LMFDB - Embedded newform 99.4.a.f.1.1

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			1.1
Root			$5.42443$ 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)

$f(q)$	$=$	$q-5.42443 q^{2} +21.4244 q^{4} +16.8489 q^{5} -7.69772 q^{7} -72.8199 q^{8} +O(q^{10})$	\(q-5.42443 q^{2} +21.4244 q^{4} +16.8489 q^{5} -7.69772 q^{7} -72.8199 q^{8} -91.3954 q^{10} +11.0000 q^{11} +24.8489 q^{13} +41.7557 q^{14} +223.611 q^{16} +15.9420 q^{17} +15.1511 q^{19} +360.977 q^{20} -59.6687 q^{22} -17.7557 q^{23} +158.884 q^{25} -134.791 q^{26} -164.919 q^{28} +128.547 q^{29} +219.395 q^{31} -630.402 q^{32} -86.4763 q^{34} -129.698 q^{35} +92.0703 q^{37} -82.1863 q^{38} -1226.93 q^{40} +459.942 q^{41} +64.9648 q^{43} +235.669 q^{44} +96.3146 q^{46} -497.408 q^{47} -283.745 q^{49} -861.855 q^{50} +532.373 q^{52} +526.919 q^{53} +185.337 q^{55} +560.547 q^{56} -697.292 q^{58} +578.443 q^{59} -221.569 q^{61} -1190.09 q^{62} +1630.68 q^{64} +418.675 q^{65} -860.745 q^{67} +341.548 q^{68} +703.536 q^{70} -580.919 q^{71} +510.116 q^{73} -499.429 q^{74} +324.605 q^{76} -84.6749 q^{77} +1035.12 q^{79} +3767.59 q^{80} -2494.92 q^{82} -606.211 q^{83} +268.605 q^{85} -352.397 q^{86} -801.018 q^{88} +23.4411 q^{89} -191.279 q^{91} -380.406 q^{92} +2698.15 q^{94} +255.279 q^{95} +719.490 q^{97} +1539.16 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−5.42443	−1.91783	−0.958913	−	0.283702i	$-0.908437\pi$
$2$	−5.42443	−1.91783	−0.958913	+	0.283702i	$0.908437\pi$
$3$	0	0
$3$	0	0
$4$	21.4244	2.67805
$5$	16.8489	1.50701	0.753504	−	0.657444i	$-0.228362\pi$
$5$	16.8489	1.50701	0.753504	+	0.657444i	$0.228362\pi$
$6$	0	0
$7$	−7.69772	−0.415638	−0.207819	−	0.978167i	$-0.566636\pi$
$7$	−7.69772	−0.415638	−0.207819	+	0.978167i	$0.566636\pi$
$8$	−72.8199	−3.21821
$9$	0	0
$10$	−91.3954	−2.89018
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	24.8489	0.530141	0.265071	−	0.964229i	$-0.414605\pi$
$13$	24.8489	0.530141	0.265071	+	0.964229i	$0.414605\pi$
$14$	41.7557	0.797120
$15$	0	0
$16$	223.611	3.49392
$17$	15.9420	0.227441	0.113721	−	0.993513i	$-0.463723\pi$
$17$	15.9420	0.227441	0.113721	+	0.993513i	$0.463723\pi$
$18$	0	0
$19$	15.1511	0.182943	0.0914713	−	0.995808i	$-0.470843\pi$
$19$	15.1511	0.182943	0.0914713	+	0.995808i	$0.470843\pi$
$20$	360.977	4.03585
$21$	0	0
$22$	−59.6687	−0.578246
$23$	−17.7557	−0.160971	−0.0804853	−	0.996756i	$-0.525647\pi$
$23$	−17.7557	−0.160971	−0.0804853	+	0.996756i	$0.525647\pi$
$24$	0	0
$25$	158.884	1.27107
$26$	−134.791	−1.01672
$27$	0	0
$28$	−164.919	−1.11310
$29$	128.547	0.823121	0.411560	−	0.911383i	$-0.364984\pi$
$29$	128.547	0.823121	0.411560	+	0.911383i	$0.364984\pi$
$30$	0	0
$31$	219.395	1.27112	0.635558	−	0.772053i	$-0.280770\pi$
$31$	219.395	1.27112	0.635558	+	0.772053i	$0.280770\pi$
$32$	−630.402	−3.48251
$33$	0	0
$34$	−86.4763	−0.436193
$35$	−129.698	−0.626369
$36$	0	0
$37$	92.0703	0.409088	0.204544	−	0.978857i	$-0.434429\pi$
$37$	92.0703	0.409088	0.204544	+	0.978857i	$0.434429\pi$
$38$	−82.1863	−0.350852
$39$	0	0
$40$	−1226.93	−4.84987
$41$	459.942	1.75197	0.875986	−	0.482336i	$-0.160212\pi$
$41$	459.942	1.75197	0.875986	+	0.482336i	$0.160212\pi$
$42$	0	0
$43$	64.9648	0.230396	0.115198	−	0.993343i	$-0.463250\pi$
$43$	64.9648	0.230396	0.115198	+	0.993343i	$0.463250\pi$
$44$	235.669	0.807464
$45$	0	0
$46$	96.3146	0.308713
$47$	−497.408	−1.54371	−0.771855	−	0.635799i	$-0.780671\pi$
$47$	−497.408	−1.54371	−0.771855	+	0.635799i	$0.780671\pi$
$48$	0	0
$49$	−283.745	−0.827245
$50$	−861.855	−2.43769
$51$	0	0
$52$	532.373	1.41975
$53$	526.919	1.36562	0.682811	−	0.730596i	$-0.260757\pi$
$53$	526.919	1.36562	0.682811	+	0.730596i	$0.260757\pi$
$54$	0	0
$55$	185.337	0.454380
$56$	560.547	1.33761
$57$	0	0
$58$	−697.292	−1.57860
$59$	578.443	1.27639	0.638194	−	0.769876i	$-0.279682\pi$
$59$	578.443	1.27639	0.638194	+	0.769876i	$0.279682\pi$
$60$	0	0
$61$	−221.569	−0.465067	−0.232533	−	0.972588i	$-0.574701\pi$
$61$	−221.569	−0.465067	−0.232533	+	0.972588i	$0.574701\pi$
$62$	−1190.09	−2.43778
$63$	0	0
$64$	1630.68	3.18493
$65$	418.675	0.798927
$66$	0	0
$67$	−860.745	−1.56950	−0.784752	−	0.619810i	$-0.787210\pi$
$67$	−860.745	−1.56950	−0.784752	+	0.619810i	$0.787210\pi$
$68$	341.548	0.609100
$69$	0	0
$70$	703.536	1.20127
$71$	−580.919	−0.971020	−0.485510	−	0.874231i	$-0.661366\pi$
$71$	−580.919	−0.971020	−0.485510	+	0.874231i	$0.661366\pi$
$72$	0	0
$73$	510.116	0.817871	0.408935	−	0.912563i	$-0.365900\pi$
$73$	510.116	0.817871	0.408935	+	0.912563i	$0.365900\pi$
$74$	−499.429	−0.784560
$75$	0	0
$76$	324.605	0.489930
$77$	−84.6749	−0.125319
$78$	0	0
$79$	1035.12	1.47418	0.737088	−	0.675797i	$-0.236200\pi$
$79$	1035.12	1.47418	0.737088	+	0.675797i	$0.236200\pi$
$80$	3767.59	5.26536
$81$	0	0
$82$	−2494.92	−3.35998
$83$	−606.211	−0.801690	−0.400845	−	0.916146i	$-0.631283\pi$
$83$	−606.211	−0.801690	−0.400845	+	0.916146i	$0.631283\pi$
