LMFDB - Embedded newform 1368.4.a.l.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1368,4,Mod(1,1368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1368, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1368.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1368 = 2^{3} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1368.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:	$80.7146128879$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 429x^{3} + 1657x^{2} + 46980x - 289104 $ x^5 - x^4 - 429x^3 + 1657x^2 + 46980*x - 289104
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 456)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$12.0276$ of defining polynomial
Character	$\chi$	$=$	1368.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-13.0276 q^{5} +31.0001 q^{7} +O(q^{10})$	$q-13.0276 q^{5} +31.0001 q^{7} -59.5005 q^{11} +62.8566 q^{13} +97.4381 q^{17} -19.0000 q^{19} +160.137 q^{23} +44.7185 q^{25} -300.940 q^{29} -62.6862 q^{31} -403.857 q^{35} +91.7129 q^{37} -394.042 q^{41} -395.387 q^{43} +515.283 q^{47} +618.005 q^{49} +402.412 q^{53} +775.150 q^{55} +229.471 q^{59} +16.8809 q^{61} -818.871 q^{65} +32.1239 q^{67} +141.969 q^{71} +667.595 q^{73} -1844.52 q^{77} +510.085 q^{79} -201.011 q^{83} -1269.39 q^{85} -346.868 q^{89} +1948.56 q^{91} +247.525 q^{95} -490.086 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 6 q^{5} + 10 q^{7}+O(q^{10}) $ 5 * q - 6 * q^5 + 10 * q^7	$ 5 q - 6 q^{5} + 10 q^{7} + 32 q^{11} + 36 q^{13} + 8 q^{17} - 95 q^{19} + 152 q^{23} + 241 q^{25} - 248 q^{29} + 118 q^{31} - 162 q^{35} + 472 q^{37} - 944 q^{41} + 150 q^{43} - 38 q^{47} + 1719 q^{49} - 788 q^{53} + 1270 q^{55} - 396 q^{59} + 1724 q^{61} - 2200 q^{65} + 204 q^{67} - 480 q^{71} + 2608 q^{73} - 2458 q^{77} + 786 q^{79} + 1118 q^{83} + 1798 q^{85} - 792 q^{89} + 1040 q^{91} + 114 q^{95} + 4638 q^{97}+O(q^{100}) $ 5 * q - 6 * q^5 + 10 * q^7 + 32 * q^11 + 36 * q^13 + 8 * q^17 - 95 * q^19 + 152 * q^23 + 241 * q^25 - 248 * q^29 + 118 * q^31 - 162 * q^35 + 472 * q^37 - 944 * q^41 + 150 * q^43 - 38 * q^47 + 1719 * q^49 - 788 * q^53 + 1270 * q^55 - 396 * q^59 + 1724 * q^61 - 2200 * q^65 + 204 * q^67 - 480 * q^71 + 2608 * q^73 - 2458 * q^77 + 786 * q^79 + 1118 * q^83 + 1798 * q^85 - 792 * q^89 + 1040 * q^91 + 114 * q^95 + 4638 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−13.0276	−1.16522	−0.582612	−	0.812750i	$-0.697969\pi$
$5$	−13.0276	−1.16522	−0.582612	+	0.812750i	$0.697969\pi$
$6$	0	0
$7$	31.0001	1.67385	0.836924	−	0.547320i	$-0.184352\pi$
$7$	31.0001	1.67385	0.836924	+	0.547320i	$0.184352\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−59.5005	−1.63092	−0.815459	−	0.578815i	$-0.803515\pi$
$11$	−59.5005	−1.63092	−0.815459	+	0.578815i	$0.803515\pi$
$12$	0	0
$13$	62.8566	1.34102	0.670511	−	0.741900i	$-0.266075\pi$
$13$	62.8566	1.34102	0.670511	+	0.741900i	$0.266075\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	97.4381	1.39013	0.695065	−	0.718947i	$-0.255376\pi$
$17$	97.4381	1.39013	0.695065	+	0.718947i	$0.255376\pi$
$18$	0	0
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	160.137	1.45178	0.725890	−	0.687811i	$-0.241428\pi$
$23$	160.137	1.45178	0.725890	+	0.687811i	$0.241428\pi$
$24$	0	0
$25$	44.7185	0.357748
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−300.940	−1.92701	−0.963504	−	0.267695i	$-0.913738\pi$
$29$	−300.940	−1.92701	−0.963504	+	0.267695i	$0.913738\pi$
$30$	0	0
$31$	−62.6862	−0.363186	−0.181593	−	0.983374i	$-0.558125\pi$
$31$	−62.6862	−0.363186	−0.181593	+	0.983374i	$0.558125\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−403.857	−1.95041
$36$	0	0
$37$	91.7129	0.407500	0.203750	−	0.979023i	$-0.434687\pi$
$37$	91.7129	0.407500	0.203750	+	0.979023i	$0.434687\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−394.042	−1.50095	−0.750476	−	0.660898i	$-0.770176\pi$
$41$	−394.042	−1.50095	−0.750476	+	0.660898i	$0.770176\pi$
$42$	0	0
$43$	−395.387	−1.40223	−0.701115	−	0.713048i	$-0.747314\pi$
$43$	−395.387	−1.40223	−0.701115	+	0.713048i	$0.747314\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	515.283	1.59919	0.799593	−	0.600543i	$-0.205049\pi$
$47$	515.283	1.59919	0.799593	+	0.600543i	$0.205049\pi$
$48$	0	0
$49$	618.005	1.80176
$50$	0	0
$51$	0	0
$52$	0	0
$53$	402.412	1.04293	0.521467	−	0.853271i	$-0.325385\pi$
$53$	402.412	1.04293	0.521467	+	0.853271i	$0.325385\pi$
$54$	0	0
$55$	775.150	1.90038
$56$	0	0
$57$	0	0
$58$	0	0
$59$	229.471	0.506350	0.253175	−	0.967421i	$-0.418525\pi$
$59$	229.471	0.506350	0.253175	+	0.967421i	$0.418525\pi$
$60$	0	0
