LMFDB - Embedded newform 1472.4.a.bg.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1472,4,Mod(1,1472)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1472, 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("1472.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1472 = 2^{6} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1472.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:	$86.8508115285$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 137x^{6} + 344x^{5} + 6175x^{4} - 7924x^{3} - 89643x^{2} + 45072x + 51084 $ x^8 - 4x^7 - 137x^6 + 344x^5 + 6175x^4 - 7924x^3 - 89643x^2 + 45072*x + 51084
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 736)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-6.05402$ of defining polynomial
Character	$\chi$	$=$	1472.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-8.05402 q^{3} -11.6466 q^{5} +24.4224 q^{7} +37.8672 q^{9} -18.4129 q^{11} -51.7005 q^{13} +93.8018 q^{15} -10.4582 q^{17} +22.3482 q^{19} -196.698 q^{21} -23.0000 q^{23} +10.6430 q^{25} -87.5244 q^{27} +167.380 q^{29} -176.061 q^{31} +148.298 q^{33} -284.438 q^{35} +35.8712 q^{37} +416.397 q^{39} +412.361 q^{41} +192.021 q^{43} -441.023 q^{45} -591.953 q^{47} +253.454 q^{49} +84.2303 q^{51} +750.312 q^{53} +214.447 q^{55} -179.993 q^{57} -198.433 q^{59} +264.657 q^{61} +924.807 q^{63} +602.135 q^{65} +155.057 q^{67} +185.242 q^{69} +1082.55 q^{71} -224.277 q^{73} -85.7190 q^{75} -449.687 q^{77} +261.911 q^{79} -317.491 q^{81} +15.8477 q^{83} +121.802 q^{85} -1348.08 q^{87} +200.192 q^{89} -1262.65 q^{91} +1417.99 q^{93} -260.281 q^{95} -113.155 q^{97} -697.244 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 12 q^{3} + 12 q^{5} - 14 q^{7} + 90 q^{9} - 88 q^{11} + 30 q^{13} + 30 q^{15} + 58 q^{17} - 190 q^{19} + 66 q^{21} - 184 q^{23} + 28 q^{25} - 432 q^{27} - 190 q^{29} - 60 q^{31} + 346 q^{33} - 192 q^{35}+ \cdots - 5986 q^{99}+O(q^{100}) $ 8 * q - 12 * q^3 + 12 * q^5 - 14 * q^7 + 90 * q^9 - 88 * q^11 + 30 * q^13 + 30 * q^15 + 58 * q^17 - 190 * q^19 + 66 * q^21 - 184 * q^23 + 28 * q^25 - 432 * q^27 - 190 * q^29 - 60 * q^31 + 346 * q^33 - 192 * q^35 + 156 * q^37 + 156 * q^39 + 282 * q^41 - 810 * q^43 - 358 * q^45 + 564 * q^47 + 752 * q^49 - 1614 * q^51 + 230 * q^53 + 1292 * q^55 + 796 * q^57 - 1916 * q^59 - 22 * q^61 + 1036 * q^63 + 222 * q^65 - 2292 * q^67 + 276 * q^69 + 2376 * q^71 - 630 * q^73 - 3952 * q^75 + 948 * q^77 + 892 * q^79 - 1044 * q^81 - 2800 * q^83 - 288 * q^85 + 2756 * q^87 - 308 * q^89 - 4610 * q^91 + 982 * q^93 + 1300 * q^95 + 854 * q^97 - 5986 * q^99

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−8.05402	−1.55000	−0.774998	−	0.631964i	$-0.782249\pi$
$3$	−8.05402	−1.55000	−0.774998	+	0.631964i	$0.782249\pi$
$4$	0	0
$5$	−11.6466	−1.04170	−0.520851	−	0.853647i	$-0.674385\pi$
$5$	−11.6466	−1.04170	−0.520851	+	0.853647i	$0.674385\pi$
$6$	0	0
$7$	24.4224	1.31869	0.659343	−	0.751842i	$-0.270835\pi$
$7$	24.4224	1.31869	0.659343	+	0.751842i	$0.270835\pi$
$8$	0	0
$9$	37.8672	1.40249
$10$	0	0
$11$	−18.4129	−0.504700	−0.252350	−	0.967636i	$-0.581203\pi$
$11$	−18.4129	−0.504700	−0.252350	+	0.967636i	$0.581203\pi$
$12$	0	0
$13$	−51.7005	−1.10301	−0.551506	−	0.834171i	$-0.685947\pi$
$13$	−51.7005	−1.10301	−0.551506	+	0.834171i	$0.685947\pi$
$14$	0	0
$15$	93.8018	1.61463
$16$	0	0
$17$	−10.4582	−0.149205	−0.0746023	−	0.997213i	$-0.523769\pi$
$17$	−10.4582	−0.149205	−0.0746023	+	0.997213i	$0.523769\pi$
$18$	0	0
$19$	22.3482	0.269844	0.134922	−	0.990856i	$-0.456922\pi$
$19$	22.3482	0.269844	0.134922	+	0.990856i	$0.456922\pi$
$20$	0	0
$21$	−196.698	−2.04396
$22$	0	0
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	0	0
$25$	10.6430	0.0851441
$26$	0	0
$27$	−87.5244	−0.623854
$28$	0	0
$29$	167.380	1.07178	0.535891	−	0.844287i	$-0.319976\pi$
$29$	167.380	1.07178	0.535891	+	0.844287i	$0.319976\pi$
$30$	0	0
$31$	−176.061	−1.02005	−0.510023	−	0.860161i	$-0.670363\pi$
$31$	−176.061	−1.02005	−0.510023	+	0.860161i	$0.670363\pi$
$32$	0	0
$33$	148.298	0.782283
$34$	0	0
$35$	−284.438	−1.37368
$36$	0	0
$37$	35.8712	0.159384	0.0796918	−	0.996820i	$-0.474606\pi$
$37$	35.8712	0.159384	0.0796918	+	0.996820i	$0.474606\pi$
$38$	0	0
$39$	416.397	1.70966
$40$	0	0
$41$	412.361	1.57073	0.785365	−	0.619033i	$-0.212475\pi$
$41$	412.361	1.57073	0.785365	+	0.619033i	$0.212475\pi$
$42$	0	0
$43$	192.021	0.680999	0.340500	−	0.940245i	$-0.389404\pi$
$43$	192.021	0.680999	0.340500	+	0.940245i	$0.389404\pi$
$44$	0	0
$45$	−441.023	−1.46097
$46$	0	0
$47$	−591.953	−1.83713	−0.918567	−	0.395266i	$-0.870652\pi$
$47$	−591.953	−1.83713	−0.918567	+	0.395266i	$0.870652\pi$
$48$	0	0
$49$	253.454	0.738932
$50$	0	0
$51$	84.2303	0.231267
$52$	0	0
$53$	750.312	1.94459	0.972295	−	0.233756i	$-0.0751016\pi$
