LMFDB - Embedded newform 99.4.a.c.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:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 11)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ 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)

$f(q)$	$=$	$q+0.732051 q^{2} -7.46410 q^{4} +12.8564 q^{5} +16.9282 q^{7} -11.3205 q^{8} +O(q^{10})$	\(q+0.732051 q^{2} -7.46410 q^{4} +12.8564 q^{5} +16.9282 q^{7} -11.3205 q^{8} +9.41154 q^{10} +11.0000 q^{11} +74.6410 q^{13} +12.3923 q^{14} +51.4256 q^{16} +82.7846 q^{17} -67.9230 q^{19} -95.9615 q^{20} +8.05256 q^{22} -13.3538 q^{23} +40.2872 q^{25} +54.6410 q^{26} -126.354 q^{28} -168.995 q^{29} -65.4974 q^{31} +128.210 q^{32} +60.6025 q^{34} +217.636 q^{35} +40.8564 q^{37} -49.7231 q^{38} -145.541 q^{40} -274.928 q^{41} -2.28719 q^{43} -82.1051 q^{44} -9.77568 q^{46} -71.8461 q^{47} -56.4359 q^{49} +29.4923 q^{50} -557.128 q^{52} +149.005 q^{53} +141.420 q^{55} -191.636 q^{56} -123.713 q^{58} -545.631 q^{59} +101.303 q^{61} -47.9474 q^{62} -317.549 q^{64} +959.615 q^{65} +411.641 q^{67} -617.913 q^{68} +159.321 q^{70} +470.636 q^{71} +610.600 q^{73} +29.9090 q^{74} +506.985 q^{76} +186.210 q^{77} -978.225 q^{79} +661.149 q^{80} -201.261 q^{82} -26.1539 q^{83} +1064.31 q^{85} -1.67434 q^{86} -124.526 q^{88} +352.887 q^{89} +1263.54 q^{91} +99.6743 q^{92} -52.5950 q^{94} -873.246 q^{95} +847.585 q^{97} -41.3140 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 8 q^{4} - 2 q^{5} + 20 q^{7} + 12 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 - 8 * q^4 - 2 * q^5 + 20 * q^7 + 12 * q^8	$ 2 q - 2 q^{2} - 8 q^{4} - 2 q^{5} + 20 q^{7} + 12 q^{8} + 50 q^{10} + 22 q^{11} + 80 q^{13} + 4 q^{14} - 8 q^{16} + 124 q^{17} + 72 q^{19} - 88 q^{20} - 22 q^{22} + 98 q^{23} + 136 q^{25} + 40 q^{26} - 128 q^{28} - 144 q^{29} - 34 q^{31} + 104 q^{32} - 52 q^{34} + 172 q^{35} + 54 q^{37} - 432 q^{38} - 492 q^{40} - 536 q^{41} - 60 q^{43} - 88 q^{44} - 314 q^{46} + 272 q^{47} - 390 q^{49} - 232 q^{50} - 560 q^{52} + 492 q^{53} - 22 q^{55} - 120 q^{56} - 192 q^{58} - 634 q^{59} + 840 q^{61} - 134 q^{62} + 224 q^{64} + 880 q^{65} + 754 q^{67} - 640 q^{68} + 284 q^{70} + 678 q^{71} - 400 q^{73} - 6 q^{74} + 432 q^{76} + 220 q^{77} + 316 q^{79} + 1544 q^{80} + 512 q^{82} - 468 q^{83} + 452 q^{85} + 156 q^{86} + 132 q^{88} + 1842 q^{89} + 1280 q^{91} + 40 q^{92} - 992 q^{94} - 2952 q^{95} + 2194 q^{97} + 870 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 - 8 * q^4 - 2 * q^5 + 20 * q^7 + 12 * q^8 + 50 * q^10 + 22 * q^11 + 80 * q^13 + 4 * q^14 - 8 * q^16 + 124 * q^17 + 72 * q^19 - 88 * q^20 - 22 * q^22 + 98 * q^23 + 136 * q^25 + 40 * q^26 - 128 * q^28 - 144 * q^29 - 34 * q^31 + 104 * q^32 - 52 * q^34 + 172 * q^35 + 54 * q^37 - 432 * q^38 - 492 * q^40 - 536 * q^41 - 60 * q^43 - 88 * q^44 - 314 * q^46 + 272 * q^47 - 390 * q^49 - 232 * q^50 - 560 * q^52 + 492 * q^53 - 22 * q^55 - 120 * q^56 - 192 * q^58 - 634 * q^59 + 840 * q^61 - 134 * q^62 + 224 * q^64 + 880 * q^65 + 754 * q^67 - 640 * q^68 + 284 * q^70 + 678 * q^71 - 400 * q^73 - 6 * q^74 + 432 * q^76 + 220 * q^77 + 316 * q^79 + 1544 * q^80 + 512 * q^82 - 468 * q^83 + 452 * q^85 + 156 * q^86 + 132 * q^88 + 1842 * q^89 + 1280 * q^91 + 40 * q^92 - 992 * q^94 - 2952 * q^95 + 2194 * q^97 + 870 * 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 $

$2$	0.732051	0.258819	0.129410	−	0.991591i	$-0.458692\pi$
$2$	0.732051	0.258819	0.129410	+	0.991591i	$0.458692\pi$
$3$	0	0
$3$	0	0
$4$	−7.46410	−0.933013
$5$	12.8564	1.14991	0.574956	−	0.818184i	$-0.305019\pi$
$5$	12.8564	1.14991	0.574956	+	0.818184i	$0.305019\pi$
$6$	0	0
$7$	16.9282	0.914037	0.457019	−	0.889457i	$-0.348917\pi$
$7$	16.9282	0.914037	0.457019	+	0.889457i	$0.348917\pi$
$8$	−11.3205	−0.500301
$9$	0	0
$10$	9.41154	0.297619
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	74.6410	1.59244	0.796219	−	0.605009i	$-0.206830\pi$
$13$	74.6410	1.59244	0.796219	+	0.605009i	$0.206830\pi$
$14$	12.3923	0.236570
$15$	0	0
$16$	51.4256	0.803525
$17$	82.7846	1.18107	0.590536	−	0.807011i	$-0.298916\pi$
$17$	82.7846	1.18107	0.590536	+	0.807011i	$0.298916\pi$
$18$	0	0
$19$	−67.9230	−0.820138	−0.410069	−	0.912055i	$-0.634495\pi$
$19$	−67.9230	−0.820138	−0.410069	+	0.912055i	$0.634495\pi$
$20$	−95.9615	−1.07288
$21$	0	0
$22$	8.05256	0.0780369
$23$	−13.3538	−0.121064	−0.0605319	−	0.998166i	$-0.519280\pi$
$23$	−13.3538	−0.121064	−0.0605319	+	0.998166i	$0.519280\pi$
$24$	0	0
$25$	40.2872	0.322297
$26$	54.6410	0.412153
$27$	0	0
$28$	−126.354	−0.852808
$29$	−168.995	−1.08212	−0.541061	−	0.840983i	$-0.681977\pi$
$29$	−168.995	−1.08212	−0.541061	+	0.840983i	$0.681977\pi$
$30$	0	0
$31$	−65.4974	−0.379474	−0.189737	−	0.981835i	$-0.560763\pi$
$31$	−65.4974	−0.379474	−0.189737	+	0.981835i	$0.560763\pi$
$32$	128.210	0.708268
$33$	0	0
$34$	60.6025	0.305684
$35$	217.636	1.05106
$36$	0	0
$37$	40.8564	0.181534	0.0907669	−	0.995872i	$-0.471068\pi$
$37$	40.8564	0.181534	0.0907669	+	0.995872i	$0.471068\pi$
$38$	−49.7231	−0.212267
$39$	0	0
$40$	−145.541	−0.575302
$41$	−274.928	−1.04723	−0.523617	−	0.851954i	$-0.675418\pi$
$41$	−274.928	−1.04723	−0.523617	+	0.851954i	$0.675418\pi$
$42$	0	0
$43$	−2.28719	−0.00811146	−0.00405573	−	0.999992i	$-0.501291\pi$
$43$	−2.28719	−0.00811146	−0.00405573	+	0.999992i	$0.501291\pi$
$44$	−82.1051	−0.281314
$45$	0	0
$46$	−9.77568	−0.0313336
$47$	−71.8461	−0.222975	−0.111488	−	0.993766i	$-0.535562\pi$
$47$	−71.8461	−0.222975	−0.111488	+	0.993766i	$0.535562\pi$
