LMFDB - Embedded newform 124.4.a.b.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [124,4,Mod(1,124)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(124, 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("124.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 124 = 2^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	124.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:	$7.31623684071$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.4000044.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 21x^{2} + 16x + 62 $ x^4 - x^3 - 21x^2 + 16x + 62
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.37430$ of defining polynomial
Character	$\chi$	$=$	124.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.74860 q^{3} +20.7669 q^{5} +3.45568 q^{7} -4.45080 q^{9} +O(q^{10})$	$q+4.74860 q^{3} +20.7669 q^{5} +3.45568 q^{7} -4.45080 q^{9} +1.02880 q^{11} -34.4597 q^{13} +98.6138 q^{15} -51.2460 q^{17} +141.288 q^{19} +16.4096 q^{21} -79.2715 q^{23} +306.265 q^{25} -149.347 q^{27} +19.9689 q^{29} +31.0000 q^{31} +4.88537 q^{33} +71.7638 q^{35} -300.085 q^{37} -163.635 q^{39} +41.7875 q^{41} +142.781 q^{43} -92.4293 q^{45} +275.321 q^{47} -331.058 q^{49} -243.347 q^{51} -122.993 q^{53} +21.3650 q^{55} +670.920 q^{57} -753.373 q^{59} -702.543 q^{61} -15.3805 q^{63} -715.622 q^{65} -152.569 q^{67} -376.429 q^{69} +209.083 q^{71} -885.944 q^{73} +1454.33 q^{75} +3.55521 q^{77} +1127.08 q^{79} -589.019 q^{81} -74.3029 q^{83} -1064.22 q^{85} +94.8241 q^{87} +1028.18 q^{89} -119.082 q^{91} +147.207 q^{93} +2934.11 q^{95} +941.006 q^{97} -4.57899 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{3} + 6 q^{5} + 16 q^{7} + 64 q^{9}+O(q^{10}) $ 4 * q + 2 * q^3 + 6 * q^5 + 16 * q^7 + 64 * q^9	$ 4 q + 2 q^{3} + 6 q^{5} + 16 q^{7} + 64 q^{9} + 96 q^{11} + 42 q^{13} + 182 q^{15} - 2 q^{17} + 180 q^{19} + 150 q^{21} + 142 q^{23} + 150 q^{25} + 20 q^{27} + 262 q^{29} + 124 q^{31} - 444 q^{33} + 224 q^{35} - 284 q^{37} + 60 q^{39} - 526 q^{41} - 326 q^{43} - 870 q^{45} + 468 q^{47} - 978 q^{49} + 356 q^{51} - 252 q^{53} - 876 q^{55} - 1298 q^{57} - 164 q^{59} - 1066 q^{61} - 236 q^{63} - 598 q^{65} - 956 q^{69} + 1504 q^{71} - 732 q^{73} + 188 q^{75} - 288 q^{77} + 822 q^{79} - 536 q^{81} + 1408 q^{83} + 482 q^{85} - 208 q^{87} + 250 q^{89} + 946 q^{91} + 62 q^{93} + 2292 q^{95} + 526 q^{97} + 2952 q^{99}+O(q^{100}) $ 4 * q + 2 * q^3 + 6 * q^5 + 16 * q^7 + 64 * q^9 + 96 * q^11 + 42 * q^13 + 182 * q^15 - 2 * q^17 + 180 * q^19 + 150 * q^21 + 142 * q^23 + 150 * q^25 + 20 * q^27 + 262 * q^29 + 124 * q^31 - 444 * q^33 + 224 * q^35 - 284 * q^37 + 60 * q^39 - 526 * q^41 - 326 * q^43 - 870 * q^45 + 468 * q^47 - 978 * q^49 + 356 * q^51 - 252 * q^53 - 876 * q^55 - 1298 * q^57 - 164 * q^59 - 1066 * q^61 - 236 * q^63 - 598 * q^65 - 956 * q^69 + 1504 * q^71 - 732 * q^73 + 188 * q^75 - 288 * q^77 + 822 * q^79 - 536 * q^81 + 1408 * q^83 + 482 * q^85 - 208 * q^87 + 250 * q^89 + 946 * q^91 + 62 * q^93 + 2292 * q^95 + 526 * q^97 + 2952 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	4.74860	0.913869	0.456934	−	0.889500i	$-0.348947\pi$
$3$	4.74860	0.913869	0.456934	+	0.889500i	$0.348947\pi$
$4$	0	0
$5$	20.7669	1.85745	0.928725	−	0.370770i	$-0.120906\pi$
$5$	20.7669	1.85745	0.928725	+	0.370770i	$0.120906\pi$
$6$	0	0
$7$	3.45568	0.186589	0.0932946	−	0.995639i	$-0.470260\pi$
$7$	3.45568	0.186589	0.0932946	+	0.995639i	$0.470260\pi$
$8$	0	0
$9$	−4.45080	−0.164844
$10$	0	0
$11$	1.02880	0.0281996	0.0140998	−	0.999901i	$-0.495512\pi$
$11$	1.02880	0.0281996	0.0140998	+	0.999901i	$0.495512\pi$
$12$	0	0
$13$	−34.4597	−0.735185	−0.367593	−	0.929987i	$-0.619818\pi$
$13$	−34.4597	−0.735185	−0.367593	+	0.929987i	$0.619818\pi$
$14$	0	0
$15$	98.6138	1.69746
$16$	0	0
$17$	−51.2460	−0.731117	−0.365558	−	0.930788i	$-0.619122\pi$
$17$	−51.2460	−0.731117	−0.365558	+	0.930788i	$0.619122\pi$
$18$	0	0
$19$	141.288	1.70598	0.852991	−	0.521925i	$-0.174786\pi$
$19$	141.288	1.70598	0.852991	+	0.521925i	$0.174786\pi$
$20$	0	0
$21$	16.4096	0.170518
$22$	0	0
$23$	−79.2715	−0.718663	−0.359331	−	0.933210i	$-0.616995\pi$
$23$	−79.2715	−0.718663	−0.359331	+	0.933210i	$0.616995\pi$
$24$	0	0
$25$	306.265	2.45012
$26$	0	0
$27$	−149.347	−1.06451
$28$	0	0
$29$	19.9689	0.127866	0.0639331	−	0.997954i	$-0.479636\pi$
$29$	19.9689	0.127866	0.0639331	+	0.997954i	$0.479636\pi$
$30$	0	0
$31$	31.0000	0.179605
$31$	31.0000	0.179605
$32$	0	0
$33$	4.88537	0.0257707
$34$	0	0
$35$	71.7638	0.346580
$36$	0	0
$37$	−300.085	−1.33334	−0.666671	−	0.745352i	$-0.732281\pi$
$37$	−300.085	−1.33334	−0.666671	+	0.745352i	$0.732281\pi$
$38$	0	0
$39$	−163.635	−0.671862
$40$	0	0
$41$	41.7875	0.159173	0.0795866	−	0.996828i	$-0.474640\pi$
$41$	41.7875	0.159173	0.0795866	+	0.996828i	$0.474640\pi$
$42$	0	0
$43$	142.781	0.506371	0.253186	−	0.967418i	$-0.418522\pi$
$43$	142.781	0.506371	0.253186	+	0.967418i	$0.418522\pi$
