LMFDB - Embedded newform 1344.4.a.bt.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1344,4,Mod(1,1344)]

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

lf = mfeigenbasis(mf)

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")

//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: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$79.2985670477$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.22700.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 28x + 12 $ x^3 - x^2 - 28*x + 12
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
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			1.1
Root			$0.424864$ 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)

$f(q)$	$=$	$q-3.00000 q^{3} -15.3946 q^{5} -7.00000 q^{7} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} -15.3946 q^{5} -7.00000 q^{7} +9.00000 q^{9} -17.9957 q^{11} +17.3989 q^{13} +46.1839 q^{15} -84.3720 q^{17} -159.767 q^{19} +21.0000 q^{21} +115.968 q^{23} +111.995 q^{25} -27.0000 q^{27} -215.362 q^{29} -144.160 q^{31} +53.9872 q^{33} +107.762 q^{35} -150.960 q^{37} -52.1967 q^{39} -229.492 q^{41} -411.157 q^{43} -138.552 q^{45} +45.2107 q^{47} +49.0000 q^{49} +253.116 q^{51} +604.016 q^{53} +277.037 q^{55} +479.300 q^{57} -315.649 q^{59} -595.147 q^{61} -63.0000 q^{63} -267.850 q^{65} -311.632 q^{67} -347.903 q^{69} -358.260 q^{71} -816.660 q^{73} -335.984 q^{75} +125.970 q^{77} -137.873 q^{79} +81.0000 q^{81} -64.5399 q^{83} +1298.88 q^{85} +646.087 q^{87} +487.439 q^{89} -121.792 q^{91} +432.480 q^{93} +2459.55 q^{95} +1285.20 q^{97} -161.961 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 9 q^{3} + 10 q^{5} - 21 q^{7} + 27 q^{9}+O(q^{10}) $ 3 * q - 9 * q^3 + 10 * q^5 - 21 * q^7 + 27 * q^9	$ 3 q - 9 q^{3} + 10 q^{5} - 21 q^{7} + 27 q^{9} + 50 q^{13} - 30 q^{15} + 30 q^{17} - 140 q^{19} + 63 q^{21} + 56 q^{23} + 325 q^{25} - 81 q^{27} - 298 q^{29} - 80 q^{31} - 70 q^{35} - 10 q^{37} - 150 q^{39} + 390 q^{41} - 784 q^{43} + 90 q^{45} + 248 q^{47} + 147 q^{49} - 90 q^{51} - 10 q^{53} + 1360 q^{55} + 420 q^{57} - 1500 q^{59} + 810 q^{61} - 189 q^{63} + 860 q^{65} - 1272 q^{67} - 168 q^{69} + 160 q^{71} - 1170 q^{73} - 975 q^{75} + 840 q^{79} + 243 q^{81} - 1564 q^{83} + 2740 q^{85} + 894 q^{87} - 178 q^{89} - 350 q^{91} + 240 q^{93} + 2840 q^{95} - 130 q^{97}+O(q^{100}) $ 3 * q - 9 * q^3 + 10 * q^5 - 21 * q^7 + 27 * q^9 + 50 * q^13 - 30 * q^15 + 30 * q^17 - 140 * q^19 + 63 * q^21 + 56 * q^23 + 325 * q^25 - 81 * q^27 - 298 * q^29 - 80 * q^31 - 70 * q^35 - 10 * q^37 - 150 * q^39 + 390 * q^41 - 784 * q^43 + 90 * q^45 + 248 * q^47 + 147 * q^49 - 90 * q^51 - 10 * q^53 + 1360 * q^55 + 420 * q^57 - 1500 * q^59 + 810 * q^61 - 189 * q^63 + 860 * q^65 - 1272 * q^67 - 168 * q^69 + 160 * q^71 - 1170 * q^73 - 975 * q^75 + 840 * q^79 + 243 * q^81 - 1564 * q^83 + 2740 * q^85 + 894 * q^87 - 178 * q^89 - 350 * q^91 + 240 * q^93 + 2840 * q^95 - 130 * 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
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	−15.3946	−1.37694	−0.688469	−	0.725266i	$-0.741717\pi$
$5$	−15.3946	−1.37694	−0.688469	+	0.725266i	$0.741717\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−17.9957	−0.493265	−0.246632	−	0.969109i	$-0.579324\pi$
$11$	−17.9957	−0.493265	−0.246632	+	0.969109i	$0.579324\pi$
$12$	0	0
$13$	17.3989	0.371199	0.185600	−	0.982625i	$-0.440577\pi$
$13$	17.3989	0.371199	0.185600	+	0.982625i	$0.440577\pi$
$14$	0	0
$15$	46.1839	0.794975
$16$	0	0
$17$	−84.3720	−1.20372	−0.601860	−	0.798602i	$-0.705573\pi$
$17$	−84.3720	−1.20372	−0.601860	+	0.798602i	$0.705573\pi$
$18$	0	0
$19$	−159.767	−1.92910	−0.964552	−	0.263892i	$-0.914994\pi$
$19$	−159.767	−1.92910	−0.964552	+	0.263892i	$0.914994\pi$
$20$	0	0
$21$	21.0000	0.218218
$22$	0	0
$23$	115.968	1.05134	0.525672	−	0.850687i	$-0.323814\pi$
$23$	115.968	1.05134	0.525672	+	0.850687i	$0.323814\pi$
$24$	0	0
$25$	111.995	0.895956
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	−215.362	−1.37903	−0.689513	−	0.724273i	$-0.742176\pi$
$29$	−215.362	−1.37903	−0.689513	+	0.724273i	$0.742176\pi$
$30$	0	0
$31$	−144.160	−0.835223	−0.417612	−	0.908626i	$-0.637133\pi$
$31$	−144.160	−0.835223	−0.417612	+	0.908626i	$0.637133\pi$
$32$	0	0
$33$	53.9872	0.284787
$34$	0	0
$35$	107.762	0.520433
$36$	0	0
$37$	−150.960	−0.670749	−0.335375	−	0.942085i	$-0.608863\pi$
$37$	−150.960	−0.670749	−0.335375	+	0.942085i	$0.608863\pi$
$38$	0	0
$39$	−52.1967	−0.214312
$40$	0	0
$41$	−229.492	−0.874163	−0.437082	−	0.899422i	$-0.643988\pi$
$41$	−229.492	−0.874163	−0.437082	+	0.899422i	$0.643988\pi$
$42$	0	0
$43$	−411.157	−1.45816	−0.729080	−	0.684429i	$-0.760052\pi$
$43$	−411.157	−1.45816	−0.729080	+	0.684429i	$0.760052\pi$
$44$	0	0
$45$	−138.552	−0.458979
$46$	0	0
$47$	45.2107	0.140312	0.0701560	−	0.997536i	$-0.477650\pi$