$48$	0	0
$49$	−56.4359	−0.164536
$50$	29.4923	0.0834167
$51$	0	0
$52$	−557.128	−1.48576
$53$	149.005	0.386178	0.193089	−	0.981181i	$-0.438149\pi$
$53$	149.005	0.386178	0.193089	+	0.981181i	$0.438149\pi$
$54$	0	0
$55$	141.420	0.346711
$56$	−191.636	−0.457293
$57$	0	0
$58$	−123.713	−0.280074
$59$	−545.631	−1.20398	−0.601992	−	0.798502i	$-0.705626\pi$
$59$	−545.631	−1.20398	−0.601992	+	0.798502i	$0.705626\pi$
$60$	0	0
$61$	101.303	0.212631	0.106315	−	0.994332i	$-0.466095\pi$
$61$	101.303	0.212631	0.106315	+	0.994332i	$0.466095\pi$
$62$	−47.9474	−0.0982150
$63$	0	0
$64$	−317.549	−0.620212
$65$	959.615	1.83116
$66$	0	0
$67$	411.641	0.750596	0.375298	−	0.926904i	$-0.377540\pi$
$67$	411.641	0.750596	0.375298	+	0.926904i	$0.377540\pi$
$68$	−617.913	−1.10195
$69$	0	0
$70$	159.321	0.272035
$71$	470.636	0.786679	0.393339	−	0.919393i	$-0.371320\pi$
$71$	470.636	0.786679	0.393339	+	0.919393i	$0.371320\pi$
$72$	0	0
$73$	610.600	0.978977	0.489488	−	0.872010i	$-0.337184\pi$
$73$	610.600	0.978977	0.489488	+	0.872010i	$0.337184\pi$
$74$	29.9090	0.0469844
$75$	0	0
$76$	506.985	0.765199
$77$	186.210	0.275593
$78$	0	0
$79$	−978.225	−1.39315	−0.696576	−	0.717483i	$-0.745294\pi$
$79$	−978.225	−1.39315	−0.696576	+	0.717483i	$0.745294\pi$
$80$	661.149	0.923983
$81$	0	0
$82$	−201.261	−0.271044
$83$	−26.1539	−0.0345875	−0.0172938	−	0.999850i	$-0.505505\pi$
$83$	−26.1539	−0.0345875	−0.0172938	+	0.999850i	$0.505505\pi$
$84$	0	0
$85$	1064.31	1.35813
$86$	−1.67434	−0.00209940
$87$	0	0
$88$	−124.526	−0.150846
$89$	352.887	0.420292	0.210146	−	0.977670i	$-0.432606\pi$
$89$	352.887	0.420292	0.210146	+	0.977670i	$0.432606\pi$
$90$	0	0
$91$	1263.54	1.45555
$92$	99.6743	0.112954
$93$	0	0
$94$	−52.5950	−0.0577102
$95$	−873.246	−0.943086
$96$	0	0
$97$	847.585	0.887208	0.443604	−	0.896223i	$-0.353700\pi$
$97$	847.585	0.887208	0.443604	+	0.896223i	$0.353700\pi$
$98$	−41.3140	−0.0425851
$99$	0	0
$100$	−300.708	−0.300708
$101$	−1293.46	−1.27430	−0.637150	−	0.770740i	$-0.719887\pi$
$101$	−1293.46	−1.27430	−0.637150	+	0.770740i	$0.719887\pi$
$102$	0	0
$103$	−1725.24	−1.65042	−0.825209	−	0.564828i	$-0.808943\pi$
$103$	−1725.24	−1.65042	−0.825209	+	0.564828i	$0.808943\pi$
$104$	−844.974	−0.796697
$105$	0	0
$106$	109.079	0.0999502
$107$	484.179	0.437452	0.218726	−	0.975786i	$-0.429810\pi$
$107$	484.179	0.437452	0.218726	+	0.975786i	$0.429810\pi$
$108$	0	0
$109$	−64.2563	−0.0564645	−0.0282323	−	0.999601i	$-0.508988\pi$
$109$	−64.2563	−0.0564645	−0.0282323	+	0.999601i	$0.508988\pi$
$110$	103.527	0.0897355
$111$	0	0
$112$	870.543	0.734452
$113$	2005.08	1.66922	0.834612	−	0.550839i	$-0.185692\pi$
$113$	2005.08	1.66922	0.834612	+	0.550839i	$0.185692\pi$
$114$	0	0
$115$	−171.682	−0.139213
$116$	1261.39	1.00963
$117$	0	0
$118$	−399.429	−0.311614
$119$	1401.39	1.07954
$120$	0	0
$121$	121.000	0.0909091
$122$	74.1587	0.0550329
$123$	0	0
$124$	488.879	0.354054
$125$	−1089.10	−0.779298
$126$	0	0
$127$	109.605	0.0765816	0.0382908	−	0.999267i	$-0.487809\pi$
$127$	109.605	0.0765816	0.0382908	+	0.999267i	$0.487809\pi$
$128$	−1258.14	−0.868791
$129$	0	0
$130$	702.487	0.473940
$131$	−1156.71	−0.771469	−0.385734	−	0.922610i	$-0.626052\pi$
$131$	−1156.71	−0.771469	−0.385734	+	0.922610i	$0.626052\pi$
$132$	0	0
$133$	−1149.82	−0.749636
$134$	301.342	0.194269
$135$	0	0
$136$	−937.164	−0.590891
$137$	−198.323	−0.123678	−0.0618391	−	0.998086i	$-0.519697\pi$
$137$	−198.323	−0.123678	−0.0618391	+	0.998086i	$0.519697\pi$
$138$	0	0
$139$	−2900.14	−1.76969	−0.884844	−	0.465888i	$-0.845735\pi$
$139$	−2900.14	−1.76969	−0.884844	+	0.465888i	$0.845735\pi$
$140$	−1624.46	−0.980654
$141$	0	0
$142$	344.529	0.203607
$143$	821.051	0.480138
$144$	0	0
$145$	−2172.67	−1.24435
$146$	446.990	0.253378
$147$	0	0
$148$	−304.956	−0.169373
$149$	−3488.34	−1.91796	−0.958980	−	0.283472i	$-0.908514\pi$
$149$	−3488.34	−1.91796	−0.958980	+	0.283472i	$0.908514\pi$
$150$	0	0
$151$	−1163.32	−0.626953	−0.313477	−	0.949596i	$-0.601494\pi$
$151$	−1163.32	−0.626953	−0.313477	+	0.949596i	$0.601494\pi$
$152$	768.923	0.410315
$153$	0	0
$154$	136.315	0.0713286
$155$	−842.061	−0.436361
$156$	0	0
$157$	342.057	0.173880	0.0869398	−	0.996214i	$-0.472291\pi$
$157$	342.057	0.173880	0.0869398	+	0.996214i	$0.472291\pi$
$158$	−716.111	−0.360574
$159$	0	0
$160$	1648.32	0.814446
$161$	−226.056	−0.110657
$162$	0	0
$163$	−1394.89	−0.670285	−0.335142	−	0.942167i	$-0.608784\pi$
$163$	−1394.89	−0.670285	−0.335142	+	0.942167i	$0.608784\pi$
$164$	2052.09	0.977082
$165$	0	0
$166$	−19.1460	−0.00895191
$167$	−478.703	−0.221815	−0.110908	−	0.993831i	$-0.535376\pi$
$167$	−478.703	−0.221815	−0.110908	+	0.993831i	$0.535376\pi$
$168$	0	0
$169$	3374.28	1.53586
$170$	779.131	0.351509
$171$	0	0
$172$	17.0718	0.00756809
$173$	−1808.58	−0.794822	−0.397411	−	0.917641i	$-0.630091\pi$
$173$	−1808.58	−0.794822	−0.397411	+	0.917641i	$0.630091\pi$
$174$	0	0
$175$	681.990	0.294592
$176$	565.682	0.242272
$177$	0	0
$178$	258.331	0.108780
$179$	4429.85	1.84973	0.924867	−	0.380292i	$-0.124176\pi$
$179$	4429.85	1.84973	0.924867	+	0.380292i	$0.124176\pi$
$180$	0	0
$181$	3409.17	1.40001	0.700005	−	0.714138i	$-0.253181\pi$
$181$	3409.17	1.40001	0.700005	+	0.714138i	$0.253181\pi$
$182$	924.974	0.376723
$183$	0	0
$184$	151.172	0.0605682
$185$	525.267	0.208748
$186$	0	0
$187$	910.631	0.356106
$188$	536.267	0.208039
$189$	0	0
$190$	−639.261	−0.244089
$191$	−2923.75	−1.10762	−0.553810	−	0.832643i	$-0.686827\pi$
