LMFDB - Embedded newform 464.8.a.g.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [464,8,Mod(1,464)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(464, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("464.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 464 = 2^{4} \cdot 29 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	464.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:	$144.946651825$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 1101 x^{8} - 1540 x^{7} + 405148 x^{6} + 870160 x^{5} - 54569376 x^{4} - 87078400 x^{3} + \cdots - 9372051456 $ x^10 - 1101x^8 - 1540x^7 + 405148x^6 + 870160x^5 - 54569376x^4 - 87078400x^3 + 2140673280x^2 - 1918315520x - 9372051456
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{26} $
Twist minimal:	no (minimal twist has level 29)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-19.0438$ of defining polynomial
Character	$\chi$	$=$	464.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.836957 q^{3} +287.010 q^{5} +124.374 q^{7} -2186.30 q^{9} +O(q^{10})$	$q-0.836957 q^{3} +287.010 q^{5} +124.374 q^{7} -2186.30 q^{9} -5429.60 q^{11} +3568.37 q^{13} -240.215 q^{15} -1040.46 q^{17} +50653.1 q^{19} -104.095 q^{21} +14861.2 q^{23} +4249.65 q^{25} +3660.26 q^{27} -24389.0 q^{29} +116764. q^{31} +4544.34 q^{33} +35696.4 q^{35} -62770.9 q^{37} -2986.57 q^{39} -56748.3 q^{41} -605447. q^{43} -627489. q^{45} -987536. q^{47} -808074. q^{49} +870.816 q^{51} +1.67863e6 q^{53} -1.55835e6 q^{55} -42394.5 q^{57} -1.60050e6 q^{59} +105261. q^{61} -271918. q^{63} +1.02416e6 q^{65} +3.78326e6 q^{67} -12438.1 q^{69} -3.67796e6 q^{71} +719137. q^{73} -3556.78 q^{75} -675299. q^{77} -1.69002e6 q^{79} +4.77837e6 q^{81} -2.75859e6 q^{83} -298621. q^{85} +20412.5 q^{87} +1.13447e7 q^{89} +443811. q^{91} -97726.3 q^{93} +1.45379e7 q^{95} -1.45314e7 q^{97} +1.18707e7 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 80 q^{3} + 180 q^{5} - 1040 q^{7} + 10986 q^{9}+O(q^{10}) $ 10 * q - 80 * q^3 + 180 * q^5 - 1040 * q^7 + 10986 * q^9	$ 10 q - 80 q^{3} + 180 q^{5} - 1040 q^{7} + 10986 q^{9} - 7384 q^{11} + 20820 q^{13} - 43516 q^{15} - 11620 q^{17} - 75068 q^{19} + 51480 q^{21} - 62040 q^{23} + 261022 q^{25} + 28060 q^{27} - 243890 q^{29} - 200600 q^{31} - 1068000 q^{33} - 107528 q^{35} - 367740 q^{37} - 392692 q^{39} + 932764 q^{41} - 1443560 q^{43} + 4245684 q^{45} + 286960 q^{47} + 4713194 q^{49} - 1451016 q^{51} + 3953220 q^{53} - 3981316 q^{55} + 2050640 q^{57} - 6712320 q^{59} + 1905196 q^{61} - 3643800 q^{63} + 4667544 q^{65} + 2718200 q^{67} + 1109064 q^{69} - 3447736 q^{71} - 2554460 q^{73} - 1088084 q^{75} - 3967800 q^{77} - 4187744 q^{79} + 5161402 q^{81} - 3498720 q^{83} + 1817072 q^{85} + 1951120 q^{87} - 303268 q^{89} - 27215080 q^{91} + 1097360 q^{93} + 8810536 q^{95} + 4908620 q^{97} + 14408716 q^{99}+O(q^{100}) $ 10 * q - 80 * q^3 + 180 * q^5 - 1040 * q^7 + 10986 * q^9 - 7384 * q^11 + 20820 * q^13 - 43516 * q^15 - 11620 * q^17 - 75068 * q^19 + 51480 * q^21 - 62040 * q^23 + 261022 * q^25 + 28060 * q^27 - 243890 * q^29 - 200600 * q^31 - 1068000 * q^33 - 107528 * q^35 - 367740 * q^37 - 392692 * q^39 + 932764 * q^41 - 1443560 * q^43 + 4245684 * q^45 + 286960 * q^47 + 4713194 * q^49 - 1451016 * q^51 + 3953220 * q^53 - 3981316 * q^55 + 2050640 * q^57 - 6712320 * q^59 + 1905196 * q^61 - 3643800 * q^63 + 4667544 * q^65 + 2718200 * q^67 + 1109064 * q^69 - 3447736 * q^71 - 2554460 * q^73 - 1088084 * q^75 - 3967800 * q^77 - 4187744 * q^79 + 5161402 * q^81 - 3498720 * q^83 + 1817072 * q^85 + 1951120 * q^87 - 303268 * q^89 - 27215080 * q^91 + 1097360 * q^93 + 8810536 * q^95 + 4908620 * q^97 + 14408716 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−0.836957	−0.0178969	−0.00894847	−	0.999960i	$-0.502848\pi$
$3$	−0.836957	−0.0178969	−0.00894847	+	0.999960i	$0.502848\pi$
$4$	0	0
$5$	287.010	1.02684	0.513419	−	0.858138i	$-0.328379\pi$
$5$	287.010	1.02684	0.513419	+	0.858138i	$0.328379\pi$
$6$	0	0
$7$	124.374	0.137052	0.0685259	−	0.997649i	$-0.478170\pi$
$7$	124.374	0.137052	0.0685259	+	0.997649i	$0.478170\pi$
$8$	0	0
$9$	−2186.30	−0.999680
$10$	0	0
$11$	−5429.60	−1.22997	−0.614984	−	0.788540i	$-0.710838\pi$
$11$	−5429.60	−1.22997	−0.614984	+	0.788540i	$0.710838\pi$
$12$	0	0
$13$	3568.37	0.450472	0.225236	−	0.974304i	$-0.427685\pi$
$13$	3568.37	0.450472	0.225236	+	0.974304i	$0.427685\pi$
$14$	0	0
$15$	−240.215	−0.0183772
$16$	0	0
$17$	−1040.46	−0.0513632	−0.0256816	−	0.999670i	$-0.508176\pi$
$17$	−1040.46	−0.0513632	−0.0256816	+	0.999670i	$0.508176\pi$
$18$	0	0
$19$	50653.1	1.69422	0.847108	−	0.531421i	$-0.178342\pi$
$19$	50653.1	1.69422	0.847108	+	0.531421i	$0.178342\pi$
$20$	0	0
$21$	−104.095	−0.00245281
$22$	0	0
$23$	14861.2	0.254686	0.127343	−	0.991859i	$-0.459355\pi$
$23$	14861.2	0.254686	0.127343	+	0.991859i	$0.459355\pi$
$24$	0	0
$25$	4249.65	0.0543955
$26$	0	0
$27$	3660.26	0.0357881
$28$	0	0
$29$	−24389.0	−0.185695
$29$	−24389.0	−0.185695
$30$	0	0
$31$	116764.	0.703951	0.351976	−	0.936009i	$-0.385510\pi$
$31$	116764.	0.703951	0.351976	+	0.936009i	$0.385510\pi$
$32$	0	0
$33$	4544.34	0.0220127
$34$	0	0
$35$	35696.4	0.140730
$36$	0	0
$37$	−62770.9	−0.203729	−0.101864	−	0.994798i	$-0.532481\pi$
