LMFDB - Embedded newform 8037.2.a.o.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8037,2,Mod(1,8037)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8037, 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("8037.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8037 = 3^{2} \cdot 19 \cdot 47 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8037.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:	$64.1757681045$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - x^{17} - 28 x^{16} + 25 x^{15} + 322 x^{14} - 247 x^{13} - 1971 x^{12} + 1231 x^{11} + 6953 x^{10} + \cdots + 9 $ x^18 - x^17 - 28x^16 + 25x^15 + 322x^14 - 247x^13 - 1971x^12 + 1231x^11 + 6953x^10 - 3283x^9 - 14235x^8 + 4562x^7 + 15962x^6 - 2882x^5 - 8159x^4 + 606x^3 + 890x^2 - 179x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 893)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.72555$ of defining polynomial
Character	$\chi$	$=$	8037.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.72555 q^{2} +5.42860 q^{4} +0.150400 q^{5} -4.90968 q^{7} -9.34479 q^{8} +O(q^{10})$	\(q-2.72555 q^{2} +5.42860 q^{4} +0.150400 q^{5} -4.90968 q^{7} -9.34479 q^{8} -0.409922 q^{10} +0.576590 q^{11} +2.66771 q^{13} +13.3815 q^{14} +14.6125 q^{16} +5.36245 q^{17} -1.00000 q^{19} +0.816460 q^{20} -1.57152 q^{22} +6.79740 q^{23} -4.97738 q^{25} -7.27097 q^{26} -26.6527 q^{28} -0.917682 q^{29} +7.50359 q^{31} -21.1373 q^{32} -14.6156 q^{34} -0.738415 q^{35} +5.23653 q^{37} +2.72555 q^{38} -1.40546 q^{40} -6.67956 q^{41} +7.17355 q^{43} +3.13008 q^{44} -18.5266 q^{46} -1.00000 q^{47} +17.1049 q^{49} +13.5661 q^{50} +14.4819 q^{52} +6.54945 q^{53} +0.0867191 q^{55} +45.8799 q^{56} +2.50118 q^{58} -5.85850 q^{59} -7.05645 q^{61} -20.4514 q^{62} +28.3858 q^{64} +0.401224 q^{65} +14.8820 q^{67} +29.1106 q^{68} +2.01258 q^{70} +8.96691 q^{71} -8.37376 q^{73} -14.2724 q^{74} -5.42860 q^{76} -2.83087 q^{77} -5.07972 q^{79} +2.19771 q^{80} +18.2054 q^{82} -8.02822 q^{83} +0.806512 q^{85} -19.5518 q^{86} -5.38812 q^{88} -5.89294 q^{89} -13.0976 q^{91} +36.9003 q^{92} +2.72555 q^{94} -0.150400 q^{95} -6.19575 q^{97} -46.6203 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - q^{2} + 21 q^{4} - 5 q^{5} + 3 q^{7} - 6 q^{8}+O(q^{10}) $ 18 * q - q^2 + 21 * q^4 - 5 * q^5 + 3 * q^7 - 6 * q^8	$ 18 q - q^{2} + 21 q^{4} - 5 q^{5} + 3 q^{7} - 6 q^{8} + 7 q^{10} - 6 q^{11} + 21 q^{13} + 9 q^{14} + 23 q^{16} + 4 q^{17} - 18 q^{19} - 9 q^{20} + 28 q^{22} - 9 q^{23} + 11 q^{25} + q^{26} + 7 q^{28} - 12 q^{29} + 26 q^{31} + 7 q^{32} + 26 q^{34} - 9 q^{35} + 8 q^{37} + q^{38} + 16 q^{40} - 12 q^{41} + 28 q^{43} - 2 q^{44} - 33 q^{46} - 18 q^{47} + 17 q^{49} + 29 q^{50} + 30 q^{52} - 5 q^{53} + 28 q^{55} + 77 q^{56} - 6 q^{58} - 30 q^{59} - 16 q^{61} - 16 q^{62} + 28 q^{64} + 22 q^{65} + 45 q^{67} + 96 q^{68} - 2 q^{70} + q^{71} - 24 q^{73} + 19 q^{74} - 21 q^{76} - 2 q^{77} + 33 q^{79} + 25 q^{80} + 18 q^{82} + 13 q^{83} - 7 q^{85} - 3 q^{86} + 27 q^{88} + 6 q^{89} + 42 q^{91} + 11 q^{92} + q^{94} + 5 q^{95} + 44 q^{97} - 58 q^{98}+O(q^{100}) $ 18 * q - q^2 + 21 * q^4 - 5 * q^5 + 3 * q^7 - 6 * q^8 + 7 * q^10 - 6 * q^11 + 21 * q^13 + 9 * q^14 + 23 * q^16 + 4 * q^17 - 18 * q^19 - 9 * q^20 + 28 * q^22 - 9 * q^23 + 11 * q^25 + q^26 + 7 * q^28 - 12 * q^29 + 26 * q^31 + 7 * q^32 + 26 * q^34 - 9 * q^35 + 8 * q^37 + q^38 + 16 * q^40 - 12 * q^41 + 28 * q^43 - 2 * q^44 - 33 * q^46 - 18 * q^47 + 17 * q^49 + 29 * q^50 + 30 * q^52 - 5 * q^53 + 28 * q^55 + 77 * q^56 - 6 * q^58 - 30 * q^59 - 16 * q^61 - 16 * q^62 + 28 * q^64 + 22 * q^65 + 45 * q^67 + 96 * q^68 - 2 * q^70 + q^71 - 24 * q^73 + 19 * q^74 - 21 * q^76 - 2 * q^77 + 33 * q^79 + 25 * q^80 + 18 * q^82 + 13 * q^83 - 7 * q^85 - 3 * q^86 + 27 * q^88 + 6 * q^89 + 42 * q^91 + 11 * q^92 + q^94 + 5 * q^95 + 44 * q^97 - 58 * 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 $

$2$	−2.72555	−1.92725	−0.963626	−	0.267256i	$-0.913883\pi$
$2$	−2.72555	−1.92725	−0.963626	+	0.267256i	$0.913883\pi$
$3$	0	0
$3$	0	0
$4$	5.42860	2.71430
$5$	0.150400	0.0672609	0.0336304	−	0.999434i	$-0.489293\pi$
$5$	0.150400	0.0672609	0.0336304	+	0.999434i	$0.489293\pi$
$6$	0	0
$7$	−4.90968	−1.85568	−0.927842	−	0.372974i	$-0.878338\pi$
$7$	−4.90968	−1.85568	−0.927842	+	0.372974i	$0.878338\pi$
$8$	−9.34479	−3.30388
$9$	0	0
$10$	−0.409922	−0.129629
$11$	0.576590	0.173849	0.0869243	−	0.996215i	$-0.472296\pi$
$11$	0.576590	0.173849	0.0869243	+	0.996215i	$0.472296\pi$
$12$	0	0
$13$	2.66771	0.739890	0.369945	−	0.929054i	$-0.379376\pi$
$13$	2.66771	0.739890	0.369945	+	0.929054i	$0.379376\pi$
$14$	13.3815	3.57637
$15$	0	0
$16$	14.6125	3.65312
$17$	5.36245	1.30059	0.650293	−	0.759683i	$-0.274646\pi$
$17$	5.36245	1.30059	0.650293	+	0.759683i	$0.274646\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0.816460	0.182566
$21$	0	0
$22$	−1.57152	−0.335050
$23$	6.79740	1.41736	0.708678	−	0.705532i	$-0.249292\pi$
$23$	6.79740	1.41736	0.708678	+	0.705532i	$0.249292\pi$
$24$	0	0
$25$	−4.97738	−0.995476
$26$	−7.27097	−1.42595
$27$	0	0
$28$	−26.6527	−5.03688
$29$	−0.917682	−0.170409	−0.0852046	−	0.996363i	$-0.527154\pi$
$29$	−0.917682	−0.170409	−0.0852046	+	0.996363i	$0.527154\pi$
$30$	0	0
$31$	7.50359	1.34768	0.673842	−	0.738875i	$-0.264643\pi$
$31$	7.50359	1.34768	0.673842	+	0.738875i	$0.264643\pi$
