LMFDB - Embedded newform 1280.3.h.m.1279.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1280,3,Mod(1279,1280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1280, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("1280.1279");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1280 = 2^{8} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1280.h (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:	$34.8774738381$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	16.0.741637881856000000000000.5
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 9x^{12} + 41x^{8} - 144x^{4} + 256 $ x^16 - 9x^12 + 41x^8 - 144*x^4 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{42} $
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1279.4
Root			$-0.0676000 - 1.41260i$ of defining polynomial
Character	$\chi$	$=$	1280.1279
Dual form			1280.3.h.m.1279.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.79002 q^{3} +(2.46084 + 4.35250i) q^{5} -7.67752 q^{7} +13.9443 q^{9} +0.472136i q^{11} +4.10995i q^{13} +(-11.7875 - 20.8486i) q^{15} +2.26154i q^{17} +26.3607i q^{19} +36.7754 q^{21} -9.73249 q^{23} +(-12.8885 + 21.4216i) q^{25} -23.6832 q^{27} +41.6971 q^{29} +22.0104i q^{31} -2.26154i q^{33} +(-18.8931 - 33.4164i) q^{35} +51.7449i q^{37} -19.6867i q^{39} -15.0557 q^{41} +9.31310 q^{43} +(34.3146 + 60.6925i) q^{45} +8.76226 q^{47} +9.94427 q^{49} -10.8328i q^{51} -39.9002i q^{53} +(-2.05497 + 1.16185i) q^{55} -126.268i q^{57} +77.1935i q^{59} +14.7650 q^{61} -107.057 q^{63} +(-17.8885 + 10.1139i) q^{65} +75.8395 q^{67} +46.6188 q^{69} -81.0705i q^{71} +83.4249i q^{73} +(61.7364 - 102.610i) q^{75} -3.62483i q^{77} -100.757i q^{79} -12.0557 q^{81} +0.266939 q^{83} +(-9.84336 + 5.56528i) q^{85} -199.730 q^{87} -85.5542 q^{89} -31.5542i q^{91} -105.430i q^{93} +(-114.735 + 64.8694i) q^{95} -99.1297i q^{97} +6.58359i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 80 q^{9} + 80 q^{25} - 384 q^{41} + 16 q^{49} - 336 q^{81} - 224 q^{89}+O(q^{100}) $ 16 * q + 80 * q^9 + 80 * q^25 - 384 * q^41 + 16 * q^49 - 336 * q^81 - 224 * q^89

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1280\mathbb{Z}\right)^\times$.

$n$	$257$	$261$	$511$
$\chi(n)$	$-1$	$1$	$-1$

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
$3$	−4.79002			−1.59667			−0.798336	−	0.602212i	$-0.794286\pi$
$3$	−4.79002			−1.59667			−0.798336	+	0.602212i	$0.794286\pi$
$4$		0			0
$5$	2.46084	+	4.35250i	0.492168	+	0.870500i
$5$	2.46084	+	4.35250i	0.492168	+	0.870500i
$6$		0			0
$7$	−7.67752			−1.09679			−0.548394	−	0.836220i	$-0.684761\pi$
$7$	−7.67752			−1.09679			−0.548394	+	0.836220i	$0.684761\pi$
$8$		0			0
$9$	13.9443			1.54936
$10$		0			0
$11$			0.472136i			0.0429215i	0.999770	+	0.0214607i	$0.00683169\pi$
$11$			0.472136i			0.0429215i	−0.999770	+	0.0214607i	$0.993168\pi$
$12$		0			0
$13$			4.10995i			0.316150i	0.987427	+	0.158075i	$0.0505287\pi$
$13$			4.10995i			0.316150i	−0.987427	+	0.158075i	$0.949471\pi$
$14$		0			0
$15$	−11.7875	−	20.8486i	−0.785831	−	1.38990i
$16$		0			0
$17$			2.26154i			0.133032i	0.997785	+	0.0665159i	$0.0211883\pi$
$17$			2.26154i			0.133032i	−0.997785	+	0.0665159i	$0.978812\pi$
$18$		0			0
$19$			26.3607i			1.38740i	0.720262	+	0.693702i	$0.244022\pi$
$19$			26.3607i			1.38740i	−0.720262	+	0.693702i	$0.755978\pi$
$20$		0			0
$21$	36.7754			1.75121
$22$		0			0
$23$	−9.73249			−0.423152			−0.211576	−	0.977362i	$-0.567860\pi$
$23$	−9.73249			−0.423152			−0.211576	+	0.977362i	$0.567860\pi$
$24$		0			0
$25$	−12.8885	+	21.4216i	−0.515542	+	0.856864i
$26$		0			0
$27$	−23.6832			−0.877154
$28$		0			0
$29$	41.6971			1.43783			0.718916	−	0.695097i	$-0.244639\pi$
$29$	41.6971			1.43783			0.718916	+	0.695097i	$0.244639\pi$
$30$		0			0
$31$			22.0104i			0.710013i	0.934864	+	0.355007i	$0.115521\pi$
$31$			22.0104i			0.710013i	−0.934864	+	0.355007i	$0.884479\pi$
$32$		0			0
$33$		−	2.26154i		−	0.0685315i
$34$		0			0
$35$	−18.8931	−	33.4164i	−0.539804	−	0.954755i
$36$		0			0
$37$			51.7449i			1.39851i	0.714872	+	0.699256i	$0.246485\pi$
$37$			51.7449i			1.39851i	−0.714872	+	0.699256i	$0.753515\pi$
$38$		0			0
$39$		−	19.6867i		−	0.504787i
$40$		0			0
$41$	−15.0557			−0.367213			−0.183606	−	0.983000i	$-0.558777\pi$
$41$	−15.0557			−0.367213			−0.183606	+	0.983000i	$0.558777\pi$
$42$		0			0
$43$	9.31310			0.216584			0.108292	−	0.994119i	$-0.465462\pi$
$43$	9.31310			0.216584			0.108292	+	0.994119i	$0.465462\pi$
$44$		0			0
$45$	34.3146	+	60.6925i	0.762547	+	1.34872i
$46$		0			0
$47$	8.76226			0.186431			0.0932156	−	0.995646i	$-0.470285\pi$
$47$	8.76226			0.186431			0.0932156	+	0.995646i	$0.470285\pi$
$48$		0			0
$49$	9.94427			0.202944
$50$		0			0
$51$		−	10.8328i		−	0.212408i
$52$		0			0
$53$		−	39.9002i		−	0.752834i	−0.926450	−	0.376417i	$-0.877156\pi$
$53$		−	39.9002i		−	0.752834i	0.926450	−	0.376417i	$-0.122844\pi$
$54$		0			0
$55$	−2.05497	+	1.16185i	−0.0373631	+	0.0211246i
$56$		0			0
$57$		−	126.268i		−	2.21523i
$58$		0			0
$59$			77.1935i			1.30836i	0.756337	+	0.654182i	$0.226987\pi$
$59$			77.1935i			1.30836i	−0.756337	+	0.654182i	$0.773013\pi$
$60$		0			0
$61$	14.7650			0.242050			0.121025	−	0.992649i	$-0.461382\pi$
$61$	14.7650			0.242050			0.121025	+	0.992649i	$0.461382\pi$
$62$		0			0
$63$	−107.057			−1.69932
$64$		0			0
$65$	−17.8885	+	10.1139i	−0.275208	+	0.155599i
$66$		0			0
$67$	75.8395			1.13193			0.565966	−	0.824428i	$-0.308503\pi$
$67$	75.8395			1.13193			0.565966	+	0.824428i	$0.308503\pi$
$68$		0			0
$69$	46.6188			0.675635
$70$		0			0
$71$		−	81.0705i		−	1.14184i	−0.821006	−	0.570919i	$-0.806587\pi$
$71$		−	81.0705i		−	1.14184i	0.821006	−	0.570919i	$-0.193413\pi$
$72$		0			0
$73$			83.4249i			1.14281i	0.820669	+	0.571403i	$0.193601\pi$
