LMFDB - Embedded newform 1764.4.k.f.361.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1764,4,Mod(361,1764)]

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, 4]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1764.361");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1764.k (of order $3$, 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:	$104.079369250$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 84)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			361.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1764.361
Dual form			1764.4.k.f.1549.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-3.00000 - 5.19615i) q^{5} +(18.0000 - 31.1769i) q^{11} -62.0000 q^{13} +(-57.0000 + 98.7269i) q^{17} +(-38.0000 - 65.8179i) q^{19} +(-12.0000 - 20.7846i) q^{23} +(44.5000 - 77.0763i) q^{25} -54.0000 q^{29} +(-56.0000 + 96.9948i) q^{31} +(89.0000 + 154.153i) q^{37} +378.000 q^{41} -172.000 q^{43} +(96.0000 + 166.277i) q^{47} +(-201.000 + 348.142i) q^{53} -216.000 q^{55} +(-198.000 + 342.946i) q^{59} +(127.000 + 219.970i) q^{61} +(186.000 + 322.161i) q^{65} +(506.000 - 876.418i) q^{67} -840.000 q^{71} +(445.000 - 770.763i) q^{73} +(-40.0000 - 69.2820i) q^{79} -108.000 q^{83} +684.000 q^{85} +(819.000 + 1418.55i) q^{89} +(-228.000 + 394.908i) q^{95} -1010.00 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{5} + 36 q^{11} - 124 q^{13} - 114 q^{17} - 76 q^{19} - 24 q^{23} + 89 q^{25} - 108 q^{29} - 112 q^{31} + 178 q^{37} + 756 q^{41} - 344 q^{43} + 192 q^{47} - 402 q^{53} - 432 q^{55} - 396 q^{59}+ \cdots - 2020 q^{97}+O(q^{100}) $ 2 * q - 6 * q^5 + 36 * q^11 - 124 * q^13 - 114 * q^17 - 76 * q^19 - 24 * q^23 + 89 * q^25 - 108 * q^29 - 112 * q^31 + 178 * q^37 + 756 * q^41 - 344 * q^43 + 192 * q^47 - 402 * q^53 - 432 * q^55 - 396 * q^59 + 254 * q^61 + 372 * q^65 + 1012 * q^67 - 1680 * q^71 + 890 * q^73 - 80 * q^79 - 216 * q^83 + 1368 * q^85 + 1638 * q^89 - 456 * q^95 - 2020 * q^97

Character values

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

$n$	$785$	$883$	$1081$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−3.00000	−	5.19615i	−0.268328	−	0.464758i	0.700102	−	0.714043i	$-0.253138\pi$
$5$	−3.00000	−	5.19615i	−0.268328	−	0.464758i	−0.968430	+	0.249285i	$0.919804\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	18.0000	−	31.1769i	0.493382	−	0.854563i	−0.506589	−	0.862188i	$-0.669094\pi$
$11$	18.0000	−	31.1769i	0.493382	−	0.854563i	0.999971	+	0.00762479i	$0.00242707\pi$
$12$		0			0
$13$	−62.0000			−1.32275			−0.661373	−	0.750057i	$-0.730026\pi$
$13$	−62.0000			−1.32275			−0.661373	+	0.750057i	$0.730026\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−57.0000	+	98.7269i	−0.813208	+	1.40852i	0.0974001	+	0.995245i	$0.468947\pi$
$17$	−57.0000	+	98.7269i	−0.813208	+	1.40852i	−0.910608	+	0.413272i	$0.864386\pi$
$18$		0			0
$19$	−38.0000	−	65.8179i	−0.458831	−	0.794719i	0.540068	−	0.841621i	$-0.318398\pi$
$19$	−38.0000	−	65.8179i	−0.458831	−	0.794719i	−0.998899	+	0.0469020i	$0.985065\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−12.0000	−	20.7846i	−0.108790	−	0.188430i	0.806490	−	0.591247i	$-0.201364\pi$
$23$	−12.0000	−	20.7846i	−0.108790	−	0.188430i	−0.915280	+	0.402817i	$0.868031\pi$
$24$		0			0
$25$	44.5000	−	77.0763i	0.356000	−	0.616610i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−54.0000			−0.345778			−0.172889	−	0.984941i	$-0.555310\pi$
$29$	−54.0000			−0.345778			−0.172889	+	0.984941i	$0.555310\pi$
$30$		0			0
$31$	−56.0000	+	96.9948i	−0.324448	+	0.561961i	−0.981401	−	0.191971i	$-0.938512\pi$
$31$	−56.0000	+	96.9948i	−0.324448	+	0.561961i	0.656952	+	0.753932i	$0.271845\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	89.0000	+	154.153i	0.395446	+	0.684933i	0.993158	−	0.116778i	$-0.0372566\pi$
$37$	89.0000	+	154.153i	0.395446	+	0.684933i	−0.597712	+	0.801711i	$0.703923\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	378.000			1.43985			0.719923	−	0.694054i	$-0.244177\pi$
$41$	378.000			1.43985			0.719923	+	0.694054i	$0.244177\pi$
$42$		0			0
$43$	−172.000			−0.609994			−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−172.000			−0.609994			−0.304997	+	0.952353i	$0.598656\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	96.0000	+	166.277i	0.297937	+	0.516042i	0.975664	−	0.219272i	$-0.0703681\pi$
$47$	96.0000	+	166.277i	0.297937	+	0.516042i	−0.677727	+	0.735314i	$0.737035\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−201.000	+	348.142i	−0.520933	+	0.902283i	0.478770	+	0.877940i	$0.341083\pi$
