LMFDB - Embedded newform 684.3.y.h.145.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [684,3,Mod(145,684)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(684, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 5]))

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

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

chi := DirichletCharacter("684.145");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 684 = 2^{2} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	684.y (of order $6$, degree $2$, 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:	$18.6376500822$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 56x^{6} - 154x^{5} + 917x^{4} - 1582x^{3} + 4294x^{2} - 3528x + 4971 $ x^8 - 4x^7 + 56x^6 - 154x^5 + 917x^4 - 1582x^3 + 4294x^2 - 3528*x + 4971
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 76)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			145.3
Root			$0.500000 - 4.68383i$ of defining polynomial
Character	$\chi$	$=$	684.145
Dual form			684.3.y.h.217.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.55796 + 2.69846i) q^{5} -11.1883 q^{7} +O(q^{10})$	$q+(1.55796 + 2.69846i) q^{5} -11.1883 q^{7} +5.69156 q^{11} +(-6.74507 - 3.89427i) q^{13} +(-9.15210 - 15.8519i) q^{17} +(17.9748 + 6.15688i) q^{19} +(17.0743 - 29.5736i) q^{23} +(7.64552 - 13.2424i) q^{25} +(33.1640 + 19.1473i) q^{29} -10.4564i q^{31} +(-17.4309 - 30.1912i) q^{35} +30.1044i q^{37} +(33.0491 - 19.0809i) q^{41} +(-29.7055 - 51.4514i) q^{43} +(30.9846 - 53.6669i) q^{47} +76.1775 q^{49} +(45.3846 + 26.2028i) q^{53} +(8.86722 + 15.3585i) q^{55} +(-73.7726 + 42.5926i) q^{59} +(1.82516 - 3.16127i) q^{61} -24.2685i q^{65} +(-96.7411 - 55.8535i) q^{67} +(55.4406 - 32.0086i) q^{71} +(37.7842 + 65.4442i) q^{73} -63.6787 q^{77} +(85.4278 - 49.3218i) q^{79} -60.0975 q^{83} +(28.5172 - 49.3932i) q^{85} +(-17.4003 - 10.0461i) q^{89} +(75.4657 + 43.5701i) q^{91} +(11.3899 + 58.0965i) q^{95} +(-50.1828 + 28.9730i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + q^{5} - 12 q^{7}+O(q^{10}) $ 8 * q + q^5 - 12 * q^7	$ 8 q + q^{5} - 12 q^{7} + 10 q^{11} + 9 q^{13} - 23 q^{17} - 33 q^{19} + 31 q^{23} - 73 q^{25} + 105 q^{29} + 68 q^{35} - 18 q^{41} - 41 q^{43} - 107 q^{47} + 312 q^{49} - 39 q^{53} + 70 q^{55} - 348 q^{59} - 45 q^{61} - 432 q^{67} + 243 q^{71} + 16 q^{73} - 544 q^{77} + 75 q^{79} + 82 q^{83} + 109 q^{85} + 213 q^{89} + 222 q^{91} + 385 q^{95} + 144 q^{97}+O(q^{100}) $ 8 * q + q^5 - 12 * q^7 + 10 * q^11 + 9 * q^13 - 23 * q^17 - 33 * q^19 + 31 * q^23 - 73 * q^25 + 105 * q^29 + 68 * q^35 - 18 * q^41 - 41 * q^43 - 107 * q^47 + 312 * q^49 - 39 * q^53 + 70 * q^55 - 348 * q^59 - 45 * q^61 - 432 * q^67 + 243 * q^71 + 16 * q^73 - 544 * q^77 + 75 * q^79 + 82 * q^83 + 109 * q^85 + 213 * q^89 + 222 * q^91 + 385 * q^95 + 144 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/684\mathbb{Z}\right)^\times$.

$n$	$325$	$343$	$533$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$1$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	1.55796	+	2.69846i	0.311592	+	0.539693i	0.978707	−	0.205262i	$-0.0658045\pi$
$5$	1.55796	+	2.69846i	0.311592	+	0.539693i	−0.667115	+	0.744955i	$0.732471\pi$
$6$		0			0
$7$	−11.1883			−1.59832			−0.799162	−	0.601115i	$-0.794723\pi$
$7$	−11.1883			−1.59832			−0.799162	+	0.601115i	$0.794723\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	5.69156			0.517414			0.258707	−	0.965956i	$-0.416704\pi$
$11$	5.69156			0.517414			0.258707	+	0.965956i	$0.416704\pi$
$12$		0			0
$13$	−6.74507	−	3.89427i	−0.518852	−	0.299559i	0.217613	−	0.976035i	$-0.430173\pi$
$13$	−6.74507	−	3.89427i	−0.518852	−	0.299559i	−0.736465	+	0.676476i	$0.763506\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−9.15210	−	15.8519i	−0.538359	−	0.932464i	−0.998993	−	0.0448743i	$-0.985711\pi$
$17$	−9.15210	−	15.8519i	−0.538359	−	0.932464i	0.460634	−	0.887590i	$-0.347622\pi$
$18$		0			0
$19$	17.9748	+	6.15688i	0.946041	+	0.324046i
$19$	17.9748	+	6.15688i	0.946041	+	0.324046i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	17.0743	−	29.5736i	0.742362	−	1.28581i	−0.209055	−	0.977904i	$-0.567039\pi$
$23$	17.0743	−	29.5736i	0.742362	−	1.28581i	0.951417	−	0.307905i	$-0.0996280\pi$
$24$		0			0
$25$	7.64552	−	13.2424i	0.305821	−	0.529697i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	33.1640	+	19.1473i	1.14359	+	0.660250i	0.947316	−	0.320300i	$-0.103784\pi$
$29$	33.1640	+	19.1473i	1.14359	+	0.660250i	0.196271	+	0.980550i	$0.437117\pi$
$30$		0			0
$31$		−	10.4564i		−	0.337303i	−0.985676	−	0.168651i	$-0.946059\pi$
$31$		−	10.4564i		−	0.337303i	0.985676	−	0.168651i	$-0.0539412\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−17.4309	−	30.1912i	−0.498025	−	0.862605i
$36$		0			0
$37$			30.1044i			0.813633i	0.913510	+	0.406816i	$0.133361\pi$
$37$			30.1044i			0.813633i	−0.913510	+	0.406816i	$0.866639\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	33.0491	−	19.0809i	0.806076	−	0.465388i	−0.0395153	−	0.999219i	$-0.512581\pi$
$41$	33.0491	−	19.0809i	0.806076	−	0.465388i	0.845591	+	0.533831i	$0.179248\pi$
$42$		0			0
$43$	−29.7055	−	51.4514i	−0.690825	−	1.19654i	−0.971568	−	0.236761i	$-0.923914\pi$
$43$	−29.7055	−	51.4514i	−0.690825	−	1.19654i	0.280743	−	0.959783i	$-0.409419\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	30.9846	−	53.6669i	0.659246	−	1.14185i	−0.321565	−	0.946888i	$-0.604209\pi$
$47$	30.9846	−	53.6669i	0.659246	−	1.14185i	0.980811	−	0.194961i	$-0.0624580\pi$
$48$		0			0
$49$	76.1775			1.55464
$50$		0			0
$51$		0			0
$52$		0			0
$53$	45.3846	+	26.2028i	0.856313	+	0.494392i	0.862776	−	0.505587i	$-0.168724\pi$
$53$	45.3846	+	26.2028i	0.856313	+	0.494392i	−0.00646316	+	0.999979i	$0.502057\pi$
$54$		0			0
$55$	8.86722	+	15.3585i	0.161222	+	0.279245i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−73.7726	+	42.5926i	−1.25038	+	0.721909i	−0.971185	−	0.238325i	$-0.923402\pi$
$59$	−73.7726	+	42.5926i	−1.25038	+	0.721909i	−0.279197	+	0.960234i	$0.590068\pi$
$60$		0			0
