LMFDB - Embedded newform 1449.4.a.i.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1449,4,Mod(1,1449)] 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("1449.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage: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")

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

Newform invariants

comment:select newform

sage:traces = [8,0,0,24,24] 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:	$85.4937675983$
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} - 44x^{6} - 23x^{5} + 587x^{4} + 594x^{3} - 2430x^{2} - 3403x + 110 $ x^8 - 44x^6 - 23x^5 + 587x^4 + 594x^3 - 2430x^2 - 3403x + 110
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 161)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-3.02588$ 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)

$f(q)$	$=$	\(q-3.02588 q^{2} +1.15594 q^{4} -18.5973 q^{5} -7.00000 q^{7} +20.7093 q^{8} +56.2732 q^{10} +33.2549 q^{11} -64.4205 q^{13} +21.1812 q^{14} -71.9113 q^{16} +52.2394 q^{17} -88.3614 q^{19} -21.4974 q^{20} -100.625 q^{22} -23.0000 q^{23} +220.859 q^{25} +194.928 q^{26} -8.09159 q^{28} -118.211 q^{29} -255.347 q^{31} +51.9206 q^{32} -158.070 q^{34} +130.181 q^{35} +57.9031 q^{37} +267.371 q^{38} -385.137 q^{40} +347.026 q^{41} +477.979 q^{43} +38.4407 q^{44} +69.5952 q^{46} -239.666 q^{47} +49.0000 q^{49} -668.294 q^{50} -74.4663 q^{52} +402.094 q^{53} -618.452 q^{55} -144.965 q^{56} +357.691 q^{58} +587.617 q^{59} +698.764 q^{61} +772.649 q^{62} +418.185 q^{64} +1198.05 q^{65} +126.625 q^{67} +60.3857 q^{68} -393.912 q^{70} +844.084 q^{71} +712.052 q^{73} -175.208 q^{74} -102.141 q^{76} -232.784 q^{77} -589.839 q^{79} +1337.36 q^{80} -1050.06 q^{82} -1009.86 q^{83} -971.511 q^{85} -1446.31 q^{86} +688.686 q^{88} -35.6685 q^{89} +450.943 q^{91} -26.5866 q^{92} +725.201 q^{94} +1643.28 q^{95} -127.177 q^{97} -148.268 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 24 q^{4} + 24 q^{5} - 56 q^{7} + 69 q^{8} - 30 q^{10} + 98 q^{11} - 145 q^{13} - 76 q^{16} + 96 q^{17} - 226 q^{19} + 22 q^{20} - 98 q^{22} - 184 q^{23} + 60 q^{25} + 185 q^{26} - 168 q^{28} - 73 q^{29}+ \cdots - 1452 q^{97}+O(q^{100}) $ 8 * q + 24 * q^4 + 24 * q^5 - 56 * q^7 + 69 * q^8 - 30 * q^10 + 98 * q^11 - 145 * q^13 - 76 * q^16 + 96 * q^17 - 226 * q^19 + 22 * q^20 - 98 * q^22 - 184 * q^23 + 60 * q^25 + 185 * q^26 - 168 * q^28 - 73 * q^29 - 987 * q^31 - 118 * q^32 - 846 * q^34 - 168 * q^35 - 2 * q^37 + 454 * q^38 - 640 * q^40 + 907 * q^41 + 636 * q^43 - 126 * q^44 - 421 * q^47 + 392 * q^49 - 2308 * q^50 - 931 * q^52 - 138 * q^53 - 90 * q^55 - 483 * q^56 + 1329 * q^58 + 924 * q^59 - 890 * q^61 - 2175 * q^62 + 303 * q^64 + 392 * q^65 - 1392 * q^67 - 2592 * q^68 + 210 * q^70 - 259 * q^71 - 297 * q^73 - 2314 * q^74 - 428 * q^76 - 686 * q^77 - 1788 * q^79 + 2104 * q^80 - 219 * q^82 - 2114 * q^83 + 528 * q^85 + 2360 * q^86 + 2144 * q^88 + 1040 * q^89 + 1015 * q^91 - 552 * q^92 - 1009 * q^94 - 1268 * q^95 - 1452 * q^97

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 $

$2$	−3.02588	−1.06981	−0.534905	−	0.844912i	$-0.679653\pi$
$2$	−3.02588	−1.06981	−0.534905	+	0.844912i	$0.679653\pi$
$3$	0	0
$3$	0	0
$4$	1.15594	0.144493
$5$	−18.5973	−1.66339	−0.831696	−	0.555231i	$-0.812630\pi$
$5$	−18.5973	−1.66339	−0.831696	+	0.555231i	$0.812630\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	20.7093	0.915230
$9$	0	0
$10$	56.2732	1.77951
$11$	33.2549	0.911522	0.455761	−	0.890102i	$-0.349367\pi$
$11$	33.2549	0.911522	0.455761	+	0.890102i	$0.349367\pi$
$12$	0	0
$13$	−64.4205	−1.37439	−0.687193	−	0.726475i	$-0.741157\pi$
$13$	−64.4205	−1.37439	−0.687193	+	0.726475i	$0.741157\pi$
$14$	21.1812	0.404350
$15$	0	0
$16$	−71.9113	−1.12361
$17$	52.2394	0.745289	0.372644	−	0.927974i	$-0.378451\pi$
$17$	52.2394	0.745289	0.372644	+	0.927974i	$0.378451\pi$
$18$	0	0
$19$	−88.3614	−1.06692	−0.533460	−	0.845825i	$-0.679109\pi$
$19$	−88.3614	−1.06692	−0.533460	+	0.845825i	$0.679109\pi$
$20$	−21.4974	−0.240348
$21$	0	0
$22$	−100.625	−0.975155
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	0	0
$25$	220.859	1.76687
$26$	194.928	1.47033
$27$	0	0
$28$	−8.09159	−0.0546131
$29$	−118.211	−0.756936	−0.378468	−	0.925614i	$-0.623549\pi$
$29$	−118.211	−0.756936	−0.378468	+	0.925614i	$0.623549\pi$
$30$	0	0
$31$	−255.347	−1.47941	−0.739704	−	0.672932i	$-0.765035\pi$
$31$	−255.347	−1.47941	−0.739704	+	0.672932i	$0.765035\pi$
$32$	51.9206	0.286824
$33$	0	0
$34$	−158.070	−0.797317
$35$	130.181	0.628703
$36$	0	0
$37$	57.9031	0.257276	0.128638	−	0.991692i	$-0.458939\pi$
$37$	57.9031	0.257276	0.128638	+	0.991692i	$0.458939\pi$
$38$	267.371	1.14140
$39$	0	0
$40$	−385.137	−1.52239
$41$	347.026	1.32186	0.660931	−	0.750446i	$-0.270161\pi$
$41$	347.026	1.32186	0.660931	+	0.750446i	$0.270161\pi$
$42$	0	0
$43$	477.979	1.69514	0.847571	−	0.530682i	$-0.178064\pi$
$43$	477.979	1.69514	0.847571	+	0.530682i	$0.178064\pi$
$44$	38.4407	0.131708
$45$	0	0
$46$	69.5952	0.223071
$47$	−239.666	−0.743807	−0.371903	−	0.928271i	$-0.621295\pi$
$47$	−239.666	−0.743807	−0.371903	+	0.928271i	$0.621295\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	−668.294	−1.89022
