LMFDB - Embedded newform 2583.2.a.r.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2583,2,Mod(1,2583)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2583, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2583.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2583 = 3^{2} \cdot 7 \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2583.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:	$20.6253588421$
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:	$\Z[a_1, a_2]$
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	$\chi$	$=$	2583.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.20098 q^{2} -0.557652 q^{4} +3.21704 q^{5} +1.00000 q^{7} -3.07168 q^{8} +O(q^{10})$	\(q+1.20098 q^{2} -0.557652 q^{4} +3.21704 q^{5} +1.00000 q^{7} -3.07168 q^{8} +3.86360 q^{10} -4.57695 q^{11} +0.703013 q^{13} +1.20098 q^{14} -2.57372 q^{16} -4.25337 q^{17} +8.04736 q^{19} -1.79399 q^{20} -5.49681 q^{22} +5.34842 q^{23} +5.34937 q^{25} +0.844303 q^{26} -0.557652 q^{28} +5.39204 q^{29} +7.61900 q^{31} +3.05239 q^{32} -5.10820 q^{34} +3.21704 q^{35} +5.19272 q^{37} +9.66470 q^{38} -9.88174 q^{40} +1.00000 q^{41} +10.3241 q^{43} +2.55235 q^{44} +6.42333 q^{46} -12.1160 q^{47} +1.00000 q^{49} +6.42448 q^{50} -0.392037 q^{52} +12.2837 q^{53} -14.7242 q^{55} -3.07168 q^{56} +6.47572 q^{58} +7.73023 q^{59} -2.48971 q^{61} +9.15025 q^{62} +8.81329 q^{64} +2.26162 q^{65} -3.09767 q^{67} +2.37190 q^{68} +3.86360 q^{70} -5.11581 q^{71} +4.13640 q^{73} +6.23634 q^{74} -4.48763 q^{76} -4.57695 q^{77} -13.7414 q^{79} -8.27977 q^{80} +1.20098 q^{82} -4.90626 q^{83} -13.6833 q^{85} +12.3990 q^{86} +14.0589 q^{88} -5.97567 q^{89} +0.703013 q^{91} -2.98256 q^{92} -14.5511 q^{94} +25.8887 q^{95} +1.45550 q^{97} +1.20098 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + q^{2} + 3 q^{4} + 5 q^{5} + 5 q^{7} + 3 q^{8}+O(q^{10}) $ 5 * q + q^2 + 3 * q^4 + 5 * q^5 + 5 * q^7 + 3 * q^8	$ 5 q + q^{2} + 3 q^{4} + 5 q^{5} + 5 q^{7} + 3 q^{8} - 2 q^{11} + 5 q^{13} + q^{14} - q^{16} - 13 q^{17} + 23 q^{20} + q^{22} - 2 q^{23} + 22 q^{25} + 3 q^{28} + 5 q^{29} + 17 q^{31} + 12 q^{32} - 8 q^{34} + 5 q^{35} - 7 q^{37} + 3 q^{38} + 7 q^{40} + 5 q^{41} + q^{43} + 47 q^{44} - 24 q^{46} - 9 q^{47} + 5 q^{49} - 2 q^{50} + 20 q^{52} - 5 q^{53} + 33 q^{55} + 3 q^{56} - 27 q^{58} - 7 q^{59} + 22 q^{61} + 28 q^{62} - 3 q^{64} + 31 q^{65} - 3 q^{67} - 17 q^{68} + 24 q^{71} + 40 q^{73} + 5 q^{74} - 19 q^{76} - 2 q^{77} - 42 q^{79} - 24 q^{80} + q^{82} + 12 q^{83} - 23 q^{85} - 16 q^{86} + 26 q^{88} - 8 q^{89} + 5 q^{91} - 12 q^{92} - 23 q^{94} + 17 q^{95} + 16 q^{97} + q^{98}+O(q^{100}) $ 5 * q + q^2 + 3 * q^4 + 5 * q^5 + 5 * q^7 + 3 * q^8 - 2 * q^11 + 5 * q^13 + q^14 - q^16 - 13 * q^17 + 23 * q^20 + q^22 - 2 * q^23 + 22 * q^25 + 3 * q^28 + 5 * q^29 + 17 * q^31 + 12 * q^32 - 8 * q^34 + 5 * q^35 - 7 * q^37 + 3 * q^38 + 7 * q^40 + 5 * q^41 + q^43 + 47 * q^44 - 24 * q^46 - 9 * q^47 + 5 * q^49 - 2 * q^50 + 20 * q^52 - 5 * q^53 + 33 * q^55 + 3 * q^56 - 27 * q^58 - 7 * q^59 + 22 * q^61 + 28 * q^62 - 3 * q^64 + 31 * q^65 - 3 * q^67 - 17 * q^68 + 24 * q^71 + 40 * q^73 + 5 * q^74 - 19 * q^76 - 2 * q^77 - 42 * q^79 - 24 * q^80 + q^82 + 12 * q^83 - 23 * q^85 - 16 * q^86 + 26 * q^88 - 8 * q^89 + 5 * q^91 - 12 * q^92 - 23 * q^94 + 17 * q^95 + 16 * q^97 + 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 $

$2$	1.20098	0.849220	0.424610	−	0.905376i	$-0.360411\pi$
$2$	1.20098	0.849220	0.424610	+	0.905376i	$0.360411\pi$
$3$	0	0
$3$	0	0
$4$	−0.557652	−0.278826
$5$	3.21704	1.43871	0.719353	−	0.694645i	$-0.244438\pi$
$5$	3.21704	1.43871	0.719353	+	0.694645i	$0.244438\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−3.07168	−1.08600
$9$	0	0
$10$	3.86360	1.22178
$11$	−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	$0.468918\pi$
$14$	1.20098	0.320975
$15$	0	0
$16$	−2.57372	−0.643430
$17$	−4.25337	−1.03159	−0.515796	−	0.856711i	$-0.672504\pi$
$17$	−4.25337	−1.03159	−0.515796	+	0.856711i	$0.672504\pi$
$18$	0	0
$19$	8.04736	1.84619	0.923095	−	0.384571i	$-0.125651\pi$
$19$	8.04736	1.84619	0.923095	+	0.384571i	$0.125651\pi$
$20$	−1.79399	−0.401149
$21$	0	0
$22$	−5.49681	−1.17192
$23$	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
$26$	0.844303	0.165581
$27$	0	0
$28$	−0.557652	−0.105386
$29$	5.39204	1.00128	0.500638	−	0.865657i	$-0.333099\pi$
$29$	5.39204	1.00128	0.500638	+	0.865657i	$0.333099\pi$
$30$	0	0
