LMFDB - Embedded newform 768.4.c.v.767.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,4,Mod(767,768)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(768, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("768.767");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 768 = 2^{8} \cdot 3 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	768.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$45.3134668844$
Analytic rank:	$0$
Dimension:	$16$
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} - 35x^{12} + 612x^{8} - 8960x^{4} + 65536 $ x^16 - 35x^12 + 612x^8 - 8960*x^4 + 65536
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{50} $
Twist minimal:	no (minimal twist has level 24)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			767.6
Root			$-1.99946 + 0.0465478i$ of defining polynomial
Character	$\chi$	$=$	768.767
Dual form			768.4.c.v.767.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.80770 - 4.37228i) q^{3} +13.1715i q^{5} -26.9490i q^{7} +(-11.2337 + 24.5521i) q^{9} +O(q^{10})$	\(q+(-2.80770 - 4.37228i) q^{3} +13.1715i q^{5} -26.9490i q^{7} +(-11.2337 + 24.5521i) q^{9} +21.9834 q^{11} -10.0326 q^{13} +(57.5896 - 36.9816i) q^{15} -6.09352i q^{17} -40.1902i q^{19} +(-117.829 + 75.6647i) q^{21} +9.80703 q^{23} -48.4891 q^{25} +(138.889 - 19.8179i) q^{27} +164.501i q^{29} +47.0143i q^{31} +(-61.7228 - 96.1178i) q^{33} +354.960 q^{35} +205.560 q^{37} +(28.1685 + 43.8654i) q^{39} -419.120i q^{41} -205.038i q^{43} +(-323.388 - 147.965i) q^{45} -566.089 q^{47} -383.250 q^{49} +(-26.6426 + 17.1087i) q^{51} -342.173i q^{53} +289.555i q^{55} +(-175.723 + 112.842i) q^{57} +3.70288 q^{59} +717.005 q^{61} +(661.654 + 302.737i) q^{63} -132.145i q^{65} -238.701i q^{67} +(-27.5351 - 42.8791i) q^{69} -517.054 q^{71} -984.195 q^{73} +(136.143 + 212.008i) q^{75} -592.432i q^{77} -329.100i q^{79} +(-476.609 - 551.621i) q^{81} -625.333 q^{83} +80.2609 q^{85} +(719.244 - 461.868i) q^{87} +238.583i q^{89} +270.369i q^{91} +(205.560 - 132.002i) q^{93} +529.366 q^{95} -1001.01 q^{97} +(-246.955 + 539.739i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 96 q^{9}+O(q^{10}) $ 16 * q + 96 * q^9	$ 16 q + 96 q^{9} - 592 q^{25} - 528 q^{33} - 1904 q^{49} - 2352 q^{57} - 2144 q^{73} - 1008 q^{81} - 7744 q^{97}+O(q^{100}) $ 16 * q + 96 * q^9 - 592 * q^25 - 528 * q^33 - 1904 * q^49 - 2352 * q^57 - 2144 * q^73 - 1008 * q^81 - 7744 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/768\mathbb{Z}\right)^\times$.

$n$	$257$	$511$	$517$
$\chi(n)$	$-1$	$-1$	$1$

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
$3$	−2.80770	−	4.37228i	−0.540341	−	0.841446i
$3$	−2.80770	−	4.37228i	−0.540341	−	0.841446i
$4$		0			0
$5$			13.1715i			1.17810i	0.808098	+	0.589049i	$0.200497\pi$
$5$			13.1715i			1.17810i	−0.808098	+	0.589049i	$0.799503\pi$
$6$		0			0
$7$		−	26.9490i		−	1.45511i	−0.686049	−	0.727555i	$-0.740656\pi$
$7$		−	26.9490i		−	1.45511i	0.686049	−	0.727555i	$-0.259344\pi$
$8$		0			0
$9$	−11.2337	+	24.5521i	−0.416063	+	0.909336i
$10$		0			0
$11$	21.9834			0.602569			0.301284	−	0.953534i	$-0.402585\pi$
$11$	21.9834			0.602569			0.301284	+	0.953534i	$0.402585\pi$
$12$		0			0
$13$	−10.0326			−0.214042			−0.107021	−	0.994257i	$-0.534131\pi$
$13$	−10.0326			−0.214042			−0.107021	+	0.994257i	$0.534131\pi$
$14$		0			0
$15$	57.5896	−	36.9816i	0.991305	−	0.636575i
$16$		0			0
$17$		−	6.09352i		−	0.0869350i	−0.999055	−	0.0434675i	$-0.986160\pi$
$17$		−	6.09352i		−	0.0869350i	0.999055	−	0.0434675i	$-0.0138405\pi$
$18$		0			0
$19$		−	40.1902i		−	0.485277i	−0.970117	−	0.242638i	$-0.921987\pi$
$19$		−	40.1902i		−	0.485277i	0.970117	−	0.242638i	$-0.0780129\pi$
$20$		0			0
$21$	−117.829	+	75.6647i	−1.22440	+	0.786256i
$22$		0			0
$23$	9.80703			0.0889090			0.0444545	−	0.999011i	$-0.485845\pi$
$23$	9.80703			0.0889090			0.0444545	+	0.999011i	$0.485845\pi$
$24$		0			0
$25$	−48.4891			−0.387913
$26$		0			0
$27$	138.889	−	19.8179i	0.989973	−	0.141258i
$28$		0			0
$29$			164.501i			1.05335i	0.850068	+	0.526673i	$0.176561\pi$
$29$			164.501i			1.05335i	−0.850068	+	0.526673i	$0.823439\pi$
$30$		0			0
$31$			47.0143i			0.272387i	0.990682	+	0.136194i	$0.0434870\pi$
$31$			47.0143i			0.272387i	−0.990682	+	0.136194i	$0.956513\pi$
$32$		0			0
$33$	−61.7228	−	96.1178i	−0.325593	−	0.507029i
$34$		0			0
$35$	354.960			1.71426
$36$		0			0
$37$	205.560			0.913346			0.456673	−	0.889635i	$-0.349041\pi$
$37$	205.560			0.913346			0.456673	+	0.889635i	$0.349041\pi$
$38$		0			0
$39$	28.1685	+	43.8654i	0.115656	+	0.180105i
$40$		0			0
$41$		−	419.120i		−	1.59648i	−0.602342	−	0.798238i	$-0.705766\pi$
$41$		−	419.120i		−	1.59648i	0.602342	−	0.798238i	$-0.294234\pi$
$42$		0			0
$43$		−	205.038i		−	0.727163i	−0.931562	−	0.363581i	$-0.881554\pi$
$43$		−	205.038i		−	0.727163i	0.931562	−	0.363581i	$-0.118446\pi$
$44$		0			0
$45$	−323.388	−	147.965i	−1.07129	−	0.490162i
$46$		0			0
$47$	−566.089			−1.75686			−0.878432	−	0.477868i	$-0.841410\pi$
$47$	−566.089			−1.75686			−0.878432	+	0.477868i	$0.841410\pi$
$48$		0			0
$49$	−383.250			−1.11735
$50$		0			0
$51$	−26.6426	+	17.1087i	−0.0731511	+	0.0469746i
$52$		0			0
$53$		−	342.173i		−	0.886813i	−0.896321	−	0.443407i	$-0.853770\pi$
$53$		−	342.173i		−	0.886813i	0.896321	−	0.443407i	$-0.146230\pi$
$54$		0			0
$55$			289.555i			0.709885i
$56$		0			0
$57$	−175.723	+	112.842i	−0.408334	+	0.262215i
$58$		0			0
$59$	3.70288			0.00817075			0.00408538	−	0.999992i	$-0.498700\pi$
$59$	3.70288			0.00817075			0.00408538	+	0.999992i	$0.498700\pi$
$60$		0			0
$61$	717.005			1.50497			0.752484	−	0.658610i	$-0.228855\pi$
$61$	717.005			1.50497			0.752484	+	0.658610i	$0.228855\pi$
$62$		0			0
$63$	661.654	+	302.737i	1.32318	+	0.605417i
$64$		0			0
$65$		−	132.145i		−	0.252162i
$66$		0			0
$67$		−	238.701i		−	0.435253i	−0.976032	−	0.217627i	$-0.930168\pi$