$51$	0	0
$52$	−74.4663	−0.198589
$53$	402.094	1.04211	0.521056	−	0.853523i	$-0.325538\pi$
$53$	402.094	1.04211	0.521056	+	0.853523i	$0.325538\pi$
$54$	0	0
$55$	−618.452	−1.51622
$56$	−144.965	−0.345924
$57$	0	0
$58$	357.691	0.809777
$59$	587.617	1.29663	0.648315	−	0.761372i	$-0.275474\pi$
$59$	587.617	1.29663	0.648315	+	0.761372i	$0.275474\pi$
$60$	0	0
$61$	698.764	1.46668	0.733341	−	0.679861i	$-0.237960\pi$
$61$	698.764	1.46668	0.733341	+	0.679861i	$0.237960\pi$
$62$	772.649	1.58269
$63$	0	0
$64$	418.185	0.816768
$65$	1198.05	2.28614
$66$	0	0
$67$	126.625	0.230890	0.115445	−	0.993314i	$-0.463171\pi$
$67$	126.625	0.230890	0.115445	+	0.993314i	$0.463171\pi$
$68$	60.3857	0.107689
$69$	0	0
$70$	−393.912	−0.672593
$71$	844.084	1.41091	0.705453	−	0.708757i	$-0.250744\pi$
$71$	844.084	1.41091	0.705453	+	0.708757i	$0.250744\pi$
$72$	0	0
$73$	712.052	1.14164	0.570818	−	0.821077i	$-0.306626\pi$
$73$	712.052	1.14164	0.570818	+	0.821077i	$0.306626\pi$
$74$	−175.208	−0.275236
$75$	0	0
$76$	−102.141	−0.154162
$77$	−232.784	−0.344523
$78$	0	0
$79$	−589.839	−0.840026	−0.420013	−	0.907518i	$-0.637974\pi$
$79$	−589.839	−0.840026	−0.420013	+	0.907518i	$0.637974\pi$
$80$	1337.36	1.86901
$81$	0	0
$82$	−1050.06	−1.41414
$83$	−1009.86	−1.33550	−0.667752	−	0.744384i	$-0.732743\pi$
$83$	−1009.86	−1.33550	−0.667752	+	0.744384i	$0.732743\pi$
$84$	0	0
$85$	−971.511	−1.23971
$86$	−1446.31	−1.81348
$87$	0	0
$88$	688.686	0.834252
$89$	−35.6685	−0.0424814	−0.0212407	−	0.999774i	$-0.506762\pi$
$89$	−35.6685	−0.0424814	−0.0212407	+	0.999774i	$0.506762\pi$
$90$	0	0
$91$	450.943	0.519469
$92$	−26.5866	−0.0301288
$93$	0	0
$94$	725.201	0.795731
$95$	1643.28	1.77471
$96$	0	0
$97$	−127.177	−0.133123	−0.0665613	−	0.997782i	$-0.521203\pi$
$97$	−127.177	−0.133123	−0.0665613	+	0.997782i	$0.521203\pi$
$98$	−148.268	−0.152830
$99$	0	0
$100$	255.300	0.255300
$101$	1376.80	1.35641	0.678203	−	0.734875i	$-0.262759\pi$
$101$	1376.80	1.35641	0.678203	+	0.734875i	$0.262759\pi$
$102$	0	0
$103$	−508.618	−0.486559	−0.243280	−	0.969956i	$-0.578223\pi$
$103$	−508.618	−0.486559	−0.243280	+	0.969956i	$0.578223\pi$
$104$	−1334.10	−1.25788
$105$	0	0
$106$	−1216.69	−1.11486
$107$	−435.724	−0.393673	−0.196837	−	0.980436i	$-0.563067\pi$
$107$	−435.724	−0.393673	−0.196837	+	0.980436i	$0.563067\pi$
$108$	0	0
$109$	426.303	0.374610	0.187305	−	0.982302i	$-0.440025\pi$
$109$	426.303	0.374610	0.187305	+	0.982302i	$0.440025\pi$
$110$	1871.36	1.62206
$111$	0	0
$112$	503.379	0.424686
$113$	−692.534	−0.576532	−0.288266	−	0.957550i	$-0.593079\pi$
$113$	−692.534	−0.576532	−0.288266	+	0.957550i	$0.593079\pi$
$114$	0	0
$115$	427.738	0.346841
$116$	−136.644	−0.109372
$117$	0	0
$118$	−1778.06	−1.38715
$119$	−365.676	−0.281693
$120$	0	0
$121$	−225.110	−0.169128
$122$	−2114.38	−1.56907
$123$	0	0
$124$	−295.166	−0.213764
$125$	−1782.72	−1.27561
$126$	0	0
$127$	−656.335	−0.458585	−0.229293	−	0.973358i	$-0.573641\pi$
$127$	−656.335	−0.458585	−0.229293	+	0.973358i	$0.573641\pi$
$128$	−1680.74	−1.16061
$129$	0	0
$130$	−3625.14	−2.44574
$131$	−1575.99	−1.05111	−0.525553	−	0.850761i	$-0.676142\pi$
$131$	−1575.99	−1.05111	−0.525553	+	0.850761i	$0.676142\pi$
$132$	0	0
$133$	618.530	0.403258
$134$	−383.151	−0.247009
$135$	0	0
$136$	1081.84	0.682111
$137$	−792.294	−0.494089	−0.247045	−	0.969004i	$-0.579459\pi$
$137$	−792.294	−0.494089	−0.247045	+	0.969004i	$0.579459\pi$
$138$	0	0
$139$	−2675.35	−1.63252	−0.816261	−	0.577684i	$-0.803957\pi$
$139$	−2675.35	−1.63252	−0.816261	+	0.577684i	$0.803957\pi$
$140$	150.482	0.0908430
$141$	0	0
$142$	−2554.10	−1.50940
$143$	−2142.30	−1.25278
$144$	0	0
$145$	2198.40	1.25908
$146$	−2154.58	−1.22133
$147$	0	0
$148$	66.9326	0.0371745
$149$	−291.225	−0.160121	−0.0800607	−	0.996790i	$-0.525511\pi$
$149$	−291.225	−0.160121	−0.0800607	+	0.996790i	$0.525511\pi$
$150$	0	0
$151$	2204.34	1.18799	0.593995	−	0.804469i	$-0.297550\pi$
$151$	2204.34	1.18799	0.593995	+	0.804469i	$0.297550\pi$
$152$	−1829.90	−0.976478
$153$	0	0
$154$	704.378	0.368574
$155$	4748.76	2.46084
$156$	0	0
$157$	−1615.90	−0.821422	−0.410711	−	0.911766i	$-0.634719\pi$
$157$	−1615.90	−0.821422	−0.410711	+	0.911766i	$0.634719\pi$
$158$	1784.78	0.898668
$159$	0	0
$160$	−965.583	−0.477100
$161$	161.000	0.0788110
$162$	0	0
$163$	336.417	0.161658	0.0808288	−	0.996728i	$-0.474243\pi$
$163$	336.417	0.161658	0.0808288	+	0.996728i	$0.474243\pi$
$164$	401.142	0.190999
$165$	0	0
$166$	3055.72	1.42874
$167$	27.3144	0.0126566	0.00632831	−	0.999980i	$-0.497986\pi$
$167$	27.3144	0.0126566	0.00632831	+	0.999980i	$0.497986\pi$
$168$	0	0
$169$	1953.00	0.888938
$170$	2939.68	1.32625
$171$	0	0
$172$	552.516	0.244936
$173$	2690.54	1.18242	0.591208	−	0.806519i	$-0.298651\pi$
$173$	2690.54	1.18242	0.591208	+	0.806519i	$0.298651\pi$
$174$	0	0
$175$	−1546.02	−0.667816
$176$	−2391.41	−1.02420
$177$	0	0
$178$	107.928	0.0454471
$179$	1251.42	0.522546	0.261273	−	0.965265i	$-0.415858\pi$
$179$	1251.42	0.522546	0.261273	+	0.965265i	$0.415858\pi$
$180$	0	0
$181$	193.457	0.0794449	0.0397224	−	0.999211i	$-0.487353\pi$
$181$	193.457	0.0794449	0.0397224	+	0.999211i	$0.487353\pi$
