LMFDB - Embedded newform 1850.2.a.ba.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1850,2,Mod(1,1850)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1850, 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("1850.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1850 = 2 \cdot 5^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1850.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:	$14.7723243739$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.17009$ of defining polynomial
Character	$\chi$	$=$	1850.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} +1.53919 q^{3} +1.00000 q^{4} -1.53919 q^{6} -2.87936 q^{7} -1.00000 q^{8} -0.630898 q^{9} -1.09171 q^{11} +1.53919 q^{12} +4.53919 q^{13} +2.87936 q^{14} +1.00000 q^{16} +2.80098 q^{17} +0.630898 q^{18} -5.04945 q^{19} -4.43188 q^{21} +1.09171 q^{22} +7.41855 q^{23} -1.53919 q^{24} -4.53919 q^{26} -5.58864 q^{27} -2.87936 q^{28} +6.68035 q^{29} +3.51026 q^{31} -1.00000 q^{32} -1.68035 q^{33} -2.80098 q^{34} -0.630898 q^{36} +1.00000 q^{37} +5.04945 q^{38} +6.98667 q^{39} -8.07838 q^{41} +4.43188 q^{42} +10.2329 q^{43} -1.09171 q^{44} -7.41855 q^{46} +8.68035 q^{47} +1.53919 q^{48} +1.29072 q^{49} +4.31124 q^{51} +4.53919 q^{52} -10.0989 q^{53} +5.58864 q^{54} +2.87936 q^{56} -7.77205 q^{57} -6.68035 q^{58} +10.2329 q^{59} +6.29791 q^{61} -3.51026 q^{62} +1.81658 q^{63} +1.00000 q^{64} +1.68035 q^{66} +13.2979 q^{67} +2.80098 q^{68} +11.4186 q^{69} +6.29791 q^{71} +0.630898 q^{72} +12.7093 q^{73} -1.00000 q^{74} -5.04945 q^{76} +3.14342 q^{77} -6.98667 q^{78} +2.58145 q^{79} -6.70928 q^{81} +8.07838 q^{82} +8.48360 q^{83} -4.43188 q^{84} -10.2329 q^{86} +10.2823 q^{87} +1.09171 q^{88} -6.51026 q^{89} -13.0700 q^{91} +7.41855 q^{92} +5.40295 q^{93} -8.68035 q^{94} -1.53919 q^{96} +3.07838 q^{97} -1.29072 q^{98} +0.688756 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} - 3 q^{6} + 4 q^{7} - 3 q^{8} + 2 q^{9} - q^{11} + 3 q^{12} + 12 q^{13} - 4 q^{14} + 3 q^{16} - q^{17} - 2 q^{18} + 3 q^{19} + q^{22} + 8 q^{23} - 3 q^{24} - 12 q^{26}+ \cdots + 28 q^{99}+O(q^{100}) $ 3 * q - 3 * q^2 + 3 * q^3 + 3 * q^4 - 3 * q^6 + 4 * q^7 - 3 * q^8 + 2 * q^9 - q^11 + 3 * q^12 + 12 * q^13 - 4 * q^14 + 3 * q^16 - q^17 - 2 * q^18 + 3 * q^19 + q^22 + 8 * q^23 - 3 * q^24 - 12 * q^26 + 3 * q^27 + 4 * q^28 - 2 * q^29 - 6 * q^31 - 3 * q^32 + 17 * q^33 + q^34 + 2 * q^36 + 3 * q^37 - 3 * q^38 + 20 * q^39 - 21 * q^41 + 8 * q^43 - q^44 - 8 * q^46 + 4 * q^47 + 3 * q^48 + 11 * q^49 - 13 * q^51 + 12 * q^52 + 6 * q^53 - 3 * q^54 - 4 * q^56 + q^57 + 2 * q^58 + 8 * q^59 - 8 * q^61 + 6 * q^62 + 10 * q^63 + 3 * q^64 - 17 * q^66 + 13 * q^67 - q^68 + 20 * q^69 - 8 * q^71 - 2 * q^72 + 31 * q^73 - 3 * q^74 + 3 * q^76 + 2 * q^77 - 20 * q^78 + 22 * q^79 - 13 * q^81 + 21 * q^82 + 7 * q^83 - 8 * q^86 - 10 * q^87 + q^88 - 3 * q^89 + 12 * q^91 + 8 * q^92 - 12 * q^93 - 4 * q^94 - 3 * q^96 + 6 * q^97 - 11 * q^98 + 28 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.53919	0.888651	0.444326	−	0.895865i	$-0.353443\pi$
$3$	1.53919	0.888651	0.444326	+	0.895865i	$0.353443\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−1.53919	−0.628371
$7$	−2.87936	−1.08830	−0.544148	−	0.838989i	$-0.683147\pi$
$7$	−2.87936	−1.08830	−0.544148	+	0.838989i	$0.683147\pi$
$8$	−1.00000	−0.353553
$9$	−0.630898	−0.210299
$10$	0	0
$11$	−1.09171	−0.329163	−0.164581	−	0.986364i	$-0.552627\pi$
$11$	−1.09171	−0.329163	−0.164581	+	0.986364i	$0.552627\pi$
$12$	1.53919	0.444326
$13$	4.53919	1.25894	0.629472	−	0.777023i	$-0.283271\pi$
$13$	4.53919	1.25894	0.629472	+	0.777023i	$0.283271\pi$
$14$	2.87936	0.769542
$15$	0	0
$16$	1.00000	0.250000
$17$	2.80098	0.679338	0.339669	−	0.940545i	$-0.389685\pi$
$17$	2.80098	0.679338	0.339669	+	0.940545i	$0.389685\pi$
$18$	0.630898	0.148704
$19$	−5.04945	−1.15842	−0.579211	−	0.815177i	$-0.696639\pi$
$19$	−5.04945	−1.15842	−0.579211	+	0.815177i	$0.696639\pi$
$20$	0	0
$21$	−4.43188	−0.967116
$22$	1.09171	0.232753
$23$	7.41855	1.54687	0.773437	−	0.633873i	$-0.218536\pi$
$23$	7.41855	1.54687	0.773437	+	0.633873i	$0.218536\pi$
$24$	−1.53919	−0.314186
$25$	0	0
$26$	−4.53919	−0.890208
$27$	−5.58864	−1.07553
$28$	−2.87936	−0.544148
$29$	6.68035	1.24051	0.620255	−	0.784401i	$-0.287029\pi$
$29$	6.68035	1.24051	0.620255	+	0.784401i	$0.287029\pi$
$30$	0	0
$31$	3.51026	0.630461	0.315231	−	0.949015i	$-0.397918\pi$
$31$	3.51026	0.630461	0.315231	+	0.949015i	$0.397918\pi$
$32$	−1.00000	−0.176777
$33$	−1.68035	−0.292511
$34$	−2.80098	−0.480365
$35$	0	0
$36$	−0.630898	−0.105150
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	5.04945	0.819129
$39$	6.98667	1.11876
$40$	0	0
$41$	−8.07838	−1.26163	−0.630815	−	0.775933i	$-0.717279\pi$
$41$	−8.07838	−1.26163	−0.630815	+	0.775933i	$0.717279\pi$
$42$	4.43188	0.683854
$43$	10.2329	1.56050	0.780249	−	0.625469i	$-0.215092\pi$
$43$	10.2329	1.56050	0.780249	+	0.625469i	$0.215092\pi$
$44$	−1.09171	−0.164581
$45$	0	0
$46$	−7.41855	−1.09381
$47$	8.68035	1.26616	0.633079	−	0.774087i	$-0.281791\pi$
$47$	8.68035	1.26616	0.633079	+	0.774087i	$0.281791\pi$
$48$	1.53919	0.222163
$49$	1.29072	0.184389
$50$	0	0
$51$	4.31124	0.603695
$52$	4.53919	0.629472
$53$	−10.0989	−1.38719	−0.693595	−	0.720365i	$-0.743974\pi$
$53$	−10.0989	−1.38719	−0.693595	+	0.720365i	$0.743974\pi$
$54$	5.58864	0.760517
$55$	0	0
$56$	2.87936	0.384771
$57$	−7.77205	−1.02943
$58$	−6.68035	−0.877172
