LMFDB - Embedded newform 432.4.a.p.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [432,4,Mod(1,432)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(432, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("432.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 432 = 2^{4} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	432.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:	$25.4888251225$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2}\cdot 3 $
Twist minimal:	no (minimal twist has level 216)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	432.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+11.4164 q^{5} +29.8328 q^{7} +O(q^{10})$	$q+11.4164 q^{5} +29.8328 q^{7} +66.2492 q^{11} +39.8328 q^{13} -107.416 q^{17} -70.3313 q^{19} +6.91796 q^{23} +5.33437 q^{25} -36.6687 q^{29} +231.331 q^{31} +340.584 q^{35} +36.8359 q^{37} -429.325 q^{41} +74.3344 q^{43} +52.5836 q^{47} +546.997 q^{49} -288.170 q^{53} +756.328 q^{55} -783.745 q^{59} +439.158 q^{61} +454.748 q^{65} +218.337 q^{67} +790.492 q^{71} +1098.00 q^{73} +1976.40 q^{77} -439.827 q^{79} -50.8452 q^{83} -1226.31 q^{85} -719.745 q^{89} +1188.33 q^{91} -802.930 q^{95} -1199.64 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{5} + 6 q^{7}+O(q^{10}) $ 2 * q - 4 * q^5 + 6 * q^7	$ 2 q - 4 q^{5} + 6 q^{7} + 52 q^{11} + 26 q^{13} - 188 q^{17} + 74 q^{19} + 148 q^{23} + 118 q^{25} - 288 q^{29} + 248 q^{31} + 708 q^{35} + 342 q^{37} + 256 q^{43} + 132 q^{47} + 772 q^{49} - 952 q^{53} + 976 q^{55} - 1004 q^{59} - 34 q^{61} + 668 q^{65} + 866 q^{67} + 776 q^{71} + 1874 q^{73} + 2316 q^{77} - 182 q^{79} - 1336 q^{83} + 16 q^{85} - 876 q^{89} + 1518 q^{91} - 3028 q^{95} - 38 q^{97}+O(q^{100}) $ 2 * q - 4 * q^5 + 6 * q^7 + 52 * q^11 + 26 * q^13 - 188 * q^17 + 74 * q^19 + 148 * q^23 + 118 * q^25 - 288 * q^29 + 248 * q^31 + 708 * q^35 + 342 * q^37 + 256 * q^43 + 132 * q^47 + 772 * q^49 - 952 * q^53 + 976 * q^55 - 1004 * q^59 - 34 * q^61 + 668 * q^65 + 866 * q^67 + 776 * q^71 + 1874 * q^73 + 2316 * q^77 - 182 * q^79 - 1336 * q^83 + 16 * q^85 - 876 * q^89 + 1518 * q^91 - 3028 * q^95 - 38 * 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
$5$	11.4164	1.02111	0.510557	−	0.859844i	$-0.329439\pi$
$5$	11.4164	1.02111	0.510557	+	0.859844i	$0.329439\pi$
$6$	0	0
$7$	29.8328	1.61082	0.805410	−	0.592718i	$-0.201945\pi$
$7$	29.8328	1.61082	0.805410	+	0.592718i	$0.201945\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	66.2492	1.81590	0.907950	−	0.419079i	$-0.137647\pi$
$11$	66.2492	1.81590	0.907950	+	0.419079i	$0.137647\pi$
$12$	0	0
$13$	39.8328	0.849818	0.424909	−	0.905236i	$-0.360306\pi$
$13$	39.8328	0.849818	0.424909	+	0.905236i	$0.360306\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−107.416	−1.53249	−0.766244	−	0.642549i	$-0.777877\pi$
$17$	−107.416	−1.53249	−0.766244	+	0.642549i	$0.777877\pi$
$18$	0	0
$19$	−70.3313	−0.849216	−0.424608	−	0.905377i	$-0.639588\pi$
$19$	−70.3313	−0.849216	−0.424608	+	0.905377i	$0.639588\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	6.91796	0.0627172	0.0313586	−	0.999508i	$-0.490017\pi$
$23$	6.91796	0.0627172	0.0313586	+	0.999508i	$0.490017\pi$
$24$	0	0
$25$	5.33437	0.0426749
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−36.6687	−0.234800	−0.117400	−	0.993085i	$-0.537456\pi$
$29$	−36.6687	−0.234800	−0.117400	+	0.993085i	$0.537456\pi$
$30$	0	0
$31$	231.331	1.34027	0.670134	−	0.742240i	$-0.266237\pi$
$31$	231.331	1.34027	0.670134	+	0.742240i	$0.266237\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	340.584	1.64483
$36$	0	0
$37$	36.8359	0.163670	0.0818350	−	0.996646i	$-0.473922\pi$
$37$	36.8359	0.163670	0.0818350	+	0.996646i	$0.473922\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−429.325	−1.63535	−0.817674	−	0.575681i	$-0.804737\pi$
$41$	−429.325	−1.63535	−0.817674	+	0.575681i	$0.804737\pi$
$42$	0	0
$43$	74.3344	0.263625	0.131813	−	0.991275i	$-0.457920\pi$
$43$	74.3344	0.263625	0.131813	+	0.991275i	$0.457920\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	52.5836	0.163194	0.0815969	−	0.996665i	$-0.473998\pi$
$47$	52.5836	0.163194	0.0815969	+	0.996665i	$0.473998\pi$
$48$	0	0
$49$	546.997	1.59474
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−288.170	−0.746853	−0.373427	−	0.927660i	$-0.621817\pi$
$53$	−288.170	−0.746853	−0.373427	+	0.927660i	$0.621817\pi$
$54$	0	0
$55$	756.328	1.85424
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−783.745	−1.72940	−0.864702	−	0.502285i	$-0.832493\pi$
$59$	−783.745	−1.72940	−0.864702	+	0.502285i	$0.832493\pi$
$60$	0	0
$61$	439.158	0.921777	0.460889	−	0.887458i	$-0.347531\pi$