$182$	−1364.50	−0.555733
$183$	0	0
$184$	−476.314	−0.190839
$185$	−1076.84	−0.427951
$186$	0	0
$187$	1737.22	0.679347
$188$	−277.040	−0.107475
$189$	0	0
$190$	−4972.37	−1.89860
$191$	1795.26	0.680107	0.340054	−	0.940406i	$-0.389555\pi$
$191$	1795.26	0.680107	0.340054	+	0.940406i	$0.389555\pi$
$192$	0	0
$193$	−861.357	−0.321253	−0.160626	−	0.987015i	$-0.551351\pi$
$193$	−861.357	−0.321253	−0.160626	+	0.987015i	$0.551351\pi$
$194$	384.823	0.142416
$195$	0	0
$196$	56.6411	0.0206418
$197$	−225.099	−0.0814092	−0.0407046	−	0.999171i	$-0.512960\pi$
$197$	−225.099	−0.0814092	−0.0407046	+	0.999171i	$0.512960\pi$
$198$	0	0
$199$	−223.345	−0.0795603	−0.0397801	−	0.999208i	$-0.512666\pi$
$199$	−223.345	−0.0795603	−0.0397801	+	0.999208i	$0.512666\pi$
$200$	4573.84	1.61710
$201$	0	0
$202$	−4166.04	−1.45110
$203$	827.474	0.286095
$204$	0	0
$205$	−6453.75	−2.19878
$206$	1539.02	0.520526
$207$	0	0
$208$	4632.56	1.54428
$209$	−2938.45	−0.972521
$210$	0	0
$211$	1347.64	0.439694	0.219847	−	0.975534i	$-0.429444\pi$
$211$	1347.64	0.439694	0.219847	+	0.975534i	$0.429444\pi$
$212$	464.797	0.150577
$213$	0	0
$214$	1318.45	0.421155
$215$	−8889.11	−2.81969
$216$	0	0
$217$	1787.43	0.559164
$218$	−1289.94	−0.400761
$219$	0	0
$220$	−714.894	−0.219082
$221$	−3365.29	−1.02431
$222$	0	0
$223$	1209.59	0.363228	0.181614	−	0.983370i	$-0.441868\pi$
$223$	1209.59	0.363228	0.181614	+	0.983370i	$0.441868\pi$
$224$	−363.444	−0.108409
$225$	0	0
$226$	2095.52	0.616780
$227$	3411.17	0.997390	0.498695	−	0.866777i	$-0.333813\pi$
$227$	3411.17	0.997390	0.498695	+	0.866777i	$0.333813\pi$
$228$	0	0
$229$	512.486	0.147886	0.0739432	−	0.997262i	$-0.476442\pi$
$229$	512.486	0.147886	0.0739432	+	0.997262i	$0.476442\pi$
$230$	−1294.28	−0.371054
$231$	0	0
$232$	−2448.06	−0.692770
$233$	3903.84	1.09763	0.548817	−	0.835942i	$-0.315078\pi$
$233$	3903.84	1.09763	0.548817	+	0.835942i	$0.315078\pi$
$234$	0	0
$235$	4457.14	1.23724
$236$	679.251	0.187354
$237$	0	0
$238$	1106.49	0.301358
$239$	−1424.61	−0.385567	−0.192784	−	0.981241i	$-0.561752\pi$
$239$	−1424.61	−0.385567	−0.192784	+	0.981241i	$0.561752\pi$
$240$	0	0
$241$	−6444.69	−1.72257	−0.861285	−	0.508123i	$-0.830340\pi$
$241$	−6444.69	−1.72257	−0.861285	+	0.508123i	$0.830340\pi$
$242$	681.155	0.180935
$243$	0	0
$244$	807.731	0.211925
$245$	−911.267	−0.237628
$246$	0	0
$247$	5692.28	1.46636
$248$	−5288.05	−1.35400
$249$	0	0
$250$	5394.31	1.36466
$251$	−7062.49	−1.77602	−0.888008	−	0.459827i	$-0.847911\pi$
$251$	−7062.49	−1.77602	−0.888008	+	0.459827i	$0.847911\pi$
$252$	0	0
$253$	−764.863	−0.190065
$254$	1985.99	0.490599
$255$	0	0
$256$	1740.24	0.424864
$257$	5900.90	1.43225	0.716125	−	0.697972i	$-0.245914\pi$
$257$	5900.90	1.43225	0.716125	+	0.697972i	$0.245914\pi$
$258$	0	0
$259$	−405.322	−0.0972412
$260$	1384.87	0.330331
$261$	0	0
$262$	4768.75	1.12448
$263$	1634.09	0.383127	0.191564	−	0.981480i	$-0.438644\pi$
$263$	1634.09	0.383127	0.191564	+	0.981480i	$0.438644\pi$
$264$	0	0
$265$	−7477.86	−1.73344
$266$	−1871.60	−0.431409
$267$	0	0
$268$	146.371	0.0333620
$269$	2747.85	0.622824	0.311412	−	0.950275i	$-0.399198\pi$
$269$	2747.85	0.622824	0.311412	+	0.950275i	$0.399198\pi$
$270$	0	0
$271$	−7298.61	−1.63601	−0.818005	−	0.575211i	$-0.804920\pi$
$271$	−7298.61	−1.63601	−0.818005	+	0.575211i	$0.804920\pi$
$272$	−3756.60	−0.837418
$273$	0	0
$274$	2397.39	0.528582
$275$	7344.66	1.61054
$276$	0	0
$277$	−6341.50	−1.37554	−0.687769	−	0.725930i	$-0.741410\pi$
$277$	−6341.50	−1.37554	−0.687769	+	0.725930i	$0.741410\pi$
$278$	8095.29	1.74649
$279$	0	0
$280$	2695.96	0.575408
$281$	5340.75	1.13382	0.566908	−	0.823781i	$-0.308139\pi$
$281$	5340.75	1.13382	0.566908	+	0.823781i	$0.308139\pi$
$282$	0	0
$283$	−4744.66	−0.996611	−0.498305	−	0.867002i	$-0.666044\pi$
$283$	−4744.66	−0.996611	−0.498305	+	0.867002i	$0.666044\pi$
$284$	975.711	0.203866
$285$	0	0
$286$	6482.33	1.34024
$287$	−2429.18	−0.499617
$288$	0	0
$289$	−2184.05	−0.444544
$290$	−6652.08	−1.34698
$291$	0	0
$292$	823.090	0.164958
$293$	−7990.45	−1.59320	−0.796599	−	0.604508i	$-0.793370\pi$
$293$	−7990.45	−1.59320	−0.796599	+	0.604508i	$0.793370\pi$
$294$	0	0
$295$	−10928.1	−2.15681
$296$	1199.13	0.235467
$297$	0	0
$298$	881.212	0.171300
$299$	1481.67	0.286579
$300$	0	0
$301$	−3345.85	−0.640703
$302$	−6670.06	−1.27092
$303$	0	0
$304$	6354.18	1.19881
$305$	−12995.1	−2.43967
$306$	0	0
$307$	4925.41	0.915662	0.457831	−	0.889039i	$-0.348627\pi$
$307$	4925.41	0.915662	0.457831	+	0.889039i	$0.348627\pi$
$308$	−269.085	−0.0497810
$309$	0	0
$310$	−14369.2	−2.63263
$311$	−2905.57	−0.529774	−0.264887	−	0.964279i	$-0.585335\pi$
$311$	−2905.57	−0.529774	−0.264887	+	0.964279i	$0.585335\pi$
$312$	0	0
$313$	−2970.69	−0.536464	−0.268232	−	0.963354i	$-0.586439\pi$
$313$	−2970.69	−0.536464	−0.268232	+	0.963354i	$0.586439\pi$
$314$	4889.53	0.878765
$315$	0	0
$316$	−681.819	−0.121378
$317$	799.754	0.141699	0.0708497	−	0.997487i	$-0.477429\pi$
$317$	799.754	0.141699	0.0708497	+	0.997487i	$0.477429\pi$
$318$	0	0
$319$	−3931.08	−0.689963
$320$	−7777.11	−1.35861
$321$	0	0
$322$	−487.166	−0.0843128
$323$	−4615.94	−0.795164
