LMFDB - Embedded newform 1344.4.a.bu.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1344,4,Mod(1,1344)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1344, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1344.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 1344 = 2^{6} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1344.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,9,0,-6,0,-21,0,27,0,48] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$79.2985670477$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.37341.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 57x - 148 $ x^3 - 57*x - 148
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 672)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-3.13924$ of defining polynomial
Character	$\chi$	$=$	1344.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{3} -8.27848 q^{5} -7.00000 q^{7} +9.00000 q^{9} +72.0744 q^{11} -89.4668 q^{13} -24.8354 q^{15} -38.0744 q^{17} -46.9098 q^{19} -21.0000 q^{21} +65.1883 q^{23} -56.4668 q^{25} +27.0000 q^{27} +118.274 q^{29} +175.616 q^{31} +216.223 q^{33} +57.9494 q^{35} -157.467 q^{37} -268.400 q^{39} +293.666 q^{41} +230.934 q^{43} -74.5063 q^{45} +139.240 q^{47} +49.0000 q^{49} -114.223 q^{51} +128.400 q^{53} -596.666 q^{55} -140.729 q^{57} -58.5822 q^{59} -514.116 q^{61} -63.0000 q^{63} +740.649 q^{65} +129.876 q^{67} +195.565 q^{69} +221.495 q^{71} -700.658 q^{73} -169.400 q^{75} -504.521 q^{77} +791.582 q^{79} +81.0000 q^{81} -854.668 q^{83} +315.198 q^{85} +354.821 q^{87} +921.574 q^{89} +626.267 q^{91} +526.847 q^{93} +388.342 q^{95} +1561.47 q^{97} +648.669 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 9 q^{3} - 6 q^{5} - 21 q^{7} + 27 q^{9} + 48 q^{11} - 6 q^{13} - 18 q^{15} + 54 q^{17} + 84 q^{19} - 63 q^{21} - 48 q^{23} + 93 q^{25} + 81 q^{27} - 18 q^{29} - 72 q^{31} + 144 q^{33} + 42 q^{35}+ \cdots + 432 q^{99}+O(q^{100}) $ 3 * q + 9 * q^3 - 6 * q^5 - 21 * q^7 + 27 * q^9 + 48 * q^11 - 6 * q^13 - 18 * q^15 + 54 * q^17 + 84 * q^19 - 63 * q^21 - 48 * q^23 + 93 * q^25 + 81 * q^27 - 18 * q^29 - 72 * q^31 + 144 * q^33 + 42 * q^35 - 210 * q^37 - 18 * q^39 + 414 * q^41 + 168 * q^43 - 54 * q^45 - 72 * q^47 + 147 * q^49 + 162 * q^51 - 402 * q^53 + 456 * q^55 + 252 * q^57 + 540 * q^59 - 798 * q^61 - 189 * q^63 + 1740 * q^65 + 48 * q^67 - 144 * q^69 + 456 * q^71 + 1230 * q^73 + 279 * q^75 - 336 * q^77 + 1368 * q^79 + 243 * q^81 + 60 * q^83 - 660 * q^85 - 54 * q^87 + 2742 * q^89 + 42 * q^91 - 216 * q^93 + 648 * q^95 + 1950 * q^97 + 432 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	−8.27848	−0.740450	−0.370225	−	0.928942i	$-0.620719\pi$
$5$	−8.27848	−0.740450	−0.370225	+	0.928942i	$0.620719\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	72.0744	1.97557	0.987784	−	0.155830i	$-0.0498053\pi$
$11$	72.0744	1.97557	0.987784	+	0.155830i	$0.0498053\pi$
$12$	0	0
$13$	−89.4668	−1.90874	−0.954370	−	0.298627i	$-0.903471\pi$
$13$	−89.4668	−1.90874	−0.954370	+	0.298627i	$0.903471\pi$
$14$	0	0
$15$	−24.8354	−0.427499
$16$	0	0
$17$	−38.0744	−0.543200	−0.271600	−	0.962410i	$-0.587553\pi$
$17$	−38.0744	−0.543200	−0.271600	+	0.962410i	$0.587553\pi$
$18$	0	0
$19$	−46.9098	−0.566413	−0.283207	−	0.959059i	$-0.591398\pi$
$19$	−46.9098	−0.566413	−0.283207	+	0.959059i	$0.591398\pi$
$20$	0	0
$21$	−21.0000	−0.218218
$22$	0	0
$23$	65.1883	0.590987	0.295493	−	0.955345i	$-0.404516\pi$
$23$	65.1883	0.590987	0.295493	+	0.955345i	$0.404516\pi$
$24$	0	0
$25$	−56.4668	−0.451734
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	118.274	0.757341	0.378671	−	0.925532i	$-0.376381\pi$
$29$	118.274	0.757341	0.378671	+	0.925532i	$0.376381\pi$
$30$	0	0
$31$	175.616	1.01747	0.508734	−	0.860924i	$-0.330114\pi$
$31$	175.616	1.01747	0.508734	+	0.860924i	$0.330114\pi$
$32$	0	0
$33$	216.223	1.14059
$34$	0	0
$35$	57.9494	0.279864
$36$	0	0
$37$	−157.467	−0.699659	−0.349829	−	0.936813i	$-0.613760\pi$
$37$	−157.467	−0.699659	−0.349829	+	0.936813i	$0.613760\pi$
$38$	0	0
$39$	−268.400	−1.10201
$40$	0	0
$41$	293.666	1.11861	0.559304	−	0.828962i	$-0.311068\pi$
$41$	293.666	1.11861	0.559304	+	0.828962i	$0.311068\pi$
$42$	0	0
$43$	230.934	0.819001	0.409500	−	0.912310i	$-0.365703\pi$
$43$	230.934	0.819001	0.409500	+	0.912310i	$0.365703\pi$
$44$	0	0
$45$	−74.5063	−0.246817
$46$	0	0
$47$	139.240	0.432134	0.216067	−	0.976379i	$-0.430677\pi$
$47$	139.240	0.432134	0.216067	+	0.976379i	$0.430677\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−114.223	−0.313616
$52$	0	0
$53$	128.400	0.332776	0.166388	−	0.986060i	$-0.446790\pi$
$53$	128.400	0.332776	0.166388	+	0.986060i	$0.446790\pi$
$54$	0	0
$55$	−596.666	−1.46281
$56$	0	0
$57$	−140.729	−0.327019
$58$	0	0
$59$	−58.5822	−0.129267	−0.0646335	−	0.997909i	$-0.520588\pi$