$84$	0	0
$85$	268.605	0.342756
$86$	−352.397	−0.441860
$87$	0	0
$88$	−801.018	−0.970328
$89$	23.4411	0.0279186	0.0139593	−	0.999903i	$-0.495556\pi$
$89$	23.4411	0.0279186	0.0139593	+	0.999903i	$0.495556\pi$
$90$	0	0
$91$	−191.279	−0.220347
$92$	−380.406	−0.431088
$93$	0	0
$94$	2698.15	2.96057
$95$	255.279	0.275696
$96$	0	0
$97$	719.490	0.753126	0.376563	−	0.926391i	$-0.377106\pi$
$97$	719.490	0.753126	0.376563	+	0.926391i	$0.377106\pi$
$98$	1539.16	1.58651
$99$	0	0
$100$	3404.00	3.40400
$101$	−1871.27	−1.84355	−0.921774	−	0.387727i	$-0.873260\pi$
$101$	−1871.27	−1.84355	−0.921774	+	0.387727i	$0.873260\pi$
$102$	0	0
$103$	−428.745	−0.410151	−0.205075	−	0.978746i	$-0.565744\pi$
$103$	−428.745	−0.410151	−0.205075	+	0.978746i	$0.565744\pi$
$104$	−1809.49	−1.70611
$105$	0	0
$106$	−2858.24	−2.61902
$107$	−1148.02	−1.03723	−0.518616	−	0.855008i	$-0.673552\pi$
$107$	−1148.02	−1.03723	−0.518616	+	0.855008i	$0.673552\pi$
$108$	0	0
$109$	−1828.32	−1.60662	−0.803308	−	0.595564i	$-0.796929\pi$
$109$	−1828.32	−1.60662	−0.803308	+	0.595564i	$0.796929\pi$
$110$	−1005.35	−0.871421
$111$	0	0
$112$	−1721.29	−1.45220
$113$	−1126.40	−0.937722	−0.468861	−	0.883272i	$-0.655335\pi$
$113$	−1126.40	−0.937722	−0.468861	+	0.883272i	$0.655335\pi$
$114$	0	0
$115$	−299.163	−0.242584
$116$	2754.04	2.20436
$117$	0	0
$118$	−3137.72	−2.44789
$119$	−122.717	−0.0945332
$120$	0	0
$121$	121.000	0.0909091
$122$	1201.89	0.891916
$123$	0	0
$124$	4700.42	3.40412
$125$	570.907	0.408508
$126$	0	0
$127$	661.304	0.462057	0.231029	−	0.972947i	$-0.425791\pi$
$127$	661.304	0.462057	0.231029	+	0.972947i	$0.425791\pi$
$128$	−3802.31	−2.62562
$129$	0	0
$130$	−2271.07	−1.53220
$131$	−622.186	−0.414967	−0.207483	−	0.978239i	$-0.566527\pi$
$131$	−622.186	−0.414967	−0.207483	+	0.978239i	$0.566527\pi$
$132$	0	0
$133$	−116.629	−0.0760378
$134$	4669.05	3.01003
$135$	0	0
$136$	−1160.89	−0.731955
$137$	1872.84	1.16794	0.583969	−	0.811776i	$-0.301499\pi$
$137$	1872.84	1.16794	0.583969	+	0.811776i	$0.301499\pi$
$138$	0	0
$139$	−954.058	−0.582174	−0.291087	−	0.956697i	$-0.594017\pi$
$139$	−954.058	−0.582174	−0.291087	+	0.956697i	$0.594017\pi$
$140$	−2778.70	−1.67745
$141$	0	0
$142$	3151.15	1.86225
$143$	273.337	0.159844
$144$	0	0
$145$	2165.86	1.24045
$146$	−2767.09	−1.56853
$147$	0	0
$148$	1972.55	1.09556
$149$	2047.01	1.12549	0.562745	−	0.826631i	$-0.309745\pi$
$149$	2047.01	1.12549	0.562745	+	0.826631i	$0.309745\pi$
$150$	0	0
$151$	475.863	0.256458	0.128229	−	0.991745i	$-0.459071\pi$
$151$	475.863	0.256458	0.128229	+	0.991745i	$0.459071\pi$
$152$	−1103.30	−0.588749
$153$	0	0
$154$	459.313	0.240341
$155$	3696.56	1.91558
$156$	0	0
$157$	−647.466	−0.329130	−0.164565	−	0.986366i	$-0.552622\pi$
$157$	−647.466	−0.329130	−0.164565	+	0.986366i	$0.552622\pi$
$158$	−5614.92	−2.82721
$159$	0	0
$160$	−10621.5	−5.24817
$161$	136.678	0.0669054
$162$	0	0
$163$	1093.23	0.525329	0.262665	−	0.964887i	$-0.415399\pi$
$163$	1093.23	0.525329	0.262665	+	0.964887i	$0.415399\pi$
$164$	9853.99	4.69188
$165$	0	0
$166$	3288.35	1.53750
$167$	−1123.25	−0.520479	−0.260240	−	0.965544i	$-0.583802\pi$
$167$	−1123.25	−0.520479	−0.260240	+	0.965544i	$0.583802\pi$
$168$	0	0
$169$	−1579.53	−0.718951
$170$	−1457.03	−0.657346
$171$	0	0
$172$	1391.83	0.617014
$173$	−46.0123	−0.0202211	−0.0101106	−	0.999949i	$-0.503218\pi$
$173$	−46.0123	−0.0202211	−0.0101106	+	0.999949i	$0.503218\pi$
$174$	0	0
$175$	−1223.04	−0.528305
$176$	2459.72	1.05346
$177$	0	0
$178$	−127.155	−0.0535430
$179$	831.975	0.347401	0.173700	−	0.984799i	$-0.444428\pi$
$179$	831.975	0.347401	0.173700	+	0.984799i	$0.444428\pi$
$180$	0	0
$181$	−1810.63	−0.743553	−0.371776	−	0.928322i	$-0.621251\pi$
$181$	−1810.63	−0.743553	−0.371776	+	0.928322i	$0.621251\pi$
$182$	1037.58	0.422586
$183$	0	0
$184$	1292.97	0.518037
$185$	1551.28	0.616499
$186$	0	0
$187$	175.362	0.0685762
$188$	−10656.7	−4.13414
$189$	0	0
$190$	−1384.75	−0.528737
$191$	−458.898	−0.173847	−0.0869233	−	0.996215i	$-0.527704\pi$
$191$	−458.898	−0.173847	−0.0869233	+	0.996215i	$0.527704\pi$
$192$	0	0
$193$	1778.91	0.663465	0.331733	−	0.943373i	$-0.392367\pi$
$193$	1778.91	0.663465	0.331733	+	0.943373i	$0.392367\pi$
$194$	−3902.82	−1.44436
$195$	0	0
$196$	−6079.08	−2.21541
$197$	−5304.53	−1.91844	−0.959218	−	0.282666i	$-0.908781\pi$
$197$	−5304.53	−1.91844	−0.959218	+	0.282666i	$0.908781\pi$
$198$	0	0
$199$	−5138.40	−1.83041	−0.915205	−	0.402989i	$-0.867971\pi$
$199$	−5138.40	−1.83041	−0.915205	+	0.402989i	$0.867971\pi$
$200$	−11569.9	−4.09058
$201$	0	0
$202$	10150.6	3.53560
$203$	−989.515	−0.342120
$204$	0	0
$205$	7749.50	2.64024
$206$	2325.70	0.786597
$207$	0	0
$208$	5556.47	1.85227
$209$	166.663	0.0551593
$210$	0	0
$211$	−4262.36	−1.39068	−0.695339	−	0.718682i	$-0.744746\pi$
$211$	−4262.36	−1.39068	−0.695339	+	0.718682i	$0.744746\pi$
$212$	11288.9	3.65721
$213$	0	0
$214$	6227.38	1.98923