$32$	−21.1373	−3.73659
$33$	0	0
$34$	−14.6156	−2.50656
$35$	−0.738415	−0.124815
$36$	0	0
$37$	5.23653	0.860881	0.430440	−	0.902619i	$-0.358358\pi$
$37$	5.23653	0.860881	0.430440	+	0.902619i	$0.358358\pi$
$38$	2.72555	0.442142
$39$	0	0
$40$	−1.40546	−0.222222
$41$	−6.67956	−1.04317	−0.521586	−	0.853199i	$-0.674659\pi$
$41$	−6.67956	−1.04317	−0.521586	+	0.853199i	$0.674659\pi$
$42$	0	0
$43$	7.17355	1.09396	0.546978	−	0.837147i	$-0.315778\pi$
$43$	7.17355	1.09396	0.546978	+	0.837147i	$0.315778\pi$
$44$	3.13008	0.471877
$45$	0	0
$46$	−18.5266	−2.73160
$47$	−1.00000	−0.145865
$47$	−1.00000	−0.145865
$48$	0	0
$49$	17.1049	2.44356
$50$	13.5661	1.91853
$51$	0	0
$52$	14.4819	2.00828
$53$	6.54945	0.899636	0.449818	−	0.893120i	$-0.351489\pi$
$53$	6.54945	0.899636	0.449818	+	0.893120i	$0.351489\pi$
$54$	0	0
$55$	0.0867191	0.0116932
$56$	45.8799	6.13096
$57$	0	0
$58$	2.50118	0.328421
$59$	−5.85850	−0.762712	−0.381356	−	0.924428i	$-0.624543\pi$
$59$	−5.85850	−0.762712	−0.381356	+	0.924428i	$0.624543\pi$
$60$	0	0
$61$	−7.05645	−0.903486	−0.451743	−	0.892148i	$-0.649198\pi$
$61$	−7.05645	−0.903486	−0.451743	+	0.892148i	$0.649198\pi$
$62$	−20.4514	−2.59733
$63$	0	0
$64$	28.3858	3.54823
$65$	0.401224	0.0497657
$66$	0	0
$67$	14.8820	1.81813	0.909065	−	0.416655i	$-0.136798\pi$
$67$	14.8820	1.81813	0.909065	+	0.416655i	$0.136798\pi$
$68$	29.1106	3.53018
$69$	0	0
$70$	2.01258	0.240550
$71$	8.96691	1.06418	0.532088	−	0.846689i	$-0.321407\pi$
$71$	8.96691	1.06418	0.532088	+	0.846689i	$0.321407\pi$
$72$	0	0
$73$	−8.37376	−0.980074	−0.490037	−	0.871702i	$-0.663017\pi$
$73$	−8.37376	−0.980074	−0.490037	+	0.871702i	$0.663017\pi$
$74$	−14.2724	−1.65913
$75$	0	0
$76$	−5.42860	−0.622703
$77$	−2.83087	−0.322608
$78$	0	0
$79$	−5.07972	−0.571513	−0.285757	−	0.958302i	$-0.592245\pi$
$79$	−5.07972	−0.571513	−0.285757	+	0.958302i	$0.592245\pi$
$80$	2.19771	0.245712
$81$	0	0
$82$	18.2054	2.01045
$83$	−8.02822	−0.881212	−0.440606	−	0.897701i	$-0.645236\pi$
$83$	−8.02822	−0.881212	−0.440606	+	0.897701i	$0.645236\pi$
$84$	0	0
$85$	0.806512	0.0874785
$86$	−19.5518	−2.10833
$87$	0	0
$88$	−5.38812	−0.574375
$89$	−5.89294	−0.624650	−0.312325	−	0.949975i	$-0.601108\pi$
$89$	−5.89294	−0.624650	−0.312325	+	0.949975i	$0.601108\pi$
$90$	0	0
$91$	−13.0976	−1.37300
$92$	36.9003	3.84713
$93$	0	0
$94$	2.72555	0.281119
$95$	−0.150400	−0.0154307
$96$	0	0
$97$	−6.19575	−0.629083	−0.314541	−	0.949244i	$-0.601851\pi$
$97$	−6.19575	−0.629083	−0.314541	+	0.949244i	$0.601851\pi$
$98$	−46.6203	−4.70936
$99$	0	0
$100$	−27.0202	−2.70202
$101$	6.06232	0.603223	0.301612	−	0.953431i	$-0.402475\pi$
$101$	6.06232	0.603223	0.301612	+	0.953431i	$0.402475\pi$
$102$	0	0
$103$	8.95097	0.881966	0.440983	−	0.897516i	$-0.354630\pi$
$103$	8.95097	0.881966	0.440983	+	0.897516i	$0.354630\pi$
$104$	−24.9292	−2.44451
$105$	0	0
$106$	−17.8508	−1.73382
$107$	−9.75982	−0.943517	−0.471759	−	0.881728i	$-0.656381\pi$
$107$	−9.75982	−0.943517	−0.471759	+	0.881728i	$0.656381\pi$
$108$	0	0
$109$	−8.99870	−0.861919	−0.430960	−	0.902371i	$-0.641825\pi$
$109$	−8.99870	−0.861919	−0.430960	+	0.902371i	$0.641825\pi$
$110$	−0.236357	−0.0225357
$111$	0	0
$112$	−71.7425	−6.77903
$113$	8.81986	0.829702	0.414851	−	0.909889i	$-0.363834\pi$
$113$	8.81986	0.829702	0.414851	+	0.909889i	$0.363834\pi$
$114$	0	0
$115$	1.02233	0.0953326
$116$	−4.98172	−0.462541
$117$	0	0
$118$	15.9676	1.46994
$119$	−26.3279	−2.41348
$120$	0	0
$121$	−10.6675	−0.969777
$122$	19.2327	1.74124
$123$	0	0
$124$	40.7339	3.65802
$125$	−1.50060	−0.134217
$126$	0	0
$127$	12.7609	1.13235	0.566174	−	0.824286i	$-0.308423\pi$
$127$	12.7609	1.13235	0.566174	+	0.824286i	$0.308423\pi$
$128$	−35.0922	−3.10174
$129$	0	0
$130$	−1.09355	−0.0959109
$131$	−13.3100	−1.16290	−0.581450	−	0.813582i	$-0.697515\pi$
$131$	−13.3100	−1.16290	−0.581450	+	0.813582i	$0.697515\pi$
$132$	0	0
$133$	4.90968	0.425723
$134$	−40.5616	−3.50399
$135$	0	0
$136$	−50.1110	−4.29698
$137$	−11.8026	−1.00836	−0.504182	−	0.863598i	$-0.668206\pi$
$137$	−11.8026	−1.00836	−0.504182	+	0.863598i	$0.668206\pi$
$138$	0	0
$139$	−2.15989	−0.183200	−0.0915999	−	0.995796i	$-0.529198\pi$
$139$	−2.15989	−0.183200	−0.0915999	+	0.995796i	$0.529198\pi$
$140$	−4.00856	−0.338785
$141$	0	0
$142$	−24.4397	−2.05094
$143$	1.53818	0.128629
$144$	0	0
$145$	−0.138019	−0.0114619
$146$	22.8230	1.88885
$147$	0	0
$148$	28.4270	2.33669
$149$	6.80170	0.557217	0.278608	−	0.960405i	$-0.410127\pi$
$149$	6.80170	0.557217	0.278608	+	0.960405i	$0.410127\pi$
$150$	0	0
$151$	−23.7931	−1.93625	−0.968127	−	0.250461i	$-0.919418\pi$
$151$	−23.7931	−1.93625	−0.968127	+	0.250461i	$0.919418\pi$
$152$	9.34479	0.757963
$153$	0	0
$154$	7.71567	0.621747
$155$	1.12854	0.0906464
$156$	0	0
$157$	14.0774	1.12350	0.561750	−	0.827307i	$-0.310128\pi$
$157$	14.0774	1.12350	0.561750	+	0.827307i	$0.310128\pi$
$158$	13.8450	1.10145
$159$	0	0
$160$	−3.17905	−0.251326
$161$	−33.3730	−2.63016
$162$	0	0
$163$	−6.78192	−0.531201	−0.265601	−	0.964083i	$-0.585570\pi$
$163$	−6.78192	−0.531201	−0.265601	+	0.964083i	$0.585570\pi$
$164$	−36.2606	−2.83148
$165$	0	0
$166$	21.8813	1.69832
$167$	3.12813	0.242062	0.121031	−	0.992649i	$-0.461380\pi$
$167$	3.12813	0.242062	0.121031	+	0.992649i	$0.461380\pi$
$168$	0	0
$169$	−5.88331	−0.452562
$170$	−2.19819	−0.168593
$171$	0	0
$172$	38.9423	2.96932
