LMFDB - Embedded newform 460.3.f.a.321.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [460,3,Mod(321,460)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(460, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("460.321");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 460 = 2^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	460.f (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$12.5340921606$
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} - 64 x^{14} - 16 x^{13} + 2252 x^{12} + 648 x^{11} - 30106 x^{10} + 12360 x^{9} + \cdots + 1535848276 $ x^16 - 64x^14 - 16x^13 + 2252x^12 + 648x^11 - 30106x^10 + 12360x^9 + 374528x^8 + 196544x^7 + 1261236x^6 - 4237944x^5 + 38013345x^4 - 8913096x^3 + 327235800x^2 + 72566336x + 1535848276
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			321.1
Root			$-5.14043 + 2.23607i$ of defining polynomial
Character	$\chi$	$=$	460.321
Dual form			460.3.f.a.321.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.14043 q^{3} -2.23607i q^{5} -7.88014i q^{7} +17.4240 q^{9} +O(q^{10})$	$q-5.14043 q^{3} -2.23607i q^{5} -7.88014i q^{7} +17.4240 q^{9} +3.98380i q^{11} +13.5122 q^{13} +11.4943i q^{15} -8.06277i q^{17} +2.96508i q^{19} +40.5073i q^{21} +(-4.80075 - 22.4934i) q^{23} -5.00000 q^{25} -43.3029 q^{27} +0.688711 q^{29} -36.6485 q^{31} -20.4784i q^{33} -17.6205 q^{35} +3.63322i q^{37} -69.4587 q^{39} -32.4636 q^{41} -42.5054i q^{43} -38.9612i q^{45} -59.3870 q^{47} -13.0966 q^{49} +41.4461i q^{51} -57.5176i q^{53} +8.90804 q^{55} -15.2418i q^{57} -25.3686 q^{59} +31.8220i q^{61} -137.303i q^{63} -30.2143i q^{65} +108.405i q^{67} +(24.6779 + 115.626i) q^{69} -67.6191 q^{71} -102.151 q^{73} +25.7021 q^{75} +31.3929 q^{77} +48.5263i q^{79} +65.7793 q^{81} -73.0389i q^{83} -18.0289 q^{85} -3.54027 q^{87} +92.9465i q^{89} -106.478i q^{91} +188.389 q^{93} +6.63011 q^{95} +138.539i q^{97} +69.4136i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 64 q^{9}+O(q^{10}) $ 16 * q + 64 * q^9	$ 16 q + 64 q^{9} - 12 q^{13} - 14 q^{23} - 80 q^{25} + 48 q^{27} + 90 q^{29} + 10 q^{31} + 30 q^{35} + 20 q^{39} + 186 q^{41} - 320 q^{47} + 2 q^{49} - 120 q^{55} - 90 q^{59} - 232 q^{69} - 238 q^{71} - 280 q^{73} + 324 q^{77} + 704 q^{81} - 30 q^{85} + 724 q^{87} - 380 q^{93} + 80 q^{95}+O(q^{100}) $ 16 * q + 64 * q^9 - 12 * q^13 - 14 * q^23 - 80 * q^25 + 48 * q^27 + 90 * q^29 + 10 * q^31 + 30 * q^35 + 20 * q^39 + 186 * q^41 - 320 * q^47 + 2 * q^49 - 120 * q^55 - 90 * q^59 - 232 * q^69 - 238 * q^71 - 280 * q^73 + 324 * q^77 + 704 * q^81 - 30 * q^85 + 724 * q^87 - 380 * q^93 + 80 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/460\mathbb{Z}\right)^\times$.

$n$	$231$	$277$	$281$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	−5.14043			−1.71348			−0.856738	−	0.515752i	$-0.827512\pi$
$3$	−5.14043			−1.71348			−0.856738	+	0.515752i	$0.827512\pi$
$4$		0			0
$5$		−	2.23607i		−	0.447214i
$5$		−	2.23607i		−	0.447214i
$6$		0			0
$7$		−	7.88014i		−	1.12573i	−0.826547	−	0.562867i	$-0.809698\pi$
$7$		−	7.88014i		−	1.12573i	0.826547	−	0.562867i	$-0.190302\pi$
$8$		0			0
$9$	17.4240			1.93600
$10$		0			0
$11$			3.98380i			0.362163i	0.983468	+	0.181082i	$0.0579598\pi$
$11$			3.98380i			0.362163i	−0.983468	+	0.181082i	$0.942040\pi$
$12$		0			0
$13$	13.5122			1.03940			0.519702	−	0.854348i	$-0.326043\pi$
$13$	13.5122			1.03940			0.519702	+	0.854348i	$0.326043\pi$
$14$		0			0
$15$			11.4943i			0.766290i
$16$		0			0
$17$		−	8.06277i		−	0.474280i	−0.971475	−	0.237140i	$-0.923790\pi$
$17$		−	8.06277i		−	0.474280i	0.971475	−	0.237140i	$-0.0762101\pi$
$18$		0			0
$19$			2.96508i			0.156057i	0.996951	+	0.0780284i	$0.0248625\pi$
$19$			2.96508i			0.156057i	−0.996951	+	0.0780284i	$0.975138\pi$
$20$		0			0
$21$			40.5073i			1.92892i
$22$		0			0
$23$	−4.80075	−	22.4934i	−0.208728	−	0.977974i
$23$	−4.80075	−	22.4934i	−0.208728	−	0.977974i
$24$		0			0
$25$	−5.00000			−0.200000
$26$		0			0
$27$	−43.3029			−1.60381
$28$		0			0
$29$	0.688711			0.0237487			0.0118743	−	0.999929i	$-0.496220\pi$
$29$	0.688711			0.0237487			0.0118743	+	0.999929i	$0.496220\pi$
$30$		0			0
$31$	−36.6485			−1.18221			−0.591105	−	0.806595i	$-0.701308\pi$
$31$	−36.6485			−1.18221			−0.591105	+	0.806595i	$0.701308\pi$
$32$		0			0
$33$		−	20.4784i		−	0.620558i
$34$		0			0
$35$	−17.6205			−0.503444
$36$		0			0
$37$			3.63322i			0.0981952i	0.998794	+	0.0490976i	$0.0156345\pi$
$37$			3.63322i			0.0981952i	−0.998794	+	0.0490976i	$0.984365\pi$
$38$		0			0
$39$	−69.4587			−1.78099
$40$		0			0
$41$	−32.4636			−0.791796			−0.395898	−	0.918294i	$-0.629567\pi$
$41$	−32.4636			−0.791796			−0.395898	+	0.918294i	$0.629567\pi$
$42$		0			0
$43$		−	42.5054i		−	0.988497i	−0.869321	−	0.494248i	$-0.835443\pi$
$43$		−	42.5054i		−	0.988497i	0.869321	−	0.494248i	$-0.164557\pi$
$44$		0			0
$45$		−	38.9612i		−	0.865805i
$46$		0			0
$47$	−59.3870			−1.26355			−0.631777	−	0.775151i	$-0.717674\pi$
$47$	−59.3870			−1.26355			−0.631777	+	0.775151i	$0.717674\pi$
$48$		0			0
$49$	−13.0966			−0.267279
$50$		0			0
$51$			41.4461i			0.812668i
$52$		0			0
$53$		−	57.5176i		−	1.08524i	−0.839979	−	0.542618i	$-0.817433\pi$
$53$		−	57.5176i		−	1.08524i	0.839979	−	0.542618i	$-0.182567\pi$
$54$		0			0
$55$	8.90804			0.161964
$56$		0			0
$57$		−	15.2418i		−	0.267399i
$58$		0			0
$59$	−25.3686			−0.429977			−0.214989	−	0.976617i	$-0.568971\pi$
$59$	−25.3686			−0.429977			−0.214989	+	0.976617i	$0.568971\pi$
$60$		0			0