$61$	439.158	0.921777	0.460889	+	0.887458i	$0.347531\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	454.748	0.867762
$66$	0	0
$67$	218.337	0.398122	0.199061	−	0.979987i	$-0.436211\pi$
$67$	218.337	0.398122	0.199061	+	0.979987i	$0.436211\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	790.492	1.32133	0.660663	−	0.750682i	$-0.270275\pi$
$71$	790.492	1.32133	0.660663	+	0.750682i	$0.270275\pi$
$72$	0	0
$73$	1098.00	1.76042	0.880211	−	0.474582i	$-0.157401\pi$
$73$	1098.00	1.76042	0.880211	+	0.474582i	$0.157401\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1976.40	2.92509
$78$	0	0
$79$	−439.827	−0.626384	−0.313192	−	0.949690i	$-0.601398\pi$
$79$	−439.827	−0.626384	−0.313192	+	0.949690i	$0.601398\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−50.8452	−0.0672408	−0.0336204	−	0.999435i	$-0.510704\pi$
$83$	−50.8452	−0.0672408	−0.0336204	+	0.999435i	$0.510704\pi$
$84$	0	0
$85$	−1226.31	−1.56485
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−719.745	−0.857222	−0.428611	−	0.903489i	$-0.640997\pi$
$89$	−719.745	−0.857222	−0.428611	+	0.903489i	$0.640997\pi$
$90$	0	0
$91$	1188.33	1.36890
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−802.930	−0.867147
$96$	0	0
$97$	−1199.64	−1.25573	−0.627863	−	0.778324i	$-0.716070\pi$
$97$	−1199.64	−1.25573	−0.627863	+	0.778324i	$0.716070\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−245.981	−0.242337	−0.121169	−	0.992632i	$-0.538664\pi$
$101$	−245.981	−0.242337	−0.121169	+	0.992632i	$0.538664\pi$
$102$	0	0
$103$	−575.820	−0.550847	−0.275424	−	0.961323i	$-0.588818\pi$
$103$	−575.820	−0.550847	−0.275424	+	0.961323i	$0.588818\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1015.89	−0.917849	−0.458924	−	0.888475i	$-0.651765\pi$
$107$	−1015.89	−0.917849	−0.458924	+	0.888475i	$0.651765\pi$
$108$	0	0
$109$	1471.64	1.29319	0.646595	−	0.762834i	$-0.276193\pi$
$109$	1471.64	1.29319	0.646595	+	0.762834i	$0.276193\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1903.57	1.58472	0.792359	−	0.610055i	$-0.208853\pi$
$113$	1903.57	1.58472	0.792359	+	0.610055i	$0.208853\pi$
$114$	0	0
$115$	78.9783	0.0640414
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−3204.53	−2.46856
$120$	0	0
$121$	3057.96	2.29749
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1366.15	−0.977539
$126$	0	0
$127$	−230.310	−0.160919	−0.0804593	−	0.996758i	$-0.525639\pi$
$127$	−230.310	−0.160919	−0.0804593	+	0.996758i	$0.525639\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	794.310	0.529764	0.264882	−	0.964281i	$-0.414667\pi$
$131$	794.310	0.529764	0.264882	+	0.964281i	$0.414667\pi$
$132$	0	0
$133$	−2098.18	−1.36793
$134$	0	0
$135$	0	0
$136$	0	0
$137$	683.076	0.425979	0.212989	−	0.977055i	$-0.431680\pi$
$137$	683.076	0.425979	0.212989	+	0.977055i	$0.431680\pi$
$138$	0	0
$139$	1512.64	0.923025	0.461513	−	0.887134i	$-0.347307\pi$
$139$	1512.64	0.923025	0.461513	+	0.887134i	$0.347307\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	2638.89	1.54318
$144$	0	0
$145$	−418.625	−0.239758
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2307.14	−1.26851	−0.634255	−	0.773124i	$-0.718693\pi$
$149$	−2307.14	−1.26851	−0.634255	+	0.773124i	$0.718693\pi$
$150$	0	0
$151$	−2523.17	−1.35982	−0.679910	−	0.733296i	$-0.737981\pi$
$151$	−2523.17	−1.35982	−0.679910	+	0.733296i	$0.737981\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2640.97	1.36857
$156$	0	0
$157$	−2918.98	−1.48382	−0.741912	−	0.670497i	$-0.766081\pi$
$157$	−2918.98	−1.48382	−0.741912	+	0.670497i	$0.766081\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	206.382	0.101026
$162$	0	0
$163$	−1096.99	−0.527133	−0.263567	−	0.964641i	$-0.584899\pi$
$163$	−1096.99	−0.527133	−0.263567	+	0.964641i	$0.584899\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−732.729	−0.339523	−0.169761	−	0.985485i	$-0.554300\pi$
$167$	−732.729	−0.339523	−0.169761	+	0.985485i	$0.554300\pi$
$168$	0	0
$169$	−610.347	−0.277809
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−722.675	−0.317595	−0.158798	−	0.987311i	$-0.550762\pi$
$173$	−722.675	−0.317595	−0.158798	+	0.987311i	$0.550762\pi$
$174$	0	0
$175$	159.139	0.0687417
$176$	0	0
$177$	0	0
$178$	0	0
$179$	1010.16	0.421806	0.210903	−	0.977507i	$-0.432360\pi$
$179$	1010.16	0.421806	0.210903	+	0.977507i	$0.432360\pi$
$180$	0	0
$181$	−4256.79	−1.74809	−0.874045	−	0.485844i	$-0.838512\pi$
$181$	−4256.79	−1.74809	−0.874045	+	0.485844i	$0.838512\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	420.534	0.167126
$186$	0	0
$187$	−7116.25	−2.78284
$188$	0	0
$189$	0	0
$190$	0	0
$191$	3429.39	1.29917	0.649585	−	0.760289i	$-0.274943\pi$
$191$	3429.39	1.29917	0.649585	+	0.760289i	$0.274943\pi$
$192$	0	0
$193$	967.341	0.360781	0.180390	−	0.983595i	$-0.442264\pi$
