LMFDB - Embedded newform 160.5.e.c.79.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [160,5,Mod(79,160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(160, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("160.79");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 160 = 2^{5} \cdot 5 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	160.e (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:	$16.5391940934$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 10 x^{18} - 20 x^{16} + 2640 x^{14} + 22400 x^{12} - 652288 x^{10} + 5734400 x^{8} + \cdots + 1099511627776 $ x^20 + 10x^18 - 20x^16 + 2640x^14 + 22400x^12 - 652288x^10 + 5734400x^8 + 173015040x^6 - 335544320x^4 + 42949672960*x^2 + 1099511627776
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{70}\cdot 5^{5} $
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			79.16
Root			$-0.495691 - 3.96917i$ of defining polynomial
Character	$\chi$	$=$	160.79
Dual form			160.5.e.c.79.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+10.2033i q^{3} +(17.9141 + 17.4381i) q^{5} -84.8600 q^{7} -23.1073 q^{9} -71.3055 q^{11} -109.279 q^{13} +(-177.926 + 182.783i) q^{15} -151.344i q^{17} +368.470 q^{19} -865.852i q^{21} -358.031 q^{23} +(16.8280 + 624.773i) q^{25} +590.697i q^{27} -387.610i q^{29} -1687.73i q^{31} -727.551i q^{33} +(-1520.19 - 1479.79i) q^{35} -1158.77 q^{37} -1115.01i q^{39} -2545.99 q^{41} +1354.04i q^{43} +(-413.946 - 402.946i) q^{45} -901.424 q^{47} +4800.23 q^{49} +1544.21 q^{51} -3409.47 q^{53} +(-1277.37 - 1243.43i) q^{55} +3759.61i q^{57} +1454.37 q^{59} +5324.84i q^{61} +1960.88 q^{63} +(-1957.64 - 1905.62i) q^{65} -657.048i q^{67} -3653.09i q^{69} +6741.79i q^{71} -4130.26i q^{73} +(-6374.75 + 171.701i) q^{75} +6050.98 q^{77} +2307.14i q^{79} -7898.74 q^{81} -2146.14i q^{83} +(2639.14 - 2711.18i) q^{85} +3954.90 q^{87} +5117.03 q^{89} +9273.44 q^{91} +17220.4 q^{93} +(6600.80 + 6425.40i) q^{95} +9169.66i q^{97} +1647.68 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 652 q^{9} + 168 q^{11} - 728 q^{19} - 1420 q^{25} - 4440 q^{35} - 4728 q^{41} + 540 q^{49} - 4352 q^{51} - 8280 q^{59} + 9480 q^{65} + 16000 q^{75} - 10956 q^{81} + 22248 q^{89} + 39760 q^{91} + 62504 q^{99}+O(q^{100}) $ 20 * q - 652 * q^9 + 168 * q^11 - 728 * q^19 - 1420 * q^25 - 4440 * q^35 - 4728 * q^41 + 540 * q^49 - 4352 * q^51 - 8280 * q^59 + 9480 * q^65 + 16000 * q^75 - 10956 * q^81 + 22248 * q^89 + 39760 * q^91 + 62504 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/160\mathbb{Z}\right)^\times$.

$n$	$31$	$97$	$101$
$\chi(n)$	$-1$	$-1$	$-1$

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
$3$			10.2033i			1.13370i	0.823821	+	0.566850i	$0.191838\pi$
$3$			10.2033i			1.13370i	−0.823821	+	0.566850i	$0.808162\pi$
$4$		0			0
$5$	17.9141	+	17.4381i	0.716563	+	0.697522i
$5$	17.9141	+	17.4381i	0.716563	+	0.697522i
$6$		0			0
$7$	−84.8600			−1.73184			−0.865919	−	0.500185i	$-0.833265\pi$
$7$	−84.8600			−1.73184			−0.865919	+	0.500185i	$0.833265\pi$
$8$		0			0
$9$	−23.1073			−0.285275
$10$		0			0
$11$	−71.3055			−0.589301			−0.294651	−	0.955605i	$-0.595203\pi$
$11$	−71.3055			−0.589301			−0.294651	+	0.955605i	$0.595203\pi$
$12$		0			0
$13$	−109.279			−0.646623			−0.323311	−	0.946293i	$-0.604796\pi$
$13$	−109.279			−0.646623			−0.323311	+	0.946293i	$0.604796\pi$
$14$		0			0
$15$	−177.926	+	182.783i	−0.790781	+	0.812367i
$16$		0			0
$17$		−	151.344i		−	0.523681i	−0.965111	−	0.261840i	$-0.915671\pi$
$17$		−	151.344i		−	0.523681i	0.965111	−	0.261840i	$-0.0843294\pi$
$18$		0			0
$19$	368.470			1.02069			0.510346	−	0.859969i	$-0.329517\pi$
$19$	368.470			1.02069			0.510346	+	0.859969i	$0.329517\pi$
$20$		0			0
$21$		−	865.852i		−	1.96338i
$22$		0			0
$23$	−358.031			−0.676807			−0.338403	−	0.941001i	$-0.609887\pi$
$23$	−358.031			−0.676807			−0.338403	+	0.941001i	$0.609887\pi$
$24$		0			0
$25$	16.8280	+	624.773i	0.0269248	+	0.999637i
$26$		0			0
$27$			590.697i			0.810283i
$28$		0			0
$29$		−	387.610i		−	0.460892i	−0.973085	−	0.230446i	$-0.925982\pi$
$29$		−	387.610i		−	0.460892i	0.973085	−	0.230446i	$-0.0740185\pi$
$30$		0			0
$31$		−	1687.73i		−	1.75622i	−0.478457	−	0.878111i	$-0.658804\pi$
$31$		−	1687.73i		−	1.75622i	0.478457	−	0.878111i	$-0.341196\pi$
$32$		0			0
$33$		−	727.551i		−	0.668091i
$34$		0			0
$35$	−1520.19	−	1479.79i	−1.24097	−	1.20800i
$36$		0			0
$37$	−1158.77			−0.846434			−0.423217	−	0.906028i	$-0.639099\pi$
$37$	−1158.77			−0.846434			−0.423217	+	0.906028i	$0.639099\pi$
$38$		0			0
$39$		−	1115.01i		−	0.733076i
$40$		0			0
$41$	−2545.99			−1.51457			−0.757285	−	0.653084i	$-0.773475\pi$
$41$	−2545.99			−1.51457			−0.757285	+	0.653084i	$0.773475\pi$
$42$		0			0
$43$			1354.04i			0.732308i	0.930554	+	0.366154i	$0.119326\pi$
$43$			1354.04i			0.732308i	−0.930554	+	0.366154i	$0.880674\pi$
$44$		0			0
$45$	−413.946	−	402.946i	−0.204418	−	0.198986i
$46$		0			0
$47$	−901.424			−0.408069			−0.204034	−	0.978964i	$-0.565405\pi$
$47$	−901.424			−0.408069			−0.204034	+	0.978964i	$0.565405\pi$
$48$		0			0
$49$	4800.23			1.99926
$50$		0			0
$51$	1544.21			0.593697
$52$		0			0
$53$	−3409.47			−1.21377			−0.606883	−	0.794791i	$-0.707580\pi$
$53$	−3409.47			−1.21377			−0.606883	+	0.794791i	$0.707580\pi$
$54$		0			0
$55$	−1277.37	−	1243.43i	−0.422271	−	0.411051i
$56$		0			0
$57$			3759.61i			1.15716i
$58$		0			0
$59$	1454.37			0.417803			0.208902	−	0.977937i	$-0.433011\pi$
$59$	1454.37			0.417803			0.208902	+	0.977937i	$0.433011\pi$
$60$		0			0
$61$			5324.84i			1.43102i	0.698601	+	0.715512i	$0.253806\pi$
$61$			5324.84i			1.43102i	−0.698601	+	0.715512i	$0.746194\pi$
$62$		0			0
$63$	1960.88			0.494050