$324$	0	0
$325$	−14227.9	−2.42837
$326$	−1017.96	−0.172943
$327$	0	0
$328$	7186.67	1.20981
$329$	1677.66	0.281132
$330$	0	0
$331$	7735.22	1.28449	0.642245	−	0.766500i	$-0.278003\pi$
$331$	7735.22	1.28449	0.642245	+	0.766500i	$0.278003\pi$
$332$	−1167.34	−0.192971
$333$	0	0
$334$	−82.6502	−0.0135402
$335$	−2354.88	−0.384061
$336$	0	0
$337$	11779.0	1.90398	0.951990	−	0.306128i	$-0.0990337\pi$
$337$	11779.0	1.90398	0.951990	+	0.306128i	$0.0990337\pi$
$338$	−5909.53	−0.950994
$339$	0	0
$340$	−1123.01	−0.179129
$341$	−8491.54	−1.34851
$342$	0	0
$343$	−343.000	−0.0539949
$344$	9898.60	1.55144
$345$	0	0
$346$	−8141.25	−1.26496
$347$	−5624.50	−0.870141	−0.435071	−	0.900396i	$-0.643277\pi$
$347$	−5624.50	−0.870141	−0.435071	+	0.900396i	$0.643277\pi$
$348$	0	0
$349$	−3023.16	−0.463684	−0.231842	−	0.972753i	$-0.574475\pi$
$349$	−3023.16	−0.463684	−0.231842	+	0.972753i	$0.574475\pi$
$350$	4678.05	0.714436
$351$	0	0
$352$	1726.62	0.261446
$353$	−404.183	−0.0609419	−0.0304709	−	0.999536i	$-0.509701\pi$
$353$	−404.183	−0.0609419	−0.0304709	+	0.999536i	$0.509701\pi$
$354$	0	0
$355$	−15697.7	−2.34689
$356$	−41.2306	−0.00613826
$357$	0	0
$358$	−3786.65	−0.559024
$359$	−2268.39	−0.333485	−0.166742	−	0.986001i	$-0.553325\pi$
$359$	−2268.39	−0.333485	−0.166742	+	0.986001i	$0.553325\pi$
$360$	0	0
$361$	948.734	0.138320
$362$	−585.377	−0.0849909
$363$	0	0
$364$	521.264	0.0750595
$365$	−13242.2	−1.89899
$366$	0	0
$367$	−129.076	−0.0183589	−0.00917945	−	0.999958i	$-0.502922\pi$
$367$	−129.076	−0.0183589	−0.00917945	+	0.999958i	$0.502922\pi$
$368$	1653.96	0.234290
$369$	0	0
$370$	3258.39	0.457826
$371$	−2814.66	−0.393881
$372$	0	0
$373$	1354.27	0.187994	0.0939969	−	0.995572i	$-0.470036\pi$
$373$	1354.27	0.187994	0.0939969	+	0.995572i	$0.470036\pi$
$374$	−5256.61	−0.726772
$375$	0	0
$376$	−4963.32	−0.680754
$377$	7615.18	1.04032
$378$	0	0
$379$	−10233.1	−1.38691	−0.693455	−	0.720500i	$-0.743912\pi$
$379$	−10233.1	−1.38691	−0.693455	+	0.720500i	$0.743912\pi$
$380$	1899.54	0.256432
$381$	0	0
$382$	−5432.24	−0.727585
$383$	−5780.40	−0.771188	−0.385594	−	0.922669i	$-0.626003\pi$
$383$	−5780.40	−0.771188	−0.385594	+	0.922669i	$0.626003\pi$
$384$	0	0
$385$	4329.16	0.573077
$386$	2606.36	0.343679
$387$	0	0
$388$	−147.009	−0.0192352
$389$	−11762.5	−1.53311	−0.766556	−	0.642177i	$-0.778031\pi$
$389$	−11762.5	−1.53311	−0.766556	+	0.642177i	$0.778031\pi$
$390$	0	0
$391$	−1201.51	−0.155403
$392$	1014.76	0.130747
$393$	0	0
$394$	681.121	0.0870924
$395$	10969.4	1.39729
$396$	0	0
$397$	−454.539	−0.0574626	−0.0287313	−	0.999587i	$-0.509147\pi$
$397$	−454.539	−0.0574626	−0.0287313	+	0.999587i	$0.509147\pi$
$398$	675.814	0.0851144
$399$	0	0
$400$	−15882.3	−1.98529
$401$	11330.2	1.41098	0.705491	−	0.708719i	$-0.250727\pi$
$401$	11330.2	1.41098	0.705491	+	0.708719i	$0.250727\pi$
$402$	0	0
$403$	16449.6	2.03328
$404$	1591.50	0.195991
$405$	0	0
$406$	−2503.83	−0.306067
$407$	1925.56	0.234513
$408$	0	0
$409$	−1009.32	−0.122024	−0.0610121	−	0.998137i	$-0.519433\pi$
$409$	−1009.32	−0.122024	−0.0610121	+	0.998137i	$0.519433\pi$
$410$	19528.3	2.35227
$411$	0	0
$412$	−587.932	−0.0703042
$413$	−4113.32	−0.490080
$414$	0	0
$415$	18780.7	2.22147
$416$	−3344.75	−0.394206
$417$	0	0
$418$	8891.40	1.04041
$419$	14056.5	1.63891	0.819455	−	0.573143i	$-0.194276\pi$
$419$	14056.5	1.63891	0.819455	+	0.573143i	$0.194276\pi$
$420$	0	0
$421$	−3217.99	−0.372531	−0.186265	−	0.982499i	$-0.559638\pi$
$421$	−3217.99	−0.372531	−0.186265	+	0.982499i	$0.559638\pi$
$422$	−4077.80	−0.470389
$423$	0	0
$424$	8327.09	0.953771
$425$	11537.6	1.31683
$426$	0	0
$427$	−4891.35	−0.554354
$428$	−503.671	−0.0568829
$429$	0	0
$430$	26897.4	3.01653
$431$	−872.562	−0.0975170	−0.0487585	−	0.998811i	$-0.515526\pi$
$431$	−872.562	−0.0975170	−0.0487585	+	0.998811i	$0.515526\pi$
$432$	0	0
$433$	−9665.82	−1.07277	−0.536385	−	0.843973i	$-0.680211\pi$
$433$	−9665.82	−1.07277	−0.536385	+	0.843973i	$0.680211\pi$
$434$	−5408.54	−0.598199
$435$	0	0
$436$	492.782	0.0541284
$437$	2032.31	0.222468
$438$	0	0
$439$	15457.9	1.68055	0.840277	−	0.542158i	$-0.182392\pi$
$439$	15457.9	1.68055	0.840277	+	0.542158i	$0.182392\pi$
$440$	−12807.7	−1.38769
$441$	0	0
$442$	10182.9	1.09582
$443$	10487.1	1.12473	0.562365	−	0.826889i	$-0.309892\pi$
$443$	10487.1	1.12473	0.562365	+	0.826889i	$0.309892\pi$
$444$	0	0
$445$	663.337	0.0706633
$446$	−3660.06	−0.388585
$447$	0	0
$448$	−2927.30	−0.308709
$449$	−6169.92	−0.648500	−0.324250	−	0.945971i	$-0.605112\pi$
$449$	−6169.92	−0.648500	−0.324250	+	0.945971i	$0.605112\pi$
$450$	0	0
$451$	11540.3	1.20491
$452$	−800.529	−0.0833047
$453$	0	0
$454$	−10321.8	−1.06702
$455$	−8386.32	−0.864081
$456$	0	0
$457$	−16296.4	−1.66808	−0.834039	−	0.551706i	$-0.813977\pi$
$457$	−16296.4	−1.66808	−0.834039	+	0.551706i	$0.813977\pi$
$458$	−1550.72	−0.158210
$459$	0	0
$460$	494.440	0.0501160
$461$	−18404.0	−1.85935	−0.929673	−	0.368386i	$-0.879911\pi$
$461$	−18404.0	−1.85935	−0.929673	+	0.368386i	$0.879911\pi$
$462$	0	0
$463$	−13024.1	−1.30730	−0.653650	−	0.756797i	$-0.726763\pi$
$463$	−13024.1	−1.30730	−0.653650	+	0.756797i	$0.726763\pi$
$464$	8500.68	0.850504