$61$	16.8809	0.0354324	0.0177162	−	0.999843i	$-0.494360\pi$
$61$	16.8809	0.0354324	0.0177162	+	0.999843i	$0.494360\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−818.871	−1.56259
$66$	0	0
$67$	32.1239	0.0585755	0.0292877	−	0.999571i	$-0.490676\pi$
$67$	32.1239	0.0585755	0.0292877	+	0.999571i	$0.490676\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	141.969	0.237304	0.118652	−	0.992936i	$-0.462143\pi$
$71$	141.969	0.237304	0.118652	+	0.992936i	$0.462143\pi$
$72$	0	0
$73$	667.595	1.07036	0.535179	−	0.844739i	$-0.320244\pi$
$73$	667.595	1.07036	0.535179	+	0.844739i	$0.320244\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1844.52	−2.72991
$78$	0	0
$79$	510.085	0.726444	0.363222	−	0.931703i	$-0.381677\pi$
$79$	510.085	0.726444	0.363222	+	0.931703i	$0.381677\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−201.011	−0.265829	−0.132915	−	0.991127i	$-0.542434\pi$
$83$	−201.011	−0.265829	−0.132915	+	0.991127i	$0.542434\pi$
$84$	0	0
$85$	−1269.39	−1.61981
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−346.868	−0.413123	−0.206561	−	0.978434i	$-0.566227\pi$
$89$	−346.868	−0.413123	−0.206561	+	0.978434i	$0.566227\pi$
$90$	0	0
$91$	1948.56	2.24467
$92$	0	0
$93$	0	0
$94$	0	0
$95$	247.525	0.267321
$96$	0	0
$97$	−490.086	−0.512996	−0.256498	−	0.966545i	$-0.582569\pi$
$97$	−490.086	−0.512996	−0.256498	+	0.966545i	$0.582569\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1674.74	1.64993	0.824965	−	0.565183i	$-0.191194\pi$
$101$	1674.74	1.64993	0.824965	+	0.565183i	$0.191194\pi$
$102$	0	0
$103$	1578.63	1.51017	0.755084	−	0.655628i	$-0.227596\pi$
$103$	1578.63	1.51017	0.755084	+	0.655628i	$0.227596\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−690.036	−0.623442	−0.311721	−	0.950174i	$-0.600905\pi$
$107$	−690.036	−0.623442	−0.311721	+	0.950174i	$0.600905\pi$
$108$	0	0
$109$	−1404.55	−1.23424	−0.617118	−	0.786871i	$-0.711700\pi$
$109$	−1404.55	−1.23424	−0.617118	+	0.786871i	$0.711700\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1068.15	−0.889228	−0.444614	−	0.895722i	$-0.646659\pi$
$113$	−1068.15	−0.889228	−0.444614	+	0.895722i	$0.646659\pi$
$114$	0	0
$115$	−2086.21	−1.69165
$116$	0	0
$117$	0	0
$118$	0	0
$119$	3020.59	2.32687
$120$	0	0
$121$	2209.32	1.65989
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1045.88	0.748367
$126$	0	0
$127$	−226.490	−0.158250	−0.0791249	−	0.996865i	$-0.525213\pi$
$127$	−226.490	−0.158250	−0.0791249	+	0.996865i	$0.525213\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1390.37	0.927307	0.463653	−	0.886017i	$-0.346538\pi$
$131$	1390.37	0.927307	0.463653	+	0.886017i	$0.346538\pi$
$132$	0	0
$133$	−589.002	−0.384007
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−332.868	−0.207583	−0.103791	−	0.994599i	$-0.533097\pi$
$137$	−332.868	−0.207583	−0.103791	+	0.994599i	$0.533097\pi$
$138$	0	0
$139$	793.925	0.484459	0.242230	−	0.970219i	$-0.422121\pi$
$139$	793.925	0.484459	0.242230	+	0.970219i	$0.422121\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−3740.00	−2.18710
$144$	0	0
$145$	3920.53	2.24540
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1933.53	1.06309	0.531547	−	0.847029i	$-0.321611\pi$
$149$	1933.53	1.06309	0.531547	+	0.847029i	$0.321611\pi$
$150$	0	0
$151$	−1788.38	−0.963818	−0.481909	−	0.876221i	$-0.660056\pi$
$151$	−1788.38	−0.963818	−0.481909	+	0.876221i	$0.660056\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	816.651	0.423194
$156$	0	0
$157$	3805.65	1.93455	0.967274	−	0.253736i	$-0.0816594\pi$
$157$	3805.65	1.93455	0.967274	+	0.253736i	$0.0816594\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	4964.27	2.43006
$162$	0	0
$163$	−1780.90	−0.855770	−0.427885	−	0.903833i	$-0.640741\pi$
$163$	−1780.90	−0.855770	−0.427885	+	0.903833i	$0.640741\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3408.05	1.57918	0.789588	−	0.613637i	$-0.210294\pi$
$167$	3408.05	1.57918	0.789588	+	0.613637i	$0.210294\pi$
$168$	0	0
$169$	1753.95	0.798339
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2688.43	1.18149	0.590744	−	0.806859i	$-0.298834\pi$
$173$	2688.43	1.18149	0.590744	+	0.806859i	$0.298834\pi$
$174$	0	0
$175$	1386.28	0.598816
$176$	0	0
$177$	0	0
$178$	0	0
$179$	233.937	0.0976832	0.0488416	−	0.998807i	$-0.484447\pi$
$179$	233.937	0.0976832	0.0488416	+	0.998807i	$0.484447\pi$
$180$	0	0
$181$	2403.03	0.986828	0.493414	−	0.869795i	$-0.335749\pi$
$181$	2403.03	0.986828	0.493414	+	0.869795i	$0.335749\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−1194.80	−0.474829
$186$	0	0
$187$	−5797.62	−2.26719
$188$	0	0
$189$	0	0
$190$	0	0
$191$	3249.46	1.23101	0.615505	−	0.788133i	$-0.288952\pi$
