LMFDB - Embedded newform 8037.2.a.w.1.3

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:	$34$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
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)

$f(q)$	$=$	$q-2.42394 q^{2} +3.87547 q^{4} +3.99749 q^{5} -0.305320 q^{7} -4.54602 q^{8} +O(q^{10})$	\(q-2.42394 q^{2} +3.87547 q^{4} +3.99749 q^{5} -0.305320 q^{7} -4.54602 q^{8} -9.68967 q^{10} -1.33550 q^{11} -2.39980 q^{13} +0.740078 q^{14} +3.26833 q^{16} -2.71547 q^{17} -1.00000 q^{19} +15.4922 q^{20} +3.23718 q^{22} -3.63280 q^{23} +10.9800 q^{25} +5.81696 q^{26} -1.18326 q^{28} -3.99602 q^{29} +0.552364 q^{31} +1.16982 q^{32} +6.58212 q^{34} -1.22052 q^{35} -0.819242 q^{37} +2.42394 q^{38} -18.1727 q^{40} +9.50876 q^{41} +4.82147 q^{43} -5.17570 q^{44} +8.80568 q^{46} +1.00000 q^{47} -6.90678 q^{49} -26.6147 q^{50} -9.30034 q^{52} +10.1106 q^{53} -5.33867 q^{55} +1.38799 q^{56} +9.68609 q^{58} +0.680548 q^{59} -12.7218 q^{61} -1.33890 q^{62} -9.37223 q^{64} -9.59317 q^{65} +2.58225 q^{67} -10.5237 q^{68} +2.95846 q^{70} +4.97858 q^{71} -14.6674 q^{73} +1.98579 q^{74} -3.87547 q^{76} +0.407757 q^{77} -0.138047 q^{79} +13.0651 q^{80} -23.0486 q^{82} +7.62417 q^{83} -10.8551 q^{85} -11.6869 q^{86} +6.07123 q^{88} +16.1292 q^{89} +0.732707 q^{91} -14.0788 q^{92} -2.42394 q^{94} -3.99749 q^{95} +5.73965 q^{97} +16.7416 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 34 q + 5 q^{2} + 31 q^{4} + 6 q^{5} + 15 q^{8}+O(q^{10}) $ 34 * q + 5 * q^2 + 31 * q^4 + 6 * q^5 + 15 * q^8	$ 34 q + 5 q^{2} + 31 q^{4} + 6 q^{5} + 15 q^{8} + 4 q^{10} + 6 q^{11} + 2 q^{13} + 12 q^{14} + 21 q^{16} + 4 q^{17} - 34 q^{19} + 20 q^{20} - 8 q^{22} + 26 q^{23} + 32 q^{25} + 29 q^{26} - 4 q^{28} + 14 q^{29} + 2 q^{31} + 35 q^{32} - 18 q^{34} + 50 q^{35} - 10 q^{37} - 5 q^{38} + 17 q^{40} + 18 q^{41} + 6 q^{43} + 6 q^{44} + 18 q^{46} + 34 q^{47} + 28 q^{49} + 41 q^{50} + 10 q^{52} + 40 q^{53} - 8 q^{55} + 76 q^{56} + 4 q^{58} + 62 q^{59} - 2 q^{61} + 50 q^{62} + 11 q^{64} + 32 q^{65} + 20 q^{67} + 28 q^{68} + 22 q^{70} + 52 q^{71} - 8 q^{73} + 10 q^{74} - 31 q^{76} + 36 q^{77} - 12 q^{79} + 92 q^{80} + 10 q^{82} + 82 q^{83} - 4 q^{85} + 40 q^{86} - 16 q^{88} + 58 q^{89} + 100 q^{92} + 5 q^{94} - 6 q^{95} - 6 q^{97} + 45 q^{98}+O(q^{100}) $ 34 * q + 5 * q^2 + 31 * q^4 + 6 * q^5 + 15 * q^8 + 4 * q^10 + 6 * q^11 + 2 * q^13 + 12 * q^14 + 21 * q^16 + 4 * q^17 - 34 * q^19 + 20 * q^20 - 8 * q^22 + 26 * q^23 + 32 * q^25 + 29 * q^26 - 4 * q^28 + 14 * q^29 + 2 * q^31 + 35 * q^32 - 18 * q^34 + 50 * q^35 - 10 * q^37 - 5 * q^38 + 17 * q^40 + 18 * q^41 + 