LMFDB - Embedded newform 160.3.g.a.111.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [160,3,Mod(111,160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("160.111");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 160 = 2^{5} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	160.g (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:	$4.35968422976$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.148996000000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 7x^{6} - 2x^{5} + 12x^{3} + 47x^{2} + 114x + 81 $ x^8 - 4x^7 + 7x^6 - 2x^5 + 12x^3 + 47x^2 + 114x + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			111.6
Root			$2.49601 - 1.32738i$ of defining polynomial
Character	$\chi$	$=$	160.111
Dual form			160.3.g.a.111.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.20470 q^{3} +2.23607i q^{5} +12.3974i q^{7} -7.54870 q^{9} +O(q^{10})$	$q+1.20470 q^{3} +2.23607i q^{5} +12.3974i q^{7} -7.54870 q^{9} +9.41939 q^{11} +4.18033i q^{13} +2.69379i q^{15} +21.3875 q^{17} -0.990008 q^{19} +14.9351i q^{21} -0.480059i q^{23} -5.00000 q^{25} -19.9362 q^{27} +32.5969i q^{29} -23.2761i q^{31} +11.3475 q^{33} -27.7215 q^{35} -55.8927i q^{37} +5.03604i q^{39} +16.5487 q^{41} +57.3157 q^{43} -16.8794i q^{45} -17.9921i q^{47} -104.696 q^{49} +25.7654 q^{51} -26.8978i q^{53} +21.0624i q^{55} -1.19266 q^{57} -76.8224 q^{59} -104.235i q^{61} -93.5845i q^{63} -9.34751 q^{65} +2.30211 q^{67} -0.578326i q^{69} +54.2243i q^{71} +73.9447 q^{73} -6.02349 q^{75} +116.776i q^{77} -39.5803i q^{79} +43.9213 q^{81} +58.1742 q^{83} +47.8239i q^{85} +39.2694i q^{87} -28.3395 q^{89} -51.8254 q^{91} -28.0407i q^{93} -2.21373i q^{95} -4.84731 q^{97} -71.1042 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 24 q^{9}+O(q^{10}) $ 8 * q + 24 * q^9	$ 8 q + 24 q^{9} + 32 q^{11} - 32 q^{19} - 40 q^{25} + 96 q^{27} + 16 q^{33} + 48 q^{41} - 96 q^{43} - 88 q^{49} - 64 q^{51} - 176 q^{57} - 224 q^{59} - 160 q^{67} + 160 q^{73} - 56 q^{81} + 480 q^{83} - 48 q^{89} + 224 q^{97} + 352 q^{99}+O(q^{100}) $ 8 * q + 24 * q^9 + 32 * q^11 - 32 * q^19 - 40 * q^25 + 96 * q^27 + 16 * q^33 + 48 * q^41 - 96 * q^43 - 88 * q^49 - 64 * q^51 - 176 * q^57 - 224 * q^59 - 160 * q^67 + 160 * q^73 - 56 * q^81 + 480 * q^83 - 48 * q^89 + 224 * q^97 + 352 * q^99

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$	0	0
$2$	0	0
$3$	1.20470	0.401566	0.200783	−	0.979636i	$-0.435651\pi$
$3$	1.20470	0.401566	0.200783	+	0.979636i	$0.435651\pi$
$4$	0	0
$5$	2.23607i	0.447214i
$5$	2.23607i	0.447214i
$6$	0	0
$7$	12.3974i	1.77106i	0.464581	+	0.885531i	$0.346205\pi$
$7$	12.3974i	1.77106i	−0.464581	+	0.885531i	$0.653795\pi$
$8$	0	0
$9$	−7.54870	−0.838745
$10$	0	0
$11$	9.41939	0.856308	0.428154	−	0.903706i	$-0.359164\pi$
$11$	9.41939	0.856308	0.428154	+	0.903706i	$0.359164\pi$
$12$	0	0
$13$	4.18033i	0.321564i	0.986990	+	0.160782i	$0.0514016\pi$
$13$	4.18033i	0.321564i	−0.986990	+	0.160782i	$0.948598\pi$
$14$	0	0
$15$	2.69379i	0.179586i
$16$	0	0
$17$	21.3875	1.25809	0.629043	−	0.777370i	$-0.283447\pi$
$17$	21.3875	1.25809	0.629043	+	0.777370i	$0.283447\pi$
$18$	0	0
$19$	−0.990008	−0.0521057	−0.0260529	−	0.999661i	$-0.508294\pi$
$19$	−0.990008	−0.0521057	−0.0260529	+	0.999661i	$0.508294\pi$
$20$	0	0
$21$	14.9351i	0.711198i
$22$	0	0
$23$	− 0.480059i	− 0.0208721i	−0.999946	−	0.0104361i	$-0.996678\pi$
$23$	− 0.480059i	− 0.0208721i	0.999946	−	0.0104361i	$-0.00332197\pi$
$24$	0	0
$25$	−5.00000	−0.200000
$26$	0	0
$27$	−19.9362	−0.738377
$28$	0	0
$29$	32.5969i	1.12403i	0.827127	+	0.562016i	$0.189974\pi$
$29$	32.5969i	1.12403i	−0.827127	+	0.562016i	$0.810026\pi$
$30$	0	0
$31$	− 23.2761i	− 0.750842i	−0.926854	−	0.375421i	$-0.877498\pi$
$31$	− 23.2761i	− 0.750842i	0.926854	−	0.375421i	$-0.122502\pi$
$32$	0	0
$33$	11.3475	0.343864
$34$	0	0
$35$	−27.7215	−0.792043
$36$	0	0
$37$	− 55.8927i	− 1.51061i	−0.655371	−	0.755307i	$-0.727488\pi$
$37$	− 55.8927i	− 1.51061i	0.655371	−	0.755307i	$-0.272512\pi$
$38$	0	0
$39$	5.03604i	0.129129i
$40$	0	0
$41$	16.5487	0.403627	0.201813	−	0.979424i	$-0.435317\pi$
$41$	16.5487	0.403627	0.201813	+	0.979424i	$0.435317\pi$
$42$	0	0
$43$	57.3157	1.33292	0.666462	−	0.745539i	$-0.267808\pi$
$43$	57.3157	1.33292	0.666462	+	0.745539i	$0.267808\pi$
$44$	0	0
$45$	− 16.8794i	− 0.375098i
$46$	0	0
$47$	− 17.9921i	− 0.382810i	−0.981511	−	0.191405i	$-0.938696\pi$
$47$	− 17.9921i	− 0.382810i	0.981511	−	0.191405i	$-0.0613045\pi$
$48$	0	0
$49$	−104.696	−2.13666
$50$	0	0
$51$	25.7654	0.505205
$52$	0	0
$53$	− 26.8978i	− 0.507506i	−0.967269	−	0.253753i	$-0.918335\pi$
$53$	− 26.8978i	− 0.507506i	0.967269	−	0.253753i	$-0.0816651\pi$
$54$	0	0
$55$	21.0624i	0.382953i
$56$	0	0
$57$	−1.19266	−0.0209239
$58$	0	0
$59$	−76.8224	−1.30207	−0.651037	−	0.759046i	$-0.725666\pi$
$59$	−76.8224	−1.30207	−0.651037	+	0.759046i	$0.725666\pi$
$60$	0	0
$61$	− 104.235i	− 1.70877i	−0.519638	−	0.854387i	$-0.673933\pi$
$61$	− 104.235i	− 1.70877i	0.519638	−	0.854387i	$-0.326067\pi$
$62$	0	0
$63$	− 93.5845i	− 1.48547i
