LMFDB - Embedded newform 1458.4.a.i.1.9

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1458,4,Mod(1,1458)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1458.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1458, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1458 = 2 \cdot 3^{6} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1458.a (trivial)

Newform invariants

comment:select newform

sage:traces = [15,-30,0,60,-15] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$86.0247847884$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 339 x^{13} - 1151 x^{12} + 35865 x^{11} + 180141 x^{10} - 1644266 x^{9} - 10786662 x^{8} + \cdots + 33185995624 $ x^15 - 339x^13 - 1151x^12 + 35865x^11 + 180141x^10 - 1644266x^9 - 10786662x^8 + 31430970x^7 + 304598356x^6 - 45265230x^5 - 3933959154x^4 - 5634080801x^3 + 16258950054x^2 + 47732969400*x + 33185995624
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3^{24} $
Twist minimal:	no (minimal twist has level 54)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$4.59599$ of defining polynomial
Character	$\chi$	$=$	1458.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.00000 q^{2} +4.00000 q^{4} +0.857990 q^{5} +21.8463 q^{7} -8.00000 q^{8} -1.71598 q^{10} -48.5698 q^{11} +55.0596 q^{13} -43.6926 q^{14} +16.0000 q^{16} -79.2082 q^{17} +110.031 q^{19} +3.43196 q^{20} +97.1396 q^{22} +45.2289 q^{23} -124.264 q^{25} -110.119 q^{26} +87.3852 q^{28} +124.902 q^{29} -133.698 q^{31} -32.0000 q^{32} +158.416 q^{34} +18.7439 q^{35} +168.323 q^{37} -220.062 q^{38} -6.86392 q^{40} -21.3943 q^{41} -168.177 q^{43} -194.279 q^{44} -90.4578 q^{46} +318.097 q^{47} +134.260 q^{49} +248.528 q^{50} +220.238 q^{52} +463.685 q^{53} -41.6724 q^{55} -174.770 q^{56} -249.804 q^{58} +542.432 q^{59} +690.176 q^{61} +267.396 q^{62} +64.0000 q^{64} +47.2406 q^{65} +643.049 q^{67} -316.833 q^{68} -37.4878 q^{70} -1049.54 q^{71} -960.314 q^{73} -336.646 q^{74} +440.125 q^{76} -1061.07 q^{77} +229.118 q^{79} +13.7278 q^{80} +42.7886 q^{82} -170.123 q^{83} -67.9598 q^{85} +336.355 q^{86} +388.558 q^{88} -759.358 q^{89} +1202.85 q^{91} +180.916 q^{92} -636.193 q^{94} +94.4057 q^{95} +1608.49 q^{97} -268.521 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - 30 q^{2} + 60 q^{4} - 15 q^{5} + 42 q^{7} - 120 q^{8} + 30 q^{10} - 33 q^{11} + 117 q^{13} - 84 q^{14} + 240 q^{16} - 102 q^{17} + 171 q^{19} - 60 q^{20} + 66 q^{22} - 174 q^{23} + 600 q^{25} - 234 q^{26}+ \cdots - 4002 q^{98}+O(q^{100}) $ 15 * q - 30 * q^2 + 60 * q^4 - 15 * q^5 + 42 * q^7 - 120 * q^8 + 30 * q^10 - 33 * q^11 + 117 * q^13 - 84 * q^14 + 240 * q^16 - 102 * q^17 + 171 * q^19 - 60 * q^20 + 66 * q^22 - 174 * q^23 + 600 * q^25 - 234 * q^26 + 168 * q^28 - 573 * q^29 + 372 * q^31 - 480 * q^32 + 204 * q^34 - 624 * q^35 + 555 * q^37 - 342 * q^38 + 120 * q^40 - 852 * q^41 + 1002 * q^43 - 132 * q^44 + 348 * q^46 - 306 * q^47 + 2001 * q^49 - 1200 * q^50 + 468 * q^52 - 897 * q^53 + 1953 * q^55 - 336 * q^56 + 1146 * q^58 - 795 * q^59 + 1587 * q^61 - 744 * q^62 + 960 * q^64 - 1020 * q^65 + 1884 * q^67 - 408 * q^68 + 1248 * q^70 - 120 * q^71 + 2604 * q^73 - 1110 * q^74 + 684 * q^76 - 984 * q^77 + 1944 * q^79 - 240 * q^80 + 1704 * q^82 - 2079 * q^83 + 2916 * q^85 - 2004 * q^86 + 264 * q^88 - 2604 * q^89 + 3399 * q^91 - 696 * q^92 + 612 * q^94 - 1038 * q^95 + 3690 * q^97 - 4002 * q^98

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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	0.857990	0.0767410	0.0383705	−	0.999264i	$-0.487783\pi$
$5$	0.857990	0.0767410	0.0383705	+	0.999264i	$0.487783\pi$
$6$	0	0
$7$	21.8463	1.17959	0.589794	−	0.807553i	$-0.299209\pi$
$7$	21.8463	1.17959	0.589794	+	0.807553i	$0.299209\pi$
$8$	−8.00000	−0.353553
$9$	0	0
$10$	−1.71598	−0.0542641
$11$	−48.5698	−1.33130	−0.665652	−	0.746262i	$-0.731846\pi$
$11$	−48.5698	−1.33130	−0.665652	+	0.746262i	$0.731846\pi$
$12$	0	0
$13$	55.0596	1.17468	0.587338	−	0.809342i	$-0.300176\pi$
$13$	55.0596	1.17468	0.587338	+	0.809342i	$0.300176\pi$
$14$	−43.6926	−0.834095
$15$	0	0
$16$	16.0000	0.250000
$17$	−79.2082	−1.13005	−0.565024	−	0.825075i	$-0.691133\pi$
$17$	−79.2082	−1.13005	−0.565024	+	0.825075i	$0.691133\pi$
$18$	0	0
$19$	110.031	1.32857	0.664287	−	0.747478i	$-0.268735\pi$
$19$	110.031	1.32857	0.664287	+	0.747478i	$0.268735\pi$
$20$	3.43196	0.0383705
$21$	0	0
$22$	97.1396	0.941374
$23$	45.2289	0.410038	0.205019	−	0.978758i	$-0.434274\pi$
$23$	45.2289	0.410038	0.205019	+	0.978758i	$0.434274\pi$
$24$	0	0
$25$	−124.264	−0.994111
$26$	−110.119	−0.830621
$27$	0	0
$28$	87.3852	0.589794
$29$	124.902	0.799784	0.399892	−	0.916562i	$-0.369048\pi$
$29$	124.902	0.799784	0.399892	+	0.916562i	$0.369048\pi$
$30$	0	0
$31$	−133.698	−0.774608	−0.387304	−	0.921952i	$-0.626594\pi$
$31$	−133.698	−0.774608	−0.387304	+	0.921952i	$0.626594\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	158.416	0.799064
$35$	18.7439	0.0905228
$36$	0	0
$37$	168.323	0.747895	0.373947	−	0.927450i	$-0.378004\pi$
$37$	168.323	0.747895	0.373947	+	0.927450i	$0.378004\pi$
$38$	−220.062	−0.939443
$39$	0	0
$40$	−6.86392	−0.0271320
$41$	−21.3943	−0.0814934	−0.0407467	−	0.999170i	$-0.512974\pi$
$41$	−21.3943	−0.0814934	−0.0407467	+	0.999170i	$0.512974\pi$
