LMFDB - Embedded newform 1200.3.bg.m.193.1

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			193.1
Root			$-1.22474 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	1200.193
Dual form			1200.3.bg.m.1057.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.22474 - 1.22474i) q^{3} +(6.44949 - 6.44949i) q^{7} +3.00000i q^{9} +O(q^{10})$	$q+(-1.22474 - 1.22474i) q^{3} +(6.44949 - 6.44949i) q^{7} +3.00000i q^{9} +11.7980 q^{11} +(2.44949 + 2.44949i) q^{13} +(0.898979 - 0.898979i) q^{17} +33.5959i q^{19} -15.7980 q^{21} +(21.7980 + 21.7980i) q^{23} +(3.67423 - 3.67423i) q^{27} -3.59592i q^{29} +25.1918 q^{31} +(-14.4495 - 14.4495i) q^{33} +(9.14643 - 9.14643i) q^{37} -6.00000i q^{39} -60.7878 q^{41} +(-22.2020 - 22.2020i) q^{43} +(4.49490 - 4.49490i) q^{47} -34.1918i q^{49} -2.20204 q^{51} +(52.0908 + 52.0908i) q^{53} +(41.1464 - 41.1464i) q^{57} +86.9898i q^{59} -108.384 q^{61} +(19.3485 + 19.3485i) q^{63} +(77.8888 - 77.8888i) q^{67} -53.3939i q^{69} +66.7878 q^{71} +(78.6969 + 78.6969i) q^{73} +(76.0908 - 76.0908i) q^{77} -66.0000i q^{79} -9.00000 q^{81} +(84.0908 + 84.0908i) q^{83} +(-4.40408 + 4.40408i) q^{87} -9.59592i q^{89} +31.5959 q^{91} +(-30.8536 - 30.8536i) q^{93} +(53.0806 - 53.0806i) q^{97} +35.3939i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	6.44949	−	6.44949i	0.921356	−	0.921356i	−0.0757697	−	0.997125i	$-0.524141\pi$
$7$	6.44949	−	6.44949i	0.921356	−	0.921356i	0.997125	+	0.0757697i	$0.0241414\pi$
$8$		0			0
$9$			3.00000i			0.333333i
$10$		0			0
$11$	11.7980			1.07254			0.536271	−	0.844046i	$-0.319833\pi$
$11$	11.7980			1.07254			0.536271	+	0.844046i	$0.319833\pi$
$12$		0			0
$13$	2.44949	+	2.44949i	0.188422	+	0.188422i	0.795014	−	0.606591i	$-0.207464\pi$
$13$	2.44949	+	2.44949i	0.188422	+	0.188422i	−0.606591	+	0.795014i	$0.707464\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	0.898979	−	0.898979i	0.0528811	−	0.0528811i	−0.680172	−	0.733053i	$-0.738095\pi$
$17$	0.898979	−	0.898979i	0.0528811	−	0.0528811i	0.733053	+	0.680172i	$0.238095\pi$
$18$		0			0
$19$			33.5959i			1.76821i	0.467292	+	0.884103i	$0.345230\pi$
$19$			33.5959i			1.76821i	−0.467292	+	0.884103i	$0.654770\pi$
$20$		0			0
$21$	−15.7980			−0.752284
$22$		0			0
$23$	21.7980	+	21.7980i	0.947737	+	0.947737i	0.998701	−	0.0509632i	$-0.0162291\pi$
$23$	21.7980	+	21.7980i	0.947737	+	0.947737i	−0.0509632	+	0.998701i	$0.516229\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	3.67423	−	3.67423i	0.136083	−	0.136083i
$28$		0			0
$29$		−	3.59592i		−	0.123997i	−0.998076	−	0.0619986i	$-0.980253\pi$
$29$		−	3.59592i		−	0.123997i	0.998076	−	0.0619986i	$-0.0197474\pi$
$30$		0			0
$31$	25.1918			0.812640			0.406320	−	0.913731i	$-0.366812\pi$
$31$	25.1918			0.812640			0.406320	+	0.913731i	$0.366812\pi$
$32$		0			0
$33$	−14.4495	−	14.4495i	−0.437863	−	0.437863i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	9.14643	−	9.14643i	0.247201	−	0.247201i	−0.572620	−	0.819821i	$-0.694073\pi$
$37$	9.14643	−	9.14643i	0.247201	−	0.247201i	0.819821	+	0.572620i	$0.194073\pi$
$38$		0			0
$39$		−	6.00000i		−	0.153846i
$40$		0			0
$41$	−60.7878			−1.48263			−0.741314	−	0.671158i	$-0.765797\pi$
$41$	−60.7878			−1.48263			−0.741314	+	0.671158i	$0.765797\pi$
$42$		0			0
$43$	−22.2020	−	22.2020i	−0.516327	−	0.516327i	0.400131	−	0.916458i	$-0.368965\pi$
$43$	−22.2020	−	22.2020i	−0.516327	−	0.516327i	−0.916458	+	0.400131i	$0.868965\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	4.49490	−	4.49490i	0.0956361	−	0.0956361i	−0.657670	−	0.753306i	$-0.728458\pi$
$47$	4.49490	−	4.49490i	0.0956361	−	0.0956361i	0.753306	+	0.657670i	$0.228458\pi$
$48$		0			0
$49$		−	34.1918i		−	0.697793i
$50$		0			0
$51$	−2.20204			−0.0431773
$52$		0			0
$53$	52.0908	+	52.0908i	0.982846	+	0.982846i	0.999855	−	0.0170098i	$-0.00541463\pi$
$53$	52.0908	+	52.0908i	0.982846	+	0.982846i	−0.0170098	+	0.999855i	$0.505415\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	41.1464	−	41.1464i	0.721867	−	0.721867i
$58$		0			0
$59$			86.9898i			1.47440i	0.675673	+	0.737202i	$0.263853\pi$
$59$			86.9898i			1.47440i	−0.675673	+	0.737202i	$0.736147\pi$
$60$		0			0
$61$	−108.384			−1.77678			−0.888391	−	0.459088i	$-0.848176\pi$
$61$	−108.384			−1.77678			−0.888391	+	0.459088i	$0.848176\pi$
$62$		0			0
$63$	19.3485	+	19.3485i	0.307119	+	0.307119i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	77.8888	−	77.8888i	1.16252	−	1.16252i	0.178597	−	0.983922i	$-0.442844\pi$
$67$	77.8888	−	77.8888i	1.16252	−	1.16252i	0.983922	−	0.178597i	$-0.0571557\pi$
$68$		0			0
$69$		−	53.3939i		−	0.773824i
$70$		0			0
$71$	66.7878			0.940673			0.470336	−	0.882487i	$-0.344133\pi$
$71$	66.7878			0.940673			0.470336	+	0.882487i	$0.344133\pi$
$72$		0			0
$73$	78.6969	+	78.6969i	1.07804	+	1.07804i	0.996685	+	0.0813551i	$0.0259248\pi$
$73$	78.6969	+	78.6969i	1.07804	+	1.07804i	0.0813551	+	0.996685i	$0.474075\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	76.0908	−	76.0908i	0.988192	−	0.988192i
$78$		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
$83$	84.0908	+	84.0908i	1.01314	+	1.01314i	0.999912	+	0.0132299i	$0.00421132\pi$
$83$	84.0908	+	84.0908i	1.01314	+	1.01314i	0.0132299	+	0.999912i	$0.495789\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−4.40408	+	4.40408i	−0.0506216	+	0.0506216i
$88$		0			0
$89$		−	9.59592i		−	0.107819i	−0.998546	−	0.0539097i	$-0.982832\pi$
$89$		−	9.59592i		−	0.107819i	0.998546	−	0.0539097i	$-0.0171683\pi$
$90$		0			0
$91$	31.5959			0.347208
$92$		0			0
