LMFDB - Embedded newform 990.4.c.i.199.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [990,4,Mod(199,990)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(990, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("990.199");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$990 = 2 \cdot 3^{2} \cdot 5 \cdot 11$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	990.c (of order $2$ , degree $1$ , minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$58.4118909057$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{8} + 151x^{6} + 7935x^{4} + 171721x^{2} + 1308736$ x^8 + 151x^6 + 7935x^4 + 171721*x^2 + 1308736
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$2^{3}$
Twist minimal:	no (minimal twist has level 110)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.8
Root			$-4.82127i$ of defining polynomial
Character	$\chi$	$=$	990.199
Dual form			990.4.c.i.199.4

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000i q^{2} -4.00000 q^{4} +(8.37442 + 7.40737i) q^{5} -13.6360i q^{7} -8.00000i q^{8} +(-14.8147 + 16.7488i) q^{10} +11.0000 q^{11} -66.9953i q^{13} +27.2720 q^{14} +16.0000 q^{16} +136.870i q^{17} -126.061 q^{19} +(-33.4977 - 29.6295i) q^{20} +22.0000i q^{22} -62.5641i q^{23} +(15.2617 + 124.065i) q^{25} +133.991 q^{26} +54.5440i q^{28} +144.857 q^{29} +229.288 q^{31} +32.0000i q^{32} -273.740 q^{34} +(101.007 - 114.194i) q^{35} -6.56930i q^{37} -252.123i q^{38} +(59.2590 - 66.9953i) q^{40} +276.684 q^{41} -233.339i q^{43} -44.0000 q^{44} +125.128 q^{46} -381.735i q^{47} +157.059 q^{49} +(-248.130 + 30.5234i) q^{50} +267.981i q^{52} +182.225i q^{53} +(92.1186 + 81.4811i) q^{55} -109.088 q^{56} +289.714i q^{58} +111.466 q^{59} +116.977 q^{61} +458.575i q^{62} -64.0000 q^{64} +(496.259 - 561.047i) q^{65} -480.882i q^{67} -547.480i q^{68} +(228.387 + 202.014i) q^{70} +602.696 q^{71} +622.178i q^{73} +13.1386 q^{74} +504.245 q^{76} -149.996i q^{77} +768.900 q^{79} +(133.991 + 118.518i) q^{80} +553.369i q^{82} +953.137i q^{83} +(-1013.85 + 1146.21i) q^{85} +466.678 q^{86} -88.0000i q^{88} -757.114 q^{89} -913.549 q^{91} +250.256i q^{92} +763.469 q^{94} +(-1055.69 - 933.783i) q^{95} -1118.29i q^{97} +314.118i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$8 q - 32 q^{4} - 16 q^{5} - 28 q^{10} + 88 q^{11} - 44 q^{14} + 128 q^{16} - 302 q^{19} + 64 q^{20} - 162 q^{25} - 256 q^{26} + 58 q^{29} + 1022 q^{31} - 660 q^{34} + 1058 q^{35} + 112 q^{40} - 452 q^{41}+ \cdots - 1268 q^{95}+O(q^{100})$ 8 * q - 32 * q^4 - 16 * q^5 - 28 * q^10 + 88 * q^11 - 44 * q^14 + 128 * q^16 - 302 * q^19 + 64 * q^20 - 162 * q^25 - 256 * q^26 + 58 * q^29 + 1022 * q^31 - 660 * q^34 + 1058 * q^35 + 112 * q^40 - 452 * q^41 - 352 * q^44 + 888 * q^46 + 222 * q^49 - 704 * q^50 - 176 * q^55 + 176 * q^56 - 268 * q^59 - 490 * q^61 - 512 * q^64 + 1408 * q^65 + 880 * q^70 - 2034 * q^71 + 708 * q^74 + 1208 * q^76 + 272 * q^79 - 256 * q^80 + 330 * q^85 + 1568 * q^86 - 5782 * q^89 - 3520 * q^91 - 2536 * q^94 - 1268 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/990\mathbb{Z}\right)^\times$ .

$n$	$397$	$541$	$551$
$\chi(n)$	$-1$	$1$	$1$

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

)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$\alpha_n$			$\theta_n$
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$\alpha_p$			$\theta_p$

$2$			2.00000i			0.707107i
$2$			2.00000i			0.707107i
$3$		0			0
$3$		0			0
$4$	−4.00000			−0.500000
$5$	8.37442	+	7.40737i	0.749031	+	0.662535i
$5$	8.37442	+	7.40737i	0.749031	+	0.662535i
$6$		0			0
$7$		−	13.6360i		−	0.736275i	−0.929771	−	0.368138i	$-0.879995\pi$
$7$		−	13.6360i		−	0.736275i	0.929771	−	0.368138i	$-0.120005\pi$
$8$		−	8.00000i		−	0.353553i
$9$		0			0
$10$	−14.8147	+	16.7488i	−0.468483	+	0.529645i
$11$	11.0000			0.301511
$11$	11.0000			0.301511
$12$		0			0
$13$		−	66.9953i		−	1.42932i	−0.699472	−	0.714660i	$-0.746581\pi$
$13$		−	66.9953i		−	1.42932i	0.699472	−	0.714660i	$-0.253419\pi$
$14$	27.2720			0.520625
$15$		0			0
$16$	16.0000			0.250000
$17$			136.870i			1.95270i	0.216201	+	0.976349i	$0.430633\pi$
$17$			136.870i			1.95270i	−0.216201	+	0.976349i	$0.569367\pi$
$18$		0			0
$19$	−126.061			−1.52213			−0.761064	−	0.648676i	$-0.775323\pi$
$19$	−126.061			−1.52213			−0.761064	+	0.648676i	$0.775323\pi$
$20$	−33.4977	−	29.6295i	−0.374515	−	0.331268i
$21$		0			0
$22$			22.0000i			0.213201i
$23$		−	62.5641i		−	0.567196i	−0.958943	−	0.283598i	$-0.908472\pi$
$23$		−	62.5641i		−	0.567196i	0.958943	−	0.283598i	$-0.0915281\pi$
$24$		0			0
$25$	15.2617	+	124.065i	0.122094	+	0.992519i
$26$	133.991			1.01068
$27$		0			0
$28$			54.5440i			0.368138i
$29$	144.857			0.927562			0.463781	−	0.885950i	$-0.346493\pi$
$29$	144.857			0.927562			0.463781	+	0.885950i	$0.346493\pi$
$30$		0			0
$31$	229.288			1.32843			0.664214	−	0.747543i	$-0.268766\pi$
$31$	229.288			1.32843			0.664214	+	0.747543i	$0.268766\pi$
$32$			32.0000i			0.176777i
$33$		0			0
$34$	−273.740			−1.38077
$35$	101.007	−	114.194i	0.487809	−	0.551493i
$36$		0			0
$37$		−	6.56930i		−	0.0291888i	−0.999893	−	0.0145944i	$-0.995354\pi$
$37$		−	6.56930i		−	0.0291888i	0.999893	−	0.0145944i	$-0.00464571\pi$
$38$		−	252.123i		−	1.07631i
$39$		0			0
$40$	59.2590	−	66.9953i	0.234242	−	0.264822i
$41$	276.684			1.05392			0.526961	−	0.849889i	$-0.323331\pi$
$41$	276.684			1.05392			0.526961	+	0.849889i	$0.323331\pi$
$42$		0			0
$43$		−	233.339i		−	0.827531i	−0.910384	−	0.413765i	$-0.864213\pi$
$43$		−	233.339i		−	0.827531i	0.910384	−	0.413765i	$-0.135787\pi$
$44$	−44.0000			−0.150756
$45$		0			0
$46$	125.128			0.401068
$47$		−	381.735i		−	1.18472i	−0.805674	−	0.592359i	$-0.798197\pi$
$47$		−	381.735i		−	1.18472i	0.805674	−	0.592359i	$-0.201803\pi$
$48$		0			0
$49$	157.059			0.457898
$50$	−248.130	+	30.5234i	−0.701817	+	0.0863332i
$51$		0			0
$52$			267.981i			0.714660i
$53$			182.225i			0.472275i	0.971720	+	0.236137i	$0.0758815\pi$
