LMFDB - Embedded newform 160.2.d.a.81.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [160,2,Mod(81,160)]

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([0, 1, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("160.81");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 160 = 2^{5} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	160.d (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:	$1.27760643234$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			81.2
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	160.81
Dual form			160.2.d.a.81.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.732051i q^{3} +1.00000i q^{5} +2.73205 q^{7} +2.46410 q^{9} +O(q^{10})$	$q-0.732051i q^{3} +1.00000i q^{5} +2.73205 q^{7} +2.46410 q^{9} -2.00000i q^{11} -3.46410i q^{13} +0.732051 q^{15} -3.46410 q^{17} +7.46410i q^{19} -2.00000i q^{21} -4.19615 q^{23} -1.00000 q^{25} -4.00000i q^{27} +6.92820i q^{29} -1.46410 q^{31} -1.46410 q^{33} +2.73205i q^{35} +2.00000i q^{37} -2.53590 q^{39} -5.46410 q^{41} -8.73205i q^{43} +2.46410i q^{45} -6.73205 q^{47} +0.464102 q^{49} +2.53590i q^{51} -4.53590i q^{53} +2.00000 q^{55} +5.46410 q^{57} -0.535898i q^{59} -4.92820i q^{61} +6.73205 q^{63} +3.46410 q^{65} +7.26795i q^{67} +3.07180i q^{69} +1.46410 q^{71} +0.535898 q^{73} +0.732051i q^{75} -5.46410i q^{77} +14.9282 q^{79} +4.46410 q^{81} +4.73205i q^{83} -3.46410i q^{85} +5.07180 q^{87} -4.92820 q^{89} -9.46410i q^{91} +1.07180i q^{93} -7.46410 q^{95} +6.39230 q^{97} -4.92820i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{7} - 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^7 - 4 * q^9	$ 4 q + 4 q^{7} - 4 q^{9} - 4 q^{15} + 4 q^{23} - 4 q^{25} + 8 q^{31} + 8 q^{33} - 24 q^{39} - 8 q^{41} - 20 q^{47} - 12 q^{49} + 8 q^{55} + 8 q^{57} + 20 q^{63} - 8 q^{71} + 16 q^{73} + 32 q^{79} + 4 q^{81} + 48 q^{87} + 8 q^{89} - 16 q^{95} - 16 q^{97}+O(q^{100}) $ 4 * q + 4 * q^7 - 4 * q^9 - 4 * q^15 + 4 * q^23 - 4 * q^25 + 8 * q^31 + 8 * q^33 - 24 * q^39 - 8 * q^41 - 20 * q^47 - 12 * q^49 + 8 * q^55 + 8 * q^57 + 20 * q^63 - 8 * q^71 + 16 * q^73 + 32 * q^79 + 4 * q^81 + 48 * q^87 + 8 * q^89 - 16 * q^95 - 16 * q^97

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
$3$	− 0.732051i	− 0.422650i	−0.977416	−	0.211325i	$-0.932222\pi$
$3$	− 0.732051i	− 0.422650i	0.977416	−	0.211325i	$-0.0677778\pi$
$4$	0	0
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	0	0
$7$	2.73205	1.03262	0.516309	−	0.856402i	$-0.327306\pi$
$7$	2.73205	1.03262	0.516309	+	0.856402i	$0.327306\pi$
$8$	0	0
$9$	2.46410	0.821367
$10$	0	0
$11$	− 2.00000i	− 0.603023i	−0.953463	−	0.301511i	$-0.902509\pi$
$11$	− 2.00000i	− 0.603023i	0.953463	−	0.301511i	$-0.0974911\pi$
$12$	0	0
$13$	− 3.46410i	− 0.960769i	−0.877058	−	0.480384i	$-0.840497\pi$
$13$	− 3.46410i	− 0.960769i	0.877058	−	0.480384i	$-0.159503\pi$
$14$	0	0
$15$	0.732051	0.189015
$16$	0	0
$17$	−3.46410	−0.840168	−0.420084	−	0.907485i	$-0.637999\pi$
$17$	−3.46410	−0.840168	−0.420084	+	0.907485i	$0.637999\pi$
$18$	0	0
$19$	7.46410i	1.71238i	0.516659	+	0.856191i	$0.327175\pi$
$19$	7.46410i	1.71238i	−0.516659	+	0.856191i	$0.672825\pi$
$20$	0	0
$21$	− 2.00000i	− 0.436436i
$22$	0	0
$23$	−4.19615	−0.874958	−0.437479	−	0.899229i	$-0.644129\pi$
$23$	−4.19615	−0.874958	−0.437479	+	0.899229i	$0.644129\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	− 4.00000i	− 0.769800i
$28$	0	0
$29$	6.92820i	1.28654i	0.765641	+	0.643268i	$0.222422\pi$
$29$	6.92820i	1.28654i	−0.765641	+	0.643268i	$0.777578\pi$
$30$	0	0
$31$	−1.46410	−0.262960	−0.131480	−	0.991319i	$-0.541973\pi$
$31$	−1.46410	−0.262960	−0.131480	+	0.991319i	$0.541973\pi$
$32$	0	0
$33$	−1.46410	−0.254867
$34$	0	0
$35$	2.73205i	0.461801i
$36$	0	0
$37$	2.00000i	0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$	2.00000i	0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$	0	0
$39$	−2.53590	−0.406069
$40$	0	0
$41$	−5.46410	−0.853349	−0.426675	−	0.904405i	$-0.640315\pi$
$41$	−5.46410	−0.853349	−0.426675	+	0.904405i	$0.640315\pi$
$42$	0	0
$43$	− 8.73205i	− 1.33163i	−0.746119	−	0.665813i	$-0.768085\pi$
$43$	− 8.73205i	− 1.33163i	0.746119	−	0.665813i	$-0.231915\pi$
$44$	0	0
$45$	2.46410i	0.367327i
$46$	0	0
$47$	−6.73205	−0.981971	−0.490985	−	0.871168i	$-0.663363\pi$
$47$	−6.73205	−0.981971	−0.490985	+	0.871168i	$0.663363\pi$
$48$	0	0
$49$	0.464102	0.0663002
$50$	0	0
$51$	2.53590i	0.355097i
$52$	0	0
$53$	− 4.53590i	− 0.623054i	−0.950237	−	0.311527i	$-0.899160\pi$
$53$	− 4.53590i	− 0.623054i	0.950237	−	0.311527i	$-0.100840\pi$
$54$	0	0
$55$	2.00000	0.269680
$56$	0	0
$57$	5.46410	0.723738
$58$	0	0
$59$	− 0.535898i	− 0.0697680i	−0.999391	−	0.0348840i	$-0.988894\pi$
$59$	− 0.535898i	− 0.0697680i	0.999391	−	0.0348840i	$-0.0111062\pi$
$60$	0	0
$61$	− 4.92820i	− 0.630992i	−0.948927	−	0.315496i	$-0.897829\pi$
$61$	− 4.92820i	− 0.630992i	0.948927	−	0.315496i	$-0.102171\pi$
$62$	0	0
$63$	6.73205	0.848159
$64$	0	0
$65$	3.46410	0.429669
$66$	0	0
$67$	7.26795i	0.887921i	0.896046	+	0.443961i	$0.146427\pi$
$67$	7.26795i	0.887921i	−0.896046	+	0.443961i	$0.853573\pi$
$68$	0	0
$69$	3.07180i	0.369801i
$70$	0	0