$59$	10.2329	1.33221	0.666103	−	0.745860i	$-0.267961\pi$
$59$	10.2329	1.33221	0.666103	+	0.745860i	$0.267961\pi$
$60$	0	0
$61$	6.29791	0.806365	0.403183	−	0.915120i	$-0.367904\pi$
$61$	6.29791	0.806365	0.403183	+	0.915120i	$0.367904\pi$
$62$	−3.51026	−0.445803
$63$	1.81658	0.228868
$64$	1.00000	0.125000
$65$	0	0
$66$	1.68035	0.206836
$67$	13.2979	1.62460	0.812299	−	0.583241i	$-0.198216\pi$
$67$	13.2979	1.62460	0.812299	+	0.583241i	$0.198216\pi$
$68$	2.80098	0.339669
$69$	11.4186	1.37463
$70$	0	0
$71$	6.29791	0.747425	0.373712	−	0.927545i	$-0.378085\pi$
$71$	6.29791	0.747425	0.373712	+	0.927545i	$0.378085\pi$
$72$	0.630898	0.0743520
$73$	12.7093	1.48751	0.743754	−	0.668453i	$-0.233043\pi$
$73$	12.7093	1.48751	0.743754	+	0.668453i	$0.233043\pi$
$74$	−1.00000	−0.116248
$75$	0	0
$76$	−5.04945	−0.579211
$77$	3.14342	0.358226
$78$	−6.98667	−0.791084
$79$	2.58145	0.290436	0.145218	−	0.989400i	$-0.453612\pi$
$79$	2.58145	0.290436	0.145218	+	0.989400i	$0.453612\pi$
$80$	0	0
$81$	−6.70928	−0.745475
$82$	8.07838	0.892108
$83$	8.48360	0.931196	0.465598	−	0.884996i	$-0.345839\pi$
$83$	8.48360	0.931196	0.465598	+	0.884996i	$0.345839\pi$
$84$	−4.43188	−0.483558
$85$	0	0
$86$	−10.2329	−1.10344
$87$	10.2823	1.10238
$88$	1.09171	0.116377
$89$	−6.51026	−0.690086	−0.345043	−	0.938587i	$-0.612136\pi$
$89$	−6.51026	−0.690086	−0.345043	+	0.938587i	$0.612136\pi$
$90$	0	0
$91$	−13.0700	−1.37010
$92$	7.41855	0.773437
$93$	5.40295	0.560260
$94$	−8.68035	−0.895309
$95$	0	0
$96$	−1.53919	−0.157093
$97$	3.07838	0.312562	0.156281	−	0.987713i	$-0.450049\pi$
$97$	3.07838	0.312562	0.156281	+	0.987713i	$0.450049\pi$
$98$	−1.29072	−0.130383
$99$	0.688756	0.0692226
$100$	0	0
$101$	−15.9155	−1.58365	−0.791825	−	0.610748i	$-0.790869\pi$
$101$	−15.9155	−1.58365	−0.791825	+	0.610748i	$0.790869\pi$
$102$	−4.31124	−0.426877
$103$	−10.5886	−1.04333	−0.521665	−	0.853151i	$-0.674689\pi$
$103$	−10.5886	−1.04333	−0.521665	+	0.853151i	$0.674689\pi$
$104$	−4.53919	−0.445104
$105$	0	0
$106$	10.0989	0.980892
$107$	−4.03612	−0.390186	−0.195093	−	0.980785i	$-0.562501\pi$
$107$	−4.03612	−0.390186	−0.195093	+	0.980785i	$0.562501\pi$
$108$	−5.58864	−0.537767
$109$	−4.49693	−0.430728	−0.215364	−	0.976534i	$-0.569094\pi$
$109$	−4.49693	−0.430728	−0.215364	+	0.976534i	$0.569094\pi$
$110$	0	0
$111$	1.53919	0.146093
$112$	−2.87936	−0.272074
$113$	3.58864	0.337591	0.168795	−	0.985651i	$-0.446012\pi$
$113$	3.58864	0.337591	0.168795	+	0.985651i	$0.446012\pi$
$114$	7.77205	0.727920
$115$	0	0
$116$	6.68035	0.620255
$117$	−2.86376	−0.264755
$118$	−10.2329	−0.942012
$119$	−8.06505	−0.739322
$120$	0	0
$121$	−9.80817	−0.891652
$122$	−6.29791	−0.570186
$123$	−12.4341	−1.12115
$124$	3.51026	0.315231
$125$	0	0
$126$	−1.81658	−0.161834
$127$	8.14116	0.722411	0.361205	−	0.932486i	$-0.382365\pi$
$127$	8.14116	0.722411	0.361205	+	0.932486i	$0.382365\pi$
$128$	−1.00000	−0.0883883
$129$	15.7503	1.38674
$130$	0	0
$131$	−1.46800	−0.128260	−0.0641298	−	0.997942i	$-0.520427\pi$
$131$	−1.46800	−0.128260	−0.0641298	+	0.997942i	$0.520427\pi$
$132$	−1.68035	−0.146255
$133$	14.5392	1.26071
$134$	−13.2979	−1.14876
$135$	0	0
$136$	−2.80098	−0.240182
$137$	−6.17727	−0.527760	−0.263880	−	0.964555i	$-0.585002\pi$
$137$	−6.17727	−0.527760	−0.263880	+	0.964555i	$0.585002\pi$
$138$	−11.4186	−0.972012
$139$	13.0338	1.10552	0.552758	−	0.833342i	$-0.313575\pi$
$139$	13.0338	1.10552	0.552758	+	0.833342i	$0.313575\pi$
$140$	0	0
$141$	13.3607	1.12517
$142$	−6.29791	−0.528509
$143$	−4.95547	−0.414397
$144$	−0.630898	−0.0525748
$145$	0	0
$146$	−12.7093	−1.05183
$147$	1.98667	0.163858
$148$	1.00000	0.0821995
$149$	−4.40522	−0.360890	−0.180445	−	0.983585i	$-0.557754\pi$
$149$	−4.40522	−0.360890	−0.180445	+	0.983585i	$0.557754\pi$
$150$	0	0
$151$	2.92162	0.237758	0.118879	−	0.992909i	$-0.462070\pi$
$151$	2.92162	0.237758	0.118879	+	0.992909i	$0.462070\pi$
$152$	5.04945	0.409564
$153$	−1.76713	−0.142864
$154$	−3.14342	−0.253304
$155$	0	0
$156$	6.98667	0.559381
$157$	22.1906	1.77100	0.885502	−	0.464636i	$-0.153815\pi$
$157$	22.1906	1.77100	0.885502	+	0.464636i	$0.153815\pi$
$158$	−2.58145	−0.205369
$159$	−15.5441	−1.23273
$160$	0	0
$161$	−21.3607	−1.68346
$162$	6.70928	0.527130
$163$	−14.6248	−1.14550	−0.572750	−	0.819730i	$-0.694123\pi$
$163$	−14.6248	−1.14550	−0.572750	+	0.819730i	$0.694123\pi$
$164$	−8.07838	−0.630815
$165$	0	0
$166$	−8.48360	−0.658455
$167$	−6.34736	−0.491174	−0.245587	−	0.969375i	$-0.578981\pi$
$167$	−6.34736	−0.491174	−0.245587	+	0.969375i	$0.578981\pi$
$168$	4.43188	0.341927
$169$	7.60424	0.584941
$170$	0	0
$171$	3.18568	0.243615
$172$	10.2329	0.780249
$173$	−1.23513	−0.0939054	−0.0469527	−	0.998897i	$-0.514951\pi$
$173$	−1.23513	−0.0939054	−0.0469527	+	0.998897i	$0.514951\pi$
$174$	−10.2823	−0.779500
$175$	0	0
$176$	−1.09171	−0.0822906
$177$	15.7503	1.18387
$178$	6.51026	0.487965
$179$	1.76487	0.131912	0.0659562	−	0.997823i	$-0.478990\pi$
$179$	1.76487	0.131912	0.0659562	+	0.997823i	$0.478990\pi$
$180$	0	0
$181$	6.49693	0.482913	0.241456	−	0.970412i	$-0.422375\pi$
$181$	6.49693	0.482913	0.241456	+	0.970412i	$0.422375\pi$
$182$	13.0700	0.968810
$183$	9.69368	0.716577
$184$	−7.41855	−0.546903
$185$	0	0
$186$	−5.40295	−0.396164
$187$	−3.05786	−0.223613
$188$	8.68035	0.633079
$189$	16.0917	1.17050
$190$	0	0
$191$	21.2039	1.53426	0.767131	−	0.641490i	$-0.221683\pi$
$191$	21.2039	1.53426	0.767131	+	0.641490i	$0.221683\pi$
$192$	1.53919	0.111081