$64$	0	0
$65$	−9.34751	−0.143808
$66$	0	0
$67$	2.30211	0.0343598	0.0171799	−	0.999852i	$-0.494531\pi$
$67$	2.30211	0.0343598	0.0171799	+	0.999852i	$0.494531\pi$
$68$	0	0
$69$	− 0.578326i	− 0.00838154i
$70$	0	0
$71$	54.2243i	0.763722i	0.924220	+	0.381861i	$0.124717\pi$
$71$	54.2243i	0.763722i	−0.924220	+	0.381861i	$0.875283\pi$
$72$	0	0
$73$	73.9447	1.01294	0.506471	−	0.862257i	$-0.330950\pi$
$73$	73.9447	1.01294	0.506471	+	0.862257i	$0.330950\pi$
$74$	0	0
$75$	−6.02349	−0.0803132
$76$	0	0
$77$	116.776i	1.51657i
$78$	0	0
$79$	− 39.5803i	− 0.501016i	−0.968114	−	0.250508i	$-0.919402\pi$
$79$	− 39.5803i	− 0.501016i	0.968114	−	0.250508i	$-0.0805977\pi$
$80$	0	0
$81$	43.9213	0.542238
$82$	0	0
$83$	58.1742	0.700895	0.350447	−	0.936582i	$-0.386030\pi$
$83$	58.1742	0.700895	0.350447	+	0.936582i	$0.386030\pi$
$84$	0	0
$85$	47.8239i	0.562634i
$86$	0	0
$87$	39.2694i	0.451372i
$88$	0	0
$89$	−28.3395	−0.318422	−0.159211	−	0.987245i	$-0.550895\pi$
$89$	−28.3395	−0.318422	−0.159211	+	0.987245i	$0.550895\pi$
$90$	0	0
$91$	−51.8254	−0.569510
$92$	0	0
$93$	− 28.0407i	− 0.301513i
$94$	0	0
$95$	− 2.21373i	− 0.0233024i
$96$	0	0
$97$	−4.84731	−0.0499722	−0.0249861	−	0.999688i	$-0.507954\pi$
$97$	−4.84731	−0.0499722	−0.0249861	+	0.999688i	$0.507954\pi$
$98$	0	0
$99$	−71.1042	−0.718224
$100$	0	0
$101$	152.613i	1.51102i	0.655137	+	0.755510i	$0.272611\pi$
$101$	152.613i	1.51102i	−0.655137	+	0.755510i	$0.727389\pi$
$102$	0	0
$103$	112.926i	1.09637i	0.836357	+	0.548185i	$0.184681\pi$
$103$	112.926i	1.09637i	−0.836357	+	0.548185i	$0.815319\pi$
$104$	0	0
$105$	−33.3960	−0.318057
$106$	0	0
$107$	131.573	1.22966	0.614829	−	0.788660i	$-0.289225\pi$
$107$	131.573	1.22966	0.614829	+	0.788660i	$0.289225\pi$
$108$	0	0
$109$	58.7214i	0.538728i	0.963038	+	0.269364i	$0.0868135\pi$
$109$	58.7214i	0.538728i	−0.963038	+	0.269364i	$0.913186\pi$
$110$	0	0
$111$	− 67.3338i	− 0.606611i
$112$	0	0
$113$	155.198	1.37343	0.686715	−	0.726927i	$-0.259052\pi$
$113$	155.198	1.37343	0.686715	+	0.726927i	$0.259052\pi$
$114$	0	0
$115$	1.07345	0.00933431
$116$	0	0
$117$	− 31.5561i	− 0.269710i
$118$	0	0
$119$	265.150i	2.22815i
$120$	0	0
$121$	−32.2752	−0.266737
$122$	0	0
$123$	19.9362	0.162083
$124$	0	0
$125$	− 11.1803i	− 0.0894427i
$126$	0	0
$127$	− 134.542i	− 1.05939i	−0.848189	−	0.529693i	$-0.822307\pi$
$127$	− 134.542i	− 1.05939i	0.848189	−	0.529693i	$-0.177693\pi$
$128$	0	0
$129$	69.0481	0.535257
$130$	0	0
$131$	139.446	1.06447	0.532235	−	0.846596i	$-0.321352\pi$
$131$	139.446	1.06447	0.532235	+	0.846596i	$0.321352\pi$
$132$	0	0
$133$	− 12.2736i	− 0.0922824i
$134$	0	0
$135$	− 44.5787i	− 0.330212i
$136$	0	0
$137$	−87.3194	−0.637368	−0.318684	−	0.947861i	$-0.603241\pi$
$137$	−87.3194	−0.637368	−0.318684	+	0.947861i	$0.603241\pi$
$138$	0	0
$139$	−101.245	−0.728380	−0.364190	−	0.931325i	$-0.618654\pi$
$139$	−101.245	−0.728380	−0.364190	+	0.931325i	$0.618654\pi$
$140$	0	0
$141$	− 21.6750i	− 0.153724i
$142$	0	0
$143$	39.3762i	0.275358i
$144$	0	0
$145$	−72.8889	−0.502682
$146$	0	0
$147$	−126.127	−0.858009
$148$	0	0
$149$	− 95.8745i	− 0.643453i	−0.946833	−	0.321727i	$-0.895737\pi$
$149$	− 95.8745i	− 0.643453i	0.946833	−	0.321727i	$-0.104263\pi$
$150$	0	0
$151$	27.7535i	0.183798i	0.995768	+	0.0918991i	$0.0292937\pi$
$151$	27.7535i	0.183798i	−0.995768	+	0.0918991i	$0.970706\pi$
$152$	0	0
$153$	−161.448	−1.05521
$154$	0	0
$155$	52.0470	0.335787
$156$	0	0
$157$	− 72.4052i	− 0.461180i	−0.973051	−	0.230590i	$-0.925934\pi$
$157$	− 72.4052i	− 0.461180i	0.973051	−	0.230590i	$-0.0740656\pi$
$158$	0	0
$159$	− 32.4037i	− 0.203797i
$160$	0	0
$161$	5.95150	0.0369658
$162$	0	0
$163$	−66.4519	−0.407680	−0.203840	−	0.979004i	$-0.565342\pi$
$163$	−66.4519	−0.407680	−0.203840	+	0.979004i	$0.565342\pi$
$164$	0	0
$165$	25.3738i	0.153781i
$166$	0	0
$167$	51.6512i	0.309288i	0.987970	+	0.154644i	$0.0494231\pi$
$167$	51.6512i	0.309288i	−0.987970	+	0.154644i	$0.950577\pi$
$168$	0	0
$169$	151.525	0.896597
$170$	0	0
$171$	7.47328	0.0437034
$172$	0	0
$173$	18.5308i	0.107114i	0.998565	+	0.0535571i	$0.0170559\pi$
$173$	18.5308i	0.107114i	−0.998565	+	0.0535571i	$0.982944\pi$
$174$	0	0
$175$	− 61.9871i	− 0.354212i
$176$	0	0
$177$	−92.5478	−0.522869
$178$	0	0
$179$	152.782	0.853533	0.426767	−	0.904362i	$-0.359653\pi$
$179$	152.782	0.853533	0.426767	+	0.904362i	$0.359653\pi$
$180$	0	0
$181$	− 139.334i	− 0.769801i	−0.922958	−	0.384901i	$-0.874236\pi$
$181$	− 139.334i	− 0.769801i	0.922958	−	0.384901i	$-0.125764\pi$
$182$	0	0
$183$	− 125.572i	− 0.686185i
$184$	0	0
$185$	124.980	0.675567
$186$	0	0
$187$	201.457	1.07731
$188$	0	0
$189$	− 247.157i	− 1.30771i
$190$	0	0
$191$	− 127.507i	− 0.667577i	−0.942648	−	0.333789i	$-0.891673\pi$
$191$	− 127.507i	− 0.667577i	0.942648	−	0.333789i	$-0.108327\pi$
$192$	0	0
$193$	−160.944	−0.833908	−0.416954	−	0.908928i	$-0.636902\pi$
$193$	−160.944	−0.833908	−0.416954	+	0.908928i	$0.636902\pi$
$194$	0	0
$195$	−11.2609	−0.0577483
$196$	0	0
$197$	149.338i	0.758059i	0.925385	+	0.379030i	$0.123742\pi$
