LMFDB - Embedded newform 1449.4.a.m.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1449,4,Mod(1,1449)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1449, 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("1449.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1449 = 3^{2} \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1449.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:	$85.4937675983$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 57x^{7} - 13x^{6} + 1042x^{5} + 331x^{4} - 6570x^{3} - 1782x^{2} + 9424x + 5112 $ x^9 - 57x^7 - 13x^6 + 1042x^5 + 331x^4 - 6570x^3 - 1782x^2 + 9424*x + 5112
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 483)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$4.38536$ of defining polynomial
Character	$\chi$	$=$	1449.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.38536 q^{2} +11.2314 q^{4} +12.0618 q^{5} +7.00000 q^{7} +14.1708 q^{8} +O(q^{10})$	\(q+4.38536 q^{2} +11.2314 q^{4} +12.0618 q^{5} +7.00000 q^{7} +14.1708 q^{8} +52.8954 q^{10} +1.44787 q^{11} +49.8433 q^{13} +30.6975 q^{14} -27.7072 q^{16} +71.7055 q^{17} +38.5034 q^{19} +135.471 q^{20} +6.34942 q^{22} -23.0000 q^{23} +20.4873 q^{25} +218.581 q^{26} +78.6196 q^{28} +166.975 q^{29} +4.42781 q^{31} -234.872 q^{32} +314.454 q^{34} +84.4327 q^{35} -267.412 q^{37} +168.851 q^{38} +170.925 q^{40} -148.225 q^{41} +70.2323 q^{43} +16.2616 q^{44} -100.863 q^{46} +527.600 q^{47} +49.0000 q^{49} +89.8441 q^{50} +559.809 q^{52} +123.364 q^{53} +17.4639 q^{55} +99.1953 q^{56} +732.244 q^{58} +446.325 q^{59} -34.0113 q^{61} +19.4175 q^{62} -808.341 q^{64} +601.201 q^{65} +191.170 q^{67} +805.352 q^{68} +370.268 q^{70} -767.142 q^{71} +413.998 q^{73} -1172.70 q^{74} +432.447 q^{76} +10.1351 q^{77} +488.008 q^{79} -334.199 q^{80} -650.019 q^{82} -958.312 q^{83} +864.898 q^{85} +307.994 q^{86} +20.5174 q^{88} +1234.50 q^{89} +348.903 q^{91} -258.322 q^{92} +2313.72 q^{94} +464.421 q^{95} +817.516 q^{97} +214.883 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 42 q^{4} - 29 q^{5} + 63 q^{7} + 39 q^{8}+O(q^{10}) $ 9 * q + 42 * q^4 - 29 * q^5 + 63 * q^7 + 39 * q^8	$ 9 q + 42 q^{4} - 29 q^{5} + 63 q^{7} + 39 q^{8} - 55 q^{10} - 12 q^{11} + 199 q^{13} + 170 q^{16} - 116 q^{17} + 260 q^{19} - 324 q^{20} - 265 q^{22} - 207 q^{23} + 438 q^{25} + 270 q^{26} + 294 q^{28} + 107 q^{29} + 440 q^{31} + 802 q^{32} + 295 q^{34} - 203 q^{35} + 563 q^{37} + 569 q^{38} - 640 q^{40} - 243 q^{41} + 435 q^{43} - 1025 q^{44} + 133 q^{47} + 441 q^{49} - 104 q^{50} + 2693 q^{52} - 958 q^{53} + 1846 q^{55} + 273 q^{56} + 2796 q^{58} - 538 q^{59} + 1374 q^{61} - 1263 q^{62} - 83 q^{64} - 745 q^{65} + 752 q^{67} - 5593 q^{68} - 385 q^{70} + 418 q^{71} + 2406 q^{73} - 352 q^{74} + 2765 q^{76} - 84 q^{77} - 486 q^{79} - 5709 q^{80} + 2726 q^{82} - 106 q^{83} + 4130 q^{85} + 2576 q^{86} + 1270 q^{88} - 234 q^{89} + 1393 q^{91} - 966 q^{92} + 4967 q^{94} + 3074 q^{95} + 2409 q^{97}+O(q^{100}) $ 9 * q + 42 * q^4 - 29 * q^5 + 63 * q^7 + 39 * q^8 - 55 * q^10 - 12 * q^11 + 199 * q^13 + 170 * q^16 - 116 * q^17 + 260 * q^19 - 324 * q^20 - 265 * q^22 - 207 * q^23 + 438 * q^25 + 270 * q^26 + 294 * q^28 + 107 * q^29 + 440 * q^31 + 802 * q^32 + 295 * q^34 - 203 * q^35 + 563 * q^37 + 569 * q^38 - 640 * q^40 - 243 * q^41 + 435 * q^43 - 1025 * q^44 + 133 * q^47 + 441 * q^49 - 104 * q^50 + 2693 * q^52 - 958 * q^53 + 1846 * q^55 + 273 * q^56 + 2796 * q^58 - 538 * q^59 + 1374 * q^61 - 1263 * q^62 - 83 * q^64 - 745 * q^65 + 752 * q^67 - 5593 * q^68 - 385 * q^70 + 418 * q^71 + 2406 * q^73 - 352 * q^74 + 2765 * q^76 - 84 * q^77 - 486 * q^79 - 5709 * q^80 + 2726 * q^82 - 106 * q^83 + 4130 * q^85 + 2576 * q^86 + 1270 * q^88 - 234 * q^89 + 1393 * q^91 - 966 * q^92 + 4967 * q^94 + 3074 * q^95 + 2409 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	4.38536	1.55046	0.775229	−	0.631680i	$-0.217634\pi$
$2$	4.38536	1.55046	0.775229	+	0.631680i	$0.217634\pi$
$3$	0	0
$3$	0	0
$4$	11.2314	1.40392
$5$	12.0618	1.07884	0.539421	−	0.842036i	$-0.318643\pi$
$5$	12.0618	1.07884	0.539421	+	0.842036i	$0.318643\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	14.1708	0.626265
$9$	0	0
$10$	52.8954	1.67270
$11$	1.44787	0.0396862	0.0198431	−	0.999803i	$-0.493683\pi$
$11$	1.44787	0.0396862	0.0198431	+	0.999803i	$0.493683\pi$
$12$	0	0
$13$	49.8433	1.06339	0.531694	−	0.846936i	$-0.321556\pi$
$13$	49.8433	1.06339	0.531694	+	0.846936i	$0.321556\pi$
$14$	30.6975	0.586018
$15$	0	0
$16$	−27.7072	−0.432925
$17$	71.7055	1.02301	0.511504	−	0.859281i	$-0.329089\pi$
$17$	71.7055	1.02301	0.511504	+	0.859281i	$0.329089\pi$
$18$	0	0
$19$	38.5034	0.464910	0.232455	−	0.972607i	$-0.425324\pi$
$19$	38.5034	0.464910	0.232455	+	0.972607i	$0.425324\pi$
$20$	135.471	1.51461
$21$	0	0
$22$	6.34942	0.0615319
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	0	0
$25$	20.4873	0.163898
$26$	218.581	1.64874
$27$	0	0
$28$	78.6196	0.530633
$29$	166.975	1.06919	0.534593	−	0.845109i	$-0.320465\pi$
$29$	166.975	1.06919	0.534593	+	0.845109i	$0.320465\pi$
$30$	0	0
$31$	4.42781	0.0256535	0.0128267	−	0.999918i	$-0.495917\pi$
$31$	4.42781	0.0256535	0.0128267	+	0.999918i	$0.495917\pi$
$32$	−234.872	−1.29750
$33$	0	0
$34$	314.454	1.58613
$35$	84.4327	0.407764
$36$	0	0
$37$	−267.412	−1.18817	−0.594086	−	0.804402i	$-0.702486\pi$
