LMFDB - Embedded newform 1764.3.z.n.325.1

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			325.1
Root			$-1.45341 - 0.839125i$ of defining polynomial
Character	$\chi$	$=$	1764.325
Dual form			1764.3.z.n.901.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-5.33147 + 3.07813i) q^{5} +O(q^{10})$	$q+(-5.33147 + 3.07813i) q^{5} +(-0.975845 + 1.69021i) q^{11} -0.317025i q^{13} +(-15.9944 - 9.23438i) q^{17} +(6.01255 - 3.47135i) q^{19} +(18.4430 + 31.9442i) q^{23} +(6.44975 - 11.1713i) q^{25} -40.7893 q^{29} +(-26.2190 - 15.1376i) q^{31} +(6.84924 + 11.8632i) q^{37} -30.1625i q^{41} +41.7990 q^{43} +(-42.1160 + 24.3157i) q^{47} +(1.95169 - 3.38043i) q^{53} -12.0151i q^{55} +(-20.7901 - 12.0031i) q^{59} +(69.1305 - 39.9125i) q^{61} +(0.975845 + 1.69021i) q^{65} +(45.6985 - 79.1521i) q^{67} +75.7236 q^{71} +(-114.430 - 66.0663i) q^{73} +(12.5980 + 21.8203i) q^{79} +84.9501i q^{83} +113.698 q^{85} +(110.889 - 64.0220i) q^{89} +(-21.3705 + 37.0148i) q^{95} +4.05760i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times$.

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

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	$-0.877763\pi$
$5$	−5.33147	+	3.07813i	−1.06629	+	0.615626i	−0.139128	+	0.990274i	$0.544430\pi$
$6$		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	$-0.861609\pi$
$11$	−0.975845	+	1.69021i	−0.0887132	+	0.153656i	0.818254	+	0.574856i	$0.194942\pi$
$12$		0			0
$13$		−	0.317025i		−	0.0243866i	−0.999926	−	0.0121933i	$-0.996119\pi$
$13$		−	0.317025i		−	0.0243866i	0.999926	−	0.0121933i	$-0.00388134\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−15.9944	−	9.23438i	−0.940848	−	0.543199i	−0.0506223	−	0.998718i	$-0.516120\pi$
$17$	−15.9944	−	9.23438i	−0.940848	−	0.543199i	−0.890226	+	0.455519i	$0.849454\pi$
$18$		0			0
$19$	6.01255	−	3.47135i	0.316450	−	0.182702i	−0.333359	−	0.942800i	$-0.608182\pi$
$19$	6.01255	−	3.47135i	0.316450	−	0.182702i	0.649809	+	0.760097i	$0.274849\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	18.4430	+	31.9442i	0.801868	+	1.38888i	0.918385	+	0.395688i	$0.129494\pi$
$23$	18.4430	+	31.9442i	0.801868	+	1.38888i	−0.116517	+	0.993189i	$0.537173\pi$
$24$		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	$-0.748275\pi$
$29$	−40.7893			−1.40653			−0.703264	+	0.710929i	$0.748275\pi$
$30$		0			0
$31$	−26.2190	−	15.1376i	−0.845775	−	0.488308i	0.0134481	−	0.999910i	$-0.495719\pi$
$31$	−26.2190	−	15.1376i	−0.845775	−	0.488308i	−0.859223	+	0.511601i	$0.829053\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	6.84924	+	11.8632i	0.185115	+	0.320628i	0.943615	−	0.331044i	$-0.107401\pi$
$37$	6.84924	+	11.8632i	0.185115	+	0.320628i	−0.758500	+	0.651672i	$0.774068\pi$
$38$		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	$-0.839750\pi$
$47$	−42.1160	+	24.3157i	−0.896084	+	0.517354i	−0.0201565	+	0.999797i	$0.506416\pi$
$48$		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	$-0.821609\pi$
$53$	1.95169	−	3.38043i	0.0368243	−	0.0637816i	0.883850	+	0.467770i	$0.154942\pi$
$54$		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	$-0.398547\pi$
$59$	−20.7901	−	12.0031i	−0.352374	−	0.203443i	−0.665730	+	0.746192i	$0.731880\pi$
$60$		0			0
$61$	69.1305	−	39.9125i	1.13329	−	0.654304i	0.188528	−	0.982068i	$-0.439628\pi$