$61$	1.82516	−	3.16127i	0.0299206	−	0.0518240i	−0.850677	−	0.525688i	$-0.823808\pi$
$61$	1.82516	−	3.16127i	0.0299206	−	0.0518240i	0.880598	+	0.473864i	$0.157141\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	24.2685i		−	0.373361i
$66$		0			0
$67$	−96.7411	−	55.8535i	−1.44390	−	0.833635i	−0.445791	−	0.895137i	$-0.647077\pi$
$67$	−96.7411	−	55.8535i	−1.44390	−	0.833635i	−0.998107	+	0.0615024i	$0.980411\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	55.4406	−	32.0086i	0.780853	−	0.450826i	−0.0558793	−	0.998438i	$-0.517796\pi$
$71$	55.4406	−	32.0086i	0.780853	−	0.450826i	0.836733	+	0.547612i	$0.184463\pi$
$72$		0			0
$73$	37.7842	+	65.4442i	0.517592	+	0.896496i	0.999791	+	0.0204344i	$0.00650493\pi$
$73$	37.7842	+	65.4442i	0.517592	+	0.896496i	−0.482199	+	0.876062i	$0.660162\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−63.6787			−0.826996
$78$		0			0
$79$	85.4278	−	49.3218i	1.08137	−	0.624326i	0.150101	−	0.988671i	$-0.452040\pi$
$79$	85.4278	−	49.3218i	1.08137	−	0.624326i	0.931264	+	0.364344i	$0.118707\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−60.0975			−0.724067			−0.362033	−	0.932165i	$-0.617917\pi$
$83$	−60.0975			−0.724067			−0.362033	+	0.932165i	$0.617917\pi$
$84$		0			0
$85$	28.5172	−	49.3932i	0.335496	−	0.581097i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−17.4003	−	10.0461i	−0.195509	−	0.112877i	0.399050	−	0.916929i	$-0.369340\pi$
$89$	−17.4003	−	10.0461i	−0.195509	−	0.112877i	−0.594559	+	0.804052i	$0.702673\pi$
$90$		0			0
$91$	75.4657	+	43.5701i	0.829293	+	0.478793i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	11.3899	+	58.0965i	0.119893	+	0.611542i
$96$		0			0
$97$	−50.1828	+	28.9730i	−0.517348	+	0.298691i	−0.735849	−	0.677146i	$-0.763217\pi$
$97$	−50.1828	+	28.9730i	−0.517348	+	0.298691i	0.218501	+	0.975837i	$0.429883\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	64.2010	−	111.199i	0.635654	−	1.10098i	−0.350723	−	0.936479i	$-0.614064\pi$
$101$	64.2010	−	111.199i	0.635654	−	1.10098i	0.986376	−	0.164505i	$-0.0526027\pi$
$102$		0			0
$103$		−	192.337i		−	1.86735i	−0.358123	−	0.933674i	$-0.616583\pi$
$103$		−	192.337i		−	1.86735i	0.358123	−	0.933674i	$-0.383417\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	81.0841i		−	0.757795i	−0.925439	−	0.378898i	$-0.876303\pi$
$107$		−	81.0841i		−	0.757795i	0.925439	−	0.378898i	$-0.123697\pi$
$108$		0			0
$109$	3.27848	−	1.89283i	0.0300778	−	0.0173655i	−0.484886	−	0.874578i	$-0.661139\pi$
$109$	3.27848	−	1.89283i	0.0300778	−	0.0173655i	0.514964	+	0.857212i	$0.327805\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	149.144i		−	1.31986i	−0.751328	−	0.659929i	$-0.770586\pi$
$113$		−	149.144i		−	1.31986i	0.751328	−	0.659929i	$-0.229414\pi$
$114$		0			0
$115$	106.404			0.925256
$116$		0			0
$117$		0			0
$118$		0			0
$119$	102.396	+	177.355i	0.860472	+	1.49038i
$120$		0			0
$121$	−88.6062			−0.732282
$122$		0			0
$123$		0			0
$124$		0			0
$125$	125.544			1.00435
$126$		0			0
$127$	19.5249	+	11.2727i	0.153739	+	0.0887614i	0.574896	−	0.818226i	$-0.305042\pi$
$127$	19.5249	+	11.2727i	0.153739	+	0.0887614i	−0.421157	+	0.906988i	$0.638376\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−75.0267	−	129.950i	−0.572723	−	0.991985i	−0.996285	−	0.0861177i	$-0.972554\pi$
$131$	−75.0267	−	129.950i	−0.572723	−	0.991985i	0.423562	−	0.905867i	$-0.360779\pi$
$132$		0			0
$133$	−201.107	−	68.8848i	−1.51208	−	0.517931i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	36.0632	−	62.4633i	0.263235	−	0.455936i	−0.703865	−	0.710334i	$-0.748544\pi$
$137$	36.0632	−	62.4633i	0.263235	−	0.455936i	0.967100	+	0.254398i	$0.0818772\pi$
$138$		0			0
$139$	−63.7916	+	110.490i	−0.458932	+	0.794894i	−0.998905	−	0.0467887i	$-0.985101\pi$
$139$	−63.7916	+	110.490i	−0.458932	+	0.794894i	0.539973	+	0.841683i	$0.318435\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−38.3900	−	22.1645i	−0.268461	−	0.154996i
$144$		0			0
$145$			119.323i			0.822914i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	107.009	+	185.345i	0.718180	+	1.24392i	0.961720	+	0.274033i	$0.0883578\pi$
$149$	107.009	+	185.345i	0.718180	+	1.24392i	−0.243541	+	0.969891i	$0.578309\pi$
$150$		0			0
$151$			67.1874i			0.444950i	0.974938	+	0.222475i	$0.0714135\pi$
$151$			67.1874i			0.444950i	−0.974938	+	0.222475i	$0.928586\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	28.2162	−	16.2906i	0.182040	−	0.105101i
$156$		0			0
$157$	110.750	+	191.824i	0.705411	+	1.22181i	0.966543	+	0.256505i	$0.0825711\pi$
$157$	110.750	+	191.824i	0.705411	+	1.22181i	−0.261131	+	0.965303i	$0.584096\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−191.032	+	330.878i	−1.18654	+	2.05514i
$162$		0			0
$163$	−126.587			−0.776606			−0.388303	−	0.921532i	$-0.626939\pi$
$163$	−126.587			−0.776606			−0.388303	+	0.921532i	$0.626939\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	161.012	+	92.9605i	0.964146	+	0.556650i	0.897447	−	0.441123i	$-0.145420\pi$
$167$	161.012	+	92.9605i	0.964146	+	0.556650i	0.0666994	+	0.997773i	$0.478753\pi$
$168$		0			0
$169$	−54.1693	−	93.8240i	−0.320529	−	0.555172i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−235.750	+	136.110i	−1.36272	+	0.786765i	−0.989985	−	0.141174i	$-0.954912\pi$
$173$	−235.750	+	136.110i	−1.36272	+	0.786765i	−0.372732	+	0.927939i	$0.621579\pi$
$174$		0			0
$175$	−85.5402	+	148.160i	−0.488801	+	0.846629i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			23.6065i			0.131880i	0.997824	+	0.0659400i	$0.0210046\pi$
$179$			23.6065i			0.131880i	−0.997824	+	0.0659400i	$0.978995\pi$
$180$		0			0
$181$	49.3722	+	28.5050i	0.272774	+	0.157486i	0.630148	−	0.776475i	$-0.282994\pi$
$181$	49.3722	+	28.5050i	0.272774	+	0.157486i	−0.357373	+	0.933962i	$0.616328\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−81.2357	+	46.9015i	−0.439112	+	0.253521i
$186$		0			0
$187$	−52.0897	−	90.2220i	−0.278554	−	0.482471i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−158.384			−0.829238			−0.414619	−	0.909995i	$-0.636085\pi$
