LMFDB - Embedded newform 684.3.y.h.145.2

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			145.2
Root			$0.500000 - 1.68338i$ of defining polynomial
Character	$\chi$	$=$	684.145
Dual form			684.3.y.h.217.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.50973 - 4.34699i) q^{5} +7.91625 q^{7} +O(q^{10})$	$q+(-2.50973 - 4.34699i) q^{5} +7.91625 q^{7} -16.3514 q^{11} +(-11.1527 - 6.43904i) q^{13} +(4.46786 + 7.73856i) q^{17} +(-18.5298 + 4.20096i) q^{19} +(-7.27948 + 12.6084i) q^{23} +(-0.0975349 + 0.168935i) q^{25} +(29.7763 + 17.1913i) q^{29} -25.5436i q^{31} +(-19.8677 - 34.4118i) q^{35} +49.2849i q^{37} +(-27.8092 + 16.0557i) q^{41} +(-4.51003 - 7.81160i) q^{43} +(-19.9292 + 34.5184i) q^{47} +13.6670 q^{49} +(-73.6502 - 42.5220i) q^{53} +(41.0377 + 71.0794i) q^{55} +(-53.9311 + 31.1371i) q^{59} +(-38.1931 + 66.1524i) q^{61} +64.6411i q^{65} +(-69.8101 - 40.3049i) q^{67} +(11.0359 - 6.37160i) q^{71} +(-42.2833 - 73.2369i) q^{73} -129.442 q^{77} +(-106.825 + 61.6757i) q^{79} -21.9310 q^{83} +(22.4263 - 38.8434i) q^{85} +(98.0897 + 56.6321i) q^{89} +(-88.2879 - 50.9730i) q^{91} +(64.7663 + 70.0053i) q^{95} +(146.478 - 84.5692i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−2.50973	−	4.34699i	−0.501947	−	0.869398i	−0.999997	−	0.00224956i	$-0.999284\pi$
$5$	−2.50973	−	4.34699i	−0.501947	−	0.869398i	0.498051	−	0.867148i	$-0.334049\pi$
$6$		0			0
$7$	7.91625			1.13089			0.565446	−	0.824785i	$-0.308704\pi$
$7$	7.91625			1.13089			0.565446	+	0.824785i	$0.308704\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−16.3514			−1.48649			−0.743246	−	0.669018i	$-0.766715\pi$
$11$	−16.3514			−1.48649			−0.743246	+	0.669018i	$0.766715\pi$
$12$		0			0
$13$	−11.1527	−	6.43904i	−0.857903	−	0.495311i	0.00540634	−	0.999985i	$-0.498279\pi$
$13$	−11.1527	−	6.43904i	−0.857903	−	0.495311i	−0.863310	+	0.504675i	$0.831612\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	4.46786	+	7.73856i	0.262815	+	0.455209i	0.966989	−	0.254819i	$-0.0820157\pi$
$17$	4.46786	+	7.73856i	0.262815	+	0.455209i	−0.704174	+	0.710028i	$0.748682\pi$
$18$		0			0
$19$	−18.5298	+	4.20096i	−0.975250	+	0.221103i
$19$	−18.5298	+	4.20096i	−0.975250	+	0.221103i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−7.27948	+	12.6084i	−0.316499	+	0.548193i	−0.979755	−	0.200200i	$-0.935841\pi$
$23$	−7.27948	+	12.6084i	−0.316499	+	0.548193i	0.663256	+	0.748393i	$0.269174\pi$
$24$		0			0
$25$	−0.0975349	+	0.168935i	−0.00390140	+	0.00675742i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	29.7763	+	17.1913i	1.02677	+	0.592805i	0.916057	−	0.401048i	$-0.131354\pi$
$29$	29.7763	+	17.1913i	1.02677	+	0.592805i	0.110711	+	0.993853i	$0.464687\pi$
$30$		0			0
$31$		−	25.5436i		−	0.823989i	−0.911186	−	0.411994i	$-0.864832\pi$
$31$		−	25.5436i		−	0.823989i	0.911186	−	0.411994i	$-0.135168\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−19.8677	−	34.4118i	−0.567648	−	0.983195i
$36$		0			0
$37$			49.2849i			1.33202i	0.745941	+	0.666012i	$0.232000\pi$
$37$			49.2849i			1.33202i	−0.745941	+	0.666012i	$0.768000\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−27.8092	+	16.0557i	−0.678274	+	0.391602i	−0.799204	−	0.601059i	$-0.794746\pi$
$41$	−27.8092	+	16.0557i	−0.678274	+	0.391602i	0.120930	+	0.992661i	$0.461412\pi$
$42$		0			0
$43$	−4.51003	−	7.81160i	−0.104884	−	0.181665i	0.808807	−	0.588075i	$-0.200114\pi$
$43$	−4.51003	−	7.81160i	−0.104884	−	0.181665i	−0.913691	+	0.406410i	$0.866781\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−19.9292	+	34.5184i	−0.424025	+	0.734433i	−0.996329	−	0.0856087i	$-0.972717\pi$
$47$	−19.9292	+	34.5184i	−0.424025	+	0.734433i	0.572304	+	0.820042i	$0.306050\pi$
$48$		0			0
$49$	13.6670			0.278917
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−73.6502	−	42.5220i	−1.38963	−	0.802301i	−0.396354	−	0.918098i	$-0.629725\pi$
$53$	−73.6502	−	42.5220i	−1.38963	−	0.802301i	−0.993273	+	0.115796i	$0.963058\pi$
$54$		0			0
$55$	41.0377	+	71.0794i	0.746140	+	1.29235i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−53.9311	+	31.1371i	−0.914087	+	0.527748i	−0.881744	−	0.471729i	$-0.843630\pi$
$59$	−53.9311	+	31.1371i	−0.914087	+	0.527748i	−0.0323429	+	0.999477i	$0.510297\pi$
$60$		0			0
$61$	−38.1931	+	66.1524i	−0.626116	+	1.08446i	0.362208	+	0.932097i	$0.382023\pi$
$61$	−38.1931	+	66.1524i	−0.626116	+	1.08446i	−0.988324	+	0.152367i	$0.951310\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			64.6411i			0.994479i
$66$		0			0
$67$	−69.8101	−	40.3049i	−1.04194	−	0.601566i	−0.121559	−	0.992584i	$-0.538789\pi$
$67$	−69.8101	−	40.3049i	−1.04194	−	0.601566i	−0.920383	+	0.391019i	$0.872123\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	11.0359	−	6.37160i	0.155436	−	0.0897408i	−0.420265	−	0.907402i	$-0.638063\pi$
$71$	11.0359	−	6.37160i	0.155436	−	0.0897408i	0.575700	+	0.817661i	$0.304729\pi$
$72$		0			0
$73$	−42.2833	−	73.2369i	−0.579224	−	1.00325i	−0.995569	−	0.0940382i	$-0.970022\pi$
$73$	−42.2833	−	73.2369i	−0.579224	−	1.00325i	0.416345	−	0.909207i	$-0.363311\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−129.442			−1.68106
$78$		0			0
$79$	−106.825	+	61.6757i	−1.35222	+	0.780705i	−0.988560	−	0.150827i	$-0.951807\pi$
$79$	−106.825	+	61.6757i	−1.35222	+	0.780705i	−0.363661	+	0.931532i	$0.618473\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−21.9310			−0.264229			−0.132115	−	0.991234i	$-0.542177\pi$
$83$	−21.9310			−0.264229			−0.132115	+	0.991234i	$0.542177\pi$
$84$		0			0
$85$	22.4263	−	38.8434i	0.263839	−	0.456982i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	98.0897	+	56.6321i	1.10213	+	0.636316i	0.936780	−	0.349919i	$-0.113791\pi$
$89$	98.0897	+	56.6321i	1.10213	+	0.636316i	0.165352	+	0.986235i	$0.447124\pi$
