LMFDB - Embedded newform 189.3.d.a.55.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [189,3,Mod(55,189)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(189, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("189.55");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 189 = 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	189.d (of order $2$, degree $1$, 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:	$5.14987699641$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			55.2
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	189.55
Dual form			189.3.d.a.55.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} -3.00000 q^{4} +1.73205i q^{5} +(-3.50000 - 6.06218i) q^{7} +7.00000 q^{8} -1.73205i q^{10} +17.0000 q^{11} +17.3205i q^{13} +(3.50000 + 6.06218i) q^{14} +5.00000 q^{16} +31.1769i q^{17} -20.7846i q^{19} -5.19615i q^{20} -17.0000 q^{22} +32.0000 q^{23} +22.0000 q^{25} -17.3205i q^{26} +(10.5000 + 18.1865i) q^{28} +2.00000 q^{29} +22.5167i q^{31} -33.0000 q^{32} -31.1769i q^{34} +(10.5000 - 6.06218i) q^{35} +8.00000 q^{37} +20.7846i q^{38} +12.1244i q^{40} -13.8564i q^{41} +26.0000 q^{43} -51.0000 q^{44} -32.0000 q^{46} -38.1051i q^{47} +(-24.5000 + 42.4352i) q^{49} -22.0000 q^{50} -51.9615i q^{52} -1.00000 q^{53} +29.4449i q^{55} +(-24.5000 - 42.4352i) q^{56} -2.00000 q^{58} +90.0666i q^{59} -69.2820i q^{61} -22.5167i q^{62} +13.0000 q^{64} -30.0000 q^{65} -34.0000 q^{67} -93.5307i q^{68} +(-10.5000 + 6.06218i) q^{70} -10.0000 q^{71} -57.1577i q^{73} -8.00000 q^{74} +62.3538i q^{76} +(-59.5000 - 103.057i) q^{77} -46.0000 q^{79} +8.66025i q^{80} +13.8564i q^{82} +102.191i q^{83} -54.0000 q^{85} -26.0000 q^{86} +119.000 q^{88} -51.9615i q^{89} +(105.000 - 60.6218i) q^{91} -96.0000 q^{92} +38.1051i q^{94} +36.0000 q^{95} +91.7987i q^{97} +(24.5000 - 42.4352i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 6 q^{4} - 7 q^{7} + 14 q^{8} + 34 q^{11} + 7 q^{14} + 10 q^{16} - 34 q^{22} + 64 q^{23} + 44 q^{25} + 21 q^{28} + 4 q^{29} - 66 q^{32} + 21 q^{35} + 16 q^{37} + 52 q^{43} - 102 q^{44} - 64 q^{46}+ \cdots + 49 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 - 6 * q^4 - 7 * q^7 + 14 * q^8 + 34 * q^11 + 7 * q^14 + 10 * q^16 - 34 * q^22 + 64 * q^23 + 44 * q^25 + 21 * q^28 + 4 * q^29 - 66 * q^32 + 21 * q^35 + 16 * q^37 + 52 * q^43 - 102 * q^44 - 64 * q^46 - 49 * q^49 - 44 * q^50 - 2 * q^53 - 49 * q^56 - 4 * q^58 + 26 * q^64 - 60 * q^65 - 68 * q^67 - 21 * q^70 - 20 * q^71 - 16 * q^74 - 119 * q^77 - 92 * q^79 - 108 * q^85 - 52 * q^86 + 238 * q^88 + 210 * q^91 - 192 * q^92 + 72 * q^95 + 49 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/189\mathbb{Z}\right)^\times$.

$n$	$29$	$136$
$\chi(n)$	$1$	$-1$

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 $

$2$	−1.00000			−0.500000			−0.250000	−	0.968246i	$-0.580431\pi$
$2$	−1.00000			−0.500000			−0.250000	+	0.968246i	$0.580431\pi$
$3$		0			0
$3$		0			0
$4$	−3.00000			−0.750000
$5$			1.73205i			0.346410i	0.984886	+	0.173205i	$0.0554123\pi$
$5$			1.73205i			0.346410i	−0.984886	+	0.173205i	$0.944588\pi$
$6$		0			0
$7$	−3.50000	−	6.06218i	−0.500000	−	0.866025i
$7$	−3.50000	−	6.06218i	−0.500000	−	0.866025i
$8$	7.00000			0.875000
$9$		0			0
$10$		−	1.73205i		−	0.173205i
$11$	17.0000			1.54545			0.772727	−	0.634738i	$-0.218892\pi$
$11$	17.0000			1.54545			0.772727	+	0.634738i	$0.218892\pi$
$12$		0			0
$13$			17.3205i			1.33235i	0.745797	+	0.666173i	$0.232069\pi$
$13$			17.3205i			1.33235i	−0.745797	+	0.666173i	$0.767931\pi$
$14$	3.50000	+	6.06218i	0.250000	+	0.433013i
$15$		0			0
$16$	5.00000			0.312500
$17$			31.1769i			1.83394i	0.398961	+	0.916968i	$0.369371\pi$
$17$			31.1769i			1.83394i	−0.398961	+	0.916968i	$0.630629\pi$
$18$		0			0
$19$		−	20.7846i		−	1.09393i	−0.837157	−	0.546963i	$-0.815784\pi$
$19$		−	20.7846i		−	1.09393i	0.837157	−	0.546963i	$-0.184216\pi$
$20$		−	5.19615i		−	0.259808i
$21$		0			0
$22$	−17.0000			−0.772727
$23$	32.0000			1.39130			0.695652	−	0.718379i	$-0.255116\pi$
$23$	32.0000			1.39130			0.695652	+	0.718379i	$0.255116\pi$
$24$		0			0
$25$	22.0000			0.880000
$26$		−	17.3205i		−	0.666173i
$27$		0			0
$28$	10.5000	+	18.1865i	0.375000	+	0.649519i
$29$	2.00000			0.0689655			0.0344828	−	0.999405i	$-0.489022\pi$
$29$	2.00000			0.0689655			0.0344828	+	0.999405i	$0.489022\pi$
$30$		0			0
$31$			22.5167i			0.726344i	0.931722	+	0.363172i	$0.118306\pi$
$31$			22.5167i			0.726344i	−0.931722	+	0.363172i	$0.881694\pi$
$32$	−33.0000			−1.03125
$33$		0			0
$34$		−	31.1769i		−	0.916968i
$35$	10.5000	−	6.06218i	0.300000	−	0.173205i
$36$		0			0
$37$	8.00000			0.216216			0.108108	−	0.994139i	$-0.465521\pi$
$37$	8.00000			0.216216			0.108108	+	0.994139i	$0.465521\pi$
$38$			20.7846i			0.546963i
$39$		0			0
$40$			12.1244i			0.303109i
$41$		−	13.8564i		−	0.337961i	−0.985619	−	0.168981i	$-0.945952\pi$
$41$		−	13.8564i		−	0.337961i	0.985619	−	0.168981i	$-0.0540475\pi$
$42$		0			0
$43$	26.0000			0.604651			0.302326	−	0.953205i	$-0.402237\pi$
$43$	26.0000			0.604651			0.302326	+	0.953205i	$0.402237\pi$
$44$	−51.0000			−1.15909
$45$		0			0
$46$	−32.0000			−0.695652
$47$		−	38.1051i		−	0.810747i	−0.914151	−	0.405374i	$-0.867141\pi$
$47$		−	38.1051i		−	0.810747i	0.914151	−	0.405374i	$-0.132859\pi$
$48$		0			0
$49$	−24.5000	+	42.4352i	−0.500000	+	0.866025i
$50$	−22.0000			−0.440000
$51$		0			0
$52$		−	51.9615i		−	0.999260i
$53$	−1.00000			−0.0188679			−0.00943396	−	0.999955i	$-0.503003\pi$
$53$	−1.00000			−0.0188679			−0.00943396	+	0.999955i	$0.503003\pi$
$54$		0			0
$55$			29.4449i			0.535361i
$56$	−24.5000	−	42.4352i	−0.437500	−	0.757772i
$57$		0			0
$58$	−2.00000			−0.0344828
$59$			90.0666i			1.52655i	0.646072	+	0.763277i	$0.276411\pi$
$59$			90.0666i			1.52655i	−0.646072	+	0.763277i	$0.723589\pi$
$60$		0			0
$61$		−	69.2820i		−	1.13577i	−0.823108	−	0.567886i	$-0.807762\pi$
$61$		−	69.2820i		−	1.13577i	0.823108	−	0.567886i	$-0.192238\pi$
$62$		−	22.5167i		−	0.363172i
