LMFDB - Embedded newform 4592.2.a.bb.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4592,2,Mod(1,4592)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4592, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("4592.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4592 = 2^{4} \cdot 7 \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4592.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:	$36.6673046082$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.633117.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 6x^{3} + 4x^{2} + 6x - 3 $ x^5 - x^4 - 6x^3 + 4x^2 + 6*x - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 287)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.20098$ of defining polynomial
Character	$\chi$	$=$	4592.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.200978 q^{3} -3.21704 q^{5} -1.00000 q^{7} -2.95961 q^{9} +O(q^{10})$	$q+0.200978 q^{3} -3.21704 q^{5} -1.00000 q^{7} -2.95961 q^{9} -4.57695 q^{11} +0.703013 q^{13} -0.646554 q^{15} +4.25337 q^{17} -8.04736 q^{19} -0.200978 q^{21} +5.34842 q^{23} +5.34937 q^{25} -1.19775 q^{27} -5.39204 q^{29} -7.61900 q^{31} -0.919865 q^{33} +3.21704 q^{35} +5.19272 q^{37} +0.141290 q^{39} -1.00000 q^{41} -10.3241 q^{43} +9.52119 q^{45} -12.1160 q^{47} +1.00000 q^{49} +0.854832 q^{51} -12.2837 q^{53} +14.7242 q^{55} -1.61734 q^{57} +7.73023 q^{59} -2.48971 q^{61} +2.95961 q^{63} -2.26162 q^{65} +3.09767 q^{67} +1.07491 q^{69} -5.11581 q^{71} +4.13640 q^{73} +1.07511 q^{75} +4.57695 q^{77} +13.7414 q^{79} +8.63810 q^{81} -4.90626 q^{83} -13.6833 q^{85} -1.08368 q^{87} +5.97567 q^{89} -0.703013 q^{91} -1.53125 q^{93} +25.8887 q^{95} +1.45550 q^{97} +13.5460 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 4 q^{3} - 5 q^{5} - 5 q^{7} + q^{9}+O(q^{10}) $ 5 * q - 4 * q^3 - 5 * q^5 - 5 * q^7 + q^9	$ 5 q - 4 q^{3} - 5 q^{5} - 5 q^{7} + q^{9} - 2 q^{11} + 5 q^{13} + 5 q^{15} + 13 q^{17} + 4 q^{21} - 2 q^{23} + 22 q^{25} - 10 q^{27} - 5 q^{29} - 17 q^{31} + 3 q^{33} + 5 q^{35} - 7 q^{37} - 5 q^{39} - 5 q^{41} - q^{43} - 23 q^{45} - 9 q^{47} + 5 q^{49} - 5 q^{51} + 5 q^{53} - 33 q^{55} - 3 q^{57} - 7 q^{59} + 22 q^{61} - q^{63} - 31 q^{65} + 3 q^{67} - 22 q^{69} + 24 q^{71} + 40 q^{73} - 24 q^{75} + 2 q^{77} + 42 q^{79} + 9 q^{81} + 12 q^{83} - 23 q^{85} + 32 q^{87} + 8 q^{89} - 5 q^{91} - 11 q^{93} + 17 q^{95} + 16 q^{97} + 45 q^{99}+O(q^{100}) $ 5 * q - 4 * q^3 - 5 * q^5 - 5 * q^7 + q^9 - 2 * q^11 + 5 * q^13 + 5 * q^15 + 13 * q^17 + 4 * q^21 - 2 * q^23 + 22 * q^25 - 10 * q^27 - 5 * q^29 - 17 * q^31 + 3 * q^33 + 5 * q^35 - 7 * q^37 - 5 * q^39 - 5 * q^41 - q^43 - 23 * q^45 - 9 * q^47 + 5 * q^49 - 5 * q^51 + 5 * q^53 - 33 * q^55 - 3 * q^57 - 7 * q^59 + 22 * q^61 - q^63 - 31 * q^65 + 3 * q^67 - 22 * q^69 + 24 * q^71 + 40 * q^73 - 24 * q^75 + 2 * q^77 + 42 * q^79 + 9 * q^81 + 12 * q^83 - 23 * q^85 + 32 * q^87 + 8 * q^89 - 5 * q^91 - 11 * q^93 + 17 * q^95 + 16 * q^97 + 45 * 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
$3$	0.200978	0.116035	0.0580173	−	0.998316i	$-0.481522\pi$
$3$	0.200978	0.116035	0.0580173	+	0.998316i	$0.481522\pi$
$4$	0	0
$5$	−3.21704	−1.43871	−0.719353	−	0.694645i	$-0.755562\pi$
$5$	−3.21704	−1.43871	−0.719353	+	0.694645i	$0.755562\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−2.95961	−0.986536
$10$	0	0
$11$	−4.57695	−1.38000	−0.690001	−	0.723809i	$-0.742390\pi$
$11$	−4.57695	−1.38000	−0.690001	+	0.723809i	$0.742390\pi$
$12$	0	0
$13$	0.703013	0.194981	0.0974904	−	0.995236i	$-0.468918\pi$
$13$	0.703013	0.194981	0.0974904	+	0.995236i	$0.468918\pi$
$14$	0	0
$15$	−0.646554	−0.166940
$16$	0	0
$17$	4.25337	1.03159	0.515796	−	0.856711i	$-0.327496\pi$
$17$	4.25337	1.03159	0.515796	+	0.856711i	$0.327496\pi$
$18$	0	0
$19$	−8.04736	−1.84619	−0.923095	−	0.384571i	$-0.874349\pi$
$19$	−8.04736	−1.84619	−0.923095	+	0.384571i	$0.874349\pi$
$20$	0	0
$21$	−0.200978	−0.0438569
$22$	0	0
$23$	5.34842	1.11522	0.557611	−	0.830102i	$-0.311718\pi$
$23$	5.34842	1.11522	0.557611	+	0.830102i	$0.311718\pi$
$24$	0	0
$25$	5.34937	1.06987
$26$	0	0
$27$	−1.19775	−0.230507
$28$	0	0
$29$	−5.39204	−1.00128	−0.500638	−	0.865657i	$-0.666901\pi$
$29$	−5.39204	−1.00128	−0.500638	+	0.865657i	$0.666901\pi$
$30$	0	0
$31$	−7.61900	−1.36841	−0.684206	−	0.729288i	$-0.739851\pi$
$31$	−7.61900	−1.36841	−0.684206	+	0.729288i	$0.739851\pi$
$32$	0	0
$33$	−0.919865	−0.160128
$34$	0	0
$35$	3.21704	0.543780
$36$	0	0
$37$	5.19272	0.853678	0.426839	−	0.904328i	$-0.359627\pi$
$37$	5.19272	0.853678	0.426839	+	0.904328i	$0.359627\pi$
$38$	0	0
$39$	0.141290	0.0226245
$40$	0	0
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	0	0
$43$	−10.3241	−1.57441	−0.787205	−	0.616692i	$-0.788472\pi$
$43$	−10.3241	−1.57441	−0.787205	+	0.616692i	$0.788472\pi$
$44$	0	0
$45$	9.52119	1.41934
$46$	0	0
$47$	−12.1160	−1.76730	−0.883651	−	0.468147i	$-0.844922\pi$