$197$	149.338i	0.758059i	−0.925385	+	0.379030i	$0.876258\pi$
$198$	0	0
$199$	− 350.488i	− 1.76125i	−0.473817	−	0.880624i	$-0.657124\pi$
$199$	− 350.488i	− 1.76125i	0.473817	−	0.880624i	$-0.342876\pi$
$200$	0	0
$201$	2.77334	0.0137977
$202$	0	0
$203$	−404.118	−1.99073
$204$	0	0
$205$	37.0040i	0.180507i
$206$	0	0
$207$	3.62383i	0.0175064i
$208$	0	0
$209$	−9.32527	−0.0446185
$210$	0	0
$211$	−384.668	−1.82307	−0.911535	−	0.411224i	$-0.865102\pi$
$211$	−384.668	−1.82307	−0.911535	+	0.411224i	$0.865102\pi$
$212$	0	0
$213$	65.3238i	0.306685i
$214$	0	0
$215$	128.162i	0.596102i
$216$	0	0
$217$	288.564	1.32979
$218$	0	0
$219$	89.0810	0.406763
$220$	0	0
$221$	89.4068i	0.404556i
$222$	0	0
$223$	− 59.1979i	− 0.265461i	−0.991152	−	0.132731i	$-0.957625\pi$
$223$	− 59.1979i	− 0.265461i	0.991152	−	0.132731i	$-0.0423745\pi$
$224$	0	0
$225$	37.7435	0.167749
$226$	0	0
$227$	−225.689	−0.994226	−0.497113	−	0.867686i	$-0.665607\pi$
$227$	−225.689	−0.994226	−0.497113	+	0.867686i	$0.665607\pi$
$228$	0	0
$229$	147.030i	0.642051i	0.947070	+	0.321026i	$0.104028\pi$
$229$	147.030i	0.642051i	−0.947070	+	0.321026i	$0.895972\pi$
$230$	0	0
$231$	140.680i	0.609004i
$232$	0	0
$233$	−154.448	−0.662868	−0.331434	−	0.943478i	$-0.607532\pi$
$233$	−154.448	−0.662868	−0.331434	+	0.943478i	$0.607532\pi$
$234$	0	0
$235$	40.2315	0.171198
$236$	0	0
$237$	− 47.6823i	− 0.201191i
$238$	0	0
$239$	− 194.606i	− 0.814249i	−0.913373	−	0.407125i	$-0.866531\pi$
$239$	− 194.606i	− 0.814249i	0.913373	−	0.407125i	$-0.133469\pi$
$240$	0	0
$241$	−101.503	−0.421176	−0.210588	−	0.977575i	$-0.567538\pi$
$241$	−101.503	−0.421176	−0.210588	+	0.977575i	$0.567538\pi$
$242$	0	0
$243$	232.337	0.956121
$244$	0	0
$245$	− 234.108i	− 0.955542i
$246$	0	0
$247$	− 4.13857i	− 0.0167553i
$248$	0	0
$249$	70.0824	0.281455
$250$	0	0
$251$	206.393	0.822282	0.411141	−	0.911572i	$-0.365130\pi$
$251$	206.393	0.822282	0.411141	+	0.911572i	$0.365130\pi$
$252$	0	0
$253$	− 4.52186i	− 0.0178730i
$254$	0	0
$255$	57.6133i	0.225934i
$256$	0	0
$257$	−74.3497	−0.289298	−0.144649	−	0.989483i	$-0.546205\pi$
$257$	−74.3497	−0.289298	−0.144649	+	0.989483i	$0.546205\pi$
$258$	0	0
$259$	692.926	2.67539
$260$	0	0
$261$	− 246.064i	− 0.942775i
$262$	0	0
$263$	391.694i	1.48933i	0.667439	+	0.744665i	$0.267391\pi$
$263$	391.694i	1.48933i	−0.667439	+	0.744665i	$0.732609\pi$
$264$	0	0
$265$	60.1454	0.226964
$266$	0	0
$267$	−34.1405	−0.127867
$268$	0	0
$269$	268.340i	0.997548i	0.866732	+	0.498774i	$0.166216\pi$
$269$	268.340i	0.997548i	−0.866732	+	0.498774i	$0.833784\pi$
$270$	0	0
$271$	426.407i	1.57346i	0.617300	+	0.786728i	$0.288227\pi$
$271$	426.407i	1.57346i	−0.617300	+	0.786728i	$0.711773\pi$
$272$	0	0
$273$	−62.4339	−0.228696
$274$	0	0
$275$	−47.0969	−0.171262
$276$	0	0
$277$	− 3.48615i	− 0.0125854i	−0.999980	−	0.00629269i	$-0.997997\pi$
$277$	− 3.48615i	− 0.0125854i	0.999980	−	0.00629269i	$-0.00200304\pi$
$278$	0	0
$279$	175.705i	0.629765i
$280$	0	0
$281$	−142.361	−0.506621	−0.253311	−	0.967385i	$-0.581519\pi$
$281$	−142.361	−0.506621	−0.253311	+	0.967385i	$0.581519\pi$
$282$	0	0
$283$	−127.113	−0.449162	−0.224581	−	0.974455i	$-0.572101\pi$
$283$	−127.113	−0.449162	−0.224581	+	0.974455i	$0.572101\pi$
$284$	0	0
$285$	− 2.66687i	− 0.00935744i
$286$	0	0
$287$	205.161i	0.714848i
$288$	0	0
$289$	168.424	0.582783
$290$	0	0
$291$	−5.83954	−0.0200671
$292$	0	0
$293$	213.860i	0.729898i	0.931027	+	0.364949i	$0.118914\pi$
$293$	213.860i	0.729898i	−0.931027	+	0.364949i	$0.881086\pi$
$294$	0	0
$295$	− 171.780i	− 0.582305i
$296$	0	0
$297$	−187.787	−0.632278
$298$	0	0
$299$	2.00681	0.00671173
$300$	0	0
$301$	710.567i	2.36069i
$302$	0	0
$303$	183.852i	0.606774i
$304$	0	0
$305$	233.077	0.764187
$306$	0	0
$307$	−341.127	−1.11116	−0.555582	−	0.831462i	$-0.687504\pi$
$307$	−341.127	−1.11116	−0.555582	+	0.831462i	$0.687504\pi$
$308$	0	0
$309$	136.042i	0.440265i
$310$	0	0
$311$	− 384.458i	− 1.23620i	−0.786100	−	0.618100i	$-0.787903\pi$
$311$	− 384.458i	− 1.23620i	0.786100	−	0.618100i	$-0.212097\pi$
$312$	0	0
$313$	−198.197	−0.633216	−0.316608	−	0.948557i	$-0.602544\pi$
$313$	−198.197	−0.633216	−0.316608	+	0.948557i	$0.602544\pi$
$314$	0	0
$315$	209.261	0.664322
$316$	0	0
$317$	− 506.683i	− 1.59837i	−0.601087	−	0.799184i	$-0.705265\pi$
$317$	− 506.683i	− 1.59837i	0.601087	−	0.799184i	$-0.294735\pi$
$318$	0	0
$319$	307.043i	0.962517i
$320$	0	0
$321$	158.506	0.493789
$322$	0	0
$323$	−21.1738	−0.0655535
$324$	0	0
$325$	− 20.9017i	− 0.0643128i
$326$	0	0
$327$	70.7415i	0.216335i
$328$	0	0
$329$	223.056	0.677981
$330$	0	0
$331$	−341.770	−1.03254	−0.516268	−	0.856427i	$-0.672679\pi$
$331$	−341.770	−1.03254	−0.516268	+	0.856427i	$0.672679\pi$
$332$	0	0
$333$	421.918i	1.26702i
$334$	0	0
$335$	5.14766i	0.0153662i
$336$	0	0
$337$	483.617	1.43506	0.717532	−	0.696525i	$-0.245271\pi$
$337$	483.617	1.43506	0.717532	+	0.696525i	$0.245271\pi$
$338$	0	0
$339$	186.966	0.551523
$340$	0	0
$341$	− 219.247i	− 0.642952i
$342$	0	0
$343$	− 690.490i	− 2.01309i
$344$	0	0
$345$	1.29318	0.00374834
$346$	0	0