6 * q^43 + 6 * q^44 + 18 * q^46 + 34 * q^47 + 28 * q^49 + 41 * q^50 + 10 * q^52 + 40 * q^53 - 8 * q^55 + 76 * q^56 + 4 * q^58 + 62 * q^59 - 2 * q^61 + 50 * q^62 + 11 * q^64 + 32 * q^65 + 20 * q^67 + 28 * q^68 + 22 * q^70 + 52 * q^71 - 8 * q^73 + 10 * q^74 - 31 * q^76 + 36 * q^77 - 12 * q^79 + 92 * q^80 + 10 * q^82 + 82 * q^83 - 4 * q^85 + 40 * q^86 - 16 * q^88 + 58 * q^89 + 100 * q^92 + 5 * q^94 - 6 * q^95 - 6 * q^97 + 45 * 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 $

$2$	−2.42394	−1.71398	−0.856991	−	0.515331i	$-0.827669\pi$
$2$	−2.42394	−1.71398	−0.856991	+	0.515331i	$0.827669\pi$
$3$	0	0
$3$	0	0
$4$	3.87547	1.93774
$5$	3.99749	1.78773	0.893867	−	0.448333i	$-0.147982\pi$
$5$	3.99749	1.78773	0.893867	+	0.448333i	$0.147982\pi$
$6$	0	0
$7$	−0.305320	−0.115400	−0.0577001	−	0.998334i	$-0.518377\pi$
$7$	−0.305320	−0.115400	−0.0577001	+	0.998334i	$0.518377\pi$
$8$	−4.54602	−1.60726
$9$	0	0
$10$	−9.68967	−3.06414
$11$	−1.33550	−0.402669	−0.201335	−	0.979523i	$-0.564528\pi$
$11$	−1.33550	−0.402669	−0.201335	+	0.979523i	$0.564528\pi$
$12$	0	0
$13$	−2.39980	−0.665584	−0.332792	−	0.943000i	$-0.607991\pi$
$13$	−2.39980	−0.665584	−0.332792	+	0.943000i	$0.607991\pi$
$14$	0.740078	0.197794
$15$	0	0
$16$	3.26833	0.817082
$17$	−2.71547	−0.658598	−0.329299	−	0.944226i	$-0.606812\pi$
$17$	−2.71547	−0.658598	−0.329299	+	0.944226i	$0.606812\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	15.4922	3.46415
$21$	0	0
$22$	3.23718	0.690168
$23$	−3.63280	−0.757492	−0.378746	−	0.925501i	$-0.623645\pi$
$23$	−3.63280	−0.757492	−0.378746	+	0.925501i	$0.623645\pi$
$24$	0	0
$25$	10.9800	2.19599
$26$	5.81696	1.14080
$27$	0	0
$28$	−1.18326	−0.223615
$29$	−3.99602	−0.742042	−0.371021	−	0.928625i	$-0.620992\pi$
$29$	−3.99602	−0.742042	−0.371021	+	0.928625i	$0.620992\pi$
$30$	0	0
$31$	0.552364	0.0992076	0.0496038	−	0.998769i	$-0.484204\pi$
$31$	0.552364	0.0992076	0.0496038	+	0.998769i	$0.484204\pi$
$32$	1.16982	0.206797
$33$	0	0
$34$	6.58212	1.12882
$35$	−1.22052	−0.206305
$36$	0	0
$37$	−0.819242	−0.134683	−0.0673413	−	0.997730i	$-0.521452\pi$
$37$	−0.819242	−0.134683	−0.0673413	+	0.997730i	$0.521452\pi$
$38$	2.42394	0.393214
$39$	0	0
$40$	−18.1727	−2.87335
$41$	9.50876	1.48502	0.742509	−	0.669836i	$-0.233636\pi$
$41$	9.50876	1.48502	0.742509	+	0.669836i	$0.233636\pi$
$42$	0	0
$43$	4.82147	0.735268	0.367634	−	0.929971i	$-0.380168\pi$
$43$	4.82147	0.735268	0.367634	+	0.929971i	$0.380168\pi$
$44$	−5.17570	−0.780267
$45$	0	0
$46$	8.80568	1.29833
$47$	1.00000	0.145865
$47$	1.00000	0.145865
$48$	0	0
$49$	−6.90678	−0.986683
$50$	−26.6147	−3.76389
