LMFDB - Embedded newform 1755.2.a.u.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1755,2,Mod(1,1755)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1755, 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("1755.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1755 = 3^{3} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1755.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:	$14.0137455547$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 12x^{5} + 9x^{4} + 39x^{3} - 16x^{2} - 30x + 12 $ x^7 - x^6 - 12x^5 + 9x^4 + 39x^3 - 16x^2 - 30*x + 12
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.72441$ of defining polynomial
Character	$\chi$	$=$	1755.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.72441 q^{2} +5.42239 q^{4} -1.00000 q^{5} -4.53684 q^{7} -9.32397 q^{8} +O(q^{10})$	\(q-2.72441 q^{2} +5.42239 q^{4} -1.00000 q^{5} -4.53684 q^{7} -9.32397 q^{8} +2.72441 q^{10} -4.77545 q^{11} +1.00000 q^{13} +12.3602 q^{14} +14.5575 q^{16} -4.35966 q^{17} -7.13511 q^{19} -5.42239 q^{20} +13.0103 q^{22} -6.97590 q^{23} +1.00000 q^{25} -2.72441 q^{26} -24.6005 q^{28} +0.650087 q^{29} -2.33639 q^{31} -21.0126 q^{32} +11.8775 q^{34} +4.53684 q^{35} +2.41467 q^{37} +19.4389 q^{38} +9.32397 q^{40} +6.66229 q^{41} -10.5490 q^{43} -25.8943 q^{44} +19.0052 q^{46} +4.79872 q^{47} +13.5829 q^{49} -2.72441 q^{50} +5.42239 q^{52} -2.54223 q^{53} +4.77545 q^{55} +42.3013 q^{56} -1.77110 q^{58} +2.05957 q^{59} +8.41251 q^{61} +6.36527 q^{62} +28.1318 q^{64} -1.00000 q^{65} +4.76571 q^{67} -23.6398 q^{68} -12.3602 q^{70} -3.01407 q^{71} -4.07907 q^{73} -6.57853 q^{74} -38.6893 q^{76} +21.6654 q^{77} -13.1473 q^{79} -14.5575 q^{80} -18.1508 q^{82} +0.290340 q^{83} +4.35966 q^{85} +28.7399 q^{86} +44.5262 q^{88} +11.1868 q^{89} -4.53684 q^{91} -37.8260 q^{92} -13.0737 q^{94} +7.13511 q^{95} -8.90090 q^{97} -37.0053 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - q^{2} + 11 q^{4} - 7 q^{5} + 2 q^{7} - 6 q^{8}+O(q^{10}) $ 7 * q - q^2 + 11 * q^4 - 7 * q^5 + 2 * q^7 - 6 * q^8	$ 7 q - q^{2} + 11 q^{4} - 7 q^{5} + 2 q^{7} - 6 q^{8} + q^{10} - q^{11} + 7 q^{13} + 12 q^{14} + 23 q^{16} - 11 q^{17} + 2 q^{19} - 11 q^{20} + 16 q^{22} + q^{23} + 7 q^{25} - q^{26} + 10 q^{28} + 4 q^{29} - 13 q^{32} + q^{34} - 2 q^{35} + 23 q^{37} + 15 q^{38} + 6 q^{40} - 2 q^{41} + 8 q^{43} - 10 q^{44} + 37 q^{46} - 2 q^{47} + 43 q^{49} - q^{50} + 11 q^{52} - 10 q^{53} + q^{55} + 68 q^{56} - 26 q^{58} + 13 q^{59} + 21 q^{61} - 9 q^{62} + 46 q^{64} - 7 q^{65} + 21 q^{67} - 53 q^{68} - 12 q^{70} + 10 q^{71} + 13 q^{73} + 68 q^{74} - 41 q^{76} - 6 q^{77} + 8 q^{79} - 23 q^{80} - 26 q^{82} + 4 q^{83} + 11 q^{85} + 12 q^{86} + 44 q^{88} + 27 q^{89} + 2 q^{91} + 9 q^{92} - 24 q^{94} - 2 q^{95} - 15 q^{97} - 7 q^{98}+O(q^{100}) $ 7 * q - q^2 + 11 * q^4 - 7 * q^5 + 2 * q^7 - 6 * q^8 + q^10 - q^11 + 7 * q^13 + 12 * q^14 + 23 * q^16 - 11 * q^17 + 2 * q^19 - 11 * q^20 + 16 * q^22 + q^23 + 7 * q^25 - q^26 + 10 * q^28 + 4 * q^29 - 13 * q^32 + q^34 - 2 * q^35 + 23 * q^37 + 15 * q^38 + 6 * q^40 - 2 * q^41 + 8 * q^43 - 10 * q^44 + 37 * q^46 - 2 * q^47 + 43 * q^49 - q^50 + 11 * q^52 - 10 * q^53 + q^55 + 68 * q^56 - 26 * q^58 + 13 * q^59 + 21 * q^61 - 9 * q^62 + 46 * q^64 - 7 * q^65 + 21 * q^67 - 53 * q^68 - 12 * q^70 + 10 * q^71 + 13 * q^73 + 68 * q^74 - 41 * q^76 - 6 * q^77 + 8 * q^79 - 23 * q^80 - 26 * q^82 + 4 * q^83 + 11 * q^85 + 12 * q^86 + 44 * q^88 + 27 * q^89 + 2 * q^91 + 9 * q^92 - 24 * q^94 - 2 * q^95 - 15 * q^97 - 7 * 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 $

$2$	−2.72441	−1.92645	−0.963223	−	0.268704i	$-0.913405\pi$
$2$	−2.72441	−1.92645	−0.963223	+	0.268704i	$0.913405\pi$
$3$	0	0
$3$	0	0
$4$	5.42239	2.71119
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−4.53684	−1.71476	−0.857381	−	0.514681i	$-0.827910\pi$
$7$	−4.53684	−1.71476	−0.857381	+	0.514681i	$0.827910\pi$
$8$	−9.32397	−3.29652
$9$	0	0
$10$	2.72441	0.861533
$11$	−4.77545	−1.43985	−0.719927	−	0.694050i	$-0.755825\pi$
$11$	−4.77545	−1.43985	−0.719927	+	0.694050i	$0.755825\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	12.3602	3.30340
$15$	0	0
$16$	14.5575	3.63937
$17$	−4.35966	−1.05737	−0.528686	−	0.848817i	$-0.677315\pi$
$17$	−4.35966	−1.05737	−0.528686	+	0.848817i	$0.677315\pi$
$18$	0	0
$19$	−7.13511	−1.63691	−0.818454	−	0.574573i	$-0.805168\pi$
$19$	−7.13511	−1.63691	−0.818454	+	0.574573i	$0.805168\pi$
$20$	−5.42239	−1.21248
$21$	0	0
$22$	13.0103	2.77380
$23$	−6.97590	−1.45458	−0.727288	−	0.686332i	$-0.759220\pi$
$23$	−6.97590	−1.45458	−0.727288	+	0.686332i	$0.759220\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−2.72441	−0.534300
$27$	0	0
$28$	−24.6005	−4.64905
$29$	0.650087	0.120718	0.0603590	−	0.998177i	$-0.480775\pi$
$29$	0.650087	0.120718	0.0603590	+	0.998177i	$0.480775\pi$
$30$	0	0
$31$	−2.33639	−0.419628	−0.209814	−	0.977741i	$-0.567286\pi$
$31$	−2.33639	−0.419628	−0.209814	+	0.977741i	$0.567286\pi$
$32$	−21.0126	−3.71454
$33$	0	0
$34$	11.8775	2.03697
$35$	4.53684	0.766865
$36$	0	0
$37$	2.41467	0.396969	0.198484	−	0.980104i	$-0.436398\pi$
$37$	2.41467	0.396969	0.198484	+	0.980104i	$0.436398\pi$
$38$	19.4389	3.15341
$39$	0	0
$40$	9.32397	1.47425
$41$	6.66229	1.04047	0.520237	−	0.854022i	$-0.325843\pi$
$41$	6.66229	1.04047	0.520237	+	0.854022i	$0.325843\pi$