$47$	45.2107	0.140312	0.0701560	+	0.997536i	$0.477650\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	253.116	0.694968
$52$	0	0
$53$	604.016	1.56543	0.782717	−	0.622378i	$-0.213833\pi$
$53$	604.016	1.56543	0.782717	+	0.622378i	$0.213833\pi$
$54$	0	0
$55$	277.037	0.679195
$56$	0	0
$57$	479.300	1.11377
$58$	0	0
$59$	−315.649	−0.696509	−0.348255	−	0.937400i	$-0.613226\pi$
$59$	−315.649	−0.696509	−0.348255	+	0.937400i	$0.613226\pi$
$60$	0	0
$61$	−595.147	−1.24919	−0.624597	−	0.780947i	$-0.714737\pi$
$61$	−595.147	−1.24919	−0.624597	+	0.780947i	$0.714737\pi$
$62$	0	0
$63$	−63.0000	−0.125988
$64$	0	0
$65$	−267.850	−0.511118
$66$	0	0
$67$	−311.632	−0.568238	−0.284119	−	0.958789i	$-0.591701\pi$
$67$	−311.632	−0.568238	−0.284119	+	0.958789i	$0.591701\pi$
$68$	0	0
$69$	−347.903	−0.606994
$70$	0	0
$71$	−358.260	−0.598840	−0.299420	−	0.954121i	$-0.596793\pi$
$71$	−358.260	−0.598840	−0.299420	+	0.954121i	$0.596793\pi$
$72$	0	0
$73$	−816.660	−1.30935	−0.654677	−	0.755909i	$-0.727195\pi$
$73$	−816.660	−1.30935	−0.654677	+	0.755909i	$0.727195\pi$
$74$	0	0
$75$	−335.984	−0.517281
$76$	0	0
$77$	125.970	0.186437
$78$	0	0
$79$	−137.873	−0.196354	−0.0981768	−	0.995169i	$-0.531301\pi$
$79$	−137.873	−0.196354	−0.0981768	+	0.995169i	$0.531301\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−64.5399	−0.0853514	−0.0426757	−	0.999089i	$-0.513588\pi$
$83$	−64.5399	−0.0853514	−0.0426757	+	0.999089i	$0.513588\pi$
$84$	0	0
$85$	1298.88	1.65745
$86$	0	0
$87$	646.087	0.796181
$88$	0	0
$89$	487.439	0.580544	0.290272	−	0.956944i	$-0.406254\pi$
$89$	487.439	0.580544	0.290272	+	0.956944i	$0.406254\pi$
$90$	0	0
$91$	−121.792	−0.140300
$92$	0	0
$93$	432.480	0.482216
$94$	0	0
$95$	2459.55	2.65626
$96$	0	0
$97$	1285.20	1.34528	0.672639	−	0.739970i	$-0.265161\pi$
$97$	1285.20	1.34528	0.672639	+	0.739970i	$0.265161\pi$
$98$	0	0
$99$	−161.961	−0.164422
$100$	0	0
$101$	1114.36	1.09785	0.548925	−	0.835872i	$-0.315037\pi$
$101$	1114.36	1.09785	0.548925	+	0.835872i	$0.315037\pi$
$102$	0	0
$103$	−860.304	−0.822993	−0.411497	−	0.911411i	$-0.634994\pi$
$103$	−860.304	−0.822993	−0.411497	+	0.911411i	$0.634994\pi$
$104$	0	0
$105$	−323.287	−0.300472
$106$	0	0
$107$	−1241.76	−1.12192	−0.560962	−	0.827842i	$-0.689569\pi$
$107$	−1241.76	−1.12192	−0.560962	+	0.827842i	$0.689569\pi$
$108$	0	0
$109$	1640.96	1.44198	0.720990	−	0.692945i	$-0.243687\pi$
$109$	1640.96	1.44198	0.720990	+	0.692945i	$0.243687\pi$
$110$	0	0
$111$	452.881	0.387257
$112$	0	0
$113$	1945.07	1.61926	0.809632	−	0.586938i	$-0.199667\pi$
$113$	1945.07	1.61926	0.809632	+	0.586938i	$0.199667\pi$
$114$	0	0
$115$	−1785.28	−1.44764
$116$	0	0
$117$	156.590	0.123733
$118$	0	0
$119$	590.604	0.454963
$120$	0	0
$121$	−1007.15	−0.756690
$122$	0	0
$123$	688.477	0.504699
$124$	0	0
$125$	200.214	0.143262
$126$	0	0
$127$	1423.59	0.994672	0.497336	−	0.867558i	$-0.334312\pi$
$127$	1423.59	0.994672	0.497336	+	0.867558i	$0.334312\pi$
$128$	0	0
$129$	1233.47	0.841869
$130$	0	0
$131$	−861.290	−0.574437	−0.287219	−	0.957865i	$-0.592731\pi$
$131$	−861.290	−0.574437	−0.287219	+	0.957865i	$0.592731\pi$
$132$	0	0
$133$	1118.37	0.729133
$134$	0	0
$135$	415.655	0.264992
$136$	0	0
$137$	1812.47	1.13029	0.565145	−	0.824992i	$-0.308820\pi$
$137$	1812.47	1.13029	0.565145	+	0.824992i	$0.308820\pi$
$138$	0	0
$139$	−2220.22	−1.35479	−0.677397	−	0.735618i	$-0.736892\pi$
$139$	−2220.22	−1.35479	−0.677397	+	0.735618i	$0.736892\pi$
$140$	0	0
$141$	−135.632	−0.0810092
$142$	0	0
$143$	−313.106	−0.183099
$144$	0	0
$145$	3315.42	1.89883
$146$	0	0
$147$	−147.000	−0.0824786
$148$	0	0
$149$	830.283	0.456506	0.228253	−	0.973602i	$-0.426699\pi$
$149$	830.283	0.456506	0.228253	+	0.973602i	$0.426699\pi$
$150$	0	0
$151$	−249.611	−0.134523	−0.0672617	−	0.997735i	$-0.521426\pi$
$151$	−249.611	−0.134523	−0.0672617	+	0.997735i	$0.521426\pi$
$152$	0	0
$153$	−759.348	−0.401240
$154$	0	0
$155$	2219.29	1.15005
$156$	0	0
$157$	2479.96	1.26065	0.630326	−	0.776330i	$-0.282921\pi$
$157$	2479.96	1.26065	0.630326	+	0.776330i	$0.282921\pi$
$158$	0	0
$159$	−1812.05	−0.903804
$160$	0	0
$161$	−811.774	−0.397371
$162$	0	0
$163$	−924.909	−0.444444	−0.222222	−	0.974996i	$-0.571331\pi$
$163$	−924.909	−0.444444	−0.222222	+	0.974996i	$0.571331\pi$
$164$	0	0
$165$	−831.112	−0.392133
$166$	0	0
$167$	2508.91	1.16255	0.581273	−	0.813708i	$-0.302555\pi$
$167$	2508.91	1.16255	0.581273	+	0.813708i	$0.302555\pi$
$168$	0	0
$169$	−1894.28	−0.862211
$170$	0	0
$171$	−1437.90	−0.643035
$172$	0	0
$173$	−3741.87	−1.64445	−0.822224	−	0.569164i	$-0.807267\pi$
$173$	−3741.87	−1.64445	−0.822224	+	0.569164i	$0.807267\pi$
$174$	0	0
$175$	−783.962	−0.338640
$176$	0	0
$177$	946.948	0.402130
$178$	0	0
$179$	2806.16	1.17174	0.585872	−	0.810404i	$-0.300752\pi$
$179$	2806.16	1.17174	0.585872	+	0.810404i	$0.300752\pi$
$180$	0	0
$181$	2078.69	0.853635	0.426817	−	0.904338i	$-0.359635\pi$
$181$	2078.69	0.853635	0.426817	+	0.904338i	$0.359635\pi$