$193$	967.341	0.360781	0.180390	+	0.983595i	$0.442264\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	468.413	0.169406	0.0847032	−	0.996406i	$-0.473006\pi$
$197$	468.413	0.169406	0.0847032	+	0.996406i	$0.473006\pi$
$198$	0	0
$199$	2673.47	0.952347	0.476174	−	0.879351i	$-0.342023\pi$
$199$	2673.47	0.952347	0.476174	+	0.879351i	$0.342023\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−1093.93	−0.378221
$204$	0	0
$205$	−4901.35	−1.66988
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−4659.39	−1.54209
$210$	0	0
$211$	−2870.62	−0.936594	−0.468297	−	0.883571i	$-0.655132\pi$
$211$	−2870.62	−0.936594	−0.468297	+	0.883571i	$0.655132\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	848.631	0.269192
$216$	0	0
$217$	6901.26	2.15893
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−4278.70	−1.30234
$222$	0	0
$223$	1246.60	0.374343	0.187172	−	0.982327i	$-0.440068\pi$
$223$	1246.60	0.374343	0.187172	+	0.982327i	$0.440068\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	560.461	0.163873	0.0819364	−	0.996638i	$-0.473890\pi$
$227$	560.461	0.163873	0.0819364	+	0.996638i	$0.473890\pi$
$228$	0	0
$229$	−1145.38	−0.330519	−0.165260	−	0.986250i	$-0.552846\pi$
$229$	−1145.38	−0.330519	−0.165260	+	0.986250i	$0.552846\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	623.709	0.175367	0.0876836	−	0.996148i	$-0.472054\pi$
$233$	623.709	0.175367	0.0876836	+	0.996148i	$0.472054\pi$
$234$	0	0
$235$	600.316	0.166639
$236$	0	0
$237$	0	0
$238$	0	0
$239$	6265.80	1.69582	0.847910	−	0.530141i	$-0.177861\pi$
$239$	6265.80	1.69582	0.847910	+	0.530141i	$0.177861\pi$
$240$	0	0
$241$	−640.653	−0.171237	−0.0856185	−	0.996328i	$-0.527287\pi$
$241$	−640.653	−0.171237	−0.0856185	+	0.996328i	$0.527287\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	6244.74	1.62842
$246$	0	0
$247$	−2801.49	−0.721679
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−7748.91	−1.94863	−0.974316	−	0.225183i	$-0.927702\pi$
$251$	−7748.91	−1.94863	−0.974316	+	0.225183i	$0.927702\pi$
$252$	0	0
$253$	458.310	0.113888
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−3191.76	−0.774693	−0.387347	−	0.921934i	$-0.626608\pi$
$257$	−3191.76	−0.774693	−0.387347	+	0.921934i	$0.626608\pi$
$258$	0	0
$259$	1098.92	0.263643
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−3776.41	−0.885413	−0.442706	−	0.896667i	$-0.645982\pi$
$263$	−3776.41	−0.885413	−0.442706	+	0.896667i	$0.645982\pi$
$264$	0	0
$265$	−3289.87	−0.762623
$266$	0	0
$267$	0	0
$268$	0	0
$269$	4065.91	0.921572	0.460786	−	0.887511i	$-0.347568\pi$
$269$	4065.91	0.921572	0.460786	+	0.887511i	$0.347568\pi$
$270$	0	0
$271$	5508.17	1.23468	0.617339	−	0.786697i	$-0.288211\pi$
$271$	5508.17	1.23468	0.617339	+	0.786697i	$0.288211\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	353.398	0.0774934
$276$	0	0
$277$	−8306.91	−1.80186	−0.900928	−	0.433970i	$-0.857113\pi$
$277$	−8306.91	−1.80186	−0.900928	+	0.433970i	$0.857113\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	2962.67	0.628962	0.314481	−	0.949264i	$-0.398169\pi$
$281$	2962.67	0.628962	0.314481	+	0.949264i	$0.398169\pi$
$282$	0	0
$283$	4509.67	0.947250	0.473625	−	0.880727i	$-0.342945\pi$
$283$	4509.67	0.947250	0.473625	+	0.880727i	$0.342945\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−12808.0	−2.63425
$288$	0	0
$289$	6625.28	1.34852
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−1814.46	−0.361781	−0.180890	−	0.983503i	$-0.557898\pi$
$293$	−1814.46	−0.361781	−0.180890	+	0.983503i	$0.557898\pi$
$294$	0	0
$295$	−8947.55	−1.76592
$296$	0	0
$297$	0	0
$298$	0	0
$299$	275.562	0.0532982
$300$	0	0
$301$	2217.60	0.424653
$302$	0	0
$303$	0	0
$304$	0	0
$305$	5013.61	0.941240
$306$	0	0
$307$	−4395.62	−0.817171	−0.408585	−	0.912720i	$-0.633978\pi$
$307$	−4395.62	−0.817171	−0.408585	+	0.912720i	$0.633978\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−1377.23	−0.251111	−0.125555	−	0.992087i	$-0.540071\pi$
$311$	−1377.23	−0.251111	−0.125555	+	0.992087i	$0.540071\pi$
$312$	0	0
$313$	7347.95	1.32693	0.663467	−	0.748205i	$-0.269084\pi$
$313$	7347.95	1.32693	0.663467	+	0.748205i	$0.269084\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−2607.78	−0.462043	−0.231021	−	0.972949i	$-0.574207\pi$
$317$	−2607.78	−0.462043	−0.231021	+	0.972949i	$0.574207\pi$
$318$	0	0
$319$	−2429.28	−0.426374
$320$	0	0
$321$	0	0
$322$	0	0
$323$	7554.73	1.30141
$324$	0	0
$325$	212.483	0.0362659
$326$	0	0
$327$	0	0
$328$	0	0
$329$	1568.72	0.262876
$330$	0	0
$331$	6541.33	1.08624	0.543118	−	0.839656i	$-0.317244\pi$
$331$	6541.33	1.08624	0.543118	+	0.839656i	$0.317244\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	2492.63	0.406528