$42$	0	0
$43$	−10.5490	−1.60871	−0.804357	−	0.594147i	$-0.797490\pi$
$43$	−10.5490	−1.60871	−0.804357	+	0.594147i	$0.797490\pi$
$44$	−25.8943	−3.90372
$45$	0	0
$46$	19.0052	2.80216
$47$	4.79872	0.699966	0.349983	−	0.936756i	$-0.386187\pi$
$47$	4.79872	0.699966	0.349983	+	0.936756i	$0.386187\pi$
$48$	0	0
$49$	13.5829	1.94041
$50$	−2.72441	−0.385289
$51$	0	0
$52$	5.42239	0.751950
$53$	−2.54223	−0.349202	−0.174601	−	0.984639i	$-0.555864\pi$
$53$	−2.54223	−0.349202	−0.174601	+	0.984639i	$0.555864\pi$
$54$	0	0
$55$	4.77545	0.643922
$56$	42.3013	5.65275
$57$	0	0
$58$	−1.77110	−0.232557
$59$	2.05957	0.268134	0.134067	−	0.990972i	$-0.457196\pi$
$59$	2.05957	0.268134	0.134067	+	0.990972i	$0.457196\pi$
$60$	0	0
$61$	8.41251	1.07711	0.538556	−	0.842590i	$-0.318970\pi$
$61$	8.41251	1.07711	0.538556	+	0.842590i	$0.318970\pi$
$62$	6.36527	0.808390
$63$	0	0
$64$	28.1318	3.51648
$65$	−1.00000	−0.124035
$66$	0	0
$67$	4.76571	0.582224	0.291112	−	0.956689i	$-0.405975\pi$
$67$	4.76571	0.582224	0.291112	+	0.956689i	$0.405975\pi$
$68$	−23.6398	−2.86674
$69$	0	0
$70$	−12.3602	−1.47732
$71$	−3.01407	−0.357704	−0.178852	−	0.983876i	$-0.557238\pi$
$71$	−3.01407	−0.357704	−0.178852	+	0.983876i	$0.557238\pi$
$72$	0	0
$73$	−4.07907	−0.477419	−0.238709	−	0.971091i	$-0.576724\pi$
$73$	−4.07907	−0.477419	−0.238709	+	0.971091i	$0.576724\pi$
$74$	−6.57853	−0.764739
$75$	0	0
$76$	−38.6893	−4.43797
$77$	21.6654	2.46901
$78$	0	0
$79$	−13.1473	−1.47919	−0.739594	−	0.673053i	$-0.764983\pi$
$79$	−13.1473	−1.47919	−0.739594	+	0.673053i	$0.764983\pi$
$80$	−14.5575	−1.62758
$81$	0	0
$82$	−18.1508	−2.00442
$83$	0.290340	0.0318690	0.0159345	−	0.999873i	$-0.494928\pi$
$83$	0.290340	0.0318690	0.0159345	+	0.999873i	$0.494928\pi$
$84$	0	0
$85$	4.35966	0.472872
$86$	28.7399	3.09910
$87$	0	0
$88$	44.5262	4.74650
$89$	11.1868	1.18580	0.592901	−	0.805275i	$-0.297982\pi$
$89$	11.1868	1.18580	0.592901	+	0.805275i	$0.297982\pi$
$90$	0	0
$91$	−4.53684	−0.475590
$92$	−37.8260	−3.94364
$93$	0	0
$94$	−13.0737	−1.34845
$95$	7.13511	0.732047
$96$	0	0
$97$	−8.90090	−0.903750	−0.451875	−	0.892081i	$-0.649245\pi$
$97$	−8.90090	−0.903750	−0.451875	+	0.892081i	$0.649245\pi$
$98$	−37.0053	−3.73810
$99$	0	0
$100$	5.42239	0.542239
$101$	−1.62486	−0.161680	−0.0808399	−	0.996727i	$-0.525760\pi$
$101$	−1.62486	−0.161680	−0.0808399	+	0.996727i	$0.525760\pi$
$102$	0	0
$103$	−11.6435	−1.14727	−0.573634	−	0.819112i	$-0.694467\pi$
$103$	−11.6435	−1.14727	−0.573634	+	0.819112i	$0.694467\pi$
$104$	−9.32397	−0.914290
$105$	0	0
$106$	6.92607	0.672719
$107$	7.17008	0.693158	0.346579	−	0.938021i	$-0.387343\pi$
$107$	7.17008	0.693158	0.346579	+	0.938021i	$0.387343\pi$
$108$	0	0
$109$	−20.3808	−1.95212	−0.976062	−	0.217491i	$-0.930213\pi$
$109$	−20.3808	−1.95212	−0.976062	+	0.217491i	$0.930213\pi$
$110$	−13.0103	−1.24048
$111$	0	0
$112$	−66.0450	−6.24066
$113$	−8.98780	−0.845501	−0.422751	−	0.906246i	$-0.638935\pi$
$113$	−8.98780	−0.845501	−0.422751	+	0.906246i	$0.638935\pi$
$114$	0	0
$115$	6.97590	0.650506
$116$	3.52502	0.327290
$117$	0	0
$118$	−5.61111	−0.516545
$119$	19.7791	1.81314
$120$	0	0
$121$	11.8049	1.07318
$122$	−22.9191	−2.07500
$123$	0	0
$124$	−12.6688	−1.13769
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	7.57201	0.671907	0.335954	−	0.941879i	$-0.390941\pi$
$127$	7.57201	0.671907	0.335954	+	0.941879i	$0.390941\pi$
$128$	−34.6173	−3.05977
$129$	0	0
$130$	2.72441	0.238946
$131$	0.397827	0.0347583	0.0173792	−	0.999849i	$-0.494468\pi$
$131$	0.397827	0.0347583	0.0173792	+	0.999849i	$0.494468\pi$
$132$	0	0
$133$	32.3708	2.80691
$134$	−12.9837	−1.12162
$135$	0	0
$136$	40.6493	3.48565
$137$	7.11175	0.607598	0.303799	−	0.952736i	$-0.401745\pi$
$137$	7.11175	0.607598	0.303799	+	0.952736i	$0.401745\pi$
$138$	0	0
$139$	1.37319	0.116472	0.0582361	−	0.998303i	$-0.481452\pi$
$139$	1.37319	0.116472	0.0582361	+	0.998303i	$0.481452\pi$
$140$	24.6005	2.07912
$141$	0	0
$142$	8.21154	0.689097
$143$	−4.77545	−0.399343
$144$	0	0
$145$	−0.650087	−0.0539868
$146$	11.1130	0.919721
$147$	0	0
$148$	13.0933	1.07626
$149$	−3.53248	−0.289392	−0.144696	−	0.989476i	$-0.546220\pi$
$149$	−3.53248	−0.289392	−0.144696	+	0.989476i	$0.546220\pi$
$150$	0	0
$151$	−2.37074	−0.192928	−0.0964639	−	0.995336i	$-0.530753\pi$
$151$	−2.37074	−0.192928	−0.0964639	+	0.995336i	$0.530753\pi$
$152$	66.5276	5.39610
$153$	0	0
$154$	−59.0255	−4.75641
$155$	2.33639	0.187663
$156$	0	0
$157$	16.4469	1.31261	0.656305	−	0.754496i	$-0.272119\pi$
$157$	16.4469	1.31261	0.656305	+	0.754496i	$0.272119\pi$
$158$	35.8186	2.84958
$159$	0	0
$160$	21.0126	1.66119
$161$	31.6485	2.49425
$162$	0	0
$163$	6.90090	0.540521	0.270260	−	0.962787i	$-0.412890\pi$
$163$	6.90090	0.540521	0.270260	+	0.962787i	$0.412890\pi$
$164$	36.1255	2.82093
$165$	0	0
$166$	−0.791005	−0.0613939
$167$	14.0032	1.08360	0.541802	−	0.840506i	$-0.317742\pi$
$167$	14.0032	1.08360	0.541802	+	0.840506i	$0.317742\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−11.8775	−0.910961
$171$	0	0
$172$	−57.2010	−4.36153
$173$	−19.4793	−1.48098	−0.740492	−	0.672066i	$-0.765407\pi$
$173$	−19.4793	−1.48098	−0.740492	+	0.672066i	$0.765407\pi$
$174$	0	0
$175$	−4.53684	−0.342953
$176$	−69.5186	−5.24016
$177$	0	0
$178$	−30.4775	−2.28438
$179$	−6.24073	−0.466454	−0.233227	−	0.972422i	$-0.574929\pi$