$215$	1094.58	0.347209
$216$	0	0
$217$	−1688.84	−0.528323
$218$	9917.58	3.08121
$219$	0	0
$220$	3970.75	1.21685
$221$	396.141	0.120576
$222$	0	0
$223$	−1377.80	−0.413740	−0.206870	−	0.978368i	$-0.566328\pi$
$223$	−1377.80	−0.413740	−0.206870	+	0.978368i	$0.566328\pi$
$224$	4852.65	1.44746
$225$	0	0
$226$	6110.06	1.79839
$227$	1227.28	0.358843	0.179422	−	0.983772i	$-0.442577\pi$
$227$	1227.28	0.358843	0.179422	+	0.983772i	$0.442577\pi$
$228$	0	0
$229$	3890.28	1.12261	0.561304	−	0.827610i	$-0.310300\pi$
$229$	3890.28	1.12261	0.561304	+	0.827610i	$0.310300\pi$
$230$	1622.79	0.465233
$231$	0	0
$232$	−9360.74	−2.64898
$233$	3218.14	0.904837	0.452419	−	0.891806i	$-0.350561\pi$
$233$	3218.14	0.904837	0.452419	+	0.891806i	$0.350561\pi$
$234$	0	0
$235$	−8380.75	−2.32638
$236$	12392.8	3.41823
$237$	0	0
$238$	665.670	0.181298
$239$	428.098	0.115864	0.0579318	−	0.998321i	$-0.481549\pi$
$239$	428.098	0.115864	0.0579318	+	0.998321i	$0.481549\pi$
$240$	0	0
$241$	1231.16	0.329070	0.164535	−	0.986371i	$-0.447388\pi$
$241$	1231.16	0.329070	0.164535	+	0.986371i	$0.447388\pi$
$242$	−656.356	−0.174348
$243$	0	0
$244$	−4747.00	−1.24547
$245$	−4780.78	−1.24667
$246$	0	0
$247$	376.489	0.0969854
$248$	−15976.3	−4.09072
$249$	0	0
$250$	−3096.84	−0.783446
$251$	−2838.22	−0.713732	−0.356866	−	0.934156i	$-0.616155\pi$
$251$	−2838.22	−0.713732	−0.356866	+	0.934156i	$0.616155\pi$
$252$	0	0
$253$	−195.313	−0.0485344
$254$	−3587.20	−0.886145
$255$	0	0
$256$	7579.90	1.85056
$257$	−342.007	−0.0830110	−0.0415055	−	0.999138i	$-0.513215\pi$
$257$	−342.007	−0.0830110	−0.0415055	+	0.999138i	$0.513215\pi$
$258$	0	0
$259$	−708.731	−0.170032
$260$	8969.87	2.13957
$261$	0	0
$262$	3375.01	0.795834
$263$	−5895.00	−1.38213	−0.691067	−	0.722791i	$-0.742859\pi$
$263$	−5895.00	−1.38213	−0.691067	+	0.722791i	$0.742859\pi$
$264$	0	0
$265$	8877.99	2.05800
$266$	632.647	0.145827
$267$	0	0
$268$	−18441.0	−4.20322
$269$	2496.18	0.565779	0.282890	−	0.959152i	$-0.408707\pi$
$269$	2496.18	0.565779	0.282890	+	0.959152i	$0.408707\pi$
$270$	0	0
$271$	−2249.68	−0.504274	−0.252137	−	0.967692i	$-0.581133\pi$
$271$	−2249.68	−0.504274	−0.252137	+	0.967692i	$0.581133\pi$
$272$	3564.80	0.794662
$273$	0	0
$274$	−10159.1	−2.23990
$275$	1747.72	0.383243
$276$	0	0
$277$	4082.59	0.885556	0.442778	−	0.896631i	$-0.353993\pi$
$277$	4082.59	0.885556	0.442778	+	0.896631i	$0.353993\pi$
$278$	5175.22	1.11651
$279$	0	0
$280$	9444.57	2.01579
$281$	1033.79	0.219468	0.109734	−	0.993961i	$-0.465000\pi$
$281$	1033.79	0.219468	0.109734	+	0.993961i	$0.465000\pi$
$282$	0	0
$283$	−7809.14	−1.64030	−0.820150	−	0.572148i	$-0.806110\pi$
$283$	−7809.14	−1.64030	−0.820150	+	0.572148i	$0.806110\pi$
$284$	−12445.9	−2.60044
$285$	0	0
$286$	−1482.70	−0.306552
$287$	−3540.50	−0.728186
$288$	0	0
$289$	−4658.85	−0.948270
$290$	−11748.6	−2.37896
$291$	0	0
$292$	10928.9	2.19030
$293$	1949.19	0.388645	0.194323	−	0.980938i	$-0.437749\pi$
$293$	1949.19	0.388645	0.194323	+	0.980938i	$0.437749\pi$
$294$	0	0
$295$	9746.10	1.92353
$296$	−6704.55	−1.31653
$297$	0	0
$298$	−11103.9	−2.15849
$299$	−441.209	−0.0853371
$300$	0	0
$301$	−500.081	−0.0957614
$302$	−2581.28	−0.491842
$303$	0	0
$304$	3387.96	0.639187
$305$	−3733.19	−0.700859
$306$	0	0
$307$	2364.09	0.439497	0.219748	−	0.975557i	$-0.429476\pi$
$307$	2364.09	0.439497	0.219748	+	0.975557i	$0.429476\pi$
$308$	−1814.11	−0.335612
$309$	0	0
$310$	−20051.7	−3.67375
$311$	1989.17	0.362686	0.181343	−	0.983420i	$-0.441956\pi$
$311$	1989.17	0.362686	0.181343	+	0.983420i	$0.441956\pi$
$312$	0	0
$313$	−3878.67	−0.700433	−0.350216	−	0.936669i	$-0.613892\pi$
$313$	−3878.67	−0.700433	−0.350216	+	0.936669i	$0.613892\pi$
$314$	3512.13	0.631214
$315$	0	0
$316$	22176.8	3.94792
$317$	−2913.73	−0.516251	−0.258126	−	0.966111i	$-0.583105\pi$
$317$	−2913.73	−0.516251	−0.258126	+	0.966111i	$0.583105\pi$
$318$	0	0
$319$	1414.01	0.248180
$320$	27475.1	4.79971
$321$	0	0
$322$	−741.402	−0.128313
$323$	241.540	0.0416087
$324$	0	0
$325$	3948.09	0.673847
$326$	−5930.17	−1.00749
$327$	0	0
$328$	−33492.9	−5.63822
$329$	3828.90	0.641624
$330$	0	0
$331$	8104.46	1.34580	0.672902	−	0.739731i	$-0.265047\pi$
$331$	8104.46	1.34580	0.672902	+	0.739731i	$0.265047\pi$
$332$	−12987.7	−2.14697
$333$	0	0
$334$	6093.02	0.998189
$335$	−14502.6	−2.36525
$336$	0	0
$337$	5919.19	0.956792	0.478396	−	0.878144i	$-0.341218\pi$
$337$	5919.19	0.956792	0.478396	+	0.878144i	$0.341218\pi$
$338$	8568.07	1.37882
$339$	0	0
$340$	5754.70	0.917919
$341$	2413.35	0.383256
$342$	0	0
$343$	4824.51	0.759472
$344$	−4730.73	−0.741465
$345$	0	0
$346$	249.590	0.0387805
$347$	8540.59	1.32128	0.660638	−	0.750705i	$-0.270286\pi$
$347$	8540.59	1.32128	0.660638	+	0.750705i	$0.270286\pi$
$348$	0	0
$349$	937.337	0.143767	0.0718833	−	0.997413i	$-0.477099\pi$
$349$	937.337	0.143767	0.0718833	+	0.997413i	$0.477099\pi$
$350$	6634.31	1.01320
$351$	0	0
