LMFDB - Embedded newform 1764.3.z.n.901.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1764,3,Mod(325,1764)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1764, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

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

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

chi := DirichletCharacter("1764.325");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1764.z (of order $6$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$48.0655186332$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 12x^{14} + 114x^{12} - 336x^{10} + 755x^{8} - 336x^{6} + 114x^{4} - 12x^{2} + 1 $ x^16 - 12x^14 + 114x^12 - 336x^10 + 755x^8 - 336x^6 + 114x^4 - 12*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{6}\cdot 7^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			901.8
Root			$0.516029 - 0.297929i$ of defining polynomial
Character	$\chi$	$=$	1764.901
Dual form			1764.3.z.n.325.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(5.33147 + 3.07813i) q^{5} +O(q^{10})$	$q+(5.33147 + 3.07813i) q^{5} +(0.975845 + 1.69021i) q^{11} +0.317025i q^{13} +(15.9944 - 9.23438i) q^{17} +(6.01255 + 3.47135i) q^{19} +(-18.4430 + 31.9442i) q^{23} +(6.44975 + 11.1713i) q^{25} +40.7893 q^{29} +(-26.2190 + 15.1376i) q^{31} +(6.84924 - 11.8632i) q^{37} -30.1625i q^{41} +41.7990 q^{43} +(42.1160 + 24.3157i) q^{47} +(-1.95169 - 3.38043i) q^{53} +12.0151i q^{55} +(20.7901 - 12.0031i) q^{59} +(69.1305 + 39.9125i) q^{61} +(-0.975845 + 1.69021i) q^{65} +(45.6985 + 79.1521i) q^{67} -75.7236 q^{71} +(-114.430 + 66.0663i) q^{73} +(12.5980 - 21.8203i) q^{79} +84.9501i q^{83} +113.698 q^{85} +(-110.889 - 64.0220i) q^{89} +(21.3705 + 37.0148i) q^{95} -4.05760i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q + 24 q^{25} - 128 q^{37} + 352 q^{43} + 256 q^{67} - 432 q^{79} + 1344 q^{85}+O(q^{100}) $ 16 * q + 24 * q^25 - 128 * q^37 + 352 * q^43 + 256 * q^67 - 432 * q^79 + 1344 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times$.

$n$	$785$	$883$	$1081$
$\chi(n)$	$1$	$1$	$e\left(\frac{5}{6}\right)$

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
$5$	5.33147	+	3.07813i	1.06629	+	0.615626i	0.927167	−	0.374649i	$-0.122237\pi$
$5$	5.33147	+	3.07813i	1.06629	+	0.615626i	0.139128	+	0.990274i	$0.455570\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	0.975845	+	1.69021i	0.0887132	+	0.153656i	0.906967	−	0.421201i	$-0.138391\pi$
$11$	0.975845	+	1.69021i	0.0887132	+	0.153656i	−0.818254	+	0.574856i	$0.805058\pi$
$12$		0			0
$13$			0.317025i			0.0243866i	0.999926	+	0.0121933i	$0.00388134\pi$
$13$			0.317025i			0.0243866i	−0.999926	+	0.0121933i	$0.996119\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	15.9944	−	9.23438i	0.940848	−	0.543199i	0.0506223	−	0.998718i	$-0.483880\pi$
$17$	15.9944	−	9.23438i	0.940848	−	0.543199i	0.890226	+	0.455519i	$0.150546\pi$
$18$		0			0
$19$	6.01255	+	3.47135i	0.316450	+	0.182702i	0.649809	−	0.760097i	$-0.274849\pi$
$19$	6.01255	+	3.47135i	0.316450	+	0.182702i	−0.333359	+	0.942800i	$0.608182\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−18.4430	+	31.9442i	−0.801868	+	1.38888i	0.116517	+	0.993189i	$0.462827\pi$
$23$	−18.4430	+	31.9442i	−0.801868	+	1.38888i	−0.918385	+	0.395688i	$0.870506\pi$
$24$		0			0
$25$	6.44975	+	11.1713i	0.257990	+	0.446852i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	40.7893			1.40653			0.703264	−	0.710929i	$-0.251725\pi$
$29$	40.7893			1.40653			0.703264	+	0.710929i	$0.251725\pi$
$30$		0			0
$31$	−26.2190	+	15.1376i	−0.845775	+	0.488308i	−0.859223	−	0.511601i	$-0.829053\pi$
$31$	−26.2190	+	15.1376i	−0.845775	+	0.488308i	0.0134481	+	0.999910i	$0.495719\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	6.84924	−	11.8632i	0.185115	−	0.320628i	−0.758500	−	0.651672i	$-0.774068\pi$
$37$	6.84924	−	11.8632i	0.185115	−	0.320628i	0.943615	+	0.331044i	$0.107401\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	30.1625i		−	0.735672i	−0.929891	−	0.367836i	$-0.880099\pi$
$41$		−	30.1625i		−	0.735672i	0.929891	−	0.367836i	$-0.119901\pi$
$42$		0			0
$43$	41.7990			0.972070			0.486035	−	0.873939i	$-0.338443\pi$
$43$	41.7990			0.972070			0.486035	+	0.873939i	$0.338443\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	42.1160	+	24.3157i	0.896084	+	0.517354i	0.875928	−	0.482442i	$-0.160250\pi$
$47$	42.1160	+	24.3157i	0.896084	+	0.517354i	0.0201565	+	0.999797i	$0.493584\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−1.95169	−	3.38043i	−0.0368243	−	0.0637816i	0.847026	−	0.531552i	$-0.178391\pi$
$53$	−1.95169	−	3.38043i	−0.0368243	−	0.0637816i	−0.883850	+	0.467770i	$0.845058\pi$
$54$		0			0
$55$			12.0151i			0.218456i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	20.7901	−	12.0031i	0.352374	−	0.203443i	−0.313356	−	0.949636i	$-0.601453\pi$
$59$	20.7901	−	12.0031i	0.352374	−	0.203443i	0.665730	+	0.746192i	$0.268120\pi$
$60$		0			0
$61$	69.1305	+	39.9125i	1.13329	+	0.654304i	0.944760	−	0.327764i	$-0.106295\pi$
$61$	69.1305	+	39.9125i	1.13329	+	0.654304i	0.188528	+	0.982068i	$0.439628\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−0.975845	+	1.69021i	−0.0150130	+	0.0260033i
$66$		0			0
$67$	45.6985	+	79.1521i	0.682067	+	1.18137i	0.974349	+	0.225042i	$0.0722521\pi$
$67$	45.6985	+	79.1521i	0.682067	+	1.18137i	−0.292282	+	0.956332i	$0.594415\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−75.7236			−1.06653			−0.533265	−	0.845948i	$-0.679035\pi$
$71$	−75.7236			−1.06653			−0.533265	+	0.845948i	$0.679035\pi$
