LMFDB - Embedded newform 1472.4.a.bg.1.1

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			1.1
Root			$-6.70977$ 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)

$f(q)$	$=$	$q-8.70977 q^{3} +18.2706 q^{5} +11.7096 q^{7} +48.8601 q^{9} +2.01666 q^{11} +23.3256 q^{13} -159.133 q^{15} +55.7210 q^{17} -105.455 q^{19} -101.988 q^{21} -23.0000 q^{23} +208.814 q^{25} -190.396 q^{27} -275.935 q^{29} -282.019 q^{31} -17.5646 q^{33} +213.941 q^{35} -229.974 q^{37} -203.161 q^{39} -75.0934 q^{41} -125.927 q^{43} +892.702 q^{45} +349.795 q^{47} -205.885 q^{49} -485.317 q^{51} -682.616 q^{53} +36.8455 q^{55} +918.487 q^{57} -604.301 q^{59} +502.787 q^{61} +572.132 q^{63} +426.172 q^{65} -120.360 q^{67} +200.325 q^{69} +381.204 q^{71} +583.989 q^{73} -1818.72 q^{75} +23.6143 q^{77} +714.302 q^{79} +339.085 q^{81} -26.3693 q^{83} +1018.05 q^{85} +2403.33 q^{87} -108.995 q^{89} +273.134 q^{91} +2456.32 q^{93} -1926.72 q^{95} +1365.94 q^{97} +98.5340 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−8.70977	−1.67620	−0.838098	−	0.545520i	$-0.816332\pi$
$3$	−8.70977	−1.67620	−0.838098	+	0.545520i	$0.816332\pi$
$4$	0	0
$5$	18.2706	1.63417	0.817085	−	0.576517i	$-0.195588\pi$
$5$	18.2706	1.63417	0.817085	+	0.576517i	$0.195588\pi$
$6$	0	0
$7$	11.7096	0.632259	0.316130	−	0.948716i	$-0.397617\pi$
$7$	11.7096	0.632259	0.316130	+	0.948716i	$0.397617\pi$
$8$	0	0
$9$	48.8601	1.80963
$10$	0	0
$11$	2.01666	0.0552768	0.0276384	−	0.999618i	$-0.491201\pi$
$11$	2.01666	0.0552768	0.0276384	+	0.999618i	$0.491201\pi$
$12$	0	0
$13$	23.3256	0.497643	0.248821	−	0.968549i	$-0.419957\pi$
$13$	23.3256	0.497643	0.248821	+	0.968549i	$0.419957\pi$
$14$	0	0
$15$	−159.133	−2.73919
$16$	0	0
$17$	55.7210	0.794960	0.397480	−	0.917611i	$-0.369885\pi$
$17$	55.7210	0.794960	0.397480	+	0.917611i	$0.369885\pi$
$18$	0	0
$19$	−105.455	−1.27332	−0.636658	−	0.771146i	$-0.719684\pi$
$19$	−105.455	−1.27332	−0.636658	+	0.771146i	$0.719684\pi$
$20$	0	0
$21$	−101.988	−1.05979
$22$	0	0
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	0	0
$25$	208.814	1.67051
$26$	0	0
$27$	−190.396	−1.35710
$28$	0	0
$29$	−275.935	−1.76689	−0.883446	−	0.468532i	$-0.844783\pi$
$29$	−275.935	−1.76689	−0.883446	+	0.468532i	$0.844783\pi$
$30$	0	0
$31$	−282.019	−1.63394	−0.816970	−	0.576680i	$-0.804348\pi$
$31$	−282.019	−1.63394	−0.816970	+	0.576680i	$0.804348\pi$
$32$	0	0
$33$	−17.5646	−0.0926547
$34$	0	0
$35$	213.941	1.03322
$36$	0	0
$37$	−229.974	−1.02183	−0.510913	−	0.859632i	$-0.670693\pi$
$37$	−229.974	−1.02183	−0.510913	+	0.859632i	$0.670693\pi$
$38$	0	0
$39$	−203.161	−0.834147
$40$	0	0
$41$	−75.0934	−0.286039	−0.143020	−	0.989720i	$-0.545681\pi$
$41$	−75.0934	−0.286039	−0.143020	+	0.989720i	$0.545681\pi$
$42$	0	0
$43$	−125.927	−0.446598	−0.223299	−	0.974750i	$-0.571683\pi$
$43$	−125.927	−0.446598	−0.223299	+	0.974750i	$0.571683\pi$
$44$	0	0
$45$	892.702	2.95725
$46$	0	0
$47$	349.795	1.08559	0.542796	−	0.839864i	$-0.317366\pi$
$47$	349.795	1.08559	0.542796	+	0.839864i	$0.317366\pi$
$48$	0	0
$49$	−205.885	−0.600248
$50$	0	0
$51$	−485.317	−1.33251
$52$	0	0
$53$	−682.616	−1.76914	−0.884571	−	0.466406i	$-0.845549\pi$
$53$	−682.616	−1.76914	−0.884571	+	0.466406i	$0.845549\pi$
$54$	0	0
$55$	36.8455	0.0903317
$56$	0	0
$57$	918.487	2.13433
$58$	0	0
$59$	−604.301	−1.33345	−0.666723	−	0.745305i	$-0.732304\pi$
$59$	−604.301	−1.33345	−0.666723	+	0.745305i	$0.732304\pi$
$60$	0	0
$61$	502.787	1.05533	0.527666	−	0.849452i	$-0.323067\pi$
$61$	502.787	1.05533	0.527666	+	0.849452i	$0.323067\pi$
$62$	0	0
$63$	572.132	1.14416
$64$	0	0
$65$	426.172	0.813233
$66$	0	0
$67$	−120.360	−0.219467	−0.109734	−	0.993961i	$-0.535000\pi$
$67$	−120.360	−0.219467	−0.109734	+	0.993961i	$0.535000\pi$
$68$	0	0
$69$	200.325	0.349511
$70$	0	0
$71$	381.204	0.637191	0.318595	−	0.947891i	$-0.396789\pi$
$71$	381.204	0.637191	0.318595	+	0.947891i	$0.396789\pi$
$72$	0	0
$73$	583.989	0.936312	0.468156	−	0.883646i	$-0.344919\pi$
$73$	583.989	0.936312	0.468156	+	0.883646i	$0.344919\pi$
$74$	0	0
$75$	−1818.72	−2.80011
$76$	0	0
$77$	23.6143	0.0349493
$78$	0	0
$79$	714.302	1.01728	0.508641	−	0.860979i	$-0.330148\pi$
$79$	714.302	1.01728	0.508641	+	0.860979i	$0.330148\pi$
$80$	0	0
$81$	339.085	0.465137
$82$	0	0
$83$	−26.3693	−0.0348724	−0.0174362	−	0.999848i	$-0.505550\pi$
$83$	−26.3693	−0.0348724	−0.0174362	+	0.999848i	$0.505550\pi$
$84$	0	0
$85$	1018.05	1.29910
$86$	0	0
$87$	2403.33	2.96166
$88$	0	0