$352$	−6934.42	−1.05002
$353$	211.118	0.0318319	0.0159160	−	0.999873i	$-0.494934\pi$
$353$	211.118	0.0318319	0.0159160	+	0.999873i	$0.494934\pi$
$354$	0	0
$355$	−9787.82	−1.46333
$356$	502.213	0.0747675
$357$	0	0
$358$	−4512.99	−0.666254
$359$	1376.31	0.202337	0.101169	−	0.994869i	$-0.467742\pi$
$359$	1376.31	0.202337	0.101169	+	0.994869i	$0.467742\pi$
$360$	0	0
$361$	−6629.44	−0.966532
$362$	9821.63	1.42600
$363$	0	0
$364$	−4098.05	−0.590100
$365$	8594.87	1.23254
$366$	0	0
$367$	1030.45	0.146564	0.0732821	−	0.997311i	$-0.476653\pi$
$367$	1030.45	0.146564	0.0732821	+	0.997311i	$0.476653\pi$
$368$	−3970.37	−0.562418
$369$	0	0
$370$	−8414.81	−1.18234
$371$	−4056.07	−0.567603
$372$	0	0
$373$	9365.39	1.30006	0.650029	−	0.759909i	$-0.274757\pi$
$373$	9365.39	1.30006	0.650029	+	0.759909i	$0.274757\pi$
$374$	−951.239	−0.131517
$375$	0	0
$376$	36221.2	4.96799
$377$	3194.24	0.436370
$378$	0	0
$379$	−7120.23	−0.965017	−0.482509	−	0.875891i	$-0.660274\pi$
$379$	−7120.23	−0.965017	−0.482509	+	0.875891i	$0.660274\pi$
$380$	5469.22	0.738329
$381$	0	0
$382$	2489.26	0.333407
$383$	1163.56	0.155235	0.0776176	−	0.996983i	$-0.475269\pi$
$383$	1163.56	0.155235	0.0776176	+	0.996983i	$0.475269\pi$
$384$	0	0
$385$	−1426.67	−0.188857
$386$	−9649.57	−1.27241
$387$	0	0
$388$	15414.7	2.01691
$389$	10958.9	1.42838	0.714188	−	0.699954i	$-0.246796\pi$
$389$	10958.9	1.42838	0.714188	+	0.699954i	$0.246796\pi$
$390$	0	0
$391$	−283.062	−0.0366114
$392$	20662.3	2.66225
$393$	0	0
$394$	28774.0	3.67923
$395$	17440.6	2.22159
$396$	0	0
$397$	−2172.09	−0.274595	−0.137298	−	0.990530i	$-0.543842\pi$
$397$	−2172.09	−0.274595	−0.137298	+	0.990530i	$0.543842\pi$
$398$	27872.9	3.51041
$399$	0	0
$400$	35528.2	4.44102
$401$	−7830.71	−0.975180	−0.487590	−	0.873073i	$-0.662124\pi$
$401$	−7830.71	−0.975180	−0.487590	+	0.873073i	$0.662124\pi$
$402$	0	0
$403$	5451.73	0.673870
$404$	−40090.9	−4.93712
$405$	0	0
$406$	5367.55	0.656126
$407$	1012.77	0.123345
$408$	0	0
$409$	−10731.2	−1.29736	−0.648682	−	0.761060i	$-0.724679\pi$
$409$	−10731.2	−1.29736	−0.648682	+	0.761060i	$0.724679\pi$
$410$	−42036.6	−5.06351
$411$	0	0
$412$	−9185.62	−1.09841
$413$	−4452.69	−0.530515
$414$	0	0
$415$	−10214.0	−1.20815
$416$	−15664.8	−1.84622
$417$	0	0
$418$	−904.049	−0.105786
$419$	7315.88	0.852994	0.426497	−	0.904489i	$-0.359748\pi$
$419$	7315.88	0.852994	0.426497	+	0.904489i	$0.359748\pi$
$420$	0	0
$421$	−12495.7	−1.44657	−0.723284	−	0.690551i	$-0.757368\pi$
$421$	−12495.7	−1.44657	−0.723284	+	0.690551i	$0.757368\pi$
$422$	23120.9	2.66708
$423$	0	0
$424$	−38370.2	−4.39486
$425$	2532.93	0.289094
$426$	0	0
$427$	1705.58	0.193299
$428$	−24595.8	−2.77776
$429$	0	0
$430$	−5937.49	−0.665887
$431$	−6075.01	−0.678939	−0.339470	−	0.940617i	$-0.610248\pi$
$431$	−6075.01	−0.678939	−0.339470	+	0.940617i	$0.610248\pi$
$432$	0	0
$433$	5641.79	0.626160	0.313080	−	0.949727i	$-0.398639\pi$
$433$	5641.79	0.626160	0.313080	+	0.949727i	$0.398639\pi$
$434$	9161.01	1.01323
$435$	0	0
$436$	−39170.7	−4.30260
$437$	−269.019	−0.0294484
$438$	0	0
$439$	10897.0	1.18470	0.592351	−	0.805680i	$-0.298200\pi$
$439$	10897.0	1.18470	0.592351	+	0.805680i	$0.298200\pi$
$440$	−13496.2	−1.46229
$441$	0	0
$442$	−2148.84	−0.231244
$443$	7720.83	0.828054	0.414027	−	0.910265i	$-0.364122\pi$
$443$	7720.83	0.828054	0.414027	+	0.910265i	$0.364122\pi$
$444$	0	0
$445$	394.956	0.0420735
$446$	7473.76	0.793481
$447$	0	0
$448$	−12552.5	−1.32378
$449$	7473.86	0.785553	0.392776	−	0.919634i	$-0.371515\pi$
$449$	7473.86	0.785553	0.392776	+	0.919634i	$0.371515\pi$
$450$	0	0
$451$	5059.36	0.528240
$452$	−24132.4	−2.51127
$453$	0	0
$454$	−6657.29	−0.688198
$455$	−3222.84	−0.332064
$456$	0	0
$457$	11140.5	1.14033	0.570167	−	0.821529i	$-0.306879\pi$
$457$	11140.5	1.14033	0.570167	+	0.821529i	$0.306879\pi$
$458$	−21102.6	−2.15296
$459$	0	0
$460$	−6409.41	−0.649652
$461$	−14328.8	−1.44763	−0.723817	−	0.689992i	$-0.757614\pi$
$461$	−14328.8	−1.44763	−0.723817	+	0.689992i	$0.757614\pi$
$462$	0	0
$463$	11760.7	1.18049	0.590246	−	0.807223i	$-0.299031\pi$
$463$	11760.7	1.18049	0.590246	+	0.807223i	$0.299031\pi$
$464$	28744.4	2.87592
$465$	0	0
$466$	−17456.5	−1.73532
$467$	−11854.9	−1.17469	−0.587343	−	0.809338i	$-0.699826\pi$
$467$	−11854.9	−1.17469	−0.587343	+	0.809338i	$0.699826\pi$
$468$	0	0
$469$	6625.77	0.652345
$470$	45460.8	4.46160
$471$	0	0
$472$	−42122.1	−4.10769
$473$	714.613	0.0694671
$474$	0	0
$475$	2407.27	0.232533
$476$	−2629.14	−0.253165
$477$	0	0
$478$	−2322.19	−0.222206
$479$	1324.68	0.126359	0.0631796	−	0.998002i	$-0.479876\pi$
$479$	1324.68	0.126359	0.0631796	+	0.998002i	$0.479876\pi$
$480$	0	0
$481$	2287.84	0.216874
$482$	−6678.34	−0.631100
$483$	0	0
$484$	2592.36	0.243459
$485$	12122.6	1.13497
$486$	0	0
$487$	18636.4	1.73408	0.867040	−	0.498239i	$-0.166020\pi$
$487$	18636.4	1.73408	0.867040	+	0.498239i	$0.166020\pi$
$488$	16134.7	1.49668
