LMFDB - Embedded newform 1728.3.n.d.1313.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1728,3,Mod(737,1728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1728, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 1]))

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

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

chi := DirichletCharacter("1728.737");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1728.n (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:	$47.0845896815$
Analytic rank:	$0$
Dimension:	$32$
Relative dimension:	$16$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 576)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1313.8
Character	$\chi$	$=$	1728.1313
Dual form			1728.3.n.d.737.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.920717 + 1.59473i) q^{5} +(4.56979 + 7.91511i) q^{7} +O(q^{10})$	$q+(-0.920717 + 1.59473i) q^{5} +(4.56979 + 7.91511i) q^{7} +(-3.81937 - 6.61534i) q^{11} +(15.7478 + 9.09201i) q^{13} -30.7770i q^{17} -10.9172i q^{19} +(12.5177 + 7.22708i) q^{23} +(10.8046 + 18.7140i) q^{25} +(-18.9375 - 32.8007i) q^{29} +(19.9319 - 34.5231i) q^{31} -16.8299 q^{35} -7.72657i q^{37} +(43.1646 + 24.9211i) q^{41} +(-7.75029 + 4.47463i) q^{43} +(12.9978 - 7.50427i) q^{47} +(-17.2659 + 29.9055i) q^{49} -20.1038 q^{53} +14.0662 q^{55} +(-19.0494 + 32.9945i) q^{59} +(90.9853 - 52.5304i) q^{61} +(-28.9986 + 16.7423i) q^{65} +(18.1407 + 10.4735i) q^{67} +132.111i q^{71} -23.9882 q^{73} +(34.9074 - 60.4614i) q^{77} +(21.1445 + 36.6233i) q^{79} +(18.7004 + 32.3901i) q^{83} +(49.0810 + 28.3369i) q^{85} -45.6700i q^{89} +166.194i q^{91} +(17.4099 + 10.0516i) q^{95} +(41.6001 + 72.0535i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 18 q^{5}+O(q^{10}) $ 32 * q + 18 * q^5	$ 32 q + 18 q^{5} + 30 q^{13} - 74 q^{25} + 54 q^{29} + 216 q^{41} - 86 q^{49} - 144 q^{53} - 42 q^{61} - 306 q^{65} + 196 q^{73} + 414 q^{77} - 180 q^{85} - 64 q^{97}+O(q^{100}) $ 32 * q + 18 * q^5 + 30 * q^13 - 74 * q^25 + 54 * q^29 + 216 * q^41 - 86 * q^49 - 144 * q^53 - 42 * q^61 - 306 * q^65 + 196 * q^73 + 414 * q^77 - 180 * q^85 - 64 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times$.

$n$	$325$	$703$	$1217$
$\chi(n)$	$-1$	$1$	$e\left(\frac{5}{6}\right)$

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
$5$	−0.920717	+	1.59473i	−0.184143	+	0.318946i	−0.943288	−	0.331977i	$-0.892284\pi$
$5$	−0.920717	+	1.59473i	−0.184143	+	0.318946i	0.759144	+	0.650923i	$0.225618\pi$
$6$		0			0
$7$	4.56979	+	7.91511i	0.652827	+	1.13073i	0.982434	+	0.186611i	$0.0597504\pi$
$7$	4.56979	+	7.91511i	0.652827	+	1.13073i	−0.329607	+	0.944118i	$0.606916\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−3.81937	−	6.61534i	−0.347215	−	0.601395i	0.638538	−	0.769590i	$-0.279539\pi$
$11$	−3.81937	−	6.61534i	−0.347215	−	0.601395i	−0.985754	+	0.168195i	$0.946206\pi$
$12$		0			0
$13$	15.7478	+	9.09201i	1.21137	+	0.699385i	0.963058	−	0.269294i	$-0.0867905\pi$
$13$	15.7478	+	9.09201i	1.21137	+	0.699385i	0.248313	+	0.968680i	$0.420124\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	30.7770i		−	1.81041i	−0.424971	−	0.905207i	$-0.639716\pi$
$17$		−	30.7770i		−	1.81041i	0.424971	−	0.905207i	$-0.360284\pi$
$18$		0			0
$19$		−	10.9172i		−	0.574587i	−0.957843	−	0.287294i	$-0.907244\pi$
$19$		−	10.9172i		−	0.574587i	0.957843	−	0.287294i	$-0.0927556\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	12.5177	+	7.22708i	0.544247	+	0.314221i	0.746798	−	0.665051i	$-0.231590\pi$
$23$	12.5177	+	7.22708i	0.544247	+	0.314221i	−0.202552	+	0.979272i	$0.564923\pi$
$24$		0			0
$25$	10.8046	+	18.7140i	0.432182	+	0.748562i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−18.9375	−	32.8007i	−0.653017	−	1.13106i	−0.982387	−	0.186858i	$-0.940169\pi$
$29$	−18.9375	−	32.8007i	−0.653017	−	1.13106i	0.329369	−	0.944201i	$-0.393164\pi$
$30$		0			0
$31$	19.9319	−	34.5231i	0.642965	−	1.11365i	−0.341802	−	0.939772i	$-0.611037\pi$
$31$	19.9319	−	34.5231i	0.642965	−	1.11365i	0.984767	−	0.173877i	$-0.0556295\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−16.8299			−0.480855
$36$		0			0
$37$		−	7.72657i		−	0.208826i	−0.994534	−	0.104413i	$-0.966704\pi$
$37$		−	7.72657i		−	0.208826i	0.994534	−	0.104413i	$-0.0332964\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	43.1646	+	24.9211i	1.05279	+	0.607831i	0.923430	−	0.383766i	$-0.125373\pi$
$41$	43.1646	+	24.9211i	1.05279	+	0.607831i	0.129364	+	0.991597i	$0.458706\pi$
$42$		0			0
$43$	−7.75029	+	4.47463i	−0.180239	+	0.104061i	−0.587405	−	0.809293i	$-0.699850\pi$
$43$	−7.75029	+	4.47463i	−0.180239	+	0.104061i	0.407166	+	0.913354i	$0.366517\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	12.9978	−	7.50427i	0.276548	−	0.159665i	−0.355311	−	0.934748i	$-0.615625\pi$
$47$	12.9978	−	7.50427i	0.276548	−	0.159665i	0.631860	+	0.775083i	$0.282292\pi$
$48$		0			0
$49$	−17.2659	+	29.9055i	−0.352366	+	0.610316i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−20.1038			−0.379316			−0.189658	−	0.981850i	$-0.560738\pi$
$53$	−20.1038			−0.379316			−0.189658	+	0.981850i	$0.560738\pi$
$54$		0			0
$55$	14.0662			0.255750
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−19.0494	+	32.9945i	−0.322871	+	0.559228i	−0.981079	−	0.193607i	$-0.937981\pi$
$59$	−19.0494	+	32.9945i	−0.322871	+	0.559228i	0.658208	+	0.752836i	$0.271315\pi$
$60$		0			0
$61$	90.9853	−	52.5304i	1.49156	−	0.861154i	0.491608	−	0.870816i	$-0.336409\pi$
$61$	90.9853	−	52.5304i	1.49156	−	0.861154i	0.999953	+	0.00966279i	$0.00307581\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−28.9986	+	16.7423i	−0.446132	+	0.257574i
$66$		0			0
$67$	18.1407	+	10.4735i	0.270756	+	0.156321i	0.629231	−	0.777218i	$-0.283370\pi$
$67$	18.1407	+	10.4735i	0.270756	+	0.156321i	−0.358475	+	0.933539i	$0.616703\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			132.111i			1.86072i	0.366652	+	0.930358i	$0.380504\pi$
$71$			132.111i			1.86072i	−0.366652	+	0.930358i	$0.619496\pi$
