LMFDB - Embedded newform 1200.3.bg.m.193.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1200,3,Mod(193,1200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1200, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 0, 0, 3]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1200.193");

S:= CuspForms(chi, 3);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$32.6976317232$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 600)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			193.2
Root			$1.22474 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	1200.193
Dual form			1200.3.bg.m.1057.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.22474 + 1.22474i) q^{3} +(1.55051 - 1.55051i) q^{7} +3.00000i q^{9} +O(q^{10})$	$q+(1.22474 + 1.22474i) q^{3} +(1.55051 - 1.55051i) q^{7} +3.00000i q^{9} -7.79796 q^{11} +(-2.44949 - 2.44949i) q^{13} +(-8.89898 + 8.89898i) q^{17} -5.59592i q^{19} +3.79796 q^{21} +(2.20204 + 2.20204i) q^{23} +(-3.67423 + 3.67423i) q^{27} +35.5959i q^{29} -53.1918 q^{31} +(-9.55051 - 9.55051i) q^{33} +(-25.1464 + 25.1464i) q^{37} -6.00000i q^{39} +56.7878 q^{41} +(-41.7980 - 41.7980i) q^{43} +(-44.4949 + 44.4949i) q^{47} +44.1918i q^{49} -21.7980 q^{51} +(-36.0908 - 36.0908i) q^{53} +(6.85357 - 6.85357i) q^{57} -10.9898i q^{59} +48.3837 q^{61} +(4.65153 + 4.65153i) q^{63} +(-29.8888 + 29.8888i) q^{67} +5.39388i q^{69} -50.7878 q^{71} +(49.3031 + 49.3031i) q^{73} +(-12.0908 + 12.0908i) q^{77} -66.0000i q^{79} -9.00000 q^{81} +(-4.09082 - 4.09082i) q^{83} +(-43.5959 + 43.5959i) q^{87} +29.5959i q^{89} -7.59592 q^{91} +(-65.1464 - 65.1464i) q^{93} +(-133.081 + 133.081i) q^{97} -23.3939i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 16 q^{7}+O(q^{10}) $ 4 * q + 16 * q^7	$ 4 q + 16 q^{7} + 8 q^{11} - 16 q^{17} - 24 q^{21} + 48 q^{23} - 56 q^{31} - 48 q^{33} - 32 q^{37} - 8 q^{41} - 128 q^{43} - 80 q^{47} - 48 q^{51} + 32 q^{53} + 96 q^{57} - 120 q^{61} + 48 q^{63} + 96 q^{67} + 32 q^{71} + 256 q^{73} + 128 q^{77} - 36 q^{81} + 160 q^{83} - 96 q^{87} + 48 q^{91} - 192 q^{93} - 160 q^{97}+O(q^{100}) $ 4 * q + 16 * q^7 + 8 * q^11 - 16 * q^17 - 24 * q^21 + 48 * q^23 - 56 * q^31 - 48 * q^33 - 32 * q^37 - 8 * q^41 - 128 * q^43 - 80 * q^47 - 48 * q^51 + 32 * q^53 + 96 * q^57 - 120 * q^61 + 48 * q^63 + 96 * q^67 + 32 * q^71 + 256 * q^73 + 128 * q^77 - 36 * q^81 + 160 * q^83 - 96 * q^87 + 48 * q^91 - 192 * q^93 - 160 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$401$	$577$	$751$	$901$
$\chi(n)$	$1$	$e\left(\frac{3}{4}\right)$	$1$	$1$

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
$3$	1.22474	+	1.22474i	0.408248	+	0.408248i
$3$	1.22474	+	1.22474i	0.408248	+	0.408248i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	1.55051	−	1.55051i	0.221501	−	0.221501i	−0.587629	−	0.809130i	$-0.699939\pi$
$7$	1.55051	−	1.55051i	0.221501	−	0.221501i	0.809130	+	0.587629i	$0.199939\pi$
$8$		0			0
$9$			3.00000i			0.333333i
$10$		0			0
$11$	−7.79796			−0.708905			−0.354453	−	0.935074i	$-0.615333\pi$
$11$	−7.79796			−0.708905			−0.354453	+	0.935074i	$0.615333\pi$
$12$		0			0
$13$	−2.44949	−	2.44949i	−0.188422	−	0.188422i	0.606591	−	0.795014i	$-0.292536\pi$
$13$	−2.44949	−	2.44949i	−0.188422	−	0.188422i	−0.795014	+	0.606591i	$0.792536\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−8.89898	+	8.89898i	−0.523469	+	0.523469i	−0.918617	−	0.395148i	$-0.870693\pi$
$17$	−8.89898	+	8.89898i	−0.523469	+	0.523469i	0.395148	+	0.918617i	$0.370693\pi$
$18$		0			0
$19$		−	5.59592i		−	0.294522i	−0.989098	−	0.147261i	$-0.952954\pi$
$19$		−	5.59592i		−	0.294522i	0.989098	−	0.147261i	$-0.0470457\pi$
$20$		0			0
$21$	3.79796			0.180855
$22$		0			0
$23$	2.20204	+	2.20204i	0.0957409	+	0.0957409i	0.753355	−	0.657614i	$-0.228434\pi$
$23$	2.20204	+	2.20204i	0.0957409	+	0.0957409i	−0.657614	+	0.753355i	$0.728434\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−3.67423	+	3.67423i	−0.136083	+	0.136083i
$28$		0			0
$29$			35.5959i			1.22745i	0.789522	+	0.613723i	$0.210329\pi$
$29$			35.5959i			1.22745i	−0.789522	+	0.613723i	$0.789671\pi$
$30$		0			0
$31$	−53.1918			−1.71587			−0.857933	−	0.513762i	$-0.828251\pi$
$31$	−53.1918			−1.71587			−0.857933	+	0.513762i	$0.828251\pi$
$32$		0			0
$33$	−9.55051	−	9.55051i	−0.289409	−	0.289409i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−25.1464	+	25.1464i	−0.679633	+	0.679633i	−0.959917	−	0.280284i	$-0.909571\pi$
$37$	−25.1464	+	25.1464i	−0.679633	+	0.679633i	0.280284	+	0.959917i	$0.409571\pi$
$38$		0			0
$39$		−	6.00000i		−	0.153846i
$40$		0			0
$41$	56.7878			1.38507			0.692534	−	0.721386i	$-0.256494\pi$
$41$	56.7878			1.38507			0.692534	+	0.721386i	$0.256494\pi$
$42$		0			0
$43$	−41.7980	−	41.7980i	−0.972046	−	0.972046i	0.0275742	−	0.999620i	$-0.491222\pi$
$43$	−41.7980	−	41.7980i	−0.972046	−	0.972046i	−0.999620	+	0.0275742i	$0.991222\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−44.4949	+	44.4949i	−0.946700	+	0.946700i	−0.998650	−	0.0519498i	$-0.983456\pi$
$47$	−44.4949	+	44.4949i	−0.946700	+	0.946700i	0.0519498	+	0.998650i	$0.483456\pi$
$48$		0			0
$49$			44.1918i			0.901874i
$50$		0			0
$51$	−21.7980			−0.427411
$52$		0			0
$53$	−36.0908	−	36.0908i	−0.680959	−	0.680959i	0.279257	−	0.960216i	$-0.409912\pi$
$53$	−36.0908	−	36.0908i	−0.680959	−	0.680959i	−0.960216	+	0.279257i	$0.909912\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	6.85357	−	6.85357i	0.120238	−	0.120238i
$58$		0			0
$59$		−	10.9898i		−	0.186268i	−0.995654	−	0.0931339i	$-0.970312\pi$
$59$		−	10.9898i		−	0.186268i	0.995654	−	0.0931339i	$-0.0296885\pi$
$60$		0			0
$61$	48.3837			0.793175			0.396587	−	0.917997i	$-0.370194\pi$
$61$	48.3837			0.793175			0.396587	+	0.917997i	$0.370194\pi$
$62$		0			0
$63$	4.65153	+	4.65153i	0.0738338	+	0.0738338i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−29.8888	+	29.8888i	−0.446101	+	0.446101i	−0.894056	−	0.447955i	$-0.852152\pi$
$67$	−29.8888	+	29.8888i	−0.446101	+	0.446101i	0.447955	+	0.894056i	$0.352152\pi$