$179$	−6.24073	−0.466454	−0.233227	+	0.972422i	$0.574929\pi$
$180$	0	0
$181$	−12.5067	−0.929618	−0.464809	−	0.885411i	$-0.653877\pi$
$181$	−12.5067	−0.929618	−0.464809	+	0.885411i	$0.653877\pi$
$182$	12.3602	0.916198
$183$	0	0
$184$	65.0431	4.79504
$185$	−2.41467	−0.177530
$186$	0	0
$187$	20.8193	1.52246
$188$	26.0205	1.89774
$189$	0	0
$190$	−19.4389	−1.41025
$191$	−5.41865	−0.392080	−0.196040	−	0.980596i	$-0.562808\pi$
$191$	−5.41865	−0.392080	−0.196040	+	0.980596i	$0.562808\pi$
$192$	0	0
$193$	9.23969	0.665087	0.332544	−	0.943088i	$-0.392093\pi$
$193$	9.23969	0.665087	0.332544	+	0.943088i	$0.392093\pi$
$194$	24.2497	1.74103
$195$	0	0
$196$	73.6516	5.26083
$197$	25.1230	1.78994	0.894969	−	0.446129i	$-0.147198\pi$
$197$	25.1230	1.78994	0.894969	+	0.446129i	$0.147198\pi$
$198$	0	0
$199$	−3.98242	−0.282306	−0.141153	−	0.989988i	$-0.545081\pi$
$199$	−3.98242	−0.282306	−0.141153	+	0.989988i	$0.545081\pi$
$200$	−9.32397	−0.659304
$201$	0	0
$202$	4.42678	0.311467
$203$	−2.94934	−0.207003
$204$	0	0
$205$	−6.66229	−0.465314
$206$	31.7216	2.21015
$207$	0	0
$208$	14.5575	1.00938
$209$	34.0734	2.35691
$210$	0	0
$211$	−17.8340	−1.22774	−0.613871	−	0.789406i	$-0.710389\pi$
$211$	−17.8340	−1.22774	−0.613871	+	0.789406i	$0.710389\pi$
$212$	−13.7850	−0.946755
$213$	0	0
$214$	−19.5342	−1.33533
$215$	10.5490	0.719438
$216$	0	0
$217$	10.5998	0.719562
$218$	55.5255	3.76066
$219$	0	0
$220$	25.8943	1.74580
$221$	−4.35966	−0.293262
$222$	0	0
$223$	27.7749	1.85995	0.929973	−	0.367628i	$-0.119830\pi$
$223$	27.7749	1.85995	0.929973	+	0.367628i	$0.119830\pi$
$224$	95.3307	6.36955
$225$	0	0
$226$	24.4864	1.62881
$227$	−9.44342	−0.626782	−0.313391	−	0.949624i	$-0.601465\pi$
$227$	−9.44342	−0.626782	−0.313391	+	0.949624i	$0.601465\pi$
$228$	0	0
$229$	−2.22957	−0.147334	−0.0736671	−	0.997283i	$-0.523470\pi$
$229$	−2.22957	−0.147334	−0.0736671	+	0.997283i	$0.523470\pi$
$230$	−19.0052	−1.25316
$231$	0	0
$232$	−6.06139	−0.397950
$233$	−7.10666	−0.465573	−0.232786	−	0.972528i	$-0.574784\pi$
$233$	−7.10666	−0.465573	−0.232786	+	0.972528i	$0.574784\pi$
$234$	0	0
$235$	−4.79872	−0.313034
$236$	11.1678	0.726962
$237$	0	0
$238$	−53.8862	−3.49292
$239$	−16.6343	−1.07598	−0.537991	−	0.842950i	$-0.680817\pi$
$239$	−16.6343	−1.07598	−0.537991	+	0.842950i	$0.680817\pi$
$240$	0	0
$241$	6.63519	0.427410	0.213705	−	0.976898i	$-0.431447\pi$
$241$	6.63519	0.427410	0.213705	+	0.976898i	$0.431447\pi$
$242$	−32.1615	−2.06742
$243$	0	0
$244$	45.6159	2.92026
$245$	−13.5829	−0.867779
$246$	0	0
$247$	−7.13511	−0.453996
$248$	21.7844	1.38331
$249$	0	0
$250$	2.72441	0.172307
$251$	19.8492	1.25287	0.626436	−	0.779473i	$-0.284513\pi$
$251$	19.8492	1.25287	0.626436	+	0.779473i	$0.284513\pi$
$252$	0	0
$253$	33.3131	2.09438
$254$	−20.6292	−1.29439
$255$	0	0
$256$	38.0480	2.37800
$257$	−24.6764	−1.53927	−0.769634	−	0.638485i	$-0.779562\pi$
$257$	−24.6764	−1.53927	−0.769634	+	0.638485i	$0.779562\pi$
$258$	0	0
$259$	−10.9549	−0.680707
$260$	−5.42239	−0.336282
$261$	0	0
$262$	−1.08384	−0.0669601
$263$	12.8052	0.789602	0.394801	−	0.918767i	$-0.370814\pi$
$263$	12.8052	0.789602	0.394801	+	0.918767i	$0.370814\pi$
$264$	0	0
$265$	2.54223	0.156168
$266$	−88.1913	−5.40735
$267$	0	0
$268$	25.8415	1.57852
$269$	0.246491	0.0150288	0.00751442	−	0.999972i	$-0.497608\pi$
$269$	0.246491	0.0150288	0.00751442	+	0.999972i	$0.497608\pi$
$270$	0	0
$271$	−9.44102	−0.573501	−0.286750	−	0.958005i	$-0.592575\pi$
$271$	−9.44102	−0.573501	−0.286750	+	0.958005i	$0.592575\pi$
$272$	−63.4657	−3.84818
$273$	0	0
$274$	−19.3753	−1.17050
$275$	−4.77545	−0.287971
$276$	0	0
$277$	1.60217	0.0962652	0.0481326	−	0.998841i	$-0.484673\pi$
$277$	1.60217	0.0962652	0.0481326	+	0.998841i	$0.484673\pi$
$278$	−3.74112	−0.224377
$279$	0	0
$280$	−42.3013	−2.52799
$281$	−25.6855	−1.53227	−0.766134	−	0.642681i	$-0.777822\pi$
$281$	−25.6855	−1.53227	−0.766134	+	0.642681i	$0.777822\pi$
$282$	0	0
$283$	18.2904	1.08725	0.543625	−	0.839328i	$-0.317051\pi$
$283$	18.2904	1.08725	0.543625	+	0.839328i	$0.317051\pi$
$284$	−16.3434	−0.969805
$285$	0	0
$286$	13.0103	0.769313
$287$	−30.2257	−1.78417
$288$	0	0
$289$	2.00664	0.118037
$290$	1.77110	0.104003
$291$	0	0
$292$	−22.1183	−1.29437
$293$	20.7353	1.21137	0.605683	−	0.795706i	$-0.292900\pi$
$293$	20.7353	1.21137	0.605683	+	0.795706i	$0.292900\pi$
$294$	0	0
$295$	−2.05957	−0.119913
$296$	−22.5143	−1.30862
$297$	0	0
$298$	9.62392	0.557499
$299$	−6.97590	−0.403427
$300$	0	0
$301$	47.8593	2.75856
$302$	6.45885	0.371665
$303$	0	0
$304$	−103.869	−5.95732
$305$	−8.41251	−0.481699
$306$	0	0
$307$	17.7893	1.01529	0.507646	−	0.861566i	$-0.330516\pi$
$307$	17.7893	1.01529	0.507646	+	0.861566i	$0.330516\pi$
$308$	117.478	6.69395
$309$	0	0
$310$	−6.36527	−0.361523
$311$	18.9469	1.07438	0.537189	−	0.843462i	$-0.319486\pi$
$311$	18.9469	1.07438	0.537189	+	0.843462i	$0.319486\pi$
$312$	0	0
$313$	4.20260	0.237545	0.118773	−	0.992921i	$-0.462104\pi$
$313$	4.20260	0.237545	0.118773	+	0.992921i	$0.462104\pi$
$314$	−44.8081	−2.52867
$315$	0	0
$316$	−71.2898	−4.01037
$317$	−10.9786	−0.616617	−0.308309	−	0.951286i	$-0.599763\pi$
$317$	−10.9786	−0.616617	−0.308309	+	0.951286i	$0.599763\pi$
$318$	0	0
$319$	−3.10446	−0.173816
$320$	−28.1318	−1.57262
$321$	0	0
$322$	−86.2234	−4.80504