$47$	−12.1160	−1.76730	−0.883651	+	0.468147i	$0.844922\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0.854832	0.119700
$52$	0	0
$53$	−12.2837	−1.68730	−0.843648	−	0.536897i	$-0.819596\pi$
$53$	−12.2837	−1.68730	−0.843648	+	0.536897i	$0.819596\pi$
$54$	0	0
$55$	14.7242	1.98542
$56$	0	0
$57$	−1.61734	−0.214222
$58$	0	0
$59$	7.73023	1.00639	0.503195	−	0.864173i	$-0.332158\pi$
$59$	7.73023	1.00639	0.503195	+	0.864173i	$0.332158\pi$
$60$	0	0
$61$	−2.48971	−0.318774	−0.159387	−	0.987216i	$-0.550952\pi$
$61$	−2.48971	−0.318774	−0.159387	+	0.987216i	$0.550952\pi$
$62$	0	0
$63$	2.95961	0.372876
$64$	0	0
$65$	−2.26162	−0.280520
$66$	0	0
$67$	3.09767	0.378440	0.189220	−	0.981935i	$-0.439404\pi$
$67$	3.09767	0.378440	0.189220	+	0.981935i	$0.439404\pi$
$68$	0	0
$69$	1.07491	0.129404
$70$	0	0
$71$	−5.11581	−0.607135	−0.303568	−	0.952810i	$-0.598178\pi$
$71$	−5.11581	−0.607135	−0.303568	+	0.952810i	$0.598178\pi$
$72$	0	0
$73$	4.13640	0.484129	0.242065	−	0.970260i	$-0.422175\pi$
$73$	4.13640	0.484129	0.242065	+	0.970260i	$0.422175\pi$
$74$	0	0
$75$	1.07511	0.124142
$76$	0	0
$77$	4.57695	0.521592
$78$	0	0
$79$	13.7414	1.54603	0.773015	−	0.634388i	$-0.218748\pi$
$79$	13.7414	1.54603	0.773015	+	0.634388i	$0.218748\pi$
$80$	0	0
$81$	8.63810	0.959789
$82$	0	0
$83$	−4.90626	−0.538532	−0.269266	−	0.963066i	$-0.586781\pi$
$83$	−4.90626	−0.538532	−0.269266	+	0.963066i	$0.586781\pi$
$84$	0	0
$85$	−13.6833	−1.48416
$86$	0	0
$87$	−1.08368	−0.116183
$88$	0	0
$89$	5.97567	0.633420	0.316710	−	0.948522i	$-0.397422\pi$
$89$	5.97567	0.633420	0.316710	+	0.948522i	$0.397422\pi$
$90$	0	0
$91$	−0.703013	−0.0736958
$92$	0	0
$93$	−1.53125	−0.158783
$94$	0	0
$95$	25.8887	2.65613
$96$	0	0
$97$	1.45550	0.147783	0.0738916	−	0.997266i	$-0.476458\pi$
$97$	1.45550	0.147783	0.0738916	+	0.997266i	$0.476458\pi$
$98$	0	0
$99$	13.5460	1.36142
$100$	0	0
$101$	1.92425	0.191470	0.0957348	−	0.995407i	$-0.469480\pi$
$101$	1.92425	0.191470	0.0957348	+	0.995407i	$0.469480\pi$
$102$	0	0
$103$	5.35990	0.528127	0.264063	−	0.964505i	$-0.414937\pi$
$103$	5.35990	0.528127	0.264063	+	0.964505i	$0.414937\pi$
$104$	0	0
$105$	0.646554	0.0630973
$106$	0	0
$107$	−2.83031	−0.273617	−0.136808	−	0.990598i	$-0.543684\pi$
$107$	−2.83031	−0.273617	−0.136808	+	0.990598i	$0.543684\pi$
$108$	0	0
$109$	11.3960	1.09154	0.545768	−	0.837936i	$-0.316238\pi$
$109$	11.3960	1.09154	0.545768	+	0.837936i	$0.316238\pi$
$110$	0	0
$111$	1.04362	0.0990561
$112$	0	0
$113$	18.4852	1.73894	0.869470	−	0.493986i	$-0.164461\pi$
$113$	18.4852	1.73894	0.869470	+	0.493986i	$0.164461\pi$
$114$	0	0
$115$	−17.2061	−1.60448
$116$	0	0
$117$	−2.08064	−0.192356
$118$	0	0
$119$	−4.25337	−0.389905
$120$	0	0
$121$	9.94845	0.904405
$122$	0	0
$123$	−0.200978	−0.0181216
$124$	0	0
$125$	−1.12395	−0.100530
$126$	0	0
$127$	−9.98152	−0.885717	−0.442858	−	0.896592i	$-0.646036\pi$
$127$	−9.98152	−0.885717	−0.442858	+	0.896592i	$0.646036\pi$
$128$	0	0
$129$	−2.07491	−0.182686
$130$	0	0
$131$	11.2796	0.985506	0.492753	−	0.870169i	$-0.335991\pi$
$131$	11.2796	0.985506	0.492753	+	0.870169i	$0.335991\pi$
$132$	0	0
$133$	8.04736	0.697794
$134$	0	0
$135$	3.85321	0.331632
$136$	0	0
$137$	9.73838	0.832006	0.416003	−	0.909363i	$-0.363431\pi$
$137$	9.73838	0.832006	0.416003	+	0.909363i	$0.363431\pi$
$138$	0	0
$139$	4.85307	0.411632	0.205816	−	0.978591i	$-0.434015\pi$
$139$	4.85307	0.411632	0.205816	+	0.978591i	$0.434015\pi$
$140$	0	0
$141$	−2.43505	−0.205068
$142$	0	0
$143$	−3.21765	−0.269074
$144$	0	0
$145$	17.3464	1.44054
$146$	0	0
$147$	0.200978	0.0165764
$148$	0	0
$149$	−6.47268	−0.530263	−0.265131	−	0.964212i	$-0.585415\pi$
$149$	−6.47268	−0.530263	−0.265131	+	0.964212i	$0.585415\pi$
$150$	0	0
$151$	16.6124	1.35190	0.675949	−	0.736948i	$-0.263734\pi$
$151$	16.6124	1.35190	0.675949	+	0.736948i	$0.263734\pi$
$152$	0	0
$153$	−12.5883	−1.01770
$154$	0	0
$155$	24.5107	1.96874
$156$	0	0
$157$	21.5294	1.71824	0.859119	−	0.511777i	$-0.171012\pi$
$157$	21.5294	1.71824	0.859119	+	0.511777i	$0.171012\pi$
$158$	0	0
$159$	−2.46875	−0.195785
$160$	0	0
$161$	−5.34842	−0.421514
$162$	0	0
$163$	0.00553783	0.000433757 0	0.000216878	−	1.00000i	$-0.499931\pi$
$163$	0.00553783	0.000433757 0	0.000216878		1.00000i	$0.499931\pi$
$164$	0	0
$165$	2.95925	0.230377
$166$	0	0
$167$	−22.4914	−1.74044	−0.870220	−	0.492663i	$-0.836024\pi$
$167$	−22.4914	−1.74044	−0.870220	+	0.492663i	$0.836024\pi$
$168$	0	0
$169$	−12.5058	−0.961983
$170$	0	0
$171$	23.8170	1.82133
$172$	0	0
$173$	6.07961	0.462224	0.231112	−	0.972927i	$-0.425764\pi$
$173$	6.07961	0.462224	0.231112	+	0.972927i	$0.425764\pi$
$174$	0	0
$175$	−5.34937	−0.404375
$176$	0	0
$177$	1.55361	0.116776
$178$	0	0
$179$	7.91618	0.591683	0.295842	−	0.955237i	$-0.404400\pi$
$179$	7.91618	0.591683	0.295842	+	0.955237i	$0.404400\pi$
$180$	0	0
$181$	−2.20401	−0.163823	−0.0819115	−	0.996640i	$-0.526102\pi$