$93$	−30.8536	−	30.8536i	−0.331759	−	0.331759i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	53.0806	−	53.0806i	0.547223	−	0.547223i	−0.378414	−	0.925637i	$-0.623530\pi$
$97$	53.0806	−	53.0806i	0.547223	−	0.547223i	0.925637	+	0.378414i	$0.123530\pi$
$98$		0			0
$99$			35.3939i			0.357514i
$100$		0			0
$101$	155.192			1.53655			0.768276	−	0.640118i	$-0.221115\pi$
$101$	155.192			1.53655			0.768276	+	0.640118i	$0.221115\pi$
$102$		0			0
$103$	−42.3587	−	42.3587i	−0.411249	−	0.411249i	0.470924	−	0.882174i	$-0.343920\pi$
$103$	−42.3587	−	42.3587i	−0.411249	−	0.411249i	−0.882174	+	0.470924i	$0.843920\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	29.8888	−	29.8888i	0.279334	−	0.279334i	−0.553509	−	0.832843i	$-0.686712\pi$
$107$	29.8888	−	29.8888i	0.279334	−	0.279334i	0.832843	+	0.553509i	$0.186712\pi$
$108$		0			0
$109$			17.1918i			0.157723i	0.996886	+	0.0788616i	$0.0251285\pi$
$109$			17.1918i			0.157723i	−0.996886	+	0.0788616i	$0.974871\pi$
$110$		0			0
$111$	−22.4041			−0.201839
$112$		0			0
$113$	−134.293	−	134.293i	−1.18843	−	1.18843i	−0.977501	−	0.210932i	$-0.932350\pi$
$113$	−134.293	−	134.293i	−1.18843	−	1.18843i	−0.210932	−	0.977501i	$-0.567650\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−7.34847	+	7.34847i	−0.0628074	+	0.0628074i
$118$		0			0
$119$		−	11.5959i		−	0.0974447i
$120$		0			0
$121$	18.1918			0.150346
$122$		0			0
$123$	74.4495	+	74.4495i	0.605280	+	0.605280i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	153.237	−	153.237i	1.20659	−	1.20659i	0.234469	−	0.972124i	$-0.424665\pi$
$127$	153.237	−	153.237i	1.20659	−	1.20659i	0.972124	−	0.234469i	$-0.0753351\pi$
$128$		0			0
$129$			54.3837i			0.421579i
$130$		0			0
$131$	196.606			1.50081			0.750405	−	0.660978i	$-0.229858\pi$
$131$	196.606			1.50081			0.750405	+	0.660978i	$0.229858\pi$
$132$		0			0
$133$	216.677	+	216.677i	1.62915	+	1.62915i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	81.0806	−	81.0806i	0.591829	−	0.591829i	−0.346296	−	0.938125i	$-0.612561\pi$
$137$	81.0806	−	81.0806i	0.591829	−	0.591829i	0.938125	+	0.346296i	$0.112561\pi$
$138$		0			0
$139$		−	37.1918i		−	0.267567i	−0.991011	−	0.133784i	$-0.957287\pi$
$139$		−	37.1918i		−	0.267567i	0.991011	−	0.133784i	$-0.0427127\pi$
$140$		0			0
$141$	−11.0102			−0.0780866
$142$		0			0
$143$	28.8990	+	28.8990i	0.202091	+	0.202091i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−41.8763	+	41.8763i	−0.284873	+	0.284873i
$148$		0			0
$149$			86.7878i			0.582468i	0.956652	+	0.291234i	$0.0940658\pi$
$149$			86.7878i			0.582468i	−0.956652	+	0.291234i	$0.905934\pi$
$150$		0			0
$151$	−266.767			−1.76667			−0.883336	−	0.468741i	$-0.844708\pi$
$151$	−266.767			−1.76667			−0.883336	+	0.468741i	$0.844708\pi$
$152$		0			0
$153$	2.69694	+	2.69694i	0.0176270	+	0.0176270i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−203.439	+	203.439i	−1.29579	+	1.29579i	−0.364645	+	0.931147i	$0.618810\pi$
$157$	−203.439	+	203.439i	−1.29579	+	1.29579i	−0.931147	+	0.364645i	$0.881190\pi$
$158$		0			0
$159$		−	127.596i		−	0.802490i
$160$		0			0
$161$	281.171			1.74641
$162$		0			0
$163$	−29.2122	−	29.2122i	−0.179216	−	0.179216i	0.611798	−	0.791014i	$-0.290447\pi$
$163$	−29.2122	−	29.2122i	−0.179216	−	0.179216i	−0.791014	+	0.611798i	$0.790447\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	116.182	−	116.182i	0.695698	−	0.695698i	−0.267781	−	0.963480i	$-0.586290\pi$
$167$	116.182	−	116.182i	0.695698	−	0.695698i	0.963480	+	0.267781i	$0.0862905\pi$
$168$		0			0
$169$		−	157.000i		−	0.928994i
$170$		0			0
$171$	−100.788			−0.589402
$172$		0			0
$173$	60.2724	+	60.2724i	0.348396	+	0.348396i	0.859512	−	0.511116i	$-0.170768\pi$
$173$	60.2724	+	60.2724i	0.348396	+	0.348396i	−0.511116	+	0.859512i	$0.670768\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	106.540	−	106.540i	0.601923	−	0.601923i
$178$		0			0
$179$		−	176.161i		−	0.984141i	−0.870555	−	0.492070i	$-0.836240\pi$
$179$		−	176.161i		−	0.984141i	0.870555	−	0.492070i	$-0.163760\pi$
$180$		0			0
$181$	161.959			0.894802			0.447401	−	0.894333i	$-0.352350\pi$
$181$	161.959			0.894802			0.447401	+	0.894333i	$0.352350\pi$
$182$		0			0
$183$	132.742	+	132.742i	0.725368	+	0.725368i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	10.6061	−	10.6061i	0.0567172	−	0.0567172i
$188$		0			0
$189$		−	47.3939i		−	0.250761i
$190$		0			0
$191$	−250.747			−1.31281			−0.656406	−	0.754408i	$-0.727924\pi$
$191$	−250.747			−1.31281			−0.656406	+	0.754408i	$0.727924\pi$
$192$		0			0
$193$	8.62653	+	8.62653i	0.0446971	+	0.0446971i	0.729102	−	0.684405i	$-0.239938\pi$
$193$	8.62653	+	8.62653i	0.0446971	+	0.0446971i	−0.684405	+	0.729102i	$0.739938\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	182.202	−	182.202i	0.924883	−	0.924883i	−0.0724860	−	0.997369i	$-0.523093\pi$
$197$	182.202	−	182.202i	0.924883	−	0.924883i	0.997369	+	0.0724860i	$0.0230933\pi$
$198$		0			0
$199$		−	81.1918i		−	0.407999i	−0.978971	−	0.204000i	$-0.934606\pi$
$199$		−	81.1918i		−	0.407999i	0.978971	−	0.204000i	$-0.0653941\pi$
$200$		0			0
$201$	−190.788			−0.949193
$202$		0			0
$203$	−23.1918	−	23.1918i	−0.114245	−	0.114245i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−65.3939	+	65.3939i	−0.315912	+	0.315912i
$208$		0			0
$209$			396.363i			1.89647i
$210$		0			0
$211$	−199.576			−0.945855			−0.472928	−	0.881101i	$-0.656803\pi$
$211$	−199.576			−0.945855			−0.472928	+	0.881101i	$0.656803\pi$
$212$		0			0
$213$	−81.7980	−	81.7980i	−0.384028	−	0.384028i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	162.474	−	162.474i	0.748730	−	0.748730i
$218$		0			0