$182$	0	0
$183$	1785.44	0.721222
$184$	0	0
$185$	2323.98	0.923580
$186$	0	0
$187$	1518.34	0.593752
$188$	0	0
$189$	189.000	0.0727393
$190$	0	0
$191$	−3860.97	−1.46267	−0.731334	−	0.682019i	$-0.761102\pi$
$191$	−3860.97	−1.46267	−0.731334	+	0.682019i	$0.761102\pi$
$192$	0	0
$193$	−2486.23	−0.927268	−0.463634	−	0.886027i	$-0.653455\pi$
$193$	−2486.23	−0.927268	−0.463634	+	0.886027i	$0.653455\pi$
$194$	0	0
$195$	803.549	0.295094
$196$	0	0
$197$	−2256.81	−0.816200	−0.408100	−	0.912937i	$-0.633809\pi$
$197$	−2256.81	−0.816200	−0.408100	+	0.912937i	$0.633809\pi$
$198$	0	0
$199$	−3025.28	−1.07767	−0.538835	−	0.842411i	$-0.681135\pi$
$199$	−3025.28	−1.07767	−0.538835	+	0.842411i	$0.681135\pi$
$200$	0	0
$201$	934.897	0.328072
$202$	0	0
$203$	1507.54	0.521223
$204$	0	0
$205$	3532.95	1.20367
$206$	0	0
$207$	1043.71	0.350448
$208$	0	0
$209$	2875.12	0.951559
$210$	0	0
$211$	4415.84	1.44075	0.720377	−	0.693583i	$-0.243969\pi$
$211$	4415.84	1.44075	0.720377	+	0.693583i	$0.243969\pi$
$212$	0	0
$213$	1074.78	0.345740
$214$	0	0
$215$	6329.61	2.00779
$216$	0	0
$217$	1009.12	0.315685
$218$	0	0
$219$	2449.98	0.755956
$220$	0	0
$221$	−1467.98	−0.446820
$222$	0	0
$223$	1159.22	0.348105	0.174052	−	0.984736i	$-0.444314\pi$
$223$	1159.22	0.348105	0.174052	+	0.984736i	$0.444314\pi$
$224$	0	0
$225$	1007.95	0.298652
$226$	0	0
$227$	3861.19	1.12897	0.564485	−	0.825443i	$-0.309075\pi$
$227$	3861.19	1.12897	0.564485	+	0.825443i	$0.309075\pi$
$228$	0	0
$229$	3141.40	0.906504	0.453252	−	0.891382i	$-0.350264\pi$
$229$	3141.40	0.906504	0.453252	+	0.891382i	$0.350264\pi$
$230$	0	0
$231$	−377.910	−0.107639
$232$	0	0
$233$	3317.81	0.932863	0.466431	−	0.884557i	$-0.345539\pi$
$233$	3317.81	0.932863	0.466431	+	0.884557i	$0.345539\pi$
$234$	0	0
$235$	−696.003	−0.193201
$236$	0	0
$237$	413.619	0.113365
$238$	0	0
$239$	−3309.41	−0.895681	−0.447840	−	0.894114i	$-0.647807\pi$
$239$	−3309.41	−0.895681	−0.447840	+	0.894114i	$0.647807\pi$
$240$	0	0
$241$	−278.021	−0.0743108	−0.0371554	−	0.999309i	$-0.511830\pi$
$241$	−278.021	−0.0743108	−0.0371554	+	0.999309i	$0.511830\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	−754.337	−0.196705
$246$	0	0
$247$	−2779.77	−0.716082
$248$	0	0
$249$	193.620	0.0492777
$250$	0	0
$251$	−2957.60	−0.743753	−0.371877	−	0.928282i	$-0.621286\pi$
$251$	−2957.60	−0.743753	−0.371877	+	0.928282i	$0.621286\pi$
$252$	0	0
$253$	−2086.92	−0.518591
$254$	0	0
$255$	−3896.63	−0.956927
$256$	0	0
$257$	3008.66	0.730254	0.365127	−	0.930958i	$-0.381026\pi$
$257$	3008.66	0.730254	0.365127	+	0.930958i	$0.381026\pi$
$258$	0	0
$259$	1056.72	0.253519
$260$	0	0
$261$	−1938.26	−0.459676
$262$	0	0
$263$	−219.859	−0.0515480	−0.0257740	−	0.999668i	$-0.508205\pi$
$263$	−219.859	−0.0515480	−0.0257740	+	0.999668i	$0.508205\pi$
$264$	0	0
$265$	−9298.60	−2.15550
$266$	0	0
$267$	−1462.32	−0.335177
$268$	0	0
$269$	5831.35	1.32172	0.660862	−	0.750507i	$-0.270191\pi$
$269$	5831.35	1.32172	0.660862	+	0.750507i	$0.270191\pi$
$270$	0	0
$271$	−3306.95	−0.741266	−0.370633	−	0.928779i	$-0.620859\pi$
$271$	−3306.95	−0.741266	−0.370633	+	0.928779i	$0.620859\pi$
$272$	0	0
$273$	365.377	0.0810023
$274$	0	0
$275$	−2015.42	−0.441944
$276$	0	0
$277$	−729.382	−0.158210	−0.0791052	−	0.996866i	$-0.525206\pi$
$277$	−729.382	−0.158210	−0.0791052	+	0.996866i	$0.525206\pi$
$278$	0	0
$279$	−1297.44	−0.278408
$280$	0	0
$281$	−4339.86	−0.921333	−0.460666	−	0.887573i	$-0.652390\pi$
$281$	−4339.86	−0.921333	−0.460666	+	0.887573i	$0.652390\pi$
$282$	0	0
$283$	−9156.94	−1.92341	−0.961703	−	0.274095i	$-0.911622\pi$
$283$	−9156.94	−1.92341	−0.961703	+	0.274095i	$0.911622\pi$
$284$	0	0
$285$	−7378.65	−1.53359
$286$	0	0
$287$	1606.45	0.330403
$288$	0	0
$289$	2205.64	0.448940
$290$	0	0
$291$	−3855.59	−0.776697
$292$	0	0
$293$	603.692	0.120369	0.0601844	−	0.998187i	$-0.480831\pi$
$293$	603.692	0.120369	0.0601844	+	0.998187i	$0.480831\pi$
$294$	0	0
$295$	4859.30	0.959050
$296$	0	0
$297$	485.884	0.0949289
$298$	0	0
$299$	2017.71	0.390258
$300$	0	0
$301$	2878.10	0.551132
$302$	0	0
$303$	−3343.08	−0.633844
$304$	0	0
$305$	9162.07	1.72006
$306$	0	0
$307$	9123.78	1.69616	0.848081	−	0.529867i	$-0.177758\pi$
$307$	9123.78	1.69616	0.848081	+	0.529867i	$0.177758\pi$
$308$	0	0
$309$	2580.91	0.475155
$310$	0	0
$311$	−3679.09	−0.670811	−0.335405	−	0.942074i	$-0.608873\pi$
$311$	−3679.09	−0.670811	−0.335405	+	0.942074i	$0.608873\pi$
$312$	0	0
$313$	−6446.62	−1.16417	−0.582084	−	0.813128i	$-0.697763\pi$
$313$	−6446.62	−1.16417	−0.582084	+	0.813128i	$0.697763\pi$
$314$	0	0
$315$	969.862	0.173478
$316$	0	0
$317$	4458.32	0.789919	0.394960	−	0.918698i	$-0.370759\pi$
$317$	4458.32	0.789919	0.394960	+	0.918698i	$0.370759\pi$
$318$	0	0
$319$	3875.60	0.680225
$320$	0	0
$321$	3725.29	0.647743
$322$	0	0
$323$	13479.8	2.32210
$324$	0	0
$325$	1948.58	0.332578
$326$	0	0