$191$	−158.384			−0.829238			−0.414619	+	0.909995i	$0.636085\pi$
$192$		0			0
$193$	160.689	−	92.7739i	0.832586	−	0.480694i	−0.0221514	−	0.999755i	$-0.507052\pi$
$193$	160.689	−	92.7739i	0.832586	−	0.480694i	0.854737	+	0.519061i	$0.173718\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−19.8590			−0.100807			−0.0504035	−	0.998729i	$-0.516051\pi$
$197$	−19.8590			−0.100807			−0.0504035	+	0.998729i	$0.516051\pi$
$198$		0			0
$199$	−67.6494	+	117.172i	−0.339947	+	0.588805i	−0.984422	−	0.175820i	$-0.943742\pi$
$199$	−67.6494	+	117.172i	−0.339947	+	0.588805i	0.644475	+	0.764625i	$0.277076\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−371.048	−	214.225i	−1.82782	−	1.05529i
$204$		0			0
$205$	102.978	+	59.4546i	0.502333	+	0.290022i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	102.305	+	35.0422i	0.489495	+	0.167666i
$210$		0			0
$211$	−94.9925	+	54.8439i	−0.450201	+	0.259924i	−0.707915	−	0.706297i	$-0.750364\pi$
$211$	−94.9925	+	54.8439i	−0.450201	+	0.259924i	0.257714	+	0.966221i	$0.417031\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	92.5598	−	160.318i	0.430511	−	0.745666i
$216$		0			0
$217$			116.989i			0.539119i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			142.563i			0.645081i
$222$		0			0
$223$	44.5718	−	25.7335i	0.199874	−	0.115397i	−0.396723	−	0.917938i	$-0.629853\pi$
$223$	44.5718	−	25.7335i	0.199874	−	0.115397i	0.596597	+	0.802541i	$0.296519\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			153.761i			0.677361i	0.940901	+	0.338681i	$0.109981\pi$
$227$			153.761i			0.677361i	−0.940901	+	0.338681i	$0.890019\pi$
$228$		0			0
$229$	−58.5939			−0.255869			−0.127934	−	0.991783i	$-0.540835\pi$
$229$	−58.5939			−0.255869			−0.127934	+	0.991783i	$0.540835\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	140.348	+	243.089i	0.602351	+	1.04330i	0.992464	+	0.122535i	$0.0391024\pi$
$233$	140.348	+	243.089i	0.602351	+	1.04330i	−0.390113	+	0.920767i	$0.627564\pi$
$234$		0			0
$235$	193.091			0.821663
$236$		0			0
$237$		0			0
$238$		0			0
$239$	46.5330			0.194699			0.0973494	−	0.995250i	$-0.468964\pi$
$239$	46.5330			0.194699			0.0973494	+	0.995250i	$0.468964\pi$
$240$		0			0
$241$	−217.033	−	125.304i	−0.900553	−	0.519934i	−0.0231734	−	0.999731i	$-0.507377\pi$
$241$	−217.033	−	125.304i	−0.900553	−	0.519934i	−0.877380	+	0.479797i	$0.840710\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	118.681	+	205.562i	0.484414	+	0.839029i
$246$		0			0
$247$	−97.2647	−	111.527i	−0.393784	−	0.451527i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	54.5360	−	94.4592i	0.217275	−	0.376331i	−0.736699	−	0.676221i	$-0.763617\pi$
$251$	54.5360	−	94.4592i	0.217275	−	0.376331i	0.953974	+	0.299890i	$0.0969498\pi$
$252$		0			0
$253$	97.1796	−	168.320i	0.384109	−	0.665296i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−103.211	−	59.5890i	−0.401600	−	0.231864i	0.285574	−	0.958357i	$-0.407816\pi$
$257$	−103.211	−	59.5890i	−0.401600	−	0.231864i	−0.687174	+	0.726493i	$0.741149\pi$
$258$		0			0
$259$		−	336.816i		−	1.30045i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−229.768	−	397.969i	−0.873641	−	1.51319i	−0.858203	−	0.513311i	$-0.828419\pi$
$263$	−229.768	−	397.969i	−0.873641	−	1.51319i	−0.0154385	−	0.999881i	$-0.504914\pi$
$264$		0			0
$265$			163.292i			0.616195i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−296.027	+	170.911i	−1.10047	+	0.635358i	−0.936345	−	0.351082i	$-0.885814\pi$
$269$	−296.027	+	170.911i	−1.10047	+	0.635358i	−0.164127	+	0.986439i	$0.552481\pi$
$270$		0			0
$271$	−58.9909	−	102.175i	−0.217679	−	0.377031i	0.736419	−	0.676526i	$-0.236515\pi$
$271$	−58.9909	−	102.175i	−0.217679	−	0.377031i	−0.954098	+	0.299495i	$0.903182\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	43.5150	−	75.3701i	0.158236	−	0.274073i
$276$		0			0
$277$	−181.314			−0.654563			−0.327281	−	0.944927i	$-0.606133\pi$
$277$	−181.314			−0.654563			−0.327281	+	0.944927i	$0.606133\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	196.670	+	113.547i	0.699892	+	0.404083i	0.807307	−	0.590131i	$-0.200924\pi$
$281$	196.670	+	113.547i	0.699892	+	0.404083i	−0.107415	+	0.994214i	$0.534257\pi$
$282$		0			0
$283$	162.854	+	282.071i	0.575455	+	0.996718i	0.995992	+	0.0894423i	$0.0285085\pi$
$283$	162.854	+	282.071i	0.575455	+	0.996718i	−0.420537	+	0.907276i	$0.638158\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−369.763	+	213.482i	−1.28837	+	0.743841i
$288$		0			0
$289$	−23.0217	+	39.8748i	−0.0796599	+	0.137975i
$290$		0			0
$291$		0			0
$292$		0			0
$293$			51.8203i			0.176861i	0.996082	+	0.0884305i	$0.0281851\pi$
$293$			51.8203i			0.176861i	−0.996082	+	0.0884305i	$0.971815\pi$
$294$		0			0
$295$	−229.869	−	132.715i	−0.779218	−	0.449882i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−230.335	+	132.984i	−0.770352	+	0.444763i
$300$		0			0
$301$	332.353	+	575.652i	1.10416	+	1.91246i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	11.3741			0.0372921
$306$		0			0
$307$	−9.83046	+	5.67562i	−0.0320211	+	0.0184874i	−0.515925	−	0.856634i	$-0.672552\pi$
$307$	−9.83046	+	5.67562i	−0.0320211	+	0.0184874i	0.483904	+	0.875121i	$0.339218\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	120.753			0.388274			0.194137	−	0.980974i	$-0.437809\pi$
$311$	120.753			0.388274			0.194137	+	0.980974i	$0.437809\pi$
$312$		0			0
$313$	64.8317	−	112.292i	0.207130	−	0.358760i	−0.743679	−	0.668537i	$-0.766921\pi$
$313$	64.8317	−	112.292i	0.207130	−	0.358760i	0.950809	+	0.309777i	$0.100254\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	438.037	+	252.901i	1.38182	+	0.797794i	0.992375	−	0.123255i	$-0.0393333\pi$
$317$	438.037	+	252.901i	1.38182	+	0.797794i	0.389445	+	0.921050i	$0.372667\pi$
$318$		0			0
$319$	188.755	+	108.978i	0.591708	+	0.341623i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−66.9088	−	341.283i	−0.207148	−	1.05660i
$324$		0			0
$325$	−103.139	+	59.5475i	−0.317351	+	0.183223i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−346.664	+	600.440i	−1.05369	+	1.82504i
$330$		0			0
