LMFDB - Embedded newform 3080.2.a.r.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3080,2,Mod(1,3080)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3080, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3080.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3080 = 2^{3} \cdot 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3080.a (trivial)

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:	yes
Analytic conductor:	$24.5939238226$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.549616.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 7x^{3} + 4x^{2} + 4x - 2 $ x^5 - 2x^4 - 7x^3 + 4x^2 + 4x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.718866$ of defining polynomial
Character	$\chi$	$=$	3080.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.33189 q^{3} -1.00000 q^{5} -1.00000 q^{7} +2.43773 q^{9} +O(q^{10})$	$q+2.33189 q^{3} -1.00000 q^{5} -1.00000 q^{7} +2.43773 q^{9} +1.00000 q^{11} +1.04175 q^{13} -2.33189 q^{15} +6.88368 q^{17} -6.55179 q^{19} -2.33189 q^{21} -0.836442 q^{23} +1.00000 q^{25} -1.31115 q^{27} +8.55179 q^{29} +3.19278 q^{31} +2.33189 q^{33} +1.00000 q^{35} +7.50023 q^{37} +2.42926 q^{39} -7.72012 q^{41} +10.8739 q^{43} -2.43773 q^{45} +13.5475 q^{47} +1.00000 q^{49} +16.0520 q^{51} +3.85241 q^{53} -1.00000 q^{55} -15.2781 q^{57} -7.16834 q^{59} +2.29287 q^{61} -2.43773 q^{63} -1.04175 q^{65} -10.0771 q^{67} -1.95050 q^{69} +3.89143 q^{71} +0.547006 q^{73} +2.33189 q^{75} -1.00000 q^{77} +8.83644 q^{79} -10.3707 q^{81} -7.89143 q^{83} -6.88368 q^{85} +19.9419 q^{87} +7.67288 q^{89} -1.04175 q^{91} +7.44522 q^{93} +6.55179 q^{95} -6.52672 q^{97} +2.43773 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 2 q^{3} - 5 q^{5} - 5 q^{7} + 9 q^{9}+O(q^{10}) $ 5 * q - 2 * q^3 - 5 * q^5 - 5 * q^7 + 9 * q^9	$ 5 q - 2 q^{3} - 5 q^{5} - 5 q^{7} + 9 q^{9} + 5 q^{11} + 2 q^{13} + 2 q^{15} - 6 q^{17} - 6 q^{19} + 2 q^{21} + 2 q^{23} + 5 q^{25} - 20 q^{27} + 16 q^{29} + 12 q^{31} - 2 q^{33} + 5 q^{35} + 4 q^{37} - 8 q^{39} + 8 q^{41} + 4 q^{43} - 9 q^{45} + 5 q^{49} + 20 q^{51} + 12 q^{53} - 5 q^{55} + 8 q^{57} - 16 q^{59} - 2 q^{61} - 9 q^{63} - 2 q^{65} - 4 q^{67} + 20 q^{69} + 12 q^{71} + 2 q^{73} - 2 q^{75} - 5 q^{77} + 38 q^{79} + 13 q^{81} - 32 q^{83} + 6 q^{85} - 12 q^{87} + 26 q^{89} - 2 q^{91} + 16 q^{93} + 6 q^{95} + 8 q^{97} + 9 q^{99}+O(q^{100}) $ 5 * q - 2 * q^3 - 5 * q^5 - 5 * q^7 + 9 * q^9 + 5 * q^11 + 2 * q^13 + 2 * q^15 - 6 * q^17 - 6 * q^19 + 2 * q^21 + 2 * q^23 + 5 * q^25 - 20 * q^27 + 16 * q^29 + 12 * q^31 - 2 * q^33 + 5 * q^35 + 4 * q^37 - 8 * q^39 + 8 * q^41 + 4 * q^43 - 9 * q^45 + 5 * q^49 + 20 * q^51 + 12 * q^53 - 5 * q^55 + 8 * q^57 - 16 * q^59 - 2 * q^61 - 9 * q^63 - 2 * q^65 - 4 * q^67 + 20 * q^69 + 12 * q^71 + 2 * q^73 - 2 * q^75 - 5 * q^77 + 38 * q^79 + 13 * q^81 - 32 * q^83 + 6 * q^85 - 12 * q^87 + 26 * q^89 - 2 * q^91 + 16 * q^93 + 6 * q^95 + 8 * q^97 + 9 * q^99

Display 100 coefficients 10 coefficients

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
$3$	2.33189	1.34632	0.673160	−	0.739497i	$-0.264937\pi$
$3$	2.33189	1.34632	0.673160	+	0.739497i	$0.264937\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	2.43773	0.812578
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	1.04175	0.288930	0.144465	−	0.989510i	$-0.453854\pi$
$13$	1.04175	0.288930	0.144465	+	0.989510i	$0.453854\pi$
$14$	0	0
$15$	−2.33189	−0.602093
$16$	0	0
$17$	6.88368	1.66954	0.834769	−	0.550601i	$-0.185601\pi$
$17$	6.88368	1.66954	0.834769	+	0.550601i	$0.185601\pi$
$18$	0	0
$19$	−6.55179	−1.50308	−0.751541	−	0.659686i	$-0.770689\pi$
$19$	−6.55179	−1.50308	−0.751541	+	0.659686i	$0.770689\pi$
$20$	0	0
$21$	−2.33189	−0.508861
$22$	0	0
$23$	−0.836442	−0.174410	−0.0872051	−	0.996190i	$-0.527794\pi$
$23$	−0.836442	−0.174410	−0.0872051	+	0.996190i	$0.527794\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.31115	−0.252331
$28$	0	0
$29$	8.55179	1.58803	0.794013	−	0.607900i	$-0.207988\pi$
$29$	8.55179	1.58803	0.794013	+	0.607900i	$0.207988\pi$
$30$	0	0
$31$	3.19278	0.573440	0.286720	−	0.958014i	$-0.407435\pi$
$31$	3.19278	0.573440	0.286720	+	0.958014i	$0.407435\pi$
$32$	0	0
$33$	2.33189	0.405931
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	7.50023	1.23303	0.616515	−	0.787343i	$-0.288544\pi$
$37$	7.50023	1.23303	0.616515	+	0.787343i	$0.288544\pi$
$38$	0	0
$39$	2.42926	0.388993
$40$	0	0
$41$	−7.72012	−1.20568	−0.602840	−	0.797862i	$-0.705964\pi$
$41$	−7.72012	−1.20568	−0.602840	+	0.797862i	$0.705964\pi$
$42$	0	0
$43$	10.8739	1.65825	0.829125	−	0.559063i	$-0.188839\pi$
$43$	10.8739	1.65825	0.829125	+	0.559063i	$0.188839\pi$
$44$	0	0
$45$	−2.43773	−0.363396
$46$	0	0
$47$	13.5475	1.97610	0.988051	−	0.154129i	$-0.0492571\pi$
$47$	13.5475	1.97610	0.988051	+	0.154129i	$0.0492571\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	16.0520	2.24773