$63$		0			0
$64$	13.0000			0.203125
$65$	−30.0000			−0.461538
$66$		0			0
$67$	−34.0000			−0.507463			−0.253731	−	0.967275i	$-0.581658\pi$
$67$	−34.0000			−0.507463			−0.253731	+	0.967275i	$0.581658\pi$
$68$		−	93.5307i		−	1.37545i
$69$		0			0
$70$	−10.5000	+	6.06218i	−0.150000	+	0.0866025i
$71$	−10.0000			−0.140845			−0.0704225	−	0.997517i	$-0.522435\pi$
$71$	−10.0000			−0.140845			−0.0704225	+	0.997517i	$0.522435\pi$
$72$		0			0
$73$		−	57.1577i		−	0.782982i	−0.920182	−	0.391491i	$-0.871959\pi$
$73$		−	57.1577i		−	0.782982i	0.920182	−	0.391491i	$-0.128041\pi$
$74$	−8.00000			−0.108108
$75$		0			0
$76$			62.3538i			0.820445i
$77$	−59.5000	−	103.057i	−0.772727	−	1.33840i
$78$		0			0
$79$	−46.0000			−0.582278			−0.291139	−	0.956681i	$-0.594034\pi$
$79$	−46.0000			−0.582278			−0.291139	+	0.956681i	$0.594034\pi$
$80$			8.66025i			0.108253i
$81$		0			0
$82$			13.8564i			0.168981i
$83$			102.191i			1.23122i	0.788052	+	0.615608i	$0.211090\pi$
$83$			102.191i			1.23122i	−0.788052	+	0.615608i	$0.788910\pi$
$84$		0			0
$85$	−54.0000			−0.635294
$86$	−26.0000			−0.302326
$87$		0			0
$88$	119.000			1.35227
$89$		−	51.9615i		−	0.583837i	−0.956443	−	0.291919i	$-0.905706\pi$
$89$		−	51.9615i		−	0.583837i	0.956443	−	0.291919i	$-0.0942937\pi$
$90$		0			0
$91$	105.000	−	60.6218i	1.15385	−	0.666173i
$92$	−96.0000			−1.04348
$93$		0			0
$94$			38.1051i			0.405374i
$95$	36.0000			0.378947
$96$		0			0
$97$			91.7987i			0.946378i	0.880961	+	0.473189i	$0.156897\pi$
$97$			91.7987i			0.946378i	−0.880961	+	0.473189i	$0.843103\pi$
$98$	24.5000	−	42.4352i	0.250000	−	0.433013i
$99$		0			0
$100$	−66.0000			−0.660000
$101$			122.976i			1.21758i	0.793331	+	0.608790i	$0.208345\pi$
$101$			122.976i			1.21758i	−0.793331	+	0.608790i	$0.791655\pi$
$102$		0			0
$103$			6.92820i			0.0672641i	0.999434	+	0.0336321i	$0.0107074\pi$
$103$			6.92820i			0.0672641i	−0.999434	+	0.0336321i	$0.989293\pi$
$104$			121.244i			1.16580i
$105$		0			0
$106$	1.00000			0.00943396
$107$	107.000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$107$	107.000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$108$		0			0
$109$	−88.0000			−0.807339			−0.403670	−	0.914905i	$-0.632266\pi$
$109$	−88.0000			−0.807339			−0.403670	+	0.914905i	$0.632266\pi$
$110$		−	29.4449i		−	0.267681i
$111$		0			0
$112$	−17.5000	−	30.3109i	−0.156250	−	0.270633i
$113$	200.000			1.76991			0.884956	−	0.465675i	$-0.154188\pi$
$113$	200.000			1.76991			0.884956	+	0.465675i	$0.154188\pi$
$114$		0			0
$115$			55.4256i			0.481962i
$116$	−6.00000			−0.0517241
$117$		0			0
$118$		−	90.0666i		−	0.763277i
$119$	189.000	−	109.119i	1.58824	−	0.916968i
$120$		0			0
$121$	168.000			1.38843
$122$			69.2820i			0.567886i
$123$		0			0
$124$		−	67.5500i		−	0.544758i
$125$			81.4064i			0.651251i
$126$		0			0
$127$	143.000			1.12598			0.562992	−	0.826462i	$-0.309650\pi$
$127$	143.000			1.12598			0.562992	+	0.826462i	$0.309650\pi$
$128$	119.000			0.929688
$129$		0			0
$130$	30.0000			0.230769
$131$			12.1244i			0.0925523i	0.998929	+	0.0462762i	$0.0147354\pi$
$131$			12.1244i			0.0925523i	−0.998929	+	0.0462762i	$0.985265\pi$
$132$		0			0
$133$	−126.000	+	72.7461i	−0.947368	+	0.546963i
$134$	34.0000			0.253731
$135$		0			0
$136$			218.238i			1.60469i
$137$	−10.0000			−0.0729927			−0.0364964	−	0.999334i	$-0.511620\pi$
$137$	−10.0000			−0.0729927			−0.0364964	+	0.999334i	$0.511620\pi$
$138$		0			0
$139$		−	138.564i		−	0.996864i	−0.866929	−	0.498432i	$-0.833909\pi$
$139$		−	138.564i		−	0.996864i	0.866929	−	0.498432i	$-0.166091\pi$
$140$	−31.5000	+	18.1865i	−0.225000	+	0.129904i
$141$		0			0
$142$	10.0000			0.0704225
$143$			294.449i			2.05908i
$144$		0			0
$145$			3.46410i			0.0238904i
$146$			57.1577i			0.391491i
$147$		0			0
$148$	−24.0000			−0.162162
$149$	−79.0000			−0.530201			−0.265101	−	0.964221i	$-0.585405\pi$
$149$	−79.0000			−0.530201			−0.265101	+	0.964221i	$0.585405\pi$
$150$		0			0
$151$	−73.0000			−0.483444			−0.241722	−	0.970346i	$-0.577712\pi$
$151$	−73.0000			−0.483444			−0.241722	+	0.970346i	$0.577712\pi$
$152$		−	145.492i		−	0.957186i
$153$		0			0
$154$	59.5000	+	103.057i	0.386364	+	0.669201i
$155$	−39.0000			−0.251613
$156$		0			0
$157$		−	200.918i		−	1.27973i	−0.768487	−	0.639866i	$-0.778990\pi$
$157$		−	200.918i		−	1.27973i	0.768487	−	0.639866i	$-0.221010\pi$
$158$	46.0000			0.291139
$159$		0			0
$160$		−	57.1577i		−	0.357235i
$161$	−112.000	−	193.990i	−0.695652	−	1.20490i
$162$		0			0
$163$	−226.000			−1.38650			−0.693252	−	0.720696i	$-0.743823\pi$
$163$	−226.000			−1.38650			−0.693252	+	0.720696i	$0.743823\pi$
$164$			41.5692i			0.253471i
$165$		0			0
$166$		−	102.191i		−	0.615608i
$167$		−	86.6025i		−	0.518578i	−0.965800	−	0.259289i	$-0.916512\pi$
$167$		−	86.6025i		−	0.518578i	0.965800	−	0.259289i	$-0.0834882\pi$
$168$		0			0
$169$	−131.000			−0.775148
$170$	54.0000			0.317647
$171$		0			0
$172$	−78.0000			−0.453488
$173$		−	74.4782i		−	0.430510i	−0.976558	−	0.215255i	$-0.930942\pi$
$173$		−	74.4782i		−	0.430510i	0.976558	−	0.215255i	$-0.0690582\pi$
$174$		0			0
$175$	−77.0000	−	133.368i	−0.440000	−	0.762102i
$176$	85.0000			0.482955
$177$		0			0
$178$			51.9615i			0.291919i
$179$	119.000			0.664804			0.332402	−	0.943138i	$-0.392141\pi$
$179$	119.000			0.664804			0.332402	+	0.943138i	$0.392141\pi$
$180$		0			0
$181$			62.3538i			0.344496i	0.985054	+	0.172248i	$0.0551031\pi$
$181$			62.3538i			0.344496i	−0.985054	+	0.172248i	$0.944897\pi$
$182$	−105.000	+	60.6218i	−0.576923	+	0.333087i
$183$		0			0
$184$	224.000			1.21739
$185$			13.8564i			0.0748995i
$186$		0			0
$187$			530.008i			2.83426i
$188$			114.315i			0.608060i
$189$		0			0
$190$	−36.0000			−0.189474
$191$	−190.000			−0.994764			−0.497382	−	0.867532i	$-0.665705\pi$
$191$	−190.000			−0.994764			−0.497382	+	0.867532i	$0.665705\pi$
$192$		0			0
$193$	311.000			1.61140			0.805699	−	0.592325i	$-0.201790\pi$
$193$	311.000			1.61140			0.805699	+	0.592325i	$0.201790\pi$
