LMFDB - Embedded newform 3432.2.a.w.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3432,2,Mod(1,3432)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3432, 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("3432.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3432 = 2^{3} \cdot 3 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3432.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:	$27.4046579737$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.2172244.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 8x^{3} + 16x^{2} + 5x - 8 $ x^5 - 2x^4 - 8x^3 + 16x^2 + 5x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.679950$ of defining polynomial
Character	$\chi$	$=$	3432.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.05398 q^{5} +4.24902 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.05398 q^{5} +4.24902 q^{7} +1.00000 q^{9} -1.00000 q^{11} +1.00000 q^{13} +1.05398 q^{15} -4.55494 q^{17} -0.977426 q^{19} -4.24902 q^{21} -2.82631 q^{23} -3.88912 q^{25} -1.00000 q^{27} +4.75121 q^{29} -8.28043 q^{31} +1.00000 q^{33} -4.47839 q^{35} -6.14251 q^{37} -1.00000 q^{39} -6.92368 q^{41} +7.56208 q^{43} -1.05398 q^{45} +0.967375 q^{47} +11.0542 q^{49} +4.55494 q^{51} -5.76058 q^{53} +1.05398 q^{55} +0.977426 q^{57} -5.16364 q^{59} +12.8023 q^{61} +4.24902 q^{63} -1.05398 q^{65} +8.76942 q^{67} +2.82631 q^{69} +3.74999 q^{71} +0.141064 q^{73} +3.88912 q^{75} -4.24902 q^{77} -12.2704 q^{79} +1.00000 q^{81} -8.73747 q^{83} +4.80082 q^{85} -4.75121 q^{87} -16.5309 q^{89} +4.24902 q^{91} +8.28043 q^{93} +1.03019 q^{95} -7.42271 q^{97} -1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{3} + q^{5} - 5 q^{7} + 5 q^{9}+O(q^{10}) $ 5 * q - 5 * q^3 + q^5 - 5 * q^7 + 5 * q^9	$ 5 q - 5 q^{3} + q^{5} - 5 q^{7} + 5 q^{9} - 5 q^{11} + 5 q^{13} - q^{15} - 4 q^{19} + 5 q^{21} - 5 q^{23} + 4 q^{25} - 5 q^{27} + 11 q^{29} - 8 q^{31} + 5 q^{33} - 5 q^{35} - 8 q^{37} - 5 q^{39} - q^{41} + q^{43} + q^{45} - 18 q^{47} + 10 q^{49} + 2 q^{53} - q^{55} + 4 q^{57} - 13 q^{59} + 9 q^{61} - 5 q^{63} + q^{65} + 5 q^{67} + 5 q^{69} - 24 q^{71} - 13 q^{73} - 4 q^{75} + 5 q^{77} - 6 q^{79} + 5 q^{81} - 22 q^{83} - 22 q^{85} - 11 q^{87} - 14 q^{89} - 5 q^{91} + 8 q^{93} - 32 q^{95} - 20 q^{97} - 5 q^{99}+O(q^{100}) $ 5 * q - 5 * q^3 + q^5 - 5 * q^7 + 5 * q^9 - 5 * q^11 + 5 * q^13 - q^15 - 4 * q^19 + 5 * q^21 - 5 * q^23 + 4 * q^25 - 5 * q^27 + 11 * q^29 - 8 * q^31 + 5 * q^33 - 5 * q^35 - 8 * q^37 - 5 * q^39 - q^41 + q^43 + q^45 - 18 * q^47 + 10 * q^49 + 2 * q^53 - q^55 + 4 * q^57 - 13 * q^59 + 9 * q^61 - 5 * q^63 + q^65 + 5 * q^67 + 5 * q^69 - 24 * q^71 - 13 * q^73 - 4 * q^75 + 5 * q^77 - 6 * q^79 + 5 * q^81 - 22 * q^83 - 22 * q^85 - 11 * q^87 - 14 * q^89 - 5 * q^91 + 8 * q^93 - 32 * q^95 - 20 * q^97 - 5 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−1.05398	−0.471354	−0.235677	−	0.971831i	$-0.575731\pi$
$5$	−1.05398	−0.471354	−0.235677	+	0.971831i	$0.575731\pi$
$6$	0	0
$7$	4.24902	1.60598	0.802990	−	0.595992i	$-0.203241\pi$
$7$	4.24902	1.60598	0.802990	+	0.595992i	$0.203241\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	1.05398	0.272137
$16$	0	0
$17$	−4.55494	−1.10474	−0.552368	−	0.833600i	$-0.686276\pi$
$17$	−4.55494	−1.10474	−0.552368	+	0.833600i	$0.686276\pi$
$18$	0	0
$19$	−0.977426	−0.224237	−0.112118	−	0.993695i	$-0.535764\pi$
$19$	−0.977426	−0.224237	−0.112118	+	0.993695i	$0.535764\pi$
$20$	0	0
$21$	−4.24902	−0.927213
$22$	0	0
$23$	−2.82631	−0.589327	−0.294663	−	0.955601i	$-0.595207\pi$
$23$	−2.82631	−0.589327	−0.294663	+	0.955601i	$0.595207\pi$
$24$	0	0
$25$	−3.88912	−0.777825
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	4.75121	0.882277	0.441139	−	0.897439i	$-0.354575\pi$
$29$	4.75121	0.882277	0.441139	+	0.897439i	$0.354575\pi$
$30$	0	0
$31$	−8.28043	−1.48721	−0.743605	−	0.668619i	$-0.766886\pi$
$31$	−8.28043	−1.48721	−0.743605	+	0.668619i	$0.766886\pi$
$32$	0	0
$33$	1.00000	0.174078
$34$	0	0
$35$	−4.47839	−0.756986
$36$	0	0
$37$	−6.14251	−1.00982	−0.504912	−	0.863171i	$-0.668475\pi$
$37$	−6.14251	−1.00982	−0.504912	+	0.863171i	$0.668475\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−6.92368	−1.08130	−0.540648	−	0.841249i	$-0.681821\pi$
$41$	−6.92368	−1.08130	−0.540648	+	0.841249i	$0.681821\pi$
$42$	0	0
$43$	7.56208	1.15321	0.576603	−	0.817024i	$-0.304378\pi$
$43$	7.56208	1.15321	0.576603	+	0.817024i	$0.304378\pi$
$44$	0	0
$45$	−1.05398	−0.157118
$46$	0	0
$47$	0.967375	0.141106	0.0705530	−	0.997508i	$-0.477524\pi$
$47$	0.967375	0.141106	0.0705530	+	0.997508i	$0.477524\pi$
$48$	0	0
$49$	11.0542	1.57917
$50$	0	0
$51$	4.55494	0.637820
$52$	0	0
$53$	−5.76058	−0.791277	−0.395638	−	0.918406i	$-0.629477\pi$
$53$	−5.76058	−0.791277	−0.395638	+	0.918406i	$0.629477\pi$
$54$	0	0
$55$	1.05398	0.142119