$191$	3249.46	1.23101	0.615505	+	0.788133i	$0.288952\pi$
$192$	0	0
$193$	−644.839	−0.240500	−0.120250	−	0.992744i	$-0.538370\pi$
$193$	−644.839	−0.240500	−0.120250	+	0.992744i	$0.538370\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	664.368	0.240276	0.120138	−	0.992757i	$-0.461666\pi$
$197$	664.368	0.240276	0.120138	+	0.992757i	$0.461666\pi$
$198$	0	0
$199$	178.095	0.0634413	0.0317206	−	0.999497i	$-0.489901\pi$
$199$	178.095	0.0634413	0.0317206	+	0.999497i	$0.489901\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−9329.18	−3.22552
$204$	0	0
$205$	5133.42	1.74895
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1130.51	0.374158
$210$	0	0
$211$	3530.49	1.15189	0.575945	−	0.817489i	$-0.304634\pi$
$211$	3530.49	1.15189	0.575945	+	0.817489i	$0.304634\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	5150.94	1.63391
$216$	0	0
$217$	−1943.28	−0.607919
$218$	0	0
$219$	0	0
$220$	0	0
$221$	6124.63	1.86419
$222$	0	0
$223$	4510.24	1.35438	0.677192	−	0.735806i	$-0.263197\pi$
$223$	4510.24	1.35438	0.677192	+	0.735806i	$0.263197\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−996.009	−0.291222	−0.145611	−	0.989342i	$-0.546515\pi$
$227$	−996.009	−0.291222	−0.145611	+	0.989342i	$0.546515\pi$
$228$	0	0
$229$	6120.31	1.76612	0.883060	−	0.469260i	$-0.155479\pi$
$229$	6120.31	1.76612	0.883060	+	0.469260i	$0.155479\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	2071.81	0.582526	0.291263	−	0.956643i	$-0.405925\pi$
$233$	2071.81	0.582526	0.291263	+	0.956643i	$0.405925\pi$
$234$	0	0
$235$	−6712.90	−1.86341
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−4365.41	−1.18148	−0.590742	−	0.806860i	$-0.701165\pi$
$239$	−4365.41	−1.18148	−0.590742	+	0.806860i	$0.701165\pi$
$240$	0	0
$241$	3367.75	0.900148	0.450074	−	0.892991i	$-0.351398\pi$
$241$	3367.75	0.900148	0.450074	+	0.892991i	$0.351398\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−8051.13	−2.09946
$246$	0	0
$247$	−1194.28	−0.307651
$248$	0	0
$249$	0	0
$250$	0	0
$251$	45.5268	0.0114487	0.00572436	−	0.999984i	$-0.498178\pi$
$251$	45.5268	0.0114487	0.00572436	+	0.999984i	$0.498178\pi$
$252$	0	0
$253$	−9528.26	−2.36773
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2487.81	0.603835	0.301917	−	0.953334i	$-0.402373\pi$
$257$	2487.81	0.603835	0.301917	+	0.953334i	$0.402373\pi$
$258$	0	0
$259$	2843.11	0.682093
$260$	0	0
$261$	0	0
$262$	0	0
$263$	442.448	0.103736	0.0518680	−	0.998654i	$-0.483483\pi$
$263$	442.448	0.103736	0.0518680	+	0.998654i	$0.483483\pi$
$264$	0	0
$265$	−5242.46	−1.21525
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−6900.44	−1.56404	−0.782021	−	0.623252i	$-0.785811\pi$
$269$	−6900.44	−1.56404	−0.782021	+	0.623252i	$0.785811\pi$
$270$	0	0
$271$	−7017.95	−1.57310	−0.786550	−	0.617527i	$-0.788135\pi$
$271$	−7017.95	−1.57310	−0.786550	+	0.617527i	$0.788135\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−2660.78	−0.583458
$276$	0	0
$277$	1902.97	0.412773	0.206386	−	0.978471i	$-0.433830\pi$
$277$	1902.97	0.412773	0.206386	+	0.978471i	$0.433830\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	3948.25	0.838196	0.419098	−	0.907941i	$-0.362346\pi$
$281$	3948.25	0.838196	0.419098	+	0.907941i	$0.362346\pi$
$282$	0	0
$283$	−4772.76	−1.00251	−0.501256	−	0.865299i	$-0.667129\pi$
$283$	−4772.76	−1.00251	−0.501256	+	0.865299i	$0.667129\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−12215.3	−2.51236
$288$	0	0
$289$	4581.18	0.932462
$290$	0	0
$291$	0	0
$292$	0	0
$293$	4860.24	0.969072	0.484536	−	0.874771i	$-0.338988\pi$
$293$	4860.24	0.969072	0.484536	+	0.874771i	$0.338988\pi$
$294$	0	0
$295$	−2989.46	−0.590011
$296$	0	0
$297$	0	0
$298$	0	0
$299$	10065.7	1.94687
$300$	0	0
$301$	−12257.0	−2.34712
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−219.918	−0.0412867
$306$	0	0
$307$	−1100.65	−0.204617	−0.102309	−	0.994753i	$-0.532623\pi$
$307$	−1100.65	−0.204617	−0.102309	+	0.994753i	$0.532623\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−3585.77	−0.653796	−0.326898	−	0.945060i	$-0.606003\pi$
$311$	−3585.77	−0.653796	−0.326898	+	0.945060i	$0.606003\pi$
$312$	0	0
$313$	−144.081	−0.0260190	−0.0130095	−	0.999915i	$-0.504141\pi$
$313$	−144.081	−0.0260190	−0.0130095	+	0.999915i	$0.504141\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−2165.75	−0.383725	−0.191862	−	0.981422i	$-0.561453\pi$
$317$	−2165.75	−0.383725	−0.191862	+	0.981422i	$0.561453\pi$
$318$	0	0
$319$	17906.1	3.14279
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−1851.32	−0.318918
$324$	0	0
$325$	2810.85	0.479748
$326$	0	0
$327$	0	0
$328$	0	0
$329$	15973.8	2.67679