$489$	0	0
$490$	25933.0	2.39089
$491$	−124.552	−0.0114480	−0.00572398	−	0.999984i	$-0.501822\pi$
$491$	−124.552	−0.0114480	−0.00572398	+	0.999984i	$0.501822\pi$
$492$	0	0
$493$	2049.29	0.187212
$494$	−2042.24	−0.186001
$495$	0	0
$496$	49059.2	4.44117
$497$	4471.75	0.403592
$498$	0	0
$499$	10230.2	0.917768	0.458884	−	0.888496i	$-0.348249\pi$
$499$	10230.2	0.917768	0.458884	+	0.888496i	$0.348249\pi$
$500$	12231.4	1.09401
$501$	0	0
$502$	15395.7	1.36881
$503$	−5150.81	−0.456587	−0.228294	−	0.973592i	$-0.573315\pi$
$503$	−5150.81	−0.456587	−0.228294	+	0.973592i	$0.573315\pi$
$504$	0	0
$505$	−31528.8	−2.77824
$506$	1059.46	0.0930806
$507$	0	0
$508$	14168.1	1.23741
$509$	22.7715	0.00198296	0.000991481	−	1.00000i	$-0.499684\pi$
$509$	22.7715	0.00198296	0.000991481		1.00000i	$0.499684\pi$
$510$	0	0
$511$	−3926.73	−0.339938
$512$	−10698.1	−0.923429
$513$	0	0
$514$	1855.19	0.159201
$515$	−7223.87	−0.618100
$516$	0	0
$517$	−5471.49	−0.465446
$518$	3844.46	0.326093
$519$	0	0
$520$	−30487.8	−2.57112
$521$	−21521.7	−1.80976	−0.904879	−	0.425669i	$-0.860039\pi$
$521$	−21521.7	−1.80976	−0.904879	+	0.425669i	$0.860039\pi$
$522$	0	0
$523$	2923.36	0.244416	0.122208	−	0.992504i	$-0.461002\pi$
$523$	2923.36	0.244416	0.122208	+	0.992504i	$0.461002\pi$
$524$	−13330.0	−1.11130
$525$	0	0
$526$	31977.0	2.65069
$527$	3497.60	0.289104
$528$	0	0
$529$	−11851.7	−0.974088
$530$	−48158.0	−3.94689
$531$	0	0
$532$	−2498.71	−0.203633
$533$	11429.0	0.928792
$534$	0	0
$535$	−19342.9	−1.56312
$536$	62679.3	5.05100
$537$	0	0
$538$	−13540.3	−1.08507
$539$	−3121.20	−0.249424
$540$	0	0
$541$	21272.8	1.69056	0.845278	−	0.534327i	$-0.179435\pi$
$541$	21272.8	1.69056	0.845278	+	0.534327i	$0.179435\pi$
$542$	12203.2	0.967108
$543$	0	0
$544$	−10049.9	−0.792067
$545$	−30805.1	−2.42118
$546$	0	0
$547$	−18730.5	−1.46409	−0.732046	−	0.681256i	$-0.761434\pi$
$547$	−18730.5	−1.46409	−0.732046	+	0.681256i	$0.761434\pi$
$548$	40124.5	3.12780
$549$	0	0
$550$	−9480.41	−0.734992
$551$	1947.63	0.150584
$552$	0	0
$553$	−7968.04	−0.612723
$554$	−22145.7	−1.69834
$555$	0	0
$556$	−20440.1	−1.55909
$557$	18885.0	1.43659	0.718297	−	0.695736i	$-0.244922\pi$
$557$	18885.0	1.43659	0.718297	+	0.695736i	$0.244922\pi$
$558$	0	0
$559$	1614.30	0.122143
$560$	−29001.8	−2.18848
$561$	0	0
$562$	−5607.71	−0.420902
$563$	−10285.1	−0.769922	−0.384961	−	0.922933i	$-0.625785\pi$
$563$	−10285.1	−0.769922	−0.384961	+	0.922933i	$0.625785\pi$
$564$	0	0
$565$	−18978.5	−1.41315
$566$	42360.1	3.14581
$567$	0	0
$568$	42302.5	3.12495
$569$	−18008.8	−1.32683	−0.663415	−	0.748251i	$-0.730894\pi$
$569$	−18008.8	−1.32683	−0.663415	+	0.748251i	$0.730894\pi$
$570$	0	0
$571$	−7010.79	−0.513822	−0.256911	−	0.966435i	$-0.582705\pi$
$571$	−7010.79	−0.513822	−0.256911	+	0.966435i	$0.582705\pi$
$572$	5856.10	0.428070
$573$	0	0
$574$	19205.2	1.39653
$575$	−2821.10	−0.204605
$576$	0	0
$577$	−16398.9	−1.18318	−0.591589	−	0.806240i	$-0.701499\pi$
$577$	−16398.9	−1.18318	−0.591589	+	0.806240i	$0.701499\pi$
$578$	25271.6	1.81862
$579$	0	0
$580$	46402.4	3.32199
$581$	4666.44	0.333213
$582$	0	0
$583$	5796.11	0.411750
$584$	−37146.6	−2.63208
$585$	0	0
$586$	−10573.3	−0.745354
$587$	−12823.5	−0.901671	−0.450836	−	0.892607i	$-0.648874\pi$
$587$	−12823.5	−0.901671	−0.450836	+	0.892607i	$0.648874\pi$
$588$	0	0
$589$	3324.09	0.232541
$590$	−52867.0	−3.68899
$591$	0	0
$592$	20587.9	1.42932
$593$	−16899.5	−1.17029	−0.585144	−	0.810929i	$-0.698962\pi$
$593$	−16899.5	−1.17029	−0.585144	+	0.810929i	$0.698962\pi$
$594$	0	0
$595$	−2067.64	−0.142462
$596$	43856.1	3.01412
$597$	0	0
$598$	2393.31	0.163662
$599$	15074.9	1.02829	0.514143	−	0.857704i	$-0.328110\pi$
$599$	15074.9	1.02829	0.514143	+	0.857704i	$0.328110\pi$
$600$	0	0
$601$	−11418.8	−0.775014	−0.387507	−	0.921867i	$-0.626664\pi$
$601$	−11418.8	−0.775014	−0.387507	+	0.921867i	$0.626664\pi$
$602$	2712.65	0.183654
$603$	0	0
$604$	10195.1	0.686809
$605$	2038.71	0.137001
$606$	0	0
$607$	−17952.8	−1.20046	−0.600232	−	0.799826i	$-0.704925\pi$
$607$	−17952.8	−1.20046	−0.600232	+	0.799826i	$0.704925\pi$
$608$	−9551.30	−0.637100
$609$	0	0
$610$	20250.4	1.34412
$611$	−12360.0	−0.818384
$612$	0	0
$613$	12528.9	0.825507	0.412753	−	0.910843i	$-0.364567\pi$
$613$	12528.9	0.825507	0.412753	+	0.910843i	$0.364567\pi$
$614$	−12823.8	−0.842878
$615$	0	0
$616$	6166.01	0.403305
$617$	8586.10	0.560232	0.280116	−	0.959966i	$-0.409627\pi$
$617$	8586.10	0.560232	0.280116	+	0.959966i	$0.409627\pi$
$618$	0	0
$619$	18415.4	1.19576	0.597882	−	0.801584i	$-0.296009\pi$
$619$	18415.4	1.19576	0.597882	+	0.801584i	$0.296009\pi$
$620$	79196.7	5.13003
$621$	0	0
$622$	−10790.1	−0.695569
$623$	−180.443	−0.0116040
$624$	0	0
$625$	−10241.4	−0.655448
$626$	21039.6	1.34331
$627$	0	0
$628$	−13871.6	−0.881427
$629$	1467.79	0.0930436
$630$	0	0
$631$	2374.38	0.149798	0.0748989	−	0.997191i	$-0.476137\pi$
