LMFDB - Embedded newform 1764.4.k.e.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.e.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	−0.968430	−	0.249285i	$-0.919804\pi$
$5$	−3.00000	+	5.19615i	−0.268328	+	0.464758i	0.700102	+	0.714043i	$0.253138\pi$
$6$		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	$-0.161046\pi$
$13$	82.0000			1.74944			0.874720	+	0.484629i	$0.161046\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	15.0000	+	25.9808i	0.214002	+	0.370662i	0.952963	−	0.303085i	$-0.0980168\pi$
$17$	15.0000	+	25.9808i	0.214002	+	0.370662i	−0.738961	+	0.673748i	$0.764683\pi$
$18$		0			0
$19$	34.0000	−	58.8897i	0.410533	−	0.711065i	−0.584415	−	0.811455i	$-0.698676\pi$
$19$	34.0000	−	58.8897i	0.410533	−	0.711065i	0.994948	+	0.100390i	$0.0320092\pi$
$20$		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	0.656952	−	0.753932i	$-0.271845\pi$
$31$	−56.0000	−	96.9948i	−0.324448	−	0.561961i	−0.981401	+	0.191971i	$0.938512\pi$
$32$		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	$-0.655213\pi$
$41$	−246.000			−0.937043			−0.468521	+	0.883452i	$0.655213\pi$
$42$		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	$-0.929632\pi$
$47$	−96.0000	+	166.277i	−0.297937	+	0.516042i	0.677727	+	0.735314i	$0.262965\pi$
$48$		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	$-0.963510\pi$
$59$	−270.000	−	467.654i	−0.595780	−	1.03192i	0.397657	−	0.917534i	$-0.369824\pi$
$60$		0			0
$61$	55.0000	−	95.2628i	0.115443	−	0.199953i	−0.802514	−	0.596634i	$-0.796505\pi$
$61$	55.0000	−	95.2628i	0.115443	−	0.199953i	0.917957	+	0.396680i	$0.129838\pi$
$62$		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	0.556849	−	0.830614i	$-0.312010\pi$
$73$	−275.000	−	476.314i	−0.440908	−	0.763676i	−0.997757	+	0.0669381i	$0.978677\pi$
$74$		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	$-0.389168\pi$
$83$	516.000			0.682390			0.341195	+	0.939993i	$0.389168\pi$
$84$		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	$-0.520233\pi$
$89$	699.000	−	1210.70i	0.832515	−	1.44196i	0.896038	−	0.443978i	$-0.146433\pi$
$90$		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	$-0.811701\pi$
$97$	−1586.00			−1.66014			−0.830072	+	0.557657i	$0.811701\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	621.000	+	1075.60i	0.611800	+	1.05967i	0.990937	+	0.134328i	$0.0428876\pi$
$101$	621.000	+	1075.60i	0.611800	+	1.05967i	−0.379137	+	0.925341i	$0.623779\pi$
$102$		0			0
$103$	340.000	−	588.897i	0.325254	−	0.563357i	−0.656309	−	0.754492i	$-0.727883\pi$
$103$	340.000	−	588.897i	0.325254	−	0.563357i	0.981564	+	0.191135i	$0.0612168\pi$
$104$		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	$-0.660672\pi$
$131$	774.000	−	1340.61i	0.516219	−	0.894118i	0.999823	−	0.0188305i	$-0.00599429\pi$
$132$		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	$-0.223844\pi$
$139$	2500.00			1.52552			0.762760	+	0.646682i	$0.223844\pi$
$140$		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	0.675020	−	0.737799i	$-0.264135\pi$
$157$	−593.000	−	1027.11i	−0.301443	−	0.522115i	−0.976463	+	0.215685i	$0.930802\pi$
$158$		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	$-0.648716\pi$
$167$	−1944.00			−0.900786			−0.450393	+	0.892830i	$0.648716\pi$
$168$		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	−0.676691	−	0.736267i	$-0.736587\pi$
$173$	681.000	−	1179.53i	0.299280	−	0.518368i	0.975971	+	0.217898i	$0.0699201\pi$
$174$		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	$-0.387198\pi$