$53$	−201.000	+	348.142i	−0.520933	+	0.902283i	−0.999704	+	0.0243430i	$0.992251\pi$
$54$		0			0
$55$	−216.000			−0.529553
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−198.000	+	342.946i	−0.436905	+	0.756742i	−0.997449	−	0.0713828i	$-0.977259\pi$
$59$	−198.000	+	342.946i	−0.436905	+	0.756742i	0.560544	+	0.828125i	$0.310592\pi$
$60$		0			0
$61$	127.000	+	219.970i	0.266569	+	0.461710i	0.967973	−	0.251053i	$-0.0807768\pi$
$61$	127.000	+	219.970i	0.266569	+	0.461710i	−0.701405	+	0.712763i	$0.747443\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	186.000	+	322.161i	0.354930	+	0.614757i
$66$		0			0
$67$	506.000	−	876.418i	0.922653	−	1.59808i	0.127360	−	0.991857i	$-0.459350\pi$
$67$	506.000	−	876.418i	0.922653	−	1.59808i	0.795293	−	0.606225i	$-0.207317\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−840.000			−1.40408			−0.702040	−	0.712138i	$-0.747727\pi$
$71$	−840.000			−1.40408			−0.702040	+	0.712138i	$0.747727\pi$
$72$		0			0
$73$	445.000	−	770.763i	0.713470	−	1.23577i	−0.250077	−	0.968226i	$-0.580456\pi$
$73$	445.000	−	770.763i	0.713470	−	1.23577i	0.963547	−	0.267540i	$-0.0862108\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−40.0000	−	69.2820i	−0.0569665	−	0.0986688i	0.836136	−	0.548522i	$-0.184809\pi$
$79$	−40.0000	−	69.2820i	−0.0569665	−	0.0986688i	−0.893102	+	0.449854i	$0.851476\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−108.000			−0.142826			−0.0714129	−	0.997447i	$-0.522751\pi$
$83$	−108.000			−0.142826			−0.0714129	+	0.997447i	$0.522751\pi$
$84$		0			0
$85$	684.000			0.872826
$86$		0			0
$87$		0			0
$88$		0			0
$89$	819.000	+	1418.55i	0.975436	+	1.68951i	0.678488	+	0.734612i	$0.262636\pi$
$89$	819.000	+	1418.55i	0.975436	+	1.68951i	0.296948	+	0.954894i	$0.404031\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−228.000	+	394.908i	−0.246235	+	0.426491i
$96$		0			0
$97$	−1010.00			−1.05722			−0.528608	−	0.848866i	$-0.677286\pi$
$97$	−1010.00			−1.05722			−0.528608	+	0.848866i	$0.677286\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−3.00000	+	5.19615i	−0.00295556	+	0.00511917i	−0.867499	−	0.497438i	$-0.834274\pi$
$101$	−3.00000	+	5.19615i	−0.00295556	+	0.00511917i	0.864544	+	0.502557i	$0.167607\pi$
$102$		0			0
$103$	−236.000	−	408.764i	−0.225765	−	0.391036i	0.730784	−	0.682609i	$-0.239155\pi$
$103$	−236.000	−	408.764i	−0.225765	−	0.391036i	−0.956549	+	0.291573i	$0.905821\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−486.000	−	841.777i	−0.439097	−	0.760539i	0.558523	−	0.829489i	$-0.311368\pi$
$107$	−486.000	−	841.777i	−0.439097	−	0.760539i	−0.997620	+	0.0689505i	$0.978035\pi$
$108$		0			0
$109$	893.000	−	1546.72i	0.784715	−	1.35917i	−0.144455	−	0.989511i	$-0.546143\pi$
$109$	893.000	−	1546.72i	0.784715	−	1.35917i	0.929169	−	0.369654i	$-0.120524\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	2286.00			1.90309			0.951543	−	0.307515i	$-0.0994973\pi$
$113$	2286.00			1.90309			0.951543	+	0.307515i	$0.0994973\pi$
$114$		0			0
$115$	−72.0000	+	124.708i	−0.0583829	+	0.101122i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	17.5000	+	30.3109i	0.0131480	+	0.0227730i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1284.00			−0.918756
$126$		0			0
$127$	1328.00			0.927881			0.463941	−	0.885866i	$-0.346435\pi$
$127$	1328.00			0.927881			0.463941	+	0.885866i	$0.346435\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	606.000	+	1049.62i	0.404171	+	0.700046i	0.994225	−	0.107319i	$-0.0342266\pi$
$131$	606.000	+	1049.62i	0.404171	+	0.700046i	−0.590053	+	0.807364i	$0.700893\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−627.000	+	1086.00i	−0.391009	+	0.677247i	−0.992583	−	0.121570i	$-0.961207\pi$
$137$	−627.000	+	1086.00i	−0.391009	+	0.677247i	0.601574	+	0.798817i	$0.294541\pi$
$138$		0			0
$139$	340.000			0.207471			0.103735	−	0.994605i	$-0.466921\pi$
$139$	340.000			0.207471			0.103735	+	0.994605i	$0.466921\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1116.00	+	1932.97i	−0.652620	+	1.13037i
$144$		0			0
$145$	162.000	+	280.592i	0.0927818	+	0.160703i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	519.000	+	898.934i	0.285357	+	0.494252i	0.972696	−	0.232085i	$-0.0745546\pi$
$149$	519.000	+	898.934i	0.285357	+	0.494252i	−0.687339	+	0.726337i	$0.741221\pi$
$150$		0			0
$151$	−1468.00	+	2542.65i	−0.791153	+	1.37032i	0.134100	+	0.990968i	$0.457186\pi$