$631$	2374.38	0.149798	0.0748989	+	0.997191i	$0.476137\pi$
$632$	−75377.1	−4.74421
$633$	0	0
$634$	15805.3	0.990080
$635$	11142.2	0.696324
$636$	0	0
$637$	−7050.74	−0.438557
$638$	−7670.21	−0.475966
$639$	0	0
$640$	−64064.6	−3.95684
$641$	−11086.0	−0.683104	−0.341552	−	0.939863i	$-0.610953\pi$
$641$	−11086.0	−0.683104	−0.341552	+	0.939863i	$0.610953\pi$
$642$	0	0
$643$	19934.1	1.22259	0.611294	−	0.791403i	$-0.290649\pi$
$643$	19934.1	1.22259	0.611294	+	0.791403i	$0.290649\pi$
$644$	2928.26	0.179176
$645$	0	0
$646$	−1310.21	−0.0797983
$647$	30634.8	1.86148	0.930739	−	0.365684i	$-0.119165\pi$
$647$	30634.8	1.86148	0.930739	+	0.365684i	$0.119165\pi$
$648$	0	0
$649$	6362.87	0.384845
$650$	−21416.1	−1.29232
$651$	0	0
$652$	23421.9	1.40686
$653$	9818.07	0.588378	0.294189	−	0.955747i	$-0.404951\pi$
$653$	9818.07	0.588378	0.294189	+	0.955747i	$0.404951\pi$
$654$	0	0
$655$	−10483.1	−0.625358
$656$	102848.	6.12125
$657$	0	0
$658$	−20769.6	−1.23052
$659$	−16478.5	−0.974070	−0.487035	−	0.873383i	$-0.661922\pi$
$659$	−16478.5	−0.974070	−0.487035	+	0.873383i	$0.661922\pi$
$660$	0	0
$661$	2958.12	0.174066	0.0870328	−	0.996205i	$-0.472262\pi$
$661$	2958.12	0.174066	0.0870328	+	0.996205i	$0.472262\pi$
$662$	−43962.1	−2.58102
$663$	0	0
$664$	44144.2	2.58001
$665$	−1965.07	−0.114590
$666$	0	0
$667$	−2282.44	−0.132498
$668$	−24065.1	−1.39387
$669$	0	0
$670$	78668.2	4.53614
$671$	−2437.26	−0.140223
$672$	0	0
$673$	−29960.3	−1.71602	−0.858012	−	0.513630i	$-0.828300\pi$
$673$	−29960.3	−1.71602	−0.858012	+	0.513630i	$0.828300\pi$
$674$	−32108.2	−1.83496
$675$	0	0
$676$	−33840.6	−1.92539
$677$	4514.73	0.256300	0.128150	−	0.991755i	$-0.459096\pi$
$677$	4514.73	0.256300	0.128150	+	0.991755i	$0.459096\pi$
$678$	0	0
$679$	−5538.43	−0.313027
$680$	−19559.7	−1.10306
$681$	0	0
$682$	−13091.0	−0.735018
$683$	13555.7	0.759438	0.379719	−	0.925102i	$-0.376021\pi$
$683$	13555.7	0.759438	0.379719	+	0.925102i	$0.376021\pi$
$684$	0	0
$685$	31555.2	1.76009
$686$	−26170.2	−1.45653
$687$	0	0
$688$	14526.8	0.804986
$689$	13093.3	0.723972
$690$	0	0
$691$	−11471.3	−0.631535	−0.315768	−	0.948837i	$-0.602262\pi$
$691$	−11471.3	−0.631535	−0.315768	+	0.948837i	$0.602262\pi$
$692$	−985.787	−0.0541532
$693$	0	0
$694$	−46327.8	−2.53398
$695$	−16074.8	−0.877340
$696$	0	0
$697$	7332.40	0.398471
$698$	−5084.52	−0.275719
$699$	0	0
$700$	−26203.0	−1.41483
$701$	−22229.0	−1.19769	−0.598843	−	0.800866i	$-0.704373\pi$
$701$	−22229.0	−1.19769	−0.598843	+	0.800866i	$0.704373\pi$
$702$	0	0
$703$	1394.97	0.0748397
$704$	17937.5	0.960292
$705$	0	0
$706$	−1145.19	−0.0610480
$707$	14404.5	0.766248
$708$	0	0
$709$	−15081.2	−0.798851	−0.399426	−	0.916766i	$-0.630790\pi$
$709$	−15081.2	−0.798851	−0.399426	+	0.916766i	$0.630790\pi$
$710$	53093.4	2.80642
$711$	0	0
$712$	−1706.98	−0.0898480
$713$	−3895.52	−0.204612
$714$	0	0
$715$	4605.42	0.240885
$716$	17824.6	0.930358
$717$	0	0
$718$	−7465.71	−0.388047
$719$	7399.80	0.383819	0.191910	−	0.981413i	$-0.438532\pi$
$719$	7399.80	0.383819	0.191910	+	0.981413i	$0.438532\pi$
$720$	0	0
$721$	3300.36	0.170474
$722$	35960.9	1.85364
$723$	0	0
$724$	−38791.7	−1.99127
$725$	20424.0	1.04625
$726$	0	0
$727$	1705.77	0.0870202	0.0435101	−	0.999053i	$-0.486146\pi$
$727$	1705.77	0.0870202	0.0435101	+	0.999053i	$0.486146\pi$
$728$	13928.9	0.709122
$729$	0	0
$730$	−46622.3	−2.36379
$731$	1035.67	0.0524017
$732$	0	0
$733$	37122.6	1.87061	0.935303	−	0.353847i	$-0.115127\pi$
$733$	37122.6	1.87061	0.935303	+	0.353847i	$0.115127\pi$
$734$	−5589.60	−0.281084
$735$	0	0
$736$	11193.2	0.560581
$737$	−9468.20	−0.473223
$738$	0	0
$739$	34256.3	1.70520	0.852598	−	0.522568i	$-0.175026\pi$
$739$	34256.3	1.70520	0.852598	+	0.522568i	$0.175026\pi$
$740$	33235.3	1.65102
$741$	0	0
$742$	22001.9	1.08856
$743$	−1567.88	−0.0774160	−0.0387080	−	0.999251i	$-0.512324\pi$
$743$	−1567.88	−0.0774160	−0.0387080	+	0.999251i	$0.512324\pi$
$744$	0	0
$745$	34489.8	1.69612
$746$	−50801.9	−2.49328
$747$	0	0
$748$	3757.03	0.183651
$749$	8837.17	0.431112
$750$	0	0
$751$	−955.613	−0.0464325	−0.0232163	−	0.999730i	$-0.507391\pi$
$751$	−955.613	−0.0464325	−0.0232163	+	0.999730i	$0.507391\pi$
$752$	−111226.	−5.39360
$753$	0	0
$754$	−17326.9	−0.836881
$755$	8017.75	0.386484
$756$	0	0
$757$	−14015.4	−0.672918	−0.336459	−	0.941698i	$-0.609229\pi$
$757$	−14015.4	−0.672918	−0.336459	+	0.941698i	$0.609229\pi$
$758$	38623.2	1.85073
$759$	0	0
$760$	−18589.4	−0.887249
$761$	36271.0	1.72776	0.863879	−	0.503699i	$-0.168028\pi$
$761$	36271.0	1.72776	0.863879	+	0.503699i	$0.168028\pi$
$762$	0	0
$763$	14073.9	0.667770
$764$	−9831.63	−0.465571
$765$	0	0
$766$	−6311.64	−0.297714
$767$	14373.6	0.676665
$768$	0	0
$769$	−18163.6	−0.851749	−0.425874	−	0.904782i	$-0.640033\pi$
$769$	−18163.6	−0.851749	−0.425874	+	0.904782i	$0.640033\pi$
$770$	7738.90	0.362195
$771$	0	0
$772$	38112.1	1.77680
$773$	8345.65	0.388321	0.194160	−	0.980970i	$-0.437802\pi$