$181$	−2.20401	−0.163823	−0.0819115	+	0.996640i	$0.526102\pi$
$182$	0	0
$183$	−0.500376	−0.0369888
$184$	0	0
$185$	−16.7052	−1.22819
$186$	0	0
$187$	−19.4674	−1.42360
$188$	0	0
$189$	1.19775	0.0871234
$190$	0	0
$191$	1.75601	0.127061	0.0635303	−	0.997980i	$-0.479764\pi$
$191$	1.75601	0.127061	0.0635303	+	0.997980i	$0.479764\pi$
$192$	0	0
$193$	2.06376	0.148553	0.0742763	−	0.997238i	$-0.476335\pi$
$193$	2.06376	0.148553	0.0742763	+	0.997238i	$0.476335\pi$
$194$	0	0
$195$	−0.454536	−0.0325500
$196$	0	0
$197$	−21.2674	−1.51524	−0.757621	−	0.652695i	$-0.773638\pi$
$197$	−21.2674	−1.51524	−0.757621	+	0.652695i	$0.773638\pi$
$198$	0	0
$199$	6.50285	0.460975	0.230488	−	0.973075i	$-0.425968\pi$
$199$	6.50285	0.460975	0.230488	+	0.973075i	$0.425968\pi$
$200$	0	0
$201$	0.622563	0.0439122
$202$	0	0
$203$	5.39204	0.378447
$204$	0	0
$205$	3.21704	0.224688
$206$	0	0
$207$	−15.8292	−1.10021
$208$	0	0
$209$	36.8323	2.54775
$210$	0	0
$211$	10.4931	0.722377	0.361188	−	0.932493i	$-0.382371\pi$
$211$	10.4931	0.722377	0.361188	+	0.932493i	$0.382371\pi$
$212$	0	0
$213$	−1.02816	−0.0704487
$214$	0	0
$215$	33.2131	2.26511
$216$	0	0
$217$	7.61900	0.517211
$218$	0	0
$219$	0.831325	0.0561757
$220$	0	0
$221$	2.99017	0.201141
$222$	0	0
$223$	−14.4757	−0.969366	−0.484683	−	0.874690i	$-0.661065\pi$
$223$	−14.4757	−0.969366	−0.484683	+	0.874690i	$0.661065\pi$
$224$	0	0
$225$	−15.8321	−1.05547
$226$	0	0
$227$	17.5499	1.16483	0.582414	−	0.812892i	$-0.302108\pi$
$227$	17.5499	1.16483	0.582414	+	0.812892i	$0.302108\pi$
$228$	0	0
$229$	−0.189800	−0.0125423	−0.00627117	−	0.999980i	$-0.501996\pi$
$229$	−0.189800	−0.0125423	−0.00627117	+	0.999980i	$0.501996\pi$
$230$	0	0
$231$	0.919865	0.0605227
$232$	0	0
$233$	2.03466	0.133295	0.0666476	−	0.997777i	$-0.478770\pi$
$233$	2.03466	0.133295	0.0666476	+	0.997777i	$0.478770\pi$
$234$	0	0
$235$	38.9777	2.54263
$236$	0	0
$237$	2.76172	0.179393
$238$	0	0
$239$	−11.6053	−0.750684	−0.375342	−	0.926886i	$-0.622475\pi$
$239$	−11.6053	−0.750684	−0.375342	+	0.926886i	$0.622475\pi$
$240$	0	0
$241$	−4.33539	−0.279267	−0.139633	−	0.990203i	$-0.544592\pi$
$241$	−4.33539	−0.279267	−0.139633	+	0.990203i	$0.544592\pi$
$242$	0	0
$243$	5.32931	0.341876
$244$	0	0
$245$	−3.21704	−0.205529
$246$	0	0
$247$	−5.65740	−0.359972
$248$	0	0
$249$	−0.986049	−0.0624883
$250$	0	0
$251$	−23.3501	−1.47385	−0.736923	−	0.675976i	$-0.763722\pi$
$251$	−23.3501	−1.47385	−0.736923	+	0.675976i	$0.763722\pi$
$252$	0	0
$253$	−24.4794	−1.53901
$254$	0	0
$255$	−2.75003	−0.172214
$256$	0	0
$257$	−19.7199	−1.23009	−0.615046	−	0.788491i	$-0.710862\pi$
$257$	−19.7199	−1.23009	−0.615046	+	0.788491i	$0.710862\pi$
$258$	0	0
$259$	−5.19272	−0.322660
$260$	0	0
$261$	15.9583	0.987795
$262$	0	0
$263$	−1.67120	−0.103050	−0.0515252	−	0.998672i	$-0.516408\pi$
$263$	−1.67120	−0.103050	−0.0515252	+	0.998672i	$0.516408\pi$
$264$	0	0
$265$	39.5172	2.42752
$266$	0	0
$267$	1.20098	0.0734986
$268$	0	0
$269$	−24.4059	−1.48805	−0.744027	−	0.668150i	$-0.767086\pi$
$269$	−24.4059	−1.48805	−0.744027	+	0.668150i	$0.767086\pi$
$270$	0	0
$271$	−7.83315	−0.475830	−0.237915	−	0.971286i	$-0.576464\pi$
$271$	−7.83315	−0.475830	−0.237915	+	0.971286i	$0.576464\pi$
$272$	0	0
$273$	−0.141290	−0.00855126
$274$	0	0
$275$	−24.4838	−1.47643
$276$	0	0
$277$	−6.53052	−0.392381	−0.196190	−	0.980566i	$-0.562857\pi$
$277$	−6.53052	−0.392381	−0.196190	+	0.980566i	$0.562857\pi$
$278$	0	0
$279$	22.5493	1.34999
$280$	0	0
$281$	−2.23112	−0.133097	−0.0665487	−	0.997783i	$-0.521199\pi$
$281$	−2.23112	−0.133097	−0.0665487	+	0.997783i	$0.521199\pi$
$282$	0	0
$283$	18.2949	1.08752	0.543759	−	0.839242i	$-0.317001\pi$
$283$	18.2949	1.08752	0.543759	+	0.839242i	$0.317001\pi$
$284$	0	0
$285$	5.20306	0.308202
$286$	0	0
$287$	1.00000	0.0590281
$288$	0	0
$289$	1.09112	0.0641835
$290$	0	0
$291$	0.292522	0.0171480
$292$	0	0
$293$	−7.37440	−0.430817	−0.215409	−	0.976524i	$-0.569108\pi$
$293$	−7.37440	−0.430817	−0.215409	+	0.976524i	$0.569108\pi$
$294$	0	0
$295$	−24.8685	−1.44790
$296$	0	0
$297$	5.48203	0.318100
$298$	0	0
$299$	3.76001	0.217447
$300$	0	0
$301$	10.3241	0.595071
$302$	0	0
$303$	0.386731	0.0222171
$304$	0	0
$305$	8.00949	0.458622
$306$	0	0
$307$	11.6587	0.665398	0.332699	−	0.943033i	$-0.392041\pi$
$307$	11.6587	0.665398	0.332699	+	0.943033i	$0.392041\pi$
$308$	0	0
$309$	1.07722	0.0612810
$310$	0	0
$311$	19.2508	1.09161	0.545806	−	0.837911i	$-0.316223\pi$
$311$	19.2508	1.09161	0.545806	+	0.837911i	$0.316223\pi$
$312$	0	0
$313$	−3.73012	−0.210839	−0.105419	−	0.994428i	$-0.533618\pi$
$313$	−3.73012	−0.210839	−0.105419	+	0.994428i	$0.533618\pi$
$314$	0	0
$315$	−9.52119	−0.536458
$316$	0	0
$317$	−19.6027	−1.10100	−0.550499	−	0.834836i	$-0.685562\pi$
$317$	−19.6027	−1.10100	−0.550499	+	0.834836i	$0.685562\pi$
$318$	0	0
$319$	24.6791	1.38176
$320$	0	0
$321$	−0.568830	−0.0317490
$322$	0	0
$323$	−34.2284	−1.90452
$324$	0	0
$325$	3.76068	0.208605