$53$			182.225i			0.472275i	−0.971720	+	0.236137i	$0.924118\pi$
$54$		0			0
$55$	92.1186	+	81.4811i	0.225841	+	0.199762i
$56$	−109.088			−0.260313
$57$		0			0
$58$			289.714i			0.655885i
$59$	111.466			0.245961			0.122980	−	0.992409i	$-0.460755\pi$
$59$	111.466			0.245961			0.122980	+	0.992409i	$0.460755\pi$
$60$		0			0
$61$	116.977			0.245530			0.122765	−	0.992436i	$-0.460824\pi$
$61$	116.977			0.245530			0.122765	+	0.992436i	$0.460824\pi$
$62$			458.575i			0.939340i
$63$		0			0
$64$	−64.0000			−0.125000
$65$	496.259	−	561.047i	0.946975	−	1.07060i
$66$		0			0
$67$		−	480.882i		−	0.876853i	−0.898767	−	0.438426i	$-0.855536\pi$
$67$		−	480.882i		−	0.876853i	0.898767	−	0.438426i	$-0.144464\pi$
$68$		−	547.480i		−	0.976349i
$69$		0			0
$70$	228.387	+	202.014i	0.389964	+	0.344933i
$71$	602.696			1.00742			0.503710	−	0.863873i	$-0.331968\pi$
$71$	602.696			1.00742			0.503710	+	0.863873i	$0.331968\pi$
$72$		0			0
$73$			622.178i			0.997540i	0.866734	+	0.498770i	$0.166215\pi$
$73$			622.178i			0.997540i	−0.866734	+	0.498770i	$0.833785\pi$
$74$	13.1386			0.0206396
$75$		0			0
$76$	504.245			0.761064
$77$		−	149.996i		−	0.221995i
$78$		0			0
$79$	768.900			1.09504			0.547519	−	0.836793i	$-0.315572\pi$
$79$	768.900			1.09504			0.547519	+	0.836793i	$0.315572\pi$
$80$	133.991	+	118.518i	0.187258	+	0.165634i
$81$		0			0
$82$			553.369i			0.745236i
$83$			953.137i			1.26049i	0.776398	+	0.630243i	$0.217045\pi$
$83$			953.137i			1.26049i	−0.776398	+	0.630243i	$0.782955\pi$
$84$		0			0
$85$	−1013.85	+	1146.21i	−1.29373	+	1.46263i
$86$	466.678			0.585153
$87$		0			0
$88$		−	88.0000i		−	0.106600i
$89$	−757.114			−0.901730			−0.450865	−	0.892592i	$-0.648884\pi$
$89$	−757.114			−0.901730			−0.450865	+	0.892592i	$0.648884\pi$
$90$		0			0
$91$	−913.549			−1.05237
$92$			250.256i			0.283598i
$93$		0			0
$94$	763.469			0.837722
$95$	−1055.69	−	933.783i	−1.14012	−	1.00846i
$96$		0			0
$97$		−	1118.29i		−	1.17057i	−0.810827	−	0.585285i	$-0.800982\pi$
$97$		−	1118.29i		−	1.17057i	0.810827	−	0.585285i	$-0.199018\pi$
$98$			314.118i			0.323783i
$99$		0			0
$100$	−61.0468	−	496.259i	−0.0610468	−	0.496259i
$101$	1053.05			1.03745			0.518723	−	0.854942i	$-0.326408\pi$
$101$	1053.05			1.03745			0.518723	+	0.854942i	$0.326408\pi$
$102$		0			0
$103$		−	728.192i		−	0.696610i	−0.937381	−	0.348305i	$-0.886757\pi$
$103$		−	728.192i		−	0.696610i	0.937381	−	0.348305i	$-0.113243\pi$
$104$	−535.963			−0.505341
$105$		0			0
$106$	−364.450			−0.333949
$107$			988.919i			0.893480i	0.894664	+	0.446740i	$0.147415\pi$
$107$			988.919i			0.893480i	−0.894664	+	0.446740i	$0.852585\pi$
$108$		0			0
$109$	1736.51			1.52594			0.762970	−	0.646434i	$-0.223740\pi$
$109$	1736.51			1.52594			0.762970	+	0.646434i	$0.223740\pi$
$110$	−162.962	+	184.237i	−0.141253	+	0.159694i
$111$		0			0
$112$		−	218.176i		−	0.184069i
$113$		−	1115.15i		−	0.928360i	−0.885741	−	0.464180i	$-0.846349\pi$
$113$		−	1115.15i		−	0.928360i	0.885741	−	0.464180i	$-0.153651\pi$
$114$		0			0
$115$	463.435	−	523.937i	0.375787	−	0.424847i
$116$	−579.428			−0.463781
$117$		0			0
$118$			222.933i			0.173920i
$119$	1866.36			1.43772
$120$		0			0
$121$	121.000			0.0909091
$122$			233.954i			0.173616i
$123$		0			0
$124$	−917.150			−0.664214
$125$	−791.186	+	1152.02i	−0.566127	+	0.824318i
$126$		0			0
$127$			1558.13i			1.08868i	0.838866	+	0.544338i	$0.183219\pi$
$127$			1558.13i			1.08868i	−0.838866	+	0.544338i	$0.816781\pi$
$128$		−	128.000i		−	0.0883883i
$129$		0			0
$130$	1122.09	+	992.519i	0.757032	+	0.669613i
$131$	620.634			0.413932			0.206966	−	0.978348i	$-0.433641\pi$
$131$	620.634			0.413932			0.206966	+	0.978348i	$0.433641\pi$
$132$		0			0
$133$			1718.97i			1.12071i
$134$	961.765			0.620028
$135$		0			0
$136$	1094.96			0.690383
$137$			1138.81i			0.710185i	0.934831	+	0.355093i	$0.115551\pi$
$137$			1138.81i			0.710185i	−0.934831	+	0.355093i	$0.884449\pi$
$138$		0			0
$139$	1363.18			0.831826			0.415913	−	0.909404i	$-0.363462\pi$
$139$	1363.18			0.831826			0.415913	+	0.909404i	$0.363462\pi$
$140$	−404.028	+	456.775i	−0.243904	+	0.275746i
$141$		0			0
$142$			1205.39i			0.712353i
$143$		−	736.949i		−	0.430956i
$144$		0			0
$145$	1213.09	+	1073.01i	0.694772	+	0.614542i
$146$	−1244.36			−0.705367
$147$		0			0
$148$			26.2772i			0.0145944i
$149$	2823.58			1.55246			0.776232	−	0.630448i	$-0.217129\pi$
$149$	2823.58			1.55246			0.776232	+	0.630448i	$0.217129\pi$
$150$		0			0
$151$	975.281			0.525611			0.262806	−	0.964849i	$-0.415352\pi$
$151$	975.281			0.525611			0.262806	+	0.964849i	$0.415352\pi$
$152$			1008.49i			0.538154i
$153$		0			0
$154$	299.992			0.156974
$155$	1920.15	+	1698.42i	0.995033	+	0.880130i
$156$		0			0
$157$			842.402i			0.428223i	0.976809	+	0.214111i	$0.0686856\pi$
$157$			842.402i			0.428223i	−0.976809	+	0.214111i	$0.931314\pi$
$158$			1537.80i			0.774309i
$159$		0			0
$160$	−237.036	+	267.981i	−0.117121	+	0.132411i
$161$	−853.124			−0.417612
$162$		0			0
$163$			376.648i			0.180990i	0.995897	+	0.0904950i	$0.0288449\pi$
$163$			376.648i			0.180990i	−0.995897	+	0.0904950i	$0.971155\pi$
$164$	−1106.74			−0.526961
$165$		0			0
$166$	−1906.27			−0.891298
$167$			1298.93i			0.601882i	0.953643	+	0.300941i	$0.0973008\pi$
$167$			1298.93i			0.601882i	−0.953643	+	0.300941i	$0.902699\pi$
$168$		0			0
$169$	−2291.37			−1.04296
$170$	−2292.41	−	2027.69i	−1.03424	−	0.914806i
$171$		0			0
$172$			933.355i			0.413765i
$173$			2066.43i			0.908138i	0.890967	+	0.454069i	$0.150028\pi$
$173$			2066.43i			0.908138i	−0.890967	+	0.454069i	$0.849972\pi$
$174$		0			0
$175$	1691.75	−	208.109i	0.730767	−	0.0898945i
$176$	176.000			0.0753778
$177$		0			0
$178$		−	1514.23i		−	0.637619i
$179$	−2046.17			−0.854400			−0.427200	−	0.904157i	$-0.640500\pi$
$179$	−2046.17			−0.854400			−0.427200	+	0.904157i	$0.640500\pi$
$180$		0			0
$181$	2561.44			1.05188			0.525941	−	0.850521i	$-0.323713\pi$