$37$	−267.412	−1.18817	−0.594086	+	0.804402i	$0.702486\pi$
$38$	168.851	0.720824
$39$	0	0
$40$	170.925	0.675640
$41$	−148.225	−0.564605	−0.282303	−	0.959325i	$-0.591098\pi$
$41$	−148.225	−0.564605	−0.282303	+	0.959325i	$0.591098\pi$
$42$	0	0
$43$	70.2323	0.249077	0.124539	−	0.992215i	$-0.460255\pi$
$43$	70.2323	0.249077	0.124539	+	0.992215i	$0.460255\pi$
$44$	16.2616	0.0557164
$45$	0	0
$46$	−100.863	−0.323293
$47$	527.600	1.63741	0.818706	−	0.574213i	$-0.194692\pi$
$47$	527.600	1.63741	0.818706	+	0.574213i	$0.194692\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	89.8441	0.254117
$51$	0	0
$52$	559.809	1.49291
$53$	123.364	0.319724	0.159862	−	0.987139i	$-0.448895\pi$
$53$	123.364	0.319724	0.159862	+	0.987139i	$0.448895\pi$
$54$	0	0
$55$	17.4639	0.0428152
$56$	99.1953	0.236706
$57$	0	0
$58$	732.244	1.65773
$59$	446.325	0.984857	0.492428	−	0.870353i	$-0.336109\pi$
$59$	446.325	0.984857	0.492428	+	0.870353i	$0.336109\pi$
$60$	0	0
$61$	−34.0113	−0.0713886	−0.0356943	−	0.999363i	$-0.511364\pi$
$61$	−34.0113	−0.0713886	−0.0356943	+	0.999363i	$0.511364\pi$
$62$	19.4175	0.0397746
$63$	0	0
$64$	−808.341	−1.57879
$65$	601.201	1.14723
$66$	0	0
$67$	191.170	0.348584	0.174292	−	0.984694i	$-0.444236\pi$
$67$	191.170	0.348584	0.174292	+	0.984694i	$0.444236\pi$
$68$	805.352	1.43622
$69$	0	0
$70$	370.268	0.632221
$71$	−767.142	−1.28230	−0.641148	−	0.767417i	$-0.721542\pi$
$71$	−767.142	−1.28230	−0.641148	+	0.767417i	$0.721542\pi$
$72$	0	0
$73$	413.998	0.663765	0.331882	−	0.943321i	$-0.392316\pi$
$73$	413.998	0.663765	0.331882	+	0.943321i	$0.392316\pi$
$74$	−1172.70	−1.84221
$75$	0	0
$76$	432.447	0.652698
$77$	10.1351	0.0150000
$78$	0	0
$79$	488.008	0.695002	0.347501	−	0.937680i	$-0.387030\pi$
$79$	488.008	0.695002	0.347501	+	0.937680i	$0.387030\pi$
$80$	−334.199	−0.467057
$81$	0	0
$82$	−650.019	−0.875397
$83$	−958.312	−1.26733	−0.633665	−	0.773607i	$-0.718450\pi$
$83$	−958.312	−1.26733	−0.633665	+	0.773607i	$0.718450\pi$
$84$	0	0
$85$	864.898	1.10366
$86$	307.994	0.386184
$87$	0	0
$88$	20.5174	0.0248541
$89$	1234.50	1.47030	0.735151	−	0.677904i	$-0.237111\pi$
$89$	1234.50	1.47030	0.735151	+	0.677904i	$0.237111\pi$
$90$	0	0
$91$	348.903	0.401923
$92$	−258.322	−0.292738
$93$	0	0
$94$	2313.72	2.53874
$95$	464.421	0.501564
$96$	0	0
$97$	817.516	0.855734	0.427867	−	0.903842i	$-0.359265\pi$
$97$	817.516	0.855734	0.427867	+	0.903842i	$0.359265\pi$
$98$	214.883	0.221494
$99$	0	0
$100$	230.100	0.230100
$101$	−1949.94	−1.92105	−0.960525	−	0.278194i	$-0.910264\pi$
$101$	−1949.94	−1.92105	−0.960525	+	0.278194i	$0.910264\pi$
$102$	0	0
$103$	767.975	0.734668	0.367334	−	0.930089i	$-0.380271\pi$
$103$	767.975	0.734668	0.367334	+	0.930089i	$0.380271\pi$
$104$	706.317	0.665963
$105$	0	0
$106$	540.996	0.495719
$107$	−284.797	−0.257312	−0.128656	−	0.991689i	$-0.541066\pi$
$107$	−284.797	−0.257312	−0.128656	+	0.991689i	$0.541066\pi$
$108$	0	0
$109$	−28.9882	−0.0254730	−0.0127365	−	0.999919i	$-0.504054\pi$
$109$	−28.9882	−0.0254730	−0.0127365	+	0.999919i	$0.504054\pi$
$110$	76.5855	0.0663831
$111$	0	0
$112$	−193.950	−0.163630
$113$	−1029.28	−0.856875	−0.428438	−	0.903571i	$-0.640936\pi$
$113$	−1029.28	−0.856875	−0.428438	+	0.903571i	$0.640936\pi$
$114$	0	0
$115$	−277.422	−0.224954
$116$	1875.36	1.50105
$117$	0	0
$118$	1957.29	1.52698
$119$	501.938	0.386661
$120$	0	0
$121$	−1328.90	−0.998425
$122$	−149.152	−0.110685
$123$	0	0
$124$	49.7304	0.0360155
$125$	−1260.61	−0.902021
$126$	0	0
$127$	2060.27	1.43952	0.719762	−	0.694221i	$-0.244251\pi$
$127$	2060.27	1.43952	0.719762	+	0.694221i	$0.244251\pi$
$128$	−1665.89	−1.15035
$129$	0	0
$130$	2636.48	1.77873
$131$	−913.286	−0.609116	−0.304558	−	0.952494i	$-0.598509\pi$
$131$	−913.286	−0.609116	−0.304558	+	0.952494i	$0.598509\pi$
$132$	0	0
$133$	269.524	0.175720
$134$	838.349	0.540465
$135$	0	0
$136$	1016.12	0.640674
$137$	1434.55	0.894613	0.447307	−	0.894381i	$-0.352383\pi$
$137$	1434.55	0.894613	0.447307	+	0.894381i	$0.352383\pi$
$138$	0	0
$139$	−921.342	−0.562210	−0.281105	−	0.959677i	$-0.590701\pi$
$139$	−921.342	−0.562210	−0.281105	+	0.959677i	$0.590701\pi$
$140$	948.295	0.572468
$141$	0	0
$142$	−3364.19	−1.98815
$143$	72.1666	0.0422019
$144$	0	0
$145$	2014.02	1.15348
$146$	1815.53	1.02914
$147$	0	0
$148$	−3003.41	−1.66810
$149$	1073.20	0.590064	0.295032	−	0.955487i	$-0.404670\pi$
$149$	1073.20	0.590064	0.295032	+	0.955487i	$0.404670\pi$
$150$	0	0
$151$	−1051.25	−0.566555	−0.283277	−	0.959038i	$-0.591422\pi$
$151$	−1051.25	−0.566555	−0.283277	+	0.959038i	$0.591422\pi$
$152$	545.623	0.291157
$153$	0	0
$154$	44.4460	0.0232569
$155$	53.4073	0.0276760
$156$	0	0
$157$	3252.41	1.65332	0.826658	−	0.562705i	$-0.190239\pi$
$157$	3252.41	1.65332	0.826658	+	0.562705i	$0.190239\pi$
$158$	2140.09	1.07757
$159$	0	0
$160$	−2832.98	−1.39979
$161$	−161.000	−0.0788110
$162$	0	0
$163$	54.9590	0.0264094	0.0132047	−	0.999913i	$-0.495797\pi$
$163$	54.9590	0.0264094	0.0132047	+	0.999913i	$0.495797\pi$
$164$	−1664.77	−0.792662
$165$	0	0
$166$	−4202.54	−1.96494
$167$	644.264	0.298531	0.149265	−	0.988797i	$-0.452309\pi$
$167$	644.264	0.298531	0.149265	+	0.988797i	$0.452309\pi$
$168$	0	0
$169$	287.356	0.130795
$170$	3792.89	1.71118
$171$	0	0
$172$	788.806	0.349685
$173$	−3866.86	−1.69937	−0.849687	−	0.527288i	$-0.823209\pi$
$173$	−3866.86	−1.69937	−0.849687	+	0.527288i	$0.823209\pi$