$336$	0	0
$337$	4315.66	0.697594	0.348797	−	0.937198i	$-0.386590\pi$
$337$	4315.66	0.697594	0.348797	+	0.937198i	$0.386590\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	15325.5	2.43379
$342$	0	0
$343$	6085.80	0.958025
$344$	0	0
$345$	0	0
$346$	0	0
$347$	3895.76	0.602695	0.301347	−	0.953514i	$-0.402564\pi$
$347$	3895.76	0.602695	0.301347	+	0.953514i	$0.402564\pi$
$348$	0	0
$349$	−4877.42	−0.748087	−0.374044	−	0.927411i	$-0.622029\pi$
$349$	−4877.42	−0.748087	−0.374044	+	0.927411i	$0.622029\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−1739.79	−0.262322	−0.131161	−	0.991361i	$-0.541871\pi$
$353$	−1739.79	−0.262322	−0.131161	+	0.991361i	$0.541871\pi$
$354$	0	0
$355$	9024.58	1.34923
$356$	0	0
$357$	0	0
$358$	0	0
$359$	188.828	0.0277604	0.0138802	−	0.999904i	$-0.495582\pi$
$359$	188.828	0.0277604	0.0138802	+	0.999904i	$0.495582\pi$
$360$	0	0
$361$	−1912.51	−0.278833
$362$	0	0
$363$	0	0
$364$	0	0
$365$	12535.2	1.79759
$366$	0	0
$367$	6338.15	0.901495	0.450747	−	0.892651i	$-0.351157\pi$
$367$	6338.15	0.901495	0.450747	+	0.892651i	$0.351157\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−8596.93	−1.20305
$372$	0	0
$373$	8641.41	1.19956	0.599779	−	0.800166i	$-0.295255\pi$
$373$	8641.41	1.19956	0.599779	+	0.800166i	$0.295255\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−1460.62	−0.199538
$378$	0	0
$379$	4143.88	0.561627	0.280814	−	0.959762i	$-0.409396\pi$
$379$	4143.88	0.561627	0.280814	+	0.959762i	$0.409396\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−4193.94	−0.559531	−0.279766	−	0.960068i	$-0.590257\pi$
$383$	−4193.94	−0.559531	−0.279766	+	0.960068i	$0.590257\pi$
$384$	0	0
$385$	22563.4	2.98685
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−10870.4	−1.41684	−0.708422	−	0.705789i	$-0.750593\pi$
$389$	−10870.4	−1.41684	−0.708422	+	0.705789i	$0.750593\pi$
$390$	0	0
$391$	−743.102	−0.0961133
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−5021.24	−0.639610
$396$	0	0
$397$	−8890.87	−1.12398	−0.561990	−	0.827144i	$-0.689964\pi$
$397$	−8890.87	−1.12398	−0.561990	+	0.827144i	$0.689964\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−15095.4	−1.87987	−0.939936	−	0.341349i	$-0.889116\pi$
$401$	−15095.4	−1.87987	−0.939936	+	0.341349i	$0.889116\pi$
$402$	0	0
$403$	9214.58	1.13898
$404$	0	0
$405$	0	0
$406$	0	0
$407$	2440.35	0.297208
$408$	0	0
$409$	1618.34	0.195652	0.0978260	−	0.995204i	$-0.468811\pi$
$409$	1618.34	0.195652	0.0978260	+	0.995204i	$0.468811\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−23381.3	−2.78576
$414$	0	0
$415$	−580.470	−0.0686606
$416$	0	0
$417$	0	0
$418$	0	0
$419$	10011.9	1.16733	0.583666	−	0.811994i	$-0.301618\pi$
$419$	10011.9	1.16733	0.583666	+	0.811994i	$0.301618\pi$
$420$	0	0
$421$	7234.43	0.837493	0.418747	−	0.908103i	$-0.362470\pi$
$421$	7234.43	0.837493	0.418747	+	0.908103i	$0.362470\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−572.999	−0.0653989
$426$	0	0
$427$	13101.3	1.48482
$428$	0	0
$429$	0	0
$430$	0	0
$431$	5379.03	0.601157	0.300579	−	0.953757i	$-0.402820\pi$
$431$	5379.03	0.601157	0.300579	+	0.953757i	$0.402820\pi$
$432$	0	0
$433$	−1602.96	−0.177906	−0.0889530	−	0.996036i	$-0.528352\pi$
$433$	−1602.96	−0.177906	−0.0889530	+	0.996036i	$0.528352\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−486.549	−0.0532604
$438$	0	0
$439$	−6725.87	−0.731226	−0.365613	−	0.930767i	$-0.619141\pi$
$439$	−6725.87	−0.731226	−0.365613	+	0.930767i	$0.619141\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	1169.92	0.125473	0.0627367	−	0.998030i	$-0.480017\pi$
$443$	1169.92	0.125473	0.0627367	+	0.998030i	$0.480017\pi$
$444$	0	0
$445$	−8216.90	−0.875322
$446$	0	0
$447$	0	0
$448$	0	0
$449$	2996.56	0.314958	0.157479	−	0.987522i	$-0.449663\pi$
$449$	2996.56	0.314958	0.157479	+	0.987522i	$0.449663\pi$
$450$	0	0
$451$	−28442.5	−2.96963
$452$	0	0
$453$	0	0
$454$	0	0
$455$	13566.4	1.39781
$456$	0	0
$457$	−11152.1	−1.14151	−0.570757	−	0.821119i	$-0.693350\pi$
$457$	−11152.1	−1.14151	−0.570757	+	0.821119i	$0.693350\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	2947.17	0.297752	0.148876	−	0.988856i	$-0.452435\pi$
$461$	2947.17	0.297752	0.148876	+	0.988856i	$0.452435\pi$
$462$	0	0
$463$	−6563.38	−0.658804	−0.329402	−	0.944190i	$-0.606847\pi$
$463$	−6563.38	−0.658804	−0.329402	+	0.944190i	$0.606847\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−2727.33	−0.270248	−0.135124	−	0.990829i	$-0.543143\pi$
$467$	−2727.33	−0.270248	−0.135124	+	0.990829i	$0.543143\pi$
$468$	0	0
$469$	6513.62	0.641303
$470$	0	0
$471$	0	0
$472$	0	0
$473$	4924.59	0.478717
$474$	0	0
$475$	−375.173	−0.0362402
$476$	0	0
$477$	0	0
$478$	0	0
$479$	14512.1	1.38429	0.692146	−	0.721758i	$-0.256665\pi$
