LMFDB - Embedded newform 3344.2.a.r.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3344,2,Mod(1,3344)]

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

lf = mfeigenbasis(mf)

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")

//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);

Level:	$ N $	$=$	$ 3344 = 2^{4} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3344.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:	$26.7019744359$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
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			1.2
Root			$-0.254102$ 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)

$f(q)$	$=$	$q+0.254102 q^{3} -0.254102 q^{5} -0.745898 q^{7} -2.93543 q^{9} +O(q^{10})$	$q+0.254102 q^{3} -0.254102 q^{5} -0.745898 q^{7} -2.93543 q^{9} +1.00000 q^{11} +4.61676 q^{13} -0.0645677 q^{15} -0.508203 q^{17} +1.00000 q^{19} -0.189534 q^{21} -4.68133 q^{23} -4.93543 q^{25} -1.50820 q^{27} +1.57277 q^{29} -9.29809 q^{31} +0.254102 q^{33} +0.189534 q^{35} +0.810466 q^{37} +1.17313 q^{39} +7.97942 q^{41} +4.80630 q^{43} +0.745898 q^{45} -4.37907 q^{47} -6.44364 q^{49} -0.129135 q^{51} +3.87086 q^{53} -0.254102 q^{55} +0.254102 q^{57} -12.5522 q^{59} +3.36266 q^{61} +2.18953 q^{63} -1.17313 q^{65} -1.58918 q^{67} -1.18953 q^{69} -5.74590 q^{71} +2.50820 q^{73} -1.25410 q^{75} -0.745898 q^{77} +2.85446 q^{79} +8.42306 q^{81} -0.210110 q^{83} +0.129135 q^{85} +0.399644 q^{87} +11.4395 q^{89} -3.44364 q^{91} -2.36266 q^{93} -0.254102 q^{95} -12.5522 q^{97} -2.93543 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{7} - q^{9}+O(q^{10}) $ 3 * q - 3 * q^7 - q^9	$ 3 q - 3 q^{7} - q^{9} + 3 q^{11} - q^{13} - 8 q^{15} + 3 q^{19} + 8 q^{21} - 7 q^{23} - 7 q^{25} - 3 q^{27} + 11 q^{29} - 6 q^{31} - 8 q^{35} + 11 q^{37} - 2 q^{39} - 5 q^{41} - 9 q^{43} + 3 q^{45} + 4 q^{47} - 10 q^{49} - 16 q^{51} - 4 q^{53} - 15 q^{59} - 4 q^{61} - 2 q^{63} + 2 q^{65} - 8 q^{67} + 5 q^{69} - 18 q^{71} + 6 q^{73} - 3 q^{75} - 3 q^{77} - 4 q^{79} - 13 q^{81} - 21 q^{83} + 16 q^{85} + 13 q^{87} - 7 q^{89} - q^{91} + 7 q^{93} - 15 q^{97} - q^{99}+O(q^{100}) $ 3 * q - 3 * q^7 - q^9 + 3 * q^11 - q^13 - 8 * q^15 + 3 * q^19 + 8 * q^21 - 7 * q^23 - 7 * q^25 - 3 * q^27 + 11 * q^29 - 6 * q^31 - 8 * q^35 + 11 * q^37 - 2 * q^39 - 5 * q^41 - 9 * q^43 + 3 * q^45 + 4 * q^47 - 10 * q^49 - 16 * q^51 - 4 * q^53 - 15 * q^59 - 4 * q^61 - 2 * q^63 + 2 * q^65 - 8 * q^67 + 5 * q^69 - 18 * q^71 + 6 * q^73 - 3 * q^75 - 3 * q^77 - 4 * q^79 - 13 * q^81 - 21 * q^83 + 16 * q^85 + 13 * q^87 - 7 * q^89 - q^91 + 7 * q^93 - 15 * q^97 - 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
$3$	0.254102	0.146706	0.0733528	−	0.997306i	$-0.476630\pi$
$3$	0.254102	0.146706	0.0733528	+	0.997306i	$0.476630\pi$
$4$	0	0
$5$	−0.254102	−0.113638	−0.0568189	−	0.998385i	$-0.518096\pi$
$5$	−0.254102	−0.113638	−0.0568189	+	0.998385i	$0.518096\pi$
$6$	0	0
$7$	−0.745898	−0.281923	−0.140962	−	0.990015i	$-0.545019\pi$
$7$	−0.745898	−0.281923	−0.140962	+	0.990015i	$0.545019\pi$
$8$	0	0
$9$	−2.93543	−0.978477
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	4.61676	1.28046	0.640230	−	0.768183i	$-0.278839\pi$
$13$	4.61676	1.28046	0.640230	+	0.768183i	$0.278839\pi$
$14$	0	0
$15$	−0.0645677	−0.0166713
$16$	0	0
$17$	−0.508203	−0.123257	−0.0616287	−	0.998099i	$-0.519629\pi$
$17$	−0.508203	−0.123257	−0.0616287	+	0.998099i	$0.519629\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−0.189534	−0.0413597
$22$	0	0
$23$	−4.68133	−0.976125	−0.488062	−	0.872809i	$-0.662296\pi$
$23$	−4.68133	−0.976125	−0.488062	+	0.872809i	$0.662296\pi$
$24$	0	0
$25$	−4.93543	−0.987086
$26$	0	0
$27$	−1.50820	−0.290254
$28$	0	0
$29$	1.57277	0.292056	0.146028	−	0.989280i	$-0.453351\pi$
$29$	1.57277	0.292056	0.146028	+	0.989280i	$0.453351\pi$
$30$	0	0
$31$	−9.29809	−1.66999	−0.834993	−	0.550260i	$-0.814529\pi$
$31$	−9.29809	−1.66999	−0.834993	+	0.550260i	$0.814529\pi$
$32$	0	0
$33$	0.254102	0.0442334
$34$	0	0
$35$	0.189534	0.0320371
$36$	0	0
$37$	0.810466	0.133240	0.0666199	−	0.997778i	$-0.478779\pi$
$37$	0.810466	0.133240	0.0666199	+	0.997778i	$0.478779\pi$
$38$	0	0
$39$	1.17313	0.187851
$40$	0	0
$41$	7.97942	1.24618	0.623088	−	0.782151i	$-0.285878\pi$
$41$	7.97942	1.24618	0.623088	+	0.782151i	$0.285878\pi$
$42$	0	0
$43$	4.80630	0.732953	0.366477	−	0.930427i	$-0.380564\pi$
$43$	4.80630	0.732953	0.366477	+	0.930427i	$0.380564\pi$
$44$	0	0
$45$	0.745898	0.111192
$46$	0	0
$47$	−4.37907	−0.638753	−0.319376	−	0.947628i	$-0.603473\pi$
$47$	−4.37907	−0.638753	−0.319376	+	0.947628i	$0.603473\pi$
$48$	0	0
$49$	−6.44364	−0.920519
$50$	0	0
$51$	−0.129135	−0.0180826
$52$	0	0
$53$	3.87086	0.531704	0.265852	−	0.964014i	$-0.414347\pi$
$53$	3.87086	0.531704	0.265852	+	0.964014i	$0.414347\pi$
$54$	0	0