$326$	0	0
$327$	2.29034	0.126656
$328$	0	0
$329$	12.1160	0.667977
$330$	0	0
$331$	23.4828	1.29073	0.645366	−	0.763873i	$-0.276705\pi$
$331$	23.4828	1.29073	0.645366	+	0.763873i	$0.276705\pi$
$332$	0	0
$333$	−15.3684	−0.842184
$334$	0	0
$335$	−9.96534	−0.544465
$336$	0	0
$337$	1.20201	0.0654778	0.0327389	−	0.999464i	$-0.489577\pi$
$337$	1.20201	0.0654778	0.0327389	+	0.999464i	$0.489577\pi$
$338$	0	0
$339$	3.71511	0.201777
$340$	0	0
$341$	34.8718	1.88841
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−3.45804	−0.186175
$346$	0	0
$347$	3.01800	0.162015	0.0810074	−	0.996713i	$-0.474186\pi$
$347$	3.01800	0.162015	0.0810074	+	0.996713i	$0.474186\pi$
$348$	0	0
$349$	−30.2795	−1.62082	−0.810412	−	0.585861i	$-0.800757\pi$
$349$	−30.2795	−1.62082	−0.810412	+	0.585861i	$0.800757\pi$
$350$	0	0
$351$	−0.842033	−0.0449444
$352$	0	0
$353$	4.59065	0.244336	0.122168	−	0.992509i	$-0.461015\pi$
$353$	4.59065	0.244336	0.122168	+	0.992509i	$0.461015\pi$
$354$	0	0
$355$	16.4578	0.873489
$356$	0	0
$357$	−0.854832	−0.0452425
$358$	0	0
$359$	−18.4050	−0.971381	−0.485691	−	0.874131i	$-0.661432\pi$
$359$	−18.4050	−0.971381	−0.485691	+	0.874131i	$0.661432\pi$
$360$	0	0
$361$	45.7600	2.40842
$362$	0	0
$363$	1.99942	0.104942
$364$	0	0
$365$	−13.3070	−0.696519
$366$	0	0
$367$	−17.0270	−0.888802	−0.444401	−	0.895828i	$-0.646583\pi$
$367$	−17.0270	−0.888802	−0.444401	+	0.895828i	$0.646583\pi$
$368$	0	0
$369$	2.95961	0.154071
$370$	0	0
$371$	12.2837	0.637738
$372$	0	0
$373$	26.3381	1.36373	0.681867	−	0.731476i	$-0.261168\pi$
$373$	26.3381	1.36373	0.681867	+	0.731476i	$0.261168\pi$
$374$	0	0
$375$	−0.225890	−0.0116649
$376$	0	0
$377$	−3.79067	−0.195230
$378$	0	0
$379$	25.4631	1.30795	0.653975	−	0.756516i	$-0.273100\pi$
$379$	25.4631	1.30795	0.653975	+	0.756516i	$0.273100\pi$
$380$	0	0
$381$	−2.00606	−0.102774
$382$	0	0
$383$	−26.6465	−1.36157	−0.680786	−	0.732483i	$-0.738361\pi$
$383$	−26.6465	−1.36157	−0.680786	+	0.732483i	$0.738361\pi$
$384$	0	0
$385$	−14.7242	−0.750417
$386$	0	0
$387$	30.5553	1.55321
$388$	0	0
$389$	−33.7017	−1.70874	−0.854372	−	0.519662i	$-0.826058\pi$
$389$	−33.7017	−1.70874	−0.854372	+	0.519662i	$0.826058\pi$
$390$	0	0
$391$	22.7488	1.15045
$392$	0	0
$393$	2.26695	0.114353
$394$	0	0
$395$	−44.2067	−2.22428
$396$	0	0
$397$	−24.7242	−1.24087	−0.620437	−	0.784256i	$-0.713045\pi$
$397$	−24.7242	−1.24087	−0.620437	+	0.784256i	$0.713045\pi$
$398$	0	0
$399$	1.61734	0.0809683
$400$	0	0
$401$	14.7100	0.734581	0.367291	−	0.930106i	$-0.380285\pi$
$401$	14.7100	0.734581	0.367291	+	0.930106i	$0.380285\pi$
$402$	0	0
$403$	−5.35626	−0.266814
$404$	0	0
$405$	−27.7892	−1.38085
$406$	0	0
$407$	−23.7668	−1.17808
$408$	0	0
$409$	18.8735	0.933233	0.466616	−	0.884460i	$-0.345473\pi$
$409$	18.8735	0.933233	0.466616	+	0.884460i	$0.345473\pi$
$410$	0	0
$411$	1.95720	0.0965414
$412$	0	0
$413$	−7.73023	−0.380380
$414$	0	0
$415$	15.7837	0.774789
$416$	0	0
$417$	0.975359	0.0477635
$418$	0	0
$419$	−28.7795	−1.40597	−0.702985	−	0.711205i	$-0.748150\pi$
$419$	−28.7795	−1.40597	−0.702985	+	0.711205i	$0.748150\pi$
$420$	0	0
$421$	36.0120	1.75512	0.877559	−	0.479468i	$-0.159171\pi$
$421$	36.0120	1.75512	0.877559	+	0.479468i	$0.159171\pi$
$422$	0	0
$423$	35.8586	1.74351
$424$	0	0
$425$	22.7528	1.10368
$426$	0	0
$427$	2.48971	0.120485
$428$	0	0
$429$	−0.646677	−0.0312219
$430$	0	0
$431$	−9.23519	−0.444843	−0.222422	−	0.974951i	$-0.571396\pi$
$431$	−9.23519	−0.444843	−0.222422	+	0.974951i	$0.571396\pi$
$432$	0	0
$433$	−18.1355	−0.871537	−0.435768	−	0.900059i	$-0.643523\pi$
$433$	−18.1355	−0.871537	−0.435768	+	0.900059i	$0.643523\pi$
$434$	0	0
$435$	3.48625	0.167153
$436$	0	0
$437$	−43.0406	−2.05891
$438$	0	0
$439$	6.79012	0.324075	0.162037	−	0.986785i	$-0.448193\pi$
$439$	6.79012	0.324075	0.162037	+	0.986785i	$0.448193\pi$
$440$	0	0
$441$	−2.95961	−0.140934
$442$	0	0
$443$	−34.2284	−1.62624	−0.813119	−	0.582097i	$-0.802232\pi$
$443$	−34.2284	−1.62624	−0.813119	+	0.582097i	$0.802232\pi$
$444$	0	0
$445$	−19.2240	−0.911306
$446$	0	0
$447$	−1.30086	−0.0615288
$448$	0	0
$449$	−2.19148	−0.103422	−0.0517112	−	0.998662i	$-0.516468\pi$
$449$	−2.19148	−0.103422	−0.0517112	+	0.998662i	$0.516468\pi$
$450$	0	0
$451$	4.57695	0.215520
$452$	0	0
$453$	3.33872	0.156867
$454$	0	0
$455$	2.26162	0.106027
$456$	0	0
$457$	15.2932	0.715387	0.357694	−	0.933839i	$-0.383563\pi$
$457$	15.2932	0.715387	0.357694	+	0.933839i	$0.383563\pi$
$458$	0	0
$459$	−5.09446	−0.237789
$460$	0	0
$461$	−7.37440	−0.343460	−0.171730	−	0.985144i	$-0.554936\pi$
$461$	−7.37440	−0.343460	−0.171730	+	0.985144i	$0.554936\pi$
$462$	0	0
$463$	25.3329	1.17732	0.588661	−	0.808380i	$-0.299655\pi$
$463$	25.3329	1.17732	0.588661	+	0.808380i	$0.299655\pi$
$464$	0	0
$465$	4.92610	0.228442
$466$	0	0
$467$	−9.05902	−0.419201	−0.209601	−	0.977787i	$-0.567216\pi$
$467$	−9.05902	−0.419201	−0.209601	+	0.977787i	$0.567216\pi$
