LMFDB - Embedded newform 1764.3.z.m.325.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("1764.325");

S:= CuspForms(chi, 3);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$48.0655186332$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.339738624.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{6} + 14x^{4} - 8x^{2} + 4 $ x^8 - 4x^6 + 14x^4 - 8*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 7^{2} $
Twist minimal:	no (minimal twist has level 588)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			325.3
Root			$-1.60021 - 0.923880i$ of defining polynomial
Character	$\chi$	$=$	1764.325
Dual form			1764.3.z.m.901.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.804540 - 0.464502i) q^{5} +O(q^{10})$	$q+(0.804540 - 0.464502i) q^{5} +(4.84490 - 8.39161i) q^{11} -15.9753i q^{13} +(-9.14154 - 5.27787i) q^{17} +(6.25313 - 3.61025i) q^{19} +(-5.65003 - 9.78614i) q^{23} +(-12.0685 + 20.9032i) q^{25} -46.3148 q^{29} +(-0.418333 - 0.241525i) q^{31} +(-1.24065 - 2.14887i) q^{37} +55.8520i q^{41} +60.6786 q^{43} +(-31.6850 + 18.2933i) q^{47} +(-14.2615 + 24.7016i) q^{53} -9.00185i q^{55} +(-81.4683 - 47.0358i) q^{59} +(95.4301 - 55.0966i) q^{61} +(-7.42055 - 12.8528i) q^{65} +(-41.0155 + 71.0409i) q^{67} -127.349 q^{71} +(-40.0577 - 23.1273i) q^{73} +(-9.35016 - 16.1949i) q^{79} -59.6357i q^{83} -9.80632 q^{85} +(61.5988 - 35.5641i) q^{89} +(3.35393 - 5.80917i) q^{95} -102.239i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 48 q^{17} + 96 q^{19} - 8 q^{23} - 36 q^{25} - 80 q^{29} - 48 q^{31} - 64 q^{37} - 112 q^{43} - 264 q^{47} - 72 q^{53} + 168 q^{59} + 144 q^{61} + 120 q^{65} + 32 q^{67} - 224 q^{71} - 336 q^{73} + 216 q^{79} - 96 q^{85} + 96 q^{89} - 136 q^{95}+O(q^{100}) $ 8 * q + 48 * q^17 + 96 * q^19 - 8 * q^23 - 36 * q^25 - 80 * q^29 - 48 * q^31 - 64 * q^37 - 112 * q^43 - 264 * q^47 - 72 * q^53 + 168 * q^59 + 144 * q^61 + 120 * q^65 + 32 * q^67 - 224 * q^71 - 336 * q^73 + 216 * q^79 - 96 * q^85 + 96 * q^89 - 136 * q^95

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	0.804540	−	0.464502i	0.160908	−	0.0929003i	−0.417384	−	0.908730i	$-0.637053\pi$
$5$	0.804540	−	0.464502i	0.160908	−	0.0929003i	0.578292	+	0.815830i	$0.303720\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	4.84490	−	8.39161i	0.440445	−	0.762874i	−0.557277	−	0.830327i	$-0.688154\pi$
$11$	4.84490	−	8.39161i	0.440445	−	0.762874i	0.997722	+	0.0674529i	$0.0214872\pi$
$12$		0			0
$13$		−	15.9753i		−	1.22887i	−0.788968	−	0.614434i	$-0.789384\pi$
$13$		−	15.9753i		−	1.22887i	0.788968	−	0.614434i	$-0.210616\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−9.14154	−	5.27787i	−0.537738	−	0.310463i	0.206424	−	0.978463i	$-0.433817\pi$
$17$	−9.14154	−	5.27787i	−0.537738	−	0.310463i	−0.744162	+	0.668000i	$0.767151\pi$
$18$		0			0
$19$	6.25313	−	3.61025i	0.329112	−	0.190013i	−0.326335	−	0.945254i	$-0.605814\pi$
$19$	6.25313	−	3.61025i	0.329112	−	0.190013i	0.655447	+	0.755241i	$0.272480\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−5.65003	−	9.78614i	−0.245654	−	0.425484i	0.716662	−	0.697421i	$-0.245669\pi$
$23$	−5.65003	−	9.78614i	−0.245654	−	0.425484i	−0.962315	+	0.271937i	$0.912336\pi$
$24$		0			0
$25$	−12.0685	+	20.9032i	−0.482739	+	0.836129i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−46.3148			−1.59706			−0.798532	−	0.601953i	$-0.794390\pi$
$29$	−46.3148			−1.59706			−0.798532	+	0.601953i	$0.794390\pi$
$30$		0			0
$31$	−0.418333	−	0.241525i	−0.0134946	−	0.00779111i	0.493238	−	0.869895i	$-0.335813\pi$
$31$	−0.418333	−	0.241525i	−0.0134946	−	0.00779111i	−0.506732	+	0.862104i	$0.669147\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−1.24065	−	2.14887i	−0.0335312	−	0.0580777i	0.848773	−	0.528758i	$-0.177342\pi$
$37$	−1.24065	−	2.14887i	−0.0335312	−	0.0580777i	−0.882304	+	0.470680i	$0.844009\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			55.8520i			1.36224i	0.732170	+	0.681122i	$0.238508\pi$
$41$			55.8520i			1.36224i	−0.732170	+	0.681122i	$0.761492\pi$
$42$		0			0
$43$	60.6786			1.41113			0.705566	−	0.708645i	$-0.250693\pi$
$43$	60.6786			1.41113			0.705566	+	0.708645i	$0.250693\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−31.6850	+	18.2933i	−0.674149	+	0.389220i	−0.797647	−	0.603125i	$-0.793922\pi$
$47$	−31.6850	+	18.2933i	−0.674149	+	0.389220i	0.123498	+	0.992345i	$0.460589\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−14.2615	+	24.7016i	−0.269084	+	0.466068i	−0.968626	−	0.248524i	$-0.920054\pi$
$53$	−14.2615	+	24.7016i	−0.269084	+	0.466068i	0.699541	+	0.714592i	$0.253388\pi$
$54$		0			0
$55$		−	9.00185i		−	0.163670i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−81.4683	−	47.0358i	−1.38082	−	0.797216i	−0.388563	−	0.921422i	$-0.627028\pi$
$59$	−81.4683	−	47.0358i	−1.38082	−	0.797216i	−0.992256	+	0.124206i	$0.960362\pi$
$60$		0			0
$61$	95.4301	−	55.0966i	1.56443	−	0.903223i	0.567628	−	0.823285i	$-0.307861\pi$
$61$	95.4301	−	55.0966i	1.56443	−	0.903223i	0.996800	−	0.0799382i	$-0.0254723\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−7.42055	−	12.8528i	−0.114162	−	0.197735i
$66$		0			0
$67$	−41.0155	+	71.0409i	−0.612171	+	1.06031i	0.378702	+	0.925518i	$0.376370\pi$
$67$	−41.0155	+	71.0409i	−0.612171	+	1.06031i	−0.990874	+	0.134793i	$0.956963\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−127.349			−1.79365			−0.896827	−	0.442382i	$-0.854133\pi$
$71$	−127.349			−1.79365			−0.896827	+	0.442382i	$0.854133\pi$
$72$		0			0
$73$	−40.0577	−	23.1273i	−0.548735	−	0.316812i	0.199877	−	0.979821i	$-0.435946\pi$