$73$			83.4249i			1.14281i	−0.820669	+	0.571403i	$0.806399\pi$
$74$		0			0
$75$	61.7364	−	102.610i	0.823151	−	1.36813i
$76$		0			0
$77$		−	3.62483i		−	0.0470757i
$78$		0			0
$79$		−	100.757i		−	1.27541i	−0.770281	−	0.637704i	$-0.779884\pi$
$79$		−	100.757i		−	1.27541i	0.770281	−	0.637704i	$-0.220116\pi$
$80$		0			0
$81$	−12.0557			−0.148836
$82$		0			0
$83$	0.266939			0.00321613			0.00160806	−	0.999999i	$-0.499488\pi$
$83$	0.266939			0.00321613			0.00160806	+	0.999999i	$0.499488\pi$
$84$		0			0
$85$	−9.84336	+	5.56528i	−0.115804	+	0.0654739i
$86$		0			0
$87$	−199.730			−2.29575
$88$		0			0
$89$	−85.5542			−0.961283			−0.480641	−	0.876917i	$-0.659596\pi$
$89$	−85.5542			−0.961283			−0.480641	+	0.876917i	$0.659596\pi$
$90$		0			0
$91$		−	31.5542i		−	0.346749i
$92$		0			0
$93$		−	105.430i		−	1.13366i
$94$		0			0
$95$	−114.735	+	64.8694i	−1.20774	+	0.682836i
$96$		0			0
$97$		−	99.1297i		−	1.02196i	−0.859594	−	0.510978i	$-0.829284\pi$
$97$		−	99.1297i		−	1.02196i	0.859594	−	0.510978i	$-0.170716\pi$
$98$		0			0
$99$			6.58359i			0.0665009i
$100$		0			0
$101$	−105.405			−1.04361			−0.521805	−	0.853065i	$-0.674741\pi$
$101$	−105.405			−1.04361			−0.521805	+	0.853065i	$0.674741\pi$
$102$		0			0
$103$	−101.863			−0.988958			−0.494479	−	0.869190i	$-0.664641\pi$
$103$	−101.863			−0.988958			−0.494479	+	0.869190i	$0.664641\pi$
$104$		0			0
$105$	90.4984	+	160.065i	0.861890	+	1.52443i
$106$		0			0
$107$	−32.9962			−0.308376			−0.154188	−	0.988042i	$-0.549276\pi$
$107$	−32.9962			−0.308376			−0.154188	+	0.988042i	$0.549276\pi$
$108$		0			0
$109$	−181.554			−1.66563			−0.832814	−	0.553552i	$-0.813272\pi$
$109$	−181.554			−1.66563			−0.832814	+	0.553552i	$0.813272\pi$
$110$		0			0
$111$		−	247.859i		−	2.23296i
$112$		0			0
$113$			76.6403i			0.678233i	0.940744	+	0.339116i	$0.110128\pi$
$113$			76.6403i			0.678233i	−0.940744	+	0.339116i	$0.889872\pi$
$114$		0			0
$115$	−23.9501	−	42.3607i	−0.208262	−	0.368354i
$116$		0			0
$117$			57.3102i			0.489831i
$118$		0			0
$119$		−	17.3630i		−	0.145908i
$120$		0			0
$121$	120.777			0.998158
$122$		0			0
$123$	72.1172			0.586319
$124$		0			0
$125$	−124.954	−	3.38228i	−0.999634	−	0.0270582i
$126$		0			0
$127$	−142.962			−1.12569			−0.562843	−	0.826564i	$-0.690292\pi$
$127$	−142.962			−1.12569			−0.562843	+	0.826564i	$0.690292\pi$
$128$		0			0
$129$	−44.6099			−0.345813
$130$		0			0
$131$			110.138i			0.840746i	0.907351	+	0.420373i	$0.138101\pi$
$131$			110.138i			0.840746i	−0.907351	+	0.420373i	$0.861899\pi$
$132$		0			0
$133$		−	202.385i		−	1.52169i
$134$		0			0
$135$	−58.2804	−	103.081i	−0.431707	−	0.763563i
$136$		0			0
$137$		−	225.398i		−	1.64524i	−0.568592	−	0.822620i	$-0.692512\pi$
$137$		−	225.398i		−	1.64524i	0.568592	−	0.822620i	$-0.307488\pi$
$138$		0			0
$139$		−	97.4164i		−	0.700837i	−0.936593	−	0.350419i	$-0.886039\pi$
$139$		−	97.4164i		−	0.700837i	0.936593	−	0.350419i	$-0.113961\pi$
$140$		0			0
$141$	−41.9714			−0.297669
$142$		0			0
$143$	−1.94045			−0.0135696
$144$		0			0
$145$	102.610	+	181.487i	0.707655	+	1.25163i
$146$		0			0
$147$	−47.6332			−0.324036
$148$		0			0
$149$	−191.397			−1.28454			−0.642271	−	0.766477i	$-0.722008\pi$
$149$	−191.397			−1.28454			−0.642271	+	0.766477i	$0.722008\pi$
$150$		0			0
$151$			68.3549i			0.452682i	0.974048	+	0.226341i	$0.0726763\pi$
$151$			68.3549i			0.452682i	−0.974048	+	0.226341i	$0.927324\pi$
$152$		0			0
$153$			31.5355i			0.206115i
$154$		0			0
$155$	−95.8004	+	54.1641i	−0.618067	+	0.349446i
$156$		0			0
$157$		−	155.235i		−	0.988756i	−0.869247	−	0.494378i	$-0.835396\pi$
$157$		−	155.235i		−	0.988756i	0.869247	−	0.494378i	$-0.164604\pi$
$158$		0			0
$159$			191.123i			1.20203i
$160$		0			0
$161$	74.7214			0.464108
$162$		0			0
$163$	−90.4765			−0.555070			−0.277535	−	0.960715i	$-0.589518\pi$
$163$	−90.4765			−0.555070			−0.277535	+	0.960715i	$0.589518\pi$
$164$		0			0
$165$	9.84336	−	5.56528i	0.0596567	−	0.0337290i
$166$		0			0
$167$	100.778			0.603461			0.301730	−	0.953393i	$-0.402436\pi$
$167$	100.778			0.603461			0.301730	+	0.953393i	$0.402436\pi$
$168$		0			0
$169$	152.108			0.900049
$170$		0			0
$171$			367.580i			2.14959i
$172$		0			0
$173$			54.6556i			0.315928i	0.987445	+	0.157964i	$0.0504930\pi$
$173$			54.6556i			0.315928i	−0.987445	+	0.157964i	$0.949507\pi$
$174$		0			0
$175$	98.9520	−	164.465i	0.565440	−	0.939799i
$176$		0			0
$177$		−	369.758i		−	2.08903i
$178$		0			0
$179$		−	106.807i		−	0.596684i	−0.954459	−	0.298342i	$-0.903566\pi$
$179$		−	106.807i		−	0.596684i	0.954459	−	0.298342i	$-0.0964337\pi$
$180$		0			0
$181$	−144.778			−0.799879			−0.399939	−	0.916542i	$-0.630969\pi$
$181$	−144.778			−0.799879			−0.399939	+	0.916542i	$0.630969\pi$
$182$		0			0
$183$	−70.7248			−0.386474
$184$		0			0
$185$	−225.220	+	127.336i	−1.21740	+	0.688302i
$186$		0			0
$187$	−1.06775			−0.00570992
$188$		0			0
$189$	181.828			0.962052
$190$		0			0
$191$			210.809i			1.10371i	0.833939	+	0.551857i	$0.186081\pi$
$191$			210.809i			1.10371i	−0.833939	+	0.551857i	$0.813919\pi$
$192$		0			0
$193$			108.176i			0.560496i	0.959928	+	0.280248i	$0.0904168\pi$
$193$			108.176i			0.560496i	−0.959928	+	0.280248i	$0.909583\pi$
$194$		0			0
$195$	85.6864	−	48.4458i	0.439418	−	0.248440i
$196$		0			0
$197$		−	308.300i		−	1.56497i	−0.622667	−	0.782487i	$-0.713951\pi$
$197$		−	308.300i		−	1.56497i	0.622667	−	0.782487i	$-0.286049\pi$
$198$		0			0
$199$		−	247.859i		−	1.24552i	−0.782412	−	0.622761i	$-0.786011\pi$
$199$		−	247.859i		−	1.24552i	0.782412	−	0.622761i	$-0.213989\pi$
$200$		0			0
$201$	−363.272			−1.80733
$202$		0			0
$203$	−320.130			−1.57700
$204$		0			0
$205$	−37.0497	−	65.5301i	−0.180730	−	0.319659i
$206$		0			0
$207$	−135.712			−0.655616