$191$	−2923.75	−1.10762	−0.553810	+	0.832643i	$0.686827\pi$
$192$	0	0
$193$	−2484.18	−0.926505	−0.463253	−	0.886226i	$-0.653318\pi$
$193$	−2484.18	−0.926505	−0.463253	+	0.886226i	$0.653318\pi$
$194$	620.475	0.229626
$195$	0	0
$196$	421.244	0.153514
$197$	5125.67	1.85375	0.926876	−	0.375369i	$-0.122484\pi$
$197$	5125.67	1.85375	0.926876	+	0.375369i	$0.122484\pi$
$198$	0	0
$199$	−7.69219	−0.00274013	−0.00137006	−	0.999999i	$-0.500436\pi$
$199$	−7.69219	−0.00274013	−0.00137006	+	0.999999i	$0.500436\pi$
$200$	−456.071	−0.161246
$201$	0	0
$202$	−946.879	−0.329813
$203$	−2860.78	−0.989100
$204$	0	0
$205$	−3534.59	−1.20423
$206$	−1262.96	−0.427160
$207$	0	0
$208$	3838.46	1.27956
$209$	−747.154	−0.247281
$210$	0	0
$211$	3107.34	1.01383	0.506915	−	0.861996i	$-0.330786\pi$
$211$	3107.34	1.01383	0.506915	+	0.861996i	$0.330786\pi$
$212$	−1112.19	−0.360309
$213$	0	0
$214$	354.444	0.113221
$215$	−29.4050	−0.00932746
$216$	0	0
$217$	−1108.75	−0.346853
$218$	−47.0388	−0.0146141
$219$	0	0
$220$	−1055.58	−0.323486
$221$	6179.13	1.88078
$222$	0	0
$223$	−12.3185	−0.00369913	−0.00184957	−	0.999998i	$-0.500589\pi$
$223$	−12.3185	−0.00369913	−0.00184957	+	0.999998i	$0.500589\pi$
$224$	2170.37	0.647383
$225$	0	0
$226$	1467.82	0.432027
$227$	−4615.90	−1.34964	−0.674820	−	0.737983i	$-0.735779\pi$
$227$	−4615.90	−1.34964	−0.674820	+	0.737983i	$0.735779\pi$
$228$	0	0
$229$	5074.63	1.46437	0.732186	−	0.681105i	$-0.238500\pi$
$229$	5074.63	1.46437	0.732186	+	0.681105i	$0.238500\pi$
$230$	−125.680	−0.0360309
$231$	0	0
$232$	1913.11	0.541386
$233$	−211.683	−0.0595184	−0.0297592	−	0.999557i	$-0.509474\pi$
$233$	−211.683	−0.0595184	−0.0297592	+	0.999557i	$0.509474\pi$
$234$	0	0
$235$	−923.683	−0.256402
$236$	4072.64	1.12333
$237$	0	0
$238$	1025.89	0.279406
$239$	−4312.49	−1.16716	−0.583581	−	0.812055i	$-0.698349\pi$
$239$	−4312.49	−1.16716	−0.583581	+	0.812055i	$0.698349\pi$
$240$	0	0
$241$	−996.584	−0.266372	−0.133186	−	0.991091i	$-0.542521\pi$
$241$	−996.584	−0.266372	−0.133186	+	0.991091i	$0.542521\pi$
$242$	88.5781	0.0235290
$243$	0	0
$244$	−756.133	−0.198387
$245$	−725.563	−0.189202
$246$	0	0
$247$	−5069.85	−1.30602
$248$	741.464	0.189851
$249$	0	0
$250$	−797.278	−0.201697
$251$	276.892	0.0696306	0.0348153	−	0.999394i	$-0.488916\pi$
$251$	276.892	0.0696306	0.0348153	+	0.999394i	$0.488916\pi$
$252$	0	0
$253$	−146.892	−0.0365021
$254$	80.2364	0.0198208
$255$	0	0
$256$	1619.36	0.395352
$257$	3235.18	0.785233	0.392617	−	0.919702i	$-0.371570\pi$
$257$	3235.18	0.785233	0.392617	+	0.919702i	$0.371570\pi$
$258$	0	0
$259$	691.626	0.165929
$260$	−7162.67	−1.70850
$261$	0	0
$262$	−846.772	−0.199671
$263$	−207.944	−0.0487544	−0.0243772	−	0.999703i	$-0.507760\pi$
$263$	−207.944	−0.0487544	−0.0243772	+	0.999703i	$0.507760\pi$
$264$	0	0
$265$	1915.67	0.444071
$266$	−841.723	−0.194020
$267$	0	0
$268$	−3072.53	−0.700316
$269$	−5033.04	−1.14078	−0.570390	−	0.821374i	$-0.693208\pi$
$269$	−5033.04	−1.14078	−0.570390	+	0.821374i	$0.693208\pi$
$270$	0	0
$271$	1487.01	0.333319	0.166660	−	0.986015i	$-0.446702\pi$
$271$	1487.01	0.333319	0.166660	+	0.986015i	$0.446702\pi$
$272$	4257.25	0.949021
$273$	0	0
$274$	−145.183	−0.0320102
$275$	443.159	0.0971764
$276$	0	0
$277$	−235.836	−0.0511552	−0.0255776	−	0.999673i	$-0.508142\pi$
$277$	−235.836	−0.0511552	−0.0255776	+	0.999673i	$0.508142\pi$
$278$	−2123.05	−0.458029
$279$	0	0
$280$	−2463.75	−0.525847
$281$	4915.01	1.04343	0.521717	−	0.853118i	$-0.325292\pi$
$281$	4915.01	1.04343	0.521717	+	0.853118i	$0.325292\pi$
$282$	0	0
$283$	−5199.56	−1.09216	−0.546081	−	0.837733i	$-0.683881\pi$
$283$	−5199.56	−1.09216	−0.546081	+	0.837733i	$0.683881\pi$
$284$	−3512.87	−0.733981
$285$	0	0
$286$	601.051	0.124269
$287$	−4654.04	−0.957210
$288$	0	0
$289$	1940.29	0.394930
$290$	−1590.50	−0.322060
$291$	0	0
$292$	−4557.58	−0.913398
$293$	8880.92	1.77075	0.885373	−	0.464881i	$-0.153903\pi$
$293$	8880.92	1.77075	0.885373	+	0.464881i	$0.153903\pi$
$294$	0	0
$295$	−7014.85	−1.38448
$296$	−462.515	−0.0908215
$297$	0	0
$298$	−2553.64	−0.496405
$299$	−996.743	−0.192786
$300$	0	0
$301$	−38.7180	−0.00741417
$302$	−851.612	−0.162267
$303$	0	0
$304$	−3492.99	−0.659001
$305$	1302.39	0.244507
$306$	0	0
$307$	−1497.93	−0.278474	−0.139237	−	0.990259i	$-0.544465\pi$
$307$	−1497.93	−0.278474	−0.139237	+	0.990259i	$0.544465\pi$
$308$	−1389.89	−0.257131
$309$	0	0
$310$	−616.432	−0.112939
$311$	7484.71	1.36469	0.682345	−	0.731030i	$-0.260960\pi$
$311$	7484.71	1.36469	0.682345	+	0.731030i	$0.260960\pi$
$312$	0	0
$313$	−658.363	−0.118891	−0.0594455	−	0.998232i	$-0.518933\pi$
$313$	−658.363	−0.118891	−0.0594455	+	0.998232i	$0.518933\pi$
$314$	250.403	0.0450034
$315$	0	0
$316$	7301.57	1.29983
$317$	−233.708	−0.0414080	−0.0207040	−	0.999786i	$-0.506591\pi$
$317$	−233.708	−0.0414080	−0.0207040	+	0.999786i	$0.506591\pi$
$318$	0	0
$319$	−1858.94	−0.326272
$320$	−4082.53	−0.713189
$321$	0	0
$322$	−165.485	−0.0286401
$323$	−5622.98	−0.968641
$324$	0	0
$325$	3007.08	0.513239
$326$	−1021.13	−0.173482
$327$	0	0
$328$	3112.33	0.523931
$329$	−1216.23	−0.203808
$330$	0	0
$331$	8532.95	1.41696	0.708480	−	0.705731i	$-0.249381\pi$
$331$	8532.95	1.41696	0.708480	+	0.705731i	$0.249381\pi$
$332$	195.215	0.0322706
$333$	0	0
$334$	−350.435	−0.0574100
$335$	5292.22	0.863120
$336$	0	0
$337$	11691.2	1.88979	0.944895	−	0.327373i	$-0.106163\pi$
$337$	11691.2	1.88979	0.944895	+	0.327373i	$0.106163\pi$
$338$	2470.15	0.397509
$339$	0	0
$340$	−7944.14	−1.26715
$341$	−720.472	−0.114416