$31$	7.61900	1.36841	0.684206	−	0.729288i	$-0.260149\pi$
$31$	7.61900	1.36841	0.684206	+	0.729288i	$0.260149\pi$
$32$	3.05239	0.539591
$33$	0	0
$34$	−5.10820	−0.876049
$35$	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	$0.359627\pi$
$38$	9.66470	1.56782
$39$	0	0
$40$	−9.88174	−1.56244
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	10.3241	1.57441	0.787205	−	0.616692i	$-0.211528\pi$
$43$	10.3241	1.57441	0.787205	+	0.616692i	$0.211528\pi$
$44$	2.55235	0.384781
$45$	0	0
$46$	6.42333	0.947068
$47$	−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
$50$	6.42448	0.908559
$51$	0	0
$52$	−0.392037	−0.0543657
$53$	12.2837	1.68730	0.843648	−	0.536897i	$-0.180404\pi$
$53$	12.2837	1.68730	0.843648	+	0.536897i	$0.180404\pi$
$54$	0	0
$55$	−14.7242	−1.98542
$56$	−3.07168	−0.410471
$57$	0	0
$58$	6.47572	0.850303
$59$	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	$0.550952\pi$
$62$	9.15025	1.16208
$63$	0	0
$64$	8.81329	1.10166
$65$	2.26162	0.280520
$66$	0	0
$67$	−3.09767	−0.378440	−0.189220	−	0.981935i	$-0.560596\pi$
$67$	−3.09767	−0.378440	−0.189220	+	0.981935i	$0.560596\pi$
$68$	2.37190	0.287635
$69$	0	0
$70$	3.86360	0.461788
$71$	−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	$0.422175\pi$
$74$	6.23634	0.724960
$75$	0	0
$76$	−4.48763	−0.514766
$77$	−4.57695	−0.521592
$78$	0	0
$79$	−13.7414	−1.54603	−0.773015	−	0.634388i	$-0.781252\pi$
$79$	−13.7414	−1.54603	−0.773015	+	0.634388i	$0.781252\pi$
$80$	−8.27977	−0.925706
$81$	0	0
$82$	1.20098	0.132626
$83$	−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
$86$	12.3990	1.33702
$87$	0	0
$88$	14.0589	1.49869
$89$	−5.97567	−0.633420	−0.316710	−	0.948522i	$-0.602578\pi$
$89$	−5.97567	−0.633420	−0.316710	+	0.948522i	$0.602578\pi$
$90$	0	0
$91$	0.703013	0.0736958
$92$	−2.98256	−0.310953
$93$	0	0
$94$	−14.5511	−1.50083
$95$	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	$0.476458\pi$
$98$	1.20098	0.121317
$99$	0	0
$100$	−2.98309	−0.298309
$101$	−1.92425	−0.191470	−0.0957348	−	0.995407i	$-0.530520\pi$
$101$	−1.92425	−0.191470	−0.0957348	+	0.995407i	$0.530520\pi$
$102$	0	0
$103$	−5.35990	−0.528127	−0.264063	−	0.964505i	$-0.585063\pi$
$103$	−5.35990	−0.528127	−0.264063	+	0.964505i	$0.585063\pi$
$104$	−2.15943	−0.211750
$105$	0	0
$106$	14.7524	1.43288
$107$	−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	$0.316238\pi$
$110$	−17.6835	−1.68605
$111$	0	0
$112$	−2.57372	−0.243194
$113$	−18.4852	−1.73894	−0.869470	−	0.493986i	$-0.835539\pi$
$113$	−18.4852	−1.73894	−0.869470	+	0.493986i	$0.835539\pi$
$114$	0	0
$115$	17.2061	1.60448
$116$	−3.00688	−0.279182
$117$	0	0
$118$	9.28384	0.854647
$119$	−4.25337	−0.389905
$120$	0	0
$121$	9.94845	0.904405
$122$	−2.99008	−0.270709
$123$	0	0
$124$	−4.24875	−0.381549
$125$	1.12395	0.100530
$126$	0	0
$127$	9.98152	0.885717	0.442858	−	0.896592i	$-0.353964\pi$
$127$	9.98152	0.885717	0.442858	+	0.896592i	$0.353964\pi$
$128$	4.47979	0.395961
$129$	0	0
$130$	2.71616	0.238223
$131$	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
$134$	−3.72023	−0.321379
$135$	0	0
$136$	13.0650	1.12031
$137$	−9.73838	−0.832006	−0.416003	−	0.909363i	$-0.636569\pi$
$137$	−9.73838	−0.832006	−0.416003	+	0.909363i	$0.636569\pi$
$138$	0	0
$139$	−4.85307	−0.411632	−0.205816	−	0.978591i	$-0.565985\pi$
$139$	−4.85307	−0.411632	−0.205816	+	0.978591i	$0.565985\pi$
$140$	−1.79399	−0.151620
$141$	0	0
$142$	−6.14398	−0.515591
$143$	−3.21765	−0.269074
$144$	0	0
$145$	17.3464	1.44054
$146$	4.96773	0.411132
$147$	0	0
$148$	−2.89573	−0.238028
$149$	6.47268	0.530263	0.265131	−	0.964212i	$-0.414585\pi$
$149$	6.47268	0.530263	0.265131	+	0.964212i	$0.414585\pi$
$150$	0	0
$151$	−16.6124	−1.35190	−0.675949	−	0.736948i	$-0.736266\pi$
$151$	−16.6124	−1.35190	−0.675949	+	0.736948i	$0.736266\pi$
$152$	−24.7189	−2.00497
$153$	0	0
$154$	−5.49681	−0.442946
$155$	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	$0.171012\pi$
$158$	−16.5031	−1.31292
$159$	0	0
$160$	9.81967	0.776313
$161$	5.34842	0.421514
$162$	0	0
$163$	−0.00553783	−0.000433757 0	−0.000216878	−	1.00000i	$-0.500069\pi$
$163$	−0.00553783	−0.000433757 0	−0.000216878		1.00000i	$0.500069\pi$
$164$	−0.557652	−0.0435453
$165$	0	0
$166$	−5.89231	−0.457332
$167$	−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
$170$	−16.4333	−1.26038
$171$	0	0
$172$	−5.75725	−0.438986
$173$	−6.07961	−0.462224	−0.231112	−	0.972927i	$-0.574236\pi$