$55$	−0.254102	−0.0342631
$56$	0	0
$57$	0.254102	0.0336566
$58$	0	0
$59$	−12.5522	−1.63416	−0.817078	−	0.576527i	$-0.804408\pi$
$59$	−12.5522	−1.63416	−0.817078	+	0.576527i	$0.804408\pi$
$60$	0	0
$61$	3.36266	0.430545	0.215272	−	0.976554i	$-0.430936\pi$
$61$	3.36266	0.430545	0.215272	+	0.976554i	$0.430936\pi$
$62$	0	0
$63$	2.18953	0.275855
$64$	0	0
$65$	−1.17313	−0.145509
$66$	0	0
$67$	−1.58918	−0.194149	−0.0970745	−	0.995277i	$-0.530949\pi$
$67$	−1.58918	−0.194149	−0.0970745	+	0.995277i	$0.530949\pi$
$68$	0	0
$69$	−1.18953	−0.143203
$70$	0	0
$71$	−5.74590	−0.681913	−0.340956	−	0.940079i	$-0.610751\pi$
$71$	−5.74590	−0.681913	−0.340956	+	0.940079i	$0.610751\pi$
$72$	0	0
$73$	2.50820	0.293563	0.146782	−	0.989169i	$-0.453109\pi$
$73$	2.50820	0.293563	0.146782	+	0.989169i	$0.453109\pi$
$74$	0	0
$75$	−1.25410	−0.144811
$76$	0	0
$77$	−0.745898	−0.0850030
$78$	0	0
$79$	2.85446	0.321152	0.160576	−	0.987024i	$-0.448665\pi$
$79$	2.85446	0.321152	0.160576	+	0.987024i	$0.448665\pi$
$80$	0	0
$81$	8.42306	0.935896
$82$	0	0
$83$	−0.210110	−0.0230625	−0.0115313	−	0.999934i	$-0.503671\pi$
$83$	−0.210110	−0.0230625	−0.0115313	+	0.999934i	$0.503671\pi$
$84$	0	0
$85$	0.129135	0.0140067
$86$	0	0
$87$	0.399644	0.0428463
$88$	0	0
$89$	11.4395	1.21258	0.606291	−	0.795243i	$-0.292657\pi$
$89$	11.4395	1.21258	0.606291	+	0.795243i	$0.292657\pi$
$90$	0	0
$91$	−3.44364	−0.360991
$92$	0	0
$93$	−2.36266	−0.244997
$94$	0	0
$95$	−0.254102	−0.0260703
$96$	0	0
$97$	−12.5522	−1.27448	−0.637241	−	0.770664i	$-0.719924\pi$
$97$	−12.5522	−1.27448	−0.637241	+	0.770664i	$0.719924\pi$
$98$	0	0
$99$	−2.93543	−0.295022
$100$	0	0
$101$	−1.83805	−0.182893	−0.0914465	−	0.995810i	$-0.529149\pi$
$101$	−1.83805	−0.182893	−0.0914465	+	0.995810i	$0.529149\pi$
$102$	0	0
$103$	−12.6772	−1.24912	−0.624559	−	0.780978i	$-0.714721\pi$
$103$	−12.6772	−1.24912	−0.624559	+	0.780978i	$0.714721\pi$
$104$	0	0
$105$	0.0481609	0.00470002
$106$	0	0
$107$	−10.0000	−0.966736	−0.483368	−	0.875417i	$-0.660587\pi$
$107$	−10.0000	−0.966736	−0.483368	+	0.875417i	$0.660587\pi$
$108$	0	0
$109$	−17.7417	−1.69935	−0.849675	−	0.527307i	$-0.823202\pi$
$109$	−17.7417	−1.69935	−0.849675	+	0.527307i	$0.823202\pi$
$110$	0	0
$111$	0.205941	0.0195470
$112$	0	0
$113$	−16.2939	−1.53280	−0.766402	−	0.642362i	$-0.777955\pi$
$113$	−16.2939	−1.53280	−0.766402	+	0.642362i	$0.777955\pi$
$114$	0	0
$115$	1.18953	0.110925
$116$	0	0
$117$	−13.5522	−1.25290
$118$	0	0
$119$	0.379068	0.0347491
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	2.02759	0.182821
$124$	0	0
$125$	2.52461	0.225808
$126$	0	0
$127$	−16.9753	−1.50631	−0.753155	−	0.657843i	$-0.771469\pi$
$127$	−16.9753	−1.50631	−0.753155	+	0.657843i	$0.771469\pi$
$128$	0	0
$129$	1.22129	0.107528
$130$	0	0
$131$	−20.3257	−1.77586	−0.887931	−	0.459976i	$-0.847858\pi$
$131$	−20.3257	−1.77586	−0.887931	+	0.459976i	$0.847858\pi$
$132$	0	0
$133$	−0.745898	−0.0646776
$134$	0	0
$135$	0.383237	0.0329838
$136$	0	0
$137$	0.701906	0.0599679	0.0299840	−	0.999550i	$-0.490454\pi$
$137$	0.701906	0.0599679	0.0299840	+	0.999550i	$0.490454\pi$
$138$	0	0
$139$	−6.08097	−0.515782	−0.257891	−	0.966174i	$-0.583027\pi$
$139$	−6.08097	−0.515782	−0.257891	+	0.966174i	$0.583027\pi$
$140$	0	0
$141$	−1.11273	−0.0937087
$142$	0	0
$143$	4.61676	0.386073
$144$	0	0
$145$	−0.399644	−0.0331886
$146$	0	0
$147$	−1.63734	−0.135045
$148$	0	0
$149$	7.32985	0.600485	0.300242	−	0.953863i	$-0.402932\pi$
$149$	7.32985	0.600485	0.300242	+	0.953863i	$0.402932\pi$
$150$	0	0
$151$	10.7581	0.875485	0.437742	−	0.899100i	$-0.355778\pi$
$151$	10.7581	0.875485	0.437742	+	0.899100i	$0.355778\pi$
$152$	0	0
$153$	1.49180	0.120605
$154$	0	0
$155$	2.36266	0.189774
$156$	0	0
$157$	−1.36683	−0.109085	−0.0545425	−	0.998511i	$-0.517370\pi$
$157$	−1.36683	−0.109085	−0.0545425	+	0.998511i	$0.517370\pi$
$158$	0	0
$159$	0.983593	0.0780040
$160$	0	0
$161$	3.49180	0.275192
$162$	0	0
$163$	−8.85446	−0.693535	−0.346767	−	0.937951i	$-0.612721\pi$
$163$	−8.85446	−0.693535	−0.346767	+	0.937951i	$0.612721\pi$
$164$	0	0
$165$	−0.0645677	−0.00502659
$166$	0	0
$167$	2.34625	0.181559	0.0907793	−	0.995871i	$-0.471064\pi$
$167$	2.34625	0.181559	0.0907793	+	0.995871i	$0.471064\pi$
$168$	0	0
$169$	8.31450	0.639577
$170$	0	0
$171$	−2.93543	−0.224478
$172$	0	0
$173$	4.80630	0.365416	0.182708	−	0.983167i	$-0.441514\pi$
$173$	4.80630	0.365416	0.182708	+	0.983167i	$0.441514\pi$
$174$	0	0
$175$	3.68133	0.278282
$176$	0	0
$177$	−3.18953	−0.239740
$178$	0	0
$179$	−5.80630	−0.433983	−0.216992	−	0.976173i	$-0.569624\pi$
$179$	−5.80630	−0.433983	−0.216992	+	0.976173i	$0.569624\pi$
$180$	0	0
$181$	22.1648	1.64750	0.823748	−	0.566956i	$-0.191879\pi$
$181$	22.1648	1.64750	0.823748	+	0.566956i	$0.191879\pi$
$182$	0	0
$183$	0.854458	0.0631633
$184$	0	0
$185$	−0.205941	−0.0151411
$186$	0	0
