41$			33.9411i			0.827832i	−0.910315	+	0.413916i	$0.864161\pi$
$42$		0			0
$43$		−	82.0000i		−	1.90698i	−0.301430	−	0.953488i	$-0.597464\pi$
$43$		−	82.0000i		−	1.90698i	0.301430	−	0.953488i	$-0.402536\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−67.8823			−1.44430			−0.722152	−	0.691735i	$-0.756847\pi$
$47$	−67.8823			−1.44430			−0.722152	+	0.691735i	$0.756847\pi$
$48$		0			0
$49$	13.0000			0.265306
$50$		0			0
$51$	64.0000	+	22.6274i	1.25490	+	0.443675i
$52$		0			0
$53$	62.2254			1.17406			0.587032	−	0.809564i	$-0.300296\pi$
$53$	62.2254			1.17406			0.587032	+	0.809564i	$0.300296\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−5.65685	−	2.00000i	−0.0992431	−	0.0350877i
$58$		0			0
$59$			73.5391i			1.24643i	0.782052	+	0.623213i	$0.214173\pi$
$59$			73.5391i			1.24643i	−0.782052	+	0.623213i	$0.785827\pi$
$60$		0			0
$61$	−86.0000			−1.40984			−0.704918	−	0.709289i	$-0.749016\pi$
$61$	−86.0000			−1.40984			−0.704918	+	0.709289i	$0.749016\pi$
$62$		0			0
$63$	33.9411	−	42.0000i	0.538748	−	0.666667i
$64$		0			0
$65$		0			0
$66$		0			0
$67$			2.00000i			0.0298507i	0.999889	+	0.0149254i	$0.00475107\pi$
$67$			2.00000i			0.0298507i	−0.999889	+	0.0149254i	$0.995249\pi$
$68$		0			0
$69$	32.0000	+	11.3137i	0.463768	+	0.163967i
$70$		0			0
$71$			124.451i			1.75283i	0.481558	+	0.876414i	$0.340071\pi$
$71$			124.451i			1.75283i	−0.481558	+	0.876414i	$0.659929\pi$
$72$		0			0
$73$			82.0000i			1.12329i	0.827379	+	0.561644i	$0.189831\pi$
$73$			82.0000i			1.12329i	−0.827379	+	0.561644i	$0.810169\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−33.9411			−0.440794
$78$		0			0
$79$	10.0000			0.126582			0.0632911	−	0.997995i	$-0.479840\pi$
$79$	10.0000			0.126582			0.0632911	+	0.997995i	$0.479840\pi$
$80$		0			0
$81$	17.0000	+	79.1960i	0.209877	+	0.977728i
$82$		0			0
$83$	−73.5391			−0.886013			−0.443007	−	0.896518i	$-0.646088\pi$
$83$	−73.5391			−0.886013			−0.443007	+	0.896518i	$0.646088\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−16.9706	+	48.0000i	−0.195064	+	0.551724i
$88$		0			0
$89$			33.9411i			0.381361i	0.981652	+	0.190680i	$0.0610694\pi$
$89$			33.9411i			0.381361i	−0.981652	+	0.190680i	$0.938931\pi$
$90$		0			0
$91$	60.0000			0.659341
$92$		0			0
$93$	−62.2254	−	22.0000i	−0.669090	−	0.236559i
$94$		0			0
$95$		0			0
$96$		0			0
$97$			94.0000i			0.969072i	0.874771	+	0.484536i	$0.161012\pi$
$97$			94.0000i			0.969072i	−0.874771	+	0.484536i	$0.838988\pi$
$98$		0			0
$99$	32.0000	−	39.5980i	0.323232	−	0.399980i
$100$		0			0
$101$			50.9117i			0.504076i	0.967717	+	0.252038i	$0.0811008\pi$
$101$			50.9117i			0.504076i	−0.967717	+	0.252038i	$0.918899\pi$
$102$		0			0
$103$			134.000i			1.30097i	0.759519	+	0.650485i	$0.225435\pi$
$103$			134.000i			1.30097i	−0.759519	+	0.650485i	$0.774565\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−50.9117			−0.475810			−0.237905	−	0.971288i	$-0.576461\pi$
$107$	−50.9117			−0.475810			−0.237905	+	0.971288i	$0.576461\pi$
$108$		0			0
$109$	−10.0000			−0.0917431			−0.0458716	−	0.998947i	$-0.514606\pi$
$109$	−10.0000			−0.0917431			−0.0458716	+	0.998947i	$0.514606\pi$
$110$		0			0
$111$	6.00000	−	16.9706i	0.0540541	−	0.152888i
$112$		0			0
$113$	67.8823			0.600728			0.300364	−	0.953825i	$-0.402892\pi$
$113$	67.8823			0.600728			0.300364	+	0.953825i	$0.402892\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−56.5685	+	70.0000i	−0.483492	+	0.598291i
$118$		0			0
$119$			135.765i			1.14088i
$120$		0			0
$121$	89.0000			0.735537
$122$		0			0
$123$	33.9411	−	96.0000i	0.275944	−	0.780488i
$124$		0			0
$125$		0			0
$126$		0			0
$127$			106.000i			0.834646i	0.908758	+	0.417323i	$0.137032\pi$
$127$			106.000i			0.834646i	−0.908758	+	0.417323i	$0.862968\pi$
$128$		0			0
$129$	−82.0000	+	231.931i	−0.635659	+	1.79791i
$130$		0			0
$131$			5.65685i			0.0431821i	0.999767	+	0.0215910i	$0.00687318\pi$
$131$			5.65685i			0.0431821i	−0.999767	+	0.0215910i	$0.993127\pi$
$132$		0			0
$133$		−	12.0000i		−	0.0902256i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−101.823			−0.743236			−0.371618	−	0.928386i	$-0.621197\pi$
$137$	−101.823			−0.743236			−0.371618	+	0.928386i	$0.621197\pi$
$138$		0			0
$139$	−78.0000			−0.561151			−0.280576	−	0.959832i	$-0.590525\pi$
$139$	−78.0000			−0.561151			−0.280576	+	0.959832i	$0.590525\pi$
$140$		0			0
$141$	192.000	+	67.8823i	1.36170	+	0.481434i
$142$		0			0
$143$	56.5685			0.395584
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−36.7696	−	13.0000i	−0.250133	−	0.0884354i
$148$		0			0
$149$		−	164.049i		−	1.10100i	−0.834836	−	0.550499i	$-0.814437\pi$
$149$		−	164.049i		−	1.10100i	0.834836	−	0.550499i	$-0.185563\pi$
$150$		0			0
$151$	−218.000			−1.44371			−0.721854	−	0.692045i	$-0.756710\pi$
$151$	−218.000			−1.44371			−0.721854	+	0.692045i	$0.756710\pi$
$152$		0			0
$153$	−158.392	−	128.000i	−1.03524	−	0.836601i
$154$		0			0
$155$		0			0
$156$		0			0
$157$			86.0000i			0.547771i	0.961762	+	0.273885i	$0.0883089\pi$
$157$			86.0000i			0.547771i	−0.961762	+	0.273885i	$0.911691\pi$
$158$		0			0
$159$	−176.000	−	62.2254i	−1.10692	−	0.391355i
$160$		0			0
$161$			67.8823i			0.421629i
$162$		0			0