$219$		−	192.767i		−	0.880216i
$220$		0			0
$221$	4.40408			0.0199280
$222$		0			0
$223$	−134.944	−	134.944i	−0.605132	−	0.605132i	0.336538	−	0.941670i	$-0.390744\pi$
$223$	−134.944	−	134.944i	−0.605132	−	0.605132i	−0.941670	+	0.336538i	$0.890744\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−60.8990	+	60.8990i	−0.268277	+	0.268277i	−0.828406	−	0.560128i	$-0.810752\pi$
$227$	−60.8990	+	60.8990i	−0.268277	+	0.268277i	0.560128	+	0.828406i	$0.310752\pi$
$228$		0			0
$229$			138.767i			0.605971i	0.952995	+	0.302985i	$0.0979834\pi$
$229$			138.767i			0.605971i	−0.952995	+	0.302985i	$0.902017\pi$
$230$		0			0
$231$	−186.384			−0.806856
$232$		0			0
$233$	−254.656	−	254.656i	−1.09294	−	1.09294i	−0.995213	−	0.0977319i	$-0.968841\pi$
$233$	−254.656	−	254.656i	−1.09294	−	1.09294i	−0.0977319	−	0.995213i	$-0.531159\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−80.8332	+	80.8332i	−0.341068	+	0.341068i
$238$		0			0
$239$			422.788i			1.76899i	0.466553	+	0.884493i	$0.345496\pi$
$239$			422.788i			1.76899i	−0.466553	+	0.884493i	$0.654504\pi$
$240$		0			0
$241$	10.7673			0.0446778			0.0223389	−	0.999750i	$-0.492889\pi$
$241$	10.7673			0.0446778			0.0223389	+	0.999750i	$0.492889\pi$
$242$		0			0
$243$	11.0227	+	11.0227i	0.0453609	+	0.0453609i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−82.2929	+	82.2929i	−0.333169	+	0.333169i
$248$		0			0
$249$		−	205.980i		−	0.827227i
$250$		0			0
$251$	177.818			0.708440			0.354220	−	0.935162i	$-0.384746\pi$
$251$	177.818			0.708440			0.354220	+	0.935162i	$0.384746\pi$
$252$		0			0
$253$	257.171	+	257.171i	1.01649	+	1.01649i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−142.293	+	142.293i	−0.553669	+	0.553669i	−0.927498	−	0.373829i	$-0.878045\pi$
$257$	−142.293	+	142.293i	−0.553669	+	0.553669i	0.373829	+	0.927498i	$0.378045\pi$
$258$		0			0
$259$		−	117.980i		−	0.455520i
$260$		0			0
$261$	10.7878			0.0413324
$262$		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
$267$	−11.7526	+	11.7526i	−0.0440170	+	0.0440170i
$268$		0			0
$269$			38.8286i			0.144344i	0.997392	+	0.0721721i	$0.0229931\pi$
$269$			38.8286i			0.144344i	−0.997392	+	0.0721721i	$0.977007\pi$
$270$		0			0
$271$	11.5755			0.0427141			0.0213570	−	0.999772i	$-0.493201\pi$
$271$	11.5755			0.0427141			0.0213570	+	0.999772i	$0.493201\pi$
$272$		0			0
$273$	−38.6969	−	38.6969i	−0.141747	−	0.141747i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−282.499	+	282.499i	−1.01985	+	1.01985i	−0.0200549	+	0.999799i	$0.506384\pi$
$277$	−282.499	+	282.499i	−1.01985	+	1.01985i	−0.999799	+	0.0200549i	$0.993616\pi$
$278$		0			0
$279$			75.5755i			0.270880i
$280$		0			0
$281$	432.384			1.53873			0.769366	−	0.638808i	$-0.220572\pi$
$281$	432.384			1.53873			0.769366	+	0.638808i	$0.220572\pi$
$282$		0			0
$283$	−223.868	−	223.868i	−0.791054	−	0.791054i	0.190611	−	0.981666i	$-0.438953\pi$
$283$	−223.868	−	223.868i	−0.791054	−	0.791054i	−0.981666	+	0.190611i	$0.938953\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−392.050	+	392.050i	−1.36603	+	1.36603i
$288$		0			0
$289$			287.384i			0.994407i
$290$		0			0
$291$	−130.020			−0.446806
$292$		0			0
$293$	196.767	+	196.767i	0.671561	+	0.671561i	0.958076	−	0.286515i	$-0.0924968\pi$
$293$	196.767	+	196.767i	0.671561	+	0.671561i	−0.286515	+	0.958076i	$0.592497\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	43.3485	−	43.3485i	0.145954	−	0.145954i
$298$		0			0
$299$			106.788i			0.357150i
$300$		0			0
$301$	−286.384			−0.951441
$302$		0			0
$303$	−190.070	−	190.070i	−0.627295	−	0.627295i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	141.707	−	141.707i	0.461587	−	0.461587i	−0.437589	−	0.899175i	$-0.644167\pi$
$307$	141.707	−	141.707i	0.461587	−	0.461587i	0.899175	+	0.437589i	$0.144167\pi$
$308$		0			0
$309$			103.757i			0.335784i
$310$		0			0
$311$	411.555			1.32333			0.661664	−	0.749800i	$-0.269850\pi$
$311$	411.555			1.32333			0.661664	+	0.749800i	$0.269850\pi$
$312$		0			0
$313$	−365.939	−	365.939i	−1.16913	−	1.16913i	−0.982413	−	0.186720i	$-0.940214\pi$
$313$	−365.939	−	365.939i	−1.16913	−	1.16913i	−0.186720	−	0.982413i	$-0.559786\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−338.343	+	338.343i	−1.06733	+	1.06733i	−0.0697641	+	0.997564i	$0.522225\pi$
$317$	−338.343	+	338.343i	−1.06733	+	1.06733i	−0.997564	+	0.0697641i	$0.977775\pi$
$318$		0			0
$319$		−	42.4245i		−	0.132992i
$320$		0			0
$321$	−73.2122			−0.228076
$322$		0			0
$323$	30.2020	+	30.2020i	0.0935048	+	0.0935048i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	21.0556	−	21.0556i	0.0643903	−	0.0643903i
$328$		0			0
$329$		−	57.9796i		−	0.176230i
$330$		0			0
$331$	97.5551			0.294728			0.147364	−	0.989082i	$-0.452921\pi$
$331$	97.5551			0.294728			0.147364	+	0.989082i	$0.452921\pi$
$332$		0			0
$333$	27.4393	+	27.4393i	0.0824003	+	0.0824003i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−72.4949	+	72.4949i	−0.215118	+	0.215118i	−0.806438	−	0.591319i	$-0.798607\pi$
$337$	−72.4949	+	72.4949i	−0.215118	+	0.215118i	0.591319	+	0.806438i	$0.298607\pi$
$338$		0			0
$339$			328.949i			0.970351i
$340$		0			0
$341$	297.212			0.871590
$342$		0			0
$343$	95.5051	+	95.5051i	0.278441	+	0.278441i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−40.8582	+	40.8582i	−0.117747	+	0.117747i	−0.763525	−	0.645778i	$-0.776533\pi$
$347$	−40.8582	+	40.8582i	−0.117747	+	0.117747i	0.645778	+	0.763525i	$0.276533\pi$
$348$		0			0
$349$		−	332.343i		−	0.952272i	−0.879372	−	0.476136i	$-0.842037\pi$
$349$		−	332.343i		−	0.952272i	0.879372	−	0.476136i	$-0.157963\pi$