$61$			31.8220i			0.521672i	0.965383	+	0.260836i	$0.0839981\pi$
$61$			31.8220i			0.521672i	−0.965383	+	0.260836i	$0.916002\pi$
$62$		0			0
$63$		−	137.303i		−	2.17942i
$64$		0			0
$65$		−	30.2143i		−	0.464835i
$66$		0			0
$67$			108.405i			1.61799i	0.587819	+	0.808993i	$0.299987\pi$
$67$			108.405i			1.61799i	−0.587819	+	0.808993i	$0.700013\pi$
$68$		0			0
$69$	24.6779	+	115.626i	0.357651	+	1.67573i
$70$		0			0
$71$	−67.6191			−0.952382			−0.476191	−	0.879342i	$-0.657983\pi$
$71$	−67.6191			−0.952382			−0.476191	+	0.879342i	$0.657983\pi$
$72$		0			0
$73$	−102.151			−1.39933			−0.699663	−	0.714473i	$-0.746667\pi$
$73$	−102.151			−1.39933			−0.699663	+	0.714473i	$0.746667\pi$
$74$		0			0
$75$	25.7021			0.342695
$76$		0			0
$77$	31.3929			0.407700
$78$		0			0
$79$			48.5263i			0.614257i	0.951668	+	0.307128i	$0.0993681\pi$
$79$			48.5263i			0.614257i	−0.951668	+	0.307128i	$0.900632\pi$
$80$		0			0
$81$	65.7793			0.812091
$82$		0			0
$83$		−	73.0389i		−	0.879987i	−0.898001	−	0.439993i	$-0.854981\pi$
$83$		−	73.0389i		−	0.879987i	0.898001	−	0.439993i	$-0.145019\pi$
$84$		0			0
$85$	−18.0289			−0.212105
$86$		0			0
$87$	−3.54027			−0.0406928
$88$		0			0
$89$			92.9465i			1.04434i	0.852840	+	0.522172i	$0.174878\pi$
$89$			92.9465i			1.04434i	−0.852840	+	0.522172i	$0.825122\pi$
$90$		0			0
$91$		−	106.478i		−	1.17009i
$92$		0			0
$93$	188.389			2.02569
$94$		0			0
$95$	6.63011			0.0697907
$96$		0			0
$97$			138.539i			1.42824i	0.700025	+	0.714119i	$0.253172\pi$
$97$			138.539i			1.42824i	−0.700025	+	0.714119i	$0.746828\pi$
$98$		0			0
$99$			69.4136i			0.701148i
$100$		0			0
$101$	71.2312			0.705259			0.352630	−	0.935763i	$-0.385288\pi$
$101$	71.2312			0.705259			0.352630	+	0.935763i	$0.385288\pi$
$102$		0			0
$103$		−	173.582i		−	1.68526i	−0.538492	−	0.842630i	$-0.681006\pi$
$103$		−	173.582i		−	1.68526i	0.538492	−	0.842630i	$-0.318994\pi$
$104$		0			0
$105$	90.5771			0.862639
$106$		0			0
$107$			88.3635i			0.825827i	0.910770	+	0.412914i	$0.135489\pi$
$107$			88.3635i			0.825827i	−0.910770	+	0.412914i	$0.864511\pi$
$108$		0			0
$109$		−	142.645i		−	1.30867i	−0.756203	−	0.654337i	$-0.772948\pi$
$109$		−	142.645i		−	1.30867i	0.756203	−	0.654337i	$-0.227052\pi$
$110$		0			0
$111$		−	18.6763i		−	0.168255i
$112$		0			0
$113$			91.5083i			0.809808i	0.914359	+	0.404904i	$0.132695\pi$
$113$			91.5083i			0.809808i	−0.914359	+	0.404904i	$0.867305\pi$
$114$		0			0
$115$	−50.2968	+	10.7348i	−0.437363	+	0.0933462i
$116$		0			0
$117$	235.437			2.01228
$118$		0			0
$119$	−63.5358			−0.533914
$120$		0			0
$121$	105.129			0.868838
$122$		0			0
$123$	166.877			1.35672
$124$		0			0
$125$			11.1803i			0.0894427i
$126$		0			0
$127$	101.492			0.799147			0.399574	−	0.916701i	$-0.369158\pi$
$127$	101.492			0.799147			0.399574	+	0.916701i	$0.369158\pi$
$128$		0			0
$129$			218.496i			1.69377i
$130$		0			0
$131$	−43.9095			−0.335187			−0.167594	−	0.985856i	$-0.553600\pi$
$131$	−43.9095			−0.335187			−0.167594	+	0.985856i	$0.553600\pi$
$132$		0			0
$133$	23.3652			0.175678
$134$		0			0
$135$			96.8282i			0.717246i
$136$		0			0
$137$		−	215.195i		−	1.57077i	−0.619009	−	0.785384i	$-0.712465\pi$
$137$		−	215.195i		−	1.57077i	0.619009	−	0.785384i	$-0.287535\pi$
$138$		0			0
$139$	63.6545			0.457946			0.228973	−	0.973433i	$-0.426463\pi$
$139$	63.6545			0.457946			0.228973	+	0.973433i	$0.426463\pi$
$140$		0			0
$141$	305.275			2.16507
$142$		0			0
$143$			53.8300i			0.376434i
$144$		0			0
$145$		−	1.54001i		−	0.0106207i
$146$		0			0
$147$	67.3223			0.457975
$148$		0			0
$149$		−	97.7394i		−	0.655969i	−0.944683	−	0.327985i	$-0.893631\pi$
$149$		−	97.7394i		−	0.655969i	0.944683	−	0.327985i	$-0.106369\pi$
$150$		0			0
$151$	−209.737			−1.38899			−0.694493	−	0.719500i	$-0.744371\pi$
$151$	−209.737			−1.38899			−0.694493	+	0.719500i	$0.744371\pi$
$152$		0			0
$153$		−	140.486i		−	0.918206i
$154$		0			0
$155$			81.9485i			0.528700i
$156$		0			0
$157$		−	287.459i		−	1.83095i	−0.402378	−	0.915474i	$-0.631816\pi$
$157$		−	287.459i		−	1.83095i	0.402378	−	0.915474i	$-0.368184\pi$
$158$		0			0
$159$			295.665i			1.85953i
$160$		0			0
$161$	−177.251	+	37.8306i	−1.10094	+	0.234973i
$162$		0			0
$163$	112.054			0.687446			0.343723	−	0.939071i	$-0.388312\pi$
$163$	112.054			0.687446			0.343723	+	0.939071i	$0.388312\pi$
$164$		0			0
$165$	−45.7911			−0.277522
$166$		0			0
$167$	−192.709			−1.15394			−0.576972	−	0.816764i	$-0.695766\pi$
$167$	−192.709			−1.15394			−0.576972	+	0.816764i	$0.695766\pi$
$168$		0			0
$169$	13.5807			0.0803593
$170$		0			0
$171$			51.6635i			0.302125i
$172$		0			0
$173$	−303.454			−1.75407			−0.877034	−	0.480429i	$-0.840481\pi$
$173$	−303.454			−1.75407			−0.877034	+	0.480429i	$0.840481\pi$
$174$		0			0
$175$			39.4007i			0.225147i
$176$		0			0
$177$	130.406			0.736755
$178$		0			0
$179$	−94.6527			−0.528786			−0.264393	−	0.964415i	$-0.585172\pi$
$179$	−94.6527			−0.528786			−0.264393	+	0.964415i	$0.585172\pi$
$180$		0			0
$181$			251.014i			1.38682i	0.720545	+	0.693408i	$0.243892\pi$
$181$			251.014i			1.38682i	−0.720545	+	0.693408i	$0.756108\pi$
$182$		0			0
$183$		−	163.578i		−	0.893871i
$184$		0			0
$185$	8.12413			0.0439142
$186$		0			0
$187$	32.1204			0.171767
$188$		0			0
$189$			341.233i			1.80546i
$190$		0			0
$191$		−	74.3125i		−	0.389071i	−0.980895	−	0.194535i	$-0.937680\pi$
$191$		−	74.3125i		−	0.389071i	0.980895	−	0.194535i	$-0.0623199\pi$
$192$		0			0
$193$	−279.864			−1.45007			−0.725037	−	0.688710i	$-0.758177\pi$
