LMFDB - Embedded newform 3344.2.a.u.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3344,2,Mod(1,3344)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3344.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3344, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3344 = 2^{4} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3344.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,0,-1,0,7,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$26.7019744359$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.3979184.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 9x^{3} + 18x^{2} + 5x - 12 $ x^5 - 2x^4 - 9x^3 + 18x^2 + 5x - 12
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1672)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.915789$ of defining polynomial
Character	$\chi$	$=$	3344.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.85680 q^{3} +1.91579 q^{5} -1.08421 q^{7} +5.16133 q^{9} +1.00000 q^{11} +5.31769 q^{13} +5.47303 q^{15} +2.00000 q^{17} -1.00000 q^{19} -3.09738 q^{21} +3.07712 q^{23} -1.32975 q^{25} +6.17450 q^{27} +1.29854 q^{29} -4.99189 q^{31} +2.85680 q^{33} -2.07712 q^{35} -3.24554 q^{37} +15.1916 q^{39} +2.99503 q^{41} +8.87494 q^{43} +9.88802 q^{45} -6.32266 q^{47} -5.82449 q^{49} +5.71361 q^{51} +2.00000 q^{53} +1.91579 q^{55} -2.85680 q^{57} -3.72669 q^{59} -14.3731 q^{61} -5.59597 q^{63} +10.1876 q^{65} -5.78576 q^{67} +8.79073 q^{69} -12.3102 q^{71} +12.9461 q^{73} -3.79884 q^{75} -1.08421 q^{77} -0.168422 q^{79} +2.15534 q^{81} -1.58998 q^{83} +3.83158 q^{85} +3.70966 q^{87} +6.32376 q^{89} -5.76550 q^{91} -14.2608 q^{93} -1.91579 q^{95} +14.5305 q^{97} +5.16133 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - q^{3} + 7 q^{5} - 8 q^{7} + 8 q^{9} + 5 q^{11} + 4 q^{13} - q^{15} + 10 q^{17} - 5 q^{19} + 2 q^{21} - 5 q^{23} + 6 q^{25} - 7 q^{27} + 16 q^{29} - q^{31} - q^{33} + 10 q^{35} - q^{37} + 4 q^{39}+ \cdots + 8 q^{99}+O(q^{100}) $ 5 * q - q^3 + 7 * q^5 - 8 * q^7 + 8 * q^9 + 5 * q^11 + 4 * q^13 - q^15 + 10 * q^17 - 5 * q^19 + 2 * q^21 - 5 * q^23 + 6 * q^25 - 7 * q^27 + 16 * q^29 - q^31 - q^33 + 10 * q^35 - q^37 + 4 * q^39 + 28 * q^41 - 4 * q^43 + 12 * q^45 + 4 * q^47 - q^49 - 2 * q^51 + 10 * q^53 + 7 * q^55 + q^57 + q^59 - 16 * q^61 - 12 * q^63 + 24 * q^65 + 9 * q^67 - 7 * q^69 - 7 * q^71 + 8 * q^73 + 8 * q^75 - 8 * q^77 - 6 * q^79 + 5 * q^81 - 4 * q^83 + 14 * q^85 + 2 * q^87 + 31 * q^89 + 12 * q^91 - 7 * q^93 - 7 * q^95 + 13 * q^97 + 8 * q^99

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
$3$	2.85680	1.64938	0.824688	−	0.565587i	$-0.191351\pi$
$3$	2.85680	1.64938	0.824688	+	0.565587i	$0.191351\pi$
$4$	0	0
$5$	1.91579	0.856767	0.428383	−	0.903597i	$-0.359083\pi$
$5$	1.91579	0.856767	0.428383	+	0.903597i	$0.359083\pi$
$6$	0	0
$7$	−1.08421	−0.409793	−0.204897	−	0.978784i	$-0.565686\pi$
$7$	−1.08421	−0.409793	−0.204897	+	0.978784i	$0.565686\pi$
$8$	0	0
$9$	5.16133	1.72044
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	5.31769	1.47486	0.737431	−	0.675422i	$-0.236038\pi$
$13$	5.31769	1.47486	0.737431	+	0.675422i	$0.236038\pi$
$14$	0	0
$15$	5.47303	1.41313
$16$	0	0
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−3.09738	−0.675903
$22$	0	0
$23$	3.07712	0.641624	0.320812	−	0.947143i	$-0.396044\pi$
$23$	3.07712	0.641624	0.320812	+	0.947143i	$0.396044\pi$
$24$	0	0
$25$	−1.32975	−0.265950
$26$	0	0
$27$	6.17450	1.18828
$28$	0	0
$29$	1.29854	0.241132	0.120566	−	0.992705i	$-0.461529\pi$
$29$	1.29854	0.241132	0.120566	+	0.992705i	$0.461529\pi$
$30$	0	0
$31$	−4.99189	−0.896569	−0.448285	−	0.893891i	$-0.647965\pi$
$31$	−4.99189	−0.896569	−0.448285	+	0.893891i	$0.647965\pi$
$32$	0	0
$33$	2.85680	0.497306
$34$	0	0
$35$	−2.07712	−0.351097
$36$	0	0
$37$	−3.24554	−0.533564	−0.266782	−	0.963757i	$-0.585960\pi$
$37$	−3.24554	−0.533564	−0.266782	+	0.963757i	$0.585960\pi$
$38$	0	0
$39$	15.1916	2.43260
$40$	0	0
$41$	2.99503	0.467746	0.233873	−	0.972267i	$-0.424860\pi$
$41$	2.99503	0.467746	0.233873	+	0.972267i	$0.424860\pi$
$42$	0	0
$43$	8.87494	1.35342	0.676708	−	0.736252i	$-0.263406\pi$
$43$	8.87494	1.35342	0.676708	+	0.736252i	$0.263406\pi$
$44$	0	0
$45$	9.88802	1.47402
$46$	0	0
$47$	−6.32266	−0.922255	−0.461127	−	0.887334i	$-0.652555\pi$
$47$	−6.32266	−0.922255	−0.461127	+	0.887334i	$0.652555\pi$
$48$	0	0
$49$	−5.82449	−0.832070
$50$	0	0
$51$	5.71361	0.800065
$52$	0	0
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	0	0
$55$	1.91579	0.258325
$56$	0	0
$57$	−2.85680	−0.378393
$58$	0	0
$59$	−3.72669	−0.485174	−0.242587	−	0.970130i	$-0.577996\pi$
$59$	−3.72669	−0.485174	−0.242587	+	0.970130i	$0.577996\pi$
$60$	0	0
$61$	−14.3731	−1.84029	−0.920144	−	0.391579i	$-0.871929\pi$