$68$		0			0
$69$			5.39388i			0.0781721i
$70$		0			0
$71$	−50.7878			−0.715320			−0.357660	−	0.933852i	$-0.616425\pi$
$71$	−50.7878			−0.715320			−0.357660	+	0.933852i	$0.616425\pi$
$72$		0			0
$73$	49.3031	+	49.3031i	0.675384	+	0.675384i	0.958952	−	0.283568i	$-0.0915181\pi$
$73$	49.3031	+	49.3031i	0.675384	+	0.675384i	−0.283568	+	0.958952i	$0.591518\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−12.0908	+	12.0908i	−0.157024	+	0.157024i
$78$		0			0
$79$		−	66.0000i		−	0.835443i	−0.908575	−	0.417722i	$-0.862829\pi$
$79$		−	66.0000i		−	0.835443i	0.908575	−	0.417722i	$-0.137171\pi$
$80$		0			0
$81$	−9.00000			−0.111111
$82$		0			0
$83$	−4.09082	−	4.09082i	−0.0492869	−	0.0492869i	0.682034	−	0.731321i	$-0.261096\pi$
$83$	−4.09082	−	4.09082i	−0.0492869	−	0.0492869i	−0.731321	+	0.682034i	$0.761096\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−43.5959	+	43.5959i	−0.501103	+	0.501103i
$88$		0			0
$89$			29.5959i			0.332538i	0.986080	+	0.166269i	$0.0531721\pi$
$89$			29.5959i			0.332538i	−0.986080	+	0.166269i	$0.946828\pi$
$90$		0			0
$91$	−7.59592			−0.0834716
$92$		0			0
$93$	−65.1464	−	65.1464i	−0.700499	−	0.700499i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−133.081	+	133.081i	−1.37197	+	1.37197i	−0.514437	+	0.857528i	$0.671999\pi$
$97$	−133.081	+	133.081i	−1.37197	+	1.37197i	−0.857528	+	0.514437i	$0.828001\pi$
$98$		0			0
$99$		−	23.3939i		−	0.236302i
$100$		0			0
$101$	76.8082			0.760477			0.380238	−	0.924888i	$-0.375842\pi$
$101$	76.8082			0.760477			0.380238	+	0.924888i	$0.375842\pi$
$102$		0			0
$103$	−125.641	−	125.641i	−1.21982	−	1.21982i	−0.967695	−	0.252124i	$-0.918871\pi$
$103$	−125.641	−	125.641i	−1.21982	−	1.21982i	−0.252124	−	0.967695i	$-0.581129\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−77.8888	+	77.8888i	−0.727932	+	0.727932i	−0.970208	−	0.242275i	$-0.922106\pi$
$107$	−77.8888	+	77.8888i	−0.727932	+	0.727932i	0.242275	+	0.970208i	$0.422106\pi$
$108$		0			0
$109$		−	61.1918i		−	0.561393i	−0.959797	−	0.280696i	$-0.909435\pi$
$109$		−	61.1918i		−	0.561393i	0.959797	−	0.280696i	$-0.0905654\pi$
$110$		0			0
$111$	−61.5959			−0.554918
$112$		0			0
$113$	−65.7071	−	65.7071i	−0.581479	−	0.581479i	0.353830	−	0.935310i	$-0.384879\pi$
$113$	−65.7071	−	65.7071i	−0.581479	−	0.581479i	−0.935310	+	0.353830i	$0.884879\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	7.34847	−	7.34847i	0.0628074	−	0.0628074i
$118$		0			0
$119$			27.5959i			0.231898i
$120$		0			0
$121$	−60.1918			−0.497453
$122$		0			0
$123$	69.5505	+	69.5505i	0.565451	+	0.565451i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	30.7628	−	30.7628i	0.242226	−	0.242226i	−0.575544	−	0.817771i	$-0.695210\pi$
$127$	30.7628	−	30.7628i	0.242226	−	0.242226i	0.817771	+	0.575544i	$0.195210\pi$
$128$		0			0
$129$		−	102.384i		−	0.793672i
$130$		0			0
$131$	255.394			1.94957			0.974786	−	0.223143i	$-0.0716316\pi$
$131$	255.394			1.94957			0.974786	+	0.223143i	$0.0716316\pi$
$132$		0			0
$133$	−8.67653	−	8.67653i	−0.0652371	−	0.0652371i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−105.081	+	105.081i	−0.767012	+	0.767012i	−0.977579	−	0.210568i	$-0.932469\pi$
$137$	−105.081	+	105.081i	−0.767012	+	0.767012i	0.210568	+	0.977579i	$0.432469\pi$
$138$		0			0
$139$			41.1918i			0.296344i	0.988962	+	0.148172i	$0.0473389\pi$
$139$			41.1918i			0.296344i	−0.988962	+	0.148172i	$0.952661\pi$
$140$		0			0
$141$	−108.990			−0.772977
$142$		0			0
$143$	19.1010	+	19.1010i	0.133574	+	0.133574i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−54.1237	+	54.1237i	−0.368189	+	0.368189i
$148$		0			0
$149$		−	30.7878i		−	0.206629i	−0.994649	−	0.103315i	$-0.967055\pi$
$149$		−	30.7878i		−	0.206629i	0.994649	−	0.103315i	$-0.0329448\pi$
$150$		0			0
$151$	46.7673			0.309718			0.154859	−	0.987937i	$-0.450508\pi$
$151$	46.7673			0.309718			0.154859	+	0.987937i	$0.450508\pi$
$152$		0			0
$153$	−26.6969	−	26.6969i	−0.174490	−	0.174490i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−100.561	+	100.561i	−0.640514	+	0.640514i	−0.950682	−	0.310168i	$-0.899615\pi$
$157$	−100.561	+	100.561i	−0.640514	+	0.640514i	0.310168	+	0.950682i	$0.399615\pi$
$158$		0			0
$159$		−	88.4041i		−	0.556001i
$160$		0			0
$161$	6.82857			0.0424135
$162$		0			0
$163$	−146.788	−	146.788i	−0.900538	−	0.900538i	0.0949442	−	0.995483i	$-0.469733\pi$
$163$	−146.788	−	146.788i	−0.900538	−	0.900538i	−0.995483	+	0.0949442i	$0.969733\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−60.1816	+	60.1816i	−0.360369	+	0.360369i	−0.863949	−	0.503580i	$-0.832016\pi$
$167$	−60.1816	+	60.1816i	−0.360369	+	0.360369i	0.503580	+	0.863949i	$0.332016\pi$
$168$		0			0
$169$		−	157.000i		−	0.928994i
$170$		0			0
$171$	16.7878			0.0981740
$172$		0			0
$173$	−204.272	−	204.272i	−1.18077	−	1.18077i	−0.979546	−	0.201219i	$-0.935510\pi$
$173$	−204.272	−	204.272i	−1.18077	−	1.18077i	−0.201219	−	0.979546i	$-0.564490\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	13.4597	−	13.4597i	0.0760435	−	0.0760435i
$178$		0			0
$179$			196.161i			1.09587i	0.836520	+	0.547936i	$0.184586\pi$
$179$			196.161i			1.09587i	−0.836520	+	0.547936i	$0.815414\pi$
$180$		0			0
$181$	−229.959			−1.27049			−0.635246	−	0.772310i	$-0.719101\pi$
$181$	−229.959			−1.27049			−0.635246	+	0.772310i	$0.719101\pi$
$182$		0			0
$183$	59.2577	+	59.2577i	0.323812	+	0.323812i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	69.3939	−	69.3939i	0.371090	−	0.371090i
$188$		0			0
$189$			11.3939i			0.0602851i
$190$		0			0
$191$	258.747			1.35470			0.677348	−	0.735663i	$-0.263129\pi$
$191$	258.747			1.35470			0.677348	+	0.735663i	$0.263129\pi$
$192$		0			0
$193$	263.373	+	263.373i	1.36463	+	1.36463i	0.867913	+	0.496717i	$0.165461\pi$
$193$	263.373	+	263.373i	1.36463	+	1.36463i	0.496717	+	0.867913i	$0.334539\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	201.798	−	201.798i	1.02436	−	1.02436i	0.0246592	−	0.999696i	$-0.492150\pi$