$331$			80.9684i			0.244618i	0.992492	+	0.122309i	$0.0390298\pi$
$331$			80.9684i			0.244618i	−0.992492	+	0.122309i	$0.960970\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	348.070i		−	1.03902i
$336$		0			0
$337$	392.289	−	226.488i	1.16406	−	0.672071i	0.211787	−	0.977316i	$-0.432072\pi$
$337$	392.289	−	226.488i	1.16406	−	0.672071i	0.952274	+	0.305245i	$0.0987384\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	59.5131i		−	0.174525i
$342$		0			0
$343$	−304.069			−0.886498
$344$		0			0
$345$		0			0
$346$		0			0
$347$	132.251	+	229.065i	0.381126	+	0.660130i	0.991224	−	0.132197i	$-0.0422030\pi$
$347$	132.251	+	229.065i	0.381126	+	0.660130i	−0.610097	+	0.792326i	$0.708870\pi$
$348$		0			0
$349$	−252.830			−0.724440			−0.362220	−	0.932093i	$-0.617981\pi$
$349$	−252.830			−0.724440			−0.362220	+	0.932093i	$0.617981\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−49.2202			−0.139434			−0.0697170	−	0.997567i	$-0.522210\pi$
$353$	−49.2202			−0.139434			−0.0697170	+	0.997567i	$0.522210\pi$
$354$		0			0
$355$	172.748	+	99.7363i	0.486615	+	0.280947i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−2.28868	−	3.96411i	−0.00637515	−	0.0110421i	0.862820	−	0.505511i	$-0.168696\pi$
$359$	−2.28868	−	3.96411i	−0.00637515	−	0.0110421i	−0.869195	+	0.494469i	$0.835363\pi$
$360$		0			0
$361$	285.186	+	221.337i	0.789988	+	0.613122i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−117.733	+	203.919i	−0.322555	+	0.558682i
$366$		0			0
$367$	248.170	−	429.843i	0.676212	−	1.17123i	−0.299901	−	0.953970i	$-0.596954\pi$
$367$	248.170	−	429.843i	0.676212	−	1.17123i	0.976113	−	0.217264i	$-0.0697132\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−507.775	−	293.164i	−1.36867	−	0.790199i
$372$		0			0
$373$			4.20799i			0.0112815i	0.999984	+	0.00564073i	$0.00179551\pi$
$373$			4.20799i			0.0112815i	−0.999984	+	0.00564073i	$0.998204\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−149.129	−	258.299i	−0.395568	−	0.685144i
$378$		0			0
$379$			248.684i			0.656157i	0.944650	+	0.328079i	$0.106401\pi$
$379$			248.684i			0.656157i	−0.944650	+	0.328079i	$0.893599\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	527.607	−	304.614i	1.37756	−	0.795337i	0.385699	−	0.922625i	$-0.373960\pi$
$383$	527.607	−	304.614i	1.37756	−	0.795337i	0.991866	+	0.127288i	$0.0406271\pi$
$384$		0			0
$385$	−99.2088	−	171.835i	−0.257685	−	0.446324i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	229.333	−	397.216i	0.589544	−	1.02112i	−0.404748	−	0.914428i	$-0.632641\pi$
$389$	229.333	−	397.216i	0.589544	−	1.02112i	0.994292	−	0.106692i	$-0.0340259\pi$
$390$		0			0
$391$	−625.064			−1.59863
$392$		0			0
$393$		0			0
$394$		0			0
$395$	266.186	+	153.683i	0.673889	+	0.389070i
$396$		0			0
$397$	−51.9676	−	90.0104i	−0.130901	−	0.226727i	0.793123	−	0.609061i	$-0.208454\pi$
$397$	−51.9676	−	90.0104i	−0.130901	−	0.226727i	−0.924024	+	0.382334i	$0.875120\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	640.104	−	369.564i	1.59627	−	0.921606i	0.604071	−	0.796930i	$-0.293544\pi$
$401$	640.104	−	369.564i	1.59627	−	0.921606i	0.992198	−	0.124676i	$-0.0397891\pi$
$402$		0			0
$403$	−40.7200	+	70.5290i	−0.101042	+	0.175010i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			171.341i			0.420985i
$408$		0			0
$409$	−244.207	−	140.993i	−0.597084	−	0.344726i	0.170810	−	0.985304i	$-0.445362\pi$
$409$	−244.207	−	140.993i	−0.597084	−	0.344726i	−0.767893	+	0.640578i	$0.778695\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	825.387	−	476.538i	1.99852	−	1.15384i
$414$		0			0
$415$	−93.6295	−	162.171i	−0.225613	−	0.390774i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	491.659			1.17341			0.586705	−	0.809801i	$-0.300425\pi$
$419$	491.659			1.17341			0.586705	+	0.809801i	$0.300425\pi$
$420$		0			0
$421$	185.865	−	107.309i	0.441484	−	0.254891i	−0.262743	−	0.964866i	$-0.584627\pi$
$421$	185.865	−	107.309i	0.441484	−	0.254891i	0.704227	+	0.709975i	$0.251294\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−279.890			−0.658565
$426$		0			0
$427$	−20.4204	+	35.3691i	−0.0478229	+	0.0828316i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−81.8737	−	47.2698i	−0.189962	−	0.109675i	0.402003	−	0.915638i	$-0.368314\pi$
$431$	−81.8737	−	47.2698i	−0.189962	−	0.109675i	−0.591965	+	0.805964i	$0.701648\pi$
$432$		0			0
$433$	−329.028	−	189.964i	−0.759880	−	0.438717i	0.0693729	−	0.997591i	$-0.477900\pi$
$433$	−329.028	−	189.964i	−0.759880	−	0.438717i	−0.829253	+	0.558874i	$0.811233\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	488.988	−	426.455i	1.11897	−	0.975869i
$438$		0			0
$439$	239.645	−	138.359i	0.545887	−	0.315168i	−0.201574	−	0.979473i	$-0.564606\pi$
$439$	239.645	−	138.359i	0.545887	−	0.315168i	0.747462	+	0.664305i	$0.231272\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−7.38439	+	12.7901i	−0.0166691	+	0.0288717i	−0.874240	−	0.485495i	$-0.838640\pi$
$443$	−7.38439	+	12.7901i	−0.0166691	+	0.0288717i	0.857571	+	0.514366i	$0.171973\pi$
$444$		0			0
$445$		−	62.6054i		−	0.140686i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	402.117i		−	0.895584i	−0.894138	−	0.447792i	$-0.852211\pi$
$449$		−	402.117i		−	0.895584i	0.894138	−	0.447792i	$-0.147789\pi$
$450$		0			0
$451$	188.101	−	108.600i	0.417075	−	0.240799i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			271.522i			0.596752i
$456$		0			0
$457$	−645.600			−1.41269			−0.706345	−	0.707867i	$-0.749657\pi$
$457$	−645.600			−1.41269			−0.706345	+	0.707867i	$0.749657\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−114.286	−	197.948i	−0.247908	−	0.429389i	0.715037	−	0.699086i	$-0.246410\pi$
$461$	−114.286	−	197.948i	−0.247908	−	0.429389i	−0.962945	+	0.269697i	$0.913076\pi$
$462$		0			0
$463$	56.6097			0.122267			0.0611336	−	0.998130i	$-0.480528\pi$
$463$	56.6097			0.122267			0.0611336	+	0.998130i	$0.480528\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	440.857			0.944020			0.472010	−	0.881593i	$-0.343529\pi$
$467$	440.857			0.944020			0.472010	+	0.881593i	$0.343529\pi$
$468$		0			0