$44$	0	0
$45$	−92.4293	−0.306190
$46$	0	0
$47$	275.321	0.854461	0.427231	−	0.904143i	$-0.359489\pi$
$47$	275.321	0.854461	0.427231	+	0.904143i	$0.359489\pi$
$48$	0	0
$49$	−331.058	−0.965184
$50$	0	0
$51$	−243.347	−0.668145
$52$	0	0
$53$	−122.993	−0.318761	−0.159380	−	0.987217i	$-0.550950\pi$
$53$	−122.993	−0.318761	−0.159380	+	0.987217i	$0.550950\pi$
$54$	0	0
$55$	21.3650	0.0523793
$56$	0	0
$57$	670.920	1.55904
$58$	0	0
$59$	−753.373	−1.66239	−0.831193	−	0.555984i	$-0.812342\pi$
$59$	−753.373	−1.66239	−0.831193	+	0.555984i	$0.812342\pi$
$60$	0	0
$61$	−702.543	−1.47461	−0.737307	−	0.675558i	$-0.763903\pi$
$61$	−702.543	−1.47461	−0.737307	+	0.675558i	$0.763903\pi$
$62$	0	0
$63$	−15.3805	−0.0307582
$64$	0	0
$65$	−715.622	−1.36557
$66$	0	0
$67$	−152.569	−0.278197	−0.139099	−	0.990279i	$-0.544421\pi$
$67$	−152.569	−0.278197	−0.139099	+	0.990279i	$0.544421\pi$
$68$	0	0
$69$	−376.429	−0.656763
$70$	0	0
$71$	209.083	0.349487	0.174743	−	0.984614i	$-0.444090\pi$
$71$	209.083	0.349487	0.174743	+	0.984614i	$0.444090\pi$
$72$	0	0
$73$	−885.944	−1.42044	−0.710218	−	0.703981i	$-0.751404\pi$
$73$	−885.944	−1.42044	−0.710218	+	0.703981i	$0.751404\pi$
$74$	0	0
$75$	1454.33	2.23909
$76$	0	0
$77$	3.55521	0.00526174
$78$	0	0
$79$	1127.08	1.60514	0.802571	−	0.596557i	$-0.203465\pi$
$79$	1127.08	1.60514	0.802571	+	0.596557i	$0.203465\pi$
$80$	0	0
$81$	−589.019	−0.807982
$82$	0	0
$83$	−74.3029	−0.0982627	−0.0491314	−	0.998792i	$-0.515645\pi$
$83$	−74.3029	−0.0982627	−0.0491314	+	0.998792i	$0.515645\pi$
$84$	0	0
$85$	−1064.22	−1.35801
$86$	0	0
$87$	94.8241	0.116853
$88$	0	0
$89$	1028.18	1.22458	0.612289	−	0.790634i	$-0.290249\pi$
$89$	1028.18	1.22458	0.612289	+	0.790634i	$0.290249\pi$
$90$	0	0
$91$	−119.082	−0.137178
$92$	0	0
$93$	147.207	0.164136
$94$	0	0
$95$	2934.11	3.16878
$96$	0	0
$97$	941.006	0.984997	0.492499	−	0.870313i	$-0.336084\pi$
$97$	941.006	0.984997	0.492499	+	0.870313i	$0.336084\pi$
$98$	0	0
$99$	−4.57899	−0.00464854
$100$	0	0
$101$	1285.41	1.26637	0.633185	−	0.774000i	$-0.281747\pi$
$101$	1285.41	1.26637	0.633185	+	0.774000i	$0.281747\pi$
$102$	0	0
$103$	1073.49	1.02693	0.513467	−	0.858109i	$-0.328361\pi$
$103$	1073.49	1.02693	0.513467	+	0.858109i	$0.328361\pi$
$104$	0	0
$105$	340.778	0.316729
$106$	0	0
$107$	826.932	0.747127	0.373563	−	0.927605i	$-0.378136\pi$
$107$	826.932	0.747127	0.373563	+	0.927605i	$0.378136\pi$
$108$	0	0
$109$	−2093.64	−1.83977	−0.919883	−	0.392192i	$-0.871717\pi$
$109$	−2093.64	−1.83977	−0.919883	+	0.392192i	$0.871717\pi$
$110$	0	0
$111$	−1424.98	−1.21850
$112$	0	0
$113$	−1270.85	−1.05797	−0.528987	−	0.848630i	$-0.677428\pi$
$113$	−1270.85	−1.05797	−0.528987	+	0.848630i	$0.677428\pi$
$114$	0	0
$115$	−1646.22	−1.33488
$116$	0	0
$117$	153.373	0.121191
$118$	0	0
$119$	−177.090	−0.136419
$120$	0	0
$121$	−1329.94	−0.999205
$122$	0	0
$123$	198.432	0.145463
$124$	0	0
$125$	3764.31	2.69352
$126$	0	0
$127$	−422.725	−0.295361	−0.147680	−	0.989035i	$-0.547181\pi$
$127$	−422.725	−0.295361	−0.147680	+	0.989035i	$0.547181\pi$
$128$	0	0
$129$	678.012	0.462757
$130$	0	0
$131$	51.6703	0.0344615	0.0172308	−	0.999852i	$-0.494515\pi$
$131$	51.6703	0.0344615	0.0172308	+	0.999852i	$0.494515\pi$
$132$	0	0
$133$	488.246	0.318318
$134$	0	0
$135$	−3101.48	−1.97728
$136$	0	0
$137$	−1312.64	−0.818584	−0.409292	−	0.912403i	$-0.634224\pi$
$137$	−1312.64	−0.818584	−0.409292	+	0.912403i	$0.634224\pi$
$138$	0	0
$139$	1842.17	1.12411	0.562053	−	0.827101i	$-0.310012\pi$
$139$	1842.17	1.12411	0.562053	+	0.827101i	$0.310012\pi$
$140$	0	0
$141$	1307.39	0.780865
$142$	0	0
$143$	−35.4522	−0.0207319
$144$	0	0
$145$	414.691	0.237505
$146$	0	0
$147$	−1572.06	−0.882052
$148$	0	0
$149$	3165.36	1.74038	0.870189	−	0.492719i	$-0.163997\pi$
$149$	3165.36	1.74038	0.870189	+	0.492719i	$0.163997\pi$
$150$	0	0
$151$	−1107.13	−0.596667	−0.298333	−	0.954462i	$-0.596431\pi$
$151$	−1107.13	−0.596667	−0.298333	+	0.954462i	$0.596431\pi$
$152$	0	0
$153$	228.086	0.120520
$154$	0	0
$155$	643.774	0.333608
$156$	0	0
$157$	2439.86	1.24027	0.620134	−	0.784496i	$-0.287078\pi$
$157$	2439.86	1.24027	0.620134	+	0.784496i	$0.287078\pi$
$158$	0	0
$159$	−584.043	−0.291306
$160$	0	0
$161$	−273.937	−0.134095
$162$	0	0
$163$	−3089.75	−1.48471	−0.742356	−	0.670005i	$-0.766292\pi$
$163$	−3089.75	−1.48471	−0.742356	+	0.670005i	$0.766292\pi$
$164$	0	0
$165$	101.454	0.0478678
$166$	0	0
$167$	2237.05	1.03658	0.518288	−	0.855206i	$-0.326570\pi$
$167$	2237.05	1.03658	0.518288	+	0.855206i	$0.326570\pi$
$168$	0	0
$169$	−1009.53	−0.459503
$170$	0	0
$171$	−628.844	−0.281222
$172$	0	0
$173$	−517.644	−0.227490	−0.113745	−	0.993510i	$-0.536285\pi$
$173$	−517.644	−0.227490	−0.113745	+	0.993510i	$0.536285\pi$
$174$	0	0
$175$	1058.35	0.457166
$176$	0	0
$177$	−3577.47	−1.51920
$178$	0	0
$179$	−2028.34	−0.846958	−0.423479	−	0.905906i	$-0.639191\pi$
$179$	−2028.34	−0.846958	−0.423479	+	0.905906i	$0.639191\pi$