$73$	−40.0577	−	23.1273i	−0.548735	−	0.316812i	−0.748612	+	0.663009i	$0.769279\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−9.35016	−	16.1949i	−0.118356	−	0.204999i	0.800760	−	0.598985i	$-0.204429\pi$
$79$	−9.35016	−	16.1949i	−0.118356	−	0.204999i	−0.919116	+	0.393986i	$0.871096\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	59.6357i		−	0.718502i	−0.933241	−	0.359251i	$-0.883032\pi$
$83$		−	59.6357i		−	0.718502i	0.933241	−	0.359251i	$-0.116968\pi$
$84$		0			0
$85$	−9.80632			−0.115368
$86$		0			0
$87$		0			0
$88$		0			0
$89$	61.5988	−	35.5641i	0.692121	−	0.399596i	−0.112285	−	0.993676i	$-0.535817\pi$
$89$	61.5988	−	35.5641i	0.692121	−	0.399596i	0.804406	+	0.594080i	$0.202484\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	3.35393	−	5.80917i	0.0353045	−	0.0611492i
$96$		0			0
$97$		−	102.239i		−	1.05401i	−0.849861	−	0.527007i	$-0.823314\pi$
$97$		−	102.239i		−	1.05401i	0.849861	−	0.527007i	$-0.176686\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−94.3357	−	54.4647i	−0.934017	−	0.539255i	−0.0459371	−	0.998944i	$-0.514627\pi$
$101$	−94.3357	−	54.4647i	−0.934017	−	0.539255i	−0.888080	+	0.459690i	$0.847961\pi$
$102$		0			0
$103$	−0.647083	+	0.373594i	−0.00628236	+	0.00362712i	−0.503138	−	0.864206i	$-0.667821\pi$
$103$	−0.647083	+	0.373594i	−0.00628236	+	0.00362712i	0.496856	+	0.867833i	$0.334488\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	7.64630	+	13.2438i	0.0714608	+	0.123774i	0.899542	−	0.436835i	$-0.143901\pi$
$107$	7.64630	+	13.2438i	0.0714608	+	0.123774i	−0.828081	+	0.560609i	$0.810567\pi$
$108$		0			0
$109$	27.1116	−	46.9587i	0.248730	−	0.430814i	−0.714443	−	0.699693i	$-0.753320\pi$
$109$	27.1116	−	46.9587i	0.248730	−	0.430814i	0.963174	+	0.268879i	$0.0866533\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−65.2511			−0.577444			−0.288722	−	0.957413i	$-0.593230\pi$
$113$	−65.2511			−0.577444			−0.288722	+	0.957413i	$0.593230\pi$
$114$		0			0
$115$	−9.09136	−	5.24890i	−0.0790553	−	0.0456426i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	13.5539	+	23.4761i	0.112016	+	0.194017i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			45.6484i			0.365187i
$126$		0			0
$127$	−235.761			−1.85639			−0.928193	−	0.372098i	$-0.878639\pi$
$127$	−235.761			−1.85639			−0.928193	+	0.372098i	$0.878639\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	196.180	−	113.265i	1.49756	−	0.864616i	0.497563	−	0.867428i	$-0.334228\pi$
$131$	196.180	−	113.265i	1.49756	−	0.864616i	0.999996	+	0.00281117i	$0.000894824\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−126.872	+	219.748i	−0.926070	+	1.60400i	−0.136238	+	0.990676i	$0.543501\pi$
$137$	−126.872	+	219.748i	−0.926070	+	1.60400i	−0.789832	+	0.613323i	$0.789832\pi$
$138$		0			0
$139$			148.040i			1.06503i	0.846419	+	0.532517i	$0.178754\pi$
$139$			148.040i			1.06503i	−0.846419	+	0.532517i	$0.821246\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−134.058	−	77.3986i	−0.937471	−	0.541249i
$144$		0			0
$145$	−37.2622	+	21.5133i	−0.256980	+	0.148368i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−122.732	−	212.579i	−0.823708	−	1.42670i	−0.902903	−	0.429845i	$-0.858568\pi$
$149$	−122.732	−	212.579i	−0.823708	−	1.42670i	0.0791950	−	0.996859i	$-0.474765\pi$
$150$		0			0
$151$	−88.1270	+	152.640i	−0.583623	+	1.01086i	0.411423	+	0.911445i	$0.365032\pi$
$151$	−88.1270	+	152.640i	−0.583623	+	1.01086i	−0.995046	+	0.0994194i	$0.968301\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−0.448754			−0.00289519
$156$		0			0
$157$	−179.836	−	103.828i	−1.14545	−	0.661326i	−0.197675	−	0.980268i	$-0.563339\pi$
$157$	−179.836	−	103.828i	−1.14545	−	0.661326i	−0.947774	+	0.318942i	$0.896673\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−47.5121	−	82.2933i	−0.291485	−	0.504867i	0.682676	−	0.730721i	$-0.260816\pi$
$163$	−47.5121	−	82.2933i	−0.291485	−	0.504867i	−0.974161	+	0.225854i	$0.927483\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	158.478i		−	0.948972i	−0.880263	−	0.474486i	$-0.842634\pi$
$167$		−	158.478i		−	0.948972i	0.880263	−	0.474486i	$-0.157366\pi$
$168$		0			0
$169$	−86.2098			−0.510117
$170$		0			0
$171$		0			0
$172$		0			0
$173$	187.054	−	107.995i	1.08123	−	0.624251i	0.150005	−	0.988685i	$-0.452071\pi$
$173$	187.054	−	107.995i	1.08123	−	0.624251i	0.931229	+	0.364434i	$0.118738\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−131.029	+	226.950i	−0.732008	+	1.26787i	0.224016	+	0.974585i	$0.428083\pi$
$179$	−131.029	+	226.950i	−0.732008	+	1.26787i	−0.956024	+	0.293289i	$0.905250\pi$
$180$		0			0
$181$			83.8554i			0.463290i	0.972800	+	0.231645i	$0.0744107\pi$
$181$			83.8554i			0.463290i	−0.972800	+	0.231645i	$0.925589\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−1.99631	−	1.15257i	−0.0107909	−	0.00623011i
$186$		0			0
$187$	−88.5797	+	51.1415i	−0.473688	+	0.273484i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	116.846	+	202.384i	0.611760	+	1.05960i	0.990944	+	0.134278i	$0.0428716\pi$
$191$	116.846	+	202.384i	0.611760	+	1.05960i	−0.379183	+	0.925322i	$0.623795\pi$
$192$		0			0
$193$	111.819	−	193.677i	0.579375	−	1.00351i	−0.416176	−	0.909284i	$-0.636630\pi$
$193$	111.819	−	193.677i	0.579375	−	1.00351i	0.995551	−	0.0942227i	$-0.0300366\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	133.006			0.675158			0.337579	−	0.941297i	$-0.390392\pi$
$197$	133.006			0.675158			0.337579	+	0.941297i	$0.390392\pi$
$198$		0			0
$199$	−52.5277	−	30.3269i	−0.263959	−	0.152397i	0.362180	−	0.932108i	$-0.382032\pi$
