LMFDB - Embedded newform 4023.2.a.f.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4023,2,Mod(1,4023)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4023, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4023.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4023 = 3^{3} \cdot 149 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4023.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:	$32.1238167332$
Analytic rank:	$0$
Dimension:	$25$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Character	$\chi$	$=$	4023.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.964175 q^{2} -1.07037 q^{4} -0.686967 q^{5} -3.77336 q^{7} -2.96037 q^{8} +O(q^{10})$	\(q+0.964175 q^{2} -1.07037 q^{4} -0.686967 q^{5} -3.77336 q^{7} -2.96037 q^{8} -0.662356 q^{10} +2.33234 q^{11} -5.60879 q^{13} -3.63818 q^{14} -0.713581 q^{16} -0.836479 q^{17} +3.09439 q^{19} +0.735306 q^{20} +2.24878 q^{22} -7.69646 q^{23} -4.52808 q^{25} -5.40785 q^{26} +4.03888 q^{28} -8.52286 q^{29} +7.78128 q^{31} +5.23272 q^{32} -0.806512 q^{34} +2.59217 q^{35} +8.23554 q^{37} +2.98353 q^{38} +2.03368 q^{40} -8.33445 q^{41} +5.70950 q^{43} -2.49646 q^{44} -7.42073 q^{46} +9.23459 q^{47} +7.23822 q^{49} -4.36586 q^{50} +6.00346 q^{52} +14.1692 q^{53} -1.60224 q^{55} +11.1705 q^{56} -8.21753 q^{58} +4.88546 q^{59} -12.7269 q^{61} +7.50252 q^{62} +6.47242 q^{64} +3.85305 q^{65} +6.81042 q^{67} +0.895339 q^{68} +2.49931 q^{70} +3.58727 q^{71} -3.65812 q^{73} +7.94050 q^{74} -3.31213 q^{76} -8.80075 q^{77} -15.2640 q^{79} +0.490206 q^{80} -8.03587 q^{82} -3.01862 q^{83} +0.574633 q^{85} +5.50496 q^{86} -6.90459 q^{88} -10.6104 q^{89} +21.1640 q^{91} +8.23803 q^{92} +8.90375 q^{94} -2.12574 q^{95} +0.288109 q^{97} +6.97891 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q + 7 q^{2} + 27 q^{4} + 12 q^{5} - 2 q^{7} + 21 q^{8}+O(q^{10}) $ 25 * q + 7 * q^2 + 27 * q^4 + 12 * q^5 - 2 * q^7 + 21 * q^8	$ 25 q + 7 q^{2} + 27 q^{4} + 12 q^{5} - 2 q^{7} + 21 q^{8} - 4 q^{10} + 12 q^{11} + 10 q^{14} + 35 q^{16} + 26 q^{17} + 30 q^{20} + 8 q^{22} + 26 q^{23} + 27 q^{25} + 16 q^{26} + 4 q^{28} + 20 q^{29} - 6 q^{31} + 49 q^{32} - 14 q^{34} + 16 q^{35} - 2 q^{37} + 27 q^{38} + 2 q^{40} + 35 q^{41} + 4 q^{43} + 22 q^{44} + 6 q^{46} + 38 q^{47} + 19 q^{49} + 22 q^{50} + 4 q^{52} + 36 q^{53} + 10 q^{55} + 79 q^{56} - 22 q^{58} + 15 q^{59} + 10 q^{61} - 14 q^{62} + 41 q^{64} + 80 q^{65} - 6 q^{67} + 33 q^{68} + 8 q^{70} + 26 q^{71} - 6 q^{73} + 75 q^{74} - 10 q^{76} + 47 q^{77} - 6 q^{79} + 66 q^{80} + 12 q^{82} + 40 q^{83} - 12 q^{85} + 4 q^{86} + 12 q^{88} + 30 q^{89} + 158 q^{92} + 18 q^{94} + 22 q^{95} - 20 q^{97} + 35 q^{98}+O(q^{100}) $ 25 * q + 7 * q^2 + 27 * q^4 + 12 * q^5 - 2 * q^7 + 21 * q^8 - 4 * q^10 + 12 * q^11 + 10 * q^14 + 35 * q^16 + 26 * q^17 + 30 * q^20 + 8 * q^22 + 26 * q^23 + 27 * q^25 + 16 * q^26 + 4 * q^28 + 20 * q^29 - 6 * q^31 + 49 * q^32 - 14 * q^34 + 16 * q^35 - 2 * q^37 + 27 * q^38 + 2 * q^40 + 35 * q^41 + 4 * q^43 + 22 * q^44 + 6 * q^46 + 38 * q^47 + 19 * q^49 + 22 * q^50 + 4 * q^52 + 36 * q^53 + 10 * q^55 + 79 * q^56 - 22 * q^58 + 15 * q^59 + 10 * q^61 - 14 * q^62 + 41 * q^64 + 80 * q^65 - 6 * q^67 + 33 * q^68 + 8 * q^70 + 26 * q^71 - 6 * q^73 + 75 * q^74 - 10 * q^76 + 47 * q^77 - 6 * q^79 + 66 * q^80 + 12 * q^82 + 40 * q^83 - 12 * q^85 + 4 * q^86 + 12 * q^88 + 30 * q^89 + 158 * q^92 + 18 * q^94 + 22 * q^95 - 20 * q^97 + 35 * 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 $

$2$	0.964175	0.681775	0.340887	−	0.940104i	$-0.389273\pi$
$2$	0.964175	0.681775	0.340887	+	0.940104i	$0.389273\pi$
$3$	0	0
$3$	0	0
$4$	−1.07037	−0.535183
$5$	−0.686967	−0.307221	−0.153610	−	0.988131i	$-0.549090\pi$
$5$	−0.686967	−0.307221	−0.153610	+	0.988131i	$0.549090\pi$
$6$	0	0
$7$	−3.77336	−1.42619	−0.713097	−	0.701065i	$-0.752708\pi$
$7$	−3.77336	−1.42619	−0.713097	+	0.701065i	$0.752708\pi$
$8$	−2.96037	−1.04665
$9$	0	0
$10$	−0.662356	−0.209455
$11$	2.33234	0.703227	0.351614	−	0.936145i	$-0.385633\pi$
$11$	2.33234	0.703227	0.351614	+	0.936145i	$0.385633\pi$
$12$	0	0
$13$	−5.60879	−1.55560	−0.777799	−	0.628513i	$-0.783664\pi$
$13$	−5.60879	−1.55560	−0.777799	+	0.628513i	$0.783664\pi$
$14$	−3.63818	−0.972343
$15$	0	0
$16$	−0.713581	−0.178395
$17$	−0.836479	−0.202876	−0.101438	−	0.994842i	$-0.532344\pi$
$17$	−0.836479	−0.202876	−0.101438	+	0.994842i	$0.532344\pi$
$18$	0	0
$19$	3.09439	0.709902	0.354951	−	0.934885i	$-0.384498\pi$
$19$	3.09439	0.709902	0.354951	+	0.934885i	$0.384498\pi$
$20$	0.735306	0.164420
$21$	0	0
$22$	2.24878	0.479442
$23$	−7.69646	−1.60482	−0.802411	−	0.596771i	$-0.796450\pi$
$23$	−7.69646	−1.60482	−0.802411	+	0.596771i	$0.796450\pi$
$24$	0	0
$25$	−4.52808	−0.905615
$26$	−5.40785	−1.06057
$27$	0	0
$28$	4.03888	0.763276
$29$	−8.52286	−1.58266	−0.791328	−	0.611392i	$-0.790610\pi$
$29$	−8.52286	−1.58266	−0.791328	+	0.611392i	$0.790610\pi$
$30$	0	0
$31$	7.78128	1.39756	0.698780	−	0.715337i	$-0.253727\pi$
$31$	7.78128	1.39756	0.698780	+	0.715337i	$0.253727\pi$
$32$	5.23272	0.925024
$33$	0	0
$34$	−0.806512	−0.138316
$35$	2.59217	0.438157
$36$	0	0
$37$	8.23554	1.35391	0.676957	−	0.736022i	$-0.263298\pi$
$37$	8.23554	1.35391	0.676957	+	0.736022i	$0.263298\pi$
$38$	2.98353	0.483993
$39$	0	0
$40$	2.03368	0.321552