$72$		0			0
$73$	−114.430	+	66.0663i	−1.56754	+	0.905018i	−0.571082	+	0.820893i	$0.693476\pi$
$73$	−114.430	+	66.0663i	−1.56754	+	0.905018i	−0.996455	+	0.0841251i	$0.973190\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	12.5980	−	21.8203i	0.159468	−	0.276207i	−0.775209	−	0.631705i	$-0.782355\pi$
$79$	12.5980	−	21.8203i	0.159468	−	0.276207i	0.934677	+	0.355498i	$0.115689\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			84.9501i			1.02350i	0.859136	+	0.511748i	$0.171002\pi$
$83$			84.9501i			1.02350i	−0.859136	+	0.511748i	$0.828998\pi$
$84$		0			0
$85$	113.698			1.33763
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−110.889	−	64.0220i	−1.24595	−	0.719348i	−0.275649	−	0.961259i	$-0.588893\pi$
$89$	−110.889	−	64.0220i	−1.24595	−	0.719348i	−0.970299	+	0.241911i	$0.922226\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	21.3705	+	37.0148i	0.224953	+	0.389629i
$96$		0			0
$97$		−	4.05760i		−	0.0418310i	−0.999781	−	0.0209155i	$-0.993342\pi$
$97$		−	4.05760i		−	0.0418310i	0.999781	−	0.0209155i	$-0.00665809\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−131.679	+	76.0251i	−1.30376	+	0.752724i	−0.981046	−	0.193774i	$-0.937927\pi$
$101$	−131.679	+	76.0251i	−1.30376	+	0.752724i	−0.322710	+	0.946498i	$0.604594\pi$
$102$		0			0
$103$	82.0069	+	47.3467i	0.796183	+	0.459677i	0.842135	−	0.539267i	$-0.181299\pi$
$103$	82.0069	+	47.3467i	0.796183	+	0.459677i	−0.0459515	+	0.998944i	$0.514632\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	94.1665	−	163.101i	0.880061	−	1.52431i	0.0287895	−	0.999585i	$-0.490835\pi$
$107$	94.1665	−	163.101i	0.880061	−	1.52431i	0.851272	−	0.524725i	$-0.175832\pi$
$108$		0			0
$109$	59.4472	+	102.966i	0.545387	+	0.944639i	0.998582	+	0.0532272i	$0.0169508\pi$
$109$	59.4472	+	102.966i	0.545387	+	0.944639i	−0.453195	+	0.891411i	$0.649716\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	19.3207			0.170980			0.0854900	−	0.996339i	$-0.472754\pi$
$113$	19.3207			0.170980			0.0854900	+	0.996339i	$0.472754\pi$
$114$		0			0
$115$	−196.656	+	113.540i	−1.71006	+	0.987301i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	58.5955	−	101.490i	0.484260	−	0.838763i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	74.4938i		−	0.595951i
$126$		0			0
$127$	174.995			1.37791			0.688956	−	0.724803i	$-0.258069\pi$
$127$	174.995			1.37791			0.688956	+	0.724803i	$0.258069\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−30.9172	−	17.8500i	−0.236009	−	0.136260i	0.377332	−	0.926078i	$-0.376842\pi$
$131$	−30.9172	−	17.8500i	−0.236009	−	0.136260i	−0.613341	+	0.789818i	$0.710175\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−100.022	−	173.243i	−0.730085	−	1.26454i	−0.956847	−	0.290594i	$-0.906147\pi$
$137$	−100.022	−	173.243i	−0.730085	−	1.26454i	0.226762	−	0.973950i	$-0.427186\pi$
$138$		0			0
$139$			92.8231i			0.667792i	0.942610	+	0.333896i	$0.108363\pi$
$139$			92.8231i			0.667792i	−0.942610	+	0.333896i	$0.891637\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−0.535840	+	0.309367i	−0.00374713	+	0.00216341i
$144$		0			0
$145$	217.467	+	125.555i	1.49977	+	0.865895i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	42.7410	−	74.0296i	0.286852	−	0.496843i	−0.686204	−	0.727409i	$-0.740724\pi$
$149$	42.7410	−	74.0296i	0.286852	−	0.496843i	0.973057	+	0.230566i	$0.0740577\pi$
$150$		0			0
$151$	122.296	+	211.824i	0.809910	+	1.40281i	0.912926	+	0.408126i	$0.133818\pi$
$151$	122.296	+	211.824i	0.809910	+	1.40281i	−0.103015	+	0.994680i	$0.532849\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−186.381			−1.20246
$156$		0			0
$157$	−196.327	+	113.349i	−1.25049	+	0.721970i	−0.971207	−	0.238239i	$-0.923430\pi$
$157$	−196.327	+	113.349i	−1.25049	+	0.721970i	−0.279282	+	0.960209i	$0.590096\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−22.5025	+	38.9755i	−0.138052	+	0.239114i	−0.926759	−	0.375656i	$-0.877418\pi$
$163$	−22.5025	+	38.9755i	−0.138052	+	0.239114i	0.788707	+	0.614769i	$0.210751\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			86.8063i			0.519798i	0.965636	+	0.259899i	$0.0836893\pi$
$167$			86.8063i			0.519798i	−0.965636	+	0.259899i	$0.916311\pi$
$168$		0			0
$169$	168.899			0.999405
$170$		0			0
$171$		0			0
$172$		0			0
$173$	3.18811	+	1.84066i	0.0184284	+	0.0106396i	0.509186	−	0.860657i	$-0.329947\pi$
$173$	3.18811	+	1.84066i	0.0184284	+	0.0106396i	−0.490757	+	0.871296i	$0.663280\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	148.520	+	257.244i	0.829719	+	1.43711i	0.898259	+	0.439466i	$0.144832\pi$
$179$	148.520	+	257.244i	0.829719	+	1.43711i	−0.0685406	+	0.997648i	$0.521834\pi$
$180$		0			0
$181$		−	119.483i		−	0.660130i	−0.943958	−	0.330065i	$-0.892929\pi$
$181$		−	119.483i		−	0.660130i	0.943958	−	0.330065i	$-0.107071\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	73.0331	−	42.1657i	0.394774	−	0.227923i
$186$		0			0
$187$	31.2161	+	18.0227i	0.166931	+	0.0963778i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−168.037	+	291.048i	−0.879772	+	1.52381i	−0.0281817	+	0.999603i	$0.508972\pi$
$191$	−168.037	+	291.048i	−0.879772	+	1.52381i	−0.851591	+	0.524207i	$0.824362\pi$
$192$		0			0
$193$	24.7990	+	42.9531i	0.128492	+	0.222555i	0.923093	−	0.384578i	$-0.125653\pi$
$193$	24.7990	+	42.9531i	0.128492	+	0.222555i	−0.794600	+	0.607133i	$0.792320\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	217.608			1.10461			0.552306	−	0.833642i	$-0.313748\pi$
$197$	217.608			1.10461			0.552306	+	0.833642i	$0.313748\pi$
$198$		0			0
