LMFDB - Embedded newform 99.4.a.f.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [99,4,Mod(1,99)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(99, 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("99.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 99 = 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	99.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:	$5.84118909057$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{97}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 24 $ x^2 - x - 24
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 33)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-4.42443$ of defining polynomial
Character	$\chi$	$=$	99.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.42443 q^{2} +11.5756 q^{4} -2.84886 q^{5} +31.6977 q^{7} +15.8199 q^{8} +O(q^{10})$	\(q+4.42443 q^{2} +11.5756 q^{4} -2.84886 q^{5} +31.6977 q^{7} +15.8199 q^{8} -12.6046 q^{10} +11.0000 q^{11} +5.15114 q^{13} +140.244 q^{14} -22.6107 q^{16} -121.942 q^{17} +34.8489 q^{19} -32.9772 q^{20} +48.6687 q^{22} -116.244 q^{23} -116.884 q^{25} +22.7909 q^{26} +366.919 q^{28} +69.4534 q^{29} +140.605 q^{31} -226.598 q^{32} -539.524 q^{34} -90.3023 q^{35} -420.070 q^{37} +154.186 q^{38} -45.0685 q^{40} +322.058 q^{41} +321.035 q^{43} +127.331 q^{44} -514.315 q^{46} +231.408 q^{47} +661.745 q^{49} -517.145 q^{50} +59.6274 q^{52} -4.91916 q^{53} -31.3374 q^{55} +501.453 q^{56} +307.292 q^{58} -406.443 q^{59} -556.431 q^{61} +622.095 q^{62} -821.683 q^{64} -14.6749 q^{65} +84.7452 q^{67} -1411.55 q^{68} -399.536 q^{70} -49.0808 q^{71} +785.884 q^{73} -1858.57 q^{74} +403.395 q^{76} +348.675 q^{77} -383.118 q^{79} +64.4147 q^{80} +1424.92 q^{82} +930.211 q^{83} +347.395 q^{85} +1420.40 q^{86} +174.018 q^{88} +732.559 q^{89} +163.279 q^{91} -1345.59 q^{92} +1023.85 q^{94} -99.2794 q^{95} -1171.49 q^{97} +2927.84 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} + 33 q^{4} + 14 q^{5} + 24 q^{7} - 57 q^{8}+O(q^{10}) $ 2 * q - q^2 + 33 * q^4 + 14 * q^5 + 24 * q^7 - 57 * q^8	$ 2 q - q^{2} + 33 q^{4} + 14 q^{5} + 24 q^{7} - 57 q^{8} - 104 q^{10} + 22 q^{11} + 30 q^{13} + 182 q^{14} + 201 q^{16} - 106 q^{17} + 50 q^{19} + 328 q^{20} - 11 q^{22} - 134 q^{23} + 42 q^{25} - 112 q^{26} + 202 q^{28} + 198 q^{29} + 360 q^{31} - 857 q^{32} - 626 q^{34} - 220 q^{35} - 328 q^{37} + 72 q^{38} - 1272 q^{40} + 782 q^{41} + 386 q^{43} + 363 q^{44} - 418 q^{46} - 266 q^{47} + 378 q^{49} - 1379 q^{50} + 592 q^{52} + 522 q^{53} + 154 q^{55} + 1062 q^{56} - 390 q^{58} + 172 q^{59} - 778 q^{61} - 568 q^{62} + 809 q^{64} + 404 q^{65} - 776 q^{67} - 1070 q^{68} + 304 q^{70} - 630 q^{71} + 1296 q^{73} - 2358 q^{74} + 728 q^{76} + 264 q^{77} + 652 q^{79} + 3832 q^{80} - 1070 q^{82} + 324 q^{83} + 616 q^{85} + 1068 q^{86} - 627 q^{88} + 756 q^{89} - 28 q^{91} - 1726 q^{92} + 3722 q^{94} + 156 q^{95} - 452 q^{97} + 4467 q^{98}+O(q^{100}) $ 2 * q - q^2 + 33 * q^4 + 14 * q^5 + 24 * q^7 - 57 * q^8 - 104 * q^10 + 22 * q^11 + 30 * q^13 + 182 * q^14 + 201 * q^16 - 106 * q^17 + 50 * q^19 + 328 * q^20 - 11 * q^22 - 134 * q^23 + 42 * q^25 - 112 * q^26 + 202 * q^28 + 198 * q^29 + 360 * q^31 - 857 * q^32 - 626 * q^34 - 220 * q^35 - 328 * q^37 + 72 * q^38 - 1272 * q^40 + 782 * q^41 + 386 * q^43 + 363 * q^44 - 418 * q^46 - 266 * q^47 + 378 * q^49 - 1379 * q^50 + 592 * q^52 + 522 * q^53 + 154 * q^55 + 1062 * q^56 - 390 * q^58 + 172 * q^59 - 778 * q^61 - 568 * q^62 + 809 * q^64 + 404 * q^65 - 776 * q^67 - 1070 * q^68 + 304 * q^70 - 630 * q^71 + 1296 * q^73 - 2358 * q^74 + 728 * q^76 + 264 * q^77 + 652 * q^79 + 3832 * q^80 - 1070 * q^82 + 324 * q^83 + 616 * q^85 + 1068 * q^86 - 627 * q^88 + 756 * q^89 - 28 * q^91 - 1726 * q^92 + 3722 * q^94 + 156 * q^95 - 452 * q^97 + 4467 * q^98

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 $

$2$	4.42443	1.56427	0.782136	−	0.623108i	$-0.214130\pi$
$2$	4.42443	1.56427	0.782136	+	0.623108i	$0.214130\pi$
$3$	0	0
$3$	0	0
$4$	11.5756	1.44695
$5$	−2.84886	−0.254810	−0.127405	−	0.991851i	$-0.540665\pi$
$5$	−2.84886	−0.254810	−0.127405	+	0.991851i	$0.540665\pi$
$6$	0	0
$7$	31.6977	1.71152	0.855758	−	0.517377i	$-0.173091\pi$
$7$	31.6977	1.71152	0.855758	+	0.517377i	$0.173091\pi$
$8$	15.8199	0.699146
$9$	0	0
$10$	−12.6046	−0.398591
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	5.15114	0.109898	0.0549488	−	0.998489i	$-0.482500\pi$
$13$	5.15114	0.109898	0.0549488	+	0.998489i	$0.482500\pi$
$14$	140.244	2.67728
$15$	0	0
$16$	−22.6107	−0.353293
$17$	−121.942	−1.73972	−0.869861	−	0.493297i	$-0.835792\pi$
$17$	−121.942	−1.73972	−0.869861	+	0.493297i	$0.835792\pi$
$18$	0	0
$19$	34.8489	0.420783	0.210391	−	0.977617i	$-0.432526\pi$
$19$	34.8489	0.420783	0.210391	+	0.977617i	$0.432526\pi$
$20$	−32.9772	−0.368696
$21$	0	0
$22$	48.6687	0.471646
$23$	−116.244	−1.05385	−0.526926	−	0.849911i	$-0.676656\pi$
$23$	−116.244	−1.05385	−0.526926	+	0.849911i	$0.676656\pi$
$24$	0	0
$25$	−116.884	−0.935072
$26$	22.7909	0.171910
$27$	0	0
$28$	366.919	2.47647
$29$	69.4534	0.444730	0.222365	−	0.974963i	$-0.428622\pi$
$29$	69.4534	0.444730	0.222365	+	0.974963i	$0.428622\pi$
$30$	0	0
$31$	140.605	0.814623	0.407312	−	0.913289i	$-0.366466\pi$
$31$	140.605	0.814623	0.407312	+	0.913289i	$0.366466\pi$
$32$	−226.598	−1.25179
$33$	0	0
$34$	−539.524	−2.72140
$35$	−90.3023	−0.436111
$36$	0	0
$37$	−420.070	−1.86646	−0.933232	−	0.359276i	$-0.883024\pi$
