LMFDB - Embedded newform 1472.4.a.bg.1.7

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			1.7
Root			$7.48973$ 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)

$f(q)$	$=$	$q+5.48973 q^{3} +7.85529 q^{5} +7.65099 q^{7} +3.13714 q^{9} +O(q^{10})$	$q+5.48973 q^{3} +7.85529 q^{5} +7.65099 q^{7} +3.13714 q^{9} +2.42526 q^{11} -62.4046 q^{13} +43.1234 q^{15} -117.017 q^{17} +76.2008 q^{19} +42.0019 q^{21} -23.0000 q^{23} -63.2945 q^{25} -131.001 q^{27} -39.2568 q^{29} -171.793 q^{31} +13.3140 q^{33} +60.1007 q^{35} +280.327 q^{37} -342.584 q^{39} -280.321 q^{41} -393.268 q^{43} +24.6431 q^{45} +467.968 q^{47} -284.462 q^{49} -642.393 q^{51} -253.167 q^{53} +19.0511 q^{55} +418.322 q^{57} -850.800 q^{59} -176.962 q^{61} +24.0022 q^{63} -490.206 q^{65} +684.006 q^{67} -126.264 q^{69} +1110.22 q^{71} -510.525 q^{73} -347.470 q^{75} +18.5556 q^{77} -535.994 q^{79} -803.861 q^{81} -323.934 q^{83} -919.204 q^{85} -215.509 q^{87} +327.742 q^{89} -477.457 q^{91} -943.095 q^{93} +598.579 q^{95} +1658.33 q^{97} +7.60838 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 12 q^{3} + 12 q^{5} - 14 q^{7} + 90 q^{9}+O(q^{10}) $ 8 * q - 12 * q^3 + 12 * q^5 - 14 * q^7 + 90 * q^9	$ 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}+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

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
$3$	5.48973	1.05650	0.528250	−	0.849089i	$-0.322849\pi$
$3$	5.48973	1.05650	0.528250	+	0.849089i	$0.322849\pi$
$4$	0	0
$5$	7.85529	0.702598	0.351299	−	0.936263i	$-0.385740\pi$
$5$	7.85529	0.702598	0.351299	+	0.936263i	$0.385740\pi$
$6$	0	0
$7$	7.65099	0.413115	0.206557	−	0.978434i	$-0.433774\pi$
$7$	7.65099	0.413115	0.206557	+	0.978434i	$0.433774\pi$
$8$	0	0
$9$	3.13714	0.116190
$10$	0	0
$11$	2.42526	0.0664767	0.0332383	−	0.999447i	$-0.489418\pi$
$11$	2.42526	0.0664767	0.0332383	+	0.999447i	$0.489418\pi$
$12$	0	0
$13$	−62.4046	−1.33138	−0.665689	−	0.746229i	$-0.731862\pi$
$13$	−62.4046	−1.33138	−0.665689	+	0.746229i	$0.731862\pi$
$14$	0	0
$15$	43.1234	0.742294
$16$	0	0
$17$	−117.017	−1.66946	−0.834731	−	0.550658i	$-0.814377\pi$
$17$	−117.017	−1.66946	−0.834731	+	0.550658i	$0.814377\pi$
$18$	0	0
$19$	76.2008	0.920088	0.460044	−	0.887896i	$-0.347834\pi$
$19$	76.2008	0.920088	0.460044	+	0.887896i	$0.347834\pi$
$20$	0	0
$21$	42.0019	0.436455
$22$	0	0
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	0	0
$25$	−63.2945	−0.506356
$26$	0	0
$27$	−131.001	−0.933744
$28$	0	0
$29$	−39.2568	−0.251372	−0.125686	−	0.992070i	$-0.540113\pi$
$29$	−39.2568	−0.251372	−0.125686	+	0.992070i	$0.540113\pi$
$30$	0	0
$31$	−171.793	−0.995318	−0.497659	−	0.867373i	$-0.665807\pi$
$31$	−171.793	−0.995318	−0.497659	+	0.867373i	$0.665807\pi$
$32$	0	0
$33$	13.3140	0.0702325
$34$	0	0
$35$	60.1007	0.290254
$36$	0	0
$37$	280.327	1.24555	0.622776	−	0.782400i	$-0.286005\pi$
$37$	280.327	1.24555	0.622776	+	0.782400i	$0.286005\pi$
$38$	0	0
$39$	−342.584	−1.40660
$40$	0	0
$41$	−280.321	−1.06778	−0.533888	−	0.845555i	$-0.679269\pi$
$41$	−280.321	−1.06778	−0.533888	+	0.845555i	$0.679269\pi$
$42$	0	0
$43$	−393.268	−1.39472	−0.697359	−	0.716722i	$-0.745642\pi$
$43$	−393.268	−1.39472	−0.697359	+	0.716722i	$0.745642\pi$
$44$	0	0
$45$	24.6431	0.0816352
$46$	0	0
$47$	467.968	1.45234	0.726172	−	0.687513i	$-0.241297\pi$
$47$	467.968	1.45234	0.726172	+	0.687513i	$0.241297\pi$
$48$	0	0
$49$	−284.462	−0.829336
$50$	0	0
$51$	−642.393	−1.76378
$52$	0	0
$53$	−253.167	−0.656134	−0.328067	−	0.944654i	$-0.606397\pi$
$53$	−253.167	−0.656134	−0.328067	+	0.944654i	$0.606397\pi$
$54$	0	0
$55$	19.0511	0.0467064
$56$	0	0
$57$	418.322	0.972072
$58$	0	0
$59$	−850.800	−1.87737	−0.938685	−	0.344777i	$-0.887955\pi$
$59$	−850.800	−1.87737	−0.938685	+	0.344777i	$0.887955\pi$
$60$	0	0
$61$	−176.962	−0.371437	−0.185719	−	0.982603i	$-0.559461\pi$
$61$	−176.962	−0.371437	−0.185719	+	0.982603i	$0.559461\pi$
$62$	0	0
$63$	24.0022	0.0480000
$64$	0	0
$65$	−490.206	−0.935424
$66$	0	0
$67$	684.006	1.24723	0.623616	−	0.781731i	$-0.285663\pi$
$67$	684.006	1.24723	0.623616	+	0.781731i	$0.285663\pi$
$68$	0	0
$69$	−126.264	−0.220295
$70$	0	0
$71$	1110.22	1.85576	0.927879	−	0.372883i	$-0.121631\pi$
$71$	1110.22	1.85576	0.927879	+	0.372883i	$0.121631\pi$
$72$	0	0
$73$	−510.525	−0.818527	−0.409264	−	0.912416i	$-0.634214\pi$
$73$	−510.525	−0.818527	−0.409264	+	0.912416i	$0.634214\pi$
$74$	0	0
$75$	−347.470	−0.534965
$76$	0	0
$77$	18.5556	0.0274625
$78$	0	0
$79$	−535.994	−0.763343	−0.381671	−	0.924298i	$-0.624651\pi$
$79$	−535.994	−0.763343	−0.381671	+	0.924298i	$0.624651\pi$