$71$	1.46410	0.173757	0.0868784	−	0.996219i	$-0.472311\pi$
$71$	1.46410	0.173757	0.0868784	+	0.996219i	$0.472311\pi$
$72$	0	0
$73$	0.535898	0.0627222	0.0313611	−	0.999508i	$-0.490016\pi$
$73$	0.535898	0.0627222	0.0313611	+	0.999508i	$0.490016\pi$
$74$	0	0
$75$	0.732051i	0.0845299i
$76$	0	0
$77$	− 5.46410i	− 0.622692i
$78$	0	0
$79$	14.9282	1.67955	0.839777	−	0.542931i	$-0.182686\pi$
$79$	14.9282	1.67955	0.839777	+	0.542931i	$0.182686\pi$
$80$	0	0
$81$	4.46410	0.496011
$82$	0	0
$83$	4.73205i	0.519410i	0.965688	+	0.259705i	$0.0836253\pi$
$83$	4.73205i	0.519410i	−0.965688	+	0.259705i	$0.916375\pi$
$84$	0	0
$85$	− 3.46410i	− 0.375735i
$86$	0	0
$87$	5.07180	0.543754
$88$	0	0
$89$	−4.92820	−0.522388	−0.261194	−	0.965286i	$-0.584116\pi$
$89$	−4.92820	−0.522388	−0.261194	+	0.965286i	$0.584116\pi$
$90$	0	0
$91$	− 9.46410i	− 0.992107i
$92$	0	0
$93$	1.07180i	0.111140i
$94$	0	0
$95$	−7.46410	−0.765801
$96$	0	0
$97$	6.39230	0.649040	0.324520	−	0.945879i	$-0.394797\pi$
$97$	6.39230	0.649040	0.324520	+	0.945879i	$0.394797\pi$
$98$	0	0
$99$	− 4.92820i	− 0.495303i
$100$	0	0
$101$	10.9282i	1.08740i	0.839281	+	0.543698i	$0.182976\pi$
$101$	10.9282i	1.08740i	−0.839281	+	0.543698i	$0.817024\pi$
$102$	0	0
$103$	1.66025	0.163590	0.0817948	−	0.996649i	$-0.473935\pi$
$103$	1.66025	0.163590	0.0817948	+	0.996649i	$0.473935\pi$
$104$	0	0
$105$	2.00000	0.195180
$106$	0	0
$107$	− 0.732051i	− 0.0707700i	−0.999374	−	0.0353850i	$-0.988734\pi$
$107$	− 0.732051i	− 0.0707700i	0.999374	−	0.0353850i	$-0.0112658\pi$
$108$	0	0
$109$	− 3.07180i	− 0.294225i	−0.989120	−	0.147112i	$-0.953002\pi$
$109$	− 3.07180i	− 0.294225i	0.989120	−	0.147112i	$-0.0469979\pi$
$110$	0	0
$111$	1.46410	0.138966
$112$	0	0
$113$	0.928203	0.0873180	0.0436590	−	0.999046i	$-0.486098\pi$
$113$	0.928203	0.0873180	0.0436590	+	0.999046i	$0.486098\pi$
$114$	0	0
$115$	− 4.19615i	− 0.391293i
$116$	0	0
$117$	− 8.53590i	− 0.789144i
$118$	0	0
$119$	−9.46410	−0.867573
$120$	0	0
$121$	7.00000	0.636364
$122$	0	0
$123$	4.00000i	0.360668i
$124$	0	0
$125$	− 1.00000i	− 0.0894427i
$126$	0	0
$127$	−13.2679	−1.17734	−0.588670	−	0.808373i	$-0.700348\pi$
$127$	−13.2679	−1.17734	−0.588670	+	0.808373i	$0.700348\pi$
$128$	0	0
$129$	−6.39230	−0.562811
$130$	0	0
$131$	7.85641i	0.686417i	0.939259	+	0.343209i	$0.111514\pi$
$131$	7.85641i	0.686417i	−0.939259	+	0.343209i	$0.888486\pi$
$132$	0	0
$133$	20.3923i	1.76824i
$134$	0	0
$135$	4.00000	0.344265
$136$	0	0
$137$	−8.92820	−0.762788	−0.381394	−	0.924413i	$-0.624556\pi$
$137$	−8.92820	−0.762788	−0.381394	+	0.924413i	$0.624556\pi$
$138$	0	0
$139$	− 7.46410i	− 0.633097i	−0.948576	−	0.316548i	$-0.897476\pi$
$139$	− 7.46410i	− 0.633097i	0.948576	−	0.316548i	$-0.102524\pi$
$140$	0	0
$141$	4.92820i	0.415030i
$142$	0	0
$143$	−6.92820	−0.579365
$144$	0	0
$145$	−6.92820	−0.575356
$146$	0	0
$147$	− 0.339746i	− 0.0280218i
$148$	0	0
$149$	− 19.8564i	− 1.62670i	−0.581775	−	0.813350i	$-0.697641\pi$
$149$	− 19.8564i	− 1.62670i	0.581775	−	0.813350i	$-0.302359\pi$
$150$	0	0
$151$	−8.39230	−0.682956	−0.341478	−	0.939890i	$-0.610927\pi$
$151$	−8.39230	−0.682956	−0.341478	+	0.939890i	$0.610927\pi$
$152$	0	0
$153$	−8.53590	−0.690086
$154$	0	0
$155$	− 1.46410i	− 0.117599i
$156$	0	0
$157$	16.9282i	1.35102i	0.737352	+	0.675509i	$0.236076\pi$
$157$	16.9282i	1.35102i	−0.737352	+	0.675509i	$0.763924\pi$
$158$	0	0
$159$	−3.32051	−0.263333
$160$	0	0
$161$	−11.4641	−0.903498
$162$	0	0
$163$	10.1962i	0.798624i	0.916815	+	0.399312i	$0.130751\pi$
$163$	10.1962i	0.798624i	−0.916815	+	0.399312i	$0.869249\pi$
$164$	0	0
$165$	− 1.46410i	− 0.113980i
$166$	0	0
$167$	20.1962	1.56283	0.781413	−	0.624015i	$-0.214499\pi$
$167$	20.1962	1.56283	0.781413	+	0.624015i	$0.214499\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	18.3923i	1.40649i
$172$	0	0
$173$	− 2.00000i	− 0.152057i	−0.997106	−	0.0760286i	$-0.975776\pi$
$173$	− 2.00000i	− 0.152057i	0.997106	−	0.0760286i	$-0.0242240\pi$
$174$	0	0
$175$	−2.73205	−0.206524
$176$	0	0
$177$	−0.392305	−0.0294874
$178$	0	0
$179$	− 15.4641i	− 1.15584i	−0.816093	−	0.577921i	$-0.803864\pi$
$179$	− 15.4641i	− 1.15584i	0.816093	−	0.577921i	$-0.196136\pi$
$180$	0	0
$181$	− 16.0000i	− 1.18927i	−0.803996	−	0.594635i	$-0.797296\pi$
$181$	− 16.0000i	− 1.18927i	0.803996	−	0.594635i	$-0.202704\pi$
$182$	0	0
$183$	−3.60770	−0.266688
$184$	0	0
$185$	−2.00000	−0.147043
$186$	0	0
$187$	6.92820i	0.506640i
$188$	0	0
$189$	− 10.9282i	− 0.794910i
$190$	0	0
$191$	19.3205	1.39798	0.698991	−	0.715130i	$-0.253633\pi$
$191$	19.3205	1.39798	0.698991	+	0.715130i	$0.253633\pi$
$192$	0	0
$193$	7.46410	0.537278	0.268639	−	0.963241i	$-0.413426\pi$
$193$	7.46410	0.537278	0.268639	+	0.963241i	$0.413426\pi$
$194$	0	0
$195$	− 2.53590i	− 0.181599i
$196$	0	0
$197$	12.5359i	0.893146i	0.894747	+	0.446573i	$0.147356\pi$
$197$	12.5359i	0.893146i	−0.894747	+	0.446573i	$0.852644\pi$
$198$	0	0
$199$	−25.8564	−1.83291	−0.916456	−	0.400135i	$-0.868963\pi$
$199$	−25.8564	−1.83291	−0.916456	+	0.400135i	$0.868963\pi$
$200$	0	0
$201$	5.32051	0.375280
$202$	0	0
$203$	18.9282i	1.32850i
$204$	0	0
$205$	− 5.46410i	− 0.381629i
$206$	0	0
$207$	−10.3397	−0.718662
$208$	0	0
