LMFDB - Embedded newform 1450.2.a.q.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1450,2,Mod(1,1450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1450, 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("1450.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1450 = 2 \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1450.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:	$11.5783082931$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.316.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
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			1.3
Root			$2.34292$ of defining polynomial
Character	$\chi$	$=$	1450.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} +2.34292 q^{3} +1.00000 q^{4} -2.34292 q^{6} +3.83221 q^{7} -1.00000 q^{8} +2.48929 q^{9} +3.19656 q^{11} +2.34292 q^{12} -1.14637 q^{13} -3.83221 q^{14} +1.00000 q^{16} -0.803442 q^{17} -2.48929 q^{18} -2.34292 q^{19} +8.97858 q^{21} -3.19656 q^{22} +8.12494 q^{23} -2.34292 q^{24} +1.14637 q^{26} -1.19656 q^{27} +3.83221 q^{28} +1.00000 q^{29} +1.90383 q^{31} -1.00000 q^{32} +7.48929 q^{33} +0.803442 q^{34} +2.48929 q^{36} -7.63565 q^{37} +2.34292 q^{38} -2.68585 q^{39} +8.63565 q^{41} -8.97858 q^{42} -7.66442 q^{43} +3.19656 q^{44} -8.12494 q^{46} -11.4679 q^{47} +2.34292 q^{48} +7.68585 q^{49} -1.88240 q^{51} -1.14637 q^{52} +0.560904 q^{53} +1.19656 q^{54} -3.83221 q^{56} -5.48929 q^{57} -1.00000 q^{58} +2.36435 q^{59} +10.7146 q^{61} -1.90383 q^{62} +9.53948 q^{63} +1.00000 q^{64} -7.48929 q^{66} +5.12494 q^{67} -0.803442 q^{68} +19.0361 q^{69} +8.12494 q^{71} -2.48929 q^{72} -5.71462 q^{73} +7.63565 q^{74} -2.34292 q^{76} +12.2499 q^{77} +2.68585 q^{78} -14.9357 q^{79} -10.2713 q^{81} -8.63565 q^{82} +3.09617 q^{83} +8.97858 q^{84} +7.66442 q^{86} +2.34292 q^{87} -3.19656 q^{88} -3.32150 q^{89} -4.39312 q^{91} +8.12494 q^{92} +4.46052 q^{93} +11.4679 q^{94} -2.34292 q^{96} -11.5395 q^{97} -7.68585 q^{98} +7.95715 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + q^{3} + 3 q^{4} - q^{6} - 2 q^{7} - 3 q^{8} + 5 q^{11} + q^{12} - 2 q^{13} + 2 q^{14} + 3 q^{16} - 7 q^{17} - q^{19} + 12 q^{21} - 5 q^{22} + 8 q^{23} - q^{24} + 2 q^{26} + q^{27}+ \cdots - 6 q^{99}+O(q^{100}) $ 3 * q - 3 * q^2 + q^3 + 3 * q^4 - q^6 - 2 * q^7 - 3 * q^8 + 5 * q^11 + q^12 - 2 * q^13 + 2 * q^14 + 3 * q^16 - 7 * q^17 - q^19 + 12 * q^21 - 5 * q^22 + 8 * q^23 - q^24 + 2 * q^26 + q^27 - 2 * q^28 + 3 * q^29 + 4 * q^31 - 3 * q^32 + 15 * q^33 + 7 * q^34 - 14 * q^37 + q^38 + 4 * q^39 + 17 * q^41 - 12 * q^42 + 4 * q^43 + 5 * q^44 - 8 * q^46 - 12 * q^47 + q^48 + 11 * q^49 + 11 * q^51 - 2 * q^52 + 6 * q^53 - q^54 + 2 * q^56 - 9 * q^57 - 3 * q^58 + 16 * q^59 + 2 * q^61 - 4 * q^62 + 18 * q^63 + 3 * q^64 - 15 * q^66 - q^67 - 7 * q^68 + 6 * q^69 + 8 * q^71 + 13 * q^73 + 14 * q^74 - q^76 + 4 * q^77 - 4 * q^78 - 13 * q^81 - 17 * q^82 + 11 * q^83 + 12 * q^84 - 4 * q^86 + q^87 - 5 * q^88 + 11 * q^89 - 4 * q^91 + 8 * q^92 + 24 * q^93 + 12 * q^94 - q^96 - 24 * q^97 - 11 * q^98 - 6 * 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
$3$	2.34292	1.35269	0.676344	−	0.736586i	$-0.263563\pi$
$3$	2.34292	1.35269	0.676344	+	0.736586i	$0.263563\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−2.34292	−0.956494
$7$	3.83221	1.44844	0.724220	−	0.689569i	$-0.242200\pi$
$7$	3.83221	1.44844	0.724220	+	0.689569i	$0.242200\pi$
$8$	−1.00000	−0.353553
$9$	2.48929	0.829763
$10$	0	0
$11$	3.19656	0.963798	0.481899	−	0.876227i	$-0.339947\pi$
$11$	3.19656	0.963798	0.481899	+	0.876227i	$0.339947\pi$
$12$	2.34292	0.676344
$13$	−1.14637	−0.317945	−0.158972	−	0.987283i	$-0.550818\pi$
$13$	−1.14637	−0.317945	−0.158972	+	0.987283i	$0.550818\pi$
$14$	−3.83221	−1.02420
$15$	0	0
$16$	1.00000	0.250000
$17$	−0.803442	−0.194863	−0.0974317	−	0.995242i	$-0.531063\pi$
$17$	−0.803442	−0.194863	−0.0974317	+	0.995242i	$0.531063\pi$
$18$	−2.48929	−0.586731
$19$	−2.34292	−0.537503	−0.268752	−	0.963209i	$-0.586611\pi$
$19$	−2.34292	−0.537503	−0.268752	+	0.963209i	$0.586611\pi$
$20$	0	0
$21$	8.97858	1.95929
$22$	−3.19656	−0.681508
$23$	8.12494	1.69417	0.847084	−	0.531459i	$-0.178356\pi$
$23$	8.12494	1.69417	0.847084	+	0.531459i	$0.178356\pi$
$24$	−2.34292	−0.478247
$25$	0	0
$26$	1.14637	0.224821
$27$	−1.19656	−0.230278
$28$	3.83221	0.724220
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	1.90383	0.341937	0.170969	−	0.985276i	$-0.445310\pi$
$31$	1.90383	0.341937	0.170969	+	0.985276i	$0.445310\pi$
$32$	−1.00000	−0.176777
$33$	7.48929	1.30372
$34$	0.803442	0.137789
$35$	0	0
$36$	2.48929	0.414881
$37$	−7.63565	−1.25529	−0.627647	−	0.778498i	$-0.715982\pi$
$37$	−7.63565	−1.25529	−0.627647	+	0.778498i	$0.715982\pi$
$38$	2.34292	0.380072
$39$	−2.68585	−0.430080
$40$	0	0
$41$	8.63565	1.34866	0.674331	−	0.738429i	$-0.264432\pi$
$41$	8.63565	1.34866	0.674331	+	0.738429i	$0.264432\pi$
$42$	−8.97858	−1.38542
$43$	−7.66442	−1.16881	−0.584407	−	0.811461i	$-0.698673\pi$
$43$	−7.66442	−1.16881	−0.584407	+	0.811461i	$0.698673\pi$
$44$	3.19656	0.481899
$45$	0	0
$46$	−8.12494	−1.19796
$47$	−11.4679	−1.67276	−0.836380	−	0.548150i	$-0.815332\pi$
$47$	−11.4679	−1.67276	−0.836380	+	0.548150i	$0.815332\pi$
$48$	2.34292	0.338172
$49$	7.68585	1.09798
$50$	0	0
$51$	−1.88240	−0.263589
$52$	−1.14637	−0.158972
$53$	0.560904	0.0770460	0.0385230	−	0.999258i	$-0.487735\pi$
$53$	0.560904	0.0770460	0.0385230	+	0.999258i	$0.487735\pi$