$90$		0			0
$91$	−88.2879	−	50.9730i	−0.970196	−	0.560143i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	64.7663	+	70.0053i	0.681751	+	0.736898i
$96$		0			0
$97$	146.478	−	84.5692i	1.51008	−	0.871848i	0.510154	−	0.860083i	$-0.329589\pi$
$97$	146.478	−	84.5692i	1.51008	−	0.871848i	0.999931	−	0.0117643i	$-0.00374477\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	69.5239	−	120.419i	0.688356	−	1.19227i	−0.284014	−	0.958820i	$-0.591666\pi$
$101$	69.5239	−	120.419i	0.688356	−	1.19227i	0.972370	−	0.233447i	$-0.0750004\pi$
$102$		0			0
$103$		−	82.3278i		−	0.799299i	−0.916668	−	0.399650i	$-0.869132\pi$
$103$		−	82.3278i		−	0.799299i	0.916668	−	0.399650i	$-0.130868\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			0.743185i			0.00694565i	0.999994	+	0.00347283i	$0.00110544\pi$
$107$			0.743185i			0.00694565i	−0.999994	+	0.00347283i	$0.998895\pi$
$108$		0			0
$109$	−56.7170	+	32.7456i	−0.520339	+	0.300418i	−0.737074	−	0.675813i	$-0.763793\pi$
$109$	−56.7170	+	32.7456i	−0.520339	+	0.300418i	0.216734	+	0.976231i	$0.430460\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	158.219i		−	1.40017i	−0.714060	−	0.700084i	$-0.753146\pi$
$113$		−	158.219i		−	1.40017i	0.714060	−	0.700084i	$-0.246854\pi$
$114$		0			0
$115$	73.0783			0.635463
$116$		0			0
$117$		0			0
$118$		0			0
$119$	35.3687	+	61.2603i	0.297216	+	0.514793i
$120$		0			0
$121$	146.369			1.20966
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−124.508			−0.996061
$126$		0			0
$127$	184.362	+	106.442i	1.45167	+	0.838122i	0.998576	−	0.0533406i	$-0.0169869\pi$
$127$	184.362	+	106.442i	1.45167	+	0.838122i	0.453094	+	0.891463i	$0.350320\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−36.5824	−	63.3625i	−0.279255	−	0.483683i	0.691945	−	0.721950i	$-0.256754\pi$
$131$	−36.5824	−	63.3625i	−0.279255	−	0.483683i	−0.971200	+	0.238267i	$0.923421\pi$
$132$		0			0
$133$	−146.686	+	33.2559i	−1.10290	+	0.250044i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−49.6408	+	85.9803i	−0.362341	+	0.627594i	−0.988346	−	0.152226i	$-0.951356\pi$
$137$	−49.6408	+	85.9803i	−0.362341	+	0.627594i	0.626004	+	0.779820i	$0.284689\pi$
$138$		0			0
$139$	−109.870	+	190.300i	−0.790429	+	1.36906i	0.135273	+	0.990808i	$0.456809\pi$
$139$	−109.870	+	190.300i	−0.790429	+	1.36906i	−0.925702	+	0.378255i	$0.876524\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	182.363	+	105.287i	1.27527	+	0.736275i
$144$		0			0
$145$		−	172.583i		−	1.19023i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−88.6369	−	153.524i	−0.594878	−	1.03036i	−0.993564	−	0.113272i	$-0.963867\pi$
$149$	−88.6369	−	153.524i	−0.594878	−	1.03036i	0.398686	−	0.917088i	$-0.369466\pi$
$150$		0			0
$151$			213.367i			1.41303i	0.707699	+	0.706514i	$0.249733\pi$
$151$			213.367i			1.41303i	−0.707699	+	0.706514i	$0.750267\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−111.038	+	64.1078i	−0.716374	+	0.413599i
$156$		0			0
$157$	−6.35003	−	10.9986i	−0.0404460	−	0.0700546i	0.845094	−	0.534618i	$-0.179545\pi$
$157$	−6.35003	−	10.9986i	−0.0404460	−	0.0700546i	−0.885540	+	0.464564i	$0.846211\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−57.6262	+	99.8114i	−0.357926	+	0.619947i
$162$		0			0
$163$	130.921			0.803197			0.401598	−	0.915816i	$-0.368455\pi$
$163$	130.921			0.803197			0.401598	+	0.915816i	$0.368455\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−106.185	−	61.3060i	−0.635839	−	0.367102i	0.147171	−	0.989111i	$-0.452983\pi$
$167$	−106.185	−	61.3060i	−0.635839	−	0.367102i	−0.783010	+	0.622009i	$0.786317\pi$
$168$		0			0
$169$	−1.57756	−	2.73242i	−0.00933469	−	0.0161682i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	276.607	−	159.699i	1.59888	−	0.923115i	0.607179	−	0.794565i	$-0.292301\pi$
$173$	276.607	−	159.699i	1.59888	−	0.923115i	0.991703	−	0.128550i	$-0.0410322\pi$
$174$		0			0
$175$	−0.772110	+	1.33733i	−0.00441206	+	0.00764191i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			204.943i			1.14493i	0.819929	+	0.572466i	$0.194013\pi$
$179$			204.943i			1.14493i	−0.819929	+	0.572466i	$0.805987\pi$
$180$		0			0
$181$	−172.729	−	99.7250i	−0.954302	−	0.550967i	−0.0598875	−	0.998205i	$-0.519074\pi$
$181$	−172.729	−	99.7250i	−0.954302	−	0.550967i	−0.894415	+	0.447238i	$0.852408\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	214.241	−	123.692i	1.15806	−	0.668605i
$186$		0			0
$187$	−73.0558	−	126.536i	−0.390673	−	0.676665i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−66.6367			−0.348883			−0.174442	−	0.984668i	$-0.555812\pi$
$191$	−66.6367			−0.348883			−0.174442	+	0.984668i	$0.555812\pi$
$192$		0			0
$193$	−31.3434	+	18.0961i	−0.162401	+	0.0937623i	−0.578998	−	0.815329i	$-0.696556\pi$
$193$	−31.3434	+	18.0961i	−0.162401	+	0.0937623i	0.416597	+	0.909091i	$0.363223\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	180.463			0.916055			0.458028	−	0.888938i	$-0.348556\pi$
$197$	180.463			0.916055			0.458028	+	0.888938i	$0.348556\pi$
$198$		0			0
$199$	32.1761	−	55.7307i	0.161689	−	0.280054i	−0.773786	−	0.633448i	$-0.781639\pi$
$199$	32.1761	−	55.7307i	0.161689	−	0.280054i	0.935475	+	0.353394i	$0.114973\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	235.716	+	136.091i	1.16116	+	0.670399i
$204$		0			0
$205$	139.588	+	80.5909i	0.680915	+	0.393127i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	302.988	−	68.6916i	1.44970	−	0.328668i
$210$		0			0
$211$	171.356	−	98.9324i	0.812114	−	0.468874i	−0.0355756	−	0.999367i	$-0.511326\pi$
$211$	171.356	−	98.9324i	0.812114	−	0.468874i	0.847689	+	0.530493i	$0.177993\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−22.6379	+	39.2101i	−0.105293	+	0.182372i