$197$	201.798	−	201.798i	1.02436	−	1.02436i	0.999696	−	0.0246592i	$-0.00785006\pi$
$198$		0			0
$199$		−	2.80816i		−	0.0141114i	−0.999975	−	0.00705569i	$-0.997754\pi$
$199$		−	2.80816i		−	0.0141114i	0.999975	−	0.00705569i	$-0.00224591\pi$
$200$		0			0
$201$	−73.2122			−0.364240
$202$		0			0
$203$	55.1918	+	55.1918i	0.271881	+	0.271881i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−6.60612	+	6.60612i	−0.0319136	+	0.0319136i
$208$		0			0
$209$			43.6367i			0.208788i
$210$		0			0
$211$	35.5755			0.168604			0.0843022	−	0.996440i	$-0.473134\pi$
$211$	35.5755			0.168604			0.0843022	+	0.996440i	$0.473134\pi$
$212$		0			0
$213$	−62.2020	−	62.2020i	−0.292028	−	0.292028i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−82.4745	+	82.4745i	−0.380067	+	0.380067i
$218$		0			0
$219$			120.767i			0.551449i
$220$		0			0
$221$	43.5959			0.197267
$222$		0			0
$223$	−81.0556	−	81.0556i	−0.363478	−	0.363478i	0.501614	−	0.865092i	$-0.332740\pi$
$223$	−81.0556	−	81.0556i	−0.363478	−	0.363478i	−0.865092	+	0.501614i	$0.832740\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−51.1010	+	51.1010i	−0.225115	+	0.225115i	−0.810648	−	0.585534i	$-0.800885\pi$
$227$	−51.1010	+	51.1010i	−0.225115	+	0.225115i	0.585534	+	0.810648i	$0.300885\pi$
$228$		0			0
$229$		−	174.767i		−	0.763176i	−0.924332	−	0.381588i	$-0.875377\pi$
$229$		−	174.767i		−	0.763176i	0.924332	−	0.381588i	$-0.124623\pi$
$230$		0			0
$231$	−29.6163			−0.128209
$232$		0			0
$233$	166.656	+	166.656i	0.715262	+	0.715262i	0.967631	−	0.252369i	$-0.0812096\pi$
$233$	166.656	+	166.656i	0.715262	+	0.715262i	−0.252369	+	0.967631i	$0.581210\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	80.8332	−	80.8332i	0.341068	−	0.341068i
$238$		0			0
$239$			305.212i			1.27704i	0.769606	+	0.638519i	$0.220453\pi$
$239$			305.212i			1.27704i	−0.769606	+	0.638519i	$0.779547\pi$
$240$		0			0
$241$	−302.767			−1.25630			−0.628148	−	0.778094i	$-0.716187\pi$
$241$	−302.767			−1.25630			−0.628148	+	0.778094i	$0.716187\pi$
$242$		0			0
$243$	−11.0227	−	11.0227i	−0.0453609	−	0.0453609i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−13.7071	+	13.7071i	−0.0554945	+	0.0554945i
$248$		0			0
$249$		−	10.0204i		−	0.0402426i
$250$		0			0
$251$	354.182			1.41108			0.705541	−	0.708669i	$-0.250704\pi$
$251$	354.182			1.41108			0.705541	+	0.708669i	$0.250704\pi$
$252$		0			0
$253$	−17.1714	−	17.1714i	−0.0678712	−	0.0678712i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−73.7071	+	73.7071i	−0.286798	+	0.286798i	−0.835813	−	0.549015i	$-0.815003\pi$
$257$	−73.7071	+	73.7071i	−0.286798	+	0.286798i	0.549015	+	0.835813i	$0.315003\pi$
$258$		0			0
$259$			77.9796i			0.301079i
$260$		0			0
$261$	−106.788			−0.409148
$262$		0			0
$263$	36.0000	+	36.0000i	0.136882	+	0.136882i	0.772228	−	0.635346i	$-0.219142\pi$
$263$	36.0000	+	36.0000i	0.136882	+	0.136882i	−0.635346	+	0.772228i	$0.719142\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−36.2474	+	36.2474i	−0.135758	+	0.135758i
$268$		0			0
$269$			313.171i			1.16421i	0.813115	+	0.582103i	$0.197770\pi$
$269$			313.171i			1.16421i	−0.813115	+	0.582103i	$0.802230\pi$
$270$		0			0
$271$	−223.576			−0.825002			−0.412501	−	0.910957i	$-0.635345\pi$
$271$	−223.576			−0.825002			−0.412501	+	0.910957i	$0.635345\pi$
$272$		0			0
$273$	−9.30306	−	9.30306i	−0.0340771	−	0.0340771i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	202.499	−	202.499i	0.731045	−	0.731045i	−0.239782	−	0.970827i	$-0.577076\pi$
$277$	202.499	−	202.499i	0.731045	−	0.731045i	0.970827	+	0.239782i	$0.0770759\pi$
$278$		0			0
$279$		−	159.576i		−	0.571955i
$280$		0			0
$281$	275.616			0.980841			0.490421	−	0.871486i	$-0.336843\pi$
$281$	275.616			0.980841			0.490421	+	0.871486i	$0.336843\pi$
$282$		0			0
$283$	79.8684	+	79.8684i	0.282220	+	0.282220i	0.833994	−	0.551774i	$-0.186049\pi$
$283$	79.8684	+	79.8684i	0.282220	+	0.282220i	−0.551774	+	0.833994i	$0.686049\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	88.0500	−	88.0500i	0.306794	−	0.306794i
$288$		0			0
$289$			130.616i			0.451960i
$290$		0			0
$291$	−325.980			−1.12020
$292$		0			0
$293$	−116.767	−	116.767i	−0.398523	−	0.398523i	0.479189	−	0.877712i	$-0.340931\pi$
$293$	−116.767	−	116.767i	−0.398523	−	0.398523i	−0.877712	+	0.479189i	$0.840931\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	28.6515	−	28.6515i	0.0964698	−	0.0964698i
$298$		0			0
$299$		−	10.7878i		−	0.0360794i
$300$		0			0
$301$	−129.616			−0.430619
$302$		0			0
$303$	94.0704	+	94.0704i	0.310463	+	0.310463i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	210.293	−	210.293i	0.684993	−	0.684993i	−0.276128	−	0.961121i	$-0.589051\pi$
$307$	210.293	−	210.293i	0.684993	−	0.684993i	0.961121	+	0.276128i	$0.0890513\pi$
$308$		0			0
$309$		−	307.757i		−	0.995978i
$310$		0			0
$311$	−19.5551			−0.0628781			−0.0314391	−	0.999506i	$-0.510009\pi$
$311$	−19.5551			−0.0628781			−0.0314391	+	0.999506i	$0.510009\pi$
$312$		0			0
$313$	221.939	+	221.939i	0.709070	+	0.709070i	0.966340	−	0.257270i	$-0.0828230\pi$
$313$	221.939	+	221.939i	0.709070	+	0.709070i	−0.257270	+	0.966340i	$0.582823\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	210.343	−	210.343i	0.663542	−	0.663542i	−0.292671	−	0.956213i	$-0.594544\pi$
$317$	210.343	−	210.343i	0.663542	−	0.663542i	0.956213	+	0.292671i	$0.0945441\pi$
$318$		0			0
$319$		−	277.576i		−	0.870143i
$320$		0			0
$321$	−190.788			−0.594354
$322$		0			0
$323$	49.7980	+	49.7980i	0.154173	+	0.154173i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	74.9444	−	74.9444i	0.229188	−	0.229188i
$328$		0			0
$329$			137.980i			0.419391i
$330$		0			0
$331$	−333.555			−1.00772			−0.503860	−	0.863786i	$-0.668087\pi$
$331$	−333.555			−1.00772			−0.503860	+	0.863786i	$0.668087\pi$
$332$		0			0
$333$	−75.4393	−	75.4393i	−0.226544	−	0.226544i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−23.5051	+	23.5051i	−0.0697481	+	0.0697481i	−0.741120	−	0.671372i	$-0.765705\pi$
$337$	−23.5051	+	23.5051i	−0.0697481	+	0.0697481i	0.671372	+	0.741120i	$0.265705\pi$