$37$	−62770.9	−0.203729	−0.101864	+	0.994798i	$0.532481\pi$
$38$	0	0
$39$	−2986.57	−0.00806207
$40$	0	0
$41$	−56748.3	−0.128591	−0.0642954	−	0.997931i	$-0.520480\pi$
$41$	−56748.3	−0.128591	−0.0642954	+	0.997931i	$0.520480\pi$
$42$	0	0
$43$	−605447.	−1.16128	−0.580640	−	0.814161i	$-0.697197\pi$
$43$	−605447.	−1.16128	−0.580640	+	0.814161i	$0.697197\pi$
$44$	0	0
$45$	−627489.	−1.02651
$46$	0	0
$47$	−987536.	−1.38743	−0.693714	−	0.720250i	$-0.744027\pi$
$47$	−987536.	−1.38743	−0.693714	+	0.720250i	$0.744027\pi$
$48$	0	0
$49$	−808074.	−0.981217
$50$	0	0
$51$	870.816	0.000919244 0
$52$	0	0
$53$	1.67863e6	1.54878	0.774388	−	0.632711i	$-0.218058\pi$
$53$	1.67863e6	1.54878	0.774388	+	0.632711i	$0.218058\pi$
$54$	0	0
$55$	−1.55835e6	−1.26298
$56$	0	0
$57$	−42394.5	−0.0303213
$58$	0	0
$59$	−1.60050e6	−1.01455	−0.507276	−	0.861784i	$-0.669348\pi$
$59$	−1.60050e6	−1.01455	−0.507276	+	0.861784i	$0.669348\pi$
$60$	0	0
$61$	105261.	0.0593762	0.0296881	−	0.999559i	$-0.490549\pi$
$61$	105261.	0.0593762	0.0296881	+	0.999559i	$0.490549\pi$
$62$	0	0
$63$	−271918.	−0.137008
$64$	0	0
$65$	1.02416e6	0.462562
$66$	0	0
$67$	3.78326e6	1.53676	0.768378	−	0.639996i	$-0.221064\pi$
$67$	3.78326e6	1.53676	0.768378	+	0.639996i	$0.221064\pi$
$68$	0	0
$69$	−12438.1	−0.00455810
$70$	0	0
$71$	−3.67796e6	−1.21956	−0.609779	−	0.792572i	$-0.708742\pi$
$71$	−3.67796e6	−1.21956	−0.609779	+	0.792572i	$0.708742\pi$
$72$	0	0
$73$	719137.	0.216362	0.108181	−	0.994131i	$-0.465497\pi$
$73$	719137.	0.216362	0.108181	+	0.994131i	$0.465497\pi$
$74$	0	0
$75$	−3556.78	−0.000973514 0
$76$	0	0
$77$	−675299.	−0.168569
$78$	0	0
$79$	−1.69002e6	−0.385654	−0.192827	−	0.981233i	$-0.561766\pi$
$79$	−1.69002e6	−0.385654	−0.192827	+	0.981233i	$0.561766\pi$
$80$	0	0
$81$	4.77837e6	0.999039
$82$	0	0
$83$	−2.75859e6	−0.529558	−0.264779	−	0.964309i	$-0.585299\pi$
$83$	−2.75859e6	−0.529558	−0.264779	+	0.964309i	$0.585299\pi$
$84$	0	0
$85$	−298621.	−0.0527417
$86$	0	0
$87$	20412.5	0.00332338
$88$	0	0
$89$	1.13447e7	1.70580	0.852898	−	0.522077i	$-0.174843\pi$
$89$	1.13447e7	1.70580	0.852898	+	0.522077i	$0.174843\pi$
$90$	0	0
$91$	443811.	0.0617380
$92$	0	0
$93$	−97726.3	−0.0125986
$94$	0	0
$95$	1.45379e7	1.73968
$96$	0	0
$97$	−1.45314e7	−1.61661	−0.808304	−	0.588765i	$-0.799614\pi$
$97$	−1.45314e7	−1.61661	−0.808304	+	0.588765i	$0.799614\pi$
$98$	0	0
$99$	1.18707e7	1.22957
$100$	0	0
$101$	−7.18921e6	−0.694315	−0.347157	−	0.937807i	$-0.612853\pi$
$101$	−7.18921e6	−0.694315	−0.347157	+	0.937807i	$0.612853\pi$
$102$	0	0
$103$	−1.57131e7	−1.41687	−0.708437	−	0.705774i	$-0.750600\pi$
$103$	−1.57131e7	−1.41687	−0.708437	+	0.705774i	$0.750600\pi$
$104$	0	0
$105$	−29876.4	−0.00251863
$106$	0	0
$107$	1.80106e7	1.42130	0.710649	−	0.703547i	$-0.248402\pi$
$107$	1.80106e7	1.42130	0.710649	+	0.703547i	$0.248402\pi$
$108$	0	0
$109$	−1.74463e7	−1.29036	−0.645178	−	0.764032i	$-0.723217\pi$
$109$	−1.74463e7	−1.29036	−0.645178	+	0.764032i	$0.723217\pi$
$110$	0	0
$111$	52536.5	0.00364612
$112$	0	0
$113$	1.65701e7	1.08031	0.540157	−	0.841564i	$-0.318365\pi$
$113$	1.65701e7	1.08031	0.540157	+	0.841564i	$0.318365\pi$
$114$	0	0
$115$	4.26530e6	0.261521
$116$	0	0
$117$	−7.80152e6	−0.450328
$118$	0	0
$119$	−129405.	−0.00703942
$120$	0	0
$121$	9.99343e6	0.512821
$122$	0	0
$123$	47495.9	0.00230138
$124$	0	0
$125$	−2.12030e7	−0.970982
$126$	0	0
$127$	−3.24510e7	−1.40577	−0.702886	−	0.711303i	$-0.748106\pi$
$127$	−3.24510e7	−1.40577	−0.702886	+	0.711303i	$0.748106\pi$
$128$	0	0
$129$	506733.	0.0207833
$130$	0	0
$131$	9.21009e6	0.357944	0.178972	−	0.983854i	$-0.442723\pi$
$131$	9.21009e6	0.357944	0.178972	+	0.983854i	$0.442723\pi$
$132$	0	0
$133$	6.29991e6	0.232195
$134$	0	0
$135$	1.05053e6	0.0367486
$136$	0	0
$137$	−5.63936e7	−1.87373	−0.936867	−	0.349687i	$-0.886288\pi$
$137$	−5.63936e7	−1.87373	−0.936867	+	0.349687i	$0.886288\pi$
$138$	0	0
$139$	−3.59342e7	−1.13490	−0.567448	−	0.823409i	$-0.692069\pi$
$139$	−3.59342e7	−1.13490	−0.567448	+	0.823409i	$0.692069\pi$
$140$	0	0
$141$	826525.	0.0248307
$142$	0	0
$143$	−1.93748e7	−0.554066
$144$	0	0
$145$	−6.99988e6	−0.190679
$146$	0	0
$147$	676323.	0.0175608
$148$	0	0
$149$	−1.44990e7	−0.359076	−0.179538	−	0.983751i	$-0.557460\pi$
$149$	−1.44990e7	−0.359076	−0.179538	+	0.983751i	$0.557460\pi$
$150$	0	0
$151$	6.72088e6	0.158857	0.0794286	−	0.996841i	$-0.474690\pi$
$151$	6.72088e6	0.158857	0.0794286	+	0.996841i	$0.474690\pi$
$152$	0	0
$153$	2.27475e6	0.0513468
$154$	0	0
$155$	3.35124e7	0.722844
$156$	0	0
$157$	5.45966e7	1.12594	0.562972	−	0.826476i	$-0.309658\pi$
$157$	5.45966e7	1.12594	0.562972	+	0.826476i	$0.309658\pi$
$158$	0	0
$159$	−1.40494e6	−0.0277184
$160$	0	0
$161$	1.84833e6	0.0349052
$162$	0	0
$163$	2.24322e7	0.405709	0.202855	−	0.979209i	$-0.434978\pi$
$163$	2.24322e7	0.405709	0.202855	+	0.979209i	$0.434978\pi$
$164$	0	0
$165$	1.30427e6	0.0226034
$166$	0	0
$167$	−9.01363e7	−1.49759	−0.748794	−	0.662803i	$-0.769367\pi$
$167$	−9.01363e7	−1.49759	−0.748794	+	0.662803i	$0.769367\pi$
$168$	0	0
$169$	−5.00153e7	−0.797075
$170$	0	0
$171$	−1.10743e8	−1.69367
$172$	0	0
$173$	5.38454e7	0.790655	0.395327	−	0.918540i	$-0.370631\pi$
$173$	5.38454e7	0.790655	0.395327	+	0.918540i	$0.370631\pi$