$181$	1690.00			0.694015			0.347007	+	0.937862i	$0.387198\pi$
$182$		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	$-0.989371\pi$
$199$	−1484.00	−	2570.36i	−0.528633	−	0.915619i	0.470810	−	0.882235i	$-0.343962\pi$
$200$		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	$-0.697481\pi$
$223$	−3872.00			−1.16273			−0.581364	+	0.813644i	$0.697481\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−2682.00	−	4645.36i	−0.784188	−	1.35825i	−0.929483	−	0.368865i	$-0.879747\pi$
$227$	−2682.00	−	4645.36i	−0.784188	−	1.35825i	0.145296	−	0.989388i	$-0.453587\pi$
$228$		0			0
$229$	−437.000	+	756.906i	−0.126104	+	0.218418i	−0.922164	−	0.386799i	$-0.873581\pi$
$229$	−437.000	+	756.906i	−0.126104	+	0.218418i	0.796060	+	0.605218i	$0.206914\pi$
$230$		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	$0.0293450\pi$
$241$	2161.00	+	3742.96i	0.577603	+	1.00044i	−0.418151	+	0.908378i	$0.637322\pi$
$242$		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	$-0.268263\pi$
$251$	5292.00			1.33079			0.665395	+	0.746492i	$0.268263\pi$
$252$		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	$-0.620014\pi$
$257$	2559.00	−	4432.32i	0.621113	−	1.07580i	0.989279	−	0.146039i	$-0.0466525\pi$
$258$		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	0.553960	−	0.832543i	$-0.313116\pi$
$269$	−1959.00	−	3393.09i	−0.444024	−	0.769071i	−0.997984	+	0.0634719i	$0.979783\pi$
$270$		0			0
$271$	2440.00	−	4226.20i	0.546935	−	0.947320i	−0.451547	−	0.892247i	$-0.649128\pi$
$271$	2440.00	−	4226.20i	0.546935	−	0.947320i	0.998482	−	0.0550723i	$-0.0175389\pi$
$272$		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	$-0.930406\pi$
$283$	−3218.00	−	5573.74i	−0.675937	−	1.17076i	0.300257	−	0.953858i	$-0.402928\pi$
$284$		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	$-0.457029\pi$
$293$	1350.00			0.269174			0.134587	+	0.990902i	$0.457029\pi$
$294$		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	$-0.600235\pi$
$307$	−3332.00			−0.619437			−0.309719	+	0.950828i	$0.600235\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	2364.00	+	4094.57i	0.431029	+	0.746565i	0.996962	−	0.0778865i	$-0.0248172\pi$
$311$	2364.00	+	4094.57i	0.431029	+	0.746565i	−0.565933	+	0.824451i	$0.691484\pi$
$312$		0			0
$313$	2557.00	−	4428.85i	0.461758	−	0.799788i	−0.537291	−	0.843397i	$-0.680552\pi$
$313$	2557.00	−	4428.85i	0.461758	−	0.799788i	0.999049	+	0.0436091i	$0.0138856\pi$
$314$		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	$-0.577260\pi$
$349$	−3134.00			−0.480685			−0.240343	+	0.970688i	$0.577260\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−609.000	−	1054.82i	−0.0918238	−	0.159043i	0.816455	−	0.577409i	$-0.195936\pi$
$353$	−609.000	−	1054.82i	−0.0918238	−	0.159043i	−0.908279	+	0.418366i	$0.862603\pi$
$354$		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	0.825387	−	0.564568i	$-0.190957\pi$
$367$	−536.000	−	928.379i	−0.0762370	−	0.132046i	−0.901624	+	0.432522i	$0.857624\pi$
$368$		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	−0.929032	−	0.369999i	$-0.879358\pi$
$383$	−1080.00	+	1870.61i	−0.144087	+	0.249566i	0.784945	+	0.619566i	$0.212691\pi$
$384$		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	−0.559728	−	0.828676i	$-0.689094\pi$
$397$	3463.00	−	5998.09i	0.437791	−	0.758276i	0.997519	+	0.0704004i	$0.0224277\pi$
$398$		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	$-0.970214\pi$
$409$	−4787.00	−	8291.33i	−0.578733	−	1.00240i	0.416892	−	0.908956i	$-0.363119\pi$