$37$	−420.070	−1.86646	−0.933232	+	0.359276i	$0.883024\pi$
$38$	154.186	0.658219
$39$	0	0
$40$	−45.0685	−0.178149
$41$	322.058	1.22676	0.613378	−	0.789789i	$-0.289810\pi$
$41$	322.058	1.22676	0.613378	+	0.789789i	$0.289810\pi$
$42$	0	0
$43$	321.035	1.13854	0.569272	−	0.822149i	$-0.307225\pi$
$43$	321.035	1.13854	0.569272	+	0.822149i	$0.307225\pi$
$44$	127.331	0.436271
$45$	0	0
$46$	−514.315	−1.64851
$47$	231.408	0.718176	0.359088	−	0.933304i	$-0.383088\pi$
$47$	231.408	0.718176	0.359088	+	0.933304i	$0.383088\pi$
$48$	0	0
$49$	661.745	1.92929
$50$	−517.145	−1.46271
$51$	0	0
$52$	59.6274	0.159016
$53$	−4.91916	−0.0127490	−0.00637452	−	0.999980i	$-0.502029\pi$
$53$	−4.91916	−0.0127490	−0.00637452	+	0.999980i	$0.502029\pi$
$54$	0	0
$55$	−31.3374	−0.0768280
$56$	501.453	1.19660
$57$	0	0
$58$	307.292	0.695679
$59$	−406.443	−0.896854	−0.448427	−	0.893820i	$-0.648016\pi$
$59$	−406.443	−0.896854	−0.448427	+	0.893820i	$0.648016\pi$
$60$	0	0
$61$	−556.431	−1.16793	−0.583964	−	0.811779i	$-0.698499\pi$
$61$	−556.431	−1.16793	−0.583964	+	0.811779i	$0.698499\pi$
$62$	622.095	1.27429
$63$	0	0
$64$	−821.683	−1.60485
$65$	−14.6749	−0.0280030
$66$	0	0
$67$	84.7452	0.154526	0.0772632	−	0.997011i	$-0.475382\pi$
$67$	84.7452	0.154526	0.0772632	+	0.997011i	$0.475382\pi$
$68$	−1411.55	−2.51728
$69$	0	0
$70$	−399.536	−0.682196
$71$	−49.0808	−0.0820398	−0.0410199	−	0.999158i	$-0.513061\pi$
$71$	−49.0808	−0.0820398	−0.0410199	+	0.999158i	$0.513061\pi$
$72$	0	0
$73$	785.884	1.26001	0.630005	−	0.776591i	$-0.283053\pi$
$73$	785.884	1.26001	0.630005	+	0.776591i	$0.283053\pi$
$74$	−1858.57	−2.91966
$75$	0	0
$76$	403.395	0.608850
$77$	348.675	0.516041
$78$	0	0
$79$	−383.118	−0.545622	−0.272811	−	0.962068i	$-0.587953\pi$
$79$	−383.118	−0.545622	−0.272811	+	0.962068i	$0.587953\pi$
$80$	64.4147	0.0900223
$81$	0	0
$82$	1424.92	1.91898
$83$	930.211	1.23017	0.615084	−	0.788462i	$-0.289122\pi$
$83$	930.211	1.23017	0.615084	+	0.788462i	$0.289122\pi$
$84$	0	0
$85$	347.395	0.443298
$86$	1420.40	1.78099
$87$	0	0
$88$	174.018	0.210800
$89$	732.559	0.872484	0.436242	−	0.899829i	$-0.356309\pi$
$89$	732.559	0.872484	0.436242	+	0.899829i	$0.356309\pi$
$90$	0	0
$91$	163.279	0.188092
$92$	−1345.59	−1.52487
$93$	0	0
$94$	1023.85	1.12342
$95$	−99.2794	−0.107220
$96$	0	0
$97$	−1171.49	−1.22626	−0.613128	−	0.789984i	$-0.710089\pi$
$97$	−1171.49	−1.22626	−0.613128	+	0.789984i	$0.710089\pi$
$98$	2927.84	3.01793
$99$	0	0
$100$	−1353.00	−1.35300
$101$	1221.27	1.20318	0.601589	−	0.798806i	$-0.294535\pi$
$101$	1221.27	1.20318	0.601589	+	0.798806i	$0.294535\pi$
$102$	0	0
$103$	516.745	0.494334	0.247167	−	0.968973i	$-0.420500\pi$
$103$	516.745	0.494334	0.247167	+	0.968973i	$0.420500\pi$
$104$	81.4903	0.0768345
$105$	0	0
$106$	−21.7645	−0.0199430
$107$	152.025	0.137353	0.0686765	−	0.997639i	$-0.478122\pi$
$107$	152.025	0.137353	0.0686765	+	0.997639i	$0.478122\pi$
$108$	0	0
$109$	2170.32	1.90714	0.953572	−	0.301164i	$-0.0973752\pi$
$109$	2170.32	1.90714	0.953572	+	0.301164i	$0.0973752\pi$
$110$	−138.650	−0.120180
$111$	0	0
$112$	−716.708	−0.604666
$113$	646.397	0.538123	0.269062	−	0.963123i	$-0.413286\pi$
$113$	646.397	0.538123	0.269062	+	0.963123i	$0.413286\pi$
$114$	0	0
$115$	331.163	0.268532
$116$	803.963	0.643501
$117$	0	0
$118$	−1798.28	−1.40292
$119$	−3865.28	−2.97756
$120$	0	0
$121$	121.000	0.0909091
$122$	−2461.89	−1.82696
$123$	0	0
$124$	1627.58	1.17872
$125$	689.093	0.493075
$126$	0	0
$127$	−993.304	−0.694027	−0.347014	−	0.937860i	$-0.612804\pi$
$127$	−993.304	−0.694027	−0.347014	+	0.937860i	$0.612804\pi$
$128$	−1822.69	−1.25863
$129$	0	0
$130$	−64.9279	−0.0438043
$131$	−385.814	−0.257318	−0.128659	−	0.991689i	$-0.541067\pi$
$131$	−385.814	−0.257318	−0.128659	+	0.991689i	$0.541067\pi$
$132$	0	0
$133$	1104.63	0.720177
$134$	374.949	0.241721
$135$	0	0
$136$	−1929.11	−1.21632
$137$	−884.840	−0.551803	−0.275901	−	0.961186i	$-0.588976\pi$
$137$	−884.840	−0.551803	−0.275901	+	0.961186i	$0.588976\pi$
$138$	0	0
$139$	−1091.94	−0.666312	−0.333156	−	0.942872i	$-0.608114\pi$
$139$	−1091.94	−0.666312	−0.333156	+	0.942872i	$0.608114\pi$
$140$	−1045.30	−0.631029
$141$	0	0
$142$	−217.155	−0.128333
$143$	56.6626	0.0331354
$144$	0	0
$145$	−197.863	−0.113322
$146$	3477.09	1.97100
$147$	0	0
$148$	−4862.55	−2.70067
$149$	−297.014	−0.163304	−0.0816522	−	0.996661i	$-0.526020\pi$
$149$	−297.014	−0.163304	−0.0816522	+	0.996661i	$0.526020\pi$
$150$	0	0
$151$	−1887.86	−1.01743	−0.508716	−	0.860935i	$-0.669880\pi$
$151$	−1887.86	−1.01743	−0.508716	+	0.860935i	$0.669880\pi$
$152$	551.304	0.294189
$153$	0	0
$154$	1542.69	0.807229
$155$	−400.562	−0.207574
$156$	0	0
$157$	−56.5343	−0.0287384	−0.0143692	−	0.999897i	$-0.504574\pi$
$157$	−56.5343	−0.0287384	−0.0143692	+	0.999897i	$0.504574\pi$
$158$	−1695.08	−0.853501
$159$	0	0
$160$	645.547	0.318968
$161$	−3684.68	−1.80369
$162$	0	0
$163$	−49.2338	−0.0236582	−0.0118291	−	0.999930i	$-0.503765\pi$
$163$	−49.2338	−0.0236582	−0.0118291	+	0.999930i	$0.503765\pi$
$164$	3728.01	1.77505
$165$	0	0
$166$	4115.65	1.92432
$167$	−2068.75	−0.958589	−0.479294	−	0.877654i	$-0.659107\pi$
$167$	−2068.75	−0.958589	−0.479294	+	0.877654i	$0.659107\pi$
$168$	0	0
$169$	−2170.47	−0.987923
$170$	1537.03	0.693438
$171$	0	0
$172$	3716.17	1.64741
$173$	604.012	0.265446	0.132723	−	0.991153i	$-0.457628\pi$
$173$	604.012	0.265446	0.132723	+	0.991153i	$0.457628\pi$
$174$	0	0
$175$	−3704.96	−1.60039
$176$	−248.718	−0.106522
$177$	0	0
$178$	3241.15	1.36480