$773$	8345.65	0.388321	0.194160	+	0.980970i	$0.437802\pi$
$774$	0	0
$775$	34858.4	1.61568
$776$	−52393.2	−2.42372
$777$	0	0
$778$	−59445.7	−2.73937
$779$	6968.65	0.320510
$780$	0	0
$781$	−6390.11	−0.292774
$782$	1535.45	0.0702142
$783$	0	0
$784$	−63448.5	−2.89033
$785$	−10909.1	−0.496001
$786$	0	0
$787$	−22996.2	−1.04158	−0.520791	−	0.853684i	$-0.674363\pi$
$787$	−22996.2	−1.04158	−0.520791	+	0.853684i	$0.674363\pi$
$788$	−113647.	−5.13768
$789$	0	0
$790$	−94605.0	−4.26063
$791$	8670.69	0.389752
$792$	0	0
$793$	−5505.75	−0.246551
$794$	11782.4	0.526626
$795$	0	0
$796$	−110087.	−4.90194
$797$	2743.82	0.121946	0.0609730	−	0.998139i	$-0.480580\pi$
$797$	2743.82	0.121946	0.0609730	+	0.998139i	$0.480580\pi$
$798$	0	0
$799$	−7929.68	−0.351104
$800$	−100161.	−4.42652
$801$	0	0
$802$	42477.1	1.87022
$803$	5611.28	0.246597
$804$	0	0
$805$	2302.88	0.100827
$806$	−29572.5	−1.29237
$807$	0	0
$808$	136266.	5.93293
$809$	−41241.7	−1.79231	−0.896156	−	0.443738i	$-0.853652\pi$
$809$	−41241.7	−1.79231	−0.896156	+	0.443738i	$0.853652\pi$
$810$	0	0
$811$	12832.9	0.555641	0.277820	−	0.960633i	$-0.410388\pi$
$811$	12832.9	0.555641	0.277820	+	0.960633i	$0.410388\pi$
$812$	−21199.8	−0.916215
$813$	0	0
$814$	−5493.72	−0.236554
$815$	18419.7	0.791675
$816$	0	0
$817$	984.292	0.0421493
$818$	58210.4	2.48812
$819$	0	0
$820$	166029.	7.07069
$821$	−16368.5	−0.695817	−0.347908	−	0.937529i	$-0.613108\pi$
$821$	−16368.5	−0.695817	−0.347908	+	0.937529i	$0.613108\pi$
$822$	0	0
$823$	3869.53	0.163892	0.0819461	−	0.996637i	$-0.473886\pi$
$823$	3869.53	0.163892	0.0819461	+	0.996637i	$0.473886\pi$
$824$	31221.2	1.31995
$825$	0	0
$826$	24153.3	1.01743
$827$	−7388.69	−0.310677	−0.155339	−	0.987861i	$-0.549647\pi$
$827$	−7388.69	−0.310677	−0.155339	+	0.987861i	$0.549647\pi$
$828$	0	0
$829$	23990.1	1.00508	0.502539	−	0.864554i	$-0.332399\pi$
$829$	23990.1	1.00508	0.502539	+	0.864554i	$0.332399\pi$
$830$	55404.9	2.31703
$831$	0	0
$832$	40520.6	1.68846
$833$	−4523.47	−0.188150
$834$	0	0
$835$	−18925.6	−0.784367
$836$	3570.65	0.147720
$837$	0	0
$838$	−39684.5	−1.63589
$839$	18228.3	0.750074	0.375037	−	0.927010i	$-0.377630\pi$
$839$	18228.3	0.750074	0.375037	+	0.927010i	$0.377630\pi$
$840$	0	0
$841$	−7864.78	−0.322472
$842$	67782.3	2.77427
$843$	0	0
$844$	−91318.7	−3.72431
$845$	−26613.3	−1.08346
$846$	0	0
$847$	−931.424	−0.0377852
$848$	117825.	4.77137
$849$	0	0
$850$	−13739.7	−0.554433
$851$	−1634.77	−0.0658511
$852$	0	0
$853$	−21737.3	−0.872534	−0.436267	−	0.899817i	$-0.643700\pi$
$853$	−21737.3	−0.872534	−0.436267	+	0.899817i	$0.643700\pi$
$854$	−9251.79	−0.370714
$855$	0	0
$856$	83599.0	3.33803
$857$	18712.2	0.745852	0.372926	−	0.927861i	$-0.378355\pi$
$857$	18712.2	0.745852	0.372926	+	0.927861i	$0.378355\pi$
$858$	0	0
$859$	30527.6	1.21256	0.606279	−	0.795252i	$-0.292661\pi$
$859$	30527.6	1.21256	0.606279	+	0.795252i	$0.292661\pi$
$860$	23450.8	0.929845
$861$	0	0
$862$	32953.4	1.30209
$863$	−10906.4	−0.430196	−0.215098	−	0.976592i	$-0.569007\pi$
$863$	−10906.4	−0.430196	−0.215098	+	0.976592i	$0.569007\pi$
$864$	0	0
$865$	−775.255	−0.0304734
$866$	−30603.5	−1.20087
$867$	0	0
$868$	−36182.5	−1.41488
$869$	11386.3	0.444481
$870$	0	0
$871$	−21388.5	−0.832058
$872$	133138.	5.17043
$873$	0	0
$874$	1459.28	0.0564768
$875$	−4394.68	−0.169791
$876$	0	0
$877$	21770.9	0.838256	0.419128	−	0.907927i	$-0.362336\pi$
$877$	21770.9	0.838256	0.419128	+	0.907927i	$0.362336\pi$
$878$	−59109.8	−2.27205
$879$	0	0
$880$	41443.4	1.58757
$881$	−47206.9	−1.80527	−0.902634	−	0.430409i	$-0.858369\pi$
$881$	−47206.9	−1.80527	−0.902634	+	0.430409i	$0.858369\pi$
$882$	0	0
$883$	−6059.68	−0.230945	−0.115473	−	0.993311i	$-0.536838\pi$
$883$	−6059.68	−0.230945	−0.115473	+	0.993311i	$0.536838\pi$
$884$	8487.09	0.322909
$885$	0	0
$886$	−41881.1	−1.58806
$887$	37130.2	1.40553	0.702767	−	0.711420i	$-0.251948\pi$
$887$	37130.2	1.40553	0.702767	+	0.711420i	$0.251948\pi$
$888$	0	0
$889$	−5090.53	−0.192048
$890$	−2142.41	−0.0806896
$891$	0	0
$892$	−29518.5	−1.10802
$893$	−7536.30	−0.282410
$894$	0	0
$895$	14017.8	0.523536
$896$	29269.1	1.09131
$897$	0	0
$898$	−40541.4	−1.50655
$899$	28202.5	1.04628
$900$	0	0
$901$	8400.15	0.310599
$902$	−27444.1	−1.01307
$903$	0	0
$904$	82024.1	3.01779
$905$	−30507.0	−1.12054
$906$	0	0
$907$	1182.94	0.0433064	0.0216532	−	0.999766i	$-0.493107\pi$
$907$	1182.94	0.0433064	0.0216532	+	0.999766i	$0.493107\pi$
$908$	26293.8	0.961001
$909$	0	0
$910$	17482.1	0.636841
$911$	−37676.4	−1.37022	−0.685112	−	0.728438i	$-0.740247\pi$
$911$	−37676.4	−1.37022	−0.685112	+	0.728438i	$0.740247\pi$
$912$	0	0
$913$	−6668.32	−0.241719
$914$	−60431.1	−2.18696
$915$	0	0
$916$	83347.1	3.00640
$917$	4789.41	0.172476
$918$	0	0
$919$	8697.82	0.312203	0.156101	−	0.987741i	$-0.450107\pi$
$919$	8697.82	0.312203	0.156101	+	0.987741i	$0.450107\pi$
$920$	21785.0	0.780686
$921$	0	0
$922$	77725.7	2.77631