$410$		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	$-0.595159\pi$
$419$	−5052.00			−0.589037			−0.294518	+	0.955646i	$0.595159\pi$
$420$		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	$-0.376567\pi$
$433$	6814.00			0.756259			0.378129	+	0.925753i	$0.376567\pi$
$434$		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	$-0.593548\pi$
$439$	6292.00	−	10898.1i	0.684056	−	1.18482i	0.973732	−	0.227696i	$-0.0731191\pi$
$440$		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	$-0.444437\pi$
$461$	3438.00			0.347340			0.173670	+	0.984804i	$0.444437\pi$
$462$		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	−0.716742	−	0.697339i	$-0.754367\pi$
$467$	2478.00	−	4292.02i	0.245542	−	0.425291i	0.962284	+	0.272047i	$0.0877007\pi$
$468$		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	$0.273053\pi$
$479$	10296.0	+	17833.2i	0.982122	+	1.70108i	0.328034	+	0.944666i	$0.393614\pi$
$480$		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	$-0.229064\pi$
$503$	16968.0			1.50411			0.752053	+	0.659102i	$0.229064\pi$
$504$		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	−0.956924	−	0.290340i	$-0.906232\pi$
$509$	−2607.00	+	4515.46i	−0.227020	+	0.393210i	0.729903	+	0.683550i	$0.239565\pi$
$510$		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	0.893917	−	0.448232i	$-0.147946\pi$
$521$	699.000	+	1210.70i	0.0587788	+	0.101808i	−0.835139	+	0.550039i	$0.814613\pi$
$522$		0			0
$523$	−9290.00	+	16090.8i	−0.776718	+	1.34531i	0.157106	+	0.987582i	$0.449783\pi$
$523$	−9290.00	+	16090.8i	−0.776718	+	1.34531i	−0.933824	+	0.357733i	$0.883550\pi$
$524$		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	0.702234	−	0.711946i	$-0.252186\pi$
$563$	−3546.00	−	6141.85i	−0.265446	−	0.459766i	−0.967680	+	0.252180i	$0.918853\pi$
$564$		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	$0.121297\pi$
$577$	10897.0	+	18874.2i	0.786218	+	1.36177i	−0.142050	+	0.989859i	$0.545369\pi$
$578$		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	$-0.388559\pi$
$587$	9756.00			0.685985			0.342993	+	0.939338i	$0.388559\pi$
$588$		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	−0.946380	−	0.323057i	$-0.895290\pi$
$593$	−2793.00	+	4837.62i	−0.193414	+	0.335004i	0.752965	+	0.658060i	$0.228623\pi$
$594$		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	$-0.546594\pi$
$601$	−4298.00			−0.291712			−0.145856	+	0.989306i	$0.546594\pi$
$602$		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	−0.688730	−	0.725018i	$-0.741831\pi$
$607$	4240.00	−	7343.90i	0.283519	−	0.491070i	0.972249	+	0.233948i	$0.0751646\pi$
$608$		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	0.721739	−	0.692165i	$-0.243343\pi$
$619$	−3674.00	−	6363.55i	−0.238563	−	0.413203i	−0.960302	+	0.278962i	$0.910010\pi$
$620$		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	$-0.450803\pi$
$643$	5020.00			0.307884			0.153942	+	0.988080i	$0.450803\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−14196.0	−	24588.2i	−0.862600	−	1.49407i	−0.869410	−	0.494091i	$-0.835501\pi$
$647$	−14196.0	−	24588.2i	−0.862600	−	1.49407i	0.00681018	−	0.999977i	$-0.497832\pi$
$648$		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	$-0.933091\pi$
$661$	−11381.0	−	19712.5i	−0.669697	−	1.15995i	0.308292	−	0.951292i	$-0.400243\pi$
$662$		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	$0.376270\pi$
$677$	−10779.0	+	18669.8i	−0.611921	+	1.05988i	−0.990916	+	0.134480i	$0.957064\pi$
$678$		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	−0.931741	−	0.363123i	$-0.881710\pi$
