LMFDB - Embedded newform 1764.3.bk.e.1745.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1764,3,Mod(557,1764)]

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, 3, 4]))

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

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

chi := DirichletCharacter("1764.557");

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.bk (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:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.12745506816.5
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 8x^{6} + 55x^{4} - 72x^{2} + 81 $ x^8 - 8x^6 + 55x^4 - 72*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 252)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1745.2
Root			$-1.00781 - 0.581861i$ of defining polynomial
Character	$\chi$	$=$	1764.1745
Dual form			1764.3.bk.e.557.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.01563 - 1.16372i) q^{5} +O(q^{10})$	$q+(-2.01563 - 1.16372i) q^{5} +(-7.70549 + 4.44876i) q^{11} -8.58301 q^{13} +(17.4266 - 10.0613i) q^{17} +(2.93725 - 5.08747i) q^{19} +(15.0540 + 8.69140i) q^{23} +(-9.79150 - 16.9594i) q^{25} -19.0318i q^{29} +(2.35425 + 4.07768i) q^{31} +(-19.8745 + 34.4237i) q^{37} +6.98233i q^{41} -35.7490 q^{43} +(63.5330 + 36.6808i) q^{47} +(-40.4166 + 23.3345i) q^{53} +20.7085 q^{55} +(-20.8703 + 12.0495i) q^{59} +(-36.6458 + 63.4723i) q^{61} +(17.3001 + 9.98823i) q^{65} +(16.0405 + 27.7830i) q^{67} +114.843i q^{71} +(20.6458 + 35.7595i) q^{73} +(13.3320 - 23.0917i) q^{79} -136.057i q^{83} -46.8340 q^{85} +(126.226 + 72.8764i) q^{89} +(-11.8408 + 6.83629i) q^{95} +65.9555 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 16 q^{13} - 40 q^{19} - 36 q^{25} + 40 q^{31} - 32 q^{37} - 32 q^{43} + 208 q^{55} - 272 q^{61} - 168 q^{67} + 144 q^{73} - 232 q^{79} - 544 q^{85} - 192 q^{97}+O(q^{100}) $ 8 * q + 16 * q^13 - 40 * q^19 - 36 * q^25 + 40 * q^31 - 32 * q^37 - 32 * q^43 + 208 * q^55 - 272 * q^61 - 168 * q^67 + 144 * q^73 - 232 * q^79 - 544 * q^85 - 192 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$785$	$883$	$1081$
$\chi(n)$	$-1$	$1$	$e\left(\frac{1}{3}\right)$

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
$5$	−2.01563	−	1.16372i	−0.403125	−	0.232744i	0.284706	−	0.958615i	$-0.408104\pi$
$5$	−2.01563	−	1.16372i	−0.403125	−	0.232744i	−0.687832	+	0.725870i	$0.741437\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−7.70549	+	4.44876i	−0.700499	+	0.404433i	−0.807533	−	0.589822i	$-0.799198\pi$
$11$	−7.70549	+	4.44876i	−0.700499	+	0.404433i	0.107034	+	0.994255i	$0.465864\pi$
$12$		0			0
$13$	−8.58301			−0.660231			−0.330116	−	0.943941i	$-0.607088\pi$
$13$	−8.58301			−0.660231			−0.330116	+	0.943941i	$0.607088\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	17.4266	−	10.0613i	1.02509	−	0.591838i	0.109519	−	0.993985i	$-0.465069\pi$
$17$	17.4266	−	10.0613i	1.02509	−	0.591838i	0.915575	+	0.402146i	$0.131736\pi$
$18$		0			0
$19$	2.93725	−	5.08747i	0.154592	−	0.267762i	−0.778318	−	0.627870i	$-0.783927\pi$
$19$	2.93725	−	5.08747i	0.154592	−	0.267762i	0.932911	+	0.360108i	$0.117260\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	15.0540	+	8.69140i	0.654520	+	0.377887i	0.790186	−	0.612867i	$-0.209984\pi$
$23$	15.0540	+	8.69140i	0.654520	+	0.377887i	−0.135666	+	0.990755i	$0.543317\pi$
$24$		0			0
$25$	−9.79150	−	16.9594i	−0.391660	−	0.678375i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	19.0318i		−	0.656269i	−0.944631	−	0.328134i	$-0.893580\pi$
$29$		−	19.0318i		−	0.656269i	0.944631	−	0.328134i	$-0.106420\pi$
$30$		0			0
$31$	2.35425	+	4.07768i	0.0759435	+	0.131538i	0.901496	−	0.432787i	$-0.142470\pi$
$31$	2.35425	+	4.07768i	0.0759435	+	0.131538i	−0.825553	+	0.564325i	$0.809136\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−19.8745	+	34.4237i	−0.537149	+	0.930369i	0.461907	+	0.886928i	$0.347165\pi$
$37$	−19.8745	+	34.4237i	−0.537149	+	0.930369i	−0.999056	+	0.0434408i	$0.986168\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			6.98233i			0.170301i	0.996368	+	0.0851504i	$0.0271371\pi$
$41$			6.98233i			0.170301i	−0.996368	+	0.0851504i	$0.972863\pi$
$42$		0			0
$43$	−35.7490			−0.831372			−0.415686	−	0.909508i	$-0.636459\pi$
$43$	−35.7490			−0.831372			−0.415686	+	0.909508i	$0.636459\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	63.5330	+	36.6808i	1.35177	+	0.780443i	0.988497	−	0.151241i	$-0.0483270\pi$
$47$	63.5330	+	36.6808i	1.35177	+	0.780443i	0.363270	+	0.931684i	$0.381660\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−40.4166	+	23.3345i	−0.762577	+	0.440274i	−0.830220	−	0.557436i	$-0.811785\pi$
$53$	−40.4166	+	23.3345i	−0.762577	+	0.440274i	0.0676432	+	0.997710i	$0.478452\pi$
$54$		0			0
$55$	20.7085			0.376518
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−20.8703	+	12.0495i	−0.353734	+	0.204228i	−0.666328	−	0.745658i	$-0.732135\pi$
$59$	−20.8703	+	12.0495i	−0.353734	+	0.204228i	0.312595	+	0.949887i	$0.398802\pi$
$60$		0			0
$61$	−36.6458	+	63.4723i	−0.600750	+	1.04053i	0.391958	+	0.919983i	$0.371798\pi$
$61$	−36.6458	+	63.4723i	−0.600750	+	1.04053i	−0.992708	+	0.120546i	$0.961535\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	17.3001	+	9.98823i	0.266156	+	0.153665i
$66$		0			0
$67$	16.0405	+	27.7830i	0.239411	+	0.414672i	0.960545	−	0.278123i	$-0.0897124\pi$
$67$	16.0405	+	27.7830i	0.239411	+	0.414672i	−0.721135	+	0.692795i	$0.756379\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			114.843i			1.61751i	0.588144	+	0.808756i	$0.299859\pi$
$71$			114.843i			1.61751i	−0.588144	+	0.808756i	$0.700141\pi$
$72$		0			0