$173$	−6.07961	−0.462224	−0.231112	+	0.972927i	$0.574236\pi$
$174$	0	0
$175$	5.34937	0.404375
$176$	11.7798	0.887934
$177$	0	0
$178$	−7.17665	−0.537913
$179$	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	$0.526102\pi$
$182$	0.844303	0.0625839
$183$	0	0
$184$	−16.4286	−1.21114
$185$	16.7052	1.22819
$186$	0	0
$187$	19.4674	1.42360
$188$	6.75652	0.492770
$189$	0	0
$190$	31.0918	2.25563
$191$	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	$0.476335\pi$
$194$	1.74802	0.125500
$195$	0	0
$196$	−0.557652	−0.0398323
$197$	21.2674	1.51524	0.757621	−	0.652695i	$-0.226362\pi$
$197$	21.2674	1.51524	0.757621	+	0.652695i	$0.226362\pi$
$198$	0	0
$199$	−6.50285	−0.460975	−0.230488	−	0.973075i	$-0.574032\pi$
$199$	−6.50285	−0.460975	−0.230488	+	0.973075i	$0.574032\pi$
$200$	−16.4316	−1.16189
$201$	0	0
$202$	−2.31098	−0.162600
$203$	5.39204	0.378447
$204$	0	0
$205$	3.21704	0.224688
$206$	−6.43713	−0.448496
$207$	0	0
$208$	−1.80936	−0.125456
$209$	−36.8323	−2.54775
$210$	0	0
$211$	−10.4931	−0.722377	−0.361188	−	0.932493i	$-0.617629\pi$
$211$	−10.4931	−0.722377	−0.361188	+	0.932493i	$0.617629\pi$
$212$	−6.85003	−0.470462
$213$	0	0
$214$	−3.39914	−0.232361
$215$	33.2131	2.26511
$216$	0	0
$217$	7.61900	0.517211
$218$	13.6863	0.926953
$219$	0	0
$220$	8.21101	0.553586
$221$	−2.99017	−0.201141
$222$	0	0
$223$	14.4757	0.969366	0.484683	−	0.874690i	$-0.338935\pi$
$223$	14.4757	0.969366	0.484683	+	0.874690i	$0.338935\pi$
$224$	3.05239	0.203946
$225$	0	0
$226$	−22.2003	−1.47674
$227$	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	$0.501996\pi$
$230$	20.6641	1.36255
$231$	0	0
$232$	−16.5626	−1.08739
$233$	−2.03466	−0.133295	−0.0666476	−	0.997777i	$-0.521230\pi$
$233$	−2.03466	−0.133295	−0.0666476	+	0.997777i	$0.521230\pi$
$234$	0	0
$235$	−38.9777	−2.54263
$236$	−4.31078	−0.280608
$237$	0	0
$238$	−5.10820	−0.331115
$239$	−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	$0.544592\pi$
$242$	11.9479	0.768038
$243$	0	0
$244$	1.38839	0.0888826
$245$	3.21704	0.205529
$246$	0	0
$247$	5.65740	0.359972
$248$	−23.4032	−1.48610
$249$	0	0
$250$	1.34984	0.0853716
$251$	−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
$254$	11.9876	0.752168
$255$	0	0
$256$	−12.2465	−0.765403
$257$	19.7199	1.23009	0.615046	−	0.788491i	$-0.289138\pi$
$257$	19.7199	1.23009	0.615046	+	0.788491i	$0.289138\pi$
$258$	0	0
$259$	5.19272	0.322660
$260$	−1.26120	−0.0782163
$261$	0	0
$262$	13.5466	0.836911
$263$	−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
$266$	9.66470	0.592581
$267$	0	0
$268$	1.72742	0.105519
$269$	24.4059	1.48805	0.744027	−	0.668150i	$-0.232914\pi$
$269$	24.4059	1.48805	0.744027	+	0.668150i	$0.232914\pi$
$270$	0	0
$271$	7.83315	0.475830	0.237915	−	0.971286i	$-0.423536\pi$
$271$	7.83315	0.475830	0.237915	+	0.971286i	$0.423536\pi$
$272$	10.9470	0.663757
$273$	0	0
$274$	−11.6956	−0.706555
$275$	−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	$0.562857\pi$
$278$	−5.82843	−0.349566
$279$	0	0
$280$	−9.88174	−0.590547
$281$	2.23112	0.133097	0.0665487	−	0.997783i	$-0.478801\pi$
$281$	2.23112	0.133097	0.0665487	+	0.997783i	$0.478801\pi$
$282$	0	0
$283$	−18.2949	−1.08752	−0.543759	−	0.839242i	$-0.682999\pi$
$283$	−18.2949	−1.08752	−0.543759	+	0.839242i	$0.682999\pi$
$284$	2.85285	0.169285
$285$	0	0
$286$	−3.86433	−0.228503
$287$	1.00000	0.0590281
$288$	0	0
$289$	1.09112	0.0641835
$290$	20.8327	1.22334
$291$	0	0
$292$	−2.30667	−0.134988
$293$	7.37440	0.430817	0.215409	−	0.976524i	$-0.430892\pi$
$293$	7.37440	0.430817	0.215409	+	0.976524i	$0.430892\pi$
$294$	0	0
$295$	24.8685	1.44790
$296$	−15.9504	−0.927098
$297$	0	0
$298$	7.77355	0.450309
$299$	3.76001	0.217447
$300$	0	0
$301$	10.3241	0.595071
$302$	−19.9511	−1.14806
$303$	0	0
$304$	−20.7116	−1.18789
$305$	−8.00949	−0.458622
$306$	0	0
$307$	−11.6587	−0.665398	−0.332699	−	0.943033i	$-0.607959\pi$
$307$	−11.6587	−0.665398	−0.332699	+	0.943033i	$0.607959\pi$
$308$	2.55235	0.145433
$309$	0	0
$310$	29.4368	1.67190
$311$	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	$0.533618\pi$
$314$	25.8564	1.45916
$315$	0	0
$316$	7.66293	0.431074
$317$	19.6027	1.10100	0.550499	−	0.834836i	$-0.314438\pi$
$317$	19.6027	1.10100	0.550499	+	0.834836i	$0.314438\pi$
$318$	0	0
$319$	−24.6791	−1.38176
$320$	28.3527	1.58497
$321$	0	0
$322$	6.42333	0.357958
$323$	−34.2284	−1.90452
$324$	0	0
$325$	3.76068	0.208605
$326$	−0.00665081	−0.000368355 0
$327$	0	0
$328$	−3.07168	−0.169605