$342$	0	0
$343$	−6761.73	−1.06443
$344$	25.8921	0.00405817
$345$	0	0
$346$	−1323.98	−0.205715
$347$	−4598.79	−0.711459	−0.355729	−	0.934589i	$-0.615768\pi$
$347$	−4598.79	−0.711459	−0.355729	+	0.934589i	$0.615768\pi$
$348$	0	0
$349$	6720.27	1.03074	0.515369	−	0.856968i	$-0.327655\pi$
$349$	6720.27	1.03074	0.515369	+	0.856968i	$0.327655\pi$
$350$	499.251	0.0762460
$351$	0	0
$352$	1410.31	0.213551
$353$	−5738.70	−0.865270	−0.432635	−	0.901569i	$-0.642416\pi$
$353$	−5738.70	−0.865270	−0.432635	+	0.901569i	$0.642416\pi$
$354$	0	0
$355$	6050.69	0.904611
$356$	−2633.99	−0.392138
$357$	0	0
$358$	3242.87	0.478746
$359$	4115.27	0.605001	0.302501	−	0.953149i	$-0.402179\pi$
$359$	4115.27	0.605001	0.302501	+	0.953149i	$0.402179\pi$
$360$	0	0
$361$	−2245.46	−0.327374
$362$	2495.69	0.362349
$363$	0	0
$364$	−9431.18	−1.35804
$365$	7850.12	1.12574
$366$	0	0
$367$	9662.99	1.37440	0.687199	−	0.726469i	$-0.258840\pi$
$367$	9662.99	1.37440	0.687199	+	0.726469i	$0.258840\pi$
$368$	−686.729	−0.0972778
$369$	0	0
$370$	384.522	0.0540279
$371$	2522.39	0.352981
$372$	0	0
$373$	−141.780	−0.0196812	−0.00984062	−	0.999952i	$-0.503132\pi$
$373$	−141.780	−0.0196812	−0.00984062	+	0.999952i	$0.503132\pi$
$374$	666.628	0.0921671
$375$	0	0
$376$	813.334	0.111555
$377$	−12613.9	−1.72321
$378$	0	0
$379$	−2819.73	−0.382163	−0.191082	−	0.981574i	$-0.561200\pi$
$379$	−2819.73	−0.382163	−0.191082	+	0.981574i	$0.561200\pi$
$380$	6518.00	0.879911
$381$	0	0
$382$	−2140.34	−0.286673
$383$	6337.84	0.845557	0.422778	−	0.906233i	$-0.361055\pi$
$383$	6337.84	0.845557	0.422778	+	0.906233i	$0.361055\pi$
$384$	0	0
$385$	2393.99	0.316907
$386$	−1818.55	−0.239797
$387$	0	0
$388$	−6326.46	−0.827776
$389$	8805.25	1.14767	0.573836	−	0.818970i	$-0.305455\pi$
$389$	8805.25	1.14767	0.573836	+	0.818970i	$0.305455\pi$
$390$	0	0
$391$	−1105.49	−0.142985
$392$	638.883	0.0823176
$393$	0	0
$394$	3752.25	0.479786
$395$	−12576.5	−1.60200
$396$	0	0
$397$	4315.26	0.545534	0.272767	−	0.962080i	$-0.412061\pi$
$397$	4315.26	0.545534	0.272767	+	0.962080i	$0.412061\pi$
$398$	−5.63108	−0.000709197 0
$399$	0	0
$400$	2071.79	0.258974
$401$	−361.681	−0.0450411	−0.0225206	−	0.999746i	$-0.507169\pi$
$401$	−361.681	−0.0450411	−0.0225206	+	0.999746i	$0.507169\pi$
$402$	0	0
$403$	−4888.79	−0.604288
$404$	9654.53	1.18894
$405$	0	0
$406$	−2094.24	−0.255998
$407$	449.420	0.0547345
$408$	0	0
$409$	9220.50	1.11473	0.557365	−	0.830268i	$-0.311812\pi$
$409$	9220.50	1.11473	0.557365	+	0.830268i	$0.311812\pi$
$410$	−2587.50	−0.311677
$411$	0	0
$412$	12877.4	1.53986
$413$	−9236.55	−1.10049
$414$	0	0
$415$	−336.245	−0.0397726
$416$	9569.74	1.12787
$417$	0	0
$418$	−546.954	−0.0640010
$419$	14912.9	1.73876	0.869380	−	0.494144i	$-0.164519\pi$
$419$	14912.9	1.73876	0.869380	+	0.494144i	$0.164519\pi$
$420$	0	0
$421$	−13486.0	−1.56121	−0.780603	−	0.625027i	$-0.785088\pi$
$421$	−13486.0	−1.56121	−0.780603	+	0.625027i	$0.785088\pi$
$422$	2274.73	0.262399
$423$	0	0
$424$	−1686.81	−0.193205
$425$	3335.16	0.380656
$426$	0	0
$427$	1714.87	0.194352
$428$	−3613.96	−0.408148
$429$	0	0
$430$	−21.5260	−0.00241413
$431$	−406.334	−0.0454116	−0.0227058	−	0.999742i	$-0.507228\pi$
$431$	−406.334	−0.0454116	−0.0227058	+	0.999742i	$0.507228\pi$
$432$	0	0
$433$	−1766.69	−0.196078	−0.0980391	−	0.995183i	$-0.531257\pi$
$433$	−1766.69	−0.196078	−0.0980391	+	0.995183i	$0.531257\pi$
$434$	−811.664	−0.0897722
$435$	0	0
$436$	479.615	0.0526821
$437$	907.033	0.0992889
$438$	0	0
$439$	7824.19	0.850634	0.425317	−	0.905044i	$-0.360163\pi$
$439$	7824.19	0.850634	0.425317	+	0.905044i	$0.360163\pi$
$440$	−1600.95	−0.173460
$441$	0	0
$442$	4523.44	0.486783
$443$	−11667.9	−1.25137	−0.625686	−	0.780075i	$-0.715181\pi$
$443$	−11667.9	−1.25137	−0.625686	+	0.780075i	$0.715181\pi$
$444$	0	0
$445$	4536.86	0.483299
$446$	−9.01776	−0.000957406 0
$447$	0	0
$448$	−5375.53	−0.566897
$449$	−16975.3	−1.78421	−0.892107	−	0.451825i	$-0.850773\pi$
$449$	−16975.3	−1.78421	−0.892107	+	0.451825i	$0.850773\pi$
$450$	0	0
$451$	−3024.21	−0.315753
$452$	−14966.1	−1.55741
$453$	0	0
$454$	−3379.07	−0.349312
$455$	16244.6	1.67375
$456$	0	0
$457$	−16192.9	−1.65748	−0.828741	−	0.559632i	$-0.810943\pi$
$457$	−16192.9	−1.65748	−0.828741	+	0.559632i	$0.810943\pi$
$458$	3714.89	0.379007
$459$	0	0
$460$	1281.45	0.129887
$461$	−8586.04	−0.867444	−0.433722	−	0.901047i	$-0.642800\pi$
$461$	−8586.04	−0.867444	−0.433722	+	0.901047i	$0.642800\pi$
$462$	0	0
$463$	−7917.20	−0.794694	−0.397347	−	0.917668i	$-0.630069\pi$
$463$	−7917.20	−0.794694	−0.397347	+	0.917668i	$0.630069\pi$
$464$	−8690.67	−0.869513
$465$	0	0
$466$	−154.962	−0.0154045
$467$	15155.0	1.50169	0.750844	−	0.660480i	$-0.229647\pi$
$467$	15155.0	1.50169	0.750844	+	0.660480i	$0.229647\pi$
$468$	0	0
$469$	6968.34	0.686073
$470$	−676.183	−0.0663617
$471$	0	0
$472$	6176.82	0.602354
$473$	−25.1591	−0.00244570
$474$	0	0
$475$	−2736.43	−0.264328
$476$	−10460.2	−1.00723
$477$	0	0
$478$	−3156.96	−0.302084
$479$	−10001.1	−0.953993	−0.476996	−	0.878905i	$-0.658275\pi$
$479$	−10001.1	−0.953993	−0.476996	+	0.878905i	$0.658275\pi$
$480$	0	0
$481$	3049.56	0.289081
$482$	−729.550	−0.0689421
$483$	0	0
$484$	−903.156	−0.0848193
$485$	10896.9	1.02021
$486$	0	0
$487$	7044.54	0.655480	0.327740	−	0.944768i	$-0.393713\pi$
$487$	7044.54	0.655480	0.327740	+	0.944768i	$0.393713\pi$
$488$	−1146.80	−0.106379
$489$	0	0
$490$	−531.149	−0.0489691
$491$	13326.4	1.22487	0.612437	−	0.790520i	$-0.290189\pi$