$327$	−4922.89	−0.832528
$328$	0	0
$329$	−316.475	−0.0530330
$330$	0	0
$331$	4027.46	0.668789	0.334394	−	0.942433i	$-0.391468\pi$
$331$	4027.46	0.668789	0.334394	+	0.942433i	$0.391468\pi$
$332$	0	0
$333$	−1358.64	−0.223583
$334$	0	0
$335$	4797.46	0.782428
$336$	0	0
$337$	−6213.49	−1.00436	−0.502182	−	0.864762i	$-0.667469\pi$
$337$	−6213.49	−1.00436	−0.502182	+	0.864762i	$0.667469\pi$
$338$	0	0
$339$	−5835.21	−0.934882
$340$	0	0
$341$	2594.27	0.411986
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	5355.84	0.835793
$346$	0	0
$347$	7962.17	1.23179	0.615895	−	0.787828i	$-0.288794\pi$
$347$	7962.17	1.23179	0.615895	+	0.787828i	$0.288794\pi$
$348$	0	0
$349$	9690.48	1.48630	0.743151	−	0.669124i	$-0.233330\pi$
$349$	9690.48	1.48630	0.743151	+	0.669124i	$0.233330\pi$
$350$	0	0
$351$	−469.771	−0.0714373
$352$	0	0
$353$	−6649.02	−1.00253	−0.501263	−	0.865295i	$-0.667131\pi$
$353$	−6649.02	−1.00253	−0.501263	+	0.865295i	$0.667131\pi$
$354$	0	0
$355$	5515.28	0.824565
$356$	0	0
$357$	−1771.81	−0.262673
$358$	0	0
$359$	−889.154	−0.130718	−0.0653590	−	0.997862i	$-0.520819\pi$
$359$	−889.154	−0.130718	−0.0653590	+	0.997862i	$0.520819\pi$
$360$	0	0
$361$	18666.4	2.72144
$362$	0	0
$363$	3021.46	0.436875
$364$	0	0
$365$	12572.2	1.80290
$366$	0	0
$367$	−5047.48	−0.717920	−0.358960	−	0.933353i	$-0.616868\pi$
$367$	−5047.48	−0.717920	−0.358960	+	0.933353i	$0.616868\pi$
$368$	0	0
$369$	−2065.43	−0.291388
$370$	0	0
$371$	−4228.11	−0.591678
$372$	0	0
$373$	−1435.50	−0.199270	−0.0996348	−	0.995024i	$-0.531767\pi$
$373$	−1435.50	−0.199270	−0.0996348	+	0.995024i	$0.531767\pi$
$374$	0	0
$375$	−600.642	−0.0827121
$376$	0	0
$377$	−3747.07	−0.511894
$378$	0	0
$379$	3653.10	0.495112	0.247556	−	0.968874i	$-0.420373\pi$
$379$	3653.10	0.495112	0.247556	+	0.968874i	$0.420373\pi$
$380$	0	0
$381$	−4270.77	−0.574274
$382$	0	0
$383$	−12685.9	−1.69248	−0.846239	−	0.532804i	$-0.821138\pi$
$383$	−12685.9	−1.69248	−0.846239	+	0.532804i	$0.821138\pi$
$384$	0	0
$385$	−1939.26	−0.256711
$386$	0	0
$387$	−3700.41	−0.486053
$388$	0	0
$389$	−8256.58	−1.07616	−0.538079	−	0.842895i	$-0.680850\pi$
$389$	−8256.58	−1.07616	−0.538079	+	0.842895i	$0.680850\pi$
$390$	0	0
$391$	−9784.43	−1.26552
$392$	0	0
$393$	2583.87	0.331652
$394$	0	0
$395$	2122.50	0.270367
$396$	0	0
$397$	8693.84	1.09907	0.549536	−	0.835470i	$-0.314805\pi$
$397$	8693.84	1.09907	0.549536	+	0.835470i	$0.314805\pi$
$398$	0	0
$399$	−3355.10	−0.420965
$400$	0	0
$401$	8905.25	1.10899	0.554497	−	0.832185i	$-0.312911\pi$
$401$	8905.25	1.10899	0.554497	+	0.832185i	$0.312911\pi$
$402$	0	0
$403$	−2508.23	−0.310034
$404$	0	0
$405$	−1246.96	−0.152993
$406$	0	0
$407$	2716.64	0.330857
$408$	0	0
$409$	−2343.87	−0.283367	−0.141683	−	0.989912i	$-0.545251\pi$
$409$	−2343.87	−0.283367	−0.141683	+	0.989912i	$0.545251\pi$
$410$	0	0
$411$	−5437.41	−0.652573
$412$	0	0
$413$	2209.55	0.263256
$414$	0	0
$415$	993.567	0.117524
$416$	0	0
$417$	6660.65	0.782191
$418$	0	0
$419$	−9025.01	−1.05227	−0.526134	−	0.850402i	$-0.676359\pi$
$419$	−9025.01	−1.05227	−0.526134	+	0.850402i	$0.676359\pi$
$420$	0	0
$421$	−4426.02	−0.512378	−0.256189	−	0.966627i	$-0.582467\pi$
$421$	−4426.02	−0.512378	−0.256189	+	0.966627i	$0.582467\pi$
$422$	0	0
$423$	406.897	0.0467707
$424$	0	0
$425$	−9449.21	−1.07848
$426$	0	0
$427$	4166.03	0.472151
$428$	0	0
$429$	939.318	0.105713
$430$	0	0
$431$	−347.465	−0.0388325	−0.0194162	−	0.999811i	$-0.506181\pi$
$431$	−347.465	−0.0388325	−0.0194162	+	0.999811i	$0.506181\pi$
$432$	0	0
$433$	−1984.35	−0.220235	−0.110117	−	0.993919i	$-0.535123\pi$
$433$	−1984.35	−0.220235	−0.110117	+	0.993919i	$0.535123\pi$
$434$	0	0
$435$	−9946.27	−1.09629
$436$	0	0
$437$	−18527.8	−2.02815
$438$	0	0
$439$	−10528.3	−1.14462	−0.572310	−	0.820037i	$-0.693953\pi$
$439$	−10528.3	−1.14462	−0.572310	+	0.820037i	$0.693953\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	0	0
$443$	−3723.08	−0.399297	−0.199649	−	0.979868i	$-0.563980\pi$
$443$	−3723.08	−0.399297	−0.199649	+	0.979868i	$0.563980\pi$
$444$	0	0
$445$	−7503.94	−0.799372
$446$	0	0
$447$	−2490.85	−0.263564
$448$	0	0
$449$	501.935	0.0527567	0.0263784	−	0.999652i	$-0.491603\pi$
$449$	501.935	0.0527567	0.0263784	+	0.999652i	$0.491603\pi$
$450$	0	0
$451$	4129.88	0.431194
$452$	0	0
$453$	748.832	0.0776671
$454$	0	0
$455$	1874.95	0.193184
$456$	0	0
$457$	−7205.28	−0.737525	−0.368763	−	0.929524i	$-0.620218\pi$
$457$	−7205.28	−0.737525	−0.368763	+	0.929524i	$0.620218\pi$
$458$	0	0
$459$	2278.05	0.231656
$460$	0	0
$461$	2264.90	0.228822	0.114411	−	0.993434i	$-0.463502\pi$
$461$	2264.90	0.228822	0.114411	+	0.993434i	$0.463502\pi$
$462$	0	0
$463$	9703.82	0.974027	0.487014	−	0.873394i	$-0.338086\pi$
$463$	9703.82	0.974027	0.487014	+	0.873394i	$0.338086\pi$
$464$	0	0
$465$	−6657.87	−0.663982
$466$	0	0
$467$	−13418.1	−1.32959	−0.664794	−	0.747027i	$-0.731481\pi$
