LMFDB - Embedded newform 1764.4.k.k.1549.1

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 28)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1549.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1764.1549
Dual form			1764.4.k.k.361.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(3.00000 - 5.19615i) q^{5} +O(q^{10})$	$q+(3.00000 - 5.19615i) q^{5} +(-6.00000 - 10.3923i) q^{11} -82.0000 q^{13} +(-15.0000 - 25.9808i) q^{17} +(-34.0000 + 58.8897i) q^{19} +(108.000 - 187.061i) q^{23} +(44.5000 + 77.0763i) q^{25} -246.000 q^{29} +(56.0000 + 96.9948i) q^{31} +(-55.0000 + 95.2628i) q^{37} +246.000 q^{41} -172.000 q^{43} +(96.0000 - 166.277i) q^{47} +(279.000 + 483.242i) q^{53} -72.0000 q^{55} +(270.000 + 467.654i) q^{59} +(-55.0000 + 95.2628i) q^{61} +(-246.000 + 426.084i) q^{65} +(-70.0000 - 121.244i) q^{67} +840.000 q^{71} +(275.000 + 476.314i) q^{73} +(104.000 - 180.133i) q^{79} -516.000 q^{83} -180.000 q^{85} +(-699.000 + 1210.70i) q^{89} +(204.000 + 353.338i) q^{95} +1586.00 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{5}+O(q^{10}) $ 2 * q + 6 * q^5	$ 2 q + 6 q^{5} - 12 q^{11} - 164 q^{13} - 30 q^{17} - 68 q^{19} + 216 q^{23} + 89 q^{25} - 492 q^{29} + 112 q^{31} - 110 q^{37} + 492 q^{41} - 344 q^{43} + 192 q^{47} + 558 q^{53} - 144 q^{55} + 540 q^{59} - 110 q^{61} - 492 q^{65} - 140 q^{67} + 1680 q^{71} + 550 q^{73} + 208 q^{79} - 1032 q^{83} - 360 q^{85} - 1398 q^{89} + 408 q^{95} + 3172 q^{97}+O(q^{100}) $ 2 * q + 6 * q^5 - 12 * q^11 - 164 * q^13 - 30 * q^17 - 68 * q^19 + 216 * q^23 + 89 * q^25 - 492 * q^29 + 112 * q^31 - 110 * q^37 + 492 * q^41 - 344 * q^43 + 192 * q^47 + 558 * q^53 - 144 * q^55 + 540 * q^59 - 110 * q^61 - 492 * q^65 - 140 * q^67 + 1680 * q^71 + 550 * q^73 + 208 * q^79 - 1032 * q^83 - 360 * q^85 - 1398 * q^89 + 408 * q^95 + 3172 * q^97

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	3.00000	−	5.19615i	0.268328	−	0.464758i	−0.700102	−	0.714043i	$-0.746862\pi$
$5$	3.00000	−	5.19615i	0.268328	−	0.464758i	0.968430	+	0.249285i	$0.0801955\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−6.00000	−	10.3923i	−0.164461	−	0.284854i	0.772003	−	0.635619i	$-0.219255\pi$
$11$	−6.00000	−	10.3923i	−0.164461	−	0.284854i	−0.936464	+	0.350765i	$0.885922\pi$
$12$		0			0
$13$	−82.0000			−1.74944			−0.874720	−	0.484629i	$-0.838954\pi$
$13$	−82.0000			−1.74944			−0.874720	+	0.484629i	$0.838954\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−15.0000	−	25.9808i	−0.214002	−	0.370662i	0.738961	−	0.673748i	$-0.235317\pi$
$17$	−15.0000	−	25.9808i	−0.214002	−	0.370662i	−0.952963	+	0.303085i	$0.901983\pi$
$18$		0			0
$19$	−34.0000	+	58.8897i	−0.410533	+	0.711065i	−0.994948	−	0.100390i	$-0.967991\pi$
$19$	−34.0000	+	58.8897i	−0.410533	+	0.711065i	0.584415	+	0.811455i	$0.301324\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	108.000	−	187.061i	0.979111	−	1.69587i	0.313470	−	0.949598i	$-0.398508\pi$
$23$	108.000	−	187.061i	0.979111	−	1.69587i	0.665641	−	0.746272i	$-0.268158\pi$
$24$		0			0
$25$	44.5000	+	77.0763i	0.356000	+	0.616610i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−246.000			−1.57521			−0.787604	−	0.616181i	$-0.788679\pi$
$29$	−246.000			−1.57521			−0.787604	+	0.616181i	$0.788679\pi$
$30$		0			0
$31$	56.0000	+	96.9948i	0.324448	+	0.561961i	0.981401	−	0.191971i	$-0.0614880\pi$
$31$	56.0000	+	96.9948i	0.324448	+	0.561961i	−0.656952	+	0.753932i	$0.728155\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−55.0000	+	95.2628i	−0.244377	+	0.423273i	−0.961956	−	0.273204i	$-0.911917\pi$
$37$	−55.0000	+	95.2628i	−0.244377	+	0.423273i	0.717579	+	0.696477i	$0.245250\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	246.000			0.937043			0.468521	−	0.883452i	$-0.344787\pi$
$41$	246.000			0.937043			0.468521	+	0.883452i	$0.344787\pi$
$42$		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	$-0.737035\pi$
$47$	96.0000	−	166.277i	0.297937	−	0.516042i	0.975664	+	0.219272i	$0.0703681\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	279.000	+	483.242i	0.723087	+	1.25242i	0.959757	+	0.280833i	$0.0906107\pi$
$53$	279.000	+	483.242i	0.723087	+	1.25242i	−0.236670	+	0.971590i	$0.576056\pi$
$54$		0			0
$55$	−72.0000			−0.176518
$56$		0			0
$57$		0			0
$58$		0			0
$59$	270.000	+	467.654i	0.595780	+	1.03192i	0.993436	+	0.114386i	$0.0364902\pi$
$59$	270.000	+	467.654i	0.595780	+	1.03192i	−0.397657	+	0.917534i	$0.630176\pi$