$41$	−8.33445	−1.30162	−0.650811	−	0.759239i	$-0.725571\pi$
$41$	−8.33445	−1.30162	−0.650811	+	0.759239i	$0.725571\pi$
$42$	0	0
$43$	5.70950	0.870691	0.435345	−	0.900264i	$-0.356626\pi$
$43$	5.70950	0.870691	0.435345	+	0.900264i	$0.356626\pi$
$44$	−2.49646	−0.376356
$45$	0	0
$46$	−7.42073	−1.09413
$47$	9.23459	1.34700	0.673501	−	0.739186i	$-0.264790\pi$
$47$	9.23459	1.34700	0.673501	+	0.739186i	$0.264790\pi$
$48$	0	0
$49$	7.23822	1.03403
$50$	−4.36586	−0.617425
$51$	0	0
$52$	6.00346	0.832530
$53$	14.1692	1.94628	0.973142	−	0.230205i	$-0.0739396\pi$
$53$	14.1692	1.94628	0.973142	+	0.230205i	$0.0739396\pi$
$54$	0	0
$55$	−1.60224	−0.216046
$56$	11.1705	1.49273
$57$	0	0
$58$	−8.21753	−1.07901
$59$	4.88546	0.636033	0.318016	−	0.948085i	$-0.396983\pi$
$59$	4.88546	0.636033	0.318016	+	0.948085i	$0.396983\pi$
$60$	0	0
$61$	−12.7269	−1.62951	−0.814757	−	0.579803i	$-0.803130\pi$
$61$	−12.7269	−1.62951	−0.814757	+	0.579803i	$0.803130\pi$
$62$	7.50252	0.952820
$63$	0	0
$64$	6.47242	0.809053
$65$	3.85305	0.477912
$66$	0	0
$67$	6.81042	0.832026	0.416013	−	0.909359i	$-0.363427\pi$
$67$	6.81042	0.832026	0.416013	+	0.909359i	$0.363427\pi$
$68$	0.895339	0.108576
$69$	0	0
$70$	2.49931	0.298724
$71$	3.58727	0.425731	0.212865	−	0.977082i	$-0.431720\pi$
$71$	3.58727	0.425731	0.212865	+	0.977082i	$0.431720\pi$
$72$	0	0
$73$	−3.65812	−0.428150	−0.214075	−	0.976817i	$-0.568674\pi$
$73$	−3.65812	−0.428150	−0.214075	+	0.976817i	$0.568674\pi$
$74$	7.94050	0.923065
$75$	0	0
$76$	−3.31213	−0.379928
$77$	−8.80075	−1.00294
$78$	0	0
$79$	−15.2640	−1.71733	−0.858666	−	0.512535i	$-0.828706\pi$
$79$	−15.2640	−1.71733	−0.858666	+	0.512535i	$0.828706\pi$
$80$	0.490206	0.0548067
$81$	0	0
$82$	−8.03587	−0.887413
$83$	−3.01862	−0.331337	−0.165668	−	0.986182i	$-0.552978\pi$
$83$	−3.01862	−0.331337	−0.165668	+	0.986182i	$0.552978\pi$
$84$	0	0
$85$	0.574633	0.0623277
$86$	5.50496	0.593615
$87$	0	0
$88$	−6.90459	−0.736032
$89$	−10.6104	−1.12470	−0.562351	−	0.826898i	$-0.690103\pi$
$89$	−10.6104	−1.12470	−0.562351	+	0.826898i	$0.690103\pi$
$90$	0	0
$91$	21.1640	2.21859
$92$	8.23803	0.858875
$93$	0	0
$94$	8.90375	0.918352
$95$	−2.12574	−0.218097
$96$	0	0
$97$	0.288109	0.0292531	0.0146265	−	0.999893i	$-0.495344\pi$
$97$	0.288109	0.0292531	0.0146265	+	0.999893i	$0.495344\pi$
$98$	6.97891	0.704976
$99$	0	0
$100$	4.84670	0.484670
$101$	−3.13247	−0.311693	−0.155846	−	0.987781i	$-0.549810\pi$
$101$	−3.13247	−0.311693	−0.155846	+	0.987781i	$0.549810\pi$
$102$	0	0
$103$	−1.31488	−0.129559	−0.0647796	−	0.997900i	$-0.520634\pi$
$103$	−1.31488	−0.129559	−0.0647796	+	0.997900i	$0.520634\pi$
$104$	16.6041	1.62816
$105$	0	0
$106$	13.6616	1.32693
$107$	8.43900	0.815829	0.407914	−	0.913020i	$-0.366256\pi$
$107$	8.43900	0.815829	0.407914	+	0.913020i	$0.366256\pi$
$108$	0	0
$109$	10.5881	1.01415	0.507076	−	0.861901i	$-0.330726\pi$
$109$	10.5881	1.01415	0.507076	+	0.861901i	$0.330726\pi$
$110$	−1.54484	−0.147295
$111$	0	0
$112$	2.69259	0.254426
$113$	6.51631	0.613003	0.306502	−	0.951870i	$-0.400842\pi$
$113$	6.51631	0.613003	0.306502	+	0.951870i	$0.400842\pi$
$114$	0	0
$115$	5.28721	0.493035
$116$	9.12259	0.847011
$117$	0	0
$118$	4.71044	0.433631
$119$	3.15633	0.289341
$120$	0	0
$121$	−5.56019	−0.505472
$122$	−12.2710	−1.11096
$123$	0	0
$124$	−8.32883	−0.747951
$125$	6.54547	0.585445
$126$	0	0
$127$	0.474581	0.0421123	0.0210561	−	0.999778i	$-0.493297\pi$
$127$	0.474581	0.0421123	0.0210561	+	0.999778i	$0.493297\pi$
$128$	−4.22490	−0.373432
$129$	0	0
$130$	3.71501	0.325828
$131$	18.7235	1.63588	0.817941	−	0.575302i	$-0.195115\pi$
$131$	18.7235	1.63588	0.817941	+	0.575302i	$0.195115\pi$
$132$	0	0
$133$	−11.6762	−1.01246
$134$	6.56644	0.567254
$135$	0	0
$136$	2.47629	0.212340
$137$	13.0452	1.11453	0.557263	−	0.830336i	$-0.311852\pi$
$137$	13.0452	1.11453	0.557263	+	0.830336i	$0.311852\pi$
$138$	0	0
$139$	2.87847	0.244149	0.122074	−	0.992521i	$-0.461045\pi$
$139$	2.87847	0.244149	0.122074	+	0.992521i	$0.461045\pi$
$140$	−2.77457	−0.234494
$141$	0	0
$142$	3.45876	0.290252
$143$	−13.0816	−1.09394
$144$	0	0
$145$	5.85492	0.486225
$146$	−3.52706	−0.291902
$147$	0	0
$148$	−8.81505	−0.724593
$149$	1.00000	0.0819232
$149$	1.00000	0.0819232
$150$	0	0
$151$	21.0872	1.71605	0.858025	−	0.513607i	$-0.171691\pi$
$151$	21.0872	1.71605	0.858025	+	0.513607i	$0.171691\pi$
$152$	−9.16055	−0.743018
$153$	0	0
$154$	−8.48546	−0.683778
$155$	−5.34548	−0.429359
$156$	0	0
$157$	4.91671	0.392396	0.196198	−	0.980564i	$-0.437140\pi$
$157$	4.91671	0.392396	0.196198	+	0.980564i	$0.437140\pi$
$158$	−14.7171	−1.17083
$159$	0	0
$160$	−3.59471	−0.284187
$161$	29.0415	2.28879
$162$	0	0
$163$	−19.1541	−1.50026	−0.750131	−	0.661289i	$-0.770010\pi$
$163$	−19.1541	−1.50026	−0.750131	+	0.661289i	$0.770010\pi$
$164$	8.92092	0.696607
$165$	0	0
$166$	−2.91048	−0.225897
$167$	3.34581	0.258907	0.129453	−	0.991586i	$-0.458678\pi$
$167$	3.34581	0.258907	0.129453	+	0.991586i	$0.458678\pi$
$168$	0	0
$169$	18.4585	1.41988
$170$	0.554047	0.0424934
$171$	0	0
$172$	−6.11126	−0.465979
$173$	8.69183	0.660827	0.330414	−	0.943836i	$-0.392812\pi$
$173$	8.69183	0.660827	0.330414	+	0.943836i	$0.392812\pi$
$174$	0	0
$175$	17.0860	1.29158
$176$	−1.66431	−0.125452
$177$	0	0
$178$	−10.2303	−0.766794
$179$	8.51681	0.636576	0.318288	−	0.947994i	$-0.396892\pi$