$199$	−52.5277	−	30.3269i	−0.263959	−	0.152397i	−0.626139	+	0.779711i	$0.715366\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	25.9433	+	44.9352i	0.126553	+	0.219196i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	69.9651i		−	0.334761i
$210$		0			0
$211$	−169.145			−0.801637			−0.400819	−	0.916157i	$-0.631274\pi$
$211$	−169.145			−0.801637			−0.400819	+	0.916157i	$0.631274\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	48.8184	−	28.1853i	0.227062	−	0.131095i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−84.3155	+	146.039i	−0.381518	+	0.660809i
$222$		0			0
$223$			162.093i			0.726874i	0.931619	+	0.363437i	$0.118397\pi$
$223$			162.093i			0.726874i	−0.931619	+	0.363437i	$0.881603\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−306.755	−	177.105i	−1.35134	−	0.780198i	−0.362905	−	0.931826i	$-0.618215\pi$
$227$	−306.755	−	177.105i	−1.35134	−	0.780198i	−0.988438	+	0.151628i	$0.951548\pi$
$228$		0			0
$229$	113.844	−	65.7279i	0.497136	−	0.287021i	−0.230394	−	0.973097i	$-0.574002\pi$
$229$	113.844	−	65.7279i	0.497136	−	0.287021i	0.727530	+	0.686076i	$0.240668\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−71.4744	−	123.797i	−0.306757	−	0.531319i	0.670894	−	0.741553i	$-0.265911\pi$
$233$	−71.4744	−	123.797i	−0.306757	−	0.531319i	−0.977651	+	0.210234i	$0.932577\pi$
$234$		0			0
$235$	−16.9946	+	29.4354i	−0.0723173	+	0.125257i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−47.5259			−0.198853			−0.0994266	−	0.995045i	$-0.531701\pi$
$239$	−47.5259			−0.198853			−0.0994266	+	0.995045i	$0.531701\pi$
$240$		0			0
$241$	−205.380	−	118.576i	−0.852198	−	0.492017i	0.00919389	−	0.999958i	$-0.497073\pi$
$241$	−205.380	−	118.576i	−0.852198	−	0.492017i	−0.861392	+	0.507941i	$0.830407\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−57.6747	−	99.8955i	−0.233501	−	0.404435i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	309.248i		−	1.23206i	−0.787722	−	0.616031i	$-0.788739\pi$
$251$		−	309.248i		−	1.23206i	0.787722	−	0.616031i	$-0.211261\pi$
$252$		0			0
$253$	−109.495			−0.432788
$254$		0			0
$255$		0			0
$256$		0			0
$257$	155.317	−	89.6724i	0.604347	−	0.348920i	−0.166403	−	0.986058i	$-0.553215\pi$
$257$	155.317	−	89.6724i	0.604347	−	0.348920i	0.770750	+	0.637138i	$0.219882\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	37.3848	−	64.7523i	0.142147	−	0.246206i	−0.786158	−	0.618026i	$-0.787933\pi$
$263$	37.3848	−	64.7523i	0.142147	−	0.246206i	0.928305	+	0.371820i	$0.121266\pi$
$264$		0			0
$265$			26.4979i			0.0999921i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	49.2164	+	28.4151i	0.182961	+	0.105632i	0.588683	−	0.808364i	$-0.299647\pi$
$269$	49.2164	+	28.4151i	0.182961	+	0.105632i	−0.405722	+	0.913996i	$0.632980\pi$
$270$		0			0
$271$	−11.6951	+	6.75218i	−0.0431554	+	0.0249158i	−0.521422	−	0.853299i	$-0.674598\pi$
$271$	−11.6951	+	6.75218i	−0.0431554	+	0.0249158i	0.478267	+	0.878214i	$0.341265\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	116.941	+	202.548i	0.425240	+	0.736538i
$276$		0			0
$277$	124.726	−	216.032i	0.450274	−	0.779897i	−0.548129	−	0.836394i	$-0.684660\pi$
$277$	124.726	−	216.032i	0.450274	−	0.779897i	0.998403	+	0.0564965i	$0.0179930\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−200.268			−0.712696			−0.356348	−	0.934353i	$-0.615978\pi$
$281$	−200.268			−0.712696			−0.356348	+	0.934353i	$0.615978\pi$
$282$		0			0
$283$	59.5549	+	34.3840i	0.210441	+	0.121498i	0.601516	−	0.798860i	$-0.294563\pi$
$283$	59.5549	+	34.3840i	0.210441	+	0.121498i	−0.391075	+	0.920359i	$0.627897\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−88.7881	−	153.786i	−0.307225	−	0.532130i
$290$		0			0
$291$		0			0
$292$		0			0
$293$			253.164i			0.864040i	0.901864	+	0.432020i	$0.142199\pi$
$293$			253.164i			0.864040i	−0.901864	+	0.432020i	$0.857801\pi$
$294$		0			0
$295$	−87.3927			−0.296247
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−156.336	+	90.2609i	−0.522864	+	0.301876i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	51.1849	−	88.6549i	0.167819	−	0.290672i
$306$		0			0
$307$		−	529.913i		−	1.72610i	−0.505116	−	0.863051i	$-0.668550\pi$
$307$		−	529.913i		−	1.72610i	0.505116	−	0.863051i	$-0.331450\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−26.4186	−	15.2528i	−0.0849473	−	0.0490444i	0.456925	−	0.889505i	$-0.348951\pi$
$311$	−26.4186	−	15.2528i	−0.0849473	−	0.0490444i	−0.541872	+	0.840461i	$0.682284\pi$
$312$		0			0
$313$	−7.44956	+	4.30101i	−0.0238005	+	0.0137412i	−0.511853	−	0.859073i	$-0.671041\pi$
$313$	−7.44956	+	4.30101i	−0.0238005	+	0.0137412i	0.488053	+	0.872814i	$0.337707\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−240.076	−	415.824i	−0.757337	−	1.31175i	−0.944204	−	0.329362i	$-0.893167\pi$
$317$	−240.076	−	415.824i	−0.757337	−	1.31175i	0.186866	−	0.982385i	$-0.440167\pi$
$318$		0			0
$319$	−224.391	+	388.656i	−0.703419	+	1.21836i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−76.2176			−0.235968
$324$		0			0
$325$	333.935	+	192.797i	1.02749	+	0.593223i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	44.5098	+	77.0932i	0.134471	+	0.232910i	0.925395	−	0.379004i	$-0.123733\pi$
$331$	44.5098	+	77.0932i	0.134471	+	0.232910i	−0.790924	+	0.611914i	$0.790400\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			76.2070i			0.227484i
$336$		0			0
$337$	495.701			1.47092			0.735461	−	0.677567i	$-0.236966\pi$
$337$	495.701			1.47092			0.735461	+	0.677567i	$0.236966\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−4.05356	+	2.34032i	−0.0118873	+	0.00686312i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−30.1767	+	52.2676i	−0.0869645	+	0.150627i	−0.906227	−	0.422792i	$-0.861050\pi$