$180$	0	0
$181$	−163.275	−0.0670506	−0.0335253	−	0.999438i	$-0.510673\pi$
$181$	−163.275	−0.0670506	−0.0335253	+	0.999438i	$0.510673\pi$
$182$	0	0
$183$	−3336.10	−1.34760
$184$	0	0
$185$	−6231.84	−2.47662
$186$	0	0
$187$	−52.7220	−0.0206172
$188$	0	0
$189$	−516.096	−0.198627
$190$	0	0
$191$	−2690.61	−1.01930	−0.509648	−	0.860383i	$-0.670224\pi$
$191$	−2690.61	−1.01930	−0.509648	+	0.860383i	$0.670224\pi$
$192$	0	0
$193$	−1416.40	−0.528262	−0.264131	−	0.964487i	$-0.585085\pi$
$193$	−1416.40	−0.528262	−0.264131	+	0.964487i	$0.585085\pi$
$194$	0	0
$195$	−3398.20	−1.24795
$196$	0	0
$197$	4559.06	1.64883	0.824415	−	0.565985i	$-0.191504\pi$
$197$	4559.06	1.64883	0.824415	+	0.565985i	$0.191504\pi$
$198$	0	0
$199$	5004.67	1.78277	0.891386	−	0.453245i	$-0.149734\pi$
$199$	5004.67	1.78277	0.891386	+	0.453245i	$0.149734\pi$
$200$	0	0
$201$	−724.488	−0.254236
$202$	0	0
$203$	69.0060	0.0238585
$204$	0	0
$205$	867.797	0.295656
$206$	0	0
$207$	352.821	0.118467
$208$	0	0
$209$	145.357	0.0481080
$210$	0	0
$211$	2150.55	0.701660	0.350830	−	0.936439i	$-0.385899\pi$
$211$	2150.55	0.701660	0.350830	+	0.936439i	$0.385899\pi$
$212$	0	0
$213$	992.851	0.319385
$214$	0	0
$215$	2965.13	0.940559
$216$	0	0
$217$	107.126	0.0335124
$218$	0	0
$219$	−4206.99	−1.29809
$220$	0	0
$221$	1765.92	0.537506
$222$	0	0
$223$	−1332.68	−0.400192	−0.200096	−	0.979776i	$-0.564125\pi$
$223$	−1332.68	−0.400192	−0.200096	+	0.979776i	$0.564125\pi$
$224$	0	0
$225$	−1363.12	−0.403888
$226$	0	0
$227$	268.976	0.0786456	0.0393228	−	0.999227i	$-0.487480\pi$
$227$	268.976	0.0786456	0.0393228	+	0.999227i	$0.487480\pi$
$228$	0	0
$229$	−3410.49	−0.984155	−0.492077	−	0.870551i	$-0.663762\pi$
$229$	−3410.49	−0.984155	−0.492077	+	0.870551i	$0.663762\pi$
$230$	0	0
$231$	16.8823	0.00480853
$232$	0	0
$233$	4242.38	1.19282	0.596411	−	0.802679i	$-0.296593\pi$
$233$	4242.38	1.19282	0.596411	+	0.802679i	$0.296593\pi$
$234$	0	0
$235$	5717.57	1.58712
$236$	0	0
$237$	5352.04	1.46689
$238$	0	0
$239$	5723.09	1.54894	0.774468	−	0.632613i	$-0.218018\pi$
$239$	5723.09	1.54894	0.774468	+	0.632613i	$0.218018\pi$
$240$	0	0
$241$	3592.24	0.960151	0.480076	−	0.877227i	$-0.340609\pi$
$241$	3592.24	0.960151	0.480076	+	0.877227i	$0.340609\pi$
$242$	0	0
$243$	1235.36	0.326125
$244$	0	0
$245$	−6875.06	−1.79278
$246$	0	0
$247$	−4868.74	−1.25421
$248$	0	0
$249$	−352.835	−0.0897992
$250$	0	0
$251$	6635.80	1.66872	0.834358	−	0.551223i	$-0.185839\pi$
$251$	6635.80	1.66872	0.834358	+	0.551223i	$0.185839\pi$
$252$	0	0
$253$	−81.5546	−0.0202660
$254$	0	0
$255$	−5053.57	−1.24105
$256$	0	0
$257$	5329.16	1.29348	0.646739	−	0.762711i	$-0.276132\pi$
$257$	5329.16	1.29348	0.646739	+	0.762711i	$0.276132\pi$
$258$	0	0
$259$	−1037.00	−0.248787
$260$	0	0
$261$	−88.8773	−0.0210780
$262$	0	0
$263$	−7609.66	−1.78415	−0.892076	−	0.451886i	$-0.850751\pi$
$263$	−7609.66	−1.78415	−0.892076	+	0.451886i	$0.850751\pi$
$264$	0	0
$265$	−2554.18	−0.592082
$266$	0	0
$267$	4882.44	1.11910
$268$	0	0
$269$	4719.57	1.06973	0.534865	−	0.844937i	$-0.320362\pi$
$269$	4719.57	1.06973	0.534865	+	0.844937i	$0.320362\pi$
$270$	0	0
$271$	1812.08	0.406185	0.203093	−	0.979160i	$-0.434901\pi$
$271$	1812.08	0.406185	0.203093	+	0.979160i	$0.434901\pi$
$272$	0	0
$273$	−565.472	−0.125362
$274$	0	0
$275$	315.086	0.0690923
$276$	0	0
$277$	4535.40	0.983776	0.491888	−	0.870658i	$-0.336307\pi$
$277$	4535.40	0.983776	0.491888	+	0.870658i	$0.336307\pi$
$278$	0	0
$279$	−137.975	−0.0296069
$280$	0	0
$281$	1530.52	0.324922	0.162461	−	0.986715i	$-0.448057\pi$
$281$	1530.52	0.324922	0.162461	+	0.986715i	$0.448057\pi$
$282$	0	0
$283$	−4129.58	−0.867413	−0.433706	−	0.901054i	$-0.642794\pi$
$283$	−4129.58	−0.867413	−0.433706	+	0.901054i	$0.642794\pi$
$284$	0	0
$285$	13932.9	2.89585
$286$	0	0
$287$	144.404	0.0297000
$288$	0	0
$289$	−2286.84	−0.465468
$290$	0	0
$291$	4468.46	0.900158
$292$	0	0
$293$	−470.920	−0.0938957	−0.0469478	−	0.998897i	$-0.514949\pi$
$293$	−470.920	−0.0938957	−0.0469478	+	0.998897i	$0.514949\pi$
$294$	0	0
$295$	−15645.2	−3.08780
$296$	0	0
$297$	−153.649	−0.0300189
$298$	0	0
$299$	2731.67	0.528350
$300$	0	0
$301$	493.407	0.0944834
$302$	0	0
$303$	6103.91	1.15730
$304$	0	0
$305$	−14589.6	−2.73902
$306$	0	0
$307$	2568.37	0.477474	0.238737	−	0.971084i	$-0.423267\pi$
$307$	2568.37	0.477474	0.238737	+	0.971084i	$0.423267\pi$
$308$	0	0
$309$	5097.58	0.938483
$310$	0	0
$311$	1874.62	0.341801	0.170900	−	0.985288i	$-0.445332\pi$
$311$	1874.62	0.341801	0.170900	+	0.985288i	$0.445332\pi$
$312$	0	0
$313$	7478.18	1.35045	0.675226	−	0.737611i	$-0.264046\pi$
$313$	7478.18	1.35045	0.675226	+	0.737611i	$0.264046\pi$
$314$	0	0
$315$	−319.406	−0.0571318
$316$	0	0
$317$	1687.36	0.298964	0.149482	−	0.988764i	$-0.452239\pi$
$317$	1687.36	0.298964	0.149482	+	0.988764i	$0.452239\pi$
$318$	0	0
$319$	20.5440	0.00360577
$320$	0	0
$321$	3926.77	0.682775
$322$	0	0
$323$	−7240.44	−1.24727
$324$	0	0