$67$		−	238.701i		−	0.435253i	0.976032	−	0.217627i	$-0.0698315\pi$
$68$		0			0
$69$	−27.5351	−	42.8791i	−0.0480412	−	0.0748121i
$70$		0			0
$71$	−517.054			−0.864268			−0.432134	−	0.901809i	$-0.642239\pi$
$71$	−517.054			−0.864268			−0.432134	+	0.901809i	$0.642239\pi$
$72$		0			0
$73$	−984.195			−1.57796			−0.788982	−	0.614417i	$-0.789391\pi$
$73$	−984.195			−1.57796			−0.788982	+	0.614417i	$0.789391\pi$
$74$		0			0
$75$	136.143	+	212.008i	0.209605	+	0.326408i
$76$		0			0
$77$		−	592.432i		−	0.876804i
$78$		0			0
$79$		−	329.100i		−	0.468691i	−0.972153	−	0.234346i	$-0.924705\pi$
$79$		−	329.100i		−	0.468691i	0.972153	−	0.234346i	$-0.0752948\pi$
$80$		0			0
$81$	−476.609	−	551.621i	−0.653784	−	0.756681i
$82$		0			0
$83$	−625.333			−0.826978			−0.413489	−	0.910509i	$-0.635690\pi$
$83$	−625.333			−0.826978			−0.413489	+	0.910509i	$0.635690\pi$
$84$		0			0
$85$	80.2609			0.102418
$86$		0			0
$87$	719.244	−	461.868i	0.886334	−	0.569167i
$88$		0			0
$89$			238.583i			0.284154i	0.989856	+	0.142077i	$0.0453781\pi$
$89$			238.583i			0.284154i	−0.989856	+	0.142077i	$0.954622\pi$
$90$		0			0
$91$			270.369i			0.311455i
$92$		0			0
$93$	205.560	−	132.002i	0.229199	−	0.147182i
$94$		0			0
$95$	529.366			0.571703
$96$		0			0
$97$	−1001.01			−1.04781			−0.523903	−	0.851778i	$-0.675525\pi$
$97$	−1001.01			−1.04781			−0.523903	+	0.851778i	$0.675525\pi$
$98$		0			0
$99$	−246.955	+	539.739i	−0.250706	+	0.547937i
$100$		0			0
$101$			598.875i			0.590003i	0.955497	+	0.295001i	$0.0953201\pi$
$101$			598.875i			0.590003i	−0.955497	+	0.295001i	$0.904680\pi$
$102$		0			0
$103$		−	1225.30i		−	1.17216i	−0.810253	−	0.586080i	$-0.800670\pi$
$103$		−	1225.30i		−	1.17216i	0.810253	−	0.586080i	$-0.199330\pi$
$104$		0			0
$105$	−996.619	−	1551.98i	−0.926286	−	1.44246i
$106$		0			0
$107$	−324.370			−0.293066			−0.146533	−	0.989206i	$-0.546811\pi$
$107$	−324.370			−0.293066			−0.146533	+	0.989206i	$0.546811\pi$
$108$		0			0
$109$	−1208.38			−1.06186			−0.530928	−	0.847417i	$-0.678157\pi$
$109$	−1208.38			−1.06186			−0.530928	+	0.847417i	$0.678157\pi$
$110$		0			0
$111$	−577.149	−	898.764i	−0.493518	−	0.768531i
$112$		0			0
$113$		−	1823.44i		−	1.51800i	−0.651089	−	0.759002i	$-0.725687\pi$
$113$		−	1823.44i		−	1.51800i	0.651089	−	0.759002i	$-0.274313\pi$
$114$		0			0
$115$			129.174i			0.104743i
$116$		0			0
$117$	112.703	−	246.322i	0.0890549	−	0.194636i
$118$		0			0
$119$	−164.214			−0.126500
$120$		0			0
$121$	−847.728			−0.636911
$122$		0			0
$123$	−1832.51	+	1176.76i	−1.34335	+	0.862642i
$124$		0			0
$125$			1007.77i			0.721098i
$126$		0			0
$127$			543.520i			0.379760i	0.981807	+	0.189880i	$0.0608100\pi$
$127$			543.520i			0.379760i	−0.981807	+	0.189880i	$0.939190\pi$
$128$		0			0
$129$	−896.484	+	575.684i	−0.611868	+	0.392916i
$130$		0			0
$131$	−1246.47			−0.831335			−0.415668	−	0.909517i	$-0.636452\pi$
$131$	−1246.47			−0.831335			−0.415668	+	0.909517i	$0.636452\pi$
$132$		0			0
$133$	−1083.09			−0.706132
$134$		0			0
$135$	261.032	+	1829.38i	0.166415	+	1.16628i
$136$		0			0
$137$		−	2246.38i		−	1.40088i	−0.713709	−	0.700442i	$-0.752986\pi$
$137$		−	2246.38i		−	1.40088i	0.713709	−	0.700442i	$-0.247014\pi$
$138$		0			0
$139$		−	1733.77i		−	1.05796i	−0.848635	−	0.528979i	$-0.822575\pi$
$139$		−	1733.77i		−	1.05796i	0.848635	−	0.528979i	$-0.177425\pi$
$140$		0			0
$141$	1589.41	+	2475.10i	0.949306	+	1.47831i
$142$		0			0
$143$	−220.551			−0.128975
$144$		0			0
$145$	−2166.73			−1.24094
$146$		0			0
$147$	1076.05	+	1675.68i	0.603749	+	0.940187i
$148$		0			0
$149$			1196.89i			0.658074i	0.944317	+	0.329037i	$0.106724\pi$
$149$			1196.89i			0.658074i	−0.944317	+	0.329037i	$0.893276\pi$
$150$		0			0
$151$			2505.09i			1.35007i	0.737784	+	0.675037i	$0.235872\pi$
$151$			2505.09i			1.35007i	−0.737784	+	0.675037i	$0.764128\pi$
$152$		0			0
$153$	149.609	+	68.4527i	0.0790531	+	0.0361704i
$154$		0			0
$155$	−619.250			−0.320899
$156$		0			0
$157$	−2717.53			−1.38142			−0.690709	−	0.723133i	$-0.742701\pi$
$157$	−2717.53			−1.38142			−0.690709	+	0.723133i	$0.742701\pi$
$158$		0			0
$159$	−1496.08	+	960.718i	−0.746205	+	0.479182i
$160$		0			0
$161$		−	264.290i		−	0.129372i
$162$		0			0
$163$		−	2009.71i		−	0.965723i	−0.875697	−	0.482861i	$-0.839598\pi$
$163$		−	2009.71i		−	0.965723i	0.875697	−	0.482861i	$-0.160402\pi$
$164$		0			0
$165$	1266.02	−	812.984i	0.597329	−	0.383580i
$166$		0			0
$167$	4.79686			0.00222271			0.00111135	−	0.999999i	$-0.499646\pi$
$167$	4.79686			0.00222271			0.00111135	+	0.999999i	$0.499646\pi$
$168$		0			0
$169$	−2096.35			−0.954186
$170$		0			0
$171$	986.752	+	451.484i	0.441280	+	0.201906i
$172$		0			0
$173$		−	1501.84i		−	0.660016i	−0.943978	−	0.330008i	$-0.892948\pi$
$173$		−	1501.84i		−	0.660016i	0.943978	−	0.330008i	$-0.107052\pi$
$174$		0			0
$175$			1306.73i			0.564456i
$176$		0			0
$177$	−10.3966	−	16.1901i	−0.00441500	−	0.00687525i
$178$		0			0
$179$	360.443			0.150507			0.0752535	−	0.997164i	$-0.476023\pi$
$179$	360.443			0.150507			0.0752535	+	0.997164i	$0.476023\pi$
$180$		0			0
$181$	−1143.06			−0.469410			−0.234705	−	0.972067i	$-0.575412\pi$
$181$	−1143.06			−0.469410			−0.234705	+	0.972067i	$0.575412\pi$
$182$		0			0
$183$	−2013.13	−	3134.95i	−0.813197	−	1.26635i
$184$		0			0
$185$			2707.53i			1.07601i
$186$		0			0
$187$		−	133.957i		−	0.0523843i
$188$		0			0
$189$	−534.073	−	3742.93i	−0.205546	−	1.44052i
$190$		0			0
$191$	1904.00			0.721302			0.360651	−	0.932701i	$-0.382554\pi$
$191$	1904.00			0.721302			0.360651	+	0.932701i	$0.382554\pi$
$192$		0			0
$193$	934.152			0.348403			0.174201	−	0.984710i	$-0.444266\pi$
$193$	934.152			0.348403			0.174201	+	0.984710i	$0.444266\pi$
$194$		0			0
$195$	−577.775	+	371.023i	−0.212181	+	0.136254i
$196$		0			0
$197$		−	1268.62i		−	0.458808i	−0.973331	−	0.229404i	$-0.926322\pi$