$468$	0	0
$469$	−3.09767	−0.143037
$470$	0	0
$471$	4.32694	0.199375
$472$	0	0
$473$	47.2528	2.17269
$474$	0	0
$475$	−43.0483	−1.97519
$476$	0	0
$477$	36.3549	1.66458
$478$	0	0
$479$	−15.7297	−0.718708	−0.359354	−	0.933201i	$-0.617003\pi$
$479$	−15.7297	−0.718708	−0.359354	+	0.933201i	$0.617003\pi$
$480$	0	0
$481$	3.65055	0.166451
$482$	0	0
$483$	−1.07491	−0.0489102
$484$	0	0
$485$	−4.68239	−0.212617
$486$	0	0
$487$	11.5758	0.524549	0.262275	−	0.964993i	$-0.415527\pi$
$487$	11.5758	0.524549	0.262275	+	0.964993i	$0.415527\pi$
$488$	0	0
$489$	0.00111298	5.03308e−5 0
$490$	0	0
$491$	15.2590	0.688628	0.344314	−	0.938855i	$-0.388111\pi$
$491$	15.2590	0.688628	0.344314	+	0.938855i	$0.388111\pi$
$492$	0	0
$493$	−22.9343	−1.03291
$494$	0	0
$495$	−43.5780	−1.95869
$496$	0	0
$497$	5.11581	0.229476
$498$	0	0
$499$	−8.35644	−0.374086	−0.187043	−	0.982352i	$-0.559890\pi$
$499$	−8.35644	−0.374086	−0.187043	+	0.982352i	$0.559890\pi$
$500$	0	0
$501$	−4.52028	−0.201951
$502$	0	0
$503$	−2.53984	−0.113246	−0.0566230	−	0.998396i	$-0.518033\pi$
$503$	−2.53984	−0.113246	−0.0566230	+	0.998396i	$0.518033\pi$
$504$	0	0
$505$	−6.19038	−0.275468
$506$	0	0
$507$	−2.51338	−0.111623
$508$	0	0
$509$	−6.48190	−0.287305	−0.143653	−	0.989628i	$-0.545885\pi$
$509$	−6.48190	−0.287305	−0.143653	+	0.989628i	$0.545885\pi$
$510$	0	0
$511$	−4.13640	−0.182984
$512$	0	0
$513$	9.63871	0.425560
$514$	0	0
$515$	−17.2430	−0.759819
$516$	0	0
$517$	55.4543	2.43888
$518$	0	0
$519$	1.22187	0.0536340
$520$	0	0
$521$	−1.36749	−0.0599107	−0.0299553	−	0.999551i	$-0.509537\pi$
$521$	−1.36749	−0.0599107	−0.0299553	+	0.999551i	$0.509537\pi$
$522$	0	0
$523$	−12.9198	−0.564944	−0.282472	−	0.959276i	$-0.591154\pi$
$523$	−12.9198	−0.564944	−0.282472	+	0.959276i	$0.591154\pi$
$524$	0	0
$525$	−1.07511	−0.0469214
$526$	0	0
$527$	−32.4064	−1.41164
$528$	0	0
$529$	5.60555	0.243720
$530$	0	0
$531$	−22.8785	−0.992841
$532$	0	0
$533$	−0.703013	−0.0304509
$534$	0	0
$535$	9.10525	0.393654
$536$	0	0
$537$	1.59098	0.0686557
$538$	0	0
$539$	−4.57695	−0.197143
$540$	0	0
$541$	−4.47397	−0.192351	−0.0961756	−	0.995364i	$-0.530661\pi$
$541$	−4.47397	−0.192351	−0.0961756	+	0.995364i	$0.530661\pi$
$542$	0	0
$543$	−0.442958	−0.0190091
$544$	0	0
$545$	−36.6613	−1.57040
$546$	0	0
$547$	22.3133	0.954048	0.477024	−	0.878890i	$-0.341715\pi$
$547$	22.3133	0.954048	0.477024	+	0.878890i	$0.341715\pi$
$548$	0	0
$549$	7.36855	0.314482
$550$	0	0
$551$	43.3917	1.84855
$552$	0	0
$553$	−13.7414	−0.584344
$554$	0	0
$555$	−3.35738	−0.142513
$556$	0	0
$557$	15.3755	0.651482	0.325741	−	0.945459i	$-0.394386\pi$
$557$	15.3755	0.651482	0.325741	+	0.945459i	$0.394386\pi$
$558$	0	0
$559$	−7.25797	−0.306979
$560$	0	0
$561$	−3.91252	−0.165187
$562$	0	0
$563$	4.13910	0.174442	0.0872212	−	0.996189i	$-0.472201\pi$
$563$	4.13910	0.174442	0.0872212	+	0.996189i	$0.472201\pi$
$564$	0	0
$565$	−59.4677	−2.50182
$566$	0	0
$567$	−8.63810	−0.362766
$568$	0	0
$569$	20.9479	0.878182	0.439091	−	0.898443i	$-0.355301\pi$
$569$	20.9479	0.878182	0.439091	+	0.898443i	$0.355301\pi$
$570$	0	0
$571$	20.5728	0.860947	0.430473	−	0.902603i	$-0.358347\pi$
$571$	20.5728	0.860947	0.430473	+	0.902603i	$0.358347\pi$
$572$	0	0
$573$	0.352919	0.0147434
$574$	0	0
$575$	28.6107	1.19315
$576$	0	0
$577$	−26.3061	−1.09514	−0.547568	−	0.836761i	$-0.684446\pi$
$577$	−26.3061	−1.09514	−0.547568	+	0.836761i	$0.684446\pi$
$578$	0	0
$579$	0.414770	0.0172372
$580$	0	0
$581$	4.90626	0.203546
$582$	0	0
$583$	56.2218	2.32847
$584$	0	0
$585$	6.69352	0.276743
$586$	0	0
$587$	0.835878	0.0345004	0.0172502	−	0.999851i	$-0.494509\pi$
$587$	0.835878	0.0345004	0.0172502	+	0.999851i	$0.494509\pi$
$588$	0	0
$589$	61.3128	2.52635
$590$	0	0
$591$	−4.27428	−0.175820
$592$	0	0
$593$	−5.90293	−0.242404	−0.121202	−	0.992628i	$-0.538675\pi$
$593$	−5.90293	−0.242404	−0.121202	+	0.992628i	$0.538675\pi$
$594$	0	0
$595$	13.6833	0.560959
$596$	0	0
$597$	1.30693	0.0534890
$598$	0	0
$599$	20.7273	0.846896	0.423448	−	0.905920i	$-0.360820\pi$
$599$	20.7273	0.846896	0.423448	+	0.905920i	$0.360820\pi$
$600$	0	0
$601$	4.69241	0.191407	0.0957037	−	0.995410i	$-0.469490\pi$
$601$	4.69241	0.191407	0.0957037	+	0.995410i	$0.469490\pi$
$602$	0	0
$603$	−9.16789	−0.373345
$604$	0	0
$605$	−32.0046	−1.30117
$606$	0	0
$607$	−27.8857	−1.13184	−0.565922	−	0.824459i	$-0.691480\pi$
$607$	−27.8857	−1.13184	−0.565922	+	0.824459i	$0.691480\pi$
$608$	0	0
$609$	1.08368	0.0439129
$610$	0	0
$611$	−8.51771	−0.344590
$612$	0	0
$613$	−15.6076	−0.630383	−0.315192	−	0.949028i	$-0.602069\pi$
$613$	−15.6076	−0.630383	−0.315192	+	0.949028i	$0.602069\pi$
$614$	0	0
$615$	0.646554	0.0260716
$616$	0	0
$617$	12.2414	0.492820	0.246410	−	0.969166i	$-0.420749\pi$
$617$	12.2414	0.492820	0.246410	+	0.969166i	$0.420749\pi$
$618$	0	0
$619$	−23.4809	−0.943778	−0.471889	−	0.881658i	$-0.656428\pi$
$619$	−23.4809	−0.943778	−0.471889	+	0.881658i	$0.656428\pi$