$174$	0	0
$175$	143.411	0.0619477
$176$	−40.1163	−0.0171811
$177$	0	0
$178$	5413.73	2.27964
$179$	337.689	0.141006	0.0705030	−	0.997512i	$-0.477540\pi$
$179$	337.689	0.141006	0.0705030	+	0.997512i	$0.477540\pi$
$180$	0	0
$181$	−2629.96	−1.08002	−0.540010	−	0.841659i	$-0.681579\pi$
$181$	−2629.96	−1.08002	−0.540010	+	0.841659i	$0.681579\pi$
$182$	1530.07	0.623165
$183$	0	0
$184$	−325.927	−0.130585
$185$	−3225.48	−1.28185
$186$	0	0
$187$	103.820	0.0405994
$188$	5925.67	2.29880
$189$	0	0
$190$	2036.65	0.777655
$191$	1380.11	0.522834	0.261417	−	0.965226i	$-0.415810\pi$
$191$	1380.11	0.522834	0.261417	+	0.965226i	$0.415810\pi$
$192$	0	0
$193$	−3754.76	−1.40038	−0.700190	−	0.713957i	$-0.746901\pi$
$193$	−3754.76	−1.40038	−0.700190	+	0.713957i	$0.746901\pi$
$194$	3585.10	1.32678
$195$	0	0
$196$	550.338	0.200560
$197$	264.979	0.0958325	0.0479163	−	0.998851i	$-0.484742\pi$
$197$	264.979	0.0958325	0.0479163	+	0.998851i	$0.484742\pi$
$198$	0	0
$199$	332.298	0.118372	0.0591859	−	0.998247i	$-0.481150\pi$
$199$	332.298	0.118372	0.0591859	+	0.998247i	$0.481150\pi$
$200$	290.320	0.102644
$201$	0	0
$202$	−8551.18	−2.97851
$203$	1168.82	0.404115
$204$	0	0
$205$	−1787.86	−0.609119
$206$	3367.85	1.13907
$207$	0	0
$208$	−1381.02	−0.460367
$209$	55.7479	0.0184505
$210$	0	0
$211$	−4.29261	−0.00140055	−0.000700274	−	1.00000i	$-0.500223\pi$
$211$	−4.29261	−0.00140055	−0.000700274		1.00000i	$0.500223\pi$
$212$	1385.55	0.448867
$213$	0	0
$214$	−1248.94	−0.398951
$215$	847.129	0.268715
$216$	0	0
$217$	30.9946	0.00969610
$218$	−127.123	−0.0394949
$219$	0	0
$220$	196.144	0.0601092
$221$	3574.04	1.08785
$222$	0	0
$223$	1329.32	0.399183	0.199592	−	0.979879i	$-0.436038\pi$
$223$	1329.32	0.399183	0.199592	+	0.979879i	$0.436038\pi$
$224$	−1644.10	−0.490408
$225$	0	0
$226$	−4513.78	−1.32855
$227$	−4978.77	−1.45574	−0.727869	−	0.685717i	$-0.759489\pi$
$227$	−4978.77	−1.45574	−0.727869	+	0.685717i	$0.759489\pi$
$228$	0	0
$229$	−1182.55	−0.341245	−0.170623	−	0.985336i	$-0.554578\pi$
$229$	−1182.55	−0.341245	−0.170623	+	0.985336i	$0.554578\pi$
$230$	−1216.59	−0.348782
$231$	0	0
$232$	2366.16	0.669594
$233$	−1222.12	−0.343622	−0.171811	−	0.985130i	$-0.554962\pi$
$233$	−1222.12	−0.343622	−0.171811	+	0.985130i	$0.554962\pi$
$234$	0	0
$235$	6363.81	1.76651
$236$	5012.84	1.38266
$237$	0	0
$238$	2201.18	0.599501
$239$	3967.06	1.07367	0.536836	−	0.843687i	$-0.319619\pi$
$239$	3967.06	1.07367	0.536836	+	0.843687i	$0.319619\pi$
$240$	0	0
$241$	2386.82	0.637960	0.318980	−	0.947762i	$-0.396660\pi$
$241$	2386.82	0.637960	0.318980	+	0.947762i	$0.396660\pi$
$242$	−5827.72	−1.54802
$243$	0	0
$244$	−381.994	−0.100224
$245$	591.029	0.154120
$246$	0	0
$247$	1919.14	0.494380
$248$	62.7453	0.0160659
$249$	0	0
$250$	−5528.24	−1.39855
$251$	2664.40	0.670022	0.335011	−	0.942214i	$-0.391260\pi$
$251$	2664.40	0.670022	0.335011	+	0.942214i	$0.391260\pi$
$252$	0	0
$253$	−33.3010	−0.00827515
$254$	9035.03	2.23192
$255$	0	0
$256$	−838.795	−0.204784
$257$	−4717.25	−1.14496	−0.572479	−	0.819919i	$-0.694018\pi$
$257$	−4717.25	−1.14496	−0.572479	+	0.819919i	$0.694018\pi$
$258$	0	0
$259$	−1871.89	−0.449087
$260$	6752.31	1.61062
$261$	0	0
$262$	−4005.09	−0.944409
$263$	−3875.68	−0.908687	−0.454344	−	0.890827i	$-0.650126\pi$
$263$	−3875.68	−0.908687	−0.454344	+	0.890827i	$0.650126\pi$
$264$	0	0
$265$	1487.99	0.344931
$266$	1181.96	0.272446
$267$	0	0
$268$	2147.10	0.489385
$269$	−2659.30	−0.602752	−0.301376	−	0.953505i	$-0.597446\pi$
$269$	−2659.30	−0.602752	−0.301376	+	0.953505i	$0.597446\pi$
$270$	0	0
$271$	2661.36	0.596553	0.298277	−	0.954479i	$-0.403588\pi$
$271$	2661.36	0.596553	0.298277	+	0.954479i	$0.403588\pi$
$272$	−1986.76	−0.442885
$273$	0	0
$274$	6291.03	1.38706
$275$	29.6629	0.00650450
$276$	0	0
$277$	4783.85	1.03767	0.518833	−	0.854876i	$-0.326367\pi$
$277$	4783.85	1.03767	0.518833	+	0.854876i	$0.326367\pi$
$278$	−4040.42	−0.871684
$279$	0	0
$280$	1196.47	0.255368
$281$	−7979.98	−1.69411	−0.847056	−	0.531504i	$-0.821627\pi$
$281$	−7979.98	−1.69411	−0.847056	+	0.531504i	$0.821627\pi$
$282$	0	0
$283$	5908.29	1.24103	0.620515	−	0.784195i	$-0.286924\pi$
$283$	5908.29	1.24103	0.620515	+	0.784195i	$0.286924\pi$
$284$	−8616.06	−1.80024
$285$	0	0
$286$	316.476	0.0654323
$287$	−1037.57	−0.213401
$288$	0	0
$289$	228.678	0.0465456
$290$	8832.19	1.78843
$291$	0	0
$292$	4649.77	0.931874
$293$	−6604.93	−1.31694	−0.658472	−	0.752606i	$-0.728797\pi$
$293$	−6604.93	−1.31694	−0.658472	+	0.752606i	$0.728797\pi$
$294$	0	0
$295$	5383.48	1.06250
$296$	−3789.44	−0.744110
$297$	0	0
$298$	4706.35	0.914870
$299$	−1146.40	−0.221732
$300$	0	0
$301$	491.626	0.0941424
$302$	−4610.12	−0.878420
$303$	0	0
$304$	−1066.82	−0.201271
$305$	−410.238	−0.0770170
$306$	0	0
$307$	4681.03	0.870230	0.435115	−	0.900375i	$-0.356708\pi$
$307$	4681.03	0.870230	0.435115	+	0.900375i	$0.356708\pi$
$308$	113.831	0.0210588
$309$	0	0
$310$	234.210	0.0429105
$311$	2758.35	0.502932	0.251466	−	0.967866i	$-0.419087\pi$
$311$	2758.35	0.502932	0.251466	+	0.967866i	$0.419087\pi$
$312$	0	0
$313$	−5675.55	−1.02492	−0.512462	−	0.858710i	$-0.671267\pi$
$313$	−5675.55	−1.02492	−0.512462	+	0.858710i	$0.671267\pi$
$314$	14263.0	2.56340
$315$	0	0
$316$	5481.00	0.975729
$317$	1829.95	0.324227	0.162114	−	0.986772i	$-0.448169\pi$
$317$	1829.95	0.324227	0.162114	+	0.986772i	$0.448169\pi$