$52$	0	0
$53$	3.85241	0.529169	0.264585	−	0.964362i	$-0.414765\pi$
$53$	3.85241	0.529169	0.264585	+	0.964362i	$0.414765\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	0	0
$57$	−15.2781	−2.02363
$58$	0	0
$59$	−7.16834	−0.933238	−0.466619	−	0.884458i	$-0.654528\pi$
$59$	−7.16834	−0.933238	−0.466619	+	0.884458i	$0.654528\pi$
$60$	0	0
$61$	2.29287	0.293572	0.146786	−	0.989168i	$-0.453107\pi$
$61$	2.29287	0.293572	0.146786	+	0.989168i	$0.453107\pi$
$62$	0	0
$63$	−2.43773	−0.307125
$64$	0	0
$65$	−1.04175	−0.129214
$66$	0	0
$67$	−10.0771	−1.23111	−0.615556	−	0.788093i	$-0.711068\pi$
$67$	−10.0771	−1.23111	−0.615556	+	0.788093i	$0.711068\pi$
$68$	0	0
$69$	−1.95050	−0.234812
$70$	0	0
$71$	3.89143	0.461828	0.230914	−	0.972974i	$-0.425828\pi$
$71$	3.89143	0.461828	0.230914	+	0.972974i	$0.425828\pi$
$72$	0	0
$73$	0.547006	0.0640223	0.0320111	−	0.999488i	$-0.489809\pi$
$73$	0.547006	0.0640223	0.0320111	+	0.999488i	$0.489809\pi$
$74$	0	0
$75$	2.33189	0.269264
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	8.83644	0.994177	0.497089	−	0.867700i	$-0.334402\pi$
$79$	8.83644	0.994177	0.497089	+	0.867700i	$0.334402\pi$
$80$	0	0
$81$	−10.3707	−1.15230
$82$	0	0
$83$	−7.89143	−0.866197	−0.433099	−	0.901347i	$-0.642580\pi$
$83$	−7.89143	−0.866197	−0.433099	+	0.901347i	$0.642580\pi$
$84$	0	0
$85$	−6.88368	−0.746640
$86$	0	0
$87$	19.9419	2.13799
$88$	0	0
$89$	7.67288	0.813324	0.406662	−	0.913579i	$-0.366693\pi$
$89$	7.67288	0.813324	0.406662	+	0.913579i	$0.366693\pi$
$90$	0	0
$91$	−1.04175	−0.109205
$92$	0	0
$93$	7.44522	0.772034
$94$	0	0
$95$	6.55179	0.672199
$96$	0	0
$97$	−6.52672	−0.662688	−0.331344	−	0.943510i	$-0.607502\pi$
$97$	−6.52672	−0.662688	−0.331344	+	0.943510i	$0.607502\pi$
$98$	0	0
$99$	2.43773	0.245001
$100$	0	0
$101$	11.8572	1.17983	0.589917	−	0.807464i	$-0.299160\pi$
$101$	11.8572	1.17983	0.589917	+	0.807464i	$0.299160\pi$
$102$	0	0
$103$	16.3449	1.61051	0.805255	−	0.592929i	$-0.202028\pi$
$103$	16.3449	1.61051	0.805255	+	0.592929i	$0.202028\pi$
$104$	0	0
$105$	2.33189	0.227570
$106$	0	0
$107$	0.796799	0.0770295	0.0385147	−	0.999258i	$-0.487737\pi$
$107$	0.796799	0.0770295	0.0385147	+	0.999258i	$0.487737\pi$
$108$	0	0
$109$	−11.9044	−1.14024	−0.570119	−	0.821562i	$-0.693103\pi$
$109$	−11.9044	−1.14024	−0.570119	+	0.821562i	$0.693103\pi$
$110$	0	0
$111$	17.4898	1.66005
$112$	0	0
$113$	13.6478	1.28388	0.641940	−	0.766755i	$-0.278130\pi$
$113$	13.6478	1.28388	0.641940	+	0.766755i	$0.278130\pi$
$114$	0	0
$115$	0.836442	0.0779987
$116$	0	0
$117$	2.53951	0.234778
$118$	0	0
$119$	−6.88368	−0.631026
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−18.0025	−1.62323
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−21.2286	−1.88373	−0.941865	−	0.335990i	$-0.890929\pi$
$127$	−21.2286	−1.88373	−0.941865	+	0.335990i	$0.890929\pi$
$128$	0	0
$129$	25.3567	2.23254
$130$	0	0
$131$	−3.91306	−0.341886	−0.170943	−	0.985281i	$-0.554681\pi$
$131$	−3.91306	−0.341886	−0.170943	+	0.985281i	$0.554681\pi$
$132$	0	0
$133$	6.55179	0.568112
$134$	0	0
$135$	1.31115	0.112846
$136$	0	0
$137$	0.0606495	0.00518164	0.00259082	−	0.999997i	$-0.499175\pi$
$137$	0.0606495	0.00518164	0.00259082	+	0.999997i	$0.499175\pi$
$138$	0	0
$139$	−15.5272	−1.31700	−0.658499	−	0.752581i	$-0.728808\pi$
$139$	−15.5272	−1.31700	−0.658499	+	0.752581i	$0.728808\pi$
$140$	0	0
$141$	31.5913	2.66047
$142$	0	0
$143$	1.04175	0.0871157
$144$	0	0
$145$	−8.55179	−0.710187
$146$	0	0
$147$	2.33189	0.192331
$148$	0	0
$149$	−10.8885	−0.892017	−0.446009	−	0.895029i	$-0.647155\pi$
$149$	−10.8885	−0.892017	−0.446009	+	0.895029i	$0.647155\pi$
$150$	0	0
$151$	19.0645	1.55145	0.775725	−	0.631071i	$-0.217384\pi$
$151$	19.0645	1.55145	0.775725	+	0.631071i	$0.217384\pi$
$152$	0	0
$153$	16.7806	1.35663
$154$	0	0
$155$	−3.19278	−0.256450
$156$	0	0
$157$	4.58028	0.365546	0.182773	−	0.983155i	$-0.441493\pi$
$157$	4.58028	0.365546	0.182773	+	0.983155i	$0.441493\pi$
$158$	0	0
$159$	8.98341	0.712431
$160$	0	0
$161$	0.836442	0.0659209
$162$	0	0
$163$	6.94845	0.544244	0.272122	−	0.962263i	$-0.412275\pi$
$163$	6.94845	0.544244	0.272122	+	0.962263i	$0.412275\pi$
$164$	0	0
$165$	−2.33189	−0.181538
$166$	0	0
$167$	−10.9713	−0.848985	−0.424492	−	0.905431i	$-0.639547\pi$
$167$	−10.9713	−0.848985	−0.424492	+	0.905431i	$0.639547\pi$
$168$	0	0
$169$	−11.9148	−0.916519
$170$	0	0
$171$	−15.9715	−1.22137
$172$	0	0
$173$	7.34443	0.558386	0.279193	−	0.960235i	$-0.409933\pi$
$173$	7.34443	0.558386	0.279193	+	0.960235i	$0.409933\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	−16.7158	−1.25644
$178$	0	0
$179$	23.8264	1.78087	0.890435	−	0.455110i	$-0.150400\pi$
$179$	23.8264	1.78087	0.890435	+	0.455110i	$0.150400\pi$
$180$	0	0
$181$	3.31115	0.246116	0.123058	−	0.992399i	$-0.460730\pi$
$181$	3.31115	0.246116	0.123058	+	0.992399i	$0.460730\pi$