$620$	0	0
$621$	−6.40606	−0.257066
$622$	0	0
$623$	−5.97567	−0.239410
$624$	0	0
$625$	−23.1311	−0.925243
$626$	0	0
$627$	7.40248	0.295627
$628$	0	0
$629$	22.0865	0.880648
$630$	0	0
$631$	0.0226990	0.000903631 0	0.000451816	−	1.00000i	$-0.499856\pi$
$631$	0.0226990	0.000903631 0	0.000451816		1.00000i	$0.499856\pi$
$632$	0	0
$633$	2.10889	0.0838207
$634$	0	0
$635$	32.1110	1.27429
$636$	0	0
$637$	0.703013	0.0278544
$638$	0	0
$639$	15.1408	0.598961
$640$	0	0
$641$	38.0777	1.50398	0.751989	−	0.659175i	$-0.229095\pi$
$641$	38.0777	1.50398	0.751989	+	0.659175i	$0.229095\pi$
$642$	0	0
$643$	−24.2224	−0.955239	−0.477619	−	0.878567i	$-0.658500\pi$
$643$	−24.2224	−0.955239	−0.477619	+	0.878567i	$0.658500\pi$
$644$	0	0
$645$	6.67509	0.262831
$646$	0	0
$647$	44.1321	1.73501	0.867507	−	0.497426i	$-0.165721\pi$
$647$	44.1321	1.73501	0.867507	+	0.497426i	$0.165721\pi$
$648$	0	0
$649$	−35.3809	−1.38882
$650$	0	0
$651$	1.53125	0.0600144
$652$	0	0
$653$	−3.76579	−0.147367	−0.0736833	−	0.997282i	$-0.523475\pi$
$653$	−3.76579	−0.147367	−0.0736833	+	0.997282i	$0.523475\pi$
$654$	0	0
$655$	−36.2871	−1.41785
$656$	0	0
$657$	−12.2421	−0.477611
$658$	0	0
$659$	21.1980	0.825757	0.412878	−	0.910786i	$-0.364523\pi$
$659$	21.1980	0.825757	0.412878	+	0.910786i	$0.364523\pi$
$660$	0	0
$661$	−11.3027	−0.439624	−0.219812	−	0.975542i	$-0.570544\pi$
$661$	−11.3027	−0.439624	−0.219812	+	0.975542i	$0.570544\pi$
$662$	0	0
$663$	0.600958	0.0233393
$664$	0	0
$665$	−25.8887	−1.00392
$666$	0	0
$667$	−28.8389	−1.11664
$668$	0	0
$669$	−2.90930	−0.112480
$670$	0	0
$671$	11.3953	0.439909
$672$	0	0
$673$	18.3138	0.705946	0.352973	−	0.935634i	$-0.385171\pi$
$673$	18.3138	0.705946	0.352973	+	0.935634i	$0.385171\pi$
$674$	0	0
$675$	−6.40721	−0.246614
$676$	0	0
$677$	−15.9963	−0.614788	−0.307394	−	0.951582i	$-0.599457\pi$
$677$	−15.9963	−0.614788	−0.307394	+	0.951582i	$0.599457\pi$
$678$	0	0
$679$	−1.45550	−0.0558568
$680$	0	0
$681$	3.52714	0.135160
$682$	0	0
$683$	−6.75437	−0.258449	−0.129224	−	0.991615i	$-0.541249\pi$
$683$	−6.75437	−0.258449	−0.129224	+	0.991615i	$0.541249\pi$
$684$	0	0
$685$	−31.3288	−1.19701
$686$	0	0
$687$	−0.0381456	−0.00145534
$688$	0	0
$689$	−8.63560	−0.328990
$690$	0	0
$691$	−8.11399	−0.308671	−0.154335	−	0.988019i	$-0.549324\pi$
$691$	−8.11399	−0.308671	−0.154335	+	0.988019i	$0.549324\pi$
$692$	0	0
$693$	−13.5460	−0.514569
$694$	0	0
$695$	−15.6125	−0.592217
$696$	0	0
$697$	−4.25337	−0.161108
$698$	0	0
$699$	0.408922	0.0154668
$700$	0	0
$701$	−49.0400	−1.85221	−0.926107	−	0.377260i	$-0.876866\pi$
$701$	−49.0400	−1.85221	−0.926107	+	0.377260i	$0.876866\pi$
$702$	0	0
$703$	−41.7877	−1.57605
$704$	0	0
$705$	7.83366	0.295033
$706$	0	0
$707$	−1.92425	−0.0723687
$708$	0	0
$709$	10.8982	0.409292	0.204646	−	0.978836i	$-0.434396\pi$
$709$	10.8982	0.409292	0.204646	+	0.978836i	$0.434396\pi$
$710$	0	0
$711$	−40.6692	−1.52521
$712$	0	0
$713$	−40.7496	−1.52608
$714$	0	0
$715$	10.3513	0.387118
$716$	0	0
$717$	−2.33241	−0.0871053
$718$	0	0
$719$	−0.0450043	−0.00167838	−0.000839188	−	1.00000i	$-0.500267\pi$
$719$	−0.0450043	−0.00167838	−0.000839188		1.00000i	$0.500267\pi$
$720$	0	0
$721$	−5.35990	−0.199613
$722$	0	0
$723$	−0.871316	−0.0324046
$724$	0	0
$725$	−28.8440	−1.07124
$726$	0	0
$727$	29.3527	1.08863	0.544315	−	0.838881i	$-0.316790\pi$
$727$	29.3527	1.08863	0.544315	+	0.838881i	$0.316790\pi$
$728$	0	0
$729$	−24.8432	−0.920120
$730$	0	0
$731$	−43.9121	−1.62415
$732$	0	0
$733$	19.0948	0.705281	0.352640	−	0.935759i	$-0.385284\pi$
$733$	19.0948	0.705281	0.352640	+	0.935759i	$0.385284\pi$
$734$	0	0
$735$	−0.646554	−0.0238485
$736$	0	0
$737$	−14.1779	−0.522249
$738$	0	0
$739$	29.0981	1.07039	0.535196	−	0.844728i	$-0.320238\pi$
$739$	29.0981	1.07039	0.535196	+	0.844728i	$0.320238\pi$
$740$	0	0
$741$	−1.13701	−0.0417692
$742$	0	0
$743$	10.9674	0.402354	0.201177	−	0.979555i	$-0.435523\pi$
$743$	10.9674	0.402354	0.201177	+	0.979555i	$0.435523\pi$
$744$	0	0
$745$	20.8229	0.762892
$746$	0	0
$747$	14.5206	0.531281
$748$	0	0
$749$	2.83031	0.103417
$750$	0	0
$751$	−33.9703	−1.23959	−0.619797	−	0.784762i	$-0.712785\pi$
$751$	−33.9703	−1.23959	−0.619797	+	0.784762i	$0.712785\pi$
$752$	0	0
$753$	−4.69286	−0.171017
$754$	0	0
$755$	−53.4428	−1.94498
$756$	0	0
$757$	7.52333	0.273440	0.136720	−	0.990610i	$-0.456344\pi$
$757$	7.52333	0.273440	0.136720	+	0.990610i	$0.456344\pi$
$758$	0	0
$759$	−4.91982	−0.178578
$760$	0	0
$761$	−23.8739	−0.865428	−0.432714	−	0.901531i	$-0.642444\pi$
$761$	−23.8739	−0.865428	−0.432714	+	0.901531i	$0.642444\pi$
$762$	0	0
$763$	−11.3960	−0.412562
$764$	0	0
$765$	40.4971	1.46418
$766$	0	0
$767$	5.43446	0.196227
$768$	0	0
$769$	2.21793	0.0799805	0.0399903	−	0.999200i	$-0.487267\pi$
$769$	2.21793	0.0799805	0.0399903	+	0.999200i	$0.487267\pi$
$770$	0	0
$771$	−3.96325	−0.142733
$772$	0	0
$773$	18.4207	0.662547	0.331274	−	0.943535i	$-0.392522\pi$