$350$		0			0
$351$	18.0000			0.0512821
$352$		0			0
$353$	−260.232	−	260.232i	−0.737200	−	0.737200i	0.234835	−	0.972035i	$-0.424545\pi$
$353$	−260.232	−	260.232i	−0.737200	−	0.737200i	−0.972035	+	0.234835i	$0.924545\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−14.2020	+	14.2020i	−0.0397816	+	0.0397816i
$358$		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
$361$	−767.686			−2.12655
$362$		0			0
$363$	−22.2804	−	22.2804i	−0.0613784	−	0.0613784i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−210.409	+	210.409i	−0.573321	+	0.573321i	−0.933055	−	0.359734i	$-0.882867\pi$
$367$	−210.409	+	210.409i	−0.573321	+	0.573321i	0.359734	+	0.933055i	$0.382867\pi$
$368$		0			0
$369$		−	182.363i		−	0.494209i
$370$		0			0
$371$	671.918			1.81110
$372$		0			0
$373$	−222.672	−	222.672i	−0.596976	−	0.596976i	0.342531	−	0.939507i	$-0.388716\pi$
$373$	−222.672	−	222.672i	−0.596976	−	0.596976i	−0.939507	+	0.342531i	$0.888716\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	8.80816	−	8.80816i	0.0233638	−	0.0233638i
$378$		0			0
$379$			222.767i			0.587777i	0.955840	+	0.293888i	$0.0949494\pi$
$379$			222.767i			0.587777i	−0.955840	+	0.293888i	$0.905051\pi$
$380$		0			0
$381$	−375.353			−0.985179
$382$		0			0
$383$	120.717	+	120.717i	0.315189	+	0.315189i	0.846916	−	0.531727i	$-0.178457\pi$
$383$	120.717	+	120.717i	0.315189	+	0.315189i	−0.531727	+	0.846916i	$0.678457\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	66.6061	−	66.6061i	0.172109	−	0.172109i
$388$		0			0
$389$			80.8082i			0.207733i	0.994591	+	0.103867i	$0.0331215\pi$
$389$			80.8082i			0.207733i	−0.994591	+	0.103867i	$0.966879\pi$
$390$		0			0
$391$	39.1918			0.100235
$392$		0			0
$393$	−240.792	−	240.792i	−0.612703	−	0.612703i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	21.9954	−	21.9954i	0.0554041	−	0.0554041i	−0.678862	−	0.734266i	$-0.737527\pi$
$397$	21.9954	−	21.9954i	0.0554041	−	0.0554041i	0.734266	+	0.678862i	$0.237527\pi$
$398$		0			0
$399$		−	530.747i		−	1.33019i
$400$		0			0
$401$	−267.131			−0.666161			−0.333081	−	0.942898i	$-0.608088\pi$
$401$	−267.131			−0.666161			−0.333081	+	0.942898i	$0.608088\pi$
$402$		0			0
$403$	61.7071	+	61.7071i	0.153119	+	0.153119i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	107.909	−	107.909i	0.265133	−	0.265133i
$408$		0			0
$409$		−	510.767i		−	1.24882i	−0.781097	−	0.624410i	$-0.785339\pi$
$409$		−	510.767i		−	1.24882i	0.781097	−	0.624410i	$-0.214661\pi$
$410$		0			0
$411$	−198.606			−0.483227
$412$		0			0
$413$	561.040	+	561.040i	1.35845	+	1.35845i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−45.5505	+	45.5505i	−0.109234	+	0.109234i
$418$		0			0
$419$			40.1204i			0.0957527i	0.998853	+	0.0478764i	$0.0152454\pi$
$419$			40.1204i			0.0957527i	−0.998853	+	0.0478764i	$0.984755\pi$
$420$		0			0
$421$	−132.343			−0.314354			−0.157177	−	0.987570i	$-0.550239\pi$
$421$	−132.343			−0.314354			−0.157177	+	0.987570i	$0.550239\pi$
$422$		0			0
$423$	13.4847	+	13.4847i	0.0318787	+	0.0318787i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−699.019	+	699.019i	−1.63705	+	1.63705i
$428$		0			0
$429$		−	70.7878i		−	0.165006i
$430$		0			0
$431$	510.343			1.18409			0.592045	−	0.805905i	$-0.298321\pi$
$431$	510.343			1.18409			0.592045	+	0.805905i	$0.298321\pi$
$432$		0			0
$433$	−443.828	−	443.828i	−1.02501	−	1.02501i	−0.999679	−	0.0253266i	$-0.991937\pi$
$433$	−443.828	−	443.828i	−1.02501	−	1.02501i	−0.0253266	−	0.999679i	$-0.508063\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−732.322	+	732.322i	−1.67580	+	1.67580i
$438$		0			0
$439$		−	306.808i		−	0.698880i	−0.936959	−	0.349440i	$-0.886372\pi$
$439$		−	306.808i		−	0.698880i	0.936959	−	0.349440i	$-0.113628\pi$
$440$		0			0
$441$	102.576			0.232598
$442$		0			0
$443$	−443.646	−	443.646i	−1.00146	−	1.00146i	−0.999999	−	0.00145911i	$-0.999536\pi$
$443$	−443.646	−	443.646i	−1.00146	−	1.00146i	−0.00145911	−	0.999999i	$-0.500464\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	106.293	−	106.293i	0.237792	−	0.237792i
$448$		0			0
$449$		−	230.727i		−	0.513868i	−0.966429	−	0.256934i	$-0.917288\pi$
$449$		−	230.727i		−	0.513868i	0.966429	−	0.256934i	$-0.0827122\pi$
$450$		0			0
$451$	−717.171			−1.59018
$452$		0			0
$453$	326.722	+	326.722i	0.721240	+	0.721240i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−27.2827	+	27.2827i	−0.0596995	+	0.0596995i	−0.736326	−	0.676627i	$-0.763441\pi$
$457$	−27.2827	+	27.2827i	−0.0596995	+	0.0596995i	0.676627	+	0.736326i	$0.263441\pi$
$458$		0			0
$459$		−	6.60612i		−	0.0143924i
$460$		0			0
$461$	−256.322			−0.556014			−0.278007	−	0.960579i	$-0.589674\pi$
$461$	−256.322			−0.556014			−0.278007	+	0.960579i	$0.589674\pi$
$462$		0			0
$463$	−173.196	−	173.196i	−0.374074	−	0.374074i	0.494884	−	0.868959i	$-0.335210\pi$
$463$	−173.196	−	173.196i	−0.374074	−	0.374074i	−0.868959	+	0.494884i	$0.835210\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−285.707	+	285.707i	−0.611793	+	0.611793i	−0.943413	−	0.331620i	$-0.892405\pi$
$467$	−285.707	+	285.707i	−0.611793	+	0.611793i	0.331620	+	0.943413i	$0.392405\pi$
$468$		0			0
$469$		−	1004.69i		−	2.14219i
$470$		0			0
$471$	498.322			1.05801
$472$		0			0
$473$	−261.939	−	261.939i	−0.553782	−	0.553782i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−156.272	+	156.272i	−0.327615	+	0.327615i
$478$		0			0
$479$			356.404i			0.744059i	0.928221	+	0.372029i	$0.121338\pi$
$479$			356.404i			0.744059i	−0.928221	+	0.372029i	$0.878662\pi$
$480$		0			0
$481$	44.8082			0.0931563
$482$		0			0
