LMFDB - Embedded newform 684.2.bk.a.449.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [684,2,Mod(449,684)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(684, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 5]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("684.449");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 684 = 2^{2} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	684.bk (of order $6$, degree $2$, 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:	$5.46176749826$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 10x^{14} + 46x^{12} + 126x^{10} + 315x^{8} + 1134x^{6} + 3726x^{4} + 7290x^{2} + 6561 $ x^16 + 10x^14 + 46x^12 + 126x^10 + 315x^8 + 1134x^6 + 3726x^4 + 7290*x^2 + 6561
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{7} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			449.2
Root			$0.750719 - 1.56090i$ of defining polynomial
Character	$\chi$	$=$	684.449
Dual form			684.2.bk.a.521.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.47786 + 1.43059i) q^{5} +3.93208 q^{7} +O(q^{10})$	$q+(-2.47786 + 1.43059i) q^{5} +3.93208 q^{7} +1.80663i q^{11} +(-0.947142 - 0.546833i) q^{13} +(-1.36468 + 0.787896i) q^{17} +(-2.09320 + 3.82342i) q^{19} +(4.33601 + 2.50340i) q^{23} +(1.59320 - 2.75950i) q^{25} +(-0.451406 + 0.781859i) q^{29} +3.02361i q^{31} +(-9.74314 + 5.62521i) q^{35} +5.27640i q^{37} +(4.04245 + 7.00173i) q^{41} +(3.05924 + 5.29875i) q^{43} +(3.22283 + 1.86070i) q^{47} +8.46123 q^{49} +(-6.06891 + 10.5117i) q^{53} +(-2.58456 - 4.47659i) q^{55} +(-6.06891 - 10.5117i) q^{59} +(7.27096 - 12.5937i) q^{61} +3.12918 q^{65} +(3.17138 + 1.83100i) q^{67} +(-3.67423 - 6.36396i) q^{71} +(1.10826 + 1.91957i) q^{73} +7.10383i q^{77} +(-12.5205 + 7.22870i) q^{79} -10.9114i q^{83} +(2.25432 - 3.90459i) q^{85} +(8.37847 - 14.5119i) q^{89} +(-3.72424 - 2.15019i) q^{91} +(-0.283104 - 12.4684i) q^{95} +(2.78990 - 1.61075i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 12 q^{19} + 4 q^{25} - 4 q^{43} + 20 q^{55} + 12 q^{61} + 36 q^{67} + 44 q^{73} - 12 q^{79} + 56 q^{85} - 60 q^{91}+O(q^{100}) $ 16 * q - 12 * q^19 + 4 * q^25 - 4 * q^43 + 20 * q^55 + 12 * q^61 + 36 * q^67 + 44 * q^73 - 12 * q^79 + 56 * q^85 - 60 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/684\mathbb{Z}\right)^\times$.

$n$	$325$	$343$	$533$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$1$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−2.47786	+	1.43059i	−1.10813	+	0.639781i	−0.938345	−	0.345700i	$-0.887642\pi$
$5$	−2.47786	+	1.43059i	−1.10813	+	0.639781i	−0.169788	+	0.985481i	$0.554308\pi$
$6$		0			0
$7$	3.93208			1.48619			0.743093	−	0.669188i	$-0.233358\pi$
$7$	3.93208			1.48619			0.743093	+	0.669188i	$0.233358\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			1.80663i			0.544721i	0.962195	+	0.272360i	$0.0878043\pi$
$11$			1.80663i			0.544721i	−0.962195	+	0.272360i	$0.912196\pi$
$12$		0			0
$13$	−0.947142	−	0.546833i	−0.262690	−	0.151664i	0.362871	−	0.931839i	$-0.381797\pi$
$13$	−0.947142	−	0.546833i	−0.262690	−	0.151664i	−0.625561	+	0.780175i	$0.715130\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.36468	+	0.787896i	−0.330983	+	0.191093i	−0.656277	−	0.754520i	$-0.727870\pi$
$17$	−1.36468	+	0.787896i	−0.330983	+	0.191093i	0.325295	+	0.945613i	$0.394536\pi$
$18$		0			0
$19$	−2.09320	+	3.82342i	−0.480213	+	0.877152i
$19$	−2.09320	+	3.82342i	−0.480213	+	0.877152i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	4.33601	+	2.50340i	0.904121	+	0.521995i	0.878535	−	0.477678i	$-0.158521\pi$
$23$	4.33601	+	2.50340i	0.904121	+	0.521995i	0.0255863	+	0.999673i	$0.491855\pi$
$24$		0			0
$25$	1.59320	−	2.75950i	0.318640	−	0.551900i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−0.451406	+	0.781859i	−0.0838240	+	0.145188i	−0.904890	−	0.425647i	$-0.860047\pi$
$29$	−0.451406	+	0.781859i	−0.0838240	+	0.145188i	0.821065	+	0.570834i	$0.193380\pi$
$30$		0			0
$31$			3.02361i			0.543056i	0.962431	+	0.271528i	$0.0875290\pi$
$31$			3.02361i			0.543056i	−0.962431	+	0.271528i	$0.912471\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−9.74314	+	5.62521i	−1.64689	+	0.950833i
$36$		0			0
$37$			5.27640i			0.867435i	0.901049	+	0.433717i	$0.142798\pi$
$37$			5.27640i			0.867435i	−0.901049	+	0.433717i	$0.857202\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	4.04245	+	7.00173i	0.631325	+	1.09349i	0.987281	+	0.158984i	$0.0508220\pi$
$41$	4.04245	+	7.00173i	0.631325	+	1.09349i	−0.355956	+	0.934503i	$0.615845\pi$
$42$		0			0
$43$	3.05924	+	5.29875i	0.466529	+	0.808052i	0.999269	−	0.0382267i	$-0.0121709\pi$
$43$	3.05924	+	5.29875i	0.466529	+	0.808052i	−0.532740	+	0.846279i	$0.678838\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.22283	+	1.86070i	0.470098	+	0.271411i	0.716281	−	0.697812i	$-0.245843\pi$
$47$	3.22283	+	1.86070i	0.470098	+	0.271411i	−0.246183	+	0.969223i	$0.579176\pi$
$48$		0			0
$49$	8.46123			1.20875
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−6.06891	+	10.5117i	−0.833629	+	1.44389i	0.0615134	+	0.998106i	$0.480407\pi$
$53$	−6.06891	+	10.5117i	−0.833629	+	1.44389i	−0.895142	+	0.445781i	$0.852926\pi$
$54$		0			0
$55$	−2.58456	−	4.47659i	−0.348502	−	0.603623i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−6.06891	−	10.5117i	−0.790105	−	1.36850i	−0.925902	−	0.377765i	$-0.876693\pi$
$59$	−6.06891	−	10.5117i	−0.790105	−	1.36850i	0.135797	−	0.990737i	$-0.456640\pi$
$60$		0			0
$61$	7.27096	−	12.5937i	0.930951	−	1.61245i	0.149250	−	0.988799i	$-0.452314\pi$
$61$	7.27096	−	12.5937i	0.930951	−	1.61245i	0.781700	−	0.623654i	$-0.214353\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	3.12918			0.388127
$66$		0			0
$67$	3.17138	+	1.83100i	0.387446	+	0.223692i	0.681053	−	0.732234i	$-0.261522\pi$
$67$	3.17138	+	1.83100i	0.387446	+	0.223692i	−0.293607	+	0.955926i	$0.594856\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−3.67423	−	6.36396i	−0.436051	−	0.755263i	0.561329	−	0.827592i	$-0.310290\pi$