$329$	−12.1160	−0.667977
$330$	0	0
$331$	−23.4828	−1.29073	−0.645366	−	0.763873i	$-0.723295\pi$
$331$	−23.4828	−1.29073	−0.645366	+	0.763873i	$0.723295\pi$
$332$	2.73599	0.150157
$333$	0	0
$334$	−27.0117	−1.47802
$335$	−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	$0.489577\pi$
$338$	−15.0192	−0.816934
$339$	0	0
$340$	7.63051	0.413822
$341$	−34.8718	−1.88841
$342$	0	0
$343$	1.00000	0.0539949
$344$	−31.7123	−1.70981
$345$	0	0
$346$	−7.30148	−0.392530
$347$	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	$0.800757\pi$
$350$	6.42448	0.343403
$351$	0	0
$352$	−13.9706	−0.744637
$353$	−4.59065	−0.244336	−0.122168	−	0.992509i	$-0.538985\pi$
$353$	−4.59065	−0.244336	−0.122168	+	0.992509i	$0.538985\pi$
$354$	0	0
$355$	−16.4578	−0.873489
$356$	3.33235	0.176614
$357$	0	0
$358$	9.50716	0.502469
$359$	−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
$362$	−2.64697	−0.139122
$363$	0	0
$364$	−0.392037	−0.0205483
$365$	13.3070	0.696519
$366$	0	0
$367$	17.0270	0.888802	0.444401	−	0.895828i	$-0.353417\pi$
$367$	17.0270	0.888802	0.444401	+	0.895828i	$0.353417\pi$
$368$	−13.7653	−0.717567
$369$	0	0
$370$	20.0626	1.04300
$371$	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	$0.261168\pi$
$374$	23.3800	1.20895
$375$	0	0
$376$	37.2165	1.91930
$377$	3.79067	0.195230
$378$	0	0
$379$	−25.4631	−1.30795	−0.653975	−	0.756516i	$-0.726900\pi$
$379$	−25.4631	−1.30795	−0.653975	+	0.756516i	$0.726900\pi$
$380$	−14.4369	−0.740597
$381$	0	0
$382$	2.10893	0.107902
$383$	−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
$386$	2.47853	0.126154
$387$	0	0
$388$	−0.811660	−0.0412058
$389$	33.7017	1.70874	0.854372	−	0.519662i	$-0.173942\pi$
$389$	33.7017	1.70874	0.854372	+	0.519662i	$0.173942\pi$
$390$	0	0
$391$	−22.7488	−1.15045
$392$	−3.07168	−0.155143
$393$	0	0
$394$	25.5417	1.28677
$395$	−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	$0.713045\pi$
$398$	−7.80978	−0.391469
$399$	0	0
$400$	−13.7678	−0.688389
$401$	−14.7100	−0.734581	−0.367291	−	0.930106i	$-0.619715\pi$
$401$	−14.7100	−0.734581	−0.367291	+	0.930106i	$0.619715\pi$
$402$	0	0
$403$	5.35626	0.266814
$404$	1.07306	0.0533867
$405$	0	0
$406$	6.47572	0.321384
$407$	−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	$0.345473\pi$
$410$	3.86360	0.190810
$411$	0	0
$412$	2.98896	0.147256
$413$	7.73023	0.380380
$414$	0	0
$415$	−15.7837	−0.774789
$416$	2.14587	0.105210
$417$	0	0
$418$	−44.2348	−2.16360
$419$	−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	$0.159171\pi$
$422$	−12.6020	−0.613456
$423$	0	0
$424$	−37.7316	−1.83241
$425$	−22.7528	−1.10368
$426$	0	0
$427$	−2.48971	−0.120485
$428$	1.57833	0.0762915
$429$	0	0
$430$	39.8881	1.92358
$431$	−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	$0.643523\pi$
$434$	9.15025	0.439226
$435$	0	0
$436$	−6.35499	−0.304349
$437$	43.0406	2.05891
$438$	0	0
$439$	−6.79012	−0.324075	−0.162037	−	0.986785i	$-0.551807\pi$
$439$	−6.79012	−0.324075	−0.162037	+	0.986785i	$0.551807\pi$
$440$	45.2282	2.15617
$441$	0	0
$442$	−3.59113	−0.170813
$443$	−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
$446$	17.3850	0.823204
$447$	0	0
$448$	8.81329	0.416389
$449$	2.19148	0.103422	0.0517112	−	0.998662i	$-0.483532\pi$
$449$	2.19148	0.103422	0.0517112	+	0.998662i	$0.483532\pi$
$450$	0	0
$451$	−4.57695	−0.215520
$452$	10.3083	0.484862
$453$	0	0
$454$	21.0770	0.989195
$455$	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	$0.383563\pi$
$458$	−0.227946	−0.0106512
$459$	0	0
$460$	−9.59502	−0.447370
$461$	7.37440	0.343460	0.171730	−	0.985144i	$-0.445064\pi$
$461$	7.37440	0.343460	0.171730	+	0.985144i	$0.445064\pi$
$462$	0	0
$463$	−25.3329	−1.17732	−0.588661	−	0.808380i	$-0.700345\pi$
$463$	−25.3329	−1.17732	−0.588661	+	0.808380i	$0.700345\pi$
$464$	−13.8776	−0.644251
$465$	0	0
$466$	−2.44358	−0.113197
$467$	−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
$470$	−46.8114	−2.15925
$471$	0	0
$472$	−23.7448	−1.09294
$473$	−47.2528	−2.17269
$474$	0	0
$475$	43.0483	1.97519
$476$	2.37190	0.108716
$477$	0	0
$478$	−13.9377	−0.637496
$479$	−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
$482$	−5.20670	−0.237159
$483$	0	0
$484$	−5.54778	−0.252172
$485$	4.68239	0.212617
$486$	0	0
$487$	−11.5758	−0.524549	−0.262275	−	0.964993i	$-0.584473\pi$
$487$	−11.5758	−0.524549	−0.262275	+	0.964993i	$0.584473\pi$
$488$	7.64759	0.346190
$489$	0	0
$490$	3.86360	0.174540
