LMFDB - Embedded newform 603.2.a.h.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [603,2,Mod(1,603)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(603, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("603.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 603 = 3^{2} \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	603.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:	$4.81497924188$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 67)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	603.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.61803 q^{2} +4.85410 q^{4} +3.00000 q^{5} -3.85410 q^{7} +7.47214 q^{8} +O(q^{10})$	\(q+2.61803 q^{2} +4.85410 q^{4} +3.00000 q^{5} -3.85410 q^{7} +7.47214 q^{8} +7.85410 q^{10} -2.23607 q^{11} -0.145898 q^{13} -10.0902 q^{14} +9.85410 q^{16} +0.763932 q^{17} -2.85410 q^{19} +14.5623 q^{20} -5.85410 q^{22} -7.47214 q^{23} +4.00000 q^{25} -0.381966 q^{26} -18.7082 q^{28} +7.47214 q^{29} -1.00000 q^{31} +10.8541 q^{32} +2.00000 q^{34} -11.5623 q^{35} -3.85410 q^{37} -7.47214 q^{38} +22.4164 q^{40} +0.381966 q^{41} +4.85410 q^{43} -10.8541 q^{44} -19.5623 q^{46} +8.61803 q^{47} +7.85410 q^{49} +10.4721 q^{50} -0.708204 q^{52} +9.00000 q^{53} -6.70820 q^{55} -28.7984 q^{56} +19.5623 q^{58} -6.00000 q^{59} -13.5623 q^{61} -2.61803 q^{62} +8.70820 q^{64} -0.437694 q^{65} -1.00000 q^{67} +3.70820 q^{68} -30.2705 q^{70} -3.76393 q^{71} -4.00000 q^{73} -10.0902 q^{74} -13.8541 q^{76} +8.61803 q^{77} +6.56231 q^{79} +29.5623 q^{80} +1.00000 q^{82} +15.3262 q^{83} +2.29180 q^{85} +12.7082 q^{86} -16.7082 q^{88} +2.23607 q^{89} +0.562306 q^{91} -36.2705 q^{92} +22.5623 q^{94} -8.56231 q^{95} +14.4164 q^{97} +20.5623 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{2} + 3 q^{4} + 6 q^{5} - q^{7} + 6 q^{8}+O(q^{10}) $ 2 * q + 3 * q^2 + 3 * q^4 + 6 * q^5 - q^7 + 6 * q^8	$ 2 q + 3 q^{2} + 3 q^{4} + 6 q^{5} - q^{7} + 6 q^{8} + 9 q^{10} - 7 q^{13} - 9 q^{14} + 13 q^{16} + 6 q^{17} + q^{19} + 9 q^{20} - 5 q^{22} - 6 q^{23} + 8 q^{25} - 3 q^{26} - 24 q^{28} + 6 q^{29} - 2 q^{31} + 15 q^{32} + 4 q^{34} - 3 q^{35} - q^{37} - 6 q^{38} + 18 q^{40} + 3 q^{41} + 3 q^{43} - 15 q^{44} - 19 q^{46} + 15 q^{47} + 9 q^{49} + 12 q^{50} + 12 q^{52} + 18 q^{53} - 33 q^{56} + 19 q^{58} - 12 q^{59} - 7 q^{61} - 3 q^{62} + 4 q^{64} - 21 q^{65} - 2 q^{67} - 6 q^{68} - 27 q^{70} - 12 q^{71} - 8 q^{73} - 9 q^{74} - 21 q^{76} + 15 q^{77} - 7 q^{79} + 39 q^{80} + 2 q^{82} + 15 q^{83} + 18 q^{85} + 12 q^{86} - 20 q^{88} - 19 q^{91} - 39 q^{92} + 25 q^{94} + 3 q^{95} + 2 q^{97} + 21 q^{98}+O(q^{100}) $ 2 * q + 3 * q^2 + 3 * q^4 + 6 * q^5 - q^7 + 6 * q^8 + 9 * q^10 - 7 * q^13 - 9 * q^14 + 13 * q^16 + 6 * q^17 + q^19 + 9 * q^20 - 5 * q^22 - 6 * q^23 + 8 * q^25 - 3 * q^26 - 24 * q^28 + 6 * q^29 - 2 * q^31 + 15 * q^32 + 4 * q^34 - 3 * q^35 - q^37 - 6 * q^38 + 18 * q^40 + 3 * q^41 + 3 * q^43 - 15 * q^44 - 19 * q^46 + 15 * q^47 + 9 * q^49 + 12 * q^50 + 12 * q^52 + 18 * q^53 - 33 * q^56 + 19 * q^58 - 12 * q^59 - 7 * q^61 - 3 * q^62 + 4 * q^64 - 21 * q^65 - 2 * q^67 - 6 * q^68 - 27 * q^70 - 12 * q^71 - 8 * q^73 - 9 * q^74 - 21 * q^76 + 15 * q^77 - 7 * q^79 + 39 * q^80 + 2 * q^82 + 15 * q^83 + 18 * q^85 + 12 * q^86 - 20 * q^88 - 19 * q^91 - 39 * q^92 + 25 * q^94 + 3 * q^95 + 2 * q^97 + 21 * 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 $

$2$	2.61803	1.85123	0.925615	−	0.378467i	$-0.123549\pi$
$2$	2.61803	1.85123	0.925615	+	0.378467i	$0.123549\pi$
$3$	0	0
$3$	0	0
$4$	4.85410	2.42705
$5$	3.00000	1.34164	0.670820	−	0.741620i	$-0.265942\pi$
$5$	3.00000	1.34164	0.670820	+	0.741620i	$0.265942\pi$
$6$	0	0
$7$	−3.85410	−1.45671	−0.728357	−	0.685198i	$-0.759716\pi$
$7$	−3.85410	−1.45671	−0.728357	+	0.685198i	$0.759716\pi$
$8$	7.47214	2.64180
$9$	0	0
$10$	7.85410	2.48369
$11$	−2.23607	−0.674200	−0.337100	−	0.941469i	$-0.609446\pi$
$11$	−2.23607	−0.674200	−0.337100	+	0.941469i	$0.609446\pi$
$12$	0	0
$13$	−0.145898	−0.0404648	−0.0202324	−	0.999795i	$-0.506441\pi$
$13$	−0.145898	−0.0404648	−0.0202324	+	0.999795i	$0.506441\pi$
$14$	−10.0902	−2.69671
$15$	0	0
$16$	9.85410	2.46353
$17$	0.763932	0.185281	0.0926404	−	0.995700i	$-0.470469\pi$
$17$	0.763932	0.185281	0.0926404	+	0.995700i	$0.470469\pi$
$18$	0	0
$19$	−2.85410	−0.654776	−0.327388	−	0.944890i	$-0.606168\pi$
$19$	−2.85410	−0.654776	−0.327388	+	0.944890i	$0.606168\pi$
$20$	14.5623	3.25623
$21$	0	0
$22$	−5.85410	−1.24810
$23$	−7.47214	−1.55805	−0.779024	−	0.626994i	$-0.784285\pi$
$23$	−7.47214	−1.55805	−0.779024	+	0.626994i	$0.784285\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	−0.381966	−0.0749097
$27$	0	0
$28$	−18.7082	−3.53552
$29$	7.47214	1.38754	0.693770	−	0.720196i	$-0.255948\pi$
$29$	7.47214	1.38754	0.693770	+	0.720196i	$0.255948\pi$
$30$	0	0
$31$	−1.00000	−0.179605	−0.0898027	−	0.995960i	$-0.528624\pi$
$31$	−1.00000	−0.179605	−0.0898027	+	0.995960i	$0.528624\pi$
$32$	10.8541	1.91875
$33$	0	0
$34$	2.00000	0.342997
$35$	−11.5623	−1.95439
$36$	0	0
$37$	−3.85410	−0.633610	−0.316805	−	0.948491i	$-0.602610\pi$
$37$	−3.85410	−0.633610	−0.316805	+	0.948491i	$0.602610\pi$
$38$	−7.47214	−1.21214
$39$	0	0
$40$	22.4164	3.54435
$41$	0.381966	0.0596531	0.0298265	−	0.999555i	$-0.490505\pi$
$41$	0.381966	0.0596531	0.0298265	+	0.999555i	$0.490505\pi$
$42$	0	0