$61$	−14.3731	−1.84029	−0.920144	+	0.391579i	$0.871929\pi$
$62$	0	0
$63$	−5.59597	−0.705026
$64$	0	0
$65$	10.1876	1.26361
$66$	0	0
$67$	−5.78576	−0.706843	−0.353422	−	0.935464i	$-0.614982\pi$
$67$	−5.78576	−0.706843	−0.353422	+	0.935464i	$0.614982\pi$
$68$	0	0
$69$	8.79073	1.05828
$70$	0	0
$71$	−12.3102	−1.46095	−0.730475	−	0.682939i	$-0.760701\pi$
$71$	−12.3102	−1.46095	−0.730475	+	0.682939i	$0.760701\pi$
$72$	0	0
$73$	12.9461	1.51522	0.757611	−	0.652706i	$-0.226366\pi$
$73$	12.9461	1.51522	0.757611	+	0.652706i	$0.226366\pi$
$74$	0	0
$75$	−3.79884	−0.438653
$76$	0	0
$77$	−1.08421	−0.123557
$78$	0	0
$79$	−0.168422	−0.0189489	−0.00947447	−	0.999955i	$-0.503016\pi$
$79$	−0.168422	−0.0189489	−0.00947447	+	0.999955i	$0.503016\pi$
$80$	0	0
$81$	2.15534	0.239482
$82$	0	0
$83$	−1.58998	−0.174523	−0.0872616	−	0.996185i	$-0.527812\pi$
$83$	−1.58998	−0.174523	−0.0872616	+	0.996185i	$0.527812\pi$
$84$	0	0
$85$	3.83158	0.415593
$86$	0	0
$87$	3.70966	0.397718
$88$	0	0
$89$	6.32376	0.670317	0.335159	−	0.942162i	$-0.391210\pi$
$89$	6.32376	0.670317	0.335159	+	0.942162i	$0.391210\pi$
$90$	0	0
$91$	−5.76550	−0.604389
$92$	0	0
$93$	−14.2608	−1.47878
$94$	0	0
$95$	−1.91579	−0.196556
$96$	0	0
$97$	14.5305	1.47535	0.737674	−	0.675157i	$-0.235924\pi$
$97$	14.5305	1.47535	0.737674	+	0.675157i	$0.235924\pi$
$98$	0	0
$99$	5.16133	0.518733
$100$	0	0
$101$	11.1407	1.10854	0.554268	−	0.832338i	$-0.312998\pi$
$101$	11.1407	1.10854	0.554268	+	0.832338i	$0.312998\pi$
$102$	0	0
$103$	−7.18057	−0.707522	−0.353761	−	0.935336i	$-0.615097\pi$
$103$	−7.18057	−0.707522	−0.353761	+	0.935336i	$0.615097\pi$
$104$	0	0
$105$	−5.93392	−0.579092
$106$	0	0
$107$	2.49108	0.240822	0.120411	−	0.992724i	$-0.461579\pi$
$107$	2.49108	0.240822	0.120411	+	0.992724i	$0.461579\pi$
$108$	0	0
$109$	4.48115	0.429216	0.214608	−	0.976700i	$-0.431153\pi$
$109$	4.48115	0.429216	0.214608	+	0.976700i	$0.431153\pi$
$110$	0	0
$111$	−9.27188	−0.880048
$112$	0	0
$113$	15.7510	1.48173	0.740864	−	0.671655i	$-0.234416\pi$
$113$	15.7510	1.48173	0.740864	+	0.671655i	$0.234416\pi$
$114$	0	0
$115$	5.89511	0.549722
$116$	0	0
$117$	27.4464	2.53742
$118$	0	0
$119$	−2.16842	−0.198779
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	8.55622	0.771489
$124$	0	0
$125$	−12.1265	−1.08462
$126$	0	0
$127$	2.62341	0.232790	0.116395	−	0.993203i	$-0.462866\pi$
$127$	2.62341	0.232790	0.116395	+	0.993203i	$0.462866\pi$
$128$	0	0
$129$	25.3540	2.23229
$130$	0	0
$131$	11.4987	1.00464	0.502322	−	0.864681i	$-0.332479\pi$
$131$	11.4987	1.00464	0.502322	+	0.864681i	$0.332479\pi$
$132$	0	0
$133$	1.08421	0.0940130
$134$	0	0
$135$	11.8290	1.01808
$136$	0	0
$137$	0.161330	0.0137834	0.00689169	−	0.999976i	$-0.497806\pi$
$137$	0.161330	0.0137834	0.00689169	+	0.999976i	$0.497806\pi$
$138$	0	0
$139$	−14.3697	−1.21882	−0.609410	−	0.792855i	$-0.708594\pi$
$139$	−14.3697	−1.21882	−0.609410	+	0.792855i	$0.708594\pi$
$140$	0	0
$141$	−18.0626	−1.52115
$142$	0	0
$143$	5.31769	0.444688
$144$	0	0
$145$	2.48772	0.206594
$146$	0	0
$147$	−16.6394	−1.37240
$148$	0	0
$149$	3.09020	0.253159	0.126580	−	0.991956i	$-0.459600\pi$
$149$	3.09020	0.253159	0.126580	+	0.991956i	$0.459600\pi$
$150$	0	0
$151$	3.09020	0.251477	0.125738	−	0.992063i	$-0.459870\pi$
$151$	3.09020	0.251477	0.125738	+	0.992063i	$0.459870\pi$
$152$	0	0
$153$	10.3227	0.834538
$154$	0	0
$155$	−9.56340	−0.768151
$156$	0	0
$157$	7.03232	0.561241	0.280620	−	0.959819i	$-0.409460\pi$
$157$	7.03232	0.561241	0.280620	+	0.959819i	$0.409460\pi$
$158$	0	0
$159$	5.71361	0.453119
$160$	0	0
$161$	−3.33625	−0.262933
$162$	0	0
$163$	−4.36318	−0.341751	−0.170875	−	0.985293i	$-0.554660\pi$
$163$	−4.36318	−0.341751	−0.170875	+	0.985293i	$0.554660\pi$
$164$	0	0
$165$	5.47303	0.426075
$166$	0	0
$167$	−2.75131	−0.212903	−0.106451	−	0.994318i	$-0.533949\pi$
$167$	−2.75131	−0.212903	−0.106451	+	0.994318i	$0.533949\pi$
$168$	0	0
$169$	15.2779	1.17522
$170$	0	0
$171$	−5.16133	−0.394697
$172$	0	0
$173$	−9.20690	−0.699988	−0.349994	−	0.936752i	$-0.613816\pi$
$173$	−9.20690	−0.699988	−0.349994	+	0.936752i	$0.613816\pi$
$174$	0	0
$175$	1.44173	0.108985
$176$	0	0
$177$	−10.6464	−0.800234
$178$	0	0
$179$	1.18647	0.0886810	0.0443405	−	0.999016i	$-0.485881\pi$
$179$	1.18647	0.0886810	0.0443405	+	0.999016i	$0.485881\pi$
$180$	0	0
$181$	−24.9593	−1.85521	−0.927606	−	0.373560i	$-0.878137\pi$
$181$	−24.9593	−1.85521	−0.927606	+	0.373560i	$0.878137\pi$
$182$	0	0
$183$	−41.0612	−3.03533
$184$	0	0
$185$	−6.21777	−0.457140
$186$	0	0
$187$	2.00000	0.146254
$188$	0	0
$189$	−6.69446	−0.486950
$190$	0	0
$191$	10.3096	0.745975	0.372987	−	0.927836i	$-0.378333\pi$
$191$	10.3096	0.745975	0.372987	+	0.927836i	$0.378333\pi$