$209$	14.9282	1.03261
$210$	0	0
$211$	− 14.7846i	− 1.01781i	−0.860821	−	0.508907i	$-0.830050\pi$
$211$	− 14.7846i	− 1.01781i	0.860821	−	0.508907i	$-0.169950\pi$
$212$	0	0
$213$	− 1.07180i	− 0.0734383i
$214$	0	0
$215$	8.73205	0.595521
$216$	0	0
$217$	−4.00000	−0.271538
$218$	0	0
$219$	− 0.392305i	− 0.0265095i
$220$	0	0
$221$	12.0000i	0.807207i
$222$	0	0
$223$	16.1962	1.08457	0.542287	−	0.840193i	$-0.317558\pi$
$223$	16.1962	1.08457	0.542287	+	0.840193i	$0.317558\pi$
$224$	0	0
$225$	−2.46410	−0.164273
$226$	0	0
$227$	− 28.0526i	− 1.86191i	−0.365129	−	0.930957i	$-0.618975\pi$
$227$	− 28.0526i	− 1.86191i	0.365129	−	0.930957i	$-0.381025\pi$
$228$	0	0
$229$	4.00000i	0.264327i	0.991228	+	0.132164i	$0.0421925\pi$
$229$	4.00000i	0.264327i	−0.991228	+	0.132164i	$0.957808\pi$
$230$	0	0
$231$	−4.00000	−0.263181
$232$	0	0
$233$	29.3205	1.92085	0.960425	−	0.278538i	$-0.0898499\pi$
$233$	29.3205	1.92085	0.960425	+	0.278538i	$0.0898499\pi$
$234$	0	0
$235$	− 6.73205i	− 0.439151i
$236$	0	0
$237$	− 10.9282i	− 0.709863i
$238$	0	0
$239$	20.0000	1.29369	0.646846	−	0.762620i	$-0.276088\pi$
$239$	20.0000	1.29369	0.646846	+	0.762620i	$0.276088\pi$
$240$	0	0
$241$	−4.39230	−0.282933	−0.141467	−	0.989943i	$-0.545182\pi$
$241$	−4.39230	−0.282933	−0.141467	+	0.989943i	$0.545182\pi$
$242$	0	0
$243$	− 15.2679i	− 0.979439i
$244$	0	0
$245$	0.464102i	0.0296504i
$246$	0	0
$247$	25.8564	1.64520
$248$	0	0
$249$	3.46410	0.219529
$250$	0	0
$251$	11.0718i	0.698846i	0.936965	+	0.349423i	$0.113622\pi$
$251$	11.0718i	0.698846i	−0.936965	+	0.349423i	$0.886378\pi$
$252$	0	0
$253$	8.39230i	0.527620i
$254$	0	0
$255$	−2.53590	−0.158804
$256$	0	0
$257$	−2.00000	−0.124757	−0.0623783	−	0.998053i	$-0.519869\pi$
$257$	−2.00000	−0.124757	−0.0623783	+	0.998053i	$0.519869\pi$
$258$	0	0
$259$	5.46410i	0.339523i
$260$	0	0
$261$	17.0718i	1.05672i
$262$	0	0
$263$	−5.66025	−0.349026	−0.174513	−	0.984655i	$-0.555835\pi$
$263$	−5.66025	−0.349026	−0.174513	+	0.984655i	$0.555835\pi$
$264$	0	0
$265$	4.53590	0.278638
$266$	0	0
$267$	3.60770i	0.220787i
$268$	0	0
$269$	− 4.92820i	− 0.300478i	−0.988650	−	0.150239i	$-0.951996\pi$
$269$	− 4.92820i	− 0.300478i	0.988650	−	0.150239i	$-0.0480043\pi$
$270$	0	0
$271$	−15.3205	−0.930655	−0.465327	−	0.885139i	$-0.654063\pi$
$271$	−15.3205	−0.930655	−0.465327	+	0.885139i	$0.654063\pi$
$272$	0	0
$273$	−6.92820	−0.419314
$274$	0	0
$275$	2.00000i	0.120605i
$276$	0	0
$277$	− 2.00000i	− 0.120168i	−0.998193	−	0.0600842i	$-0.980863\pi$
$277$	− 2.00000i	− 0.120168i	0.998193	−	0.0600842i	$-0.0191369\pi$
$278$	0	0
$279$	−3.60770	−0.215987
$280$	0	0
$281$	17.4641	1.04182	0.520910	−	0.853611i	$-0.325593\pi$
$281$	17.4641	1.04182	0.520910	+	0.853611i	$0.325593\pi$
$282$	0	0
$283$	7.66025i	0.455355i	0.973737	+	0.227677i	$0.0731132\pi$
$283$	7.66025i	0.455355i	−0.973737	+	0.227677i	$0.926887\pi$
$284$	0	0
$285$	5.46410i	0.323665i
$286$	0	0
$287$	−14.9282	−0.881184
$288$	0	0
$289$	−5.00000	−0.294118
$290$	0	0
$291$	− 4.67949i	− 0.274317i
$292$	0	0
$293$	− 11.8564i	− 0.692659i	−0.938113	−	0.346329i	$-0.887428\pi$
$293$	− 11.8564i	− 0.692659i	0.938113	−	0.346329i	$-0.112572\pi$
$294$	0	0
$295$	0.535898	0.0312012
$296$	0	0
$297$	−8.00000	−0.464207
$298$	0	0
$299$	14.5359i	0.840633i
$300$	0	0
$301$	− 23.8564i	− 1.37506i
$302$	0	0
$303$	8.00000	0.459588
$304$	0	0
$305$	4.92820	0.282188
$306$	0	0
$307$	26.9808i	1.53987i	0.638120	+	0.769937i	$0.279712\pi$
$307$	26.9808i	1.53987i	−0.638120	+	0.769937i	$0.720288\pi$
$308$	0	0
$309$	− 1.21539i	− 0.0691411i
$310$	0	0
$311$	3.32051	0.188289	0.0941444	−	0.995559i	$-0.469988\pi$
$311$	3.32051	0.188289	0.0941444	+	0.995559i	$0.469988\pi$
$312$	0	0
$313$	−31.8564	−1.80063	−0.900315	−	0.435238i	$-0.856664\pi$
$313$	−31.8564	−1.80063	−0.900315	+	0.435238i	$0.856664\pi$
$314$	0	0
$315$	6.73205i	0.379308i
$316$	0	0
$317$	− 15.4641i	− 0.868550i	−0.900780	−	0.434275i	$-0.857005\pi$
$317$	− 15.4641i	− 0.868550i	0.900780	−	0.434275i	$-0.142995\pi$
$318$	0	0
$319$	13.8564	0.775810
$320$	0	0
$321$	−0.535898	−0.0299109
$322$	0	0
$323$	− 25.8564i	− 1.43869i
$324$	0	0
$325$	3.46410i	0.192154i
$326$	0	0
$327$	−2.24871	−0.124354
$328$	0	0
$329$	−18.3923	−1.01400
$330$	0	0
$331$	− 14.0000i	− 0.769510i	−0.923019	−	0.384755i	$-0.874286\pi$
$331$	− 14.0000i	− 0.769510i	0.923019	−	0.384755i	$-0.125714\pi$
$332$	0	0
$333$	4.92820i	0.270064i
$334$	0	0
$335$	−7.26795	−0.397090
$336$	0	0
$337$	7.85641	0.427966	0.213983	−	0.976837i	$-0.431356\pi$
$337$	7.85641	0.427966	0.213983	+	0.976837i	$0.431356\pi$
$338$	0	0
$339$	− 0.679492i	− 0.0369049i
$340$	0	0
$341$	2.92820i	0.158571i
$342$	0	0
$343$	−17.8564	−0.964155
$344$	0	0
$345$	−3.07180	−0.165380
$346$	0	0
$347$	15.6603i	0.840686i	0.907365	+	0.420343i	$0.138090\pi$
$347$	15.6603i	0.840686i	−0.907365	+	0.420343i	$0.861910\pi$
$348$	0	0
$349$	28.0000i	1.49881i	0.662114	+	0.749403i	$0.269659\pi$
$349$	28.0000i	1.49881i	−0.662114	+	0.749403i	$0.730341\pi$
$350$	0	0
$351$	−13.8564	−0.739600
$352$	0	0
$353$	−0.928203	−0.0494033	−0.0247016	−	0.999695i	$-0.507864\pi$
$353$	−0.928203	−0.0494033	−0.0247016	+	0.999695i	$0.507864\pi$
$354$	0	0
$355$	1.46410i	0.0777064i