$53$	750.312	1.94459	0.972295	+	0.233756i	$0.0751016\pi$
$54$	0	0
$55$	214.447	0.525747
$56$	0	0
$57$	−179.993	−0.418257
$58$	0	0
$59$	−198.433	−0.437860	−0.218930	−	0.975741i	$-0.570257\pi$
$59$	−198.433	−0.437860	−0.218930	+	0.975741i	$0.570257\pi$
$60$	0	0
$61$	264.657	0.555506	0.277753	−	0.960653i	$-0.410410\pi$
$61$	264.657	0.555506	0.277753	+	0.960653i	$0.410410\pi$
$62$	0	0
$63$	924.807	1.84944
$64$	0	0
$65$	602.135	1.14901
$66$	0	0
$67$	155.057	0.282734	0.141367	−	0.989957i	$-0.454850\pi$
$67$	155.057	0.282734	0.141367	+	0.989957i	$0.454850\pi$
$68$	0	0
$69$	185.242	0.323197
$70$	0	0
$71$	1082.55	1.80952	0.904758	−	0.425926i	$-0.140052\pi$
$71$	1082.55	1.80952	0.904758	+	0.425926i	$0.140052\pi$
$72$	0	0
$73$	−224.277	−0.359585	−0.179792	−	0.983705i	$-0.557543\pi$
$73$	−224.277	−0.359585	−0.179792	+	0.983705i	$0.557543\pi$
$74$	0	0
$75$	−85.7190	−0.131973
$76$	0	0
$77$	−449.687	−0.665541
$78$	0	0
$79$	261.911	0.373004	0.186502	−	0.982455i	$-0.440285\pi$
$79$	261.911	0.373004	0.186502	+	0.982455i	$0.440285\pi$
$80$	0	0
$81$	−317.491	−0.435516
$82$	0	0
$83$	15.8477	0.0209579	0.0104790	−	0.999945i	$-0.496664\pi$
$83$	15.8477	0.0209579	0.0104790	+	0.999945i	$0.496664\pi$
$84$	0	0
$85$	121.802	0.155427
$86$	0	0
$87$	−1348.08	−1.66126
$88$	0	0
$89$	200.192	0.238431	0.119215	−	0.992868i	$-0.461962\pi$
$89$	200.192	0.238431	0.119215	+	0.992868i	$0.461962\pi$
$90$	0	0
$91$	−1262.65	−1.45453
$92$	0	0
$93$	1417.99	1.58107
$94$	0	0
$95$	−260.281	−0.281097
$96$	0	0
$97$	−113.155	−0.118445	−0.0592227	−	0.998245i	$-0.518862\pi$
$97$	−113.155	−0.118445	−0.0592227	+	0.998245i	$0.518862\pi$
$98$	0	0
$99$	−697.244	−0.707835
$100$	0	0
$101$	950.072	0.935997	0.467999	−	0.883729i	$-0.344975\pi$
$101$	950.072	0.935997	0.467999	+	0.883729i	$0.344975\pi$
$102$	0	0
$103$	385.330	0.368618	0.184309	−	0.982868i	$-0.440995\pi$
$103$	385.330	0.368618	0.184309	+	0.982868i	$0.440995\pi$
$104$	0	0
$105$	2290.87	2.12920
$106$	0	0
$107$	−1014.00	−0.916139	−0.458070	−	0.888916i	$-0.651459\pi$
$107$	−1014.00	−0.916139	−0.458070	+	0.888916i	$0.651459\pi$
$108$	0	0
$109$	2065.44	1.81499	0.907493	−	0.420067i	$-0.137994\pi$
$109$	2065.44	1.81499	0.907493	+	0.420067i	$0.137994\pi$
$110$	0	0
$111$	−288.908	−0.247044
$112$	0	0
$113$	1217.12	1.01325	0.506625	−	0.862166i	$-0.330893\pi$
$113$	1217.12	1.01325	0.506625	+	0.862166i	$0.330893\pi$
$114$	0	0
$115$	267.872	0.217210
$116$	0	0
$117$	−1957.75	−1.54696
$118$	0	0
$119$	−255.414	−0.196754
$120$	0	0
$121$	−991.965	−0.745278
$122$	0	0
$123$	−3321.16	−2.43462
$124$	0	0
$125$	1331.87	0.953008
$126$	0	0
$127$	−138.778	−0.0969652	−0.0484826	−	0.998824i	$-0.515439\pi$
$127$	−138.778	−0.0969652	−0.0484826	+	0.998824i	$0.515439\pi$
$128$	0	0
$129$	−1546.54	−1.05555
$130$	0	0
$131$	−1370.77	−0.914234	−0.457117	−	0.889406i	$-0.651118\pi$
$131$	−1370.77	−0.914234	−0.457117	+	0.889406i	$0.651118\pi$
$132$	0	0
$133$	545.798	0.355840
$134$	0	0
$135$	1019.36	0.649871
$136$	0	0
$137$	−1687.59	−1.05242	−0.526208	−	0.850356i	$-0.676387\pi$
$137$	−1687.59	−1.05242	−0.526208	+	0.850356i	$0.676387\pi$
$138$	0	0
$139$	−2507.17	−1.52989	−0.764947	−	0.644093i	$-0.777235\pi$
$139$	−2507.17	−1.52989	−0.764947	+	0.644093i	$0.777235\pi$
$140$	0	0
$141$	4767.60	2.84755
$142$	0	0
$143$	951.957	0.556690
$144$	0	0
$145$	−1949.40	−1.11648
$146$	0	0
$147$	−2041.32	−1.14534
$148$	0	0
$149$	−3399.54	−1.86913	−0.934567	−	0.355787i	$-0.884213\pi$
$149$	−3399.54	−1.86913	−0.934567	+	0.355787i	$0.884213\pi$
$150$	0	0
$151$	985.734	0.531244	0.265622	−	0.964077i	$-0.414423\pi$
$151$	985.734	0.531244	0.265622	+	0.964077i	$0.414423\pi$
$152$	0	0
$153$	−396.021	−0.209258
$154$	0	0
$155$	2050.50	1.06258
$156$	0	0
$157$	−201.810	−0.102587	−0.0512936	−	0.998684i	$-0.516334\pi$
$157$	−201.810	−0.102587	−0.0512936	+	0.998684i	$0.516334\pi$
$158$	0	0
$159$	−6043.03	−3.01411
$160$	0	0
$161$	−561.715	−0.274965
$162$	0	0
$163$	−3446.39	−1.65609	−0.828043	−	0.560664i	$-0.810546\pi$
$163$	−3446.39	−1.65609	−0.828043	+	0.560664i	$0.810546\pi$
$164$	0	0
$165$	−1727.16	−0.814906
$166$	0	0
$167$	−971.671	−0.450240	−0.225120	−	0.974331i	$-0.572277\pi$
$167$	−971.671	−0.450240	−0.225120	+	0.974331i	$0.572277\pi$
$168$	0	0
$169$	475.945	0.216634
$170$	0	0
$171$	846.265	0.378453
$172$	0	0
$173$	−1589.84	−0.698689	−0.349345	−	0.936994i	$-0.613596\pi$
$173$	−1589.84	−0.698689	−0.349345	+	0.936994i	$0.613596\pi$
$174$	0	0
$175$	259.928	0.112278
$176$	0	0
$177$	1598.18	0.678681
$178$	0	0
$179$	−2393.41	−0.999395	−0.499697	−	0.866200i	$-0.666555\pi$
$179$	−2393.41	−0.999395	−0.499697	+	0.866200i	$0.666555\pi$
$180$	0	0
$181$	−4104.06	−1.68537	−0.842687	−	0.538404i	$-0.819027\pi$
$181$	−4104.06	−1.68537	−0.842687	+	0.538404i	$0.819027\pi$
$182$	0	0
$183$	−2131.55	−0.861032