$192$	0	0
$193$	26.1125	1.87962	0.939808	−	0.341704i	$-0.111004\pi$
$193$	26.1125	1.87962	0.939808	+	0.341704i	$0.111004\pi$
$194$	0	0
$195$	29.1039	2.08417
$196$	0	0
$197$	−10.0768	−0.717941	−0.358971	−	0.933349i	$-0.616872\pi$
$197$	−10.0768	−0.717941	−0.358971	+	0.933349i	$0.616872\pi$
$198$	0	0
$199$	−10.0002	−0.708893	−0.354447	−	0.935076i	$-0.615331\pi$
$199$	−10.0002	−0.708893	−0.354447	+	0.935076i	$0.615331\pi$
$200$	0	0
$201$	−16.5288	−1.16585
$202$	0	0
$203$	−1.40789	−0.0988143
$204$	0	0
$205$	5.73785	0.400749
$206$	0	0
$207$	15.8820	1.10388
$208$	0	0
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	3.10438	0.213715	0.106857	−	0.994274i	$-0.465921\pi$
$211$	3.10438	0.213715	0.106857	+	0.994274i	$0.465921\pi$
$212$	0	0
$213$	−35.1678	−2.40966
$214$	0	0
$215$	17.0025	1.15956
$216$	0	0
$217$	5.41226	0.367408
$218$	0	0
$219$	36.9844	2.49917
$220$	0	0
$221$	10.6354	0.715414
$222$	0	0
$223$	15.4097	1.03191	0.515955	−	0.856615i	$-0.327437\pi$
$223$	15.4097	1.03191	0.515955	+	0.856615i	$0.327437\pi$
$224$	0	0
$225$	−6.86329	−0.457553
$226$	0	0
$227$	−13.9183	−0.923790	−0.461895	−	0.886935i	$-0.652830\pi$
$227$	−13.9183	−0.923790	−0.461895	+	0.886935i	$0.652830\pi$
$228$	0	0
$229$	−18.4742	−1.22081	−0.610406	−	0.792089i	$-0.708994\pi$
$229$	−18.4742	−1.22081	−0.610406	+	0.792089i	$0.708994\pi$
$230$	0	0
$231$	−3.09738	−0.203793
$232$	0	0
$233$	28.7221	1.88165	0.940824	−	0.338896i	$-0.110053\pi$
$233$	28.7221	1.88165	0.940824	+	0.338896i	$0.110053\pi$
$234$	0	0
$235$	−12.1129	−0.790157
$236$	0	0
$237$	−0.481148	−0.0312540
$238$	0	0
$239$	−18.9274	−1.22431	−0.612157	−	0.790736i	$-0.709698\pi$
$239$	−18.9274	−1.22431	−0.612157	+	0.790736i	$0.709698\pi$
$240$	0	0
$241$	−23.5821	−1.51905	−0.759527	−	0.650476i	$-0.774570\pi$
$241$	−23.5821	−1.51905	−0.759527	+	0.650476i	$0.774570\pi$
$242$	0	0
$243$	−12.3661	−0.793286
$244$	0	0
$245$	−11.1585	−0.712890
$246$	0	0
$247$	−5.31769	−0.338357
$248$	0	0
$249$	−4.54226	−0.287854
$250$	0	0
$251$	−14.7745	−0.932558	−0.466279	−	0.884638i	$-0.654406\pi$
$251$	−14.7745	−0.932558	−0.466279	+	0.884638i	$0.654406\pi$
$252$	0	0
$253$	3.07712	0.193457
$254$	0	0
$255$	10.9461	0.685469
$256$	0	0
$257$	−8.53160	−0.532187	−0.266093	−	0.963947i	$-0.585733\pi$
$257$	−8.53160	−0.532187	−0.266093	+	0.963947i	$0.585733\pi$
$258$	0	0
$259$	3.51885	0.218651
$260$	0	0
$261$	6.70217	0.414854
$262$	0	0
$263$	−23.8573	−1.47110	−0.735551	−	0.677469i	$-0.763077\pi$
$263$	−23.8573	−1.47110	−0.735551	+	0.677469i	$0.763077\pi$
$264$	0	0
$265$	3.83158	0.235372
$266$	0	0
$267$	18.0658	1.10561
$268$	0	0
$269$	14.7534	0.899528	0.449764	−	0.893147i	$-0.351508\pi$
$269$	14.7534	0.899528	0.449764	+	0.893147i	$0.351508\pi$
$270$	0	0
$271$	−3.61665	−0.219696	−0.109848	−	0.993948i	$-0.535036\pi$
$271$	−3.61665	−0.219696	−0.109848	+	0.993948i	$0.535036\pi$
$272$	0	0
$273$	−16.4709	−0.996865
$274$	0	0
$275$	−1.32975	−0.0801871
$276$	0	0
$277$	−0.871327	−0.0523529	−0.0261765	−	0.999657i	$-0.508333\pi$
$277$	−0.871327	−0.0523529	−0.0261765	+	0.999657i	$0.508333\pi$
$278$	0	0
$279$	−25.7648	−1.54250
$280$	0	0
$281$	−25.5658	−1.52513	−0.762565	−	0.646912i	$-0.776060\pi$
$281$	−25.5658	−1.52513	−0.762565	+	0.646912i	$0.776060\pi$
$282$	0	0
$283$	21.5513	1.28109	0.640547	−	0.767919i	$-0.278708\pi$
$283$	21.5513	1.28109	0.640547	+	0.767919i	$0.278708\pi$
$284$	0	0
$285$	−5.47303	−0.324195
$286$	0	0
$287$	−3.24725	−0.191679
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	41.5108	2.43341
$292$	0	0
$293$	2.07530	0.121240	0.0606201	−	0.998161i	$-0.480692\pi$
$293$	2.07530	0.121240	0.0606201	+	0.998161i	$0.480692\pi$
$294$	0	0
$295$	−7.13955	−0.415681
$296$	0	0
$297$	6.17450	0.358281
$298$	0	0
$299$	16.3632	0.946307
$300$	0	0
$301$	−9.62231	−0.554620
$302$	0	0
$303$	31.8267	1.82839
$304$	0	0
$305$	−27.5359	−1.57670
$306$	0	0
$307$	−9.47971	−0.541036	−0.270518	−	0.962715i	$-0.587195\pi$
$307$	−9.47971	−0.541036	−0.270518	+	0.962715i	$0.587195\pi$
$308$	0	0
$309$	−20.5135	−1.16697
$310$	0	0
$311$	−8.49108	−0.481485	−0.240743	−	0.970589i	$-0.577391\pi$
$311$	−8.49108	−0.481485	−0.240743	+	0.970589i	$0.577391\pi$
$312$	0	0
$313$	−9.21577	−0.520906	−0.260453	−	0.965487i	$-0.583872\pi$
$313$	−9.21577	−0.520906	−0.260453	+	0.965487i	$0.583872\pi$
$314$	0	0
$315$	−10.7207	−0.604043
$316$	0	0
$317$	−28.2279	−1.58544	−0.792718	−	0.609589i	$-0.791335\pi$
$317$	−28.2279	−1.58544	−0.792718	+	0.609589i	$0.791335\pi$
$318$	0	0
$319$	1.29854	0.0727041
$320$	0	0
$321$	7.11654	0.397206
$322$	0	0
$323$	−2.00000	−0.111283
$324$	0	0
$325$	−7.07122	−0.392240
$326$	0	0
$327$	12.8018	0.707939
$328$	0	0
$329$	6.85510	0.377934
$330$	0	0