$179$	2132.02	0.890251	0.445126	−	0.895468i	$-0.353159\pi$
$179$	2132.02	0.890251	0.445126	+	0.895468i	$0.353159\pi$
$180$	0	0
$181$	−589.371	−0.242031	−0.121015	−	0.992651i	$-0.538615\pi$
$181$	−589.371	−0.242031	−0.121015	+	0.992651i	$0.538615\pi$
$182$	722.418	0.294226
$183$	0	0
$184$	−1838.97	−0.736796
$185$	1196.72	0.475593
$186$	0	0
$187$	−1341.36	−0.524546
$188$	2678.68	1.03916
$189$	0	0
$190$	−439.255	−0.167720
$191$	2160.90	0.818624	0.409312	−	0.912395i	$-0.365769\pi$
$191$	2160.90	0.818624	0.409312	+	0.912395i	$0.365769\pi$
$192$	0	0
$193$	−1490.91	−0.556052	−0.278026	−	0.960574i	$-0.589680\pi$
$193$	−1490.91	−0.556052	−0.278026	+	0.960574i	$0.589680\pi$
$194$	−5183.18	−1.91820
$195$	0	0
$196$	7660.08	2.79157
$197$	230.529	0.0833732	0.0416866	−	0.999131i	$-0.486727\pi$
$197$	230.529	0.0833732	0.0416866	+	0.999131i	$0.486727\pi$
$198$	0	0
$199$	22.4007	0.00797963	0.00398982	−	0.999992i	$-0.498730\pi$
$199$	22.4007	0.00797963	0.00398982	+	0.999992i	$0.498730\pi$
$200$	−1849.09	−0.653752
$201$	0	0
$202$	5403.43	1.88210
$203$	2201.51	0.761163
$204$	0	0
$205$	−917.497	−0.312589
$206$	2286.30	0.773273
$207$	0	0
$208$	−116.471	−0.0388260
$209$	383.337	0.126871
$210$	0	0
$211$	−1051.64	−0.343117	−0.171558	−	0.985174i	$-0.554880\pi$
$211$	−1051.64	−0.343117	−0.171558	+	0.985174i	$0.554880\pi$
$212$	−56.9421	−0.0184472
$213$	0	0
$214$	672.622	0.214857
$215$	−914.583	−0.290112
$216$	0	0
$217$	4456.84	1.39424
$218$	9602.42	2.98329
$219$	0	0
$220$	−362.749	−0.111166
$221$	−628.141	−0.191191
$222$	0	0
$223$	3861.80	1.15966	0.579832	−	0.814736i	$-0.303118\pi$
$223$	3861.80	1.15966	0.579832	+	0.814736i	$0.303118\pi$
$224$	−7182.65	−2.14246
$225$	0	0
$226$	2859.94	0.841771
$227$	872.721	0.255174	0.127587	−	0.991827i	$-0.459277\pi$
$227$	872.721	0.255174	0.127587	+	0.991827i	$0.459277\pi$
$228$	0	0
$229$	1841.72	0.531459	0.265730	−	0.964048i	$-0.414387\pi$
$229$	1841.72	0.531459	0.265730	+	0.964048i	$0.414387\pi$
$230$	1465.21	0.420057
$231$	0	0
$232$	1098.74	0.310931
$233$	−3932.14	−1.10559	−0.552796	−	0.833317i	$-0.686439\pi$
$233$	−3932.14	−1.10559	−0.552796	+	0.833317i	$0.686439\pi$
$234$	0	0
$235$	−659.248	−0.182998
$236$	−4704.81	−1.29770
$237$	0	0
$238$	−17101.7	−4.65772
$239$	−4772.10	−1.29155	−0.645777	−	0.763526i	$-0.723466\pi$
$239$	−4772.10	−1.29155	−0.645777	+	0.763526i	$0.723466\pi$
$240$	0	0
$241$	3988.84	1.06616	0.533078	−	0.846066i	$-0.321035\pi$
$241$	3988.84	1.06616	0.533078	+	0.846066i	$0.321035\pi$
$242$	535.356	0.142207
$243$	0	0
$244$	−6441.00	−1.68993
$245$	−1885.22	−0.491601
$246$	0	0
$247$	179.511	0.0462431
$248$	2224.34	0.569540
$249$	0	0
$250$	3048.84	0.771303
$251$	5474.22	1.37661	0.688306	−	0.725421i	$-0.258355\pi$
$251$	5474.22	1.37661	0.688306	+	0.725421i	$0.258355\pi$
$252$	0	0
$253$	−1278.69	−0.317749
$254$	−4394.80	−1.08565
$255$	0	0
$256$	−1490.90	−0.363989
$257$	6434.01	1.56164	0.780822	−	0.624754i	$-0.214801\pi$
$257$	6434.01	1.56164	0.780822	+	0.624754i	$0.214801\pi$
$258$	0	0
$259$	−13315.3	−3.19448
$260$	−169.870	−0.0405188
$261$	0	0
$262$	−1707.01	−0.402516
$263$	−7589.00	−1.77931	−0.889654	−	0.456636i	$-0.849054\pi$
$263$	−7589.00	−1.77931	−0.889654	+	0.456636i	$0.849054\pi$
$264$	0	0
$265$	14.0140	0.00324858
$266$	4887.35	1.12655
$267$	0	0
$268$	980.974	0.223591
$269$	−478.178	−0.108383	−0.0541914	−	0.998531i	$-0.517258\pi$
$269$	−478.178	−0.108383	−0.0541914	+	0.998531i	$0.517258\pi$
$270$	0	0
$271$	−122.323	−0.0274192	−0.0137096	−	0.999906i	$-0.504364\pi$
$271$	−122.323	−0.0274192	−0.0137096	+	0.999906i	$0.504364\pi$
$272$	2757.20	0.614631
$273$	0	0
$274$	−3914.91	−0.863170
$275$	−1285.72	−0.281935
$276$	0	0
$277$	8199.41	1.77854	0.889269	−	0.457385i	$-0.151214\pi$
$277$	8199.41	1.77854	0.889269	+	0.457385i	$0.151214\pi$
$278$	−4831.22	−1.04229
$279$	0	0
$280$	−1428.57	−0.304905
$281$	−6943.79	−1.47413	−0.737067	−	0.675820i	$-0.763790\pi$
$281$	−6943.79	−1.47413	−0.737067	+	0.675820i	$0.763790\pi$
$282$	0	0
$283$	1035.14	0.217429	0.108715	−	0.994073i	$-0.465327\pi$
$283$	1035.14	0.217429	0.108715	+	0.994073i	$0.465327\pi$
$284$	−568.139	−0.118707
$285$	0	0
$286$	250.699	0.0518328
$287$	10208.5	2.09961
$288$	0	0
$289$	9956.85	2.02663
$290$	−875.430	−0.177266
$291$	0	0
$292$	9097.06	1.82317
$293$	6144.81	1.22520	0.612600	−	0.790393i	$-0.290124\pi$
$293$	6144.81	1.22520	0.612600	+	0.790393i	$0.290124\pi$
$294$	0	0
$295$	1157.90	0.228527
$296$	−6645.45	−1.30493
$297$	0	0
$298$	−1314.12	−0.255452
$299$	−598.791	−0.115816
$300$	0	0
$301$	10176.1	1.94864
$302$	−8352.72	−1.59154
$303$	0	0
$304$	−787.958	−0.148659
$305$	1585.19	0.297599
$306$	0	0
$307$	−2186.09	−0.406406	−0.203203	−	0.979137i	$-0.565135\pi$
$307$	−2186.09	−0.406406	−0.203203	+	0.979137i	$0.565135\pi$
$308$	4036.11	0.746684
$309$	0	0
$310$	−1772.26	−0.324702
$311$	7484.83	1.36471	0.682357	−	0.731019i	$-0.260955\pi$
$311$	7484.83	1.36471	0.682357	+	0.731019i	$0.260955\pi$
$312$	0	0
$313$	−6833.33	−1.23400	−0.617001	−	0.786962i	$-0.711653\pi$
$313$	−6833.33	−1.23400	−0.617001	+	0.786962i	$0.711653\pi$
$314$	−250.132	−0.0449546
$315$	0	0
$316$	−4434.81	−0.789485
$317$	−924.265	−0.163760	−0.0818800	−	0.996642i	$-0.526092\pi$
$317$	−924.265	−0.163760	−0.0818800	+	0.996642i	$0.526092\pi$
$318$	0	0
$319$	763.988	0.134091
$320$	2340.86	0.408931
$321$	0	0
$322$	−16302.6	−2.82145
$323$	−4249.54	−0.732046
$324$	0	0
$325$	−602.086	−0.102762
$326$	−217.831	−0.0370078