$318$	0	0
$319$	241.757	0.0424320
$320$	−9750.05	−1.70326
$321$	0	0
$322$	−706.043	−0.122193
$323$	2760.91	0.475607
$324$	0	0
$325$	1021.15	0.174287
$326$	241.015	0.0409466
$327$	0	0
$328$	−2100.46	−0.353592
$329$	3693.20	0.618883
$330$	0	0
$331$	6010.63	0.998109	0.499055	−	0.866571i	$-0.333681\pi$
$331$	6010.63	0.998109	0.499055	+	0.866571i	$0.333681\pi$
$332$	−10763.2	−1.77923
$333$	0	0
$334$	2825.33	0.462859
$335$	2305.86	0.376067
$336$	0	0
$337$	3716.64	0.600766	0.300383	−	0.953819i	$-0.402886\pi$
$337$	3716.64	0.600766	0.300383	+	0.953819i	$0.402886\pi$
$338$	1260.16	0.202792
$339$	0	0
$340$	9714.00	1.54946
$341$	6.41088	0.00101809
$342$	0	0
$343$	343.000	0.0539949
$344$	995.245	0.155988
$345$	0	0
$346$	−16957.6	−2.63481
$347$	−2752.77	−0.425869	−0.212935	−	0.977066i	$-0.568302\pi$
$347$	−2752.77	−0.425869	−0.212935	+	0.977066i	$0.568302\pi$
$348$	0	0
$349$	−7133.71	−1.09415	−0.547076	−	0.837083i	$-0.684259\pi$
$349$	−7133.71	−1.09415	−0.547076	+	0.837083i	$0.684259\pi$
$350$	628.908	0.0960473
$351$	0	0
$352$	−340.064	−0.0514928
$353$	−9890.20	−1.49122	−0.745612	−	0.666380i	$-0.767843\pi$
$353$	−9890.20	−1.49122	−0.745612	+	0.666380i	$0.767843\pi$
$354$	0	0
$355$	−9253.12	−1.38339
$356$	13865.1	2.06419
$357$	0	0
$358$	1480.89	0.218624
$359$	1330.58	0.195614	0.0978070	−	0.995205i	$-0.468817\pi$
$359$	1330.58	0.195614	0.0978070	+	0.995205i	$0.468817\pi$
$360$	0	0
$361$	−5376.49	−0.783859
$362$	−11533.3	−1.67453
$363$	0	0
$364$	3918.66	0.564269
$365$	4993.57	0.716097
$366$	0	0
$367$	−8527.15	−1.21284	−0.606422	−	0.795143i	$-0.707396\pi$
$367$	−8527.15	−1.21284	−0.606422	+	0.795143i	$0.707396\pi$
$368$	637.265	0.0902710
$369$	0	0
$370$	−14144.9	−1.98745
$371$	863.549	0.120844
$372$	0	0
$373$	−1777.82	−0.246788	−0.123394	−	0.992358i	$-0.539378\pi$
$373$	−1777.82	−0.246788	−0.123394	+	0.992358i	$0.539378\pi$
$374$	455.289	0.0629476
$375$	0	0
$376$	7476.49	1.02545
$377$	8322.57	1.13696
$378$	0	0
$379$	545.049	0.0738714	0.0369357	−	0.999318i	$-0.488240\pi$
$379$	545.049	0.0738714	0.0369357	+	0.999318i	$0.488240\pi$
$380$	5216.09	0.704157
$381$	0	0
$382$	6052.28	0.810633
$383$	−7061.98	−0.942168	−0.471084	−	0.882088i	$-0.656137\pi$
$383$	−7061.98	−0.942168	−0.471084	+	0.882088i	$0.656137\pi$
$384$	0	0
$385$	122.247	0.0161826
$386$	−16466.0	−2.17123
$387$	0	0
$388$	9181.83	1.20138
$389$	2554.78	0.332988	0.166494	−	0.986042i	$-0.446755\pi$
$389$	2554.78	0.332988	0.166494	+	0.986042i	$0.446755\pi$
$390$	0	0
$391$	−1649.23	−0.213312
$392$	694.367	0.0894664
$393$	0	0
$394$	1162.03	0.148584
$395$	5886.26	0.749797
$396$	0	0
$397$	−12815.6	−1.62014	−0.810071	−	0.586331i	$-0.800572\pi$
$397$	−12815.6	−1.62014	−0.810071	+	0.586331i	$0.800572\pi$
$398$	1457.25	0.183531
$399$	0	0
$400$	−567.644	−0.0709555
$401$	−13843.3	−1.72394	−0.861969	−	0.506961i	$-0.830769\pi$
$401$	−13843.3	−1.72394	−0.861969	+	0.506961i	$0.830769\pi$
$402$	0	0
$403$	220.696	0.0272796
$404$	−21900.5	−2.69700
$405$	0	0
$406$	5125.71	0.626563
$407$	−387.178	−0.0471541
$408$	0	0
$409$	10910.7	1.31907	0.659537	−	0.751672i	$-0.270752\pi$
$409$	10910.7	1.31907	0.659537	+	0.751672i	$0.270752\pi$
$410$	−7840.40	−0.944414
$411$	0	0
$412$	8625.42	1.03142
$413$	3124.27	0.372241
$414$	0	0
$415$	−11559.0	−1.36725
$416$	−11706.8	−1.37974
$417$	0	0
$418$	244.475	0.0286068
$419$	−9293.90	−1.08362	−0.541810	−	0.840501i	$-0.682261\pi$
$419$	−9293.90	−1.08362	−0.541810	+	0.840501i	$0.682261\pi$
$420$	0	0
$421$	7051.31	0.816294	0.408147	−	0.912916i	$-0.366175\pi$
$421$	7051.31	0.816294	0.408147	+	0.912916i	$0.366175\pi$
$422$	−18.8247	−0.00217149
$423$	0	0
$424$	1748.16	0.200232
$425$	1469.05	0.167669
$426$	0	0
$427$	−238.079	−0.0269824
$428$	−3198.66	−0.361246
$429$	0	0
$430$	3714.96	0.416631
$431$	−13714.7	−1.53275	−0.766374	−	0.642395i	$-0.777941\pi$
$431$	−13714.7	−1.53275	−0.766374	+	0.642395i	$0.777941\pi$
$432$	0	0
$433$	718.839	0.0797810	0.0398905	−	0.999204i	$-0.487299\pi$
$433$	718.839	0.0797810	0.0398905	+	0.999204i	$0.487299\pi$
$434$	135.923	0.0150334
$435$	0	0
$436$	−325.577	−0.0357622
$437$	−885.579	−0.0969405
$438$	0	0
$439$	−8638.69	−0.939185	−0.469593	−	0.882883i	$-0.655599\pi$
$439$	−8638.69	−0.939185	−0.469593	+	0.882883i	$0.655599\pi$
$440$	247.477	0.0268136
$441$	0	0
$442$	15673.4	1.68667
$443$	2024.07	0.217080	0.108540	−	0.994092i	$-0.465382\pi$
$443$	2024.07	0.217080	0.108540	+	0.994092i	$0.465382\pi$
$444$	0	0
$445$	14890.3	1.58622
$446$	5829.55	0.618917
$447$	0	0
$448$	−5658.38	−0.596727
$449$	3077.85	0.323502	0.161751	−	0.986832i	$-0.448286\pi$
$449$	3077.85	0.323502	0.161751	+	0.986832i	$0.448286\pi$
$450$	0	0
$451$	−214.610	−0.0224071
$452$	−11560.3	−1.20299
$453$	0	0
$454$	−21833.7	−2.25706
$455$	4208.40	0.433611
$456$	0	0
$457$	16054.7	1.64334	0.821672	−	0.569960i	$-0.193041\pi$
$457$	16054.7	1.64334	0.821672	+	0.569960i	$0.193041\pi$
$458$	−5185.91	−0.529087
$459$	0	0
$460$	−3115.83	−0.315818
$461$	−4450.97	−0.449680	−0.224840	−	0.974396i	$-0.572186\pi$
$461$	−4450.97	−0.449680	−0.224840	+	0.974396i	$0.572186\pi$
$462$	0	0
$463$	−280.593	−0.0281647	−0.0140823	−	0.999901i	$-0.504483\pi$
$463$	−280.593	−0.0281647	−0.0140823	+	0.999901i	$0.504483\pi$
$464$	−4626.39	−0.462877
$465$	0	0
$466$	−5359.45	−0.532772
$467$	−5157.88	−0.511088	−0.255544	−	0.966797i	$-0.582255\pi$