$73$	20.6458	+	35.7595i	0.282819	+	0.489856i	0.972078	−	0.234659i	$-0.0753973\pi$
$73$	20.6458	+	35.7595i	0.282819	+	0.489856i	−0.689259	+	0.724515i	$0.742064\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	13.3320	−	23.0917i	0.168760	−	0.292300i	−0.769224	−	0.638979i	$-0.779357\pi$
$79$	13.3320	−	23.0917i	0.168760	−	0.292300i	0.937984	+	0.346678i	$0.112690\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	136.057i		−	1.63924i	−0.572910	−	0.819618i	$-0.694186\pi$
$83$		−	136.057i		−	1.63924i	0.572910	−	0.819618i	$-0.305814\pi$
$84$		0			0
$85$	−46.8340			−0.550988
$86$		0			0
$87$		0			0
$88$		0			0
$89$	126.226	+	72.8764i	1.41826	+	0.818836i	0.996146	−	0.0877052i	$-0.0279533\pi$
$89$	126.226	+	72.8764i	1.41826	+	0.818836i	0.422118	+	0.906541i	$0.361287\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−11.8408	+	6.83629i	−0.124640	+	0.0719610i
$96$		0			0
$97$	65.9555			0.679954			0.339977	−	0.940434i	$-0.389581\pi$
$97$	65.9555			0.679954			0.339977	+	0.940434i	$0.389581\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	80.2456	−	46.3298i	0.794511	−	0.458711i	−0.0470373	−	0.998893i	$-0.514978\pi$
$101$	80.2456	−	46.3298i	0.794511	−	0.458711i	0.841548	+	0.540182i	$0.181645\pi$
$102$		0			0
$103$	−96.5608	+	167.248i	−0.937483	+	1.62377i	−0.167339	+	0.985899i	$0.553517\pi$
$103$	−96.5608	+	167.248i	−0.937483	+	1.62377i	−0.770145	+	0.637869i	$0.779816\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	40.8777	+	23.6007i	0.382034	+	0.220568i	0.678703	−	0.734413i	$-0.262542\pi$
$107$	40.8777	+	23.6007i	0.382034	+	0.220568i	−0.296669	+	0.954980i	$0.595876\pi$
$108$		0			0
$109$	−104.081	−	180.274i	−0.954872	−	1.65389i	−0.734661	−	0.678434i	$-0.762659\pi$
$109$	−104.081	−	180.274i	−0.954872	−	1.65389i	−0.220211	−	0.975452i	$-0.570675\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			124.686i			1.10341i	0.834039	+	0.551706i	$0.186023\pi$
$113$			124.686i			1.10341i	−0.834039	+	0.551706i	$0.813977\pi$
$114$		0			0
$115$	−20.2288	−	35.0372i	−0.175902	−	0.304672i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−20.9170	+	36.2293i	−0.172868	+	0.299416i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			103.764i			0.830116i
$126$		0			0
$127$	51.5791			0.406134			0.203067	−	0.979165i	$-0.434909\pi$
$127$	51.5791			0.406134			0.203067	+	0.979165i	$0.434909\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	153.604	+	88.6832i	1.17255	+	0.676971i	0.954279	−	0.298918i	$-0.0966255\pi$
$131$	153.604	+	88.6832i	1.17255	+	0.676971i	0.218269	+	0.975889i	$0.429959\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−139.264	+	80.4040i	−1.01652	+	0.586891i	−0.913095	−	0.407746i	$-0.866315\pi$
$137$	−139.264	+	80.4040i	−1.01652	+	0.586891i	−0.103429	+	0.994637i	$0.532982\pi$
$138$		0			0
$139$	74.8340			0.538374			0.269187	−	0.963088i	$-0.413245\pi$
$139$	74.8340			0.538374			0.269187	+	0.963088i	$0.413245\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	66.1362	−	38.1838i	0.462491	−	0.267019i
$144$		0			0
$145$	−22.1477	+	38.3610i	−0.152743	+	0.264558i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	227.490	+	131.342i	1.52678	+	0.881487i	0.999494	+	0.0317997i	$0.0101239\pi$
$149$	227.490	+	131.342i	1.52678	+	0.881487i	0.527286	+	0.849688i	$0.323209\pi$
$150$		0			0
$151$	−16.9595	−	29.3747i	−0.112314	−	0.194534i	0.804389	−	0.594104i	$-0.202493\pi$
$151$	−16.9595	−	29.3747i	−0.112314	−	0.194534i	−0.916703	+	0.399569i	$0.869160\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	10.9588i		−	0.0707017i
$156$		0			0
$157$	81.8118	+	141.702i	0.521094	+	0.902561i	0.999699	+	0.0245310i	$0.00780925\pi$
$157$	81.8118	+	141.702i	0.521094	+	0.902561i	−0.478605	+	0.878030i	$0.658857\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−140.373	+	243.132i	−0.861181	+	1.49161i	0.00960841	+	0.999954i	$0.496942\pi$
$163$	−140.373	+	243.132i	−0.861181	+	1.49161i	−0.870790	+	0.491656i	$0.836392\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	76.0755i		−	0.455542i	−0.973715	−	0.227771i	$-0.926856\pi$
$167$		−	76.0755i		−	0.455542i	0.973715	−	0.227771i	$-0.0731437\pi$
$168$		0			0
$169$	−95.3320			−0.564095
$170$		0			0
$171$		0			0
$172$		0			0
$173$	133.321	+	76.9730i	0.770642	+	0.444930i	0.833104	−	0.553117i	$-0.186562\pi$
$173$	133.321	+	76.9730i	0.770642	+	0.444930i	−0.0624615	+	0.998047i	$0.519895\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−230.094	+	132.845i	−1.28544	+	0.742148i	−0.977837	−	0.209367i	$-0.932860\pi$
$179$	−230.094	+	132.845i	−1.28544	+	0.742148i	−0.307601	+	0.951515i	$0.599526\pi$
$180$		0			0
$181$	−203.328			−1.12336			−0.561680	−	0.827355i	$-0.689845\pi$
$181$	−203.328			−1.12336			−0.561680	+	0.827355i	$0.689845\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	80.1191	−	46.2568i	0.433076	−	0.250037i
$186$		0			0
$187$	−89.5203	+	155.054i	−0.478718	+	0.829164i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	184.783	+	106.684i	0.967449	+	0.558557i	0.898458	−	0.439060i	$-0.144689\pi$
$191$	184.783	+	106.684i	0.967449	+	0.558557i	0.0689915	+	0.997617i	$0.478022\pi$
$192$		0			0
$193$	−8.25098	−	14.2911i	−0.0427512	−	0.0740473i	0.843858	−	0.536567i	$-0.180279\pi$
$193$	−8.25098	−	14.2911i	−0.0427512	−	0.0740473i	−0.886609	+	0.462519i	$0.846946\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			57.0437i			0.289562i	0.989464	+	0.144781i	$0.0462477\pi$
$197$			57.0437i			0.289562i	−0.989464	+	0.144781i	$0.953752\pi$
$198$		0			0
$199$	151.122	+	261.750i	0.759405	+	1.31533i	0.943154	+	0.332355i	$0.107843\pi$