$43$	4.85410	0.740244	0.370122	−	0.928983i	$-0.379316\pi$
$43$	4.85410	0.740244	0.370122	+	0.928983i	$0.379316\pi$
$44$	−10.8541	−1.63632
$45$	0	0
$46$	−19.5623	−2.88430
$47$	8.61803	1.25707	0.628535	−	0.777782i	$-0.283655\pi$
$47$	8.61803	1.25707	0.628535	+	0.777782i	$0.283655\pi$
$48$	0	0
$49$	7.85410	1.12201
$50$	10.4721	1.48098
$51$	0	0
$52$	−0.708204	−0.0982102
$53$	9.00000	1.23625	0.618123	−	0.786082i	$-0.287894\pi$
$53$	9.00000	1.23625	0.618123	+	0.786082i	$0.287894\pi$
$54$	0	0
$55$	−6.70820	−0.904534
$56$	−28.7984	−3.84834
$57$	0	0
$58$	19.5623	2.56866
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	0	0
$61$	−13.5623	−1.73648	−0.868238	−	0.496149i	$-0.834747\pi$
$61$	−13.5623	−1.73648	−0.868238	+	0.496149i	$0.834747\pi$
$62$	−2.61803	−0.332491
$63$	0	0
$64$	8.70820	1.08853
$65$	−0.437694	−0.0542893
$66$	0	0
$67$	−1.00000	−0.122169
$67$	−1.00000	−0.122169
$68$	3.70820	0.449686
$69$	0	0
$70$	−30.2705	−3.61802
$71$	−3.76393	−0.446697	−0.223348	−	0.974739i	$-0.571699\pi$
$71$	−3.76393	−0.446697	−0.223348	+	0.974739i	$0.571699\pi$
$72$	0	0
$73$	−4.00000	−0.468165	−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	−10.0902	−1.17296
$75$	0	0
$76$	−13.8541	−1.58917
$77$	8.61803	0.982116
$78$	0	0
$79$	6.56231	0.738317	0.369159	−	0.929366i	$-0.379646\pi$
$79$	6.56231	0.738317	0.369159	+	0.929366i	$0.379646\pi$
$80$	29.5623	3.30517
$81$	0	0
$82$	1.00000	0.110432
$83$	15.3262	1.68227	0.841137	−	0.540823i	$-0.181887\pi$
$83$	15.3262	1.68227	0.841137	+	0.540823i	$0.181887\pi$
$84$	0	0
$85$	2.29180	0.248580
$86$	12.7082	1.37036
$87$	0	0
$88$	−16.7082	−1.78110
$89$	2.23607	0.237023	0.118511	−	0.992953i	$-0.462188\pi$
$89$	2.23607	0.237023	0.118511	+	0.992953i	$0.462188\pi$
$90$	0	0
$91$	0.562306	0.0589457
$92$	−36.2705	−3.78146
$93$	0	0
$94$	22.5623	2.32712
$95$	−8.56231	−0.878474
$96$	0	0
$97$	14.4164	1.46376	0.731882	−	0.681431i	$-0.238642\pi$
$97$	14.4164	1.46376	0.731882	+	0.681431i	$0.238642\pi$
$98$	20.5623	2.07711
$99$	0	0
$100$	19.4164	1.94164
$101$	10.0902	1.00401	0.502005	−	0.864865i	$-0.332596\pi$
$101$	10.0902	1.00401	0.502005	+	0.864865i	$0.332596\pi$
$102$	0	0
$103$	7.56231	0.745136	0.372568	−	0.928005i	$-0.378477\pi$
$103$	7.56231	0.745136	0.372568	+	0.928005i	$0.378477\pi$
$104$	−1.09017	−0.106900
$105$	0	0
$106$	23.5623	2.28857
$107$	−5.29180	−0.511577	−0.255789	−	0.966733i	$-0.582335\pi$
$107$	−5.29180	−0.511577	−0.255789	+	0.966733i	$0.582335\pi$
$108$	0	0
$109$	−4.85410	−0.464939	−0.232469	−	0.972604i	$-0.574681\pi$
$109$	−4.85410	−0.464939	−0.232469	+	0.972604i	$0.574681\pi$
$110$	−17.5623	−1.67450
$111$	0	0
$112$	−37.9787	−3.58865
$113$	−15.3820	−1.44701	−0.723507	−	0.690317i	$-0.757471\pi$
$113$	−15.3820	−1.44701	−0.723507	+	0.690317i	$0.757471\pi$
$114$	0	0
$115$	−22.4164	−2.09034
$116$	36.2705	3.36763
$117$	0	0
$118$	−15.7082	−1.44606
$119$	−2.94427	−0.269901
$120$	0	0
$121$	−6.00000	−0.545455
$122$	−35.5066	−3.21461
$123$	0	0
$124$	−4.85410	−0.435911
$125$	−3.00000	−0.268328
$126$	0	0
$127$	−5.85410	−0.519468	−0.259734	−	0.965680i	$-0.583635\pi$
$127$	−5.85410	−0.519468	−0.259734	+	0.965680i	$0.583635\pi$
$128$	1.09017	0.0963583
$129$	0	0
$130$	−1.14590	−0.100502
$131$	3.00000	0.262111	0.131056	−	0.991375i	$-0.458163\pi$
$131$	3.00000	0.262111	0.131056	+	0.991375i	$0.458163\pi$
$132$	0	0
$133$	11.0000	0.953821
$134$	−2.61803	−0.226164
$135$	0	0
$136$	5.70820	0.489474
$137$	−0.708204	−0.0605059	−0.0302530	−	0.999542i	$-0.509631\pi$
$137$	−0.708204	−0.0605059	−0.0302530	+	0.999542i	$0.509631\pi$
$138$	0	0
$139$	−3.00000	−0.254457	−0.127228	−	0.991873i	$-0.540608\pi$
$139$	−3.00000	−0.254457	−0.127228	+	0.991873i	$0.540608\pi$
$140$	−56.1246	−4.74340
$141$	0	0
$142$	−9.85410	−0.826938
$143$	0.326238	0.0272814
$144$	0	0
$145$	22.4164	1.86158
$146$	−10.4721	−0.866680
$147$	0	0
$148$	−18.7082	−1.53780
$149$	−19.0344	−1.55936	−0.779681	−	0.626177i	$-0.784619\pi$
$149$	−19.0344	−1.55936	−0.779681	+	0.626177i	$0.784619\pi$
$150$	0	0
$151$	−1.00000	−0.0813788	−0.0406894	−	0.999172i	$-0.512955\pi$
$151$	−1.00000	−0.0813788	−0.0406894	+	0.999172i	$0.512955\pi$
$152$	−21.3262	−1.72979
$153$	0	0
$154$	22.5623	1.81812
$155$	−3.00000	−0.240966
$156$	0	0
$157$	−3.56231	−0.284303	−0.142151	−	0.989845i	$-0.545402\pi$
$157$	−3.56231	−0.284303	−0.142151	+	0.989845i	$0.545402\pi$
$158$	17.1803	1.36679
$159$	0	0
$160$	32.5623	2.57428
$161$	28.7984	2.26963
$162$	0	0
$163$	−6.85410	−0.536855	−0.268427	−	0.963300i	$-0.586504\pi$
$163$	−6.85410	−0.536855	−0.268427	+	0.963300i	$0.586504\pi$
$164$	1.85410	0.144781
$165$	0	0
$166$	40.1246	3.11427
$167$	11.8885	0.919963	0.459982	−	0.887928i	$-0.347856\pi$
$167$	11.8885	0.919963	0.459982	+	0.887928i	$0.347856\pi$
$168$	0	0
$169$	−12.9787	−0.998363
$170$	6.00000	0.460179
$171$	0	0
$172$	23.5623	1.79661
$173$	13.9098	1.05754	0.528772	−	0.848764i	$-0.322653\pi$
$173$	13.9098	1.05754	0.528772	+	0.848764i	$0.322653\pi$
$174$	0	0
$175$	−15.4164	−1.16537
$176$	−22.0344	−1.66091
$177$	0	0
$178$	5.85410	0.438783
$179$	6.76393	0.505560	0.252780	−	0.967524i	$-0.418655\pi$
$179$	6.76393	0.505560	0.252780	+	0.967524i	$0.418655\pi$
$180$	0	0
$181$	6.56231	0.487772	0.243886	−	0.969804i	$-0.421578\pi$
$181$	6.56231	0.487772	0.243886	+	0.969804i	$0.421578\pi$
$182$	1.47214	0.109122
$183$	0	0
$184$	−55.8328	−4.11605