$151$	−1468.00	+	2542.65i	−0.791153	+	1.37032i	−0.925253	+	0.379350i	$0.876148\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	672.000			0.348234
$156$		0			0
$157$	−665.000	+	1151.81i	−0.338043	+	0.585508i	−0.984065	−	0.177811i	$-0.943098\pi$
$157$	−665.000	+	1151.81i	−0.338043	+	0.585508i	0.646021	+	0.763319i	$0.276432\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	1682.00	+	2913.31i	0.808248	+	1.39993i	0.914076	+	0.405542i	$0.132917\pi$
$163$	1682.00	+	2913.31i	0.808248	+	1.39993i	−0.105828	+	0.994384i	$0.533749\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	3048.00			1.41234			0.706172	−	0.708041i	$-0.250421\pi$
$167$	3048.00			1.41234			0.706172	+	0.708041i	$0.250421\pi$
$168$		0			0
$169$	1647.00			0.749659
$170$		0			0
$171$		0			0
$172$		0			0
$173$	1353.00	+	2343.46i	0.594605	+	1.02989i	0.993602	+	0.112934i	$0.0360249\pi$
$173$	1353.00	+	2343.46i	0.594605	+	1.02989i	−0.398997	+	0.916952i	$0.630642\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	2358.00	−	4084.18i	0.984610	−	1.70539i	0.340953	−	0.940080i	$-0.389250\pi$
$179$	2358.00	−	4084.18i	0.984610	−	1.70539i	0.643657	−	0.765314i	$-0.277416\pi$
$180$		0			0
$181$	−1910.00			−0.784360			−0.392180	−	0.919888i	$-0.628279\pi$
$181$	−1910.00			−0.784360			−0.392180	+	0.919888i	$0.628279\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	534.000	−	924.915i	0.212219	−	0.367574i
$186$		0			0
$187$	2052.00	+	3554.17i	0.802444	+	1.38987i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	2040.00	+	3533.38i	0.772823	+	1.33857i	0.936010	+	0.351974i	$0.114489\pi$
$191$	2040.00	+	3533.38i	0.772823	+	1.33857i	−0.163187	+	0.986595i	$0.552177\pi$
$192$		0			0
$193$	1343.00	−	2326.14i	0.500887	−	0.867562i	−0.499112	−	0.866537i	$-0.666340\pi$
$193$	1343.00	−	2326.14i	0.500887	−	0.867562i	0.999999	−	0.00102491i	$-0.000326238\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−510.000			−0.184447			−0.0922233	−	0.995738i	$-0.529397\pi$
$197$	−510.000			−0.184447			−0.0922233	+	0.995738i	$0.529397\pi$
$198$		0			0
$199$	676.000	−	1170.87i	0.240806	−	0.417088i	−0.720138	−	0.693831i	$-0.755922\pi$
$199$	676.000	−	1170.87i	0.240806	−	0.417088i	0.960944	+	0.276743i	$0.0892550\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−1134.00	−	1964.15i	−0.386351	−	0.669180i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−2736.00			−0.905517
$210$		0			0
$211$	−3364.00			−1.09757			−0.548785	−	0.835963i	$-0.684909\pi$
$211$	−3364.00			−1.09757			−0.548785	+	0.835963i	$0.684909\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	516.000	+	893.738i	0.163679	+	0.283500i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	3534.00	−	6121.07i	1.07567	−	1.86311i
$222$		0			0
$223$	4768.00			1.43179			0.715894	−	0.698209i	$-0.246019\pi$
$223$	4768.00			1.43179			0.715894	+	0.698209i	$0.246019\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−210.000	+	363.731i	−0.0614017	+	0.106351i	−0.895092	−	0.445881i	$-0.852890\pi$
$227$	−210.000	+	363.731i	−0.0614017	+	0.106351i	0.833690	+	0.552232i	$0.186224\pi$
$228$		0			0
$229$	−941.000	−	1629.86i	−0.271542	−	0.470324i	0.697715	−	0.716375i	$-0.254200\pi$
$229$	−941.000	−	1629.86i	−0.271542	−	0.470324i	−0.969257	+	0.246051i	$0.920867\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	2541.00	+	4401.14i	0.714448	+	1.23746i	0.963172	+	0.268886i	$0.0866556\pi$
$233$	2541.00	+	4401.14i	0.714448	+	1.23746i	−0.248724	+	0.968574i	$0.580011\pi$
$234$		0			0
$235$	576.000	−	997.661i	0.159890	−	0.276937i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	5424.00			1.46799			0.733995	−	0.679155i	$-0.237654\pi$
$239$	5424.00			1.46799			0.733995	+	0.679155i	$0.237654\pi$
$240$		0			0
$241$	−1295.00	+	2243.01i	−0.346134	+	0.599522i	−0.985559	−	0.169332i	$-0.945839\pi$
$241$	−1295.00	+	2243.01i	−0.346134	+	0.599522i	0.639425	+	0.768853i	$0.279172\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	2356.00	+	4080.71i	0.606918	+	1.05121i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−4932.00			−1.24026			−0.620130	−	0.784499i	$-0.712920\pi$
$251$	−4932.00			−1.24026			−0.620130	+	0.784499i	$0.712920\pi$
$252$		0			0
$253$	−864.000			−0.214700
$254$		0			0
$255$		0			0
$256$		0			0
$257$	1719.00	+	2977.40i	0.417231	+	0.722665i	0.995660	−	0.0930680i	$-0.0296674\pi$
$257$	1719.00	+	2977.40i	0.417231	+	0.722665i	−0.578429	+	0.815733i	$0.696334\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	3060.00	−	5300.08i	0.717444	−	1.24265i	−0.244566	−	0.969633i	$-0.578645\pi$