$60$		0			0
$61$	−55.0000	+	95.2628i	−0.115443	+	0.199953i	−0.917957	−	0.396680i	$-0.870162\pi$
$61$	−55.0000	+	95.2628i	−0.115443	+	0.199953i	0.802514	+	0.596634i	$0.203495\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−246.000	+	426.084i	−0.469424	+	0.813066i
$66$		0			0
$67$	−70.0000	−	121.244i	−0.127640	−	0.221078i	0.795122	−	0.606450i	$-0.207407\pi$
$67$	−70.0000	−	121.244i	−0.127640	−	0.221078i	−0.922762	+	0.385371i	$0.874073\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	840.000			1.40408			0.702040	−	0.712138i	$-0.252273\pi$
$71$	840.000			1.40408			0.702040	+	0.712138i	$0.252273\pi$
$72$		0			0
$73$	275.000	+	476.314i	0.440908	+	0.763676i	0.997757	−	0.0669381i	$-0.0213230\pi$
$73$	275.000	+	476.314i	0.440908	+	0.763676i	−0.556849	+	0.830614i	$0.687990\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	104.000	−	180.133i	0.148113	−	0.256539i	−0.782417	−	0.622755i	$-0.786013\pi$
$79$	104.000	−	180.133i	0.148113	−	0.256539i	0.930530	+	0.366216i	$0.119347\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−516.000			−0.682390			−0.341195	−	0.939993i	$-0.610832\pi$
$83$	−516.000			−0.682390			−0.341195	+	0.939993i	$0.610832\pi$
$84$		0			0
$85$	−180.000			−0.229691
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−699.000	+	1210.70i	−0.832515	+	1.44196i	0.0635224	+	0.997980i	$0.479767\pi$
$89$	−699.000	+	1210.70i	−0.832515	+	1.44196i	−0.896038	+	0.443978i	$0.853567\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	204.000	+	353.338i	0.220315	+	0.381597i
$96$		0			0
$97$	1586.00			1.66014			0.830072	−	0.557657i	$-0.188299\pi$
$97$	1586.00			1.66014			0.830072	+	0.557657i	$0.188299\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−621.000	−	1075.60i	−0.611800	−	1.05967i	−0.990937	−	0.134328i	$-0.957112\pi$
$101$	−621.000	−	1075.60i	−0.611800	−	1.05967i	0.379137	−	0.925341i	$-0.376221\pi$
$102$		0			0
$103$	−340.000	+	588.897i	−0.325254	+	0.563357i	−0.981564	−	0.191135i	$-0.938783\pi$
$103$	−340.000	+	588.897i	−0.325254	+	0.563357i	0.656309	+	0.754492i	$0.272117\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	498.000	−	862.561i	0.449939	−	0.779317i	−0.548442	−	0.836188i	$-0.684779\pi$
$107$	498.000	−	862.561i	0.449939	−	0.779317i	0.998382	+	0.0568710i	$0.0181124\pi$
$108$		0			0
$109$	−691.000	−	1196.85i	−0.607209	−	1.05172i	−0.991698	−	0.128588i	$-0.958956\pi$
$109$	−691.000	−	1196.85i	−0.607209	−	1.05172i	0.384489	−	0.923130i	$-0.374378\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	750.000			0.624372			0.312186	−	0.950021i	$-0.398939\pi$
$113$	750.000			0.624372			0.312186	+	0.950021i	$0.398939\pi$
$114$		0			0
$115$	−648.000	−	1122.37i	−0.525446	−	0.910099i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	593.500	−	1027.97i	0.445905	−	0.772331i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	1284.00			0.918756
$126$		0			0
$127$	176.000			0.122972			0.0614861	−	0.998108i	$-0.480416\pi$
$127$	176.000			0.122972			0.0614861	+	0.998108i	$0.480416\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−774.000	+	1340.61i	−0.516219	+	0.894118i	0.483604	+	0.875287i	$0.339328\pi$
$131$	−774.000	+	1340.61i	−0.516219	+	0.894118i	−0.999823	+	0.0188305i	$0.994006\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	189.000	+	327.358i	0.117864	+	0.204146i	0.918921	−	0.394442i	$-0.129062\pi$
$137$	189.000	+	327.358i	0.117864	+	0.204146i	−0.801057	+	0.598588i	$0.795729\pi$
$138$		0			0
$139$	−2500.00			−1.52552			−0.762760	−	0.646682i	$-0.776156\pi$
$139$	−2500.00			−1.52552			−0.762760	+	0.646682i	$0.776156\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	492.000	+	852.169i	0.287714	+	0.498335i
$144$		0			0
$145$	−738.000	+	1278.25i	−0.422673	+	0.732091i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	423.000	−	732.657i	0.232574	−	0.402830i	−0.725991	−	0.687704i	$-0.758619\pi$
$149$	423.000	−	732.657i	0.232574	−	0.402830i	0.958565	+	0.284874i	$0.0919519\pi$
$150$		0			0
$151$	1268.00	+	2196.24i	0.683367	+	1.18363i	0.973947	+	0.226775i	$0.0728183\pi$
$151$	1268.00	+	2196.24i	0.683367	+	1.18363i	−0.290580	+	0.956851i	$0.593848\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	672.000			0.348234
$156$		0			0
$157$	593.000	+	1027.11i	0.301443	+	0.522115i	0.976463	−	0.215685i	$-0.0691984\pi$