$327$	0	0
$328$	5094.91	0.857681
$329$	7335.10	1.22917
$330$	0	0
$331$	−9820.46	−1.63076	−0.815380	−	0.578927i	$-0.803472\pi$
$331$	−9820.46	−1.63076	−0.815380	+	0.578927i	$0.803472\pi$
$332$	10767.7	1.77999
$333$	0	0
$334$	−9153.02	−1.49949
$335$	−241.427	−0.0393748
$336$	0	0
$337$	600.808	0.0971161	0.0485580	−	0.998820i	$-0.484537\pi$
$337$	600.808	0.0971161	0.0485580	+	0.998820i	$0.484537\pi$
$338$	−9603.07	−1.54538
$339$	0	0
$340$	4021.30	0.641428
$341$	1546.65	0.245618
$342$	0	0
$343$	10103.5	1.59049
$344$	5078.73	0.796008
$345$	0	0
$346$	2672.41	0.415230
$347$	3143.41	0.486303	0.243152	−	0.969988i	$-0.421819\pi$
$347$	3143.41	0.486303	0.243152	+	0.969988i	$0.421819\pi$
$348$	0	0
$349$	720.663	0.110533	0.0552667	−	0.998472i	$-0.482399\pi$
$349$	720.663	0.110533	0.0552667	+	0.998472i	$0.482399\pi$
$350$	−16392.3	−2.50345
$351$	0	0
$352$	−2492.58	−0.377429
$353$	−1207.12	−0.182007	−0.0910034	−	0.995851i	$-0.529007\pi$
$353$	−1207.12	−0.182007	−0.0910034	+	0.995851i	$0.529007\pi$
$354$	0	0
$355$	139.824	0.0209045
$356$	8479.79	1.26244
$357$	0	0
$358$	9432.99	1.39260
$359$	−8748.31	−1.28612	−0.643062	−	0.765814i	$-0.722336\pi$
$359$	−8748.31	−1.28612	−0.643062	+	0.765814i	$0.722336\pi$
$360$	0	0
$361$	−5644.56	−0.822942
$362$	−2607.63	−0.378602
$363$	0	0
$364$	1890.05	0.272158
$365$	−2238.87	−0.321063
$366$	0	0
$367$	−6730.45	−0.957293	−0.478647	−	0.878008i	$-0.658872\pi$
$367$	−6730.45	−0.957293	−0.478647	+	0.878008i	$0.658872\pi$
$368$	2628.37	0.372318
$369$	0	0
$370$	5294.81	0.743956
$371$	−155.926	−0.0218202
$372$	0	0
$373$	−227.394	−0.0315657	−0.0157828	−	0.999875i	$-0.505024\pi$
$373$	−227.394	−0.0315657	−0.0157828	+	0.999875i	$0.505024\pi$
$374$	−5934.76	−0.820533
$375$	0	0
$376$	3660.84	0.502110
$377$	357.764	0.0488748
$378$	0	0
$379$	11356.2	1.53913	0.769565	−	0.638568i	$-0.220473\pi$
$379$	11356.2	1.53913	0.769565	+	0.638568i	$0.220473\pi$
$380$	−1149.22	−0.155141
$381$	0	0
$382$	9560.74	1.28055
$383$	−10753.6	−1.43468	−0.717338	−	0.696725i	$-0.754640\pi$
$383$	−10753.6	−1.43468	−0.717338	+	0.696725i	$0.754640\pi$
$384$	0	0
$385$	−993.325	−0.131492
$386$	−6596.43	−0.869817
$387$	0	0
$388$	−13560.7	−1.77433
$389$	11727.1	1.52850	0.764252	−	0.644918i	$-0.223109\pi$
$389$	11727.1	1.52850	0.764252	+	0.644918i	$0.223109\pi$
$390$	0	0
$391$	14175.1	1.83341
$392$	10468.7	1.34885
$393$	0	0
$394$	1019.96	0.130418
$395$	1091.45	0.139030
$396$	0	0
$397$	−359.905	−0.0454990	−0.0227495	−	0.999741i	$-0.507242\pi$
$397$	−359.905	−0.0454990	−0.0227495	+	0.999741i	$0.507242\pi$
$398$	99.1105	0.0124823
$399$	0	0
$400$	2642.83	0.330354
$401$	4066.71	0.506438	0.253219	−	0.967409i	$-0.418511\pi$
$401$	4066.71	0.506438	0.253219	+	0.967409i	$0.418511\pi$
$402$	0	0
$403$	724.274	0.0895252
$404$	14136.9	1.74093
$405$	0	0
$406$	9740.45	1.19067
$407$	−4620.77	−0.562760
$408$	0	0
$409$	−13488.8	−1.63076	−0.815379	−	0.578927i	$-0.803472\pi$
$409$	−13488.8	−1.63076	−0.815379	+	0.578927i	$0.803472\pi$
$410$	−4059.40	−0.488975
$411$	0	0
$412$	5981.62	0.715275
$413$	−12883.3	−1.53498
$414$	0	0
$415$	−2650.04	−0.313459
$416$	−1167.24	−0.137569
$417$	0	0
$418$	1696.05	0.198460
$419$	7040.12	0.820841	0.410420	−	0.911896i	$-0.365382\pi$
$419$	7040.12	0.820841	0.410420	+	0.911896i	$0.365382\pi$
$420$	0	0
$421$	9171.74	1.06177	0.530883	−	0.847445i	$-0.321860\pi$
$421$	9171.74	1.06177	0.530883	+	0.847445i	$0.321860\pi$
$422$	−4652.89	−0.536728
$423$	0	0
$424$	−77.8204	−0.00891343
$425$	14253.1	1.62677
$426$	0	0
$427$	−17637.6	−1.99893
$428$	1759.77	0.198742
$429$	0	0
$430$	−4046.51	−0.453814
$431$	−992.995	−0.110976	−0.0554882	−	0.998459i	$-0.517672\pi$
$431$	−992.995	−0.110976	−0.0554882	+	0.998459i	$0.517672\pi$
$432$	0	0
$433$	3790.21	0.420660	0.210330	−	0.977630i	$-0.432546\pi$
$433$	3790.21	0.420660	0.210330	+	0.977630i	$0.432546\pi$
$434$	19719.0	2.18097
$435$	0	0
$436$	25122.7	2.75954
$437$	−4050.98	−0.443443
$438$	0	0
$439$	−5136.97	−0.558483	−0.279242	−	0.960221i	$-0.590083\pi$
$439$	−5136.97	−0.558483	−0.279242	+	0.960221i	$0.590083\pi$
$440$	−495.754	−0.0537139
$441$	0	0
$442$	−2779.16	−0.299075
$443$	−10676.8	−1.14508	−0.572541	−	0.819876i	$-0.694042\pi$
$443$	−10676.8	−1.14508	−0.572541	+	0.819876i	$0.694042\pi$
$444$	0	0
$445$	−2086.96	−0.222317
$446$	17086.2	1.81403
$447$	0	0
$448$	−26045.5	−2.74672
$449$	−10529.9	−1.10676	−0.553379	−	0.832929i	$-0.686662\pi$
$449$	−10529.9	−1.10676	−0.553379	+	0.832929i	$0.686662\pi$
$450$	0	0
$451$	3542.64	0.369881
$452$	7482.42	0.778636
$453$	0	0
$454$	3861.29	0.399162
$455$	−465.160	−0.0479275
$456$	0	0
$457$	−14072.5	−1.44045	−0.720225	−	0.693741i	$-0.755961\pi$
$457$	−14072.5	−1.44045	−0.720225	+	0.693741i	$0.755961\pi$
$458$	8148.55	0.831347
$459$	0	0
$460$	3833.41	0.388551
$461$	30.8173	0.00311346	0.00155673	−	0.999999i	$-0.499504\pi$
$461$	30.8173	0.00311346	0.00155673	+	0.999999i	$0.499504\pi$
$462$	0	0
$463$	17591.3	1.76573	0.882867	−	0.469622i	$-0.155610\pi$
$463$	17591.3	1.76573	0.882867	+	0.469622i	$0.155610\pi$
$464$	−1570.39	−0.157120
$465$	0	0
$466$	−17397.5	−1.72945
$467$	−13273.1	−1.31522	−0.657609	−	0.753360i	$-0.728432\pi$
$467$	−13273.1	−1.31522	−0.657609	+	0.753360i	$0.728432\pi$
$468$	0	0
$469$	2686.23	0.264474
$470$	−2916.79	−0.286259
$471$	0	0
$472$	−6429.87	−0.627031
$473$	3531.39	0.343284
$474$	0	0
$475$	−4073.27	−0.393462
$476$	−44742.9	−4.30837
$477$	0	0
$478$	−21113.8	−2.02034