$347$	−30.1767	+	52.2676i	−0.0869645	+	0.150627i	0.819262	+	0.573419i	$0.194383\pi$
$348$		0			0
$349$			72.2171i			0.206926i	0.994633	+	0.103463i	$0.0329923\pi$
$349$			72.2171i			0.206926i	−0.994633	+	0.103463i	$0.967008\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	332.000	+	191.680i	0.940510	+	0.543004i	0.890120	−	0.455726i	$-0.150620\pi$
$353$	332.000	+	191.680i	0.940510	+	0.543004i	0.0503901	+	0.998730i	$0.483954\pi$
$354$		0			0
$355$	−102.458	+	59.1540i	−0.288613	+	0.166631i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−40.0224	−	69.3209i	−0.111483	−	0.193094i	0.804885	−	0.593430i	$-0.202227\pi$
$359$	−40.0224	−	69.3209i	−0.111483	−	0.193094i	−0.916368	+	0.400336i	$0.868893\pi$
$360$		0			0
$361$	−154.432	+	267.485i	−0.427790	+	0.740954i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−42.9707			−0.117728
$366$		0			0
$367$	70.5218	+	40.7158i	0.192157	+	0.110942i	0.592992	−	0.805208i	$-0.297946\pi$
$367$	70.5218	+	40.7158i	0.192157	+	0.110942i	−0.400835	+	0.916150i	$0.631280\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	266.126	+	460.944i	0.713475	+	1.23577i	0.963545	+	0.267547i	$0.0862131\pi$
$373$	266.126	+	460.944i	0.713475	+	1.23577i	−0.250070	+	0.968228i	$0.580454\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			739.893i			1.96258i
$378$		0			0
$379$	440.518			1.16232			0.581159	−	0.813790i	$-0.302600\pi$
$379$	440.518			1.16232			0.581159	+	0.813790i	$0.302600\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−605.603	+	349.645i	−1.58121	+	0.912911i	−0.586525	+	0.809931i	$0.699504\pi$
$383$	−605.603	+	349.645i	−1.58121	+	0.912911i	−0.994683	+	0.102980i	$0.967162\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	312.231	−	540.800i	0.802650	−	1.39023i	−0.115216	−	0.993340i	$-0.536756\pi$
$389$	312.231	−	540.800i	0.802650	−	1.39023i	0.917866	−	0.396891i	$-0.129911\pi$
$390$		0			0
$391$			119.281i			0.305065i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−15.0452	−	8.68632i	−0.0380890	−	0.0219907i
$396$		0			0
$397$	547.801	−	316.273i	1.37985	−	0.796658i	0.387711	−	0.921781i	$-0.373266\pi$
$397$	547.801	−	316.273i	1.37985	−	0.796658i	0.992141	+	0.125123i	$0.0399325\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	233.671	+	404.731i	0.582721	+	1.00930i	0.995155	+	0.0983156i	$0.0313455\pi$
$401$	233.671	+	404.731i	0.582721	+	1.00930i	−0.412434	+	0.910988i	$0.635321\pi$
$402$		0			0
$403$	−3.85842	+	6.68298i	−0.00957425	+	0.0165831i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−24.0434			−0.0590746
$408$		0			0
$409$	−280.848	−	162.148i	−0.686671	−	0.396450i	0.115693	−	0.993285i	$-0.463091\pi$
$409$	−280.848	−	162.148i	−0.686671	−	0.396450i	−0.802364	+	0.596835i	$0.796425\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−27.7009	−	47.9793i	−0.0667491	−	0.115613i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	28.8211i		−	0.0687854i	−0.999408	−	0.0343927i	$-0.989050\pi$
$419$		−	28.8211i		−	0.0687854i	0.999408	−	0.0343927i	$-0.0109497\pi$
$420$		0			0
$421$	−0.326830			−0.000776319			−0.000388160	−	1.00000i	$-0.500124\pi$
$421$	−0.326830			−0.000776319			−0.000388160		1.00000i	$0.500124\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	220.649	−	127.392i	0.519174	−	0.299745i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	195.669	−	338.908i	0.453988	−	0.786329i	−0.544642	−	0.838669i	$-0.683334\pi$
$431$	195.669	−	338.908i	0.453988	−	0.786329i	0.998629	+	0.0523393i	$0.0166677\pi$
$432$		0			0
$433$		−	470.579i		−	1.08679i	−0.839478	−	0.543394i	$-0.817139\pi$
$433$		−	470.579i		−	1.08679i	0.839478	−	0.543394i	$-0.182861\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−70.6607	−	40.7960i	−0.161695	−	0.0933547i
$438$		0			0
$439$	−116.395	+	67.2006i	−0.265136	+	0.153077i	−0.626675	−	0.779280i	$-0.715585\pi$
$439$	−116.395	+	67.2006i	−0.265136	+	0.153077i	0.361539	+	0.932357i	$0.382251\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	164.528	+	284.971i	0.371395	+	0.643275i	0.989780	−	0.142600i	$-0.0455462\pi$
$443$	164.528	+	284.971i	0.371395	+	0.643275i	−0.618385	+	0.785875i	$0.712213\pi$
$444$		0			0
$445$	33.0391	−	57.2254i	0.0742452	−	0.128596i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−130.592			−0.290851			−0.145426	−	0.989369i	$-0.546455\pi$
$449$	−130.592			−0.290851			−0.145426	+	0.989369i	$0.546455\pi$
$450$		0			0
$451$	468.688	+	270.597i	1.03922	+	0.599994i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−158.141	−	273.908i	−0.346041	−	0.599361i	0.639501	−	0.768790i	$-0.279141\pi$
$457$	−158.141	−	273.908i	−0.346041	−	0.599361i	−0.985542	+	0.169429i	$0.945808\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	31.4853i		−	0.0682979i	−0.999417	−	0.0341490i	$-0.989128\pi$
$461$		−	31.4853i		−	0.0682979i	0.999417	−	0.0341490i	$-0.0108721\pi$
$462$		0			0
$463$	667.424			1.44152			0.720761	−	0.693184i	$-0.243793\pi$
$463$	667.424			1.44152			0.720761	+	0.693184i	$0.243793\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−748.502	+	432.148i	−1.60279	+	0.925370i	−0.611861	+	0.790965i	$0.709579\pi$
$467$	−748.502	+	432.148i	−1.60279	+	0.925370i	−0.990927	+	0.134405i	$0.957088\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	293.982	−	509.192i	0.621526	−	1.07651i
$474$		0			0
$475$			174.281i			0.366907i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	542.476	+	313.198i	1.13252	+	0.653859i	0.944567	−	0.328320i	$-0.106482\pi$
$479$	542.476	+	313.198i	1.13252	+	0.653859i	0.187950	+	0.982179i	$0.439816\pi$