$483$	−344.363	−	344.363i	−0.712967	−	0.712967i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	55.6209	−	55.6209i	0.114211	−	0.114211i	−0.647691	−	0.761903i	$-0.724265\pi$
$487$	55.6209	−	55.6209i	0.114211	−	0.114211i	0.761903	+	0.647691i	$0.224265\pi$
$488$		0			0
$489$			71.5551i			0.146329i
$490$		0			0
$491$	−106.141			−0.216173			−0.108086	−	0.994142i	$-0.534472\pi$
$491$	−106.141			−0.216173			−0.108086	+	0.994142i	$0.534472\pi$
$492$		0			0
$493$	−3.23266	−	3.23266i	−0.00655711	−	0.00655711i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	430.747	−	430.747i	0.866694	−	0.866694i
$498$		0			0
$499$			298.363i			0.597922i	0.954265	+	0.298961i	$0.0966401\pi$
$499$			298.363i			0.597922i	−0.954265	+	0.298961i	$0.903360\pi$
$500$		0			0
$501$	−284.586			−0.568035
$502$		0			0
$503$	−655.101	−	655.101i	−1.30239	−	1.30239i	−0.926778	−	0.375610i	$-0.877433\pi$
$503$	−655.101	−	655.101i	−1.30239	−	1.30239i	−0.375610	−	0.926778i	$-0.622567\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−192.285	+	192.285i	−0.379260	+	0.379260i
$508$		0			0
$509$			683.110i			1.34206i	0.741429	+	0.671032i	$0.234149\pi$
$509$			683.110i			1.34206i	−0.741429	+	0.671032i	$0.765851\pi$
$510$		0			0
$511$	1015.11			1.98652
$512$		0			0
$513$	123.439	+	123.439i	0.240622	+	0.240622i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	53.0306	−	53.0306i	0.102574	−	0.102574i
$518$		0			0
$519$		−	147.637i		−	0.284464i
$520$		0			0
$521$	−6.76734			−0.0129891			−0.00649457	−	0.999979i	$-0.502067\pi$
$521$	−6.76734			−0.0129891			−0.00649457	+	0.999979i	$0.502067\pi$
$522$		0			0
$523$	600.990	+	600.990i	1.14912	+	1.14912i	0.986726	+	0.162394i	$0.0519216\pi$
$523$	600.990	+	600.990i	1.14912	+	1.14912i	0.162394	+	0.986726i	$0.448078\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	22.6469	−	22.6469i	0.0429733	−	0.0429733i
$528$		0			0
$529$			421.302i			0.796412i
$530$		0			0
$531$	−260.969			−0.491468
$532$		0			0
$533$	−148.899	−	148.899i	−0.279360	−	0.279360i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−215.753	+	215.753i	−0.401774	+	0.401774i
$538$		0			0
$539$		−	403.394i		−	0.748412i
$540$		0			0
$541$	177.110			0.327376			0.163688	−	0.986512i	$-0.447661\pi$
$541$	177.110			0.327376			0.163688	+	0.986512i	$0.447661\pi$
$542$		0			0
$543$	−198.359	−	198.359i	−0.365301	−	0.365301i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	35.3326	−	35.3326i	0.0645935	−	0.0645935i	−0.674072	−	0.738666i	$-0.735456\pi$
$547$	35.3326	−	35.3326i	0.0645935	−	0.0645935i	0.738666	+	0.674072i	$0.235456\pi$
$548$		0			0
$549$		−	325.151i		−	0.592261i
$550$		0			0
$551$	120.808			0.219253
$552$		0			0
$553$	−425.666	−	425.666i	−0.769740	−	0.769740i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	121.930	−	121.930i	0.218904	−	0.218904i	−0.589132	−	0.808037i	$-0.700530\pi$
$557$	121.930	−	121.930i	0.218904	−	0.218904i	0.808037	+	0.589132i	$0.200530\pi$
$558$		0			0
$559$		−	108.767i		−	0.194575i
$560$		0			0
$561$	−25.9796			−0.0463094
$562$		0			0
$563$	436.636	+	436.636i	0.775552	+	0.775552i	0.979071	−	0.203519i	$-0.0652379\pi$
$563$	436.636	+	436.636i	0.775552	+	0.775552i	−0.203519	+	0.979071i	$0.565238\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−58.0454	+	58.0454i	−0.102373	+	0.102373i
$568$		0			0
$569$		−	394.363i		−	0.693081i	−0.938035	−	0.346541i	$-0.887356\pi$
$569$		−	394.363i		−	0.693081i	0.938035	−	0.346541i	$-0.112644\pi$
$570$		0			0
$571$	−32.0612			−0.0561493			−0.0280746	−	0.999606i	$-0.508938\pi$
$571$	−32.0612			−0.0561493			−0.0280746	+	0.999606i	$0.508938\pi$
$572$		0			0
$573$	307.101	+	307.101i	0.535953	+	0.535953i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	545.848	−	545.848i	0.946010	−	0.946010i	−0.0526051	−	0.998615i	$-0.516752\pi$
$577$	545.848	−	545.848i	0.946010	−	0.946010i	0.998615	+	0.0526051i	$0.0167524\pi$
$578$		0			0
$579$		−	21.1306i		−	0.0364950i
$580$		0			0
$581$	1084.69			1.86693
$582$		0			0
$583$	614.565	+	614.565i	1.05414	+	1.05414i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	307.728	−	307.728i	0.524238	−	0.524238i	−0.394611	−	0.918848i	$-0.629121\pi$
$587$	307.728	−	307.728i	0.524238	−	0.524238i	0.918848	+	0.394611i	$0.129121\pi$
$588$		0			0
$589$			846.343i			1.43691i
$590$		0			0
$591$	−446.302			−0.755164
$592$		0			0
$593$	228.940	+	228.940i	0.386070	+	0.386070i	0.873283	−	0.487213i	$-0.161986\pi$
$593$	228.940	+	228.940i	0.386070	+	0.386070i	−0.487213	+	0.873283i	$0.661986\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−99.4393	+	99.4393i	−0.166565	+	0.166565i
$598$		0			0
$599$			345.657i			0.577057i	0.957471	+	0.288529i	$0.0931660\pi$
$599$			345.657i			0.577057i	−0.957471	+	0.288529i	$0.906834\pi$
$600$		0			0
$601$	893.069			1.48597			0.742986	−	0.669307i	$-0.233409\pi$
$601$	893.069			1.48597			0.742986	+	0.669307i	$0.233409\pi$
$602$		0			0
$603$	233.666	+	233.666i	0.387506	+	0.387506i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−560.974	+	560.974i	−0.924175	+	0.924175i	−0.997321	−	0.0731466i	$-0.976696\pi$
$607$	−560.974	+	560.974i	−0.924175	+	0.924175i	0.0731466	+	0.997321i	$0.476696\pi$
$608$		0			0
$609$			56.8082i			0.0932811i
$610$		0			0
$611$	22.0204			0.0360400
$612$		0			0
$613$	439.448	+	439.448i	0.716882	+	0.716882i	0.967965	−	0.251084i	$-0.0807869\pi$
$613$	439.448	+	439.448i	0.716882	+	0.716882i	−0.251084	+	0.967965i	$0.580787\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	199.201	−	199.201i	0.322854	−	0.322854i	−0.527007	−	0.849861i	$-0.676686\pi$