$467$	−5157.88	−0.511088	−0.255544	+	0.966797i	$0.582255\pi$
$468$	0	0
$469$	1338.19	0.131752
$470$	27907.6	2.73890
$471$	0	0
$472$	6324.76	0.616781
$473$	101.687	0.00988495
$474$	0	0
$475$	788.830	0.0761979
$476$	5637.46	0.542842
$477$	0	0
$478$	17397.0	1.66468
$479$	−18688.8	−1.78270	−0.891348	−	0.453319i	$-0.850240\pi$
$479$	−18688.8	−1.78270	−0.891348	+	0.453319i	$0.850240\pi$
$480$	0	0
$481$	−13328.7	−1.26349
$482$	10467.0	0.989130
$483$	0	0
$484$	−14925.4	−1.40171
$485$	9860.72	0.923201
$486$	0	0
$487$	−14630.4	−1.36133	−0.680663	−	0.732597i	$-0.738308\pi$
$487$	−14630.4	−1.36133	−0.680663	+	0.732597i	$0.738308\pi$
$488$	−481.966	−0.0447082
$489$	0	0
$490$	2591.87	0.238957
$491$	−8127.57	−0.747031	−0.373515	−	0.927624i	$-0.621848\pi$
$491$	−8127.57	−0.747031	−0.373515	+	0.927624i	$0.621848\pi$
$492$	0	0
$493$	11973.0	1.09379
$494$	8416.11	0.766516
$495$	0	0
$496$	−122.682	−0.0111060
$497$	−5369.99	−0.484662
$498$	0	0
$499$	−18936.8	−1.69885	−0.849426	−	0.527709i	$-0.823051\pi$
$499$	−18936.8	−1.69885	−0.849426	+	0.527709i	$0.823051\pi$
$500$	−14158.4	−1.26637
$501$	0	0
$502$	11684.4	1.03884
$503$	6847.58	0.606995	0.303498	−	0.952832i	$-0.401846\pi$
$503$	6847.58	0.606995	0.303498	+	0.952832i	$0.401846\pi$
$504$	0	0
$505$	−23519.8	−2.07251
$506$	−146.037	−0.0128303
$507$	0	0
$508$	23139.7	2.02098
$509$	−7662.78	−0.667283	−0.333641	−	0.942700i	$-0.608277\pi$
$509$	−7662.78	−0.667283	−0.333641	+	0.942700i	$0.608277\pi$
$510$	0	0
$511$	2897.99	0.250880
$512$	9648.69	0.832843
$513$	0	0
$514$	−20686.8	−1.77521
$515$	9263.17	0.792590
$516$	0	0
$517$	763.895	0.0649827
$518$	−8208.90	−0.696290
$519$	0	0
$520$	8519.47	0.718468
$521$	−934.035	−0.0785428	−0.0392714	−	0.999229i	$-0.512504\pi$
$521$	−934.035	−0.0785428	−0.0392714	+	0.999229i	$0.512504\pi$
$522$	0	0
$523$	4106.44	0.343331	0.171666	−	0.985155i	$-0.445085\pi$
$523$	4106.44	0.343331	0.171666	+	0.985155i	$0.445085\pi$
$524$	−10257.5	−0.855152
$525$	0	0
$526$	−16996.3	−1.40888
$527$	317.498	0.0262437
$528$	0	0
$529$	529.000	0.0434783
$530$	6525.39	0.534802
$531$	0	0
$532$	3027.13	0.246697
$533$	−7388.01	−0.600395
$534$	0	0
$535$	−3435.17	−0.277598
$536$	2709.02	0.218306
$537$	0	0
$538$	−11662.0	−0.934542
$539$	70.9455	0.00566946
$540$	0	0
$541$	−142.254	−0.0113050	−0.00565248	−	0.999984i	$-0.501799\pi$
$541$	−142.254	−0.0113050	−0.00565248	+	0.999984i	$0.501799\pi$
$542$	11671.0	0.924931
$543$	0	0
$544$	−16841.6	−1.32735
$545$	−349.650	−0.0274814
$546$	0	0
$547$	−6349.34	−0.496304	−0.248152	−	0.968721i	$-0.579823\pi$
$547$	−6349.34	−0.496304	−0.248152	+	0.968721i	$0.579823\pi$
$548$	16112.0	1.25597
$549$	0	0
$550$	130.082	0.0100850
$551$	6429.10	0.497076
$552$	0	0
$553$	3416.05	0.262686
$554$	20978.9	1.60886
$555$	0	0
$556$	−10347.9	−0.789299
$557$	−9758.11	−0.742306	−0.371153	−	0.928572i	$-0.621037\pi$
$557$	−9758.11	−0.742306	−0.371153	+	0.928572i	$0.621037\pi$
$558$	0	0
$559$	3500.61	0.264866
$560$	−2339.39	−0.176531
$561$	0	0
$562$	−34995.1	−2.62665
$563$	−11219.1	−0.839834	−0.419917	−	0.907562i	$-0.637941\pi$
$563$	−11219.1	−0.839834	−0.419917	+	0.907562i	$0.637941\pi$
$564$	0	0
$565$	−12415.0	−0.924432
$566$	25910.0	1.92416
$567$	0	0
$568$	−10871.0	−0.803057
$569$	−21521.9	−1.58567	−0.792834	−	0.609438i	$-0.791395\pi$
$569$	−21521.9	−1.58567	−0.792834	+	0.609438i	$0.791395\pi$
$570$	0	0
$571$	−14972.5	−1.09734	−0.548670	−	0.836039i	$-0.684866\pi$
$571$	−14972.5	−1.09734	−0.548670	+	0.836039i	$0.684866\pi$
$572$	810.530	0.0592482
$573$	0	0
$574$	−4550.13	−0.330869
$575$	−471.207	−0.0341751
$576$	0	0
$577$	−1491.88	−0.107639	−0.0538197	−	0.998551i	$-0.517140\pi$
$577$	−1491.88	−0.107639	−0.0538197	+	0.998551i	$0.517140\pi$
$578$	1002.84	0.0721670
$579$	0	0
$580$	22620.2	1.61940
$581$	−6708.19	−0.479006
$582$	0	0
$583$	178.615	0.0126886
$584$	5866.67	0.415693
$585$	0	0
$586$	−28965.0	−2.04187
$587$	12343.1	0.867896	0.433948	−	0.900938i	$-0.357120\pi$
$587$	12343.1	0.867896	0.433948	+	0.900938i	$0.357120\pi$
$588$	0	0
$589$	170.486	0.0119266
$590$	23608.5	1.64737
$591$	0	0
$592$	7409.24	0.514389
$593$	−4562.03	−0.315920	−0.157960	−	0.987446i	$-0.550492\pi$
$593$	−4562.03	−0.315920	−0.157960	+	0.987446i	$0.550492\pi$
$594$	0	0
$595$	6054.29	0.417145
$596$	12053.5	0.828404
$597$	0	0
$598$	−5027.36	−0.343786
$599$	26111.3	1.78110	0.890549	−	0.454887i	$-0.150320\pi$
$599$	26111.3	1.78110	0.890549	+	0.454887i	$0.150320\pi$
$600$	0	0
$601$	16551.4	1.12337	0.561687	−	0.827350i	$-0.310153\pi$
$601$	16551.4	1.12337	0.561687	+	0.827350i	$0.310153\pi$
$602$	2155.96	0.145964
$603$	0	0
$604$	−11807.0	−0.795399
$605$	−16029.0	−1.07714
$606$	0	0
$607$	16986.1	1.13582	0.567911	−	0.823090i	$-0.307752\pi$
$607$	16986.1	1.13582	0.567911	+	0.823090i	$0.307752\pi$
$608$	−9043.38	−0.603219
$609$	0	0
$610$	−1799.04	−0.119412
$611$	26297.3	1.74120
$612$	0	0
$613$	−886.142	−0.0583865	−0.0291932	−	0.999574i	$-0.509294\pi$
$613$	−886.142	−0.0583865	−0.0291932	+	0.999574i	$0.509294\pi$
$614$	20528.0	1.34926
$615$	0	0
$616$	143.622	0.00939397
$617$	7008.47	0.457294	0.228647	−	0.973509i	$-0.426570\pi$
$617$	7008.47	0.457294	0.228647	+	0.973509i	$0.426570\pi$
$618$	0	0
$619$	2235.39	0.145150	0.0725749	−	0.997363i	$-0.476878\pi$
$619$	2235.39	0.145150	0.0725749	+	0.997363i	$0.476878\pi$
$620$	599.838	0.0388550
$621$	0	0