$51$	0	0
$52$	−9.30034	−1.28973
$53$	10.1106	1.38880	0.694400	−	0.719590i	$-0.255670\pi$
$53$	10.1106	1.38880	0.694400	+	0.719590i	$0.255670\pi$
$54$	0	0
$55$	−5.33867	−0.719866
$56$	1.38799	0.185478
$57$	0	0
$58$	9.68609	1.27185
$59$	0.680548	0.0885998	0.0442999	−	0.999018i	$-0.485894\pi$
$59$	0.680548	0.0885998	0.0442999	+	0.999018i	$0.485894\pi$
$60$	0	0
$61$	−12.7218	−1.62885	−0.814427	−	0.580265i	$-0.802949\pi$
$61$	−12.7218	−1.62885	−0.814427	+	0.580265i	$0.802949\pi$
$62$	−1.33890	−0.170040
$63$	0	0
$64$	−9.37223	−1.17153
$65$	−9.59317	−1.18989
$66$	0	0
$67$	2.58225	0.315472	0.157736	−	0.987481i	$-0.449580\pi$
$67$	2.58225	0.315472	0.157736	+	0.987481i	$0.449580\pi$
$68$	−10.5237	−1.27619
$69$	0	0
$70$	2.95846	0.353603
$71$	4.97858	0.590849	0.295425	−	0.955366i	$-0.404539\pi$
$71$	4.97858	0.590849	0.295425	+	0.955366i	$0.404539\pi$
$72$	0	0
$73$	−14.6674	−1.71669	−0.858345	−	0.513073i	$-0.828507\pi$
$73$	−14.6674	−1.71669	−0.858345	+	0.513073i	$0.828507\pi$
$74$	1.98579	0.230843
$75$	0	0
$76$	−3.87547	−0.444547
$77$	0.407757	0.0464682
$78$	0	0
$79$	−0.138047	−0.0155315	−0.00776573	−	0.999970i	$-0.502472\pi$
$79$	−0.138047	−0.0155315	−0.00776573	+	0.999970i	$0.502472\pi$
$80$	13.0651	1.46072
$81$	0	0
$82$	−23.0486	−2.54529
$83$	7.62417	0.836861	0.418430	−	0.908249i	$-0.362580\pi$
$83$	7.62417	0.836861	0.418430	+	0.908249i	$0.362580\pi$
$84$	0	0
$85$	−10.8551	−1.17740
$86$	−11.6869	−1.26024
$87$	0	0
$88$	6.07123	0.647195
$89$	16.1292	1.70969	0.854847	−	0.518881i	$-0.173651\pi$
$89$	16.1292	1.70969	0.854847	+	0.518881i	$0.173651\pi$
$90$	0	0
$91$	0.732707	0.0768086
$92$	−14.0788	−1.46782
$93$	0	0
$94$	−2.42394	−0.250010
$95$	−3.99749	−0.410134
$96$	0	0
$97$	5.73965	0.582773	0.291387	−	0.956605i	$-0.405883\pi$
$97$	5.73965	0.582773	0.291387	+	0.956605i	$0.405883\pi$
$98$	16.7416	1.69116
$99$	0	0
$100$	42.5525	4.25525
$101$	14.2243	1.41537	0.707685	−	0.706528i	$-0.249740\pi$
$101$	14.2243	1.41537	0.707685	+	0.706528i	$0.249740\pi$
$102$	0	0
$103$	0.656658	0.0647025	0.0323512	−	0.999477i	$-0.489700\pi$
$103$	0.656658	0.0647025	0.0323512	+	0.999477i	$0.489700\pi$
$104$	10.9095	1.06977
$105$	0	0
$106$	−24.5075	−2.38038
$107$	18.3429	1.77328	0.886638	−	0.462464i	$-0.153034\pi$
$107$	18.3429	1.77328	0.886638	+	0.462464i	$0.153034\pi$
$108$	0	0
$109$	−2.74238	−0.262672	−0.131336	−	0.991338i	$-0.541927\pi$
$109$	−2.74238	−0.262672	−0.131336	+	0.991338i	$0.541927\pi$
$110$	12.9406	1.23384
$111$	0	0
$112$	−0.997888	−0.0942915
$113$	−15.7938	−1.48576	−0.742878	−	0.669426i	$-0.766540\pi$
$113$	−15.7938	−1.48576	−0.742878	+	0.669426i	$0.766540\pi$