$194$		−	91.7987i		−	0.473189i
$195$		0			0
$196$	73.5000	−	127.306i	0.375000	−	0.649519i
$197$	−307.000			−1.55838			−0.779188	−	0.626791i	$-0.784368\pi$
$197$	−307.000			−1.55838			−0.779188	+	0.626791i	$0.784368\pi$
$198$		0			0
$199$		−	254.611i		−	1.27945i	−0.768602	−	0.639727i	$-0.779047\pi$
$199$		−	254.611i		−	1.27945i	0.768602	−	0.639727i	$-0.220953\pi$
$200$	154.000			0.770000
$201$		0			0
$202$		−	122.976i		−	0.608790i
$203$	−7.00000	−	12.1244i	−0.0344828	−	0.0597259i
$204$		0			0
$205$	24.0000			0.117073
$206$		−	6.92820i		−	0.0336321i
$207$		0			0
$208$			86.6025i			0.416358i
$209$		−	353.338i		−	1.69061i
$210$		0			0
$211$	−196.000			−0.928910			−0.464455	−	0.885597i	$-0.653750\pi$
$211$	−196.000			−0.928910			−0.464455	+	0.885597i	$0.653750\pi$
$212$	3.00000			0.0141509
$213$		0			0
$214$	−107.000			−0.500000
$215$			45.0333i			0.209457i
$216$		0			0
$217$	136.500	−	78.8083i	0.629032	−	0.363172i
$218$	88.0000			0.403670
$219$		0			0
$220$		−	88.3346i		−	0.401521i
$221$	−540.000			−2.44344
$222$		0			0
$223$			76.2102i			0.341750i	0.985293	+	0.170875i	$0.0546594\pi$
$223$			76.2102i			0.341750i	−0.985293	+	0.170875i	$0.945341\pi$
$224$	115.500	+	200.052i	0.515625	+	0.893089i
$225$		0			0
$226$	−200.000			−0.884956
$227$			76.2102i			0.335728i	0.985810	+	0.167864i	$0.0536869\pi$
$227$			76.2102i			0.335728i	−0.985810	+	0.167864i	$0.946313\pi$
$228$		0			0
$229$		−	24.2487i		−	0.105890i	−0.998597	−	0.0529448i	$-0.983139\pi$
$229$		−	24.2487i		−	0.105890i	0.998597	−	0.0529448i	$-0.0168607\pi$
$230$		−	55.4256i		−	0.240981i
$231$		0			0
$232$	14.0000			0.0603448
$233$	−178.000			−0.763948			−0.381974	−	0.924173i	$-0.624756\pi$
$233$	−178.000			−0.763948			−0.381974	+	0.924173i	$0.624756\pi$
$234$		0			0
$235$	66.0000			0.280851
$236$		−	270.200i		−	1.14491i
$237$		0			0
$238$	−189.000	+	109.119i	−0.794118	+	0.458484i
$239$	−88.0000			−0.368201			−0.184100	−	0.982907i	$-0.558937\pi$
$239$	−88.0000			−0.368201			−0.184100	+	0.982907i	$0.558937\pi$
$240$		0			0
$241$			200.918i			0.833684i	0.908979	+	0.416842i	$0.136863\pi$
$241$			200.918i			0.833684i	−0.908979	+	0.416842i	$0.863137\pi$
$242$	−168.000			−0.694215
$243$		0			0
$244$			207.846i			0.851828i
$245$	−73.5000	−	42.4352i	−0.300000	−	0.173205i
$246$		0			0
$247$	360.000			1.45749
$248$			157.617i			0.635551i
$249$		0			0
$250$		−	81.4064i		−	0.325626i
$251$		−	311.769i		−	1.24211i	−0.783768	−	0.621054i	$-0.786705\pi$
$251$		−	311.769i		−	1.24211i	0.783768	−	0.621054i	$-0.213295\pi$
$252$		0			0
$253$	544.000			2.15020
$254$	−143.000			−0.562992
$255$		0			0
$256$	−171.000			−0.667969
$257$		−	3.46410i		−	0.0134790i	−0.999977	−	0.00673950i	$-0.997855\pi$
$257$		−	3.46410i		−	0.0134790i	0.999977	−	0.00673950i	$-0.00214526\pi$
$258$		0			0
$259$	−28.0000	−	48.4974i	−0.108108	−	0.187249i
$260$	90.0000			0.346154
$261$		0			0
$262$		−	12.1244i		−	0.0462762i
$263$	116.000			0.441065			0.220532	−	0.975380i	$-0.429221\pi$
$263$	116.000			0.441065			0.220532	+	0.975380i	$0.429221\pi$
$264$		0			0
$265$		−	1.73205i		−	0.00653604i
$266$	126.000	−	72.7461i	0.473684	−	0.273482i
$267$		0			0
$268$	102.000			0.380597
$269$		−	353.338i		−	1.31353i	−0.754097	−	0.656763i	$-0.771925\pi$
$269$		−	353.338i		−	1.31353i	0.754097	−	0.656763i	$-0.228075\pi$
$270$		0			0
$271$		−	67.5500i		−	0.249262i	−0.992203	−	0.124631i	$-0.960225\pi$
$271$		−	67.5500i		−	0.249262i	0.992203	−	0.124631i	$-0.0397747\pi$
$272$			155.885i			0.573105i
$273$		0			0
$274$	10.0000			0.0364964
$275$	374.000			1.36000
$276$		0			0
$277$	−136.000			−0.490975			−0.245487	−	0.969400i	$-0.578948\pi$
$277$	−136.000			−0.490975			−0.245487	+	0.969400i	$0.578948\pi$
$278$			138.564i			0.498432i
$279$		0			0
$280$	73.5000	−	42.4352i	0.262500	−	0.151554i
$281$	−370.000			−1.31673			−0.658363	−	0.752701i	$-0.728751\pi$
$281$	−370.000			−1.31673			−0.658363	+	0.752701i	$0.728751\pi$
$282$		0			0
$283$			401.836i			1.41991i	0.704245	+	0.709957i	$0.251286\pi$
$283$			401.836i			1.41991i	−0.704245	+	0.709957i	$0.748714\pi$
$284$	30.0000			0.105634
$285$		0			0
$286$		−	294.449i		−	1.02954i
$287$	−84.0000	+	48.4974i	−0.292683	+	0.168981i
$288$		0			0
$289$	−683.000			−2.36332
$290$		−	3.46410i		−	0.0119452i
$291$		0			0
$292$			171.473i			0.587236i
$293$			48.4974i			0.165520i	0.996569	+	0.0827601i	$0.0263735\pi$
$293$			48.4974i			0.165520i	−0.996569	+	0.0827601i	$0.973626\pi$
$294$		0			0
$295$	−156.000			−0.528814
$296$	56.0000			0.189189
$297$		0			0
$298$	79.0000			0.265101
$299$			554.256i			1.85370i
$300$		0			0
$301$	−91.0000	−	157.617i	−0.302326	−	0.523643i
$302$	73.0000			0.241722
$303$		0			0
$304$		−	103.923i		−	0.341852i
$305$	120.000			0.393443
$306$		0			0
$307$		−	426.084i		−	1.38790i	−0.720024	−	0.693949i	$-0.755869\pi$
$307$		−	426.084i		−	1.38790i	0.720024	−	0.693949i	$-0.244131\pi$
$308$	178.500	+	309.171i	0.579545	+	1.00380i
$309$		0			0
$310$	39.0000			0.125806
$311$		−	159.349i		−	0.512375i	−0.966627	−	0.256188i	$-0.917534\pi$
$311$		−	159.349i		−	0.512375i	0.966627	−	0.256188i	$-0.0824665\pi$
$312$		0			0
$313$		−	386.247i		−	1.23402i	−0.786956	−	0.617009i	$-0.788344\pi$
$313$		−	386.247i		−	1.23402i	0.786956	−	0.617009i	$-0.211656\pi$
$314$			200.918i			0.639866i
$315$		0			0
$316$	138.000			0.436709
$317$	227.000			0.716088			0.358044	−	0.933705i	$-0.383444\pi$
$317$	227.000			0.716088			0.358044	+	0.933705i	$0.383444\pi$
$318$		0			0
$319$	34.0000			0.106583
$320$			22.5167i			0.0703646i
$321$		0			0
$322$	112.000	+	193.990i	0.347826	+	0.602452i
$323$	648.000			2.00619
$324$		0			0
$325$			381.051i			1.17247i
$326$	226.000			0.693252
$327$		0			0
$328$		−	96.9948i		−	0.295716i
$329$	−231.000	+	133.368i	−0.702128	+	0.405374i
$330$		0			0
$331$	398.000			1.20242			0.601208	−	0.799092i	$-0.294686\pi$
$331$	398.000			1.20242			0.601208	+	0.799092i	$0.294686\pi$
$332$		−	306.573i		−	0.923413i
$333$		0			0
$334$			86.6025i			0.259289i