$89$	−108.995	−0.129814	−0.0649069	−	0.997891i	$-0.520675\pi$
$89$	−108.995	−0.129814	−0.0649069	+	0.997891i	$0.520675\pi$
$90$	0	0
$91$	273.134	0.314639
$92$	0	0
$93$	2456.32	2.73880
$94$	0	0
$95$	−1926.72	−2.08082
$96$	0	0
$97$	1365.94	1.42979	0.714897	−	0.699229i	$-0.246473\pi$
$97$	1365.94	1.42979	0.714897	+	0.699229i	$0.246473\pi$
$98$	0	0
$99$	98.5340	0.100031
$100$	0	0
$101$	−228.537	−0.225151	−0.112576	−	0.993643i	$-0.535910\pi$
$101$	−228.537	−0.225151	−0.112576	+	0.993643i	$0.535910\pi$
$102$	0	0
$103$	−1580.80	−1.51224	−0.756121	−	0.654431i	$-0.772908\pi$
$103$	−1580.80	−1.51224	−0.756121	+	0.654431i	$0.772908\pi$
$104$	0	0
$105$	−1863.38	−1.73188
$106$	0	0
$107$	319.446	0.288617	0.144309	−	0.989533i	$-0.453904\pi$
$107$	319.446	0.288617	0.144309	+	0.989533i	$0.453904\pi$
$108$	0	0
$109$	554.358	0.487136	0.243568	−	0.969884i	$-0.421682\pi$
$109$	554.358	0.487136	0.243568	+	0.969884i	$0.421682\pi$
$110$	0	0
$111$	2003.02	1.71278
$112$	0	0
$113$	−636.882	−0.530202	−0.265101	−	0.964221i	$-0.585405\pi$
$113$	−636.882	−0.530202	−0.265101	+	0.964221i	$0.585405\pi$
$114$	0	0
$115$	−420.223	−0.340748
$116$	0	0
$117$	1139.69	0.900551
$118$	0	0
$119$	652.471	0.502621
$120$	0	0
$121$	−1326.93	−0.996944
$122$	0	0
$123$	654.046	0.479458
$124$	0	0
$125$	1531.33	1.09573
$126$	0	0
$127$	−2634.23	−1.84055	−0.920277	−	0.391267i	$-0.872037\pi$
$127$	−2634.23	−1.84055	−0.920277	+	0.391267i	$0.872037\pi$
$128$	0	0
$129$	1096.80	0.748585
$130$	0	0
$131$	−605.725	−0.403988	−0.201994	−	0.979387i	$-0.564742\pi$
$131$	−605.725	−0.403988	−0.201994	+	0.979387i	$0.564742\pi$
$132$	0	0
$133$	−1234.84	−0.805066
$134$	0	0
$135$	−3478.65	−2.21774
$136$	0	0
$137$	−262.980	−0.163999	−0.0819996	−	0.996632i	$-0.526131\pi$
$137$	−262.980	−0.163999	−0.0819996	+	0.996632i	$0.526131\pi$
$138$	0	0
$139$	−1428.75	−0.871835	−0.435917	−	0.899987i	$-0.643576\pi$
$139$	−1428.75	−0.871835	−0.435917	+	0.899987i	$0.643576\pi$
$140$	0	0
$141$	−3046.63	−1.81967
$142$	0	0
$143$	47.0397	0.0275081
$144$	0	0
$145$	−5041.50	−2.88740
$146$	0	0
$147$	1793.21	1.00613
$148$	0	0
$149$	−1255.23	−0.690152	−0.345076	−	0.938575i	$-0.612147\pi$
$149$	−1255.23	−0.690152	−0.345076	+	0.938575i	$0.612147\pi$
$150$	0	0
$151$	−552.311	−0.297658	−0.148829	−	0.988863i	$-0.547550\pi$
$151$	−552.311	−0.297658	−0.148829	+	0.988863i	$0.547550\pi$
$152$	0	0
$153$	2722.53	1.43859
$154$	0	0
$155$	−5152.65	−2.67014
$156$	0	0
$157$	−2800.10	−1.42339	−0.711695	−	0.702488i	$-0.752072\pi$
$157$	−2800.10	−1.42339	−0.711695	+	0.702488i	$0.752072\pi$
$158$	0	0
$159$	5945.43	2.96543
$160$	0	0
$161$	−269.321	−0.131835
$162$	0	0
$163$	3701.73	1.77879	0.889393	−	0.457144i	$-0.151127\pi$
$163$	3701.73	1.77879	0.889393	+	0.457144i	$0.151127\pi$
$164$	0	0
$165$	−320.916	−0.151414
$166$	0	0
$167$	3038.22	1.40781	0.703905	−	0.710294i	$-0.251438\pi$
$167$	3038.22	1.40781	0.703905	+	0.710294i	$0.251438\pi$
$168$	0	0
$169$	−1652.92	−0.752352
$170$	0	0
$171$	−5152.53	−2.30423
$172$	0	0
$173$	2984.30	1.31152	0.655758	−	0.754971i	$-0.272349\pi$
$173$	2984.30	1.31152	0.655758	+	0.754971i	$0.272349\pi$
$174$	0	0
$175$	2445.13	1.05620
$176$	0	0
$177$	5263.33	2.23512
$178$	0	0
$179$	−897.226	−0.374647	−0.187323	−	0.982298i	$-0.559981\pi$
$179$	−897.226	−0.374647	−0.187323	+	0.982298i	$0.559981\pi$
$180$	0	0
$181$	2049.03	0.841453	0.420726	−	0.907188i	$-0.361775\pi$
$181$	2049.03	0.841453	0.420726	+	0.907188i	$0.361775\pi$
$182$	0	0
$183$	−4379.16	−1.76894
$184$	0	0
$185$	−4201.77	−1.66984
$186$	0	0
$187$	112.370	0.0439429
$188$	0	0
$189$	−2229.46	−0.858041
$190$	0	0
$191$	−1024.04	−0.387941	−0.193971	−	0.981007i	$-0.562137\pi$
$191$	−1024.04	−0.387941	−0.193971	+	0.981007i	$0.562137\pi$
$192$	0	0
$193$	−1616.95	−0.603061	−0.301530	−	0.953457i	$-0.597497\pi$
$193$	−1616.95	−0.603061	−0.301530	+	0.953457i	$0.597497\pi$
$194$	0	0
$195$	−3711.86	−1.36314
$196$	0	0
$197$	−1786.83	−0.646224	−0.323112	−	0.946361i	$-0.604729\pi$
$197$	−1786.83	−0.646224	−0.323112	+	0.946361i	$0.604729\pi$
$198$	0	0
$199$	−3092.49	−1.10161	−0.550805	−	0.834634i	$-0.685679\pi$
$199$	−3092.49	−1.10161	−0.550805	+	0.834634i	$0.685679\pi$
$200$	0	0
$201$	1048.31	0.367870
$202$	0	0
$203$	−3231.09	−1.11713
$204$	0	0
$205$	−1372.00	−0.467437
$206$	0	0
$207$	−1123.78	−0.377334
$208$	0	0
$209$	−212.666	−0.0703848
$210$	0	0
$211$	3733.03	1.21797	0.608987	−	0.793180i	$-0.291576\pi$
$211$	3733.03	1.21797	0.608987	+	0.793180i	$0.291576\pi$
$212$	0	0
$213$	−3320.19	−1.06806
$214$	0	0