$465$	0	0
$466$	−11812.5	−1.17426
$467$	2332.68	0.231142	0.115571	−	0.993299i	$-0.463130\pi$
$467$	2332.68	0.231142	0.115571	+	0.993299i	$0.463130\pi$
$468$	0	0
$469$	−886.372	−0.0872684
$470$	−13486.8	−1.32361
$471$	0	0
$472$	12169.1	1.18672
$473$	15895.2	1.54516
$474$	0	0
$475$	−19515.4	−1.88512
$476$	−422.700	−0.0407025
$477$	0	0
$478$	4310.71	0.412484
$479$	−16019.5	−1.52808	−0.764040	−	0.645169i	$-0.776787\pi$
$479$	−16019.5	−1.52808	−0.764040	+	0.645169i	$0.776787\pi$
$480$	0	0
$481$	−3730.15	−0.353597
$482$	19500.9	1.84282
$483$	0	0
$484$	−260.214	−0.0244378
$485$	2365.15	0.221435
$486$	0	0
$487$	−1579.85	−0.147002	−0.0735010	−	0.997295i	$-0.523417\pi$
$487$	−1579.85	−0.147002	−0.0735010	+	0.997295i	$0.523417\pi$
$488$	14470.9	1.34235
$489$	0	0
$490$	2757.38	0.254216
$491$	20545.3	1.88838	0.944191	−	0.329399i	$-0.106846\pi$
$491$	20545.3	1.88838	0.944191	+	0.329399i	$0.106846\pi$
$492$	0	0
$493$	−6175.24	−0.564136
$494$	−17224.2	−1.56873
$495$	0	0
$496$	18362.3	1.66228
$497$	−5908.59	−0.533272
$498$	0	0
$499$	−1331.47	−0.119448	−0.0597242	−	0.998215i	$-0.519022\pi$
$499$	−1331.47	−0.119448	−0.0597242	+	0.998215i	$0.519022\pi$
$500$	−2060.72	−0.184317
$501$	0	0
$502$	21370.2	1.90000
$503$	927.686	0.0822335	0.0411168	−	0.999154i	$-0.486908\pi$
$503$	927.686	0.0822335	0.0411168	+	0.999154i	$0.486908\pi$
$504$	0	0
$505$	−25604.8	−2.25624
$506$	2314.38	0.203334
$507$	0	0
$508$	−758.685	−0.0662622
$509$	−6991.85	−0.608857	−0.304429	−	0.952535i	$-0.598466\pi$
$509$	−6991.85	−0.608857	−0.304429	+	0.952535i	$0.598466\pi$
$510$	0	0
$511$	−4984.36	−0.431498
$512$	8180.18	0.706086
$513$	0	0
$514$	−17855.4	−1.53223
$515$	9458.91	0.809339
$516$	0	0
$517$	−7970.08	−0.677996
$518$	1226.45	0.104030
$519$	0	0
$520$	24810.7	2.09235
$521$	−15300.3	−1.28660	−0.643302	−	0.765613i	$-0.722436\pi$
$521$	−15300.3	−1.28660	−0.643302	+	0.765613i	$0.722436\pi$
$522$	0	0
$523$	13089.8	1.09441	0.547206	−	0.836998i	$-0.315691\pi$
$523$	13089.8	1.09441	0.547206	+	0.836998i	$0.315691\pi$
$524$	−1821.75	−0.151877
$525$	0	0
$526$	−4944.57	−0.409873
$527$	−13339.2	−1.10259
$528$	0	0
$529$	529.000	0.0434783
$530$	22627.1	1.85445
$531$	0	0
$532$	714.984	0.0582678
$533$	−22355.6	−1.81675
$534$	0	0
$535$	8103.29	0.654833
$536$	2622.31	0.211318
$537$	0	0
$538$	−8314.68	−0.666303
$539$	1629.49	0.130217
$540$	0	0
$541$	1599.69	0.127128	0.0635638	−	0.997978i	$-0.479753\pi$
$541$	1599.69	0.127128	0.0635638	+	0.997978i	$0.479753\pi$
$542$	22084.7	1.75022
$543$	0	0
$544$	2712.30	0.213767
$545$	−7928.09	−0.623123
$546$	0	0
$547$	18730.5	1.46409	0.732045	−	0.681256i	$-0.238566\pi$
$547$	18730.5	1.46409	0.732045	+	0.681256i	$0.238566\pi$
$548$	−915.845	−0.0713923
$549$	0	0
$550$	−22224.1	−1.72298
$551$	10445.2	0.807591
$552$	0	0
$553$	4128.87	0.317500
$554$	19188.6	1.47156
$555$	0	0
$556$	−3092.55	−0.235887
$557$	5488.44	0.417509	0.208755	−	0.977968i	$-0.433059\pi$
$557$	5488.44	0.417509	0.208755	+	0.977968i	$0.433059\pi$
$558$	0	0
$559$	−30791.6	−2.32978
$560$	−9361.49	−0.706420
$561$	0	0
$562$	−16160.5	−1.21297
$563$	8122.20	0.608010	0.304005	−	0.952670i	$-0.401676\pi$
$563$	8122.20	0.608010	0.304005	+	0.952670i	$0.401676\pi$
$564$	0	0
$565$	12879.3	0.958999
$566$	14356.8	1.06618
$567$	0	0
$568$	17480.4	1.29130
$569$	15394.8	1.13424	0.567120	−	0.823635i	$-0.308057\pi$
$569$	15394.8	1.13424	0.567120	+	0.823635i	$0.308057\pi$
$570$	0	0
$571$	1704.64	0.124934	0.0624668	−	0.998047i	$-0.480103\pi$
$571$	1704.64	0.124934	0.0624668	+	0.998047i	$0.480103\pi$
$572$	−2476.37	−0.181018
$573$	0	0
$574$	7350.41	0.534495
$575$	−5079.76	−0.368419
$576$	0	0
$577$	−2603.58	−0.187848	−0.0939241	−	0.995579i	$-0.529941\pi$
$577$	−2603.58	−0.187848	−0.0939241	+	0.995579i	$0.529941\pi$
$578$	6608.66	0.475578
$579$	0	0
$580$	2541.22	0.181928
$581$	7069.04	0.504773
$582$	0	0
$583$	13371.6	0.949907
$584$	14746.1	1.04486
$585$	0	0
$586$	24178.1	1.70442
$587$	−8901.60	−0.625909	−0.312954	−	0.949768i	$-0.601319\pi$
$587$	−8901.60	−0.625909	−0.312954	+	0.949768i	$0.601319\pi$
$588$	0	0
$589$	22562.8	1.57841
$590$	33067.1	2.30737
$591$	0	0
$592$	−4163.89	−0.289079
$593$	642.837	0.0445163	0.0222581	−	0.999752i	$-0.492914\pi$
$593$	642.837	0.0445163	0.0222581	+	0.999752i	$0.492914\pi$
$594$	0	0
$595$	6800.58	0.468566
$596$	−336.639	−0.0231364
$597$	0	0
$598$	−4483.36	−0.306585
$599$	14923.7	1.01797	0.508987	−	0.860774i	$-0.330020\pi$
$599$	14923.7	1.01797	0.508987	+	0.860774i	$0.330020\pi$
$600$	0	0
$601$	−9190.75	−0.623791	−0.311896	−	0.950116i	$-0.600964\pi$
$601$	−9190.75	−0.623791	−0.311896	+	0.950116i	$0.600964\pi$
$602$	10124.1	0.685431
$603$	0	0
$604$	2548.09	0.171656
$605$	4186.43	0.281327
$606$	0	0
$607$	−16137.4	−1.07907	−0.539535	−	0.841963i	$-0.681400\pi$
$607$	−16137.4	−1.07907	−0.539535	+	0.841963i	$0.681400\pi$
$608$	−4587.78	−0.306018
$609$	0	0
$610$	39321.7	2.60998
$611$	15439.4	1.02228
$612$	0	0
$613$	−12316.5	−0.811516	−0.405758	−	0.913981i	$-0.632992\pi$
$613$	−12316.5	−0.811516	−0.405758	+	0.913981i	$0.632992\pi$
$614$	−14903.7	−0.979584
$615$	0	0
$616$	−4820.80	−0.315318
$617$	11398.5	0.743735	0.371868	−	0.928286i	$-0.378718\pi$