$185$	−11.5623	−0.850078
$186$	0	0
$187$	−1.70820	−0.124916
$188$	41.8328	3.05097
$189$	0	0
$190$	−22.4164	−1.62626
$191$	−1.47214	−0.106520	−0.0532600	−	0.998581i	$-0.516961\pi$
$191$	−1.47214	−0.106520	−0.0532600	+	0.998581i	$0.516961\pi$
$192$	0	0
$193$	16.8541	1.21318	0.606592	−	0.795013i	$-0.292536\pi$
$193$	16.8541	1.21318	0.606592	+	0.795013i	$0.292536\pi$
$194$	37.7426	2.70976
$195$	0	0
$196$	38.1246	2.72319
$197$	4.52786	0.322597	0.161298	−	0.986906i	$-0.448432\pi$
$197$	4.52786	0.322597	0.161298	+	0.986906i	$0.448432\pi$
$198$	0	0
$199$	10.2705	0.728057	0.364029	−	0.931388i	$-0.381401\pi$
$199$	10.2705	0.728057	0.364029	+	0.931388i	$0.381401\pi$
$200$	29.8885	2.11344
$201$	0	0
$202$	26.4164	1.85865
$203$	−28.7984	−2.02125
$204$	0	0
$205$	1.14590	0.0800330
$206$	19.7984	1.37942
$207$	0	0
$208$	−1.43769	−0.0996862
$209$	6.38197	0.441450
$210$	0	0
$211$	−2.14590	−0.147730	−0.0738649	−	0.997268i	$-0.523533\pi$
$211$	−2.14590	−0.147730	−0.0738649	+	0.997268i	$0.523533\pi$
$212$	43.6869	3.00043
$213$	0	0
$214$	−13.8541	−0.947047
$215$	14.5623	0.993141
$216$	0	0
$217$	3.85410	0.261633
$218$	−12.7082	−0.860708
$219$	0	0
$220$	−32.5623	−2.19535
$221$	−0.111456	−0.00749735
$222$	0	0
$223$	15.7082	1.05190	0.525950	−	0.850516i	$-0.323710\pi$
$223$	15.7082	1.05190	0.525950	+	0.850516i	$0.323710\pi$
$224$	−41.8328	−2.79507
$225$	0	0
$226$	−40.2705	−2.67875
$227$	20.5066	1.36107	0.680535	−	0.732716i	$-0.261748\pi$
$227$	20.5066	1.36107	0.680535	+	0.732716i	$0.261748\pi$
$228$	0	0
$229$	−18.8541	−1.24591	−0.622957	−	0.782256i	$-0.714069\pi$
$229$	−18.8541	−1.24591	−0.622957	+	0.782256i	$0.714069\pi$
$230$	−58.6869	−3.86970
$231$	0	0
$232$	55.8328	3.66560
$233$	−1.52786	−0.100094	−0.0500469	−	0.998747i	$-0.515937\pi$
$233$	−1.52786	−0.100094	−0.0500469	+	0.998747i	$0.515937\pi$
$234$	0	0
$235$	25.8541	1.68654
$236$	−29.1246	−1.89585
$237$	0	0
$238$	−7.70820	−0.499649
$239$	19.0902	1.23484	0.617420	−	0.786634i	$-0.288178\pi$
$239$	19.0902	1.23484	0.617420	+	0.786634i	$0.288178\pi$
$240$	0	0
$241$	25.8541	1.66541	0.832705	−	0.553718i	$-0.186791\pi$
$241$	25.8541	1.66541	0.832705	+	0.553718i	$0.186791\pi$
$242$	−15.7082	−1.00976
$243$	0	0
$244$	−65.8328	−4.21451
$245$	23.5623	1.50534
$246$	0	0
$247$	0.416408	0.0264954
$248$	−7.47214	−0.474481
$249$	0	0
$250$	−7.85410	−0.496737
$251$	−11.2361	−0.709214	−0.354607	−	0.935015i	$-0.615385\pi$
$251$	−11.2361	−0.709214	−0.354607	+	0.935015i	$0.615385\pi$
$252$	0	0
$253$	16.7082	1.05044
$254$	−15.3262	−0.961654
$255$	0	0
$256$	−14.5623	−0.910144
$257$	26.8885	1.67726	0.838631	−	0.544701i	$-0.183357\pi$
$257$	26.8885	1.67726	0.838631	+	0.544701i	$0.183357\pi$
$258$	0	0
$259$	14.8541	0.922989
$260$	−2.12461	−0.131763
$261$	0	0
$262$	7.85410	0.485228
$263$	−8.56231	−0.527974	−0.263987	−	0.964526i	$-0.585038\pi$
$263$	−8.56231	−0.527974	−0.263987	+	0.964526i	$0.585038\pi$
$264$	0	0
$265$	27.0000	1.65860
$266$	28.7984	1.76574
$267$	0	0
$268$	−4.85410	−0.296511
$269$	−16.7984	−1.02421	−0.512107	−	0.858921i	$-0.671135\pi$
$269$	−16.7984	−1.02421	−0.512107	+	0.858921i	$0.671135\pi$
$270$	0	0
$271$	24.7082	1.50092	0.750458	−	0.660918i	$-0.229833\pi$
$271$	24.7082	1.50092	0.750458	+	0.660918i	$0.229833\pi$
$272$	7.52786	0.456444
$273$	0	0
$274$	−1.85410	−0.112010
$275$	−8.94427	−0.539360
$276$	0	0
$277$	32.8328	1.97273	0.986366	−	0.164564i	$-0.0526218\pi$
$277$	32.8328	1.97273	0.986366	+	0.164564i	$0.0526218\pi$
$278$	−7.85410	−0.471058
$279$	0	0
$280$	−86.3951	−5.16310
$281$	−8.61803	−0.514109	−0.257054	−	0.966397i	$-0.582752\pi$
$281$	−8.61803	−0.514109	−0.257054	+	0.966397i	$0.582752\pi$
$282$	0	0
$283$	−22.4164	−1.33252	−0.666259	−	0.745721i	$-0.732105\pi$
$283$	−22.4164	−1.33252	−0.666259	+	0.745721i	$0.732105\pi$
$284$	−18.2705	−1.08416
$285$	0	0
$286$	0.854102	0.0505041
$287$	−1.47214	−0.0868974
$288$	0	0
$289$	−16.4164	−0.965671
$290$	58.6869	3.44621
$291$	0	0
$292$	−19.4164	−1.13626
$293$	−4.41641	−0.258009	−0.129005	−	0.991644i	$-0.541178\pi$
$293$	−4.41641	−0.258009	−0.129005	+	0.991644i	$0.541178\pi$
$294$	0	0
$295$	−18.0000	−1.04800
$296$	−28.7984	−1.67387
$297$	0	0
$298$	−49.8328	−2.88674
$299$	1.09017	0.0630462
$300$	0	0
$301$	−18.7082	−1.07832
$302$	−2.61803	−0.150651
$303$	0	0
$304$	−28.1246	−1.61306
$305$	−40.6869	−2.32973
$306$	0	0
$307$	17.4164	0.994007	0.497003	−	0.867749i	$-0.334434\pi$
$307$	17.4164	0.994007	0.497003	+	0.867749i	$0.334434\pi$
$308$	41.8328	2.38365
$309$	0	0
$310$	−7.85410	−0.446083
$311$	26.8328	1.52155	0.760775	−	0.649016i	$-0.224819\pi$
$311$	26.8328	1.52155	0.760775	+	0.649016i	$0.224819\pi$
$312$	0	0
$313$	−13.8541	−0.783080	−0.391540	−	0.920161i	$-0.628058\pi$
$313$	−13.8541	−0.783080	−0.391540	+	0.920161i	$0.628058\pi$
$314$	−9.32624	−0.526310
$315$	0	0
$316$	31.8541	1.79193
$317$	−18.5967	−1.04450	−0.522249	−	0.852793i	$-0.674907\pi$
$317$	−18.5967	−1.04450	−0.522249	+	0.852793i	$0.674907\pi$
$318$	0	0
$319$	−16.7082	−0.935480
$320$	26.1246	1.46041
$321$	0	0
$322$	75.3951	4.20161
$323$	−2.18034	−0.121317
$324$	0	0
$325$	−0.583592	−0.0323719
$326$	−17.9443	−0.993841
$327$	0	0
$328$	2.85410	0.157591
$329$	−33.2148	−1.83119
$330$	0	0
$331$	−3.41641	−0.187783	−0.0938914	−	0.995582i	$-0.529931\pi$
$331$	−3.41641	−0.187783	−0.0938914	+	0.995582i	$0.529931\pi$
$332$	74.3951	4.08296