$179$	8.51681	0.636576	0.318288	+	0.947994i	$0.396892\pi$
$180$	0	0
$181$	17.3708	1.29116	0.645582	−	0.763691i	$-0.276615\pi$
$181$	17.3708	1.29116	0.645582	+	0.763691i	$0.276615\pi$
$182$	20.4058	1.51258
$183$	0	0
$184$	22.7844	1.67969
$185$	−5.65754	−0.415951
$186$	0	0
$187$	−1.95095	−0.142668
$188$	−9.88440	−0.720894
$189$	0	0
$190$	−2.04959	−0.148693
$191$	−2.81770	−0.203881	−0.101941	−	0.994790i	$-0.532505\pi$
$191$	−2.81770	−0.203881	−0.101941	+	0.994790i	$0.532505\pi$
$192$	0	0
$193$	3.10583	0.223562	0.111781	−	0.993733i	$-0.464344\pi$
$193$	3.10583	0.223562	0.111781	+	0.993733i	$0.464344\pi$
$194$	0.277788	0.0199440
$195$	0	0
$196$	−7.74755	−0.553397
$197$	−21.1902	−1.50974	−0.754868	−	0.655876i	$-0.772299\pi$
$197$	−21.1902	−1.50974	−0.754868	+	0.655876i	$0.772299\pi$
$198$	0	0
$199$	2.58371	0.183154	0.0915770	−	0.995798i	$-0.470809\pi$
$199$	2.58371	0.183154	0.0915770	+	0.995798i	$0.470809\pi$
$200$	13.4048	0.947861
$201$	0	0
$202$	−3.02025	−0.212504
$203$	32.1598	2.25717
$204$	0	0
$205$	5.72549	0.399886
$206$	−1.26778	−0.0883302
$207$	0	0
$208$	4.00232	0.277511
$209$	7.21718	0.499222
$210$	0	0
$211$	−2.51245	−0.172964	−0.0864820	−	0.996253i	$-0.527562\pi$
$211$	−2.51245	−0.172964	−0.0864820	+	0.996253i	$0.527562\pi$
$212$	−15.1662	−1.04162
$213$	0	0
$214$	8.13667	0.556211
$215$	−3.92224	−0.267494
$216$	0	0
$217$	−29.3616	−1.99319
$218$	10.2087	0.691423
$219$	0	0
$220$	1.71499	0.115624
$221$	4.69163	0.315593
$222$	0	0
$223$	−22.2458	−1.48969	−0.744845	−	0.667237i	$-0.767477\pi$
$223$	−22.2458	−1.48969	−0.744845	+	0.667237i	$0.767477\pi$
$224$	−19.7449	−1.31926
$225$	0	0
$226$	6.28286	0.417930
$227$	5.80082	0.385014	0.192507	−	0.981296i	$-0.438338\pi$
$227$	5.80082	0.385014	0.192507	+	0.981296i	$0.438338\pi$
$228$	0	0
$229$	−19.3986	−1.28189	−0.640947	−	0.767585i	$-0.721458\pi$
$229$	−19.3986	−1.28189	−0.640947	+	0.767585i	$0.721458\pi$
$230$	5.09780	0.336139
$231$	0	0
$232$	25.2308	1.65648
$233$	21.9567	1.43843	0.719215	−	0.694787i	$-0.244501\pi$
$233$	21.9567	1.43843	0.719215	+	0.694787i	$0.244501\pi$
$234$	0	0
$235$	−6.34385	−0.413827
$236$	−5.22923	−0.340394
$237$	0	0
$238$	3.04326	0.197265
$239$	−25.3549	−1.64007	−0.820037	−	0.572310i	$-0.806047\pi$
$239$	−25.3549	−1.64007	−0.820037	+	0.572310i	$0.806047\pi$
$240$	0	0
$241$	1.84545	0.118876	0.0594378	−	0.998232i	$-0.481069\pi$
$241$	1.84545	0.118876	0.0594378	+	0.998232i	$0.481069\pi$
$242$	−5.36099	−0.344618
$243$	0	0
$244$	13.6225	0.872089
$245$	−4.97242	−0.317676
$246$	0	0
$247$	−17.3558	−1.10432
$248$	−23.0355	−1.46275
$249$	0	0
$250$	6.31098	0.399141
$251$	4.95328	0.312648	0.156324	−	0.987706i	$-0.450036\pi$
$251$	4.95328	0.312648	0.156324	+	0.987706i	$0.450036\pi$
$252$	0	0
$253$	−17.9508	−1.12855
$254$	0.457579	0.0287111
$255$	0	0
$256$	−17.0184	−1.06365
$257$	15.7408	0.981882	0.490941	−	0.871193i	$-0.336653\pi$
$257$	15.7408	0.981882	0.490941	+	0.871193i	$0.336653\pi$
$258$	0	0
$259$	−31.0756	−1.93095
$260$	−4.12418	−0.255771
$261$	0	0
$262$	18.0528	1.11530
$263$	23.5394	1.45150	0.725752	−	0.687956i	$-0.241492\pi$
$263$	23.5394	1.45150	0.725752	+	0.687956i	$0.241492\pi$
$264$	0	0
$265$	−9.73374	−0.597939
$266$	−11.2579	−0.690269
$267$	0	0
$268$	−7.28965	−0.445286
$269$	10.9520	0.667754	0.333877	−	0.942617i	$-0.391643\pi$
$269$	10.9520	0.667754	0.333877	+	0.942617i	$0.391643\pi$
$270$	0	0
$271$	−22.6413	−1.37536	−0.687681	−	0.726013i	$-0.741371\pi$
$271$	−22.6413	−1.37536	−0.687681	+	0.726013i	$0.741371\pi$
$272$	0.596895	0.0361921
$273$	0	0
$274$	12.5778	0.759855
$275$	−10.5610	−0.636853
$276$	0	0
$277$	8.84537	0.531467	0.265734	−	0.964047i	$-0.414386\pi$
$277$	8.84537	0.531467	0.265734	+	0.964047i	$0.414386\pi$
$278$	2.77535	0.166454
$279$	0	0
$280$	−7.67378	−0.458596
$281$	−17.9911	−1.07326	−0.536629	−	0.843819i	$-0.680302\pi$
$281$	−17.9911	−1.07326	−0.536629	+	0.843819i	$0.680302\pi$
$282$	0	0
$283$	12.2334	0.727198	0.363599	−	0.931556i	$-0.381548\pi$
$283$	12.2334	0.727198	0.363599	+	0.931556i	$0.381548\pi$
$284$	−3.83970	−0.227844
$285$	0	0
$286$	−12.6130	−0.745820
$287$	31.4489	1.85637
$288$	0	0
$289$	−16.3003	−0.958841
$290$	5.64517	0.331496
$291$	0	0
$292$	3.91553	0.229139
$293$	27.8569	1.62742	0.813708	−	0.581274i	$-0.197446\pi$
$293$	27.8569	1.62742	0.813708	+	0.581274i	$0.197446\pi$
$294$	0	0
$295$	−3.35615	−0.195403
$296$	−24.3803	−1.41707
$297$	0	0
$298$	0.964175	0.0558531
$299$	43.1678	2.49646
$300$	0	0
$301$	−21.5440	−1.24177
$302$	20.3317	1.16996
$303$	0	0
$304$	−2.20810	−0.126643
$305$	8.74296	0.500621
$306$	0	0
$307$	20.8340	1.18906	0.594530	−	0.804074i	$-0.297338\pi$
$307$	20.8340	1.18906	0.594530	+	0.804074i	$0.297338\pi$
$308$	9.42004	0.536756
$309$	0	0
$310$	−5.15398	−0.292726
$311$	−20.4072	−1.15719	−0.578594	−	0.815616i	$-0.696398\pi$
$311$	−20.4072	−1.15719	−0.578594	+	0.815616i	$0.696398\pi$
$312$	0	0
$313$	5.18070	0.292831	0.146415	−	0.989223i	$-0.453226\pi$
$313$	5.18070	0.292831	0.146415	+	0.989223i	$0.453226\pi$
$314$	4.74057	0.267526
$315$	0	0
$316$	16.3381	0.919088
$317$	8.61505	0.483870	0.241935	−	0.970293i	$-0.422218\pi$
$317$	8.61505	0.483870	0.241935	+	0.970293i	$0.422218\pi$
$318$	0	0
$319$	−19.8782	−1.11297
$320$	−4.44634	−0.248558
$321$	0	0
$322$	28.0011	1.56044
$323$	−2.58839	−0.144022