$59$	−58.5822	−0.129267	−0.0646335	+	0.997909i	$0.520588\pi$
$60$	0	0
$61$	−514.116	−1.07911	−0.539555	−	0.841950i	$-0.681408\pi$
$61$	−514.116	−1.07911	−0.539555	+	0.841950i	$0.681408\pi$
$62$	0	0
$63$	−63.0000	−0.125988
$64$	0	0
$65$	740.649	1.41333
$66$	0	0
$67$	129.876	0.236820	0.118410	−	0.992965i	$-0.462220\pi$
$67$	129.876	0.236820	0.118410	+	0.992965i	$0.462220\pi$
$68$	0	0
$69$	195.565	0.341206
$70$	0	0
$71$	221.495	0.370234	0.185117	−	0.982716i	$-0.440734\pi$
$71$	221.495	0.370234	0.185117	+	0.982716i	$0.440734\pi$
$72$	0	0
$73$	−700.658	−1.12337	−0.561684	−	0.827352i	$-0.689846\pi$
$73$	−700.658	−1.12337	−0.561684	+	0.827352i	$0.689846\pi$
$74$	0	0
$75$	−169.400	−0.260809
$76$	0	0
$77$	−504.521	−0.746694
$78$	0	0
$79$	791.582	1.12734	0.563671	−	0.826000i	$-0.309389\pi$
$79$	791.582	1.12734	0.563671	+	0.826000i	$0.309389\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−854.668	−1.13026	−0.565132	−	0.825000i	$-0.691175\pi$
$83$	−854.668	−1.13026	−0.565132	+	0.825000i	$0.691175\pi$
$84$	0	0
$85$	315.198	0.402212
$86$	0	0
$87$	354.821	0.437251
$88$	0	0
$89$	921.574	1.09760	0.548802	−	0.835953i	$-0.315084\pi$
$89$	921.574	1.09760	0.548802	+	0.835953i	$0.315084\pi$
$90$	0	0
$91$	626.267	0.721436
$92$	0	0
$93$	526.847	0.587435
$94$	0	0
$95$	388.342	0.419401
$96$	0	0
$97$	1561.47	1.63447	0.817236	−	0.576303i	$-0.195505\pi$
$97$	1561.47	1.63447	0.817236	+	0.576303i	$0.195505\pi$
$98$	0	0
$99$	648.669	0.658523
$100$	0	0
$101$	−1599.03	−1.57534	−0.787671	−	0.616096i	$-0.788713\pi$
$101$	−1599.03	−1.57534	−0.787671	+	0.616096i	$0.788713\pi$
$102$	0	0
$103$	1213.55	1.16092	0.580459	−	0.814290i	$-0.302873\pi$
$103$	1213.55	1.16092	0.580459	+	0.814290i	$0.302873\pi$
$104$	0	0
$105$	173.848	0.161579
$106$	0	0
$107$	1370.27	1.23803	0.619014	−	0.785380i	$-0.287532\pi$
$107$	1370.27	1.23803	0.619014	+	0.785380i	$0.287532\pi$
$108$	0	0
$109$	−371.867	−0.326774	−0.163387	−	0.986562i	$-0.552242\pi$
$109$	−371.867	−0.326774	−0.163387	+	0.986562i	$0.552242\pi$
$110$	0	0
$111$	−472.400	−0.403948
$112$	0	0
$113$	731.585	0.609042	0.304521	−	0.952506i	$-0.401504\pi$
$113$	731.585	0.609042	0.304521	+	0.952506i	$0.401504\pi$
$114$	0	0
$115$	−539.660	−0.437596
$116$	0	0
$117$	−805.201	−0.636247
$118$	0	0
$119$	266.521	0.205310
$120$	0	0
$121$	3863.72	2.90287
$122$	0	0
$123$	880.999	0.645829
$124$	0	0
$125$	1502.27	1.07494
$126$	0	0
$127$	2386.05	1.66715	0.833575	−	0.552406i	$-0.186290\pi$
$127$	2386.05	1.66715	0.833575	+	0.552406i	$0.186290\pi$
$128$	0	0
$129$	692.801	0.472850
$130$	0	0
$131$	−65.1831	−0.0434739	−0.0217369	−	0.999764i	$-0.506920\pi$
$131$	−65.1831	−0.0434739	−0.0217369	+	0.999764i	$0.506920\pi$
$132$	0	0
$133$	328.369	0.214084
$134$	0	0
$135$	−223.519	−0.142500
$136$	0	0
$137$	1515.27	0.944950	0.472475	−	0.881344i	$-0.343361\pi$
$137$	1515.27	0.944950	0.472475	+	0.881344i	$0.343361\pi$
$138$	0	0
$139$	−2173.33	−1.32618	−0.663092	−	0.748538i	$-0.730756\pi$
$139$	−2173.33	−1.32618	−0.663092	+	0.748538i	$0.730756\pi$
$140$	0	0
$141$	417.721	0.249493
$142$	0	0
$143$	−6448.26	−3.77085
$144$	0	0
$145$	−979.127	−0.560773
$146$	0	0
$147$	147.000	0.0824786
$148$	0	0
$149$	3098.15	1.70343	0.851714	−	0.524007i	$-0.175564\pi$
$149$	3098.15	1.70343	0.851714	+	0.524007i	$0.175564\pi$
$150$	0	0
$151$	793.085	0.427420	0.213710	−	0.976897i	$-0.431445\pi$
$151$	793.085	0.427420	0.213710	+	0.976897i	$0.431445\pi$
$152$	0	0
$153$	−342.669	−0.181067
$154$	0	0
$155$	−1453.83	−0.753383
$156$	0	0
$157$	1586.47	0.806459	0.403230	−	0.915099i	$-0.367888\pi$
$157$	1586.47	0.806459	0.403230	+	0.915099i	$0.367888\pi$
$158$	0	0
$159$	385.201	0.192128
$160$	0	0
$161$	−456.318	−0.223372
$162$	0	0
$163$	945.643	0.454408	0.227204	−	0.973847i	$-0.427042\pi$
$163$	945.643	0.454408	0.227204	+	0.973847i	$0.427042\pi$
$164$	0	0
$165$	−1790.00	−0.844553
$166$	0	0
$167$	878.013	0.406843	0.203421	−	0.979091i	$-0.434794\pi$
$167$	878.013	0.406843	0.203421	+	0.979091i	$0.434794\pi$
$168$	0	0
$169$	5807.30	2.64329
$170$	0	0
$171$	−422.188	−0.188804
$172$	0	0
$173$	1704.42	0.749046	0.374523	−	0.927218i	$-0.377806\pi$
$173$	1704.42	0.749046	0.374523	+	0.927218i	$0.377806\pi$
$174$	0	0
$175$	395.267	0.170739
$176$	0	0
$177$	−175.747	−0.0746323
$178$	0	0
$179$	−2742.78	−1.14528	−0.572639	−	0.819807i	$-0.694080\pi$
$179$	−2742.78	−1.14528	−0.572639	+	0.819807i	$0.694080\pi$
$180$	0	0
$181$	3462.15	1.42177	0.710883	−	0.703310i	$-0.248296\pi$
$181$	3462.15	1.42177	0.710883	+	0.703310i	$0.248296\pi$
$182$	0	0
$183$	−1542.35	−0.623025
$184$	0	0
$185$	1303.59	0.518062
$186$	0	0
$187$	−2744.19	−1.07313
$188$	0	0
$189$	−189.000	−0.0727393
$190$	0	0
$191$	−4674.18	−1.77074	−0.885372	−	0.464884i	$-0.846096\pi$