$335$		−	58.8897i		−	0.175790i
$336$		0			0
$337$	−610.000			−1.81009			−0.905045	−	0.425317i	$-0.860163\pi$
$337$	−610.000			−1.81009			−0.905045	+	0.425317i	$0.860163\pi$
$338$	131.000			0.387574
$339$		0			0
$340$	162.000			0.476471
$341$			382.783i			1.12253i
$342$		0			0
$343$	343.000			1.00000
$344$	182.000			0.529070
$345$		0			0
$346$			74.4782i			0.215255i
$347$	515.000			1.48415			0.742075	−	0.670317i	$-0.233842\pi$
$347$	515.000			1.48415			0.742075	+	0.670317i	$0.233842\pi$
$348$		0			0
$349$			325.626i			0.933025i	0.884515	+	0.466512i	$0.154490\pi$
$349$			325.626i			0.933025i	−0.884515	+	0.466512i	$0.845510\pi$
$350$	77.0000	+	133.368i	0.220000	+	0.381051i
$351$		0			0
$352$	−561.000			−1.59375
$353$		−	235.559i		−	0.667306i	−0.942696	−	0.333653i	$-0.891719\pi$
$353$		−	235.559i		−	0.667306i	0.942696	−	0.333653i	$-0.108281\pi$
$354$		0			0
$355$		−	17.3205i		−	0.0487902i
$356$			155.885i			0.437878i
$357$		0			0
$358$	−119.000			−0.332402
$359$	−226.000			−0.629526			−0.314763	−	0.949170i	$-0.601925\pi$
$359$	−226.000			−0.629526			−0.314763	+	0.949170i	$0.601925\pi$
$360$		0			0
$361$	−71.0000			−0.196676
$362$		−	62.3538i		−	0.172248i
$363$		0			0
$364$	−315.000	+	181.865i	−0.865385	+	0.499630i
$365$	99.0000			0.271233
$366$		0			0
$367$		−	531.740i		−	1.44888i	−0.689337	−	0.724441i	$-0.742098\pi$
$367$		−	531.740i		−	1.44888i	0.689337	−	0.724441i	$-0.257902\pi$
$368$	160.000			0.434783
$369$		0			0
$370$		−	13.8564i		−	0.0374497i
$371$	3.50000	+	6.06218i	0.00943396	+	0.0163401i
$372$		0			0
$373$	218.000			0.584450			0.292225	−	0.956350i	$-0.405604\pi$
$373$	218.000			0.584450			0.292225	+	0.956350i	$0.405604\pi$
$374$		−	530.008i		−	1.41713i
$375$		0			0
$376$		−	266.736i		−	0.709404i
$377$			34.6410i			0.0918860i
$378$		0			0
$379$	296.000			0.781003			0.390501	−	0.920602i	$-0.372302\pi$
$379$	296.000			0.781003			0.390501	+	0.920602i	$0.372302\pi$
$380$	−108.000			−0.284211
$381$		0			0
$382$	190.000			0.497382
$383$			640.859i			1.67326i	0.547768	+	0.836630i	$0.315478\pi$
$383$			640.859i			1.67326i	−0.547768	+	0.836630i	$0.684522\pi$
$384$		0			0
$385$	178.500	−	103.057i	0.463636	−	0.267681i
$386$	−311.000			−0.805699
$387$		0			0
$388$		−	275.396i		−	0.709784i
$389$	−457.000			−1.17481			−0.587404	−	0.809294i	$-0.699850\pi$
$389$	−457.000			−1.17481			−0.587404	+	0.809294i	$0.699850\pi$
$390$		0			0
$391$			997.661i			2.55156i
$392$	−171.500	+	297.047i	−0.437500	+	0.757772i
$393$		0			0
$394$	307.000			0.779188
$395$		−	79.6743i		−	0.201707i
$396$		0			0
$397$			114.315i			0.287948i	0.989581	+	0.143974i	$0.0459882\pi$
$397$			114.315i			0.287948i	−0.989581	+	0.143974i	$0.954012\pi$
$398$			254.611i			0.639727i
$399$		0			0
$400$	110.000			0.275000
$401$	−448.000			−1.11721			−0.558603	−	0.829435i	$-0.688663\pi$
$401$	−448.000			−1.11721			−0.558603	+	0.829435i	$0.688663\pi$
$402$		0			0
$403$	−390.000			−0.967742
$404$		−	368.927i		−	0.913185i
$405$		0			0
$406$	7.00000	+	12.1244i	0.0172414	+	0.0298629i
$407$	136.000			0.334152
$408$		0			0
$409$		−	476.314i		−	1.16458i	−0.812980	−	0.582291i	$-0.802156\pi$
$409$		−	476.314i		−	1.16458i	0.812980	−	0.582291i	$-0.197844\pi$
$410$	−24.0000			−0.0585366
$411$		0			0
$412$		−	20.7846i		−	0.0504481i
$413$	546.000	−	315.233i	1.32203	−	0.763277i
$414$		0			0
$415$	−177.000			−0.426506
$416$		−	571.577i		−	1.37398i
$417$		0			0
$418$			353.338i			0.845307i
$419$		−	242.487i		−	0.578728i	−0.957219	−	0.289364i	$-0.906556\pi$
$419$		−	242.487i		−	0.578728i	0.957219	−	0.289364i	$-0.0934438\pi$
$420$		0			0
$421$	548.000			1.30166			0.650831	−	0.759222i	$-0.274420\pi$
$421$	548.000			1.30166			0.650831	+	0.759222i	$0.274420\pi$
$422$	196.000			0.464455
$423$		0			0
$424$	−7.00000			−0.0165094
$425$			685.892i			1.61386i
$426$		0			0
$427$	−420.000	+	242.487i	−0.983607	+	0.567886i
$428$	−321.000			−0.750000
$429$		0			0
$430$		−	45.0333i		−	0.104729i
$431$	−358.000			−0.830626			−0.415313	−	0.909678i	$-0.636328\pi$
$431$	−358.000			−0.830626			−0.415313	+	0.909678i	$0.636328\pi$
$432$		0			0
$433$			161.081i			0.372011i	0.982549	+	0.186005i	$0.0595542\pi$
$433$			161.081i			0.372011i	−0.982549	+	0.186005i	$0.940446\pi$
$434$	−136.500	+	78.8083i	−0.314516	+	0.181586i
$435$		0			0
$436$	264.000			0.605505
$437$		−	665.108i		−	1.52199i
$438$		0			0
$439$		−	334.286i		−	0.761471i	−0.924684	−	0.380736i	$-0.875671\pi$
$439$		−	334.286i		−	0.761471i	0.924684	−	0.380736i	$-0.124329\pi$
$440$			206.114i			0.468441i
$441$		0			0
$442$	540.000			1.22172
$443$	650.000			1.46727			0.733634	−	0.679544i	$-0.237823\pi$
$443$	650.000			1.46727			0.733634	+	0.679544i	$0.237823\pi$
$444$		0			0
$445$	90.0000			0.202247
$446$		−	76.2102i		−	0.170875i
$447$		0			0
$448$	−45.5000	−	78.8083i	−0.101562	−	0.175911i
$449$	−316.000			−0.703786			−0.351893	−	0.936040i	$-0.614462\pi$
$449$	−316.000			−0.703786			−0.351893	+	0.936040i	$0.614462\pi$
$450$		0			0
$451$		−	235.559i		−	0.522304i
$452$	−600.000			−1.32743
$453$		0			0
$454$		−	76.2102i		−	0.167864i
$455$	105.000	+	181.865i	0.230769	+	0.399704i
$456$		0			0
$457$	−19.0000			−0.0415755			−0.0207877	−	0.999784i	$-0.506617\pi$
$457$	−19.0000			−0.0415755			−0.0207877	+	0.999784i	$0.506617\pi$
$458$			24.2487i			0.0529448i
$459$		0			0
$460$		−	166.277i		−	0.361471i
$461$		−	355.070i		−	0.770218i	−0.922871	−	0.385109i	$-0.874164\pi$
$461$		−	355.070i		−	0.770218i	0.922871	−	0.385109i	$-0.125836\pi$
$462$		0			0
$463$	−835.000			−1.80346			−0.901728	−	0.432304i	$-0.857701\pi$
$463$	−835.000			−1.80346			−0.901728	+	0.432304i	$0.857701\pi$
$464$	10.0000			0.0215517
$465$		0			0
$466$	178.000			0.381974
$467$			431.281i			0.923513i	0.887007	+	0.461757i	$0.152781\pi$
$467$			431.281i			0.923513i	−0.887007	+	0.461757i	$0.847219\pi$
$468$		0			0
$469$	119.000	+	206.114i	0.253731	+	0.439476i
$470$	−66.0000			−0.140426
$471$		0			0
$472$			630.466i			1.33573i