$263$	3060.00	−	5300.08i	0.717444	−	1.24265i	0.962009	−	0.273016i	$-0.0880213\pi$
$264$		0			0
$265$	2412.00			0.559124
$266$		0			0
$267$		0			0
$268$		0			0
$269$	9.00000	−	15.5885i	0.00203992	−	0.00353325i	−0.865004	−	0.501766i	$-0.832684\pi$
$269$	9.00000	−	15.5885i	0.00203992	−	0.00353325i	0.867044	+	0.498232i	$0.166017\pi$
$270$		0			0
$271$	3448.00	+	5972.11i	0.772882	+	1.33867i	0.935977	+	0.352061i	$0.114519\pi$
$271$	3448.00	+	5972.11i	0.772882	+	1.33867i	−0.163095	+	0.986610i	$0.552148\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1602.00	−	2774.75i	−0.351288	−	0.608449i
$276$		0			0
$277$	−3127.00	+	5416.12i	−0.678279	+	1.17481i	0.297220	+	0.954809i	$0.403940\pi$
$277$	−3127.00	+	5416.12i	−0.678279	+	1.17481i	−0.975499	+	0.220004i	$0.929393\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−1194.00			−0.253481			−0.126740	−	0.991936i	$-0.540451\pi$
$281$	−1194.00			−0.253481			−0.126740	+	0.991936i	$0.540451\pi$
$282$		0			0
$283$	−3578.00	+	6197.28i	−0.751555	+	1.30173i	0.195514	+	0.980701i	$0.437362\pi$
$283$	−3578.00	+	6197.28i	−0.751555	+	1.30173i	−0.947069	+	0.321030i	$0.895971\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−4041.50	−	7000.08i	−0.822613	−	1.42481i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−3738.00			−0.745312			−0.372656	−	0.927970i	$-0.621553\pi$
$293$	−3738.00			−0.745312			−0.372656	+	0.927970i	$0.621553\pi$
$294$		0			0
$295$	2376.00			0.468936
$296$		0			0
$297$		0			0
$298$		0			0
$299$	744.000	+	1288.65i	0.143902	+	0.249245i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	762.000	−	1319.82i	0.143056	−	0.247780i
$306$		0			0
$307$	844.000			0.156904			0.0784522	−	0.996918i	$-0.475002\pi$
$307$	844.000			0.156904			0.0784522	+	0.996918i	$0.475002\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−3156.00	+	5466.35i	−0.575435	+	0.996683i	0.420559	+	0.907265i	$0.361834\pi$
$311$	−3156.00	+	5466.35i	−0.575435	+	0.996683i	−0.995994	+	0.0894178i	$0.971499\pi$
$312$		0			0
$313$	4141.00	+	7172.42i	0.747806	+	1.29524i	0.948872	+	0.315660i	$0.102226\pi$
$313$	4141.00	+	7172.42i	0.747806	+	1.29524i	−0.201067	+	0.979578i	$0.564441\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	4659.00	+	8069.62i	0.825475	+	1.42976i	0.901556	+	0.432663i	$0.142426\pi$
$317$	4659.00	+	8069.62i	0.825475	+	1.42976i	−0.0760811	+	0.997102i	$0.524241\pi$
$318$		0			0
$319$	−972.000	+	1683.55i	−0.170600	+	0.295489i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	8664.00			1.49250
$324$		0			0
$325$	−2759.00	+	4778.73i	−0.470898	+	0.815619i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−826.000	−	1430.67i	−0.137163	−	0.237574i	0.789258	−	0.614061i	$-0.210465\pi$
$331$	−826.000	−	1430.67i	−0.137163	−	0.237574i	−0.926422	+	0.376487i	$0.877132\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−6072.00			−0.990295
$336$		0			0
$337$	−1294.00			−0.209165			−0.104583	−	0.994516i	$-0.533351\pi$
$337$	−1294.00			−0.209165			−0.104583	+	0.994516i	$0.533351\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	2016.00	+	3491.81i	0.320154	+	0.554523i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	1818.00	−	3148.87i	0.281255	−	0.487147i	−0.690439	−	0.723390i	$-0.742583\pi$
$347$	1818.00	−	3148.87i	0.281255	−	0.487147i	0.971694	+	0.236243i	$0.0759161\pi$
$348$		0			0
$349$	−10478.0			−1.60709			−0.803545	−	0.595244i	$-0.797055\pi$
$349$	−10478.0			−1.60709			−0.803545	+	0.595244i	$0.797055\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	3783.00	−	6552.35i	0.570393	−	0.987950i	−0.426132	−	0.904661i	$-0.640124\pi$
$353$	3783.00	−	6552.35i	0.570393	−	0.987950i	0.996525	−	0.0832890i	$-0.0265425\pi$
$354$		0			0
$355$	2520.00	+	4364.77i	0.376754	+	0.652557i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−4020.00	−	6962.84i	−0.590996	−	1.02363i	−0.994099	−	0.108480i	$-0.965402\pi$
$359$	−4020.00	−	6962.84i	−0.590996	−	1.02363i	0.403103	−	0.915155i	$-0.367932\pi$
$360$		0			0
$361$	541.500	−	937.906i	0.0789474	−	0.136741i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−5340.00			−0.765776
$366$		0			0
$367$	3784.00	−	6554.08i	0.538210	−	0.932208i	−0.460790	−	0.887509i	$-0.652434\pi$
$367$	3784.00	−	6554.08i	0.538210	−	0.932208i	0.999001	−	0.0446985i	$-0.0142327\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	6761.00	+	11710.4i	0.938529	+	1.62558i	0.768217	+	0.640190i	$0.221144\pi$