$333$	0	0
$334$	31.1246	1.70306
$335$	−3.00000	−0.163908
$336$	0	0
$337$	−8.27051	−0.450523	−0.225262	−	0.974298i	$-0.572324\pi$
$337$	−8.27051	−0.450523	−0.225262	+	0.974298i	$0.572324\pi$
$338$	−33.9787	−1.84820
$339$	0	0
$340$	11.1246	0.603317
$341$	2.23607	0.121090
$342$	0	0
$343$	−3.29180	−0.177740
$344$	36.2705	1.95557
$345$	0	0
$346$	36.4164	1.95776
$347$	11.1803	0.600192	0.300096	−	0.953909i	$-0.402981\pi$
$347$	11.1803	0.600192	0.300096	+	0.953909i	$0.402981\pi$
$348$	0	0
$349$	−6.00000	−0.321173	−0.160586	−	0.987022i	$-0.551338\pi$
$349$	−6.00000	−0.321173	−0.160586	+	0.987022i	$0.551338\pi$
$350$	−40.3607	−2.15737
$351$	0	0
$352$	−24.2705	−1.29362
$353$	17.5066	0.931781	0.465891	−	0.884842i	$-0.345734\pi$
$353$	17.5066	0.931781	0.465891	+	0.884842i	$0.345734\pi$
$354$	0	0
$355$	−11.2918	−0.599306
$356$	10.8541	0.575266
$357$	0	0
$358$	17.7082	0.935908
$359$	18.0000	0.950004	0.475002	−	0.879985i	$-0.342447\pi$
$359$	18.0000	0.950004	0.475002	+	0.879985i	$0.342447\pi$
$360$	0	0
$361$	−10.8541	−0.571269
$362$	17.1803	0.902979
$363$	0	0
$364$	2.72949	0.143064
$365$	−12.0000	−0.628109
$366$	0	0
$367$	−19.1246	−0.998297	−0.499148	−	0.866517i	$-0.666354\pi$
$367$	−19.1246	−0.998297	−0.499148	+	0.866517i	$0.666354\pi$
$368$	−73.6312	−3.83829
$369$	0	0
$370$	−30.2705	−1.57369
$371$	−34.6869	−1.80086
$372$	0	0
$373$	−18.5623	−0.961120	−0.480560	−	0.876962i	$-0.659567\pi$
$373$	−18.5623	−0.961120	−0.480560	+	0.876962i	$0.659567\pi$
$374$	−4.47214	−0.231249
$375$	0	0
$376$	64.3951	3.32092
$377$	−1.09017	−0.0561466
$378$	0	0
$379$	4.00000	0.205466	0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000	0.205466	0.102733	+	0.994709i	$0.467241\pi$
$380$	−41.5623	−2.13210
$381$	0	0
$382$	−3.85410	−0.197193
$383$	7.20163	0.367986	0.183993	−	0.982928i	$-0.441098\pi$
$383$	7.20163	0.367986	0.183993	+	0.982928i	$0.441098\pi$
$384$	0	0
$385$	25.8541	1.31765
$386$	44.1246	2.24588
$387$	0	0
$388$	69.9787	3.55263
$389$	18.6525	0.945718	0.472859	−	0.881138i	$-0.343222\pi$
$389$	18.6525	0.945718	0.472859	+	0.881138i	$0.343222\pi$
$390$	0	0
$391$	−5.70820	−0.288676
$392$	58.6869	2.96414
$393$	0	0
$394$	11.8541	0.597201
$395$	19.6869	0.990556
$396$	0	0
$397$	−35.5623	−1.78482	−0.892410	−	0.451224i	$-0.850987\pi$
$397$	−35.5623	−1.78482	−0.892410	+	0.451224i	$0.850987\pi$
$398$	26.8885	1.34780
$399$	0	0
$400$	39.4164	1.97082
$401$	−7.85410	−0.392215	−0.196108	−	0.980582i	$-0.562830\pi$
$401$	−7.85410	−0.392215	−0.196108	+	0.980582i	$0.562830\pi$
$402$	0	0
$403$	0.145898	0.00726770
$404$	48.9787	2.43678
$405$	0	0
$406$	−75.3951	−3.74180
$407$	8.61803	0.427180
$408$	0	0
$409$	13.1246	0.648970	0.324485	−	0.945891i	$-0.394809\pi$
$409$	13.1246	0.648970	0.324485	+	0.945891i	$0.394809\pi$
$410$	3.00000	0.148159
$411$	0	0
$412$	36.7082	1.80848
$413$	23.1246	1.13789
$414$	0	0
$415$	45.9787	2.25701
$416$	−1.58359	−0.0776420
$417$	0	0
$418$	16.7082	0.817225
$419$	−22.3607	−1.09239	−0.546195	−	0.837658i	$-0.683924\pi$
$419$	−22.3607	−1.09239	−0.546195	+	0.837658i	$0.683924\pi$
$420$	0	0
$421$	−30.8541	−1.50374	−0.751868	−	0.659313i	$-0.770847\pi$
$421$	−30.8541	−1.50374	−0.751868	+	0.659313i	$0.770847\pi$
$422$	−5.61803	−0.273482
$423$	0	0
$424$	67.2492	3.26591
$425$	3.05573	0.148225
$426$	0	0
$427$	52.2705	2.52955
$428$	−25.6869	−1.24162
$429$	0	0
$430$	38.1246	1.83853
$431$	−27.7639	−1.33734	−0.668671	−	0.743559i	$-0.733136\pi$
$431$	−27.7639	−1.33734	−0.668671	+	0.743559i	$0.733136\pi$
$432$	0	0
$433$	−40.3951	−1.94127	−0.970633	−	0.240566i	$-0.922667\pi$
$433$	−40.3951	−1.94127	−0.970633	+	0.240566i	$0.922667\pi$
$434$	10.0902	0.484344
$435$	0	0
$436$	−23.5623	−1.12843
$437$	21.3262	1.02017
$438$	0	0
$439$	−38.6869	−1.84643	−0.923213	−	0.384289i	$-0.874447\pi$
$439$	−38.6869	−1.84643	−0.923213	+	0.384289i	$0.874447\pi$
$440$	−50.1246	−2.38960
$441$	0	0
$442$	−0.291796	−0.0138793
$443$	10.5279	0.500194	0.250097	−	0.968221i	$-0.419538\pi$
$443$	10.5279	0.500194	0.250097	+	0.968221i	$0.419538\pi$
$444$	0	0
$445$	6.70820	0.317999
$446$	41.1246	1.94731
$447$	0	0
$448$	−33.5623	−1.58567
$449$	−8.67376	−0.409340	−0.204670	−	0.978831i	$-0.565612\pi$
$449$	−8.67376	−0.409340	−0.204670	+	0.978831i	$0.565612\pi$
$450$	0	0
$451$	−0.854102	−0.0402181
$452$	−74.6656	−3.51198
$453$	0	0
$454$	53.6869	2.51965
$455$	1.68692	0.0790839
$456$	0	0
$457$	−9.58359	−0.448302	−0.224151	−	0.974554i	$-0.571961\pi$
$457$	−9.58359	−0.448302	−0.224151	+	0.974554i	$0.571961\pi$
$458$	−49.3607	−2.30647
$459$	0	0
$460$	−108.812	−5.07336
$461$	13.8541	0.645250	0.322625	−	0.946527i	$-0.395435\pi$
$461$	13.8541	0.645250	0.322625	+	0.946527i	$0.395435\pi$
$462$	0	0
$463$	−27.5623	−1.28093	−0.640464	−	0.767988i	$-0.721258\pi$
$463$	−27.5623	−1.28093	−0.640464	+	0.767988i	$0.721258\pi$
$464$	73.6312	3.41824
$465$	0	0
$466$	−4.00000	−0.185296
$467$	−23.1803	−1.07266	−0.536329	−	0.844009i	$-0.680189\pi$
$467$	−23.1803	−1.07266	−0.536329	+	0.844009i	$0.680189\pi$
$468$	0	0
$469$	3.85410	0.177966
$470$	67.6869	3.12216
$471$	0	0
$472$	−44.8328	−2.06360
$473$	−10.8541	−0.499072
$474$	0	0
$475$	−11.4164	−0.523821
$476$	−14.2918	−0.655063
$477$	0	0
$478$	49.9787	2.28597
$479$	−30.6525	−1.40055	−0.700274	−	0.713874i	$-0.746939\pi$
$479$	−30.6525	−1.40055	−0.700274	+	0.713874i	$0.746939\pi$
$480$	0	0