$338$		0			0
$339$		−	160.949i		−	0.474776i
$340$		0			0
$341$	414.788			1.21639
$342$		0			0
$343$	144.495	+	144.495i	0.421268	+	0.421268i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	360.858	−	360.858i	1.03994	−	1.03994i	0.0407684	−	0.999169i	$-0.487019\pi$
$347$	360.858	−	360.858i	1.03994	−	1.03994i	0.999169	−	0.0407684i	$-0.0129806\pi$
$348$		0			0
$349$			216.343i			0.619894i	0.950754	+	0.309947i	$0.100311\pi$
$349$			216.343i			0.619894i	−0.950754	+	0.309947i	$0.899689\pi$
$350$		0			0
$351$	18.0000			0.0512821
$352$		0			0
$353$	396.232	+	396.232i	1.12247	+	1.12247i	0.991369	+	0.131100i	$0.0418509\pi$
$353$	396.232	+	396.232i	1.12247	+	1.12247i	0.131100	+	0.991369i	$0.458149\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−33.7980	+	33.7980i	−0.0946722	+	0.0946722i
$358$		0			0
$359$			332.000i			0.924791i	0.886674	+	0.462396i	$0.153010\pi$
$359$			332.000i			0.924791i	−0.886674	+	0.462396i	$0.846990\pi$
$360$		0			0
$361$	329.686			0.913257
$362$		0			0
$363$	−73.7196	−	73.7196i	−0.203084	−	0.203084i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	186.409	−	186.409i	0.507926	−	0.507926i	−0.405964	−	0.913889i	$-0.633064\pi$
$367$	186.409	−	186.409i	0.507926	−	0.507926i	0.913889	+	0.405964i	$0.133064\pi$
$368$		0			0
$369$			170.363i			0.461689i
$370$		0			0
$371$	−111.918			−0.301667
$372$		0			0
$373$	−433.328	−	433.328i	−1.16174	−	1.16174i	−0.984095	−	0.177642i	$-0.943153\pi$
$373$	−433.328	−	433.328i	−1.16174	−	1.16174i	−0.177642	−	0.984095i	$-0.556847\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	87.1918	−	87.1918i	0.231278	−	0.231278i
$378$		0			0
$379$		−	90.7673i		−	0.239492i	−0.992805	−	0.119746i	$-0.961792\pi$
$379$		−	90.7673i		−	0.239492i	0.992805	−	0.119746i	$-0.0382080\pi$
$380$		0			0
$381$	75.3531			0.197777
$382$		0			0
$383$	287.283	+	287.283i	0.750085	+	0.750085i	0.974495	−	0.224410i	$-0.0720454\pi$
$383$	287.283	+	287.283i	0.750085	+	0.750085i	−0.224410	+	0.974495i	$0.572045\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	125.394	−	125.394i	0.324015	−	0.324015i
$388$		0			0
$389$			159.192i			0.409234i	0.978842	+	0.204617i	$0.0655948\pi$
$389$			159.192i			0.409234i	−0.978842	+	0.204617i	$0.934405\pi$
$390$		0			0
$391$	−39.1918			−0.100235
$392$		0			0
$393$	312.792	+	312.792i	0.795909	+	0.795909i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	458.005	−	458.005i	1.15366	−	1.15366i	0.167852	−	0.985812i	$-0.446317\pi$
$397$	458.005	−	458.005i	1.15366	−	1.15366i	0.985812	−	0.167852i	$-0.0536830\pi$
$398$		0			0
$399$		−	21.2531i		−	0.0532658i
$400$		0			0
$401$	399.131			0.995338			0.497669	−	0.867367i	$-0.334189\pi$
$401$	399.131			0.995338			0.497669	+	0.867367i	$0.334189\pi$
$402$		0			0
$403$	130.293	+	130.293i	0.323307	+	0.323307i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	196.091	−	196.091i	0.481796	−	0.481796i
$408$		0			0
$409$		−	197.233i		−	0.482231i	−0.970496	−	0.241116i	$-0.922487\pi$
$409$		−	197.233i		−	0.482231i	0.970496	−	0.241116i	$-0.0775133\pi$
$410$		0			0
$411$	−257.394			−0.626262
$412$		0			0
$413$	−17.0398	−	17.0398i	−0.0412586	−	0.0412586i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−50.4495	+	50.4495i	−0.120982	+	0.120982i
$418$		0			0
$419$		−	724.120i		−	1.72821i	−0.503311	−	0.864105i	$-0.667885\pi$
$419$		−	724.120i		−	1.72821i	0.503311	−	0.864105i	$-0.332115\pi$
$420$		0			0
$421$	416.343			0.988938			0.494469	−	0.869195i	$-0.335363\pi$
$421$	416.343			0.988938			0.494469	+	0.869195i	$0.335363\pi$
$422$		0			0
$423$	−133.485	−	133.485i	−0.315567	−	0.315567i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	75.0194	−	75.0194i	0.175689	−	0.175689i
$428$		0			0
$429$			46.7878i			0.109062i
$430$		0			0
$431$	−38.3429			−0.0889625			−0.0444813	−	0.999010i	$-0.514163\pi$
$431$	−38.3429			−0.0889625			−0.0444813	+	0.999010i	$0.514163\pi$
$432$		0			0
$433$	251.828	+	251.828i	0.581588	+	0.581588i	0.935339	−	0.353752i	$-0.115094\pi$
$433$	251.828	+	251.828i	0.581588	+	0.581588i	−0.353752	+	0.935339i	$0.615094\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	12.3224	−	12.3224i	0.0281978	−	0.0281978i
$438$		0			0
$439$		−	385.192i		−	0.877430i	−0.898626	−	0.438715i	$-0.855434\pi$
$439$		−	385.192i		−	0.877430i	0.898626	−	0.438715i	$-0.144566\pi$
$440$		0			0
$441$	−132.576			−0.300625
$442$		0			0
$443$	75.6459	+	75.6459i	0.170758	+	0.170758i	0.787312	−	0.616554i	$-0.211472\pi$
$443$	75.6459	+	75.6459i	0.170758	+	0.170758i	−0.616554	+	0.787312i	$0.711472\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	37.7071	−	37.7071i	0.0843560	−	0.0843560i
$448$		0			0
$449$			474.727i			1.05730i	0.848841	+	0.528649i	$0.177301\pi$
$449$			474.727i			1.05730i	−0.848841	+	0.528649i	$0.822699\pi$
$450$		0			0
$451$	−442.829			−0.981882
$452$		0			0
$453$	57.2781	+	57.2781i	0.126442	+	0.126442i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	139.283	−	139.283i	0.304776	−	0.304776i	−0.538103	−	0.842879i	$-0.680859\pi$
$457$	139.283	−	139.283i	0.304776	−	0.304776i	0.842879	+	0.538103i	$0.180859\pi$
$458$		0			0
$459$		−	65.3939i		−	0.142470i
$460$		0			0
$461$	488.322			1.05927			0.529634	−	0.848226i	$-0.322329\pi$
$461$	488.322			1.05927			0.529634	+	0.848226i	$0.322329\pi$
$462$		0			0
$463$	341.196	+	341.196i	0.736925	+	0.736925i	0.971982	−	0.235056i	$-0.0755275\pi$
$463$	341.196	+	341.196i	0.736925	+	0.736925i	−0.235056	+	0.971982i	$0.575527\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−354.293	+	354.293i	−0.758657	+	0.758657i	−0.976078	−	0.217421i	$-0.930236\pi$
$467$	−354.293	+	354.293i	−0.758657	+	0.758657i	0.217421	+	0.976078i	$0.430236\pi$
$468$		0			0
$469$			92.6857i			0.197624i
$470$		0			0
$471$	−246.322			−0.522978
$472$		0			0
$473$	325.939	+	325.939i	0.689088	+	0.689088i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	108.272	−	108.272i	0.226986	−	0.226986i
$478$		0			0
$479$			395.596i			0.825879i	0.910759	+	0.412939i	$0.135498\pi$