$174$	0	0
$175$	528544.	0.00745501
$176$	0	0
$177$	1.33955e6	0.0181574
$178$	0	0
$179$	1.86781e7	0.243415	0.121707	−	0.992566i	$-0.461163\pi$
$179$	1.86781e7	0.243415	0.121707	+	0.992566i	$0.461163\pi$
$180$	0	0
$181$	−1.29869e8	−1.62790	−0.813952	−	0.580931i	$-0.802688\pi$
$181$	−1.29869e8	−1.62790	−0.813952	+	0.580931i	$0.802688\pi$
$182$	0	0
$183$	−88098.8	−0.00106265
$184$	0	0
$185$	−1.80159e7	−0.209196
$186$	0	0
$187$	5.64926e6	0.0631751
$188$	0	0
$189$	455240.	0.00490483
$190$	0	0
$191$	7.18882e7	0.746519	0.373259	−	0.927727i	$-0.378240\pi$
$191$	7.18882e7	0.746519	0.373259	+	0.927727i	$0.378240\pi$
$192$	0	0
$193$	−7.68746e7	−0.769719	−0.384859	−	0.922975i	$-0.625750\pi$
$193$	−7.68746e7	−0.769719	−0.384859	+	0.922975i	$0.625750\pi$
$194$	0	0
$195$	−857175.	−0.00827844
$196$	0	0
$197$	7.82417e7	0.729132	0.364566	−	0.931177i	$-0.381217\pi$
$197$	7.82417e7	0.729132	0.364566	+	0.931177i	$0.381217\pi$
$198$	0	0
$199$	−1.59291e8	−1.43287	−0.716435	−	0.697654i	$-0.754227\pi$
$199$	−1.59291e8	−1.43287	−0.716435	+	0.697654i	$0.754227\pi$
$200$	0	0
$201$	−3.16643e6	−0.0275032
$202$	0	0
$203$	−3.03335e6	−0.0254499
$204$	0	0
$205$	−1.62873e7	−0.132042
$206$	0	0
$207$	−3.24909e7	−0.254604
$208$	0	0
$209$	−2.75026e8	−2.08383
$210$	0	0
$211$	−8.79435e7	−0.644488	−0.322244	−	0.946657i	$-0.604437\pi$
$211$	−8.79435e7	−0.644488	−0.322244	+	0.946657i	$0.604437\pi$
$212$	0	0
$213$	3.07829e6	0.0218263
$214$	0	0
$215$	−1.73769e8	−1.19245
$216$	0	0
$217$	1.45223e7	0.0964778
$218$	0	0
$219$	−601886.	−0.00387222
$220$	0	0
$221$	−3.71273e6	−0.0231377
$222$	0	0
$223$	−1.55197e8	−0.937166	−0.468583	−	0.883419i	$-0.655235\pi$
$223$	−1.55197e8	−0.937166	−0.468583	+	0.883419i	$0.655235\pi$
$224$	0	0
$225$	−9.29101e6	−0.0543781
$226$	0	0
$227$	2.60816e8	1.47994	0.739968	−	0.672642i	$-0.234840\pi$
$227$	2.60816e8	1.47994	0.739968	+	0.672642i	$0.234840\pi$
$228$	0	0
$229$	2.08807e7	0.114900	0.0574501	−	0.998348i	$-0.481703\pi$
$229$	2.08807e7	0.114900	0.0574501	+	0.998348i	$0.481703\pi$
$230$	0	0
$231$	565196.	0.00301687
$232$	0	0
$233$	−3.99218e7	−0.206759	−0.103380	−	0.994642i	$-0.532966\pi$
$233$	−3.99218e7	−0.206759	−0.103380	+	0.994642i	$0.532966\pi$
$234$	0	0
$235$	−2.83433e8	−1.42466
$236$	0	0
$237$	1.41448e6	0.00690202
$238$	0	0
$239$	3.19559e8	1.51411	0.757057	−	0.653349i	$-0.226637\pi$
$239$	3.19559e8	1.51411	0.757057	+	0.653349i	$0.226637\pi$
$240$	0	0
$241$	2.89494e8	1.33223	0.666117	−	0.745847i	$-0.267955\pi$
$241$	2.89494e8	1.33223	0.666117	+	0.745847i	$0.267955\pi$
$242$	0	0
$243$	−1.20043e7	−0.0536679
$244$	0	0
$245$	−2.31925e8	−1.00755
$246$	0	0
$247$	1.80749e8	0.763197
$248$	0	0
$249$	2.30882e6	0.00947746
$250$	0	0
$251$	−2.65544e8	−1.05993	−0.529967	−	0.848018i	$-0.677796\pi$
$251$	−2.65544e8	−1.05993	−0.529967	+	0.848018i	$0.677796\pi$
$252$	0	0
$253$	−8.06902e7	−0.313256
$254$	0	0
$255$	249933.	0.000943915 0
$256$	0	0
$257$	1.45081e8	0.533145	0.266573	−	0.963815i	$-0.414109\pi$
$257$	1.45081e8	0.533145	0.266573	+	0.963815i	$0.414109\pi$
$258$	0	0
$259$	−7.80704e6	−0.0279214
$260$	0	0
$261$	5.33217e7	0.185636
$262$	0	0
$263$	−3.55310e8	−1.20438	−0.602189	−	0.798354i	$-0.705704\pi$
$263$	−3.55310e8	−1.20438	−0.602189	+	0.798354i	$0.705704\pi$
$264$	0	0
$265$	4.81783e8	1.59034
$266$	0	0
$267$	−9.49501e6	−0.0305285
$268$	0	0
$269$	−3.23141e8	−1.01218	−0.506091	−	0.862480i	$-0.668910\pi$
$269$	−3.23141e8	−1.01218	−0.506091	+	0.862480i	$0.668910\pi$
$270$	0	0
$271$	−3.16979e7	−0.0967472	−0.0483736	−	0.998829i	$-0.515404\pi$
$271$	−3.16979e7	−0.0967472	−0.0483736	+	0.998829i	$0.515404\pi$
$272$	0	0
$273$	−371450.	−0.00110492
$274$	0	0
$275$	−2.30739e7	−0.0669048
$276$	0	0
$277$	−1.44682e6	−0.00409012	−0.00204506	−	0.999998i	$-0.500651\pi$
$277$	−1.44682e6	−0.00409012	−0.00204506	+	0.999998i	$0.500651\pi$
$278$	0	0
$279$	−2.55281e8	−0.703726
$280$	0	0
$281$	−4.42261e8	−1.18907	−0.594534	−	0.804070i	$-0.702664\pi$
$281$	−4.42261e8	−1.18907	−0.594534	+	0.804070i	$0.702664\pi$
$282$	0	0
$283$	2.61084e7	0.0684743	0.0342372	−	0.999414i	$-0.489100\pi$
$283$	2.61084e7	0.0684743	0.0342372	+	0.999414i	$0.489100\pi$
$284$	0	0
$285$	−1.21676e7	−0.0311350
$286$	0	0
$287$	−7.05799e6	−0.0176236
$288$	0	0
$289$	−4.09256e8	−0.997362
$290$	0	0
$291$	1.21621e7	0.0289323
$292$	0	0
$293$	−2.70813e8	−0.628974	−0.314487	−	0.949262i	$-0.601833\pi$
$293$	−2.70813e8	−0.628974	−0.314487	+	0.949262i	$0.601833\pi$
$294$	0	0
$295$	−4.59360e8	−1.04178
$296$	0	0
$297$	−1.98738e7	−0.0440183
$298$	0	0
$299$	5.30301e7	0.114729
$300$	0	0
$301$	−7.53016e7	−0.159155
$302$	0	0
$303$	6.01706e6	0.0124261
$304$	0	0
$305$	3.02109e7	0.0609697
$306$	0	0
$307$	−4.32282e7	−0.0852674	−0.0426337	−	0.999091i	$-0.513575\pi$
$307$	−4.32282e7	−0.0852674	−0.0426337	+	0.999091i	$0.513575\pi$
$308$	0	0
$309$	1.31512e7	0.0253577
$310$	0	0
$311$	1.47802e8	0.278624	0.139312	−	0.990249i	$-0.455511\pi$
$311$	1.47802e8	0.278624	0.139312	+	0.990249i	$0.455511\pi$
$312$	0	0
$313$	−4.47020e8	−0.823990	−0.411995	−	0.911186i	$-0.635168\pi$
$313$	−4.47020e8	−0.823990	−0.411995	+	0.911186i	$0.635168\pi$
$314$	0	0
$315$	−7.80431e7	−0.140685
$316$	0	0
$317$	−8.05083e8	−1.41949	−0.709746	−	0.704457i	$-0.751190\pi$