$491$	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
$494$	6.79441	0.305695
$495$	0	0
$496$	−19.6092	−0.880478
$497$	−5.11581	−0.229476
$498$	0	0
$499$	8.35644	0.374086	0.187043	−	0.982352i	$-0.440110\pi$
$499$	8.35644	0.374086	0.187043	+	0.982352i	$0.440110\pi$
$500$	−0.626776	−0.0280303
$501$	0	0
$502$	−28.0430	−1.25162
$503$	−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
$506$	−29.3992	−1.30696
$507$	0	0
$508$	−5.56622	−0.246961
$509$	6.48190	0.287305	0.143653	−	0.989628i	$-0.454115\pi$
$509$	6.48190	0.287305	0.143653	+	0.989628i	$0.454115\pi$
$510$	0	0
$511$	4.13640	0.182984
$512$	−23.6673	−1.04596
$513$	0	0
$514$	23.6831	1.04462
$515$	−17.2430	−0.759819
$516$	0	0
$517$	55.4543	2.43888
$518$	6.23634	0.274009
$519$	0	0
$520$	−6.94700	−0.304646
$521$	1.36749	0.0599107	0.0299553	−	0.999551i	$-0.490463\pi$
$521$	1.36749	0.0599107	0.0299553	+	0.999551i	$0.490463\pi$
$522$	0	0
$523$	12.9198	0.564944	0.282472	−	0.959276i	$-0.408846\pi$
$523$	12.9198	0.564944	0.282472	+	0.959276i	$0.408846\pi$
$524$	−6.29011	−0.274785
$525$	0	0
$526$	−2.00707	−0.0875123
$527$	−32.4064	−1.41164
$528$	0	0
$529$	5.60555	0.243720
$530$	47.4593	2.06150
$531$	0	0
$532$	−4.48763	−0.194563
$533$	0.703013	0.0304509
$534$	0	0
$535$	−9.10525	−0.393654
$536$	9.51506	0.410988
$537$	0	0
$538$	29.3109	1.26368
$539$	−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	$0.530661\pi$
$542$	9.40744	0.404084
$543$	0	0
$544$	−12.9829	−0.556638
$545$	36.6613	1.57040
$546$	0	0
$547$	−22.3133	−0.954048	−0.477024	−	0.878890i	$-0.658285\pi$
$547$	−22.3133	−0.954048	−0.477024	+	0.878890i	$0.658285\pi$
$548$	5.43063	0.231985
$549$	0	0
$550$	−29.4045	−1.25381
$551$	43.3917	1.84855
$552$	0	0
$553$	−13.7414	−0.584344
$554$	−7.84300	−0.333217
$555$	0	0
$556$	2.70633	0.114774
$557$	−15.3755	−0.651482	−0.325741	−	0.945459i	$-0.605614\pi$
$557$	−15.3755	−0.651482	−0.325741	+	0.945459i	$0.605614\pi$
$558$	0	0
$559$	7.25797	0.306979
$560$	−8.27977	−0.349884
$561$	0	0
$562$	2.67952	0.113029
$563$	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
$566$	−21.9717	−0.923541
$567$	0	0
$568$	15.7142	0.659351
$569$	−20.9479	−0.878182	−0.439091	−	0.898443i	$-0.644699\pi$
$569$	−20.9479	−0.878182	−0.439091	+	0.898443i	$0.644699\pi$
$570$	0	0
$571$	−20.5728	−0.860947	−0.430473	−	0.902603i	$-0.641653\pi$
$571$	−20.5728	−0.860947	−0.430473	+	0.902603i	$0.641653\pi$
$572$	1.79433	0.0750248
$573$	0	0
$574$	1.20098	0.0501278
$575$	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	$0.684446\pi$
$578$	1.31041	0.0545059
$579$	0	0
$580$	−9.67327	−0.401661
$581$	−4.90626	−0.203546
$582$	0	0
$583$	−56.2218	−2.32847
$584$	−12.7057	−0.525766
$585$	0	0
$586$	8.85649	0.365858
$587$	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
$590$	29.8665	1.22959
$591$	0	0
$592$	−13.3646	−0.549282
$593$	5.90293	0.242404	0.121202	−	0.992628i	$-0.461325\pi$
$593$	5.90293	0.242404	0.121202	+	0.992628i	$0.461325\pi$
$594$	0	0
$595$	−13.6833	−0.560959
$596$	−3.60951	−0.147851
$597$	0	0
$598$	4.51568	0.184660
$599$	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	$0.469490\pi$
$602$	12.3990	0.505346
$603$	0	0
$604$	9.26395	0.376945
$605$	32.0046	1.30117
$606$	0	0
$607$	27.8857	1.13184	0.565922	−	0.824459i	$-0.308520\pi$
$607$	27.8857	1.13184	0.565922	+	0.824459i	$0.308520\pi$
$608$	24.5637	0.996188
$609$	0	0
$610$	−9.61922	−0.389471
$611$	−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	$0.602069\pi$
$614$	−14.0019	−0.565069
$615$	0	0
$616$	14.0589	0.566451
$617$	−12.2414	−0.492820	−0.246410	−	0.969166i	$-0.579251\pi$
$617$	−12.2414	−0.492820	−0.246410	+	0.969166i	$0.579251\pi$
$618$	0	0
$619$	23.4809	0.943778	0.471889	−	0.881658i	$-0.343572\pi$
$619$	23.4809	0.943778	0.471889	+	0.881658i	$0.343572\pi$
$620$	−13.6684	−0.548937
$621$	0	0
$622$	23.1198	0.927019
$623$	−5.97567	−0.239410
$624$	0	0
$625$	−23.1311	−0.925243
$626$	−4.47979	−0.179048
$627$	0	0
$628$	−12.0059	−0.479090
$629$	−22.0865	−0.880648
$630$	0	0
$631$	−0.0226990	−0.000903631 0	−0.000451816	−	1.00000i	$-0.500144\pi$
$631$	−0.0226990	−0.000903631 0	−0.000451816		1.00000i	$0.500144\pi$
$632$	42.2093	1.67899
$633$	0	0
$634$	23.5424	0.934989
$635$	32.1110	1.27429
$636$	0	0
$637$	0.703013	0.0278544
$638$	−29.6390	−1.17342
$639$	0	0
$640$	14.4117	0.569671
$641$	−38.0777	−1.50398	−0.751989	−	0.659175i	$-0.770905\pi$
$641$	−38.0777	−1.50398	−0.751989	+	0.659175i	$0.770905\pi$
$642$	0	0