$479$	14512.1	1.38429	0.692146	+	0.721758i	$0.256665\pi$
$480$	0	0
$481$	1467.28	0.139090
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−13695.6	−1.28224
$486$	0	0
$487$	14189.9	1.32034	0.660170	−	0.751116i	$-0.270484\pi$
$487$	14189.9	1.32034	0.660170	+	0.751116i	$0.270484\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−7215.91	−0.663238	−0.331619	−	0.943413i	$-0.607595\pi$
$491$	−7215.91	−0.663238	−0.331619	+	0.943413i	$0.607595\pi$
$492$	0	0
$493$	3938.82	0.359829
$494$	0	0
$495$	0	0
$496$	0	0
$497$	23582.6	2.12842
$498$	0	0
$499$	−13032.6	−1.16917	−0.584587	−	0.811331i	$-0.698743\pi$
$499$	−13032.6	−1.16917	−0.584587	+	0.811331i	$0.698743\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	9502.84	0.842367	0.421184	−	0.906975i	$-0.361615\pi$
$503$	9502.84	0.842367	0.421184	+	0.906975i	$0.361615\pi$
$504$	0	0
$505$	−2808.22	−0.247454
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−3794.30	−0.330411	−0.165206	−	0.986259i	$-0.552829\pi$
$509$	−3794.30	−0.330411	−0.165206	+	0.986259i	$0.552829\pi$
$510$	0	0
$511$	32756.3	2.83572
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−6573.80	−0.562478
$516$	0	0
$517$	3483.62	0.296343
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−14811.2	−1.24547	−0.622735	−	0.782432i	$-0.713979\pi$
$521$	−14811.2	−1.24547	−0.622735	+	0.782432i	$0.713979\pi$
$522$	0	0
$523$	345.532	0.0288892	0.0144446	−	0.999896i	$-0.495402\pi$
$523$	345.532	0.0288892	0.0144446	+	0.999896i	$0.495402\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−24848.8	−2.05395
$528$	0	0
$529$	−12119.1	−0.996067
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−17101.2	−1.38975
$534$	0	0
$535$	−11597.8	−0.937229
$536$	0	0
$537$	0	0
$538$	0	0
$539$	36238.1	2.89589
$540$	0	0
$541$	−5474.45	−0.435056	−0.217528	−	0.976054i	$-0.569799\pi$
$541$	−5474.45	−0.435056	−0.217528	+	0.976054i	$0.569799\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	16800.9	1.32049
$546$	0	0
$547$	−977.278	−0.0763901	−0.0381951	−	0.999270i	$-0.512161\pi$
$547$	−977.278	−0.0763901	−0.0381951	+	0.999270i	$0.512161\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	2578.96	0.199396
$552$	0	0
$553$	−13121.3	−1.00899
$554$	0	0
$555$	0	0
$556$	0	0
$557$	18833.3	1.43266	0.716330	−	0.697762i	$-0.245821\pi$
$557$	18833.3	1.43266	0.716330	+	0.697762i	$0.245821\pi$
$558$	0	0
$559$	2960.95	0.224033
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−20967.0	−1.56954	−0.784772	−	0.619784i	$-0.787220\pi$
$563$	−20967.0	−1.56954	−0.784772	+	0.619784i	$0.787220\pi$
$564$	0	0
$565$	21732.0	1.61818
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−13488.7	−0.993804	−0.496902	−	0.867807i	$-0.665529\pi$
$569$	−13488.7	−0.993804	−0.496902	+	0.867807i	$0.665529\pi$
$570$	0	0
$571$	2248.90	0.164822	0.0824110	−	0.996598i	$-0.473738\pi$
$571$	2248.90	0.164822	0.0824110	+	0.996598i	$0.473738\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	36.9030	0.00267645
$576$	0	0
$577$	−1968.87	−0.142054	−0.0710270	−	0.997474i	$-0.522628\pi$
$577$	−1968.87	−0.142054	−0.0710270	+	0.997474i	$0.522628\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−1516.86	−0.108313
$582$	0	0
$583$	−19091.1	−1.35621
$584$	0	0
$585$	0	0
$586$	0	0
$587$	4107.66	0.288827	0.144413	−	0.989517i	$-0.453871\pi$
$587$	4107.66	0.288827	0.144413	+	0.989517i	$0.453871\pi$
$588$	0	0
$589$	−16269.8	−1.13818
$590$	0	0
$591$	0	0
$592$	0	0
$593$	7497.21	0.519180	0.259590	−	0.965719i	$-0.416413\pi$
$593$	7497.21	0.519180	0.259590	+	0.965719i	$0.416413\pi$
$594$	0	0
$595$	−36584.3	−2.52069
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−27107.1	−1.84903	−0.924513	−	0.381151i	$-0.875528\pi$
$599$	−27107.1	−1.84903	−0.924513	+	0.381151i	$0.875528\pi$
$600$	0	0
$601$	6802.17	0.461674	0.230837	−	0.972992i	$-0.425854\pi$
$601$	6802.17	0.461674	0.230837	+	0.972992i	$0.425854\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	34910.9	2.34600
$606$	0	0
$607$	−16992.4	−1.13625	−0.568123	−	0.822943i	$-0.692330\pi$
$607$	−16992.4	−1.13625	−0.568123	+	0.822943i	$0.692330\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	2094.55	0.138685
$612$	0	0
$613$	−13766.6	−0.907062	−0.453531	−	0.891241i	$-0.649836\pi$
$613$	−13766.6	−0.907062	−0.453531	+	0.891241i	$0.649836\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	17356.0	1.13246	0.566229	−	0.824248i	$-0.308402\pi$
$617$	17356.0	1.13246	0.566229	+	0.824248i	$0.308402\pi$
$618$	0	0
$619$	−2924.89	−0.189922	−0.0949608	−	0.995481i	$-0.530273\pi$
$619$	−2924.89	−0.189922	−0.0949608	+	0.995481i	$0.530273\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−21472.0	−1.38083
$624$	0	0