$187$	−0.508203	−0.0371635
$188$	0	0
$189$	1.12497	0.0818293
$190$	0	0
$191$	13.2775	0.960727	0.480364	−	0.877069i	$-0.340505\pi$
$191$	13.2775	0.960727	0.480364	+	0.877069i	$0.340505\pi$
$192$	0	0
$193$	−26.5480	−1.91097	−0.955484	−	0.295042i	$-0.904666\pi$
$193$	−26.5480	−1.91097	−0.955484	+	0.295042i	$0.904666\pi$
$194$	0	0
$195$	−0.298094	−0.0213469
$196$	0	0
$197$	6.75814	0.481497	0.240749	−	0.970588i	$-0.422607\pi$
$197$	6.75814	0.481497	0.240749	+	0.970588i	$0.422607\pi$
$198$	0	0
$199$	0.161949	0.0114802	0.00574012	−	0.999984i	$-0.498173\pi$
$199$	0.161949	0.0114802	0.00574012	+	0.999984i	$0.498173\pi$
$200$	0	0
$201$	−0.403813	−0.0284828
$202$	0	0
$203$	−1.17313	−0.0823374
$204$	0	0
$205$	−2.02759	−0.141613
$206$	0	0
$207$	13.7417	0.955116
$208$	0	0
$209$	1.00000	0.0691714
$210$	0	0
$211$	−13.8709	−0.954910	−0.477455	−	0.878656i	$-0.658441\pi$
$211$	−13.8709	−0.954910	−0.477455	+	0.878656i	$0.658441\pi$
$212$	0	0
$213$	−1.46004	−0.100040
$214$	0	0
$215$	−1.22129	−0.0832912
$216$	0	0
$217$	6.93543	0.470808
$218$	0	0
$219$	0.637339	0.0430674
$220$	0	0
$221$	−2.34625	−0.157826
$222$	0	0
$223$	11.5358	0.772494	0.386247	−	0.922395i	$-0.373771\pi$
$223$	11.5358	0.772494	0.386247	+	0.922395i	$0.373771\pi$
$224$	0	0
$225$	14.4876	0.965842
$226$	0	0
$227$	−1.20071	−0.0796941	−0.0398470	−	0.999206i	$-0.512687\pi$
$227$	−1.20071	−0.0796941	−0.0398470	+	0.999206i	$0.512687\pi$
$228$	0	0
$229$	−12.0646	−0.797249	−0.398625	−	0.917114i	$-0.630512\pi$
$229$	−12.0646	−0.797249	−0.398625	+	0.917114i	$0.630512\pi$
$230$	0	0
$231$	−0.189534	−0.0124704
$232$	0	0
$233$	2.03281	0.133174	0.0665870	−	0.997781i	$-0.478789\pi$
$233$	2.03281	0.133174	0.0665870	+	0.997781i	$0.478789\pi$
$234$	0	0
$235$	1.11273	0.0725864
$236$	0	0
$237$	0.725323	0.0471148
$238$	0	0
$239$	16.0726	1.03965	0.519826	−	0.854272i	$-0.325997\pi$
$239$	16.0726	1.03965	0.519826	+	0.854272i	$0.325997\pi$
$240$	0	0
$241$	−23.6608	−1.52412	−0.762062	−	0.647505i	$-0.775813\pi$
$241$	−23.6608	−1.52412	−0.762062	+	0.647505i	$0.775813\pi$
$242$	0	0
$243$	6.66492	0.427555
$244$	0	0
$245$	1.63734	0.104606
$246$	0	0
$247$	4.61676	0.293758
$248$	0	0
$249$	−0.0533892	−0.00338341
$250$	0	0
$251$	−1.06874	−0.0674581	−0.0337290	−	0.999431i	$-0.510738\pi$
$251$	−1.06874	−0.0674581	−0.0337290	+	0.999431i	$0.510738\pi$
$252$	0	0
$253$	−4.68133	−0.294313
$254$	0	0
$255$	0.0328135	0.00205486
$256$	0	0
$257$	−12.9424	−0.807327	−0.403664	−	0.914907i	$-0.632263\pi$
$257$	−12.9424	−0.807327	−0.403664	+	0.914907i	$0.632263\pi$
$258$	0	0
$259$	−0.604525	−0.0375634
$260$	0	0
$261$	−4.61676	−0.285770
$262$	0	0
$263$	20.3309	1.25366	0.626829	−	0.779157i	$-0.284353\pi$
$263$	20.3309	1.25366	0.626829	+	0.779157i	$0.284353\pi$
$264$	0	0
$265$	−0.983593	−0.0604217
$266$	0	0
$267$	2.90679	0.177893
$268$	0	0
$269$	−9.14554	−0.557614	−0.278807	−	0.960347i	$-0.589939\pi$
$269$	−9.14554	−0.557614	−0.278807	+	0.960347i	$0.589939\pi$
$270$	0	0
$271$	−16.3585	−0.993708	−0.496854	−	0.867834i	$-0.665511\pi$
$271$	−16.3585	−0.993708	−0.496854	+	0.867834i	$0.665511\pi$
$272$	0	0
$273$	−0.875034	−0.0529594
$274$	0	0
$275$	−4.93543	−0.297618
$276$	0	0
$277$	−15.7745	−0.947800	−0.473900	−	0.880579i	$-0.657154\pi$
$277$	−15.7745	−0.947800	−0.473900	+	0.880579i	$0.657154\pi$
$278$	0	0
$279$	27.2939	1.63404
$280$	0	0
$281$	3.15255	0.188065	0.0940327	−	0.995569i	$-0.470024\pi$
$281$	3.15255	0.188065	0.0940327	+	0.995569i	$0.470024\pi$
$282$	0	0
$283$	−9.25410	−0.550099	−0.275050	−	0.961430i	$-0.588694\pi$
$283$	−9.25410	−0.550099	−0.275050	+	0.961430i	$0.588694\pi$
$284$	0	0
$285$	−0.0645677	−0.00382466
$286$	0	0
$287$	−5.95184	−0.351326
$288$	0	0
$289$	−16.7417	−0.984808
$290$	0	0
$291$	−3.18953	−0.186974
$292$	0	0
$293$	21.1578	1.23605	0.618025	−	0.786158i	$-0.287933\pi$
$293$	21.1578	1.23605	0.618025	+	0.786158i	$0.287933\pi$
$294$	0	0
$295$	3.18953	0.185702
$296$	0	0
$297$	−1.50820	−0.0875148
$298$	0	0
$299$	−21.6126	−1.24989
$300$	0	0
$301$	−3.58501	−0.206636
$302$	0	0
$303$	−0.467052	−0.0268314
$304$	0	0
$305$	−0.854458	−0.0489261
$306$	0	0
$307$	15.9037	0.907671	0.453835	−	0.891086i	$-0.350055\pi$
$307$	15.9037	0.907671	0.453835	+	0.891086i	$0.350055\pi$
$308$	0	0
$309$	−3.22129	−0.183253
$310$	0	0
$311$	28.5962	1.62154	0.810771	−	0.585364i	$-0.199048\pi$
$311$	28.5962	1.62154	0.810771	+	0.585364i	$0.199048\pi$
$312$	0	0
$313$	−3.70475	−0.209405	−0.104702	−	0.994504i	$-0.533389\pi$
$313$	−3.70475	−0.209405	−0.104702	+	0.994504i	$0.533389\pi$
$314$	0	0
$315$	−0.556364	−0.0313476
$316$	0	0
$317$	−17.4067	−0.977655	−0.488828	−	0.872380i	$-0.662575\pi$
$317$	−17.4067	−0.977655	−0.488828	+	0.872380i	$0.662575\pi$
$318$	0	0
$319$	1.57277	0.0880583
$320$	0	0
$321$	−2.54102	−0.141826
$322$	0	0
$323$	−0.508203	−0.0282772
$324$	0	0