$215$	−2300.76	−0.729817
$216$	0	0
$217$	−3302.33	−1.03307
$218$	0	0
$219$	−5086.41	−1.56944
$220$	0	0
$221$	1299.73	0.395606
$222$	0	0
$223$	1145.85	0.344090	0.172045	−	0.985089i	$-0.444963\pi$
$223$	1145.85	0.344090	0.172045	+	0.985089i	$0.444963\pi$
$224$	0	0
$225$	10202.7	3.02302
$226$	0	0
$227$	−5916.71	−1.72998	−0.864990	−	0.501789i	$-0.832676\pi$
$227$	−5916.71	−1.72998	−0.864990	+	0.501789i	$0.832676\pi$
$228$	0	0
$229$	−4894.62	−1.41243	−0.706214	−	0.707999i	$-0.749598\pi$
$229$	−4894.62	−1.41243	−0.706214	+	0.707999i	$0.749598\pi$
$230$	0	0
$231$	−205.675	−0.0585818
$232$	0	0
$233$	1921.15	0.540167	0.270083	−	0.962837i	$-0.412949\pi$
$233$	1921.15	0.540167	0.270083	+	0.962837i	$0.412949\pi$
$234$	0	0
$235$	6390.96	1.77404
$236$	0	0
$237$	−6221.41	−1.70516
$238$	0	0
$239$	−3806.42	−1.03020	−0.515098	−	0.857131i	$-0.672244\pi$
$239$	−3806.42	−1.03020	−0.515098	+	0.857131i	$0.672244\pi$
$240$	0	0
$241$	6887.20	1.84085	0.920423	−	0.390925i	$-0.127845\pi$
$241$	6887.20	1.84085	0.920423	+	0.390925i	$0.127845\pi$
$242$	0	0
$243$	2187.35	0.577442
$244$	0	0
$245$	−3761.64	−0.980908
$246$	0	0
$247$	−2459.80	−0.633657
$248$	0	0
$249$	229.671	0.0584529
$250$	0	0
$251$	3961.61	0.996234	0.498117	−	0.867110i	$-0.334025\pi$
$251$	3961.61	0.996234	0.498117	+	0.867110i	$0.334025\pi$
$252$	0	0
$253$	−46.3831	−0.0115260
$254$	0	0
$255$	−8867.02	−2.17755
$256$	0	0
$257$	−6821.86	−1.65578	−0.827891	−	0.560890i	$-0.810459\pi$
$257$	−6821.86	−1.65578	−0.827891	+	0.560890i	$0.810459\pi$
$258$	0	0
$259$	−2692.91	−0.646059
$260$	0	0
$261$	−13482.2	−3.19743
$262$	0	0
$263$	5109.63	1.19800	0.598999	−	0.800750i	$-0.295565\pi$
$263$	5109.63	1.19800	0.598999	+	0.800750i	$0.295565\pi$
$264$	0	0
$265$	−12471.8	−2.89108
$266$	0	0
$267$	949.320	0.217593
$268$	0	0
$269$	2427.02	0.550105	0.275052	−	0.961429i	$-0.411305\pi$
$269$	2427.02	0.550105	0.275052	+	0.961429i	$0.411305\pi$
$270$	0	0
$271$	6618.07	1.48346	0.741732	−	0.670696i	$-0.234004\pi$
$271$	6618.07	1.48346	0.741732	+	0.670696i	$0.234004\pi$
$272$	0	0
$273$	−2378.93	−0.527397
$274$	0	0
$275$	421.107	0.0923406
$276$	0	0
$277$	5142.17	1.11539	0.557695	−	0.830046i	$-0.311686\pi$
$277$	5142.17	1.11539	0.557695	+	0.830046i	$0.311686\pi$
$278$	0	0
$279$	−13779.5	−2.95683
$280$	0	0
$281$	−6603.95	−1.40199	−0.700994	−	0.713167i	$-0.747260\pi$
$281$	−6603.95	−1.40199	−0.700994	+	0.713167i	$0.747260\pi$
$282$	0	0
$283$	6283.79	1.31990	0.659951	−	0.751309i	$-0.270577\pi$
$283$	6283.79	1.31990	0.659951	+	0.751309i	$0.270577\pi$
$284$	0	0
$285$	16781.3	3.48785
$286$	0	0
$287$	−879.314	−0.180851
$288$	0	0
$289$	−1808.17	−0.368038
$290$	0	0
$291$	−11897.0	−2.39662
$292$	0	0
$293$	−5960.96	−1.18854	−0.594272	−	0.804264i	$-0.702560\pi$
$293$	−5960.96	−1.18854	−0.594272	+	0.804264i	$0.702560\pi$
$294$	0	0
$295$	−11040.9	−2.17908
$296$	0	0
$297$	−383.964	−0.0750163
$298$	0	0
$299$	−536.489	−0.103766
$300$	0	0
$301$	−1474.56	−0.282366
$302$	0	0
$303$	1990.51	0.377398
$304$	0	0
$305$	9186.21	1.72459
$306$	0	0
$307$	−4006.77	−0.744881	−0.372440	−	0.928056i	$-0.621479\pi$
$307$	−4006.77	−0.744881	−0.372440	+	0.928056i	$0.621479\pi$
$308$	0	0
$309$	13768.4	2.53481
$310$	0	0
$311$	−3461.38	−0.631115	−0.315558	−	0.948906i	$-0.602192\pi$
$311$	−3461.38	−0.631115	−0.315558	+	0.948906i	$0.602192\pi$
$312$	0	0
$313$	−4013.93	−0.724858	−0.362429	−	0.932011i	$-0.618053\pi$
$313$	−4013.93	−0.724858	−0.362429	+	0.932011i	$0.618053\pi$
$314$	0	0
$315$	10453.2	1.86975
$316$	0	0
$317$	8418.82	1.49164	0.745818	−	0.666150i	$-0.232059\pi$
$317$	8418.82	1.49164	0.745818	+	0.666150i	$0.232059\pi$
$318$	0	0
$319$	−556.467	−0.0976682
$320$	0	0
$321$	−2782.30	−0.483779
$322$	0	0
$323$	−5876.05	−1.01224
$324$	0	0
$325$	4870.72	0.831319
$326$	0	0
$327$	−4828.33	−0.816535
$328$	0	0
$329$	4095.96	0.686376
$330$	0	0
$331$	7515.07	1.24793	0.623966	−	0.781452i	$-0.285520\pi$
$331$	7515.07	1.24793	0.623966	+	0.781452i	$0.285520\pi$
$332$	0	0
$333$	−11236.6	−1.84913
$334$	0	0
$335$	−2199.05	−0.358647
$336$	0	0
$337$	402.803	0.0651100	0.0325550	−	0.999470i	$-0.489636\pi$
$337$	402.803	0.0651100	0.0325550	+	0.999470i	$0.489636\pi$
$338$	0	0
$339$	5547.10	0.888723
$340$	0	0
$341$	−568.736	−0.0903190
$342$	0	0
$343$	−6427.23	−1.01177
$344$	0	0
$345$	3660.05	0.571161
$346$	0	0
$347$	−9222.48	−1.42677	−0.713384	−	0.700773i	$-0.752839\pi$
$347$	−9222.48	−1.42677	−0.713384	+	0.700773i	$0.752839\pi$
$348$	0	0
$349$	2246.37	0.344542	0.172271	−	0.985050i	$-0.444889\pi$