$80$	0	0
$81$	−803.861	−1.10269
$82$	0	0
$83$	−323.934	−0.428390	−0.214195	−	0.976791i	$-0.568713\pi$
$83$	−323.934	−0.428390	−0.214195	+	0.976791i	$0.568713\pi$
$84$	0	0
$85$	−919.204	−1.17296
$86$	0	0
$87$	−215.509	−0.265575
$88$	0	0
$89$	327.742	0.390343	0.195172	−	0.980769i	$-0.437474\pi$
$89$	327.742	0.390343	0.195172	+	0.980769i	$0.437474\pi$
$90$	0	0
$91$	−477.457	−0.550012
$92$	0	0
$93$	−943.095	−1.05155
$94$	0	0
$95$	598.579	0.646452
$96$	0	0
$97$	1658.33	1.73585	0.867926	−	0.496693i	$-0.165452\pi$
$97$	1658.33	1.73585	0.867926	+	0.496693i	$0.165452\pi$
$98$	0	0
$99$	7.60838	0.00772395
$100$	0	0
$101$	446.685	0.440067	0.220034	−	0.975492i	$-0.429383\pi$
$101$	446.685	0.440067	0.220034	+	0.975492i	$0.429383\pi$
$102$	0	0
$103$	281.783	0.269563	0.134781	−	0.990875i	$-0.456967\pi$
$103$	281.783	0.269563	0.134781	+	0.990875i	$0.456967\pi$
$104$	0	0
$105$	329.937	0.306653
$106$	0	0
$107$	−308.125	−0.278389	−0.139194	−	0.990265i	$-0.544451\pi$
$107$	−308.125	−0.278389	−0.139194	+	0.990265i	$0.544451\pi$
$108$	0	0
$109$	−308.232	−0.270856	−0.135428	−	0.990787i	$-0.543241\pi$
$109$	−308.232	−0.270856	−0.135428	+	0.990787i	$0.543241\pi$
$110$	0	0
$111$	1538.92	1.31593
$112$	0	0
$113$	1988.11	1.65509	0.827547	−	0.561396i	$-0.189736\pi$
$113$	1988.11	1.65509	0.827547	+	0.561396i	$0.189736\pi$
$114$	0	0
$115$	−180.672	−0.146502
$116$	0	0
$117$	−195.772	−0.154693
$118$	0	0
$119$	−895.298	−0.689679
$120$	0	0
$121$	−1325.12	−0.995581
$122$	0	0
$123$	−1538.89	−1.12810
$124$	0	0
$125$	−1479.11	−1.05836
$126$	0	0
$127$	1004.98	0.702184	0.351092	−	0.936341i	$-0.385810\pi$
$127$	1004.98	0.702184	0.351092	+	0.936341i	$0.385810\pi$
$128$	0	0
$129$	−2158.94	−1.47352
$130$	0	0
$131$	−2395.25	−1.59751	−0.798754	−	0.601658i	$-0.794507\pi$
$131$	−2395.25	−1.59751	−0.798754	+	0.601658i	$0.794507\pi$
$132$	0	0
$133$	583.012	0.380102
$134$	0	0
$135$	−1029.05	−0.656047
$136$	0	0
$137$	2300.47	1.43462	0.717309	−	0.696755i	$-0.245374\pi$
$137$	2300.47	1.43462	0.717309	+	0.696755i	$0.245374\pi$
$138$	0	0
$139$	1229.62	0.750323	0.375162	−	0.926959i	$-0.377587\pi$
$139$	1229.62	0.750323	0.375162	+	0.926959i	$0.377587\pi$
$140$	0	0
$141$	2569.02	1.53440
$142$	0	0
$143$	−151.347	−0.0885056
$144$	0	0
$145$	−308.373	−0.176614
$146$	0	0
$147$	−1561.62	−0.876193
$148$	0	0
$149$	−531.897	−0.292448	−0.146224	−	0.989252i	$-0.546712\pi$
$149$	−531.897	−0.292448	−0.146224	+	0.989252i	$0.546712\pi$
$150$	0	0
$151$	−844.608	−0.455187	−0.227594	−	0.973756i	$-0.573086\pi$
$151$	−844.608	−0.455187	−0.227594	+	0.973756i	$0.573086\pi$
$152$	0	0
$153$	−367.099	−0.193975
$154$	0	0
$155$	−1349.48	−0.699309
$156$	0	0
$157$	492.127	0.250166	0.125083	−	0.992146i	$-0.460080\pi$
$157$	492.127	0.250166	0.125083	+	0.992146i	$0.460080\pi$
$158$	0	0
$159$	−1389.82	−0.693205
$160$	0	0
$161$	−175.973	−0.0861404
$162$	0	0
$163$	−1380.19	−0.663220	−0.331610	−	0.943417i	$-0.607592\pi$
$163$	−1380.19	−0.663220	−0.331610	+	0.943417i	$0.607592\pi$
$164$	0	0
$165$	104.585	0.0493452
$166$	0	0
$167$	−2773.16	−1.28499	−0.642495	−	0.766290i	$-0.722101\pi$
$167$	−2773.16	−1.28499	−0.642495	+	0.766290i	$0.722101\pi$
$168$	0	0
$169$	1697.33	0.772569
$170$	0	0
$171$	239.053	0.106905
$172$	0	0
$173$	480.280	0.211069	0.105535	−	0.994416i	$-0.466345\pi$
$173$	480.280	0.211069	0.105535	+	0.994416i	$0.466345\pi$
$174$	0	0
$175$	−484.266	−0.209183
$176$	0	0
$177$	−4670.67	−1.98344
$178$	0	0
$179$	1929.00	0.805476	0.402738	−	0.915315i	$-0.368059\pi$
$179$	1929.00	0.805476	0.402738	+	0.915315i	$0.368059\pi$
$180$	0	0
$181$	−1779.61	−0.730815	−0.365407	−	0.930848i	$-0.619070\pi$
$181$	−1779.61	−0.730815	−0.365407	+	0.930848i	$0.619070\pi$
$182$	0	0
$183$	−971.474	−0.392423
$184$	0	0
$185$	2202.05	0.875123
$186$	0	0
$187$	−283.797	−0.110980
$188$	0	0
$189$	−1002.29	−0.385743
$190$	0	0
$191$	3234.01	1.22516	0.612578	−	0.790410i	$-0.290132\pi$
$191$	3234.01	1.22516	0.612578	+	0.790410i	$0.290132\pi$
$192$	0	0
$193$	−2011.95	−0.750378	−0.375189	−	0.926948i	$-0.622422\pi$
$193$	−2011.95	−0.750378	−0.375189	+	0.926948i	$0.622422\pi$
$194$	0	0
$195$	−2691.10	−0.988275
$196$	0	0
$197$	1638.43	0.592553	0.296277	−	0.955102i	$-0.404255\pi$
$197$	1638.43	0.592553	0.296277	+	0.955102i	$0.404255\pi$
$198$	0	0
$199$	4660.98	1.66034	0.830171	−	0.557508i	$-0.188243\pi$
$199$	4660.98	1.66034	0.830171	+	0.557508i	$0.188243\pi$
$200$	0	0
$201$	3755.01	1.31770
$202$	0	0
$203$	−300.353	−0.103846
$204$	0	0
$205$	−2202.00	−0.750217
$206$	0	0
$207$	−72.1542	−0.0242274
$208$	0	0
$209$	184.807	0.0611643
$210$	0	0