$64$		0			0
$65$	−1957.64	−	1905.62i	−0.463346	−	0.451034i
$66$		0			0
$67$		−	657.048i		−	0.146368i	−0.997318	−	0.0731842i	$-0.976684\pi$
$67$		−	657.048i		−	0.146368i	0.997318	−	0.0731842i	$-0.0233161\pi$
$68$		0			0
$69$		−	3653.09i		−	0.767295i
$70$		0			0
$71$			6741.79i			1.33739i	0.743536	+	0.668696i	$0.233147\pi$
$71$			6741.79i			1.33739i	−0.743536	+	0.668696i	$0.766853\pi$
$72$		0			0
$73$		−	4130.26i		−	0.775054i	−0.921858	−	0.387527i	$-0.873329\pi$
$73$		−	4130.26i		−	0.775054i	0.921858	−	0.387527i	$-0.126671\pi$
$74$		0			0
$75$	−6374.75	+	171.701i	−1.13329	+	0.0305246i
$76$		0			0
$77$	6050.98			1.02057
$78$		0			0
$79$			2307.14i			0.369675i	0.982769	+	0.184837i	$0.0591758\pi$
$79$			2307.14i			0.369675i	−0.982769	+	0.184837i	$0.940824\pi$
$80$		0			0
$81$	−7898.74			−1.20389
$82$		0			0
$83$		−	2146.14i		−	0.311532i	−0.987794	−	0.155766i	$-0.950215\pi$
$83$		−	2146.14i		−	0.311532i	0.987794	−	0.155766i	$-0.0497846\pi$
$84$		0			0
$85$	2639.14	−	2711.18i	0.365279	−	0.375250i
$86$		0			0
$87$	3954.90			0.522513
$88$		0			0
$89$	5117.03			0.646008			0.323004	−	0.946398i	$-0.395307\pi$
$89$	5117.03			0.646008			0.323004	+	0.946398i	$0.395307\pi$
$90$		0			0
$91$	9273.44			1.11985
$92$		0			0
$93$	17220.4			1.99103
$94$		0			0
$95$	6600.80	+	6425.40i	0.731391	+	0.711956i
$96$		0			0
$97$			9169.66i			0.974563i	0.873245	+	0.487281i	$0.162011\pi$
$97$			9169.66i			0.974563i	−0.873245	+	0.487281i	$0.837989\pi$
$98$		0			0
$99$	1647.68			0.168113
$100$		0			0
$101$			536.675i			0.0526100i	0.999654	+	0.0263050i	$0.00837411\pi$
$101$			536.675i			0.0526100i	−0.999654	+	0.0263050i	$0.991626\pi$
$102$		0			0
$103$	−3140.33			−0.296007			−0.148003	−	0.988987i	$-0.547285\pi$
$103$	−3140.33			−0.296007			−0.148003	+	0.988987i	$0.547285\pi$
$104$		0			0
$105$	15098.8	−	15510.9i	1.36950	−	1.40689i
$106$		0			0
$107$			16703.3i			1.45893i	0.684017	+	0.729466i	$0.260231\pi$
$107$			16703.3i			1.45893i	−0.684017	+	0.729466i	$0.739769\pi$
$108$		0			0
$109$			15790.1i			1.32902i	0.747279	+	0.664510i	$0.231360\pi$
$109$			15790.1i			1.32902i	−0.747279	+	0.664510i	$0.768640\pi$
$110$		0			0
$111$		−	11823.3i		−	0.959602i
$112$		0			0
$113$			4460.76i			0.349343i	0.984627	+	0.174672i	$0.0558864\pi$
$113$			4460.76i			0.349343i	−0.984627	+	0.174672i	$0.944114\pi$
$114$		0			0
$115$	−6413.79	−	6243.36i	−0.484974	−	0.472088i
$116$		0			0
$117$	2525.15			0.184465
$118$		0			0
$119$			12843.0i			0.906930i
$120$		0			0
$121$	−9556.53			−0.652724
$122$		0			0
$123$		−	25977.5i		−	1.71707i
$124$		0			0
$125$	−10593.4	+	11485.7i	−0.677976	+	0.735084i
$126$		0			0
$127$	10888.2			0.675070			0.337535	−	0.941313i	$-0.390407\pi$
$127$	10888.2			0.675070			0.337535	+	0.941313i	$0.390407\pi$
$128$		0			0
$129$	−13815.7			−0.830218
$130$		0			0
$131$	9105.25			0.530578			0.265289	−	0.964169i	$-0.414533\pi$
$131$	9105.25			0.530578			0.265289	+	0.964169i	$0.414533\pi$
$132$		0			0
$133$	−31268.4			−1.76767
$134$		0			0
$135$	−10300.6	+	10581.8i	−0.565191	+	0.580619i
$136$		0			0
$137$			14976.2i			0.797921i	0.916968	+	0.398960i	$0.130629\pi$
$137$			14976.2i			0.797921i	−0.916968	+	0.398960i	$0.869371\pi$
$138$		0			0
$139$	−25598.6			−1.32491			−0.662456	−	0.749101i	$-0.730486\pi$
$139$	−25598.6			−1.32491			−0.662456	+	0.749101i	$0.730486\pi$
$140$		0			0
$141$		−	9197.49i		−	0.462627i
$142$		0			0
$143$	7792.21			0.381056
$144$		0			0
$145$	6759.17	−	6943.68i	0.321482	−	0.330258i
$146$		0			0
$147$			48978.1i			2.26656i
$148$		0			0
$149$		−	10269.8i		−	0.462585i	−0.972884	−	0.231292i	$-0.925705\pi$
$149$		−	10269.8i		−	0.462585i	0.972884	−	0.231292i	$-0.0742954\pi$
$150$		0			0
$151$		−	14739.4i		−	0.646435i	−0.946325	−	0.323217i	$-0.895236\pi$
$151$		−	14739.4i		−	0.646435i	0.946325	−	0.323217i	$-0.104764\pi$
$152$		0			0
$153$			3497.14i			0.149393i
$154$		0			0
$155$	29430.7	−	30234.1i	1.22500	−	1.25844i
$156$		0			0
$157$	−27740.2			−1.12541			−0.562705	−	0.826658i	$-0.690239\pi$
$157$	−27740.2			−1.12541			−0.562705	+	0.826658i	$0.690239\pi$
$158$		0			0
$159$		−	34787.8i		−	1.37605i
$160$		0			0
$161$	30382.5			1.17212
$162$		0			0
$163$		−	3732.97i		−	0.140501i	−0.997529	−	0.0702505i	$-0.977620\pi$
$163$		−	3732.97i		−	0.140501i	0.997529	−	0.0702505i	$-0.0223799\pi$
$164$		0			0
$165$	12687.1	−	13033.4i	0.466008	−	0.478729i
$166$		0			0
$167$	−20717.4			−0.742853			−0.371427	−	0.928462i	$-0.621131\pi$
$167$	−20717.4			−0.742853			−0.371427	+	0.928462i	$0.621131\pi$
$168$		0			0
$169$	−16619.0			−0.581879
$170$		0			0
$171$	−8514.34			−0.291178
$172$		0			0
$173$	35335.7			1.18065			0.590326	−	0.807165i	$-0.298999\pi$
$173$	35335.7			1.18065			0.590326	+	0.807165i	$0.298999\pi$
$174$		0			0
$175$	−1428.02	−	53018.3i	−0.0466293	−	1.73121i
$176$		0			0
$177$			14839.4i			0.473664i
$178$		0			0
$179$	−23669.8			−0.738734			−0.369367	−	0.929284i	$-0.620425\pi$
$179$	−23669.8			−0.738734			−0.369367	+	0.929284i	$0.620425\pi$
$180$		0			0
$181$		−	7525.59i		−	0.229712i	−0.993382	−	0.114856i	$-0.963359\pi$
$181$		−	7525.59i		−	0.229712i	0.993382	−	0.114856i	$-0.0366406\pi$
$182$		0			0
$183$	−54330.9			−1.62235
$184$		0			0
$185$	−20758.2	−	20206.7i	−0.606523	−	0.590407i
$186$		0			0
$187$			10791.6i			0.308606i
$188$		0			0
$189$		−	50126.5i		−	1.40328i
$190$		0			0
$191$			28925.8i			0.792901i	0.918056	+	0.396450i	$0.129758\pi$
$191$			28925.8i			0.792901i	−0.918056	+	0.396450i	$0.870242\pi$
$192$		0			0
$193$		−	48944.4i		−	1.31398i	−0.753900	−	0.656990i	$-0.771829\pi$
$193$		−	48944.4i		−	1.31398i	0.753900	−	0.656990i	$-0.228171\pi$
$194$		0			0
$195$	19443.6	−	19974.3i	0.511337	−	0.525295i