$324$	0	0
$325$	25.3970	1.40877
$326$	−18.4679	−1.02284
$327$	0	0
$328$	24.6731	1.36234
$329$	−34.8454	−1.92109
$330$	0	0
$331$	−26.9283	−1.48011	−0.740056	−	0.672545i	$-0.765201\pi$
$331$	−26.9283	−1.48011	−0.740056	+	0.672545i	$0.765201\pi$
$332$	3.23103	0.177326
$333$	0	0
$334$	3.22595	0.176516
$335$	−4.67853	−0.255616
$336$	0	0
$337$	−26.5032	−1.44372	−0.721860	−	0.692039i	$-0.756713\pi$
$337$	−26.5032	−1.44372	−0.721860	+	0.692039i	$0.756713\pi$
$338$	17.7972	0.968041
$339$	0	0
$340$	−0.615068	−0.0333568
$341$	18.1486	0.982802
$342$	0	0
$343$	−0.898892	−0.0485356
$344$	−16.9022	−0.911308
$345$	0	0
$346$	8.38044	0.450535
$347$	−12.1206	−0.650669	−0.325334	−	0.945599i	$-0.605477\pi$
$347$	−12.1206	−0.650669	−0.325334	+	0.945599i	$0.605477\pi$
$348$	0	0
$349$	3.47985	0.186272	0.0931360	−	0.995653i	$-0.470311\pi$
$349$	3.47985	0.186272	0.0931360	+	0.995653i	$0.470311\pi$
$350$	16.4739	0.880569
$351$	0	0
$352$	12.2045	0.650502
$353$	−30.8820	−1.64368	−0.821842	−	0.569716i	$-0.807053\pi$
$353$	−30.8820	−1.64368	−0.821842	+	0.569716i	$0.807053\pi$
$354$	0	0
$355$	−2.46434	−0.130793
$356$	11.3570	0.601922
$357$	0	0
$358$	8.21169	0.434001
$359$	−12.8413	−0.677736	−0.338868	−	0.940834i	$-0.610044\pi$
$359$	−12.8413	−0.677736	−0.338868	+	0.940834i	$0.610044\pi$
$360$	0	0
$361$	−9.42474	−0.496039
$362$	16.7485	0.880283
$363$	0	0
$364$	−22.6532	−1.18735
$365$	2.51300	0.131537
$366$	0	0
$367$	27.7737	1.44977	0.724887	−	0.688868i	$-0.241892\pi$
$367$	27.7737	1.44977	0.724887	+	0.688868i	$0.241892\pi$
$368$	5.49204	0.286293
$369$	0	0
$370$	−5.45486	−0.283585
$371$	−53.4653	−2.77578
$372$	0	0
$373$	−18.7343	−0.970025	−0.485012	−	0.874507i	$-0.661185\pi$
$373$	−18.7343	−0.970025	−0.485012	+	0.874507i	$0.661185\pi$
$374$	−1.88106	−0.0972673
$375$	0	0
$376$	−27.3378	−1.40984
$377$	47.8029	2.46198
$378$	0	0
$379$	28.0555	1.44112	0.720558	−	0.693394i	$-0.243886\pi$
$379$	28.0555	1.44112	0.720558	+	0.693394i	$0.243886\pi$
$380$	2.27533	0.116722
$381$	0	0
$382$	−2.71675	−0.139001
$383$	−25.4378	−1.29981	−0.649905	−	0.760016i	$-0.725191\pi$
$383$	−25.4378	−1.29981	−0.649905	+	0.760016i	$0.725191\pi$
$384$	0	0
$385$	6.04582	0.308124
$386$	2.99456	0.152419
$387$	0	0
$388$	−0.308383	−0.0156558
$389$	15.9649	0.809451	0.404726	−	0.914438i	$-0.367367\pi$
$389$	15.9649	0.809451	0.404726	+	0.914438i	$0.367367\pi$
$390$	0	0
$391$	6.43792	0.325580
$392$	−21.4278	−1.08227
$393$	0	0
$394$	−20.4310	−1.02930
$395$	10.4858	0.527600
$396$	0	0
$397$	10.2199	0.512922	0.256461	−	0.966555i	$-0.417443\pi$
$397$	10.2199	0.512922	0.256461	+	0.966555i	$0.417443\pi$
$398$	2.49114	0.124870
$399$	0	0
$400$	3.23115	0.161557
$401$	12.9252	0.645452	0.322726	−	0.946493i	$-0.395401\pi$
$401$	12.9252	0.645452	0.322726	+	0.946493i	$0.395401\pi$
$402$	0	0
$403$	−43.6436	−2.17404
$404$	3.35290	0.166813
$405$	0	0
$406$	31.0077	1.53888
$407$	19.2081	0.952110
$408$	0	0
$409$	−12.6527	−0.625633	−0.312817	−	0.949814i	$-0.601273\pi$
$409$	−12.6527	−0.625633	−0.312817	+	0.949814i	$0.601273\pi$
$410$	5.52037	0.272632
$411$	0	0
$412$	1.40741	0.0693380
$413$	−18.4346	−0.907106
$414$	0	0
$415$	2.07369	0.101794
$416$	−29.3492	−1.43896
$417$	0	0
$418$	6.95862	0.340357
$419$	17.5538	0.857560	0.428780	−	0.903409i	$-0.358944\pi$
$419$	17.5538	0.857560	0.428780	+	0.903409i	$0.358944\pi$
$420$	0	0
$421$	−7.42373	−0.361811	−0.180905	−	0.983501i	$-0.557903\pi$
$421$	−7.42373	−0.361811	−0.180905	+	0.983501i	$0.557903\pi$
$422$	−2.42244	−0.117922
$423$	0	0
$424$	−41.9460	−2.03708
$425$	3.78764	0.183728
$426$	0	0
$427$	48.0232	2.32400
$428$	−9.03283	−0.436618
$429$	0	0
$430$	−3.78172	−0.182371
$431$	−8.06728	−0.388587	−0.194293	−	0.980943i	$-0.562241\pi$
$431$	−8.06728	−0.388587	−0.194293	+	0.980943i	$0.562241\pi$
$432$	0	0
$433$	10.2599	0.493057	0.246529	−	0.969136i	$-0.420710\pi$
$433$	10.2599	0.493057	0.246529	+	0.969136i	$0.420710\pi$
$434$	−28.3097	−1.35891
$435$	0	0
$436$	−11.3331	−0.542757
$437$	−23.8159	−1.13927
$438$	0	0
$439$	−13.4065	−0.639856	−0.319928	−	0.947442i	$-0.603659\pi$
$439$	−13.4065	−0.639856	−0.319928	+	0.947442i	$0.603659\pi$
$440$	4.74323	0.226124
$441$	0	0
$442$	4.52355	0.215163
$443$	34.1005	1.62016	0.810081	−	0.586318i	$-0.199423\pi$
$443$	34.1005	1.62016	0.810081	+	0.586318i	$0.199423\pi$
$444$	0	0
$445$	7.28901	0.345532
$446$	−21.4489	−1.01563
$447$	0	0
$448$	−24.4228	−1.15387
$449$	30.2056	1.42549	0.712745	−	0.701424i	$-0.247452\pi$
$449$	30.2056	1.42549	0.712745	+	0.701424i	$0.247452\pi$
$450$	0	0
$451$	−19.4388	−0.915337
$452$	−6.97485	−0.328069
$453$	0	0
$454$	5.59301	0.262493
$455$	−14.5389	−0.681596
$456$	0	0
$457$	−22.6171	−1.05798	−0.528991	−	0.848628i	$-0.677429\pi$
$457$	−22.6171	−1.05798	−0.528991	+	0.848628i	$0.677429\pi$
$458$	−18.7036	−0.873962
$459$	0	0
$460$	−5.65926	−0.263864
$461$	−6.08843	−0.283566	−0.141783	−	0.989898i	$-0.545284\pi$
$461$	−6.08843	−0.283566	−0.141783	+	0.989898i	$0.545284\pi$
$462$	0	0
$463$	−27.8122	−1.29254	−0.646272	−	0.763107i	$-0.723673\pi$
$463$	−27.8122	−1.29254	−0.646272	+	0.763107i	$0.723673\pi$
$464$	6.08175	0.282338
$465$	0	0
$466$	21.1701	0.980685
$467$	−11.2073	−0.518614	−0.259307	−	0.965795i	$-0.583494\pi$
$467$	−11.2073	−0.518614	−0.259307	+	0.965795i	$0.583494\pi$
$468$	0	0
$469$	−25.6982	−1.18663