$211$	−3479.16	−1.13514	−0.567571	−	0.823324i	$-0.692117\pi$
$211$	−3479.16	−1.13514	−0.567571	+	0.823324i	$0.692117\pi$
$212$	0	0
$213$	6094.80	1.96061
$214$	0	0
$215$	−3089.24	−0.979926
$216$	0	0
$217$	−1314.38	−0.411181
$218$	0	0
$219$	−2802.65	−0.864773
$220$	0	0
$221$	7302.41	2.22269
$222$	0	0
$223$	129.110	0.0387706	0.0193853	−	0.999812i	$-0.493829\pi$
$223$	129.110	0.0387706	0.0193853	+	0.999812i	$0.493829\pi$
$224$	0	0
$225$	−198.564	−0.0588337
$226$	0	0
$227$	−1345.95	−0.393542	−0.196771	−	0.980449i	$-0.563046\pi$
$227$	−1345.95	−0.393542	−0.196771	+	0.980449i	$0.563046\pi$
$228$	0	0
$229$	3539.19	1.02129	0.510647	−	0.859791i	$-0.329406\pi$
$229$	3539.19	1.02129	0.510647	+	0.859791i	$0.329406\pi$
$230$	0	0
$231$	101.865	0.0290141
$232$	0	0
$233$	709.902	0.199602	0.0998009	−	0.995007i	$-0.468179\pi$
$233$	709.902	0.199602	0.0998009	+	0.995007i	$0.468179\pi$
$234$	0	0
$235$	3676.03	1.02041
$236$	0	0
$237$	−2942.46	−0.806471
$238$	0	0
$239$	−4538.38	−1.22830	−0.614149	−	0.789190i	$-0.710501\pi$
$239$	−4538.38	−1.22830	−0.614149	+	0.789190i	$0.710501\pi$
$240$	0	0
$241$	4621.77	1.23533	0.617665	−	0.786442i	$-0.288079\pi$
$241$	4621.77	1.23533	0.617665	+	0.786442i	$0.288079\pi$
$242$	0	0
$243$	−875.963	−0.231247
$244$	0	0
$245$	−2234.53	−0.582690
$246$	0	0
$247$	−4755.28	−1.22499
$248$	0	0
$249$	−1778.31	−0.452594
$250$	0	0
$251$	−4438.87	−1.11625	−0.558125	−	0.829757i	$-0.688479\pi$
$251$	−4438.87	−1.11625	−0.558125	+	0.829757i	$0.688479\pi$
$252$	0	0
$253$	−55.7810	−0.0138613
$254$	0	0
$255$	−5046.18	−1.23923
$256$	0	0
$257$	−2737.89	−0.664533	−0.332267	−	0.943186i	$-0.607813\pi$
$257$	−2737.89	−0.664533	−0.332267	+	0.943186i	$0.607813\pi$
$258$	0	0
$259$	2144.78	0.514556
$260$	0	0
$261$	−123.154	−0.0292070
$262$	0	0
$263$	−1402.58	−0.328847	−0.164423	−	0.986390i	$-0.552576\pi$
$263$	−1402.58	−0.328847	−0.164423	+	0.986390i	$0.552576\pi$
$264$	0	0
$265$	−1988.70	−0.460999
$266$	0	0
$267$	1799.21	0.412397
$268$	0	0
$269$	−6170.17	−1.39852	−0.699260	−	0.714868i	$-0.746487\pi$
$269$	−6170.17	−1.39852	−0.699260	+	0.714868i	$0.746487\pi$
$270$	0	0
$271$	−7394.68	−1.65755	−0.828773	−	0.559586i	$-0.810960\pi$
$271$	−7394.68	−1.65755	−0.828773	+	0.559586i	$0.810960\pi$
$272$	0	0
$273$	−2621.11	−0.581087
$274$	0	0
$275$	−153.506	−0.0336608
$276$	0	0
$277$	6011.84	1.30403	0.652015	−	0.758206i	$-0.273924\pi$
$277$	6011.84	1.30403	0.652015	+	0.758206i	$0.273924\pi$
$278$	0	0
$279$	−538.938	−0.115646
$280$	0	0
$281$	881.575	0.187154	0.0935772	−	0.995612i	$-0.470170\pi$
$281$	881.575	0.187154	0.0935772	+	0.995612i	$0.470170\pi$
$282$	0	0
$283$	−105.593	−0.0221796	−0.0110898	−	0.999939i	$-0.503530\pi$
$283$	−105.593	−0.0221796	−0.0110898	+	0.999939i	$0.503530\pi$
$284$	0	0
$285$	3286.04	0.682976
$286$	0	0
$287$	−2144.73	−0.441114
$288$	0	0
$289$	8780.03	1.78710
$290$	0	0
$291$	9103.78	1.83393
$292$	0	0
$293$	2713.76	0.541091	0.270545	−	0.962707i	$-0.412796\pi$
$293$	2713.76	0.541091	0.270545	+	0.962707i	$0.412796\pi$
$294$	0	0
$295$	−6683.28	−1.31904
$296$	0	0
$297$	−317.711	−0.0620722
$298$	0	0
$299$	1435.31	0.277612
$300$	0	0
$301$	−3008.89	−0.576179
$302$	0	0
$303$	2452.18	0.464930
$304$	0	0
$305$	−1390.09	−0.260971
$306$	0	0
$307$	−7196.59	−1.33789	−0.668943	−	0.743314i	$-0.733253\pi$
$307$	−7196.59	−1.33789	−0.668943	+	0.743314i	$0.733253\pi$
$308$	0	0
$309$	1546.92	0.284793
$310$	0	0
$311$	−6355.15	−1.15874	−0.579369	−	0.815066i	$-0.696701\pi$
$311$	−6355.15	−1.15874	−0.579369	+	0.815066i	$0.696701\pi$
$312$	0	0
$313$	9574.89	1.72909	0.864544	−	0.502557i	$-0.167607\pi$
$313$	9574.89	1.72909	0.864544	+	0.502557i	$0.167607\pi$
$314$	0	0
$315$	188.544	0.0337247
$316$	0	0
$317$	−9519.61	−1.68667	−0.843335	−	0.537388i	$-0.819411\pi$
$317$	−9519.61	−1.68667	−0.843335	+	0.537388i	$0.819411\pi$
$318$	0	0
$319$	−95.2078	−0.0167104
$320$	0	0
$321$	−1691.52	−0.294117
$322$	0	0
$323$	−8916.81	−1.53605
$324$	0	0
$325$	3949.87	0.674151
$326$	0	0
$327$	−1692.11	−0.286159
$328$	0	0
$329$	3580.42	0.599985
$330$	0	0
$331$	3307.29	0.549200	0.274600	−	0.961558i	$-0.411455\pi$
$331$	3307.29	0.549200	0.274600	+	0.961558i	$0.411455\pi$
$332$	0	0
$333$	879.425	0.144721
$334$	0	0
$335$	5373.06	0.876303
$336$	0	0
$337$	−5207.02	−0.841674	−0.420837	−	0.907136i	$-0.638264\pi$
$337$	−5207.02	−0.841674	−0.420837	+	0.907136i	$0.638264\pi$
$338$	0	0
$339$	10914.2	1.74861
$340$	0	0
$341$	−416.642	−0.0661654
$342$	0	0
$343$	−4800.71	−0.755726
$344$	0	0
$345$	−991.838	−0.154779
$346$	0	0
$347$	1463.16	0.226358	0.113179	−	0.993575i	$-0.463897\pi$
$347$	1463.16	0.226358	0.113179	+	0.993575i	$0.463897\pi$
