LMFDB - Embedded newform 3344.2.a.k.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:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 418)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.56155$ 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)

$f(q)$	$=$	$q+1.56155 q^{3} +2.00000 q^{5} -3.56155 q^{7} -0.561553 q^{9} +O(q^{10})$	$q+1.56155 q^{3} +2.00000 q^{5} -3.56155 q^{7} -0.561553 q^{9} -1.00000 q^{11} -3.56155 q^{13} +3.12311 q^{15} +3.56155 q^{17} +1.00000 q^{19} -5.56155 q^{21} -5.56155 q^{23} -1.00000 q^{25} -5.56155 q^{27} +6.68466 q^{29} -2.00000 q^{31} -1.56155 q^{33} -7.12311 q^{35} +3.12311 q^{37} -5.56155 q^{39} +2.00000 q^{41} -1.12311 q^{45} -8.00000 q^{47} +5.68466 q^{49} +5.56155 q^{51} -7.80776 q^{53} -2.00000 q^{55} +1.56155 q^{57} -4.68466 q^{59} -10.2462 q^{61} +2.00000 q^{63} -7.12311 q^{65} -4.68466 q^{67} -8.68466 q^{69} -6.00000 q^{71} +2.68466 q^{73} -1.56155 q^{75} +3.56155 q^{77} +4.00000 q^{79} -7.00000 q^{81} +2.24621 q^{83} +7.12311 q^{85} +10.4384 q^{87} -9.12311 q^{89} +12.6847 q^{91} -3.12311 q^{93} +2.00000 q^{95} +1.12311 q^{97} +0.561553 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} + 4 q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) $ 2 * q - q^3 + 4 * q^5 - 3 * q^7 + 3 * q^9	$ 2 q - q^{3} + 4 q^{5} - 3 q^{7} + 3 q^{9} - 2 q^{11} - 3 q^{13} - 2 q^{15} + 3 q^{17} + 2 q^{19} - 7 q^{21} - 7 q^{23} - 2 q^{25} - 7 q^{27} + q^{29} - 4 q^{31} + q^{33} - 6 q^{35} - 2 q^{37} - 7 q^{39} + 4 q^{41} + 6 q^{45} - 16 q^{47} - q^{49} + 7 q^{51} + 5 q^{53} - 4 q^{55} - q^{57} + 3 q^{59} - 4 q^{61} + 4 q^{63} - 6 q^{65} + 3 q^{67} - 5 q^{69} - 12 q^{71} - 7 q^{73} + q^{75} + 3 q^{77} + 8 q^{79} - 14 q^{81} - 12 q^{83} + 6 q^{85} + 25 q^{87} - 10 q^{89} + 13 q^{91} + 2 q^{93} + 4 q^{95} - 6 q^{97} - 3 q^{99}+O(q^{100}) $ 2 * q - q^3 + 4 * q^5 - 3 * q^7 + 3 * q^9 - 2 * q^11 - 3 * q^13 - 2 * q^15 + 3 * q^17 + 2 * q^19 - 7 * q^21 - 7 * q^23 - 2 * q^25 - 7 * q^27 + q^29 - 4 * q^31 + q^33 - 6 * q^35 - 2 * q^37 - 7 * q^39 + 4 * q^41 + 6 * q^45 - 16 * q^47 - q^49 + 7 * q^51 + 5 * q^53 - 4 * q^55 - q^57 + 3 * q^59 - 4 * q^61 + 4 * q^63 - 6 * q^65 + 3 * q^67 - 5 * q^69 - 12 * q^71 - 7 * q^73 + q^75 + 3 * q^77 + 8 * q^79 - 14 * q^81 - 12 * q^83 + 6 * q^85 + 25 * q^87 - 10 * q^89 + 13 * q^91 + 2 * q^93 + 4 * q^95 - 6 * q^97 - 3 * 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
$3$	1.56155	0.901563	0.450781	−	0.892634i	$-0.351145\pi$
$3$	1.56155	0.901563	0.450781	+	0.892634i	$0.351145\pi$
$4$	0	0
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	0	0
$7$	−3.56155	−1.34614	−0.673070	−	0.739579i	$-0.735025\pi$
$7$	−3.56155	−1.34614	−0.673070	+	0.739579i	$0.735025\pi$
$8$	0	0
$9$	−0.561553	−0.187184
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−3.56155	−0.987797	−0.493899	−	0.869520i	$-0.664429\pi$
$13$	−3.56155	−0.987797	−0.493899	+	0.869520i	$0.664429\pi$
$14$	0	0
$15$	3.12311	0.806382
$16$	0	0
$17$	3.56155	0.863803	0.431902	−	0.901921i	$-0.357843\pi$
$17$	3.56155	0.863803	0.431902	+	0.901921i	$0.357843\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−5.56155	−1.21363
$22$	0	0
$23$	−5.56155	−1.15966	−0.579832	−	0.814736i	$-0.696882\pi$
$23$	−5.56155	−1.15966	−0.579832	+	0.814736i	$0.696882\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−5.56155	−1.07032
$28$	0	0
$29$	6.68466	1.24131	0.620655	−	0.784084i	$-0.286867\pi$
$29$	6.68466	1.24131	0.620655	+	0.784084i	$0.286867\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	0	0
$33$	−1.56155	−0.271831
$34$	0	0
$35$	−7.12311	−1.20402
$36$	0	0
$37$	3.12311	0.513435	0.256718	−	0.966486i	$-0.417359\pi$
$37$	3.12311	0.513435	0.256718	+	0.966486i	$0.417359\pi$
$38$	0	0
$39$	−5.56155	−0.890561
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	−1.12311	−0.167423
$46$	0	0
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	0	0
$49$	5.68466	0.812094
$50$	0	0
$51$	5.56155	0.778773
$52$	0	0
$53$	−7.80776	−1.07248	−0.536239	−	0.844066i	$-0.680156\pi$
$53$	−7.80776	−1.07248	−0.536239	+	0.844066i	$0.680156\pi$
$54$	0	0
$55$	−2.00000	−0.269680
$56$	0	0
$57$	1.56155	0.206833
$58$	0	0
$59$	−4.68466	−0.609891	−0.304945	−	0.952370i	$-0.598638\pi$
$59$	−4.68466	−0.609891	−0.304945	+	0.952370i	$0.598638\pi$
$60$	0	0
$61$	−10.2462	−1.31189	−0.655946	−	0.754807i	$-0.727730\pi$
$61$	−10.2462	−1.31189	−0.655946	+	0.754807i	$0.727730\pi$
$62$	0	0
$63$	2.00000	0.251976
$64$	0	0
$65$	−7.12311	−0.883513
$66$	0	0
$67$	−4.68466	−0.572322	−0.286161	−	0.958182i	$-0.592379\pi$
$67$	−4.68466	−0.572322	−0.286161	+	0.958182i	$0.592379\pi$
$68$	0	0
$69$	−8.68466	−1.04551
$70$	0	0
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	0	0