$42$	0	0
$43$	−168.177	−0.596437	−0.298219	−	0.954498i	$-0.596392\pi$
$43$	−168.177	−0.596437	−0.298219	+	0.954498i	$0.596392\pi$
$44$	−194.279	−0.665652
$45$	0	0
$46$	−90.4578	−0.289941
$47$	318.097	0.987216	0.493608	−	0.869685i	$-0.335678\pi$
$47$	318.097	0.987216	0.493608	+	0.869685i	$0.335678\pi$
$48$	0	0
$49$	134.260	0.391430
$50$	248.528	0.702943
$51$	0	0
$52$	220.238	0.587338
$53$	463.685	1.20174	0.600869	−	0.799348i	$-0.294822\pi$
$53$	463.685	1.20174	0.600869	+	0.799348i	$0.294822\pi$
$54$	0	0
$55$	−41.6724	−0.102166
$56$	−174.770	−0.417048
$57$	0	0
$58$	−249.804	−0.565532
$59$	542.432	1.19693	0.598463	−	0.801151i	$-0.295778\pi$
$59$	542.432	1.19693	0.598463	+	0.801151i	$0.295778\pi$
$60$	0	0
$61$	690.176	1.44866	0.724328	−	0.689455i	$-0.242150\pi$
$61$	690.176	1.44866	0.724328	+	0.689455i	$0.242150\pi$
$62$	267.396	0.547730
$63$	0	0
$64$	64.0000	0.125000
$65$	47.2406	0.0901458
$66$	0	0
$67$	643.049	1.17255	0.586276	−	0.810112i	$-0.300594\pi$
$67$	643.049	1.17255	0.586276	+	0.810112i	$0.300594\pi$
$68$	−316.833	−0.565024
$69$	0	0
$70$	−37.4878	−0.0640093
$71$	−1049.54	−1.75434	−0.877169	−	0.480181i	$-0.840571\pi$
$71$	−1049.54	−1.75434	−0.877169	+	0.480181i	$0.840571\pi$
$72$	0	0
$73$	−960.314	−1.53967	−0.769837	−	0.638241i	$-0.779663\pi$
$73$	−960.314	−1.53967	−0.769837	+	0.638241i	$0.779663\pi$
$74$	−336.646	−0.528841
$75$	0	0
$76$	440.125	0.664287
$77$	−1061.07	−1.57039
$78$	0	0
$79$	229.118	0.326301	0.163150	−	0.986601i	$-0.447834\pi$
$79$	229.118	0.326301	0.163150	+	0.986601i	$0.447834\pi$
$80$	13.7278	0.0191852
$81$	0	0
$82$	42.7886	0.0576246
$83$	−170.123	−0.224981	−0.112491	−	0.993653i	$-0.535883\pi$
$83$	−170.123	−0.224981	−0.112491	+	0.993653i	$0.535883\pi$
$84$	0	0
$85$	−67.9598	−0.0867209
$86$	336.355	0.421745
$87$	0	0
$88$	388.558	0.470687
$89$	−759.358	−0.904402	−0.452201	−	0.891916i	$-0.649361\pi$
$89$	−759.358	−0.904402	−0.452201	+	0.891916i	$0.649361\pi$
$90$	0	0
$91$	1202.85	1.38563
$92$	180.916	0.205019
$93$	0	0
$94$	−636.193	−0.698067
$95$	94.4057	0.101956
$96$	0	0
$97$	1608.49	1.68368	0.841842	−	0.539724i	$-0.181471\pi$
$97$	1608.49	1.68368	0.841842	+	0.539724i	$0.181471\pi$
$98$	−268.521	−0.276783
$99$	0	0
$100$	−497.055	−0.497055
$101$	1625.32	1.60124	0.800622	−	0.599170i	$-0.204503\pi$
$101$	1625.32	1.60124	0.800622	+	0.599170i	$0.204503\pi$
$102$	0	0
$103$	−7.78227	−0.00744476	−0.00372238	−	0.999993i	$-0.501185\pi$
$103$	−7.78227	−0.00744476	−0.00372238	+	0.999993i	$0.501185\pi$
$104$	−440.477	−0.415311
$105$	0	0
$106$	−927.371	−0.849756
$107$	−1846.09	−1.66792	−0.833962	−	0.551822i	$-0.813933\pi$
$107$	−1846.09	−1.66792	−0.833962	+	0.551822i	$0.813933\pi$
$108$	0	0
$109$	−1637.48	−1.43892	−0.719461	−	0.694532i	$-0.755611\pi$
$109$	−1637.48	−1.43892	−0.719461	+	0.694532i	$0.755611\pi$
$110$	83.3448	0.0722420
$111$	0	0
$112$	349.541	0.294897
$113$	392.755	0.326967	0.163483	−	0.986546i	$-0.447727\pi$
$113$	392.755	0.326967	0.163483	+	0.986546i	$0.447727\pi$
$114$	0	0
$115$	38.8059	0.0314667
$116$	499.608	0.399892
$117$	0	0
$118$	−1084.86	−0.846354
$119$	−1730.40	−1.33299
$120$	0	0
$121$	1028.03	0.772371
$122$	−1380.35	−1.02435
$123$	0	0
$124$	−534.791	−0.387304
$125$	−213.866	−0.153030
$126$	0	0
$127$	−631.149	−0.440988	−0.220494	−	0.975388i	$-0.570767\pi$
$127$	−631.149	−0.440988	−0.220494	+	0.975388i	$0.570767\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	−94.4812	−0.0637427
$131$	1804.25	1.20334	0.601672	−	0.798743i	$-0.294501\pi$
$131$	1804.25	1.20334	0.601672	+	0.798743i	$0.294501\pi$
$132$	0	0
$133$	2403.77	1.56717
$134$	−1286.10	−0.829119
$135$	0	0
$136$	633.665	0.399532
$137$	−226.723	−0.141388	−0.0706942	−	0.997498i	$-0.522521\pi$
$137$	−226.723	−0.141388	−0.0706942	+	0.997498i	$0.522521\pi$
$138$	0	0
$139$	−1348.56	−0.822902	−0.411451	−	0.911432i	$-0.634978\pi$
$139$	−1348.56	−0.822902	−0.411451	+	0.911432i	$0.634978\pi$
$140$	74.9756	0.0452614
$141$	0	0
$142$	2099.09	1.24050
$143$	−2674.23	−1.56385
$144$	0	0
$145$	107.165	0.0613762
$146$	1920.63	1.08871
$147$	0	0
$148$	673.291	0.373947
$149$	1732.78	0.952718	0.476359	−	0.879251i	$-0.341956\pi$
$149$	1732.78	0.952718	0.476359	+	0.879251i	$0.341956\pi$
$150$	0	0
$151$	3196.76	1.72284	0.861420	−	0.507893i	$-0.169575\pi$
$151$	3196.76	1.72284	0.861420	+	0.507893i	$0.169575\pi$
$152$	−880.250	−0.469722
$153$	0	0
$154$	2122.14	1.11043
$155$	−114.711	−0.0594442
$156$	0	0
$157$	−3521.62	−1.79016	−0.895082	−	0.445902i	$-0.852883\pi$
$157$	−3521.62	−1.79016	−0.895082	+	0.445902i	$0.852883\pi$
$158$	−458.236	−0.230730
$159$	0	0
$160$	−27.4557	−0.0135660
$161$	988.083	0.483676
$162$	0	0
$163$	3109.36	1.49414	0.747068	−	0.664747i	$-0.231461\pi$
$163$	3109.36	1.49414	0.747068	+	0.664747i	$0.231461\pi$
$164$	−85.5773	−0.0407467
$165$	0	0
$166$	340.247	0.159086
$167$	−462.302	−0.214216	−0.107108	−	0.994247i	$-0.534159\pi$
$167$	−462.302	−0.214216	−0.107108	+	0.994247i	$0.534159\pi$
$168$	0	0
$169$	834.561	0.379864
$170$	135.920	0.0613209
$171$	0	0
$172$	−672.710	−0.298219
$173$	−945.039	−0.415318	−0.207659	−	0.978201i	$-0.566584\pi$
$173$	−945.039	−0.415318	−0.207659	+	0.978201i	$0.566584\pi$
$174$	0	0
$175$	−2714.70	−1.17264
$176$	−777.117	−0.332826
$177$	0	0
$178$	1518.72	0.639509