$331$	1.72813	0.0949867	0.0474933	−	0.998872i	$-0.484877\pi$
$331$	1.72813	0.0949867	0.0474933	+	0.998872i	$0.484877\pi$
$332$	0	0
$333$	−16.7513	−0.917966
$334$	0	0
$335$	−11.0843	−0.605600
$336$	0	0
$337$	13.2502	0.721783	0.360892	−	0.932608i	$-0.382472\pi$
$337$	13.2502	0.721783	0.360892	+	0.932608i	$0.382472\pi$
$338$	0	0
$339$	44.9975	2.44393
$340$	0	0
$341$	−4.99189	−0.270326
$342$	0	0
$343$	13.9044	0.750770
$344$	0	0
$345$	16.8412	0.906699
$346$	0	0
$347$	−8.88126	−0.476771	−0.238386	−	0.971171i	$-0.576618\pi$
$347$	−8.88126	−0.476771	−0.238386	+	0.971171i	$0.576618\pi$
$348$	0	0
$349$	−24.0464	−1.28717	−0.643586	−	0.765373i	$-0.722554\pi$
$349$	−24.0464	−1.28717	−0.643586	+	0.765373i	$0.722554\pi$
$350$	0	0
$351$	32.8341	1.75255
$352$	0	0
$353$	9.90522	0.527202	0.263601	−	0.964632i	$-0.415090\pi$
$353$	9.90522	0.527202	0.263601	+	0.964632i	$0.415090\pi$
$354$	0	0
$355$	−23.5837	−1.25169
$356$	0	0
$357$	−6.19476	−0.327861
$358$	0	0
$359$	12.9158	0.681669	0.340835	−	0.940123i	$-0.389290\pi$
$359$	12.9158	0.681669	0.340835	+	0.940123i	$0.389290\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	2.85680	0.149943
$364$	0	0
$365$	24.8019	1.29819
$366$	0	0
$367$	30.7858	1.60701	0.803503	−	0.595300i	$-0.202967\pi$
$367$	30.7858	1.60701	0.803503	+	0.595300i	$0.202967\pi$
$368$	0	0
$369$	15.4584	0.804730
$370$	0	0
$371$	−2.16842	−0.112579
$372$	0	0
$373$	−22.4250	−1.16112	−0.580561	−	0.814216i	$-0.697167\pi$
$373$	−22.4250	−1.16112	−0.580561	+	0.814216i	$0.697167\pi$
$374$	0	0
$375$	−34.6430	−1.78895
$376$	0	0
$377$	6.90522	0.355637
$378$	0	0
$379$	−7.51924	−0.386237	−0.193119	−	0.981175i	$-0.561860\pi$
$379$	−7.51924	−0.386237	−0.193119	+	0.981175i	$0.561860\pi$
$380$	0	0
$381$	7.49456	0.383958
$382$	0	0
$383$	16.5491	0.845617	0.422809	−	0.906219i	$-0.361044\pi$
$383$	16.5491	0.845617	0.422809	+	0.906219i	$0.361044\pi$
$384$	0	0
$385$	−2.07712	−0.105860
$386$	0	0
$387$	45.8065	2.32847
$388$	0	0
$389$	35.5840	1.80418	0.902089	−	0.431549i	$-0.142033\pi$
$389$	35.5840	1.80418	0.902089	+	0.431549i	$0.142033\pi$
$390$	0	0
$391$	6.15424	0.311233
$392$	0	0
$393$	32.8495	1.65704
$394$	0	0
$395$	−0.322661	−0.0162348
$396$	0	0
$397$	12.3529	0.619976	0.309988	−	0.950740i	$-0.399675\pi$
$397$	12.3529	0.619976	0.309988	+	0.950740i	$0.399675\pi$
$398$	0	0
$399$	3.09738	0.155063
$400$	0	0
$401$	−32.9049	−1.64319	−0.821596	−	0.570070i	$-0.806916\pi$
$401$	−32.9049	−1.64319	−0.821596	+	0.570070i	$0.806916\pi$
$402$	0	0
$403$	−26.5453	−1.32232
$404$	0	0
$405$	4.12918	0.205180
$406$	0	0
$407$	−3.24554	−0.160876
$408$	0	0
$409$	−25.8018	−1.27582	−0.637908	−	0.770113i	$-0.720200\pi$
$409$	−25.8018	−1.27582	−0.637908	+	0.770113i	$0.720200\pi$
$410$	0	0
$411$	0.460889	0.0227340
$412$	0	0
$413$	4.04052	0.198821
$414$	0	0
$415$	−3.04607	−0.149526
$416$	0	0
$417$	−41.0513	−2.01029
$418$	0	0
$419$	−21.2971	−1.04043	−0.520216	−	0.854035i	$-0.674149\pi$
$419$	−21.2971	−1.04043	−0.520216	+	0.854035i	$0.674149\pi$
$420$	0	0
$421$	29.4734	1.43645	0.718223	−	0.695813i	$-0.244956\pi$
$421$	29.4734	1.43645	0.718223	+	0.695813i	$0.244956\pi$
$422$	0	0
$423$	−32.6333	−1.58669
$424$	0	0
$425$	−2.65950	−0.129005
$426$	0	0
$427$	15.5835	0.754138
$428$	0	0
$429$	15.1916	0.733458
$430$	0	0
$431$	21.7861	1.04940	0.524701	−	0.851287i	$-0.324177\pi$
$431$	21.7861	1.04940	0.524701	+	0.851287i	$0.324177\pi$
$432$	0	0
$433$	21.3657	1.02677	0.513386	−	0.858158i	$-0.328391\pi$
$433$	21.3657	1.02677	0.513386	+	0.858158i	$0.328391\pi$
$434$	0	0
$435$	7.10693	0.340751
$436$	0	0
$437$	−3.07712	−0.147199
$438$	0	0
$439$	−8.81816	−0.420868	−0.210434	−	0.977608i	$-0.567488\pi$
$439$	−8.81816	−0.420868	−0.210434	+	0.977608i	$0.567488\pi$
$440$	0	0
$441$	−30.0621	−1.43153
$442$	0	0
$443$	27.5521	1.30904	0.654521	−	0.756044i	$-0.272870\pi$
$443$	27.5521	1.30904	0.654521	+	0.756044i	$0.272870\pi$
$444$	0	0
$445$	12.1150	0.574306
$446$	0	0
$447$	8.82810	0.417555
$448$	0	0
$449$	8.16732	0.385440	0.192720	−	0.981254i	$-0.438269\pi$
$449$	8.16732	0.385440	0.192720	+	0.981254i	$0.438269\pi$
$450$	0	0
$451$	2.99503	0.141031
$452$	0	0
$453$	8.82810	0.414780
$454$	0	0
$455$	−11.0455	−0.517820
$456$	0	0
$457$	−4.97240	−0.232599	−0.116300	−	0.993214i	$-0.537103\pi$
$457$	−4.97240	−0.232599	−0.116300	+	0.993214i	$0.537103\pi$
$458$	0	0
$459$	12.3490	0.576402
$460$	0	0
$461$	−15.9039	−0.740721	−0.370360	−	0.928888i	$-0.620766\pi$
$461$	−15.9039	−0.740721	−0.370360	+	0.928888i	$0.620766\pi$
$462$	0	0
$463$	−4.24715	−0.197382	−0.0986908	−	0.995118i	$-0.531465\pi$
$463$	−4.24715	−0.197382	−0.0986908	+	0.995118i	$0.531465\pi$
$464$	0	0
$465$	−27.3208	−1.26697
$466$	0	0
$467$	−27.2216	−1.25967	−0.629833	−	0.776731i	$-0.716877\pi$
