LMFDB - Embedded newform 3344.2.a.k.1.1

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			1.1
Root			$2.56155$ 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)

$f(q)$	$=$	$q-2.56155 q^{3} +2.00000 q^{5} +0.561553 q^{7} +3.56155 q^{9} +O(q^{10})$	$q-2.56155 q^{3} +2.00000 q^{5} +0.561553 q^{7} +3.56155 q^{9} -1.00000 q^{11} +0.561553 q^{13} -5.12311 q^{15} -0.561553 q^{17} +1.00000 q^{19} -1.43845 q^{21} -1.43845 q^{23} -1.00000 q^{25} -1.43845 q^{27} -5.68466 q^{29} -2.00000 q^{31} +2.56155 q^{33} +1.12311 q^{35} -5.12311 q^{37} -1.43845 q^{39} +2.00000 q^{41} +7.12311 q^{45} -8.00000 q^{47} -6.68466 q^{49} +1.43845 q^{51} +12.8078 q^{53} -2.00000 q^{55} -2.56155 q^{57} +7.68466 q^{59} +6.24621 q^{61} +2.00000 q^{63} +1.12311 q^{65} +7.68466 q^{67} +3.68466 q^{69} -6.00000 q^{71} -9.68466 q^{73} +2.56155 q^{75} -0.561553 q^{77} +4.00000 q^{79} -7.00000 q^{81} -14.2462 q^{83} -1.12311 q^{85} +14.5616 q^{87} -0.876894 q^{89} +0.315342 q^{91} +5.12311 q^{93} +2.00000 q^{95} -7.12311 q^{97} -3.56155 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.56155	−1.47891	−0.739457	−	0.673204i	$-0.764917\pi$
$3$	−2.56155	−1.47891	−0.739457	+	0.673204i	$0.764917\pi$
$4$	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
$7$	0.561553	0.212247	0.106124	−	0.994353i	$-0.466156\pi$
$7$	0.561553	0.212247	0.106124	+	0.994353i	$0.466156\pi$
$8$	0	0
$9$	3.56155	1.18718
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	0.561553	0.155747	0.0778734	−	0.996963i	$-0.475187\pi$
$13$	0.561553	0.155747	0.0778734	+	0.996963i	$0.475187\pi$
$14$	0	0
$15$	−5.12311	−1.32278
$16$	0	0
$17$	−0.561553	−0.136197	−0.0680983	−	0.997679i	$-0.521693\pi$
$17$	−0.561553	−0.136197	−0.0680983	+	0.997679i	$0.521693\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−1.43845	−0.313895
$22$	0	0
$23$	−1.43845	−0.299937	−0.149968	−	0.988691i	$-0.547917\pi$
$23$	−1.43845	−0.299937	−0.149968	+	0.988691i	$0.547917\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−1.43845	−0.276829
$28$	0	0
$29$	−5.68466	−1.05561	−0.527807	−	0.849364i	$-0.676986\pi$
$29$	−5.68466	−1.05561	−0.527807	+	0.849364i	$0.676986\pi$
$30$	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
$33$	2.56155	0.445909
$34$	0	0
$35$	1.12311	0.189839
$36$	0	0
$37$	−5.12311	−0.842233	−0.421117	−	0.907006i	$-0.638362\pi$
$37$	−5.12311	−0.842233	−0.421117	+	0.907006i	$0.638362\pi$
$38$	0	0
$39$	−1.43845	−0.230336
$40$	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
$45$	7.12311	1.06185
$46$	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
$49$	−6.68466	−0.954951
$50$	0	0
$51$	1.43845	0.201423
$52$	0	0
$53$	12.8078	1.75928	0.879641	−	0.475638i	$-0.157783\pi$
$53$	12.8078	1.75928	0.879641	+	0.475638i	$0.157783\pi$
$54$	0	0
$55$	−2.00000	−0.269680
$56$	0	0
$57$	−2.56155	−0.339286
$58$	0	0
$59$	7.68466	1.00046	0.500229	−	0.865893i	$-0.333249\pi$
$59$	7.68466	1.00046	0.500229	+	0.865893i	$0.333249\pi$
$60$	0	0
$61$	6.24621	0.799745	0.399873	−	0.916571i	$-0.369054\pi$
$61$	6.24621	0.799745	0.399873	+	0.916571i	$0.369054\pi$
$62$	0	0
$63$	2.00000	0.251976
$64$	0	0
$65$	1.12311	0.139304
$66$	0	0
$67$	7.68466	0.938830	0.469415	−	0.882978i	$-0.344465\pi$
$67$	7.68466	0.938830	0.469415	+	0.882978i	$0.344465\pi$
$68$	0	0
$69$	3.68466	0.443581
$70$	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
$73$	−9.68466	−1.13350	−0.566752	−	0.823889i	$-0.691800\pi$
$73$	−9.68466	−1.13350	−0.566752	+	0.823889i	$0.691800\pi$
$74$	0	0
$75$	2.56155	0.295783
$76$	0	0
$77$	−0.561553	−0.0639949
$78$	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
$83$	−14.2462	−1.56372	−0.781862	−	0.623451i	$-0.785730\pi$
$83$	−14.2462	−1.56372	−0.781862	+	0.623451i	$0.785730\pi$
$84$	0	0
$85$	−1.12311	−0.121818
$86$	0	0
$87$	14.5616	1.56116
$88$	0	0
$89$	−0.876894	−0.0929506	−0.0464753	−	0.998919i	$-0.514799\pi$
$89$	−0.876894	−0.0929506	−0.0464753	+	0.998919i	$0.514799\pi$
$90$	0	0
$91$	0.315342	0.0330568
$92$	0	0
$93$	5.12311	0.531241
$94$	0	0
$95$	2.00000	0.205196
$96$	0	0
$97$	−7.12311	−0.723242	−0.361621	−	0.932325i	$-0.617777\pi$
$97$	−7.12311	−0.723242	−0.361621	+	0.932325i	$0.617777\pi$
$98$	0	0
$99$	−3.56155	−0.357950
$100$	0	0
$101$	−6.87689	−0.684277	−0.342138	−	0.939650i	$-0.611151\pi$