$71$	−3.67423	−	6.36396i	−0.436051	−	0.755263i	−0.997381	+	0.0723293i	$0.976957\pi$
$72$		0			0
$73$	1.10826	+	1.91957i	0.129712	+	0.224668i	0.923565	−	0.383442i	$-0.125261\pi$
$73$	1.10826	+	1.91957i	0.129712	+	0.224668i	−0.793853	+	0.608110i	$0.791928\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			7.10383i			0.809556i
$78$		0			0
$79$	−12.5205	+	7.22870i	−1.40866	+	0.813292i	−0.995259	−	0.0972560i	$-0.968993\pi$
$79$	−12.5205	+	7.22870i	−1.40866	+	0.813292i	−0.413404	+	0.910548i	$0.635660\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	10.9114i		−	1.19769i	−0.800866	−	0.598843i	$-0.795627\pi$
$83$		−	10.9114i		−	1.19769i	0.800866	−	0.598843i	$-0.204373\pi$
$84$		0			0
$85$	2.25432	−	3.90459i	0.244515	−	0.423513i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	8.37847	−	14.5119i	0.888116	−	1.53826i	0.0460153	−	0.998941i	$-0.485348\pi$
$89$	8.37847	−	14.5119i	0.888116	−	1.53826i	0.842100	−	0.539321i	$-0.181319\pi$
$90$		0			0
$91$	−3.72424	−	2.15019i	−0.390406	−	0.225401i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−0.283104	−	12.4684i	−0.0290458	−	1.27923i
$96$		0			0
$97$	2.78990	−	1.61075i	0.283272	−	0.163547i	−0.351632	−	0.936138i	$-0.614373\pi$
$97$	2.78990	−	1.61075i	0.283272	−	0.163547i	0.634904	+	0.772591i	$0.281040\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	11.7696	+	6.79518i	1.17112	+	0.676146i	0.953943	−	0.299987i	$-0.0969823\pi$
$101$	11.7696	+	6.79518i	1.17112	+	0.676146i	0.217175	+	0.976133i	$0.430316\pi$
$102$		0			0
$103$		−	13.8190i		−	1.36163i	−0.732457	−	0.680814i	$-0.761626\pi$
$103$		−	13.8190i		−	1.36163i	0.732457	−	0.680814i	$-0.238374\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−4.61912			−0.446547			−0.223273	−	0.974756i	$-0.571674\pi$
$107$	−4.61912			−0.446547			−0.223273	+	0.974756i	$0.571674\pi$
$108$		0			0
$109$	−14.5166	+	8.38119i	−1.39044	+	0.802772i	−0.993364	−	0.115013i	$-0.963309\pi$
$109$	−14.5166	+	8.38119i	−1.39044	+	0.802772i	−0.397078	+	0.917785i	$0.629976\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	15.0335			1.41424			0.707118	−	0.707095i	$-0.249995\pi$
$113$	15.0335			1.41424			0.707118	+	0.707095i	$0.249995\pi$
$114$		0			0
$115$	−14.3254			−1.33585
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−5.36601	+	3.09807i	−0.491901	+	0.283999i
$120$		0			0
$121$	7.73607			0.703279
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	5.18906i		−	0.464124i
$126$		0			0
$127$	−7.17521	−	4.14261i	−0.636697	−	0.367597i	0.146644	−	0.989189i	$-0.453153\pi$
$127$	−7.17521	−	4.14261i	−0.636697	−	0.367597i	−0.783341	+	0.621592i	$0.786486\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−3.22283	+	1.86070i	−0.281580	+	0.162570i	−0.634138	−	0.773220i	$-0.718645\pi$
$131$	−3.22283	+	1.86070i	−0.281580	+	0.162570i	0.352559	+	0.935790i	$0.385312\pi$
$132$		0			0
$133$	−8.23062	+	15.0340i	−0.713685	+	1.30361i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	4.57705	+	2.64256i	0.391044	+	0.225769i	0.682612	−	0.730781i	$-0.260844\pi$
$137$	4.57705	+	2.64256i	0.391044	+	0.225769i	−0.291569	+	0.956550i	$0.594177\pi$
$138$		0			0
$139$	2.26458	−	3.92236i	0.192079	−	0.332691i	−0.753860	−	0.657035i	$-0.771810\pi$
$139$	2.26458	−	3.92236i	0.192079	−	0.332691i	0.945939	+	0.324344i	$0.105144\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	0.987927	−	1.71114i	0.0826146	−	0.143093i
$144$		0			0
$145$		−	2.58312i		−	0.214516i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−9.45811	+	5.46064i	−0.774839	+	0.447353i	−0.834598	−	0.550859i	$-0.814300\pi$
$149$	−9.45811	+	5.46064i	−0.774839	+	0.447353i	0.0597593	+	0.998213i	$0.480967\pi$
$150$		0			0
$151$			17.1655i			1.39691i	0.715656	+	0.698453i	$0.246128\pi$
$151$			17.1655i			1.39691i	−0.715656	+	0.698453i	$0.753872\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−4.32556	−	7.49208i	−0.347437	−	0.601779i
$156$		0			0
$157$	−5.68640	−	9.84913i	−0.453824	−	0.786046i	0.544796	−	0.838569i	$-0.316607\pi$
$157$	−5.68640	−	9.84913i	−0.453824	−	0.786046i	−0.998620	+	0.0525228i	$0.983274\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	17.0495	+	9.84356i	1.34369	+	0.775781i
$162$		0			0
$163$	−5.63487			−0.441357			−0.220679	−	0.975347i	$-0.570827\pi$
$163$	−5.63487			−0.441357			−0.220679	+	0.975347i	$0.570827\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	9.82633	−	17.0197i	0.760384	−	1.31702i	−0.182268	−	0.983249i	$-0.558344\pi$
$167$	9.82633	−	17.0197i	0.760384	−	1.31702i	0.942653	−	0.333775i	$-0.108323\pi$
$168$		0			0
$169$	−5.90195	−	10.2225i	−0.453996	−	0.786344i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−4.95572	−	8.58356i	−0.376777	−	0.652596i	0.613815	−	0.789450i	$-0.289634\pi$
$173$	−4.95572	−	8.58356i	−0.376777	−	0.652596i	−0.990591	+	0.136854i	$0.956301\pi$
$174$		0			0
$175$	6.26458	−	10.8506i	0.473558	−	0.820226i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	10.4986			0.784700			0.392350	−	0.919816i	$-0.371662\pi$
$179$	10.4986			0.784700			0.392350	+	0.919816i	$0.371662\pi$
$180$		0			0
$181$	8.44714	+	4.87696i	0.627871	+	0.362502i	0.779927	−	0.625870i	$-0.215256\pi$
$181$	8.44714	+	4.87696i	0.627871	+	0.362502i	−0.152056	+	0.988372i	$0.548589\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−7.54839	−	13.0742i	−0.554968	−	0.961233i
$186$		0			0
$187$	−1.42344	−	2.46547i	−0.104092	−	0.180293i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	5.85716i		−	0.423809i	−0.977290	−	0.211905i	$-0.932033\pi$
$191$		−	5.85716i		−	0.423809i	0.977290	−	0.211905i	$-0.0679666\pi$
$192$		0			0
$193$	10.8491	−	6.26372i	0.780935	−	0.450873i	−0.0558268	−	0.998440i	$-0.517779\pi$
$193$	10.8491	−	6.26372i	0.780935	−	0.450873i	0.836761	+	0.547568i	$0.184446\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	16.6557i		−	1.18667i	−0.804955	−	0.593336i	$-0.797811\pi$