$325$	−10553.8	−1.80129
$326$	0	0
$327$	−9941.87	−1.68130
$328$	0	0
$329$	951.421	0.159433
$330$	0	0
$331$	−6260.55	−1.03961	−0.519805	−	0.854285i	$-0.673996\pi$
$331$	−6260.55	−1.03961	−0.519805	+	0.854285i	$0.673996\pi$
$332$	0	0
$333$	1335.62	0.219794
$334$	0	0
$335$	−3168.38	−0.516738
$336$	0	0
$337$	−5908.52	−0.955067	−0.477534	−	0.878613i	$-0.658469\pi$
$337$	−5908.52	−0.955067	−0.477534	+	0.878613i	$0.658469\pi$
$338$	0	0
$339$	−6034.74	−0.966850
$340$	0	0
$341$	31.8928	0.00506479
$342$	0	0
$343$	−2329.33	−0.366682
$344$	0	0
$345$	−7817.26	−1.21990
$346$	0	0
$347$	10424.7	1.61276	0.806382	−	0.591395i	$-0.201423\pi$
$347$	10424.7	1.61276	0.806382	+	0.591395i	$0.201423\pi$
$348$	0	0
$349$	−143.825	−0.0220596	−0.0110298	−	0.999939i	$-0.503511\pi$
$349$	−143.825	−0.0220596	−0.0110298	+	0.999939i	$0.503511\pi$
$350$	0	0
$351$	5146.46	0.782615
$352$	0	0
$353$	−10266.2	−1.54791	−0.773957	−	0.633238i	$-0.781725\pi$
$353$	−10266.2	−1.54791	−0.773957	+	0.633238i	$0.781725\pi$
$354$	0	0
$355$	4342.01	0.649154
$356$	0	0
$357$	−840.929	−0.124669
$358$	0	0
$359$	11749.7	1.72737	0.863686	−	0.504030i	$-0.168150\pi$
$359$	11749.7	1.72737	0.863686	+	0.504030i	$0.168150\pi$
$360$	0	0
$361$	13103.3	1.91038
$362$	0	0
$363$	−6315.36	−0.913142
$364$	0	0
$365$	−18398.3	−2.63839
$366$	0	0
$367$	−2821.15	−0.401261	−0.200631	−	0.979667i	$-0.564299\pi$
$367$	−2821.15	−0.401261	−0.200631	+	0.979667i	$0.564299\pi$
$368$	0	0
$369$	−185.987	−0.0262388
$370$	0	0
$371$	−425.023	−0.0594773
$372$	0	0
$373$	3973.93	0.551642	0.275821	−	0.961209i	$-0.411050\pi$
$373$	3973.93	0.551642	0.275821	+	0.961209i	$0.411050\pi$
$374$	0	0
$375$	17875.2	2.46153
$376$	0	0
$377$	−688.121	−0.0940054
$378$	0	0
$379$	−10212.9	−1.38417	−0.692083	−	0.721818i	$-0.743307\pi$
$379$	−10212.9	−1.38417	−0.692083	+	0.721818i	$0.743307\pi$
$380$	0	0
$381$	−2007.35	−0.269921
$382$	0	0
$383$	−7264.05	−0.969126	−0.484563	−	0.874756i	$-0.661021\pi$
$383$	−7264.05	−0.969126	−0.484563	+	0.874756i	$0.661021\pi$
$384$	0	0
$385$	73.8307	0.00977341
$386$	0	0
$387$	−635.491	−0.0834724
$388$	0	0
$389$	312.404	0.0407185	0.0203592	−	0.999793i	$-0.493519\pi$
$389$	312.404	0.0407185	0.0203592	+	0.999793i	$0.493519\pi$
$390$	0	0
$391$	4062.35	0.525427
$392$	0	0
$393$	245.362	0.0314933
$394$	0	0
$395$	23405.9	2.98147
$396$	0	0
$397$	−7062.82	−0.892879	−0.446440	−	0.894814i	$-0.647308\pi$
$397$	−7062.82	−0.892879	−0.446440	+	0.894814i	$0.647308\pi$
$398$	0	0
$399$	2318.48	0.290901
$400$	0	0
$401$	3819.39	0.475639	0.237819	−	0.971309i	$-0.423567\pi$
$401$	3819.39	0.475639	0.237819	+	0.971309i	$0.423567\pi$
$402$	0	0
$403$	−1068.25	−0.132043
$404$	0	0
$405$	−12232.1	−1.50079
$406$	0	0
$407$	−308.728	−0.0375997
$408$	0	0
$409$	2497.34	0.301920	0.150960	−	0.988540i	$-0.451764\pi$
$409$	2497.34	0.301920	0.150960	+	0.988540i	$0.451764\pi$
$410$	0	0
$411$	−6233.18	−0.748078
$412$	0	0
$413$	−2603.42	−0.310183
$414$	0	0
$415$	−1543.04	−0.182518
$416$	0	0
$417$	8747.73	1.02729
$418$	0	0
$419$	−3568.62	−0.416082	−0.208041	−	0.978120i	$-0.566709\pi$
$419$	−3568.62	−0.416082	−0.208041	+	0.978120i	$0.566709\pi$
$420$	0	0
$421$	9914.73	1.14778	0.573889	−	0.818933i	$-0.305434\pi$
$421$	9914.73	1.14778	0.573889	+	0.818933i	$0.305434\pi$
$422$	0	0
$423$	−1225.40	−0.140853
$424$	0	0
$425$	−15694.9	−1.79132
$426$	0	0
$427$	−2427.76	−0.275147
$428$	0	0
$429$	−168.348	−0.0189462
$430$	0	0
$431$	−3056.87	−0.341634	−0.170817	−	0.985303i	$-0.554641\pi$
$431$	−3056.87	−0.341634	−0.170817	+	0.985303i	$0.554641\pi$
$432$	0	0
$433$	−12704.0	−1.40997	−0.704983	−	0.709224i	$-0.749045\pi$
$433$	−12704.0	−1.40997	−0.704983	+	0.709224i	$0.749045\pi$
$434$	0	0
$435$	1969.20	0.217049
$436$	0	0
$437$	−11200.1	−1.22603
$438$	0	0
$439$	5595.05	0.608285	0.304143	−	0.952627i	$-0.401630\pi$
$439$	5595.05	0.608285	0.304143	+	0.952627i	$0.401630\pi$
$440$	0	0
$441$	1473.47	0.159105
$442$	0	0
$443$	−6050.93	−0.648958	−0.324479	−	0.945893i	$-0.605189\pi$
$443$	−6050.93	−0.648958	−0.324479	+	0.945893i	$0.605189\pi$
$444$	0	0
$445$	21352.2	2.27459
$446$	0	0
$447$	15031.0	1.59048
$448$	0	0
$449$	−15290.7	−1.60715	−0.803577	−	0.595201i	$-0.797072\pi$
$449$	−15290.7	−1.60715	−0.803577	+	0.595201i	$0.797072\pi$
$450$	0	0
$451$	42.9910	0.00448862
$452$	0	0
$453$	−5257.30	−0.545275
$454$	0	0
$455$	−2472.96	−0.254800
$456$	0	0
$457$	9684.47	0.991291	0.495646	−	0.868525i	$-0.334931\pi$
$457$	9684.47	0.991291	0.495646	+	0.868525i	$0.334931\pi$
$458$	0	0
$459$	7653.45	0.778285
$460$	0	0
$461$	53.3410	0.00538902	0.00269451	−	0.999996i	$-0.499142\pi$
$461$	53.3410	0.00538902	0.00269451	+	0.999996i	$0.499142\pi$
$462$	0	0
$463$	2249.16	0.225761	0.112880	−	0.993609i	$-0.463992\pi$
$463$	2249.16	0.225761	0.112880	+	0.993609i	$0.463992\pi$
$464$	0	0
$465$	3057.03	0.304874
$466$	0	0
$467$	−15577.1	−1.54351	−0.771756	−	0.635918i	$-0.780621\pi$
$467$	−15577.1	−1.54351	−0.771756	+	0.635918i	$0.780621\pi$