$193$	8.14342	0.586177	0.293088	−	0.956085i	$-0.405317\pi$
$193$	8.14342	0.586177	0.293088	+	0.956085i	$0.405317\pi$
$194$	−3.07838	−0.221015
$195$	0	0
$196$	1.29072	0.0921946
$197$	−24.8443	−1.77008	−0.885041	−	0.465513i	$-0.845870\pi$
$197$	−24.8443	−1.77008	−0.885041	+	0.465513i	$0.845870\pi$
$198$	−0.688756	−0.0489478
$199$	−8.47027	−0.600441	−0.300221	−	0.953870i	$-0.597060\pi$
$199$	−8.47027	−0.600441	−0.300221	+	0.953870i	$0.597060\pi$
$200$	0	0
$201$	20.4680	1.44370
$202$	15.9155	1.11981
$203$	−19.2351	−1.35004
$204$	4.31124	0.301847
$205$	0	0
$206$	10.5886	0.737745
$207$	−4.68035	−0.325307
$208$	4.53919	0.314736
$209$	5.51253	0.381309
$210$	0	0
$211$	21.7370	1.49644	0.748218	−	0.663453i	$-0.230910\pi$
$211$	21.7370	1.49644	0.748218	+	0.663453i	$0.230910\pi$
$212$	−10.0989	−0.693595
$213$	9.69368	0.664200
$214$	4.03612	0.275903
$215$	0	0
$216$	5.58864	0.380259
$217$	−10.1073	−0.686129
$218$	4.49693	0.304570
$219$	19.5620	1.32188
$220$	0	0
$221$	12.7142	0.855249
$222$	−1.53919	−0.103304
$223$	16.6537	1.11521	0.557607	−	0.830105i	$-0.311720\pi$
$223$	16.6537	1.11521	0.557607	+	0.830105i	$0.311720\pi$
$224$	2.87936	0.192385
$225$	0	0
$226$	−3.58864	−0.238713
$227$	−11.2123	−0.744190	−0.372095	−	0.928195i	$-0.621360\pi$
$227$	−11.2123	−0.744190	−0.372095	+	0.928195i	$0.621360\pi$
$228$	−7.77205	−0.514717
$229$	−7.62863	−0.504114	−0.252057	−	0.967712i	$-0.581107\pi$
$229$	−7.62863	−0.504114	−0.252057	+	0.967712i	$0.581107\pi$
$230$	0	0
$231$	4.83832	0.318338
$232$	−6.68035	−0.438586
$233$	−0.0494483	−0.00323947	−0.00161973	−	0.999999i	$-0.500516\pi$
$233$	−0.0494483	−0.00323947	−0.00161973	+	0.999999i	$0.500516\pi$
$234$	2.86376	0.187210
$235$	0	0
$236$	10.2329	0.666103
$237$	3.97334	0.258096
$238$	8.06505	0.522779
$239$	−14.7070	−0.951317	−0.475659	−	0.879630i	$-0.657790\pi$
$239$	−14.7070	−0.951317	−0.475659	+	0.879630i	$0.657790\pi$
$240$	0	0
$241$	−14.8999	−0.959786	−0.479893	−	0.877327i	$-0.659324\pi$
$241$	−14.8999	−0.959786	−0.479893	+	0.877327i	$0.659324\pi$
$242$	9.80817	0.630493
$243$	6.43907	0.413067
$244$	6.29791	0.403183
$245$	0	0
$246$	12.4341	0.792772
$247$	−22.9204	−1.45839
$248$	−3.51026	−0.222902
$249$	13.0579	0.827508
$250$	0	0
$251$	6.23513	0.393558	0.196779	−	0.980448i	$-0.436952\pi$
$251$	6.23513	0.393558	0.196779	+	0.980448i	$0.436952\pi$
$252$	1.81658	0.114434
$253$	−8.09890	−0.509173
$254$	−8.14116	−0.510822
$255$	0	0
$256$	1.00000	0.0625000
$257$	16.1568	1.00783	0.503915	−	0.863753i	$-0.331892\pi$
$257$	16.1568	1.00783	0.503915	+	0.863753i	$0.331892\pi$
$258$	−15.7503	−0.980572
$259$	−2.87936	−0.178915
$260$	0	0
$261$	−4.21461	−0.260878
$262$	1.46800	0.0906933
$263$	−18.6537	−1.15024	−0.575118	−	0.818071i	$-0.695044\pi$
$263$	−18.6537	−1.15024	−0.575118	+	0.818071i	$0.695044\pi$
$264$	1.68035	0.103418
$265$	0	0
$266$	−14.5392	−0.891455
$267$	−10.0205	−0.613246
$268$	13.2979	0.812299
$269$	21.8310	1.33106	0.665529	−	0.746372i	$-0.268206\pi$
$269$	21.8310	1.33106	0.665529	+	0.746372i	$0.268206\pi$
$270$	0	0
$271$	24.9783	1.51732	0.758661	−	0.651486i	$-0.225854\pi$
$271$	24.9783	1.51732	0.758661	+	0.651486i	$0.225854\pi$
$272$	2.80098	0.169835
$273$	−20.1171	−1.21755
$274$	6.17727	0.373183
$275$	0	0
$276$	11.4186	0.687316
$277$	−0.822726	−0.0494328	−0.0247164	−	0.999695i	$-0.507868\pi$
$277$	−0.822726	−0.0494328	−0.0247164	+	0.999695i	$0.507868\pi$
$278$	−13.0338	−0.781718
$279$	−2.21461	−0.132585
$280$	0	0
$281$	−1.65142	−0.0985153	−0.0492576	−	0.998786i	$-0.515686\pi$
$281$	−1.65142	−0.0985153	−0.0492576	+	0.998786i	$0.515686\pi$
$282$	−13.3607	−0.795618
$283$	15.1340	0.899621	0.449811	−	0.893124i	$-0.351492\pi$
$283$	15.1340	0.899621	0.449811	+	0.893124i	$0.351492\pi$
$284$	6.29791	0.373712
$285$	0	0
$286$	4.95547	0.293023
$287$	23.2606	1.37303
$288$	0.630898	0.0371760
$289$	−9.15449	−0.538499
$290$	0	0
$291$	4.73820	0.277758
$292$	12.7093	0.743754
$293$	−25.4524	−1.48695	−0.743473	−	0.668766i	$-0.766823\pi$
$293$	−25.4524	−1.48695	−0.743473	+	0.668766i	$0.766823\pi$
$294$	−1.98667	−0.115865
$295$	0	0
$296$	−1.00000	−0.0581238
$297$	6.10116	0.354025
$298$	4.40522	0.255188
$299$	33.6742	1.94743
$300$	0	0
$301$	−29.4641	−1.69828
$302$	−2.92162	−0.168120
$303$	−24.4969	−1.40731
$304$	−5.04945	−0.289606
$305$	0	0
$306$	1.76713	0.101020
$307$	−27.8176	−1.58764	−0.793818	−	0.608155i	$-0.791910\pi$
$307$	−27.8176	−1.58764	−0.793818	+	0.608155i	$0.791910\pi$
$308$	3.14342	0.179113
$309$	−16.2979	−0.927156
$310$	0	0
$311$	0.0650468	0.00368846	0.00184423	−	0.999998i	$-0.499413\pi$
$311$	0.0650468	0.00368846	0.00184423	+	0.999998i	$0.499413\pi$
$312$	−6.98667	−0.395542
$313$	22.5464	1.27440	0.637198	−	0.770700i	$-0.280093\pi$
$313$	22.5464	1.27440	0.637198	+	0.770700i	$0.280093\pi$
$314$	−22.1906	−1.25229
$315$	0	0
$316$	2.58145	0.145218
$317$	−24.4775	−1.37479	−0.687395	−	0.726283i	$-0.741246\pi$
$317$	−24.4775	−1.37479	−0.687395	+	0.726283i	$0.741246\pi$
$318$	15.5441	0.871670
$319$	−7.29299	−0.408329
$320$	0	0
$321$	−6.21235	−0.346739
$322$	21.3607	1.19038
$323$	−14.1434	−0.786961
$324$	−6.70928	−0.372738
$325$	0	0
$326$	14.6248	0.809990
$327$	−6.92162	−0.382767
$328$	8.07838	0.446054
$329$	−24.9939	−1.37796
$330$	0	0
$331$	−1.91321	−0.105160	−0.0525798	−	0.998617i	$-0.516744\pi$
$331$	−1.91321	−0.105160	−0.0525798	+	0.998617i	$0.516744\pi$
$332$	8.48360	0.465598
$333$	−0.630898	−0.0345730
$334$	6.34736	0.347312
$335$	0	0