$473$	442.000			0.934461
$474$		0			0
$475$		−	457.261i		−	0.962656i
$476$	−567.000	+	327.358i	−1.19118	+	0.687726i
$477$		0			0
$478$	88.0000			0.184100
$479$		−	620.074i		−	1.29452i	−0.762270	−	0.647259i	$-0.775915\pi$
$479$		−	620.074i		−	1.29452i	0.762270	−	0.647259i	$-0.224085\pi$
$480$		0			0
$481$			138.564i			0.288075i
$482$		−	200.918i		−	0.416842i
$483$		0			0
$484$	−504.000			−1.04132
$485$	−159.000			−0.327835
$486$		0			0
$487$	98.0000			0.201232			0.100616	−	0.994925i	$-0.467919\pi$
$487$	98.0000			0.201232			0.100616	+	0.994925i	$0.467919\pi$
$488$		−	484.974i		−	0.993800i
$489$		0			0
$490$	73.5000	+	42.4352i	0.150000	+	0.0866025i
$491$	−283.000			−0.576375			−0.288187	−	0.957574i	$-0.593053\pi$
$491$	−283.000			−0.576375			−0.288187	+	0.957574i	$0.593053\pi$
$492$		0			0
$493$			62.3538i			0.126478i
$494$	−360.000			−0.728745
$495$		0			0
$496$			112.583i			0.226982i
$497$	35.0000	+	60.6218i	0.0704225	+	0.121975i
$498$		0			0
$499$	302.000			0.605210			0.302605	−	0.953116i	$-0.402144\pi$
$499$	302.000			0.605210			0.302605	+	0.953116i	$0.402144\pi$
$500$		−	244.219i		−	0.488438i
$501$		0			0
$502$			311.769i			0.621054i
$503$		−	665.108i		−	1.32228i	−0.750262	−	0.661141i	$-0.770073\pi$
$503$		−	665.108i		−	1.32228i	0.750262	−	0.661141i	$-0.229927\pi$
$504$		0			0
$505$	−213.000			−0.421782
$506$	−544.000			−1.07510
$507$		0			0
$508$	−429.000			−0.844488
$509$			583.701i			1.14676i	0.819289	+	0.573380i	$0.194368\pi$
$509$			583.701i			1.14676i	−0.819289	+	0.573380i	$0.805632\pi$
$510$		0			0
$511$	−346.500	+	200.052i	−0.678082	+	0.391491i
$512$	−305.000			−0.595703
$513$		0			0
$514$			3.46410i			0.00673950i
$515$	−12.0000			−0.0233010
$516$		0			0
$517$		−	647.787i		−	1.25297i
$518$	28.0000	+	48.4974i	0.0540541	+	0.0936244i
$519$		0			0
$520$	−210.000			−0.403846
$521$		−	280.592i		−	0.538565i	−0.963061	−	0.269282i	$-0.913214\pi$
$521$		−	280.592i		−	0.538565i	0.963061	−	0.269282i	$-0.0867865\pi$
$522$		0			0
$523$		−	966.484i		−	1.84796i	−0.382438	−	0.923981i	$-0.624915\pi$
$523$		−	966.484i		−	1.84796i	0.382438	−	0.923981i	$-0.375085\pi$
$524$		−	36.3731i		−	0.0694142i
$525$		0			0
$526$	−116.000			−0.220532
$527$	−702.000			−1.33207
$528$		0			0
$529$	495.000			0.935728
$530$			1.73205i			0.00326802i
$531$		0			0
$532$	378.000	−	218.238i	0.710526	−	0.410223i
$533$	240.000			0.450281
$534$		0			0
$535$			185.329i			0.346410i
$536$	−238.000			−0.444030
$537$		0			0
$538$			353.338i			0.656763i
$539$	−416.500	+	721.399i	−0.772727	+	1.33840i
$540$		0			0
$541$	−784.000			−1.44917			−0.724584	−	0.689186i	$-0.757968\pi$
$541$	−784.000			−1.44917			−0.724584	+	0.689186i	$0.757968\pi$
$542$			67.5500i			0.124631i
$543$		0			0
$544$		−	1028.84i		−	1.89125i
$545$		−	152.420i		−	0.279671i
$546$		0			0
$547$	26.0000			0.0475320			0.0237660	−	0.999718i	$-0.492434\pi$
$547$	26.0000			0.0475320			0.0237660	+	0.999718i	$0.492434\pi$
$548$	30.0000			0.0547445
$549$		0			0
$550$	−374.000			−0.680000
$551$		−	41.5692i		−	0.0754432i
$552$		0			0
$553$	161.000	+	278.860i	0.291139	+	0.504268i
$554$	136.000			0.245487
$555$		0			0
$556$			415.692i			0.747648i
$557$	629.000			1.12926			0.564632	−	0.825343i	$-0.309018\pi$
$557$	629.000			1.12926			0.564632	+	0.825343i	$0.309018\pi$
$558$		0			0
$559$			450.333i			0.805605i
$560$	52.5000	−	30.3109i	0.0937500	−	0.0541266i
$561$		0			0
$562$	370.000			0.658363
$563$			874.686i			1.55362i	0.629738	+	0.776808i	$0.283162\pi$
$563$			874.686i			1.55362i	−0.629738	+	0.776808i	$0.716838\pi$
$564$		0			0
$565$			346.410i			0.613115i
$566$		−	401.836i		−	0.709957i
$567$		0			0
$568$	−70.0000			−0.123239
$569$	116.000			0.203866			0.101933	−	0.994791i	$-0.467497\pi$
$569$	116.000			0.203866			0.101933	+	0.994791i	$0.467497\pi$
$570$		0			0
$571$	590.000			1.03327			0.516637	−	0.856204i	$-0.327184\pi$
$571$	590.000			1.03327			0.516637	+	0.856204i	$0.327184\pi$
$572$		−	883.346i		−	1.54431i
$573$		0			0
$574$	84.0000	−	48.4974i	0.146341	−	0.0844903i
$575$	704.000			1.22435
$576$		0			0
$577$			270.200i			0.468284i	0.972202	+	0.234142i	$0.0752281\pi$
$577$			270.200i			0.468284i	−0.972202	+	0.234142i	$0.924772\pi$
$578$	683.000			1.18166
$579$		0			0
$580$		−	10.3923i		−	0.0179178i
$581$	619.500	−	357.668i	1.06627	−	0.615608i
$582$		0			0
$583$	−17.0000			−0.0291595
$584$		−	400.104i		−	0.685109i
$585$		0			0
$586$		−	48.4974i		−	0.0827601i
$587$			174.937i			0.298019i	0.988836	+	0.149009i	$0.0476085\pi$
$587$			174.937i			0.298019i	−0.988836	+	0.149009i	$0.952392\pi$
$588$		0			0
$589$	468.000			0.794567
$590$	156.000			0.264407
$591$		0			0
$592$	40.0000			0.0675676
$593$			592.361i			0.998923i	0.866336	+	0.499462i	$0.166469\pi$
$593$			592.361i			0.998923i	−0.866336	+	0.499462i	$0.833531\pi$
$594$		0			0
$595$	189.000	+	327.358i	0.317647	+	0.550181i
$596$	237.000			0.397651
$597$		0			0
$598$		−	554.256i		−	0.926850i
$599$	−106.000			−0.176962			−0.0884808	−	0.996078i	$-0.528201\pi$
$599$	−106.000			−0.176962			−0.0884808	+	0.996078i	$0.528201\pi$
$600$		0			0
$601$		−	365.463i		−	0.608091i	−0.952658	−	0.304046i	$-0.901663\pi$
$601$		−	365.463i		−	0.608091i	0.952658	−	0.304046i	$-0.0983375\pi$
$602$	91.0000	+	157.617i	0.151163	+	0.261822i
$603$		0			0
$604$	219.000			0.362583
$605$			290.985i			0.480966i
$606$		0			0
$607$		−	96.9948i		−	0.159794i	−0.996803	−	0.0798969i	$-0.974541\pi$
$607$		−	96.9948i		−	0.159794i	0.996803	−	0.0798969i	$-0.0254591\pi$
$608$			685.892i			1.12811i
$609$		0			0
$610$	−120.000			−0.196721
$611$	660.000			1.08020
$612$		0			0
$613$	−802.000			−1.30832			−0.654160	−	0.756356i	$-0.726978\pi$
$613$	−802.000			−1.30832			−0.654160	+	0.756356i	$0.726978\pi$
$614$			426.084i			0.693949i
$615$		0			0
$616$	−416.500	−	721.399i	−0.676136	−	1.17110i
$617$	104.000			0.168558			0.0842788	−	0.996442i	$-0.473141\pi$
$617$	104.000			0.168558			0.0842788	+	0.996442i	$0.473141\pi$
$618$		0			0