$216$		0			0
$217$		−	202.210i		−	0.931842i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	115.075i		−	0.520701i
$222$		0			0
$223$	70.2326	−	40.5488i	0.314944	−	0.181833i	−0.334193	−	0.942505i	$-0.608464\pi$
$223$	70.2326	−	40.5488i	0.314944	−	0.181833i	0.649137	+	0.760672i	$0.275130\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			90.2255i			0.397469i	0.980053	+	0.198735i	$0.0636832\pi$
$227$			90.2255i			0.397469i	−0.980053	+	0.198735i	$0.936317\pi$
$228$		0			0
$229$	55.9088			0.244143			0.122072	−	0.992521i	$-0.461046\pi$
$229$	55.9088			0.244143			0.122072	+	0.992521i	$0.461046\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−196.214	−	339.853i	−0.842120	−	1.45859i	−0.888099	−	0.459653i	$-0.847974\pi$
$233$	−196.214	−	339.853i	−0.842120	−	1.45859i	0.0459786	−	0.998942i	$-0.485359\pi$
$234$		0			0
$235$	200.068			0.851352
$236$		0			0
$237$		0			0
$238$		0			0
$239$	141.732			0.593022			0.296511	−	0.955029i	$-0.404177\pi$
$239$	141.732			0.593022			0.296511	+	0.955029i	$0.404177\pi$
$240$		0			0
$241$	−135.848	−	78.4321i	−0.563686	−	0.325445i	0.190937	−	0.981602i	$-0.438847\pi$
$241$	−135.848	−	78.4321i	−0.563686	−	0.325445i	−0.754624	+	0.656158i	$0.772181\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−34.3004	−	59.4101i	−0.140002	−	0.242490i
$246$		0			0
$247$	233.708	+	72.4616i	0.946185	+	0.293367i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	155.691	−	269.665i	0.620283	−	1.07436i	−0.369150	−	0.929370i	$-0.620351\pi$
$251$	155.691	−	269.665i	0.620283	−	1.07436i	0.989433	−	0.144992i	$-0.0463155\pi$
$252$		0			0
$253$	119.030	−	206.166i	0.470473	−	0.814884i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−57.9437	−	33.4538i	−0.225462	−	0.130170i	0.383015	−	0.923742i	$-0.374886\pi$
$257$	−57.9437	−	33.4538i	−0.225462	−	0.130170i	−0.608477	+	0.793572i	$0.708219\pi$
$258$		0			0
$259$			390.151i			1.50638i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−29.7834	−	51.5863i	−0.113245	−	0.196146i	0.803832	−	0.594856i	$-0.202791\pi$
$263$	−29.7834	−	51.5863i	−0.113245	−	0.196146i	−0.917077	+	0.398711i	$0.869458\pi$
$264$		0			0
$265$			426.875i			1.61085i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	107.536	−	62.0859i	0.399762	−	0.230802i	−0.286620	−	0.958044i	$-0.592532\pi$
$269$	107.536	−	62.0859i	0.399762	−	0.230802i	0.686381	+	0.727242i	$0.259198\pi$
$270$		0			0
$271$	−226.667	−	392.599i	−0.836410	−	1.44870i	−0.892877	−	0.450300i	$-0.851317\pi$
$271$	−226.667	−	392.599i	−0.836410	−	1.44870i	0.0564669	−	0.998404i	$-0.482016\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.59483	−	2.76233i	0.00579939	−	0.0100448i
$276$		0			0
$277$	53.0523			0.191524			0.0957622	−	0.995404i	$-0.469471\pi$
$277$	53.0523			0.191524			0.0957622	+	0.995404i	$0.469471\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	119.565	+	69.0306i	0.425497	+	0.245661i	0.697426	−	0.716657i	$-0.254328\pi$
$281$	119.565	+	69.0306i	0.425497	+	0.245661i	−0.271930	+	0.962317i	$0.587662\pi$
$282$		0			0
$283$	−170.242	−	294.868i	−0.601562	−	1.04194i	−0.992585	−	0.121554i	$-0.961212\pi$
$283$	−170.242	−	294.868i	−0.601562	−	1.04194i	0.391023	−	0.920381i	$-0.372121\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−220.145	+	127.101i	−0.767055	+	0.442859i
$288$		0			0
$289$	104.576	−	181.132i	0.361856	−	0.626754i
$290$		0			0
$291$		0			0
$292$		0			0
$293$			44.8116i			0.152941i	0.997072	+	0.0764704i	$0.0243651\pi$
$293$			44.8116i			0.152941i	−0.997072	+	0.0764704i	$0.975635\pi$
$294$		0			0
$295$	270.706	+	156.292i	0.917646	+	0.529803i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	162.372	−	93.7457i	0.543051	−	0.313531i
$300$		0			0
$301$	−35.7025	−	61.8385i	−0.118613	−	0.205444i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	383.418			1.25711
$306$		0			0
$307$	81.0030	−	46.7671i	0.263853	−	0.152336i	−0.362238	−	0.932086i	$-0.617987\pi$
$307$	81.0030	−	46.7671i	0.263853	−	0.152336i	0.626091	+	0.779750i	$0.284654\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	225.416			0.724809			0.362405	−	0.932021i	$-0.381956\pi$
$311$	225.416			0.724809			0.362405	+	0.932021i	$0.381956\pi$
$312$		0			0
$313$	−169.798	+	294.098i	−0.542485	+	0.939612i	0.456276	+	0.889839i	$0.349183\pi$
$313$	−169.798	+	294.098i	−0.542485	+	0.939612i	−0.998761	+	0.0497730i	$0.984150\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−363.555	−	209.899i	−1.14686	−	0.662141i	−0.198742	−	0.980052i	$-0.563686\pi$
$317$	−363.555	−	209.899i	−1.14686	−	0.662141i	−0.948121	+	0.317910i	$0.897019\pi$
$318$		0			0
$319$	−486.884	−	281.103i	−1.52628	−	0.881200i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−115.298	−	124.624i	−0.356959	−	0.385834i
$324$		0			0
$325$	2.17556	−	1.25606i	0.00669404	−	0.00386481i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−157.764	+	273.256i	−0.479527	+	0.830565i
$330$		0			0
$331$			649.784i			1.96309i	0.191224	+	0.981546i	$0.438754\pi$
$331$			649.784i			1.96309i	−0.191224	+	0.981546i	$0.561246\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			404.618i			1.20782i
$336$		0			0
$337$	103.025	−	59.4815i	0.305712	−	0.176503i	−0.339294	−	0.940680i	$-0.610188\pi$
$337$	103.025	−	59.4815i	0.305712	−	0.176503i	0.645006	+	0.764177i	$0.276855\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			417.675i			1.22485i
$342$		0			0
$343$	−279.705			−0.815467
$344$		0			0
$345$		0			0
$346$		0			0
$347$	320.971	+	555.939i	0.924989	+	1.60213i	0.791578	+	0.611068i	$0.209260\pi$
$347$	320.971	+	555.939i	0.924989	+	1.60213i	0.133411	+	0.991061i	$0.457407\pi$
$348$		0			0
$349$	−76.1335			−0.218148			−0.109074	−	0.994034i	$-0.534788\pi$