$317$	−8.05083e8	−1.41949	−0.709746	+	0.704457i	$0.751190\pi$
$318$	0	0
$319$	1.32423e8	0.228399
$320$	0	0
$321$	−1.50741e7	−0.0254369
$322$	0	0
$323$	−5.27023e7	−0.0870204
$324$	0	0
$325$	1.51643e7	0.0245037
$326$	0	0
$327$	1.46018e7	0.0230934
$328$	0	0
$329$	−1.22823e8	−0.190150
$330$	0	0
$331$	9.07105e8	1.37486	0.687432	−	0.726249i	$-0.258738\pi$
$331$	9.07105e8	1.37486	0.687432	+	0.726249i	$0.258738\pi$
$332$	0	0
$333$	1.37236e8	0.203663
$334$	0	0
$335$	1.08583e9	1.57800
$336$	0	0
$337$	−8.31707e8	−1.18377	−0.591883	−	0.806024i	$-0.701615\pi$
$337$	−8.31707e8	−1.18377	−0.591883	+	0.806024i	$0.701615\pi$
$338$	0	0
$339$	−1.38684e7	−0.0193343
$340$	0	0
$341$	−6.33982e8	−0.865837
$342$	0	0
$343$	−2.02930e8	−0.271529
$344$	0	0
$345$	−3.56987e6	−0.00468043
$346$	0	0
$347$	−1.08397e9	−1.39272	−0.696358	−	0.717694i	$-0.745197\pi$
$347$	−1.08397e9	−1.39272	−0.696358	+	0.717694i	$0.745197\pi$
$348$	0	0
$349$	1.48607e9	1.87132	0.935662	−	0.352897i	$-0.114803\pi$
$349$	1.48607e9	1.87132	0.935662	+	0.352897i	$0.114803\pi$
$350$	0	0
$351$	1.30612e7	0.0161216
$352$	0	0
$353$	−7.07613e8	−0.856218	−0.428109	−	0.903727i	$-0.640820\pi$
$353$	−7.07613e8	−0.856218	−0.428109	+	0.903727i	$0.640820\pi$
$354$	0	0
$355$	−1.05561e9	−1.25229
$356$	0	0
$357$	108307.	0.000125984 0
$358$	0	0
$359$	−1.63812e8	−0.186859	−0.0934296	−	0.995626i	$-0.529783\pi$
$359$	−1.63812e8	−0.186859	−0.0934296	+	0.995626i	$0.529783\pi$
$360$	0	0
$361$	1.67187e9	1.87037
$362$	0	0
$363$	−8.36407e6	−0.00917792
$364$	0	0
$365$	2.06399e8	0.222169
$366$	0	0
$367$	8.55095e7	0.0902991	0.0451495	−	0.998980i	$-0.485624\pi$
$367$	8.55095e7	0.0902991	0.0451495	+	0.998980i	$0.485624\pi$
$368$	0	0
$369$	1.24069e8	0.128550
$370$	0	0
$371$	2.08777e8	0.212263
$372$	0	0
$373$	−1.92616e9	−1.92182	−0.960908	−	0.276869i	$-0.910703\pi$
$373$	−1.92616e9	−1.92182	−0.960908	+	0.276869i	$0.910703\pi$
$374$	0	0
$375$	1.77460e7	0.0173776
$376$	0	0
$377$	−8.70289e7	−0.0836506
$378$	0	0
$379$	−1.21692e9	−1.14822	−0.574109	−	0.818779i	$-0.694651\pi$
$379$	−1.21692e9	−1.14822	−0.574109	+	0.818779i	$0.694651\pi$
$380$	0	0
$381$	2.71601e7	0.0251590
$382$	0	0
$383$	−3.81325e8	−0.346817	−0.173408	−	0.984850i	$-0.555478\pi$
$383$	−3.81325e8	−0.346817	−0.173408	+	0.984850i	$0.555478\pi$
$384$	0	0
$385$	−1.93817e8	−0.173093
$386$	0	0
$387$	1.32369e9	1.16091
$388$	0	0
$389$	−4.02109e8	−0.346354	−0.173177	−	0.984891i	$-0.555403\pi$
$389$	−4.02109e8	−0.346354	−0.173177	+	0.984891i	$0.555403\pi$
$390$	0	0
$391$	−1.54624e7	−0.0130815
$392$	0	0
$393$	−7.70845e6	−0.00640609
$394$	0	0
$395$	−4.85053e8	−0.396004
$396$	0	0
$397$	1.66353e9	1.33433	0.667166	−	0.744909i	$-0.267507\pi$
$397$	1.66353e9	1.33433	0.667166	+	0.744909i	$0.267507\pi$
$398$	0	0
$399$	−5.27275e6	−0.00415558
$400$	0	0
$401$	1.21705e9	0.942549	0.471274	−	0.881987i	$-0.343794\pi$
$401$	1.21705e9	0.942549	0.471274	+	0.881987i	$0.343794\pi$
$402$	0	0
$403$	4.16656e8	0.317110
$404$	0	0
$405$	1.37144e9	1.02585
$406$	0	0
$407$	3.40821e8	0.250580
$408$	0	0
$409$	−4.54192e8	−0.328253	−0.164126	−	0.986439i	$-0.552480\pi$
$409$	−4.54192e8	−0.328253	−0.164126	+	0.986439i	$0.552480\pi$
$410$	0	0
$411$	4.71990e7	0.0335341
$412$	0	0
$413$	−1.99060e8	−0.139046
$414$	0	0
$415$	−7.91741e8	−0.543770
$416$	0	0
$417$	3.00754e7	0.0203112
$418$	0	0
$419$	−1.37516e9	−0.913282	−0.456641	−	0.889651i	$-0.650948\pi$
$419$	−1.37516e9	−0.913282	−0.456641	+	0.889651i	$0.650948\pi$
$420$	0	0
$421$	2.30644e9	1.50645	0.753227	−	0.657761i	$-0.228496\pi$
$421$	2.30644e9	1.50645	0.753227	+	0.657761i	$0.228496\pi$
$422$	0	0
$423$	2.15905e9	1.38698
$424$	0	0
$425$	−4.42157e6	−0.00279393
$426$	0	0
$427$	1.30917e7	0.00813762
$428$	0	0
$429$	1.62159e7	0.00991609
$430$	0	0
$431$	−1.16371e9	−0.700122	−0.350061	−	0.936727i	$-0.613839\pi$
$431$	−1.16371e9	−0.700122	−0.350061	+	0.936727i	$0.613839\pi$
$432$	0	0
$433$	−5.10224e8	−0.302032	−0.151016	−	0.988531i	$-0.548255\pi$
$433$	−5.10224e8	−0.302032	−0.151016	+	0.988531i	$0.548255\pi$
$434$	0	0
$435$	5.85860e6	0.00341257
$436$	0	0
$437$	7.52764e8	0.431493
$438$	0	0
$439$	1.85039e9	1.04385	0.521925	−	0.852991i	$-0.325214\pi$
$439$	1.85039e9	1.04385	0.521925	+	0.852991i	$0.325214\pi$
$440$	0	0
$441$	1.76669e9	0.980903
$442$	0	0
$443$	−1.32689e9	−0.725141	−0.362571	−	0.931956i	$-0.618101\pi$
$443$	−1.32689e9	−0.725141	−0.362571	+	0.931956i	$0.618101\pi$
$444$	0	0
$445$	3.25603e9	1.75158
$446$	0	0
$447$	1.21350e7	0.00642636
$448$	0	0
$449$	2.32628e8	0.121283	0.0606415	−	0.998160i	$-0.480685\pi$
$449$	2.32628e8	0.121283	0.0606415	+	0.998160i	$0.480685\pi$
$450$	0	0
$451$	3.08121e8	0.158162
$452$	0	0
$453$	−5.62508e6	−0.00284306
$454$	0	0
$455$	1.27378e8	0.0633949
$456$	0	0
$457$	2.13977e8	0.104872	0.0524362	−	0.998624i	$-0.483301\pi$
$457$	2.13977e8	0.104872	0.0524362	+	0.998624i	$0.483301\pi$
$458$	0	0
$459$	−3.80834e6	−0.00183819
$460$	0	0
$461$	−1.08354e9	−0.515102	−0.257551	−	0.966265i	$-0.582916\pi$
$461$	−1.08354e9	−0.515102	−0.257551	+	0.966265i	$0.582916\pi$
$462$	0	0
$463$	1.82247e9	0.853348	0.426674	−	0.904405i	$-0.359685\pi$
$463$	1.82247e9	0.853348	0.426674	+	0.904405i	$0.359685\pi$
$464$	0	0
$465$	−2.80484e7	−0.0129367
$466$	0	0