$182$	0	0
$183$	5.34674	0.395242
$184$	0	0
$185$	−7.50023	−0.551428
$186$	0	0
$187$	6.88368	0.503385
$188$	0	0
$189$	1.31115	0.0953720
$190$	0	0
$191$	4.60535	0.333231	0.166616	−	0.986022i	$-0.446716\pi$
$191$	4.60535	0.333231	0.166616	+	0.986022i	$0.446716\pi$
$192$	0	0
$193$	8.76531	0.630941	0.315470	−	0.948935i	$-0.397838\pi$
$193$	8.76531	0.630941	0.315470	+	0.948935i	$0.397838\pi$
$194$	0	0
$195$	−2.42926	−0.173963
$196$	0	0
$197$	−3.19615	−0.227717	−0.113858	−	0.993497i	$-0.536321\pi$
$197$	−3.19615	−0.227717	−0.113858	+	0.993497i	$0.536321\pi$
$198$	0	0
$199$	−8.84809	−0.627225	−0.313612	−	0.949551i	$-0.601539\pi$
$199$	−8.84809	−0.627225	−0.313612	+	0.949551i	$0.601539\pi$
$200$	0	0
$201$	−23.4987	−1.65747
$202$	0	0
$203$	−8.55179	−0.600218
$204$	0	0
$205$	7.72012	0.539197
$206$	0	0
$207$	−2.03902	−0.141722
$208$	0	0
$209$	−6.55179	−0.453196
$210$	0	0
$211$	10.0041	0.688711	0.344355	−	0.938839i	$-0.388098\pi$
$211$	10.0041	0.688711	0.344355	+	0.938839i	$0.388098\pi$
$212$	0	0
$213$	9.07441	0.621768
$214$	0	0
$215$	−10.8739	−0.741592
$216$	0	0
$217$	−3.19278	−0.216740
$218$	0	0
$219$	1.27556	0.0861944
$220$	0	0
$221$	7.17109	0.482380
$222$	0	0
$223$	5.77465	0.386699	0.193350	−	0.981130i	$-0.438065\pi$
$223$	5.77465	0.386699	0.193350	+	0.981130i	$0.438065\pi$
$224$	0	0
$225$	2.43773	0.162516
$226$	0	0
$227$	−11.1775	−0.741878	−0.370939	−	0.928657i	$-0.620964\pi$
$227$	−11.1775	−0.741878	−0.370939	+	0.928657i	$0.620964\pi$
$228$	0	0
$229$	−14.2370	−0.940810	−0.470405	−	0.882451i	$-0.655892\pi$
$229$	−14.2370	−0.940810	−0.470405	+	0.882451i	$0.655892\pi$
$230$	0	0
$231$	−2.33189	−0.153427
$232$	0	0
$233$	−18.3832	−1.20433	−0.602163	−	0.798373i	$-0.705694\pi$
$233$	−18.3832	−1.20433	−0.602163	+	0.798373i	$0.705694\pi$
$234$	0	0
$235$	−13.5475	−0.883739
$236$	0	0
$237$	20.6057	1.33848
$238$	0	0
$239$	−19.4186	−1.25609	−0.628043	−	0.778179i	$-0.716144\pi$
$239$	−19.4186	−1.25609	−0.628043	+	0.778179i	$0.716144\pi$
$240$	0	0
$241$	22.0067	1.41757	0.708787	−	0.705422i	$-0.249243\pi$
$241$	22.0067	1.41757	0.708787	+	0.705422i	$0.249243\pi$
$242$	0	0
$243$	−20.2498	−1.29903
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	−6.82534	−0.434286
$248$	0	0
$249$	−18.4020	−1.16618
$250$	0	0
$251$	−6.72855	−0.424703	−0.212351	−	0.977193i	$-0.568112\pi$
$251$	−6.72855	−0.424703	−0.212351	+	0.977193i	$0.568112\pi$
$252$	0	0
$253$	−0.836442	−0.0525867
$254$	0	0
$255$	−16.0520	−1.00522
$256$	0	0
$257$	−2.00890	−0.125311	−0.0626557	−	0.998035i	$-0.519957\pi$
$257$	−2.00890	−0.125311	−0.0626557	+	0.998035i	$0.519957\pi$
$258$	0	0
$259$	−7.50023	−0.466042
$260$	0	0
$261$	20.8470	1.29039
$262$	0	0
$263$	−29.4946	−1.81871	−0.909357	−	0.416016i	$-0.863426\pi$
$263$	−29.4946	−1.81871	−0.909357	+	0.416016i	$0.863426\pi$
$264$	0	0
$265$	−3.85241	−0.236652
$266$	0	0
$267$	17.8924	1.09499
$268$	0	0
$269$	−25.1987	−1.53639	−0.768195	−	0.640216i	$-0.778845\pi$
$269$	−25.1987	−1.53639	−0.768195	+	0.640216i	$0.778845\pi$
$270$	0	0
$271$	−19.2231	−1.16772	−0.583860	−	0.811854i	$-0.698458\pi$
$271$	−19.2231	−1.16772	−0.583860	+	0.811854i	$0.698458\pi$
$272$	0	0
$273$	−2.42926	−0.147025
$274$	0	0
$275$	1.00000	0.0603023
$276$	0	0
$277$	18.9690	1.13974	0.569868	−	0.821737i	$-0.306994\pi$
$277$	18.9690	1.13974	0.569868	+	0.821737i	$0.306994\pi$
$278$	0	0
$279$	7.78314	0.465965
$280$	0	0
$281$	−16.6679	−0.994323	−0.497161	−	0.867658i	$-0.665624\pi$
$281$	−16.6679	−0.994323	−0.497161	+	0.867658i	$0.665624\pi$
$282$	0	0
$283$	6.52329	0.387769	0.193885	−	0.981024i	$-0.437891\pi$
$283$	6.52329	0.387769	0.193885	+	0.981024i	$0.437891\pi$
$284$	0	0
$285$	15.2781	0.904995
$286$	0	0
$287$	7.72012	0.455704
$288$	0	0
$289$	30.3851	1.78736
$290$	0	0
$291$	−15.2196	−0.892191
$292$	0	0
$293$	28.1868	1.64669	0.823346	−	0.567540i	$-0.192105\pi$
$293$	28.1868	1.64669	0.823346	+	0.567540i	$0.192105\pi$
$294$	0	0
$295$	7.16834	0.417357
$296$	0	0
$297$	−1.31115	−0.0760805
$298$	0	0
$299$	−0.871366	−0.0503924
$300$	0	0
$301$	−10.8739	−0.626760
$302$	0	0
$303$	27.6497	1.58843
$304$	0	0
$305$	−2.29287	−0.131289
$306$	0	0
$307$	−23.7728	−1.35679	−0.678393	−	0.734699i	$-0.737323\pi$
$307$	−23.7728	−1.35679	−0.678393	+	0.734699i	$0.737323\pi$
$308$	0	0
$309$	38.1146	2.16826
$310$	0	0
$311$	8.08058	0.458207	0.229104	−	0.973402i	$-0.426421\pi$
$311$	8.08058	0.458207	0.229104	+	0.973402i	$0.426421\pi$
$312$	0	0
$313$	8.44322	0.477239	0.238619	−	0.971113i	$-0.423305\pi$
$313$	8.44322	0.477239	0.238619	+	0.971113i	$0.423305\pi$
$314$	0	0
$315$	2.43773	0.137351
$316$	0	0
$317$	−3.01663	−0.169431	−0.0847154	−	0.996405i	$-0.526998\pi$
$317$	−3.01663	−0.169431	−0.0847154	+	0.996405i	$0.526998\pi$
$318$	0	0
$319$	8.55179	0.478808
$320$	0	0
$321$	1.85805	0.103706
$322$	0	0
$323$	−45.1004	−2.50945
$324$	0	0
$325$	1.04175	0.0577860
$326$	0	0