$691$	−2750.00	+	4763.14i	−0.151396	+	0.262226i	0.780345	+	0.625350i	$0.215044\pi$
$692$		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	−0.961847	−	0.273586i	$-0.911790\pi$
$719$	−4704.00	+	8147.57i	−0.243991	+	0.422605i	0.717856	+	0.696191i	$0.245123\pi$
$720$		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	$-0.180562\pi$
$727$	33064.0			1.68676			0.843381	+	0.537316i	$0.180562\pi$
$728$		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	−0.934610	−	0.355674i	$-0.884251\pi$
$733$	−3161.00	+	5475.01i	−0.159283	+	0.275886i	0.775328	+	0.631559i	$0.217585\pi$
$734$		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	−0.965767	−	0.259411i	$-0.916472\pi$
$761$	−5421.00	+	9389.45i	−0.258227	+	0.447263i	0.707540	+	0.706674i	$0.249805\pi$
$762$		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	$-0.730687\pi$
$769$	−28274.0			−1.32586			−0.662930	+	0.748681i	$0.730687\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	16173.0	+	28012.5i	0.752526	+	1.30341i	0.946595	+	0.322425i	$0.104498\pi$
$773$	16173.0	+	28012.5i	0.752526	+	1.30341i	−0.194069	+	0.980988i	$0.562169\pi$
$774$		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	$0.0722372\pi$
$787$	15058.0	+	26081.2i	0.682033	+	1.18132i	−0.292327	+	0.956318i	$0.594430\pi$
$788$		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
$796$		0			0
$797$	−6594.00			−0.293063			−0.146532	−	0.989206i	$-0.546811\pi$
$797$	−6594.00			−0.293063			−0.146532	+	0.989206i	$0.546811\pi$
$798$		0			0
$799$	−5760.00			−0.255036
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−3300.00	+	5715.77i	−0.145024	+	0.251189i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−21507.0	−	37251.2i	−0.934667	−	1.61889i	−0.775226	−	0.631684i	$-0.782364\pi$
$809$	−21507.0	−	37251.2i	−0.934667	−	1.61889i	−0.159441	−	0.987207i	$-0.550969\pi$
$810$		0			0
$811$	14164.0			0.613274			0.306637	−	0.951827i	$-0.400796\pi$
$811$	14164.0			0.613274			0.306637	+	0.951827i	$0.400796\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−6324.00	−	10953.5i	−0.271804	−	0.470778i
$816$		0			0
$817$	−5848.00	+	10129.0i	−0.250423	+	0.433745i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	17415.0	−	30163.7i	0.740302	−	1.28224i	−0.212056	−	0.977257i	$-0.568016\pi$
$821$	17415.0	−	30163.7i	0.740302	−	1.28224i	0.952358	−	0.304983i	$-0.0986507\pi$
$822$		0			0
$823$	−15508.0	−	26860.6i	−0.656835	−	1.13767i	−0.981430	−	0.191818i	$-0.938562\pi$
$823$	−15508.0	−	26860.6i	−0.656835	−	1.13767i	0.324596	−	0.945853i	$-0.394772\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−9876.00			−0.415263			−0.207631	−	0.978207i	$-0.566575\pi$
$827$	−9876.00			−0.415263			−0.207631	+	0.978207i	$0.566575\pi$
$828$		0			0
$829$	−1577.00	−	2731.44i	−0.0660693	−	0.114435i	0.831099	−	0.556125i	$-0.187713\pi$
$829$	−1577.00	−	2731.44i	−0.0660693	−	0.114435i	−0.897168	+	0.441690i	$0.854379\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	5832.00	−	10101.3i	0.241706	−	0.418647i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	36936.0			1.51987			0.759936	−	0.649998i	$-0.225230\pi$
$839$	36936.0			1.51987			0.759936	+	0.649998i	$0.225230\pi$
$840$		0			0
$841$	36127.0			1.48128
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−13581.0	+	23523.0i	−0.552900	+	0.957651i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	11880.0	+	20576.8i	0.478544	+	0.828863i
$852$		0			0
$853$	−9638.00			−0.386869			−0.193434	−	0.981113i	$-0.561963\pi$