$467$	−13418.1	−1.32959	−0.664794	+	0.747027i	$0.731481\pi$
$468$	0	0
$469$	2181.43	0.214774
$470$	0	0
$471$	−7439.88	−0.727838
$472$	0	0
$473$	7399.07	0.719259
$474$	0	0
$475$	−17893.0	−1.72839
$476$	0	0
$477$	5436.14	0.521811
$478$	0	0
$479$	8508.95	0.811658	0.405829	−	0.913949i	$-0.366983\pi$
$479$	8508.95	0.811658	0.405829	+	0.913949i	$0.366983\pi$
$480$	0	0
$481$	−2626.54	−0.248982
$482$	0	0
$483$	2435.32	0.229422
$484$	0	0
$485$	−19785.1	−1.85236
$486$	0	0
$487$	3185.29	0.296385	0.148192	−	0.988959i	$-0.452655\pi$
$487$	3185.29	0.296385	0.148192	+	0.988959i	$0.452655\pi$
$488$	0	0
$489$	2774.73	0.256600
$490$	0	0
$491$	15198.0	1.39690	0.698449	−	0.715660i	$-0.253874\pi$
$491$	15198.0	1.39690	0.698449	+	0.715660i	$0.253874\pi$
$492$	0	0
$493$	18170.6	1.65996
$494$	0	0
$495$	2493.34	0.226398
$496$	0	0
$497$	2507.82	0.226340
$498$	0	0
$499$	−2209.48	−0.198216	−0.0991082	−	0.995077i	$-0.531599\pi$
$499$	−2209.48	−0.198216	−0.0991082	+	0.995077i	$0.531599\pi$
$500$	0	0
$501$	−7526.73	−0.671197
$502$	0	0
$503$	10573.7	0.937291	0.468645	−	0.883386i	$-0.344742\pi$
$503$	10573.7	0.937291	0.468645	+	0.883386i	$0.344742\pi$
$504$	0	0
$505$	−17155.1	−1.51167
$506$	0	0
$507$	5682.83	0.497798
$508$	0	0
$509$	−17028.6	−1.48287	−0.741434	−	0.671026i	$-0.765854\pi$
$509$	−17028.6	−1.48287	−0.741434	+	0.671026i	$0.765854\pi$
$510$	0	0
$511$	5716.62	0.494889
$512$	0	0
$513$	4313.70	0.371256
$514$	0	0
$515$	13244.1	1.13321
$516$	0	0
$517$	−813.600	−0.0692110
$518$	0	0
$519$	11225.6	0.949422
$520$	0	0
$521$	−10214.0	−0.858896	−0.429448	−	0.903092i	$-0.641292\pi$
$521$	−10214.0	−0.858896	−0.429448	+	0.903092i	$0.641292\pi$
$522$	0	0
$523$	−21393.1	−1.78863	−0.894316	−	0.447437i	$-0.852337\pi$
$523$	−21393.1	−1.78863	−0.894316	+	0.447437i	$0.852337\pi$
$524$	0	0
$525$	2351.89	0.195514
$526$	0	0
$527$	12163.1	1.00537
$528$	0	0
$529$	1281.50	0.105326
$530$	0	0
$531$	−2840.84	−0.232170
$532$	0	0
$533$	−3992.92	−0.324489
$534$	0	0
$535$	19116.5	1.54482
$536$	0	0
$537$	−8418.48	−0.676507
$538$	0	0
$539$	−881.790	−0.0704664
$540$	0	0
$541$	−25094.8	−1.99429	−0.997145	−	0.0755138i	$-0.975940\pi$
$541$	−25094.8	−1.99429	−0.997145	+	0.0755138i	$0.975940\pi$
$542$	0	0
$543$	−6236.07	−0.492846
$544$	0	0
$545$	−25262.0	−1.98552
$546$	0	0
$547$	−17839.9	−1.39447	−0.697237	−	0.716841i	$-0.745588\pi$
$547$	−17839.9	−1.39447	−0.697237	+	0.716841i	$0.745588\pi$
$548$	0	0
$549$	−5356.33	−0.416398
$550$	0	0
$551$	34407.7	2.66029
$552$	0	0
$553$	965.112	0.0742147
$554$	0	0
$555$	−6971.93	−0.533229
$556$	0	0
$557$	−15032.1	−1.14350	−0.571749	−	0.820428i	$-0.693735\pi$
$557$	−15032.1	−1.14350	−0.571749	+	0.820428i	$0.693735\pi$
$558$	0	0
$559$	−7153.68	−0.541268
$560$	0	0
$561$	−4555.01	−0.342803
$562$	0	0
$563$	5211.21	0.390100	0.195050	−	0.980793i	$-0.437513\pi$
$563$	5211.21	0.390100	0.195050	+	0.980793i	$0.437513\pi$
$564$	0	0
$565$	−29943.6	−2.22962
$566$	0	0
$567$	−567.000	−0.0419961
$568$	0	0
$569$	17148.5	1.26345	0.631723	−	0.775194i	$-0.282348\pi$
$569$	17148.5	1.26345	0.631723	+	0.775194i	$0.282348\pi$
$570$	0	0
$571$	−5712.23	−0.418650	−0.209325	−	0.977846i	$-0.567127\pi$
$571$	−5712.23	−0.418650	−0.209325	+	0.977846i	$0.567127\pi$
$572$	0	0
$573$	11582.9	0.844472
$574$	0	0
$575$	12987.7	0.941959
$576$	0	0
$577$	9198.45	0.663668	0.331834	−	0.943338i	$-0.392333\pi$
$577$	9198.45	0.663668	0.331834	+	0.943338i	$0.392333\pi$
$578$	0	0
$579$	7458.69	0.535359
$580$	0	0
$581$	451.779	0.0322598
$582$	0	0
$583$	−10869.7	−0.772173
$584$	0	0
$585$	−2410.65	−0.170373
$586$	0	0
$587$	20649.4	1.45194	0.725971	−	0.687725i	$-0.241391\pi$
$587$	20649.4	1.45194	0.725971	+	0.687725i	$0.241391\pi$
$588$	0	0
$589$	23032.0	1.61123
$590$	0	0
$591$	6770.44	0.471233
$592$	0	0
$593$	−5262.45	−0.364423	−0.182212	−	0.983259i	$-0.558326\pi$
$593$	−5262.45	−0.364423	−0.182212	+	0.983259i	$0.558326\pi$
$594$	0	0
$595$	−9092.13	−0.626456
$596$	0	0
$597$	9075.84	0.622193
$598$	0	0
$599$	6825.01	0.465547	0.232773	−	0.972531i	$-0.425220\pi$
$599$	6825.01	0.465547	0.232773	+	0.972531i	$0.425220\pi$
$600$	0	0
$601$	−25287.1	−1.71628	−0.858139	−	0.513418i	$-0.828379\pi$
$601$	−25287.1	−1.71628	−0.858139	+	0.513418i	$0.828379\pi$
$602$	0	0
$603$	−2804.69	−0.189413
$604$	0	0
$605$	15504.8	1.04191
$606$	0	0
$607$	−15067.9	−1.00755	−0.503777	−	0.863833i	$-0.668057\pi$
$607$	−15067.9	−1.00755	−0.503777	+	0.863833i	$0.668057\pi$
$608$	0	0
$609$	−4522.61	−0.300928
$610$	0	0
$611$	786.618	0.0520837
$612$	0	0
$613$	−17249.8	−1.13656	−0.568282	−	0.822834i	$-0.692392\pi$
$613$	−17249.8	−1.13656	−0.568282	+	0.822834i	$0.692392\pi$
$614$	0	0
$615$	−10598.9	−0.694938
$616$	0	0
$617$	−6229.57	−0.406471	−0.203236	−	0.979130i	$-0.565146\pi$
$617$	−6229.57	−0.406471	−0.203236	+	0.979130i	$0.565146\pi$
$618$	0	0
$619$	11372.9	0.738472	0.369236	−	0.929336i	$-0.379619\pi$
$619$	11372.9	0.738472	0.369236	+	0.929336i	$0.379619\pi$