$327$	−27.7599	−1.53512
$328$	0	0
$329$	−13.5475	−0.746896
$330$	0	0
$331$	28.3315	1.55724	0.778621	−	0.627495i	$-0.215920\pi$
$331$	28.3315	1.55724	0.778621	+	0.627495i	$0.215920\pi$
$332$	0	0
$333$	18.2836	1.00193
$334$	0	0
$335$	10.0771	0.550570
$336$	0	0
$337$	−31.8778	−1.73650	−0.868248	−	0.496131i	$-0.834754\pi$
$337$	−31.8778	−1.73650	−0.868248	+	0.496131i	$0.834754\pi$
$338$	0	0
$339$	31.8253	1.72851
$340$	0	0
$341$	3.19278	0.172899
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	1.95050	0.105011
$346$	0	0
$347$	−26.1376	−1.40314	−0.701569	−	0.712602i	$-0.747517\pi$
$347$	−26.1376	−1.40314	−0.701569	+	0.712602i	$0.747517\pi$
$348$	0	0
$349$	5.51364	0.295139	0.147569	−	0.989052i	$-0.452855\pi$
$349$	5.51364	0.295139	0.147569	+	0.989052i	$0.452855\pi$
$350$	0	0
$351$	−1.36589	−0.0729059
$352$	0	0
$353$	11.2115	0.596727	0.298363	−	0.954452i	$-0.403559\pi$
$353$	11.2115	0.596727	0.298363	+	0.954452i	$0.403559\pi$
$354$	0	0
$355$	−3.89143	−0.206536
$356$	0	0
$357$	−16.0520	−0.849563
$358$	0	0
$359$	−27.7238	−1.46321	−0.731603	−	0.681731i	$-0.761228\pi$
$359$	−27.7238	−1.46321	−0.731603	+	0.681731i	$0.761228\pi$
$360$	0	0
$361$	23.9259	1.25926
$362$	0	0
$363$	2.33189	0.122393
$364$	0	0
$365$	−0.547006	−0.0286316
$366$	0	0
$367$	−25.3900	−1.32535	−0.662674	−	0.748908i	$-0.730579\pi$
$367$	−25.3900	−1.32535	−0.662674	+	0.748908i	$0.730579\pi$
$368$	0	0
$369$	−18.8196	−0.979709
$370$	0	0
$371$	−3.85241	−0.200007
$372$	0	0
$373$	27.9184	1.44556	0.722780	−	0.691079i	$-0.242864\pi$
$373$	27.9184	1.44556	0.722780	+	0.691079i	$0.242864\pi$
$374$	0	0
$375$	−2.33189	−0.120419
$376$	0	0
$377$	8.90885	0.458829
$378$	0	0
$379$	12.0666	0.619817	0.309909	−	0.950766i	$-0.399702\pi$
$379$	12.0666	0.619817	0.309909	+	0.950766i	$0.399702\pi$
$380$	0	0
$381$	−49.5028	−2.53610
$382$	0	0
$383$	−15.1022	−0.771688	−0.385844	−	0.922564i	$-0.626090\pi$
$383$	−15.1022	−0.771688	−0.385844	+	0.922564i	$0.626090\pi$
$384$	0	0
$385$	1.00000	0.0509647
$386$	0	0
$387$	26.5076	1.34746
$388$	0	0
$389$	−32.2140	−1.63332	−0.816658	−	0.577122i	$-0.804176\pi$
$389$	−32.2140	−1.63332	−0.816658	+	0.577122i	$0.804176\pi$
$390$	0	0
$391$	−5.75780	−0.291185
$392$	0	0
$393$	−9.12484	−0.460287
$394$	0	0
$395$	−8.83644	−0.444610
$396$	0	0
$397$	−11.6017	−0.582273	−0.291137	−	0.956681i	$-0.594033\pi$
$397$	−11.6017	−0.582273	−0.291137	+	0.956681i	$0.594033\pi$
$398$	0	0
$399$	15.2781	0.764860
$400$	0	0
$401$	11.7475	0.586640	0.293320	−	0.956014i	$-0.405240\pi$
$401$	11.7475	0.586640	0.293320	+	0.956014i	$0.405240\pi$
$402$	0	0
$403$	3.32609	0.165684
$404$	0	0
$405$	10.3707	0.515322
$406$	0	0
$407$	7.50023	0.371773
$408$	0	0
$409$	1.74394	0.0862325	0.0431163	−	0.999070i	$-0.486271\pi$
$409$	1.74394	0.0862325	0.0431163	+	0.999070i	$0.486271\pi$
$410$	0	0
$411$	0.141428	0.00697615
$412$	0	0
$413$	7.16834	0.352731
$414$	0	0
$415$	7.89143	0.387375
$416$	0	0
$417$	−36.2078	−1.77310
$418$	0	0
$419$	−27.3213	−1.33473	−0.667365	−	0.744730i	$-0.732578\pi$
$419$	−27.3213	−1.33473	−0.667365	+	0.744730i	$0.732578\pi$
$420$	0	0
$421$	−28.4667	−1.38738	−0.693692	−	0.720272i	$-0.744017\pi$
$421$	−28.4667	−1.38738	−0.693692	+	0.720272i	$0.744017\pi$
$422$	0	0
$423$	33.0251	1.60574
$424$	0	0
$425$	6.88368	0.333908
$426$	0	0
$427$	−2.29287	−0.110960
$428$	0	0
$429$	2.42926	0.117286
$430$	0	0
$431$	−29.3287	−1.41271	−0.706357	−	0.707856i	$-0.749662\pi$
$431$	−29.3287	−1.41271	−0.706357	+	0.707856i	$0.749662\pi$
$432$	0	0
$433$	23.3561	1.12242	0.561211	−	0.827673i	$-0.310336\pi$
$433$	23.3561	1.12242	0.561211	+	0.827673i	$0.310336\pi$
$434$	0	0
$435$	−19.9419	−0.956139
$436$	0	0
$437$	5.48019	0.262153
$438$	0	0
$439$	26.9699	1.28721	0.643603	−	0.765360i	$-0.277439\pi$
$439$	26.9699	1.28721	0.643603	+	0.765360i	$0.277439\pi$
$440$	0	0
$441$	2.43773	0.116083
$442$	0	0
$443$	33.0539	1.57044	0.785219	−	0.619218i	$-0.212550\pi$
$443$	33.0539	1.57044	0.785219	+	0.619218i	$0.212550\pi$
$444$	0	0
$445$	−7.67288	−0.363730
$446$	0	0
$447$	−25.3907	−1.20094
$448$	0	0
$449$	39.0715	1.84390	0.921948	−	0.387313i	$-0.126597\pi$
$449$	39.0715	1.84390	0.921948	+	0.387313i	$0.126597\pi$
$450$	0	0
$451$	−7.72012	−0.363526
$452$	0	0
$453$	44.4565	2.08875
$454$	0	0
$455$	1.04175	0.0488381
$456$	0	0
$457$	26.1465	1.22308	0.611541	−	0.791213i	$-0.290550\pi$
$457$	26.1465	1.22308	0.611541	+	0.791213i	$0.290550\pi$
$458$	0	0
$459$	−9.02553	−0.421275
$460$	0	0
$461$	6.45708	0.300736	0.150368	−	0.988630i	$-0.451954\pi$
$461$	6.45708	0.300736	0.150368	+	0.988630i	$0.451954\pi$
$462$	0	0
$463$	−8.83644	−0.410664	−0.205332	−	0.978692i	$-0.565827\pi$
$463$	−8.83644	−0.410664	−0.205332	+	0.978692i	$0.565827\pi$
$464$	0	0
$465$	−7.44522	−0.345264
$466$	0	0
$467$	5.64714	0.261319	0.130659	−	0.991427i	$-0.458291\pi$
$467$	5.64714	0.261319	0.130659	+	0.991427i	$0.458291\pi$