$349$	−76.1335			−0.218148			−0.109074	+	0.994034i	$0.534788\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−61.9059			−0.175371			−0.0876854	−	0.996148i	$-0.527947\pi$
$353$	−61.9059			−0.175371			−0.0876854	+	0.996148i	$0.527947\pi$
$354$		0			0
$355$	−55.3945	−	31.9820i	−0.156041	−	0.0900902i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−159.790	−	276.764i	−0.445097	−	0.770930i	0.552962	−	0.833206i	$-0.313497\pi$
$359$	−159.790	−	276.764i	−0.445097	−	0.770930i	−0.998059	+	0.0622764i	$0.980164\pi$
$360$		0			0
$361$	325.704	−	155.686i	0.902227	−	0.431262i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−212.240	+	367.610i	−0.581479	+	1.00715i
$366$		0			0
$367$	−165.635	+	286.889i	−0.451323	+	0.781714i	−0.998468	−	0.0553234i	$-0.982381\pi$
$367$	−165.635	+	286.889i	−0.451323	+	0.781714i	0.547146	+	0.837037i	$0.315714\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−583.033	−	336.614i	−1.57152	−	0.907316i
$372$		0			0
$373$			81.2925i			0.217942i	0.994045	+	0.108971i	$0.0347556\pi$
$373$			81.2925i			0.217942i	−0.994045	+	0.108971i	$0.965244\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−221.391	−	383.461i	−0.587245	−	1.01714i
$378$		0			0
$379$		−	656.466i		−	1.73210i	−0.499958	−	0.866050i	$-0.666651\pi$
$379$		−	656.466i		−	1.73210i	0.499958	−	0.866050i	$-0.333349\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−79.5352	+	45.9197i	−0.207664	+	0.119895i	−0.600225	−	0.799831i	$-0.704922\pi$
$383$	−79.5352	+	45.9197i	−0.207664	+	0.119895i	0.392561	+	0.919726i	$0.371589\pi$
$384$		0			0
$385$	324.864	+	562.682i	0.843804	+	1.46151i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−53.6411	+	92.9091i	−0.137895	+	0.238841i	−0.926700	−	0.375803i	$-0.877367\pi$
$389$	−53.6411	+	92.9091i	−0.137895	+	0.238841i	0.788805	+	0.614644i	$0.210700\pi$
$390$		0			0
$391$	−130.095			−0.332723
$392$		0			0
$393$		0			0
$394$		0			0
$395$	536.207	+	309.579i	1.35749	+	0.783745i
$396$		0			0
$397$	−292.013	−	505.782i	−0.735550	−	1.27401i	−0.954482	−	0.298269i	$-0.903591\pi$
$397$	−292.013	−	505.782i	−0.735550	−	1.27401i	0.218932	−	0.975740i	$-0.429743\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	375.374	−	216.722i	0.936094	−	0.540454i	0.0473606	−	0.998878i	$-0.484919\pi$
$401$	375.374	−	216.722i	0.936094	−	0.540454i	0.888734	+	0.458423i	$0.151586\pi$
$402$		0			0
$403$	−164.477	+	284.882i	−0.408130	+	0.706902i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	805.877i		−	1.98004i
$408$		0			0
$409$	398.602	+	230.133i	0.974577	+	0.562672i	0.900629	−	0.434590i	$-0.143107\pi$
$409$	398.602	+	230.133i	0.974577	+	0.562672i	0.0739487	+	0.997262i	$0.476440\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−426.932	+	246.489i	−1.03373	+	0.596826i
$414$		0			0
$415$	55.0410	+	95.3338i	0.132629	+	0.229720i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−701.648			−1.67458			−0.837289	−	0.546760i	$-0.815861\pi$
$419$	−701.648			−1.67458			−0.837289	+	0.546760i	$0.815861\pi$
$420$		0			0
$421$	113.807	−	65.7067i	0.270326	−	0.156073i	−0.358710	−	0.933449i	$-0.616783\pi$
$421$	113.807	−	65.7067i	0.270326	−	0.156073i	0.629036	+	0.777376i	$0.283450\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1.74309			−0.00410139
$426$		0			0
$427$	−302.346	+	523.678i	−0.708070	+	1.22641i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−345.433	−	199.436i	−0.801469	−	0.462728i	0.0425156	−	0.999096i	$-0.486463\pi$
$431$	−345.433	−	199.436i	−0.801469	−	0.462728i	−0.843985	+	0.536367i	$0.819796\pi$
$432$		0			0
$433$	570.124	+	329.161i	1.31668	+	0.760187i	0.983193	−	0.182567i	$-0.0584405\pi$
$433$	570.124	+	329.161i	1.31668	+	0.760187i	0.333489	+	0.942754i	$0.391774\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	81.9195	−	264.212i	0.187459	−	0.604604i
$438$		0			0
$439$	−152.109	+	87.8200i	−0.346489	+	0.200046i	−0.663138	−	0.748497i	$-0.730776\pi$
$439$	−152.109	+	87.8200i	−0.346489	+	0.200046i	0.316649	+	0.948543i	$0.397442\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−423.361	+	733.282i	−0.955668	+	1.65527i	−0.222835	+	0.974856i	$0.571531\pi$
$443$	−423.361	+	733.282i	−0.955668	+	1.65527i	−0.732832	+	0.680409i	$0.761802\pi$
$444$		0			0
$445$		−	568.526i		−	1.27759i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	659.400i		−	1.46860i	−0.678827	−	0.734298i	$-0.737511\pi$
$449$		−	659.400i		−	1.46860i	0.678827	−	0.734298i	$-0.262489\pi$
$450$		0			0
$451$	454.720	−	262.533i	1.00825	−	0.582113i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			511.715i			1.12465i
$456$		0			0
$457$	−882.171			−1.93035			−0.965176	−	0.261602i	$-0.915749\pi$
$457$	−882.171			−1.93035			−0.965176	+	0.261602i	$0.915749\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−69.8914	−	121.055i	−0.151608	−	0.262593i	0.780211	−	0.625517i	$-0.215112\pi$
$461$	−69.8914	−	121.055i	−0.151608	−	0.262593i	−0.931819	+	0.362924i	$0.881779\pi$
$462$		0			0
$463$	73.9066			0.159625			0.0798127	−	0.996810i	$-0.474568\pi$
$463$	73.9066			0.159625			0.0798127	+	0.996810i	$0.474568\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	689.899			1.47730			0.738650	−	0.674089i	$-0.235464\pi$
$467$	689.899			1.47730			0.738650	+	0.674089i	$0.235464\pi$
$468$		0			0
$469$	−552.634	−	319.063i	−1.17832	−	0.680306i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	73.7453	+	127.731i	0.155910	+	0.270044i
$474$		0			0
$475$	1.09761	−	3.54007i	0.00231075	−	0.00745279i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−178.186	+	308.628i	−0.371996	+	0.644316i	−0.989873	−	0.141959i	$-0.954660\pi$
$479$	−178.186	+	308.628i	−0.371996	+	0.644316i	0.617876	+	0.786275i	$0.287993\pi$
$480$		0			0