$157$	593.000	+	1027.11i	0.301443	+	0.522115i	−0.675020	+	0.737799i	$0.735865\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−1054.00	+	1825.58i	−0.506476	+	0.877243i	0.493496	+	0.869748i	$0.335719\pi$
$163$	−1054.00	+	1825.58i	−0.506476	+	0.877243i	−0.999972	+	0.00749450i	$0.997614\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1944.00			0.900786			0.450393	−	0.892830i	$-0.351284\pi$
$167$	1944.00			0.900786			0.450393	+	0.892830i	$0.351284\pi$
$168$		0			0
$169$	4527.00			2.06054
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−681.000	+	1179.53i	−0.299280	+	0.518368i	−0.975971	−	0.217898i	$-0.930080\pi$
$173$	−681.000	+	1179.53i	−0.299280	+	0.518368i	0.676691	+	0.736267i	$0.263413\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	798.000	+	1382.18i	0.333214	+	0.577144i	0.983140	−	0.182854i	$-0.0585335\pi$
$179$	798.000	+	1382.18i	0.333214	+	0.577144i	−0.649926	+	0.759997i	$0.725200\pi$
$180$		0			0
$181$	−1690.00			−0.694015			−0.347007	−	0.937862i	$-0.612802\pi$
$181$	−1690.00			−0.694015			−0.347007	+	0.937862i	$0.612802\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	330.000	+	571.577i	0.131146	+	0.227152i
$186$		0			0
$187$	−180.000	+	311.769i	−0.0703899	+	0.121919i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	1776.00	−	3076.12i	0.672811	−	1.16534i	−0.304293	−	0.952579i	$-0.598420\pi$
$191$	1776.00	−	3076.12i	0.672811	−	1.16534i	0.977104	−	0.212764i	$-0.0682465\pi$
$192$		0			0
$193$	1343.00	+	2326.14i	0.500887	+	0.867562i	0.999999	+	0.00102491i	$0.000326238\pi$
$193$	1343.00	+	2326.14i	0.500887	+	0.867562i	−0.499112	+	0.866537i	$0.666340\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	1410.00			0.509941			0.254970	−	0.966949i	$-0.417934\pi$
$197$	1410.00			0.509941			0.254970	+	0.966949i	$0.417934\pi$
$198$		0			0
$199$	1484.00	+	2570.36i	0.528633	+	0.915619i	0.999443	+	0.0333844i	$0.0106286\pi$
$199$	1484.00	+	2570.36i	0.528633	+	0.915619i	−0.470810	+	0.882235i	$0.656038\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	738.000	−	1278.25i	0.251435	−	0.435498i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	816.000			0.270067
$210$		0			0
$211$	−1348.00			−0.439811			−0.219906	−	0.975521i	$-0.570575\pi$
$211$	−1348.00			−0.439811			−0.219906	+	0.975521i	$0.570575\pi$
$212$		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
$221$	1230.00	+	2130.42i	0.374384	+	0.648451i
$222$		0			0
$223$	3872.00			1.16273			0.581364	−	0.813644i	$-0.302519\pi$
$223$	3872.00			1.16273			0.581364	+	0.813644i	$0.302519\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	2682.00	+	4645.36i	0.784188	+	1.35825i	0.929483	+	0.368865i	$0.120253\pi$
$227$	2682.00	+	4645.36i	0.784188	+	1.35825i	−0.145296	+	0.989388i	$0.546413\pi$
$228$		0			0
$229$	437.000	−	756.906i	0.126104	−	0.218418i	−0.796060	−	0.605218i	$-0.793086\pi$
$229$	437.000	−	756.906i	0.126104	−	0.218418i	0.922164	+	0.386799i	$0.126419\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	189.000	−	327.358i	0.0531408	−	0.0920425i	−0.838231	−	0.545315i	$-0.816410\pi$
$233$	189.000	−	327.358i	0.0531408	−	0.0920425i	0.891372	+	0.453272i	$0.149743\pi$
$234$		0			0
$235$	−576.000	−	997.661i	−0.159890	−	0.276937i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−1920.00			−0.519642			−0.259821	−	0.965657i	$-0.583664\pi$
$239$	−1920.00			−0.519642			−0.259821	+	0.965657i	$0.583664\pi$
$240$		0			0
$241$	−2161.00	−	3742.96i	−0.577603	−	1.00044i	−0.995754	−	0.0920596i	$-0.970655\pi$
$241$	−2161.00	−	3742.96i	−0.577603	−	1.00044i	0.418151	−	0.908378i	$-0.362678\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	2788.00	−	4828.96i	0.718203	−	1.24396i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−5292.00			−1.33079			−0.665395	−	0.746492i	$-0.731737\pi$
$251$	−5292.00			−1.33079			−0.665395	+	0.746492i	$0.731737\pi$
$252$		0			0
$253$	−2592.00			−0.644101
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−2559.00	+	4432.32i	−0.621113	+	1.07580i	0.368166	+	0.929760i	$0.379986\pi$
$257$	−2559.00	+	4432.32i	−0.621113	+	1.07580i	−0.989279	+	0.146039i	$0.953347\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	1884.00	+	3263.18i	0.441720	+	0.765082i	0.997817	−	0.0660355i	$-0.0210351\pi$
$263$	1884.00	+	3263.18i	0.441720	+	0.765082i	−0.556097	+	0.831117i	$0.687702\pi$
$264$		0			0
$265$	3348.00			0.776098
$266$		0			0
$267$		0			0
$268$		0			0
$269$	1959.00	+	3393.09i	0.444024	+	0.769071i	0.997984	−	0.0634719i	$-0.0202173\pi$