$114$	0	0
$115$	−14.5221	−1.35419
$116$	−15.4864	−1.43788
$117$	0	0
$118$	−1.64961	−0.151859
$119$	0.829088	0.0760024
$120$	0	0
$121$	−9.21643	−0.837857
$122$	30.8368	2.79183
$123$	0	0
$124$	2.14067	0.192238
$125$	23.9048	2.13811
$126$	0	0
$127$	19.7031	1.74837	0.874183	−	0.485597i	$-0.161398\pi$
$127$	19.7031	1.74837	0.874183	+	0.485597i	$0.161398\pi$
$128$	20.3780	1.80118
$129$	0	0
$130$	23.2532	2.03944
$131$	−15.0917	−1.31857	−0.659283	−	0.751895i	$-0.729140\pi$
$131$	−15.0917	−1.31857	−0.659283	+	0.751895i	$0.729140\pi$
$132$	0	0
$133$	0.305320	0.0264746
$134$	−6.25922	−0.540714
$135$	0	0
$136$	12.3446	1.05854
$137$	3.01014	0.257173	0.128587	−	0.991698i	$-0.458956\pi$
$137$	3.01014	0.257173	0.128587	+	0.991698i	$0.458956\pi$
$138$	0	0
$139$	−21.2354	−1.80116	−0.900582	−	0.434687i	$-0.856859\pi$
$139$	−21.2354	−1.80116	−0.900582	+	0.434687i	$0.856859\pi$
$140$	−4.73008	−0.399764
$141$	0	0
$142$	−12.0678	−1.01270
$143$	3.20494	0.268010
$144$	0	0
$145$	−15.9740	−1.32657
$146$	35.5529	2.94238
$147$	0	0
$148$	−3.17495	−0.260979
$149$	10.6375	0.871459	0.435730	−	0.900078i	$-0.356490\pi$
$149$	10.6375	0.871459	0.435730	+	0.900078i	$0.356490\pi$
$150$	0	0
$151$	23.2803	1.89453	0.947264	−	0.320455i	$-0.103836\pi$
$151$	23.2803	1.89453	0.947264	+	0.320455i	$0.103836\pi$
$152$	4.54602	0.368731
$153$	0	0
$154$	−0.988376	−0.0796456
$155$	2.20807	0.177357
$156$	0	0
$157$	12.1006	0.965734	0.482867	−	0.875694i	$-0.339595\pi$
$157$	12.1006	0.965734	0.482867	+	0.875694i	$0.339595\pi$
$158$	0.334616	0.0266206
$159$	0	0
$160$	4.67635	0.369698
$161$	1.10917	0.0874148
$162$	0	0
$163$	22.0240	1.72505	0.862527	−	0.506011i	$-0.168881\pi$
$163$	22.0240	1.72505	0.862527	+	0.506011i	$0.168881\pi$
$164$	36.8509	2.87757
$165$	0	0
$166$	−18.4805	−1.43436
$167$	−1.14784	−0.0888227	−0.0444113	−	0.999013i	$-0.514141\pi$
$167$	−1.14784	−0.0888227	−0.0444113	+	0.999013i	$0.514141\pi$
$168$	0	0
$169$	−7.24097	−0.556998
$170$	26.3120	2.01804
$171$	0	0
$172$	18.6855	1.42475
$173$	−3.03178	−0.230502	−0.115251	−	0.993336i	$-0.536767\pi$
$173$	−3.03178	−0.230502	−0.115251	+	0.993336i	$0.536767\pi$
$174$	0	0
$175$	−3.35241	−0.253418
$176$	−4.36486	−0.329014
$177$	0	0
$178$	−39.0962	−2.93038
$179$	5.59541	0.418221	0.209110	−	0.977892i	$-0.432943\pi$
$179$	5.59541	0.418221	0.209110	+	0.977892i	$0.432943\pi$
$180$	0	0
$181$	4.14841	0.308349	0.154174	−	0.988044i	$-0.450728\pi$
$181$	4.14841	0.308349	0.154174	+	0.988044i	$0.450728\pi$
$182$	−1.77604	−0.131649
$183$	0	0
$184$	16.5148	1.21749
$185$	−3.27491	−0.240776
$186$	0	0
$187$	3.62652	0.265197
$188$	3.87547	0.282648