$197$		−	1268.62i		−	0.458808i	0.973331	−	0.229404i	$-0.0736778\pi$
$198$		0			0
$199$		−	2293.01i		−	0.816820i	−0.912798	−	0.408410i	$-0.866083\pi$
$199$		−	2293.01i		−	0.816820i	0.912798	−	0.408410i	$-0.133917\pi$
$200$		0			0
$201$	−1043.67	+	670.200i	−0.366242	+	0.235185i
$202$		0			0
$203$	4433.14			1.53274
$204$		0			0
$205$	5520.45			1.88080
$206$		0			0
$207$	−110.169	+	240.783i	−0.0369917	+	0.0808481i
$208$		0			0
$209$		−	883.519i		−	0.292413i
$210$		0			0
$211$		−	2663.70i		−	0.869084i	−0.900652	−	0.434542i	$-0.856910\pi$
$211$		−	2663.70i		−	0.869084i	0.900652	−	0.434542i	$-0.143090\pi$
$212$		0			0
$213$	1451.73	+	2260.71i	0.467000	+	0.727235i
$214$		0			0
$215$	2700.66			0.856668
$216$		0			0
$217$	1266.99			0.396354
$218$		0			0
$219$	2763.32	+	4303.18i	0.852639	+	1.32777i
$220$		0			0
$221$			61.1339i			0.0186078i
$222$		0			0
$223$		−	5227.96i		−	1.56991i	−0.619552	−	0.784956i	$-0.712686\pi$
$223$		−	5227.96i		−	1.56991i	0.619552	−	0.784956i	$-0.287314\pi$
$224$		0			0
$225$	544.712	−	1190.51i	0.161396	−	0.352743i
$226$		0			0
$227$	5531.45			1.61734			0.808669	−	0.588265i	$-0.200189\pi$
$227$	5531.45			1.61734			0.808669	+	0.588265i	$0.200189\pi$
$228$		0			0
$229$	782.326			0.225754			0.112877	−	0.993609i	$-0.463993\pi$
$229$	782.326			0.225754			0.112877	+	0.993609i	$0.463993\pi$
$230$		0			0
$231$	−2590.28	+	1663.37i	−0.737783	+	0.473774i
$232$		0			0
$233$		−	3918.64i		−	1.10180i	−0.834573	−	0.550898i	$-0.814285\pi$
$233$		−	3918.64i		−	1.10180i	0.834573	−	0.550898i	$-0.185715\pi$
$234$		0			0
$235$		−	7456.26i		−	2.06976i
$236$		0			0
$237$	−1438.92	+	924.012i	−0.394378	+	0.253253i
$238$		0			0
$239$	4134.14			1.11889			0.559446	−	0.828867i	$-0.311014\pi$
$239$	4134.14			1.11889			0.559446	+	0.828867i	$0.311014\pi$
$240$		0			0
$241$	1221.11			0.326384			0.163192	−	0.986594i	$-0.447821\pi$
$241$	1221.11			0.326384			0.163192	+	0.986594i	$0.447821\pi$
$242$		0			0
$243$	−1073.67	+	3632.65i	−0.283440	+	0.958990i
$244$		0			0
$245$		−	5047.99i		−	1.31634i
$246$		0			0
$247$			403.213i			0.103870i
$248$		0			0
$249$	1755.74	+	2734.13i	0.446850	+	0.695857i
$250$		0			0
$251$	−4399.34			−1.10631			−0.553156	−	0.833078i	$-0.686577\pi$
$251$	−4399.34			−1.10631			−0.553156	+	0.833078i	$0.686577\pi$
$252$		0			0
$253$	215.592			0.0535738
$254$		0			0
$255$	−225.348	−	350.923i	−0.0553406	−	0.0861791i
$256$		0			0
$257$			3796.23i			0.921409i	0.887553	+	0.460705i	$0.152403\pi$
$257$			3796.23i			0.921409i	−0.887553	+	0.460705i	$0.847597\pi$
$258$		0			0
$259$		−	5539.63i		−	1.32902i
$260$		0			0
$261$	−4038.84	−	1847.95i	−0.957846	−	0.438258i
$262$		0			0
$263$	6525.99			1.53007			0.765037	−	0.643986i	$-0.222720\pi$
$263$	6525.99			1.53007			0.765037	+	0.643986i	$0.222720\pi$
$264$		0			0
$265$	4506.94			1.04475
$266$		0			0
$267$	1043.15	−	669.868i	0.239100	−	0.153540i
$268$		0			0
$269$			772.537i			0.175102i	0.996160	+	0.0875509i	$0.0279040\pi$
$269$			772.537i			0.175102i	−0.996160	+	0.0875509i	$0.972096\pi$
$270$		0			0
$271$			3732.58i			0.836673i	0.908292	+	0.418336i	$0.137387\pi$
$271$			3732.58i			0.836673i	−0.908292	+	0.418336i	$0.862613\pi$
$272$		0			0
$273$	1182.13	−	759.115i	0.262072	−	0.168292i
$274$		0			0
$275$	−1065.96			−0.233744
$276$		0			0
$277$	6502.55			1.41047			0.705236	−	0.708973i	$-0.250841\pi$
$277$	6502.55			1.41047			0.705236	+	0.708973i	$0.250841\pi$
$278$		0			0
$279$	−1154.30	−	528.144i	−0.247692	−	0.113330i
$280$		0			0
$281$		−	548.243i		−	0.116389i	−0.998305	−	0.0581947i	$-0.981466\pi$
$281$		−	548.243i		−	0.116389i	0.998305	−	0.0581947i	$-0.0185344\pi$
$282$		0			0
$283$			664.623i			0.139603i	0.997561	+	0.0698017i	$0.0222366\pi$
$283$			664.623i			0.139603i	−0.997561	+	0.0698017i	$0.977763\pi$
$284$		0			0
$285$	−1486.30	−	2314.54i	−0.308915	−	0.481057i
$286$		0			0
$287$	−11294.9			−2.32305
$288$		0			0
$289$	4875.87			0.992442
$290$		0			0
$291$	2810.53	+	4376.70i	0.566173	+	0.881673i
$292$		0			0
$293$			5112.98i			1.01947i	0.860332	+	0.509733i	$0.170256\pi$
$293$			5112.98i			1.01947i	−0.860332	+	0.509733i	$0.829744\pi$
$294$		0			0
$295$			48.7726i			0.00962594i
$296$		0			0
$297$	3053.27	−	435.666i	0.596527	−	0.0851175i
$298$		0			0
$299$	−98.3902			−0.0190303
$300$		0			0
$301$	−5525.57			−1.05810
$302$		0			0
$303$	2618.45	−	1681.46i	0.496455	−	0.318803i
$304$		0			0
$305$			9444.05i			1.77300i
$306$		0			0
$307$			594.602i			0.110540i	0.998471	+	0.0552699i	$0.0176019\pi$
$307$			594.602i			0.110540i	−0.998471	+	0.0552699i	$0.982398\pi$
$308$		0			0
$309$	−5357.36	+	3440.27i	−0.986310	+	0.633367i
$310$		0			0
$311$	−7168.24			−1.30699			−0.653495	−	0.756931i	$-0.726698\pi$
$311$	−7168.24			−1.30699			−0.653495	+	0.756931i	$0.726698\pi$
$312$		0			0
$313$	3774.51			0.681623			0.340811	−	0.940132i	$-0.389298\pi$
$313$	3774.51			0.681623			0.340811	+	0.940132i	$0.389298\pi$
$314$		0			0
$315$	−3987.51	+	8715.00i	−0.713240	+	1.55884i
$316$		0			0
$317$			9400.75i			1.66561i	0.553566	+	0.832805i	$0.313267\pi$
$317$			9400.75i			1.66561i	−0.553566	+	0.832805i	$0.686733\pi$
$318$		0			0
$319$			3616.29i			0.634714i
$320$		0			0
$321$	910.733	+	1418.24i	0.158356	+	0.246599i
$322$		0			0
$323$	−244.900			−0.0421876
$324$		0			0
$325$	486.473			0.0830297
$326$		0			0
$327$	3392.78	+	5283.40i	0.573765	+	0.893494i
$328$		0			0
$329$			15255.6i			2.55643i
$330$		0			0
$331$		−	9218.16i		−	1.53074i	−0.643589	−	0.765371i	$-0.722555\pi$
$331$		−	9218.16i		−	1.53074i	0.643589	−	0.765371i	$-0.277445\pi$
$332$		0			0
$333$	−2309.19	+	5046.91i	−0.380009	+	0.830538i
$334$		0			0
$335$	3144.06			0.512771
$336$		0			0
$337$	2977.58			0.481302			0.240651	−	0.970612i	$-0.422639\pi$
$337$	2977.58			0.481302			0.240651	+	0.970612i	$0.422639\pi$
$338$		0			0