$193$	−279.864			−1.45007			−0.725037	+	0.688710i	$0.758177\pi$
$194$		0			0
$195$			155.314i			0.796484i
$196$		0			0
$197$	−239.461			−1.21554			−0.607770	−	0.794113i	$-0.707936\pi$
$197$	−239.461			−1.21554			−0.607770	+	0.794113i	$0.707936\pi$
$198$		0			0
$199$		−	302.424i		−	1.51972i	−0.650087	−	0.759860i	$-0.725267\pi$
$199$		−	302.424i		−	1.51972i	0.650087	−	0.759860i	$-0.274733\pi$
$200$		0			0
$201$		−	557.248i		−	2.77238i
$202$		0			0
$203$		−	5.42714i		−	0.0267347i
$204$		0			0
$205$			72.5909i			0.354102i
$206$		0			0
$207$	−83.6483	−	391.925i	−0.404098	−	1.89336i
$208$		0			0
$209$	−11.8123			−0.0565180
$210$		0			0
$211$	313.131			1.48403			0.742016	−	0.670382i	$-0.233870\pi$
$211$	313.131			1.48403			0.742016	+	0.670382i	$0.233870\pi$
$212$		0			0
$213$	347.591			1.63188
$214$		0			0
$215$	−95.0449			−0.442069
$216$		0			0
$217$			288.795i			1.33085i
$218$		0			0
$219$	525.099			2.39771
$220$		0			0
$221$		−	108.946i		−	0.492969i
$222$		0			0
$223$	−42.5933			−0.191001			−0.0955007	−	0.995429i	$-0.530445\pi$
$223$	−42.5933			−0.191001			−0.0955007	+	0.995429i	$0.530445\pi$
$224$		0			0
$225$	−87.1199			−0.387200
$226$		0			0
$227$			246.716i			1.08686i	0.839456	+	0.543428i	$0.182874\pi$
$227$			246.716i			1.08686i	−0.839456	+	0.543428i	$0.817126\pi$
$228$		0			0
$229$			192.536i			0.840767i	0.907347	+	0.420383i	$0.138104\pi$
$229$			192.536i			0.840767i	−0.907347	+	0.420383i	$0.861896\pi$
$230$		0			0
$231$	−161.373			−0.698584
$232$		0			0
$233$	246.508			1.05797			0.528986	−	0.848630i	$-0.322572\pi$
$233$	246.508			1.05797			0.528986	+	0.848630i	$0.322572\pi$
$234$		0			0
$235$			132.793i			0.565078i
$236$		0			0
$237$		−	249.446i		−	1.05251i
$238$		0			0
$239$	−36.1050			−0.151067			−0.0755335	−	0.997143i	$-0.524066\pi$
$239$	−36.1050			−0.151067			−0.0755335	+	0.997143i	$0.524066\pi$
$240$		0			0
$241$		−	137.672i		−	0.571253i	−0.958341	−	0.285627i	$-0.907798\pi$
$241$		−	137.672i		−	0.571253i	0.958341	−	0.285627i	$-0.0922017\pi$
$242$		0			0
$243$	51.5919			0.212312
$244$		0			0
$245$			29.2850i			0.119531i
$246$		0			0
$247$			40.0648i			0.162206i
$248$		0			0
$249$			375.451i			1.50784i
$250$		0			0
$251$			60.0114i			0.239089i	0.992829	+	0.119545i	$0.0381435\pi$
$251$			60.0114i			0.239089i	−0.992829	+	0.119545i	$0.961857\pi$
$252$		0			0
$253$	89.6091	−	19.1252i	0.354186	−	0.0755938i
$254$		0			0
$255$	92.6762			0.363436
$256$		0			0
$257$	−162.546			−0.632475			−0.316237	−	0.948680i	$-0.602420\pi$
$257$	−162.546			−0.632475			−0.316237	+	0.948680i	$0.602420\pi$
$258$		0			0
$259$	28.6303			0.110542
$260$		0			0
$261$	12.0001			0.0459774
$262$		0			0
$263$			416.956i			1.58539i	0.609621	+	0.792693i	$0.291322\pi$
$263$			416.956i			1.58539i	−0.609621	+	0.792693i	$0.708678\pi$
$264$		0			0
$265$	−128.613			−0.485333
$266$		0			0
$267$		−	477.785i		−	1.78946i
$268$		0			0
$269$	−354.383			−1.31741			−0.658704	−	0.752402i	$-0.728895\pi$
$269$	−354.383			−1.31741			−0.658704	+	0.752402i	$0.728895\pi$
$270$		0			0
$271$	430.921			1.59011			0.795056	−	0.606535i	$-0.207441\pi$
$271$	430.921			1.59011			0.795056	+	0.606535i	$0.207441\pi$
$272$		0			0
$273$			547.344i			2.00492i
$274$		0			0
$275$		−	19.9190i		−	0.0724327i
$276$		0			0
$277$	−17.8119			−0.0643029			−0.0321515	−	0.999483i	$-0.510236\pi$
$277$	−17.8119			−0.0643029			−0.0321515	+	0.999483i	$0.510236\pi$
$278$		0			0
$279$	−638.563			−2.28876
$280$		0			0
$281$		−	537.350i		−	1.91228i	−0.292915	−	0.956138i	$-0.594625\pi$
$281$		−	537.350i		−	1.91228i	0.292915	−	0.956138i	$-0.405375\pi$
$282$		0			0
$283$			227.251i			0.803007i	0.915858	+	0.401503i	$0.131512\pi$
$283$			227.251i			0.803007i	−0.915858	+	0.401503i	$0.868488\pi$
$284$		0			0
$285$	−34.0816			−0.119585
$286$		0			0
$287$			255.818i			0.891352i
$288$		0			0
$289$	223.992			0.775058
$290$		0			0
$291$		−	712.150i		−	2.44725i
$292$		0			0
$293$		−	570.580i		−	1.94737i	−0.227896	−	0.973685i	$-0.573185\pi$
$293$		−	570.580i		−	1.94737i	0.227896	−	0.973685i	$-0.426815\pi$
$294$		0			0
$295$			56.7260i			0.192292i
$296$		0			0
$297$		−	172.510i		−	0.580841i
$298$		0			0
$299$	−64.8690	−	303.936i	−0.216953	−	1.01651i
$300$		0			0
$301$	−334.948			−1.11279
$302$		0			0
$303$	−366.159			−1.20844
$304$		0			0
$305$	71.1561			0.233299
$306$		0			0
$307$	−17.4675			−0.0568973			−0.0284487	−	0.999595i	$-0.509057\pi$
$307$	−17.4675			−0.0568973			−0.0284487	+	0.999595i	$0.509057\pi$
$308$		0			0
$309$			892.285i			2.88765i
$310$		0			0
$311$	−41.5565			−0.133622			−0.0668111	−	0.997766i	$-0.521282\pi$
$311$	−41.5565			−0.133622			−0.0668111	+	0.997766i	$0.521282\pi$
$312$		0			0
$313$			193.750i			0.619011i	0.950898	+	0.309505i	$0.100164\pi$
$313$			193.750i			0.619011i	−0.950898	+	0.309505i	$0.899836\pi$
$314$		0			0
$315$	−307.020			−0.974666
$316$		0			0
$317$	292.088			0.921413			0.460707	−	0.887553i	$-0.347596\pi$
$317$	292.088			0.921413			0.460707	+	0.887553i	$0.347596\pi$
$318$		0			0
$319$			2.74369i			0.00860090i
$320$		0			0
$321$		−	454.226i		−	1.41503i
$322$		0			0
$323$	23.9067			0.0740146
$324$		0			0
$325$	−67.5612			−0.207881
$326$		0			0
$327$			733.259i			2.24238i
$328$		0			0
$329$			467.978i			1.42243i
$330$		0			0
$331$	562.886			1.70056			0.850281	−	0.526329i	$-0.176432\pi$
$331$	562.886			1.70056			0.850281	+	0.526329i	$0.176432\pi$
$332$		0			0
$333$			63.3052i			0.190106i
$334$		0			0
$335$	242.401			0.723585
$336$		0			0
$337$		−	178.192i		−	0.528759i	−0.964419	−	0.264379i	$-0.914833\pi$
