LMFDB - Embedded newform 2450.4.a.cw.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2450,4,Mod(1,2450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2450, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2450.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2450 = 2 \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2450.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:	$144.554679514$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 92x^{4} + 26x^{3} + 2116x^{2} + 80x - 7800 $ x^6 - x^5 - 92x^4 + 26x^3 + 2116x^2 + 80x - 7800
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 70)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-5.49484$ of defining polynomial
Character	$\chi$	$=$	2450.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{2} -4.49484 q^{3} +4.00000 q^{4} +8.98967 q^{6} -8.00000 q^{8} -6.79645 q^{9} +O(q^{10})$	\(q-2.00000 q^{2} -4.49484 q^{3} +4.00000 q^{4} +8.98967 q^{6} -8.00000 q^{8} -6.79645 q^{9} +13.9329 q^{11} -17.9793 q^{12} +78.0096 q^{13} +16.0000 q^{16} -105.132 q^{17} +13.5929 q^{18} -152.300 q^{19} -27.8658 q^{22} -108.116 q^{23} +35.9587 q^{24} -156.019 q^{26} +151.909 q^{27} -109.323 q^{29} +197.196 q^{31} -32.0000 q^{32} -62.6261 q^{33} +210.265 q^{34} -27.1858 q^{36} +254.290 q^{37} +304.601 q^{38} -350.641 q^{39} -193.905 q^{41} -41.5329 q^{43} +55.7316 q^{44} +216.232 q^{46} +109.886 q^{47} -71.9174 q^{48} +472.553 q^{51} +312.039 q^{52} +11.0054 q^{53} -303.819 q^{54} +684.566 q^{57} +218.645 q^{58} -192.141 q^{59} +34.9035 q^{61} -394.392 q^{62} +64.0000 q^{64} +125.252 q^{66} +374.830 q^{67} -420.530 q^{68} +485.963 q^{69} -575.722 q^{71} +54.3716 q^{72} -522.161 q^{73} -508.580 q^{74} -609.202 q^{76} +701.281 q^{78} -456.182 q^{79} -499.304 q^{81} +387.810 q^{82} +773.057 q^{83} +83.0657 q^{86} +491.387 q^{87} -111.463 q^{88} -1329.07 q^{89} -432.463 q^{92} -886.364 q^{93} -219.772 q^{94} +143.835 q^{96} +1038.18 q^{97} -94.6942 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 12 q^{2} + 7 q^{3} + 24 q^{4} - 14 q^{6} - 48 q^{8} + 31 q^{9}+O(q^{10}) $ 6 * q - 12 * q^2 + 7 * q^3 + 24 * q^4 - 14 * q^6 - 48 * q^8 + 31 * q^9	$ 6 q - 12 q^{2} + 7 q^{3} + 24 q^{4} - 14 q^{6} - 48 q^{8} + 31 q^{9} + 31 q^{11} + 28 q^{12} + 59 q^{13} + 96 q^{16} + 68 q^{17} - 62 q^{18} - 93 q^{19} - 62 q^{22} + 94 q^{23} - 56 q^{24} - 118 q^{26} + 385 q^{27} + 169 q^{29} - 326 q^{31} - 192 q^{32} + 400 q^{33} - 136 q^{34} + 124 q^{36} + 253 q^{37} + 186 q^{38} - 434 q^{39} - 198 q^{41} + 99 q^{43} + 124 q^{44} - 188 q^{46} + 901 q^{47} + 112 q^{48} + 724 q^{51} + 236 q^{52} - 233 q^{53} - 770 q^{54} + 518 q^{57} - 338 q^{58} - 668 q^{59} - 157 q^{61} + 652 q^{62} + 384 q^{64} - 800 q^{66} - 1193 q^{67} + 272 q^{68} + 45 q^{69} + 1108 q^{71} - 248 q^{72} + 1458 q^{73} - 506 q^{74} - 372 q^{76} + 868 q^{78} - 886 q^{79} - 614 q^{81} + 396 q^{82} + 1799 q^{83} - 198 q^{86} + 789 q^{87} - 248 q^{88} - 3047 q^{89} + 376 q^{92} - 1902 q^{93} - 1802 q^{94} - 224 q^{96} + 3404 q^{97} + 4273 q^{99}+O(q^{100}) $ 6 * q - 12 * q^2 + 7 * q^3 + 24 * q^4 - 14 * q^6 - 48 * q^8 + 31 * q^9 + 31 * q^11 + 28 * q^12 + 59 * q^13 + 96 * q^16 + 68 * q^17 - 62 * q^18 - 93 * q^19 - 62 * q^22 + 94 * q^23 - 56 * q^24 - 118 * q^26 + 385 * q^27 + 169 * q^29 - 326 * q^31 - 192 * q^32 + 400 * q^33 - 136 * q^34 + 124 * q^36 + 253 * q^37 + 186 * q^38 - 434 * q^39 - 198 * q^41 + 99 * q^43 + 124 * q^44 - 188 * q^46 + 901 * q^47 + 112 * q^48 + 724 * q^51 + 236 * q^52 - 233 * q^53 - 770 * q^54 + 518 * q^57 - 338 * q^58 - 668 * q^59 - 157 * q^61 + 652 * q^62 + 384 * q^64 - 800 * q^66 - 1193 * q^67 + 272 * q^68 + 45 * q^69 + 1108 * q^71 - 248 * q^72 + 1458 * q^73 - 506 * q^74 - 372 * q^76 + 868 * q^78 - 886 * q^79 - 614 * q^81 + 396 * q^82 + 1799 * q^83 - 198 * q^86 + 789 * q^87 - 248 * q^88 - 3047 * q^89 + 376 * q^92 - 1902 * q^93 - 1802 * q^94 - 224 * q^96 + 3404 * q^97 + 4273 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	−4.49484	−0.865032	−0.432516	−	0.901626i	$-0.642374\pi$
$3$	−4.49484	−0.865032	−0.432516	+	0.901626i	$0.642374\pi$
$4$	4.00000	0.500000
$5$	0	0
$5$	0	0
$6$	8.98967	0.611670
$7$	0	0
$7$	0	0
$8$	−8.00000	−0.353553
$9$	−6.79645	−0.251720
$10$	0	0
$11$	13.9329	0.381902	0.190951	−	0.981600i	$-0.438843\pi$
$11$	13.9329	0.381902	0.190951	+	0.981600i	$0.438843\pi$
$12$	−17.9793	−0.432516
$13$	78.0096	1.66431	0.832153	−	0.554546i	$-0.187108\pi$
$13$	78.0096	1.66431	0.832153	+	0.554546i	$0.187108\pi$
$14$	0	0
$15$	0	0
$16$	16.0000	0.250000
$17$	−105.132	−1.49990	−0.749952	−	0.661492i	$-0.769923\pi$
$17$	−105.132	−1.49990	−0.749952	+	0.661492i	$0.769923\pi$
$18$	13.5929	0.177993
$19$	−152.300	−1.83895	−0.919477	−	0.393144i	$-0.871387\pi$
$19$	−152.300	−1.83895	−0.919477	+	0.393144i	$0.871387\pi$
$20$	0	0
$21$	0	0
$22$	−27.8658	−0.270046
$23$	−108.116	−0.980161	−0.490081	−	0.871677i	$-0.663033\pi$
$23$	−108.116	−0.980161	−0.490081	+	0.871677i	$0.663033\pi$
$24$	35.9587	0.305835
$25$	0	0
$26$	−156.019	−1.17684
$27$	151.909	1.08278
$28$	0	0
$29$	−109.323	−0.700023	−0.350012	−	0.936745i	$-0.613822\pi$
$29$	−109.323	−0.700023	−0.350012	+	0.936745i	$0.613822\pi$
$30$	0	0
$31$	197.196	1.14250	0.571249	−	0.820777i	$-0.306459\pi$
$31$	197.196	1.14250	0.571249	+	0.820777i	$0.306459\pi$
$32$	−32.0000	−0.176777
$33$	−62.6261	−0.330358
$34$	210.265	1.06059
$35$	0	0
$36$	−27.1858	−0.125860
$37$	254.290	1.12986	0.564932	−	0.825137i	$-0.308902\pi$
$37$	254.290	1.12986	0.564932	+	0.825137i	$0.308902\pi$
$38$	304.601	1.30034