$72$		0			0
$73$	−23.9882			−0.328605			−0.164303	−	0.986410i	$-0.552537\pi$
$73$	−23.9882			−0.328605			−0.164303	+	0.986410i	$0.552537\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	34.9074	−	60.4614i	0.453343	−	0.785213i
$78$		0			0
$79$	21.1445	+	36.6233i	0.267651	+	0.463586i	0.968255	−	0.249965i	$-0.0804191\pi$
$79$	21.1445	+	36.6233i	0.267651	+	0.463586i	−0.700604	+	0.713551i	$0.747086\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	18.7004	+	32.3901i	0.225306	+	0.390242i	0.956411	−	0.292023i	$-0.0943284\pi$
$83$	18.7004	+	32.3901i	0.225306	+	0.390242i	−0.731105	+	0.682265i	$0.760995\pi$
$84$		0			0
$85$	49.0810	+	28.3369i	0.577424	+	0.333376i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	45.6700i		−	0.513146i	−0.966525	−	0.256573i	$-0.917407\pi$
$89$		−	45.6700i		−	0.513146i	0.966525	−	0.256573i	$-0.0825935\pi$
$90$		0			0
$91$			166.194i			1.82631i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	17.4099	+	10.0516i	0.183262	+	0.105806i
$96$		0			0
$97$	41.6001	+	72.0535i	0.428867	+	0.742820i	0.996773	−	0.0802740i	$-0.0255795\pi$
$97$	41.6001	+	72.0535i	0.428867	+	0.742820i	−0.567906	+	0.823094i	$0.692246\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−74.5873	−	129.189i	−0.738489	−	1.27910i	−0.953176	−	0.302417i	$-0.902207\pi$
$101$	−74.5873	−	129.189i	−0.738489	−	1.27910i	0.214687	−	0.976683i	$-0.431127\pi$
$102$		0			0
$103$	−66.5665	+	115.296i	−0.646276	+	1.11938i	0.337729	+	0.941243i	$0.390341\pi$
$103$	−66.5665	+	115.296i	−0.646276	+	1.11938i	−0.984005	+	0.178140i	$0.942992\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	104.310			0.974855			0.487428	−	0.873163i	$-0.337935\pi$
$107$	104.310			0.974855			0.487428	+	0.873163i	$0.337935\pi$
$108$		0			0
$109$		−	51.1660i		−	0.469413i	−0.972066	−	0.234706i	$-0.924587\pi$
$109$		−	51.1660i		−	0.469413i	0.972066	−	0.234706i	$-0.0754129\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	89.8153	+	51.8549i	0.794825	+	0.458893i	0.841659	−	0.540010i	$-0.181580\pi$
$113$	89.8153	+	51.8549i	0.794825	+	0.458893i	−0.0468333	+	0.998903i	$0.514913\pi$
$114$		0			0
$115$	−23.0505	+	13.3082i	−0.200439	+	0.115723i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	243.603	−	140.645i	2.04709	−	1.18189i
$120$		0			0
$121$	31.3248	−	54.2562i	0.258883	−	0.448398i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−85.8276			−0.686621
$126$		0			0
$127$	136.733			1.07664			0.538319	−	0.842741i	$-0.319059\pi$
$127$	136.733			1.07664			0.538319	+	0.842741i	$0.319059\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−93.7823	+	162.436i	−0.715895	+	1.23997i	0.246718	+	0.969087i	$0.420648\pi$
$131$	−93.7823	+	162.436i	−0.715895	+	1.23997i	−0.962613	+	0.270879i	$0.912686\pi$
$132$		0			0
$133$	86.4105	−	49.8891i	0.649703	−	0.375106i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	161.057	−	92.9862i	1.17560	−	0.678732i	0.220606	−	0.975363i	$-0.429197\pi$
$137$	161.057	−	92.9862i	1.17560	−	0.678732i	0.954992	+	0.296631i	$0.0958633\pi$
$138$		0			0
$139$	−201.731	−	116.469i	−1.45130	−	0.837908i	−0.452744	−	0.891641i	$-0.649555\pi$
$139$	−201.731	−	116.469i	−1.45130	−	0.837908i	−0.998555	+	0.0537326i	$0.982888\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	138.903i		−	0.971350i
$144$		0			0
$145$	69.7443			0.480995
$146$		0			0
$147$		0			0
$148$		0			0
$149$	108.575	−	188.057i	0.728690	−	1.26213i	−0.228747	−	0.973486i	$-0.573463\pi$
$149$	108.575	−	188.057i	0.728690	−	1.26213i	0.957437	−	0.288642i	$-0.0932039\pi$
$150$		0			0
$151$	135.092	+	233.986i	0.894648	+	1.54957i	0.834240	+	0.551401i	$0.185907\pi$
$151$	135.092	+	233.986i	0.894648	+	1.54957i	0.0604073	+	0.998174i	$0.480760\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	36.7033	+	63.5720i	0.236796	+	0.410142i
$156$		0			0
$157$	71.4453	+	41.2490i	0.455066	+	0.262732i	0.709967	−	0.704235i	$-0.248710\pi$
$157$	71.4453	+	41.2490i	0.455066	+	0.262732i	−0.254901	+	0.966967i	$0.582043\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			132.105i			0.820527i
$162$		0			0
$163$		−	170.850i		−	1.04816i	−0.851668	−	0.524081i	$-0.824409\pi$
$163$		−	170.850i		−	1.04816i	0.851668	−	0.524081i	$-0.175591\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	22.0145	+	12.7101i	0.131823	+	0.0761083i	0.564461	−	0.825459i	$-0.309084\pi$
$167$	22.0145	+	12.7101i	0.131823	+	0.0761083i	−0.432638	+	0.901568i	$0.642417\pi$
$168$		0			0
$169$	80.8293	+	140.000i	0.478280	+	0.828405i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	153.244	+	265.426i	0.885803	+	1.53426i	0.844790	+	0.535098i	$0.179725\pi$
$173$	153.244	+	265.426i	0.885803	+	1.53426i	0.0410133	+	0.999159i	$0.486941\pi$
$174$		0			0
$175$	−98.7491	+	171.038i	−0.564281	+	0.977363i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−220.996			−1.23462			−0.617308	−	0.786722i	$-0.711777\pi$
$179$	−220.996			−1.23462			−0.617308	+	0.786722i	$0.711777\pi$
$180$		0			0
$181$		−	211.405i		−	1.16798i	−0.811759	−	0.583992i	$-0.801490\pi$
$181$		−	211.405i		−	1.16798i	0.811759	−	0.583992i	$-0.198510\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	12.3218	+	7.11399i	0.0666043	+	0.0384540i
$186$		0			0
$187$	−203.601	+	117.549i	−1.08877	+	0.628604i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	210.554	−	121.563i	1.10238	−	0.636458i	0.165533	−	0.986204i	$-0.447066\pi$
$191$	210.554	−	121.563i	1.10238	−	0.636458i	0.936844	+	0.349747i	$0.113732\pi$
$192$		0			0
$193$	−55.6885	+	96.4553i	−0.288541	+	0.499768i	−0.973462	−	0.228850i	$-0.926504\pi$
$193$	−55.6885	+	96.4553i	−0.288541	+	0.499768i	0.684920	+	0.728618i	$0.259837\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	138.054			0.700782			0.350391	−	0.936604i	$-0.386049\pi$
$197$	138.054			0.700782			0.350391	+	0.936604i	$0.386049\pi$
$198$		0			0
$199$	−33.4917			−0.168300			−0.0841500	−	0.996453i	$-0.526817\pi$