$620$	0	0
$621$	−3131.13	−0.202331
$622$	0	0
$623$	−3412.07	−0.219425
$624$	0	0
$625$	−17081.5	−1.09322
$626$	0	0
$627$	−8625.35	−0.549383
$628$	0	0
$629$	12736.8	0.807394
$630$	0	0
$631$	−8041.72	−0.507347	−0.253673	−	0.967290i	$-0.581639\pi$
$631$	−8041.72	−0.507347	−0.253673	+	0.967290i	$0.581639\pi$
$632$	0	0
$633$	−13247.5	−0.831819
$634$	0	0
$635$	−21915.7	−1.36960
$636$	0	0
$637$	852.547	0.0530285
$638$	0	0
$639$	−3224.34	−0.199613
$640$	0	0
$641$	12462.1	0.767898	0.383949	−	0.923354i	$-0.374564\pi$
$641$	12462.1	0.767898	0.383949	+	0.923354i	$0.374564\pi$
$642$	0	0
$643$	7116.38	0.436458	0.218229	−	0.975898i	$-0.429972\pi$
$643$	7116.38	0.436458	0.218229	+	0.975898i	$0.429972\pi$
$644$	0	0
$645$	−18988.8	−1.15920
$646$	0	0
$647$	−12695.1	−0.771398	−0.385699	−	0.922625i	$-0.626040\pi$
$647$	−12695.1	−0.771398	−0.385699	+	0.922625i	$0.626040\pi$
$648$	0	0
$649$	5680.34	0.343564
$650$	0	0
$651$	−3027.36	−0.182261
$652$	0	0
$653$	28102.2	1.68411	0.842056	−	0.539391i	$-0.181345\pi$
$653$	28102.2	1.68411	0.842056	+	0.539391i	$0.181345\pi$
$654$	0	0
$655$	13259.2	0.790964
$656$	0	0
$657$	−7349.94	−0.436451
$658$	0	0
$659$	25203.9	1.48984	0.744921	−	0.667152i	$-0.232487\pi$
$659$	25203.9	1.48984	0.744921	+	0.667152i	$0.232487\pi$
$660$	0	0
$661$	28508.6	1.67754	0.838772	−	0.544483i	$-0.183274\pi$
$661$	28508.6	1.67754	0.838772	+	0.544483i	$0.183274\pi$
$662$	0	0
$663$	4403.94	0.257971
$664$	0	0
$665$	−17216.8	−1.00397
$666$	0	0
$667$	−24975.1	−1.44983
$668$	0	0
$669$	−3477.67	−0.200978
$670$	0	0
$671$	10710.1	0.616183
$672$	0	0
$673$	−1549.88	−0.0887720	−0.0443860	−	0.999014i	$-0.514133\pi$
$673$	−1549.88	−0.0887720	−0.0443860	+	0.999014i	$0.514133\pi$
$674$	0	0
$675$	−3023.85	−0.172427
$676$	0	0
$677$	−30216.3	−1.71537	−0.857686	−	0.514173i	$-0.828099\pi$
$677$	−30216.3	−1.71537	−0.857686	+	0.514173i	$0.828099\pi$
$678$	0	0
$679$	−8996.38	−0.508468
$680$	0	0
$681$	−11583.6	−0.651811
$682$	0	0
$683$	−8620.18	−0.482931	−0.241466	−	0.970409i	$-0.577628\pi$
$683$	−8620.18	−0.482931	−0.241466	+	0.970409i	$0.577628\pi$
$684$	0	0
$685$	−27902.3	−1.55634
$686$	0	0
$687$	−9424.20	−0.523371
$688$	0	0
$689$	10509.2	0.581088
$690$	0	0
$691$	23194.7	1.27694	0.638472	−	0.769645i	$-0.279567\pi$
$691$	23194.7	1.27694	0.638472	+	0.769645i	$0.279567\pi$
$692$	0	0
$693$	1133.73	0.0621455
$694$	0	0
$695$	34179.4	1.86547
$696$	0	0
$697$	19362.7	1.05225
$698$	0	0
$699$	−9953.43	−0.538589
$700$	0	0
$701$	−19232.3	−1.03622	−0.518111	−	0.855313i	$-0.673365\pi$
$701$	−19232.3	−1.03622	−0.518111	+	0.855313i	$0.673365\pi$
$702$	0	0
$703$	24118.4	1.29395
$704$	0	0
$705$	2088.01	0.111545
$706$	0	0
$707$	−7800.51	−0.414948
$708$	0	0
$709$	−28157.2	−1.49149	−0.745744	−	0.666233i	$-0.767906\pi$
$709$	−28157.2	−1.49149	−0.745744	+	0.666233i	$0.767906\pi$
$710$	0	0
$711$	−1240.86	−0.0654512
$712$	0	0
$713$	−16717.9	−0.878108
$714$	0	0
$715$	4820.15	0.252117
$716$	0	0
$717$	9928.22	0.517121
$718$	0	0
$719$	−5511.45	−0.285873	−0.142936	−	0.989732i	$-0.545654\pi$
$719$	−5511.45	−0.285873	−0.142936	+	0.989732i	$0.545654\pi$
$720$	0	0
$721$	6022.13	0.311062
$722$	0	0
$723$	834.063	0.0429034
$724$	0	0
$725$	−24119.4	−1.23555
$726$	0	0
$727$	−32419.8	−1.65390	−0.826948	−	0.562278i	$-0.809925\pi$
$727$	−32419.8	−1.65390	−0.826948	+	0.562278i	$0.809925\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	34690.2	1.75521
$732$	0	0
$733$	1919.18	0.0967076	0.0483538	−	0.998830i	$-0.484603\pi$
$733$	1919.18	0.0967076	0.0483538	+	0.998830i	$0.484603\pi$
$734$	0	0
$735$	2263.01	0.113568
$736$	0	0
$737$	5608.05	0.280292
$738$	0	0
$739$	−17633.0	−0.877729	−0.438865	−	0.898553i	$-0.644619\pi$
$739$	−17633.0	−0.877729	−0.438865	+	0.898553i	$0.644619\pi$
$740$	0	0
$741$	8339.30	0.413430
$742$	0	0
$743$	12128.6	0.598865	0.299433	−	0.954117i	$-0.403203\pi$
$743$	12128.6	0.598865	0.299433	+	0.954117i	$0.403203\pi$
$744$	0	0
$745$	−12781.9	−0.628581
$746$	0	0
$747$	−580.859	−0.0284505
$748$	0	0
$749$	8692.35	0.424047
$750$	0	0
$751$	26822.2	1.30327	0.651634	−	0.758533i	$-0.274084\pi$
$751$	26822.2	1.30327	0.651634	+	0.758533i	$0.274084\pi$
$752$	0	0
$753$	8872.80	0.429406
$754$	0	0
$755$	3842.66	0.185230
$756$	0	0
$757$	−10169.2	−0.488251	−0.244126	−	0.969744i	$-0.578501\pi$
$757$	−10169.2	−0.488251	−0.244126	+	0.969744i	$0.578501\pi$
$758$	0	0
$759$	6260.77	0.299409
$760$	0	0
$761$	23389.5	1.11415	0.557075	−	0.830462i	$-0.311924\pi$
$761$	23389.5	1.11415	0.557075	+	0.830462i	$0.311924\pi$
$762$	0	0
$763$	−11486.7	−0.545017
$764$	0	0
$765$	11689.9	0.552482
$766$	0	0
$767$	−5491.95	−0.258544
$768$	0	0
$769$	7009.66	0.328706	0.164353	−	0.986402i	$-0.447446\pi$
$769$	7009.66	0.328706	0.164353	+	0.986402i	$0.447446\pi$
$770$	0	0
$771$	−9025.99	−0.421613
$772$	0	0
$773$	23176.3	1.07839	0.539195	−	0.842181i	$-0.318729\pi$