$269$	1959.00	+	3393.09i	0.444024	+	0.769071i	−0.553960	+	0.832543i	$0.686884\pi$
$270$		0			0
$271$	−2440.00	+	4226.20i	−0.546935	+	0.947320i	0.451547	+	0.892247i	$0.350872\pi$
$271$	−2440.00	+	4226.20i	−0.546935	+	0.947320i	−0.998482	+	0.0550723i	$0.982461\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	534.000	−	924.915i	0.117096	−	0.202816i
$276$		0			0
$277$	1769.00	+	3064.00i	0.383714	+	0.664613i	0.991590	−	0.129419i	$-0.0413114\pi$
$277$	1769.00	+	3064.00i	0.383714	+	0.664613i	−0.607875	+	0.794032i	$0.707978\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	5430.00			1.15276			0.576382	−	0.817180i	$-0.304464\pi$
$281$	5430.00			1.15276			0.576382	+	0.817180i	$0.304464\pi$
$282$		0			0
$283$	3218.00	+	5573.74i	0.675937	+	1.17076i	0.976194	+	0.216899i	$0.0695943\pi$
$283$	3218.00	+	5573.74i	0.675937	+	1.17076i	−0.300257	+	0.953858i	$0.597072\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	2006.50	−	3475.36i	0.408406	−	0.707380i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−1350.00			−0.269174			−0.134587	−	0.990902i	$-0.542971\pi$
$293$	−1350.00			−0.269174			−0.134587	+	0.990902i	$0.542971\pi$
$294$		0			0
$295$	3240.00			0.639458
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−8856.00	+	15339.0i	−1.71290	+	2.96682i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	330.000	+	571.577i	0.0619533	+	0.107306i
$306$		0			0
$307$	3332.00			0.619437			0.309719	−	0.950828i	$-0.399765\pi$
$307$	3332.00			0.619437			0.309719	+	0.950828i	$0.399765\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−2364.00	−	4094.57i	−0.431029	−	0.746565i	0.565933	−	0.824451i	$-0.308516\pi$
$311$	−2364.00	−	4094.57i	−0.431029	−	0.746565i	−0.996962	+	0.0778865i	$0.975183\pi$
$312$		0			0
$313$	−2557.00	+	4428.85i	−0.461758	+	0.799788i	−0.999049	−	0.0436091i	$-0.986114\pi$
$313$	−2557.00	+	4428.85i	−0.461758	+	0.799788i	0.537291	+	0.843397i	$0.319448\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	3603.00	−	6240.58i	0.638374	−	1.10570i	−0.347415	−	0.937711i	$-0.612941\pi$
$317$	3603.00	−	6240.58i	0.638374	−	1.10570i	0.985790	−	0.167985i	$-0.0537261\pi$
$318$		0			0
$319$	1476.00	+	2556.51i	0.259060	+	0.448705i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	2040.00			0.351420
$324$		0			0
$325$	−3649.00	−	6320.25i	−0.622800	−	1.07872i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−3130.00	+	5421.32i	−0.519759	+	0.900250i	0.479977	+	0.877281i	$0.340645\pi$
$331$	−3130.00	+	5421.32i	−0.519759	+	0.900250i	−0.999736	+	0.0229685i	$0.992688\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−840.000			−0.136997
$336$		0			0
$337$	−5326.00			−0.860907			−0.430454	−	0.902613i	$-0.641646\pi$
$337$	−5326.00			−0.860907			−0.430454	+	0.902613i	$0.641646\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	672.000	−	1163.94i	0.106718	−	0.184841i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	18.0000	+	31.1769i	0.00278470	+	0.00482324i	0.867414	−	0.497586i	$-0.165780\pi$
$347$	18.0000	+	31.1769i	0.00278470	+	0.00482324i	−0.864630	+	0.502410i	$0.832447\pi$
$348$		0			0
$349$	3134.00			0.480685			0.240343	−	0.970688i	$-0.422740\pi$
$349$	3134.00			0.480685			0.240343	+	0.970688i	$0.422740\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	609.000	+	1054.82i	0.0918238	+	0.159043i	0.908279	−	0.418366i	$-0.137397\pi$
$353$	609.000	+	1054.82i	0.0918238	+	0.159043i	−0.816455	+	0.577409i	$0.804064\pi$
$354$		0			0
$355$	2520.00	−	4364.77i	0.376754	−	0.652557i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−5004.00	+	8667.18i	−0.735657	+	1.27420i	0.218777	+	0.975775i	$0.429793\pi$
$359$	−5004.00	+	8667.18i	−0.735657	+	1.27420i	−0.954434	+	0.298421i	$0.903540\pi$
$360$		0			0
$361$	1117.50	+	1935.57i	0.162925	+	0.282194i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	3300.00			0.473233
$366$		0			0
$367$	536.000	+	928.379i	0.0762370	+	0.132046i	0.901624	−	0.432522i	$-0.142376\pi$
$367$	536.000	+	928.379i	0.0762370	+	0.132046i	−0.825387	+	0.564568i	$0.809043\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	137.000	−	237.291i	0.0190177	−	0.0329396i	−0.856360	−	0.516379i	$-0.827279\pi$
$373$	137.000	−	237.291i	0.0190177	−	0.0329396i	0.875378	+	0.483440i	$0.160613\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	20172.0			2.75573
$378$		0			0
$379$	7652.00			1.03709			0.518545	−	0.855051i	$-0.326474\pi$