$348$	0	0
$349$	−4445.77	−0.681882	−0.340941	−	0.940085i	$-0.610746\pi$
$349$	−4445.77	−0.681882	−0.340941	+	0.940085i	$0.610746\pi$
$350$	0	0
$351$	8175.05	1.24317
$352$	0	0
$353$	−1260.86	−0.190110	−0.0950548	−	0.995472i	$-0.530303\pi$
$353$	−1260.86	−0.190110	−0.0950548	+	0.995472i	$0.530303\pi$
$354$	0	0
$355$	8721.09	1.30385
$356$	0	0
$357$	−4914.94	−0.728645
$358$	0	0
$359$	2402.90	0.353260	0.176630	−	0.984277i	$-0.443480\pi$
$359$	2402.90	0.353260	0.176630	+	0.984277i	$0.443480\pi$
$360$	0	0
$361$	−1052.44	−0.153439
$362$	0	0
$363$	−7274.54	−1.05183
$364$	0	0
$365$	−4010.32	−0.575096
$366$	0	0
$367$	−11147.0	−1.58548	−0.792739	−	0.609561i	$-0.791346\pi$
$367$	−11147.0	−1.58548	−0.792739	+	0.609561i	$0.791346\pi$
$368$	0	0
$369$	−879.407	−0.124065
$370$	0	0
$371$	−1936.98	−0.271059
$372$	0	0
$373$	−5613.44	−0.779230	−0.389615	−	0.920978i	$-0.627392\pi$
$373$	−5613.44	−0.779230	−0.389615	+	0.920978i	$0.627392\pi$
$374$	0	0
$375$	−8119.90	−1.11816
$376$	0	0
$377$	2449.80	0.334672
$378$	0	0
$379$	6192.90	0.839334	0.419667	−	0.907678i	$-0.362147\pi$
$379$	6192.90	0.839334	0.419667	+	0.907678i	$0.362147\pi$
$380$	0	0
$381$	5517.06	0.741857
$382$	0	0
$383$	−7459.94	−0.995261	−0.497631	−	0.867389i	$-0.665797\pi$
$383$	−7459.94	−0.995261	−0.497631	+	0.867389i	$0.665797\pi$
$384$	0	0
$385$	145.760	0.0192951
$386$	0	0
$387$	−1233.74	−0.162053
$388$	0	0
$389$	6925.74	0.902697	0.451349	−	0.892348i	$-0.350943\pi$
$389$	6925.74	0.902697	0.451349	+	0.892348i	$0.350943\pi$
$390$	0	0
$391$	2691.40	0.348107
$392$	0	0
$393$	−13149.3	−1.68777
$394$	0	0
$395$	−4210.39	−0.536323
$396$	0	0
$397$	−8893.75	−1.12434	−0.562172	−	0.827021i	$-0.690034\pi$
$397$	−8893.75	−1.12434	−0.562172	+	0.827021i	$0.690034\pi$
$398$	0	0
$399$	3200.58	0.401577
$400$	0	0
$401$	−3648.18	−0.454317	−0.227159	−	0.973858i	$-0.572944\pi$
$401$	−3648.18	−0.454317	−0.227159	+	0.973858i	$0.572944\pi$
$402$	0	0
$403$	10720.6	1.32515
$404$	0	0
$405$	−6314.56	−0.774748
$406$	0	0
$407$	679.865	0.0828002
$408$	0	0
$409$	5118.89	0.618858	0.309429	−	0.950923i	$-0.399862\pi$
$409$	5118.89	0.618858	0.309429	+	0.950923i	$0.399862\pi$
$410$	0	0
$411$	12629.0	1.51567
$412$	0	0
$413$	−6509.47	−0.775569
$414$	0	0
$415$	−2544.59	−0.300986
$416$	0	0
$417$	6750.28	0.792716
$418$	0	0
$419$	−8405.19	−0.980001	−0.490001	−	0.871722i	$-0.663003\pi$
$419$	−8405.19	−0.980001	−0.490001	+	0.871722i	$0.663003\pi$
$420$	0	0
$421$	−8606.11	−0.996285	−0.498142	−	0.867095i	$-0.665984\pi$
$421$	−8606.11	−0.996285	−0.498142	+	0.867095i	$0.665984\pi$
$422$	0	0
$423$	1468.08	0.168748
$424$	0	0
$425$	7406.54	0.845342
$426$	0	0
$427$	−1353.94	−0.153446
$428$	0	0
$429$	−830.856	−0.0935061
$430$	0	0
$431$	15070.8	1.68430	0.842150	−	0.539243i	$-0.181290\pi$
$431$	15070.8	1.68430	0.842150	+	0.539243i	$0.181290\pi$
$432$	0	0
$433$	5727.65	0.635689	0.317845	−	0.948143i	$-0.397041\pi$
$433$	5727.65	0.635689	0.317845	+	0.948143i	$0.397041\pi$
$434$	0	0
$435$	−1692.88	−0.186592
$436$	0	0
$437$	−1752.62	−0.191852
$438$	0	0
$439$	2314.32	0.251609	0.125805	−	0.992055i	$-0.459849\pi$
$439$	2314.32	0.251609	0.125805	+	0.992055i	$0.459849\pi$
$440$	0	0
$441$	−892.398	−0.0963609
$442$	0	0
$443$	4086.12	0.438234	0.219117	−	0.975699i	$-0.429682\pi$
$443$	4086.12	0.438234	0.219117	+	0.975699i	$0.429682\pi$
$444$	0	0
$445$	2574.50	0.274254
$446$	0	0
$447$	−2919.97	−0.308971
$448$	0	0
$449$	−4703.04	−0.494321	−0.247161	−	0.968975i	$-0.579498\pi$
$449$	−4703.04	−0.494321	−0.247161	+	0.968975i	$0.579498\pi$
$450$	0	0
$451$	−679.851	−0.0709821
$452$	0	0
$453$	−4636.67	−0.480905
$454$	0	0
$455$	−3750.56	−0.386438
$456$	0	0
$457$	−17069.8	−1.74724	−0.873621	−	0.486607i	$-0.838234\pi$
$457$	−17069.8	−1.74724	−0.873621	+	0.486607i	$0.838234\pi$
$458$	0	0
$459$	15329.3	1.55885
$460$	0	0
$461$	18748.8	1.89419	0.947093	−	0.320958i	$-0.104005\pi$
$461$	18748.8	1.89419	0.947093	+	0.320958i	$0.104005\pi$
$462$	0	0
$463$	8363.76	0.839518	0.419759	−	0.907636i	$-0.362115\pi$
$463$	8363.76	0.839518	0.419759	+	0.907636i	$0.362115\pi$
$464$	0	0
$465$	−7408.28	−0.738819
$466$	0	0
$467$	−16791.3	−1.66383	−0.831916	−	0.554902i	$-0.812756\pi$
$467$	−16791.3	−1.66383	−0.831916	+	0.554902i	$0.812756\pi$
$468$	0	0
$469$	5233.32	0.515250
$470$	0	0
$471$	2701.64	0.264300
$472$	0	0
$473$	−953.778	−0.0927162
$474$	0	0
$475$	−4823.09	−0.465892
$476$	0	0
$477$	−794.220	−0.0762365
$478$	0	0
$479$	16933.3	1.61524	0.807621	−	0.589702i	$-0.200755\pi$
$479$	16933.3	1.61524	0.807621	+	0.589702i	$0.200755\pi$
$480$	0	0
$481$	−17493.7	−1.65830