$470$	−6.11658	−0.282137
$471$	0	0
$472$	−14.4628	−0.665703
$473$	13.3165	0.612293
$474$	0	0
$475$	−14.0116	−0.642898
$476$	−3.37843	−0.154850
$477$	0	0
$478$	−24.4466	−1.11816
$479$	4.65708	0.212787	0.106394	−	0.994324i	$-0.466070\pi$
$479$	4.65708	0.212787	0.106394	+	0.994324i	$0.466070\pi$
$480$	0	0
$481$	−46.1914	−2.10615
$482$	1.77933	0.0810464
$483$	0	0
$484$	5.95144	0.270520
$485$	−0.197921	−0.00898715
$486$	0	0
$487$	26.0900	1.18225	0.591124	−	0.806580i	$-0.298684\pi$
$487$	26.0900	1.18225	0.591124	+	0.806580i	$0.298684\pi$
$488$	37.6764	1.70553
$489$	0	0
$490$	−4.79428	−0.216583
$491$	−22.4967	−1.01526	−0.507632	−	0.861574i	$-0.669479\pi$
$491$	−22.4967	−1.01526	−0.507632	+	0.861574i	$0.669479\pi$
$492$	0	0
$493$	7.12919	0.321083
$494$	−16.7340	−0.752899
$495$	0	0
$496$	−5.55257	−0.249318
$497$	−13.5361	−0.607175
$498$	0	0
$499$	3.78835	0.169590	0.0847949	−	0.996398i	$-0.472977\pi$
$499$	3.78835	0.169590	0.0847949	+	0.996398i	$0.472977\pi$
$500$	−7.00606	−0.313320
$501$	0	0
$502$	4.77583	0.213156
$503$	−15.7366	−0.701661	−0.350831	−	0.936439i	$-0.614101\pi$
$503$	−15.7366	−0.701661	−0.350831	+	0.936439i	$0.614101\pi$
$504$	0	0
$505$	2.15191	0.0957585
$506$	−17.3077	−0.769420
$507$	0	0
$508$	−0.507976	−0.0225378
$509$	−12.2173	−0.541521	−0.270761	−	0.962647i	$-0.587275\pi$
$509$	−12.2173	−0.541521	−0.270761	+	0.962647i	$0.587275\pi$
$510$	0	0
$511$	13.8034	0.610626
$512$	−7.95890	−0.351737
$513$	0	0
$514$	15.1769	0.669422
$515$	0.903281	0.0398033
$516$	0	0
$517$	21.5382	0.947249
$518$	−29.9623	−1.31647
$519$	0	0
$520$	−11.4065	−0.500206
$521$	20.0430	0.878100	0.439050	−	0.898463i	$-0.355315\pi$
$521$	20.0430	0.878100	0.439050	+	0.898463i	$0.355315\pi$
$522$	0	0
$523$	11.0612	0.483674	0.241837	−	0.970317i	$-0.422250\pi$
$523$	11.0612	0.483674	0.241837	+	0.970317i	$0.422250\pi$
$524$	−20.0410	−0.875497
$525$	0	0
$526$	22.6961	0.989598
$527$	−6.50888	−0.283531
$528$	0	0
$529$	36.2355	1.57546
$530$	−9.38503	−0.407660
$531$	0	0
$532$	12.4979	0.541851
$533$	46.7462	2.02480
$534$	0	0
$535$	−5.79731	−0.250640
$536$	−20.1614	−0.870839
$537$	0	0
$538$	10.5596	0.455258
$539$	16.8820	0.727159
$540$	0	0
$541$	17.4114	0.748575	0.374287	−	0.927313i	$-0.377887\pi$
$541$	17.4114	0.748575	0.374287	+	0.927313i	$0.377887\pi$
$542$	−21.8302	−0.937687
$543$	0	0
$544$	−4.37706	−0.187665
$545$	−7.27364	−0.311569
$546$	0	0
$547$	11.9906	0.512682	0.256341	−	0.966586i	$-0.417483\pi$
$547$	11.9906	0.512682	0.256341	+	0.966586i	$0.417483\pi$
$548$	−13.9631	−0.596475
$549$	0	0
$550$	−10.1827	−0.434190
$551$	−26.3731	−1.12353
$552$	0	0
$553$	57.5965	2.44925
$554$	8.52849	0.362341
$555$	0	0
$556$	−3.08102	−0.130664
$557$	16.9896	0.719871	0.359935	−	0.932977i	$-0.382799\pi$
$557$	16.9896	0.719871	0.359935	+	0.932977i	$0.382799\pi$
$558$	0	0
$559$	−32.0234	−1.35444
$560$	−1.84972	−0.0781651
$561$	0	0
$562$	−17.3465	−0.731719
$563$	−21.1652	−0.892008	−0.446004	−	0.895031i	$-0.647153\pi$
$563$	−21.1652	−0.892008	−0.446004	+	0.895031i	$0.647153\pi$
$564$	0	0
$565$	−4.47649	−0.188327
$566$	11.7951	0.495785
$567$	0	0
$568$	−10.6197	−0.445591
$569$	39.2653	1.64609	0.823044	−	0.567978i	$-0.192274\pi$
$569$	39.2653	1.64609	0.823044	+	0.567978i	$0.192274\pi$
$570$	0	0
$571$	−8.41392	−0.352112	−0.176056	−	0.984380i	$-0.556334\pi$
$571$	−8.41392	−0.352112	−0.176056	+	0.984380i	$0.556334\pi$
$572$	14.0021	0.585458
$573$	0	0
$574$	30.3222	1.26562
$575$	34.8502	1.45335
$576$	0	0
$577$	−26.9945	−1.12379	−0.561897	−	0.827207i	$-0.689928\pi$
$577$	−26.9945	−1.12379	−0.561897	+	0.827207i	$0.689928\pi$
$578$	−15.7163	−0.653714
$579$	0	0
$580$	−6.26691	−0.260219
$581$	11.3903	0.472551
$582$	0	0
$583$	33.0473	1.36868
$584$	10.8294	0.448123
$585$	0	0
$586$	26.8589	1.10953
$587$	−3.57875	−0.147711	−0.0738553	−	0.997269i	$-0.523530\pi$
$587$	−3.57875	−0.147711	−0.0738553	+	0.997269i	$0.523530\pi$
$588$	0	0
$589$	24.0783	0.992130
$590$	−3.23591	−0.133220
$591$	0	0
$592$	−5.87672	−0.241532
$593$	40.1501	1.64877	0.824383	−	0.566033i	$-0.191523\pi$
$593$	40.1501	1.64877	0.824383	+	0.566033i	$0.191523\pi$
$594$	0	0
$595$	−2.16830	−0.0888915
$596$	−1.07037	−0.0438439
$597$	0	0
$598$	41.6213	1.70202
$599$	0.671030	0.0274176	0.0137088	−	0.999906i	$-0.495636\pi$
$599$	0.671030	0.0274176	0.0137088	+	0.999906i	$0.495636\pi$
$600$	0	0
$601$	−1.24985	−0.0509824	−0.0254912	−	0.999675i	$-0.508115\pi$
$601$	−1.24985	−0.0509824	−0.0254912	+	0.999675i	$0.508115\pi$
$602$	−20.7722	−0.846610
$603$	0	0
$604$	−22.5710	−0.918402
$605$	3.81966	0.155291
$606$	0	0
$607$	−32.2551	−1.30919	−0.654596	−	0.755979i	$-0.727161\pi$
$607$	−32.2551	−1.30919	−0.654596	+	0.755979i	$0.727161\pi$
$608$	16.1921	0.656676
$609$	0	0
$610$	8.42975	0.341310
$611$	−51.7948	−2.09539
$612$	0	0
$613$	9.29757	0.375525	0.187763	−	0.982214i	$-0.439876\pi$
$613$	9.29757	0.375525	0.187763	+	0.982214i	$0.439876\pi$
$614$	20.0876	0.810671
$615$	0	0
$616$	26.0535	1.04973
$617$	45.0165	1.81230	0.906148	−	0.422962i	$-0.139009\pi$
$617$	45.0165	1.81230	0.906148	+	0.422962i	$0.139009\pi$
$618$	0	0
$619$	16.0904	0.646727	0.323363	−	0.946275i	$-0.395186\pi$
$619$	16.0904	0.646727	0.323363	+	0.946275i	$0.395186\pi$
$620$	5.72163	0.229786
$621$	0	0
$622$	−19.6761	−0.788942
$623$	40.0369	1.60405
$624$	0	0
$625$	18.1439	0.725755
