LMFDB - Embedded newform 1764.4.k.l.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.l.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	0.968430	−	0.249285i	$-0.0801955\pi$
$5$	3.00000	+	5.19615i	0.268328	+	0.464758i	−0.700102	+	0.714043i	$0.746862\pi$
$6$		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	$-0.269974\pi$
$13$	62.0000			1.32275			0.661373	+	0.750057i	$0.269974\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	57.0000	−	98.7269i	0.813208	−	1.40852i	−0.0974001	−	0.995245i	$-0.531053\pi$
$17$	57.0000	−	98.7269i	0.813208	−	1.40852i	0.910608	−	0.413272i	$-0.135614\pi$
$18$		0			0
$19$	38.0000	+	65.8179i	0.458831	+	0.794719i	0.998899	−	0.0469020i	$-0.0149348\pi$
$19$	38.0000	+	65.8179i	0.458831	+	0.794719i	−0.540068	+	0.841621i	$0.681602\pi$
$20$		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	−0.656952	−	0.753932i	$-0.728155\pi$
$31$	56.0000	−	96.9948i	0.324448	−	0.561961i	0.981401	+	0.191971i	$0.0614880\pi$
$32$		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	$-0.755823\pi$
$41$	−378.000			−1.43985			−0.719923	+	0.694054i	$0.755823\pi$
$42$		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	0.677727	−	0.735314i	$-0.262965\pi$
$47$	−96.0000	−	166.277i	−0.297937	−	0.516042i	−0.975664	+	0.219272i	$0.929632\pi$
$48$		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	−0.560544	−	0.828125i	$-0.689408\pi$
$59$	198.000	−	342.946i	0.436905	−	0.756742i	0.997449	+	0.0713828i	$0.0227412\pi$
$60$		0			0
$61$	−127.000	−	219.970i	−0.266569	−	0.461710i	0.701405	−	0.712763i	$-0.252557\pi$
$61$	−127.000	−	219.970i	−0.266569	−	0.461710i	−0.967973	+	0.251053i	$0.919223\pi$
$62$		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	$0.419544\pi$
$73$	−445.000	+	770.763i	−0.713470	+	1.23577i	−0.963547	+	0.267540i	$0.913789\pi$
$74$		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	$-0.477249\pi$
$83$	108.000			0.142826			0.0714129	+	0.997447i	$0.477249\pi$
$84$		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	$-0.737364\pi$
$89$	−819.000	−	1418.55i	−0.975436	−	1.68951i	−0.296948	−	0.954894i	$-0.595969\pi$
$90$		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	$-0.322714\pi$
$97$	1010.00			1.05722			0.528608	+	0.848866i	$0.322714\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	3.00000	−	5.19615i	0.00295556	−	0.00511917i	−0.864544	−	0.502557i	$-0.832393\pi$
$101$	3.00000	−	5.19615i	0.00295556	−	0.00511917i	0.867499	+	0.497438i	$0.165726\pi$
$102$		0			0
$103$	236.000	+	408.764i	0.225765	+	0.391036i	0.956549	−	0.291573i	$-0.0941786\pi$
$103$	236.000	+	408.764i	0.225765	+	0.391036i	−0.730784	+	0.682609i	$0.760845\pi$
$104$		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	0.590053	−	0.807364i	$-0.299107\pi$
$131$	−606.000	−	1049.62i	−0.404171	−	0.700046i	−0.994225	+	0.107319i	$0.965773\pi$
$132$		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	$-0.533079\pi$
$139$	−340.000			−0.207471			−0.103735	+	0.994605i	$0.533079\pi$
$140$		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	−0.646021	−	0.763319i	$-0.723568\pi$
$157$	665.000	−	1151.81i	0.338043	−	0.585508i	0.984065	+	0.177811i	$0.0569017\pi$
$158$		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	$-0.749579\pi$
$167$	−3048.00			−1.41234			−0.706172	+	0.708041i	$0.749579\pi$
$168$		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	$-0.963975\pi$
$173$	−1353.00	−	2343.46i	−0.594605	−	1.02989i	0.398997	−	0.916952i	$-0.369358\pi$
$174$		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	$-0.371721\pi$
$181$	1910.00			0.784360			0.392180	+	0.919888i	$0.371721\pi$
$182$		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	−0.960944	−	0.276743i	$-0.910745\pi$
$199$	−676.000	+	1170.87i	−0.240806	+	0.417088i	0.720138	+	0.693831i	$0.244078\pi$
$200$		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	$-0.753981\pi$
$223$	−4768.00			−1.43179			−0.715894	+	0.698209i	$0.753981\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	210.000	−	363.731i	0.0614017	−	0.106351i	−0.833690	−	0.552232i	$-0.813776\pi$