$479$			395.596i			0.825879i	−0.910759	+	0.412939i	$0.864502\pi$
$480$		0			0
$481$	123.192			0.256116
$482$		0			0
$483$	8.36326	+	8.36326i	0.0173152	+	0.0173152i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−223.621	+	223.621i	−0.459181	+	0.459181i	−0.898386	−	0.439206i	$-0.855260\pi$
$487$	−223.621	+	223.621i	−0.459181	+	0.459181i	0.439206	+	0.898386i	$0.355260\pi$
$488$		0			0
$489$		−	359.555i		−	0.735286i
$490$		0			0
$491$	462.141			0.941224			0.470612	−	0.882340i	$-0.344033\pi$
$491$	462.141			0.941224			0.470612	+	0.882340i	$0.344033\pi$
$492$		0			0
$493$	−316.767	−	316.767i	−0.642530	−	0.642530i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−78.7469	+	78.7469i	−0.158445	+	0.158445i
$498$		0			0
$499$		−	54.3633i		−	0.108944i	−0.998515	−	0.0544722i	$-0.982652\pi$
$499$		−	54.3633i		−	0.108944i	0.998515	−	0.0544722i	$-0.0173476\pi$
$500$		0			0
$501$	−147.414			−0.294240
$502$		0			0
$503$	−664.899	−	664.899i	−1.32187	−	1.32187i	−0.912264	−	0.409603i	$-0.865667\pi$
$503$	−664.899	−	664.899i	−1.32187	−	1.32187i	−0.409603	−	0.912264i	$-0.634333\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	192.285	−	192.285i	0.379260	−	0.379260i
$508$		0			0
$509$		−	179.110i		−	0.351886i	−0.984400	−	0.175943i	$-0.943702\pi$
$509$		−	179.110i		−	0.351886i	0.984400	−	0.175943i	$-0.0562975\pi$
$510$		0			0
$511$	152.890			0.299197
$512$		0			0
$513$	20.5607	+	20.5607i	0.0400794	+	0.0400794i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	346.969	−	346.969i	0.671121	−	0.671121i
$518$		0			0
$519$		−	500.363i		−	0.964091i
$520$		0			0
$521$	306.767			0.588805			0.294402	−	0.955682i	$-0.404879\pi$
$521$	306.767			0.588805			0.294402	+	0.955682i	$0.404879\pi$
$522$		0			0
$523$	503.010	+	503.010i	0.961779	+	0.961779i	0.999296	−	0.0375174i	$-0.0119450\pi$
$523$	503.010	+	503.010i	0.961779	+	0.961779i	−0.0375174	+	0.999296i	$0.511945\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	473.353	−	473.353i	0.898203	−	0.898203i
$528$		0			0
$529$		−	519.302i		−	0.981667i
$530$		0			0
$531$	32.9694			0.0620892
$532$		0			0
$533$	−139.101	−	139.101i	−0.260978	−	0.260978i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−240.247	+	240.247i	−0.447388	+	0.447388i
$538$		0			0
$539$		−	344.606i		−	0.639343i
$540$		0			0
$541$	−685.110			−1.26638			−0.633189	−	0.773997i	$-0.718254\pi$
$541$	−685.110			−1.26638			−0.633189	+	0.773997i	$0.718254\pi$
$542$		0			0
$543$	−281.641	−	281.641i	−0.518676	−	0.518676i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−611.333	+	611.333i	−1.11761	+	1.11761i	−0.125519	+	0.992091i	$0.540060\pi$
$547$	−611.333	+	611.333i	−1.11761	+	1.11761i	−0.992091	+	0.125519i	$0.959940\pi$
$548$		0			0
$549$			145.151i			0.264392i
$550$		0			0
$551$	199.192			0.361510
$552$		0			0
$553$	−102.334	−	102.334i	−0.185052	−	0.185052i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	406.070	−	406.070i	0.729031	−	0.729031i	−0.241396	−	0.970427i	$-0.577605\pi$
$557$	406.070	−	406.070i	0.729031	−	0.729031i	0.970427	+	0.241396i	$0.0776051\pi$
$558$		0			0
$559$			204.767i			0.366310i
$560$		0			0
$561$	169.980			0.302994
$562$		0			0
$563$	−180.636	−	180.636i	−0.320845	−	0.320845i	0.528246	−	0.849091i	$-0.322850\pi$
$563$	−180.636	−	180.636i	−0.320845	−	0.320845i	−0.849091	+	0.528246i	$0.822850\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−13.9546	+	13.9546i	−0.0246113	+	0.0246113i
$568$		0			0
$569$		−	41.6367i		−	0.0731753i	−0.999330	−	0.0365876i	$-0.988351\pi$
$569$		−	41.6367i		−	0.0731753i	0.999330	−	0.0365876i	$-0.0116488\pi$
$570$		0			0
$571$	−619.939			−1.08571			−0.542854	−	0.839827i	$-0.682656\pi$
$571$	−619.939			−1.08571			−0.542854	+	0.839827i	$0.682656\pi$
$572$		0			0
$573$	316.899	+	316.899i	0.553052	+	0.553052i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	46.1520	−	46.1520i	0.0799862	−	0.0799862i	−0.665982	−	0.745968i	$-0.731987\pi$
$577$	46.1520	−	46.1520i	0.0799862	−	0.0799862i	0.745968	+	0.665982i	$0.231987\pi$
$578$		0			0
$579$			645.131i			1.11422i
$580$		0			0
$581$	−12.6857			−0.0218343
$582$		0			0
$583$	281.435	+	281.435i	0.482735	+	0.482735i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	572.272	−	572.272i	0.974910	−	0.974910i	−0.0247824	−	0.999693i	$-0.507889\pi$
$587$	572.272	−	572.272i	0.974910	−	0.974910i	0.999693	+	0.0247824i	$0.00788929\pi$
$588$		0			0
$589$			297.657i			0.505360i
$590$		0			0
$591$	494.302			0.836382
$592$		0			0
$593$	611.060	+	611.060i	1.03046	+	1.03046i	0.999521	+	0.0309342i	$0.00984824\pi$
$593$	611.060	+	611.060i	1.03046	+	1.03046i	0.0309342	+	0.999521i	$0.490152\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	3.43928	−	3.43928i	0.00576095	−	0.00576095i
$598$		0			0
$599$			894.343i			1.49306i	0.665352	+	0.746530i	$0.268282\pi$
$599$			894.343i			1.49306i	−0.665352	+	0.746530i	$0.731718\pi$
$600$		0			0
$601$	−361.069			−0.600781			−0.300390	−	0.953816i	$-0.597117\pi$
$601$	−361.069			−0.600781			−0.300390	+	0.953816i	$0.597117\pi$
$602$		0			0
$603$	−89.6663	−	89.6663i	−0.148700	−	0.148700i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	168.974	−	168.974i	0.278376	−	0.278376i	−0.554085	−	0.832460i	$-0.686932\pi$
$607$	168.974	−	168.974i	0.278376	−	0.278376i	0.832460	+	0.554085i	$0.186932\pi$
$608$		0			0
$609$			135.192i			0.221990i
$610$		0			0
$611$	217.980			0.356759
$612$		0			0
$613$	−535.448	−	535.448i	−0.873489	−	0.873489i	0.119362	−	0.992851i	$-0.461915\pi$
$613$	−535.448	−	535.448i	−0.873489	−	0.873489i	−0.992851	+	0.119362i	$0.961915\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−751.201	+	751.201i	−1.21751	+	1.21751i	−0.249003	+	0.968503i	$0.580103\pi$
$617$	−751.201	+	751.201i	−1.21751	+	1.21751i	−0.968503	+	0.249003i	$0.919897\pi$
$618$		0			0
$619$		−	349.151i		−	0.564057i	−0.959406	−	0.282028i	$-0.908993\pi$
$619$		−	349.151i		−	0.564057i	0.959406	−	0.282028i	$-0.0910072\pi$
$620$		0			0
$621$	−16.1816			−0.0260574
$622$		0			0