$349$	2246.37	0.344542	0.172271	+	0.985050i	$0.444889\pi$
$350$	0	0
$351$	−4441.10	−0.675352
$352$	0	0
$353$	2710.12	0.408626	0.204313	−	0.978906i	$-0.434504\pi$
$353$	2710.12	0.408626	0.204313	+	0.978906i	$0.434504\pi$
$354$	0	0
$355$	6964.81	1.04128
$356$	0	0
$357$	−5682.87	−0.842491
$358$	0	0
$359$	325.575	0.0478641	0.0239320	−	0.999714i	$-0.492381\pi$
$359$	325.575	0.0478641	0.0239320	+	0.999714i	$0.492381\pi$
$360$	0	0
$361$	4261.73	0.621334
$362$	0	0
$363$	11557.3	1.67107
$364$	0	0
$365$	10669.8	1.53009
$366$	0	0
$367$	5863.03	0.833917	0.416959	−	0.908926i	$-0.363096\pi$
$367$	5863.03	0.833917	0.416959	+	0.908926i	$0.363096\pi$
$368$	0	0
$369$	−3669.07	−0.517626
$370$	0	0
$371$	−7993.17	−1.11856
$372$	0	0
$373$	−8236.61	−1.14337	−0.571683	−	0.820474i	$-0.693709\pi$
$373$	−8236.61	−1.14337	−0.571683	+	0.820474i	$0.693709\pi$
$374$	0	0
$375$	−13337.6	−1.83666
$376$	0	0
$377$	−6436.35	−0.879282
$378$	0	0
$379$	−9815.24	−1.33028	−0.665139	−	0.746720i	$-0.731628\pi$
$379$	−9815.24	−1.33028	−0.665139	+	0.746720i	$0.731628\pi$
$380$	0	0
$381$	22943.6	3.08513
$382$	0	0
$383$	−9650.81	−1.28755	−0.643777	−	0.765213i	$-0.722634\pi$
$383$	−9650.81	−1.28755	−0.643777	+	0.765213i	$0.722634\pi$
$384$	0	0
$385$	431.446	0.0571131
$386$	0	0
$387$	−6152.81	−0.808178
$388$	0	0
$389$	−4957.68	−0.646181	−0.323090	−	0.946368i	$-0.604722\pi$
$389$	−4957.68	−0.646181	−0.323090	+	0.946368i	$0.604722\pi$
$390$	0	0
$391$	−1281.58	−0.165761
$392$	0	0
$393$	5275.72	0.677163
$394$	0	0
$395$	13050.7	1.66241
$396$	0	0
$397$	11698.5	1.47892	0.739460	−	0.673201i	$-0.235081\pi$
$397$	11698.5	1.47892	0.739460	+	0.673201i	$0.235081\pi$
$398$	0	0
$399$	10755.1	1.34945
$400$	0	0
$401$	−8186.66	−1.01951	−0.509753	−	0.860321i	$-0.670263\pi$
$401$	−8186.66	−1.01951	−0.509753	+	0.860321i	$0.670263\pi$
$402$	0	0
$403$	−6578.27	−0.813118
$404$	0	0
$405$	6195.27	0.760113
$406$	0	0
$407$	−463.780	−0.0564833
$408$	0	0
$409$	6292.14	0.760700	0.380350	−	0.924843i	$-0.375803\pi$
$409$	6292.14	0.760700	0.380350	+	0.924843i	$0.375803\pi$
$410$	0	0
$411$	2290.49	0.274895
$412$	0	0
$413$	−7076.13	−0.843084
$414$	0	0
$415$	−481.783	−0.0569874
$416$	0	0
$417$	12444.1	1.46137
$418$	0	0
$419$	−2333.57	−0.272082	−0.136041	−	0.990703i	$-0.543438\pi$
$419$	−2333.57	−0.272082	−0.136041	+	0.990703i	$0.543438\pi$
$420$	0	0
$421$	636.134	0.0736420	0.0368210	−	0.999322i	$-0.488277\pi$
$421$	636.134	0.0736420	0.0368210	+	0.999322i	$0.488277\pi$
$422$	0	0
$423$	17091.0	1.96452
$424$	0	0
$425$	11635.3	1.32799
$426$	0	0
$427$	5887.44	0.667244
$428$	0	0
$429$	−409.705	−0.0461090
$430$	0	0
$431$	15664.1	1.75061	0.875307	−	0.483568i	$-0.160660\pi$
$431$	15664.1	1.75061	0.875307	+	0.483568i	$0.160660\pi$
$432$	0	0
$433$	−7695.84	−0.854130	−0.427065	−	0.904221i	$-0.640453\pi$
$433$	−7695.84	−0.854130	−0.427065	+	0.904221i	$0.640453\pi$
$434$	0	0
$435$	43910.3	4.83985
$436$	0	0
$437$	2425.46	0.265505
$438$	0	0
$439$	17864.3	1.94218	0.971088	−	0.238723i	$-0.0767289\pi$
$439$	17864.3	1.94218	0.971088	+	0.238723i	$0.0767289\pi$
$440$	0	0
$441$	−10059.6	−1.08623
$442$	0	0
$443$	14239.6	1.52719	0.763593	−	0.645698i	$-0.223434\pi$
$443$	14239.6	1.52719	0.763593	+	0.645698i	$0.223434\pi$
$444$	0	0
$445$	−1991.40	−0.212138
$446$	0	0
$447$	10932.8	1.15683
$448$	0	0
$449$	−17765.1	−1.86723	−0.933613	−	0.358282i	$-0.883363\pi$
$449$	−17765.1	−1.86723	−0.933613	+	0.358282i	$0.883363\pi$
$450$	0	0
$451$	−151.438	−0.0158113
$452$	0	0
$453$	4810.50	0.498934
$454$	0	0
$455$	4990.31	0.514174
$456$	0	0
$457$	16774.3	1.71700	0.858500	−	0.512814i	$-0.171397\pi$
$457$	16774.3	1.71700	0.858500	+	0.512814i	$0.171397\pi$
$458$	0	0
$459$	−10609.1	−1.07884
$460$	0	0
$461$	10129.7	1.02340	0.511699	−	0.859165i	$-0.329016\pi$
$461$	10129.7	1.02340	0.511699	+	0.859165i	$0.329016\pi$
$462$	0	0
$463$	−6375.12	−0.639907	−0.319954	−	0.947433i	$-0.603667\pi$
$463$	−6375.12	−0.639907	−0.319954	+	0.947433i	$0.603667\pi$
$464$	0	0
$465$	44878.4	4.47567
$466$	0	0
$467$	−287.766	−0.0285144	−0.0142572	−	0.999898i	$-0.504538\pi$
$467$	−287.766	−0.0285144	−0.0142572	+	0.999898i	$0.504538\pi$
$468$	0	0
$469$	−1409.37	−0.138760
$470$	0	0
$471$	24388.2	2.38588
$472$	0	0
$473$	−253.952	−0.0246865
$474$	0	0
$475$	−22020.5	−2.12709
$476$	0	0
$477$	−33352.7	−3.20150
$478$	0	0
$479$	−7525.38	−0.717836	−0.358918	−	0.933369i	$-0.616854\pi$
$479$	−7525.38	−0.717836	−0.358918	+	0.933369i	$0.616854\pi$
$480$	0	0
$481$	−5364.29	−0.508505