$467$	−27.2216	−1.25967	−0.629833	+	0.776731i	$0.716877\pi$
$468$	0	0
$469$	6.27299	0.289660
$470$	0	0
$471$	20.0900	0.925697
$472$	0	0
$473$	8.87494	0.408070
$474$	0	0
$475$	1.32975	0.0610132
$476$	0	0
$477$	10.3227	0.472642
$478$	0	0
$479$	−9.08201	−0.414968	−0.207484	−	0.978238i	$-0.566527\pi$
$479$	−9.08201	−0.414968	−0.207484	+	0.978238i	$0.566527\pi$
$480$	0	0
$481$	−17.2588	−0.786933
$482$	0	0
$483$	−9.53100	−0.433676
$484$	0	0
$485$	27.8374	1.26403
$486$	0	0
$487$	−17.6203	−0.798451	−0.399225	−	0.916853i	$-0.630721\pi$
$487$	−17.6203	−0.798451	−0.399225	+	0.916853i	$0.630721\pi$
$488$	0	0
$489$	−12.4647	−0.563675
$490$	0	0
$491$	−22.7082	−1.02481	−0.512404	−	0.858744i	$-0.671245\pi$
$491$	−22.7082	−1.02481	−0.512404	+	0.858744i	$0.671245\pi$
$492$	0	0
$493$	2.59707	0.116966
$494$	0	0
$495$	9.88802	0.444433
$496$	0	0
$497$	13.3468	0.598687
$498$	0	0
$499$	4.11592	0.184254	0.0921270	−	0.995747i	$-0.470633\pi$
$499$	4.11592	0.184254	0.0921270	+	0.995747i	$0.470633\pi$
$500$	0	0
$501$	−7.85996	−0.351157
$502$	0	0
$503$	−39.1716	−1.74658	−0.873288	−	0.487205i	$-0.838017\pi$
$503$	−39.1716	−1.74658	−0.873288	+	0.487205i	$0.838017\pi$
$504$	0	0
$505$	21.3431	0.949757
$506$	0	0
$507$	43.6459	1.93838
$508$	0	0
$509$	−17.4018	−0.771322	−0.385661	−	0.922640i	$-0.626027\pi$
$509$	−17.4018	−0.771322	−0.385661	+	0.922640i	$0.626027\pi$
$510$	0	0
$511$	−14.0363	−0.620928
$512$	0	0
$513$	−6.17450	−0.272611
$514$	0	0
$515$	−13.7565	−0.606182
$516$	0	0
$517$	−6.32266	−0.278070
$518$	0	0
$519$	−26.3023	−1.15454
$520$	0	0
$521$	20.2691	0.888004	0.444002	−	0.896026i	$-0.353558\pi$
$521$	20.2691	0.888004	0.444002	+	0.896026i	$0.353558\pi$
$522$	0	0
$523$	7.23875	0.316529	0.158264	−	0.987397i	$-0.449410\pi$
$523$	7.23875	0.316529	0.158264	+	0.987397i	$0.449410\pi$
$524$	0	0
$525$	4.11875	0.179757
$526$	0	0
$527$	−9.98377	−0.434900
$528$	0	0
$529$	−13.5313	−0.588319
$530$	0	0
$531$	−19.2347	−0.834714
$532$	0	0
$533$	15.9267	0.689861
$534$	0	0
$535$	4.77239	0.206328
$536$	0	0
$537$	3.38951	0.146268
$538$	0	0
$539$	−5.82449	−0.250878
$540$	0	0
$541$	38.1394	1.63974	0.819870	−	0.572549i	$-0.194046\pi$
$541$	38.1394	1.63974	0.819870	+	0.572549i	$0.194046\pi$
$542$	0	0
$543$	−71.3039	−3.05994
$544$	0	0
$545$	8.58494	0.367738
$546$	0	0
$547$	−1.50161	−0.0642044	−0.0321022	−	0.999485i	$-0.510220\pi$
$547$	−1.50161	−0.0642044	−0.0321022	+	0.999485i	$0.510220\pi$
$548$	0	0
$549$	−74.1844	−3.16611
$550$	0	0
$551$	−1.29854	−0.0553195
$552$	0	0
$553$	0.182605	0.00776515
$554$	0	0
$555$	−17.7630	−0.753996
$556$	0	0
$557$	26.0847	1.10524	0.552622	−	0.833432i	$-0.313627\pi$
$557$	26.0847	1.10524	0.552622	+	0.833432i	$0.313627\pi$
$558$	0	0
$559$	47.1942	1.99610
$560$	0	0
$561$	5.71361	0.241229
$562$	0	0
$563$	−27.9759	−1.17904	−0.589521	−	0.807753i	$-0.700684\pi$
$563$	−27.9759	−1.17904	−0.589521	+	0.807753i	$0.700684\pi$
$564$	0	0
$565$	30.1756	1.26950
$566$	0	0
$567$	−2.33684	−0.0981382
$568$	0	0
$569$	21.2097	0.889155	0.444578	−	0.895740i	$-0.353354\pi$
$569$	21.2097	0.889155	0.444578	+	0.895740i	$0.353354\pi$
$570$	0	0
$571$	19.1347	0.800761	0.400380	−	0.916349i	$-0.368878\pi$
$571$	19.1347	0.800761	0.400380	+	0.916349i	$0.368878\pi$
$572$	0	0
$573$	29.4525	1.23039
$574$	0	0
$575$	−4.09181	−0.170640
$576$	0	0
$577$	−45.9979	−1.91492	−0.957459	−	0.288568i	$-0.906821\pi$
$577$	−45.9979	−1.91492	−0.957459	+	0.288568i	$0.906821\pi$
$578$	0	0
$579$	74.5982	3.10019
$580$	0	0
$581$	1.72387	0.0715184
$582$	0	0
$583$	2.00000	0.0828315
$584$	0	0
$585$	52.5815	2.17398
$586$	0	0
$587$	46.3398	1.91265	0.956325	−	0.292306i	$-0.0944226\pi$
$587$	46.3398	1.91265	0.956325	+	0.292306i	$0.0944226\pi$
$588$	0	0
$589$	4.99189	0.205687
$590$	0	0
$591$	−28.7874	−1.18416
$592$	0	0
$593$	17.8898	0.734644	0.367322	−	0.930094i	$-0.380275\pi$
$593$	17.8898	0.734644	0.367322	+	0.930094i	$0.380275\pi$
$594$	0	0
$595$	−4.15424	−0.170307
$596$	0	0
$597$	−28.5685	−1.16923
$598$	0	0
$599$	18.8508	0.770222	0.385111	−	0.922870i	$-0.374163\pi$
$599$	18.8508	0.770222	0.385111	+	0.922870i	$0.374163\pi$
$600$	0	0
$601$	−9.16363	−0.373792	−0.186896	−	0.982380i	$-0.559843\pi$
$601$	−9.16363	−0.373792	−0.186896	+	0.982380i	$0.559843\pi$
$602$	0	0
$603$	−29.8622	−1.21608
$604$	0	0
$605$	1.91579	0.0778879
$606$	0	0
$607$	−1.11636	−0.0453118	−0.0226559	−	0.999743i	$-0.507212\pi$
$607$	−1.11636	−0.0453118	−0.0226559	+	0.999743i	$0.507212\pi$
$608$	0	0
$609$	−4.02206	−0.162982
$610$	0	0
$611$	−33.6220	−1.36020
$612$	0	0
$613$	9.64609	0.389602	0.194801	−	0.980843i	$-0.437594\pi$
$613$	9.64609	0.389602	0.194801	+	0.980843i	$0.437594\pi$
$614$	0	0
$615$	16.3919	0.660986
$616$	0	0