$643$	24.2224	0.955239	0.477619	−	0.878567i	$-0.341500\pi$
$643$	24.2224	0.955239	0.477619	+	0.878567i	$0.341500\pi$
$644$	−2.98256	−0.117529
$645$	0	0
$646$	−41.1075	−1.61735
$647$	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
$650$	4.51649	0.177151
$651$	0	0
$652$	0.00308819	0.000120943 0
$653$	3.76579	0.147367	0.0736833	−	0.997282i	$-0.476525\pi$
$653$	3.76579	0.147367	0.0736833	+	0.997282i	$0.476525\pi$
$654$	0	0
$655$	36.2871	1.41785
$656$	−2.57372	−0.100487
$657$	0	0
$658$	−14.5511	−0.567259
$659$	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	$0.570544\pi$
$662$	−28.2023	−1.09612
$663$	0	0
$664$	15.0705	0.584848
$665$	25.8887	1.00392
$666$	0	0
$667$	28.8389	1.11664
$668$	12.5424	0.485280
$669$	0	0
$670$	−11.9682	−0.462370
$671$	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	$0.385171\pi$
$674$	1.44359	0.0556050
$675$	0	0
$676$	6.97387	0.268226
$677$	15.9963	0.614788	0.307394	−	0.951582i	$-0.400543\pi$
$677$	15.9963	0.614788	0.307394	+	0.951582i	$0.400543\pi$
$678$	0	0
$679$	1.45550	0.0558568
$680$	42.0307	1.61180
$681$	0	0
$682$	−41.8802	−1.60368
$683$	−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
$686$	1.20098	0.0458535
$687$	0	0
$688$	−26.5713	−1.01302
$689$	8.63560	0.328990
$690$	0	0
$691$	8.11399	0.308671	0.154335	−	0.988019i	$-0.450676\pi$
$691$	8.11399	0.308671	0.154335	+	0.988019i	$0.450676\pi$
$692$	3.39031	0.128880
$693$	0	0
$694$	3.62455	0.137586
$695$	−15.6125	−0.592217
$696$	0	0
$697$	−4.25337	−0.161108
$698$	−36.3650	−1.37644
$699$	0	0
$700$	−2.98309	−0.112750
$701$	49.0400	1.85221	0.926107	−	0.377260i	$-0.123134\pi$
$701$	49.0400	1.85221	0.926107	+	0.377260i	$0.123134\pi$
$702$	0	0
$703$	41.7877	1.57605
$704$	−40.3380	−1.52029
$705$	0	0
$706$	−5.51327	−0.207495
$707$	−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	$0.434396\pi$
$710$	−19.7654	−0.741784
$711$	0	0
$712$	18.3554	0.687897
$713$	40.7496	1.52608
$714$	0	0
$715$	−10.3513	−0.387118
$716$	−4.41448	−0.164977
$717$	0	0
$718$	−22.1040	−0.824916
$719$	−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
$722$	54.9567	2.04528
$723$	0	0
$724$	1.22907	0.0456782
$725$	28.8440	1.07124
$726$	0	0
$727$	−29.3527	−1.08863	−0.544315	−	0.838881i	$-0.683210\pi$
$727$	−29.3527	−1.08863	−0.544315	+	0.838881i	$0.683210\pi$
$728$	−2.15943	−0.0800339
$729$	0	0
$730$	15.9814	0.591498
$731$	−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	$0.385284\pi$
$734$	20.4490	0.754788
$735$	0	0
$736$	16.3254	0.601764
$737$	14.1779	0.522249
$738$	0	0
$739$	−29.0981	−1.07039	−0.535196	−	0.844728i	$-0.679762\pi$
$739$	−29.0981	−1.07039	−0.535196	+	0.844728i	$0.679762\pi$
$740$	−9.31570	−0.342452
$741$	0	0
$742$	14.7524	0.541579
$743$	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
$746$	31.6314	1.15811
$747$	0	0
$748$	−10.8561	−0.396937
$749$	−2.83031	−0.103417
$750$	0	0
$751$	33.9703	1.23959	0.619797	−	0.784762i	$-0.287215\pi$
$751$	33.9703	1.23959	0.619797	+	0.784762i	$0.287215\pi$
$752$	31.1832	1.13713
$753$	0	0
$754$	4.55251	0.165793
$755$	−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	$0.456344\pi$
$758$	−30.5806	−1.11074
$759$	0	0
$760$	−79.5219	−2.88456
$761$	23.8739	0.865428	0.432714	−	0.901531i	$-0.357556\pi$
$761$	23.8739	0.865428	0.432714	+	0.901531i	$0.357556\pi$
$762$	0	0
$763$	11.3960	0.412562
$764$	−0.979244	−0.0354278
$765$	0	0
$766$	−32.0018	−1.15627
$767$	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	$0.487267\pi$
$770$	−17.6835	−0.637269
$771$	0	0
$772$	−1.15086	−0.0414204
$773$	−18.4207	−0.662547	−0.331274	−	0.943535i	$-0.607478\pi$
$773$	−18.4207	−0.662547	−0.331274	+	0.943535i	$0.607478\pi$
$774$	0	0
$775$	40.7569	1.46403
$776$	−4.47082	−0.160493
$777$	0	0
$778$	40.4750	1.45110
$779$	8.04736	0.288327
$780$	0	0
$781$	23.4148	0.837848
$782$	−27.3208	−0.976989
$783$	0	0
$784$	−2.57372	−0.0919185
$785$	69.2612	2.47204
$786$	0	0
$787$	18.9009	0.673745	0.336873	−	0.941550i	$-0.390631\pi$
$787$	18.9009	0.673745	0.336873	+	0.941550i	$0.390631\pi$
$788$	−11.8598	−0.422489
$789$	0	0
$790$	−53.0913	−1.88890
$791$	−18.4852	−0.657257
$792$	0	0
$793$	−1.75030	−0.0621548
$794$	−29.6933	−1.05377
$795$	0	0
$796$	3.62633	0.128532
$797$	1.21120	0.0429028	0.0214514	−	0.999770i	$-0.493171\pi$
$797$	1.21120	0.0429028	0.0214514	+	0.999770i	$0.493171\pi$
$798$	0	0
$799$	51.5338	1.82313
$800$	16.3284	0.577295
$801$	0	0
$802$	−17.6664	−0.623821
$803$	−18.9321	−0.668099