$323$	31.1067	1.73082
$324$	0	0
$325$	1.00000	0.0554700
$326$	−18.8009	−1.04128
$327$	0	0
$328$	−62.1190	−3.42995
$329$	−21.7710	−1.20028
$330$	0	0
$331$	−7.62743	−0.419242	−0.209621	−	0.977783i	$-0.567223\pi$
$331$	−7.62743	−0.419242	−0.209621	+	0.977783i	$0.567223\pi$
$332$	1.57434	0.0864030
$333$	0	0
$334$	−38.1505	−2.08750
$335$	−4.76571	−0.260378
$336$	0	0
$337$	0.454046	0.0247334	0.0123667	−	0.999924i	$-0.496063\pi$
$337$	0.454046	0.0247334	0.0123667	+	0.999924i	$0.496063\pi$
$338$	−2.72441	−0.148188
$339$	0	0
$340$	23.6398	1.28205
$341$	11.1573	0.604202
$342$	0	0
$343$	−29.8655	−1.61258
$344$	98.3589	5.30316
$345$	0	0
$346$	53.0695	2.85303
$347$	−9.11387	−0.489258	−0.244629	−	0.969617i	$-0.578666\pi$
$347$	−9.11387	−0.489258	−0.244629	+	0.969617i	$0.578666\pi$
$348$	0	0
$349$	−6.36024	−0.340456	−0.170228	−	0.985405i	$-0.554450\pi$
$349$	−6.36024	−0.340456	−0.170228	+	0.985405i	$0.554450\pi$
$350$	12.3602	0.660680
$351$	0	0
$352$	100.345	5.34839
$353$	−31.6508	−1.68460	−0.842302	−	0.539005i	$-0.818800\pi$
$353$	−31.6508	−1.68460	−0.842302	+	0.539005i	$0.818800\pi$
$354$	0	0
$355$	3.01407	0.159970
$356$	60.6593	3.21494
$357$	0	0
$358$	17.0023	0.898599
$359$	22.2572	1.17469	0.587345	−	0.809337i	$-0.300173\pi$
$359$	22.2572	1.17469	0.587345	+	0.809337i	$0.300173\pi$
$360$	0	0
$361$	31.9098	1.67946
$362$	34.0734	1.79086
$363$	0	0
$364$	−24.6005	−1.28942
$365$	4.07907	0.213508
$366$	0	0
$367$	24.4373	1.27562	0.637809	−	0.770195i	$-0.279841\pi$
$367$	24.4373	1.27562	0.637809	+	0.770195i	$0.279841\pi$
$368$	−101.552	−5.29375
$369$	0	0
$370$	6.57853	0.342001
$371$	11.5337	0.598799
$372$	0	0
$373$	18.5293	0.959410	0.479705	−	0.877430i	$-0.340744\pi$
$373$	18.5293	0.959410	0.479705	+	0.877430i	$0.340744\pi$
$374$	−56.7204	−2.93294
$375$	0	0
$376$	−44.7432	−2.30745
$377$	0.650087	0.0334812
$378$	0	0
$379$	5.70335	0.292962	0.146481	−	0.989214i	$-0.453205\pi$
$379$	5.70335	0.292962	0.146481	+	0.989214i	$0.453205\pi$
$380$	38.6893	1.98472
$381$	0	0
$382$	14.7626	0.755320
$383$	−1.58101	−0.0807860	−0.0403930	−	0.999184i	$-0.512861\pi$
$383$	−1.58101	−0.0807860	−0.0403930	+	0.999184i	$0.512861\pi$
$384$	0	0
$385$	−21.6654	−1.10417
$386$	−25.1727	−1.28125
$387$	0	0
$388$	−48.2641	−2.45024
$389$	−5.99631	−0.304025	−0.152012	−	0.988379i	$-0.548575\pi$
$389$	−5.99631	−0.304025	−0.152012	+	0.988379i	$0.548575\pi$
$390$	0	0
$391$	30.4126	1.53803
$392$	−126.646	−6.39661
$393$	0	0
$394$	−68.4451	−3.44822
$395$	13.1473	0.661513
$396$	0	0
$397$	−36.8977	−1.85184	−0.925921	−	0.377718i	$-0.876709\pi$
$397$	−36.8977	−1.85184	−0.925921	+	0.377718i	$0.876709\pi$
$398$	10.8497	0.543848
$399$	0	0
$400$	14.5575	0.727875
$401$	−6.66200	−0.332684	−0.166342	−	0.986068i	$-0.553196\pi$
$401$	−6.66200	−0.332684	−0.166342	+	0.986068i	$0.553196\pi$
$402$	0	0
$403$	−2.33639	−0.116384
$404$	−8.81063	−0.438345
$405$	0	0
$406$	8.03519	0.398780
$407$	−11.5311	−0.571577
$408$	0	0
$409$	8.65653	0.428038	0.214019	−	0.976830i	$-0.431345\pi$
$409$	8.65653	0.428038	0.214019	+	0.976830i	$0.431345\pi$
$410$	18.1508	0.896403
$411$	0	0
$412$	−63.1355	−3.11046
$413$	−9.34395	−0.459786
$414$	0	0
$415$	−0.290340	−0.0142522
$416$	−21.0126	−1.03023
$417$	0	0
$418$	−92.8297	−4.54045
$419$	18.8237	0.919596	0.459798	−	0.888023i	$-0.347922\pi$
$419$	18.8237	0.919596	0.459798	+	0.888023i	$0.347922\pi$
$420$	0	0
$421$	29.8260	1.45363	0.726815	−	0.686834i	$-0.241000\pi$
$421$	29.8260	1.45363	0.726815	+	0.686834i	$0.241000\pi$
$422$	48.5870	2.36518
$423$	0	0
$424$	23.7037	1.15115
$425$	−4.35966	−0.211475
$426$	0	0
$427$	−38.1662	−1.84699
$428$	38.8790	1.87929
$429$	0	0
$430$	−28.7399	−1.38596
$431$	−16.6379	−0.801421	−0.400711	−	0.916205i	$-0.631237\pi$
$431$	−16.6379	−0.801421	−0.400711	+	0.916205i	$0.631237\pi$
$432$	0	0
$433$	−16.4014	−0.788203	−0.394102	−	0.919067i	$-0.628944\pi$
$433$	−16.4014	−0.788203	−0.394102	+	0.919067i	$0.628944\pi$
$434$	−28.8782	−1.38620
$435$	0	0
$436$	−110.512	−5.29259
$437$	49.7738	2.38101
$438$	0	0
$439$	−2.09520	−0.0999984	−0.0499992	−	0.998749i	$-0.515922\pi$
$439$	−2.09520	−0.0999984	−0.0499992	+	0.998749i	$0.515922\pi$
$440$	−44.5262	−2.12270
$441$	0	0
$442$	11.8775	0.564954
$443$	−8.93629	−0.424576	−0.212288	−	0.977207i	$-0.568091\pi$
$443$	−8.93629	−0.424576	−0.212288	+	0.977207i	$0.568091\pi$
$444$	0	0
$445$	−11.1868	−0.530307
$446$	−75.6701	−3.58308
$447$	0	0
$448$	−127.630	−6.02993
$449$	34.5840	1.63212	0.816060	−	0.577967i	$-0.196154\pi$
$449$	34.5840	1.63212	0.816060	+	0.577967i	$0.196154\pi$
$450$	0	0
$451$	−31.8154	−1.49813
$452$	−48.7353	−2.29232
$453$	0	0
$454$	25.7277	1.20746
$455$	4.53684	0.212690
$456$	0	0
$457$	−30.3891	−1.42154	−0.710770	−	0.703424i	$-0.751653\pi$
$457$	−30.3891	−1.42154	−0.710770	+	0.703424i	$0.751653\pi$
$458$	6.07426	0.283831
$459$	0	0
$460$	37.8260	1.76365
$461$	−38.5102	−1.79360	−0.896800	−	0.442437i	$-0.854114\pi$
$461$	−38.5102	−1.79360	−0.896800	+	0.442437i	$0.854114\pi$
$462$	0	0
$463$	25.0858	1.16584	0.582918	−	0.812531i	$-0.301911\pi$
$463$	25.0858	1.16584	0.582918	+	0.812531i	$0.301911\pi$
$464$	9.46363	0.439338
$465$	0	0
$466$	19.3614	0.896901
$467$	−4.72517	−0.218655	−0.109327	−	0.994006i	$-0.534870\pi$
$467$	−4.72517	−0.218655	−0.109327	+	0.994006i	$0.534870\pi$
$468$	0	0
$469$	−21.6212	−0.998375