$61$	69.1305	−	39.9125i	1.13329	−	0.654304i	0.944760	+	0.327764i	$0.106295\pi$
$62$		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	−0.292282	−	0.956332i	$-0.594415\pi$
$67$	45.6985	−	79.1521i	0.682067	−	1.18137i	0.974349	−	0.225042i	$-0.0722521\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	75.7236			1.06653			0.533265	−	0.845948i	$-0.320965\pi$
$71$	75.7236			1.06653			0.533265	+	0.845948i	$0.320965\pi$
$72$		0			0
$73$	−114.430	−	66.0663i	−1.56754	−	0.905018i	−0.996455	−	0.0841251i	$-0.973190\pi$
$73$	−114.430	−	66.0663i	−1.56754	−	0.905018i	−0.571082	−	0.820893i	$-0.693476\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	12.5980	+	21.8203i	0.159468	+	0.276207i	0.934677	−	0.355498i	$-0.115689\pi$
$79$	12.5980	+	21.8203i	0.159468	+	0.276207i	−0.775209	+	0.631705i	$0.782355\pi$
$80$		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	$-0.411107\pi$
$89$	110.889	−	64.0220i	1.24595	−	0.719348i	0.970299	+	0.241911i	$0.0777741\pi$
$90$		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	$0.00665809\pi$
$97$			4.05760i			0.0418310i	−0.999781	+	0.0209155i	$0.993342\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	131.679	+	76.0251i	1.30376	+	0.752724i	0.981046	−	0.193774i	$-0.0620729\pi$
$101$	131.679	+	76.0251i	1.30376	+	0.752724i	0.322710	+	0.946498i	$0.395406\pi$
$102$		0			0
$103$	82.0069	−	47.3467i	0.796183	−	0.459677i	−0.0459515	−	0.998944i	$-0.514632\pi$
$103$	82.0069	−	47.3467i	0.796183	−	0.459677i	0.842135	+	0.539267i	$0.181299\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−94.1665	−	163.101i	−0.880061	−	1.52431i	−0.851272	−	0.524725i	$-0.824168\pi$
$107$	−94.1665	−	163.101i	−0.880061	−	1.52431i	−0.0287895	−	0.999585i	$-0.509165\pi$
$108$		0			0
$109$	59.4472	−	102.966i	0.545387	−	0.944639i	−0.453195	−	0.891411i	$-0.649716\pi$
$109$	59.4472	−	102.966i	0.545387	−	0.944639i	0.998582	−	0.0532272i	$-0.0169508\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−19.3207			−0.170980			−0.0854900	−	0.996339i	$-0.527246\pi$
$113$	−19.3207			−0.170980			−0.0854900	+	0.996339i	$0.527246\pi$
$114$		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	$-0.623158\pi$
$131$	30.9172	−	17.8500i	0.236009	−	0.136260i	0.613341	+	0.789818i	$0.289825\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	100.022	−	173.243i	0.730085	−	1.26454i	−0.226762	−	0.973950i	$-0.572814\pi$
$137$	100.022	−	173.243i	0.730085	−	1.26454i	0.956847	−	0.290594i	$-0.0938528\pi$
$138$		0			0
$139$		−	92.8231i		−	0.667792i	−0.942610	−	0.333896i	$-0.891637\pi$
$139$		−	92.8231i		−	0.667792i	0.942610	−	0.333896i	$-0.108363\pi$
$140$		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	$-0.259276\pi$
$149$	−42.7410	−	74.0296i	−0.286852	−	0.496843i	−0.973057	+	0.230566i	$0.925942\pi$
$150$		0			0
$151$	122.296	−	211.824i	0.809910	−	1.40281i	−0.103015	−	0.994680i	$-0.532849\pi$
$151$	122.296	−	211.824i	0.809910	−	1.40281i	0.912926	−	0.408126i	$-0.133818\pi$
$152$		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	−0.279282	−	0.960209i	$-0.590096\pi$
$157$	−196.327	−	113.349i	−1.25049	−	0.721970i	−0.971207	+	0.238239i	$0.923430\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−22.5025	−	38.9755i	−0.138052	−	0.239114i	0.788707	−	0.614769i	$-0.210751\pi$
$163$	−22.5025	−	38.9755i	−0.138052	−	0.239114i	−0.926759	+	0.375656i	$0.877418\pi$