$804$	0	0
$805$	17.2061	0.606435
$806$	6.43275	0.226584
$807$	0	0
$808$	5.91067	0.207937
$809$	1.52068	0.0534643	0.0267322	−	0.999643i	$-0.491490\pi$
$809$	1.52068	0.0534643	0.0267322	+	0.999643i	$0.491490\pi$
$810$	0	0
$811$	−27.6060	−0.969379	−0.484689	−	0.874686i	$-0.661067\pi$
$811$	−27.6060	−0.969379	−0.484689	+	0.874686i	$0.661067\pi$
$812$	−3.00688	−0.105521
$813$	0	0
$814$	−28.5434	−1.00045
$815$	−0.0178155	−0.000624048 0
$816$	0	0
$817$	83.0817	2.90666
$818$	22.6666	0.792519
$819$	0	0
$820$	−1.79399	−0.0626489
$821$	−50.0892	−1.74813	−0.874063	−	0.485813i	$-0.838524\pi$
$821$	−50.0892	−1.74813	−0.874063	+	0.485813i	$0.838524\pi$
$822$	0	0
$823$	−10.0645	−0.350826	−0.175413	−	0.984495i	$-0.556126\pi$
$823$	−10.0645	−0.350826	−0.175413	+	0.984495i	$0.556126\pi$
$824$	16.4639	0.573548
$825$	0	0
$826$	9.28384	0.323026
$827$	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	$0.153738\pi$
$830$	−18.9558	−0.657966
$831$	0	0
$832$	6.19586	0.214803
$833$	−4.25337	−0.147370
$834$	0	0
$835$	−72.3560	−2.50398
$836$	20.5396	0.710378
$837$	0	0
$838$	−34.5635	−1.19398
$839$	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
$842$	43.2496	1.49048
$843$	0	0
$844$	5.85152	0.201418
$845$	−40.2316	−1.38401
$846$	0	0
$847$	9.94845	0.341833
$848$	−31.6148	−1.08566
$849$	0	0
$850$	−27.3257	−0.937263
$851$	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	$0.573982\pi$
$854$	−2.99008	−0.102318
$855$	0	0
$856$	8.69383	0.297149
$857$	−39.6908	−1.35581	−0.677906	−	0.735149i	$-0.737112\pi$
$857$	−39.6908	−1.35581	−0.677906	+	0.735149i	$0.737112\pi$
$858$	0	0
$859$	23.5262	0.802705	0.401352	−	0.915924i	$-0.368540\pi$
$859$	23.5262	0.802705	0.401352	+	0.915924i	$0.368540\pi$
$860$	−18.5213	−0.631572
$861$	0	0
$862$	−11.0913	−0.377770
$863$	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
$866$	−21.7803	−0.740126
$867$	0	0
$868$	−4.24875	−0.144212
$869$	62.8937	2.13352
$870$	0	0
$871$	−2.17770	−0.0737886
$872$	−35.0048	−1.18541
$873$	0	0
$874$	51.6908	1.74847
$875$	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	$0.362577\pi$
$878$	−8.15478	−0.275211
$879$	0	0
$880$	37.8961	1.27748
$881$	−33.8534	−1.14055	−0.570275	−	0.821454i	$-0.693163\pi$
$881$	−33.8534	−1.14055	−0.570275	+	0.821454i	$0.693163\pi$
$882$	0	0
$883$	−9.96707	−0.335419	−0.167709	−	0.985837i	$-0.553637\pi$
$883$	−9.96707	−0.335419	−0.167709	+	0.985837i	$0.553637\pi$
$884$	1.66748	0.0560833
$885$	0	0
$886$	−41.1075	−1.38103
$887$	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
$890$	−23.0876	−0.773898
$891$	0	0
$892$	−8.07242	−0.270285
$893$	−97.5018	−3.26277
$894$	0	0
$895$	25.4667	0.851258
$896$	4.47979	0.149659
$897$	0	0
$898$	2.63192	0.0878284
$899$	41.0819	1.37016
$900$	0	0
$901$	−52.2471	−1.74060
$902$	−5.49681	−0.183024
$903$	0	0
$904$	56.7806	1.88850
$905$	−7.09041	−0.235693
$906$	0	0
$907$	−26.0776	−0.865894	−0.432947	−	0.901419i	$-0.642526\pi$
$907$	−26.0776	−0.865894	−0.432947	+	0.901419i	$0.642526\pi$
$908$	−9.78674	−0.324785
$909$	0	0
$910$	2.71616	0.0900399
$911$	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
$914$	18.3668	0.607521
$915$	0	0
$916$	0.105842	0.00349713
$917$	11.2796	0.372486
$918$	0	0
$919$	11.7633	0.388034	0.194017	−	0.980998i	$-0.437848\pi$
$919$	11.7633	0.388034	0.194017	+	0.980998i	$0.437848\pi$
$920$	−52.8517	−1.74247
$921$	0	0
$922$	8.85649	0.291673
$923$	−3.59648	−0.118380
$924$	0	0
$925$	27.7778	0.913328
$926$	−30.4243	−0.999805
$927$	0	0
$928$	16.4586	0.540280
$929$	21.9176	0.719093	0.359547	−	0.933127i	$-0.382931\pi$
$929$	21.9176	0.719093	0.359547	+	0.933127i	$0.382931\pi$
$930$	0	0
$931$	8.04736	0.263742
$932$	1.13463	0.0371662
$933$	0	0
$934$	−10.8797	−0.355994
$935$	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	$0.592474\pi$
$938$	−3.72023	−0.121470
$939$	0	0
$940$	21.7360	0.708951
$941$	47.1279	1.53633	0.768163	−	0.640255i	$-0.221171\pi$
$941$	47.1279	1.53633	0.768163	+	0.640255i	$0.221171\pi$
$942$	0	0
$943$	5.34842	0.174168
$944$	−19.8955	−0.647542
$945$	0	0
$946$	−56.7496	−1.84509
$947$	−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
$950$	51.7001	1.67737
$951$	0	0
$952$	13.0650	0.423439
$953$	−24.6552	−0.798660	−0.399330	−	0.916807i	$-0.630757\pi$
$953$	−24.6552	−0.798660	−0.399330	+	0.916807i	$0.630757\pi$
$954$	0	0
$955$	5.64917	0.182803
$956$	6.47172	0.209310
$957$	0	0
$958$	−18.8910	−0.610341
$959$	−9.73838	−0.314469
$960$	0	0