$468$	0	0
$469$	−527.228	−0.0519086
$470$	0	0
$471$	11585.9	1.13344
$472$	0	0
$473$	146.894	0.0142794
$474$	0	0
$475$	43271.5	4.17986
$476$	0	0
$477$	547.415	0.0525459
$478$	0	0
$479$	−14139.5	−1.34875	−0.674374	−	0.738390i	$-0.735587\pi$
$479$	−14139.5	−1.34875	−0.674374	+	0.738390i	$0.735587\pi$
$480$	0	0
$481$	10340.8	0.980253
$482$	0	0
$483$	−1300.82	−0.122545
$484$	0	0
$485$	19541.8	1.82958
$486$	0	0
$487$	2903.55	0.270170	0.135085	−	0.990834i	$-0.456869\pi$
$487$	2903.55	0.270170	0.135085	+	0.990834i	$0.456869\pi$
$488$	0	0
$489$	−14672.0	−1.35683
$490$	0	0
$491$	9420.24	0.865844	0.432922	−	0.901431i	$-0.357483\pi$
$491$	9420.24	0.865844	0.432922	+	0.901431i	$0.357483\pi$
$492$	0	0
$493$	−1023.32	−0.0934852
$494$	0	0
$495$	−95.0914	−0.00863443
$496$	0	0
$497$	722.524	0.0652105
$498$	0	0
$499$	4767.15	0.427669	0.213834	−	0.976870i	$-0.431405\pi$
$499$	4767.15	0.427669	0.213834	+	0.976870i	$0.431405\pi$
$500$	0	0
$501$	10622.9	0.947294
$502$	0	0
$503$	−2236.35	−0.198239	−0.0991193	−	0.995076i	$-0.531603\pi$
$503$	−2236.35	−0.198239	−0.0991193	+	0.995076i	$0.531603\pi$
$504$	0	0
$505$	26694.1	2.35222
$506$	0	0
$507$	−4793.85	−0.419925
$508$	0	0
$509$	7812.74	0.680341	0.340171	−	0.940364i	$-0.389515\pi$
$509$	7812.74	0.680341	0.340171	+	0.940364i	$0.389515\pi$
$510$	0	0
$511$	−3061.54	−0.265038
$512$	0	0
$513$	−21101.0	−1.81604
$514$	0	0
$515$	22293.1	1.90748
$516$	0	0
$517$	283.250	0.0240954
$518$	0	0
$519$	−2458.09	−0.207896
$520$	0	0
$521$	10163.2	0.854622	0.427311	−	0.904105i	$-0.359461\pi$
$521$	10163.2	0.854622	0.427311	+	0.904105i	$0.359461\pi$
$522$	0	0
$523$	−22414.0	−1.87399	−0.936996	−	0.349340i	$-0.886406\pi$
$523$	−22414.0	−1.87399	−0.936996	+	0.349340i	$0.886406\pi$
$524$	0	0
$525$	5025.70	0.417789
$526$	0	0
$527$	−1588.63	−0.131312
$528$	0	0
$529$	−5883.03	−0.483524
$530$	0	0
$531$	3353.11	0.274035
$532$	0	0
$533$	−1439.98	−0.117022
$534$	0	0
$535$	17172.8	1.38775
$536$	0	0
$537$	−9631.79	−0.774008
$538$	0	0
$539$	−340.593	−0.0272178
$540$	0	0
$541$	7852.12	0.624010	0.312005	−	0.950081i	$-0.398999\pi$
$541$	7852.12	0.624010	0.312005	+	0.950081i	$0.398999\pi$
$542$	0	0
$543$	−775.329	−0.0612755
$544$	0	0
$545$	−43478.5	−3.41727
$546$	0	0
$547$	−22825.1	−1.78415	−0.892077	−	0.451884i	$-0.850752\pi$
$547$	−22825.1	−1.78415	−0.892077	+	0.451884i	$0.850752\pi$
$548$	0	0
$549$	3126.88	0.243082
$550$	0	0
$551$	2821.36	0.218138
$552$	0	0
$553$	3894.82	0.299502
$554$	0	0
$555$	−29592.5	−2.26330
$556$	0	0
$557$	2674.18	0.203427	0.101713	−	0.994814i	$-0.467568\pi$
$557$	2674.18	0.203427	0.101713	+	0.994814i	$0.467568\pi$
$558$	0	0
$559$	−4920.21	−0.372276
$560$	0	0
$561$	−250.356	−0.0188414
$562$	0	0
$563$	17811.4	1.33332	0.666662	−	0.745360i	$-0.267723\pi$
$563$	17811.4	1.33332	0.666662	+	0.745360i	$0.267723\pi$
$564$	0	0
$565$	−26391.5	−1.96513
$566$	0	0
$567$	−2035.46	−0.150761
$568$	0	0
$569$	−19515.7	−1.43786	−0.718930	−	0.695082i	$-0.755368\pi$
$569$	−19515.7	−1.43786	−0.718930	+	0.695082i	$0.755368\pi$
$570$	0	0
$571$	161.694	0.0118506	0.00592530	−	0.999982i	$-0.498114\pi$
$571$	161.694	0.0118506	0.00592530	+	0.999982i	$0.498114\pi$
$572$	0	0
$573$	−12776.6	−0.931503
$574$	0	0
$575$	−24278.1	−1.76081
$576$	0	0
$577$	15008.2	1.08284	0.541422	−	0.840751i	$-0.317886\pi$
$577$	15008.2	1.08284	0.541422	+	0.840751i	$0.317886\pi$
$578$	0	0
$579$	−6725.91	−0.482762
$580$	0	0
$581$	−256.767	−0.0183348
$582$	0	0
$583$	−126.535	−0.00898892
$584$	0	0
$585$	3185.09	0.225106
$586$	0	0
$587$	−1676.62	−0.117890	−0.0589451	−	0.998261i	$-0.518774\pi$
$587$	−1676.62	−0.117890	−0.0589451	+	0.998261i	$0.518774\pi$
$588$	0	0
$589$	4379.92	0.306404
$590$	0	0
$591$	21649.2	1.50681
$592$	0	0
$593$	12616.6	0.873696	0.436848	−	0.899535i	$-0.356095\pi$
$593$	12616.6	0.873696	0.436848	+	0.899535i	$0.356095\pi$
$594$	0	0
$595$	−3677.61	−0.253391
$596$	0	0
$597$	23765.2	1.62922
$598$	0	0
$599$	26960.5	1.83902	0.919511	−	0.393063i	$-0.128585\pi$
$599$	26960.5	1.83902	0.919511	+	0.393063i	$0.128585\pi$
$600$	0	0
$601$	−18760.4	−1.27330	−0.636648	−	0.771155i	$-0.719680\pi$
$601$	−18760.4	−1.27330	−0.636648	+	0.771155i	$0.719680\pi$
$602$	0	0
$603$	679.052	0.0458593
$604$	0	0
$605$	−27618.8	−1.85597
$606$	0	0
$607$	−8692.30	−0.581235	−0.290617	−	0.956839i	$-0.593861\pi$
$607$	−8692.30	−0.581235	−0.290617	+	0.956839i	$0.593861\pi$
$608$	0	0
$609$	327.682	0.0218035
$610$	0	0
$611$	−9487.48	−0.628187
$612$	0	0
$613$	−24608.2	−1.62140	−0.810699	−	0.585463i	$-0.800913\pi$
$613$	−24608.2	−1.62140	−0.810699	+	0.585463i	$0.800913\pi$
$614$	0	0
$615$	4120.82	0.270191
$616$	0	0
$617$	−22964.1	−1.49838	−0.749188	−	0.662357i	$-0.769556\pi$
$617$	−22964.1	−1.49838	−0.749188	+	0.662357i	$0.769556\pi$
$618$	0	0
$619$	−2349.06	−0.152531	−0.0762656	−	0.997088i	$-0.524300\pi$
$619$	−2349.06	−0.152531	−0.0762656	+	0.997088i	$0.524300\pi$
$620$	0	0
$621$	11839.0	0.765027