$469$	1082.37	+	624.905i	2.30782	+	1.33242i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−169.070	−	292.838i	−0.357443	−	0.619109i
$474$		0			0
$475$	218.959	−	190.957i	0.460966	−	0.402016i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−61.8700	+	107.162i	−0.129165	+	0.223720i	−0.923353	−	0.383951i	$-0.874563\pi$
$479$	−61.8700	+	107.162i	−0.129165	+	0.223720i	0.794188	+	0.607672i	$0.207896\pi$
$480$		0			0
$481$	117.235	−	203.056i	0.243731	−	0.422155i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−156.365	−	90.2776i	−0.322403	−	0.186139i
$486$		0			0
$487$			614.532i			1.26187i	0.775834	+	0.630937i	$0.217329\pi$
$487$			614.532i			1.26187i	−0.775834	+	0.630937i	$0.782671\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−412.325	−	714.168i	−0.839766	−	1.45452i	−0.890091	−	0.455784i	$-0.849359\pi$
$491$	−412.325	−	714.168i	−0.839766	−	1.45452i	0.0503251	−	0.998733i	$-0.483974\pi$
$492$		0			0
$493$		−	700.950i		−	1.42181i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−620.284	+	358.121i	−1.24806	+	0.720566i
$498$		0			0
$499$	−223.004	−	386.253i	−0.446901	−	0.774055i	0.551282	−	0.834319i	$-0.314139\pi$
$499$	−223.004	−	386.253i	−0.446901	−	0.774055i	−0.998182	+	0.0602642i	$0.980806\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−206.009	+	356.818i	−0.409560	+	0.709380i	−0.994840	−	0.101452i	$-0.967651\pi$
$503$	−206.009	+	356.818i	−0.409560	+	0.709380i	0.585280	+	0.810831i	$0.300985\pi$
$504$		0			0
$505$	400.090			0.792258
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−115.980	−	66.9613i	−0.227859	−	0.131555i	0.381725	−	0.924276i	$-0.375330\pi$
$509$	−115.980	−	66.9613i	−0.227859	−	0.131555i	−0.609584	+	0.792721i	$0.708664\pi$
$510$		0			0
$511$	−422.740	−	732.208i	−0.827281	−	1.43289i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	519.014	−	299.653i	1.00780	−	0.581851i
$516$		0			0
$517$	176.351	−	305.448i	0.341104	−	0.590809i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	82.8330i		−	0.158989i	−0.996835	−	0.0794943i	$-0.974669\pi$
$521$		−	82.8330i		−	0.158989i	0.996835	−	0.0794943i	$-0.0253305\pi$
$522$		0			0
$523$	716.232	+	413.517i	1.36947	+	0.790663i	0.990860	−	0.134894i	$-0.0430694\pi$
$523$	716.232	+	413.517i	1.36947	+	0.790663i	0.378608	+	0.925557i	$0.376403\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−165.753	+	95.6978i	−0.314523	+	0.181590i
$528$		0			0
$529$	−318.566	−	551.772i	−0.602203	−	1.04305i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−297.225			−0.557645
$534$		0			0
$535$	218.803	−	126.326i	0.408977	−	0.236123i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	433.568			0.804394
$540$		0			0
$541$	−369.791	+	640.497i	−0.683533	+	1.18391i	0.290363	+	0.956917i	$0.406224\pi$
$541$	−369.791	+	640.497i	−0.683533	+	1.18391i	−0.973895	+	0.226997i	$0.927109\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	10.2155	+	5.89792i	0.0187440	+	0.0108219i
$546$		0			0
$547$	−187.501	−	108.254i	−0.342781	−	0.197905i	0.318720	−	0.947849i	$-0.396747\pi$
$547$	−187.501	−	108.254i	−0.342781	−	0.197905i	−0.661501	+	0.749944i	$0.730080\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	478.229	+	548.355i	0.867929	+	0.995199i
$552$		0			0
$553$	−955.790	+	551.826i	−1.72837	+	0.997876i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−339.435	+	587.919i	−0.609399	+	1.05551i	0.381941	+	0.924187i	$0.375256\pi$
$557$	−339.435	+	587.919i	−0.609399	+	1.05551i	−0.991340	+	0.131323i	$0.958078\pi$
$558$		0			0
$559$			462.724i			0.827771i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	186.539i		−	0.331331i	−0.986182	−	0.165665i	$-0.947023\pi$
$563$		−	186.539i		−	0.331331i	0.986182	−	0.165665i	$-0.0529771\pi$
$564$		0			0
$565$	402.460	−	232.360i	0.712319	−	0.411257i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	298.679i		−	0.524919i	−0.964943	−	0.262460i	$-0.915466\pi$
$569$		−	298.679i		−	0.524919i	0.964943	−	0.262460i	$-0.0845337\pi$
$570$		0			0
$571$	767.923			1.34487			0.672437	−	0.740155i	$-0.265248\pi$
$571$	767.923			1.34487			0.672437	+	0.740155i	$0.265248\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−261.084	−	452.212i	−0.454060	−	0.786455i
$576$		0			0
$577$	978.315			1.69552			0.847760	−	0.530381i	$-0.177951\pi$
$577$	978.315			1.69552			0.847760	+	0.530381i	$0.177951\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	672.388			1.15729
$582$		0			0
$583$	258.309	+	149.135i	0.443068	+	0.255806i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	53.0636	+	91.9088i	0.0903979	+	0.156574i	0.907679	−	0.419666i	$-0.137853\pi$
$587$	53.0636	+	91.9088i	0.0903979	+	0.156574i	−0.817281	+	0.576240i	$0.804519\pi$
$588$		0			0
$589$	64.3786	−	187.951i	0.109302	−	0.319102i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−505.114	+	874.883i	−0.851794	+	1.47535i	0.0277931	+	0.999614i	$0.491152\pi$
$593$	−505.114	+	874.883i	−0.851794	+	1.47535i	−0.879587	+	0.475737i	$0.842181\pi$
$594$		0			0
$595$	−319.058	+	552.625i	−0.536232	+	0.928781i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−434.029	−	250.587i	−0.724589	−	0.418342i	0.0918505	−	0.995773i	$-0.470722\pi$
$599$	−434.029	−	250.587i	−0.724589	−	0.418342i	−0.816439	+	0.577431i	$0.804055\pi$
$600$		0			0
$601$			119.762i			0.199271i	0.995024	+	0.0996354i	$0.0317676\pi$
$601$			119.762i			0.199271i	−0.995024	+	0.0996354i	$0.968232\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−138.045	−	239.101i	−0.228173	−	0.395208i
$606$		0			0
$607$		−	209.144i		−	0.344554i	−0.985049	−	0.172277i	$-0.944888\pi$
$607$		−	209.144i		−	0.344554i	0.985049	−	0.172277i	$-0.0551123\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−417.986	+	241.325i	−0.684102	+	0.394967i
$612$		0			0
$613$	219.716	+	380.559i	0.358427	+	0.620814i	0.987698	−	0.156372i	$-0.0499799\pi$
$613$	219.716	+	380.559i	0.358427	+	0.620814i	−0.629271	+	0.777186i	$0.716647\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−563.808	+	976.543i	−0.913788	+	1.58273i	−0.105123	+	0.994459i	$0.533524\pi$