$625$	−16263.3	−1.04085
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−3956.78	−0.250822
$630$	0	0
$631$	−6158.40	−0.388530	−0.194265	−	0.980949i	$-0.562232\pi$
$631$	−6158.40	−0.388530	−0.194265	+	0.980949i	$0.562232\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−2629.31	−0.164316
$636$	0	0
$637$	21788.4	1.35524
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−11814.2	−0.727979	−0.363990	−	0.931403i	$-0.618586\pi$
$641$	−11814.2	−0.727979	−0.363990	+	0.931403i	$0.618586\pi$
$642$	0	0
$643$	16914.7	1.03741	0.518703	−	0.854954i	$-0.326415\pi$
$643$	16914.7	1.03741	0.518703	+	0.854954i	$0.326415\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−28652.4	−1.74103	−0.870513	−	0.492145i	$-0.836213\pi$
$647$	−28652.4	−1.74103	−0.870513	+	0.492145i	$0.836213\pi$
$648$	0	0
$649$	−51922.5	−3.14042
$650$	0	0
$651$	0	0
$652$	0	0
$653$	5498.48	0.329513	0.164757	−	0.986334i	$-0.447316\pi$
$653$	5498.48	0.329513	0.164757	+	0.986334i	$0.447316\pi$
$654$	0	0
$655$	9068.16	0.540950
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−18785.1	−1.11042	−0.555209	−	0.831711i	$-0.687362\pi$
$659$	−18785.1	−1.11042	−0.555209	+	0.831711i	$0.687362\pi$
$660$	0	0
$661$	−17304.5	−1.01825	−0.509127	−	0.860691i	$-0.670032\pi$
$661$	−17304.5	−1.01825	−0.509127	+	0.860691i	$0.670032\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−23953.7	−1.39682
$666$	0	0
$667$	−253.673	−0.0147260
$668$	0	0
$669$	0	0
$670$	0	0
$671$	29093.9	1.67385
$672$	0	0
$673$	−13711.5	−0.785346	−0.392673	−	0.919678i	$-0.628450\pi$
$673$	−13711.5	−0.785346	−0.392673	+	0.919678i	$0.628450\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	30566.9	1.73527	0.867637	−	0.497198i	$-0.165638\pi$
$677$	30566.9	1.73527	0.867637	+	0.497198i	$0.165638\pi$
$678$	0	0
$679$	−35788.8	−2.02275
$680$	0	0
$681$	0	0
$682$	0	0
$683$	4445.72	0.249064	0.124532	−	0.992216i	$-0.460257\pi$
$683$	4445.72	0.249064	0.124532	+	0.992216i	$0.460257\pi$
$684$	0	0
$685$	7798.27	0.434973
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−11478.6	−0.634690
$690$	0	0
$691$	12198.7	0.671580	0.335790	−	0.941937i	$-0.390997\pi$
$691$	12198.7	0.671580	0.335790	+	0.941937i	$0.390997\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	17268.9	0.942515
$696$	0	0
$697$	46116.6	2.50615
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−21613.4	−1.16452	−0.582258	−	0.813004i	$-0.697831\pi$
$701$	−21613.4	−1.16452	−0.582258	+	0.813004i	$0.697831\pi$
$702$	0	0
$703$	−2590.72	−0.138991
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−7338.32	−0.390362
$708$	0	0
$709$	−4924.85	−0.260870	−0.130435	−	0.991457i	$-0.541637\pi$
$709$	−4924.85	−0.260870	−0.130435	+	0.991457i	$0.541637\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	1600.34	0.0840578
$714$	0	0
$715$	30126.7	1.57577
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−205.354	−0.0106515	−0.00532573	−	0.999986i	$-0.501695\pi$
$719$	−205.354	−0.0106515	−0.00532573	+	0.999986i	$0.501695\pi$
$720$	0	0
$721$	−17178.3	−0.887316
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−195.605	−0.0100201
$726$	0	0
$727$	−15796.0	−0.805836	−0.402918	−	0.915236i	$-0.632004\pi$
$727$	−15796.0	−0.805836	−0.402918	+	0.915236i	$0.632004\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−7984.73	−0.404003
$732$	0	0
$733$	32125.3	1.61880	0.809398	−	0.587261i	$-0.199794\pi$
$733$	32125.3	1.61880	0.809398	+	0.587261i	$0.199794\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	14464.7	0.722949
$738$	0	0
$739$	−28938.9	−1.44051	−0.720255	−	0.693710i	$-0.755975\pi$
$739$	−28938.9	−1.44051	−0.720255	+	0.693710i	$0.755975\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	36108.5	1.78289	0.891447	−	0.453124i	$-0.149691\pi$
$743$	36108.5	1.78289	0.891447	+	0.453124i	$0.149691\pi$
$744$	0	0
$745$	−26339.2	−1.29529
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−30306.9	−1.47849
$750$	0	0
$751$	23231.7	1.12881	0.564405	−	0.825498i	$-0.309106\pi$
$751$	23231.7	1.12881	0.564405	+	0.825498i	$0.309106\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−28805.5	−1.38853
$756$	0	0
$757$	22762.6	1.09289	0.546447	−	0.837494i	$-0.315980\pi$
$757$	22762.6	1.09289	0.546447	+	0.837494i	$0.315980\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	40518.6	1.93009	0.965044	−	0.262088i	$-0.0844112\pi$
$761$	40518.6	1.93009	0.965044	+	0.262088i	$0.0844112\pi$
$762$	0	0
$763$	43903.2	2.08310
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−31218.8	−1.46968
$768$	0	0
$769$	−27401.1	−1.28493	−0.642464	−	0.766316i	$-0.722088\pi$
$769$	−27401.1	−1.28493	−0.642464	+	0.766316i	$0.722088\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	35698.4	1.66104	0.830520	−	0.556989i	$-0.188043\pi$