$163$			222.000i			1.36196i	0.732301	+	0.680982i	$0.238447\pi$
$163$			222.000i			1.36196i	−0.732301	+	0.680982i	$0.761553\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	169.706			1.01620			0.508101	−	0.861298i	$-0.330348\pi$
$167$	169.706			1.01620			0.508101	+	0.861298i	$0.330348\pi$
$168$		0			0
$169$	69.0000			0.408284
$170$		0			0
$171$	14.0000	+	11.3137i	0.0818713	+	0.0661620i
$172$		0			0
$173$	186.676			1.07905			0.539527	−	0.841969i	$-0.318603\pi$
$173$	186.676			1.07905			0.539527	+	0.841969i	$0.318603\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	73.5391	−	208.000i	0.415475	−	1.17514i
$178$		0			0
$179$			152.735i			0.853269i	0.904424	+	0.426634i	$0.140301\pi$
$179$			152.735i			0.853269i	−0.904424	+	0.426634i	$0.859699\pi$
$180$		0			0
$181$	90.0000			0.497238			0.248619	−	0.968601i	$-0.420023\pi$
$181$	90.0000			0.497238			0.248619	+	0.968601i	$0.420023\pi$
$182$		0			0
$183$	243.245	+	86.0000i	1.32921	+	0.469945i
$184$		0			0
$185$		0			0
$186$		0			0
$187$			128.000i			0.684492i
$188$		0			0
$189$	−138.000	+	84.8528i	−0.730159	+	0.448957i
$190$		0			0
$191$			271.529i			1.42162i	0.703385	+	0.710809i	$0.251671\pi$
$191$			271.529i			1.42162i	−0.703385	+	0.710809i	$0.748329\pi$
$192$		0			0
$193$			2.00000i			0.0103627i	0.999987	+	0.00518135i	$0.00164928\pi$
$193$			2.00000i			0.0103627i	−0.999987	+	0.00518135i	$0.998351\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−84.8528			−0.430725			−0.215362	−	0.976534i	$-0.569093\pi$
$197$	−84.8528			−0.430725			−0.215362	+	0.976534i	$0.569093\pi$
$198$		0			0
$199$	250.000			1.25628			0.628141	−	0.778100i	$-0.283816\pi$
$199$	250.000			1.25628			0.628141	+	0.778100i	$0.283816\pi$
$200$		0			0
$201$	2.00000	−	5.65685i	0.00995025	−	0.0281436i
$202$		0			0
$203$	−101.823			−0.501593
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−79.1960	−	64.0000i	−0.382589	−	0.309179i
$208$		0			0
$209$		−	11.3137i		−	0.0541326i
$210$		0			0
$211$	−34.0000			−0.161137			−0.0805687	−	0.996749i	$-0.525674\pi$
$211$	−34.0000			−0.161137			−0.0805687	+	0.996749i	$0.525674\pi$
$212$		0			0
$213$	124.451	−	352.000i	0.584276	−	1.65258i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		−	132.000i		−	0.608295i
$218$		0			0
$219$	82.0000	−	231.931i	0.374429	−	1.05905i
$220$		0			0
$221$		−	226.274i		−	1.02387i
$222$		0			0
$223$			278.000i			1.24664i	0.781968	+	0.623318i	$0.214216\pi$
$223$			278.000i			1.24664i	−0.781968	+	0.623318i	$0.785784\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−220.617			−0.971882			−0.485941	−	0.873991i	$-0.661523\pi$
$227$	−220.617			−0.971882			−0.485941	+	0.873991i	$0.661523\pi$
$228$		0			0
$229$	−58.0000			−0.253275			−0.126638	−	0.991949i	$-0.540419\pi$
$229$	−58.0000			−0.253275			−0.126638	+	0.991949i	$0.540419\pi$
$230$		0			0
$231$	96.0000	+	33.9411i	0.415584	+	0.146931i
$232$		0			0
$233$	−395.980			−1.69948			−0.849742	−	0.527199i	$-0.823242\pi$
$233$	−395.980			−1.69948			−0.849742	+	0.527199i	$0.823242\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−28.2843	−	10.0000i	−0.119343	−	0.0421941i
$238$		0			0
$239$		−	22.6274i		−	0.0946754i	−0.998879	−	0.0473377i	$-0.984926\pi$
$239$		−	22.6274i		−	0.0946754i	0.998879	−	0.0473377i	$-0.0150737\pi$
$240$		0			0
$241$	−30.0000			−0.124481			−0.0622407	−	0.998061i	$-0.519825\pi$
$241$	−30.0000			−0.124481			−0.0622407	+	0.998061i	$0.519825\pi$
$242$		0			0
$243$	31.1127	−	241.000i	0.128036	−	0.991770i
$244$		0			0
$245$		0			0
$246$		0			0
$247$			20.0000i			0.0809717i
$248$		0			0
$249$	208.000	+	73.5391i	0.835341	+	0.295338i
$250$		0			0
$251$			107.480i			0.428208i	0.976811	+	0.214104i	$0.0686832\pi$
$251$			107.480i			0.428208i	−0.976811	+	0.214104i	$0.931317\pi$
$252$		0			0
$253$			64.0000i			0.252964i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	181.019			0.704355			0.352178	−	0.935933i	$-0.385441\pi$
$257$	181.019			0.704355			0.352178	+	0.935933i	$0.385441\pi$
$258$		0			0
$259$	36.0000			0.138996
$260$		0			0
$261$	96.0000	−	118.794i	0.367816	−	0.455149i
$262$		0			0
$263$	−214.960			−0.817340			−0.408670	−	0.912682i	$-0.634007\pi$
$263$	−214.960			−0.817340			−0.408670	+	0.912682i	$0.634007\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	33.9411	−	96.0000i	0.127120	−	0.359551i
$268$		0			0
$269$		−	401.637i		−	1.49307i	−0.665344	−	0.746537i	$-0.731715\pi$
$269$		−	401.637i		−	1.49307i	0.665344	−	0.746537i	$-0.268285\pi$
$270$		0			0
$271$	−266.000			−0.981550			−0.490775	−	0.871286i	$-0.663286\pi$
$271$	−266.000			−0.981550			−0.490775	+	0.871286i	$0.663286\pi$
$272$		0			0
$273$	−169.706	−	60.0000i	−0.621632	−	0.219780i
$274$		0			0
$275$		0			0
$276$		0			0
$277$		−	346.000i		−	1.24910i	−0.780986	−	0.624549i	$-0.785283\pi$
$277$		−	346.000i		−	1.24910i	0.780986	−	0.624549i	$-0.214717\pi$
$278$		0			0
$279$	154.000	+	124.451i	0.551971	+	0.446060i
$280$		0			0
$281$		−	124.451i		−	0.442885i	−0.975173	−	0.221443i	$-0.928923\pi$
$281$		−	124.451i		−	0.442885i	0.975173	−	0.221443i	$-0.0710766\pi$
$282$		0			0
$283$			46.0000i			0.162544i	0.996692	+	0.0812721i	$0.0258983\pi$