$184$	0	0
$185$	−417.778	−0.166030
$186$	0	0
$187$	192.565	0.0753036
$188$	0	0
$189$	−2137.55	−0.822668
$190$	0	0
$191$	320.971	0.121595	0.0607976	−	0.998150i	$-0.480636\pi$
$191$	320.971	0.121595	0.0607976	+	0.998150i	$0.480636\pi$
$192$	0	0
$193$	−688.697	−0.256858	−0.128429	−	0.991719i	$-0.540993\pi$
$193$	−688.697	−0.256858	−0.128429	+	0.991719i	$0.540993\pi$
$194$	0	0
$195$	−4849.60	−1.78096
$196$	0	0
$197$	1389.43	0.502501	0.251250	−	0.967922i	$-0.419158\pi$
$197$	1389.43	0.502501	0.251250	+	0.967922i	$0.419158\pi$
$198$	0	0
$199$	2153.82	0.767236	0.383618	−	0.923492i	$-0.374678\pi$
$199$	2153.82	0.767236	0.383618	+	0.923492i	$0.374678\pi$
$200$	0	0
$201$	−1248.83	−0.438237
$202$	0	0
$203$	4087.82	1.41334
$204$	0	0
$205$	−4802.59	−1.63623
$206$	0	0
$207$	−870.945	−0.292439
$208$	0	0
$209$	−411.496	−0.136190
$210$	0	0
$211$	−2265.82	−0.739267	−0.369634	−	0.929178i	$-0.620517\pi$
$211$	−2265.82	−0.739267	−0.369634	+	0.929178i	$0.620517\pi$
$212$	0	0
$213$	−8718.91	−2.80474
$214$	0	0
$215$	−2236.39	−0.709399
$216$	0	0
$217$	−4299.82	−1.34512
$218$	0	0
$219$	1806.33	0.557355
$220$	0	0
$221$	540.693	0.164574
$222$	0	0
$223$	−3832.94	−1.15100	−0.575499	−	0.817803i	$-0.695192\pi$
$223$	−3832.94	−1.15100	−0.575499	+	0.817803i	$0.695192\pi$
$224$	0	0
$225$	403.021	0.119414
$226$	0	0
$227$	551.580	0.161276	0.0806381	−	0.996743i	$-0.474304\pi$
$227$	551.580	0.161276	0.0806381	+	0.996743i	$0.474304\pi$
$228$	0	0
$229$	−3462.68	−0.999216	−0.499608	−	0.866252i	$-0.666523\pi$
$229$	−3462.68	−0.999216	−0.499608	+	0.866252i	$0.666523\pi$
$230$	0	0
$231$	3621.79	1.03159
$232$	0	0
$233$	2949.98	0.829441	0.414721	−	0.909949i	$-0.363879\pi$
$233$	2949.98	0.829441	0.414721	+	0.909949i	$0.363879\pi$
$234$	0	0
$235$	6894.24	1.91375
$236$	0	0
$237$	−2109.44	−0.578154
$238$	0	0
$239$	−1063.71	−0.287890	−0.143945	−	0.989586i	$-0.545979\pi$
$239$	−1063.71	−0.287890	−0.143945	+	0.989586i	$0.545979\pi$
$240$	0	0
$241$	160.174	0.0428121	0.0214061	−	0.999771i	$-0.493186\pi$
$241$	160.174	0.0428121	0.0214061	+	0.999771i	$0.493186\pi$
$242$	0	0
$243$	4920.24	1.29890
$244$	0	0
$245$	−2951.87	−0.769747
$246$	0	0
$247$	−1155.42	−0.297641
$248$	0	0
$249$	−127.637	−0.0324847
$250$	0	0
$251$	2892.16	0.727297	0.363648	−	0.931536i	$-0.381531\pi$
$251$	2892.16	0.727297	0.363648	+	0.931536i	$0.381531\pi$
$252$	0	0
$253$	423.497	0.105237
$254$	0	0
$255$	−980.995	−0.240911
$256$	0	0
$257$	5780.85	1.40311	0.701555	−	0.712615i	$-0.252489\pi$
$257$	5780.85	1.40311	0.701555	+	0.712615i	$0.252489\pi$
$258$	0	0
$259$	876.062	0.210177
$260$	0	0
$261$	6338.20	1.50316
$262$	0	0
$263$	−3054.14	−0.716071	−0.358035	−	0.933708i	$-0.616553\pi$
$263$	−3054.14	−0.716071	−0.358035	+	0.933708i	$0.616553\pi$
$264$	0	0
$265$	−8738.58	−2.02569
$266$	0	0
$267$	−1612.35	−0.369567
$268$	0	0
$269$	4171.83	0.945579	0.472789	−	0.881175i	$-0.343247\pi$
$269$	4171.83	0.945579	0.472789	+	0.881175i	$0.343247\pi$
$270$	0	0
$271$	1333.89	0.298996	0.149498	−	0.988762i	$-0.452234\pi$
$271$	1333.89	0.298996	0.149498	+	0.988762i	$0.452234\pi$
$272$	0	0
$273$	10169.4	2.25451
$274$	0	0
$275$	−195.969	−0.0429722
$276$	0	0
$277$	4171.55	0.904852	0.452426	−	0.891802i	$-0.350559\pi$
$277$	4171.55	0.904852	0.452426	+	0.891802i	$0.350559\pi$
$278$	0	0
$279$	−6666.92	−1.43060
$280$	0	0
$281$	4657.91	0.988852	0.494426	−	0.869220i	$-0.335378\pi$
$281$	4657.91	0.988852	0.494426	+	0.869220i	$0.335378\pi$
$282$	0	0
$283$	4182.24	0.878474	0.439237	−	0.898371i	$-0.355249\pi$
$283$	4182.24	0.878474	0.439237	+	0.898371i	$0.355249\pi$
$284$	0	0
$285$	2096.31	0.435700
$286$	0	0
$287$	10070.8	2.07130
$288$	0	0
$289$	−4803.63	−0.977738
$290$	0	0
$291$	911.356	0.183590
$292$	0	0
$293$	−678.395	−0.135264	−0.0676318	−	0.997710i	$-0.521544\pi$
$293$	−678.395	−0.135264	−0.0676318	+	0.997710i	$0.521544\pi$
$294$	0	0
$295$	2311.06	0.456120
$296$	0	0
$297$	1611.58	0.314859
$298$	0	0
$299$	1189.11	0.229994
$300$	0	0
$301$	4689.62	0.898024
$302$	0	0
$303$	−7651.90	−1.45079
$304$	0	0
$305$	−3082.35	−0.578672
$306$	0	0
$307$	−7676.72	−1.42715	−0.713573	−	0.700581i	$-0.752924\pi$
$307$	−7676.72	−1.42715	−0.713573	+	0.700581i	$0.752924\pi$
$308$	0	0
$309$	−3103.45	−0.571357
$310$	0	0
$311$	4718.78	0.860377	0.430188	−	0.902739i	$-0.358447\pi$
$311$	4718.78	0.860377	0.430188	+	0.902739i	$0.358447\pi$
$312$	0	0
$313$	−7640.96	−1.37985	−0.689924	−	0.723882i	$-0.742356\pi$
$313$	−7640.96	−1.37985	−0.689924	+	0.723882i	$0.742356\pi$
$314$	0	0
$315$	−10770.8	−1.92657
$316$	0	0
$317$	8981.33	1.59130	0.795649	−	0.605758i	$-0.207130\pi$
$317$	8981.33	1.59130	0.795649	+	0.605758i	$0.207130\pi$
$318$	0	0
$319$	−3081.95	−0.540928
$320$	0	0
$321$	8166.76	1.42001
$322$	0	0
$323$	−233.722	−0.0402620
$324$	0	0
$325$	−550.249	−0.0939149