$56$	0	0
$57$	0.977426	0.129463
$58$	0	0
$59$	−5.16364	−0.672248	−0.336124	−	0.941818i	$-0.609116\pi$
$59$	−5.16364	−0.672248	−0.336124	+	0.941818i	$0.609116\pi$
$60$	0	0
$61$	12.8023	1.63916	0.819582	−	0.572962i	$-0.194206\pi$
$61$	12.8023	1.63916	0.819582	+	0.572962i	$0.194206\pi$
$62$	0	0
$63$	4.24902	0.535327
$64$	0	0
$65$	−1.05398	−0.130730
$66$	0	0
$67$	8.76942	1.07135	0.535677	−	0.844423i	$-0.320056\pi$
$67$	8.76942	1.07135	0.535677	+	0.844423i	$0.320056\pi$
$68$	0	0
$69$	2.82631	0.340248
$70$	0	0
$71$	3.74999	0.445042	0.222521	−	0.974928i	$-0.428571\pi$
$71$	3.74999	0.445042	0.222521	+	0.974928i	$0.428571\pi$
$72$	0	0
$73$	0.141064	0.0165102	0.00825512	−	0.999966i	$-0.497372\pi$
$73$	0.141064	0.0165102	0.00825512	+	0.999966i	$0.497372\pi$
$74$	0	0
$75$	3.88912	0.449077
$76$	0	0
$77$	−4.24902	−0.484221
$78$	0	0
$79$	−12.2704	−1.38053	−0.690263	−	0.723559i	$-0.742505\pi$
$79$	−12.2704	−1.38053	−0.690263	+	0.723559i	$0.742505\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−8.73747	−0.959062	−0.479531	−	0.877525i	$-0.659193\pi$
$83$	−8.73747	−0.959062	−0.479531	+	0.877525i	$0.659193\pi$
$84$	0	0
$85$	4.80082	0.520722
$86$	0	0
$87$	−4.75121	−0.509383
$88$	0	0
$89$	−16.5309	−1.75227	−0.876136	−	0.482063i	$-0.839887\pi$
$89$	−16.5309	−1.75227	−0.876136	+	0.482063i	$0.839887\pi$
$90$	0	0
$91$	4.24902	0.445419
$92$	0	0
$93$	8.28043	0.858641
$94$	0	0
$95$	1.03019	0.105695
$96$	0	0
$97$	−7.42271	−0.753662	−0.376831	−	0.926282i	$-0.622986\pi$
$97$	−7.42271	−0.753662	−0.376831	+	0.926282i	$0.622986\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	0	0
$101$	10.2076	1.01569	0.507845	−	0.861448i	$-0.330442\pi$
$101$	10.2076	1.01569	0.507845	+	0.861448i	$0.330442\pi$
$102$	0	0
$103$	−7.27160	−0.716492	−0.358246	−	0.933627i	$-0.616625\pi$
$103$	−7.27160	−0.716492	−0.358246	+	0.933627i	$0.616625\pi$
$104$	0	0
$105$	4.47839	0.437046
$106$	0	0
$107$	−16.9870	−1.64220	−0.821099	−	0.570785i	$-0.806639\pi$
$107$	−16.9870	−1.64220	−0.821099	+	0.570785i	$0.806639\pi$
$108$	0	0
$109$	20.3032	1.94470	0.972349	−	0.233534i	$-0.0750289\pi$
$109$	20.3032	1.94470	0.972349	+	0.233534i	$0.0750289\pi$
$110$	0	0
$111$	6.14251	0.583022
$112$	0	0
$113$	−1.72404	−0.162184	−0.0810919	−	0.996707i	$-0.525841\pi$
$113$	−1.72404	−0.162184	−0.0810919	+	0.996707i	$0.525841\pi$
$114$	0	0
$115$	2.97888	0.277782
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	−19.3541	−1.77418
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	6.92368	0.624287
$124$	0	0
$125$	9.36897	0.837986
$126$	0	0
$127$	11.7382	1.04160	0.520800	−	0.853679i	$-0.325634\pi$
$127$	11.7382	1.04160	0.520800	+	0.853679i	$0.325634\pi$
$128$	0	0
$129$	−7.56208	−0.665804
$130$	0	0
$131$	−5.61898	−0.490932	−0.245466	−	0.969405i	$-0.578941\pi$
$131$	−5.61898	−0.490932	−0.245466	+	0.969405i	$0.578941\pi$
$132$	0	0
$133$	−4.15311	−0.360120
$134$	0	0
$135$	1.05398	0.0907122
$136$	0	0
$137$	3.70315	0.316381	0.158191	−	0.987409i	$-0.449434\pi$
$137$	3.70315	0.316381	0.158191	+	0.987409i	$0.449434\pi$
$138$	0	0
$139$	−20.3501	−1.72607	−0.863036	−	0.505143i	$-0.831440\pi$
$139$	−20.3501	−1.72607	−0.863036	+	0.505143i	$0.831440\pi$
$140$	0	0
$141$	−0.967375	−0.0814676
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	−5.00768	−0.415865
$146$	0	0
$147$	−11.0542	−0.911736
$148$	0	0
$149$	−2.30277	−0.188651	−0.0943253	−	0.995541i	$-0.530069\pi$
$149$	−2.30277	−0.188651	−0.0943253	+	0.995541i	$0.530069\pi$
$150$	0	0
$151$	8.19528	0.666922	0.333461	−	0.942764i	$-0.391783\pi$
$151$	8.19528	0.666922	0.333461	+	0.942764i	$0.391783\pi$
$152$	0	0
$153$	−4.55494	−0.368245
$154$	0	0
$155$	8.72741	0.701003
$156$	0	0
$157$	−11.9477	−0.953530	−0.476765	−	0.879031i	$-0.658191\pi$
$157$	−11.9477	−0.953530	−0.476765	+	0.879031i	$0.658191\pi$
$158$	0	0
$159$	5.76058	0.456844
$160$	0	0
$161$	−12.0091	−0.946447
$162$	0	0
$163$	−20.5822	−1.61212	−0.806062	−	0.591831i	$-0.798405\pi$
$163$	−20.5822	−1.61212	−0.806062	+	0.591831i	$0.798405\pi$
$164$	0	0
$165$	−1.05398	−0.0820523
$166$	0	0
$167$	−11.9463	−0.924429	−0.462214	−	0.886768i	$-0.652945\pi$
$167$	−11.9463	−0.924429	−0.462214	+	0.886768i	$0.652945\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−0.977426	−0.0747457
$172$	0	0
$173$	18.8879	1.43602	0.718010	−	0.696033i	$-0.245053\pi$
$173$	18.8879	1.43602	0.718010	+	0.696033i	$0.245053\pi$
$174$	0	0
$175$	−16.5250	−1.24917
$176$	0	0
$177$	5.16364	0.388123
$178$	0	0
$179$	−8.38247	−0.626536	−0.313268	−	0.949665i	$-0.601424\pi$
$179$	−8.38247	−0.626536	−0.313268	+	0.949665i	$0.601424\pi$
$180$	0	0
$181$	−10.2721	−0.763517	−0.381759	−	0.924262i	$-0.624681\pi$
$181$	−10.2721	−0.763517	−0.381759	+	0.924262i	$0.624681\pi$
$182$	0	0
$183$	−12.8023	−0.946371
$184$	0	0
$185$	6.47409	0.475985
$186$	0	0
$187$	4.55494	0.333091