$479$	−2496.68	−0.238155	−0.119077	−	0.992885i	$-0.537994\pi$
$479$	−2496.68	−0.238155	−0.119077	+	0.992885i	$0.537994\pi$
$480$	0	0
$481$	−2163.84	−0.205120
$482$	17648.3	1.66776
$483$	0	0
$484$	1400.64	0.131541
$485$	3337.41	0.312462
$486$	0	0
$487$	−3464.42	−0.322357	−0.161178	−	0.986925i	$-0.551529\pi$
$487$	−3464.42	−0.322357	−0.161178	+	0.986925i	$0.551529\pi$
$488$	−8802.65	−0.816552
$489$	0	0
$490$	−8341.01	−0.768997
$491$	16224.6	1.49125	0.745625	−	0.666366i	$-0.232151\pi$
$491$	16224.6	1.49125	0.745625	+	0.666366i	$0.232151\pi$
$492$	0	0
$493$	−8469.29	−0.773707
$494$	794.236	0.0723367
$495$	0	0
$496$	−3179.17	−0.287800
$497$	−1555.75	−0.140412
$498$	0	0
$499$	9993.81	0.896562	0.448281	−	0.893893i	$-0.352036\pi$
$499$	9993.81	0.896562	0.448281	+	0.893893i	$0.352036\pi$
$500$	7976.65	0.713453
$501$	0	0
$502$	24220.3	2.15340
$503$	15334.8	1.35933	0.679667	−	0.733520i	$-0.262124\pi$
$503$	15334.8	1.35933	0.679667	+	0.733520i	$0.262124\pi$
$504$	0	0
$505$	−3479.23	−0.306581
$506$	−5657.46	−0.497045
$507$	0	0
$508$	−11498.1	−1.00422
$509$	7291.23	0.634927	0.317464	−	0.948270i	$-0.397169\pi$
$509$	7291.23	0.634927	0.317464	+	0.948270i	$0.397169\pi$
$510$	0	0
$511$	24910.7	2.15653
$512$	7985.14	0.689251
$513$	0	0
$514$	28466.8	2.44283
$515$	−1472.13	−0.125961
$516$	0	0
$517$	2545.49	0.216538
$518$	−58912.5	−4.99704
$519$	0	0
$520$	−232.154	−0.0195782
$521$	−16794.3	−1.41223	−0.706114	−	0.708098i	$-0.749553\pi$
$521$	−16794.3	−1.41223	−0.706114	+	0.708098i	$0.749553\pi$
$522$	0	0
$523$	−21009.4	−1.75655	−0.878275	−	0.478157i	$-0.841305\pi$
$523$	−21009.4	−1.75655	−0.878275	+	0.478157i	$0.841305\pi$
$524$	−4466.01	−0.372326
$525$	0	0
$526$	−33577.0	−2.78332
$527$	−17145.6	−1.41722
$528$	0	0
$529$	1345.73	0.110605
$530$	62.0039	0.00508166
$531$	0	0
$532$	12786.7	1.04206
$533$	1658.97	0.134818
$534$	0	0
$535$	−433.097	−0.0349989
$536$	1340.66	0.108036
$537$	0	0
$538$	−2115.66	−0.169540
$539$	7279.20	0.581702
$540$	0	0
$541$	−16802.8	−1.33532	−0.667662	−	0.744464i	$-0.732705\pi$
$541$	−16802.8	−1.33532	−0.667662	+	0.744464i	$0.732705\pi$
$542$	−541.211	−0.0428911
$543$	0	0
$544$	27631.9	2.17777
$545$	−6182.93	−0.485959
$546$	0	0
$547$	16784.5	1.31198	0.655990	−	0.754770i	$-0.272251\pi$
$547$	16784.5	1.31198	0.655990	+	0.754770i	$0.272251\pi$
$548$	−10242.5	−0.798429
$549$	0	0
$550$	−5688.59	−0.441023
$551$	2420.37	0.187135
$552$	0	0
$553$	−12144.0	−0.933840
$554$	36277.7	2.78212
$555$	0	0
$556$	−12639.9	−0.964117
$557$	−18127.0	−1.37893	−0.689467	−	0.724317i	$-0.742155\pi$
$557$	−18127.0	−1.37893	−0.689467	+	0.724317i	$0.742155\pi$
$558$	0	0
$559$	1653.70	0.125123
$560$	2041.80	0.154075
$561$	0	0
$562$	−30722.3	−2.30595
$563$	−2090.88	−0.156518	−0.0782592	−	0.996933i	$-0.524936\pi$
$563$	−2090.88	−0.156518	−0.0782592	+	0.996933i	$0.524936\pi$
$564$	0	0
$565$	−1841.49	−0.137119
$566$	4579.89	0.340119
$567$	0	0
$568$	−776.452	−0.0573578
$569$	−6249.23	−0.460424	−0.230212	−	0.973140i	$-0.573942\pi$
$569$	−6249.23	−0.460424	−0.230212	+	0.973140i	$0.573942\pi$
$570$	0	0
$571$	6048.79	0.443317	0.221659	−	0.975124i	$-0.428853\pi$
$571$	6048.79	0.443317	0.221659	+	0.975124i	$0.428853\pi$
$572$	655.902	0.0479451
$573$	0	0
$574$	45166.8	3.28437
$575$	13587.1	0.985428
$576$	0	0
$577$	−15729.1	−1.13486	−0.567429	−	0.823423i	$-0.692062\pi$
$577$	−15729.1	−1.13486	−0.567429	+	0.823423i	$0.692062\pi$
$578$	44053.4	3.17021
$579$	0	0
$580$	−2290.38	−0.163970
$581$	29485.6	2.10545
$582$	0	0
$583$	−54.1108	−0.00384398
$584$	12432.6	0.880931
$585$	0	0
$586$	27187.3	1.91655
$587$	−15620.5	−1.09835	−0.549173	−	0.835709i	$-0.685057\pi$
$587$	−15620.5	−1.09835	−0.549173	+	0.835709i	$0.685057\pi$
$588$	0	0
$589$	4899.91	0.342780
$590$	5123.04	0.357478
$591$	0	0
$592$	9498.09	0.659407
$593$	493.541	0.0341776	0.0170888	−	0.999854i	$-0.494560\pi$
$593$	493.541	0.0341776	0.0170888	+	0.999854i	$0.494560\pi$
$594$	0	0
$595$	11011.6	0.758711
$596$	−3438.11	−0.236293
$597$	0	0
$598$	−2649.31	−0.181168
$599$	12455.1	0.849585	0.424793	−	0.905291i	$-0.360347\pi$
$599$	12455.1	0.849585	0.424793	+	0.905291i	$0.360347\pi$
$600$	0	0
$601$	12454.8	0.845329	0.422664	−	0.906286i	$-0.361095\pi$
$601$	12454.8	0.845329	0.422664	+	0.906286i	$0.361095\pi$
$602$	45023.3	3.04820
$603$	0	0
$604$	−21853.1	−1.47217
$605$	−344.712	−0.0231645
$606$	0	0
$607$	−4243.19	−0.283733	−0.141867	−	0.989886i	$-0.545310\pi$
$607$	−4243.19	−0.283733	−0.141867	+	0.989886i	$0.545310\pi$
$608$	−7896.70	−0.526732
$609$	0	0
$610$	7013.57	0.465526
$611$	1192.01	0.0789259
$612$	0	0
$613$	5733.14	0.377748	0.188874	−	0.982001i	$-0.439516\pi$
$613$	5733.14	0.377748	0.188874	+	0.982001i	$0.439516\pi$
$614$	−9672.18	−0.635729
$615$	0	0
$616$	5515.99	0.360788
$617$	−15642.1	−1.02063	−0.510314	−	0.859988i	$-0.670471\pi$
$617$	−15642.1	−1.02063	−0.510314	+	0.859988i	$0.670471\pi$
$618$	0	0
$619$	−7467.40	−0.484879	−0.242440	−	0.970167i	$-0.577948\pi$
$619$	−7467.40	−0.484879	−0.242440	+	0.970167i	$0.577948\pi$
$620$	−4636.74	−0.300348
$621$	0	0
$622$	33116.1	2.13478
$623$	23220.4	1.49327
$624$	0	0
$625$	12647.4	0.809432
$626$	−30233.6	−1.93031
$627$	0	0
$628$	−654.416	−0.0415829
$629$	51224.2	3.24713
$630$	0	0
$631$	−1486.38	−0.0937745	−0.0468872	−	0.998900i	$-0.514930\pi$
$631$	−1486.38	−0.0937745	−0.0468872	+	0.998900i	$0.514930\pi$
$632$	−6060.87	−0.381469
$633$	0	0
$634$	−4089.35	−0.256165
$635$	2829.78	0.176845
$636$	0	0
$637$	3408.74	0.212024
$638$	3380.21	0.209755