$173$	14.0524	1.06838	0.534191	−	0.845364i	$-0.320616\pi$
$173$	14.0524	1.06838	0.534191	+	0.845364i	$0.320616\pi$
$174$	0	0
$175$	24.4373	1.84729
$176$	8.42541	0.635089
$177$	0	0
$178$	16.0615	1.20386
$179$	−11.6375	−0.869831	−0.434916	−	0.900471i	$-0.643222\pi$
$179$	−11.6375	−0.869831	−0.434916	+	0.900471i	$0.643222\pi$
$180$	0	0
$181$	−15.4347	−1.14725	−0.573625	−	0.819118i	$-0.694463\pi$
$181$	−15.4347	−1.14725	−0.573625	+	0.819118i	$0.694463\pi$
$182$	35.6981	2.64612
$183$	0	0
$184$	−63.5203	−4.68278
$185$	0.787574	0.0579036
$186$	0	0
$187$	3.09194	0.226105
$188$	−5.42860	−0.395921
$189$	0	0
$190$	0.409922	0.0297388
$191$	13.2246	0.956900	0.478450	−	0.878115i	$-0.341199\pi$
$191$	13.2246	0.956900	0.478450	+	0.878115i	$0.341199\pi$
$192$	0	0
$193$	7.70213	0.554412	0.277206	−	0.960811i	$-0.410592\pi$
$193$	7.70213	0.554412	0.277206	+	0.960811i	$0.410592\pi$
$194$	16.8868	1.21240
$195$	0	0
$196$	92.8558	6.63256
$197$	7.80260	0.555912	0.277956	−	0.960594i	$-0.410343\pi$
$197$	7.80260	0.555912	0.277956	+	0.960594i	$0.410343\pi$
$198$	0	0
$199$	23.4900	1.66516	0.832580	−	0.553904i	$-0.186863\pi$
$199$	23.4900	1.66516	0.832580	+	0.553904i	$0.186863\pi$
$200$	46.5126	3.28894
$201$	0	0
$202$	−16.5231	−1.16256
$203$	4.50552	0.316226
$204$	0	0
$205$	−1.00461	−0.0701647
$206$	−24.3963	−1.69977
$207$	0	0
$208$	38.9818	2.70290
$209$	−0.576590	−0.0398836
$210$	0	0
$211$	−9.47121	−0.652025	−0.326013	−	0.945365i	$-0.605705\pi$
$211$	−9.47121	−0.652025	−0.326013	+	0.945365i	$0.605705\pi$
$212$	35.5543	2.44188
$213$	0	0
$214$	26.6008	1.81839
$215$	1.07890	0.0735804
$216$	0	0
$217$	−36.8402	−2.50088
$218$	24.5264	1.66114
$219$	0	0
$220$	0.470763	0.0317388
$221$	14.3055	0.962291
$222$	0	0
$223$	15.6599	1.04866	0.524332	−	0.851514i	$-0.324315\pi$
$223$	15.6599	1.04866	0.524332	+	0.851514i	$0.324315\pi$
$224$	103.778	6.93393
$225$	0	0
$226$	−24.0389	−1.59904
$227$	14.8629	0.986487	0.493243	−	0.869891i	$-0.335811\pi$
$227$	14.8629	0.986487	0.493243	+	0.869891i	$0.335811\pi$
$228$	0	0
$229$	−3.61726	−0.239035	−0.119518	−	0.992832i	$-0.538135\pi$
$229$	−3.61726	−0.239035	−0.119518	+	0.992832i	$0.538135\pi$
$230$	−2.78640	−0.183730
$231$	0	0
$232$	8.57555	0.563012
$233$	−3.30974	−0.216828	−0.108414	−	0.994106i	$-0.534577\pi$
$233$	−3.30974	−0.216828	−0.108414	+	0.994106i	$0.534577\pi$
$234$	0	0
$235$	−0.150400	−0.00981101
$236$	−31.8034	−2.07023
$237$	0	0
$238$	71.7579	4.65138
$239$	6.70281	0.433568	0.216784	−	0.976220i	$-0.430443\pi$
$239$	6.70281	0.433568	0.216784	+	0.976220i	$0.430443\pi$
$240$	0	0
$241$	24.3098	1.56593	0.782967	−	0.622063i	$-0.213705\pi$
$241$	24.3098	1.56593	0.782967	+	0.622063i	$0.213705\pi$
$242$	29.0749	1.86900
$243$	0	0
$244$	−38.3066	−2.45233
$245$	2.57258	0.164356
$246$	0	0
$247$	−2.66771	−0.169742
$248$	−70.1195	−4.45259
$249$	0	0
$250$	4.08994	0.258671
$251$	19.5329	1.23290	0.616452	−	0.787392i	$-0.288569\pi$
$251$	19.5329	1.23290	0.616452	+	0.787392i	$0.288569\pi$
$252$	0	0
$253$	3.91932	0.246405
$254$	−34.7804	−2.18232
$255$	0	0
$256$	38.8737	2.42961
$257$	−24.9410	−1.55577	−0.777887	−	0.628404i	$-0.783709\pi$
$257$	−24.9410	−1.55577	−0.777887	+	0.628404i	$0.783709\pi$
$258$	0	0
$259$	−25.7097	−1.59752
$260$	2.17808	0.135079
$261$	0	0
$262$	36.2770	2.24120
$263$	−26.1936	−1.61517	−0.807583	−	0.589754i	$-0.799225\pi$
$263$	−26.1936	−1.61517	−0.807583	+	0.589754i	$0.799225\pi$
$264$	0	0
$265$	0.985036	0.0605103
$266$	−13.3815	−0.820475
$267$	0	0
$268$	80.7885	4.93495
$269$	−0.649877	−0.0396237	−0.0198119	−	0.999804i	$-0.506307\pi$
$269$	−0.649877	−0.0396237	−0.0198119	+	0.999804i	$0.506307\pi$
$270$	0	0
$271$	−0.106761	−0.00648524	−0.00324262	−	0.999995i	$-0.501032\pi$
$271$	−0.106761	−0.00648524	−0.00324262	+	0.999995i	$0.501032\pi$
$272$	78.3586	4.75119
$273$	0	0
$274$	32.1685	1.94337
$275$	−2.86991	−0.173062
$276$	0	0
$277$	14.5495	0.874197	0.437098	−	0.899414i	$-0.356006\pi$
$277$	14.5495	0.874197	0.437098	+	0.899414i	$0.356006\pi$
$278$	5.88689	0.353072
$279$	0	0
$280$	6.90033	0.412374
$281$	32.4022	1.93295	0.966477	−	0.256754i	$-0.0826529\pi$
$281$	32.4022	1.93295	0.966477	+	0.256754i	$0.0826529\pi$
$282$	0	0
$283$	1.30176	0.0773818	0.0386909	−	0.999251i	$-0.487681\pi$
$283$	1.30176	0.0773818	0.0386909	+	0.999251i	$0.487681\pi$
$284$	48.6778	2.88849
$285$	0	0
$286$	−4.19237	−0.247900
$287$	32.7945	1.93580
$288$	0	0
$289$	11.7559	0.691524
$290$	0.376178	0.0220899
$291$	0	0
$292$	−45.4577	−2.66021
$293$	9.60672	0.561231	0.280615	−	0.959820i	$-0.409461\pi$
$293$	9.60672	0.561231	0.280615	+	0.959820i	$0.409461\pi$
$294$	0	0
$295$	−0.881118	−0.0513007
$296$	−48.9343	−2.84425
$297$	0	0
$298$	−18.5383	−1.07390
$299$	18.1335	1.04869
$300$	0	0
$301$	−35.2198	−2.03004
$302$	64.8491	3.73165
$303$	0	0
$304$	−14.6125	−0.838082
$305$	−1.06129	−0.0607692
$306$	0	0
$307$	8.00901	0.457099	0.228549	−	0.973532i	$-0.426602\pi$
$307$	8.00901	0.457099	0.228549	+	0.973532i	$0.426602\pi$
$308$	−15.3677	−0.875654
$309$	0	0
$310$	−3.07588	−0.174698
$311$	−28.5302	−1.61780	−0.808899	−	0.587947i	$-0.799936\pi$
$311$	−28.5302	−1.61780	−0.808899	+	0.587947i	$0.799936\pi$
$312$	0	0
$313$	14.5880	0.824563	0.412282	−	0.911056i	$-0.364732\pi$
$313$	14.5880	0.824563	0.412282	+	0.911056i	$0.364732\pi$
$314$	−38.3686	−2.16527
$315$	0	0
$316$	−27.5757	−1.55126