$179$	4224.40	1.76395	0.881974	−	0.471297i	$-0.156214\pi$
$179$	4224.40	1.76395	0.881974	+	0.471297i	$0.156214\pi$
$180$	0	0
$181$	3161.42	1.29827	0.649135	−	0.760674i	$-0.275131\pi$
$181$	3161.42	1.29827	0.649135	+	0.760674i	$0.275131\pi$
$182$	−2405.70	−0.979792
$183$	0	0
$184$	−361.831	−0.144970
$185$	144.419	0.0573942
$186$	0	0
$187$	3847.12	1.50444
$188$	1272.39	0.493608
$189$	0	0
$190$	−188.811	−0.0720938
$191$	−587.087	−0.222409	−0.111205	−	0.993798i	$-0.535471\pi$
$191$	−587.087	−0.222409	−0.111205	+	0.993798i	$0.535471\pi$
$192$	0	0
$193$	1326.86	0.494866	0.247433	−	0.968905i	$-0.420413\pi$
$193$	1326.86	0.494866	0.247433	+	0.968905i	$0.420413\pi$
$194$	−3216.98	−1.19054
$195$	0	0
$196$	537.042	0.195715
$197$	2240.64	0.810348	0.405174	−	0.914239i	$-0.367211\pi$
$197$	2240.64	0.810348	0.405174	+	0.914239i	$0.367211\pi$
$198$	0	0
$199$	−546.299	−0.194604	−0.0973019	−	0.995255i	$-0.531021\pi$
$199$	−546.299	−0.194604	−0.0973019	+	0.995255i	$0.531021\pi$
$200$	994.111	0.351471
$201$	0	0
$202$	−3250.64	−1.13225
$203$	2728.65	0.943416
$204$	0	0
$205$	−18.3561	−0.00625389
$206$	15.5645	0.00526424
$207$	0	0
$208$	880.954	0.293669
$209$	−5344.20	−1.76874
$210$	0	0
$211$	3007.57	0.981279	0.490639	−	0.871363i	$-0.336763\pi$
$211$	3007.57	0.981279	0.490639	+	0.871363i	$0.336763\pi$
$212$	1854.74	0.600869
$213$	0	0
$214$	3692.17	1.17940
$215$	−144.295	−0.0457712
$216$	0	0
$217$	−2920.80	−0.913719
$218$	3274.97	1.01747
$219$	0	0
$220$	−166.690	−0.0510828
$221$	−4361.17	−1.32744
$222$	0	0
$223$	−491.208	−0.147506	−0.0737528	−	0.997277i	$-0.523498\pi$
$223$	−491.208	−0.147506	−0.0737528	+	0.997277i	$0.523498\pi$
$224$	−699.081	−0.208524
$225$	0	0
$226$	−785.509	−0.231200
$227$	3374.83	0.986764	0.493382	−	0.869813i	$-0.335760\pi$
$227$	3374.83	0.986764	0.493382	+	0.869813i	$0.335760\pi$
$228$	0	0
$229$	516.263	0.148976	0.0744882	−	0.997222i	$-0.476268\pi$
$229$	516.263	0.148976	0.0744882	+	0.997222i	$0.476268\pi$
$230$	−77.6119	−0.0222503
$231$	0	0
$232$	−999.216	−0.282766
$233$	−1319.83	−0.371096	−0.185548	−	0.982635i	$-0.559406\pi$
$233$	−1319.83	−0.371096	−0.185548	+	0.982635i	$0.559406\pi$
$234$	0	0
$235$	272.924	0.0757599
$236$	2169.73	0.598463
$237$	0	0
$238$	3460.81	0.942567
$239$	−732.750	−0.198316	−0.0991582	−	0.995072i	$-0.531615\pi$
$239$	−732.750	−0.198316	−0.0991582	+	0.995072i	$0.531615\pi$
$240$	0	0
$241$	3906.21	1.04407	0.522036	−	0.852924i	$-0.325173\pi$
$241$	3906.21	1.04407	0.522036	+	0.852924i	$0.325173\pi$
$242$	−2056.05	−0.546148
$243$	0	0
$244$	2760.71	0.724328
$245$	115.194	0.0300387
$246$	0	0
$247$	6058.28	1.56064
$248$	1069.58	0.273865
$249$	0	0
$250$	427.732	0.108209
$251$	−2766.24	−0.695631	−0.347816	−	0.937563i	$-0.613077\pi$
$251$	−2766.24	−0.695631	−0.347816	+	0.937563i	$0.613077\pi$
$252$	0	0
$253$	−2196.76	−0.545885
$254$	1262.30	0.311825
$255$	0	0
$256$	256.000	0.0625000
$257$	4670.37	1.13358	0.566789	−	0.823863i	$-0.308185\pi$
$257$	4670.37	1.13358	0.566789	+	0.823863i	$0.308185\pi$
$258$	0	0
$259$	3677.23	0.882208
$260$	188.962	0.0450729
$261$	0	0
$262$	−3608.50	−0.850893
$263$	3145.49	0.737487	0.368744	−	0.929531i	$-0.379788\pi$
$263$	3145.49	0.737487	0.368744	+	0.929531i	$0.379788\pi$
$264$	0	0
$265$	397.837	0.0922225
$266$	−4807.55	−1.10816
$267$	0	0
$268$	2572.20	0.586276
$269$	6454.02	1.46286	0.731428	−	0.681919i	$-0.238854\pi$
$269$	6454.02	1.46286	0.731428	+	0.681919i	$0.238854\pi$
$270$	0	0
$271$	535.232	0.119974	0.0599871	−	0.998199i	$-0.480894\pi$
$271$	535.232	0.119974	0.0599871	+	0.998199i	$0.480894\pi$
$272$	−1267.33	−0.282512
$273$	0	0
$274$	453.445	0.0999767
$275$	6035.47	1.32346
$276$	0	0
$277$	1338.49	0.290333	0.145166	−	0.989407i	$-0.453628\pi$
$277$	1338.49	0.290333	0.145166	+	0.989407i	$0.453628\pi$
$278$	2697.12	0.581879
$279$	0	0
$280$	−149.951	−0.0320046
$281$	1711.37	0.363316	0.181658	−	0.983362i	$-0.441854\pi$
$281$	1711.37	0.363316	0.181658	+	0.983362i	$0.441854\pi$
$282$	0	0
$283$	514.418	0.108053	0.0540265	−	0.998540i	$-0.482794\pi$
$283$	514.418	0.108053	0.0540265	+	0.998540i	$0.482794\pi$
$284$	−4198.18	−0.877169
$285$	0	0
$286$	5348.47	1.10581
$287$	−467.386	−0.0961287
$288$	0	0
$289$	1360.93	0.277006
$290$	−214.329	−0.0433995
$291$	0	0
$292$	−3841.25	−0.769837
$293$	6528.27	1.30166	0.650829	−	0.759225i	$-0.274422\pi$
$293$	6528.27	1.30166	0.650829	+	0.759225i	$0.274422\pi$
$294$	0	0
$295$	465.401	0.0918532
$296$	−1346.58	−0.264421
$297$	0	0
$298$	−3465.56	−0.673674
$299$	2490.28	0.481662
$300$	0	0
$301$	−3674.05	−0.703551
$302$	−6393.53	−1.21823
$303$	0	0
$304$	1760.50	0.332143
$305$	592.165	0.111171
$306$	0	0
$307$	−4274.03	−0.794566	−0.397283	−	0.917696i	$-0.630047\pi$
$307$	−4274.03	−0.794566	−0.397283	+	0.917696i	$0.630047\pi$
$308$	−4244.28	−0.785196
$309$	0	0
$310$	229.423	0.0420334
$311$	509.667	0.0929279	0.0464639	−	0.998920i	$-0.485205\pi$
$311$	509.667	0.0929279	0.0464639	+	0.998920i	$0.485205\pi$
$312$	0	0
$313$	2994.56	0.540775	0.270387	−	0.962752i	$-0.412848\pi$
$313$	2994.56	0.540775	0.270387	+	0.962752i	$0.412848\pi$
$314$	7043.23	1.26584
$315$	0	0
$316$	916.471	0.163150
$317$	−5505.61	−0.975476	−0.487738	−	0.872990i	$-0.662178\pi$
$317$	−5505.61	−0.975476	−0.487738	+	0.872990i	$0.662178\pi$
$318$	0	0
$319$	−6066.47	−1.06476