$197$		−	16.6557i		−	1.18667i	0.804955	−	0.593336i	$-0.202189\pi$
$198$		0			0
$199$	2.22679	−	3.85691i	0.157853	−	0.273409i	−0.776241	−	0.630436i	$-0.782876\pi$
$199$	2.22679	−	3.85691i	0.157853	−	0.273409i	0.934094	+	0.357027i	$0.116210\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−1.77496	+	3.07433i	−0.124578	+	0.215776i
$204$		0			0
$205$	−20.0333	−	11.5662i	−1.39918	−	0.807820i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−6.90752	−	3.78164i	−0.477803	−	0.261582i
$210$		0			0
$211$	13.6262	−	7.86708i	0.938065	−	0.541592i	0.0487120	−	0.998813i	$-0.484488\pi$
$211$	13.6262	−	7.86708i	0.938065	−	0.541592i	0.889353	+	0.457221i	$0.151155\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−15.1607	−	8.75305i	−1.03395	−	0.596953i
$216$		0			0
$217$			11.8891i			0.807082i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	1.72339			0.115928
$222$		0			0
$223$	−16.5567	+	9.55901i	−1.10872	+	0.640119i	−0.938497	−	0.345287i	$-0.887781\pi$
$223$	−16.5567	+	9.55901i	−1.10872	+	0.640119i	−0.170221	+	0.985406i	$0.554448\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	19.4863			1.29335			0.646675	−	0.762766i	$-0.276159\pi$
$227$	19.4863			1.29335			0.646675	+	0.762766i	$0.276159\pi$
$228$		0			0
$229$	25.8340			1.70716			0.853580	−	0.520961i	$-0.174426\pi$
$229$	25.8340			1.70716			0.853580	+	0.520961i	$0.174426\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	18.6667	−	10.7772i	1.22289	−	0.706038i	0.257360	−	0.966316i	$-0.417147\pi$
$233$	18.6667	−	10.7772i	1.22289	−	0.706038i	0.965534	+	0.260278i	$0.0838141\pi$
$234$		0			0
$235$	−10.6476			−0.694575
$236$		0			0
$237$		0			0
$238$		0			0
$239$			6.58259i			0.425792i	0.977075	+	0.212896i	$0.0682896\pi$
$239$			6.58259i			0.425792i	−0.977075	+	0.212896i	$0.931710\pi$
$240$		0			0
$241$	−10.2267	−	5.90441i	−0.658762	−	0.380336i	0.133043	−	0.991110i	$-0.457525\pi$
$241$	−10.2267	−	5.90441i	−0.658762	−	0.380336i	−0.791805	+	0.610774i	$0.790858\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−20.9658	+	12.1046i	−1.33945	+	0.773334i
$246$		0			0
$247$	4.07333	−	2.47669i	0.259180	−	0.157588i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	7.88306	+	4.55129i	0.497575	+	0.287275i	0.727711	−	0.685883i	$-0.240584\pi$
$251$	7.88306	+	4.55129i	0.497575	+	0.287275i	−0.230137	+	0.973158i	$0.573917\pi$
$252$		0			0
$253$	−4.52273	+	7.83359i	−0.284341	+	0.492494i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−10.7396	+	18.6015i	−0.669918	+	1.16033i	0.308009	+	0.951383i	$0.400337\pi$
$257$	−10.7396	+	18.6015i	−0.669918	+	1.16033i	−0.977927	+	0.208948i	$0.932996\pi$
$258$		0			0
$259$			20.7472i			1.28917i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−20.0104	+	11.5530i	−1.23390	+	0.712390i	−0.967840	−	0.251568i	$-0.919054\pi$
$263$	−20.0104	+	11.5530i	−1.23390	+	0.712390i	−0.266056	+	0.963958i	$0.585721\pi$
$264$		0			0
$265$		−	34.7286i		−	2.13336i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	4.33601	+	7.51020i	0.264371	+	0.457905i	0.967399	−	0.253258i	$-0.0815022\pi$
$269$	4.33601	+	7.51020i	0.264371	+	0.457905i	−0.703027	+	0.711163i	$0.748169\pi$
$270$		0			0
$271$	−1.00000	−	1.73205i	−0.0607457	−	0.105215i	0.834053	−	0.551684i	$-0.186015\pi$
$271$	−1.00000	−	1.73205i	−0.0607457	−	0.105215i	−0.894799	+	0.446469i	$0.852681\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	4.98541	+	2.87833i	0.300631	+	0.173570i
$276$		0			0
$277$	10.8340			0.650953			0.325477	−	0.945550i	$-0.394475\pi$
$277$	10.8340			0.650953			0.325477	+	0.945550i	$0.394475\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	7.09698	−	12.2923i	0.423370	−	0.733299i	−0.572896	−	0.819628i	$-0.694180\pi$
$281$	7.09698	−	12.2923i	0.423370	−	0.733299i	0.996267	+	0.0863290i	$0.0275136\pi$
$282$		0			0
$283$	9.35552	+	16.2042i	0.556128	+	0.963242i	0.997815	+	0.0660724i	$0.0210468\pi$
$283$	9.35552	+	16.2042i	0.556128	+	0.963242i	−0.441687	+	0.897169i	$0.645620\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	15.8952	+	27.5314i	0.938266	+	1.62512i
$288$		0			0
$289$	−7.25844	+	12.5720i	−0.426967	+	0.739529i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−27.9710			−1.63408			−0.817042	−	0.576578i	$-0.804388\pi$
$293$	−27.9710			−1.63408			−0.817042	+	0.576578i	$0.804388\pi$
$294$		0			0
$295$	30.0758	+	17.3643i	1.75108	+	1.01099i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−2.73788	−	4.74215i	−0.158336	−	0.274246i
$300$		0			0
$301$	12.0292	+	20.8351i	0.693349	+	1.20092i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			41.6071i			2.38242i
$306$		0			0
$307$	−16.7537	+	9.67274i	−0.956183	+	0.552053i	−0.894996	−	0.446074i	$-0.852822\pi$
$307$	−16.7537	+	9.67274i	−0.956183	+	0.552053i	−0.0611870	+	0.998126i	$0.519489\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			15.8697i			0.899886i	0.893057	+	0.449943i	$0.148556\pi$
$311$			15.8697i			0.899886i	−0.893057	+	0.449943i	$0.851444\pi$
$312$		0			0
$313$	−3.70303	+	6.41384i	−0.209308	+	0.362532i	−0.951497	−	0.307659i	$-0.900454\pi$
$313$	−3.70303	+	6.41384i	−0.209308	+	0.362532i	0.742189	+	0.670191i	$0.233788\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−9.19809	+	15.9316i	−0.516616	+	0.894806i	0.483198	+	0.875511i	$0.339475\pi$
$317$	−9.19809	+	15.9316i	−0.516616	+	0.894806i	−0.999814	+	0.0192943i	$0.993858\pi$
$318$		0			0
$319$	−1.41253	−	0.815526i	−0.0790867	−	0.0456607i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−0.155919	−	6.86695i	−0.00867554	−	0.382087i
$324$		0			0
$325$	−3.01797	+	1.74243i	−0.167407	+	0.0966524i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	12.6724	+	7.31642i	0.698653	+	0.403367i
$330$		0			0
$331$		−	16.2489i		−	0.893123i	−0.894753	−	0.446561i	$-0.852648\pi$
$331$		−	16.2489i		−	0.893123i	0.894753	−	0.446561i	$-0.147352\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−10.4777			−0.572455