$325$	−22.7857	−1.26392
$326$	0	0
$327$	−4.50820	−0.249304
$328$	0	0
$329$	3.26634	0.180079
$330$	0	0
$331$	18.7212	1.02901	0.514504	−	0.857488i	$-0.327976\pi$
$331$	18.7212	1.02901	0.514504	+	0.857488i	$0.327976\pi$
$332$	0	0
$333$	−2.37907	−0.130372
$334$	0	0
$335$	0.403813	0.0220626
$336$	0	0
$337$	32.8255	1.78812	0.894061	−	0.447946i	$-0.147844\pi$
$337$	32.8255	1.78812	0.894061	+	0.447946i	$0.147844\pi$
$338$	0	0
$339$	−4.14031	−0.224871
$340$	0	0
$341$	−9.29809	−0.503520
$342$	0	0
$343$	10.0276	0.541439
$344$	0	0
$345$	0.302263	0.0162733
$346$	0	0
$347$	12.4999	0.671028	0.335514	−	0.942035i	$-0.391090\pi$
$347$	12.4999	0.671028	0.335514	+	0.942035i	$0.391090\pi$
$348$	0	0
$349$	16.7253	0.895286	0.447643	−	0.894212i	$-0.352264\pi$
$349$	16.7253	0.895286	0.447643	+	0.894212i	$0.352264\pi$
$350$	0	0
$351$	−6.96302	−0.371658
$352$	0	0
$353$	−5.53579	−0.294640	−0.147320	−	0.989089i	$-0.547065\pi$
$353$	−5.53579	−0.294640	−0.147320	+	0.989089i	$0.547065\pi$
$354$	0	0
$355$	1.46004	0.0774910
$356$	0	0
$357$	0.0963218	0.00509789
$358$	0	0
$359$	−25.1578	−1.32778	−0.663888	−	0.747832i	$-0.731095\pi$
$359$	−25.1578	−1.32778	−0.663888	+	0.747832i	$0.731095\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0.254102	0.0133369
$364$	0	0
$365$	−0.637339	−0.0333598
$366$	0	0
$367$	−10.2939	−0.537338	−0.268669	−	0.963232i	$-0.586584\pi$
$367$	−10.2939	−0.537338	−0.268669	+	0.963232i	$0.586584\pi$
$368$	0	0
$369$	−23.4231	−1.21936
$370$	0	0
$371$	−2.88727	−0.149900
$372$	0	0
$373$	20.2898	1.05056	0.525282	−	0.850928i	$-0.323960\pi$
$373$	20.2898	1.05056	0.525282	+	0.850928i	$0.323960\pi$
$374$	0	0
$375$	0.641508	0.0331273
$376$	0	0
$377$	7.26111	0.373966
$378$	0	0
$379$	8.57277	0.440354	0.220177	−	0.975460i	$-0.429337\pi$
$379$	8.57277	0.440354	0.220177	+	0.975460i	$0.429337\pi$
$380$	0	0
$381$	−4.31344	−0.220984
$382$	0	0
$383$	−0.0234162	−0.00119651	−0.000598256	−	1.00000i	$-0.500190\pi$
$383$	−0.0234162	−0.00119651	−0.000598256		1.00000i	$0.500190\pi$
$384$	0	0
$385$	0.189534	0.00965955
$386$	0	0
$387$	−14.1086	−0.717178
$388$	0	0
$389$	6.03698	0.306087	0.153044	−	0.988219i	$-0.451093\pi$
$389$	6.03698	0.306087	0.153044	+	0.988219i	$0.451093\pi$
$390$	0	0
$391$	2.37907	0.120315
$392$	0	0
$393$	−5.16479	−0.260529
$394$	0	0
$395$	−0.725323	−0.0364949
$396$	0	0
$397$	−26.1965	−1.31477	−0.657383	−	0.753556i	$-0.728337\pi$
$397$	−26.1965	−1.31477	−0.657383	+	0.753556i	$0.728337\pi$
$398$	0	0
$399$	−0.189534	−0.00948857
$400$	0	0
$401$	−6.92008	−0.345573	−0.172786	−	0.984959i	$-0.555277\pi$
$401$	−6.92008	−0.345573	−0.172786	+	0.984959i	$0.555277\pi$
$402$	0	0
$403$	−42.9271	−2.13835
$404$	0	0
$405$	−2.14031	−0.106353
$406$	0	0
$407$	0.810466	0.0401733
$408$	0	0
$409$	−2.55636	−0.126404	−0.0632020	−	0.998001i	$-0.520131\pi$
$409$	−2.55636	−0.126404	−0.0632020	+	0.998001i	$0.520131\pi$
$410$	0	0
$411$	0.178356	0.00879763
$412$	0	0
$413$	9.36266	0.460706
$414$	0	0
$415$	0.0533892	0.00262077
$416$	0	0
$417$	−1.54519	−0.0756681
$418$	0	0
$419$	−23.8625	−1.16576	−0.582880	−	0.812558i	$-0.698074\pi$
$419$	−23.8625	−1.16576	−0.582880	+	0.812558i	$0.698074\pi$
$420$	0	0
$421$	−1.40381	−0.0684176	−0.0342088	−	0.999415i	$-0.510891\pi$
$421$	−1.40381	−0.0684176	−0.0342088	+	0.999415i	$0.510891\pi$
$422$	0	0
$423$	12.8545	0.625005
$424$	0	0
$425$	2.50820	0.121666
$426$	0	0
$427$	−2.50820	−0.121380
$428$	0	0
$429$	1.17313	0.0566391
$430$	0	0
$431$	−14.1208	−0.680175	−0.340087	−	0.940394i	$-0.610457\pi$
$431$	−14.1208	−0.680175	−0.340087	+	0.940394i	$0.610457\pi$
$432$	0	0
$433$	32.8021	1.57637	0.788185	−	0.615439i	$-0.211021\pi$
$433$	32.8021	1.57637	0.788185	+	0.615439i	$0.211021\pi$
$434$	0	0
$435$	−0.101550	−0.00486896
$436$	0	0
$437$	−4.68133	−0.223938
$438$	0	0
$439$	37.4506	1.78742	0.893711	−	0.448643i	$-0.148093\pi$
$439$	37.4506	1.78742	0.893711	+	0.448643i	$0.148093\pi$
$440$	0	0
$441$	18.9149	0.900707
$442$	0	0
$443$	9.41499	0.447320	0.223660	−	0.974667i	$-0.428200\pi$
$443$	9.41499	0.447320	0.223660	+	0.974667i	$0.428200\pi$
$444$	0	0
$445$	−2.90679	−0.137795
$446$	0	0
$447$	1.86253	0.0880945
$448$	0	0
$449$	−36.8573	−1.73940	−0.869702	−	0.493578i	$-0.835689\pi$
$449$	−36.8573	−1.73940	−0.869702	+	0.493578i	$0.835689\pi$
$450$	0	0
$451$	7.97942	0.375736
$452$	0	0
$453$	2.73366	0.128439
$454$	0	0
$455$	0.875034	0.0410222
$456$	0	0
$457$	−18.8873	−0.883509	−0.441755	−	0.897136i	$-0.645644\pi$
$457$	−18.8873	−0.883509	−0.441755	+	0.897136i	$0.645644\pi$
$458$	0	0
$459$	0.766474	0.0357759
$460$	0	0
$461$	−20.4342	−0.951717	−0.475859	−	0.879522i	$-0.657863\pi$
$461$	−20.4342	−0.951717	−0.475859	+	0.879522i	$0.657863\pi$
$462$	0	0
$463$	37.3327	1.73500	0.867499	−	0.497440i	$-0.165726\pi$
$463$	37.3327	1.73500	0.867499	+	0.497440i	$0.165726\pi$
$464$	0	0
$465$	0.600356	0.0278409