$330$	0	0
$331$	−6186.35	−1.02729	−0.513644	−	0.858003i	$-0.671705\pi$
$331$	−6186.35	−1.02729	−0.513644	+	0.858003i	$0.671705\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−418.497	−0.0682536
$336$	0	0
$337$	4898.50	0.791805	0.395903	−	0.918293i	$-0.370432\pi$
$337$	4898.50	0.791805	0.395903	+	0.918293i	$0.370432\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	3729.86	0.592327
$342$	0	0
$343$	8525.19	1.34203
$344$	0	0
$345$	0	0
$346$	0	0
$347$	7844.62	1.21361	0.606803	−	0.794852i	$-0.292452\pi$
$347$	7844.62	1.21361	0.606803	+	0.794852i	$0.292452\pi$
$348$	0	0
$349$	6475.56	0.993206	0.496603	−	0.867978i	$-0.334581\pi$
$349$	6475.56	0.993206	0.496603	+	0.867978i	$0.334581\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	8038.95	1.21210	0.606049	−	0.795428i	$-0.292754\pi$
$353$	8038.95	1.21210	0.606049	+	0.795428i	$0.292754\pi$
$354$	0	0
$355$	−1849.51	−0.276512
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−8588.71	−1.26266	−0.631330	−	0.775514i	$-0.717491\pi$
$359$	−8588.71	−1.26266	−0.631330	+	0.775514i	$0.717491\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−8697.17	−1.24721
$366$	0	0
$367$	5958.68	0.847521	0.423761	−	0.905774i	$-0.360710\pi$
$367$	5958.68	0.847521	0.423761	+	0.905774i	$0.360710\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	12474.8	1.74571
$372$	0	0
$373$	−934.427	−0.129713	−0.0648563	−	0.997895i	$-0.520659\pi$
$373$	−934.427	−0.129713	−0.0648563	+	0.997895i	$0.520659\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−18916.1	−2.58416
$378$	0	0
$379$	9436.55	1.27895	0.639476	−	0.768811i	$-0.279151\pi$
$379$	9436.55	1.27895	0.639476	+	0.768811i	$0.279151\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	6866.98	0.916152	0.458076	−	0.888913i	$-0.348539\pi$
$383$	6866.98	0.916152	0.458076	+	0.888913i	$0.348539\pi$
$384$	0	0
$385$	24029.7	3.18095
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−4845.57	−0.631569	−0.315784	−	0.948831i	$-0.602268\pi$
$389$	−4845.57	−0.631569	−0.315784	+	0.948831i	$0.602268\pi$
$390$	0	0
$391$	15603.5	2.01816
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−6645.19	−0.846471
$396$	0	0
$397$	−5372.79	−0.679226	−0.339613	−	0.940565i	$-0.610296\pi$
$397$	−5372.79	−0.679226	−0.339613	+	0.940565i	$0.610296\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	1898.65	0.236444	0.118222	−	0.992987i	$-0.462281\pi$
$401$	1898.65	0.236444	0.118222	+	0.992987i	$0.462281\pi$
$402$	0	0
$403$	−3940.24	−0.487041
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−5456.97	−0.664599
$408$	0	0
$409$	−5162.67	−0.624151	−0.312075	−	0.950057i	$-0.601024\pi$
$409$	−5162.67	−0.624151	−0.312075	+	0.950057i	$0.601024\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	7113.63	0.847552
$414$	0	0
$415$	2618.69	0.309751
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−5585.78	−0.651273	−0.325636	−	0.945495i	$-0.605579\pi$
$419$	−5585.78	−0.651273	−0.325636	+	0.945495i	$0.605579\pi$
$420$	0	0
$421$	−7494.01	−0.867543	−0.433771	−	0.901023i	$-0.642817\pi$
$421$	−7494.01	−0.867543	−0.433771	+	0.901023i	$0.642817\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	4357.29	0.497317
$426$	0	0
$427$	523.309	0.0593085
$428$	0	0
$429$	0	0
$430$	0	0
$431$	1681.39	0.187911	0.0939555	−	0.995576i	$-0.470049\pi$
$431$	1681.39	0.187911	0.0939555	+	0.995576i	$0.470049\pi$
$432$	0	0
$433$	−2831.97	−0.314309	−0.157154	−	0.987574i	$-0.550232\pi$
$433$	−2831.97	−0.314309	−0.157154	+	0.987574i	$0.550232\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−3042.61	−0.333061
$438$	0	0
$439$	−12181.2	−1.32432	−0.662160	−	0.749362i	$-0.730360\pi$
$439$	−12181.2	−1.32432	−0.662160	+	0.749362i	$0.730360\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−5359.23	−0.574774	−0.287387	−	0.957815i	$-0.592787\pi$
$443$	−5359.23	−0.574774	−0.287387	+	0.957815i	$0.592787\pi$
$444$	0	0
$445$	4518.86	0.481381
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−1067.88	−0.112241	−0.0561204	−	0.998424i	$-0.517873\pi$
$449$	−1067.88	−0.112241	−0.0561204	+	0.998424i	$0.517873\pi$
$450$	0	0
$451$	23445.7	2.44793
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−25385.1	−2.61554
$456$	0	0
$457$	6609.03	0.676493	0.338246	−	0.941058i	$-0.390166\pi$
$457$	6609.03	0.676493	0.338246	+	0.941058i	$0.390166\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	10389.8	1.04968	0.524839	−	0.851201i	$-0.324126\pi$
$461$	10389.8	1.04968	0.524839	+	0.851201i	$0.324126\pi$
$462$	0	0
$463$	−8045.64	−0.807587	−0.403794	−	0.914850i	$-0.632309\pi$