$199$	151.122	+	261.750i	0.759405	+	1.31533i	−0.183750	+	0.982973i	$0.558824\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	8.12549	−	14.0738i	0.0396365	−	0.0686525i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			52.2686i			0.250089i
$210$		0			0
$211$	−142.996			−0.677706			−0.338853	−	0.940839i	$-0.610039\pi$
$211$	−142.996			−0.677706			−0.338853	+	0.940839i	$0.610039\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	72.0566	+	41.6019i	0.335147	+	0.193497i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−149.573	+	86.3558i	−0.676799	+	0.390750i
$222$		0			0
$223$	167.749			0.752238			0.376119	−	0.926571i	$-0.377258\pi$
$223$	167.749			0.752238			0.376119	+	0.926571i	$0.377258\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	53.1203	−	30.6690i	0.234010	−	0.135106i	−0.378411	−	0.925638i	$-0.623529\pi$
$227$	53.1203	−	30.6690i	0.234010	−	0.135106i	0.612421	+	0.790532i	$0.290196\pi$
$228$		0			0
$229$	−22.7935	+	39.4795i	−0.0995348	+	0.172399i	−0.911492	−	0.411317i	$-0.865069\pi$
$229$	−22.7935	+	39.4795i	−0.0995348	+	0.172399i	0.811957	+	0.583717i	$0.198402\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−259.279	−	149.695i	−1.11279	−	0.642468i	−0.173237	−	0.984880i	$-0.555423\pi$
$233$	−259.279	−	149.695i	−1.11279	−	0.642468i	−0.939550	+	0.342413i	$0.888756\pi$
$234$		0			0
$235$	−85.3725	−	147.870i	−0.363287	−	0.629232i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	146.895i		−	0.614624i	−0.951609	−	0.307312i	$-0.900570\pi$
$239$		−	146.895i		−	0.614624i	0.951609	−	0.307312i	$-0.0994295\pi$
$240$		0			0
$241$	−176.808	−	306.240i	−0.733642	−	1.27071i	−0.955316	−	0.295585i	$-0.904485\pi$
$241$	−176.808	−	306.240i	−0.733642	−	1.27071i	0.221674	−	0.975121i	$-0.428848\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−25.2105	+	43.6658i	−0.102067	+	0.176785i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	289.908i		−	1.15501i	−0.816386	−	0.577506i	$-0.804026\pi$
$251$		−	289.908i		−	1.15501i	0.816386	−	0.577506i	$-0.195974\pi$
$252$		0			0
$253$	−154.664			−0.611320
$254$		0			0
$255$		0			0
$256$		0			0
$257$	420.001	+	242.488i	1.63425	+	0.943532i	0.982763	+	0.184867i	$0.0591856\pi$
$257$	420.001	+	242.488i	1.63425	+	0.943532i	0.651482	+	0.758664i	$0.274148\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	181.719	−	104.915i	0.690945	−	0.398917i	−0.113021	−	0.993593i	$-0.536053\pi$
$263$	181.719	−	104.915i	0.690945	−	0.398917i	0.803966	+	0.594675i	$0.202719\pi$
$264$		0			0
$265$	108.620			0.409885
$266$		0			0
$267$		0			0
$268$		0			0
$269$	135.969	−	78.5018i	0.505461	−	0.291828i	−0.225505	−	0.974242i	$-0.572403\pi$
$269$	135.969	−	78.5018i	0.505461	−	0.291828i	0.730966	+	0.682414i	$0.239070\pi$
$270$		0			0
$271$	79.6013	−	137.873i	0.293732	−	0.508758i	−0.680957	−	0.732323i	$-0.738436\pi$
$271$	79.6013	−	137.873i	0.293732	−	0.508758i	0.974689	+	0.223565i	$0.0717694\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	150.897	+	87.1202i	0.548715	+	0.316801i
$276$		0			0
$277$	−13.7530	−	23.8208i	−0.0496496	−	0.0859957i	0.840133	−	0.542381i	$-0.182477\pi$
$277$	−13.7530	−	23.8208i	−0.0496496	−	0.0859957i	−0.889782	+	0.456385i	$0.849144\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			503.037i			1.79017i	0.445900	+	0.895083i	$0.352884\pi$
$281$			503.037i			1.79017i	−0.445900	+	0.895083i	$0.647116\pi$
$282$		0			0
$283$	−101.225	−	175.327i	−0.357685	−	0.619528i	0.629889	−	0.776685i	$-0.283100\pi$
$283$	−101.225	−	175.327i	−0.357685	−	0.619528i	−0.987574	+	0.157157i	$0.949767\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	57.9575	−	100.385i	0.200545	−	0.347354i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	11.3969i		−	0.0388972i	−0.999811	−	0.0194486i	$-0.993809\pi$
$293$		−	11.3969i		−	0.0388972i	0.999811	−	0.0194486i	$-0.00619107\pi$
$294$		0			0
$295$	56.0889			0.190132
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−129.208	−	74.5984i	−0.432134	−	0.249493i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	147.728	−	85.2909i	0.484355	−	0.279642i
$306$		0			0
$307$	9.96342			0.0324541			0.0162271	−	0.999868i	$-0.494835\pi$
$307$	9.96342			0.0324541			0.0162271	+	0.999868i	$0.494835\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−132.317	+	76.3934i	−0.425457	+	0.245638i	−0.697410	−	0.716673i	$-0.745664\pi$
$311$	−132.317	+	76.3934i	−0.425457	+	0.245638i	0.271952	+	0.962311i	$0.412331\pi$
$312$		0			0
$313$	−254.535	+	440.867i	−0.813210	+	1.40852i	0.0973968	+	0.995246i	$0.468948\pi$
$313$	−254.535	+	440.867i	−0.813210	+	1.40852i	−0.910606	+	0.413275i	$0.864385\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−228.413	−	131.874i	−0.720544	−	0.416006i	0.0944088	−	0.995534i	$-0.469904\pi$
$317$	−228.413	−	131.874i	−0.720544	−	0.416006i	−0.814953	+	0.579527i	$0.803237\pi$
$318$		0			0
$319$	84.6680	+	146.649i	0.265417	+	0.459715i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	118.210i		−	0.365975i
$324$		0			0
$325$	84.0405	+	145.562i	0.258586	+	0.447884i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	48.2510	−	83.5732i	0.145773	−	0.252487i	−0.783888	−	0.620902i	$-0.786766\pi$
$331$	48.2510	−	83.5732i	0.145773	−	0.252487i	0.929661	+	0.368416i	$0.120100\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	74.6668i		−	0.222886i
$336$		0			0
$337$	−213.417			−0.633285			−0.316642	−	0.948545i	$-0.602556\pi$
$337$	−213.417			−0.633285			−0.316642	+	0.948545i	$0.602556\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−36.2813	−	20.9470i	−0.106397	−	0.0614281i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	111.298	−	64.2580i	0.320744	−	0.185182i	−0.330980	−	0.943638i	$-0.607379\pi$