$623$	45.8888	+	45.8888i	0.0736577	+	0.0736577i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−53.4439	+	53.4439i	−0.0852374	+	0.0852374i
$628$		0			0
$629$		−	447.555i		−	0.711534i
$630$		0			0
$631$	771.494			1.22265			0.611326	−	0.791379i	$-0.290636\pi$
$631$	771.494			1.22265			0.611326	+	0.791379i	$0.290636\pi$
$632$		0			0
$633$	43.5709	+	43.5709i	0.0688324	+	0.0688324i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	108.247	−	108.247i	0.169933	−	0.169933i
$638$		0			0
$639$		−	152.363i		−	0.238440i
$640$		0			0
$641$	−1026.69			−1.60169			−0.800847	−	0.598869i	$-0.795617\pi$
$641$	−1026.69			−1.60169			−0.800847	+	0.598869i	$0.795617\pi$
$642$		0			0
$643$	−23.3735	−	23.3735i	−0.0363506	−	0.0363506i	0.688698	−	0.725048i	$-0.258183\pi$
$643$	−23.3735	−	23.3735i	−0.0363506	−	0.0363506i	−0.725048	+	0.688698i	$0.758183\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−860.958	+	860.958i	−1.33069	+	1.33069i	−0.425942	+	0.904750i	$0.640057\pi$
$647$	−860.958	+	860.958i	−1.33069	+	1.33069i	−0.904750	+	0.425942i	$0.859943\pi$
$648$		0			0
$649$			85.6980i			0.132046i
$650$		0			0
$651$	−202.020			−0.310323
$652$		0			0
$653$	−554.969	−	554.969i	−0.849877	−	0.849877i	0.140241	−	0.990117i	$-0.455212\pi$
$653$	−554.969	−	554.969i	−0.849877	−	0.849877i	−0.990117	+	0.140241i	$0.955212\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−147.909	+	147.909i	−0.225128	+	0.225128i
$658$		0			0
$659$		−	75.4755i		−	0.114530i	−0.998359	−	0.0572652i	$-0.981762\pi$
$659$		−	75.4755i		−	0.114530i	0.998359	−	0.0572652i	$-0.0182381\pi$
$660$		0			0
$661$	−234.808			−0.355232			−0.177616	−	0.984100i	$-0.556838\pi$
$661$	−234.808			−0.355232			−0.177616	+	0.984100i	$0.556838\pi$
$662$		0			0
$663$	53.3939	+	53.3939i	0.0805338	+	0.0805338i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−78.3837	+	78.3837i	−0.117517	+	0.117517i
$668$		0			0
$669$		−	198.545i		−	0.296779i
$670$		0			0
$671$	−377.294			−0.562286
$672$		0			0
$673$	−463.323	−	463.323i	−0.688445	−	0.688445i	0.273443	−	0.961888i	$-0.411837\pi$
$673$	−463.323	−	463.323i	−0.688445	−	0.688445i	−0.961888	+	0.273443i	$0.911837\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−347.233	+	347.233i	−0.512899	+	0.512899i	−0.915414	−	0.402515i	$-0.868136\pi$
$677$	−347.233	+	347.233i	−0.512899	+	0.512899i	0.402515	+	0.915414i	$0.368136\pi$
$678$		0			0
$679$			412.686i			0.607785i
$680$		0			0
$681$	−125.171			−0.183805
$682$		0			0
$683$	−295.768	−	295.768i	−0.433043	−	0.433043i	0.456619	−	0.889662i	$-0.349060\pi$
$683$	−295.768	−	295.768i	−0.433043	−	0.433043i	−0.889662	+	0.456619i	$0.849060\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	214.045	−	214.045i	0.311565	−	0.311565i
$688$		0			0
$689$			176.808i			0.256616i
$690$		0			0
$691$	724.665			1.04872			0.524360	−	0.851497i	$-0.324305\pi$
$691$	724.665			1.04872			0.524360	+	0.851497i	$0.324305\pi$
$692$		0			0
$693$	−36.2724	−	36.2724i	−0.0523412	−	0.0523412i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−505.353	+	505.353i	−0.725040	+	0.725040i
$698$		0			0
$699$			408.222i			0.584009i
$700$		0			0
$701$	−1066.30			−1.52112			−0.760558	−	0.649270i	$-0.775074\pi$
$701$	−1066.30			−1.52112			−0.760558	+	0.649270i	$0.775074\pi$
$702$		0			0
$703$	140.717	+	140.717i	0.200167	+	0.200167i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	119.092	−	119.092i	0.168447	−	0.168447i
$708$		0			0
$709$		−	1053.07i		−	1.48529i	−0.669686	−	0.742644i	$-0.733571\pi$
$709$		−	1053.07i		−	1.48529i	0.669686	−	0.742644i	$-0.266429\pi$
$710$		0			0
$711$	198.000			0.278481
$712$		0			0
$713$	−117.131	−	117.131i	−0.164279	−	0.164279i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−373.807	+	373.807i	−0.521349	+	0.521349i
$718$		0			0
$719$			550.424i			0.765542i	0.923843	+	0.382771i	$0.125030\pi$
$719$			550.424i			0.765542i	−0.923843	+	0.382771i	$0.874970\pi$
$720$		0			0
$721$	−389.616			−0.540383
$722$		0			0
$723$	−370.813	−	370.813i	−0.512881	−	0.512881i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−594.459	+	594.459i	−0.817687	+	0.817687i	−0.985772	−	0.168085i	$-0.946242\pi$
$727$	−594.459	+	594.459i	−0.817687	+	0.817687i	0.168085	+	0.985772i	$0.446242\pi$
$728$		0			0
$729$		−	27.0000i		−	0.0370370i
$730$		0			0
$731$	743.918			1.01767
$732$		0			0
$733$	−390.904	−	390.904i	−0.533293	−	0.533293i	0.388258	−	0.921551i	$-0.373077\pi$
$733$	−390.904	−	390.904i	−0.533293	−	0.533293i	−0.921551	+	0.388258i	$0.873077\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	233.071	−	233.071i	0.316243	−	0.316243i
$738$		0			0
$739$			1408.62i			1.90612i	0.302778	+	0.953061i	$0.402086\pi$
$739$			1408.62i			1.90612i	−0.302778	+	0.953061i	$0.597914\pi$
$740$		0			0
$741$	−33.5755			−0.0453111
$742$		0			0
$743$	152.899	+	152.899i	0.205786	+	0.205786i	0.802474	−	0.596688i	$-0.203517\pi$
$743$	152.899	+	152.899i	0.205786	+	0.205786i	−0.596688	+	0.802474i	$0.703517\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	12.2724	−	12.2724i	0.0164290	−	0.0164290i
$748$		0			0
$749$			241.535i			0.322476i
$750$		0			0
$751$	539.453			0.718313			0.359156	−	0.933277i	$-0.383064\pi$
$751$	539.453			0.718313			0.359156	+	0.933277i	$0.383064\pi$
$752$		0			0
$753$	433.782	+	433.782i	0.576072	+	0.576072i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−160.702	+	160.702i	−0.212287	+	0.212287i	−0.805238	−	0.592951i	$-0.797963\pi$
$757$	−160.702	+	160.702i	−0.212287	+	0.212287i	0.592951	+	0.805238i	$0.297963\pi$
$758$		0			0
$759$		−	42.0612i		−	0.0554166i
$760$		0			0
$761$	222.041			0.291775			0.145888	−	0.989301i	$-0.453396\pi$
$761$	222.041			0.291775			0.145888	+	0.989301i	$0.453396\pi$
$762$		0			0
$763$	−94.8786	−	94.8786i	−0.124349	−	0.124349i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−26.9194	+	26.9194i	−0.0350970	+	0.0350970i
$768$		0			0
$769$			915.412i			1.19039i	0.803580	+	0.595197i	$0.202926\pi$
$769$			915.412i			1.19039i	−0.803580	+	0.595197i	$0.797074\pi$