$283$			46.0000i			0.162544i	−0.996692	+	0.0812721i	$0.974102\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	203.647			0.709571
$288$		0			0
$289$	223.000			0.771626
$290$		0			0
$291$	94.0000	−	265.872i	0.323024	−	0.913650i
$292$		0			0
$293$	220.617			0.752960			0.376480	−	0.926425i	$-0.377134\pi$
$293$	220.617			0.752960			0.376480	+	0.926425i	$0.377134\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−130.108	+	80.0000i	−0.438073	+	0.269360i
$298$		0			0
$299$		−	113.137i		−	0.378385i
$300$		0			0
$301$	−492.000			−1.63455
$302$		0			0
$303$	50.9117	−	144.000i	0.168025	−	0.475248i
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	30.0000i		−	0.0977199i	−0.998806	−	0.0488599i	$-0.984441\pi$
$307$		−	30.0000i		−	0.0977199i	0.998806	−	0.0488599i	$-0.0155588\pi$
$308$		0			0
$309$	134.000	−	379.009i	0.433657	−	1.22657i
$310$		0			0
$311$		−	576.999i		−	1.85530i	−0.373447	−	0.927651i	$-0.621824\pi$
$311$		−	576.999i		−	1.85530i	0.373447	−	0.927651i	$-0.378176\pi$
$312$		0			0
$313$			210.000i			0.670927i	0.942053	+	0.335463i	$0.108893\pi$
$313$			210.000i			0.670927i	−0.942053	+	0.335463i	$0.891107\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	152.735			0.481814			0.240907	−	0.970548i	$-0.422555\pi$
$317$	152.735			0.481814			0.240907	+	0.970548i	$0.422555\pi$
$318$		0			0
$319$	−96.0000			−0.300940
$320$		0			0
$321$	144.000	+	50.9117i	0.448598	+	0.158603i
$322$		0			0
$323$	−45.2548			−0.140108
$324$		0			0
$325$		0			0
$326$		0			0
$327$	28.2843	+	10.0000i	0.0864962	+	0.0305810i
$328$		0			0
$329$			407.294i			1.23797i
$330$		0			0
$331$	−434.000			−1.31118			−0.655589	−	0.755118i	$-0.727580\pi$
$331$	−434.000			−1.31118			−0.655589	+	0.755118i	$0.727580\pi$
$332$		0			0
$333$	−33.9411	+	42.0000i	−0.101925	+	0.126126i
$334$		0			0
$335$		0			0
$336$		0			0
$337$			510.000i			1.51335i	0.653789	+	0.756677i	$0.273178\pi$
$337$			510.000i			1.51335i	−0.653789	+	0.756677i	$0.726822\pi$
$338$		0			0
$339$	−192.000	−	67.8823i	−0.566372	−	0.200243i
$340$		0			0
$341$		−	124.451i		−	0.364958i
$342$		0			0
$343$		−	372.000i		−	1.08455i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	152.735			0.440159			0.220079	−	0.975482i	$-0.429368\pi$
$347$	152.735			0.440159			0.220079	+	0.975482i	$0.429368\pi$
$348$		0			0
$349$	−426.000			−1.22063			−0.610315	−	0.792159i	$-0.708957\pi$
$349$	−426.000			−1.22063			−0.610315	+	0.792159i	$0.708957\pi$
$350$		0			0
$351$	230.000	−	141.421i	0.655271	−	0.402910i
$352$		0			0
$353$	−45.2548			−0.128201			−0.0641003	−	0.997943i	$-0.520418\pi$
$353$	−45.2548			−0.128201			−0.0641003	+	0.997943i	$0.520418\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	135.765	−	384.000i	0.380293	−	1.07563i
$358$		0			0
$359$		−	441.235i		−	1.22907i	−0.788891	−	0.614533i	$-0.789345\pi$
$359$		−	441.235i		−	1.22907i	0.788891	−	0.614533i	$-0.210655\pi$
$360$		0			0
$361$	−357.000			−0.988920
$362$		0			0
$363$	−251.730	−	89.0000i	−0.693471	−	0.245179i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		−	566.000i		−	1.54223i	−0.636693	−	0.771117i	$-0.719698\pi$
$367$		−	566.000i		−	1.54223i	0.636693	−	0.771117i	$-0.280302\pi$
$368$		0			0
$369$	−192.000	+	237.588i	−0.520325	+	0.643870i
$370$		0			0
$371$		−	373.352i		−	1.00634i
$372$		0			0
$373$			218.000i			0.584450i	0.956350	+	0.292225i	$0.0943957\pi$
$373$			218.000i			0.584450i	−0.956350	+	0.292225i	$0.905604\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	169.706			0.450148
$378$		0			0
$379$	−142.000			−0.374670			−0.187335	−	0.982296i	$-0.559985\pi$
$379$	−142.000			−0.374670			−0.187335	+	0.982296i	$0.559985\pi$
$380$		0			0
$381$	106.000	−	299.813i	0.278215	−	0.786911i
$382$		0			0
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	463.862	−	574.000i	1.19861	−	1.48320i
$388$		0			0
$389$		−	548.715i		−	1.41058i	−0.708920	−	0.705289i	$-0.750817\pi$
$389$		−	548.715i		−	1.41058i	0.708920	−	0.705289i	$-0.249183\pi$
$390$		0			0
$391$	256.000			0.654731
$392$		0			0
$393$	5.65685	−	16.0000i	0.0143940	−	0.0407125i
$394$		0			0
$395$		0			0
$396$		0			0
$397$			310.000i			0.780856i	0.920633	+	0.390428i	$0.127673\pi$
$397$			310.000i			0.780856i	−0.920633	+	0.390428i	$0.872327\pi$
$398$		0			0
$399$	−12.0000	+	33.9411i	−0.0300752	+	0.0850655i
$400$		0			0
$401$			339.411i			0.846412i	0.906033	+	0.423206i	$0.139095\pi$
$401$			339.411i			0.846412i	−0.906033	+	0.423206i	$0.860905\pi$
$402$		0			0
$403$			220.000i			0.545906i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	33.9411			0.0833934
$408$		0			0
$409$	270.000			0.660147			0.330073	−	0.943955i	$-0.392927\pi$
$409$	270.000			0.660147			0.330073	+	0.943955i	$0.392927\pi$
$410$		0			0
$411$	288.000	+	101.823i	0.700730	+	0.247745i
$412$		0			0
$413$	441.235			1.06836
$414$		0			0
$415$		0			0
$416$		0			0
$417$	220.617	+	78.0000i	0.529058	+	0.187050i
$418$		0			0
$419$		−	50.9117i		−	0.121508i	−0.998153	−	0.0607538i	$-0.980650\pi$
$419$		−	50.9117i		−	0.121508i	0.998153	−	0.0607538i	$-0.0193505\pi$
$420$		0			0
$421$	−454.000			−1.07838			−0.539192	−	0.842183i	$-0.681270\pi$
$421$	−454.000			−1.07838			−0.539192	+	0.842183i	$0.681270\pi$
$422$		0			0
$423$	−475.176	−	384.000i	−1.12335	−	0.907801i