$617$	−563.808	+	976.543i	−0.913788	+	1.58273i	−0.808665	+	0.588269i	$0.799810\pi$
$618$		0			0
$619$	−620.601			−1.00259			−0.501293	−	0.865278i	$-0.667142\pi$
$619$	−620.601			−1.00259			−0.501293	+	0.865278i	$0.667142\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	194.679	+	112.398i	0.312486	+	0.180414i
$624$		0			0
$625$	4.45378	+	7.71418i	0.00712605	+	0.0123427i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	477.212	−	275.519i	0.758684	−	0.438026i
$630$		0			0
$631$	−463.672	+	803.103i	−0.734820	+	1.27275i	0.219982	+	0.975504i	$0.429400\pi$
$631$	−463.672	+	803.103i	−0.734820	+	1.27275i	−0.954802	+	0.297242i	$0.903933\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			70.2496i			0.110629i
$636$		0			0
$637$	−513.822	−	296.655i	−0.806629	−	0.465707i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	1104.18	−	637.501i	1.72259	−	0.994541i	0.809122	−	0.587641i	$-0.199943\pi$
$641$	1104.18	−	637.501i	1.72259	−	0.994541i	0.913473	−	0.406900i	$-0.133390\pi$
$642$		0			0
$643$	−472.236	−	817.936i	−0.734426	−	1.27206i	−0.954975	−	0.296687i	$-0.904118\pi$
$643$	−472.236	−	817.936i	−0.734426	−	1.27206i	0.220549	−	0.975376i	$-0.429215\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	360.339			0.556938			0.278469	−	0.960445i	$-0.410173\pi$
$647$	360.339			0.556938			0.278469	+	0.960445i	$0.410173\pi$
$648$		0			0
$649$	−419.881	+	242.418i	−0.646966	+	0.373526i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	39.7164			0.0608214			0.0304107	−	0.999537i	$-0.490318\pi$
$653$	39.7164			0.0608214			0.0304107	+	0.999537i	$0.490318\pi$
$654$		0			0
$655$	233.777	−	404.914i	0.356911	−	0.618189i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	628.656	+	362.954i	0.953954	+	0.550765i	0.894307	−	0.447454i	$-0.147669\pi$
$659$	628.656	+	362.954i	0.953954	+	0.550765i	0.0596469	+	0.998220i	$0.481003\pi$
$660$		0			0
$661$	103.278	+	59.6276i	0.156245	+	0.0902082i	0.576084	−	0.817390i	$-0.304580\pi$
$661$	103.278	+	59.6276i	0.156245	+	0.0902082i	−0.419839	+	0.907599i	$0.637913\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−127.433	−	649.999i	−0.191629	−	0.977443i
$666$		0			0
$667$	1132.51	−	653.853i	1.69791	−	0.980290i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	10.3880	−	17.9925i	0.0154814	−	0.0268145i
$672$		0			0
$673$		−	667.240i		−	0.991442i	−0.868482	−	0.495721i	$-0.834904\pi$
$673$		−	667.240i		−	0.991442i	0.868482	−	0.495721i	$-0.165096\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	1141.27i		−	1.68577i	−0.538091	−	0.842887i	$-0.680854\pi$
$677$		−	1141.27i		−	1.68577i	0.538091	−	0.842887i	$-0.319146\pi$
$678$		0			0
$679$	561.458	−	324.158i	0.826890	−	0.477405i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			2.99894i			0.00439083i	0.999998	+	0.00219542i	$0.000698823\pi$
$683$			2.99894i			0.00439083i	−0.999998	+	0.00219542i	$0.999301\pi$
$684$		0			0
$685$	224.740			0.328087
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−204.081	−	353.479i	−0.296199	−	0.513033i
$690$		0			0
$691$	−85.0487			−0.123081			−0.0615403	−	0.998105i	$-0.519601\pi$
$691$	−85.0487			−0.123081			−0.0615403	+	0.998105i	$0.519601\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−397.539			−0.571998
$696$		0			0
$697$	−604.937	−	349.261i	−0.867916	−	0.501091i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	317.774	+	550.401i	0.453316	+	0.785166i	0.998590	−	0.0530922i	$-0.0169077\pi$
$701$	317.774	+	550.401i	0.453316	+	0.785166i	−0.545274	+	0.838258i	$0.683574\pi$
$702$		0			0
$703$	−185.349	+	541.120i	−0.263655	+	0.769730i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−718.298	+	1244.13i	−1.01598	+	1.75973i
$708$		0			0
$709$	86.7540	−	150.262i	0.122361	−	0.211936i	−0.798337	−	0.602211i	$-0.794287\pi$
$709$	86.7540	−	150.262i	0.122361	−	0.211936i	0.920698	+	0.390275i	$0.127620\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−309.233	−	178.536i	−0.433707	−	0.250401i
$714$		0			0
$715$		−	138.125i		−	0.193182i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−453.267	−	785.082i	−0.630413	−	1.09191i	−0.987467	−	0.157824i	$-0.949552\pi$
$719$	−453.267	−	785.082i	−0.630413	−	1.09191i	0.357054	−	0.934084i	$-0.383781\pi$
$720$		0			0
$721$			2151.92i			2.98463i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	507.113	−	292.782i	0.699466	−	0.403837i
$726$		0			0
$727$	−129.255	−	223.876i	−0.177792	−	0.307944i	0.763332	−	0.646006i	$-0.223562\pi$
$727$	−129.255	−	223.876i	−0.177792	−	0.307944i	−0.941124	+	0.338062i	$0.890229\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−543.734	+	941.776i	−0.743823	+	1.28834i
$732$		0			0
$733$	−11.6270			−0.0158621			−0.00793107	−	0.999969i	$-0.502525\pi$
$733$	−11.6270			−0.0158621			−0.00793107	+	0.999969i	$0.502525\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−550.608	−	317.894i	−0.747093	−	0.431335i
$738$		0			0
$739$	−192.783	−	333.910i	−0.260870	−	0.451840i	0.705603	−	0.708607i	$-0.250676\pi$
$739$	−192.783	−	333.910i	−0.260870	−	0.451840i	−0.966473	+	0.256767i	$0.917343\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−781.647	+	451.284i	−1.05202	+	0.607381i	−0.923213	−	0.384289i	$-0.874447\pi$
$743$	−781.647	+	451.284i	−1.05202	+	0.607381i	−0.128802	+	0.991670i	$0.541113\pi$
$744$		0			0
$745$	−333.431	+	577.519i	−0.447558	+	0.775193i
$746$		0			0
$747$		0			0
$748$		0			0
$749$			907.191i			1.21120i
$750$		0			0
$751$	446.578	+	257.832i	0.594645	+	0.343318i	0.766932	−	0.641728i	$-0.221782\pi$
$751$	446.578	+	257.832i	0.594645	+	0.343318i	−0.172287	+	0.985047i	$0.555116\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−181.303	+	104.675i	−0.240136	+	0.138643i
$756$		0			0
$757$	−70.3485	−	121.847i	−0.0929306	−	0.160961i	0.815812	−	0.578317i	$-0.196290\pi$
$757$	−70.3485	−	121.847i	−0.0929306	−	0.160961i	−0.908743	+	0.417356i	$0.862957\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−918.492			−1.20695			−0.603477	−	0.797381i	$-0.706218\pi$
$761$	−918.492			−1.20695			−0.603477	+	0.797381i	$0.706218\pi$