$208$		0			0
$209$	−12.4458			−0.0595494
$210$		0			0
$211$			230.584i			1.09281i	0.837520	+	0.546407i	$0.184005\pi$
$211$			230.584i			1.09281i	−0.837520	+	0.546407i	$0.815995\pi$
$212$		0			0
$213$			388.329i			1.82314i
$214$		0			0
$215$	22.9180	+	40.5353i	0.106595	+	0.188536i
$216$		0			0
$217$		−	168.985i		−	0.778734i
$218$		0			0
$219$		−	399.607i		−	1.82469i
$220$		0			0
$221$	−9.29480			−0.0420579
$222$		0			0
$223$	−177.782			−0.797229			−0.398615	−	0.917118i	$-0.630509\pi$
$223$	−177.782			−0.797229			−0.398615	+	0.917118i	$0.630509\pi$
$224$		0			0
$225$	−179.721	+	298.709i	−0.798762	+	1.32759i
$226$		0			0
$227$	−203.301			−0.895601			−0.447800	−	0.894134i	$-0.647792\pi$
$227$	−203.301			−0.895601			−0.447800	+	0.894134i	$0.647792\pi$
$228$		0			0
$229$	81.0705			0.354020			0.177010	−	0.984209i	$-0.443358\pi$
$229$	81.0705			0.354020			0.177010	+	0.984209i	$0.443358\pi$
$230$		0			0
$231$			17.3630i			0.0751645i
$232$		0			0
$233$			56.2864i			0.241573i	0.992679	+	0.120786i	$0.0385416\pi$
$233$			56.2864i			0.241573i	−0.992679	+	0.120786i	$0.961458\pi$
$234$		0			0
$235$	21.5625	+	38.1378i	0.0917554	+	0.162288i
$236$		0			0
$237$			482.629i			2.03641i
$238$		0			0
$239$		−	360.235i		−	1.50726i	−0.657300	−	0.753629i	$-0.728301\pi$
$239$		−	360.235i		−	1.50726i	0.657300	−	0.753629i	$-0.271699\pi$
$240$		0			0
$241$	−197.495			−0.819483			−0.409741	−	0.912202i	$-0.634381\pi$
$241$	−197.495			−0.819483			−0.409741	+	0.912202i	$0.634381\pi$
$242$		0			0
$243$	270.896			1.11480
$244$		0			0
$245$	24.4713	+	43.2825i	0.0998827	+	0.176663i
$246$		0			0
$247$	−108.341			−0.438627
$248$		0			0
$249$	−1.27864			−0.00513510
$250$		0			0
$251$			178.361i			0.710600i	0.934752	+	0.355300i	$0.115621\pi$
$251$			178.361i			0.710600i	−0.934752	+	0.355300i	$0.884379\pi$
$252$		0			0
$253$		−	4.59506i		−	0.0181623i
$254$		0			0
$255$	47.1498	−	26.6578i	0.184901	−	0.104540i
$256$		0			0
$257$		−	453.183i		−	1.76336i	−0.471849	−	0.881679i	$-0.656413\pi$
$257$		−	453.183i		−	1.76336i	0.471849	−	0.881679i	$-0.343587\pi$
$258$		0			0
$259$		−	397.272i		−	1.53387i
$260$		0			0
$261$	581.436			2.22772
$262$		0			0
$263$	21.0921			0.0801981			0.0400990	−	0.999196i	$-0.487233\pi$
$263$	21.0921			0.0801981			0.0400990	+	0.999196i	$0.487233\pi$
$264$		0			0
$265$	173.666	−	98.1879i	0.655342	−	0.370520i
$266$		0			0
$267$	409.806			1.53485
$268$		0			0
$269$	92.4148			0.343549			0.171775	−	0.985136i	$-0.445050\pi$
$269$	92.4148			0.343549			0.171775	+	0.985136i	$0.445050\pi$
$270$		0			0
$271$			188.799i			0.696675i	0.937369	+	0.348337i	$0.113254\pi$
$271$			188.799i			0.696675i	−0.937369	+	0.348337i	$0.886746\pi$
$272$		0			0
$273$			151.145i			0.553645i
$274$		0			0
$275$	−10.1139	−	6.08514i	−0.0367779	−	0.0221278i
$276$		0			0
$277$			70.1251i			0.253159i	0.991956	+	0.126580i	$0.0403999\pi$
$277$			70.1251i			0.253159i	−0.991956	+	0.126580i	$0.959600\pi$
$278$		0			0
$279$			306.919i			1.10007i
$280$		0			0
$281$	506.269			1.80167			0.900835	−	0.434161i	$-0.142955\pi$
$281$	506.269			1.80167			0.900835	+	0.434161i	$0.142955\pi$
$282$		0			0
$283$	501.350			1.77156			0.885778	−	0.464110i	$-0.153626\pi$
$283$	501.350			1.77156			0.885778	+	0.464110i	$0.153626\pi$
$284$		0			0
$285$	549.582	−	310.726i	1.92836	−	1.09026i
$286$		0			0
$287$	115.591			0.402755
$288$		0			0
$289$	283.885			0.982303
$290$		0			0
$291$			474.833i			1.63173i
$292$		0			0
$293$			324.026i			1.10589i	0.833218	+	0.552945i	$0.186496\pi$
$293$			324.026i			1.10589i	−0.833218	+	0.552945i	$0.813504\pi$
$294$		0			0
$295$	−335.985	+	189.961i	−1.13893	+	0.643935i
$296$		0			0
$297$		−	11.1817i		−	0.0376487i
$298$		0			0
$299$		−	40.0000i		−	0.133779i
$300$		0			0
$301$	−71.5015			−0.237546
$302$		0			0
$303$	504.890			1.66630
$304$		0			0
$305$	36.3344	+	64.2648i	0.119129	+	0.210704i
$306$		0			0
$307$	−9.84697			−0.0320748			−0.0160374	−	0.999871i	$-0.505105\pi$
$307$	−9.84697			−0.0320748			−0.0160374	+	0.999871i	$0.505105\pi$
$308$		0			0
$309$	487.924			1.57904
$310$		0			0
$311$		−	200.417i		−	0.644429i	−0.946667	−	0.322214i	$-0.895573\pi$
$311$		−	200.417i		−	0.644429i	0.946667	−	0.322214i	$-0.104427\pi$
$312$		0			0
$313$			261.582i			0.835727i	0.908510	+	0.417863i	$0.137221\pi$
$313$			261.582i			0.835727i	−0.908510	+	0.417863i	$0.862779\pi$
$314$		0			0
$315$	−263.451	−	465.967i	−0.836352	−	1.47926i
$316$		0			0
$317$		−	322.341i		−	1.01685i	−0.861107	−	0.508425i	$-0.830228\pi$
$317$		−	322.341i		−	1.01685i	0.861107	−	0.508425i	$-0.169772\pi$
$318$		0			0
$319$			19.6867i			0.0617138i
$320$		0			0
$321$	158.053			0.492376
$322$		0			0
$323$	−59.6157			−0.184569
$324$		0			0
$325$	−88.0416	−	52.9712i	−0.270897	−	0.162988i
$326$		0			0
$327$	869.645			2.65946
$328$		0			0
$329$	−67.2724			−0.204475
$330$		0			0
$331$			61.0883i			0.184557i	0.995733	+	0.0922783i	$0.0294150\pi$
$331$			61.0883i			0.184557i	−0.995733	+	0.0922783i	$0.970585\pi$
$332$		0			0
$333$			721.545i			2.16680i
$334$		0			0
$335$	186.629	+	330.091i	0.557101	+	0.985348i
$336$		0			0
$337$		−	114.960i		−	0.341129i	−0.985347	−	0.170564i	$-0.945441\pi$
$337$		−	114.960i		−	0.341129i	0.985347	−	0.170564i	$-0.0545591\pi$
$338$		0			0
$339$		−	367.108i		−	1.08292i
$340$		0			0
$341$	−10.3919			−0.0304748
$342$		0			0
$343$	299.851			0.874201
$344$		0			0
$345$	114.721	+	202.908i	0.332526	+	0.588140i
$346$		0			0
$347$	−143.967			−0.414892			−0.207446	−	0.978246i	$-0.566515\pi$
$347$	−143.967			−0.414892			−0.207446	+	0.978246i	$0.566515\pi$
$348$		0			0
$349$	−35.9526			−0.103016			−0.0515080	−	0.998673i	$-0.516403\pi$
$349$	−35.9526			−0.103016			−0.0515080	+	0.998673i	$0.516403\pi$
$350$		0			0