$373$	6761.00	+	11710.4i	0.938529	+	1.62558i	0.170312	+	0.985390i	$0.445522\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	3348.00			0.457376
$378$		0			0
$379$	2468.00			0.334492			0.167246	−	0.985915i	$-0.446513\pi$
$379$	2468.00			0.334492			0.167246	+	0.985915i	$0.446513\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	6168.00	+	10683.3i	0.822898	+	1.42530i	0.903515	+	0.428557i	$0.140978\pi$
$383$	6168.00	+	10683.3i	0.822898	+	1.42530i	−0.0806166	+	0.996745i	$0.525689\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−1881.00	+	3257.99i	−0.245168	+	0.424644i	−0.962179	−	0.272418i	$-0.912177\pi$
$389$	−1881.00	+	3257.99i	−0.245168	+	0.424644i	0.717011	+	0.697062i	$0.245510\pi$
$390$		0			0
$391$	2736.00			0.353876
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−240.000	+	415.692i	−0.0305714	+	0.0529513i
$396$		0			0
$397$	−4385.00	−	7595.04i	−0.554350	−	0.960162i	−0.997954	−	0.0639390i	$-0.979634\pi$
$397$	−4385.00	−	7595.04i	−0.554350	−	0.960162i	0.443604	−	0.896223i	$-0.353700\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	3321.00	+	5752.14i	0.413573	+	0.716330i	0.995277	−	0.0970706i	$-0.0309473\pi$
$401$	3321.00	+	5752.14i	0.413573	+	0.716330i	−0.581704	+	0.813400i	$0.697614\pi$
$402$		0			0
$403$	3472.00	−	6013.68i	0.429163	−	0.743332i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	6408.00			0.780424
$408$		0			0
$409$	−755.000	+	1307.70i	−0.0912771	+	0.158097i	−0.908049	−	0.418864i	$-0.862428\pi$
$409$	−755.000	+	1307.70i	−0.0912771	+	0.158097i	0.816772	+	0.576961i	$0.195762\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	324.000	+	561.184i	0.0383242	+	0.0663794i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−1260.00			−0.146909			−0.0734547	−	0.997299i	$-0.523402\pi$
$419$	−1260.00			−0.146909			−0.0734547	+	0.997299i	$0.523402\pi$
$420$		0			0
$421$	3998.00			0.462828			0.231414	−	0.972855i	$-0.425665\pi$
$421$	3998.00			0.462828			0.231414	+	0.972855i	$0.425665\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	5073.00	+	8786.69i	0.579004	+	1.00286i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−1368.00	+	2369.45i	−0.152887	+	0.264808i	−0.932288	−	0.361718i	$-0.882190\pi$
$431$	−1368.00	+	2369.45i	−0.152887	+	0.264808i	0.779401	+	0.626526i	$0.215524\pi$
$432$		0			0
$433$	−2690.00			−0.298552			−0.149276	−	0.988796i	$-0.547694\pi$
$433$	−2690.00			−0.298552			−0.149276	+	0.988796i	$0.547694\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−912.000	+	1579.63i	−0.0998327	+	0.172915i
$438$		0			0
$439$	−620.000	−	1073.87i	−0.0674054	−	0.116750i	0.830353	−	0.557238i	$-0.188139\pi$
$439$	−620.000	−	1073.87i	−0.0674054	−	0.116750i	−0.897758	+	0.440488i	$0.854805\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−1950.00	−	3377.50i	−0.209136	−	0.362234i	0.742307	−	0.670060i	$-0.233732\pi$
$443$	−1950.00	−	3377.50i	−0.209136	−	0.362234i	−0.951443	+	0.307826i	$0.900399\pi$
$444$		0			0
$445$	4914.00	−	8511.30i	0.523474	−	0.906684i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	10878.0			1.14335			0.571675	−	0.820480i	$-0.306294\pi$
$449$	10878.0			1.14335			0.571675	+	0.820480i	$0.306294\pi$
$450$		0			0
$451$	6804.00	−	11784.9i	0.710394	−	1.23044i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−1165.00	−	2017.84i	−0.119248	−	0.206544i	0.800222	−	0.599704i	$-0.204715\pi$
$457$	−1165.00	−	2017.84i	−0.119248	−	0.206544i	−0.919470	+	0.393160i	$0.871382\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	15150.0			1.53060			0.765299	−	0.643675i	$-0.222591\pi$
$461$	15150.0			1.53060			0.765299	+	0.643675i	$0.222591\pi$
$462$		0			0
$463$	−2992.00			−0.300324			−0.150162	−	0.988661i	$-0.547980\pi$
$463$	−2992.00			−0.300324			−0.150162	+	0.988661i	$0.547980\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−4362.00	−	7555.21i	−0.432225	−	0.748636i	0.564839	−	0.825201i	$-0.308938\pi$
$467$	−4362.00	−	7555.21i	−0.432225	−	0.748636i	−0.997065	+	0.0765646i	$0.975605\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−3096.00	+	5362.43i	−0.300960	+	0.521279i
$474$		0			0
$475$	−6764.00			−0.653376
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−4872.00	+	8438.55i	−0.464734	+	0.804942i	−0.999189	−	0.0402542i	$-0.987183\pi$
$479$	−4872.00	+	8438.55i	−0.464734	+	0.804942i	0.534456	+	0.845196i	$0.320517\pi$
$480$		0			0
$481$	−5518.00	−	9557.46i	−0.523075	−	0.905993i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	3030.00	+	5248.11i	0.283681	+	0.491350i