$347$	153.403	0.442084	0.221042	−	0.975264i	$-0.429054\pi$
$347$	153.403	0.442084	0.221042	+	0.975264i	$0.429054\pi$
$348$	0	0
$349$	− 470.630i	− 1.34851i	−0.738499	−	0.674254i	$-0.764465\pi$
$349$	− 470.630i	− 1.34851i	0.738499	−	0.674254i	$-0.235535\pi$
$350$	0	0
$351$	− 83.3399i	− 0.237436i
$352$	0	0
$353$	60.9892	0.172774	0.0863870	−	0.996262i	$-0.472468\pi$
$353$	60.9892	0.172774	0.0863870	+	0.996262i	$0.472468\pi$
$354$	0	0
$355$	−121.249	−0.341547
$356$	0	0
$357$	319.425i	0.894748i
$358$	0	0
$359$	− 659.963i	− 1.83834i	−0.393865	−	0.919168i	$-0.628862\pi$
$359$	− 659.963i	− 1.83834i	0.393865	−	0.919168i	$-0.371138\pi$
$360$	0	0
$361$	−360.020	−0.997285
$362$	0	0
$363$	−38.8818	−0.107112
$364$	0	0
$365$	165.345i	0.453001i
$366$	0	0
$367$	371.867i	1.01326i	0.862163	+	0.506630i	$0.169109\pi$
$367$	371.867i	1.01326i	−0.862163	+	0.506630i	$0.830891\pi$
$368$	0	0
$369$	−124.921	−0.338540
$370$	0	0
$371$	333.464	0.898824
$372$	0	0
$373$	− 292.195i	− 0.783364i	−0.920101	−	0.391682i	$-0.871893\pi$
$373$	− 292.195i	− 0.783364i	0.920101	−	0.391682i	$-0.128107\pi$
$374$	0	0
$375$	− 13.4689i	− 0.0359171i
$376$	0	0
$377$	−136.266	−0.361448
$378$	0	0
$379$	−574.767	−1.51654	−0.758268	−	0.651943i	$-0.773954\pi$
$379$	−574.767	−1.51654	−0.758268	+	0.651943i	$0.773954\pi$
$380$	0	0
$381$	− 162.083i	− 0.425413i
$382$	0	0
$383$	80.7976i	0.210960i	0.994421	+	0.105480i	$0.0336379\pi$
$383$	80.7976i	0.210960i	−0.994421	+	0.105480i	$0.966362\pi$
$384$	0	0
$385$	−261.119	−0.678232
$386$	0	0
$387$	−432.659	−1.11798
$388$	0	0
$389$	− 295.056i	− 0.758498i	−0.925295	−	0.379249i	$-0.876182\pi$
$389$	− 295.056i	− 0.758498i	0.925295	−	0.379249i	$-0.123818\pi$
$390$	0	0
$391$	− 10.2673i	− 0.0262590i
$392$	0	0
$393$	167.990	0.427455
$394$	0	0
$395$	88.5042	0.224061
$396$	0	0
$397$	119.742i	0.301618i	0.988563	+	0.150809i	$0.0481878\pi$
$397$	119.742i	0.301618i	−0.988563	+	0.150809i	$0.951812\pi$
$398$	0	0
$399$	− 14.7859i	− 0.0370575i
$400$	0	0
$401$	−332.970	−0.830350	−0.415175	−	0.909742i	$-0.636280\pi$
$401$	−332.970	−0.830350	−0.415175	+	0.909742i	$0.636280\pi$
$402$	0	0
$403$	97.3019	0.241444
$404$	0	0
$405$	98.2109i	0.242496i
$406$	0	0
$407$	− 526.475i	− 1.29355i
$408$	0	0
$409$	−14.2484	−0.0348373	−0.0174186	−	0.999848i	$-0.505545\pi$
$409$	−14.2484	−0.0348373	−0.0174186	+	0.999848i	$0.505545\pi$
$410$	0	0
$411$	−105.194	−0.255945
$412$	0	0
$413$	− 952.400i	− 2.30605i
$414$	0	0
$415$	130.082i	0.313450i
$416$	0	0
$417$	−121.969	−0.292492
$418$	0	0
$419$	−159.814	−0.381417	−0.190709	−	0.981647i	$-0.561079\pi$
$419$	−159.814	−0.381417	−0.190709	+	0.981647i	$0.561079\pi$
$420$	0	0
$421$	89.1070i	0.211656i	0.994384	+	0.105828i	$0.0337492\pi$
$421$	89.1070i	0.211656i	−0.994384	+	0.105828i	$0.966251\pi$
$422$	0	0
$423$	135.817i	0.321080i
$424$	0	0
$425$	−106.937	−0.251617
$426$	0	0
$427$	1292.25	3.02634
$428$	0	0
$429$	47.4364i	0.110574i
$430$	0	0
$431$	− 279.464i	− 0.648408i	−0.945987	−	0.324204i	$-0.894904\pi$
$431$	− 279.464i	− 0.648408i	0.945987	−	0.324204i	$-0.105096\pi$
$432$	0	0
$433$	−71.3709	−0.164829	−0.0824145	−	0.996598i	$-0.526263\pi$
$433$	−71.3709	−0.164829	−0.0824145	+	0.996598i	$0.526263\pi$
$434$	0	0
$435$	−87.8091	−0.201860
$436$	0	0
$437$	0.475263i	0.00108756i
$438$	0	0
$439$	385.168i	0.877375i	0.898640	+	0.438688i	$0.144556\pi$
$439$	385.168i	0.877375i	−0.898640	+	0.438688i	$0.855444\pi$
$440$	0	0
$441$	790.321	1.79211
$442$	0	0
$443$	−194.993	−0.440165	−0.220082	−	0.975481i	$-0.570633\pi$
$443$	−194.993	−0.440165	−0.220082	+	0.975481i	$0.570633\pi$
$444$	0	0
$445$	− 63.3691i	− 0.142402i
$446$	0	0
$447$	− 115.500i	− 0.258389i
$448$	0	0
$449$	−458.520	−1.02120	−0.510602	−	0.859817i	$-0.670577\pi$
$449$	−458.520	−1.02120	−0.510602	+	0.859817i	$0.670577\pi$
$450$	0	0
$451$	155.879	0.345629
$452$	0	0
$453$	33.4346i	0.0738071i
$454$	0	0
$455$	− 115.885i	− 0.254693i
$456$	0	0
$457$	−283.142	−0.619567	−0.309784	−	0.950807i	$-0.600257\pi$
$457$	−283.142	−0.619567	−0.309784	+	0.950807i	$0.600257\pi$
$458$	0	0
$459$	−426.385	−0.928943
$460$	0	0
$461$	− 332.355i	− 0.720943i	−0.932770	−	0.360472i	$-0.882616\pi$
$461$	− 332.355i	− 0.720943i	0.932770	−	0.360472i	$-0.117384\pi$
$462$	0	0
$463$	426.370i	0.920886i	0.887689	+	0.460443i	$0.152309\pi$
$463$	426.370i	0.920886i	−0.887689	+	0.460443i	$0.847691\pi$
$464$	0	0
$465$	62.7009	0.134841
$466$	0	0
$467$	−365.575	−0.782815	−0.391407	−	0.920217i	$-0.628012\pi$
$467$	−365.575	−0.782815	−0.391407	+	0.920217i	$0.628012\pi$
$468$	0	0
$469$	28.5402i	0.0608533i
$470$	0	0
$471$	− 87.2264i	− 0.185194i
$472$	0	0
$473$	539.879	1.14139
$474$	0	0
$475$	4.95004	0.0104211
$476$	0	0
$477$	203.044i	0.425668i
$478$	0	0
$479$	657.674i	1.37301i	0.727123	+	0.686507i	$0.240857\pi$
$479$	657.674i	1.37301i	−0.727123	+	0.686507i	$0.759143\pi$
$480$	0	0
$481$	233.650	0.485759
$482$	0	0
$483$	7.16976	0.0148442
$484$	0	0
$485$	− 10.8389i	− 0.0223483i
$486$	0	0
$487$	− 599.006i	− 1.22999i	−0.788530	−	0.614996i	$-0.789157\pi$