$326$	0	0
$327$	−16635.1	−2.81322
$328$	0	0
$329$	−14456.9	−2.42260
$330$	0	0
$331$	−3408.01	−0.565925	−0.282962	−	0.959131i	$-0.591317\pi$
$331$	−3408.01	−0.565925	−0.282962	+	0.959131i	$0.591317\pi$
$332$	0	0
$333$	1358.34	0.223534
$334$	0	0
$335$	−1805.88	−0.294525
$336$	0	0
$337$	7728.87	1.24931	0.624657	−	0.780900i	$-0.285239\pi$
$337$	7728.87	1.24931	0.624657	+	0.780900i	$0.285239\pi$
$338$	0	0
$339$	−9802.73	−1.57053
$340$	0	0
$341$	3241.79	0.514817
$342$	0	0
$343$	−2186.94	−0.344267
$344$	0	0
$345$	−2157.44	−0.336675
$346$	0	0
$347$	−1300.42	−0.201183	−0.100591	−	0.994928i	$-0.532073\pi$
$347$	−1300.42	−0.201183	−0.100591	+	0.994928i	$0.532073\pi$
$348$	0	0
$349$	1959.24	0.300503	0.150251	−	0.988648i	$-0.451992\pi$
$349$	1959.24	0.300503	0.150251	+	0.988648i	$0.451992\pi$
$350$	0	0
$351$	4525.06	0.688119
$352$	0	0
$353$	−12393.2	−1.86862	−0.934311	−	0.356460i	$-0.883984\pi$
$353$	−12393.2	−1.86862	−0.934311	+	0.356460i	$0.883984\pi$
$354$	0	0
$355$	−12608.1	−1.88498
$356$	0	0
$357$	2057.11	0.304968
$358$	0	0
$359$	−10544.6	−1.55021	−0.775104	−	0.631834i	$-0.782302\pi$
$359$	−10544.6	−1.55021	−0.775104	+	0.631834i	$0.782302\pi$
$360$	0	0
$361$	−6359.56	−0.927184
$362$	0	0
$363$	7989.30	1.15518
$364$	0	0
$365$	2612.07	0.374580
$366$	0	0
$367$	−6179.93	−0.878991	−0.439496	−	0.898245i	$-0.644843\pi$
$367$	−6179.93	−0.878991	−0.439496	+	0.898245i	$0.644843\pi$
$368$	0	0
$369$	15614.9	2.20293
$370$	0	0
$371$	18324.4	2.56430
$372$	0	0
$373$	10557.8	1.46558	0.732792	−	0.680452i	$-0.238217\pi$
$373$	10557.8	1.46558	0.732792	+	0.680452i	$0.238217\pi$
$374$	0	0
$375$	−10726.9	−1.47716
$376$	0	0
$377$	−8653.63	−1.18219
$378$	0	0
$379$	6308.79	0.855041	0.427521	−	0.904006i	$-0.359387\pi$
$379$	6308.79	0.855041	0.427521	+	0.904006i	$0.359387\pi$
$380$	0	0
$381$	1117.72	0.150296
$382$	0	0
$383$	7532.19	1.00490	0.502450	−	0.864606i	$-0.332432\pi$
$383$	7532.19	1.00490	0.502450	+	0.864606i	$0.332432\pi$
$384$	0	0
$385$	5237.32	0.693295
$386$	0	0
$387$	7271.30	0.955093
$388$	0	0
$389$	5798.06	0.755716	0.377858	−	0.925864i	$-0.376661\pi$
$389$	5798.06	0.755716	0.377858	+	0.925864i	$0.376661\pi$
$390$	0	0
$391$	240.538	0.0311113
$392$	0	0
$393$	11040.2	1.41706
$394$	0	0
$395$	−3050.37	−0.388559
$396$	0	0
$397$	−13688.8	−1.73053	−0.865264	−	0.501317i	$-0.832849\pi$
$397$	−13688.8	−1.73053	−0.865264	+	0.501317i	$0.832849\pi$
$398$	0	0
$399$	−4395.86	−0.551550
$400$	0	0
$401$	−6132.31	−0.763673	−0.381836	−	0.924230i	$-0.624708\pi$
$401$	−6132.31	−0.763673	−0.381836	+	0.924230i	$0.624708\pi$
$402$	0	0
$403$	9102.43	1.12512
$404$	0	0
$405$	3697.69	0.453678
$406$	0	0
$407$	−660.494	−0.0804409
$408$	0	0
$409$	−6991.14	−0.845207	−0.422604	−	0.906315i	$-0.638884\pi$
$409$	−6991.14	−0.845207	−0.422604	+	0.906315i	$0.638884\pi$
$410$	0	0
$411$	13591.9	1.63124
$412$	0	0
$413$	−4846.20	−0.577400
$414$	0	0
$415$	−184.571	−0.0218319
$416$	0	0
$417$	20192.8	2.37133
$418$	0	0
$419$	−12124.1	−1.41360	−0.706801	−	0.707412i	$-0.749863\pi$
$419$	−12124.1	−1.41360	−0.706801	+	0.707412i	$0.749863\pi$
$420$	0	0
$421$	−11372.3	−1.31651	−0.658256	−	0.752794i	$-0.728706\pi$
$421$	−11372.3	−1.31651	−0.658256	+	0.752794i	$0.728706\pi$
$422$	0	0
$423$	−22415.6	−2.57656
$424$	0	0
$425$	−111.306	−0.0127039
$426$	0	0
$427$	6463.56	0.732538
$428$	0	0
$429$	−7667.08	−0.862867
$430$	0	0
$431$	12259.9	1.37016	0.685079	−	0.728469i	$-0.259768\pi$
$431$	12259.9	1.37016	0.685079	+	0.728469i	$0.259768\pi$
$432$	0	0
$433$	4860.00	0.539391	0.269696	−	0.962946i	$-0.413077\pi$
$433$	4860.00	0.539391	0.269696	+	0.962946i	$0.413077\pi$
$434$	0	0
$435$	15700.5	1.73054
$436$	0	0
$437$	−514.010	−0.0562664
$438$	0	0
$439$	−11660.7	−1.26773	−0.633866	−	0.773443i	$-0.718533\pi$
$439$	−11660.7	−1.26773	−0.633866	+	0.773443i	$0.718533\pi$
$440$	0	0
$441$	9597.57	1.03634
$442$	0	0
$443$	−15197.5	−1.62992	−0.814958	−	0.579520i	$-0.803240\pi$
$443$	−15197.5	−1.62992	−0.814958	+	0.579520i	$0.803240\pi$
$444$	0	0
$445$	−2331.56	−0.248374
$446$	0	0
$447$	27379.9	2.89715
$448$	0	0
$449$	−7803.19	−0.820168	−0.410084	−	0.912048i	$-0.634501\pi$
$449$	−7803.19	−0.820168	−0.410084	+	0.912048i	$0.634501\pi$
$450$	0	0
$451$	−7592.76	−0.792747
$452$	0	0
$453$	−7939.11	−0.823426
$454$	0	0
$455$	14705.6	1.51518
$456$	0	0
$457$	4159.89	0.425802	0.212901	−	0.977074i	$-0.431709\pi$
$457$	4159.89	0.425802	0.212901	+	0.977074i	$0.431709\pi$
$458$	0	0
$459$	915.345	0.0930820
$460$	0	0
$461$	1308.17	0.132163	0.0660817	−	0.997814i	$-0.478950\pi$
$461$	1308.17	0.132163	0.0660817	+	0.997814i	$0.478950\pi$
$462$	0	0
$463$	−19412.5	−1.94855	−0.974273	−	0.225372i	$-0.927640\pi$
$463$	−19412.5	−1.94855	−0.974273	+	0.225372i	$0.927640\pi$
$464$	0	0
$465$	−16514.8	−1.64700
$466$	0	0