$339$	−7972.57	+	5119.65i	−1.27732	+	0.820240i
$340$		0			0
$341$			1033.54i			0.164132i
$342$		0			0
$343$			1084.70i			0.170752i
$344$		0			0
$345$	564.783	−	362.680i	0.0881359	−	0.0565972i
$346$		0			0
$347$	−2497.80			−0.386424			−0.193212	−	0.981157i	$-0.561890\pi$
$347$	−2497.80			−0.386424			−0.193212	+	0.981157i	$0.561890\pi$
$348$		0			0
$349$	8874.07			1.36108			0.680541	−	0.732710i	$-0.261745\pi$
$349$	8874.07			1.36108			0.680541	+	0.732710i	$0.261745\pi$
$350$		0			0
$351$	−1393.42	+	198.826i	−0.211896	+	0.0302351i
$352$		0			0
$353$		−	2525.66i		−	0.380814i	−0.981705	−	0.190407i	$-0.939019\pi$
$353$		−	2525.66i		−	0.380814i	0.981705	−	0.190407i	$-0.0609808\pi$
$354$		0			0
$355$		−	6810.39i		−	1.01819i
$356$		0			0
$357$	461.064	+	717.991i	0.0683532	+	0.106443i
$358$		0			0
$359$	−9422.15			−1.38519			−0.692594	−	0.721328i	$-0.743532\pi$
$359$	−9422.15			−1.38519			−0.692594	+	0.721328i	$0.743532\pi$
$360$		0			0
$361$	5243.75			0.764506
$362$		0			0
$363$	2380.16	+	3706.51i	0.344149	+	0.535926i
$364$		0			0
$365$		−	12963.4i		−	1.85899i
$366$		0			0
$367$			613.530i			0.0872643i	0.999048	+	0.0436322i	$0.0138930\pi$
$367$			613.530i			0.0872643i	−0.999048	+	0.0436322i	$0.986107\pi$
$368$		0			0
$369$	10290.3	+	4708.26i	1.45173	+	0.664234i
$370$		0			0
$371$	−9221.23			−1.29041
$372$		0			0
$373$	−10759.8			−1.49362			−0.746809	−	0.665039i	$-0.768415\pi$
$373$	−10759.8			−1.49362			−0.746809	+	0.665039i	$0.768415\pi$
$374$		0			0
$375$	4406.23	−	2829.50i	0.606765	−	0.389639i
$376$		0			0
$377$		−	1650.37i		−	0.225460i
$378$		0			0
$379$			10132.1i			1.37322i	0.727026	+	0.686610i	$0.240902\pi$
$379$			10132.1i			1.37322i	−0.727026	+	0.686610i	$0.759098\pi$
$380$		0			0
$381$	2376.42	−	1526.04i	0.319548	−	0.205200i
$382$		0			0
$383$	9452.43			1.26109			0.630544	−	0.776154i	$-0.282832\pi$
$383$	9452.43			1.26109			0.630544	+	0.776154i	$0.282832\pi$
$384$		0			0
$385$	7803.24			1.03296
$386$		0			0
$387$	5034.11	+	2303.33i	0.661235	+	0.302545i
$388$		0			0
$389$			3385.08i			0.441209i	0.975363	+	0.220605i	$0.0708031\pi$
$389$			3385.08i			0.441209i	−0.975363	+	0.220605i	$0.929197\pi$
$390$		0			0
$391$		−	59.7593i		−	0.00772930i
$392$		0			0
$393$	3499.72	+	5449.94i	0.449205	+	0.699524i
$394$		0			0
$395$	4334.75			0.552164
$396$		0			0
$397$	−4125.91			−0.521596			−0.260798	−	0.965393i	$-0.583986\pi$
$397$	−4125.91			−0.521596			−0.260798	+	0.965393i	$0.583986\pi$
$398$		0			0
$399$	3040.98	+	4735.56i	0.381552	+	0.594172i
$400$		0			0
$401$		−	8829.27i		−	1.09953i	−0.835318	−	0.549767i	$-0.814717\pi$
$401$		−	8829.27i		−	1.09953i	0.835318	−	0.549767i	$-0.185283\pi$
$402$		0			0
$403$		−	471.676i		−	0.0583024i
$404$		0			0
$405$	7265.69	−	6277.66i	0.891444	−	0.770221i
$406$		0			0
$407$	4518.91			0.550354
$408$		0			0
$409$	−1648.22			−0.199264			−0.0996320	−	0.995024i	$-0.531767\pi$
$409$	−1648.22			−0.199264			−0.0996320	+	0.995024i	$0.531767\pi$
$410$		0			0
$411$	−9821.81	+	6307.15i	−1.17877	+	0.756956i
$412$		0			0
$413$		−	99.7891i		−	0.0118893i
$414$		0			0
$415$		−	8236.59i		−	0.974261i
$416$		0			0
$417$	−7580.51	+	4867.89i	−0.890214	+	0.571658i
$418$		0			0
$419$	−11553.7			−1.34710			−0.673549	−	0.739143i	$-0.735231\pi$
$419$	−11553.7			−1.34710			−0.673549	+	0.739143i	$0.735231\pi$
$420$		0			0
$421$	7651.40			0.885763			0.442882	−	0.896580i	$-0.353956\pi$
$421$	7651.40			0.885763			0.442882	+	0.896580i	$0.353956\pi$
$422$		0			0
$423$	6359.27	−	13898.7i	0.730965	−	1.59758i
$424$		0			0
$425$			295.469i			0.0337232i
$426$		0			0
$427$		−	19322.6i		−	2.18990i
$428$		0			0
$429$	619.241	+	964.313i	0.0696906	+	0.108526i
$430$		0			0
$431$	−14803.7			−1.65445			−0.827224	−	0.561873i	$-0.810081\pi$
$431$	−14803.7			−1.65445			−0.827224	+	0.561873i	$0.810081\pi$
$432$		0			0
$433$	1938.03			0.215094			0.107547	−	0.994200i	$-0.465700\pi$
$433$	1938.03			0.215094			0.107547	+	0.994200i	$0.465700\pi$
$434$		0			0
$435$	6083.51	+	9473.54i	0.670533	+	1.04419i
$436$		0			0
$437$		−	394.146i		−	0.0431455i
$438$		0			0
$439$		−	5276.73i		−	0.573678i	−0.957979	−	0.286839i	$-0.907395\pi$
$439$		−	5276.73i		−	0.573678i	0.957979	−	0.286839i	$-0.0926045\pi$
$440$		0			0
$441$	4305.31	−	9409.58i	0.464886	−	1.01604i
$442$		0			0
$443$	−9203.98			−0.987121			−0.493560	−	0.869712i	$-0.664305\pi$
$443$	−9203.98			−0.987121			−0.493560	+	0.869712i	$0.664305\pi$
$444$		0			0
$445$	−3142.50			−0.334761
$446$		0			0
$447$	5233.14	−	3360.50i	0.553734	−	0.355585i
$448$		0			0
$449$			3221.17i			0.338567i	0.985567	+	0.169283i	$0.0541453\pi$
$449$			3221.17i			0.338567i	−0.985567	+	0.169283i	$0.945855\pi$
$450$		0			0
$451$		−	9213.69i		−	0.961986i
$452$		0			0
$453$	10952.9	−	7033.52i	1.13601	−	0.729500i
$454$		0			0
$455$	−3561.18			−0.366924
$456$		0			0
$457$	12015.1			1.22985			0.614925	−	0.788585i	$-0.289186\pi$
$457$	12015.1			1.22985			0.614925	+	0.788585i	$0.289186\pi$
$458$		0			0
$459$	−120.761	−	846.325i	−0.0122802	−	0.0860633i
$460$		0			0
$461$		−	4758.79i		−	0.480778i	−0.970677	−	0.240389i	$-0.922725\pi$
$461$		−	4758.79i		−	0.480778i	0.970677	−	0.240389i	$-0.0772750\pi$
$462$		0			0
$463$		−	12568.3i		−	1.26156i	−0.775963	−	0.630778i	$-0.782736\pi$
$463$		−	12568.3i		−	1.26156i	0.775963	−	0.630778i	$-0.217264\pi$
$464$		0			0
$465$	1738.66	+	2707.53i	0.173395	+	0.270019i
$466$		0			0
$467$	15360.6			1.52207			0.761034	−	0.648712i	$-0.224692\pi$
$467$	15360.6			1.52207			0.761034	+	0.648712i	$0.224692\pi$
$468$		0			0
$469$	−6432.76			−0.633342
$470$		0			0
$471$	7630.00	+	11881.8i	0.746437	+	1.16239i
$472$		0			0
$473$		−	4507.44i		−	0.438165i
$474$		0			0
$475$			1948.79i			0.188245i
$476$		0			0
$477$	8401.06	+	3843.87i	0.806411	+	0.368970i
$478$		0			0
$479$	2268.94			0.216431			0.108216	−	0.994127i	$-0.465486\pi$