$486$		0			0
$487$	−2068.00	+	3581.88i	−0.192423	+	0.333286i	−0.946053	−	0.324013i	$-0.894968\pi$
$487$	−2068.00	+	3581.88i	−0.192423	+	0.333286i	0.753630	+	0.657299i	$0.228301\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−16212.0			−1.49010			−0.745048	−	0.667011i	$-0.767574\pi$
$491$	−16212.0			−1.49010			−0.745048	+	0.667011i	$0.767574\pi$
$492$		0			0
$493$	3078.00	−	5331.25i	0.281189	−	0.487034i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−1198.00	−	2075.00i	−0.107475	−	0.186152i	0.807272	−	0.590180i	$-0.200943\pi$
$499$	−1198.00	−	2075.00i	−0.107475	−	0.186152i	−0.914747	+	0.404028i	$0.867610\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	13128.0			1.16371			0.581857	−	0.813291i	$-0.302326\pi$
$503$	13128.0			1.16371			0.581857	+	0.813291i	$0.302326\pi$
$504$		0			0
$505$	36.0000			0.00317224
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−6399.00	−	11083.4i	−0.557231	−	0.965153i	−0.997726	−	0.0673977i	$-0.978530\pi$
$509$	−6399.00	−	11083.4i	−0.557231	−	0.965153i	0.440495	−	0.897755i	$-0.354803\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−1416.00	+	2452.58i	−0.121158	+	0.209852i
$516$		0			0
$517$	6912.00			0.587987
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−3693.00	+	6396.46i	−0.310544	+	0.537877i	−0.978480	−	0.206341i	$-0.933844\pi$
$521$	−3693.00	+	6396.46i	−0.310544	+	0.537877i	0.667936	+	0.744218i	$0.267178\pi$
$522$		0			0
$523$	2590.00	+	4486.01i	0.216545	+	0.375066i	0.953749	−	0.300603i	$-0.0971880\pi$
$523$	2590.00	+	4486.01i	0.216545	+	0.375066i	−0.737205	+	0.675669i	$0.763855\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−6384.00	−	11057.4i	−0.527688	−	0.913982i
$528$		0			0
$529$	5795.50	−	10038.1i	0.476329	−	0.825027i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−23436.0			−1.90455
$534$		0			0
$535$	−2916.00	+	5050.66i	−0.235644	+	0.408148i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−2035.00	−	3524.72i	−0.161722	−	0.280110i	0.773764	−	0.633473i	$-0.218371\pi$
$541$	−2035.00	−	3524.72i	−0.161722	−	0.280110i	−0.935486	+	0.353363i	$0.885038\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−10716.0			−0.842244
$546$		0			0
$547$	14780.0			1.15530			0.577648	−	0.816286i	$-0.303971\pi$
$547$	14780.0			1.15530			0.577648	+	0.816286i	$0.303971\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	2052.00	+	3554.17i	0.158654	+	0.274796i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−3429.00	+	5939.20i	−0.260846	+	0.451799i	−0.966467	−	0.256791i	$-0.917335\pi$
$557$	−3429.00	+	5939.20i	−0.260846	+	0.451799i	0.705621	+	0.708590i	$0.250668\pi$
$558$		0			0
$559$	10664.0			0.806868
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−3330.00	+	5767.73i	−0.249277	+	0.431760i	−0.963325	−	0.268336i	$-0.913526\pi$
$563$	−3330.00	+	5767.73i	−0.249277	+	0.431760i	0.714049	+	0.700096i	$0.246860\pi$
$564$		0			0
$565$	−6858.00	−	11878.4i	−0.510652	−	0.884475i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−75.0000	−	129.904i	−0.00552577	−	0.00957092i	0.863249	−	0.504778i	$-0.168426\pi$
$569$	−75.0000	−	129.904i	−0.00552577	−	0.00957092i	−0.868775	+	0.495207i	$0.835092\pi$
$570$		0			0
$571$	4094.00	−	7091.02i	0.300050	−	0.519702i	−0.676097	−	0.736813i	$-0.736330\pi$
$571$	4094.00	−	7091.02i	0.300050	−	0.519702i	0.976147	+	0.217111i	$0.0696633\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−2136.00			−0.154917
$576$		0			0
$577$	−2927.00	+	5069.71i	−0.211183	+	0.365780i	−0.952085	−	0.305833i	$-0.901065\pi$
$577$	−2927.00	+	5069.71i	−0.211183	+	0.365780i	0.740902	+	0.671613i	$0.234398\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	7236.00	+	12533.1i	0.514039	+	0.890341i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	17580.0			1.23612			0.618062	−	0.786130i	$-0.287918\pi$
$587$	17580.0			1.23612			0.618062	+	0.786130i	$0.287918\pi$
$588$		0			0
$589$	8512.00			0.595468
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−8577.00	−	14855.8i	−0.593955	−	1.02876i	−0.993693	−	0.112131i	$-0.964232\pi$
$593$	−8577.00	−	14855.8i	−0.593955	−	1.02876i	0.399739	−	0.916629i	$-0.369101\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−9060.00	+	15692.4i	−0.617999	+	1.07041i	0.371851	+	0.928292i	$0.378723\pi$
$599$	−9060.00	+	15692.4i	−0.617999	+	1.07041i	−0.989850	+	0.142114i	$0.954610\pi$
$600$		0			0
$601$	−17546.0			−1.19088			−0.595438	−	0.803401i	$-0.703021\pi$