$356$	0	0
$357$	6.92820i	0.366679i
$358$	0	0
$359$	−5.07180	−0.267679	−0.133840	−	0.991003i	$-0.542731\pi$
$359$	−5.07180	−0.267679	−0.133840	+	0.991003i	$0.542731\pi$
$360$	0	0
$361$	−36.7128	−1.93225
$362$	0	0
$363$	− 5.12436i	− 0.268959i
$364$	0	0
$365$	0.535898i	0.0280502i
$366$	0	0
$367$	−27.1244	−1.41588	−0.707940	−	0.706273i	$-0.750375\pi$
$367$	−27.1244	−1.41588	−0.707940	+	0.706273i	$0.750375\pi$
$368$	0	0
$369$	−13.4641	−0.700913
$370$	0	0
$371$	− 12.3923i	− 0.643376i
$372$	0	0
$373$	− 29.7128i	− 1.53847i	−0.638965	−	0.769236i	$-0.720637\pi$
$373$	− 29.7128i	− 1.53847i	0.638965	−	0.769236i	$-0.279363\pi$
$374$	0	0
$375$	−0.732051	−0.0378029
$376$	0	0
$377$	24.0000	1.23606
$378$	0	0
$379$	12.2487i	0.629174i	0.949229	+	0.314587i	$0.101866\pi$
$379$	12.2487i	0.629174i	−0.949229	+	0.314587i	$0.898134\pi$
$380$	0	0
$381$	9.71281i	0.497602i
$382$	0	0
$383$	3.12436	0.159647	0.0798236	−	0.996809i	$-0.474564\pi$
$383$	3.12436	0.159647	0.0798236	+	0.996809i	$0.474564\pi$
$384$	0	0
$385$	5.46410	0.278476
$386$	0	0
$387$	− 21.5167i	− 1.09375i
$388$	0	0
$389$	34.7846i	1.76365i	0.471577	+	0.881825i	$0.343685\pi$
$389$	34.7846i	1.76365i	−0.471577	+	0.881825i	$0.656315\pi$
$390$	0	0
$391$	14.5359	0.735112
$392$	0	0
$393$	5.75129	0.290114
$394$	0	0
$395$	14.9282i	0.751119i
$396$	0	0
$397$	16.2487i	0.815499i	0.913094	+	0.407750i	$0.133686\pi$
$397$	16.2487i	0.815499i	−0.913094	+	0.407750i	$0.866314\pi$
$398$	0	0
$399$	14.9282	0.747345
$400$	0	0
$401$	19.8564	0.991582	0.495791	−	0.868442i	$-0.334878\pi$
$401$	19.8564	0.991582	0.495791	+	0.868442i	$0.334878\pi$
$402$	0	0
$403$	5.07180i	0.252644i
$404$	0	0
$405$	4.46410i	0.221823i
$406$	0	0
$407$	4.00000	0.198273
$408$	0	0
$409$	23.3205	1.15312	0.576562	−	0.817053i	$-0.304394\pi$
$409$	23.3205	1.15312	0.576562	+	0.817053i	$0.304394\pi$
$410$	0	0
$411$	6.53590i	0.322392i
$412$	0	0
$413$	− 1.46410i	− 0.0720437i
$414$	0	0
$415$	−4.73205	−0.232287
$416$	0	0
$417$	−5.46410	−0.267578
$418$	0	0
$419$	− 2.39230i	− 0.116872i	−0.998291	−	0.0584359i	$-0.981389\pi$
$419$	− 2.39230i	− 0.116872i	0.998291	−	0.0584359i	$-0.0186113\pi$
$420$	0	0
$421$	− 27.8564i	− 1.35764i	−0.734306	−	0.678819i	$-0.762492\pi$
$421$	− 27.8564i	− 1.35764i	0.734306	−	0.678819i	$-0.237508\pi$
$422$	0	0
$423$	−16.5885	−0.806558
$424$	0	0
$425$	3.46410	0.168034
$426$	0	0
$427$	− 13.4641i	− 0.651574i
$428$	0	0
$429$	5.07180i	0.244869i
$430$	0	0
$431$	14.5359	0.700170	0.350085	−	0.936718i	$-0.386153\pi$
$431$	14.5359	0.700170	0.350085	+	0.936718i	$0.386153\pi$
$432$	0	0
$433$	−12.5359	−0.602437	−0.301218	−	0.953555i	$-0.597393\pi$
$433$	−12.5359	−0.602437	−0.301218	+	0.953555i	$0.597393\pi$
$434$	0	0
$435$	5.07180i	0.243174i
$436$	0	0
$437$	− 31.3205i	− 1.49826i
$438$	0	0
$439$	0.784610	0.0374474	0.0187237	−	0.999825i	$-0.494040\pi$
$439$	0.784610	0.0374474	0.0187237	+	0.999825i	$0.494040\pi$
$440$	0	0
$441$	1.14359	0.0544568
$442$	0	0
$443$	− 30.9808i	− 1.47194i	−0.677014	−	0.735970i	$-0.736726\pi$
$443$	− 30.9808i	− 1.47194i	0.677014	−	0.735970i	$-0.263274\pi$
$444$	0	0
$445$	− 4.92820i	− 0.233619i
$446$	0	0
$447$	−14.5359	−0.687524
$448$	0	0
$449$	11.3205	0.534248	0.267124	−	0.963662i	$-0.413927\pi$
$449$	11.3205	0.534248	0.267124	+	0.963662i	$0.413927\pi$
$450$	0	0
$451$	10.9282i	0.514589i
$452$	0	0
$453$	6.14359i	0.288651i
$454$	0	0
$455$	9.46410	0.443684
$456$	0	0
$457$	14.7846	0.691595	0.345797	−	0.938309i	$-0.387608\pi$
$457$	14.7846	0.691595	0.345797	+	0.938309i	$0.387608\pi$
$458$	0	0
$459$	13.8564i	0.646762i
$460$	0	0
$461$	− 2.92820i	− 0.136380i	−0.997672	−	0.0681900i	$-0.978278\pi$
$461$	− 2.92820i	− 0.136380i	0.997672	−	0.0681900i	$-0.0217224\pi$
$462$	0	0
$463$	14.7321	0.684656	0.342328	−	0.939580i	$-0.388785\pi$
$463$	14.7321	0.684656	0.342328	+	0.939580i	$0.388785\pi$
$464$	0	0
$465$	−1.07180	−0.0497034
$466$	0	0
$467$	− 8.33975i	− 0.385917i	−0.981207	−	0.192959i	$-0.938192\pi$
$467$	− 8.33975i	− 0.385917i	0.981207	−	0.192959i	$-0.0618083\pi$
$468$	0	0
$469$	19.8564i	0.916884i
$470$	0	0
$471$	12.3923	0.571007
$472$	0	0
$473$	−17.4641	−0.803000
$474$	0	0
$475$	− 7.46410i	− 0.342476i
$476$	0	0
$477$	− 11.1769i	− 0.511756i
$478$	0	0
$479$	21.8564	0.998645	0.499322	−	0.866416i	$-0.333582\pi$
$479$	21.8564	0.998645	0.499322	+	0.866416i	$0.333582\pi$
$480$	0	0
$481$	6.92820	0.315899
$482$	0	0
$483$	8.39230i	0.381863i
$484$	0	0
$485$	6.39230i	0.290260i
$486$	0	0
$487$	24.5885	1.11421	0.557105	−	0.830442i	$-0.311912\pi$
$487$	24.5885	1.11421	0.557105	+	0.830442i	$0.311912\pi$
$488$	0	0
$489$	7.46410	0.337538
$490$	0	0
$491$	− 3.07180i	− 0.138628i	−0.997595	−	0.0693141i	$-0.977919\pi$
$491$	− 3.07180i	− 0.138628i	0.997595	−	0.0693141i	$-0.0220811\pi$
$492$	0	0
$493$	− 24.0000i	− 1.08091i
$494$	0	0
$495$	4.92820	0.221506
$496$	0	0
$497$	4.00000	0.179425
$498$	0	0
$499$	24.5359i	1.09838i	0.835698	+	0.549189i	$0.185063\pi$
$499$	24.5359i	1.09838i	−0.835698	+	0.549189i	$0.814937\pi$
$500$	0	0
$501$	− 14.7846i	− 0.660528i
$502$	0	0
$503$	17.6603	0.787432	0.393716	−	0.919232i	$-0.371189\pi$
$503$	17.6603	0.787432	0.393716	+	0.919232i	$0.371189\pi$
$504$	0	0
