LMFDB - Embedded newform 2583.2.a.r.1.3

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			1.3
Root			$0.460315$ 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)

$f(q)$	$=$	$q+0.460315 q^{2} -1.78811 q^{4} -4.10136 q^{5} +1.00000 q^{7} -1.74372 q^{8} +O(q^{10})$	\(q+0.460315 q^{2} -1.78811 q^{4} -4.10136 q^{5} +1.00000 q^{7} -1.74372 q^{8} -1.88791 q^{10} -2.76768 q^{11} -4.05697 q^{13} +0.460315 q^{14} +2.77356 q^{16} -5.22447 q^{17} -0.109209 q^{19} +7.33368 q^{20} -1.27400 q^{22} -6.08681 q^{23} +11.8211 q^{25} -1.86748 q^{26} -1.78811 q^{28} -2.25431 q^{29} -1.18073 q^{31} +4.76415 q^{32} -2.40490 q^{34} -4.10136 q^{35} -8.95429 q^{37} -0.0502704 q^{38} +7.15163 q^{40} +1.00000 q^{41} -7.93974 q^{43} +4.94891 q^{44} -2.80185 q^{46} +10.5714 q^{47} +1.00000 q^{49} +5.44143 q^{50} +7.25431 q^{52} -6.23100 q^{53} +11.3512 q^{55} -1.74372 q^{56} -1.03769 q^{58} +9.43006 q^{59} +6.89732 q^{61} -0.543506 q^{62} -3.35411 q^{64} +16.6391 q^{65} -1.35699 q^{67} +9.34193 q^{68} -1.88791 q^{70} +7.90672 q^{71} +9.88791 q^{73} -4.12179 q^{74} +0.195277 q^{76} -2.76768 q^{77} -12.5668 q^{79} -11.3754 q^{80} +0.460315 q^{82} +14.9633 q^{83} +21.4274 q^{85} -3.65478 q^{86} +4.82606 q^{88} +0.852931 q^{89} -4.05697 q^{91} +10.8839 q^{92} +4.86616 q^{94} +0.447904 q^{95} +9.92806 q^{97} +0.460315 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.460315	0.325492	0.162746	−	0.986668i	$-0.447965\pi$
$2$	0.460315	0.325492	0.162746	+	0.986668i	$0.447965\pi$
$3$	0	0
$3$	0	0
$4$	−1.78811	−0.894055
$5$	−4.10136	−1.83418	−0.917091	−	0.398678i	$-0.869469\pi$
$5$	−4.10136	−1.83418	−0.917091	+	0.398678i	$0.869469\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.74372	−0.616499
$9$	0	0
$10$	−1.88791	−0.597011
$11$	−2.76768	−0.834486	−0.417243	−	0.908795i	$-0.637004\pi$
$11$	−2.76768	−0.834486	−0.417243	+	0.908795i	$0.637004\pi$
$12$	0	0
$13$	−4.05697	−1.12520	−0.562600	−	0.826729i	$-0.690199\pi$
$13$	−4.05697	−1.12520	−0.562600	+	0.826729i	$0.690199\pi$
$14$	0.460315	0.123024
$15$	0	0
$16$	2.77356	0.693390
$17$	−5.22447	−1.26712	−0.633560	−	0.773694i	$-0.718407\pi$
$17$	−5.22447	−1.26712	−0.633560	+	0.773694i	$0.718407\pi$
$18$	0	0
$19$	−0.109209	−0.0250542	−0.0125271	−	0.999922i	$-0.503988\pi$
$19$	−0.109209	−0.0250542	−0.0125271	+	0.999922i	$0.503988\pi$
$20$	7.33368	1.63986
$21$	0	0
$22$	−1.27400	−0.271618
$23$	−6.08681	−1.26919	−0.634593	−	0.772846i	$-0.718832\pi$
$23$	−6.08681	−1.26919	−0.634593	+	0.772846i	$0.718832\pi$
$24$	0	0
$25$	11.8211	2.36422
$26$	−1.86748	−0.366243
$27$	0	0
$28$	−1.78811	−0.337921
$29$	−2.25431	−0.418614	−0.209307	−	0.977850i	$-0.567121\pi$
$29$	−2.25431	−0.418614	−0.209307	+	0.977850i	$0.567121\pi$
$30$	0	0
$31$	−1.18073	−0.212065	−0.106032	−	0.994363i	$-0.533815\pi$
$31$	−1.18073	−0.212065	−0.106032	+	0.994363i	$0.533815\pi$
$32$	4.76415	0.842192
$33$	0	0
$34$	−2.40490	−0.412437
$35$	−4.10136	−0.693256
$36$	0	0
$37$	−8.95429	−1.47208	−0.736038	−	0.676940i	$-0.763305\pi$
$37$	−8.95429	−1.47208	−0.736038	+	0.676940i	$0.763305\pi$
$38$	−0.0502704	−0.00815494
$39$	0	0
$40$	7.15163	1.13077
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	−7.93974	−1.21080	−0.605399	−	0.795922i	$-0.706987\pi$
$43$	−7.93974	−1.21080	−0.605399	+	0.795922i	$0.706987\pi$
$44$	4.94891	0.746077
$45$	0	0
$46$	−2.80185	−0.413110
$47$	10.5714	1.54199	0.770997	−	0.636839i	$-0.219758\pi$
$47$	10.5714	1.54199	0.770997	+	0.636839i	$0.219758\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	5.44143	0.769535
$51$	0	0
$52$	7.25431	1.00599
$53$	−6.23100	−0.855893	−0.427947	−	0.903804i	$-0.640763\pi$
$53$	−6.23100	−0.855893	−0.427947	+	0.903804i	$0.640763\pi$
$54$	0	0
$55$	11.3512	1.53060
$56$	−1.74372	−0.233015
$57$	0	0
$58$	−1.03769	−0.136255
$59$	9.43006	1.22769	0.613845	−	0.789427i	$-0.289622\pi$
$59$	9.43006	1.22769	0.613845	+	0.789427i	$0.289622\pi$
$60$	0	0
$61$	6.89732	0.883111	0.441556	−	0.897234i	$-0.354427\pi$
$61$	6.89732	0.883111	0.441556	+	0.897234i	$0.354427\pi$
$62$	−0.543506	−0.0690253
$63$	0	0
$64$	−3.35411	−0.419264
$65$	16.6391	2.06382
$66$	0	0
$67$	−1.35699	−0.165782	−0.0828912	−	0.996559i	$-0.526415\pi$
$67$	−1.35699	−0.165782	−0.0828912	+	0.996559i	$0.526415\pi$
$68$	9.34193	1.13288
$69$	0	0
$70$	−1.88791	−0.225649
$71$	7.90672	0.938356	0.469178	−	0.883104i	$-0.344550\pi$
$71$	7.90672	0.938356	0.469178	+	0.883104i	$0.344550\pi$
$72$	0	0
$73$	9.88791	1.15729	0.578646	−	0.815579i	$-0.303581\pi$
$73$	9.88791	1.15729	0.578646	+	0.815579i	$0.303581\pi$
$74$	−4.12179	−0.479148
$75$	0	0
$76$	0.195277	0.0223999
$77$	−2.76768	−0.315406
$78$	0	0
$79$	−12.5668	−1.41388	−0.706939	−	0.707275i	$-0.749924\pi$
$79$	−12.5668	−1.41388	−0.706939	+	0.707275i	$0.749924\pi$
$80$	−11.3754	−1.27180
$81$	0	0
$82$	0.460315	0.0508332
$83$	14.9633	1.64244	0.821220	−	0.570611i	$-0.193294\pi$