$188$	0	0
$189$	−4.24902	−0.309071
$190$	0	0
$191$	−18.2672	−1.32177	−0.660885	−	0.750487i	$-0.729819\pi$
$191$	−18.2672	−1.32177	−0.660885	+	0.750487i	$0.729819\pi$
$192$	0	0
$193$	3.13054	0.225341	0.112670	−	0.993632i	$-0.464060\pi$
$193$	3.13054	0.225341	0.112670	+	0.993632i	$0.464060\pi$
$194$	0	0
$195$	1.05398	0.0754771
$196$	0	0
$197$	14.7380	1.05004	0.525020	−	0.851090i	$-0.324058\pi$
$197$	14.7380	1.05004	0.525020	+	0.851090i	$0.324058\pi$
$198$	0	0
$199$	−24.7859	−1.75702	−0.878511	−	0.477722i	$-0.841463\pi$
$199$	−24.7859	−1.75702	−0.878511	+	0.477722i	$0.841463\pi$
$200$	0	0
$201$	−8.76942	−0.618547
$202$	0	0
$203$	20.1880	1.41692
$204$	0	0
$205$	7.29742	0.509674
$206$	0	0
$207$	−2.82631	−0.196442
$208$	0	0
$209$	0.977426	0.0676100
$210$	0	0
$211$	−5.46319	−0.376101	−0.188051	−	0.982159i	$-0.560217\pi$
$211$	−5.46319	−0.376101	−0.188051	+	0.982159i	$0.560217\pi$
$212$	0	0
$213$	−3.74999	−0.256945
$214$	0	0
$215$	−7.97029	−0.543569
$216$	0	0
$217$	−35.1838	−2.38843
$218$	0	0
$219$	−0.141064	−0.00953220
$220$	0	0
$221$	−4.55494	−0.306399
$222$	0	0
$223$	25.2765	1.69264	0.846321	−	0.532673i	$-0.178813\pi$
$223$	25.2765	1.69264	0.846321	+	0.532673i	$0.178813\pi$
$224$	0	0
$225$	−3.88912	−0.259275
$226$	0	0
$227$	0.869929	0.0577392	0.0288696	−	0.999583i	$-0.490809\pi$
$227$	0.869929	0.0577392	0.0288696	+	0.999583i	$0.490809\pi$
$228$	0	0
$229$	−18.3646	−1.21357	−0.606783	−	0.794867i	$-0.707541\pi$
$229$	−18.3646	−1.21357	−0.606783	+	0.794867i	$0.707541\pi$
$230$	0	0
$231$	4.24902	0.279565
$232$	0	0
$233$	−23.2407	−1.52255	−0.761273	−	0.648432i	$-0.775425\pi$
$233$	−23.2407	−1.52255	−0.761273	+	0.648432i	$0.775425\pi$
$234$	0	0
$235$	−1.01959	−0.0665110
$236$	0	0
$237$	12.2704	0.797047
$238$	0	0
$239$	−22.1838	−1.43495	−0.717475	−	0.696585i	$-0.754702\pi$
$239$	−22.1838	−1.43495	−0.717475	+	0.696585i	$0.754702\pi$
$240$	0	0
$241$	22.7567	1.46589	0.732943	−	0.680290i	$-0.238146\pi$
$241$	22.7567	1.46589	0.732943	+	0.680290i	$0.238146\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−11.6509	−0.744350
$246$	0	0
$247$	−0.977426	−0.0621921
$248$	0	0
$249$	8.73747	0.553714
$250$	0	0
$251$	13.3920	0.845297	0.422648	−	0.906294i	$-0.361101\pi$
$251$	13.3920	0.845297	0.422648	+	0.906294i	$0.361101\pi$
$252$	0	0
$253$	2.82631	0.177689
$254$	0	0
$255$	−4.80082	−0.300639
$256$	0	0
$257$	−17.1612	−1.07049	−0.535243	−	0.844698i	$-0.679780\pi$
$257$	−17.1612	−1.07049	−0.535243	+	0.844698i	$0.679780\pi$
$258$	0	0
$259$	−26.0997	−1.62176
$260$	0	0
$261$	4.75121	0.294092
$262$	0	0
$263$	−15.2116	−0.937985	−0.468992	−	0.883202i	$-0.655383\pi$
$263$	−15.2116	−0.937985	−0.468992	+	0.883202i	$0.655383\pi$
$264$	0	0
$265$	6.07154	0.372972
$266$	0	0
$267$	16.5309	1.01168
$268$	0	0
$269$	9.36806	0.571181	0.285590	−	0.958352i	$-0.407810\pi$
$269$	9.36806	0.571181	0.285590	+	0.958352i	$0.407810\pi$
$270$	0	0
$271$	−21.4721	−1.30434	−0.652168	−	0.758074i	$-0.726140\pi$
$271$	−21.4721	−1.30434	−0.652168	+	0.758074i	$0.726140\pi$
$272$	0	0
$273$	−4.24902	−0.257163
$274$	0	0
$275$	3.88912	0.234523
$276$	0	0
$277$	24.4894	1.47143	0.735714	−	0.677292i	$-0.236847\pi$
$277$	24.4894	1.47143	0.735714	+	0.677292i	$0.236847\pi$
$278$	0	0
$279$	−8.28043	−0.495736
$280$	0	0
$281$	17.6553	1.05323	0.526614	−	0.850105i	$-0.323461\pi$
$281$	17.6553	1.05323	0.526614	+	0.850105i	$0.323461\pi$
$282$	0	0
$283$	13.9691	0.830374	0.415187	−	0.909736i	$-0.363716\pi$
$283$	13.9691	0.830374	0.415187	+	0.909736i	$0.363716\pi$
$284$	0	0
$285$	−1.03019	−0.0610231
$286$	0	0
$287$	−29.4189	−1.73654
$288$	0	0
$289$	3.74752	0.220442
$290$	0	0
$291$	7.42271	0.435127
$292$	0	0
$293$	1.06528	0.0622346	0.0311173	−	0.999516i	$-0.490093\pi$
$293$	1.06528	0.0622346	0.0311173	+	0.999516i	$0.490093\pi$
$294$	0	0
$295$	5.44237	0.316867
$296$	0	0
$297$	1.00000	0.0580259
$298$	0	0
$299$	−2.82631	−0.163450
$300$	0	0
$301$	32.1315	1.85203
$302$	0	0
$303$	−10.2076	−0.586409
$304$	0	0
$305$	−13.4933	−0.772627
$306$	0	0
$307$	17.5349	1.00077	0.500385	−	0.865803i	$-0.333192\pi$
$307$	17.5349	1.00077	0.500385	+	0.865803i	$0.333192\pi$
$308$	0	0
$309$	7.27160	0.413667
$310$	0	0
$311$	−7.55761	−0.428553	−0.214276	−	0.976773i	$-0.568739\pi$
$311$	−7.55761	−0.428553	−0.214276	+	0.976773i	$0.568739\pi$
$312$	0	0
$313$	−10.2350	−0.578519	−0.289259	−	0.957251i	$-0.593409\pi$
$313$	−10.2350	−0.578519	−0.289259	+	0.957251i	$0.593409\pi$
$314$	0	0
$315$	−4.47839	−0.252329
$316$	0	0
$317$	−3.76319	−0.211362	−0.105681	−	0.994400i	$-0.533702\pi$
$317$	−3.76319	−0.211362	−0.105681	+	0.994400i	$0.533702\pi$
$318$	0	0
$319$	−4.75121	−0.266017
$320$	0	0
$321$	16.9870	0.948124
$322$	0	0
$323$	4.45212	0.247723
$324$	0	0
$325$	−3.88912	−0.215730
$326$	0	0
$327$	−20.3032	−1.12277
$328$	0	0
$329$	4.11040	0.226614
$330$	0	0