$482$	0	0
$483$	2345.72	0.220982
$484$	0	0
$485$	24956.5	2.33653
$486$	0	0
$487$	26.7238	0.00248659	0.00124330	−	0.999999i	$-0.499604\pi$
$487$	26.7238	0.00248659	0.00124330	+	0.999999i	$0.499604\pi$
$488$	0	0
$489$	−32241.2	−2.98159
$490$	0	0
$491$	11096.8	1.01994	0.509970	−	0.860192i	$-0.329657\pi$
$491$	11096.8	1.01994	0.509970	+	0.860192i	$0.329657\pi$
$492$	0	0
$493$	−15375.4	−1.40461
$494$	0	0
$495$	1800.27	0.163467
$496$	0	0
$497$	4463.74	0.402870
$498$	0	0
$499$	243.921	0.0218826	0.0109413	−	0.999940i	$-0.496517\pi$
$499$	243.921	0.0218826	0.0109413	+	0.999940i	$0.496517\pi$
$500$	0	0
$501$	−26462.2	−2.35977
$502$	0	0
$503$	5095.04	0.451643	0.225822	−	0.974169i	$-0.427493\pi$
$503$	5095.04	0.451643	0.225822	+	0.974169i	$0.427493\pi$
$504$	0	0
$505$	−4175.51	−0.367936
$506$	0	0
$507$	14396.5	1.26109
$508$	0	0
$509$	434.330	0.0378219	0.0189109	−	0.999821i	$-0.493980\pi$
$509$	434.330	0.0378219	0.0189109	+	0.999821i	$0.493980\pi$
$510$	0	0
$511$	6838.28	0.591992
$512$	0	0
$513$	20078.2	1.72802
$514$	0	0
$515$	−28882.2	−2.47126
$516$	0	0
$517$	705.417	0.0600081
$518$	0	0
$519$	−25992.6	−2.19836
$520$	0	0
$521$	4903.38	0.412325	0.206162	−	0.978518i	$-0.433902\pi$
$521$	4903.38	0.412325	0.206162	+	0.978518i	$0.433902\pi$
$522$	0	0
$523$	−6138.01	−0.513186	−0.256593	−	0.966520i	$-0.582600\pi$
$523$	−6138.01	−0.513186	−0.256593	+	0.966520i	$0.582600\pi$
$524$	0	0
$525$	−21296.5	−1.77039
$526$	0	0
$527$	−15714.4	−1.29892
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	−29526.2	−2.41305
$532$	0	0
$533$	−1751.60	−0.142345
$534$	0	0
$535$	5836.47	0.471650
$536$	0	0
$537$	7814.63	0.627982
$538$	0	0
$539$	−415.199	−0.0331798
$540$	0	0
$541$	−1752.95	−0.139307	−0.0696535	−	0.997571i	$-0.522189\pi$
$541$	−1752.95	−0.139307	−0.0696535	+	0.997571i	$0.522189\pi$
$542$	0	0
$543$	−17846.5	−1.41044
$544$	0	0
$545$	10128.4	0.796063
$546$	0	0
$547$	−10633.7	−0.831193	−0.415597	−	0.909549i	$-0.636427\pi$
$547$	−10633.7	−0.831193	−0.415597	+	0.909549i	$0.636427\pi$
$548$	0	0
$549$	24566.2	1.90976
$550$	0	0
$551$	29098.7	2.24981
$552$	0	0
$553$	8364.20	0.643186
$554$	0	0
$555$	36596.4	2.79898
$556$	0	0
$557$	4952.02	0.376704	0.188352	−	0.982102i	$-0.439685\pi$
$557$	4952.02	0.376704	0.188352	+	0.982102i	$0.439685\pi$
$558$	0	0
$559$	−2937.33	−0.222246
$560$	0	0
$561$	−978.718	−0.0736568
$562$	0	0
$563$	3532.30	0.264421	0.132210	−	0.991222i	$-0.457793\pi$
$563$	3532.30	0.264421	0.132210	+	0.991222i	$0.457793\pi$
$564$	0	0
$565$	−11636.2	−0.866441
$566$	0	0
$567$	3970.55	0.294087
$568$	0	0
$569$	−16297.4	−1.20074	−0.600372	−	0.799721i	$-0.704981\pi$
$569$	−16297.4	−1.20074	−0.600372	+	0.799721i	$0.704981\pi$
$570$	0	0
$571$	25981.9	1.90422	0.952110	−	0.305756i	$-0.0989091\pi$
$571$	25981.9	1.90422	0.952110	+	0.305756i	$0.0989091\pi$
$572$	0	0
$573$	8919.13	0.650265
$574$	0	0
$575$	−4802.73	−0.348326
$576$	0	0
$577$	−22813.0	−1.64596	−0.822981	−	0.568069i	$-0.807690\pi$
$577$	−22813.0	−1.64596	−0.822981	+	0.568069i	$0.807690\pi$
$578$	0	0
$579$	14083.3	1.01085
$580$	0	0
$581$	−308.774	−0.0220484
$582$	0	0
$583$	−1376.60	−0.0977925
$584$	0	0
$585$	20822.8	1.47165
$586$	0	0
$587$	−18286.9	−1.28583	−0.642916	−	0.765937i	$-0.722275\pi$
$587$	−18286.9	−1.28583	−0.642916	+	0.765937i	$0.722275\pi$
$588$	0	0
$589$	29740.3	2.08052
$590$	0	0
$591$	15562.8	1.08320
$592$	0	0
$593$	4978.89	0.344786	0.172393	−	0.985028i	$-0.444850\pi$
$593$	4978.89	0.344786	0.172393	+	0.985028i	$0.444850\pi$
$594$	0	0
$595$	11921.0	0.821369
$596$	0	0
$597$	26934.8	1.84652
$598$	0	0
$599$	9230.75	0.629646	0.314823	−	0.949150i	$-0.398055\pi$
$599$	9230.75	0.629646	0.314823	+	0.949150i	$0.398055\pi$
$600$	0	0
$601$	−2956.20	−0.200642	−0.100321	−	0.994955i	$-0.531987\pi$
$601$	−2956.20	−0.200642	−0.100321	+	0.994955i	$0.531987\pi$
$602$	0	0
$603$	−5880.80	−0.397155
$604$	0	0
$605$	−24243.8	−1.62918
$606$	0	0
$607$	−13356.1	−0.893093	−0.446547	−	0.894760i	$-0.647346\pi$
$607$	−13356.1	−0.893093	−0.446547	+	0.894760i	$0.647346\pi$
$608$	0	0
$609$	28142.1	1.87254
$610$	0	0
$611$	8159.18	0.540238
$612$	0	0
$613$	−8588.14	−0.565859	−0.282930	−	0.959141i	$-0.591306\pi$
$613$	−8588.14	−0.565859	−0.282930	+	0.959141i	$0.591306\pi$
$614$	0	0
$615$	11949.8	0.783516
$616$	0	0
$617$	−6308.36	−0.411612	−0.205806	−	0.978593i	$-0.565982\pi$
$617$	−6308.36	−0.411612	−0.205806	+	0.978593i	$0.565982\pi$
$618$	0	0
$619$	9910.23	0.643499	0.321749	−	0.946825i	$-0.395729\pi$