$83$	14.9633	1.64244	0.821220	+	0.570611i	$0.193294\pi$
$84$	0	0
$85$	21.4274	2.32413
$86$	−3.65478	−0.394105
$87$	0	0
$88$	4.82606	0.514460
$89$	0.852931	0.0904105	0.0452053	−	0.998978i	$-0.485606\pi$
$89$	0.852931	0.0904105	0.0452053	+	0.998978i	$0.485606\pi$
$90$	0	0
$91$	−4.05697	−0.425286
$92$	10.8839	1.13472
$93$	0	0
$94$	4.86616	0.501906
$95$	0.447904	0.0459540
$96$	0	0
$97$	9.92806	1.00804	0.504021	−	0.863691i	$-0.331853\pi$
$97$	9.92806	1.00804	0.504021	+	0.863691i	$0.331853\pi$
$98$	0.460315	0.0464988
$99$	0	0
$100$	−21.1375	−2.11375
$101$	−11.2908	−1.12348	−0.561740	−	0.827314i	$-0.689868\pi$
$101$	−11.2908	−1.12348	−0.561740	+	0.827314i	$0.689868\pi$
$102$	0	0
$103$	−10.8690	−1.07096	−0.535479	−	0.844549i	$-0.679869\pi$
$103$	−10.8690	−1.07096	−0.535479	+	0.844549i	$0.679869\pi$
$104$	7.07423	0.693685
$105$	0	0
$106$	−2.86822	−0.278586
$107$	−1.99215	−0.192588	−0.0962941	−	0.995353i	$-0.530699\pi$
$107$	−1.99215	−0.192588	−0.0962941	+	0.995353i	$0.530699\pi$
$108$	0	0
$109$	−17.8607	−1.71074	−0.855371	−	0.518016i	$-0.826671\pi$
$109$	−17.8607	−1.71074	−0.855371	+	0.518016i	$0.826671\pi$
$110$	5.22514	0.498197
$111$	0	0
$112$	2.77356	0.262077
$113$	16.2536	1.52901	0.764506	−	0.644616i	$-0.222983\pi$
$113$	16.2536	1.52901	0.764506	+	0.644616i	$0.222983\pi$
$114$	0	0
$115$	24.9642	2.32792
$116$	4.03095	0.374264
$117$	0	0
$118$	4.34079	0.399602
$119$	−5.22447	−0.478926
$120$	0	0
$121$	−3.33996	−0.303633
$122$	3.17494	0.287445
$123$	0	0
$124$	2.11127	0.189598
$125$	−27.9759	−2.50224
$126$	0	0
$127$	17.3566	1.54015	0.770073	−	0.637955i	$-0.220220\pi$
$127$	17.3566	1.54015	0.770073	+	0.637955i	$0.220220\pi$
$128$	−11.0723	−0.978658
$129$	0	0
$130$	7.65921	0.671757
$131$	0.803568	0.0702080	0.0351040	−	0.999384i	$-0.488824\pi$
$131$	0.803568	0.0702080	0.0351040	+	0.999384i	$0.488824\pi$
$132$	0	0
$133$	−0.109209	−0.00946960
$134$	−0.624641	−0.0539608
$135$	0	0
$136$	9.11002	0.781178
$137$	4.63907	0.396343	0.198171	−	0.980167i	$-0.436500\pi$
$137$	4.63907	0.396343	0.198171	+	0.980167i	$0.436500\pi$
$138$	0	0
$139$	−0.0641748	−0.00544323	−0.00272162	−	0.999996i	$-0.500866\pi$
$139$	−0.0641748	−0.00544323	−0.00272162	+	0.999996i	$0.500866\pi$
$140$	7.33368	0.619809
$141$	0	0
$142$	3.63958	0.305427
$143$	11.2284	0.938964
$144$	0	0
$145$	9.24572	0.767815
$146$	4.55155	0.376689
$147$	0	0
$148$	16.0113	1.31612
$149$	−14.2436	−1.16688	−0.583440	−	0.812156i	$-0.698294\pi$
$149$	−14.2436	−1.16688	−0.583440	+	0.812156i	$0.698294\pi$
$150$	0	0
$151$	12.8350	1.04450	0.522249	−	0.852793i	$-0.325093\pi$
$151$	12.8350	1.04450	0.522249	+	0.852793i	$0.325093\pi$
$152$	0.190430	0.0154459
$153$	0	0
$154$	−1.27400	−0.102662
$155$	4.84258	0.388966
$156$	0	0
$157$	14.3051	1.14167	0.570835	−	0.821064i	$-0.306619\pi$
$157$	14.3051	1.14167	0.570835	+	0.821064i	$0.306619\pi$
$158$	−5.78469	−0.460205
$159$	0	0
$160$	−19.5395	−1.54473
$161$	−6.08681	−0.479708
$162$	0	0
$163$	5.60958	0.439376	0.219688	−	0.975570i	$-0.429496\pi$
$163$	5.60958	0.439376	0.219688	+	0.975570i	$0.429496\pi$
$164$	−1.78811	−0.139628
$165$	0	0
$166$	6.88785	0.534600
$167$	−4.49350	−0.347717	−0.173859	−	0.984771i	$-0.555624\pi$
$167$	−4.49350	−0.347717	−0.173859	+	0.984771i	$0.555624\pi$
$168$	0	0
$169$	3.45899	0.266076
$170$	9.86335	0.756484
$171$	0	0
$172$	14.1971	1.08252
$173$	−14.2512	−1.08350	−0.541749	−	0.840541i	$-0.682238\pi$
$173$	−14.2512	−1.08350	−0.541749	+	0.840541i	$0.682238\pi$
$174$	0	0
$175$	11.8211	0.893593
$176$	−7.67632	−0.578625
$177$	0	0
$178$	0.392617	0.0294279
$179$	−5.78841	−0.432646	−0.216323	−	0.976322i	$-0.569406\pi$
$179$	−5.78841	−0.432646	−0.216323	+	0.976322i	$0.569406\pi$
$180$	0	0
$181$	−14.6662	−1.09013	−0.545065	−	0.838394i	$-0.683495\pi$
$181$	−14.6662	−1.09013	−0.545065	+	0.838394i	$0.683495\pi$
$182$	−1.86748	−0.138427
$183$	0	0
$184$	10.6137	0.782452
$185$	36.7247	2.70006
$186$	0	0
$187$	14.4596	1.05739
$188$	−18.9028	−1.37863
$189$	0	0
$190$	0.206177	0.0149576
$191$	−8.41984	−0.609238	−0.304619	−	0.952474i	$-0.598529\pi$
$191$	−8.41984	−0.609238	−0.304619	+	0.952474i	$0.598529\pi$
$192$	0	0
$193$	−8.76374	−0.630828	−0.315414	−	0.948954i	$-0.602143\pi$
$193$	−8.76374	−0.630828	−0.315414	+	0.948954i	$0.602143\pi$
$194$	4.57003	0.328109
$195$	0	0
$196$	−1.78811	−0.127722
$197$	18.9953	1.35336	0.676680	−	0.736277i	$-0.263418\pi$
$197$	18.9953	1.35336	0.676680	+	0.736277i	$0.263418\pi$
$198$	0	0
$199$	−6.81886	−0.483376	−0.241688	−	0.970354i	$-0.577701\pi$
$199$	−6.81886	−0.483376	−0.241688	+	0.970354i	$0.577701\pi$
$200$	−20.6128	−1.45754
$201$	0	0
$202$	−5.19734	−0.365684
$203$	−2.25431	−0.158221
$204$	0	0
$205$	−4.10136	−0.286451
$206$	−5.00317	−0.348588
$207$	0	0
$208$	−11.2522	−0.780203
$209$	0.302255	0.0209074
$210$	0	0
$211$	5.02073	0.345641	0.172821	−	0.984953i	$-0.444712\pi$
$211$	5.02073	0.345641	0.172821	+	0.984953i	$0.444712\pi$