$199$	−33.4917			−0.168300			−0.0841500	+	0.996453i	$0.526817\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	173.081	−	299.785i	0.852615	−	1.47677i
$204$		0			0
$205$	−79.4847	+	45.8905i	−0.387730	+	0.223856i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−72.2207	+	41.6967i	−0.345554	+	0.199506i
$210$		0			0
$211$	85.4798	+	49.3518i	0.405117	+	0.233895i	0.688690	−	0.725056i	$-0.258186\pi$
$211$	85.4798	+	49.3518i	0.405117	+	0.233895i	−0.283572	+	0.958951i	$0.591520\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	16.4795i		−	0.0766487i
$216$		0			0
$217$	364.339			1.67898
$218$		0			0
$219$		0			0
$220$		0			0
$221$	279.825	−	484.671i	1.26618	−	2.19308i
$222$		0			0
$223$	93.0832	+	161.225i	0.417414	+	0.722981i	0.995678	−	0.0928676i	$-0.0296033\pi$
$223$	93.0832	+	161.225i	0.417414	+	0.722981i	−0.578265	+	0.815849i	$0.696270\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	222.402	+	385.212i	0.979746	+	1.69697i	0.663291	+	0.748361i	$0.269159\pi$
$227$	222.402	+	385.212i	0.979746	+	1.69697i	0.316454	+	0.948608i	$0.397508\pi$
$228$		0			0
$229$	−73.3756	−	42.3634i	−0.320418	−	0.184993i	0.331161	−	0.943574i	$-0.392560\pi$
$229$	−73.3756	−	42.3634i	−0.320418	−	0.184993i	−0.651579	+	0.758581i	$0.725893\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	181.796i		−	0.780242i	−0.920764	−	0.390121i	$-0.872433\pi$
$233$		−	181.796i		−	0.780242i	0.920764	−	0.390121i	$-0.127567\pi$
$234$		0			0
$235$			27.6372i			0.117605i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	200.031	+	115.488i	0.836951	+	0.483214i	0.856227	−	0.516600i	$-0.172803\pi$
$239$	200.031	+	115.488i	0.836951	+	0.483214i	−0.0192757	+	0.999814i	$0.506136\pi$
$240$		0			0
$241$	−206.202	−	357.153i	−0.855612	−	1.48196i	−0.876076	−	0.482173i	$-0.839848\pi$
$241$	−206.202	−	357.153i	−0.855612	−	1.48196i	0.0204644	−	0.999791i	$-0.493486\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−31.7941	−	55.0689i	−0.129772	−	0.224771i
$246$		0			0
$247$	99.2589	−	171.921i	0.401858	−	0.696038i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−49.9321			−0.198933			−0.0994663	−	0.995041i	$-0.531714\pi$
$251$	−49.9321			−0.198933			−0.0994663	+	0.995041i	$0.531714\pi$
$252$		0			0
$253$		−	110.412i		−	0.436409i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	113.337	+	65.4352i	0.441000	+	0.254612i	0.704022	−	0.710178i	$-0.251386\pi$
$257$	113.337	+	65.4352i	0.441000	+	0.254612i	−0.263022	+	0.964790i	$0.584719\pi$
$258$		0			0
$259$	61.1566	−	35.3088i	0.236126	−	0.136327i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	77.4773	−	44.7316i	0.294591	−	0.170082i	−0.345420	−	0.938448i	$-0.612263\pi$
$263$	77.4773	−	44.7316i	0.294591	−	0.170082i	0.640010	+	0.768366i	$0.278930\pi$
$264$		0			0
$265$	18.5099	−	32.0600i	0.0698486	−	0.120981i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−428.087			−1.59140			−0.795701	−	0.605690i	$-0.792897\pi$
$269$	−428.087			−1.59140			−0.795701	+	0.605690i	$0.792897\pi$
$270$		0			0
$271$	−36.2602			−0.133802			−0.0669008	−	0.997760i	$-0.521311\pi$
$271$	−36.2602			−0.133802			−0.0669008	+	0.997760i	$0.521311\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	82.5332	−	142.952i	0.300121	−	0.519824i
$276$		0			0
$277$	130.359	−	75.2631i	0.470612	−	0.271708i	−0.245884	−	0.969299i	$-0.579078\pi$
$277$	130.359	−	75.2631i	0.470612	−	0.271708i	0.716496	+	0.697591i	$0.245745\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−132.933	+	76.7492i	−0.473073	+	0.273129i	−0.717525	−	0.696533i	$-0.754725\pi$
$281$	−132.933	+	76.7492i	−0.473073	+	0.273129i	0.244452	+	0.969661i	$0.421392\pi$
$282$		0			0
$283$	61.5307	+	35.5247i	0.217423	+	0.125529i	0.604756	−	0.796411i	$-0.293270\pi$
$283$	61.5307	+	35.5247i	0.217423	+	0.125529i	−0.387334	+	0.921940i	$0.626604\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			455.536i			1.58723i
$288$		0			0
$289$	−658.226			−2.27760
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−111.469	+	193.070i	−0.380441	+	0.658942i	−0.991125	−	0.132931i	$-0.957561\pi$
$293$	−111.469	+	193.070i	−0.380441	+	0.658942i	0.610685	+	0.791874i	$0.290894\pi$
$294$		0			0
$295$	−35.0782	−	60.7571i	−0.118909	−	0.205956i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	131.417	+	227.622i	0.439523	+	0.761276i
$300$		0			0
$301$	−70.8344	−	40.8962i	−0.235330	−	0.135868i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			193.462i			0.634303i
$306$		0			0
$307$			285.699i			0.930614i	0.885149	+	0.465307i	$0.154056\pi$
$307$			285.699i			0.930614i	−0.885149	+	0.465307i	$0.845944\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−369.736	−	213.467i	−1.18886	−	0.686390i	−0.230814	−	0.972998i	$-0.574139\pi$
$311$	−369.736	−	213.467i	−1.18886	−	0.686390i	−0.958048	+	0.286608i	$0.907472\pi$
$312$		0			0
$313$	225.712	+	390.944i	0.721124	+	1.24902i	0.960550	+	0.278108i	$0.0897074\pi$
$313$	225.712	+	390.944i	0.721124	+	1.24902i	−0.239426	+	0.970915i	$0.576959\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−2.83099	−	4.90342i	−0.00893057	−	0.0154682i	0.861526	−	0.507714i	$-0.169509\pi$
$317$	−2.83099	−	4.90342i	−0.00893057	−	0.0154682i	−0.870456	+	0.492246i	$0.836176\pi$
$318$		0			0
$319$	−144.659	+	250.556i	−0.453475	+	0.785443i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−335.998			−1.04024
$324$		0			0
$325$			392.941i			1.20905i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	118.794	+	68.5858i	0.361076	+	0.208468i
$330$		0			0
$331$	−255.584	+	147.561i	−0.772157	+	0.445805i	−0.833643	−	0.552303i	$-0.813749\pi$
$331$	−255.584	+	147.561i	−0.772157	+	0.445805i	0.0614867	+	0.998108i	$0.480416\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−33.4048	+	19.2863i	−0.0997160	+	0.0575710i
$336$		0			0
$337$	−156.434	+	270.952i	−0.464196	+	0.804012i	−0.999165	−	0.0408603i	$-0.986990\pi$
$337$	−156.434	+	270.952i	−0.464196	+	0.804012i	0.534968	+	0.844872i	$0.320323\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−304.510			−0.892990