$491$	13326.4	1.22487	0.612437	+	0.790520i	$0.290189\pi$
$492$	0	0
$493$	−13990.2	−1.27806
$494$	−3711.38	−0.338022
$495$	0	0
$496$	−3368.25	−0.304917
$497$	7967.02	0.719054
$498$	0	0
$499$	−20069.1	−1.80044	−0.900218	−	0.435440i	$-0.856593\pi$
$499$	−20069.1	−1.80044	−0.900218	+	0.435440i	$0.856593\pi$
$500$	8129.17	0.727095
$501$	0	0
$502$	202.699	0.0180217
$503$	−7782.35	−0.689856	−0.344928	−	0.938629i	$-0.612097\pi$
$503$	−7782.35	−0.689856	−0.344928	+	0.938629i	$0.612097\pi$
$504$	0	0
$505$	−16629.3	−1.46533
$506$	−107.532	−0.00944744
$507$	0	0
$508$	−818.102	−0.0714516
$509$	1475.93	0.128526	0.0642628	−	0.997933i	$-0.479530\pi$
$509$	1475.93	0.128526	0.0642628	+	0.997933i	$0.479530\pi$
$510$	0	0
$511$	10336.4	0.894821
$512$	11250.6	0.971116
$513$	0	0
$514$	2368.32	0.203233
$515$	−22180.4	−1.89784
$516$	0	0
$517$	−790.307	−0.0672295
$518$	506.305	0.0429455
$519$	0	0
$520$	−10863.3	−0.916132
$521$	−7609.43	−0.639875	−0.319938	−	0.947439i	$-0.603662\pi$
$521$	−7609.43	−0.639875	−0.319938	+	0.947439i	$0.603662\pi$
$522$	0	0
$523$	12452.9	1.04116	0.520581	−	0.853812i	$-0.325715\pi$
$523$	12452.9	1.04116	0.520581	+	0.853812i	$0.325715\pi$
$524$	8633.82	0.719790
$525$	0	0
$526$	−152.226	−0.0126186
$527$	−5422.18	−0.448186
$528$	0	0
$529$	−11988.7	−0.985344
$530$	1402.37	0.114934
$531$	0	0
$532$	8582.34	0.699420
$533$	−20520.9	−1.66765
$534$	0	0
$535$	6224.81	0.503031
$536$	−4659.99	−0.375524
$537$	0	0
$538$	−3684.44	−0.295255
$539$	−620.795	−0.0496095
$540$	0	0
$541$	9312.17	0.740039	0.370020	−	0.929024i	$-0.379351\pi$
$541$	9312.17	0.740039	0.370020	+	0.929024i	$0.379351\pi$
$542$	1088.57	0.0862693
$543$	0	0
$544$	10613.8	0.836515
$545$	−826.105	−0.0649292
$546$	0	0
$547$	−11018.6	−0.861278	−0.430639	−	0.902524i	$-0.641712\pi$
$547$	−11018.6	−0.861278	−0.430639	+	0.902524i	$0.641712\pi$
$548$	1480.31	0.115393
$549$	0	0
$550$	324.415	0.0251511
$551$	11478.6	0.887490
$552$	0	0
$553$	−16559.6	−1.27339
$554$	−172.644	−0.0132399
$555$	0	0
$556$	21646.9	1.65114
$557$	12018.4	0.914250	0.457125	−	0.889403i	$-0.348879\pi$
$557$	12018.4	0.914250	0.457125	+	0.889403i	$0.348879\pi$
$558$	0	0
$559$	−170.718	−0.0129170
$560$	11192.1	0.844555
$561$	0	0
$562$	3598.04	0.270061
$563$	8763.89	0.656046	0.328023	−	0.944670i	$-0.393618\pi$
$563$	8763.89	0.656046	0.328023	+	0.944670i	$0.393618\pi$
$564$	0	0
$565$	25778.1	1.91946
$566$	−3806.34	−0.282672
$567$	0	0
$568$	−5327.84	−0.393576
$569$	10273.2	0.756895	0.378447	−	0.925623i	$-0.376458\pi$
$569$	10273.2	0.756895	0.378447	+	0.925623i	$0.376458\pi$
$570$	0	0
$571$	2602.62	0.190747	0.0953734	−	0.995442i	$-0.469596\pi$
$571$	2602.62	0.190747	0.0953734	+	0.995442i	$0.469596\pi$
$572$	−6128.41	−0.447975
$573$	0	0
$574$	−3406.99	−0.247744
$575$	−537.988	−0.0390185
$576$	0	0
$577$	−19727.0	−1.42331	−0.711653	−	0.702532i	$-0.752053\pi$
$577$	−19727.0	−1.42331	−0.711653	+	0.702532i	$0.752053\pi$
$578$	1420.39	0.102215
$579$	0	0
$580$	16217.0	1.16099
$581$	−442.739	−0.0316143
$582$	0	0
$583$	1639.06	0.116437
$584$	−6912.30	−0.489783
$585$	0	0
$586$	6501.28	0.458303
$587$	10116.2	0.711309	0.355654	−	0.934618i	$-0.384258\pi$
$587$	10116.2	0.711309	0.355654	+	0.934618i	$0.384258\pi$
$588$	0	0
$589$	4448.78	0.311221
$590$	−5135.23	−0.358329
$591$	0	0
$592$	2101.07	0.145867
$593$	−3130.32	−0.216774	−0.108387	−	0.994109i	$-0.534569\pi$
$593$	−3130.32	−0.216774	−0.108387	+	0.994109i	$0.534569\pi$
$594$	0	0
$595$	18016.9	1.24138
$596$	26037.3	1.78948
$597$	0	0
$598$	−729.667	−0.0498968
$599$	−10080.1	−0.687581	−0.343790	−	0.939046i	$-0.611711\pi$
$599$	−10080.1	−0.687581	−0.343790	+	0.939046i	$0.611711\pi$
$600$	0	0
$601$	4777.02	0.324224	0.162112	−	0.986772i	$-0.448169\pi$
$601$	4777.02	0.324224	0.162112	+	0.986772i	$0.448169\pi$
$602$	−28.3435	−0.00191893
$603$	0	0
$604$	8683.16	0.584955
$605$	1555.63	0.104537
$606$	0	0
$607$	−2571.35	−0.171941	−0.0859703	−	0.996298i	$-0.527399\pi$
$607$	−2571.35	−0.171941	−0.0859703	+	0.996298i	$0.527399\pi$
$608$	−8708.43	−0.580877
$609$	0	0
$610$	953.414	0.0632830
$611$	−5362.67	−0.355074
$612$	0	0
$613$	12711.9	0.837564	0.418782	−	0.908087i	$-0.362457\pi$
$613$	12711.9	0.837564	0.418782	+	0.908087i	$0.362457\pi$
$614$	−1096.56	−0.0720744
$615$	0	0
$616$	−2107.99	−0.137879
$617$	−16236.1	−1.05939	−0.529693	−	0.848189i	$-0.677693\pi$
$617$	−16236.1	−1.05939	−0.529693	+	0.848189i	$0.677693\pi$
$618$	0	0
$619$	12657.3	0.821874	0.410937	−	0.911664i	$-0.365202\pi$
$619$	12657.3	0.821874	0.410937	+	0.911664i	$0.365202\pi$
$620$	6285.23	0.407131
$621$	0	0
$622$	5479.19	0.353208
$623$	5973.75	0.384162
$624$	0	0
$625$	−19037.8	−1.21842
$626$	−481.955	−0.0307713
$627$	0	0
$628$	−2553.15	−0.162232
$629$	3382.28	0.214404
$630$	0	0
$631$	−3949.97	−0.249201	−0.124600	−	0.992207i	$-0.539765\pi$
$631$	−3949.97	−0.249201	−0.124600	+	0.992207i	$0.539765\pi$
$632$	11074.0	0.696994
$633$	0	0
$634$	−171.086	−0.0107172
$635$	1409.13	0.0880621
$636$	0	0
$637$	−4212.44	−0.262014
$638$	−1360.84	−0.0844455
$639$	0	0
$640$	−16175.2	−0.999033
$641$	7398.27	0.455872	0.227936	−	0.973676i	$-0.426802\pi$
$641$	7398.27	0.455872	0.227936	+	0.973676i	$0.426802\pi$
$642$	0	0
$643$	−12491.7	−0.766134	−0.383067	−	0.923721i	$-0.625132\pi$
$643$	−12491.7	−0.766134	−0.383067	+	0.923721i	$0.625132\pi$
$644$	1687.31	0.103244
$645$	0	0
$646$	−4116.31	−0.250703
$647$	10472.0	0.636315	0.318158	−	0.948038i	$-0.396936\pi$
$647$	10472.0	0.636315	0.318158	+	0.948038i	$0.396936\pi$
$648$	0	0