$320$	54.9114	0.00959262
$321$	0	0
$322$	−1976.17	−0.342011
$323$	−8715.37	−1.50135
$324$	0	0
$325$	−6841.92	−1.16776
$326$	−6218.73	−1.05651
$327$	0	0
$328$	171.155	0.0288123
$329$	6949.23	1.16451
$330$	0	0
$331$	10777.9	1.78975	0.894875	−	0.446316i	$-0.147264\pi$
$331$	10777.9	1.78975	0.894875	+	0.446316i	$0.147264\pi$
$332$	−680.493	−0.112491
$333$	0	0
$334$	924.604	0.151473
$335$	551.730	0.0899827
$336$	0	0
$337$	4808.34	0.777232	0.388616	−	0.921400i	$-0.372953\pi$
$337$	4808.34	0.777232	0.388616	+	0.921400i	$0.372953\pi$
$338$	−1669.12	−0.268604
$339$	0	0
$340$	−271.839	−0.0433605
$341$	6493.68	1.03124
$342$	0	0
$343$	−4560.19	−0.717863
$344$	1345.42	0.210872
$345$	0	0
$346$	1890.08	0.293674
$347$	343.166	0.0530897	0.0265449	−	0.999648i	$-0.491550\pi$
$347$	343.166	0.0530897	0.0265449	+	0.999648i	$0.491550\pi$
$348$	0	0
$349$	−1971.17	−0.302334	−0.151167	−	0.988508i	$-0.548303\pi$
$349$	−1971.17	−0.302334	−0.151167	+	0.988508i	$0.548303\pi$
$350$	5429.41	0.829183
$351$	0	0
$352$	1554.23	0.235344
$353$	764.740	0.115306	0.0576530	−	0.998337i	$-0.481638\pi$
$353$	764.740	0.115306	0.0576530	+	0.998337i	$0.481638\pi$
$354$	0	0
$355$	−900.499	−0.134630
$356$	−3037.43	−0.452201
$357$	0	0
$358$	−8448.81	−1.24730
$359$	8153.31	1.19865	0.599325	−	0.800506i	$-0.295436\pi$
$359$	8153.31	1.19865	0.599325	+	0.800506i	$0.295436\pi$
$360$	0	0
$361$	5247.88	0.765108
$362$	−6322.85	−0.918015
$363$	0	0
$364$	4811.39	0.692817
$365$	−823.940	−0.118156
$366$	0	0
$367$	2939.77	0.418132	0.209066	−	0.977902i	$-0.432958\pi$
$367$	2939.77	0.418132	0.209066	+	0.977902i	$0.432958\pi$
$368$	723.662	0.102509
$369$	0	0
$370$	−288.839	−0.0405838
$371$	10129.8	1.41756
$372$	0	0
$373$	2305.63	0.320056	0.160028	−	0.987112i	$-0.448842\pi$
$373$	2305.63	0.320056	0.160028	+	0.987112i	$0.448842\pi$
$374$	−7694.25	−1.06380
$375$	0	0
$376$	−2544.77	−0.349034
$377$	6877.06	0.939487
$378$	0	0
$379$	10193.5	1.38154	0.690769	−	0.723075i	$-0.257272\pi$
$379$	10193.5	1.38154	0.690769	+	0.723075i	$0.257272\pi$
$380$	377.623	0.0509780
$381$	0	0
$382$	1174.17	0.157267
$383$	5885.48	0.785206	0.392603	−	0.919708i	$-0.371575\pi$
$383$	5885.48	0.785206	0.392603	+	0.919708i	$0.371575\pi$
$384$	0	0
$385$	−910.388	−0.120513
$386$	−2653.71	−0.349923
$387$	0	0
$388$	6433.96	0.841842
$389$	−2991.11	−0.389859	−0.194929	−	0.980817i	$-0.562448\pi$
$389$	−2991.11	−0.389859	−0.194929	+	0.980817i	$0.562448\pi$
$390$	0	0
$391$	−3582.50	−0.463362
$392$	−1074.08	−0.138391
$393$	0	0
$394$	−4481.27	−0.573003
$395$	196.581	0.0250407
$396$	0	0
$397$	−2919.36	−0.369064	−0.184532	−	0.982826i	$-0.559077\pi$
$397$	−2919.36	−0.369064	−0.184532	+	0.982826i	$0.559077\pi$
$398$	1092.60	0.137606
$399$	0	0
$400$	−1988.22	−0.248528
$401$	5914.85	0.736592	0.368296	−	0.929709i	$-0.379941\pi$
$401$	5914.85	0.736592	0.368296	+	0.929709i	$0.379941\pi$
$402$	0	0
$403$	−7361.35	−0.909913
$404$	6501.29	0.800622
$405$	0	0
$406$	−5457.29	−0.667096
$407$	−8175.41	−0.995675
$408$	0	0
$409$	−3084.71	−0.372931	−0.186466	−	0.982461i	$-0.559703\pi$
$409$	−3084.71	−0.372931	−0.186466	+	0.982461i	$0.559703\pi$
$410$	36.7122	0.00442217
$411$	0	0
$412$	−31.1291	−0.00372238
$413$	11850.1	1.41188
$414$	0	0
$415$	−145.964	−0.0172653
$416$	−1761.91	−0.207655
$417$	0	0
$418$	10688.4	1.25068
$419$	−13634.8	−1.58975	−0.794873	−	0.606776i	$-0.792463\pi$
$419$	−13634.8	−1.58975	−0.794873	+	0.606776i	$0.792463\pi$
$420$	0	0
$421$	−5233.11	−0.605811	−0.302905	−	0.953021i	$-0.597957\pi$
$421$	−5233.11	−0.605811	−0.302905	+	0.953021i	$0.597957\pi$
$422$	−6015.14	−0.693869
$423$	0	0
$424$	−3709.48	−0.424878
$425$	9842.71	1.12339
$426$	0	0
$427$	15077.8	1.70882
$428$	−7384.34	−0.833962
$429$	0	0
$430$	288.589	0.0323651
$431$	1533.38	0.171370	0.0856851	−	0.996322i	$-0.472692\pi$
$431$	1533.38	0.171370	0.0856851	+	0.996322i	$0.472692\pi$
$432$	0	0
$433$	11955.1	1.32684	0.663422	−	0.748245i	$-0.269103\pi$
$433$	11955.1	1.32684	0.663422	+	0.748245i	$0.269103\pi$
$434$	5841.60	0.646097
$435$	0	0
$436$	−6549.94	−0.719461
$437$	4976.59	0.544766
$438$	0	0
$439$	14824.4	1.61168	0.805841	−	0.592132i	$-0.201714\pi$
$439$	14824.4	1.61168	0.805841	+	0.592132i	$0.201714\pi$
$440$	333.379	0.0361210
$441$	0	0
$442$	8722.34	0.938641
$443$	9719.32	1.04239	0.521195	−	0.853438i	$-0.325486\pi$
$443$	9719.32	1.04239	0.521195	+	0.853438i	$0.325486\pi$
$444$	0	0
$445$	−651.522	−0.0694047
$446$	982.417	0.104302
$447$	0	0
$448$	1398.16	0.147449
$449$	−12202.0	−1.28251	−0.641255	−	0.767328i	$-0.721586\pi$
$449$	−12202.0	−1.28251	−0.641255	+	0.767328i	$0.721586\pi$
$450$	0	0
$451$	1039.12	0.108493
$452$	1571.02	0.163483
$453$	0	0
$454$	−6749.67	−0.697748
$455$	1032.03	0.106335
$456$	0	0
$457$	7031.97	0.719784	0.359892	−	0.932994i	$-0.382813\pi$
$457$	7031.97	0.719784	0.359892	+	0.932994i	$0.382813\pi$
$458$	−1032.53	−0.105342
$459$	0	0
$460$	155.224	0.0157334
$461$	9091.30	0.918490	0.459245	−	0.888310i	$-0.348120\pi$
$461$	9091.30	0.918490	0.459245	+	0.888310i	$0.348120\pi$
$462$	0	0
$463$	−991.188	−0.0994912	−0.0497456	−	0.998762i	$-0.515841\pi$
$463$	−991.188	−0.0994912	−0.0497456	+	0.998762i	$0.515841\pi$
$464$	1998.43	0.199946
$465$	0	0
$466$	2639.67	0.262404
$467$	6514.04	0.645469	0.322734	−	0.946490i	$-0.395398\pi$