$39$	−350.641	−1.43968
$40$	0	0
$41$	−193.905	−0.738606	−0.369303	−	0.929309i	$-0.620404\pi$
$41$	−193.905	−0.738606	−0.369303	+	0.929309i	$0.620404\pi$
$42$	0	0
$43$	−41.5329	−0.147295	−0.0736477	−	0.997284i	$-0.523464\pi$
$43$	−41.5329	−0.147295	−0.0736477	+	0.997284i	$0.523464\pi$
$44$	55.7316	0.190951
$45$	0	0
$46$	216.232	0.693079
$47$	109.886	0.341032	0.170516	−	0.985355i	$-0.445457\pi$
$47$	109.886	0.341032	0.170516	+	0.985355i	$0.445457\pi$
$48$	−71.9174	−0.216258
$49$	0	0
$50$	0	0
$51$	472.553	1.29746
$52$	312.039	0.832153
$53$	11.0054	0.0285228	0.0142614	−	0.999898i	$-0.495460\pi$
$53$	11.0054	0.0285228	0.0142614	+	0.999898i	$0.495460\pi$
$54$	−303.819	−0.765639
$55$	0	0
$56$	0	0
$57$	684.566	1.59075
$58$	218.645	0.494991
$59$	−192.141	−0.423978	−0.211989	−	0.977272i	$-0.567994\pi$
$59$	−192.141	−0.423978	−0.211989	+	0.977272i	$0.567994\pi$
$60$	0	0
$61$	34.9035	0.0732611	0.0366306	−	0.999329i	$-0.488338\pi$
$61$	34.9035	0.0732611	0.0366306	+	0.999329i	$0.488338\pi$
$62$	−394.392	−0.807868
$63$	0	0
$64$	64.0000	0.125000
$65$	0	0
$66$	125.252	0.233598
$67$	374.830	0.683474	0.341737	−	0.939796i	$-0.388985\pi$
$67$	374.830	0.683474	0.341737	+	0.939796i	$0.388985\pi$
$68$	−420.530	−0.749952
$69$	485.963	0.847870
$70$	0	0
$71$	−575.722	−0.962333	−0.481167	−	0.876629i	$-0.659787\pi$
$71$	−575.722	−0.962333	−0.481167	+	0.876629i	$0.659787\pi$
$72$	54.3716	0.0889966
$73$	−522.161	−0.837182	−0.418591	−	0.908175i	$-0.637476\pi$
$73$	−522.161	−0.837182	−0.418591	+	0.908175i	$0.637476\pi$
$74$	−508.580	−0.798935
$75$	0	0
$76$	−609.202	−0.919477
$77$	0	0
$78$	701.281	1.01801
$79$	−456.182	−0.649676	−0.324838	−	0.945770i	$-0.605310\pi$
$79$	−456.182	−0.649676	−0.324838	+	0.945770i	$0.605310\pi$
$80$	0	0
$81$	−499.304	−0.684917
$82$	387.810	0.522273
$83$	773.057	1.02234	0.511169	−	0.859480i	$-0.329213\pi$
$83$	773.057	1.02234	0.511169	+	0.859480i	$0.329213\pi$
$84$	0	0
$85$	0	0
$86$	83.0657	0.104154
$87$	491.387	0.605542
$88$	−111.463	−0.135023
$89$	−1329.07	−1.58294	−0.791470	−	0.611208i	$-0.790684\pi$
$89$	−1329.07	−1.58294	−0.791470	+	0.611208i	$0.790684\pi$
$90$	0	0
$91$	0	0
$92$	−432.463	−0.490081
$93$	−886.364	−0.988297
$94$	−219.772	−0.241146
$95$	0	0
$96$	143.835	0.152917
$97$	1038.18	1.08671	0.543356	−	0.839503i	$-0.317153\pi$
$97$	1038.18	1.08671	0.543356	+	0.839503i	$0.317153\pi$
$98$	0	0
$99$	−94.6942	−0.0961325
$100$	0	0
$101$	489.559	0.482307	0.241153	−	0.970487i	$-0.422474\pi$
$101$	489.559	0.482307	0.241153	+	0.970487i	$0.422474\pi$
$102$	−945.107	−0.917446
$103$	1185.15	1.13375	0.566877	−	0.823802i	$-0.308151\pi$
$103$	1185.15	1.13375	0.566877	+	0.823802i	$0.308151\pi$
$104$	−624.077	−0.588421
$105$	0	0
$106$	−22.0108	−0.0201686
$107$	−1192.49	−1.07741	−0.538703	−	0.842496i	$-0.681085\pi$
$107$	−1192.49	−1.07741	−0.538703	+	0.842496i	$0.681085\pi$
$108$	607.638	0.541389
$109$	−285.431	−0.250820	−0.125410	−	0.992105i	$-0.540025\pi$
$109$	−285.431	−0.250820	−0.125410	+	0.992105i	$0.540025\pi$
$110$	0	0
$111$	−1142.99	−0.977369
$112$	0	0
$113$	−350.019	−0.291389	−0.145695	−	0.989330i	$-0.546542\pi$
$113$	−350.019	−0.291389	−0.145695	+	0.989330i	$0.546542\pi$
$114$	−1369.13	−1.12483
$115$	0	0
$116$	−437.290	−0.350012
$117$	−530.188	−0.418940
$118$	384.283	0.299798
$119$	0	0
$120$	0	0
$121$	−1136.87	−0.854151
$122$	−69.8069	−0.0518034
$123$	871.570	0.638917
$124$	788.784	0.571249
$125$	0	0
$126$	0	0
$127$	−1112.45	−0.777274	−0.388637	−	0.921391i	$-0.627054\pi$
$127$	−1112.45	−0.777274	−0.388637	+	0.921391i	$0.627054\pi$
$128$	−128.000	−0.0883883
$129$	186.683	0.127415
$130$	0	0
$131$	−605.974	−0.404154	−0.202077	−	0.979370i	$-0.564769\pi$
$131$	−605.974	−0.404154	−0.202077	+	0.979370i	$0.564769\pi$
$132$	−250.504	−0.165179
$133$	0	0
$134$	−749.659	−0.483289
$135$	0	0
$136$	841.060	0.530296
$137$	403.891	0.251874	0.125937	−	0.992038i	$-0.459806\pi$
$137$	403.891	0.251874	0.125937	+	0.992038i	$0.459806\pi$
$138$	−971.926	−0.599535
$139$	−525.532	−0.320683	−0.160342	−	0.987062i	$-0.551260\pi$
$139$	−525.532	−0.320683	−0.160342	+	0.987062i	$0.551260\pi$
$140$	0	0
$141$	−493.918	−0.295003
$142$	1151.44	0.680472
$143$	1086.90	0.635602
$144$	−108.743	−0.0629301
$145$	0	0
$146$	1044.32	0.591977
$147$	0	0
$148$	1017.16	0.564932
$149$	−2034.87	−1.11881	−0.559407	−	0.828893i	$-0.688971\pi$
$149$	−2034.87	−1.11881	−0.559407	+	0.828893i	$0.688971\pi$
$150$	0	0
$151$	1748.70	0.942430	0.471215	−	0.882018i	$-0.343816\pi$
$151$	1748.70	0.942430	0.471215	+	0.882018i	$0.343816\pi$
$152$	1218.40	0.650168
$153$	714.527	0.377556
$154$	0	0
$155$	0	0
$156$	−1402.56	−0.719839
$157$	448.507	0.227992	0.113996	−	0.993481i	$-0.463635\pi$
$157$	448.507	0.227992	0.113996	+	0.993481i	$0.463635\pi$
$158$	912.363	0.459391
$159$	−49.4674	−0.0246731
$160$	0	0
$161$	0	0
$162$	998.608	0.484309
$163$	−911.681	−0.438088	−0.219044	−	0.975715i	$-0.570294\pi$
$163$	−911.681	−0.438088	−0.219044	+	0.975715i	$0.570294\pi$
$164$	−775.619	−0.369303
$165$	0	0
$166$	−1546.11	−0.722902
$167$	1131.12	0.524125	0.262063	−	0.965051i	$-0.415597\pi$
$167$	1131.12	0.524125	0.262063	+	0.965051i	$0.415597\pi$
$168$	0	0
$169$	3888.51	1.76992
$170$	0	0
$171$	1035.10	0.462902
$172$	−166.131	−0.0736477
$173$	2138.58	0.939844	0.469922	−	0.882708i	$-0.344282\pi$
$173$	2138.58	0.939844	0.469922	+	0.882708i	$0.344282\pi$
$174$	−982.773	−0.428183
$175$	0	0
$176$	222.926	0.0954756
$177$	863.644	0.366754
$178$	2658.15	1.11931
$179$	1739.07	0.726170	0.363085	−	0.931756i	$-0.381724\pi$
$179$	1739.07	0.726170	0.363085	+	0.931756i	$0.381724\pi$