$622$	12096.4	0.779775
$623$	8641.50	0.555722
$624$	0	0
$625$	−17766.2	−1.13704
$626$	−24889.3	−1.58910
$627$	0	0
$628$	36529.1	2.32113
$629$	−19174.9	−1.21551
$630$	0	0
$631$	−25788.3	−1.62696	−0.813482	−	0.581590i	$-0.802431\pi$
$631$	−25788.3	−1.62696	−0.813482	+	0.581590i	$0.802431\pi$
$632$	6915.44	0.435255
$633$	0	0
$634$	8024.98	0.502701
$635$	24850.6	1.55302
$636$	0	0
$637$	2442.32	0.151913
$638$	1060.19	0.0657891
$639$	0	0
$640$	−20093.6	−1.24105
$641$	16947.4	1.04428	0.522139	−	0.852860i	$-0.325134\pi$
$641$	16947.4	1.04428	0.522139	+	0.852860i	$0.325134\pi$
$642$	0	0
$643$	3816.96	0.234100	0.117050	−	0.993126i	$-0.462656\pi$
$643$	3816.96	0.234100	0.117050	+	0.993126i	$0.462656\pi$
$644$	−1808.25	−0.110645
$645$	0	0
$646$	12107.6	0.737409
$647$	23895.9	1.45200	0.726002	−	0.687692i	$-0.241376\pi$
$647$	23895.9	1.45200	0.726002	+	0.687692i	$0.241376\pi$
$648$	0	0
$649$	646.219	0.0390853
$650$	4478.13	0.270225
$651$	0	0
$652$	617.266	0.0370767
$653$	−22903.3	−1.37255	−0.686276	−	0.727341i	$-0.740756\pi$
$653$	−22903.3	−1.37255	−0.686276	+	0.727341i	$0.740756\pi$
$654$	0	0
$655$	−11015.9	−0.657139
$656$	4106.89	0.244431
$657$	0	0
$658$	16196.0	0.959553
$659$	13680.4	0.808668	0.404334	−	0.914612i	$-0.367503\pi$
$659$	13680.4	0.808668	0.404334	+	0.914612i	$0.367503\pi$
$660$	0	0
$661$	1348.04	0.0793232	0.0396616	−	0.999213i	$-0.487372\pi$
$661$	1348.04	0.0793232	0.0396616	+	0.999213i	$0.487372\pi$
$662$	26358.8	1.54753
$663$	0	0
$664$	−13580.0	−0.793685
$665$	3250.95	0.189573
$666$	0	0
$667$	−3840.42	−0.222941
$668$	7235.97	0.419114
$669$	0	0
$670$	10112.0	0.583076
$671$	−49.2439	−0.00283315
$672$	0	0
$673$	32816.2	1.87960	0.939799	−	0.341727i	$-0.111012\pi$
$673$	32816.2	1.87960	0.939799	+	0.341727i	$0.111012\pi$
$674$	16298.8	0.931463
$675$	0	0
$676$	3227.40	0.183626
$677$	−21382.7	−1.21389	−0.606946	−	0.794743i	$-0.707606\pi$
$677$	−21382.7	−1.21389	−0.606946	+	0.794743i	$0.707606\pi$
$678$	0	0
$679$	5722.61	0.323437
$680$	12256.3	0.691185
$681$	0	0
$682$	28.1140	0.00157851
$683$	32937.1	1.84525	0.922624	−	0.385701i	$-0.126040\pi$
$683$	32937.1	1.84525	0.922624	+	0.385701i	$0.126040\pi$
$684$	0	0
$685$	17303.3	0.965146
$686$	1504.18	0.0837169
$687$	0	0
$688$	−1945.94	−0.107832
$689$	6148.88	0.339991
$690$	0	0
$691$	2994.69	0.164868	0.0824338	−	0.996597i	$-0.473731\pi$
$691$	2994.69	0.164868	0.0824338	+	0.996597i	$0.473731\pi$
$692$	−43430.1	−2.38579
$693$	0	0
$694$	−12071.9	−0.660293
$695$	−11113.1	−0.606535
$696$	0	0
$697$	−10628.5	−0.577596
$698$	−31283.9	−1.69644
$699$	0	0
$700$	1610.70	0.0869697
$701$	27379.4	1.47519	0.737593	−	0.675245i	$-0.235962\pi$
$701$	27379.4	1.47519	0.737593	+	0.675245i	$0.235962\pi$
$702$	0	0
$703$	−10296.3	−0.552393
$704$	−1170.37	−0.0626563
$705$	0	0
$706$	−43372.1	−2.31208
$707$	−13649.6	−0.726089
$708$	0	0
$709$	17869.8	0.946565	0.473282	−	0.880911i	$-0.343069\pi$
$709$	17869.8	0.946565	0.473282	+	0.880911i	$0.343069\pi$
$710$	−40578.3	−2.14489
$711$	0	0
$712$	17493.8	0.920798
$713$	−101.840	−0.00534912
$714$	0	0
$715$	870.459	0.0455291
$716$	3792.71	0.197961
$717$	0	0
$718$	5835.08	0.303291
$719$	6637.25	0.344267	0.172133	−	0.985074i	$-0.444934\pi$
$719$	6637.25	0.344267	0.172133	+	0.985074i	$0.444934\pi$
$720$	0	0
$721$	5375.83	0.277679
$722$	−23577.8	−1.21534
$723$	0	0
$724$	−29538.1	−1.51626
$725$	3420.85	0.175238
$726$	0	0
$727$	1527.99	0.0779505	0.0389752	−	0.999240i	$-0.487591\pi$
$727$	1527.99	0.0779505	0.0389752	+	0.999240i	$0.487591\pi$
$728$	4944.22	0.251710
$729$	0	0
$730$	21898.6	1.11028
$731$	5036.04	0.254808
$732$	0	0
$733$	32912.8	1.65848	0.829238	−	0.558895i	$-0.188775\pi$
$733$	32912.8	1.65848	0.829238	+	0.558895i	$0.188775\pi$
$734$	−37394.6	−1.88046
$735$	0	0
$736$	5402.05	0.270547
$737$	276.789	0.0138340
$738$	0	0
$739$	25540.8	1.27136	0.635679	−	0.771953i	$-0.280720\pi$
$739$	25540.8	1.27136	0.635679	+	0.771953i	$0.280720\pi$
$740$	−36226.6	−1.79962
$741$	0	0
$742$	3786.97	0.187364
$743$	5849.78	0.288839	0.144420	−	0.989517i	$-0.453868\pi$
$743$	5849.78	0.288839	0.144420	+	0.989517i	$0.453868\pi$
$744$	0	0
$745$	12944.7	0.636586
$746$	−7796.36	−0.382634
$747$	0	0
$748$	1166.04	0.0569983
$749$	−1993.58	−0.0972547
$750$	0	0
$751$	−17940.5	−0.871717	−0.435858	−	0.900015i	$-0.643555\pi$
$751$	−17940.5	−0.871717	−0.435858	+	0.900015i	$0.643555\pi$
$752$	−14618.3	−0.708876
$753$	0	0
$754$	36497.5	1.76281
$755$	−12680.0	−0.611222
$756$	0	0
$757$	9988.21	0.479561	0.239780	−	0.970827i	$-0.422925\pi$
$757$	9988.21	0.479561	0.239780	+	0.970827i	$0.422925\pi$
$758$	2390.23	0.114535
$759$	0	0
$760$	6581.20	0.314112
$761$	−8181.82	−0.389738	−0.194869	−	0.980829i	$-0.562428\pi$
$761$	−8181.82	−0.389738	−0.194869	+	0.980829i	$0.562428\pi$
$762$	0	0
$763$	−202.917	−0.00962790
$764$	15500.5	0.734018
$765$	0	0
$766$	−30969.3	−1.46079
$767$	22246.3	1.04728
$768$	0	0
$769$	−8052.26	−0.377597	−0.188798	−	0.982016i	$-0.560459\pi$
$769$	−8052.26	−0.377597	−0.188798	+	0.982016i	$0.560459\pi$
$770$	536.099	0.0250905
$771$	0	0
$772$	−42171.1	−1.96602
$773$	10348.5	0.481512	0.240756	−	0.970586i	$-0.422605\pi$
$773$	10348.5	0.481512	0.240756	+	0.970586i	$0.422605\pi$
$774$	0	0
$775$	90.7137	0.00420456
$776$	11584.8	0.535916
$777$	0	0
$778$	11203.6	0.516284
$779$	−5707.16	−0.262491
$780$	0	0
$781$	−1110.72	−0.0508895