$181$	2561.44			1.05188			0.525941	+	0.850521i	$0.323713\pi$
$182$		−	1827.10i		−	0.744140i
$183$		0			0
$184$	−500.512			−0.200534
$185$	48.6613	−	55.0141i	0.0193386	−	0.0218633i
$186$		0			0
$187$			1505.57i			0.588761i
$188$			1526.94i			0.592359i
$189$		0			0
$190$	1867.57	−	2111.38i	0.713092	−	0.806187i
$191$	−3474.08			−1.31610			−0.658051	−	0.752973i	$-0.728619\pi$
$191$	−3474.08			−1.31610			−0.658051	+	0.752973i	$0.728619\pi$
$192$		0			0
$193$		−	1741.67i		−	0.649577i	−0.945787	−	0.324788i	$-0.894707\pi$
$193$		−	1741.67i		−	0.649577i	0.945787	−	0.324788i	$-0.105293\pi$
$194$	2236.58			0.827719
$195$		0			0
$196$	−628.237			−0.228949
$197$			112.400i			0.0406507i	0.999793	+	0.0203253i	$0.00647020\pi$
$197$			112.400i			0.0406507i	−0.999793	+	0.0203253i	$0.993530\pi$
$198$		0			0
$199$	4526.79			1.61254			0.806270	−	0.591548i	$-0.201483\pi$
$199$	4526.79			1.61254			0.806270	+	0.591548i	$0.201483\pi$
$200$	992.519	−	122.094i	0.350908	−	0.0431666i
$201$		0			0
$202$			2106.09i			0.733585i
$203$		−	1975.27i		−	0.682941i
$204$		0			0
$205$	2317.07	+	2049.50i	0.789420	+	0.698261i
$206$	1456.38			0.492578
$207$		0			0
$208$		−	1071.93i		−	0.357330i
$209$	−1386.67			−0.458939
$210$		0			0
$211$	−3743.57			−1.22141			−0.610707	−	0.791857i	$-0.709115\pi$
$211$	−3743.57			−1.22141			−0.610707	+	0.791857i	$0.709115\pi$
$212$		−	728.901i		−	0.236137i
$213$		0			0
$214$	−1977.84			−0.631786
$215$	1728.43	−	1954.08i	0.548269	−	0.619846i
$216$		0			0
$217$		−	3126.57i		−	0.978089i
$218$			3473.02i			1.07900i
$219$		0			0
$220$	−368.474	−	325.924i	−0.112921	−	0.0998810i
$221$	9169.65			2.79103
$222$		0			0
$223$		−	2773.89i		−	0.832975i	−0.909141	−	0.416487i	$-0.863261\pi$
$223$		−	2773.89i		−	0.832975i	0.909141	−	0.416487i	$-0.136739\pi$
$224$	436.352			0.130156
$225$		0			0
$226$	2230.30			0.656450
$227$			175.125i			0.0512046i	0.999672	+	0.0256023i	$0.00815036\pi$
$227$			175.125i			0.0512046i	−0.999672	+	0.0256023i	$0.991850\pi$
$228$		0			0
$229$	−2789.77			−0.805035			−0.402518	−	0.915412i	$-0.631865\pi$
$229$	−2789.77			−0.805035			−0.402518	+	0.915412i	$0.631865\pi$
$230$	1047.87	+	926.870i	0.300412	+	0.265722i
$231$		0			0
$232$		−	1158.86i		−	0.327943i
$233$		−	3107.83i		−	0.873822i	−0.899505	−	0.436911i	$-0.856072\pi$
$233$		−	3107.83i		−	0.873822i	0.899505	−	0.436911i	$-0.143928\pi$
$234$		0			0
$235$	2827.65	−	3196.80i	0.784917	−	0.887389i
$236$	−445.865			−0.122980
$237$		0			0
$238$			3732.72i			1.01662i
$239$	−5263.47			−1.42454			−0.712271	−	0.701904i	$-0.752333\pi$
$239$	−5263.47			−1.42454			−0.712271	+	0.701904i	$0.752333\pi$
$240$		0			0
$241$	−2063.46			−0.551533			−0.275767	−	0.961225i	$-0.588932\pi$
$241$	−2063.46			−0.551533			−0.275767	+	0.961225i	$0.588932\pi$
$242$			242.000i			0.0642824i
$243$		0			0
$244$	−467.908			−0.122765
$245$	1315.28	+	1163.40i	0.342980	+	0.303374i
$246$		0			0
$247$			8445.52i			2.17561i
$248$		−	1834.30i		−	0.469670i
$249$		0			0
$250$	−2304.04	−	1582.37i	−0.582881	−	0.400312i
$251$	−2087.66			−0.524988			−0.262494	−	0.964934i	$-0.584545\pi$
$251$	−2087.66			−0.524988			−0.262494	+	0.964934i	$0.584545\pi$
$252$		0			0
$253$		−	688.205i		−	0.171016i
$254$	−3116.26			−0.769810
$255$		0			0
$256$	256.000			0.0625000
$257$			4901.68i			1.18972i	0.803829	+	0.594861i	$0.202793\pi$
$257$			4901.68i			1.18972i	−0.803829	+	0.594861i	$0.797207\pi$
$258$		0			0
$259$	−89.5791			−0.0214910
$260$	−1985.04	+	2244.19i	−0.473488	+	0.535302i
$261$		0			0
$262$			1241.27i			0.292694i
$263$		−	5075.49i		−	1.18999i	−0.803728	−	0.594997i	$-0.797153\pi$
$263$		−	5075.49i		−	1.18999i	0.803728	−	0.594997i	$-0.202847\pi$
$264$		0			0
$265$	−1349.81	+	1526.03i	−0.312899	+	0.353748i
$266$	−3437.95			−0.792459
$267$		0			0
$268$			1923.53i			0.438426i
$269$	−3906.32			−0.885400			−0.442700	−	0.896670i	$-0.645979\pi$
$269$	−3906.32			−0.885400			−0.442700	+	0.896670i	$0.645979\pi$
$270$		0			0
$271$	2849.44			0.638714			0.319357	−	0.947635i	$-0.396533\pi$
$271$	2849.44			0.638714			0.319357	+	0.947635i	$0.396533\pi$
$272$			2189.92i			0.488174i
$273$		0			0
$274$	−2277.63			−0.502177
$275$	167.879	+	1364.71i	0.0368126	+	0.299256i
$276$		0			0
$277$			5076.18i			1.10108i	0.834810	+	0.550539i	$0.185578\pi$
$277$			5076.18i			1.10108i	−0.834810	+	0.550539i	$0.814422\pi$
$278$			2726.37i			0.588190i
$279$		0			0
$280$	−913.549	−	808.056i	−0.194982	−	0.172466i
$281$	2815.17			0.597647			0.298823	−	0.954308i	$-0.403406\pi$
$281$	2815.17			0.597647			0.298823	+	0.954308i	$0.403406\pi$
$282$		0			0
$283$		−	2420.95i		−	0.508519i	−0.967136	−	0.254259i	$-0.918168\pi$
$283$		−	2420.95i		−	0.508519i	0.967136	−	0.254259i	$-0.0818317\pi$
$284$	−2410.78			−0.503710
$285$		0			0
$286$	1473.90			0.304732
$287$		−	3772.87i		−	0.775978i
$288$		0			0
$289$	−13820.4			−2.81303
$290$	−2146.02	+	2426.19i	−0.434547	+	0.491278i
$291$		0			0
$292$		−	2488.71i		−	0.498770i
$293$			2429.28i			0.484370i	0.970230	+	0.242185i	$0.0778640\pi$
$293$			2429.28i			0.484370i	−0.970230	+	0.242185i	$0.922136\pi$
$294$		0			0
$295$	933.465	+	825.672i	0.184232	+	0.162958i
$296$	−52.5544			−0.0103198
$297$		0			0
$298$			5647.17i			1.09776i
$299$	−4191.50			−0.810705
$300$		0			0
$301$	−3181.81			−0.609291
$302$			1950.56i			0.371663i
$303$		0			0
$304$	−2016.98			−0.380532
$305$	979.613	+	866.491i	0.183910	+	0.162673i
$306$		0			0
$307$			162.937i			0.0302910i	0.999885	+	0.0151455i	$0.00482114\pi$
$307$			162.937i			0.0302910i	−0.999885	+	0.0151455i	$0.995179\pi$
$308$			599.985i			0.110998i
$309$		0			0
$310$	−3396.84	+	3840.30i	−0.622346	+	0.703594i
$311$	6155.28			1.12230			0.561148	−	0.827715i	$-0.310360\pi$
$311$	6155.28			1.12230			0.561148	+	0.827715i	$0.310360\pi$
$312$		0			0
$313$			6165.59i			1.11342i	0.830708	+	0.556709i	$0.187936\pi$
$313$			6165.59i			1.11342i	−0.830708	+	0.556709i	$0.812064\pi$