$481$	0.562306	0.0256389
$482$	67.6869	3.08305
$483$	0	0
$484$	−29.1246	−1.32385
$485$	43.2492	1.96385
$486$	0	0
$487$	39.8328	1.80500	0.902499	−	0.430693i	$-0.141731\pi$
$487$	39.8328	1.80500	0.902499	+	0.430693i	$0.141731\pi$
$488$	−101.339	−4.58742
$489$	0	0
$490$	61.6869	2.78673
$491$	−28.4164	−1.28241	−0.641207	−	0.767368i	$-0.721566\pi$
$491$	−28.4164	−1.28241	−0.641207	+	0.767368i	$0.721566\pi$
$492$	0	0
$493$	5.70820	0.257085
$494$	1.09017	0.0490491
$495$	0	0
$496$	−9.85410	−0.442462
$497$	14.5066	0.650709
$498$	0	0
$499$	−25.0000	−1.11915	−0.559577	−	0.828778i	$-0.689036\pi$
$499$	−25.0000	−1.11915	−0.559577	+	0.828778i	$0.689036\pi$
$500$	−14.5623	−0.651246
$501$	0	0
$502$	−29.4164	−1.31292
$503$	−41.9443	−1.87020	−0.935101	−	0.354380i	$-0.884692\pi$
$503$	−41.9443	−1.87020	−0.935101	+	0.354380i	$0.884692\pi$
$504$	0	0
$505$	30.2705	1.34702
$506$	43.7426	1.94460
$507$	0	0
$508$	−28.4164	−1.26077
$509$	−7.41641	−0.328726	−0.164363	−	0.986400i	$-0.552557\pi$
$509$	−7.41641	−0.328726	−0.164363	+	0.986400i	$0.552557\pi$
$510$	0	0
$511$	15.4164	0.681982
$512$	−40.3050	−1.78124
$513$	0	0
$514$	70.3951	3.10500
$515$	22.6869	0.999705
$516$	0	0
$517$	−19.2705	−0.847516
$518$	38.8885	1.70866
$519$	0	0
$520$	−3.27051	−0.143421
$521$	18.0000	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$522$	0	0
$523$	−1.56231	−0.0683149	−0.0341574	−	0.999416i	$-0.510875\pi$
$523$	−1.56231	−0.0683149	−0.0341574	+	0.999416i	$0.510875\pi$
$524$	14.5623	0.636157
$525$	0	0
$526$	−22.4164	−0.977402
$527$	−0.763932	−0.0332774
$528$	0	0
$529$	32.8328	1.42751
$530$	70.6869	3.07044
$531$	0	0
$532$	53.3951	2.31497
$533$	−0.0557281	−0.00241385
$534$	0	0
$535$	−15.8754	−0.686353
$536$	−7.47214	−0.322747
$537$	0	0
$538$	−43.9787	−1.89606
$539$	−17.5623	−0.756462
$540$	0	0
$541$	31.4164	1.35070	0.675348	−	0.737499i	$-0.263993\pi$
$541$	31.4164	1.35070	0.675348	+	0.737499i	$0.263993\pi$
$542$	64.6869	2.77854
$543$	0	0
$544$	8.29180	0.355508
$545$	−14.5623	−0.623781
$546$	0	0
$547$	5.43769	0.232499	0.116250	−	0.993220i	$-0.462913\pi$
$547$	5.43769	0.232499	0.116250	+	0.993220i	$0.462913\pi$
$548$	−3.43769	−0.146851
$549$	0	0
$550$	−23.4164	−0.998479
$551$	−21.3262	−0.908528
$552$	0	0
$553$	−25.2918	−1.07552
$554$	85.9574	3.65198
$555$	0	0
$556$	−14.5623	−0.617579
$557$	30.3820	1.28733	0.643663	−	0.765309i	$-0.277414\pi$
$557$	30.3820	1.28733	0.643663	+	0.765309i	$0.277414\pi$
$558$	0	0
$559$	−0.708204	−0.0299538
$560$	−113.936	−4.81468
$561$	0	0
$562$	−22.5623	−0.951733
$563$	9.70820	0.409152	0.204576	−	0.978851i	$-0.434418\pi$
$563$	9.70820	0.409152	0.204576	+	0.978851i	$0.434418\pi$
$564$	0	0
$565$	−46.1459	−1.94137
$566$	−58.6869	−2.46680
$567$	0	0
$568$	−28.1246	−1.18008
$569$	−20.1803	−0.846004	−0.423002	−	0.906129i	$-0.639024\pi$
$569$	−20.1803	−0.846004	−0.423002	+	0.906129i	$0.639024\pi$
$570$	0	0
$571$	25.9787	1.08718	0.543588	−	0.839352i	$-0.317066\pi$
$571$	25.9787	1.08718	0.543588	+	0.839352i	$0.317066\pi$
$572$	1.58359	0.0662133
$573$	0	0
$574$	−3.85410	−0.160867
$575$	−29.8885	−1.24644
$576$	0	0
$577$	9.29180	0.386823	0.193411	−	0.981118i	$-0.438045\pi$
$577$	9.29180	0.386823	0.193411	+	0.981118i	$0.438045\pi$
$578$	−42.9787	−1.78768
$579$	0	0
$580$	108.812	4.51815
$581$	−59.0689	−2.45059
$582$	0	0
$583$	−20.1246	−0.833476
$584$	−29.8885	−1.23680
$585$	0	0
$586$	−11.5623	−0.477634
$587$	−22.9098	−0.945590	−0.472795	−	0.881172i	$-0.656755\pi$
$587$	−22.9098	−0.945590	−0.472795	+	0.881172i	$0.656755\pi$
$588$	0	0
$589$	2.85410	0.117601
$590$	−47.1246	−1.94009
$591$	0	0
$592$	−37.9787	−1.56092
$593$	34.4721	1.41560	0.707800	−	0.706412i	$-0.249688\pi$
$593$	34.4721	1.41560	0.707800	+	0.706412i	$0.249688\pi$
$594$	0	0
$595$	−8.83282	−0.362110
$596$	−92.3951	−3.78465
$597$	0	0
$598$	2.85410	0.116713
$599$	−10.4721	−0.427880	−0.213940	−	0.976847i	$-0.568630\pi$
$599$	−10.4721	−0.427880	−0.213940	+	0.976847i	$0.568630\pi$
$600$	0	0
$601$	−7.12461	−0.290619	−0.145309	−	0.989386i	$-0.546418\pi$
$601$	−7.12461	−0.290619	−0.145309	+	0.989386i	$0.546418\pi$
$602$	−48.9787	−1.99622
$603$	0	0
$604$	−4.85410	−0.197511
$605$	−18.0000	−0.731804
$606$	0	0
$607$	23.4377	0.951307	0.475653	−	0.879633i	$-0.342212\pi$
$607$	23.4377	0.951307	0.475653	+	0.879633i	$0.342212\pi$
$608$	−30.9787	−1.25635
$609$	0	0
$610$	−106.520	−4.31286
$611$	−1.25735	−0.0508671
$612$	0	0
$613$	−14.4164	−0.582273	−0.291137	−	0.956681i	$-0.594033\pi$
$613$	−14.4164	−0.582273	−0.291137	+	0.956681i	$0.594033\pi$
$614$	45.5967	1.84013
$615$	0	0
$616$	64.3951	2.59455
$617$	−0.978714	−0.0394015	−0.0197008	−	0.999806i	$-0.506271\pi$
$617$	−0.978714	−0.0394015	−0.0197008	+	0.999806i	$0.506271\pi$
$618$	0	0
$619$	−34.2918	−1.37830	−0.689152	−	0.724617i	$-0.742017\pi$
$619$	−34.2918	−1.37830	−0.689152	+	0.724617i	$0.742017\pi$
$620$	−14.5623	−0.584836
$621$	0	0
$622$	70.2492	2.81674
$623$	−8.61803	−0.345274
$624$	0	0
$625$	−29.0000	−1.16000
$626$	−36.2705	−1.44966
$627$	0	0
$628$	−17.2918	−0.690018
$629$	−2.94427	−0.117396
$630$	0	0
$631$	−4.43769	−0.176662	−0.0883309	−	0.996091i	$-0.528153\pi$
$631$	−4.43769	−0.176662	−0.0883309	+	0.996091i	$0.528153\pi$
$632$	49.0344	1.95049
$633$	0	0
$634$	−48.6869	−1.93360
$635$	−17.5623	−0.696939
$636$	0	0
$637$	−1.14590	−0.0454021
$638$	−43.7426	−1.73179
$639$	0	0
$640$	3.27051	0.129278