$601$	−17546.0			−1.19088			−0.595438	+	0.803401i	$0.703021\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	105.000	−	181.865i	0.00705596	−	0.0122213i
$606$		0			0
$607$	−7280.00	−	12609.3i	−0.486798	−	0.843158i	0.513087	−	0.858336i	$-0.328502\pi$
$607$	−7280.00	−	12609.3i	−0.486798	−	0.843158i	−0.999885	+	0.0151784i	$0.995168\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−5952.00	−	10309.2i	−0.394095	−	0.682593i
$612$		0			0
$613$	2249.00	−	3895.38i	0.148183	−	0.256661i	−0.782373	−	0.622810i	$-0.785991\pi$
$613$	2249.00	−	3895.38i	0.148183	−	0.256661i	0.930556	+	0.366150i	$0.119324\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	5478.00			0.357433			0.178716	−	0.983901i	$-0.442806\pi$
$617$	5478.00			0.357433			0.178716	+	0.983901i	$0.442806\pi$
$618$		0			0
$619$	3022.00	−	5234.26i	0.196227	−	0.339875i	−0.751075	−	0.660217i	$-0.770464\pi$
$619$	3022.00	−	5234.26i	0.196227	−	0.339875i	0.947302	+	0.320342i	$0.103798\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−1710.50	−	2962.67i	−0.109472	−	0.189611i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−20292.0			−1.28632
$630$		0			0
$631$	−15352.0			−0.968547			−0.484274	−	0.874917i	$-0.660916\pi$
$631$	−15352.0			−0.968547			−0.484274	+	0.874917i	$0.660916\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−3984.00	−	6900.49i	−0.248977	−	0.431240i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−11199.0	+	19397.2i	−0.690068	+	1.19523i	0.281746	+	0.959489i	$0.409086\pi$
$641$	−11199.0	+	19397.2i	−0.690068	+	1.19523i	−0.971815	+	0.235745i	$0.924247\pi$
$642$		0			0
$643$	−3764.00			−0.230852			−0.115426	−	0.993316i	$-0.536823\pi$
$643$	−3764.00			−0.230852			−0.115426	+	0.993316i	$0.536823\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	8844.00	−	15318.3i	0.537393	−	0.930793i	−0.461650	−	0.887062i	$-0.652742\pi$
$647$	8844.00	−	15318.3i	0.537393	−	0.930793i	0.999043	−	0.0437305i	$-0.0139243\pi$
$648$		0			0
$649$	7128.00	+	12346.1i	0.431122	+	0.746726i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	9939.00	+	17214.9i	0.595625	+	1.03165i	0.993458	+	0.114195i	$0.0364289\pi$
$653$	9939.00	+	17214.9i	0.595625	+	1.03165i	−0.397833	+	0.917458i	$0.630238\pi$
$654$		0			0
$655$	3636.00	−	6297.74i	0.216901	−	0.375684i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	20004.0			1.18247			0.591233	−	0.806501i	$-0.298641\pi$
$659$	20004.0			1.18247			0.591233	+	0.806501i	$0.298641\pi$
$660$		0			0
$661$	−653.000	+	1131.03i	−0.0384247	+	0.0665536i	−0.884598	−	0.466354i	$-0.845567\pi$
$661$	−653.000	+	1131.03i	−0.0384247	+	0.0665536i	0.846173	+	0.532908i	$0.178901\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	648.000	+	1122.37i	0.0376172	+	0.0651549i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	9144.00			0.526081
$672$		0			0
$673$	−13054.0			−0.747689			−0.373845	−	0.927491i	$-0.621961\pi$
$673$	−13054.0			−0.747689			−0.373845	+	0.927491i	$0.621961\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−2523.00	−	4369.96i	−0.143230	−	0.248082i	0.785481	−	0.618886i	$-0.212416\pi$
$677$	−2523.00	−	4369.96i	−0.143230	−	0.248082i	−0.928711	+	0.370804i	$0.879082\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	6234.00	−	10797.6i	0.349249	−	0.604918i	−0.636867	−	0.770974i	$-0.719770\pi$
$683$	6234.00	−	10797.6i	0.349249	−	0.604918i	0.986116	+	0.166056i	$0.0531032\pi$
$684$		0			0
$685$	7524.00			0.419675
$686$		0			0
$687$		0			0
$688$		0			0
$689$	12462.0	−	21584.8i	0.689063	−	1.19349i
$690$		0			0
$691$	−11606.0	−	20102.2i	−0.638948	−	1.10669i	−0.985664	−	0.168720i	$-0.946037\pi$
$691$	−11606.0	−	20102.2i	−0.638948	−	1.10669i	0.346716	−	0.937970i	$-0.387297\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−1020.00	−	1766.69i	−0.0556702	−	0.0964237i
$696$		0			0
$697$	−21546.0	+	37318.8i	−1.17089	+	2.02805i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−35958.0			−1.93740			−0.968698	−	0.248241i	$-0.920147\pi$
$701$	−35958.0			−1.93740			−0.968698	+	0.248241i	$0.920147\pi$
$702$		0			0
$703$	6764.00	−	11715.6i	0.362886	−	0.628538i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−3223.00	−	5582.40i	−0.170723	−	0.295700i	0.767950	−	0.640510i	$-0.221277\pi$
$709$	−3223.00	−	5582.40i	−0.170723	−	0.295700i	−0.938673	+	0.344809i	$0.887944\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	2688.00			0.141187
$714$		0			0
$715$	13392.0			0.700465
$716$		0			0
$717$		0			0