$199$	−251.868	+	145.416i	−1.26567	+	0.730733i	−0.974165	−	0.225838i	$-0.927488\pi$
$199$	−251.868	+	145.416i	−1.26567	+	0.730733i	−0.291501	+	0.956570i	$0.594155\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	92.8442	−	160.811i	0.452898	−	0.784443i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			13.5500i			0.0648325i
$210$		0			0
$211$	237.990			1.12791			0.563957	−	0.825804i	$-0.309278\pi$
$211$	237.990			1.12791			0.563957	+	0.825804i	$0.309278\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	222.850	+	128.663i	1.03651	+	0.598431i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	2.92753	+	5.07064i	0.0132468	+	0.0229441i
$222$		0			0
$223$			179.242i			0.803776i	0.915689	+	0.401888i	$0.131646\pi$
$223$			179.242i			0.803776i	−0.915689	+	0.401888i	$0.868354\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	75.1765	−	43.4032i	0.331174	−	0.191203i	−0.325188	−	0.945649i	$-0.605428\pi$
$227$	75.1765	−	43.4032i	0.331174	−	0.191203i	0.656362	+	0.754446i	$0.272094\pi$
$228$		0			0
$229$	274.024	+	158.208i	1.19661	+	0.690863i	0.959798	−	0.280693i	$-0.0905642\pi$
$229$	274.024	+	158.208i	1.19661	+	0.690863i	0.236812	+	0.971556i	$0.423898\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−74.6497	+	129.297i	−0.320385	+	0.554923i	−0.980567	−	0.196182i	$-0.937146\pi$
$233$	−74.6497	+	129.297i	−0.320385	+	0.554923i	0.660183	+	0.751105i	$0.270479\pi$
$234$		0			0
$235$	149.693	+	259.277i	0.636993	+	1.10330i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	196.140			0.820669			0.410334	−	0.911935i	$-0.365412\pi$
$239$	196.140			0.820669			0.410334	+	0.911935i	$0.365412\pi$
$240$		0			0
$241$	−110.559	+	63.8312i	−0.458751	+	0.264860i	−0.711519	−	0.702667i	$-0.751992\pi$
$241$	−110.559	+	63.8312i	−0.458751	+	0.264860i	0.252768	+	0.967527i	$0.418659\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−1.10051	+	1.90613i	−0.00445549	+	0.00771713i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	329.344i		−	1.31213i	−0.754705	−	0.656064i	$-0.772220\pi$
$251$		−	329.344i		−	1.31213i	0.754705	−	0.656064i	$-0.227780\pi$
$252$		0			0
$253$	−71.9899			−0.284545
$254$		0			0
$255$		0			0
$256$		0			0
$257$	80.5080	+	46.4813i	0.313261	+	0.180861i	0.648385	−	0.761313i	$-0.275445\pi$
$257$	80.5080	+	46.4813i	0.313261	+	0.180861i	−0.335124	+	0.942174i	$0.608778\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−247.467	−	428.626i	−0.940940	−	1.62976i	−0.763683	−	0.645592i	$-0.776611\pi$
$263$	−247.467	−	428.626i	−0.940940	−	1.62976i	−0.177258	−	0.984164i	$-0.556723\pi$
$264$		0			0
$265$		−	24.0302i		−	0.0906800i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	335.347	−	193.613i	1.24664	−	0.719750i	0.276205	−	0.961099i	$-0.410923\pi$
$269$	335.347	−	193.613i	1.24664	−	0.719750i	0.970438	+	0.241349i	$0.0775898\pi$
$270$		0			0
$271$	−192.813	−	111.320i	−0.711486	−	0.410777i	0.100125	−	0.994975i	$-0.468076\pi$
$271$	−192.813	−	111.320i	−0.711486	−	0.410777i	−0.811611	+	0.584198i	$0.801409\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−12.5879	+	21.8029i	−0.0457742	+	0.0792832i
$276$		0			0
$277$	−69.6985	−	120.721i	−0.251619	−	0.435817i	0.712353	−	0.701822i	$-0.247630\pi$
$277$	−69.6985	−	120.721i	−0.251619	−	0.435817i	−0.963972	+	0.266005i	$0.914296\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	91.1409			0.324345			0.162172	−	0.986762i	$-0.448150\pi$
$281$	91.1409			0.324345			0.162172	+	0.986762i	$0.448150\pi$
$282$		0			0
$283$	−0.0275939	+	0.0159313i	−9.75048e−5	+	5.62944e-5i	−0.500049	−	0.865997i	$-0.666685\pi$
$283$	−0.0275939	+	0.0159313i	−9.75048e−5	+	5.62944e-5i	0.499951	+	0.866054i	$0.333351\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	26.0477	−	45.1160i	0.0901305	−	0.156111i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	32.6375i		−	0.111391i	−0.998448	−	0.0556954i	$-0.982262\pi$
$293$		−	32.6375i		−	0.111391i	0.998448	−	0.0556954i	$-0.0177376\pi$
$294$		0			0
$295$	147.789			0.500979
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−10.1271	−	5.84689i	−0.0338699	−	0.0195548i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	245.712	+	425.585i	0.805612	+	1.39536i
$306$		0			0
$307$		−	93.2341i		−	0.303694i	−0.988404	−	0.151847i	$-0.951478\pi$
$307$		−	93.2341i		−	0.303694i	0.988404	−	0.151847i	$-0.0485221\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−157.265	+	90.7970i	−0.505675	+	0.291952i	−0.731054	−	0.682319i	$-0.760971\pi$
$311$	−157.265	+	90.7970i	−0.505675	+	0.291952i	0.225379	+	0.974271i	$0.427638\pi$
$312$		0			0
$313$	257.523	+	148.681i	0.822757	+	0.475019i	0.851366	−	0.524572i	$-0.175775\pi$
$313$	257.523	+	148.681i	0.822757	+	0.475019i	−0.0286095	+	0.999591i	$0.509108\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−73.8700	+	127.947i	−0.233028	+	0.403617i	−0.958698	−	0.284427i	$-0.908197\pi$
$317$	−73.8700	+	127.947i	−0.233028	+	0.403617i	0.725670	+	0.688043i	$0.241530\pi$
$318$		0			0
$319$	39.8040	+	68.9426i	0.124778	+	0.216121i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	128.223			0.396975
$324$		0			0
$325$	−3.54158	+	2.04473i	−0.0108972	+	0.00629149i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−160.593	+	278.155i	−0.485175	+	0.840348i	−0.999855	−	0.0170346i	$-0.994577\pi$
$331$	−160.593	+	278.155i	−0.485175	+	0.840348i	0.514680	+	0.857382i	$0.327911\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			562.663i			1.67959i
$336$		0			0
$337$	−44.8944			−0.133218			−0.0666090	−	0.997779i	$-0.521218\pi$