$73$	2.68466	0.314216	0.157108	−	0.987581i	$-0.449783\pi$
$73$	2.68466	0.314216	0.157108	+	0.987581i	$0.449783\pi$
$74$	0	0
$75$	−1.56155	−0.180313
$76$	0	0
$77$	3.56155	0.405877
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	2.24621	0.246554	0.123277	−	0.992372i	$-0.460660\pi$
$83$	2.24621	0.246554	0.123277	+	0.992372i	$0.460660\pi$
$84$	0	0
$85$	7.12311	0.772609
$86$	0	0
$87$	10.4384	1.11912
$88$	0	0
$89$	−9.12311	−0.967047	−0.483524	−	0.875331i	$-0.660643\pi$
$89$	−9.12311	−0.967047	−0.483524	+	0.875331i	$0.660643\pi$
$90$	0	0
$91$	12.6847	1.32971
$92$	0	0
$93$	−3.12311	−0.323851
$94$	0	0
$95$	2.00000	0.205196
$96$	0	0
$97$	1.12311	0.114034	0.0570170	−	0.998373i	$-0.481841\pi$
$97$	1.12311	0.114034	0.0570170	+	0.998373i	$0.481841\pi$
$98$	0	0
$99$	0.561553	0.0564382
$100$	0	0
$101$	−15.1231	−1.50481	−0.752403	−	0.658703i	$-0.771105\pi$
$101$	−15.1231	−1.50481	−0.752403	+	0.658703i	$0.771105\pi$
$102$	0	0
$103$	−11.3693	−1.12025	−0.560126	−	0.828407i	$-0.689247\pi$
$103$	−11.3693	−1.12025	−0.560126	+	0.828407i	$0.689247\pi$
$104$	0	0
$105$	−11.1231	−1.08550
$106$	0	0
$107$	−18.9309	−1.83012	−0.915058	−	0.403322i	$-0.867855\pi$
$107$	−18.9309	−1.83012	−0.915058	+	0.403322i	$0.867855\pi$
$108$	0	0
$109$	−18.6847	−1.78967	−0.894833	−	0.446401i	$-0.852705\pi$
$109$	−18.6847	−1.78967	−0.894833	+	0.446401i	$0.852705\pi$
$110$	0	0
$111$	4.87689	0.462894
$112$	0	0
$113$	5.12311	0.481941	0.240971	−	0.970532i	$-0.422534\pi$
$113$	5.12311	0.481941	0.240971	+	0.970532i	$0.422534\pi$
$114$	0	0
$115$	−11.1231	−1.03723
$116$	0	0
$117$	2.00000	0.184900
$118$	0	0
$119$	−12.6847	−1.16280
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	3.12311	0.281601
$124$	0	0
$125$	−12.0000	−1.07331
$126$	0	0
$127$	14.2462	1.26415	0.632073	−	0.774909i	$-0.282204\pi$
$127$	14.2462	1.26415	0.632073	+	0.774909i	$0.282204\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	7.12311	0.622349	0.311174	−	0.950353i	$-0.399278\pi$
$131$	7.12311	0.622349	0.311174	+	0.950353i	$0.399278\pi$
$132$	0	0
$133$	−3.56155	−0.308826
$134$	0	0
$135$	−11.1231	−0.957325
$136$	0	0
$137$	8.43845	0.720945	0.360473	−	0.932770i	$-0.382615\pi$
$137$	8.43845	0.720945	0.360473	+	0.932770i	$0.382615\pi$
$138$	0	0
$139$	17.3693	1.47325	0.736623	−	0.676303i	$-0.236419\pi$
$139$	17.3693	1.47325	0.736623	+	0.676303i	$0.236419\pi$
$140$	0	0
$141$	−12.4924	−1.05205
$142$	0	0
$143$	3.56155	0.297832
$144$	0	0
$145$	13.3693	1.11026
$146$	0	0
$147$	8.87689	0.732154
$148$	0	0
$149$	−15.1231	−1.23893	−0.619467	−	0.785023i	$-0.712651\pi$
$149$	−15.1231	−1.23893	−0.619467	+	0.785023i	$0.712651\pi$
$150$	0	0
$151$	−4.00000	−0.325515	−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000	−0.325515	−0.162758	+	0.986666i	$0.552039\pi$
$152$	0	0
$153$	−2.00000	−0.161690
$154$	0	0
$155$	−4.00000	−0.321288
$156$	0	0
$157$	18.4924	1.47586	0.737928	−	0.674879i	$-0.235804\pi$
$157$	18.4924	1.47586	0.737928	+	0.674879i	$0.235804\pi$
$158$	0	0
$159$	−12.1922	−0.966907
$160$	0	0
$161$	19.8078	1.56107
$162$	0	0
$163$	13.3693	1.04717	0.523583	−	0.851975i	$-0.324595\pi$
$163$	13.3693	1.04717	0.523583	+	0.851975i	$0.324595\pi$
$164$	0	0
$165$	−3.12311	−0.243133
$166$	0	0
$167$	25.3693	1.96314	0.981568	−	0.191111i	$-0.0612092\pi$
$167$	25.3693	1.96314	0.981568	+	0.191111i	$0.0612092\pi$
$168$	0	0
$169$	−0.315342	−0.0242570
$170$	0	0
$171$	−0.561553	−0.0429430
$172$	0	0
$173$	18.0000	1.36851	0.684257	−	0.729241i	$-0.260127\pi$
$173$	18.0000	1.36851	0.684257	+	0.729241i	$0.260127\pi$
$174$	0	0
$175$	3.56155	0.269228
$176$	0	0
$177$	−7.31534	−0.549855
$178$	0	0
$179$	−22.2462	−1.66276	−0.831380	−	0.555704i	$-0.812449\pi$
$179$	−22.2462	−1.66276	−0.831380	+	0.555704i	$0.812449\pi$
$180$	0	0
$181$	−19.1231	−1.42141	−0.710705	−	0.703491i	$-0.751624\pi$
$181$	−19.1231	−1.42141	−0.710705	+	0.703491i	$0.751624\pi$
$182$	0	0
$183$	−16.0000	−1.18275
$184$	0	0
$185$	6.24621	0.459231
$186$	0	0
$187$	−3.56155	−0.260447
$188$	0	0
$189$	19.8078	1.44080
$190$	0	0
$191$	20.6847	1.49669	0.748345	−	0.663310i	$-0.230849\pi$
$191$	20.6847	1.49669	0.748345	+	0.663310i	$0.230849\pi$
$192$	0	0
$193$	−1.12311	−0.0808429	−0.0404215	−	0.999183i	$-0.512870\pi$
$193$	−1.12311	−0.0808429	−0.0404215	+	0.999183i	$0.512870\pi$
$194$	0	0
$195$	−11.1231	−0.796542
$196$	0	0
$197$	1.75379	0.124952	0.0624761	−	0.998046i	$-0.480100\pi$
$197$	1.75379	0.124952	0.0624761	+	0.998046i	$0.480100\pi$
$198$	0	0
$199$	0.684658	0.0485341	0.0242671	−	0.999706i	$-0.492275\pi$
$199$	0.684658	0.0485341	0.0242671	+	0.999706i	$0.492275\pi$
$200$	0	0
$201$	−7.31534	−0.515984
$202$	0	0
$203$	−23.8078	−1.67098
$204$	0	0
$205$	4.00000	0.279372
$206$	0	0
$207$	3.12311	0.217071
$208$	0	0