$342$		0			0
$343$	132.233			0.385518
$344$		0			0
$345$		0			0
$346$		0			0
$347$	30.2492	−	52.3931i	0.0871734	−	0.150989i	−0.819142	−	0.573591i	$-0.805550\pi$
$347$	30.2492	−	52.3931i	0.0871734	−	0.150989i	0.906315	+	0.422602i	$0.138883\pi$
$348$		0			0
$349$	445.021	−	256.933i	1.27513	−	0.736198i	0.299184	−	0.954196i	$-0.403286\pi$
$349$	445.021	−	256.933i	1.27513	−	0.736198i	0.975949	+	0.217997i	$0.0699523\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−238.086	+	137.459i	−0.674466	+	0.389403i	−0.797767	−	0.602966i	$-0.793985\pi$
$353$	−238.086	+	137.459i	−0.674466	+	0.389403i	0.123301	+	0.992369i	$0.460652\pi$
$354$		0			0
$355$	−210.681	−	121.637i	−0.593468	−	0.342639i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			2.40904i			0.00671042i	0.999994	+	0.00335521i	$0.00106800\pi$
$359$			2.40904i			0.00671042i	−0.999994	+	0.00335521i	$0.998932\pi$
$360$		0			0
$361$	241.816			0.669849
$362$		0			0
$363$		0			0
$364$		0			0
$365$	22.0863	−	38.2547i	0.0605105	−	0.104807i
$366$		0			0
$367$	−195.126	−	337.968i	−0.531678	−	0.920894i	−0.999316	−	0.0369739i	$-0.988228\pi$
$367$	−195.126	−	337.968i	−0.531678	−	0.920894i	0.467638	−	0.883920i	$-0.345105\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−91.8699	−	159.123i	−0.247628	−	0.428904i
$372$		0			0
$373$	98.7732	+	57.0267i	0.264807	+	0.152887i	0.626526	−	0.779401i	$-0.284476\pi$
$373$	98.7732	+	57.0267i	0.264807	+	0.152887i	−0.361718	+	0.932287i	$0.617810\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	688.720i		−	1.82684i
$378$		0			0
$379$		−	294.626i		−	0.777377i	−0.921369	−	0.388689i	$-0.872928\pi$
$379$		−	294.626i		−	0.777377i	0.921369	−	0.388689i	$-0.127072\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−413.559	−	238.769i	−1.07979	−	0.623417i	−0.148950	−	0.988845i	$-0.547589\pi$
$383$	−413.559	−	238.769i	−1.07979	−	0.623417i	−0.930840	+	0.365428i	$0.880923\pi$
$384$		0			0
$385$	64.2797	+	111.336i	0.166960	+	0.289184i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	200.807	+	347.808i	0.516213	+	0.894108i	0.999823	+	0.0188238i	$0.00599215\pi$
$389$	200.807	+	347.808i	0.516213	+	0.894108i	−0.483610	+	0.875284i	$0.660675\pi$
$390$		0			0
$391$	222.428	−	385.257i	0.568870	−	0.985311i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−77.8722			−0.197145
$396$		0			0
$397$			296.399i			0.746598i	0.927711	+	0.373299i	$0.121773\pi$
$397$			296.399i			0.746598i	−0.927711	+	0.373299i	$0.878227\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−86.2049	−	49.7704i	−0.214975	−	0.124116i	0.388646	−	0.921387i	$-0.372943\pi$
$401$	−86.2049	−	49.7704i	−0.214975	−	0.124116i	−0.603621	+	0.797271i	$0.706276\pi$
$402$		0			0
$403$	627.769	−	362.443i	1.55774	−	0.899361i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−51.1139	+	29.5106i	−0.125587	+	0.0725077i
$408$		0			0
$409$	216.978	−	375.816i	0.530508	−	0.918867i	−0.468858	−	0.883273i	$-0.655335\pi$
$409$	216.978	−	375.816i	0.530508	−	0.918867i	0.999366	−	0.0355933i	$-0.0113321\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−348.206			−0.843114
$414$		0			0
$415$	−68.8711			−0.165955
$416$		0			0
$417$		0			0
$418$		0			0
$419$	88.3811	−	153.081i	0.210933	−	0.365347i	−0.741073	−	0.671424i	$-0.765683\pi$
$419$	88.3811	−	153.081i	0.210933	−	0.365347i	0.952007	+	0.306076i	$0.0990163\pi$
$420$		0			0
$421$	−296.515	+	171.193i	−0.704312	+	0.406635i	−0.808951	−	0.587876i	$-0.799964\pi$
$421$	−296.515	+	171.193i	−0.704312	+	0.406635i	0.104639	+	0.994510i	$0.466631\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	575.963	−	332.532i	1.35521	−	0.782429i
$426$		0			0
$427$	831.567	+	480.105i	1.94746	+	1.12437i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			613.169i			1.42267i	0.702855	+	0.711333i	$0.251908\pi$
$431$			613.169i			1.42267i	−0.702855	+	0.711333i	$0.748092\pi$
$432$		0			0
$433$	172.934			0.399386			0.199693	−	0.979859i	$-0.436006\pi$
$433$	172.934			0.399386			0.199693	+	0.979859i	$0.436006\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	78.8992	−	136.657i	0.180547	−	0.312717i
$438$		0			0
$439$	−318.835	−	552.239i	−0.726276	−	1.25795i	−0.958447	−	0.285272i	$-0.907916\pi$
$439$	−318.835	−	552.239i	−0.726276	−	1.25795i	0.232171	−	0.972675i	$-0.425417\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−353.126	−	611.632i	−0.797123	−	1.38066i	−0.921482	−	0.388421i	$-0.873021\pi$
$443$	−353.126	−	611.632i	−0.797123	−	1.38066i	0.124359	−	0.992237i	$-0.460313\pi$
$444$		0			0
$445$	72.8313	+	42.0492i	0.163666	+	0.0944925i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			88.2356i			0.196516i	0.995161	+	0.0982579i	$0.0313270\pi$
$449$			88.2356i			0.196516i	−0.995161	+	0.0982579i	$0.968673\pi$
$450$		0			0
$451$		−	380.731i		−	0.844193i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−265.035	−	153.018i	−0.582494	−	0.336303i
$456$		0			0
$457$	−348.107	−	602.939i	−0.761722	−	1.31934i	−0.941962	−	0.335719i	$-0.891021\pi$
$457$	−348.107	−	602.939i	−0.761722	−	1.31934i	0.180240	−	0.983623i	$-0.442313\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	312.264	+	540.857i	0.677363	+	1.17323i	0.975772	+	0.218788i	$0.0702105\pi$
$461$	312.264	+	540.857i	0.677363	+	1.17323i	−0.298410	+	0.954438i	$0.596456\pi$
$462$		0			0
$463$	261.798	−	453.447i	0.565438	−	0.979367i	−0.431571	−	0.902079i	$-0.642040\pi$
$463$	261.798	−	453.447i	0.565438	−	0.979367i	0.997009	−	0.0772884i	$-0.0246262\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−106.125			−0.227248			−0.113624	−	0.993524i	$-0.536246\pi$
$467$	−106.125			−0.227248			−0.113624	+	0.993524i	$0.536246\pi$
$468$		0			0
$469$			191.447i			0.408203i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	59.2024	+	34.1805i	0.125164	+	0.0722633i
$474$		0			0
$475$	204.304	−	117.955i	0.430114	−	0.248327i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	291.982	−	168.576i	0.609565	−	0.351933i	−0.163230	−	0.986588i	$-0.552191\pi$