$481$	317.347	−	549.661i	0.659765	−	1.14275i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−735.243	−	424.493i	−1.51596	−	0.875242i
$486$		0			0
$487$			200.667i			0.412047i	0.978547	+	0.206024i	$0.0660523\pi$
$487$			200.667i			0.412047i	−0.978547	+	0.206024i	$0.933948\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	172.203	+	298.265i	0.350719	+	0.607464i	0.986376	−	0.164508i	$-0.0526037\pi$
$491$	172.203	+	298.265i	0.350719	+	0.607464i	−0.635656	+	0.771972i	$0.719270\pi$
$492$		0			0
$493$			307.234i			0.623192i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	87.3631	−	50.4391i	0.175781	−	0.101487i
$498$		0			0
$499$	35.6715	+	61.7848i	0.0714859	+	0.123817i	0.899553	−	0.436812i	$-0.143893\pi$
$499$	35.6715	+	61.7848i	0.0714859	+	0.123817i	−0.828067	+	0.560629i	$0.810559\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−316.476	+	548.153i	−0.629177	+	1.08977i	0.358540	+	0.933515i	$0.383275\pi$
$503$	−316.476	+	548.153i	−0.629177	+	1.08977i	−0.987717	+	0.156253i	$0.950059\pi$
$504$		0			0
$505$	−697.946			−1.38207
$506$		0			0
$507$		0			0
$508$		0			0
$509$	86.8416	+	50.1380i	0.170612	+	0.0985030i	0.582874	−	0.812562i	$-0.301928\pi$
$509$	86.8416	+	50.1380i	0.170612	+	0.0985030i	−0.412262	+	0.911065i	$0.635261\pi$
$510$		0			0
$511$	−334.725	−	579.761i	−0.655040	−	1.13456i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−357.878	+	206.621i	−0.694909	+	0.401206i
$516$		0			0
$517$	325.870	−	564.424i	0.630310	−	1.09173i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			436.922i			0.838622i	0.907843	+	0.419311i	$0.137728\pi$
$521$			436.922i			0.838622i	−0.907843	+	0.419311i	$0.862272\pi$
$522$		0			0
$523$	−22.1418	−	12.7836i	−0.0423362	−	0.0244428i	0.478683	−	0.877988i	$-0.341114\pi$
$523$	−22.1418	−	12.7836i	−0.0423362	−	0.0244428i	−0.521019	+	0.853545i	$0.674448\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	197.671	−	114.125i	0.375087	−	0.216557i
$528$		0			0
$529$	158.518	+	274.562i	0.299657	+	0.519020i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	413.532			0.775858
$534$		0			0
$535$	3.23061	−	1.86520i	0.00603853	−	0.00348635i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−223.474			−0.414608
$540$		0			0
$541$	−34.8666	+	60.3908i	−0.0644485	+	0.111628i	−0.896449	−	0.443146i	$-0.853862\pi$
$541$	−34.8666	+	60.3908i	−0.0644485	+	0.111628i	0.832001	+	0.554775i	$0.187195\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	284.689	+	164.365i	0.522366	+	0.301588i
$546$		0			0
$547$	194.009	+	112.011i	0.354678	+	0.204774i	0.666744	−	0.745287i	$-0.267688\pi$
$547$	194.009	+	112.011i	0.354678	+	0.204774i	−0.312065	+	0.950061i	$0.601021\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−623.967	−	193.462i	−1.13243	−	0.351111i
$552$		0			0
$553$	−845.656	+	488.240i	−1.52922	+	0.882893i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	72.8716	−	126.217i	0.130829	−	0.226602i	−0.793168	−	0.609003i	$-0.791570\pi$
$557$	72.8716	−	126.217i	0.130829	−	0.226602i	0.923996	+	0.382401i	$0.124903\pi$
$558$		0			0
$559$			116.161i			0.207801i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	136.626i		−	0.242675i	−0.992611	−	0.121338i	$-0.961282\pi$
$563$		−	136.626i		−	0.242675i	0.992611	−	0.121338i	$-0.0387184\pi$
$564$		0			0
$565$	−687.776	+	397.088i	−1.21730	+	0.702810i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			519.059i			0.912231i	0.889921	+	0.456116i	$0.150760\pi$
$569$			519.059i			0.912231i	−0.889921	+	0.456116i	$0.849240\pi$
$570$		0			0
$571$	−81.2673			−0.142325			−0.0711623	−	0.997465i	$-0.522671\pi$
$571$	−81.2673			−0.142325			−0.0711623	+	0.997465i	$0.522671\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−1.42001	−	2.45952i	−0.00246958	−	0.00427743i
$576$		0			0
$577$	−570.591			−0.988893			−0.494447	−	0.869208i	$-0.664629\pi$
$577$	−570.591			−0.988893			−0.494447	+	0.869208i	$0.664629\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−173.611			−0.298815
$582$		0			0
$583$	1204.28	+	695.294i	2.06567	+	1.19261i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−170.998	−	296.178i	−0.291309	−	0.504562i	0.682811	−	0.730595i	$-0.260757\pi$
$587$	−170.998	−	296.178i	−0.291309	−	0.504562i	−0.974120	+	0.226034i	$0.927424\pi$
$588$		0			0
$589$	107.308	+	473.318i	0.182187	+	0.803595i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−380.900	+	659.737i	−0.642326	+	1.11254i	0.342586	+	0.939487i	$0.388697\pi$
$593$	−380.900	+	659.737i	−0.642326	+	1.11254i	−0.984912	+	0.173055i	$0.944636\pi$
$594$		0			0
$595$	177.532	−	307.494i	0.298373	−	0.516797i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−607.037	−	350.473i	−1.01342	−	0.585097i	−0.101227	−	0.994863i	$-0.532277\pi$
$599$	−607.037	−	350.473i	−1.01342	−	0.585097i	−0.912191	+	0.409766i	$0.865610\pi$
$600$		0			0
$601$		−	256.836i		−	0.427348i	−0.976905	−	0.213674i	$-0.931457\pi$
$601$		−	256.836i		−	0.427348i	0.976905	−	0.213674i	$-0.0685431\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−367.346	−	636.262i	−0.607184	−	1.05167i
$606$		0			0
$607$		−	28.6249i		−	0.0471580i	−0.999722	−	0.0235790i	$-0.992494\pi$
$607$		−	28.6249i		−	0.0471580i	0.999722	−	0.0235790i	$-0.00750612\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	444.530	−	256.650i	0.727545	−	0.420048i
$612$		0			0
$613$	216.448	+	374.899i	0.353096	+	0.611581i	0.986790	−	0.162003i	$-0.0517954\pi$
$613$	216.448	+	374.899i	0.353096	+	0.611581i	−0.633694	+	0.773584i	$0.718462\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−153.340	+	265.593i	−0.248526	+	0.430459i	−0.963117	−	0.269083i	$-0.913279\pi$
$617$	−153.340	+	265.593i	−0.248526	+	0.430459i	0.714591	+	0.699542i	$0.246613\pi$
$618$		0			0