$470$	13.0737	0.603044
$471$	0	0
$472$	−19.2034	−0.883908
$473$	50.3764	2.31631
$474$	0	0
$475$	−7.13511	−0.327381
$476$	107.250	4.91578
$477$	0	0
$478$	45.3186	2.07282
$479$	−16.4466	−0.751463	−0.375732	−	0.926729i	$-0.622609\pi$
$479$	−16.4466	−0.751463	−0.375732	+	0.926729i	$0.622609\pi$
$480$	0	0
$481$	2.41467	0.110099
$482$	−18.0770	−0.823383
$483$	0	0
$484$	64.0110	2.90959
$485$	8.90090	0.404169
$486$	0	0
$487$	−14.6256	−0.662749	−0.331375	−	0.943499i	$-0.607512\pi$
$487$	−14.6256	−0.662749	−0.331375	+	0.943499i	$0.607512\pi$
$488$	−78.4380	−3.55072
$489$	0	0
$490$	37.0053	1.67173
$491$	36.9223	1.66628	0.833141	−	0.553061i	$-0.186540\pi$
$491$	36.9223	1.66628	0.833141	+	0.553061i	$0.186540\pi$
$492$	0	0
$493$	−2.83416	−0.127644
$494$	19.4389	0.874599
$495$	0	0
$496$	−34.0120	−1.52718
$497$	13.6743	0.613378
$498$	0	0
$499$	−19.8678	−0.889405	−0.444702	−	0.895678i	$-0.646691\pi$
$499$	−19.8678	−0.889405	−0.444702	+	0.895678i	$0.646691\pi$
$500$	−5.42239	−0.242496
$501$	0	0
$502$	−54.0773	−2.41359
$503$	−9.06638	−0.404250	−0.202125	−	0.979360i	$-0.564785\pi$
$503$	−9.06638	−0.404250	−0.202125	+	0.979360i	$0.564785\pi$
$504$	0	0
$505$	1.62486	0.0723054
$506$	−90.7583	−4.03470
$507$	0	0
$508$	41.0584	1.82167
$509$	−30.6433	−1.35824	−0.679120	−	0.734027i	$-0.737638\pi$
$509$	−30.6433	−1.35824	−0.679120	+	0.734027i	$0.737638\pi$
$510$	0	0
$511$	18.5061	0.818660
$512$	−34.4235	−1.52132
$513$	0	0
$514$	67.2284	2.96532
$515$	11.6435	0.513074
$516$	0	0
$517$	−22.9161	−1.00785
$518$	29.8457	1.31135
$519$	0	0
$520$	9.32397	0.408883
$521$	−10.3801	−0.454761	−0.227380	−	0.973806i	$-0.573016\pi$
$521$	−10.3801	−0.454761	−0.227380	+	0.973806i	$0.573016\pi$
$522$	0	0
$523$	−33.2303	−1.45306	−0.726530	−	0.687134i	$-0.758868\pi$
$523$	−33.2303	−1.45306	−0.726530	+	0.687134i	$0.758868\pi$
$524$	2.15717	0.0942366
$525$	0	0
$526$	−34.8865	−1.52113
$527$	10.1859	0.443703
$528$	0	0
$529$	25.6632	1.11579
$530$	−6.92607	−0.300849
$531$	0	0
$532$	175.527	7.61007
$533$	6.66229	0.288576
$534$	0	0
$535$	−7.17008	−0.309990
$536$	−44.4353	−1.91931
$537$	0	0
$538$	−0.671543	−0.0289523
$539$	−64.8644	−2.79391
$540$	0	0
$541$	−37.9781	−1.63281	−0.816403	−	0.577482i	$-0.804035\pi$
$541$	−37.9781	−1.63281	−0.816403	+	0.577482i	$0.804035\pi$
$542$	25.7212	1.10482
$543$	0	0
$544$	91.6078	3.92765
$545$	20.3808	0.873017
$546$	0	0
$547$	25.0129	1.06947	0.534737	−	0.845018i	$-0.320411\pi$
$547$	25.0129	1.06947	0.534737	+	0.845018i	$0.320411\pi$
$548$	38.5627	1.64732
$549$	0	0
$550$	13.0103	0.554760
$551$	−4.63844	−0.197604
$552$	0	0
$553$	59.6472	2.53646
$554$	−4.36497	−0.185450
$555$	0	0
$556$	7.44594	0.315778
$557$	−38.8205	−1.64488	−0.822438	−	0.568854i	$-0.807387\pi$
$557$	−38.8205	−1.64488	−0.822438	+	0.568854i	$0.807387\pi$
$558$	0	0
$559$	−10.5490	−0.446177
$560$	66.0450	2.79091
$561$	0	0
$562$	69.9777	2.95183
$563$	−19.3949	−0.817397	−0.408698	−	0.912669i	$-0.634017\pi$
$563$	−19.3949	−0.817397	−0.408698	+	0.912669i	$0.634017\pi$
$564$	0	0
$565$	8.98780	0.378120
$566$	−49.8304	−2.09453
$567$	0	0
$568$	28.1031	1.17918
$569$	−14.9178	−0.625386	−0.312693	−	0.949854i	$-0.601231\pi$
$569$	−14.9178	−0.625386	−0.312693	+	0.949854i	$0.601231\pi$
$570$	0	0
$571$	9.10135	0.380880	0.190440	−	0.981699i	$-0.439009\pi$
$571$	9.10135	0.380880	0.190440	+	0.981699i	$0.439009\pi$
$572$	−25.8943	−1.08270
$573$	0	0
$574$	82.3471	3.43710
$575$	−6.97590	−0.290915
$576$	0	0
$577$	5.31889	0.221428	0.110714	−	0.993852i	$-0.464686\pi$
$577$	5.31889	0.221428	0.110714	+	0.993852i	$0.464686\pi$
$578$	−5.46689	−0.227393
$579$	0	0
$580$	−3.52502	−0.146369
$581$	−1.31723	−0.0546478
$582$	0	0
$583$	12.1403	0.502800
$584$	38.0331	1.57382
$585$	0	0
$586$	−56.4913	−2.33363
$587$	27.8915	1.15121	0.575603	−	0.817729i	$-0.304767\pi$
$587$	27.8915	1.15121	0.575603	+	0.817729i	$0.304767\pi$
$588$	0	0
$589$	16.6704	0.686891
$590$	5.61111	0.231006
$591$	0	0
$592$	35.1515	1.44472
$593$	4.91638	0.201892	0.100946	−	0.994892i	$-0.467813\pi$
$593$	4.91638	0.201892	0.100946	+	0.994892i	$0.467813\pi$
$594$	0	0
$595$	−19.7791	−0.810863
$596$	−19.1545	−0.784598
$597$	0	0
$598$	19.0052	0.777180
$599$	−7.20206	−0.294268	−0.147134	−	0.989117i	$-0.547005\pi$
$599$	−7.20206	−0.294268	−0.147134	+	0.989117i	$0.547005\pi$
$600$	0	0
$601$	−28.3928	−1.15817	−0.579083	−	0.815268i	$-0.696589\pi$
$601$	−28.3928	−1.15817	−0.579083	+	0.815268i	$0.696589\pi$
$602$	−130.388	−5.31422
$603$	0	0
$604$	−12.8550	−0.523064
$605$	−11.8049	−0.479939
$606$	0	0
$607$	30.6780	1.24518	0.622592	−	0.782547i	$-0.286080\pi$
$607$	30.6780	1.24518	0.622592	+	0.782547i	$0.286080\pi$
$608$	149.927	6.08035
$609$	0	0
$610$	22.9191	0.927967
$611$	4.79872	0.194136
$612$	0	0
$613$	34.6044	1.39766	0.698829	−	0.715289i	$-0.253705\pi$
$613$	34.6044	1.39766	0.698829	+	0.715289i	$0.253705\pi$
$614$	−48.4654	−1.95590
$615$	0	0
$616$	−202.008	−8.13913
$617$	3.62491	0.145934	0.0729668	−	0.997334i	$-0.476753\pi$
$617$	3.62491	0.145934	0.0729668	+	0.997334i	$0.476753\pi$
$618$	0	0
$619$	−19.7087	−0.792161	−0.396081	−	0.918216i	$-0.629630\pi$
$619$	−19.7087	−0.792161	−0.396081	+	0.918216i	$0.629630\pi$
$620$	12.6688	0.508791
$621$	0	0
$622$	−51.6189	−2.06973
$623$	−50.7528	−2.03337
$624$	0	0
$625$	1.00000	0.0400000
$626$	−11.4496	−0.457618