$639$	0	0
$640$	5192.58	0.320711
$641$	−12386.0	−0.763211	−0.381606	−	0.924325i	$-0.624629\pi$
$641$	−12386.0	−0.763211	−0.381606	+	0.924325i	$0.624629\pi$
$642$	0	0
$643$	−14458.1	−0.886737	−0.443369	−	0.896339i	$-0.646217\pi$
$643$	−14458.1	−0.886737	−0.443369	+	0.896339i	$0.646217\pi$
$644$	−42652.3	−2.60984
$645$	0	0
$646$	−18801.8	−1.14512
$647$	−15792.8	−0.959625	−0.479813	−	0.877371i	$-0.659295\pi$
$647$	−15792.8	−0.959625	−0.479813	+	0.877371i	$0.659295\pi$
$648$	0	0
$649$	−4470.87	−0.270412
$650$	−2663.89	−0.160748
$651$	0	0
$652$	−569.909	−0.0342321
$653$	3179.93	0.190567	0.0952837	−	0.995450i	$-0.469624\pi$
$653$	3179.93	0.190567	0.0952837	+	0.995450i	$0.469624\pi$
$654$	0	0
$655$	1099.13	0.0655672
$656$	−7281.96	−0.433404
$657$	0	0
$658$	32453.6	1.92276
$659$	−11593.5	−0.685308	−0.342654	−	0.939462i	$-0.611326\pi$
$659$	−11593.5	−0.685308	−0.342654	+	0.939462i	$0.611326\pi$
$660$	0	0
$661$	3233.88	0.190293	0.0951464	−	0.995463i	$-0.469668\pi$
$661$	3233.88	0.190293	0.0951464	+	0.995463i	$0.469668\pi$
$662$	−43449.9	−2.55095
$663$	0	0
$664$	14715.8	0.860066
$665$	−3146.93	−0.183508
$666$	0	0
$667$	−8073.56	−0.468680
$668$	−23946.9	−1.38703
$669$	0	0
$670$	−1068.18	−0.0615929
$671$	−6120.74	−0.352144
$672$	0	0
$673$	−5495.72	−0.314776	−0.157388	−	0.987537i	$-0.550307\pi$
$673$	−5495.72	−0.314776	−0.157388	+	0.987537i	$0.550307\pi$
$674$	2658.23	0.151916
$675$	0	0
$676$	−25124.4	−1.42947
$677$	−33836.7	−1.92090	−0.960451	−	0.278448i	$-0.910180\pi$
$677$	−33836.7	−1.92090	−0.960451	+	0.278448i	$0.910180\pi$
$678$	0	0
$679$	−37133.6	−2.09876
$680$	5495.75	0.309930
$681$	0	0
$682$	6843.04	0.384214
$683$	21080.3	1.18099	0.590493	−	0.807043i	$-0.298933\pi$
$683$	21080.3	1.18099	0.590493	+	0.807043i	$0.298933\pi$
$684$	0	0
$685$	2520.78	0.140605
$686$	44702.2	2.48796
$687$	0	0
$688$	−7258.84	−0.402239
$689$	−25.3393	−0.00140109
$690$	0	0
$691$	11811.3	0.650253	0.325127	−	0.945671i	$-0.394593\pi$
$691$	11811.3	0.650253	0.325127	+	0.945671i	$0.394593\pi$
$692$	6991.79	0.384087
$693$	0	0
$694$	13907.8	0.760711
$695$	3110.79	0.169783
$696$	0	0
$697$	−39272.4	−2.13422
$698$	3188.52	0.172904
$699$	0	0
$700$	−42887.0	−2.31568
$701$	−4244.99	−0.228718	−0.114359	−	0.993440i	$-0.536481\pi$
$701$	−4244.99	−0.228718	−0.114359	+	0.993440i	$0.536481\pi$
$702$	0	0
$703$	−14639.0	−0.785376
$704$	−9038.51	−0.483880
$705$	0	0
$706$	−5340.81	−0.284708
$707$	38711.5	2.05926
$708$	0	0
$709$	−898.822	−0.0476107	−0.0238053	−	0.999717i	$-0.507578\pi$
$709$	−898.822	−0.0476107	−0.0238053	+	0.999717i	$0.507578\pi$
$710$	618.643	0.0327004
$711$	0	0
$712$	11589.0	0.609993
$713$	−16344.5	−0.858493
$714$	0	0
$715$	−161.424	−0.00844322
$716$	24679.4	1.28815
$717$	0	0
$718$	−38706.3	−2.01185
$719$	−10741.8	−0.557165	−0.278582	−	0.960412i	$-0.589865\pi$
$719$	−10741.8	−0.557165	−0.278582	+	0.960412i	$0.589865\pi$
$720$	0	0
$721$	16379.6	0.846061
$722$	−24973.9	−1.28730
$723$	0	0
$724$	−6822.30	−0.350206
$725$	−8117.99	−0.415855
$726$	0	0
$727$	16794.2	0.856758	0.428379	−	0.903599i	$-0.359085\pi$
$727$	16794.2	0.856758	0.428379	+	0.903599i	$0.359085\pi$
$728$	2583.06	0.131503
$729$	0	0
$730$	−9905.73	−0.502229
$731$	−39147.7	−1.98075
$732$	0	0
$733$	8659.40	0.436347	0.218173	−	0.975910i	$-0.429990\pi$
$733$	8659.40	0.436347	0.218173	+	0.975910i	$0.429990\pi$
$734$	−29778.4	−1.49747
$735$	0	0
$736$	26340.8	1.31920
$737$	932.197	0.0465915
$738$	0	0
$739$	16705.7	0.831567	0.415783	−	0.909464i	$-0.363507\pi$
$739$	16705.7	0.831567	0.415783	+	0.909464i	$0.363507\pi$
$740$	13852.7	0.688157
$741$	0	0
$742$	−689.884	−0.0341327
$743$	−1292.12	−0.0637996	−0.0318998	−	0.999491i	$-0.510156\pi$
$743$	−1292.12	−0.0637996	−0.0318998	+	0.999491i	$0.510156\pi$
$744$	0	0
$745$	846.151	0.0416115
$746$	−1006.09	−0.0493773
$747$	0	0
$748$	−15527.0	−0.758990
$749$	4818.83	0.235082
$750$	0	0
$751$	−14980.4	−0.727886	−0.363943	−	0.931421i	$-0.618570\pi$
$751$	−14980.4	−0.727886	−0.363943	+	0.931421i	$0.618570\pi$
$752$	−5232.30	−0.253726
$753$	0	0
$754$	1582.90	0.0764535
$755$	5378.25	0.259251
$756$	0	0
$757$	3003.41	0.144202	0.0721010	−	0.997397i	$-0.477030\pi$
$757$	3003.41	0.144202	0.0721010	+	0.997397i	$0.477030\pi$
$758$	50244.8	2.40762
$759$	0	0
$760$	−1570.59	−0.0749621
$761$	20375.0	0.970555	0.485277	−	0.874360i	$-0.338719\pi$
$761$	20375.0	0.970555	0.485277	+	0.874360i	$0.338719\pi$
$762$	0	0
$763$	68794.1	3.26411
$764$	25013.6	1.18450
$765$	0	0
$766$	−47578.4	−2.24422
$767$	−2093.65	−0.0985621
$768$	0	0
$769$	−12372.4	−0.580184	−0.290092	−	0.956999i	$-0.593686\pi$
$769$	−12372.4	−0.580184	−0.290092	+	0.956999i	$0.593686\pi$
$770$	−4394.90	−0.205690
$771$	0	0
$772$	−17258.1	−0.804578
$773$	−21023.6	−0.978225	−0.489113	−	0.872221i	$-0.662679\pi$
$773$	−21023.6	−0.978225	−0.489113	+	0.872221i	$0.662679\pi$
$774$	0	0
$775$	−16434.4	−0.761732
$776$	−18532.8	−0.857331
$777$	0	0
$778$	51885.7	2.39099
$779$	11223.4	0.516198
$780$	0	0
$781$	−539.889	−0.0247359
$782$	62716.6	2.86795
$783$	0	0
$784$	−14962.5	−0.681602
$785$	161.058	0.00732281
$786$	0	0
$787$	30286.2	1.37177	0.685886	−	0.727709i	$-0.259415\pi$
$787$	30286.2	1.37177	0.685886	+	0.727709i	$0.259415\pi$
$788$	2668.51	0.120637
$789$	0	0
$790$	4829.03	0.217480
$791$	20489.3	0.921007
$792$	0	0
$793$	−2866.25	−0.128353
$794$	−1592.37	−0.0711729
$795$	0	0
$796$	259.301	0.0115461
$797$	−32337.8	−1.43722	−0.718610	−	0.695413i	$-0.755221\pi$