$347$	111.298	−	64.2580i	0.320744	−	0.185182i	0.651724	+	0.758456i	$0.274046\pi$
$348$		0			0
$349$	−373.041			−1.06888			−0.534442	−	0.845205i	$-0.679478\pi$
$349$	−373.041			−1.06888			−0.534442	+	0.845205i	$0.679478\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−348.658	+	201.298i	−0.987701	+	0.570249i	−0.904586	−	0.426291i	$-0.859820\pi$
$353$	−348.658	+	201.298i	−0.987701	+	0.570249i	−0.0831146	+	0.996540i	$0.526487\pi$
$354$		0			0
$355$	133.646	−	231.481i	0.376467	−	0.652060i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	328.480	+	189.648i	0.914985	+	0.528267i	0.882032	−	0.471190i	$-0.156175\pi$
$359$	328.480	+	189.648i	0.914985	+	0.528267i	0.0329535	+	0.999457i	$0.489509\pi$
$360$		0			0
$361$	163.245	+	282.749i	0.452202	+	0.783238i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	96.1037i		−	0.263298i
$366$		0			0
$367$	−82.8784	−	143.550i	−0.225827	−	0.391144i	0.730740	−	0.682655i	$-0.239175\pi$
$367$	−82.8784	−	143.550i	−0.225827	−	0.391144i	−0.956567	+	0.291512i	$0.905842\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	42.1660	−	73.0337i	0.113046	−	0.195801i	−0.803951	−	0.594695i	$-0.797273\pi$
$373$	42.1660	−	73.0337i	0.113046	−	0.195801i	0.916997	+	0.398895i	$0.130606\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			163.350i			0.433289i
$378$		0			0
$379$	537.652			1.41861			0.709304	−	0.704903i	$-0.249009\pi$
$379$	537.652			1.41861			0.709304	+	0.704903i	$0.249009\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−364.567	−	210.483i	−0.951873	−	0.549564i	−0.0582107	−	0.998304i	$-0.518540\pi$
$383$	−364.567	−	210.483i	−0.951873	−	0.549564i	−0.893662	+	0.448740i	$0.851873\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−276.118	+	159.417i	−0.709816	+	0.409812i	−0.810993	−	0.585056i	$-0.801072\pi$
$389$	−276.118	+	159.417i	−0.709816	+	0.409812i	0.101177	+	0.994868i	$0.467739\pi$
$390$		0			0
$391$	349.786			0.894592
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−53.7447	+	31.0295i	−0.136063	+	0.0785558i
$396$		0			0
$397$	67.8562	−	117.530i	0.170922	−	0.296046i	−0.767820	−	0.640665i	$-0.778659\pi$
$397$	67.8562	−	117.530i	0.170922	−	0.296046i	0.938743	+	0.344619i	$0.111992\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−369.417	−	213.283i	−0.921239	−	0.531877i	−0.0372086	−	0.999308i	$-0.511847\pi$
$401$	−369.417	−	213.283i	−0.921239	−	0.531877i	−0.884030	+	0.467430i	$0.845180\pi$
$402$		0			0
$403$	−20.2065	−	34.9987i	−0.0501403	−	0.0868455i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	353.668i		−	0.868963i
$408$		0			0
$409$	270.557	+	468.618i	0.661508	+	1.14577i	0.980219	+	0.197914i	$0.0634166\pi$
$409$	270.557	+	468.618i	0.661508	+	1.14577i	−0.318711	+	0.947852i	$0.603250\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−158.332	+	274.239i	−0.381523	+	0.660817i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			337.625i			0.805789i	0.915247	+	0.402894i	$0.131996\pi$
$419$			337.625i			0.805789i	−0.915247	+	0.402894i	$0.868004\pi$
$420$		0			0
$421$	366.324			0.870129			0.435064	−	0.900399i	$-0.356726\pi$
$421$	366.324			0.870129			0.435064	+	0.900399i	$0.356726\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−341.265	−	197.030i	−0.802977	−	0.463599i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	261.332	−	150.880i	0.606338	−	0.350069i	−0.165193	−	0.986261i	$-0.552825\pi$
$431$	261.332	−	150.880i	0.606338	−	0.350069i	0.771531	+	0.636192i	$0.219491\pi$
$432$		0			0
$433$	40.2431			0.0929402			0.0464701	−	0.998920i	$-0.485203\pi$
$433$	40.2431			0.0929402			0.0464701	+	0.998920i	$0.485203\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	88.4346	−	51.0577i	0.202367	−	0.116837i
$438$		0			0
$439$	395.409	−	684.869i	0.900704	−	1.56007i	0.0741221	−	0.997249i	$-0.476385\pi$
$439$	395.409	−	684.869i	0.900704	−	1.56007i	0.826582	−	0.562816i	$-0.190282\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−166.813	−	96.3098i	−0.376554	−	0.217404i	0.299764	−	0.954013i	$-0.403092\pi$
$443$	−166.813	−	96.3098i	−0.376554	−	0.217404i	−0.676318	+	0.736610i	$0.736425\pi$
$444$		0			0
$445$	−169.616	−	293.783i	−0.381159	−	0.660186i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	196.690i		−	0.438063i	−0.975718	−	0.219032i	$-0.929710\pi$
$449$		−	196.690i		−	0.438063i	0.975718	−	0.219032i	$-0.0702897\pi$
$450$		0			0
$451$	−31.0627	−	53.8023i	−0.0688753	−	0.119295i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	324.664	−	562.335i	0.710425	−	1.23049i	−0.254273	−	0.967132i	$-0.581836\pi$
$457$	324.664	−	562.335i	0.710425	−	1.23049i	0.964698	−	0.263359i	$-0.0848304\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			846.252i			1.83569i	0.396941	+	0.917844i	$0.370072\pi$
$461$			846.252i			1.83569i	−0.396941	+	0.917844i	$0.629928\pi$
$462$		0			0
$463$	3.91896			0.00846428			0.00423214	−	0.999991i	$-0.498653\pi$
$463$	3.91896			0.00846428			0.00423214	+	0.999991i	$0.498653\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−723.304	−	417.600i	−1.54883	−	0.894218i	−0.998231	−	0.0594479i	$-0.981066\pi$
$467$	−723.304	−	417.600i	−1.54883	−	0.894218i	−0.550599	−	0.834770i	$-0.685601\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	275.464	−	159.039i	0.582375	−	0.336235i
$474$		0			0
$475$	−115.041			−0.242191
$476$		0			0
$477$		0			0
$478$		0			0
$479$	353.530	−	204.111i	0.738059	−	0.426118i	−0.0833043	−	0.996524i	$-0.526547\pi$
$479$	353.530	−	204.111i	0.738059	−	0.426118i	0.821363	+	0.570406i	$0.193214\pi$
$480$		0			0
$481$	170.583	−	295.458i	0.354642	−	0.614259i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−132.942	−	76.7539i	−0.274107	−	0.158255i
$486$		0			0