$627$	0	0
$628$	89.1817	3.55874
$629$	−10.5271	−0.419744
$630$	0	0
$631$	28.9457	1.15231	0.576155	−	0.817340i	$-0.304552\pi$
$631$	28.9457	1.15231	0.576155	+	0.817340i	$0.304552\pi$
$632$	122.585	4.87617
$633$	0	0
$634$	29.9100	1.18788
$635$	−7.57201	−0.300486
$636$	0	0
$637$	13.5829	0.538173
$638$	8.45780	0.334848
$639$	0	0
$640$	34.6173	1.36837
$641$	−10.1731	−0.401813	−0.200906	−	0.979610i	$-0.564389\pi$
$641$	−10.1731	−0.401813	−0.200906	+	0.979610i	$0.564389\pi$
$642$	0	0
$643$	−25.7920	−1.01714	−0.508568	−	0.861022i	$-0.669825\pi$
$643$	−25.7920	−1.01714	−0.508568	+	0.861022i	$0.669825\pi$
$644$	171.610	6.76240
$645$	0	0
$646$	−84.7472	−3.33433
$647$	−0.966674	−0.0380039	−0.0190019	−	0.999819i	$-0.506049\pi$
$647$	−0.966674	−0.0380039	−0.0190019	+	0.999819i	$0.506049\pi$
$648$	0	0
$649$	−9.83539	−0.386073
$650$	−2.72441	−0.106860
$651$	0	0
$652$	37.4194	1.46546
$653$	−27.1124	−1.06099	−0.530496	−	0.847687i	$-0.677994\pi$
$653$	−27.1124	−1.06099	−0.530496	+	0.847687i	$0.677994\pi$
$654$	0	0
$655$	−0.397827	−0.0155444
$656$	96.9862	3.78668
$657$	0	0
$658$	59.3131	2.31227
$659$	29.2239	1.13840	0.569201	−	0.822199i	$-0.307253\pi$
$659$	29.2239	1.13840	0.569201	+	0.822199i	$0.307253\pi$
$660$	0	0
$661$	29.4138	1.14406	0.572032	−	0.820231i	$-0.306155\pi$
$661$	29.4138	1.14406	0.572032	+	0.820231i	$0.306155\pi$
$662$	20.7802	0.807646
$663$	0	0
$664$	−2.70712	−0.105057
$665$	−32.3708	−1.25529
$666$	0	0
$667$	−4.53494	−0.175594
$668$	75.9310	2.93786
$669$	0	0
$670$	12.9837	0.501605
$671$	−40.1735	−1.55088
$672$	0	0
$673$	−45.9219	−1.77016	−0.885079	−	0.465440i	$-0.845896\pi$
$673$	−45.9219	−1.77016	−0.885079	+	0.465440i	$0.845896\pi$
$674$	−1.23700	−0.0476476
$675$	0	0
$676$	5.42239	0.208553
$677$	21.3792	0.821671	0.410836	−	0.911709i	$-0.365237\pi$
$677$	21.3792	0.821671	0.410836	+	0.911709i	$0.365237\pi$
$678$	0	0
$679$	40.3819	1.54972
$680$	−40.6493	−1.55883
$681$	0	0
$682$	−30.3970	−1.16396
$683$	−37.8764	−1.44930	−0.724651	−	0.689116i	$-0.757999\pi$
$683$	−37.8764	−1.44930	−0.724651	+	0.689116i	$0.757999\pi$
$684$	0	0
$685$	−7.11175	−0.271726
$686$	81.3656	3.10655
$687$	0	0
$688$	−153.568	−5.85471
$689$	−2.54223	−0.0968513
$690$	0	0
$691$	−27.7986	−1.05751	−0.528755	−	0.848775i	$-0.677341\pi$
$691$	−27.7986	−1.05751	−0.528755	+	0.848775i	$0.677341\pi$
$692$	−105.624	−4.01523
$693$	0	0
$694$	24.8299	0.942529
$695$	−1.37319	−0.0520879
$696$	0	0
$697$	−29.0453	−1.10017
$698$	17.3279	0.655870
$699$	0	0
$700$	−24.6005	−0.929811
$701$	8.98692	0.339431	0.169716	−	0.985493i	$-0.445715\pi$
$701$	8.98692	0.339431	0.169716	+	0.985493i	$0.445715\pi$
$702$	0	0
$703$	−17.2289	−0.649801
$704$	−134.342	−5.06321
$705$	0	0
$706$	86.2297	3.24530
$707$	7.37173	0.277242
$708$	0	0
$709$	−13.1134	−0.492485	−0.246242	−	0.969208i	$-0.579196\pi$
$709$	−13.1134	−0.492485	−0.246242	+	0.969208i	$0.579196\pi$
$710$	−8.21154	−0.308174
$711$	0	0
$712$	−104.306	−3.90902
$713$	16.2984	0.610380
$714$	0	0
$715$	4.77545	0.178592
$716$	−33.8397	−1.26465
$717$	0	0
$718$	−60.6376	−2.26298
$719$	23.4933	0.876153	0.438077	−	0.898938i	$-0.355660\pi$
$719$	23.4933	0.876153	0.438077	+	0.898938i	$0.355660\pi$
$720$	0	0
$721$	52.8246	1.96729
$722$	−86.9353	−3.23540
$723$	0	0
$724$	−67.8163	−2.52037
$725$	0.650087	0.0241436
$726$	0	0
$727$	7.73497	0.286874	0.143437	−	0.989659i	$-0.454185\pi$
$727$	7.73497	0.286874	0.143437	+	0.989659i	$0.454185\pi$
$728$	42.3013	1.56779
$729$	0	0
$730$	−11.1130	−0.411312
$731$	45.9902	1.70101
$732$	0	0
$733$	22.2927	0.823399	0.411699	−	0.911320i	$-0.364935\pi$
$733$	22.2927	0.823399	0.411699	+	0.911320i	$0.364935\pi$
$734$	−66.5772	−2.45741
$735$	0	0
$736$	146.582	5.40308
$737$	−22.7584	−0.838316
$738$	0	0
$739$	7.99357	0.294048	0.147024	−	0.989133i	$-0.453031\pi$
$739$	7.99357	0.294048	0.147024	+	0.989133i	$0.453031\pi$
$740$	−13.0933	−0.481317
$741$	0	0
$742$	−31.4224	−1.15355
$743$	16.8137	0.616834	0.308417	−	0.951251i	$-0.400201\pi$
$743$	16.8137	0.616834	0.308417	+	0.951251i	$0.400201\pi$
$744$	0	0
$745$	3.53248	0.129420
$746$	−50.4813	−1.84825
$747$	0	0
$748$	112.891	4.12769
$749$	−32.5295	−1.18860
$750$	0	0
$751$	31.2709	1.14109	0.570546	−	0.821266i	$-0.306732\pi$
$751$	31.2709	1.14109	0.570546	+	0.821266i	$0.306732\pi$
$752$	69.8574	2.54744
$753$	0	0
$754$	−1.77110	−0.0644996
$755$	2.37074	0.0862799
$756$	0	0
$757$	−0.439392	−0.0159700	−0.00798499	−	0.999968i	$-0.502542\pi$
$757$	−0.439392	−0.0159700	−0.00798499	+	0.999968i	$0.502542\pi$
$758$	−15.5383	−0.564375
$759$	0	0
$760$	−66.5276	−2.41321
$761$	−15.1922	−0.550718	−0.275359	−	0.961341i	$-0.588797\pi$
$761$	−15.1922	−0.550718	−0.275359	+	0.961341i	$0.588797\pi$
$762$	0	0
$763$	92.4643	3.34743
$764$	−29.3820	−1.06300
$765$	0	0
$766$	4.30732	0.155630
$767$	2.05957	0.0743669
$768$	0	0
$769$	−26.4060	−0.952224	−0.476112	−	0.879385i	$-0.657954\pi$
$769$	−26.4060	−0.952224	−0.476112	+	0.879385i	$0.657954\pi$
$770$	59.0255	2.12713
$771$	0	0
$772$	50.1012	1.80318
$773$	−43.0738	−1.54926	−0.774628	−	0.632417i	$-0.782063\pi$
$773$	−43.0738	−1.54926	−0.774628	+	0.632417i	$0.782063\pi$
$774$	0	0
$775$	−2.33639	−0.0839255
$776$	82.9917	2.97923
$777$	0	0
$778$	16.3364	0.585687
$779$	−47.5362	−1.70316
$780$	0	0
$781$	14.3935	0.515041
$782$	−82.8561	−2.96293
$783$	0	0
$784$	197.733	7.06189