$101$	−6.87689	−0.684277	−0.342138	+	0.939650i	$0.611151\pi$
$102$	0	0
$103$	13.3693	1.31732	0.658659	−	0.752442i	$-0.271124\pi$
$103$	13.3693	1.31732	0.658659	+	0.752442i	$0.271124\pi$
$104$	0	0
$105$	−2.87689	−0.280756
$106$	0	0
$107$	9.93087	0.960053	0.480027	−	0.877254i	$-0.340627\pi$
$107$	9.93087	0.960053	0.480027	+	0.877254i	$0.340627\pi$
$108$	0	0
$109$	−6.31534	−0.604900	−0.302450	−	0.953165i	$-0.597805\pi$
$109$	−6.31534	−0.604900	−0.302450	+	0.953165i	$0.597805\pi$
$110$	0	0
$111$	13.1231	1.24559
$112$	0	0
$113$	−3.12311	−0.293797	−0.146899	−	0.989152i	$-0.546929\pi$
$113$	−3.12311	−0.293797	−0.146899	+	0.989152i	$0.546929\pi$
$114$	0	0
$115$	−2.87689	−0.268272
$116$	0	0
$117$	2.00000	0.184900
$118$	0	0
$119$	−0.315342	−0.0289073
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−5.12311	−0.461935
$124$	0	0
$125$	−12.0000	−1.07331
$126$	0	0
$127$	−2.24621	−0.199319	−0.0996595	−	0.995022i	$-0.531775\pi$
$127$	−2.24621	−0.199319	−0.0996595	+	0.995022i	$0.531775\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−1.12311	−0.0981262	−0.0490631	−	0.998796i	$-0.515624\pi$
$131$	−1.12311	−0.0981262	−0.0490631	+	0.998796i	$0.515624\pi$
$132$	0	0
$133$	0.561553	0.0486928
$134$	0	0
$135$	−2.87689	−0.247604
$136$	0	0
$137$	12.5616	1.07321	0.536603	−	0.843835i	$-0.319707\pi$
$137$	12.5616	1.07321	0.536603	+	0.843835i	$0.319707\pi$
$138$	0	0
$139$	−7.36932	−0.625057	−0.312529	−	0.949908i	$-0.601176\pi$
$139$	−7.36932	−0.625057	−0.312529	+	0.949908i	$0.601176\pi$
$140$	0	0
$141$	20.4924	1.72577
$142$	0	0
$143$	−0.561553	−0.0469594
$144$	0	0
$145$	−11.3693	−0.944170
$146$	0	0
$147$	17.1231	1.41229
$148$	0	0
$149$	−6.87689	−0.563377	−0.281689	−	0.959506i	$-0.590894\pi$
$149$	−6.87689	−0.563377	−0.281689	+	0.959506i	$0.590894\pi$
$150$	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
$157$	−14.4924	−1.15662	−0.578311	−	0.815817i	$-0.696288\pi$
$157$	−14.4924	−1.15662	−0.578311	+	0.815817i	$0.696288\pi$
$158$	0	0
$159$	−32.8078	−2.60182
$160$	0	0
$161$	−0.807764	−0.0636607
$162$	0	0
$163$	−11.3693	−0.890514	−0.445257	−	0.895403i	$-0.646888\pi$
$163$	−11.3693	−0.890514	−0.445257	+	0.895403i	$0.646888\pi$
$164$	0	0
$165$	5.12311	0.398833
$166$	0	0
$167$	0.630683	0.0488037	0.0244019	−	0.999702i	$-0.492232\pi$
$167$	0.630683	0.0488037	0.0244019	+	0.999702i	$0.492232\pi$
$168$	0	0
$169$	−12.6847	−0.975743
$170$	0	0
$171$	3.56155	0.272359
$172$	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
$175$	−0.561553	−0.0424494
$176$	0	0
$177$	−19.6847	−1.47959
$178$	0	0
$179$	−5.75379	−0.430058	−0.215029	−	0.976608i	$-0.568985\pi$
$179$	−5.75379	−0.430058	−0.215029	+	0.976608i	$0.568985\pi$
$180$	0	0
$181$	−10.8769	−0.808473	−0.404237	−	0.914654i	$-0.632463\pi$
$181$	−10.8769	−0.808473	−0.404237	+	0.914654i	$0.632463\pi$
$182$	0	0
$183$	−16.0000	−1.18275
$184$	0	0
$185$	−10.2462	−0.753316
$186$	0	0
$187$	0.561553	0.0410648
$188$	0	0
$189$	−0.807764	−0.0587562
$190$	0	0
$191$	8.31534	0.601677	0.300838	−	0.953675i	$-0.402733\pi$
$191$	8.31534	0.601677	0.300838	+	0.953675i	$0.402733\pi$
$192$	0	0
$193$	7.12311	0.512732	0.256366	−	0.966580i	$-0.417475\pi$
$193$	7.12311	0.512732	0.256366	+	0.966580i	$0.417475\pi$
$194$	0	0
$195$	−2.87689	−0.206019
$196$	0	0
$197$	18.2462	1.29999	0.649994	−	0.759939i	$-0.274771\pi$
$197$	18.2462	1.29999	0.649994	+	0.759939i	$0.274771\pi$
$198$	0	0
$199$	−11.6847	−0.828303	−0.414152	−	0.910208i	$-0.635922\pi$
$199$	−11.6847	−0.828303	−0.414152	+	0.910208i	$0.635922\pi$
$200$	0	0
$201$	−19.6847	−1.38845
$202$	0	0
$203$	−3.19224	−0.224051
$204$	0	0
$205$	4.00000	0.279372
$206$	0	0
$207$	−5.12311	−0.356080
$208$	0	0
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	0.315342	0.0217090	0.0108545	−	0.999941i	$-0.496545\pi$
$211$	0.315342	0.0217090	0.0108545	+	0.999941i	$0.496545\pi$
$212$	0	0
$213$	15.3693	1.05309
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−1.12311	−0.0762414
$218$	0	0
$219$	24.8078	1.67635
$220$	0	0
$221$	−0.315342	−0.0212122
$222$	0	0
$223$	−24.7386	−1.65662	−0.828311	−	0.560269i	$-0.810698\pi$
$223$	−24.7386	−1.65662	−0.828311	+	0.560269i	$0.810698\pi$
$224$	0	0
$225$	−3.56155	−0.237437
$226$	0	0
$227$	−28.1771	−1.87018	−0.935089	−	0.354412i	$-0.884681\pi$