$773$	18.4207	0.662547	0.331274	+	0.943535i	$0.392522\pi$
$774$	0	0
$775$	−40.7569	−1.46403
$776$	0	0
$777$	−1.04362	−0.0374397
$778$	0	0
$779$	8.04736	0.288327
$780$	0	0
$781$	23.4148	0.837848
$782$	0	0
$783$	6.45831	0.230801
$784$	0	0
$785$	−69.2612	−2.47204
$786$	0	0
$787$	−18.9009	−0.673745	−0.336873	−	0.941550i	$-0.609369\pi$
$787$	−18.9009	−0.673745	−0.336873	+	0.941550i	$0.609369\pi$
$788$	0	0
$789$	−0.335873	−0.0119574
$790$	0	0
$791$	−18.4852	−0.657257
$792$	0	0
$793$	−1.75030	−0.0621548
$794$	0	0
$795$	7.94208	0.281677
$796$	0	0
$797$	−1.21120	−0.0429028	−0.0214514	−	0.999770i	$-0.506829\pi$
$797$	−1.21120	−0.0429028	−0.0214514	+	0.999770i	$0.506829\pi$
$798$	0	0
$799$	−51.5338	−1.82313
$800$	0	0
$801$	−17.6857	−0.624892
$802$	0	0
$803$	−18.9321	−0.668099
$804$	0	0
$805$	17.2061	0.606435
$806$	0	0
$807$	−4.90504	−0.172666
$808$	0	0
$809$	−1.52068	−0.0534643	−0.0267322	−	0.999643i	$-0.508510\pi$
$809$	−1.52068	−0.0534643	−0.0267322	+	0.999643i	$0.508510\pi$
$810$	0	0
$811$	27.6060	0.969379	0.484689	−	0.874686i	$-0.338933\pi$
$811$	27.6060	0.969379	0.484689	+	0.874686i	$0.338933\pi$
$812$	0	0
$813$	−1.57429	−0.0552127
$814$	0	0
$815$	−0.0178155	−0.000624048 0
$816$	0	0
$817$	83.0817	2.90666
$818$	0	0
$819$	2.08064	0.0727036
$820$	0	0
$821$	50.0892	1.74813	0.874063	−	0.485813i	$-0.161476\pi$
$821$	50.0892	1.74813	0.874063	+	0.485813i	$0.161476\pi$
$822$	0	0
$823$	10.0645	0.350826	0.175413	−	0.984495i	$-0.443874\pi$
$823$	10.0645	0.350826	0.175413	+	0.984495i	$0.443874\pi$
$824$	0	0
$825$	−4.92070	−0.171317
$826$	0	0
$827$	7.95416	0.276593	0.138297	−	0.990391i	$-0.455837\pi$
$827$	7.95416	0.276593	0.138297	+	0.990391i	$0.455837\pi$
$828$	0	0
$829$	50.9978	1.77123	0.885614	−	0.464422i	$-0.153738\pi$
$829$	50.9978	1.77123	0.885614	+	0.464422i	$0.153738\pi$
$830$	0	0
$831$	−1.31249	−0.0455297
$832$	0	0
$833$	4.25337	0.147370
$834$	0	0
$835$	72.3560	2.50398
$836$	0	0
$837$	9.12565	0.315429
$838$	0	0
$839$	10.8839	0.375753	0.187877	−	0.982193i	$-0.439839\pi$
$839$	10.8839	0.375753	0.187877	+	0.982193i	$0.439839\pi$
$840$	0	0
$841$	0.0740626	0.00255388
$842$	0	0
$843$	−0.448405	−0.0154439
$844$	0	0
$845$	40.2316	1.38401
$846$	0	0
$847$	−9.94845	−0.341833
$848$	0	0
$849$	3.67686	0.126190
$850$	0	0
$851$	27.7728	0.952040
$852$	0	0
$853$	−13.4544	−0.460670	−0.230335	−	0.973111i	$-0.573982\pi$
$853$	−13.4544	−0.460670	−0.230335	+	0.973111i	$0.573982\pi$
$854$	0	0
$855$	−76.6204	−2.62036
$856$	0	0
$857$	39.6908	1.35581	0.677906	−	0.735149i	$-0.262888\pi$
$857$	39.6908	1.35581	0.677906	+	0.735149i	$0.262888\pi$
$858$	0	0
$859$	−23.5262	−0.802705	−0.401352	−	0.915924i	$-0.631460\pi$
$859$	−23.5262	−0.802705	−0.401352	+	0.915924i	$0.631460\pi$
$860$	0	0
$861$	0.200978	0.00684930
$862$	0	0
$863$	39.1998	1.33438	0.667189	−	0.744888i	$-0.267497\pi$
$863$	39.1998	1.33438	0.667189	+	0.744888i	$0.267497\pi$
$864$	0	0
$865$	−19.5584	−0.665005
$866$	0	0
$867$	0.219291	0.00744751
$868$	0	0
$869$	−62.8937	−2.13352
$870$	0	0
$871$	2.17770	0.0737886
$872$	0	0
$873$	−4.30770	−0.145793
$874$	0	0
$875$	1.12395	0.0379966
$876$	0	0
$877$	24.7835	0.836879	0.418439	−	0.908245i	$-0.362577\pi$
$877$	24.7835	0.836879	0.418439	+	0.908245i	$0.362577\pi$
$878$	0	0
$879$	−1.48209	−0.0499897
$880$	0	0
$881$	33.8534	1.14055	0.570275	−	0.821454i	$-0.306837\pi$
$881$	33.8534	1.14055	0.570275	+	0.821454i	$0.306837\pi$
$882$	0	0
$883$	9.96707	0.335419	0.167709	−	0.985837i	$-0.446363\pi$
$883$	9.96707	0.335419	0.167709	+	0.985837i	$0.446363\pi$
$884$	0	0
$885$	−4.99802	−0.168007
$886$	0	0
$887$	26.9208	0.903910	0.451955	−	0.892041i	$-0.350727\pi$
$887$	26.9208	0.903910	0.451955	+	0.892041i	$0.350727\pi$
$888$	0	0
$889$	9.98152	0.334770
$890$	0	0
$891$	−39.5361	−1.32451
$892$	0	0
$893$	97.5018	3.26277
$894$	0	0
$895$	−25.4667	−0.851258
$896$	0	0
$897$	0.755678	0.0252313
$898$	0	0
$899$	41.0819	1.37016
$900$	0	0
$901$	−52.2471	−1.74060
$902$	0	0
$903$	2.07491	0.0690488
$904$	0	0
$905$	7.09041	0.235693
$906$	0	0
$907$	26.0776	0.865894	0.432947	−	0.901419i	$-0.357474\pi$
$907$	26.0776	0.865894	0.432947	+	0.901419i	$0.357474\pi$
$908$	0	0
$909$	−5.69501	−0.188892
$910$	0	0
$911$	34.1407	1.13113	0.565566	−	0.824703i	$-0.308658\pi$
$911$	34.1407	1.13113	0.565566	+	0.824703i	$0.308658\pi$
$912$	0	0
$913$	22.4557	0.743175
$914$	0	0
$915$	1.60973	0.0532160
$916$	0	0
$917$	−11.2796	−0.372486
$918$	0	0
$919$	−11.7633	−0.388034	−0.194017	−	0.980998i	$-0.562152\pi$
$919$	−11.7633	−0.388034	−0.194017	+	0.980998i	$0.562152\pi$
$920$	0	0
$921$	2.34314	0.0772091
$922$	0	0
$923$	−3.59648	−0.118380
$924$	0	0
$925$	27.7778	0.913328
$926$	0	0
$927$	−15.8632	−0.521016
$928$	0	0
$929$	−21.9176	−0.719093	−0.359547	−	0.933127i	$-0.617069\pi$
$929$	−21.9176	−0.719093	−0.359547	+	0.933127i	$0.617069\pi$
$930$	0	0
$931$	−8.04736	−0.263742
$932$	0	0
$933$	3.86898	0.126665