$191$	−4674.18	−1.77074	−0.885372	+	0.464884i	$0.846096\pi$
$192$	0	0
$193$	−3094.15	−1.15400	−0.576999	−	0.816745i	$-0.695777\pi$
$193$	−3094.15	−1.15400	−0.576999	+	0.816745i	$0.695777\pi$
$194$	0	0
$195$	2221.95	0.815984
$196$	0	0
$197$	−4366.68	−1.57925	−0.789626	−	0.613588i	$-0.789726\pi$
$197$	−4366.68	−1.57925	−0.789626	+	0.613588i	$0.789726\pi$
$198$	0	0
$199$	1313.07	0.467744	0.233872	−	0.972267i	$-0.424860\pi$
$199$	1313.07	0.467744	0.233872	+	0.972267i	$0.424860\pi$
$200$	0	0
$201$	389.629	0.136728
$202$	0	0
$203$	−827.917	−0.286248
$204$	0	0
$205$	−2431.11	−0.828273
$206$	0	0
$207$	586.695	0.196996
$208$	0	0
$209$	−3381.00	−1.11899
$210$	0	0
$211$	1938.29	0.632403	0.316202	−	0.948692i	$-0.397592\pi$
$211$	1938.29	0.632403	0.316202	+	0.948692i	$0.397592\pi$
$212$	0	0
$213$	664.486	0.213755
$214$	0	0
$215$	−1911.78	−0.606429
$216$	0	0
$217$	−1229.31	−0.384566
$218$	0	0
$219$	−2101.98	−0.648577
$220$	0	0
$221$	3406.39	1.03683
$222$	0	0
$223$	−2810.59	−0.843995	−0.421997	−	0.906597i	$-0.638671\pi$
$223$	−2810.59	−0.843995	−0.421997	+	0.906597i	$0.638671\pi$
$224$	0	0
$225$	−508.201	−0.150578
$226$	0	0
$227$	5677.70	1.66010	0.830049	−	0.557690i	$-0.188312\pi$
$227$	5677.70	1.66010	0.830049	+	0.557690i	$0.188312\pi$
$228$	0	0
$229$	2754.87	0.794965	0.397482	−	0.917610i	$-0.369884\pi$
$229$	2754.87	0.794965	0.397482	+	0.917610i	$0.369884\pi$
$230$	0	0
$231$	−1513.56	−0.431104
$232$	0	0
$233$	−1908.29	−0.536550	−0.268275	−	0.963342i	$-0.586454\pi$
$233$	−1908.29	−0.536550	−0.268275	+	0.963342i	$0.586454\pi$
$234$	0	0
$235$	−1152.70	−0.319974
$236$	0	0
$237$	2374.75	0.650871
$238$	0	0
$239$	1120.91	0.303371	0.151685	−	0.988429i	$-0.451530\pi$
$239$	1120.91	0.303371	0.151685	+	0.988429i	$0.451530\pi$
$240$	0	0
$241$	−593.838	−0.158724	−0.0793620	−	0.996846i	$-0.525288\pi$
$241$	−593.838	−0.158724	−0.0793620	+	0.996846i	$0.525288\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	−405.645	−0.105779
$246$	0	0
$247$	4196.87	1.08114
$248$	0	0
$249$	−2564.00	−0.652559
$250$	0	0
$251$	7256.88	1.82490	0.912450	−	0.409188i	$-0.134188\pi$
$251$	7256.88	1.82490	0.912450	+	0.409188i	$0.134188\pi$
$252$	0	0
$253$	4698.41	1.16753
$254$	0	0
$255$	945.594	0.232217
$256$	0	0
$257$	3120.96	0.757511	0.378755	−	0.925497i	$-0.376352\pi$
$257$	3120.96	0.757511	0.378755	+	0.925497i	$0.376352\pi$
$258$	0	0
$259$	1102.27	0.264446
$260$	0	0
$261$	1064.46	0.252447
$262$	0	0
$263$	3114.51	0.730224	0.365112	−	0.930964i	$-0.381031\pi$
$263$	3114.51	0.730224	0.365112	+	0.930964i	$0.381031\pi$
$264$	0	0
$265$	−1062.96	−0.246404
$266$	0	0
$267$	2764.72	0.633701
$268$	0	0
$269$	803.254	0.182064	0.0910321	−	0.995848i	$-0.470983\pi$
$269$	803.254	0.182064	0.0910321	+	0.995848i	$0.470983\pi$
$270$	0	0
$271$	5221.42	1.17040	0.585201	−	0.810889i	$-0.301016\pi$
$271$	5221.42	1.17040	0.585201	+	0.810889i	$0.301016\pi$
$272$	0	0
$273$	1878.80	0.416521
$274$	0	0
$275$	−4069.81	−0.892432
$276$	0	0
$277$	3276.37	0.710678	0.355339	−	0.934737i	$-0.384365\pi$
$277$	3276.37	0.710678	0.355339	+	0.934737i	$0.384365\pi$
$278$	0	0
$279$	1580.54	0.339156
$280$	0	0
$281$	3207.50	0.680937	0.340469	−	0.940256i	$-0.389414\pi$
$281$	3207.50	0.680937	0.340469	+	0.940256i	$0.389414\pi$
$282$	0	0
$283$	−9161.80	−1.92443	−0.962213	−	0.272299i	$-0.912216\pi$
$283$	−9161.80	−1.92443	−0.962213	+	0.272299i	$0.912216\pi$
$284$	0	0
$285$	1165.03	0.242141
$286$	0	0
$287$	−2055.66	−0.422794
$288$	0	0
$289$	−3463.34	−0.704934
$290$	0	0
$291$	4684.42	0.943663
$292$	0	0
$293$	3277.09	0.653412	0.326706	−	0.945126i	$-0.394061\pi$
$293$	3277.09	0.653412	0.326706	+	0.945126i	$0.394061\pi$
$294$	0	0
$295$	484.972	0.0957157
$296$	0	0
$297$	1946.01	0.380198
$298$	0	0
$299$	−5832.19	−1.12804
$300$	0	0
$301$	−1616.53	−0.309553
$302$	0	0
$303$	−4797.10	−0.909525
$304$	0	0
$305$	4256.10	0.799027
$306$	0	0
$307$	−2748.81	−0.511020	−0.255510	−	0.966806i	$-0.582243\pi$
$307$	−2748.81	−0.511020	−0.255510	+	0.966806i	$0.582243\pi$
$308$	0	0
$309$	3640.65	0.670256
$310$	0	0
$311$	1800.80	0.328341	0.164170	−	0.986432i	$-0.447505\pi$
$311$	1800.80	0.328341	0.164170	+	0.986432i	$0.447505\pi$
$312$	0	0
$313$	4904.55	0.885692	0.442846	−	0.896598i	$-0.353969\pi$
$313$	4904.55	0.885692	0.442846	+	0.896598i	$0.353969\pi$
$314$	0	0
$315$	521.544	0.0932879
$316$	0	0
$317$	−5017.04	−0.888912	−0.444456	−	0.895801i	$-0.646603\pi$
$317$	−5017.04	−0.888912	−0.444456	+	0.895801i	$0.646603\pi$
$318$	0	0
$319$	8524.51	1.49618
$320$	0	0
$321$	4110.81	0.714776
$322$	0	0
$323$	1786.06	0.307675
$324$	0	0
$325$	5051.90	0.862243
$326$	0	0
$327$	−1115.60	−0.188663
$328$	0	0
$329$	−974.683	−0.163331
$330$	0	0
$331$	−10026.8	−1.66502	−0.832508	−	0.554013i	$-0.813096\pi$
$331$	−10026.8	−1.66502	−0.832508	+	0.554013i	$0.813096\pi$