$331$	−24.5851	−1.35132	−0.675660	−	0.737213i	$-0.736141\pi$
$331$	−24.5851	−1.35132	−0.675660	+	0.737213i	$0.736141\pi$
$332$	0	0
$333$	−6.14251	−0.336608
$334$	0	0
$335$	−9.24280	−0.504988
$336$	0	0
$337$	17.4966	0.953100	0.476550	−	0.879147i	$-0.341887\pi$
$337$	17.4966	0.953100	0.476550	+	0.879147i	$0.341887\pi$
$338$	0	0
$339$	1.72404	0.0936368
$340$	0	0
$341$	8.28043	0.448411
$342$	0	0
$343$	17.2265	0.930141
$344$	0	0
$345$	−2.97888	−0.160377
$346$	0	0
$347$	−3.03602	−0.162982	−0.0814909	−	0.996674i	$-0.525968\pi$
$347$	−3.03602	−0.162982	−0.0814909	+	0.996674i	$0.525968\pi$
$348$	0	0
$349$	−0.587880	−0.0314685	−0.0157343	−	0.999876i	$-0.505009\pi$
$349$	−0.587880	−0.0314685	−0.0157343	+	0.999876i	$0.505009\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	−0.259864	−0.0138312	−0.00691558	−	0.999976i	$-0.502201\pi$
$353$	−0.259864	−0.0138312	−0.00691558	+	0.999976i	$0.502201\pi$
$354$	0	0
$355$	−3.95242	−0.209772
$356$	0	0
$357$	19.3541	1.02433
$358$	0	0
$359$	−6.59640	−0.348145	−0.174072	−	0.984733i	$-0.555693\pi$
$359$	−6.59640	−0.348145	−0.174072	+	0.984733i	$0.555693\pi$
$360$	0	0
$361$	−18.0446	−0.949718
$362$	0	0
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	−0.148678	−0.00778218
$366$	0	0
$367$	−5.13108	−0.267840	−0.133920	−	0.990992i	$-0.542757\pi$
$367$	−5.13108	−0.267840	−0.133920	+	0.990992i	$0.542757\pi$
$368$	0	0
$369$	−6.92368	−0.360432
$370$	0	0
$371$	−24.4769	−1.27078
$372$	0	0
$373$	2.66849	0.138169	0.0690846	−	0.997611i	$-0.477992\pi$
$373$	2.66849	0.138169	0.0690846	+	0.997611i	$0.477992\pi$
$374$	0	0
$375$	−9.36897	−0.483811
$376$	0	0
$377$	4.75121	0.244700
$378$	0	0
$379$	−17.2900	−0.888129	−0.444065	−	0.895995i	$-0.646464\pi$
$379$	−17.2900	−0.888129	−0.444065	+	0.895995i	$0.646464\pi$
$380$	0	0
$381$	−11.7382	−0.601368
$382$	0	0
$383$	−26.7826	−1.36853	−0.684264	−	0.729235i	$-0.739876\pi$
$383$	−26.7826	−1.36853	−0.684264	+	0.729235i	$0.739876\pi$
$384$	0	0
$385$	4.47839	0.228240
$386$	0	0
$387$	7.56208	0.384402
$388$	0	0
$389$	25.8661	1.31146	0.655732	−	0.754994i	$-0.272360\pi$
$389$	25.8661	1.31146	0.655732	+	0.754994i	$0.272360\pi$
$390$	0	0
$391$	12.8737	0.651050
$392$	0	0
$393$	5.61898	0.283440
$394$	0	0
$395$	12.9327	0.650717
$396$	0	0
$397$	−0.913162	−0.0458303	−0.0229151	−	0.999737i	$-0.507295\pi$
$397$	−0.913162	−0.0458303	−0.0229151	+	0.999737i	$0.507295\pi$
$398$	0	0
$399$	4.15311	0.207915
$400$	0	0
$401$	11.1048	0.554547	0.277274	−	0.960791i	$-0.410569\pi$
$401$	11.1048	0.554547	0.277274	+	0.960791i	$0.410569\pi$
$402$	0	0
$403$	−8.28043	−0.412478
$404$	0	0
$405$	−1.05398	−0.0523727
$406$	0	0
$407$	6.14251	0.304473
$408$	0	0
$409$	6.25242	0.309162	0.154581	−	0.987980i	$-0.450597\pi$
$409$	6.25242	0.309162	0.154581	+	0.987980i	$0.450597\pi$
$410$	0	0
$411$	−3.70315	−0.182663
$412$	0	0
$413$	−21.9404	−1.07962
$414$	0	0
$415$	9.20912	0.452058
$416$	0	0
$417$	20.3501	0.996548
$418$	0	0
$419$	21.7682	1.06345	0.531723	−	0.846918i	$-0.321545\pi$
$419$	21.7682	1.06345	0.531723	+	0.846918i	$0.321545\pi$
$420$	0	0
$421$	−38.3099	−1.86711	−0.933556	−	0.358433i	$-0.883311\pi$
$421$	−38.3099	−1.86711	−0.933556	+	0.358433i	$0.883311\pi$
$422$	0	0
$423$	0.967375	0.0470354
$424$	0	0
$425$	17.7147	0.859291
$426$	0	0
$427$	54.3972	2.63246
$428$	0	0
$429$	1.00000	0.0482805
$430$	0	0
$431$	32.3906	1.56020	0.780099	−	0.625656i	$-0.215169\pi$
$431$	32.3906	1.56020	0.780099	+	0.625656i	$0.215169\pi$
$432$	0	0
$433$	13.2025	0.634471	0.317236	−	0.948347i	$-0.397245\pi$
$433$	13.2025	0.634471	0.317236	+	0.948347i	$0.397245\pi$
$434$	0	0
$435$	5.00768	0.240100
$436$	0	0
$437$	2.76251	0.132149
$438$	0	0
$439$	36.2684	1.73100	0.865498	−	0.500912i	$-0.167002\pi$
$439$	36.2684	1.73100	0.865498	+	0.500912i	$0.167002\pi$
$440$	0	0
$441$	11.0542	0.526391
$442$	0	0
$443$	−24.3416	−1.15650	−0.578251	−	0.815859i	$-0.696265\pi$
$443$	−24.3416	−1.15650	−0.578251	+	0.815859i	$0.696265\pi$
$444$	0	0
$445$	17.4233	0.825942
$446$	0	0
$447$	2.30277	0.108917
$448$	0	0
$449$	34.8932	1.64671	0.823356	−	0.567525i	$-0.192099\pi$
$449$	34.8932	1.64671	0.823356	+	0.567525i	$0.192099\pi$
$450$	0	0
$451$	6.92368	0.326023
$452$	0	0
$453$	−8.19528	−0.385048
$454$	0	0
$455$	−4.47839	−0.209950
$456$	0	0
$457$	−22.1410	−1.03571	−0.517856	−	0.855468i	$-0.673270\pi$
$457$	−22.1410	−1.03571	−0.517856	+	0.855468i	$0.673270\pi$
$458$	0	0
$459$	4.55494	0.212607
$460$	0	0
$461$	−30.8891	−1.43865	−0.719325	−	0.694673i	$-0.755549\pi$
$461$	−30.8891	−1.43865	−0.719325	+	0.694673i	$0.755549\pi$
$462$	0	0
$463$	13.4244	0.623885	0.311943	−	0.950101i	$-0.399020\pi$
$463$	13.4244	0.623885	0.311943	+	0.950101i	$0.399020\pi$
$464$	0	0
$465$	−8.72741	−0.404724
$466$	0	0
$467$	14.4030	0.666493	0.333247	−	0.942840i	$-0.391856\pi$
$467$	14.4030	0.666493	0.333247	+	0.942840i	$0.391856\pi$