$619$			523.079i			0.845039i	0.906354	+	0.422520i	$0.138854\pi$
$619$			523.079i			0.845039i	−0.906354	+	0.422520i	$0.861146\pi$
$620$	117.000			0.188710
$621$		0			0
$622$			159.349i			0.256188i
$623$	−315.000	+	181.865i	−0.505618	+	0.291919i
$624$		0			0
$625$	409.000			0.654400
$626$			386.247i			0.617009i
$627$		0			0
$628$			602.754i			0.959799i
$629$			249.415i			0.396527i
$630$		0			0
$631$	767.000			1.21553			0.607765	−	0.794117i	$-0.292066\pi$
$631$	767.000			1.21553			0.607765	+	0.794117i	$0.292066\pi$
$632$	−322.000			−0.509494
$633$		0			0
$634$	−227.000			−0.358044
$635$			247.683i			0.390052i
$636$		0			0
$637$	−735.000	−	424.352i	−1.15385	−	0.666173i
$638$	−34.0000			−0.0532915
$639$		0			0
$640$			206.114i			0.322053i
$641$	260.000			0.405616			0.202808	−	0.979218i	$-0.434993\pi$
$641$	260.000			0.405616			0.202808	+	0.979218i	$0.434993\pi$
$642$		0			0
$643$		−	159.349i		−	0.247821i	−0.992293	−	0.123910i	$-0.960456\pi$
$643$		−	159.349i		−	0.247821i	0.992293	−	0.123910i	$-0.0395435\pi$
$644$	336.000	+	581.969i	0.521739	+	0.903679i
$645$		0			0
$646$	−648.000			−1.00310
$647$			519.615i			0.803115i	0.915834	+	0.401557i	$0.131531\pi$
$647$			519.615i			0.803115i	−0.915834	+	0.401557i	$0.868469\pi$
$648$		0			0
$649$			1531.13i			2.35922i
$650$		−	381.051i		−	0.586233i
$651$		0			0
$652$	678.000			1.03988
$653$	1073.00			1.64319			0.821593	−	0.570075i	$-0.193086\pi$
$653$	1073.00			1.64319			0.821593	+	0.570075i	$0.193086\pi$
$654$		0			0
$655$	−21.0000			−0.0320611
$656$		−	69.2820i		−	0.105613i
$657$		0			0
$658$	231.000	−	133.368i	0.351064	−	0.202687i
$659$	503.000			0.763278			0.381639	−	0.924312i	$-0.375360\pi$
$659$	503.000			0.763278			0.381639	+	0.924312i	$0.375360\pi$
$660$		0			0
$661$		−	990.733i		−	1.49884i	−0.662095	−	0.749420i	$-0.730333\pi$
$661$		−	990.733i		−	1.49884i	0.662095	−	0.749420i	$-0.269667\pi$
$662$	−398.000			−0.601208
$663$		0			0
$664$			715.337i			1.07731i
$665$	−126.000	−	218.238i	−0.189474	−	0.328178i
$666$		0			0
$667$	64.0000			0.0959520
$668$			259.808i			0.388934i
$669$		0			0
$670$			58.8897i			0.0878951i
$671$		−	1177.79i		−	1.75528i
$672$		0			0
$673$	305.000			0.453195			0.226597	−	0.973989i	$-0.427240\pi$
$673$	305.000			0.453195			0.226597	+	0.973989i	$0.427240\pi$
$674$	610.000			0.905045
$675$		0			0
$676$	393.000			0.581361
$677$		−	6.92820i		−	0.0102337i	−0.999987	−	0.00511684i	$-0.998371\pi$
$677$		−	6.92820i		−	0.0102337i	0.999987	−	0.00511684i	$-0.00162875\pi$
$678$		0			0
$679$	556.500	−	321.295i	0.819588	−	0.473189i
$680$	−378.000			−0.555882
$681$		0			0
$682$		−	382.783i		−	0.561266i
$683$	−154.000			−0.225476			−0.112738	−	0.993625i	$-0.535962\pi$
$683$	−154.000			−0.225476			−0.112738	+	0.993625i	$0.535962\pi$
$684$		0			0
$685$		−	17.3205i		−	0.0252854i
$686$	−343.000			−0.500000
$687$		0			0
$688$	130.000			0.188953
$689$		−	17.3205i		−	0.0251386i
$690$		0			0
$691$			13.8564i			0.0200527i	0.999950	+	0.0100263i	$0.00319154\pi$
$691$			13.8564i			0.0200527i	−0.999950	+	0.0100263i	$0.996808\pi$
$692$			223.435i			0.322882i
$693$		0			0
$694$	−515.000			−0.742075
$695$	240.000			0.345324
$696$		0			0
$697$	432.000			0.619799
$698$		−	325.626i		−	0.466512i
$699$		0			0
$700$	231.000	+	400.104i	0.330000	+	0.571577i
$701$	−469.000			−0.669044			−0.334522	−	0.942388i	$-0.608575\pi$
$701$	−469.000			−0.669044			−0.334522	+	0.942388i	$0.608575\pi$
$702$		0			0
$703$		−	166.277i		−	0.236525i
$704$	221.000			0.313920
$705$		0			0
$706$			235.559i			0.333653i
$707$	745.500	−	430.415i	1.05446	−	0.608790i
$708$		0			0
$709$	80.0000			0.112835			0.0564175	−	0.998407i	$-0.482032\pi$
$709$	80.0000			0.112835			0.0564175	+	0.998407i	$0.482032\pi$
$710$			17.3205i			0.0243951i
$711$		0			0
$712$		−	363.731i		−	0.510858i
$713$			720.533i			1.01057i
$714$		0			0
$715$	−510.000			−0.713287
$716$	−357.000			−0.498603
$717$		0			0
$718$	226.000			0.314763
$719$		−	769.031i		−	1.06958i	−0.844984	−	0.534792i	$-0.820390\pi$
$719$		−	769.031i		−	1.06958i	0.844984	−	0.534792i	$-0.179610\pi$
$720$		0			0
$721$	42.0000	−	24.2487i	0.0582524	−	0.0336321i
$722$	71.0000			0.0983380
$723$		0			0
$724$		−	187.061i		−	0.258372i
$725$	44.0000			0.0606897
$726$		0			0
$727$			549.060i			0.755241i	0.925960	+	0.377620i	$0.123258\pi$
$727$			549.060i			0.755241i	−0.925960	+	0.377620i	$0.876742\pi$
$728$	735.000	−	424.352i	1.00962	−	0.582902i
$729$		0			0
$730$	−99.0000			−0.135616
$731$			810.600i			1.10889i
$732$		0			0
$733$			27.7128i			0.0378074i	0.999821	+	0.0189037i	$0.00601759\pi$
$733$			27.7128i			0.0378074i	−0.999821	+	0.0189037i	$0.993982\pi$
$734$			531.740i			0.724441i
$735$		0			0
$736$	−1056.00			−1.43478
$737$	−578.000			−0.784261
$738$		0			0
$739$	−730.000			−0.987821			−0.493911	−	0.869513i	$-0.664433\pi$
$739$	−730.000			−0.987821			−0.493911	+	0.869513i	$0.664433\pi$
$740$		−	41.5692i		−	0.0561746i
$741$		0			0
$742$	−3.50000	−	6.06218i	−0.00471698	−	0.00817005i
$743$	−934.000			−1.25707			−0.628533	−	0.777783i	$-0.716344\pi$
$743$	−934.000			−1.25707			−0.628533	+	0.777783i	$0.716344\pi$
$744$		0			0
$745$		−	136.832i		−	0.183667i
$746$	−218.000			−0.292225
$747$		0			0
$748$		−	1590.02i		−	2.12570i
$749$	−374.500	−	648.653i	−0.500000	−	0.866025i
$750$		0			0
$751$	−763.000			−1.01598			−0.507989	−	0.861363i	$-0.669611\pi$
$751$	−763.000			−1.01598			−0.507989	+	0.861363i	$0.669611\pi$
$752$		−	190.526i		−	0.253358i
$753$		0			0
$754$		−	34.6410i		−	0.0459430i
$755$		−	126.440i		−	0.167470i
$756$		0			0
$757$	−682.000			−0.900925			−0.450462	−	0.892795i	$-0.648741\pi$
$757$	−682.000			−0.900925			−0.450462	+	0.892795i	$0.648741\pi$
$758$	−296.000			−0.390501
$759$		0			0
$760$	252.000			0.331579
$761$			193.990i			0.254914i	0.991844	+	0.127457i	$0.0406815\pi$
$761$			193.990i			0.254914i	−0.991844	+	0.127457i	$0.959318\pi$
$762$		0			0
$763$	308.000	+	533.472i	0.403670	+	0.699176i
$764$	570.000			0.746073
$765$		0			0