$505$	−10.9282	−0.486299
$506$	0	0
$507$	− 0.732051i	− 0.0325115i
$508$	0	0
$509$	− 25.8564i	− 1.14607i	−0.819533	−	0.573033i	$-0.805767\pi$
$509$	− 25.8564i	− 1.14607i	0.819533	−	0.573033i	$-0.194233\pi$
$510$	0	0
$511$	1.46410	0.0647680
$512$	0	0
$513$	29.8564	1.31819
$514$	0	0
$515$	1.66025i	0.0731595i
$516$	0	0
$517$	13.4641i	0.592151i
$518$	0	0
$519$	−1.46410	−0.0642669
$520$	0	0
$521$	−16.1436	−0.707264	−0.353632	−	0.935385i	$-0.615053\pi$
$521$	−16.1436	−0.707264	−0.353632	+	0.935385i	$0.615053\pi$
$522$	0	0
$523$	− 22.1962i	− 0.970570i	−0.874356	−	0.485285i	$-0.838716\pi$
$523$	− 22.1962i	− 0.970570i	0.874356	−	0.485285i	$-0.161284\pi$
$524$	0	0
$525$	2.00000i	0.0872872i
$526$	0	0
$527$	5.07180	0.220931
$528$	0	0
$529$	−5.39230	−0.234448
$530$	0	0
$531$	− 1.32051i	− 0.0573052i
$532$	0	0
$533$	18.9282i	0.819871i
$534$	0	0
$535$	0.732051	0.0316493
$536$	0	0
$537$	−11.3205	−0.488516
$538$	0	0
$539$	− 0.928203i	− 0.0399805i
$540$	0	0
$541$	13.0718i	0.562000i	0.959708	+	0.281000i	$0.0906662\pi$
$541$	13.0718i	0.562000i	−0.959708	+	0.281000i	$0.909334\pi$
$542$	0	0
$543$	−11.7128	−0.502645
$544$	0	0
$545$	3.07180	0.131581
$546$	0	0
$547$	36.7321i	1.57055i	0.619148	+	0.785275i	$0.287478\pi$
$547$	36.7321i	1.57055i	−0.619148	+	0.785275i	$0.712522\pi$
$548$	0	0
$549$	− 12.1436i	− 0.518276i
$550$	0	0
$551$	−51.7128	−2.20304
$552$	0	0
$553$	40.7846	1.73434
$554$	0	0
$555$	1.46410i	0.0621477i
$556$	0	0
$557$	26.7846i	1.13490i	0.823408	+	0.567450i	$0.192070\pi$
$557$	26.7846i	1.13490i	−0.823408	+	0.567450i	$0.807930\pi$
$558$	0	0
$559$	−30.2487	−1.27938
$560$	0	0
$561$	5.07180	0.214131
$562$	0	0
$563$	16.0526i	0.676535i	0.941050	+	0.338267i	$0.109841\pi$
$563$	16.0526i	0.676535i	−0.941050	+	0.338267i	$0.890159\pi$
$564$	0	0
$565$	0.928203i	0.0390498i
$566$	0	0
$567$	12.1962	0.512190
$568$	0	0
$569$	−6.53590	−0.273999	−0.137000	−	0.990571i	$-0.543746\pi$
$569$	−6.53590	−0.273999	−0.137000	+	0.990571i	$0.543746\pi$
$570$	0	0
$571$	− 34.7846i	− 1.45569i	−0.685741	−	0.727845i	$-0.740522\pi$
$571$	− 34.7846i	− 1.45569i	0.685741	−	0.727845i	$-0.259478\pi$
$572$	0	0
$573$	− 14.1436i	− 0.590857i
$574$	0	0
$575$	4.19615	0.174992
$576$	0	0
$577$	−43.5692	−1.81381	−0.906905	−	0.421335i	$-0.861562\pi$
$577$	−43.5692	−1.81381	−0.906905	+	0.421335i	$0.861562\pi$
$578$	0	0
$579$	− 5.46410i	− 0.227080i
$580$	0	0
$581$	12.9282i	0.536352i
$582$	0	0
$583$	−9.07180	−0.375715
$584$	0	0
$585$	8.53590	0.352916
$586$	0	0
$587$	− 14.1962i	− 0.585938i	−0.956122	−	0.292969i	$-0.905357\pi$
$587$	− 14.1962i	− 0.585938i	0.956122	−	0.292969i	$-0.0946433\pi$
$588$	0	0
$589$	− 10.9282i	− 0.450289i
$590$	0	0
$591$	9.17691	0.377488
$592$	0	0
$593$	−36.6410	−1.50467	−0.752333	−	0.658783i	$-0.771072\pi$
$593$	−36.6410	−1.50467	−0.752333	+	0.658783i	$0.771072\pi$
$594$	0	0
$595$	− 9.46410i	− 0.387990i
$596$	0	0
$597$	18.9282i	0.774680i
$598$	0	0
$599$	−34.6410	−1.41539	−0.707697	−	0.706516i	$-0.750266\pi$
$599$	−34.6410	−1.41539	−0.707697	+	0.706516i	$0.750266\pi$
$600$	0	0
$601$	25.4641	1.03870	0.519351	−	0.854561i	$-0.326174\pi$
$601$	25.4641	1.03870	0.519351	+	0.854561i	$0.326174\pi$
$602$	0	0
$603$	17.9090i	0.729309i
$604$	0	0
$605$	7.00000i	0.284590i
$606$	0	0
$607$	20.9808	0.851583	0.425791	−	0.904821i	$-0.359996\pi$
$607$	20.9808	0.851583	0.425791	+	0.904821i	$0.359996\pi$
$608$	0	0
$609$	13.8564	0.561490
$610$	0	0
$611$	23.3205i	0.943447i
$612$	0	0
$613$	− 5.60770i	− 0.226493i	−0.993567	−	0.113246i	$-0.963875\pi$
$613$	− 5.60770i	− 0.226493i	0.993567	−	0.113246i	$-0.0361249\pi$
$614$	0	0
$615$	−4.00000	−0.161296
$616$	0	0
$617$	−27.4641	−1.10566	−0.552832	−	0.833293i	$-0.686453\pi$
$617$	−27.4641	−1.10566	−0.552832	+	0.833293i	$0.686453\pi$
$618$	0	0
$619$	− 33.3205i	− 1.33926i	−0.742693	−	0.669632i	$-0.766452\pi$
$619$	− 33.3205i	− 1.33926i	0.742693	−	0.669632i	$-0.233548\pi$
$620$	0	0
$621$	16.7846i	0.673543i
$622$	0	0
$623$	−13.4641	−0.539428
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	− 10.9282i	− 0.436430i
$628$	0	0
$629$	− 6.92820i	− 0.276246i
$630$	0	0
$631$	−11.3205	−0.450662	−0.225331	−	0.974282i	$-0.572346\pi$
$631$	−11.3205	−0.450662	−0.225331	+	0.974282i	$0.572346\pi$
$632$	0	0
$633$	−10.8231	−0.430179
$634$	0	0
$635$	− 13.2679i	− 0.526523i
$636$	0	0
$637$	− 1.60770i	− 0.0636992i
$638$	0	0
$639$	3.60770	0.142718
$640$	0	0
$641$	−20.3923	−0.805448	−0.402724	−	0.915322i	$-0.631936\pi$
$641$	−20.3923	−0.805448	−0.402724	+	0.915322i	$0.631936\pi$
$642$	0	0
$643$	14.8756i	0.586638i	0.956015	+	0.293319i	$0.0947598\pi$
$643$	14.8756i	0.586638i	−0.956015	+	0.293319i	$0.905240\pi$
$644$	0	0
$645$	− 6.39230i	− 0.251697i
$646$	0	0
$647$	−13.2679	−0.521617	−0.260808	−	0.965391i	$-0.583989\pi$
$647$	−13.2679	−0.521617	−0.260808	+	0.965391i	$0.583989\pi$
$648$	0	0
$649$	−1.07180	−0.0420717
$650$	0	0
$651$	2.92820i	0.114765i
$652$	0	0
$653$	36.2487i	1.41852i	0.704946	+	0.709261i	$0.250971\pi$
$653$	36.2487i	1.41852i	−0.704946	+	0.709261i	$0.749029\pi$
$654$	0	0
$655$	−7.85641	−0.306975
$656$	0	0
$657$	1.32051	0.0515179
$658$	0	0
$659$	17.3205i	0.674711i	0.941377	+	0.337356i	$0.109532\pi$