$463$	−8045.64	−0.807587	−0.403794	+	0.914850i	$0.632309\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	6017.45	0.596262	0.298131	−	0.954525i	$-0.403637\pi$
$467$	6017.45	0.596262	0.298131	+	0.954525i	$0.403637\pi$
$468$	0	0
$469$	995.843	0.0980464
$470$	0	0
$471$	0	0
$472$	0	0
$473$	23525.7	2.28692
$474$	0	0
$475$	−849.652	−0.0820731
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−10882.9	−1.03811	−0.519053	−	0.854742i	$-0.673715\pi$
$479$	−10882.9	−1.03811	−0.519053	+	0.854742i	$0.673715\pi$
$480$	0	0
$481$	5764.76	0.546466
$482$	0	0
$483$	0	0
$484$	0	0
$485$	6384.64	0.597756
$486$	0	0
$487$	−16031.6	−1.49171	−0.745853	−	0.666110i	$-0.767958\pi$
$487$	−16031.6	−1.49171	−0.745853	+	0.666110i	$0.767958\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−950.970	−0.0874066	−0.0437033	−	0.999045i	$-0.513916\pi$
$491$	−950.970	−0.0874066	−0.0437033	+	0.999045i	$0.513916\pi$
$492$	0	0
$493$	−29323.1	−2.67879
$494$	0	0
$495$	0	0
$496$	0	0
$497$	4401.04	0.397211
$498$	0	0
$499$	−10181.9	−0.913433	−0.456717	−	0.889612i	$-0.650975\pi$
$499$	−10181.9	−0.913433	−0.456717	+	0.889612i	$0.650975\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−18051.2	−1.60013	−0.800064	−	0.599914i	$-0.795201\pi$
$503$	−18051.2	−1.60013	−0.800064	+	0.599914i	$0.795201\pi$
$504$	0	0
$505$	−21817.9	−1.92254
$506$	0	0
$507$	0	0
$508$	0	0
$509$	11909.7	1.03711	0.518553	−	0.855045i	$-0.326471\pi$
$509$	11909.7	1.03711	0.518553	+	0.855045i	$0.326471\pi$
$510$	0	0
$511$	20695.5	1.79162
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−20565.8	−1.75969
$516$	0	0
$517$	−30659.6	−2.60814
$518$	0	0
$519$	0	0
$520$	0	0
$521$	471.147	0.0396186	0.0198093	−	0.999804i	$-0.493694\pi$
$521$	471.147	0.0396186	0.0198093	+	0.999804i	$0.493694\pi$
$522$	0	0
$523$	−13357.3	−1.11677	−0.558387	−	0.829581i	$-0.688579\pi$
$523$	−13357.3	−1.11677	−0.558387	+	0.829581i	$0.688579\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−6108.03	−0.504876
$528$	0	0
$529$	13476.9	1.10766
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−24768.1	−2.01281
$534$	0	0
$535$	8989.52	0.726450
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−36771.7	−2.93853
$540$	0	0
$541$	−12005.7	−0.954092	−0.477046	−	0.878878i	$-0.658292\pi$
$541$	−12005.7	−0.954092	−0.477046	+	0.878878i	$0.658292\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	18298.0	1.43816
$546$	0	0
$547$	6281.27	0.490983	0.245492	−	0.969399i	$-0.421051\pi$
$547$	6281.27	0.490983	0.245492	+	0.969399i	$0.421051\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	5717.87	0.442086
$552$	0	0
$553$	15812.7	1.21596
$554$	0	0
$555$	0	0
$556$	0	0
$557$	13224.7	1.00601	0.503006	−	0.864283i	$-0.332228\pi$
$557$	13224.7	1.00601	0.503006	+	0.864283i	$0.332228\pi$
$558$	0	0
$559$	−24852.7	−1.88042
$560$	0	0
$561$	0	0
$562$	0	0
$563$	4519.29	0.338305	0.169152	−	0.985590i	$-0.445897\pi$
$563$	4519.29	0.338305	0.169152	+	0.985590i	$0.445897\pi$
$564$	0	0
$565$	13915.4	1.03615
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−24662.4	−1.81705	−0.908527	−	0.417826i	$-0.862792\pi$
$569$	−24662.4	−1.81705	−0.908527	+	0.417826i	$0.862792\pi$
$570$	0	0
$571$	1895.50	0.138922	0.0694609	−	0.997585i	$-0.477872\pi$
$571$	1895.50	0.138922	0.0694609	+	0.997585i	$0.477872\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	7161.11	0.519372
$576$	0	0
$577$	−20854.8	−1.50467	−0.752335	−	0.658780i	$-0.771073\pi$
$577$	−20854.8	−1.50467	−0.752335	+	0.658780i	$0.771073\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−6231.36	−0.444958
$582$	0	0
$583$	−23943.7	−1.70094
$584$	0	0
$585$	0	0
$586$	0	0
$587$	17405.4	1.22384	0.611922	−	0.790918i	$-0.290396\pi$
$587$	17405.4	1.22384	0.611922	+	0.790918i	$0.290396\pi$
$588$	0	0
$589$	1191.04	0.0833207
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−1411.87	−0.0977713	−0.0488856	−	0.998804i	$-0.515567\pi$
$593$	−1411.87	−0.0977713	−0.0488856	+	0.998804i	$0.515567\pi$
$594$	0	0
$595$	−39351.1	−2.71132
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−10497.0	−0.716017	−0.358009	−	0.933718i	$-0.616544\pi$
$599$	−10497.0	−0.716017	−0.358009	+	0.933718i	$0.616544\pi$
$600$	0	0
$601$	−14567.9	−0.988745	−0.494373	−	0.869250i	$-0.664602\pi$
$601$	−14567.9	−0.988745	−0.494373	+	0.869250i	$0.664602\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−28782.1	−1.93415
$606$	0	0
$607$	10020.5	0.670047	0.335024	−	0.942210i	$-0.391256\pi$
$607$	10020.5	0.670047	0.335024	+	0.942210i	$0.391256\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	32388.9	2.14454
$612$	0	0
$613$	18080.2	1.19128	0.595639	−	0.803252i	$-0.296899\pi$