$212$	11.1417	0.765216
$213$	0	0
$214$	−0.917014	−0.0626858
$215$	32.5637	2.22083
$216$	0	0
$217$	−1.18073	−0.0801530
$218$	−8.22152	−0.556832
$219$	0	0
$220$	−20.2973	−1.36844
$221$	21.1955	1.42576
$222$	0	0
$223$	6.96231	0.466231	0.233115	−	0.972449i	$-0.425108\pi$
$223$	6.96231	0.466231	0.233115	+	0.972449i	$0.425108\pi$
$224$	4.76415	0.318318
$225$	0	0
$226$	7.48178	0.497681
$227$	−10.1094	−0.670987	−0.335493	−	0.942043i	$-0.608903\pi$
$227$	−10.1094	−0.670987	−0.335493	+	0.942043i	$0.608903\pi$
$228$	0	0
$229$	−2.32356	−0.153545	−0.0767725	−	0.997049i	$-0.524462\pi$
$229$	−2.32356	−0.153545	−0.0767725	+	0.997049i	$0.524462\pi$
$230$	11.4914	0.757718
$231$	0	0
$232$	3.93089	0.258075
$233$	−17.5655	−1.15075	−0.575377	−	0.817888i	$-0.695145\pi$
$233$	−17.5655	−1.15075	−0.575377	+	0.817888i	$0.695145\pi$
$234$	0	0
$235$	−43.3570	−2.82830
$236$	−16.8620	−1.09762
$237$	0	0
$238$	−2.40490	−0.155886
$239$	23.0063	1.48815	0.744077	−	0.668093i	$-0.232889\pi$
$239$	23.0063	1.48815	0.744077	+	0.668093i	$0.232889\pi$
$240$	0	0
$241$	−12.6808	−0.816839	−0.408420	−	0.912794i	$-0.633920\pi$
$241$	−12.6808	−0.816839	−0.408420	+	0.912794i	$0.633920\pi$
$242$	−1.53743	−0.0988298
$243$	0	0
$244$	−12.3332	−0.789550
$245$	−4.10136	−0.262026
$246$	0	0
$247$	0.443057	0.0281910
$248$	2.05886	0.130738
$249$	0	0
$250$	−12.8777	−0.814457
$251$	18.5629	1.17168	0.585840	−	0.810427i	$-0.300765\pi$
$251$	18.5629	1.17168	0.585840	+	0.810427i	$0.300765\pi$
$252$	0	0
$253$	16.8463	1.05912
$254$	7.98948	0.501305
$255$	0	0
$256$	1.61150	0.100719
$257$	−18.1358	−1.13128	−0.565639	−	0.824653i	$-0.691370\pi$
$257$	−18.1358	−1.13128	−0.565639	+	0.824653i	$0.691370\pi$
$258$	0	0
$259$	−8.95429	−0.556392
$260$	−29.7525	−1.84517
$261$	0	0
$262$	0.369894	0.0228521
$263$	−4.24808	−0.261948	−0.130974	−	0.991386i	$-0.541810\pi$
$263$	−4.24808	−0.261948	−0.130974	+	0.991386i	$0.541810\pi$
$264$	0	0
$265$	25.5555	1.56986
$266$	−0.0502704	−0.00308228
$267$	0	0
$268$	2.42644	0.148219
$269$	1.31427	0.0801326	0.0400663	−	0.999197i	$-0.487243\pi$
$269$	1.31427	0.0801326	0.0400663	+	0.999197i	$0.487243\pi$
$270$	0	0
$271$	−7.91618	−0.480874	−0.240437	−	0.970665i	$-0.577291\pi$
$271$	−7.91618	−0.480874	−0.240437	+	0.970665i	$0.577291\pi$
$272$	−14.4904	−0.878608
$273$	0	0
$274$	2.13543	0.129006
$275$	−32.7171	−1.97291
$276$	0	0
$277$	0.0684353	0.00411188	0.00205594	−	0.999998i	$-0.499346\pi$
$277$	0.0684353	0.00411188	0.00205594	+	0.999998i	$0.499346\pi$
$278$	−0.0295406	−0.00177173
$279$	0	0
$280$	7.15163	0.427391
$281$	−8.33050	−0.496956	−0.248478	−	0.968638i	$-0.579930\pi$
$281$	−8.33050	−0.496956	−0.248478	+	0.968638i	$0.579930\pi$
$282$	0	0
$283$	3.09424	0.183934	0.0919668	−	0.995762i	$-0.470685\pi$
$283$	3.09424	0.183934	0.0919668	+	0.995762i	$0.470685\pi$
$284$	−14.1381	−0.838942
$285$	0	0
$286$	5.16859	0.305625
$287$	1.00000	0.0590281
$288$	0	0
$289$	10.2951	0.605593
$290$	4.25594	0.249917
$291$	0	0
$292$	−17.6807	−1.03468
$293$	−4.47354	−0.261347	−0.130674	−	0.991425i	$-0.541714\pi$
$293$	−4.47354	−0.261347	−0.130674	+	0.991425i	$0.541714\pi$
$294$	0	0
$295$	−38.6760	−2.25181
$296$	15.6138	0.907533
$297$	0	0
$298$	−6.55653	−0.379809
$299$	24.6940	1.42809
$300$	0	0
$301$	−7.93974	−0.457639
$302$	5.90813	0.339975
$303$	0	0
$304$	−0.302897	−0.0173723
$305$	−28.2884	−1.61979
$306$	0	0
$307$	25.8316	1.47429	0.737143	−	0.675737i	$-0.236175\pi$
$307$	25.8316	1.47429	0.737143	+	0.675737i	$0.236175\pi$
$308$	4.94891	0.281991
$309$	0	0
$310$	2.22911	0.126605
$311$	−13.6802	−0.775735	−0.387867	−	0.921715i	$-0.626788\pi$
$311$	−13.6802	−0.775735	−0.387867	+	0.921715i	$0.626788\pi$
$312$	0	0
$313$	24.0537	1.35959	0.679797	−	0.733400i	$-0.262068\pi$
$313$	24.0537	1.35959	0.679797	+	0.733400i	$0.262068\pi$
$314$	6.58484	0.371604
$315$	0	0
$316$	22.4709	1.26408
$317$	−3.80766	−0.213859	−0.106930	−	0.994267i	$-0.534102\pi$
$317$	−3.80766	−0.213859	−0.106930	+	0.994267i	$0.534102\pi$
$318$	0	0
$319$	6.23920	0.349328
$320$	13.7564	0.769006
$321$	0	0
$322$	−2.80185	−0.156141
$323$	0.570558	0.0317467
$324$	0	0
$325$	−47.9579	−2.66023
$326$	2.58217	0.143013
$327$	0	0
$328$	−1.74372	−0.0962810
$329$	10.5714	0.582819
$330$	0	0
$331$	−21.1336	−1.16161	−0.580805	−	0.814043i	$-0.697262\pi$
$331$	−21.1336	−1.16161	−0.580805	+	0.814043i	$0.697262\pi$
$332$	−26.7561	−1.46843
$333$	0	0
$334$	−2.06842	−0.113179
$335$	5.56549	0.304075
$336$	0	0
$337$	−20.7801	−1.13197	−0.565983	−	0.824417i	$-0.691503\pi$
$337$	−20.7801	−1.13197	−0.565983	+	0.824417i	$0.691503\pi$
$338$	1.59222	0.0866055
$339$	0	0
$340$	−38.3146	−2.07790
$341$	3.26787	0.176965
$342$	0	0
$343$	1.00000	0.0539949
$344$	13.8447	0.746456
$345$	0	0
$346$	−6.56003	−0.352669
$347$	−21.8355	−1.17219	−0.586096	−	0.810242i	$-0.699336\pi$
$347$	−21.8355	−1.17219	−0.586096	+	0.810242i	$0.699336\pi$
$348$	0	0