$619$	9910.23	0.643499	0.321749	+	0.946825i	$0.395729\pi$
$620$	0	0
$621$	4379.11	0.282975
$622$	0	0
$623$	−1276.29	−0.0820760
$624$	0	0
$625$	1876.59	0.120102
$626$	0	0
$627$	1852.27	0.117979
$628$	0	0
$629$	−12814.4	−0.812311
$630$	0	0
$631$	−8551.40	−0.539502	−0.269751	−	0.962930i	$-0.586941\pi$
$631$	−8551.40	−0.539502	−0.269751	+	0.962930i	$0.586941\pi$
$632$	0	0
$633$	−32513.8	−2.04156
$634$	0	0
$635$	−48129.0	−3.00778
$636$	0	0
$637$	−4802.39	−0.298709
$638$	0	0
$639$	18625.6	1.15308
$640$	0	0
$641$	−22921.3	−1.41238	−0.706192	−	0.708020i	$-0.749589\pi$
$641$	−22921.3	−1.41238	−0.706192	+	0.708020i	$0.749589\pi$
$642$	0	0
$643$	−29765.5	−1.82556	−0.912782	−	0.408446i	$-0.866071\pi$
$643$	−29765.5	−1.82556	−0.912782	+	0.408446i	$0.866071\pi$
$644$	0	0
$645$	20039.1	1.22332
$646$	0	0
$647$	−13184.5	−0.801139	−0.400569	−	0.916266i	$-0.631188\pi$
$647$	−13184.5	−0.801139	−0.400569	+	0.916266i	$0.631188\pi$
$648$	0	0
$649$	−1218.67	−0.0737087
$650$	0	0
$651$	28762.6	1.73163
$652$	0	0
$653$	−26389.8	−1.58149	−0.790744	−	0.612147i	$-0.790306\pi$
$653$	−26389.8	−1.58149	−0.790744	+	0.612147i	$0.790306\pi$
$654$	0	0
$655$	−11066.9	−0.660185
$656$	0	0
$657$	28533.7	1.69438
$658$	0	0
$659$	−10789.1	−0.637761	−0.318881	−	0.947795i	$-0.603307\pi$
$659$	−10789.1	−0.637761	−0.318881	+	0.947795i	$0.603307\pi$
$660$	0	0
$661$	−12018.5	−0.707209	−0.353605	−	0.935395i	$-0.615044\pi$
$661$	−12018.5	−0.707209	−0.353605	+	0.935395i	$0.615044\pi$
$662$	0	0
$663$	−11320.3	−0.663114
$664$	0	0
$665$	−22561.2	−1.31562
$666$	0	0
$667$	6346.51	0.368423
$668$	0	0
$669$	−9980.13	−0.576762
$670$	0	0
$671$	1013.95	0.0583354
$672$	0	0
$673$	−28401.4	−1.62674	−0.813369	−	0.581747i	$-0.802369\pi$
$673$	−28401.4	−1.62674	−0.813369	+	0.581747i	$0.802369\pi$
$674$	0	0
$675$	−39757.4	−2.26706
$676$	0	0
$677$	−15573.7	−0.884114	−0.442057	−	0.896987i	$-0.645751\pi$
$677$	−15573.7	−0.884114	−0.442057	+	0.896987i	$0.645751\pi$
$678$	0	0
$679$	15994.6	0.904001
$680$	0	0
$681$	51533.1	2.89979
$682$	0	0
$683$	−3646.89	−0.204311	−0.102156	−	0.994768i	$-0.532574\pi$
$683$	−3646.89	−0.204311	−0.102156	+	0.994768i	$0.532574\pi$
$684$	0	0
$685$	−4804.79	−0.268003
$686$	0	0
$687$	42631.0	2.36750
$688$	0	0
$689$	−15922.4	−0.880401
$690$	0	0
$691$	10621.3	0.584736	0.292368	−	0.956306i	$-0.405557\pi$
$691$	10621.3	0.584736	0.292368	+	0.956306i	$0.405557\pi$
$692$	0	0
$693$	1153.79	0.0632454
$694$	0	0
$695$	−26104.1	−1.42473
$696$	0	0
$697$	−4184.28	−0.227390
$698$	0	0
$699$	−16732.8	−0.905425
$700$	0	0
$701$	11654.1	0.627919	0.313959	−	0.949436i	$-0.398344\pi$
$701$	11654.1	0.627919	0.313959	+	0.949436i	$0.398344\pi$
$702$	0	0
$703$	24251.9	1.30111
$704$	0	0
$705$	−55663.8	−2.97364
$706$	0	0
$707$	−2676.08	−0.142354
$708$	0	0
$709$	26137.6	1.38451	0.692257	−	0.721652i	$-0.256617\pi$
$709$	26137.6	1.38451	0.692257	+	0.721652i	$0.256617\pi$
$710$	0	0
$711$	34900.9	1.84091
$712$	0	0
$713$	6486.44	0.340700
$714$	0	0
$715$	859.443	0.0449529
$716$	0	0
$717$	33153.0	1.72681
$718$	0	0
$719$	3021.94	0.156744	0.0783722	−	0.996924i	$-0.475028\pi$
$719$	3021.94	0.156744	0.0783722	+	0.996924i	$0.475028\pi$
$720$	0	0
$721$	−18510.6	−0.956130
$722$	0	0
$723$	−59985.9	−3.08562
$724$	0	0
$725$	−57619.2	−2.95162
$726$	0	0
$727$	10162.0	0.518415	0.259208	−	0.965822i	$-0.416539\pi$
$727$	10162.0	0.518415	0.259208	+	0.965822i	$0.416539\pi$
$728$	0	0
$729$	−28206.6	−1.43304
$730$	0	0
$731$	−7016.78	−0.355028
$732$	0	0
$733$	−12288.4	−0.619210	−0.309605	−	0.950865i	$-0.600197\pi$
$733$	−12288.4	−0.619210	−0.309605	+	0.950865i	$0.600197\pi$
$734$	0	0
$735$	32763.0	1.64419
$736$	0	0
$737$	−242.725	−0.0121315
$738$	0	0
$739$	−16349.4	−0.813834	−0.406917	−	0.913465i	$-0.633396\pi$
$739$	−16349.4	−0.813834	−0.406917	+	0.913465i	$0.633396\pi$
$740$	0	0
$741$	21424.3	1.06213
$742$	0	0
$743$	18786.9	0.927624	0.463812	−	0.885934i	$-0.346481\pi$
$743$	18786.9	0.927624	0.463812	+	0.885934i	$0.346481\pi$
$744$	0	0
$745$	−22933.8	−1.12783
$746$	0	0
$747$	−1288.41	−0.0631062
$748$	0	0
$749$	3740.59	0.182481
$750$	0	0
$751$	32006.4	1.55517	0.777584	−	0.628779i	$-0.216445\pi$
$751$	32006.4	1.55517	0.777584	+	0.628779i	$0.216445\pi$
$752$	0	0
$753$	−34504.7	−1.66988
$754$	0	0
$755$	−10091.0	−0.486425
$756$	0	0
$757$	−10352.3	−0.497042	−0.248521	−	0.968627i	$-0.579945\pi$
$757$	−10352.3	−0.497042	−0.248521	+	0.968627i	$0.579945\pi$
$758$	0	0
$759$	403.986	0.0193199
$760$	0	0
$761$	14560.4	0.693579	0.346790	−	0.937943i	$-0.387272\pi$