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	12.6847	0.873248	0.436624	−	0.899644i	$-0.356174\pi$
$211$	12.6847	0.873248	0.436624	+	0.899644i	$0.356174\pi$
$212$	0	0
$213$	−9.36932	−0.641975
$214$	0	0
$215$	0	0
$216$	0	0
$217$	7.12311	0.483548
$218$	0	0
$219$	4.19224	0.283285
$220$	0	0
$221$	−12.6847	−0.853262
$222$	0	0
$223$	24.7386	1.65662	0.828311	−	0.560269i	$-0.189302\pi$
$223$	24.7386	1.65662	0.828311	+	0.560269i	$0.189302\pi$
$224$	0	0
$225$	0.561553	0.0374369
$226$	0	0
$227$	17.1771	1.14008	0.570041	−	0.821616i	$-0.306927\pi$
$227$	17.1771	1.14008	0.570041	+	0.821616i	$0.306927\pi$
$228$	0	0
$229$	21.1231	1.39585	0.697927	−	0.716169i	$-0.254106\pi$
$229$	21.1231	1.39585	0.697927	+	0.716169i	$0.254106\pi$
$230$	0	0
$231$	5.56155	0.365923
$232$	0	0
$233$	20.7386	1.35863	0.679317	−	0.733845i	$-0.262276\pi$
$233$	20.7386	1.35863	0.679317	+	0.733845i	$0.262276\pi$
$234$	0	0
$235$	−16.0000	−1.04372
$236$	0	0
$237$	6.24621	0.405735
$238$	0	0
$239$	4.93087	0.318951	0.159476	−	0.987202i	$-0.449020\pi$
$239$	4.93087	0.318951	0.159476	+	0.987202i	$0.449020\pi$
$240$	0	0
$241$	21.6155	1.39238	0.696189	−	0.717858i	$-0.254877\pi$
$241$	21.6155	1.39238	0.696189	+	0.717858i	$0.254877\pi$
$242$	0	0
$243$	5.75379	0.369106
$244$	0	0
$245$	11.3693	0.726359
$246$	0	0
$247$	−3.56155	−0.226616
$248$	0	0
$249$	3.50758	0.222284
$250$	0	0
$251$	−8.87689	−0.560305	−0.280152	−	0.959956i	$-0.590385\pi$
$251$	−8.87689	−0.560305	−0.280152	+	0.959956i	$0.590385\pi$
$252$	0	0
$253$	5.56155	0.349652
$254$	0	0
$255$	11.1231	0.696556
$256$	0	0
$257$	−22.0000	−1.37232	−0.686161	−	0.727450i	$-0.740706\pi$
$257$	−22.0000	−1.37232	−0.686161	+	0.727450i	$0.740706\pi$
$258$	0	0
$259$	−11.1231	−0.691156
$260$	0	0
$261$	−3.75379	−0.232354
$262$	0	0
$263$	−17.1231	−1.05586	−0.527928	−	0.849289i	$-0.677031\pi$
$263$	−17.1231	−1.05586	−0.527928	+	0.849289i	$0.677031\pi$
$264$	0	0
$265$	−15.6155	−0.959254
$266$	0	0
$267$	−14.2462	−0.871854
$268$	0	0
$269$	11.1231	0.678188	0.339094	−	0.940753i	$-0.389880\pi$
$269$	11.1231	0.678188	0.339094	+	0.940753i	$0.389880\pi$
$270$	0	0
$271$	−13.3153	−0.808849	−0.404425	−	0.914571i	$-0.632528\pi$
$271$	−13.3153	−0.808849	−0.404425	+	0.914571i	$0.632528\pi$
$272$	0	0
$273$	19.8078	1.19882
$274$	0	0
$275$	1.00000	0.0603023
$276$	0	0
$277$	−24.4924	−1.47161	−0.735804	−	0.677195i	$-0.763195\pi$
$277$	−24.4924	−1.47161	−0.735804	+	0.677195i	$0.763195\pi$
$278$	0	0
$279$	1.12311	0.0672386
$280$	0	0
$281$	−0.630683	−0.0376234	−0.0188117	−	0.999823i	$-0.505988\pi$
$281$	−0.630683	−0.0376234	−0.0188117	+	0.999823i	$0.505988\pi$
$282$	0	0
$283$	13.3693	0.794723	0.397362	−	0.917662i	$-0.369926\pi$
$283$	13.3693	0.794723	0.397362	+	0.917662i	$0.369926\pi$
$284$	0	0
$285$	3.12311	0.184997
$286$	0	0
$287$	−7.12311	−0.420464
$288$	0	0
$289$	−4.31534	−0.253844
$290$	0	0
$291$	1.75379	0.102809
$292$	0	0
$293$	−24.9309	−1.45648	−0.728238	−	0.685324i	$-0.759661\pi$
$293$	−24.9309	−1.45648	−0.728238	+	0.685324i	$0.759661\pi$
$294$	0	0
$295$	−9.36932	−0.545503
$296$	0	0
$297$	5.56155	0.322714
$298$	0	0
$299$	19.8078	1.14551
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−23.6155	−1.35668
$304$	0	0
$305$	−20.4924	−1.17339
$306$	0	0
$307$	5.75379	0.328386	0.164193	−	0.986428i	$-0.447498\pi$
$307$	5.75379	0.328386	0.164193	+	0.986428i	$0.447498\pi$
$308$	0	0
$309$	−17.7538	−1.00998
$310$	0	0
$311$	−15.8078	−0.896376	−0.448188	−	0.893939i	$-0.647930\pi$
$311$	−15.8078	−0.896376	−0.448188	+	0.893939i	$0.647930\pi$
$312$	0	0
$313$	3.56155	0.201311	0.100655	−	0.994921i	$-0.467906\pi$
$313$	3.56155	0.201311	0.100655	+	0.994921i	$0.467906\pi$
$314$	0	0
$315$	4.00000	0.225374
$316$	0	0
$317$	30.0540	1.68800	0.844000	−	0.536344i	$-0.180195\pi$
$317$	30.0540	1.68800	0.844000	+	0.536344i	$0.180195\pi$
$318$	0	0
$319$	−6.68466	−0.374269
$320$	0	0
$321$	−29.5616	−1.64996
$322$	0	0
$323$	3.56155	0.198170
$324$	0	0
$325$	3.56155	0.197559
$326$	0	0
$327$	−29.1771	−1.61350
$328$	0	0
$329$	28.4924	1.57084
$330$	0	0
$331$	10.4384	0.573749	0.286874	−	0.957968i	$-0.407384\pi$
$331$	10.4384	0.573749	0.286874	+	0.957968i	$0.407384\pi$
$332$	0	0
$333$	−1.75379	−0.0961070
$334$	0	0
$335$	−9.36932	−0.511900
$336$	0	0
$337$	2.87689	0.156714	0.0783572	−	0.996925i	$-0.475033\pi$
$337$	2.87689	0.156714	0.0783572	+	0.996925i	$0.475033\pi$
$338$	0	0
$339$	8.00000	0.434500
$340$	0	0
$341$	2.00000	0.108306
$342$	0	0
$343$	4.68466	0.252948
$344$	0	0
$345$	−17.3693	−0.935133
$346$	0	0
$347$	31.6155	1.69721	0.848605	−	0.529027i	$-0.177443\pi$
$347$	31.6155	1.69721	0.848605	+	0.529027i	$0.177443\pi$
$348$	0	0
$349$	−19.6155	−1.05000	−0.524998	−	0.851104i	$-0.675934\pi$