$349$	−6.23178	−0.333579	−0.166790	−	0.985992i	$-0.553340\pi$
$349$	−6.23178	−0.333579	−0.166790	+	0.985992i	$0.553340\pi$
$350$	5.44143	0.290857
$351$	0	0
$352$	−13.1856	−0.702797
$353$	−28.5933	−1.52187	−0.760933	−	0.648830i	$-0.775258\pi$
$353$	−28.5933	−1.52187	−0.760933	+	0.648830i	$0.775258\pi$
$354$	0	0
$355$	−32.4283	−1.72112
$356$	−1.52514	−0.0808320
$357$	0	0
$358$	−2.66449	−0.140823
$359$	−6.89008	−0.363644	−0.181822	−	0.983331i	$-0.558200\pi$
$359$	−6.89008	−0.363644	−0.181822	+	0.983331i	$0.558200\pi$
$360$	0	0
$361$	−18.9881	−0.999372
$362$	−6.75107	−0.354828
$363$	0	0
$364$	7.25431	0.380229
$365$	−40.5539	−2.12269
$366$	0	0
$367$	27.9358	1.45824	0.729119	−	0.684387i	$-0.239930\pi$
$367$	27.9358	1.45824	0.729119	+	0.684387i	$0.239930\pi$
$368$	−16.8821	−0.880042
$369$	0	0
$370$	16.9049	0.878845
$371$	−6.23100	−0.323497
$372$	0	0
$373$	6.20063	0.321057	0.160528	−	0.987031i	$-0.448680\pi$
$373$	6.20063	0.321057	0.160528	+	0.987031i	$0.448680\pi$
$374$	6.65599	0.344173
$375$	0	0
$376$	−18.4335	−0.950637
$377$	9.14565	0.471025
$378$	0	0
$379$	−18.4006	−0.945174	−0.472587	−	0.881284i	$-0.656680\pi$
$379$	−18.4006	−0.945174	−0.472587	+	0.881284i	$0.656680\pi$
$380$	−0.800902	−0.0410854
$381$	0	0
$382$	−3.87577	−0.198302
$383$	34.2322	1.74918	0.874591	−	0.484861i	$-0.161130\pi$
$383$	34.2322	1.74918	0.874591	+	0.484861i	$0.161130\pi$
$384$	0	0
$385$	11.3512	0.578512
$386$	−4.03408	−0.205329
$387$	0	0
$388$	−17.7525	−0.901245
$389$	23.0399	1.16817	0.584085	−	0.811693i	$-0.301454\pi$
$389$	23.0399	1.16817	0.584085	+	0.811693i	$0.301454\pi$
$390$	0	0
$391$	31.8003	1.60821
$392$	−1.74372	−0.0880713
$393$	0	0
$394$	8.74382	0.440507
$395$	51.5410	2.59331
$396$	0	0
$397$	1.35123	0.0678165	0.0339082	−	0.999425i	$-0.489205\pi$
$397$	1.35123	0.0678165	0.0339082	+	0.999425i	$0.489205\pi$
$398$	−3.13882	−0.157335
$399$	0	0
$400$	32.7866	1.63933
$401$	−1.54256	−0.0770319	−0.0385160	−	0.999258i	$-0.512263\pi$
$401$	−1.54256	−0.0770319	−0.0385160	+	0.999258i	$0.512263\pi$
$402$	0	0
$403$	4.79017	0.238615
$404$	20.1893	1.00445
$405$	0	0
$406$	−1.03769	−0.0514997
$407$	24.7826	1.22843
$408$	0	0
$409$	4.34571	0.214882	0.107441	−	0.994211i	$-0.465734\pi$
$409$	4.34571	0.214882	0.107441	+	0.994211i	$0.465734\pi$
$410$	−1.88791	−0.0932374
$411$	0	0
$412$	19.4350	0.957495
$413$	9.43006	0.464023
$414$	0	0
$415$	−61.3700	−3.01254
$416$	−19.3280	−0.947634
$417$	0	0
$418$	0.139132	0.00680518
$419$	−27.2835	−1.33288	−0.666442	−	0.745557i	$-0.732184\pi$
$419$	−27.2835	−1.33288	−0.666442	+	0.745557i	$0.732184\pi$
$420$	0	0
$421$	−16.6168	−0.809853	−0.404927	−	0.914349i	$-0.632703\pi$
$421$	−16.6168	−0.809853	−0.404927	+	0.914349i	$0.632703\pi$
$422$	2.31111	0.112503
$423$	0	0
$424$	10.8651	0.527657
$425$	−61.7591	−2.99576
$426$	0	0
$427$	6.89732	0.333785
$428$	3.56218	0.172184
$429$	0	0
$430$	14.9895	0.722860
$431$	9.36507	0.451099	0.225550	−	0.974232i	$-0.427582\pi$
$431$	9.36507	0.451099	0.225550	+	0.974232i	$0.427582\pi$
$432$	0	0
$433$	−3.87136	−0.186046	−0.0930228	−	0.995664i	$-0.529653\pi$
$433$	−3.87136	−0.186046	−0.0930228	+	0.995664i	$0.529653\pi$
$434$	−0.543506	−0.0260891
$435$	0	0
$436$	31.9368	1.52950
$437$	0.664733	0.0317985
$438$	0	0
$439$	29.6011	1.41279	0.706393	−	0.707820i	$-0.250321\pi$
$439$	29.6011	1.41279	0.706393	+	0.707820i	$0.250321\pi$
$440$	−19.7934	−0.943613
$441$	0	0
$442$	9.75660	0.464074
$443$	0.570558	0.0271080	0.0135540	−	0.999908i	$-0.495685\pi$
$443$	0.570558	0.0271080	0.0135540	+	0.999908i	$0.495685\pi$
$444$	0	0
$445$	−3.49817	−0.165829
$446$	3.20485	0.151754
$447$	0	0
$448$	−3.35411	−0.158467
$449$	−18.8280	−0.888551	−0.444275	−	0.895890i	$-0.646539\pi$
$449$	−18.8280	−0.888551	−0.444275	+	0.895890i	$0.646539\pi$
$450$	0	0
$451$	−2.76768	−0.130325
$452$	−29.0633	−1.36702
$453$	0	0
$454$	−4.65352	−0.218401
$455$	16.6391	0.780052
$456$	0	0
$457$	22.2732	1.04190	0.520949	−	0.853588i	$-0.325578\pi$
$457$	22.2732	1.04190	0.520949	+	0.853588i	$0.325578\pi$
$458$	−1.06957	−0.0499776
$459$	0	0
$460$	−44.6387	−2.08129
$461$	−4.47354	−0.208354	−0.104177	−	0.994559i	$-0.533221\pi$
$461$	−4.47354	−0.208354	−0.104177	+	0.994559i	$0.533221\pi$
$462$	0	0
$463$	−3.62240	−0.168347	−0.0841736	−	0.996451i	$-0.526825\pi$
$463$	−3.62240	−0.168347	−0.0841736	+	0.996451i	$0.526825\pi$
$464$	−6.25246	−0.290263
$465$	0	0
$466$	−8.08565	−0.374561
$467$	1.54346	0.0714226	0.0357113	−	0.999362i	$-0.488630\pi$
$467$	1.54346	0.0714226	0.0357113	+	0.999362i	$0.488630\pi$
$468$	0	0
$469$	−1.35699	−0.0626599
$470$	−19.9578	−0.920587
$471$	0	0
$472$	−16.4434	−0.756869
$473$	21.9746	1.01039
$474$	0	0
$475$	−1.29097	−0.0592338
$476$	9.34193	0.428187
$477$	0	0
$478$	10.5901	0.484382
$479$	14.5913	0.666693	0.333347	−	0.942804i	$-0.391822\pi$
$479$	14.5913	0.666693	0.333347	+	0.942804i	$0.391822\pi$
$480$	0	0
$481$	36.3273	1.65638