$467$	6514.04	0.645469	0.322734	+	0.946490i	$0.395398\pi$
$468$	0	0
$469$	14048.2	1.38313
$470$	−545.847	−0.0535704
$471$	0	0
$472$	−4339.45	−0.423177
$473$	8168.34	0.794040
$474$	0	0
$475$	−13672.9	−1.32075
$476$	−6921.62	−0.666495
$477$	0	0
$478$	1465.50	0.140231
$479$	−18379.3	−1.75318	−0.876589	−	0.481240i	$-0.840187\pi$
$479$	−18379.3	−1.75318	−0.876589	+	0.481240i	$0.840187\pi$
$480$	0	0
$481$	9267.79	0.878534
$482$	−7812.42	−0.738270
$483$	0	0
$484$	4112.10	0.386185
$485$	1380.07	0.129208
$486$	0	0
$487$	−13583.7	−1.26394	−0.631968	−	0.774994i	$-0.717753\pi$
$487$	−13583.7	−1.26394	−0.631968	+	0.774994i	$0.717753\pi$
$488$	−5521.41	−0.512177
$489$	0	0
$490$	−230.388	−0.0212406
$491$	8330.94	0.765723	0.382862	−	0.923806i	$-0.374939\pi$
$491$	8330.94	0.765723	0.382862	+	0.923806i	$0.374939\pi$
$492$	0	0
$493$	−9893.26	−0.903793
$494$	−12116.6	−1.10354
$495$	0	0
$496$	−2139.17	−0.193652
$497$	−22928.7	−2.06940
$498$	0	0
$499$	−16678.2	−1.49623	−0.748117	−	0.663567i	$-0.769042\pi$
$499$	−16678.2	−1.49623	−0.748117	+	0.663567i	$0.769042\pi$
$500$	−855.464	−0.0765150
$501$	0	0
$502$	5532.48	0.491886
$503$	−11078.4	−0.982032	−0.491016	−	0.871151i	$-0.663374\pi$
$503$	−11078.4	−0.982032	−0.491016	+	0.871151i	$0.663374\pi$
$504$	0	0
$505$	1394.51	0.122881
$506$	4393.51	0.385999
$507$	0	0
$508$	−2524.60	−0.220494
$509$	9443.60	0.822358	0.411179	−	0.911555i	$-0.365117\pi$
$509$	9443.60	0.822358	0.411179	+	0.911555i	$0.365117\pi$
$510$	0	0
$511$	−20979.3	−1.81618
$512$	−512.000	−0.0441942
$513$	0	0
$514$	−9340.74	−0.801561
$515$	−6.67711	−0.000571318 0
$516$	0	0
$517$	−15449.9	−1.31428
$518$	−7354.46	−0.623815
$519$	0	0
$520$	−377.925	−0.0318714
$521$	−8981.78	−0.755276	−0.377638	−	0.925953i	$-0.623264\pi$
$521$	−8981.78	−0.755276	−0.377638	+	0.925953i	$0.623264\pi$
$522$	0	0
$523$	−10294.3	−0.860686	−0.430343	−	0.902665i	$-0.641607\pi$
$523$	−10294.3	−0.860686	−0.430343	+	0.902665i	$0.641607\pi$
$524$	7217.00	0.601672
$525$	0	0
$526$	−6290.98	−0.521482
$527$	10590.0	0.875343
$528$	0	0
$529$	−10121.3	−0.831869
$530$	−795.675	−0.0652111
$531$	0	0
$532$	9615.10	0.783585
$533$	−1177.96	−0.0957284
$534$	0	0
$535$	−1583.92	−0.127998
$536$	−5144.39	−0.414560
$537$	0	0
$538$	−12908.0	−1.03440
$539$	−6521.00	−0.521112
$540$	0	0
$541$	11967.5	0.951059	0.475530	−	0.879700i	$-0.342256\pi$
$541$	11967.5	0.951059	0.475530	+	0.879700i	$0.342256\pi$
$542$	−1070.46	−0.0848346
$543$	0	0
$544$	2534.66	0.199766
$545$	−1404.95	−0.110424
$546$	0	0
$547$	−16769.6	−1.31082	−0.655409	−	0.755274i	$-0.727504\pi$
$547$	−16769.6	−1.31082	−0.655409	+	0.755274i	$0.727504\pi$
$548$	−906.890	−0.0706942
$549$	0	0
$550$	−12070.9	−0.935830
$551$	13743.1	1.06257
$552$	0	0
$553$	5005.37	0.384901
$554$	−2676.98	−0.205296
$555$	0	0
$556$	−5394.24	−0.411451
$557$	−7558.54	−0.574983	−0.287491	−	0.957783i	$-0.592821\pi$
$557$	−7558.54	−0.574983	−0.287491	+	0.957783i	$0.592821\pi$
$558$	0	0
$559$	−9259.78	−0.700621
$560$	299.902	0.0226307
$561$	0	0
$562$	−3422.74	−0.256903
$563$	8559.54	0.640749	0.320374	−	0.947291i	$-0.396191\pi$
$563$	8559.54	0.640749	0.320374	+	0.947291i	$0.396191\pi$
$564$	0	0
$565$	336.980	0.0250918
$566$	−1028.84	−0.0764050
$567$	0	0
$568$	8396.36	0.620252
$569$	9600.85	0.707361	0.353680	−	0.935366i	$-0.384930\pi$
$569$	9600.85	0.707361	0.353680	+	0.935366i	$0.384930\pi$
$570$	0	0
$571$	−1609.58	−0.117966	−0.0589831	−	0.998259i	$-0.518786\pi$
$571$	−1609.58	−0.117966	−0.0589831	+	0.998259i	$0.518786\pi$
$572$	−10696.9	−0.781926
$573$	0	0
$574$	934.773	0.0679733
$575$	−5620.31	−0.407623
$576$	0	0
$577$	−13381.3	−0.965462	−0.482731	−	0.875769i	$-0.660355\pi$
$577$	−13381.3	−0.965462	−0.482731	+	0.875769i	$0.660355\pi$
$578$	−2721.86	−0.195873
$579$	0	0
$580$	428.659	0.0306881
$581$	−3716.56	−0.265386
$582$	0	0
$583$	−22521.1	−1.59988
$584$	7682.51	0.544357
$585$	0	0
$586$	−13056.5	−0.920411
$587$	1506.84	0.105952	0.0529762	−	0.998596i	$-0.483129\pi$
$587$	1506.84	0.105952	0.0529762	+	0.998596i	$0.483129\pi$
$588$	0	0
$589$	−14710.9	−1.02912
$590$	−930.802	−0.0649500
$591$	0	0
$592$	2693.17	0.186974
$593$	−14995.5	−1.03844	−0.519218	−	0.854642i	$-0.673777\pi$
$593$	−14995.5	−1.03844	−0.519218	+	0.854642i	$0.673777\pi$
$594$	0	0
$595$	−1484.67	−0.102295
$596$	6931.13	0.476359
$597$	0	0
$598$	−4980.57	−0.340586
$599$	2722.83	0.185729	0.0928645	−	0.995679i	$-0.470398\pi$
$599$	2722.83	0.185729	0.0928645	+	0.995679i	$0.470398\pi$
$600$	0	0
$601$	−14127.2	−0.958834	−0.479417	−	0.877587i	$-0.659152\pi$
$601$	−14127.2	−0.958834	−0.479417	+	0.877587i	$0.659152\pi$
$602$	7348.10	0.497486
$603$	0	0
$604$	12787.1	0.861420
$605$	882.036	0.0592725
$606$	0	0
$607$	8664.24	0.579358	0.289679	−	0.957124i	$-0.406451\pi$
$607$	8664.24	0.579358	0.289679	+	0.957124i	$0.406451\pi$
$608$	−3521.00	−0.234861
$609$	0	0
$610$	−1184.33	−0.0786100
$611$	17514.3	1.15966
$612$	0	0
$613$	15899.8	1.04761	0.523805	−	0.851838i	$-0.324512\pi$
$613$	15899.8	1.04761	0.523805	+	0.851838i	$0.324512\pi$
$614$	8548.06	0.561843
$615$	0	0
$616$	8488.56	0.555217
$617$	27133.8	1.77045	0.885223	−	0.465166i	$-0.154005\pi$
$617$	27133.8	1.77045	0.885223	+	0.465166i	$0.154005\pi$
$618$	0	0
$619$	−9864.00	−0.640497	−0.320249	−	0.947334i	$-0.603766\pi$