$773$	23176.3	1.07839	0.539195	+	0.842181i	$0.318729\pi$
$774$	0	0
$775$	−16145.1	−0.748324
$776$	0	0
$777$	−3170.17	−0.146369
$778$	0	0
$779$	36665.2	1.68635
$780$	0	0
$781$	6447.14	0.295387
$782$	0	0
$783$	5814.78	0.265394
$784$	0	0
$785$	−38178.1	−1.73584
$786$	0	0
$787$	−24261.8	−1.09891	−0.549454	−	0.835524i	$-0.685164\pi$
$787$	−24261.8	−1.09891	−0.549454	+	0.835524i	$0.685164\pi$
$788$	0	0
$789$	659.578	0.0297612
$790$	0	0
$791$	−13615.5	−0.612024
$792$	0	0
$793$	−10354.9	−0.463700
$794$	0	0
$795$	27895.8	1.24448
$796$	0	0
$797$	8259.57	0.367088	0.183544	−	0.983012i	$-0.441243\pi$
$797$	8259.57	0.367088	0.183544	+	0.983012i	$0.441243\pi$
$798$	0	0
$799$	−3814.52	−0.168896
$800$	0	0
$801$	4386.95	0.193515
$802$	0	0
$803$	14696.4	0.645858
$804$	0	0
$805$	12497.0	0.547155
$806$	0	0
$807$	−17494.1	−0.763098
$808$	0	0
$809$	−32589.7	−1.41631	−0.708155	−	0.706057i	$-0.750472\pi$
$809$	−32589.7	−1.41631	−0.708155	+	0.706057i	$0.750472\pi$
$810$	0	0
$811$	−20365.5	−0.881785	−0.440893	−	0.897560i	$-0.645338\pi$
$811$	−20365.5	−0.881785	−0.440893	+	0.897560i	$0.645338\pi$
$812$	0	0
$813$	9920.86	0.427970
$814$	0	0
$815$	14238.6	0.611972
$816$	0	0
$817$	65689.2	2.81294
$818$	0	0
$819$	−1096.13	−0.0467667
$820$	0	0
$821$	44152.7	1.87691	0.938453	−	0.345407i	$-0.112259\pi$
$821$	44152.7	1.87691	0.938453	+	0.345407i	$0.112259\pi$
$822$	0	0
$823$	−35201.7	−1.49095	−0.745477	−	0.666531i	$-0.767778\pi$
$823$	−35201.7	−1.49095	−0.745477	+	0.666531i	$0.767778\pi$
$824$	0	0
$825$	6046.27	0.255156
$826$	0	0
$827$	−38376.9	−1.61366	−0.806830	−	0.590784i	$-0.798819\pi$
$827$	−38376.9	−1.61366	−0.806830	+	0.590784i	$0.798819\pi$
$828$	0	0
$829$	−20449.0	−0.856722	−0.428361	−	0.903608i	$-0.640909\pi$
$829$	−20449.0	−0.856722	−0.428361	+	0.903608i	$0.640909\pi$
$830$	0	0
$831$	2188.15	0.0913429
$832$	0	0
$833$	−4134.23	−0.171960
$834$	0	0
$835$	−38623.7	−1.60075
$836$	0	0
$837$	3892.32	0.160739
$838$	0	0
$839$	−11967.4	−0.492443	−0.246221	−	0.969214i	$-0.579189\pi$
$839$	−11967.4	−0.492443	−0.246221	+	0.969214i	$0.579189\pi$
$840$	0	0
$841$	21991.9	0.901715
$842$	0	0
$843$	13019.6	0.531932
$844$	0	0
$845$	29161.7	1.18721
$846$	0	0
$847$	7050.08	0.286002
$848$	0	0
$849$	27470.8	1.11048
$850$	0	0
$851$	−17506.5	−0.705189
$852$	0	0
$853$	−18703.4	−0.750754	−0.375377	−	0.926872i	$-0.622487\pi$
$853$	−18703.4	−0.750754	−0.375377	+	0.926872i	$0.622487\pi$
$854$	0	0
$855$	22135.9	0.885419
$856$	0	0
$857$	−31430.5	−1.25279	−0.626397	−	0.779504i	$-0.715471\pi$
$857$	−31430.5	−1.25279	−0.626397	+	0.779504i	$0.715471\pi$
$858$	0	0
$859$	6183.40	0.245605	0.122803	−	0.992431i	$-0.460812\pi$
$859$	6183.40	0.245605	0.122803	+	0.992431i	$0.460812\pi$
$860$	0	0
$861$	−4819.34	−0.190758
$862$	0	0
$863$	−6480.22	−0.255608	−0.127804	−	0.991799i	$-0.540793\pi$
$863$	−6480.22	−0.255608	−0.127804	+	0.991799i	$0.540793\pi$
$864$	0	0
$865$	57604.8	2.26430
$866$	0	0
$867$	−6616.93	−0.259196
$868$	0	0
$869$	2481.12	0.0968543
$870$	0	0
$871$	−5422.06	−0.210929
$872$	0	0
$873$	11566.8	0.448426
$874$	0	0
$875$	−1401.50	−0.0541478
$876$	0	0
$877$	−14697.9	−0.565922	−0.282961	−	0.959131i	$-0.591317\pi$
$877$	−14697.9	−0.565922	−0.282961	+	0.959131i	$0.591317\pi$
$878$	0	0
$879$	−1811.07	−0.0694949
$880$	0	0
$881$	47733.9	1.82542	0.912711	−	0.408605i	$-0.133985\pi$
$881$	47733.9	1.82542	0.912711	+	0.408605i	$0.133985\pi$
$882$	0	0
$883$	43241.6	1.64801	0.824007	−	0.566579i	$-0.191733\pi$
$883$	43241.6	1.64801	0.824007	+	0.566579i	$0.191733\pi$
$884$	0	0
$885$	−14577.9	−0.553708
$886$	0	0
$887$	20412.9	0.772717	0.386358	−	0.922349i	$-0.373733\pi$
$887$	20412.9	0.772717	0.386358	+	0.922349i	$0.373733\pi$
$888$	0	0
$889$	−9965.14	−0.375951
$890$	0	0
$891$	−1457.65	−0.0548072
$892$	0	0
$893$	−7223.17	−0.270677
$894$	0	0
$895$	−43199.8	−1.61342
$896$	0	0
$897$	−6053.13	−0.225316
$898$	0	0
$899$	31046.7	1.15180
$900$	0	0
$901$	−50962.1	−1.88434
$902$	0	0
$903$	−8634.30	−0.318196
$904$	0	0
$905$	−32000.7	−1.17540
$906$	0	0
$907$	35891.1	1.31394	0.656971	−	0.753916i	$-0.271838\pi$
$907$	35891.1	1.31394	0.656971	+	0.753916i	$0.271838\pi$
$908$	0	0
$909$	10029.2	0.365950
$910$	0	0
$911$	36282.8	1.31954	0.659770	−	0.751467i	$-0.270654\pi$
$911$	36282.8	1.31954	0.659770	+	0.751467i	$0.270654\pi$
$912$	0	0
$913$	1161.44	0.0421009
$914$	0	0
$915$	−27486.2	−0.993078
$916$	0	0
$917$	6029.03	0.217117
$918$	0	0
$919$	−48078.2	−1.72574	−0.862869	−	0.505427i	$-0.831335\pi$
$919$	−48078.2	−1.72574	−0.862869	+	0.505427i	$0.831335\pi$
$920$	0	0
$921$	−27371.3	−0.979279
$922$	0	0
$923$	−6233.33	−0.222289
$924$	0	0
$925$	−16906.7	−0.600962
$926$	0	0
$927$	−7742.74	−0.274331
$928$	0	0
$929$	10154.0	0.358603	0.179302	−	0.983794i	$-0.442616\pi$
$929$	10154.0	0.358603	0.179302	+	0.983794i	$0.442616\pi$
$930$	0	0
$931$	−7828.57	−0.275586
$932$	0	0
$933$	11037.3	0.387293
$934$	0	0
$935$	−23374.2	−0.817560