$479$	2268.94			0.216431			0.108216	+	0.994127i	$0.465486\pi$
$480$		0			0
$481$	−2062.30			−0.195494
$482$		0			0
$483$	−1155.55	+	742.045i	−0.108860	+	0.0699052i
$484$		0			0
$485$		−	13184.8i		−	1.23442i
$486$		0			0
$487$			11478.8i			1.06808i	0.845459	+	0.534040i	$0.179327\pi$
$487$			11478.8i			1.06808i	−0.845459	+	0.534040i	$0.820673\pi$
$488$		0			0
$489$	−8787.02	+	5642.66i	−0.812603	+	0.521820i
$490$		0			0
$491$	20306.4			1.86642			0.933212	−	0.359325i	$-0.116993\pi$
$491$	20306.4			1.86642			0.933212	+	0.359325i	$0.116993\pi$
$492$		0			0
$493$	1002.39			0.0915727
$494$		0			0
$495$	−7109.19	−	3252.78i	−0.645524	−	0.295356i
$496$		0			0
$497$			13934.1i			1.25761i
$498$		0			0
$499$			1024.04i			0.0918681i	0.998944	+	0.0459340i	$0.0146264\pi$
$499$			1024.04i			0.0918681i	−0.998944	+	0.0459340i	$0.985374\pi$
$500$		0			0
$501$	−13.4681	−	20.9732i	−0.00120102	−	0.00187029i
$502$		0			0
$503$	−7799.62			−0.691388			−0.345694	−	0.938347i	$-0.612356\pi$
$503$	−7799.62			−0.691388			−0.345694	+	0.938347i	$0.612356\pi$
$504$		0			0
$505$	−7888.10			−0.695080
$506$		0			0
$507$	5885.90	+	9165.82i	0.515586	+	0.802896i
$508$		0			0
$509$		−	298.935i		−	0.0260315i	−0.999915	−	0.0130158i	$-0.995857\pi$
$509$		−	298.935i		−	0.0260315i	0.999915	−	0.0130158i	$-0.00414316\pi$
$510$		0			0
$511$			26523.1i			2.29611i
$512$		0			0
$513$	−796.485	−	5581.99i	−0.0685491	−	0.480411i
$514$		0			0
$515$	16139.1			1.38092
$516$		0			0
$517$	−12444.6			−1.05863
$518$		0			0
$519$	−6566.47	+	4216.71i	−0.555368	+	0.356634i
$520$		0			0
$521$			9553.38i			0.803342i	0.915784	+	0.401671i	$0.131571\pi$
$521$			9553.38i			0.803342i	−0.915784	+	0.401671i	$0.868429\pi$
$522$		0			0
$523$		−	13485.0i		−	1.12745i	−0.825962	−	0.563726i	$-0.809367\pi$
$523$		−	13485.0i		−	1.12745i	0.825962	−	0.563726i	$-0.190633\pi$
$524$		0			0
$525$	5713.41	−	3668.91i	0.474959	−	0.304999i
$526$		0			0
$527$	286.482			0.0236800
$528$		0			0
$529$	−12070.8			−0.992095
$530$		0			0
$531$	−41.5970	+	90.9135i	−0.00339954	+	0.00742996i
$532$		0			0
$533$			4204.87i			0.341713i
$534$		0			0
$535$		−	4272.45i		−	0.345260i
$536$		0			0
$537$	−1012.01	−	1575.96i	−0.0813251	−	0.126643i
$538$		0			0
$539$	−8425.15			−0.673278
$540$		0			0
$541$	−14897.0			−1.18387			−0.591934	−	0.805986i	$-0.701635\pi$
$541$	−14897.0			−1.18387			−0.591934	+	0.805986i	$0.701635\pi$
$542$		0			0
$543$	3209.38	+	4997.80i	0.253642	+	0.394983i
$544$		0			0
$545$		−	15916.3i		−	1.25097i
$546$		0			0
$547$		−	8674.79i		−	0.678076i	−0.940773	−	0.339038i	$-0.889899\pi$
$547$		−	8674.79i		−	0.678076i	0.940773	−	0.339038i	$-0.110101\pi$
$548$		0			0
$549$	−8054.61	+	17604.0i	−0.626161	+	1.36852i
$550$		0			0
$551$	6611.32			0.511165
$552$		0			0
$553$	−8868.92			−0.681998
$554$		0			0
$555$	11838.1	−	7601.93i	0.905404	−	0.581413i
$556$		0			0
$557$			22222.8i			1.69050i	0.534369	+	0.845251i	$0.320549\pi$
$557$			22222.8i			1.69050i	−0.534369	+	0.845251i	$0.679451\pi$
$558$		0			0
$559$			2057.07i			0.155643i
$560$		0			0
$561$	−585.696	+	376.109i	−0.0440786	+	0.0283054i
$562$		0			0
$563$	−8321.78			−0.622950			−0.311475	−	0.950254i	$-0.600823\pi$
$563$	−8321.78			−0.622950			−0.311475	+	0.950254i	$0.600823\pi$
$564$		0			0
$565$	24017.4			1.78836
$566$		0			0
$567$	−14865.6	+	12844.1i	−1.10105	+	0.951328i
$568$		0			0
$569$		−	12051.5i		−	0.887915i	−0.896048	−	0.443957i	$-0.853574\pi$
$569$		−	12051.5i		−	0.887915i	0.896048	−	0.443957i	$-0.146426\pi$
$570$		0			0
$571$			11319.5i			0.829608i	0.909911	+	0.414804i	$0.136150\pi$
$571$			11319.5i			0.829608i	−0.909911	+	0.414804i	$0.863850\pi$
$572$		0			0
$573$	−5345.86	−	8324.83i	−0.389749	−	0.606937i
$574$		0			0
$575$	−475.534			−0.0344889
$576$		0			0
$577$	3145.14			0.226922			0.113461	−	0.993542i	$-0.463806\pi$
$577$	3145.14			0.226922			0.113461	+	0.993542i	$0.463806\pi$
$578$		0			0
$579$	−2622.81	−	4084.37i	−0.188256	−	0.293162i
$580$		0			0
$581$			16852.1i			1.20334i
$582$		0			0
$583$		−	7522.14i		−	0.534366i
$584$		0			0
$585$	3244.43	+	1484.47i	0.229300	+	0.104915i
$586$		0			0
$587$	8484.84			0.596604			0.298302	−	0.954472i	$-0.403580\pi$
$587$	8484.84			0.596604			0.298302	+	0.954472i	$0.403580\pi$
$588$		0			0
$589$	1889.51			0.132183
$590$		0			0
$591$	−5546.75	+	3561.89i	−0.386062	+	0.247913i
$592$		0			0
$593$		−	6571.68i		−	0.455087i	−0.973768	−	0.227543i	$-0.926931\pi$
$593$		−	6571.68i		−	0.455087i	0.973768	−	0.227543i	$-0.0730693\pi$
$594$		0			0
$595$		−	2162.95i		−	0.149029i
$596$		0			0
$597$	−10025.7	+	6438.08i	−0.687310	+	0.441362i
$598$		0			0
$599$	17732.4			1.20956			0.604779	−	0.796393i	$-0.293261\pi$
$599$	17732.4			1.20956			0.604779	+	0.796393i	$0.293261\pi$
$600$		0			0
$601$	−22182.6			−1.50557			−0.752785	−	0.658266i	$-0.771290\pi$
$601$	−22182.6			−1.50557			−0.752785	+	0.658266i	$0.771290\pi$
$602$		0			0
$603$	5860.61	+	2681.49i	0.395792	+	0.181093i
$604$		0			0
$605$		−	11165.9i		−	0.750343i
$606$		0			0
$607$		−	4360.03i		−	0.291546i	−0.989318	−	0.145773i	$-0.953433\pi$
$607$		−	4360.03i		−	0.291546i	0.989318	−	0.145773i	$-0.0465668\pi$
$608$		0			0
$609$	−12446.9	−	19382.9i	−0.828200	−	1.28971i
$610$		0			0
$611$	5679.36			0.376043
$612$		0			0
$613$	−14917.5			−0.982892			−0.491446	−	0.870908i	$-0.663532\pi$
$613$	−14917.5			−0.982892			−0.491446	+	0.870908i	$0.663532\pi$
$614$		0			0
$615$	−15499.7	−	24136.9i	−1.01628	−	1.58259i
$616$		0			0
$617$		−	16400.2i		−	1.07010i	−0.844822	−	0.535048i	$-0.820294\pi$
$617$		−	16400.2i		−	1.07010i	0.844822	−	0.535048i	$-0.179706\pi$
$618$		0			0
$619$		−	3457.20i		−	0.224486i	−0.993681	−	0.112243i	$-0.964197\pi$
$619$		−	3457.20i		−	0.224486i	0.993681	−	0.112243i	$-0.0358035\pi$
$620$		0			0
$621$	1362.09	−	194.355i	0.0880175	−	0.0125591i
$622$		0			0
$623$	6429.57			0.413476
$624$		0			0