$467$	3742.18	0.370808	0.185404	−	0.982662i	$-0.440641\pi$
$467$	3742.18	0.370808	0.185404	+	0.982662i	$0.440641\pi$
$468$	0	0
$469$	3786.86	0.372838
$470$	0	0
$471$	1625.38	0.159010
$472$	0	0
$473$	−3535.67	−0.343700
$474$	0	0
$475$	237.853	0.0229756
$476$	0	0
$477$	28412.2	2.72726
$478$	0	0
$479$	11322.5	1.08004	0.540021	−	0.841652i	$-0.318416\pi$
$479$	11322.5	1.08004	0.540021	+	0.841652i	$0.318416\pi$
$480$	0	0
$481$	−1854.56	−0.175802
$482$	0	0
$483$	4524.06	0.426195
$484$	0	0
$485$	1317.88	0.123385
$486$	0	0
$487$	−6688.99	−0.622397	−0.311198	−	0.950345i	$-0.600730\pi$
$487$	−6688.99	−0.622397	−0.311198	+	0.950345i	$0.600730\pi$
$488$	0	0
$489$	27757.3	2.56693
$490$	0	0
$491$	−2058.53	−0.189206	−0.0946030	−	0.995515i	$-0.530158\pi$
$491$	−2058.53	−0.189206	−0.0946030	+	0.995515i	$0.530158\pi$
$492$	0	0
$493$	−1750.49	−0.159915
$494$	0	0
$495$	8120.52	0.737354
$496$	0	0
$497$	26438.6	2.38618
$498$	0	0
$499$	19788.5	1.77526	0.887632	−	0.460553i	$-0.152349\pi$
$499$	19788.5	1.77526	0.887632	+	0.460553i	$0.152349\pi$
$500$	0	0
$501$	7825.85	0.697871
$502$	0	0
$503$	17381.5	1.54076	0.770381	−	0.637584i	$-0.220066\pi$
$503$	17381.5	1.54076	0.770381	+	0.637584i	$0.220066\pi$
$504$	0	0
$505$	−11065.1	−0.975030
$506$	0	0
$507$	−3833.27	−0.335782
$508$	0	0
$509$	11595.2	1.00972	0.504862	−	0.863200i	$-0.331543\pi$
$509$	11595.2	1.00972	0.504862	+	0.863200i	$0.331543\pi$
$510$	0	0
$511$	−5477.39	−0.474179
$512$	0	0
$513$	−1956.02	−0.168343
$514$	0	0
$515$	−4487.78	−0.383991
$516$	0	0
$517$	10899.6	0.927201
$518$	0	0
$519$	12804.6	1.08297
$520$	0	0
$521$	−2597.34	−0.218409	−0.109205	−	0.994019i	$-0.534830\pi$
$521$	−2597.34	−0.218409	−0.109205	+	0.994019i	$0.534830\pi$
$522$	0	0
$523$	−7487.17	−0.625987	−0.312993	−	0.949755i	$-0.601332\pi$
$523$	−7487.17	−0.625987	−0.312993	+	0.949755i	$0.601332\pi$
$524$	0	0
$525$	−2093.46	−0.174031
$526$	0	0
$527$	1841.27	0.152196
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	−7514.08	−0.614093
$532$	0	0
$533$	−21319.3	−1.73253
$534$	0	0
$535$	11809.6	0.954345
$536$	0	0
$537$	19276.5	1.54906
$538$	0	0
$539$	−4666.82	−0.372939
$540$	0	0
$541$	−10070.5	−0.800302	−0.400151	−	0.916449i	$-0.631042\pi$
$541$	−10070.5	−0.800302	−0.400151	+	0.916449i	$0.631042\pi$
$542$	0	0
$543$	33054.2	2.61232
$544$	0	0
$545$	−24055.4	−1.89068
$546$	0	0
$547$	−21474.4	−1.67857	−0.839284	−	0.543693i	$-0.817026\pi$
$547$	−21474.4	−1.67857	−0.839284	+	0.543693i	$0.817026\pi$
$548$	0	0
$549$	10021.8	0.779091
$550$	0	0
$551$	3740.65	0.289214
$552$	0	0
$553$	6396.50	0.491875
$554$	0	0
$555$	3364.79	0.257346
$556$	0	0
$557$	−17358.6	−1.32048	−0.660242	−	0.751053i	$-0.729546\pi$
$557$	−17358.6	−1.32048	−0.660242	+	0.751053i	$0.729546\pi$
$558$	0	0
$559$	−9927.60	−0.751150
$560$	0	0
$561$	−1550.92	−0.116720
$562$	0	0
$563$	−17977.9	−1.34579	−0.672895	−	0.739738i	$-0.734950\pi$
$563$	−17977.9	−1.34579	−0.672895	+	0.739738i	$0.734950\pi$
$564$	0	0
$565$	−14175.3	−1.05551
$566$	0	0
$567$	−7753.89	−0.574309
$568$	0	0
$569$	−13529.5	−0.996810	−0.498405	−	0.866944i	$-0.666081\pi$
$569$	−13529.5	−0.996810	−0.498405	+	0.866944i	$0.666081\pi$
$570$	0	0
$571$	−3866.06	−0.283344	−0.141672	−	0.989914i	$-0.545248\pi$
$571$	−3866.06	−0.283344	−0.141672	+	0.989914i	$0.545248\pi$
$572$	0	0
$573$	−2585.11	−0.188472
$574$	0	0
$575$	−244.789	−0.0177538
$576$	0	0
$577$	21500.0	1.55123	0.775614	−	0.631208i	$-0.217441\pi$
$577$	21500.0	1.55123	0.775614	+	0.631208i	$0.217441\pi$
$578$	0	0
$579$	5546.78	0.398128
$580$	0	0
$581$	387.038	0.0276369
$582$	0	0
$583$	−13815.4	−0.981435
$584$	0	0
$585$	22801.1	1.61147
$586$	0	0
$587$	4927.12	0.346446	0.173223	−	0.984883i	$-0.444582\pi$
$587$	4927.12	0.346446	0.173223	+	0.984883i	$0.444582\pi$
$588$	0	0
$589$	−3934.64	−0.275253
$590$	0	0
$591$	−11190.5	−0.778874
$592$	0	0
$593$	17675.5	1.22402	0.612011	−	0.790849i	$-0.290361\pi$
$593$	17675.5	1.22402	0.612011	+	0.790849i	$0.290361\pi$
$594$	0	0
$595$	2974.70	0.204959
$596$	0	0
$597$	−17346.9	−1.18921
$598$	0	0
$599$	22302.6	1.52130	0.760651	−	0.649161i	$-0.224880\pi$
$599$	22302.6	1.52130	0.760651	+	0.649161i	$0.224880\pi$
$600$	0	0
$601$	−25526.8	−1.73255	−0.866273	−	0.499570i	$-0.833491\pi$
$601$	−25526.8	−1.73255	−0.866273	+	0.499570i	$0.833491\pi$
$602$	0	0
$603$	5871.56	0.396531
$604$	0	0
$605$	11553.0	0.776358
$606$	0	0
$607$	−9865.32	−0.659672	−0.329836	−	0.944038i	$-0.606994\pi$
$607$	−9865.32	−0.659672	−0.329836	+	0.944038i	$0.606994\pi$
$608$	0	0
$609$	−32923.3	−2.19068
$610$	0	0
$611$	30604.3	2.02638
$612$	0	0
$613$	−3760.57	−0.247778	−0.123889	−	0.992296i	$-0.539537\pi$
$613$	−3760.57	−0.247778	−0.123889	+	0.992296i	$0.539537\pi$
$614$	0	0
$615$	38680.2	2.53615
$616$	0	0
$617$	−4170.47	−0.272118	−0.136059	−	0.990701i	$-0.543444\pi$