$770$		0			0
$771$	−180.545			−0.234170
$772$		0			0
$773$	194.243	+	194.243i	0.251284	+	0.251284i	0.821497	−	0.570213i	$-0.193139\pi$
$773$	194.243	+	194.243i	0.251284	+	0.251284i	−0.570213	+	0.821497i	$0.693139\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−95.5051	+	95.5051i	−0.122915	+	0.122915i
$778$		0			0
$779$		−	317.780i		−	0.407933i
$780$		0			0
$781$	396.041			0.507095
$782$		0			0
$783$	−130.788	−	130.788i	−0.167034	−	0.167034i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−696.463	+	696.463i	−0.884960	+	0.884960i	−0.994034	−	0.109074i	$-0.965211\pi$
$787$	−696.463	+	696.463i	−0.884960	+	0.884960i	0.109074	+	0.994034i	$0.465211\pi$
$788$		0			0
$789$			88.1816i			0.111764i
$790$		0			0
$791$	−203.759			−0.257597
$792$		0			0
$793$	−118.515	−	118.515i	−0.149452	−	0.149452i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−904.232	+	904.232i	−1.13454	+	1.13454i	−0.145132	+	0.989412i	$0.546361\pi$
$797$	−904.232	+	904.232i	−1.13454	+	1.13454i	−0.989412	+	0.145132i	$0.953639\pi$
$798$		0			0
$799$		−	791.918i		−	0.991137i
$800$		0			0
$801$	−88.7878			−0.110846
$802$		0			0
$803$	−384.463	−	384.463i	−0.478784	−	0.478784i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−383.555	+	383.555i	−0.475285	+	0.475285i
$808$		0			0
$809$		−	1344.10i		−	1.66143i	−0.556697	−	0.830716i	$-0.687932\pi$
$809$		−	1344.10i		−	1.66143i	0.556697	−	0.830716i	$-0.312068\pi$
$810$		0			0
$811$	−198.808			−0.245140			−0.122570	−	0.992460i	$-0.539114\pi$
$811$	−198.808			−0.245140			−0.122570	+	0.992460i	$0.539114\pi$
$812$		0			0
$813$	−273.823	−	273.823i	−0.336806	−	0.336806i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−233.898	+	233.898i	−0.286289	+	0.286289i
$818$		0			0
$819$		−	22.7878i		−	0.0278239i
$820$		0			0
$821$	−114.143			−0.139029			−0.0695145	−	0.997581i	$-0.522145\pi$
$821$	−114.143			−0.139029			−0.0695145	+	0.997581i	$0.522145\pi$
$822$		0			0
$823$	−449.519	−	449.519i	−0.546195	−	0.546195i	0.379143	−	0.925338i	$-0.376219\pi$
$823$	−449.519	−	449.519i	−0.546195	−	0.546195i	−0.925338	+	0.379143i	$0.876219\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−701.344	+	701.344i	−0.848058	+	0.848058i	−0.989891	−	0.141833i	$-0.954700\pi$
$827$	−701.344	+	701.344i	−0.848058	+	0.848058i	0.141833	+	0.989891i	$0.454700\pi$
$828$		0			0
$829$			378.849i			0.456995i	0.973544	+	0.228498i	$0.0733813\pi$
$829$			378.849i			0.456995i	−0.973544	+	0.228498i	$0.926619\pi$
$830$		0			0
$831$	496.020			0.596896
$832$		0			0
$833$	−393.262	−	393.262i	−0.472104	−	0.472104i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	195.439	−	195.439i	0.233500	−	0.233500i
$838$		0			0
$839$			273.739i			0.326268i	0.986604	+	0.163134i	$0.0521603\pi$
$839$			273.739i			0.326268i	−0.986604	+	0.163134i	$0.947840\pi$
$840$		0			0
$841$	−426.069			−0.506622
$842$		0			0
$843$	337.560	+	337.560i	0.400427	+	0.400427i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−93.3281	+	93.3281i	−0.110187	+	0.110187i
$848$		0			0
$849$			195.637i			0.230432i
$850$		0			0
$851$	−110.747			−0.130137
$852$		0			0
$853$	167.298	+	167.298i	0.196130	+	0.196130i	0.798338	−	0.602209i	$-0.205713\pi$
$853$	167.298	+	167.298i	0.196130	+	0.196130i	−0.602209	+	0.798338i	$0.705713\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	53.1214	−	53.1214i	0.0619853	−	0.0619853i	−0.675435	−	0.737420i	$-0.736044\pi$
$857$	53.1214	−	53.1214i	0.0619853	−	0.0619853i	0.737420	+	0.675435i	$0.236044\pi$
$858$		0			0
$859$			1317.92i			1.53425i	0.641499	+	0.767124i	$0.278313\pi$
$859$			1317.92i			1.53425i	−0.641499	+	0.767124i	$0.721687\pi$
$860$		0			0
$861$	215.678			0.250497
$862$		0			0
$863$	−463.878	−	463.878i	−0.537517	−	0.537517i	0.385282	−	0.922799i	$-0.374104\pi$
$863$	−463.878	−	463.878i	−0.537517	−	0.537517i	−0.922799	+	0.385282i	$0.874104\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−159.972	+	159.972i	−0.184512	+	0.184512i
$868$		0			0
$869$			514.665i			0.592250i
$870$		0			0
$871$	146.424			0.168111
$872$		0			0
$873$	−399.242	−	399.242i	−0.457322	−	0.457322i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	169.410	−	169.410i	0.193170	−	0.193170i	−0.603895	−	0.797064i	$-0.706385\pi$
$877$	169.410	−	169.410i	0.193170	−	0.193170i	0.797064	+	0.603895i	$0.206385\pi$
$878$		0			0
$879$		−	286.020i		−	0.325393i
$880$		0			0
$881$	−491.253			−0.557608			−0.278804	−	0.960348i	$-0.589938\pi$
$881$	−491.253			−0.557608			−0.278804	+	0.960348i	$0.589938\pi$
$882$		0			0
$883$	−429.212	−	429.212i	−0.486084	−	0.486084i	0.420984	−	0.907068i	$-0.361685\pi$
$883$	−429.212	−	429.212i	−0.486084	−	0.486084i	−0.907068	+	0.420984i	$0.861685\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	21.5163	−	21.5163i	0.0242574	−	0.0242574i	−0.694874	−	0.719131i	$-0.744540\pi$
$887$	21.5163	−	21.5163i	0.0242574	−	0.0242574i	0.719131	+	0.694874i	$0.244540\pi$
$888$		0			0
$889$		−	95.3959i		−	0.107307i
$890$		0			0
$891$	70.1816			0.0787673
$892$		0			0
$893$	248.990	+	248.990i	0.278824	+	0.278824i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	13.2122	−	13.2122i	0.0147294	−	0.0147294i
$898$		0			0
$899$		−	1893.41i		−	2.10613i
$900$		0			0
$901$	642.343			0.712922
$902$		0			0
$903$	−158.747	−	158.747i	−0.175799	−	0.175799i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	738.706	−	738.706i	0.814450	−	0.814450i	−0.170848	−	0.985297i	$-0.554651\pi$
$907$	738.706	−	738.706i	0.814450	−	0.814450i	0.985297	+	0.170848i	$0.0546506\pi$
$908$		0			0
$909$			230.424i			0.253492i
$910$		0			0
$911$	−1650.99			−1.81228			−0.906140	−	0.422977i	$-0.860985\pi$
$911$	−1650.99			−1.81228			−0.906140	+	0.422977i	$0.860985\pi$
$912$		0			0
$913$	31.9000	+	31.9000i	0.0349398	+	0.0349398i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	395.991	−	395.991i	0.431833	−	0.431833i
$918$		0			0
$919$		−	857.796i		−	0.933401i	−0.884415	−	0.466701i	$-0.845443\pi$
$919$		−	857.796i		−	0.933401i	0.884415	−	0.466701i	$-0.154557\pi$