$336$		0			0
$337$	13.9186	−	8.03590i	0.758194	−	0.437743i	−0.0704531	−	0.997515i	$-0.522445\pi$
$337$	13.9186	−	8.03590i	0.758194	−	0.437743i	0.828647	+	0.559772i	$0.189111\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−5.46256			−0.295814
$342$		0			0
$343$	5.74568			0.310238
$344$		0			0
$345$		0			0
$346$		0			0
$347$	15.2240	−	8.78955i	0.817265	−	0.471848i	−0.0322076	−	0.999481i	$-0.510254\pi$
$347$	15.2240	−	8.78955i	0.817265	−	0.471848i	0.849472	+	0.527633i	$0.176920\pi$
$348$		0			0
$349$	8.13123			0.435255			0.217627	−	0.976032i	$-0.430168\pi$
$349$	8.13123			0.435255			0.217627	+	0.976032i	$0.430168\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	26.4166i		−	1.40601i	−0.711184	−	0.703006i	$-0.751841\pi$
$353$		−	26.4166i		−	1.40601i	0.711184	−	0.703006i	$-0.248159\pi$
$354$		0			0
$355$	18.2085	+	10.5127i	0.966406	+	0.557955i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−11.2657	+	6.50423i	−0.594579	+	0.343280i	−0.766906	−	0.641759i	$-0.778205\pi$
$359$	−11.2657	+	6.50423i	−0.594579	+	0.343280i	0.172327	+	0.985040i	$0.444871\pi$
$360$		0			0
$361$	−10.2370	−	16.0063i	−0.538792	−	0.842439i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−5.49224	−	3.17095i	−0.287477	−	0.165975i
$366$		0			0
$367$	−6.34526	+	10.9903i	−0.331220	+	0.573690i	−0.982751	−	0.184932i	$-0.940793\pi$
$367$	−6.34526	+	10.9903i	−0.331220	+	0.573690i	0.651532	+	0.758622i	$0.274127\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−23.8634	+	41.3327i	−1.23893	+	2.14588i
$372$		0			0
$373$			0.873676i			0.0452372i	0.999744	+	0.0226186i	$0.00720034\pi$
$373$			0.873676i			0.0452372i	−0.999744	+	0.0226186i	$0.992800\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	0.855092	−	0.493688i	0.0440395	−	0.0254262i
$378$		0			0
$379$		−	25.8405i		−	1.32734i	−0.748027	−	0.663669i	$-0.768998\pi$
$379$		−	25.8405i		−	1.32734i	0.748027	−	0.663669i	$-0.231002\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−13.8792	−	24.0396i	−0.709196	−	1.22836i	−0.965156	−	0.261676i	$-0.915725\pi$
$383$	−13.8792	−	24.0396i	−0.709196	−	1.22836i	0.255960	−	0.966687i	$-0.417609\pi$
$384$		0			0
$385$	−10.1627	−	17.6023i	−0.517939	−	0.897096i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	20.8911	+	12.0615i	1.05922	+	0.611542i	0.925217	−	0.379439i	$-0.123883\pi$
$389$	20.8911	+	12.0615i	1.05922	+	0.611542i	0.134004	+	0.990981i	$0.457216\pi$
$390$		0			0
$391$	−7.88967			−0.398998
$392$		0			0
$393$		0			0
$394$		0			0
$395$	20.6827	−	35.8234i	1.04066	−	1.80247i
$396$		0			0
$397$	16.9548	+	29.3666i	0.850937	+	1.47387i	0.880363	+	0.474300i	$0.157299\pi$
$397$	16.9548	+	29.3666i	0.850937	+	1.47387i	−0.0294262	+	0.999567i	$0.509368\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	18.2899	+	31.6791i	0.913355	+	1.58198i	0.809293	+	0.587406i	$0.199851\pi$
$401$	18.2899	+	31.6791i	0.913355	+	1.58198i	0.104062	+	0.994571i	$0.466816\pi$
$402$		0			0
$403$	1.65341	−	2.86379i	0.0823622	−	0.142655i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−9.53253			−0.472510
$408$		0			0
$409$	−4.41092	−	2.54665i	−0.218106	−	0.125924i	0.386967	−	0.922094i	$-0.373523\pi$
$409$	−4.41092	−	2.54665i	−0.218106	−	0.125924i	−0.605073	+	0.796170i	$0.706856\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−23.8634	−	41.3327i	−1.17424	−	2.03385i
$414$		0			0
$415$	15.6098	+	27.0370i	0.766257	+	1.32720i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			4.55747i			0.222647i	0.993784	+	0.111323i	$0.0355089\pi$
$419$			4.55747i			0.222647i	−0.993784	+	0.111323i	$0.964491\pi$
$420$		0			0
$421$	8.85940	−	5.11498i	0.431781	−	0.249289i	−0.268324	−	0.963329i	$-0.586470\pi$
$421$	8.85940	−	5.11498i	0.431781	−	0.249289i	0.700105	+	0.714040i	$0.253137\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			5.02110i			0.243559i
$426$		0			0
$427$	28.5900	−	49.5193i	1.38357	−	2.39641i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−6.68862	+	11.5850i	−0.322179	+	0.558031i	−0.980937	−	0.194324i	$-0.937749\pi$
$431$	−6.68862	+	11.5850i	−0.322179	+	0.558031i	0.658758	+	0.752355i	$0.271082\pi$
$432$		0			0
$433$	−7.33378	−	4.23416i	−0.352439	−	0.203481i	0.313320	−	0.949648i	$-0.398559\pi$
$433$	−7.33378	−	4.23416i	−0.352439	−	0.203481i	−0.665759	+	0.746167i	$0.731892\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−18.6477	+	11.3383i	−0.892039	+	0.542384i
$438$		0			0
$439$	32.0783	−	18.5204i	1.53101	−	0.883932i	0.531700	−	0.846933i	$-0.321554\pi$
$439$	32.0783	−	18.5204i	1.53101	−	0.883932i	0.999315	−	0.0369989i	$-0.0117798\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−12.9891	−	7.49923i	−0.617129	−	0.356299i	0.158622	−	0.987339i	$-0.449295\pi$
$443$	−12.9891	−	7.49923i	−0.617129	−	0.356299i	−0.775750	+	0.631040i	$0.782628\pi$
$444$		0			0
$445$			47.9447i			2.27280i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−17.3441			−0.818517			−0.409258	−	0.912419i	$-0.634212\pi$
$449$	−17.3441			−0.818517			−0.409258	+	0.912419i	$0.634212\pi$
$450$		0			0
$451$	−12.6496	+	7.30324i	−0.595645	+	0.343896i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	12.3042			0.576829
$456$		0			0
$457$	22.0333			1.03067			0.515337	−	0.856988i	$-0.327667\pi$
$457$	22.0333			1.03067			0.515337	+	0.856988i	$0.327667\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	15.7374	−	9.08599i	0.732963	−	0.423177i	−0.0865419	−	0.996248i	$-0.527582\pi$
$461$	15.7374	−	9.08599i	0.732963	−	0.423177i	0.819505	+	0.573072i	$0.194248\pi$
$462$		0			0
$463$	−42.6463			−1.98194			−0.990970	−	0.134085i	$-0.957191\pi$
$463$	−42.6463			−1.98194			−0.990970	+	0.134085i	$0.957191\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			34.3222i			1.58824i	0.607760	+	0.794121i	$0.292068\pi$
$467$			34.3222i			1.58824i	−0.607760	+	0.794121i	$0.707932\pi$
$468$		0			0
$469$	12.4701	+	7.19962i	0.575816	+	0.332448i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−9.57291	+	5.52692i	−0.440163	+	0.254128i