$466$	0	0
$467$	23.4723	1.08617	0.543084	−	0.839679i	$-0.317257\pi$
$467$	23.4723	1.08617	0.543084	+	0.839679i	$0.317257\pi$
$468$	0	0
$469$	1.18537	0.0547351
$470$	0	0
$471$	−0.347314	−0.0160034
$472$	0	0
$473$	4.80630	0.220994
$474$	0	0
$475$	−4.93543	−0.226453
$476$	0	0
$477$	−11.3627	−0.520261
$478$	0	0
$479$	29.8091	1.36201	0.681007	−	0.732277i	$-0.261542\pi$
$479$	29.8091	1.36201	0.681007	+	0.732277i	$0.261542\pi$
$480$	0	0
$481$	3.74173	0.170608
$482$	0	0
$483$	0.887271	0.0403722
$484$	0	0
$485$	3.18953	0.144829
$486$	0	0
$487$	22.6004	1.02412	0.512060	−	0.858950i	$-0.328883\pi$
$487$	22.6004	1.02412	0.512060	+	0.858950i	$0.328883\pi$
$488$	0	0
$489$	−2.24993	−0.101745
$490$	0	0
$491$	5.21295	0.235257	0.117629	−	0.993058i	$-0.462471\pi$
$491$	5.21295	0.235257	0.117629	+	0.993058i	$0.462471\pi$
$492$	0	0
$493$	−0.799288	−0.0359981
$494$	0	0
$495$	0.745898	0.0335256
$496$	0	0
$497$	4.28586	0.192247
$498$	0	0
$499$	14.1536	0.633603	0.316801	−	0.948492i	$-0.397391\pi$
$499$	14.1536	0.633603	0.316801	+	0.948492i	$0.397391\pi$
$500$	0	0
$501$	0.596187	0.0266357
$502$	0	0
$503$	−16.1033	−0.718012	−0.359006	−	0.933335i	$-0.616884\pi$
$503$	−16.1033	−0.718012	−0.359006	+	0.933335i	$0.616884\pi$
$504$	0	0
$505$	0.467052	0.0207835
$506$	0	0
$507$	2.11273	0.0938296
$508$	0	0
$509$	−0.205941	−0.00912816	−0.00456408	−	0.999990i	$-0.501453\pi$
$509$	−0.205941	−0.00912816	−0.00456408	+	0.999990i	$0.501453\pi$
$510$	0	0
$511$	−1.87086	−0.0827622
$512$	0	0
$513$	−1.50820	−0.0665888
$514$	0	0
$515$	3.22129	0.141947
$516$	0	0
$517$	−4.37907	−0.192591
$518$	0	0
$519$	1.22129	0.0536086
$520$	0	0
$521$	−0.672993	−0.0294843	−0.0147422	−	0.999891i	$-0.504693\pi$
$521$	−0.672993	−0.0294843	−0.0147422	+	0.999891i	$0.504693\pi$
$522$	0	0
$523$	−44.4035	−1.94163	−0.970816	−	0.239827i	$-0.922909\pi$
$523$	−44.4035	−1.94163	−0.970816	+	0.239827i	$0.922909\pi$
$524$	0	0
$525$	0.935432	0.0408256
$526$	0	0
$527$	4.72532	0.205838
$528$	0	0
$529$	−1.08514	−0.0471801
$530$	0	0
$531$	36.8461	1.59899
$532$	0	0
$533$	36.8391	1.59568
$534$	0	0
$535$	2.54102	0.109858
$536$	0	0
$537$	−1.47539	−0.0636678
$538$	0	0
$539$	−6.44364	−0.277547
$540$	0	0
$541$	42.2499	1.81647	0.908233	−	0.418464i	$-0.137432\pi$
$541$	42.2499	1.81647	0.908233	+	0.418464i	$0.137432\pi$
$542$	0	0
$543$	5.63211	0.241697
$544$	0	0
$545$	4.50820	0.193110
$546$	0	0
$547$	25.1924	1.07715	0.538574	−	0.842578i	$-0.318963\pi$
$547$	25.1924	1.07715	0.538574	+	0.842578i	$0.318963\pi$
$548$	0	0
$549$	−9.87086	−0.421278
$550$	0	0
$551$	1.57277	0.0670023
$552$	0	0
$553$	−2.12914	−0.0905400
$554$	0	0
$555$	−0.0523299	−0.00222128
$556$	0	0
$557$	−9.10439	−0.385765	−0.192883	−	0.981222i	$-0.561784\pi$
$557$	−9.10439	−0.385765	−0.192883	+	0.981222i	$0.561784\pi$
$558$	0	0
$559$	22.1895	0.938517
$560$	0	0
$561$	−0.129135	−0.00545210
$562$	0	0
$563$	−9.65375	−0.406857	−0.203428	−	0.979090i	$-0.565208\pi$
$563$	−9.65375	−0.406857	−0.203428	+	0.979090i	$0.565208\pi$
$564$	0	0
$565$	4.14031	0.174184
$566$	0	0
$567$	−6.28275	−0.263851
$568$	0	0
$569$	36.7376	1.54012	0.770059	−	0.637972i	$-0.220227\pi$
$569$	36.7376	1.54012	0.770059	+	0.637972i	$0.220227\pi$
$570$	0	0
$571$	13.4712	0.563753	0.281877	−	0.959451i	$-0.409043\pi$
$571$	13.4712	0.563753	0.281877	+	0.959451i	$0.409043\pi$
$572$	0	0
$573$	3.37384	0.140944
$574$	0	0
$575$	23.1044	0.963520
$576$	0	0
$577$	−22.8227	−0.950122	−0.475061	−	0.879953i	$-0.657574\pi$
$577$	−22.8227	−0.950122	−0.475061	+	0.879953i	$0.657574\pi$
$578$	0	0
$579$	−6.74590	−0.280350
$580$	0	0
$581$	0.156721	0.00650186
$582$	0	0
$583$	3.87086	0.160315
$584$	0	0
$585$	3.44364	0.142377
$586$	0	0
$587$	−46.4671	−1.91790	−0.958950	−	0.283574i	$-0.908480\pi$
$587$	−46.4671	−1.91790	−0.958950	+	0.283574i	$0.908480\pi$
$588$	0	0
$589$	−9.29809	−0.383121
$590$	0	0
$591$	1.71725	0.0706384
$592$	0	0
$593$	28.1208	1.15478	0.577391	−	0.816468i	$-0.304071\pi$
$593$	28.1208	1.15478	0.577391	+	0.816468i	$0.304071\pi$
$594$	0	0
$595$	−0.0963218	−0.00394881
$596$	0	0
$597$	0.0411515	0.00168422
$598$	0	0
$599$	−20.5152	−0.838229	−0.419114	−	0.907933i	$-0.637659\pi$
$599$	−20.5152	−0.838229	−0.419114	+	0.907933i	$0.637659\pi$
$600$	0	0
$601$	−35.5509	−1.45015	−0.725075	−	0.688670i	$-0.758195\pi$
$601$	−35.5509	−1.45015	−0.725075	+	0.688670i	$0.758195\pi$
$602$	0	0
$603$	4.66492	0.189970
$604$	0	0
$605$	−0.254102	−0.0103307
$606$	0	0
$607$	−0.629001	−0.0255304	−0.0127652	−	0.999919i	$-0.504063\pi$
$607$	−0.629001	−0.0255304	−0.0127652	+	0.999919i	$0.504063\pi$
$608$	0	0
$609$	−0.298094	−0.0120794
$610$	0	0
$611$	−20.2171	−0.817897
$612$	0	0
$613$	30.4119	1.22832	0.614162	−	0.789180i	$-0.289494\pi$
$613$	30.4119	1.22832	0.614162	+	0.789180i	$0.289494\pi$
$614$	0	0
$615$	−0.515213	−0.0207754
$616$	0	0