$337$	−44.8944			−0.133218			−0.0666090	+	0.997779i	$0.521218\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−51.1714	−	29.5438i	−0.150063	−	0.0866387i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	72.6980	+	125.917i	0.209504	+	0.362872i	0.951558	−	0.307468i	$-0.0994817\pi$
$347$	72.6980	+	125.917i	0.209504	+	0.362872i	−0.742054	+	0.670340i	$0.766148\pi$
$348$		0			0
$349$		−	187.199i		−	0.536388i	−0.963365	−	0.268194i	$-0.913573\pi$
$349$		−	187.199i		−	0.536388i	0.963365	−	0.268194i	$-0.0864268\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−400.396	+	231.169i	−1.13427	+	0.654870i	−0.945005	−	0.327057i	$-0.893943\pi$
$353$	−400.396	+	231.169i	−1.13427	+	0.654870i	−0.189263	+	0.981926i	$0.560610\pi$
$354$		0			0
$355$	−403.718	−	233.087i	−1.13723	−	0.656583i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	177.697	−	307.780i	0.494977	−	0.857326i	−0.505006	−	0.863116i	$-0.668510\pi$
$359$	177.697	−	307.780i	0.494977	−	0.857326i	0.999983	+	0.00578996i	$0.00184301\pi$
$360$		0			0
$361$	−156.399	−	270.892i	−0.433240	−	0.750393i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−813.442			−2.22861
$366$		0			0
$367$	285.636	−	164.912i	0.778300	−	0.449352i	−0.0575274	−	0.998344i	$-0.518322\pi$
$367$	285.636	−	164.912i	0.778300	−	0.449352i	0.835827	+	0.548992i	$0.184988\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	203.296	−	352.120i	0.545031	−	0.944021i	−0.453574	−	0.891218i	$-0.649851\pi$
$373$	203.296	−	352.120i	0.545031	−	0.944021i	0.998605	−	0.0528024i	$-0.0168154\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			12.9312i			0.0343004i
$378$		0			0
$379$	−435.196			−1.14827			−0.574137	−	0.818759i	$-0.694662\pi$
$379$	−435.196			−1.14827			−0.574137	+	0.818759i	$0.694662\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−407.335	−	235.175i	−1.06354	−	0.614035i	−0.137130	−	0.990553i	$-0.543788\pi$
$383$	−407.335	−	235.175i	−1.06354	−	0.614035i	−0.926409	+	0.376519i	$0.877121\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−117.587	−	203.666i	−0.302280	−	0.523564i	0.674372	−	0.738392i	$-0.264414\pi$
$389$	−117.587	−	203.666i	−0.302280	−	0.523564i	−0.976652	+	0.214828i	$0.931081\pi$
$390$		0			0
$391$			681.238i			1.74230i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	134.332	−	77.5564i	0.340080	−	0.196345i
$396$		0			0
$397$	54.1392	+	31.2573i	0.136371	+	0.0787336i	0.566633	−	0.823970i	$-0.308246\pi$
$397$	54.1392	+	31.2573i	0.136371	+	0.0787336i	−0.430263	+	0.902704i	$0.641579\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	255.176	−	441.978i	0.636349	−	1.10219i	−0.349879	−	0.936795i	$-0.613777\pi$
$401$	255.176	−	441.978i	0.636349	−	1.10219i	0.986228	−	0.165394i	$-0.0528895\pi$
$402$		0			0
$403$	−4.79899	−	8.31209i	−0.0119082	−	0.0206255i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	26.7352			0.0656884
$408$		0			0
$409$	529.213	−	305.541i	1.29392	−	0.747045i	0.314574	−	0.949233i	$-0.398139\pi$
$409$	529.213	−	305.541i	1.29392	−	0.747045i	0.979347	+	0.202188i	$0.0648052\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−261.487	+	452.909i	−0.630090	+	1.09135i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	639.632i		−	1.52657i	−0.646063	−	0.763284i	$-0.723586\pi$
$419$		−	639.632i		−	1.52657i	0.646063	−	0.763284i	$-0.276414\pi$
$420$		0			0
$421$	−369.186			−0.876926			−0.438463	−	0.898749i	$-0.644477\pi$
$421$	−369.186			−0.876926			−0.438463	+	0.898749i	$0.644477\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	206.320	+	119.119i	0.485459	+	0.280280i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−230.294	−	398.882i	−0.534326	−	0.925479i	−0.999196	−	0.0401004i	$-0.987232\pi$
$431$	−230.294	−	398.882i	−0.534326	−	0.925479i	0.464870	−	0.885379i	$-0.346101\pi$
$432$		0			0
$433$		−	801.072i		−	1.85005i	−0.379905	−	0.925025i	$-0.624043\pi$
$433$		−	801.072i		−	1.85005i	0.379905	−	0.925025i	$-0.375957\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−221.779	+	128.044i	−0.507502	+	0.293007i
$438$		0			0
$439$	−632.057	−	364.918i	−1.43977	−	0.831249i	−0.441933	−	0.897048i	$-0.645707\pi$
$439$	−632.057	−	364.918i	−1.43977	−	0.831249i	−0.997833	+	0.0657988i	$0.979040\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	191.457	−	331.613i	0.432182	−	0.748562i	−0.564879	−	0.825174i	$-0.691077\pi$
$443$	191.457	−	331.613i	0.432182	−	0.748562i	0.997061	+	0.0766122i	$0.0244103\pi$
$444$		0			0
$445$	−394.136	−	682.663i	−0.885698	−	1.53407i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−271.863			−0.605486			−0.302743	−	0.953072i	$-0.597902\pi$
$449$	−271.863			−0.605486			−0.302743	+	0.953072i	$0.597902\pi$
$450$		0			0
$451$	50.9811	−	29.4340i	0.113040	−	0.0652638i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	98.9045	−	171.308i	0.216421	−	0.374853i	−0.737290	−	0.675576i	$-0.763895\pi$
$457$	98.9045	−	171.308i	0.216421	−	0.374853i	0.953711	+	0.300724i	$0.0972282\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			486.344i			1.05498i	0.849562	+	0.527488i	$0.176866\pi$
$461$			486.344i			1.05498i	−0.849562	+	0.527488i	$0.823134\pi$
$462$		0			0
$463$	73.1859			0.158069			0.0790344	−	0.996872i	$-0.474816\pi$
$463$	73.1859			0.158069			0.0790344	+	0.996872i	$0.474816\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−676.535	−	390.597i	−1.44868	−	0.836397i	−0.450279	−	0.892888i	$-0.648675\pi$
$467$	−676.535	−	390.597i	−1.44868	−	0.836397i	−0.998403	+	0.0564910i	$0.982009\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	40.7893	+	70.6492i	0.0862354	+	0.149364i