$641$	17.0689	0.674180	0.337090	−	0.941472i	$-0.390557\pi$
$641$	17.0689	0.674180	0.337090	+	0.941472i	$0.390557\pi$
$642$	0	0
$643$	−20.5623	−0.810898	−0.405449	−	0.914118i	$-0.632885\pi$
$643$	−20.5623	−0.810898	−0.405449	+	0.914118i	$0.632885\pi$
$644$	139.790	5.50851
$645$	0	0
$646$	−5.70820	−0.224586
$647$	24.7639	0.973571	0.486785	−	0.873522i	$-0.338169\pi$
$647$	24.7639	0.973571	0.486785	+	0.873522i	$0.338169\pi$
$648$	0	0
$649$	13.4164	0.526640
$650$	−1.52786	−0.0599278
$651$	0	0
$652$	−33.2705	−1.30297
$653$	−4.47214	−0.175008	−0.0875041	−	0.996164i	$-0.527889\pi$
$653$	−4.47214	−0.175008	−0.0875041	+	0.996164i	$0.527889\pi$
$654$	0	0
$655$	9.00000	0.351659
$656$	3.76393	0.146957
$657$	0	0
$658$	−86.9574	−3.38995
$659$	29.9443	1.16646	0.583232	−	0.812306i	$-0.301788\pi$
$659$	29.9443	1.16646	0.583232	+	0.812306i	$0.301788\pi$
$660$	0	0
$661$	50.1246	1.94962	0.974811	−	0.223034i	$-0.0715960\pi$
$661$	50.1246	1.94962	0.974811	+	0.223034i	$0.0715960\pi$
$662$	−8.94427	−0.347629
$663$	0	0
$664$	114.520	4.44423
$665$	33.0000	1.27969
$666$	0	0
$667$	−55.8328	−2.16186
$668$	57.7082	2.23280
$669$	0	0
$670$	−7.85410	−0.303430
$671$	30.3262	1.17073
$672$	0	0
$673$	−35.6869	−1.37563	−0.687815	−	0.725886i	$-0.741430\pi$
$673$	−35.6869	−1.37563	−0.687815	+	0.725886i	$0.741430\pi$
$674$	−21.6525	−0.834022
$675$	0	0
$676$	−63.0000	−2.42308
$677$	−11.1803	−0.429695	−0.214848	−	0.976648i	$-0.568926\pi$
$677$	−11.1803	−0.429695	−0.214848	+	0.976648i	$0.568926\pi$
$678$	0	0
$679$	−55.5623	−2.13229
$680$	17.1246	0.656699
$681$	0	0
$682$	5.85410	0.224165
$683$	−9.38197	−0.358991	−0.179495	−	0.983759i	$-0.557447\pi$
$683$	−9.38197	−0.358991	−0.179495	+	0.983759i	$0.557447\pi$
$684$	0	0
$685$	−2.12461	−0.0811772
$686$	−8.61803	−0.329038
$687$	0	0
$688$	47.8328	1.82361
$689$	−1.31308	−0.0500245
$690$	0	0
$691$	8.00000	0.304334	0.152167	−	0.988355i	$-0.451375\pi$
$691$	8.00000	0.304334	0.152167	+	0.988355i	$0.451375\pi$
$692$	67.5197	2.56672
$693$	0	0
$694$	29.2705	1.11109
$695$	−9.00000	−0.341389
$696$	0	0
$697$	0.291796	0.0110526
$698$	−15.7082	−0.594564
$699$	0	0
$700$	−74.8328	−2.82841
$701$	−40.3607	−1.52440	−0.762201	−	0.647341i	$-0.775881\pi$
$701$	−40.3607	−1.52440	−0.762201	+	0.647341i	$0.775881\pi$
$702$	0	0
$703$	11.0000	0.414873
$704$	−19.4721	−0.733884
$705$	0	0
$706$	45.8328	1.72494
$707$	−38.8885	−1.46255
$708$	0	0
$709$	−5.41641	−0.203417	−0.101709	−	0.994814i	$-0.532431\pi$
$709$	−5.41641	−0.203417	−0.101709	+	0.994814i	$0.532431\pi$
$710$	−29.5623	−1.10945
$711$	0	0
$712$	16.7082	0.626166
$713$	7.47214	0.279834
$714$	0	0
$715$	0.978714	0.0366018
$716$	32.8328	1.22702
$717$	0	0
$718$	47.1246	1.75867
$719$	−40.5279	−1.51143	−0.755717	−	0.654898i	$-0.772712\pi$
$719$	−40.5279	−1.51143	−0.755717	+	0.654898i	$0.772712\pi$
$720$	0	0
$721$	−29.1459	−1.08545
$722$	−28.4164	−1.05755
$723$	0	0
$724$	31.8541	1.18385
$725$	29.8885	1.11003
$726$	0	0
$727$	28.0000	1.03846	0.519231	−	0.854634i	$-0.326218\pi$
$727$	28.0000	1.03846	0.519231	+	0.854634i	$0.326218\pi$
$728$	4.20163	0.155723
$729$	0	0
$730$	−31.4164	−1.16277
$731$	3.70820	0.137153
$732$	0	0
$733$	4.00000	0.147743	0.0738717	−	0.997268i	$-0.476464\pi$
$733$	4.00000	0.147743	0.0738717	+	0.997268i	$0.476464\pi$
$734$	−50.0689	−1.84808
$735$	0	0
$736$	−81.1033	−2.98951
$737$	2.23607	0.0823666
$738$	0	0
$739$	−11.2705	−0.414592	−0.207296	−	0.978278i	$-0.566466\pi$
$739$	−11.2705	−0.414592	−0.207296	+	0.978278i	$0.566466\pi$
$740$	−56.1246	−2.06318
$741$	0	0
$742$	−90.8115	−3.33380
$743$	−24.5967	−0.902367	−0.451184	−	0.892431i	$-0.648998\pi$
$743$	−24.5967	−0.902367	−0.451184	+	0.892431i	$0.648998\pi$
$744$	0	0
$745$	−57.1033	−2.09210
$746$	−48.5967	−1.77925
$747$	0	0
$748$	−8.29180	−0.303178
$749$	20.3951	0.745222
$750$	0	0
$751$	44.1246	1.61013	0.805065	−	0.593187i	$-0.202130\pi$
$751$	44.1246	1.61013	0.805065	+	0.593187i	$0.202130\pi$
$752$	84.9230	3.09682
$753$	0	0
$754$	−2.85410	−0.103940
$755$	−3.00000	−0.109181
$756$	0	0
$757$	38.4164	1.39627	0.698134	−	0.715967i	$-0.254014\pi$
$757$	38.4164	1.39627	0.698134	+	0.715967i	$0.254014\pi$
$758$	10.4721	0.380365
$759$	0	0
$760$	−63.9787	−2.32075
$761$	12.0000	0.435000	0.217500	−	0.976060i	$-0.430210\pi$
$761$	12.0000	0.435000	0.217500	+	0.976060i	$0.430210\pi$
$762$	0	0
$763$	18.7082	0.677282
$764$	−7.14590	−0.258530
$765$	0	0
$766$	18.8541	0.681226
$767$	0.875388	0.0316084
$768$	0	0
$769$	−10.7082	−0.386148	−0.193074	−	0.981184i	$-0.561846\pi$
$769$	−10.7082	−0.386148	−0.193074	+	0.981184i	$0.561846\pi$
$770$	67.6869	2.43927
$771$	0	0
$772$	81.8115	2.94446
$773$	25.3607	0.912160	0.456080	−	0.889939i	$-0.349253\pi$
$773$	25.3607	0.912160	0.456080	+	0.889939i	$0.349253\pi$
$774$	0	0
$775$	−4.00000	−0.143684
$776$	107.721	3.86697
$777$	0	0
$778$	48.8328	1.75074
$779$	−1.09017	−0.0390594
$780$	0	0
$781$	8.41641	0.301163
$782$	−14.9443	−0.534406
$783$	0	0
$784$	77.3951	2.76411
$785$	−10.6869	−0.381432
$786$	0	0
$787$	6.41641	0.228720	0.114360	−	0.993439i	$-0.463518\pi$
$787$	6.41641	0.228720	0.114360	+	0.993439i	$0.463518\pi$
$788$	21.9787	0.782959
$789$	0	0
$790$	51.5410	1.83375
$791$	59.2837	2.10788
$792$	0	0
$793$	1.97871	0.0702662
$794$	−93.1033	−3.30411
$795$	0	0
$796$	49.8541	1.76703
$797$	34.6312	1.22670	0.613350	−	0.789811i	$-0.289822\pi$