$332$	0	0
$333$	−1417.20	−0.233220
$334$	0	0
$335$	−1075.18	−0.175353
$336$	0	0
$337$	−10196.6	−1.64820	−0.824100	−	0.566444i	$-0.808319\pi$
$337$	−10196.6	−1.64820	−0.824100	+	0.566444i	$0.808319\pi$
$338$	0	0
$339$	2194.76	0.351631
$340$	0	0
$341$	12657.4	2.01008
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	−1618.98	−0.252646
$346$	0	0
$347$	732.813	0.113370	0.0566851	−	0.998392i	$-0.481947\pi$
$347$	732.813	0.113370	0.0566851	+	0.998392i	$0.481947\pi$
$348$	0	0
$349$	−4097.74	−0.628502	−0.314251	−	0.949340i	$-0.601753\pi$
$349$	−4097.74	−0.628502	−0.314251	+	0.949340i	$0.601753\pi$
$350$	0	0
$351$	−2415.60	−0.367337
$352$	0	0
$353$	−7741.61	−1.16727	−0.583633	−	0.812018i	$-0.698369\pi$
$353$	−7741.61	−1.16727	−0.583633	+	0.812018i	$0.698369\pi$
$354$	0	0
$355$	−1833.64	−0.274140
$356$	0	0
$357$	799.562	0.118536
$358$	0	0
$359$	−2204.34	−0.324068	−0.162034	−	0.986785i	$-0.551805\pi$
$359$	−2204.34	−0.324068	−0.162034	+	0.986785i	$0.551805\pi$
$360$	0	0
$361$	−4658.47	−0.679176
$362$	0	0
$363$	11591.2	1.67597
$364$	0	0
$365$	5800.39	0.831798
$366$	0	0
$367$	−3272.01	−0.465388	−0.232694	−	0.972550i	$-0.574754\pi$
$367$	−3272.01	−0.465388	−0.232694	+	0.972550i	$0.574754\pi$
$368$	0	0
$369$	2643.00	0.372870
$370$	0	0
$371$	−898.802	−0.125778
$372$	0	0
$373$	2004.07	0.278195	0.139097	−	0.990279i	$-0.455580\pi$
$373$	2004.07	0.278195	0.139097	+	0.990279i	$0.455580\pi$
$374$	0	0
$375$	4506.81	0.620615
$376$	0	0
$377$	−10581.6	−1.44557
$378$	0	0
$379$	14014.4	1.89940	0.949699	−	0.313164i	$-0.101389\pi$
$379$	14014.4	1.89940	0.949699	+	0.313164i	$0.101389\pi$
$380$	0	0
$381$	7158.16	0.962529
$382$	0	0
$383$	−10252.1	−1.36778	−0.683889	−	0.729587i	$-0.739713\pi$
$383$	−10252.1	−1.36778	−0.683889	+	0.729587i	$0.739713\pi$
$384$	0	0
$385$	4176.66	0.552890
$386$	0	0
$387$	2078.40	0.273000
$388$	0	0
$389$	10263.6	1.33776	0.668878	−	0.743372i	$-0.266775\pi$
$389$	10263.6	1.33776	0.668878	+	0.743372i	$0.266775\pi$
$390$	0	0
$391$	−2482.00	−0.321024
$392$	0	0
$393$	−195.549	−0.0250997
$394$	0	0
$395$	−6553.10	−0.834740
$396$	0	0
$397$	−10292.5	−1.30117	−0.650585	−	0.759433i	$-0.725476\pi$
$397$	−10292.5	−1.30117	−0.650585	+	0.759433i	$0.725476\pi$
$398$	0	0
$399$	985.106	0.123602
$400$	0	0
$401$	561.891	0.0699738	0.0349869	−	0.999388i	$-0.488861\pi$
$401$	561.891	0.0699738	0.0349869	+	0.999388i	$0.488861\pi$
$402$	0	0
$403$	−15711.8	−1.94208
$404$	0	0
$405$	−670.557	−0.0822722
$406$	0	0
$407$	−11349.3	−1.38222
$408$	0	0
$409$	2653.32	0.320778	0.160389	−	0.987054i	$-0.448725\pi$
$409$	2653.32	0.320778	0.160389	+	0.987054i	$0.448725\pi$
$410$	0	0
$411$	4545.81	0.545567
$412$	0	0
$413$	410.075	0.0488583
$414$	0	0
$415$	7075.35	0.836904
$416$	0	0
$417$	−6520.00	−0.765673
$418$	0	0
$419$	4597.68	0.536065	0.268033	−	0.963410i	$-0.413626\pi$
$419$	4597.68	0.536065	0.268033	+	0.963410i	$0.413626\pi$
$420$	0	0
$421$	2681.37	0.310408	0.155204	−	0.987882i	$-0.450396\pi$
$421$	2681.37	0.310408	0.155204	+	0.987882i	$0.450396\pi$
$422$	0	0
$423$	1253.16	0.144045
$424$	0	0
$425$	2149.94	0.245382
$426$	0	0
$427$	3598.81	0.407866
$428$	0	0
$429$	−19344.8	−2.17710
$430$	0	0
$431$	13557.6	1.51519	0.757594	−	0.652726i	$-0.226375\pi$
$431$	13557.6	1.51519	0.757594	+	0.652726i	$0.226375\pi$
$432$	0	0
$433$	2024.50	0.224691	0.112346	−	0.993669i	$-0.464164\pi$
$433$	2024.50	0.224691	0.112346	+	0.993669i	$0.464164\pi$
$434$	0	0
$435$	−2937.38	−0.323762
$436$	0	0
$437$	−3057.97	−0.334743
$438$	0	0
$439$	1179.52	0.128235	0.0641177	−	0.997942i	$-0.479577\pi$
$439$	1179.52	0.128235	0.0641177	+	0.997942i	$0.479577\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	0	0
$443$	−15506.8	−1.66309	−0.831547	−	0.555454i	$-0.812544\pi$
$443$	−15506.8	−1.66309	−0.831547	+	0.555454i	$0.812544\pi$
$444$	0	0
$445$	−7629.23	−0.812720
$446$	0	0
$447$	9294.46	0.983474
$448$	0	0
$449$	−7666.24	−0.805774	−0.402887	−	0.915250i	$-0.631993\pi$
$449$	−7666.24	−0.805774	−0.402887	+	0.915250i	$0.631993\pi$
$450$	0	0
$451$	21165.8	2.20989
$452$	0	0
$453$	2379.26	0.246771
$454$	0	0
$455$	−5184.54	−0.534187
$456$	0	0
$457$	−11940.9	−1.22226	−0.611128	−	0.791532i	$-0.709284\pi$
$457$	−11940.9	−1.22226	−0.611128	+	0.791532i	$0.709284\pi$
$458$	0	0
$459$	−1028.01	−0.104539
$460$	0	0
$461$	−13964.4	−1.41082	−0.705411	−	0.708799i	$-0.749237\pi$
$461$	−13964.4	−1.41082	−0.705411	+	0.708799i	$0.749237\pi$
$462$	0	0
$463$	14419.6	1.44738	0.723691	−	0.690124i	$-0.242444\pi$
$463$	14419.6	1.44738	0.723691	+	0.690124i	$0.242444\pi$
$464$	0	0
$465$	−4361.49	−0.434966
$466$	0	0
$467$	−8781.18	−0.870117	−0.435058	−	0.900402i	$-0.643272\pi$
$467$	−8781.18	−0.870117	−0.435058	+	0.900402i	$0.643272\pi$
$468$	0	0
$469$	−909.135	−0.0895095