$336$	−4.43188	−0.241779
$337$	−12.6286	−0.687925	−0.343963	−	0.938983i	$-0.611769\pi$
$337$	−12.6286	−0.687925	−0.343963	+	0.938983i	$0.611769\pi$
$338$	−7.60424	−0.413616
$339$	5.52359	0.300000
$340$	0	0
$341$	−3.83218	−0.207524
$342$	−3.18568	−0.172262
$343$	16.4391	0.887626
$344$	−10.2329	−0.551719
$345$	0	0
$346$	1.23513	0.0664012
$347$	−8.33403	−0.447394	−0.223697	−	0.974659i	$-0.571813\pi$
$347$	−8.33403	−0.447394	−0.223697	+	0.974659i	$0.571813\pi$
$348$	10.2823	0.551190
$349$	2.11837	0.113394	0.0566969	−	0.998391i	$-0.481943\pi$
$349$	2.11837	0.113394	0.0566969	+	0.998391i	$0.481943\pi$
$350$	0	0
$351$	−25.3679	−1.35404
$352$	1.09171	0.0581883
$353$	28.4534	1.51442	0.757212	−	0.653169i	$-0.226561\pi$
$353$	28.4534	1.51442	0.757212	+	0.653169i	$0.226561\pi$
$354$	−15.7503	−0.837120
$355$	0	0
$356$	−6.51026	−0.345043
$357$	−12.4136	−0.656999
$358$	−1.76487	−0.0932761
$359$	2.65368	0.140056	0.0700280	−	0.997545i	$-0.477691\pi$
$359$	2.65368	0.140056	0.0700280	+	0.997545i	$0.477691\pi$
$360$	0	0
$361$	6.49693	0.341944
$362$	−6.49693	−0.341471
$363$	−15.0966	−0.792368
$364$	−13.0700	−0.685052
$365$	0	0
$366$	−9.69368	−0.506697
$367$	−31.7575	−1.65773	−0.828864	−	0.559450i	$-0.811012\pi$
$367$	−31.7575	−1.65773	−0.828864	+	0.559450i	$0.811012\pi$
$368$	7.41855	0.386719
$369$	5.09663	0.265320
$370$	0	0
$371$	29.0784	1.50967
$372$	5.40295	0.280130
$373$	25.1122	1.30026	0.650131	−	0.759822i	$-0.274714\pi$
$373$	25.1122	1.30026	0.650131	+	0.759822i	$0.274714\pi$
$374$	3.05786	0.158118
$375$	0	0
$376$	−8.68035	−0.447655
$377$	30.3234	1.56173
$378$	−16.0917	−0.827668
$379$	22.5330	1.15744	0.578722	−	0.815525i	$-0.303551\pi$
$379$	22.5330	1.15744	0.578722	+	0.815525i	$0.303551\pi$
$380$	0	0
$381$	12.5308	0.641971
$382$	−21.2039	−1.08489
$383$	−12.5886	−0.643249	−0.321625	−	0.946867i	$-0.604229\pi$
$383$	−12.5886	−0.643249	−0.321625	+	0.946867i	$0.604229\pi$
$384$	−1.53919	−0.0785464
$385$	0	0
$386$	−8.14342	−0.414489
$387$	−6.45589	−0.328171
$388$	3.07838	0.156281
$389$	−0.879362	−0.0445854	−0.0222927	−	0.999751i	$-0.507097\pi$
$389$	−0.879362	−0.0445854	−0.0222927	+	0.999751i	$0.507097\pi$
$390$	0	0
$391$	20.7792	1.05085
$392$	−1.29072	−0.0651914
$393$	−2.25953	−0.113978
$394$	24.8443	1.25164
$395$	0	0
$396$	0.688756	0.0346113
$397$	−29.6814	−1.48967	−0.744833	−	0.667251i	$-0.767471\pi$
$397$	−29.6814	−1.48967	−0.744833	+	0.667251i	$0.767471\pi$
$398$	8.47027	0.424576
$399$	22.3786	1.12033
$400$	0	0
$401$	−13.9867	−0.698461	−0.349230	−	0.937037i	$-0.613557\pi$
$401$	−13.9867	−0.698461	−0.349230	+	0.937037i	$0.613557\pi$
$402$	−20.4680	−1.02085
$403$	15.9337	0.793716
$404$	−15.9155	−0.791825
$405$	0	0
$406$	19.2351	0.954624
$407$	−1.09171	−0.0541140
$408$	−4.31124	−0.213438
$409$	−21.9060	−1.08318	−0.541592	−	0.840642i	$-0.682178\pi$
$409$	−21.9060	−1.08318	−0.541592	+	0.840642i	$0.682178\pi$
$410$	0	0
$411$	−9.50799	−0.468995
$412$	−10.5886	−0.521665
$413$	−29.4641	−1.44983
$414$	4.68035	0.230026
$415$	0	0
$416$	−4.53919	−0.222552
$417$	20.0616	0.982419
$418$	−5.51253	−0.269627
$419$	−34.8648	−1.70326	−0.851629	−	0.524146i	$-0.824385\pi$
$419$	−34.8648	−1.70326	−0.851629	+	0.524146i	$0.824385\pi$
$420$	0	0
$421$	−4.76487	−0.232225	−0.116113	−	0.993236i	$-0.537043\pi$
$421$	−4.76487	−0.232225	−0.116113	+	0.993236i	$0.537043\pi$
$422$	−21.7370	−1.05814
$423$	−5.47641	−0.266272
$424$	10.0989	0.490446
$425$	0	0
$426$	−9.69368	−0.469660
$427$	−18.1340	−0.877564
$428$	−4.03612	−0.195093
$429$	−7.62741	−0.368255
$430$	0	0
$431$	2.66597	0.128415	0.0642076	−	0.997937i	$-0.479548\pi$
$431$	2.66597	0.128415	0.0642076	+	0.997937i	$0.479548\pi$
$432$	−5.58864	−0.268883
$433$	32.3074	1.55259	0.776297	−	0.630368i	$-0.217096\pi$
$433$	32.3074	1.55259	0.776297	+	0.630368i	$0.217096\pi$
$434$	10.1073	0.485166
$435$	0	0
$436$	−4.49693	−0.215364
$437$	−37.4596	−1.79194
$438$	−19.5620	−0.934707
$439$	−22.1568	−1.05748	−0.528742	−	0.848783i	$-0.677336\pi$
$439$	−22.1568	−1.05748	−0.528742	+	0.848783i	$0.677336\pi$
$440$	0	0
$441$	−0.814315	−0.0387769
$442$	−12.7142	−0.604753
$443$	39.1061	1.85799	0.928993	−	0.370097i	$-0.120676\pi$
$443$	39.1061	1.85799	0.928993	+	0.370097i	$0.120676\pi$
$444$	1.53919	0.0730467
$445$	0	0
$446$	−16.6537	−0.788575
$447$	−6.78047	−0.320705
$448$	−2.87936	−0.136037
$449$	30.8915	1.45786	0.728929	−	0.684589i	$-0.240018\pi$
$449$	30.8915	1.45786	0.728929	+	0.684589i	$0.240018\pi$
$450$	0	0
$451$	8.81924	0.415282
$452$	3.58864	0.168795
$453$	4.49693	0.211284
$454$	11.2123	0.526222
$455$	0	0
$456$	7.77205	0.363960
$457$	−0.438025	−0.0204899	−0.0102450	−	0.999948i	$-0.503261\pi$
$457$	−0.438025	−0.0204899	−0.0102450	+	0.999948i	$0.503261\pi$
$458$	7.62863	0.356462
$459$	−15.6537	−0.730651
$460$	0	0
$461$	0.523590	0.0243860	0.0121930	−	0.999926i	$-0.496119\pi$
$461$	0.523590	0.0243860	0.0121930	+	0.999926i	$0.496119\pi$
$462$	−4.83832	−0.225099
$463$	−29.1845	−1.35632	−0.678158	−	0.734916i	$-0.737222\pi$
$463$	−29.1845	−1.35632	−0.678158	+	0.734916i	$0.737222\pi$
$464$	6.68035	0.310127
$465$	0	0
$466$	0.0494483	0.00229065
$467$	9.49079	0.439181	0.219591	−	0.975592i	$-0.429528\pi$
$467$	9.49079	0.439181	0.219591	+	0.975592i	$0.429528\pi$
$468$	−2.86376	−0.132378
$469$	−38.2895	−1.76804
$470$	0	0
$471$	34.1555	1.57380
$472$	−10.2329	−0.471006
$473$	−11.1713	−0.513657
$474$	−3.97334	−0.182501
$475$	0	0
$476$	−8.06505	−0.369661
$477$	6.37137	0.291725