$468$	0	0
$469$	10.0771	0.465316
$470$	0	0
$471$	10.6807	0.492142
$472$	0	0
$473$	10.8739	0.499981
$474$	0	0
$475$	−6.55179	−0.300617
$476$	0	0
$477$	9.39114	0.429991
$478$	0	0
$479$	2.44388	0.111664	0.0558319	−	0.998440i	$-0.482219\pi$
$479$	2.44388	0.111664	0.0558319	+	0.998440i	$0.482219\pi$
$480$	0	0
$481$	7.81339	0.356260
$482$	0	0
$483$	1.95050	0.0887506
$484$	0	0
$485$	6.52672	0.296363
$486$	0	0
$487$	−1.21778	−0.0551830	−0.0275915	−	0.999619i	$-0.508784\pi$
$487$	−1.21778	−0.0551830	−0.0275915	+	0.999619i	$0.508784\pi$
$488$	0	0
$489$	16.2030	0.732727
$490$	0	0
$491$	−29.2057	−1.31804	−0.659018	−	0.752127i	$-0.729028\pi$
$491$	−29.2057	−1.31804	−0.659018	+	0.752127i	$0.729028\pi$
$492$	0	0
$493$	58.8678	2.65127
$494$	0	0
$495$	−2.43773	−0.109568
$496$	0	0
$497$	−3.89143	−0.174555
$498$	0	0
$499$	3.18677	0.142659	0.0713297	−	0.997453i	$-0.477276\pi$
$499$	3.18677	0.142659	0.0713297	+	0.997453i	$0.477276\pi$
$500$	0	0
$501$	−25.5839	−1.14301
$502$	0	0
$503$	−20.9238	−0.932947	−0.466474	−	0.884535i	$-0.654476\pi$
$503$	−20.9238	−0.932947	−0.466474	+	0.884535i	$0.654476\pi$
$504$	0	0
$505$	−11.8572	−0.527638
$506$	0	0
$507$	−27.7839	−1.23393
$508$	0	0
$509$	5.46121	0.242064	0.121032	−	0.992649i	$-0.461380\pi$
$509$	5.46121	0.242064	0.121032	+	0.992649i	$0.461380\pi$
$510$	0	0
$511$	−0.547006	−0.0241981
$512$	0	0
$513$	8.59036	0.379274
$514$	0	0
$515$	−16.3449	−0.720242
$516$	0	0
$517$	13.5475	0.595817
$518$	0	0
$519$	17.1264	0.751767
$520$	0	0
$521$	19.7965	0.867301	0.433651	−	0.901081i	$-0.357225\pi$
$521$	19.7965	0.867301	0.433651	+	0.901081i	$0.357225\pi$
$522$	0	0
$523$	−8.94748	−0.391246	−0.195623	−	0.980679i	$-0.562673\pi$
$523$	−8.94748	−0.391246	−0.195623	+	0.980679i	$0.562673\pi$
$524$	0	0
$525$	−2.33189	−0.101772
$526$	0	0
$527$	21.9781	0.957380
$528$	0	0
$529$	−22.3004	−0.969581
$530$	0	0
$531$	−17.4745	−0.758328
$532$	0	0
$533$	−8.04246	−0.348358
$534$	0	0
$535$	−0.796799	−0.0344486
$536$	0	0
$537$	55.5607	2.39762
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	−19.5687	−0.841323	−0.420662	−	0.907218i	$-0.638202\pi$
$541$	−19.5687	−0.841323	−0.420662	+	0.907218i	$0.638202\pi$
$542$	0	0
$543$	7.72125	0.331351
$544$	0	0
$545$	11.9044	0.509930
$546$	0	0
$547$	41.8550	1.78959	0.894795	−	0.446478i	$-0.147322\pi$
$547$	41.8550	1.78959	0.894795	+	0.446478i	$0.147322\pi$
$548$	0	0
$549$	5.58941	0.238550
$550$	0	0
$551$	−56.0295	−2.38694
$552$	0	0
$553$	−8.83644	−0.375764
$554$	0	0
$555$	−17.4898	−0.742399
$556$	0	0
$557$	−0.180622	−0.00765319	−0.00382660	−	0.999993i	$-0.501218\pi$
$557$	−0.180622	−0.00765319	−0.00382660	+	0.999993i	$0.501218\pi$
$558$	0	0
$559$	11.3279	0.479119
$560$	0	0
$561$	16.0520	0.677717
$562$	0	0
$563$	−46.6716	−1.96697	−0.983486	−	0.180982i	$-0.942072\pi$
$563$	−46.6716	−1.96697	−0.983486	+	0.180982i	$0.942072\pi$
$564$	0	0
$565$	−13.6478	−0.574168
$566$	0	0
$567$	10.3707	0.435527
$568$	0	0
$569$	−15.9845	−0.670105	−0.335052	−	0.942199i	$-0.608754\pi$
$569$	−15.9845	−0.670105	−0.335052	+	0.942199i	$0.608754\pi$
$570$	0	0
$571$	14.1792	0.593382	0.296691	−	0.954974i	$-0.404117\pi$
$571$	14.1792	0.593382	0.296691	+	0.954974i	$0.404117\pi$
$572$	0	0
$573$	10.7392	0.448636
$574$	0	0
$575$	−0.836442	−0.0348821
$576$	0	0
$577$	−13.8794	−0.577805	−0.288903	−	0.957358i	$-0.593290\pi$
$577$	−13.8794	−0.577805	−0.288903	+	0.957358i	$0.593290\pi$
$578$	0	0
$579$	20.4398	0.849448
$580$	0	0
$581$	7.89143	0.327392
$582$	0	0
$583$	3.85241	0.159550
$584$	0	0
$585$	−2.53951	−0.104996
$586$	0	0
$587$	5.86428	0.242045	0.121022	−	0.992650i	$-0.461383\pi$
$587$	5.86428	0.242045	0.121022	+	0.992650i	$0.461383\pi$
$588$	0	0
$589$	−20.9184	−0.861928
$590$	0	0
$591$	−7.45310	−0.306579
$592$	0	0
$593$	2.07763	0.0853180	0.0426590	−	0.999090i	$-0.486417\pi$
$593$	2.07763	0.0853180	0.0426590	+	0.999090i	$0.486417\pi$
$594$	0	0
$595$	6.88368	0.282203
$596$	0	0
$597$	−20.6328	−0.844445
$598$	0	0
$599$	28.4106	1.16083	0.580413	−	0.814322i	$-0.302891\pi$
$599$	28.4106	1.16083	0.580413	+	0.814322i	$0.302891\pi$
$600$	0	0
$601$	−30.1513	−1.22990	−0.614948	−	0.788568i	$-0.710823\pi$
$601$	−30.1513	−1.22990	−0.614948	+	0.788568i	$0.710823\pi$
$602$	0	0
$603$	−24.5652	−1.00037
$604$	0	0
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	25.7804	1.04639	0.523197	−	0.852212i	$-0.324739\pi$
$607$	25.7804	1.04639	0.523197	+	0.852212i	$0.324739\pi$
$608$	0	0
$609$	−19.9419	−0.808085
$610$	0	0
$611$	14.1131	0.570955
$612$	0	0
$613$	−11.5967	−0.468388	−0.234194	−	0.972190i	$-0.575245\pi$
$613$	−11.5967	−0.468388	−0.234194	+	0.972190i	$0.575245\pi$
$614$	0	0
$615$	18.0025	0.725931
$616$	0	0
$617$	−1.35907	−0.0547140	−0.0273570	−	0.999626i	$-0.508709\pi$
$617$	−1.35907	−0.0547140	−0.0273570	+	0.999626i	$0.508709\pi$
$618$	0	0
$619$	13.5703	0.545437	0.272719	−	0.962094i	$-0.412077\pi$