$54$	1.19656	0.162831
$55$	0	0
$56$	−3.83221	−0.512101
$57$	−5.48929	−0.727074
$58$	−1.00000	−0.131306
$59$	2.36435	0.307812	0.153906	−	0.988086i	$-0.450815\pi$
$59$	2.36435	0.307812	0.153906	+	0.988086i	$0.450815\pi$
$60$	0	0
$61$	10.7146	1.37187	0.685933	−	0.727665i	$-0.259394\pi$
$61$	10.7146	1.37187	0.685933	+	0.727665i	$0.259394\pi$
$62$	−1.90383	−0.241786
$63$	9.53948	1.20186
$64$	1.00000	0.125000
$65$	0	0
$66$	−7.48929	−0.921868
$67$	5.12494	0.626111	0.313056	−	0.949735i	$-0.398647\pi$
$67$	5.12494	0.626111	0.313056	+	0.949735i	$0.398647\pi$
$68$	−0.803442	−0.0974317
$69$	19.0361	2.29168
$70$	0	0
$71$	8.12494	0.964253	0.482127	−	0.876102i	$-0.339865\pi$
$71$	8.12494	0.964253	0.482127	+	0.876102i	$0.339865\pi$
$72$	−2.48929	−0.293365
$73$	−5.71462	−0.668845	−0.334423	−	0.942423i	$-0.608541\pi$
$73$	−5.71462	−0.668845	−0.334423	+	0.942423i	$0.608541\pi$
$74$	7.63565	0.887627
$75$	0	0
$76$	−2.34292	−0.268752
$77$	12.2499	1.39600
$78$	2.68585	0.304112
$79$	−14.9357	−1.68040	−0.840201	−	0.542276i	$-0.817563\pi$
$79$	−14.9357	−1.68040	−0.840201	+	0.542276i	$0.817563\pi$
$80$	0	0
$81$	−10.2713	−1.14126
$82$	−8.63565	−0.953648
$83$	3.09617	0.339849	0.169925	−	0.985457i	$-0.445648\pi$
$83$	3.09617	0.339849	0.169925	+	0.985457i	$0.445648\pi$
$84$	8.97858	0.979643
$85$	0	0
$86$	7.66442	0.826476
$87$	2.34292	0.251188
$88$	−3.19656	−0.340754
$89$	−3.32150	−0.352078	−0.176039	−	0.984383i	$-0.556329\pi$
$89$	−3.32150	−0.352078	−0.176039	+	0.984383i	$0.556329\pi$
$90$	0	0
$91$	−4.39312	−0.460524
$92$	8.12494	0.847084
$93$	4.46052	0.462534
$94$	11.4679	1.18282
$95$	0	0
$96$	−2.34292	−0.239124
$97$	−11.5395	−1.17166	−0.585828	−	0.810435i	$-0.699231\pi$
$97$	−11.5395	−1.17166	−0.585828	+	0.810435i	$0.699231\pi$
$98$	−7.68585	−0.776388
$99$	7.95715	0.799724
$100$	0	0
$101$	5.63565	0.560769	0.280384	−	0.959888i	$-0.409538\pi$
$101$	5.63565	0.560769	0.280384	+	0.959888i	$0.409538\pi$
$102$	1.88240	0.186386
$103$	−17.5970	−1.73389	−0.866943	−	0.498407i	$-0.833918\pi$
$103$	−17.5970	−1.73389	−0.866943	+	0.498407i	$0.833918\pi$
$104$	1.14637	0.112410
$105$	0	0
$106$	−0.560904	−0.0544798
$107$	1.56090	0.150898	0.0754491	−	0.997150i	$-0.475961\pi$
$107$	1.56090	0.150898	0.0754491	+	0.997150i	$0.475961\pi$
$108$	−1.19656	−0.115139
$109$	11.4966	1.10118	0.550589	−	0.834776i	$-0.314403\pi$
$109$	11.4966	1.10118	0.550589	+	0.834776i	$0.314403\pi$
$110$	0	0
$111$	−17.8898	−1.69802
$112$	3.83221	0.362110
$113$	5.36435	0.504635	0.252318	−	0.967644i	$-0.418807\pi$
$113$	5.36435	0.504635	0.252318	+	0.967644i	$0.418807\pi$
$114$	5.48929	0.514119
$115$	0	0
$116$	1.00000	0.0928477
$117$	−2.85363	−0.263819
$118$	−2.36435	−0.217656
$119$	−3.07896	−0.282248
$120$	0	0
$121$	−0.782020	−0.0710927
$122$	−10.7146	−0.970056
$123$	20.2327	1.82432
$124$	1.90383	0.170969
$125$	0	0
$126$	−9.53948	−0.849844
$127$	7.90383	0.701351	0.350676	−	0.936497i	$-0.385952\pi$
$127$	7.90383	0.701351	0.350676	+	0.936497i	$0.385952\pi$
$128$	−1.00000	−0.0883883
$129$	−17.9572	−1.58104
$130$	0	0
$131$	−3.10352	−0.271156	−0.135578	−	0.990767i	$-0.543289\pi$
$131$	−3.10352	−0.271156	−0.135578	+	0.990767i	$0.543289\pi$
$132$	7.48929	0.651859
$133$	−8.97858	−0.778541
$134$	−5.12494	−0.442728
$135$	0	0
$136$	0.803442	0.0688946
$137$	19.2541	1.64499	0.822494	−	0.568773i	$-0.192582\pi$
$137$	19.2541	1.64499	0.822494	+	0.568773i	$0.192582\pi$
$138$	−19.0361	−1.62046
$139$	6.43910	0.546157	0.273079	−	0.961992i	$-0.411958\pi$
$139$	6.43910	0.546157	0.273079	+	0.961992i	$0.411958\pi$
$140$	0	0
$141$	−26.8683	−2.26272
$142$	−8.12494	−0.681830
$143$	−3.66442	−0.306434
$144$	2.48929	0.207441
$145$	0	0
$146$	5.71462	0.472945
$147$	18.0073	1.48522
$148$	−7.63565	−0.627647
$149$	2.00000	0.163846	0.0819232	−	0.996639i	$-0.473894\pi$
$149$	2.00000	0.163846	0.0819232	+	0.996639i	$0.473894\pi$
$150$	0	0
$151$	−10.8536	−0.883256	−0.441628	−	0.897198i	$-0.645599\pi$
$151$	−10.8536	−0.883256	−0.441628	+	0.897198i	$0.645599\pi$
$152$	2.34292	0.190036
$153$	−2.00000	−0.161690
$154$	−12.2499	−0.987124
$155$	0	0
$156$	−2.68585	−0.215040
$157$	−7.69319	−0.613984	−0.306992	−	0.951712i	$-0.599322\pi$
$157$	−7.69319	−0.613984	−0.306992	+	0.951712i	$0.599322\pi$
$158$	14.9357	1.18822
$159$	1.31415	0.104219
$160$	0	0
$161$	31.1365	2.45390
$162$	10.2713	0.806990
$163$	−1.61423	−0.126436	−0.0632182	−	0.998000i	$-0.520136\pi$
$163$	−1.61423	−0.126436	−0.0632182	+	0.998000i	$0.520136\pi$
$164$	8.63565	0.674331
$165$	0	0
$166$	−3.09617	−0.240310
$167$	−13.7220	−1.06184	−0.530919	−	0.847423i	$-0.678153\pi$
$167$	−13.7220	−1.06184	−0.530919	+	0.847423i	$0.678153\pi$
$168$	−8.97858	−0.692712
$169$	−11.6858	−0.898911
$170$	0	0
$171$	−5.83221	−0.446000
$172$	−7.66442	−0.584407
$173$	18.3074	1.39189	0.695944	−	0.718096i	$-0.254986\pi$
$173$	18.3074	1.39189	0.695944	+	0.718096i	$0.254986\pi$
$174$	−2.34292	−0.177617
$175$	0	0
$176$	3.19656	0.240950
$177$	5.53948	0.416373
$178$	3.32150	0.248957
$179$	2.43910	0.182307	0.0911533	−	0.995837i	$-0.470945\pi$
$179$	2.43910	0.182307	0.0911533	+	0.995837i	$0.470945\pi$
$180$	0	0
$181$	4.02456	0.299143	0.149572	−	0.988751i	$-0.452211\pi$
$181$	4.02456	0.299143	0.149572	+	0.988751i	$0.452211\pi$
$182$	4.39312	0.325639
$183$	25.1035	1.85571
$184$	−8.12494	−0.598979
$185$	0	0
$186$	−4.46052	−0.327061
$187$	−2.56825	−0.187809
$188$	−11.4679	−0.836380
$189$	−4.58546	−0.333543
$190$	0	0