$617$	−38.4415	−1.54760	−0.773799	−	0.633431i	$-0.781646\pi$
$617$	−38.4415	−1.54760	−0.773799	+	0.633431i	$0.781646\pi$
$618$	0	0
$619$	39.2599	1.57799	0.788995	−	0.614400i	$-0.210602\pi$
$619$	39.2599	1.57799	0.788995	+	0.614400i	$0.210602\pi$
$620$	0	0
$621$	18.9997	0.762430
$622$	0	0
$623$	−6.85629	−0.274692
$624$	0	0
$625$	−16.5830	−0.663320
$626$	0	0
$627$	−2.85680	−0.114090
$628$	0	0
$629$	−6.49108	−0.258816
$630$	0	0
$631$	−14.9952	−0.596951	−0.298476	−	0.954417i	$-0.596478\pi$
$631$	−14.9952	−0.596951	−0.298476	+	0.954417i	$0.596478\pi$
$632$	0	0
$633$	8.86862	0.352496
$634$	0	0
$635$	5.02590	0.199447
$636$	0	0
$637$	−30.9728	−1.22719
$638$	0	0
$639$	−63.5369	−2.51348
$640$	0	0
$641$	15.3586	0.606628	0.303314	−	0.952891i	$-0.401907\pi$
$641$	15.3586	0.606628	0.303314	+	0.952891i	$0.401907\pi$
$642$	0	0
$643$	−36.0013	−1.41975	−0.709876	−	0.704327i	$-0.751249\pi$
$643$	−36.0013	−1.41975	−0.709876	+	0.704327i	$0.751249\pi$
$644$	0	0
$645$	48.5728	1.91255
$646$	0	0
$647$	5.30041	0.208381	0.104190	−	0.994557i	$-0.466775\pi$
$647$	5.30041	0.208381	0.104190	+	0.994557i	$0.466775\pi$
$648$	0	0
$649$	−3.72669	−0.146285
$650$	0	0
$651$	15.4618	0.605994
$652$	0	0
$653$	25.0255	0.979325	0.489662	−	0.871912i	$-0.337120\pi$
$653$	25.0255	0.979325	0.489662	+	0.871912i	$0.337120\pi$
$654$	0	0
$655$	22.0290	0.860746
$656$	0	0
$657$	66.8189	2.60686
$658$	0	0
$659$	28.7896	1.12148	0.560742	−	0.827990i	$-0.310516\pi$
$659$	28.7896	1.12148	0.560742	+	0.827990i	$0.310516\pi$
$660$	0	0
$661$	−15.8286	−0.615661	−0.307831	−	0.951441i	$-0.599603\pi$
$661$	−15.8286	−0.615661	−0.307831	+	0.951441i	$0.599603\pi$
$662$	0	0
$663$	30.3832	1.17999
$664$	0	0
$665$	2.07712	0.0805472
$666$	0	0
$667$	3.99575	0.154716
$668$	0	0
$669$	44.0225	1.70201
$670$	0	0
$671$	−14.3731	−0.554868
$672$	0	0
$673$	19.5104	0.752071	0.376035	−	0.926605i	$-0.377287\pi$
$673$	19.5104	0.752071	0.376035	+	0.926605i	$0.377287\pi$
$674$	0	0
$675$	−8.21055	−0.316024
$676$	0	0
$677$	19.3832	0.744957	0.372479	−	0.928041i	$-0.378508\pi$
$677$	19.3832	0.744957	0.372479	+	0.928041i	$0.378508\pi$
$678$	0	0
$679$	−15.7541	−0.604588
$680$	0	0
$681$	−39.7619	−1.52368
$682$	0	0
$683$	−4.67657	−0.178944	−0.0894720	−	0.995989i	$-0.528518\pi$
$683$	−4.67657	−0.178944	−0.0894720	+	0.995989i	$0.528518\pi$
$684$	0	0
$685$	0.309075	0.0118092
$686$	0	0
$687$	−52.7773	−2.01358
$688$	0	0
$689$	10.6354	0.405176
$690$	0	0
$691$	−13.7096	−0.521539	−0.260769	−	0.965401i	$-0.583976\pi$
$691$	−13.7096	−0.521539	−0.260769	+	0.965401i	$0.583976\pi$
$692$	0	0
$693$	−5.59597	−0.212573
$694$	0	0
$695$	−27.5293	−1.04424
$696$	0	0
$697$	5.99007	0.226890
$698$	0	0
$699$	82.0534	3.10355
$700$	0	0
$701$	−40.8786	−1.54396	−0.771981	−	0.635646i	$-0.780734\pi$
$701$	−40.8786	−1.54396	−0.771981	+	0.635646i	$0.780734\pi$
$702$	0	0
$703$	3.24554	0.122408
$704$	0	0
$705$	−34.6041	−1.30327
$706$	0	0
$707$	−12.0788	−0.454271
$708$	0	0
$709$	−33.3293	−1.25171	−0.625855	−	0.779940i	$-0.715250\pi$
$709$	−33.3293	−1.25171	−0.625855	+	0.779940i	$0.715250\pi$
$710$	0	0
$711$	−0.869281	−0.0326006
$712$	0	0
$713$	−15.3606	−0.575260
$714$	0	0
$715$	10.1876	0.380994
$716$	0	0
$717$	−54.0720	−2.01936
$718$	0	0
$719$	24.5647	0.916110	0.458055	−	0.888924i	$-0.348546\pi$
$719$	24.5647	0.916110	0.458055	+	0.888924i	$0.348546\pi$
$720$	0	0
$721$	7.78525	0.289938
$722$	0	0
$723$	−67.3693	−2.50549
$724$	0	0
$725$	−1.72673	−0.0641292
$726$	0	0
$727$	−8.42755	−0.312560	−0.156280	−	0.987713i	$-0.549950\pi$
$727$	−8.42755	−0.312560	−0.156280	+	0.987713i	$0.549950\pi$
$728$	0	0
$729$	−41.7936	−1.54791
$730$	0	0
$731$	17.7499	0.656503
$732$	0	0
$733$	−11.3192	−0.418084	−0.209042	−	0.977907i	$-0.567034\pi$
$733$	−11.3192	−0.418084	−0.209042	+	0.977907i	$0.567034\pi$
$734$	0	0
$735$	−31.8776	−1.17582
$736$	0	0
$737$	−5.78576	−0.213121
$738$	0	0
$739$	47.8757	1.76114	0.880568	−	0.473919i	$-0.157161\pi$
$739$	47.8757	1.76114	0.880568	+	0.473919i	$0.157161\pi$
$740$	0	0
$741$	−15.1916	−0.558078
$742$	0	0
$743$	−49.9347	−1.83193	−0.915963	−	0.401262i	$-0.868572\pi$
$743$	−49.9347	−1.83193	−0.915963	+	0.401262i	$0.868572\pi$
$744$	0	0
$745$	5.92017	0.216898
$746$	0	0
$747$	−8.20642	−0.300257
$748$	0	0
$749$	−2.70086	−0.0986872
$750$	0	0
$751$	16.9473	0.618417	0.309209	−	0.950994i	$-0.399936\pi$
$751$	16.9473	0.618417	0.309209	+	0.950994i	$0.399936\pi$
$752$	0	0
$753$	−42.2079	−1.53814
$754$	0	0
$755$	5.92017	0.215457
$756$	0	0
$757$	0.939402	0.0341431	0.0170716	−	0.999854i	$-0.494566\pi$
$757$	0.939402	0.0341431	0.0170716	+	0.999854i	$0.494566\pi$
$758$	0	0
$759$	8.79073	0.319083
$760$	0	0
$761$	10.1954	0.369584	0.184792	−	0.982778i	$-0.440839\pi$