$619$	−9864.00	−0.640497	−0.320249	+	0.947334i	$0.603766\pi$
$620$	−458.846	−0.0297221
$621$	0	0
$622$	−1019.33	−0.0657099
$623$	−16589.2	−1.06682
$624$	0	0
$625$	15349.5	0.982367
$626$	−5989.12	−0.382385
$627$	0	0
$628$	−14086.5	−0.895082
$629$	−13332.5	−0.845156
$630$	0	0
$631$	−17778.4	−1.12163	−0.560814	−	0.827942i	$-0.689512\pi$
$631$	−17778.4	−1.12163	−0.560814	+	0.827942i	$0.689512\pi$
$632$	−1832.94	−0.115365
$633$	0	0
$634$	11011.2	0.689766
$635$	−541.520	−0.0338418
$636$	0	0
$637$	7392.33	0.459803
$638$	12132.9	0.752896
$639$	0	0
$640$	−109.823	−0.00678301
$641$	−7935.87	−0.488999	−0.244499	−	0.969649i	$-0.578624\pi$
$641$	−7935.87	−0.488999	−0.244499	+	0.969649i	$0.578624\pi$
$642$	0	0
$643$	−17880.0	−1.09661	−0.548304	−	0.836279i	$-0.684726\pi$
$643$	−17880.0	−1.09661	−0.548304	+	0.836279i	$0.684726\pi$
$644$	3952.33	0.241838
$645$	0	0
$646$	17430.7	1.06162
$647$	2356.45	0.143186	0.0715931	−	0.997434i	$-0.477192\pi$
$647$	2356.45	0.143186	0.0715931	+	0.997434i	$0.477192\pi$
$648$	0	0
$649$	−26345.8	−1.59347
$650$	13683.8	0.825730
$651$	0	0
$652$	12437.5	0.747068
$653$	−26279.6	−1.57488	−0.787442	−	0.616389i	$-0.788595\pi$
$653$	−26279.6	−1.57488	−0.787442	+	0.616389i	$0.788595\pi$
$654$	0	0
$655$	1548.03	0.0923458
$656$	−342.309	−0.0203734
$657$	0	0
$658$	−13898.5	−0.823432
$659$	6300.47	0.372430	0.186215	−	0.982509i	$-0.440378\pi$
$659$	6300.47	0.372430	0.186215	+	0.982509i	$0.440378\pi$
$660$	0	0
$661$	−7369.15	−0.433626	−0.216813	−	0.976213i	$-0.569566\pi$
$661$	−7369.15	−0.433626	−0.216813	+	0.976213i	$0.569566\pi$
$662$	−21555.8	−1.26554
$663$	0	0
$664$	1360.99	0.0795430
$665$	2062.42	0.120266
$666$	0	0
$667$	5649.18	0.327942
$668$	−1849.21	−0.107108
$669$	0	0
$670$	−1103.46	−0.0636274
$671$	−33521.7	−1.92860
$672$	0	0
$673$	−19432.7	−1.11304	−0.556519	−	0.830835i	$-0.687863\pi$
$673$	−19432.7	−1.11304	−0.556519	+	0.830835i	$0.687863\pi$
$674$	−9616.68	−0.549586
$675$	0	0
$676$	3338.24	0.189932
$677$	−17716.4	−1.00576	−0.502878	−	0.864357i	$-0.667726\pi$
$677$	−17716.4	−1.00576	−0.502878	+	0.864357i	$0.667726\pi$
$678$	0	0
$679$	35139.5	1.98606
$680$	543.679	0.0306605
$681$	0	0
$682$	−12987.4	−0.729196
$683$	−6992.05	−0.391718	−0.195859	−	0.980632i	$-0.562749\pi$
$683$	−6992.05	−0.391718	−0.195859	+	0.980632i	$0.562749\pi$
$684$	0	0
$685$	−194.526	−0.0108503
$686$	9120.37	0.507606
$687$	0	0
$688$	−2690.84	−0.149109
$689$	25530.3	1.41165
$690$	0	0
$691$	16798.4	0.924804	0.462402	−	0.886670i	$-0.346988\pi$
$691$	16798.4	0.924804	0.462402	+	0.886670i	$0.346988\pi$
$692$	−3780.16	−0.207659
$693$	0	0
$694$	−686.332	−0.0375401
$695$	−1157.05	−0.0631503
$696$	0	0
$697$	1694.60	0.0920914
$698$	3942.35	0.213782
$699$	0	0
$700$	−10858.8	−0.586321
$701$	−18157.5	−0.978315	−0.489158	−	0.872195i	$-0.662696\pi$
$701$	−18157.5	−0.978315	−0.489158	+	0.872195i	$0.662696\pi$
$702$	0	0
$703$	18520.8	0.993633
$704$	−3108.47	−0.166413
$705$	0	0
$706$	−1529.48	−0.0815336
$707$	35507.3	1.88881
$708$	0	0
$709$	−25466.7	−1.34897	−0.674487	−	0.738286i	$-0.735635\pi$
$709$	−25466.7	−1.34897	−0.674487	+	0.738286i	$0.735635\pi$
$710$	1801.00	0.0951975
$711$	0	0
$712$	6074.86	0.319754
$713$	−6047.00	−0.317619
$714$	0	0
$715$	−2294.47	−0.120011
$716$	16897.6	0.881974
$717$	0	0
$718$	−16306.6	−0.847573
$719$	30910.3	1.60328	0.801642	−	0.597805i	$-0.203960\pi$
$719$	30910.3	1.60328	0.801642	+	0.597805i	$0.203960\pi$
$720$	0	0
$721$	−170.014	−0.00878175
$722$	−10495.8	−0.541013
$723$	0	0
$724$	12645.7	0.649135
$725$	−15520.8	−0.795073
$726$	0	0
$727$	−14411.0	−0.735176	−0.367588	−	0.929989i	$-0.619816\pi$
$727$	−14411.0	−0.735176	−0.367588	+	0.929989i	$0.619816\pi$
$728$	−9622.79	−0.489896
$729$	0	0
$730$	1647.88	0.0835490
$731$	13321.0	0.674002
$732$	0	0
$733$	−10183.6	−0.513151	−0.256576	−	0.966524i	$-0.582594\pi$
$733$	−10183.6	−0.513151	−0.256576	+	0.966524i	$0.582594\pi$
$734$	−5879.53	−0.295664
$735$	0	0
$736$	−1447.32	−0.0724852
$737$	−31232.8	−1.56102
$738$	0	0
$739$	9608.59	0.478292	0.239146	−	0.970984i	$-0.423133\pi$
$739$	9608.59	0.478292	0.239146	+	0.970984i	$0.423133\pi$
$740$	577.677	0.0286971
$741$	0	0
$742$	−20259.6	−1.00236
$743$	28540.2	1.40921	0.704603	−	0.709602i	$-0.251125\pi$
$743$	28540.2	1.40921	0.704603	+	0.709602i	$0.251125\pi$
$744$	0	0
$745$	1486.71	0.0731125
$746$	−4611.26	−0.226314
$747$	0	0
$748$	15388.5	0.752218
$749$	−40330.1	−1.96746
$750$	0	0
$751$	−3850.82	−0.187108	−0.0935542	−	0.995614i	$-0.529823\pi$
$751$	−3850.82	−0.187108	−0.0935542	+	0.995614i	$0.529823\pi$
$752$	5089.54	0.246804
$753$	0	0
$754$	−13754.1	−0.664317
$755$	2742.79	0.132212
$756$	0	0
$757$	−22515.6	−1.08104	−0.540518	−	0.841332i	$-0.681772\pi$
$757$	−22515.6	−1.08104	−0.540518	+	0.841332i	$0.681772\pi$
$758$	−20386.9	−0.976895
$759$	0	0
$760$	−755.246	−0.0360469
$761$	−20128.1	−0.958795	−0.479397	−	0.877598i	$-0.659145\pi$
$761$	−20128.1	−0.958795	−0.479397	+	0.877598i	$0.659145\pi$
$762$	0	0
$763$	−35773.0	−1.69734
$764$	−2348.35	−0.111205
$765$	0	0
$766$	−11771.0	−0.555225
$767$	29866.1	1.40600
$768$	0	0
$769$	28898.9	1.35516	0.677582	−	0.735447i	$-0.263028\pi$
$769$	28898.9	1.35516	0.677582	+	0.735447i	$0.263028\pi$
$770$	1820.78	0.0852158
$771$	0	0
$772$	5307.42	0.247433