$626$	4.99510	0.199644
$627$	0	0
$628$	−5.26268	−0.210004
$629$	−6.88886	−0.274677
$630$	0	0
$631$	−34.8799	−1.38855	−0.694273	−	0.719712i	$-0.744274\pi$
$631$	−34.8799	−1.38855	−0.694273	+	0.719712i	$0.744274\pi$
$632$	45.1870	1.79744
$633$	0	0
$634$	8.30642	0.329890
$635$	−0.326021	−0.0129378
$636$	0	0
$637$	−40.5976	−1.60854
$638$	−19.1661	−0.758792
$639$	0	0
$640$	2.90237	0.114726
$641$	17.9038	0.707156	0.353578	−	0.935405i	$-0.384965\pi$
$641$	17.9038	0.707156	0.353578	+	0.935405i	$0.384965\pi$
$642$	0	0
$643$	38.7385	1.52770	0.763849	−	0.645395i	$-0.223307\pi$
$643$	38.7385	1.52770	0.763849	+	0.645395i	$0.223307\pi$
$644$	−31.0850	−1.22492
$645$	0	0
$646$	−2.49566	−0.0981906
$647$	−0.104257	−0.00409877	−0.00204939	−	0.999998i	$-0.500652\pi$
$647$	−0.104257	−0.00409877	−0.00204939	+	0.999998i	$0.500652\pi$
$648$	0	0
$649$	11.3946	0.447275
$650$	24.4872	0.960466
$651$	0	0
$652$	20.5019	0.802916
$653$	17.1417	0.670807	0.335404	−	0.942075i	$-0.391127\pi$
$653$	17.1417	0.670807	0.335404	+	0.942075i	$0.391127\pi$
$654$	0	0
$655$	−12.8624	−0.502577
$656$	5.94730	0.232203
$657$	0	0
$658$	−33.5970	−1.30975
$659$	14.1771	0.552263	0.276132	−	0.961120i	$-0.410947\pi$
$659$	14.1771	0.552263	0.276132	+	0.961120i	$0.410947\pi$
$660$	0	0
$661$	−10.7351	−0.417546	−0.208773	−	0.977964i	$-0.566947\pi$
$661$	−10.7351	−0.417546	−0.208773	+	0.977964i	$0.566947\pi$
$662$	−25.9636	−1.00910
$663$	0	0
$664$	8.93624	0.346793
$665$	8.02119	0.311048
$666$	0	0
$667$	65.5958	2.53988
$668$	−3.58125	−0.138563
$669$	0	0
$670$	−4.51093	−0.174272
$671$	−29.6835	−1.14592
$672$	0	0
$673$	−22.4768	−0.866418	−0.433209	−	0.901293i	$-0.642619\pi$
$673$	−22.4768	−0.866418	−0.433209	+	0.901293i	$0.642619\pi$
$674$	−25.5537	−0.984292
$675$	0	0
$676$	−19.7574	−0.759899
$677$	−5.78331	−0.222271	−0.111135	−	0.993805i	$-0.535449\pi$
$677$	−5.78331	−0.222271	−0.111135	+	0.993805i	$0.535449\pi$
$678$	0	0
$679$	−1.08714	−0.0417206
$680$	−1.70113	−0.0652352
$681$	0	0
$682$	17.4984	0.670049
$683$	−35.0435	−1.34090	−0.670451	−	0.741954i	$-0.733899\pi$
$683$	−35.0435	−1.34090	−0.670451	+	0.741954i	$0.733899\pi$
$684$	0	0
$685$	−8.96161	−0.342405
$686$	−0.866689	−0.0330903
$687$	0	0
$688$	−4.07419	−0.155327
$689$	−79.4718	−3.02764
$690$	0	0
$691$	−10.1453	−0.385947	−0.192974	−	0.981204i	$-0.561813\pi$
$691$	−10.1453	−0.385947	−0.192974	+	0.981204i	$0.561813\pi$
$692$	−9.30344	−0.353664
$693$	0	0
$694$	−11.6864	−0.443609
$695$	−1.97741	−0.0750076
$696$	0	0
$697$	6.97159	0.264068
$698$	3.35518	0.126995
$699$	0	0
$700$	−18.2883	−0.691234
$701$	−11.2837	−0.426181	−0.213091	−	0.977032i	$-0.568353\pi$
$701$	−11.2837	−0.426181	−0.213091	+	0.977032i	$0.568353\pi$
$702$	0	0
$703$	25.4840	0.961147
$704$	15.0959	0.568948
$705$	0	0
$706$	−29.7757	−1.12062
$707$	11.8199	0.444535
$708$	0	0
$709$	−7.75800	−0.291358	−0.145679	−	0.989332i	$-0.546537\pi$
$709$	−7.75800	−0.291358	−0.145679	+	0.989332i	$0.546537\pi$
$710$	−2.37605	−0.0891716
$711$	0	0
$712$	31.4108	1.17717
$713$	−59.8883	−2.24283
$714$	0	0
$715$	8.98663	0.336081
$716$	−9.11611	−0.340685
$717$	0	0
$718$	−12.3812	−0.462063
$719$	−41.7570	−1.55727	−0.778637	−	0.627475i	$-0.784089\pi$
$719$	−41.7570	−1.55727	−0.778637	+	0.627475i	$0.784089\pi$
$720$	0	0
$721$	4.96152	0.184777
$722$	−9.08710	−0.338187
$723$	0	0
$724$	−18.5932	−0.691010
$725$	38.5922	1.43328
$726$	0	0
$727$	−6.45892	−0.239548	−0.119774	−	0.992801i	$-0.538217\pi$
$727$	−6.45892	−0.239548	−0.119774	+	0.992801i	$0.538217\pi$
$728$	−62.6531	−2.32208
$729$	0	0
$730$	2.42298	0.0896784
$731$	−4.77588	−0.176642
$732$	0	0
$733$	−49.9636	−1.84545	−0.922723	−	0.385464i	$-0.874042\pi$
$733$	−49.9636	−1.84545	−0.922723	+	0.385464i	$0.874042\pi$
$734$	26.7787	0.988419
$735$	0	0
$736$	−40.2734	−1.48450
$737$	15.8842	0.585103
$738$	0	0
$739$	−37.6626	−1.38544	−0.692720	−	0.721206i	$-0.743588\pi$
$739$	−37.6626	−1.38544	−0.692720	+	0.721206i	$0.743588\pi$
$740$	6.05565	0.222610
$741$	0	0
$742$	−51.5499	−1.89246
$743$	23.1027	0.847556	0.423778	−	0.905766i	$-0.360704\pi$
$743$	23.1027	0.847556	0.423778	+	0.905766i	$0.360704\pi$
$744$	0	0
$745$	−0.686967	−0.0251685
$746$	−18.0631	−0.661338
$747$	0	0
$748$	2.08824	0.0763535
$749$	−31.8434	−1.16353
$750$	0	0
$751$	19.8862	0.725657	0.362829	−	0.931856i	$-0.381811\pi$
$751$	19.8862	0.725657	0.362829	+	0.931856i	$0.381811\pi$
$752$	−6.58962	−0.240299
$753$	0	0
$754$	46.0904	1.67851
$755$	−14.4862	−0.527207
$756$	0	0
$757$	−7.15157	−0.259928	−0.129964	−	0.991519i	$-0.541486\pi$
$757$	−7.15157	−0.259928	−0.129964	+	0.991519i	$0.541486\pi$
$758$	27.0505	0.982517
$759$	0	0
$760$	6.29299	0.228271
$761$	44.4771	1.61229	0.806147	−	0.591715i	$-0.201549\pi$
$761$	44.4771	1.61229	0.806147	+	0.591715i	$0.201549\pi$
$762$	0	0
$763$	−39.9525	−1.44638
$764$	3.01597	0.109114
$765$	0	0
$766$	−24.5265	−0.886177
$767$	−27.4015	−0.989411
$768$	0	0
$769$	13.8312	0.498765	0.249382	−	0.968405i	$-0.419772\pi$
$769$	13.8312	0.498765	0.249382	+	0.968405i	$0.419772\pi$
$770$	5.82923	0.210071
$771$	0	0
$772$	−3.32437	−0.119647
$773$	52.9859	1.90577	0.952885	−	0.303332i	$-0.0980991\pi$
$773$	52.9859	1.90577	0.952885	+	0.303332i	$0.0980991\pi$
$774$	0	0
$775$	−35.2342	−1.26565
$776$	−0.852910	−0.0306177
$777$	0	0
$778$	15.3929	0.551863