$317$	−29.7970	−1.67357	−0.836783	−	0.547535i	$-0.815566\pi$
$317$	−29.7970	−1.67357	−0.836783	+	0.547535i	$0.815566\pi$
$318$	0	0
$319$	−0.529127	−0.0296254
$320$	4.26923	0.238657
$321$	0	0
$322$	90.9597	5.06899
$323$	−5.36245	−0.298375
$324$	0	0
$325$	−13.2782	−0.736543
$326$	18.4844	1.02376
$327$	0	0
$328$	62.4191	3.44652
$329$	4.90968	0.270679
$330$	0	0
$331$	−16.5798	−0.911309	−0.455654	−	0.890157i	$-0.650595\pi$
$331$	−16.5798	−0.911309	−0.455654	+	0.890157i	$0.650595\pi$
$332$	−43.5820	−2.39187
$333$	0	0
$334$	−8.52585	−0.466514
$335$	2.23826	0.122289
$336$	0	0
$337$	15.9919	0.871133	0.435567	−	0.900157i	$-0.356548\pi$
$337$	15.9919	0.871133	0.435567	+	0.900157i	$0.356548\pi$
$338$	16.0352	0.872202
$339$	0	0
$340$	4.37823	0.237443
$341$	4.32650	0.234293
$342$	0	0
$343$	−49.6120	−2.67879
$344$	−67.0353	−3.61430
$345$	0	0
$346$	−38.3004	−2.05904
$347$	−2.61716	−0.140496	−0.0702481	−	0.997530i	$-0.522379\pi$
$347$	−2.61716	−0.140496	−0.0702481	+	0.997530i	$0.522379\pi$
$348$	0	0
$349$	−18.8220	−1.00752	−0.503760	−	0.863843i	$-0.668051\pi$
$349$	−18.8220	−1.00752	−0.503760	+	0.863843i	$0.668051\pi$
$350$	−66.6050	−3.56019
$351$	0	0
$352$	−12.1876	−0.649601
$353$	−25.0444	−1.33298	−0.666490	−	0.745514i	$-0.732204\pi$
$353$	−25.0444	−1.33298	−0.666490	+	0.745514i	$0.732204\pi$
$354$	0	0
$355$	1.34862	0.0715775
$356$	−31.9904	−1.69549
$357$	0	0
$358$	31.7187	1.67638
$359$	11.0657	0.584024	0.292012	−	0.956415i	$-0.405675\pi$
$359$	11.0657	0.584024	0.292012	+	0.956415i	$0.405675\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	42.0679	2.21104
$363$	0	0
$364$	−71.1016	−3.72674
$365$	−1.25941	−0.0659206
$366$	0	0
$367$	10.3000	0.537654	0.268827	−	0.963188i	$-0.413364\pi$
$367$	10.3000	0.537654	0.268827	+	0.963188i	$0.413364\pi$
$368$	99.3267	5.17776
$369$	0	0
$370$	−2.14657	−0.111595
$371$	−32.1557	−1.66944
$372$	0	0
$373$	27.8101	1.43995	0.719976	−	0.693999i	$-0.244153\pi$
$373$	27.8101	1.43995	0.719976	+	0.693999i	$0.244153\pi$
$374$	−8.42722	−0.435761
$375$	0	0
$376$	9.34479	0.481921
$377$	−2.44811	−0.126084
$378$	0	0
$379$	13.4386	0.690296	0.345148	−	0.938548i	$-0.387829\pi$
$379$	13.4386	0.690296	0.345148	+	0.938548i	$0.387829\pi$
$380$	−0.816460	−0.0418835
$381$	0	0
$382$	−36.0443	−1.84419
$383$	10.7472	0.549155	0.274578	−	0.961565i	$-0.411462\pi$
$383$	10.7472	0.549155	0.274578	+	0.961565i	$0.411462\pi$
$384$	0	0
$385$	−0.425763	−0.0216989
$386$	−20.9925	−1.06849
$387$	0	0
$388$	−33.6342	−1.70752
$389$	23.3030	1.18151	0.590754	−	0.806852i	$-0.298830\pi$
$389$	23.3030	1.18151	0.590754	+	0.806852i	$0.298830\pi$
$390$	0	0
$391$	36.4507	1.84339
$392$	−159.842	−8.07324
$393$	0	0
$394$	−21.2663	−1.07138
$395$	−0.763989	−0.0384405
$396$	0	0
$397$	−4.14694	−0.208129	−0.104064	−	0.994571i	$-0.533185\pi$
$397$	−4.14694	−0.208129	−0.104064	+	0.994571i	$0.533185\pi$
$398$	−64.0230	−3.20918
$399$	0	0
$400$	−72.7318	−3.63659
$401$	31.9442	1.59522	0.797609	−	0.603175i	$-0.206098\pi$
$401$	31.9442	1.59522	0.797609	+	0.603175i	$0.206098\pi$
$402$	0	0
$403$	20.0174	0.997138
$404$	32.9099	1.63733
$405$	0	0
$406$	−12.2800	−0.609446
$407$	3.01934	0.149663
$408$	0	0
$409$	16.5716	0.819415	0.409708	−	0.912217i	$-0.365631\pi$
$409$	16.5716	0.819415	0.409708	+	0.912217i	$0.365631\pi$
$410$	2.73810	0.135225
$411$	0	0
$412$	48.5912	2.39392
$413$	28.7634	1.41535
$414$	0	0
$415$	−1.20744	−0.0592711
$416$	−56.3883	−2.76467
$417$	0	0
$418$	1.57152	0.0768657
$419$	17.0605	0.833462	0.416731	−	0.909030i	$-0.363176\pi$
$419$	17.0605	0.833462	0.416731	+	0.909030i	$0.363176\pi$
$420$	0	0
$421$	−20.4376	−0.996070	−0.498035	−	0.867157i	$-0.665945\pi$
$421$	−20.4376	−0.996070	−0.498035	+	0.867157i	$0.665945\pi$
$422$	25.8142	1.25662
$423$	0	0
$424$	−61.2032	−2.97229
$425$	−26.6910	−1.29470
$426$	0	0
$427$	34.6449	1.67658
$428$	−52.9821	−2.56099
$429$	0	0
$430$	−2.94059	−0.141808
$431$	19.7900	0.953250	0.476625	−	0.879107i	$-0.341860\pi$
$431$	19.7900	0.953250	0.476625	+	0.879107i	$0.341860\pi$
$432$	0	0
$433$	−6.77633	−0.325650	−0.162825	−	0.986655i	$-0.552061\pi$
$433$	−6.77633	−0.325650	−0.162825	+	0.986655i	$0.552061\pi$
$434$	100.410	4.81982
$435$	0	0
$436$	−48.8503	−2.33951
$437$	−6.79740	−0.325164
$438$	0	0
$439$	0.341887	0.0163174	0.00815869	−	0.999967i	$-0.497403\pi$
$439$	0.341887	0.0163174	0.00815869	+	0.999967i	$0.497403\pi$
$440$	−0.810372	−0.0386330
$441$	0	0
$442$	−38.9902	−1.85458
$443$	15.0553	0.715299	0.357650	−	0.933856i	$-0.383578\pi$
$443$	15.0553	0.715299	0.357650	+	0.933856i	$0.383578\pi$
$444$	0	0
$445$	−0.886297	−0.0420145
$446$	−42.6817	−2.02104
$447$	0	0
$448$	−139.365	−6.58439
$449$	0.647613	0.0305627	0.0152814	−	0.999883i	$-0.495136\pi$
$449$	0.647613	0.0305627	0.0152814	+	0.999883i	$0.495136\pi$
$450$	0	0
$451$	−3.85137	−0.181354
$452$	47.8794	2.25206
$453$	0	0
$454$	−40.5096	−1.90121
$455$	−1.96988	−0.0923493
$456$	0	0
$457$	−21.7670	−1.01822	−0.509110	−	0.860702i	$-0.670025\pi$
$457$	−21.7670	−1.01822	−0.509110	+	0.860702i	$0.670025\pi$
$458$	9.85901	0.460681
$459$	0	0
$460$	5.54981	0.258761
$461$	28.9646	1.34902	0.674508	−	0.738267i	$-0.264356\pi$
$461$	28.9646	1.34902	0.674508	+	0.738267i	$0.264356\pi$
$462$	0	0
$463$	−25.0440	−1.16389	−0.581947	−	0.813226i	$-0.697709\pi$
$463$	−25.0440	−1.16389	−0.581947	+	0.813226i	$0.697709\pi$
$464$	−13.4096	−0.622525