$180$	0	0
$181$	2304.17	0.946231	0.473115	−	0.881001i	$-0.343129\pi$
$181$	2304.17	0.946231	0.473115	+	0.881001i	$0.343129\pi$
$182$	0	0
$183$	−156.885	−0.0633732
$184$	864.927	0.346539
$185$	0	0
$186$	1772.73	0.698832
$187$	−1464.80	−0.572817
$188$	439.543	0.170516
$189$	0	0
$190$	0	0
$191$	−3611.07	−1.36800	−0.684000	−	0.729482i	$-0.739761\pi$
$191$	−3611.07	−1.36800	−0.684000	+	0.729482i	$0.739761\pi$
$192$	−287.670	−0.108129
$193$	−4340.90	−1.61899	−0.809494	−	0.587128i	$-0.800259\pi$
$193$	−4340.90	−1.61899	−0.809494	+	0.587128i	$0.800259\pi$
$194$	−2076.36	−0.768421
$195$	0	0
$196$	0	0
$197$	380.943	0.137772	0.0688859	−	0.997625i	$-0.478056\pi$
$197$	380.943	0.137772	0.0688859	+	0.997625i	$0.478056\pi$
$198$	189.388	0.0679760
$199$	−3476.70	−1.23848	−0.619238	−	0.785203i	$-0.712559\pi$
$199$	−3476.70	−1.23848	−0.619238	+	0.785203i	$0.712559\pi$
$200$	0	0
$201$	−1684.80	−0.591226
$202$	−979.119	−0.341042
$203$	0	0
$204$	1890.21	0.648732
$205$	0	0
$206$	−2370.31	−0.801685
$207$	734.804	0.246726
$208$	1248.15	0.416077
$209$	−2121.99	−0.702301
$210$	0	0
$211$	1693.38	0.552497	0.276248	−	0.961086i	$-0.410909\pi$
$211$	1693.38	0.552497	0.276248	+	0.961086i	$0.410909\pi$
$212$	44.0216	0.0142614
$213$	2587.78	0.832449
$214$	2384.98	0.761841
$215$	0	0
$216$	−1215.28	−0.382820
$217$	0	0
$218$	570.862	0.177356
$219$	2347.03	0.724189
$220$	0	0
$221$	−8201.35	−2.49630
$222$	2285.98	0.691104
$223$	−260.918	−0.0783514	−0.0391757	−	0.999232i	$-0.512473\pi$
$223$	−260.918	−0.0783514	−0.0391757	+	0.999232i	$0.512473\pi$
$224$	0	0
$225$	0	0
$226$	700.038	0.206043
$227$	279.657	0.0817688	0.0408844	−	0.999164i	$-0.486982\pi$
$227$	279.657	0.0817688	0.0408844	+	0.999164i	$0.486982\pi$
$228$	2738.26	0.795377
$229$	1722.39	0.497024	0.248512	−	0.968629i	$-0.420058\pi$
$229$	1722.39	0.497024	0.248512	+	0.968629i	$0.420058\pi$
$230$	0	0
$231$	0	0
$232$	874.580	0.247496
$233$	−6659.62	−1.87247	−0.936237	−	0.351370i	$-0.885716\pi$
$233$	−6659.62	−1.87247	−0.936237	+	0.351370i	$0.885716\pi$
$234$	1060.38	0.296235
$235$	0	0
$236$	−768.566	−0.211989
$237$	2050.46	0.561991
$238$	0	0
$239$	1018.63	0.275688	0.137844	−	0.990454i	$-0.455983\pi$
$239$	1018.63	0.275688	0.137844	+	0.990454i	$0.455983\pi$
$240$	0	0
$241$	−5853.84	−1.56464	−0.782322	−	0.622875i	$-0.785965\pi$
$241$	−5853.84	−1.56464	−0.782322	+	0.622875i	$0.785965\pi$
$242$	2273.75	0.603976
$243$	−1857.27	−0.490303
$244$	139.614	0.0366306
$245$	0	0
$246$	−1743.14	−0.451783
$247$	−11880.9	−3.06058
$248$	−1577.57	−0.403934
$249$	−3474.77	−0.884355
$250$	0	0
$251$	1101.55	0.277008	0.138504	−	0.990362i	$-0.455771\pi$
$251$	1101.55	0.277008	0.138504	+	0.990362i	$0.455771\pi$
$252$	0	0
$253$	−1506.37	−0.374326
$254$	2224.90	0.549616
$255$	0	0
$256$	256.000	0.0625000
$257$	−1442.47	−0.350111	−0.175056	−	0.984559i	$-0.556011\pi$
$257$	−1442.47	−0.350111	−0.175056	+	0.984559i	$0.556011\pi$
$258$	−373.367	−0.0900961
$259$	0	0
$260$	0	0
$261$	743.005	0.176210
$262$	1211.95	0.285780
$263$	5430.39	1.27320	0.636601	−	0.771194i	$-0.280340\pi$
$263$	5430.39	1.27320	0.636601	+	0.771194i	$0.280340\pi$
$264$	501.009	0.116799
$265$	0	0
$266$	0	0
$267$	5973.97	1.36929
$268$	1499.32	0.341737
$269$	5011.89	1.13599	0.567993	−	0.823033i	$-0.307720\pi$
$269$	5011.89	1.13599	0.567993	+	0.823033i	$0.307720\pi$
$270$	0	0
$271$	3317.40	0.743607	0.371803	−	0.928311i	$-0.378740\pi$
$271$	3317.40	0.743607	0.371803	+	0.928311i	$0.378740\pi$
$272$	−1682.12	−0.374976
$273$	0	0
$274$	−807.783	−0.178102
$275$	0	0
$276$	1943.85	0.423935
$277$	−4160.35	−0.902424	−0.451212	−	0.892417i	$-0.649008\pi$
$277$	−4160.35	−0.902424	−0.451212	+	0.892417i	$0.649008\pi$
$278$	1051.06	0.226757
$279$	−1340.23	−0.287590
$280$	0	0
$281$	−2423.12	−0.514418	−0.257209	−	0.966356i	$-0.582803\pi$
$281$	−2423.12	−0.514418	−0.257209	+	0.966356i	$0.582803\pi$
$282$	987.837	0.208599
$283$	8370.14	1.75814	0.879069	−	0.476695i	$-0.158165\pi$
$283$	8370.14	1.75814	0.879069	+	0.476695i	$0.158165\pi$
$284$	−2302.89	−0.481167
$285$	0	0
$286$	−2173.80	−0.449439
$287$	0	0
$288$	217.486	0.0444983
$289$	6139.84	1.24971
$290$	0	0
$291$	−4666.44	−0.940040
$292$	−2088.64	−0.418591
$293$	5292.21	1.05520	0.527601	−	0.849492i	$-0.323092\pi$
$293$	5292.21	1.05520	0.527601	+	0.849492i	$0.323092\pi$
$294$	0	0
$295$	0	0
$296$	−2034.32	−0.399467
$297$	2116.54	0.413515
$298$	4069.75	0.791121
$299$	−8434.08	−1.63129
$300$	0	0
$301$	0	0
$302$	−3497.39	−0.666398
$303$	−2200.49	−0.417211
$304$	−2436.81	−0.459738
$305$	0	0
$306$	−1429.05	−0.266973
$307$	3852.07	0.716121	0.358061	−	0.933698i	$-0.383438\pi$
$307$	3852.07	0.716121	0.358061	+	0.933698i	$0.383438\pi$
$308$	0	0
$309$	−5327.07	−0.980733
$310$	0	0
$311$	1092.48	0.199192	0.0995962	−	0.995028i	$-0.468245\pi$
$311$	1092.48	0.199192	0.0995962	+	0.995028i	$0.468245\pi$
$312$	2805.12	0.509003
$313$	−1324.90	−0.239257	−0.119629	−	0.992819i	$-0.538170\pi$
$313$	−1324.90	−0.239257	−0.119629	+	0.992819i	$0.538170\pi$
$314$	−897.014	−0.161215
$315$	0	0
$316$	−1824.73	−0.324838
$317$	4985.47	0.883319	0.441659	−	0.897183i	$-0.354390\pi$
$317$	4985.47	0.883319	0.441659	+	0.897183i	$0.354390\pi$
$318$	98.9349	0.0174465
$319$	−1523.18	−0.267341
$320$	0	0
$321$	5360.05	0.931990
$322$	0	0
$323$	16011.7	2.75825
$324$	−1997.22	−0.342458
$325$	0	0
$326$	1823.36	0.309775
$327$	1282.97	0.216967
$328$	1551.24	0.261137
$329$	0	0
$330$	0	0
$331$	10041.0	1.66738	0.833691	−	0.552231i	$-0.186223\pi$
$331$	10041.0	1.66738	0.833691	+	0.552231i	$0.186223\pi$