$619$	13.5703	0.545437	0.272719	+	0.962094i	$0.412077\pi$
$620$	0	0
$621$	1.09670	0.0440091
$622$	0	0
$623$	−7.67288	−0.307408
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−15.2781	−0.610148
$628$	0	0
$629$	51.6292	2.05859
$630$	0	0
$631$	34.2889	1.36502	0.682510	−	0.730877i	$-0.260889\pi$
$631$	34.2889	1.36502	0.682510	+	0.730877i	$0.260889\pi$
$632$	0	0
$633$	23.3285	0.927225
$634$	0	0
$635$	21.2286	0.842430
$636$	0	0
$637$	1.04175	0.0412757
$638$	0	0
$639$	9.48627	0.375271
$640$	0	0
$641$	27.1720	1.07323	0.536615	−	0.843827i	$-0.319703\pi$
$641$	27.1720	1.07323	0.536615	+	0.843827i	$0.319703\pi$
$642$	0	0
$643$	−46.7251	−1.84266	−0.921330	−	0.388782i	$-0.872896\pi$
$643$	−46.7251	−1.84266	−0.921330	+	0.388782i	$0.872896\pi$
$644$	0	0
$645$	−25.3567	−0.998421
$646$	0	0
$647$	−29.8896	−1.17508	−0.587541	−	0.809195i	$-0.699904\pi$
$647$	−29.8896	−1.17508	−0.587541	+	0.809195i	$0.699904\pi$
$648$	0	0
$649$	−7.16834	−0.281382
$650$	0	0
$651$	−7.44522	−0.291801
$652$	0	0
$653$	−14.4473	−0.565367	−0.282684	−	0.959213i	$-0.591225\pi$
$653$	−14.4473	−0.565367	−0.282684	+	0.959213i	$0.591225\pi$
$654$	0	0
$655$	3.91306	0.152896
$656$	0	0
$657$	1.33346	0.0520230
$658$	0	0
$659$	−21.3339	−0.831050	−0.415525	−	0.909582i	$-0.636402\pi$
$659$	−21.3339	−0.831050	−0.415525	+	0.909582i	$0.636402\pi$
$660$	0	0
$661$	49.5315	1.92655	0.963277	−	0.268510i	$-0.0865311\pi$
$661$	49.5315	1.92655	0.963277	+	0.268510i	$0.0865311\pi$
$662$	0	0
$663$	16.7222	0.649438
$664$	0	0
$665$	−6.55179	−0.254067
$666$	0	0
$667$	−7.15308	−0.276968
$668$	0	0
$669$	13.4659	0.520621
$670$	0	0
$671$	2.29287	0.0885153
$672$	0	0
$673$	−25.9586	−1.00063	−0.500316	−	0.865843i	$-0.666783\pi$
$673$	−25.9586	−1.00063	−0.500316	+	0.865843i	$0.666783\pi$
$674$	0	0
$675$	−1.31115	−0.0504661
$676$	0	0
$677$	−33.6670	−1.29393	−0.646964	−	0.762520i	$-0.723962\pi$
$677$	−33.6670	−1.29393	−0.646964	+	0.762520i	$0.723962\pi$
$678$	0	0
$679$	6.52672	0.250473
$680$	0	0
$681$	−26.0648	−0.998805
$682$	0	0
$683$	3.68024	0.140820	0.0704102	−	0.997518i	$-0.477569\pi$
$683$	3.68024	0.140820	0.0704102	+	0.997518i	$0.477569\pi$
$684$	0	0
$685$	−0.0606495	−0.00231730
$686$	0	0
$687$	−33.1993	−1.26663
$688$	0	0
$689$	4.01326	0.152893
$690$	0	0
$691$	−29.2473	−1.11262	−0.556310	−	0.830975i	$-0.687783\pi$
$691$	−29.2473	−1.11262	−0.556310	+	0.830975i	$0.687783\pi$
$692$	0	0
$693$	−2.43773	−0.0926018
$694$	0	0
$695$	15.5272	0.588980
$696$	0	0
$697$	−53.1429	−2.01293
$698$	0	0
$699$	−42.8678	−1.62141
$700$	0	0
$701$	−3.75847	−0.141955	−0.0709776	−	0.997478i	$-0.522612\pi$
$701$	−3.75847	−0.141955	−0.0709776	+	0.997478i	$0.522612\pi$
$702$	0	0
$703$	−49.1399	−1.85335
$704$	0	0
$705$	−31.5913	−1.18980
$706$	0	0
$707$	−11.8572	−0.445935
$708$	0	0
$709$	−2.05700	−0.0772521	−0.0386261	−	0.999254i	$-0.512298\pi$
$709$	−2.05700	−0.0772521	−0.0386261	+	0.999254i	$0.512298\pi$
$710$	0	0
$711$	21.5409	0.807846
$712$	0	0
$713$	−2.67058	−0.100014
$714$	0	0
$715$	−1.04175	−0.0389593
$716$	0	0
$717$	−45.2822	−1.69109
$718$	0	0
$719$	28.7851	1.07350	0.536752	−	0.843740i	$-0.319651\pi$
$719$	28.7851	1.07350	0.536752	+	0.843740i	$0.319651\pi$
$720$	0	0
$721$	−16.3449	−0.608715
$722$	0	0
$723$	51.3172	1.90851
$724$	0	0
$725$	8.55179	0.317605
$726$	0	0
$727$	−12.4508	−0.461773	−0.230887	−	0.972981i	$-0.574163\pi$
$727$	−12.4508	−0.461773	−0.230887	+	0.972981i	$0.574163\pi$
$728$	0	0
$729$	−16.1085	−0.596612
$730$	0	0
$731$	74.8523	2.76851
$732$	0	0
$733$	2.81849	0.104103	0.0520516	−	0.998644i	$-0.483424\pi$
$733$	2.81849	0.104103	0.0520516	+	0.998644i	$0.483424\pi$
$734$	0	0
$735$	−2.33189	−0.0860132
$736$	0	0
$737$	−10.0771	−0.371194
$738$	0	0
$739$	−43.5526	−1.60211	−0.801054	−	0.598592i	$-0.795727\pi$
$739$	−43.5526	−1.60211	−0.801054	+	0.598592i	$0.795727\pi$
$740$	0	0
$741$	−15.9160	−0.584688
$742$	0	0
$743$	−0.0849172	−0.00311531	−0.00155766	−	0.999999i	$-0.500496\pi$
$743$	−0.0849172	−0.00311531	−0.00155766	+	0.999999i	$0.500496\pi$
$744$	0	0
$745$	10.8885	0.398922
$746$	0	0
$747$	−19.2372	−0.703852
$748$	0	0
$749$	−0.796799	−0.0291144
$750$	0	0
$751$	52.0412	1.89901	0.949505	−	0.313753i	$-0.101586\pi$
$751$	52.0412	1.89901	0.949505	+	0.313753i	$0.101586\pi$
$752$	0	0
$753$	−15.6903	−0.571786
$754$	0	0
$755$	−19.0645	−0.693830
$756$	0	0
$757$	10.0869	0.366616	0.183308	−	0.983056i	$-0.441319\pi$
$757$	10.0869	0.366616	0.183308	+	0.983056i	$0.441319\pi$
$758$	0	0
$759$	−1.95050	−0.0707985
$760$	0	0
$761$	−47.9897	−1.73963	−0.869813	−	0.493381i	$-0.835761\pi$
$761$	−47.9897	−1.73963	−0.869813	+	0.493381i	$0.835761\pi$
$762$	0	0
$763$	11.9044	0.430969
$764$	0	0
$765$	−16.7806	−0.606703
$766$	0	0
$767$	−7.46763	−0.269641
$768$	0	0
$769$	18.8164	0.678538	0.339269	−	0.940689i	$-0.389820\pi$
$769$	18.8164	0.678538	0.339269	+	0.940689i	$0.389820\pi$
$770$	0	0