$961$	27.0492	0.872554
$962$	4.38423	0.141353
$963$	0	0
$964$	2.41764	0.0778669
$965$	6.63920	0.213723
$966$	0	0
$967$	−28.2925	−0.909826	−0.454913	−	0.890536i	$-0.650330\pi$
$967$	−28.2925	−0.909826	−0.454913	+	0.890536i	$0.650330\pi$
$968$	−30.5585	−0.982187
$969$	0	0
$970$	5.62345	0.180558
$971$	−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
$974$	−13.9023	−0.445458
$975$	0	0
$976$	6.40780	0.205109
$977$	−45.7403	−1.46336	−0.731682	−	0.681647i	$-0.761264\pi$
$977$	−45.7403	−1.46336	−0.731682	+	0.681647i	$0.761264\pi$
$978$	0	0
$979$	27.3504	0.874121
$980$	−1.79399	−0.0573070
$981$	0	0
$982$	18.3257	0.584797
$983$	−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
$986$	−27.5436	−0.877167
$987$	0	0
$988$	−3.15486	−0.100370
$989$	55.2175	1.75582
$990$	0	0
$991$	−40.8830	−1.29869	−0.649346	−	0.760493i	$-0.724957\pi$
$991$	−40.8830	−1.29869	−0.649346	+	0.760493i	$0.724957\pi$
$992$	23.2561	0.738383
$993$	0	0
$994$	−6.14398	−0.194875
$995$	−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	$0.478912\pi$
$998$	10.0359	0.317681
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2583.2.a.r.1.4		5
3.2	odd	2		287.2.a.e.1.2	✓	5
12.11	even	2		4592.2.a.bb.1.4		5
15.14	odd	2		7175.2.a.n.1.4		5
21.20	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	3.2	odd	2
2009.2.a.n.1.2		5	21.20	even	2
2583.2.a.r.1.4		5	1.1	even	1	trivial
4592.2.a.bb.1.4		5	12.11	even	2
7175.2.a.n.1.4		5	15.14	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:	\( N \)	\(=\)	\( 2583 = 3^{2} \cdot 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2583.a (trivial)

\(f(q)\)	\(=\)	\(q+1.20098 q^{2} -0.557652 q^{4} +3.21704 q^{5} +1.00000 q^{7} -3.07168 q^{8} +O(q^{10})\)	\(q+1.20098 q^{2} -0.557652 q^{4} +3.21704 q^{5} +1.00000 q^{7} -3.07168 q^{8} +3.86360 q^{10} -4.57695 q^{11} +0.703013 q^{13} +1.20098 q^{14} -2.57372 q^{16} -4.25337 q^{17} +8.04736 q^{19} -1.79399 q^{20} -5.49681 q^{22} +5.34842 q^{23} +5.34937 q^{25} +0.844303 q^{26} -0.557652 q^{28} +5.39204 q^{29} +7.61900 q^{31} +3.05239 q^{32} -5.10820 q^{34} +3.21704 q^{35} +5.19272 q^{37} +9.66470 q^{38} -9.88174 q^{40} +1.00000 q^{41} +10.3241 q^{43} +2.55235 q^{44} +6.42333 q^{46} -12.1160 q^{47} +1.00000 q^{49} +6.42448 q^{50} -0.392037 q^{52} +12.2837 q^{53} -14.7242 q^{55} -3.07168 q^{56} +6.47572 q^{58} +7.73023 q^{59} -2.48971 q^{61} +9.15025 q^{62} +8.81329 q^{64} +2.26162 q^{65} -3.09767 q^{67} +2.37190 q^{68} +3.86360 q^{70} -5.11581 q^{71} +4.13640 q^{73} +6.23634 q^{74} -4.48763 q^{76} -4.57695 q^{77} -13.7414 q^{79} -8.27977 q^{80} +1.20098 q^{82} -4.90626 q^{83} -13.6833 q^{85} +12.3990 q^{86} +14.0589 q^{88} -5.97567 q^{89} +0.703013 q^{91} -2.98256 q^{92} -14.5511 q^{94} +25.8887 q^{95} +1.45550 q^{97} +1.20098 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + q^{2} + 3 q^{4} + 5 q^{5} + 5 q^{7} + 3 q^{8}+O(q^{10}) \) 5 * q + q^2 + 3 * q^4 + 5 * q^5 + 5 * q^7 + 3 * q^8	\( 5 q + q^{2} + 3 q^{4} + 5 q^{5} + 5 q^{7} + 3 q^{8} - 2 q^{11} + 5 q^{13} + q^{14} - q^{16} - 13 q^{17} + 23 q^{20} + q^{22} - 2 q^{23} + 22 q^{25} + 3 q^{28} + 5 q^{29} + 17 q^{31} + 12 q^{32} - 8 q^{34} + 5 q^{35} - 7 q^{37} + 3 q^{38} + 7 q^{40} + 5 q^{41} + q^{43} + 47 q^{44} - 24 q^{46} - 9 q^{47} + 5 q^{49} - 2 q^{50} + 20 q^{52} - 5 q^{53} + 33 q^{55} + 3 q^{56} - 27 q^{58} - 7 q^{59} + 22 q^{61} + 28 q^{62} - 3 q^{64} + 31 q^{65} - 3 q^{67} - 17 q^{68} + 24 q^{71} + 40 q^{73} + 5 q^{74} - 19 q^{76} - 2 q^{77} - 42 q^{79} - 24 q^{80} + q^{82} + 12 q^{83} - 23 q^{85} - 16 q^{86} + 26 q^{88} - 8 q^{89} + 5 q^{91} - 12 q^{92} - 23 q^{94} + 17 q^{95} + 16 q^{97} + q^{98}+O(q^{100}) \) 5 * q + q^2 + 3 * q^4 + 5 * q^5 + 5 * q^7 + 3 * q^8 - 2 * q^11 + 5 * q^13 + q^14 - q^16 - 13 * q^17 + 23 * q^20 + q^22 - 2 * q^23 + 22 * q^25 + 3 * q^28 + 5 * q^29 + 17 * q^31 + 12 * q^32 - 8 * q^34 + 5 * q^35 - 7 * q^37 + 3 * q^38 + 7 * q^40 + 5 * q^41 + q^43 + 47 * q^44 - 24 * q^46 - 9 * q^47 + 5 * q^49 - 2 * q^50 + 20 * q^52 - 5 * q^53 + 33 * q^55 + 3 * q^56 - 27 * q^58 - 7 * q^59 + 22 * q^61 + 28 * q^62 - 3 * q^64 + 31 * q^65 - 3 * q^67 - 17 * q^68 + 24 * q^71 + 40 * q^73 + 5 * q^74 - 19 * q^76 - 2 * q^77 - 42 * q^79 - 24 * q^80 + q^82 + 12 * q^83 - 23 * q^85 - 16 * q^86 + 26 * q^88 - 8 * q^89 + 5 * q^91 - 12 * q^92 - 23 * q^94 + 17 * q^95 + 16 * q^97 + q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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