$332$	3092.23	0.511169
$333$	−1728.27	−0.284410
$334$	−2262.25	−0.370613
$335$	0	0
$336$	0	0
$337$	−7227.40	−1.16825	−0.584127	−	0.811662i	$-0.698563\pi$
$337$	−7227.40	−1.16825	−0.584127	+	0.811662i	$0.698563\pi$
$338$	−7777.01	−1.25152
$339$	1573.28	0.252061
$340$	0	0
$341$	2747.51	0.436323
$342$	−2070.20	−0.327321
$343$	0	0
$344$	332.263	0.0520768
$345$	0	0
$346$	−4277.16	−0.664570
$347$	3425.67	0.529970	0.264985	−	0.964252i	$-0.414633\pi$
$347$	3425.67	0.529970	0.264985	+	0.964252i	$0.414633\pi$
$348$	1965.55	0.302771
$349$	7092.03	1.08776	0.543879	−	0.839164i	$-0.316955\pi$
$349$	7092.03	1.08776	0.543879	+	0.839164i	$0.316955\pi$
$350$	0	0
$351$	11850.4	1.80207
$352$	−445.853	−0.0675114
$353$	11964.8	1.80402	0.902011	−	0.431714i	$-0.142091\pi$
$353$	11964.8	1.80402	0.902011	+	0.431714i	$0.142091\pi$
$354$	−1727.29	−0.259334
$355$	0	0
$356$	−5316.30	−0.791470
$357$	0	0
$358$	−3478.14	−0.513479
$359$	−1556.01	−0.228755	−0.114377	−	0.993437i	$-0.536487\pi$
$359$	−1556.01	−0.228755	−0.114377	+	0.993437i	$0.536487\pi$
$360$	0	0
$361$	16336.4	2.38175
$362$	−4608.34	−0.669086
$363$	5110.06	0.738867
$364$	0	0
$365$	0	0
$366$	313.771	0.0448116
$367$	−2418.56	−0.343999	−0.172000	−	0.985097i	$-0.555023\pi$
$367$	−2418.56	−0.343999	−0.172000	+	0.985097i	$0.555023\pi$
$368$	−1729.85	−0.245040
$369$	1317.86	0.185922
$370$	0	0
$371$	0	0
$372$	−3545.45	−0.494149
$373$	−2737.30	−0.379979	−0.189989	−	0.981786i	$-0.560845\pi$
$373$	−2737.30	−0.379979	−0.189989	+	0.981786i	$0.560845\pi$
$374$	2929.60	0.405043
$375$	0	0
$376$	−879.086	−0.120573
$377$	−8528.21	−1.16505
$378$	0	0
$379$	3489.69	0.472964	0.236482	−	0.971636i	$-0.424006\pi$
$379$	3489.69	0.472964	0.236482	+	0.971636i	$0.424006\pi$
$380$	0	0
$381$	5000.27	0.672367
$382$	7222.15	0.967323
$383$	−180.481	−0.0240787	−0.0120394	−	0.999928i	$-0.503832\pi$
$383$	−180.481	−0.0240787	−0.0120394	+	0.999928i	$0.503832\pi$
$384$	575.339	0.0764587
$385$	0	0
$386$	8681.80	1.14480
$387$	282.276	0.0370772
$388$	4152.71	0.543356
$389$	6220.77	0.810811	0.405406	−	0.914137i	$-0.367130\pi$
$389$	6220.77	0.810811	0.405406	+	0.914137i	$0.367130\pi$
$390$	0	0
$391$	11366.5	1.47015
$392$	0	0
$393$	2723.75	0.349606
$394$	−761.885	−0.0974194
$395$	0	0
$396$	−378.777	−0.0480663
$397$	−651.379	−0.0823470	−0.0411735	−	0.999152i	$-0.513110\pi$
$397$	−651.379	−0.0823470	−0.0411735	+	0.999152i	$0.513110\pi$
$398$	6953.40	0.875735
$399$	0	0
$400$	0	0
$401$	12791.2	1.59292	0.796460	−	0.604692i	$-0.206704\pi$
$401$	12791.2	1.59292	0.796460	+	0.604692i	$0.206704\pi$
$402$	3369.60	0.418060
$403$	15383.2	1.90147
$404$	1958.24	0.241153
$405$	0	0
$406$	0	0
$407$	3542.99	0.431498
$408$	−3780.43	−0.458723
$409$	−12347.5	−1.49278	−0.746389	−	0.665510i	$-0.768214\pi$
$409$	−12347.5	−1.49278	−0.746389	+	0.665510i	$0.768214\pi$
$410$	0	0
$411$	−1815.43	−0.217879
$412$	4740.61	0.566877
$413$	0	0
$414$	−1469.61	−0.174462
$415$	0	0
$416$	−2496.31	−0.294211
$417$	2362.18	0.277401
$418$	4243.97	0.496602
$419$	9286.60	1.08277	0.541385	−	0.840775i	$-0.317900\pi$
$419$	9286.60	1.08277	0.541385	+	0.840775i	$0.317900\pi$
$420$	0	0
$421$	9253.33	1.07121	0.535605	−	0.844469i	$-0.320084\pi$
$421$	9253.33	1.07121	0.535605	+	0.844469i	$0.320084\pi$
$422$	−3386.75	−0.390674
$423$	−746.833	−0.0858446
$424$	−88.0432	−0.0100843
$425$	0	0
$426$	−5175.55	−0.588630
$427$	0	0
$428$	−4769.96	−0.538703
$429$	−4885.44	−0.549816
$430$	0	0
$431$	−3385.57	−0.378369	−0.189185	−	0.981942i	$-0.560584\pi$
$431$	−3385.57	−0.378369	−0.189185	+	0.981942i	$0.560584\pi$
$432$	2430.55	0.270694
$433$	12073.8	1.34002	0.670010	−	0.742352i	$-0.266290\pi$
$433$	12073.8	1.34002	0.670010	+	0.742352i	$0.266290\pi$
$434$	0	0
$435$	0	0
$436$	−1141.72	−0.125410
$437$	16466.1	1.80247
$438$	−4694.05	−0.512079
$439$	13382.5	1.45492	0.727462	−	0.686148i	$-0.240700\pi$
$439$	13382.5	1.45492	0.727462	+	0.686148i	$0.240700\pi$
$440$	0	0
$441$	0	0
$442$	16402.7	1.76515
$443$	1205.17	0.129254	0.0646271	−	0.997909i	$-0.479414\pi$
$443$	1205.17	0.129254	0.0646271	+	0.997909i	$0.479414\pi$
$444$	−4571.96	−0.488684
$445$	0	0
$446$	521.836	0.0554028
$447$	9146.42	0.967810
$448$	0	0
$449$	7957.88	0.836427	0.418214	−	0.908349i	$-0.362656\pi$
$449$	7957.88	0.836427	0.418214	+	0.908349i	$0.362656\pi$
$450$	0	0
$451$	−2701.65	−0.282075
$452$	−1400.08	−0.145695
$453$	−7860.10	−0.815231
$454$	−559.315	−0.0578193
$455$	0	0
$456$	−5476.53	−0.562416
$457$	1697.84	0.173789	0.0868943	−	0.996218i	$-0.472306\pi$
$457$	1697.84	0.173789	0.0868943	+	0.996218i	$0.472306\pi$
$458$	−3444.77	−0.351449
$459$	−15970.6	−1.62406
$460$	0	0
$461$	−14516.2	−1.46657	−0.733283	−	0.679924i	$-0.762013\pi$
$461$	−14516.2	−1.46657	−0.733283	+	0.679924i	$0.762013\pi$
$462$	0	0
$463$	6570.80	0.659549	0.329774	−	0.944060i	$-0.393027\pi$
$463$	6570.80	0.659549	0.329774	+	0.944060i	$0.393027\pi$
$464$	−1749.16	−0.175006
$465$	0	0
$466$	13319.2	1.32404
$467$	18664.4	1.84943	0.924715	−	0.380659i	$-0.124303\pi$
$467$	18664.4	1.84943	0.924715	+	0.380659i	$0.124303\pi$
$468$	−2120.75	−0.209470
$469$	0	0
$470$	0	0
$471$	−2015.97	−0.197220
$472$	1537.13	0.149899
$473$	−578.673	−0.0562524
$474$	−4100.92	−0.397387
$475$	0	0
$476$	0	0
$477$	−74.7976	−0.00717976
$478$	−2037.25	−0.194941
$479$	5296.28	0.505205	0.252602	−	0.967570i	$-0.418714\pi$
$479$	5296.28	0.505205	0.252602	+	0.967570i	$0.418714\pi$
$480$	0	0
$481$	19837.1	1.88044
$482$	11707.7	1.10637
$483$	0	0
$484$	−4547.50	−0.427075
$485$	0	0
$486$	3714.53	0.346697
$487$	−13257.2	−1.23355	−0.616777	−	0.787138i	$-0.711562\pi$