$467$	2.62038e9	1.19057	0.595284	−	0.803515i	$-0.297039\pi$
$467$	2.62038e9	1.19057	0.595284	+	0.803515i	$0.297039\pi$
$468$	0	0
$469$	4.70538e8	0.210615
$470$	0	0
$471$	−4.56950e7	−0.0201509
$472$	0	0
$473$	3.28734e9	1.42834
$474$	0	0
$475$	2.15258e8	0.0921578
$476$	0	0
$477$	−3.66998e9	−1.54828
$478$	0	0
$479$	2.54441e9	1.05782	0.528910	−	0.848678i	$-0.322601\pi$
$479$	2.54441e9	1.05782	0.528910	+	0.848678i	$0.322601\pi$
$480$	0	0
$481$	−2.23990e8	−0.0917741
$482$	0	0
$483$	−1.54698e6	−0.000624696 0
$484$	0	0
$485$	−4.17064e9	−1.65999
$486$	0	0
$487$	2.15742e9	0.846415	0.423207	−	0.906033i	$-0.360904\pi$
$487$	2.15742e9	0.846415	0.423207	+	0.906033i	$0.360904\pi$
$488$	0	0
$489$	−1.87748e7	−0.00726095
$490$	0	0
$491$	1.25122e9	0.477033	0.238516	−	0.971138i	$-0.423339\pi$
$491$	1.25122e9	0.477033	0.238516	+	0.971138i	$0.423339\pi$
$492$	0	0
$493$	2.53757e7	0.00953791
$494$	0	0
$495$	3.40702e9	1.26257
$496$	0	0
$497$	−4.57440e8	−0.167143
$498$	0	0
$499$	−3.48226e9	−1.25461	−0.627306	−	0.778773i	$-0.715843\pi$
$499$	−3.48226e9	−1.25461	−0.627306	+	0.778773i	$0.715843\pi$
$500$	0	0
$501$	7.54402e7	0.0268022
$502$	0	0
$503$	5.73533e8	0.200942	0.100471	−	0.994940i	$-0.467965\pi$
$503$	5.73533e8	0.200942	0.100471	+	0.994940i	$0.467965\pi$
$504$	0	0
$505$	−2.06337e9	−0.712948
$506$	0	0
$507$	4.18606e7	0.0142652
$508$	0	0
$509$	9.51755e8	0.319899	0.159950	−	0.987125i	$-0.448867\pi$
$509$	9.51755e8	0.319899	0.159950	+	0.987125i	$0.448867\pi$
$510$	0	0
$511$	8.94416e7	0.0296528
$512$	0	0
$513$	1.85404e8	0.0606328
$514$	0	0
$515$	−4.50981e9	−1.45490
$516$	0	0
$517$	5.36193e9	1.70649
$518$	0	0
$519$	−4.50662e7	−0.0141503
$520$	0	0
$521$	3.07880e9	0.953782	0.476891	−	0.878962i	$-0.341764\pi$
$521$	3.07880e9	0.953782	0.476891	+	0.878962i	$0.341764\pi$
$522$	0	0
$523$	1.47884e9	0.452029	0.226014	−	0.974124i	$-0.427430\pi$
$523$	1.47884e9	0.452029	0.226014	+	0.974124i	$0.427430\pi$
$524$	0	0
$525$	−442369.	−0.000133422 0
$526$	0	0
$527$	−1.21488e8	−0.0361572
$528$	0	0
$529$	−3.18397e9	−0.935135
$530$	0	0
$531$	3.49918e9	1.01423
$532$	0	0
$533$	−2.02499e8	−0.0579265
$534$	0	0
$535$	5.16922e9	1.45944
$536$	0	0
$537$	−1.56328e7	−0.00435638
$538$	0	0
$539$	4.38752e9	1.20687
$540$	0	0
$541$	−2.15744e9	−0.585798	−0.292899	−	0.956143i	$-0.594620\pi$
$541$	−2.15744e9	−0.585798	−0.292899	+	0.956143i	$0.594620\pi$
$542$	0	0
$543$	1.08694e8	0.0291345
$544$	0	0
$545$	−5.00725e9	−1.32499
$546$	0	0
$547$	−3.53337e9	−0.923067	−0.461534	−	0.887123i	$-0.652701\pi$
$547$	−3.53337e9	−0.923067	−0.461534	+	0.887123i	$0.652701\pi$
$548$	0	0
$549$	−2.30132e8	−0.0593572
$550$	0	0
$551$	−1.23538e9	−0.314608
$552$	0	0
$553$	−2.10194e8	−0.0528546
$554$	0	0
$555$	1.50785e7	0.00374397
$556$	0	0
$557$	−4.32894e9	−1.06142	−0.530711	−	0.847553i	$-0.678075\pi$
$557$	−4.32894e9	−1.06142	−0.530711	+	0.847553i	$0.678075\pi$
$558$	0	0
$559$	−2.16046e9	−0.523124
$560$	0	0
$561$	−4.72819e6	−0.00113064
$562$	0	0
$563$	−3.08897e9	−0.729515	−0.364758	−	0.931102i	$-0.618848\pi$
$563$	−3.08897e9	−0.729515	−0.364758	+	0.931102i	$0.618848\pi$
$564$	0	0
$565$	4.75578e9	1.10931
$566$	0	0
$567$	5.94303e8	0.136920
$568$	0	0
$569$	6.69281e9	1.52305	0.761527	−	0.648133i	$-0.224450\pi$
$569$	6.69281e9	1.52305	0.761527	+	0.648133i	$0.224450\pi$
$570$	0	0
$571$	1.24645e9	0.280188	0.140094	−	0.990138i	$-0.455260\pi$
$571$	1.24645e9	0.280188	0.140094	+	0.990138i	$0.455260\pi$
$572$	0	0
$573$	−6.01673e7	−0.0133604
$574$	0	0
$575$	6.31547e7	0.0138538
$576$	0	0
$577$	−1.22638e9	−0.265772	−0.132886	−	0.991131i	$-0.542424\pi$
$577$	−1.22638e9	−0.265772	−0.132886	+	0.991131i	$0.542424\pi$
$578$	0	0
$579$	6.43407e7	0.0137756
$580$	0	0
$581$	−3.43095e8	−0.0725768
$582$	0	0
$583$	−9.11428e9	−1.90495
$584$	0	0
$585$	−2.23911e9	−0.462414
$586$	0	0
$587$	−4.79761e8	−0.0979020	−0.0489510	−	0.998801i	$-0.515588\pi$
$587$	−4.79761e8	−0.0979020	−0.0489510	+	0.998801i	$0.515588\pi$
$588$	0	0
$589$	5.91445e9	1.19264
$590$	0	0
$591$	−6.54849e7	−0.0130492
$592$	0	0
$593$	3.49156e9	0.687587	0.343794	−	0.939045i	$-0.388288\pi$
$593$	3.49156e9	0.687587	0.343794	+	0.939045i	$0.388288\pi$
$594$	0	0
$595$	−3.71405e7	−0.00722835
$596$	0	0
$597$	1.33320e8	0.0256440
$598$	0	0
$599$	8.43205e9	1.60302	0.801511	−	0.597980i	$-0.204030\pi$
$599$	8.43205e9	1.60302	0.801511	+	0.597980i	$0.204030\pi$
$600$	0	0
$601$	9.99881e9	1.87883	0.939415	−	0.342781i	$-0.111369\pi$
$601$	9.99881e9	1.87883	0.939415	+	0.342781i	$0.111369\pi$
$602$	0	0
$603$	−8.27135e9	−1.53626
$604$	0	0
$605$	2.86821e9	0.526584
$606$	0	0
$607$	8.44900e9	1.53336	0.766681	−	0.642028i	$-0.221907\pi$
$607$	8.44900e9	1.53336	0.766681	+	0.642028i	$0.221907\pi$
$608$	0	0
$609$	2.53878e6	0.000455475 0
$610$	0	0
$611$	−3.52389e9	−0.624998
$612$	0	0
$613$	−7.14246e8	−0.125238	−0.0626190	−	0.998038i	$-0.519945\pi$
$613$	−7.14246e8	−0.125238	−0.0626190	+	0.998038i	$0.519945\pi$
$614$	0	0
$615$	1.36318e7	0.00236314
$616$	0	0
$617$	−6.38724e9	−1.09475	−0.547375	−	0.836887i	$-0.684373\pi$
$617$	−6.38724e9	−1.09475	−0.547375	+	0.836887i	$0.684373\pi$
$618$	0	0
$619$	8.47356e9	1.43598	0.717991	−	0.696053i	$-0.245062\pi$
$619$	8.47356e9	1.43598	0.717991	+	0.696053i	$0.245062\pi$
$620$	0	0
$621$	5.43957e7	0.00911474
$622$	0	0