$470$	0	0
$471$	4759.41	0.465609
$472$	0	0
$473$	16644.4	1.61799
$474$	0	0
$475$	2648.85	0.255868
$476$	0	0
$477$	1155.60	0.110925
$478$	0	0
$479$	−10270.5	−0.979686	−0.489843	−	0.871811i	$-0.662946\pi$
$479$	−10270.5	−0.979686	−0.489843	+	0.871811i	$0.662946\pi$
$480$	0	0
$481$	14088.0	1.33547
$482$	0	0
$483$	−1368.95	−0.128964
$484$	0	0
$485$	−12926.6	−1.21024
$486$	0	0
$487$	12516.1	1.16459	0.582297	−	0.812976i	$-0.302154\pi$
$487$	12516.1	1.16459	0.582297	+	0.812976i	$0.302154\pi$
$488$	0	0
$489$	2836.93	0.262352
$490$	0	0
$491$	12534.1	1.15205	0.576024	−	0.817433i	$-0.304603\pi$
$491$	12534.1	1.15205	0.576024	+	0.817433i	$0.304603\pi$
$492$	0	0
$493$	−4503.20	−0.411387
$494$	0	0
$495$	−5370.00	−0.487603
$496$	0	0
$497$	−1550.47	−0.139935
$498$	0	0
$499$	14357.3	1.28801	0.644007	−	0.765019i	$-0.277271\pi$
$499$	14357.3	1.28801	0.644007	+	0.765019i	$0.277271\pi$
$500$	0	0
$501$	2634.04	0.234891
$502$	0	0
$503$	−16412.8	−1.45489	−0.727446	−	0.686165i	$-0.759293\pi$
$503$	−16412.8	−1.45489	−0.727446	+	0.686165i	$0.759293\pi$
$504$	0	0
$505$	13237.6	1.16646
$506$	0	0
$507$	17421.9	1.52610
$508$	0	0
$509$	3581.31	0.311864	0.155932	−	0.987768i	$-0.450162\pi$
$509$	3581.31	0.311864	0.155932	+	0.987768i	$0.450162\pi$
$510$	0	0
$511$	4904.61	0.424593
$512$	0	0
$513$	−1266.57	−0.109006
$514$	0	0
$515$	−10046.3	−0.859601
$516$	0	0
$517$	10035.7	0.853710
$518$	0	0
$519$	5113.27	0.432462
$520$	0	0
$521$	6838.64	0.575060	0.287530	−	0.957772i	$-0.407166\pi$
$521$	6838.64	0.575060	0.287530	+	0.957772i	$0.407166\pi$
$522$	0	0
$523$	330.872	0.0276635	0.0138318	−	0.999904i	$-0.495597\pi$
$523$	330.872	0.0276635	0.0138318	+	0.999904i	$0.495597\pi$
$524$	0	0
$525$	1185.80	0.0985765
$526$	0	0
$527$	−6686.45	−0.552688
$528$	0	0
$529$	−7917.49	−0.650734
$530$	0	0
$531$	−527.240	−0.0430890
$532$	0	0
$533$	−26273.4	−2.13513
$534$	0	0
$535$	−11343.8	−0.916698
$536$	0	0
$537$	−8228.33	−0.661227
$538$	0	0
$539$	3531.64	0.282224
$540$	0	0
$541$	10233.6	0.813265	0.406633	−	0.913592i	$-0.366703\pi$
$541$	10233.6	0.813265	0.406633	+	0.913592i	$0.366703\pi$
$542$	0	0
$543$	10386.5	0.820857
$544$	0	0
$545$	3078.49	0.241960
$546$	0	0
$547$	−5374.48	−0.420103	−0.210052	−	0.977690i	$-0.567363\pi$
$547$	−5374.48	−0.420103	−0.210052	+	0.977690i	$0.567363\pi$
$548$	0	0
$549$	−4627.04	−0.359704
$550$	0	0
$551$	−5548.20	−0.428968
$552$	0	0
$553$	−5541.08	−0.426095
$554$	0	0
$555$	3910.76	0.299103
$556$	0	0
$557$	−23749.7	−1.80665	−0.903327	−	0.428952i	$-0.858883\pi$
$557$	−23749.7	−1.80665	−0.903327	+	0.428952i	$0.858883\pi$
$558$	0	0
$559$	−20660.9	−1.56326
$560$	0	0
$561$	−8232.56	−0.619571
$562$	0	0
$563$	4551.40	0.340708	0.170354	−	0.985383i	$-0.445509\pi$
$563$	4551.40	0.340708	0.170354	+	0.985383i	$0.445509\pi$
$564$	0	0
$565$	−6056.41	−0.450965
$566$	0	0
$567$	−567.000	−0.0419961
$568$	0	0
$569$	−13864.8	−1.02152	−0.510758	−	0.859725i	$-0.670635\pi$
$569$	−13864.8	−1.02152	−0.510758	+	0.859725i	$0.670635\pi$
$570$	0	0
$571$	−26668.4	−1.95453	−0.977267	−	0.212011i	$-0.931999\pi$
$571$	−26668.4	−1.95453	−0.977267	+	0.212011i	$0.931999\pi$
$572$	0	0
$573$	−14022.5	−1.02234
$574$	0	0
$575$	−3680.97	−0.266969
$576$	0	0
$577$	723.611	0.0522085	0.0261043	−	0.999659i	$-0.491690\pi$
$577$	723.611	0.0522085	0.0261043	+	0.999659i	$0.491690\pi$
$578$	0	0
$579$	−9282.45	−0.666262
$580$	0	0
$581$	5982.67	0.427200
$582$	0	0
$583$	9254.38	0.657422
$584$	0	0
$585$	6665.84	0.471109
$586$	0	0
$587$	−5753.39	−0.404545	−0.202273	−	0.979329i	$-0.564833\pi$
$587$	−5753.39	−0.404545	−0.202273	+	0.979329i	$0.564833\pi$
$588$	0	0
$589$	−8238.09	−0.576307
$590$	0	0
$591$	−13100.0	−0.911782
$592$	0	0
$593$	−9298.15	−0.643894	−0.321947	−	0.946758i	$-0.604337\pi$
$593$	−9298.15	−0.643894	−0.321947	+	0.946758i	$0.604337\pi$
$594$	0	0
$595$	−2206.39	−0.152022
$596$	0	0
$597$	3939.21	0.270052
$598$	0	0
$599$	−7480.54	−0.510261	−0.255131	−	0.966907i	$-0.582118\pi$
$599$	−7480.54	−0.510261	−0.255131	+	0.966907i	$0.582118\pi$
$600$	0	0
$601$	8449.27	0.573466	0.286733	−	0.958011i	$-0.407431\pi$
$601$	8449.27	0.573466	0.286733	+	0.958011i	$0.407431\pi$
$602$	0	0
$603$	1168.89	0.0789400
$604$	0	0
$605$	−31985.7	−2.14943
$606$	0	0
$607$	12385.4	0.828182	0.414091	−	0.910236i	$-0.364100\pi$
$607$	12385.4	0.828182	0.414091	+	0.910236i	$0.364100\pi$
$608$	0	0
$609$	−2483.75	−0.165265
$610$	0	0
$611$	−12457.4	−0.824832
$612$	0	0
$613$	−28991.8	−1.91023	−0.955113	−	0.296240i	$-0.904267\pi$
$613$	−28991.8	−1.91023	−0.955113	+	0.296240i	$0.904267\pi$
$614$	0	0
$615$	−7293.33	−0.478204
$616$	0	0
$617$	28726.3	1.87435	0.937177	−	0.348853i	$-0.113429\pi$
$617$	28726.3	1.87435	0.937177	+	0.348853i	$0.113429\pi$
$618$	0	0
$619$	7766.73	0.504316	0.252158	−	0.967686i	$-0.418860\pi$