$337$		−	178.192i		−	0.528759i	0.964419	−	0.264379i	$-0.0851671\pi$
$338$		0			0
$339$		−	470.392i		−	1.38759i
$340$		0			0
$341$		−	146.000i		−	0.428153i
$342$		0			0
$343$		−	282.924i		−	0.824850i
$344$		0			0
$345$	258.547	−	55.1815i	0.749411	−	0.159946i
$346$		0			0
$347$	−462.453			−1.33272			−0.666359	−	0.745631i	$-0.732148\pi$
$347$	−462.453			−1.33272			−0.666359	+	0.745631i	$0.732148\pi$
$348$		0			0
$349$	159.441			0.456850			0.228425	−	0.973562i	$-0.426642\pi$
$349$	159.441			0.456850			0.228425	+	0.973562i	$0.426642\pi$
$350$		0			0
$351$	−585.119			−1.66701
$352$		0			0
$353$	−474.031			−1.34286			−0.671432	−	0.741067i	$-0.734320\pi$
$353$	−474.031			−1.34286			−0.671432	+	0.741067i	$0.734320\pi$
$354$		0			0
$355$			151.201i			0.425918i
$356$		0			0
$357$	326.601			0.914848
$358$		0			0
$359$		−	462.027i		−	1.28698i	−0.765453	−	0.643492i	$-0.777485\pi$
$359$		−	462.027i		−	1.28698i	0.765453	−	0.643492i	$-0.222515\pi$
$360$		0			0
$361$	352.208			0.975646
$362$		0			0
$363$	−540.410			−1.48873
$364$		0			0
$365$			228.416i			0.625798i
$366$		0			0
$367$			215.737i			0.587839i	0.955830	+	0.293919i	$0.0949597\pi$
$367$			215.737i			0.587839i	−0.955830	+	0.293919i	$0.905040\pi$
$368$		0			0
$369$	−565.646			−1.53292
$370$		0			0
$371$	−453.247			−1.22169
$372$		0			0
$373$		−	137.933i		−	0.369795i	−0.982758	−	0.184897i	$-0.940805\pi$
$373$		−	137.933i		−	0.369795i	0.982758	−	0.184897i	$-0.0591952\pi$
$374$		0			0
$375$		−	57.4717i		−	0.153258i
$376$		0			0
$377$	9.30604			0.0246844
$378$		0			0
$379$		−	15.4281i		−	0.0407073i	−0.999793	−	0.0203536i	$-0.993521\pi$
$379$		−	15.4281i		−	0.0407073i	0.999793	−	0.0203536i	$-0.00647921\pi$
$380$		0			0
$381$	−521.711			−1.36932
$382$		0			0
$383$		−	713.842i		−	1.86382i	−0.362694	−	0.931908i	$-0.618143\pi$
$383$		−	713.842i		−	1.86382i	0.362694	−	0.931908i	$-0.381857\pi$
$384$		0			0
$385$		−	70.1966i		−	0.182329i
$386$		0			0
$387$		−	740.613i		−	1.91373i
$388$		0			0
$389$		−	165.475i		−	0.425385i	−0.977119	−	0.212692i	$-0.931777\pi$
$389$		−	165.475i		−	0.425385i	0.977119	−	0.212692i	$-0.0682232\pi$
$390$		0			0
$391$	−181.359	+	38.7074i	−0.463834	+	0.0989958i
$392$		0			0
$393$	225.714			0.574335
$394$		0			0
$395$	108.508			0.274704
$396$		0			0
$397$	−92.3570			−0.232637			−0.116319	−	0.993212i	$-0.537109\pi$
$397$	−92.3570			−0.232637			−0.116319	+	0.993212i	$0.537109\pi$
$398$		0			0
$399$	−120.107			−0.301021
$400$		0			0
$401$			162.145i			0.404352i	0.979349	+	0.202176i	$0.0648013\pi$
$401$			162.145i			0.404352i	−0.979349	+	0.202176i	$0.935199\pi$
$402$		0			0
$403$	−495.203			−1.22879
$404$		0			0
$405$		−	147.087i		−	0.363178i
$406$		0			0
$407$	−14.4740			−0.0355627
$408$		0			0
$409$	−91.8175			−0.224493			−0.112246	−	0.993680i	$-0.535805\pi$
$409$	−91.8175			−0.224493			−0.112246	+	0.993680i	$0.535805\pi$
$410$		0			0
$411$			1106.20i			2.69147i
$412$		0			0
$413$			199.909i			0.484040i
$414$		0			0
$415$	−163.320			−0.393542
$416$		0			0
$417$	−327.211			−0.784679
$418$		0			0
$419$			210.857i			0.503238i	0.967826	+	0.251619i	$0.0809629\pi$
$419$			210.857i			0.503238i	−0.967826	+	0.251619i	$0.919037\pi$
$420$		0			0
$421$			557.635i			1.32455i	0.749261	+	0.662274i	$0.230409\pi$
$421$			557.635i			1.32455i	−0.749261	+	0.662274i	$0.769591\pi$
$422$		0			0
$423$	−1034.76			−2.44624
$424$		0			0
$425$			40.3138i			0.0948561i
$426$		0			0
$427$	250.762			0.587264
$428$		0			0
$429$		−	276.709i		−	0.645010i
$430$		0			0
$431$			162.258i			0.376469i	0.982124	+	0.188234i	$0.0602765\pi$
$431$			162.258i			0.376469i	−0.982124	+	0.188234i	$0.939724\pi$
$432$		0			0
$433$		−	465.859i		−	1.07589i	−0.842981	−	0.537944i	$-0.819201\pi$
$433$		−	465.859i		−	1.07589i	0.842981	−	0.537944i	$-0.180799\pi$
$434$		0			0
$435$			7.91629i			0.0181984i
$436$		0			0
$437$	66.6947	−	14.2346i	0.152619	−	0.0325735i
$438$		0			0
$439$	−112.072			−0.255289			−0.127644	−	0.991820i	$-0.540742\pi$
$439$	−112.072			−0.255289			−0.127644	+	0.991820i	$0.540742\pi$
$440$		0			0
$441$	−228.196			−0.517451
$442$		0			0
$443$	765.565			1.72814			0.864069	−	0.503373i	$-0.167908\pi$
$443$	765.565			1.72814			0.864069	+	0.503373i	$0.167908\pi$
$444$		0			0
$445$	207.835			0.467044
$446$		0			0
$447$			502.422i			1.12399i
$448$		0			0
$449$	−167.225			−0.372439			−0.186220	−	0.982508i	$-0.559624\pi$
$449$	−167.225			−0.372439			−0.186220	+	0.982508i	$0.559624\pi$
$450$		0			0
$451$		−	129.329i		−	0.286759i
$452$		0			0
$453$	1078.14			2.37999
$454$		0			0
$455$	−238.093			−0.523281
$456$		0			0
$457$			283.250i			0.619804i	0.950768	+	0.309902i	$0.100296\pi$
$457$			283.250i			0.619804i	−0.950768	+	0.309902i	$0.899704\pi$
$458$		0			0
$459$			349.141i			0.760656i
$460$		0			0
$461$	−98.1109			−0.212822			−0.106411	−	0.994322i	$-0.533936\pi$
$461$	−98.1109			−0.212822			−0.106411	+	0.994322i	$0.533936\pi$
$462$		0			0
$463$	749.068			1.61786			0.808929	−	0.587907i	$-0.200048\pi$
$463$	749.068			1.61786			0.808929	+	0.587907i	$0.200048\pi$
$464$		0			0
$465$		−	421.250i		−	0.905915i
$466$		0			0
$467$			71.8733i			0.153904i	0.997035	+	0.0769522i	$0.0245189\pi$
$467$			71.8733i			0.153904i	−0.997035	+	0.0769522i	$0.975481\pi$
$468$		0			0
$469$	854.247			1.82142
$470$		0			0
$471$			1477.66i			3.13728i
$472$		0			0
$473$	169.333			0.357997
$474$		0			0
$475$		−	14.8254i		−	0.0312113i
$476$		0			0
$477$		−	1002.18i		−	2.10102i
$478$		0			0