$766$		−	640.859i		−	0.836630i
$767$	−1560.00			−2.03390
$768$		0			0
$769$		−	1255.74i		−	1.63295i	−0.577382	−	0.816474i	$-0.695926\pi$
$769$		−	1255.74i		−	1.63295i	0.577382	−	0.816474i	$-0.304074\pi$
$770$	−178.500	+	103.057i	−0.231818	+	0.133840i
$771$		0			0
$772$	−933.000			−1.20855
$773$			665.108i			0.860424i	0.902728	+	0.430212i	$0.141561\pi$
$773$			665.108i			0.860424i	−0.902728	+	0.430212i	$0.858439\pi$
$774$		0			0
$775$			495.367i			0.639183i
$776$			642.591i			0.828081i
$777$		0			0
$778$	457.000			0.587404
$779$	−288.000			−0.369705
$780$		0			0
$781$	−170.000			−0.217670
$782$		−	997.661i		−	1.27578i
$783$		0			0
$784$	−122.500	+	212.176i	−0.156250	+	0.270633i
$785$	348.000			0.443312
$786$		0			0
$787$			1150.08i			1.46135i	0.682726	+	0.730675i	$0.260794\pi$
$787$			1150.08i			1.46135i	−0.682726	+	0.730675i	$0.739206\pi$
$788$	921.000			1.16878
$789$		0			0
$790$			79.6743i			0.100854i
$791$	−700.000	−	1212.44i	−0.884956	−	1.53279i
$792$		0			0
$793$	1200.00			1.51324
$794$		−	114.315i		−	0.143974i
$795$		0			0
$796$			763.834i			0.959591i
$797$			968.216i			1.21483i	0.794386	+	0.607413i	$0.207793\pi$
$797$			968.216i			1.21483i	−0.794386	+	0.607413i	$0.792207\pi$
$798$		0			0
$799$	1188.00			1.48686
$800$	−726.000			−0.907500
$801$		0			0
$802$	448.000			0.558603
$803$		−	971.681i		−	1.21006i
$804$		0			0
$805$	336.000	−	193.990i	0.417391	−	0.240981i
$806$	390.000			0.483871
$807$		0			0
$808$			860.829i			1.06538i
$809$	638.000			0.788628			0.394314	−	0.918976i	$-0.370982\pi$
$809$	638.000			0.788628			0.394314	+	0.918976i	$0.370982\pi$
$810$		0			0
$811$		−	581.969i		−	0.717594i	−0.933416	−	0.358797i	$-0.883187\pi$
$811$		−	581.969i		−	0.717594i	0.933416	−	0.358797i	$-0.116813\pi$
$812$	21.0000	+	36.3731i	0.0258621	+	0.0447944i
$813$		0			0
$814$	−136.000			−0.167076
$815$		−	391.443i		−	0.480299i
$816$		0			0
$817$		−	540.400i		−	0.661444i
$818$			476.314i			0.582291i
$819$		0			0
$820$	−72.0000			−0.0878049
$821$	−262.000			−0.319123			−0.159562	−	0.987188i	$-0.551008\pi$
$821$	−262.000			−0.319123			−0.159562	+	0.987188i	$0.551008\pi$
$822$		0			0
$823$	−769.000			−0.934386			−0.467193	−	0.884155i	$-0.654735\pi$
$823$	−769.000			−0.934386			−0.467193	+	0.884155i	$0.654735\pi$
$824$			48.4974i			0.0588561i
$825$		0			0
$826$	−546.000	+	315.233i	−0.661017	+	0.381638i
$827$	−442.000			−0.534462			−0.267231	−	0.963633i	$-0.586109\pi$
$827$	−442.000			−0.534462			−0.267231	+	0.963633i	$0.586109\pi$
$828$		0			0
$829$			1340.61i			1.61714i	0.588401	+	0.808569i	$0.299757\pi$
$829$			1340.61i			1.61714i	−0.588401	+	0.808569i	$0.700243\pi$
$830$	177.000			0.213253
$831$		0			0
$832$			225.167i			0.270633i
$833$	−1323.00	−	763.834i	−1.58824	−	0.916968i
$834$		0			0
$835$	150.000			0.179641
$836$			1060.02i			1.26796i
$837$		0			0
$838$			242.487i			0.289364i
$839$		−	1118.90i		−	1.33362i	−0.745229	−	0.666809i	$-0.767660\pi$
$839$		−	1118.90i		−	1.33362i	0.745229	−	0.666809i	$-0.232340\pi$
$840$		0			0
$841$	−837.000			−0.995244
$842$	−548.000			−0.650831
$843$		0			0
$844$	588.000			0.696682
$845$		−	226.899i		−	0.268519i
$846$		0			0
$847$	−588.000	−	1018.45i	−0.694215	−	1.20242i
$848$	−5.00000			−0.00589623
$849$		0			0
$850$		−	685.892i		−	0.806932i
$851$	256.000			0.300823
$852$		0			0
$853$		−	1077.34i		−	1.26300i	−0.775377	−	0.631498i	$-0.782440\pi$
$853$		−	1077.34i		−	1.26300i	0.775377	−	0.631498i	$-0.217560\pi$
$854$	420.000	−	242.487i	0.491803	−	0.283943i
$855$		0			0
$856$	749.000			0.875000
$857$		−	100.459i		−	0.117222i	−0.998281	−	0.0586108i	$-0.981333\pi$
$857$		−	100.459i		−	0.117222i	0.998281	−	0.0586108i	$-0.0186671\pi$
$858$		0			0
$859$		−	96.9948i		−	0.112916i	−0.998405	−	0.0564580i	$-0.982019\pi$
$859$		−	96.9948i		−	0.112916i	0.998405	−	0.0564580i	$-0.0179807\pi$
$860$		−	135.100i		−	0.157093i
$861$		0			0
$862$	358.000			0.415313
$863$	−1216.00			−1.40904			−0.704519	−	0.709685i	$-0.748837\pi$
$863$	−1216.00			−1.40904			−0.704519	+	0.709685i	$0.748837\pi$
$864$		0			0
$865$	129.000			0.149133
$866$		−	161.081i		−	0.186005i
$867$		0			0
$868$	−409.500	+	236.425i	−0.471774	+	0.272379i
$869$	−782.000			−0.899885
$870$		0			0
$871$		−	588.897i		−	0.676116i
$872$	−616.000			−0.706422
$873$		0			0
$874$			665.108i			0.760993i
$875$	493.500	−	284.922i	0.564000	−	0.325626i
$876$		0			0
$877$	254.000			0.289624			0.144812	−	0.989459i	$-0.453742\pi$
$877$	254.000			0.289624			0.144812	+	0.989459i	$0.453742\pi$
$878$			334.286i			0.380736i
$879$		0			0
$880$			147.224i			0.167300i
$881$			654.715i			0.743150i	0.928403	+	0.371575i	$0.121182\pi$
$881$			654.715i			0.743150i	−0.928403	+	0.371575i	$0.878818\pi$
$882$		0			0
$883$	80.0000			0.0906002			0.0453001	−	0.998973i	$-0.485576\pi$
$883$	80.0000			0.0906002			0.0453001	+	0.998973i	$0.485576\pi$
$884$	1620.00			1.83258
$885$		0			0
$886$	−650.000			−0.733634
$887$		−	1063.48i		−	1.19896i	−0.800389	−	0.599481i	$-0.795374\pi$
$887$		−	1063.48i		−	1.19896i	0.800389	−	0.599481i	$-0.204626\pi$
$888$		0			0
$889$	−500.500	−	866.891i	−0.562992	−	0.975131i
$890$	−90.0000			−0.101124
$891$		0			0
$892$		−	228.631i		−	0.256312i
$893$	−792.000			−0.886898
$894$		0			0
$895$			206.114i			0.230295i
$896$	−416.500	−	721.399i	−0.464844	−	0.805133i
$897$		0			0
$898$	316.000			0.351893
$899$			45.0333i			0.0500927i
$900$		0			0
$901$		−	31.1769i		−	0.0346026i
$902$			235.559i			0.261152i
$903$		0			0
$904$	1400.00			1.54867
$905$	−108.000			−0.119337
$906$		0			0
$907$	−208.000			−0.229327			−0.114664	−	0.993404i	$-0.536579\pi$
$907$	−208.000			−0.229327			−0.114664	+	0.993404i	$0.536579\pi$
$908$		−	228.631i		−	0.251796i
$909$		0			0
$910$	−105.000	−	181.865i	−0.115385	−	0.199852i
$911$	−1654.00			−1.81559			−0.907794	−	0.419417i	$-0.862234\pi$
$911$	−1654.00			−1.81559			−0.907794	+	0.419417i	$0.862234\pi$
$912$		0			0
$913$			1737.25i			1.90279i
$914$	19.0000			0.0207877
$915$		0			0