$314$	−1684.80			−0.302799
$315$		0			0
$316$	−3075.60			−0.547519
$317$			3099.89i			0.549234i	0.961554	+	0.274617i	$0.0885510\pi$
$317$			3099.89i			0.549234i	−0.961554	+	0.274617i	$0.911449\pi$
$318$		0			0
$319$	1593.43			0.279670
$320$	−535.963	−	474.072i	−0.0936288	−	0.0828169i
$321$		0			0
$322$		−	1706.25i		−	0.295297i
$323$		−	17254.0i		−	2.97226i
$324$		0			0
$325$	8311.76	−	1022.46i	1.41863	−	0.174511i
$326$	−753.296			−0.127979
$327$		0			0
$328$		−	2213.47i		−	0.372618i
$329$	−5205.34			−0.872278
$330$		0			0
$331$	−7423.91			−1.23279			−0.616397	−	0.787436i	$-0.711408\pi$
$331$	−7423.91			−1.23279			−0.616397	+	0.787436i	$0.711408\pi$
$332$		−	3812.55i		−	0.630243i
$333$		0			0
$334$	−2597.86			−0.425595
$335$	3562.07	−	4027.11i	0.580946	−	0.656789i
$336$		0			0
$337$		−	1808.87i		−	0.292391i	−0.989256	−	0.146195i	$-0.953297\pi$
$337$		−	1808.87i		−	0.292391i	0.989256	−	0.146195i	$-0.0467028\pi$
$338$		−	4582.75i		−	0.737481i
$339$		0			0
$340$	4055.39	−	4584.83i	0.646866	−	0.731315i
$341$	2522.16			0.400536
$342$		0			0
$343$		−	6818.81i		−	1.07341i
$344$	−1866.71			−0.292576
$345$		0			0
$346$	−4132.86			−0.642150
$347$		−	5850.60i		−	0.905120i	−0.891734	−	0.452560i	$-0.850511\pi$
$347$		−	5850.60i		−	0.905120i	0.891734	−	0.452560i	$-0.149489\pi$
$348$		0			0
$349$	10231.6			1.56930			0.784648	−	0.619941i	$-0.212844\pi$
$349$	10231.6			1.56930			0.784648	+	0.619941i	$0.212844\pi$
$350$	416.218	+	3383.50i	0.0635650	+	0.516730i
$351$		0			0
$352$			352.000i			0.0533002i
$353$			438.848i			0.0661687i	0.999453	+	0.0330843i	$0.0105330\pi$
$353$			438.848i			0.0661687i	−0.999453	+	0.0330843i	$0.989467\pi$
$354$		0			0
$355$	5047.22	+	4464.39i	0.754588	+	0.667451i
$356$	3028.46			0.450865
$357$		0			0
$358$		−	4092.33i		−	0.604152i
$359$	9743.06			1.43236			0.716182	−	0.697913i	$-0.245888\pi$
$359$	9743.06			1.43236			0.716182	+	0.697913i	$0.245888\pi$
$360$		0			0
$361$	9032.45			1.31688
$362$			5122.88i			0.743792i
$363$		0			0
$364$	3654.20			0.526187
$365$	−4608.70	+	5210.38i	−0.660905	+	0.747188i
$366$		0			0
$367$		−	11674.8i		−	1.66054i	−0.557363	−	0.830269i	$-0.688187\pi$
$367$		−	11674.8i		−	1.66054i	0.557363	−	0.830269i	$-0.311813\pi$
$368$		−	1001.02i		−	0.141799i
$369$		0			0
$370$	110.028	+	97.3225i	0.0154597	+	0.0136745i
$371$	2484.82			0.347724
$372$		0			0
$373$			537.719i			0.0746435i	0.999303	+	0.0373218i	$0.0118826\pi$
$373$			537.719i			0.0746435i	−0.999303	+	0.0373218i	$0.988117\pi$
$374$	−3011.14			−0.416317
$375$		0			0
$376$	−3053.88			−0.418861
$377$		−	9704.75i		−	1.32578i
$378$		0			0
$379$	−9268.43			−1.25617			−0.628084	−	0.778146i	$-0.716160\pi$
$379$	−9268.43			−1.25617			−0.628084	+	0.778146i	$0.716160\pi$
$380$	4222.76	+	3735.13i	0.570060	+	0.504232i
$381$		0			0
$382$		−	6948.16i		−	0.930625i
$383$		−	5038.62i		−	0.672223i	−0.941822	−	0.336111i	$-0.890888\pi$
$383$		−	5038.62i		−	0.672223i	0.941822	−	0.336111i	$-0.109112\pi$
$384$		0			0
$385$	1111.08	−	1256.13i	0.147080	−	0.166281i
$386$	3483.35			0.459320
$387$		0			0
$388$			4473.17i			0.585285i
$389$	12545.6			1.63518			0.817590	−	0.575800i	$-0.195309\pi$
$389$	12545.6			1.63518			0.817590	+	0.575800i	$0.195309\pi$
$390$		0			0
$391$	8563.14			1.10756
$392$		−	1256.47i		−	0.161892i
$393$		0			0
$394$	−224.800			−0.0287444
$395$	6439.09	+	5695.53i	0.820217	+	0.725502i
$396$		0			0
$397$		−	9116.35i		−	1.15248i	−0.817279	−	0.576242i	$-0.804518\pi$
$397$		−	9116.35i		−	1.15248i	0.817279	−	0.576242i	$-0.195482\pi$
$398$			9053.58i			1.14024i
$399$		0			0
$400$	244.187	+	1985.04i	0.0305234	+	0.248130i
$401$	3052.85			0.380179			0.190090	−	0.981767i	$-0.439122\pi$
$401$	3052.85			0.380179			0.190090	+	0.981767i	$0.439122\pi$
$402$		0			0
$403$		−	15361.2i		−	1.89875i
$404$	−4212.19			−0.518723
$405$		0			0
$406$	3950.55			0.482912
$407$		−	72.2623i		−	0.00880077i
$408$		0			0
$409$	4417.02			0.534004			0.267002	−	0.963696i	$-0.413967\pi$
$409$	4417.02			0.534004			0.267002	+	0.963696i	$0.413967\pi$
$410$	−4099.01	+	4634.14i	−0.493745	+	0.558205i
$411$		0			0
$412$			2912.77i			0.348305i
$413$		−	1519.96i		−	0.181095i
$414$		0			0
$415$	−7060.24	+	7981.96i	−0.835117	+	0.944143i
$416$	2143.85			0.252670
$417$		0			0
$418$		−	2773.35i		−	0.324519i
$419$	6184.20			0.721045			0.360522	−	0.932751i	$-0.382598\pi$
$419$	6184.20			0.721045			0.360522	+	0.932751i	$0.382598\pi$
$420$		0			0
$421$	−14153.4			−1.63846			−0.819231	−	0.573464i	$-0.805599\pi$
$421$	−14153.4			−1.63846			−0.819231	+	0.573464i	$0.805599\pi$
$422$		−	7487.15i		−	0.863670i
$423$		0			0
$424$	1457.80			0.166974
$425$	−16980.8	+	2088.87i	−1.93809	+	0.238412i
$426$		0			0
$427$		−	1595.10i		−	0.180778i
$428$		−	3955.68i		−	0.446740i
$429$		0			0
$430$	3908.15	+	3456.85i	0.438297	+	0.387684i
$431$	−539.740			−0.0603210			−0.0301605	−	0.999545i	$-0.509602\pi$
$431$	−539.740			−0.0603210			−0.0301605	+	0.999545i	$0.509602\pi$
$432$		0			0
$433$		−	13447.3i		−	1.49246i	−0.665687	−	0.746231i	$-0.731861\pi$
$433$		−	13447.3i		−	1.49246i	0.665687	−	0.746231i	$-0.268139\pi$
$434$	6253.13			0.691613
$435$		0			0
$436$	−6946.04			−0.762970
$437$			7886.91i			0.863345i
$438$		0			0
$439$	−9970.49			−1.08398			−0.541988	−	0.840386i	$-0.682328\pi$
$439$	−9970.49			−1.08398			−0.541988	+	0.840386i	$0.682328\pi$
$440$	651.849	−	736.949i	0.0706265	−	0.0798469i
$441$		0			0
$442$			18339.3i			1.97356i
$443$		−	11600.7i		−	1.24416i	−0.782952	−	0.622082i	$-0.786287\pi$
$443$		−	11600.7i		−	1.24416i	0.782952	−	0.622082i	$-0.213713\pi$
$444$		0			0
$445$	−6340.39	−	5608.23i	−0.675423	−	0.597428i
$446$	5547.78			0.589002
$447$		0			0
$448$			872.705i			0.0920344i
$449$	−12957.9			−1.36196			−0.680982	−	0.732300i	$-0.738447\pi$
$449$	−12957.9			−1.36196			−0.680982	+	0.732300i	$0.738447\pi$
$450$		0			0
$451$	3043.53			0.317770
$452$			4460.61i			0.464180i
$453$		0			0