$227$	−28.1771	−1.87018	−0.935089	+	0.354412i	$0.884681\pi$
$228$	0	0
$229$	12.8769	0.850929	0.425465	−	0.904975i	$-0.360111\pi$
$229$	12.8769	0.850929	0.425465	+	0.904975i	$0.360111\pi$
$230$	0	0
$231$	1.43845	0.0946429
$232$	0	0
$233$	−28.7386	−1.88273	−0.941365	−	0.337389i	$-0.890456\pi$
$233$	−28.7386	−1.88273	−0.941365	+	0.337389i	$0.890456\pi$
$234$	0	0
$235$	−16.0000	−1.04372
$236$	0	0
$237$	−10.2462	−0.665563
$238$	0	0
$239$	−23.9309	−1.54796	−0.773980	−	0.633210i	$-0.781737\pi$
$239$	−23.9309	−1.54796	−0.773980	+	0.633210i	$0.781737\pi$
$240$	0	0
$241$	−19.6155	−1.26355	−0.631774	−	0.775153i	$-0.717673\pi$
$241$	−19.6155	−1.26355	−0.631774	+	0.775153i	$0.717673\pi$
$242$	0	0
$243$	22.2462	1.42710
$244$	0	0
$245$	−13.3693	−0.854134
$246$	0	0
$247$	0.561553	0.0357307
$248$	0	0
$249$	36.4924	2.31261
$250$	0	0
$251$	−17.1231	−1.08080	−0.540400	−	0.841408i	$-0.681727\pi$
$251$	−17.1231	−1.08080	−0.540400	+	0.841408i	$0.681727\pi$
$252$	0	0
$253$	1.43845	0.0904344
$254$	0	0
$255$	2.87689	0.180158
$256$	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
$259$	−2.87689	−0.178762
$260$	0	0
$261$	−20.2462	−1.25321
$262$	0	0
$263$	−8.87689	−0.547373	−0.273686	−	0.961819i	$-0.588243\pi$
$263$	−8.87689	−0.547373	−0.273686	+	0.961819i	$0.588243\pi$
$264$	0	0
$265$	25.6155	1.57355
$266$	0	0
$267$	2.24621	0.137466
$268$	0	0
$269$	2.87689	0.175407	0.0877037	−	0.996147i	$-0.472047\pi$
$269$	2.87689	0.175407	0.0877037	+	0.996147i	$0.472047\pi$
$270$	0	0
$271$	−25.6847	−1.56023	−0.780116	−	0.625635i	$-0.784840\pi$
$271$	−25.6847	−1.56023	−0.780116	+	0.625635i	$0.784840\pi$
$272$	0	0
$273$	−0.807764	−0.0488881
$274$	0	0
$275$	1.00000	0.0603023
$276$	0	0
$277$	8.49242	0.510260	0.255130	−	0.966907i	$-0.417882\pi$
$277$	8.49242	0.510260	0.255130	+	0.966907i	$0.417882\pi$
$278$	0	0
$279$	−7.12311	−0.426449
$280$	0	0
$281$	−25.3693	−1.51341	−0.756703	−	0.653758i	$-0.773191\pi$
$281$	−25.3693	−1.51341	−0.756703	+	0.653758i	$0.773191\pi$
$282$	0	0
$283$	−11.3693	−0.675836	−0.337918	−	0.941176i	$-0.609723\pi$
$283$	−11.3693	−0.675836	−0.337918	+	0.941176i	$0.609723\pi$
$284$	0	0
$285$	−5.12311	−0.303467
$286$	0	0
$287$	1.12311	0.0662948
$288$	0	0
$289$	−16.6847	−0.981450
$290$	0	0
$291$	18.2462	1.06961
$292$	0	0
$293$	3.93087	0.229644	0.114822	−	0.993386i	$-0.463370\pi$
$293$	3.93087	0.229644	0.114822	+	0.993386i	$0.463370\pi$
$294$	0	0
$295$	15.3693	0.894836
$296$	0	0
$297$	1.43845	0.0834672
$298$	0	0
$299$	−0.807764	−0.0467142
$300$	0	0
$301$	0	0
$302$	0	0
$303$	17.6155	1.01199
$304$	0	0
$305$	12.4924	0.715314
$306$	0	0
$307$	22.2462	1.26966	0.634829	−	0.772653i	$-0.281070\pi$
$307$	22.2462	1.26966	0.634829	+	0.772653i	$0.281070\pi$
$308$	0	0
$309$	−34.2462	−1.94820
$310$	0	0
$311$	4.80776	0.272623	0.136312	−	0.990666i	$-0.456475\pi$
$311$	4.80776	0.272623	0.136312	+	0.990666i	$0.456475\pi$
$312$	0	0
$313$	−0.561553	−0.0317408	−0.0158704	−	0.999874i	$-0.505052\pi$
$313$	−0.561553	−0.0317408	−0.0158704	+	0.999874i	$0.505052\pi$
$314$	0	0
$315$	4.00000	0.225374
$316$	0	0
$317$	−7.05398	−0.396191	−0.198095	−	0.980183i	$-0.563476\pi$
$317$	−7.05398	−0.396191	−0.198095	+	0.980183i	$0.563476\pi$
$318$	0	0
$319$	5.68466	0.318280
$320$	0	0
$321$	−25.4384	−1.41984
$322$	0	0
$323$	−0.561553	−0.0312456
$324$	0	0
$325$	−0.561553	−0.0311493
$326$	0	0
$327$	16.1771	0.894595
$328$	0	0
$329$	−4.49242	−0.247675
$330$	0	0
$331$	14.5616	0.800375	0.400188	−	0.916433i	$-0.368945\pi$
$331$	14.5616	0.800375	0.400188	+	0.916433i	$0.368945\pi$
$332$	0	0
$333$	−18.2462	−0.999886
$334$	0	0
$335$	15.3693	0.839715
$336$	0	0
$337$	11.1231	0.605914	0.302957	−	0.953004i	$-0.402026\pi$
$337$	11.1231	0.605914	0.302957	+	0.953004i	$0.402026\pi$
$338$	0	0
$339$	8.00000	0.434500
$340$	0	0
$341$	2.00000	0.108306
$342$	0	0
$343$	−7.68466	−0.414933
$344$	0	0
$345$	7.36932	0.396751
$346$	0	0
$347$	−9.61553	−0.516189	−0.258094	−	0.966120i	$-0.583095\pi$
$347$	−9.61553	−0.516189	−0.258094	+	0.966120i	$0.583095\pi$
$348$	0	0
$349$	21.6155	1.15705	0.578526	−	0.815664i	$-0.303628\pi$
$349$	21.6155	1.15705	0.578526	+	0.815664i	$0.303628\pi$
$350$	0	0
$351$	−0.807764	−0.0431153
$352$	0	0
$353$	22.1771	1.18037	0.590183	−	0.807269i	$-0.299055\pi$
$353$	22.1771	1.18037	0.590183	+	0.807269i	$0.299055\pi$