$474$		0			0
$475$			89.5573i			0.188542i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−613.628	+	354.279i	−1.28106	+	0.739621i	−0.977042	−	0.213045i	$-0.931662\pi$
$479$	−613.628	+	354.279i	−1.28106	+	0.739621i	−0.304019	+	0.952666i	$0.598329\pi$
$480$		0			0
$481$	3.76095	+	2.17138i	0.00781902	+	0.00451431i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	12.4898	−	21.6330i	0.0257522	−	0.0446042i
$486$		0			0
$487$	23.1005	+	40.0112i	0.0474343	+	0.0821586i	0.888768	−	0.458358i	$-0.151562\pi$
$487$	23.1005	+	40.0112i	0.0474343	+	0.0821586i	−0.841333	+	0.540516i	$0.818229\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	627.453			1.27791			0.638955	−	0.769245i	$-0.279367\pi$
$491$	627.453			1.27791			0.638955	+	0.769245i	$0.279367\pi$
$492$		0			0
$493$	652.402	−	376.664i	1.32333	−	0.764025i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−205.784	+	356.428i	−0.412392	+	0.714285i	−0.995151	−	0.0983608i	$-0.968640\pi$
$499$	−205.784	+	356.428i	−0.412392	+	0.714285i	0.582758	+	0.812645i	$0.301973\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	137.913i		−	0.274180i	−0.990559	−	0.137090i	$-0.956225\pi$
$503$		−	137.913i		−	0.274180i	0.990559	−	0.137090i	$-0.0437750\pi$
$504$		0			0
$505$	−936.060			−1.85358
$506$		0			0
$507$		0			0
$508$		0			0
$509$	229.789	+	132.669i	0.451452	+	0.260646i	0.708443	−	0.705768i	$-0.249398\pi$
$509$	229.789	+	132.669i	0.451452	+	0.260646i	−0.256991	+	0.966414i	$0.582731\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	291.478	+	504.855i	0.565977	+	0.980302i
$516$		0			0
$517$			94.9132i			0.183585i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	315.093	−	181.919i	0.604785	−	0.349173i	−0.166137	−	0.986103i	$-0.553129\pi$
$521$	315.093	−	181.919i	0.604785	−	0.349173i	0.770922	+	0.636930i	$0.219796\pi$
$522$		0			0
$523$	−679.391	−	392.246i	−1.29903	−	0.749993i	−0.318790	−	0.947825i	$-0.603276\pi$
$523$	−679.391	−	392.246i	−1.29903	−	0.749993i	−0.980236	+	0.197832i	$0.936610\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−279.572	+	484.233i	−0.530497	+	0.918848i
$528$		0			0
$529$	−415.786	−	720.163i	−0.785986	−	1.36137i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	9.56229			0.0179405
$534$		0			0
$535$	1004.09	−	579.713i	1.87681	−	1.08358i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−201.090	+	348.299i	−0.371701	+	0.643806i	−0.989827	−	0.142273i	$-0.954559\pi$
$541$	−201.090	+	348.299i	−0.371701	+	0.643806i	0.618126	+	0.786079i	$0.287892\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			731.945i			1.34302i
$546$		0			0
$547$	−411.206			−0.751748			−0.375874	−	0.926671i	$-0.622657\pi$
$547$	−411.206			−0.751748			−0.375874	+	0.926671i	$0.622657\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	245.248	+	141.594i	0.445096	+	0.256976i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	103.145	+	178.653i	0.185180	+	0.320741i	0.943637	−	0.330982i	$-0.107380\pi$
$557$	103.145	+	178.653i	0.185180	+	0.320741i	−0.758457	+	0.651723i	$0.774046\pi$
$558$		0			0
$559$			13.2513i			0.0237054i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	841.837	−	486.035i	1.49527	−	0.863295i	0.495285	−	0.868731i	$-0.335064\pi$
$563$	841.837	−	486.035i	1.49527	−	0.863295i	0.999985	+	0.00543587i	$0.00173030\pi$
$564$		0			0
$565$	103.008	+	59.4717i	0.182315	+	0.105260i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−284.451	+	492.684i	−0.499914	+	0.865877i	−1.00000		9.87795e-5i	$-0.999969\pi$
$569$	−284.451	+	492.684i	−0.499914	+	0.865877i	0.500086	+	0.865976i	$0.333302\pi$
$570$		0			0
$571$	−21.6934	−	37.5741i	−0.0379920	−	0.0658041i	0.846404	−	0.532541i	$-0.178763\pi$
$571$	−21.6934	−	37.5741i	−0.0379920	−	0.0658041i	−0.884396	+	0.466737i	$0.845429\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−475.810			−0.827496
$576$		0			0
$577$	215.380	−	124.350i	0.373275	−	0.215511i	−0.301613	−	0.953430i	$-0.597525\pi$
$577$	215.380	−	124.350i	0.373275	−	0.215511i	0.674888	+	0.737920i	$0.264192\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	3.80909	−	6.59754i	0.00653360	−	0.0113165i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	804.582i		−	1.37067i	−0.728229	−	0.685334i	$-0.759656\pi$
$587$		−	804.582i		−	1.37067i	0.728229	−	0.685334i	$-0.240344\pi$
$588$		0			0
$589$	−210.191			−0.356861
$590$		0			0
$591$		0			0
$592$		0			0
$593$	503.275	+	290.566i	0.848693	+	0.489993i	0.860210	−	0.509940i	$-0.170333\pi$
$593$	503.275	+	290.566i	0.848693	+	0.489993i	−0.0115165	+	0.999934i	$0.503666\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−358.125	−	620.291i	−0.597872	−	1.03554i	−0.993135	−	0.116976i	$-0.962680\pi$
$599$	−358.125	−	620.291i	−0.597872	−	1.03554i	0.395263	−	0.918568i	$-0.370653\pi$
$600$		0			0
$601$		−	135.622i		−	0.225660i	−0.993614	−	0.112830i	$-0.964008\pi$
$601$		−	135.622i		−	0.225660i	0.993614	−	0.112830i	$-0.0359915\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	624.800	−	360.729i	1.03273	−	0.596246i
$606$		0			0
$607$	−865.749	−	499.841i	−1.42628	−	0.823461i	−0.429451	−	0.903090i	$-0.641293\pi$
$607$	−865.749	−	499.841i	−1.42628	−	0.823461i	−0.996825	+	0.0796296i	$0.974626\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−7.70868	+	13.3518i	−0.0126165	+	0.0218524i
$612$		0			0
$613$	−407.246	−	705.371i	−0.664349	−	1.15069i	−0.979461	−	0.201632i	$-0.935375\pi$
$613$	−407.246	−	705.371i	−0.664349	−	1.15069i	0.315112	−	0.949055i	$-0.397958\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	417.848			0.677225			0.338612	−	0.940926i	$-0.390042\pi$