$773$	35698.4	1.66104	0.830520	+	0.556989i	$0.188043\pi$
$774$	0	0
$775$	1234.01	0.0571959
$776$	0	0
$777$	0	0
$778$	0	0
$779$	30195.0	1.38876
$780$	0	0
$781$	52369.5	2.39940
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−33324.3	−1.51515
$786$	0	0
$787$	9566.36	0.433296	0.216648	−	0.976250i	$-0.430488\pi$
$787$	9566.36	0.433296	0.216648	+	0.976250i	$0.430488\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	56789.0	2.55270
$792$	0	0
$793$	17492.9	0.783343
$794$	0	0
$795$	0	0
$796$	0	0
$797$	5306.01	0.235820	0.117910	−	0.993024i	$-0.462381\pi$
$797$	5306.01	0.235820	0.117910	+	0.993024i	$0.462381\pi$
$798$	0	0
$799$	−5648.34	−0.250093
$800$	0	0
$801$	0	0
$802$	0	0
$803$	72741.4	3.19675
$804$	0	0
$805$	2356.14	0.103159
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−9362.75	−0.406893	−0.203447	−	0.979086i	$-0.565214\pi$
$809$	−9362.75	−0.406893	−0.203447	+	0.979086i	$0.565214\pi$
$810$	0	0
$811$	−32610.2	−1.41196	−0.705981	−	0.708231i	$-0.749493\pi$
$811$	−32610.2	−1.41196	−0.705981	+	0.708231i	$0.749493\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−12523.7	−0.538263
$816$	0	0
$817$	−5228.03	−0.223875
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−18930.7	−0.804732	−0.402366	−	0.915479i	$-0.631812\pi$
$821$	−18930.7	−0.804732	−0.402366	+	0.915479i	$0.631812\pi$
$822$	0	0
$823$	−40673.1	−1.72269	−0.861346	−	0.508018i	$-0.830378\pi$
$823$	−40673.1	−1.72269	−0.861346	+	0.508018i	$0.830378\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−30770.2	−1.29382	−0.646908	−	0.762568i	$-0.723938\pi$
$827$	−30770.2	−1.29382	−0.646908	+	0.762568i	$0.723938\pi$
$828$	0	0
$829$	10464.9	0.438434	0.219217	−	0.975676i	$-0.429650\pi$
$829$	10464.9	0.438434	0.219217	+	0.975676i	$0.429650\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−58756.4	−2.44393
$834$	0	0
$835$	−8365.13	−0.346691
$836$	0	0
$837$	0	0
$838$	0	0
$839$	22716.4	0.934752	0.467376	−	0.884059i	$-0.345200\pi$
$839$	22716.4	0.934752	0.467376	+	0.884059i	$0.345200\pi$
$840$	0	0
$841$	−23044.4	−0.944869
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−6967.97	−0.283675
$846$	0	0
$847$	91227.5	3.70084
$848$	0	0
$849$	0	0
$850$	0	0
$851$	254.829	0.0102649
$852$	0	0
$853$	−10181.2	−0.408671	−0.204335	−	0.978901i	$-0.565503\pi$
$853$	−10181.2	−0.408671	−0.204335	+	0.978901i	$0.565503\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−3885.59	−0.154877	−0.0774384	−	0.996997i	$-0.524674\pi$
$857$	−3885.59	−0.154877	−0.0774384	+	0.996997i	$0.524674\pi$
$858$	0	0
$859$	17734.9	0.704433	0.352217	−	0.935919i	$-0.385428\pi$
$859$	17734.9	0.704433	0.352217	+	0.935919i	$0.385428\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	19658.7	0.775421	0.387711	−	0.921781i	$-0.373266\pi$
$863$	19658.7	0.775421	0.387711	+	0.921781i	$0.373266\pi$
$864$	0	0
$865$	−8250.35	−0.324301
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−29138.2	−1.13745
$870$	0	0
$871$	8697.00	0.338331
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−40756.2	−1.57464
$876$	0	0
$877$	5586.82	0.215112	0.107556	−	0.994199i	$-0.465697\pi$
$877$	5586.82	0.215112	0.107556	+	0.994199i	$0.465697\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	8050.03	0.307846	0.153923	−	0.988083i	$-0.450809\pi$
$881$	8050.03	0.307846	0.153923	+	0.988083i	$0.450809\pi$
$882$	0	0
$883$	2326.88	0.0886815	0.0443408	−	0.999016i	$-0.485881\pi$
$883$	2326.88	0.0886815	0.0443408	+	0.999016i	$0.485881\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−19343.5	−0.732233	−0.366116	−	0.930569i	$-0.619313\pi$
$887$	−19343.5	−0.732233	−0.366116	+	0.930569i	$0.619313\pi$
$888$	0	0
$889$	−6870.78	−0.259211
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−3698.27	−0.138587
$894$	0	0
$895$	11532.4	0.430712
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−8482.63	−0.314696
$900$	0	0
$901$	30954.2	1.14454
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−48597.2	−1.78500
$906$	0	0
$907$	299.317	0.0109577	0.00547886	−	0.999985i	$-0.498256\pi$
$907$	299.317	0.0109577	0.00547886	+	0.999985i	$0.498256\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−21022.5	−0.764551	−0.382275	−	0.924048i	$-0.624859\pi$
$911$	−21022.5	−0.764551	−0.382275	+	0.924048i	$0.624859\pi$
$912$	0	0
$913$	−3368.46	−0.122103
$914$	0	0
$915$	0	0
$916$	0	0
$917$	23696.5	0.853356
$918$	0	0
$919$	18375.8	0.659587	0.329794	−	0.944053i	$-0.393021\pi$
$919$	18375.8	0.659587	0.329794	+	0.944053i	$0.393021\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	31487.5	1.12289
$924$	0	0
$925$	196.496	0.00698461
$926$	0	0
$927$	0	0
$928$	0	0
$929$	29548.0	1.04353	0.521764	−	0.853090i	$-0.325274\pi$