$465$	0	0
$466$	9.02083	0.417882
$467$	−9.65674	−0.446861	−0.223430	−	0.974720i	$-0.571726\pi$
$467$	−9.65674	−0.446861	−0.223430	+	0.974720i	$0.571726\pi$
$468$	0	0
$469$	−73.0660	−3.37387
$470$	0.409922	0.0189083
$471$	0	0
$472$	54.7465	2.51991
$473$	4.13620	0.190183
$474$	0	0
$475$	4.97738	0.228378
$476$	−142.924	−6.55089
$477$	0	0
$478$	−18.2688	−0.835595
$479$	3.98535	0.182095	0.0910476	−	0.995847i	$-0.470978\pi$
$479$	3.98535	0.182095	0.0910476	+	0.995847i	$0.470978\pi$
$480$	0	0
$481$	13.9696	0.636957
$482$	−66.2576	−3.01795
$483$	0	0
$484$	−57.9098	−2.63226
$485$	−0.931839	−0.0423126
$486$	0	0
$487$	−3.02795	−0.137209	−0.0686047	−	0.997644i	$-0.521855\pi$
$487$	−3.02795	−0.137209	−0.0686047	+	0.997644i	$0.521855\pi$
$488$	65.9410	2.98501
$489$	0	0
$490$	−7.01168	−0.316756
$491$	−13.9202	−0.628210	−0.314105	−	0.949388i	$-0.601704\pi$
$491$	−13.9202	−0.628210	−0.314105	+	0.949388i	$0.601704\pi$
$492$	0	0
$493$	−4.92103	−0.221632
$494$	7.27097	0.327136
$495$	0	0
$496$	109.646	4.92325
$497$	−44.0247	−1.97478
$498$	0	0
$499$	−30.2586	−1.35456	−0.677280	−	0.735725i	$-0.736841\pi$
$499$	−30.2586	−1.35456	−0.677280	+	0.735725i	$0.736841\pi$
$500$	−8.14613	−0.364306
$501$	0	0
$502$	−53.2377	−2.37612
$503$	13.8609	0.618026	0.309013	−	0.951058i	$-0.400001\pi$
$503$	13.8609	0.618026	0.309013	+	0.951058i	$0.400001\pi$
$504$	0	0
$505$	0.911772	0.0405733
$506$	−10.6823	−0.474885
$507$	0	0
$508$	69.2738	3.07353
$509$	3.15617	0.139895	0.0699473	−	0.997551i	$-0.477717\pi$
$509$	3.15617	0.139895	0.0699473	+	0.997551i	$0.477717\pi$
$510$	0	0
$511$	41.1124	1.81871
$512$	−35.7677	−1.58072
$513$	0	0
$514$	67.9777	2.99837
$515$	1.34623	0.0593218
$516$	0	0
$517$	−0.576590	−0.0253584
$518$	70.0729	3.07883
$519$	0	0
$520$	−3.74935	−0.164420
$521$	23.2268	1.01758	0.508791	−	0.860890i	$-0.330092\pi$
$521$	23.2268	1.01758	0.508791	+	0.860890i	$0.330092\pi$
$522$	0	0
$523$	4.48011	0.195902	0.0979508	−	0.995191i	$-0.468771\pi$
$523$	4.48011	0.195902	0.0979508	+	0.995191i	$0.468771\pi$
$524$	−72.2547	−3.15646
$525$	0	0
$526$	71.3918	3.11283
$527$	40.2376	1.75278
$528$	0	0
$529$	23.2046	1.00890
$530$	−2.68476	−0.116619
$531$	0	0
$532$	26.6527	1.15554
$533$	−17.8191	−0.771833
$534$	0	0
$535$	−1.46788	−0.0634618
$536$	−139.069	−6.00689
$537$	0	0
$538$	1.77127	0.0763648
$539$	9.86254	0.424810
$540$	0	0
$541$	−31.5345	−1.35577	−0.677887	−	0.735166i	$-0.737104\pi$
$541$	−31.5345	−1.35577	−0.677887	+	0.735166i	$0.737104\pi$
$542$	0.290981	0.0124987
$543$	0	0
$544$	−113.348	−4.85976
$545$	−1.35340	−0.0579735
$546$	0	0
$547$	41.9320	1.79288	0.896442	−	0.443161i	$-0.146143\pi$
$547$	41.9320	1.79288	0.896442	+	0.443161i	$0.146143\pi$
$548$	−64.0715	−2.73700
$549$	0	0
$550$	7.82207	0.333534
$551$	0.917682	0.0390946
$552$	0	0
$553$	24.9398	1.06055
$554$	−39.6554	−1.68480
$555$	0	0
$556$	−11.7252	−0.497259
$557$	8.40131	0.355975	0.177987	−	0.984033i	$-0.443041\pi$
$557$	8.40131	0.355975	0.177987	+	0.984033i	$0.443041\pi$
$558$	0	0
$559$	19.1370	0.809407
$560$	−10.7901	−0.455963
$561$	0	0
$562$	−88.3137	−3.72529
$563$	29.5302	1.24455	0.622275	−	0.782798i	$-0.286208\pi$
$563$	29.5302	1.24455	0.622275	+	0.782798i	$0.286208\pi$
$564$	0	0
$565$	1.32651	0.0558065
$566$	−3.54801	−0.149134
$567$	0	0
$568$	−83.7939	−3.51592
$569$	−30.9069	−1.29569	−0.647843	−	0.761774i	$-0.724329\pi$
$569$	−30.9069	−1.29569	−0.647843	+	0.761774i	$0.724329\pi$
$570$	0	0
$571$	36.5200	1.52832	0.764158	−	0.645029i	$-0.223155\pi$
$571$	36.5200	1.52832	0.764158	+	0.645029i	$0.223155\pi$
$572$	8.35014	0.349137
$573$	0	0
$574$	−89.3829	−3.73077
$575$	−33.8332	−1.41094
$576$	0	0
$577$	18.8465	0.784590	0.392295	−	0.919839i	$-0.371681\pi$
$577$	18.8465	0.784590	0.392295	+	0.919839i	$0.371681\pi$
$578$	−32.0413	−1.33274
$579$	0	0
$580$	−0.749251	−0.0311109
$581$	39.4160	1.63525
$582$	0	0
$583$	3.77635	0.156400
$584$	78.2510	3.23805
$585$	0	0
$586$	−26.1836	−1.08163
$587$	−18.3602	−0.757806	−0.378903	−	0.925436i	$-0.623699\pi$
$587$	−18.3602	−0.757806	−0.378903	+	0.925436i	$0.623699\pi$
$588$	0	0
$589$	−7.50359	−0.309180
$590$	2.40153	0.0988693
$591$	0	0
$592$	76.5186	3.14490
$593$	1.95865	0.0804321	0.0402160	−	0.999191i	$-0.487195\pi$
$593$	1.95865	0.0804321	0.0402160	+	0.999191i	$0.487195\pi$
$594$	0	0
$595$	−3.95972	−0.162333
$596$	36.9237	1.51245
$597$	0	0
$598$	−49.4237	−2.02109
$599$	19.3426	0.790319	0.395160	−	0.918612i	$-0.370689\pi$
$599$	19.3426	0.790319	0.395160	+	0.918612i	$0.370689\pi$
$600$	0	0
$601$	−40.7713	−1.66310	−0.831548	−	0.555453i	$-0.812545\pi$
$601$	−40.7713	−1.66310	−0.831548	+	0.555453i	$0.812545\pi$
$602$	95.9932	3.91239
$603$	0	0
$604$	−129.163	−5.25557
$605$	−1.60440	−0.0652280
$606$	0	0
$607$	−16.5781	−0.672883	−0.336442	−	0.941704i	$-0.609223\pi$
$607$	−16.5781	−0.672883	−0.336442	+	0.941704i	$0.609223\pi$
$608$	21.1373	0.857232
$609$	0	0
$610$	2.89259	0.117118
$611$	−2.66771	−0.107924
$612$	0	0
$613$	−31.9328	−1.28976	−0.644878	−	0.764286i	$-0.723092\pi$
$613$	−31.9328	−1.28976	−0.644878	+	0.764286i	$0.723092\pi$
$614$	−21.8289	−0.880944
$615$	0	0
$616$	26.4539	1.06586
$617$	−43.7384	−1.76084	−0.880420	−	0.474194i	$-0.842739\pi$
$617$	−43.7384	−1.76084	−0.880420	+	0.474194i	$0.842739\pi$
$618$	0	0
$619$	32.0259	1.28723	0.643615	−	0.765349i	$-0.277434\pi$
$619$	32.0259	1.28723	0.643615	+	0.765349i	$0.277434\pi$