$487$	− 599.006i	− 1.22999i	0.788530	−	0.614996i	$-0.210843\pi$
$488$	0	0
$489$	−80.0544	−0.163711
$490$	0	0
$491$	−81.4721	−0.165931	−0.0829655	−	0.996552i	$-0.526439\pi$
$491$	−81.4721	−0.165931	−0.0829655	+	0.996552i	$0.526439\pi$
$492$	0	0
$493$	697.165i	1.41413i
$494$	0	0
$495$	− 158.994i	− 0.321199i
$496$	0	0
$497$	−672.241	−1.35260
$498$	0	0
$499$	263.876	0.528809	0.264404	−	0.964412i	$-0.414825\pi$
$499$	263.876	0.528809	0.264404	+	0.964412i	$0.414825\pi$
$500$	0	0
$501$	62.2240i	0.124200i
$502$	0	0
$503$	− 173.548i	− 0.345026i	−0.985007	−	0.172513i	$-0.944811\pi$
$503$	− 173.548i	− 0.345026i	0.985007	−	0.172513i	$-0.0551887\pi$
$504$	0	0
$505$	−341.253	−0.675749
$506$	0	0
$507$	182.542	0.360042
$508$	0	0
$509$	459.387i	0.902528i	0.892390	+	0.451264i	$0.149027\pi$
$509$	459.387i	0.902528i	−0.892390	+	0.451264i	$0.850973\pi$
$510$	0	0
$511$	916.724i	1.79398i
$512$	0	0
$513$	19.7370	0.0384737
$514$	0	0
$515$	−252.511	−0.490312
$516$	0	0
$517$	− 169.474i	− 0.327804i
$518$	0	0
$519$	22.3240i	0.0430134i
$520$	0	0
$521$	430.141	0.825606	0.412803	−	0.910820i	$-0.364550\pi$
$521$	430.141	0.825606	0.412803	+	0.910820i	$0.364550\pi$
$522$	0	0
$523$	745.353	1.42515	0.712574	−	0.701596i	$-0.247529\pi$
$523$	745.353	1.42515	0.712574	+	0.701596i	$0.247529\pi$
$524$	0	0
$525$	− 74.6757i	− 0.142240i
$526$	0	0
$527$	− 497.817i	− 0.944625i
$528$	0	0
$529$	528.770	0.999564
$530$	0	0
$531$	579.910	1.09211
$532$	0	0
$533$	69.1791i	0.129792i
$534$	0	0
$535$	294.207i	0.549920i
$536$	0	0
$537$	184.057	0.342750
$538$	0	0
$539$	−986.174	−1.82964
$540$	0	0
$541$	− 725.703i	− 1.34141i	−0.741724	−	0.670705i	$-0.765992\pi$
$541$	− 725.703i	− 1.34141i	0.741724	−	0.670705i	$-0.234008\pi$
$542$	0	0
$543$	− 167.855i	− 0.309126i
$544$	0	0
$545$	−131.305	−0.240927
$546$	0	0
$547$	−92.4741	−0.169057	−0.0845284	−	0.996421i	$-0.526938\pi$
$547$	−92.4741	−0.169057	−0.0845284	+	0.996421i	$0.526938\pi$
$548$	0	0
$549$	786.840i	1.43322i
$550$	0	0
$551$	− 32.2712i	− 0.0585684i
$552$	0	0
$553$	490.694	0.887330
$554$	0	0
$555$	150.563	0.271285
$556$	0	0
$557$	1043.87i	1.87409i	0.349215	+	0.937043i	$0.386448\pi$
$557$	1043.87i	1.87409i	−0.349215	+	0.937043i	$0.613552\pi$
$558$	0	0
$559$	239.599i	0.428620i
$560$	0	0
$561$	242.695	0.432611
$562$	0	0
$563$	149.723	0.265939	0.132969	−	0.991120i	$-0.457549\pi$
$563$	149.723	0.265939	0.132969	+	0.991120i	$0.457549\pi$
$564$	0	0
$565$	347.032i	0.614217i
$566$	0	0
$567$	544.511i	0.960337i
$568$	0	0
$569$	−814.440	−1.43135	−0.715677	−	0.698432i	$-0.753881\pi$
$569$	−814.440	−1.43135	−0.715677	+	0.698432i	$0.753881\pi$
$570$	0	0
$571$	−250.759	−0.439157	−0.219578	−	0.975595i	$-0.570468\pi$
$571$	−250.759	−0.439157	−0.219578	+	0.975595i	$0.570468\pi$
$572$	0	0
$573$	− 153.608i	− 0.268076i
$574$	0	0
$575$	2.40030i	0.00417443i
$576$	0	0
$577$	851.781	1.47622	0.738112	−	0.674678i	$-0.235718\pi$
$577$	851.781	1.47622	0.738112	+	0.674678i	$0.235718\pi$
$578$	0	0
$579$	−193.889	−0.334869
$580$	0	0
$581$	721.211i	1.24133i
$582$	0	0
$583$	− 253.361i	− 0.434581i
$584$	0	0
$585$	70.5616	0.120618
$586$	0	0
$587$	426.708	0.726929	0.363465	−	0.931608i	$-0.381594\pi$
$587$	426.708	0.726929	0.363465	+	0.931608i	$0.381594\pi$
$588$	0	0
$589$	23.0435i	0.0391232i
$590$	0	0
$591$	179.907i	0.304411i
$592$	0	0
$593$	668.977	1.12812	0.564061	−	0.825733i	$-0.309238\pi$
$593$	668.977	1.12812	0.564061	+	0.825733i	$0.309238\pi$
$594$	0	0
$595$	−592.893	−0.996458
$596$	0	0
$597$	− 422.232i	− 0.707257i
$598$	0	0
$599$	75.4852i	0.126019i	0.998013	+	0.0630093i	$0.0200698\pi$
$599$	75.4852i	0.126019i	−0.998013	+	0.0630093i	$0.979930\pi$
$600$	0	0
$601$	−887.728	−1.47709	−0.738543	−	0.674207i	$-0.764486\pi$
$601$	−887.728	−1.47709	−0.738543	+	0.674207i	$0.764486\pi$
$602$	0	0
$603$	−17.3779	−0.0288191
$604$	0	0
$605$	− 72.1694i	− 0.119288i
$606$	0	0
$607$	560.008i	0.922583i	0.887249	+	0.461292i	$0.152614\pi$
$607$	560.008i	0.922583i	−0.887249	+	0.461292i	$0.847386\pi$
$608$	0	0
$609$	−486.840	−0.799408
$610$	0	0
$611$	75.2129	0.123098
$612$	0	0
$613$	− 28.9794i	− 0.0472747i	−0.999721	−	0.0236374i	$-0.992475\pi$
$613$	− 28.9794i	− 0.0472747i	0.999721	−	0.0236374i	$-0.00752471\pi$
$614$	0	0
$615$	44.5787i	0.0724856i
$616$	0	0
$617$	345.379	0.559771	0.279886	−	0.960033i	$-0.409703\pi$
$617$	345.379	0.559771	0.279886	+	0.960033i	$0.409703\pi$
$618$	0	0
$619$	−624.297	−1.00856	−0.504279	−	0.863541i	$-0.668242\pi$
$619$	−624.297	−1.00856	−0.504279	+	0.863541i	$0.668242\pi$
$620$	0	0
$621$	9.57055i	0.0154115i
$622$	0	0
$623$	− 351.337i	− 0.563944i
$624$	0	0
$625$	25.0000	0.0400000
$626$	0	0
$627$	−11.2341	−0.0179173
$628$	0	0
$629$	− 1195.40i	− 1.90048i
$630$	0	0
$631$	− 717.185i	− 1.13658i	−0.822827	−	0.568292i	$-0.807604\pi$
$631$	− 717.185i	− 1.13658i	0.822827	−	0.568292i	$-0.192396\pi$
$632$	0	0
$633$	−463.408	−0.732082
$634$	0	0
$635$	300.845	0.473772
$636$	0	0
$637$	− 437.665i	− 0.687072i
$638$	0	0
$639$	− 409.323i	− 0.640568i
$640$	0	0
$641$	538.058	0.839404	0.419702	−	0.907662i	$-0.362135\pi$