$617$	11398.5	0.743735	0.371868	+	0.928286i	$0.378718\pi$
$618$	0	0
$619$	7815.13	0.507458	0.253729	−	0.967275i	$-0.418343\pi$
$619$	7815.13	0.507458	0.253729	+	0.967275i	$0.418343\pi$
$620$	5489.29	0.355573
$621$	0	0
$622$	8791.90	0.566758
$623$	249.679	0.0160565
$624$	0	0
$625$	5546.43	0.354971
$626$	8988.94	0.573914
$627$	0	0
$628$	−1867.89	−0.118689
$629$	3024.82	0.191745
$630$	0	0
$631$	−30593.9	−1.93015	−0.965073	−	0.261982i	$-0.915624\pi$
$631$	−30593.9	−1.93015	−0.965073	+	0.261982i	$0.915624\pi$
$632$	−12215.1	−0.768817
$633$	0	0
$634$	−2419.96	−0.151591
$635$	12206.1	0.762807
$636$	0	0
$637$	−3156.60	−0.196341
$638$	11895.0	0.738130
$639$	0	0
$640$	31257.3	1.93055
$641$	−22556.1	−1.38988	−0.694938	−	0.719069i	$-0.744568\pi$
$641$	−22556.1	−1.38988	−0.694938	+	0.719069i	$0.744568\pi$
$642$	0	0
$643$	17917.3	1.09890	0.549448	−	0.835528i	$-0.314838\pi$
$643$	17917.3	1.09890	0.549448	+	0.835528i	$0.314838\pi$
$644$	186.107	0.0113876
$645$	0	0
$646$	13967.3	0.850674
$647$	13444.1	0.816911	0.408455	−	0.912778i	$-0.366068\pi$
$647$	13444.1	0.816911	0.408455	+	0.912778i	$0.366068\pi$
$648$	0	0
$649$	19541.2	1.18191
$650$	43051.8	2.59789
$651$	0	0
$652$	388.878	0.0233583
$653$	−14177.3	−0.849616	−0.424808	−	0.905283i	$-0.639658\pi$
$653$	−14177.3	−0.849616	−0.424808	+	0.905283i	$0.639658\pi$
$654$	0	0
$655$	29309.1	1.74840
$656$	−24955.1	−1.48526
$657$	0	0
$658$	−5076.41	−0.300758
$659$	−12726.7	−0.752292	−0.376146	−	0.926560i	$-0.622751\pi$
$659$	−12726.7	−0.752292	−0.376146	+	0.926560i	$0.622751\pi$
$660$	0	0
$661$	18860.6	1.10982	0.554911	−	0.831909i	$-0.312752\pi$
$661$	18860.6	1.10982	0.554911	+	0.831909i	$0.312752\pi$
$662$	−23405.8	−1.37416
$663$	0	0
$664$	−20913.5	−1.22229
$665$	−11503.0	−0.670776
$666$	0	0
$667$	2718.84	0.157832
$668$	31.5739	0.00182879
$669$	0	0
$670$	7125.57	0.410873
$671$	23237.4	1.33691
$672$	0	0
$673$	22067.4	1.26395	0.631974	−	0.774989i	$-0.282245\pi$
$673$	22067.4	1.26395	0.631974	+	0.774989i	$0.282245\pi$
$674$	−35641.7	−2.03690
$675$	0	0
$676$	2257.55	0.128445
$677$	−19136.3	−1.08636	−0.543181	−	0.839616i	$-0.682780\pi$
$677$	−19136.3	−1.08636	−0.543181	+	0.839616i	$0.682780\pi$
$678$	0	0
$679$	890.240	0.0503156
$680$	−20119.3	−1.13462
$681$	0	0
$682$	25694.4	1.44265
$683$	−27945.8	−1.56562	−0.782808	−	0.622263i	$-0.786213\pi$
$683$	−27945.8	−1.56562	−0.782808	+	0.622263i	$0.786213\pi$
$684$	0	0
$685$	14734.5	0.821865
$686$	1037.88	0.0577643
$687$	0	0
$688$	−34372.1	−1.90469
$689$	−25903.1	−1.43226
$690$	0	0
$691$	−28123.0	−1.54826	−0.774131	−	0.633025i	$-0.781813\pi$
$691$	−28123.0	−1.54826	−0.774131	+	0.633025i	$0.781813\pi$
$692$	3110.11	0.170850
$693$	0	0
$694$	17019.1	0.930885
$695$	49754.3	2.71552
$696$	0	0
$697$	18128.4	0.985170
$698$	9147.70	0.496054
$699$	0	0
$700$	−1787.10	−0.0964945
$701$	−3602.16	−0.194082	−0.0970410	−	0.995280i	$-0.530938\pi$
$701$	−3602.16	−0.194082	−0.0970410	+	0.995280i	$0.530938\pi$
$702$	0	0
$703$	−5116.40	−0.274493
$704$	13906.7	0.744502
$705$	0	0
$706$	1223.01	0.0651962
$707$	−9637.62	−0.512673
$708$	0	0
$709$	−2994.24	−0.158605	−0.0793025	−	0.996851i	$-0.525269\pi$
$709$	−2994.24	−0.158605	−0.0793025	+	0.996851i	$0.525269\pi$
$710$	47499.3	2.51073
$711$	0	0
$712$	−738.668	−0.0388803
$713$	5872.98	0.308478
$714$	0	0
$715$	39840.9	2.08387
$716$	1446.57	0.0755040
$717$	0	0
$718$	6863.87	0.356765
$719$	−29085.7	−1.50864	−0.754321	−	0.656506i	$-0.772034\pi$
$719$	−29085.7	−1.50864	−0.754321	+	0.656506i	$0.772034\pi$
$720$	0	0
$721$	3560.32	0.183902
$722$	−2870.75	−0.147976
$723$	0	0
$724$	223.625	0.0114792
$725$	−26107.9	−1.33741
$726$	0	0
$727$	−15224.7	−0.776691	−0.388346	−	0.921514i	$-0.626953\pi$
$727$	−15224.7	−0.776691	−0.388346	+	0.921514i	$0.626953\pi$
$728$	9338.71	0.475434
$729$	0	0
$730$	40069.4	2.03156
$731$	24969.3	1.26337
$732$	0	0
$733$	26889.5	1.35496	0.677480	−	0.735542i	$-0.263072\pi$
$733$	26889.5	1.35496	0.677480	+	0.735542i	$0.263072\pi$
$734$	390.569	0.0196405
$735$	0	0
$736$	−1194.17	−0.0598069
$737$	4210.89	0.210462
$738$	0	0
$739$	−26240.9	−1.30621	−0.653104	−	0.757269i	$-0.726533\pi$
$739$	−26240.9	−1.30621	−0.653104	+	0.757269i	$0.726533\pi$
$740$	−1244.77	−0.0618358
$741$	0	0
$742$	8516.82	0.421378
$743$	−16294.7	−0.804571	−0.402285	−	0.915514i	$-0.631784\pi$
$743$	−16294.7	−0.804571	−0.402285	+	0.915514i	$0.631784\pi$
$744$	0	0
$745$	5416.00	0.266345
$746$	−4097.87	−0.201118
$747$	0	0
$748$	2008.12	0.0981607
$749$	3050.07	0.148795
$750$	0	0
$751$	10383.4	0.504522	0.252261	−	0.967659i	$-0.418826\pi$
$751$	10383.4	0.504522	0.252261	+	0.967659i	$0.418826\pi$
$752$	17234.7	0.835752
$753$	0	0
$754$	−23042.6	−1.11295
$755$	−40994.7	−1.97609
$756$	0	0
$757$	7102.44	0.341008	0.170504	−	0.985357i	$-0.445460\pi$
$757$	7102.44	0.341008	0.170504	+	0.985357i	$0.445460\pi$
$758$	30964.1	1.48373
$759$	0	0
$760$	34031.2	1.62427
$761$	7096.71	0.338049	0.169025	−	0.985612i	$-0.445938\pi$
$761$	7096.71	0.338049	0.169025	+	0.985612i	$0.445938\pi$
$762$	0	0
$763$	−2984.12	−0.141589
$764$	2075.22	0.0982705
$765$	0	0
$766$	17490.8	0.825024
$767$	−37854.5	−1.78207
$768$	0	0