$354$	0	0
$355$	−12.0000	−0.636894
$356$	0	0
$357$	0.807764	0.0427514
$358$	0	0
$359$	15.9309	0.840799	0.420400	−	0.907339i	$-0.361890\pi$
$359$	15.9309	0.840799	0.420400	+	0.907339i	$0.361890\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−2.56155	−0.134447
$364$	0	0
$365$	−19.3693	−1.01384
$366$	0	0
$367$	−30.2462	−1.57884	−0.789420	−	0.613854i	$-0.789618\pi$
$367$	−30.2462	−1.57884	−0.789420	+	0.613854i	$0.789618\pi$
$368$	0	0
$369$	7.12311	0.370814
$370$	0	0
$371$	7.19224	0.373402
$372$	0	0
$373$	3.43845	0.178036	0.0890180	−	0.996030i	$-0.471627\pi$
$373$	3.43845	0.178036	0.0890180	+	0.996030i	$0.471627\pi$
$374$	0	0
$375$	30.7386	1.58734
$376$	0	0
$377$	−3.19224	−0.164409
$378$	0	0
$379$	−6.56155	−0.337044	−0.168522	−	0.985698i	$-0.553899\pi$
$379$	−6.56155	−0.337044	−0.168522	+	0.985698i	$0.553899\pi$
$380$	0	0
$381$	5.75379	0.294776
$382$	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
$385$	−1.12311	−0.0572388
$386$	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
$391$	0.807764	0.0408504
$392$	0	0
$393$	2.87689	0.145120
$394$	0	0
$395$	8.00000	0.402524
$396$	0	0
$397$	38.9848	1.95659	0.978297	−	0.207209i	$-0.0664381\pi$
$397$	38.9848	1.95659	0.978297	+	0.207209i	$0.0664381\pi$
$398$	0	0
$399$	−1.43845	−0.0720124
$400$	0	0
$401$	−7.75379	−0.387206	−0.193603	−	0.981080i	$-0.562017\pi$
$401$	−7.75379	−0.387206	−0.193603	+	0.981080i	$0.562017\pi$
$402$	0	0
$403$	−1.12311	−0.0559459
$404$	0	0
$405$	−14.0000	−0.695666
$406$	0	0
$407$	5.12311	0.253943
$408$	0	0
$409$	16.2462	0.803323	0.401662	−	0.915788i	$-0.368433\pi$
$409$	16.2462	0.803323	0.401662	+	0.915788i	$0.368433\pi$
$410$	0	0
$411$	−32.1771	−1.58718
$412$	0	0
$413$	4.31534	0.212344
$414$	0	0
$415$	−28.4924	−1.39864
$416$	0	0
$417$	18.8769	0.924405
$418$	0	0
$419$	14.8769	0.726784	0.363392	−	0.931636i	$-0.381619\pi$
$419$	14.8769	0.726784	0.363392	+	0.931636i	$0.381619\pi$
$420$	0	0
$421$	−29.9309	−1.45874	−0.729371	−	0.684119i	$-0.760187\pi$
$421$	−29.9309	−1.45874	−0.729371	+	0.684119i	$0.760187\pi$
$422$	0	0
$423$	−28.4924	−1.38535
$424$	0	0
$425$	0.561553	0.0272393
$426$	0	0
$427$	3.50758	0.169744
$428$	0	0
$429$	1.43845	0.0694489
$430$	0	0
$431$	−31.8617	−1.53473	−0.767363	−	0.641213i	$-0.778432\pi$
$431$	−31.8617	−1.53473	−0.767363	+	0.641213i	$0.778432\pi$
$432$	0	0
$433$	6.63068	0.318650	0.159325	−	0.987226i	$-0.449068\pi$
$433$	6.63068	0.318650	0.159325	+	0.987226i	$0.449068\pi$
$434$	0	0
$435$	29.1231	1.39635
$436$	0	0
$437$	−1.43845	−0.0688103
$438$	0	0
$439$	9.61553	0.458924	0.229462	−	0.973318i	$-0.426303\pi$
$439$	9.61553	0.458924	0.229462	+	0.973318i	$0.426303\pi$
$440$	0	0
$441$	−23.8078	−1.13370
$442$	0	0
$443$	−39.8617	−1.89389	−0.946944	−	0.321398i	$-0.895847\pi$
$443$	−39.8617	−1.89389	−0.946944	+	0.321398i	$0.895847\pi$
$444$	0	0
$445$	−1.75379	−0.0831376
$446$	0	0
$447$	17.6155	0.833186
$448$	0	0
$449$	23.6155	1.11449	0.557243	−	0.830350i	$-0.311859\pi$
$449$	23.6155	1.11449	0.557243	+	0.830350i	$0.311859\pi$
$450$	0	0
$451$	−2.00000	−0.0941763
$452$	0	0
$453$	10.2462	0.481409
$454$	0	0
$455$	0.630683	0.0295669
$456$	0	0
$457$	36.5616	1.71028	0.855139	−	0.518399i	$-0.173472\pi$
$457$	36.5616	1.71028	0.855139	+	0.518399i	$0.173472\pi$
$458$	0	0
$459$	0.807764	0.0377032
$460$	0	0
$461$	26.7386	1.24534	0.622671	−	0.782484i	$-0.286047\pi$
$461$	26.7386	1.24534	0.622671	+	0.782484i	$0.286047\pi$
$462$	0	0
$463$	−3.50758	−0.163011	−0.0815055	−	0.996673i	$-0.525973\pi$
$463$	−3.50758	−0.163011	−0.0815055	+	0.996673i	$0.525973\pi$
$464$	0	0
$465$	10.2462	0.475157
$466$	0	0
$467$	−9.75379	−0.451352	−0.225676	−	0.974202i	$-0.572459\pi$
$467$	−9.75379	−0.451352	−0.225676	+	0.974202i	$0.572459\pi$
$468$	0	0
$469$	4.31534	0.199264
$470$	0	0
$471$	37.1231	1.71054
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	45.6155	2.08859
$478$	0	0
$479$	13.3693	0.610860	0.305430	−	0.952215i	$-0.401200\pi$
$479$	13.3693	0.610860	0.305430	+	0.952215i	$0.401200\pi$
$480$	0	0
$481$	−2.87689	−0.131175
$482$	0	0
$483$	2.06913	0.0941487
$484$	0	0
$485$	−14.2462	−0.646887
$486$	0	0
$487$	−31.1231	−1.41032	−0.705161	−	0.709047i	$-0.749125\pi$
$487$	−31.1231	−1.41032	−0.705161	+	0.709047i	$0.749125\pi$
$488$	0	0
$489$	29.1231	1.31699
$490$	0	0