$480$		0			0
$481$	−34.3289	+	19.8198i	−0.0713698	+	0.0412054i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−47.4904	−	82.2557i	−0.0979183	−	0.169599i
$486$		0			0
$487$	242.924	−	420.756i	0.498817	−	0.863976i	−0.501182	−	0.865342i	$-0.667101\pi$
$487$	242.924	−	420.756i	0.498817	−	0.863976i	0.999999	+	0.00136550i	$0.000434653\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	6.00051			0.0122210			0.00611049	−	0.999981i	$-0.498055\pi$
$491$	6.00051			0.0122210			0.00611049	+	0.999981i	$0.498055\pi$
$492$		0			0
$493$	423.389	+	244.444i	0.858801	+	0.495829i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	343.775	+	595.437i	0.688929	+	1.19326i	0.972185	+	0.234215i	$0.0752520\pi$
$499$	343.775	+	595.437i	0.688929	+	1.19326i	−0.283256	+	0.959044i	$0.591415\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	435.270i		−	0.865348i	−0.901550	−	0.432674i	$-0.857570\pi$
$503$		−	435.270i		−	0.865348i	0.901550	−	0.432674i	$-0.142430\pi$
$504$		0			0
$505$	−101.196			−0.200388
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−418.381	+	241.553i	−0.821967	+	0.474563i	−0.851094	−	0.525013i	$-0.824061\pi$
$509$	−418.381	+	241.553i	−0.821967	+	0.474563i	0.0291272	+	0.999576i	$0.490727\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−0.347070	+	0.601142i	−0.000673921	+	0.00116727i
$516$		0			0
$517$			354.517i			0.685720i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	272.425	+	157.285i	0.522888	+	0.301890i	0.738116	−	0.674674i	$-0.235716\pi$
$521$	272.425	+	157.285i	0.522888	+	0.301890i	−0.215227	+	0.976564i	$0.569049\pi$
$522$		0			0
$523$	−135.591	+	78.2835i	−0.259256	+	0.149682i	−0.623995	−	0.781428i	$-0.714492\pi$
$523$	−135.591	+	78.2835i	−0.259256	+	0.149682i	0.364739	+	0.931110i	$0.381158\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	2.54947	+	4.41581i	0.00483771	+	0.00837915i
$528$		0			0
$529$	200.654	−	347.543i	0.379309	−	0.656982i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	892.251			1.67402
$534$		0			0
$535$	12.3035	+	7.10344i	0.0229972	+	0.0132775i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−386.033	−	668.628i	−0.713554	−	1.23591i	−0.963515	−	0.267656i	$-0.913751\pi$
$541$	−386.033	−	668.628i	−0.713554	−	1.23591i	0.249960	−	0.968256i	$-0.419582\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	50.3736i		−	0.0924285i
$546$		0			0
$547$	48.0113			0.0877721			0.0438860	−	0.999037i	$-0.486026\pi$
$547$	48.0113			0.0877721			0.0438860	+	0.999037i	$0.486026\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−289.613	+	167.208i	−0.525613	+	0.303463i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	75.7027	−	131.121i	0.135911	−	0.235406i	−0.790034	−	0.613063i	$-0.789937\pi$
$557$	75.7027	−	131.121i	0.135911	−	0.235406i	0.925945	+	0.377658i	$0.123270\pi$
$558$		0			0
$559$		−	969.359i		−	1.73409i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	732.918	+	423.150i	1.30181	+	0.751599i	0.980714	−	0.195448i	$-0.0626161\pi$
$563$	732.918	+	423.150i	1.30181	+	0.751599i	0.321094	+	0.947047i	$0.395949\pi$
$564$		0			0
$565$	−52.4972	+	30.3093i	−0.0929153	+	0.0536447i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−10.7887	−	18.6865i	−0.0189608	−	0.0328410i	0.856389	−	0.516331i	$-0.172702\pi$
$569$	−10.7887	−	18.6865i	−0.0189608	−	0.0328410i	−0.875350	+	0.483490i	$0.839369\pi$
$570$		0			0
$571$	−141.623	+	245.298i	−0.248026	+	0.429593i	−0.962978	−	0.269580i	$-0.913115\pi$
$571$	−141.623	+	245.298i	−0.248026	+	0.429593i	0.714952	+	0.699173i	$0.246448\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	272.749			0.474346
$576$		0			0
$577$	406.431	+	234.653i	0.704387	+	0.406678i	0.808979	−	0.587837i	$-0.200021\pi$
$577$	406.431	+	234.653i	0.704387	+	0.406678i	−0.104593	+	0.994515i	$0.533354\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	138.191	+	239.353i	0.237034	+	0.410555i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			554.500i			0.944634i	0.881429	+	0.472317i	$0.156582\pi$
$587$			554.500i			0.944634i	−0.881429	+	0.472317i	$0.843418\pi$
$588$		0			0
$589$	−3.48785			−0.00592165
$590$		0			0
$591$		0			0
$592$		0			0
$593$	707.094	−	408.241i	1.19240	−	0.688433i	0.233550	−	0.972345i	$-0.424966\pi$
$593$	707.094	−	408.241i	1.19240	−	0.688433i	0.958850	+	0.283912i	$0.0916323\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−434.341	+	752.301i	−0.725111	+	1.25593i	0.233818	+	0.972280i	$0.424878\pi$
$599$	−434.341	+	752.301i	−0.725111	+	1.25593i	−0.958928	+	0.283648i	$0.908455\pi$
$600$		0			0
$601$			705.861i			1.17448i	0.809413	+	0.587239i	$0.199785\pi$
$601$			705.861i			1.17448i	−0.809413	+	0.587239i	$0.800215\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	21.8093	+	12.5916i	0.0360485	+	0.0208126i
$606$		0			0
$607$	−464.026	+	267.906i	−0.764458	+	0.441360i	−0.830894	−	0.556431i	$-0.812170\pi$
$607$	−464.026	+	267.906i	−0.764458	+	0.441360i	0.0664361	+	0.997791i	$0.478837\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	292.241	+	506.177i	0.478300	+	0.828440i
$612$		0			0
$613$	142.247	−	246.379i	0.232050	−	0.401923i	−0.726361	−	0.687313i	$-0.758790\pi$
$613$	142.247	−	246.379i	0.232050	−	0.401923i	0.958411	+	0.285390i	$0.0921232\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	766.565			1.24241			0.621203	−	0.783650i	$-0.286644\pi$
$617$	766.565			1.24241			0.621203	+	0.783650i	$0.286644\pi$
$618$		0			0
$619$	−470.257	−	271.503i	−0.759705	−	0.438616i	0.0694849	−	0.997583i	$-0.477864\pi$