$929$	29548.0	1.04353	0.521764	+	0.853090i	$0.325274\pi$
$930$	0	0
$931$	−38471.0	−1.35428
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−81242.1	−2.84160
$936$	0	0
$937$	36751.2	1.28133	0.640667	−	0.767819i	$-0.278658\pi$
$937$	36751.2	1.28133	0.640667	+	0.767819i	$0.278658\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−26024.7	−0.901575	−0.450788	−	0.892631i	$-0.648857\pi$
$941$	−26024.7	−0.901575	−0.450788	+	0.892631i	$0.648857\pi$
$942$	0	0
$943$	−2970.05	−0.102564
$944$	0	0
$945$	0	0
$946$	0	0
$947$	40238.6	1.38076	0.690379	−	0.723448i	$-0.257444\pi$
$947$	40238.6	1.38076	0.690379	+	0.723448i	$0.257444\pi$
$948$	0	0
$949$	43736.3	1.49604
$950$	0	0
$951$	0	0
$952$	0	0
$953$	52396.4	1.78099	0.890496	−	0.454991i	$-0.150357\pi$
$953$	52396.4	1.78099	0.890496	+	0.454991i	$0.150357\pi$
$954$	0	0
$955$	39151.3	1.32660
$956$	0	0
$957$	0	0
$958$	0	0
$959$	20378.1	0.686176
$960$	0	0
$961$	23723.2	0.796319
$962$	0	0
$963$	0	0
$964$	0	0
$965$	11043.6	0.368399
$966$	0	0
$967$	9007.03	0.299531	0.149765	−	0.988722i	$-0.452148\pi$
$967$	9007.03	0.299531	0.149765	+	0.988722i	$0.452148\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−1110.63	−0.0367062	−0.0183531	−	0.999832i	$-0.505842\pi$
$971$	−1110.63	−0.0367062	−0.0183531	+	0.999832i	$0.505842\pi$
$972$	0	0
$973$	45126.3	1.48683
$974$	0	0
$975$	0	0
$976$	0	0
$977$	27544.4	0.901969	0.450984	−	0.892532i	$-0.351073\pi$
$977$	27544.4	0.901969	0.450984	+	0.892532i	$0.351073\pi$
$978$	0	0
$979$	−47682.5	−1.55663
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−8657.48	−0.280906	−0.140453	−	0.990087i	$-0.544856\pi$
$983$	−8657.48	−0.280906	−0.140453	+	0.990087i	$0.544856\pi$
$984$	0	0
$985$	5347.60	0.172983
$986$	0	0
$987$	0	0
$988$	0	0
$989$	514.242	0.0165338
$990$	0	0
$991$	54045.9	1.73242	0.866208	−	0.499683i	$-0.166550\pi$
$991$	54045.9	1.73242	0.866208	+	0.499683i	$0.166550\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	30521.4	0.972456
$996$	0	0
$997$	−32591.1	−1.03528	−0.517638	−	0.855600i	$-0.673189\pi$
$997$	−32591.1	−1.03528	−0.517638	+	0.855600i	$0.673189\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	432.4.a.p.1.2		2
3.2	odd	2		432.4.a.r.1.1		2
4.3	odd	2		216.4.a.f.1.2	✓	2
8.3	odd	2		1728.4.a.bq.1.1		2
8.5	even	2		1728.4.a.br.1.1		2
12.11	even	2		216.4.a.g.1.1	yes	2
24.5	odd	2		1728.4.a.bj.1.2		2
24.11	even	2		1728.4.a.bi.1.2		2
36.7	odd	6		648.4.i.r.433.1		4
36.11	even	6		648.4.i.o.433.2		4
36.23	even	6		648.4.i.o.217.2		4
36.31	odd	6		648.4.i.r.217.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.4.a.f.1.2	✓	2	4.3	odd	2
216.4.a.g.1.1	yes	2	12.11	even	2
432.4.a.p.1.2		2	1.1	even	1	trivial
432.4.a.r.1.1		2	3.2	odd	2
648.4.i.o.217.2		4	36.23	even	6
648.4.i.o.433.2		4	36.11	even	6
648.4.i.r.217.1		4	36.31	odd	6
648.4.i.r.433.1		4	36.7	odd	6
1728.4.a.bi.1.2		2	24.11	even	2
1728.4.a.bj.1.2		2	24.5	odd	2
1728.4.a.bq.1.1		2	8.3	odd	2
1728.4.a.br.1.1		2	8.5	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:	\( N \)	\(=\)	\( 432 = 2^{4} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	432.a (trivial)

\(f(q)\)	\(=\)	\(q+11.4164 q^{5} +29.8328 q^{7} +O(q^{10})\)	\(q+11.4164 q^{5} +29.8328 q^{7} +66.2492 q^{11} +39.8328 q^{13} -107.416 q^{17} -70.3313 q^{19} +6.91796 q^{23} +5.33437 q^{25} -36.6687 q^{29} +231.331 q^{31} +340.584 q^{35} +36.8359 q^{37} -429.325 q^{41} +74.3344 q^{43} +52.5836 q^{47} +546.997 q^{49} -288.170 q^{53} +756.328 q^{55} -783.745 q^{59} +439.158 q^{61} +454.748 q^{65} +218.337 q^{67} +790.492 q^{71} +1098.00 q^{73} +1976.40 q^{77} -439.827 q^{79} -50.8452 q^{83} -1226.31 q^{85} -719.745 q^{89} +1188.33 q^{91} -802.930 q^{95} -1199.64 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{5} + 6 q^{7}+O(q^{10}) \) 2 * q - 4 * q^5 + 6 * q^7	\( 2 q - 4 q^{5} + 6 q^{7} + 52 q^{11} + 26 q^{13} - 188 q^{17} + 74 q^{19} + 148 q^{23} + 118 q^{25} - 288 q^{29} + 248 q^{31} + 708 q^{35} + 342 q^{37} + 256 q^{43} + 132 q^{47} + 772 q^{49} - 952 q^{53} + 976 q^{55} - 1004 q^{59} - 34 q^{61} + 668 q^{65} + 866 q^{67} + 776 q^{71} + 1874 q^{73} + 2316 q^{77} - 182 q^{79} - 1336 q^{83} + 16 q^{85} - 876 q^{89} + 1518 q^{91} - 3028 q^{95} - 38 q^{97}+O(q^{100}) \) 2 * q - 4 * q^5 + 6 * q^7 + 52 * q^11 + 26 * q^13 - 188 * q^17 + 74 * q^19 + 148 * q^23 + 118 * q^25 - 288 * q^29 + 248 * q^31 + 708 * q^35 + 342 * q^37 + 256 * q^43 + 132 * q^47 + 772 * q^49 - 952 * q^53 + 976 * q^55 - 1004 * q^59 - 34 * q^61 + 668 * q^65 + 866 * q^67 + 776 * q^71 + 1874 * q^73 + 2316 * q^77 - 182 * q^79 - 1336 * q^83 + 16 * q^85 - 876 * q^89 + 1518 * q^91 - 3028 * q^95 - 38 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more