$769$	17765.5	0.833080	0.416540	−	0.909117i	$-0.363243\pi$
$769$	17765.5	0.833080	0.416540	+	0.909117i	$0.363243\pi$
$770$	−13099.5	−0.613083
$771$	0	0
$772$	−995.678	−0.0464187
$773$	22105.9	1.02858	0.514291	−	0.857616i	$-0.328055\pi$
$773$	22105.9	1.02858	0.514291	+	0.857616i	$0.328055\pi$
$774$	0	0
$775$	−56395.7	−2.61393
$776$	−2633.75	−0.121838
$777$	0	0
$778$	35591.8	1.64014
$779$	−30663.7	−1.41032
$780$	0	0
$781$	28069.9	1.28607
$782$	3635.61	0.166252
$783$	0	0
$784$	−3523.66	−0.160516
$785$	30051.5	1.36635
$786$	0	0
$787$	42605.5	1.92976	0.964881	−	0.262686i	$-0.0846085\pi$
$787$	42605.5	1.92976	0.964881	+	0.262686i	$0.0846085\pi$
$788$	−260.201	−0.0117630
$789$	0	0
$790$	−33192.1	−1.49484
$791$	4847.74	0.217909
$792$	0	0
$793$	−45014.7	−2.01579
$794$	1375.38	0.0614741
$795$	0	0
$796$	−258.174	−0.0114959
$797$	24457.1	1.08697	0.543485	−	0.839419i	$-0.317105\pi$
$797$	24457.1	1.08697	0.543485	+	0.839419i	$0.317105\pi$
$798$	0	0
$799$	−12520.0	−0.554351
$800$	11467.2	0.506781
$801$	0	0
$802$	−34283.8	−1.50948
$803$	23679.2	1.04063
$804$	0	0
$805$	−2994.16	−0.131094
$806$	−49774.4	−2.17522
$807$	0	0
$808$	28512.6	1.24142
$809$	−15019.4	−0.652725	−0.326362	−	0.945245i	$-0.605823\pi$
$809$	−15019.4	−0.652725	−0.326362	+	0.945245i	$0.605823\pi$
$810$	0	0
$811$	−4964.25	−0.214943	−0.107471	−	0.994208i	$-0.534275\pi$
$811$	−4964.25	−0.214943	−0.107471	+	0.994208i	$0.534275\pi$
$812$	956.511	0.0413386
$813$	0	0
$814$	−5826.52	−0.250884
$815$	−6256.44	−0.268900
$816$	0	0
$817$	−42234.9	−1.80858
$818$	3054.09	0.130543
$819$	0	0
$820$	−7460.15	−0.317707
$821$	40616.6	1.72659	0.863293	−	0.504703i	$-0.168398\pi$
$821$	40616.6	1.72659	0.863293	+	0.504703i	$0.168398\pi$
$822$	0	0
$823$	1222.42	0.0517752	0.0258876	−	0.999665i	$-0.491759\pi$
$823$	1222.42	0.0517752	0.0258876	+	0.999665i	$0.491759\pi$
$824$	−10533.1	−0.445313
$825$	0	0
$826$	12446.4	0.524293
$827$	−726.223	−0.0305360	−0.0152680	−	0.999883i	$-0.504860\pi$
$827$	−726.223	−0.0305360	−0.0152680	+	0.999883i	$0.504860\pi$
$828$	0	0
$829$	36946.4	1.54789	0.773945	−	0.633253i	$-0.218281\pi$
$829$	36946.4	1.54789	0.773945	+	0.633253i	$0.218281\pi$
$830$	−56828.2	−2.37655
$831$	0	0
$832$	−26939.7	−1.12255
$833$	2559.73	0.106470
$834$	0	0
$835$	−507.975	−0.0210529
$836$	−3396.68	−0.140522
$837$	0	0
$838$	−42533.2	−1.75332
$839$	3592.55	0.147829	0.0739145	−	0.997265i	$-0.476451\pi$
$839$	3592.55	0.147829	0.0739145	+	0.997265i	$0.476451\pi$
$840$	0	0
$841$	−10415.3	−0.427048
$842$	9737.26	0.398537
$843$	0	0
$844$	1557.79	0.0635325
$845$	−36320.4	−1.47865
$846$	0	0
$847$	1575.77	0.0639245
$848$	−28915.1	−1.17093
$849$	0	0
$850$	−34911.2	−1.40876
$851$	−1331.77	−0.0536458
$852$	0	0
$853$	18955.7	0.760880	0.380440	−	0.924806i	$-0.375773\pi$
$853$	18955.7	0.760880	0.380440	+	0.924806i	$0.375773\pi$
$854$	14800.6	0.593053
$855$	0	0
$856$	−9023.54	−0.360302
$857$	−26150.7	−1.04235	−0.521174	−	0.853451i	$-0.674506\pi$
$857$	−26150.7	−1.04235	−0.521174	+	0.853451i	$0.674506\pi$
$858$	0	0
$859$	18422.9	0.731757	0.365879	−	0.930663i	$-0.380768\pi$
$859$	18422.9	0.731757	0.365879	+	0.930663i	$0.380768\pi$
$860$	−10275.3	−0.407424
$861$	0	0
$862$	2640.27	0.104325
$863$	−23731.6	−0.936077	−0.468038	−	0.883708i	$-0.655039\pi$
$863$	−23731.6	−0.936077	−0.468038	+	0.883708i	$0.655039\pi$
$864$	0	0
$865$	−50036.8	−1.96682
$866$	29247.6	1.14766
$867$	0	0
$868$	2066.16	0.0807951
$869$	−19615.0	−0.765702
$870$	0	0
$871$	−8157.22	−0.317333
$872$	8828.44	0.342854
$873$	0	0
$874$	−6149.53	−0.237999
$875$	12479.1	0.482137
$876$	0	0
$877$	−27340.9	−1.05272	−0.526360	−	0.850262i	$-0.676444\pi$
$877$	−27340.9	−1.05272	−0.526360	+	0.850262i	$0.676444\pi$
$878$	−46773.6	−1.79787
$879$	0	0
$880$	44473.7	1.70364
$881$	29829.5	1.14073	0.570363	−	0.821393i	$-0.306802\pi$
$881$	29829.5	1.14073	0.570363	+	0.821393i	$0.306802\pi$
$882$	0	0
$883$	28971.8	1.10417	0.552084	−	0.833789i	$-0.313833\pi$
$883$	28971.8	1.10417	0.552084	+	0.833789i	$0.313833\pi$
$884$	−3890.07	−0.148006
$885$	0	0
$886$	−31732.6	−1.20325
$887$	−13412.2	−0.507708	−0.253854	−	0.967243i	$-0.581698\pi$
$887$	−13412.2	−0.507708	−0.253854	+	0.967243i	$0.581698\pi$
$888$	0	0
$889$	4594.34	0.173329
$890$	−2007.18	−0.0755963
$891$	0	0
$892$	1398.21	0.0524838
$893$	21177.2	0.793583
$894$	0	0
$895$	−23273.1	−0.869198
$896$	11765.2	0.438669
$897$	0	0
$898$	18669.4	0.693772
$899$	30184.7	1.11982
$900$	0	0
$901$	21005.2	0.776674
$902$	−34919.6	−1.28902
$903$	0	0
$904$	−14341.9	−0.527660
$905$	−3597.77	−0.132148
$906$	0	0
$907$	−21692.1	−0.794129	−0.397064	−	0.917791i	$-0.629971\pi$
$907$	−21692.1	−0.794129	−0.397064	+	0.917791i	$0.629971\pi$
$908$	3943.12	0.144116
$909$	0	0
$910$	25376.0	0.924402
$911$	6922.04	0.251742	0.125871	−	0.992047i	$-0.459827\pi$
$911$	6922.04	0.251742	0.125871	+	0.992047i	$0.459827\pi$
$912$	0	0
$913$	−33582.9	−1.21734
$914$	49310.8	1.78453
$915$	0	0
$916$	592.403	0.0213685
$917$	11031.9	0.397281
$918$	0	0
$919$	−32716.2	−1.17433	−0.587164	−	0.809468i	$-0.699755\pi$
$919$	−32716.2	−1.17433	−0.587164	+	0.809468i	$0.699755\pi$
$920$	8858.15	0.317440
$921$	0	0
$922$	55688.2	1.98915