$349$	−19.6155	−1.05000	−0.524998	+	0.851104i	$0.675934\pi$
$350$	0	0
$351$	19.8078	1.05726
$352$	0	0
$353$	−23.1771	−1.23359	−0.616796	−	0.787123i	$-0.711570\pi$
$353$	−23.1771	−1.23359	−0.616796	+	0.787123i	$0.711570\pi$
$354$	0	0
$355$	−12.0000	−0.636894
$356$	0	0
$357$	−19.8078	−1.04834
$358$	0	0
$359$	−12.9309	−0.682465	−0.341233	−	0.939979i	$-0.610844\pi$
$359$	−12.9309	−0.682465	−0.341233	+	0.939979i	$0.610844\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	1.56155	0.0819603
$364$	0	0
$365$	5.36932	0.281043
$366$	0	0
$367$	−13.7538	−0.717942	−0.358971	−	0.933349i	$-0.616872\pi$
$367$	−13.7538	−0.717942	−0.358971	+	0.933349i	$0.616872\pi$
$368$	0	0
$369$	−1.12311	−0.0584665
$370$	0	0
$371$	27.8078	1.44371
$372$	0	0
$373$	7.56155	0.391522	0.195761	−	0.980652i	$-0.437282\pi$
$373$	7.56155	0.391522	0.195761	+	0.980652i	$0.437282\pi$
$374$	0	0
$375$	−18.7386	−0.967659
$376$	0	0
$377$	−23.8078	−1.22616
$378$	0	0
$379$	−2.43845	−0.125255	−0.0626273	−	0.998037i	$-0.519948\pi$
$379$	−2.43845	−0.125255	−0.0626273	+	0.998037i	$0.519948\pi$
$380$	0	0
$381$	22.2462	1.13971
$382$	0	0
$383$	2.00000	0.102195	0.0510976	−	0.998694i	$-0.483728\pi$
$383$	2.00000	0.102195	0.0510976	+	0.998694i	$0.483728\pi$
$384$	0	0
$385$	7.12311	0.363027
$386$	0	0
$387$	0	0
$388$	0	0
$389$	18.0000	0.912636	0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000	0.912636	0.456318	+	0.889817i	$0.349168\pi$
$390$	0	0
$391$	−19.8078	−1.00172
$392$	0	0
$393$	11.1231	0.561086
$394$	0	0
$395$	8.00000	0.402524
$396$	0	0
$397$	−26.9848	−1.35433	−0.677165	−	0.735831i	$-0.736792\pi$
$397$	−26.9848	−1.35433	−0.677165	+	0.735831i	$0.736792\pi$
$398$	0	0
$399$	−5.56155	−0.278426
$400$	0	0
$401$	−24.2462	−1.21080	−0.605399	−	0.795922i	$-0.706986\pi$
$401$	−24.2462	−1.21080	−0.605399	+	0.795922i	$0.706986\pi$
$402$	0	0
$403$	7.12311	0.354827
$404$	0	0
$405$	−14.0000	−0.695666
$406$	0	0
$407$	−3.12311	−0.154807
$408$	0	0
$409$	−0.246211	−0.0121744	−0.00608718	−	0.999981i	$-0.501938\pi$
$409$	−0.246211	−0.0121744	−0.00608718	+	0.999981i	$0.501938\pi$
$410$	0	0
$411$	13.1771	0.649977
$412$	0	0
$413$	16.6847	0.820998
$414$	0	0
$415$	4.49242	0.220524
$416$	0	0
$417$	27.1231	1.32822
$418$	0	0
$419$	23.1231	1.12964	0.564819	−	0.825215i	$-0.308946\pi$
$419$	23.1231	1.12964	0.564819	+	0.825215i	$0.308946\pi$
$420$	0	0
$421$	−1.06913	−0.0521062	−0.0260531	−	0.999661i	$-0.508294\pi$
$421$	−1.06913	−0.0521062	−0.0260531	+	0.999661i	$0.508294\pi$
$422$	0	0
$423$	4.49242	0.218429
$424$	0	0
$425$	−3.56155	−0.172761
$426$	0	0
$427$	36.4924	1.76599
$428$	0	0
$429$	5.56155	0.268514
$430$	0	0
$431$	25.8617	1.24572	0.622858	−	0.782335i	$-0.285971\pi$
$431$	25.8617	1.24572	0.622858	+	0.782335i	$0.285971\pi$
$432$	0	0
$433$	31.3693	1.50751	0.753757	−	0.657154i	$-0.228240\pi$
$433$	31.3693	1.50751	0.753757	+	0.657154i	$0.228240\pi$
$434$	0	0
$435$	20.8769	1.00097
$436$	0	0
$437$	−5.56155	−0.266045
$438$	0	0
$439$	−31.6155	−1.50893	−0.754463	−	0.656342i	$-0.772103\pi$
$439$	−31.6155	−1.50893	−0.754463	+	0.656342i	$0.772103\pi$
$440$	0	0
$441$	−3.19224	−0.152011
$442$	0	0
$443$	17.8617	0.848637	0.424318	−	0.905513i	$-0.360514\pi$
$443$	17.8617	0.848637	0.424318	+	0.905513i	$0.360514\pi$
$444$	0	0
$445$	−18.2462	−0.864953
$446$	0	0
$447$	−23.6155	−1.11698
$448$	0	0
$449$	−17.6155	−0.831328	−0.415664	−	0.909518i	$-0.636451\pi$
$449$	−17.6155	−0.831328	−0.415664	+	0.909518i	$0.636451\pi$
$450$	0	0
$451$	−2.00000	−0.0941763
$452$	0	0
$453$	−6.24621	−0.293473
$454$	0	0
$455$	25.3693	1.18933
$456$	0	0
$457$	32.4384	1.51741	0.758703	−	0.651436i	$-0.225833\pi$
$457$	32.4384	1.51741	0.758703	+	0.651436i	$0.225833\pi$
$458$	0	0
$459$	−19.8078	−0.924547
$460$	0	0
$461$	−22.7386	−1.05904	−0.529522	−	0.848296i	$-0.677629\pi$
$461$	−22.7386	−1.05904	−0.529522	+	0.848296i	$0.677629\pi$
$462$	0	0
$463$	−36.4924	−1.69595	−0.847973	−	0.530039i	$-0.822177\pi$
$463$	−36.4924	−1.69595	−0.847973	+	0.530039i	$0.822177\pi$
$464$	0	0
$465$	−6.24621	−0.289661
$466$	0	0
$467$	−26.2462	−1.21453	−0.607265	−	0.794499i	$-0.707733\pi$
$467$	−26.2462	−1.21453	−0.607265	+	0.794499i	$0.707733\pi$
$468$	0	0
$469$	16.6847	0.770426
$470$	0	0
$471$	28.8769	1.33058
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	4.38447	0.200751
$478$	0	0
$479$	−11.3693	−0.519477	−0.259739	−	0.965679i	$-0.583636\pi$
$479$	−11.3693	−0.519477	−0.259739	+	0.965679i	$0.583636\pi$
$480$	0	0
$481$	−11.1231	−0.507170
$482$	0	0
$483$	30.9309	1.40740
$484$	0	0
$485$	2.24621	0.101995
$486$	0	0
$487$	−22.8769	−1.03665	−0.518326	−	0.855183i	$-0.673444\pi$
$487$	−22.8769	−1.03665	−0.518326	+	0.855183i	$0.673444\pi$
$488$	0	0
$489$	20.8769	0.944086
$490$	0	0