$619$	−470.257	−	271.503i	−0.759705	−	0.438616i	−0.829190	+	0.558967i	$0.811198\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−280.508	−	485.854i	−0.448813	−	0.777367i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			26.1920i			0.0416408i
$630$		0			0
$631$	−967.080			−1.53261			−0.766307	−	0.642474i	$-0.777908\pi$
$631$	−967.080			−1.53261			−0.766307	+	0.642474i	$0.777908\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−189.679	+	109.511i	−0.298708	+	0.172459i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	489.512	−	847.860i	0.763669	−	1.32271i	−0.177278	−	0.984161i	$-0.556729\pi$
$641$	489.512	−	847.860i	0.763669	−	1.32271i	0.940947	−	0.338553i	$-0.109937\pi$
$642$		0			0
$643$			991.244i			1.54159i	0.637081	+	0.770797i	$0.280142\pi$
$643$			991.244i			1.54159i	−0.637081	+	0.770797i	$0.719858\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	1067.31	+	616.209i	1.64962	+	0.952409i	0.977222	+	0.212222i	$0.0680698\pi$
$647$	1067.31	+	616.209i	1.64962	+	0.952409i	0.672400	+	0.740188i	$0.265264\pi$
$648$		0			0
$649$	−789.412	+	455.767i	−1.21635	+	0.702260i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−390.535	−	676.426i	−0.598062	−	1.03587i	−0.993107	−	0.117212i	$-0.962604\pi$
$653$	−390.535	−	676.426i	−0.598062	−	1.03587i	0.395045	−	0.918662i	$-0.370729\pi$
$654$		0			0
$655$	105.223	−	182.252i	0.160646	−	0.278247i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	549.328			0.833578			0.416789	−	0.909003i	$-0.363155\pi$
$659$	549.328			0.833578			0.416789	+	0.909003i	$0.363155\pi$
$660$		0			0
$661$	358.214	+	206.815i	0.541927	+	0.312882i	0.745860	−	0.666103i	$-0.232039\pi$
$661$	358.214	+	206.815i	0.541927	+	0.312882i	−0.203932	+	0.978985i	$0.565372\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	261.680	+	453.244i	0.392324	+	0.679526i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	1067.75i		−	1.59128i
$672$		0			0
$673$	553.924			0.823067			0.411533	−	0.911395i	$-0.364993\pi$
$673$	553.924			0.823067			0.411533	+	0.911395i	$0.364993\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−95.5972	+	55.1930i	−0.141207	+	0.0815259i	−0.568939	−	0.822380i	$-0.692646\pi$
$677$	−95.5972	+	55.1930i	−0.141207	+	0.0815259i	0.427732	+	0.903906i	$0.359313\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−97.8156	+	169.422i	−0.143215	+	0.248055i	−0.928705	−	0.370818i	$-0.879077\pi$
$683$	−97.8156	+	169.422i	−0.143215	+	0.248055i	0.785491	+	0.618873i	$0.212411\pi$
$684$		0			0
$685$			235.728i			0.344129i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	394.615	+	227.831i	0.572736	+	0.330669i
$690$		0			0
$691$	−574.132	+	331.475i	−0.830871	+	0.479703i	−0.854151	−	0.520025i	$-0.825922\pi$
$691$	−574.132	+	331.475i	−0.830871	+	0.479703i	0.0232799	+	0.999729i	$0.492589\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	68.7647	+	119.104i	0.0989421	+	0.171373i
$696$		0			0
$697$	294.780	−	510.573i	0.422926	−	0.732530i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	1236.29			1.76361			0.881807	−	0.471610i	$-0.156327\pi$
$701$	1236.29			1.76361			0.881807	+	0.471610i	$0.156327\pi$
$702$		0			0
$703$	−15.5159	−	8.95812i	−0.0220710	−	0.0127427i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	450.250	+	779.856i	0.635050	+	1.09994i	0.986505	+	0.163734i	$0.0523537\pi$
$709$	450.250	+	779.856i	0.635050	+	1.09994i	−0.351455	+	0.936205i	$0.614313\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			5.45848i			0.00765566i
$714$		0			0
$715$	−143.807			−0.201129
$716$		0			0
$717$		0			0
$718$		0			0
$719$	103.145	−	59.5509i	0.143456	−	0.0828246i	−0.426554	−	0.904462i	$-0.640273\pi$
$719$	103.145	−	59.5509i	0.143456	−	0.0828246i	0.570010	+	0.821638i	$0.306939\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	558.950	−	968.129i	0.770965	−	1.33535i
$726$		0			0
$727$		−	815.672i		−	1.12197i	−0.827826	−	0.560985i	$-0.810423\pi$
$727$		−	815.672i		−	1.12197i	0.827826	−	0.560985i	$-0.189577\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−554.696	−	320.254i	−0.758819	−	0.438104i
$732$		0			0
$733$	−193.766	+	111.871i	−0.264346	+	0.152620i	−0.626316	−	0.779570i	$-0.715438\pi$
$733$	−193.766	+	111.871i	−0.264346	+	0.152620i	0.361969	+	0.932190i	$0.382104\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	397.432	+	688.372i	0.539256	+	0.934019i
$738$		0			0
$739$	636.897	−	1103.14i	0.861836	−	1.49274i	−0.00831892	−	0.999965i	$-0.502648\pi$
$739$	636.897	−	1103.14i	0.861836	−	1.49274i	0.870155	−	0.492778i	$-0.164019\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	204.339			0.275019			0.137510	−	0.990500i	$-0.456090\pi$
$743$	204.339			0.275019			0.137510	+	0.990500i	$0.456090\pi$
$744$		0			0
$745$	−197.486	−	114.019i	−0.265082	−	0.153045i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	193.280	+	334.770i	0.257363	+	0.445766i	0.965535	−	0.260275i	$-0.0838131\pi$
$751$	193.280	+	334.770i	0.257363	+	0.445766i	−0.708172	+	0.706040i	$0.750480\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			163.741i			0.216875i
$756$		0			0
$757$	−1254.03			−1.65658			−0.828290	−	0.560300i	$-0.810686\pi$
$757$	−1254.03			−1.65658			−0.828290	+	0.560300i	$0.810686\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	702.009	−	405.305i	0.922483	−	0.532596i	0.0380564	−	0.999276i	$-0.487883\pi$
$761$	702.009	−	405.305i	0.922483	−	0.532596i	0.884426	+	0.466680i	$0.154550\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−751.410	+	1301.48i	−0.979674	+	1.69684i
$768$		0			0
$769$			1338.07i			1.74002i	0.493036	+	0.870009i	$0.335887\pi$