$164$		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	$-0.670053\pi$
$173$	−3.18811	+	1.84066i	−0.0184284	+	0.0106396i	0.490757	+	0.871296i	$0.336720\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−148.520	+	257.244i	−0.829719	+	1.43711i	0.0685406	+	0.997648i	$0.478166\pi$
$179$	−148.520	+	257.244i	−0.829719	+	1.43711i	−0.898259	+	0.439466i	$0.855168\pi$
$180$		0			0
$181$			119.483i			0.660130i	0.943958	+	0.330065i	$0.107071\pi$
$181$			119.483i			0.660130i	−0.943958	+	0.330065i	$0.892929\pi$
$182$		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	0.851591	+	0.524207i	$0.175638\pi$
$191$	168.037	+	291.048i	0.879772	+	1.52381i	0.0281817	+	0.999603i	$0.491028\pi$
$192$		0			0
$193$	24.7990	−	42.9531i	0.128492	−	0.222555i	−0.794600	−	0.607133i	$-0.792320\pi$
$193$	24.7990	−	42.9531i	0.128492	−	0.222555i	0.923093	+	0.384578i	$0.125653\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−217.608			−1.10461			−0.552306	−	0.833642i	$-0.686252\pi$
$197$	−217.608			−1.10461			−0.552306	+	0.833642i	$0.686252\pi$
$198$		0			0
$199$	−251.868	−	145.416i	−1.26567	−	0.730733i	−0.291501	−	0.956570i	$-0.594155\pi$
$199$	−251.868	−	145.416i	−1.26567	−	0.730733i	−0.974165	+	0.225838i	$0.927488\pi$
$200$		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	$-0.868354\pi$
$223$		−	179.242i		−	0.803776i	0.915689	−	0.401888i	$-0.131646\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−75.1765	−	43.4032i	−0.331174	−	0.191203i	0.325188	−	0.945649i	$-0.394572\pi$
$227$	−75.1765	−	43.4032i	−0.331174	−	0.191203i	−0.656362	+	0.754446i	$0.727906\pi$
$228$		0			0
$229$	274.024	−	158.208i	1.19661	−	0.690863i	0.236812	−	0.971556i	$-0.423898\pi$
$229$	274.024	−	158.208i	1.19661	−	0.690863i	0.959798	+	0.280693i	$0.0905642\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	74.6497	+	129.297i	0.320385	+	0.554923i	0.980567	−	0.196182i	$-0.0628545\pi$
$233$	74.6497	+	129.297i	0.320385	+	0.554923i	−0.660183	+	0.751105i	$0.729521\pi$
$234$		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	$-0.634588\pi$
$239$	−196.140			−0.820669			−0.410334	+	0.911935i	$0.634588\pi$
$240$		0			0
$241$	−110.559	−	63.8312i	−0.458751	−	0.264860i	0.252768	−	0.967527i	$-0.418659\pi$
$241$	−110.559	−	63.8312i	−0.458751	−	0.264860i	−0.711519	+	0.702667i	$0.751992\pi$
$242$		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	$-0.724555\pi$
$257$	−80.5080	+	46.4813i	−0.313261	+	0.180861i	0.335124	+	0.942174i	$0.391222\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	247.467	−	428.626i	0.940940	−	1.62976i	0.177258	−	0.984164i	$-0.443277\pi$
$263$	247.467	−	428.626i	0.940940	−	1.62976i	0.763683	−	0.645592i	$-0.223389\pi$
$264$		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	$-0.589077\pi$
$269$	−335.347	−	193.613i	−1.24664	−	0.719750i	−0.970438	+	0.241349i	$0.922410\pi$
$270$		0			0
$271$	−192.813	+	111.320i	−0.711486	+	0.410777i	−0.811611	−	0.584198i	$-0.801409\pi$
$271$	−192.813	+	111.320i	−0.711486	+	0.410777i	0.100125	+	0.994975i	$0.468076\pi$
$272$		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	−0.963972	−	0.266005i	$-0.914296\pi$
$277$	−69.6985	+	120.721i	−0.251619	+	0.435817i	0.712353	+	0.701822i	$0.247630\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−91.1409			−0.324345			−0.162172	−	0.986762i	$-0.551850\pi$