$487$	−13257.2	−1.23355	−0.616777	+	0.787138i	$0.711562\pi$
$488$	−279.228	−0.0259017
$489$	4097.86	0.378960
$490$	0	0
$491$	−11404.4	−1.04821	−0.524107	−	0.851653i	$-0.675601\pi$
$491$	−11404.4	−1.04821	−0.524107	+	0.851653i	$0.675601\pi$
$492$	3486.28	0.319459
$493$	11493.3	1.04997
$494$	23761.8	2.16416
$495$	0	0
$496$	3155.14	0.285625
$497$	0	0
$498$	6949.53	0.625333
$499$	−18988.6	−1.70350	−0.851751	−	0.523947i	$-0.824459\pi$
$499$	−18988.6	−1.70350	−0.851751	+	0.523947i	$0.824459\pi$
$500$	0	0
$501$	−5084.21	−0.453385
$502$	−2203.09	−0.195874
$503$	−5173.78	−0.458623	−0.229311	−	0.973353i	$-0.573647\pi$
$503$	−5173.78	−0.458623	−0.229311	+	0.973353i	$0.573647\pi$
$504$	0	0
$505$	0	0
$506$	3012.73	0.264688
$507$	−17478.2	−1.53103
$508$	−4449.79	−0.388637
$509$	15987.0	1.39216	0.696080	−	0.717964i	$-0.254926\pi$
$509$	15987.0	1.39216	0.696080	+	0.717964i	$0.254926\pi$
$510$	0	0
$511$	0	0
$512$	−512.000	−0.0441942
$513$	−23135.9	−1.99118
$514$	2884.93	0.247566
$515$	0	0
$516$	746.734	0.0637076
$517$	1531.03	0.130241
$518$	0	0
$519$	−9612.56	−0.812995
$520$	0	0
$521$	6278.45	0.527954	0.263977	−	0.964529i	$-0.414966\pi$
$521$	6278.45	0.527954	0.263977	+	0.964529i	$0.414966\pi$
$522$	−1486.01	−0.124599
$523$	16580.9	1.38629	0.693147	−	0.720797i	$-0.256224\pi$
$523$	16580.9	1.38629	0.693147	+	0.720797i	$0.256224\pi$
$524$	−2423.90	−0.202077
$525$	0	0
$526$	−10860.8	−0.900289
$527$	−20731.7	−1.71364
$528$	−1002.02	−0.0825894
$529$	−477.968	−0.0392840
$530$	0	0
$531$	1305.88	0.106724
$532$	0	0
$533$	−15126.4	−1.22927
$534$	−11947.9	−0.968236
$535$	0	0
$536$	−2998.64	−0.241644
$537$	−7816.84	−0.628160
$538$	−10023.8	−0.803263
$539$	0	0
$540$	0	0
$541$	−10593.3	−0.841851	−0.420925	−	0.907095i	$-0.638295\pi$
$541$	−10593.3	−0.841851	−0.420925	+	0.907095i	$0.638295\pi$
$542$	−6634.79	−0.525809
$543$	−10356.9	−0.818519
$544$	3364.24	0.265148
$545$	0	0
$546$	0	0
$547$	163.918	0.0128129	0.00640644	−	0.999979i	$-0.497961\pi$
$547$	163.918	0.0128129	0.00640644	+	0.999979i	$0.497961\pi$
$548$	1615.57	0.125937
$549$	−237.219	−0.0184413
$550$	0	0
$551$	16649.9	1.28731
$552$	−3887.70	−0.299767
$553$	0	0
$554$	8320.71	0.638110
$555$	0	0
$556$	−2102.13	−0.160342
$557$	−3673.20	−0.279423	−0.139712	−	0.990192i	$-0.544618\pi$
$557$	−3673.20	−0.279423	−0.139712	+	0.990192i	$0.544618\pi$
$558$	2680.46	0.203357
$559$	−3239.96	−0.245145
$560$	0	0
$561$	6584.03	0.495505
$562$	4846.24	0.363748
$563$	10985.5	0.822353	0.411176	−	0.911556i	$-0.365118\pi$
$563$	10985.5	0.822353	0.411176	+	0.911556i	$0.365118\pi$
$564$	−1975.67	−0.147502
$565$	0	0
$566$	−16740.3	−1.24319
$567$	0	0
$568$	4605.78	0.340236
$569$	26436.8	1.94778	0.973891	−	0.227016i	$-0.0728969\pi$
$569$	26436.8	1.94778	0.973891	+	0.227016i	$0.0728969\pi$
$570$	0	0
$571$	−4688.41	−0.343614	−0.171807	−	0.985131i	$-0.554961\pi$
$571$	−4688.41	−0.343614	−0.171807	+	0.985131i	$0.554961\pi$
$572$	4347.60	0.317801
$573$	16231.2	1.18336
$574$	0	0
$575$	0	0
$576$	−434.973	−0.0314650
$577$	21915.1	1.58118	0.790588	−	0.612349i	$-0.209775\pi$
$577$	21915.1	1.58118	0.790588	+	0.612349i	$0.209775\pi$
$578$	−12279.7	−0.883680
$579$	19511.6	1.40048
$580$	0	0
$581$	0	0
$582$	9332.88	0.664709
$583$	153.337	0.0108929
$584$	4177.29	0.295989
$585$	0	0
$586$	−10584.4	−0.746141
$587$	−10705.4	−0.752742	−0.376371	−	0.926469i	$-0.622828\pi$
$587$	−10705.4	−0.752742	−0.376371	+	0.926469i	$0.622828\pi$
$588$	0	0
$589$	−30033.0	−2.10100
$590$	0	0
$591$	−1712.28	−0.119177
$592$	4068.64	0.282466
$593$	−17638.7	−1.22148	−0.610739	−	0.791832i	$-0.709128\pi$
$593$	−17638.7	−1.22148	−0.610739	+	0.791832i	$0.709128\pi$
$594$	−4233.08	−0.292399
$595$	0	0
$596$	−8139.49	−0.559407
$597$	15627.2	1.07132
$598$	16868.2	1.15350
$599$	12921.6	0.881407	0.440703	−	0.897653i	$-0.354729\pi$
$599$	12921.6	0.881407	0.440703	+	0.897653i	$0.354729\pi$
$600$	0	0
$601$	25860.7	1.75521	0.877603	−	0.479388i	$-0.159141\pi$
$601$	25860.7	1.75521	0.877603	+	0.479388i	$0.159141\pi$
$602$	0	0
$603$	−2547.51	−0.172044
$604$	6994.78	0.471215
$605$	0	0
$606$	4400.98	0.295012
$607$	6462.68	0.432145	0.216073	−	0.976377i	$-0.430675\pi$
$607$	6462.68	0.432145	0.216073	+	0.976377i	$0.430675\pi$
$608$	4873.62	0.325084
$609$	0	0
$610$	0	0
$611$	8572.15	0.567581
$612$	2858.11	0.188778
$613$	13103.7	0.863382	0.431691	−	0.902022i	$-0.357917\pi$
$613$	13103.7	0.863382	0.431691	+	0.902022i	$0.357917\pi$
$614$	−7704.14	−0.506374
$615$	0	0
$616$	0	0
$617$	−1840.69	−0.120103	−0.0600514	−	0.998195i	$-0.519126\pi$
$617$	−1840.69	−0.120103	−0.0600514	+	0.998195i	$0.519126\pi$
$618$	10654.1	0.693483
$619$	20894.4	1.35673	0.678365	−	0.734725i	$-0.262689\pi$
$619$	20894.4	1.35673	0.678365	+	0.734725i	$0.262689\pi$
$620$	0	0
$621$	−16423.8	−1.06130
$622$	−2184.96	−0.140850
$623$	0	0
$624$	−5610.25	−0.359919
$625$	0	0
$626$	2649.79	0.169181
$627$	9537.98	0.607512
$628$	1794.03	0.113996
$629$	−26734.1	−1.69469
$630$	0	0
$631$	−11128.4	−0.702086	−0.351043	−	0.936359i	$-0.614173\pi$
$631$	−11128.4	−0.702086	−0.351043	+	0.936359i	$0.614173\pi$
$632$	3649.45	0.229695
$633$	−7611.44	−0.477927
$634$	−9970.95	−0.624601
$635$	0	0
$636$	−197.870	−0.0123366
$637$	0	0
$638$	3046.36	0.189038
$639$	3912.87	0.242239
$640$	0	0
$641$	−31784.4	−1.95851	−0.979257	−	0.202620i	$-0.935054\pi$
$641$	−31784.4	−1.95851	−0.979257	+	0.202620i	$0.935054\pi$
$642$	−10720.1	−0.659016
$643$	1781.74	0.109277	0.0546384	−	0.998506i	$-0.482599\pi$
$643$	1781.74	0.109277	0.0546384	+	0.998506i	$0.482599\pi$
$644$	0	0
$645$	0	0