$617$	417.848			0.677225			0.338612	+	0.940926i	$0.390042\pi$
$618$		0			0
$619$	249.395	−	143.988i	0.402900	−	0.232615i	−0.284834	−	0.958577i	$-0.591939\pi$
$619$	249.395	−	143.988i	0.402900	−	0.232615i	0.687735	+	0.725962i	$0.258605\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	390.545	−	676.444i	0.624872	−	1.08231i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	252.994i		−	0.402216i
$630$		0			0
$631$	−474.412			−0.751842			−0.375921	−	0.926652i	$-0.622674\pi$
$631$	−474.412			−0.751842			−0.375921	+	0.926652i	$0.622674\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	932.981	+	538.657i	1.46926	+	0.848279i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	35.8120	+	62.0282i	0.0558690	+	0.0967679i	0.892607	−	0.450835i	$-0.148874\pi$
$641$	35.8120	+	62.0282i	0.0558690	+	0.0967679i	−0.836738	+	0.547603i	$0.815540\pi$
$642$		0			0
$643$			256.561i			0.399006i	0.979897	+	0.199503i	$0.0639328\pi$
$643$			256.561i			0.399006i	−0.979897	+	0.199503i	$0.936067\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−614.700	+	354.897i	−0.950078	+	0.548528i	−0.893105	−	0.449848i	$-0.851478\pi$
$647$	−614.700	+	354.897i	−0.950078	+	0.548528i	−0.0569725	+	0.998376i	$0.518145\pi$
$648$		0			0
$649$	40.5757	+	23.4264i	0.0625204	+	0.0360962i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−208.532	+	361.187i	−0.319344	+	0.553120i	−0.980351	−	0.197259i	$-0.936796\pi$
$653$	−208.532	+	361.187i	−0.319344	+	0.553120i	0.661007	+	0.750379i	$0.270129\pi$
$654$		0			0
$655$	−109.889	−	190.334i	−0.167770	−	0.290586i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	755.676			1.14670			0.573351	−	0.819310i	$-0.305643\pi$
$659$	755.676			1.14670			0.573351	+	0.819310i	$0.305643\pi$
$660$		0			0
$661$	746.681	−	431.096i	1.12962	−	0.652188i	0.185783	−	0.982591i	$-0.440518\pi$
$661$	746.681	−	431.096i	1.12962	−	0.652188i	0.943840	+	0.330403i	$0.107185\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−752.276	+	1302.98i	−1.12785	+	1.95349i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			155.794i			0.232181i
$672$		0			0
$673$	−529.266			−0.786428			−0.393214	−	0.919447i	$-0.628637\pi$
$673$	−529.266			−0.786428			−0.393214	+	0.919447i	$0.628637\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−390.269	−	225.322i	−0.576469	−	0.332824i	0.183260	−	0.983064i	$-0.441335\pi$
$677$	−390.269	−	225.322i	−0.576469	−	0.332824i	−0.759729	+	0.650240i	$0.774668\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	60.8897	+	105.464i	0.0891504	+	0.154413i	0.907152	−	0.420803i	$-0.138251\pi$
$683$	60.8897	+	105.464i	0.0891504	+	0.154413i	−0.818002	+	0.575216i	$0.804918\pi$
$684$		0			0
$685$		−	1231.52i		−	1.79784i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	1.07168	−	0.618735i	0.00155541	−	0.000898019i
$690$		0			0
$691$	−933.123	−	538.739i	−1.35040	−	0.779651i	−0.362091	−	0.932143i	$-0.617937\pi$
$691$	−933.123	−	538.739i	−1.35040	−	0.779651i	−0.988305	+	0.152491i	$0.951270\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−285.721	+	494.884i	−0.411110	+	0.712063i
$696$		0			0
$697$	−278.533	−	482.433i	−0.399616	−	0.692156i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	770.701			1.09943			0.549716	−	0.835352i	$-0.314736\pi$
$701$	770.701			1.09943			0.549716	+	0.835352i	$0.314736\pi$
$702$		0			0
$703$	82.3628	−	47.5522i	0.117159	−	0.0676418i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	124.251	−	215.209i	0.175249	−	0.303539i	−0.764999	−	0.644032i	$-0.777260\pi$
$709$	124.251	−	215.209i	0.175249	−	0.303539i	0.940247	+	0.340492i	$0.110594\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	1116.73i		−	1.56624i
$714$		0			0
$715$	−3.80909			−0.00532740
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−261.269	−	150.844i	−0.363379	−	0.209797i	0.307183	−	0.951650i	$-0.400614\pi$
$719$	−261.269	−	150.844i	−0.363379	−	0.209797i	−0.670562	+	0.741854i	$0.733947\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	263.081	+	455.669i	0.362870	+	0.628509i
$726$		0			0
$727$			1082.52i			1.48902i	0.667610	+	0.744511i	$0.267317\pi$
$727$			1082.52i			1.48902i	−0.667610	+	0.744511i	$0.732683\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	668.551	−	385.988i	0.914570	−	0.528027i
$732$		0			0
$733$	813.779	+	469.835i	1.11020	+	0.640976i	0.938882	−	0.344239i	$-0.111863\pi$
$733$	813.779	+	469.835i	1.11020	+	0.640976i	0.171321	+	0.985215i	$0.445196\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−89.1892	+	154.480i	−0.121017	+	0.209607i
$738$		0			0
$739$	281.176	+	487.011i	0.380481	+	0.659013i	0.991131	−	0.132888i	$-0.0424250\pi$
$739$	281.176	+	487.011i	0.380481	+	0.659013i	−0.610650	+	0.791901i	$0.709092\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	52.3033			0.0703948			0.0351974	−	0.999380i	$-0.488794\pi$
$743$	52.3033			0.0703948			0.0351974	+	0.999380i	$0.488794\pi$
$744$		0			0
$745$	455.745	−	263.125i	0.611739	−	0.353187i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	159.709	−	276.623i	0.212661	−	0.368340i	−0.739885	−	0.672733i	$-0.765120\pi$
$751$	159.709	−	276.623i	0.212661	−	0.368340i	0.952547	+	0.304393i	$0.0984536\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			1505.78i			1.99441i
$756$		0			0
$757$	−784.060			−1.03575			−0.517873	−	0.855457i	$-0.673276\pi$
$757$	−784.060			−1.03575			−0.517873	+	0.855457i	$0.673276\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−793.318	−	458.022i	−1.04247	−	0.601869i	−0.121937	−	0.992538i	$-0.538910\pi$