$191$	−5.56825	−0.402904	−0.201452	−	0.979498i	$-0.564566\pi$
$191$	−5.56825	−0.402904	−0.201452	+	0.979498i	$0.564566\pi$
$192$	2.34292	0.169086
$193$	13.7146	0.987200	0.493600	−	0.869689i	$-0.335681\pi$
$193$	13.7146	0.987200	0.493600	+	0.869689i	$0.335681\pi$
$194$	11.5395	0.828486
$195$	0	0
$196$	7.68585	0.548989
$197$	−25.4292	−1.81176	−0.905879	−	0.423537i	$-0.860788\pi$
$197$	−25.4292	−1.81176	−0.905879	+	0.423537i	$0.860788\pi$
$198$	−7.95715	−0.565490
$199$	15.1035	1.07066	0.535330	−	0.844643i	$-0.320187\pi$
$199$	15.1035	1.07066	0.535330	+	0.844643i	$0.320187\pi$
$200$	0	0
$201$	12.0073	0.846933
$202$	−5.63565	−0.396523
$203$	3.83221	0.268969
$204$	−1.88240	−0.131795
$205$	0	0
$206$	17.5970	1.22604
$207$	20.2253	1.40576
$208$	−1.14637	−0.0794861
$209$	−7.48929	−0.518045
$210$	0	0
$211$	19.7967	1.36286	0.681431	−	0.731882i	$-0.261358\pi$
$211$	19.7967	1.36286	0.681431	+	0.731882i	$0.261358\pi$
$212$	0.560904	0.0385230
$213$	19.0361	1.30433
$214$	−1.56090	−0.106701
$215$	0	0
$216$	1.19656	0.0814154
$217$	7.29587	0.495276
$218$	−11.4966	−0.778650
$219$	−13.3889	−0.904738
$220$	0	0
$221$	0.921039	0.0619558
$222$	17.8898	1.20068
$223$	5.37169	0.359715	0.179858	−	0.983693i	$-0.442436\pi$
$223$	5.37169	0.359715	0.179858	+	0.983693i	$0.442436\pi$
$224$	−3.83221	−0.256050
$225$	0	0
$226$	−5.36435	−0.356831
$227$	28.3215	1.87976	0.939882	−	0.341499i	$-0.110935\pi$
$227$	28.3215	1.87976	0.939882	+	0.341499i	$0.110935\pi$
$228$	−5.48929	−0.363537
$229$	10.0147	0.661790	0.330895	−	0.943668i	$-0.392649\pi$
$229$	10.0147	0.661790	0.330895	+	0.943668i	$0.392649\pi$
$230$	0	0
$231$	28.7005	1.88836
$232$	−1.00000	−0.0656532
$233$	−23.3246	−1.52805	−0.764024	−	0.645188i	$-0.776779\pi$
$233$	−23.3246	−1.52805	−0.764024	+	0.645188i	$0.776779\pi$
$234$	2.85363	0.186548
$235$	0	0
$236$	2.36435	0.153906
$237$	−34.9933	−2.27306
$238$	3.07896	0.199579
$239$	−21.9143	−1.41752	−0.708759	−	0.705450i	$-0.750745\pi$
$239$	−21.9143	−1.41752	−0.708759	+	0.705450i	$0.750745\pi$
$240$	0	0
$241$	9.45065	0.608770	0.304385	−	0.952549i	$-0.401549\pi$
$241$	9.45065	0.608770	0.304385	+	0.952549i	$0.401549\pi$
$242$	0.782020	0.0502701
$243$	−20.4752	−1.31349
$244$	10.7146	0.685933
$245$	0	0
$246$	−20.2327	−1.28999
$247$	2.68585	0.170896
$248$	−1.90383	−0.120893
$249$	7.25410	0.459710
$250$	0	0
$251$	−26.8855	−1.69700	−0.848500	−	0.529195i	$-0.822494\pi$
$251$	−26.8855	−1.69700	−0.848500	+	0.529195i	$0.822494\pi$
$252$	9.53948	0.600931
$253$	25.9718	1.63284
$254$	−7.90383	−0.495930
$255$	0	0
$256$	1.00000	0.0625000
$257$	6.39312	0.398792	0.199396	−	0.979919i	$-0.436102\pi$
$257$	6.39312	0.398792	0.199396	+	0.979919i	$0.436102\pi$
$258$	17.9572	1.11796
$259$	−29.2614	−1.81822
$260$	0	0
$261$	2.48929	0.154083
$262$	3.10352	0.191736
$263$	−27.4679	−1.69374	−0.846871	−	0.531799i	$-0.821516\pi$
$263$	−27.4679	−1.69374	−0.846871	+	0.531799i	$0.821516\pi$
$264$	−7.48929	−0.460934
$265$	0	0
$266$	8.97858	0.550512
$267$	−7.78202	−0.476252
$268$	5.12494	0.313056
$269$	9.59281	0.584884	0.292442	−	0.956283i	$-0.405532\pi$
$269$	9.59281	0.584884	0.292442	+	0.956283i	$0.405532\pi$
$270$	0	0
$271$	−29.9614	−1.82002	−0.910012	−	0.414583i	$-0.863928\pi$
$271$	−29.9614	−1.82002	−0.910012	+	0.414583i	$0.863928\pi$
$272$	−0.803442	−0.0487159
$273$	−10.2927	−0.622944
$274$	−19.2541	−1.16318
$275$	0	0
$276$	19.0361	1.14584
$277$	−15.7894	−0.948691	−0.474346	−	0.880339i	$-0.657315\pi$
$277$	−15.7894	−0.948691	−0.474346	+	0.880339i	$0.657315\pi$
$278$	−6.43910	−0.386191
$279$	4.73917	0.283727
$280$	0	0
$281$	12.1390	0.724153	0.362077	−	0.932148i	$-0.382068\pi$
$281$	12.1390	0.724153	0.362077	+	0.932148i	$0.382068\pi$
$282$	26.8683	1.59999
$283$	−4.17513	−0.248186	−0.124093	−	0.992271i	$-0.539602\pi$
$283$	−4.17513	−0.248186	−0.124093	+	0.992271i	$0.539602\pi$
$284$	8.12494	0.482127
$285$	0	0
$286$	3.66442	0.216682
$287$	33.0937	1.95346
$288$	−2.48929	−0.146683
$289$	−16.3545	−0.962028
$290$	0	0
$291$	−27.0361	−1.58489
$292$	−5.71462	−0.334423
$293$	9.09931	0.531587	0.265794	−	0.964030i	$-0.414366\pi$
$293$	9.09931	0.531587	0.265794	+	0.964030i	$0.414366\pi$
$294$	−18.0073	−1.05021
$295$	0	0
$296$	7.63565	0.443813
$297$	−3.82487	−0.221941
$298$	−2.00000	−0.115857
$299$	−9.31415	−0.538651
$300$	0	0
$301$	−29.3717	−1.69296
$302$	10.8536	0.624556
$303$	13.2039	0.758544
$304$	−2.34292	−0.134376
$305$	0	0
$306$	2.00000	0.114332
$307$	33.3116	1.90120	0.950598	−	0.310426i	$-0.100472\pi$
$307$	33.3116	1.90120	0.950598	+	0.310426i	$0.100472\pi$
$308$	12.2499	0.698002
$309$	−41.2285	−2.34541
$310$	0	0
$311$	19.7648	1.12076	0.560380	−	0.828236i	$-0.310655\pi$
$311$	19.7648	1.12076	0.560380	+	0.828236i	$0.310655\pi$
$312$	2.68585	0.152056
$313$	19.1898	1.08467	0.542337	−	0.840161i	$-0.317540\pi$
$313$	19.1898	1.08467	0.542337	+	0.840161i	$0.317540\pi$
$314$	7.69319	0.434152
$315$	0	0
$316$	−14.9357	−0.840201
$317$	−30.7722	−1.72834	−0.864168	−	0.503203i	$-0.832155\pi$
$317$	−30.7722	−1.72834	−0.864168	+	0.503203i	$0.832155\pi$
$318$	−1.31415	−0.0736941
$319$	3.19656	0.178973
$320$	0	0
$321$	3.65708	0.204118
$322$	−31.1365	−1.73517
$323$	1.88240	0.104740
$324$	−10.2713	−0.570628
$325$	0	0
$326$	1.61423	0.0894040
$327$	26.9357	1.48955
$328$	−8.63565	−0.476824
$329$	−43.9473	−2.42289
$330$	0	0
$331$	−31.5714	−1.73532	−0.867660	−	0.497158i	$-0.834377\pi$
$331$	−31.5714	−1.73532	−0.867660	+	0.497158i	$0.834377\pi$
$332$	3.09617	0.169925