$617$	199.201	−	199.201i	0.322854	−	0.322854i	0.849861	+	0.527007i	$0.176686\pi$
$618$		0			0
$619$			121.151i			0.195721i	0.995200	+	0.0978603i	$0.0311998\pi$
$619$			121.151i			0.195721i	−0.995200	+	0.0978603i	$0.968800\pi$
$620$		0			0
$621$	160.182			0.257941
$622$		0			0
$623$	−61.8888	−	61.8888i	−0.0993399	−	0.0993399i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	485.444	−	485.444i	0.774233	−	0.774233i
$628$		0			0
$629$		−	16.4449i		−	0.0261445i
$630$		0			0
$631$	−247.494			−0.392225			−0.196112	−	0.980581i	$-0.562832\pi$
$631$	−247.494			−0.392225			−0.196112	+	0.980581i	$0.562832\pi$
$632$		0			0
$633$	244.429	+	244.429i	0.386144	+	0.386144i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	83.7526	−	83.7526i	0.131480	−	0.131480i
$638$		0			0
$639$			200.363i			0.313558i
$640$		0			0
$641$	70.6857			0.110274			0.0551371	−	0.998479i	$-0.482440\pi$
$641$	70.6857			0.110274			0.0551371	+	0.998479i	$0.482440\pi$
$642$		0			0
$643$	231.373	+	231.373i	0.359834	+	0.359834i	0.863752	−	0.503917i	$-0.168108\pi$
$643$	231.373	+	231.373i	0.359834	+	0.359834i	−0.503917	+	0.863752i	$0.668108\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	500.958	−	500.958i	0.774278	−	0.774278i	−0.204573	−	0.978851i	$-0.565581\pi$
$647$	500.958	−	500.958i	0.774278	−	0.774278i	0.978851	+	0.204573i	$0.0655805\pi$
$648$		0			0
$649$			1026.30i			1.58136i
$650$		0			0
$651$	−397.980			−0.611336
$652$		0			0
$653$	−261.031	−	261.031i	−0.399741	−	0.399741i	0.478401	−	0.878142i	$-0.341217\pi$
$653$	−261.031	−	261.031i	−0.399741	−	0.399741i	−0.878142	+	0.478401i	$0.841217\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−236.091	+	236.091i	−0.359347	+	0.359347i
$658$		0			0
$659$		−	800.524i		−	1.21476i	−0.794413	−	0.607378i	$-0.792221\pi$
$659$		−	800.524i		−	1.21476i	0.794413	−	0.607378i	$-0.207779\pi$
$660$		0			0
$661$	−313.192			−0.473815			−0.236908	−	0.971532i	$-0.576134\pi$
$661$	−313.192			−0.473815			−0.236908	+	0.971532i	$0.576134\pi$
$662$		0			0
$663$	−5.39388	−	5.39388i	−0.00813556	−	0.00813556i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	78.3837	−	78.3837i	0.117517	−	0.117517i
$668$		0			0
$669$			330.545i			0.494088i
$670$		0			0
$671$	−1278.71			−1.90567
$672$		0			0
$673$	−688.677	−	688.677i	−1.02329	−	1.02329i	−0.999722	−	0.0235713i	$-0.992496\pi$
$673$	−688.677	−	688.677i	−1.02329	−	1.02329i	−0.0235713	−	0.999722i	$-0.507504\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−660.767	+	660.767i	−0.976023	+	0.976023i	−0.999719	−	0.0236965i	$-0.992456\pi$
$677$	−660.767	+	660.767i	−0.976023	+	0.976023i	0.0236965	+	0.999719i	$0.492456\pi$
$678$		0			0
$679$		−	684.686i		−	1.00837i
$680$		0			0
$681$	149.171			0.219048
$682$		0			0
$683$	−952.232	−	952.232i	−1.39419	−	1.39419i	−0.815664	−	0.578525i	$-0.803628\pi$
$683$	−952.232	−	952.232i	−1.39419	−	1.39419i	−0.578525	−	0.815664i	$-0.696372\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	169.955	−	169.955i	0.247387	−	0.247387i
$688$		0			0
$689$			255.192i			0.370380i
$690$		0			0
$691$	−568.665			−0.822960			−0.411480	−	0.911419i	$-0.634988\pi$
$691$	−568.665			−0.822960			−0.411480	+	0.911419i	$0.634988\pi$
$692$		0			0
$693$	228.272	+	228.272i	0.329397	+	0.329397i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−54.6469	+	54.6469i	−0.0784031	+	0.0784031i
$698$		0			0
$699$			623.778i			0.892386i
$700$		0			0
$701$	−125.698			−0.179312			−0.0896562	−	0.995973i	$-0.528577\pi$
$701$	−125.698			−0.179312			−0.0896562	+	0.995973i	$0.528577\pi$
$702$		0			0
$703$	307.283	+	307.283i	0.437102	+	0.437102i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	1000.91	−	1000.91i	1.41571	−	1.41571i
$708$		0			0
$709$			201.069i			0.283596i	0.989896	+	0.141798i	$0.0452883\pi$
$709$			201.069i			0.283596i	−0.989896	+	0.141798i	$0.954712\pi$
$710$		0			0
$711$	198.000			0.278481
$712$		0			0
$713$	549.131	+	549.131i	0.770169	+	0.770169i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	517.807	−	517.807i	0.722186	−	0.722186i
$718$		0			0
$719$			785.576i			1.09259i	0.837591	+	0.546297i	$0.183963\pi$
$719$			785.576i			1.09259i	−0.837591	+	0.546297i	$0.816037\pi$
$720$		0			0
$721$	−546.384			−0.757814
$722$		0			0
$723$	−13.1872	−	13.1872i	−0.0182396	−	0.0182396i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	282.459	−	282.459i	0.388526	−	0.388526i	−0.485635	−	0.874162i	$-0.661412\pi$
$727$	282.459	−	282.459i	0.388526	−	0.388526i	0.874162	+	0.485635i	$0.161412\pi$
$728$		0			0
$729$		−	27.0000i		−	0.0370370i
$730$		0			0
$731$	−39.9184			−0.0546079
$732$		0			0
$733$	54.9036	+	54.9036i	0.0749025	+	0.0749025i	0.743566	−	0.668663i	$-0.233133\pi$
$733$	54.9036	+	54.9036i	0.0749025	+	0.0749025i	−0.668663	+	0.743566i	$0.733133\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	918.929	−	918.929i	1.24685	−	1.24685i
$738$		0			0
$739$		−	276.624i		−	0.374323i	−0.982329	−	0.187161i	$-0.940071\pi$
$739$		−	276.624i		−	0.374323i	0.982329	−	0.187161i	$-0.0599287\pi$
$740$		0			0
$741$	201.576			0.272032
$742$		0			0
$743$	143.101	+	143.101i	0.192599	+	0.192599i	0.796818	−	0.604219i	$-0.206515\pi$
$743$	143.101	+	143.101i	0.192599	+	0.192599i	−0.604219	+	0.796818i	$0.706515\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−252.272	+	252.272i	−0.337714	+	0.337714i
$748$		0			0
$749$		−	385.535i		−	0.514733i
$750$		0			0
$751$	−871.453			−1.16039			−0.580195	−	0.814478i	$-0.697024\pi$
$751$	−871.453			−1.16039			−0.580195	+	0.814478i	$0.697024\pi$
$752$		0			0