$379$	7652.00			1.03709			0.518545	+	0.855051i	$0.326474\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	1080.00	−	1870.61i	0.144087	−	0.249566i	−0.784945	−	0.619566i	$-0.787309\pi$
$383$	1080.00	−	1870.61i	0.144087	−	0.249566i	0.929032	+	0.369999i	$0.120642\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−537.000	−	930.111i	−0.0699922	−	0.121230i	0.828905	−	0.559389i	$-0.188964\pi$
$389$	−537.000	−	930.111i	−0.0699922	−	0.121230i	−0.898898	+	0.438159i	$0.855631\pi$
$390$		0			0
$391$	−6480.00			−0.838127
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−624.000	−	1080.80i	−0.0794857	−	0.137673i
$396$		0			0
$397$	−3463.00	+	5998.09i	−0.437791	+	0.758276i	−0.997519	−	0.0704004i	$-0.977572\pi$
$397$	−3463.00	+	5998.09i	−0.437791	+	0.758276i	0.559728	+	0.828676i	$0.310906\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	969.000	−	1678.36i	0.120672	−	0.209010i	−0.799361	−	0.600851i	$-0.794828\pi$
$401$	969.000	−	1678.36i	0.120672	−	0.209010i	0.920033	+	0.391841i	$0.128162\pi$
$402$		0			0
$403$	−4592.00	−	7953.58i	−0.567603	−	0.983116i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	1320.00			0.160762
$408$		0			0
$409$	4787.00	+	8291.33i	0.578733	+	1.00240i	0.995625	+	0.0934393i	$0.0297861\pi$
$409$	4787.00	+	8291.33i	0.578733	+	1.00240i	−0.416892	+	0.908956i	$0.636881\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−1548.00	+	2681.21i	−0.183104	+	0.317146i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	5052.00			0.589037			0.294518	−	0.955646i	$-0.404841\pi$
$419$	5052.00			0.589037			0.294518	+	0.955646i	$0.404841\pi$
$420$		0			0
$421$	3422.00			0.396147			0.198074	−	0.980187i	$-0.436531\pi$
$421$	3422.00			0.396147			0.198074	+	0.980187i	$0.436531\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	1335.00	−	2312.29i	0.152369	−	0.263912i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	1104.00	+	1912.18i	0.123382	+	0.213705i	0.921099	−	0.389327i	$-0.127292\pi$
$431$	1104.00	+	1912.18i	0.123382	+	0.213705i	−0.797717	+	0.603032i	$0.793959\pi$
$432$		0			0
$433$	−6814.00			−0.756259			−0.378129	−	0.925753i	$-0.623433\pi$
$433$	−6814.00			−0.756259			−0.378129	+	0.925753i	$0.623433\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	7344.00	+	12720.2i	0.803916	+	1.39242i
$438$		0			0
$439$	−6292.00	+	10898.1i	−0.684056	+	1.18482i	0.289676	+	0.957125i	$0.406452\pi$
$439$	−6292.00	+	10898.1i	−0.684056	+	1.18482i	−0.973732	+	0.227696i	$0.926881\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	3498.00	−	6058.71i	0.375158	−	0.649793i	−0.615193	−	0.788377i	$-0.710922\pi$
$443$	3498.00	−	6058.71i	0.375158	−	0.649793i	0.990351	+	0.138584i	$0.0442551\pi$
$444$		0			0
$445$	4194.00	+	7264.22i	0.446775	+	0.773836i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−9474.00			−0.995781			−0.497891	−	0.867240i	$-0.665892\pi$
$449$	−9474.00			−0.995781			−0.497891	+	0.867240i	$0.665892\pi$
$450$		0			0
$451$	−1476.00	−	2556.51i	−0.154107	−	0.266921i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−2893.00	+	5010.82i	−0.296124	+	0.512902i	−0.975246	−	0.221123i	$-0.929028\pi$
$457$	−2893.00	+	5010.82i	−0.296124	+	0.512902i	0.679121	+	0.734026i	$0.262361\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−3438.00			−0.347340			−0.173670	−	0.984804i	$-0.555563\pi$
$461$	−3438.00			−0.347340			−0.173670	+	0.984804i	$0.555563\pi$
$462$		0			0
$463$	9392.00			0.942728			0.471364	−	0.881939i	$-0.343762\pi$
$463$	9392.00			0.942728			0.471364	+	0.881939i	$0.343762\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−2478.00	+	4292.02i	−0.245542	+	0.425291i	−0.962284	−	0.272047i	$-0.912299\pi$
$467$	−2478.00	+	4292.02i	−0.245542	+	0.425291i	0.716742	+	0.697339i	$0.245633\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	1032.00	+	1787.48i	0.100320	+	0.173760i
$474$		0			0
$475$	−6052.00			−0.584600
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−10296.0	−	17833.2i	−0.982122	−	1.70108i	−0.654088	−	0.756418i	$-0.726947\pi$
$479$	−10296.0	−	17833.2i	−0.982122	−	1.70108i	−0.328034	−	0.944666i	$-0.606386\pi$
$480$		0			0
$481$	4510.00	−	7811.55i	0.427522	−	0.740491i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	4758.00	−	8241.10i	0.445463	−	0.771565i
$486$		0			0
$487$	6716.00	+	11632.5i	0.624910	+	1.08238i	0.988558	+	0.150839i	$0.0481975\pi$