$482$	−5.83713	−0.265874
$483$	0	0
$484$	5.97221	0.271464
$485$	−40.7185	−1.84893
$486$	0	0
$487$	−1.52440	−0.0690771	−0.0345385	−	0.999403i	$-0.510996\pi$
$487$	−1.52440	−0.0690771	−0.0345385	+	0.999403i	$0.510996\pi$
$488$	−12.0270	−0.544437
$489$	0	0
$490$	−1.88791	−0.0852873
$491$	34.8827	1.57424	0.787118	−	0.616802i	$-0.211572\pi$
$491$	34.8827	1.57424	0.787118	+	0.616802i	$0.211572\pi$
$492$	0	0
$493$	11.7776	0.530435
$494$	0.203945	0.00917594
$495$	0	0
$496$	−3.27482	−0.147044
$497$	7.90672	0.354665
$498$	0	0
$499$	14.7766	0.661490	0.330745	−	0.943720i	$-0.392700\pi$
$499$	14.7766	0.661490	0.330745	+	0.943720i	$0.392700\pi$
$500$	50.0239	2.23714
$501$	0	0
$502$	8.54478	0.381372
$503$	−18.9590	−0.845338	−0.422669	−	0.906284i	$-0.638907\pi$
$503$	−18.9590	−0.845338	−0.422669	+	0.906284i	$0.638907\pi$
$504$	0	0
$505$	46.3078	2.06067
$506$	7.75461	0.344734
$507$	0	0
$508$	−31.0355	−1.37698
$509$	17.0790	0.757011	0.378506	−	0.925599i	$-0.376438\pi$
$509$	17.0790	0.757011	0.378506	+	0.925599i	$0.376438\pi$
$510$	0	0
$511$	9.88791	0.437416
$512$	22.8863	1.01144
$513$	0	0
$514$	−8.34815	−0.368221
$515$	44.5778	1.96433
$516$	0	0
$517$	−29.2582	−1.28677
$518$	−4.12179	−0.181101
$519$	0	0
$520$	−29.0139	−1.27234
$521$	−25.3095	−1.10883	−0.554415	−	0.832240i	$-0.687058\pi$
$521$	−25.3095	−1.10883	−0.554415	+	0.832240i	$0.687058\pi$
$522$	0	0
$523$	−4.43499	−0.193928	−0.0969642	−	0.995288i	$-0.530913\pi$
$523$	−4.43499	−0.193928	−0.0969642	+	0.995288i	$0.530913\pi$
$524$	−1.43687	−0.0627699
$525$	0	0
$526$	−1.95545	−0.0852617
$527$	6.16867	0.268712
$528$	0	0
$529$	14.0492	0.610835
$530$	11.7636	0.510978
$531$	0	0
$532$	0.195277	0.00846635
$533$	−4.05697	−0.175727
$534$	0	0
$535$	8.17051	0.353242
$536$	2.36621	0.102205
$537$	0	0
$538$	0.604979	0.0260825
$539$	−2.76768	−0.119212
$540$	0	0
$541$	−5.57149	−0.239537	−0.119769	−	0.992802i	$-0.538215\pi$
$541$	−5.57149	−0.239537	−0.119769	+	0.992802i	$0.538215\pi$
$542$	−3.64393	−0.156520
$543$	0	0
$544$	−24.8902	−1.06716
$545$	73.2529	3.13781
$546$	0	0
$547$	38.0432	1.62661	0.813304	−	0.581839i	$-0.197666\pi$
$547$	38.0432	1.62661	0.813304	+	0.581839i	$0.197666\pi$
$548$	−8.29517	−0.354352
$549$	0	0
$550$	−15.0601	−0.642167
$551$	0.246190	0.0104881
$552$	0	0
$553$	−12.5668	−0.534395
$554$	0.0315018	0.00133838
$555$	0	0
$556$	0.114752	0.00486655
$557$	19.0835	0.808594	0.404297	−	0.914628i	$-0.367516\pi$
$557$	19.0835	0.808594	0.404297	+	0.914628i	$0.367516\pi$
$558$	0	0
$559$	32.2113	1.36239
$560$	−11.3754	−0.480697
$561$	0	0
$562$	−3.83465	−0.161755
$563$	19.1867	0.808621	0.404311	−	0.914622i	$-0.367511\pi$
$563$	19.1867	0.808621	0.404311	+	0.914622i	$0.367511\pi$
$564$	0	0
$565$	−66.6619	−2.80449
$566$	1.42432	0.0598688
$567$	0	0
$568$	−13.7871	−0.578495
$569$	−29.3295	−1.22956	−0.614778	−	0.788700i	$-0.710754\pi$
$569$	−29.3295	−1.22956	−0.614778	+	0.788700i	$0.710754\pi$
$570$	0	0
$571$	−21.5864	−0.903362	−0.451681	−	0.892180i	$-0.649175\pi$
$571$	−21.5864	−0.903362	−0.451681	+	0.892180i	$0.649175\pi$
$572$	−20.0776	−0.839486
$573$	0	0
$574$	0.460315	0.0192132
$575$	−71.9529	−3.00064
$576$	0	0
$577$	5.28198	0.219892	0.109946	−	0.993938i	$-0.464932\pi$
$577$	5.28198	0.219892	0.109946	+	0.993938i	$0.464932\pi$
$578$	4.73898	0.197115
$579$	0	0
$580$	−16.5324	−0.686469
$581$	14.9633	0.620784
$582$	0	0
$583$	17.2454	0.714231
$584$	−17.2418	−0.713470
$585$	0	0
$586$	−2.05924	−0.0850662
$587$	41.8869	1.72886	0.864428	−	0.502757i	$-0.167681\pi$
$587$	41.8869	1.72886	0.864428	+	0.502757i	$0.167681\pi$
$588$	0	0
$589$	0.128946	0.00531312
$590$	−17.8031	−0.732944
$591$	0	0
$592$	−24.8353	−1.02072
$593$	−25.8605	−1.06196	−0.530982	−	0.847383i	$-0.678177\pi$
$593$	−25.8605	−1.06196	−0.530982	+	0.847383i	$0.678177\pi$
$594$	0	0
$595$	21.4274	0.878438
$596$	25.4691	1.04325
$597$	0	0
$598$	11.3670	0.464831
$599$	39.4470	1.61176	0.805881	−	0.592078i	$-0.201692\pi$
$599$	39.4470	1.61176	0.805881	+	0.592078i	$0.201692\pi$
$600$	0	0
$601$	40.0435	1.63341	0.816704	−	0.577056i	$-0.195799\pi$
$601$	40.0435	1.63341	0.816704	+	0.577056i	$0.195799\pi$
$602$	−3.65478	−0.148958
$603$	0	0
$604$	−22.9504	−0.933838
$605$	13.6984	0.556918
$606$	0	0
$607$	−10.7580	−0.436653	−0.218327	−	0.975876i	$-0.570060\pi$
$607$	−10.7580	−0.436653	−0.218327	+	0.975876i	$0.570060\pi$
$608$	−0.520288	−0.0211004
$609$	0	0
$610$	−13.0215	−0.527227
$611$	−42.8877	−1.73505
$612$	0	0
$613$	16.7220	0.675393	0.337697	−	0.941255i	$-0.390352\pi$
$613$	16.7220	0.675393	0.337697	+	0.941255i	$0.390352\pi$
$614$	11.8906	0.479867
$615$	0	0
$616$	4.82606	0.194448
$617$	42.3381	1.70447	0.852234	−	0.523160i	$-0.175247\pi$
$617$	42.3381	1.70447	0.852234	+	0.523160i	$0.175247\pi$
$618$	0	0
$619$	32.0830	1.28952	0.644762	−	0.764384i	$-0.276957\pi$
$619$	32.0830	1.28952	0.644762	+	0.764384i	$0.276957\pi$
$620$	−8.65907	−0.347757