$761$	−793.318	−	458.022i	−1.04247	−	0.601869i	−0.920531	+	0.390669i	$0.872244\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	3.80530	+	6.59098i	0.00496128	+	0.00859319i
$768$		0			0
$769$			452.414i			0.588315i	0.955757	+	0.294157i	$0.0950390\pi$
$769$			452.414i			0.588315i	−0.955757	+	0.294157i	$0.904961\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−717.606	+	414.310i	−0.928339	+	0.535977i	−0.886286	−	0.463139i	$-0.846723\pi$
$773$	−717.606	+	414.310i	−0.928339	+	0.535977i	−0.0420528	+	0.999115i	$0.513390\pi$
$774$		0			0
$775$	−338.212	−	195.267i	−0.436403	−	0.251957i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	104.705	−	181.354i	0.134409	−	0.232803i
$780$		0			0
$781$	−73.8944	−	127.989i	−0.0946152	−	0.163878i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−1395.61			−1.77785
$786$		0			0
$787$	213.430	−	123.224i	0.271195	−	0.156574i	−0.358236	−	0.933631i	$-0.616622\pi$
$787$	213.430	−	123.224i	0.271195	−	0.156574i	0.629431	+	0.777057i	$0.283288\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−12.6533	+	21.9161i	−0.0159562	+	0.0276370i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	426.638i		−	0.535305i	−0.963516	−	0.267652i	$-0.913752\pi$
$797$		−	426.638i		−	0.535305i	0.963516	−	0.267652i	$-0.0862479\pi$
$798$		0			0
$799$	898.161			1.12411
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−223.332	−	128.941i	−0.278122	−	0.160574i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−553.289	−	958.325i	−0.683917	−	1.18458i	−0.973776	−	0.227510i	$-0.926942\pi$
$809$	−553.289	−	958.325i	−0.683917	−	1.18458i	0.289858	−	0.957070i	$-0.406392\pi$
$810$		0			0
$811$		−	1002.47i		−	1.23609i	−0.786143	−	0.618045i	$-0.787925\pi$
$811$		−	1002.47i		−	1.23609i	0.786143	−	0.618045i	$-0.212075\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−239.943	+	138.531i	−0.294409	+	0.169977i
$816$		0			0
$817$	251.319	+	145.099i	0.307611	+	0.177600i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	765.337	−	1325.60i	0.932201	−	1.61462i	0.152649	−	0.988280i	$-0.451219\pi$
$821$	765.337	−	1325.60i	0.932201	−	1.61462i	0.779551	−	0.626338i	$-0.215447\pi$
$822$		0			0
$823$	−252.276	−	436.955i	−0.306533	−	0.530930i	0.671069	−	0.741395i	$-0.265835\pi$
$823$	−252.276	−	436.955i	−0.306533	−	0.530930i	−0.977601	+	0.210465i	$0.932502\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1105.22			−1.33641			−0.668207	−	0.743975i	$-0.732938\pi$
$827$	−1105.22			−1.33641			−0.668207	+	0.743975i	$0.732938\pi$
$828$		0			0
$829$	−717.661	+	414.342i	−0.865695	+	0.499809i	−0.865915	−	0.500191i	$-0.833263\pi$
$829$	−717.661	+	414.342i	−0.865695	+	0.499809i	0.000220471		1.00000i	$0.499930\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−267.201	+	462.806i	−0.320001	+	0.554258i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			594.032i			0.708024i	0.935241	+	0.354012i	$0.115183\pi$
$839$			594.032i			0.708024i	−0.935241	+	0.354012i	$0.884817\pi$
$840$		0			0
$841$	822.769			0.978322
$842$		0			0
$843$		0			0
$844$		0			0
$845$	900.483	+	519.894i	1.06566	+	0.615260i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	252.641	+	437.587i	0.296875	+	0.514203i
$852$		0			0
$853$		−	481.329i		−	0.564278i	−0.959374	−	0.282139i	$-0.908956\pi$
$853$		−	481.329i		−	0.564278i	0.959374	−	0.282139i	$-0.0910439\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	332.722	−	192.097i	0.388240	−	0.224150i	−0.293157	−	0.956064i	$-0.594706\pi$
$857$	332.722	−	192.097i	0.388240	−	0.224150i	0.681397	+	0.731914i	$0.261373\pi$
$858$		0			0
$859$	−348.973	−	201.480i	−0.406255	−	0.234552i	0.282924	−	0.959142i	$-0.408696\pi$
$859$	−348.973	−	201.480i	−0.406255	−	0.234552i	−0.689180	+	0.724591i	$0.742029\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	297.721	−	515.668i	0.344984	−	0.597529i	−0.640367	−	0.768069i	$-0.721218\pi$
$863$	297.721	−	515.668i	0.344984	−	0.597529i	0.985351	+	0.170540i	$0.0545511\pi$
$864$		0			0
$865$	11.3316	+	19.6268i	0.0131001	+	0.0226900i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	49.1747			0.0565877
$870$		0			0
$871$	−25.0932	+	14.4876i	−0.0288097	+	0.0166333i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	499.829	−	865.729i	0.569930	−	0.987149i	−0.426642	−	0.904421i	$-0.640303\pi$
$877$	499.829	−	865.729i	0.569930	−	0.987149i	0.996572	−	0.0827278i	$-0.0263632\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	39.4436i		−	0.0447714i	−0.999749	−	0.0223857i	$-0.992874\pi$
$881$		−	39.4436i		−	0.0447714i	0.999749	−	0.0223857i	$-0.00712618\pi$
$882$		0			0
$883$	−1168.75			−1.32362			−0.661808	−	0.749673i	$-0.730211\pi$
$883$	−1168.75			−1.32362			−0.661808	+	0.749673i	$0.730211\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	367.363	+	212.097i	0.414163	+	0.239117i	0.692577	−	0.721344i	$-0.256475\pi$
$887$	367.363	+	212.097i	0.414163	+	0.239117i	−0.278414	+	0.960461i	$0.589809\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	168.816	+	292.398i	0.189044	+	0.327434i
$894$		0			0
$895$			1828.65i			2.04318i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−1069.46	+	617.451i	−1.18961	+	0.686820i
$900$		0			0
$901$	−62.4323	−	36.0453i	−0.0692922	−	0.0400059i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	367.785	−	637.023i	0.406393	−	0.703893i
$906$		0			0
$907$	629.080	+	1089.60i	0.693584	+	1.20132i	0.970656	+	0.240473i	$0.0773025\pi$