$491$	14.6307	0.660273	0.330137	−	0.943933i	$-0.392905\pi$
$491$	14.6307	0.660273	0.330137	+	0.943933i	$0.392905\pi$
$492$	0	0
$493$	23.8078	1.07225
$494$	0	0
$495$	1.12311	0.0504798
$496$	0	0
$497$	21.3693	0.958545
$498$	0	0
$499$	−26.2462	−1.17494	−0.587471	−	0.809245i	$-0.699876\pi$
$499$	−26.2462	−1.17494	−0.587471	+	0.809245i	$0.699876\pi$
$500$	0	0
$501$	39.6155	1.76989
$502$	0	0
$503$	−9.31534	−0.415351	−0.207675	−	0.978198i	$-0.566590\pi$
$503$	−9.31534	−0.415351	−0.207675	+	0.978198i	$0.566590\pi$
$504$	0	0
$505$	−30.2462	−1.34594
$506$	0	0
$507$	−0.492423	−0.0218693
$508$	0	0
$509$	−6.63068	−0.293900	−0.146950	−	0.989144i	$-0.546946\pi$
$509$	−6.63068	−0.293900	−0.146950	+	0.989144i	$0.546946\pi$
$510$	0	0
$511$	−9.56155	−0.422978
$512$	0	0
$513$	−5.56155	−0.245549
$514$	0	0
$515$	−22.7386	−1.00198
$516$	0	0
$517$	8.00000	0.351840
$518$	0	0
$519$	28.1080	1.23380
$520$	0	0
$521$	1.12311	0.0492042	0.0246021	−	0.999697i	$-0.492168\pi$
$521$	1.12311	0.0492042	0.0246021	+	0.999697i	$0.492168\pi$
$522$	0	0
$523$	6.05398	0.264722	0.132361	−	0.991202i	$-0.457744\pi$
$523$	6.05398	0.264722	0.132361	+	0.991202i	$0.457744\pi$
$524$	0	0
$525$	5.56155	0.242726
$526$	0	0
$527$	−7.12311	−0.310287
$528$	0	0
$529$	7.93087	0.344820
$530$	0	0
$531$	2.63068	0.114162
$532$	0	0
$533$	−7.12311	−0.308536
$534$	0	0
$535$	−37.8617	−1.63691
$536$	0	0
$537$	−34.7386	−1.49908
$538$	0	0
$539$	−5.68466	−0.244856
$540$	0	0
$541$	43.2311	1.85865	0.929324	−	0.369265i	$-0.120391\pi$
$541$	43.2311	1.85865	0.929324	+	0.369265i	$0.120391\pi$
$542$	0	0
$543$	−29.8617	−1.28149
$544$	0	0
$545$	−37.3693	−1.60073
$546$	0	0
$547$	−6.73863	−0.288123	−0.144062	−	0.989569i	$-0.546016\pi$
$547$	−6.73863	−0.288123	−0.144062	+	0.989569i	$0.546016\pi$
$548$	0	0
$549$	5.75379	0.245566
$550$	0	0
$551$	6.68466	0.284776
$552$	0	0
$553$	−14.2462	−0.605811
$554$	0	0
$555$	9.75379	0.414025
$556$	0	0
$557$	8.87689	0.376126	0.188063	−	0.982157i	$-0.439779\pi$
$557$	8.87689	0.376126	0.188063	+	0.982157i	$0.439779\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−5.56155	−0.234809
$562$	0	0
$563$	−6.73863	−0.284000	−0.142000	−	0.989867i	$-0.545353\pi$
$563$	−6.73863	−0.284000	−0.142000	+	0.989867i	$0.545353\pi$
$564$	0	0
$565$	10.2462	0.431061
$566$	0	0
$567$	24.9309	1.04700
$568$	0	0
$569$	3.36932	0.141249	0.0706246	−	0.997503i	$-0.477501\pi$
$569$	3.36932	0.141249	0.0706246	+	0.997503i	$0.477501\pi$
$570$	0	0
$571$	−35.6155	−1.49046	−0.745232	−	0.666806i	$-0.767661\pi$
$571$	−35.6155	−1.49046	−0.745232	+	0.666806i	$0.767661\pi$
$572$	0	0
$573$	32.3002	1.34936
$574$	0	0
$575$	5.56155	0.231933
$576$	0	0
$577$	3.94602	0.164275	0.0821376	−	0.996621i	$-0.473825\pi$
$577$	3.94602	0.164275	0.0821376	+	0.996621i	$0.473825\pi$
$578$	0	0
$579$	−1.75379	−0.0728850
$580$	0	0
$581$	−8.00000	−0.331896
$582$	0	0
$583$	7.80776	0.323365
$584$	0	0
$585$	4.00000	0.165380
$586$	0	0
$587$	7.50758	0.309871	0.154935	−	0.987925i	$-0.450483\pi$
$587$	7.50758	0.309871	0.154935	+	0.987925i	$0.450483\pi$
$588$	0	0
$589$	−2.00000	−0.0824086
$590$	0	0
$591$	2.73863	0.112652
$592$	0	0
$593$	34.0000	1.39621	0.698106	−	0.715994i	$-0.254026\pi$
$593$	34.0000	1.39621	0.698106	+	0.715994i	$0.254026\pi$
$594$	0	0
$595$	−25.3693	−1.04004
$596$	0	0
$597$	1.06913	0.0437566
$598$	0	0
$599$	−39.8617	−1.62871	−0.814353	−	0.580370i	$-0.802908\pi$
$599$	−39.8617	−1.62871	−0.814353	+	0.580370i	$0.802908\pi$
$600$	0	0
$601$	−3.36932	−0.137437	−0.0687187	−	0.997636i	$-0.521891\pi$
$601$	−3.36932	−0.137437	−0.0687187	+	0.997636i	$0.521891\pi$
$602$	0	0
$603$	2.63068	0.107130
$604$	0	0
$605$	2.00000	0.0813116
$606$	0	0
$607$	−27.6155	−1.12088	−0.560440	−	0.828195i	$-0.689368\pi$
$607$	−27.6155	−1.12088	−0.560440	+	0.828195i	$0.689368\pi$
$608$	0	0
$609$	−37.1771	−1.50649
$610$	0	0
$611$	28.4924	1.15268
$612$	0	0
$613$	1.75379	0.0708349	0.0354174	−	0.999373i	$-0.488724\pi$
$613$	1.75379	0.0708349	0.0354174	+	0.999373i	$0.488724\pi$
$614$	0	0
$615$	6.24621	0.251872
$616$	0	0
$617$	−4.73863	−0.190770	−0.0953851	−	0.995440i	$-0.530408\pi$
$617$	−4.73863	−0.190770	−0.0953851	+	0.995440i	$0.530408\pi$
$618$	0	0
$619$	26.2462	1.05492	0.527462	−	0.849579i	$-0.323144\pi$
$619$	26.2462	1.05492	0.527462	+	0.849579i	$0.323144\pi$
$620$	0	0
$621$	30.9309	1.24121
$622$	0	0
$623$	32.4924	1.30178
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	−1.56155	−0.0623624
$628$	0	0
$629$	11.1231	0.443507
$630$	0	0
$631$	−36.4924	−1.45274	−0.726370	−	0.687304i	$-0.758794\pi$
$631$	−36.4924	−1.45274	−0.726370	+	0.687304i	$0.758794\pi$
$632$	0	0
$633$	19.8078	0.787288
$634$	0	0
$635$	28.4924	1.13069
$636$	0	0
$637$	−20.2462	−0.802184
$638$	0	0
$639$	3.36932	0.133288