$479$		−	624.966i		−	1.30473i	−0.757905	−	0.652365i	$-0.773777\pi$
$479$		−	624.966i		−	1.30473i	0.757905	−	0.652365i	$-0.226223\pi$
$480$		0			0
$481$			49.0930i			0.102064i
$482$		0			0
$483$	911.146	−	194.466i	1.88643	−	0.402620i
$484$		0			0
$485$	309.783			0.638727
$486$		0			0
$487$	879.504			1.80596			0.902982	−	0.429679i	$-0.141373\pi$
$487$	879.504			1.80596			0.902982	+	0.429679i	$0.141373\pi$
$488$		0			0
$489$	−576.004			−1.17792
$490$		0			0
$491$	716.334			1.45893			0.729464	−	0.684019i	$-0.239770\pi$
$491$	716.334			1.45893			0.729464	+	0.684019i	$0.239770\pi$
$492$		0			0
$493$		−	5.55292i		−	0.0112635i
$494$		0			0
$495$	155.214			0.313563
$496$		0			0
$497$			532.848i			1.07213i
$498$		0			0
$499$	−34.6000			−0.0693387			−0.0346693	−	0.999399i	$-0.511038\pi$
$499$	−34.6000			−0.0693387			−0.0346693	+	0.999399i	$0.511038\pi$
$500$		0			0
$501$	990.604			1.97725
$502$		0			0
$503$		−	287.726i		−	0.572020i	−0.958227	−	0.286010i	$-0.907671\pi$
$503$		−	287.726i		−	0.572020i	0.958227	−	0.286010i	$-0.0923291\pi$
$504$		0			0
$505$		−	159.278i		−	0.315402i
$506$		0			0
$507$	−69.8107			−0.137694
$508$		0			0
$509$	284.848			0.559624			0.279812	−	0.960055i	$-0.409728\pi$
$509$	284.848			0.559624			0.279812	+	0.960055i	$0.409728\pi$
$510$		0			0
$511$			804.963i			1.57527i
$512$		0			0
$513$		−	128.396i		−	0.250285i
$514$		0			0
$515$	−388.141			−0.753672
$516$		0			0
$517$		−	236.586i		−	0.457613i
$518$		0			0
$519$	1559.88			3.00555
$520$		0			0
$521$		−	195.190i		−	0.374645i	−0.982298	−	0.187323i	$-0.940019\pi$
$521$		−	195.190i		−	0.374645i	0.982298	−	0.187323i	$-0.0599810\pi$
$522$		0			0
$523$		−	0.505041i		−	0.000965661i	−1.00000		0.000482831i	$-0.999846\pi$
$523$		−	0.505041i		−	0.000965661i	1.00000		0.000482831i	$-0.000153690\pi$
$524$		0			0
$525$		−	202.536i		−	0.385784i
$526$		0			0
$527$			295.488i			0.560699i
$528$		0			0
$529$	−482.906	+	215.970i	−0.912865	+	0.408262i
$530$		0			0
$531$	−442.023			−0.832435
$532$		0			0
$533$	−438.657			−0.822995
$534$		0			0
$535$	197.587			0.369321
$536$		0			0
$537$	486.555			0.906062
$538$		0			0
$539$		−	52.1744i		−	0.0967985i
$540$		0			0
$541$	579.191			1.07059			0.535296	−	0.844664i	$-0.320200\pi$
$541$	579.191			1.07059			0.535296	+	0.844664i	$0.320200\pi$
$542$		0			0
$543$		−	1290.32i		−	2.37628i
$544$		0			0
$545$	−318.965			−0.585257
$546$		0			0
$547$	623.934			1.14065			0.570324	−	0.821420i	$-0.306818\pi$
$547$	623.934			1.14065			0.570324	+	0.821420i	$0.306818\pi$
$548$		0			0
$549$			554.465i			1.00996i
$550$		0			0
$551$			2.04208i			0.00370614i
$552$		0			0
$553$	382.394			0.691490
$554$		0			0
$555$	−41.7615			−0.0752460
$556$		0			0
$557$			445.491i			0.799805i	0.916558	+	0.399903i	$0.130956\pi$
$557$			445.491i			0.799805i	−0.916558	+	0.399903i	$0.869044\pi$
$558$		0			0
$559$		−	574.343i		−	1.02745i
$560$		0			0
$561$	−165.113			−0.294319
$562$		0			0
$563$			315.518i			0.560422i	0.959938	+	0.280211i	$0.0904045\pi$
$563$			315.518i			0.560422i	−0.959938	+	0.280211i	$0.909595\pi$
$564$		0			0
$565$	204.619			0.362157
$566$		0			0
$567$		−	518.351i		−	0.914199i
$568$		0			0
$569$		−	954.594i		−	1.67767i	−0.544386	−	0.838835i	$-0.683237\pi$
$569$		−	954.594i		−	1.67767i	0.544386	−	0.838835i	$-0.316763\pi$
$570$		0			0
$571$			266.616i			0.466929i	0.972365	+	0.233464i	$0.0750062\pi$
$571$			266.616i			0.466929i	−0.972365	+	0.233464i	$0.924994\pi$
$572$		0			0
$573$			381.998i			0.666663i
$574$		0			0
$575$	24.0038	+	112.467i	0.0417457	+	0.195595i
$576$		0			0
$577$	−1069.07			−1.85280			−0.926401	−	0.376539i	$-0.877114\pi$
$577$	−1069.07			−1.85280			−0.926401	+	0.376539i	$0.877114\pi$
$578$		0			0
$579$	1438.62			2.48466
$580$		0			0
$581$	−575.557			−0.990632
$582$		0			0
$583$	229.138			0.393033
$584$		0			0
$585$		−	526.453i		−	0.899920i
$586$		0			0
$587$	−1065.52			−1.81520			−0.907600	−	0.419835i	$-0.862088\pi$
$587$	−1065.52			−1.81520			−0.907600	+	0.419835i	$0.862088\pi$
$588$		0			0
$589$		−	108.666i		−	0.184492i
$590$		0			0
$591$	1230.93			2.08280
$592$		0			0
$593$	−337.936			−0.569875			−0.284938	−	0.958546i	$-0.591973\pi$
$593$	−337.936			−0.569875			−0.284938	+	0.958546i	$0.591973\pi$
$594$		0			0
$595$			142.070i			0.238774i
$596$		0			0
$597$			1554.59i			2.60400i
$598$		0			0
$599$	549.708			0.917709			0.458854	−	0.888511i	$-0.348260\pi$
$599$	549.708			0.917709			0.458854	+	0.888511i	$0.348260\pi$
$600$		0			0
$601$	−375.387			−0.624605			−0.312302	−	0.949983i	$-0.601100\pi$
$601$	−375.387			−0.624605			−0.312302	+	0.949983i	$0.601100\pi$
$602$		0			0
$603$			1888.85i			3.13242i
$604$		0			0
$605$		−	235.076i		−	0.388556i
$606$		0			0
$607$	440.263			0.725309			0.362654	−	0.931924i	$-0.381871\pi$
$607$	440.263			0.725309			0.362654	+	0.931924i	$0.381871\pi$
$608$		0			0
$609$			27.8978i			0.0458093i
$610$		0			0
$611$	−802.452			−1.31334
$612$		0			0
$613$		−	49.5676i		−	0.0808607i	−0.999182	−	0.0404303i	$-0.987127\pi$
$613$		−	49.5676i		−	0.0808607i	0.999182	−	0.0404303i	$-0.0128729\pi$
$614$		0			0
$615$		−	373.148i		−	0.606745i
$616$		0			0
$617$			785.550i			1.27318i	0.771204	+	0.636588i	$0.219655\pi$
$617$			785.550i			1.27318i	−0.771204	+	0.636588i	$0.780345\pi$
$618$		0			0
$619$		−	693.014i		−	1.11957i	−0.828638	−	0.559785i	$-0.810884\pi$
$619$		−	693.014i		−	1.11957i	0.828638	−	0.559785i	$-0.189116\pi$
$620$		0			0
$621$	207.886	+	974.028i	0.334761	+	1.56848i
$622$		0			0
$623$	732.432			1.17565