$333$	−19.0073	−1.04160
$334$	13.7220	0.750832
$335$	0	0
$336$	8.97858	0.489822
$337$	−15.1966	−0.827809	−0.413905	−	0.910320i	$-0.635835\pi$
$337$	−15.1966	−0.827809	−0.413905	+	0.910320i	$0.635835\pi$
$338$	11.6858	0.635626
$339$	12.5682	0.682614
$340$	0	0
$341$	6.08569	0.329559
$342$	5.83221	0.315370
$343$	2.62831	0.141915
$344$	7.66442	0.413238
$345$	0	0
$346$	−18.3074	−0.984213
$347$	17.0246	0.913926	0.456963	−	0.889486i	$-0.348937\pi$
$347$	17.0246	0.913926	0.456963	+	0.889486i	$0.348937\pi$
$348$	2.34292	0.125594
$349$	−7.20390	−0.385616	−0.192808	−	0.981236i	$-0.561759\pi$
$349$	−7.20390	−0.385616	−0.192808	+	0.981236i	$0.561759\pi$
$350$	0	0
$351$	1.37169	0.0732155
$352$	−3.19656	−0.170377
$353$	−24.3116	−1.29398	−0.646989	−	0.762499i	$-0.723972\pi$
$353$	−24.3116	−1.29398	−0.646989	+	0.762499i	$0.723972\pi$
$354$	−5.53948	−0.294420
$355$	0	0
$356$	−3.32150	−0.176039
$357$	−7.21377	−0.381793
$358$	−2.43910	−0.128910
$359$	33.2902	1.75699	0.878495	−	0.477751i	$-0.158548\pi$
$359$	33.2902	1.75699	0.878495	+	0.477751i	$0.158548\pi$
$360$	0	0
$361$	−13.5107	−0.711090
$362$	−4.02456	−0.211526
$363$	−1.83221	−0.0961662
$364$	−4.39312	−0.230262
$365$	0	0
$366$	−25.1035	−1.31218
$367$	−18.9315	−0.988217	−0.494109	−	0.869400i	$-0.664506\pi$
$367$	−18.9315	−0.988217	−0.494109	+	0.869400i	$0.664506\pi$
$368$	8.12494	0.423542
$369$	21.4966	1.11907
$370$	0	0
$371$	2.14950	0.111597
$372$	4.46052	0.231267
$373$	−11.8996	−0.616139	−0.308069	−	0.951364i	$-0.599683\pi$
$373$	−11.8996	−0.616139	−0.308069	+	0.951364i	$0.599683\pi$
$374$	2.56825	0.132801
$375$	0	0
$376$	11.4679	0.591410
$377$	−1.14637	−0.0590408
$378$	4.58546	0.235851
$379$	−20.4005	−1.04790	−0.523951	−	0.851749i	$-0.675542\pi$
$379$	−20.4005	−1.04790	−0.523951	+	0.851749i	$0.675542\pi$
$380$	0	0
$381$	18.5181	0.948709
$382$	5.56825	0.284896
$383$	−36.8929	−1.88514	−0.942569	−	0.334011i	$-0.891598\pi$
$383$	−36.8929	−1.88514	−0.942569	+	0.334011i	$0.891598\pi$
$384$	−2.34292	−0.119562
$385$	0	0
$386$	−13.7146	−0.698056
$387$	−19.0790	−0.969838
$388$	−11.5395	−0.585828
$389$	−22.7146	−1.15168	−0.575838	−	0.817564i	$-0.695324\pi$
$389$	−22.7146	−1.15168	−0.575838	+	0.817564i	$0.695324\pi$
$390$	0	0
$391$	−6.52792	−0.330131
$392$	−7.68585	−0.388194
$393$	−7.27131	−0.366789
$394$	25.4292	1.28111
$395$	0	0
$396$	7.95715	0.399862
$397$	33.5113	1.68189	0.840943	−	0.541124i	$-0.182001\pi$
$397$	33.5113	1.68189	0.840943	+	0.541124i	$0.182001\pi$
$398$	−15.1035	−0.757071
$399$	−21.0361	−1.05312
$400$	0	0
$401$	−17.0147	−0.849673	−0.424837	−	0.905270i	$-0.639668\pi$
$401$	−17.0147	−0.849673	−0.424837	+	0.905270i	$0.639668\pi$
$402$	−12.0073	−0.598872
$403$	−2.18248	−0.108717
$404$	5.63565	0.280384
$405$	0	0
$406$	−3.83221	−0.190189
$407$	−24.4078	−1.20985
$408$	1.88240	0.0931929
$409$	−21.3461	−1.05549	−0.527747	−	0.849401i	$-0.676963\pi$
$409$	−21.3461	−1.05549	−0.527747	+	0.849401i	$0.676963\pi$
$410$	0	0
$411$	45.1109	2.22515
$412$	−17.5970	−0.866943
$413$	9.06067	0.445847
$414$	−20.2253	−0.994021
$415$	0	0
$416$	1.14637	0.0562052
$417$	15.0863	0.738780
$418$	7.48929	0.366313
$419$	3.81079	0.186169	0.0930846	−	0.995658i	$-0.470327\pi$
$419$	3.81079	0.186169	0.0930846	+	0.995658i	$0.470327\pi$
$420$	0	0
$421$	−23.9859	−1.16900	−0.584501	−	0.811393i	$-0.698710\pi$
$421$	−23.9859	−1.16900	−0.584501	+	0.811393i	$0.698710\pi$
$422$	−19.7967	−0.963689
$423$	−28.5468	−1.38799
$424$	−0.560904	−0.0272399
$425$	0	0
$426$	−19.0361	−0.922303
$427$	41.0607	1.98707
$428$	1.56090	0.0754491
$429$	−8.58546	−0.414510
$430$	0	0
$431$	7.00314	0.337329	0.168665	−	0.985674i	$-0.446055\pi$
$431$	7.00314	0.337329	0.168665	+	0.985674i	$0.446055\pi$
$432$	−1.19656	−0.0575694
$433$	−23.3790	−1.12352	−0.561762	−	0.827299i	$-0.689877\pi$
$433$	−23.3790	−1.12352	−0.561762	+	0.827299i	$0.689877\pi$
$434$	−7.29587	−0.350213
$435$	0	0
$436$	11.4966	0.550589
$437$	−19.0361	−0.910621
$438$	13.3889	0.639747
$439$	13.7073	0.654212	0.327106	−	0.944988i	$-0.393927\pi$
$439$	13.7073	0.654212	0.327106	+	0.944988i	$0.393927\pi$
$440$	0	0
$441$	19.1323	0.911061
$442$	−0.921039	−0.0438093
$443$	−9.25410	−0.439675	−0.219838	−	0.975536i	$-0.570553\pi$
$443$	−9.25410	−0.439675	−0.219838	+	0.975536i	$0.570553\pi$
$444$	−17.8898	−0.849010
$445$	0	0
$446$	−5.37169	−0.254357
$447$	4.68585	0.221633
$448$	3.83221	0.181055
$449$	25.8971	1.22216	0.611080	−	0.791569i	$-0.290735\pi$
$449$	25.8971	1.22216	0.611080	+	0.791569i	$0.290735\pi$
$450$	0	0
$451$	27.6044	1.29984
$452$	5.36435	0.252318
$453$	−25.4292	−1.19477
$454$	−28.3215	−1.32919
$455$	0	0
$456$	5.48929	0.257059
$457$	−26.5829	−1.24350	−0.621749	−	0.783217i	$-0.713578\pi$
$457$	−26.5829	−1.24350	−0.621749	+	0.783217i	$0.713578\pi$
$458$	−10.0147	−0.467956
$459$	0.961365	0.0448727
$460$	0	0
$461$	0.527923	0.0245878	0.0122939	−	0.999924i	$-0.496087\pi$
$461$	0.527923	0.0245878	0.0122939	+	0.999924i	$0.496087\pi$
$462$	−28.7005	−1.33527
$463$	−38.0575	−1.76868	−0.884342	−	0.466840i	$-0.845392\pi$
$463$	−38.0575	−1.76868	−0.884342	+	0.466840i	$0.845392\pi$
$464$	1.00000	0.0464238
$465$	0	0
$466$	23.3246	1.08049
$467$	−10.5181	−0.486718	−0.243359	−	0.969936i	$-0.578249\pi$
$467$	−10.5181	−0.486718	−0.243359	+	0.969936i	$0.578249\pi$
$468$	−2.85363	−0.131909
$469$	19.6399	0.906885
$470$	0	0
$471$	−18.0246	−0.830528
$472$	−2.36435	−0.108828
$473$	−24.4998	−1.12650
$474$	34.9933	1.60729