$478$	14.7070	0.672683
$479$	−6.12291	−0.279763	−0.139881	−	0.990168i	$-0.544672\pi$
$479$	−6.12291	−0.279763	−0.139881	+	0.990168i	$0.544672\pi$
$480$	0	0
$481$	4.53919	0.206969
$482$	14.8999	0.678671
$483$	−32.8781	−1.49601
$484$	−9.80817	−0.445826
$485$	0	0
$486$	−6.43907	−0.292082
$487$	10.3018	0.466819	0.233409	−	0.972379i	$-0.425012\pi$
$487$	10.3018	0.466819	0.233409	+	0.972379i	$0.425012\pi$
$488$	−6.29791	−0.285093
$489$	−22.5103	−1.01795
$490$	0	0
$491$	−10.5380	−0.475572	−0.237786	−	0.971318i	$-0.576422\pi$
$491$	−10.5380	−0.475572	−0.237786	+	0.971318i	$0.576422\pi$
$492$	−12.4341	−0.560575
$493$	18.7115	0.842726
$494$	22.9204	1.03124
$495$	0	0
$496$	3.51026	0.157615
$497$	−18.1340	−0.813420
$498$	−13.0579	−0.585137
$499$	−31.2123	−1.39726	−0.698628	−	0.715485i	$-0.746206\pi$
$499$	−31.2123	−1.39726	−0.698628	+	0.715485i	$0.746206\pi$
$500$	0	0
$501$	−9.76979	−0.436482
$502$	−6.23513	−0.278288
$503$	−36.1978	−1.61398	−0.806990	−	0.590565i	$-0.798905\pi$
$503$	−36.1978	−1.61398	−0.806990	+	0.590565i	$0.798905\pi$
$504$	−1.81658	−0.0809170
$505$	0	0
$506$	8.09890	0.360040
$507$	11.7044	0.519809
$508$	8.14116	0.361205
$509$	−33.6092	−1.48970	−0.744850	−	0.667232i	$-0.767479\pi$
$509$	−33.6092	−1.48970	−0.744850	+	0.667232i	$0.767479\pi$
$510$	0	0
$511$	−36.5946	−1.61885
$512$	−1.00000	−0.0441942
$513$	28.2195	1.24592
$514$	−16.1568	−0.712644
$515$	0	0
$516$	15.7503	0.693369
$517$	−9.47641	−0.416772
$518$	2.87936	0.126512
$519$	−1.90110	−0.0834492
$520$	0	0
$521$	5.42082	0.237490	0.118745	−	0.992925i	$-0.462113\pi$
$521$	5.42082	0.237490	0.118745	+	0.992925i	$0.462113\pi$
$522$	4.21461	0.184469
$523$	−31.3545	−1.37104	−0.685519	−	0.728054i	$-0.740425\pi$
$523$	−31.3545	−1.37104	−0.685519	+	0.728054i	$0.740425\pi$
$524$	−1.46800	−0.0641298
$525$	0	0
$526$	18.6537	0.813339
$527$	9.83218	0.428297
$528$	−1.68035	−0.0731277
$529$	32.0349	1.39282
$530$	0	0
$531$	−6.45589	−0.280162
$532$	14.5392	0.630354
$533$	−36.6693	−1.58832
$534$	10.0205	0.433630
$535$	0	0
$536$	−13.2979	−0.574382
$537$	2.71646	0.117224
$538$	−21.8310	−0.941199
$539$	−1.40910	−0.0606940
$540$	0	0
$541$	−9.98440	−0.429263	−0.214631	−	0.976695i	$-0.568855\pi$
$541$	−9.98440	−0.429263	−0.214631	+	0.976695i	$0.568855\pi$
$542$	−24.9783	−1.07291
$543$	10.0000	0.429141
$544$	−2.80098	−0.120091
$545$	0	0
$546$	20.1171	0.860934
$547$	32.9649	1.40948	0.704739	−	0.709466i	$-0.251064\pi$
$547$	32.9649	1.40948	0.704739	+	0.709466i	$0.251064\pi$
$548$	−6.17727	−0.263880
$549$	−3.97334	−0.169578
$550$	0	0
$551$	−33.7321	−1.43703
$552$	−11.4186	−0.486006
$553$	−7.43293	−0.316080
$554$	0.822726	0.0349543
$555$	0	0
$556$	13.0338	0.552758
$557$	−21.7009	−0.919495	−0.459748	−	0.888050i	$-0.652060\pi$
$557$	−21.7009	−0.919495	−0.459748	+	0.888050i	$0.652060\pi$
$558$	2.21461	0.0937521
$559$	46.4489	1.96458
$560$	0	0
$561$	−4.70662	−0.198714
$562$	1.65142	0.0696608
$563$	−1.05559	−0.0444879	−0.0222439	−	0.999753i	$-0.507081\pi$
$563$	−1.05559	−0.0444879	−0.0222439	+	0.999753i	$0.507081\pi$
$564$	13.3607	0.562587
$565$	0	0
$566$	−15.1340	−0.636128
$567$	19.3184	0.811298
$568$	−6.29791	−0.264255
$569$	−41.4196	−1.73640	−0.868200	−	0.496215i	$-0.834723\pi$
$569$	−41.4196	−1.73640	−0.868200	+	0.496215i	$0.834723\pi$
$570$	0	0
$571$	21.4452	0.897454	0.448727	−	0.893669i	$-0.351878\pi$
$571$	21.4452	0.897454	0.448727	+	0.893669i	$0.351878\pi$
$572$	−4.95547	−0.207199
$573$	32.6369	1.36342
$574$	−23.2606	−0.970878
$575$	0	0
$576$	−0.630898	−0.0262874
$577$	17.2667	0.718823	0.359411	−	0.933179i	$-0.382977\pi$
$577$	17.2667	0.718823	0.359411	+	0.933179i	$0.382977\pi$
$578$	9.15449	0.380777
$579$	12.5343	0.520906
$580$	0	0
$581$	−24.4273	−1.01342
$582$	−4.73820	−0.196405
$583$	11.0251	0.456611
$584$	−12.7093	−0.525914
$585$	0	0
$586$	25.4524	1.05143
$587$	−9.63090	−0.397510	−0.198755	−	0.980049i	$-0.563690\pi$
$587$	−9.63090	−0.397510	−0.198755	+	0.980049i	$0.563690\pi$
$588$	1.98667	0.0819288
$589$	−17.7249	−0.730341
$590$	0	0
$591$	−38.2401	−1.57299
$592$	1.00000	0.0410997
$593$	2.78992	0.114568	0.0572842	−	0.998358i	$-0.481756\pi$
$593$	2.78992	0.114568	0.0572842	+	0.998358i	$0.481756\pi$
$594$	−6.10116	−0.250334
$595$	0	0
$596$	−4.40522	−0.180445
$597$	−13.0373	−0.533583
$598$	−33.6742	−1.37704
$599$	25.2606	1.03212	0.516060	−	0.856553i	$-0.327398\pi$
$599$	25.2606	1.03212	0.516060	+	0.856553i	$0.327398\pi$
$600$	0	0
$601$	35.4040	1.44416	0.722080	−	0.691810i	$-0.243186\pi$
$601$	35.4040	1.44416	0.722080	+	0.691810i	$0.243186\pi$
$602$	29.4641	1.20087
$603$	−8.38962	−0.341652
$604$	2.92162	0.118879
$605$	0	0
$606$	24.4969	0.995120
$607$	38.2485	1.55246	0.776229	−	0.630452i	$-0.217130\pi$
$607$	38.2485	1.55246	0.776229	+	0.630452i	$0.217130\pi$
$608$	5.04945	0.204782
$609$	−29.6065	−1.19972
$610$	0	0
$611$	39.4017	1.59402
$612$	−1.76713	−0.0714322
$613$	−46.3279	−1.87117	−0.935583	−	0.353107i	$-0.885125\pi$
$613$	−46.3279	−1.87117	−0.935583	+	0.353107i	$0.885125\pi$
$614$	27.8176	1.12263
$615$	0	0
$616$	−3.14342	−0.126652
$617$	23.7503	0.956152	0.478076	−	0.878319i	$-0.341334\pi$
$617$	23.7503	0.956152	0.478076	+	0.878319i	$0.341334\pi$
$618$	16.2979	0.655598
$619$	−6.53797	−0.262783	−0.131392	−	0.991331i	$-0.541945\pi$
$619$	−6.53797	−0.262783	−0.131392	+	0.991331i	$0.541945\pi$
$620$	0	0
$621$	−41.4596	−1.66372
$622$	−0.0650468	−0.00260814
$623$	18.7454	0.751018
$624$	6.98667	0.279691
$625$	0	0