$934$	0	0
$935$	62.6276	2.04814
$936$	0	0
$937$	−17.5365	−0.572892	−0.286446	−	0.958096i	$-0.592474\pi$
$937$	−17.5365	−0.572892	−0.286446	+	0.958096i	$0.592474\pi$
$938$	0	0
$939$	−0.749671	−0.0244646
$940$	0	0
$941$	−47.1279	−1.53633	−0.768163	−	0.640255i	$-0.778829\pi$
$941$	−47.1279	−1.53633	−0.768163	+	0.640255i	$0.778829\pi$
$942$	0	0
$943$	−5.34842	−0.174168
$944$	0	0
$945$	−3.85321	−0.125345
$946$	0	0
$947$	−15.4167	−0.500976	−0.250488	−	0.968120i	$-0.580591\pi$
$947$	−15.4167	−0.500976	−0.250488	+	0.968120i	$0.580591\pi$
$948$	0	0
$949$	2.90794	0.0943959
$950$	0	0
$951$	−3.93971	−0.127754
$952$	0	0
$953$	24.6552	0.798660	0.399330	−	0.916807i	$-0.369243\pi$
$953$	24.6552	0.798660	0.399330	+	0.916807i	$0.369243\pi$
$954$	0	0
$955$	−5.64917	−0.182803
$956$	0	0
$957$	4.95995	0.160332
$958$	0	0
$959$	−9.73838	−0.314469
$960$	0	0
$961$	27.0492	0.872554
$962$	0	0
$963$	8.37662	0.269933
$964$	0	0
$965$	−6.63920	−0.213723
$966$	0	0
$967$	28.2925	0.909826	0.454913	−	0.890536i	$-0.349670\pi$
$967$	28.2925	0.909826	0.454913	+	0.890536i	$0.349670\pi$
$968$	0	0
$969$	−6.87914	−0.220990
$970$	0	0
$971$	−48.5600	−1.55837	−0.779183	−	0.626797i	$-0.784366\pi$
$971$	−48.5600	−1.55837	−0.779183	+	0.626797i	$0.784366\pi$
$972$	0	0
$973$	−4.85307	−0.155582
$974$	0	0
$975$	0.755813	0.0242054
$976$	0	0
$977$	45.7403	1.46336	0.731682	−	0.681647i	$-0.238736\pi$
$977$	45.7403	1.46336	0.731682	+	0.681647i	$0.238736\pi$
$978$	0	0
$979$	−27.3504	−0.874121
$980$	0	0
$981$	−33.7276	−1.07684
$982$	0	0
$983$	−15.9534	−0.508834	−0.254417	−	0.967095i	$-0.581884\pi$
$983$	−15.9534	−0.508834	−0.254417	+	0.967095i	$0.581884\pi$
$984$	0	0
$985$	68.4183	2.17999
$986$	0	0
$987$	2.43505	0.0775084
$988$	0	0
$989$	−55.2175	−1.75582
$990$	0	0
$991$	40.8830	1.29869	0.649346	−	0.760493i	$-0.275043\pi$
$991$	40.8830	1.29869	0.649346	+	0.760493i	$0.275043\pi$
$992$	0	0
$993$	4.71953	0.149770
$994$	0	0
$995$	−20.9200	−0.663208
$996$	0	0
$997$	4.18063	0.132402	0.0662010	−	0.997806i	$-0.478912\pi$
$997$	4.18063	0.132402	0.0662010	+	0.997806i	$0.478912\pi$
$998$	0	0
$999$	−6.21957	−0.196779

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4592.2.a.bb.1.4		5
4.3	odd	2		287.2.a.e.1.2	✓	5
12.11	even	2		2583.2.a.r.1.4		5
20.19	odd	2		7175.2.a.n.1.4		5
28.27	even	2		2009.2.a.n.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.a.e.1.2	✓	5	4.3	odd	2
2009.2.a.n.1.2		5	28.27	even	2
2583.2.a.r.1.4		5	12.11	even	2
4592.2.a.bb.1.4		5	1.1	even	1	trivial
7175.2.a.n.1.4		5	20.19	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:	\( N \)	\(=\)	\( 4592 = 2^{4} \cdot 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4592.a (trivial)

\(f(q)\)	\(=\)	\(q+0.200978 q^{3} -3.21704 q^{5} -1.00000 q^{7} -2.95961 q^{9} +O(q^{10})\)	\(q+0.200978 q^{3} -3.21704 q^{5} -1.00000 q^{7} -2.95961 q^{9} -4.57695 q^{11} +0.703013 q^{13} -0.646554 q^{15} +4.25337 q^{17} -8.04736 q^{19} -0.200978 q^{21} +5.34842 q^{23} +5.34937 q^{25} -1.19775 q^{27} -5.39204 q^{29} -7.61900 q^{31} -0.919865 q^{33} +3.21704 q^{35} +5.19272 q^{37} +0.141290 q^{39} -1.00000 q^{41} -10.3241 q^{43} +9.52119 q^{45} -12.1160 q^{47} +1.00000 q^{49} +0.854832 q^{51} -12.2837 q^{53} +14.7242 q^{55} -1.61734 q^{57} +7.73023 q^{59} -2.48971 q^{61} +2.95961 q^{63} -2.26162 q^{65} +3.09767 q^{67} +1.07491 q^{69} -5.11581 q^{71} +4.13640 q^{73} +1.07511 q^{75} +4.57695 q^{77} +13.7414 q^{79} +8.63810 q^{81} -4.90626 q^{83} -13.6833 q^{85} -1.08368 q^{87} +5.97567 q^{89} -0.703013 q^{91} -1.53125 q^{93} +25.8887 q^{95} +1.45550 q^{97} +13.5460 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 4 q^{3} - 5 q^{5} - 5 q^{7} + q^{9}+O(q^{10}) \) 5 * q - 4 * q^3 - 5 * q^5 - 5 * q^7 + q^9	\( 5 q - 4 q^{3} - 5 q^{5} - 5 q^{7} + q^{9} - 2 q^{11} + 5 q^{13} + 5 q^{15} + 13 q^{17} + 4 q^{21} - 2 q^{23} + 22 q^{25} - 10 q^{27} - 5 q^{29} - 17 q^{31} + 3 q^{33} + 5 q^{35} - 7 q^{37} - 5 q^{39} - 5 q^{41} - q^{43} - 23 q^{45} - 9 q^{47} + 5 q^{49} - 5 q^{51} + 5 q^{53} - 33 q^{55} - 3 q^{57} - 7 q^{59} + 22 q^{61} - q^{63} - 31 q^{65} + 3 q^{67} - 22 q^{69} + 24 q^{71} + 40 q^{73} - 24 q^{75} + 2 q^{77} + 42 q^{79} + 9 q^{81} + 12 q^{83} - 23 q^{85} + 32 q^{87} + 8 q^{89} - 5 q^{91} - 11 q^{93} + 17 q^{95} + 16 q^{97} + 45 q^{99}+O(q^{100}) \) 5 * q - 4 * q^3 - 5 * q^5 - 5 * q^7 + q^9 - 2 * q^11 + 5 * q^13 + 5 * q^15 + 13 * q^17 + 4 * q^21 - 2 * q^23 + 22 * q^25 - 10 * q^27 - 5 * q^29 - 17 * q^31 + 3 * q^33 + 5 * q^35 - 7 * q^37 - 5 * q^39 - 5 * q^41 - q^43 - 23 * q^45 - 9 * q^47 + 5 * q^49 - 5 * q^51 + 5 * q^53 - 33 * q^55 - 3 * q^57 - 7 * q^59 + 22 * q^61 - q^63 - 31 * q^65 + 3 * q^67 - 22 * q^69 + 24 * q^71 + 40 * q^73 - 24 * q^75 + 2 * q^77 + 42 * q^79 + 9 * q^81 + 12 * q^83 - 23 * q^85 + 32 * q^87 + 8 * q^89 - 5 * q^91 - 11 * q^93 + 17 * q^95 + 16 * q^97 + 45 * q^99

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