$479$	291.982	−	168.576i	0.609565	−	0.351933i	0.772795	+	0.634655i	$0.218858\pi$
$480$		0			0
$481$	70.2501	−	121.677i	0.146050	−	0.252966i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−153.208			−0.315892
$486$		0			0
$487$	−89.4910			−0.183760			−0.0918799	−	0.995770i	$-0.529288\pi$
$487$	−89.4910			−0.183760			−0.0918799	+	0.995770i	$0.529288\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−255.696	+	442.879i	−0.520766	+	0.901993i	0.478942	+	0.877846i	$0.341020\pi$
$491$	−255.696	+	442.879i	−0.520766	+	0.901993i	−0.999708	+	0.0241469i	$0.992313\pi$
$492$		0			0
$493$	−1009.51	+	582.840i	−2.04769	+	1.18223i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−1045.67	+	603.719i	−2.10397	+	1.21473i
$498$		0			0
$499$	−1.97568	−	1.14066i	−0.00395929	−	0.00228590i	0.498019	−	0.867166i	$-0.334061\pi$
$499$	−1.97568	−	1.14066i	−0.00395929	−	0.00228590i	−0.501978	+	0.864880i	$0.667394\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			702.224i			1.39607i	0.716063	+	0.698036i	$0.245942\pi$
$503$			702.224i			1.39607i	−0.716063	+	0.698036i	$0.754058\pi$
$504$		0			0
$505$	274.695			0.543951
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−56.9044	+	98.5614i	−0.111797	+	0.193637i	−0.916495	−	0.400047i	$-0.868994\pi$
$509$	−56.9044	+	98.5614i	−0.111797	+	0.193637i	0.804698	+	0.593684i	$0.202327\pi$
$510$		0			0
$511$	−109.621	−	189.869i	−0.214522	−	0.371564i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−122.578	−	212.311i	−0.238015	−	0.412254i
$516$		0			0
$517$	−99.2866	−	57.3231i	−0.192044	−	0.110876i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	766.079i		−	1.47040i	−0.677849	−	0.735201i	$-0.737088\pi$
$521$		−	766.079i		−	1.47040i	0.677849	−	0.735201i	$-0.262912\pi$
$522$		0			0
$523$			28.1668i			0.0538562i	0.999637	+	0.0269281i	$0.00857252\pi$
$523$			28.1668i			0.0538562i	−0.999637	+	0.0269281i	$0.991427\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−1062.52	−	613.446i	−2.01616	−	1.16403i
$528$		0			0
$529$	−160.039	−	277.195i	−0.302530	−	0.523998i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	453.165	+	784.905i	0.850216	+	1.47262i
$534$		0			0
$535$	−96.0396	+	166.345i	−0.179513	+	0.310926i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	263.780			0.489387
$540$		0			0
$541$		−	65.6575i		−	0.121363i	−0.998157	−	0.0606816i	$-0.980673\pi$
$541$		−	65.6575i		−	0.121363i	0.998157	−	0.0606816i	$-0.0193274\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	81.5959	+	47.1094i	0.149717	+	0.0864393i
$546$		0			0
$547$	−442.899	+	255.708i	−0.809687	+	0.467473i	−0.846847	−	0.531836i	$-0.821502\pi$
$547$	−442.899	+	255.708i	−0.809687	+	0.467473i	0.0371602	+	0.999309i	$0.488169\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−358.091	+	206.744i	−0.649892	+	0.375216i
$552$		0			0
$553$	−193.251	+	334.721i	−0.349460	+	0.605282i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−70.9792			−0.127431			−0.0637156	−	0.997968i	$-0.520295\pi$
$557$	−70.9792			−0.127431			−0.0637156	+	0.997968i	$0.520295\pi$
$558$		0			0
$559$	−162.734			−0.291116
$560$		0			0
$561$		0			0
$562$		0			0
$563$	484.898	−	839.869i	0.861276	−	1.49177i	−0.00942163	−	0.999956i	$-0.502999\pi$
$563$	484.898	−	839.869i	0.861276	−	1.49177i	0.870698	−	0.491818i	$-0.163668\pi$
$564$		0			0
$565$	−165.389	+	95.4873i	−0.292724	+	0.169004i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−547.463	+	316.078i	−0.962150	+	0.555498i	−0.896834	−	0.442367i	$-0.854139\pi$
$569$	−547.463	+	316.078i	−0.962150	+	0.555498i	−0.0653159	+	0.997865i	$0.520806\pi$
$570$		0			0
$571$	−805.680	−	465.159i	−1.41100	−	0.814640i	−0.415515	−	0.909586i	$-0.636399\pi$
$571$	−805.680	−	465.159i	−1.41100	−	0.814640i	−0.995482	+	0.0949465i	$0.969732\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			312.342i			0.543203i
$576$		0			0
$577$	−1027.79			−1.78127			−0.890636	−	0.454717i	$-0.849741\pi$
$577$	−1027.79			−1.78127			−0.890636	+	0.454717i	$0.849741\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−170.914	+	296.031i	−0.294172	+	0.509521i
$582$		0			0
$583$	76.7837	+	132.993i	0.131704	+	0.228119i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−65.3603	−	113.207i	−0.111346	−	0.192857i	0.804967	−	0.593320i	$-0.202183\pi$
$587$	−65.3603	−	113.207i	−0.111346	−	0.192857i	−0.916313	+	0.400462i	$0.868850\pi$
$588$		0			0
$589$	−376.894	−	217.600i	−0.639889	−	0.369440i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			379.858i			0.640570i	0.947321	+	0.320285i	$0.103779\pi$
$593$			379.858i			0.640570i	−0.947321	+	0.320285i	$0.896221\pi$
$594$		0			0
$595$			517.975i			0.870547i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−309.923	−	178.934i	−0.517401	−	0.298721i	0.218470	−	0.975844i	$-0.429893\pi$
$599$	−309.923	−	178.934i	−0.517401	−	0.298721i	−0.735871	+	0.677122i	$0.763227\pi$
$600$		0			0
$601$	1.64207	+	2.84415i	0.00273223	+	0.00473236i	0.867388	−	0.497632i	$-0.165797\pi$
$601$	1.64207	+	2.84415i	0.00273223	+	0.00473236i	−0.864656	+	0.502364i	$0.832464\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	57.6826	+	99.9092i	0.0953432	+	0.165139i
$606$		0			0
$607$	226.365	−	392.076i	0.372925	−	0.645925i	−0.617089	−	0.786893i	$-0.711688\pi$
$607$	226.365	−	392.076i	0.372925	−	0.645925i	0.990014	+	0.140969i	$0.0450216\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	272.915			0.446670
$612$		0			0
$613$		−	736.259i		−	1.20108i	−0.799597	−	0.600538i	$-0.794953\pi$
$613$		−	736.259i		−	1.20108i	0.799597	−	0.600538i	$-0.205047\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	384.563	+	222.027i	0.623278	+	0.359850i	0.778144	−	0.628086i	$-0.216161\pi$
$617$	384.563	+	222.027i	0.623278	+	0.359850i	−0.154866	+	0.987935i	$0.549495\pi$
$618$		0			0
$619$	974.353	−	562.543i	1.57408	−	0.908793i	0.578414	−	0.815743i	$-0.303672\pi$
$619$	974.353	−	562.543i	1.57408	−	0.908793i	0.995661	−	0.0930494i	$-0.0296614\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	361.483	−	208.702i	0.580230	−	0.334996i