$769$			1338.07i			1.74002i	−0.493036	+	0.870009i	$0.664113\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−446.977	−	258.062i	−0.578236	−	0.333845i	0.182196	−	0.983262i	$-0.441680\pi$
$773$	−446.977	−	258.062i	−0.578236	−	0.333845i	−0.760432	+	0.649417i	$0.775013\pi$
$774$		0			0
$775$	10.0973	−	5.82967i	0.0130287	−	0.00752215i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	201.639	+	349.250i	0.258844	+	0.448331i
$780$		0			0
$781$	−616.995	+	1068.67i	−0.790006	+	1.36833i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−192.913			−0.245749
$786$		0			0
$787$	−1302.69	−	752.107i	−1.65526	−	0.955663i	−0.974860	−	0.222819i	$-0.928474\pi$
$787$	−1302.69	−	752.107i	−1.65526	−	0.955663i	−0.680396	−	0.732844i	$-0.738192\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−880.184	−	1524.52i	−1.10994	−	1.92248i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	147.647i		−	0.185254i	−0.995701	−	0.0926268i	$-0.970474\pi$
$797$		−	147.647i		−	0.185254i	0.995701	−	0.0926268i	$-0.0295263\pi$
$798$		0			0
$799$	386.200			0.483354
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−388.151	+	224.099i	−0.483375	+	0.279077i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	85.5922	−	148.250i	0.105800	−	0.183251i	−0.808265	−	0.588819i	$-0.799593\pi$
$809$	85.5922	−	148.250i	0.105800	−	0.183251i	0.914065	+	0.405568i	$0.132926\pi$
$810$		0			0
$811$		−	668.261i		−	0.823997i	−0.911185	−	0.411998i	$-0.864831\pi$
$811$		−	668.261i		−	0.823997i	0.911185	−	0.411998i	$-0.135169\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−76.4507	−	44.1388i	−0.0938046	−	0.0541581i
$816$		0			0
$817$	379.431	−	219.065i	0.464420	−	0.268133i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−8.32054	−	14.4116i	−0.0101346	−	0.0175537i	0.860914	−	0.508751i	$-0.169893\pi$
$821$	−8.32054	−	14.4116i	−0.0101346	−	0.0175537i	−0.871048	+	0.491197i	$0.836559\pi$
$822$		0			0
$823$	155.347	−	269.070i	0.188758	−	0.326938i	−0.756079	−	0.654481i	$-0.772887\pi$
$823$	155.347	−	269.070i	0.188758	−	0.326938i	0.944836	+	0.327543i	$0.106221\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−139.891			−0.169155			−0.0845777	−	0.996417i	$-0.526954\pi$
$827$	−139.891			−0.169155			−0.0845777	+	0.996417i	$0.526954\pi$
$828$		0			0
$829$	1249.13	+	721.184i	1.50679	+	0.869944i	0.999969	+	0.00789201i	$0.00251213\pi$
$829$	1249.13	+	721.184i	1.50679	+	0.869944i	0.506819	+	0.862052i	$0.330821\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−73.6134	−	127.502i	−0.0881598	−	0.152697i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	657.634i		−	0.783831i	−0.920001	−	0.391915i	$-0.871813\pi$
$839$		−	657.634i		−	0.783831i	0.920001	−	0.391915i	$-0.128187\pi$
$840$		0			0
$841$	1304.06			1.55061
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−69.3592	+	40.0446i	−0.0820819	+	0.0473900i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−14.0195	+	24.2824i	−0.0164741	+	0.0285340i
$852$		0			0
$853$			121.230i			0.142122i	0.997472	+	0.0710609i	$0.0226385\pi$
$853$			121.230i			0.142122i	−0.997472	+	0.0710609i	$0.977362\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	1078.70	+	622.790i	1.25870	+	0.726710i	0.972821	−	0.231557i	$-0.0743820\pi$
$857$	1078.70	+	622.790i	1.25870	+	0.726710i	0.285876	+	0.958266i	$0.407715\pi$
$858$		0			0
$859$	675.130	−	389.787i	0.785949	−	0.453768i	−0.0525855	−	0.998616i	$-0.516746\pi$
$859$	675.130	−	389.787i	0.785949	−	0.453768i	0.838534	+	0.544849i	$0.183413\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−420.375	−	728.111i	−0.487109	−	0.843697i	0.512781	−	0.858519i	$-0.328615\pi$
$863$	−420.375	−	728.111i	−0.487109	−	0.843697i	−0.999890	+	0.0148220i	$0.995282\pi$
$864$		0			0
$865$	100.328	−	173.773i	0.115986	−	0.200894i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−181.202			−0.208518
$870$		0			0
$871$	1134.90	+	655.234i	1.30298	+	0.752278i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	737.122	+	1276.73i	0.840504	+	1.45580i	0.889469	+	0.456995i	$0.151074\pi$
$877$	737.122	+	1276.73i	0.840504	+	1.45580i	−0.0489658	+	0.998800i	$0.515593\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	583.982i		−	0.662863i	−0.943479	−	0.331431i	$-0.892468\pi$
$881$		−	583.982i		−	0.662863i	0.943479	−	0.331431i	$-0.107532\pi$
$882$		0			0
$883$	−1226.32			−1.38882			−0.694408	−	0.719581i	$-0.744334\pi$
$883$	−1226.32			−1.38882			−0.694408	+	0.719581i	$0.744334\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−431.607	+	249.188i	−0.486591	+	0.280934i	−0.723159	−	0.690681i	$-0.757311\pi$
$887$	−431.607	+	249.188i	−0.486591	+	0.280934i	0.236568	+	0.971615i	$0.423977\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−132.087	+	228.781i	−0.147914	+	0.256194i
$894$		0			0
$895$			243.453i			0.272015i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	19.3750	+	11.1862i	0.0215517	+	0.0124429i
$900$		0			0
$901$	260.744	−	150.540i	0.289394	−	0.167081i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	38.9510	+	67.4651i	0.0430398	+	0.0745470i
$906$		0			0
$907$	−36.9123	+	63.9339i	−0.0406971	+	0.0704895i	−0.885656	−	0.464341i	$-0.846291\pi$
$907$	−36.9123	+	63.9339i	−0.0406971	+	0.0704895i	0.844959	+	0.534830i	$0.179625\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−1102.46			−1.21017			−0.605084	−	0.796162i	$-0.706860\pi$
$911$	−1102.46			−1.21017			−0.605084	+	0.796162i	$0.706860\pi$
$912$		0			0
$913$	−500.439	−	288.929i	−0.548126	−	0.316461i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−491.086	−	850.586i	−0.534370	−	0.925556i	−0.999194	−	0.0401527i	$-0.987216\pi$