$487$	6716.00	+	11632.5i	0.624910	+	1.08238i	−0.363649	+	0.931536i	$0.618469\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	14172.0			1.30259			0.651297	−	0.758823i	$-0.274225\pi$
$491$	14172.0			1.30259			0.651297	+	0.758823i	$0.274225\pi$
$492$		0			0
$493$	3690.00	+	6391.27i	0.337098	+	0.583871i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	2978.00	−	5158.05i	0.267162	−	0.462737i	−0.700966	−	0.713195i	$-0.747248\pi$
$499$	2978.00	−	5158.05i	0.267162	−	0.462737i	0.968128	+	0.250457i	$0.0805809\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−16968.0			−1.50411			−0.752053	−	0.659102i	$-0.770936\pi$
$503$	−16968.0			−1.50411			−0.752053	+	0.659102i	$0.770936\pi$
$504$		0			0
$505$	−7452.00			−0.656653
$506$		0			0
$507$		0			0
$508$		0			0
$509$	2607.00	−	4515.46i	0.227020	−	0.393210i	−0.729903	−	0.683550i	$-0.760435\pi$
$509$	2607.00	−	4515.46i	0.227020	−	0.393210i	0.956924	+	0.290340i	$0.0937683\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	2040.00	+	3533.38i	0.174550	+	0.302329i
$516$		0			0
$517$	−2304.00			−0.195996
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−699.000	−	1210.70i	−0.0587788	−	0.101808i	0.835139	−	0.550039i	$-0.185387\pi$
$521$	−699.000	−	1210.70i	−0.0587788	−	0.101808i	−0.893917	+	0.448232i	$0.852054\pi$
$522$		0			0
$523$	9290.00	−	16090.8i	0.776718	−	1.34531i	−0.157106	−	0.987582i	$-0.550217\pi$
$523$	9290.00	−	16090.8i	0.776718	−	1.34531i	0.933824	−	0.357733i	$-0.116450\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	1680.00	−	2909.85i	0.138865	−	0.240522i
$528$		0			0
$529$	−17244.5	−	29868.4i	−1.41732	−	2.45487i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−20172.0			−1.63930
$534$		0			0
$535$	−2988.00	−	5175.37i	−0.241463	−	0.418226i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	9485.00	−	16428.5i	0.753774	−	1.30558i	−0.192207	−	0.981354i	$-0.561564\pi$
$541$	9485.00	−	16428.5i	0.753774	−	1.30558i	0.945981	−	0.324221i	$-0.105102\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−8292.00			−0.651725
$546$		0			0
$547$	−16036.0			−1.25347			−0.626737	−	0.779231i	$-0.715610\pi$
$547$	−16036.0			−1.25347			−0.626737	+	0.779231i	$0.715610\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	8364.00	−	14486.9i	0.646676	−	1.12008i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	4155.00	+	7196.67i	0.316074	+	0.547456i	0.979665	−	0.200640i	$-0.0643020\pi$
$557$	4155.00	+	7196.67i	0.316074	+	0.547456i	−0.663592	+	0.748095i	$0.730969\pi$
$558$		0			0
$559$	14104.0			1.06715
$560$		0			0
$561$		0			0
$562$		0			0
$563$	3546.00	+	6141.85i	0.265446	+	0.459766i	0.967680	−	0.252180i	$-0.0811475\pi$
$563$	3546.00	+	6141.85i	0.265446	+	0.459766i	−0.702234	+	0.711946i	$0.747814\pi$
$564$		0			0
$565$	2250.00	−	3897.11i	0.167537	−	0.290182i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−3579.00	+	6199.01i	−0.263690	+	0.456724i	−0.967220	−	0.253942i	$-0.918273\pi$
$569$	−3579.00	+	6199.01i	−0.263690	+	0.456724i	0.703530	+	0.710666i	$0.251606\pi$
$570$		0			0
$571$	−3250.00	−	5629.17i	−0.238193	−	0.412563i	0.722003	−	0.691890i	$-0.243222\pi$
$571$	−3250.00	−	5629.17i	−0.238193	−	0.412563i	−0.960196	+	0.279328i	$0.909888\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	19224.0			1.39425
$576$		0			0
$577$	−10897.0	−	18874.2i	−0.786218	−	1.36177i	−0.928268	−	0.371911i	$-0.878703\pi$
$577$	−10897.0	−	18874.2i	−0.786218	−	1.36177i	0.142050	−	0.989859i	$-0.454631\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	3348.00	−	5798.91i	0.237839	−	0.411949i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−9756.00			−0.685985			−0.342993	−	0.939338i	$-0.611441\pi$
$587$	−9756.00			−0.685985			−0.342993	+	0.939338i	$0.611441\pi$
$588$		0			0
$589$	−7616.00			−0.532787
$590$		0			0
$591$		0			0
$592$		0			0
$593$	2793.00	−	4837.62i	0.193414	−	0.335004i	−0.752965	−	0.658060i	$-0.771377\pi$
$593$	2793.00	−	4837.62i	0.193414	−	0.335004i	0.946380	+	0.323057i	$0.104710\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−12.0000	−	20.7846i	−0.000818542	−	0.00141776i	0.865616	−	0.500709i	$-0.166927\pi$
$599$	−12.0000	−	20.7846i	−0.000818542	−	0.00141776i	−0.866434	+	0.499291i	$0.833594\pi$
$600$		0			0
$601$	4298.00			0.291712			0.145856	−	0.989306i	$-0.453406\pi$
$601$	4298.00			0.291712			0.145856	+	0.989306i	$0.453406\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−3561.00	−	6167.83i	−0.239298	−	0.414476i