$351$		−	97.3365i		−	0.277312i
$352$		0			0
$353$			252.536i			0.715400i	0.933837	+	0.357700i	$0.116439\pi$
$353$			252.536i			0.715400i	−0.933837	+	0.357700i	$0.883561\pi$
$354$		0			0
$355$	352.860	−	199.502i	0.993971	−	0.561976i
$356$		0			0
$357$			83.1691i			0.232967i
$358$		0			0
$359$			594.152i			1.65502i	0.561452	+	0.827509i	$0.310243\pi$
$359$			594.152i			1.65502i	−0.561452	+	0.827509i	$0.689757\pi$
$360$		0			0
$361$	−333.885			−0.924890
$362$		0			0
$363$	−578.524			−1.59373
$364$		0			0
$365$	−363.107	+	205.295i	−0.994814	+	0.562453i
$366$		0			0
$367$	88.9062			0.242251			0.121126	−	0.992637i	$-0.461350\pi$
$367$	88.9062			0.242251			0.121126	+	0.992637i	$0.461350\pi$
$368$		0			0
$369$	−209.941			−0.568946
$370$		0			0
$371$			306.334i			0.825699i
$372$		0			0
$373$		−	83.6542i		−	0.224274i	−0.993693	−	0.112137i	$-0.964230\pi$
$373$		−	83.6542i		−	0.224274i	0.993693	−	0.112137i	$-0.0357696\pi$
$374$		0			0
$375$	598.533	+	16.2012i	1.59609	+	0.0432031i
$376$		0			0
$377$			171.373i			0.454570i
$378$		0			0
$379$			135.135i			0.356556i	0.983980	+	0.178278i	$0.0570526\pi$
$379$			135.135i			0.356556i	−0.983980	+	0.178278i	$0.942947\pi$
$380$		0			0
$381$	684.791			1.79735
$382$		0			0
$383$	498.526			1.30164			0.650818	−	0.759234i	$-0.274426\pi$
$383$	498.526			1.30164			0.650818	+	0.759234i	$0.274426\pi$
$384$		0			0
$385$	15.7771	−	8.92013i	0.0409794	−	0.0231692i
$386$		0			0
$387$	129.864			0.335567
$388$		0			0
$389$	−308.420			−0.792854			−0.396427	−	0.918066i	$-0.629750\pi$
$389$	−308.420			−0.792854			−0.396427	+	0.918066i	$0.629750\pi$
$390$		0			0
$391$		−	22.0104i		−	0.0562926i
$392$		0			0
$393$		−	527.562i		−	1.34240i
$394$		0			0
$395$	438.546	−	247.947i	1.11024	−	0.627715i
$396$		0			0
$397$			297.682i			0.749828i	0.927060	+	0.374914i	$0.122328\pi$
$397$			297.682i			0.749828i	−0.927060	+	0.374914i	$0.877672\pi$
$398$		0			0
$399$			969.426i			2.42964i
$400$		0			0
$401$	148.663			0.370729			0.185365	−	0.982670i	$-0.440653\pi$
$401$	148.663			0.370729			0.185365	+	0.982670i	$0.440653\pi$
$402$		0			0
$403$	−90.4616			−0.224470
$404$		0			0
$405$	−29.6672	−	52.4726i	−0.0732524	−	0.129562i
$406$		0			0
$407$	−24.4306			−0.0600261
$408$		0			0
$409$	−442.387			−1.08163			−0.540815	−	0.841141i	$-0.681884\pi$
$409$	−442.387			−1.08163			−0.540815	+	0.841141i	$0.681884\pi$
$410$		0			0
$411$			1079.66i			2.62691i
$412$		0			0
$413$		−	592.654i		−	1.43500i
$414$		0			0
$415$	0.656893	+	1.16185i	0.00158287	+	0.00279964i
$416$		0			0
$417$			466.626i			1.11901i
$418$		0			0
$419$			536.184i			1.27968i	0.768510	+	0.639838i	$0.220999\pi$
$419$			536.184i			1.27968i	−0.768510	+	0.639838i	$0.779001\pi$
$420$		0			0
$421$	−514.582			−1.22228			−0.611142	−	0.791521i	$-0.709290\pi$
$421$	−514.582			−1.22228			−0.611142	+	0.791521i	$0.709290\pi$
$422$		0			0
$423$	122.183			0.288850
$424$		0			0
$425$	−48.4458	−	29.1480i	−0.113990	−	0.0685834i
$426$		0			0
$427$	−113.359			−0.265477
$428$		0			0
$429$	9.29480			0.0216662
$430$		0			0
$431$			86.8151i			0.201427i	0.994915	+	0.100714i	$0.0321126\pi$
$431$			86.8151i			0.201427i	−0.994915	+	0.100714i	$0.967887\pi$
$432$		0			0
$433$		−	494.017i		−	1.14092i	−0.821327	−	0.570458i	$-0.806766\pi$
$433$		−	494.017i		−	1.14092i	0.821327	−	0.570458i	$-0.193234\pi$
$434$		0			0
$435$	−491.503	−	869.325i	−1.12989	−	1.99845i
$436$		0			0
$437$		−	256.555i		−	0.587082i
$438$		0			0
$439$		−	524.700i		−	1.19522i	−0.801789	−	0.597608i	$-0.796118\pi$
$439$		−	524.700i		−	1.19522i	0.801789	−	0.597608i	$-0.203882\pi$
$440$		0			0
$441$	138.666			0.314435
$442$		0			0
$443$	496.575			1.12094			0.560469	−	0.828176i	$-0.310621\pi$
$443$	496.575			1.12094			0.560469	+	0.828176i	$0.310621\pi$
$444$		0			0
$445$	−210.535	−	372.375i	−0.473112	−	0.836797i
$446$		0			0
$447$	916.795			2.05099
$448$		0			0
$449$	−332.158			−0.739773			−0.369886	−	0.929077i	$-0.620603\pi$
$449$	−332.158			−0.739773			−0.369886	+	0.929077i	$0.620603\pi$
$450$		0			0
$451$		−	7.10835i		−	0.0157613i
$452$		0			0
$453$		−	327.421i		−	0.722785i
$454$		0			0
$455$	137.340	−	77.6497i	0.301845	−	0.170659i
$456$		0			0
$457$		−	252.788i		−	0.553147i	−0.960993	−	0.276574i	$-0.910801\pi$
$457$		−	252.788i		−	0.553147i	0.960993	−	0.276574i	$-0.0891990\pi$
$458$		0			0
$459$		−	53.5604i		−	0.116689i
$460$		0			0
$461$	−832.716			−1.80633			−0.903163	−	0.429299i	$-0.858761\pi$
$461$	−832.716			−1.80633			−0.903163	+	0.429299i	$0.858761\pi$
$462$		0			0
$463$	−854.205			−1.84494			−0.922468	−	0.386074i	$-0.873831\pi$
$463$	−854.205			−1.84494			−0.922468	+	0.386074i	$0.873831\pi$
$464$		0			0
$465$	458.885	−	259.447i	0.986850	−	0.557950i
$466$		0			0
$467$	−102.726			−0.219970			−0.109985	−	0.993933i	$-0.535080\pi$
$467$	−102.726			−0.219970			−0.109985	+	0.993933i	$0.535080\pi$
$468$		0			0
$469$	−582.259			−1.24149
$470$		0			0
$471$			743.577i			1.57872i
$472$		0			0
$473$			4.39705i			0.00929608i
$474$		0			0
$475$	−564.688	−	339.751i	−1.18882	−	0.715265i
$476$		0			0
$477$		−	556.379i		−	1.16641i
$478$		0			0
$479$		−	268.772i		−	0.561111i	−0.959838	−	0.280556i	$-0.909481\pi$
$479$		−	268.772i		−	0.561111i	0.959838	−	0.280556i	$-0.0905187\pi$
$480$		0			0
$481$	−212.669			−0.442139
$482$		0			0
$483$	−357.917			−0.741028
$484$		0			0
$485$	431.462	−	243.942i	0.889612	−	0.502973i
$486$		0			0
$487$	−504.065			−1.03504			−0.517520	−	0.855671i	$-0.673145\pi$
$487$	−504.065			−1.03504			−0.517520	+	0.855671i	$0.673145\pi$
$488$		0			0
$489$	433.384			0.886266
$490$		0			0
$491$		−	130.020i		−	0.264807i	−0.991196	−	0.132403i	$-0.957731\pi$
$491$		−	130.020i		−	0.264807i	0.991196	−	0.132403i	$-0.0422694\pi$
$492$		0			0
$493$			94.2997i			0.191277i