$487$	186.627	+	323.248i	0.383219	+	0.663754i	0.991520	−	0.129952i	$-0.0414823\pi$
$487$	186.627	+	323.248i	0.383219	+	0.663754i	−0.608302	+	0.793706i	$0.708149\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	328.281i		−	0.668596i	−0.942467	−	0.334298i	$-0.891501\pi$
$491$		−	328.281i		−	0.668596i	0.942467	−	0.334298i	$-0.108499\pi$
$492$		0			0
$493$	−191.484	−	331.659i	−0.388405	−	0.672737i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−143.203	+	248.034i	−0.286979	+	0.497062i	−0.973087	−	0.230437i	$-0.925984\pi$
$499$	−143.203	+	248.034i	−0.286979	+	0.497062i	0.686108	+	0.727500i	$0.259318\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	798.493i		−	1.58746i	−0.608270	−	0.793730i	$-0.708136\pi$
$503$		−	798.493i		−	1.58746i	0.608270	−	0.793730i	$-0.291864\pi$
$504$		0			0
$505$	−215.660			−0.427050
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−732.921	−	423.152i	−1.43992	−	0.831340i	−0.442080	−	0.896976i	$-0.645759\pi$
$509$	−732.921	−	423.152i	−1.43992	−	0.831340i	−0.997844	+	0.0656358i	$0.979092\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	389.261	−	224.740i	0.755846	−	0.436388i
$516$		0			0
$517$	−652.737			−1.26255
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−222.515	+	128.469i	−0.427091	+	0.246581i	−0.698107	−	0.715994i	$-0.745974\pi$
$521$	−222.515	+	128.469i	−0.427091	+	0.246581i	0.271015	+	0.962575i	$0.412641\pi$
$522$		0			0
$523$	21.7569	−	37.6840i	0.0416002	−	0.0720536i	−0.844476	−	0.535594i	$-0.820088\pi$
$523$	21.7569	−	37.6840i	0.0416002	−	0.0720536i	0.886076	+	0.463540i	$0.153421\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	82.0531	+	47.3734i	0.155698	+	0.0898925i
$528$		0			0
$529$	−113.419	−	196.447i	−0.214403	−	0.371356i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	59.9294i		−	0.112438i
$534$		0			0
$535$	−54.9294	−	95.1405i	−0.102672	−	0.177833i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	304.830	−	527.981i	0.563457	−	0.975936i	−0.433735	−	0.901041i	$-0.642805\pi$
$541$	304.830	−	527.981i	0.563457	−	0.975936i	0.997191	−	0.0748950i	$-0.0238622\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			484.486i			0.888964i
$546$		0			0
$547$	1003.83			1.83516			0.917578	−	0.397556i	$-0.130142\pi$
$547$	1003.83			1.83516			0.917578	+	0.397556i	$0.130142\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−96.8238	−	55.9012i	−0.175724	−	0.101454i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	9.08875	−	5.24739i	0.0163173	−	0.00942081i	−0.491819	−	0.870697i	$-0.663668\pi$
$557$	9.08875	−	5.24739i	0.0163173	−	0.00942081i	0.508136	+	0.861277i	$0.330335\pi$
$558$		0			0
$559$	306.834			0.548898
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−833.992	+	481.505i	−1.48134	+	0.855250i	−0.999776	−	0.0211638i	$-0.993263\pi$
$563$	−833.992	+	481.505i	−1.48134	+	0.855250i	−0.481560	+	0.876413i	$0.659930\pi$
$564$		0			0
$565$	145.099	−	251.319i	0.256813	−	0.444813i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−381.972	−	220.531i	−0.671303	−	0.387577i	0.125267	−	0.992123i	$-0.460021\pi$
$569$	−381.972	−	220.531i	−0.671303	−	0.387577i	−0.796570	+	0.604546i	$0.793355\pi$
$570$		0			0
$571$	481.575	+	834.113i	0.843389	+	1.46079i	0.887013	+	0.461745i	$0.152776\pi$
$571$	481.575	+	834.113i	0.843389	+	1.46079i	−0.0436239	+	0.999048i	$0.513890\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	340.408i		−	0.592013i
$576$		0			0
$577$	−204.033	−	353.395i	−0.353609	−	0.612470i	0.633270	−	0.773931i	$-0.281712\pi$
$577$	−204.033	−	353.395i	−0.353609	−	0.612470i	−0.986879	+	0.161462i	$0.948379\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	207.620	−	359.608i	0.356123	−	0.616823i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	643.388i		−	1.09606i	−0.836458	−	0.548030i	$-0.815378\pi$
$587$		−	643.388i		−	1.09606i	0.836458	−	0.548030i	$-0.184622\pi$
$588$		0			0
$589$	27.6601			0.0469611
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−82.7592	−	47.7810i	−0.139560	−	0.0805751i	0.428594	−	0.903497i	$-0.359009\pi$
$593$	−82.7592	−	47.7810i	−0.139560	−	0.0805751i	−0.568154	+	0.822922i	$0.692342\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	627.833	−	362.480i	1.04814	−	0.605141i	0.126009	−	0.992029i	$-0.459783\pi$
$599$	627.833	−	362.480i	1.04814	−	0.605141i	0.922127	+	0.386888i	$0.126450\pi$
$600$		0			0
$601$	−999.239			−1.66263			−0.831314	−	0.555803i	$-0.812411\pi$
$601$	−999.239			−1.66263			−0.831314	+	0.555803i	$0.812411\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	84.3217	−	48.6831i	0.139375	−	0.0804680i
$606$		0			0
$607$	231.793	−	401.478i	0.381867	−	0.661414i	−0.609462	−	0.792815i	$-0.708614\pi$
$607$	231.793	−	401.478i	0.381867	−	0.661414i	0.991329	+	0.131402i	$0.0419478\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−545.304	−	314.832i	−0.892479	−	0.515273i
$612$		0			0
$613$	472.697	+	818.735i	0.771120	+	1.33562i	0.936949	+	0.349465i	$0.113637\pi$
$613$	472.697	+	818.735i	0.771120	+	1.33562i	−0.165829	+	0.986154i	$0.553030\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			346.419i			0.561458i	0.959787	+	0.280729i	$0.0905762\pi$
$617$			346.419i			0.561458i	−0.959787	+	0.280729i	$0.909424\pi$
$618$		0			0
$619$	−569.292	−	986.042i	−0.919695	−	1.59296i	−0.799877	−	0.600163i	$-0.795102\pi$
$619$	−569.292	−	986.042i	−0.919695	−	1.59296i	−0.119818	−	0.992796i	$-0.538231\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−124.035	+	214.834i	−0.198455	+	0.343735i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			799.850i			1.27162i
$630$		0			0
$631$	−1006.22			−1.59464			−0.797321	−	0.603555i	$-0.793750\pi$