$424$		0			0
$425$		0			0
$426$		0			0
$427$			516.000i			1.20843i
$428$		0			0
$429$	−160.000	−	56.5685i	−0.372960	−	0.131861i
$430$		0			0
$431$		−	248.902i		−	0.577498i	−0.957405	−	0.288749i	$-0.906761\pi$
$431$		−	248.902i		−	0.577498i	0.957405	−	0.288749i	$-0.0932393\pi$
$432$		0			0
$433$			706.000i			1.63048i	0.579120	+	0.815242i	$0.303396\pi$
$433$			706.000i			1.63048i	−0.579120	+	0.815242i	$0.696604\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−22.6274			−0.0517790
$438$		0			0
$439$	−486.000			−1.10706			−0.553531	−	0.832829i	$-0.686720\pi$
$439$	−486.000			−1.10706			−0.553531	+	0.832829i	$0.686720\pi$
$440$		0			0
$441$	91.0000	+	73.5391i	0.206349	+	0.166755i
$442$		0			0
$443$	707.107			1.59618			0.798089	−	0.602540i	$-0.205844\pi$
$443$	707.107			1.59618			0.798089	+	0.602540i	$0.205844\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−164.049	+	464.000i	−0.366999	+	1.03803i
$448$		0			0
$449$			724.077i			1.61264i	0.591477	+	0.806322i	$0.298545\pi$
$449$			724.077i			1.61264i	−0.591477	+	0.806322i	$0.701455\pi$
$450$		0			0
$451$	192.000			0.425721
$452$		0			0
$453$	616.597	+	218.000i	1.36114	+	0.481236i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	338.000i		−	0.739606i	−0.929110	−	0.369803i	$-0.879425\pi$
$457$		−	338.000i		−	0.739606i	0.929110	−	0.369803i	$-0.120575\pi$
$458$		0			0
$459$	320.000	+	520.431i	0.697168	+	1.13384i
$460$		0			0
$461$		−	774.989i		−	1.68110i	−0.541731	−	0.840552i	$-0.682231\pi$
$461$		−	774.989i		−	1.68110i	0.541731	−	0.840552i	$-0.317769\pi$
$462$		0			0
$463$		−	74.0000i		−	0.159827i	−0.996802	−	0.0799136i	$-0.974536\pi$
$463$		−	74.0000i		−	0.159827i	0.996802	−	0.0799136i	$-0.0254644\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	797.616			1.70796			0.853979	−	0.520307i	$-0.174183\pi$
$467$	797.616			1.70796			0.853979	+	0.520307i	$0.174183\pi$
$468$		0			0
$469$	12.0000			0.0255864
$470$		0			0
$471$	86.0000	−	243.245i	0.182590	−	0.516443i
$472$		0			0
$473$	−463.862			−0.980681
$474$		0			0
$475$		0			0
$476$		0			0
$477$	435.578	+	352.000i	0.913161	+	0.737945i
$478$		0			0
$479$			316.784i			0.661344i	0.943746	+	0.330672i	$0.107275\pi$
$479$			316.784i			0.661344i	−0.943746	+	0.330672i	$0.892725\pi$
$480$		0			0
$481$	−60.0000			−0.124740
$482$		0			0
$483$	67.8823	−	192.000i	0.140543	−	0.397516i
$484$		0			0
$485$		0			0
$486$		0			0
$487$		−	134.000i		−	0.275154i	−0.990491	−	0.137577i	$-0.956069\pi$
$487$		−	134.000i		−	0.275154i	0.990491	−	0.137577i	$-0.0439315\pi$
$488$		0			0
$489$	222.000	−	627.911i	0.453988	−	1.28407i
$490$		0			0
$491$		−	50.9117i		−	0.103690i	−0.998655	−	0.0518449i	$-0.983490\pi$
$491$		−	50.9117i		−	0.103690i	0.998655	−	0.0518449i	$-0.0165101\pi$
$492$		0			0
$493$			384.000i			0.778905i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	746.705			1.50242
$498$		0			0
$499$	−30.0000			−0.0601202			−0.0300601	−	0.999548i	$-0.509570\pi$
$499$	−30.0000			−0.0601202			−0.0300601	+	0.999548i	$0.509570\pi$
$500$		0			0
$501$	−480.000	−	169.706i	−0.958084	−	0.338734i
$502$		0			0
$503$	−237.588			−0.472342			−0.236171	−	0.971712i	$-0.575893\pi$
$503$	−237.588			−0.472342			−0.236171	+	0.971712i	$0.575893\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−195.161	−	69.0000i	−0.384934	−	0.136095i
$508$		0			0
$509$			118.794i			0.233387i	0.993168	+	0.116693i	$0.0372295\pi$
$509$			118.794i			0.233387i	−0.993168	+	0.116693i	$0.962770\pi$
$510$		0			0
$511$	492.000			0.962818
$512$		0			0
$513$	−28.2843	−	46.0000i	−0.0551350	−	0.0896686i
$514$		0			0
$515$		0			0
$516$		0			0
$517$			384.000i			0.742747i
$518$		0			0
$519$	−528.000	−	186.676i	−1.01734	−	0.359684i
$520$		0			0
$521$			79.1960i			0.152008i	0.997108	+	0.0760038i	$0.0242161\pi$
$521$			79.1960i			0.152008i	−0.997108	+	0.0760038i	$0.975784\pi$
$522$		0			0
$523$			494.000i			0.944551i	0.881451	+	0.472275i	$0.156567\pi$
$523$			494.000i			0.944551i	−0.881451	+	0.472275i	$0.843433\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−497.803			−0.944598
$528$		0			0
$529$	−401.000			−0.758034
$530$		0			0
$531$	−416.000	+	514.774i	−0.783427	+	0.969442i
$532$		0			0
$533$	−339.411			−0.636794
$534$		0			0
$535$		0			0
$536$		0			0
$537$	152.735	−	432.000i	0.284423	−	0.804469i
$538$		0			0
$539$		−	73.5391i		−	0.136436i
$540$		0			0
$541$	234.000			0.432532			0.216266	−	0.976334i	$-0.430612\pi$
$541$	234.000			0.432532			0.216266	+	0.976334i	$0.430612\pi$
$542$		0			0
$543$	−254.558	−	90.0000i	−0.468800	−	0.165746i
$544$		0			0
$545$		0			0
$546$		0			0
$547$			290.000i			0.530165i	0.964226	+	0.265082i	$0.0853991\pi$
$547$			290.000i			0.530165i	−0.964226	+	0.265082i	$0.914601\pi$
$548$		0			0
$549$	−602.000	−	486.489i	−1.09654	−	0.886137i
$550$		0			0
$551$		−	33.9411i		−	0.0615991i
$552$		0			0
$553$		−	60.0000i		−	0.108499i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	175.362			0.314834			0.157417	−	0.987532i	$-0.449683\pi$
$557$	175.362			0.314834			0.157417	+	0.987532i	$0.449683\pi$
$558$		0			0
$559$	820.000			1.46691
$560$		0			0
$561$	128.000	−	362.039i	0.228164	−	0.645345i
$562$		0			0