$468$	0	0
$469$	37.2615	1.72057
$470$	0	0
$471$	11.9477	0.550521
$472$	0	0
$473$	−7.56208	−0.347705
$474$	0	0
$475$	3.80133	0.174417
$476$	0	0
$477$	−5.76058	−0.263759
$478$	0	0
$479$	36.4321	1.66463	0.832313	−	0.554306i	$-0.187016\pi$
$479$	36.4321	1.66463	0.832313	+	0.554306i	$0.187016\pi$
$480$	0	0
$481$	−6.14251	−0.280075
$482$	0	0
$483$	12.0091	0.546431
$484$	0	0
$485$	7.82340	0.355242
$486$	0	0
$487$	−16.2709	−0.737306	−0.368653	−	0.929567i	$-0.620181\pi$
$487$	−16.2709	−0.737306	−0.368653	+	0.929567i	$0.620181\pi$
$488$	0	0
$489$	20.5822	0.930760
$490$	0	0
$491$	40.6615	1.83503	0.917513	−	0.397705i	$-0.130193\pi$
$491$	40.6615	1.83503	0.917513	+	0.397705i	$0.130193\pi$
$492$	0	0
$493$	−21.6415	−0.974683
$494$	0	0
$495$	1.05398	0.0473729
$496$	0	0
$497$	15.9338	0.714728
$498$	0	0
$499$	7.96515	0.356569	0.178285	−	0.983979i	$-0.442945\pi$
$499$	7.96515	0.356569	0.178285	+	0.983979i	$0.442945\pi$
$500$	0	0
$501$	11.9463	0.533719
$502$	0	0
$503$	−36.6300	−1.63325	−0.816625	−	0.577169i	$-0.804157\pi$
$503$	−36.6300	−1.63325	−0.816625	+	0.577169i	$0.804157\pi$
$504$	0	0
$505$	−10.7586	−0.478750
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	−23.2934	−1.03246	−0.516231	−	0.856449i	$-0.672666\pi$
$509$	−23.2934	−1.03246	−0.516231	+	0.856449i	$0.672666\pi$
$510$	0	0
$511$	0.599383	0.0265151
$512$	0	0
$513$	0.977426	0.0431544
$514$	0	0
$515$	7.66412	0.337722
$516$	0	0
$517$	−0.967375	−0.0425451
$518$	0	0
$519$	−18.8879	−0.829087
$520$	0	0
$521$	−2.85802	−0.125212	−0.0626060	−	0.998038i	$-0.519941\pi$
$521$	−2.85802	−0.125212	−0.0626060	+	0.998038i	$0.519941\pi$
$522$	0	0
$523$	26.1624	1.14400	0.572001	−	0.820253i	$-0.306167\pi$
$523$	26.1624	1.14400	0.572001	+	0.820253i	$0.306167\pi$
$524$	0	0
$525$	16.5250	0.721210
$526$	0	0
$527$	37.7169	1.64297
$528$	0	0
$529$	−15.0120	−0.652694
$530$	0	0
$531$	−5.16364	−0.224083
$532$	0	0
$533$	−6.92368	−0.299898
$534$	0	0
$535$	17.9040	0.774058
$536$	0	0
$537$	8.38247	0.361730
$538$	0	0
$539$	−11.0542	−0.476139
$540$	0	0
$541$	−41.4318	−1.78129	−0.890646	−	0.454697i	$-0.849747\pi$
$541$	−41.4318	−1.78129	−0.890646	+	0.454697i	$0.849747\pi$
$542$	0	0
$543$	10.2721	0.440817
$544$	0	0
$545$	−21.3992	−0.916642
$546$	0	0
$547$	15.9239	0.680857	0.340429	−	0.940270i	$-0.389428\pi$
$547$	15.9239	0.680857	0.340429	+	0.940270i	$0.389428\pi$
$548$	0	0
$549$	12.8023	0.546388
$550$	0	0
$551$	−4.64395	−0.197839
$552$	0	0
$553$	−52.1372	−2.21710
$554$	0	0
$555$	−6.47409	−0.274810
$556$	0	0
$557$	33.6251	1.42474	0.712370	−	0.701804i	$-0.247622\pi$
$557$	33.6251	1.42474	0.712370	+	0.701804i	$0.247622\pi$
$558$	0	0
$559$	7.56208	0.319842
$560$	0	0
$561$	−4.55494	−0.192310
$562$	0	0
$563$	15.4510	0.651181	0.325590	−	0.945511i	$-0.394437\pi$
$563$	15.4510	0.651181	0.325590	+	0.945511i	$0.394437\pi$
$564$	0	0
$565$	1.81710	0.0764460
$566$	0	0
$567$	4.24902	0.178442
$568$	0	0
$569$	15.2238	0.638214	0.319107	−	0.947719i	$-0.396617\pi$
$569$	15.2238	0.638214	0.319107	+	0.947719i	$0.396617\pi$
$570$	0	0
$571$	2.18083	0.0912651	0.0456325	−	0.998958i	$-0.485470\pi$
$571$	2.18083	0.0912651	0.0456325	+	0.998958i	$0.485470\pi$
$572$	0	0
$573$	18.2672	0.763125
$574$	0	0
$575$	10.9919	0.458393
$576$	0	0
$577$	15.3464	0.638879	0.319440	−	0.947607i	$-0.396505\pi$
$577$	15.3464	0.638879	0.319440	+	0.947607i	$0.396505\pi$
$578$	0	0
$579$	−3.13054	−0.130101
$580$	0	0
$581$	−37.1257	−1.54023
$582$	0	0
$583$	5.76058	0.238579
$584$	0	0
$585$	−1.05398	−0.0435767
$586$	0	0
$587$	−33.8295	−1.39629	−0.698147	−	0.715954i	$-0.745992\pi$
$587$	−33.8295	−1.39629	−0.698147	+	0.715954i	$0.745992\pi$
$588$	0	0
$589$	8.09351	0.333487
$590$	0	0
$591$	−14.7380	−0.606241
$592$	0	0
$593$	0.689961	0.0283333	0.0141666	−	0.999900i	$-0.495490\pi$
$593$	0.689961	0.0283333	0.0141666	+	0.999900i	$0.495490\pi$
$594$	0	0
$595$	20.3988	0.836270
$596$	0	0
$597$	24.7859	1.01442
$598$	0	0
$599$	4.15034	0.169578	0.0847891	−	0.996399i	$-0.472978\pi$
$599$	4.15034	0.169578	0.0847891	+	0.996399i	$0.472978\pi$
$600$	0	0
$601$	6.88767	0.280954	0.140477	−	0.990084i	$-0.455136\pi$
$601$	6.88767	0.280954	0.140477	+	0.990084i	$0.455136\pi$
$602$	0	0
$603$	8.76942	0.357118
$604$	0	0
$605$	−1.05398	−0.0428504
$606$	0	0
$607$	11.7382	0.476441	0.238220	−	0.971211i	$-0.423436\pi$
$607$	11.7382	0.476441	0.238220	+	0.971211i	$0.423436\pi$
$608$	0	0
$609$	−20.1880	−0.818059
$610$	0	0
$611$	0.967375	0.0391358
$612$	0	0
$613$	−14.6477	−0.591616	−0.295808	−	0.955247i	$-0.595589\pi$
$613$	−14.6477	−0.591616	−0.295808	+	0.955247i	$0.595589\pi$
$614$	0	0
$615$	−7.29742	−0.294260
$616$	0	0
$617$	−10.9752	−0.441844	−0.220922	−	0.975291i	$-0.570907\pi$
$617$	−10.9752	−0.441844	−0.220922	+	0.975291i	$0.570907\pi$
$618$	0	0
$619$	32.2138	1.29478	0.647391	−	0.762158i	$-0.275860\pi$