$761$	10.1954	0.369584	0.184792	+	0.982778i	$0.440839\pi$
$762$	0	0
$763$	−4.85851	−0.175890
$764$	0	0
$765$	19.7760	0.715004
$766$	0	0
$767$	−19.8174	−0.715565
$768$	0	0
$769$	−2.22600	−0.0802718	−0.0401359	−	0.999194i	$-0.512779\pi$
$769$	−2.22600	−0.0802718	−0.0401359	+	0.999194i	$0.512779\pi$
$770$	0	0
$771$	−24.3731	−0.877776
$772$	0	0
$773$	−24.2189	−0.871092	−0.435546	−	0.900166i	$-0.643445\pi$
$773$	−24.2189	−0.871092	−0.435546	+	0.900166i	$0.643445\pi$
$774$	0	0
$775$	6.63797	0.238443
$776$	0	0
$777$	10.0527	0.360638
$778$	0	0
$779$	−2.99503	−0.107308
$780$	0	0
$781$	−12.3102	−0.440493
$782$	0	0
$783$	8.01781	0.286533
$784$	0	0
$785$	13.4725	0.480852
$786$	0	0
$787$	36.7756	1.31091	0.655454	−	0.755235i	$-0.272477\pi$
$787$	36.7756	1.31091	0.655454	+	0.755235i	$0.272477\pi$
$788$	0	0
$789$	−68.1556	−2.42640
$790$	0	0
$791$	−17.0774	−0.607202
$792$	0	0
$793$	−76.4318	−2.71417
$794$	0	0
$795$	10.9461	0.388217
$796$	0	0
$797$	42.1391	1.49264	0.746322	−	0.665585i	$-0.231818\pi$
$797$	42.1391	1.49264	0.746322	+	0.665585i	$0.231818\pi$
$798$	0	0
$799$	−12.6453	−0.447359
$800$	0	0
$801$	32.6390	1.15324
$802$	0	0
$803$	12.9461	0.456857
$804$	0	0
$805$	−6.39154	−0.225272
$806$	0	0
$807$	42.1475	1.48366
$808$	0	0
$809$	−11.6056	−0.408030	−0.204015	−	0.978968i	$-0.565399\pi$
$809$	−11.6056	−0.408030	−0.204015	+	0.978968i	$0.565399\pi$
$810$	0	0
$811$	52.3945	1.83982	0.919910	−	0.392128i	$-0.128261\pi$
$811$	52.3945	1.83982	0.919910	+	0.392128i	$0.128261\pi$
$812$	0	0
$813$	−10.3321	−0.362361
$814$	0	0
$815$	−8.35893	−0.292801
$816$	0	0
$817$	−8.87494	−0.310495
$818$	0	0
$819$	−29.7577	−1.03982
$820$	0	0
$821$	−49.2156	−1.71764	−0.858818	−	0.512281i	$-0.828801\pi$
$821$	−49.2156	−1.71764	−0.858818	+	0.512281i	$0.828801\pi$
$822$	0	0
$823$	−20.6059	−0.718277	−0.359138	−	0.933284i	$-0.616929\pi$
$823$	−20.6059	−0.718277	−0.359138	+	0.933284i	$0.616929\pi$
$824$	0	0
$825$	−3.79884	−0.132259
$826$	0	0
$827$	−28.6595	−0.996589	−0.498295	−	0.867008i	$-0.666040\pi$
$827$	−28.6595	−0.996589	−0.498295	+	0.867008i	$0.666040\pi$
$828$	0	0
$829$	−10.1774	−0.353476	−0.176738	−	0.984258i	$-0.556555\pi$
$829$	−10.1774	−0.353476	−0.176738	+	0.984258i	$0.556555\pi$
$830$	0	0
$831$	−2.48921	−0.0863497
$832$	0	0
$833$	−11.6490	−0.403613
$834$	0	0
$835$	−5.27093	−0.182408
$836$	0	0
$837$	−30.8224	−1.06538
$838$	0	0
$839$	18.4526	0.637055	0.318528	−	0.947914i	$-0.396812\pi$
$839$	18.4526	0.637055	0.318528	+	0.947914i	$0.396812\pi$
$840$	0	0
$841$	−27.3138	−0.941855
$842$	0	0
$843$	−73.0366	−2.51551
$844$	0	0
$845$	29.2692	1.00689
$846$	0	0
$847$	−1.08421	−0.0372539
$848$	0	0
$849$	61.5680	2.11301
$850$	0	0
$851$	−9.98692	−0.342347
$852$	0	0
$853$	31.8680	1.09114	0.545570	−	0.838065i	$-0.316313\pi$
$853$	31.8680	1.09114	0.545570	+	0.838065i	$0.316313\pi$
$854$	0	0
$855$	−9.88802	−0.338163
$856$	0	0
$857$	−2.31725	−0.0791559	−0.0395779	−	0.999216i	$-0.512601\pi$
$857$	−2.31725	−0.0791559	−0.0395779	+	0.999216i	$0.512601\pi$
$858$	0	0
$859$	44.0900	1.50433	0.752166	−	0.658973i	$-0.229009\pi$
$859$	44.0900	1.50433	0.752166	+	0.658973i	$0.229009\pi$
$860$	0	0
$861$	−9.27675	−0.316151
$862$	0	0
$863$	5.09373	0.173393	0.0866964	−	0.996235i	$-0.472369\pi$
$863$	5.09373	0.173393	0.0866964	+	0.996235i	$0.472369\pi$
$864$	0	0
$865$	−17.6385	−0.599726
$866$	0	0
$867$	−37.1385	−1.26129
$868$	0	0
$869$	−0.168422	−0.00571332
$870$	0	0
$871$	−30.7669	−1.04250
$872$	0	0
$873$	74.9967	2.53825
$874$	0	0
$875$	13.1477	0.444472
$876$	0	0
$877$	13.8573	0.467928	0.233964	−	0.972245i	$-0.424830\pi$
$877$	13.8573	0.467928	0.233964	+	0.972245i	$0.424830\pi$
$878$	0	0
$879$	5.92872	0.199971
$880$	0	0
$881$	−32.6376	−1.09959	−0.549794	−	0.835300i	$-0.685294\pi$
$881$	−32.6376	−1.09959	−0.549794	+	0.835300i	$0.685294\pi$
$882$	0	0
$883$	31.3091	1.05363	0.526817	−	0.849979i	$-0.323385\pi$
$883$	31.3091	1.05363	0.526817	+	0.849979i	$0.323385\pi$
$884$	0	0
$885$	−20.3963	−0.685614
$886$	0	0
$887$	35.6199	1.19600	0.598000	−	0.801496i	$-0.295962\pi$
$887$	35.6199	1.19600	0.598000	+	0.801496i	$0.295962\pi$
$888$	0	0
$889$	−2.84433	−0.0953957
$890$	0	0
$891$	2.15534	0.0722066
$892$	0	0
$893$	6.32266	0.211580
$894$	0	0
$895$	2.27303	0.0759789
$896$	0	0
$897$	46.7464	1.56082
$898$	0	0
$899$	−6.48214	−0.216192
$900$	0	0
$901$	4.00000	0.133259
$902$	0	0
$903$	−27.4890	−0.914778
$904$	0	0
$905$	−47.8168	−1.58948
$906$	0	0
$907$	41.1629	1.36679	0.683396	−	0.730048i	$-0.260502\pi$
$907$	41.1629	1.36679	0.683396	+	0.730048i	$0.260502\pi$
$908$	0	0
$909$	57.5006	1.90717
$910$	0	0
$911$	8.30560	0.275177	0.137588	−	0.990490i	$-0.456065\pi$
$911$	8.30560	0.275177	0.137588	+	0.990490i	$0.456065\pi$
$912$	0	0