$623$	1.41098e9	0.233782
$624$	0	0
$625$	−6.41746e9	−1.05144
$626$	0	0
$627$	2.30185e8	0.0372942
$628$	0	0
$629$	6.53103e7	0.0104642
$630$	0	0
$631$	−1.34035e9	−0.212381	−0.106191	−	0.994346i	$-0.533865\pi$
$631$	−1.34035e9	−0.212381	−0.106191	+	0.994346i	$0.533865\pi$
$632$	0	0
$633$	7.36049e7	0.0115344
$634$	0	0
$635$	−9.31375e9	−1.44350
$636$	0	0
$637$	−2.88351e9	−0.442011
$638$	0	0
$639$	8.04112e9	1.21917
$640$	0	0
$641$	9.56239e9	1.43405	0.717023	−	0.697049i	$-0.245504\pi$
$641$	9.56239e9	1.43405	0.717023	+	0.697049i	$0.245504\pi$
$642$	0	0
$643$	7.89728e9	1.17149	0.585746	−	0.810495i	$-0.300802\pi$
$643$	7.89728e9	1.17149	0.585746	+	0.810495i	$0.300802\pi$
$644$	0	0
$645$	1.45437e8	0.0213411
$646$	0	0
$647$	−1.45293e9	−0.210901	−0.105451	−	0.994425i	$-0.533628\pi$
$647$	−1.45293e9	−0.210901	−0.105451	+	0.994425i	$0.533628\pi$
$648$	0	0
$649$	8.69010e9	1.24787
$650$	0	0
$651$	−1.21546e7	−0.00172666
$652$	0	0
$653$	4.94004e9	0.694279	0.347140	−	0.937813i	$-0.387153\pi$
$653$	4.94004e9	0.694279	0.347140	+	0.937813i	$0.387153\pi$
$654$	0	0
$655$	2.64339e9	0.367550
$656$	0	0
$657$	−1.57225e9	−0.216293
$658$	0	0
$659$	1.42643e10	1.94157	0.970784	−	0.239953i	$-0.0771321\pi$
$659$	1.42643e10	1.94157	0.970784	+	0.239953i	$0.0771321\pi$
$660$	0	0
$661$	8.41697e9	1.13358	0.566788	−	0.823864i	$-0.308186\pi$
$661$	8.41697e9	1.13358	0.566788	+	0.823864i	$0.308186\pi$
$662$	0	0
$663$	3.10739e6	0.000414094 0
$664$	0	0
$665$	1.80814e9	0.238427
$666$	0	0
$667$	−3.62449e8	−0.0472940
$668$	0	0
$669$	1.29893e8	0.0167724
$670$	0	0
$671$	−5.71525e8	−0.0730308
$672$	0	0
$673$	1.60907e9	0.203481	0.101740	−	0.994811i	$-0.467559\pi$
$673$	1.60907e9	0.203481	0.101740	+	0.994811i	$0.467559\pi$
$674$	0	0
$675$	1.55548e7	0.00194672
$676$	0	0
$677$	−6.26194e9	−0.775620	−0.387810	−	0.921739i	$-0.626768\pi$
$677$	−6.26194e9	−0.775620	−0.387810	+	0.921739i	$0.626768\pi$
$678$	0	0
$679$	−1.80732e9	−0.221559
$680$	0	0
$681$	−2.18291e8	−0.0264863
$682$	0	0
$683$	−8.28070e8	−0.0994477	−0.0497239	−	0.998763i	$-0.515834\pi$
$683$	−8.28070e8	−0.0994477	−0.0497239	+	0.998763i	$0.515834\pi$
$684$	0	0
$685$	−1.61855e10	−1.92402
$686$	0	0
$687$	−1.74762e7	−0.00205636
$688$	0	0
$689$	5.98996e9	0.697681
$690$	0	0
$691$	−9.95391e9	−1.14768	−0.573839	−	0.818968i	$-0.694547\pi$
$691$	−9.95391e9	−1.14768	−0.573839	+	0.818968i	$0.694547\pi$
$692$	0	0
$693$	1.47641e9	0.168515
$694$	0	0
$695$	−1.03135e10	−1.16535
$696$	0	0
$697$	5.90441e7	0.00660483
$698$	0	0
$699$	3.34129e7	0.00370036
$700$	0	0
$701$	−1.11577e9	−0.122338	−0.0611692	−	0.998127i	$-0.519483\pi$
$701$	−1.11577e9	−0.122338	−0.0611692	+	0.998127i	$0.519483\pi$
$702$	0	0
$703$	−3.17954e9	−0.345160
$704$	0	0
$705$	2.37221e8	0.0254971
$706$	0	0
$707$	−8.94147e8	−0.0951571
$708$	0	0
$709$	−1.92104e9	−0.202430	−0.101215	−	0.994865i	$-0.532273\pi$
$709$	−1.92104e9	−0.202430	−0.101215	+	0.994865i	$0.532273\pi$
$710$	0	0
$711$	3.69490e9	0.385530
$712$	0	0
$713$	1.73525e9	0.179286
$714$	0	0
$715$	−5.56077e9	−0.568936
$716$	0	0
$717$	−2.67457e8	−0.0270980
$718$	0	0
$719$	−8.64628e9	−0.867518	−0.433759	−	0.901029i	$-0.642813\pi$
$719$	−8.64628e9	−0.867518	−0.433759	+	0.901029i	$0.642813\pi$
$720$	0	0
$721$	−1.95429e9	−0.194185
$722$	0	0
$723$	−2.42294e8	−0.0238429
$724$	0	0
$725$	−1.03645e8	−0.0101010
$726$	0	0
$727$	−8.59715e9	−0.829821	−0.414910	−	0.909862i	$-0.636187\pi$
$727$	−8.59715e9	−0.829821	−0.414910	+	0.909862i	$0.636187\pi$
$728$	0	0
$729$	−1.04403e10	−0.998079
$730$	0	0
$731$	6.29941e8	0.0596471
$732$	0	0
$733$	−1.59376e10	−1.49471	−0.747357	−	0.664423i	$-0.768677\pi$
$733$	−1.59376e10	−1.49471	−0.747357	+	0.664423i	$0.768677\pi$
$734$	0	0
$735$	1.94111e8	0.0180321
$736$	0	0
$737$	−2.05416e10	−1.89016
$738$	0	0
$739$	−1.69117e10	−1.54146	−0.770729	−	0.637163i	$-0.780108\pi$
$739$	−1.69117e10	−1.54146	−0.770729	+	0.637163i	$0.780108\pi$
$740$	0	0
$741$	−1.51279e8	−0.0136589
$742$	0	0
$743$	1.78167e10	1.59355	0.796776	−	0.604275i	$-0.206537\pi$
$743$	1.78167e10	1.59355	0.796776	+	0.604275i	$0.206537\pi$
$744$	0	0
$745$	−4.16136e9	−0.368713
$746$	0	0
$747$	6.03110e9	0.529388
$748$	0	0
$749$	2.24004e9	0.194791
$750$	0	0
$751$	−1.11904e10	−0.964067	−0.482034	−	0.876153i	$-0.660102\pi$
$751$	−1.11904e10	−0.964067	−0.482034	+	0.876153i	$0.660102\pi$
$752$	0	0
$753$	2.22249e8	0.0189696
$754$	0	0
$755$	1.92896e9	0.163120
$756$	0	0
$757$	2.33314e9	0.195482	0.0977408	−	0.995212i	$-0.468838\pi$
$757$	2.33314e9	0.195482	0.0977408	+	0.995212i	$0.468838\pi$
$758$	0	0
$759$	6.75342e7	0.00560631
$760$	0	0
$761$	4.76111e9	0.391618	0.195809	−	0.980642i	$-0.437267\pi$
$761$	4.76111e9	0.391618	0.195809	+	0.980642i	$0.437267\pi$
$762$	0	0
$763$	−2.16985e9	−0.176846
$764$	0	0
$765$	6.52875e8	0.0527248
$766$	0	0
$767$	−5.71118e9	−0.457027
$768$	0	0
$769$	−9.44013e9	−0.748576	−0.374288	−	0.927312i	$-0.622113\pi$
$769$	−9.44013e9	−0.748576	−0.374288	+	0.927312i	$0.622113\pi$
$770$	0	0
$771$	−1.21427e8	−0.00954167
$772$	0	0
$773$	4.76243e9	0.370851	0.185426	−	0.982658i	$-0.440634\pi$
$773$	4.76243e9	0.370851	0.185426	+	0.982658i	$0.440634\pi$
$774$	0	0
$775$	4.96206e8	0.0382918
$776$	0	0
$777$	6.53415e6	0.000499707 0
$778$	0	0
$779$	−2.87448e9	−0.217860