$624$		0			0
$625$	−191.091	+	330.979i	−0.305746	+	0.529567i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−237.801			−0.378062
$630$		0			0
$631$	−1131.60			−1.79334			−0.896671	−	0.442698i	$-0.854022\pi$
$631$	−1131.60			−1.79334			−0.896671	+	0.442698i	$0.854022\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−125.893	+	218.052i	−0.198256	+	0.343389i
$636$		0			0
$637$	−543.802	+	313.964i	−0.853692	+	0.492879i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−869.162	+	501.811i	−1.35595	+	0.782857i	−0.989075	−	0.147414i	$-0.952905\pi$
$641$	−869.162	+	501.811i	−1.35595	+	0.782857i	−0.366873	+	0.930271i	$0.619572\pi$
$642$		0			0
$643$	202.633	+	116.990i	0.315137	+	0.181944i	0.649223	−	0.760598i	$-0.275094\pi$
$643$	202.633	+	116.990i	0.315137	+	0.181944i	−0.334086	+	0.942543i	$0.608428\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	597.659i		−	0.923740i	−0.886948	−	0.461870i	$-0.847179\pi$
$647$		−	597.659i		−	0.923740i	0.886948	−	0.461870i	$-0.152821\pi$
$648$		0			0
$649$	291.026			0.448423
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−162.139	+	280.833i	−0.248299	+	0.430067i	−0.963054	−	0.269308i	$-0.913205\pi$
$653$	−162.139	+	280.833i	−0.248299	+	0.430067i	0.714755	+	0.699375i	$0.246538\pi$
$654$		0			0
$655$	−172.694	−	299.115i	−0.263655	−	0.456663i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	596.518	+	1033.20i	0.905186	+	1.56783i	0.820667	+	0.571407i	$0.193602\pi$
$659$	596.518	+	1033.20i	0.905186	+	1.56783i	0.0845191	+	0.996422i	$0.473065\pi$
$660$		0			0
$661$	−84.0657	−	48.5353i	−0.127180	−	0.0734271i	0.435061	−	0.900401i	$-0.356727\pi$
$661$	−84.0657	−	48.5353i	−0.127180	−	0.0734271i	−0.562240	+	0.826974i	$0.690060\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			183.735i			0.276293i
$666$		0			0
$667$		−	547.452i		−	0.820767i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−695.013	−	401.266i	−1.03579	−	0.598012i
$672$		0			0
$673$	−299.292	−	518.390i	−0.444714	−	0.770267i	0.553318	−	0.832970i	$-0.313361\pi$
$673$	−299.292	−	518.390i	−0.444714	−	0.770267i	−0.998032	+	0.0627029i	$0.980028\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	12.3011	+	21.3061i	0.0181700	+	0.0314714i	0.874967	−	0.484182i	$-0.160883\pi$
$677$	12.3011	+	21.3061i	0.0181700	+	0.0314714i	−0.856797	+	0.515653i	$0.827549\pi$
$678$		0			0
$679$	−380.207	+	658.538i	−0.559952	+	0.969865i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	156.845			0.229641			0.114821	−	0.993386i	$-0.463371\pi$
$683$	156.845			0.229641			0.114821	+	0.993386i	$0.463371\pi$
$684$		0			0
$685$			342.456i			0.499936i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−316.590	−	182.784i	−0.459493	−	0.265288i
$690$		0			0
$691$	−653.119	+	377.079i	−0.945180	+	0.545700i	−0.891580	−	0.452863i	$-0.850403\pi$
$691$	−653.119	+	377.079i	−0.945180	+	0.545700i	−0.0535995	+	0.998563i	$0.517069\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	371.474	−	214.470i	0.534494	−	0.308590i
$696$		0			0
$697$	766.997	−	1328.48i	1.10043	−	1.90599i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	587.766			0.838468			0.419234	−	0.907878i	$-0.362299\pi$
$701$	587.766			0.838468			0.419234	+	0.907878i	$0.362299\pi$
$702$		0			0
$703$	−84.3522			−0.119989
$704$		0			0
$705$		0			0
$706$		0			0
$707$	681.697	−	1180.73i	0.964210	−	1.67006i
$708$		0			0
$709$	−394.808	+	227.942i	−0.556851	+	0.321498i	−0.751881	−	0.659299i	$-0.770853\pi$
$709$	−394.808	+	227.942i	−0.556851	+	0.321498i	0.195029	+	0.980797i	$0.437520\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	499.003	−	288.099i	0.699864	−	0.404066i
$714$		0			0
$715$	221.513	+	127.890i	0.309808	+	0.178868i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	1142.22i		−	1.58862i	−0.607509	−	0.794312i	$-0.707831\pi$
$719$		−	1142.22i		−	1.58862i	0.607509	−	0.794312i	$-0.292169\pi$
$720$		0			0
$721$	−1216.78			−1.68763
$722$		0			0
$723$		0			0
$724$		0			0
$725$	409.223	−	708.795i	0.564445	−	0.977648i
$726$		0			0
$727$	48.3798	+	83.7963i	0.0665472	+	0.115263i	0.897379	−	0.441260i	$-0.145468\pi$
$727$	48.3798	+	83.7963i	0.0665472	+	0.115263i	−0.830832	+	0.556523i	$0.812135\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	137.716	+	238.531i	0.188394	+	0.326308i
$732$		0			0
$733$	−845.937	−	488.402i	−1.15408	−	0.666306i	−0.204199	−	0.978929i	$-0.565459\pi$
$733$	−845.937	−	488.402i	−1.15408	−	0.666306i	−0.949877	+	0.312624i	$0.898792\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	160.009i		−	0.217109i
$738$		0			0
$739$			88.0366i			0.119129i	0.998224	+	0.0595647i	$0.0189713\pi$
$739$			88.0366i			0.119129i	−0.998224	+	0.0595647i	$0.981029\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	787.884	+	454.885i	1.06041	+	0.612228i	0.925546	−	0.378636i	$-0.123607\pi$
$743$	787.884	+	454.885i	1.06041	+	0.612228i	0.134864	+	0.990864i	$0.456940\pi$
$744$		0			0
$745$	199.933	+	346.295i	0.268367	+	0.464825i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	476.672	+	825.621i	0.636412	+	1.10230i
$750$		0			0
$751$	−246.537	+	427.015i	−0.328279	+	0.568595i	−0.982170	−	0.187993i	$-0.939802\pi$
$751$	−246.537	+	427.015i	−0.328279	+	0.568595i	0.653892	+	0.756588i	$0.273135\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−497.525			−0.658974
$756$		0			0
$757$			477.734i			0.631089i	0.948911	+	0.315544i	$0.102187\pi$
$757$			477.734i			0.631089i	−0.948911	+	0.315544i	$0.897813\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−777.129	−	448.676i	−1.02120	−	0.589587i	−0.106745	−	0.994286i	$-0.534043\pi$
$761$	−777.129	−	448.676i	−1.02120	−	0.589587i	−0.914450	+	0.404699i	$0.867376\pi$
$762$		0			0
$763$	404.984	−	233.818i	0.530779	−	0.306445i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−599.972	+	346.394i	−0.782232	+	0.451622i
$768$		0			0
$769$	226.721	−	392.692i	0.294826	−	0.510653i	−0.680119	−	0.733102i	$-0.738072\pi$