$617$	−4170.47	−0.272118	−0.136059	+	0.990701i	$0.543444\pi$
$618$	0	0
$619$	24392.4	1.58387	0.791935	−	0.610606i	$-0.209074\pi$
$619$	24392.4	1.58387	0.791935	+	0.610606i	$0.209074\pi$
$620$	0	0
$621$	2013.06	0.130083
$622$	0	0
$623$	4889.18	0.314415
$624$	0	0
$625$	−16842.1	−1.07789
$626$	0	0
$627$	3314.20	0.211094
$628$	0	0
$629$	−375.148	−0.0237808
$630$	0	0
$631$	10776.6	0.679886	0.339943	−	0.940446i	$-0.389592\pi$
$631$	10776.6	0.679886	0.339943	+	0.940446i	$0.389592\pi$
$632$	0	0
$633$	18248.9	1.14586
$634$	0	0
$635$	1616.29	0.101009
$636$	0	0
$637$	−13103.7	−0.815050
$638$	0	0
$639$	40993.3	2.53782
$640$	0	0
$641$	27979.4	1.72405	0.862027	−	0.506863i	$-0.169195\pi$
$641$	27979.4	1.72405	0.862027	+	0.506863i	$0.169195\pi$
$642$	0	0
$643$	−18519.5	−1.13583	−0.567915	−	0.823087i	$-0.692250\pi$
$643$	−18519.5	−1.13583	−0.567915	+	0.823087i	$0.692250\pi$
$644$	0	0
$645$	18011.9	1.09957
$646$	0	0
$647$	−20048.6	−1.21822	−0.609112	−	0.793084i	$-0.708474\pi$
$647$	−20048.6	−1.21822	−0.609112	+	0.793084i	$0.708474\pi$
$648$	0	0
$649$	3653.72	0.220988
$650$	0	0
$651$	34630.8	2.08493
$652$	0	0
$653$	−30798.2	−1.84568	−0.922839	−	0.385185i	$-0.874138\pi$
$653$	−30798.2	−1.84568	−0.922839	+	0.385185i	$0.874138\pi$
$654$	0	0
$655$	15964.8	0.952360
$656$	0	0
$657$	−8492.75	−0.504313
$658$	0	0
$659$	−25007.5	−1.47823	−0.739116	−	0.673578i	$-0.764757\pi$
$659$	−25007.5	−1.47823	−0.739116	+	0.673578i	$0.764757\pi$
$660$	0	0
$661$	9196.16	0.541133	0.270567	−	0.962701i	$-0.412789\pi$
$661$	9196.16	0.541133	0.270567	+	0.962701i	$0.412789\pi$
$662$	0	0
$663$	−4354.75	−0.255090
$664$	0	0
$665$	−6356.68	−0.370679
$666$	0	0
$667$	−3849.74	−0.223482
$668$	0	0
$669$	30870.5	1.78404
$670$	0	0
$671$	−4873.11	−0.280364
$672$	0	0
$673$	−8440.22	−0.483427	−0.241714	−	0.970348i	$-0.577709\pi$
$673$	−8440.22	−0.483427	−0.241714	+	0.970348i	$0.577709\pi$
$674$	0	0
$675$	−931.522	−0.0531175
$676$	0	0
$677$	−7557.24	−0.429023	−0.214511	−	0.976721i	$-0.568816\pi$
$677$	−7557.24	−0.429023	−0.214511	+	0.976721i	$0.568816\pi$
$678$	0	0
$679$	−2763.53	−0.156192
$680$	0	0
$681$	−4442.44	−0.249977
$682$	0	0
$683$	−8410.59	−0.471189	−0.235595	−	0.971851i	$-0.575704\pi$
$683$	−8410.59	−0.471189	−0.235595	+	0.971851i	$0.575704\pi$
$684$	0	0
$685$	19654.7	1.09630
$686$	0	0
$687$	27888.5	1.54878
$688$	0	0
$689$	−38791.5	−2.14491
$690$	0	0
$691$	−15913.2	−0.876072	−0.438036	−	0.898958i	$-0.644326\pi$
$691$	−15913.2	−0.876072	−0.438036	+	0.898958i	$0.644326\pi$
$692$	0	0
$693$	−17028.4	−0.933412
$694$	0	0
$695$	29200.0	1.59369
$696$	0	0
$697$	−4312.54	−0.234360
$698$	0	0
$699$	−23759.2	−1.28563
$700$	0	0
$701$	29374.3	1.58267	0.791335	−	0.611383i	$-0.209386\pi$
$701$	29374.3	1.58267	0.791335	+	0.611383i	$0.209386\pi$
$702$	0	0
$703$	801.659	0.0430087
$704$	0	0
$705$	−55526.3	−2.96630
$706$	0	0
$707$	23203.0	1.23429
$708$	0	0
$709$	−23319.2	−1.23522	−0.617609	−	0.786485i	$-0.711899\pi$
$709$	−23319.2	−1.23522	−0.617609	+	0.786485i	$0.711899\pi$
$710$	0	0
$711$	9917.83	0.523133
$712$	0	0
$713$	4049.39	0.212694
$714$	0	0
$715$	−11087.0	−0.579905
$716$	0	0
$717$	8567.13	0.446228
$718$	0	0
$719$	25216.5	1.30795	0.653975	−	0.756517i	$-0.273100\pi$
$719$	25216.5	1.30795	0.653975	+	0.756517i	$0.273100\pi$
$720$	0	0
$721$	9410.69	0.486092
$722$	0	0
$723$	−1290.05	−0.0663586
$724$	0	0
$725$	1781.42	0.0912558
$726$	0	0
$727$	−36727.3	−1.87364	−0.936822	−	0.349806i	$-0.886247\pi$
$727$	−36727.3	−1.87364	−0.936822	+	0.349806i	$0.886247\pi$
$728$	0	0
$729$	−31055.4	−1.57778
$730$	0	0
$731$	−2008.19	−0.101608
$732$	0	0
$733$	−3401.78	−0.171416	−0.0857079	−	0.996320i	$-0.527315\pi$
$733$	−3401.78	−0.171416	−0.0857079	+	0.996320i	$0.527315\pi$
$734$	0	0
$735$	23774.4	1.19311
$736$	0	0
$737$	−2855.04	−0.142696
$738$	0	0
$739$	11982.1	0.596442	0.298221	−	0.954497i	$-0.403607\pi$
$739$	11982.1	0.596442	0.298221	+	0.954497i	$0.403607\pi$
$740$	0	0
$741$	9305.74	0.461343
$742$	0	0
$743$	−32449.8	−1.60225	−0.801123	−	0.598499i	$-0.795764\pi$
$743$	−32449.8	−1.60225	−0.801123	+	0.598499i	$0.795764\pi$
$744$	0	0
$745$	39593.0	1.94708
$746$	0	0
$747$	600.107	0.0293933
$748$	0	0
$749$	−24764.3	−1.20810
$750$	0	0
$751$	−4340.97	−0.210925	−0.105462	−	0.994423i	$-0.533632\pi$
$751$	−4340.97	−0.210925	−0.105462	+	0.994423i	$0.533632\pi$
$752$	0	0
$753$	−23293.5	−1.12731
$754$	0	0
$755$	−11480.4	−0.553398
$756$	0	0
$757$	36757.0	1.76480	0.882401	−	0.470498i	$-0.155926\pi$
$757$	36757.0	1.76480	0.882401	+	0.470498i	$0.155926\pi$
$758$	0	0
$759$	−3410.85	−0.163117
$760$	0	0
$761$	2408.45	0.114726	0.0573629	−	0.998353i	$-0.481731\pi$
$761$	2408.45	0.114726	0.0573629	+	0.998353i	$0.481731\pi$
$762$	0	0
$763$	50443.1	2.39340
$764$	0	0
$765$	4612.30	0.217984
$766$	0	0
$767$	10259.1	0.482964
$768$	0	0