$779$	−25.7901	−0.924025
$780$	0	0
$781$	8.36674	0.299385
$782$	6.20729	0.221972
$783$	0	0
$784$	−5.16505	−0.184466
$785$	−3.37762	−0.120552
$786$	0	0
$787$	27.7178	0.988034	0.494017	−	0.869452i	$-0.335528\pi$
$787$	27.7178	0.988034	0.494017	+	0.869452i	$0.335528\pi$
$788$	22.6813	0.807986
$789$	0	0
$790$	10.1102	0.359704
$791$	−24.5884	−0.874262
$792$	0	0
$793$	71.3825	2.53487
$794$	9.85377	0.349697
$795$	0	0
$796$	−2.76551	−0.0980210
$797$	13.4311	0.475752	0.237876	−	0.971295i	$-0.423549\pi$
$797$	13.4311	0.475752	0.237876	+	0.971295i	$0.423549\pi$
$798$	0	0
$799$	−7.72454	−0.273274
$800$	−23.6942	−0.837716
$801$	0	0
$802$	12.4621	0.440052
$803$	−8.53198	−0.301087
$804$	0	0
$805$	−19.9505	−0.703164
$806$	−42.0800	−1.48221
$807$	0	0
$808$	9.27328	0.326233
$809$	−16.1134	−0.566516	−0.283258	−	0.959044i	$-0.591415\pi$
$809$	−16.1134	−0.566516	−0.283258	+	0.959044i	$0.591415\pi$
$810$	0	0
$811$	31.2479	1.09726	0.548631	−	0.836065i	$-0.315149\pi$
$811$	31.2479	1.09726	0.548631	+	0.836065i	$0.315149\pi$
$812$	−34.4228	−1.20800
$813$	0	0
$814$	18.5200	0.649124
$815$	13.1582	0.460912
$816$	0	0
$817$	17.6674	0.618105
$818$	−12.1994	−0.426541
$819$	0	0
$820$	−6.12838	−0.214012
$821$	−30.9385	−1.07976	−0.539880	−	0.841742i	$-0.681530\pi$
$821$	−30.9385	−1.07976	−0.539880	+	0.841742i	$0.681530\pi$
$822$	0	0
$823$	−48.7951	−1.70089	−0.850445	−	0.526063i	$-0.823667\pi$
$823$	−48.7951	−1.70089	−0.850445	+	0.526063i	$0.823667\pi$
$824$	3.89254	0.135603
$825$	0	0
$826$	−17.7742	−0.618442
$827$	−18.0232	−0.626729	−0.313365	−	0.949633i	$-0.601456\pi$
$827$	−18.0232	−0.626729	−0.313365	+	0.949633i	$0.601456\pi$
$828$	0	0
$829$	−29.5662	−1.02688	−0.513439	−	0.858126i	$-0.671629\pi$
$829$	−29.5662	−1.02688	−0.513439	+	0.858126i	$0.671629\pi$
$830$	1.99940	0.0694003
$831$	0	0
$832$	−36.3024	−1.25856
$833$	−6.05462	−0.209780
$834$	0	0
$835$	−2.29846	−0.0795415
$836$	−7.72503	−0.267176
$837$	0	0
$838$	16.9249	0.584663
$839$	35.0949	1.21161	0.605806	−	0.795613i	$-0.292851\pi$
$839$	35.0949	1.21161	0.605806	+	0.795613i	$0.292851\pi$
$840$	0	0
$841$	43.6391	1.50480
$842$	−7.15778	−0.246673
$843$	0	0
$844$	2.68924	0.0925675
$845$	−12.6804	−0.436218
$846$	0	0
$847$	20.9806	0.720901
$848$	−10.1108	−0.347208
$849$	0	0
$850$	3.65195	0.125261
$851$	−63.3845	−2.17279
$852$	0	0
$853$	−43.9184	−1.50374	−0.751868	−	0.659314i	$-0.770847\pi$
$853$	−43.9184	−1.50374	−0.751868	+	0.659314i	$0.770847\pi$
$854$	46.3027	1.58445
$855$	0	0
$856$	−24.9826	−0.853887
$857$	41.2850	1.41027	0.705135	−	0.709073i	$-0.250887\pi$
$857$	41.2850	1.41027	0.705135	+	0.709073i	$0.250887\pi$
$858$	0	0
$859$	25.2775	0.862457	0.431229	−	0.902243i	$-0.358080\pi$
$859$	25.2775	0.862457	0.431229	+	0.902243i	$0.358080\pi$
$860$	4.19823	0.143159
$861$	0	0
$862$	−7.77826	−0.264929
$863$	24.6202	0.838082	0.419041	−	0.907967i	$-0.362366\pi$
$863$	24.6202	0.838082	0.419041	+	0.907967i	$0.362366\pi$
$864$	0	0
$865$	−5.97100	−0.203020
$866$	9.89230	0.336154
$867$	0	0
$868$	31.4276	1.06672
$869$	−35.6008	−1.20767
$870$	0	0
$871$	−38.1982	−1.29430
$872$	−31.3446	−1.06146
$873$	0	0
$874$	−22.9626	−0.776723
$875$	−24.6984	−0.834958
$876$	0	0
$877$	−3.89020	−0.131363	−0.0656813	−	0.997841i	$-0.520922\pi$
$877$	−3.89020	−0.131363	−0.0656813	+	0.997841i	$0.520922\pi$
$878$	−12.9262	−0.436237
$879$	0	0
$880$	1.14333	0.0385416
$881$	5.58705	0.188232	0.0941162	−	0.995561i	$-0.469997\pi$
$881$	5.58705	0.188232	0.0941162	+	0.995561i	$0.469997\pi$
$882$	0	0
$883$	20.2941	0.682951	0.341476	−	0.939891i	$-0.389073\pi$
$883$	20.2941	0.682951	0.341476	+	0.939891i	$0.389073\pi$
$884$	−5.02177	−0.168900
$885$	0	0
$886$	32.8788	1.10458
$887$	−47.8968	−1.60822	−0.804108	−	0.594484i	$-0.797357\pi$
$887$	−47.8968	−1.60822	−0.804108	+	0.594484i	$0.797357\pi$
$888$	0	0
$889$	−1.79076	−0.0600603
$890$	7.02788	0.235575
$891$	0	0
$892$	23.8112	0.797258
$893$	28.5754	0.956240
$894$	0	0
$895$	−5.85076	−0.195569
$896$	15.9421	0.532587
$897$	0	0
$898$	29.1235	0.971862
$899$	−66.3188	−2.21185
$900$	0	0
$901$	−11.8522	−0.394854
$902$	−18.7424	−0.624053
$903$	0	0
$904$	−19.2907	−0.641599
$905$	−11.9332	−0.396673
$906$	0	0
$907$	39.8596	1.32352	0.661758	−	0.749718i	$-0.269811\pi$
$907$	39.8596	1.32352	0.661758	+	0.749718i	$0.269811\pi$
$908$	−6.20901	−0.206053
$909$	0	0
$910$	−14.0181	−0.464695
$911$	4.21309	0.139586	0.0697929	−	0.997561i	$-0.477766\pi$
$911$	4.21309	0.139586	0.0697929	+	0.997561i	$0.477766\pi$
$912$	0	0
$913$	−7.04046	−0.233005
$914$	−21.8068	−0.721305
$915$	0	0
$916$	20.7636	0.686048
$917$	−70.6506	−2.33309
$918$	0	0
$919$	35.2639	1.16325	0.581625	−	0.813457i	$-0.302417\pi$
$919$	35.2639	1.16325	0.581625	+	0.813457i	$0.302417\pi$
$920$	−15.6521	−0.516035
$921$	0	0
$922$	−5.87031	−0.193328
$923$	−20.1202	−0.662266
$924$	0	0
$925$	−37.2912	−1.22613
$926$	−26.8158	−0.881223
$927$	0	0
$928$	−44.5978	−1.46399
$929$	26.7055	0.876180	0.438090	−	0.898931i	$-0.355655\pi$
$929$	26.7055	0.876180	0.438090	+	0.898931i	$0.355655\pi$
$930$	0	0
$931$	22.3979	0.734061
$932$	−23.5017	−0.769824
$933$	0	0
$934$	−10.8058	−0.353578
$935$	1.34024	0.0438305
$936$	0	0
$937$	−4.11828	−0.134538	−0.0672691	−	0.997735i	$-0.521429\pi$
$937$	−4.11828	−0.134538	−0.0672691	+	0.997735i	$0.521429\pi$
$938$	−24.7775	−0.809015
$939$	0	0