$621$	0	0
$622$	−6.29721	−0.252495
$623$	0.852931	0.0341720
$624$	0	0
$625$	55.6333	2.22533
$626$	11.0723	0.442536
$627$	0	0
$628$	−25.5791	−1.02072
$629$	46.7814	1.86530
$630$	0	0
$631$	−44.9458	−1.78927	−0.894633	−	0.446802i	$-0.852563\pi$
$631$	−44.9458	−1.78927	−0.894633	+	0.446802i	$0.852563\pi$
$632$	21.9130	0.871654
$633$	0	0
$634$	−1.75272	−0.0696094
$635$	−71.1855	−2.82491
$636$	0	0
$637$	−4.05697	−0.160743
$638$	2.87199	0.113703
$639$	0	0
$640$	45.4113	1.79504
$641$	−10.4353	−0.412171	−0.206085	−	0.978534i	$-0.566072\pi$
$641$	−10.4353	−0.412171	−0.206085	+	0.978534i	$0.566072\pi$
$642$	0	0
$643$	−33.1360	−1.30676	−0.653378	−	0.757032i	$-0.726649\pi$
$643$	−33.1360	−1.30676	−0.653378	+	0.757032i	$0.726649\pi$
$644$	10.8839	0.428885
$645$	0	0
$646$	0.262636	0.0103333
$647$	29.4094	1.15620	0.578101	−	0.815965i	$-0.303794\pi$
$647$	29.4094	1.15620	0.578101	+	0.815965i	$0.303794\pi$
$648$	0	0
$649$	−26.0994	−1.02449
$650$	−22.0757	−0.865881
$651$	0	0
$652$	−10.0306	−0.392827
$653$	41.0121	1.60493	0.802464	−	0.596701i	$-0.203522\pi$
$653$	41.0121	1.60493	0.802464	+	0.596701i	$0.203522\pi$
$654$	0	0
$655$	−3.29572	−0.128774
$656$	2.77356	0.108289
$657$	0	0
$658$	4.86616	0.189703
$659$	−41.6194	−1.62126	−0.810631	−	0.585558i	$-0.800876\pi$
$659$	−41.6194	−1.62126	−0.810631	+	0.585558i	$0.800876\pi$
$660$	0	0
$661$	−43.0712	−1.67527	−0.837637	−	0.546227i	$-0.816064\pi$
$661$	−43.0712	−1.67527	−0.837637	+	0.546227i	$0.816064\pi$
$662$	−9.72812	−0.378094
$663$	0	0
$664$	−26.0919	−1.01256
$665$	0.447904	0.0173690
$666$	0	0
$667$	13.7215	0.531300
$668$	8.03487	0.310878
$669$	0	0
$670$	2.56188	0.0989739
$671$	−19.0896	−0.736944
$672$	0	0
$673$	44.0679	1.69869	0.849346	−	0.527837i	$-0.176997\pi$
$673$	44.0679	1.69869	0.849346	+	0.527837i	$0.176997\pi$
$674$	−9.56540	−0.368445
$675$	0	0
$676$	−6.18506	−0.237887
$677$	−5.02017	−0.192941	−0.0964704	−	0.995336i	$-0.530755\pi$
$677$	−5.02017	−0.192941	−0.0964704	+	0.995336i	$0.530755\pi$
$678$	0	0
$679$	9.92806	0.381004
$680$	−37.3635	−1.43282
$681$	0	0
$682$	1.50425	0.0576007
$683$	−24.9476	−0.954595	−0.477297	−	0.878742i	$-0.658384\pi$
$683$	−24.9476	−0.954595	−0.477297	+	0.878742i	$0.658384\pi$
$684$	0	0
$685$	−19.0265	−0.726965
$686$	0.460315	0.0175749
$687$	0	0
$688$	−22.0213	−0.839556
$689$	25.2790	0.963052
$690$	0	0
$691$	42.8508	1.63012	0.815061	−	0.579375i	$-0.196703\pi$
$691$	42.8508	1.63012	0.815061	+	0.579375i	$0.196703\pi$
$692$	25.4827	0.968706
$693$	0	0
$694$	−10.0512	−0.381538
$695$	0.263204	0.00998389
$696$	0	0
$697$	−5.22447	−0.197891
$698$	−2.86858	−0.108577
$699$	0	0
$700$	−21.1375	−0.798921
$701$	46.3548	1.75079	0.875397	−	0.483404i	$-0.160600\pi$
$701$	46.3548	1.75079	0.875397	+	0.483404i	$0.160600\pi$
$702$	0	0
$703$	0.977887	0.0368817
$704$	9.28310	0.349870
$705$	0	0
$706$	−13.1619	−0.495355
$707$	−11.2908	−0.424636
$708$	0	0
$709$	10.9521	0.411316	0.205658	−	0.978624i	$-0.434067\pi$
$709$	10.9521	0.411316	0.205658	+	0.978624i	$0.434067\pi$
$710$	−14.9272	−0.560208
$711$	0	0
$712$	−1.48728	−0.0557380
$713$	7.18686	0.269150
$714$	0	0
$715$	−46.0516	−1.72223
$716$	10.3503	0.386809
$717$	0	0
$718$	−3.17160	−0.118363
$719$	−7.62700	−0.284439	−0.142220	−	0.989835i	$-0.545424\pi$
$719$	−7.62700	−0.284439	−0.142220	+	0.989835i	$0.545424\pi$
$720$	0	0
$721$	−10.8690	−0.404784
$722$	−8.74049	−0.325287
$723$	0	0
$724$	26.2248	0.974637
$725$	−26.6484	−0.989698
$726$	0	0
$727$	−37.2537	−1.38166	−0.690832	−	0.723015i	$-0.742756\pi$
$727$	−37.2537	−1.38166	−0.690832	+	0.723015i	$0.742756\pi$
$728$	7.07423	0.262188
$729$	0	0
$730$	−18.6675	−0.690916
$731$	41.4809	1.53423
$732$	0	0
$733$	7.99745	0.295393	0.147696	−	0.989033i	$-0.452814\pi$
$733$	7.99745	0.295393	0.147696	+	0.989033i	$0.452814\pi$
$734$	12.8593	0.474644
$735$	0	0
$736$	−28.9985	−1.06890
$737$	3.75571	0.138343
$738$	0	0
$739$	19.7183	0.725350	0.362675	−	0.931916i	$-0.381864\pi$
$739$	19.7183	0.725350	0.362675	+	0.931916i	$0.381864\pi$
$740$	−65.6679	−2.41400
$741$	0	0
$742$	−2.86822	−0.105296
$743$	−31.1345	−1.14221	−0.571106	−	0.820876i	$-0.693486\pi$
$743$	−31.1345	−1.14221	−0.571106	+	0.820876i	$0.693486\pi$
$744$	0	0
$745$	58.4180	2.14027
$746$	2.85424	0.104501
$747$	0	0
$748$	−25.8555	−0.945369
$749$	−1.99215	−0.0727915
$750$	0	0
$751$	−40.7231	−1.48601	−0.743004	−	0.669287i	$-0.766600\pi$
$751$	−40.7231	−1.48601	−0.743004	+	0.669287i	$0.766600\pi$
$752$	29.3203	1.06920
$753$	0	0
$754$	4.20988	0.153315
$755$	−52.6409	−1.91580
$756$	0	0
$757$	34.9086	1.26878	0.634388	−	0.773015i	$-0.281252\pi$
$757$	34.9086	1.26878	0.634388	+	0.773015i	$0.281252\pi$
$758$	−8.47005	−0.307646
$759$	0	0
$760$	−0.781021	−0.0283306
$761$	−4.76771	−0.172830	−0.0864148	−	0.996259i	$-0.527541\pi$
$761$	−4.76771	−0.172830	−0.0864148	+	0.996259i	$0.527541\pi$
$762$	0	0
$763$	−17.8607	−0.646599