$619$	7766.73	0.504316	0.252158	+	0.967686i	$0.418860\pi$
$620$	0	0
$621$	1760.08	0.113735
$622$	0	0
$623$	−6451.02	−0.414855
$624$	0	0
$625$	−5378.15	−0.344202
$626$	0	0
$627$	−10143.0	−0.646048
$628$	0	0
$629$	5995.45	0.380055
$630$	0	0
$631$	17284.6	1.09047	0.545236	−	0.838283i	$-0.316440\pi$
$631$	17284.6	1.09047	0.545236	+	0.838283i	$0.316440\pi$
$632$	0	0
$633$	5814.86	0.365118
$634$	0	0
$635$	−19752.9	−1.23444
$636$	0	0
$637$	−4383.87	−0.272677
$638$	0	0
$639$	1993.46	0.123411
$640$	0	0
$641$	−31764.4	−1.95729	−0.978643	−	0.205568i	$-0.934096\pi$
$641$	−31764.4	−1.95729	−0.978643	+	0.205568i	$0.934096\pi$
$642$	0	0
$643$	−20740.6	−1.27205	−0.636026	−	0.771668i	$-0.719423\pi$
$643$	−20740.6	−1.27205	−0.636026	+	0.771668i	$0.719423\pi$
$644$	0	0
$645$	−5735.34	−0.350122
$646$	0	0
$647$	23613.3	1.43483	0.717415	−	0.696646i	$-0.245325\pi$
$647$	23613.3	1.43483	0.717415	+	0.696646i	$0.245325\pi$
$648$	0	0
$649$	−4222.28	−0.255376
$650$	0	0
$651$	−3687.93	−0.222030
$652$	0	0
$653$	1400.83	0.0839489	0.0419745	−	0.999119i	$-0.486635\pi$
$653$	1400.83	0.0839489	0.0419745	+	0.999119i	$0.486635\pi$
$654$	0	0
$655$	539.617	0.0321902
$656$	0	0
$657$	−6305.93	−0.374456
$658$	0	0
$659$	19793.0	1.16999	0.584997	−	0.811036i	$-0.301096\pi$
$659$	19793.0	1.16999	0.584997	+	0.811036i	$0.301096\pi$
$660$	0	0
$661$	−29180.8	−1.71710	−0.858549	−	0.512731i	$-0.828634\pi$
$661$	−29180.8	−1.71710	−0.858549	+	0.512731i	$0.828634\pi$
$662$	0	0
$663$	10219.2	0.598612
$664$	0	0
$665$	−2718.39	−0.158518
$666$	0	0
$667$	7710.07	0.447579
$668$	0	0
$669$	−8431.76	−0.487281
$670$	0	0
$671$	−37054.6	−2.13186
$672$	0	0
$673$	−9691.22	−0.555081	−0.277540	−	0.960714i	$-0.589519\pi$
$673$	−9691.22	−0.555081	−0.277540	+	0.960714i	$0.589519\pi$
$674$	0	0
$675$	−1524.60	−0.0869363
$676$	0	0
$677$	24254.8	1.37694	0.688469	−	0.725266i	$-0.258283\pi$
$677$	24254.8	1.37694	0.688469	+	0.725266i	$0.258283\pi$
$678$	0	0
$679$	−10930.3	−0.617772
$680$	0	0
$681$	17033.1	0.958459
$682$	0	0
$683$	−27067.1	−1.51639	−0.758193	−	0.652030i	$-0.773918\pi$
$683$	−27067.1	−1.51639	−0.758193	+	0.652030i	$0.773918\pi$
$684$	0	0
$685$	−12544.1	−0.699688
$686$	0	0
$687$	8264.61	0.458973
$688$	0	0
$689$	−11487.6	−0.635183
$690$	0	0
$691$	6908.11	0.380314	0.190157	−	0.981754i	$-0.439100\pi$
$691$	6908.11	0.380314	0.190157	+	0.981754i	$0.439100\pi$
$692$	0	0
$693$	−4540.69	−0.248898
$694$	0	0
$695$	17991.9	0.981973
$696$	0	0
$697$	−11181.2	−0.607628
$698$	0	0
$699$	−5724.86	−0.309777
$700$	0	0
$701$	14121.5	0.760856	0.380428	−	0.924811i	$-0.375777\pi$
$701$	14121.5	0.760856	0.380428	+	0.924811i	$0.375777\pi$
$702$	0	0
$703$	7386.74	0.396296
$704$	0	0
$705$	−3458.10	−0.184737
$706$	0	0
$707$	11193.2	0.595424
$708$	0	0
$709$	27122.6	1.43669	0.718343	−	0.695690i	$-0.244901\pi$
$709$	27122.6	1.43669	0.718343	+	0.695690i	$0.244901\pi$
$710$	0	0
$711$	7124.24	0.375781
$712$	0	0
$713$	11448.1	0.601310
$714$	0	0
$715$	53381.8	2.79212
$716$	0	0
$717$	3362.73	0.175151
$718$	0	0
$719$	29577.0	1.53413	0.767064	−	0.641571i	$-0.221717\pi$
$719$	29577.0	1.53413	0.767064	+	0.641571i	$0.221717\pi$
$720$	0	0
$721$	−8494.84	−0.438786
$722$	0	0
$723$	−1781.51	−0.0916393
$724$	0	0
$725$	−6678.54	−0.342117
$726$	0	0
$727$	−34389.1	−1.75436	−0.877181	−	0.480160i	$-0.840579\pi$
$727$	−34389.1	−1.75436	−0.877181	+	0.480160i	$0.840579\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−8792.65	−0.444881
$732$	0	0
$733$	−21859.4	−1.10150	−0.550748	−	0.834672i	$-0.685657\pi$
$733$	−21859.4	−1.10150	−0.550748	+	0.834672i	$0.685657\pi$
$734$	0	0
$735$	−1216.94	−0.0610713
$736$	0	0
$737$	9360.77	0.467854
$738$	0	0
$739$	16828.4	0.837677	0.418838	−	0.908061i	$-0.362437\pi$
$739$	16828.4	0.837677	0.418838	+	0.908061i	$0.362437\pi$
$740$	0	0
$741$	12590.6	0.624194
$742$	0	0
$743$	−15612.2	−0.770871	−0.385436	−	0.922735i	$-0.625949\pi$
$743$	−15612.2	−0.770871	−0.385436	+	0.922735i	$0.625949\pi$
$744$	0	0
$745$	−25648.0	−1.26130
$746$	0	0
$747$	−7692.01	−0.376755
$748$	0	0
$749$	−9591.90	−0.467931
$750$	0	0
$751$	−5405.70	−0.262659	−0.131330	−	0.991339i	$-0.541925\pi$
$751$	−5405.70	−0.262659	−0.131330	+	0.991339i	$0.541925\pi$
$752$	0	0
$753$	21770.6	1.05361
$754$	0	0
$755$	−6565.54	−0.316483
$756$	0	0
$757$	−7432.52	−0.356855	−0.178428	−	0.983953i	$-0.557101\pi$
$757$	−7432.52	−0.356855	−0.178428	+	0.983953i	$0.557101\pi$
$758$	0	0
$759$	14095.2	0.674077
$760$	0	0
$761$	−10851.9	−0.516927	−0.258463	−	0.966021i	$-0.583216\pi$
$761$	−10851.9	−0.516927	−0.258463	+	0.966021i	$0.583216\pi$
$762$	0	0
$763$	2603.07	0.123509
$764$	0	0
$765$	2836.78	0.134071
$766$	0	0
$767$	5241.16	0.246737
$768$	0	0
$769$	24016.7	1.12622	0.563112	−	0.826381i	$-0.309604\pi$