$649$	−6001.94	−0.363015
$650$	2201.33	0.132836
$651$	0	0
$652$	10411.6	0.625384
$653$	−6337.94	−0.379820	−0.189910	−	0.981801i	$-0.560820\pi$
$653$	−6337.94	−0.379820	−0.189910	+	0.981801i	$0.560820\pi$
$654$	0	0
$655$	−14871.2	−0.887121
$656$	−14138.4	−0.841479
$657$	0	0
$658$	−890.339	−0.0527493
$659$	−15196.7	−0.898302	−0.449151	−	0.893456i	$-0.648274\pi$
$659$	−15196.7	−0.898302	−0.449151	+	0.893456i	$0.648274\pi$
$660$	0	0
$661$	2298.17	0.135232	0.0676161	−	0.997711i	$-0.478461\pi$
$661$	2298.17	0.135232	0.0676161	+	0.997711i	$0.478461\pi$
$662$	6246.55	0.366736
$663$	0	0
$664$	296.075	0.0173042
$665$	−14782.5	−0.862016
$666$	0	0
$667$	2256.73	0.131006
$668$	3573.09	0.206956
$669$	0	0
$670$	3874.18	0.223392
$671$	1114.33	0.0641106
$672$	0	0
$673$	23199.6	1.32880	0.664398	−	0.747379i	$-0.268688\pi$
$673$	23199.6	1.32880	0.664398	+	0.747379i	$0.268688\pi$
$674$	8558.54	0.489114
$675$	0	0
$676$	−25186.0	−1.43298
$677$	2145.38	0.121793	0.0608963	−	0.998144i	$-0.480604\pi$
$677$	2145.38	0.121793	0.0608963	+	0.998144i	$0.480604\pi$
$678$	0	0
$679$	14348.1	0.810941
$680$	−12048.6	−0.679472
$681$	0	0
$682$	−527.422	−0.0296129
$683$	29544.6	1.65519	0.827593	−	0.561329i	$-0.189710\pi$
$683$	29544.6	1.65519	0.827593	+	0.561329i	$0.189710\pi$
$684$	0	0
$685$	−2549.72	−0.142219
$686$	−4949.93	−0.275495
$687$	0	0
$688$	−117.620	−0.00651776
$689$	11121.9	0.614964
$690$	0	0
$691$	27803.1	1.53065	0.765325	−	0.643644i	$-0.222578\pi$
$691$	27803.1	1.53065	0.765325	+	0.643644i	$0.222578\pi$
$692$	13499.5	0.741579
$693$	0	0
$694$	−3366.55	−0.184139
$695$	−37285.4	−2.03498
$696$	0	0
$697$	−22759.8	−1.23686
$698$	4919.58	0.266775
$699$	0	0
$700$	−5090.44	−0.274858
$701$	19697.8	1.06130	0.530652	−	0.847590i	$-0.321947\pi$
$701$	19697.8	1.06130	0.530652	+	0.847590i	$0.321947\pi$
$702$	0	0
$703$	−2775.09	−0.148883
$704$	−3493.03	−0.187001
$705$	0	0
$706$	−4201.02	−0.223948
$707$	−21896.0	−1.16476
$708$	0	0
$709$	19122.5	1.01292	0.506460	−	0.862263i	$-0.330954\pi$
$709$	19122.5	1.01292	0.506460	+	0.862263i	$0.330954\pi$
$710$	4429.41	0.234131
$711$	0	0
$712$	−3994.86	−0.210272
$713$	874.641	0.0459405
$714$	0	0
$715$	10555.8	0.552117
$716$	−33064.8	−1.72582
$717$	0	0
$718$	3012.59	0.156586
$719$	−1837.44	−0.0953060	−0.0476530	−	0.998864i	$-0.515174\pi$
$719$	−1837.44	−0.0953060	−0.0476530	+	0.998864i	$0.515174\pi$
$720$	0	0
$721$	−29205.2	−1.50854
$722$	−1643.79	−0.0847307
$723$	0	0
$724$	−25446.4	−1.30623
$725$	−6808.33	−0.348765
$726$	0	0
$727$	−7555.46	−0.385442	−0.192721	−	0.981254i	$-0.561731\pi$
$727$	−7555.46	−0.385442	−0.192721	+	0.981254i	$0.561731\pi$
$728$	−14303.9	−0.728211
$729$	0	0
$730$	5746.69	0.291362
$731$	−189.344	−0.00958021
$732$	0	0
$733$	11984.6	0.603905	0.301952	−	0.953323i	$-0.402362\pi$
$733$	11984.6	0.603905	0.301952	+	0.953323i	$0.402362\pi$
$734$	7073.80	0.355720
$735$	0	0
$736$	−1712.10	−0.0857456
$737$	4528.05	0.226313
$738$	0	0
$739$	−27142.5	−1.35109	−0.675543	−	0.737321i	$-0.736091\pi$
$739$	−27142.5	−1.35109	−0.675543	+	0.737321i	$0.736091\pi$
$740$	−3920.64	−0.194764
$741$	0	0
$742$	1846.52	0.0913582
$743$	29222.6	1.44290	0.721450	−	0.692467i	$-0.243476\pi$
$743$	29222.6	1.44290	0.721450	+	0.692467i	$0.243476\pi$
$744$	0	0
$745$	−44847.6	−2.20549
$746$	−103.790	−0.00509388
$747$	0	0
$748$	−6797.04	−0.332252
$749$	8196.29	0.399847
$750$	0	0
$751$	−8859.39	−0.430471	−0.215236	−	0.976562i	$-0.569052\pi$
$751$	−8859.39	−0.430471	−0.215236	+	0.976562i	$0.569052\pi$
$752$	−3694.73	−0.179166
$753$	0	0
$754$	−9234.05	−0.446000
$755$	−14956.2	−0.720941
$756$	0	0
$757$	35734.4	1.71571	0.857853	−	0.513896i	$-0.171798\pi$
$757$	35734.4	1.71571	0.857853	+	0.513896i	$0.171798\pi$
$758$	−2064.19	−0.0989112
$759$	0	0
$760$	9885.59	0.471826
$761$	−34394.7	−1.63838	−0.819189	−	0.573524i	$-0.805576\pi$
$761$	−34394.7	−1.63838	−0.819189	+	0.573524i	$0.805576\pi$
$762$	0	0
$763$	−1087.74	−0.0516107
$764$	21823.2	1.03342
$765$	0	0
$766$	4639.62	0.218846
$767$	−40726.4	−1.91727
$768$	0	0
$769$	−11602.7	−0.544091	−0.272045	−	0.962284i	$-0.587700\pi$
$769$	−11602.7	−0.544091	−0.272045	+	0.962284i	$0.587700\pi$
$770$	1752.53	0.0820216
$771$	0	0
$772$	18542.2	0.864441
$773$	12680.6	0.590026	0.295013	−	0.955493i	$-0.404676\pi$
$773$	12680.6	0.590026	0.295013	+	0.955493i	$0.404676\pi$
$774$	0	0
$775$	−2638.71	−0.122303
$776$	−9595.09	−0.443871
$777$	0	0
$778$	6445.89	0.297039
$779$	18674.0	0.858876
$780$	0	0
$781$	5176.99	0.237193
$782$	−809.276	−0.0370072
$783$	0	0
$784$	−2902.25	−0.132209
$785$	4397.62	0.199946
$786$	0	0
$787$	−4417.61	−0.200090	−0.100045	−	0.994983i	$-0.531899\pi$
$787$	−4417.61	−0.200090	−0.100045	+	0.994983i	$0.531899\pi$
$788$	−38258.5	−1.72957
$789$	0	0
$790$	−9206.61	−0.414628
$791$	33942.4	1.52573
$792$	0	0
$793$	7561.33	0.338601
$794$	3158.99	0.141194
$795$	0	0
$796$	57.4153	0.00255657
$797$	27030.1	1.20132	0.600661	−	0.799504i	$-0.294904\pi$
$797$	27030.1	1.20132	0.600661	+	0.799504i	$0.294904\pi$
$798$	0	0
$799$	−5947.75	−0.263350
$800$	5165.23	0.228273
$801$	0	0
$802$	−264.769	−0.0116575
$803$	6716.60	0.295173
$804$	0	0
$805$	−2906.27	−0.127246
$806$	−3578.85	−0.156401
$807$	0	0
$808$	14642.6	0.637533
$809$	−23647.0	−1.02767	−0.513835	−	0.857889i	$-0.671776\pi$
$809$	−23647.0	−1.02767	−0.513835	+	0.857889i	$0.671776\pi$
$810$	0	0
$811$	33486.1	1.44988	0.724941	−	0.688811i	$-0.241867\pi$
$811$	33486.1	1.44988	0.724941	+	0.688811i	$0.241867\pi$
$812$	21353.1	0.922843
$813$	0	0