$622$	0	0
$623$	3553.08	0.228493
$624$	0	0
$625$	39890.1	2.55296
$626$	0	0
$627$	690.243	0.0439644
$628$	0	0
$629$	15378.2	0.974829
$630$	0	0
$631$	−12074.2	−0.761753	−0.380876	−	0.924626i	$-0.624378\pi$
$631$	−12074.2	−0.761753	−0.380876	+	0.924626i	$0.624378\pi$
$632$	0	0
$633$	10212.1	0.641225
$634$	0	0
$635$	−8778.70	−0.548618
$636$	0	0
$637$	11408.2	0.709589
$638$	0	0
$639$	−930.585	−0.0576109
$640$	0	0
$641$	1269.04	0.0781965	0.0390983	−	0.999235i	$-0.487551\pi$
$641$	1269.04	0.0781965	0.0390983	+	0.999235i	$0.487551\pi$
$642$	0	0
$643$	−18368.4	−1.12656	−0.563281	−	0.826266i	$-0.690461\pi$
$643$	−18368.4	−1.12656	−0.563281	+	0.826266i	$0.690461\pi$
$644$	0	0
$645$	14080.2	0.859547
$646$	0	0
$647$	−22345.8	−1.35781	−0.678906	−	0.734225i	$-0.737546\pi$
$647$	−22345.8	−1.35781	−0.678906	+	0.734225i	$0.737546\pi$
$648$	0	0
$649$	−775.071	−0.0468786
$650$	0	0
$651$	508.699	0.0306259
$652$	0	0
$653$	16293.9	0.976461	0.488231	−	0.872715i	$-0.337643\pi$
$653$	16293.9	0.976461	0.488231	+	0.872715i	$0.337643\pi$
$654$	0	0
$655$	1073.03	0.0640105
$656$	0	0
$657$	3943.16	0.234151
$658$	0	0
$659$	−1578.08	−0.0932824	−0.0466412	−	0.998912i	$-0.514852\pi$
$659$	−1578.08	−0.0932824	−0.0466412	+	0.998912i	$0.514852\pi$
$660$	0	0
$661$	4243.36	0.249693	0.124847	−	0.992176i	$-0.460156\pi$
$661$	4243.36	0.249693	0.124847	+	0.992176i	$0.460156\pi$
$662$	0	0
$663$	8385.67	0.491210
$664$	0	0
$665$	10139.4	0.591260
$666$	0	0
$667$	−1582.96	−0.0918927
$668$	0	0
$669$	−6328.37	−0.365723
$670$	0	0
$671$	−722.777	−0.0415835
$672$	0	0
$673$	−21299.1	−1.21994	−0.609970	−	0.792424i	$-0.708819\pi$
$673$	−21299.1	−1.21994	−0.609970	+	0.792424i	$0.708819\pi$
$674$	0	0
$675$	−45739.8	−2.60819
$676$	0	0
$677$	31866.5	1.80905	0.904525	−	0.426420i	$-0.140225\pi$
$677$	31866.5	1.80905	0.904525	+	0.426420i	$0.140225\pi$
$678$	0	0
$679$	3251.82	0.183790
$680$	0	0
$681$	1277.26	0.0718718
$682$	0	0
$683$	−21008.5	−1.17697	−0.588483	−	0.808509i	$-0.700275\pi$
$683$	−21008.5	−1.17697	−0.588483	+	0.808509i	$0.700275\pi$
$684$	0	0
$685$	−27259.4	−1.52048
$686$	0	0
$687$	−16195.0	−0.899388
$688$	0	0
$689$	4238.29	0.234348
$690$	0	0
$691$	11981.2	0.659606	0.329803	−	0.944050i	$-0.393018\pi$
$691$	11981.2	0.659606	0.329803	+	0.944050i	$0.393018\pi$
$692$	0	0
$693$	−15.8235	−0.000867367 0
$694$	0	0
$695$	38256.2	2.08797
$696$	0	0
$697$	−2141.44	−0.116374
$698$	0	0
$699$	20145.4	1.09008
$700$	0	0
$701$	−13177.2	−0.709979	−0.354990	−	0.934870i	$-0.615516\pi$
$701$	−13177.2	−0.709979	−0.354990	+	0.934870i	$0.615516\pi$
$702$	0	0
$703$	−42398.4	−2.27466
$704$	0	0
$705$	27150.4	1.45042
$706$	0	0
$707$	4441.98	0.236291
$708$	0	0
$709$	−18688.6	−0.989934	−0.494967	−	0.868912i	$-0.664820\pi$
$709$	−18688.6	−0.989934	−0.494967	+	0.868912i	$0.664820\pi$
$710$	0	0
$711$	−5016.39	−0.264598
$712$	0	0
$713$	−2457.42	−0.129076
$714$	0	0
$715$	−736.233	−0.0385085
$716$	0	0
$717$	27176.6	1.41552
$718$	0	0
$719$	26534.0	1.37629	0.688145	−	0.725573i	$-0.258425\pi$
$719$	26534.0	1.37629	0.688145	+	0.725573i	$0.258425\pi$
$720$	0	0
$721$	3709.64	0.191615
$722$	0	0
$723$	17058.1	0.877452
$724$	0	0
$725$	6115.76	0.313288
$726$	0	0
$727$	25956.2	1.32416	0.662078	−	0.749435i	$-0.269675\pi$
$727$	25956.2	1.32416	0.662078	+	0.749435i	$0.269675\pi$
$728$	0	0
$729$	21769.7	1.10602
$730$	0	0
$731$	−7316.98	−0.370216
$732$	0	0
$733$	13018.2	0.655986	0.327993	−	0.944680i	$-0.393628\pi$
$733$	13018.2	0.655986	0.327993	+	0.944680i	$0.393628\pi$
$734$	0	0
$735$	−32646.9	−1.63837
$736$	0	0
$737$	−156.963	−0.00784505
$738$	0	0
$739$	26134.0	1.30089	0.650444	−	0.759554i	$-0.274583\pi$
$739$	26134.0	1.30089	0.650444	+	0.759554i	$0.274583\pi$
$740$	0	0
$741$	−23119.7	−1.14619
$742$	0	0
$743$	−10380.9	−0.512569	−0.256285	−	0.966601i	$-0.582498\pi$
$743$	−10380.9	−0.512569	−0.256285	+	0.966601i	$0.582498\pi$
$744$	0	0
$745$	65734.7	3.23266
$746$	0	0
$747$	330.707	0.0161981
$748$	0	0
$749$	2857.61	0.139406
$750$	0	0
$751$	−28295.4	−1.37485	−0.687425	−	0.726255i	$-0.741259\pi$
$751$	−28295.4	−1.37485	−0.687425	+	0.726255i	$0.741259\pi$
$752$	0	0
$753$	31510.7	1.52499
$754$	0	0
$755$	−22991.6	−1.10828
$756$	0	0
$757$	−4719.47	−0.226594	−0.113297	−	0.993561i	$-0.536141\pi$
$757$	−4719.47	−0.226594	−0.113297	+	0.993561i	$0.536141\pi$
$758$	0	0
$759$	−387.270	−0.0185204
$760$	0	0
$761$	9630.13	0.458728	0.229364	−	0.973341i	$-0.426335\pi$
$761$	9630.13	0.458728	0.229364	+	0.973341i	$0.426335\pi$
$762$	0	0
$763$	−7234.96	−0.343281
$764$	0	0
$765$	4736.64	0.223861
$766$	0	0
$767$	25961.0	1.22216
$768$	0	0
$769$	−17078.6	−0.800872	−0.400436	−	0.916325i	$-0.631141\pi$
$769$	−17078.6	−0.800872	−0.400436	+	0.916325i	$0.631141\pi$
$770$	0	0
$771$	25306.1	1.18207
$772$	0	0
$773$	−35075.2	−1.63204	−0.816021	−	0.578022i	$-0.803825\pi$
$773$	−35075.2	−1.63204	−0.816021	+	0.578022i	$0.803825\pi$
$774$	0	0
$775$	9494.21	0.440054