$782$	−7232.45	−0.330731
$783$	0	0
$784$	−1357.65	−0.0618464
$785$	39230.0	1.78367
$786$	0	0
$787$	−12675.3	−0.574112	−0.287056	−	0.957914i	$-0.592677\pi$
$787$	−12675.3	−0.574112	−0.287056	+	0.957914i	$0.592677\pi$
$788$	2976.08	0.134541
$789$	0	0
$790$	25813.4	1.16253
$791$	−7204.99	−0.323868
$792$	0	0
$793$	−1695.24	−0.0759138
$794$	−56201.0	−2.51196
$795$	0	0
$796$	3732.16	0.166185
$797$	−21422.2	−0.952088	−0.476044	−	0.879422i	$-0.657930\pi$
$797$	−21422.2	−0.952088	−0.476044	+	0.879422i	$0.657930\pi$
$798$	0	0
$799$	37831.8	1.67509
$800$	−4811.89	−0.212657
$801$	0	0
$802$	−60707.7	−2.67290
$803$	599.415	0.0263423
$804$	0	0
$805$	−1941.95	−0.0850246
$806$	967.833	0.0422959
$807$	0	0
$808$	−27632.1	−1.20309
$809$	10490.2	0.455892	0.227946	−	0.973674i	$-0.426799\pi$
$809$	10490.2	0.455892	0.227946	+	0.973674i	$0.426799\pi$
$810$	0	0
$811$	−7250.12	−0.313916	−0.156958	−	0.987605i	$-0.550169\pi$
$811$	−7250.12	−0.313916	−0.156958	+	0.987605i	$0.550169\pi$
$812$	13127.5	0.567345
$813$	0	0
$814$	−1697.91	−0.0731104
$815$	662.906	0.0284915
$816$	0	0
$817$	2704.19	0.115799
$818$	47847.5	2.04517
$819$	0	0
$820$	−20080.1	−0.855156
$821$	−29354.3	−1.24783	−0.623917	−	0.781491i	$-0.714460\pi$
$821$	−29354.3	−1.24783	−0.623917	+	0.781491i	$0.714460\pi$
$822$	0	0
$823$	1878.75	0.0795737	0.0397869	−	0.999208i	$-0.487332\pi$
$823$	1878.75	0.0795737	0.0397869	+	0.999208i	$0.487332\pi$
$824$	10882.8	0.460097
$825$	0	0
$826$	13701.1	0.577144
$827$	20411.9	0.858274	0.429137	−	0.903239i	$-0.358818\pi$
$827$	20411.9	0.858274	0.429137	+	0.903239i	$0.358818\pi$
$828$	0	0
$829$	24179.9	1.01303	0.506516	−	0.862230i	$-0.330933\pi$
$829$	24179.9	1.01303	0.506516	+	0.862230i	$0.330933\pi$
$830$	−50690.3	−2.11986
$831$	0	0
$832$	−40290.4	−1.67887
$833$	3513.57	0.146144
$834$	0	0
$835$	7770.99	0.322067
$836$	626.126	0.0259031
$837$	0	0
$838$	−40757.1	−1.68011
$839$	6771.79	0.278651	0.139325	−	0.990247i	$-0.455507\pi$
$839$	6771.79	0.278651	0.139325	+	0.990247i	$0.455507\pi$
$840$	0	0
$841$	3491.52	0.143160
$842$	30922.5	1.26563
$843$	0	0
$844$	−48.2120	−0.00196626
$845$	3466.03	0.141107
$846$	0	0
$847$	−9302.33	−0.377369
$848$	−3418.07	−0.138416
$849$	0	0
$850$	6442.31	0.259964
$851$	6150.49	0.247751
$852$	0	0
$853$	34029.9	1.36596	0.682979	−	0.730438i	$-0.260684\pi$
$853$	34029.9	1.36596	0.682979	+	0.730438i	$0.260684\pi$
$854$	−1044.06	−0.0418350
$855$	0	0
$856$	−4035.79	−0.161145
$857$	21630.1	0.862158	0.431079	−	0.902314i	$-0.358133\pi$
$857$	21630.1	0.862158	0.431079	+	0.902314i	$0.358133\pi$
$858$	0	0
$859$	35922.0	1.42682	0.713412	−	0.700744i	$-0.247149\pi$
$859$	35922.0	1.42682	0.713412	+	0.700744i	$0.247149\pi$
$860$	9514.42	0.377255
$861$	0	0
$862$	−60143.9	−2.37646
$863$	−21128.7	−0.833405	−0.416703	−	0.909043i	$-0.636814\pi$
$863$	−21128.7	−0.833405	−0.416703	+	0.909043i	$0.636814\pi$
$864$	0	0
$865$	−46641.3	−1.83335
$866$	3152.37	0.123697
$867$	0	0
$868$	348.112	0.0136126
$869$	706.571	0.0275820
$870$	0	0
$871$	9528.54	0.370680
$872$	−410.784	−0.0159529
$873$	0	0
$874$	−3883.58	−0.150302
$875$	−8824.29	−0.340932
$876$	0	0
$877$	−13831.6	−0.532566	−0.266283	−	0.963895i	$-0.585796\pi$
$877$	−13831.6	−0.532566	−0.266283	+	0.963895i	$0.585796\pi$
$878$	−37883.8	−1.45617
$879$	0	0
$880$	−483.876	−0.0185357
$881$	15533.3	0.594019	0.297009	−	0.954875i	$-0.404011\pi$
$881$	15533.3	0.594019	0.297009	+	0.954875i	$0.404011\pi$
$882$	0	0
$883$	38905.1	1.48274	0.741370	−	0.671096i	$-0.234176\pi$
$883$	38905.1	1.48274	0.741370	+	0.671096i	$0.234176\pi$
$884$	40141.4	1.52726
$885$	0	0
$886$	8876.27	0.336573
$887$	42655.1	1.61468	0.807338	−	0.590089i	$-0.200907\pi$
$887$	42655.1	1.61468	0.807338	+	0.590089i	$0.200907\pi$
$888$	0	0
$889$	14421.9	0.544089
$890$	65299.4	2.45937
$891$	0	0
$892$	14930.1	0.560422
$893$	20314.4	0.761249
$894$	0	0
$895$	4073.14	0.152123
$896$	−11661.2	−0.434792
$897$	0	0
$898$	13497.5	0.501577
$899$	739.331	0.0274283
$900$	0	0
$901$	8845.88	0.327080
$902$	−941.142	−0.0347412
$903$	0	0
$904$	−14585.7	−0.536631
$905$	−31722.1	−1.16517
$906$	0	0
$907$	−30658.6	−1.12238	−0.561192	−	0.827685i	$-0.689657\pi$
$907$	−30658.6	−1.12238	−0.561192	+	0.827685i	$0.689657\pi$
$908$	−55918.4	−2.04374
$909$	0	0
$910$	18455.4	0.672296
$911$	19369.9	0.704448	0.352224	−	0.935916i	$-0.385425\pi$
$911$	19369.9	0.704448	0.352224	+	0.935916i	$0.385425\pi$
$912$	0	0
$913$	−1387.51	−0.0502956
$914$	70405.8	2.54794
$915$	0	0
$916$	−13281.7	−0.479082
$917$	−6393.00	−0.230224
$918$	0	0
$919$	−33948.7	−1.21857	−0.609284	−	0.792952i	$-0.708543\pi$
$919$	−33948.7	−1.21857	−0.609284	+	0.792952i	$0.708543\pi$
$920$	−3931.27	−0.140881
$921$	0	0
$922$	−19519.1	−0.697210
$923$	−38236.9	−1.36358
$924$	0	0
$925$	−5478.55	−0.194739
$926$	−1230.50	−0.0436682
$927$	0	0
$928$	−39217.7	−1.38727
$929$	−49043.3	−1.73203	−0.866017	−	0.500015i	$-0.833328\pi$
$929$	−49043.3	−1.73203	−0.866017	+	0.500015i	$0.833328\pi$
$930$	0	0
$931$	1886.67	0.0664157
$932$	−13726.1	−0.482419
$933$	0	0
$934$	−22619.2	−0.792421
$935$	1252.26	0.0438003
$936$	0	0
$937$	−23112.6	−0.805824	−0.402912	−	0.915239i	$-0.632002\pi$
$937$	−23112.6	−0.805824	−0.402912	+	0.915239i	$0.632002\pi$
$938$	5868.44	0.204277
$939$	0	0
$940$	71474.4	2.48004
$941$	−44600.2	−1.54508	−0.772542	−	0.634963i	$-0.781015\pi$
$941$	−44600.2	−1.54508	−0.772542	+	0.634963i	$0.781015\pi$