$853$	−9638.00			−0.386869			−0.193434	+	0.981113i	$0.561963\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−5133.00	−	8890.62i	−0.204597	−	0.354373i	0.745407	−	0.666610i	$-0.232255\pi$
$857$	−5133.00	−	8890.62i	−0.204597	−	0.354373i	−0.950004	+	0.312237i	$0.898922\pi$
$858$		0			0
$859$	−2042.00	+	3536.85i	−0.0811084	+	0.140484i	−0.903726	−	0.428111i	$-0.859179\pi$
$859$	−2042.00	+	3536.85i	−0.0811084	+	0.140484i	0.822618	+	0.568595i	$0.192513\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−96.0000	+	166.277i	−0.00378665	+	0.00655867i	−0.867913	−	0.496717i	$-0.834539\pi$
$863$	−96.0000	+	166.277i	−0.00378665	+	0.00655867i	0.864126	+	0.503276i	$0.167872\pi$
$864$		0			0
$865$	4086.00	+	7077.16i	0.160611	+	0.278186i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−2496.00			−0.0974350
$870$		0			0
$871$	−5740.00	−	9941.97i	−0.223298	−	0.386763i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−9955.00	+	17242.6i	−0.383303	+	0.663900i	−0.991532	−	0.129862i	$-0.958547\pi$
$877$	−9955.00	+	17242.6i	−0.383303	+	0.663900i	0.608229	+	0.793761i	$0.291880\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	14802.0			0.566052			0.283026	−	0.959112i	$-0.408662\pi$
$881$	14802.0			0.566052			0.283026	+	0.959112i	$0.408662\pi$
$882$		0			0
$883$	−32548.0			−1.24046			−0.620231	−	0.784419i	$-0.712961\pi$
$883$	−32548.0			−1.24046			−0.620231	+	0.784419i	$0.712961\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−732.000	+	1267.86i	−0.0277093	+	0.0479939i	−0.879548	−	0.475811i	$-0.842155\pi$
$887$	−732.000	+	1267.86i	−0.0277093	+	0.0479939i	0.851838	+	0.523805i	$0.175488\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	6528.00	+	11306.8i	0.244626	+	0.423705i
$894$		0			0
$895$	−9576.00			−0.357643
$896$		0			0
$897$		0			0
$898$		0			0
$899$	13776.0	+	23860.7i	0.511074	+	0.885206i
$900$		0			0
$901$	−8370.00	+	14497.3i	−0.309484	+	0.536042i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−5070.00	+	8781.50i	−0.186224	+	0.322549i
$906$		0			0
$907$	24782.0	+	42923.7i	0.907247	+	1.57140i	0.817873	+	0.575399i	$0.195153\pi$
$907$	24782.0	+	42923.7i	0.907247	+	1.57140i	0.0893742	+	0.995998i	$0.471513\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−8448.00			−0.307239			−0.153619	−	0.988130i	$-0.549093\pi$
$911$	−8448.00			−0.307239			−0.153619	+	0.988130i	$0.549093\pi$
$912$		0			0
$913$	−3096.00	−	5362.43i	−0.112226	−	0.194382i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−7300.00	+	12644.0i	−0.262029	+	0.453848i	−0.966781	−	0.255606i	$-0.917725\pi$
$919$	−7300.00	+	12644.0i	−0.262029	+	0.453848i	0.704752	+	0.709454i	$0.251058\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	68880.0			2.45635
$924$		0			0
$925$	−9790.00			−0.347993
$926$		0			0
$927$		0			0
$928$		0			0
$929$	10551.0	−	18274.9i	0.372623	−	0.645403i	−0.617345	−	0.786693i	$-0.711792\pi$
$929$	10551.0	−	18274.9i	0.372623	−	0.645403i	0.989968	+	0.141290i	$0.0451250\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	1080.00	+	1870.61i	0.0377752	+	0.0654285i
$936$		0			0
$937$	20806.0			0.725403			0.362701	−	0.931905i	$-0.381854\pi$
$937$	20806.0			0.725403			0.362701	+	0.931905i	$0.381854\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−12255.0	−	21226.3i	−0.424550	−	0.735342i	0.571828	−	0.820373i	$-0.306234\pi$
$941$	−12255.0	−	21226.3i	−0.424550	−	0.735342i	−0.996378	+	0.0850311i	$0.972901\pi$
$942$		0			0
$943$	−26568.0	+	46017.1i	−0.917469	+	1.58910i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−22074.0	+	38233.3i	−0.757454	+	1.31195i	0.186692	+	0.982419i	$0.440223\pi$