$454$	−350.250			−0.0362071
$455$	−7650.44	−	6767.00i	−0.788260	−	0.697235i
$456$		0			0
$457$		−	16667.8i		−	1.70610i	−0.521831	−	0.853049i	$-0.674751\pi$
$457$		−	16667.8i		−	1.70610i	0.521831	−	0.853049i	$-0.325249\pi$
$458$		−	5579.54i		−	0.569246i
$459$		0			0
$460$	−1853.74	+	2095.75i	−0.187894	+	0.212424i
$461$	−5625.88			−0.568380			−0.284190	−	0.958768i	$-0.591725\pi$
$461$	−5625.88			−0.568380			−0.284190	+	0.958768i	$0.591725\pi$
$462$		0			0
$463$		−	3395.81i		−	0.340857i	−0.985370	−	0.170429i	$-0.945485\pi$
$463$		−	3395.81i		−	0.340857i	0.985370	−	0.170429i	$-0.0545152\pi$
$464$	2317.71			0.231890
$465$		0			0
$466$	6215.65			0.617885
$467$			12556.6i			1.24422i	0.782930	+	0.622110i	$0.213724\pi$
$467$			12556.6i			1.24422i	−0.782930	+	0.622110i	$0.786276\pi$
$468$		0			0
$469$	−6557.32			−0.645605
$470$	6393.61	+	5655.30i	0.627479	+	0.555020i
$471$		0			0
$472$		−	891.730i		−	0.0869602i
$473$		−	2566.73i		−	0.249510i
$474$		0			0
$475$	−1923.91	−	15639.8i	−0.185842	−	1.51074i
$476$	−7465.45			−0.718862
$477$		0			0
$478$		−	10526.9i		−	1.00730i
$479$	7841.71			0.748010			0.374005	−	0.927427i	$-0.377984\pi$
$479$	7841.71			0.748010			0.374005	+	0.927427i	$0.377984\pi$
$480$		0			0
$481$	−440.113			−0.0417202
$482$		−	4126.93i		−	0.389993i
$483$		0			0
$484$	−484.000			−0.0454545
$485$	8283.61	−	9365.05i	0.775545	−	0.876793i
$486$		0			0
$487$		−	1205.34i		−	0.112154i	−0.998426	−	0.0560772i	$-0.982141\pi$
$487$		−	1205.34i		−	0.112154i	0.998426	−	0.0560772i	$-0.0178593\pi$
$488$		−	935.815i		−	0.0868081i
$489$		0			0
$490$	−2326.79	+	2630.56i	−0.214518	+	0.242523i
$491$	2708.46			0.248944			0.124472	−	0.992223i	$-0.460276\pi$
$491$	2708.46			0.248944			0.124472	+	0.992223i	$0.460276\pi$
$492$		0			0
$493$			19826.6i			1.81125i
$494$	−16891.0			−1.53839
$495$		0			0
$496$	3668.60			0.332107
$497$		−	8218.36i		−	0.741739i
$498$		0			0
$499$	17160.4			1.53949			0.769745	−	0.638351i	$-0.220383\pi$
$499$	17160.4			1.53949			0.769745	+	0.638351i	$0.220383\pi$
$500$	3164.75	−	4608.08i	0.283063	−	0.412159i
$501$		0			0
$502$		−	4175.32i		−	0.371222i
$503$			263.269i			0.0233371i	0.999932	+	0.0116686i	$0.00371431\pi$
$503$			263.269i			0.0233371i	−0.999932	+	0.0116686i	$0.996286\pi$
$504$		0			0
$505$	8818.65	+	7800.31i	0.777079	+	0.687345i
$506$	1376.41			0.120927
$507$		0			0
$508$		−	6232.52i		−	0.544338i
$509$	−21083.7			−1.83599			−0.917997	−	0.396588i	$-0.870194\pi$
$509$	−21083.7			−1.83599			−0.917997	+	0.396588i	$0.870194\pi$
$510$		0			0
$511$	8484.02			0.734464
$512$			512.000i			0.0441942i
$513$		0			0
$514$	−9803.37			−0.841261
$515$	5393.99	−	6098.18i	0.461529	−	0.521782i
$516$		0			0
$517$		−	4199.08i		−	0.357206i
$518$		−	179.158i		−	0.0151964i
$519$		0			0
$520$	−4488.37	−	3970.07i	−0.378516	−	0.334806i
$521$	−20745.1			−1.74445			−0.872226	−	0.489102i	$-0.837324\pi$
$521$	−20745.1			−1.74445			−0.872226	+	0.489102i	$0.837324\pi$
$522$		0			0
$523$			19060.9i			1.59364i	0.604214	+	0.796822i	$0.293487\pi$
$523$			19060.9i			1.59364i	−0.604214	+	0.796822i	$0.706513\pi$
$524$	−2482.54			−0.206966
$525$		0			0
$526$	10151.0			0.841452
$527$			31382.6i			2.59402i
$528$		0			0
$529$	8252.74			0.678289
$530$	−3052.06	−	2699.62i	−0.250138	−	0.221253i
$531$		0			0
$532$		−	6875.89i		−	0.560353i
$533$		−	18536.6i		−	1.50639i
$534$		0			0
$535$	−7325.29	+	8281.62i	−0.591962	+	0.669244i
$536$	−3847.06			−0.310014
$537$		0			0
$538$		−	7812.64i		−	0.626072i
$539$	1727.65			0.138062
$540$		0			0
$541$	−3953.81			−0.314210			−0.157105	−	0.987582i	$-0.550216\pi$
$541$	−3953.81			−0.314210			−0.157105	+	0.987582i	$0.550216\pi$
$542$			5698.89i			0.451639i
$543$		0			0
$544$	−4379.84			−0.345191
$545$	14542.3	+	12863.0i	1.14298	+	1.01099i
$546$		0			0
$547$			10226.2i			0.799340i	0.916659	+	0.399670i	$0.130875\pi$
$547$			10226.2i			0.799340i	−0.916659	+	0.399670i	$0.869125\pi$
$548$		−	4555.25i		−	0.355093i
$549$		0			0
$550$	−2729.43	+	335.757i	−0.211606	+	0.0260304i
$551$	−18260.9			−1.41187
$552$		0			0
$553$		−	10484.7i		−	0.806250i
$554$	−10152.4			−0.778579
$555$		0			0
$556$	−5452.74			−0.415913
$557$		−	17378.7i		−	1.32201i	−0.750381	−	0.661005i	$-0.770130\pi$
$557$		−	17378.7i		−	1.32201i	0.750381	−	0.661005i	$-0.229870\pi$
$558$		0			0
$559$	−15632.6			−1.18281
$560$	1616.11	−	1827.10i	0.121952	−	0.137873i
$561$		0			0
$562$			5630.33i			0.422600i
$563$		−	12073.4i		−	0.903791i	−0.892071	−	0.451895i	$-0.850748\pi$
$563$		−	12073.4i		−	0.903791i	0.892071	−	0.451895i	$-0.149252\pi$
$564$		0			0
$565$	8260.35	−	9338.75i	0.615072	−	0.695370i
$566$	4841.91			0.359577
$567$		0			0
$568$		−	4821.56i		−	0.356177i
$569$	11995.8			0.883811			0.441906	−	0.897062i	$-0.354303\pi$
$569$	11995.8			0.883811			0.441906	+	0.897062i	$0.354303\pi$
$570$		0			0
$571$	−6422.26			−0.470688			−0.235344	−	0.971912i	$-0.575622\pi$
$571$	−6422.26			−0.470688			−0.235344	+	0.971912i	$0.575622\pi$
$572$			2947.79i			0.215478i
$573$		0			0
$574$	7545.74			0.548699
$575$	7762.00	−	954.834i	0.562953	−	0.0692510i
$576$		0			0
$577$		−	21036.3i		−	1.51777i	−0.651227	−	0.758883i	$-0.725745\pi$
$577$		−	21036.3i		−	1.51777i	0.651227	−	0.758883i	$-0.274255\pi$
$578$		−	27640.8i		−	1.98911i
$579$		0			0
$580$	−4852.37	−	4292.04i	−0.347386	−	0.307271i
$581$	12997.0			0.928065
$582$		0			0
$583$			2004.48i			0.142396i
$584$	4977.42			0.352684
$585$		0			0
$586$	−4858.57			−0.342501
$587$			5970.28i			0.419795i	0.977723	+	0.209898i	$0.0673131\pi$
$587$			5970.28i			0.419795i	−0.977723	+	0.209898i	$0.932687\pi$
$588$		0			0
$589$	−28904.3			−2.02204
$590$	−1651.34	+	1866.93i	−0.115228	+	0.130272i
$591$		0			0
$592$		−	105.109i		−	0.00729721i
$593$			25442.7i			1.76190i	0.473209	+	0.880950i	$0.343096\pi$
$593$			25442.7i			1.76190i	−0.473209	+	0.880950i	$0.656904\pi$
$594$		0			0
$595$	15629.7	+	13824.8i	1.07690	+	0.952543i