$769$	24016.7	1.12622	0.563112	+	0.826381i	$0.309604\pi$
$770$	0	0
$771$	9362.89	0.437349
$772$	0	0
$773$	26716.0	1.24309	0.621544	−	0.783379i	$-0.286506\pi$
$773$	26716.0	1.24309	0.621544	+	0.783379i	$0.286506\pi$
$774$	0	0
$775$	−9916.45	−0.459625
$776$	0	0
$777$	3306.80	0.152678
$778$	0	0
$779$	−13775.8	−0.633595
$780$	0	0
$781$	15964.1	0.731423
$782$	0	0
$783$	3193.39	0.145750
$784$	0	0
$785$	−13133.6	−0.597142
$786$	0	0
$787$	35503.3	1.60808	0.804038	−	0.594578i	$-0.202681\pi$
$787$	35503.3	1.60808	0.804038	+	0.594578i	$0.202681\pi$
$788$	0	0
$789$	9343.53	0.421595
$790$	0	0
$791$	−5121.10	−0.230196
$792$	0	0
$793$	45996.3	2.05974
$794$	0	0
$795$	−3188.88	−0.142261
$796$	0	0
$797$	−9626.24	−0.427828	−0.213914	−	0.976852i	$-0.568621\pi$
$797$	−9626.24	−0.427828	−0.213914	+	0.976852i	$0.568621\pi$
$798$	0	0
$799$	−5301.49	−0.234735
$800$	0	0
$801$	8294.17	0.365868
$802$	0	0
$803$	−50499.5	−2.21929
$804$	0	0
$805$	3777.62	0.165396
$806$	0	0
$807$	2409.76	0.105115
$808$	0	0
$809$	14915.6	0.648215	0.324107	−	0.946020i	$-0.394936\pi$
$809$	14915.6	0.648215	0.324107	+	0.946020i	$0.394936\pi$
$810$	0	0
$811$	39232.2	1.69868	0.849340	−	0.527845i	$-0.177000\pi$
$811$	39232.2	1.69868	0.849340	+	0.527845i	$0.177000\pi$
$812$	0	0
$813$	15664.3	0.675731
$814$	0	0
$815$	−7828.48	−0.336466
$816$	0	0
$817$	−10833.1	−0.463893
$818$	0	0
$819$	5636.41	0.240479
$820$	0	0
$821$	−1312.17	−0.0557797	−0.0278898	−	0.999611i	$-0.508879\pi$
$821$	−1312.17	−0.0557797	−0.0278898	+	0.999611i	$0.508879\pi$
$822$	0	0
$823$	32578.1	1.37983	0.689916	−	0.723890i	$-0.257648\pi$
$823$	32578.1	1.37983	0.689916	+	0.723890i	$0.257648\pi$
$824$	0	0
$825$	−12209.4	−0.515246
$826$	0	0
$827$	−15486.0	−0.651149	−0.325574	−	0.945516i	$-0.605558\pi$
$827$	−15486.0	−0.651149	−0.325574	+	0.945516i	$0.605558\pi$
$828$	0	0
$829$	4875.60	0.204266	0.102133	−	0.994771i	$-0.467433\pi$
$829$	4875.60	0.204266	0.102133	+	0.994771i	$0.467433\pi$
$830$	0	0
$831$	9829.11	0.410310
$832$	0	0
$833$	−1865.64	−0.0776000
$834$	0	0
$835$	−7268.61	−0.301246
$836$	0	0
$837$	4741.62	0.195812
$838$	0	0
$839$	−11091.5	−0.456403	−0.228201	−	0.973614i	$-0.573284\pi$
$839$	−11091.5	−0.456403	−0.228201	+	0.973614i	$0.573284\pi$
$840$	0	0
$841$	−10400.3	−0.426434
$842$	0	0
$843$	9622.50	0.393139
$844$	0	0
$845$	−48075.7	−1.95722
$846$	0	0
$847$	−27046.0	−1.09718
$848$	0	0
$849$	−27485.4	−1.11107
$850$	0	0
$851$	−10265.0	−0.413489
$852$	0	0
$853$	−1127.04	−0.0452393	−0.0226197	−	0.999744i	$-0.507201\pi$
$853$	−1127.04	−0.0452393	−0.0226197	+	0.999744i	$0.507201\pi$
$854$	0	0
$855$	3495.08	0.139800
$856$	0	0
$857$	31542.5	1.25726	0.628630	−	0.777705i	$-0.283616\pi$
$857$	31542.5	1.25726	0.628630	+	0.777705i	$0.283616\pi$
$858$	0	0
$859$	−21430.0	−0.851201	−0.425600	−	0.904911i	$-0.639937\pi$
$859$	−21430.0	−0.851201	−0.425600	+	0.904911i	$0.639937\pi$
$860$	0	0
$861$	−6166.99	−0.244100
$862$	0	0
$863$	−29408.3	−1.15999	−0.579994	−	0.814621i	$-0.696945\pi$
$863$	−29408.3	−1.15999	−0.579994	+	0.814621i	$0.696945\pi$
$864$	0	0
$865$	−14110.0	−0.554631
$866$	0	0
$867$	−10390.0	−0.406994
$868$	0	0
$869$	57052.8	2.22714
$870$	0	0
$871$	−11619.6	−0.452028
$872$	0	0
$873$	14053.3	0.544824
$874$	0	0
$875$	−10515.9	−0.406288
$876$	0	0
$877$	3247.51	0.125041	0.0625203	−	0.998044i	$-0.480086\pi$
$877$	3247.51	0.125041	0.0625203	+	0.998044i	$0.480086\pi$
$878$	0	0
$879$	9831.27	0.377248
$880$	0	0
$881$	17925.3	0.685492	0.342746	−	0.939428i	$-0.388643\pi$
$881$	17925.3	0.685492	0.342746	+	0.939428i	$0.388643\pi$
$882$	0	0
$883$	649.227	0.0247432	0.0123716	−	0.999923i	$-0.496062\pi$
$883$	649.227	0.0247432	0.0123716	+	0.999923i	$0.496062\pi$
$884$	0	0
$885$	1454.91	0.0552615
$886$	0	0
$887$	281.201	0.0106446	0.00532232	−	0.999986i	$-0.498306\pi$
$887$	281.201	0.0106446	0.00532232	+	0.999986i	$0.498306\pi$
$888$	0	0
$889$	−16702.4	−0.630123
$890$	0	0
$891$	5838.03	0.219508
$892$	0	0
$893$	−6531.75	−0.244767
$894$	0	0
$895$	22706.0	0.848021
$896$	0	0
$897$	−17496.6	−0.651275
$898$	0	0
$899$	20770.7	0.770570
$900$	0	0
$901$	−4888.76	−0.180764
$902$	0	0
$903$	−4849.60	−0.178721
$904$	0	0
$905$	−28661.3	−1.05275
$906$	0	0
$907$	22119.6	0.809780	0.404890	−	0.914365i	$-0.367310\pi$
$907$	22119.6	0.809780	0.404890	+	0.914365i	$0.367310\pi$
$908$	0	0
$909$	−14391.3	−0.525114
$910$	0	0
$911$	23370.6	0.849948	0.424974	−	0.905206i	$-0.360283\pi$
$911$	23370.6	0.849948	0.424974	+	0.905206i	$0.360283\pi$
$912$	0	0
$913$	−61599.7	−2.23291
$914$	0	0
$915$	12768.3	0.461319
$916$	0	0
$917$	456.282	0.0164316
$918$	0	0
$919$	−44334.6	−1.59136	−0.795682	−	0.605715i	$-0.792887\pi$
$919$	−44334.6	−1.59136	−0.795682	+	0.605715i	$0.792887\pi$
$920$	0	0
$921$	−8246.44	−0.295037
$922$	0	0
$923$	−19816.5	−0.706681