$753$	−217.782	−	217.782i	−0.289219	−	0.289219i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	304.702	−	304.702i	0.402512	−	0.402512i	−0.476605	−	0.879117i	$-0.658133\pi$
$757$	304.702	−	304.702i	0.402512	−	0.402512i	0.879117	+	0.476605i	$0.158133\pi$
$758$		0			0
$759$		−	629.939i		−	0.829959i
$760$		0			0
$761$	613.959			0.806779			0.403390	−	0.915028i	$-0.367832\pi$
$761$	613.959			0.806779			0.403390	+	0.915028i	$0.367832\pi$
$762$		0			0
$763$	110.879	+	110.879i	0.145319	+	0.145319i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−213.081	+	213.081i	−0.277810	+	0.277810i
$768$		0			0
$769$		−	887.412i		−	1.15398i	−0.816750	−	0.576991i	$-0.804227\pi$
$769$		−	887.412i		−	1.15398i	0.816750	−	0.576991i	$-0.195773\pi$
$770$		0			0
$771$	348.545			0.452069
$772$		0			0
$773$	605.757	+	605.757i	0.783644	+	0.783644i	0.980444	−	0.196799i	$-0.0630548\pi$
$773$	605.757	+	605.757i	0.783644	+	0.783644i	−0.196799	+	0.980444i	$0.563055\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−144.495	+	144.495i	−0.185965	+	0.185965i
$778$		0			0
$779$		−	2042.22i		−	2.62159i
$780$		0			0
$781$	787.959			1.00891
$782$		0			0
$783$	−13.2122	−	13.2122i	−0.0168739	−	0.0168739i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	616.463	−	616.463i	0.783308	−	0.783308i	−0.197080	−	0.980387i	$-0.563146\pi$
$787$	616.463	−	616.463i	0.783308	−	0.783308i	0.980387	+	0.197080i	$0.0631458\pi$
$788$		0			0
$789$		−	88.1816i		−	0.111764i
$790$		0			0
$791$	−1732.24			−2.18994
$792$		0			0
$793$	−265.485	−	265.485i	−0.334785	−	0.334785i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−247.768	+	247.768i	−0.310876	+	0.310876i	−0.845249	−	0.534373i	$-0.820548\pi$
$797$	−247.768	+	247.768i	−0.310876	+	0.310876i	0.534373	+	0.845249i	$0.320548\pi$
$798$		0			0
$799$		−	8.08164i		−	0.0101147i
$800$		0			0
$801$	28.7878			0.0359398
$802$		0			0
$803$	928.463	+	928.463i	1.15624	+	1.15624i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	47.5551	−	47.5551i	0.0589282	−	0.0589282i
$808$		0			0
$809$			1556.10i			1.92348i	0.273958	+	0.961742i	$0.411667\pi$
$809$			1556.10i			1.92348i	−0.273958	+	0.961742i	$0.588333\pi$
$810$		0			0
$811$	−277.192			−0.341790			−0.170895	−	0.985289i	$-0.554666\pi$
$811$	−277.192			−0.341790			−0.170895	+	0.985289i	$0.554666\pi$
$812$		0			0
$813$	−14.1770	−	14.1770i	−0.0174379	−	0.0174379i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	745.898	−	745.898i	0.912972	−	0.912972i
$818$		0			0
$819$			94.7878i			0.115736i
$820$		0			0
$821$	−1485.86			−1.80981			−0.904907	−	0.425610i	$-0.860060\pi$
$821$	−1485.86			−1.80981			−0.904907	+	0.425610i	$0.860060\pi$
$822$		0			0
$823$	809.519	+	809.519i	0.983620	+	0.983620i	0.999868	−	0.0162485i	$-0.00517228\pi$
$823$	809.519	+	809.519i	0.983620	+	0.983620i	−0.0162485	+	0.999868i	$0.505172\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1122.66	+	1122.66i	−1.35750	+	1.35750i	−0.480521	+	0.876983i	$0.659553\pi$
$827$	−1122.66	+	1122.66i	−1.35750	+	1.35750i	−0.876983	+	0.480521i	$0.840447\pi$
$828$		0			0
$829$			849.151i			1.02431i	0.858894	+	0.512154i	$0.171152\pi$
$829$			849.151i			1.02431i	−0.858894	+	0.512154i	$0.828848\pi$
$830$		0			0
$831$	691.980			0.832707
$832$		0			0
$833$	−30.7378	−	30.7378i	−0.0369001	−	0.0369001i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	92.5607	−	92.5607i	0.110586	−	0.110586i
$838$		0			0
$839$			1606.26i			1.91449i	0.289270	+	0.957247i	$0.406587\pi$
$839$			1606.26i			1.91449i	−0.289270	+	0.957247i	$0.593413\pi$
$840$		0			0
$841$	828.069			0.984625
$842$		0			0
$843$	−529.560	−	529.560i	−0.628185	−	0.628185i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	117.328	−	117.328i	0.138522	−	0.138522i
$848$		0			0
$849$			548.363i			0.645893i
$850$		0			0
$851$	398.747			0.468563
$852$		0			0
$853$	632.702	+	632.702i	0.741737	+	0.741737i	0.972912	−	0.231175i	$-0.0742571\pi$
$853$	632.702	+	632.702i	0.741737	+	0.741737i	−0.231175	+	0.972912i	$0.574257\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	258.879	−	258.879i	0.302075	−	0.302075i	−0.539750	−	0.841825i	$-0.681481\pi$
$857$	258.879	−	258.879i	0.302075	−	0.302075i	0.841825	+	0.539750i	$0.181481\pi$
$858$		0			0
$859$			534.082i			0.621748i	0.950451	+	0.310874i	$0.100622\pi$
$859$			534.082i			0.621748i	−0.950451	+	0.310874i	$0.899378\pi$
$860$		0			0
$861$	960.322			1.11536
$862$		0			0
$863$	711.878	+	711.878i	0.824887	+	0.824887i	0.986804	−	0.161917i	$-0.0517678\pi$
$863$	711.878	+	711.878i	0.824887	+	0.824887i	−0.161917	+	0.986804i	$0.551768\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	351.972	−	351.972i	0.405965	−	0.405965i
$868$		0			0
$869$		−	778.665i		−	0.896048i
$870$		0			0
$871$	381.576			0.438089
$872$		0			0
$873$	159.242	+	159.242i	0.182408	+	0.182408i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	742.590	−	742.590i	0.846739	−	0.846739i	−0.142986	−	0.989725i	$-0.545670\pi$
$877$	742.590	−	742.590i	0.846739	−	0.846739i	0.989725	+	0.142986i	$0.0456702\pi$
$878$		0			0
$879$		−	481.980i		−	0.548327i
$880$		0			0
$881$	−1000.75			−1.13592			−0.567961	−	0.823056i	$-0.692267\pi$
$881$	−1000.75			−1.13592			−0.567961	+	0.823056i	$0.692267\pi$
$882$		0			0
$883$	−546.788	−	546.788i	−0.619239	−	0.619239i	0.326097	−	0.945336i	$-0.394266\pi$
$883$	−546.788	−	546.788i	−0.619239	−	0.619239i	−0.945336	+	0.326097i	$0.894266\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1138.48	−	1138.48i	1.28352	−	1.28352i	0.344872	−	0.938650i	$-0.387922\pi$
$887$	1138.48	−	1138.48i	1.28352	−	1.28352i	0.938650	−	0.344872i	$-0.112078\pi$
$888$		0			0