$771$	−4.68453	−0.168709
$772$	0	0
$773$	−21.3610	−0.768300	−0.384150	−	0.923271i	$-0.625506\pi$
$773$	−21.3610	−0.768300	−0.384150	+	0.923271i	$0.625506\pi$
$774$	0	0
$775$	3.19278	0.114688
$776$	0	0
$777$	−17.4898	−0.627441
$778$	0	0
$779$	50.5806	1.81224
$780$	0	0
$781$	3.89143	0.139246
$782$	0	0
$783$	−11.2127	−0.400708
$784$	0	0
$785$	−4.58028	−0.163477
$786$	0	0
$787$	−9.66333	−0.344460	−0.172230	−	0.985057i	$-0.555097\pi$
$787$	−9.66333	−0.344460	−0.172230	+	0.985057i	$0.555097\pi$
$788$	0	0
$789$	−68.7783	−2.44857
$790$	0	0
$791$	−13.6478	−0.485261
$792$	0	0
$793$	2.38861	0.0848219
$794$	0	0
$795$	−8.98341	−0.318609
$796$	0	0
$797$	−4.48127	−0.158735	−0.0793675	−	0.996845i	$-0.525290\pi$
$797$	−4.48127	−0.158735	−0.0793675	+	0.996845i	$0.525290\pi$
$798$	0	0
$799$	93.2565	3.29918
$800$	0	0
$801$	18.7044	0.660889
$802$	0	0
$803$	0.547006	0.0193034
$804$	0	0
$805$	−0.836442	−0.0294807
$806$	0	0
$807$	−58.7606	−2.06847
$808$	0	0
$809$	51.6638	1.81640	0.908202	−	0.418533i	$-0.137456\pi$
$809$	51.6638	1.81640	0.908202	+	0.418533i	$0.137456\pi$
$810$	0	0
$811$	−4.85243	−0.170392	−0.0851959	−	0.996364i	$-0.527152\pi$
$811$	−4.85243	−0.170392	−0.0851959	+	0.996364i	$0.527152\pi$
$812$	0	0
$813$	−44.8263	−1.57213
$814$	0	0
$815$	−6.94845	−0.243393
$816$	0	0
$817$	−71.2433	−2.49249
$818$	0	0
$819$	−2.53951	−0.0887378
$820$	0	0
$821$	−13.6135	−0.475113	−0.237557	−	0.971374i	$-0.576347\pi$
$821$	−13.6135	−0.475113	−0.237557	+	0.971374i	$0.576347\pi$
$822$	0	0
$823$	−42.4114	−1.47837	−0.739185	−	0.673503i	$-0.764789\pi$
$823$	−42.4114	−1.47837	−0.739185	+	0.673503i	$0.764789\pi$
$824$	0	0
$825$	2.33189	0.0811862
$826$	0	0
$827$	−45.0844	−1.56774	−0.783869	−	0.620926i	$-0.786757\pi$
$827$	−45.0844	−1.56774	−0.783869	+	0.620926i	$0.786757\pi$
$828$	0	0
$829$	−45.2139	−1.57034	−0.785171	−	0.619279i	$-0.787425\pi$
$829$	−45.2139	−1.57034	−0.785171	+	0.619279i	$0.787425\pi$
$830$	0	0
$831$	44.2337	1.53445
$832$	0	0
$833$	6.88368	0.238505
$834$	0	0
$835$	10.9713	0.379678
$836$	0	0
$837$	−4.18621	−0.144696
$838$	0	0
$839$	0.935511	0.0322974	0.0161487	−	0.999870i	$-0.494859\pi$
$839$	0.935511	0.0322974	0.0161487	+	0.999870i	$0.494859\pi$
$840$	0	0
$841$	44.1330	1.52183
$842$	0	0
$843$	−38.8678	−1.33868
$844$	0	0
$845$	11.9148	0.409880
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	15.2116	0.522061
$850$	0	0
$851$	−6.27351	−0.215053
$852$	0	0
$853$	6.48591	0.222073	0.111037	−	0.993816i	$-0.464583\pi$
$853$	6.48591	0.222073	0.111037	+	0.993816i	$0.464583\pi$
$854$	0	0
$855$	15.9715	0.546214
$856$	0	0
$857$	−5.14560	−0.175770	−0.0878852	−	0.996131i	$-0.528011\pi$
$857$	−5.14560	−0.175770	−0.0878852	+	0.996131i	$0.528011\pi$
$858$	0	0
$859$	20.1437	0.687295	0.343648	−	0.939099i	$-0.388337\pi$
$859$	20.1437	0.687295	0.343648	+	0.939099i	$0.388337\pi$
$860$	0	0
$861$	18.0025	0.613524
$862$	0	0
$863$	33.3580	1.13552	0.567759	−	0.823195i	$-0.307810\pi$
$863$	33.3580	1.13552	0.567759	+	0.823195i	$0.307810\pi$
$864$	0	0
$865$	−7.34443	−0.249718
$866$	0	0
$867$	70.8547	2.40635
$868$	0	0
$869$	8.83644	0.299756
$870$	0	0
$871$	−10.4978	−0.355705
$872$	0	0
$873$	−15.9104	−0.538486
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	−2.50003	−0.0844201	−0.0422100	−	0.999109i	$-0.513440\pi$
$877$	−2.50003	−0.0844201	−0.0422100	+	0.999109i	$0.513440\pi$
$878$	0	0
$879$	65.7287	2.21697
$880$	0	0
$881$	27.5154	0.927018	0.463509	−	0.886092i	$-0.346590\pi$
$881$	27.5154	0.927018	0.463509	+	0.886092i	$0.346590\pi$
$882$	0	0
$883$	−51.9181	−1.74718	−0.873591	−	0.486661i	$-0.838215\pi$
$883$	−51.9181	−1.74718	−0.873591	+	0.486661i	$0.838215\pi$
$884$	0	0
$885$	16.7158	0.561896
$886$	0	0
$887$	−16.9590	−0.569427	−0.284713	−	0.958613i	$-0.591898\pi$
$887$	−16.9590	−0.569427	−0.284713	+	0.958613i	$0.591898\pi$
$888$	0	0
$889$	21.2286	0.711983
$890$	0	0
$891$	−10.3707	−0.347430
$892$	0	0
$893$	−88.7601	−2.97024
$894$	0	0
$895$	−23.8264	−0.796430
$896$	0	0
$897$	−2.03193	−0.0678443
$898$	0	0
$899$	27.3040	0.910638
$900$	0	0
$901$	26.5188	0.883468
$902$	0	0
$903$	−25.3567	−0.843819
$904$	0	0
$905$	−3.31115	−0.110066
$906$	0	0
$907$	25.8809	0.859361	0.429681	−	0.902981i	$-0.358626\pi$
$907$	25.8809	0.859361	0.429681	+	0.902981i	$0.358626\pi$
$908$	0	0
$909$	28.9047	0.958707
$910$	0	0
$911$	6.86397	0.227414	0.113707	−	0.993514i	$-0.463728\pi$
$911$	6.86397	0.227414	0.113707	+	0.993514i	$0.463728\pi$
$912$	0	0
$913$	−7.89143	−0.261168
$914$	0	0
$915$	−5.34674	−0.176758
$916$	0	0
$917$	3.91306	0.129221
$918$	0	0
$919$	−14.4400	−0.476333	−0.238167	−	0.971224i	$-0.576546\pi$
$919$	−14.4400	−0.476333	−0.238167	+	0.971224i	$0.576546\pi$
$920$	0	0
$921$	−55.4357	−1.82667
$922$	0	0
$923$	4.05391	0.133436
$924$	0	0
$925$	7.50023	0.246606
$926$	0	0
$927$	39.8445	1.30866
$928$	0	0
$929$	−47.7350	−1.56614	−0.783068	−	0.621936i	$-0.786346\pi$