$491$	39.3693	1.77671	0.888356	−	0.459155i	$-0.151848\pi$
$491$	39.3693	1.77671	0.888356	+	0.459155i	$0.151848\pi$
$492$	0	0
$493$	3.19224	0.143771
$494$	0	0
$495$	−7.12311	−0.320160
$496$	0	0
$497$	−3.36932	−0.151135
$498$	0	0
$499$	−9.75379	−0.436640	−0.218320	−	0.975877i	$-0.570058\pi$
$499$	−9.75379	−0.436640	−0.218320	+	0.975877i	$0.570058\pi$
$500$	0	0
$501$	−1.61553	−0.0721765
$502$	0	0
$503$	−21.6847	−0.966871	−0.483436	−	0.875380i	$-0.660611\pi$
$503$	−21.6847	−0.966871	−0.483436	+	0.875380i	$0.660611\pi$
$504$	0	0
$505$	−13.7538	−0.612036
$506$	0	0
$507$	32.4924	1.44304
$508$	0	0
$509$	−31.3693	−1.39042	−0.695210	−	0.718806i	$-0.744689\pi$
$509$	−31.3693	−1.39042	−0.695210	+	0.718806i	$0.744689\pi$
$510$	0	0
$511$	−5.43845	−0.240583
$512$	0	0
$513$	−1.43845	−0.0635090
$514$	0	0
$515$	26.7386	1.17824
$516$	0	0
$517$	8.00000	0.351840
$518$	0	0
$519$	−46.1080	−2.02391
$520$	0	0
$521$	−7.12311	−0.312069	−0.156034	−	0.987752i	$-0.549871\pi$
$521$	−7.12311	−0.312069	−0.156034	+	0.987752i	$0.549871\pi$
$522$	0	0
$523$	−31.0540	−1.35790	−0.678948	−	0.734187i	$-0.737564\pi$
$523$	−31.0540	−1.35790	−0.678948	+	0.734187i	$0.737564\pi$
$524$	0	0
$525$	1.43845	0.0627790
$526$	0	0
$527$	1.12311	0.0489232
$528$	0	0
$529$	−20.9309	−0.910038
$530$	0	0
$531$	27.3693	1.18773
$532$	0	0
$533$	1.12311	0.0486471
$534$	0	0
$535$	19.8617	0.858698
$536$	0	0
$537$	14.7386	0.636019
$538$	0	0
$539$	6.68466	0.287929
$540$	0	0
$541$	−39.2311	−1.68667	−0.843337	−	0.537384i	$-0.819412\pi$
$541$	−39.2311	−1.68667	−0.843337	+	0.537384i	$0.819412\pi$
$542$	0	0
$543$	27.8617	1.19566
$544$	0	0
$545$	−12.6307	−0.541039
$546$	0	0
$547$	42.7386	1.82737	0.913686	−	0.406421i	$-0.133223\pi$
$547$	42.7386	1.82737	0.913686	+	0.406421i	$0.133223\pi$
$548$	0	0
$549$	22.2462	0.949445
$550$	0	0
$551$	−5.68466	−0.242175
$552$	0	0
$553$	2.24621	0.0955186
$554$	0	0
$555$	26.2462	1.11409
$556$	0	0
$557$	17.1231	0.725529	0.362765	−	0.931881i	$-0.381833\pi$
$557$	17.1231	0.725529	0.362765	+	0.931881i	$0.381833\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−1.43845	−0.0607313
$562$	0	0
$563$	42.7386	1.80122	0.900609	−	0.434630i	$-0.143121\pi$
$563$	42.7386	1.80122	0.900609	+	0.434630i	$0.143121\pi$
$564$	0	0
$565$	−6.24621	−0.262780
$566$	0	0
$567$	−3.93087	−0.165081
$568$	0	0
$569$	−21.3693	−0.895848	−0.447924	−	0.894072i	$-0.647837\pi$
$569$	−21.3693	−0.895848	−0.447924	+	0.894072i	$0.647837\pi$
$570$	0	0
$571$	5.61553	0.235003	0.117501	−	0.993073i	$-0.462512\pi$
$571$	5.61553	0.235003	0.117501	+	0.993073i	$0.462512\pi$
$572$	0	0
$573$	−21.3002	−0.889828
$574$	0	0
$575$	1.43845	0.0599874
$576$	0	0
$577$	41.0540	1.70910	0.854550	−	0.519370i	$-0.173833\pi$
$577$	41.0540	1.70910	0.854550	+	0.519370i	$0.173833\pi$
$578$	0	0
$579$	−18.2462	−0.758287
$580$	0	0
$581$	−8.00000	−0.331896
$582$	0	0
$583$	−12.8078	−0.530443
$584$	0	0
$585$	4.00000	0.165380
$586$	0	0
$587$	40.4924	1.67130	0.835651	−	0.549261i	$-0.185091\pi$
$587$	40.4924	1.67130	0.835651	+	0.549261i	$0.185091\pi$
$588$	0	0
$589$	−2.00000	−0.0824086
$590$	0	0
$591$	−46.7386	−1.92257
$592$	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
$595$	−0.630683	−0.0258555
$596$	0	0
$597$	29.9309	1.22499
$598$	0	0
$599$	17.8617	0.729811	0.364905	−	0.931045i	$-0.381101\pi$
$599$	17.8617	0.729811	0.364905	+	0.931045i	$0.381101\pi$
$600$	0	0
$601$	21.3693	0.871673	0.435836	−	0.900026i	$-0.356453\pi$
$601$	21.3693	0.871673	0.435836	+	0.900026i	$0.356453\pi$
$602$	0	0
$603$	27.3693	1.11456
$604$	0	0
$605$	2.00000	0.0813116
$606$	0	0
$607$	13.6155	0.552637	0.276319	−	0.961066i	$-0.410885\pi$
$607$	13.6155	0.552637	0.276319	+	0.961066i	$0.410885\pi$
$608$	0	0
$609$	8.17708	0.331352
$610$	0	0
$611$	−4.49242	−0.181744
$612$	0	0
$613$	18.2462	0.736958	0.368479	−	0.929636i	$-0.379879\pi$
$613$	18.2462	0.736958	0.368479	+	0.929636i	$0.379879\pi$
$614$	0	0
$615$	−10.2462	−0.413167
$616$	0	0
$617$	44.7386	1.80111	0.900555	−	0.434743i	$-0.143161\pi$
$617$	44.7386	1.80111	0.900555	+	0.434743i	$0.143161\pi$
$618$	0	0
$619$	9.75379	0.392038	0.196019	−	0.980600i	$-0.437199\pi$
$619$	9.75379	0.392038	0.196019	+	0.980600i	$0.437199\pi$
$620$	0	0
$621$	2.06913	0.0830313
$622$	0	0
$623$	−0.492423	−0.0197285