$606$		0			0
$607$	−4240.00	+	7343.90i	−0.283519	+	0.491070i	−0.972249	−	0.233948i	$-0.924835\pi$
$607$	−4240.00	+	7343.90i	−0.283519	+	0.491070i	0.688730	+	0.725018i	$0.258169\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−7872.00	+	13634.7i	−0.521223	+	0.902784i
$612$		0			0
$613$	953.000	+	1650.64i	0.0627917	+	0.108758i	0.895712	−	0.444634i	$-0.146666\pi$
$613$	953.000	+	1650.64i	0.0627917	+	0.108758i	−0.832921	+	0.553393i	$0.813333\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−7482.00			−0.488191			−0.244096	−	0.969751i	$-0.578491\pi$
$617$	−7482.00			−0.488191			−0.244096	+	0.969751i	$0.578491\pi$
$618$		0			0
$619$	3674.00	+	6363.55i	0.238563	+	0.413203i	0.960302	−	0.278962i	$-0.0899903\pi$
$619$	3674.00	+	6363.55i	0.238563	+	0.413203i	−0.721739	+	0.692165i	$0.756657\pi$
$620$		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
$629$	3300.00			0.209189
$630$		0			0
$631$	4520.00			0.285164			0.142582	−	0.989783i	$-0.454460\pi$
$631$	4520.00			0.285164			0.142582	+	0.989783i	$0.454460\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	528.000	−	914.523i	0.0329969	−	0.0571523i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−9903.00	−	17152.5i	−0.610211	−	1.05692i	−0.991205	−	0.132338i	$-0.957751\pi$
$641$	−9903.00	−	17152.5i	−0.610211	−	1.05692i	0.380994	−	0.924577i	$-0.375582\pi$
$642$		0			0
$643$	−5020.00			−0.307884			−0.153942	−	0.988080i	$-0.549197\pi$
$643$	−5020.00			−0.307884			−0.153942	+	0.988080i	$0.549197\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	14196.0	+	24588.2i	0.862600	+	1.49407i	0.869410	+	0.494091i	$0.164499\pi$
$647$	14196.0	+	24588.2i	0.862600	+	1.49407i	−0.00681018	+	0.999977i	$0.502168\pi$
$648$		0			0
$649$	3240.00	−	5611.84i	0.195965	−	0.339421i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−8781.00	+	15209.1i	−0.526228	+	0.911454i	0.473305	+	0.880899i	$0.343061\pi$
$653$	−8781.00	+	15209.1i	−0.526228	+	0.911454i	−0.999533	+	0.0305554i	$0.990272\pi$
$654$		0			0
$655$	4644.00	+	8043.64i	0.277032	+	0.479834i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−4716.00			−0.278770			−0.139385	−	0.990238i	$-0.544513\pi$
$659$	−4716.00			−0.278770			−0.139385	+	0.990238i	$0.544513\pi$
$660$		0			0
$661$	11381.0	+	19712.5i	0.669697	+	1.15995i	0.977989	+	0.208657i	$0.0669093\pi$
$661$	11381.0	+	19712.5i	0.669697	+	1.15995i	−0.308292	+	0.951292i	$0.599757\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−26568.0	+	46017.1i	−1.54230	+	2.67135i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	1320.00			0.0759434
$672$		0			0
$673$	4802.00			0.275042			0.137521	−	0.990499i	$-0.456086\pi$
$673$	4802.00			0.275042			0.137521	+	0.990499i	$0.456086\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	10779.0	−	18669.8i	0.611921	−	1.05988i	−0.378995	−	0.925399i	$-0.623730\pi$
$677$	10779.0	−	18669.8i	0.611921	−	1.05988i	0.990916	−	0.134480i	$-0.0429364\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	1890.00	+	3273.58i	0.105884	+	0.183397i	0.914099	−	0.405491i	$-0.132899\pi$
$683$	1890.00	+	3273.58i	0.105884	+	0.183397i	−0.808215	+	0.588888i	$0.799566\pi$
$684$		0			0
$685$	2268.00			0.126505
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−22878.0	−	39625.9i	−1.26500	−	2.19104i
$690$		0			0
$691$	2750.00	−	4763.14i	0.151396	−	0.262226i	−0.780345	−	0.625350i	$-0.784956\pi$
$691$	2750.00	−	4763.14i	0.151396	−	0.262226i	0.931741	+	0.363123i	$0.118290\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−7500.00	+	12990.4i	−0.409340	+	0.708997i
$696$		0			0
$697$	−3690.00	−	6391.27i	−0.200529	−	0.347326i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−10230.0			−0.551187			−0.275593	−	0.961274i	$-0.588874\pi$
$701$	−10230.0			−0.551187			−0.275593	+	0.961274i	$0.588874\pi$
$702$		0			0
$703$	−3740.00	−	6477.87i	−0.200650	−	0.347536i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−5095.00	+	8824.80i	−0.269883	+	0.467450i	−0.968831	−	0.247721i	$-0.920318\pi$
$709$	−5095.00	+	8824.80i	−0.269883	+	0.467450i	0.698949	+	0.715172i	$0.253652\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	24192.0			1.27068
$714$		0			0
$715$	5904.00			0.308807
$716$		0			0
$717$		0			0
$718$		0			0
$719$	4704.00	−	8147.57i	0.243991	−	0.422605i	−0.717856	−	0.696191i	$-0.754877\pi$