$619$	106.458			0.171983			0.0859916	−	0.996296i	$-0.472594\pi$
$619$	106.458			0.171983			0.0859916	+	0.996296i	$0.472594\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	776.502	+	448.314i	1.24639	+	0.719605i
$624$		0			0
$625$	314.919	+	545.456i	0.503871	+	0.872730i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−381.394	+	220.198i	−0.606349	+	0.350076i
$630$		0			0
$631$	−191.653	+	331.952i	−0.303728	+	0.526073i	−0.976977	−	0.213343i	$-0.931565\pi$
$631$	−191.653	+	331.952i	−0.303728	+	0.526073i	0.673249	+	0.739416i	$0.264898\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	1068.56i		−	1.68277i
$636$		0			0
$637$	−152.424	−	88.0020i	−0.239284	−	0.138151i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	483.399	−	279.090i	0.754132	−	0.435398i	−0.0730531	−	0.997328i	$-0.523274\pi$
$641$	483.399	−	279.090i	0.754132	−	0.435398i	0.827185	+	0.561930i	$0.189941\pi$
$642$		0			0
$643$	−457.645	−	792.665i	−0.711734	−	1.23276i	−0.964206	−	0.265156i	$-0.914577\pi$
$643$	−457.645	−	792.665i	−0.711734	−	1.23276i	0.252471	−	0.967604i	$-0.418757\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	194.941			0.301300			0.150650	−	0.988587i	$-0.451863\pi$
$647$	194.941			0.301300			0.150650	+	0.988587i	$0.451863\pi$
$648$		0			0
$649$	881.850	−	509.136i	1.35878	−	0.784493i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−555.988			−0.851437			−0.425718	−	0.904856i	$-0.639979\pi$
$653$	−555.988			−0.851437			−0.425718	+	0.904856i	$0.639979\pi$
$654$		0			0
$655$	−183.624	+	318.046i	−0.280342	+	0.485567i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	61.4668	+	35.4879i	0.0932728	+	0.0538511i	0.545911	−	0.837843i	$-0.316184\pi$
$659$	61.4668	+	35.4879i	0.0932728	+	0.0538511i	−0.452638	+	0.891694i	$0.649517\pi$
$660$		0			0
$661$	242.120	+	139.788i	0.366294	+	0.211480i	0.671838	−	0.740698i	$-0.265505\pi$
$661$	242.120	+	139.788i	0.366294	+	0.211480i	−0.305544	+	0.952178i	$0.598838\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	512.706	+	554.179i	0.770986	+	0.833352i
$666$		0			0
$667$	−433.512	+	250.288i	−0.649943	+	0.375245i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	624.511	−	1081.68i	0.930716	−	1.61205i
$672$		0			0
$673$			666.977i			0.991051i	0.868593	+	0.495525i	$0.165024\pi$
$673$			666.977i			0.991051i	−0.868593	+	0.495525i	$0.834976\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	544.859i		−	0.804814i	−0.915461	−	0.402407i	$-0.868174\pi$
$677$		−	544.859i		−	0.804814i	0.915461	−	0.402407i	$-0.131826\pi$
$678$		0			0
$679$	1159.56	−	669.471i	1.70774	−	0.985966i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			782.237i			1.14530i	0.819801	+	0.572648i	$0.194084\pi$
$683$			782.237i			1.14530i	−0.819801	+	0.572648i	$0.805916\pi$
$684$		0			0
$685$	498.341			0.727505
$686$		0			0
$687$		0			0
$688$		0			0
$689$	547.601	+	948.473i	0.794777	+	1.37659i
$690$		0			0
$691$	205.312			0.297124			0.148562	−	0.988903i	$-0.452536\pi$
$691$	205.312			0.297124			0.148562	+	0.988903i	$0.452536\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	1102.97			1.58701
$696$		0			0
$697$	−248.495	−	143.469i	−0.356521	−	0.205838i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	279.434	+	483.995i	0.398622	+	0.690434i	0.993556	−	0.113340i	$-0.0361550\pi$
$701$	279.434	+	483.995i	0.398622	+	0.690434i	−0.594934	+	0.803775i	$0.702822\pi$
$702$		0			0
$703$	−207.044	−	913.237i	−0.294515	−	1.29906i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	550.368	−	953.266i	0.778456	−	1.34833i
$708$		0			0
$709$	288.187	−	499.154i	0.406469	−	0.704025i	−0.588022	−	0.808845i	$-0.700093\pi$
$709$	288.187	−	499.154i	0.406469	−	0.704025i	0.994491	+	0.104820i	$0.0334265\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	322.065	+	185.944i	0.451704	+	0.260792i
$714$		0			0
$715$		−	1056.97i		−	1.47828i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−509.468	−	882.425i	−0.708579	−	1.22729i	−0.965384	−	0.260832i	$-0.916003\pi$
$719$	−509.468	−	882.425i	−0.708579	−	1.22729i	0.256805	−	0.966463i	$-0.417330\pi$
$720$		0			0
$721$		−	651.727i		−	0.903921i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−5.80845	+	3.35351i	−0.00801166	+	0.00462553i
$726$		0			0
$727$	−91.2836	−	158.108i	−0.125562	−	0.217480i	0.796390	−	0.604783i	$-0.206740\pi$
$727$	−91.2836	−	158.108i	−0.125562	−	0.217480i	−0.921953	+	0.387303i	$0.873407\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	40.3003	−	69.8022i	0.0551304	−	0.0954886i
$732$		0			0
$733$	−144.986			−0.197798			−0.0988990	−	0.995097i	$-0.531532\pi$
$733$	−144.986			−0.197798			−0.0988990	+	0.995097i	$0.531532\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	1141.49	+	659.042i	1.54884	+	0.894222i
$738$		0			0
$739$	410.289	+	710.641i	0.555194	+	0.961625i	0.997888	+	0.0649521i	$0.0206895\pi$
$739$	410.289	+	710.641i	0.555194	+	0.961625i	−0.442694	+	0.896673i	$0.645977\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−1096.80	+	633.239i	−1.47618	+	0.852274i	−0.999639	−	0.0268779i	$-0.991443\pi$
$743$	−1096.80	+	633.239i	−1.47618	+	0.852274i	−0.476542	+	0.879151i	$0.658110\pi$
$744$		0			0
$745$	−444.910	+	770.607i	−0.597195	+	1.03437i
$746$		0			0
$747$		0			0
$748$		0			0
$749$			5.88323i			0.00785478i
$750$		0			0
$751$	−738.952	−	426.634i	−0.983957	−	0.568088i	−0.0804946	−	0.996755i	$-0.525650\pi$
$751$	−738.952	−	426.634i	−0.983957	−	0.568088i	−0.903462	+	0.428667i	$0.858983\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	927.505	−	535.495i	1.22848	−	0.709265i
$756$		0			0
$757$	−170.760	−	295.765i	−0.225574	−	0.390706i	0.730917	−	0.682466i	$-0.239093\pi$