$596$	−11294.3			−0.776232
$597$		0			0
$598$		−	8383.00i		−	0.573255i
$599$	15519.2			1.05860			0.529298	−	0.848436i	$-0.322456\pi$
$599$	15519.2			1.05860			0.529298	+	0.848436i	$0.322456\pi$
$600$		0			0
$601$	−12438.2			−0.844203			−0.422102	−	0.906549i	$-0.638707\pi$
$601$	−12438.2			−0.844203			−0.422102	+	0.906549i	$0.638707\pi$
$602$		−	6363.62i		−	0.430834i
$603$		0			0
$604$	−3901.13			−0.262806
$605$	1013.30	+	896.292i	0.0680937	+	0.0602305i
$606$		0			0
$607$			16809.7i			1.12403i	0.827128	+	0.562014i	$0.189973\pi$
$607$			16809.7i			1.12403i	−0.827128	+	0.562014i	$0.810027\pi$
$608$		−	4033.96i		−	0.269077i
$609$		0			0
$610$	−1732.98	+	1959.23i	−0.115027	+	0.130044i
$611$	−25574.4			−1.69334
$612$		0			0
$613$		−	18549.3i		−	1.22218i	−0.791560	−	0.611092i	$-0.790731\pi$
$613$		−	18549.3i		−	1.22218i	0.791560	−	0.611092i	$-0.209269\pi$
$614$	−325.875			−0.0214190
$615$		0			0
$616$	−1199.97			−0.0784872
$617$			21973.9i			1.43377i	0.697191	+	0.716885i	$0.254433\pi$
$617$			21973.9i			1.43377i	−0.697191	+	0.716885i	$0.745567\pi$
$618$		0			0
$619$	−3457.29			−0.224492			−0.112246	−	0.993680i	$-0.535804\pi$
$619$	−3457.29			−0.224492			−0.112246	+	0.993680i	$0.535804\pi$
$620$	−7680.60	−	6793.67i	−0.497516	−	0.440065i
$621$		0			0
$622$			12310.6i			0.793583i
$623$			10324.0i			0.663922i
$624$		0			0
$625$	−15159.2	+	3786.88i	−0.970186	+	0.242360i
$626$	−12331.2			−0.787305
$627$		0			0
$628$		−	3369.61i		−	0.214111i
$629$	899.141			0.0569970
$630$		0			0
$631$	4266.16			0.269149			0.134575	−	0.990903i	$-0.457033\pi$
$631$	4266.16			0.269149			0.134575	+	0.990903i	$0.457033\pi$
$632$		−	6151.20i		−	0.387155i
$633$		0			0
$634$	−6199.78			−0.388367
$635$	−11541.7	+	13048.4i	−0.721286	+	0.815451i
$636$		0			0
$637$		−	10522.2i		−	0.654484i
$638$			3186.86i			0.197757i
$639$		0			0
$640$	948.144	−	1071.93i	0.0585604	−	0.0662056i
$641$	4556.59			0.280771			0.140386	−	0.990097i	$-0.455166\pi$
$641$	4556.59			0.280771			0.140386	+	0.990097i	$0.455166\pi$
$642$		0			0
$643$			4616.92i			0.283162i	0.989927	+	0.141581i	$0.0452186\pi$
$643$			4616.92i			0.283162i	−0.989927	+	0.141581i	$0.954781\pi$
$644$	3412.50			0.208806
$645$		0			0
$646$	34508.0			2.10170
$647$		−	1201.79i		−	0.0730252i	−0.999333	−	0.0365126i	$-0.988375\pi$
$647$		−	1201.79i		−	0.0730252i	0.999333	−	0.0365126i	$-0.0116249\pi$
$648$		0			0
$649$	1226.13			0.0741599
$650$	2044.93	+	16623.5i	0.123398	+	1.00312i
$651$		0			0
$652$		−	1506.59i		−	0.0904950i
$653$			8658.24i			0.518871i	0.965760	+	0.259436i	$0.0835366\pi$
$653$			8658.24i			0.518871i	−0.965760	+	0.259436i	$0.916463\pi$
$654$		0			0
$655$	5197.45	+	4597.27i	0.310047	+	0.274244i
$656$	4426.95			0.263481
$657$		0			0
$658$		−	10410.7i		−	0.616794i
$659$	15311.7			0.905096			0.452548	−	0.891740i	$-0.350515\pi$
$659$	15311.7			0.905096			0.452548	+	0.891740i	$0.350515\pi$
$660$		0			0
$661$	−24981.1			−1.46997			−0.734987	−	0.678081i	$-0.762812\pi$
$661$	−24981.1			−1.46997			−0.734987	+	0.678081i	$0.762812\pi$
$662$		−	14847.8i		−	0.871717i
$663$		0			0
$664$	7625.09			0.445649
$665$	−12733.1	+	14395.4i	−0.742507	+	0.839443i
$666$		0			0
$667$		−	9062.85i		−	0.526109i
$668$		−	5195.73i		−	0.300941i
$669$		0			0
$670$	8054.22	+	7124.15i	0.464420	+	0.410791i
$671$	1286.75			0.0740302
$672$		0			0
$673$			7248.66i			0.415179i	0.978216	+	0.207589i	$0.0665618\pi$
$673$			7248.66i			0.415179i	−0.978216	+	0.207589i	$0.933438\pi$
$674$	3617.75			0.206752
$675$		0			0
$676$	9165.50			0.521478
$677$		−	10382.1i		−	0.589388i	−0.955592	−	0.294694i	$-0.904782\pi$
$677$		−	10382.1i		−	0.589388i	0.955592	−	0.294694i	$-0.0952178\pi$
$678$		0			0
$679$	−15249.0			−0.861863
$680$	9169.65	+	8110.78i	0.517118	+	0.457403i
$681$		0			0
$682$			5044.33i			0.283222i
$683$		−	32874.9i		−	1.84176i	−0.389844	−	0.920881i	$-0.627471\pi$
$683$		−	32874.9i		−	1.84176i	0.389844	−	0.920881i	$-0.372529\pi$
$684$		0			0
$685$	−8435.61	+	9536.89i	−0.470523	+	0.531950i
$686$	13637.6			0.759019
$687$		0			0
$688$		−	3733.42i		−	0.206883i
$689$	12208.2			0.675032
$690$		0			0
$691$	20337.8			1.11966			0.559830	−	0.828607i	$-0.310866\pi$
$691$	20337.8			1.11966			0.559830	+	0.828607i	$0.310866\pi$
$692$		−	8265.72i		−	0.454069i
$693$		0			0
$694$	11701.2			0.640016
$695$	11415.9	+	10097.6i	0.623063	+	0.551114i
$696$		0			0
$697$			37869.8i			2.05799i
$698$			20463.2i			1.10966i
$699$		0			0
$700$	−6767.00	+	832.435i	−0.365384	+	0.0449473i
$701$	−12930.7			−0.696699			−0.348349	−	0.937365i	$-0.613258\pi$
$701$	−12930.7			−0.696699			−0.348349	+	0.937365i	$0.613258\pi$
$702$		0			0
$703$			828.135i			0.0444292i
$704$	−704.000			−0.0376889
$705$		0			0
$706$	−877.696			−0.0467883
$707$		−	14359.4i		−	0.763846i
$708$		0			0
$709$	−7078.40			−0.374944			−0.187472	−	0.982270i	$-0.560029\pi$
$709$	−7078.40			−0.374944			−0.187472	+	0.982270i	$0.560029\pi$
$710$	−8928.78	+	10094.4i	−0.471959	+	0.533575i
$711$		0			0
$712$			6056.91i			0.318810i
$713$		−	14345.2i		−	0.753479i
$714$		0			0
$715$	5458.85	−	6171.51i	0.285524	−	0.322799i
$716$	8184.67			0.427200
$717$		0			0
$718$			19486.1i			1.01283i
$719$	−9201.25			−0.477258			−0.238629	−	0.971111i	$-0.576698\pi$
$719$	−9201.25			−0.477258			−0.238629	+	0.971111i	$0.576698\pi$
$720$		0			0
$721$	−9929.63			−0.512897
$722$			18064.9i			0.931172i
$723$		0			0
$724$	−10245.8			−0.525941
$725$	2210.77	+	17971.7i	0.113249	+	0.920622i
$726$		0			0
$727$		−	1522.05i		−	0.0776477i	−0.999246	−	0.0388239i	$-0.987639\pi$
$727$		−	1522.05i		−	0.0776477i	0.999246	−	0.0388239i	$-0.0123611\pi$
$728$			7308.39i			0.372070i
$729$		0			0
$730$	−10420.8	−	9217.40i	−0.528342	−	0.467331i
$731$	31937.1			1.61592
$732$		0			0
$733$			21264.1i			1.07150i	0.844377	+	0.535749i	$0.179971\pi$
$733$			21264.1i			1.07150i	−0.844377	+	0.535749i	$0.820029\pi$
$734$	23349.5			1.17418
$735$		0			0
$736$	2002.05			0.100267