$769$	−20323.2	−0.953022	−0.476511	−	0.879168i	$-0.658099\pi$
$769$	−20323.2	−0.953022	−0.476511	+	0.879168i	$0.658099\pi$
$770$	0	0
$771$	−46559.0	−2.17482
$772$	0	0
$773$	−23521.1	−1.09443	−0.547214	−	0.836992i	$-0.684312\pi$
$773$	−23521.1	−1.09443	−0.547214	+	0.836992i	$0.684312\pi$
$774$	0	0
$775$	−1873.81	−0.0868508
$776$	0	0
$777$	−7055.82	−0.325773
$778$	0	0
$779$	9215.54	0.423852
$780$	0	0
$781$	−19933.0	−0.913262
$782$	0	0
$783$	−14649.8	−0.668635
$784$	0	0
$785$	2350.40	0.106865
$786$	0	0
$787$	32043.4	1.45136	0.725682	−	0.688030i	$-0.241524\pi$
$787$	32043.4	1.45136	0.725682	+	0.688030i	$0.241524\pi$
$788$	0	0
$789$	24598.1	1.10991
$790$	0	0
$791$	29725.1	1.33616
$792$	0	0
$793$	−13682.9	−0.612730
$794$	0	0
$795$	70380.6	3.13980
$796$	0	0
$797$	−21774.0	−0.967720	−0.483860	−	0.875145i	$-0.660766\pi$
$797$	−21774.0	−0.967720	−0.483860	+	0.875145i	$0.660766\pi$
$798$	0	0
$799$	6190.75	0.274109
$800$	0	0
$801$	7580.71	0.334396
$802$	0	0
$803$	4129.60	0.181482
$804$	0	0
$805$	6542.07	0.286432
$806$	0	0
$807$	−33599.9	−1.46564
$808$	0	0
$809$	14693.1	0.638543	0.319271	−	0.947663i	$-0.396562\pi$
$809$	14693.1	0.638543	0.319271	+	0.947663i	$0.396562\pi$
$810$	0	0
$811$	41634.1	1.80268	0.901338	−	0.433117i	$-0.142586\pi$
$811$	41634.1	1.80268	0.901338	+	0.433117i	$0.142586\pi$
$812$	0	0
$813$	−10743.1	−0.463442
$814$	0	0
$815$	40138.7	1.72515
$816$	0	0
$817$	4291.34	0.183764
$818$	0	0
$819$	−47813.0	−2.03995
$820$	0	0
$821$	−311.819	−0.0132553	−0.00662763	−	0.999978i	$-0.502110\pi$
$821$	−311.819	−0.0132553	−0.00662763	+	0.999978i	$0.502110\pi$
$822$	0	0
$823$	18804.4	0.796451	0.398225	−	0.917288i	$-0.369626\pi$
$823$	18804.4	0.796451	0.398225	+	0.917288i	$0.369626\pi$
$824$	0	0
$825$	1578.33	0.0666067
$826$	0	0
$827$	−40839.7	−1.71721	−0.858607	−	0.512635i	$-0.828670\pi$
$827$	−40839.7	−1.71721	−0.858607	+	0.512635i	$0.828670\pi$
$828$	0	0
$829$	−21272.1	−0.891206	−0.445603	−	0.895231i	$-0.647011\pi$
$829$	−21272.1	−0.891206	−0.445603	+	0.895231i	$0.647011\pi$
$830$	0	0
$831$	−33597.7	−1.40252
$832$	0	0
$833$	−2650.66	−0.110252
$834$	0	0
$835$	11316.7	0.469017
$836$	0	0
$837$	15409.6	0.636360
$838$	0	0
$839$	17124.4	0.704648	0.352324	−	0.935878i	$-0.385391\pi$
$839$	17124.4	0.704648	0.352324	+	0.935878i	$0.385391\pi$
$840$	0	0
$841$	3627.01	0.148715
$842$	0	0
$843$	−37514.9	−1.53272
$844$	0	0
$845$	−5543.14	−0.225668
$846$	0	0
$847$	−24226.2	−0.982787
$848$	0	0
$849$	−33683.8	−1.36163
$850$	0	0
$851$	−825.039	−0.0332338
$852$	0	0
$853$	41037.7	1.64725	0.823624	−	0.567136i	$-0.191948\pi$
$853$	41037.7	1.64725	0.823624	+	0.567136i	$0.191948\pi$
$854$	0	0
$855$	−9856.10	−0.394236
$856$	0	0
$857$	−43891.1	−1.74946	−0.874732	−	0.484607i	$-0.838963\pi$
$857$	−43891.1	−1.74946	−0.874732	+	0.484607i	$0.838963\pi$
$858$	0	0
$859$	−27562.0	−1.09477	−0.547383	−	0.836882i	$-0.684376\pi$
$859$	−27562.0	−1.09477	−0.547383	+	0.836882i	$0.684376\pi$
$860$	0	0
$861$	−81110.7	−3.21050
$862$	0	0
$863$	24931.7	0.983412	0.491706	−	0.870761i	$-0.336373\pi$
$863$	24931.7	0.983412	0.491706	+	0.870761i	$0.336373\pi$
$864$	0	0
$865$	18516.2	0.727826
$866$	0	0
$867$	38688.5	1.51549
$868$	0	0
$869$	−4822.54	−0.188255
$870$	0	0
$871$	−8016.52	−0.311859
$872$	0	0
$873$	−4284.88	−0.166118
$874$	0	0
$875$	32527.4	1.25672
$876$	0	0
$877$	27169.3	1.04612	0.523058	−	0.852297i	$-0.324791\pi$
$877$	27169.3	1.04612	0.523058	+	0.852297i	$0.324791\pi$
$878$	0	0
$879$	5463.80	0.209658
$880$	0	0
$881$	41469.9	1.58587	0.792937	−	0.609303i	$-0.208551\pi$
$881$	41469.9	1.58587	0.792937	+	0.609303i	$0.208551\pi$
$882$	0	0
$883$	−20433.7	−0.778763	−0.389382	−	0.921076i	$-0.627311\pi$
$883$	−20433.7	−0.778763	−0.389382	+	0.921076i	$0.627311\pi$
$884$	0	0
$885$	−18613.3	−0.706984
$886$	0	0
$887$	−47298.0	−1.79043	−0.895215	−	0.445634i	$-0.852978\pi$
$887$	−47298.0	−1.79043	−0.895215	+	0.445634i	$0.852978\pi$
$888$	0	0
$889$	−3389.30	−0.127867
$890$	0	0
$891$	5845.93	0.219805
$892$	0	0
$893$	−13229.1	−0.495740
$894$	0	0
$895$	27875.0	1.04107
$896$	0	0
$897$	−9577.13	−0.356489
$898$	0	0
$899$	−29469.0	−1.09327
$900$	0	0
$901$	−7846.89	−0.290142
$902$	0	0
$903$	−37770.3	−1.39193
$904$	0	0
$905$	47798.3	1.75566
$906$	0	0
$907$	−38147.2	−1.39653	−0.698267	−	0.715838i	$-0.746045\pi$
$907$	−38147.2	−1.39653	−0.698267	+	0.715838i	$0.746045\pi$
$908$	0	0
$909$	35976.5	1.31272
$910$	0	0
$911$	2307.84	0.0839320	0.0419660	−	0.999119i	$-0.486638\pi$
$911$	2307.84	0.0839320	0.0419660	+	0.999119i	$0.486638\pi$
$912$	0	0
$913$	−291.802	−0.0105775
$914$	0	0
$915$	24825.3	0.896940
$916$	0	0
$917$	−33477.5	−1.20559
$918$	0	0
$919$	47460.3	1.70356	0.851780	−	0.523900i	$-0.175523\pi$
$919$	47460.3	1.70356	0.851780	+	0.523900i	$0.175523\pi$
$920$	0	0
$921$	61828.5	2.21207