$474$		0			0
$475$	7.21584	+	11.8676i	0.331086	+	0.544525i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	30.5606	+	17.6441i	1.39635	+	0.806182i	0.994008	−	0.109309i	$-0.0348637\pi$
$479$	30.5606	+	17.6441i	1.39635	+	0.806182i	0.402340	+	0.915490i	$0.368197\pi$
$480$		0			0
$481$	2.88531	−	4.99750i	0.131559	−	0.227866i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−4.60866	+	7.98243i	−0.209268	+	0.362464i
$486$		0			0
$487$			9.59157i			0.434635i	0.976101	+	0.217318i	$0.0697308\pi$
$487$			9.59157i			0.434635i	−0.976101	+	0.217318i	$0.930269\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−6.99071	+	4.03609i	−0.315486	+	0.182146i	−0.649379	−	0.760465i	$-0.724971\pi$
$491$	−6.99071	+	4.03609i	−0.315486	+	0.182146i	0.333893	+	0.942611i	$0.391638\pi$
$492$		0			0
$493$		−	1.42264i		−	0.0640727i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−14.4474	−	25.0236i	−0.648053	−	1.12246i
$498$		0			0
$499$	−11.5743	−	20.0473i	−0.518137	−	0.897439i	−0.999778	−	0.0210709i	$-0.993292\pi$
$499$	−11.5743	−	20.0473i	−0.518137	−	0.897439i	0.481641	−	0.876369i	$-0.340041\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	4.67069	+	2.69663i	0.208256	+	0.120237i	0.600501	−	0.799624i	$-0.294968\pi$
$503$	4.67069	+	2.69663i	0.208256	+	0.120237i	−0.392245	+	0.919861i	$0.628301\pi$
$504$		0			0
$505$	−38.8846			−1.73034
$506$		0			0
$507$		0			0
$508$		0			0
$509$	5.95218	−	10.3095i	0.263826	−	0.456960i	−0.703430	−	0.710765i	$-0.748349\pi$
$509$	5.95218	−	10.3095i	0.263826	−	0.456960i	0.967255	+	0.253805i	$0.0816823\pi$
$510$		0			0
$511$	4.35778	+	7.54789i	0.192777	+	0.333899i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	19.7694	+	34.2416i	0.871143	+	1.50886i
$516$		0			0
$517$	−3.36161	+	5.82247i	−0.147843	+	0.256072i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	21.9651			0.962309			0.481155	−	0.876636i	$-0.340218\pi$
$521$	21.9651			0.962309			0.481155	+	0.876636i	$0.340218\pi$
$522$		0			0
$523$	−0.0371082	−	0.0214244i	−0.00162263	−	0.000936824i	0.499188	−	0.866493i	$-0.333632\pi$
$523$	−0.0371082	−	0.0214244i	−0.00162263	−	0.000936824i	−0.500811	+	0.865557i	$0.666965\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−2.38229	−	4.12625i	−0.103774	−	0.179742i
$528$		0			0
$529$	1.03401	+	1.79096i	0.0449570	+	0.0778677i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	8.84219i		−	0.382998i
$534$		0			0
$535$	11.4455	−	6.60808i	0.494834	−	0.285692i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			15.2864i			0.658430i
$540$		0			0
$541$	3.15724	−	5.46850i	0.135740	−	0.235109i	−0.790140	−	0.612927i	$-0.789992\pi$
$541$	3.15724	−	5.46850i	0.135740	−	0.235109i	0.925880	+	0.377818i	$0.123325\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	23.9801	−	41.5348i	1.02720	−	1.77916i
$546$		0			0
$547$	−13.9330	−	8.04422i	−0.595732	−	0.343946i	0.171629	−	0.985162i	$-0.445097\pi$
$547$	−13.9330	−	8.04422i	−0.595732	−	0.343946i	−0.767361	+	0.641216i	$0.778430\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−2.04449	−	3.36250i	−0.0870982	−	0.143247i
$552$		0			0
$553$	−49.2315	+	28.4238i	−2.09353	+	1.20870i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−28.0014	−	16.1666i	−1.18646	−	0.685003i	−0.228959	−	0.973436i	$-0.573532\pi$
$557$	−28.0014	−	16.1666i	−1.18646	−	0.685003i	−0.957500	+	0.288433i	$0.906866\pi$
$558$		0			0
$559$		−	6.69157i		−	0.283023i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−28.3247			−1.19374			−0.596872	−	0.802337i	$-0.703590\pi$
$563$	−28.3247			−1.19374			−0.596872	+	0.802337i	$0.703590\pi$
$564$		0			0
$565$	−37.2510	+	21.5069i	−1.56716	+	0.904802i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	36.5760			1.53334			0.766672	−	0.642039i	$-0.221911\pi$
$569$	36.5760			1.53334			0.766672	+	0.642039i	$0.221911\pi$
$570$		0			0
$571$	26.4818			1.10823			0.554113	−	0.832441i	$-0.313057\pi$
$571$	26.4818			1.10823			0.554113	+	0.832441i	$0.313057\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	13.8163	−	7.97682i	0.576178	−	0.332656i
$576$		0			0
$577$	−33.1089			−1.37834			−0.689170	−	0.724600i	$-0.742025\pi$
$577$	−33.1089			−1.37834			−0.689170	+	0.724600i	$0.742025\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	42.9046i		−	1.77998i
$582$		0			0
$583$	−18.9907	−	10.9643i	−0.786516	−	0.454095i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	38.9220	−	22.4716i	1.60648	−	0.927503i	0.616333	−	0.787486i	$-0.288618\pi$
$587$	38.9220	−	22.4716i	1.60648	−	0.927503i	0.990149	−	0.140017i	$-0.0447156\pi$
$588$		0			0
$589$	−11.5605	−	6.32901i	−0.476343	−	0.260782i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−15.8247	−	9.13638i	−0.649842	−	0.375186i	0.138554	−	0.990355i	$-0.455755\pi$
$593$	−15.8247	−	9.13638i	−0.649842	−	0.375186i	−0.788396	+	0.615169i	$0.789088\pi$
$594$		0			0
$595$	8.86415	−	15.3532i	0.363395	−	0.629418i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−11.1805	+	19.3653i	−0.456825	+	0.791244i	−0.998791	−	0.0491564i	$-0.984347\pi$
$599$	−11.1805	+	19.3653i	−0.456825	+	0.791244i	0.541966	+	0.840400i	$0.317680\pi$
$600$		0			0
$601$		−	24.6726i		−	1.00641i	−0.864166	−	0.503207i	$-0.832153\pi$
$601$		−	24.6726i		−	1.00641i	0.864166	−	0.503207i	$-0.167847\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−19.1689	+	11.0672i	−0.779327	+	0.449945i
$606$		0			0
$607$		−	47.4017i		−	1.92398i	−0.273090	−	0.961988i	$-0.588046\pi$
$607$		−	47.4017i		−	1.92398i	0.273090	−	0.961988i	$-0.411954\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−2.03498	−	3.52470i	−0.0823267	−	0.142594i
$612$		0			0
$613$	0.807176	+	1.39807i	0.0326015	+	0.0564675i	0.881866	−	0.471500i	$-0.156287\pi$
$613$	0.807176	+	1.39807i	0.0326015	+	0.0564675i	−0.849264	+	0.527968i	$0.822954\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	19.0845	+	11.0185i	0.768314	+	0.443586i	0.832273	−	0.554366i	$-0.187039\pi$