$640$	0	0
$641$	−9.61553	−0.379791	−0.189895	−	0.981804i	$-0.560815\pi$
$641$	−9.61553	−0.379791	−0.189895	+	0.981804i	$0.560815\pi$
$642$	0	0
$643$	37.3693	1.47370	0.736851	−	0.676055i	$-0.236312\pi$
$643$	37.3693	1.47370	0.736851	+	0.676055i	$0.236312\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−22.5464	−0.886390	−0.443195	−	0.896425i	$-0.646155\pi$
$647$	−22.5464	−0.886390	−0.443195	+	0.896425i	$0.646155\pi$
$648$	0	0
$649$	4.68466	0.183889
$650$	0	0
$651$	11.1231	0.435949
$652$	0	0
$653$	25.6155	1.00241	0.501207	−	0.865328i	$-0.332890\pi$
$653$	25.6155	1.00241	0.501207	+	0.865328i	$0.332890\pi$
$654$	0	0
$655$	14.2462	0.556646
$656$	0	0
$657$	−1.50758	−0.0588162
$658$	0	0
$659$	−47.4233	−1.84735	−0.923675	−	0.383178i	$-0.874830\pi$
$659$	−47.4233	−1.84735	−0.923675	+	0.383178i	$0.874830\pi$
$660$	0	0
$661$	−22.4384	−0.872754	−0.436377	−	0.899764i	$-0.643739\pi$
$661$	−22.4384	−0.872754	−0.436377	+	0.899764i	$0.643739\pi$
$662$	0	0
$663$	−19.8078	−0.769270
$664$	0	0
$665$	−7.12311	−0.276222
$666$	0	0
$667$	−37.1771	−1.43950
$668$	0	0
$669$	38.6307	1.49355
$670$	0	0
$671$	10.2462	0.395551
$672$	0	0
$673$	−18.8769	−0.727651	−0.363825	−	0.931467i	$-0.618530\pi$
$673$	−18.8769	−0.727651	−0.363825	+	0.931467i	$0.618530\pi$
$674$	0	0
$675$	5.56155	0.214064
$676$	0	0
$677$	−39.1771	−1.50570	−0.752849	−	0.658194i	$-0.771321\pi$
$677$	−39.1771	−1.50570	−0.752849	+	0.658194i	$0.771321\pi$
$678$	0	0
$679$	−4.00000	−0.153506
$680$	0	0
$681$	26.8229	1.02786
$682$	0	0
$683$	4.49242	0.171898	0.0859489	−	0.996300i	$-0.472608\pi$
$683$	4.49242	0.171898	0.0859489	+	0.996300i	$0.472608\pi$
$684$	0	0
$685$	16.8769	0.644833
$686$	0	0
$687$	32.9848	1.25845
$688$	0	0
$689$	27.8078	1.05939
$690$	0	0
$691$	43.6155	1.65921	0.829606	−	0.558349i	$-0.188565\pi$
$691$	43.6155	1.65921	0.829606	+	0.558349i	$0.188565\pi$
$692$	0	0
$693$	−2.00000	−0.0759737
$694$	0	0
$695$	34.7386	1.31771
$696$	0	0
$697$	7.12311	0.269807
$698$	0	0
$699$	32.3845	1.22489
$700$	0	0
$701$	24.9848	0.943665	0.471832	−	0.881688i	$-0.343593\pi$
$701$	24.9848	0.943665	0.471832	+	0.881688i	$0.343593\pi$
$702$	0	0
$703$	3.12311	0.117790
$704$	0	0
$705$	−24.9848	−0.940984
$706$	0	0
$707$	53.8617	2.02568
$708$	0	0
$709$	−9.50758	−0.357065	−0.178532	−	0.983934i	$-0.557135\pi$
$709$	−9.50758	−0.357065	−0.178532	+	0.983934i	$0.557135\pi$
$710$	0	0
$711$	−2.24621	−0.0842395
$712$	0	0
$713$	11.1231	0.416564
$714$	0	0
$715$	7.12311	0.266389
$716$	0	0
$717$	7.69981	0.287555
$718$	0	0
$719$	6.43845	0.240114	0.120057	−	0.992767i	$-0.461692\pi$
$719$	6.43845	0.240114	0.120057	+	0.992767i	$0.461692\pi$
$720$	0	0
$721$	40.4924	1.50802
$722$	0	0
$723$	33.7538	1.25532
$724$	0	0
$725$	−6.68466	−0.248262
$726$	0	0
$727$	39.4233	1.46213	0.731064	−	0.682308i	$-0.239024\pi$
$727$	39.4233	1.46213	0.731064	+	0.682308i	$0.239024\pi$
$728$	0	0
$729$	29.9848	1.11055
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−48.1080	−1.77691	−0.888454	−	0.458966i	$-0.848220\pi$
$733$	−48.1080	−1.77691	−0.888454	+	0.458966i	$0.848220\pi$
$734$	0	0
$735$	17.7538	0.654858
$736$	0	0
$737$	4.68466	0.172562
$738$	0	0
$739$	−20.9848	−0.771940	−0.385970	−	0.922511i	$-0.626133\pi$
$739$	−20.9848	−0.771940	−0.385970	+	0.922511i	$0.626133\pi$
$740$	0	0
$741$	−5.56155	−0.204309
$742$	0	0
$743$	25.7538	0.944815	0.472407	−	0.881380i	$-0.343385\pi$
$743$	25.7538	0.944815	0.472407	+	0.881380i	$0.343385\pi$
$744$	0	0
$745$	−30.2462	−1.10814
$746$	0	0
$747$	−1.26137	−0.0461510
$748$	0	0
$749$	67.4233	2.46359
$750$	0	0
$751$	41.2311	1.50454	0.752271	−	0.658853i	$-0.228958\pi$
$751$	41.2311	1.50454	0.752271	+	0.658853i	$0.228958\pi$
$752$	0	0
$753$	−13.8617	−0.505150
$754$	0	0
$755$	−8.00000	−0.291150
$756$	0	0
$757$	−9.50758	−0.345559	−0.172779	−	0.984961i	$-0.555275\pi$
$757$	−9.50758	−0.345559	−0.172779	+	0.984961i	$0.555275\pi$
$758$	0	0
$759$	8.68466	0.315233
$760$	0	0
$761$	−22.1922	−0.804468	−0.402234	−	0.915537i	$-0.631766\pi$
$761$	−22.1922	−0.804468	−0.402234	+	0.915537i	$0.631766\pi$
$762$	0	0
$763$	66.5464	2.40914
$764$	0	0
$765$	−4.00000	−0.144620
$766$	0	0
$767$	16.6847	0.602448
$768$	0	0
$769$	−1.31534	−0.0474324	−0.0237162	−	0.999719i	$-0.507550\pi$
$769$	−1.31534	−0.0474324	−0.0237162	+	0.999719i	$0.507550\pi$
$770$	0	0
$771$	−34.3542	−1.23723
$772$	0	0
$773$	9.06913	0.326194	0.163097	−	0.986610i	$-0.447852\pi$
$773$	9.06913	0.326194	0.163097	+	0.986610i	$0.447852\pi$
$774$	0	0
$775$	2.00000	0.0718421
$776$	0	0
$777$	−17.3693	−0.623121
$778$	0	0
$779$	2.00000	0.0716574
$780$	0	0
$781$	6.00000	0.214697
$782$	0	0
$783$	−37.1771	−1.32860
$784$	0	0
$785$	36.9848	1.32005
$786$	0	0
$787$	−3.31534	−0.118179	−0.0590896	−	0.998253i	$-0.518820\pi$