$797$	−32337.8	−1.43722	−0.718610	+	0.695413i	$0.755221\pi$
$798$	0	0
$799$	−28218.3	−1.24943
$800$	26485.7	1.17052
$801$	0	0
$802$	17992.9	0.792207
$803$	8644.72	0.379907
$804$	0	0
$805$	10497.1	0.459596
$806$	3204.50	0.140042
$807$	0	0
$808$	19320.3	0.841197
$809$	891.707	0.0387525	0.0193762	−	0.999812i	$-0.493832\pi$
$809$	891.707	0.0387525	0.0193762	+	0.999812i	$0.493832\pi$
$810$	0	0
$811$	−10114.9	−0.437957	−0.218978	−	0.975730i	$-0.570272\pi$
$811$	−10114.9	−0.437957	−0.218978	+	0.975730i	$0.570272\pi$
$812$	25483.8	1.10136
$813$	0	0
$814$	−20444.3	−0.880309
$815$	140.260	0.00602833
$816$	0	0
$817$	11187.7	0.479080
$818$	−59680.4	−2.55095
$819$	0	0
$820$	−10620.6	−0.452300
$821$	−10833.5	−0.460525	−0.230262	−	0.973129i	$-0.573958\pi$
$821$	−10833.5	−0.460525	−0.230262	+	0.973129i	$0.573958\pi$
$822$	0	0
$823$	31958.5	1.35359	0.676794	−	0.736173i	$-0.263369\pi$
$823$	31958.5	1.35359	0.676794	+	0.736173i	$0.263369\pi$
$824$	8174.84	0.345612
$825$	0	0
$826$	−57001.3	−2.40112
$827$	−34847.3	−1.46525	−0.732624	−	0.680634i	$-0.761704\pi$
$827$	−34847.3	−1.46525	−0.732624	+	0.680634i	$0.761704\pi$
$828$	0	0
$829$	6537.91	0.273910	0.136955	−	0.990577i	$-0.456268\pi$
$829$	6537.91	0.273910	0.136955	+	0.990577i	$0.456268\pi$
$830$	−11724.9	−0.490334
$831$	0	0
$832$	−4232.60	−0.176369
$833$	−80694.5	−3.35642
$834$	0	0
$835$	5893.56	0.244258
$836$	4437.35	0.183575
$837$	0	0
$838$	31148.5	1.28402
$839$	−2710.34	−0.111527	−0.0557635	−	0.998444i	$-0.517759\pi$
$839$	−2710.34	−0.111527	−0.0557635	+	0.998444i	$0.517759\pi$
$840$	0	0
$841$	−19565.2	−0.802215
$842$	40579.7	1.66089
$843$	0	0
$844$	−12173.3	−0.496471
$845$	6183.35	0.251732
$846$	0	0
$847$	3835.42	0.155592
$848$	111.226	0.00450414
$849$	0	0
$850$	63061.7	2.54470
$851$	48830.8	1.96698
$852$	0	0
$853$	9759.32	0.391738	0.195869	−	0.980630i	$-0.437247\pi$
$853$	9759.32	0.391738	0.195869	+	0.980630i	$0.437247\pi$
$854$	−78036.2	−3.12687
$855$	0	0
$856$	2405.01	0.0960298
$857$	13649.8	0.544072	0.272036	−	0.962287i	$-0.412303\pi$
$857$	13649.8	0.544072	0.272036	+	0.962287i	$0.412303\pi$
$858$	0	0
$859$	7796.42	0.309674	0.154837	−	0.987940i	$-0.450515\pi$
$859$	7796.42	0.309674	0.154837	+	0.987940i	$0.450515\pi$
$860$	−10586.8	−0.419776
$861$	0	0
$862$	−4393.43	−0.173597
$863$	−7183.57	−0.283350	−0.141675	−	0.989913i	$-0.545249\pi$
$863$	−7183.57	−0.283350	−0.141675	+	0.989913i	$0.545249\pi$
$864$	0	0
$865$	−1720.75	−0.0676383
$866$	16769.5	0.658026
$867$	0	0
$868$	51590.5	2.01739
$869$	−4214.30	−0.164511
$870$	0	0
$871$	436.534	0.0169821
$872$	34334.1	1.33337
$873$	0	0
$874$	−17923.3	−0.693666
$875$	21842.7	0.843905
$876$	0	0
$877$	17063.1	0.656991	0.328495	−	0.944506i	$-0.393458\pi$
$877$	17063.1	0.656991	0.328495	+	0.944506i	$0.393458\pi$
$878$	−22728.2	−0.873620
$879$	0	0
$880$	708.562	0.0271428
$881$	32174.9	1.23042	0.615210	−	0.788363i	$-0.289071\pi$
$881$	32174.9	1.23042	0.615210	+	0.788363i	$0.289071\pi$
$882$	0	0
$883$	2843.68	0.108378	0.0541889	−	0.998531i	$-0.482743\pi$
$883$	2843.68	0.108378	0.0541889	+	0.998531i	$0.482743\pi$
$884$	−7271.09	−0.276644
$885$	0	0
$886$	−47238.9	−1.79122
$887$	31417.8	1.18930	0.594649	−	0.803985i	$-0.297291\pi$
$887$	31417.8	1.18930	0.594649	+	0.803985i	$0.297291\pi$
$888$	0	0
$889$	−31485.5	−1.18784
$890$	−9233.59	−0.347765
$891$	0	0
$892$	44702.5	1.67797
$893$	8064.30	0.302196
$894$	0	0
$895$	−6073.83	−0.226845
$896$	−57775.1	−2.15416
$897$	0	0
$898$	−46588.6	−1.73127
$899$	9765.47	0.362288
$900$	0	0
$901$	599.852	0.0221798
$902$	15674.1	0.578594
$903$	0	0
$904$	10225.9	0.376227
$905$	1679.03	0.0616718
$906$	0	0
$907$	12253.1	0.448573	0.224287	−	0.974523i	$-0.427995\pi$
$907$	12253.1	0.448573	0.224287	+	0.974523i	$0.427995\pi$
$908$	10102.2	0.369223
$909$	0	0
$910$	−2058.07	−0.0749717
$911$	48422.4	1.76104	0.880518	−	0.474012i	$-0.157195\pi$
$911$	48422.4	1.76104	0.880518	+	0.474012i	$0.157195\pi$
$912$	0	0
$913$	10232.3	0.370909
$914$	−62262.9	−2.25326
$915$	0	0
$916$	21318.9	0.768993
$917$	−12229.4	−0.440404
$918$	0	0
$919$	5546.18	0.199077	0.0995385	−	0.995034i	$-0.468263\pi$
$919$	5546.18	0.199077	0.0995385	+	0.995034i	$0.468263\pi$
$920$	5238.96	0.187743
$921$	0	0
$922$	136.349	0.00487030
$923$	−252.822	−0.00901598
$924$	0	0
$925$	49099.5	1.74528
$926$	77831.3	2.76209
$927$	0	0
$928$	−15738.0	−0.556709
$929$	35684.5	1.26025	0.630125	−	0.776494i	$-0.283004\pi$
$929$	35684.5	1.26025	0.630125	+	0.776494i	$0.283004\pi$
$930$	0	0
$931$	23061.1	0.811811
$932$	−45516.7	−1.59973
$933$	0	0
$934$	−58726.0	−2.05736
$935$	3821.35	0.133659
$936$	0	0
$937$	−48903.6	−1.70503	−0.852514	−	0.522705i	$-0.824923\pi$
$937$	−48903.6	−1.70503	−0.852514	+	0.522705i	$0.824923\pi$
$938$	11885.0	0.413710
$939$	0	0
$940$	−7631.17	−0.264789
$941$	23741.9	0.822490	0.411245	−	0.911525i	$-0.365094\pi$
$941$	23741.9	0.822490	0.411245	+	0.911525i	$0.365094\pi$
$942$	0	0
$943$	−37437.4	−1.29282
$944$	9189.97	0.316852
$945$	0	0
$946$	15624.4	0.536989
$947$	−37612.4	−1.29064	−0.645321	−	0.763911i	$-0.723276\pi$
$947$	−37612.4	−1.29064	−0.645321	+	0.763911i	$0.723276\pi$
$948$	0	0
$949$	4048.20	0.138472
$950$	−18021.9	−0.615482
$951$	0	0
$952$	−61148.2	−2.08175
$953$	48294.3	1.64156	0.820779	−	0.571246i	$-0.193540\pi$
$953$	48294.3	1.64156	0.820779	+	0.571246i	$0.193540\pi$
$954$	0	0
$955$	−6156.09	−0.208593
$956$	−55239.8	−1.86881
$957$	0	0
$958$	−11046.4	−0.372539
$959$	−28047.4	−0.944419
$960$	0	0
$961$	−10021.4	−0.336389