$641$	538.058	0.839404	0.419702	+	0.907662i	$0.362135\pi$
$642$	0	0
$643$	1266.66	1.96993	0.984963	−	0.172767i	$-0.0552708\pi$
$643$	1266.66	1.96993	0.984963	+	0.172767i	$0.0552708\pi$
$644$	0	0
$645$	154.396i	0.239374i
$646$	0	0
$647$	816.424i	1.26186i	0.775839	+	0.630930i	$0.217327\pi$
$647$	816.424i	1.26186i	−0.775839	+	0.630930i	$0.782673\pi$
$648$	0	0
$649$	−723.620	−1.11498
$650$	0	0
$651$	347.632	0.533997
$652$	0	0
$653$	− 692.042i	− 1.05979i	−0.848064	−	0.529894i	$-0.822232\pi$
$653$	− 692.042i	− 1.05979i	0.848064	−	0.529894i	$-0.177768\pi$
$654$	0	0
$655$	311.810i	0.476046i
$656$	0	0
$657$	−558.187	−0.849599
$658$	0	0
$659$	890.532	1.35134	0.675669	−	0.737205i	$-0.263855\pi$
$659$	890.532	1.35134	0.675669	+	0.737205i	$0.263855\pi$
$660$	0	0
$661$	656.736i	0.993549i	0.867880	+	0.496774i	$0.165482\pi$
$661$	656.736i	0.993549i	−0.867880	+	0.496774i	$0.834518\pi$
$662$	0	0
$663$	107.708i	0.162456i
$664$	0	0
$665$	27.4445	0.0412699
$666$	0	0
$667$	15.6484	0.0234609
$668$	0	0
$669$	− 71.3155i	− 0.106600i
$670$	0	0
$671$	− 981.831i	− 1.46324i
$672$	0	0
$673$	−242.150	−0.359807	−0.179904	−	0.983684i	$-0.557579\pi$
$673$	−242.150	−0.359807	−0.179904	+	0.983684i	$0.557579\pi$
$674$	0	0
$675$	99.6809	0.147675
$676$	0	0
$677$	− 592.051i	− 0.874521i	−0.899335	−	0.437261i	$-0.855949\pi$
$677$	− 592.051i	− 0.874521i	0.899335	−	0.437261i	$-0.144051\pi$
$678$	0	0
$679$	− 60.0941i	− 0.0885039i
$680$	0	0
$681$	−271.887	−0.399247
$682$	0	0
$683$	−640.413	−0.937647	−0.468824	−	0.883292i	$-0.655322\pi$
$683$	−640.413	−0.937647	−0.468824	+	0.883292i	$0.655322\pi$
$684$	0	0
$685$	− 195.252i	− 0.285040i
$686$	0	0
$687$	177.126i	0.257826i
$688$	0	0
$689$	112.442	0.163196
$690$	0	0
$691$	−848.214	−1.22752	−0.613759	−	0.789494i	$-0.710343\pi$
$691$	−848.214	−1.22752	−0.613759	+	0.789494i	$0.710343\pi$
$692$	0	0
$693$	− 881.509i	− 1.27202i
$694$	0	0
$695$	− 226.390i	− 0.325741i
$696$	0	0
$697$	353.935	0.507798
$698$	0	0
$699$	−186.063	−0.266185
$700$	0	0
$701$	− 724.318i	− 1.03326i	−0.856207	−	0.516632i	$-0.827185\pi$
$701$	− 724.318i	− 1.03326i	0.856207	−	0.516632i	$-0.172815\pi$
$702$	0	0
$703$	55.3343i	0.0787116i
$704$	0	0
$705$	48.4668	0.0687473
$706$	0	0
$707$	−1892.01	−2.67611
$708$	0	0
$709$	109.289i	0.154145i	0.997026	+	0.0770723i	$0.0245572\pi$
$709$	109.289i	0.154145i	−0.997026	+	0.0770723i	$0.975443\pi$
$710$	0	0
$711$	298.780i	0.420225i
$712$	0	0
$713$	−11.1739	−0.0156717
$714$	0	0
$715$	−88.0478	−0.123144
$716$	0	0
$717$	− 234.441i	− 0.326975i
$718$	0	0
$719$	− 794.119i	− 1.10448i	−0.833686	−	0.552238i	$-0.813774\pi$
$719$	− 794.119i	− 1.10448i	0.833686	−	0.552238i	$-0.186226\pi$
$720$	0	0
$721$	−1399.99	−1.94174
$722$	0	0
$723$	−122.281	−0.169130
$724$	0	0
$725$	− 162.985i	− 0.224806i
$726$	0	0
$727$	− 756.542i	− 1.04064i	−0.853973	−	0.520318i	$-0.825813\pi$
$727$	− 756.542i	− 1.04064i	0.853973	−	0.520318i	$-0.174187\pi$
$728$	0	0
$729$	−115.395	−0.158292
$730$	0	0
$731$	1225.84	1.67693
$732$	0	0
$733$	884.232i	1.20632i	0.797621	+	0.603160i	$0.206092\pi$
$733$	884.232i	1.20632i	−0.797621	+	0.603160i	$0.793908\pi$
$734$	0	0
$735$	− 282.029i	− 0.383713i
$736$	0	0
$737$	21.6844	0.0294226
$738$	0	0
$739$	746.061	1.00956	0.504778	−	0.863249i	$-0.331574\pi$
$739$	746.061	1.00956	0.504778	+	0.863249i	$0.331574\pi$
$740$	0	0
$741$	− 4.98572i	− 0.00672837i
$742$	0	0
$743$	− 47.1955i	− 0.0635202i	−0.999496	−	0.0317601i	$-0.989889\pi$
$743$	− 47.1955i	− 0.0635202i	0.999496	−	0.0317601i	$-0.0101113\pi$
$744$	0	0
$745$	214.382	0.287761
$746$	0	0
$747$	−439.140	−0.587872
$748$	0	0
$749$	1631.17i	2.17780i
$750$	0	0
$751$	− 863.233i	− 1.14945i	−0.818348	−	0.574723i	$-0.805110\pi$
$751$	− 863.233i	− 1.14945i	0.818348	−	0.574723i	$-0.194890\pi$
$752$	0	0
$753$	248.641	0.330200
$754$	0	0
$755$	−62.0588	−0.0821971
$756$	0	0
$757$	− 980.034i	− 1.29463i	−0.762223	−	0.647314i	$-0.775892\pi$
$757$	− 980.034i	− 1.29463i	0.762223	−	0.647314i	$-0.224108\pi$
$758$	0	0
$759$	− 5.44748i	− 0.00717718i
$760$	0	0
$761$	1247.89	1.63981	0.819903	−	0.572502i	$-0.194027\pi$
$761$	1247.89	1.63981	0.819903	+	0.572502i	$0.194027\pi$
$762$	0	0
$763$	−727.994	−0.954121
$764$	0	0
$765$	− 361.008i	− 0.471906i
$766$	0	0
$767$	− 321.143i	− 0.418701i
$768$	0	0
$769$	782.117	1.01706	0.508528	−	0.861045i	$-0.330190\pi$
$769$	782.117	1.01706	0.508528	+	0.861045i	$0.330190\pi$
$770$	0	0
$771$	−89.5689	−0.116172
$772$	0	0
$773$	673.425i	0.871184i	0.900144	+	0.435592i	$0.143461\pi$
$773$	673.425i	0.871184i	−0.900144	+	0.435592i	$0.856539\pi$
$774$	0	0
$775$	116.381i	0.150168i
$776$	0	0
$777$	834.766	1.07435
$778$	0	0
$779$	−16.3834	−0.0210313
$780$	0	0
$781$	510.759i	0.653981i
$782$	0	0
$783$	− 649.858i	− 0.829959i
$784$	0	0
$785$	161.903	0.206246
$786$	0	0
$787$	−96.9208	−0.123152	−0.0615761	−	0.998102i	$-0.519613\pi$
$787$	−96.9208	−0.123152	−0.0615761	+	0.998102i	$0.519613\pi$
$788$	0	0
$789$	471.872i	0.598064i
$790$	0	0
$791$	1924.05i	2.43243i
$792$	0	0
$793$	435.738	0.549480
$794$	0	0