$785$	−16.4469	−0.587017
$786$	0	0
$787$	−41.9806	−1.49645	−0.748223	−	0.663447i	$-0.769093\pi$
$787$	−41.9806	−1.49645	−0.748223	+	0.663447i	$0.769093\pi$
$788$	136.226	4.85286
$789$	0	0
$790$	−35.8186	−1.27437
$791$	40.7762	1.44983
$792$	0	0
$793$	8.41251	0.298737
$794$	100.524	3.56747
$795$	0	0
$796$	−21.5942	−0.765387
$797$	3.07263	0.108838	0.0544191	−	0.998518i	$-0.482669\pi$
$797$	3.07263	0.108838	0.0544191	+	0.998518i	$0.482669\pi$
$798$	0	0
$799$	−20.9208	−0.740125
$800$	−21.0126	−0.742907
$801$	0	0
$802$	18.1500	0.640898
$803$	19.4794	0.687413
$804$	0	0
$805$	−31.6485	−1.11546
$806$	6.36527	0.224207
$807$	0	0
$808$	15.1502	0.532981
$809$	18.9489	0.666208	0.333104	−	0.942890i	$-0.391904\pi$
$809$	18.9489	0.666208	0.333104	+	0.942890i	$0.391904\pi$
$810$	0	0
$811$	23.2117	0.815075	0.407537	−	0.913189i	$-0.366388\pi$
$811$	23.2117	0.815075	0.407537	+	0.913189i	$0.366388\pi$
$812$	−15.9924	−0.561225
$813$	0	0
$814$	31.4155	1.10111
$815$	−6.90090	−0.241728
$816$	0	0
$817$	75.2686	2.63331
$818$	−23.5839	−0.824591
$819$	0	0
$820$	−36.1255	−1.26156
$821$	−34.5746	−1.20666	−0.603331	−	0.797491i	$-0.706160\pi$
$821$	−34.5746	−1.20666	−0.603331	+	0.797491i	$0.706160\pi$
$822$	0	0
$823$	24.2652	0.845831	0.422916	−	0.906169i	$-0.361007\pi$
$823$	24.2652	0.845831	0.422916	+	0.906169i	$0.361007\pi$
$824$	108.564	3.78199
$825$	0	0
$826$	25.4567	0.885752
$827$	43.8801	1.52586	0.762930	−	0.646482i	$-0.223760\pi$
$827$	43.8801	1.52586	0.762930	+	0.646482i	$0.223760\pi$
$828$	0	0
$829$	39.0820	1.35737	0.678687	−	0.734428i	$-0.262549\pi$
$829$	39.0820	1.35737	0.678687	+	0.734428i	$0.262549\pi$
$830$	0.791005	0.0274562
$831$	0	0
$832$	28.1318	0.975296
$833$	−59.2168	−2.05174
$834$	0	0
$835$	−14.0032	−0.484602
$836$	184.759	6.39003
$837$	0	0
$838$	−51.2833	−1.77155
$839$	−10.9849	−0.379239	−0.189620	−	0.981858i	$-0.560726\pi$
$839$	−10.9849	−0.379239	−0.189620	+	0.981858i	$0.560726\pi$
$840$	0	0
$841$	−28.5774	−0.985427
$842$	−81.2581	−2.80034
$843$	0	0
$844$	−96.7027	−3.32865
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	−53.5571	−1.84024
$848$	−37.0085	−1.27088
$849$	0	0
$850$	11.8775	0.407394
$851$	−16.8445	−0.577421
$852$	0	0
$853$	−34.4403	−1.17921	−0.589607	−	0.807690i	$-0.700717\pi$
$853$	−34.4403	−1.17921	−0.589607	+	0.807690i	$0.700717\pi$
$854$	103.980	3.55813
$855$	0	0
$856$	−66.8536	−2.28501
$857$	−15.1321	−0.516904	−0.258452	−	0.966024i	$-0.583212\pi$
$857$	−15.1321	−0.516904	−0.258452	+	0.966024i	$0.583212\pi$
$858$	0	0
$859$	1.83576	0.0626353	0.0313177	−	0.999509i	$-0.490030\pi$
$859$	1.83576	0.0626353	0.0313177	+	0.999509i	$0.490030\pi$
$860$	57.2010	1.95054
$861$	0	0
$862$	45.3285	1.54389
$863$	−45.6326	−1.55335	−0.776676	−	0.629901i	$-0.783096\pi$
$863$	−45.6326	−1.55335	−0.776676	+	0.629901i	$0.783096\pi$
$864$	0	0
$865$	19.4793	0.662316
$866$	44.6842	1.51843
$867$	0	0
$868$	57.4763	1.95087
$869$	62.7844	2.12981
$870$	0	0
$871$	4.76571	0.161480
$872$	190.030	6.43522
$873$	0	0
$874$	−135.604	−4.58688
$875$	4.53684	0.153373
$876$	0	0
$877$	41.5289	1.40233	0.701166	−	0.712998i	$-0.252663\pi$
$877$	41.5289	1.40233	0.701166	+	0.712998i	$0.252663\pi$
$878$	5.70817	0.192642
$879$	0	0
$880$	69.5186	2.34347
$881$	−45.1513	−1.52119	−0.760593	−	0.649229i	$-0.775092\pi$
$881$	−45.1513	−1.52119	−0.760593	+	0.649229i	$0.775092\pi$
$882$	0	0
$883$	−25.0763	−0.843886	−0.421943	−	0.906622i	$-0.638652\pi$
$883$	−25.0763	−0.843886	−0.421943	+	0.906622i	$0.638652\pi$
$884$	−23.6398	−0.795091
$885$	0	0
$886$	24.3461	0.817922
$887$	−41.5913	−1.39650	−0.698249	−	0.715855i	$-0.746037\pi$
$887$	−41.5913	−1.39650	−0.698249	+	0.715855i	$0.746037\pi$
$888$	0	0
$889$	−34.3530	−1.15216
$890$	30.4775	1.02161
$891$	0	0
$892$	150.606	5.04267
$893$	−34.2394	−1.14578
$894$	0	0
$895$	6.24073	0.208605
$896$	157.053	5.24678
$897$	0	0
$898$	−94.2209	−3.14419
$899$	−1.51885	−0.0506566
$900$	0	0
$901$	11.0833	0.369237
$902$	86.6782	2.88607
$903$	0	0
$904$	83.8020	2.78721
$905$	12.5067	0.415738
$906$	0	0
$907$	−25.9178	−0.860585	−0.430293	−	0.902690i	$-0.641590\pi$
$907$	−25.9178	−0.860585	−0.430293	+	0.902690i	$0.641590\pi$
$908$	−51.2059	−1.69933
$909$	0	0
$910$	−12.3602	−0.409736
$911$	19.6577	0.651290	0.325645	−	0.945492i	$-0.394419\pi$
$911$	19.6577	0.651290	0.325645	+	0.945492i	$0.394419\pi$
$912$	0	0
$913$	−1.38651	−0.0458867
$914$	82.7922	2.73852
$915$	0	0
$916$	−12.0896	−0.399452
$917$	−1.80488	−0.0596023
$918$	0	0
$919$	−44.8424	−1.47921	−0.739606	−	0.673040i	$-0.764988\pi$
$919$	−44.8424	−1.47921	−0.739606	+	0.673040i	$0.764988\pi$
$920$	−65.0431	−2.14441
$921$	0	0
$922$	104.917	3.45527
$923$	−3.01407	−0.0992093
$924$	0	0
$925$	2.41467	0.0793937
$926$	−68.3439	−2.24592
$927$	0	0
$928$	−13.6600	−0.448412
$929$	8.02378	0.263252	0.131626	−	0.991299i	$-0.457980\pi$
$929$	8.02378	0.263252	0.131626	+	0.991299i	$0.457980\pi$
$930$	0	0
$931$	−96.9154	−3.17627
$932$	−38.5351	−1.26226
$933$	0	0
$934$	12.8733	0.421227
$935$	−20.8193	−0.680866
$936$	0	0
$937$	−5.38461	−0.175907	−0.0879537	−	0.996125i	$-0.528033\pi$
$937$	−5.38461	−0.175907	−0.0879537	+	0.996125i	$0.528033\pi$
$938$	58.9050	1.92332
$939$	0	0
$940$	−26.0205	−0.848696
$941$	28.3147	0.923033	0.461516	−	0.887132i	$-0.347306\pi$
$941$	28.3147	0.923033	0.461516	+	0.887132i	$0.347306\pi$