$659$	17.3205i	0.674711i	−0.941377	+	0.337356i	$0.890468\pi$
$660$	0	0
$661$	35.8564i	1.39465i	0.716754	+	0.697326i	$0.245627\pi$
$661$	35.8564i	1.39465i	−0.716754	+	0.697326i	$0.754373\pi$
$662$	0	0
$663$	8.78461	0.341166
$664$	0	0
$665$	−20.3923	−0.790780
$666$	0	0
$667$	− 29.0718i	− 1.12566i
$668$	0	0
$669$	− 11.8564i	− 0.458395i
$670$	0	0
$671$	−9.85641	−0.380502
$672$	0	0
$673$	19.4641	0.750286	0.375143	−	0.926967i	$-0.377594\pi$
$673$	19.4641	0.750286	0.375143	+	0.926967i	$0.377594\pi$
$674$	0	0
$675$	4.00000i	0.153960i
$676$	0	0
$677$	− 38.3923i	− 1.47554i	−0.675054	−	0.737768i	$-0.735880\pi$
$677$	− 38.3923i	− 1.47554i	0.675054	−	0.737768i	$-0.264120\pi$
$678$	0	0
$679$	17.4641	0.670211
$680$	0	0
$681$	−20.5359	−0.786937
$682$	0	0
$683$	34.9808i	1.33850i	0.743037	+	0.669251i	$0.233385\pi$
$683$	34.9808i	1.33850i	−0.743037	+	0.669251i	$0.766615\pi$
$684$	0	0
$685$	− 8.92820i	− 0.341129i
$686$	0	0
$687$	2.92820	0.111718
$688$	0	0
$689$	−15.7128	−0.598610
$690$	0	0
$691$	18.0000i	0.684752i	0.939563	+	0.342376i	$0.111232\pi$
$691$	18.0000i	0.684752i	−0.939563	+	0.342376i	$0.888768\pi$
$692$	0	0
$693$	− 13.4641i	− 0.511459i
$694$	0	0
$695$	7.46410	0.283130
$696$	0	0
$697$	18.9282	0.716957
$698$	0	0
$699$	− 21.4641i	− 0.811847i
$700$	0	0
$701$	− 32.9282i	− 1.24368i	−0.783144	−	0.621841i	$-0.786385\pi$
$701$	− 32.9282i	− 1.24368i	0.783144	−	0.621841i	$-0.213615\pi$
$702$	0	0
$703$	−14.9282	−0.563028
$704$	0	0
$705$	−4.92820	−0.185607
$706$	0	0
$707$	29.8564i	1.12287i
$708$	0	0
$709$	− 28.7846i	− 1.08103i	−0.841335	−	0.540514i	$-0.818230\pi$
$709$	− 28.7846i	− 1.08103i	0.841335	−	0.540514i	$-0.181770\pi$
$710$	0	0
$711$	36.7846	1.37953
$712$	0	0
$713$	6.14359	0.230079
$714$	0	0
$715$	− 6.92820i	− 0.259100i
$716$	0	0
$717$	− 14.6410i	− 0.546779i
$718$	0	0
$719$	−25.8564	−0.964281	−0.482141	−	0.876094i	$-0.660141\pi$
$719$	−25.8564	−0.964281	−0.482141	+	0.876094i	$0.660141\pi$
$720$	0	0
$721$	4.53590	0.168926
$722$	0	0
$723$	3.21539i	0.119582i
$724$	0	0
$725$	− 6.92820i	− 0.257307i
$726$	0	0
$727$	−14.0526	−0.521181	−0.260590	−	0.965449i	$-0.583917\pi$
$727$	−14.0526	−0.521181	−0.260590	+	0.965449i	$0.583917\pi$
$728$	0	0
$729$	2.21539	0.0820515
$730$	0	0
$731$	30.2487i	1.11879i
$732$	0	0
$733$	48.9282i	1.80720i	0.428373	+	0.903602i	$0.359087\pi$
$733$	48.9282i	1.80720i	−0.428373	+	0.903602i	$0.640913\pi$
$734$	0	0
$735$	0.339746	0.0125317
$736$	0	0
$737$	14.5359	0.535437
$738$	0	0
$739$	− 5.32051i	− 0.195718i	−0.995200	−	0.0978590i	$-0.968801\pi$
$739$	− 5.32051i	− 0.195718i	0.995200	−	0.0978590i	$-0.0311994\pi$
$740$	0	0
$741$	− 18.9282i	− 0.695345i
$742$	0	0
$743$	40.9808	1.50344	0.751719	−	0.659483i	$-0.229225\pi$
$743$	40.9808	1.50344	0.751719	+	0.659483i	$0.229225\pi$
$744$	0	0
$745$	19.8564	0.727482
$746$	0	0
$747$	11.6603i	0.426626i
$748$	0	0
$749$	− 2.00000i	− 0.0730784i
$750$	0	0
$751$	22.2487	0.811867	0.405934	−	0.913903i	$-0.366946\pi$
$751$	22.2487	0.811867	0.405934	+	0.913903i	$0.366946\pi$
$752$	0	0
$753$	8.10512	0.295367
$754$	0	0
$755$	− 8.39230i	− 0.305427i
$756$	0	0
$757$	32.9282i	1.19680i	0.801199	+	0.598398i	$0.204196\pi$
$757$	32.9282i	1.19680i	−0.801199	+	0.598398i	$0.795804\pi$
$758$	0	0
$759$	6.14359	0.222998
$760$	0	0
$761$	49.7128	1.80209	0.901044	−	0.433728i	$-0.142802\pi$
$761$	49.7128	1.80209	0.901044	+	0.433728i	$0.142802\pi$
$762$	0	0
$763$	− 8.39230i	− 0.303822i
$764$	0	0
$765$	− 8.53590i	− 0.308616i
$766$	0	0
$767$	−1.85641	−0.0670310
$768$	0	0
$769$	−0.928203	−0.0334719	−0.0167359	−	0.999860i	$-0.505327\pi$
$769$	−0.928203	−0.0334719	−0.0167359	+	0.999860i	$0.505327\pi$
$770$	0	0
$771$	1.46410i	0.0527283i
$772$	0	0
$773$	1.60770i	0.0578248i	0.999582	+	0.0289124i	$0.00920438\pi$
$773$	1.60770i	0.0578248i	−0.999582	+	0.0289124i	$0.990796\pi$
$774$	0	0
$775$	1.46410	0.0525921
$776$	0	0
$777$	4.00000	0.143499
$778$	0	0
$779$	− 40.7846i	− 1.46126i
$780$	0	0
$781$	− 2.92820i	− 0.104779i
$782$	0	0
$783$	27.7128	0.990375
$784$	0	0
$785$	−16.9282	−0.604193
$786$	0	0
$787$	− 14.5885i	− 0.520022i	−0.965606	−	0.260011i	$-0.916274\pi$
$787$	− 14.5885i	− 0.520022i	0.965606	−	0.260011i	$-0.0837262\pi$
$788$	0	0
$789$	4.14359i	0.147516i
$790$	0	0
$791$	2.53590	0.0901662
$792$	0	0
$793$	−17.0718	−0.606237
$794$	0	0
$795$	− 3.32051i	− 0.117766i
$796$	0	0
$797$	26.1051i	0.924691i	0.886700	+	0.462345i	$0.152992\pi$
$797$	26.1051i	0.924691i	−0.886700	+	0.462345i	$0.847008\pi$
$798$	0	0
$799$	23.3205	0.825020
$800$	0	0
$801$	−12.1436	−0.429073
$802$	0	0
$803$	− 1.07180i	− 0.0378229i
$804$	0	0
$805$	− 11.4641i	− 0.404056i
$806$	0	0
$807$	−3.60770	−0.126997
$808$	0	0
$809$	−3.85641	−0.135584	−0.0677920	−	0.997699i	$-0.521595\pi$
$809$	−3.85641	−0.135584	−0.0677920	+	0.997699i	$0.521595\pi$
$810$	0	0
$811$	− 15.0718i	− 0.529242i	−0.964352	−	0.264621i	$-0.914753\pi$
$811$	− 15.0718i	− 0.529242i	0.964352	−	0.264621i	$-0.0852469\pi$
$812$	0	0
$813$	11.2154i	0.393341i
$814$	0	0
$815$	−10.1962	−0.357156
$816$	0	0
$817$	65.1769	2.28025
$818$	0	0
$819$	− 23.3205i	− 0.814885i
$820$	0	0
$821$	6.78461i	0.236785i	0.992967	+	0.118392i	$0.0377740\pi$