$757$	−170.760	−	295.765i	−0.225574	−	0.390706i	−0.956492	+	0.291760i	$0.905759\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−1213.79			−1.59499			−0.797497	−	0.603323i	$-0.793843\pi$
$761$	−1213.79			−1.59499			−0.797497	+	0.603323i	$0.793843\pi$
$762$		0			0
$763$	−448.986	+	259.222i	−0.588448	+	0.339741i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	801.973			1.04560
$768$		0			0
$769$	384.753	−	666.412i	0.500329	−	0.866596i	−0.499670	−	0.866216i	$-0.666546\pi$
$769$	384.753	−	666.412i	0.500329	−	0.866596i	1.00000		0.000380508i	$-0.000121119\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−144.274	−	83.2965i	−0.186641	−	0.107757i	0.403768	−	0.914861i	$-0.367700\pi$
$773$	−144.274	−	83.2965i	−0.186641	−	0.107757i	−0.590409	+	0.807104i	$0.701034\pi$
$774$		0			0
$775$	4.31523	+	2.49140i	0.00556804	+	0.00321471i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	447.849	−	414.333i	0.574903	−	0.531878i
$780$		0			0
$781$	−180.453	+	104.185i	−0.231054	+	0.133399i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−31.8738	+	55.2070i	−0.0406035	+	0.0703274i
$786$		0			0
$787$			79.4979i			0.101014i	0.998724	+	0.0505069i	$0.0160837\pi$
$787$			79.4979i			0.101014i	−0.998724	+	0.0505069i	$0.983916\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	1252.50i		−	1.58344i
$792$		0			0
$793$	851.915	−	491.853i	1.07429	−	0.620244i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	1047.76i		−	1.31463i	−0.753618	−	0.657313i	$-0.771693\pi$
$797$		−	1047.76i		−	1.31463i	0.753618	−	0.657313i	$-0.228307\pi$
$798$		0			0
$799$	−356.163			−0.445761
$800$		0			0
$801$		0			0
$802$		0			0
$803$	691.392	+	1197.53i	0.861011	+	1.49132i
$804$		0			0
$805$	578.505			0.718640
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−945.929			−1.16926			−0.584628	−	0.811301i	$-0.698760\pi$
$809$	−945.929			−1.16926			−0.584628	+	0.811301i	$0.698760\pi$
$810$		0			0
$811$	200.806	+	115.935i	0.247602	+	0.142953i	0.618666	−	0.785654i	$-0.287673\pi$
$811$	200.806	+	115.935i	0.247602	+	0.142953i	−0.371063	+	0.928608i	$0.621007\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−328.577	−	569.112i	−0.403162	−	0.698297i
$816$		0			0
$817$	116.386	+	125.801i	0.142455	+	0.153979i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−480.029	+	831.434i	−0.584688	+	1.01271i	0.410227	+	0.911984i	$0.365450\pi$
$821$	−480.029	+	831.434i	−0.584688	+	1.01271i	−0.994914	+	0.100725i	$0.967884\pi$
$822$		0			0
$823$	−519.453	+	899.719i	−0.631170	+	1.09322i	0.356143	+	0.934432i	$0.384092\pi$
$823$	−519.453	+	899.719i	−0.631170	+	1.09322i	−0.987313	+	0.158787i	$0.949242\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−416.992	−	240.751i	−0.504223	−	0.291113i	0.226233	−	0.974073i	$-0.427359\pi$
$827$	−416.992	−	240.751i	−0.504223	−	0.291113i	−0.730456	+	0.682960i	$0.760692\pi$
$828$		0			0
$829$		−	635.461i		−	0.766540i	−0.923636	−	0.383270i	$-0.874798\pi$
$829$		−	635.461i		−	0.766540i	0.923636	−	0.383270i	$-0.125202\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	61.0620	+	105.762i	0.0733037	+	0.126966i
$834$		0			0
$835$			615.447i			0.737063i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	777.925	−	449.135i	0.927205	−	0.535322i	0.0412788	−	0.999148i	$-0.486857\pi$
$839$	777.925	−	449.135i	0.927205	−	0.535322i	0.885927	+	0.463825i	$0.153523\pi$
$840$		0			0
$841$	170.585	+	295.461i	0.202835	+	0.351321i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−7.91853	+	13.7153i	−0.00937104	+	0.0162311i
$846$		0			0
$847$	1158.69			1.36799
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−621.405	−	358.768i	−0.730205	−	0.421584i
$852$		0			0
$853$	725.701	+	1256.95i	0.850763	+	1.47356i	0.880521	+	0.474007i	$0.157193\pi$
$853$	725.701	+	1256.95i	0.850763	+	1.47356i	−0.0297581	+	0.999557i	$0.509474\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−455.016	+	262.704i	−0.530941	+	0.306539i	−0.741399	−	0.671064i	$-0.765838\pi$
$857$	−455.016	+	262.704i	−0.530941	+	0.306539i	0.210459	+	0.977603i	$0.432504\pi$
$858$		0			0
$859$	60.5838	−	104.934i	0.0705283	−	0.122159i	−0.828605	−	0.559834i	$-0.810865\pi$
$859$	60.5838	−	104.934i	0.0705283	−	0.122159i	0.899133	+	0.437676i	$0.144198\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			1093.87i			1.26752i	0.773529	+	0.633760i	$0.218490\pi$
$863$			1093.87i			1.26752i	−0.773529	+	0.633760i	$0.781510\pi$
$864$		0			0
$865$	−1388.42	−	801.604i	−1.60511	−	0.926709i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	1746.75	−	1008.48i	2.01006	−	1.16051i
$870$		0			0
$871$	519.049	+	899.020i	0.595924	+	1.03217i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−985.633			−1.12644
$876$		0			0
$877$	−394.174	+	227.576i	−0.449457	+	0.259494i	−0.707601	−	0.706612i	$-0.750223\pi$
$877$	−394.174	+	227.576i	−0.449457	+	0.259494i	0.258144	+	0.966107i	$0.416889\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	691.715			0.785147			0.392574	−	0.919721i	$-0.371585\pi$
$881$	691.715			0.785147			0.392574	+	0.919721i	$0.371585\pi$
$882$		0			0
$883$	620.410	−	1074.58i	0.702616	−	1.21697i	−0.264929	−	0.964268i	$-0.585348\pi$
$883$	620.410	−	1074.58i	0.702616	−	1.21697i	0.967545	−	0.252699i	$-0.0813182\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1148.21	+	662.918i	1.29448	+	0.747371i	0.979446	−	0.201708i	$-0.0646491\pi$
$887$	1148.21	+	662.918i	1.29448	+	0.747371i	0.315039	+	0.949079i	$0.397982\pi$
$888$		0			0
$889$	1459.46	+	842.617i	1.64168	+	0.947826i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	224.273	−	723.338i	0.251145	−	0.810009i
$894$		0			0
$895$	890.884	−	514.352i	0.995401	−	0.574695i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	439.130	−	760.595i	0.488465	−	0.846045i
$900$		0			0