$631$	−1006.22			−1.59464			−0.797321	+	0.603555i	$0.793750\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−103.964	−	60.0237i	−0.163723	−	0.0945255i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	103.741	−	59.8952i	0.161843	−	0.0934402i	−0.416891	−	0.908957i	$-0.636880\pi$
$641$	103.741	−	59.8952i	0.161843	−	0.0934402i	0.578734	+	0.815516i	$0.303547\pi$
$642$		0			0
$643$	935.616			1.45508			0.727539	−	0.686066i	$-0.240664\pi$
$643$	935.616			1.45508			0.727539	+	0.686066i	$0.240664\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−494.624	+	285.571i	−0.764488	+	0.441378i	−0.830905	−	0.556414i	$-0.812177\pi$
$647$	−494.624	+	285.571i	−0.764488	+	0.441378i	0.0664166	+	0.997792i	$0.478843\pi$
$648$		0			0
$649$	107.210	−	185.694i	0.165193	−	0.286123i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−532.601	−	307.497i	−0.815621	−	0.470899i	0.0332829	−	0.999446i	$-0.489404\pi$
$653$	−532.601	−	307.497i	−0.815621	−	0.470899i	−0.848904	+	0.528547i	$0.822737\pi$
$654$		0			0
$655$	−206.405	−	357.504i	−0.315122	−	0.545808i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	958.528i		−	1.45452i	−0.686363	−	0.727259i	$-0.740794\pi$
$659$		−	958.528i		−	1.45452i	0.686363	−	0.727259i	$-0.259206\pi$
$660$		0			0
$661$	397.893	+	689.171i	0.601956	+	1.04262i	0.992525	+	0.122044i	$0.0389450\pi$
$661$	397.893	+	689.171i	0.601956	+	1.04262i	−0.390569	+	0.920574i	$0.627722\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	165.413	−	286.504i	0.247996	−	0.429541i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	652.113i		−	0.971853i
$672$		0			0
$673$	127.514			0.189471			0.0947353	−	0.995502i	$-0.469800\pi$
$673$	127.514			0.189471			0.0947353	+	0.995502i	$0.469800\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−26.4113	−	15.2486i	−0.0390122	−	0.0225237i	0.480367	−	0.877068i	$-0.340503\pi$
$677$	−26.4113	−	15.2486i	−0.0390122	−	0.0225237i	−0.519379	+	0.854544i	$0.673837\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	947.477	−	547.026i	1.38723	−	0.800917i	0.394226	−	0.919013i	$-0.371013\pi$
$683$	947.477	−	547.026i	1.38723	−	0.800917i	0.993002	+	0.118097i	$0.0376793\pi$
$684$		0			0
$685$	374.272			0.546382
$686$		0			0
$687$		0			0
$688$		0			0
$689$	346.896	−	200.280i	0.503477	−	0.290683i
$690$		0			0
$691$	55.8745	−	96.7775i	0.0808604	−	0.140054i	−0.822759	−	0.568390i	$-0.807567\pi$
$691$	55.8745	−	96.7775i	0.0808604	−	0.140054i	0.903620	+	0.428336i	$0.140900\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−150.837	−	87.0860i	−0.217032	−	0.125304i
$696$		0			0
$697$	70.2510	+	121.678i	0.100791	+	0.174574i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			71.8328i			0.102472i	0.998687	+	0.0512360i	$0.0163161\pi$
$701$			71.8328i			0.102472i	−0.998687	+	0.0512360i	$0.983684\pi$
$702$		0			0
$703$	116.753	+	202.222i	0.166078	+	0.287656i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−90.9882	+	157.596i	−0.128333	+	0.222280i	−0.923031	−	0.384726i	$-0.874296\pi$
$709$	−90.9882	+	157.596i	−0.128333	+	0.222280i	0.794698	+	0.607005i	$0.207629\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			81.8469i			0.114792i
$714$		0			0
$715$	−177.741			−0.248589
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−635.330	−	366.808i	−0.883631	−	0.510164i	−0.0117768	−	0.999931i	$-0.503749\pi$
$719$	−635.330	−	366.808i	−0.883631	−	0.510164i	−0.871854	+	0.489766i	$0.837082\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−322.767	+	186.350i	−0.445197	+	0.257034i
$726$		0			0
$727$	130.450			0.179436			0.0897178	−	0.995967i	$-0.471403\pi$
$727$	130.450			0.179436			0.0897178	+	0.995967i	$0.471403\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−622.984	+	359.680i	−0.852235	+	0.492038i
$732$		0			0
$733$	419.907	−	727.301i	0.572861	−	0.992224i	−0.423409	−	0.905938i	$-0.639167\pi$
$733$	419.907	−	727.301i	0.572861	−	0.992224i	0.996270	−	0.0862860i	$-0.0274999\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−247.200	−	142.721i	−0.335414	−	0.193651i
$738$		0			0
$739$	130.454	+	225.952i	0.176527	+	0.305754i	0.940689	−	0.339271i	$-0.110180\pi$
$739$	130.454	+	225.952i	0.176527	+	0.305754i	−0.764162	+	0.645025i	$0.776847\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			143.701i			0.193406i	0.995313	+	0.0967030i	$0.0308297\pi$
$743$			143.701i			0.193406i	−0.995313	+	0.0967030i	$0.969170\pi$
$744$		0			0
$745$	−305.690	−	529.471i	−0.410322	−	0.710699i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	44.0732	−	76.3370i	0.0586860	−	0.101647i	−0.835190	−	0.549962i	$-0.814642\pi$
$751$	44.0732	−	76.3370i	0.0586860	−	0.101647i	0.893876	+	0.448315i	$0.147976\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			78.9445i			0.104562i
$756$		0			0
$757$	−519.660			−0.686473			−0.343237	−	0.939249i	$-0.611523\pi$
$757$	−519.660			−0.686473			−0.343237	+	0.939249i	$0.611523\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−221.637	−	127.962i	−0.291245	−	0.168150i	0.347258	−	0.937769i	$-0.387113\pi$
$761$	−221.637	−	127.962i	−0.291245	−	0.168150i	−0.638503	+	0.769619i	$0.720446\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	179.130	−	103.421i	0.233546	−	0.134838i
$768$		0			0
$769$	−1202.63			−1.56389			−0.781946	−	0.623347i	$-0.785773\pi$
$769$	−1202.63			−1.56389			−0.781946	+	0.623347i	$0.785773\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	87.1330	−	50.3063i	0.112721	−	0.0650792i	−0.442580	−	0.896729i	$-0.645937\pi$
$773$	87.1330	−	50.3063i	0.112721	−	0.0650792i	0.555300	+	0.831650i	$0.312603\pi$
$774$		0			0