$494$		0			0
$495$	−28.6551	+	16.2012i	−0.0578891	+	0.0327296i
$496$		0			0
$497$			622.421i			1.25236i
$498$		0			0
$499$			664.184i			1.33103i	0.746384	+	0.665515i	$0.231788\pi$
$499$			664.184i			1.33103i	−0.746384	+	0.665515i	$0.768212\pi$
$500$		0			0
$501$	−482.728			−0.963529
$502$		0			0
$503$	550.015			1.09347			0.546735	−	0.837306i	$-0.315871\pi$
$503$	550.015			1.09347			0.546735	+	0.837306i	$0.315871\pi$
$504$		0			0
$505$	−259.384	−	458.774i	−0.513631	−	0.908463i
$506$		0			0
$507$	−728.602			−1.43708
$508$		0			0
$509$	−349.843			−0.687314			−0.343657	−	0.939095i	$-0.611666\pi$
$509$	−349.843			−0.687314			−0.343657	+	0.939095i	$0.611666\pi$
$510$		0			0
$511$		−	640.496i		−	1.25342i
$512$		0			0
$513$		−	624.304i		−	1.21697i
$514$		0			0
$515$	−250.668	−	443.358i	−0.486733	−	0.860888i
$516$		0			0
$517$			4.13698i			0.00800189i
$518$		0			0
$519$		−	261.801i		−	0.504434i
$520$		0			0
$521$	−29.7771			−0.0571537			−0.0285769	−	0.999592i	$-0.509098\pi$
$521$	−29.7771			−0.0571537			−0.0285769	+	0.999592i	$0.509098\pi$
$522$		0			0
$523$	−603.023			−1.15301			−0.576504	−	0.817094i	$-0.695584\pi$
$523$	−603.023			−1.15301			−0.576504	+	0.817094i	$0.695584\pi$
$524$		0			0
$525$	−473.982	+	787.789i	−0.902823	+	1.50055i
$526$		0			0
$527$	−49.7774			−0.0944543
$528$		0			0
$529$	−434.279			−0.820943
$530$		0			0
$531$			1076.41i			2.02713i
$532$		0			0
$533$		−	61.8782i		−	0.116094i
$534$		0			0
$535$	−81.1985	−	143.616i	−0.151773	−	0.268442i
$536$		0			0
$537$			511.605i			0.952710i
$538$		0			0
$539$			4.69505i			0.00871066i
$540$		0			0
$541$	−163.368			−0.301974			−0.150987	−	0.988536i	$-0.548245\pi$
$541$	−163.368			−0.301974			−0.150987	+	0.988536i	$0.548245\pi$
$542$		0			0
$543$	693.490			1.27714
$544$		0			0
$545$	−446.774	−	790.212i	−0.819769	−	1.44993i
$546$		0			0
$547$	524.218			0.958350			0.479175	−	0.877719i	$-0.340936\pi$
$547$	524.218			0.958350			0.479175	+	0.877719i	$0.340936\pi$
$548$		0			0
$549$	205.888			0.375023
$550$		0			0
$551$			1099.16i			1.99485i
$552$		0			0
$553$			773.566i			1.39885i
$554$		0			0
$555$	1078.81	−	609.941i	1.94380	−	1.09899i
$556$		0			0
$557$		−	589.515i		−	1.05837i	−0.848505	−	0.529187i	$-0.822497\pi$
$557$		−	589.515i		−	1.05837i	0.848505	−	0.529187i	$-0.177503\pi$
$558$		0			0
$559$			38.2763i			0.0684728i
$560$		0			0
$561$	5.11456			0.00911687
$562$		0			0
$563$	557.763			0.990698			0.495349	−	0.868694i	$-0.335040\pi$
$563$	557.763			0.990698			0.495349	+	0.868694i	$0.335040\pi$
$564$		0			0
$565$	−333.577	+	188.599i	−0.590402	+	0.333804i
$566$		0			0
$567$	92.5581			0.163242
$568$		0			0
$569$	542.715			0.953805			0.476903	−	0.878956i	$-0.341759\pi$
$569$	542.715			0.953805			0.476903	+	0.878956i	$0.341759\pi$
$570$		0			0
$571$			476.695i			0.834842i	0.908713	+	0.417421i	$0.137066\pi$
$571$			476.695i			0.834842i	−0.908713	+	0.417421i	$0.862934\pi$
$572$		0			0
$573$		−	1009.78i		−	1.76227i
$574$		0			0
$575$	125.438	−	208.486i	0.218152	−	0.362584i
$576$		0			0
$577$			865.659i			1.50027i	0.661282	+	0.750137i	$0.270013\pi$
$577$			865.659i			1.50027i	−0.661282	+	0.750137i	$0.729987\pi$
$578$		0			0
$579$		−	518.164i		−	0.894929i
$580$		0			0
$581$	−2.04943			−0.00352741
$582$		0			0
$583$	18.8383			0.0323127
$584$		0			0
$585$	−249.443	+	141.031i	−0.426398	+	0.241079i
$586$		0			0
$587$	567.091			0.966083			0.483041	−	0.875597i	$-0.339532\pi$
$587$	567.091			0.966083			0.483041	+	0.875597i	$0.339532\pi$
$588$		0			0
$589$	−580.209			−0.985075
$590$		0			0
$591$			1476.76i			2.49875i
$592$		0			0
$593$			421.774i			0.711254i	0.934628	+	0.355627i	$0.115733\pi$
$593$			421.774i			0.711254i	−0.934628	+	0.355627i	$0.884267\pi$
$594$		0			0
$595$	75.5725	−	42.7276i	0.127013	−	0.0718110i
$596$		0			0
$597$			1187.25i			1.98869i
$598$		0			0
$599$			120.444i			0.201075i	0.994933	+	0.100538i	$0.0320563\pi$
$599$			120.444i			0.201075i	−0.994933	+	0.100538i	$0.967944\pi$
$600$		0			0
$601$	−340.158			−0.565986			−0.282993	−	0.959122i	$-0.591327\pi$
$601$	−340.158			−0.565986			−0.282993	+	0.959122i	$0.591327\pi$
$602$		0			0
$603$	1057.53			1.75377
$604$		0			0
$605$	297.213	+	525.682i	0.491261	+	0.868897i
$606$		0			0
$607$	115.621			0.190479			0.0952396	−	0.995454i	$-0.469638\pi$
$607$	115.621			0.190479			0.0952396	+	0.995454i	$0.469638\pi$
$608$		0			0
$609$	1533.43			2.51795
$610$		0			0
$611$			36.0124i			0.0589401i
$612$		0			0
$613$		−	1095.63i		−	1.78733i	−0.448738	−	0.893663i	$-0.648126\pi$
$613$		−	1095.63i		−	1.78733i	0.448738	−	0.893663i	$-0.351874\pi$
$614$		0			0
$615$	177.469	+	313.890i	0.288567	+	0.510391i
$616$		0			0
$617$			716.775i			1.16171i	0.814007	+	0.580855i	$0.197282\pi$
$617$			716.775i			1.16171i	−0.814007	+	0.580855i	$0.802718\pi$
$618$		0			0
$619$		−	118.584i		−	0.191573i	−0.995402	−	0.0957864i	$-0.969463\pi$
$619$		−	118.584i		−	0.191573i	0.995402	−	0.0957864i	$-0.0305366\pi$
$620$		0			0
$621$	230.496			0.371169
$622$		0			0
$623$	656.844			1.05432
$624$		0			0
$625$	−292.771	−	552.187i	−0.468433	−	0.883499i
$626$		0			0
$627$	59.6157			0.0950809
$628$		0			0
$629$	−117.023			−0.186046
$630$		0			0
$631$		−	1168.62i		−	1.85201i	−0.377515	−	0.926004i	$-0.623221\pi$
$631$		−	1168.62i		−	1.85201i	0.377515	−	0.926004i	$-0.376779\pi$
$632$		0			0
$633$		−	1104.50i		−	1.74486i
$634$		0			0
$635$	−351.807	−	622.243i	−0.554026	−	0.979910i
$636$		0			0
$637$			40.8704i			0.0641608i
$638$		0			0
$639$		−	1130.47i		−	1.76912i
$640$		0			0
$641$	−244.158			−0.380902			−0.190451	−	0.981697i	$-0.560995\pi$
$641$	−244.158			−0.380902			−0.190451	+	0.981697i	$0.560995\pi$
$642$		0			0
$643$	−868.847			−1.35124			−0.675620	−	0.737250i	$-0.736124\pi$