$196$		0			0
$197$	47883.5			1.23382			0.616912	−	0.787032i	$-0.288383\pi$
$197$	47883.5			1.23382			0.616912	+	0.787032i	$0.288383\pi$
$198$		0			0
$199$			29644.2i			0.748573i	0.927313	+	0.374286i	$0.122112\pi$
$199$			29644.2i			0.748573i	−0.927313	+	0.374286i	$0.877888\pi$
$200$		0			0
$201$	6704.06			0.165938
$202$		0			0
$203$			32892.6i			0.798190i
$204$		0			0
$205$	−45609.1	−	44397.2i	−1.08529	−	1.05645i
$206$		0			0
$207$	8273.12			0.193076
$208$		0			0
$209$	−26273.9			−0.601496
$210$		0			0
$211$	−19653.5			−0.441444			−0.220722	−	0.975337i	$-0.570841\pi$
$211$	−19653.5			−0.441444			−0.220722	+	0.975337i	$0.570841\pi$
$212$		0			0
$213$	−68788.5			−1.51620
$214$		0			0
$215$	−23611.8	+	24256.3i	−0.510802	+	0.524745i
$216$		0			0
$217$			143221.i			3.04149i
$218$		0			0
$219$	42142.3			0.878678
$220$		0			0
$221$			16538.7i			0.338624i
$222$		0			0
$223$	60977.7			1.22620			0.613100	−	0.790005i	$-0.289922\pi$
$223$	60977.7			1.22620			0.613100	+	0.790005i	$0.289922\pi$
$224$		0			0
$225$	−388.849	−	14436.8i	−0.00768097	−	0.285172i
$226$		0			0
$227$			2698.14i			0.0523616i	0.999657	+	0.0261808i	$0.00833456\pi$
$227$			2698.14i			0.0523616i	−0.999657	+	0.0261808i	$0.991665\pi$
$228$		0			0
$229$			68901.3i			1.31388i	0.753942	+	0.656941i	$0.228150\pi$
$229$			68901.3i			1.31388i	−0.753942	+	0.656941i	$0.771850\pi$
$230$		0			0
$231$			61740.0i			1.15702i
$232$		0			0
$233$		−	83770.9i		−	1.54305i	−0.636196	−	0.771527i	$-0.719493\pi$
$233$		−	83770.9i		−	1.54305i	0.636196	−	0.771527i	$-0.280507\pi$
$234$		0			0
$235$	−16148.2	−	15719.1i	−0.292407	−	0.284637i
$236$		0			0
$237$	−23540.4			−0.419100
$238$		0			0
$239$		−	36602.8i		−	0.640795i	−0.947283	−	0.320397i	$-0.896184\pi$
$239$		−	36602.8i		−	0.640795i	0.947283	−	0.320397i	$-0.103816\pi$
$240$		0			0
$241$	48755.9			0.839447			0.419724	−	0.907652i	$-0.362127\pi$
$241$	48755.9			0.839447			0.419724	+	0.907652i	$0.362127\pi$
$242$		0			0
$243$		−	32746.8i		−	0.554570i
$244$		0			0
$245$	85991.6	+	83706.6i	1.43260	+	1.39453i
$246$		0			0
$247$	−40266.1			−0.660003
$248$		0			0
$249$	21897.8			0.353184
$250$		0			0
$251$	−75319.3			−1.19553			−0.597763	−	0.801673i	$-0.703944\pi$
$251$	−75319.3			−1.19553			−0.597763	+	0.801673i	$0.703944\pi$
$252$		0			0
$253$	25529.5			0.398843
$254$		0			0
$255$	27663.0	+	26928.0i	0.425421	+	0.414117i
$256$		0			0
$257$		−	27190.5i		−	0.411672i	−0.978587	−	0.205836i	$-0.934009\pi$
$257$		−	27190.5i		−	0.411672i	0.978587	−	0.205836i	$-0.0659913\pi$
$258$		0			0
$259$	98333.1			1.46589
$260$		0			0
$261$			8956.62i			0.131481i
$262$		0			0
$263$	48135.0			0.695905			0.347952	−	0.937512i	$-0.386877\pi$
$263$	48135.0			0.695905			0.347952	+	0.937512i	$0.386877\pi$
$264$		0			0
$265$	−61077.5	−	59454.6i	−0.869740	−	0.846630i
$266$		0			0
$267$			52210.6i			0.732379i
$268$		0			0
$269$			57856.4i			0.799553i	0.916613	+	0.399776i	$0.130912\pi$
$269$			57856.4i			0.799553i	−0.916613	+	0.399776i	$0.869088\pi$
$270$		0			0
$271$		−	135141.i		−	1.84013i	−0.391765	−	0.920065i	$-0.628136\pi$
$271$		−	135141.i		−	1.84013i	0.391765	−	0.920065i	$-0.371864\pi$
$272$		0			0
$273$			94619.7i			1.26957i
$274$		0			0
$275$	−1199.93	−	44549.8i	−0.0158668	−	0.589088i
$276$		0			0
$277$	66626.3			0.868333			0.434166	−	0.900833i	$-0.357043\pi$
$277$	66626.3			0.868333			0.434166	+	0.900833i	$0.357043\pi$
$278$		0			0
$279$			38998.8i			0.501006i
$280$		0			0
$281$	70518.3			0.893078			0.446539	−	0.894764i	$-0.352656\pi$
$281$	70518.3			0.893078			0.446539	+	0.894764i	$0.352656\pi$
$282$		0			0
$283$			5580.69i			0.0696811i	0.999393	+	0.0348406i	$0.0110923\pi$
$283$			5580.69i			0.0696811i	−0.999393	+	0.0348406i	$0.988908\pi$
$284$		0			0
$285$	−65560.3	+	67349.9i	−0.807144	+	0.829177i
$286$		0			0
$287$	216053.			2.62299
$288$		0			0
$289$	60616.1			0.725758
$290$		0			0
$291$	−93560.8			−1.10486
$292$		0			0
$293$	85817.1			0.999628			0.499814	−	0.866133i	$-0.333402\pi$
$293$	85817.1			0.999628			0.499814	+	0.866133i	$0.333402\pi$
$294$		0			0
$295$	26053.8	+	25361.5i	0.299382	+	0.291427i
$296$		0			0
$297$		−	42119.9i		−	0.477501i
$298$		0			0
$299$	39125.3			0.437639
$300$		0			0
$301$		−	114904.i		−	1.26824i
$302$		0			0
$303$	−5475.85			−0.0596440
$304$		0			0
$305$	−92854.9	+	95389.5i	−0.998171	+	1.02542i
$306$		0			0
$307$		−	24380.3i		−	0.258680i	−0.991600	−	0.129340i	$-0.958714\pi$
$307$		−	24380.3i		−	0.258680i	0.991600	−	0.129340i	$-0.0412858\pi$
$308$		0			0
$309$		−	32041.8i		−	0.335583i
$310$		0			0
$311$			145304.i			1.50230i	0.660133	+	0.751149i	$0.270500\pi$
$311$			145304.i			1.50230i	−0.660133	+	0.751149i	$0.729500\pi$
$312$		0			0
$313$		−	15.3803i		−	0.000156991i	−1.00000		7.84956e-5i	$-0.999975\pi$
$313$		−	15.3803i		−	0.000156991i	1.00000		7.84956e-5i	$-2.49859e-5\pi$
$314$		0			0
$315$	35127.4	+	34194.0i	0.354018	+	0.344611i
$316$		0			0
$317$	−19873.1			−0.197764			−0.0988819	−	0.995099i	$-0.531527\pi$
$317$	−19873.1			−0.197764			−0.0988819	+	0.995099i	$0.531527\pi$
$318$		0			0
$319$			27638.7i			0.271604i
$320$		0			0
$321$	−170429.			−1.65399
$322$		0			0
$323$		−	55765.7i		−	0.534517i
$324$		0			0
$325$	−1838.95	−	68274.8i	−0.0174102	−	0.646388i
$326$		0			0
$327$	−161111.			−1.50671
$328$		0			0
$329$	76494.8			0.706708
$330$		0			0
$331$	55183.9			0.503682			0.251841	−	0.967769i	$-0.418964\pi$
$331$	55183.9			0.503682			0.251841	+	0.967769i	$0.418964\pi$
$332$		0			0
$333$	26776.0			0.241466
$334$		0			0
$335$	11457.6	−	11770.4i	0.102095	−	0.104882i
$336$		0			0
$337$		−	172509.i		−	1.51898i	−0.650520	−	0.759489i	$-0.725449\pi$
$337$		−	172509.i		−	1.51898i	0.650520	−	0.759489i	$-0.274551\pi$