$775$	46.1033	−	79.8532i	0.0594881	−	0.103036i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	35.5224	+	20.5089i	0.0456000	+	0.0263272i
$780$		0			0
$781$	−510.911	−	884.924i	−0.654176	−	1.13307i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	380.825i		−	0.485127i
$786$		0			0
$787$	368.907	+	638.966i	0.468751	+	0.811901i	0.999362	−	0.0357147i	$-0.0113708\pi$
$787$	368.907	+	638.966i	0.468751	+	0.811901i	−0.530611	+	0.847616i	$0.678037\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	314.531	−	544.783i	0.396634	−	0.686990i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			747.143i			0.937444i	0.883346	+	0.468722i	$0.155285\pi$
$797$			747.143i			0.937444i	−0.883346	+	0.468722i	$0.844715\pi$
$798$		0			0
$799$	1476.22			1.84758
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−318.171	−	183.696i	−0.396228	−	0.228762i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−1001.31	+	578.107i	−1.23771	+	0.714595i	−0.968627	−	0.248521i	$-0.920056\pi$
$809$	−1001.31	+	578.107i	−1.23771	+	0.714595i	−0.269088	+	0.963116i	$0.586722\pi$
$810$		0			0
$811$	194.081			0.239311			0.119655	−	0.992815i	$-0.461821\pi$
$811$	194.081			0.239311			0.119655	+	0.992815i	$0.461821\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	565.877	−	326.709i	0.694328	−	0.400870i
$816$		0			0
$817$	−105.004	+	181.872i	−0.128524	+	0.222610i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	93.8794	+	54.2013i	0.114348	+	0.0660187i	0.556083	−	0.831127i	$-0.312304\pi$
$821$	93.8794	+	54.2013i	0.114348	+	0.0660187i	−0.441735	+	0.897145i	$0.645637\pi$
$822$		0			0
$823$	546.409	+	946.408i	0.663924	+	1.14995i	0.979576	+	0.201074i	$0.0644433\pi$
$823$	546.409	+	946.408i	0.663924	+	1.14995i	−0.315652	+	0.948875i	$0.602223\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	18.9283i		−	0.0228879i	−0.999935	−	0.0114440i	$-0.996357\pi$
$827$		−	18.9283i		−	0.0228879i	0.999935	−	0.0114440i	$-0.00364281\pi$
$828$		0			0
$829$	9.08891	+	15.7425i	0.0109637	+	0.0189897i	0.871455	−	0.490475i	$-0.163177\pi$
$829$	9.08891	+	15.7425i	0.0109637	+	0.0189897i	−0.860492	+	0.509465i	$0.829843\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−88.5307	+	153.340i	−0.106025	+	0.183640i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	297.466i		−	0.354548i	−0.984162	−	0.177274i	$-0.943272\pi$
$839$		−	297.466i		−	0.354548i	0.984162	−	0.177274i	$-0.0567278\pi$
$840$		0			0
$841$	478.791			0.569311
$842$		0			0
$843$		0			0
$844$		0			0
$845$	192.154	+	110.940i	0.227401	+	0.131290i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−598.380	+	345.475i	−0.703149	+	0.405963i
$852$		0			0
$853$	−280.125			−0.328400			−0.164200	−	0.986427i	$-0.552504\pi$
$853$	−280.125			−0.328400			−0.164200	+	0.986427i	$0.552504\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	370.867	−	214.120i	0.432750	−	0.249849i	−0.267767	−	0.963484i	$-0.586286\pi$
$857$	370.867	−	214.120i	0.432750	−	0.249849i	0.700518	+	0.713635i	$0.252952\pi$
$858$		0			0
$859$	−267.549	+	463.408i	−0.311466	+	0.539474i	−0.978680	−	0.205392i	$-0.934153\pi$
$859$	−267.549	+	463.408i	−0.311466	+	0.539474i	0.667214	+	0.744866i	$0.267487\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	1410.52	+	814.364i	1.63444	+	0.943643i	0.982701	+	0.185198i	$0.0592927\pi$
$863$	1410.52	+	814.364i	1.63444	+	0.943643i	0.651737	+	0.758445i	$0.274041\pi$
$864$		0			0
$865$	−179.150	−	310.297i	−0.207110	−	0.358725i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			237.244i			0.273008i
$870$		0			0
$871$	−137.676	−	238.462i	−0.158066	−	0.273779i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−310.705	+	538.156i	−0.354281	+	0.613633i	−0.986995	−	0.160753i	$-0.948608\pi$
$877$	−310.705	+	538.156i	−0.354281	+	0.613633i	0.632714	+	0.774386i	$0.281941\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			1011.60i			1.14824i	0.818773	+	0.574118i	$0.194655\pi$
$881$			1011.60i			1.14824i	−0.818773	+	0.574118i	$0.805345\pi$
$882$		0			0
$883$	400.235			0.453268			0.226634	−	0.973980i	$-0.427228\pi$
$883$	400.235			0.453268			0.226634	+	0.973980i	$0.427228\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−396.030	−	228.648i	−0.446482	−	0.257777i	0.259861	−	0.965646i	$-0.416323\pi$
$887$	−396.030	−	228.648i	−0.446482	−	0.257777i	−0.706343	+	0.707869i	$0.749656\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	373.225	−	215.482i	0.417945	−	0.241301i
$894$		0			0
$895$	618.376			0.690923
$896$		0			0
$897$		0			0
$898$		0			0
$899$	77.6055	−	44.8056i	0.0863243	−	0.0498394i
$900$		0			0
$901$	−469.549	+	813.283i	−0.521142	+	0.902644i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	409.833	+	236.617i	0.452854	+	0.261456i
$906$		0			0
$907$	−543.793	−	941.878i	−0.599552	−	1.03845i	−0.992887	−	0.119059i	$-0.962012\pi$
$907$	−543.793	−	941.878i	−0.599552	−	1.03845i	0.393335	−	0.919395i	$-0.371321\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			740.521i			0.812866i	0.913681	+	0.406433i	$0.133228\pi$
$911$			740.521i			0.812866i	−0.913681	+	0.406433i	$0.866772\pi$
$912$		0			0
$913$	605.284	+	1048.38i	0.662961	+	1.14828i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−539.956	+	935.230i	−0.587547	+	1.01766i	0.407006	+	0.913426i	$0.366573\pi$
$919$	−539.956	+	935.230i	−0.587547	+	1.01766i	−0.994553	+	0.104235i	$0.966760\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	985.701i		−	1.06793i
$924$		0			0
$925$	778.405			0.841519
$926$		0			0
$927$		0			0
$928$		0			0
$929$	126.389	+	72.9707i	0.136048	+	0.0785475i	0.566479	−	0.824076i	$-0.308305\pi$
$929$	126.389	+	72.9707i	0.136048	+	0.0785475i	−0.430431	+	0.902623i	$0.641638\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	360.879	−	208.353i	0.385966	−	0.222838i