$475$	0	0
$476$	−3.07896	−0.141124
$477$	1.39625	0.0639299
$478$	21.9143	1.00234
$479$	5.54935	0.253556	0.126778	−	0.991931i	$-0.459536\pi$
$479$	5.54935	0.253556	0.126778	+	0.991931i	$0.459536\pi$
$480$	0	0
$481$	8.75325	0.399114
$482$	−9.45065	−0.430465
$483$	72.9504	3.31936
$484$	−0.782020	−0.0355463
$485$	0	0
$486$	20.4752	0.928774
$487$	−26.6858	−1.20925	−0.604626	−	0.796510i	$-0.706677\pi$
$487$	−26.6858	−1.20925	−0.604626	+	0.796510i	$0.706677\pi$
$488$	−10.7146	−0.485028
$489$	−3.78202	−0.171029
$490$	0	0
$491$	24.0575	1.08570	0.542851	−	0.839829i	$-0.317345\pi$
$491$	24.0575	1.08570	0.542851	+	0.839829i	$0.317345\pi$
$492$	20.2327	0.912159
$493$	−0.803442	−0.0361852
$494$	−2.68585	−0.120842
$495$	0	0
$496$	1.90383	0.0854843
$497$	31.1365	1.39666
$498$	−7.25410	−0.325064
$499$	3.87085	0.173283	0.0866414	−	0.996240i	$-0.472387\pi$
$499$	3.87085	0.173283	0.0866414	+	0.996240i	$0.472387\pi$
$500$	0	0
$501$	−32.1495	−1.43633
$502$	26.8855	1.19996
$503$	−43.1751	−1.92508	−0.962542	−	0.271132i	$-0.912602\pi$
$503$	−43.1751	−1.92508	−0.962542	+	0.271132i	$0.912602\pi$
$504$	−9.53948	−0.424922
$505$	0	0
$506$	−25.9718	−1.15459
$507$	−27.3790	−1.21595
$508$	7.90383	0.350676
$509$	−21.5725	−0.956183	−0.478091	−	0.878310i	$-0.658671\pi$
$509$	−21.5725	−0.956183	−0.478091	+	0.878310i	$0.658671\pi$
$510$	0	0
$511$	−21.8996	−0.968782
$512$	−1.00000	−0.0441942
$513$	2.80344	0.123775
$514$	−6.39312	−0.281988
$515$	0	0
$516$	−17.9572	−0.790520
$517$	−36.6577	−1.61220
$518$	29.2614	1.28567
$519$	42.8929	1.88279
$520$	0	0
$521$	32.0790	1.40540	0.702702	−	0.711484i	$-0.251977\pi$
$521$	32.0790	1.40540	0.702702	+	0.711484i	$0.251977\pi$
$522$	−2.48929	−0.108953
$523$	31.6247	1.38285	0.691426	−	0.722447i	$-0.256983\pi$
$523$	31.6247	1.38285	0.691426	+	0.722447i	$0.256983\pi$
$524$	−3.10352	−0.135578
$525$	0	0
$526$	27.4679	1.19766
$527$	−1.52962	−0.0666311
$528$	7.48929	0.325929
$529$	43.0147	1.87020
$530$	0	0
$531$	5.88554	0.255411
$532$	−8.97858	−0.389271
$533$	−9.89962	−0.428800
$534$	7.78202	0.336761
$535$	0	0
$536$	−5.12494	−0.221364
$537$	5.71462	0.246604
$538$	−9.59281	−0.413575
$539$	24.5682	1.05823
$540$	0	0
$541$	−21.9368	−0.943137	−0.471568	−	0.881829i	$-0.656312\pi$
$541$	−21.9368	−0.943137	−0.471568	+	0.881829i	$0.656312\pi$
$542$	29.9614	1.28695
$543$	9.42923	0.404647
$544$	0.803442	0.0344473
$545$	0	0
$546$	10.2927	0.440488
$547$	13.6901	0.585345	0.292672	−	0.956213i	$-0.405456\pi$
$547$	13.6901	0.585345	0.292672	+	0.956213i	$0.405456\pi$
$548$	19.2541	0.822494
$549$	26.6718	1.13832
$550$	0	0
$551$	−2.34292	−0.0998119
$552$	−19.0361	−0.810231
$553$	−57.2369	−2.43396
$554$	15.7894	0.670826
$555$	0	0
$556$	6.43910	0.273079
$557$	1.66442	0.0705239	0.0352619	−	0.999378i	$-0.488773\pi$
$557$	1.66442	0.0705239	0.0352619	+	0.999378i	$0.488773\pi$
$558$	−4.73917	−0.200625
$559$	8.78623	0.371618
$560$	0	0
$561$	−6.01721	−0.254047
$562$	−12.1390	−0.512054
$563$	−21.5725	−0.909171	−0.454585	−	0.890703i	$-0.650213\pi$
$563$	−21.5725	−0.909171	−0.454585	+	0.890703i	$0.650213\pi$
$564$	−26.8683	−1.13136
$565$	0	0
$566$	4.17513	0.175494
$567$	−39.3618	−1.65304
$568$	−8.12494	−0.340915
$569$	7.88240	0.330448	0.165224	−	0.986256i	$-0.447165\pi$
$569$	7.88240	0.330448	0.165224	+	0.986256i	$0.447165\pi$
$570$	0	0
$571$	−29.0796	−1.21694	−0.608471	−	0.793576i	$-0.708217\pi$
$571$	−29.0796	−1.21694	−0.608471	+	0.793576i	$0.708217\pi$
$572$	−3.66442	−0.153217
$573$	−13.0460	−0.545004
$574$	−33.0937	−1.38130
$575$	0	0
$576$	2.48929	0.103720
$577$	38.5155	1.60342	0.801711	−	0.597711i	$-0.203923\pi$
$577$	38.5155	1.60342	0.801711	+	0.597711i	$0.203923\pi$
$578$	16.3545	0.680257
$579$	32.1323	1.33537
$580$	0	0
$581$	11.8652	0.492251
$582$	27.0361	1.12068
$583$	1.79296	0.0742568
$584$	5.71462	0.236472
$585$	0	0
$586$	−9.09931	−0.375889
$587$	15.6184	0.644642	0.322321	−	0.946630i	$-0.395537\pi$
$587$	15.6184	0.644642	0.322321	+	0.946630i	$0.395537\pi$
$588$	18.0073	0.742610
$589$	−4.46052	−0.183792
$590$	0	0
$591$	−59.5787	−2.45074
$592$	−7.63565	−0.313823
$593$	1.72869	0.0709889	0.0354944	−	0.999370i	$-0.488699\pi$
$593$	1.72869	0.0709889	0.0354944	+	0.999370i	$0.488699\pi$
$594$	3.82487	0.156936
$595$	0	0
$596$	2.00000	0.0819232
$597$	35.3864	1.44827
$598$	9.31415	0.380884
$599$	20.0189	0.817950	0.408975	−	0.912546i	$-0.365886\pi$
$599$	20.0189	0.817950	0.408975	+	0.912546i	$0.365886\pi$
$600$	0	0
$601$	11.6388	0.474756	0.237378	−	0.971417i	$-0.423712\pi$
$601$	11.6388	0.474756	0.237378	+	0.971417i	$0.423712\pi$
$602$	29.3717	1.19710
$603$	12.7575	0.519524
$604$	−10.8536	−0.441628
$605$	0	0
$606$	−13.2039	−0.536372
$607$	36.4036	1.47758	0.738788	−	0.673938i	$-0.235398\pi$
$607$	36.4036	1.47758	0.738788	+	0.673938i	$0.235398\pi$
$608$	2.34292	0.0950181
$609$	8.97858	0.363830
$610$	0	0
$611$	13.1464	0.531845
$612$	−2.00000	−0.0808452
$613$	33.4292	1.35019	0.675097	−	0.737729i	$-0.264102\pi$
$613$	33.4292	1.35019	0.675097	+	0.737729i	$0.264102\pi$
$614$	−33.3116	−1.34435
$615$	0	0
$616$	−12.2499	−0.493562
$617$	16.2400	0.653799	0.326899	−	0.945059i	$-0.393996\pi$
$617$	16.2400	0.653799	0.326899	+	0.945059i	$0.393996\pi$
$618$	41.2285	1.65845
$619$	30.1004	1.20984	0.604918	−	0.796288i	$-0.293206\pi$
$619$	30.1004	1.20984	0.604918	+	0.796288i	$0.293206\pi$
$620$	0	0
$621$	−9.72196	−0.390129
$622$	−19.7648	−0.792497
$623$	−12.7287	−0.509964
$624$	−2.68585	−0.107520