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	2.56155	0.102299
$628$	0	0
$629$	2.87689	0.114709
$630$	0	0
$631$	−3.50758	−0.139634	−0.0698172	−	0.997560i	$-0.522242\pi$
$631$	−3.50758	−0.139634	−0.0698172	+	0.997560i	$0.522242\pi$
$632$	0	0
$633$	−0.807764	−0.0321057
$634$	0	0
$635$	−4.49242	−0.178276
$636$	0	0
$637$	−3.75379	−0.148731
$638$	0	0
$639$	−21.3693	−0.845357
$640$	0	0
$641$	31.6155	1.24874	0.624369	−	0.781129i	$-0.285356\pi$
$641$	31.6155	1.24874	0.624369	+	0.781129i	$0.285356\pi$
$642$	0	0
$643$	12.6307	0.498106	0.249053	−	0.968490i	$-0.419881\pi$
$643$	12.6307	0.498106	0.249053	+	0.968490i	$0.419881\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	47.5464	1.86924	0.934621	−	0.355646i	$-0.115739\pi$
$647$	47.5464	1.86924	0.934621	+	0.355646i	$0.115739\pi$
$648$	0	0
$649$	−7.68466	−0.301649
$650$	0	0
$651$	2.87689	0.112754
$652$	0	0
$653$	−15.6155	−0.611083	−0.305541	−	0.952179i	$-0.598837\pi$
$653$	−15.6155	−0.611083	−0.305541	+	0.952179i	$0.598837\pi$
$654$	0	0
$655$	−2.24621	−0.0877667
$656$	0	0
$657$	−34.4924	−1.34568
$658$	0	0
$659$	14.4233	0.561852	0.280926	−	0.959729i	$-0.409359\pi$
$659$	14.4233	0.561852	0.280926	+	0.959729i	$0.409359\pi$
$660$	0	0
$661$	−26.5616	−1.03312	−0.516562	−	0.856250i	$-0.672789\pi$
$661$	−26.5616	−1.03312	−0.516562	+	0.856250i	$0.672789\pi$
$662$	0	0
$663$	0.807764	0.0313710
$664$	0	0
$665$	1.12311	0.0435522
$666$	0	0
$667$	8.17708	0.316618
$668$	0	0
$669$	63.3693	2.45000
$670$	0	0
$671$	−6.24621	−0.241132
$672$	0	0
$673$	−27.1231	−1.04552	−0.522759	−	0.852480i	$-0.675097\pi$
$673$	−27.1231	−1.04552	−0.522759	+	0.852480i	$0.675097\pi$
$674$	0	0
$675$	1.43845	0.0553659
$676$	0	0
$677$	6.17708	0.237405	0.118702	−	0.992930i	$-0.462127\pi$
$677$	6.17708	0.237405	0.118702	+	0.992930i	$0.462127\pi$
$678$	0	0
$679$	−4.00000	−0.153506
$680$	0	0
$681$	72.1771	2.76583
$682$	0	0
$683$	−28.4924	−1.09023	−0.545116	−	0.838361i	$-0.683514\pi$
$683$	−28.4924	−1.09023	−0.545116	+	0.838361i	$0.683514\pi$
$684$	0	0
$685$	25.1231	0.959905
$686$	0	0
$687$	−32.9848	−1.25845
$688$	0	0
$689$	7.19224	0.274002
$690$	0	0
$691$	2.38447	0.0907096	0.0453548	−	0.998971i	$-0.485558\pi$
$691$	2.38447	0.0907096	0.0453548	+	0.998971i	$0.485558\pi$
$692$	0	0
$693$	−2.00000	−0.0759737
$694$	0	0
$695$	−14.7386	−0.559068
$696$	0	0
$697$	−1.12311	−0.0425407
$698$	0	0
$699$	73.6155	2.78439
$700$	0	0
$701$	−40.9848	−1.54798	−0.773988	−	0.633200i	$-0.781741\pi$
$701$	−40.9848	−1.54798	−0.773988	+	0.633200i	$0.781741\pi$
$702$	0	0
$703$	−5.12311	−0.193222
$704$	0	0
$705$	40.9848	1.54358
$706$	0	0
$707$	−3.86174	−0.145236
$708$	0	0
$709$	−42.4924	−1.59584	−0.797918	−	0.602766i	$-0.794065\pi$
$709$	−42.4924	−1.59584	−0.797918	+	0.602766i	$0.794065\pi$
$710$	0	0
$711$	14.2462	0.534275
$712$	0	0
$713$	2.87689	0.107741
$714$	0	0
$715$	−1.12311	−0.0420018
$716$	0	0
$717$	61.3002	2.28930
$718$	0	0
$719$	10.5616	0.393879	0.196940	−	0.980416i	$-0.436900\pi$
$719$	10.5616	0.393879	0.196940	+	0.980416i	$0.436900\pi$
$720$	0	0
$721$	7.50758	0.279597
$722$	0	0
$723$	50.2462	1.86868
$724$	0	0
$725$	5.68466	0.211123
$726$	0	0
$727$	−22.4233	−0.831634	−0.415817	−	0.909448i	$-0.636504\pi$
$727$	−22.4233	−0.831634	−0.415817	+	0.909448i	$0.636504\pi$
$728$	0	0
$729$	−35.9848	−1.33277
$730$	0	0
$731$	0	0
$732$	0	0
$733$	26.1080	0.964319	0.482160	−	0.876083i	$-0.339853\pi$
$733$	26.1080	0.964319	0.482160	+	0.876083i	$0.339853\pi$
$734$	0	0
$735$	34.2462	1.26319
$736$	0	0
$737$	−7.68466	−0.283068
$738$	0	0
$739$	44.9848	1.65479	0.827397	−	0.561617i	$-0.189821\pi$
$739$	44.9848	1.65479	0.827397	+	0.561617i	$0.189821\pi$
$740$	0	0
$741$	−1.43845	−0.0528427
$742$	0	0
$743$	42.2462	1.54986	0.774932	−	0.632045i	$-0.217784\pi$
$743$	42.2462	1.54986	0.774932	+	0.632045i	$0.217784\pi$
$744$	0	0
$745$	−13.7538	−0.503900
$746$	0	0
$747$	−50.7386	−1.85643
$748$	0	0
$749$	5.57671	0.203768
$750$	0	0
$751$	−41.2311	−1.50454	−0.752271	−	0.658853i	$-0.771042\pi$
$751$	−41.2311	−1.50454	−0.752271	+	0.658853i	$0.771042\pi$
$752$	0	0
$753$	43.8617	1.59841
$754$	0	0
$755$	−8.00000	−0.291150
$756$	0	0
$757$	−42.4924	−1.54441	−0.772207	−	0.635371i	$-0.780847\pi$
$757$	−42.4924	−1.54441	−0.772207	+	0.635371i	$0.780847\pi$
$758$	0	0
$759$	−3.68466	−0.133745
$760$	0	0