$936$		0			0
$937$	406.972			0.434336			0.217168	−	0.976134i	$-0.430318\pi$
$937$	406.972			0.434336			0.217168	+	0.976134i	$0.430318\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	359.026	−	207.284i	0.381537	−	0.220281i	−0.296950	−	0.954893i	$-0.595969\pi$
$941$	359.026	−	207.284i	0.381537	−	0.220281i	0.678487	+	0.734613i	$0.262636\pi$
$942$		0			0
$943$	−60.6863	+	105.112i	−0.0643545	+	0.111465i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	582.910	+	336.543i	0.615533	+	0.355378i	0.775128	−	0.631805i	$-0.217685\pi$
$947$	582.910	+	336.543i	0.615533	+	0.355378i	−0.159595	+	0.987183i	$0.551019\pi$
$948$		0			0
$949$	−177.203	−	306.924i	−0.186726	−	0.323418i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			68.0024i			0.0713562i	0.999363	+	0.0356781i	$0.0113591\pi$
$953$			68.0024i			0.0713562i	−0.999363	+	0.0356781i	$0.988641\pi$
$954$		0			0
$955$	−248.302	−	430.072i	−0.260002	−	0.450337i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	469.415	−	813.051i	0.488465	−	0.846046i
$962$		0			0
$963$		0			0
$964$		0			0
$965$			38.4074i			0.0398004i
$966$		0			0
$967$	−1209.25			−1.25051			−0.625257	−	0.780419i	$-0.715006\pi$
$967$	−1209.25			−1.25051			−0.625257	+	0.780419i	$0.715006\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	1248.15	+	720.622i	1.28543	+	0.742144i	0.977836	−	0.209373i	$-0.0671422\pi$
$971$	1248.15	+	720.622i	1.28543	+	0.742144i	0.307596	+	0.951517i	$0.400476\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−41.7998	+	24.1332i	−0.0427839	+	0.0247013i	−0.521239	−	0.853411i	$-0.674530\pi$
$977$	−41.7998	+	24.1332i	−0.0427839	+	0.0247013i	0.478456	+	0.878112i	$0.341197\pi$
$978$		0			0
$979$	−1296.84			−1.32466
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−16.0513	+	9.26720i	−0.0163289	+	0.00942747i	−0.508142	−	0.861273i	$-0.669668\pi$
$983$	−16.0513	+	9.26720i	−0.0163289	+	0.00942747i	0.491813	+	0.870701i	$0.336334\pi$
$984$		0			0
$985$	66.3830	−	114.979i	0.0673939	−	0.116730i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−538.164	−	310.709i	−0.544150	−	0.314165i
$990$		0			0
$991$	−38.6275	−	66.9047i	−0.0389783	−	0.0675123i	0.845878	−	0.533376i	$-0.179077\pi$
$991$	−38.6275	−	66.9047i	−0.0389783	−	0.0675123i	−0.884856	+	0.465864i	$0.845744\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	703.454i		−	0.706989i
$996$		0			0
$997$	−758.674	−	1314.06i	−0.760957	−	1.31802i	−0.942358	−	0.334607i	$-0.891396\pi$
$997$	−758.674	−	1314.06i	−0.760957	−	1.31802i	0.181400	−	0.983409i	$-0.441937\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.3.bk.e.1745.2		8
3.2	odd	2	inner	1764.3.bk.e.1745.3		8
7.2	even	3		1764.3.c.f.197.3		4
7.3	odd	6		1764.3.bk.d.557.2		8
7.4	even	3	inner	1764.3.bk.e.557.3		8
7.5	odd	6		252.3.c.a.197.2	✓	4
7.6	odd	2		1764.3.bk.d.1745.3		8
21.2	odd	6		1764.3.c.f.197.2		4
21.5	even	6		252.3.c.a.197.3	yes	4
21.11	odd	6	inner	1764.3.bk.e.557.2		8
21.17	even	6		1764.3.bk.d.557.3		8
21.20	even	2		1764.3.bk.d.1745.2		8
28.19	even	6		1008.3.d.c.449.2		4
56.5	odd	6		4032.3.d.h.449.3		4
56.19	even	6		4032.3.d.e.449.3		4
63.5	even	6		2268.3.bg.c.2213.3		8
63.40	odd	6		2268.3.bg.c.2213.2		8
63.47	even	6		2268.3.bg.c.701.2		8
63.61	odd	6		2268.3.bg.c.701.3		8
84.47	odd	6		1008.3.d.c.449.3		4
168.5	even	6		4032.3.d.h.449.2		4
168.131	odd	6		4032.3.d.e.449.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.3.c.a.197.2	✓	4	7.5	odd	6
252.3.c.a.197.3	yes	4	21.5	even	6
1008.3.d.c.449.2		4	28.19	even	6
1008.3.d.c.449.3		4	84.47	odd	6
1764.3.c.f.197.2		4	21.2	odd	6
1764.3.c.f.197.3		4	7.2	even	3
1764.3.bk.d.557.2		8	7.3	odd	6
1764.3.bk.d.557.3		8	21.17	even	6
1764.3.bk.d.1745.2		8	21.20	even	2
1764.3.bk.d.1745.3		8	7.6	odd	2
1764.3.bk.e.557.2		8	21.11	odd	6	inner
1764.3.bk.e.557.3		8	7.4	even	3	inner
1764.3.bk.e.1745.2		8	1.1	even	1	trivial
1764.3.bk.e.1745.3		8	3.2	odd	2	inner
2268.3.bg.c.701.2		8	63.47	even	6
2268.3.bg.c.701.3		8	63.61	odd	6
2268.3.bg.c.2213.2		8	63.40	odd	6
2268.3.bg.c.2213.3		8	63.5	even	6
4032.3.d.e.449.2		4	168.131	odd	6
4032.3.d.e.449.3		4	56.19	even	6
4032.3.d.h.449.2		4	168.5	even	6
4032.3.d.h.449.3		4	56.5	odd	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:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1764.bk (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.01563 - 1.16372i) q^{5} +O(q^{10})\)	\(q+(-2.01563 - 1.16372i) q^{5} +(-7.70549 + 4.44876i) q^{11} -8.58301 q^{13} +(17.4266 - 10.0613i) q^{17} +(2.93725 - 5.08747i) q^{19} +(15.0540 + 8.69140i) q^{23} +(-9.79150 - 16.9594i) q^{25} -19.0318i q^{29} +(2.35425 + 4.07768i) q^{31} +(-19.8745 + 34.4237i) q^{37} +6.98233i q^{41} -35.7490 q^{43} +(63.5330 + 36.6808i) q^{47} +(-40.4166 + 23.3345i) q^{53} +20.7085 q^{55} +(-20.8703 + 12.0495i) q^{59} +(-36.6458 + 63.4723i) q^{61} +(17.3001 + 9.98823i) q^{65} +(16.0405 + 27.7830i) q^{67} +114.843i q^{71} +(20.6458 + 35.7595i) q^{73} +(13.3320 - 23.0917i) q^{79} -136.057i q^{83} -46.8340 q^{85} +(126.226 + 72.8764i) q^{89} +(-11.8408 + 6.83629i) q^{95} +65.9555 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 16 q^{13} - 40 q^{19} - 36 q^{25} + 40 q^{31} - 32 q^{37} - 32 q^{43} + 208 q^{55} - 272 q^{61} - 168 q^{67} + 144 q^{73} - 232 q^{79} - 544 q^{85} - 192 q^{97}+O(q^{100}) \) 8 * q + 16 * q^13 - 40 * q^19 - 36 * q^25 + 40 * q^31 - 32 * q^37 - 32 * q^43 + 208 * q^55 - 272 * q^61 - 168 * q^67 + 144 * q^73 - 232 * q^79 - 544 * q^85 - 192 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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