$338$		0			0
$339$	−45514.5			−0.396050
$340$		0			0
$341$			120344.i			1.03494i
$342$		0			0
$343$	−203598.			−1.73056
$344$		0			0
$345$	63702.9	−	65441.8i	0.535206	−	0.549815i
$346$		0			0
$347$			53756.2i			0.446447i	0.974767	+	0.223223i	$0.0716579\pi$
$347$			53756.2i			0.446447i	−0.974767	+	0.223223i	$0.928342\pi$
$348$		0			0
$349$		−	101720.i		−	0.835136i	−0.908646	−	0.417568i	$-0.862883\pi$
$349$		−	101720.i		−	0.835136i	0.908646	−	0.417568i	$-0.137117\pi$
$350$		0			0
$351$		−	64550.9i		−	0.523948i
$352$		0			0
$353$			119472.i			0.958778i	0.877603	+	0.479389i	$0.159142\pi$
$353$			119472.i			0.958778i	−0.877603	+	0.479389i	$0.840858\pi$
$354$		0			0
$355$	−117564.	+	120773.i	−0.932861	+	0.958325i
$356$		0			0
$357$	−131041.			−1.02819
$358$		0			0
$359$			66348.7i			0.514806i	0.966304	+	0.257403i	$0.0828668\pi$
$359$			66348.7i			0.514806i	−0.966304	+	0.257403i	$0.917133\pi$
$360$		0			0
$361$	5449.20			0.0418137
$362$		0			0
$363$		−	97508.1i		−	0.739993i
$364$		0			0
$365$	72023.8	−	73989.8i	0.540618	−	0.555375i
$366$		0			0
$367$	−146492.			−1.08763			−0.543817	−	0.839204i	$-0.683021\pi$
$367$	−146492.			−1.08763			−0.543817	+	0.839204i	$0.683021\pi$
$368$		0			0
$369$	58831.0			0.432069
$370$		0			0
$371$	289328.			2.10205
$372$		0			0
$373$	−113125.			−0.813092			−0.406546	−	0.913630i	$-0.633267\pi$
$373$	−113125.			−0.813092			−0.406546	+	0.913630i	$0.633267\pi$
$374$		0			0
$375$	−117192.	−	108087.i	−0.833364	−	0.768622i
$376$		0			0
$377$			42357.7i			0.298023i
$378$		0			0
$379$	−43592.1			−0.303480			−0.151740	−	0.988420i	$-0.548488\pi$
$379$	−43592.1			−0.303480			−0.151740	+	0.988420i	$0.548488\pi$
$380$		0			0
$381$			111096.i			0.765327i
$382$		0			0
$383$	99396.2			0.677598			0.338799	−	0.940859i	$-0.389979\pi$
$383$	99396.2			0.677598			0.338799	+	0.940859i	$0.389979\pi$
$384$		0			0
$385$	108398.	+	105517.i	0.731306	+	0.711873i
$386$		0			0
$387$		−	31288.2i		−	0.208909i
$388$		0			0
$389$		−	99896.0i		−	0.660159i	−0.943953	−	0.330080i	$-0.892924\pi$
$389$		−	99896.0i		−	0.660159i	0.943953	−	0.330080i	$-0.107076\pi$
$390$		0			0
$391$			54185.7i			0.354431i
$392$		0			0
$393$			92903.6i			0.601516i
$394$		0			0
$395$	−40232.0	+	41330.3i	−0.257856	+	0.264895i
$396$		0			0
$397$	55234.5			0.350453			0.175226	−	0.984528i	$-0.443934\pi$
$397$	55234.5			0.350453			0.175226	+	0.984528i	$0.443934\pi$
$398$		0			0
$399$		−	319041.i		−	2.00401i
$400$		0			0
$401$	−40688.8			−0.253038			−0.126519	−	0.991964i	$-0.540380\pi$
$401$	−40688.8			−0.253038			−0.126519	+	0.991964i	$0.540380\pi$
$402$		0			0
$403$			184434.i			1.13561i
$404$		0			0
$405$	−141499.	−	137739.i	−0.862665	−	0.839743i
$406$		0			0
$407$	82626.5			0.498804
$408$		0			0
$409$	−313804.			−1.87591			−0.937955	−	0.346757i	$-0.887283\pi$
$409$	−313804.			−1.87591			−0.937955	+	0.346757i	$0.887283\pi$
$410$		0			0
$411$	−152806.			−0.904603
$412$		0			0
$413$	−123418.			−0.723567
$414$		0			0
$415$	37424.6	−	38446.2i	0.217301	−	0.223232i
$416$		0			0
$417$		−	261190.i		−	1.50205i
$418$		0			0
$419$	−264287.			−1.50538			−0.752692	−	0.658373i	$-0.771245\pi$
$419$	−264287.			−1.50538			−0.752692	+	0.658373i	$0.771245\pi$
$420$		0			0
$421$		−	179764.i		−	1.01424i	−0.861877	−	0.507118i	$-0.830711\pi$
$421$		−	179764.i		−	1.01424i	0.861877	−	0.507118i	$-0.169289\pi$
$422$		0			0
$423$	20829.4			0.116412
$424$		0			0
$425$	94555.6	−	2546.81i	0.523491	−	0.0141000i
$426$		0			0
$427$		−	451866.i		−	2.47830i
$428$		0			0
$429$			79506.2i			0.432003i
$430$		0			0
$431$		−	151073.i		−	0.813266i	−0.913592	−	0.406633i	$-0.866703\pi$
$431$		−	151073.i		−	0.813266i	0.913592	−	0.406633i	$-0.133297\pi$
$432$		0			0
$433$		−	133641.i		−	0.712796i	−0.934334	−	0.356398i	$-0.884005\pi$
$433$		−	133641.i		−	0.712796i	0.934334	−	0.356398i	$-0.115995\pi$
$434$		0			0
$435$	70848.4	+	68965.8i	0.374413	+	0.364465i
$436$		0			0
$437$	−131924.			−0.690812
$438$		0			0
$439$		−	74613.0i		−	0.387156i	−0.981085	−	0.193578i	$-0.937991\pi$
$439$		−	74613.0i		−	0.387156i	0.981085	−	0.193578i	$-0.0620092\pi$
$440$		0			0
$441$	−110920.			−0.570339
$442$		0			0
$443$			110554.i			0.563334i	0.959512	+	0.281667i	$0.0908874\pi$
$443$			110554.i			0.563334i	−0.959512	+	0.281667i	$0.909113\pi$
$444$		0			0
$445$	91666.8	+	89231.1i	0.462905	+	0.450605i
$446$		0			0
$447$	104786.			0.524432
$448$		0			0
$449$	9722.73			0.0482276			0.0241138	−	0.999709i	$-0.492324\pi$
$449$	9722.73			0.0482276			0.0241138	+	0.999709i	$0.492324\pi$
$450$		0			0
$451$	181543.			0.892539
$452$		0			0
$453$	150390.			0.732863
$454$		0			0
$455$	166125.	+	161711.i	0.802440	+	0.781117i
$456$		0			0
$457$			331629.i			1.58789i	0.607990	+	0.793945i	$0.291976\pi$
$457$			331629.i			1.58789i	−0.607990	+	0.793945i	$0.708024\pi$
$458$		0			0
$459$	89398.3			0.424330
$460$		0			0
$461$			86774.7i			0.408311i	0.978938	+	0.204156i	$0.0654449\pi$
$461$			86774.7i			0.408311i	−0.978938	+	0.204156i	$0.934555\pi$
$462$		0			0
$463$	35664.8			0.166371			0.0831856	−	0.996534i	$-0.473491\pi$
$463$	35664.8			0.166371			0.0831856	+	0.996534i	$0.473491\pi$
$464$		0			0
$465$	308487.	+	300290.i	1.42670	+	1.38879i
$466$		0			0
$467$		−	293718.i		−	1.34678i	−0.739288	−	0.673389i	$-0.764838\pi$
$467$		−	293718.i		−	1.34678i	0.739288	−	0.673389i	$-0.235162\pi$
$468$		0			0
$469$			55757.1i			0.253486i
$470$		0			0
$471$		−	283042.i		−	1.27588i
$472$		0			0
$473$		−	96550.3i		−	0.431550i
$474$		0			0
$475$	6200.61	+	230210.i	0.0274819	+	1.02032i
$476$		0			0
$477$	78783.6			0.346257
$478$		0			0
$479$			435529.i			1.89822i	0.314945	+	0.949110i	$0.398014\pi$