$821$	6.78461i	0.236785i	−0.992967	+	0.118392i	$0.962226\pi$
$822$	0	0
$823$	−15.1244	−0.527202	−0.263601	−	0.964632i	$-0.584910\pi$
$823$	−15.1244	−0.527202	−0.263601	+	0.964632i	$0.584910\pi$
$824$	0	0
$825$	1.46410	0.0509735
$826$	0	0
$827$	− 1.12436i	− 0.0390977i	−0.999809	−	0.0195488i	$-0.993777\pi$
$827$	− 1.12436i	− 0.0390977i	0.999809	−	0.0195488i	$-0.00622298\pi$
$828$	0	0
$829$	− 15.0718i	− 0.523465i	−0.965140	−	0.261733i	$-0.915706\pi$
$829$	− 15.0718i	− 0.523465i	0.965140	−	0.261733i	$-0.0842938\pi$
$830$	0	0
$831$	−1.46410	−0.0507891
$832$	0	0
$833$	−1.60770	−0.0557033
$834$	0	0
$835$	20.1962i	0.698917i
$836$	0	0
$837$	5.85641i	0.202427i
$838$	0	0
$839$	16.7846	0.579469	0.289735	−	0.957107i	$-0.406433\pi$
$839$	16.7846	0.579469	0.289735	+	0.957107i	$0.406433\pi$
$840$	0	0
$841$	−19.0000	−0.655172
$842$	0	0
$843$	− 12.7846i	− 0.440325i
$844$	0	0
$845$	1.00000i	0.0344010i
$846$	0	0
$847$	19.1244	0.657121
$848$	0	0
$849$	5.60770	0.192456
$850$	0	0
$851$	− 8.39230i	− 0.287685i
$852$	0	0
$853$	42.3923i	1.45148i	0.687967	+	0.725742i	$0.258504\pi$
$853$	42.3923i	1.45148i	−0.687967	+	0.725742i	$0.741496\pi$
$854$	0	0
$855$	−18.3923	−0.629004
$856$	0	0
$857$	7.85641	0.268370	0.134185	−	0.990956i	$-0.457158\pi$
$857$	7.85641	0.268370	0.134185	+	0.990956i	$0.457158\pi$
$858$	0	0
$859$	− 20.2487i	− 0.690877i	−0.938441	−	0.345439i	$-0.887730\pi$
$859$	− 20.2487i	− 0.690877i	0.938441	−	0.345439i	$-0.112270\pi$
$860$	0	0
$861$	10.9282i	0.372432i
$862$	0	0
$863$	−30.3397	−1.03278	−0.516388	−	0.856354i	$-0.672724\pi$
$863$	−30.3397	−1.03278	−0.516388	+	0.856354i	$0.672724\pi$
$864$	0	0
$865$	2.00000	0.0680020
$866$	0	0
$867$	3.66025i	0.124309i
$868$	0	0
$869$	− 29.8564i	− 1.01281i
$870$	0	0
$871$	25.1769	0.853087
$872$	0	0
$873$	15.7513	0.533100
$874$	0	0
$875$	− 2.73205i	− 0.0923602i
$876$	0	0
$877$	− 53.7128i	− 1.81375i	−0.421397	−	0.906876i	$-0.638460\pi$
$877$	− 53.7128i	− 1.81375i	0.421397	−	0.906876i	$-0.361540\pi$
$878$	0	0
$879$	−8.67949	−0.292752
$880$	0	0
$881$	2.53590	0.0854366	0.0427183	−	0.999087i	$-0.486398\pi$
$881$	2.53590	0.0854366	0.0427183	+	0.999087i	$0.486398\pi$
$882$	0	0
$883$	37.9090i	1.27574i	0.770145	+	0.637869i	$0.220184\pi$
$883$	37.9090i	1.27574i	−0.770145	+	0.637869i	$0.779816\pi$
$884$	0	0
$885$	− 0.392305i	− 0.0131872i
$886$	0	0
$887$	−51.9090	−1.74293	−0.871466	−	0.490455i	$-0.836830\pi$
$887$	−51.9090	−1.74293	−0.871466	+	0.490455i	$0.836830\pi$
$888$	0	0
$889$	−36.2487	−1.21574
$890$	0	0
$891$	− 8.92820i	− 0.299106i
$892$	0	0
$893$	− 50.2487i	− 1.68151i
$894$	0	0
$895$	15.4641	0.516908
$896$	0	0
$897$	10.6410	0.355293
$898$	0	0
$899$	− 10.1436i	− 0.338308i
$900$	0	0
$901$	15.7128i	0.523470i
$902$	0	0
$903$	−17.4641	−0.581169
$904$	0	0
$905$	16.0000	0.531858
$906$	0	0
$907$	29.1244i	0.967058i	0.875328	+	0.483529i	$0.160645\pi$
$907$	29.1244i	0.967058i	−0.875328	+	0.483529i	$0.839355\pi$
$908$	0	0
$909$	26.9282i	0.893152i
$910$	0	0
$911$	−13.1769	−0.436571	−0.218285	−	0.975885i	$-0.570046\pi$
$911$	−13.1769	−0.436571	−0.218285	+	0.975885i	$0.570046\pi$
$912$	0	0
$913$	9.46410	0.313216
$914$	0	0
$915$	− 3.60770i	− 0.119267i
$916$	0	0
$917$	21.4641i	0.708807i
$918$	0	0
$919$	25.0718	0.827042	0.413521	−	0.910495i	$-0.364299\pi$
$919$	25.0718	0.827042	0.413521	+	0.910495i	$0.364299\pi$
$920$	0	0
$921$	19.7513	0.650827
$922$	0	0
$923$	− 5.07180i	− 0.166940i
$924$	0	0
$925$	− 2.00000i	− 0.0657596i
$926$	0	0
$927$	4.09103	0.134367
$928$	0	0
$929$	−10.5359	−0.345672	−0.172836	−	0.984951i	$-0.555293\pi$
$929$	−10.5359	−0.345672	−0.172836	+	0.984951i	$0.555293\pi$
$930$	0	0
$931$	3.46410i	0.113531i
$932$	0	0
$933$	− 2.43078i	− 0.0795802i
$934$	0	0
$935$	−6.92820	−0.226576
$936$	0	0
$937$	−44.2487	−1.44554	−0.722771	−	0.691087i	$-0.757132\pi$
$937$	−44.2487	−1.44554	−0.722771	+	0.691087i	$0.757132\pi$
$938$	0	0
$939$	23.3205i	0.761036i
$940$	0	0
$941$	32.0000i	1.04317i	0.853199	+	0.521585i	$0.174659\pi$
$941$	32.0000i	1.04317i	−0.853199	+	0.521585i	$0.825341\pi$
$942$	0	0
$943$	22.9282	0.746645
$944$	0	0
$945$	10.9282	0.355494
$946$	0	0
$947$	21.1244i	0.686449i	0.939253	+	0.343225i	$0.111519\pi$
$947$	21.1244i	0.686449i	−0.939253	+	0.343225i	$0.888481\pi$
$948$	0	0
$949$	− 1.85641i	− 0.0602615i
$950$	0	0
$951$	−11.3205	−0.367093
$952$	0	0
$953$	−58.7846	−1.90422	−0.952110	−	0.305755i	$-0.901091\pi$
$953$	−58.7846	−1.90422	−0.952110	+	0.305755i	$0.901091\pi$
$954$	0	0
$955$	19.3205i	0.625197i
$956$	0	0
$957$	− 10.1436i	− 0.327896i
$958$	0	0
$959$	−24.3923	−0.787669
$960$	0	0
$961$	−28.8564	−0.930852
$962$	0	0
$963$	− 1.80385i	− 0.0581282i
$964$	0	0
$965$	7.46410i	0.240278i
$966$	0	0
$967$	−33.6603	−1.08244	−0.541220	−	0.840881i	$-0.682038\pi$
$967$	−33.6603	−1.08244	−0.541220	+	0.840881i	$0.682038\pi$
$968$	0	0
$969$	−18.9282	−0.608061
$970$	0	0
$971$	23.0718i	0.740409i	0.928950	+	0.370205i	$0.120712\pi$
$971$	23.0718i	0.740409i	−0.928950	+	0.370205i	$0.879288\pi$
$972$	0	0
$973$	− 20.3923i	− 0.653747i
$974$	0	0
$975$	2.53590	0.0812137
$976$	0	0
$977$	−31.4641	−1.00663	−0.503313	−	0.864104i	$-0.667886\pi$
$977$	−31.4641	−1.00663	−0.503313	+	0.864104i	$0.667886\pi$
$978$	0	0