$281$	−91.1409			−0.324345			−0.162172	+	0.986762i	$0.551850\pi$
$282$		0			0
$283$	−0.0275939	−	0.0159313i	−9.75048e−5	−	5.62944e-5i	0.499951	−	0.866054i	$-0.333351\pi$
$283$	−0.0275939	−	0.0159313i	−9.75048e−5	−	5.62944e-5i	−0.500049	+	0.865997i	$0.666685\pi$
$284$		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	$0.0485221\pi$
$307$			93.2341i			0.303694i	−0.988404	+	0.151847i	$0.951478\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	157.265	+	90.7970i	0.505675	+	0.291952i	0.731054	−	0.682319i	$-0.239029\pi$
$311$	157.265	+	90.7970i	0.505675	+	0.291952i	−0.225379	+	0.974271i	$0.572362\pi$
$312$		0			0
$313$	257.523	−	148.681i	0.822757	−	0.475019i	−0.0286095	−	0.999591i	$-0.509108\pi$
$313$	257.523	−	148.681i	0.822757	−	0.475019i	0.851366	+	0.524572i	$0.175775\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	73.8700	+	127.947i	0.233028	+	0.403617i	0.958698	−	0.284427i	$-0.0918032\pi$
$317$	73.8700	+	127.947i	0.233028	+	0.403617i	−0.725670	+	0.688043i	$0.758470\pi$
$318$		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	0.514680	−	0.857382i	$-0.327911\pi$
$331$	−160.593	−	278.155i	−0.485175	−	0.840348i	−0.999855	+	0.0170346i	$0.994577\pi$
$332$		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	$-0.900518\pi$
$347$	−72.6980	+	125.917i	−0.209504	+	0.362872i	0.742054	+	0.670340i	$0.233852\pi$
$348$		0			0
$349$			187.199i			0.536388i	0.963365	+	0.268194i	$0.0864268\pi$
$349$			187.199i			0.536388i	−0.963365	+	0.268194i	$0.913573\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	400.396	+	231.169i	1.13427	+	0.654870i	0.945005	−	0.327057i	$-0.106057\pi$
$353$	400.396	+	231.169i	1.13427	+	0.654870i	0.189263	+	0.981926i	$0.439390\pi$
$354$		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	$-0.331490\pi$
$359$	−177.697	−	307.780i	−0.494977	−	0.857326i	−0.999983	+	0.00578996i	$0.998157\pi$
$360$		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	0.835827	−	0.548992i	$-0.184988\pi$
$367$	285.636	+	164.912i	0.778300	+	0.449352i	−0.0575274	+	0.998344i	$0.518322\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	203.296	+	352.120i	0.545031	+	0.944021i	0.998605	+	0.0528024i	$0.0168154\pi$
$373$	203.296	+	352.120i	0.545031	+	0.944021i	−0.453574	+	0.891218i	$0.649851\pi$
$374$		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	$-0.456212\pi$
$383$	407.335	−	235.175i	1.06354	−	0.614035i	0.926409	+	0.376519i	$0.122879\pi$
$384$		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	$-0.735586\pi$
$389$	117.587	−	203.666i	0.302280	−	0.523564i	0.976652	+	0.214828i	$0.0689190\pi$
$390$		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	−0.430263	−	0.902704i	$-0.641579\pi$
$397$	54.1392	−	31.2573i	0.136371	−	0.0787336i	0.566633	+	0.823970i	$0.308246\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−255.176	−	441.978i	−0.636349	−	1.10219i	−0.986228	−	0.165394i	$-0.947110\pi$
$401$	−255.176	−	441.978i	−0.636349	−	1.10219i	0.349879	−	0.936795i	$-0.386223\pi$
$402$		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	0.979347	−	0.202188i	$-0.0648052\pi$
$409$	529.213	+	305.541i	1.29392	+	0.747045i	0.314574	+	0.949233i	$0.398139\pi$
$410$		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	−0.464870	−	0.885379i	$-0.653899\pi$