$643$	−868.847			−1.35124			−0.675620	+	0.737250i	$0.736124\pi$
$644$		0			0
$645$	−109.778	−	194.165i	−0.170198	−	0.301031i
$646$		0			0
$647$	35.1333			0.0543019			0.0271510	−	0.999631i	$-0.491357\pi$
$647$	35.1333			0.0543019			0.0271510	+	0.999631i	$0.491357\pi$
$648$		0			0
$649$	−36.4458			−0.0561569
$650$		0			0
$651$			809.443i			1.24338i
$652$		0			0
$653$			580.068i			0.888313i	0.895949	+	0.444157i	$0.146497\pi$
$653$			580.068i			0.888313i	−0.895949	+	0.444157i	$0.853503\pi$
$654$		0			0
$655$	−479.375	+	271.031i	−0.731870	+	0.413788i
$656$		0			0
$657$			1163.30i			1.77062i
$658$		0			0
$659$		−	992.788i		−	1.50651i	−0.657731	−	0.753253i	$-0.728483\pi$
$659$		−	992.788i		−	1.50651i	0.657731	−	0.753253i	$-0.271517\pi$
$660$		0			0
$661$	719.647			1.08872			0.544362	−	0.838850i	$-0.316772\pi$
$661$	719.647			1.08872			0.544362	+	0.838850i	$0.316772\pi$
$662$		0			0
$663$	44.5223			0.0671528
$664$		0			0
$665$	880.879	−	498.036i	1.32463	−	0.748926i
$666$		0			0
$667$	−405.817			−0.608421
$668$		0			0
$669$	851.580			1.27291
$670$		0			0
$671$			6.97110i			0.0103891i
$672$		0			0
$673$		−	189.591i		−	0.281711i	−0.990030	−	0.140855i	$-0.955015\pi$
$673$		−	189.591i		−	0.281711i	0.990030	−	0.140855i	$-0.0449852\pi$
$674$		0			0
$675$	305.241	−	507.331i	0.452209	−	0.751602i
$676$		0			0
$677$		−	327.192i		−	0.483297i	−0.970364	−	0.241649i	$-0.922312\pi$
$677$		−	327.192i		−	0.483297i	0.970364	−	0.241649i	$-0.0776882\pi$
$678$		0			0
$679$			761.070i			1.12087i
$680$		0			0
$681$	973.817			1.42998
$682$		0			0
$683$	−673.539			−0.986148			−0.493074	−	0.869987i	$-0.664127\pi$
$683$	−673.539			−0.986148			−0.493074	+	0.869987i	$0.664127\pi$
$684$		0			0
$685$	981.044	−	554.668i	1.43218	−	0.809734i
$686$		0			0
$687$	−388.329			−0.565254
$688$		0			0
$689$	163.988			0.238008
$690$		0			0
$691$			237.036i			0.343033i	0.985181	+	0.171516i	$0.0548666\pi$
$691$			237.036i			0.343033i	−0.985181	+	0.171516i	$0.945133\pi$
$692$		0			0
$693$		−	50.5456i		−	0.0729374i
$694$		0			0
$695$	424.005	−	239.726i	0.610079	−	0.344930i
$696$		0			0
$697$		−	34.0491i		−	0.0488510i
$698$		0			0
$699$		−	269.613i		−	0.385712i
$700$		0			0
$701$	476.306			0.679466			0.339733	−	0.940522i	$-0.389663\pi$
$701$	476.306			0.679466			0.339733	+	0.940522i	$0.389663\pi$
$702$		0			0
$703$	−1364.03			−1.94030
$704$		0			0
$705$	−103.285	−	182.681i	−0.146503	−	0.259121i
$706$		0			0
$707$	809.246			1.14462
$708$		0			0
$709$	964.778			1.36076			0.680380	−	0.732860i	$-0.261815\pi$
$709$	964.778			1.36076			0.680380	+	0.732860i	$0.261815\pi$
$710$		0			0
$711$		−	1404.99i		−	1.97607i
$712$		0			0
$713$		−	214.216i		−	0.300443i
$714$		0			0
$715$	−4.77514	−	8.44582i	−0.00667852	−	0.0118123i
$716$		0			0
$717$			1725.53i			2.40660i
$718$		0			0
$719$			899.974i			1.25170i	0.779943	+	0.625851i	$0.215248\pi$
$719$			899.974i			1.25170i	−0.779943	+	0.625851i	$0.784752\pi$
$720$		0			0
$721$	782.053			1.08468
$722$		0			0
$723$	946.006			1.30845
$724$		0			0
$725$	−537.415	+	893.220i	−0.741262	+	1.23203i
$726$		0			0
$727$	−908.888			−1.25019			−0.625095	−	0.780549i	$-0.714940\pi$
$727$	−908.888			−1.25019			−0.625095	+	0.780549i	$0.714940\pi$
$728$		0			0
$729$	−1189.09			−1.63113
$730$		0			0
$731$			21.0619i			0.0288125i
$732$		0			0
$733$			433.283i			0.591109i	0.955326	+	0.295554i	$0.0955043\pi$
$733$			433.283i			0.591109i	−0.955326	+	0.295554i	$0.904496\pi$
$734$		0			0
$735$	−117.218	−	207.324i	−0.159480	−	0.282073i
$736$		0			0
$737$			35.8065i			0.0485842i
$738$		0			0
$739$		−	886.138i		−	1.19910i	−0.800336	−	0.599552i	$-0.795345\pi$
$739$		−	886.138i		−	1.19910i	0.800336	−	0.599552i	$-0.204655\pi$
$740$		0			0
$741$	518.955			0.700344
$742$		0			0
$743$	−895.305			−1.20499			−0.602493	−	0.798124i	$-0.705826\pi$
$743$	−895.305			−1.20499			−0.602493	+	0.798124i	$0.705826\pi$
$744$		0			0
$745$	−470.997	−	833.055i	−0.632211	−	1.11820i
$746$		0			0
$747$	3.72226			0.00498295
$748$		0			0
$749$	253.329			0.338223
$750$		0			0
$751$		−	1297.13i		−	1.72720i	−0.504176	−	0.863601i	$-0.668204\pi$
$751$		−	1297.13i		−	1.72720i	0.504176	−	0.863601i	$-0.331796\pi$
$752$		0			0
$753$		−	854.351i		−	1.13460i
$754$		0			0
$755$	−297.515	+	168.210i	−0.394060	+	0.222795i
$756$		0			0
$757$			421.694i			0.557060i	0.960428	+	0.278530i	$0.0898471\pi$
$757$			421.694i			0.557060i	−0.960428	+	0.278530i	$0.910153\pi$
$758$		0			0
$759$			22.0104i			0.0289992i
$760$		0			0
$761$	−1415.76			−1.86039			−0.930196	−	0.367063i	$-0.880363\pi$
$761$	−1415.76			−1.86039			−0.930196	+	0.367063i	$0.880363\pi$
$762$		0			0
$763$	1393.88			1.82684
$764$		0			0
$765$	−137.258	+	77.6038i	−0.179423	+	0.101443i
$766$		0			0
$767$	−317.261			−0.413639
$768$		0			0
$769$	−414.210			−0.538635			−0.269318	−	0.963051i	$-0.586798\pi$
$769$	−414.210			−0.538635			−0.269318	+	0.963051i	$0.586798\pi$
$770$		0			0
$771$			2170.76i			2.81551i
$772$		0			0
$773$			727.056i			0.940564i	0.882516	+	0.470282i	$0.155848\pi$
$773$			727.056i			0.940564i	−0.882516	+	0.470282i	$0.844152\pi$
$774$		0			0
$775$	−471.498	−	283.682i	−0.608385	−	0.366042i
$776$		0			0
$777$			1902.94i			2.44909i
$778$		0			0
$779$		−	396.879i		−	0.509473i
$780$		0			0
$781$	38.2763			0.0490094
$782$		0			0
$783$	−987.519			−1.26120
$784$		0			0
$785$	675.659	−	382.008i	0.860713	−	0.486634i
$786$		0			0
$787$	−1030.11			−1.30890			−0.654451	−	0.756104i	$-0.727100\pi$
$787$	−1030.11			−1.30890			−0.654451	+	0.756104i	$0.727100\pi$
$788$		0			0
$789$	−101.032			−0.128050
$790$		0			0
$791$		−	588.407i		−	0.743878i
$792$		0			0
$793$			60.6835i			0.0765239i
$794$		0			0
$795$	−831.861	+	470.322i	−1.04637	+	0.591600i
$796$		0			0