$719$	4704.00	−	8147.57i	0.243991	−	0.422605i	0.961847	+	0.273586i	$0.0882099\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−10947.0	−	18960.8i	−0.560774	−	0.971290i
$726$		0			0
$727$	−33064.0			−1.68676			−0.843381	−	0.537316i	$-0.819438\pi$
$727$	−33064.0			−1.68676			−0.843381	+	0.537316i	$0.819438\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	2580.00	+	4468.69i	0.130540	+	0.226102i
$732$		0			0
$733$	3161.00	−	5475.01i	0.159283	−	0.275886i	−0.775328	−	0.631559i	$-0.782415\pi$
$733$	3161.00	−	5475.01i	0.159283	−	0.275886i	0.934610	+	0.355674i	$0.115749\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−840.000	+	1454.92i	−0.0419834	+	0.0727175i
$738$		0			0
$739$	10370.0	+	17961.4i	0.516193	+	0.894072i	0.999823	+	0.0188001i	$0.00598461\pi$
$739$	10370.0	+	17961.4i	0.516193	+	0.894072i	−0.483630	+	0.875272i	$0.660682\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−32040.0			−1.58201			−0.791005	−	0.611810i	$-0.790442\pi$
$743$	−32040.0			−1.58201			−0.791005	+	0.611810i	$0.790442\pi$
$744$		0			0
$745$	−2538.00	−	4395.94i	−0.124812	−	0.216181i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	6416.00	−	11112.8i	0.311749	−	0.539964i	−0.666992	−	0.745064i	$-0.732419\pi$
$751$	6416.00	−	11112.8i	0.311749	−	0.539964i	0.978741	+	0.205100i	$0.0657520\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	15216.0			0.733466
$756$		0			0
$757$	−19906.0			−0.955741			−0.477870	−	0.878430i	$-0.658591\pi$
$757$	−19906.0			−0.955741			−0.477870	+	0.878430i	$0.658591\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	5421.00	−	9389.45i	0.258227	−	0.447263i	−0.707540	−	0.706674i	$-0.750195\pi$
$761$	5421.00	−	9389.45i	0.258227	−	0.447263i	0.965767	+	0.259411i	$0.0835283\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−22140.0	−	38347.6i	−1.04228	−	1.80528i
$768$		0			0
$769$	28274.0			1.32586			0.662930	−	0.748681i	$-0.269313\pi$
$769$	28274.0			1.32586			0.662930	+	0.748681i	$0.269313\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−16173.0	−	28012.5i	−0.752526	−	1.30341i	−0.946595	−	0.322425i	$-0.895502\pi$
$773$	−16173.0	−	28012.5i	−0.752526	−	1.30341i	0.194069	−	0.980988i	$-0.437831\pi$
$774$		0			0
$775$	−4984.00	+	8632.54i	−0.231007	+	0.400116i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−8364.00	+	14486.9i	−0.384687	+	0.666298i
$780$		0			0
$781$	−5040.00	−	8729.54i	−0.230916	−	0.399958i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	7116.00			0.323543
$786$		0			0
$787$	−15058.0	−	26081.2i	−0.682033	−	1.18132i	−0.974360	−	0.224997i	$-0.927763\pi$
$787$	−15058.0	−	26081.2i	−0.682033	−	1.18132i	0.292327	−	0.956318i	$-0.405570\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	4510.00	−	7811.55i	0.201961	−	0.349806i
$794$		0			0
$795$		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)

\(f(q)\)	\(=\)	\(q+(3.00000 - 5.19615i) q^{5} +O(q^{10})\)	\(q+(3.00000 - 5.19615i) q^{5} +(-6.00000 - 10.3923i) q^{11} -82.0000 q^{13} +(-15.0000 - 25.9808i) q^{17} +(-34.0000 + 58.8897i) q^{19} +(108.000 - 187.061i) q^{23} +(44.5000 + 77.0763i) q^{25} -246.000 q^{29} +(56.0000 + 96.9948i) q^{31} +(-55.0000 + 95.2628i) q^{37} +246.000 q^{41} -172.000 q^{43} +(96.0000 - 166.277i) q^{47} +(279.000 + 483.242i) q^{53} -72.0000 q^{55} +(270.000 + 467.654i) q^{59} +(-55.0000 + 95.2628i) q^{61} +(-246.000 + 426.084i) q^{65} +(-70.0000 - 121.244i) q^{67} +840.000 q^{71} +(275.000 + 476.314i) q^{73} +(104.000 - 180.133i) q^{79} -516.000 q^{83} -180.000 q^{85} +(-699.000 + 1210.70i) q^{89} +(204.000 + 353.338i) q^{95} +1586.00 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{5}+O(q^{10}) \) 2 * q + 6 * q^5	\( 2 q + 6 q^{5} - 12 q^{11} - 164 q^{13} - 30 q^{17} - 68 q^{19} + 216 q^{23} + 89 q^{25} - 492 q^{29} + 112 q^{31} - 110 q^{37} + 492 q^{41} - 344 q^{43} + 192 q^{47} + 558 q^{53} - 144 q^{55} + 540 q^{59} - 110 q^{61} - 492 q^{65} - 140 q^{67} + 1680 q^{71} + 550 q^{73} + 208 q^{79} - 1032 q^{83} - 360 q^{85} - 1398 q^{89} + 408 q^{95} + 3172 q^{97}+O(q^{100}) \) 2 * q + 6 * q^5 - 12 * q^11 - 164 * q^13 - 30 * q^17 - 68 * q^19 + 216 * q^23 + 89 * q^25 - 492 * q^29 + 112 * q^31 - 110 * q^37 + 492 * q^41 - 344 * q^43 + 192 * q^47 + 558 * q^53 - 144 * q^55 + 540 * q^59 - 110 * q^61 - 492 * q^65 - 140 * q^67 + 1680 * q^71 + 550 * q^73 + 208 * q^79 - 1032 * q^83 - 360 * q^85 - 1398 * q^89 + 408 * q^95 + 3172 * q^97

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