$479$			435529.i			1.89822i	−0.314945	+	0.949110i	$0.601986\pi$
$480$		0			0
$481$	126629.			0.547323
$482$		0			0
$483$			310002.i			1.32883i
$484$		0			0
$485$	−159901.	+	164266.i	−0.679779	+	0.698335i
$486$		0			0
$487$	−187286.			−0.789672			−0.394836	−	0.918752i	$-0.629199\pi$
$487$	−187286.			−0.789672			−0.394836	+	0.918752i	$0.629199\pi$
$488$		0			0
$489$	38088.6			0.159286
$490$		0			0
$491$	287674.			1.19327			0.596634	−	0.802513i	$-0.296504\pi$
$491$	287674.			1.19327			0.596634	+	0.802513i	$0.296504\pi$
$492$		0			0
$493$	−58662.4			−0.241360
$494$		0			0
$495$	29516.6	+	28732.3i	0.120464	+	0.117263i
$496$		0			0
$497$		−	572109.i		−	2.31615i
$498$		0			0
$499$	−175430.			−0.704537			−0.352268	−	0.935899i	$-0.614590\pi$
$499$	−175430.			−0.704537			−0.352268	+	0.935899i	$0.614590\pi$
$500$		0			0
$501$		−	211386.i		−	0.842172i
$502$		0			0
$503$	95682.5			0.378178			0.189089	−	0.981960i	$-0.439447\pi$
$503$	95682.5			0.378178			0.189089	+	0.981960i	$0.439447\pi$
$504$		0			0
$505$	−9358.57	+	9614.03i	−0.0366967	+	0.0376984i
$506$		0			0
$507$		−	169569.i		−	0.659676i
$508$		0			0
$509$			370388.i			1.42962i	0.699318	+	0.714811i	$0.253487\pi$
$509$			370388.i			1.42962i	−0.699318	+	0.714811i	$0.746513\pi$
$510$		0			0
$511$			350494.i			1.34227i
$512$		0			0
$513$			217654.i			0.827050i
$514$		0			0
$515$	−56256.2	−	54761.3i	−0.212107	−	0.206471i
$516$		0			0
$517$	64276.4			0.240475
$518$		0			0
$519$			360541.i			1.33850i
$520$		0			0
$521$	−259622.			−0.956458			−0.478229	−	0.878235i	$-0.658721\pi$
$521$	−259622.			−0.956458			−0.478229	+	0.878235i	$0.658721\pi$
$522$		0			0
$523$			108562.i			0.396895i	0.980111	+	0.198448i	$0.0635900\pi$
$523$			108562.i			0.396895i	−0.980111	+	0.198448i	$0.936410\pi$
$524$		0			0
$525$	540961.	−	14570.5i	1.96267	−	0.0528637i
$526$		0			0
$527$	−255427.			−0.919700
$528$		0			0
$529$	−151655.			−0.541933
$530$		0			0
$531$	−33606.6			−0.119189
$532$		0			0
$533$	278224.			0.979356
$534$		0			0
$535$	−291273.	+	299224.i	−1.01764	+	1.04542i
$536$		0			0
$537$		−	241510.i		−	0.837502i
$538$		0			0
$539$	−342282.			−1.17817
$540$		0			0
$541$			280006.i			0.956694i	0.878171	+	0.478347i	$0.158764\pi$
$541$			280006.i			0.956694i	−0.878171	+	0.478347i	$0.841236\pi$
$542$		0			0
$543$	76785.8			0.260424
$544$		0			0
$545$	−275348.	+	282865.i	−0.927021	+	0.952326i
$546$		0			0
$547$			72377.7i			0.241897i	0.992659	+	0.120948i	$0.0385936\pi$
$547$			72377.7i			0.241897i	−0.992659	+	0.120948i	$0.961406\pi$
$548$		0			0
$549$		−	123043.i		−	0.408235i
$550$		0			0
$551$		−	142823.i		−	0.470429i
$552$		0			0
$553$		−	195784.i		−	0.640216i
$554$		0			0
$555$	206175.	−	211803.i	0.669344	−	0.687615i
$556$		0			0
$557$	402974.			1.29887			0.649437	−	0.760416i	$-0.275005\pi$
$557$	402974.			1.29887			0.649437	+	0.760416i	$0.275005\pi$
$558$		0			0
$559$		−	147968.i		−	0.473527i
$560$		0			0
$561$	−110110.			−0.349866
$562$		0			0
$563$		−	244892.i		−	0.772606i	−0.922372	−	0.386303i	$-0.873752\pi$
$563$		−	244892.i		−	0.772606i	0.922372	−	0.386303i	$-0.126248\pi$
$564$		0			0
$565$	−77787.0	+	79910.4i	−0.243675	+	0.250326i
$566$		0			0
$567$	670288.			2.08495
$568$		0			0
$569$	644997.			1.99220			0.996100	−	0.0882331i	$-0.0281220\pi$
$569$	644997.			1.99220			0.996100	+	0.0882331i	$0.0281220\pi$
$570$		0			0
$571$	−198216.			−0.607947			−0.303973	−	0.952681i	$-0.598313\pi$
$571$	−198216.			−0.607947			−0.303973	+	0.952681i	$0.598313\pi$
$572$		0			0
$573$	−295139.			−0.898911
$574$		0			0
$575$	−6024.94	−	223688.i	−0.0182229	−	0.676561i
$576$		0			0
$577$		−	218459.i		−	0.656172i	−0.944648	−	0.328086i	$-0.893596\pi$
$577$		−	218459.i		−	0.656172i	0.944648	−	0.328086i	$-0.106404\pi$
$578$		0			0
$579$	499394.			1.48966
$580$		0			0
$581$			182122.i			0.539523i
$582$		0			0
$583$	243114.			0.715274
$584$		0			0
$585$	45235.7	+	44033.7i	0.132181	+	0.128669i
$586$		0			0
$587$			330673.i			0.959673i	0.877358	+	0.479836i	$0.159304\pi$
$587$			330673.i			0.959673i	−0.877358	+	0.479836i	$0.840696\pi$
$588$		0			0
$589$		−	621878.i		−	1.79256i
$590$		0			0
$591$			488569.i			1.39879i
$592$		0			0
$593$			499960.i			1.42176i	0.703314	+	0.710879i	$0.251703\pi$
$593$			499960.i			1.42176i	−0.703314	+	0.710879i	$0.748297\pi$
$594$		0			0
$595$	−223958.	+	230071.i	−0.632604	+	0.649873i
$596$		0			0
$597$	−302469.			−0.848657
$598$		0			0
$599$		−	407848.i		−	1.13670i	−0.822788	−	0.568348i	$-0.807583\pi$
$599$		−	407848.i		−	1.13670i	0.822788	−	0.568348i	$-0.192417\pi$
$600$		0			0
$601$	441217.			1.22153			0.610763	−	0.791813i	$-0.290863\pi$
$601$	441217.			1.22153			0.610763	+	0.791813i	$0.290863\pi$
$602$		0			0
$603$			15182.6i			0.0417553i
$604$		0			0
$605$	−171196.	−	166647.i	−0.467718	−	0.455290i
$606$		0			0
$607$	71921.7			0.195201			0.0976006	−	0.995226i	$-0.468883\pi$
$607$	71921.7			0.195201			0.0976006	+	0.995226i	$0.468883\pi$
$608$		0			0
$609$	−335613.			−0.904908
$610$		0			0
$611$	98506.9			0.263866
$612$		0			0
$613$	527274.			1.40319			0.701593	−	0.712577i	$-0.252472\pi$
$613$	527274.			1.40319			0.701593	+	0.712577i	$0.252472\pi$
$614$		0			0
$615$	452998.	−	465363.i	1.19769	−	1.23039i
$616$		0			0
$617$			91867.0i			0.241318i	0.992694	+	0.120659i	$0.0385007\pi$
$617$			91867.0i			0.241318i	−0.992694	+	0.120659i	$0.961499\pi$
$618$		0			0
$619$	633381.			1.65304			0.826521	−	0.562906i	$-0.190317\pi$
$619$	633381.			1.65304			0.826521	+	0.562906i	$0.190317\pi$
$620$		0			0
$621$		−	211488.i		−	0.548405i
$622$		0			0
$623$	−434231.			−1.11878
$624$		0			0
$625$	−390059.	+	21027.4i	−0.998550	+	0.0538300i
$626$		0			0
$627$		−	268081.i		−	0.681915i
$628$		0			0