$795$	72.4570	0.0911408
$796$	0	0
$797$	100.960i	0.126675i	0.997992	+	0.0633373i	$0.0201744\pi$
$797$	100.960i	0.126675i	−0.997992	+	0.0633373i	$0.979826\pi$
$798$	0	0
$799$	− 384.805i	− 0.481609i
$800$	0	0
$801$	213.927	0.267074
$802$	0	0
$803$	696.514	0.867390
$804$	0	0
$805$	13.3080i	0.0165316i
$806$	0	0
$807$	323.269i	0.400581i
$808$	0	0
$809$	−125.149	−0.154696	−0.0773480	−	0.997004i	$-0.524645\pi$
$809$	−125.149	−0.154696	−0.0773480	+	0.997004i	$0.524645\pi$
$810$	0	0
$811$	−1504.84	−1.85554	−0.927769	−	0.373156i	$-0.878276\pi$
$811$	−1504.84	−1.85554	−0.927769	+	0.373156i	$0.878276\pi$
$812$	0	0
$813$	513.691i	0.631846i
$814$	0	0
$815$	− 148.591i	− 0.182320i
$816$	0	0
$817$	−56.7430	−0.0694529
$818$	0	0
$819$	391.215	0.477673
$820$	0	0
$821$	− 386.633i	− 0.470929i	−0.971883	−	0.235464i	$-0.924339\pi$
$821$	− 386.633i	− 0.470929i	0.971883	−	0.235464i	$-0.0756611\pi$
$822$	0	0
$823$	737.692i	0.896345i	0.893947	+	0.448173i	$0.147925\pi$
$823$	737.692i	0.896345i	−0.893947	+	0.448173i	$0.852075\pi$
$824$	0	0
$825$	−56.7376	−0.0687728
$826$	0	0
$827$	1339.34	1.61952	0.809758	−	0.586764i	$-0.199598\pi$
$827$	1339.34	1.61952	0.809758	+	0.586764i	$0.199598\pi$
$828$	0	0
$829$	− 1103.12i	− 1.33066i	−0.746548	−	0.665331i	$-0.768290\pi$
$829$	− 1103.12i	− 1.33066i	0.746548	−	0.665331i	$-0.231710\pi$
$830$	0	0
$831$	− 4.19976i	− 0.00505386i
$832$	0	0
$833$	−2239.19	−2.68810
$834$	0	0
$835$	−115.496	−0.138318
$836$	0	0
$837$	464.037i	0.554405i
$838$	0	0
$839$	1401.49i	1.67043i	0.549927	+	0.835213i	$0.314655\pi$
$839$	1401.49i	1.67043i	−0.549927	+	0.835213i	$0.685345\pi$
$840$	0	0
$841$	−221.558	−0.263446
$842$	0	0
$843$	−171.501	−0.203442
$844$	0	0
$845$	338.820i	0.400970i
$846$	0	0
$847$	− 400.129i	− 0.472407i
$848$	0	0
$849$	−153.133	−0.180368
$850$	0	0
$851$	−26.8318	−0.0315298
$852$	0	0
$853$	493.813i	0.578913i	0.957191	+	0.289456i	$0.0934745\pi$
$853$	493.813i	0.578913i	−0.957191	+	0.289456i	$0.906525\pi$
$854$	0	0
$855$	16.7108i	0.0195448i
$856$	0	0
$857$	−1179.48	−1.37628	−0.688142	−	0.725576i	$-0.741573\pi$
$857$	−1179.48	−1.37628	−0.688142	+	0.725576i	$0.741573\pi$
$858$	0	0
$859$	1057.63	1.23123	0.615616	−	0.788046i	$-0.288907\pi$
$859$	1057.63	1.23123	0.615616	+	0.788046i	$0.288907\pi$
$860$	0	0
$861$	247.157i	0.287059i
$862$	0	0
$863$	29.0807i	0.0336972i	0.999858	+	0.0168486i	$0.00536333\pi$
$863$	29.0807i	0.0336972i	−0.999858	+	0.0168486i	$0.994637\pi$
$864$	0	0
$865$	−41.4360	−0.0479029
$866$	0	0
$867$	202.900	0.234026
$868$	0	0
$869$	− 372.822i	− 0.429024i
$870$	0	0
$871$	9.62357i	0.0110489i
$872$	0	0
$873$	36.5909	0.0419139
$874$	0	0
$875$	138.607	0.158409
$876$	0	0
$877$	656.995i	0.749139i	0.927199	+	0.374570i	$0.122210\pi$
$877$	656.995i	0.749139i	−0.927199	+	0.374570i	$0.877790\pi$
$878$	0	0
$879$	257.637i	0.293102i
$880$	0	0
$881$	−151.559	−0.172030	−0.0860152	−	0.996294i	$-0.527413\pi$
$881$	−151.559	−0.172030	−0.0860152	+	0.996294i	$0.527413\pi$
$882$	0	0
$883$	−1139.07	−1.29001	−0.645003	−	0.764180i	$-0.723144\pi$
$883$	−1139.07	−1.29001	−0.645003	+	0.764180i	$0.723144\pi$
$884$	0	0
$885$	− 206.943i	− 0.233834i
$886$	0	0
$887$	1559.13i	1.75776i	0.477041	+	0.878881i	$0.341709\pi$
$887$	1559.13i	1.75776i	−0.477041	+	0.878881i	$0.658291\pi$
$888$	0	0
$889$	1667.98	1.87624
$890$	0	0
$891$	413.711	0.464323
$892$	0	0
$893$	17.8123i	0.0199466i
$894$	0	0
$895$	341.632i	0.381712i
$896$	0	0
$897$	2.41760	0.00269520
$898$	0	0
$899$	758.729	0.843970
$900$	0	0
$901$	− 575.277i	− 0.638487i
$902$	0	0
$903$	856.019i	0.947972i
$904$	0	0
$905$	311.560	0.344266
$906$	0	0
$907$	1295.44	1.42827	0.714133	−	0.700010i	$-0.246821\pi$
$907$	1295.44	1.42827	0.714133	+	0.700010i	$0.246821\pi$
$908$	0	0
$909$	− 1152.03i	− 1.26736i
$910$	0	0
$911$	1144.77i	1.25661i	0.777969	+	0.628303i	$0.216250\pi$
$911$	1144.77i	1.25661i	−0.777969	+	0.628303i	$0.783750\pi$
$912$	0	0
$913$	547.966	0.600182
$914$	0	0
$915$	280.787	0.306871
$916$	0	0
$917$	1728.77i	1.88524i
$918$	0	0
$919$	− 1735.96i	− 1.88897i	−0.328557	−	0.944484i	$-0.606562\pi$
$919$	− 1735.96i	− 1.88897i	0.328557	−	0.944484i	$-0.393438\pi$
$920$	0	0
$921$	−410.955	−0.446205
$922$	0	0
$923$	−226.675	−0.245586
$924$	0	0
$925$	279.464i	0.302123i
$926$	0	0
$927$	− 852.446i	− 0.919575i
$928$	0	0
$929$	1547.16	1.66540	0.832701	−	0.553724i	$-0.186794\pi$
$929$	1547.16	1.66540	0.832701	+	0.553724i	$0.186794\pi$
$930$	0	0
$931$	103.650	0.111332
$932$	0	0
$933$	− 463.156i	− 0.496416i
$934$	0	0
$935$	450.471i	0.481788i
$936$	0	0
$937$	11.0709	0.0118153	0.00590764	−	0.999983i	$-0.498120\pi$
$937$	11.0709	0.0118153	0.00590764	+	0.999983i	$0.498120\pi$
$938$	0	0
$939$	−238.767	−0.254278
$940$	0	0
$941$	237.253i	0.252129i	0.992022	+	0.126064i	$0.0402346\pi$
$941$	237.253i	0.252129i	−0.992022	+	0.126064i	$0.959765\pi$
$942$	0	0
$943$	− 7.94436i	− 0.00842456i
$944$	0	0
$945$	552.661	0.584826
$946$	0	0
$947$	−923.233	−0.974903	−0.487451	−	0.873150i	$-0.662073\pi$
$947$	−923.233	−0.974903	−0.487451	+	0.873150i	$0.662073\pi$