$780$	0	0
$781$	1.99699e10	1.50002
$782$	0	0
$783$	−8.92701e7	−0.00664569
$784$	0	0
$785$	1.56698e10	1.15616
$786$	0	0
$787$	−9.01011e9	−0.658899	−0.329449	−	0.944173i	$-0.606863\pi$
$787$	−9.01011e9	−0.658899	−0.329449	+	0.944173i	$0.606863\pi$
$788$	0	0
$789$	2.97379e8	0.0215547
$790$	0	0
$791$	2.06088e9	0.148059
$792$	0	0
$793$	3.75609e8	0.0267473
$794$	0	0
$795$	−4.03231e8	−0.0284623
$796$	0	0
$797$	4.47431e9	0.313056	0.156528	−	0.987674i	$-0.449970\pi$
$797$	4.47431e9	0.313056	0.156528	+	0.987674i	$0.449970\pi$
$798$	0	0
$799$	1.02749e9	0.0712628
$800$	0	0
$801$	−2.48029e10	−1.70525
$802$	0	0
$803$	−3.90463e9	−0.266119
$804$	0	0
$805$	5.30490e8	0.0358419
$806$	0	0
$807$	2.70455e8	0.0181150
$808$	0	0
$809$	−3.59948e9	−0.239012	−0.119506	−	0.992833i	$-0.538131\pi$
$809$	−3.59948e9	−0.239012	−0.119506	+	0.992833i	$0.538131\pi$
$810$	0	0
$811$	−4.19700e9	−0.276290	−0.138145	−	0.990412i	$-0.544114\pi$
$811$	−4.19700e9	−0.276290	−0.138145	+	0.990412i	$0.544114\pi$
$812$	0	0
$813$	2.65298e7	0.00173148
$814$	0	0
$815$	6.43826e9	0.416598
$816$	0	0
$817$	−3.06678e10	−1.96746
$818$	0	0
$819$	−9.70303e8	−0.0617183
$820$	0	0
$821$	2.95484e10	1.86351	0.931756	−	0.363084i	$-0.118276\pi$
$821$	2.95484e10	1.86351	0.931756	+	0.363084i	$0.118276\pi$
$822$	0	0
$823$	1.65648e10	1.03583	0.517913	−	0.855433i	$-0.326709\pi$
$823$	1.65648e10	1.03583	0.517913	+	0.855433i	$0.326709\pi$
$824$	0	0
$825$	1.93119e7	0.00119739
$826$	0	0
$827$	−1.56492e10	−0.962108	−0.481054	−	0.876691i	$-0.659746\pi$
$827$	−1.56492e10	−0.962108	−0.481054	+	0.876691i	$0.659746\pi$
$828$	0	0
$829$	−1.89289e10	−1.15395	−0.576973	−	0.816763i	$-0.695766\pi$
$829$	−1.89289e10	−1.15395	−0.576973	+	0.816763i	$0.695766\pi$
$830$	0	0
$831$	1.21093e6	7.32006e−5 0
$832$	0	0
$833$	8.40765e8	0.0503985
$834$	0	0
$835$	−2.58700e10	−1.53778
$836$	0	0
$837$	4.27386e8	0.0251931
$838$	0	0
$839$	3.11150e10	1.81888	0.909439	−	0.415838i	$-0.136512\pi$
$839$	3.11150e10	1.81888	0.909439	+	0.415838i	$0.136512\pi$
$840$	0	0
$841$	5.94823e8	0.0344828
$842$	0	0
$843$	3.70154e8	0.0212807
$844$	0	0
$845$	−1.43549e10	−0.818466
$846$	0	0
$847$	1.24292e9	0.0702830
$848$	0	0
$849$	−2.18516e7	−0.00122548
$850$	0	0
$851$	−9.32848e8	−0.0518869
$852$	0	0
$853$	−1.39347e10	−0.768736	−0.384368	−	0.923180i	$-0.625581\pi$
$853$	−1.39347e10	−0.768736	−0.384368	+	0.923180i	$0.625581\pi$
$854$	0	0
$855$	−3.17843e10	−1.73913
$856$	0	0
$857$	−3.21238e10	−1.74339	−0.871693	−	0.490052i	$-0.836978\pi$
$857$	−3.21238e10	−1.74339	−0.871693	+	0.490052i	$0.836978\pi$
$858$	0	0
$859$	8.24123e9	0.443625	0.221813	−	0.975089i	$-0.428803\pi$
$859$	8.24123e9	0.443625	0.221813	+	0.975089i	$0.428803\pi$
$860$	0	0
$861$	5.90723e6	0.000315408 0
$862$	0	0
$863$	−1.07922e10	−0.571575	−0.285788	−	0.958293i	$-0.592255\pi$
$863$	−1.07922e10	−0.571575	−0.285788	+	0.958293i	$0.592255\pi$
$864$	0	0
$865$	1.54541e10	0.811874
$866$	0	0
$867$	3.42530e8	0.0178497
$868$	0	0
$869$	9.17615e9	0.474342
$870$	0	0
$871$	1.35001e10	0.692266
$872$	0	0
$873$	3.17699e10	1.61609
$874$	0	0
$875$	−2.63709e9	−0.133075
$876$	0	0
$877$	−2.32533e10	−1.16409	−0.582043	−	0.813158i	$-0.697747\pi$
$877$	−2.32533e10	−1.16409	−0.582043	+	0.813158i	$0.697747\pi$
$878$	0	0
$879$	2.26659e8	0.0112567
$880$	0	0
$881$	−1.88747e10	−0.929962	−0.464981	−	0.885321i	$-0.653939\pi$
$881$	−1.88747e10	−0.929962	−0.464981	+	0.885321i	$0.653939\pi$
$882$	0	0
$883$	7.37985e9	0.360732	0.180366	−	0.983600i	$-0.442272\pi$
$883$	7.37985e9	0.360732	0.180366	+	0.983600i	$0.442272\pi$
$884$	0	0
$885$	3.84464e8	0.0186447
$886$	0	0
$887$	2.27600e10	1.09506	0.547531	−	0.836785i	$-0.315568\pi$
$887$	2.27600e10	1.09506	0.547531	+	0.836785i	$0.315568\pi$
$888$	0	0
$889$	−4.03604e9	−0.192664
$890$	0	0
$891$	−2.59447e10	−1.22879
$892$	0	0
$893$	−5.00218e10	−2.35060
$894$	0	0
$895$	5.36080e9	0.249947
$896$	0	0
$897$	−4.43839e7	−0.00205330
$898$	0	0
$899$	−2.84775e9	−0.130720
$900$	0	0
$901$	−1.74654e9	−0.0795502
$902$	0	0
$903$	6.30242e7	0.00284839
$904$	0	0
$905$	−3.72736e10	−1.67159
$906$	0	0
$907$	−2.05239e10	−0.913345	−0.456673	−	0.889635i	$-0.650959\pi$
$907$	−2.05239e10	−0.913345	−0.456673	+	0.889635i	$0.650959\pi$
$908$	0	0
$909$	1.57178e10	0.694092
$910$	0	0
$911$	6.50900e9	0.285233	0.142616	−	0.989778i	$-0.454448\pi$
$911$	6.50900e9	0.285233	0.142616	+	0.989778i	$0.454448\pi$
$912$	0	0
$913$	1.49780e10	0.651339
$914$	0	0
$915$	−2.52852e7	−0.00109117
$916$	0	0
$917$	1.14549e9	0.0490568
$918$	0	0
$919$	2.04744e10	0.870174	0.435087	−	0.900388i	$-0.356718\pi$
$919$	2.04744e10	0.870174	0.435087	+	0.900388i	$0.356718\pi$
$920$	0	0
$921$	3.61801e7	0.00152602
$922$	0	0
$923$	−1.31243e10	−0.549377
$924$	0	0
$925$	−2.66754e8	−0.0110819
$926$	0	0
$927$	3.43535e10	1.41642
$928$	0	0
$929$	4.58067e10	1.87445	0.937224	−	0.348727i	$-0.113386\pi$
$929$	4.58067e10	1.87445	0.937224	+	0.348727i	$0.113386\pi$
$930$	0	0
$931$	−4.09315e10	−1.66239
$932$	0	0
$933$	−1.23704e8	−0.00498652
$934$	0	0
$935$	1.62139e9	0.0648706
$936$	0	0
$937$	1.88271e10	0.747645	0.373822	−	0.927500i	$-0.378047\pi$
$937$	1.88271e10	0.747645	0.373822	+	0.927500i	$0.378047\pi$
$938$	0	0
$939$	3.74137e8	0.0147469
$940$	0	0
$941$	−2.59965e10	−1.01707	−0.508535	−	0.861041i	$-0.669813\pi$