$629$			175372.i			0.443261i
$630$		0			0
$631$			128753.i			0.323368i	0.986843	+	0.161684i	$0.0516926\pi$
$631$			128753.i			0.323368i	−0.986843	+	0.161684i	$0.948307\pi$
$632$		0			0
$633$		−	200531.i		−	0.500465i
$634$		0			0
$635$	195052.	+	189869.i	0.483730	+	0.470877i
$636$		0			0
$637$	−524565.			−1.29277
$638$		0			0
$639$		−	155785.i		−	0.381525i
$640$		0			0
$641$	391151.			0.951980			0.475990	−	0.879451i	$-0.342090\pi$
$641$	391151.			0.951980			0.475990	+	0.879451i	$0.342090\pi$
$642$		0			0
$643$		−	720581.i		−	1.74285i	−0.490525	−	0.871427i	$-0.663195\pi$
$643$		−	720581.i		−	1.74285i	0.490525	−	0.871427i	$-0.336805\pi$
$644$		0			0
$645$	−247495.	−	240918.i	−0.594903	−	0.579096i
$646$		0			0
$647$	−403682.			−0.964341			−0.482171	−	0.876077i	$-0.660151\pi$
$647$	−403682.			−0.964341			−0.482171	+	0.876077i	$0.660151\pi$
$648$		0			0
$649$	−103705.			−0.246212
$650$		0			0
$651$	−1.46132e6			−3.44814
$652$		0			0
$653$	−404377.			−0.948332			−0.474166	−	0.880436i	$-0.657250\pi$
$653$	−404377.			−0.948332			−0.474166	+	0.880436i	$0.657250\pi$
$654$		0			0
$655$	163112.	+	158778.i	0.380192	+	0.370090i
$656$		0			0
$657$			95439.1i			0.221104i
$658$		0			0
$659$	−179720.			−0.413834			−0.206917	−	0.978359i	$-0.566343\pi$
$659$	−179720.			−0.413834			−0.206917	+	0.978359i	$0.566343\pi$
$660$		0			0
$661$			512413.i			1.17278i	0.810028	+	0.586391i	$0.199452\pi$
$661$			512413.i			1.17278i	−0.810028	+	0.586391i	$0.800548\pi$
$662$		0			0
$663$	−168750.			−0.383898
$664$		0			0
$665$	−560144.	−	545260.i	−1.26665	−	1.23299i
$666$		0			0
$667$			138776.i			0.311935i
$668$		0			0
$669$			622174.i			1.39014i
$670$		0			0
$671$		−	379690.i		−	0.843304i
$672$		0			0
$673$			136431.i			0.301220i	0.988593	+	0.150610i	$0.0481238\pi$
$673$			136431.i			0.301220i	−0.988593	+	0.150610i	$0.951876\pi$
$674$		0			0
$675$	−369052.	+	9940.24i	−0.809990	+	0.0218167i
$676$		0			0
$677$	−185710.			−0.405190			−0.202595	−	0.979263i	$-0.564937\pi$
$677$	−185710.			−0.405190			−0.202595	+	0.979263i	$0.564937\pi$
$678$		0			0
$679$		−	778138.i		−	1.68778i
$680$		0			0
$681$	−27529.9			−0.0593623
$682$		0			0
$683$			521505.i			1.11794i	0.829189	+	0.558968i	$0.188802\pi$
$683$			521505.i			1.11794i	−0.829189	+	0.558968i	$0.811198\pi$
$684$		0			0
$685$	−261156.	+	268284.i	−0.556568	+	0.571761i
$686$		0			0
$687$	−703020.			−1.48955
$688$		0			0
$689$	372584.			0.784849
$690$		0			0
$691$	159485.			0.334012			0.167006	−	0.985956i	$-0.446590\pi$
$691$	159485.			0.334012			0.167006	+	0.985956i	$0.446590\pi$
$692$		0			0
$693$	−139822.			−0.291144
$694$		0			0
$695$	−458576.	−	446391.i	−0.949383	−	0.924156i
$696$		0			0
$697$			385320.i			0.793152i
$698$		0			0
$699$	854739.			1.74936
$700$		0			0
$701$		−	924458.i		−	1.88127i	−0.339418	−	0.940635i	$-0.610230\pi$
$701$		−	924458.i		−	1.88127i	0.339418	−	0.940635i	$-0.389770\pi$
$702$		0			0
$703$	−426971.			−0.863949
$704$		0			0
$705$	160386.	−	164765.i	0.322693	−	0.331502i
$706$		0			0
$707$		−	45542.2i		−	0.0911120i
$708$		0			0
$709$		−	49979.9i		−	0.0994267i	−0.998764	−	0.0497134i	$-0.984169\pi$
$709$		−	49979.9i		−	0.0994267i	0.998764	−	0.0497134i	$-0.0158308\pi$
$710$		0			0
$711$		−	53311.7i		−	0.105459i
$712$		0			0
$713$			604259.i			1.18862i
$714$		0			0
$715$	139590.	+	135881.i	0.273050	+	0.265795i
$716$		0			0
$717$	373470.			0.726469
$718$		0			0
$719$			823502.i			1.59297i	0.604659	+	0.796484i	$0.293309\pi$
$719$			823502.i			1.59297i	−0.604659	+	0.796484i	$0.706691\pi$
$720$		0			0
$721$	266489.			0.512635
$722$		0			0
$723$			497471.i			0.951681i
$724$		0			0
$725$	242168.	−	6522.70i	0.460725	−	0.0124094i
$726$		0			0
$727$	791865.			1.49824			0.749122	−	0.662432i	$-0.230476\pi$
$727$	791865.			1.49824			0.749122	+	0.662432i	$0.230476\pi$
$728$		0			0
$729$	−305673.			−0.575177
$730$		0			0
$731$	204925.			0.383496
$732$		0			0
$733$	−586208.			−1.09105			−0.545524	−	0.838095i	$-0.683669\pi$
$733$	−586208.			−1.09105			−0.545524	+	0.838095i	$0.683669\pi$
$734$		0			0
$735$	−854084.	+	877398.i	−1.58098	+	1.62413i
$736$		0			0
$737$			46851.1i			0.0862551i
$738$		0			0
$739$	−51496.3			−0.0942947			−0.0471474	−	0.998888i	$-0.515013\pi$
$739$	−51496.3			−0.0942947			−0.0471474	+	0.998888i	$0.515013\pi$
$740$		0			0
$741$		−	410847.i		−	0.748245i
$742$		0			0
$743$	333510.			0.604132			0.302066	−	0.953287i	$-0.402324\pi$
$743$	333510.			0.604132			0.302066	+	0.953287i	$0.402324\pi$
$744$		0			0
$745$	179086.	−	183975.i	0.322663	−	0.331471i
$746$		0			0
$747$			49591.6i			0.0888724i
$748$		0			0
$749$		−	1.41744e6i		−	2.52663i
$750$		0			0
$751$		−	315513.i		−	0.559419i	−0.960085	−	0.279710i	$-0.909762\pi$
$751$		−	315513.i		−	0.559419i	0.960085	−	0.279710i	$-0.0902382\pi$
$752$		0			0
$753$		−	768505.i		−	1.35537i
$754$		0			0
$755$	257026.	−	264042.i	0.450903	−	0.463211i
$756$		0			0
$757$	559184.			0.975806			0.487903	−	0.872898i	$-0.337762\pi$
$757$	559184.			0.975806			0.487903	+	0.872898i	$0.337762\pi$
$758$		0			0
$759$			260486.i			0.452168i
$760$		0			0
$761$	−432702.			−0.747170			−0.373585	−	0.927596i	$-0.621872\pi$
$761$	−432702.			−0.747170			−0.373585	+	0.927596i	$0.621872\pi$
$762$		0			0
$763$		−	1.33995e6i		−	2.30165i
$764$		0			0
$765$	−60983.4	+	62648.1i	−0.104205	+	0.107050i
$766$		0			0
$767$	−158933.			−0.270161
$768$		0			0
$769$	258310.			0.436805			0.218403	−	0.975859i	$-0.429915\pi$
$769$	258310.			0.436805			0.218403	+	0.975859i	$0.429915\pi$
$770$		0			0
$771$	277433.			0.466712
$772$		0			0
$773$	280428.			0.469312			0.234656	−	0.972078i	$-0.424604\pi$
$773$	280428.			0.469312			0.234656	+	0.972078i	$0.424604\pi$