$761$	14560.4	0.693579	0.346790	+	0.937943i	$0.387272\pi$
$762$	0	0
$763$	6491.31	0.307996
$764$	0	0
$765$	49742.2	2.35089
$766$	0	0
$767$	−14095.7	−0.663580
$768$	0	0
$769$	−23352.2	−1.09506	−0.547529	−	0.836786i	$-0.684432\pi$
$769$	−23352.2	−1.09506	−0.547529	+	0.836786i	$0.684432\pi$
$770$	0	0
$771$	59416.8	2.77541
$772$	0	0
$773$	−1738.74	−0.0809032	−0.0404516	−	0.999181i	$-0.512880\pi$
$773$	−1738.74	−0.0809032	−0.0404516	+	0.999181i	$0.512880\pi$
$774$	0	0
$775$	−58889.6	−2.72952
$776$	0	0
$777$	23454.6	1.08292
$778$	0	0
$779$	7918.96	0.364219
$780$	0	0
$781$	768.757	0.0352219
$782$	0	0
$783$	52537.0	2.39785
$784$	0	0
$785$	−51159.5	−2.32606
$786$	0	0
$787$	21512.5	0.974380	0.487190	−	0.873296i	$-0.338022\pi$
$787$	21512.5	0.974380	0.487190	+	0.873296i	$0.338022\pi$
$788$	0	0
$789$	−44503.7	−2.00808
$790$	0	0
$791$	−7457.64	−0.335225
$792$	0	0
$793$	11727.8	0.525178
$794$	0	0
$795$	108626.	4.84602
$796$	0	0
$797$	6508.32	0.289255	0.144628	−	0.989486i	$-0.453802\pi$
$797$	6508.32	0.289255	0.144628	+	0.989486i	$0.453802\pi$
$798$	0	0
$799$	19490.9	0.863003
$800$	0	0
$801$	−5325.49	−0.234915
$802$	0	0
$803$	1177.71	0.0517563
$804$	0	0
$805$	−4920.65	−0.215441
$806$	0	0
$807$	−21138.8	−0.922083
$808$	0	0
$809$	−3638.30	−0.158116	−0.0790580	−	0.996870i	$-0.525191\pi$
$809$	−3638.30	−0.158116	−0.0790580	+	0.996870i	$0.525191\pi$
$810$	0	0
$811$	−45098.6	−1.95268	−0.976341	−	0.216237i	$-0.930621\pi$
$811$	−45098.6	−1.95268	−0.976341	+	0.216237i	$0.930621\pi$
$812$	0	0
$813$	−57641.8	−2.48658
$814$	0	0
$815$	67632.8	2.90684
$816$	0	0
$817$	13279.6	0.568660
$818$	0	0
$819$	13345.3	0.569382
$820$	0	0
$821$	11064.2	0.470332	0.235166	−	0.971955i	$-0.424437\pi$
$821$	11064.2	0.470332	0.235166	+	0.971955i	$0.424437\pi$
$822$	0	0
$823$	41270.8	1.74801	0.874004	−	0.485918i	$-0.161515\pi$
$823$	41270.8	1.74801	0.874004	+	0.485918i	$0.161515\pi$
$824$	0	0
$825$	−3667.74	−0.154781
$826$	0	0
$827$	23569.3	0.991035	0.495517	−	0.868598i	$-0.334978\pi$
$827$	23569.3	0.991035	0.495517	+	0.868598i	$0.334978\pi$
$828$	0	0
$829$	−6327.92	−0.265112	−0.132556	−	0.991176i	$-0.542318\pi$
$829$	−6327.92	−0.265112	−0.132556	+	0.991176i	$0.542318\pi$
$830$	0	0
$831$	−44787.1	−1.86961
$832$	0	0
$833$	−11472.1	−0.477173
$834$	0	0
$835$	55510.0	2.30060
$836$	0	0
$837$	53695.4	2.21742
$838$	0	0
$839$	45428.3	1.86932	0.934659	−	0.355545i	$-0.115705\pi$
$839$	45428.3	1.86932	0.934659	+	0.355545i	$0.115705\pi$
$840$	0	0
$841$	51751.3	2.12191
$842$	0	0
$843$	57518.9	2.35001
$844$	0	0
$845$	−30199.7	−1.22947
$846$	0	0
$847$	−15537.9	−0.630328
$848$	0	0
$849$	−54730.3	−2.21241
$850$	0	0
$851$	5289.41	0.213066
$852$	0	0
$853$	−3650.11	−0.146515	−0.0732575	−	0.997313i	$-0.523340\pi$
$853$	−3650.11	−0.146515	−0.0732575	+	0.997313i	$0.523340\pi$
$854$	0	0
$855$	−94139.8	−3.76551
$856$	0	0
$857$	21525.6	0.857993	0.428997	−	0.903306i	$-0.358867\pi$
$857$	21525.6	0.857993	0.428997	+	0.903306i	$0.358867\pi$
$858$	0	0
$859$	−20494.7	−0.814052	−0.407026	−	0.913417i	$-0.633434\pi$
$859$	−20494.7	−0.814052	−0.407026	+	0.913417i	$0.633434\pi$
$860$	0	0
$861$	7658.62	0.303142
$862$	0	0
$863$	34600.9	1.36481	0.682403	−	0.730976i	$-0.260935\pi$
$863$	34600.9	1.36481	0.682403	+	0.730976i	$0.260935\pi$
$864$	0	0
$865$	54524.9	2.14324
$866$	0	0
$867$	15748.8	0.616904
$868$	0	0
$869$	1440.50	0.0562321
$870$	0	0
$871$	−2807.47	−0.109216
$872$	0	0
$873$	66739.9	2.58740
$874$	0	0
$875$	17931.3	0.692788
$876$	0	0
$877$	−4815.58	−0.185417	−0.0927085	−	0.995693i	$-0.529552\pi$
$877$	−4815.58	−0.185417	−0.0927085	+	0.995693i	$0.529552\pi$
$878$	0	0
$879$	51918.6	1.99223
$880$	0	0
$881$	45060.2	1.72318	0.861588	−	0.507609i	$-0.169470\pi$
$881$	45060.2	1.72318	0.861588	+	0.507609i	$0.169470\pi$
$882$	0	0
$883$	−30384.6	−1.15801	−0.579005	−	0.815324i	$-0.696559\pi$
$883$	−30384.6	−1.15801	−0.579005	+	0.815324i	$0.696559\pi$
$884$	0	0
$885$	96164.0	3.65256
$886$	0	0
$887$	−22951.5	−0.868810	−0.434405	−	0.900718i	$-0.643041\pi$
$887$	−22951.5	−0.868810	−0.434405	+	0.900718i	$0.643041\pi$
$888$	0	0
$889$	−30845.8	−1.16371
$890$	0	0
$891$	683.817	0.0257113
$892$	0	0
$893$	−36887.6	−1.38230
$894$	0	0
$895$	−16392.8	−0.612237
$896$	0	0
$897$	4672.69	0.173932
$898$	0	0
$899$	77819.0	2.88700
$900$	0	0
$901$	−38036.0	−1.40640
$902$	0	0
$903$	12843.1	0.473300
$904$	0	0
$905$	37436.9	1.37508
$906$	0	0
$907$	1091.19	0.0399474	0.0199737	−	0.999801i	$-0.493642\pi$