$737$		−	5289.71i		−	0.264381i
$738$		0			0
$739$	−21555.5			−1.07298			−0.536490	−	0.843907i	$-0.680250\pi$
$739$	−21555.5			−1.07298			−0.536490	+	0.843907i	$0.680250\pi$
$740$	−194.645	+	220.056i	−0.00966932	+	0.0109317i
$741$		0			0
$742$			4969.65i			0.245878i
$743$			18974.7i			0.936895i	0.883491	+	0.468447i	$0.155186\pi$
$743$			18974.7i			0.936895i	−0.883491	+	0.468447i	$0.844814\pi$
$744$		0			0
$745$	23645.9	+	20915.3i	1.16284	+	1.02856i
$746$	−1075.44			−0.0527810
$747$		0			0
$748$		−	6022.28i		−	0.294380i
$749$	13484.9			0.657848
$750$		0			0
$751$	24260.6			1.17880			0.589402	−	0.807840i	$-0.299363\pi$
$751$	24260.6			1.17880			0.589402	+	0.807840i	$0.299363\pi$
$752$		−	6107.75i		−	0.296179i
$753$		0			0
$754$	19409.5			0.937470
$755$	8167.41	+	7224.27i	0.393699	+	0.348236i
$756$		0			0
$757$		−	14128.2i		−	0.678333i	−0.940726	−	0.339166i	$-0.889855\pi$
$757$		−	14128.2i		−	0.678333i	0.940726	−	0.339166i	$-0.110145\pi$
$758$		−	18536.9i		−	0.888244i
$759$		0			0
$760$	−7470.26	+	8445.52i	−0.356546	+	0.403094i
$761$	917.035			0.0436826			0.0218413	−	0.999761i	$-0.493047\pi$
$761$	917.035			0.0436826			0.0218413	+	0.999761i	$0.493047\pi$
$762$		0			0
$763$		−	23679.1i		−	1.12351i
$764$	13896.3			0.658051
$765$		0			0
$766$	10077.2			0.475333
$767$		−	7467.72i		−	0.351556i
$768$		0			0
$769$	21609.1			1.01332			0.506661	−	0.862145i	$-0.330880\pi$
$769$	21609.1			1.01332			0.506661	+	0.862145i	$0.330880\pi$
$770$	2512.26	+	2222.15i	0.117579	+	0.104001i
$771$		0			0
$772$			6966.69i			0.324788i
$773$		−	16205.3i		−	0.754029i	−0.926207	−	0.377014i	$-0.876951\pi$
$773$		−	16205.3i		−	0.754029i	0.926207	−	0.377014i	$-0.123049\pi$
$774$		0			0
$775$	3499.32	+	28446.5i	0.162193	+	1.31849i
$776$	−8946.34			−0.413859
$777$		0			0
$778$			25091.1i			1.15625i
$779$	−34879.2			−1.60421
$780$		0			0
$781$	6629.65			0.303749
$782$			17126.3i			0.783165i
$783$		0			0
$784$	2512.95			0.114475
$785$	−6239.98	+	7054.63i	−0.283713	+	0.320752i
$786$		0			0
$787$			6106.91i			0.276605i	0.990390	+	0.138302i	$0.0441645\pi$
$787$			6106.91i			0.276605i	−0.990390	+	0.138302i	$0.955835\pi$
$788$		−	449.601i		−	0.0203253i
$789$		0			0
$790$	−11391.1	+	12878.2i	−0.513007	+	0.579981i
$791$	−15206.2			−0.683529
$792$		0			0
$793$		−	7836.91i		−	0.350942i
$794$	18232.7			0.814930
$795$		0			0
$796$	−18107.2			−0.806270
$797$			1702.01i			0.0756442i	0.999284	+	0.0378221i	$0.0120420\pi$
$797$			1702.01i			0.0756442i	−0.999284	+	0.0378221i	$0.987958\pi$
$798$		0			0
$799$	52248.0			2.31339
$800$	−3970.07	+	488.374i	−0.175454	+	0.0215833i
$801$		0			0
$802$			6105.69i			0.268827i
$803$			6843.96i			0.300770i
$804$		0			0
$805$	−7144.42	−	6319.41i	−0.312804	−	0.276683i
$806$	30722.4			1.34262
$807$		0			0
$808$		−	8424.37i		−	0.366793i
$809$	−22987.7			−0.999016			−0.499508	−	0.866309i	$-0.666486\pi$
$809$	−22987.7			−0.999016			−0.499508	+	0.866309i	$0.666486\pi$
$810$		0			0
$811$	24705.7			1.06971			0.534855	−	0.844944i	$-0.320366\pi$
$811$	24705.7			1.06971			0.534855	+	0.844944i	$0.320366\pi$
$812$			7901.09i			0.341470i
$813$		0			0
$814$	144.525			0.00622308
$815$	−2789.97	+	3154.21i	−0.119912	+	0.135567i
$816$		0			0
$817$			29415.0i			1.25961i
$818$			8834.04i			0.377598i
$819$		0			0
$820$	−9268.28	−	8198.02i	−0.394710	−	0.349131i
$821$	−37165.6			−1.57989			−0.789945	−	0.613178i	$-0.789891\pi$
$821$	−37165.6			−1.57989			−0.789945	+	0.613178i	$0.789891\pi$
$822$		0			0
$823$		−	1333.29i		−	0.0564707i	−0.999601	−	0.0282354i	$-0.991011\pi$
$823$		−	1333.29i		−	0.0564707i	0.999601	−	0.0282354i	$-0.00898879\pi$
$824$	−5825.53			−0.246289
$825$		0			0
$826$	3039.91			0.128053
$827$			19441.3i			0.817459i	0.912656	+	0.408729i	$0.134028\pi$
$827$			19441.3i			0.817459i	−0.912656	+	0.408729i	$0.865972\pi$
$828$		0			0
$829$	30978.1			1.29785			0.648923	−	0.760854i	$-0.275220\pi$
$829$	30978.1			1.29785			0.648923	+	0.760854i	$0.275220\pi$
$830$	−15963.9	−	14120.5i	−0.667610	−	0.590517i
$831$		0			0
$832$			4287.70i			0.178665i
$833$			21496.7i			0.894137i
$834$		0			0
$835$	−9621.67	+	10877.8i	−0.398768	+	0.450828i
$836$	5546.70			0.229470
$837$		0			0
$838$			12368.4i			0.509856i
$839$	−19548.7			−0.804406			−0.402203	−	0.915551i	$-0.631755\pi$
$839$	−19548.7			−0.804406			−0.402203	+	0.915551i	$0.631755\pi$
$840$		0			0
$841$	−3405.42			−0.139629
$842$		−	28306.7i		−	1.15857i
$843$		0			0
$844$	14974.3			0.610707
$845$	−19188.9	−	16973.1i	−0.781206	−	0.690995i
$846$		0			0
$847$		−	1649.96i		−	0.0669341i
$848$			2915.60i			0.118069i
$849$		0			0
$850$	−4177.74	−	33961.5i	−0.168583	−	1.37044i
$851$	−411.002			−0.0165558
$852$		0			0
$853$		−	27242.9i		−	1.09353i	−0.837287	−	0.546764i	$-0.815860\pi$
$853$		−	27242.9i		−	1.09353i	0.837287	−	0.546764i	$-0.184140\pi$
$854$	3190.20			0.127829
$855$		0			0
$856$	7911.35			0.315893
$857$			16567.2i			0.660356i	0.943919	+	0.330178i	$0.107109\pi$
$857$			16567.2i			0.660356i	−0.943919	+	0.330178i	$0.892891\pi$
$858$		0			0
$859$	2689.42			0.106824			0.0534121	−	0.998573i	$-0.482990\pi$
$859$	2689.42			0.106824			0.0534121	+	0.998573i	$0.482990\pi$
$860$	−6913.71	+	7816.30i	−0.274134	+	0.309923i
$861$		0			0
$862$		−	1079.48i		−	0.0426534i
$863$		−	17196.1i		−	0.678289i	−0.940734	−	0.339144i	$-0.889862\pi$
$863$		−	17196.1i		−	0.678289i	0.940734	−	0.339144i	$-0.110138\pi$
$864$		0			0
$865$	−15306.8	+	17305.2i	−0.601673	+	0.680223i
$866$	26894.6			1.05533
$867$		0			0
$868$			12506.3i			0.489044i
$869$	8457.90			0.330167
$870$		0			0
$871$	−32216.9			−1.25330
$872$		−	13892.1i		−	0.539501i
$873$		0			0
$874$	−15773.8			−0.610477
$875$	15709.0	+	10788.6i	0.606925	+	0.416825i
$876$		0			0
$877$			43591.8i			1.67844i	0.543794	+	0.839219i	$0.316987\pi$
$877$			43591.8i			1.67844i	−0.543794	+	0.839219i	$0.683013\pi$
$878$		−	19941.0i		−	0.766487i
$879$		0			0