$620$	6.12638	0.246041
$621$	0	0
$622$	77.7603	3.11791
$623$	28.9324	1.15915
$624$	0	0
$625$	24.6612	0.986448
$626$	−39.7603	−1.58914
$627$	0	0
$628$	76.4206	3.04951
$629$	28.0807	1.11965
$630$	0	0
$631$	7.09851	0.282587	0.141294	−	0.989968i	$-0.454874\pi$
$631$	7.09851	0.282587	0.141294	+	0.989968i	$0.454874\pi$
$632$	47.4689	1.88821
$633$	0	0
$634$	81.2130	3.22538
$635$	1.91924	0.0761626
$636$	0	0
$637$	45.6310	1.80797
$638$	1.44216	0.0570956
$639$	0	0
$640$	−5.27787	−0.208626
$641$	−11.6907	−0.461755	−0.230877	−	0.972983i	$-0.574160\pi$
$641$	−11.6907	−0.461755	−0.230877	+	0.972983i	$0.574160\pi$
$642$	0	0
$643$	44.8214	1.76759	0.883793	−	0.467879i	$-0.154982\pi$
$643$	44.8214	1.76759	0.883793	+	0.467879i	$0.154982\pi$
$644$	−181.169	−7.13905
$645$	0	0
$646$	14.6156	0.575043
$647$	−6.59540	−0.259292	−0.129646	−	0.991560i	$-0.541384\pi$
$647$	−6.59540	−0.259292	−0.129646	+	0.991560i	$0.541384\pi$
$648$	0	0
$649$	−3.37796	−0.132596
$650$	36.1904	1.41950
$651$	0	0
$652$	−36.8163	−1.44184
$653$	−9.99783	−0.391245	−0.195623	−	0.980679i	$-0.562673\pi$
$653$	−9.99783	−0.391245	−0.195623	+	0.980679i	$0.562673\pi$
$654$	0	0
$655$	−2.00182	−0.0782177
$656$	−97.6048	−3.81083
$657$	0	0
$658$	−13.3815	−0.521667
$659$	28.2517	1.10053	0.550265	−	0.834990i	$-0.314527\pi$
$659$	28.2517	1.10053	0.550265	+	0.834990i	$0.314527\pi$
$660$	0	0
$661$	−8.05210	−0.313191	−0.156595	−	0.987663i	$-0.550052\pi$
$661$	−8.05210	−0.313191	−0.156595	+	0.987663i	$0.550052\pi$
$662$	45.1890	1.75632
$663$	0	0
$664$	75.0221	2.91142
$665$	0.738415	0.0286345
$666$	0	0
$667$	−6.23785	−0.241531
$668$	16.9813	0.657028
$669$	0	0
$670$	−6.10047	−0.235682
$671$	−4.06868	−0.157070
$672$	0	0
$673$	17.5755	0.677485	0.338743	−	0.940879i	$-0.389998\pi$
$673$	17.5755	0.677485	0.338743	+	0.940879i	$0.389998\pi$
$674$	−43.5866	−1.67889
$675$	0	0
$676$	−31.9381	−1.22839
$677$	23.5866	0.906505	0.453253	−	0.891382i	$-0.350264\pi$
$677$	23.5866	0.906505	0.453253	+	0.891382i	$0.350264\pi$
$678$	0	0
$679$	30.4191	1.16738
$680$	−7.53669	−0.289019
$681$	0	0
$682$	−11.7921	−0.451541
$683$	46.6782	1.78609	0.893046	−	0.449965i	$-0.148564\pi$
$683$	46.6782	1.78609	0.893046	+	0.449965i	$0.148564\pi$
$684$	0	0
$685$	−1.77511	−0.0678234
$686$	135.220	5.16271
$687$	0	0
$688$	104.823	3.99635
$689$	17.4720	0.665632
$690$	0	0
$691$	25.8472	0.983273	0.491637	−	0.870800i	$-0.336399\pi$
$691$	25.8472	0.983273	0.491637	+	0.870800i	$0.336399\pi$
$692$	76.2847	2.89991
$693$	0	0
$694$	7.13318	0.270772
$695$	−0.324848	−0.0123222
$696$	0	0
$697$	−35.8188	−1.35673
$698$	51.3003	1.94175
$699$	0	0
$700$	132.660	5.01409
$701$	−45.1821	−1.70650	−0.853252	−	0.521498i	$-0.825373\pi$
$701$	−45.1821	−1.70650	−0.853252	+	0.521498i	$0.825373\pi$
$702$	0	0
$703$	−5.23653	−0.197500
$704$	16.3670	0.616855
$705$	0	0
$706$	68.2597	2.56899
$707$	−29.7640	−1.11939
$708$	0	0
$709$	4.07098	0.152889	0.0764444	−	0.997074i	$-0.475643\pi$
$709$	4.07098	0.152889	0.0764444	+	0.997074i	$0.475643\pi$
$710$	−3.67573	−0.137948
$711$	0	0
$712$	55.0683	2.06377
$713$	51.0049	1.91015
$714$	0	0
$715$	0.231342	0.00865169
$716$	−63.1755	−2.36098
$717$	0	0
$718$	−30.1600	−1.12556
$719$	−24.6380	−0.918842	−0.459421	−	0.888219i	$-0.651943\pi$
$719$	−24.6380	−0.918842	−0.459421	+	0.888219i	$0.651943\pi$
$720$	0	0
$721$	−43.9464	−1.63665
$722$	−2.72555	−0.101434
$723$	0	0
$724$	−83.7886	−3.11398
$725$	4.56765	0.169638
$726$	0	0
$727$	38.5344	1.42916	0.714580	−	0.699553i	$-0.246618\pi$
$727$	38.5344	1.42916	0.714580	+	0.699553i	$0.246618\pi$
$728$	122.394	4.53624
$729$	0	0
$730$	3.43258	0.127046
$731$	38.4678	1.42278
$732$	0	0
$733$	−37.9082	−1.40017	−0.700085	−	0.714059i	$-0.746855\pi$
$733$	−37.9082	−1.40017	−0.700085	+	0.714059i	$0.746855\pi$
$734$	−28.0731	−1.03620
$735$	0	0
$736$	−143.679	−5.29608
$737$	8.58084	0.316079
$738$	0	0
$739$	25.8327	0.950271	0.475135	−	0.879913i	$-0.342399\pi$
$739$	25.8327	0.950271	0.475135	+	0.879913i	$0.342399\pi$
$740$	4.27542	0.157168
$741$	0	0
$742$	87.6417	3.21743
$743$	−34.4126	−1.26248	−0.631239	−	0.775589i	$-0.717453\pi$
$743$	−34.4126	−1.26248	−0.631239	+	0.775589i	$0.717453\pi$
$744$	0	0
$745$	1.02297	0.0374789
$746$	−75.7977	−2.77515
$747$	0	0
$748$	16.7849	0.613716
$749$	47.9176	1.75087
$750$	0	0
$751$	−5.06153	−0.184698	−0.0923489	−	0.995727i	$-0.529438\pi$
$751$	−5.06153	−0.184698	−0.0923489	+	0.995727i	$0.529438\pi$
$752$	−14.6125	−0.532862
$753$	0	0
$754$	6.67244	0.242996
$755$	−3.57848	−0.130234
$756$	0	0
$757$	−18.0574	−0.656306	−0.328153	−	0.944625i	$-0.606426\pi$
$757$	−18.0574	−0.656306	−0.328153	+	0.944625i	$0.606426\pi$
$758$	−36.6276	−1.33037
$759$	0	0
$760$	1.40546	0.0509812
$761$	−9.00837	−0.326553	−0.163277	−	0.986580i	$-0.552206\pi$
$761$	−9.00837	−0.326553	−0.163277	+	0.986580i	$0.552206\pi$
$762$	0	0
$763$	44.1807	1.59945
$764$	71.7911	2.59731
$765$	0	0
$766$	−29.2919	−1.05836
$767$	−15.6288	−0.564323
$768$	0	0
$769$	42.2105	1.52215	0.761074	−	0.648665i	$-0.224672\pi$
$769$	42.2105	1.52215	0.761074	+	0.648665i	$0.224672\pi$
$770$	1.16044	0.0418192
$771$	0	0
$772$	41.8118	1.50484
$773$	−29.8136	−1.07232	−0.536161	−	0.844116i	$-0.680126\pi$
$773$	−29.8136	−1.07232	−0.536161	+	0.844116i	$0.680126\pi$
$774$	0	0
$775$	−37.3482	−1.34159
$776$	57.8980	2.07842
$777$	0	0
$778$	−63.5133	−2.27706