$619$	32.2138	1.29478	0.647391	+	0.762158i	$0.275860\pi$
$620$	0	0
$621$	2.82631	0.113416
$622$	0	0
$623$	−70.2402	−2.81412
$624$	0	0
$625$	9.57092	0.382837
$626$	0	0
$627$	−0.977426	−0.0390346
$628$	0	0
$629$	27.9788	1.11559
$630$	0	0
$631$	45.0712	1.79426	0.897128	−	0.441770i	$-0.145649\pi$
$631$	45.0712	1.79426	0.897128	+	0.441770i	$0.145649\pi$
$632$	0	0
$633$	5.46319	0.217142
$634$	0	0
$635$	−12.3719	−0.490963
$636$	0	0
$637$	11.0542	0.437984
$638$	0	0
$639$	3.74999	0.148347
$640$	0	0
$641$	28.1430	1.11158	0.555791	−	0.831322i	$-0.312415\pi$
$641$	28.1430	1.11158	0.555791	+	0.831322i	$0.312415\pi$
$642$	0	0
$643$	−7.10237	−0.280090	−0.140045	−	0.990145i	$-0.544725\pi$
$643$	−7.10237	−0.280090	−0.140045	+	0.990145i	$0.544725\pi$
$644$	0	0
$645$	7.97029	0.313830
$646$	0	0
$647$	−28.2768	−1.11167	−0.555837	−	0.831291i	$-0.687602\pi$
$647$	−28.2768	−1.11167	−0.555837	+	0.831291i	$0.687602\pi$
$648$	0	0
$649$	5.16364	0.202690
$650$	0	0
$651$	35.1838	1.37896
$652$	0	0
$653$	12.1708	0.476279	0.238140	−	0.971231i	$-0.423462\pi$
$653$	12.1708	0.476279	0.238140	+	0.971231i	$0.423462\pi$
$654$	0	0
$655$	5.92229	0.231403
$656$	0	0
$657$	0.141064	0.00550342
$658$	0	0
$659$	−31.9183	−1.24336	−0.621681	−	0.783271i	$-0.713550\pi$
$659$	−31.9183	−1.24336	−0.621681	+	0.783271i	$0.713550\pi$
$660$	0	0
$661$	−33.6021	−1.30697	−0.653485	−	0.756939i	$-0.726694\pi$
$661$	−33.6021	−1.30697	−0.653485	+	0.756939i	$0.726694\pi$
$662$	0	0
$663$	4.55494	0.176899
$664$	0	0
$665$	4.37730	0.169744
$666$	0	0
$667$	−13.4284	−0.519949
$668$	0	0
$669$	−25.2765	−0.977247
$670$	0	0
$671$	−12.8023	−0.494226
$672$	0	0
$673$	28.3314	1.09210	0.546048	−	0.837754i	$-0.316132\pi$
$673$	28.3314	1.09210	0.546048	+	0.837754i	$0.316132\pi$
$674$	0	0
$675$	3.88912	0.149692
$676$	0	0
$677$	2.74310	0.105426	0.0527129	−	0.998610i	$-0.483213\pi$
$677$	2.74310	0.105426	0.0527129	+	0.998610i	$0.483213\pi$
$678$	0	0
$679$	−31.5393	−1.21037
$680$	0	0
$681$	−0.869929	−0.0333358
$682$	0	0
$683$	−23.3066	−0.891803	−0.445901	−	0.895082i	$-0.647117\pi$
$683$	−23.3066	−0.891803	−0.445901	+	0.895082i	$0.647117\pi$
$684$	0	0
$685$	−3.90304	−0.149128
$686$	0	0
$687$	18.3646	0.700653
$688$	0	0
$689$	−5.76058	−0.219461
$690$	0	0
$691$	−9.97295	−0.379389	−0.189694	−	0.981843i	$-0.560750\pi$
$691$	−9.97295	−0.379389	−0.189694	+	0.981843i	$0.560750\pi$
$692$	0	0
$693$	−4.24902	−0.161407
$694$	0	0
$695$	21.4486	0.813591
$696$	0	0
$697$	31.5370	1.19455
$698$	0	0
$699$	23.2407	0.879042
$700$	0	0
$701$	13.1096	0.495144	0.247572	−	0.968869i	$-0.420367\pi$
$701$	13.1096	0.495144	0.247572	+	0.968869i	$0.420367\pi$
$702$	0	0
$703$	6.00386	0.226440
$704$	0	0
$705$	1.01959	0.0384001
$706$	0	0
$707$	43.3722	1.63118
$708$	0	0
$709$	23.7733	0.892826	0.446413	−	0.894827i	$-0.352701\pi$
$709$	23.7733	0.892826	0.446413	+	0.894827i	$0.352701\pi$
$710$	0	0
$711$	−12.2704	−0.460175
$712$	0	0
$713$	23.4031	0.876452
$714$	0	0
$715$	1.05398	0.0394166
$716$	0	0
$717$	22.1838	0.828468
$718$	0	0
$719$	−21.1219	−0.787712	−0.393856	−	0.919172i	$-0.628859\pi$
$719$	−21.1219	−0.787712	−0.393856	+	0.919172i	$0.628859\pi$
$720$	0	0
$721$	−30.8972	−1.15067
$722$	0	0
$723$	−22.7567	−0.846330
$724$	0	0
$725$	−18.4780	−0.686257
$726$	0	0
$727$	10.5157	0.390006	0.195003	−	0.980803i	$-0.437528\pi$
$727$	10.5157	0.390006	0.195003	+	0.980803i	$0.437528\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−34.4449	−1.27399
$732$	0	0
$733$	−49.4643	−1.82701	−0.913503	−	0.406833i	$-0.866633\pi$
$733$	−49.4643	−1.82701	−0.913503	+	0.406833i	$0.866633\pi$
$734$	0	0
$735$	11.6509	0.429751
$736$	0	0
$737$	−8.76942	−0.323026
$738$	0	0
$739$	−9.74040	−0.358306	−0.179153	−	0.983821i	$-0.557336\pi$
$739$	−9.74040	−0.358306	−0.179153	+	0.983821i	$0.557336\pi$
$740$	0	0
$741$	0.977426	0.0359067
$742$	0	0
$743$	−13.7182	−0.503272	−0.251636	−	0.967822i	$-0.580969\pi$
$743$	−13.7182	−0.503272	−0.251636	+	0.967822i	$0.580969\pi$
$744$	0	0
$745$	2.42708	0.0889213
$746$	0	0
$747$	−8.73747	−0.319687
$748$	0	0
$749$	−72.1783	−2.63734
$750$	0	0
$751$	22.6627	0.826972	0.413486	−	0.910510i	$-0.364311\pi$
$751$	22.6627	0.826972	0.413486	+	0.910510i	$0.364311\pi$
$752$	0	0
$753$	−13.3920	−0.488032
$754$	0	0
$755$	−8.63766	−0.314357
$756$	0	0
$757$	−21.1376	−0.768259	−0.384130	−	0.923279i	$-0.625498\pi$
$757$	−21.1376	−0.768259	−0.384130	+	0.923279i	$0.625498\pi$
$758$	0	0
$759$	−2.82631	−0.102589
$760$	0	0
$761$	2.32152	0.0841551	0.0420775	−	0.999114i	$-0.486602\pi$
$761$	2.32152	0.0841551	0.0420775	+	0.999114i	$0.486602\pi$
$762$	0	0
$763$	86.2690	3.12315
$764$	0	0
$765$	4.80082	0.173574
$766$	0	0
$767$	−5.16364	−0.186448
$768$	0	0
$769$	12.1170	0.436951	0.218476	−	0.975842i	$-0.429892\pi$
$769$	12.1170	0.436951	0.218476	+	0.975842i	$0.429892\pi$