$613$	18080.2	1.19128	0.595639	+	0.803252i	$0.296899\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−8530.16	−0.556583	−0.278291	−	0.960497i	$-0.589768\pi$
$617$	−8530.16	−0.556583	−0.278291	+	0.960497i	$0.589768\pi$
$618$	0	0
$619$	−24032.5	−1.56050	−0.780248	−	0.625471i	$-0.784907\pi$
$619$	−24032.5	−1.56050	−0.780248	+	0.625471i	$0.784907\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−10752.9	−0.691504
$624$	0	0
$625$	−19215.1	−1.22976
$626$	0	0
$627$	0	0
$628$	0	0
$629$	8936.33	0.566478
$630$	0	0
$631$	−25978.7	−1.63898	−0.819491	−	0.573092i	$-0.805744\pi$
$631$	−25978.7	−1.63898	−0.819491	+	0.573092i	$0.805744\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	2950.62	0.184396
$636$	0	0
$637$	38845.7	2.41621
$638$	0	0
$639$	0	0
$640$	0	0
$641$	18647.2	1.14902	0.574508	−	0.818499i	$-0.305193\pi$
$641$	18647.2	1.14902	0.574508	+	0.818499i	$0.305193\pi$
$642$	0	0
$643$	9675.48	0.593412	0.296706	−	0.954969i	$-0.404112\pi$
$643$	9675.48	0.593412	0.296706	+	0.954969i	$0.404112\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	4099.69	0.249112	0.124556	−	0.992213i	$-0.460249\pi$
$647$	4099.69	0.249112	0.124556	+	0.992213i	$0.460249\pi$
$648$	0	0
$649$	−13653.7	−0.825815
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−19905.3	−1.19289	−0.596444	−	0.802654i	$-0.703420\pi$
$653$	−19905.3	−1.19289	−0.596444	+	0.802654i	$0.703420\pi$
$654$	0	0
$655$	−18113.2	−1.08052
$656$	0	0
$657$	0	0
$658$	0	0
$659$	15735.3	0.930137	0.465069	−	0.885275i	$-0.346030\pi$
$659$	15735.3	0.930137	0.465069	+	0.885275i	$0.346030\pi$
$660$	0	0
$661$	20697.3	1.21790	0.608950	−	0.793208i	$-0.291591\pi$
$661$	20697.3	1.21790	0.608950	+	0.793208i	$0.291591\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	7673.28	0.447454
$666$	0	0
$667$	−48191.8	−2.79759
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−1004.42	−0.0577873
$672$	0	0
$673$	2020.56	0.115731	0.0578653	−	0.998324i	$-0.481571\pi$
$673$	2020.56	0.115731	0.0578653	+	0.998324i	$0.481571\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	16230.7	0.921412	0.460706	−	0.887553i	$-0.347596\pi$
$677$	16230.7	0.921412	0.460706	+	0.887553i	$0.347596\pi$
$678$	0	0
$679$	−15192.7	−0.858678
$680$	0	0
$681$	0	0
$682$	0	0
$683$	20687.2	1.15897	0.579483	−	0.814984i	$-0.303254\pi$
$683$	20687.2	1.15897	0.579483	+	0.814984i	$0.303254\pi$
$684$	0	0
$685$	4336.47	0.241880
$686$	0	0
$687$	0	0
$688$	0	0
$689$	25294.2	1.39860
$690$	0	0
$691$	32568.0	1.79298	0.896488	−	0.443069i	$-0.146110\pi$
$691$	32568.0	1.79298	0.896488	+	0.443069i	$0.146110\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−10342.9	−0.564504
$696$	0	0
$697$	−38394.7	−2.08652
$698$	0	0
$699$	0	0
$700$	0	0
$701$	32340.2	1.74247	0.871235	−	0.490867i	$-0.163320\pi$
$701$	32340.2	1.74247	0.871235	+	0.490867i	$0.163320\pi$
$702$	0	0
$703$	−1742.54	−0.0934870
$704$	0	0
$705$	0	0
$706$	0	0
$707$	51917.1	2.76173
$708$	0	0
$709$	−23319.0	−1.23521	−0.617604	−	0.786490i	$-0.711896\pi$
$709$	−23319.0	−1.23521	−0.617604	+	0.786490i	$0.711896\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−10038.4	−0.527267
$714$	0	0
$715$	48723.3	2.54846
$716$	0	0
$717$	0	0
$718$	0	0
$719$	148.353	0.00769490	0.00384745	−	0.999993i	$-0.498775\pi$
$719$	148.353	0.00769490	0.00384745	+	0.999993i	$0.498775\pi$
$720$	0	0
$721$	48937.8	2.52779
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−13457.6	−0.689384
$726$	0	0
$727$	13673.8	0.697568	0.348784	−	0.937203i	$-0.386595\pi$
$727$	13673.8	0.697568	0.348784	+	0.937203i	$0.386595\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−38525.7	−1.94928
$732$	0	0
$733$	−8125.20	−0.409428	−0.204714	−	0.978822i	$-0.565626\pi$
$733$	−8125.20	−0.409428	−0.204714	+	0.978822i	$0.565626\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1911.39	−0.0955317
$738$	0	0
$739$	24462.8	1.21770	0.608848	−	0.793287i	$-0.291632\pi$
$739$	24462.8	1.21770	0.608848	+	0.793287i	$0.291632\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1250.21	0.0617307	0.0308653	−	0.999524i	$-0.490174\pi$
$743$	1250.21	0.0617307	0.0308653	+	0.999524i	$0.490174\pi$
$744$	0	0
$745$	−25189.3	−1.23874
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−21391.2	−1.04355
$750$	0	0
$751$	−33036.8	−1.60523	−0.802615	−	0.596497i	$-0.796559\pi$
$751$	−33036.8	−1.60523	−0.802615	+	0.596497i	$0.796559\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	23298.3	1.12306
$756$	0	0
$757$	21676.1	1.04073	0.520364	−	0.853944i	$-0.325796\pi$
$757$	21676.1	1.04073	0.520364	+	0.853944i	$0.325796\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−24782.3	−1.18050	−0.590248	−	0.807222i	$-0.700970\pi$