$797$	34.6312	1.22670	0.613350	+	0.789811i	$0.289822\pi$
$798$	0	0
$799$	6.58359	0.232911
$800$	43.4164	1.53500
$801$	0	0
$802$	−20.5623	−0.726080
$803$	8.94427	0.315637
$804$	0	0
$805$	86.3951	3.04503
$806$	0.381966	0.0134542
$807$	0	0
$808$	75.3951	2.65239
$809$	−2.56231	−0.0900859	−0.0450429	−	0.998985i	$-0.514342\pi$
$809$	−2.56231	−0.0900859	−0.0450429	+	0.998985i	$0.514342\pi$
$810$	0	0
$811$	−48.1246	−1.68988	−0.844942	−	0.534858i	$-0.820365\pi$
$811$	−48.1246	−1.68988	−0.844942	+	0.534858i	$0.820365\pi$
$812$	−139.790	−4.90568
$813$	0	0
$814$	22.5623	0.790808
$815$	−20.5623	−0.720266
$816$	0	0
$817$	−13.8541	−0.484694
$818$	34.3607	1.20139
$819$	0	0
$820$	5.56231	0.194244
$821$	15.7639	0.550165	0.275083	−	0.961421i	$-0.411295\pi$
$821$	15.7639	0.550165	0.275083	+	0.961421i	$0.411295\pi$
$822$	0	0
$823$	−33.6869	−1.17425	−0.587126	−	0.809496i	$-0.699741\pi$
$823$	−33.6869	−1.17425	−0.587126	+	0.809496i	$0.699741\pi$
$824$	56.5066	1.96850
$825$	0	0
$826$	60.5410	2.10649
$827$	−12.7639	−0.443845	−0.221923	−	0.975064i	$-0.571233\pi$
$827$	−12.7639	−0.443845	−0.221923	+	0.975064i	$0.571233\pi$
$828$	0	0
$829$	−4.12461	−0.143254	−0.0716268	−	0.997431i	$-0.522819\pi$
$829$	−4.12461	−0.143254	−0.0716268	+	0.997431i	$0.522819\pi$
$830$	120.374	4.17824
$831$	0	0
$832$	−1.27051	−0.0440470
$833$	6.00000	0.207888
$834$	0	0
$835$	35.6656	1.23426
$836$	30.9787	1.07142
$837$	0	0
$838$	−58.5410	−2.02227
$839$	19.3050	0.666481	0.333240	−	0.942842i	$-0.391858\pi$
$839$	19.3050	0.666481	0.333240	+	0.942842i	$0.391858\pi$
$840$	0	0
$841$	26.8328	0.925270
$842$	−80.7771	−2.78376
$843$	0	0
$844$	−10.4164	−0.358548
$845$	−38.9361	−1.33944
$846$	0	0
$847$	23.1246	0.794571
$848$	88.6869	3.04552
$849$	0	0
$850$	8.00000	0.274398
$851$	28.7984	0.987196
$852$	0	0
$853$	39.6869	1.35885	0.679427	−	0.733743i	$-0.262228\pi$
$853$	39.6869	1.35885	0.679427	+	0.733743i	$0.262228\pi$
$854$	136.846	4.68277
$855$	0	0
$856$	−39.5410	−1.35148
$857$	22.4164	0.765730	0.382865	−	0.923804i	$-0.374937\pi$
$857$	22.4164	0.765730	0.382865	+	0.923804i	$0.374937\pi$
$858$	0	0
$859$	−1.97871	−0.0675128	−0.0337564	−	0.999430i	$-0.510747\pi$
$859$	−1.97871	−0.0675128	−0.0337564	+	0.999430i	$0.510747\pi$
$860$	70.6869	2.41040
$861$	0	0
$862$	−72.6869	−2.47573
$863$	−35.5623	−1.21055	−0.605277	−	0.796015i	$-0.706938\pi$
$863$	−35.5623	−1.21055	−0.605277	+	0.796015i	$0.706938\pi$
$864$	0	0
$865$	41.7295	1.41885
$866$	−105.756	−3.59373
$867$	0	0
$868$	18.7082	0.634998
$869$	−14.6738	−0.497773
$870$	0	0
$871$	0.145898	0.00494357
$872$	−36.2705	−1.22827
$873$	0	0
$874$	55.8328	1.88857
$875$	11.5623	0.390877
$876$	0	0
$877$	−12.1246	−0.409419	−0.204710	−	0.978823i	$-0.565625\pi$
$877$	−12.1246	−0.409419	−0.204710	+	0.978823i	$0.565625\pi$
$878$	−101.284	−3.41816
$879$	0	0
$880$	−66.1033	−2.22834
$881$	26.2361	0.883916	0.441958	−	0.897036i	$-0.354284\pi$
$881$	26.2361	0.883916	0.441958	+	0.897036i	$0.354284\pi$
$882$	0	0
$883$	20.7082	0.696887	0.348443	−	0.937330i	$-0.386710\pi$
$883$	20.7082	0.696887	0.348443	+	0.937330i	$0.386710\pi$
$884$	−0.541020	−0.0181965
$885$	0	0
$886$	27.5623	0.925974
$887$	−23.0689	−0.774577	−0.387289	−	0.921959i	$-0.626588\pi$
$887$	−23.0689	−0.774577	−0.387289	+	0.921959i	$0.626588\pi$
$888$	0	0
$889$	22.5623	0.756715
$890$	17.5623	0.588690
$891$	0	0
$892$	76.2492	2.55301
$893$	−24.5967	−0.823099
$894$	0	0
$895$	20.2918	0.678280
$896$	−4.20163	−0.140366
$897$	0	0
$898$	−22.7082	−0.757783
$899$	−7.47214	−0.249210
$900$	0	0
$901$	6.87539	0.229052
$902$	−2.23607	−0.0744529
$903$	0	0
$904$	−114.936	−3.82272
$905$	19.6869	0.654415
$906$	0	0
$907$	−21.1459	−0.702138	−0.351069	−	0.936350i	$-0.614182\pi$
$907$	−21.1459	−0.702138	−0.351069	+	0.936350i	$0.614182\pi$
$908$	99.5410	3.30338
$909$	0	0
$910$	4.41641	0.146402
$911$	43.1459	1.42949	0.714744	−	0.699386i	$-0.246543\pi$
$911$	43.1459	1.42949	0.714744	+	0.699386i	$0.246543\pi$
$912$	0	0
$913$	−34.2705	−1.13419
$914$	−25.0902	−0.829909
$915$	0	0
$916$	−91.5197	−3.02390
$917$	−11.5623	−0.381821
$918$	0	0
$919$	−50.4164	−1.66308	−0.831542	−	0.555462i	$-0.812541\pi$
$919$	−50.4164	−1.66308	−0.831542	+	0.555462i	$0.812541\pi$
$920$	−167.498	−5.52226
$921$	0	0
$922$	36.2705	1.19451
$923$	0.549150	0.0180755
$924$	0	0
$925$	−15.4164	−0.506888
$926$	−72.1591	−2.37129
$927$	0	0
$928$	81.1033	2.66235
$929$	−17.9443	−0.588732	−0.294366	−	0.955693i	$-0.595109\pi$
$929$	−17.9443	−0.588732	−0.294366	+	0.955693i	$0.595109\pi$
$930$	0	0
$931$	−22.4164	−0.734668
$932$	−7.41641	−0.242933
$933$	0	0
$934$	−60.6869	−1.98574
$935$	−5.12461	−0.167593
$936$	0	0
$937$	11.2918	0.368887	0.184443	−	0.982843i	$-0.440952\pi$
$937$	11.2918	0.368887	0.184443	+	0.982843i	$0.440952\pi$
$938$	10.0902	0.329456
$939$	0	0
$940$	125.498	4.09331
$941$	0.167184	0.00545005	0.00272503	−	0.999996i	$-0.499133\pi$
$941$	0.167184	0.00545005	0.00272503	+	0.999996i	$0.499133\pi$
$942$	0	0
$943$	−2.85410	−0.0929423
$944$	−59.1246	−1.92434
$945$	0	0
$946$	−28.4164	−0.923897
$947$	2.72949	0.0886965	0.0443483	−	0.999016i	$-0.485879\pi$
$947$	2.72949	0.0886965	0.0443483	+	0.999016i	$0.485879\pi$
$948$	0	0
$949$	0.583592	0.0189442
$950$	−29.8885	−0.969712
$951$	0	0
$952$	−22.0000	−0.713024
$953$	−31.3607	−1.01587	−0.507936	−	0.861395i	$-0.669591\pi$