$646$	−32023.5	−1.95038
$647$	−13936.7	−0.846843	−0.423422	−	0.905933i	$-0.639171\pi$
$647$	−13936.7	−0.846843	−0.423422	+	0.905933i	$0.639171\pi$
$648$	3994.43	0.242155
$649$	−2677.09	−0.161918
$650$	0	0
$651$	0	0
$652$	−3646.72	−0.219044
$653$	−856.900	−0.0513523	−0.0256762	−	0.999670i	$-0.508174\pi$
$653$	−856.900	−0.0513523	−0.0256762	+	0.999670i	$0.508174\pi$
$654$	−2565.93	−0.153419
$655$	0	0
$656$	−3102.48	−0.184651
$657$	3548.84	0.210736
$658$	0	0
$659$	11560.8	0.683374	0.341687	−	0.939814i	$-0.389002\pi$
$659$	11560.8	0.683374	0.341687	+	0.939814i	$0.389002\pi$
$660$	0	0
$661$	9094.31	0.535140	0.267570	−	0.963538i	$-0.413779\pi$
$661$	9094.31	0.535140	0.267570	+	0.963538i	$0.413779\pi$
$662$	−20082.0	−1.17902
$663$	36863.7	2.15938
$664$	−6184.46	−0.361451
$665$	0	0
$666$	3456.54	0.201108
$667$	11819.5	0.686136
$668$	4524.49	0.262063
$669$	1172.78	0.0677765
$670$	0	0
$671$	486.306	0.0279786
$672$	0	0
$673$	5216.21	0.298767	0.149384	−	0.988779i	$-0.452271\pi$
$673$	5216.21	0.298767	0.149384	+	0.988779i	$0.452271\pi$
$674$	14454.8	0.826080
$675$	0	0
$676$	15554.0	0.884958
$677$	−5631.52	−0.319700	−0.159850	−	0.987141i	$-0.551101\pi$
$677$	−5631.52	−0.319700	−0.159850	+	0.987141i	$0.551101\pi$
$678$	−3146.56	−0.178234
$679$	0	0
$680$	0	0
$681$	−1257.01	−0.0707326
$682$	−5495.02	−0.308527
$683$	20993.9	1.17615	0.588074	−	0.808807i	$-0.299886\pi$
$683$	20993.9	1.17615	0.588074	+	0.808807i	$0.299886\pi$
$684$	4140.41	0.231451
$685$	0	0
$686$	0	0
$687$	−7741.84	−0.429941
$688$	−664.526	−0.0368238
$689$	858.527	0.0474706
$690$	0	0
$691$	−20881.2	−1.14958	−0.574790	−	0.818301i	$-0.694916\pi$
$691$	−20881.2	−1.14958	−0.574790	+	0.818301i	$0.694916\pi$
$692$	8554.31	0.469922
$693$	0	0
$694$	−6851.34	−0.374746
$695$	0	0
$696$	−3931.09	−0.214092
$697$	20385.7	1.10784
$698$	−14184.1	−0.769161
$699$	29933.9	1.61975
$700$	0	0
$701$	−14493.3	−0.780889	−0.390444	−	0.920626i	$-0.627679\pi$
$701$	−14493.3	−0.780889	−0.390444	+	0.920626i	$0.627679\pi$
$702$	−23700.8	−1.27426
$703$	−38728.5	−2.07777
$704$	891.705	0.0477378
$705$	0	0
$706$	−23929.5	−1.27564
$707$	0	0
$708$	3454.58	0.183377
$709$	19466.9	1.03116	0.515580	−	0.856841i	$-0.327576\pi$
$709$	19466.9	1.03116	0.515580	+	0.856841i	$0.327576\pi$
$710$	0	0
$711$	3100.41	0.163537
$712$	10632.6	0.559654
$713$	−21320.0	−1.11983
$714$	0	0
$715$	0	0
$716$	6956.29	0.363085
$717$	−4578.56	−0.238479
$718$	3112.01	0.161754
$719$	−31341.8	−1.62566	−0.812831	−	0.582499i	$-0.802075\pi$
$719$	−31341.8	−1.62566	−0.812831	+	0.582499i	$0.802075\pi$
$720$	0	0
$721$	0	0
$722$	−32672.9	−1.68415
$723$	26312.1	1.35347
$724$	9216.69	0.473115
$725$	0	0
$726$	−10220.1	−0.522458
$727$	21225.4	1.08282	0.541408	−	0.840760i	$-0.317892\pi$
$727$	21225.4	1.08282	0.541408	+	0.840760i	$0.317892\pi$
$728$	0	0
$729$	21829.3	1.10904
$730$	0	0
$731$	4366.45	0.220929
$732$	−627.541	−0.0316866
$733$	−10542.1	−0.531217	−0.265609	−	0.964081i	$-0.585573\pi$
$733$	−10542.1	−0.531217	−0.265609	+	0.964081i	$0.585573\pi$
$734$	4837.12	0.243244
$735$	0	0
$736$	3459.71	0.173270
$737$	5222.46	0.261020
$738$	−2635.73	−0.131467
$739$	13623.8	0.678158	0.339079	−	0.940758i	$-0.389885\pi$
$739$	13623.8	0.678158	0.339079	+	0.940758i	$0.389885\pi$
$740$	0	0
$741$	53402.7	2.64750
$742$	0	0
$743$	8366.07	0.413084	0.206542	−	0.978438i	$-0.433779\pi$
$743$	8366.07	0.413084	0.206542	+	0.978438i	$0.433779\pi$
$744$	7090.91	0.349416
$745$	0	0
$746$	5474.60	0.268685
$747$	−5254.04	−0.257343
$748$	−5859.20	−0.286408
$749$	0	0
$750$	0	0
$751$	−4026.96	−0.195667	−0.0978334	−	0.995203i	$-0.531191\pi$
$751$	−4026.96	−0.195667	−0.0978334	+	0.995203i	$0.531191\pi$
$752$	1758.17	0.0852579
$753$	−4951.27	−0.239621
$754$	17056.4	0.823817
$755$	0	0
$756$	0	0
$757$	−7909.33	−0.379748	−0.189874	−	0.981808i	$-0.560808\pi$
$757$	−7909.33	−0.379748	−0.189874	+	0.981808i	$0.560808\pi$
$758$	−6979.38	−0.334436
$759$	6770.87	0.323804
$760$	0	0
$761$	12265.2	0.584247	0.292124	−	0.956381i	$-0.405638\pi$
$761$	12265.2	0.584247	0.292124	+	0.956381i	$0.405638\pi$
$762$	−10000.5	−0.475435
$763$	0	0
$764$	−14444.3	−0.684000
$765$	0	0
$766$	360.962	0.0170262
$767$	−14988.9	−0.705629
$768$	−1150.68	−0.0540645
$769$	−7111.92	−0.333501	−0.166750	−	0.985999i	$-0.553327\pi$
$769$	−7111.92	−0.333501	−0.166750	+	0.985999i	$0.553327\pi$
$770$	0	0
$771$	6483.65	0.302857
$772$	−17363.6	−0.809494
$773$	−7131.22	−0.331814	−0.165907	−	0.986141i	$-0.553055\pi$
$773$	−7131.22	−0.331814	−0.165907	+	0.986141i	$0.553055\pi$
$774$	−564.552	−0.0262176
$775$	0	0
$776$	−8305.43	−0.384211
$777$	0	0
$778$	−12441.5	−0.573330
$779$	29531.8	1.35826
$780$	0	0
$781$	−8021.48	−0.367517
$782$	−22733.0	−1.03955
$783$	−16607.1	−0.757970
$784$	0	0
$785$	0	0
$786$	−5447.51	−0.247209
$787$	12062.3	0.546345	0.273172	−	0.961965i	$-0.411927\pi$
$787$	12062.3	0.546345	0.273172	+	0.961965i	$0.411927\pi$
$788$	1523.77	0.0688859
$789$	−24408.7	−1.10136
$790$	0	0
$791$	0	0
$792$	757.553	0.0339880
$793$	2722.81	0.121929
$794$	1302.76	0.0582281
$795$	0	0
$796$	−13906.8	−0.619238
$797$	−15849.5	−0.704415	−0.352207	−	0.935922i	$-0.614569\pi$
$797$	−15849.5	−0.704415	−0.352207	+	0.935922i	$0.614569\pi$
$798$	0	0
$799$	−11552.6	−0.511515
$800$	0	0
$801$	9032.98	0.398458
$802$	−25582.3	−1.12636
$803$	−7275.21	−0.319722
$804$	−6739.19	−0.295613
$805$	0	0
$806$	−30766.4	−1.34454
$807$	−22527.6	−0.982664
$808$	−3916.47	−0.170521
$809$	−4367.34	−0.189799	−0.0948995	−	0.995487i	$-0.530253\pi$
$809$	−4367.34	−0.189799	−0.0948995	+	0.995487i	$0.530253\pi$