$617$	41.4077	1.66701	0.833506	−	0.552511i	$-0.186330\pi$
$617$	41.4077	1.66701	0.833506	+	0.552511i	$0.186330\pi$
$618$	0	0
$619$	26.8984	1.08114	0.540570	−	0.841299i	$-0.318209\pi$
$619$	26.8984	1.08114	0.540570	+	0.841299i	$0.318209\pi$
$620$	0	0
$621$	7.06040	0.283324
$622$	0	0
$623$	−8.53268	−0.341855
$624$	0	0
$625$	24.0357	0.961426
$626$	0	0
$627$	0.254102	0.0101478
$628$	0	0
$629$	−0.411882	−0.0164228
$630$	0	0
$631$	−17.3738	−0.691642	−0.345821	−	0.938301i	$-0.612399\pi$
$631$	−17.3738	−0.691642	−0.345821	+	0.938301i	$0.612399\pi$
$632$	0	0
$633$	−3.52461	−0.140091
$634$	0	0
$635$	4.31344	0.171174
$636$	0	0
$637$	−29.7487	−1.17869
$638$	0	0
$639$	16.8667	0.667236
$640$	0	0
$641$	26.3574	1.04106	0.520528	−	0.853845i	$-0.325735\pi$
$641$	26.3574	1.04106	0.520528	+	0.853845i	$0.325735\pi$
$642$	0	0
$643$	−11.6014	−0.457515	−0.228758	−	0.973483i	$-0.573466\pi$
$643$	−11.6014	−0.457515	−0.228758	+	0.973483i	$0.573466\pi$
$644$	0	0
$645$	−0.310331	−0.0122193
$646$	0	0
$647$	−23.9393	−0.941152	−0.470576	−	0.882359i	$-0.655954\pi$
$647$	−23.9393	−0.941152	−0.470576	+	0.882359i	$0.655954\pi$
$648$	0	0
$649$	−12.5522	−0.492717
$650$	0	0
$651$	1.76231	0.0690702
$652$	0	0
$653$	33.3861	1.30650	0.653249	−	0.757143i	$-0.273405\pi$
$653$	33.3861	1.30650	0.653249	+	0.757143i	$0.273405\pi$
$654$	0	0
$655$	5.16479	0.201805
$656$	0	0
$657$	−7.36266	−0.287245
$658$	0	0
$659$	41.3215	1.60966	0.804829	−	0.593507i	$-0.202257\pi$
$659$	41.3215	1.60966	0.804829	+	0.593507i	$0.202257\pi$
$660$	0	0
$661$	−44.7938	−1.74228	−0.871138	−	0.491038i	$-0.836618\pi$
$661$	−44.7938	−1.74228	−0.871138	+	0.491038i	$0.836618\pi$
$662$	0	0
$663$	−0.596187	−0.0231540
$664$	0	0
$665$	0.189534	0.00734981
$666$	0	0
$667$	−7.36266	−0.285083
$668$	0	0
$669$	2.93126	0.113329
$670$	0	0
$671$	3.36266	0.129814
$672$	0	0
$673$	4.36683	0.168329	0.0841645	−	0.996452i	$-0.473178\pi$
$673$	4.36683	0.168329	0.0841645	+	0.996452i	$0.473178\pi$
$674$	0	0
$675$	7.44364	0.286506
$676$	0	0
$677$	37.0786	1.42505	0.712523	−	0.701649i	$-0.247552\pi$
$677$	37.0786	1.42505	0.712523	+	0.701649i	$0.247552\pi$
$678$	0	0
$679$	9.36266	0.359306
$680$	0	0
$681$	−0.305103	−0.0116916
$682$	0	0
$683$	21.4506	0.820786	0.410393	−	0.911909i	$-0.365392\pi$
$683$	21.4506	0.820786	0.410393	+	0.911909i	$0.365392\pi$
$684$	0	0
$685$	−0.178356	−0.00681462
$686$	0	0
$687$	−3.06563	−0.116961
$688$	0	0
$689$	17.8709	0.680826
$690$	0	0
$691$	−8.10750	−0.308424	−0.154212	−	0.988038i	$-0.549284\pi$
$691$	−8.10750	−0.308424	−0.154212	+	0.988038i	$0.549284\pi$
$692$	0	0
$693$	2.18953	0.0831735
$694$	0	0
$695$	1.54519	0.0586122
$696$	0	0
$697$	−4.05517	−0.153601
$698$	0	0
$699$	0.516541	0.0195374
$700$	0	0
$701$	21.0388	0.794623	0.397312	−	0.917684i	$-0.369943\pi$
$701$	21.0388	0.794623	0.397312	+	0.917684i	$0.369943\pi$
$702$	0	0
$703$	0.810466	0.0305673
$704$	0	0
$705$	0.282746	0.0106488
$706$	0	0
$707$	1.37100	0.0515617
$708$	0	0
$709$	13.4517	0.505189	0.252595	−	0.967572i	$-0.418716\pi$
$709$	13.4517	0.505189	0.252595	+	0.967572i	$0.418716\pi$
$710$	0	0
$711$	−8.37907	−0.314240
$712$	0	0
$713$	43.5275	1.63012
$714$	0	0
$715$	−1.17313	−0.0438725
$716$	0	0
$717$	4.08408	0.152523
$718$	0	0
$719$	9.40665	0.350809	0.175404	−	0.984496i	$-0.443877\pi$
$719$	9.40665	0.350809	0.175404	+	0.984496i	$0.443877\pi$
$720$	0	0
$721$	9.45587	0.352155
$722$	0	0
$723$	−6.01224	−0.223598
$724$	0	0
$725$	−7.76231	−0.288285
$726$	0	0
$727$	−32.1236	−1.19140	−0.595700	−	0.803207i	$-0.703125\pi$
$727$	−32.1236	−1.19140	−0.595700	+	0.803207i	$0.703125\pi$
$728$	0	0
$729$	−23.5756	−0.873171
$730$	0	0
$731$	−2.44258	−0.0903420
$732$	0	0
$733$	−6.34625	−0.234404	−0.117202	−	0.993108i	$-0.537393\pi$
$733$	−6.34625	−0.234404	−0.117202	+	0.993108i	$0.537393\pi$
$734$	0	0
$735$	0.416051	0.0153463
$736$	0	0
$737$	−1.58918	−0.0585381
$738$	0	0
$739$	−2.59752	−0.0955512	−0.0477756	−	0.998858i	$-0.515213\pi$
$739$	−2.59752	−0.0955512	−0.0477756	+	0.998858i	$0.515213\pi$
$740$	0	0
$741$	1.17313	0.0430959
$742$	0	0
$743$	−6.34625	−0.232821	−0.116411	−	0.993201i	$-0.537139\pi$
$743$	−6.34625	−0.232821	−0.116411	+	0.993201i	$0.537139\pi$
$744$	0	0
$745$	−1.86253	−0.0682377
$746$	0	0
$747$	0.616763	0.0225662
$748$	0	0
$749$	7.45898	0.272545
$750$	0	0
$751$	−23.3431	−0.851803	−0.425902	−	0.904769i	$-0.640043\pi$
$751$	−23.3431	−0.851803	−0.425902	+	0.904769i	$0.640043\pi$
$752$	0	0
$753$	−0.271568	−0.00989648
$754$	0	0
$755$	−2.73366	−0.0994881
$756$	0	0
$757$	−12.1138	−0.440283	−0.220142	−	0.975468i	$-0.570652\pi$
$757$	−12.1138	−0.440283	−0.220142	+	0.975468i	$0.570652\pi$
$758$	0	0
$759$	−1.18953	−0.0431773
$760$	0	0
$761$	26.8216	0.972284	0.486142	−	0.873880i	$-0.338404\pi$
$761$	26.8216	0.972284	0.486142	+	0.873880i	$0.338404\pi$
$762$	0	0
$763$	13.2335	0.479086