$769$	226.721	−	392.692i	0.294826	−	0.510653i	0.974944	+	0.222449i	$0.0714051\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	429.732			0.555927			0.277964	−	0.960592i	$-0.410341\pi$
$773$	429.732			0.555927			0.277964	+	0.960592i	$0.410341\pi$
$774$		0			0
$775$	861.423			1.11151
$776$		0			0
$777$		0			0
$778$		0			0
$779$	272.067	−	471.234i	0.349252	−	0.604922i
$780$		0			0
$781$	873.959	−	504.580i	1.11903	−	0.646069i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−131.562	+	75.9573i	−0.167595	+	0.0967609i
$786$		0			0
$787$	−127.451	−	73.5837i	−0.161945	−	0.0934990i	0.416837	−	0.908981i	$-0.363139\pi$
$787$	−127.451	−	73.5837i	−0.161945	−	0.0934990i	−0.578782	+	0.815482i	$0.696472\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			947.863i			1.19831i
$792$		0			0
$793$	1910.43			2.40911
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−256.948	+	445.047i	−0.322394	+	0.558403i	−0.980982	−	0.194101i	$-0.937821\pi$
$797$	−256.948	+	445.047i	−0.322394	+	0.558403i	0.658587	+	0.752504i	$0.271154\pi$
$798$		0			0
$799$	−230.959	−	400.033i	−0.289060	−	0.500667i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	91.6198	+	158.690i	0.114097	+	0.197622i
$804$		0			0
$805$	−210.672	−	121.631i	−0.261704	−	0.151095i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	369.964i		−	0.457311i	−0.973507	−	0.228655i	$-0.926567\pi$
$809$		−	369.964i		−	0.457311i	0.973507	−	0.228655i	$-0.0734329\pi$
$810$		0			0
$811$			42.1364i			0.0519561i	0.999663	+	0.0259781i	$0.00827001\pi$
$811$			42.1364i			0.0519561i	−0.999663	+	0.0259781i	$0.991730\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	272.460	+	157.305i	0.334307	+	0.193012i
$816$		0			0
$817$	48.8503	+	84.6112i	0.0597923	+	0.103563i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	258.078	+	447.005i	0.314346	+	0.544464i	0.979298	−	0.202422i	$-0.0648814\pi$
$821$	258.078	+	447.005i	0.314346	+	0.544464i	−0.664952	+	0.746886i	$0.731548\pi$
$822$		0			0
$823$	169.184	−	293.035i	0.205569	−	0.356057i	−0.744745	−	0.667350i	$-0.767429\pi$
$823$	169.184	−	293.035i	0.205569	−	0.356057i	0.950314	+	0.311293i	$0.100762\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1283.33			−1.55178			−0.775892	−	0.630866i	$-0.782700\pi$
$827$	−1283.33			−1.55178			−0.775892	+	0.630866i	$0.782700\pi$
$828$		0			0
$829$			664.918i			0.802072i	0.916062	+	0.401036i	$0.131350\pi$
$829$			664.918i			0.802072i	−0.916062	+	0.401036i	$0.868650\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	920.401	+	531.394i	1.10492	+	0.637928i
$834$		0			0
$835$	−40.5383	+	23.4048i	−0.0485488	+	0.0280297i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−145.241	+	83.8547i	−0.173112	+	0.0999460i	−0.584052	−	0.811716i	$-0.698534\pi$
$839$	−145.241	+	83.8547i	−0.173112	+	0.0999460i	0.410941	+	0.911662i	$0.365200\pi$
$840$		0			0
$841$	−296.758	+	514.000i	−0.352864	+	0.611178i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−297.684			−0.352288
$846$		0			0
$847$	572.591			0.676023
$848$		0			0
$849$		0			0
$850$		0			0
$851$	55.8406	−	96.7187i	0.0656176	−	0.113653i
$852$		0			0
$853$	−211.425	+	122.066i	−0.247861	+	0.143102i	−0.618784	−	0.785561i	$-0.712374\pi$
$853$	−211.425	+	122.066i	−0.247861	+	0.143102i	0.370924	+	0.928663i	$0.379041\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	579.385	−	334.508i	0.676061	−	0.390324i	−0.122308	−	0.992492i	$-0.539030\pi$
$857$	579.385	−	334.508i	0.676061	−	0.390324i	0.798369	+	0.602168i	$0.205696\pi$
$858$		0			0
$859$	−626.013	−	361.429i	−0.728769	−	0.420755i	0.0892025	−	0.996014i	$-0.471568\pi$
$859$	−626.013	−	361.429i	−0.728769	−	0.420755i	−0.817972	+	0.575258i	$0.804902\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			311.048i			0.360426i	0.983628	+	0.180213i	$0.0576787\pi$
$863$			311.048i			0.360426i	−0.983628	+	0.180213i	$0.942321\pi$
$864$		0			0
$865$	−564.377			−0.652459
$866$		0			0
$867$		0			0
$868$		0			0
$869$	161.517	−	279.756i	0.185865	−	0.321928i
$870$		0			0
$871$	190.451	+	329.870i	0.218658	+	0.378726i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−392.214	−	679.335i	−0.448245	−	0.776382i
$876$		0			0
$877$	931.826	+	537.990i	1.06252	+	0.613444i	0.926127	−	0.377212i	$-0.123117\pi$
$877$	931.826	+	537.990i	1.06252	+	0.613444i	0.136388	+	0.990655i	$0.456450\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			1149.33i			1.30457i	0.757973	+	0.652286i	$0.226190\pi$
$881$			1149.33i			1.30457i	−0.757973	+	0.652286i	$0.773810\pi$
$882$		0			0
$883$		−	457.946i		−	0.518625i	−0.965793	−	0.259313i	$-0.916504\pi$
$883$		−	457.946i		−	0.518625i	0.965793	−	0.259313i	$-0.0834960\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1052.34	−	607.568i	−1.18640	−	0.684969i	−0.228915	−	0.973447i	$-0.573518\pi$
$887$	−1052.34	−	607.568i	−1.18640	−	0.684969i	−0.957487	+	0.288477i	$0.906851\pi$
$888$		0			0
$889$	624.841	+	1082.26i	0.702859	+	1.21739i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−81.9253	−	141.899i	−0.0917416	−	0.158901i
$894$		0			0
$895$	203.475	−	352.429i	0.227346	−	0.393775i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−1509.84			−1.67947
$900$		0			0
$901$			618.734i			0.686719i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	337.134	+	194.644i	0.372524	+	0.215077i
$906$		0			0
$907$	157.538	−	90.9546i	0.173691	−	0.100281i	−0.410634	−	0.911800i	$-0.634693\pi$
$907$	157.538	−	90.9546i	0.173691	−	0.100281i	0.584325	+	0.811520i	$0.301359\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	208.841	−	120.575i	0.229244	−	0.132354i	−0.380979	−	0.924584i	$-0.624413\pi$
$911$	208.841	−	120.575i	0.229244	−	0.132354i	0.610223	+	0.792229i	$0.291080\pi$
$912$		0			0
$913$	142.848	−	247.419i	0.156460	−	0.270996i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−1714.26			−1.86942
$918$		0			0
$919$	−501.923			−0.546163			−0.273081	−	0.961991i	$-0.588043\pi$