$776$	0	0
$777$	−4924.29	−0.227359
$778$	0	0
$779$	5904.06	0.271547
$780$	0	0
$781$	215.105	0.00985538
$782$	0	0
$783$	−2982.29	−0.136116
$784$	0	0
$785$	50668.4	2.30374
$786$	0	0
$787$	1704.64	0.0772096	0.0386048	−	0.999255i	$-0.487709\pi$
$787$	1704.64	0.0772096	0.0386048	+	0.999255i	$0.487709\pi$
$788$	0	0
$789$	−36135.2	−1.63048
$790$	0	0
$791$	−4391.64	−0.197407
$792$	0	0
$793$	24209.4	1.08411
$794$	0	0
$795$	−12128.8	−0.541085
$796$	0	0
$797$	−28317.4	−1.25854	−0.629268	−	0.777188i	$-0.716645\pi$
$797$	−28317.4	−1.25854	−0.629268	+	0.777188i	$0.716645\pi$
$798$	0	0
$799$	−14109.1	−0.624711
$800$	0	0
$801$	−4576.24	−0.201865
$802$	0	0
$803$	−911.460	−0.0400557
$804$	0	0
$805$	−5688.82	−0.249074
$806$	0	0
$807$	22411.4	0.977593
$808$	0	0
$809$	−2028.75	−0.0881667	−0.0440834	−	0.999028i	$-0.514037\pi$
$809$	−2028.75	−0.0881667	−0.0440834	+	0.999028i	$0.514037\pi$
$810$	0	0
$811$	−66.7220	−0.00288894	−0.00144447	−	0.999999i	$-0.500460\pi$
$811$	−66.7220	−0.00288894	−0.00144447	+	0.999999i	$0.500460\pi$
$812$	0	0
$813$	8604.86	0.371200
$814$	0	0
$815$	−64164.6	−2.75778
$816$	0	0
$817$	20173.3	0.863860
$818$	0	0
$819$	530.009	0.0226129
$820$	0	0
$821$	1399.55	0.0594940	0.0297470	−	0.999557i	$-0.490530\pi$
$821$	1399.55	0.0594940	0.0297470	+	0.999557i	$0.490530\pi$
$822$	0	0
$823$	28695.9	1.21540	0.607701	−	0.794166i	$-0.292092\pi$
$823$	28695.9	1.21540	0.607701	+	0.794166i	$0.292092\pi$
$824$	0	0
$825$	1496.22	0.0631413
$826$	0	0
$827$	17607.3	0.740344	0.370172	−	0.928963i	$-0.379299\pi$
$827$	17607.3	0.740344	0.370172	+	0.928963i	$0.379299\pi$
$828$	0	0
$829$	−27785.1	−1.16407	−0.582037	−	0.813163i	$-0.697744\pi$
$829$	−27785.1	−1.16407	−0.582037	+	0.813163i	$0.697744\pi$
$830$	0	0
$831$	21536.8	0.899042
$832$	0	0
$833$	16965.4	0.705663
$834$	0	0
$835$	46456.6	1.92539
$836$	0	0
$837$	−4629.76	−0.191192
$838$	0	0
$839$	−16185.1	−0.665998	−0.332999	−	0.942927i	$-0.608061\pi$
$839$	−16185.1	−0.665998	−0.332999	+	0.942927i	$0.608061\pi$
$840$	0	0
$841$	−23990.2	−0.983650
$842$	0	0
$843$	7267.81	0.296936
$844$	0	0
$845$	−20964.8	−0.853504
$846$	0	0
$847$	−4595.85	−0.186441
$848$	0	0
$849$	−19609.7	−0.792701
$850$	0	0
$851$	23788.2	0.958224
$852$	0	0
$853$	−14232.7	−0.571301	−0.285651	−	0.958334i	$-0.592210\pi$
$853$	−14232.7	−0.571301	−0.285651	+	0.958334i	$0.592210\pi$
$854$	0	0
$855$	−13059.1	−0.522355
$856$	0	0
$857$	3028.79	0.120725	0.0603627	−	0.998177i	$-0.480774\pi$
$857$	3028.79	0.120725	0.0603627	+	0.998177i	$0.480774\pi$
$858$	0	0
$859$	17614.2	0.699636	0.349818	−	0.936818i	$-0.386243\pi$
$859$	17614.2	0.699636	0.349818	+	0.936818i	$0.386243\pi$
$860$	0	0
$861$	685.717	0.0271419
$862$	0	0
$863$	−7914.33	−0.312175	−0.156087	−	0.987743i	$-0.549888\pi$
$863$	−7914.33	−0.312175	−0.156087	+	0.987743i	$0.549888\pi$
$864$	0	0
$865$	−10749.9	−0.422551
$866$	0	0
$867$	−10859.3	−0.425377
$868$	0	0
$869$	1159.54	0.0452643
$870$	0	0
$871$	5257.47	0.204527
$872$	0	0
$873$	−4188.23	−0.162371
$874$	0	0
$875$	13008.3	0.502582
$876$	0	0
$877$	−4103.70	−0.158007	−0.0790036	−	0.996874i	$-0.525174\pi$
$877$	−4103.70	−0.158007	−0.0790036	+	0.996874i	$0.525174\pi$
$878$	0	0
$879$	−2236.21	−0.0858083
$880$	0	0
$881$	−2945.45	−0.112639	−0.0563194	−	0.998413i	$-0.517937\pi$
$881$	−2945.45	−0.112639	−0.0563194	+	0.998413i	$0.517937\pi$
$882$	0	0
$883$	2693.78	0.102665	0.0513323	−	0.998682i	$-0.483653\pi$
$883$	2693.78	0.102665	0.0513323	+	0.998682i	$0.483653\pi$
$884$	0	0
$885$	−74292.9	−2.82184
$886$	0	0
$887$	28751.0	1.08835	0.544174	−	0.838972i	$-0.316843\pi$
$887$	28751.0	1.08835	0.544174	+	0.838972i	$0.316843\pi$
$888$	0	0
$889$	−1460.80	−0.0551111
$890$	0	0
$891$	−605.983	−0.0227847
$892$	0	0
$893$	38899.5	1.45770
$894$	0	0
$895$	−42122.4	−1.57318
$896$	0	0
$897$	12971.6	0.482843
$898$	0	0
$899$	619.034	0.0229655
$900$	0	0
$901$	6302.88	0.233052
$902$	0	0
$903$	2342.99	0.0863454
$904$	0	0
$905$	−3390.73	−0.124543
$906$	0	0
$907$	29209.6	1.06934	0.534669	−	0.845062i	$-0.320436\pi$
$907$	29209.6	1.06934	0.534669	+	0.845062i	$0.320436\pi$
$908$	0	0
$909$	−5721.11	−0.208754
$910$	0	0
$911$	12618.5	0.458914	0.229457	−	0.973319i	$-0.426305\pi$
$911$	12618.5	0.458914	0.229457	+	0.973319i	$0.426305\pi$
$912$	0	0
$913$	−76.4430	−0.00277097
$914$	0	0
$915$	−69280.4	−2.50310
$916$	0	0
$917$	178.556	0.00643015
$918$	0	0
$919$	33307.1	1.19554	0.597769	−	0.801668i	$-0.296054\pi$
$919$	33307.1	1.19554	0.597769	+	0.801668i	$0.296054\pi$
$920$	0	0
$921$	12196.2	0.436349
$922$	0	0
$923$	−7204.94	−0.256938
$924$	0	0
$925$	−91905.5	−3.26685
$926$	0	0
$927$	−4777.89	−0.169284
$928$	0	0
$929$	14169.9	0.500431	0.250215	−	0.968190i	$-0.419499\pi$
$929$	14169.9	0.500431	0.250215	+	0.968190i	$0.419499\pi$
$930$	0	0
$931$	−46774.5	−1.64659
$932$	0	0
$933$	8901.83	0.312361
$934$	0	0
$935$	−1094.87	−0.0382954
$936$	0	0
$937$	−48123.7	−1.67784	−0.838919	−	0.544256i	$-0.816812\pi$