$482$	0	0
$483$	−966.043	−0.0910072
$484$	0	0
$485$	13026.6	1.21961
$486$	0	0
$487$	10890.1	1.01330	0.506648	−	0.862153i	$-0.330884\pi$
$487$	10890.1	1.01330	0.506648	+	0.862153i	$0.330884\pi$
$488$	0	0
$489$	−7576.87	−0.700691
$490$	0	0
$491$	19997.8	1.83806	0.919029	−	0.394190i	$-0.128975\pi$
$491$	19997.8	1.83806	0.919029	+	0.394190i	$0.128975\pi$
$492$	0	0
$493$	4593.72	0.419656
$494$	0	0
$495$	59.7660	0.00542683
$496$	0	0
$497$	8494.28	0.766641
$498$	0	0
$499$	−17327.8	−1.55451	−0.777254	−	0.629187i	$-0.783388\pi$
$499$	−17327.8	−1.55451	−0.777254	+	0.629187i	$0.783388\pi$
$500$	0	0
$501$	−15223.9	−1.35759
$502$	0	0
$503$	−12292.5	−1.08965	−0.544824	−	0.838550i	$-0.683404\pi$
$503$	−12292.5	−1.08965	−0.544824	+	0.838550i	$0.683404\pi$
$504$	0	0
$505$	3508.83	0.309190
$506$	0	0
$507$	9317.91	0.816219
$508$	0	0
$509$	−15951.6	−1.38908	−0.694540	−	0.719454i	$-0.744392\pi$
$509$	−15951.6	−1.38908	−0.694540	+	0.719454i	$0.744392\pi$
$510$	0	0
$511$	−3906.03	−0.338146
$512$	0	0
$513$	−9982.36	−0.859126
$514$	0	0
$515$	2213.49	0.189394
$516$	0	0
$517$	1134.94	0.0965470
$518$	0	0
$519$	2636.61	0.222995
$520$	0	0
$521$	−9467.89	−0.796153	−0.398077	−	0.917352i	$-0.630322\pi$
$521$	−9467.89	−0.796153	−0.398077	+	0.917352i	$0.630322\pi$
$522$	0	0
$523$	−4081.75	−0.341266	−0.170633	−	0.985335i	$-0.554581\pi$
$523$	−4081.75	−0.341266	−0.170633	+	0.985335i	$0.554581\pi$
$524$	0	0
$525$	−2658.49	−0.221002
$526$	0	0
$527$	20102.7	1.66165
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	−2669.08	−0.218132
$532$	0	0
$533$	17493.3	1.42161
$534$	0	0
$535$	−2420.41	−0.195595
$536$	0	0
$537$	10589.7	0.850985
$538$	0	0
$539$	−689.895	−0.0551315
$540$	0	0
$541$	−4111.48	−0.326740	−0.163370	−	0.986565i	$-0.552236\pi$
$541$	−4111.48	−0.326740	−0.163370	+	0.986565i	$0.552236\pi$
$542$	0	0
$543$	−9769.59	−0.772105
$544$	0	0
$545$	−2421.25	−0.190303
$546$	0	0
$547$	12040.9	0.941189	0.470595	−	0.882350i	$-0.344039\pi$
$547$	12040.9	0.941189	0.470595	+	0.882350i	$0.344039\pi$
$548$	0	0
$549$	−555.155	−0.0431574
$550$	0	0
$551$	−2991.40	−0.231285
$552$	0	0
$553$	−4100.89	−0.315348
$554$	0	0
$555$	12088.6	0.924567
$556$	0	0
$557$	5052.74	0.384365	0.192183	−	0.981359i	$-0.438443\pi$
$557$	5052.74	0.384365	0.192183	+	0.981359i	$0.438443\pi$
$558$	0	0
$559$	24541.8	1.85690
$560$	0	0
$561$	−1557.97	−0.117250
$562$	0	0
$563$	985.606	0.0737803	0.0368902	−	0.999319i	$-0.488255\pi$
$563$	985.606	0.0737803	0.0368902	+	0.999319i	$0.488255\pi$
$564$	0	0
$565$	15617.2	1.16287
$566$	0	0
$567$	−6150.34	−0.455538
$568$	0	0
$569$	11356.1	0.836681	0.418340	−	0.908290i	$-0.362612\pi$
$569$	11356.1	0.836681	0.418340	+	0.908290i	$0.362612\pi$
$570$	0	0
$571$	7142.79	0.523496	0.261748	−	0.965136i	$-0.415701\pi$
$571$	7142.79	0.523496	0.261748	+	0.965136i	$0.415701\pi$
$572$	0	0
$573$	17753.9	1.29438
$574$	0	0
$575$	1455.77	0.105583
$576$	0	0
$577$	16039.3	1.15723	0.578617	−	0.815599i	$-0.303593\pi$
$577$	16039.3	1.15723	0.578617	+	0.815599i	$0.303593\pi$
$578$	0	0
$579$	−11045.0	−0.792774
$580$	0	0
$581$	−2478.42	−0.176974
$582$	0	0
$583$	−613.995	−0.0436176
$584$	0	0
$585$	−1537.85	−0.108687
$586$	0	0
$587$	1617.10	0.113705	0.0568527	−	0.998383i	$-0.481893\pi$
$587$	1617.10	0.113705	0.0568527	+	0.998383i	$0.481893\pi$
$588$	0	0
$589$	−13090.7	−0.915780
$590$	0	0
$591$	8994.51	0.626032
$592$	0	0
$593$	−1874.90	−0.129836	−0.0649182	−	0.997891i	$-0.520679\pi$
$593$	−1874.90	−0.129836	−0.0649182	+	0.997891i	$0.520679\pi$
$594$	0	0
$595$	−7032.82	−0.484567
$596$	0	0
$597$	25587.5	1.75415
$598$	0	0
$599$	13139.5	0.896270	0.448135	−	0.893966i	$-0.352088\pi$
$599$	13139.5	0.896270	0.448135	+	0.893966i	$0.352088\pi$
$600$	0	0
$601$	11841.7	0.803718	0.401859	−	0.915701i	$-0.368364\pi$
$601$	11841.7	0.803718	0.401859	+	0.915701i	$0.368364\pi$
$602$	0	0
$603$	2145.82	0.144916
$604$	0	0
$605$	−10409.2	−0.699493
$606$	0	0
$607$	−11573.8	−0.773915	−0.386958	−	0.922097i	$-0.626474\pi$
$607$	−11573.8	−0.773915	−0.386958	+	0.922097i	$0.626474\pi$
$608$	0	0
$609$	−1648.86	−0.109713
$610$	0	0
$611$	−29203.4	−1.93362
$612$	0	0
$613$	1661.20	0.109454	0.0547270	−	0.998501i	$-0.482571\pi$
$613$	1661.20	0.109454	0.0547270	+	0.998501i	$0.482571\pi$
$614$	0	0
$615$	−12088.4	−0.792604
$616$	0	0
$617$	9195.09	0.599968	0.299984	−	0.953944i	$-0.403019\pi$
$617$	9195.09	0.599968	0.299984	+	0.953944i	$0.403019\pi$
$618$	0	0
$619$	−14699.0	−0.954448	−0.477224	−	0.878782i	$-0.658357\pi$
$619$	−14699.0	−0.954448	−0.477224	+	0.878782i	$0.658357\pi$
$620$	0	0
$621$	3013.02	0.194699
$622$	0	0