$936$	0	0
$937$	−51206.2	−1.78531	−0.892654	−	0.450743i	$-0.851159\pi$
$937$	−51206.2	−1.78531	−0.892654	+	0.450743i	$0.851159\pi$
$938$	0	0
$939$	19339.9	0.672133
$940$	0	0
$941$	22523.6	0.780286	0.390143	−	0.920754i	$-0.372426\pi$
$941$	22523.6	0.780286	0.390143	+	0.920754i	$0.372426\pi$
$942$	0	0
$943$	−26613.7	−0.919047
$944$	0	0
$945$	−2909.58	−0.100157
$946$	0	0
$947$	24323.4	0.834641	0.417321	−	0.908759i	$-0.362969\pi$
$947$	24323.4	0.834641	0.417321	+	0.908759i	$0.362969\pi$
$948$	0	0
$949$	−14209.0	−0.486031
$950$	0	0
$951$	−13375.0	−0.456060
$952$	0	0
$953$	−11966.4	−0.406747	−0.203374	−	0.979101i	$-0.565191\pi$
$953$	−11966.4	−0.406747	−0.203374	+	0.979101i	$0.565191\pi$
$954$	0	0
$955$	59438.1	2.01400
$956$	0	0
$957$	−11626.8	−0.392728
$958$	0	0
$959$	−12687.3	−0.427209
$960$	0	0
$961$	−9008.86	−0.302402
$962$	0	0
$963$	−11175.9	−0.373975
$964$	0	0
$965$	38274.6	1.27679
$966$	0	0
$967$	30595.1	1.01745	0.508724	−	0.860930i	$-0.330118\pi$
$967$	30595.1	1.01745	0.508724	+	0.860930i	$0.330118\pi$
$968$	0	0
$969$	−40439.5	−1.34067
$970$	0	0
$971$	−14498.5	−0.479174	−0.239587	−	0.970875i	$-0.577012\pi$
$971$	−14498.5	−0.479174	−0.239587	+	0.970875i	$0.577012\pi$
$972$	0	0
$973$	15541.5	0.512064
$974$	0	0
$975$	−5845.75	−0.192014
$976$	0	0
$977$	49564.0	1.62302	0.811511	−	0.584337i	$-0.198645\pi$
$977$	49564.0	1.62302	0.811511	+	0.584337i	$0.198645\pi$
$978$	0	0
$979$	−8771.81	−0.286362
$980$	0	0
$981$	14768.7	0.480660
$982$	0	0
$983$	55113.5	1.78825	0.894125	−	0.447818i	$-0.147799\pi$
$983$	55113.5	1.78825	0.894125	+	0.447818i	$0.147799\pi$
$984$	0	0
$985$	34742.8	1.12386
$986$	0	0
$987$	949.426	0.0306186
$988$	0	0
$989$	−47680.9	−1.53303
$990$	0	0
$991$	3342.51	0.107143	0.0535714	−	0.998564i	$-0.482940\pi$
$991$	3342.51	0.107143	0.0535714	+	0.998564i	$0.482940\pi$
$992$	0	0
$993$	−12082.4	−0.386125
$994$	0	0
$995$	46573.0	1.48388
$996$	0	0
$997$	13993.3	0.444505	0.222252	−	0.974989i	$-0.428659\pi$
$997$	13993.3	0.444505	0.222252	+	0.974989i	$0.428659\pi$
$998$	0	0
$999$	4075.93	0.129086

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.4.a.bt.1.1		3
4.3	odd	2		1344.4.a.bv.1.1		3
8.3	odd	2		672.4.a.o.1.3	✓	3
8.5	even	2		672.4.a.q.1.3	yes	3
24.5	odd	2		2016.4.a.y.1.1		3
24.11	even	2		2016.4.a.z.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.4.a.o.1.3	✓	3	8.3	odd	2
672.4.a.q.1.3	yes	3	8.5	even	2
1344.4.a.bt.1.1		3	1.1	even	1	trivial
1344.4.a.bv.1.1		3	4.3	odd	2
2016.4.a.y.1.1		3	24.5	odd	2
2016.4.a.z.1.1		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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} -15.3946 q^{5} -7.00000 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} -15.3946 q^{5} -7.00000 q^{7} +9.00000 q^{9} -17.9957 q^{11} +17.3989 q^{13} +46.1839 q^{15} -84.3720 q^{17} -159.767 q^{19} +21.0000 q^{21} +115.968 q^{23} +111.995 q^{25} -27.0000 q^{27} -215.362 q^{29} -144.160 q^{31} +53.9872 q^{33} +107.762 q^{35} -150.960 q^{37} -52.1967 q^{39} -229.492 q^{41} -411.157 q^{43} -138.552 q^{45} +45.2107 q^{47} +49.0000 q^{49} +253.116 q^{51} +604.016 q^{53} +277.037 q^{55} +479.300 q^{57} -315.649 q^{59} -595.147 q^{61} -63.0000 q^{63} -267.850 q^{65} -311.632 q^{67} -347.903 q^{69} -358.260 q^{71} -816.660 q^{73} -335.984 q^{75} +125.970 q^{77} -137.873 q^{79} +81.0000 q^{81} -64.5399 q^{83} +1298.88 q^{85} +646.087 q^{87} +487.439 q^{89} -121.792 q^{91} +432.480 q^{93} +2459.55 q^{95} +1285.20 q^{97} -161.961 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 9 q^{3} + 10 q^{5} - 21 q^{7} + 27 q^{9}+O(q^{10}) \) 3 * q - 9 * q^3 + 10 * q^5 - 21 * q^7 + 27 * q^9	\( 3 q - 9 q^{3} + 10 q^{5} - 21 q^{7} + 27 q^{9} + 50 q^{13} - 30 q^{15} + 30 q^{17} - 140 q^{19} + 63 q^{21} + 56 q^{23} + 325 q^{25} - 81 q^{27} - 298 q^{29} - 80 q^{31} - 70 q^{35} - 10 q^{37} - 150 q^{39} + 390 q^{41} - 784 q^{43} + 90 q^{45} + 248 q^{47} + 147 q^{49} - 90 q^{51} - 10 q^{53} + 1360 q^{55} + 420 q^{57} - 1500 q^{59} + 810 q^{61} - 189 q^{63} + 860 q^{65} - 1272 q^{67} - 168 q^{69} + 160 q^{71} - 1170 q^{73} - 975 q^{75} + 840 q^{79} + 243 q^{81} - 1564 q^{83} + 2740 q^{85} + 894 q^{87} - 178 q^{89} - 350 q^{91} + 240 q^{93} + 2840 q^{95} - 130 q^{97}+O(q^{100}) \) 3 * q - 9 * q^3 + 10 * q^5 - 21 * q^7 + 27 * q^9 + 50 * q^13 - 30 * q^15 + 30 * q^17 - 140 * q^19 + 63 * q^21 + 56 * q^23 + 325 * q^25 - 81 * q^27 - 298 * q^29 - 80 * q^31 - 70 * q^35 - 10 * q^37 - 150 * q^39 + 390 * q^41 - 784 * q^43 + 90 * q^45 + 248 * q^47 + 147 * q^49 - 90 * q^51 - 10 * q^53 + 1360 * q^55 + 420 * q^57 - 1500 * q^59 + 810 * q^61 - 189 * q^63 + 860 * q^65 - 1272 * q^67 - 168 * q^69 + 160 * q^71 - 1170 * q^73 - 975 * q^75 + 840 * q^79 + 243 * q^81 - 1564 * q^83 + 2740 * q^85 + 894 * q^87 - 178 * q^89 - 350 * q^91 + 240 * q^93 + 2840 * q^95 - 130 * q^97

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