$929$	−47.7350	−1.56614	−0.783068	+	0.621936i	$0.786346\pi$
$930$	0	0
$931$	−6.55179	−0.214726
$932$	0	0
$933$	18.8431	0.616894
$934$	0	0
$935$	−6.88368	−0.225120
$936$	0	0
$937$	25.2026	0.823332	0.411666	−	0.911335i	$-0.364947\pi$
$937$	25.2026	0.823332	0.411666	+	0.911335i	$0.364947\pi$
$938$	0	0
$939$	19.6887	0.642516
$940$	0	0
$941$	32.6447	1.06419	0.532094	−	0.846685i	$-0.321405\pi$
$941$	32.6447	1.06419	0.532094	+	0.846685i	$0.321405\pi$
$942$	0	0
$943$	6.45744	0.210283
$944$	0	0
$945$	−1.31115	−0.0426517
$946$	0	0
$947$	−54.3257	−1.76535	−0.882674	−	0.469985i	$-0.844259\pi$
$947$	−54.3257	−1.76535	−0.882674	+	0.469985i	$0.844259\pi$
$948$	0	0
$949$	0.569845	0.0184980
$950$	0	0
$951$	−7.03446	−0.228108
$952$	0	0
$953$	−19.5837	−0.634378	−0.317189	−	0.948362i	$-0.602739\pi$
$953$	−19.5837	−0.634378	−0.317189	+	0.948362i	$0.602739\pi$
$954$	0	0
$955$	−4.60535	−0.149026
$956$	0	0
$957$	19.9419	0.644629
$958$	0	0
$959$	−0.0606495	−0.00195848
$960$	0	0
$961$	−20.8062	−0.671167
$962$	0	0
$963$	1.94238	0.0625924
$964$	0	0
$965$	−8.76531	−0.282165
$966$	0	0
$967$	41.8637	1.34625	0.673123	−	0.739530i	$-0.264952\pi$
$967$	41.8637	1.34625	0.673123	+	0.739530i	$0.264952\pi$
$968$	0	0
$969$	−105.169	−3.37853
$970$	0	0
$971$	14.6310	0.469531	0.234765	−	0.972052i	$-0.424568\pi$
$971$	14.6310	0.469531	0.234765	+	0.972052i	$0.424568\pi$
$972$	0	0
$973$	15.5272	0.497779
$974$	0	0
$975$	2.42926	0.0777985
$976$	0	0
$977$	−34.9335	−1.11762	−0.558810	−	0.829295i	$-0.688742\pi$
$977$	−34.9335	−1.11762	−0.558810	+	0.829295i	$0.688742\pi$
$978$	0	0
$979$	7.67288	0.245226
$980$	0	0
$981$	−29.0198	−0.926531
$982$	0	0
$983$	−43.0654	−1.37357	−0.686786	−	0.726860i	$-0.740979\pi$
$983$	−43.0654	−1.37357	−0.686786	+	0.726860i	$0.740979\pi$
$984$	0	0
$985$	3.19615	0.101838
$986$	0	0
$987$	−31.5913	−1.00556
$988$	0	0
$989$	−9.09537	−0.289216
$990$	0	0
$991$	42.8418	1.36091	0.680457	−	0.732788i	$-0.261781\pi$
$991$	42.8418	1.36091	0.680457	+	0.732788i	$0.261781\pi$
$992$	0	0
$993$	66.0661	2.09654
$994$	0	0
$995$	8.84809	0.280503
$996$	0	0
$997$	−36.0734	−1.14246	−0.571228	−	0.820791i	$-0.693533\pi$
$997$	−36.0734	−1.14246	−0.571228	+	0.820791i	$0.693533\pi$
$998$	0	0
$999$	−9.83392	−0.311131

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3080.2.a.r.1.5	✓	5
4.3	odd	2		6160.2.a.by.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3080.2.a.r.1.5	✓	5	1.1	even	1	trivial
6160.2.a.by.1.1		5	4.3	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:	\( N \)	\(=\)	\( 3080 = 2^{3} \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3080.a (trivial)

\(f(q)\)	\(=\)	\(q+2.33189 q^{3} -1.00000 q^{5} -1.00000 q^{7} +2.43773 q^{9} +O(q^{10})\)	\(q+2.33189 q^{3} -1.00000 q^{5} -1.00000 q^{7} +2.43773 q^{9} +1.00000 q^{11} +1.04175 q^{13} -2.33189 q^{15} +6.88368 q^{17} -6.55179 q^{19} -2.33189 q^{21} -0.836442 q^{23} +1.00000 q^{25} -1.31115 q^{27} +8.55179 q^{29} +3.19278 q^{31} +2.33189 q^{33} +1.00000 q^{35} +7.50023 q^{37} +2.42926 q^{39} -7.72012 q^{41} +10.8739 q^{43} -2.43773 q^{45} +13.5475 q^{47} +1.00000 q^{49} +16.0520 q^{51} +3.85241 q^{53} -1.00000 q^{55} -15.2781 q^{57} -7.16834 q^{59} +2.29287 q^{61} -2.43773 q^{63} -1.04175 q^{65} -10.0771 q^{67} -1.95050 q^{69} +3.89143 q^{71} +0.547006 q^{73} +2.33189 q^{75} -1.00000 q^{77} +8.83644 q^{79} -10.3707 q^{81} -7.89143 q^{83} -6.88368 q^{85} +19.9419 q^{87} +7.67288 q^{89} -1.04175 q^{91} +7.44522 q^{93} +6.55179 q^{95} -6.52672 q^{97} +2.43773 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{3} - 5 q^{5} - 5 q^{7} + 9 q^{9}+O(q^{10}) \) 5 * q - 2 * q^3 - 5 * q^5 - 5 * q^7 + 9 * q^9	\( 5 q - 2 q^{3} - 5 q^{5} - 5 q^{7} + 9 q^{9} + 5 q^{11} + 2 q^{13} + 2 q^{15} - 6 q^{17} - 6 q^{19} + 2 q^{21} + 2 q^{23} + 5 q^{25} - 20 q^{27} + 16 q^{29} + 12 q^{31} - 2 q^{33} + 5 q^{35} + 4 q^{37} - 8 q^{39} + 8 q^{41} + 4 q^{43} - 9 q^{45} + 5 q^{49} + 20 q^{51} + 12 q^{53} - 5 q^{55} + 8 q^{57} - 16 q^{59} - 2 q^{61} - 9 q^{63} - 2 q^{65} - 4 q^{67} + 20 q^{69} + 12 q^{71} + 2 q^{73} - 2 q^{75} - 5 q^{77} + 38 q^{79} + 13 q^{81} - 32 q^{83} + 6 q^{85} - 12 q^{87} + 26 q^{89} - 2 q^{91} + 16 q^{93} + 6 q^{95} + 8 q^{97} + 9 q^{99}+O(q^{100}) \) 5 * q - 2 * q^3 - 5 * q^5 - 5 * q^7 + 9 * q^9 + 5 * q^11 + 2 * q^13 + 2 * q^15 - 6 * q^17 - 6 * q^19 + 2 * q^21 + 2 * q^23 + 5 * q^25 - 20 * q^27 + 16 * q^29 + 12 * q^31 - 2 * q^33 + 5 * q^35 + 4 * q^37 - 8 * q^39 + 8 * q^41 + 4 * q^43 - 9 * q^45 + 5 * q^49 + 20 * q^51 + 12 * q^53 - 5 * q^55 + 8 * q^57 - 16 * q^59 - 2 * q^61 - 9 * q^63 - 2 * q^65 - 4 * q^67 + 20 * q^69 + 12 * q^71 + 2 * q^73 - 2 * q^75 - 5 * q^77 + 38 * q^79 + 13 * q^81 - 32 * q^83 + 6 * q^85 - 12 * q^87 + 26 * q^89 - 2 * q^91 + 16 * q^93 + 6 * q^95 + 8 * q^97 + 9 * q^99

Introduction

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