$941$	−2.59965e10	−1.01707	−0.508535	+	0.861041i	$0.669813\pi$
$942$	0	0
$943$	−8.43346e8	−0.0327503
$944$	0	0
$945$	1.30658e8	0.00503646
$946$	0	0
$947$	2.57319e10	0.984572	0.492286	−	0.870434i	$-0.336161\pi$
$947$	2.57319e10	0.984572	0.492286	+	0.870434i	$0.336161\pi$
$948$	0	0
$949$	2.56614e9	0.0974652
$950$	0	0
$951$	6.73820e8	0.0254046
$952$	0	0
$953$	2.08096e10	0.778821	0.389411	−	0.921064i	$-0.372679\pi$
$953$	2.08096e10	0.778821	0.389411	+	0.921064i	$0.372679\pi$
$954$	0	0
$955$	2.06326e10	0.766554
$956$	0	0
$957$	−1.10832e8	−0.00408765
$958$	0	0
$959$	−7.01387e9	−0.256799
$960$	0	0
$961$	−1.38788e10	−0.504453
$962$	0	0
$963$	−3.93766e10	−1.42084
$964$	0	0
$965$	−2.20638e10	−0.790376
$966$	0	0
$967$	−9.77904e8	−0.0347779	−0.0173890	−	0.999849i	$-0.505535\pi$
$967$	−9.77904e8	−0.0347779	−0.0173890	+	0.999849i	$0.505535\pi$
$968$	0	0
$969$	4.41096e7	0.00155740
$970$	0	0
$971$	−2.77032e10	−0.971096	−0.485548	−	0.874210i	$-0.661380\pi$
$971$	−2.77032e10	−0.971096	−0.485548	+	0.874210i	$0.661380\pi$
$972$	0	0
$973$	−4.46926e9	−0.155540
$974$	0	0
$975$	−1.26919e7	−0.000438541 0
$976$	0	0
$977$	2.66223e10	0.913302	0.456651	−	0.889646i	$-0.349049\pi$
$977$	2.66223e10	0.913302	0.456651	+	0.889646i	$0.349049\pi$
$978$	0	0
$979$	−6.15971e10	−2.09807
$980$	0	0
$981$	3.81428e10	1.28994
$982$	0	0
$983$	2.14323e10	0.719666	0.359833	−	0.933017i	$-0.382834\pi$
$983$	2.14323e10	0.719666	0.359833	+	0.933017i	$0.382834\pi$
$984$	0	0
$985$	2.24561e10	0.748701
$986$	0	0
$987$	1.02798e8	0.00340309
$988$	0	0
$989$	−8.99764e9	−0.295762
$990$	0	0
$991$	3.33513e10	1.08857	0.544283	−	0.838902i	$-0.316802\pi$
$991$	3.33513e10	1.08857	0.544283	+	0.838902i	$0.316802\pi$
$992$	0	0
$993$	−7.59208e8	−0.0246058
$994$	0	0
$995$	−4.57182e10	−1.47132
$996$	0	0
$997$	5.53807e10	1.76980	0.884902	−	0.465778i	$-0.154225\pi$
$997$	5.53807e10	1.76980	0.884902	+	0.465778i	$0.154225\pi$
$998$	0	0
$999$	−2.29758e8	−0.00729107

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	464.8.a.g.1.6		10
4.3	odd	2		29.8.a.b.1.9	✓	10
12.11	even	2		261.8.a.f.1.2		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.8.a.b.1.9	✓	10	4.3	odd	2
261.8.a.f.1.2		10	12.11	even	2
464.8.a.g.1.6		10	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:	\( N \)	\(=\)	\( 464 = 2^{4} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	464.a (trivial)

\(f(q)\)	\(=\)	\(q-0.836957 q^{3} +287.010 q^{5} +124.374 q^{7} -2186.30 q^{9} +O(q^{10})\)	\(q-0.836957 q^{3} +287.010 q^{5} +124.374 q^{7} -2186.30 q^{9} -5429.60 q^{11} +3568.37 q^{13} -240.215 q^{15} -1040.46 q^{17} +50653.1 q^{19} -104.095 q^{21} +14861.2 q^{23} +4249.65 q^{25} +3660.26 q^{27} -24389.0 q^{29} +116764. q^{31} +4544.34 q^{33} +35696.4 q^{35} -62770.9 q^{37} -2986.57 q^{39} -56748.3 q^{41} -605447. q^{43} -627489. q^{45} -987536. q^{47} -808074. q^{49} +870.816 q^{51} +1.67863e6 q^{53} -1.55835e6 q^{55} -42394.5 q^{57} -1.60050e6 q^{59} +105261. q^{61} -271918. q^{63} +1.02416e6 q^{65} +3.78326e6 q^{67} -12438.1 q^{69} -3.67796e6 q^{71} +719137. q^{73} -3556.78 q^{75} -675299. q^{77} -1.69002e6 q^{79} +4.77837e6 q^{81} -2.75859e6 q^{83} -298621. q^{85} +20412.5 q^{87} +1.13447e7 q^{89} +443811. q^{91} -97726.3 q^{93} +1.45379e7 q^{95} -1.45314e7 q^{97} +1.18707e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 80 q^{3} + 180 q^{5} - 1040 q^{7} + 10986 q^{9}+O(q^{10}) \) 10 * q - 80 * q^3 + 180 * q^5 - 1040 * q^7 + 10986 * q^9	\( 10 q - 80 q^{3} + 180 q^{5} - 1040 q^{7} + 10986 q^{9} - 7384 q^{11} + 20820 q^{13} - 43516 q^{15} - 11620 q^{17} - 75068 q^{19} + 51480 q^{21} - 62040 q^{23} + 261022 q^{25} + 28060 q^{27} - 243890 q^{29} - 200600 q^{31} - 1068000 q^{33} - 107528 q^{35} - 367740 q^{37} - 392692 q^{39} + 932764 q^{41} - 1443560 q^{43} + 4245684 q^{45} + 286960 q^{47} + 4713194 q^{49} - 1451016 q^{51} + 3953220 q^{53} - 3981316 q^{55} + 2050640 q^{57} - 6712320 q^{59} + 1905196 q^{61} - 3643800 q^{63} + 4667544 q^{65} + 2718200 q^{67} + 1109064 q^{69} - 3447736 q^{71} - 2554460 q^{73} - 1088084 q^{75} - 3967800 q^{77} - 4187744 q^{79} + 5161402 q^{81} - 3498720 q^{83} + 1817072 q^{85} + 1951120 q^{87} - 303268 q^{89} - 27215080 q^{91} + 1097360 q^{93} + 8810536 q^{95} + 4908620 q^{97} + 14408716 q^{99}+O(q^{100}) \) 10 * q - 80 * q^3 + 180 * q^5 - 1040 * q^7 + 10986 * q^9 - 7384 * q^11 + 20820 * q^13 - 43516 * q^15 - 11620 * q^17 - 75068 * q^19 + 51480 * q^21 - 62040 * q^23 + 261022 * q^25 + 28060 * q^27 - 243890 * q^29 - 200600 * q^31 - 1068000 * q^33 - 107528 * q^35 - 367740 * q^37 - 392692 * q^39 + 932764 * q^41 - 1443560 * q^43 + 4245684 * q^45 + 286960 * q^47 + 4713194 * q^49 - 1451016 * q^51 + 3953220 * q^53 - 3981316 * q^55 + 2050640 * q^57 - 6712320 * q^59 + 1905196 * q^61 - 3643800 * q^63 + 4667544 * q^65 + 2718200 * q^67 + 1109064 * q^69 - 3447736 * q^71 - 2554460 * q^73 - 1088084 * q^75 - 3967800 * q^77 - 4187744 * q^79 + 5161402 * q^81 - 3498720 * q^83 + 1817072 * q^85 + 1951120 * q^87 - 303268 * q^89 - 27215080 * q^91 + 1097360 * q^93 + 8810536 * q^95 + 4908620 * q^97 + 14408716 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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