$617$	19.0845	+	11.0185i	0.768314	+	0.443586i	−0.0639588	+	0.997953i	$0.520373\pi$
$618$		0			0
$619$	−16.1090			−0.647474			−0.323737	−	0.946147i	$-0.604939\pi$
$619$	−16.1090			−0.647474			−0.323737	+	0.946147i	$0.604939\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	32.9448	−	57.0620i	1.31990	−	2.28614i
$624$		0			0
$625$	15.3894	+	26.6553i	0.615577	+	1.06621i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−4.15725	−	7.20058i	−0.165761	−	0.287106i
$630$		0			0
$631$	3.35620	−	5.81311i	0.133608	−	0.231416i	−0.791457	−	0.611225i	$-0.790677\pi$
$631$	3.35620	−	5.81311i	0.133608	−	0.231416i	0.925065	+	0.379809i	$0.124010\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	23.7056			0.940727
$636$		0			0
$637$	−8.01399	−	4.62688i	−0.317526	−	0.183324i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−9.20662	−	15.9463i	−0.363640	−	0.629842i	0.624917	−	0.780691i	$-0.285133\pi$
$641$	−9.20662	−	15.9463i	−0.363640	−	0.629842i	−0.988557	+	0.150849i	$0.951799\pi$
$642$		0			0
$643$	7.40066	+	12.8183i	0.291854	+	0.505505i	0.974248	−	0.225479i	$-0.0723946\pi$
$643$	7.40066	+	12.8183i	0.291854	+	0.505505i	−0.682394	+	0.730984i	$0.739061\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			13.5539i			0.532857i	0.963855	+	0.266429i	$0.0858436\pi$
$647$			13.5539i			0.532857i	−0.963855	+	0.266429i	$0.914156\pi$
$648$		0			0
$649$	18.9907	−	10.9643i	0.745451	−	0.430386i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	41.4603i		−	1.62247i	−0.584723	−	0.811233i	$-0.698797\pi$
$653$		−	41.4603i		−	1.62247i	0.584723	−	0.811233i	$-0.301203\pi$
$654$		0			0
$655$	5.32381	−	9.22112i	0.208019	−	0.360299i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	8.46165	−	14.6560i	0.329619	−	0.570917i	−0.652817	−	0.757516i	$-0.726413\pi$
$659$	8.46165	−	14.6560i	0.329619	−	0.570917i	0.982436	+	0.186598i	$0.0597463\pi$
$660$		0			0
$661$	5.45347	+	3.14856i	0.212116	+	0.122465i	0.602294	−	0.798274i	$-0.294253\pi$
$661$	5.45347	+	3.14856i	0.212116	+	0.122465i	−0.390179	+	0.920739i	$0.627587\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−1.11319	−	49.0268i	−0.0431675	−	1.90118i
$666$		0			0
$667$	−3.91461	+	2.26010i	−0.151574	+	0.0875114i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	22.7522	+	13.1360i	0.878337	+	0.507108i
$672$		0			0
$673$			3.33167i			0.128426i	0.997936	+	0.0642132i	$0.0204538\pi$
$673$			3.33167i			0.128426i	−0.997936	+	0.0642132i	$0.979546\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−23.5183			−0.903881			−0.451940	−	0.892048i	$-0.649268\pi$
$677$	−23.5183			−0.903881			−0.451940	+	0.892048i	$0.649268\pi$
$678$		0			0
$679$	10.9701	−	6.33360i	0.420994	−	0.243061i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−39.3092			−1.50412			−0.752062	−	0.659092i	$-0.770941\pi$
$683$	−39.3092			−1.50412			−0.752062	+	0.659092i	$0.770941\pi$
$684$		0			0
$685$	−15.1217			−0.577771
$686$		0			0
$687$		0			0
$688$		0			0
$689$	11.4962	−	6.63736i	0.437972	−	0.252863i
$690$		0			0
$691$	31.5925			1.20183			0.600917	−	0.799311i	$-0.294802\pi$
$691$	31.5925			1.20183			0.600917	+	0.799311i	$0.294802\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			12.9588i			0.491554i
$696$		0			0
$697$	−11.0333	−	6.37007i	−0.417915	−	0.241283i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−25.4470	+	14.6918i	−0.961120	+	0.554903i	−0.896517	−	0.443008i	$-0.853911\pi$
$701$	−25.4470	+	14.6918i	−0.961120	+	0.554903i	−0.0646023	+	0.997911i	$0.520578\pi$
$702$		0			0
$703$	−20.1739	−	11.0445i	−0.760872	−	0.416553i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	46.2790	+	26.7192i	1.74050	+	1.00488i
$708$		0			0
$709$	−4.67388	+	8.09540i	−0.175531	+	0.304029i	−0.940345	−	0.340222i	$-0.889498\pi$
$709$	−4.67388	+	8.09540i	−0.175531	+	0.304029i	0.764814	+	0.644251i	$0.222831\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−7.56930	+	13.1104i	−0.283472	+	0.490989i
$714$		0			0
$715$			5.65329i			0.211421i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−9.86840	+	5.69752i	−0.368029	+	0.212482i	−0.672597	−	0.740009i	$-0.734821\pi$
$719$	−9.86840	+	5.69752i	−0.368029	+	0.212482i	0.304568	+	0.952491i	$0.401488\pi$
$720$		0			0
$721$		−	54.3374i		−	2.02363i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	1.43836	+	2.49131i	0.0534193	+	0.0925250i
$726$		0			0
$727$	4.61852	+	7.99952i	0.171292	+	0.296686i	0.938872	−	0.344267i	$-0.111873\pi$
$727$	4.61852	+	7.99952i	0.171292	+	0.296686i	−0.767580	+	0.640953i	$0.778539\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−8.34973	−	4.82072i	−0.308826	−	0.178301i
$732$		0			0
$733$	6.20877			0.229326			0.114663	−	0.993404i	$-0.463421\pi$
$733$	6.20877			0.229326			0.114663	+	0.993404i	$0.463421\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−3.30794	+	5.72953i	−0.121850	+	0.211050i
$738$		0			0
$739$	14.1240	+	24.4634i	0.519559	+	0.899902i	0.999742	+	0.0227338i	$0.00723703\pi$
$739$	14.1240	+	24.4634i	0.519559	+	0.899902i	−0.480183	+	0.877168i	$0.659430\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−8.17002	−	14.1509i	−0.299729	−	0.519146i	0.676345	−	0.736585i	$-0.263563\pi$
$743$	−8.17002	−	14.1509i	−0.299729	−	0.519146i	−0.976074	+	0.217439i	$0.930230\pi$
$744$		0			0
$745$	15.6239	−	27.0614i	0.572416	−	0.991454i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−18.1627			−0.663652
$750$		0			0
$751$	4.30271	+	2.48417i	0.157008	+	0.0906487i	0.576446	−	0.817136i	$-0.304439\pi$
$751$	4.30271	+	2.48417i	0.157008	+	0.0906487i	−0.419437	+	0.907784i	$0.637773\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−24.5568	−	42.5336i	−0.893714	−	1.54796i
$756$		0			0
$757$	22.4951	+	38.9627i	0.817600	+	1.41612i	0.907446	+	0.420169i	$0.138029\pi$
$757$	22.4951	+	38.9627i	0.817600	+	1.41612i	−0.0898459	+	0.995956i	$0.528637\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			10.8971i			0.395021i	0.980301	+	0.197510i	$0.0632856\pi$