$880$	1473.90	+	1303.70i	0.0564603	+	0.0499405i
$881$	28746.0			1.09929			0.549646	−	0.835398i	$-0.314763\pi$
$881$	28746.0			1.09929			0.549646	+	0.835398i	$0.314763\pi$
$882$		0			0
$883$		−	2883.18i		−	0.109883i	−0.998490	−	0.0549415i	$-0.982503\pi$
$883$		−	2883.18i		−	0.109883i	0.998490	−	0.0549415i	$-0.0174972\pi$
$884$	−36678.6			−1.39552
$885$		0			0
$886$	23201.3			0.879757
$887$		−	10797.1i		−	0.408718i	−0.978896	−	0.204359i	$-0.934489\pi$
$887$		−	10797.1i		−	0.408718i	0.978896	−	0.204359i	$-0.0655109\pi$
$888$		0			0
$889$	21246.7			0.801565
$890$	11216.5	−	12680.8i	0.422445	−	0.477596i
$891$		0			0
$892$			11095.6i			0.416487i
$893$			48122.0i			1.80329i
$894$		0			0
$895$	−17135.5	−	15156.7i	−0.639972	−	0.566071i
$896$	−1745.41			−0.0650782
$897$		0			0
$898$		−	25915.8i		−	0.963054i
$899$	33213.9			1.23220
$900$		0			0
$901$	−24941.2			−0.922209
$902$			6087.06i			0.224697i
$903$		0			0
$904$	−8921.22			−0.328225
$905$	21450.6	+	18973.6i	0.787891	+	0.696909i
$906$		0			0
$907$			6158.05i			0.225441i	0.993627	+	0.112720i	$0.0359564\pi$
$907$			6158.05i			0.225441i	−0.993627	+	0.112720i	$0.964044\pi$
$908$		−	700.500i		−	0.0256023i
$909$		0			0
$910$	13534.0	−	15300.9i	0.493019	−	0.557384i
$911$	−8119.84			−0.295304			−0.147652	−	0.989039i	$-0.547172\pi$
$911$	−8119.84			−0.295304			−0.147652	+	0.989039i	$0.547172\pi$
$912$		0			0
$913$			10484.5i			0.380051i
$914$	33335.6			1.20639
$915$		0			0
$916$	11159.1			0.402518
$917$		−	8462.97i		−	0.304768i
$918$		0			0
$919$	20945.0			0.751807			0.375904	−	0.926659i	$-0.377332\pi$
$919$	20945.0			0.751807			0.375904	+	0.926659i	$0.377332\pi$
$920$	−4191.50	−	3707.48i	−0.150206	−	0.132861i
$921$		0			0
$922$		−	11251.8i		−	0.401906i
$923$		−	40377.8i		−	1.43993i
$924$		0			0
$925$	815.020	−	100.259i	0.0289705	−	0.00356377i
$926$	6791.63			0.241022
$927$		0			0
$928$			4635.43i			0.163971i
$929$	−34175.1			−1.20694			−0.603470	−	0.797386i	$-0.706216\pi$
$929$	−34175.1			−1.20694			−0.603470	+	0.797386i	$0.706216\pi$
$930$		0			0
$931$	−19799.1			−0.696980
$932$			12431.3i			0.436911i
$933$		0			0
$934$	−25113.2			−0.879796
$935$	−11152.3	+	12608.3i	−0.390075	+	0.441000i
$936$		0			0
$937$			41106.5i			1.43318i	0.697494	+	0.716591i	$0.254298\pi$
$937$			41106.5i			1.43318i	−0.697494	+	0.716591i	$0.745702\pi$
$938$		−	13114.6i		−	0.456512i
$939$		0			0
$940$	−11310.6	+	12787.2i	−0.392459	+	0.443695i
$941$	−32287.7			−1.11854			−0.559271	−	0.828985i	$-0.688919\pi$
$941$	−32287.7			−1.11854			−0.559271	+	0.828985i	$0.688919\pi$
$942$		0			0
$943$		−	17310.5i		−	0.597781i
$944$	1783.46			0.0614902
$945$		0			0
$946$	5133.45			0.176430
$947$			57062.9i			1.95807i	0.203683	+	0.979037i	$0.434709\pi$
$947$			57062.9i			1.95807i	−0.203683	+	0.979037i	$0.565291\pi$
$948$		0			0
$949$	41683.0			1.42580
$950$	31279.5	−	3847.82i	1.06826	−	0.131410i
$951$		0			0
$952$		−	14930.9i		−	0.508312i
$953$		−	51251.6i		−	1.74208i	−0.491211	−	0.871040i	$-0.663446\pi$
$953$		−	51251.6i		−	1.74208i	0.491211	−	0.871040i	$-0.336554\pi$
$954$		0			0
$955$	−29093.4	−	25733.8i	−0.985801	−	0.871964i
$956$	21053.9			0.712271
$957$		0			0
$958$			15683.4i			0.528923i
$959$	15528.9			0.522892
$960$		0			0
$961$	22781.8			0.764720
$962$		−	880.225i		−	0.0295006i
$963$		0			0
$964$	8253.86			0.275767
$965$	12901.2	−	14585.5i	0.430368	−	0.486553i
$966$		0			0
$967$		−	23103.2i		−	0.768303i	−0.923270	−	0.384151i	$-0.874494\pi$
$967$		−	23103.2i		−	0.768303i	0.923270	−	0.384151i	$-0.125506\pi$
$968$		−	968.000i		−	0.0321412i
$969$		0			0
$970$	18730.1	+	16567.2i	0.619987	+	0.548393i
$971$	−27006.6			−0.892566			−0.446283	−	0.894892i	$-0.647253\pi$
$971$	−27006.6			−0.892566			−0.446283	+	0.894892i	$0.647253\pi$
$972$		0			0
$973$		−	18588.4i		−	0.612453i
$974$	2410.68			0.0793052
$975$		0			0
$976$	1871.63			0.0613826
$977$			50695.2i			1.66006i	0.557715	+	0.830032i	$0.311678\pi$
$977$			50695.2i			1.66006i	−0.557715	+	0.830032i	$0.688322\pi$
$978$		0			0
$979$	−8328.26			−0.271882
$980$	−5261.12	−	4653.58i	−0.171490	−	0.151687i
$981$		0			0
$982$			5416.93i			0.176030i
$983$		−	34483.1i		−	1.11886i	−0.828877	−	0.559431i	$-0.811020\pi$
$983$		−	34483.1i		−	1.11886i	0.828877	−	0.559431i	$-0.188980\pi$
$984$		0			0
$985$	−832.590	+	941.286i	−0.0269325	+	0.0304486i
$986$	−39653.2			−1.28075
$987$		0			0
$988$		−	33782.1i		−	1.08780i
$989$	−14598.6			−0.469372
$990$		0			0
$991$	22631.4			0.725439			0.362719	−	0.931898i	$-0.381848\pi$
$991$	22631.4			0.725439			0.362719	+	0.931898i	$0.381848\pi$
$992$			7337.20i			0.234835i
$993$		0			0
$994$	16436.7			0.524488
$995$	37909.2	+	33531.6i	1.20784	+	1.06836i
$996$		0			0
$997$			14507.6i			0.460844i	0.973091	+	0.230422i	$0.0740107\pi$
$997$			14507.6i			0.460844i	−0.973091	+	0.230422i	$0.925989\pi$
$998$			34320.8i			1.08858i
$999$		0			0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	990.4.c.i.199.8		8
3.2	odd	2		110.4.b.c.89.3	✓	8
5.4	even	2	inner	990.4.c.i.199.4		8
12.11	even	2		880.4.b.h.529.2		8
15.2	even	4		550.4.a.bb.1.3		4
15.8	even	4		550.4.a.ba.1.2		4
15.14	odd	2		110.4.b.c.89.6	yes	8
60.59	even	2		880.4.b.h.529.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
110.4.b.c.89.3	✓	8	3.2	odd	2
110.4.b.c.89.6	yes	8	15.14	odd	2
550.4.a.ba.1.2		4	15.8	even	4
550.4.a.bb.1.3		4	15.2	even	4
880.4.b.h.529.2		8	12.11	even	2
880.4.b.h.529.7		8	60.59	even	2
990.4.c.i.199.4		8	5.4	even	2	inner
990.4.c.i.199.8		8	1.1	even	1	trivial

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 $\F_{q}$

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Modular forms

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

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Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	990.4.c.i.199.8

Level	$990$
Weight	$4$
Character	990.199
Analytic conductor	$58.412$
Analytic rank	$0$
Dimension	$8$
Inner twists	$2$