$940$	6.79025	0.221474
$941$	28.0127	0.913187	0.456593	−	0.889676i	$-0.349069\pi$
$941$	28.0127	0.913187	0.456593	+	0.889676i	$0.349069\pi$
$942$	0	0
$943$	64.1458	2.08887
$944$	−3.48617	−0.113465
$945$	0	0
$946$	12.8394	0.417446
$947$	8.88358	0.288678	0.144339	−	0.989528i	$-0.453894\pi$
$947$	8.88358	0.288678	0.144339	+	0.989528i	$0.453894\pi$
$948$	0	0
$949$	20.5176	0.666030
$950$	−13.5097	−0.438312
$951$	0	0
$952$	−9.34392	−0.302838
$953$	−16.3569	−0.529853	−0.264926	−	0.964269i	$-0.585348\pi$
$953$	−16.3569	−0.529853	−0.264926	+	0.964269i	$0.585348\pi$
$954$	0	0
$955$	1.93566	0.0626366
$956$	27.1391	0.877741
$957$	0	0
$958$	4.49024	0.145073
$959$	−49.2241	−1.58953
$960$	0	0
$961$	29.5483	0.953172
$962$	−44.5366	−1.43592
$963$	0	0
$964$	−1.97530	−0.0636203
$965$	−2.13360	−0.0686830
$966$	0	0
$967$	−19.9424	−0.641304	−0.320652	−	0.947197i	$-0.603902\pi$
$967$	−19.9424	−0.641304	−0.320652	+	0.947197i	$0.603902\pi$
$968$	16.4602	0.529051
$969$	0	0
$970$	−0.190831	−0.00612721
$971$	28.8869	0.927023	0.463512	−	0.886091i	$-0.346589\pi$
$971$	28.8869	0.927023	0.463512	+	0.886091i	$0.346589\pi$
$972$	0	0
$973$	−10.8615	−0.348204
$974$	25.1553	0.806027
$975$	0	0
$976$	9.08168	0.290697
$977$	45.8318	1.46629	0.733145	−	0.680072i	$-0.238052\pi$
$977$	45.8318	1.46629	0.733145	+	0.680072i	$0.238052\pi$
$978$	0	0
$979$	−24.7471	−0.790922
$980$	5.32231	0.170015
$981$	0	0
$982$	−21.6908	−0.692181
$983$	6.27333	0.200088	0.100044	−	0.994983i	$-0.468102\pi$
$983$	6.27333	0.200088	0.100044	+	0.994983i	$0.468102\pi$
$984$	0	0
$985$	14.5569	0.463823
$986$	6.87379	0.218906
$987$	0	0
$988$	18.5771	0.591015
$989$	−43.9429	−1.39730
$990$	0	0
$991$	46.9589	1.49170	0.745849	−	0.666115i	$-0.232044\pi$
$991$	46.9589	1.49170	0.745849	+	0.666115i	$0.232044\pi$
$992$	40.7173	1.29278
$993$	0	0
$994$	−13.0511	−0.413956
$995$	−1.77492	−0.0562687
$996$	0	0
$997$	62.1256	1.96754	0.983769	−	0.179441i	$-0.0574290\pi$
$997$	62.1256	1.96754	0.983769	+	0.179441i	$0.0574290\pi$
$998$	3.65263	0.115622
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4023.2.a.f.1.15	yes	25
3.2	odd	2		4023.2.a.e.1.11	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4023.2.a.e.1.11	✓	25	3.2	odd	2
4023.2.a.f.1.15	yes	25	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 4023 = 3^{3} \cdot 149 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4023.a (trivial)

\(f(q)\)	\(=\)	\(q+0.964175 q^{2} -1.07037 q^{4} -0.686967 q^{5} -3.77336 q^{7} -2.96037 q^{8} +O(q^{10})\)	\(q+0.964175 q^{2} -1.07037 q^{4} -0.686967 q^{5} -3.77336 q^{7} -2.96037 q^{8} -0.662356 q^{10} +2.33234 q^{11} -5.60879 q^{13} -3.63818 q^{14} -0.713581 q^{16} -0.836479 q^{17} +3.09439 q^{19} +0.735306 q^{20} +2.24878 q^{22} -7.69646 q^{23} -4.52808 q^{25} -5.40785 q^{26} +4.03888 q^{28} -8.52286 q^{29} +7.78128 q^{31} +5.23272 q^{32} -0.806512 q^{34} +2.59217 q^{35} +8.23554 q^{37} +2.98353 q^{38} +2.03368 q^{40} -8.33445 q^{41} +5.70950 q^{43} -2.49646 q^{44} -7.42073 q^{46} +9.23459 q^{47} +7.23822 q^{49} -4.36586 q^{50} +6.00346 q^{52} +14.1692 q^{53} -1.60224 q^{55} +11.1705 q^{56} -8.21753 q^{58} +4.88546 q^{59} -12.7269 q^{61} +7.50252 q^{62} +6.47242 q^{64} +3.85305 q^{65} +6.81042 q^{67} +0.895339 q^{68} +2.49931 q^{70} +3.58727 q^{71} -3.65812 q^{73} +7.94050 q^{74} -3.31213 q^{76} -8.80075 q^{77} -15.2640 q^{79} +0.490206 q^{80} -8.03587 q^{82} -3.01862 q^{83} +0.574633 q^{85} +5.50496 q^{86} -6.90459 q^{88} -10.6104 q^{89} +21.1640 q^{91} +8.23803 q^{92} +8.90375 q^{94} -2.12574 q^{95} +0.288109 q^{97} +6.97891 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q + 7 q^{2} + 27 q^{4} + 12 q^{5} - 2 q^{7} + 21 q^{8}+O(q^{10}) \) 25 * q + 7 * q^2 + 27 * q^4 + 12 * q^5 - 2 * q^7 + 21 * q^8	\( 25 q + 7 q^{2} + 27 q^{4} + 12 q^{5} - 2 q^{7} + 21 q^{8} - 4 q^{10} + 12 q^{11} + 10 q^{14} + 35 q^{16} + 26 q^{17} + 30 q^{20} + 8 q^{22} + 26 q^{23} + 27 q^{25} + 16 q^{26} + 4 q^{28} + 20 q^{29} - 6 q^{31} + 49 q^{32} - 14 q^{34} + 16 q^{35} - 2 q^{37} + 27 q^{38} + 2 q^{40} + 35 q^{41} + 4 q^{43} + 22 q^{44} + 6 q^{46} + 38 q^{47} + 19 q^{49} + 22 q^{50} + 4 q^{52} + 36 q^{53} + 10 q^{55} + 79 q^{56} - 22 q^{58} + 15 q^{59} + 10 q^{61} - 14 q^{62} + 41 q^{64} + 80 q^{65} - 6 q^{67} + 33 q^{68} + 8 q^{70} + 26 q^{71} - 6 q^{73} + 75 q^{74} - 10 q^{76} + 47 q^{77} - 6 q^{79} + 66 q^{80} + 12 q^{82} + 40 q^{83} - 12 q^{85} + 4 q^{86} + 12 q^{88} + 30 q^{89} + 158 q^{92} + 18 q^{94} + 22 q^{95} - 20 q^{97} + 35 q^{98}+O(q^{100}) \) 25 * q + 7 * q^2 + 27 * q^4 + 12 * q^5 - 2 * q^7 + 21 * q^8 - 4 * q^10 + 12 * q^11 + 10 * q^14 + 35 * q^16 + 26 * q^17 + 30 * q^20 + 8 * q^22 + 26 * q^23 + 27 * q^25 + 16 * q^26 + 4 * q^28 + 20 * q^29 - 6 * q^31 + 49 * q^32 - 14 * q^34 + 16 * q^35 - 2 * q^37 + 27 * q^38 + 2 * q^40 + 35 * q^41 + 4 * q^43 + 22 * q^44 + 6 * q^46 + 38 * q^47 + 19 * q^49 + 22 * q^50 + 4 * q^52 + 36 * q^53 + 10 * q^55 + 79 * q^56 - 22 * q^58 + 15 * q^59 + 10 * q^61 - 14 * q^62 + 41 * q^64 + 80 * q^65 - 6 * q^67 + 33 * q^68 + 8 * q^70 + 26 * q^71 - 6 * q^73 + 75 * q^74 - 10 * q^76 + 47 * q^77 - 6 * q^79 + 66 * q^80 + 12 * q^82 + 40 * q^83 - 12 * q^85 + 4 * q^86 + 12 * q^88 + 30 * q^89 + 158 * q^92 + 18 * q^94 + 22 * q^95 - 20 * q^97 + 35 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more