$762$		0			0
$763$	−36.6806	+	21.1775i	−0.0480742	+	0.0277556i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	663.468			0.865017
$768$		0			0
$769$	−501.724	+	869.012i	−0.652437	+	1.13005i	0.330092	+	0.943949i	$0.392920\pi$
$769$	−501.724	+	869.012i	−0.652437	+	1.13005i	−0.982530	+	0.186106i	$0.940413\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	365.934	+	211.272i	0.473395	+	0.273314i	0.717660	−	0.696394i	$-0.245213\pi$
$773$	365.934	+	211.272i	0.473395	+	0.273314i	−0.244265	+	0.969708i	$0.578547\pi$
$774$		0			0
$775$	−138.468	−	79.9445i	−0.178668	−	0.103154i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	711.530	−	139.496i	0.913388	−	0.179071i
$780$		0			0
$781$	315.543	−	182.179i	0.404025	−	0.233264i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−345.087	+	597.708i	−0.439601	+	0.761411i
$786$		0			0
$787$			1279.25i			1.62547i	0.582633	+	0.812735i	$0.302022\pi$
$787$			1279.25i			1.62547i	−0.582633	+	0.812735i	$0.697978\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			1668.66i			2.10956i
$792$		0			0
$793$	−24.6216	+	14.2153i	−0.0310487	+	0.0179260i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			726.878i			0.912018i	0.889975	+	0.456009i	$0.150721\pi$
$797$			726.878i			0.912018i	−0.889975	+	0.456009i	$0.849279\pi$
$798$		0			0
$799$	−1134.30			−1.41964
$800$		0			0
$801$		0			0
$802$		0			0
$803$	215.051	+	372.480i	0.267810	+	0.463860i
$804$		0			0
$805$	−1190.48			−1.47886
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−908.429			−1.12290			−0.561452	−	0.827509i	$-0.689757\pi$
$809$	−908.429			−1.12290			−0.561452	+	0.827509i	$0.689757\pi$
$810$		0			0
$811$	−1028.49	−	593.797i	−1.26817	−	0.732179i	−0.293530	−	0.955950i	$-0.594830\pi$
$811$	−1028.49	−	593.797i	−1.26817	−	0.732179i	−0.974642	+	0.223771i	$0.928163\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−197.217	−	341.590i	−0.241984	−	0.419129i
$816$		0			0
$817$	−217.170	−	1107.72i	−0.265813	−	1.35584i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	19.4768	−	33.7348i	0.0237233	−	0.0410899i	−0.853920	−	0.520404i	$-0.825781\pi$
$821$	19.4768	−	33.7348i	0.0237233	−	0.0410899i	0.877643	+	0.479314i	$0.159115\pi$
$822$		0			0
$823$	−30.7697	+	53.2947i	−0.0373873	+	0.0647566i	−0.884114	−	0.467272i	$-0.845237\pi$
$823$	−30.7697	+	53.2947i	−0.0373873	+	0.0647566i	0.846726	+	0.532029i	$0.178570\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	811.008	+	468.236i	0.980662	+	0.566186i	0.902470	−	0.430753i	$-0.141752\pi$
$827$	811.008	+	468.236i	0.980662	+	0.566186i	0.0781923	+	0.996938i	$0.475085\pi$
$828$		0			0
$829$		−	188.937i		−	0.227909i	−0.993486	−	0.113955i	$-0.963648\pi$
$829$		−	188.937i		−	0.227909i	0.993486	−	0.113955i	$-0.0363518\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−697.183	−	1207.56i	−0.836955	−	1.44965i
$834$		0			0
$835$			579.315i			0.693790i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	543.333	−	313.694i	0.647596	−	0.373890i	−0.139939	−	0.990160i	$-0.544690\pi$
$839$	543.333	−	313.694i	0.647596	−	0.373890i	0.787535	+	0.616270i	$0.211357\pi$
$840$		0			0
$841$	312.735	+	541.673i	0.371861	+	0.644082i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	168.787	−	292.348i	0.199748	−	0.345974i
$846$		0			0
$847$	991.350			1.17042
$848$		0			0
$849$		0			0
$850$		0			0
$851$	890.296	+	514.013i	1.04618	+	0.604010i
$852$		0			0
$853$	−20.1765	−	34.9468i	−0.0236536	−	0.0409693i	0.853956	−	0.520345i	$-0.174197\pi$
$853$	−20.1765	−	34.9468i	−0.0236536	−	0.0409693i	−0.877610	+	0.479375i	$0.840863\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1068.27	+	616.764i	−1.24652	+	0.719678i	−0.970413	−	0.241449i	$-0.922377\pi$
$857$	−1068.27	+	616.764i	−1.24652	+	0.719678i	−0.276106	+	0.961127i	$0.589044\pi$
$858$		0			0
$859$	−465.744	+	806.693i	−0.542194	+	0.939107i	0.456584	+	0.889680i	$0.349073\pi$
$859$	−465.744	+	806.693i	−0.542194	+	0.939107i	−0.998778	+	0.0494265i	$0.984261\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	426.295i		−	0.493968i	−0.969020	−	0.246984i	$-0.920560\pi$
$863$		−	426.295i		−	0.493968i	0.969020	−	0.246984i	$-0.0794396\pi$
$864$		0			0
$865$	−734.578	−	424.109i	−0.849223	−	0.490299i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	486.218	−	280.718i	0.559514	−	0.323035i
$870$		0			0
$871$	435.017	+	753.472i	0.499446	+	0.865066i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−1404.62			−1.60528
$876$		0			0
$877$	1027.85	−	593.427i	1.17200	−	0.676656i	0.217851	−	0.975982i	$-0.430095\pi$
$877$	1027.85	−	593.427i	1.17200	−	0.676656i	0.954151	+	0.299326i	$0.0967619\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	310.782			0.352761			0.176380	−	0.984322i	$-0.443561\pi$
$881$	310.782			0.352761			0.176380	+	0.984322i	$0.443561\pi$
$882$		0			0
$883$	−124.778	+	216.122i	−0.141312	+	0.244759i	−0.927991	−	0.372603i	$-0.878465\pi$
$883$	−124.778	+	216.122i	−0.141312	+	0.244759i	0.786679	+	0.617362i	$0.211799\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	16.0576	+	9.27083i	0.0181032	+	0.0104519i	0.509024	−	0.860752i	$-0.330006\pi$
$887$	16.0576	+	9.27083i	0.0181032	+	0.0104519i	−0.490921	+	0.871204i	$0.663340\pi$
$888$		0			0
$889$	−218.450	−	126.122i	−0.245725	−	0.141870i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	887.362	−	773.882i	0.993686	−	0.866610i
$894$		0			0
$895$	−63.7013	+	36.7780i	−0.0711747	+	0.0410927i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	200.211	−	346.776i	0.222704	−	0.385735i
$900$		0			0
$901$		−	959.242i		−	1.06464i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			177.639i			0.196286i
$906$		0			0
$907$	−639.562	+	369.251i	−0.705140	+	0.407113i	−0.809259	−	0.587452i	$-0.800131\pi$
$907$	−639.562	+	369.251i	−0.705140	+	0.407113i	0.104119	+	0.994565i	$0.466798\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			1646.37i			1.80722i	0.428359	+	0.903609i	$0.359092\pi$
$911$			1646.37i			1.80722i	−0.428359	+	0.903609i	$0.640908\pi$
$912$		0			0
$913$	−342.049			−0.374643
$914$		0			0