$563$	243.245			0.432051			0.216026	−	0.976388i	$-0.430691\pi$
$563$	243.245			0.432051			0.216026	+	0.976388i	$0.430691\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	475.176	−	102.000i	0.838052	−	0.179894i
$568$		0			0
$569$			622.254i			1.09359i	0.837266	+	0.546796i	$0.184153\pi$
$569$			622.254i			1.09359i	−0.837266	+	0.546796i	$0.815847\pi$
$570$		0			0
$571$	−402.000			−0.704028			−0.352014	−	0.935995i	$-0.614503\pi$
$571$	−402.000			−0.704028			−0.352014	+	0.935995i	$0.614503\pi$
$572$		0			0
$573$	271.529	−	768.000i	0.473873	−	1.34031i
$574$		0			0
$575$		0			0
$576$		0			0
$577$		−	98.0000i		−	0.169844i	−0.996388	−	0.0849220i	$-0.972936\pi$
$577$		−	98.0000i		−	0.169844i	0.996388	−	0.0849220i	$-0.0270641\pi$
$578$		0			0
$579$	2.00000	−	5.65685i	0.00345423	−	0.00977004i
$580$		0			0
$581$			441.235i			0.759440i
$582$		0			0
$583$		−	352.000i		−	0.603774i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−548.715			−0.934778			−0.467389	−	0.884052i	$-0.654805\pi$
$587$	−548.715			−0.934778			−0.467389	+	0.884052i	$0.654805\pi$
$588$		0			0
$589$	44.0000			0.0747029
$590$		0			0
$591$	240.000	+	84.8528i	0.406091	+	0.143575i
$592$		0			0
$593$	−701.450			−1.18288			−0.591442	−	0.806348i	$-0.701441\pi$
$593$	−701.450			−1.18288			−0.591442	+	0.806348i	$0.701441\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−707.107	−	250.000i	−1.18443	−	0.418760i
$598$		0			0
$599$		−	644.881i		−	1.07660i	−0.842754	−	0.538298i	$-0.819067\pi$
$599$		−	644.881i		−	1.07660i	0.842754	−	0.538298i	$-0.180933\pi$
$600$		0			0
$601$	−398.000			−0.662230			−0.331115	−	0.943590i	$-0.607425\pi$
$601$	−398.000			−0.662230			−0.331115	+	0.943590i	$0.607425\pi$
$602$		0			0
$603$	−11.3137	+	14.0000i	−0.0187624	+	0.0232172i
$604$		0			0
$605$		0			0
$606$		0			0
$607$			170.000i			0.280066i	0.990147	+	0.140033i	$0.0447208\pi$
$607$			170.000i			0.280066i	−0.990147	+	0.140033i	$0.955279\pi$
$608$		0			0
$609$	288.000	+	101.823i	0.472906	+	0.167198i
$610$		0			0
$611$		−	678.823i		−	1.11100i
$612$		0			0
$613$		−	1030.00i		−	1.68026i	−0.542384	−	0.840131i	$-0.682478\pi$
$613$		−	1030.00i		−	1.68026i	0.542384	−	0.840131i	$-0.317522\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−1052.17			−1.70531			−0.852654	−	0.522476i	$-0.825008\pi$
$617$	−1052.17			−1.70531			−0.852654	+	0.522476i	$0.825008\pi$
$618$		0			0
$619$	−14.0000			−0.0226171			−0.0113086	−	0.999936i	$-0.503600\pi$
$619$	−14.0000			−0.0226171			−0.0113086	+	0.999936i	$0.503600\pi$
$620$		0			0
$621$	160.000	+	260.215i	0.257649	+	0.419026i
$622$		0			0
$623$	203.647			0.326881
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−11.3137	+	32.0000i	−0.0180442	+	0.0510367i
$628$		0			0
$629$		−	135.765i		−	0.215842i
$630$		0			0
$631$	−1114.00			−1.76545			−0.882726	−	0.469888i	$-0.844294\pi$
$631$	−1114.00			−1.76545			−0.882726	+	0.469888i	$0.844294\pi$
$632$		0			0
$633$	96.1665	+	34.0000i	0.151922	+	0.0537125i
$634$		0			0
$635$		0			0
$636$		0			0
$637$			130.000i			0.204082i
$638$		0			0
$639$	−704.000	+	871.156i	−1.10172	+	1.36331i
$640$		0			0
$641$		−	452.548i		−	0.706004i	−0.935623	−	0.353002i	$-0.885161\pi$
$641$		−	452.548i		−	0.706004i	0.935623	−	0.353002i	$-0.114839\pi$
$642$		0			0
$643$			798.000i			1.24106i	0.784184	+	0.620529i	$0.213082\pi$
$643$			798.000i			1.24106i	−0.784184	+	0.620529i	$0.786918\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−147.078			−0.227323			−0.113662	−	0.993520i	$-0.536258\pi$
$647$	−147.078			−0.227323			−0.113662	+	0.993520i	$0.536258\pi$
$648$		0			0
$649$	416.000			0.640986
$650$		0			0
$651$	−132.000	+	373.352i	−0.202765	+	0.573506i
$652$		0			0
$653$	322.441			0.493784			0.246892	−	0.969043i	$-0.420591\pi$
$653$	322.441			0.493784			0.246892	+	0.969043i	$0.420591\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−463.862	+	574.000i	−0.706031	+	0.873668i
$658$		0			0
$659$		−	797.616i		−	1.21034i	−0.796095	−	0.605172i	$-0.793104\pi$
$659$		−	797.616i		−	1.21034i	0.796095	−	0.605172i	$-0.206896\pi$
$660$		0			0
$661$	986.000			1.49168			0.745840	−	0.666126i	$-0.232049\pi$
$661$	986.000			1.49168			0.745840	+	0.666126i	$0.232049\pi$
$662$		0			0
$663$	−226.274	+	640.000i	−0.341288	+	0.965309i
$664$		0			0
$665$		0			0
$666$		0			0
$667$			192.000i			0.287856i
$668$		0			0
$669$	278.000	−	786.303i	0.415546	−	1.17534i
$670$		0			0
$671$			486.489i			0.725022i
$672$		0			0
$673$			34.0000i			0.0505201i	0.999681	+	0.0252600i	$0.00804137\pi$
$673$			34.0000i			0.0505201i	−0.999681	+	0.0252600i	$0.991959\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−401.637			−0.593259			−0.296630	−	0.954993i	$-0.595863\pi$
$677$	−401.637			−0.593259			−0.296630	+	0.954993i	$0.595863\pi$
$678$		0			0
$679$	564.000			0.830633
$680$		0			0
$681$	624.000	+	220.617i	0.916300	+	0.323961i
$682$		0			0
$683$	−130.108			−0.190494			−0.0952472	−	0.995454i	$-0.530364\pi$
$683$	−130.108			−0.190494			−0.0952472	+	0.995454i	$0.530364\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	164.049	+	58.0000i	0.238790	+	0.0844250i
$688$		0			0
$689$			622.254i			0.903126i
$690$		0			0
$691$	−578.000			−0.836469			−0.418234	−	0.908339i	$-0.637351\pi$