$773$	−40186.4	−1.86986	−0.934932	−	0.354827i	$-0.884540\pi$
$773$	−40186.4	−1.86986	−0.934932	+	0.354827i	$0.884540\pi$
$774$	0	0
$775$	16613.8	0.770046
$776$	−12867.9	−0.595272
$777$	0	0
$778$	5982.21	0.275672
$779$	−2354.04	−0.108270
$780$	0	0
$781$	50976.2	2.33556
$782$	7164.99	0.327647
$783$	0	0
$784$	2148.17	0.0978574
$785$	−3021.51	−0.137379
$786$	0	0
$787$	13239.9	0.599684	0.299842	−	0.953989i	$-0.403066\pi$
$787$	13239.9	0.599684	0.299842	+	0.953989i	$0.403066\pi$
$788$	8962.54	0.405174
$789$	0	0
$790$	−393.162	−0.0177064
$791$	8580.23	0.385686
$792$	0	0
$793$	38000.9	1.70170
$794$	5838.72	0.260968
$795$	0	0
$796$	−2185.20	−0.0973019
$797$	−25373.2	−1.12769	−0.563843	−	0.825882i	$-0.690678\pi$
$797$	−25373.2	−1.12769	−0.563843	+	0.825882i	$0.690678\pi$
$798$	0	0
$799$	−25195.8	−1.11560
$800$	3976.44	0.175736
$801$	0	0
$802$	−11829.7	−0.520849
$803$	46642.2	2.04977
$804$	0	0
$805$	847.766	0.0371178
$806$	14722.7	0.643406
$807$	0	0
$808$	−13002.6	−0.566125
$809$	13345.4	0.579976	0.289988	−	0.957030i	$-0.406349\pi$
$809$	13345.4	0.579976	0.289988	+	0.957030i	$0.406349\pi$
$810$	0	0
$811$	33175.7	1.43644	0.718222	−	0.695815i	$-0.244956\pi$
$811$	33175.7	1.43644	0.718222	+	0.695815i	$0.244956\pi$
$812$	10914.6	0.471708
$813$	0	0
$814$	16350.8	0.704049
$815$	2667.80	0.114661
$816$	0	0
$817$	−18504.8	−0.792411
$818$	6169.41	0.263702
$819$	0	0
$820$	−73.4245	−0.00312694
$821$	−18076.5	−0.768423	−0.384211	−	0.923245i	$-0.625527\pi$
$821$	−18076.5	−0.768423	−0.384211	+	0.923245i	$0.625527\pi$
$822$	0	0
$823$	−24596.2	−1.04176	−0.520880	−	0.853630i	$-0.674396\pi$
$823$	−24596.2	−1.04176	−0.520880	+	0.853630i	$0.674396\pi$
$824$	62.2582	0.00263212
$825$	0	0
$826$	−23700.2	−0.998350
$827$	−7720.57	−0.324632	−0.162316	−	0.986739i	$-0.551896\pi$
$827$	−7720.57	−0.324632	−0.162316	+	0.986739i	$0.551896\pi$
$828$	0	0
$829$	−12761.4	−0.534647	−0.267323	−	0.963607i	$-0.586139\pi$
$829$	−12761.4	−0.534647	−0.267323	+	0.963607i	$0.586139\pi$
$830$	291.928	0.0122084
$831$	0	0
$832$	3523.82	0.146835
$833$	−10634.5	−0.442334
$834$	0	0
$835$	−396.651	−0.0164391
$836$	−21376.8	−0.884368
$837$	0	0
$838$	27269.6	1.12412
$839$	1787.98	0.0735731	0.0367865	−	0.999323i	$-0.488288\pi$
$839$	1787.98	0.0735731	0.0367865	+	0.999323i	$0.488288\pi$
$840$	0	0
$841$	−8788.49	−0.360346
$842$	10466.2	0.428373
$843$	0	0
$844$	12030.3	0.490639
$845$	716.045	0.0291511
$846$	0	0
$847$	22458.5	0.911080
$848$	7418.96	0.300434
$849$	0	0
$850$	−19685.4	−0.794358
$851$	7613.05	0.306665
$852$	0	0
$853$	−23287.5	−0.934759	−0.467380	−	0.884057i	$-0.654802\pi$
$853$	−23287.5	−0.934759	−0.467380	+	0.884057i	$0.654802\pi$
$854$	−30155.6	−1.20832
$855$	0	0
$856$	14768.7	0.589700
$857$	9906.51	0.394866	0.197433	−	0.980316i	$-0.436740\pi$
$857$	9906.51	0.394866	0.197433	+	0.980316i	$0.436740\pi$
$858$	0	0
$859$	6761.43	0.268565	0.134282	−	0.990943i	$-0.457127\pi$
$859$	6761.43	0.268565	0.134282	+	0.990943i	$0.457127\pi$
$860$	−577.178	−0.0228856
$861$	0	0
$862$	−3066.77	−0.121177
$863$	6733.15	0.265584	0.132792	−	0.991144i	$-0.457606\pi$
$863$	6733.15	0.265584	0.132792	+	0.991144i	$0.457606\pi$
$864$	0	0
$865$	−810.834	−0.0318719
$866$	−23910.1	−0.938221
$867$	0	0
$868$	−11683.2	−0.456859
$869$	−11128.2	−0.434406
$870$	0	0
$871$	35406.0	1.37737
$872$	13099.9	0.508736
$873$	0	0
$874$	−9953.18	−0.385207
$875$	−4672.18	−0.180512
$876$	0	0
$877$	19284.6	0.742524	0.371262	−	0.928528i	$-0.378925\pi$
$877$	19284.6	0.742524	0.371262	+	0.928528i	$0.378925\pi$
$878$	−29648.7	−1.13963
$879$	0	0
$880$	−666.759	−0.0255414
$881$	38159.4	1.45928	0.729638	−	0.683833i	$-0.239688\pi$
$881$	38159.4	1.45928	0.729638	+	0.683833i	$0.239688\pi$
$882$	0	0
$883$	−15205.9	−0.579522	−0.289761	−	0.957099i	$-0.593576\pi$
$883$	−15205.9	−0.579522	−0.289761	+	0.957099i	$0.593576\pi$
$884$	−17444.7	−0.663720
$885$	0	0
$886$	−19438.6	−0.737081
$887$	33530.4	1.26927	0.634634	−	0.772813i	$-0.281151\pi$
$887$	33530.4	1.26927	0.634634	+	0.772813i	$0.281151\pi$
$888$	0	0
$889$	−13788.3	−0.520184
$890$	1303.04	0.0490765
$891$	0	0
$892$	−1964.83	−0.0737528
$893$	35000.6	1.31159
$894$	0	0
$895$	3624.50	0.135367
$896$	−2796.33	−0.104262
$897$	0	0
$898$	24404.0	0.906871
$899$	−16699.1	−0.619519
$900$	0	0
$901$	−36727.7	−1.35802
$902$	−2078.24	−0.0767158
$903$	0	0
$904$	−3142.04	−0.115600
$905$	2712.47	0.0996305
$906$	0	0
$907$	45928.5	1.68140	0.840700	−	0.541501i	$-0.182144\pi$
$907$	45928.5	1.68140	0.840700	+	0.541501i	$0.182144\pi$
$908$	13499.3	0.493382
$909$	0	0
$910$	−2064.06	−0.0751902
$911$	44740.0	1.62712	0.813558	−	0.581484i	$-0.197528\pi$
$911$	44740.0	1.62712	0.813558	+	0.581484i	$0.197528\pi$
$912$	0	0
$913$	8262.86	0.299519
$914$	−14063.9	−0.508964
$915$	0	0
$916$	2065.05	0.0744882
$917$	39416.2	1.41945
$918$	0	0
$919$	−24618.1	−0.883651	−0.441826	−	0.897101i	$-0.645669\pi$
$919$	−24618.1	−0.883651	−0.441826	+	0.897101i	$0.645669\pi$
$920$	−310.447	−0.0111252
$921$	0	0
$922$	−18182.6	−0.649471
$923$	−57787.5	−2.06078
$924$	0	0
$925$	−20916.4	−0.743490
$926$	1982.38	0.0703509
$927$	0	0
$928$	−3996.86	−0.141383
$929$	−17898.1	−0.632097	−0.316048	−	0.948743i	$-0.602356\pi$
$929$	−17898.1	−0.632097	−0.316048	+	0.948743i	$0.602356\pi$
$930$	0	0
$931$	14772.8	0.520043