$623$	2507.55	0.161257
$624$	0	0
$625$	−3707.00	−0.237248
$626$	0	0
$627$	1014.54	0.0646201
$628$	0	0
$629$	−32803.1	−2.07940
$630$	0	0
$631$	−1488.43	−0.0939043	−0.0469522	−	0.998897i	$-0.514951\pi$
$631$	−1488.43	−0.0939043	−0.0469522	+	0.998897i	$0.514951\pi$
$632$	0	0
$633$	−19099.6	−1.19928
$634$	0	0
$635$	7894.39	0.493353
$636$	0	0
$637$	17751.8	1.10416
$638$	0	0
$639$	3482.91	0.215621
$640$	0	0
$641$	13182.6	0.812296	0.406148	−	0.913807i	$-0.366872\pi$
$641$	13182.6	0.812296	0.406148	+	0.913807i	$0.366872\pi$
$642$	0	0
$643$	−15982.1	−0.980203	−0.490102	−	0.871665i	$-0.663040\pi$
$643$	−15982.1	−0.980203	−0.490102	+	0.871665i	$0.663040\pi$
$644$	0	0
$645$	−16959.1	−1.03529
$646$	0	0
$647$	20355.8	1.23689	0.618447	−	0.785826i	$-0.287762\pi$
$647$	20355.8	1.23689	0.618447	+	0.785826i	$0.287762\pi$
$648$	0	0
$649$	−2063.41	−0.124801
$650$	0	0
$651$	−7215.61	−0.434412
$652$	0	0
$653$	3084.67	0.184858	0.0924291	−	0.995719i	$-0.470537\pi$
$653$	3084.67	0.184858	0.0924291	+	0.995719i	$0.470537\pi$
$654$	0	0
$655$	−18815.3	−1.12241
$656$	0	0
$657$	−1601.59	−0.0951050
$658$	0	0
$659$	−2056.48	−0.121561	−0.0607807	−	0.998151i	$-0.519359\pi$
$659$	−2056.48	−0.121561	−0.0607807	+	0.998151i	$0.519359\pi$
$660$	0	0
$661$	−17916.9	−1.05429	−0.527145	−	0.849775i	$-0.676737\pi$
$661$	−17916.9	−1.05429	−0.527145	+	0.849775i	$0.676737\pi$
$662$	0	0
$663$	40088.3	2.34827
$664$	0	0
$665$	4579.72	0.267059
$666$	0	0
$667$	902.905	0.0524147
$668$	0	0
$669$	708.780	0.0409611
$670$	0	0
$671$	−429.179	−0.0246919
$672$	0	0
$673$	10308.1	0.590416	0.295208	−	0.955433i	$-0.404611\pi$
$673$	10308.1	0.590416	0.295208	+	0.955433i	$0.404611\pi$
$674$	0	0
$675$	8291.62	0.472807
$676$	0	0
$677$	−357.650	−0.0203037	−0.0101518	−	0.999948i	$-0.503231\pi$
$677$	−357.650	−0.0203037	−0.0101518	+	0.999948i	$0.503231\pi$
$678$	0	0
$679$	12687.9	0.717107
$680$	0	0
$681$	−7388.92	−0.415777
$682$	0	0
$683$	−1489.89	−0.0834688	−0.0417344	−	0.999129i	$-0.513288\pi$
$683$	−1489.89	−0.0834688	−0.0417344	+	0.999129i	$0.513288\pi$
$684$	0	0
$685$	18070.9	1.00796
$686$	0	0
$687$	19429.2	1.07900
$688$	0	0
$689$	15798.8	0.873563
$690$	0	0
$691$	3520.91	0.193838	0.0969189	−	0.995292i	$-0.469101\pi$
$691$	3520.91	0.193838	0.0969189	+	0.995292i	$0.469101\pi$
$692$	0	0
$693$	58.2117	0.00319088
$694$	0	0
$695$	9659.01	0.527176
$696$	0	0
$697$	32802.4	1.78261
$698$	0	0
$699$	3897.17	0.210879
$700$	0	0
$701$	31535.2	1.69910	0.849548	−	0.527511i	$-0.176875\pi$
$701$	31535.2	1.69910	0.849548	+	0.527511i	$0.176875\pi$
$702$	0	0
$703$	21361.1	1.14602
$704$	0	0
$705$	20180.4	1.07807
$706$	0	0
$707$	3417.58	0.181798
$708$	0	0
$709$	−8182.42	−0.433424	−0.216712	−	0.976236i	$-0.569533\pi$
$709$	−8182.42	−0.433424	−0.216712	+	0.976236i	$0.569533\pi$
$710$	0	0
$711$	−1681.49	−0.0886931
$712$	0	0
$713$	3951.23	0.207538
$714$	0	0
$715$	−1188.88	−0.0621839
$716$	0	0
$717$	−24914.5	−1.29770
$718$	0	0
$719$	29878.1	1.54974	0.774872	−	0.632119i	$-0.217814\pi$
$719$	29878.1	1.54974	0.774872	+	0.632119i	$0.217814\pi$
$720$	0	0
$721$	2155.92	0.111360
$722$	0	0
$723$	25372.3	1.30512
$724$	0	0
$725$	2484.74	0.127284
$726$	0	0
$727$	7403.24	0.377677	0.188838	−	0.982008i	$-0.439528\pi$
$727$	7403.24	0.377677	0.188838	+	0.982008i	$0.439528\pi$
$728$	0	0
$729$	16895.4	0.858378
$730$	0	0
$731$	46019.2	2.32843
$732$	0	0
$733$	33897.3	1.70808	0.854042	−	0.520204i	$-0.174144\pi$
$733$	33897.3	1.70808	0.854042	+	0.520204i	$0.174144\pi$
$734$	0	0
$735$	−12267.0	−0.615612
$736$	0	0
$737$	1658.89	0.0829119
$738$	0	0
$739$	−16685.2	−0.830546	−0.415273	−	0.909697i	$-0.636314\pi$
$739$	−16685.2	−0.830546	−0.415273	+	0.909697i	$0.636314\pi$
$740$	0	0
$741$	−26105.2	−1.29420
$742$	0	0
$743$	1995.90	0.0985500	0.0492750	−	0.998785i	$-0.484309\pi$
$743$	1995.90	0.0985500	0.0492750	+	0.998785i	$0.484309\pi$
$744$	0	0
$745$	−4178.20	−0.205473
$746$	0	0
$747$	−1016.23	−0.0497748
$748$	0	0
$749$	−2357.46	−0.115007
$750$	0	0
$751$	11144.2	0.541487	0.270743	−	0.962652i	$-0.412731\pi$
$751$	11144.2	0.541487	0.270743	+	0.962652i	$0.412731\pi$
$752$	0	0
$753$	−24368.2	−1.17932
$754$	0	0
$755$	−6634.64	−0.319814
$756$	0	0
$757$	5734.32	0.275320	0.137660	−	0.990480i	$-0.456042\pi$
$757$	5734.32	0.275320	0.137660	+	0.990480i	$0.456042\pi$
$758$	0	0
$759$	−306.222	−0.0146445
$760$	0	0
$761$	−27665.0	−1.31781	−0.658907	−	0.752224i	$-0.728981\pi$
$761$	−27665.0	−1.31781	−0.658907	+	0.752224i	$0.728981\pi$
$762$	0	0
$763$	−2358.28	−0.111895
$764$	0	0
$765$	−2883.67	−0.136287
$766$	0	0
$767$	53093.9	2.49949
$768$	0	0