$761$	−24782.3	−1.18050	−0.590248	+	0.807222i	$0.700970\pi$
$762$	0	0
$763$	−43541.2	−2.06592
$764$	0	0
$765$	0	0
$766$	0	0
$767$	14423.8	0.679026
$768$	0	0
$769$	12723.5	0.596647	0.298324	−	0.954465i	$-0.403573\pi$
$769$	12723.5	0.596647	0.298324	+	0.954465i	$0.403573\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−20885.9	−0.971815	−0.485908	−	0.874010i	$-0.661511\pi$
$773$	−20885.9	−0.971815	−0.485908	+	0.874010i	$0.661511\pi$
$774$	0	0
$775$	−2803.24	−0.129929
$776$	0	0
$777$	0	0
$778$	0	0
$779$	7486.80	0.344342
$780$	0	0
$781$	−8447.22	−0.387023
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−49578.5	−2.25418
$786$	0	0
$787$	−17419.4	−0.788990	−0.394495	−	0.918898i	$-0.629080\pi$
$787$	−17419.4	−0.788990	−0.394495	+	0.918898i	$0.629080\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−33112.6	−1.48843
$792$	0	0
$793$	1061.08	0.0475156
$794$	0	0
$795$	0	0
$796$	0	0
$797$	717.027	0.0318675	0.0159337	−	0.999873i	$-0.494928\pi$
$797$	717.027	0.0318675	0.0159337	+	0.999873i	$0.494928\pi$
$798$	0	0
$799$	50208.2	2.22308
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−39722.3	−1.74567
$804$	0	0
$805$	−64672.6	−2.83156
$806$	0	0
$807$	0	0
$808$	0	0
$809$	2647.79	0.115070	0.0575349	−	0.998343i	$-0.481676\pi$
$809$	2647.79	0.115070	0.0575349	+	0.998343i	$0.481676\pi$
$810$	0	0
$811$	33532.0	1.45187	0.725935	−	0.687763i	$-0.241407\pi$
$811$	33532.0	1.45187	0.725935	+	0.687763i	$0.241407\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	23200.8	0.997164
$816$	0	0
$817$	7512.35	0.321694
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−26557.3	−1.12893	−0.564467	−	0.825456i	$-0.690918\pi$
$821$	−26557.3	−1.12893	−0.564467	+	0.825456i	$0.690918\pi$
$822$	0	0
$823$	19710.1	0.834812	0.417406	−	0.908720i	$-0.362939\pi$
$823$	19710.1	0.834812	0.417406	+	0.908720i	$0.362939\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	308.144	0.0129567	0.00647836	−	0.999979i	$-0.497938\pi$
$827$	308.144	0.0129567	0.00647836	+	0.999979i	$0.497938\pi$
$828$	0	0
$829$	44283.9	1.85530	0.927651	−	0.373449i	$-0.121825\pi$
$829$	44283.9	1.85530	0.927651	+	0.373449i	$0.121825\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	60217.3	2.50469
$834$	0	0
$835$	−44398.7	−1.84010
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−13160.1	−0.541520	−0.270760	−	0.962647i	$-0.587275\pi$
$839$	−13160.1	−0.541520	−0.270760	+	0.962647i	$0.587275\pi$
$840$	0	0
$841$	66176.1	2.71336
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−22849.8	−0.930244
$846$	0	0
$847$	68489.0	2.77840
$848$	0	0
$849$	0	0
$850$	0	0
$851$	14686.7	0.591600
$852$	0	0
$853$	−11147.5	−0.447461	−0.223731	−	0.974651i	$-0.571824\pi$
$853$	−11147.5	−0.447461	−0.223731	+	0.974651i	$0.571824\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	17623.3	0.702451	0.351226	−	0.936291i	$-0.385765\pi$
$857$	17623.3	0.702451	0.351226	+	0.936291i	$0.385765\pi$
$858$	0	0
$859$	−16188.4	−0.643006	−0.321503	−	0.946909i	$-0.604188\pi$
$859$	−16188.4	−0.643006	−0.321503	+	0.946909i	$0.604188\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−14921.5	−0.588568	−0.294284	−	0.955718i	$-0.595081\pi$
$863$	−14921.5	−0.588568	−0.294284	+	0.955718i	$0.595081\pi$
$864$	0	0
$865$	−35023.8	−1.37670
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−30350.4	−1.18477
$870$	0	0
$871$	2019.20	0.0785509
$872$	0	0
$873$	0	0
$874$	0	0
$875$	32422.2	1.25265
$876$	0	0
$877$	−8862.90	−0.341253	−0.170627	−	0.985336i	$-0.554579\pi$
$877$	−8862.90	−0.341253	−0.170627	+	0.985336i	$0.554579\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	37762.2	1.44409	0.722044	−	0.691847i	$-0.243203\pi$
$881$	37762.2	1.44409	0.722044	+	0.691847i	$0.243203\pi$
$882$	0	0
$883$	6748.33	0.257191	0.128595	−	0.991697i	$-0.458953\pi$
$883$	6748.33	0.257191	0.128595	+	0.991697i	$0.458953\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	18432.7	0.697757	0.348878	−	0.937168i	$-0.386563\pi$
$887$	18432.7	0.697757	0.348878	+	0.937168i	$0.386563\pi$
$888$	0	0
$889$	−7021.20	−0.264886
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−9790.37	−0.366878
$894$	0	0
$895$	−3047.64	−0.113823
$896$	0	0
$897$	0	0
$898$	0	0
$899$	18864.8	0.699863
$900$	0	0
$901$	39210.2	1.44981
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−31305.7	−1.14988
$906$	0	0
$907$	−27498.5	−1.00670	−0.503348	−	0.864084i	$-0.667899\pi$
$907$	−27498.5	−1.00670	−0.503348	+	0.864084i	$0.667899\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−18214.9	−0.662444	−0.331222	−	0.943553i	$-0.607461\pi$
$911$	−18214.9	−0.662444	−0.331222	+	0.943553i	$0.607461\pi$