$624$		0			0
$625$	25.0000			0.0400000
$626$		0			0
$627$	60.7201			0.0968422
$628$		0			0
$629$	29.2938			0.0465721
$630$		0			0
$631$		−	9.68025i		−	0.0153411i	−0.999971	−	0.00767056i	$-0.997558\pi$
$631$		−	9.68025i		−	0.0153411i	0.999971	−	0.00767056i	$-0.00244164\pi$
$632$		0			0
$633$	−1609.63			−2.54285
$634$		0			0
$635$		−	226.942i		−	0.357389i
$636$		0			0
$637$	−176.965			−0.277810
$638$		0			0
$639$	−1178.19			−1.84381
$640$		0			0
$641$			174.812i			0.272718i	0.990659	+	0.136359i	$0.0435401\pi$
$641$			174.812i			0.272718i	−0.990659	+	0.136359i	$0.956460\pi$
$642$		0			0
$643$		−	285.997i		−	0.444785i	−0.974957	−	0.222393i	$-0.928613\pi$
$643$		−	285.997i		−	0.444785i	0.974957	−	0.222393i	$-0.0713867\pi$
$644$		0			0
$645$	488.571			0.757475
$646$		0			0
$647$	698.951			1.08030			0.540148	−	0.841570i	$-0.318368\pi$
$647$	698.951			1.08030			0.540148	+	0.841570i	$0.318368\pi$
$648$		0			0
$649$		−	101.064i		−	0.155722i
$650$		0			0
$651$		−	1484.53i		−	2.28039i
$652$		0			0
$653$	−639.144			−0.978781			−0.489390	−	0.872065i	$-0.662781\pi$
$653$	−639.144			−0.978781			−0.489390	+	0.872065i	$0.662781\pi$
$654$		0			0
$655$			98.1846i			0.149900i
$656$		0			0
$657$	−1779.87			−2.70909
$658$		0			0
$659$			1204.48i			1.82774i	0.406004	+	0.913871i	$0.366922\pi$
$659$			1204.48i			1.82774i	−0.406004	+	0.913871i	$0.633078\pi$
$660$		0			0
$661$		−	442.012i		−	0.668702i	−0.942449	−	0.334351i	$-0.891483\pi$
$661$		−	442.012i		−	0.668702i	0.942449	−	0.334351i	$-0.108517\pi$
$662$		0			0
$663$			560.029i			0.844690i
$664$		0			0
$665$		−	52.2462i		−	0.0785658i
$666$		0			0
$667$	−3.30633	−	15.4915i	−0.00495702	−	0.0232256i
$668$		0			0
$669$	218.948			0.327276
$670$		0			0
$671$	−126.772			−0.188930
$672$		0			0
$673$	401.763			0.596974			0.298487	−	0.954414i	$-0.403518\pi$
$673$	401.763			0.596974			0.298487	+	0.954414i	$0.403518\pi$
$674$		0			0
$675$	216.514			0.320762
$676$		0			0
$677$			575.340i			0.849838i	0.905231	+	0.424919i	$0.139697\pi$
$677$			575.340i			0.849838i	−0.905231	+	0.424919i	$0.860303\pi$
$678$		0			0
$679$	1091.71			1.60782
$680$		0			0
$681$		−	1268.23i		−	1.86230i
$682$		0			0
$683$	−148.111			−0.216854			−0.108427	−	0.994104i	$-0.534581\pi$
$683$	−148.111			−0.216854			−0.108427	+	0.994104i	$0.534581\pi$
$684$		0			0
$685$	−481.191			−0.702469
$686$		0			0
$687$		−	989.715i		−	1.44063i
$688$		0			0
$689$		−	777.191i		−	1.12800i
$690$		0			0
$691$	−176.640			−0.255629			−0.127815	−	0.991798i	$-0.540796\pi$
$691$	−176.640			−0.255629			−0.127815	+	0.991798i	$0.540796\pi$
$692$		0			0
$693$	546.989			0.789306
$694$		0			0
$695$		−	142.336i		−	0.204800i
$696$		0			0
$697$			261.747i			0.375533i
$698$		0			0
$699$	−1267.15			−1.81281
$700$		0			0
$701$		−	179.122i		−	0.255524i	−0.991805	−	0.127762i	$-0.959221\pi$
$701$		−	179.122i		−	0.255524i	0.991805	−	0.127762i	$-0.0407793\pi$
$702$		0			0
$703$	−10.7728			−0.0153240
$704$		0			0
$705$		−	682.615i		−	0.968248i
$706$		0			0
$707$		−	561.312i		−	0.793935i
$708$		0			0
$709$		−	612.124i		−	0.863363i	−0.902026	−	0.431681i	$-0.857920\pi$
$709$		−	612.124i		−	0.863363i	0.902026	−	0.431681i	$-0.142080\pi$
$710$		0			0
$711$			845.521i			1.18920i
$712$		0			0
$713$	175.940	+	824.349i	0.246761	+	1.15617i
$714$		0			0
$715$	120.368			0.168346
$716$		0			0
$717$	185.595			0.258850
$718$		0			0
$719$	−332.970			−0.463102			−0.231551	−	0.972823i	$-0.574380\pi$
$719$	−332.970			−0.463102			−0.231551	+	0.972823i	$0.574380\pi$
$720$		0			0
$721$	−1367.85			−1.89716
$722$		0			0
$723$			707.693i			0.978829i
$724$		0			0
$725$	−3.44356			−0.00474973
$726$		0			0
$727$			363.921i			0.500579i	0.968171	+	0.250290i	$0.0805258\pi$
$727$			363.921i			0.500579i	−0.968171	+	0.250290i	$0.919474\pi$
$728$		0			0
$729$	−857.219			−1.17588
$730$		0			0
$731$	−342.711			−0.468825
$732$		0			0
$733$			414.694i			0.565749i	0.959157	+	0.282875i	$0.0912880\pi$
$733$			414.694i			0.565749i	−0.959157	+	0.282875i	$0.908712\pi$
$734$		0			0
$735$		−	150.537i		−	0.204813i
$736$		0			0
$737$	−431.864			−0.585975
$738$		0			0
$739$	−750.501			−1.01556			−0.507782	−	0.861486i	$-0.669534\pi$
$739$	−750.501			−1.01556			−0.507782	+	0.861486i	$0.669534\pi$
$740$		0			0
$741$		−	205.950i		−	0.277936i
$742$		0			0
$743$		−	1301.85i		−	1.75215i	−0.482171	−	0.876077i	$-0.660152\pi$
$743$		−	1301.85i		−	1.75215i	0.482171	−	0.876077i	$-0.339848\pi$
$744$		0			0
$745$	−218.552			−0.293358
$746$		0			0
$747$		−	1272.63i		−	1.70365i
$748$		0			0
$749$	696.317			0.929662
$750$		0			0
$751$			860.771i			1.14617i	0.819497	+	0.573083i	$0.194253\pi$
$751$			860.771i			1.14617i	−0.819497	+	0.573083i	$0.805747\pi$
$752$		0			0
$753$		−	308.484i		−	0.409674i
$754$		0			0
$755$			468.986i			0.621173i
$756$		0			0
$757$		−	281.079i		−	0.371306i	−0.982615	−	0.185653i	$-0.940560\pi$
$757$		−	281.079i		−	0.371306i	0.982615	−	0.185653i	$-0.0594400\pi$
$758$		0			0
$759$	−460.629	+	98.3118i	−0.606889	+	0.129528i
$760$		0			0
$761$	155.150			0.203877			0.101938	−	0.994791i	$-0.467496\pi$
$761$	155.150			0.203877			0.101938	+	0.994791i	$0.467496\pi$
$762$		0			0
$763$	−1124.07			−1.47322
$764$		0			0
$765$	−314.135			−0.410634
$766$		0			0
$767$	−342.787			−0.446920
$768$		0			0
$769$		−	1242.19i		−	1.61533i	−0.589642	−	0.807665i	$-0.700731\pi$
$769$		−	1242.19i		−	1.61533i	0.589642	−	0.807665i	$-0.299269\pi$
$770$		0			0
$771$	835.556			1.08373
$772$		0			0
$773$			1105.56i			1.43022i	0.699014	+	0.715108i	$0.253623\pi$