$625$	0	0
$626$	−19.1898	−0.766980
$627$	−17.5468	−0.700753
$628$	−7.69319	−0.306992
$629$	6.13481	0.244611
$630$	0	0
$631$	31.1793	1.24123	0.620615	−	0.784115i	$-0.286883\pi$
$631$	31.1793	1.24123	0.620615	+	0.784115i	$0.286883\pi$
$632$	14.9357	0.594111
$633$	46.3822	1.84353
$634$	30.7722	1.22212
$635$	0	0
$636$	1.31415	0.0521096
$637$	−8.81079	−0.349096
$638$	−3.19656	−0.126553
$639$	20.2253	0.800102
$640$	0	0
$641$	5.29587	0.209174	0.104587	−	0.994516i	$-0.466648\pi$
$641$	5.29587	0.209174	0.104587	+	0.994516i	$0.466648\pi$
$642$	−3.65708	−0.144333
$643$	28.1867	1.11157	0.555787	−	0.831325i	$-0.312417\pi$
$643$	28.1867	1.11157	0.555787	+	0.831325i	$0.312417\pi$
$644$	31.1365	1.22695
$645$	0	0
$646$	−1.88240	−0.0740622
$647$	−15.6497	−0.615254	−0.307627	−	0.951507i	$-0.599535\pi$
$647$	−15.6497	−0.615254	−0.307627	+	0.951507i	$0.599535\pi$
$648$	10.2713	0.403495
$649$	7.55777	0.296668
$650$	0	0
$651$	17.0937	0.669953
$652$	−1.61423	−0.0632182
$653$	24.1292	0.944247	0.472123	−	0.881532i	$-0.343488\pi$
$653$	24.1292	0.944247	0.472123	+	0.881532i	$0.343488\pi$
$654$	−26.9357	−1.05327
$655$	0	0
$656$	8.63565	0.337166
$657$	−14.2253	−0.554983
$658$	43.9473	1.71324
$659$	28.1898	1.09812	0.549060	−	0.835783i	$-0.314986\pi$
$659$	28.1898	1.09812	0.549060	+	0.835783i	$0.314986\pi$
$660$	0	0
$661$	−16.1678	−0.628854	−0.314427	−	0.949282i	$-0.601812\pi$
$661$	−16.1678	−0.628854	−0.314427	+	0.949282i	$0.601812\pi$
$662$	31.5714	1.22706
$663$	2.15792	0.0838068
$664$	−3.09617	−0.120155
$665$	0	0
$666$	19.0073	0.736520
$667$	8.12494	0.314599
$668$	−13.7220	−0.530919
$669$	12.5855	0.486582
$670$	0	0
$671$	34.2499	1.32220
$672$	−8.97858	−0.346356
$673$	−13.7690	−0.530757	−0.265378	−	0.964144i	$-0.585497\pi$
$673$	−13.7690	−0.530757	−0.265378	+	0.964144i	$0.585497\pi$
$674$	15.1966	0.585350
$675$	0	0
$676$	−11.6858	−0.449456
$677$	−14.4710	−0.556166	−0.278083	−	0.960557i	$-0.589699\pi$
$677$	−14.4710	−0.556166	−0.278083	+	0.960557i	$0.589699\pi$
$678$	−12.5682	−0.482681
$679$	−44.2217	−1.69707
$680$	0	0
$681$	66.3551	2.54273
$682$	−6.08569	−0.233033
$683$	19.0533	0.729055	0.364528	−	0.931193i	$-0.381230\pi$
$683$	19.0533	0.729055	0.364528	+	0.931193i	$0.381230\pi$
$684$	−5.83221	−0.223000
$685$	0	0
$686$	−2.62831	−0.100349
$687$	23.4637	0.895194
$688$	−7.66442	−0.292203
$689$	−0.643000	−0.0244964
$690$	0	0
$691$	41.7679	1.58893	0.794464	−	0.607312i	$-0.207752\pi$
$691$	41.7679	1.58893	0.794464	+	0.607312i	$0.207752\pi$
$692$	18.3074	0.695944
$693$	30.4935	1.15835
$694$	−17.0246	−0.646243
$695$	0	0
$696$	−2.34292	−0.0888083
$697$	−6.93825	−0.262805
$698$	7.20390	0.272672
$699$	−54.6478	−2.06697
$700$	0	0
$701$	−0.149501	−0.00564658	−0.00282329	−	0.999996i	$-0.500899\pi$
$701$	−0.149501	−0.00564658	−0.00282329	+	0.999996i	$0.500899\pi$
$702$	−1.37169	−0.0517712
$703$	17.8898	0.674725
$704$	3.19656	0.120475
$705$	0	0
$706$	24.3116	0.914980
$707$	21.5970	0.812240
$708$	5.53948	0.208186
$709$	−11.5725	−0.434613	−0.217306	−	0.976103i	$-0.569727\pi$
$709$	−11.5725	−0.434613	−0.217306	+	0.976103i	$0.569727\pi$
$710$	0	0
$711$	−37.1793	−1.39433
$712$	3.32150	0.124478
$713$	15.4685	0.579299
$714$	7.21377	0.269969
$715$	0	0
$716$	2.43910	0.0911533
$717$	−51.3435	−1.91746
$718$	−33.2902	−1.24238
$719$	2.93933	0.109618	0.0548092	−	0.998497i	$-0.482545\pi$
$719$	2.93933	0.109618	0.0548092	+	0.998497i	$0.482545\pi$
$720$	0	0
$721$	−67.4355	−2.51143
$722$	13.5107	0.502817
$723$	22.1422	0.823476
$724$	4.02456	0.149572
$725$	0	0
$726$	1.83221	0.0679998
$727$	−29.3435	−1.08829	−0.544146	−	0.838991i	$-0.683146\pi$
$727$	−29.3435	−1.08829	−0.544146	+	0.838991i	$0.683146\pi$
$728$	4.39312	0.162820
$729$	−17.1579	−0.635479
$730$	0	0
$731$	6.15792	0.227759
$732$	25.1035	0.927853
$733$	28.2927	1.04502	0.522508	−	0.852634i	$-0.324997\pi$
$733$	28.2927	1.04502	0.522508	+	0.852634i	$0.324997\pi$
$734$	18.9315	0.698775
$735$	0	0
$736$	−8.12494	−0.299489
$737$	16.3822	0.603445
$738$	−21.4966	−0.791302
$739$	15.6069	0.574109	0.287054	−	0.957914i	$-0.407324\pi$
$739$	15.6069	0.574109	0.287054	+	0.957914i	$0.407324\pi$
$740$	0	0
$741$	6.29273	0.231169
$742$	−2.14950	−0.0789107
$743$	42.9013	1.57390	0.786948	−	0.617019i	$-0.211660\pi$
$743$	42.9013	1.57390	0.786948	+	0.617019i	$0.211660\pi$
$744$	−4.46052	−0.163531
$745$	0	0
$746$	11.8996	0.435676
$747$	7.70727	0.281994
$748$	−2.56825	−0.0939045
$749$	5.98171	0.218567
$750$	0	0
$751$	−41.0403	−1.49758	−0.748791	−	0.662806i	$-0.769366\pi$
$751$	−41.0403	−1.49758	−0.748791	+	0.662806i	$0.769366\pi$
$752$	−11.4679	−0.418190
$753$	−62.9908	−2.29551
$754$	1.14637	0.0417482
$755$	0	0
$756$	−4.58546	−0.166772
$757$	−35.9227	−1.30563	−0.652817	−	0.757516i	$-0.726413\pi$
$757$	−35.9227	−1.30563	−0.652817	+	0.757516i	$0.726413\pi$
$758$	20.4005	0.740978
$759$	60.8500	2.20872
$760$	0	0
$761$	3.77154	0.136718	0.0683591	−	0.997661i	$-0.478224\pi$
$761$	3.77154	0.136718	0.0683591	+	0.997661i	$0.478224\pi$
$762$	−18.5181	−0.670838
$763$	44.0575	1.59499
$764$	−5.56825	−0.201452
$765$	0	0
$766$	36.8929	1.33299
$767$	−2.71040	−0.0978670
$768$	2.34292	0.0845430
$769$	15.7062	0.566380	0.283190	−	0.959064i	$-0.408607\pi$
$769$	15.7062	0.566380	0.283190	+	0.959064i	$0.408607\pi$
$770$	0	0
$771$	14.9786	0.539440
$772$	13.7146	0.493600
$773$	3.13588	0.112790	0.0563949	−	0.998409i	$-0.482039\pi$
$773$	3.13588	0.112790	0.0563949	+	0.998409i	$0.482039\pi$