$912$	0	0
$913$	11960.3	0.433545
$914$	0	0
$915$	0	0
$916$	0	0
$917$	43101.6	1.55217
$918$	0	0
$919$	−24596.3	−0.882868	−0.441434	−	0.897294i	$-0.645530\pi$
$919$	−24596.3	−0.882868	−0.441434	+	0.897294i	$0.645530\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	8923.67	0.318230
$924$	0	0
$925$	4101.27	0.145783
$926$	0	0
$927$	0	0
$928$	0	0
$929$	29231.1	1.03234	0.516169	−	0.856487i	$-0.327358\pi$
$929$	29231.1	1.03234	0.516169	+	0.856487i	$0.327358\pi$
$930$	0	0
$931$	−11742.1	−0.413353
$932$	0	0
$933$	0	0
$934$	0	0
$935$	75529.1	2.64178
$936$	0	0
$937$	4892.08	0.170563	0.0852814	−	0.996357i	$-0.472821\pi$
$937$	4892.08	0.170563	0.0852814	+	0.996357i	$0.472821\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	34958.5	1.21107	0.605534	−	0.795819i	$-0.292960\pi$
$941$	34958.5	1.21107	0.605534	+	0.795819i	$0.292960\pi$
$942$	0	0
$943$	−63100.8	−2.17905
$944$	0	0
$945$	0	0
$946$	0	0
$947$	17799.6	0.610782	0.305391	−	0.952227i	$-0.401213\pi$
$947$	17799.6	0.610782	0.305391	+	0.952227i	$0.401213\pi$
$948$	0	0
$949$	41962.8	1.43537
$950$	0	0
$951$	0	0
$952$	0	0
$953$	24499.9	0.832770	0.416385	−	0.909188i	$-0.363297\pi$
$953$	24499.9	0.832770	0.416385	+	0.909188i	$0.363297\pi$
$954$	0	0
$955$	−42332.7	−1.43440
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−10318.9	−0.347462
$960$	0	0
$961$	−25861.4	−0.868096
$962$	0	0
$963$	0	0
$964$	0	0
$965$	8400.70	0.280237
$966$	0	0
$967$	−22010.6	−0.731967	−0.365983	−	0.930621i	$-0.619267\pi$
$967$	−22010.6	−0.731967	−0.365983	+	0.930621i	$0.619267\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−56047.3	−1.85236	−0.926181	−	0.377080i	$-0.876928\pi$
$971$	−56047.3	−1.85236	−0.926181	+	0.377080i	$0.876928\pi$
$972$	0	0
$973$	24611.7	0.810910
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−54839.8	−1.79578	−0.897891	−	0.440217i	$-0.854901\pi$
$977$	−54839.8	−1.79578	−0.897891	+	0.440217i	$0.854901\pi$
$978$	0	0
$979$	20638.8	0.673769
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−24595.7	−0.798047	−0.399023	−	0.916941i	$-0.630651\pi$
$983$	−24595.7	−0.798047	−0.399023	+	0.916941i	$0.630651\pi$
$984$	0	0
$985$	−8655.13	−0.279975
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−63316.2	−2.03573
$990$	0	0
$991$	1123.56	0.0360151	0.0180075	−	0.999838i	$-0.494268\pi$
$991$	1123.56	0.0360151	0.0180075	+	0.999838i	$0.494268\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−2320.15	−0.0739233
$996$	0	0
$997$	10435.8	0.331499	0.165750	−	0.986168i	$-0.446996\pi$
$997$	10435.8	0.331499	0.165750	+	0.986168i	$0.446996\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1368.4.a.l.1.2		5
3.2	odd	2		456.4.a.h.1.4	✓	5
12.11	even	2		912.4.a.w.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
456.4.a.h.1.4	✓	5	3.2	odd	2
912.4.a.w.1.4		5	12.11	even	2
1368.4.a.l.1.2		5	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1368 = 2^{3} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1368.a (trivial)

\(f(q)\)	\(=\)	\(q-13.0276 q^{5} +31.0001 q^{7} +O(q^{10})\)	\(q-13.0276 q^{5} +31.0001 q^{7} -59.5005 q^{11} +62.8566 q^{13} +97.4381 q^{17} -19.0000 q^{19} +160.137 q^{23} +44.7185 q^{25} -300.940 q^{29} -62.6862 q^{31} -403.857 q^{35} +91.7129 q^{37} -394.042 q^{41} -395.387 q^{43} +515.283 q^{47} +618.005 q^{49} +402.412 q^{53} +775.150 q^{55} +229.471 q^{59} +16.8809 q^{61} -818.871 q^{65} +32.1239 q^{67} +141.969 q^{71} +667.595 q^{73} -1844.52 q^{77} +510.085 q^{79} -201.011 q^{83} -1269.39 q^{85} -346.868 q^{89} +1948.56 q^{91} +247.525 q^{95} -490.086 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 6 q^{5} + 10 q^{7}+O(q^{10}) \) 5 * q - 6 * q^5 + 10 * q^7	\( 5 q - 6 q^{5} + 10 q^{7} + 32 q^{11} + 36 q^{13} + 8 q^{17} - 95 q^{19} + 152 q^{23} + 241 q^{25} - 248 q^{29} + 118 q^{31} - 162 q^{35} + 472 q^{37} - 944 q^{41} + 150 q^{43} - 38 q^{47} + 1719 q^{49} - 788 q^{53} + 1270 q^{55} - 396 q^{59} + 1724 q^{61} - 2200 q^{65} + 204 q^{67} - 480 q^{71} + 2608 q^{73} - 2458 q^{77} + 786 q^{79} + 1118 q^{83} + 1798 q^{85} - 792 q^{89} + 1040 q^{91} + 114 q^{95} + 4638 q^{97}+O(q^{100}) \) 5 * q - 6 * q^5 + 10 * q^7 + 32 * q^11 + 36 * q^13 + 8 * q^17 - 95 * q^19 + 152 * q^23 + 241 * q^25 - 248 * q^29 + 118 * q^31 - 162 * q^35 + 472 * q^37 - 944 * q^41 + 150 * q^43 - 38 * q^47 + 1719 * q^49 - 788 * q^53 + 1270 * q^55 - 396 * q^59 + 1724 * q^61 - 2200 * q^65 + 204 * q^67 - 480 * q^71 + 2608 * q^73 - 2458 * q^77 + 786 * q^79 + 1118 * q^83 + 1798 * q^85 - 792 * q^89 + 1040 * q^91 + 114 * q^95 + 4638 * q^97

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