$901$		−	759.929i		−	0.843428i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			1001.13i			1.10622i
$906$		0			0
$907$	−348.557	+	201.240i	−0.384297	+	0.221874i	−0.679686	−	0.733503i	$-0.737884\pi$
$907$	−348.557	+	201.240i	−0.384297	+	0.221874i	0.295389	+	0.955377i	$0.404551\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	466.329i		−	0.511887i	−0.966692	−	0.255944i	$-0.917614\pi$
$911$		−	466.329i		−	0.511887i	0.966692	−	0.255944i	$-0.0823862\pi$
$912$		0			0
$913$	358.603			0.392774
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−289.595	−	501.593i	−0.315807	−	0.546994i
$918$		0			0
$919$	−1026.65			−1.11714			−0.558570	−	0.829457i	$-0.688650\pi$
$919$	−1026.65			−1.11714			−0.558570	+	0.829457i	$0.688650\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−164.108			−0.177798
$924$		0			0
$925$	−8.32596	−	4.80700i	−0.00900104	−	0.00519675i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	35.3949	+	61.3057i	0.0381000	+	0.0659911i	0.884447	−	0.466641i	$-0.154536\pi$
$929$	35.3949	+	61.3057i	0.0381000	+	0.0659911i	−0.846347	+	0.532633i	$0.821203\pi$
$930$		0			0
$931$	−253.245	+	57.4144i	−0.272014	+	0.0616696i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−366.701	+	635.145i	−0.392194	+	0.679299i
$936$		0			0
$937$	118.322	−	204.940i	0.126278	−	0.218719i	−0.795954	−	0.605357i	$-0.793030\pi$
$937$	118.322	−	204.940i	0.126278	−	0.218719i	0.922232	+	0.386638i	$0.126364\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	317.001	+	183.021i	0.336877	+	0.194496i	0.658890	−	0.752239i	$-0.271026\pi$
$941$	317.001	+	183.021i	0.336877	+	0.194496i	−0.322013	+	0.946735i	$0.604360\pi$
$942$		0			0
$943$		−	467.508i		−	0.495766i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−126.579	−	219.241i	−0.133663	−	0.231511i	0.791423	−	0.611269i	$-0.209341\pi$
$947$	−126.579	−	219.241i	−0.133663	−	0.231511i	−0.925086	+	0.379758i	$0.876007\pi$
$948$		0			0
$949$			1089.06i			1.14758i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1094.11	−	631.687i	1.14807	−	0.662841i	0.199657	−	0.979866i	$-0.436017\pi$
$953$	1094.11	−	631.687i	1.14807	−	0.662841i	0.948417	+	0.317025i	$0.102684\pi$
$954$		0			0
$955$	167.240	+	289.669i	0.175121	+	0.303318i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−392.969	+	680.642i	−0.409769	+	0.709741i
$960$		0			0
$961$	308.522			0.321043
$962$		0			0
$963$		0			0
$964$		0			0
$965$	157.327	+	90.8329i	0.163033	+	0.0941274i
$966$		0			0
$967$	6.50042	+	11.2591i	0.00672225	+	0.0116433i	0.869367	−	0.494167i	$-0.164527\pi$
$967$	6.50042	+	11.2591i	0.00672225	+	0.0116433i	−0.862645	+	0.505810i	$0.831194\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	385.798	−	222.741i	0.397320	−	0.229393i	−0.288007	−	0.957628i	$-0.592993\pi$
$971$	385.798	−	222.741i	0.397320	−	0.229393i	0.685327	+	0.728235i	$0.259659\pi$
$972$		0			0
$973$	−869.755	+	1506.46i	−0.893890	+	1.54826i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			1388.97i			1.42167i	0.703361	+	0.710833i	$0.251682\pi$
$977$			1388.97i			1.42167i	−0.703361	+	0.710833i	$0.748318\pi$
$978$		0			0
$979$	−1603.90	−	926.015i	−1.63831	−	0.945878i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	590.123	−	340.708i	0.600329	−	0.346600i	−0.168842	−	0.985643i	$-0.554003\pi$
$983$	590.123	−	340.708i	0.600329	−	0.346600i	0.769171	+	0.639043i	$0.220669\pi$
$984$		0			0
$985$	−452.914	−	784.470i	−0.459811	−	0.796416i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	131.323			0.132783
$990$		0			0
$991$	661.844	−	382.116i	0.667855	−	0.385586i	−0.127409	−	0.991850i	$-0.540666\pi$
$991$	661.844	−	382.116i	0.667855	−	0.385586i	0.795263	+	0.606264i	$0.207333\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−323.014			−0.324637
$996$		0			0
$997$	−57.3295	+	99.2976i	−0.0575020	+	0.0995964i	−0.893343	−	0.449374i	$-0.851647\pi$
$997$	−57.3295	+	99.2976i	−0.0575020	+	0.0995964i	0.835841	+	0.548971i	$0.184980\pi$
$998$		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.145.2		8
3.2	odd	2		76.3.h.a.69.2	yes	8
12.11	even	2		304.3.r.c.145.3		8
19.8	odd	6	inner	684.3.y.h.217.2		8
57.8	even	6		76.3.h.a.65.2	✓	8
57.26	odd	6		1444.3.c.b.721.4		8
57.50	even	6		1444.3.c.b.721.5		8
228.179	odd	6		304.3.r.c.65.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.3.h.a.65.2	✓	8	57.8	even	6
76.3.h.a.69.2	yes	8	3.2	odd	2
304.3.r.c.65.3		8	228.179	odd	6
304.3.r.c.145.3		8	12.11	even	2
684.3.y.h.145.2		8	1.1	even	1	trivial
684.3.y.h.217.2		8	19.8	odd	6	inner
1444.3.c.b.721.4		8	57.26	odd	6
1444.3.c.b.721.5		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)

\(f(q)\)	\(=\)	\(q+(-2.50973 - 4.34699i) q^{5} +7.91625 q^{7} +O(q^{10})\)	\(q+(-2.50973 - 4.34699i) q^{5} +7.91625 q^{7} -16.3514 q^{11} +(-11.1527 - 6.43904i) q^{13} +(4.46786 + 7.73856i) q^{17} +(-18.5298 + 4.20096i) q^{19} +(-7.27948 + 12.6084i) q^{23} +(-0.0975349 + 0.168935i) q^{25} +(29.7763 + 17.1913i) q^{29} -25.5436i q^{31} +(-19.8677 - 34.4118i) q^{35} +49.2849i q^{37} +(-27.8092 + 16.0557i) q^{41} +(-4.51003 - 7.81160i) q^{43} +(-19.9292 + 34.5184i) q^{47} +13.6670 q^{49} +(-73.6502 - 42.5220i) q^{53} +(41.0377 + 71.0794i) q^{55} +(-53.9311 + 31.1371i) q^{59} +(-38.1931 + 66.1524i) q^{61} +64.6411i q^{65} +(-69.8101 - 40.3049i) q^{67} +(11.0359 - 6.37160i) q^{71} +(-42.2833 - 73.2369i) q^{73} -129.442 q^{77} +(-106.825 + 61.6757i) q^{79} -21.9310 q^{83} +(22.4263 - 38.8434i) q^{85} +(98.0897 + 56.6321i) q^{89} +(-88.2879 - 50.9730i) q^{91} +(64.7663 + 70.0053i) q^{95} +(146.478 - 84.5692i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + q^{5} - 12 q^{7}+O(q^{10}) \) 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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