$922$	0	0
$923$	−55968.7	−1.99592
$924$	0	0
$925$	381.778	0.0135706
$926$	0	0
$927$	14591.4	0.516983
$928$	0	0
$929$	−5371.11	−0.189688	−0.0948440	−	0.995492i	$-0.530235\pi$
$929$	−5371.11	−0.189688	−0.0948440	+	0.995492i	$0.530235\pi$
$930$	0	0
$931$	5664.24	0.199396
$932$	0	0
$933$	−38005.1	−1.33358
$934$	0	0
$935$	−2242.73	−0.0784439
$936$	0	0
$937$	37291.4	1.30017	0.650084	−	0.759863i	$-0.274734\pi$
$937$	37291.4	1.30017	0.650084	+	0.759863i	$0.274734\pi$
$938$	0	0
$939$	61540.4	2.13876
$940$	0	0
$941$	−10354.8	−0.358721	−0.179361	−	0.983783i	$-0.557403\pi$
$941$	−10354.8	−0.358721	−0.179361	+	0.983783i	$0.557403\pi$
$942$	0	0
$943$	−9484.30	−0.327520
$944$	0	0
$945$	24895.2	0.856975
$946$	0	0
$947$	−393.905	−0.0135166	−0.00675829	−	0.999977i	$-0.502151\pi$
$947$	−393.905	−0.0135166	−0.00675829	+	0.999977i	$0.502151\pi$
$948$	0	0
$949$	11595.3	0.396626
$950$	0	0
$951$	−72335.7	−2.46651
$952$	0	0
$953$	20824.5	0.707840	0.353920	−	0.935276i	$-0.384849\pi$
$953$	20824.5	0.707840	0.353920	+	0.935276i	$0.384849\pi$
$954$	0	0
$955$	−3738.22	−0.126666
$956$	0	0
$957$	24822.1	0.838436
$958$	0	0
$959$	−41215.1	−1.38781
$960$	0	0
$961$	1206.32	0.0404929
$962$	0	0
$963$	−38397.2	−1.28487
$964$	0	0
$965$	8020.97	0.267569
$966$	0	0
$967$	−36967.5	−1.22936	−0.614681	−	0.788776i	$-0.710715\pi$
$967$	−36967.5	−1.22936	−0.614681	+	0.788776i	$0.710715\pi$
$968$	0	0
$969$	1882.40	0.0624060
$970$	0	0
$971$	−5339.18	−0.176460	−0.0882299	−	0.996100i	$-0.528121\pi$
$971$	−5339.18	−0.176460	−0.0882299	+	0.996100i	$0.528121\pi$
$972$	0	0
$973$	−61231.1	−2.01745
$974$	0	0
$975$	4431.72	0.145568
$976$	0	0
$977$	8161.14	0.267245	0.133622	−	0.991032i	$-0.457339\pi$
$977$	8161.14	0.267245	0.133622	+	0.991032i	$0.457339\pi$
$978$	0	0
$979$	−3686.12	−0.120336
$980$	0	0
$981$	78212.4	2.54550
$982$	0	0
$983$	33898.0	1.09988	0.549938	−	0.835206i	$-0.314651\pi$
$983$	33898.0	1.09988	0.549938	+	0.835206i	$0.314651\pi$
$984$	0	0
$985$	−16182.1	−0.523456
$986$	0	0
$987$	116436.	3.75502
$988$	0	0
$989$	−4416.49	−0.141998
$990$	0	0
$991$	24087.4	0.772111	0.386055	−	0.922476i	$-0.373837\pi$
$991$	24087.4	0.772111	0.386055	+	0.922476i	$0.373837\pi$
$992$	0	0
$993$	27448.1	0.877181
$994$	0	0
$995$	−25084.6	−0.799232
$996$	0	0
$997$	29288.3	0.930362	0.465181	−	0.885216i	$-0.345989\pi$
$997$	29288.3	0.930362	0.465181	+	0.885216i	$0.345989\pi$
$998$	0	0
$999$	−3139.61	−0.0994322

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1472.4.a.bg.1.2		8
4.3	odd	2		1472.4.a.bh.1.7		8
8.3	odd	2		736.4.a.e.1.2	✓	8
8.5	even	2		736.4.a.f.1.7	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
736.4.a.e.1.2	✓	8	8.3	odd	2
736.4.a.f.1.7	yes	8	8.5	even	2
1472.4.a.bg.1.2		8	1.1	even	1	trivial
1472.4.a.bh.1.7		8	4.3	odd	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:	\( N \)	\(=\)	\( 1472 = 2^{6} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1472.a (trivial)

\(f(q)\)	\(=\)	\(q-8.05402 q^{3} -11.6466 q^{5} +24.4224 q^{7} +37.8672 q^{9} -18.4129 q^{11} -51.7005 q^{13} +93.8018 q^{15} -10.4582 q^{17} +22.3482 q^{19} -196.698 q^{21} -23.0000 q^{23} +10.6430 q^{25} -87.5244 q^{27} +167.380 q^{29} -176.061 q^{31} +148.298 q^{33} -284.438 q^{35} +35.8712 q^{37} +416.397 q^{39} +412.361 q^{41} +192.021 q^{43} -441.023 q^{45} -591.953 q^{47} +253.454 q^{49} +84.2303 q^{51} +750.312 q^{53} +214.447 q^{55} -179.993 q^{57} -198.433 q^{59} +264.657 q^{61} +924.807 q^{63} +602.135 q^{65} +155.057 q^{67} +185.242 q^{69} +1082.55 q^{71} -224.277 q^{73} -85.7190 q^{75} -449.687 q^{77} +261.911 q^{79} -317.491 q^{81} +15.8477 q^{83} +121.802 q^{85} -1348.08 q^{87} +200.192 q^{89} -1262.65 q^{91} +1417.99 q^{93} -260.281 q^{95} -113.155 q^{97} -697.244 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 12 q^{3} + 12 q^{5} - 14 q^{7} + 90 q^{9} - 88 q^{11} + 30 q^{13} + 30 q^{15} + 58 q^{17} - 190 q^{19} + 66 q^{21} - 184 q^{23} + 28 q^{25} - 432 q^{27} - 190 q^{29} - 60 q^{31} + 346 q^{33} - 192 q^{35}+ \cdots - 5986 q^{99}+O(q^{100}) \) 8 * q - 12 * q^3 + 12 * q^5 - 14 * q^7 + 90 * q^9 - 88 * q^11 + 30 * q^13 + 30 * q^15 + 58 * q^17 - 190 * q^19 + 66 * q^21 - 184 * q^23 + 28 * q^25 - 432 * q^27 - 190 * q^29 - 60 * q^31 + 346 * q^33 - 192 * q^35 + 156 * q^37 + 156 * q^39 + 282 * q^41 - 810 * q^43 - 358 * q^45 + 564 * q^47 + 752 * q^49 - 1614 * q^51 + 230 * q^53 + 1292 * q^55 + 796 * q^57 - 1916 * q^59 - 22 * q^61 + 1036 * q^63 + 222 * q^65 - 2292 * q^67 + 276 * q^69 + 2376 * q^71 - 630 * q^73 - 3952 * q^75 + 948 * q^77 + 892 * q^79 - 1044 * q^81 - 2800 * q^83 - 288 * q^85 + 2756 * q^87 - 308 * q^89 - 4610 * q^91 + 982 * q^93 + 1300 * q^95 + 854 * q^97 - 5986 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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