$770$	0	0
$771$	17.1612	0.618046
$772$	0	0
$773$	−40.2007	−1.44592	−0.722959	−	0.690890i	$-0.757219\pi$
$773$	−40.2007	−1.44592	−0.722959	+	0.690890i	$0.757219\pi$
$774$	0	0
$775$	32.2036	1.15679
$776$	0	0
$777$	26.0997	0.936321
$778$	0	0
$779$	6.76738	0.242467
$780$	0	0
$781$	−3.74999	−0.134185
$782$	0	0
$783$	−4.75121	−0.169794
$784$	0	0
$785$	12.5926	0.449451
$786$	0	0
$787$	−30.3882	−1.08322	−0.541611	−	0.840629i	$-0.682185\pi$
$787$	−30.3882	−1.08322	−0.541611	+	0.840629i	$0.682185\pi$
$788$	0	0
$789$	15.2116	0.541546
$790$	0	0
$791$	−7.32548	−0.260464
$792$	0	0
$793$	12.8023	0.454622
$794$	0	0
$795$	−6.07154	−0.215335
$796$	0	0
$797$	33.9490	1.20254	0.601268	−	0.799047i	$-0.294662\pi$
$797$	33.9490	1.20254	0.601268	+	0.799047i	$0.294662\pi$
$798$	0	0
$799$	−4.40634	−0.155885
$800$	0	0
$801$	−16.5309	−0.584091
$802$	0	0
$803$	−0.141064	−0.00497803
$804$	0	0
$805$	12.6573	0.446112
$806$	0	0
$807$	−9.36806	−0.329771
$808$	0	0
$809$	4.60592	0.161936	0.0809678	−	0.996717i	$-0.474199\pi$
$809$	4.60592	0.161936	0.0809678	+	0.996717i	$0.474199\pi$
$810$	0	0
$811$	15.5206	0.545003	0.272501	−	0.962155i	$-0.412149\pi$
$811$	15.5206	0.545003	0.272501	+	0.962155i	$0.412149\pi$
$812$	0	0
$813$	21.4721	0.753059
$814$	0	0
$815$	21.6933	0.759882
$816$	0	0
$817$	−7.39138	−0.258592
$818$	0	0
$819$	4.24902	0.148473
$820$	0	0
$821$	15.7361	0.549193	0.274597	−	0.961560i	$-0.411456\pi$
$821$	15.7361	0.549193	0.274597	+	0.961560i	$0.411456\pi$
$822$	0	0
$823$	−4.77401	−0.166412	−0.0832058	−	0.996532i	$-0.526516\pi$
$823$	−4.77401	−0.166412	−0.0832058	+	0.996532i	$0.526516\pi$
$824$	0	0
$825$	−3.88912	−0.135402
$826$	0	0
$827$	−0.536827	−0.0186673	−0.00933365	−	0.999956i	$-0.502971\pi$
$827$	−0.536827	−0.0186673	−0.00933365	+	0.999956i	$0.502971\pi$
$828$	0	0
$829$	26.1757	0.909120	0.454560	−	0.890716i	$-0.349796\pi$
$829$	26.1757	0.909120	0.454560	+	0.890716i	$0.349796\pi$
$830$	0	0
$831$	−24.4894	−0.849530
$832$	0	0
$833$	−50.3513	−1.74457
$834$	0	0
$835$	12.5911	0.435734
$836$	0	0
$837$	8.28043	0.286214
$838$	0	0
$839$	−20.3887	−0.703897	−0.351949	−	0.936019i	$-0.614481\pi$
$839$	−20.3887	−0.703897	−0.351949	+	0.936019i	$0.614481\pi$
$840$	0	0
$841$	−6.42603	−0.221587
$842$	0	0
$843$	−17.6553	−0.608081
$844$	0	0
$845$	−1.05398	−0.0362580
$846$	0	0
$847$	4.24902	0.145998
$848$	0	0
$849$	−13.9691	−0.479417
$850$	0	0
$851$	17.3607	0.595116
$852$	0	0
$853$	−39.4039	−1.34916	−0.674581	−	0.738201i	$-0.735676\pi$
$853$	−39.4039	−1.34916	−0.674581	+	0.738201i	$0.735676\pi$
$854$	0	0
$855$	1.03019	0.0352317
$856$	0	0
$857$	3.72146	0.127123	0.0635614	−	0.997978i	$-0.479754\pi$
$857$	3.72146	0.127123	0.0635614	+	0.997978i	$0.479754\pi$
$858$	0	0
$859$	19.8699	0.677952	0.338976	−	0.940795i	$-0.389919\pi$
$859$	19.8699	0.677952	0.338976	+	0.940795i	$0.389919\pi$
$860$	0	0
$861$	29.4189	1.00259
$862$	0	0
$863$	33.6108	1.14413	0.572063	−	0.820210i	$-0.306143\pi$
$863$	33.6108	1.14413	0.572063	+	0.820210i	$0.306143\pi$
$864$	0	0
$865$	−19.9075	−0.676874
$866$	0	0
$867$	−3.74752	−0.127272
$868$	0	0
$869$	12.2704	0.416244
$870$	0	0
$871$	8.76942	0.297140
$872$	0	0
$873$	−7.42271	−0.251221
$874$	0	0
$875$	39.8090	1.34579
$876$	0	0
$877$	52.9946	1.78950	0.894750	−	0.446567i	$-0.147354\pi$
$877$	52.9946	1.78950	0.894750	+	0.446567i	$0.147354\pi$
$878$	0	0
$879$	−1.06528	−0.0359311
$880$	0	0
$881$	16.6459	0.560816	0.280408	−	0.959881i	$-0.409530\pi$
$881$	16.6459	0.560816	0.280408	+	0.959881i	$0.409530\pi$
$882$	0	0
$883$	−42.1900	−1.41981	−0.709903	−	0.704299i	$-0.751261\pi$
$883$	−42.1900	−1.41981	−0.709903	+	0.704299i	$0.751261\pi$
$884$	0	0
$885$	−5.44237	−0.182943
$886$	0	0
$887$	−55.4874	−1.86308	−0.931542	−	0.363633i	$-0.881536\pi$
$887$	−55.4874	−1.86308	−0.931542	+	0.363633i	$0.881536\pi$
$888$	0	0
$889$	49.8761	1.67279
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	−0.945537	−0.0316412
$894$	0	0
$895$	8.83497	0.295320
$896$	0	0
$897$	2.82631	0.0943678
$898$	0	0
$899$	−39.3420	−1.31213
$900$	0	0
$901$	26.2391	0.874152
$902$	0	0
$903$	−32.1315	−1.06927
$904$	0	0
$905$	10.8266	0.359887
$906$	0	0
$907$	1.47170	0.0488670	0.0244335	−	0.999701i	$-0.492222\pi$
$907$	1.47170	0.0488670	0.0244335	+	0.999701i	$0.492222\pi$
$908$	0	0
$909$	10.2076	0.338564
$910$	0	0
$911$	−31.3498	−1.03866	−0.519332	−	0.854573i	$-0.673819\pi$
$911$	−31.3498	−1.03866	−0.519332	+	0.854573i	$0.673819\pi$
$912$	0	0
$913$	8.73747	0.289168
$914$	0	0
$915$	13.4933	0.446076
$916$	0	0
$917$	−23.8752	−0.788428
$918$	0	0
$919$	3.07327	0.101378	0.0506889	−	0.998714i	$-0.483858\pi$
$919$	3.07327	0.101378	0.0506889	+	0.998714i	$0.483858\pi$
$920$	0	0
$921$	−17.5349	−0.577795
$922$	0	0
$923$	3.74999	0.123432
$924$	0	0
$925$	23.8890	0.785466
$926$	0	0
$927$	−7.27160	−0.238831
$928$	0	0