$626$	−22.5464	−0.901134
$627$	8.48482	0.338851
$628$	22.1906	0.885502
$629$	2.80098	0.111683
$630$	0	0
$631$	27.1506	1.08085	0.540424	−	0.841393i	$-0.318264\pi$
$631$	27.1506	1.08085	0.540424	+	0.841393i	$0.318264\pi$
$632$	−2.58145	−0.102685
$633$	33.4573	1.32981
$634$	24.4775	0.972124
$635$	0	0
$636$	−15.5441	−0.616364
$637$	5.85884	0.232136
$638$	7.29299	0.288732
$639$	−3.97334	−0.157183
$640$	0	0
$641$	31.9877	1.26344	0.631719	−	0.775197i	$-0.282350\pi$
$641$	31.9877	1.26344	0.631719	+	0.775197i	$0.282350\pi$
$642$	6.21235	0.245182
$643$	26.2784	1.03632	0.518160	−	0.855284i	$-0.326617\pi$
$643$	26.2784	1.03632	0.518160	+	0.855284i	$0.326617\pi$
$644$	−21.3607	−0.841729
$645$	0	0
$646$	14.1434	0.556466
$647$	31.3679	1.23320	0.616599	−	0.787277i	$-0.288510\pi$
$647$	31.3679	1.23320	0.616599	+	0.787277i	$0.288510\pi$
$648$	6.70928	0.263565
$649$	−11.1713	−0.438512
$650$	0	0
$651$	−15.5571	−0.609729
$652$	−14.6248	−0.572750
$653$	−25.3184	−0.990787	−0.495393	−	0.868669i	$-0.664976\pi$
$653$	−25.3184	−0.990787	−0.495393	+	0.868669i	$0.664976\pi$
$654$	6.92162	0.270657
$655$	0	0
$656$	−8.07838	−0.315408
$657$	−8.01825	−0.312822
$658$	24.9939	0.974362
$659$	−38.2072	−1.48834	−0.744172	−	0.667989i	$-0.767156\pi$
$659$	−38.2072	−1.48834	−0.744172	+	0.667989i	$0.767156\pi$
$660$	0	0
$661$	13.7899	0.536366	0.268183	−	0.963368i	$-0.413577\pi$
$661$	13.7899	0.536366	0.268183	+	0.963368i	$0.413577\pi$
$662$	1.91321	0.0743591
$663$	19.5695	0.760018
$664$	−8.48360	−0.329227
$665$	0	0
$666$	0.630898	0.0244468
$667$	49.5585	1.91891
$668$	−6.34736	−0.245587
$669$	25.6332	0.991035
$670$	0	0
$671$	−6.87549	−0.265425
$672$	4.43188	0.170964
$673$	4.41628	0.170235	0.0851176	−	0.996371i	$-0.472873\pi$
$673$	4.41628	0.170235	0.0851176	+	0.996371i	$0.472873\pi$
$674$	12.6286	0.486437
$675$	0	0
$676$	7.60424	0.292471
$677$	−38.1399	−1.46584	−0.732918	−	0.680317i	$-0.761842\pi$
$677$	−38.1399	−1.46584	−0.732918	+	0.680317i	$0.761842\pi$
$678$	−5.52359	−0.212132
$679$	−8.86376	−0.340160
$680$	0	0
$681$	−17.2579	−0.661325
$682$	3.83218	0.146742
$683$	−22.6514	−0.866732	−0.433366	−	0.901218i	$-0.642674\pi$
$683$	−22.6514	−0.866732	−0.433366	+	0.901218i	$0.642674\pi$
$684$	3.18568	0.121808
$685$	0	0
$686$	−16.4391	−0.627647
$687$	−11.7419	−0.447982
$688$	10.2329	0.390124
$689$	−45.8408	−1.74640
$690$	0	0
$691$	−15.8443	−0.602745	−0.301373	−	0.953506i	$-0.597445\pi$
$691$	−15.8443	−0.602745	−0.301373	+	0.953506i	$0.597445\pi$
$692$	−1.23513	−0.0469527
$693$	−1.98318	−0.0753347
$694$	8.33403	0.316355
$695$	0	0
$696$	−10.2823	−0.389750
$697$	−22.6274	−0.857074
$698$	−2.11837	−0.0801815
$699$	−0.0761103	−0.00287876
$700$	0	0
$701$	−2.92284	−0.110394	−0.0551972	−	0.998475i	$-0.517579\pi$
$701$	−2.92284	−0.110394	−0.0551972	+	0.998475i	$0.517579\pi$
$702$	25.3679	0.957449
$703$	−5.04945	−0.190444
$704$	−1.09171	−0.0411453
$705$	0	0
$706$	−28.4534	−1.07086
$707$	45.8264	1.72348
$708$	15.7503	0.591933
$709$	−8.45136	−0.317397	−0.158699	−	0.987327i	$-0.550730\pi$
$709$	−8.45136	−0.317397	−0.158699	+	0.987327i	$0.550730\pi$
$710$	0	0
$711$	−1.62863	−0.0610784
$712$	6.51026	0.243982
$713$	26.0410	0.975245
$714$	12.4136	0.464568
$715$	0	0
$716$	1.76487	0.0659562
$717$	−22.6369	−0.845389
$718$	−2.65368	−0.0990346
$719$	−15.3763	−0.573439	−0.286719	−	0.958015i	$-0.592565\pi$
$719$	−15.3763	−0.573439	−0.286719	+	0.958015i	$0.592565\pi$
$720$	0	0
$721$	30.4885	1.13545
$722$	−6.49693	−0.241791
$723$	−22.9337	−0.852915
$724$	6.49693	0.241456
$725$	0	0
$726$	15.0966	0.560288
$727$	−15.7275	−0.583302	−0.291651	−	0.956525i	$-0.594205\pi$
$727$	−15.7275	−0.583302	−0.291651	+	0.956525i	$0.594205\pi$
$728$	13.0700	0.484405
$729$	30.0388	1.11255
$730$	0	0
$731$	28.6621	1.06011
$732$	9.69368	0.358289
$733$	15.7275	0.580909	0.290455	−	0.956889i	$-0.406193\pi$
$733$	15.7275	0.580909	0.290455	+	0.956889i	$0.406193\pi$
$734$	31.7575	1.17219
$735$	0	0
$736$	−7.41855	−0.273451
$737$	−14.5174	−0.534757
$738$	−5.09663	−0.187610
$739$	45.0472	1.65709	0.828544	−	0.559924i	$-0.189170\pi$
$739$	45.0472	1.65709	0.828544	+	0.559924i	$0.189170\pi$
$740$	0	0
$741$	−35.2788	−1.29600
$742$	−29.0784	−1.06750
$743$	−9.31965	−0.341905	−0.170952	−	0.985279i	$-0.554684\pi$
$743$	−9.31965	−0.341905	−0.170952	+	0.985279i	$0.554684\pi$
$744$	−5.40295	−0.198082
$745$	0	0
$746$	−25.1122	−0.919424
$747$	−5.35228	−0.195830
$748$	−3.05786	−0.111806
$749$	11.6214	0.424638
$750$	0	0
$751$	11.6020	0.423362	0.211681	−	0.977339i	$-0.432106\pi$
$751$	11.6020	0.423362	0.211681	+	0.977339i	$0.432106\pi$
$752$	8.68035	0.316540
$753$	9.59705	0.349736
$754$	−30.3234	−1.10431
$755$	0	0
$756$	16.0917	0.585250
$757$	38.3545	1.39402	0.697010	−	0.717062i	$-0.254513\pi$
$757$	38.3545	1.39402	0.697010	+	0.717062i	$0.254513\pi$
$758$	−22.5330	−0.818437
$759$	−12.4657	−0.452477
$760$	0	0
$761$	−45.3318	−1.64328	−0.821638	−	0.570010i	$-0.806939\pi$
$761$	−45.3318	−1.64328	−0.821638	+	0.570010i	$0.806939\pi$
$762$	−12.5308	−0.453942
$763$	12.9483	0.468759
$764$	21.2039	0.767131
$765$	0	0
$766$	12.5886	0.454846
$767$	46.4489	1.67717
$768$	1.53919	0.0555407
$769$	−31.7454	−1.14477	−0.572384	−	0.819986i	$-0.693981\pi$
$769$	−31.7454	−1.14477	−0.572384	+	0.819986i	$0.693981\pi$
$770$	0	0
$771$	24.8683	0.895610
$772$	8.14342	0.293088
$773$	−38.0482	−1.36850	−0.684250	−	0.729248i	$-0.739870\pi$
$773$	−38.0482	−1.36850	−0.684250	+	0.729248i	$0.739870\pi$