$814$	328.999	0.0141663
$815$	−17933.3	−0.770768
$816$	0	0
$817$	155.353	0.00665251
$818$	6749.88	0.288513
$819$	0	0
$820$	26382.5	1.12356
$821$	−2605.69	−0.110766	−0.0553832	−	0.998465i	$-0.517638\pi$
$821$	−2605.69	−0.110766	−0.0553832	+	0.998465i	$0.517638\pi$
$822$	0	0
$823$	31976.2	1.35434	0.677169	−	0.735828i	$-0.263207\pi$
$823$	31976.2	1.35434	0.677169	+	0.735828i	$0.263207\pi$
$824$	19530.6	0.825705
$825$	0	0
$826$	−6761.62	−0.284827
$827$	37759.0	1.58768	0.793839	−	0.608128i	$-0.208079\pi$
$827$	37759.0	1.58768	0.793839	+	0.608128i	$0.208079\pi$
$828$	0	0
$829$	−1137.55	−0.0476584	−0.0238292	−	0.999716i	$-0.507586\pi$
$829$	−1137.55	−0.0476584	−0.0238292	+	0.999716i	$0.507586\pi$
$830$	−246.149	−0.0102939
$831$	0	0
$832$	−23702.2	−0.987649
$833$	−4672.03	−0.194329
$834$	0	0
$835$	−6154.40	−0.255068
$836$	5576.83	0.230716
$837$	0	0
$838$	10917.0	0.450024
$839$	37372.2	1.53782	0.768911	−	0.639356i	$-0.220799\pi$
$839$	37372.2	1.53782	0.768911	+	0.639356i	$0.220799\pi$
$840$	0	0
$841$	4170.26	0.170989
$842$	−9872.44	−0.404070
$843$	0	0
$844$	−23193.5	−0.945917
$845$	43381.1	1.76610
$846$	0	0
$847$	2048.31	0.0830943
$848$	7662.68	0.310304
$849$	0	0
$850$	2441.51	0.0985211
$851$	−545.589	−0.0219772
$852$	0	0
$853$	−22490.8	−0.902780	−0.451390	−	0.892327i	$-0.649072\pi$
$853$	−22490.8	−0.902780	−0.451390	+	0.892327i	$0.649072\pi$
$854$	1255.37	0.0503021
$855$	0	0
$856$	−5481.16	−0.218858
$857$	−43409.5	−1.73027	−0.865135	−	0.501539i	$-0.832767\pi$
$857$	−43409.5	−1.73027	−0.865135	+	0.501539i	$0.832767\pi$
$858$	0	0
$859$	29533.2	1.17306	0.586532	−	0.809926i	$-0.300493\pi$
$859$	29533.2	1.17306	0.586532	+	0.809926i	$0.300493\pi$
$860$	219.482	0.00870264
$861$	0	0
$862$	−297.457	−0.0117534
$863$	−14351.6	−0.566090	−0.283045	−	0.959107i	$-0.591345\pi$
$863$	−14351.6	−0.566090	−0.283045	+	0.959107i	$0.591345\pi$
$864$	0	0
$865$	−23251.9	−0.913975
$866$	−1293.31	−0.0507488
$867$	0	0
$868$	8275.85	0.323618
$869$	−10760.5	−0.420051
$870$	0	0
$871$	30725.3	1.19528
$872$	727.413	0.0282492
$873$	0	0
$874$	663.994	0.0256979
$875$	−18436.5	−0.712307
$876$	0	0
$877$	−43248.7	−1.66523	−0.832614	−	0.553854i	$-0.813156\pi$
$877$	−43248.7	−1.66523	−0.832614	+	0.553854i	$0.813156\pi$
$878$	5727.71	0.220160
$879$	0	0
$880$	7272.64	0.278591
$881$	−3816.13	−0.145935	−0.0729675	−	0.997334i	$-0.523247\pi$
$881$	−3816.13	−0.145935	−0.0729675	+	0.997334i	$0.523247\pi$
$882$	0	0
$883$	48787.6	1.85938	0.929690	−	0.368343i	$-0.120075\pi$
$883$	48787.6	1.85938	0.929690	+	0.368343i	$0.120075\pi$
$884$	−46121.6	−1.75479
$885$	0	0
$886$	−8541.49	−0.323879
$887$	−41495.1	−1.57077	−0.785384	−	0.619009i	$-0.787534\pi$
$887$	−41495.1	−1.57077	−0.785384	+	0.619009i	$0.787534\pi$
$888$	0	0
$889$	1855.41	0.0699984
$890$	3321.21	0.125087
$891$	0	0
$892$	91.9464	0.00345134
$893$	4880.01	0.182870
$894$	0	0
$895$	56951.9	2.12703
$896$	−21298.1	−0.794107
$897$	0	0
$898$	−12426.7	−0.461788
$899$	11068.7	0.410637
$900$	0	0
$901$	12335.3	0.456104
$902$	−2213.88	−0.0817228
$903$	0	0
$904$	−22698.5	−0.835113
$905$	43829.7	1.60989
$906$	0	0
$907$	21615.3	0.791316	0.395658	−	0.918398i	$-0.370517\pi$
$907$	21615.3	0.791316	0.395658	+	0.918398i	$0.370517\pi$
$908$	34453.6	1.25923
$909$	0	0
$910$	11891.8	0.433199
$911$	−3646.35	−0.132611	−0.0663057	−	0.997799i	$-0.521121\pi$
$911$	−3646.35	−0.132611	−0.0663057	+	0.997799i	$0.521121\pi$
$912$	0	0
$913$	−287.693	−0.0104285
$914$	−11854.0	−0.428988
$915$	0	0
$916$	−37877.6	−1.36628
$917$	−19581.1	−0.705151
$918$	0	0
$919$	31280.0	1.12278	0.561388	−	0.827553i	$-0.310267\pi$
$919$	31280.0	1.12278	0.561388	+	0.827553i	$0.310267\pi$
$920$	1943.53	0.0696482
$921$	0	0
$922$	−6285.42	−0.224511
$923$	35128.7	1.25274
$924$	0	0
$925$	1645.99	0.0585079
$926$	−5795.79	−0.205682
$927$	0	0
$928$	−21666.9	−0.766433
$929$	−6557.92	−0.231602	−0.115801	−	0.993272i	$-0.536944\pi$
$929$	−6557.92	−0.231602	−0.115801	+	0.993272i	$0.536944\pi$
$930$	0	0
$931$	3833.30	0.134942
$932$	1580.02	0.0555314
$933$	0	0
$934$	11094.2	0.388665
$935$	11707.4	0.409491
$936$	0	0
$937$	−24473.3	−0.853265	−0.426632	−	0.904425i	$-0.640300\pi$
$937$	−24473.3	−0.853265	−0.426632	+	0.904425i	$0.640300\pi$
$938$	5101.18	0.177569
$939$	0	0
$940$	6894.46	0.239226
$941$	−15420.8	−0.534224	−0.267112	−	0.963665i	$-0.586069\pi$
$941$	−15420.8	−0.534224	−0.267112	+	0.963665i	$0.586069\pi$
$942$	0	0
$943$	3671.34	0.126782
$944$	−28059.4	−0.967432
$945$	0	0
$946$	−18.4177	−0.000632993 0
$947$	33141.2	1.13722	0.568608	−	0.822609i	$-0.307482\pi$
$947$	33141.2	1.13722	0.568608	+	0.822609i	$0.307482\pi$
$948$	0	0
$949$	45575.8	1.55896
$950$	−2003.20	−0.0684132
$951$	0	0
$952$	−15864.5	−0.540096
$953$	20735.4	0.704813	0.352406	−	0.935847i	$-0.385363\pi$
$953$	20735.4	0.704813	0.352406	+	0.935847i	$0.385363\pi$
$954$	0	0
$955$	−37589.0	−1.27367
$956$	32188.9	1.08898
$957$	0	0
$958$	−7321.32	−0.246911
$959$	−3357.26	−0.113046
$960$	0	0
$961$	−25501.1	−0.856000
$962$	2232.44	0.0748198
$963$	0	0
$964$	7438.60	0.248528
$965$	−31937.7	−1.06540
$966$	0	0
$967$	−8178.87	−0.271990	−0.135995	−	0.990710i	$-0.543423\pi$
$967$	−8178.87	−0.271990	−0.135995	+	0.990710i	$0.543423\pi$
$968$	−1369.78	−0.0454819
$969$	0	0
$970$	7977.08	0.264050
$971$	20576.1	0.680039	0.340020	−	0.940418i	$-0.389566\pi$
$971$	20576.1	0.680039	0.340020	+	0.940418i	$0.389566\pi$
$972$	0	0
$973$	−49094.1	−1.61756
$974$	5156.96	0.169651
$975$	0	0
$976$	5209.55	0.170854