$907$	1091.19	0.0399474	0.0199737	+	0.999801i	$0.493642\pi$
$908$	0	0
$909$	−11166.3	−0.407441
$910$	0	0
$911$	10998.3	0.399988	0.199994	−	0.979797i	$-0.435908\pi$
$911$	10998.3	0.399988	0.199994	+	0.979797i	$0.435908\pi$
$912$	0	0
$913$	−53.1778	−0.00192763
$914$	0	0
$915$	−80009.7	−2.89075
$916$	0	0
$917$	−7092.80	−0.255425
$918$	0	0
$919$	−30936.1	−1.11043	−0.555217	−	0.831706i	$-0.687365\pi$
$919$	−30936.1	−1.11043	−0.555217	+	0.831706i	$0.687365\pi$
$920$	0	0
$921$	34898.0	1.24857
$922$	0	0
$923$	8891.80	0.317093
$924$	0	0
$925$	−48021.9	−1.70697
$926$	0	0
$927$	−77238.1	−2.73660
$928$	0	0
$929$	28736.7	1.01488	0.507438	−	0.861688i	$-0.330593\pi$
$929$	28736.7	1.01488	0.507438	+	0.861688i	$0.330593\pi$
$930$	0	0
$931$	21711.6	0.764305
$932$	0	0
$933$	30147.8	1.05787
$934$	0	0
$935$	2053.07	0.0718101
$936$	0	0
$937$	6636.94	0.231397	0.115699	−	0.993284i	$-0.463089\pi$
$937$	6636.94	0.231397	0.115699	+	0.993284i	$0.463089\pi$
$938$	0	0
$939$	34960.4	1.21500
$940$	0	0
$941$	24771.1	0.858145	0.429073	−	0.903270i	$-0.358840\pi$
$941$	24771.1	0.858145	0.429073	+	0.903270i	$0.358840\pi$
$942$	0	0
$943$	1727.15	0.0596433
$944$	0	0
$945$	−40733.6	−1.40218
$946$	0	0
$947$	−44391.1	−1.52325	−0.761625	−	0.648018i	$-0.775598\pi$
$947$	−44391.1	−1.52325	−0.761625	+	0.648018i	$0.775598\pi$
$948$	0	0
$949$	13621.9	0.465949
$950$	0	0
$951$	−73326.0	−2.50027
$952$	0	0
$953$	43327.0	1.47272	0.736359	−	0.676591i	$-0.236543\pi$
$953$	43327.0	1.47272	0.736359	+	0.676591i	$0.236543\pi$
$954$	0	0
$955$	−18709.8	−0.633962
$956$	0	0
$957$	4846.70	0.163711
$958$	0	0
$959$	−3079.39	−0.103690
$960$	0	0
$961$	49743.8	1.66976
$962$	0	0
$963$	15608.2	0.522291
$964$	0	0
$965$	−29542.6	−0.985504
$966$	0	0
$967$	−51071.7	−1.69840	−0.849201	−	0.528070i	$-0.822916\pi$
$967$	−51071.7	−1.69840	−0.849201	+	0.528070i	$0.822916\pi$
$968$	0	0
$969$	51179.0	1.69671
$970$	0	0
$971$	20649.3	0.682459	0.341229	−	0.939980i	$-0.389157\pi$
$971$	20649.3	0.682459	0.341229	+	0.939980i	$0.389157\pi$
$972$	0	0
$973$	−16730.1	−0.551226
$974$	0	0
$975$	−42422.8	−1.39345
$976$	0	0
$977$	−38114.1	−1.24809	−0.624043	−	0.781390i	$-0.714511\pi$
$977$	−38114.1	−1.24809	−0.624043	+	0.781390i	$0.714511\pi$
$978$	0	0
$979$	−219.805	−0.00717569
$980$	0	0
$981$	27086.0	0.881537
$982$	0	0
$983$	47799.2	1.55092	0.775461	−	0.631395i	$-0.217517\pi$
$983$	47799.2	1.55092	0.775461	+	0.631395i	$0.217517\pi$
$984$	0	0
$985$	−32646.4	−1.05604
$986$	0	0
$987$	−35674.9	−1.15050
$988$	0	0
$989$	2896.32	0.0931221
$990$	0	0
$991$	−43820.0	−1.40463	−0.702314	−	0.711867i	$-0.747850\pi$
$991$	−43820.0	−1.40463	−0.702314	+	0.711867i	$0.747850\pi$
$992$	0	0
$993$	−65454.5	−2.09178
$994$	0	0
$995$	−56501.5	−1.80022
$996$	0	0
$997$	−11484.4	−0.364809	−0.182405	−	0.983224i	$-0.558388\pi$
$997$	−11484.4	−0.364809	−0.182405	+	0.983224i	$0.558388\pi$
$998$	0	0
$999$	43786.3	1.38672

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1472.4.a.bg.1.1		8
4.3	odd	2		1472.4.a.bh.1.8		8
8.3	odd	2		736.4.a.e.1.1	✓	8
8.5	even	2		736.4.a.f.1.8	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
736.4.a.e.1.1	✓	8	8.3	odd	2
736.4.a.f.1.8	yes	8	8.5	even	2
1472.4.a.bg.1.1		8	1.1	even	1	trivial
1472.4.a.bh.1.8		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)

\(f(q)\)	\(=\)	\(q-8.70977 q^{3} +18.2706 q^{5} +11.7096 q^{7} +48.8601 q^{9} +2.01666 q^{11} +23.3256 q^{13} -159.133 q^{15} +55.7210 q^{17} -105.455 q^{19} -101.988 q^{21} -23.0000 q^{23} +208.814 q^{25} -190.396 q^{27} -275.935 q^{29} -282.019 q^{31} -17.5646 q^{33} +213.941 q^{35} -229.974 q^{37} -203.161 q^{39} -75.0934 q^{41} -125.927 q^{43} +892.702 q^{45} +349.795 q^{47} -205.885 q^{49} -485.317 q^{51} -682.616 q^{53} +36.8455 q^{55} +918.487 q^{57} -604.301 q^{59} +502.787 q^{61} +572.132 q^{63} +426.172 q^{65} -120.360 q^{67} +200.325 q^{69} +381.204 q^{71} +583.989 q^{73} -1818.72 q^{75} +23.6143 q^{77} +714.302 q^{79} +339.085 q^{81} -26.3693 q^{83} +1018.05 q^{85} +2403.33 q^{87} -108.995 q^{89} +273.134 q^{91} +2456.32 q^{93} -1926.72 q^{95} +1365.94 q^{97} +98.5340 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 12 q^{3} + 12 q^{5} - 14 q^{7} + 90 q^{9} - 88 q^{11} + 30 q^{13} + 30 q^{15} + 58 q^{17} - 190 q^{19} + 66 q^{21} - 184 q^{23} + 28 q^{25} - 432 q^{27} - 190 q^{29} - 60 q^{31} + 346 q^{33} - 192 q^{35}+ \cdots - 5986 q^{99}+O(q^{100}) \) 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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