$915$		0			0
$916$		0			0
$917$	839.419	+	1453.92i	0.915397	+	1.58551i
$918$		0			0
$919$	1110.40			1.20827			0.604136	−	0.796882i	$-0.293519\pi$
$919$	1110.40			1.20827			0.604136	+	0.796882i	$0.293519\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−498.601			−0.540196
$924$		0			0
$925$	398.656	+	230.164i	0.430979	+	0.248826i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−757.979	−	1312.86i	−0.815908	−	1.41319i	−0.908674	−	0.417507i	$-0.862904\pi$
$929$	−757.979	−	1312.86i	−0.815908	−	1.41319i	0.0927656	−	0.995688i	$-0.470429\pi$
$930$		0			0
$931$	1369.27	+	469.015i	1.47076	+	0.503776i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	162.307	−	281.124i	0.173591	−	0.300668i
$936$		0			0
$937$	−496.916	+	860.684i	−0.530327	+	0.918552i	0.469047	+	0.883173i	$0.344597\pi$
$937$	−496.916	+	860.684i	−0.530327	+	0.918552i	−0.999374	+	0.0353795i	$0.988736\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−379.535	−	219.125i	−0.403332	−	0.232864i	0.284589	−	0.958650i	$-0.408143\pi$
$941$	−379.535	−	219.125i	−0.403332	−	0.232864i	−0.687921	+	0.725786i	$0.741476\pi$
$942$		0			0
$943$		−	1303.18i		−	1.38195i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−91.6405	−	158.726i	−0.0967693	−	0.167609i	0.813576	−	0.581458i	$-0.197518\pi$
$947$	−91.6405	−	158.726i	−0.0967693	−	0.167609i	−0.910346	+	0.413849i	$0.864184\pi$
$948$		0			0
$949$		−	588.568i		−	0.620198i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−158.904	+	91.7433i	−0.166741	+	0.0962679i	−0.581048	−	0.813869i	$-0.697357\pi$
$953$	−158.904	+	91.7433i	−0.166741	+	0.0962679i	0.414307	+	0.910137i	$0.364024\pi$
$954$		0			0
$955$	−246.757	−	427.395i	−0.258384	−	0.447534i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−403.485	+	698.856i	−0.420735	+	0.728734i
$960$		0			0
$961$	851.664			0.886227
$962$		0			0
$963$		0			0
$964$		0			0
$965$	500.694	+	289.076i	0.518854	+	0.299560i
$966$		0			0
$967$	712.541	+	1234.16i	0.736857	+	1.27627i	0.953904	+	0.300113i	$0.0970243\pi$
$967$	712.541	+	1234.16i	0.736857	+	1.27627i	−0.217047	+	0.976161i	$0.569642\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	143.906	−	83.0843i	0.148204	−	0.0855657i	−0.424064	−	0.905632i	$-0.639397\pi$
$971$	143.906	−	83.0843i	0.148204	−	0.0855657i	0.572268	+	0.820066i	$0.306064\pi$
$972$		0			0
$973$	713.718	−	1236.20i	0.733523	−	1.27050i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	959.257i		−	0.981839i	−0.871205	−	0.490920i	$-0.836661\pi$
$977$		−	959.257i		−	0.981839i	0.871205	−	0.490920i	$-0.163339\pi$
$978$		0			0
$979$	−99.0347	−	57.1777i	−0.101159	−	0.0584042i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	94.1725	−	54.3705i	0.0958011	−	0.0553108i	−0.451334	−	0.892355i	$-0.649052\pi$
$983$	94.1725	−	54.3705i	0.0958011	−	0.0553108i	0.547135	+	0.837044i	$0.315718\pi$
$984$		0			0
$985$	−30.9395	−	53.5888i	−0.0314107	−	0.0544049i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−2028.80			−2.05137
$990$		0			0
$991$	−1596.80	+	921.912i	−1.61130	+	0.930284i	−0.622230	+	0.782834i	$0.713773\pi$
$991$	−1596.80	+	921.912i	−1.61130	+	0.930284i	−0.989070	+	0.147450i	$0.952894\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−421.580			−0.423699
$996$		0			0
$997$	−537.881	+	931.637i	−0.539499	+	0.934440i	0.459432	+	0.888213i	$0.348053\pi$
$997$	−537.881	+	931.637i	−0.539499	+	0.934440i	−0.998931	+	0.0462269i	$0.985280\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	684.3.y.h.145.3		8
3.2	odd	2		76.3.h.a.69.1	yes	8
12.11	even	2		304.3.r.c.145.4		8
19.8	odd	6	inner	684.3.y.h.217.3		8
57.8	even	6		76.3.h.a.65.1	✓	8
57.26	odd	6		1444.3.c.b.721.2		8
57.50	even	6		1444.3.c.b.721.7		8
228.179	odd	6		304.3.r.c.65.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.3.h.a.65.1	✓	8	57.8	even	6
76.3.h.a.69.1	yes	8	3.2	odd	2
304.3.r.c.65.4		8	228.179	odd	6
304.3.r.c.145.4		8	12.11	even	2
684.3.y.h.145.3		8	1.1	even	1	trivial
684.3.y.h.217.3		8	19.8	odd	6	inner
1444.3.c.b.721.2		8	57.26	odd	6
1444.3.c.b.721.7		8	57.50	even	6

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:	\( N \)	\(=\)	\( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	684.y (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.55796 + 2.69846i) q^{5} -11.1883 q^{7} +O(q^{10})\)	\(q+(1.55796 + 2.69846i) q^{5} -11.1883 q^{7} +5.69156 q^{11} +(-6.74507 - 3.89427i) q^{13} +(-9.15210 - 15.8519i) q^{17} +(17.9748 + 6.15688i) q^{19} +(17.0743 - 29.5736i) q^{23} +(7.64552 - 13.2424i) q^{25} +(33.1640 + 19.1473i) q^{29} -10.4564i q^{31} +(-17.4309 - 30.1912i) q^{35} +30.1044i q^{37} +(33.0491 - 19.0809i) q^{41} +(-29.7055 - 51.4514i) q^{43} +(30.9846 - 53.6669i) q^{47} +76.1775 q^{49} +(45.3846 + 26.2028i) q^{53} +(8.86722 + 15.3585i) q^{55} +(-73.7726 + 42.5926i) q^{59} +(1.82516 - 3.16127i) q^{61} -24.2685i q^{65} +(-96.7411 - 55.8535i) q^{67} +(55.4406 - 32.0086i) q^{71} +(37.7842 + 65.4442i) q^{73} -63.6787 q^{77} +(85.4278 - 49.3218i) q^{79} -60.0975 q^{83} +(28.5172 - 49.3932i) q^{85} +(-17.4003 - 10.0461i) q^{89} +(75.4657 + 43.5701i) q^{91} +(11.3899 + 58.0965i) q^{95} +(-50.1828 + 28.9730i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + q^{5} - 12 q^{7}+O(q^{10}) \) 8 * q + q^5 - 12 * q^7	\( 8 q + q^{5} - 12 q^{7} + 10 q^{11} + 9 q^{13} - 23 q^{17} - 33 q^{19} + 31 q^{23} - 73 q^{25} + 105 q^{29} + 68 q^{35} - 18 q^{41} - 41 q^{43} - 107 q^{47} + 312 q^{49} - 39 q^{53} + 70 q^{55} - 348 q^{59} - 45 q^{61} - 432 q^{67} + 243 q^{71} + 16 q^{73} - 544 q^{77} + 75 q^{79} + 82 q^{83} + 109 q^{85} + 213 q^{89} + 222 q^{91} + 385 q^{95} + 144 q^{97}+O(q^{100}) \) 8 * q + q^5 - 12 * q^7 + 10 * q^11 + 9 * q^13 - 23 * q^17 - 33 * q^19 + 31 * q^23 - 73 * q^25 + 105 * q^29 + 68 * q^35 - 18 * q^41 - 41 * q^43 - 107 * q^47 + 312 * q^49 - 39 * q^53 + 70 * q^55 - 348 * q^59 - 45 * q^61 - 432 * q^67 + 243 * q^71 + 16 * q^73 - 544 * q^77 + 75 * q^79 + 82 * q^83 + 109 * q^85 + 213 * q^89 + 222 * q^91 + 385 * q^95 + 144 * q^97

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