$769$	19614.7	0.919798	0.459899	−	0.887971i	$-0.347886\pi$
$769$	19614.7	0.919798	0.459899	+	0.887971i	$0.347886\pi$
$770$	0	0
$771$	−15030.3	−0.702079
$772$	0	0
$773$	18009.7	0.837989	0.418994	−	0.907989i	$-0.362383\pi$
$773$	18009.7	0.837989	0.418994	+	0.907989i	$0.362383\pi$
$774$	0	0
$775$	10873.5	0.503985
$776$	0	0
$777$	11774.3	0.543628
$778$	0	0
$779$	−21360.7	−0.982447
$780$	0	0
$781$	2692.57	0.123365
$782$	0	0
$783$	5142.66	0.234717
$784$	0	0
$785$	3865.80	0.175766
$786$	0	0
$787$	1306.98	0.0591981	0.0295991	−	0.999562i	$-0.490577\pi$
$787$	1306.98	0.0591981	0.0295991	+	0.999562i	$0.490577\pi$
$788$	0	0
$789$	−7699.77	−0.347426
$790$	0	0
$791$	15211.0	0.683744
$792$	0	0
$793$	11043.2	0.494524
$794$	0	0
$795$	−10917.4	−0.487045
$796$	0	0
$797$	824.672	0.0366517	0.0183258	−	0.999832i	$-0.494166\pi$
$797$	824.672	0.0366517	0.0183258	+	0.999832i	$0.494166\pi$
$798$	0	0
$799$	−54760.4	−2.42463
$800$	0	0
$801$	1028.17	0.0453541
$802$	0	0
$803$	−1238.16	−0.0544129
$804$	0	0
$805$	−1382.32	−0.0605221
$806$	0	0
$807$	−33872.6	−1.47753
$808$	0	0
$809$	−6391.38	−0.277761	−0.138881	−	0.990309i	$-0.544350\pi$
$809$	−6391.38	−0.277761	−0.138881	+	0.990309i	$0.544350\pi$
$810$	0	0
$811$	38881.0	1.68347	0.841737	−	0.539887i	$-0.181533\pi$
$811$	38881.0	1.68347	0.841737	+	0.539887i	$0.181533\pi$
$812$	0	0
$813$	−40594.8	−1.75119
$814$	0	0
$815$	−10841.8	−0.465977
$816$	0	0
$817$	−29967.4	−1.28326
$818$	0	0
$819$	−1497.85	−0.0639061
$820$	0	0
$821$	30235.1	1.28528	0.642639	−	0.766169i	$-0.277840\pi$
$821$	30235.1	1.28528	0.642639	+	0.766169i	$0.277840\pi$
$822$	0	0
$823$	−11940.8	−0.505749	−0.252875	−	0.967499i	$-0.581376\pi$
$823$	−11940.8	−0.505749	−0.252875	+	0.967499i	$0.581376\pi$
$824$	0	0
$825$	−842.704	−0.0355627
$826$	0	0
$827$	34537.9	1.45224	0.726118	−	0.687570i	$-0.241322\pi$
$827$	34537.9	1.45224	0.726118	+	0.687570i	$0.241322\pi$
$828$	0	0
$829$	−25718.1	−1.07748	−0.538738	−	0.842473i	$-0.681099\pi$
$829$	−25718.1	−1.07748	−0.538738	+	0.842473i	$0.681099\pi$
$830$	0	0
$831$	33003.4	1.37771
$832$	0	0
$833$	33287.0	1.38454
$834$	0	0
$835$	−21784.0	−0.902832
$836$	0	0
$837$	22504.9	0.929372
$838$	0	0
$839$	−4305.30	−0.177158	−0.0885788	−	0.996069i	$-0.528233\pi$
$839$	−4305.30	−0.177158	−0.0885788	+	0.996069i	$0.528233\pi$
$840$	0	0
$841$	−22847.9	−0.936812
$842$	0	0
$843$	4839.61	0.197728
$844$	0	0
$845$	13333.1	0.542806
$846$	0	0
$847$	−10138.5	−0.411289
$848$	0	0
$849$	−579.675	−0.0234327
$850$	0	0
$851$	−6447.52	−0.259716
$852$	0	0
$853$	−15662.7	−0.628698	−0.314349	−	0.949307i	$-0.601786\pi$
$853$	−15662.7	−0.628698	−0.314349	+	0.949307i	$0.601786\pi$
$854$	0	0
$855$	1877.83	0.0751115
$856$	0	0
$857$	−25676.4	−1.02344	−0.511720	−	0.859152i	$-0.670991\pi$
$857$	−25676.4	−1.02344	−0.511720	+	0.859152i	$0.670991\pi$
$858$	0	0
$859$	−2442.30	−0.0970083	−0.0485041	−	0.998823i	$-0.515445\pi$
$859$	−2442.30	−0.0970083	−0.0485041	+	0.998823i	$0.515445\pi$
$860$	0	0
$861$	−11774.0	−0.466036
$862$	0	0
$863$	−13122.0	−0.517589	−0.258795	−	0.965932i	$-0.583325\pi$
$863$	−13122.0	−0.517589	−0.258795	+	0.965932i	$0.583325\pi$
$864$	0	0
$865$	3772.74	0.148297
$866$	0	0
$867$	48200.0	1.88807
$868$	0	0
$869$	−1299.93	−0.0507445
$870$	0	0
$871$	−42685.1	−1.66054
$872$	0	0
$873$	5202.41	0.201689
$874$	0	0
$875$	−11316.6	−0.437225
$876$	0	0
$877$	43482.6	1.67423	0.837117	−	0.547024i	$-0.184239\pi$
$877$	43482.6	1.67423	0.837117	+	0.547024i	$0.184239\pi$
$878$	0	0
$879$	14897.8	0.571662
$880$	0	0
$881$	−12660.8	−0.484170	−0.242085	−	0.970255i	$-0.577831\pi$
$881$	−12660.8	−0.484170	−0.242085	+	0.970255i	$0.577831\pi$
$882$	0	0
$883$	25125.1	0.957561	0.478780	−	0.877935i	$-0.341079\pi$
$883$	25125.1	0.957561	0.478780	+	0.877935i	$0.341079\pi$
$884$	0	0
$885$	−36689.4	−1.39356
$886$	0	0
$887$	−4437.31	−0.167971	−0.0839854	−	0.996467i	$-0.526765\pi$
$887$	−4437.31	−0.167971	−0.0839854	+	0.996467i	$0.526765\pi$
$888$	0	0
$889$	7689.08	0.290083
$890$	0	0
$891$	−1949.57	−0.0733032
$892$	0	0
$893$	35659.6	1.33628
$894$	0	0
$895$	15152.9	0.565926
$896$	0	0
$897$	7879.44	0.293296
$898$	0	0
$899$	6744.02	0.250195
$900$	0	0
$901$	29624.9	1.09539
$902$	0	0
$903$	−16518.0	−0.608732
$904$	0	0
$905$	−13979.4	−0.513469
$906$	0	0
$907$	37531.4	1.37399	0.686996	−	0.726661i	$-0.258929\pi$
$907$	37531.4	1.37399	0.686996	+	0.726661i	$0.258929\pi$
$908$	0	0
$909$	1401.31	0.0511316
$910$	0	0
$911$	54011.5	1.96431	0.982153	−	0.188086i	$-0.0602284\pi$
$911$	54011.5	1.96431	0.982153	+	0.188086i	$0.0602284\pi$
$912$	0	0
$913$	−785.624	−0.0284779
$914$	0	0
$915$	−7631.21	−0.275716