$227$	210.000	−	363.731i	0.0614017	−	0.106351i	0.895092	+	0.445881i	$0.147110\pi$
$228$		0			0
$229$	941.000	+	1629.86i	0.271542	+	0.470324i	0.969257	−	0.246051i	$-0.0791332\pi$
$229$	941.000	+	1629.86i	0.271542	+	0.470324i	−0.697715	+	0.716375i	$0.745800\pi$
$230$		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	−0.639425	−	0.768853i	$-0.720828\pi$
$241$	1295.00	−	2243.01i	0.346134	−	0.599522i	0.985559	+	0.169332i	$0.0541609\pi$
$242$		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	$-0.287080\pi$
$251$	4932.00			1.24026			0.620130	+	0.784499i	$0.287080\pi$
$252$		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	0.578429	−	0.815733i	$-0.303666\pi$
$257$	−1719.00	−	2977.40i	−0.417231	−	0.722665i	−0.995660	+	0.0930680i	$0.970333\pi$
$258$		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	−0.867044	−	0.498232i	$-0.833983\pi$
$269$	−9.00000	+	15.5885i	−0.00203992	+	0.00353325i	0.865004	+	0.501766i	$0.167316\pi$
$270$		0			0
$271$	−3448.00	−	5972.11i	−0.772882	−	1.33867i	−0.935977	−	0.352061i	$-0.885481\pi$
$271$	−3448.00	−	5972.11i	−0.772882	−	1.33867i	0.163095	−	0.986610i	$-0.447852\pi$
$272$		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	$-0.562638\pi$
$283$	3578.00	−	6197.28i	0.751555	−	1.30173i	0.947069	−	0.321030i	$-0.104029\pi$
$284$		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	$-0.378447\pi$
$293$	3738.00			0.745312			0.372656	+	0.927970i	$0.378447\pi$
$294$		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	$-0.524998\pi$
$307$	−844.000			−0.156904			−0.0784522	+	0.996918i	$0.524998\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	3156.00	−	5466.35i	0.575435	−	0.996683i	−0.420559	−	0.907265i	$-0.638166\pi$
$311$	3156.00	−	5466.35i	0.575435	−	0.996683i	0.995994	−	0.0894178i	$-0.0285006\pi$
$312$		0			0
$313$	−4141.00	−	7172.42i	−0.747806	−	1.29524i	−0.948872	−	0.315660i	$-0.897774\pi$
$313$	−4141.00	−	7172.42i	−0.747806	−	1.29524i	0.201067	−	0.979578i	$-0.435559\pi$
$314$		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	$-0.202945\pi$
$349$	10478.0			1.60709			0.803545	+	0.595244i	$0.202945\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−3783.00	+	6552.35i	−0.570393	+	0.987950i	0.426132	+	0.904661i	$0.359876\pi$
$353$	−3783.00	+	6552.35i	−0.570393	+	0.987950i	−0.996525	+	0.0832890i	$0.973458\pi$
$354$		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	$0.347566\pi$
$367$	−3784.00	+	6554.08i	−0.538210	+	0.932208i	−0.999001	+	0.0446985i	$0.985767\pi$
$368$		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	$-0.859022\pi$
$383$	−6168.00	−	10683.3i	−0.822898	−	1.42530i	0.0806166	−	0.996745i	$-0.474311\pi$
$384$		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	$0.0203663\pi$
$397$	4385.00	+	7595.04i	0.554350	+	0.960162i	−0.443604	+	0.896223i	$0.646300\pi$
$398$		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	−0.816772	−	0.576961i	$-0.804238\pi$
$409$	755.000	−	1307.70i	0.0912771	−	0.158097i	0.908049	+	0.418864i	$0.137572\pi$
$410$		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	$-0.476598\pi$
$419$	1260.00			0.146909			0.0734547	+	0.997299i	$0.476598\pi$
$420$		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	$-0.452306\pi$
$433$	2690.00			0.298552			0.149276	+	0.988796i	$0.452306\pi$
$434$		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	0.897758	−	0.440488i	$-0.145195\pi$
$439$	620.000	+	1073.87i	0.0674054	+	0.116750i	−0.830353	+	0.557238i	$0.811861\pi$
$440$		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	$-0.777409\pi$
$461$	−15150.0			−1.53060			−0.765299	+	0.643675i	$0.777409\pi$
$462$		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	0.997065	−	0.0765646i	$-0.0243951\pi$
$467$	4362.00	+	7555.21i	0.432225	+	0.748636i	−0.564839	+	0.825201i	$0.691062\pi$
$468$		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	−0.534456	−	0.845196i	$-0.679483\pi$
$479$	4872.00	−	8438.55i	0.464734	−	0.804942i	0.999189	+	0.0402542i	$0.0128168\pi$
$480$		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	$-0.697674\pi$