$942$	0	0
$943$	−46.4755	−1.51345
$944$	29.9822	0.975839
$945$	0	0
$946$	−137.246	−4.46225
$947$	−45.1431	−1.46696	−0.733478	−	0.679714i	$-0.762104\pi$
$947$	−45.1431	−1.46696	−0.733478	+	0.679714i	$0.762104\pi$
$948$	0	0
$949$	−4.07907	−0.132412
$950$	19.4389	0.630683
$951$	0	0
$952$	−184.419	−5.97707
$953$	13.9180	0.450848	0.225424	−	0.974261i	$-0.427623\pi$
$953$	13.9180	0.450848	0.225424	+	0.974261i	$0.427623\pi$
$954$	0	0
$955$	5.41865	0.175343
$956$	−90.1976	−2.91720
$957$	0	0
$958$	44.8071	1.44765
$959$	−32.2649	−1.04189
$960$	0	0
$961$	−25.5413	−0.823913
$962$	−6.57853	−0.212100
$963$	0	0
$964$	35.9786	1.15879
$965$	−9.23969	−0.297436
$966$	0	0
$967$	60.6064	1.94897	0.974485	−	0.224451i	$-0.0720588\pi$
$967$	60.6064	1.94897	0.974485	+	0.224451i	$0.0720588\pi$
$968$	−110.069	−3.53775
$969$	0	0
$970$	−24.2497	−0.778610
$971$	32.1286	1.03105	0.515527	−	0.856873i	$-0.327596\pi$
$971$	32.1286	1.03105	0.515527	+	0.856873i	$0.327596\pi$
$972$	0	0
$973$	−6.22992	−0.199722
$974$	39.8461	1.27675
$975$	0	0
$976$	122.465	3.92001
$977$	36.9639	1.18258	0.591290	−	0.806459i	$-0.298619\pi$
$977$	36.9639	1.18258	0.591290	+	0.806459i	$0.298619\pi$
$978$	0	0
$979$	−53.4222	−1.70738
$980$	−73.6516	−2.35272
$981$	0	0
$982$	−100.591	−3.21000
$983$	−17.1828	−0.548047	−0.274023	−	0.961723i	$-0.588355\pi$
$983$	−17.1828	−0.548047	−0.274023	+	0.961723i	$0.588355\pi$
$984$	0	0
$985$	−25.1230	−0.800484
$986$	7.72139	0.245899
$987$	0	0
$988$	−38.6893	−1.23087
$989$	73.5890	2.34000
$990$	0	0
$991$	59.1332	1.87843	0.939214	−	0.343332i	$-0.111555\pi$
$991$	59.1332	1.87843	0.939214	+	0.343332i	$0.111555\pi$
$992$	49.0936	1.55872
$993$	0	0
$994$	−37.2544	−1.18164
$995$	3.98242	0.126251
$996$	0	0
$997$	55.0571	1.74367	0.871837	−	0.489796i	$-0.162929\pi$
$997$	55.0571	1.74367	0.871837	+	0.489796i	$0.162929\pi$
$998$	54.1279	1.71339
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1755.2.a.u.1.1	✓	7
3.2	odd	2		1755.2.a.v.1.7	yes	7
5.4	even	2		8775.2.a.bx.1.7		7
15.14	odd	2		8775.2.a.bw.1.1		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1755.2.a.u.1.1	✓	7	1.1	even	1	trivial
1755.2.a.v.1.7	yes	7	3.2	odd	2
8775.2.a.bw.1.1		7	15.14	odd	2
8775.2.a.bx.1.7		7	5.4	even	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:	\( N \)	\(=\)	\( 1755 = 3^{3} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1755.a (trivial)

\(f(q)\)	\(=\)	\(q-2.72441 q^{2} +5.42239 q^{4} -1.00000 q^{5} -4.53684 q^{7} -9.32397 q^{8} +O(q^{10})\)	\(q-2.72441 q^{2} +5.42239 q^{4} -1.00000 q^{5} -4.53684 q^{7} -9.32397 q^{8} +2.72441 q^{10} -4.77545 q^{11} +1.00000 q^{13} +12.3602 q^{14} +14.5575 q^{16} -4.35966 q^{17} -7.13511 q^{19} -5.42239 q^{20} +13.0103 q^{22} -6.97590 q^{23} +1.00000 q^{25} -2.72441 q^{26} -24.6005 q^{28} +0.650087 q^{29} -2.33639 q^{31} -21.0126 q^{32} +11.8775 q^{34} +4.53684 q^{35} +2.41467 q^{37} +19.4389 q^{38} +9.32397 q^{40} +6.66229 q^{41} -10.5490 q^{43} -25.8943 q^{44} +19.0052 q^{46} +4.79872 q^{47} +13.5829 q^{49} -2.72441 q^{50} +5.42239 q^{52} -2.54223 q^{53} +4.77545 q^{55} +42.3013 q^{56} -1.77110 q^{58} +2.05957 q^{59} +8.41251 q^{61} +6.36527 q^{62} +28.1318 q^{64} -1.00000 q^{65} +4.76571 q^{67} -23.6398 q^{68} -12.3602 q^{70} -3.01407 q^{71} -4.07907 q^{73} -6.57853 q^{74} -38.6893 q^{76} +21.6654 q^{77} -13.1473 q^{79} -14.5575 q^{80} -18.1508 q^{82} +0.290340 q^{83} +4.35966 q^{85} +28.7399 q^{86} +44.5262 q^{88} +11.1868 q^{89} -4.53684 q^{91} -37.8260 q^{92} -13.0737 q^{94} +7.13511 q^{95} -8.90090 q^{97} -37.0053 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - q^{2} + 11 q^{4} - 7 q^{5} + 2 q^{7} - 6 q^{8}+O(q^{10}) \) 7 * q - q^2 + 11 * q^4 - 7 * q^5 + 2 * q^7 - 6 * q^8	\( 7 q - q^{2} + 11 q^{4} - 7 q^{5} + 2 q^{7} - 6 q^{8} + q^{10} - q^{11} + 7 q^{13} + 12 q^{14} + 23 q^{16} - 11 q^{17} + 2 q^{19} - 11 q^{20} + 16 q^{22} + q^{23} + 7 q^{25} - q^{26} + 10 q^{28} + 4 q^{29} - 13 q^{32} + q^{34} - 2 q^{35} + 23 q^{37} + 15 q^{38} + 6 q^{40} - 2 q^{41} + 8 q^{43} - 10 q^{44} + 37 q^{46} - 2 q^{47} + 43 q^{49} - q^{50} + 11 q^{52} - 10 q^{53} + q^{55} + 68 q^{56} - 26 q^{58} + 13 q^{59} + 21 q^{61} - 9 q^{62} + 46 q^{64} - 7 q^{65} + 21 q^{67} - 53 q^{68} - 12 q^{70} + 10 q^{71} + 13 q^{73} + 68 q^{74} - 41 q^{76} - 6 q^{77} + 8 q^{79} - 23 q^{80} - 26 q^{82} + 4 q^{83} + 11 q^{85} + 12 q^{86} + 44 q^{88} + 27 q^{89} + 2 q^{91} + 9 q^{92} - 24 q^{94} - 2 q^{95} - 15 q^{97} - 7 q^{98}+O(q^{100}) \) 7 * q - q^2 + 11 * q^4 - 7 * q^5 + 2 * q^7 - 6 * q^8 + q^10 - q^11 + 7 * q^13 + 12 * q^14 + 23 * q^16 - 11 * q^17 + 2 * q^19 - 11 * q^20 + 16 * q^22 + q^23 + 7 * q^25 - q^26 + 10 * q^28 + 4 * q^29 - 13 * q^32 + q^34 - 2 * q^35 + 23 * q^37 + 15 * q^38 + 6 * q^40 - 2 * q^41 + 8 * q^43 - 10 * q^44 + 37 * q^46 - 2 * q^47 + 43 * q^49 - q^50 + 11 * q^52 - 10 * q^53 + q^55 + 68 * q^56 - 26 * q^58 + 13 * q^59 + 21 * q^61 - 9 * q^62 + 46 * q^64 - 7 * q^65 + 21 * q^67 - 53 * q^68 - 12 * q^70 + 10 * q^71 + 13 * q^73 + 68 * q^74 - 41 * q^76 - 6 * q^77 + 8 * q^79 - 23 * q^80 - 26 * q^82 + 4 * q^83 + 11 * q^85 + 12 * q^86 + 44 * q^88 + 27 * q^89 + 2 * q^91 + 9 * q^92 - 24 * q^94 - 2 * q^95 - 15 * q^97 - 7 * q^98

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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