$779$	6.67956	0.239320
$780$	0	0
$781$	5.17024	0.185006
$782$	−99.3481	−3.55268
$783$	0	0
$784$	249.945	8.92661
$785$	2.11724	0.0755676
$786$	0	0
$787$	3.38498	0.120661	0.0603307	−	0.998178i	$-0.480784\pi$
$787$	3.38498	0.120661	0.0603307	+	0.998178i	$0.480784\pi$
$788$	42.3572	1.50891
$789$	0	0
$790$	2.08229	0.0740845
$791$	−43.3027	−1.53967
$792$	0	0
$793$	−18.8246	−0.668480
$794$	11.3027	0.401117
$795$	0	0
$796$	127.518	4.51974
$797$	52.9842	1.87680	0.938399	−	0.345553i	$-0.112309\pi$
$797$	52.9842	1.87680	0.938399	+	0.345553i	$0.112309\pi$
$798$	0	0
$799$	−5.36245	−0.189710
$800$	105.209	3.71968
$801$	0	0
$802$	−87.0654	−3.07438
$803$	−4.82823	−0.170384
$804$	0	0
$805$	−5.01930	−0.176907
$806$	−54.5584	−1.92174
$807$	0	0
$808$	−56.6511	−1.99298
$809$	23.1917	0.815377	0.407688	−	0.913121i	$-0.366335\pi$
$809$	23.1917	0.815377	0.407688	+	0.913121i	$0.366335\pi$
$810$	0	0
$811$	35.8267	1.25805	0.629023	−	0.777387i	$-0.283455\pi$
$811$	35.8267	1.25805	0.629023	+	0.777387i	$0.283455\pi$
$812$	24.4587	0.858331
$813$	0	0
$814$	−8.22933	−0.288438
$815$	−1.02000	−0.0357291
$816$	0	0
$817$	−7.17355	−0.250971
$818$	−45.1668	−1.57922
$819$	0	0
$820$	−5.45359	−0.190448
$821$	33.2283	1.15968	0.579838	−	0.814732i	$-0.303116\pi$
$821$	33.2283	1.15968	0.579838	+	0.814732i	$0.303116\pi$
$822$	0	0
$823$	−27.2931	−0.951378	−0.475689	−	0.879614i	$-0.657801\pi$
$823$	−27.2931	−0.951378	−0.475689	+	0.879614i	$0.657801\pi$
$824$	−83.6450	−2.91391
$825$	0	0
$826$	−78.3958	−2.72774
$827$	35.7890	1.24451	0.622253	−	0.782816i	$-0.286218\pi$
$827$	35.7890	1.24451	0.622253	+	0.782816i	$0.286218\pi$
$828$	0	0
$829$	42.9460	1.49158	0.745788	−	0.666183i	$-0.232073\pi$
$829$	42.9460	1.49158	0.745788	+	0.666183i	$0.232073\pi$
$830$	3.29094	0.114230
$831$	0	0
$832$	75.7253	2.62530
$833$	91.7244	3.17806
$834$	0	0
$835$	0.470470	0.0162813
$836$	−3.13008	−0.108256
$837$	0	0
$838$	−46.4993	−1.60629
$839$	10.6346	0.367146	0.183573	−	0.983006i	$-0.441234\pi$
$839$	10.6346	0.367146	0.183573	+	0.983006i	$0.441234\pi$
$840$	0	0
$841$	−28.1579	−0.970961
$842$	55.7037	1.91968
$843$	0	0
$844$	−51.4154	−1.76979
$845$	−0.884849	−0.0304397
$846$	0	0
$847$	52.3742	1.79960
$848$	95.7035	3.28647
$849$	0	0
$850$	72.7474	2.49522
$851$	35.5948	1.22017
$852$	0	0
$853$	−13.1142	−0.449021	−0.224510	−	0.974472i	$-0.572078\pi$
$853$	−13.1142	−0.449021	−0.224510	+	0.974472i	$0.572078\pi$
$854$	−94.4262	−3.23120
$855$	0	0
$856$	91.2035	3.11727
$857$	−21.1819	−0.723561	−0.361780	−	0.932263i	$-0.617831\pi$
$857$	−21.1819	−0.723561	−0.361780	+	0.932263i	$0.617831\pi$
$858$	0	0
$859$	−8.28397	−0.282646	−0.141323	−	0.989964i	$-0.545136\pi$
$859$	−8.28397	−0.282646	−0.141323	+	0.989964i	$0.545136\pi$
$860$	5.85692	0.199719
$861$	0	0
$862$	−53.9385	−1.83715
$863$	6.63022	0.225695	0.112848	−	0.993612i	$-0.464003\pi$
$863$	6.63022	0.225695	0.112848	+	0.993612i	$0.464003\pi$
$864$	0	0
$865$	2.11348	0.0718603
$866$	18.4692	0.627609
$867$	0	0
$868$	−199.991	−6.78812
$869$	−2.92892	−0.0993568
$870$	0	0
$871$	39.7010	1.34522
$872$	84.0910	2.84768
$873$	0	0
$874$	18.5266	0.626672
$875$	7.36745	0.249065
$876$	0	0
$877$	−44.3143	−1.49639	−0.748194	−	0.663480i	$-0.769079\pi$
$877$	−44.3143	−1.49639	−0.748194	+	0.663480i	$0.769079\pi$
$878$	−0.931829	−0.0314477
$879$	0	0
$880$	1.26718	0.0427166
$881$	−28.4032	−0.956929	−0.478465	−	0.878107i	$-0.658807\pi$
$881$	−28.4032	−0.956929	−0.478465	+	0.878107i	$0.658807\pi$
$882$	0	0
$883$	26.7052	0.898701	0.449350	−	0.893356i	$-0.351655\pi$
$883$	26.7052	0.898701	0.449350	+	0.893356i	$0.351655\pi$
$884$	77.6587	2.61194
$885$	0	0
$886$	−41.0339	−1.37856
$887$	14.5652	0.489053	0.244527	−	0.969643i	$-0.421367\pi$
$887$	14.5652	0.489053	0.244527	+	0.969643i	$0.421367\pi$
$888$	0	0
$889$	−62.6519	−2.10128
$890$	2.41564	0.0809725
$891$	0	0
$892$	85.0112	2.84639
$893$	1.00000	0.0334637
$894$	0	0
$895$	−1.75029	−0.0585056
$896$	172.291	5.75585
$897$	0	0
$898$	−1.76510	−0.0589021
$899$	−6.88591	−0.229658
$900$	0	0
$901$	35.1211	1.17005
$902$	10.4971	0.349515
$903$	0	0
$904$	−82.4197	−2.74124
$905$	−2.32137	−0.0771650
$906$	0	0
$907$	−15.4241	−0.512147	−0.256074	−	0.966657i	$-0.582429\pi$
$907$	−15.4241	−0.512147	−0.256074	+	0.966657i	$0.582429\pi$
$908$	80.6848	2.67762
$909$	0	0
$910$	5.36899	0.177980
$911$	−24.4713	−0.810769	−0.405384	−	0.914146i	$-0.632862\pi$
$911$	−24.4713	−0.810769	−0.405384	+	0.914146i	$0.632862\pi$
$912$	0	0
$913$	−4.62900	−0.153197
$914$	59.3271	1.96237
$915$	0	0
$916$	−19.6367	−0.648813
$917$	65.3479	2.15798
$918$	0	0
$919$	3.80497	0.125514	0.0627572	−	0.998029i	$-0.480011\pi$
$919$	3.80497	0.125514	0.0627572	+	0.998029i	$0.480011\pi$
$920$	−9.55344	−0.314968
$921$	0	0
$922$	−78.9444	−2.59989
$923$	23.9211	0.787374
$924$	0	0
$925$	−26.0642	−0.856986
$926$	68.2586	2.24312
$927$	0	0
$928$	19.3974	0.636749
$929$	55.9670	1.83622	0.918109	−	0.396328i	$-0.129716\pi$
$929$	55.9670	1.83622	0.918109	+	0.396328i	$0.129716\pi$
$930$	0	0
$931$	−17.1049	−0.560592
$932$	−17.9672	−0.588536
$933$	0	0
$934$	26.3199	0.861213
$935$	0.465027	0.0152080
$936$	0	0
$937$	23.9306	0.781777	0.390889	−	0.920438i	$-0.372168\pi$
$937$	23.9306	0.781777	0.390889	+	0.920438i	$0.372168\pi$
$938$	199.145	6.50230
$939$	0	0
$940$	−0.816460	−0.0266300
$941$	−33.1867	−1.08185	−0.540927	−	0.841069i	$-0.681927\pi$