$810$	0	0
$811$	−284.034	−0.0122981	−0.00614907	−	0.999981i	$-0.501957\pi$
$811$	−284.034	−0.0122981	−0.00614907	+	0.999981i	$0.501957\pi$
$812$	0	0
$813$	−14911.2	−0.643243
$814$	−7085.99	−0.305115
$815$	0	0
$816$	7560.85	0.324366
$817$	6325.47	0.270869
$818$	24695.1	1.05555
$819$	0	0
$820$	0	0
$821$	−11862.3	−0.504261	−0.252131	−	0.967693i	$-0.581131\pi$
$821$	−11862.3	−0.504261	−0.252131	+	0.967693i	$0.581131\pi$
$822$	3630.85	0.154064
$823$	−33092.7	−1.40163	−0.700814	−	0.713344i	$-0.747180\pi$
$823$	−33092.7	−1.40163	−0.700814	+	0.713344i	$0.747180\pi$
$824$	−9481.23	−0.400843
$825$	0	0
$826$	0	0
$827$	38358.2	1.61287	0.806436	−	0.591322i	$-0.201394\pi$
$827$	38358.2	1.61287	0.806436	+	0.591322i	$0.201394\pi$
$828$	2939.21	0.123363
$829$	44688.0	1.87223	0.936114	−	0.351697i	$-0.114395\pi$
$829$	44688.0	1.87223	0.936114	+	0.351697i	$0.114395\pi$
$830$	0	0
$831$	18700.1	0.780625
$832$	4992.62	0.208038
$833$	0	0
$834$	−4724.36	−0.196152
$835$	0	0
$836$	−8487.94	−0.351150
$837$	29955.9	1.23707
$838$	−18573.2	−0.765633
$839$	25673.2	1.05642	0.528211	−	0.849113i	$-0.322863\pi$
$839$	25673.2	1.05642	0.528211	+	0.849113i	$0.322863\pi$
$840$	0	0
$841$	−12437.6	−0.509967
$842$	−18506.7	−0.757460
$843$	10891.5	0.444987
$844$	6773.50	0.276248
$845$	0	0
$846$	1493.67	0.0607013
$847$	0	0
$848$	176.086	0.00713069
$849$	−37622.4	−1.52084
$850$	0	0
$851$	−27492.8	−1.10745
$852$	10351.1	0.416224
$853$	1652.74	0.0663408	0.0331704	−	0.999450i	$-0.489440\pi$
$853$	1652.74	0.0663408	0.0331704	+	0.999450i	$0.489440\pi$
$854$	0	0
$855$	0	0
$856$	9539.92	0.380920
$857$	−1566.46	−0.0624379	−0.0312190	−	0.999513i	$-0.509939\pi$
$857$	−1566.46	−0.0624379	−0.0312190	+	0.999513i	$0.509939\pi$
$858$	9770.87	0.388779
$859$	17466.8	0.693783	0.346892	−	0.937905i	$-0.387237\pi$
$859$	17466.8	0.693783	0.346892	+	0.937905i	$0.387237\pi$
$860$	0	0
$861$	0	0
$862$	6771.14	0.267547
$863$	42901.9	1.69224	0.846118	−	0.532996i	$-0.178934\pi$
$863$	42901.9	1.69224	0.846118	+	0.532996i	$0.178934\pi$
$864$	−4861.10	−0.191410
$865$	0	0
$866$	−24147.6	−0.947537
$867$	−27597.6	−1.08104
$868$	0	0
$869$	−6355.93	−0.248113
$870$	0	0
$871$	29240.3	1.13751
$872$	2283.45	0.0886781
$873$	−7055.92	−0.273547
$874$	−32932.2	−1.27454
$875$	0	0
$876$	9388.11	0.362095
$877$	35156.0	1.35363	0.676815	−	0.736153i	$-0.263360\pi$
$877$	35156.0	1.35363	0.676815	+	0.736153i	$0.263360\pi$
$878$	−26765.0	−1.02879
$879$	−23787.6	−0.912783
$880$	0	0
$881$	−26203.5	−1.00206	−0.501032	−	0.865429i	$-0.667046\pi$
$881$	−26203.5	−1.00206	−0.501032	+	0.865429i	$0.667046\pi$
$882$	0	0
$883$	−12182.9	−0.464312	−0.232156	−	0.972679i	$-0.574578\pi$
$883$	−12182.9	−0.464312	−0.232156	+	0.972679i	$0.574578\pi$
$884$	−32805.4	−1.24815
$885$	0	0
$886$	−2410.35	−0.0913965
$887$	−22769.7	−0.861931	−0.430965	−	0.902368i	$-0.641827\pi$
$887$	−22769.7	−0.861931	−0.430965	+	0.902368i	$0.641827\pi$
$888$	9143.93	0.345552
$889$	0	0
$890$	0	0
$891$	−6956.75	−0.261571
$892$	−1043.67	−0.0391757
$893$	−16735.7	−0.627141
$894$	−18292.8	−0.684345
$895$	0	0
$896$	0	0
$897$	37909.8	1.41112
$898$	−15915.8	−0.591443
$899$	−21558.0	−0.799775
$900$	0	0
$901$	−1157.02	−0.0427814
$902$	5403.31	0.199457
$903$	0	0
$904$	2800.15	0.103022
$905$	0	0
$906$	15720.2	0.576456
$907$	28419.0	1.04039	0.520197	−	0.854046i	$-0.325858\pi$
$907$	28419.0	1.04039	0.520197	+	0.854046i	$0.325858\pi$
$908$	1118.63	0.0408844
$909$	−3327.26	−0.121406
$910$	0	0
$911$	21021.6	0.764519	0.382259	−	0.924055i	$-0.375146\pi$
$911$	21021.6	0.764519	0.382259	+	0.924055i	$0.375146\pi$
$912$	10953.1	0.397688
$913$	10770.9	0.390433
$914$	−3395.67	−0.122887
$915$	0	0
$916$	6889.54	0.248512
$917$	0	0
$918$	31941.2	1.14839
$919$	−24704.5	−0.886752	−0.443376	−	0.896336i	$-0.646219\pi$
$919$	−24704.5	−0.886752	−0.443376	+	0.896336i	$0.646219\pi$
$920$	0	0
$921$	−17314.4	−0.619467
$922$	29032.4	1.03702
$923$	−44911.9	−1.60162
$924$	0	0
$925$	0	0
$926$	−13141.6	−0.466371
$927$	−8054.83	−0.285389
$928$	3498.32	0.123748
$929$	17304.1	0.611119	0.305560	−	0.952173i	$-0.401156\pi$
$929$	17304.1	0.611119	0.305560	+	0.952173i	$0.401156\pi$
$930$	0	0
$931$	0	0
$932$	−26638.5	−0.936237
$933$	−4910.52	−0.172308
$934$	−37328.7	−1.30775
$935$	0	0
$936$	4241.51	0.148118
$937$	−31154.0	−1.08619	−0.543094	−	0.839672i	$-0.682747\pi$
$937$	−31154.0	−1.08619	−0.543094	+	0.839672i	$0.682747\pi$
$938$	0	0
$939$	5955.19	0.206965
$940$	0	0
$941$	−53006.5	−1.83631	−0.918153	−	0.396227i	$-0.870319\pi$
$941$	−53006.5	−1.83631	−0.918153	+	0.396227i	$0.870319\pi$
$942$	4031.93	0.139456
$943$	20964.2	0.723953
$944$	−3074.26	−0.105994
$945$	0	0
$946$	1157.35	0.0397765
$947$	54176.8	1.85904	0.929520	−	0.368772i	$-0.120222\pi$
$947$	54176.8	1.85904	0.929520	+	0.368772i	$0.120222\pi$
$948$	8201.85	0.280995
$949$	−40733.6	−1.39333
$950$	0	0
$951$	−22408.9	−0.764099
$952$	0	0
$953$	−6214.74	−0.211244	−0.105622	−	0.994406i	$-0.533683\pi$
$953$	−6214.74	−0.211244	−0.105622	+	0.994406i	$0.533683\pi$
$954$	149.595	0.00507686
$955$	0	0
$956$	4074.51	0.137844
$957$	6846.44	0.231258
$958$	−10592.6	−0.357234
$959$	0	0
$960$	0	0
$961$	9095.25	0.305302
$962$	−39674.1	−1.32967
$963$	8104.70	0.271205
$964$	−23415.4	−0.782322
$965$	0	0
$966$	0	0
$967$	−45681.3	−1.51914	−0.759571	−	0.650424i	$-0.774591\pi$
$967$	−45681.3	−1.51914	−0.759571	+	0.650424i	$0.774591\pi$
$968$	9095.00	0.301988
$969$	−71970.1	−2.38598
$970$	0	0
$971$	52107.6	1.72215	0.861077	−	0.508474i	$-0.169790\pi$
$971$	52107.6	1.72215	0.861077	+	0.508474i	$0.169790\pi$