$764$	0	0
$765$	−0.379068	−0.0137052
$766$	0	0
$767$	−57.9505	−2.09247
$768$	0	0
$769$	33.8297	1.21993	0.609965	−	0.792428i	$-0.291183\pi$
$769$	33.8297	1.21993	0.609965	+	0.792428i	$0.291183\pi$
$770$	0	0
$771$	−3.28870	−0.118440
$772$	0	0
$773$	33.4835	1.20432	0.602158	−	0.798377i	$-0.294308\pi$
$773$	33.4835	1.20432	0.602158	+	0.798377i	$0.294308\pi$
$774$	0	0
$775$	45.8901	1.64842
$776$	0	0
$777$	−0.153611	−0.00551076
$778$	0	0
$779$	7.97942	0.285893
$780$	0	0
$781$	−5.74590	−0.205604
$782$	0	0
$783$	−2.37206	−0.0847705
$784$	0	0
$785$	0.347314	0.0123962
$786$	0	0
$787$	−27.1455	−0.967634	−0.483817	−	0.875169i	$-0.660750\pi$
$787$	−27.1455	−0.967634	−0.483817	+	0.875169i	$0.660750\pi$
$788$	0	0
$789$	5.16612	0.183919
$790$	0	0
$791$	12.1536	0.432133
$792$	0	0
$793$	15.5246	0.551295
$794$	0	0
$795$	−0.249933	−0.00886420
$796$	0	0
$797$	19.2859	0.683140	0.341570	−	0.939856i	$-0.389041\pi$
$797$	19.2859	0.683140	0.341570	+	0.939856i	$0.389041\pi$
$798$	0	0
$799$	2.22546	0.0787310
$800$	0	0
$801$	−33.5798	−1.18648
$802$	0	0
$803$	2.50820	0.0885126
$804$	0	0
$805$	−0.887271	−0.0312722
$806$	0	0
$807$	−2.32390	−0.0818051
$808$	0	0
$809$	−53.6266	−1.88541	−0.942706	−	0.333626i	$-0.891728\pi$
$809$	−53.6266	−1.88541	−0.942706	+	0.333626i	$0.891728\pi$
$810$	0	0
$811$	−0.549355	−0.0192905	−0.00964523	−	0.999953i	$-0.503070\pi$
$811$	−0.549355	−0.0192905	−0.00964523	+	0.999953i	$0.503070\pi$
$812$	0	0
$813$	−4.15672	−0.145783
$814$	0	0
$815$	2.24993	0.0788117
$816$	0	0
$817$	4.80630	0.168151
$818$	0	0
$819$	10.1086	0.353222
$820$	0	0
$821$	23.8709	0.833099	0.416549	−	0.909113i	$-0.363239\pi$
$821$	23.8709	0.833099	0.416549	+	0.909113i	$0.363239\pi$
$822$	0	0
$823$	−7.70608	−0.268617	−0.134308	−	0.990940i	$-0.542881\pi$
$823$	−7.70608	−0.268617	−0.134308	+	0.990940i	$0.542881\pi$
$824$	0	0
$825$	−1.25410	−0.0436622
$826$	0	0
$827$	−26.9836	−0.938311	−0.469156	−	0.883115i	$-0.655442\pi$
$827$	−26.9836	−0.938311	−0.469156	+	0.883115i	$0.655442\pi$
$828$	0	0
$829$	−11.2223	−0.389768	−0.194884	−	0.980826i	$-0.562433\pi$
$829$	−11.2223	−0.389768	−0.194884	+	0.980826i	$0.562433\pi$
$830$	0	0
$831$	−4.00834	−0.139048
$832$	0	0
$833$	3.27468	0.113461
$834$	0	0
$835$	−0.596187	−0.0206319
$836$	0	0
$837$	14.0234	0.484720
$838$	0	0
$839$	25.6496	0.885522	0.442761	−	0.896640i	$-0.353999\pi$
$839$	25.6496	0.885522	0.442761	+	0.896640i	$0.353999\pi$
$840$	0	0
$841$	−26.5264	−0.914703
$842$	0	0
$843$	0.801069	0.0275903
$844$	0	0
$845$	−2.11273	−0.0726801
$846$	0	0
$847$	−0.745898	−0.0256294
$848$	0	0
$849$	−2.35148	−0.0807027
$850$	0	0
$851$	−3.79406	−0.130059
$852$	0	0
$853$	1.18669	0.0406316	0.0203158	−	0.999794i	$-0.493533\pi$
$853$	1.18669	0.0406316	0.0203158	+	0.999794i	$0.493533\pi$
$854$	0	0
$855$	0.745898	0.0255092
$856$	0	0
$857$	−19.7868	−0.675904	−0.337952	−	0.941163i	$-0.609734\pi$
$857$	−19.7868	−0.675904	−0.337952	+	0.941163i	$0.609734\pi$
$858$	0	0
$859$	−56.0357	−1.91191	−0.955956	−	0.293510i	$-0.905177\pi$
$859$	−56.0357	−1.91191	−0.955956	+	0.293510i	$0.905177\pi$
$860$	0	0
$861$	−1.51237	−0.0515415
$862$	0	0
$863$	27.1494	0.924178	0.462089	−	0.886834i	$-0.347100\pi$
$863$	27.1494	0.924178	0.462089	+	0.886834i	$0.347100\pi$
$864$	0	0
$865$	−1.22129	−0.0415250
$866$	0	0
$867$	−4.25410	−0.144477
$868$	0	0
$869$	2.85446	0.0968309
$870$	0	0
$871$	−7.33686	−0.248600
$872$	0	0
$873$	36.8461	1.24705
$874$	0	0
$875$	−1.88310	−0.0636605
$876$	0	0
$877$	−44.7980	−1.51272	−0.756360	−	0.654156i	$-0.773024\pi$
$877$	−44.7980	−1.51272	−0.756360	+	0.654156i	$0.773024\pi$
$878$	0	0
$879$	5.37623	0.181336
$880$	0	0
$881$	−26.7816	−0.902293	−0.451147	−	0.892450i	$-0.648985\pi$
$881$	−26.7816	−0.902293	−0.451147	+	0.892450i	$0.648985\pi$
$882$	0	0
$883$	−1.80524	−0.0607511	−0.0303755	−	0.999539i	$-0.509670\pi$
$883$	−1.80524	−0.0607511	−0.0303755	+	0.999539i	$0.509670\pi$
$884$	0	0
$885$	0.810466	0.0272435
$886$	0	0
$887$	15.8953	0.533713	0.266857	−	0.963736i	$-0.414015\pi$
$887$	15.8953	0.533713	0.266857	+	0.963736i	$0.414015\pi$
$888$	0	0
$889$	12.6618	0.424664
$890$	0	0
$891$	8.42306	0.282183
$892$	0	0
$893$	−4.37907	−0.146540
$894$	0	0
$895$	1.47539	0.0493168
$896$	0	0
$897$	−5.49180	−0.183366
$898$	0	0
$899$	−14.6238	−0.487730
$900$	0	0
$901$	−1.96719	−0.0655365
$902$	0	0
$903$	−0.910957	−0.0303147
$904$	0	0
$905$	−5.63211	−0.187218
$906$	0	0
$907$	−31.4178	−1.04321	−0.521606	−	0.853186i	$-0.674667\pi$
$907$	−31.4178	−1.04321	−0.521606	+	0.853186i	$0.674667\pi$
$908$	0	0
$909$	5.39547	0.178957
$910$	0	0
$911$	39.2252	1.29959	0.649794	−	0.760110i	$-0.274855\pi$
$911$	39.2252	1.29959	0.649794	+	0.760110i	$0.274855\pi$
$912$	0	0
$913$	−0.210110	−0.00695362
$914$	0	0
$915$	−0.217119	−0.00717774
$916$	0	0
$917$	15.1609	0.500657
$918$	0	0