$929$	8.13209	0.266805	0.133403	−	0.991062i	$-0.457410\pi$
$929$	8.13209	0.266805	0.133403	+	0.991062i	$0.457410\pi$
$930$	0	0
$931$	−10.8047	−0.354109
$932$	0	0
$933$	7.55761	0.247425
$934$	0	0
$935$	−4.80082	−0.157004
$936$	0	0
$937$	36.7711	1.20126	0.600630	−	0.799527i	$-0.294916\pi$
$937$	36.7711	1.20126	0.600630	+	0.799527i	$0.294916\pi$
$938$	0	0
$939$	10.2350	0.334008
$940$	0	0
$941$	35.5475	1.15881	0.579407	−	0.815038i	$-0.303284\pi$
$941$	35.5475	1.15881	0.579407	+	0.815038i	$0.303284\pi$
$942$	0	0
$943$	19.5685	0.637237
$944$	0	0
$945$	4.47839	0.145682
$946$	0	0
$947$	55.7033	1.81012	0.905058	−	0.425289i	$-0.139828\pi$
$947$	55.7033	1.81012	0.905058	+	0.425289i	$0.139828\pi$
$948$	0	0
$949$	0.141064	0.00457912
$950$	0	0
$951$	3.76319	0.122030
$952$	0	0
$953$	5.19748	0.168363	0.0841815	−	0.996450i	$-0.473172\pi$
$953$	5.19748	0.168363	0.0841815	+	0.996450i	$0.473172\pi$
$954$	0	0
$955$	19.2533	0.623023
$956$	0	0
$957$	4.75121	0.153585
$958$	0	0
$959$	15.7348	0.508102
$960$	0	0
$961$	37.5656	1.21179
$962$	0	0
$963$	−16.9870	−0.547400
$964$	0	0
$965$	−3.29952	−0.106215
$966$	0	0
$967$	21.7678	0.700004	0.350002	−	0.936749i	$-0.386181\pi$
$967$	21.7678	0.700004	0.350002	+	0.936749i	$0.386181\pi$
$968$	0	0
$969$	−4.45212	−0.143023
$970$	0	0
$971$	−7.57482	−0.243087	−0.121544	−	0.992586i	$-0.538784\pi$
$971$	−7.57482	−0.243087	−0.121544	+	0.992586i	$0.538784\pi$
$972$	0	0
$973$	−86.4680	−2.77204
$974$	0	0
$975$	3.88912	0.124552
$976$	0	0
$977$	−18.0642	−0.577925	−0.288962	−	0.957340i	$-0.593310\pi$
$977$	−18.0642	−0.577925	−0.288962	+	0.957340i	$0.593310\pi$
$978$	0	0
$979$	16.5309	0.528330
$980$	0	0
$981$	20.3032	0.648232
$982$	0	0
$983$	7.91261	0.252373	0.126186	−	0.992007i	$-0.459726\pi$
$983$	7.91261	0.252373	0.126186	+	0.992007i	$0.459726\pi$
$984$	0	0
$985$	−15.5336	−0.494941
$986$	0	0
$987$	−4.11040	−0.130835
$988$	0	0
$989$	−21.3728	−0.679615
$990$	0	0
$991$	7.66751	0.243567	0.121783	−	0.992557i	$-0.461139\pi$
$991$	7.66751	0.243567	0.121783	+	0.992557i	$0.461139\pi$
$992$	0	0
$993$	24.5851	0.780185
$994$	0	0
$995$	26.1238	0.828180
$996$	0	0
$997$	−40.2163	−1.27366	−0.636831	−	0.771004i	$-0.719755\pi$
$997$	−40.2163	−1.27366	−0.636831	+	0.771004i	$0.719755\pi$
$998$	0	0
$999$	6.14251	0.194341

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3432.2.a.w.1.2	✓	5
4.3	odd	2		6864.2.a.cf.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3432.2.a.w.1.2	✓	5	1.1	even	1	trivial
6864.2.a.cf.1.2		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:	\( N \)	\(=\)	\( 3432 = 2^{3} \cdot 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3432.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.05398 q^{5} +4.24902 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.05398 q^{5} +4.24902 q^{7} +1.00000 q^{9} -1.00000 q^{11} +1.00000 q^{13} +1.05398 q^{15} -4.55494 q^{17} -0.977426 q^{19} -4.24902 q^{21} -2.82631 q^{23} -3.88912 q^{25} -1.00000 q^{27} +4.75121 q^{29} -8.28043 q^{31} +1.00000 q^{33} -4.47839 q^{35} -6.14251 q^{37} -1.00000 q^{39} -6.92368 q^{41} +7.56208 q^{43} -1.05398 q^{45} +0.967375 q^{47} +11.0542 q^{49} +4.55494 q^{51} -5.76058 q^{53} +1.05398 q^{55} +0.977426 q^{57} -5.16364 q^{59} +12.8023 q^{61} +4.24902 q^{63} -1.05398 q^{65} +8.76942 q^{67} +2.82631 q^{69} +3.74999 q^{71} +0.141064 q^{73} +3.88912 q^{75} -4.24902 q^{77} -12.2704 q^{79} +1.00000 q^{81} -8.73747 q^{83} +4.80082 q^{85} -4.75121 q^{87} -16.5309 q^{89} +4.24902 q^{91} +8.28043 q^{93} +1.03019 q^{95} -7.42271 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{3} + q^{5} - 5 q^{7} + 5 q^{9}+O(q^{10}) \) 5 * q - 5 * q^3 + q^5 - 5 * q^7 + 5 * q^9	\( 5 q - 5 q^{3} + q^{5} - 5 q^{7} + 5 q^{9} - 5 q^{11} + 5 q^{13} - q^{15} - 4 q^{19} + 5 q^{21} - 5 q^{23} + 4 q^{25} - 5 q^{27} + 11 q^{29} - 8 q^{31} + 5 q^{33} - 5 q^{35} - 8 q^{37} - 5 q^{39} - q^{41} + q^{43} + q^{45} - 18 q^{47} + 10 q^{49} + 2 q^{53} - q^{55} + 4 q^{57} - 13 q^{59} + 9 q^{61} - 5 q^{63} + q^{65} + 5 q^{67} + 5 q^{69} - 24 q^{71} - 13 q^{73} - 4 q^{75} + 5 q^{77} - 6 q^{79} + 5 q^{81} - 22 q^{83} - 22 q^{85} - 11 q^{87} - 14 q^{89} - 5 q^{91} + 8 q^{93} - 32 q^{95} - 20 q^{97} - 5 q^{99}+O(q^{100}) \) 5 * q - 5 * q^3 + q^5 - 5 * q^7 + 5 * q^9 - 5 * q^11 + 5 * q^13 - q^15 - 4 * q^19 + 5 * q^21 - 5 * q^23 + 4 * q^25 - 5 * q^27 + 11 * q^29 - 8 * q^31 + 5 * q^33 - 5 * q^35 - 8 * q^37 - 5 * q^39 - q^41 + q^43 + q^45 - 18 * q^47 + 10 * q^49 + 2 * q^53 - q^55 + 4 * q^57 - 13 * q^59 + 9 * q^61 - 5 * q^63 + q^65 + 5 * q^67 + 5 * q^69 - 24 * q^71 - 13 * q^73 - 4 * q^75 + 5 * q^77 - 6 * q^79 + 5 * q^81 - 22 * q^83 - 22 * q^85 - 11 * q^87 - 14 * q^89 - 5 * q^91 + 8 * q^93 - 32 * q^95 - 20 * q^97 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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