$913$	−1.58998	−0.0526207
$914$	0	0
$915$	−78.6645	−2.60057
$916$	0	0
$917$	−12.4670	−0.411696
$918$	0	0
$919$	17.5343	0.578402	0.289201	−	0.957268i	$-0.406610\pi$
$919$	17.5343	0.578402	0.289201	+	0.957268i	$0.406610\pi$
$920$	0	0
$921$	−27.0817	−0.892372
$922$	0	0
$923$	−65.4618	−2.15470
$924$	0	0
$925$	4.31577	0.141902
$926$	0	0
$927$	−37.0613	−1.21725
$928$	0	0
$929$	−17.8335	−0.585097	−0.292549	−	0.956251i	$-0.594503\pi$
$929$	−17.8335	−0.585097	−0.292549	+	0.956251i	$0.594503\pi$
$930$	0	0
$931$	5.82449	0.190890
$932$	0	0
$933$	−24.2574	−0.794150
$934$	0	0
$935$	3.83158	0.125306
$936$	0	0
$937$	15.8865	0.518988	0.259494	−	0.965745i	$-0.416444\pi$
$937$	15.8865	0.518988	0.259494	+	0.965745i	$0.416444\pi$
$938$	0	0
$939$	−26.3276	−0.859170
$940$	0	0
$941$	−14.0364	−0.457575	−0.228787	−	0.973476i	$-0.573476\pi$
$941$	−14.0364	−0.457575	−0.228787	+	0.973476i	$0.573476\pi$
$942$	0	0
$943$	9.21607	0.300117
$944$	0	0
$945$	−12.8252	−0.417203
$946$	0	0
$947$	30.1959	0.981234	0.490617	−	0.871375i	$-0.336771\pi$
$947$	30.1959	0.981234	0.490617	+	0.871375i	$0.336771\pi$
$948$	0	0
$949$	68.8432	2.23475
$950$	0	0
$951$	−80.6415	−2.61498
$952$	0	0
$953$	8.71300	0.282242	0.141121	−	0.989992i	$-0.454929\pi$
$953$	8.71300	0.282242	0.141121	+	0.989992i	$0.454929\pi$
$954$	0	0
$955$	19.7510	0.639127
$956$	0	0
$957$	3.70966	0.119916
$958$	0	0
$959$	−0.174916	−0.00564834
$960$	0	0
$961$	−6.08108	−0.196164
$962$	0	0
$963$	12.8573	0.414321
$964$	0	0
$965$	50.0259	1.61039
$966$	0	0
$967$	−43.1106	−1.38634	−0.693171	−	0.720773i	$-0.743787\pi$
$967$	−43.1106	−1.38634	−0.693171	+	0.720773i	$0.743787\pi$
$968$	0	0
$969$	−5.71361	−0.183548
$970$	0	0
$971$	−21.2824	−0.682985	−0.341493	−	0.939884i	$-0.610932\pi$
$971$	−21.2824	−0.682985	−0.341493	+	0.939884i	$0.610932\pi$
$972$	0	0
$973$	15.5798	0.499464
$974$	0	0
$975$	−20.2011	−0.646952
$976$	0	0
$977$	−13.2820	−0.424928	−0.212464	−	0.977169i	$-0.568149\pi$
$977$	−13.2820	−0.424928	−0.212464	+	0.977169i	$0.568149\pi$
$978$	0	0
$979$	6.32376	0.202108
$980$	0	0
$981$	23.1287	0.738442
$982$	0	0
$983$	−59.4823	−1.89719	−0.948596	−	0.316491i	$-0.897495\pi$
$983$	−59.4823	−1.89719	−0.948596	+	0.316491i	$0.897495\pi$
$984$	0	0
$985$	−19.3050	−0.615108
$986$	0	0
$987$	19.5837	0.623355
$988$	0	0
$989$	27.3092	0.868384
$990$	0	0
$991$	−56.0005	−1.77891	−0.889457	−	0.457019i	$-0.848917\pi$
$991$	−56.0005	−1.77891	−0.889457	+	0.457019i	$0.848917\pi$
$992$	0	0
$993$	4.93693	0.156669
$994$	0	0
$995$	−19.1582	−0.607356
$996$	0	0
$997$	16.3839	0.518883	0.259441	−	0.965759i	$-0.416462\pi$
$997$	16.3839	0.518883	0.259441	+	0.965759i	$0.416462\pi$
$998$	0	0
$999$	−20.0396	−0.634025

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.a.u.1.5		5
4.3	odd	2		1672.2.a.g.1.1	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1672.2.a.g.1.1	✓	5	4.3	odd	2
3344.2.a.u.1.5		5	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3344 = 2^{4} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3344.a (trivial)

\(f(q)\)	\(=\)	\(q+2.85680 q^{3} +1.91579 q^{5} -1.08421 q^{7} +5.16133 q^{9} +1.00000 q^{11} +5.31769 q^{13} +5.47303 q^{15} +2.00000 q^{17} -1.00000 q^{19} -3.09738 q^{21} +3.07712 q^{23} -1.32975 q^{25} +6.17450 q^{27} +1.29854 q^{29} -4.99189 q^{31} +2.85680 q^{33} -2.07712 q^{35} -3.24554 q^{37} +15.1916 q^{39} +2.99503 q^{41} +8.87494 q^{43} +9.88802 q^{45} -6.32266 q^{47} -5.82449 q^{49} +5.71361 q^{51} +2.00000 q^{53} +1.91579 q^{55} -2.85680 q^{57} -3.72669 q^{59} -14.3731 q^{61} -5.59597 q^{63} +10.1876 q^{65} -5.78576 q^{67} +8.79073 q^{69} -12.3102 q^{71} +12.9461 q^{73} -3.79884 q^{75} -1.08421 q^{77} -0.168422 q^{79} +2.15534 q^{81} -1.58998 q^{83} +3.83158 q^{85} +3.70966 q^{87} +6.32376 q^{89} -5.76550 q^{91} -14.2608 q^{93} -1.91579 q^{95} +14.5305 q^{97} +5.16133 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - q^{3} + 7 q^{5} - 8 q^{7} + 8 q^{9} + 5 q^{11} + 4 q^{13} - q^{15} + 10 q^{17} - 5 q^{19} + 2 q^{21} - 5 q^{23} + 6 q^{25} - 7 q^{27} + 16 q^{29} - q^{31} - q^{33} + 10 q^{35} - q^{37} + 4 q^{39}+ \cdots + 8 q^{99}+O(q^{100}) \) 5 * q - q^3 + 7 * q^5 - 8 * q^7 + 8 * q^9 + 5 * q^11 + 4 * q^13 - q^15 + 10 * q^17 - 5 * q^19 + 2 * q^21 - 5 * q^23 + 6 * q^25 - 7 * q^27 + 16 * q^29 - q^31 - q^33 + 10 * q^35 - q^37 + 4 * q^39 + 28 * q^41 - 4 * q^43 + 12 * q^45 + 4 * q^47 - q^49 - 2 * q^51 + 10 * q^53 + 7 * q^55 + q^57 + q^59 - 16 * q^61 - 12 * q^63 + 24 * q^65 + 9 * q^67 - 7 * q^69 - 7 * q^71 + 8 * q^73 + 8 * q^75 - 8 * q^77 - 6 * q^79 + 5 * q^81 - 4 * q^83 + 14 * q^85 + 2 * q^87 + 31 * q^89 + 12 * q^91 - 7 * q^93 - 7 * q^95 + 13 * q^97 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more