$761$	−42.8078	−1.55178	−0.775890	−	0.630868i	$-0.782699\pi$
$761$	−42.8078	−1.55178	−0.775890	+	0.630868i	$0.782699\pi$
$762$	0	0
$763$	−3.54640	−0.128388
$764$	0	0
$765$	−4.00000	−0.144620
$766$	0	0
$767$	4.31534	0.155818
$768$	0	0
$769$	−13.6847	−0.493481	−0.246741	−	0.969082i	$-0.579360\pi$
$769$	−13.6847	−0.493481	−0.246741	+	0.969082i	$0.579360\pi$
$770$	0	0
$771$	56.3542	2.02955
$772$	0	0
$773$	37.9309	1.36428	0.682139	−	0.731222i	$-0.261050\pi$
$773$	37.9309	1.36428	0.682139	+	0.731222i	$0.261050\pi$
$774$	0	0
$775$	2.00000	0.0718421
$776$	0	0
$777$	7.36932	0.264373
$778$	0	0
$779$	2.00000	0.0716574
$780$	0	0
$781$	6.00000	0.214697
$782$	0	0
$783$	8.17708	0.292225
$784$	0	0
$785$	−28.9848	−1.03451
$786$	0	0
$787$	−15.6847	−0.559098	−0.279549	−	0.960131i	$-0.590185\pi$
$787$	−15.6847	−0.559098	−0.279549	+	0.960131i	$0.590185\pi$
$788$	0	0
$789$	22.7386	0.809517
$790$	0	0
$791$	−1.75379	−0.0623575
$792$	0	0
$793$	3.50758	0.124558
$794$	0	0
$795$	−65.6155	−2.32714
$796$	0	0
$797$	−15.1922	−0.538137	−0.269068	−	0.963121i	$-0.586716\pi$
$797$	−15.1922	−0.538137	−0.269068	+	0.963121i	$0.586716\pi$
$798$	0	0
$799$	4.49242	0.158930
$800$	0	0
$801$	−3.12311	−0.110350
$802$	0	0
$803$	9.68466	0.341764
$804$	0	0
$805$	−1.61553	−0.0569399
$806$	0	0
$807$	−7.36932	−0.259412
$808$	0	0
$809$	8.06913	0.283696	0.141848	−	0.989888i	$-0.454696\pi$
$809$	8.06913	0.283696	0.141848	+	0.989888i	$0.454696\pi$
$810$	0	0
$811$	−8.31534	−0.291991	−0.145996	−	0.989285i	$-0.546639\pi$
$811$	−8.31534	−0.291991	−0.145996	+	0.989285i	$0.546639\pi$
$812$	0	0
$813$	65.7926	2.30745
$814$	0	0
$815$	−22.7386	−0.796500
$816$	0	0
$817$	0	0
$818$	0	0
$819$	1.12311	0.0392445
$820$	0	0
$821$	−20.9848	−0.732376	−0.366188	−	0.930541i	$-0.619337\pi$
$821$	−20.9848	−0.732376	−0.366188	+	0.930541i	$0.619337\pi$
$822$	0	0
$823$	13.9309	0.485600	0.242800	−	0.970076i	$-0.421934\pi$
$823$	13.9309	0.485600	0.242800	+	0.970076i	$0.421934\pi$
$824$	0	0
$825$	−2.56155	−0.0891818
$826$	0	0
$827$	−49.9309	−1.73627	−0.868133	−	0.496331i	$-0.834680\pi$
$827$	−49.9309	−1.73627	−0.868133	+	0.496331i	$0.834680\pi$
$828$	0	0
$829$	12.1771	0.422928	0.211464	−	0.977386i	$-0.432177\pi$
$829$	12.1771	0.422928	0.211464	+	0.977386i	$0.432177\pi$
$830$	0	0
$831$	−21.7538	−0.754631
$832$	0	0
$833$	3.75379	0.130061
$834$	0	0
$835$	1.26137	0.0436514
$836$	0	0
$837$	2.87689	0.0994400
$838$	0	0
$839$	7.12311	0.245917	0.122958	−	0.992412i	$-0.460762\pi$
$839$	7.12311	0.245917	0.122958	+	0.992412i	$0.460762\pi$
$840$	0	0
$841$	3.31534	0.114322
$842$	0	0
$843$	64.9848	2.23820
$844$	0	0
$845$	−25.3693	−0.872731
$846$	0	0
$847$	0.561553	0.0192952
$848$	0	0
$849$	29.1231	0.999502
$850$	0	0
$851$	7.36932	0.252617
$852$	0	0
$853$	−52.3542	−1.79257	−0.896286	−	0.443476i	$-0.853745\pi$
$853$	−52.3542	−1.79257	−0.896286	+	0.443476i	$0.853745\pi$
$854$	0	0
$855$	7.12311	0.243605
$856$	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
$859$	−15.8617	−0.541196	−0.270598	−	0.962692i	$-0.587221\pi$
$859$	−15.8617	−0.541196	−0.270598	+	0.962692i	$0.587221\pi$
$860$	0	0
$861$	−2.87689	−0.0980443
$862$	0	0
$863$	−0.246211	−0.00838113	−0.00419056	−	0.999991i	$-0.501334\pi$
$863$	−0.246211	−0.00838113	−0.00419056	+	0.999991i	$0.501334\pi$
$864$	0	0
$865$	36.0000	1.22404
$866$	0	0
$867$	42.7386	1.45148
$868$	0	0
$869$	−4.00000	−0.135691
$870$	0	0
$871$	4.31534	0.146220
$872$	0	0
$873$	−25.3693	−0.858621
$874$	0	0
$875$	−6.73863	−0.227807
$876$	0	0
$877$	−44.5616	−1.50474	−0.752368	−	0.658743i	$-0.771089\pi$
$877$	−44.5616	−1.50474	−0.752368	+	0.658743i	$0.771089\pi$
$878$	0	0
$879$	−10.0691	−0.339623
$880$	0	0
$881$	20.7386	0.698702	0.349351	−	0.936992i	$-0.386402\pi$
$881$	20.7386	0.698702	0.349351	+	0.936992i	$0.386402\pi$
$882$	0	0
$883$	5.26137	0.177059	0.0885295	−	0.996074i	$-0.471783\pi$
$883$	5.26137	0.177059	0.0885295	+	0.996074i	$0.471783\pi$
$884$	0	0
$885$	−39.3693	−1.32339
$886$	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
$889$	−1.26137	−0.0423049
$890$	0	0
$891$	7.00000	0.234509
$892$	0	0
$893$	−8.00000	−0.267710
$894$	0	0
$895$	−11.5076	−0.384656
$896$	0	0
$897$	2.06913	0.0690863
$898$	0	0
$899$	11.3693	0.379188
$900$	0	0
$901$	−7.19224	−0.239608
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−21.7538	−0.723120