$787$	−3.31534	−0.118179	−0.0590896	+	0.998253i	$0.518820\pi$
$788$	0	0
$789$	−26.7386	−0.951921
$790$	0	0
$791$	−18.2462	−0.648761
$792$	0	0
$793$	36.4924	1.29588
$794$	0	0
$795$	−24.3845	−0.864828
$796$	0	0
$797$	−35.8078	−1.26838	−0.634188	−	0.773179i	$-0.718665\pi$
$797$	−35.8078	−1.26838	−0.634188	+	0.773179i	$0.718665\pi$
$798$	0	0
$799$	−28.4924	−1.00799
$800$	0	0
$801$	5.12311	0.181016
$802$	0	0
$803$	−2.68466	−0.0947395
$804$	0	0
$805$	39.6155	1.39626
$806$	0	0
$807$	17.3693	0.611429
$808$	0	0
$809$	36.9309	1.29842	0.649210	−	0.760609i	$-0.275100\pi$
$809$	36.9309	1.29842	0.649210	+	0.760609i	$0.275100\pi$
$810$	0	0
$811$	−20.6847	−0.726337	−0.363168	−	0.931724i	$-0.618305\pi$
$811$	−20.6847	−0.726337	−0.363168	+	0.931724i	$0.618305\pi$
$812$	0	0
$813$	−20.7926	−0.729229
$814$	0	0
$815$	26.7386	0.936613
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−7.12311	−0.248901
$820$	0	0
$821$	44.9848	1.56998	0.784991	−	0.619507i	$-0.212668\pi$
$821$	44.9848	1.56998	0.784991	+	0.619507i	$0.212668\pi$
$822$	0	0
$823$	−14.9309	−0.520457	−0.260229	−	0.965547i	$-0.583798\pi$
$823$	−14.9309	−0.520457	−0.260229	+	0.965547i	$0.583798\pi$
$824$	0	0
$825$	1.56155	0.0543663
$826$	0	0
$827$	−21.0691	−0.732645	−0.366323	−	0.930488i	$-0.619383\pi$
$827$	−21.0691	−0.732645	−0.366323	+	0.930488i	$0.619383\pi$
$828$	0	0
$829$	−33.1771	−1.15229	−0.576144	−	0.817348i	$-0.695443\pi$
$829$	−33.1771	−1.15229	−0.576144	+	0.817348i	$0.695443\pi$
$830$	0	0
$831$	−38.2462	−1.32675
$832$	0	0
$833$	20.2462	0.701490
$834$	0	0
$835$	50.7386	1.75588
$836$	0	0
$837$	11.1231	0.384471
$838$	0	0
$839$	−1.12311	−0.0387739	−0.0193870	−	0.999812i	$-0.506171\pi$
$839$	−1.12311	−0.0387739	−0.0193870	+	0.999812i	$0.506171\pi$
$840$	0	0
$841$	15.6847	0.540850
$842$	0	0
$843$	−0.984845	−0.0339199
$844$	0	0
$845$	−0.630683	−0.0216962
$846$	0	0
$847$	−3.56155	−0.122376
$848$	0	0
$849$	20.8769	0.716493
$850$	0	0
$851$	−17.3693	−0.595413
$852$	0	0
$853$	38.3542	1.31322	0.656611	−	0.754230i	$-0.271989\pi$
$853$	38.3542	1.31322	0.656611	+	0.754230i	$0.271989\pi$
$854$	0	0
$855$	−1.12311	−0.0384094
$856$	0	0
$857$	−10.0000	−0.341593	−0.170797	−	0.985306i	$-0.554634\pi$
$857$	−10.0000	−0.341593	−0.170797	+	0.985306i	$0.554634\pi$
$858$	0	0
$859$	41.8617	1.42830	0.714152	−	0.699991i	$-0.246812\pi$
$859$	41.8617	1.42830	0.714152	+	0.699991i	$0.246812\pi$
$860$	0	0
$861$	−11.1231	−0.379074
$862$	0	0
$863$	16.2462	0.553027	0.276514	−	0.961010i	$-0.410821\pi$
$863$	16.2462	0.553027	0.276514	+	0.961010i	$0.410821\pi$
$864$	0	0
$865$	36.0000	1.22404
$866$	0	0
$867$	−6.73863	−0.228856
$868$	0	0
$869$	−4.00000	−0.135691
$870$	0	0
$871$	16.6847	0.565338
$872$	0	0
$873$	−0.630683	−0.0213454
$874$	0	0
$875$	42.7386	1.44483
$876$	0	0
$877$	−40.4384	−1.36551	−0.682755	−	0.730648i	$-0.739218\pi$
$877$	−40.4384	−1.36551	−0.682755	+	0.730648i	$0.739218\pi$
$878$	0	0
$879$	−38.9309	−1.31311
$880$	0	0
$881$	−28.7386	−0.968229	−0.484115	−	0.875005i	$-0.660858\pi$
$881$	−28.7386	−0.968229	−0.484115	+	0.875005i	$0.660858\pi$
$882$	0	0
$883$	54.7386	1.84210	0.921051	−	0.389442i	$-0.127332\pi$
$883$	54.7386	1.84210	0.921051	+	0.389442i	$0.127332\pi$
$884$	0	0
$885$	−14.6307	−0.491805
$886$	0	0
$887$	32.0000	1.07445	0.537227	−	0.843437i	$-0.319472\pi$
$887$	32.0000	1.07445	0.537227	+	0.843437i	$0.319472\pi$
$888$	0	0
$889$	−50.7386	−1.70172
$890$	0	0
$891$	7.00000	0.234509
$892$	0	0
$893$	−8.00000	−0.267710
$894$	0	0
$895$	−44.4924	−1.48722
$896$	0	0
$897$	30.9309	1.03275
$898$	0	0
$899$	−13.3693	−0.445892
$900$	0	0
$901$	−27.8078	−0.926411
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−38.2462	−1.27135
$906$	0	0
$907$	−56.7926	−1.88577	−0.942884	−	0.333122i	$-0.891898\pi$
$907$	−56.7926	−1.88577	−0.942884	+	0.333122i	$0.891898\pi$
$908$	0	0
$909$	8.49242	0.281676
$910$	0	0
$911$	46.9848	1.55668	0.778339	−	0.627845i	$-0.216063\pi$
$911$	46.9848	1.55668	0.778339	+	0.627845i	$0.216063\pi$
$912$	0	0
$913$	−2.24621	−0.0743387
$914$	0	0
$915$	−32.0000	−1.05789
$916$	0	0
$917$	−25.3693	−0.837769
$918$	0	0
$919$	−28.4384	−0.938098	−0.469049	−	0.883172i	$-0.655403\pi$
$919$	−28.4384	−0.938098	−0.469049	+	0.883172i	$0.655403\pi$
$920$	0	0
$921$	8.98485	0.296061
$922$	0	0
$923$	21.3693	0.703380
$924$	0	0
$925$	−3.12311	−0.102687
$926$	0	0
$927$	6.38447	0.209694
$928$	0	0
$929$	−16.9309	−0.555484	−0.277742	−	0.960656i	$-0.589586\pi$
$929$	−16.9309	−0.555484	−0.277742	+	0.960656i	$0.589586\pi$
$930$	0	0
$931$	5.68466	0.186307
$932$	0	0
$933$	−24.6847	−0.808139
$934$	0	0
$935$	−7.12311	−0.232950
$936$	0	0
$937$	−54.3002	−1.77391	−0.886955	−	0.461856i	$-0.847184\pi$
$937$	−54.3002	−1.77391	−0.886955	+	0.461856i	$0.847184\pi$