$773$			1105.56i			1.43022i	−0.699014	+	0.715108i	$0.746377\pi$
$774$		0			0
$775$	183.242			0.236442
$776$		0			0
$777$	−147.172			−0.189411
$778$		0			0
$779$		−	96.2572i		−	0.123565i
$780$		0			0
$781$		−	269.381i		−	0.344918i
$782$		0			0
$783$	−29.8232			−0.0380884
$784$		0			0
$785$	−642.777			−0.818825
$786$		0			0
$787$		−	166.440i		−	0.211486i	−0.994393	−	0.105743i	$-0.966278\pi$
$787$		−	166.440i		−	0.211486i	0.994393	−	0.105743i	$-0.0337221\pi$
$788$		0			0
$789$		−	2143.33i		−	2.71652i
$790$		0			0
$791$	721.098			0.911629
$792$		0			0
$793$			429.986i			0.542227i
$794$		0			0
$795$	661.126			0.831606
$796$		0			0
$797$			702.946i			0.881990i	0.897510	+	0.440995i	$0.145374\pi$
$797$			702.946i			0.881990i	−0.897510	+	0.440995i	$0.854626\pi$
$798$		0			0
$799$			478.824i			0.599279i
$800$		0			0
$801$			1619.50i			2.02185i
$802$		0			0
$803$		−	406.948i		−	0.506785i
$804$		0			0
$805$	84.5919	+	396.346i	0.105083	+	0.492355i
$806$		0			0
$807$	1821.68			2.25735
$808$		0			0
$809$	612.933			0.757643			0.378822	−	0.925470i	$-0.376329\pi$
$809$	612.933			0.757643			0.378822	+	0.925470i	$0.376329\pi$
$810$		0			0
$811$	−664.035			−0.818785			−0.409393	−	0.912358i	$-0.634259\pi$
$811$	−664.035			−0.818785			−0.409393	+	0.912358i	$0.634259\pi$
$812$		0			0
$813$	−2215.12			−2.72462
$814$		0			0
$815$		−	250.560i		−	0.307435i
$816$		0			0
$817$	126.032			0.154262
$818$		0			0
$819$		−	1855.28i		−	2.26530i
$820$		0			0
$821$	598.546			0.729046			0.364523	−	0.931194i	$-0.381232\pi$
$821$	598.546			0.729046			0.364523	+	0.931194i	$0.381232\pi$
$822$		0			0
$823$	1216.57			1.47821			0.739105	−	0.673591i	$-0.235249\pi$
$823$	1216.57			1.47821			0.739105	+	0.673591i	$0.235249\pi$
$824$		0			0
$825$			102.392i			0.124112i
$826$		0			0
$827$			719.387i			0.869875i	0.900461	+	0.434937i	$0.143230\pi$
$827$			719.387i			0.869875i	−0.900461	+	0.434937i	$0.856770\pi$
$828$		0			0
$829$	−340.645			−0.410911			−0.205455	−	0.978666i	$-0.565868\pi$
$829$	−340.645			−0.410911			−0.205455	+	0.978666i	$0.565868\pi$
$830$		0			0
$831$	91.5608			0.110181
$832$		0			0
$833$			105.595i			0.126765i
$834$		0			0
$835$			430.909i			0.516059i
$836$		0			0
$837$	1586.98			1.89604
$838$		0			0
$839$		−	9.11326i		−	0.0108621i	−0.999985	−	0.00543103i	$-0.998271\pi$
$839$		−	9.11326i		−	0.0108621i	0.999985	−	0.00543103i	$-0.00172876\pi$
$840$		0			0
$841$	−840.526			−0.999436
$842$		0			0
$843$			2762.21i			3.27664i
$844$		0			0
$845$		−	30.3674i		−	0.0359378i
$846$		0			0
$847$		−	828.434i		−	0.978081i
$848$		0			0
$849$		−	1168.17i		−	1.37593i
$850$		0			0
$851$	81.7235	−	17.4422i	0.0960323	−	0.0204961i
$852$		0			0
$853$	−1029.00			−1.20633			−0.603163	−	0.797618i	$-0.706093\pi$
$853$	−1029.00			−1.20633			−0.603163	+	0.797618i	$0.706093\pi$
$854$		0			0
$855$	115.523			0.135115
$856$		0			0
$857$	606.460			0.707654			0.353827	−	0.935311i	$-0.384880\pi$
$857$	606.460			0.707654			0.353827	+	0.935311i	$0.384880\pi$
$858$		0			0
$859$	1147.62			1.33600			0.668001	−	0.744161i	$-0.267150\pi$
$859$	1147.62			1.33600			0.668001	+	0.744161i	$0.267150\pi$
$860$		0			0
$861$		−	1315.01i		−	1.52731i
$862$		0			0
$863$	−1299.38			−1.50566			−0.752829	−	0.658216i	$-0.771311\pi$
$863$	−1299.38			−1.50566			−0.752829	+	0.658216i	$0.771311\pi$
$864$		0			0
$865$			678.543i			0.784443i
$866$		0			0
$867$	−1151.41			−1.32804
$868$		0			0
$869$	−193.319			−0.222461
$870$		0			0
$871$			1464.80i			1.68174i
$872$		0			0
$873$			2413.90i			2.76507i
$874$		0			0
$875$	88.1027			0.100689
$876$		0			0
$877$	−869.529			−0.991482			−0.495741	−	0.868470i	$-0.665103\pi$
$877$	−869.529			−0.991482			−0.495741	+	0.868470i	$0.665103\pi$
$878$		0			0
$879$			2933.02i			3.33677i
$880$		0			0
$881$		−	1210.10i		−	1.37355i	−0.726869	−	0.686776i	$-0.759025\pi$
$881$		−	1210.10i		−	1.37355i	0.726869	−	0.686776i	$-0.240975\pi$
$882$		0			0
$883$	275.674			0.312202			0.156101	−	0.987741i	$-0.450107\pi$
$883$	275.674			0.312202			0.156101	+	0.987741i	$0.450107\pi$
$884$		0			0
$885$		−	291.596i		−	0.329487i
$886$		0			0
$887$	742.524			0.837119			0.418559	−	0.908189i	$-0.362535\pi$
$887$	742.524			0.837119			0.418559	+	0.908189i	$0.362535\pi$
$888$		0			0
$889$		−	799.769i		−	0.899628i
$890$		0			0
$891$			262.052i			0.294109i
$892$		0			0
$893$		−	176.087i		−	0.197186i
$894$		0			0
$895$			211.650i			0.236480i
$896$		0			0
$897$	333.454	+	1562.36i	0.371744	+	1.74176i
$898$		0			0
$899$	−25.2402			−0.0280759
$900$		0			0
$901$	−463.751			−0.514707
$902$		0			0
$903$	1721.78			1.90673
$904$		0			0
$905$	561.284			0.620203
$906$		0			0
$907$		−	1335.53i		−	1.47247i	−0.676727	−	0.736234i	$-0.736602\pi$
$907$		−	1335.53i		−	1.47247i	0.676727	−	0.736234i	$-0.263398\pi$
$908$		0			0
$909$	1241.13			1.36538
$910$		0			0
$911$			511.312i			0.561265i	0.959815	+	0.280632i	$0.0905442\pi$
$911$			511.312i			0.561265i	−0.959815	+	0.280632i	$0.909456\pi$
$912$		0			0
$913$	290.972			0.318699
$914$		0			0
$915$	−365.773			−0.399751
$916$		0			0
$917$			346.013i			0.377332i
$918$		0			0
$919$		−	712.793i		−	0.775618i	−0.921740	−	0.387809i	$-0.873232\pi$
$919$		−	712.793i		−	0.775618i	0.921740	−	0.387809i	$-0.126768\pi$
$920$		0			0
$921$	89.7903			0.0974921
$922$		0			0
$923$	−913.686			−0.989909
$924$		0			0
$925$		−	18.1661i		−	0.0196390i
$926$		0			0
$927$		−	3024.49i		−	3.26266i
$928$		0			0
$929$	86.3665			0.0929672			0.0464836	−	0.998919i	$-0.485198\pi$