$431$	230.294	−	398.882i	0.534326	−	0.925479i	0.999196	−	0.0401004i	$-0.0127678\pi$
$432$		0			0
$433$			801.072i			1.85005i	0.379905	+	0.925025i	$0.375957\pi$
$433$			801.072i			1.85005i	−0.379905	+	0.925025i	$0.624043\pi$
$434$		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	−0.997833	−	0.0657988i	$-0.979040\pi$
$439$	−632.057	+	364.918i	−1.43977	+	0.831249i	−0.441933	+	0.897048i	$0.645707\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−191.457	−	331.613i	−0.432182	−	0.748562i	0.564879	−	0.825174i	$-0.308923\pi$
$443$	−191.457	−	331.613i	−0.432182	−	0.748562i	−0.997061	+	0.0766122i	$0.975590\pi$
$444$		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	$-0.402098\pi$
$449$	271.863			0.605486			0.302743	+	0.953072i	$0.402098\pi$
$450$		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	0.953711	−	0.300724i	$-0.0972282\pi$
$457$	98.9045	+	171.308i	0.216421	+	0.374853i	−0.737290	+	0.675576i	$0.763895\pi$
$458$		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	$-0.351325\pi$
$467$	676.535	−	390.597i	1.44868	−	0.836397i	0.998403	+	0.0564910i	$0.0179912\pi$
$468$		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	$-0.0683380\pi$
$479$	613.628	+	354.279i	1.28106	+	0.739621i	0.304019	+	0.952666i	$0.401671\pi$
$480$		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	−0.841333	−	0.540516i	$-0.818229\pi$
$487$	23.1005	−	40.0112i	0.0474343	−	0.0821586i	0.888768	+	0.458358i	$0.151562\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−627.453			−1.27791			−0.638955	−	0.769245i	$-0.720633\pi$
$491$	−627.453			−1.27791			−0.638955	+	0.769245i	$0.720633\pi$
$492$		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	0.582758	−	0.812645i	$-0.301973\pi$
$499$	−205.784	−	356.428i	−0.412392	−	0.714285i	−0.995151	+	0.0983608i	$0.968640\pi$
$500$		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	$-0.750602\pi$
$509$	−229.789	+	132.669i	−0.451452	+	0.260646i	0.256991	+	0.966414i	$0.417269\pi$
$510$		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	$-0.446871\pi$
$521$	−315.093	−	181.919i	−0.604785	−	0.349173i	−0.770922	+	0.636930i	$0.780204\pi$
$522$		0			0
$523$	−679.391	+	392.246i	−1.29903	+	0.749993i	−0.980236	−	0.197832i	$-0.936610\pi$
$523$	−679.391	+	392.246i	−1.29903	+	0.749993i	−0.318790	+	0.947825i	$0.603276\pi$
$524$		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	0.618126	−	0.786079i	$-0.287892\pi$
$541$	−201.090	−	348.299i	−0.371701	−	0.643806i	−0.989827	+	0.142273i	$0.954559\pi$
$542$		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	$-0.892620\pi$
$557$	−103.145	+	178.653i	−0.185180	+	0.320741i	0.758457	+	0.651723i	$0.225954\pi$
$558$		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	$-0.664936\pi$
$563$	−841.837	−	486.035i	−1.49527	−	0.863295i	−0.999985	+	0.00543587i	$0.998270\pi$
$564$		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	$-3.14425e-5\pi$
$569$	284.451	+	492.684i	0.499914	+	0.865877i	−0.500086	+	0.865976i	$0.666698\pi$
$570$		0			0
$571$	−21.6934	+	37.5741i	−0.0379920	+	0.0658041i	−0.884396	−	0.466737i	$-0.845429\pi$
$571$	−21.6934	+	37.5741i	−0.0379920	+	0.0658041i	0.846404	+	0.532541i	$0.178763\pi$
$572$		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	0.674888	−	0.737920i	$-0.264192\pi$