$691$	−578.000			−0.836469			−0.418234	+	0.908339i	$0.637351\pi$
$692$		0			0
$693$	−237.588	−	192.000i	−0.342840	−	0.277056i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		−	768.000i		−	1.10187i
$698$		0			0
$699$	1120.00	+	395.980i	1.60229	+	0.566495i
$700$		0			0
$701$			1238.85i			1.76726i	0.468183	+	0.883631i	$0.344909\pi$
$701$			1238.85i			1.76726i	−0.468183	+	0.883631i	$0.655091\pi$
$702$		0			0
$703$			12.0000i			0.0170697i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	305.470			0.432065
$708$		0			0
$709$	1222.00			1.72355			0.861777	−	0.507287i	$-0.169352\pi$
$709$	1222.00			1.72355			0.861777	+	0.507287i	$0.169352\pi$
$710$		0			0
$711$	70.0000	+	56.5685i	0.0984529	+	0.0795619i
$712$		0			0
$713$	−248.902			−0.349091
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−22.6274	+	64.0000i	−0.0315585	+	0.0892608i
$718$		0			0
$719$		−	248.902i		−	0.346177i	−0.984906	−	0.173089i	$-0.944625\pi$
$719$		−	248.902i		−	0.346177i	0.984906	−	0.173089i	$-0.0553747\pi$
$720$		0			0
$721$	804.000			1.11512
$722$		0			0
$723$	84.8528	+	30.0000i	0.117362	+	0.0414938i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	870.000i		−	1.19670i	−0.801235	−	0.598349i	$-0.795823\pi$
$727$		−	870.000i		−	1.19670i	0.801235	−	0.598349i	$-0.204177\pi$
$728$		0			0
$729$	−329.000	+	650.538i	−0.451303	+	0.892371i
$730$		0			0
$731$			1855.45i			2.53823i
$732$		0			0
$733$		−	214.000i		−	0.291951i	−0.989288	−	0.145975i	$-0.953368\pi$
$733$		−	214.000i		−	0.291951i	0.989288	−	0.145975i	$-0.0466321\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	11.3137			0.0153510
$738$		0			0
$739$	−958.000			−1.29635			−0.648173	−	0.761493i	$-0.724467\pi$
$739$	−958.000			−1.29635			−0.648173	+	0.761493i	$0.724467\pi$
$740$		0			0
$741$	20.0000	−	56.5685i	0.0269906	−	0.0763408i
$742$		0			0
$743$	1006.92			1.35521			0.677604	−	0.735427i	$-0.263018\pi$
$743$	1006.92			1.35521			0.677604	+	0.735427i	$0.263018\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−514.774	−	416.000i	−0.689121	−	0.556894i
$748$		0			0
$749$			305.470i			0.407837i
$750$		0			0
$751$	630.000			0.838881			0.419441	−	0.907783i	$-0.362226\pi$
$751$	630.000			0.838881			0.419441	+	0.907783i	$0.362226\pi$
$752$		0			0
$753$	107.480	−	304.000i	0.142736	−	0.403718i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	602.000i		−	0.795244i	−0.917549	−	0.397622i	$-0.869835\pi$
$757$		−	602.000i		−	0.795244i	0.917549	−	0.397622i	$-0.130165\pi$
$758$		0			0
$759$	64.0000	−	181.019i	0.0843215	−	0.238497i
$760$		0			0
$761$			1097.43i			1.44209i	0.692889	+	0.721044i	$0.256338\pi$
$761$			1097.43i			1.44209i	−0.692889	+	0.721044i	$0.743662\pi$
$762$		0			0
$763$			60.0000i			0.0786370i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−735.391			−0.958789
$768$		0			0
$769$	−770.000			−1.00130			−0.500650	−	0.865650i	$-0.666906\pi$
$769$	−770.000			−1.00130			−0.500650	+	0.865650i	$0.666906\pi$
$770$		0			0
$771$	−512.000	−	181.019i	−0.664073	−	0.234785i
$772$		0			0
$773$	−186.676			−0.241496			−0.120748	−	0.992683i	$-0.538529\pi$
$773$	−186.676			−0.241496			−0.120748	+	0.992683i	$0.538529\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−101.823	−	36.0000i	−0.131047	−	0.0463320i
$778$		0			0
$779$			67.8823i			0.0871402i
$780$		0			0
$781$	704.000			0.901408
$782$		0			0
$783$	−390.323	+	240.000i	−0.498497	+	0.306513i
$784$		0			0
$785$		0			0
$786$		0			0
$787$			514.000i			0.653113i	0.945178	+	0.326557i	$0.105888\pi$
$787$			514.000i			0.653113i	−0.945178	+	0.326557i	$0.894112\pi$
$788$		0			0
$789$	608.000	+	214.960i	0.770596	+	0.272447i
$790$		0			0
$791$		−	407.294i		−	0.514910i
$792$		0			0
$793$		−	860.000i		−	1.08449i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−707.107			−0.887211			−0.443605	−	0.896222i	$-0.646301\pi$
$797$	−707.107			−0.887211			−0.443605	+	0.896222i	$0.646301\pi$
$798$		0			0
$799$	1536.00			1.92240
$800$		0			0
$801$	−192.000	+	237.588i	−0.239700	+	0.296614i
$802$		0			0
$803$	463.862			0.577661
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−401.637	+	1136.00i	−0.497691	+	1.40768i
$808$		0			0
$809$		−	758.018i		−	0.936982i	−0.883468	−	0.468491i	$-0.844798\pi$
$809$		−	758.018i		−	0.936982i	0.883468	−	0.468491i	$-0.155202\pi$
$810$		0			0
$811$	1454.00			1.79285			0.896424	−	0.443197i	$-0.146156\pi$
$811$	1454.00			1.79285			0.896424	+	0.443197i	$0.146156\pi$
$812$		0			0
$813$	752.362	+	266.000i	0.925414	+	0.327183i
$814$		0			0
$815$		0			0
$816$		0			0
$817$		−	164.000i		−	0.200734i
$818$		0			0
$819$	420.000	+	339.411i	0.512821	+	0.414422i
$820$		0			0
$821$		−	967.322i		−	1.17822i	−0.808051	−	0.589112i	$-0.799478\pi$
$821$		−	967.322i		−	1.17822i	0.808051	−	0.589112i	$-0.200522\pi$
$822$		0			0
$823$			166.000i			0.201701i	0.994902	+	0.100851i	$0.0321564\pi$
$823$			166.000i			0.201701i	−0.994902	+	0.100851i	$0.967844\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−978.636			−1.18336			−0.591678	−	0.806174i	$-0.701534\pi$
$827$	−978.636			−1.18336			−0.591678	+	0.806174i	$0.701534\pi$
$828$		0			0
$829$	−1258.00			−1.51749			−0.758745	−	0.651387i	$-0.774187\pi$