$916$	0	0
$917$	−18326.0	−0.659954
$918$	0	0
$919$	−41699.0	−1.49676	−0.748381	−	0.663269i	$-0.769168\pi$
$919$	−41699.0	−1.49676	−0.748381	+	0.663269i	$0.769168\pi$
$920$	0	0
$921$	−39507.3	−1.41348
$922$	0	0
$923$	−69282.8	−2.47072
$924$	0	0
$925$	−17743.1	−0.630693
$926$	0	0
$927$	883.994	0.0313206
$928$	0	0
$929$	−2259.47	−0.0797965	−0.0398982	−	0.999204i	$-0.512703\pi$
$929$	−2259.47	−0.0797965	−0.0398982	+	0.999204i	$0.512703\pi$
$930$	0	0
$931$	−21676.3	−0.763062
$932$	0	0
$933$	−34888.1	−1.22421
$934$	0	0
$935$	−2229.31	−0.0779745
$936$	0	0
$937$	−45807.4	−1.59708	−0.798539	−	0.601943i	$-0.794393\pi$
$937$	−45807.4	−1.59708	−0.798539	+	0.601943i	$0.794393\pi$
$938$	0	0
$939$	52563.5	1.82678
$940$	0	0
$941$	−31202.1	−1.08093	−0.540467	−	0.841365i	$-0.681752\pi$
$941$	−31202.1	−1.08093	−0.540467	+	0.841365i	$0.681752\pi$
$942$	0	0
$943$	6447.38	0.222647
$944$	0	0
$945$	−7873.24	−0.271023
$946$	0	0
$947$	−28637.3	−0.982670	−0.491335	−	0.870971i	$-0.663491\pi$
$947$	−28637.3	−0.982670	−0.491335	+	0.870971i	$0.663491\pi$
$948$	0	0
$949$	31859.1	1.08977
$950$	0	0
$951$	−52260.1	−1.78197
$952$	0	0
$953$	42835.7	1.45602	0.728009	−	0.685568i	$-0.240446\pi$
$953$	42835.7	1.45602	0.728009	+	0.685568i	$0.240446\pi$
$954$	0	0
$955$	25404.1	0.860793
$956$	0	0
$957$	−522.665	−0.0176545
$958$	0	0
$959$	17600.9	0.592662
$960$	0	0
$961$	−278.302	−0.00934182
$962$	0	0
$963$	−966.632	−0.0323461
$964$	0	0
$965$	−15804.4	−0.527214
$966$	0	0
$967$	−36655.4	−1.21898	−0.609492	−	0.792792i	$-0.708626\pi$
$967$	−36655.4	−1.21898	−0.609492	+	0.792792i	$0.708626\pi$
$968$	0	0
$969$	−48950.9	−1.62284
$970$	0	0
$971$	−14726.2	−0.486699	−0.243350	−	0.969939i	$-0.578246\pi$
$971$	−14726.2	−0.486699	−0.243350	+	0.969939i	$0.578246\pi$
$972$	0	0
$973$	9407.80	0.309970
$974$	0	0
$975$	21683.7	0.712240
$976$	0	0
$977$	−28358.0	−0.928612	−0.464306	−	0.885675i	$-0.653696\pi$
$977$	−28358.0	−0.928612	−0.464306	+	0.885675i	$0.653696\pi$
$978$	0	0
$979$	794.859	0.0259487
$980$	0	0
$981$	−966.968	−0.0314709
$982$	0	0
$983$	1529.50	0.0496271	0.0248136	−	0.999692i	$-0.492101\pi$
$983$	1529.50	0.0496271	0.0248136	+	0.999692i	$0.492101\pi$
$984$	0	0
$985$	12870.3	0.416327
$986$	0	0
$987$	19655.6	0.633884
$988$	0	0
$989$	9045.17	0.290819
$990$	0	0
$991$	50505.7	1.61894	0.809468	−	0.587164i	$-0.199756\pi$
$991$	50505.7	1.61894	0.809468	+	0.587164i	$0.199756\pi$
$992$	0	0
$993$	18156.2	0.580230
$994$	0	0
$995$	36613.3	1.16655
$996$	0	0
$997$	−3016.06	−0.0958070	−0.0479035	−	0.998852i	$-0.515254\pi$
$997$	−3016.06	−0.0958070	−0.0479035	+	0.998852i	$0.515254\pi$
$998$	0	0
$999$	−36723.0	−1.16303

Twists

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

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

\(f(q)\)	\(=\)	\(q+5.48973 q^{3} +7.85529 q^{5} +7.65099 q^{7} +3.13714 q^{9} +O(q^{10})\)	\(q+5.48973 q^{3} +7.85529 q^{5} +7.65099 q^{7} +3.13714 q^{9} +2.42526 q^{11} -62.4046 q^{13} +43.1234 q^{15} -117.017 q^{17} +76.2008 q^{19} +42.0019 q^{21} -23.0000 q^{23} -63.2945 q^{25} -131.001 q^{27} -39.2568 q^{29} -171.793 q^{31} +13.3140 q^{33} +60.1007 q^{35} +280.327 q^{37} -342.584 q^{39} -280.321 q^{41} -393.268 q^{43} +24.6431 q^{45} +467.968 q^{47} -284.462 q^{49} -642.393 q^{51} -253.167 q^{53} +19.0511 q^{55} +418.322 q^{57} -850.800 q^{59} -176.962 q^{61} +24.0022 q^{63} -490.206 q^{65} +684.006 q^{67} -126.264 q^{69} +1110.22 q^{71} -510.525 q^{73} -347.470 q^{75} +18.5556 q^{77} -535.994 q^{79} -803.861 q^{81} -323.934 q^{83} -919.204 q^{85} -215.509 q^{87} +327.742 q^{89} -477.457 q^{91} -943.095 q^{93} +598.579 q^{95} +1658.33 q^{97} +7.60838 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 12 q^{3} + 12 q^{5} - 14 q^{7} + 90 q^{9}+O(q^{10}) \) 8 * q - 12 * q^3 + 12 * q^5 - 14 * q^7 + 90 * q^9	\( 8 q - 12 q^{3} + 12 q^{5} - 14 q^{7} + 90 q^{9} - 88 q^{11} + 30 q^{13} + 30 q^{15} + 58 q^{17} - 190 q^{19} + 66 q^{21} - 184 q^{23} + 28 q^{25} - 432 q^{27} - 190 q^{29} - 60 q^{31} + 346 q^{33} - 192 q^{35} + 156 q^{37} + 156 q^{39} + 282 q^{41} - 810 q^{43} - 358 q^{45} + 564 q^{47} + 752 q^{49} - 1614 q^{51} + 230 q^{53} + 1292 q^{55} + 796 q^{57} - 1916 q^{59} - 22 q^{61} + 1036 q^{63} + 222 q^{65} - 2292 q^{67} + 276 q^{69} + 2376 q^{71} - 630 q^{73} - 3952 q^{75} + 948 q^{77} + 892 q^{79} - 1044 q^{81} - 2800 q^{83} - 288 q^{85} + 2756 q^{87} - 308 q^{89} - 4610 q^{91} + 982 q^{93} + 1300 q^{95} + 854 q^{97} - 5986 q^{99}+O(q^{100}) \) 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

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