$972$	−7429.06	−0.245152
$973$	0	0
$974$	26514.4	0.872255
$975$	0	0
$976$	558.455	0.0183153
$977$	4174.15	0.136687	0.0683433	−	0.997662i	$-0.478229\pi$
$977$	4174.15	0.136687	0.0683433	+	0.997662i	$0.478229\pi$
$978$	−8195.71	−0.267965
$979$	−18517.8	−0.604528
$980$	0	0
$981$	1939.92	0.0631364
$982$	22808.8	0.741199
$983$	−19717.9	−0.639781	−0.319890	−	0.947455i	$-0.603646\pi$
$983$	−19717.9	−0.639781	−0.319890	+	0.947455i	$0.603646\pi$
$984$	−6972.56	−0.225891
$985$	0	0
$986$	−22986.7	−0.742440
$987$	0	0
$988$	−47523.6	−1.53029
$989$	4490.36	0.144373
$990$	0	0
$991$	24502.6	0.785419	0.392710	−	0.919663i	$-0.371538\pi$
$991$	24502.6	0.785419	0.392710	+	0.919663i	$0.371538\pi$
$992$	−6310.27	−0.201967
$993$	−45132.7	−1.44234
$994$	0	0
$995$	0	0
$996$	−13899.1	−0.442177
$997$	−22942.8	−0.728791	−0.364395	−	0.931244i	$-0.618724\pi$
$997$	−22942.8	−0.728791	−0.364395	+	0.931244i	$0.618724\pi$
$998$	37977.2	1.20456
$999$	38629.0	1.22339

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2450.4.a.cw.1.2		6
5.2	odd	4		490.4.c.e.99.5		12
5.3	odd	4		490.4.c.e.99.8		12
5.4	even	2		2450.4.a.cx.1.5		6
7.3	odd	6		350.4.e.o.51.2		12
7.5	odd	6		350.4.e.o.151.2		12
7.6	odd	2		2450.4.a.cv.1.5		6
35.3	even	12		70.4.i.a.9.5	✓	24
35.12	even	12		70.4.i.a.39.5	yes	24
35.13	even	4		490.4.c.f.99.11		12
35.17	even	12		70.4.i.a.9.8	yes	24
35.19	odd	6		350.4.e.n.151.5		12
35.24	odd	6		350.4.e.n.51.5		12
35.27	even	4		490.4.c.f.99.2		12
35.33	even	12		70.4.i.a.39.8	yes	24
35.34	odd	2		2450.4.a.cy.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
70.4.i.a.9.5	✓	24	35.3	even	12
70.4.i.a.9.8	yes	24	35.17	even	12
70.4.i.a.39.5	yes	24	35.12	even	12
70.4.i.a.39.8	yes	24	35.33	even	12
350.4.e.n.51.5		12	35.24	odd	6
350.4.e.n.151.5		12	35.19	odd	6
350.4.e.o.51.2		12	7.3	odd	6
350.4.e.o.151.2		12	7.5	odd	6
490.4.c.e.99.5		12	5.2	odd	4
490.4.c.e.99.8		12	5.3	odd	4
490.4.c.f.99.2		12	35.27	even	4
490.4.c.f.99.11		12	35.13	even	4
2450.4.a.cv.1.5		6	7.6	odd	2
2450.4.a.cw.1.2		6	1.1	even	1	trivial
2450.4.a.cx.1.5		6	5.4	even	2
2450.4.a.cy.1.2		6	35.34	odd	2

Overview	Random
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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:	\( N \)	\(=\)	\( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2450.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} -4.49484 q^{3} +4.00000 q^{4} +8.98967 q^{6} -8.00000 q^{8} -6.79645 q^{9} +O(q^{10})\)	\(q-2.00000 q^{2} -4.49484 q^{3} +4.00000 q^{4} +8.98967 q^{6} -8.00000 q^{8} -6.79645 q^{9} +13.9329 q^{11} -17.9793 q^{12} +78.0096 q^{13} +16.0000 q^{16} -105.132 q^{17} +13.5929 q^{18} -152.300 q^{19} -27.8658 q^{22} -108.116 q^{23} +35.9587 q^{24} -156.019 q^{26} +151.909 q^{27} -109.323 q^{29} +197.196 q^{31} -32.0000 q^{32} -62.6261 q^{33} +210.265 q^{34} -27.1858 q^{36} +254.290 q^{37} +304.601 q^{38} -350.641 q^{39} -193.905 q^{41} -41.5329 q^{43} +55.7316 q^{44} +216.232 q^{46} +109.886 q^{47} -71.9174 q^{48} +472.553 q^{51} +312.039 q^{52} +11.0054 q^{53} -303.819 q^{54} +684.566 q^{57} +218.645 q^{58} -192.141 q^{59} +34.9035 q^{61} -394.392 q^{62} +64.0000 q^{64} +125.252 q^{66} +374.830 q^{67} -420.530 q^{68} +485.963 q^{69} -575.722 q^{71} +54.3716 q^{72} -522.161 q^{73} -508.580 q^{74} -609.202 q^{76} +701.281 q^{78} -456.182 q^{79} -499.304 q^{81} +387.810 q^{82} +773.057 q^{83} +83.0657 q^{86} +491.387 q^{87} -111.463 q^{88} -1329.07 q^{89} -432.463 q^{92} -886.364 q^{93} -219.772 q^{94} +143.835 q^{96} +1038.18 q^{97} -94.6942 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 12 q^{2} + 7 q^{3} + 24 q^{4} - 14 q^{6} - 48 q^{8} + 31 q^{9}+O(q^{10}) \) 6 * q - 12 * q^2 + 7 * q^3 + 24 * q^4 - 14 * q^6 - 48 * q^8 + 31 * q^9	\( 6 q - 12 q^{2} + 7 q^{3} + 24 q^{4} - 14 q^{6} - 48 q^{8} + 31 q^{9} + 31 q^{11} + 28 q^{12} + 59 q^{13} + 96 q^{16} + 68 q^{17} - 62 q^{18} - 93 q^{19} - 62 q^{22} + 94 q^{23} - 56 q^{24} - 118 q^{26} + 385 q^{27} + 169 q^{29} - 326 q^{31} - 192 q^{32} + 400 q^{33} - 136 q^{34} + 124 q^{36} + 253 q^{37} + 186 q^{38} - 434 q^{39} - 198 q^{41} + 99 q^{43} + 124 q^{44} - 188 q^{46} + 901 q^{47} + 112 q^{48} + 724 q^{51} + 236 q^{52} - 233 q^{53} - 770 q^{54} + 518 q^{57} - 338 q^{58} - 668 q^{59} - 157 q^{61} + 652 q^{62} + 384 q^{64} - 800 q^{66} - 1193 q^{67} + 272 q^{68} + 45 q^{69} + 1108 q^{71} - 248 q^{72} + 1458 q^{73} - 506 q^{74} - 372 q^{76} + 868 q^{78} - 886 q^{79} - 614 q^{81} + 396 q^{82} + 1799 q^{83} - 198 q^{86} + 789 q^{87} - 248 q^{88} - 3047 q^{89} + 376 q^{92} - 1902 q^{93} - 1802 q^{94} - 224 q^{96} + 3404 q^{97} + 4273 q^{99}+O(q^{100}) \) 6 * q - 12 * q^2 + 7 * q^3 + 24 * q^4 - 14 * q^6 - 48 * q^8 + 31 * q^9 + 31 * q^11 + 28 * q^12 + 59 * q^13 + 96 * q^16 + 68 * q^17 - 62 * q^18 - 93 * q^19 - 62 * q^22 + 94 * q^23 - 56 * q^24 - 118 * q^26 + 385 * q^27 + 169 * q^29 - 326 * q^31 - 192 * q^32 + 400 * q^33 - 136 * q^34 + 124 * q^36 + 253 * q^37 + 186 * q^38 - 434 * q^39 - 198 * q^41 + 99 * q^43 + 124 * q^44 - 188 * q^46 + 901 * q^47 + 112 * q^48 + 724 * q^51 + 236 * q^52 - 233 * q^53 - 770 * q^54 + 518 * q^57 - 338 * q^58 - 668 * q^59 - 157 * q^61 + 652 * q^62 + 384 * q^64 - 800 * q^66 - 1193 * q^67 + 272 * q^68 + 45 * q^69 + 1108 * q^71 - 248 * q^72 + 1458 * q^73 - 506 * q^74 - 372 * q^76 + 868 * q^78 - 886 * q^79 - 614 * q^81 + 396 * q^82 + 1799 * q^83 - 198 * q^86 + 789 * q^87 - 248 * q^88 - 3047 * q^89 + 376 * q^92 - 1902 * q^93 - 1802 * q^94 - 224 * q^96 + 3404 * q^97 + 4273 * q^99

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