$920$		0			0
$921$	515.110			0.559294
$922$		0			0
$923$	124.404	+	124.404i	0.134782	+	0.134782i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	376.924	−	376.924i	0.406606	−	0.406606i
$928$		0			0
$929$		−	966.727i		−	1.04061i	−0.853981	−	0.520305i	$-0.825818\pi$
$929$		−	966.727i		−	1.04061i	0.853981	−	0.520305i	$-0.174182\pi$
$930$		0			0
$931$	247.294			0.265622
$932$		0			0
$933$	−23.9500	−	23.9500i	−0.0256699	−	0.0256699i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−38.2020	+	38.2020i	−0.0407706	+	0.0407706i	−0.727198	−	0.686428i	$-0.759178\pi$
$937$	−38.2020	+	38.2020i	−0.0407706	+	0.0407706i	0.686428	+	0.727198i	$0.259178\pi$
$938$		0			0
$939$			543.637i			0.578953i
$940$		0			0
$941$	−1167.84			−1.24106			−0.620530	−	0.784183i	$-0.713082\pi$
$941$	−1167.84			−1.24106			−0.620530	+	0.784183i	$0.713082\pi$
$942$		0			0
$943$	125.049	+	125.049i	0.132608	+	0.132608i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−896.958	+	896.958i	−0.947157	+	0.947157i	−0.998672	−	0.0515147i	$-0.983595\pi$
$947$	−896.958	+	896.958i	−0.947157	+	0.947157i	0.0515147	+	0.998672i	$0.483595\pi$
$948$		0			0
$949$		−	241.535i		−	0.254515i
$950$		0			0
$951$	515.233			0.541780
$952$		0			0
$953$	68.7582	+	68.7582i	0.0721492	+	0.0721492i	0.742261	−	0.670111i	$-0.233754\pi$
$953$	68.7582	+	68.7582i	0.0721492	+	0.0721492i	−0.670111	+	0.742261i	$0.733754\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	339.959	−	339.959i	0.355234	−	0.355234i
$958$		0			0
$959$			325.857i			0.339788i
$960$		0			0
$961$	1868.37			1.94420
$962$		0			0
$963$	−233.666	−	233.666i	−0.242644	−	0.242644i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−846.023	+	846.023i	−0.874894	+	0.874894i	−0.993001	−	0.118106i	$-0.962318\pi$
$967$	−846.023	+	846.023i	−0.874894	+	0.874894i	0.118106	+	0.993001i	$0.462318\pi$
$968$		0			0
$969$			121.980i			0.125882i
$970$		0			0
$971$	−725.169			−0.746827			−0.373414	−	0.927665i	$-0.621813\pi$
$971$	−725.169			−0.746827			−0.373414	+	0.927665i	$0.621813\pi$
$972$		0			0
$973$	63.8684	+	63.8684i	0.0656407	+	0.0656407i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−713.344	+	713.344i	−0.730137	+	0.730137i	−0.970647	−	0.240510i	$-0.922685\pi$
$977$	−713.344	+	713.344i	−0.730137	+	0.730137i	0.240510	+	0.970647i	$0.422685\pi$
$978$		0			0
$979$		−	230.788i		−	0.235738i
$980$		0			0
$981$	183.576			0.187131
$982$		0			0
$983$	114.515	+	114.515i	0.116496	+	0.116496i	0.762951	−	0.646456i	$-0.223750\pi$
$983$	114.515	+	114.515i	0.116496	+	0.116496i	−0.646456	+	0.762951i	$0.723750\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−168.990	+	168.990i	−0.171216	+	0.171216i
$988$		0			0
$989$		−	184.082i		−	0.186129i
$990$		0			0
$991$	−538.645			−0.543537			−0.271768	−	0.962363i	$-0.587608\pi$
$991$	−538.645			−0.543537			−0.271768	+	0.962363i	$0.587608\pi$
$992$		0			0
$993$	−408.520	−	408.520i	−0.411400	−	0.411400i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−1119.94	+	1119.94i	−1.12331	+	1.12331i	−0.132073	+	0.991240i	$0.542163\pi$
$997$	−1119.94	+	1119.94i	−1.12331	+	1.12331i	−0.991240	+	0.132073i	$0.957837\pi$
$998$		0			0
$999$		−	184.788i		−	0.184973i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1200.3.bg.m.193.2		4
4.3	odd	2		600.3.u.a.193.1	✓	4
5.2	odd	4	inner	1200.3.bg.m.1057.2		4
5.3	odd	4		1200.3.bg.b.1057.1		4
5.4	even	2		1200.3.bg.b.193.1		4
12.11	even	2		1800.3.v.j.793.2		4
20.3	even	4		600.3.u.f.457.2	yes	4
20.7	even	4		600.3.u.a.457.1	yes	4
20.19	odd	2		600.3.u.f.193.2	yes	4
60.23	odd	4		1800.3.v.q.1657.1		4
60.47	odd	4		1800.3.v.j.1657.2		4
60.59	even	2		1800.3.v.q.793.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
600.3.u.a.193.1	✓	4	4.3	odd	2
600.3.u.a.457.1	yes	4	20.7	even	4
600.3.u.f.193.2	yes	4	20.19	odd	2
600.3.u.f.457.2	yes	4	20.3	even	4
1200.3.bg.b.193.1		4	5.4	even	2
1200.3.bg.b.1057.1		4	5.3	odd	4
1200.3.bg.m.193.2		4	1.1	even	1	trivial
1200.3.bg.m.1057.2		4	5.2	odd	4	inner
1800.3.v.j.793.2		4	12.11	even	2
1800.3.v.j.1657.2		4	60.47	odd	4
1800.3.v.q.793.1		4	60.59	even	2
1800.3.v.q.1657.1		4	60.23	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(1.22474 + 1.22474i) q^{3} +(1.55051 - 1.55051i) q^{7} +3.00000i q^{9} +O(q^{10})\)	\(q+(1.22474 + 1.22474i) q^{3} +(1.55051 - 1.55051i) q^{7} +3.00000i q^{9} -7.79796 q^{11} +(-2.44949 - 2.44949i) q^{13} +(-8.89898 + 8.89898i) q^{17} -5.59592i q^{19} +3.79796 q^{21} +(2.20204 + 2.20204i) q^{23} +(-3.67423 + 3.67423i) q^{27} +35.5959i q^{29} -53.1918 q^{31} +(-9.55051 - 9.55051i) q^{33} +(-25.1464 + 25.1464i) q^{37} -6.00000i q^{39} +56.7878 q^{41} +(-41.7980 - 41.7980i) q^{43} +(-44.4949 + 44.4949i) q^{47} +44.1918i q^{49} -21.7980 q^{51} +(-36.0908 - 36.0908i) q^{53} +(6.85357 - 6.85357i) q^{57} -10.9898i q^{59} +48.3837 q^{61} +(4.65153 + 4.65153i) q^{63} +(-29.8888 + 29.8888i) q^{67} +5.39388i q^{69} -50.7878 q^{71} +(49.3031 + 49.3031i) q^{73} +(-12.0908 + 12.0908i) q^{77} -66.0000i q^{79} -9.00000 q^{81} +(-4.09082 - 4.09082i) q^{83} +(-43.5959 + 43.5959i) q^{87} +29.5959i q^{89} -7.59592 q^{91} +(-65.1464 - 65.1464i) q^{93} +(-133.081 + 133.081i) q^{97} -23.3939i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 16 q^{7}+O(q^{10}) \) 4 * q + 16 * q^7	\( 4 q + 16 q^{7} + 8 q^{11} - 16 q^{17} - 24 q^{21} + 48 q^{23} - 56 q^{31} - 48 q^{33} - 32 q^{37} - 8 q^{41} - 128 q^{43} - 80 q^{47} - 48 q^{51} + 32 q^{53} + 96 q^{57} - 120 q^{61} + 48 q^{63} + 96 q^{67} + 32 q^{71} + 256 q^{73} + 128 q^{77} - 36 q^{81} + 160 q^{83} - 96 q^{87} + 48 q^{91} - 192 q^{93} - 160 q^{97}+O(q^{100}) \) 4 * q + 16 * q^7 + 8 * q^11 - 16 * q^17 - 24 * q^21 + 48 * q^23 - 56 * q^31 - 48 * q^33 - 32 * q^37 - 8 * q^41 - 128 * q^43 - 80 * q^47 - 48 * q^51 + 32 * q^53 + 96 * q^57 - 120 * q^61 + 48 * q^63 + 96 * q^67 + 32 * q^71 + 256 * q^73 + 128 * q^77 - 36 * q^81 + 160 * q^83 - 96 * q^87 + 48 * q^91 - 192 * q^93 - 160 * q^97

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