$774$	19.0790	0.685779
$775$	0	0
$776$	11.5395	0.414243
$777$	−68.5573	−2.45948
$778$	22.7146	0.814358
$779$	−20.2327	−0.724911
$780$	0	0
$781$	25.9718	0.929346
$782$	6.52792	0.233438
$783$	−1.19656	−0.0427615
$784$	7.68585	0.274495
$785$	0	0
$786$	7.27131	0.259359
$787$	25.3858	0.904905	0.452452	−	0.891789i	$-0.350549\pi$
$787$	25.3858	0.904905	0.452452	+	0.891789i	$0.350549\pi$
$788$	−25.4292	−0.905879
$789$	−64.3551	−2.29110
$790$	0	0
$791$	20.5573	0.730934
$792$	−7.95715	−0.282745
$793$	−12.2829	−0.436177
$794$	−33.5113	−1.18927
$795$	0	0
$796$	15.1035	0.535330
$797$	0.593884	0.0210364	0.0105182	−	0.999945i	$-0.496652\pi$
$797$	0.593884	0.0210364	0.0105182	+	0.999945i	$0.496652\pi$
$798$	21.0361	0.744670
$799$	9.21377	0.325960
$800$	0	0
$801$	−8.26817	−0.292142
$802$	17.0147	0.600810
$803$	−18.2671	−0.644632
$804$	12.0073	0.423466
$805$	0	0
$806$	2.18248	0.0768746
$807$	22.4752	0.791165
$808$	−5.63565	−0.198262
$809$	−35.2776	−1.24029	−0.620147	−	0.784486i	$-0.712927\pi$
$809$	−35.2776	−1.24029	−0.620147	+	0.784486i	$0.712927\pi$
$810$	0	0
$811$	−13.1568	−0.461999	−0.231000	−	0.972954i	$-0.574200\pi$
$811$	−13.1568	−0.461999	−0.231000	+	0.972954i	$0.574200\pi$
$812$	3.83221	0.134484
$813$	−70.1972	−2.46192
$814$	24.4078	0.855493
$815$	0	0
$816$	−1.88240	−0.0658973
$817$	17.9572	0.628241
$818$	21.3461	0.746347
$819$	−10.9357	−0.382125
$820$	0	0
$821$	−48.2400	−1.68359	−0.841794	−	0.539799i	$-0.818500\pi$
$821$	−48.2400	−1.68359	−0.841794	+	0.539799i	$0.818500\pi$
$822$	−45.1109	−1.57342
$823$	32.7581	1.14187	0.570937	−	0.820994i	$-0.306580\pi$
$823$	32.7581	1.14187	0.570937	+	0.820994i	$0.306580\pi$
$824$	17.5970	0.613021
$825$	0	0
$826$	−9.06067	−0.315261
$827$	−35.9118	−1.24878	−0.624388	−	0.781115i	$-0.714651\pi$
$827$	−35.9118	−1.24878	−0.624388	+	0.781115i	$0.714651\pi$
$828$	20.2253	0.702879
$829$	23.0361	0.800077	0.400039	−	0.916498i	$-0.368997\pi$
$829$	23.0361	0.800077	0.400039	+	0.916498i	$0.368997\pi$
$830$	0	0
$831$	−36.9933	−1.28328
$832$	−1.14637	−0.0397431
$833$	−6.17513	−0.213956
$834$	−15.0863	−0.522396
$835$	0	0
$836$	−7.48929	−0.259022
$837$	−2.27804	−0.0787405
$838$	−3.81079	−0.131642
$839$	−11.8757	−0.409994	−0.204997	−	0.978763i	$-0.565718\pi$
$839$	−11.8757	−0.409994	−0.204997	+	0.978763i	$0.565718\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	23.9859	0.826610
$843$	28.4408	0.979553
$844$	19.7967	0.681431
$845$	0	0
$846$	28.5468	0.981460
$847$	−2.99686	−0.102973
$848$	0.560904	0.0192615
$849$	−9.78202	−0.335718
$850$	0	0
$851$	−62.0393	−2.12668
$852$	19.0361	0.652167
$853$	−50.6493	−1.73420	−0.867099	−	0.498136i	$-0.834018\pi$
$853$	−50.6493	−1.73420	−0.867099	+	0.498136i	$0.834018\pi$
$854$	−41.0607	−1.40507
$855$	0	0
$856$	−1.56090	−0.0533506
$857$	11.1453	0.380716	0.190358	−	0.981715i	$-0.439035\pi$
$857$	11.1453	0.380716	0.190358	+	0.981715i	$0.439035\pi$
$858$	8.58546	0.293103
$859$	−0.352789	−0.0120370	−0.00601850	−	0.999982i	$-0.501916\pi$
$859$	−0.352789	−0.0120370	−0.00601850	+	0.999982i	$0.501916\pi$
$860$	0	0
$861$	77.5359	2.64242
$862$	−7.00314	−0.238528
$863$	5.06067	0.172267	0.0861337	−	0.996284i	$-0.472549\pi$
$863$	5.06067	0.172267	0.0861337	+	0.996284i	$0.472549\pi$
$864$	1.19656	0.0407077
$865$	0	0
$866$	23.3790	0.794452
$867$	−38.3173	−1.30132
$868$	7.29587	0.247638
$869$	−47.7429	−1.61957
$870$	0	0
$871$	−5.87506	−0.199069
$872$	−11.4966	−0.389325
$873$	−28.7251	−0.972197
$874$	19.0361	0.643906
$875$	0	0
$876$	−13.3889	−0.452369
$877$	41.8370	1.41274	0.706368	−	0.707845i	$-0.250332\pi$
$877$	41.8370	1.41274	0.706368	+	0.707845i	$0.250332\pi$
$878$	−13.7073	−0.462598
$879$	21.3190	0.719071
$880$	0	0
$881$	2.10666	0.0709750	0.0354875	−	0.999370i	$-0.488702\pi$
$881$	2.10666	0.0709750	0.0354875	+	0.999370i	$0.488702\pi$
$882$	−19.1323	−0.644218
$883$	13.8389	0.465717	0.232859	−	0.972511i	$-0.425192\pi$
$883$	13.8389	0.465717	0.232859	+	0.972511i	$0.425192\pi$
$884$	0.921039	0.0309779
$885$	0	0
$886$	9.25410	0.310897
$887$	37.7795	1.26851	0.634256	−	0.773123i	$-0.281307\pi$
$887$	37.7795	1.26851	0.634256	+	0.773123i	$0.281307\pi$
$888$	17.8898	0.600341
$889$	30.2891	1.01587
$890$	0	0
$891$	−32.8328	−1.09994
$892$	5.37169	0.179858
$893$	26.8683	0.899114
$894$	−4.68585	−0.156718
$895$	0	0
$896$	−3.83221	−0.128025
$897$	−21.8223	−0.728627
$898$	−25.8971	−0.864197
$899$	1.90383	0.0634962
$900$	0	0
$901$	−0.450654	−0.0150135
$902$	−27.6044	−0.919125
$903$	−68.8156	−2.29004
$904$	−5.36435	−0.178415
$905$	0	0
$906$	25.4292	0.844830
$907$	−20.8683	−0.692921	−0.346461	−	0.938064i	$-0.612617\pi$
$907$	−20.8683	−0.692921	−0.346461	+	0.938064i	$0.612617\pi$
$908$	28.3215	0.939882
$909$	14.0288	0.465305
$910$	0	0
$911$	30.5468	1.01206	0.506031	−	0.862515i	$-0.331112\pi$
$911$	30.5468	1.01206	0.506031	+	0.862515i	$0.331112\pi$
$912$	−5.48929	−0.181769
$913$	9.89710	0.327546
$914$	26.5829	0.879286
$915$	0	0
$916$	10.0147	0.330895
$917$	−11.8933	−0.392753
$918$	−0.961365	−0.0317298
$919$	33.8898	1.11792	0.558960	−	0.829195i	$-0.311201\pi$
$919$	33.8898	1.11792	0.558960	+	0.829195i	$0.311201\pi$
$920$	0	0
$921$	78.0466	2.57172
$922$	−0.527923	−0.0173862
$923$	−9.31415	−0.306579
$924$	28.7005	0.944178
$925$	0	0
$926$	38.0575	1.25065
$927$	−43.8041	−1.43871
$928$	−1.00000	−0.0328266
$929$	44.5040	1.46013	0.730064	−	0.683379i	$-0.239490\pi$
$929$	44.5040	1.46013	0.730064	+	0.683379i	$0.239490\pi$