$942$	0	0
$943$	3409.17	0.117728
$944$	−12366.4	−0.426369
$945$	0	0
$946$	445.935	0.0153262
$947$	−54228.3	−1.86080	−0.930402	−	0.366540i	$-0.880542\pi$
$947$	−54228.3	−1.86080	−0.930402	+	0.366540i	$0.880542\pi$
$948$	0	0
$949$	20635.1	0.705840
$950$	3459.30	0.118142
$951$	0	0
$952$	7112.85	0.242152
$953$	−5311.21	−0.180532	−0.0902660	−	0.995918i	$-0.528772\pi$
$953$	−5311.21	−0.180532	−0.0902660	+	0.995918i	$0.528772\pi$
$954$	0	0
$955$	16646.6	0.564055
$956$	44555.5	1.50735
$957$	0	0
$958$	−81957.0	−2.76400
$959$	10041.9	0.338132
$960$	0	0
$961$	−29771.4	−0.999342
$962$	−58451.2	−1.95899
$963$	0	0
$964$	26807.2	0.895646
$965$	−45289.2	−1.51079
$966$	0	0
$967$	29332.5	0.975461	0.487730	−	0.872994i	$-0.337825\pi$
$967$	29332.5	0.975461	0.487730	+	0.872994i	$0.337825\pi$
$968$	−18831.6	−0.625278
$969$	0	0
$970$	43242.8	1.43138
$971$	−10198.7	−0.337066	−0.168533	−	0.985696i	$-0.553903\pi$
$971$	−10198.7	−0.337066	−0.168533	+	0.985696i	$0.553903\pi$
$972$	0	0
$973$	−6449.39	−0.212495
$974$	−64159.4	−2.11068
$975$	0	0
$976$	942.358	0.0309059
$977$	41737.3	1.36673	0.683364	−	0.730078i	$-0.260516\pi$
$977$	41737.3	1.36673	0.683364	+	0.730078i	$0.260516\pi$
$978$	0	0
$979$	1787.39	0.0583507
$980$	6638.07	0.216373
$981$	0	0
$982$	−35642.3	−1.15824
$983$	−50.2330	−0.00162989	−0.000814946	−	1.00000i	$-0.500259\pi$
$983$	−50.2330	−0.00162989	−0.000814946		1.00000i	$0.500259\pi$
$984$	0	0
$985$	3196.13	0.103388
$986$	52505.9	1.69587
$987$	0	0
$988$	21554.6	0.694071
$989$	−1615.34	−0.0519362
$990$	0	0
$991$	−16223.3	−0.520031	−0.260016	−	0.965604i	$-0.583728\pi$
$991$	−16223.3	−0.520031	−0.260016	+	0.965604i	$0.583728\pi$
$992$	−1039.97	−0.0332853
$993$	0	0
$994$	−23549.4	−0.751449
$995$	4008.12	0.127704
$996$	0	0
$997$	−46151.5	−1.46603	−0.733015	−	0.680212i	$-0.761888\pi$
$997$	−46151.5	−1.46603	−0.733015	+	0.680212i	$0.761888\pi$
$998$	−83044.6	−2.63400
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1449.4.a.m.1.8		9
3.2	odd	2		483.4.a.f.1.2	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.4.a.f.1.2	✓	9	3.2	odd	2
1449.4.a.m.1.8		9	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1449 = 3^{2} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1449.a (trivial)

\(f(q)\)	\(=\)	\(q+4.38536 q^{2} +11.2314 q^{4} +12.0618 q^{5} +7.00000 q^{7} +14.1708 q^{8} +O(q^{10})\)	\(q+4.38536 q^{2} +11.2314 q^{4} +12.0618 q^{5} +7.00000 q^{7} +14.1708 q^{8} +52.8954 q^{10} +1.44787 q^{11} +49.8433 q^{13} +30.6975 q^{14} -27.7072 q^{16} +71.7055 q^{17} +38.5034 q^{19} +135.471 q^{20} +6.34942 q^{22} -23.0000 q^{23} +20.4873 q^{25} +218.581 q^{26} +78.6196 q^{28} +166.975 q^{29} +4.42781 q^{31} -234.872 q^{32} +314.454 q^{34} +84.4327 q^{35} -267.412 q^{37} +168.851 q^{38} +170.925 q^{40} -148.225 q^{41} +70.2323 q^{43} +16.2616 q^{44} -100.863 q^{46} +527.600 q^{47} +49.0000 q^{49} +89.8441 q^{50} +559.809 q^{52} +123.364 q^{53} +17.4639 q^{55} +99.1953 q^{56} +732.244 q^{58} +446.325 q^{59} -34.0113 q^{61} +19.4175 q^{62} -808.341 q^{64} +601.201 q^{65} +191.170 q^{67} +805.352 q^{68} +370.268 q^{70} -767.142 q^{71} +413.998 q^{73} -1172.70 q^{74} +432.447 q^{76} +10.1351 q^{77} +488.008 q^{79} -334.199 q^{80} -650.019 q^{82} -958.312 q^{83} +864.898 q^{85} +307.994 q^{86} +20.5174 q^{88} +1234.50 q^{89} +348.903 q^{91} -258.322 q^{92} +2313.72 q^{94} +464.421 q^{95} +817.516 q^{97} +214.883 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 42 q^{4} - 29 q^{5} + 63 q^{7} + 39 q^{8}+O(q^{10}) \) 9 * q + 42 * q^4 - 29 * q^5 + 63 * q^7 + 39 * q^8	\( 9 q + 42 q^{4} - 29 q^{5} + 63 q^{7} + 39 q^{8} - 55 q^{10} - 12 q^{11} + 199 q^{13} + 170 q^{16} - 116 q^{17} + 260 q^{19} - 324 q^{20} - 265 q^{22} - 207 q^{23} + 438 q^{25} + 270 q^{26} + 294 q^{28} + 107 q^{29} + 440 q^{31} + 802 q^{32} + 295 q^{34} - 203 q^{35} + 563 q^{37} + 569 q^{38} - 640 q^{40} - 243 q^{41} + 435 q^{43} - 1025 q^{44} + 133 q^{47} + 441 q^{49} - 104 q^{50} + 2693 q^{52} - 958 q^{53} + 1846 q^{55} + 273 q^{56} + 2796 q^{58} - 538 q^{59} + 1374 q^{61} - 1263 q^{62} - 83 q^{64} - 745 q^{65} + 752 q^{67} - 5593 q^{68} - 385 q^{70} + 418 q^{71} + 2406 q^{73} - 352 q^{74} + 2765 q^{76} - 84 q^{77} - 486 q^{79} - 5709 q^{80} + 2726 q^{82} - 106 q^{83} + 4130 q^{85} + 2576 q^{86} + 1270 q^{88} - 234 q^{89} + 1393 q^{91} - 966 q^{92} + 4967 q^{94} + 3074 q^{95} + 2409 q^{97}+O(q^{100}) \) 9 * q + 42 * q^4 - 29 * q^5 + 63 * q^7 + 39 * q^8 - 55 * q^10 - 12 * q^11 + 199 * q^13 + 170 * q^16 - 116 * q^17 + 260 * q^19 - 324 * q^20 - 265 * q^22 - 207 * q^23 + 438 * q^25 + 270 * q^26 + 294 * q^28 + 107 * q^29 + 440 * q^31 + 802 * q^32 + 295 * q^34 - 203 * q^35 + 563 * q^37 + 569 * q^38 - 640 * q^40 - 243 * q^41 + 435 * q^43 - 1025 * q^44 + 133 * q^47 + 441 * q^49 - 104 * q^50 + 2693 * q^52 - 958 * q^53 + 1846 * q^55 + 273 * q^56 + 2796 * q^58 - 538 * q^59 + 1374 * q^61 - 1263 * q^62 - 83 * q^64 - 745 * q^65 + 752 * q^67 - 5593 * q^68 - 385 * q^70 + 418 * q^71 + 2406 * q^73 - 352 * q^74 + 2765 * q^76 - 84 * q^77 - 486 * q^79 - 5709 * q^80 + 2726 * q^82 - 106 * q^83 + 4130 * q^85 + 2576 * q^86 + 1270 * q^88 - 234 * q^89 + 1393 * q^91 - 966 * q^92 + 4967 * q^94 + 3074 * q^95 + 2409 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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