$774$		0			0
$775$	1.05445e6	−	28401.1i	1.75558	−	0.0472859i
$776$		0			0
$777$			1.00332e6i			1.66187i
$778$		0			0
$779$	−938122.			−1.54591
$780$		0			0
$781$		−	480727.i		−	0.788127i
$782$		0			0
$783$	228960.			0.373453
$784$		0			0
$785$	−496940.	−	483735.i	−0.806426	−	0.784998i
$786$		0			0
$787$			1.17964e6i			1.90459i	0.305177	+	0.952296i	$0.401284\pi$
$787$			1.17964e6i			1.90459i	−0.305177	+	0.952296i	$0.598716\pi$
$788$		0			0
$789$			491136.i			0.788947i
$790$		0			0
$791$		−	378540.i		−	0.605005i
$792$		0			0
$793$		−	581894.i		−	0.925332i
$794$		0			0
$795$	606633.	−	623192.i	0.959824	−	0.986024i
$796$		0			0
$797$	−627511.			−0.987881			−0.493940	−	0.869496i	$-0.664444\pi$
$797$	−627511.			−0.987881			−0.493940	+	0.869496i	$0.664444\pi$
$798$		0			0
$799$			136425.i			0.213698i
$800$		0			0
$801$	−118241.			−0.184290
$802$		0			0
$803$			294510.i			0.456740i
$804$		0			0
$805$	544274.	+	529812.i	0.839897	+	0.817579i
$806$		0			0
$807$	−590326.			−0.906453
$808$		0			0
$809$	−328656.			−0.502163			−0.251081	−	0.967966i	$-0.580786\pi$
$809$	−328656.			−0.502163			−0.251081	+	0.967966i	$0.580786\pi$
$810$		0			0
$811$	−510808.			−0.776634			−0.388317	−	0.921526i	$-0.626943\pi$
$811$	−510808.			−0.776634			−0.388317	+	0.921526i	$0.626943\pi$
$812$		0			0
$813$	1.37888e6			2.08616
$814$		0			0
$815$	65095.8	−	66872.7i	0.0980026	−	0.100678i
$816$		0			0
$817$			498923.i			0.747462i
$818$		0			0
$819$	−214284.			−0.319464
$820$		0			0
$821$		−	37920.6i		−	0.0562586i	−0.999604	−	0.0281293i	$-0.991045\pi$
$821$		−	37920.6i		−	0.0562586i	0.999604	−	0.0281293i	$-0.00895502\pi$
$822$		0			0
$823$	449975.			0.664337			0.332168	−	0.943220i	$-0.392220\pi$
$823$	449975.			0.664337			0.332168	+	0.943220i	$0.392220\pi$
$824$		0			0
$825$	454554.	−	12243.2i	0.667849	−	0.0179882i
$826$		0			0
$827$			41920.8i			0.0612940i	0.999530	+	0.0306470i	$0.00975678\pi$
$827$			41920.8i			0.0612940i	−0.999530	+	0.0306470i	$0.990243\pi$
$828$		0			0
$829$			1.28858e6i			1.87500i	0.347985	+	0.937500i	$0.386866\pi$
$829$			1.28858e6i			1.87500i	−0.347985	+	0.937500i	$0.613134\pi$
$830$		0			0
$831$			679808.i			0.984429i
$832$		0			0
$833$		−	726484.i		−	1.04697i
$834$		0			0
$835$	−371134.	−	361272.i	−0.532301	−	0.518157i
$836$		0			0
$837$	996936.			1.42304
$838$		0			0
$839$		−	506198.i		−	0.719112i	−0.933124	−	0.359556i	$-0.882928\pi$
$839$		−	506198.i		−	0.719112i	0.933124	−	0.359556i	$-0.117072\pi$
$840$		0			0
$841$	557039.			0.787579
$842$		0			0
$843$			719519.i			1.01248i
$844$		0			0
$845$	−297715.	−	289804.i	−0.416953	−	0.405874i
$846$		0			0
$847$	810968.			1.13041
$848$		0			0
$849$	−56941.5			−0.0789975
$850$		0			0
$851$	414874.			0.572872
$852$		0			0
$853$	−656590.			−0.902395			−0.451197	−	0.892424i	$-0.649003\pi$
$853$	−656590.			−0.902395			−0.451197	+	0.892424i	$0.649003\pi$
$854$		0			0
$855$	−152527.	−	148474.i	−0.208648	−	0.203103i
$856$		0			0
$857$			388067.i			0.528379i	0.964471	+	0.264189i	$0.0851044\pi$
$857$			388067.i			0.528379i	−0.964471	+	0.264189i	$0.914896\pi$
$858$		0			0
$859$	−282145.			−0.382372			−0.191186	−	0.981554i	$-0.561233\pi$
$859$	−282145.			−0.382372			−0.191186	+	0.981554i	$0.561233\pi$
$860$		0			0
$861$			2.20445e6i			2.97368i
$862$		0			0
$863$	−1.01110e6			−1.35761			−0.678804	−	0.734320i	$-0.737501\pi$
$863$	−1.01110e6			−1.35761			−0.678804	+	0.734320i	$0.737501\pi$
$864$		0			0
$865$	633007.	+	616186.i	0.846011	+	0.823531i
$866$		0			0
$867$			618484.i			0.822792i
$868$		0			0
$869$		−	164512.i		−	0.217850i
$870$		0			0
$871$			71801.7i			0.0946452i
$872$		0			0
$873$		−	211886.i		−	0.278018i
$874$		0			0
$875$	898955.	−	974676.i	1.17414	−	1.27305i
$876$		0			0
$877$	190192.			0.247283			0.123641	−	0.992327i	$-0.460543\pi$
$877$	190192.			0.247283			0.123641	+	0.992327i	$0.460543\pi$
$878$		0			0
$879$			875617.i			1.13328i
$880$		0			0
$881$	237801.			0.306380			0.153190	−	0.988197i	$-0.451045\pi$
$881$	237801.			0.306380			0.153190	+	0.988197i	$0.451045\pi$
$882$		0			0
$883$		−	175861.i		−	0.225552i	−0.993620	−	0.112776i	$-0.964026\pi$
$883$		−	175861.i		−	0.225552i	0.993620	−	0.112776i	$-0.0359743\pi$
$884$		0			0
$885$	−258770.	+	265834.i	−0.330391	+	0.339410i
$886$		0			0
$887$	1.36691e6			1.73738			0.868688	−	0.495360i	$-0.164964\pi$
$887$	1.36691e6			1.73738			0.868688	+	0.495360i	$0.164964\pi$
$888$		0			0
$889$	−923974.			−1.16911
$890$		0			0
$891$	563224.			0.709456
$892$		0			0
$893$	−332148.			−0.416513
$894$		0			0
$895$	−424022.	−	412755.i	−0.529349	−	0.515283i
$896$		0			0
$897$			399207.i			0.496151i
$898$		0			0
$899$	−654181.			−0.809428
$900$		0			0
$901$			516002.i			0.635626i
$902$		0			0
$903$	1.17240e6			1.43780
$904$		0			0
$905$	131232.	−	134814.i	0.160229	−	0.164603i
$906$		0			0
$907$			699123.i			0.849843i	0.905230	+	0.424922i	$0.139698\pi$
$907$			699123.i			0.849843i	−0.905230	+	0.424922i	$0.860302\pi$
$908$		0			0
$909$		−	12401.1i		−	0.0150083i
$910$		0			0
$911$			1.27217e6i			1.53288i	0.642315	+	0.766441i	$0.277974\pi$
$911$			1.27217e6i			1.53288i	−0.642315	+	0.766441i	$0.722026\pi$
$912$		0			0
$913$			153032.i			0.183586i
$914$		0			0
$915$	−973288.	−	947426.i	−1.16252	−	1.13163i
$916$		0			0
$917$	−772672.			−0.918875
$918$		0			0
$919$			507149.i			0.600488i	0.953862	+	0.300244i	$0.0970681\pi$
$919$			507149.i			0.600488i	−0.953862	+	0.300244i	$0.902932\pi$
$920$		0			0
$921$	248760.			0.293265
$922$		0			0
$923$		−	736738.i		−	0.864788i
$924$		0			0
$925$	−19499.7	−	723967.i	−0.0227900	−	0.846127i
$926$		0			0
$927$	72564.6			0.0844433
$928$		0			0
$929$	−31736.2			−0.0367725			−0.0183862	−	0.999831i	$-0.505853\pi$