$625$	−19334.9			−1.23744
$626$		0			0
$627$	−3862.99	+	2480.65i	−0.246050	+	0.158003i
$628$		0			0
$629$		−	1252.58i		−	0.0794017i
$630$		0			0
$631$		−	10766.4i		−	0.679247i	−0.940562	−	0.339623i	$-0.889700\pi$
$631$		−	10766.4i		−	0.679247i	0.940562	−	0.339623i	$-0.110300\pi$
$632$		0			0
$633$	−11646.4	+	7478.86i	−0.731287	+	0.469602i
$634$		0			0
$635$	−7158.99			−0.447395
$636$		0			0
$637$	3845.00			0.239159
$638$		0			0
$639$	5808.42	−	12694.7i	0.359590	−	0.785910i
$640$		0			0
$641$			4116.88i			0.253677i	0.991923	+	0.126839i	$0.0404830\pi$
$641$			4116.88i			0.253677i	−0.991923	+	0.126839i	$0.959517\pi$
$642$		0			0
$643$		−	7733.58i		−	0.474312i	−0.971472	−	0.237156i	$-0.923785\pi$
$643$		−	7733.58i		−	0.474312i	0.971472	−	0.237156i	$-0.0762153\pi$
$644$		0			0
$645$	−7582.64	−	11808.1i	−0.462893	−	0.720840i
$646$		0			0
$647$	2769.95			0.168312			0.0841561	−	0.996453i	$-0.473181\pi$
$647$	2769.95			0.168312			0.0841561	+	0.996453i	$0.473181\pi$
$648$		0			0
$649$	81.4021			0.00492344
$650$		0			0
$651$	−3557.32	−	5539.63i	−0.214166	−	0.333510i
$652$		0			0
$653$			13463.7i			0.806855i	0.915012	+	0.403428i	$0.132181\pi$
$653$			13463.7i			0.806855i	−0.915012	+	0.403428i	$0.867819\pi$
$654$		0			0
$655$		−	16418.0i		−	0.979394i
$656$		0			0
$657$	11056.1	−	24164.0i	0.656532	−	1.43490i
$658$		0			0
$659$	−5523.87			−0.326524			−0.163262	−	0.986583i	$-0.552202\pi$
$659$	−5523.87			−0.326524			−0.163262	+	0.986583i	$0.552202\pi$
$660$		0			0
$661$	−5189.81			−0.305386			−0.152693	−	0.988274i	$-0.548795\pi$
$661$	−5189.81			−0.305386			−0.152693	+	0.988274i	$0.548795\pi$
$662$		0			0
$663$	267.295	−	171.646i	0.0156574	−	0.0100545i
$664$		0			0
$665$		−	14265.9i		−	0.831892i
$666$		0			0
$667$			1613.26i			0.0936519i
$668$		0			0
$669$	−22858.1	+	14678.5i	−1.32100	+	0.848288i
$670$		0			0
$671$	15762.2			0.906847
$672$		0			0
$673$	−3878.74			−0.222161			−0.111081	−	0.993811i	$-0.535431\pi$
$673$	−3878.74			−0.222161			−0.111081	+	0.993811i	$0.535431\pi$
$674$		0			0
$675$	−6734.62	+	960.953i	−0.384023	+	0.0547957i
$676$		0			0
$677$		−	24431.3i		−	1.38696i	−0.720476	−	0.693480i	$-0.756077\pi$
$677$		−	24431.3i		−	1.38696i	0.720476	−	0.693480i	$-0.243923\pi$
$678$		0			0
$679$			26976.3i			1.52467i
$680$		0			0
$681$	−15530.6	−	24185.1i	−0.873914	−	1.36090i
$682$		0			0
$683$	−18958.5			−1.06212			−0.531058	−	0.847335i	$-0.678205\pi$
$683$	−18958.5			−1.06212			−0.531058	+	0.847335i	$0.678205\pi$
$684$		0			0
$685$	29588.3			1.65038
$686$		0			0
$687$	−2196.53	−	3420.55i	−0.121984	−	0.189959i
$688$		0			0
$689$			3432.89i			0.189815i
$690$		0			0
$691$			25328.5i			1.39442i	0.716868	+	0.697209i	$0.245575\pi$
$691$			25328.5i			1.39442i	−0.716868	+	0.697209i	$0.754425\pi$
$692$		0			0
$693$	14545.4	+	6655.20i	0.797310	+	0.364805i
$694$		0			0
$695$	22836.3			1.24638
$696$		0			0
$697$	−2553.91			−0.138790
$698$		0			0
$699$	−17133.4	+	11002.3i	−0.927102	+	0.595346i
$700$		0			0
$701$		−	11295.0i		−	0.608569i	−0.952581	−	0.304284i	$-0.901583\pi$
$701$		−	11295.0i		−	0.608569i	0.952581	−	0.304284i	$-0.0984173\pi$
$702$		0			0
$703$		−	8261.48i		−	0.443226i
$704$		0			0
$705$	−32600.9	+	20934.9i	−1.74159	+	1.11837i
$706$		0			0
$707$	16139.1			0.858519
$708$		0			0
$709$	29445.8			1.55975			0.779874	−	0.625936i	$-0.215283\pi$
$709$	29445.8			1.55975			0.779874	+	0.625936i	$0.215283\pi$
$710$		0			0
$711$	8080.08	+	3697.00i	0.426198	+	0.195005i
$712$		0			0
$713$			461.070i			0.0242177i
$714$		0			0
$715$		−	2905.00i		−	0.151945i
$716$		0			0
$717$	−11607.4	−	18075.6i	−0.604584	−	0.941488i
$718$		0			0
$719$	−12404.3			−0.643396			−0.321698	−	0.946842i	$-0.604254\pi$
$719$	−12404.3			−0.643396			−0.321698	+	0.946842i	$0.604254\pi$
$720$		0			0
$721$	−33020.7			−1.70562
$722$		0			0
$723$	−3428.50	−	5339.03i	−0.176359	−	0.274634i
$724$		0			0
$725$		−	7976.50i		−	0.408607i
$726$		0			0
$727$		−	7076.17i		−	0.360991i	−0.983576	−	0.180496i	$-0.942230\pi$
$727$		−	7076.17i		−	0.360991i	0.983576	−	0.180496i	$-0.0577702\pi$
$728$		0			0
$729$	18897.5	−	5504.99i	0.960093	−	0.279683i
$730$		0			0
$731$	−1249.40			−0.0632159
$732$		0			0
$733$	10430.6			0.525600			0.262800	−	0.964850i	$-0.415354\pi$
$733$	10430.6			0.525600			0.262800	+	0.964850i	$0.415354\pi$
$734$		0			0
$735$	−22071.2	+	14173.2i	−1.10763	+	0.711274i
$736$		0			0
$737$		−	5247.47i		−	0.262270i
$738$		0			0
$739$			30256.6i			1.50610i	0.657963	+	0.753050i	$0.271418\pi$
$739$			30256.6i			1.50610i	−0.657963	+	0.753050i	$0.728582\pi$
$740$		0			0
$741$	1762.96	−	1132.10i	0.0874007	−	0.0561251i
$742$		0			0
$743$	5577.21			0.275381			0.137690	−	0.990475i	$-0.456032\pi$
$743$	5577.21			0.275381			0.137690	+	0.990475i	$0.456032\pi$
$744$		0			0
$745$	−15764.9			−0.775275
$746$		0			0
$747$	7024.79	−	15353.2i	0.344075	−	0.752001i
$748$		0			0
$749$			8741.46i			0.426443i
$750$		0			0
$751$			10687.5i			0.519296i	0.965703	+	0.259648i	$0.0836066\pi$
$751$			10687.5i			0.519296i	−0.965703	+	0.259648i	$0.916393\pi$
$752$		0			0
$753$	12352.0	+	19235.2i	0.597786	+	0.930901i
$754$		0			0
$755$	−32995.8			−1.59052
$756$		0			0
$757$	9377.01			0.450215			0.225108	−	0.974334i	$-0.427727\pi$
$757$	9377.01			0.450215			0.225108	+	0.974334i	$0.427727\pi$
$758$		0			0
$759$	−605.317	−	942.630i	−0.0289481	−	0.0450794i
$760$		0			0
$761$			15663.5i			0.746124i	0.927806	+	0.373062i	$0.121692\pi$
$761$			15663.5i			0.746124i	−0.927806	+	0.373062i	$0.878308\pi$
$762$		0			0
$763$			32564.8i			1.54512i
$764$		0			0
$765$	−901.626	+	1970.57i	−0.0426123	+	0.0931323i
$766$		0			0
$767$	−37.1496			−0.00174889
$768$		0			0
$769$	18293.6			0.857848			0.428924	−	0.903341i	$-0.358893\pi$
$769$	18293.6			0.857848			0.428924	+	0.903341i	$0.358893\pi$
$770$		0			0
$771$	16598.2	−	10658.7i	0.775316	−	0.497876i