$923$	−14435.2	−0.514778
$924$	0	0
$925$	14628.5	0.519981
$926$	−63795.3	−2.26398
$927$	0	0
$928$	−81036.0	−2.86653
$929$	17247.5	0.609119	0.304559	−	0.952493i	$-0.401491\pi$
$929$	17247.5	0.609119	0.304559	+	0.952493i	$0.401491\pi$
$930$	0	0
$931$	−4299.06	−0.151338
$932$	68946.7	2.42320
$933$	0	0
$934$	64306.0	2.25284
$935$	2954.65	0.103345
$936$	0	0
$937$	−41812.4	−1.45779	−0.728896	−	0.684624i	$-0.759966\pi$
$937$	−41812.4	−1.45779	−0.728896	+	0.684624i	$0.759966\pi$
$938$	−35941.0	−1.25108
$939$	0	0
$940$	−179553.	−6.23018
$941$	−37655.9	−1.30451	−0.652257	−	0.757998i	$-0.726178\pi$
$941$	−37655.9	−1.30451	−0.652257	+	0.757998i	$0.726178\pi$
$942$	0	0
$943$	−8166.60	−0.282016
$944$	129346.	4.45959
$945$	0	0
$946$	−3876.37	−0.133226
$947$	21244.4	0.728986	0.364493	−	0.931206i	$-0.381242\pi$
$947$	21244.4	0.728986	0.364493	+	0.931206i	$0.381242\pi$
$948$	0	0
$949$	12675.8	0.433587
$950$	−13058.1	−0.445958
$951$	0	0
$952$	8936.24	0.304228
$953$	−1324.27	−0.0450130	−0.0225065	−	0.999747i	$-0.507165\pi$
$953$	−1324.27	−0.0450130	−0.0225065	+	0.999747i	$0.507165\pi$
$954$	0	0
$955$	−7731.91	−0.261988
$956$	9171.77	0.310289
$957$	0	0
$958$	−7185.62	−0.242335
$959$	−14416.6	−0.485439
$960$	0	0
$961$	18343.4	0.615735
$962$	−12410.2	−0.415927
$963$	0	0
$964$	26376.9	0.881268
$965$	29972.6	0.999847
$966$	0	0
$967$	52267.1	1.73815	0.869077	−	0.494676i	$-0.164713\pi$
$967$	52267.1	1.73815	0.869077	+	0.494676i	$0.164713\pi$
$968$	−8811.20	−0.292565
$969$	0	0
$970$	−65758.1	−2.17667
$971$	52489.8	1.73479	0.867394	−	0.497622i	$-0.165793\pi$
$971$	52489.8	1.73479	0.867394	+	0.497622i	$0.165793\pi$
$972$	0	0
$973$	7344.07	0.241973
$974$	−101092.	−3.32566
$975$	0	0
$976$	−49545.3	−1.62490
$977$	−8324.11	−0.272581	−0.136291	−	0.990669i	$-0.543518\pi$
$977$	−8324.11	−0.272581	−0.136291	+	0.990669i	$0.543518\pi$
$978$	0	0
$979$	257.852	0.00841777
$980$	−102426.	−3.33864
$981$	0	0
$982$	675.623	0.0219552
$983$	44407.1	1.44086	0.720431	−	0.693527i	$-0.243944\pi$
$983$	44407.1	1.44086	0.720431	+	0.693527i	$0.243944\pi$
$984$	0	0
$985$	−89375.3	−2.89110
$986$	−11116.2	−0.359039
$987$	0	0
$988$	8066.05	0.259732
$989$	−1153.50	−0.0370870
$990$	0	0
$991$	−45124.7	−1.44645	−0.723226	−	0.690612i	$-0.757341\pi$
$991$	−45124.7	−1.44645	−0.723226	+	0.690612i	$0.757341\pi$
$992$	−138307.	−4.42667
$993$	0	0
$994$	−24256.7	−0.774020
$995$	−86576.2	−2.75844
$996$	0	0
$997$	5480.61	0.174095	0.0870474	−	0.996204i	$-0.472257\pi$
$997$	5480.61	0.174095	0.0870474	+	0.996204i	$0.472257\pi$
$998$	−55492.9	−1.76012
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	99.4.a.f.1.1		2
3.2	odd	2		33.4.a.c.1.2	✓	2
4.3	odd	2		1584.4.a.bj.1.2		2
5.4	even	2		2475.4.a.p.1.2		2
11.10	odd	2		1089.4.a.u.1.2		2
12.11	even	2		528.4.a.p.1.1		2
15.2	even	4		825.4.c.h.199.4		4
15.8	even	4		825.4.c.h.199.1		4
15.14	odd	2		825.4.a.l.1.1		2
21.20	even	2		1617.4.a.k.1.2		2
24.5	odd	2		2112.4.a.bn.1.2		2
24.11	even	2		2112.4.a.bg.1.2		2
33.32	even	2		363.4.a.i.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.4.a.c.1.2	✓	2	3.2	odd	2
99.4.a.f.1.1		2	1.1	even	1	trivial
363.4.a.i.1.1		2	33.32	even	2
528.4.a.p.1.1		2	12.11	even	2
825.4.a.l.1.1		2	15.14	odd	2
825.4.c.h.199.1		4	15.8	even	4
825.4.c.h.199.4		4	15.2	even	4
1089.4.a.u.1.2		2	11.10	odd	2
1584.4.a.bj.1.2		2	4.3	odd	2
1617.4.a.k.1.2		2	21.20	even	2
2112.4.a.bg.1.2		2	24.11	even	2
2112.4.a.bn.1.2		2	24.5	odd	2
2475.4.a.p.1.2		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)

\(f(q)\)	\(=\)	\(q-5.42443 q^{2} +21.4244 q^{4} +16.8489 q^{5} -7.69772 q^{7} -72.8199 q^{8} +O(q^{10})\)	\(q-5.42443 q^{2} +21.4244 q^{4} +16.8489 q^{5} -7.69772 q^{7} -72.8199 q^{8} -91.3954 q^{10} +11.0000 q^{11} +24.8489 q^{13} +41.7557 q^{14} +223.611 q^{16} +15.9420 q^{17} +15.1511 q^{19} +360.977 q^{20} -59.6687 q^{22} -17.7557 q^{23} +158.884 q^{25} -134.791 q^{26} -164.919 q^{28} +128.547 q^{29} +219.395 q^{31} -630.402 q^{32} -86.4763 q^{34} -129.698 q^{35} +92.0703 q^{37} -82.1863 q^{38} -1226.93 q^{40} +459.942 q^{41} +64.9648 q^{43} +235.669 q^{44} +96.3146 q^{46} -497.408 q^{47} -283.745 q^{49} -861.855 q^{50} +532.373 q^{52} +526.919 q^{53} +185.337 q^{55} +560.547 q^{56} -697.292 q^{58} +578.443 q^{59} -221.569 q^{61} -1190.09 q^{62} +1630.68 q^{64} +418.675 q^{65} -860.745 q^{67} +341.548 q^{68} +703.536 q^{70} -580.919 q^{71} +510.116 q^{73} -499.429 q^{74} +324.605 q^{76} -84.6749 q^{77} +1035.12 q^{79} +3767.59 q^{80} -2494.92 q^{82} -606.211 q^{83} +268.605 q^{85} -352.397 q^{86} -801.018 q^{88} +23.4411 q^{89} -191.279 q^{91} -380.406 q^{92} +2698.15 q^{94} +255.279 q^{95} +719.490 q^{97} +1539.16 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} + 33 q^{4} + 14 q^{5} + 24 q^{7} - 57 q^{8}+O(q^{10}) \) 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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