$916$			72.7461i			0.0794172i
$917$	73.5000	−	42.4352i	0.0801527	−	0.0462762i
$918$		0			0
$919$	53.0000			0.0576714			0.0288357	−	0.999584i	$-0.490820\pi$
$919$	53.0000			0.0576714			0.0288357	+	0.999584i	$0.490820\pi$
$920$			387.979i			0.421717i
$921$		0			0
$922$			355.070i			0.385109i
$923$		−	173.205i		−	0.187654i
$924$		0			0
$925$	176.000			0.190270
$926$	835.000			0.901728
$927$		0			0
$928$	−66.0000			−0.0711207
$929$		−	297.913i		−	0.320681i	−0.987062	−	0.160341i	$-0.948741\pi$
$929$		−	297.913i		−	0.320681i	0.987062	−	0.160341i	$-0.0512592\pi$
$930$		0			0
$931$	882.000	+	509.223i	0.947368	+	0.546963i
$932$	534.000			0.572961
$933$		0			0
$934$		−	431.281i		−	0.461757i
$935$	−918.000			−0.981818
$936$		0			0
$937$			1834.24i			1.95757i	0.204893	+	0.978784i	$0.434315\pi$
$937$			1834.24i			1.95757i	−0.204893	+	0.978784i	$0.565685\pi$
$938$	−119.000	−	206.114i	−0.126866	−	0.219738i
$939$		0			0
$940$	−198.000			−0.210638
$941$			1373.52i			1.45963i	0.683642	+	0.729817i	$0.260395\pi$
$941$			1373.52i			1.45963i	−0.683642	+	0.729817i	$0.739605\pi$
$942$		0			0
$943$		−	443.405i		−	0.470207i
$944$			450.333i			0.477048i
$945$		0			0
$946$	−442.000			−0.467230
$947$	−385.000			−0.406547			−0.203273	−	0.979122i	$-0.565158\pi$
$947$	−385.000			−0.406547			−0.203273	+	0.979122i	$0.565158\pi$
$948$		0			0
$949$	990.000			1.04320
$950$			457.261i			0.481328i
$951$		0			0
$952$	1323.00	−	763.834i	1.38971	−	0.802347i
$953$	1244.00			1.30535			0.652676	−	0.757637i	$-0.273646\pi$
$953$	1244.00			1.30535			0.652676	+	0.757637i	$0.273646\pi$
$954$		0			0
$955$		−	329.090i		−	0.344596i
$956$	264.000			0.276151
$957$		0			0
$958$			620.074i			0.647259i
$959$	35.0000	+	60.6218i	0.0364964	+	0.0632135i
$960$		0			0
$961$	454.000			0.472425
$962$		−	138.564i		−	0.144037i
$963$		0			0
$964$		−	602.754i		−	0.625263i
$965$			538.668i			0.558205i
$966$		0			0
$967$	−535.000			−0.553257			−0.276629	−	0.960977i	$-0.589217\pi$
$967$	−535.000			−0.553257			−0.276629	+	0.960977i	$0.589217\pi$
$968$	1176.00			1.21488
$969$		0			0
$970$	159.000			0.163918
$971$			743.050i			0.765242i	0.923905	+	0.382621i	$0.124978\pi$
$971$			743.050i			0.765242i	−0.923905	+	0.382621i	$0.875022\pi$
$972$		0			0
$973$	−840.000	+	484.974i	−0.863309	+	0.498432i
$974$	−98.0000			−0.100616
$975$		0			0
$976$		−	346.410i		−	0.354928i
$977$	950.000			0.972364			0.486182	−	0.873857i	$-0.338389\pi$
$977$	950.000			0.972364			0.486182	+	0.873857i	$0.338389\pi$
$978$		0			0
$979$		−	883.346i		−	0.902294i
$980$	220.500	+	127.306i	0.225000	+	0.129904i
$981$		0			0
$982$	283.000			0.288187
$983$			1603.88i			1.63162i	0.578322	+	0.815808i	$0.303708\pi$
$983$			1603.88i			1.63162i	−0.578322	+	0.815808i	$0.696292\pi$
$984$		0			0
$985$		−	531.740i		−	0.539837i
$986$		−	62.3538i		−	0.0632392i
$987$		0			0
$988$	−1080.00			−1.09312
$989$	832.000			0.841254
$990$		0			0
$991$	1151.00			1.16145			0.580727	−	0.814099i	$-0.302769\pi$
$991$	1151.00			1.16145			0.580727	+	0.814099i	$0.302769\pi$
$992$		−	743.050i		−	0.749042i
$993$		0			0
$994$	−35.0000	−	60.6218i	−0.0352113	−	0.0609877i
$995$	441.000			0.443216
$996$		0			0
$997$		−	370.659i		−	0.371774i	−0.982571	−	0.185887i	$-0.940484\pi$
$997$		−	370.659i		−	0.371774i	0.982571	−	0.185887i	$-0.0595159\pi$
$998$	−302.000			−0.302605
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	189.3.d.a.55.2	yes	2
3.2	odd	2		189.3.d.c.55.1	yes	2
7.6	odd	2	inner	189.3.d.a.55.1	✓	2
21.20	even	2		189.3.d.c.55.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.3.d.a.55.1	✓	2	7.6	odd	2	inner
189.3.d.a.55.2	yes	2	1.1	even	1	trivial
189.3.d.c.55.1	yes	2	3.2	odd	2
189.3.d.c.55.2	yes	2	21.20	even	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:	\( N \)	\(=\)	\( 189 = 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	189.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -3.00000 q^{4} +1.73205i q^{5} +(-3.50000 - 6.06218i) q^{7} +7.00000 q^{8} -1.73205i q^{10} +17.0000 q^{11} +17.3205i q^{13} +(3.50000 + 6.06218i) q^{14} +5.00000 q^{16} +31.1769i q^{17} -20.7846i q^{19} -5.19615i q^{20} -17.0000 q^{22} +32.0000 q^{23} +22.0000 q^{25} -17.3205i q^{26} +(10.5000 + 18.1865i) q^{28} +2.00000 q^{29} +22.5167i q^{31} -33.0000 q^{32} -31.1769i q^{34} +(10.5000 - 6.06218i) q^{35} +8.00000 q^{37} +20.7846i q^{38} +12.1244i q^{40} -13.8564i q^{41} +26.0000 q^{43} -51.0000 q^{44} -32.0000 q^{46} -38.1051i q^{47} +(-24.5000 + 42.4352i) q^{49} -22.0000 q^{50} -51.9615i q^{52} -1.00000 q^{53} +29.4449i q^{55} +(-24.5000 - 42.4352i) q^{56} -2.00000 q^{58} +90.0666i q^{59} -69.2820i q^{61} -22.5167i q^{62} +13.0000 q^{64} -30.0000 q^{65} -34.0000 q^{67} -93.5307i q^{68} +(-10.5000 + 6.06218i) q^{70} -10.0000 q^{71} -57.1577i q^{73} -8.00000 q^{74} +62.3538i q^{76} +(-59.5000 - 103.057i) q^{77} -46.0000 q^{79} +8.66025i q^{80} +13.8564i q^{82} +102.191i q^{83} -54.0000 q^{85} -26.0000 q^{86} +119.000 q^{88} -51.9615i q^{89} +(105.000 - 60.6218i) q^{91} -96.0000 q^{92} +38.1051i q^{94} +36.0000 q^{95} +91.7987i q^{97} +(24.5000 - 42.4352i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 6 q^{4} - 7 q^{7} + 14 q^{8} + 34 q^{11} + 7 q^{14} + 10 q^{16} - 34 q^{22} + 64 q^{23} + 44 q^{25} + 21 q^{28} + 4 q^{29} - 66 q^{32} + 21 q^{35} + 16 q^{37} + 52 q^{43} - 102 q^{44} - 64 q^{46}+ \cdots + 49 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 - 6 * q^4 - 7 * q^7 + 14 * q^8 + 34 * q^11 + 7 * q^14 + 10 * q^16 - 34 * q^22 + 64 * q^23 + 44 * q^25 + 21 * q^28 + 4 * q^29 - 66 * q^32 + 21 * q^35 + 16 * q^37 + 52 * q^43 - 102 * q^44 - 64 * q^46 - 49 * q^49 - 44 * q^50 - 2 * q^53 - 49 * q^56 - 4 * q^58 + 26 * q^64 - 60 * q^65 - 68 * q^67 - 21 * q^70 - 20 * q^71 - 16 * q^74 - 119 * q^77 - 92 * q^79 - 108 * q^85 - 52 * q^86 + 238 * q^88 + 210 * q^91 - 192 * q^92 + 72 * q^95 + 49 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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