$772$		0			0
$773$			7495.03i			0.348742i	0.984680	+	0.174371i	$0.0557892\pi$
$773$			7495.03i			0.348742i	−0.984680	+	0.174371i	$0.944211\pi$
$774$		0			0
$775$		−	2279.68i		−	0.105663i
$776$		0			0
$777$	−24220.8	+	15553.6i	−1.11830	+	0.718124i
$778$		0			0
$779$	−16844.5			−0.774733
$780$		0			0
$781$	−11366.6			−0.520781
$782$		0			0
$783$	3260.06	+	22847.4i	0.148793	+	1.04278i
$784$		0			0
$785$		−	35794.0i		−	1.62744i
$786$		0			0
$787$		−	16884.3i		−	0.764753i	−0.924007	−	0.382377i	$-0.875106\pi$
$787$		−	16884.3i		−	0.764753i	0.924007	−	0.382377i	$-0.124894\pi$
$788$		0			0
$789$	−18323.0	−	28533.5i	−0.826763	−	1.28748i
$790$		0			0
$791$	−49139.8			−2.20886
$792$		0			0
$793$	−7193.44			−0.322127
$794$		0			0
$795$	−12654.1	−	19705.6i	−0.564523	−	0.879103i
$796$		0			0
$797$		−	29212.0i		−	1.29830i	−0.760662	−	0.649148i	$-0.775125\pi$
$797$		−	29212.0i		−	1.29830i	0.760662	−	0.649148i	$-0.224875\pi$
$798$		0			0
$799$			3449.48i			0.152733i
$800$		0			0
$801$	−5857.70	−	2680.16i	−0.258392	−	0.118226i
$802$		0			0
$803$	−21636.0			−0.950832
$804$		0			0
$805$	3481.10			0.152413
$806$		0			0
$807$	3377.75	−	2169.05i	0.147339	−	0.0946147i
$808$		0			0
$809$		−	28157.8i		−	1.22370i	−0.790973	−	0.611851i	$-0.790425\pi$
$809$		−	28157.8i		−	1.22370i	0.790973	−	0.611851i	$-0.209575\pi$
$810$		0			0
$811$			24658.2i			1.06765i	0.845594	+	0.533827i	$0.179247\pi$
$811$			24658.2i			1.06765i	−0.845594	+	0.533827i	$0.820753\pi$
$812$		0			0
$813$	16319.9	−	10480.0i	0.704015	−	0.452089i
$814$		0			0
$815$	26471.0			1.13771
$816$		0			0
$817$	−8240.51			−0.352875
$818$		0			0
$819$	−6638.13	−	3037.24i	−0.283217	−	0.129585i
$820$		0			0
$821$			23980.9i			1.01941i	0.860348	+	0.509707i	$0.170246\pi$
$821$			23980.9i			1.01941i	−0.860348	+	0.509707i	$0.829754\pi$
$822$		0			0
$823$			7533.72i			0.319087i	0.987191	+	0.159544i	$0.0510023\pi$
$823$			7533.72i			0.319087i	−0.987191	+	0.159544i	$0.948998\pi$
$824$		0			0
$825$	2992.89	+	4660.67i	0.126302	+	0.196683i
$826$		0			0
$827$	25167.2			1.05822			0.529110	−	0.848553i	$-0.322526\pi$
$827$	25167.2			1.05822			0.529110	+	0.848553i	$0.322526\pi$
$828$		0			0
$829$	4440.96			0.186057			0.0930283	−	0.995663i	$-0.470345\pi$
$829$	4440.96			0.186057			0.0930283	+	0.995663i	$0.470345\pi$
$830$		0			0
$831$	−18257.2	−	28431.0i	−0.762136	−	1.18684i
$832$		0			0
$833$			2335.34i			0.0971365i
$834$		0			0
$835$			63.1820i			0.00261857i
$836$		0			0
$837$	931.724	+	6529.78i	0.0384768	+	0.269656i
$838$		0			0
$839$	−8795.36			−0.361918			−0.180959	−	0.983491i	$-0.557920\pi$
$839$	−8795.36			−0.361918			−0.180959	+	0.983491i	$0.557920\pi$
$840$		0			0
$841$	−2671.53			−0.109538
$842$		0			0
$843$	−2397.07	+	1539.30i	−0.0979353	+	0.0628900i
$844$		0			0
$845$		−	27612.1i		−	1.12412i
$846$		0			0
$847$			22845.5i			0.926776i
$848$		0			0
$849$	2905.92	−	1866.06i	0.117469	−	0.0754335i
$850$		0			0
$851$	2015.93			0.0812046
$852$		0			0
$853$	−35919.6			−1.44181			−0.720905	−	0.693034i	$-0.756274\pi$
$853$	−35919.6			−1.44181			−0.720905	+	0.693034i	$0.756274\pi$
$854$		0			0
$855$	−5946.73	+	12997.0i	−0.237864	+	0.519870i
$856$		0			0
$857$			35984.3i			1.43431i	0.696916	+	0.717153i	$0.254555\pi$
$857$			35984.3i			1.43431i	−0.696916	+	0.717153i	$0.745445\pi$
$858$		0			0
$859$		−	19362.0i		−	0.769060i	−0.923112	−	0.384530i	$-0.874364\pi$
$859$		−	19362.0i		−	0.769060i	0.923112	−	0.384530i	$-0.125636\pi$
$860$		0			0
$861$	31712.6	+	49384.3i	1.25524	+	1.95472i
$862$		0			0
$863$	−25590.4			−1.00940			−0.504698	−	0.863296i	$-0.668396\pi$
$863$	−25590.4			−1.00940			−0.504698	+	0.863296i	$0.668396\pi$
$864$		0			0
$865$	19781.5			0.777563
$866$		0			0
$867$	−13690.0	−	21318.7i	−0.536258	−	0.835087i
$868$		0			0
$869$		−	7234.75i		−	0.282419i
$870$		0			0
$871$			2394.80i			0.0931626i
$872$		0			0
$873$	11245.0	−	24576.9i	0.435953	−	0.952808i
$874$		0			0
$875$	27158.3			1.04928
$876$		0			0
$877$	−28481.1			−1.09662			−0.548311	−	0.836275i	$-0.684729\pi$
$877$	−28481.1			−1.09662			−0.548311	+	0.836275i	$0.684729\pi$
$878$		0			0
$879$	22355.4	−	14355.7i	0.857826	−	0.550860i
$880$		0			0
$881$		−	24952.4i		−	0.954219i	−0.878844	−	0.477109i	$-0.841685\pi$
$881$		−	24952.4i		−	0.954219i	0.878844	−	0.477109i	$-0.158315\pi$
$882$		0			0
$883$			28650.5i			1.09192i	0.837811	+	0.545961i	$0.183835\pi$
$883$			28650.5i			1.09192i	−0.837811	+	0.545961i	$0.816165\pi$
$884$		0			0
$885$	213.248	−	136.939i	0.00809971	−	0.00520129i
$886$		0			0
$887$	542.531			0.0205371			0.0102686	−	0.999947i	$-0.496731\pi$
$887$	542.531			0.0205371			0.0102686	+	0.999947i	$0.496731\pi$
$888$		0			0
$889$	14647.3			0.552593
$890$		0			0
$891$	−10477.5	−	12126.5i	−0.393950	−	0.455952i
$892$		0			0
$893$			22751.2i			0.852565i
$894$		0			0
$895$			4747.58i			0.177312i
$896$		0			0
$897$	276.250	+	430.189i	0.0102828	+	0.0160129i
$898$		0			0
$899$	−7733.89			−0.286918
$900$		0			0
$901$	−2085.04			−0.0770951
$902$		0			0
$903$	15514.1	+	24159.4i	0.571736	+	0.890336i
$904$		0			0
$905$		−	15055.9i		−	0.553011i
$906$		0			0
$907$			17915.8i			0.655880i	0.944699	+	0.327940i	$0.106354\pi$
$907$			17915.8i			0.655880i	−0.944699	+	0.327940i	$0.893646\pi$
$908$		0			0
$909$	−14703.6	−	6727.57i	−0.536511	−	0.245478i
$910$		0			0
$911$	30577.4			1.11205			0.556023	−	0.831167i	$-0.312327\pi$
$911$	30577.4			1.11205			0.556023	+	0.831167i	$0.312327\pi$
$912$		0			0
$913$	−13747.0			−0.498311
$914$		0			0
$915$	41292.0	−	26516.0i	1.49188	−	0.958025i
$916$		0			0
$917$			33591.3i			1.20968i
$918$		0			0
$919$		−	21519.6i		−	0.772435i	−0.922408	−	0.386217i	$-0.873781\pi$
$919$		−	21519.6i		−	0.772435i	0.922408	−	0.386217i	$-0.126219\pi$
$920$		0			0
$921$	2599.77	−	1669.46i	0.0930133	−	0.0597292i
$922$		0			0
$923$	5187.41			0.184990
$924$		0			0