$937$	−48123.7	−1.67784	−0.838919	+	0.544256i	$0.816812\pi$
$938$	0	0
$939$	35510.9	1.23414
$940$	0	0
$941$	−8150.07	−0.282343	−0.141171	−	0.989985i	$-0.545087\pi$
$941$	−8150.07	−0.282343	−0.141171	+	0.989985i	$0.545087\pi$
$942$	0	0
$943$	−3312.55	−0.114392
$944$	0	0
$945$	−10717.7	−0.368940
$946$	0	0
$947$	−36794.4	−1.26257	−0.631286	−	0.775550i	$-0.717473\pi$
$947$	−36794.4	−1.26257	−0.631286	+	0.775550i	$0.717473\pi$
$948$	0	0
$949$	30529.4	1.04428
$950$	0	0
$951$	8012.61	0.273214
$952$	0	0
$953$	−15137.6	−0.514537	−0.257268	−	0.966340i	$-0.582822\pi$
$953$	−15137.6	−0.514537	−0.257268	+	0.966340i	$0.582822\pi$
$954$	0	0
$955$	−55875.6	−1.89329
$956$	0	0
$957$	97.5551	0.00329520
$958$	0	0
$959$	−4536.05	−0.152739
$960$	0	0
$961$	961.000	0.0322581
$962$	0	0
$963$	−3680.51	−0.123160
$964$	0	0
$965$	−29414.2	−0.981221
$966$	0	0
$967$	−10337.4	−0.343773	−0.171887	−	0.985117i	$-0.554986\pi$
$967$	−10337.4	−0.343773	−0.171887	+	0.985117i	$0.554986\pi$
$968$	0	0
$969$	−34382.0	−1.13984
$970$	0	0
$971$	−30059.1	−0.993452	−0.496726	−	0.867907i	$-0.665465\pi$
$971$	−30059.1	−0.993452	−0.496726	+	0.867907i	$0.665465\pi$
$972$	0	0
$973$	6365.95	0.209746
$974$	0	0
$975$	−50115.8	−1.64614
$976$	0	0
$977$	10488.0	0.343439	0.171720	−	0.985146i	$-0.445068\pi$
$977$	10488.0	0.343439	0.171720	+	0.985146i	$0.445068\pi$
$978$	0	0
$979$	1057.80	0.0345325
$980$	0	0
$981$	9318.38	0.303275
$982$	0	0
$983$	60097.5	1.94996	0.974981	−	0.222289i	$-0.0713530\pi$
$983$	60097.5	1.94996	0.974981	+	0.222289i	$0.0713530\pi$
$984$	0	0
$985$	94677.6	3.06262
$986$	0	0
$987$	4517.92	0.145701
$988$	0	0
$989$	−11318.5	−0.363910
$990$	0	0
$991$	−61345.2	−1.96639	−0.983197	−	0.182549i	$-0.941565\pi$
$991$	−61345.2	−1.96639	−0.983197	+	0.182549i	$0.941565\pi$
$992$	0	0
$993$	−29728.9	−0.950067
$994$	0	0
$995$	103932.	3.31141
$996$	0	0
$997$	34825.9	1.10627	0.553134	−	0.833093i	$-0.313432\pi$
$997$	34825.9	1.10627	0.553134	+	0.833093i	$0.313432\pi$
$998$	0	0
$999$	44816.9	1.41936

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	124.4.a.b.1.3	✓	4
3.2	odd	2		1116.4.a.f.1.1		4
4.3	odd	2		496.4.a.f.1.2		4
8.3	odd	2		1984.4.a.o.1.3		4
8.5	even	2		1984.4.a.m.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
124.4.a.b.1.3	✓	4	1.1	even	1	trivial
496.4.a.f.1.2		4	4.3	odd	2
1116.4.a.f.1.1		4	3.2	odd	2
1984.4.a.m.1.2		4	8.5	even	2
1984.4.a.o.1.3		4	8.3	odd	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:	\( N \)	\(=\)	\( 124 = 2^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	124.a (trivial)

\(f(q)\)	\(=\)	\(q+4.74860 q^{3} +20.7669 q^{5} +3.45568 q^{7} -4.45080 q^{9} +O(q^{10})\)	\(q+4.74860 q^{3} +20.7669 q^{5} +3.45568 q^{7} -4.45080 q^{9} +1.02880 q^{11} -34.4597 q^{13} +98.6138 q^{15} -51.2460 q^{17} +141.288 q^{19} +16.4096 q^{21} -79.2715 q^{23} +306.265 q^{25} -149.347 q^{27} +19.9689 q^{29} +31.0000 q^{31} +4.88537 q^{33} +71.7638 q^{35} -300.085 q^{37} -163.635 q^{39} +41.7875 q^{41} +142.781 q^{43} -92.4293 q^{45} +275.321 q^{47} -331.058 q^{49} -243.347 q^{51} -122.993 q^{53} +21.3650 q^{55} +670.920 q^{57} -753.373 q^{59} -702.543 q^{61} -15.3805 q^{63} -715.622 q^{65} -152.569 q^{67} -376.429 q^{69} +209.083 q^{71} -885.944 q^{73} +1454.33 q^{75} +3.55521 q^{77} +1127.08 q^{79} -589.019 q^{81} -74.3029 q^{83} -1064.22 q^{85} +94.8241 q^{87} +1028.18 q^{89} -119.082 q^{91} +147.207 q^{93} +2934.11 q^{95} +941.006 q^{97} -4.57899 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} + 6 q^{5} + 16 q^{7} + 64 q^{9}+O(q^{10}) \) 4 * q + 2 * q^3 + 6 * q^5 + 16 * q^7 + 64 * q^9	\( 4 q + 2 q^{3} + 6 q^{5} + 16 q^{7} + 64 q^{9} + 96 q^{11} + 42 q^{13} + 182 q^{15} - 2 q^{17} + 180 q^{19} + 150 q^{21} + 142 q^{23} + 150 q^{25} + 20 q^{27} + 262 q^{29} + 124 q^{31} - 444 q^{33} + 224 q^{35} - 284 q^{37} + 60 q^{39} - 526 q^{41} - 326 q^{43} - 870 q^{45} + 468 q^{47} - 978 q^{49} + 356 q^{51} - 252 q^{53} - 876 q^{55} - 1298 q^{57} - 164 q^{59} - 1066 q^{61} - 236 q^{63} - 598 q^{65} - 956 q^{69} + 1504 q^{71} - 732 q^{73} + 188 q^{75} - 288 q^{77} + 822 q^{79} - 536 q^{81} + 1408 q^{83} + 482 q^{85} - 208 q^{87} + 250 q^{89} + 946 q^{91} + 62 q^{93} + 2292 q^{95} + 526 q^{97} + 2952 q^{99}+O(q^{100}) \) 4 * q + 2 * q^3 + 6 * q^5 + 16 * q^7 + 64 * q^9 + 96 * q^11 + 42 * q^13 + 182 * q^15 - 2 * q^17 + 180 * q^19 + 150 * q^21 + 142 * q^23 + 150 * q^25 + 20 * q^27 + 262 * q^29 + 124 * q^31 - 444 * q^33 + 224 * q^35 - 284 * q^37 + 60 * q^39 - 526 * q^41 - 326 * q^43 - 870 * q^45 + 468 * q^47 - 978 * q^49 + 356 * q^51 - 252 * q^53 - 876 * q^55 - 1298 * q^57 - 164 * q^59 - 1066 * q^61 - 236 * q^63 - 598 * q^65 - 956 * q^69 + 1504 * q^71 - 732 * q^73 + 188 * q^75 - 288 * q^77 + 822 * q^79 - 536 * q^81 + 1408 * q^83 + 482 * q^85 - 208 * q^87 + 250 * q^89 + 946 * q^91 + 62 * q^93 + 2292 * q^95 + 526 * q^97 + 2952 * q^99

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