$962$	−9573.76	−0.320863
$963$	0	0
$964$	46173.1	1.54267
$965$	4247.39	0.141687
$966$	0	0
$967$	1840.92	0.0612204	0.0306102	−	0.999531i	$-0.490255\pi$
$967$	1840.92	0.0612204	0.0306102	+	0.999531i	$0.490255\pi$
$968$	1914.20	0.0635587
$969$	0	0
$970$	14766.1	0.488775
$971$	−31461.8	−1.03981	−0.519906	−	0.854223i	$-0.674033\pi$
$971$	−31461.8	−1.03981	−0.519906	+	0.854223i	$0.674033\pi$
$972$	0	0
$973$	−34612.1	−1.14040
$974$	−15328.1	−0.504254
$975$	0	0
$976$	12581.3	0.412620
$977$	7040.11	0.230535	0.115268	−	0.993334i	$-0.463227\pi$
$977$	7040.11	0.230535	0.115268	+	0.993334i	$0.463227\pi$
$978$	0	0
$979$	8058.15	0.263064
$980$	−21822.5	−0.711320
$981$	0	0
$982$	71784.4	2.33272
$983$	24610.9	0.798541	0.399270	−	0.916833i	$-0.369263\pi$
$983$	24610.9	0.798541	0.399270	+	0.916833i	$0.369263\pi$
$984$	0	0
$985$	−656.744	−0.0212443
$986$	−37471.8	−1.21029
$987$	0	0
$988$	2077.95	0.0669112
$989$	−37318.5	−1.19986
$990$	0	0
$991$	−40003.3	−1.28229	−0.641144	−	0.767421i	$-0.721540\pi$
$991$	−40003.3	−1.28229	−0.641144	+	0.767421i	$0.721540\pi$
$992$	−31860.8	−1.01974
$993$	0	0
$994$	−6883.31	−0.219643
$995$	−63.8165	−0.00203329
$996$	0	0
$997$	−7342.61	−0.233242	−0.116621	−	0.993176i	$-0.537206\pi$
$997$	−7342.61	−0.233242	−0.116621	+	0.993176i	$0.537206\pi$
$998$	44216.9	1.40247
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	99.4.a.f.1.2		2
3.2	odd	2		33.4.a.c.1.1	✓	2
4.3	odd	2		1584.4.a.bj.1.1		2
5.4	even	2		2475.4.a.p.1.1		2
11.10	odd	2		1089.4.a.u.1.1		2
12.11	even	2		528.4.a.p.1.2		2
15.2	even	4		825.4.c.h.199.2		4
15.8	even	4		825.4.c.h.199.3		4
15.14	odd	2		825.4.a.l.1.2		2
21.20	even	2		1617.4.a.k.1.1		2
24.5	odd	2		2112.4.a.bn.1.1		2
24.11	even	2		2112.4.a.bg.1.1		2
33.32	even	2		363.4.a.i.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.4.a.c.1.1	✓	2	3.2	odd	2
99.4.a.f.1.2		2	1.1	even	1	trivial
363.4.a.i.1.2		2	33.32	even	2
528.4.a.p.1.2		2	12.11	even	2
825.4.a.l.1.2		2	15.14	odd	2
825.4.c.h.199.2		4	15.2	even	4
825.4.c.h.199.3		4	15.8	even	4
1089.4.a.u.1.1		2	11.10	odd	2
1584.4.a.bj.1.1		2	4.3	odd	2
1617.4.a.k.1.1		2	21.20	even	2
2112.4.a.bg.1.1		2	24.11	even	2
2112.4.a.bn.1.1		2	24.5	odd	2
2475.4.a.p.1.1		2	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 99 = 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	99.a (trivial)

\(f(q)\)	\(=\)	\(q+4.42443 q^{2} +11.5756 q^{4} -2.84886 q^{5} +31.6977 q^{7} +15.8199 q^{8} +O(q^{10})\)	\(q+4.42443 q^{2} +11.5756 q^{4} -2.84886 q^{5} +31.6977 q^{7} +15.8199 q^{8} -12.6046 q^{10} +11.0000 q^{11} +5.15114 q^{13} +140.244 q^{14} -22.6107 q^{16} -121.942 q^{17} +34.8489 q^{19} -32.9772 q^{20} +48.6687 q^{22} -116.244 q^{23} -116.884 q^{25} +22.7909 q^{26} +366.919 q^{28} +69.4534 q^{29} +140.605 q^{31} -226.598 q^{32} -539.524 q^{34} -90.3023 q^{35} -420.070 q^{37} +154.186 q^{38} -45.0685 q^{40} +322.058 q^{41} +321.035 q^{43} +127.331 q^{44} -514.315 q^{46} +231.408 q^{47} +661.745 q^{49} -517.145 q^{50} +59.6274 q^{52} -4.91916 q^{53} -31.3374 q^{55} +501.453 q^{56} +307.292 q^{58} -406.443 q^{59} -556.431 q^{61} +622.095 q^{62} -821.683 q^{64} -14.6749 q^{65} +84.7452 q^{67} -1411.55 q^{68} -399.536 q^{70} -49.0808 q^{71} +785.884 q^{73} -1858.57 q^{74} +403.395 q^{76} +348.675 q^{77} -383.118 q^{79} +64.4147 q^{80} +1424.92 q^{82} +930.211 q^{83} +347.395 q^{85} +1420.40 q^{86} +174.018 q^{88} +732.559 q^{89} +163.279 q^{91} -1345.59 q^{92} +1023.85 q^{94} -99.2794 q^{95} -1171.49 q^{97} +2927.84 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} + 33 q^{4} + 14 q^{5} + 24 q^{7} - 57 q^{8}+O(q^{10}) \) 2 * q - q^2 + 33 * q^4 + 14 * q^5 + 24 * q^7 - 57 * q^8	\( 2 q - q^{2} + 33 q^{4} + 14 q^{5} + 24 q^{7} - 57 q^{8} - 104 q^{10} + 22 q^{11} + 30 q^{13} + 182 q^{14} + 201 q^{16} - 106 q^{17} + 50 q^{19} + 328 q^{20} - 11 q^{22} - 134 q^{23} + 42 q^{25} - 112 q^{26} + 202 q^{28} + 198 q^{29} + 360 q^{31} - 857 q^{32} - 626 q^{34} - 220 q^{35} - 328 q^{37} + 72 q^{38} - 1272 q^{40} + 782 q^{41} + 386 q^{43} + 363 q^{44} - 418 q^{46} - 266 q^{47} + 378 q^{49} - 1379 q^{50} + 592 q^{52} + 522 q^{53} + 154 q^{55} + 1062 q^{56} - 390 q^{58} + 172 q^{59} - 778 q^{61} - 568 q^{62} + 809 q^{64} + 404 q^{65} - 776 q^{67} - 1070 q^{68} + 304 q^{70} - 630 q^{71} + 1296 q^{73} - 2358 q^{74} + 728 q^{76} + 264 q^{77} + 652 q^{79} + 3832 q^{80} - 1070 q^{82} + 324 q^{83} + 616 q^{85} + 1068 q^{86} - 627 q^{88} + 756 q^{89} - 28 q^{91} - 1726 q^{92} + 3722 q^{94} + 156 q^{95} - 452 q^{97} + 4467 q^{98}+O(q^{100}) \) 2 * q - q^2 + 33 * q^4 + 14 * q^5 + 24 * q^7 - 57 * q^8 - 104 * q^10 + 22 * q^11 + 30 * q^13 + 182 * q^14 + 201 * q^16 - 106 * q^17 + 50 * q^19 + 328 * q^20 - 11 * q^22 - 134 * q^23 + 42 * q^25 - 112 * q^26 + 202 * q^28 + 198 * q^29 + 360 * q^31 - 857 * q^32 - 626 * q^34 - 220 * q^35 - 328 * q^37 + 72 * q^38 - 1272 * q^40 + 782 * q^41 + 386 * q^43 + 363 * q^44 - 418 * q^46 - 266 * q^47 + 378 * q^49 - 1379 * q^50 + 592 * q^52 + 522 * q^53 + 154 * q^55 + 1062 * q^56 - 390 * q^58 + 172 * q^59 - 778 * q^61 - 568 * q^62 + 809 * q^64 + 404 * q^65 - 776 * q^67 - 1070 * q^68 + 304 * q^70 - 630 * q^71 + 1296 * q^73 - 2358 * q^74 + 728 * q^76 + 264 * q^77 + 652 * q^79 + 3832 * q^80 - 1070 * q^82 + 324 * q^83 + 616 * q^85 + 1068 * q^86 - 627 * q^88 + 756 * q^89 - 28 * q^91 - 1726 * q^92 + 3722 * q^94 + 156 * q^95 - 452 * q^97 + 4467 * q^98

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