$718$		0			0
$719$	2352.00	+	4073.78i	0.121996	+	0.211302i	0.920555	−	0.390614i	$-0.127737\pi$
$719$	2352.00	+	4073.78i	0.121996	+	0.211302i	−0.798559	+	0.601917i	$0.794404\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−2403.00	+	4162.12i	−0.123097	+	0.213210i
$726$		0			0
$727$	10600.0			0.540760			0.270380	−	0.962754i	$-0.412851\pi$
$727$	10600.0			0.540760			0.270380	+	0.962754i	$0.412851\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	9804.00	−	16981.0i	0.496052	−	0.859187i
$732$		0			0
$733$	6271.00	+	10861.7i	0.315995	+	0.547320i	0.979649	−	0.200720i	$-0.0643282\pi$
$733$	6271.00	+	10861.7i	0.315995	+	0.547320i	−0.663653	+	0.748040i	$0.730995\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−18216.0	−	31551.0i	−0.910441	−	1.57693i
$738$		0			0
$739$	−11662.0	+	20199.2i	−0.580506	+	1.00547i	0.414914	+	0.909861i	$0.363812\pi$
$739$	−11662.0	+	20199.2i	−0.580506	+	1.00547i	−0.995419	+	0.0956044i	$0.969522\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	6312.00			0.311662			0.155831	−	0.987784i	$-0.450194\pi$
$743$	6312.00			0.311662			0.155831	+	0.987784i	$0.450194\pi$
$744$		0			0
$745$	3114.00	−	5393.61i	0.153138	−	0.265244i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−17920.0	−	31038.4i	−0.870719	−	1.50813i	−0.861254	−	0.508175i	$-0.830320\pi$
$751$	−17920.0	−	31038.4i	−0.870719	−	1.50813i	−0.00946509	−	0.999955i	$-0.503013\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	17616.0			0.849155
$756$		0			0
$757$	−34594.0			−1.66095			−0.830476	−	0.557055i	$-0.811931\pi$
$757$	−34594.0			−1.66095			−0.830476	+	0.557055i	$0.811931\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−11973.0	−	20737.8i	−0.570330	−	0.987840i	−0.996532	−	0.0832121i	$-0.973482\pi$
$761$	−11973.0	−	20737.8i	−0.570330	−	0.987840i	0.426202	−	0.904628i	$-0.359851\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	12276.0	−	21262.7i	0.577915	−	1.00098i
$768$		0			0
$769$	−18770.0			−0.880187			−0.440093	−	0.897952i	$-0.645055\pi$
$769$	−18770.0			−0.880187			−0.440093	+	0.897952i	$0.645055\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−15171.0	+	26276.9i	−0.705903	+	1.22266i	0.260462	+	0.965484i	$0.416125\pi$
$773$	−15171.0	+	26276.9i	−0.705903	+	1.22266i	−0.966365	+	0.257176i	$0.917208\pi$
$774$		0			0
$775$	4984.00	+	8632.54i	0.231007	+	0.400116i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−14364.0	−	24879.2i	−0.660647	−	1.14427i
$780$		0			0
$781$	−15120.0	+	26188.6i	−0.692748	+	1.19987i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	7980.00			0.362826
$786$		0			0
$787$	−13094.0	+	22679.5i	−0.593076	+	1.02724i	0.400739	+	0.916192i	$0.368753\pi$
$787$	−13094.0	+	22679.5i	−0.593076	+	1.02724i	−0.993815	+	0.111045i	$0.964580\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−7874.00	−	13638.2i	−0.352603	−	0.610726i
$794$		0			0
$795$		0			0
$796$		0			0

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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1764.k (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-3.00000 - 5.19615i) q^{5} +(18.0000 - 31.1769i) q^{11} -62.0000 q^{13} +(-57.0000 + 98.7269i) q^{17} +(-38.0000 - 65.8179i) q^{19} +(-12.0000 - 20.7846i) q^{23} +(44.5000 - 77.0763i) q^{25} -54.0000 q^{29} +(-56.0000 + 96.9948i) q^{31} +(89.0000 + 154.153i) q^{37} +378.000 q^{41} -172.000 q^{43} +(96.0000 + 166.277i) q^{47} +(-201.000 + 348.142i) q^{53} -216.000 q^{55} +(-198.000 + 342.946i) q^{59} +(127.000 + 219.970i) q^{61} +(186.000 + 322.161i) q^{65} +(506.000 - 876.418i) q^{67} -840.000 q^{71} +(445.000 - 770.763i) q^{73} +(-40.0000 - 69.2820i) q^{79} -108.000 q^{83} +684.000 q^{85} +(819.000 + 1418.55i) q^{89} +(-228.000 + 394.908i) q^{95} -1010.00 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{5} + 36 q^{11} - 124 q^{13} - 114 q^{17} - 76 q^{19} - 24 q^{23} + 89 q^{25} - 108 q^{29} - 112 q^{31} + 178 q^{37} + 756 q^{41} - 344 q^{43} + 192 q^{47} - 402 q^{53} - 432 q^{55} - 396 q^{59}+ \cdots - 2020 q^{97}+O(q^{100}) \) 2 * q - 6 * q^5 + 36 * q^11 - 124 * q^13 - 114 * q^17 - 76 * q^19 - 24 * q^23 + 89 * q^25 - 108 * q^29 - 112 * q^31 + 178 * q^37 + 756 * q^41 - 344 * q^43 + 192 * q^47 - 402 * q^53 - 432 * q^55 - 396 * q^59 + 254 * q^61 + 372 * q^65 + 1012 * q^67 - 1680 * q^71 + 890 * q^73 - 80 * q^79 - 216 * q^83 + 1368 * q^85 + 1638 * q^89 - 456 * q^95 - 2020 * q^97

Introduction

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