$907$	629.080	+	1089.60i	0.693584	+	1.20132i	−0.277072	+	0.960849i	$0.589364\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−467.611			−0.513294			−0.256647	−	0.966505i	$-0.582618\pi$
$911$	−467.611			−0.513294			−0.256647	+	0.966505i	$0.582618\pi$
$912$		0			0
$913$	−143.584	+	82.8981i	−0.157266	+	0.0907975i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−111.302	+	192.780i	−0.121112	+	0.209771i	−0.920206	−	0.391434i	$-0.871979\pi$
$919$	−111.302	+	192.780i	−0.121112	+	0.209771i	0.799095	+	0.601205i	$0.205313\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	24.0063i		−	0.0260090i
$924$		0			0
$925$	176.704			0.191031
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−1362.69	−	786.748i	−1.46683	−	0.846876i	−0.467521	−	0.883982i	$-0.654853\pi$
$929$	−1362.69	−	786.748i	−1.46683	−	0.846876i	−0.999311	+	0.0371060i	$0.988186\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	110.952	+	192.175i	0.118665	+	0.205534i
$936$		0			0
$937$		−	1165.80i		−	1.24418i	−0.782945	−	0.622090i	$-0.786284\pi$
$937$		−	1165.80i		−	1.24418i	0.782945	−	0.622090i	$-0.213716\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−1227.82	+	708.882i	−1.30480	+	0.753328i	−0.981224	−	0.192873i	$-0.938220\pi$
$941$	−1227.82	+	708.882i	−1.30480	+	0.753328i	−0.323579	+	0.946201i	$0.604886\pi$
$942$		0			0
$943$	963.517	+	556.287i	1.02176	+	0.589912i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	630.283	−	1091.68i	0.665557	−	1.15278i	−0.313577	−	0.949563i	$-0.601527\pi$
$947$	630.283	−	1091.68i	0.665557	−	1.15278i	0.979134	−	0.203216i	$-0.0651394\pi$
$948$		0			0
$949$	−20.9447	−	36.2773i	−0.0220703	−	0.0382268i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1479.73			1.55271			0.776355	−	0.630295i	$-0.217066\pi$
$953$	1479.73			1.55271			0.776355	+	0.630295i	$0.217066\pi$
$954$		0			0
$955$	−1791.76	+	1034.48i	−1.87619	+	1.08322i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−22.2086	+	38.4664i	−0.0231099	+	0.0400275i
$962$		0			0
$963$		0			0
$964$		0			0
$965$			305.338i			0.316412i
$966$		0			0
$967$	1725.53			1.78441			0.892206	−	0.451628i	$-0.149156\pi$
$967$	1725.53			1.78441			0.892206	+	0.451628i	$0.149156\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−781.610	−	451.263i	−0.804954	−	0.464740i	0.0402465	−	0.999190i	$-0.487186\pi$
$971$	−781.610	−	451.263i	−0.804954	−	0.464740i	−0.845200	+	0.534449i	$0.820519\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−599.448	−	1038.27i	−0.613560	−	1.06272i	−0.990635	−	0.136534i	$-0.956404\pi$
$977$	−599.448	−	1038.27i	−0.613560	−	1.06272i	0.377075	−	0.926183i	$-0.376930\pi$
$978$		0			0
$979$		−	249.902i		−	0.255262i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	937.268	−	541.132i	0.953477	−	0.550490i	0.0593177	−	0.998239i	$-0.481108\pi$
$983$	937.268	−	541.132i	0.953477	−	0.550490i	0.894159	+	0.447749i	$0.147774\pi$
$984$		0			0
$985$	1160.17	+	669.827i	1.17784	+	0.680027i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−770.898	+	1335.23i	−0.779472	+	1.35008i
$990$		0			0
$991$	824.377	+	1427.86i	0.831864	+	1.44083i	0.896559	+	0.442925i	$0.146059\pi$
$991$	824.377	+	1427.86i	0.831864	+	1.44083i	−0.0646952	+	0.997905i	$0.520608\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−1790.43			−1.79943
$996$		0			0
$997$	−340.518	+	196.598i	−0.341542	+	0.197189i	−0.660954	−	0.750427i	$-0.729848\pi$
$997$	−340.518	+	196.598i	−0.341542	+	0.197189i	0.319412	+	0.947616i	$0.396515\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.3.z.n.901.8		16
3.2	odd	2	inner	1764.3.z.n.901.1		16
7.2	even	3		1764.3.d.g.685.1	✓	8
7.3	odd	6	inner	1764.3.z.n.325.8		16
7.4	even	3	inner	1764.3.z.n.325.2		16
7.5	odd	6		1764.3.d.g.685.7	yes	8
7.6	odd	2	inner	1764.3.z.n.901.2		16
21.2	odd	6		1764.3.d.g.685.8	yes	8
21.5	even	6		1764.3.d.g.685.2	yes	8
21.11	odd	6	inner	1764.3.z.n.325.7		16
21.17	even	6	inner	1764.3.z.n.325.1		16
21.20	even	2	inner	1764.3.z.n.901.7		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1764.3.d.g.685.1	✓	8	7.2	even	3
1764.3.d.g.685.2	yes	8	21.5	even	6
1764.3.d.g.685.7	yes	8	7.5	odd	6
1764.3.d.g.685.8	yes	8	21.2	odd	6
1764.3.z.n.325.1		16	21.17	even	6	inner
1764.3.z.n.325.2		16	7.4	even	3	inner
1764.3.z.n.325.7		16	21.11	odd	6	inner
1764.3.z.n.325.8		16	7.3	odd	6	inner
1764.3.z.n.901.1		16	3.2	odd	2	inner
1764.3.z.n.901.2		16	7.6	odd	2	inner
1764.3.z.n.901.7		16	21.20	even	2	inner
1764.3.z.n.901.8		16	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1764.z (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(5.33147 + 3.07813i) q^{5} +O(q^{10})\)	\(q+(5.33147 + 3.07813i) q^{5} +(0.975845 + 1.69021i) q^{11} +0.317025i q^{13} +(15.9944 - 9.23438i) q^{17} +(6.01255 + 3.47135i) q^{19} +(-18.4430 + 31.9442i) q^{23} +(6.44975 + 11.1713i) q^{25} +40.7893 q^{29} +(-26.2190 + 15.1376i) q^{31} +(6.84924 - 11.8632i) q^{37} -30.1625i q^{41} +41.7990 q^{43} +(42.1160 + 24.3157i) q^{47} +(-1.95169 - 3.38043i) q^{53} +12.0151i q^{55} +(20.7901 - 12.0031i) q^{59} +(69.1305 + 39.9125i) q^{61} +(-0.975845 + 1.69021i) q^{65} +(45.6985 + 79.1521i) q^{67} -75.7236 q^{71} +(-114.430 + 66.0663i) q^{73} +(12.5980 - 21.8203i) q^{79} +84.9501i q^{83} +113.698 q^{85} +(-110.889 - 64.0220i) q^{89} +(21.3705 + 37.0148i) q^{95} -4.05760i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q + 24 q^{25} - 128 q^{37} + 352 q^{43} + 256 q^{67} - 432 q^{79} + 1344 q^{85}+O(q^{100}) \) 16 * q + 24 * q^25 - 128 * q^37 + 352 * q^43 + 256 * q^67 - 432 * q^79 + 1344 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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