$932$	−5279.34	−0.185548
$933$	0	0
$934$	−13028.1	−0.456415
$935$	3300.79	0.115452
$936$	0	0
$937$	−10883.8	−0.379464	−0.189732	−	0.981836i	$-0.560762\pi$
$937$	−10883.8	−0.379464	−0.189732	+	0.981836i	$0.560762\pi$
$938$	−28096.5	−0.978020
$939$	0	0
$940$	1091.69	0.0378800
$941$	−14329.9	−0.496431	−0.248216	−	0.968705i	$-0.579844\pi$
$941$	−14329.9	−0.496431	−0.248216	+	0.968705i	$0.579844\pi$
$942$	0	0
$943$	−967.641	−0.0334154
$944$	8678.91	0.299231
$945$	0	0
$946$	−16336.7	−0.561471
$947$	32112.6	1.10192	0.550960	−	0.834532i	$-0.314262\pi$
$947$	32112.6	1.10192	0.550960	+	0.834532i	$0.314262\pi$
$948$	0	0
$949$	−52874.5	−1.80862
$950$	27345.8	0.933911
$951$	0	0
$952$	13843.2	0.471283
$953$	47431.6	1.61224	0.806118	−	0.591755i	$-0.201565\pi$
$953$	47431.6	1.61224	0.806118	+	0.591755i	$0.201565\pi$
$954$	0	0
$955$	−503.715	−0.0170679
$956$	−2931.00	−0.0991582
$957$	0	0
$958$	36758.6	1.23968
$959$	−4953.05	−0.166780
$960$	0	0
$961$	−11915.9	−0.399983
$962$	−18535.6	−0.621217
$963$	0	0
$964$	15624.8	0.522036
$965$	1138.43	0.0379765
$966$	0	0
$967$	−804.773	−0.0267629	−0.0133815	−	0.999910i	$-0.504260\pi$
$967$	−804.773	−0.0267629	−0.0133815	+	0.999910i	$0.504260\pi$
$968$	−8224.20	−0.273074
$969$	0	0
$970$	−2760.14	−0.0913636
$971$	5884.04	0.194467	0.0972337	−	0.995262i	$-0.469001\pi$
$971$	5884.04	0.194467	0.0972337	+	0.995262i	$0.469001\pi$
$972$	0	0
$973$	−29461.0	−0.970686
$974$	27167.4	0.893738
$975$	0	0
$976$	11042.8	0.362164
$977$	35797.1	1.17221	0.586106	−	0.810234i	$-0.300660\pi$
$977$	35797.1	1.17221	0.586106	+	0.810234i	$0.300660\pi$
$978$	0	0
$979$	36881.9	1.20403
$980$	460.776	0.0150194
$981$	0	0
$982$	−16661.9	−0.541448
$983$	−4133.63	−0.134122	−0.0670611	−	0.997749i	$-0.521362\pi$
$983$	−4133.63	−0.134122	−0.0670611	+	0.997749i	$0.521362\pi$
$984$	0	0
$985$	1922.44	0.0621869
$986$	19786.5	0.639078
$987$	0	0
$988$	24233.1	0.780322
$989$	−7606.47	−0.244562
$990$	0	0
$991$	9307.49	0.298347	0.149174	−	0.988811i	$-0.452339\pi$
$991$	9307.49	0.298347	0.149174	+	0.988811i	$0.452339\pi$
$992$	4278.33	0.136933
$993$	0	0
$994$	45857.3	1.46329
$995$	−468.720	−0.0149341
$996$	0	0
$997$	−26708.5	−0.848413	−0.424207	−	0.905565i	$-0.639447\pi$
$997$	−26708.5	−0.848413	−0.424207	+	0.905565i	$0.639447\pi$
$998$	33356.5	1.05800
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1458.4.a.i.1.9		15
3.2	odd	2		1458.4.a.j.1.7		15
27.5	odd	18		162.4.e.b.73.3		30
27.11	odd	18		162.4.e.b.91.3		30
27.16	even	9		54.4.e.b.13.2	✓	30
27.22	even	9		54.4.e.b.25.2	yes	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
54.4.e.b.13.2	✓	30	27.16	even	9
54.4.e.b.25.2	yes	30	27.22	even	9
162.4.e.b.73.3		30	27.5	odd	18
162.4.e.b.91.3		30	27.11	odd	18
1458.4.a.i.1.9		15	1.1	even	1	trivial
1458.4.a.j.1.7		15	3.2	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1458 = 2 \cdot 3^{6} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1458.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +4.00000 q^{4} +0.857990 q^{5} +21.8463 q^{7} -8.00000 q^{8} -1.71598 q^{10} -48.5698 q^{11} +55.0596 q^{13} -43.6926 q^{14} +16.0000 q^{16} -79.2082 q^{17} +110.031 q^{19} +3.43196 q^{20} +97.1396 q^{22} +45.2289 q^{23} -124.264 q^{25} -110.119 q^{26} +87.3852 q^{28} +124.902 q^{29} -133.698 q^{31} -32.0000 q^{32} +158.416 q^{34} +18.7439 q^{35} +168.323 q^{37} -220.062 q^{38} -6.86392 q^{40} -21.3943 q^{41} -168.177 q^{43} -194.279 q^{44} -90.4578 q^{46} +318.097 q^{47} +134.260 q^{49} +248.528 q^{50} +220.238 q^{52} +463.685 q^{53} -41.6724 q^{55} -174.770 q^{56} -249.804 q^{58} +542.432 q^{59} +690.176 q^{61} +267.396 q^{62} +64.0000 q^{64} +47.2406 q^{65} +643.049 q^{67} -316.833 q^{68} -37.4878 q^{70} -1049.54 q^{71} -960.314 q^{73} -336.646 q^{74} +440.125 q^{76} -1061.07 q^{77} +229.118 q^{79} +13.7278 q^{80} +42.7886 q^{82} -170.123 q^{83} -67.9598 q^{85} +336.355 q^{86} +388.558 q^{88} -759.358 q^{89} +1202.85 q^{91} +180.916 q^{92} -636.193 q^{94} +94.4057 q^{95} +1608.49 q^{97} -268.521 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - 30 q^{2} + 60 q^{4} - 15 q^{5} + 42 q^{7} - 120 q^{8} + 30 q^{10} - 33 q^{11} + 117 q^{13} - 84 q^{14} + 240 q^{16} - 102 q^{17} + 171 q^{19} - 60 q^{20} + 66 q^{22} - 174 q^{23} + 600 q^{25} - 234 q^{26}+ \cdots - 4002 q^{98}+O(q^{100}) \) 15 * q - 30 * q^2 + 60 * q^4 - 15 * q^5 + 42 * q^7 - 120 * q^8 + 30 * q^10 - 33 * q^11 + 117 * q^13 - 84 * q^14 + 240 * q^16 - 102 * q^17 + 171 * q^19 - 60 * q^20 + 66 * q^22 - 174 * q^23 + 600 * q^25 - 234 * q^26 + 168 * q^28 - 573 * q^29 + 372 * q^31 - 480 * q^32 + 204 * q^34 - 624 * q^35 + 555 * q^37 - 342 * q^38 + 120 * q^40 - 852 * q^41 + 1002 * q^43 - 132 * q^44 + 348 * q^46 - 306 * q^47 + 2001 * q^49 - 1200 * q^50 + 468 * q^52 - 897 * q^53 + 1953 * q^55 - 336 * q^56 + 1146 * q^58 - 795 * q^59 + 1587 * q^61 - 744 * q^62 + 960 * q^64 - 1020 * q^65 + 1884 * q^67 - 408 * q^68 + 1248 * q^70 - 120 * q^71 + 2604 * q^73 - 1110 * q^74 + 684 * q^76 - 984 * q^77 + 1944 * q^79 - 240 * q^80 + 1704 * q^82 - 2079 * q^83 + 2916 * q^85 - 2004 * q^86 + 264 * q^88 - 2604 * q^89 + 3399 * q^91 - 696 * q^92 + 612 * q^94 - 1038 * q^95 + 3690 * q^97 - 4002 * q^98

Introduction

L-functions

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