$979$	9.85641i	0.315012i
$980$	0	0
$981$	− 7.56922i	− 0.241667i
$982$	0	0
$983$	45.2679	1.44382	0.721912	−	0.691985i	$-0.243264\pi$
$983$	45.2679	1.44382	0.721912	+	0.691985i	$0.243264\pi$
$984$	0	0
$985$	−12.5359	−0.399427
$986$	0	0
$987$	13.4641i	0.428567i
$988$	0	0
$989$	36.6410i	1.16512i
$990$	0	0
$991$	−34.5359	−1.09707	−0.548534	−	0.836128i	$-0.684814\pi$
$991$	−34.5359	−1.09707	−0.548534	+	0.836128i	$0.684814\pi$
$992$	0	0
$993$	−10.2487	−0.325233
$994$	0	0
$995$	− 25.8564i	− 0.819703i
$996$	0	0
$997$	− 51.1769i	− 1.62079i	−0.585884	−	0.810395i	$-0.699253\pi$
$997$	− 51.1769i	− 1.62079i	0.585884	−	0.810395i	$-0.300747\pi$
$998$	0	0
$999$	8.00000	0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	160.2.d.a.81.2		4
3.2	odd	2		1440.2.k.e.721.2		4
4.3	odd	2		40.2.d.a.21.3	✓	4
5.2	odd	4		800.2.f.e.49.2		4
5.3	odd	4		800.2.f.c.49.3		4
5.4	even	2		800.2.d.e.401.3		4
8.3	odd	2		40.2.d.a.21.4	yes	4
8.5	even	2	inner	160.2.d.a.81.3		4
12.11	even	2		360.2.k.e.181.2		4
15.2	even	4		7200.2.d.n.2449.4		4
15.8	even	4		7200.2.d.o.2449.1		4
15.14	odd	2		7200.2.k.j.3601.2		4
16.3	odd	4		1280.2.a.o.1.1		2
16.5	even	4		1280.2.a.n.1.1		2
16.11	odd	4		1280.2.a.a.1.2		2
16.13	even	4		1280.2.a.d.1.2		2
20.3	even	4		200.2.f.c.149.1		4
20.7	even	4		200.2.f.e.149.4		4
20.19	odd	2		200.2.d.f.101.2		4
24.5	odd	2		1440.2.k.e.721.4		4
24.11	even	2		360.2.k.e.181.1		4
40.3	even	4		200.2.f.e.149.3		4
40.13	odd	4		800.2.f.e.49.1		4
40.19	odd	2		200.2.d.f.101.1		4
40.27	even	4		200.2.f.c.149.2		4
40.29	even	2		800.2.d.e.401.2		4
40.37	odd	4		800.2.f.c.49.4		4
60.23	odd	4		1800.2.d.p.1549.4		4
60.47	odd	4		1800.2.d.l.1549.1		4
60.59	even	2		1800.2.k.j.901.3		4
80.19	odd	4		6400.2.a.z.1.2		2
80.29	even	4		6400.2.a.cj.1.1		2
80.59	odd	4		6400.2.a.ce.1.1		2
80.69	even	4		6400.2.a.be.1.2		2
120.29	odd	2		7200.2.k.j.3601.1		4
120.53	even	4		7200.2.d.n.2449.1		4
120.59	even	2		1800.2.k.j.901.4		4
120.77	even	4		7200.2.d.o.2449.4		4
120.83	odd	4		1800.2.d.l.1549.2		4
120.107	odd	4		1800.2.d.p.1549.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.2.d.a.21.3	✓	4	4.3	odd	2
40.2.d.a.21.4	yes	4	8.3	odd	2
160.2.d.a.81.2		4	1.1	even	1	trivial
160.2.d.a.81.3		4	8.5	even	2	inner
200.2.d.f.101.1		4	40.19	odd	2
200.2.d.f.101.2		4	20.19	odd	2
200.2.f.c.149.1		4	20.3	even	4
200.2.f.c.149.2		4	40.27	even	4
200.2.f.e.149.3		4	40.3	even	4
200.2.f.e.149.4		4	20.7	even	4
360.2.k.e.181.1		4	24.11	even	2
360.2.k.e.181.2		4	12.11	even	2
800.2.d.e.401.2		4	40.29	even	2
800.2.d.e.401.3		4	5.4	even	2
800.2.f.c.49.3		4	5.3	odd	4
800.2.f.c.49.4		4	40.37	odd	4
800.2.f.e.49.1		4	40.13	odd	4
800.2.f.e.49.2		4	5.2	odd	4
1280.2.a.a.1.2		2	16.11	odd	4
1280.2.a.d.1.2		2	16.13	even	4
1280.2.a.n.1.1		2	16.5	even	4
1280.2.a.o.1.1		2	16.3	odd	4
1440.2.k.e.721.2		4	3.2	odd	2
1440.2.k.e.721.4		4	24.5	odd	2
1800.2.d.l.1549.1		4	60.47	odd	4
1800.2.d.l.1549.2		4	120.83	odd	4
1800.2.d.p.1549.3		4	120.107	odd	4
1800.2.d.p.1549.4		4	60.23	odd	4
1800.2.k.j.901.3		4	60.59	even	2
1800.2.k.j.901.4		4	120.59	even	2
6400.2.a.z.1.2		2	80.19	odd	4
6400.2.a.be.1.2		2	80.69	even	4
6400.2.a.ce.1.1		2	80.59	odd	4
6400.2.a.cj.1.1		2	80.29	even	4
7200.2.d.n.2449.1		4	120.53	even	4
7200.2.d.n.2449.4		4	15.2	even	4
7200.2.d.o.2449.1		4	15.8	even	4
7200.2.d.o.2449.4		4	120.77	even	4
7200.2.k.j.3601.1		4	120.29	odd	2
7200.2.k.j.3601.2		4	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-0.732051i q^{3} +1.00000i q^{5} +2.73205 q^{7} +2.46410 q^{9} +O(q^{10})\)	\(q-0.732051i q^{3} +1.00000i q^{5} +2.73205 q^{7} +2.46410 q^{9} -2.00000i q^{11} -3.46410i q^{13} +0.732051 q^{15} -3.46410 q^{17} +7.46410i q^{19} -2.00000i q^{21} -4.19615 q^{23} -1.00000 q^{25} -4.00000i q^{27} +6.92820i q^{29} -1.46410 q^{31} -1.46410 q^{33} +2.73205i q^{35} +2.00000i q^{37} -2.53590 q^{39} -5.46410 q^{41} -8.73205i q^{43} +2.46410i q^{45} -6.73205 q^{47} +0.464102 q^{49} +2.53590i q^{51} -4.53590i q^{53} +2.00000 q^{55} +5.46410 q^{57} -0.535898i q^{59} -4.92820i q^{61} +6.73205 q^{63} +3.46410 q^{65} +7.26795i q^{67} +3.07180i q^{69} +1.46410 q^{71} +0.535898 q^{73} +0.732051i q^{75} -5.46410i q^{77} +14.9282 q^{79} +4.46410 q^{81} +4.73205i q^{83} -3.46410i q^{85} +5.07180 q^{87} -4.92820 q^{89} -9.46410i q^{91} +1.07180i q^{93} -7.46410 q^{95} +6.39230 q^{97} -4.92820i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{7} - 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^7 - 4 * q^9	\( 4 q + 4 q^{7} - 4 q^{9} - 4 q^{15} + 4 q^{23} - 4 q^{25} + 8 q^{31} + 8 q^{33} - 24 q^{39} - 8 q^{41} - 20 q^{47} - 12 q^{49} + 8 q^{55} + 8 q^{57} + 20 q^{63} - 8 q^{71} + 16 q^{73} + 32 q^{79} + 4 q^{81} + 48 q^{87} + 8 q^{89} - 16 q^{95} - 16 q^{97}+O(q^{100}) \) 4 * q + 4 * q^7 - 4 * q^9 - 4 * q^15 + 4 * q^23 - 4 * q^25 + 8 * q^31 + 8 * q^33 - 24 * q^39 - 8 * q^41 - 20 * q^47 - 12 * q^49 + 8 * q^55 + 8 * q^57 + 20 * q^63 - 8 * q^71 + 16 * q^73 + 32 * q^79 + 4 * q^81 + 48 * q^87 + 8 * q^89 - 16 * q^95 - 16 * q^97

Introduction

L-functions

Modular forms

Varieties

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Database

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