$764$	15.0556	0.544693
$765$	0	0
$766$	15.7576	0.569344
$767$	−38.2574	−1.38140
$768$	0	0
$769$	51.9550	1.87354	0.936772	−	0.349941i	$-0.113798\pi$
$769$	51.9550	1.87354	0.936772	+	0.349941i	$0.113798\pi$
$770$	5.22514	0.188301
$771$	0	0
$772$	15.6705	0.563995
$773$	−19.6577	−0.707038	−0.353519	−	0.935427i	$-0.615015\pi$
$773$	−19.6577	−0.707038	−0.353519	+	0.935427i	$0.615015\pi$
$774$	0	0
$775$	−13.9575	−0.501369
$776$	−17.3118	−0.621457
$777$	0	0
$778$	10.6056	0.380229
$779$	−0.109209	−0.00391281
$780$	0	0
$781$	−21.8833	−0.783045
$782$	14.6382	0.523459
$783$	0	0
$784$	2.77356	0.0990557
$785$	−58.6703	−2.09403
$786$	0	0
$787$	16.1094	0.574238	0.287119	−	0.957895i	$-0.407302\pi$
$787$	16.1094	0.574238	0.287119	+	0.957895i	$0.407302\pi$
$788$	−33.9657	−1.20998
$789$	0	0
$790$	23.7251	0.844100
$791$	16.2536	0.577913
$792$	0	0
$793$	−27.9822	−0.993677
$794$	0.621992	0.0220737
$795$	0	0
$796$	12.1929	0.432165
$797$	−20.3109	−0.719447	−0.359724	−	0.933059i	$-0.617129\pi$
$797$	−20.3109	−0.719447	−0.359724	+	0.933059i	$0.617129\pi$
$798$	0	0
$799$	−55.2298	−1.95389
$800$	56.3177	1.99113
$801$	0	0
$802$	−0.710064	−0.0250732
$803$	−27.3666	−0.965745
$804$	0	0
$805$	24.9642	0.879871
$806$	2.20499	0.0776673
$807$	0	0
$808$	19.6881	0.692625
$809$	−3.62656	−0.127503	−0.0637515	−	0.997966i	$-0.520307\pi$
$809$	−3.62656	−0.127503	−0.0637515	+	0.997966i	$0.520307\pi$
$810$	0	0
$811$	10.2870	0.361225	0.180613	−	0.983554i	$-0.442192\pi$
$811$	10.2870	0.361225	0.180613	+	0.983554i	$0.442192\pi$
$812$	4.03095	0.141459
$813$	0	0
$814$	11.4078	0.399843
$815$	−23.0069	−0.805896
$816$	0	0
$817$	0.867089	0.0303356
$818$	2.00039	0.0699422
$819$	0	0
$820$	7.33368	0.256103
$821$	−6.42929	−0.224384	−0.112192	−	0.993687i	$-0.535787\pi$
$821$	−6.42929	−0.224384	−0.112192	+	0.993687i	$0.535787\pi$
$822$	0	0
$823$	4.04441	0.140979	0.0704896	−	0.997513i	$-0.477544\pi$
$823$	4.04441	0.140979	0.0704896	+	0.997513i	$0.477544\pi$
$824$	18.9526	0.660244
$825$	0	0
$826$	4.34079	0.151036
$827$	−4.70225	−0.163513	−0.0817566	−	0.996652i	$-0.526053\pi$
$827$	−4.70225	−0.163513	−0.0817566	+	0.996652i	$0.526053\pi$
$828$	0	0
$829$	−6.74357	−0.234214	−0.117107	−	0.993119i	$-0.537362\pi$
$829$	−6.74357	−0.234214	−0.117107	+	0.993119i	$0.537362\pi$
$830$	−28.2495	−0.980555
$831$	0	0
$832$	13.6075	0.471756
$833$	−5.22447	−0.181017
$834$	0	0
$835$	18.4294	0.637777
$836$	−0.540465	−0.0186924
$837$	0	0
$838$	−12.5590	−0.433842
$839$	−50.7412	−1.75178	−0.875890	−	0.482511i	$-0.839725\pi$
$839$	−50.7412	−1.75178	−0.875890	+	0.482511i	$0.839725\pi$
$840$	0	0
$841$	−23.9181	−0.824762
$842$	−7.64895	−0.263600
$843$	0	0
$844$	−8.97762	−0.309022
$845$	−14.1866	−0.488032
$846$	0	0
$847$	−3.33996	−0.114762
$848$	−17.2820	−0.593468
$849$	0	0
$850$	−28.4286	−0.975093
$851$	54.5030	1.86834
$852$	0	0
$853$	9.55166	0.327042	0.163521	−	0.986540i	$-0.447715\pi$
$853$	9.55166	0.327042	0.163521	+	0.986540i	$0.447715\pi$
$854$	3.17494	0.108644
$855$	0	0
$856$	3.47375	0.118730
$857$	25.8962	0.884598	0.442299	−	0.896868i	$-0.354163\pi$
$857$	25.8962	0.884598	0.442299	+	0.896868i	$0.354163\pi$
$858$	0	0
$859$	−41.3258	−1.41002	−0.705010	−	0.709198i	$-0.749057\pi$
$859$	−41.3258	−1.41002	−0.705010	+	0.709198i	$0.749057\pi$
$860$	−58.2275	−1.98554
$861$	0	0
$862$	4.31088	0.146829
$863$	1.76472	0.0600719	0.0300360	−	0.999549i	$-0.490438\pi$
$863$	1.76472	0.0600719	0.0300360	+	0.999549i	$0.490438\pi$
$864$	0	0
$865$	58.4492	1.98733
$866$	−1.78204	−0.0605563
$867$	0	0
$868$	2.11127	0.0716612
$869$	34.7809	1.17986
$870$	0	0
$871$	5.50526	0.186539
$872$	31.1440	1.05467
$873$	0	0
$874$	0.305986	0.0103501
$875$	−27.9759	−0.945757
$876$	0	0
$877$	−37.4713	−1.26532	−0.632658	−	0.774432i	$-0.718036\pi$
$877$	−37.4713	−1.26532	−0.632658	+	0.774432i	$0.718036\pi$
$878$	13.6258	0.459850
$879$	0	0
$880$	31.4833	1.06130
$881$	14.9319	0.503068	0.251534	−	0.967849i	$-0.419065\pi$
$881$	14.9319	0.503068	0.251534	+	0.967849i	$0.419065\pi$
$882$	0	0
$883$	35.3025	1.18802	0.594012	−	0.804456i	$-0.297543\pi$
$883$	35.3025	1.18802	0.594012	+	0.804456i	$0.297543\pi$
$884$	−37.8999	−1.27471
$885$	0	0
$886$	0.262636	0.00882343
$887$	6.34513	0.213049	0.106524	−	0.994310i	$-0.466028\pi$
$887$	6.34513	0.213049	0.106524	+	0.994310i	$0.466028\pi$
$888$	0	0
$889$	17.3566	0.582121
$890$	−1.61026	−0.0539761
$891$	0	0
$892$	−12.4494	−0.416836
$893$	−1.15449	−0.0386334
$894$	0	0
$895$	23.7403	0.793551
$896$	−11.0723	−0.369898
$897$	0	0
$898$	−8.66682	−0.289216
$899$	2.66172	0.0887734
$900$	0	0
$901$	32.5537	1.08452
$902$	−1.27400	−0.0424196
$903$	0	0
$904$	−28.3418	−0.942635
$905$	60.1513	1.99950
$906$	0	0
$907$	−21.6718	−0.719599	−0.359800	−	0.933030i	$-0.617155\pi$
$907$	−21.6718	−0.719599	−0.359800	+	0.933030i	$0.617155\pi$
$908$	18.0768	0.599899
$909$	0	0
$910$	7.65921	0.253900
$911$	5.13876	0.170255	0.0851274	−	0.996370i	$-0.472870\pi$