$797$		−	371.202i		−	0.465750i	−0.972507	−	0.232875i	$-0.925187\pi$
$797$		−	371.202i		−	0.465750i	0.972507	−	0.232875i	$-0.0748132\pi$
$798$		0			0
$799$			19.8162i			0.0248013i
$800$		0			0
$801$	−1192.99			−1.48938
$802$		0			0
$803$	−39.3879			−0.0490509
$804$		0			0
$805$	183.877	+	325.225i	0.228419	+	0.404006i
$806$		0			0
$807$	−442.668			−0.548536
$808$		0			0
$809$	−77.7771			−0.0961398			−0.0480699	−	0.998844i	$-0.515307\pi$
$809$	−77.7771			−0.0961398			−0.0480699	+	0.998844i	$0.515307\pi$
$810$		0			0
$811$			1407.90i			1.73600i	0.496564	+	0.868000i	$0.334595\pi$
$811$			1407.90i			1.73600i	−0.496564	+	0.868000i	$0.665405\pi$
$812$		0			0
$813$		−	904.350i		−	1.11236i
$814$		0			0
$815$	−222.648	−	393.799i	−0.273188	−	0.483189i
$816$		0			0
$817$			245.500i			0.300489i
$818$		0			0
$819$		−	440.000i		−	0.537241i
$820$		0			0
$821$	820.275			0.999116			0.499558	−	0.866280i	$-0.333496\pi$
$821$	820.275			0.999116			0.499558	+	0.866280i	$0.333496\pi$
$822$		0			0
$823$	−1385.41			−1.68336			−0.841681	−	0.539976i	$-0.818433\pi$
$823$	−1385.41			−1.68336			−0.841681	+	0.539976i	$0.818433\pi$
$824$		0			0
$825$	48.4458	+	29.1480i	0.0587222	+	0.0353309i
$826$		0			0
$827$	258.898			0.313057			0.156529	−	0.987673i	$-0.449970\pi$
$827$	258.898			0.313057			0.156529	+	0.987673i	$0.449970\pi$
$828$		0			0
$829$	1299.31			1.56732			0.783660	−	0.621190i	$-0.213351\pi$
$829$	1299.31			1.56732			0.783660	+	0.621190i	$0.213351\pi$
$830$		0			0
$831$		−	335.901i		−	0.404213i
$832$		0			0
$833$			22.4894i			0.0269980i
$834$		0			0
$835$	247.998	+	438.636i	0.297004	+	0.525313i
$836$		0			0
$837$		−	521.276i		−	0.622791i
$838$		0			0
$839$		−	804.961i		−	0.959429i	−0.877425	−	0.479715i	$-0.840740\pi$
$839$		−	804.961i		−	0.959429i	0.877425	−	0.479715i	$-0.159260\pi$
$840$		0			0
$841$	897.650			1.06736
$842$		0			0
$843$	−2425.04			−2.87668
$844$		0			0
$845$	374.314	+	662.052i	0.442975	+	0.783493i
$846$		0			0
$847$	−927.268			−1.09477
$848$		0			0
$849$	−2401.48			−2.82859
$850$		0			0
$851$		−	503.607i		−	0.591782i
$852$		0			0
$853$			470.811i			0.551948i	0.961165	+	0.275974i	$0.0890003\pi$
$853$			470.811i			0.551948i	−0.961165	+	0.275974i	$0.911000\pi$
$854$		0			0
$855$	−1599.89	+	904.556i	−1.87122	+	1.05796i
$856$		0			0
$857$			266.105i			0.310508i	0.987875	+	0.155254i	$0.0496196\pi$
$857$			266.105i			0.310508i	−0.987875	+	0.155254i	$0.950380\pi$
$858$		0			0
$859$		−	1442.02i		−	1.67872i	−0.543576	−	0.839360i	$-0.682930\pi$
$859$		−	1442.02i		−	1.67872i	0.543576	−	0.839360i	$-0.317070\pi$
$860$		0			0
$861$	−553.681			−0.643067
$862$		0			0
$863$	364.495			0.422358			0.211179	−	0.977447i	$-0.432270\pi$
$863$	364.495			0.422358			0.211179	+	0.977447i	$0.432270\pi$
$864$		0			0
$865$	−237.889	+	134.499i	−0.275016	+	0.155490i
$866$		0			0
$867$	−1359.82			−1.56842
$868$		0			0
$869$	47.5711			0.0547424
$870$		0			0
$871$			311.696i			0.357860i
$872$		0			0
$873$		−	1382.29i		−	1.58338i
$874$		0			0
$875$	959.338	+	25.9675i	1.09639	+	0.0296771i
$876$		0			0
$877$		−	77.4289i		−	0.0882883i	−0.999025	−	0.0441442i	$-0.985944\pi$
$877$		−	77.4289i		−	0.0882883i	0.999025	−	0.0441442i	$-0.0140561\pi$
$878$		0			0
$879$		−	1552.09i		−	1.76574i
$880$		0			0
$881$	724.932			0.822851			0.411426	−	0.911443i	$-0.365031\pi$
$881$	724.932			0.822851			0.411426	+	0.911443i	$0.365031\pi$
$882$		0			0
$883$	−493.342			−0.558711			−0.279356	−	0.960188i	$-0.590121\pi$
$883$	−493.342			−0.558711			−0.279356	+	0.960188i	$0.590121\pi$
$884$		0			0
$885$	1609.37	−	909.915i	1.81850	−	1.02815i
$886$		0			0
$887$	1514.47			1.70741			0.853705	−	0.520758i	$-0.174351\pi$
$887$	1514.47			1.70741			0.853705	+	0.520758i	$0.174351\pi$
$888$		0			0
$889$	1097.59			1.23464
$890$		0			0
$891$		−	5.69194i		−	0.00638826i
$892$		0			0
$893$			230.979i			0.258655i
$894$		0			0
$895$	464.876	−	262.834i	0.519414	−	0.293669i
$896$		0			0
$897$			191.601i			0.213602i
$898$		0			0
$899$			917.771i			1.02088i
$900$		0			0
$901$	90.2359			0.100151
$902$		0			0
$903$	342.493			0.379284
$904$		0			0
$905$	−356.276	−	630.147i	−0.393675	−	0.696295i
$906$		0			0
$907$	851.570			0.938887			0.469443	−	0.882963i	$-0.344455\pi$
$907$	851.570			0.938887			0.469443	+	0.882963i	$0.344455\pi$
$908$		0			0
$909$	−1469.79			−1.61693
$910$		0			0
$911$		−	517.728i		−	0.568308i	−0.958779	−	0.284154i	$-0.908287\pi$
$911$		−	517.728i		−	0.568308i	0.958779	−	0.284154i	$-0.0917127\pi$
$912$		0			0
$913$			0.126031i			0.000138041i
$914$		0			0
$915$	−174.042	−	307.830i	−0.190210	−	0.336426i
$916$		0			0
$917$		−	845.585i		−	0.922121i
$918$		0			0
$919$		−	657.730i		−	0.715701i	−0.933779	−	0.357851i	$-0.883510\pi$
$919$		−	657.730i		−	0.715701i	0.933779	−	0.357851i	$-0.116490\pi$
$920$		0			0
$921$	47.1672			0.0512130
$922$		0			0
$923$	333.195			0.360992
$924$		0			0
$925$	−1108.46	−	666.917i	−1.19833	−	0.720991i
$926$		0			0
$927$	−1420.40			−1.53226
$928$		0			0
$929$	1206.28			1.29847			0.649237	−	0.760586i	$-0.275088\pi$
$929$	1206.28			1.29847			0.649237	+	0.760586i	$0.275088\pi$
$930$		0			0
$931$			262.138i			0.281566i
$932$		0			0
$933$			960.003i			1.02894i
$934$		0			0
$935$	−2.62757	−	4.64740i	−0.00281024	−	0.00497048i
$936$		0			0
$937$			216.730i			0.231302i	0.993290	+	0.115651i	$0.0368954\pi$
$937$			216.730i			0.231302i	−0.993290	+	0.115651i	$0.963105\pi$
$938$		0			0
$939$		−	1252.98i		−	1.33438i
$940$		0			0
$941$	749.322			0.796304			0.398152	−	0.917320i	$-0.369652\pi$
$941$	749.322			0.796304			0.398152	+	0.917320i	$0.369652\pi$
$942$		0			0
$943$	146.530			0.155387
$944$		0			0
$945$	447.449	+	791.406i	0.473491	+	0.837466i
$946$		0			0
$947$	114.160			0.120549			0.0602743	−	0.998182i	$-0.480802\pi$