$577$	215.380	+	124.350i	0.373275	+	0.215511i	−0.301613	+	0.953430i	$0.597525\pi$
$578$		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	$-0.829667\pi$
$593$	−503.275	+	290.566i	−0.848693	+	0.489993i	0.0115165	+	0.999934i	$0.496334\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	358.125	−	620.291i	0.597872	−	1.03554i	−0.395263	−	0.918568i	$-0.629347\pi$
$599$	358.125	−	620.291i	0.597872	−	1.03554i	0.993135	−	0.116976i	$-0.0373201\pi$
$600$		0			0
$601$			135.622i			0.225660i	0.993614	+	0.112830i	$0.0359915\pi$
$601$			135.622i			0.225660i	−0.993614	+	0.112830i	$0.964008\pi$
$602$		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	−0.996825	−	0.0796296i	$-0.974626\pi$
$607$	−865.749	+	499.841i	−1.42628	+	0.823461i	−0.429451	+	0.903090i	$0.641293\pi$
$608$		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	0.315112	+	0.949055i	$0.397958\pi$
$613$	−407.246	+	705.371i	−0.664349	+	1.15069i	−0.979461	+	0.201632i	$0.935375\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−417.848			−0.677225			−0.338612	−	0.940926i	$-0.609958\pi$
$617$	−417.848			−0.677225			−0.338612	+	0.940926i	$0.609958\pi$
$618$		0			0
$619$	249.395	+	143.988i	0.402900	+	0.232615i	0.687735	−	0.725962i	$-0.258605\pi$
$619$	249.395	+	143.988i	0.402900	+	0.232615i	−0.284834	+	0.958577i	$0.591939\pi$
$620$		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	$-0.851126\pi$
$641$	−35.8120	+	62.0282i	−0.0558690	+	0.0967679i	0.836738	+	0.547603i	$0.184460\pi$
$642$		0			0
$643$		−	256.561i		−	0.399006i	−0.979897	−	0.199503i	$-0.936067\pi$
$643$		−	256.561i		−	0.399006i	0.979897	−	0.199503i	$-0.0639328\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	614.700	+	354.897i	0.950078	+	0.548528i	0.893105	−	0.449848i	$-0.148522\pi$
$647$	614.700	+	354.897i	0.950078	+	0.548528i	0.0569725	+	0.998376i	$0.481855\pi$
$648$		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	$-0.0632041\pi$
$653$	208.532	+	361.187i	0.319344	+	0.553120i	−0.661007	+	0.750379i	$0.729871\pi$
$654$		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	$-0.694357\pi$
$659$	−755.676			−1.14670			−0.573351	+	0.819310i	$0.694357\pi$
$660$		0			0
$661$	746.681	+	431.096i	1.12962	+	0.652188i	0.943840	−	0.330403i	$-0.107185\pi$
$661$	746.681	+	431.096i	1.12962	+	0.652188i	0.185783	+	0.982591i	$0.440518\pi$
$662$		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	$-0.558665\pi$
$677$	390.269	−	225.322i	0.576469	−	0.332824i	0.759729	+	0.650240i	$0.225332\pi$
$678$		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	$-0.861749\pi$
$683$	−60.8897	+	105.464i	−0.0891504	+	0.154413i	0.818002	+	0.575216i	$0.195082\pi$
$684$		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	−0.988305	−	0.152491i	$-0.951270\pi$
$691$	−933.123	+	538.739i	−1.35040	+	0.779651i	−0.362091	+	0.932143i	$0.617937\pi$
$692$		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	$-0.685264\pi$
$701$	−770.701			−1.09943			−0.549716	+	0.835352i	$0.685264\pi$
$702$		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	0.940247	−	0.340492i	$-0.110594\pi$
$709$	124.251	+	215.209i	0.175249	+	0.303539i	−0.764999	+	0.644032i	$0.777260\pi$
$710$		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	$-0.599386\pi$
$719$	261.269	−	150.844i	0.363379	−	0.209797i	0.670562	+	0.741854i	$0.266053\pi$