$889$		−	1976.60i		−	2.22340i
$890$		0			0
$891$	−106.182			−0.119171
$892$		0			0
$893$	151.010	+	151.010i	0.169104	+	0.169104i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	130.788	−	130.788i	0.145806	−	0.145806i
$898$		0			0
$899$		−	90.5878i		−	0.100765i
$900$		0			0
$901$	93.6571			0.103948
$902$		0			0
$903$	350.747	+	350.747i	0.388424	+	0.388424i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−162.706	+	162.706i	−0.179389	+	0.179389i	−0.791090	−	0.611700i	$-0.790486\pi$
$907$	−162.706	+	162.706i	−0.179389	+	0.179389i	0.611700	+	0.791090i	$0.290486\pi$
$908$		0			0
$909$			465.576i			0.512184i
$910$		0			0
$911$	386.988			0.424794			0.212397	−	0.977183i	$-0.431873\pi$
$911$	386.988			0.424794			0.212397	+	0.977183i	$0.431873\pi$
$912$		0			0
$913$	992.100	+	992.100i	1.08664	+	1.08664i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	1268.01	−	1268.01i	1.38278	−	1.38278i
$918$		0			0
$919$			1101.80i			1.19891i	0.800409	+	0.599454i	$0.204615\pi$
$919$			1101.80i			1.19891i	−0.800409	+	0.599454i	$0.795385\pi$
$920$		0			0
$921$	−347.110			−0.376884
$922$		0			0
$923$	163.596	+	163.596i	0.177244	+	0.177244i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	127.076	−	127.076i	0.137083	−	0.137083i
$928$		0			0
$929$		−	261.273i		−	0.281242i	−0.990064	−	0.140621i	$-0.955090\pi$
$929$		−	261.273i		−	0.281242i	0.990064	−	0.140621i	$-0.0449098\pi$
$930$		0			0
$931$	1148.71			1.23384
$932$		0			0
$933$	−504.050	−	504.050i	−0.540247	−	0.540247i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−57.7980	+	57.7980i	−0.0616841	+	0.0616841i	−0.737276	−	0.675592i	$-0.763888\pi$
$937$	−57.7980	+	57.7980i	−0.0616841	+	0.0616841i	0.675592	+	0.737276i	$0.263888\pi$
$938$		0			0
$939$			896.363i			0.954593i
$940$		0			0
$941$	399.837			0.424906			0.212453	−	0.977171i	$-0.431855\pi$
$941$	399.837			0.424906			0.212453	+	0.977171i	$0.431855\pi$
$942$		0			0
$943$	−1325.05	−	1325.05i	−1.40514	−	1.40514i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	464.958	−	464.958i	0.490980	−	0.490980i	−0.417635	−	0.908615i	$-0.637141\pi$
$947$	464.958	−	464.958i	0.490980	−	0.490980i	0.908615	+	0.417635i	$0.137141\pi$
$948$		0			0
$949$			385.535i			0.406254i
$950$		0			0
$951$	828.767			0.871469
$952$		0			0
$953$	627.242	+	627.242i	0.658176	+	0.658176i	0.954948	−	0.296772i	$-0.0959102\pi$
$953$	627.242	+	627.242i	0.658176	+	0.658176i	−0.296772	+	0.954948i	$0.595910\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−51.9592	+	51.9592i	−0.0542938	+	0.0542938i
$958$		0			0
$959$		−	1045.86i		−	1.09057i
$960$		0			0
$961$	−326.371			−0.339616
$962$		0			0
$963$	89.6663	+	89.6663i	0.0931114	+	0.0931114i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	1334.02	−	1334.02i	1.37955	−	1.37955i	0.534173	−	0.845375i	$-0.320623\pi$
$967$	1334.02	−	1334.02i	1.37955	−	1.37955i	0.845375	−	0.534173i	$-0.179377\pi$
$968$		0			0
$969$		−	73.9796i		−	0.0763463i
$970$		0			0
$971$	1489.17			1.53365			0.766823	−	0.641859i	$-0.221837\pi$
$971$	1489.17			1.53365			0.766823	+	0.641859i	$0.221837\pi$
$972$		0			0
$973$	−239.868	−	239.868i	−0.246525	−	0.246525i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−1134.66	+	1134.66i	−1.16137	+	1.16137i	−0.177191	+	0.984176i	$0.556701\pi$
$977$	−1134.66	+	1134.66i	−1.16137	+	1.16137i	−0.984176	+	0.177191i	$0.943299\pi$
$978$		0			0
$979$		−	113.212i		−	0.115641i
$980$		0			0
$981$	−51.5755			−0.0525744
$982$		0			0
$983$	261.485	+	261.485i	0.266007	+	0.266007i	0.827489	−	0.561482i	$-0.189769\pi$
$983$	261.485	+	261.485i	0.266007	+	0.266007i	−0.561482	+	0.827489i	$0.689769\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−71.0102	+	71.0102i	−0.0719455	+	0.0719455i
$988$		0			0
$989$		−	967.918i		−	0.978684i
$990$		0			0
$991$	950.645			0.959278			0.479639	−	0.877466i	$-0.340768\pi$
$991$	950.645			0.959278			0.479639	+	0.877466i	$0.340768\pi$
$992$		0			0
$993$	−119.480	−	119.480i	−0.120322	−	0.120322i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−96.0566	+	96.0566i	−0.0963457	+	0.0963457i	−0.753637	−	0.657291i	$-0.771702\pi$
$997$	−96.0566	+	96.0566i	−0.0963457	+	0.0963457i	0.657291	+	0.753637i	$0.271702\pi$
$998$		0			0
$999$		−	67.2122i		−	0.0672795i

Twists

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

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

\(f(q)\)	\(=\)	\(q+(-1.22474 - 1.22474i) q^{3} +(6.44949 - 6.44949i) q^{7} +3.00000i q^{9} +O(q^{10})\)	\(q+(-1.22474 - 1.22474i) q^{3} +(6.44949 - 6.44949i) q^{7} +3.00000i q^{9} +11.7980 q^{11} +(2.44949 + 2.44949i) q^{13} +(0.898979 - 0.898979i) q^{17} +33.5959i q^{19} -15.7980 q^{21} +(21.7980 + 21.7980i) q^{23} +(3.67423 - 3.67423i) q^{27} -3.59592i q^{29} +25.1918 q^{31} +(-14.4495 - 14.4495i) q^{33} +(9.14643 - 9.14643i) q^{37} -6.00000i q^{39} -60.7878 q^{41} +(-22.2020 - 22.2020i) q^{43} +(4.49490 - 4.49490i) q^{47} -34.1918i q^{49} -2.20204 q^{51} +(52.0908 + 52.0908i) q^{53} +(41.1464 - 41.1464i) q^{57} +86.9898i q^{59} -108.384 q^{61} +(19.3485 + 19.3485i) q^{63} +(77.8888 - 77.8888i) q^{67} -53.3939i q^{69} +66.7878 q^{71} +(78.6969 + 78.6969i) q^{73} +(76.0908 - 76.0908i) q^{77} -66.0000i q^{79} -9.00000 q^{81} +(84.0908 + 84.0908i) q^{83} +(-4.40408 + 4.40408i) q^{87} -9.59592i q^{89} +31.5959 q^{91} +(-30.8536 - 30.8536i) q^{93} +(53.0806 - 53.0806i) q^{97} +35.3939i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 16 q^{7}+O(q^{10}) \) 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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more