$503$	−13128.0			−1.16371			−0.581857	+	0.813291i	$0.697674\pi$
$504$		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	$0.0214696\pi$
$509$	6399.00	+	11083.4i	0.557231	+	0.965153i	−0.440495	+	0.897755i	$0.645197\pi$
$510$		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	−0.667936	−	0.744218i	$-0.732822\pi$
$521$	3693.00	−	6396.46i	0.310544	−	0.537877i	0.978480	+	0.206341i	$0.0661556\pi$
$522$		0			0
$523$	−2590.00	−	4486.01i	−0.216545	−	0.375066i	0.737205	−	0.675669i	$-0.236145\pi$
$523$	−2590.00	−	4486.01i	−0.216545	−	0.375066i	−0.953749	+	0.300603i	$0.902812\pi$
$524$		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	−0.714049	−	0.700096i	$-0.753140\pi$
$563$	3330.00	−	5767.73i	0.249277	−	0.431760i	0.963325	+	0.268336i	$0.0864738\pi$
$564$		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	−0.740902	−	0.671613i	$-0.765602\pi$
$577$	2927.00	−	5069.71i	0.211183	−	0.365780i	0.952085	+	0.305833i	$0.0989351\pi$
$578$		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	$-0.712082\pi$
$587$	−17580.0			−1.23612			−0.618062	+	0.786130i	$0.712082\pi$
$588$		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	$0.0357675\pi$
$593$	8577.00	+	14855.8i	0.593955	+	1.02876i	−0.399739	+	0.916629i	$0.630899\pi$
$594$		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	$-0.296979\pi$
$601$	17546.0			1.19088			0.595438	+	0.803401i	$0.296979\pi$
$602$		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	0.999885	−	0.0151784i	$-0.00483163\pi$
$607$	7280.00	+	12609.3i	0.486798	+	0.843158i	−0.513087	+	0.858336i	$0.671498\pi$
$608$		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	−0.947302	−	0.320342i	$-0.896202\pi$
$619$	−3022.00	+	5234.26i	−0.196227	+	0.339875i	0.751075	+	0.660217i	$0.229536\pi$
$620$		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	$-0.463177\pi$
$643$	3764.00			0.230852			0.115426	+	0.993316i	$0.463177\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−8844.00	+	15318.3i	−0.537393	+	0.930793i	0.461650	+	0.887062i	$0.347258\pi$
$647$	−8844.00	+	15318.3i	−0.537393	+	0.930793i	−0.999043	+	0.0437305i	$0.986076\pi$
$648$		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	−0.846173	−	0.532908i	$-0.821099\pi$
$661$	653.000	−	1131.03i	0.0384247	−	0.0665536i	0.884598	+	0.466354i	$0.154433\pi$
$662$		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	0.928711	−	0.370804i	$-0.120918\pi$
$677$	2523.00	+	4369.96i	0.143230	+	0.248082i	−0.785481	+	0.618886i	$0.787584\pi$
$678$		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	$0.0539634\pi$
$691$	11606.0	+	20102.2i	0.638948	+	1.10669i	−0.346716	+	0.937970i	$0.612703\pi$
$692$		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	0.798559	−	0.601917i	$-0.205596\pi$
$719$	−2352.00	−	4073.78i	−0.121996	−	0.211302i	−0.920555	+	0.390614i	$0.872263\pi$
$720$		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	$-0.587149\pi$
$727$	−10600.0			−0.540760			−0.270380	+	0.962754i	$0.587149\pi$
$728$		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	0.663653	−	0.748040i	$-0.269005\pi$
$733$	−6271.00	−	10861.7i	−0.315995	−	0.547320i	−0.979649	+	0.200720i	$0.935672\pi$
$734$		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	$0.0265179\pi$
$761$	11973.0	+	20737.8i	0.570330	+	0.987840i	−0.426202	+	0.904628i	$0.640149\pi$
$762$		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	$-0.354945\pi$
$769$	18770.0			0.880187			0.440093	+	0.897952i	$0.354945\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	15171.0	−	26276.9i	0.705903	−	1.22266i	−0.260462	−	0.965484i	$-0.583875\pi$
$773$	15171.0	−	26276.9i	0.705903	−	1.22266i	0.966365	−	0.257176i	$-0.0827919\pi$
$774$		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	$-0.631247\pi$
$787$	13094.0	−	22679.5i	0.593076	−	1.02724i	0.993815	−	0.111045i	$-0.0354199\pi$
$788$		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
$797$	34818.0

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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