$829$	−1258.00			−1.51749			−0.758745	+	0.651387i	$0.774187\pi$
$830$		0			0
$831$	−346.000	+	978.636i	−0.416366	+	1.17766i
$832$		0			0
$833$	−294.156			−0.353129
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−311.127	−	506.000i	−0.371717	−	0.604540i
$838$		0			0
$839$			1323.70i			1.57772i	0.614575	+	0.788858i	$0.289327\pi$
$839$			1323.70i			1.57772i	−0.614575	+	0.788858i	$0.710673\pi$
$840$		0			0
$841$	553.000			0.657551
$842$		0			0
$843$	−124.451	+	352.000i	−0.147628	+	0.417556i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		−	534.000i		−	0.630460i
$848$		0			0
$849$	46.0000	−	130.108i	0.0541814	−	0.153248i
$850$		0			0
$851$		−	67.8823i		−	0.0797676i
$852$		0			0
$853$		−	742.000i		−	0.869871i	−0.900462	−	0.434936i	$-0.856771\pi$
$853$		−	742.000i		−	0.869871i	0.900462	−	0.434936i	$-0.143229\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	237.588			0.277232			0.138616	−	0.990346i	$-0.455735\pi$
$857$	237.588			0.277232			0.138616	+	0.990346i	$0.455735\pi$
$858$		0			0
$859$	−1230.00			−1.43190			−0.715949	−	0.698153i	$-0.754006\pi$
$859$	−1230.00			−1.43190			−0.715949	+	0.698153i	$0.754006\pi$
$860$		0			0
$861$	−576.000	−	203.647i	−0.668990	−	0.236524i
$862$		0			0
$863$	−45.2548			−0.0524390			−0.0262195	−	0.999656i	$-0.508347\pi$
$863$	−45.2548			−0.0524390			−0.0262195	+	0.999656i	$0.508347\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−630.739	−	223.000i	−0.727496	−	0.257209i
$868$		0			0
$869$		−	56.5685i		−	0.0650961i
$870$		0			0
$871$	−20.0000			−0.0229621
$872$		0			0
$873$	−531.744	+	658.000i	−0.609100	+	0.753723i
$874$		0			0
$875$		0			0
$876$		0			0
$877$			822.000i			0.937286i	0.883388	+	0.468643i	$0.155257\pi$
$877$			822.000i			0.937286i	−0.883388	+	0.468643i	$0.844743\pi$
$878$		0			0
$879$	−624.000	−	220.617i	−0.709898	−	0.250987i
$880$		0			0
$881$			656.195i			0.744830i	0.928066	+	0.372415i	$0.121470\pi$
$881$			656.195i			0.744830i	−0.928066	+	0.372415i	$0.878530\pi$
$882$		0			0
$883$		−	962.000i		−	1.08947i	−0.838609	−	0.544734i	$-0.816631\pi$
$883$		−	962.000i		−	1.08947i	0.838609	−	0.544734i	$-0.183369\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1142.68			1.28826			0.644129	−	0.764917i	$-0.277220\pi$
$887$	1142.68			1.28826			0.644129	+	0.764917i	$0.277220\pi$
$888$		0			0
$889$	636.000			0.715411
$890$		0			0
$891$	448.000	−	96.1665i	0.502806	−	0.107931i
$892$		0			0
$893$	−135.765			−0.152032
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−113.137	+	320.000i	−0.126128	+	0.356745i
$898$		0			0
$899$		−	373.352i		−	0.415297i
$900$		0			0
$901$	−1408.00			−1.56271
$902$		0			0
$903$	1391.59	+	492.000i	1.54107	+	0.544850i
$904$		0			0
$905$		0			0
$906$		0			0
$907$			1042.00i			1.14884i	0.818560	+	0.574421i	$0.194773\pi$
$907$			1042.00i			1.14884i	−0.818560	+	0.574421i	$0.805227\pi$
$908$		0			0
$909$	−288.000	+	356.382i	−0.316832	+	0.392059i
$910$		0			0
$911$			1606.55i			1.76350i	0.471719	+	0.881749i	$0.343634\pi$
$911$			1606.55i			1.76350i	−0.471719	+	0.881749i	$0.656366\pi$
$912$		0			0
$913$			416.000i			0.455641i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	33.9411			0.0370132
$918$		0			0
$919$	−614.000			−0.668118			−0.334059	−	0.942552i	$-0.608418\pi$
$919$	−614.000			−0.668118			−0.334059	+	0.942552i	$0.608418\pi$
$920$		0			0
$921$	−30.0000	+	84.8528i	−0.0325733	+	0.0921312i
$922$		0			0
$923$	−1244.51			−1.34833
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−758.018	+	938.000i	−0.817711	+	1.01187i
$928$		0			0
$929$		−	45.2548i		−	0.0487135i	−0.999703	−	0.0243567i	$-0.992246\pi$
$929$		−	45.2548i		−	0.0487135i	0.999703	−	0.0243567i	$-0.00775376\pi$
$930$		0			0
$931$	26.0000			0.0279270
$932$		0			0
$933$	−576.999	+	1632.00i	−0.618434	+	1.74920i
$934$		0			0
$935$		0			0
$936$		0			0
$937$			462.000i			0.493063i	0.969135	+	0.246531i	$0.0792909\pi$
$937$			462.000i			0.493063i	−0.969135	+	0.246531i	$0.920709\pi$
$938$		0			0
$939$	210.000	−	593.970i	0.223642	−	0.632556i
$940$		0			0
$941$			356.382i			0.378727i	0.981907	+	0.189363i	$0.0606424\pi$
$941$			356.382i			0.378727i	−0.981907	+	0.189363i	$0.939358\pi$
$942$		0			0
$943$		−	384.000i		−	0.407211i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−695.793			−0.734734			−0.367367	−	0.930076i	$-0.619741\pi$
$947$	−695.793			−0.734734			−0.367367	+	0.930076i	$0.619741\pi$
$948$		0			0
$949$	−820.000			−0.864067
$950$		0			0
$951$	−432.000	−	152.735i	−0.454259	−	0.160605i
$952$		0			0
$953$	1527.35			1.60268			0.801338	−	0.598212i	$-0.204122\pi$
$953$	1527.35			1.60268			0.801338	+	0.598212i	$0.204122\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	271.529	+	96.0000i	0.283729	+	0.100313i
$958$		0			0
$959$			610.940i			0.637060i
$960$		0			0
$961$	−477.000			−0.496358
$962$		0			0
$963$	−356.382	−	288.000i	−0.370075	−	0.299065i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	70.0000i		−	0.0723888i	−0.999345	−	0.0361944i	$-0.988476\pi$
$967$		−	70.0000i		−	0.0723888i	0.999345	−	0.0361944i	$-0.0115236\pi$
$968$		0			0
$969$	128.000	+	45.2548i	0.132095	+	0.0467026i
$970$		0			0
$971$			627.911i			0.646664i	0.946286	+	0.323332i	$0.104803\pi$