$929$	−31736.2			−0.0367725			−0.0183862	+	0.999831i	$0.505853\pi$
$930$		0			0
$931$	1.76874e6			2.04063
$932$		0			0
$933$	−1.48258e6			−1.70315
$934$		0			0
$935$	−188185.	+	193322.i	−0.215260	+	0.221135i
$936$		0			0
$937$		−	1.23459e6i		−	1.40619i	−0.711096	−	0.703094i	$-0.751801\pi$
$937$		−	1.23459e6i		−	1.40619i	0.711096	−	0.703094i	$-0.248199\pi$
$938$		0			0
$939$	156.930			0.000177981
$940$		0			0
$941$		−	126651.i		−	0.143031i	−0.997439	−	0.0715156i	$-0.977216\pi$
$941$		−	126651.i		−	0.143031i	0.997439	−	0.0715156i	$-0.0227836\pi$
$942$		0			0
$943$	911544.			1.02507
$944$		0			0
$945$	874110.	−	897970.i	0.978819	−	1.00554i
$946$		0			0
$947$			213933.i			0.238549i	0.992861	+	0.119275i	$0.0380569\pi$
$947$			213933.i			0.238549i	−0.992861	+	0.119275i	$0.961943\pi$
$948$		0			0
$949$			451352.i			0.501168i
$950$		0			0
$951$		−	202771.i		−	0.224205i
$952$		0			0
$953$			281174.i			0.309592i	0.987946	+	0.154796i	$0.0494720\pi$
$953$			281174.i			0.309592i	−0.987946	+	0.154796i	$0.950528\pi$
$954$		0			0
$955$	−504410.	+	518179.i	−0.553066	+	0.568163i
$956$		0			0
$957$	−282006.			−0.307918
$958$		0			0
$959$		−	1.27088e6i		−	1.38187i
$960$		0			0
$961$	−1.92491e6			−2.08431
$962$		0			0
$963$		−	385968.i		−	0.416197i
$964$		0			0
$965$	853496.	−	876794.i	0.916530	−	0.941549i
$966$		0			0
$967$	−1.15290e6			−1.23293			−0.616467	−	0.787381i	$-0.711437\pi$
$967$	−1.15290e6			−1.23293			−0.616467	+	0.787381i	$0.711437\pi$
$968$		0			0
$969$	568994.			0.605982
$970$		0			0
$971$	−857201.			−0.909169			−0.454584	−	0.890704i	$-0.650212\pi$
$971$	−857201.			−0.909169			−0.454584	+	0.890704i	$0.650212\pi$
$972$		0			0
$973$	2.17230e6			2.29453
$974$		0			0
$975$	696628.	−	18763.4i	0.732810	−	0.0197379i
$976$		0			0
$977$			1.68459e6i			1.76484i	0.470464	+	0.882419i	$0.344086\pi$
$977$			1.68459e6i			1.76484i	−0.470464	+	0.882419i	$0.655914\pi$
$978$		0			0
$979$	−364872.			−0.380693
$980$		0			0
$981$		−	364866.i		−	0.379136i
$982$		0			0
$983$	−501269.			−0.518756			−0.259378	−	0.965776i	$-0.583518\pi$
$983$	−501269.			−0.518756			−0.259378	+	0.965776i	$0.583518\pi$
$984$		0			0
$985$	857788.	+	834995.i	0.884112	+	0.860620i
$986$		0			0
$987$			780500.i			0.801195i
$988$		0			0
$989$		−	484787.i		−	0.495631i
$990$		0			0
$991$		−	650776.i		−	0.662650i	−0.943517	−	0.331325i	$-0.892504\pi$
$991$		−	650776.i		−	0.662650i	0.943517	−	0.331325i	$-0.107496\pi$
$992$		0			0
$993$			563058.i			0.571024i
$994$		0			0
$995$	−516938.	+	531049.i	−0.522146	+	0.536400i
$996$		0			0
$997$	724549.			0.728916			0.364458	−	0.931220i	$-0.381254\pi$
$997$	724549.			0.728916			0.364458	+	0.931220i	$0.381254\pi$
$998$		0			0
$999$		−	684480.i		−	0.685851i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	160.5.e.c.79.16		20
4.3	odd	2		40.5.e.c.19.10	yes	20
5.2	odd	4		800.5.g.i.751.15		20
5.3	odd	4		800.5.g.i.751.6		20
5.4	even	2	inner	160.5.e.c.79.5		20
8.3	odd	2	inner	160.5.e.c.79.15		20
8.5	even	2		40.5.e.c.19.12	yes	20
20.3	even	4		200.5.g.i.51.20		20
20.7	even	4		200.5.g.i.51.1		20
20.19	odd	2		40.5.e.c.19.11	yes	20
40.3	even	4		800.5.g.i.751.5		20
40.13	odd	4		200.5.g.i.51.19		20
40.19	odd	2	inner	160.5.e.c.79.6		20
40.27	even	4		800.5.g.i.751.16		20
40.29	even	2		40.5.e.c.19.9	✓	20
40.37	odd	4		200.5.g.i.51.2		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.5.e.c.19.9	✓	20	40.29	even	2
40.5.e.c.19.10	yes	20	4.3	odd	2
40.5.e.c.19.11	yes	20	20.19	odd	2
40.5.e.c.19.12	yes	20	8.5	even	2
160.5.e.c.79.5		20	5.4	even	2	inner
160.5.e.c.79.6		20	40.19	odd	2	inner
160.5.e.c.79.15		20	8.3	odd	2	inner
160.5.e.c.79.16		20	1.1	even	1	trivial
200.5.g.i.51.1		20	20.7	even	4
200.5.g.i.51.2		20	40.37	odd	4
200.5.g.i.51.19		20	40.13	odd	4
200.5.g.i.51.20		20	20.3	even	4
800.5.g.i.751.5		20	40.3	even	4
800.5.g.i.751.6		20	5.3	odd	4
800.5.g.i.751.15		20	5.2	odd	4
800.5.g.i.751.16		20	40.27	even	4

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:	\( N \)	\(=\)	\( 160 = 2^{5} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	160.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+10.2033i q^{3} +(17.9141 + 17.4381i) q^{5} -84.8600 q^{7} -23.1073 q^{9} -71.3055 q^{11} -109.279 q^{13} +(-177.926 + 182.783i) q^{15} -151.344i q^{17} +368.470 q^{19} -865.852i q^{21} -358.031 q^{23} +(16.8280 + 624.773i) q^{25} +590.697i q^{27} -387.610i q^{29} -1687.73i q^{31} -727.551i q^{33} +(-1520.19 - 1479.79i) q^{35} -1158.77 q^{37} -1115.01i q^{39} -2545.99 q^{41} +1354.04i q^{43} +(-413.946 - 402.946i) q^{45} -901.424 q^{47} +4800.23 q^{49} +1544.21 q^{51} -3409.47 q^{53} +(-1277.37 - 1243.43i) q^{55} +3759.61i q^{57} +1454.37 q^{59} +5324.84i q^{61} +1960.88 q^{63} +(-1957.64 - 1905.62i) q^{65} -657.048i q^{67} -3653.09i q^{69} +6741.79i q^{71} -4130.26i q^{73} +(-6374.75 + 171.701i) q^{75} +6050.98 q^{77} +2307.14i q^{79} -7898.74 q^{81} -2146.14i q^{83} +(2639.14 - 2711.18i) q^{85} +3954.90 q^{87} +5117.03 q^{89} +9273.44 q^{91} +17220.4 q^{93} +(6600.80 + 6425.40i) q^{95} +9169.66i q^{97} +1647.68 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 652 q^{9} + 168 q^{11} - 728 q^{19} - 1420 q^{25} - 4440 q^{35} - 4728 q^{41} + 540 q^{49} - 4352 q^{51} - 8280 q^{59} + 9480 q^{65} + 16000 q^{75} - 10956 q^{81} + 22248 q^{89} + 39760 q^{91} + 62504 q^{99}+O(q^{100}) \) 20 * q - 652 * q^9 + 168 * q^11 - 728 * q^19 - 1420 * q^25 - 4440 * q^35 - 4728 * q^41 + 540 * q^49 - 4352 * q^51 - 8280 * q^59 + 9480 * q^65 + 16000 * q^75 - 10956 * q^81 + 22248 * q^89 + 39760 * q^91 + 62504 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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