$761$			10.8971i			0.395021i	−0.980301	+	0.197510i	$0.936714\pi$
$762$		0			0
$763$	−57.0805	+	32.9555i	−2.06645	+	1.19307i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			13.2747i			0.479322i
$768$		0			0
$769$	4.53013	−	7.84642i	0.163361	−	0.282949i	−0.772711	−	0.634758i	$-0.781100\pi$
$769$	4.53013	−	7.84642i	0.163361	−	0.282949i	0.936072	+	0.351809i	$0.114433\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	4.45179	−	7.71073i	0.160120	−	0.277336i	−0.774792	−	0.632217i	$-0.782145\pi$
$773$	4.45179	−	7.71073i	0.160120	−	0.277336i	0.934911	+	0.354881i	$0.115479\pi$
$774$		0			0
$775$	8.34365	+	4.81721i	0.299713	+	0.173039i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−35.2322	+	0.799971i	−1.26232	+	0.0286619i
$780$		0			0
$781$	11.4974	−	6.63800i	0.411408	−	0.237526i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	28.1802	+	16.2698i	1.00579	+	0.580696i
$786$		0			0
$787$		−	25.4374i		−	0.906746i	−0.891321	−	0.453373i	$-0.850221\pi$
$787$		−	25.4374i		−	0.906746i	0.891321	−	0.453373i	$-0.149779\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	59.1131			2.10182
$792$		0			0
$793$	−13.7733	+	7.95200i	−0.489103	+	0.282384i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	9.50776			0.336782			0.168391	−	0.985720i	$-0.446143\pi$
$797$	9.50776			0.336782			0.168391	+	0.985720i	$0.446143\pi$
$798$		0			0
$799$	−5.86415			−0.207459
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−3.46796	+	2.00223i	−0.122382	+	0.0706570i
$804$		0			0
$805$	−56.3285			−1.98532
$806$		0			0
$807$		0			0
$808$		0			0
$809$			41.8158i			1.47016i	0.677978	+	0.735082i	$0.262857\pi$
$809$			41.8158i			1.47016i	−0.677978	+	0.735082i	$0.737143\pi$
$810$		0			0
$811$	−17.3840	−	10.0366i	−0.610434	−	0.352434i	0.162701	−	0.986675i	$-0.447979\pi$
$811$	−17.3840	−	10.0366i	−0.610434	−	0.352434i	−0.773135	+	0.634241i	$0.781313\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	13.9624	−	8.06121i	0.489083	−	0.282372i
$816$		0			0
$817$	−26.6629	+	0.605400i	−0.932818	+	0.0211803i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	42.3352	+	24.4422i	1.47751	+	0.853040i	0.999677	−	0.0254110i	$-0.00808945\pi$
$821$	42.3352	+	24.4422i	1.47751	+	0.853040i	0.477832	+	0.878451i	$0.341423\pi$
$822$		0			0
$823$	0.434625	−	0.752793i	0.0151501	−	0.0262407i	−0.858351	−	0.513063i	$-0.828511\pi$
$823$	0.434625	−	0.752793i	0.0151501	−	0.0262407i	0.873501	+	0.486822i	$0.161844\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	9.29366	−	16.0971i	0.323172	−	0.559751i	−0.657968	−	0.753046i	$-0.728584\pi$
$827$	9.29366	−	16.0971i	0.323172	−	0.559751i	0.981141	+	0.193295i	$0.0619173\pi$
$828$		0			0
$829$		−	13.7240i		−	0.476653i	−0.971185	−	0.238327i	$-0.923401\pi$
$829$		−	13.7240i		−	0.476653i	0.971185	−	0.238327i	$-0.0765989\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−11.5468	+	6.66657i	−0.400074	+	0.230983i
$834$		0			0
$835$			56.2300i			1.94592i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	3.43320	+	5.94648i	0.118527	+	0.205295i	0.919184	−	0.393828i	$-0.128849\pi$
$839$	3.43320	+	5.94648i	0.118527	+	0.205295i	−0.800657	+	0.599123i	$0.795516\pi$
$840$		0			0
$841$	14.0925	+	24.4089i	0.485947	+	0.841685i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	29.2484	+	16.8866i	1.00618	+	0.580916i
$846$		0			0
$847$	30.4188			1.04520
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−13.2089	+	22.8785i	−0.452796	+	0.784266i
$852$		0			0
$853$	−8.36573	−	14.4899i	−0.286437	−	0.496124i	0.686520	−	0.727111i	$-0.259138\pi$
$853$	−8.36573	−	14.4899i	−0.286437	−	0.496124i	−0.972957	+	0.230988i	$0.925804\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−26.3106	−	45.5713i	−0.898754	−	1.55669i	−0.829089	−	0.559116i	$-0.811140\pi$
$857$	−26.3106	−	45.5713i	−0.898754	−	1.55669i	−0.0696644	−	0.997570i	$-0.522193\pi$
$858$		0			0
$859$	−3.70563	+	6.41834i	−0.126435	+	0.218991i	−0.922293	−	0.386492i	$-0.873687\pi$
$859$	−3.70563	+	6.41834i	−0.126435	+	0.218991i	0.795858	+	0.605483i	$0.207020\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	1.66016			0.0565126			0.0282563	−	0.999601i	$-0.491005\pi$
$863$	1.66016			0.0565126			0.0282563	+	0.999601i	$0.491005\pi$
$864$		0			0
$865$	24.5592	+	14.1793i	0.835037	+	0.482109i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−13.0596	−	22.6199i	−0.443017	−	0.767328i
$870$		0			0
$871$	−2.00250	−	3.46843i	−0.0678521	−	0.117523i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		−	20.4038i		−	0.689774i
$876$		0			0
$877$	17.3451	−	10.0142i	0.585703	−	0.338156i	−0.177694	−	0.984086i	$-0.556864\pi$
$877$	17.3451	−	10.0142i	0.585703	−	0.338156i	0.763397	+	0.645930i	$0.223530\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			1.41033i			0.0475153i	0.999718	+	0.0237577i	$0.00756301\pi$
$881$			1.41033i			0.0475153i	−0.999718	+	0.0237577i	$0.992437\pi$
$882$		0			0
$883$	2.76203	−	4.78398i	0.0929497	−	0.160994i	−0.815801	−	0.578332i	$-0.803704\pi$
$883$	2.76203	−	4.78398i	0.0929497	−	0.160994i	0.908751	+	0.417339i	$0.137037\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−7.02330	+	12.1647i	−0.235819	+	0.408451i	−0.959510	−	0.281673i	$-0.909111\pi$
$887$	−7.02330	+	12.1647i	−0.235819	+	0.408451i	0.723691	+	0.690124i	$0.242444\pi$
$888$		0			0
$889$	−28.2135	−	16.2891i	−0.946250	−	0.546318i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−13.8603	+	8.42740i	−0.463816	+	0.282012i
$894$		0			0
$895$	−26.0140	+	15.0192i	−0.869552	+	0.502036i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−2.36403	−	1.36488i	−0.0788450	−	0.0455212i
$900$		0			0
$901$		−	19.1267i		−	0.637202i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−27.9078			−0.927687
$906$		0			0
$907$	−1.33510	+	0.770819i	−0.0443312	+	0.0255946i	−0.522002	−	0.852944i	$-0.674815\pi$