$925$	−9967.40			−0.354299
$926$		0			0
$927$	30083.7	+	13764.7i	1.06589	+	0.487692i
$928$		0			0
$929$			23763.7i			0.839247i	0.907698	+	0.419624i	$0.137838\pi$
$929$			23763.7i			0.839247i	−0.907698	+	0.419624i	$0.862162\pi$
$930$		0			0
$931$			15402.9i			0.542223i
$932$		0			0
$933$	20126.2	+	31341.6i	0.706220	+	1.09976i
$934$		0			0
$935$	1764.41			0.0617138
$936$		0			0
$937$	11835.5			0.412644			0.206322	−	0.978484i	$-0.433851\pi$
$937$	11835.5			0.412644			0.206322	+	0.978484i	$0.433851\pi$
$938$		0			0
$939$	−10597.7	−	16503.2i	−0.368309	−	0.573549i
$940$		0			0
$941$			41547.4i			1.43933i	0.694323	+	0.719664i	$0.255704\pi$
$941$			41547.4i			1.43933i	−0.694323	+	0.719664i	$0.744296\pi$
$942$		0			0
$943$		−	4110.32i		−	0.141941i
$944$		0			0
$945$	49300.1	−	7034.56i	1.69707	−	0.242153i
$946$		0			0
$947$	41519.6			1.42472			0.712358	−	0.701816i	$-0.247627\pi$
$947$	41519.6			1.42472			0.712358	+	0.701816i	$0.247627\pi$
$948$		0			0
$949$	9874.06			0.337751
$950$		0			0
$951$	41102.7	−	26394.4i	1.40152	−	0.899998i
$952$		0			0
$953$			31542.2i			1.07214i	0.844173	+	0.536071i	$0.180092\pi$
$953$			31542.2i			1.07214i	−0.844173	+	0.536071i	$0.819908\pi$
$954$		0			0
$955$			25078.6i			0.849764i
$956$		0			0
$957$	15811.5	−	10153.5i	0.534077	−	0.342962i
$958$		0			0
$959$	−60537.8			−2.03844
$960$		0			0
$961$	27580.7			0.925805
$962$		0			0
$963$	3643.87	−	7963.96i	0.121934	−	0.266495i
$964$		0			0
$965$			12304.2i			0.410452i
$966$		0			0
$967$		−	51845.5i		−	1.72414i	−0.506793	−	0.862068i	$-0.669169\pi$
$967$		−	51845.5i		−	1.72414i	0.506793	−	0.862068i	$-0.330831\pi$
$968$		0			0
$969$	687.604	+	1070.77i	0.0227957	+	0.0354986i
$970$		0			0
$971$	16303.5			0.538830			0.269415	−	0.963024i	$-0.413170\pi$
$971$	16303.5			0.538830			0.269415	+	0.963024i	$0.413170\pi$
$972$		0			0
$973$	−46723.3			−1.53945
$974$		0			0
$975$	−1365.87	−	2127.00i	−0.0448644	−	0.0698650i
$976$		0			0
$977$			46997.0i			1.53896i	0.638668	+	0.769482i	$0.279486\pi$
$977$			46997.0i			1.53896i	−0.638668	+	0.769482i	$0.720514\pi$
$978$		0			0
$979$			5244.87i			0.171222i
$980$		0			0
$981$	13574.6	−	29668.4i	0.441798	−	0.965584i
$982$		0			0
$983$	−10730.9			−0.348180			−0.174090	−	0.984730i	$-0.555698\pi$
$983$	−10730.9			−0.348180			−0.174090	+	0.984730i	$0.555698\pi$
$984$		0			0
$985$	16709.6			0.540521
$986$		0			0
$987$	66701.6	−	42832.9i	2.15110	−	1.38135i
$988$		0			0
$989$		−	2010.81i		−	0.0646513i
$990$		0			0
$991$		−	53985.5i		−	1.73048i	−0.501358	−	0.865240i	$-0.667166\pi$
$991$		−	53985.5i		−	1.73048i	0.501358	−	0.865240i	$-0.332834\pi$
$992$		0			0
$993$	−40304.4	+	25881.8i	−1.28804	+	0.827123i
$994$		0			0
$995$	30202.5			0.962294
$996$		0			0
$997$	−49281.6			−1.56546			−0.782731	−	0.622361i	$-0.786174\pi$
$997$	−49281.6			−1.56546			−0.782731	+	0.622361i	$0.786174\pi$
$998$		0			0
$999$	28550.0	−	4073.76i	0.904187	−	0.129017i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.4.c.v.767.6		16
3.2	odd	2	inner	768.4.c.v.767.9		16
4.3	odd	2	inner	768.4.c.v.767.12		16
8.3	odd	2	inner	768.4.c.v.767.5		16
8.5	even	2	inner	768.4.c.v.767.11		16
12.11	even	2	inner	768.4.c.v.767.7		16
16.3	odd	4		24.4.f.b.11.6	yes	8
16.5	even	4		24.4.f.b.11.4	yes	8
16.11	odd	4		96.4.f.b.47.5		8
16.13	even	4		96.4.f.b.47.6		8
24.5	odd	2	inner	768.4.c.v.767.8		16
24.11	even	2	inner	768.4.c.v.767.10		16
48.5	odd	4		24.4.f.b.11.5	yes	8
48.11	even	4		96.4.f.b.47.8		8
48.29	odd	4		96.4.f.b.47.7		8
48.35	even	4		24.4.f.b.11.3	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.4.f.b.11.3	✓	8	48.35	even	4
24.4.f.b.11.4	yes	8	16.5	even	4
24.4.f.b.11.5	yes	8	48.5	odd	4
24.4.f.b.11.6	yes	8	16.3	odd	4
96.4.f.b.47.5		8	16.11	odd	4
96.4.f.b.47.6		8	16.13	even	4
96.4.f.b.47.7		8	48.29	odd	4
96.4.f.b.47.8		8	48.11	even	4
768.4.c.v.767.5		16	8.3	odd	2	inner
768.4.c.v.767.6		16	1.1	even	1	trivial
768.4.c.v.767.7		16	12.11	even	2	inner
768.4.c.v.767.8		16	24.5	odd	2	inner
768.4.c.v.767.9		16	3.2	odd	2	inner
768.4.c.v.767.10		16	24.11	even	2	inner
768.4.c.v.767.11		16	8.5	even	2	inner
768.4.c.v.767.12		16	4.3	odd	2	inner

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:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	768.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.80770 - 4.37228i) q^{3} +13.1715i q^{5} -26.9490i q^{7} +(-11.2337 + 24.5521i) q^{9} +O(q^{10})\)	\(q+(-2.80770 - 4.37228i) q^{3} +13.1715i q^{5} -26.9490i q^{7} +(-11.2337 + 24.5521i) q^{9} +21.9834 q^{11} -10.0326 q^{13} +(57.5896 - 36.9816i) q^{15} -6.09352i q^{17} -40.1902i q^{19} +(-117.829 + 75.6647i) q^{21} +9.80703 q^{23} -48.4891 q^{25} +(138.889 - 19.8179i) q^{27} +164.501i q^{29} +47.0143i q^{31} +(-61.7228 - 96.1178i) q^{33} +354.960 q^{35} +205.560 q^{37} +(28.1685 + 43.8654i) q^{39} -419.120i q^{41} -205.038i q^{43} +(-323.388 - 147.965i) q^{45} -566.089 q^{47} -383.250 q^{49} +(-26.6426 + 17.1087i) q^{51} -342.173i q^{53} +289.555i q^{55} +(-175.723 + 112.842i) q^{57} +3.70288 q^{59} +717.005 q^{61} +(661.654 + 302.737i) q^{63} -132.145i q^{65} -238.701i q^{67} +(-27.5351 - 42.8791i) q^{69} -517.054 q^{71} -984.195 q^{73} +(136.143 + 212.008i) q^{75} -592.432i q^{77} -329.100i q^{79} +(-476.609 - 551.621i) q^{81} -625.333 q^{83} +80.2609 q^{85} +(719.244 - 461.868i) q^{87} +238.583i q^{89} +270.369i q^{91} +(205.560 - 132.002i) q^{93} +529.366 q^{95} -1001.01 q^{97} +(-246.955 + 539.739i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 96 q^{9}+O(q^{10}) \) 16 * q + 96 * q^9	\( 16 q + 96 q^{9} - 592 q^{25} - 528 q^{33} - 1904 q^{49} - 2352 q^{57} - 2144 q^{73} - 1008 q^{81} - 7744 q^{97}+O(q^{100}) \) 16 * q + 96 * q^9 - 592 * q^25 - 528 * q^33 - 1904 * q^49 - 2352 * q^57 - 2144 * q^73 - 1008 * q^81 - 7744 * q^97

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