$924$	0	0
$925$	8891.64	0.316060
$926$	0	0
$927$	10921.9	0.386973
$928$	0	0
$929$	−31674.4	−1.11862	−0.559312	−	0.828957i	$-0.688935\pi$
$929$	−31674.4	−1.11862	−0.559312	+	0.828957i	$0.688935\pi$
$930$	0	0
$931$	−2298.58	−0.0809162
$932$	0	0
$933$	5402.39	0.189567
$934$	0	0
$935$	22717.7	0.794597
$936$	0	0
$937$	5825.76	0.203116	0.101558	−	0.994830i	$-0.467617\pi$
$937$	5825.76	0.203116	0.101558	+	0.994830i	$0.467617\pi$
$938$	0	0
$939$	14713.7	0.511355
$940$	0	0
$941$	−698.816	−0.0242091	−0.0121045	−	0.999927i	$-0.503853\pi$
$941$	−698.816	−0.0242091	−0.0121045	+	0.999927i	$0.503853\pi$
$942$	0	0
$943$	19143.6	0.661083
$944$	0	0
$945$	1564.63	0.0538598
$946$	0	0
$947$	−10938.8	−0.375357	−0.187679	−	0.982230i	$-0.560096\pi$
$947$	−10938.8	−0.375357	−0.187679	+	0.982230i	$0.560096\pi$
$948$	0	0
$949$	62685.7	2.14422
$950$	0	0
$951$	−15051.1	−0.513214
$952$	0	0
$953$	−34467.3	−1.17157	−0.585784	−	0.810467i	$-0.699213\pi$
$953$	−34467.3	−1.17157	−0.585784	+	0.810467i	$0.699213\pi$
$954$	0	0
$955$	38695.1	1.31115
$956$	0	0
$957$	25573.5	0.863819
$958$	0	0
$959$	−10606.9	−0.357158
$960$	0	0
$961$	1049.82	0.0352396
$962$	0	0
$963$	12332.4	0.412676
$964$	0	0
$965$	25614.9	0.854478
$966$	0	0
$967$	−24386.7	−0.810985	−0.405493	−	0.914098i	$-0.632900\pi$
$967$	−24386.7	−0.810985	−0.405493	+	0.914098i	$0.632900\pi$
$968$	0	0
$969$	5358.19	0.177637
$970$	0	0
$971$	33108.9	1.09425	0.547124	−	0.837052i	$-0.315723\pi$
$971$	33108.9	1.09425	0.547124	+	0.837052i	$0.315723\pi$
$972$	0	0
$973$	15213.3	0.501251
$974$	0	0
$975$	15155.7	0.497816
$976$	0	0
$977$	−8633.92	−0.282726	−0.141363	−	0.989958i	$-0.545149\pi$
$977$	−8633.92	−0.282726	−0.141363	+	0.989958i	$0.545149\pi$
$978$	0	0
$979$	66421.9	2.16839
$980$	0	0
$981$	−3346.80	−0.108925
$982$	0	0
$983$	−22005.7	−0.714011	−0.357006	−	0.934102i	$-0.616202\pi$
$983$	−22005.7	−0.714011	−0.357006	+	0.934102i	$0.616202\pi$
$984$	0	0
$985$	36149.4	1.16936
$986$	0	0
$987$	−2924.05	−0.0942994
$988$	0	0
$989$	15054.2	0.484019
$990$	0	0
$991$	15185.7	0.486772	0.243386	−	0.969930i	$-0.421742\pi$
$991$	15185.7	0.486772	0.243386	+	0.969930i	$0.421742\pi$
$992$	0	0
$993$	−30080.3	−0.961298
$994$	0	0
$995$	−10870.2	−0.346341
$996$	0	0
$997$	41110.7	1.30591	0.652953	−	0.757398i	$-0.273530\pi$
$997$	41110.7	1.30591	0.652953	+	0.757398i	$0.273530\pi$
$998$	0	0
$999$	−4251.60	−0.134649

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.4.a.bu.1.2		3
4.3	odd	2		1344.4.a.bs.1.2		3
8.3	odd	2		672.4.a.r.1.2	yes	3
8.5	even	2		672.4.a.p.1.2	✓	3
24.5	odd	2		2016.4.a.u.1.2		3
24.11	even	2		2016.4.a.v.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.4.a.p.1.2	✓	3	8.5	even	2
672.4.a.r.1.2	yes	3	8.3	odd	2
1344.4.a.bs.1.2		3	4.3	odd	2
1344.4.a.bu.1.2		3	1.1	even	1	trivial
2016.4.a.u.1.2		3	24.5	odd	2
2016.4.a.v.1.2		3	24.11	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1344.a (trivial)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} -8.27848 q^{5} -7.00000 q^{7} +9.00000 q^{9} +72.0744 q^{11} -89.4668 q^{13} -24.8354 q^{15} -38.0744 q^{17} -46.9098 q^{19} -21.0000 q^{21} +65.1883 q^{23} -56.4668 q^{25} +27.0000 q^{27} +118.274 q^{29} +175.616 q^{31} +216.223 q^{33} +57.9494 q^{35} -157.467 q^{37} -268.400 q^{39} +293.666 q^{41} +230.934 q^{43} -74.5063 q^{45} +139.240 q^{47} +49.0000 q^{49} -114.223 q^{51} +128.400 q^{53} -596.666 q^{55} -140.729 q^{57} -58.5822 q^{59} -514.116 q^{61} -63.0000 q^{63} +740.649 q^{65} +129.876 q^{67} +195.565 q^{69} +221.495 q^{71} -700.658 q^{73} -169.400 q^{75} -504.521 q^{77} +791.582 q^{79} +81.0000 q^{81} -854.668 q^{83} +315.198 q^{85} +354.821 q^{87} +921.574 q^{89} +626.267 q^{91} +526.847 q^{93} +388.342 q^{95} +1561.47 q^{97} +648.669 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 9 q^{3} - 6 q^{5} - 21 q^{7} + 27 q^{9} + 48 q^{11} - 6 q^{13} - 18 q^{15} + 54 q^{17} + 84 q^{19} - 63 q^{21} - 48 q^{23} + 93 q^{25} + 81 q^{27} - 18 q^{29} - 72 q^{31} + 144 q^{33} + 42 q^{35}+ \cdots + 432 q^{99}+O(q^{100}) \) 3 * q + 9 * q^3 - 6 * q^5 - 21 * q^7 + 27 * q^9 + 48 * q^11 - 6 * q^13 - 18 * q^15 + 54 * q^17 + 84 * q^19 - 63 * q^21 - 48 * q^23 + 93 * q^25 + 81 * q^27 - 18 * q^29 - 72 * q^31 + 144 * q^33 + 42 * q^35 - 210 * q^37 - 18 * q^39 + 414 * q^41 + 168 * q^43 - 54 * q^45 - 72 * q^47 + 147 * q^49 + 162 * q^51 - 402 * q^53 + 456 * q^55 + 252 * q^57 + 540 * q^59 - 798 * q^61 - 189 * q^63 + 1740 * q^65 + 48 * q^67 - 144 * q^69 + 456 * q^71 + 1230 * q^73 + 279 * q^75 - 336 * q^77 + 1368 * q^79 + 243 * q^81 + 60 * q^83 - 660 * q^85 - 54 * q^87 + 2742 * q^89 + 42 * q^91 - 216 * q^93 + 648 * q^95 + 1950 * q^97 + 432 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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