$947$	−22074.0	+	38233.3i	−0.757454	+	1.31195i	−0.944145	+	0.329530i	$0.893110\pi$
$948$		0			0
$949$	−22550.0	−	39057.7i	−0.771342	−	1.33600i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−27114.0			−0.921625			−0.460812	−	0.887498i	$-0.652442\pi$
$953$	−27114.0			−0.921625			−0.460812	+	0.887498i	$0.652442\pi$
$954$		0			0
$955$	10656.0	+	18456.7i	0.361068	+	0.625388i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	8623.50	−	14936.3i	0.289467	−	0.501371i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−16116.0			−0.537609
$966$		0			0
$967$	−10264.0			−0.341332			−0.170666	−	0.985329i	$-0.554592\pi$
$967$	−10264.0			−0.341332			−0.170666	+	0.985329i	$0.554592\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−25734.0	+	44572.6i	−0.850508	+	1.47312i	0.0302424	+	0.999543i	$0.490372\pi$
$971$	−25734.0	+	44572.6i	−0.850508	+	1.47312i	−0.880750	+	0.473581i	$0.842961\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−11895.0	−	20602.7i	−0.389514	−	0.674657i	0.602871	−	0.797839i	$-0.294024\pi$
$977$	−11895.0	−	20602.7i	−0.389514	−	0.674657i	−0.992384	+	0.123182i	$0.960690\pi$
$978$		0			0
$979$	−16776.0			−0.547664
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−13212.0	−	22883.9i	−0.428685	−	0.742504i	0.568072	−	0.822979i	$-0.307690\pi$
$983$	−13212.0	−	22883.9i	−0.428685	−	0.742504i	−0.996757	+	0.0804749i	$0.974356\pi$
$984$		0			0
$985$	−4230.00	+	7326.57i	−0.136831	+	0.236999i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−18576.0	+	32174.6i	−0.597252	+	1.03447i
$990$		0			0
$991$	−19744.0	−	34197.6i	−0.632885	−	1.09619i	−0.986959	−	0.160971i	$-0.948537\pi$
$991$	−19744.0	−	34197.6i	−0.632885	−	1.09619i	0.354074	−	0.935217i	$-0.384796\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	17808.0			0.567388
$996$		0			0
$997$	15427.0	+	26720.3i	0.490048	+	0.848788i	0.999934	−	0.0114536i	$-0.00364586\pi$
$997$	15427.0	+	26720.3i	0.490048	+	0.848788i	−0.509886	+	0.860242i	$0.670313\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.4.k.e.1549.1		2
3.2	odd	2		196.4.e.d.177.1		2
7.2	even	3		1764.4.a.k.1.1		1
7.3	odd	6		1764.4.k.k.361.1		2
7.4	even	3	inner	1764.4.k.e.361.1		2
7.5	odd	6		252.4.a.c.1.1		1
7.6	odd	2		1764.4.k.k.1549.1		2
21.2	odd	6		196.4.a.b.1.1		1
21.5	even	6		28.4.a.b.1.1	✓	1
21.11	odd	6		196.4.e.d.165.1		2
21.17	even	6		196.4.e.c.165.1		2
21.20	even	2		196.4.e.c.177.1		2
28.19	even	6		1008.4.a.f.1.1		1
84.23	even	6		784.4.a.n.1.1		1
84.47	odd	6		112.4.a.c.1.1		1
105.47	odd	12		700.4.e.f.449.1		2
105.68	odd	12		700.4.e.f.449.2		2
105.89	even	6		700.4.a.e.1.1		1
168.5	even	6		448.4.a.d.1.1		1
168.131	odd	6		448.4.a.m.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.4.a.b.1.1	✓	1	21.5	even	6
112.4.a.c.1.1		1	84.47	odd	6
196.4.a.b.1.1		1	21.2	odd	6
196.4.e.c.165.1		2	21.17	even	6
196.4.e.c.177.1		2	21.20	even	2
196.4.e.d.165.1		2	21.11	odd	6
196.4.e.d.177.1		2	3.2	odd	2
252.4.a.c.1.1		1	7.5	odd	6
448.4.a.d.1.1		1	168.5	even	6
448.4.a.m.1.1		1	168.131	odd	6
700.4.a.e.1.1		1	105.89	even	6
700.4.e.f.449.1		2	105.47	odd	12
700.4.e.f.449.2		2	105.68	odd	12
784.4.a.n.1.1		1	84.23	even	6
1008.4.a.f.1.1		1	28.19	even	6
1764.4.a.k.1.1		1	7.2	even	3
1764.4.k.e.361.1		2	7.4	even	3	inner
1764.4.k.e.1549.1		2	1.1	even	1	trivial
1764.4.k.k.361.1		2	7.3	odd	6
1764.4.k.k.1549.1		2	7.6	odd	2

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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