$911$	5.13876	0.170255	0.0851274	+	0.996370i	$0.472870\pi$
$912$	0	0
$913$	−41.4137	−1.37059
$914$	10.2527	0.339129
$915$	0	0
$916$	4.15478	0.137278
$917$	0.803568	0.0265361
$918$	0	0
$919$	−34.5055	−1.13823	−0.569116	−	0.822257i	$-0.692715\pi$
$919$	−34.5055	−1.13823	−0.569116	+	0.822257i	$0.692715\pi$
$920$	−43.5306	−1.43516
$921$	0	0
$922$	−2.05924	−0.0678173
$923$	−32.0773	−1.05584
$924$	0	0
$925$	−105.850	−3.48032
$926$	−1.66744	−0.0547956
$927$	0	0
$928$	−10.7399	−0.352553
$929$	5.42071	0.177848	0.0889238	−	0.996038i	$-0.471657\pi$
$929$	5.42071	0.177848	0.0889238	+	0.996038i	$0.471657\pi$
$930$	0	0
$931$	−0.109209	−0.00357917
$932$	31.4090	1.02884
$933$	0	0
$934$	0.710475	0.0232475
$935$	−59.3042	−1.93945
$936$	0	0
$937$	46.9692	1.53442	0.767208	−	0.641398i	$-0.221645\pi$
$937$	46.9692	1.53442	0.767208	+	0.641398i	$0.221645\pi$
$938$	−0.624641	−0.0203953
$939$	0	0
$940$	77.5271	2.52865
$941$	−3.58221	−0.116777	−0.0583884	−	0.998294i	$-0.518596\pi$
$941$	−3.58221	−0.116777	−0.0583884	+	0.998294i	$0.518596\pi$
$942$	0	0
$943$	−6.08681	−0.198214
$944$	26.1548	0.851267
$945$	0	0
$946$	10.1152	0.328875
$947$	35.5839	1.15632	0.578161	−	0.815923i	$-0.303771\pi$
$947$	35.5839	1.15632	0.578161	+	0.815923i	$0.303771\pi$
$948$	0	0
$949$	−40.1150	−1.30219
$950$	−0.594253	−0.0192801
$951$	0	0
$952$	9.11002	0.295258
$953$	43.5164	1.40963	0.704817	−	0.709389i	$-0.251029\pi$
$953$	43.5164	1.40963	0.704817	+	0.709389i	$0.251029\pi$
$954$	0	0
$955$	34.5328	1.11745
$956$	−41.1378	−1.33049
$957$	0	0
$958$	6.71659	0.217003
$959$	4.63907	0.149803
$960$	0	0
$961$	−29.6059	−0.955028
$962$	16.7220	0.539138
$963$	0	0
$964$	22.6746	0.730299
$965$	35.9432	1.15705
$966$	0	0
$967$	−22.2431	−0.715289	−0.357645	−	0.933858i	$-0.616420\pi$
$967$	−22.2431	−0.715289	−0.357645	+	0.933858i	$0.616420\pi$
$968$	5.82396	0.187189
$969$	0	0
$970$	−18.7433	−0.601812
$971$	−10.0764	−0.323366	−0.161683	−	0.986843i	$-0.551692\pi$
$971$	−10.0764	−0.323366	−0.161683	+	0.986843i	$0.551692\pi$
$972$	0	0
$973$	−0.0641748	−0.00205735
$974$	−0.701702	−0.0224840
$975$	0	0
$976$	19.1301	0.612341
$977$	−60.5914	−1.93849	−0.969245	−	0.246097i	$-0.920852\pi$
$977$	−60.5914	−1.93849	−0.969245	+	0.246097i	$0.920852\pi$
$978$	0	0
$979$	−2.36064	−0.0754463
$980$	7.33368	0.234266
$981$	0	0
$982$	16.0570	0.512401
$983$	−41.9563	−1.33820	−0.669100	−	0.743173i	$-0.733320\pi$
$983$	−41.9563	−1.33820	−0.669100	+	0.743173i	$0.733320\pi$
$984$	0	0
$985$	−77.9066	−2.48231
$986$	5.42138	0.172652
$987$	0	0
$988$	−0.792234	−0.0252043
$989$	48.3276	1.53673
$990$	0	0
$991$	40.0186	1.27123	0.635616	−	0.772005i	$-0.280746\pi$
$991$	40.0186	1.27123	0.635616	+	0.772005i	$0.280746\pi$
$992$	−5.62517	−0.178599
$993$	0	0
$994$	3.63958	0.115440
$995$	27.9666	0.886599
$996$	0	0
$997$	6.59339	0.208815	0.104407	−	0.994535i	$-0.466705\pi$
$997$	6.59339	0.208815	0.104407	+	0.994535i	$0.466705\pi$
$998$	6.80187	0.215310
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2583.2.a.r.1.3		5
3.2	odd	2		287.2.a.e.1.3	✓	5
12.11	even	2		4592.2.a.bb.1.3		5
15.14	odd	2		7175.2.a.n.1.3		5
21.20	even	2		2009.2.a.n.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.a.e.1.3	✓	5	3.2	odd	2
2009.2.a.n.1.3		5	21.20	even	2
2583.2.a.r.1.3		5	1.1	even	1	trivial
4592.2.a.bb.1.3		5	12.11	even	2
7175.2.a.n.1.3		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)

\(f(q)\)	\(=\)	\(q+0.460315 q^{2} -1.78811 q^{4} -4.10136 q^{5} +1.00000 q^{7} -1.74372 q^{8} +O(q^{10})\)	\(q+0.460315 q^{2} -1.78811 q^{4} -4.10136 q^{5} +1.00000 q^{7} -1.74372 q^{8} -1.88791 q^{10} -2.76768 q^{11} -4.05697 q^{13} +0.460315 q^{14} +2.77356 q^{16} -5.22447 q^{17} -0.109209 q^{19} +7.33368 q^{20} -1.27400 q^{22} -6.08681 q^{23} +11.8211 q^{25} -1.86748 q^{26} -1.78811 q^{28} -2.25431 q^{29} -1.18073 q^{31} +4.76415 q^{32} -2.40490 q^{34} -4.10136 q^{35} -8.95429 q^{37} -0.0502704 q^{38} +7.15163 q^{40} +1.00000 q^{41} -7.93974 q^{43} +4.94891 q^{44} -2.80185 q^{46} +10.5714 q^{47} +1.00000 q^{49} +5.44143 q^{50} +7.25431 q^{52} -6.23100 q^{53} +11.3512 q^{55} -1.74372 q^{56} -1.03769 q^{58} +9.43006 q^{59} +6.89732 q^{61} -0.543506 q^{62} -3.35411 q^{64} +16.6391 q^{65} -1.35699 q^{67} +9.34193 q^{68} -1.88791 q^{70} +7.90672 q^{71} +9.88791 q^{73} -4.12179 q^{74} +0.195277 q^{76} -2.76768 q^{77} -12.5668 q^{79} -11.3754 q^{80} +0.460315 q^{82} +14.9633 q^{83} +21.4274 q^{85} -3.65478 q^{86} +4.82606 q^{88} +0.852931 q^{89} -4.05697 q^{91} +10.8839 q^{92} +4.86616 q^{94} +0.447904 q^{95} +9.92806 q^{97} +0.460315 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + q^{2} + 3 q^{4} + 5 q^{5} + 5 q^{7} + 3 q^{8}+O(q^{10}) \) 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

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Database

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