$774$	6.45589	0.232052
$775$	0	0
$776$	−3.07838	−0.110507
$777$	−4.43188	−0.158993
$778$	0.879362	0.0315266
$779$	40.7914	1.46150
$780$	0	0
$781$	−6.87549	−0.246024
$782$	−20.7792	−0.743064
$783$	−37.3340	−1.33421
$784$	1.29072	0.0460973
$785$	0	0
$786$	2.25953	0.0805947
$787$	−15.1506	−0.540061	−0.270031	−	0.962852i	$-0.587034\pi$
$787$	−15.1506	−0.540061	−0.270031	+	0.962852i	$0.587034\pi$
$788$	−24.8443	−0.885041
$789$	−28.7115	−1.02216
$790$	0	0
$791$	−10.3330	−0.367399
$792$	−0.688756	−0.0244739
$793$	28.5874	1.01517
$794$	29.6814	1.05335
$795$	0	0
$796$	−8.47027	−0.300221
$797$	43.1350	1.52792	0.763960	−	0.645263i	$-0.223252\pi$
$797$	43.1350	1.52792	0.763960	+	0.645263i	$0.223252\pi$
$798$	−22.3786	−0.792192
$799$	24.3135	0.860150
$800$	0	0
$801$	4.10731	0.145125
$802$	13.9867	0.493886
$803$	−13.8748	−0.489632
$804$	20.4680	0.721851
$805$	0	0
$806$	−15.9337	−0.561242
$807$	33.6020	1.18285
$808$	15.9155	0.559905
$809$	33.3874	1.17384	0.586918	−	0.809646i	$-0.300341\pi$
$809$	33.3874	1.17384	0.586918	+	0.809646i	$0.300341\pi$
$810$	0	0
$811$	10.9216	0.383510	0.191755	−	0.981443i	$-0.438582\pi$
$811$	10.9216	0.383510	0.191755	+	0.981443i	$0.438582\pi$
$812$	−19.2351	−0.675021
$813$	38.4463	1.34837
$814$	1.09171	0.0382644
$815$	0	0
$816$	4.31124	0.150924
$817$	−51.6703	−1.80772
$818$	21.9060	0.765926
$819$	8.24581	0.288132
$820$	0	0
$821$	−32.7910	−1.14441	−0.572206	−	0.820110i	$-0.693912\pi$
$821$	−32.7910	−1.14441	−0.572206	+	0.820110i	$0.693912\pi$
$822$	9.50799	0.331629
$823$	9.83218	0.342728	0.171364	−	0.985208i	$-0.445183\pi$
$823$	9.83218	0.342728	0.171364	+	0.985208i	$0.445183\pi$
$824$	10.5886	0.368873
$825$	0	0
$826$	29.4641	1.02519
$827$	17.0228	0.591940	0.295970	−	0.955197i	$-0.404357\pi$
$827$	17.0228	0.591940	0.295970	+	0.955197i	$0.404357\pi$
$828$	−4.68035	−0.162653
$829$	12.8260	0.445467	0.222733	−	0.974879i	$-0.428502\pi$
$829$	12.8260	0.445467	0.222733	+	0.974879i	$0.428502\pi$
$830$	0	0
$831$	−1.26633	−0.0439285
$832$	4.53919	0.157368
$833$	3.61530	0.125263
$834$	−20.0616	−0.694675
$835$	0	0
$836$	5.51253	0.190655
$837$	−19.6176	−0.678082
$838$	34.8648	1.20438
$839$	−11.5018	−0.397088	−0.198544	−	0.980092i	$-0.563621\pi$
$839$	−11.5018	−0.397088	−0.198544	+	0.980092i	$0.563621\pi$
$840$	0	0
$841$	15.6270	0.538863
$842$	4.76487	0.164208
$843$	−2.54184	−0.0875457
$844$	21.7370	0.748218
$845$	0	0
$846$	5.47641	0.188283
$847$	28.2413	0.970382
$848$	−10.0989	−0.346798
$849$	23.2940	0.799449
$850$	0	0
$851$	7.41855	0.254305
$852$	9.69368	0.332100
$853$	40.5946	1.38993	0.694966	−	0.719042i	$-0.255419\pi$
$853$	40.5946	1.38993	0.694966	+	0.719042i	$0.255419\pi$
$854$	18.1340	0.620532
$855$	0	0
$856$	4.03612	0.137952
$857$	38.0361	1.29929	0.649645	−	0.760238i	$-0.274918\pi$
$857$	38.0361	1.29929	0.649645	+	0.760238i	$0.274918\pi$
$858$	7.62741	0.260395
$859$	−40.9506	−1.39721	−0.698607	−	0.715505i	$-0.746197\pi$
$859$	−40.9506	−1.39721	−0.698607	+	0.715505i	$0.746197\pi$
$860$	0	0
$861$	35.8024	1.22014
$862$	−2.66597	−0.0908033
$863$	33.0817	1.12611	0.563057	−	0.826418i	$-0.309625\pi$
$863$	33.0817	1.12611	0.563057	+	0.826418i	$0.309625\pi$
$864$	5.58864	0.190129
$865$	0	0
$866$	−32.3074	−1.09785
$867$	−14.0905	−0.478538
$868$	−10.1073	−0.343064
$869$	−2.81819	−0.0956006
$870$	0	0
$871$	60.3617	2.04528
$872$	4.49693	0.152285
$873$	−1.94214	−0.0657315
$874$	37.4596	1.26709
$875$	0	0
$876$	19.5620	0.660938
$877$	25.5057	0.861267	0.430634	−	0.902527i	$-0.358290\pi$
$877$	25.5057	0.861267	0.430634	+	0.902527i	$0.358290\pi$
$878$	22.1568	0.747754
$879$	−39.1761	−1.32138
$880$	0	0
$881$	15.6514	0.527310	0.263655	−	0.964617i	$-0.415072\pi$
$881$	15.6514	0.527310	0.263655	+	0.964617i	$0.415072\pi$
$882$	0.814315	0.0274194
$883$	56.1071	1.88816	0.944078	−	0.329723i	$-0.106955\pi$
$883$	56.1071	1.88816	0.944078	+	0.329723i	$0.106955\pi$
$884$	12.7142	0.427625
$885$	0	0
$886$	−39.1061	−1.31379
$887$	−47.5052	−1.59507	−0.797534	−	0.603275i	$-0.793862\pi$
$887$	−47.5052	−1.59507	−0.797534	+	0.603275i	$0.793862\pi$
$888$	−1.53919	−0.0516518
$889$	−23.4413	−0.786197
$890$	0	0
$891$	7.32457	0.245382
$892$	16.6537	0.557607
$893$	−43.8310	−1.46675
$894$	6.78047	0.226773
$895$	0	0
$896$	2.87936	0.0961927
$897$	51.8310	1.73059
$898$	−30.8915	−1.03086
$899$	23.4497	0.782093
$900$	0	0
$901$	−28.2868	−0.942372
$902$	−8.81924	−0.293648
$903$	−45.3509	−1.50918
$904$	−3.58864	−0.119356
$905$	0	0
$906$	−4.49693	−0.149400
$907$	15.8394	0.525938	0.262969	−	0.964804i	$-0.415298\pi$
$907$	15.8394	0.525938	0.262969	+	0.964804i	$0.415298\pi$
$908$	−11.2123	−0.372095
$909$	10.0410	0.333040
$910$	0	0
$911$	11.9539	0.396049	0.198025	−	0.980197i	$-0.436547\pi$
$911$	11.9539	0.396049	0.198025	+	0.980197i	$0.436547\pi$
$912$	−7.77205	−0.257358
$913$	−9.26162	−0.306515
$914$	0.438025	0.0144886
$915$	0	0
$916$	−7.62863	−0.252057
$917$	4.22690	0.139585
$918$	15.6537	0.516649
$919$	−34.8781	−1.15052	−0.575262	−	0.817969i	$-0.695100\pi$
$919$	−34.8781	−1.15052	−0.575262	+	0.817969i	$0.695100\pi$
$920$	0	0
$921$	−42.8166	−1.41085
$922$	−0.523590	−0.0172435
$923$	28.5874	0.940966
$924$	4.83832	0.159169
$925$	0	0
$926$	29.1845	0.959061
$927$	6.68035	0.219411
$928$	−6.68035	−0.219293
$929$	−52.5439	−1.72391	−0.861955	−	0.506984i	$-0.830760\pi$
$929$	−52.5439	−1.72391	−0.861955	+	0.506984i	$0.830760\pi$
$930$	0	0
$931$	−6.51745	−0.213601
$932$	−0.0494483	−0.00161973