$977$	−14541.9	−0.476188	−0.238094	−	0.971242i	$-0.576523\pi$
$977$	−14541.9	−0.476188	−0.238094	+	0.971242i	$0.576523\pi$
$978$	0	0
$979$	3881.76	0.126723
$980$	5415.68	0.176528
$981$	0	0
$982$	9755.62	0.317021
$983$	29285.7	0.950223	0.475111	−	0.879926i	$-0.342408\pi$
$983$	29285.7	0.950223	0.475111	+	0.879926i	$0.342408\pi$
$984$	0	0
$985$	65897.7	2.13165
$986$	−10241.5	−0.330787
$987$	0	0
$988$	37841.8	1.21853
$989$	30.5427	0.000982004 0
$990$	0	0
$991$	−38085.9	−1.22083	−0.610413	−	0.792083i	$-0.708997\pi$
$991$	−38085.9	−1.22083	−0.610413	+	0.792083i	$0.708997\pi$
$992$	−8397.44	−0.268769
$993$	0	0
$994$	5832.26	0.186105
$995$	−98.8940	−0.00315090
$996$	0	0
$997$	−26803.6	−0.851434	−0.425717	−	0.904856i	$-0.639978\pi$
$997$	−26803.6	−0.851434	−0.425717	+	0.904856i	$0.639978\pi$
$998$	−14691.6	−0.465987
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	99.4.a.c.1.2		2
3.2	odd	2		11.4.a.a.1.1	✓	2
4.3	odd	2		1584.4.a.bc.1.2		2
5.4	even	2		2475.4.a.q.1.1		2
11.10	odd	2		1089.4.a.v.1.1		2
12.11	even	2		176.4.a.i.1.1		2
15.2	even	4		275.4.b.c.199.2		4
15.8	even	4		275.4.b.c.199.3		4
15.14	odd	2		275.4.a.b.1.2		2
21.20	even	2		539.4.a.e.1.1		2
24.5	odd	2		704.4.a.p.1.1		2
24.11	even	2		704.4.a.n.1.2		2
33.2	even	10		121.4.c.f.81.1		8
33.5	odd	10		121.4.c.c.3.2		8
33.8	even	10		121.4.c.f.9.2		8
33.14	odd	10		121.4.c.c.9.1		8
33.17	even	10		121.4.c.f.3.1		8
33.20	odd	10		121.4.c.c.81.2		8
33.26	odd	10		121.4.c.c.27.1		8
33.29	even	10		121.4.c.f.27.2		8
33.32	even	2		121.4.a.c.1.2		2
39.38	odd	2		1859.4.a.a.1.2		2
132.131	odd	2		1936.4.a.w.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.4.a.a.1.1	✓	2	3.2	odd	2
99.4.a.c.1.2		2	1.1	even	1	trivial
121.4.a.c.1.2		2	33.32	even	2
121.4.c.c.3.2		8	33.5	odd	10
121.4.c.c.9.1		8	33.14	odd	10
121.4.c.c.27.1		8	33.26	odd	10
121.4.c.c.81.2		8	33.20	odd	10
121.4.c.f.3.1		8	33.17	even	10
121.4.c.f.9.2		8	33.8	even	10
121.4.c.f.27.2		8	33.29	even	10
121.4.c.f.81.1		8	33.2	even	10
176.4.a.i.1.1		2	12.11	even	2
275.4.a.b.1.2		2	15.14	odd	2
275.4.b.c.199.2		4	15.2	even	4
275.4.b.c.199.3		4	15.8	even	4
539.4.a.e.1.1		2	21.20	even	2
704.4.a.n.1.2		2	24.11	even	2
704.4.a.p.1.1		2	24.5	odd	2
1089.4.a.v.1.1		2	11.10	odd	2
1584.4.a.bc.1.2		2	4.3	odd	2
1859.4.a.a.1.2		2	39.38	odd	2
1936.4.a.w.1.1		2	132.131	odd	2
2475.4.a.q.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)

\(f(q)\)	\(=\)	\(q+0.732051 q^{2} -7.46410 q^{4} +12.8564 q^{5} +16.9282 q^{7} -11.3205 q^{8} +O(q^{10})\)	\(q+0.732051 q^{2} -7.46410 q^{4} +12.8564 q^{5} +16.9282 q^{7} -11.3205 q^{8} +9.41154 q^{10} +11.0000 q^{11} +74.6410 q^{13} +12.3923 q^{14} +51.4256 q^{16} +82.7846 q^{17} -67.9230 q^{19} -95.9615 q^{20} +8.05256 q^{22} -13.3538 q^{23} +40.2872 q^{25} +54.6410 q^{26} -126.354 q^{28} -168.995 q^{29} -65.4974 q^{31} +128.210 q^{32} +60.6025 q^{34} +217.636 q^{35} +40.8564 q^{37} -49.7231 q^{38} -145.541 q^{40} -274.928 q^{41} -2.28719 q^{43} -82.1051 q^{44} -9.77568 q^{46} -71.8461 q^{47} -56.4359 q^{49} +29.4923 q^{50} -557.128 q^{52} +149.005 q^{53} +141.420 q^{55} -191.636 q^{56} -123.713 q^{58} -545.631 q^{59} +101.303 q^{61} -47.9474 q^{62} -317.549 q^{64} +959.615 q^{65} +411.641 q^{67} -617.913 q^{68} +159.321 q^{70} +470.636 q^{71} +610.600 q^{73} +29.9090 q^{74} +506.985 q^{76} +186.210 q^{77} -978.225 q^{79} +661.149 q^{80} -201.261 q^{82} -26.1539 q^{83} +1064.31 q^{85} -1.67434 q^{86} -124.526 q^{88} +352.887 q^{89} +1263.54 q^{91} +99.6743 q^{92} -52.5950 q^{94} -873.246 q^{95} +847.585 q^{97} -41.3140 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 8 q^{4} - 2 q^{5} + 20 q^{7} + 12 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 - 8 * q^4 - 2 * q^5 + 20 * q^7 + 12 * q^8	\( 2 q - 2 q^{2} - 8 q^{4} - 2 q^{5} + 20 q^{7} + 12 q^{8} + 50 q^{10} + 22 q^{11} + 80 q^{13} + 4 q^{14} - 8 q^{16} + 124 q^{17} + 72 q^{19} - 88 q^{20} - 22 q^{22} + 98 q^{23} + 136 q^{25} + 40 q^{26} - 128 q^{28} - 144 q^{29} - 34 q^{31} + 104 q^{32} - 52 q^{34} + 172 q^{35} + 54 q^{37} - 432 q^{38} - 492 q^{40} - 536 q^{41} - 60 q^{43} - 88 q^{44} - 314 q^{46} + 272 q^{47} - 390 q^{49} - 232 q^{50} - 560 q^{52} + 492 q^{53} - 22 q^{55} - 120 q^{56} - 192 q^{58} - 634 q^{59} + 840 q^{61} - 134 q^{62} + 224 q^{64} + 880 q^{65} + 754 q^{67} - 640 q^{68} + 284 q^{70} + 678 q^{71} - 400 q^{73} - 6 q^{74} + 432 q^{76} + 220 q^{77} + 316 q^{79} + 1544 q^{80} + 512 q^{82} - 468 q^{83} + 452 q^{85} + 156 q^{86} + 132 q^{88} + 1842 q^{89} + 1280 q^{91} + 40 q^{92} - 992 q^{94} - 2952 q^{95} + 2194 q^{97} + 870 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 - 8 * q^4 - 2 * q^5 + 20 * q^7 + 12 * q^8 + 50 * q^10 + 22 * q^11 + 80 * q^13 + 4 * q^14 - 8 * q^16 + 124 * q^17 + 72 * q^19 - 88 * q^20 - 22 * q^22 + 98 * q^23 + 136 * q^25 + 40 * q^26 - 128 * q^28 - 144 * q^29 - 34 * q^31 + 104 * q^32 - 52 * q^34 + 172 * q^35 + 54 * q^37 - 432 * q^38 - 492 * q^40 - 536 * q^41 - 60 * q^43 - 88 * q^44 - 314 * q^46 + 272 * q^47 - 390 * q^49 - 232 * q^50 - 560 * q^52 + 492 * q^53 - 22 * q^55 - 120 * q^56 - 192 * q^58 - 634 * q^59 + 840 * q^61 - 134 * q^62 + 224 * q^64 + 880 * q^65 + 754 * q^67 - 640 * q^68 + 284 * q^70 + 678 * q^71 - 400 * q^73 - 6 * q^74 + 432 * q^76 + 220 * q^77 + 316 * q^79 + 1544 * q^80 + 512 * q^82 - 468 * q^83 + 452 * q^85 + 156 * q^86 + 132 * q^88 + 1842 * q^89 + 1280 * q^91 + 40 * q^92 - 992 * q^94 - 2952 * q^95 + 2194 * q^97 + 870 * q^98

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