$930$	0	0
$931$	−18.0073	−0.590167
$932$	−23.3246	−0.764024
$933$	46.3074	1.51604
$934$	10.5181	0.344161
$935$	0	0
$936$	2.85363	0.0932740
$937$	−36.1793	−1.18193	−0.590964	−	0.806698i	$-0.701252\pi$
$937$	−36.1793	−1.18193	−0.590964	+	0.806698i	$0.701252\pi$
$938$	−19.6399	−0.641264
$939$	44.9603	1.46722
$940$	0	0
$941$	11.1464	0.363361	0.181681	−	0.983358i	$-0.441846\pi$
$941$	11.1464	0.363361	0.181681	+	0.983358i	$0.441846\pi$
$942$	18.0246	0.587272
$943$	70.1642	2.28486
$944$	2.36435	0.0769529
$945$	0	0
$946$	24.4998	0.796556
$947$	38.8255	1.26166	0.630829	−	0.775922i	$-0.282715\pi$
$947$	38.8255	1.26166	0.630829	+	0.775922i	$0.282715\pi$
$948$	−34.9933	−1.13653
$949$	6.55104	0.212656
$950$	0	0
$951$	−72.0968	−2.33790
$952$	3.07896	0.0997897
$953$	−9.83329	−0.318531	−0.159266	−	0.987236i	$-0.550913\pi$
$953$	−9.83329	−0.318531	−0.159266	+	0.987236i	$0.550913\pi$
$954$	−1.39625	−0.0452053
$955$	0	0
$956$	−21.9143	−0.708759
$957$	7.48929	0.242094
$958$	−5.54935	−0.179291
$959$	73.7858	2.38267
$960$	0	0
$961$	−27.3754	−0.883079
$962$	−8.75325	−0.282216
$963$	3.88554	0.125210
$964$	9.45065	0.304385
$965$	0	0
$966$	−72.9504	−2.34714
$967$	44.8500	1.44228	0.721140	−	0.692789i	$-0.243618\pi$
$967$	44.8500	1.44228	0.721140	+	0.692789i	$0.243618\pi$
$968$	0.782020	0.0251351
$969$	4.41033	0.141680
$970$	0	0
$971$	48.4496	1.55482	0.777410	−	0.628994i	$-0.216533\pi$
$971$	48.4496	1.55482	0.777410	+	0.628994i	$0.216533\pi$
$972$	−20.4752	−0.656743
$973$	24.6760	0.791076
$974$	26.6858	0.855070
$975$	0	0
$976$	10.7146	0.342966
$977$	33.2155	1.06266	0.531328	−	0.847166i	$-0.321693\pi$
$977$	33.2155	1.06266	0.531328	+	0.847166i	$0.321693\pi$
$978$	3.78202	0.120936
$979$	−10.6174	−0.339333
$980$	0	0
$981$	28.6184	0.913717
$982$	−24.0575	−0.767707
$983$	26.2457	0.837107	0.418554	−	0.908192i	$-0.362537\pi$
$983$	26.2457	0.837107	0.418554	+	0.908192i	$0.362537\pi$
$984$	−20.2327	−0.644994
$985$	0	0
$986$	0.803442	0.0255868
$987$	−102.965	−3.27742
$988$	2.68585	0.0854481
$989$	−62.2730	−1.98017
$990$	0	0
$991$	−40.5145	−1.28698	−0.643492	−	0.765453i	$-0.722515\pi$
$991$	−40.5145	−1.28698	−0.643492	+	0.765453i	$0.722515\pi$
$992$	−1.90383	−0.0604466
$993$	−73.9693	−2.34735
$994$	−31.1365	−0.987590
$995$	0	0
$996$	7.25410	0.229855
$997$	−14.6227	−0.463104	−0.231552	−	0.972823i	$-0.574380\pi$
$997$	−14.6227	−0.463104	−0.231552	+	0.972823i	$0.574380\pi$
$998$	−3.87085	−0.122530
$999$	9.13650	0.289066

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1450.2.a.q.1.3	✓	3
5.2	odd	4		1450.2.b.k.349.1		6
5.3	odd	4		1450.2.b.k.349.6		6
5.4	even	2		1450.2.a.s.1.1	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1450.2.a.q.1.3	✓	3	1.1	even	1	trivial
1450.2.a.s.1.1	yes	3	5.4	even	2
1450.2.b.k.349.1		6	5.2	odd	4
1450.2.b.k.349.6		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:	\( N \)	\(=\)	\( 1450 = 2 \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1450.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +2.34292 q^{3} +1.00000 q^{4} -2.34292 q^{6} +3.83221 q^{7} -1.00000 q^{8} +2.48929 q^{9} +3.19656 q^{11} +2.34292 q^{12} -1.14637 q^{13} -3.83221 q^{14} +1.00000 q^{16} -0.803442 q^{17} -2.48929 q^{18} -2.34292 q^{19} +8.97858 q^{21} -3.19656 q^{22} +8.12494 q^{23} -2.34292 q^{24} +1.14637 q^{26} -1.19656 q^{27} +3.83221 q^{28} +1.00000 q^{29} +1.90383 q^{31} -1.00000 q^{32} +7.48929 q^{33} +0.803442 q^{34} +2.48929 q^{36} -7.63565 q^{37} +2.34292 q^{38} -2.68585 q^{39} +8.63565 q^{41} -8.97858 q^{42} -7.66442 q^{43} +3.19656 q^{44} -8.12494 q^{46} -11.4679 q^{47} +2.34292 q^{48} +7.68585 q^{49} -1.88240 q^{51} -1.14637 q^{52} +0.560904 q^{53} +1.19656 q^{54} -3.83221 q^{56} -5.48929 q^{57} -1.00000 q^{58} +2.36435 q^{59} +10.7146 q^{61} -1.90383 q^{62} +9.53948 q^{63} +1.00000 q^{64} -7.48929 q^{66} +5.12494 q^{67} -0.803442 q^{68} +19.0361 q^{69} +8.12494 q^{71} -2.48929 q^{72} -5.71462 q^{73} +7.63565 q^{74} -2.34292 q^{76} +12.2499 q^{77} +2.68585 q^{78} -14.9357 q^{79} -10.2713 q^{81} -8.63565 q^{82} +3.09617 q^{83} +8.97858 q^{84} +7.66442 q^{86} +2.34292 q^{87} -3.19656 q^{88} -3.32150 q^{89} -4.39312 q^{91} +8.12494 q^{92} +4.46052 q^{93} +11.4679 q^{94} -2.34292 q^{96} -11.5395 q^{97} -7.68585 q^{98} +7.95715 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + q^{3} + 3 q^{4} - q^{6} - 2 q^{7} - 3 q^{8} + 5 q^{11} + q^{12} - 2 q^{13} + 2 q^{14} + 3 q^{16} - 7 q^{17} - q^{19} + 12 q^{21} - 5 q^{22} + 8 q^{23} - q^{24} + 2 q^{26} + q^{27}+ \cdots - 6 q^{99}+O(q^{100}) \) 3 * q - 3 * q^2 + q^3 + 3 * q^4 - q^6 - 2 * q^7 - 3 * q^8 + 5 * q^11 + q^12 - 2 * q^13 + 2 * q^14 + 3 * q^16 - 7 * q^17 - q^19 + 12 * q^21 - 5 * q^22 + 8 * q^23 - q^24 + 2 * q^26 + q^27 - 2 * q^28 + 3 * q^29 + 4 * q^31 - 3 * q^32 + 15 * q^33 + 7 * q^34 - 14 * q^37 + q^38 + 4 * q^39 + 17 * q^41 - 12 * q^42 + 4 * q^43 + 5 * q^44 - 8 * q^46 - 12 * q^47 + q^48 + 11 * q^49 + 11 * q^51 - 2 * q^52 + 6 * q^53 - q^54 + 2 * q^56 - 9 * q^57 - 3 * q^58 + 16 * q^59 + 2 * q^61 - 4 * q^62 + 18 * q^63 + 3 * q^64 - 15 * q^66 - q^67 - 7 * q^68 + 6 * q^69 + 8 * q^71 + 13 * q^73 + 14 * q^74 - q^76 + 4 * q^77 - 4 * q^78 - 13 * q^81 - 17 * q^82 + 11 * q^83 + 12 * q^84 - 4 * q^86 + q^87 - 5 * q^88 + 11 * q^89 - 4 * q^91 + 8 * q^92 + 24 * q^93 + 12 * q^94 - q^96 - 24 * q^97 - 11 * q^98 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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