$919$	−7.21818	−0.238106	−0.119053	−	0.992888i	$-0.537986\pi$
$919$	−7.21818	−0.238106	−0.119053	+	0.992888i	$0.537986\pi$
$920$	0	0
$921$	4.04115	0.133160
$922$	0	0
$923$	−26.5275	−0.873162
$924$	0	0
$925$	−4.00000	−0.131519
$926$	0	0
$927$	37.2130	1.22223
$928$	0	0
$929$	5.70713	0.187245	0.0936225	−	0.995608i	$-0.470155\pi$
$929$	5.70713	0.187245	0.0936225	+	0.995608i	$0.470155\pi$
$930$	0	0
$931$	−6.44364	−0.211182
$932$	0	0
$933$	7.26634	0.237889
$934$	0	0
$935$	0.129135	0.00422318
$936$	0	0
$937$	−30.3463	−0.991369	−0.495684	−	0.868503i	$-0.665083\pi$
$937$	−30.3463	−0.991369	−0.495684	+	0.868503i	$0.665083\pi$
$938$	0	0
$939$	−0.941382	−0.0307209
$940$	0	0
$941$	32.9669	1.07469	0.537345	−	0.843362i	$-0.319427\pi$
$941$	32.9669	1.07469	0.537345	+	0.843362i	$0.319427\pi$
$942$	0	0
$943$	−37.3543	−1.21642
$944$	0	0
$945$	−0.285856	−0.00929889
$946$	0	0
$947$	−6.18147	−0.200871	−0.100435	−	0.994944i	$-0.532024\pi$
$947$	−6.18147	−0.200871	−0.100435	+	0.994944i	$0.532024\pi$
$948$	0	0
$949$	11.5798	0.375896
$950$	0	0
$951$	−4.42306	−0.143428
$952$	0	0
$953$	31.5163	1.02091	0.510456	−	0.859904i	$-0.329477\pi$
$953$	31.5163	1.02091	0.510456	+	0.859904i	$0.329477\pi$
$954$	0	0
$955$	−3.37384	−0.109175
$956$	0	0
$957$	0.399644	0.0129186
$958$	0	0
$959$	−0.523551	−0.0169063
$960$	0	0
$961$	55.4545	1.78886
$962$	0	0
$963$	29.3543	0.945930
$964$	0	0
$965$	6.74590	0.217158
$966$	0	0
$967$	−5.01641	−0.161317	−0.0806584	−	0.996742i	$-0.525702\pi$
$967$	−5.01641	−0.161317	−0.0806584	+	0.996742i	$0.525702\pi$
$968$	0	0
$969$	−0.129135	−0.00414842
$970$	0	0
$971$	55.5756	1.78351	0.891753	−	0.452522i	$-0.149476\pi$
$971$	55.5756	1.78351	0.891753	+	0.452522i	$0.149476\pi$
$972$	0	0
$973$	4.53579	0.145411
$974$	0	0
$975$	−5.78989	−0.185425
$976$	0	0
$977$	30.0908	0.962691	0.481345	−	0.876531i	$-0.340148\pi$
$977$	30.0908	0.962691	0.481345	+	0.876531i	$0.340148\pi$
$978$	0	0
$979$	11.4395	0.365607
$980$	0	0
$981$	52.0796	1.66278
$982$	0	0
$983$	−28.2405	−0.900733	−0.450367	−	0.892844i	$-0.648707\pi$
$983$	−28.2405	−0.900733	−0.450367	+	0.892844i	$0.648707\pi$
$984$	0	0
$985$	−1.71725	−0.0547163
$986$	0	0
$987$	0.829982	0.0264186
$988$	0	0
$989$	−22.4999	−0.715454
$990$	0	0
$991$	14.1742	0.450258	0.225129	−	0.974329i	$-0.427720\pi$
$991$	14.1742	0.450258	0.225129	+	0.974329i	$0.427720\pi$
$992$	0	0
$993$	4.75708	0.150961
$994$	0	0
$995$	−0.0411515	−0.00130459
$996$	0	0
$997$	48.9976	1.55177	0.775885	−	0.630874i	$-0.217304\pi$
$997$	48.9976	1.55177	0.775885	+	0.630874i	$0.217304\pi$
$998$	0	0
$999$	−1.22235	−0.0386734

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.a.r.1.2		3
4.3	odd	2		1672.2.a.e.1.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1672.2.a.e.1.2	✓	3	4.3	odd	2
3344.2.a.r.1.2		3	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+0.254102 q^{3} -0.254102 q^{5} -0.745898 q^{7} -2.93543 q^{9} +O(q^{10})\)	\(q+0.254102 q^{3} -0.254102 q^{5} -0.745898 q^{7} -2.93543 q^{9} +1.00000 q^{11} +4.61676 q^{13} -0.0645677 q^{15} -0.508203 q^{17} +1.00000 q^{19} -0.189534 q^{21} -4.68133 q^{23} -4.93543 q^{25} -1.50820 q^{27} +1.57277 q^{29} -9.29809 q^{31} +0.254102 q^{33} +0.189534 q^{35} +0.810466 q^{37} +1.17313 q^{39} +7.97942 q^{41} +4.80630 q^{43} +0.745898 q^{45} -4.37907 q^{47} -6.44364 q^{49} -0.129135 q^{51} +3.87086 q^{53} -0.254102 q^{55} +0.254102 q^{57} -12.5522 q^{59} +3.36266 q^{61} +2.18953 q^{63} -1.17313 q^{65} -1.58918 q^{67} -1.18953 q^{69} -5.74590 q^{71} +2.50820 q^{73} -1.25410 q^{75} -0.745898 q^{77} +2.85446 q^{79} +8.42306 q^{81} -0.210110 q^{83} +0.129135 q^{85} +0.399644 q^{87} +11.4395 q^{89} -3.44364 q^{91} -2.36266 q^{93} -0.254102 q^{95} -12.5522 q^{97} -2.93543 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{7} - q^{9}+O(q^{10}) \) 3 * q - 3 * q^7 - q^9	\( 3 q - 3 q^{7} - q^{9} + 3 q^{11} - q^{13} - 8 q^{15} + 3 q^{19} + 8 q^{21} - 7 q^{23} - 7 q^{25} - 3 q^{27} + 11 q^{29} - 6 q^{31} - 8 q^{35} + 11 q^{37} - 2 q^{39} - 5 q^{41} - 9 q^{43} + 3 q^{45} + 4 q^{47} - 10 q^{49} - 16 q^{51} - 4 q^{53} - 15 q^{59} - 4 q^{61} - 2 q^{63} + 2 q^{65} - 8 q^{67} + 5 q^{69} - 18 q^{71} + 6 q^{73} - 3 q^{75} - 3 q^{77} - 4 q^{79} - 13 q^{81} - 21 q^{83} + 16 q^{85} + 13 q^{87} - 7 q^{89} - q^{91} + 7 q^{93} - 15 q^{97} - q^{99}+O(q^{100}) \) 3 * q - 3 * q^7 - q^9 + 3 * q^11 - q^13 - 8 * q^15 + 3 * q^19 + 8 * q^21 - 7 * q^23 - 7 * q^25 - 3 * q^27 + 11 * q^29 - 6 * q^31 - 8 * q^35 + 11 * q^37 - 2 * q^39 - 5 * q^41 - 9 * q^43 + 3 * q^45 + 4 * q^47 - 10 * q^49 - 16 * q^51 - 4 * q^53 - 15 * q^59 - 4 * q^61 - 2 * q^63 + 2 * q^65 - 8 * q^67 + 5 * q^69 - 18 * q^71 + 6 * q^73 - 3 * q^75 - 3 * q^77 - 4 * q^79 - 13 * q^81 - 21 * q^83 + 16 * q^85 + 13 * q^87 - 7 * q^89 - q^91 + 7 * q^93 - 15 * q^97 - q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more