$938$	0	0
$939$	5.56155	0.181494
$940$	0	0
$941$	20.4384	0.666274	0.333137	−	0.942878i	$-0.391893\pi$
$941$	20.4384	0.666274	0.333137	+	0.942878i	$0.391893\pi$
$942$	0	0
$943$	−11.1231	−0.362218
$944$	0	0
$945$	39.6155	1.28869
$946$	0	0
$947$	−28.9848	−0.941881	−0.470940	−	0.882165i	$-0.656085\pi$
$947$	−28.9848	−0.941881	−0.470940	+	0.882165i	$0.656085\pi$
$948$	0	0
$949$	−9.56155	−0.310381
$950$	0	0
$951$	46.9309	1.52184
$952$	0	0
$953$	−9.12311	−0.295526	−0.147763	−	0.989023i	$-0.547207\pi$
$953$	−9.12311	−0.295526	−0.147763	+	0.989023i	$0.547207\pi$
$954$	0	0
$955$	41.3693	1.33868
$956$	0	0
$957$	−10.4384	−0.337427
$958$	0	0
$959$	−30.0540	−0.970493
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	10.6307	0.342569
$964$	0	0
$965$	−2.24621	−0.0723081
$966$	0	0
$967$	3.86174	0.124185	0.0620926	−	0.998070i	$-0.480223\pi$
$967$	3.86174	0.124185	0.0620926	+	0.998070i	$0.480223\pi$
$968$	0	0
$969$	5.56155	0.178663
$970$	0	0
$971$	14.2462	0.457183	0.228591	−	0.973522i	$-0.426588\pi$
$971$	14.2462	0.457183	0.228591	+	0.973522i	$0.426588\pi$
$972$	0	0
$973$	−61.8617	−1.98320
$974$	0	0
$975$	5.56155	0.178112
$976$	0	0
$977$	55.3693	1.77142	0.885711	−	0.464238i	$-0.153672\pi$
$977$	55.3693	1.77142	0.885711	+	0.464238i	$0.153672\pi$
$978$	0	0
$979$	9.12311	0.291576
$980$	0	0
$981$	10.4924	0.334997
$982$	0	0
$983$	48.2462	1.53882	0.769408	−	0.638758i	$-0.220552\pi$
$983$	48.2462	1.53882	0.769408	+	0.638758i	$0.220552\pi$
$984$	0	0
$985$	3.50758	0.111761
$986$	0	0
$987$	44.4924	1.41621
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−21.1231	−0.670998	−0.335499	−	0.942041i	$-0.608905\pi$
$991$	−21.1231	−0.670998	−0.335499	+	0.942041i	$0.608905\pi$
$992$	0	0
$993$	16.3002	0.517271
$994$	0	0
$995$	1.36932	0.0434103
$996$	0	0
$997$	−53.3693	−1.69022	−0.845112	−	0.534590i	$-0.820466\pi$
$997$	−53.3693	−1.69022	−0.845112	+	0.534590i	$0.820466\pi$
$998$	0	0
$999$	−17.3693	−0.549541

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.a.k.1.2		2
4.3	odd	2		418.2.a.e.1.1	✓	2
12.11	even	2		3762.2.a.y.1.2		2
44.43	even	2		4598.2.a.bj.1.1		2
76.75	even	2		7942.2.a.x.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
418.2.a.e.1.1	✓	2	4.3	odd	2
3344.2.a.k.1.2		2	1.1	even	1	trivial
3762.2.a.y.1.2		2	12.11	even	2
4598.2.a.bj.1.1		2	44.43	even	2
7942.2.a.x.1.2		2	76.75	even	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:	\( N \)	\(=\)	\( 3344 = 2^{4} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3344.a (trivial)

\(f(q)\)	\(=\)	\(q+1.56155 q^{3} +2.00000 q^{5} -3.56155 q^{7} -0.561553 q^{9} +O(q^{10})\)	\(q+1.56155 q^{3} +2.00000 q^{5} -3.56155 q^{7} -0.561553 q^{9} -1.00000 q^{11} -3.56155 q^{13} +3.12311 q^{15} +3.56155 q^{17} +1.00000 q^{19} -5.56155 q^{21} -5.56155 q^{23} -1.00000 q^{25} -5.56155 q^{27} +6.68466 q^{29} -2.00000 q^{31} -1.56155 q^{33} -7.12311 q^{35} +3.12311 q^{37} -5.56155 q^{39} +2.00000 q^{41} -1.12311 q^{45} -8.00000 q^{47} +5.68466 q^{49} +5.56155 q^{51} -7.80776 q^{53} -2.00000 q^{55} +1.56155 q^{57} -4.68466 q^{59} -10.2462 q^{61} +2.00000 q^{63} -7.12311 q^{65} -4.68466 q^{67} -8.68466 q^{69} -6.00000 q^{71} +2.68466 q^{73} -1.56155 q^{75} +3.56155 q^{77} +4.00000 q^{79} -7.00000 q^{81} +2.24621 q^{83} +7.12311 q^{85} +10.4384 q^{87} -9.12311 q^{89} +12.6847 q^{91} -3.12311 q^{93} +2.00000 q^{95} +1.12311 q^{97} +0.561553 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} + 4 q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) \) 2 * q - q^3 + 4 * q^5 - 3 * q^7 + 3 * q^9	\( 2 q - q^{3} + 4 q^{5} - 3 q^{7} + 3 q^{9} - 2 q^{11} - 3 q^{13} - 2 q^{15} + 3 q^{17} + 2 q^{19} - 7 q^{21} - 7 q^{23} - 2 q^{25} - 7 q^{27} + q^{29} - 4 q^{31} + q^{33} - 6 q^{35} - 2 q^{37} - 7 q^{39} + 4 q^{41} + 6 q^{45} - 16 q^{47} - q^{49} + 7 q^{51} + 5 q^{53} - 4 q^{55} - q^{57} + 3 q^{59} - 4 q^{61} + 4 q^{63} - 6 q^{65} + 3 q^{67} - 5 q^{69} - 12 q^{71} - 7 q^{73} + q^{75} + 3 q^{77} + 8 q^{79} - 14 q^{81} - 12 q^{83} + 6 q^{85} + 25 q^{87} - 10 q^{89} + 13 q^{91} + 2 q^{93} + 4 q^{95} - 6 q^{97} - 3 q^{99}+O(q^{100}) \) 2 * q - q^3 + 4 * q^5 - 3 * q^7 + 3 * q^9 - 2 * q^11 - 3 * q^13 - 2 * q^15 + 3 * q^17 + 2 * q^19 - 7 * q^21 - 7 * q^23 - 2 * q^25 - 7 * q^27 + q^29 - 4 * q^31 + q^33 - 6 * q^35 - 2 * q^37 - 7 * q^39 + 4 * q^41 + 6 * q^45 - 16 * q^47 - q^49 + 7 * q^51 + 5 * q^53 - 4 * q^55 - q^57 + 3 * q^59 - 4 * q^61 + 4 * q^63 - 6 * q^65 + 3 * q^67 - 5 * q^69 - 12 * q^71 - 7 * q^73 + q^75 + 3 * q^77 + 8 * q^79 - 14 * q^81 - 12 * q^83 + 6 * q^85 + 25 * q^87 - 10 * q^89 + 13 * q^91 + 2 * q^93 + 4 * q^95 - 6 * q^97 - 3 * q^99

Introduction

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