$919$	−501.923			−0.546163			−0.273081	+	0.961991i	$0.588043\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−1201.15	+	2080.46i	−1.30136	+	2.25402i
$924$		0			0
$925$	144.595	−	83.4822i	0.156319	−	0.0902511i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	839.922	−	484.929i	0.904114	−	0.521990i	0.0255812	−	0.999673i	$-0.491856\pi$
$929$	839.922	−	484.929i	0.904114	−	0.521990i	0.878533	+	0.477682i	$0.158523\pi$
$930$		0			0
$931$	326.483	+	188.495i	0.350680	+	0.202465i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	432.917i		−	0.463013i
$936$		0			0
$937$	−700.392			−0.747483			−0.373742	−	0.927533i	$-0.621925\pi$
$937$	−700.392			−0.747483			−0.373742	+	0.927533i	$0.621925\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	679.823	−	1177.49i	0.722447	−	1.25131i	−0.237569	−	0.971371i	$-0.576351\pi$
$941$	679.823	−	1177.49i	0.722447	−	1.25131i	0.960016	−	0.279944i	$-0.0903160\pi$
$942$		0			0
$943$	360.213	+	623.908i	0.381986	+	0.661620i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	79.7517	+	138.134i	0.0842151	+	0.145865i	0.905056	−	0.425291i	$-0.139828\pi$
$947$	79.7517	+	138.134i	0.0842151	+	0.145865i	−0.820841	+	0.571156i	$0.806495\pi$
$948$		0			0
$949$	−377.762	−	218.101i	−0.398063	−	0.229822i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	941.561i		−	0.987997i	−0.869463	−	0.493998i	$-0.835535\pi$
$953$		−	941.561i		−	0.987997i	0.869463	−	0.493998i	$-0.164465\pi$
$954$		0			0
$955$			447.702i			0.468798i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	1471.99	+	849.855i	1.53492	+	0.886189i
$960$		0			0
$961$	−314.064	−	543.974i	−0.326809	−	0.566050i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−102.547	−	177.616i	−0.106266	−	0.184058i
$966$		0			0
$967$	615.116	−	1065.41i	0.636107	−	1.10177i	−0.350172	−	0.936685i	$-0.613877\pi$
$967$	615.116	−	1065.41i	0.636107	−	1.10177i	0.986279	−	0.165085i	$-0.0527899\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	1584.43			1.63175			0.815876	−	0.578227i	$-0.196255\pi$
$971$	1584.43			1.63175			0.815876	+	0.578227i	$0.196255\pi$
$972$		0			0
$973$		−	2128.96i		−	2.18804i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−572.558	−	330.567i	−0.586037	−	0.338349i	0.177492	−	0.984122i	$-0.443202\pi$
$977$	−572.558	−	330.567i	−0.586037	−	0.338349i	−0.763529	+	0.645774i	$0.776535\pi$
$978$		0			0
$979$	−302.123	+	174.431i	−0.308603	+	0.178172i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	716.470	−	413.654i	0.728861	−	0.420808i	−0.0891445	−	0.996019i	$-0.528413\pi$
$983$	716.470	−	413.654i	0.728861	−	0.420808i	0.818005	+	0.575211i	$0.195080\pi$
$984$		0			0
$985$	−127.109	+	220.159i	−0.129044	+	0.223511i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−129.354			−0.130793
$990$		0			0
$991$	−72.7970			−0.0734581			−0.0367291	−	0.999325i	$-0.511694\pi$
$991$	−72.7970			−0.0734581			−0.0367291	+	0.999325i	$0.511694\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	30.8364	−	53.4101i	0.0309913	−	0.0536785i
$996$		0			0
$997$	676.198	−	390.403i	0.678233	−	0.391578i	−0.120956	−	0.992658i	$-0.538596\pi$
$997$	676.198	−	390.403i	0.678233	−	0.391578i	0.799189	+	0.601080i	$0.205263\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.3.n.d.1313.8		32
3.2	odd	2		576.3.n.c.545.9	yes	32
4.3	odd	2	inner	1728.3.n.d.1313.7		32
8.3	odd	2		1728.3.n.c.1313.10		32
8.5	even	2		1728.3.n.c.1313.9		32
9.2	odd	6		1728.3.n.c.737.9		32
9.7	even	3		576.3.n.d.353.8	yes	32
12.11	even	2		576.3.n.c.545.8	yes	32
24.5	odd	2		576.3.n.d.545.8	yes	32
24.11	even	2		576.3.n.d.545.9	yes	32
36.7	odd	6		576.3.n.d.353.9	yes	32
36.11	even	6		1728.3.n.c.737.10		32
72.11	even	6	inner	1728.3.n.d.737.7		32
72.29	odd	6	inner	1728.3.n.d.737.8		32
72.43	odd	6		576.3.n.c.353.8	✓	32
72.61	even	6		576.3.n.c.353.9	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.3.n.c.353.8	✓	32	72.43	odd	6
576.3.n.c.353.9	yes	32	72.61	even	6
576.3.n.c.545.8	yes	32	12.11	even	2
576.3.n.c.545.9	yes	32	3.2	odd	2
576.3.n.d.353.8	yes	32	9.7	even	3
576.3.n.d.353.9	yes	32	36.7	odd	6
576.3.n.d.545.8	yes	32	24.5	odd	2
576.3.n.d.545.9	yes	32	24.11	even	2
1728.3.n.c.737.9		32	9.2	odd	6
1728.3.n.c.737.10		32	36.11	even	6
1728.3.n.c.1313.9		32	8.5	even	2
1728.3.n.c.1313.10		32	8.3	odd	2
1728.3.n.d.737.7		32	72.11	even	6	inner
1728.3.n.d.737.8		32	72.29	odd	6	inner
1728.3.n.d.1313.7		32	4.3	odd	2	inner
1728.3.n.d.1313.8		32	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:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1728.n (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.920717 + 1.59473i) q^{5} +(4.56979 + 7.91511i) q^{7} +O(q^{10})\)	\(q+(-0.920717 + 1.59473i) q^{5} +(4.56979 + 7.91511i) q^{7} +(-3.81937 - 6.61534i) q^{11} +(15.7478 + 9.09201i) q^{13} -30.7770i q^{17} -10.9172i q^{19} +(12.5177 + 7.22708i) q^{23} +(10.8046 + 18.7140i) q^{25} +(-18.9375 - 32.8007i) q^{29} +(19.9319 - 34.5231i) q^{31} -16.8299 q^{35} -7.72657i q^{37} +(43.1646 + 24.9211i) q^{41} +(-7.75029 + 4.47463i) q^{43} +(12.9978 - 7.50427i) q^{47} +(-17.2659 + 29.9055i) q^{49} -20.1038 q^{53} +14.0662 q^{55} +(-19.0494 + 32.9945i) q^{59} +(90.9853 - 52.5304i) q^{61} +(-28.9986 + 16.7423i) q^{65} +(18.1407 + 10.4735i) q^{67} +132.111i q^{71} -23.9882 q^{73} +(34.9074 - 60.4614i) q^{77} +(21.1445 + 36.6233i) q^{79} +(18.7004 + 32.3901i) q^{83} +(49.0810 + 28.3369i) q^{85} -45.6700i q^{89} +166.194i q^{91} +(17.4099 + 10.0516i) q^{95} +(41.6001 + 72.0535i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 18 q^{5}+O(q^{10}) \) 32 * q + 18 * q^5	\( 32 q + 18 q^{5} + 30 q^{13} - 74 q^{25} + 54 q^{29} + 216 q^{41} - 86 q^{49} - 144 q^{53} - 42 q^{61} - 306 q^{65} + 196 q^{73} + 414 q^{77} - 180 q^{85} - 64 q^{97}+O(q^{100}) \) 32 * q + 18 * q^5 + 30 * q^13 - 74 * q^25 + 54 * q^29 + 216 * q^41 - 86 * q^49 - 144 * q^53 - 42 * q^61 - 306 * q^65 + 196 * q^73 + 414 * q^77 - 180 * q^85 - 64 * q^97

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