$929$	86.3665			0.0929672			0.0464836	+	0.998919i	$0.485198\pi$
$930$		0			0
$931$		−	38.8326i		−	0.0417106i
$932$		0			0
$933$	213.618			0.228958
$934$		0			0
$935$		−	71.8235i		−	0.0768165i
$936$		0			0
$937$		−	994.559i		−	1.06143i	−0.847551	−	0.530714i	$-0.821924\pi$
$937$		−	994.559i		−	1.06143i	0.847551	−	0.530714i	$-0.178076\pi$
$938$		0			0
$939$		−	995.960i		−	1.06066i
$940$		0			0
$941$		−	1456.29i		−	1.54760i	−0.633429	−	0.773800i	$-0.718353\pi$
$941$		−	1456.29i		−	1.54760i	0.633429	−	0.773800i	$-0.281647\pi$
$942$		0			0
$943$	155.850	+	730.217i	0.165270	+	0.774356i
$944$		0			0
$945$	763.020			0.807428
$946$		0			0
$947$	−206.079			−0.217613			−0.108806	−	0.994063i	$-0.534703\pi$
$947$	−206.079			−0.217613			−0.108806	+	0.994063i	$0.534703\pi$
$948$		0			0
$949$	−1380.29			−1.45446
$950$		0			0
$951$	−1501.46			−1.57882
$952$		0			0
$953$		−	457.750i		−	0.480325i	−0.970733	−	0.240163i	$-0.922799\pi$
$953$		−	457.750i		−	0.480325i	0.970733	−	0.240163i	$-0.0772007\pi$
$954$		0			0
$955$	−166.168			−0.173998
$956$		0			0
$957$		−	14.1037i		−	0.0147374i
$958$		0			0
$959$	−1695.77			−1.76827
$960$		0			0
$961$	382.112			0.397619
$962$		0			0
$963$			1539.64i			1.59880i
$964$		0			0
$965$			625.795i			0.648492i
$966$		0			0
$967$	−942.336			−0.974495			−0.487247	−	0.873264i	$-0.661999\pi$
$967$	−942.336			−0.974495			−0.487247	+	0.873264i	$0.661999\pi$
$968$		0			0
$969$	−122.891			−0.126822
$970$		0			0
$971$		−	218.442i		−	0.224966i	−0.993654	−	0.112483i	$-0.964120\pi$
$971$		−	218.442i		−	0.224966i	0.993654	−	0.112483i	$-0.0358804\pi$
$972$		0			0
$973$		−	501.607i		−	0.515526i
$974$		0			0
$975$	347.293			0.356198
$976$		0			0
$977$		−	378.433i		−	0.387342i	−0.981067	−	0.193671i	$-0.937961\pi$
$977$		−	378.433i		−	0.387342i	0.981067	−	0.193671i	$-0.0620394\pi$
$978$		0			0
$979$	−370.280			−0.378223
$980$		0			0
$981$		−	2485.45i		−	2.53359i
$982$		0			0
$983$			1359.48i			1.38299i	0.722381	+	0.691495i	$0.243048\pi$
$983$			1359.48i			1.38299i	−0.722381	+	0.691495i	$0.756952\pi$
$984$		0			0
$985$			535.452i			0.543606i
$986$		0			0
$987$		−	2405.61i		−	2.43729i
$988$		0			0
$989$	−956.090	+	204.058i	−0.966724	+	0.206327i
$990$		0			0
$991$	−648.537			−0.654426			−0.327213	−	0.944951i	$-0.606110\pi$
$991$	−648.537			−0.654426			−0.327213	+	0.944951i	$0.606110\pi$
$992$		0			0
$993$	−2893.47			−2.91387
$994$		0			0
$995$	−676.241			−0.679640
$996$		0			0
$997$	−155.843			−0.156312			−0.0781562	−	0.996941i	$-0.524903\pi$
$997$	−155.843			−0.156312			−0.0781562	+	0.996941i	$0.524903\pi$
$998$		0			0
$999$		−	157.329i		−	0.157486i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	460.3.f.a.321.1	✓	16
3.2	odd	2		4140.3.d.a.2161.10		16
4.3	odd	2		1840.3.k.c.321.15		16
5.2	odd	4		2300.3.d.b.1149.4		32
5.3	odd	4		2300.3.d.b.1149.29		32
5.4	even	2		2300.3.f.e.1701.16		16
23.22	odd	2	inner	460.3.f.a.321.2	yes	16
69.68	even	2		4140.3.d.a.2161.7		16
92.91	even	2		1840.3.k.c.321.16		16
115.22	even	4		2300.3.d.b.1149.30		32
115.68	even	4		2300.3.d.b.1149.3		32
115.114	odd	2		2300.3.f.e.1701.15		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
460.3.f.a.321.1	✓	16	1.1	even	1	trivial
460.3.f.a.321.2	yes	16	23.22	odd	2	inner
1840.3.k.c.321.15		16	4.3	odd	2
1840.3.k.c.321.16		16	92.91	even	2
2300.3.d.b.1149.3		32	115.68	even	4
2300.3.d.b.1149.4		32	5.2	odd	4
2300.3.d.b.1149.29		32	5.3	odd	4
2300.3.d.b.1149.30		32	115.22	even	4
2300.3.f.e.1701.15		16	115.114	odd	2
2300.3.f.e.1701.16		16	5.4	even	2
4140.3.d.a.2161.7		16	69.68	even	2
4140.3.d.a.2161.10		16	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 460 = 2^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	460.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-5.14043 q^{3} -2.23607i q^{5} -7.88014i q^{7} +17.4240 q^{9} +O(q^{10})\)	\(q-5.14043 q^{3} -2.23607i q^{5} -7.88014i q^{7} +17.4240 q^{9} +3.98380i q^{11} +13.5122 q^{13} +11.4943i q^{15} -8.06277i q^{17} +2.96508i q^{19} +40.5073i q^{21} +(-4.80075 - 22.4934i) q^{23} -5.00000 q^{25} -43.3029 q^{27} +0.688711 q^{29} -36.6485 q^{31} -20.4784i q^{33} -17.6205 q^{35} +3.63322i q^{37} -69.4587 q^{39} -32.4636 q^{41} -42.5054i q^{43} -38.9612i q^{45} -59.3870 q^{47} -13.0966 q^{49} +41.4461i q^{51} -57.5176i q^{53} +8.90804 q^{55} -15.2418i q^{57} -25.3686 q^{59} +31.8220i q^{61} -137.303i q^{63} -30.2143i q^{65} +108.405i q^{67} +(24.6779 + 115.626i) q^{69} -67.6191 q^{71} -102.151 q^{73} +25.7021 q^{75} +31.3929 q^{77} +48.5263i q^{79} +65.7793 q^{81} -73.0389i q^{83} -18.0289 q^{85} -3.54027 q^{87} +92.9465i q^{89} -106.478i q^{91} +188.389 q^{93} +6.63011 q^{95} +138.539i q^{97} +69.4136i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 64 q^{9}+O(q^{10}) \) 16 * q + 64 * q^9	\( 16 q + 64 q^{9} - 12 q^{13} - 14 q^{23} - 80 q^{25} + 48 q^{27} + 90 q^{29} + 10 q^{31} + 30 q^{35} + 20 q^{39} + 186 q^{41} - 320 q^{47} + 2 q^{49} - 120 q^{55} - 90 q^{59} - 232 q^{69} - 238 q^{71} - 280 q^{73} + 324 q^{77} + 704 q^{81} - 30 q^{85} + 724 q^{87} - 380 q^{93} + 80 q^{95}+O(q^{100}) \) 16 * q + 64 * q^9 - 12 * q^13 - 14 * q^23 - 80 * q^25 + 48 * q^27 + 90 * q^29 + 10 * q^31 + 30 * q^35 + 20 * q^39 + 186 * q^41 - 320 * q^47 + 2 * q^49 - 120 * q^55 - 90 * q^59 - 232 * q^69 - 238 * q^71 - 280 * q^73 + 324 * q^77 + 704 * q^81 - 30 * q^85 + 724 * q^87 - 380 * q^93 + 80 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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