$947$	114.160			0.120549			0.0602743	+	0.998182i	$0.480802\pi$
$948$		0			0
$949$	−342.872			−0.361298
$950$		0			0
$951$			1544.02i			1.62358i
$952$		0			0
$953$			820.680i			0.861154i	0.902554	+	0.430577i	$0.141690\pi$
$953$			820.680i			0.861154i	−0.902554	+	0.430577i	$0.858310\pi$
$954$		0			0
$955$	−917.548	+	518.768i	−0.960783	+	0.543212i
$956$		0			0
$957$		−	94.2997i		−	0.0985368i
$958$		0			0
$959$			1730.50i			1.80448i
$960$		0			0
$961$	476.542			0.495881
$962$		0			0
$963$	−460.109			−0.477787
$964$		0			0
$965$	−470.835	+	266.203i	−0.487912	+	0.275858i
$966$		0			0
$967$	−1108.56			−1.14639			−0.573194	−	0.819420i	$-0.694296\pi$
$967$	−1108.56			−1.14639			−0.573194	+	0.819420i	$0.694296\pi$
$968$		0			0
$969$	285.560			0.294696
$970$		0			0
$971$			1372.41i			1.41340i	0.707516	+	0.706698i	$0.249816\pi$
$971$			1372.41i			1.41340i	−0.707516	+	0.706698i	$0.750184\pi$
$972$		0			0
$973$			747.916i			0.768670i
$974$		0			0
$975$	421.721	+	253.733i	0.432534	+	0.260239i
$976$		0			0
$977$		−	927.224i		−	0.949052i	−0.880241	−	0.474526i	$-0.842619\pi$
$977$		−	927.224i		−	0.949052i	0.880241	−	0.474526i	$-0.157381\pi$
$978$		0			0
$979$		−	40.3932i		−	0.0412597i
$980$		0			0
$981$	−2531.63			−2.58066
$982$		0			0
$983$	354.106			0.360230			0.180115	−	0.983646i	$-0.442353\pi$
$983$	354.106			0.360230			0.180115	+	0.983646i	$0.442353\pi$
$984$		0			0
$985$	1341.88	−	758.677i	1.36231	−	0.770230i
$986$		0			0
$987$	322.236			0.326480
$988$		0			0
$989$	−90.6396			−0.0916478
$990$		0			0
$991$			537.545i			0.542427i	0.962519	+	0.271213i	$0.0874249\pi$
$991$			537.545i			0.542427i	−0.962519	+	0.271213i	$0.912575\pi$
$992$		0			0
$993$		−	292.614i		−	0.294677i
$994$		0			0
$995$	1078.81	−	609.941i	1.08423	−	0.613006i
$996$		0			0
$997$			441.759i			0.443088i	0.975150	+	0.221544i	$0.0711096\pi$
$997$			441.759i			0.443088i	−0.975150	+	0.221544i	$0.928890\pi$
$998$		0			0
$999$		−	1225.48i		−	1.22671i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1280.3.h.m.1279.4		16
4.3	odd	2	inner	1280.3.h.m.1279.16		16
5.4	even	2	inner	1280.3.h.m.1279.15		16
8.3	odd	2	inner	1280.3.h.m.1279.1		16
8.5	even	2	inner	1280.3.h.m.1279.13		16
16.3	odd	4		40.3.e.c.19.7	yes	8
16.5	even	4		40.3.e.c.19.1	✓	8
16.11	odd	4		160.3.e.c.79.1		8
16.13	even	4		160.3.e.c.79.2		8
20.19	odd	2	inner	1280.3.h.m.1279.3		16
40.19	odd	2	inner	1280.3.h.m.1279.14		16
40.29	even	2	inner	1280.3.h.m.1279.2		16
48.5	odd	4		360.3.p.g.19.8		8
48.11	even	4		1440.3.p.g.559.7		8
48.29	odd	4		1440.3.p.g.559.2		8
48.35	even	4		360.3.p.g.19.2		8
80.3	even	4		200.3.g.h.51.3		8
80.13	odd	4		800.3.g.h.751.7		8
80.19	odd	4		40.3.e.c.19.2	yes	8
80.27	even	4		800.3.g.h.751.1		8
80.29	even	4		160.3.e.c.79.7		8
80.37	odd	4		200.3.g.h.51.5		8
80.43	even	4		800.3.g.h.751.8		8
80.53	odd	4		200.3.g.h.51.4		8
80.59	odd	4		160.3.e.c.79.8		8
80.67	even	4		200.3.g.h.51.6		8
80.69	even	4		40.3.e.c.19.8	yes	8
80.77	odd	4		800.3.g.h.751.2		8
240.29	odd	4		1440.3.p.g.559.8		8
240.59	even	4		1440.3.p.g.559.1		8
240.149	odd	4		360.3.p.g.19.1		8
240.179	even	4		360.3.p.g.19.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.3.e.c.19.1	✓	8	16.5	even	4
40.3.e.c.19.2	yes	8	80.19	odd	4
40.3.e.c.19.7	yes	8	16.3	odd	4
40.3.e.c.19.8	yes	8	80.69	even	4
160.3.e.c.79.1		8	16.11	odd	4
160.3.e.c.79.2		8	16.13	even	4
160.3.e.c.79.7		8	80.29	even	4
160.3.e.c.79.8		8	80.59	odd	4
200.3.g.h.51.3		8	80.3	even	4
200.3.g.h.51.4		8	80.53	odd	4
200.3.g.h.51.5		8	80.37	odd	4
200.3.g.h.51.6		8	80.67	even	4
360.3.p.g.19.1		8	240.149	odd	4
360.3.p.g.19.2		8	48.35	even	4
360.3.p.g.19.7		8	240.179	even	4
360.3.p.g.19.8		8	48.5	odd	4
800.3.g.h.751.1		8	80.27	even	4
800.3.g.h.751.2		8	80.77	odd	4
800.3.g.h.751.7		8	80.13	odd	4
800.3.g.h.751.8		8	80.43	even	4
1280.3.h.m.1279.1		16	8.3	odd	2	inner
1280.3.h.m.1279.2		16	40.29	even	2	inner
1280.3.h.m.1279.3		16	20.19	odd	2	inner
1280.3.h.m.1279.4		16	1.1	even	1	trivial
1280.3.h.m.1279.13		16	8.5	even	2	inner
1280.3.h.m.1279.14		16	40.19	odd	2	inner
1280.3.h.m.1279.15		16	5.4	even	2	inner
1280.3.h.m.1279.16		16	4.3	odd	2	inner
1440.3.p.g.559.1		8	240.59	even	4
1440.3.p.g.559.2		8	48.29	odd	4
1440.3.p.g.559.7		8	48.11	even	4
1440.3.p.g.559.8		8	240.29	odd	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:	\( N \)	\(=\)	\( 1280 = 2^{8} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1280.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-4.79002 q^{3} +(2.46084 + 4.35250i) q^{5} -7.67752 q^{7} +13.9443 q^{9} +0.472136i q^{11} +4.10995i q^{13} +(-11.7875 - 20.8486i) q^{15} +2.26154i q^{17} +26.3607i q^{19} +36.7754 q^{21} -9.73249 q^{23} +(-12.8885 + 21.4216i) q^{25} -23.6832 q^{27} +41.6971 q^{29} +22.0104i q^{31} -2.26154i q^{33} +(-18.8931 - 33.4164i) q^{35} +51.7449i q^{37} -19.6867i q^{39} -15.0557 q^{41} +9.31310 q^{43} +(34.3146 + 60.6925i) q^{45} +8.76226 q^{47} +9.94427 q^{49} -10.8328i q^{51} -39.9002i q^{53} +(-2.05497 + 1.16185i) q^{55} -126.268i q^{57} +77.1935i q^{59} +14.7650 q^{61} -107.057 q^{63} +(-17.8885 + 10.1139i) q^{65} +75.8395 q^{67} +46.6188 q^{69} -81.0705i q^{71} +83.4249i q^{73} +(61.7364 - 102.610i) q^{75} -3.62483i q^{77} -100.757i q^{79} -12.0557 q^{81} +0.266939 q^{83} +(-9.84336 + 5.56528i) q^{85} -199.730 q^{87} -85.5542 q^{89} -31.5542i q^{91} -105.430i q^{93} +(-114.735 + 64.8694i) q^{95} -99.1297i q^{97} +6.58359i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 80 q^{9} + 80 q^{25} - 384 q^{41} + 16 q^{49} - 336 q^{81} - 224 q^{89}+O(q^{100}) \) 16 * q + 80 * q^9 + 80 * q^25 - 384 * q^41 + 16 * q^49 - 336 * q^81 - 224 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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