$720$		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	$-0.732683\pi$
$727$		−	1082.52i		−	1.48902i	0.667610	−	0.744511i	$-0.267317\pi$
$728$		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	0.171321	−	0.985215i	$-0.445196\pi$
$733$	813.779	−	469.835i	1.11020	−	0.640976i	0.938882	+	0.344239i	$0.111863\pi$
$734$		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	−0.610650	−	0.791901i	$-0.709092\pi$
$739$	281.176	−	487.011i	0.380481	−	0.659013i	0.991131	+	0.132888i	$0.0424250\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−52.3033			−0.0703948			−0.0351974	−	0.999380i	$-0.511206\pi$
$743$	−52.3033			−0.0703948			−0.0351974	+	0.999380i	$0.511206\pi$
$744$		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	0.952547	−	0.304393i	$-0.0984536\pi$
$751$	159.709	+	276.623i	0.212661	+	0.368340i	−0.739885	+	0.672733i	$0.765120\pi$
$752$		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	$-0.461090\pi$
$761$	793.318	−	458.022i	1.04247	−	0.601869i	0.920531	+	0.390669i	$0.127756\pi$
$762$		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	$-0.904961\pi$
$769$		−	452.414i		−	0.588315i	0.955757	−	0.294157i	$-0.0950390\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	717.606	+	414.310i	0.928339	+	0.535977i	0.886286	−	0.463139i	$-0.153277\pi$
$773$	717.606	+	414.310i	0.928339	+	0.535977i	0.0420528	+	0.999115i	$0.486610\pi$
$774$		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	0.629431	−	0.777057i	$-0.283288\pi$
$787$	213.430	+	123.224i	0.271195	+	0.156574i	−0.358236	+	0.933631i	$0.616622\pi$
$788$		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	−0.289858	−	0.957070i	$-0.593608\pi$
$809$	553.289	−	958.325i	0.683917	−	1.18458i	0.973776	−	0.227510i	$-0.0730585\pi$
$810$		0			0
$811$			1002.47i			1.23609i	0.786143	+	0.618045i	$0.212075\pi$
$811$			1002.47i			1.23609i	−0.786143	+	0.618045i	$0.787925\pi$
$812$		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	−0.779551	−	0.626338i	$-0.784553\pi$
$821$	−765.337	−	1325.60i	−0.932201	−	1.61462i	−0.152649	−	0.988280i	$-0.548781\pi$
$822$		0			0
$823$	−252.276	+	436.955i	−0.306533	+	0.530930i	−0.977601	−	0.210465i	$-0.932502\pi$
$823$	−252.276	+	436.955i	−0.306533	+	0.530930i	0.671069	+	0.741395i	$0.265835\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	1105.22			1.33641			0.668207	−	0.743975i	$-0.267062\pi$
$827$	1105.22			1.33641			0.668207	+	0.743975i	$0.267062\pi$
$828$		0			0
$829$	−717.661	−	414.342i	−0.865695	−	0.499809i	0.000220471	−	1.00000i	$-0.499930\pi$
$829$	−717.661	−	414.342i	−0.865695	−	0.499809i	−0.865915	+	0.500191i	$0.833263\pi$
$830$		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	$0.0910439\pi$
$853$			481.329i			0.564278i	−0.959374	+	0.282139i	$0.908956\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−332.722	−	192.097i	−0.388240	−	0.224150i	0.293157	−	0.956064i	$-0.405294\pi$
$857$	−332.722	−	192.097i	−0.388240	−	0.224150i	−0.681397	+	0.731914i	$0.738627\pi$
$858$		0			0
$859$	−348.973	+	201.480i	−0.406255	+	0.234552i	−0.689180	−	0.724591i	$-0.742029\pi$
$859$	−348.973	+	201.480i	−0.406255	+	0.234552i	0.282924	+	0.959142i	$0.408696\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−297.721	−	515.668i	−0.344984	−	0.597529i	0.640367	−	0.768069i	$-0.278782\pi$
$863$	−297.721	−	515.668i	−0.344984	−	0.597529i	−0.985351	+	0.170540i	$0.945449\pi$