$933$	0.100119	0.00327776
$934$	−9.49079	−0.310548
$935$	0	0
$936$	2.86376	0.0936050
$937$	−13.3318	−0.435530	−0.217765	−	0.976001i	$-0.569877\pi$
$937$	−13.3318	−0.435530	−0.217765	+	0.976001i	$0.569877\pi$
$938$	38.2895	1.25020
$939$	34.7031	1.13249
$940$	0	0
$941$	14.5281	0.473603	0.236802	−	0.971558i	$-0.423901\pi$
$941$	14.5281	0.473603	0.236802	+	0.971558i	$0.423901\pi$
$942$	−34.1555	−1.11285
$943$	−59.9299	−1.95158
$944$	10.2329	0.333051
$945$	0	0
$946$	11.1713	0.363211
$947$	−24.6803	−0.802003	−0.401002	−	0.916077i	$-0.631338\pi$
$947$	−24.6803	−0.802003	−0.401002	+	0.916077i	$0.631338\pi$
$948$	3.97334	0.129048
$949$	57.6898	1.87269
$950$	0	0
$951$	−37.6754	−1.22171
$952$	8.06505	0.261390
$953$	−40.5936	−1.31495	−0.657477	−	0.753474i	$-0.728376\pi$
$953$	−40.5936	−1.31495	−0.657477	+	0.753474i	$0.728376\pi$
$954$	−6.37137	−0.206281
$955$	0	0
$956$	−14.7070	−0.475659
$957$	−11.2253	−0.362862
$958$	6.12291	0.197822
$959$	17.7866	0.574360
$960$	0	0
$961$	−18.6781	−0.602519
$962$	−4.53919	−0.146349
$963$	2.54638	0.0820558
$964$	−14.8999	−0.479893
$965$	0	0
$966$	32.8781	1.05784
$967$	−1.06400	−0.0342160	−0.0171080	−	0.999854i	$-0.505446\pi$
$967$	−1.06400	−0.0342160	−0.0171080	+	0.999854i	$0.505446\pi$
$968$	9.80817	0.315247
$969$	−21.7694	−0.699334
$970$	0	0
$971$	11.9333	0.382959	0.191480	−	0.981497i	$-0.438671\pi$
$971$	11.9333	0.382959	0.191480	+	0.981497i	$0.438671\pi$
$972$	6.43907	0.206533
$973$	−37.5292	−1.20313
$974$	−10.3018	−0.330091
$975$	0	0
$976$	6.29791	0.201591
$977$	−19.1867	−0.613838	−0.306919	−	0.951736i	$-0.599298\pi$
$977$	−19.1867	−0.613838	−0.306919	+	0.951736i	$0.599298\pi$
$978$	22.5103	0.719799
$979$	7.10731	0.227151
$980$	0	0
$981$	2.83710	0.0905817
$982$	10.5380	0.336280
$983$	−22.9171	−0.730942	−0.365471	−	0.930823i	$-0.619092\pi$
$983$	−22.9171	−0.730942	−0.365471	+	0.930823i	$0.619092\pi$
$984$	12.4341	0.396386
$985$	0	0
$986$	−18.7115	−0.595897
$987$	−38.4703	−1.22452
$988$	−22.9204	−0.729195
$989$	75.9130	2.41389
$990$	0	0
$991$	−52.2772	−1.66064	−0.830320	−	0.557287i	$-0.811842\pi$
$991$	−52.2772	−1.66064	−0.830320	+	0.557287i	$0.811842\pi$
$992$	−3.51026	−0.111451
$993$	−2.94479	−0.0934502
$994$	18.1340	0.575175
$995$	0	0
$996$	13.0579	0.413754
$997$	48.8892	1.54834	0.774168	−	0.632980i	$-0.218168\pi$
$997$	48.8892	1.54834	0.774168	+	0.632980i	$0.218168\pi$
$998$	31.2123	0.988010
$999$	−5.58864	−0.176817

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1850.2.a.ba.1.2	✓	3
5.2	odd	4		1850.2.b.n.149.2		6
5.3	odd	4		1850.2.b.n.149.5		6
5.4	even	2		1850.2.a.bb.1.2	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1850.2.a.ba.1.2	✓	3	1.1	even	1	trivial
1850.2.a.bb.1.2	yes	3	5.4	even	2
1850.2.b.n.149.2		6	5.2	odd	4
1850.2.b.n.149.5		6	5.3	odd	4

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:	\( N \)	\(=\)	\( 1850 = 2 \cdot 5^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1850.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.53919 q^{3} +1.00000 q^{4} -1.53919 q^{6} -2.87936 q^{7} -1.00000 q^{8} -0.630898 q^{9} -1.09171 q^{11} +1.53919 q^{12} +4.53919 q^{13} +2.87936 q^{14} +1.00000 q^{16} +2.80098 q^{17} +0.630898 q^{18} -5.04945 q^{19} -4.43188 q^{21} +1.09171 q^{22} +7.41855 q^{23} -1.53919 q^{24} -4.53919 q^{26} -5.58864 q^{27} -2.87936 q^{28} +6.68035 q^{29} +3.51026 q^{31} -1.00000 q^{32} -1.68035 q^{33} -2.80098 q^{34} -0.630898 q^{36} +1.00000 q^{37} +5.04945 q^{38} +6.98667 q^{39} -8.07838 q^{41} +4.43188 q^{42} +10.2329 q^{43} -1.09171 q^{44} -7.41855 q^{46} +8.68035 q^{47} +1.53919 q^{48} +1.29072 q^{49} +4.31124 q^{51} +4.53919 q^{52} -10.0989 q^{53} +5.58864 q^{54} +2.87936 q^{56} -7.77205 q^{57} -6.68035 q^{58} +10.2329 q^{59} +6.29791 q^{61} -3.51026 q^{62} +1.81658 q^{63} +1.00000 q^{64} +1.68035 q^{66} +13.2979 q^{67} +2.80098 q^{68} +11.4186 q^{69} +6.29791 q^{71} +0.630898 q^{72} +12.7093 q^{73} -1.00000 q^{74} -5.04945 q^{76} +3.14342 q^{77} -6.98667 q^{78} +2.58145 q^{79} -6.70928 q^{81} +8.07838 q^{82} +8.48360 q^{83} -4.43188 q^{84} -10.2329 q^{86} +10.2823 q^{87} +1.09171 q^{88} -6.51026 q^{89} -13.0700 q^{91} +7.41855 q^{92} +5.40295 q^{93} -8.68035 q^{94} -1.53919 q^{96} +3.07838 q^{97} -1.29072 q^{98} +0.688756 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} - 3 q^{6} + 4 q^{7} - 3 q^{8} + 2 q^{9} - q^{11} + 3 q^{12} + 12 q^{13} - 4 q^{14} + 3 q^{16} - q^{17} - 2 q^{18} + 3 q^{19} + q^{22} + 8 q^{23} - 3 q^{24} - 12 q^{26}+ \cdots + 28 q^{99}+O(q^{100}) \) 3 * q - 3 * q^2 + 3 * q^3 + 3 * q^4 - 3 * q^6 + 4 * q^7 - 3 * q^8 + 2 * q^9 - q^11 + 3 * q^12 + 12 * q^13 - 4 * q^14 + 3 * q^16 - q^17 - 2 * q^18 + 3 * q^19 + q^22 + 8 * q^23 - 3 * q^24 - 12 * q^26 + 3 * q^27 + 4 * q^28 - 2 * q^29 - 6 * q^31 - 3 * q^32 + 17 * q^33 + q^34 + 2 * q^36 + 3 * q^37 - 3 * q^38 + 20 * q^39 - 21 * q^41 + 8 * q^43 - q^44 - 8 * q^46 + 4 * q^47 + 3 * q^48 + 11 * q^49 - 13 * q^51 + 12 * q^52 + 6 * q^53 - 3 * q^54 - 4 * q^56 + q^57 + 2 * q^58 + 8 * q^59 - 8 * q^61 + 6 * q^62 + 10 * q^63 + 3 * q^64 - 17 * q^66 + 13 * q^67 - q^68 + 20 * q^69 - 8 * q^71 - 2 * q^72 + 31 * q^73 - 3 * q^74 + 3 * q^76 + 2 * q^77 - 20 * q^78 + 22 * q^79 - 13 * q^81 + 21 * q^82 + 7 * q^83 - 8 * q^86 - 10 * q^87 + q^88 - 3 * q^89 + 12 * q^91 + 8 * q^92 - 12 * q^93 - 4 * q^94 - 3 * q^96 + 6 * q^97 - 11 * q^98 + 28 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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