$953$	−31.3607	−1.01587	−0.507936	+	0.861395i	$0.669591\pi$
$954$	0	0
$955$	−4.41641	−0.142912
$956$	92.6656	2.99702
$957$	0	0
$958$	−80.2492	−2.59273
$959$	2.72949	0.0881398
$960$	0	0
$961$	−30.0000	−0.967742
$962$	1.47214	0.0474636
$963$	0	0
$964$	125.498	4.04203
$965$	50.5623	1.62766
$966$	0	0
$967$	−4.16718	−0.134008	−0.0670038	−	0.997753i	$-0.521344\pi$
$967$	−4.16718	−0.134008	−0.0670038	+	0.997753i	$0.521344\pi$
$968$	−44.8328	−1.44098
$969$	0	0
$970$	113.228	3.63553
$971$	−50.7771	−1.62951	−0.814757	−	0.579802i	$-0.803130\pi$
$971$	−50.7771	−1.62951	−0.814757	+	0.579802i	$0.803130\pi$
$972$	0	0
$973$	11.5623	0.370671
$974$	104.284	3.34146
$975$	0	0
$976$	−133.644	−4.27785
$977$	−3.65248	−0.116853	−0.0584265	−	0.998292i	$-0.518608\pi$
$977$	−3.65248	−0.116853	−0.0584265	+	0.998292i	$0.518608\pi$
$978$	0	0
$979$	−5.00000	−0.159801
$980$	114.374	3.65354
$981$	0	0
$982$	−74.3951	−2.37404
$983$	29.8328	0.951519	0.475760	−	0.879575i	$-0.342173\pi$
$983$	29.8328	0.951519	0.475760	+	0.879575i	$0.342173\pi$
$984$	0	0
$985$	13.5836	0.432809
$986$	14.9443	0.475923
$987$	0	0
$988$	2.02129	0.0643057
$989$	−36.2705	−1.15334
$990$	0	0
$991$	50.5623	1.60616	0.803082	−	0.595868i	$-0.203192\pi$
$991$	50.5623	1.60616	0.803082	+	0.595868i	$0.203192\pi$
$992$	−10.8541	−0.344618
$993$	0	0
$994$	37.9787	1.20461
$995$	30.8115	0.976791
$996$	0	0
$997$	27.5410	0.872233	0.436116	−	0.899890i	$-0.356354\pi$
$997$	27.5410	0.872233	0.436116	+	0.899890i	$0.356354\pi$
$998$	−65.4508	−2.07181
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	603.2.a.h.1.2		2
3.2	odd	2		67.2.a.b.1.1	✓	2
4.3	odd	2		9648.2.a.bi.1.2		2
12.11	even	2		1072.2.a.f.1.1		2
15.2	even	4		1675.2.c.d.1274.1		4
15.8	even	4		1675.2.c.d.1274.4		4
15.14	odd	2		1675.2.a.h.1.2		2
21.20	even	2		3283.2.a.f.1.1		2
24.5	odd	2		4288.2.a.p.1.1		2
24.11	even	2		4288.2.a.h.1.2		2
33.32	even	2		8107.2.a.i.1.2		2
201.200	even	2		4489.2.a.d.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
67.2.a.b.1.1	✓	2	3.2	odd	2
603.2.a.h.1.2		2	1.1	even	1	trivial
1072.2.a.f.1.1		2	12.11	even	2
1675.2.a.h.1.2		2	15.14	odd	2
1675.2.c.d.1274.1		4	15.2	even	4
1675.2.c.d.1274.4		4	15.8	even	4
3283.2.a.f.1.1		2	21.20	even	2
4288.2.a.h.1.2		2	24.11	even	2
4288.2.a.p.1.1		2	24.5	odd	2
4489.2.a.d.1.2		2	201.200	even	2
8107.2.a.i.1.2		2	33.32	even	2
9648.2.a.bi.1.2		2	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 603 = 3^{2} \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	603.a (trivial)

\(f(q)\)	\(=\)	\(q+2.61803 q^{2} +4.85410 q^{4} +3.00000 q^{5} -3.85410 q^{7} +7.47214 q^{8} +O(q^{10})\)	\(q+2.61803 q^{2} +4.85410 q^{4} +3.00000 q^{5} -3.85410 q^{7} +7.47214 q^{8} +7.85410 q^{10} -2.23607 q^{11} -0.145898 q^{13} -10.0902 q^{14} +9.85410 q^{16} +0.763932 q^{17} -2.85410 q^{19} +14.5623 q^{20} -5.85410 q^{22} -7.47214 q^{23} +4.00000 q^{25} -0.381966 q^{26} -18.7082 q^{28} +7.47214 q^{29} -1.00000 q^{31} +10.8541 q^{32} +2.00000 q^{34} -11.5623 q^{35} -3.85410 q^{37} -7.47214 q^{38} +22.4164 q^{40} +0.381966 q^{41} +4.85410 q^{43} -10.8541 q^{44} -19.5623 q^{46} +8.61803 q^{47} +7.85410 q^{49} +10.4721 q^{50} -0.708204 q^{52} +9.00000 q^{53} -6.70820 q^{55} -28.7984 q^{56} +19.5623 q^{58} -6.00000 q^{59} -13.5623 q^{61} -2.61803 q^{62} +8.70820 q^{64} -0.437694 q^{65} -1.00000 q^{67} +3.70820 q^{68} -30.2705 q^{70} -3.76393 q^{71} -4.00000 q^{73} -10.0902 q^{74} -13.8541 q^{76} +8.61803 q^{77} +6.56231 q^{79} +29.5623 q^{80} +1.00000 q^{82} +15.3262 q^{83} +2.29180 q^{85} +12.7082 q^{86} -16.7082 q^{88} +2.23607 q^{89} +0.562306 q^{91} -36.2705 q^{92} +22.5623 q^{94} -8.56231 q^{95} +14.4164 q^{97} +20.5623 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{2} + 3 q^{4} + 6 q^{5} - q^{7} + 6 q^{8}+O(q^{10}) \) 2 * q + 3 * q^2 + 3 * q^4 + 6 * q^5 - q^7 + 6 * q^8	\( 2 q + 3 q^{2} + 3 q^{4} + 6 q^{5} - q^{7} + 6 q^{8} + 9 q^{10} - 7 q^{13} - 9 q^{14} + 13 q^{16} + 6 q^{17} + q^{19} + 9 q^{20} - 5 q^{22} - 6 q^{23} + 8 q^{25} - 3 q^{26} - 24 q^{28} + 6 q^{29} - 2 q^{31} + 15 q^{32} + 4 q^{34} - 3 q^{35} - q^{37} - 6 q^{38} + 18 q^{40} + 3 q^{41} + 3 q^{43} - 15 q^{44} - 19 q^{46} + 15 q^{47} + 9 q^{49} + 12 q^{50} + 12 q^{52} + 18 q^{53} - 33 q^{56} + 19 q^{58} - 12 q^{59} - 7 q^{61} - 3 q^{62} + 4 q^{64} - 21 q^{65} - 2 q^{67} - 6 q^{68} - 27 q^{70} - 12 q^{71} - 8 q^{73} - 9 q^{74} - 21 q^{76} + 15 q^{77} - 7 q^{79} + 39 q^{80} + 2 q^{82} + 15 q^{83} + 18 q^{85} + 12 q^{86} - 20 q^{88} - 19 q^{91} - 39 q^{92} + 25 q^{94} + 3 q^{95} + 2 q^{97} + 21 q^{98}+O(q^{100}) \) 2 * q + 3 * q^2 + 3 * q^4 + 6 * q^5 - q^7 + 6 * q^8 + 9 * q^10 - 7 * q^13 - 9 * q^14 + 13 * q^16 + 6 * q^17 + q^19 + 9 * q^20 - 5 * q^22 - 6 * q^23 + 8 * q^25 - 3 * q^26 - 24 * q^28 + 6 * q^29 - 2 * q^31 + 15 * q^32 + 4 * q^34 - 3 * q^35 - q^37 - 6 * q^38 + 18 * q^40 + 3 * q^41 + 3 * q^43 - 15 * q^44 - 19 * q^46 + 15 * q^47 + 9 * q^49 + 12 * q^50 + 12 * q^52 + 18 * q^53 - 33 * q^56 + 19 * q^58 - 12 * q^59 - 7 * q^61 - 3 * q^62 + 4 * q^64 - 21 * q^65 - 2 * q^67 - 6 * q^68 - 27 * q^70 - 12 * q^71 - 8 * q^73 - 9 * q^74 - 21 * q^76 + 15 * q^77 - 7 * q^79 + 39 * q^80 + 2 * q^82 + 15 * q^83 + 18 * q^85 + 12 * q^86 - 20 * q^88 - 19 * q^91 - 39 * q^92 + 25 * q^94 + 3 * q^95 + 2 * q^97 + 21 * q^98

Introduction

L-functions

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