$941$	−33.1867	−1.08185	−0.540927	+	0.841069i	$0.681927\pi$
$942$	0	0
$943$	−45.4036	−1.47855
$944$	−85.6071	−2.78627
$945$	0	0
$946$	−11.2734	−0.366530
$947$	−14.3170	−0.465241	−0.232621	−	0.972568i	$-0.574730\pi$
$947$	−14.3170	−0.465241	−0.232621	+	0.972568i	$0.574730\pi$
$948$	0	0
$949$	−22.3388	−0.725147
$950$	−13.5661	−0.440142
$951$	0	0
$952$	246.029	7.97384
$953$	45.2631	1.46622	0.733109	−	0.680112i	$-0.238069\pi$
$953$	45.2631	1.46622	0.733109	+	0.680112i	$0.238069\pi$
$954$	0	0
$955$	1.98898	0.0643619
$956$	36.3868	1.17683
$957$	0	0
$958$	−10.8622	−0.350943
$959$	57.9469	1.87120
$960$	0	0
$961$	25.3038	0.816252
$962$	−38.0747	−1.22758
$963$	0	0
$964$	131.968	4.25041
$965$	1.15840	0.0372902
$966$	0	0
$967$	17.9668	0.577774	0.288887	−	0.957363i	$-0.406715\pi$
$967$	17.9668	0.577774	0.288887	+	0.957363i	$0.406715\pi$
$968$	99.6860	3.20403
$969$	0	0
$970$	2.53977	0.0815471
$971$	42.4895	1.36355	0.681776	−	0.731561i	$-0.261208\pi$
$971$	42.4895	1.36355	0.681776	+	0.731561i	$0.261208\pi$
$972$	0	0
$973$	10.6044	0.339961
$974$	8.25281	0.264437
$975$	0	0
$976$	−103.112	−3.30054
$977$	−20.4050	−0.652814	−0.326407	−	0.945229i	$-0.605838\pi$
$977$	−20.4050	−0.652814	−0.326407	+	0.945229i	$0.605838\pi$
$978$	0	0
$979$	−3.39781	−0.108594
$980$	13.9655	0.446111
$981$	0	0
$982$	37.9402	1.21072
$983$	37.0777	1.18260	0.591298	−	0.806453i	$-0.298616\pi$
$983$	37.0777	1.18260	0.591298	+	0.806453i	$0.298616\pi$
$984$	0	0
$985$	1.17351	0.0373911
$986$	13.4125	0.427140
$987$	0	0
$988$	−14.4819	−0.460732
$989$	48.7615	1.55052
$990$	0	0
$991$	−26.6463	−0.846448	−0.423224	−	0.906025i	$-0.639102\pi$
$991$	−26.6463	−0.846448	−0.423224	+	0.906025i	$0.639102\pi$
$992$	−158.606	−5.03574
$993$	0	0
$994$	119.991	3.80589
$995$	3.53289	0.112000
$996$	0	0
$997$	−42.3351	−1.34076	−0.670382	−	0.742016i	$-0.733870\pi$
$997$	−42.3351	−1.34076	−0.670382	+	0.742016i	$0.733870\pi$
$998$	82.4711	2.61058
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8037.2.a.o.1.1		18
3.2	odd	2		893.2.a.c.1.18	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
893.2.a.c.1.18	✓	18	3.2	odd	2
8037.2.a.o.1.1		18	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8037 = 3^{2} \cdot 19 \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8037.a (trivial)

\(f(q)\)	\(=\)	\(q-2.72555 q^{2} +5.42860 q^{4} +0.150400 q^{5} -4.90968 q^{7} -9.34479 q^{8} +O(q^{10})\)	\(q-2.72555 q^{2} +5.42860 q^{4} +0.150400 q^{5} -4.90968 q^{7} -9.34479 q^{8} -0.409922 q^{10} +0.576590 q^{11} +2.66771 q^{13} +13.3815 q^{14} +14.6125 q^{16} +5.36245 q^{17} -1.00000 q^{19} +0.816460 q^{20} -1.57152 q^{22} +6.79740 q^{23} -4.97738 q^{25} -7.27097 q^{26} -26.6527 q^{28} -0.917682 q^{29} +7.50359 q^{31} -21.1373 q^{32} -14.6156 q^{34} -0.738415 q^{35} +5.23653 q^{37} +2.72555 q^{38} -1.40546 q^{40} -6.67956 q^{41} +7.17355 q^{43} +3.13008 q^{44} -18.5266 q^{46} -1.00000 q^{47} +17.1049 q^{49} +13.5661 q^{50} +14.4819 q^{52} +6.54945 q^{53} +0.0867191 q^{55} +45.8799 q^{56} +2.50118 q^{58} -5.85850 q^{59} -7.05645 q^{61} -20.4514 q^{62} +28.3858 q^{64} +0.401224 q^{65} +14.8820 q^{67} +29.1106 q^{68} +2.01258 q^{70} +8.96691 q^{71} -8.37376 q^{73} -14.2724 q^{74} -5.42860 q^{76} -2.83087 q^{77} -5.07972 q^{79} +2.19771 q^{80} +18.2054 q^{82} -8.02822 q^{83} +0.806512 q^{85} -19.5518 q^{86} -5.38812 q^{88} -5.89294 q^{89} -13.0976 q^{91} +36.9003 q^{92} +2.72555 q^{94} -0.150400 q^{95} -6.19575 q^{97} -46.6203 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - q^{2} + 21 q^{4} - 5 q^{5} + 3 q^{7} - 6 q^{8}+O(q^{10}) \) 18 * q - q^2 + 21 * q^4 - 5 * q^5 + 3 * q^7 - 6 * q^8	\( 18 q - q^{2} + 21 q^{4} - 5 q^{5} + 3 q^{7} - 6 q^{8} + 7 q^{10} - 6 q^{11} + 21 q^{13} + 9 q^{14} + 23 q^{16} + 4 q^{17} - 18 q^{19} - 9 q^{20} + 28 q^{22} - 9 q^{23} + 11 q^{25} + q^{26} + 7 q^{28} - 12 q^{29} + 26 q^{31} + 7 q^{32} + 26 q^{34} - 9 q^{35} + 8 q^{37} + q^{38} + 16 q^{40} - 12 q^{41} + 28 q^{43} - 2 q^{44} - 33 q^{46} - 18 q^{47} + 17 q^{49} + 29 q^{50} + 30 q^{52} - 5 q^{53} + 28 q^{55} + 77 q^{56} - 6 q^{58} - 30 q^{59} - 16 q^{61} - 16 q^{62} + 28 q^{64} + 22 q^{65} + 45 q^{67} + 96 q^{68} - 2 q^{70} + q^{71} - 24 q^{73} + 19 q^{74} - 21 q^{76} - 2 q^{77} + 33 q^{79} + 25 q^{80} + 18 q^{82} + 13 q^{83} - 7 q^{85} - 3 q^{86} + 27 q^{88} + 6 q^{89} + 42 q^{91} + 11 q^{92} + q^{94} + 5 q^{95} + 44 q^{97} - 58 q^{98}+O(q^{100}) \) 18 * q - q^2 + 21 * q^4 - 5 * q^5 + 3 * q^7 - 6 * q^8 + 7 * q^10 - 6 * q^11 + 21 * q^13 + 9 * q^14 + 23 * q^16 + 4 * q^17 - 18 * q^19 - 9 * q^20 + 28 * q^22 - 9 * q^23 + 11 * q^25 + q^26 + 7 * q^28 - 12 * q^29 + 26 * q^31 + 7 * q^32 + 26 * q^34 - 9 * q^35 + 8 * q^37 + q^38 + 16 * q^40 - 12 * q^41 + 28 * q^43 - 2 * q^44 - 33 * q^46 - 18 * q^47 + 17 * q^49 + 29 * q^50 + 30 * q^52 - 5 * q^53 + 28 * q^55 + 77 * q^56 - 6 * q^58 - 30 * q^59 - 16 * q^61 - 16 * q^62 + 28 * q^64 + 22 * q^65 + 45 * q^67 + 96 * q^68 - 2 * q^70 + q^71 - 24 * q^73 + 19 * q^74 - 21 * q^76 - 2 * q^77 + 33 * q^79 + 25 * q^80 + 18 * q^82 + 13 * q^83 - 7 * q^85 - 3 * q^86 + 27 * q^88 + 6 * q^89 + 42 * q^91 + 11 * q^92 + q^94 + 5 * q^95 + 44 * q^97 - 58 * q^98

Introduction

L-functions

Modular forms

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Database

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