$864$		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	0.996572	+	0.0827278i	$0.0263632\pi$
$877$	499.829	+	865.729i	0.569930	+	0.987149i	−0.426642	+	0.904421i	$0.640303\pi$
$878$		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	$-0.743525\pi$
$887$	−367.363	+	212.097i	−0.414163	+	0.239117i	0.278414	+	0.960461i	$0.410191\pi$
$888$		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	−0.277072	−	0.960849i	$-0.589364\pi$
$907$	629.080	−	1089.60i	0.693584	−	1.20132i	0.970656	−	0.240473i	$-0.0773025\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	467.611			0.513294			0.256647	−	0.966505i	$-0.417382\pi$
$911$	467.611			0.513294			0.256647	+	0.966505i	$0.417382\pi$
$912$		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	0.799095	−	0.601205i	$-0.205313\pi$
$919$	−111.302	−	192.780i	−0.121112	−	0.209771i	−0.920206	+	0.391434i	$0.871979\pi$
$920$		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	$-0.345147\pi$
$929$	1362.69	−	786.748i	1.46683	−	0.846876i	0.999311	+	0.0371060i	$0.0118139\pi$
$930$		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	$0.213716\pi$
$937$			1165.80i			1.24418i	−0.782945	+	0.622090i	$0.786284\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	1227.82	+	708.882i	1.30480	+	0.753328i	0.981224	−	0.192873i	$-0.0617804\pi$
$941$	1227.82	+	708.882i	1.30480	+	0.753328i	0.323579	+	0.946201i	$0.395114\pi$
$942$		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	−0.979134	−	0.203216i	$-0.934861\pi$
$947$	−630.283	−	1091.68i	−0.665557	−	1.15278i	0.313577	−	0.949563i	$-0.398473\pi$
$948$		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	$-0.782934\pi$
$953$	−1479.73			−1.55271			−0.776355	+	0.630295i	$0.782934\pi$
$954$		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	$-0.512814\pi$
$971$	781.610	−	451.263i	0.804954	−	0.464740i	0.845200	+	0.534449i	$0.179481\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	599.448	−	1038.27i	0.613560	−	1.06272i	−0.377075	−	0.926183i	$-0.623070\pi$
$977$	599.448	−	1038.27i	0.613560	−	1.06272i	0.990635	−	0.136534i	$-0.0435964\pi$
$978$		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	$-0.518892\pi$
$983$	−937.268	−	541.132i	−0.953477	−	0.550490i	−0.894159	+	0.447749i	$0.852226\pi$
$984$		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	−0.0646952	−	0.997905i	$-0.520608\pi$
$991$	824.377	−	1427.86i	0.831864	−	1.44083i	0.896559	−	0.442925i	$-0.146059\pi$
$992$		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	0.319412	−	0.947616i	$-0.396515\pi$
$997$	−340.518	−	196.598i	−0.341542	−	0.197189i	−0.660954	+	0.750427i	$0.729848\pi$
$998$		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.325.1		16
3.2	odd	2	inner	1764.3.z.n.325.8		16
7.2	even	3	inner	1764.3.z.n.901.7		16
7.3	odd	6		1764.3.d.g.685.8	yes	8
7.4	even	3		1764.3.d.g.685.2	yes	8
7.5	odd	6	inner	1764.3.z.n.901.1		16
7.6	odd	2	inner	1764.3.z.n.325.7		16
21.2	odd	6	inner	1764.3.z.n.901.2		16
21.5	even	6	inner	1764.3.z.n.901.8		16
21.11	odd	6		1764.3.d.g.685.7	yes	8
21.17	even	6		1764.3.d.g.685.1	✓	8
21.20	even	2	inner	1764.3.z.n.325.2		16

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

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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