$923$	−54376.3	−1.93913
$924$	0	0
$925$	12788.4	0.454575
$926$	39409.2	1.39856
$927$	0	0
$928$	−6137.56	−0.217107
$929$	−9370.01	−0.330915	−0.165458	−	0.986217i	$-0.552910\pi$
$929$	−9370.01	−0.330915	−0.165458	+	0.986217i	$0.552910\pi$
$930$	0	0
$931$	−4329.71	−0.152417
$932$	4512.60	0.158600
$933$	0	0
$934$	−7058.40	−0.247278
$935$	−32307.5	−1.13002
$936$	0	0
$937$	−50755.5	−1.76960	−0.884798	−	0.465975i	$-0.845704\pi$
$937$	−50755.5	−1.76960	−0.884798	+	0.465975i	$0.845704\pi$
$938$	2682.06	0.0933606
$939$	0	0
$940$	5152.20	0.178772
$941$	−1528.30	−0.0529449	−0.0264724	−	0.999650i	$-0.508427\pi$
$941$	−1528.30	−0.0529449	−0.0264724	+	0.999650i	$0.508427\pi$
$942$	0	0
$943$	−7981.60	−0.275627
$944$	−42256.3	−1.45691
$945$	0	0
$946$	−48096.8	−1.65303
$947$	−14564.9	−0.499786	−0.249893	−	0.968274i	$-0.580395\pi$
$947$	−14564.9	−0.499786	−0.249893	+	0.968274i	$0.580395\pi$
$948$	0	0
$949$	−45870.7	−1.56905
$950$	59051.3	2.01671
$951$	0	0
$952$	−7572.88	−0.257814
$953$	26605.3	0.904334	0.452167	−	0.891933i	$-0.350651\pi$
$953$	26605.3	0.904334	0.452167	+	0.891933i	$0.350651\pi$
$954$	0	0
$955$	−33387.0	−1.13129
$956$	−1646.77	−0.0557117
$957$	0	0
$958$	48473.1	1.63475
$959$	5546.06	0.186748
$960$	0	0
$961$	35411.0	1.18865
$962$	11287.0	0.378281
$963$	0	0
$964$	−7449.69	−0.248899
$965$	16018.9	0.534370
$966$	0	0
$967$	−13268.4	−0.441244	−0.220622	−	0.975359i	$-0.570809\pi$
$967$	−13268.4	−0.441244	−0.220622	+	0.975359i	$0.570809\pi$
$968$	−4661.86	−0.154791
$969$	0	0
$970$	−7156.66	−0.236893
$971$	40126.2	1.32617	0.663085	−	0.748544i	$-0.269247\pi$
$971$	40126.2	1.32617	0.663085	+	0.748544i	$0.269247\pi$
$972$	0	0
$973$	18727.5	0.617035
$974$	4780.44	0.157264
$975$	0	0
$976$	−50249.1	−1.64799
$977$	−997.072	−0.0326501	−0.0163251	−	0.999867i	$-0.505197\pi$
$977$	−997.072	−0.0326501	−0.0163251	+	0.999867i	$0.505197\pi$
$978$	0	0
$979$	−1186.15	−0.0387228
$980$	−1053.37	−0.0343354
$981$	0	0
$982$	−62167.5	−2.02021
$983$	20022.9	0.649677	0.324838	−	0.945770i	$-0.394690\pi$
$983$	20022.9	0.649677	0.324838	+	0.945770i	$0.394690\pi$
$984$	0	0
$985$	4186.23	0.135415
$986$	18685.5	0.603518
$987$	0	0
$988$	6579.94	0.211878
$989$	−10993.5	−0.353462
$990$	0	0
$991$	1277.26	0.0409419	0.0204710	−	0.999790i	$-0.493483\pi$
$991$	1277.26	0.0409419	0.0204710	+	0.999790i	$0.493483\pi$
$992$	−13257.8	−0.424329
$993$	0	0
$994$	17878.7	0.570500
$995$	4153.61	0.132340
$996$	0	0
$997$	59113.7	1.87778	0.938891	−	0.344215i	$-0.111855\pi$
$997$	59113.7	1.87778	0.938891	+	0.344215i	$0.111855\pi$
$998$	4028.87	0.127787
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1449.4.a.i.1.2		8
3.2	odd	2		161.4.a.b.1.7	✓	8
21.20	even	2		1127.4.a.e.1.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
161.4.a.b.1.7	✓	8	3.2	odd	2
1127.4.a.e.1.7		8	21.20	even	2
1449.4.a.i.1.2		8	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-3.02588 q^{2} +1.15594 q^{4} -18.5973 q^{5} -7.00000 q^{7} +20.7093 q^{8} +56.2732 q^{10} +33.2549 q^{11} -64.4205 q^{13} +21.1812 q^{14} -71.9113 q^{16} +52.2394 q^{17} -88.3614 q^{19} -21.4974 q^{20} -100.625 q^{22} -23.0000 q^{23} +220.859 q^{25} +194.928 q^{26} -8.09159 q^{28} -118.211 q^{29} -255.347 q^{31} +51.9206 q^{32} -158.070 q^{34} +130.181 q^{35} +57.9031 q^{37} +267.371 q^{38} -385.137 q^{40} +347.026 q^{41} +477.979 q^{43} +38.4407 q^{44} +69.5952 q^{46} -239.666 q^{47} +49.0000 q^{49} -668.294 q^{50} -74.4663 q^{52} +402.094 q^{53} -618.452 q^{55} -144.965 q^{56} +357.691 q^{58} +587.617 q^{59} +698.764 q^{61} +772.649 q^{62} +418.185 q^{64} +1198.05 q^{65} +126.625 q^{67} +60.3857 q^{68} -393.912 q^{70} +844.084 q^{71} +712.052 q^{73} -175.208 q^{74} -102.141 q^{76} -232.784 q^{77} -589.839 q^{79} +1337.36 q^{80} -1050.06 q^{82} -1009.86 q^{83} -971.511 q^{85} -1446.31 q^{86} +688.686 q^{88} -35.6685 q^{89} +450.943 q^{91} -26.5866 q^{92} +725.201 q^{94} +1643.28 q^{95} -127.177 q^{97} -148.268 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 24 q^{4} + 24 q^{5} - 56 q^{7} + 69 q^{8} - 30 q^{10} + 98 q^{11} - 145 q^{13} - 76 q^{16} + 96 q^{17} - 226 q^{19} + 22 q^{20} - 98 q^{22} - 184 q^{23} + 60 q^{25} + 185 q^{26} - 168 q^{28} - 73 q^{29}+ \cdots - 1452 q^{97}+O(q^{100}) \) 8 * q + 24 * q^4 + 24 * q^5 - 56 * q^7 + 69 * q^8 - 30 * q^10 + 98 * q^11 - 145 * q^13 - 76 * q^16 + 96 * q^17 - 226 * q^19 + 22 * q^20 - 98 * q^22 - 184 * q^23 + 60 * q^25 + 185 * q^26 - 168 * q^28 - 73 * q^29 - 987 * q^31 - 118 * q^32 - 846 * q^34 - 168 * q^35 - 2 * q^37 + 454 * q^38 - 640 * q^40 + 907 * q^41 + 636 * q^43 - 126 * q^44 - 421 * q^47 + 392 * q^49 - 2308 * q^50 - 931 * q^52 - 138 * q^53 - 90 * q^55 - 483 * q^56 + 1329 * q^58 + 924 * q^59 - 890 * q^61 - 2175 * q^62 + 303 * q^64 + 392 * q^65 - 1392 * q^67 - 2592 * q^68 + 210 * q^70 - 259 * q^71 - 297 * q^73 - 2314 * q^74 - 428 * q^76 - 686 * q^77 - 1788 * q^79 + 2104 * q^80 - 219 * q^82 - 2114 * q^83 + 528 * q^85 + 2360 * q^86 + 2144 * q^88 + 1040 * q^89 + 1015 * q^91 - 552 * q^92 - 1009 * q^94 - 1268 * q^95 - 1452 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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