$971$			627.911i			0.646664i	−0.946286	+	0.323332i	$0.895197\pi$
$972$		0			0
$973$			468.000i			0.480987i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−1018.23			−1.04220			−0.521102	−	0.853494i	$-0.674479\pi$
$977$	−1018.23			−1.04220			−0.521102	+	0.853494i	$0.674479\pi$
$978$		0			0
$979$	192.000			0.196118
$980$		0			0
$981$	−70.0000	−	56.5685i	−0.0713558	−	0.0576642i
$982$		0			0
$983$	−644.881			−0.656034			−0.328017	−	0.944672i	$-0.606380\pi$
$983$	−644.881			−0.656034			−0.328017	+	0.944672i	$0.606380\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	407.294	−	1152.00i	0.412658	−	1.16717i
$988$		0			0
$989$			927.724i			0.938043i
$990$		0			0
$991$	854.000			0.861756			0.430878	−	0.902410i	$-0.358204\pi$
$991$	854.000			0.861756			0.430878	+	0.902410i	$0.358204\pi$
$992$		0			0
$993$	1227.54	+	434.000i	1.23619	+	0.437059i
$994$		0			0
$995$		0			0
$996$		0			0
$997$			518.000i			0.519559i	0.965668	+	0.259779i	$0.0836498\pi$
$997$			518.000i			0.519559i	−0.965668	+	0.259779i	$0.916350\pi$
$998$		0			0
$999$	138.000	−	84.8528i	0.138138	−	0.0849378i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1200.3.c.i.449.1		4
3.2	odd	2	inner	1200.3.c.i.449.3		4
4.3	odd	2		600.3.c.a.449.4		4
5.2	odd	4		48.3.e.b.17.2		2
5.3	odd	4		1200.3.l.n.401.1		2
5.4	even	2	inner	1200.3.c.i.449.4		4
12.11	even	2		600.3.c.a.449.2		4
15.2	even	4		48.3.e.b.17.1		2
15.8	even	4		1200.3.l.n.401.2		2
15.14	odd	2	inner	1200.3.c.i.449.2		4
20.3	even	4		600.3.l.b.401.2		2
20.7	even	4		24.3.e.a.17.1	✓	2
20.19	odd	2		600.3.c.a.449.1		4
40.27	even	4		192.3.e.c.65.2		2
40.37	odd	4		192.3.e.d.65.1		2
45.2	even	12		1296.3.q.e.1025.2		4
45.7	odd	12		1296.3.q.e.1025.1		4
45.22	odd	12		1296.3.q.e.593.2		4
45.32	even	12		1296.3.q.e.593.1		4
60.23	odd	4		600.3.l.b.401.1		2
60.47	odd	4		24.3.e.a.17.2	yes	2
60.59	even	2		600.3.c.a.449.3		4
80.27	even	4		768.3.h.d.641.1		4
80.37	odd	4		768.3.h.c.641.4		4
80.67	even	4		768.3.h.d.641.4		4
80.77	odd	4		768.3.h.c.641.1		4
120.77	even	4		192.3.e.d.65.2		2
120.107	odd	4		192.3.e.c.65.1		2
140.27	odd	4		1176.3.d.a.785.2		2
180.7	even	12		648.3.m.d.377.1		4
180.47	odd	12		648.3.m.d.377.2		4
180.67	even	12		648.3.m.d.593.2		4
180.167	odd	12		648.3.m.d.593.1		4
240.77	even	4		768.3.h.c.641.3		4
240.107	odd	4		768.3.h.d.641.3		4
240.197	even	4		768.3.h.c.641.2		4
240.227	odd	4		768.3.h.d.641.2		4
420.167	even	4		1176.3.d.a.785.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.3.e.a.17.1	✓	2	20.7	even	4
24.3.e.a.17.2	yes	2	60.47	odd	4
48.3.e.b.17.1		2	15.2	even	4
48.3.e.b.17.2		2	5.2	odd	4
192.3.e.c.65.1		2	120.107	odd	4
192.3.e.c.65.2		2	40.27	even	4
192.3.e.d.65.1		2	40.37	odd	4
192.3.e.d.65.2		2	120.77	even	4
600.3.c.a.449.1		4	20.19	odd	2
600.3.c.a.449.2		4	12.11	even	2
600.3.c.a.449.3		4	60.59	even	2
600.3.c.a.449.4		4	4.3	odd	2
600.3.l.b.401.1		2	60.23	odd	4
600.3.l.b.401.2		2	20.3	even	4
648.3.m.d.377.1		4	180.7	even	12
648.3.m.d.377.2		4	180.47	odd	12
648.3.m.d.593.1		4	180.167	odd	12
648.3.m.d.593.2		4	180.67	even	12
768.3.h.c.641.1		4	80.77	odd	4
768.3.h.c.641.2		4	240.197	even	4
768.3.h.c.641.3		4	240.77	even	4
768.3.h.c.641.4		4	80.37	odd	4
768.3.h.d.641.1		4	80.27	even	4
768.3.h.d.641.2		4	240.227	odd	4
768.3.h.d.641.3		4	240.107	odd	4
768.3.h.d.641.4		4	80.67	even	4
1176.3.d.a.785.1		2	420.167	even	4
1176.3.d.a.785.2		2	140.27	odd	4
1200.3.c.i.449.1		4	1.1	even	1	trivial
1200.3.c.i.449.2		4	15.14	odd	2	inner
1200.3.c.i.449.3		4	3.2	odd	2	inner
1200.3.c.i.449.4		4	5.4	even	2	inner
1200.3.l.n.401.1		2	5.3	odd	4
1200.3.l.n.401.2		2	15.8	even	4
1296.3.q.e.593.1		4	45.32	even	12
1296.3.q.e.593.2		4	45.22	odd	12
1296.3.q.e.1025.1		4	45.7	odd	12
1296.3.q.e.1025.2		4	45.2	even	12

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.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.82843 - 1.00000i) q^{3} -6.00000i q^{7} +(7.00000 + 5.65685i) q^{9} -5.65685i q^{11} +10.0000i q^{13} -22.6274 q^{17} +2.00000 q^{19} +(-6.00000 + 16.9706i) q^{21} -11.3137 q^{23} +(-14.1421 - 23.0000i) q^{27} -16.9706i q^{29} +22.0000 q^{31} +(-5.65685 + 16.0000i) q^{33} +6.00000i q^{37} +(10.0000 - 28.2843i) q^{39} +33.9411i q^{41} -82.0000i q^{43} -67.8823 q^{47} +13.0000 q^{49} +(64.0000 + 22.6274i) q^{51} +62.2254 q^{53} +(-5.65685 - 2.00000i) q^{57} +73.5391i q^{59} -86.0000 q^{61} +(33.9411 - 42.0000i) q^{63} +2.00000i q^{67} +(32.0000 + 11.3137i) q^{69} +124.451i q^{71} +82.0000i q^{73} -33.9411 q^{77} +10.0000 q^{79} +(17.0000 + 79.1960i) q^{81} -73.5391 q^{83} +(-16.9706 + 48.0000i) q^{87} +33.9411i q^{89} +60.0000 q^{91} +(-62.2254 - 22.0000i) q^{93} +94.0000i q^{97} +(32.0000 - 39.5980i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 28 q^{9} + 8 q^{19} - 24 q^{21} + 88 q^{31} + 40 q^{39} + 52 q^{49} + 256 q^{51} - 344 q^{61} + 128 q^{69} + 40 q^{79} + 68 q^{81} + 240 q^{91} + 128 q^{99}+O(q^{100}) \) 4 * q + 28 * q^9 + 8 * q^19 - 24 * q^21 + 88 * q^31 + 40 * q^39 + 52 * q^49 + 256 * q^51 - 344 * q^61 + 128 * q^69 + 40 * q^79 + 68 * q^81 + 240 * q^91 + 128 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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