$907$	−1.33510	+	0.770819i	−0.0443312	+	0.0255946i	0.477671	+	0.878539i	$0.341481\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	0.652297			0.0216116			0.0108058	−	0.999942i	$-0.496560\pi$
$911$	0.652297			0.0216116			0.0108058	+	0.999942i	$0.496560\pi$
$912$		0			0
$913$	19.7130			0.652405
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−12.6724	+	7.31642i	−0.418480	+	0.241609i
$918$		0			0
$919$	28.9270			0.954213			0.477107	−	0.878845i	$-0.341686\pi$
$919$	28.9270			0.954213			0.477107	+	0.878845i	$0.341686\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			8.03677i			0.264534i
$924$		0			0
$925$	14.5602	+	8.40635i	0.478737	+	0.276399i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	0.0544542	−	0.0314391i	0.00178658	−	0.00103149i	−0.499106	−	0.866541i	$-0.666338\pi$
$929$	0.0544542	−	0.0314391i	0.00178658	−	0.00103149i	0.500893	+	0.865509i	$0.333005\pi$
$930$		0			0
$931$	−17.7110	+	32.3508i	−0.580456	+	1.06026i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	7.05418	+	4.07273i	0.230696	+	0.133193i
$936$		0			0
$937$	15.8491	−	27.4514i	0.517767	−	0.896799i	−0.482020	−	0.876160i	$-0.660097\pi$
$937$	15.8491	−	27.4514i	0.517767	−	0.896799i	0.999787	−	0.0206388i	$-0.00656999\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−15.9057	+	27.5495i	−0.518511	+	0.898087i	0.481258	+	0.876579i	$0.340180\pi$
$941$	−15.9057	+	27.5495i	−0.518511	+	0.898087i	−0.999769	+	0.0215081i	$0.993153\pi$
$942$		0			0
$943$			40.4795i			1.31819i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−30.7308	+	17.7424i	−0.998617	+	0.576552i	−0.907839	−	0.419320i	$-0.862269\pi$
$947$	−30.7308	+	17.7424i	−0.998617	+	0.576552i	−0.0907779	+	0.995871i	$0.528935\pi$
$948$		0			0
$949$		−	2.42414i		−	0.0786909i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	3.47432	+	6.01770i	0.112544	+	0.194932i	0.916795	−	0.399357i	$-0.130767\pi$
$953$	3.47432	+	6.01770i	0.112544	+	0.194932i	−0.804251	+	0.594290i	$0.797433\pi$
$954$		0			0
$955$	8.37922	+	14.5132i	0.271145	+	0.469637i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	17.9973	+	10.3907i	0.581163	+	0.335535i
$960$		0			0
$961$	21.8578			0.705090
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−17.9217	+	31.0413i	−0.576920	+	0.999254i
$966$		0			0
$967$	−27.9439	−	48.4003i	−0.898616	−	1.55645i	−0.829265	−	0.558855i	$-0.811241\pi$
$967$	−27.9439	−	48.4003i	−0.898616	−	1.55645i	−0.0693505	−	0.997592i	$-0.522093\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	21.8370	+	37.8227i	0.700781	+	1.21379i	0.968192	+	0.250207i	$0.0804985\pi$
$971$	21.8370	+	37.8227i	0.700781	+	1.21379i	−0.267411	+	0.963583i	$0.586168\pi$
$972$		0			0
$973$	8.90450	−	15.4230i	0.285465	−	0.494440i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	26.4159			0.845120			0.422560	−	0.906335i	$-0.361132\pi$
$977$	26.4159			0.845120			0.422560	+	0.906335i	$0.361132\pi$
$978$		0			0
$979$	26.2178	+	15.1368i	0.837923	+	0.483775i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−7.13810	−	12.3635i	−0.227670	−	0.394336i	0.729447	−	0.684037i	$-0.239777\pi$
$983$	−7.13810	−	12.3635i	−0.227670	−	0.394336i	−0.957117	+	0.289701i	$0.906444\pi$
$984$		0			0
$985$	23.8276	+	41.2706i	0.759210	+	1.31499i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			30.6340i			0.974103i
$990$		0			0
$991$	53.1901	−	30.7093i	1.68964	−	0.975514i	0.734850	−	0.678230i	$-0.237253\pi$
$991$	53.1901	−	30.7093i	1.68964	−	0.975514i	0.954789	−	0.297284i	$-0.0960807\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			12.7425i			0.403964i
$996$		0			0
$997$	3.31360	−	5.73933i	0.104943	−	0.181766i	−0.808772	−	0.588122i	$-0.799867\pi$
$997$	3.31360	−	5.73933i	0.104943	−	0.181766i	0.913715	+	0.406356i	$0.133201\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	684.2.bk.a.449.2	✓	16
3.2	odd	2	inner	684.2.bk.a.449.7	yes	16
4.3	odd	2		2736.2.dc.d.449.2		16
12.11	even	2		2736.2.dc.d.449.7		16
19.8	odd	6	inner	684.2.bk.a.521.7	yes	16
57.8	even	6	inner	684.2.bk.a.521.2	yes	16
76.27	even	6		2736.2.dc.d.1889.7		16
228.179	odd	6		2736.2.dc.d.1889.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
684.2.bk.a.449.2	✓	16	1.1	even	1	trivial
684.2.bk.a.449.7	yes	16	3.2	odd	2	inner
684.2.bk.a.521.2	yes	16	57.8	even	6	inner
684.2.bk.a.521.7	yes	16	19.8	odd	6	inner
2736.2.dc.d.449.2		16	4.3	odd	2
2736.2.dc.d.449.7		16	12.11	even	2
2736.2.dc.d.1889.2		16	228.179	odd	6
2736.2.dc.d.1889.7		16	76.27	even	6

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:	\( N \)	\(=\)	\( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	684.bk (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.47786 + 1.43059i) q^{5} +3.93208 q^{7} +O(q^{10})\)	\(q+(-2.47786 + 1.43059i) q^{5} +3.93208 q^{7} +1.80663i q^{11} +(-0.947142 - 0.546833i) q^{13} +(-1.36468 + 0.787896i) q^{17} +(-2.09320 + 3.82342i) q^{19} +(4.33601 + 2.50340i) q^{23} +(1.59320 - 2.75950i) q^{25} +(-0.451406 + 0.781859i) q^{29} +3.02361i q^{31} +(-9.74314 + 5.62521i) q^{35} +5.27640i q^{37} +(4.04245 + 7.00173i) q^{41} +(3.05924 + 5.29875i) q^{43} +(3.22283 + 1.86070i) q^{47} +8.46123 q^{49} +(-6.06891 + 10.5117i) q^{53} +(-2.58456 - 4.47659i) q^{55} +(-6.06891 - 10.5117i) q^{59} +(7.27096 - 12.5937i) q^{61} +3.12918 q^{65} +(3.17138 + 1.83100i) q^{67} +(-3.67423 - 6.36396i) q^{71} +(1.10826 + 1.91957i) q^{73} +7.10383i q^{77} +(-12.5205 + 7.22870i) q^{79} -10.9114i q^{83} +(2.25432 - 3.90459i) q^{85} +(8.37847 - 14.5119i) q^{89} +(-3.72424 - 2.15019i) q^{91} +(-0.283104 - 12.4684i) q^{95} +(2.78990 - 1.61075i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 12 q^{19} + 4 q^{25} - 4 q^{43} + 20 q^{55} + 12 q^{61} + 36 q^{67} + 44 q^{73} - 12 q^{79} + 56 q^{85} - 60 q^{91}+O(q^{100}) \) 16 * q - 12 * q^19 + 4 * q^25 - 4 * q^43 + 20 * q^55 + 12 * q^61 + 36 * q^67 + 44 * q^73 - 12 * q^79 + 56 * q^85 - 60 * q^91

Introduction

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