$948$	0	0
$949$	309.114i	0.325726i
$950$	0	0
$951$	− 610.399i	− 0.641850i
$952$	0	0
$953$	−1616.56	−1.69629	−0.848144	−	0.529765i	$-0.822280\pi$
$953$	−1616.56	−1.69629	−0.848144	+	0.529765i	$0.822280\pi$
$954$	0	0
$955$	285.115	0.298550
$956$	0	0
$957$	369.894i	0.386514i
$958$	0	0
$959$	− 1082.54i	− 1.12882i
$960$	0	0
$961$	419.222	0.436236
$962$	0	0
$963$	−993.209	−1.03137
$964$	0	0
$965$	− 359.882i	− 0.372935i
$966$	0	0
$967$	− 714.006i	− 0.738373i	−0.929355	−	0.369186i	$-0.879636\pi$
$967$	− 714.006i	− 0.738373i	0.929355	−	0.369186i	$-0.120364\pi$
$968$	0	0
$969$	−25.5080	−0.0263240
$970$	0	0
$971$	−549.922	−0.566346	−0.283173	−	0.959069i	$-0.591387\pi$
$971$	−549.922	−0.566346	−0.283173	+	0.959069i	$0.591387\pi$
$972$	0	0
$973$	− 1255.17i	− 1.29000i
$974$	0	0
$975$	− 25.1802i	− 0.0258258i
$976$	0	0
$977$	391.158	0.400366	0.200183	−	0.979758i	$-0.435846\pi$
$977$	391.158	0.400366	0.200183	+	0.979758i	$0.435846\pi$
$978$	0	0
$979$	−266.941	−0.272667
$980$	0	0
$981$	− 443.270i	− 0.451856i
$982$	0	0
$983$	1452.94i	1.47806i	0.673670	+	0.739032i	$0.264717\pi$
$983$	1452.94i	1.47806i	−0.673670	+	0.739032i	$0.735283\pi$
$984$	0	0
$985$	−333.929	−0.339014
$986$	0	0
$987$	268.715	0.272254
$988$	0	0
$989$	− 27.5149i	− 0.0278210i
$990$	0	0
$991$	123.313i	0.124433i	0.998063	+	0.0622167i	$0.0198170\pi$
$991$	123.313i	0.124433i	−0.998063	+	0.0622167i	$0.980183\pi$
$992$	0	0
$993$	−411.729	−0.414631
$994$	0	0
$995$	783.715	0.787654
$996$	0	0
$997$	1235.61i	1.23933i	0.784868	+	0.619663i	$0.212731\pi$
$997$	1235.61i	1.23933i	−0.784868	+	0.619663i	$0.787269\pi$
$998$	0	0
$999$	1114.29i	1.11540i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	160.3.g.a.111.6		8
3.2	odd	2		1440.3.g.a.271.4		8
4.3	odd	2		40.3.g.a.11.8	yes	8
5.2	odd	4		800.3.e.d.399.7		16
5.3	odd	4		800.3.e.d.399.10		16
5.4	even	2		800.3.g.g.751.3		8
8.3	odd	2	inner	160.3.g.a.111.5		8
8.5	even	2		40.3.g.a.11.7	✓	8
12.11	even	2		360.3.g.a.91.1		8
16.3	odd	4		1280.3.b.i.511.8		16
16.5	even	4		1280.3.b.i.511.7		16
16.11	odd	4		1280.3.b.i.511.9		16
16.13	even	4		1280.3.b.i.511.10		16
20.3	even	4		200.3.e.d.99.9		16
20.7	even	4		200.3.e.d.99.8		16
20.19	odd	2		200.3.g.g.51.1		8
24.5	odd	2		360.3.g.a.91.2		8
24.11	even	2		1440.3.g.a.271.5		8
40.3	even	4		800.3.e.d.399.9		16
40.13	odd	4		200.3.e.d.99.7		16
40.19	odd	2		800.3.g.g.751.4		8
40.27	even	4		800.3.e.d.399.8		16
40.29	even	2		200.3.g.g.51.2		8
40.37	odd	4		200.3.e.d.99.10		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.3.g.a.11.7	✓	8	8.5	even	2
40.3.g.a.11.8	yes	8	4.3	odd	2
160.3.g.a.111.5		8	8.3	odd	2	inner
160.3.g.a.111.6		8	1.1	even	1	trivial
200.3.e.d.99.7		16	40.13	odd	4
200.3.e.d.99.8		16	20.7	even	4
200.3.e.d.99.9		16	20.3	even	4
200.3.e.d.99.10		16	40.37	odd	4
200.3.g.g.51.1		8	20.19	odd	2
200.3.g.g.51.2		8	40.29	even	2
360.3.g.a.91.1		8	12.11	even	2
360.3.g.a.91.2		8	24.5	odd	2
800.3.e.d.399.7		16	5.2	odd	4
800.3.e.d.399.8		16	40.27	even	4
800.3.e.d.399.9		16	40.3	even	4
800.3.e.d.399.10		16	5.3	odd	4
800.3.g.g.751.3		8	5.4	even	2
800.3.g.g.751.4		8	40.19	odd	2
1280.3.b.i.511.7		16	16.5	even	4
1280.3.b.i.511.8		16	16.3	odd	4
1280.3.b.i.511.9		16	16.11	odd	4
1280.3.b.i.511.10		16	16.13	even	4
1440.3.g.a.271.4		8	3.2	odd	2
1440.3.g.a.271.5		8	24.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 160 = 2^{5} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	160.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.20470 q^{3} +2.23607i q^{5} +12.3974i q^{7} -7.54870 q^{9} +O(q^{10})\)	\(q+1.20470 q^{3} +2.23607i q^{5} +12.3974i q^{7} -7.54870 q^{9} +9.41939 q^{11} +4.18033i q^{13} +2.69379i q^{15} +21.3875 q^{17} -0.990008 q^{19} +14.9351i q^{21} -0.480059i q^{23} -5.00000 q^{25} -19.9362 q^{27} +32.5969i q^{29} -23.2761i q^{31} +11.3475 q^{33} -27.7215 q^{35} -55.8927i q^{37} +5.03604i q^{39} +16.5487 q^{41} +57.3157 q^{43} -16.8794i q^{45} -17.9921i q^{47} -104.696 q^{49} +25.7654 q^{51} -26.8978i q^{53} +21.0624i q^{55} -1.19266 q^{57} -76.8224 q^{59} -104.235i q^{61} -93.5845i q^{63} -9.34751 q^{65} +2.30211 q^{67} -0.578326i q^{69} +54.2243i q^{71} +73.9447 q^{73} -6.02349 q^{75} +116.776i q^{77} -39.5803i q^{79} +43.9213 q^{81} +58.1742 q^{83} +47.8239i q^{85} +39.2694i q^{87} -28.3395 q^{89} -51.8254 q^{91} -28.0407i q^{93} -2.21373i q^{95} -4.84731 q^{97} -71.1042 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 24 q^{9}+O(q^{10}) \) 8 * q + 24 * q^9	\( 8 q + 24 q^{9} + 32 q^{11} - 32 q^{19} - 40 q^{25} + 96 q^{27} + 16 q^{33} + 48 q^{41} - 96 q^{43} - 88 q^{49} - 64 q^{51} - 176 q^{57} - 224 q^{59} - 160 q^{67} + 160 q^{73} - 56 q^{81} + 480 q^{83} - 48 q^{89} + 224 q^{97} + 352 q^{99}+O(q^{100}) \) 8 * q + 24 * q^9 + 32 * q^11 - 32 * q^19 - 40 * q^25 + 96 * q^27 + 16 * q^33 + 48 * q^41 - 96 * q^43 - 88 * q^49 - 64 * q^51 - 176 * q^57 - 224 * q^59 - 160 * q^67 + 160 * q^73 - 56 * q^81 + 480 * q^83 - 48 * q^89 + 224 * q^97 + 352 * q^99

Introduction

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