$906$	0	0
$907$	29.7926	0.989247	0.494624	−	0.869107i	$-0.335306\pi$
$907$	29.7926	0.989247	0.494624	+	0.869107i	$0.335306\pi$
$908$	0	0
$909$	−24.4924	−0.812362
$910$	0	0
$911$	−18.9848	−0.628996	−0.314498	−	0.949258i	$-0.601836\pi$
$911$	−18.9848	−0.628996	−0.314498	+	0.949258i	$0.601836\pi$
$912$	0	0
$913$	14.2462	0.471481
$914$	0	0
$915$	−32.0000	−1.05789
$916$	0	0
$917$	−0.630683	−0.0208270
$918$	0	0
$919$	−32.5616	−1.07411	−0.537053	−	0.843548i	$-0.680463\pi$
$919$	−32.5616	−1.07411	−0.537053	+	0.843548i	$0.680463\pi$
$920$	0	0
$921$	−56.9848	−1.87771
$922$	0	0
$923$	−3.36932	−0.110902
$924$	0	0
$925$	5.12311	0.168447
$926$	0	0
$927$	47.6155	1.56390
$928$	0	0
$929$	11.9309	0.391439	0.195720	−	0.980660i	$-0.437296\pi$
$929$	11.9309	0.391439	0.195720	+	0.980660i	$0.437296\pi$
$930$	0	0
$931$	−6.68466	−0.219081
$932$	0	0
$933$	−12.3153	−0.403186
$934$	0	0
$935$	1.12311	0.0367295
$936$	0	0
$937$	−0.699813	−0.0228619	−0.0114310	−	0.999935i	$-0.503639\pi$
$937$	−0.699813	−0.0228619	−0.0114310	+	0.999935i	$0.503639\pi$
$938$	0	0
$939$	1.43845	0.0469419
$940$	0	0
$941$	24.5616	0.800684	0.400342	−	0.916366i	$-0.368891\pi$
$941$	24.5616	0.800684	0.400342	+	0.916366i	$0.368891\pi$
$942$	0	0
$943$	−2.87689	−0.0936846
$944$	0	0
$945$	−1.61553	−0.0525531
$946$	0	0
$947$	36.9848	1.20185	0.600923	−	0.799307i	$-0.294800\pi$
$947$	36.9848	1.20185	0.600923	+	0.799307i	$0.294800\pi$
$948$	0	0
$949$	−5.43845	−0.176539
$950$	0	0
$951$	18.0691	0.585932
$952$	0	0
$953$	−0.876894	−0.0284054	−0.0142027	−	0.999899i	$-0.504521\pi$
$953$	−0.876894	−0.0284054	−0.0142027	+	0.999899i	$0.504521\pi$
$954$	0	0
$955$	16.6307	0.538156
$956$	0	0
$957$	−14.5616	−0.470708
$958$	0	0
$959$	7.05398	0.227785
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	35.3693	1.13976
$964$	0	0
$965$	14.2462	0.458602
$966$	0	0
$967$	−53.8617	−1.73208	−0.866038	−	0.499978i	$-0.833342\pi$
$967$	−53.8617	−1.73208	−0.866038	+	0.499978i	$0.833342\pi$
$968$	0	0
$969$	1.43845	0.0462096
$970$	0	0
$971$	−2.24621	−0.0720843	−0.0360422	−	0.999350i	$-0.511475\pi$
$971$	−2.24621	−0.0720843	−0.0360422	+	0.999350i	$0.511475\pi$
$972$	0	0
$973$	−4.13826	−0.132667
$974$	0	0
$975$	1.43845	0.0460672
$976$	0	0
$977$	30.6307	0.979962	0.489981	−	0.871733i	$-0.337004\pi$
$977$	30.6307	0.979962	0.489981	+	0.871733i	$0.337004\pi$
$978$	0	0
$979$	0.876894	0.0280257
$980$	0	0
$981$	−22.4924	−0.718128
$982$	0	0
$983$	31.7538	1.01279	0.506394	−	0.862302i	$-0.330978\pi$
$983$	31.7538	1.01279	0.506394	+	0.862302i	$0.330978\pi$
$984$	0	0
$985$	36.4924	1.16275
$986$	0	0
$987$	11.5076	0.366290
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−12.8769	−0.409048	−0.204524	−	0.978862i	$-0.565565\pi$
$991$	−12.8769	−0.409048	−0.204524	+	0.978862i	$0.565565\pi$
$992$	0	0
$993$	−37.3002	−1.18369
$994$	0	0
$995$	−23.3693	−0.740857
$996$	0	0
$997$	−28.6307	−0.906743	−0.453371	−	0.891322i	$-0.649779\pi$
$997$	−28.6307	−0.906743	−0.453371	+	0.891322i	$0.649779\pi$
$998$	0	0
$999$	7.36932	0.233155

Twists

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

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

\(f(q)\)	\(=\)	\(q-2.56155 q^{3} +2.00000 q^{5} +0.561553 q^{7} +3.56155 q^{9} +O(q^{10})\)	\(q-2.56155 q^{3} +2.00000 q^{5} +0.561553 q^{7} +3.56155 q^{9} -1.00000 q^{11} +0.561553 q^{13} -5.12311 q^{15} -0.561553 q^{17} +1.00000 q^{19} -1.43845 q^{21} -1.43845 q^{23} -1.00000 q^{25} -1.43845 q^{27} -5.68466 q^{29} -2.00000 q^{31} +2.56155 q^{33} +1.12311 q^{35} -5.12311 q^{37} -1.43845 q^{39} +2.00000 q^{41} +7.12311 q^{45} -8.00000 q^{47} -6.68466 q^{49} +1.43845 q^{51} +12.8078 q^{53} -2.00000 q^{55} -2.56155 q^{57} +7.68466 q^{59} +6.24621 q^{61} +2.00000 q^{63} +1.12311 q^{65} +7.68466 q^{67} +3.68466 q^{69} -6.00000 q^{71} -9.68466 q^{73} +2.56155 q^{75} -0.561553 q^{77} +4.00000 q^{79} -7.00000 q^{81} -14.2462 q^{83} -1.12311 q^{85} +14.5616 q^{87} -0.876894 q^{89} +0.315342 q^{91} +5.12311 q^{93} +2.00000 q^{95} -7.12311 q^{97} -3.56155 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} + 4 q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) \) 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

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more