$919$	−491.086	−	850.586i	−0.534370	−	0.925556i	0.464823	−	0.885403i	$-0.346118\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			2034.44i			2.20416i
$924$		0			0
$925$	59.8912			0.0647472
$926$		0			0
$927$		0			0
$928$		0			0
$929$	561.889	−	324.407i	0.604832	−	0.349200i	−0.166108	−	0.986108i	$-0.553120\pi$
$929$	561.889	−	324.407i	0.604832	−	0.349200i	0.770940	+	0.636908i	$0.219787\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−47.5106	+	82.2908i	−0.0508135	+	0.0880116i
$936$		0			0
$937$		−	1348.17i		−	1.43881i	−0.694590	−	0.719406i	$-0.744414\pi$
$937$		−	1348.17i		−	1.43881i	0.694590	−	0.719406i	$-0.255586\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−1028.20	−	593.631i	−1.09267	−	0.630851i	−0.158381	−	0.987378i	$-0.550627\pi$
$941$	−1028.20	−	593.631i	−1.09267	−	0.630851i	−0.934285	+	0.356527i	$0.883961\pi$
$942$		0			0
$943$	546.575	−	315.565i	0.579613	−	0.334640i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−66.2820	−	114.804i	−0.0699915	−	0.121229i	0.828906	−	0.559388i	$-0.188964\pi$
$947$	−66.2820	−	114.804i	−0.0699915	−	0.121229i	−0.898897	+	0.438159i	$0.855631\pi$
$948$		0			0
$949$	−369.465	+	639.932i	−0.389321	+	0.674323i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−1521.99			−1.59705			−0.798524	−	0.601963i	$-0.794385\pi$
$953$	−1521.99			−1.59705			−0.798524	+	0.601963i	$0.794385\pi$
$954$		0			0
$955$	188.015	+	108.550i	0.196874	+	0.113665i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−480.383	−	832.048i	−0.499879	−	0.865815i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	207.761i		−	0.215296i
$966$		0			0
$967$	645.014			0.667026			0.333513	−	0.942746i	$-0.391766\pi$
$967$	645.014			0.667026			0.333513	+	0.942746i	$0.391766\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−1394.17	+	804.924i	−1.43581	+	0.828964i	−0.997555	−	0.0698871i	$-0.977736\pi$
$971$	−1394.17	+	804.924i	−1.43581	+	0.828964i	−0.438253	+	0.898851i	$0.644403\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−127.572	+	220.960i	−0.130575	+	0.226162i	−0.923898	−	0.382638i	$-0.875016\pi$
$977$	−127.572	+	220.960i	−0.130575	+	0.226162i	0.793324	+	0.608800i	$0.208349\pi$
$978$		0			0
$979$		−	689.217i		−	0.704001i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−417.370	−	240.969i	−0.424588	−	0.245136i	0.272450	−	0.962170i	$-0.412166\pi$
$983$	−417.370	−	240.969i	−0.424588	−	0.245136i	−0.697038	+	0.717034i	$0.745499\pi$
$984$		0			0
$985$	107.009	−	61.7815i	0.108638	−	0.0627224i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−342.836	−	593.810i	−0.346649	−	0.600414i
$990$		0			0
$991$	−563.108	+	975.331i	−0.568222	+	0.984189i	0.428520	+	0.903532i	$0.359035\pi$
$991$	−563.108	+	975.331i	−0.568222	+	0.984189i	−0.996742	+	0.0806568i	$0.974298\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−56.3476			−0.0566307
$996$		0			0
$997$	−918.605	−	530.357i	−0.921369	−	0.531953i	−0.0372976	−	0.999304i	$-0.511875\pi$
$997$	−918.605	−	530.357i	−0.921369	−	0.531953i	−0.884072	+	0.467351i	$0.845208\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.3.z.m.325.3		8
3.2	odd	2		588.3.m.e.325.2		8
7.2	even	3		1764.3.z.l.901.2		8
7.3	odd	6		1764.3.d.h.685.3		8
7.4	even	3		1764.3.d.h.685.6		8
7.5	odd	6	inner	1764.3.z.m.901.3		8
7.6	odd	2		1764.3.z.l.325.2		8
21.2	odd	6		588.3.m.f.313.3		8
21.5	even	6		588.3.m.e.313.2		8
21.11	odd	6		588.3.d.c.97.6	yes	8
21.17	even	6		588.3.d.c.97.3	✓	8
21.20	even	2		588.3.m.f.325.3		8
84.11	even	6		2352.3.f.j.97.2		8
84.59	odd	6		2352.3.f.j.97.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
588.3.d.c.97.3	✓	8	21.17	even	6
588.3.d.c.97.6	yes	8	21.11	odd	6
588.3.m.e.313.2		8	21.5	even	6
588.3.m.e.325.2		8	3.2	odd	2
588.3.m.f.313.3		8	21.2	odd	6
588.3.m.f.325.3		8	21.20	even	2
1764.3.d.h.685.3		8	7.3	odd	6
1764.3.d.h.685.6		8	7.4	even	3
1764.3.z.l.325.2		8	7.6	odd	2
1764.3.z.l.901.2		8	7.2	even	3
1764.3.z.m.325.3		8	1.1	even	1	trivial
1764.3.z.m.901.3		8	7.5	odd	6	inner
2352.3.f.j.97.2		8	84.11	even	6
2352.3.f.j.97.7		8	84.59	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.z (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.804540 - 0.464502i) q^{5} +O(q^{10})\)	\(q+(0.804540 - 0.464502i) q^{5} +(4.84490 - 8.39161i) q^{11} -15.9753i q^{13} +(-9.14154 - 5.27787i) q^{17} +(6.25313 - 3.61025i) q^{19} +(-5.65003 - 9.78614i) q^{23} +(-12.0685 + 20.9032i) q^{25} -46.3148 q^{29} +(-0.418333 - 0.241525i) q^{31} +(-1.24065 - 2.14887i) q^{37} +55.8520i q^{41} +60.6786 q^{43} +(-31.6850 + 18.2933i) q^{47} +(-14.2615 + 24.7016i) q^{53} -9.00185i q^{55} +(-81.4683 - 47.0358i) q^{59} +(95.4301 - 55.0966i) q^{61} +(-7.42055 - 12.8528i) q^{65} +(-41.0155 + 71.0409i) q^{67} -127.349 q^{71} +(-40.0577 - 23.1273i) q^{73} +(-9.35016 - 16.1949i) q^{79} -59.6357i q^{83} -9.80632 q^{85} +(61.5988 - 35.5641i) q^{89} +(3.35393 - 5.80917i) q^{95} -102.239i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 48 q^{17} + 96 q^{19} - 8 q^{23} - 36 q^{25} - 80 q^{29} - 48 q^{31} - 64 q^{37} - 112 q^{43} - 264 q^{47} - 72 q^{53} + 168 q^{59} + 144 q^{61} + 120 q^{65} + 32 q^{67} - 224 q^{71} - 336 q^{73} + 216 q^{79} - 96 q^{85} + 96 q^{89} - 136 q^{95}+O(q^{100}) \) 8 * q + 48 * q^17 + 96 * q^19 - 8 * q^23 - 36 * q^25 - 80 * q^29 - 48 * q^31 - 64 * q^37 - 112 * q^43 - 264 * q^47 - 72 * q^53 + 168 * q^59 + 144 * q^61 + 120 * q^65 + 32 * q^67 - 224 * q^71 - 336 * q^73 + 216 * q^79 - 96 * q^85 + 96 * q^89 - 136 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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