$189$	0	0
$190$	9.68967	0.702963
$191$	22.7594	1.64681	0.823404	−	0.567455i	$-0.192072\pi$
$191$	22.7594	1.64681	0.823404	+	0.567455i	$0.192072\pi$
$192$	0	0
$193$	11.8387	0.852165	0.426082	−	0.904684i	$-0.359893\pi$
$193$	11.8387	0.852165	0.426082	+	0.904684i	$0.359893\pi$
$194$	−13.9126	−0.998863
$195$	0	0
$196$	−26.7670	−1.91193
$197$	16.1424	1.15010	0.575049	−	0.818119i	$-0.304983\pi$
$197$	16.1424	1.15010	0.575049	+	0.818119i	$0.304983\pi$
$198$	0	0
$199$	8.81319	0.624750	0.312375	−	0.949959i	$-0.398875\pi$
$199$	8.81319	0.624750	0.312375	+	0.949959i	$0.398875\pi$
$200$	−49.9151	−3.52953
$201$	0	0
$202$	−34.4788	−2.42592
$203$	1.22007	0.0856318
$204$	0	0
$205$	38.0112	2.65482
$206$	−1.59170	−0.110899
$207$	0	0
$208$	−7.84332	−0.543837
$209$	1.33550	0.0923787
$210$	0	0
$211$	−21.7186	−1.49517	−0.747583	−	0.664168i	$-0.768786\pi$
$211$	−21.7186	−1.49517	−0.747583	+	0.664168i	$0.768786\pi$
$212$	39.1834	2.69113
$213$	0	0
$214$	−44.4621	−3.03936
$215$	19.2738	1.31446
$216$	0	0
$217$	−0.168648	−0.0114486
$218$	6.64735	0.450215
$219$	0	0
$220$	−20.6898	−1.39491
$221$	6.51657	0.438352
$222$	0	0
$223$	19.0615	1.27645	0.638227	−	0.769848i	$-0.279668\pi$
$223$	19.0615	1.27645	0.638227	+	0.769848i	$0.279668\pi$
$224$	−0.357170	−0.0238644
$225$	0	0
$226$	38.2832	2.54656
$227$	20.8383	1.38309	0.691543	−	0.722335i	$-0.256931\pi$
$227$	20.8383	1.38309	0.691543	+	0.722335i	$0.256931\pi$
$228$	0	0
$229$	18.7546	1.23934	0.619671	−	0.784862i	$-0.287266\pi$
$229$	18.7546	1.23934	0.619671	+	0.784862i	$0.287266\pi$
$230$	35.2007	2.32106
$231$	0	0
$232$	18.1660	1.19265
$233$	2.37922	0.155868	0.0779341	−	0.996959i	$-0.475168\pi$
$233$	2.37922	0.155868	0.0779341	+	0.996959i	$0.475168\pi$
$234$	0	0
$235$	3.99749	0.260768
$236$	2.63744	0.171683
$237$	0	0
$238$	−2.00966	−0.130267
$239$	−5.59673	−0.362022	−0.181011	−	0.983481i	$-0.557937\pi$
$239$	−5.59673	−0.362022	−0.181011	+	0.983481i	$0.557937\pi$
$240$	0	0
$241$	17.9778	1.15805	0.579026	−	0.815309i	$-0.303433\pi$
$241$	17.9778	1.15805	0.579026	+	0.815309i	$0.303433\pi$
$242$	22.3400	1.43607
$243$	0	0
$244$	−49.3028	−3.15629
$245$	−27.6098	−1.76393
$246$	0	0
$247$	2.39980	0.152695
$248$	−2.51106	−0.159452
$249$	0	0
$250$	−57.9438	−3.66469
$251$	2.43331	0.153589	0.0767946	−	0.997047i	$-0.475531\pi$
$251$	2.43331	0.153589	0.0767946	+	0.997047i	$0.475531\pi$
$252$	0	0
$253$	4.85162	0.305019
$254$	−47.7590	−2.99667
$255$	0	0
$256$	−30.6506	−1.91567
$257$	−0.0678485	−0.00423227	−0.00211614	−	0.999998i	$-0.500674\pi$
$257$	−0.0678485	−0.00423227	−0.00211614	+	0.999998i	$0.500674\pi$
$258$	0	0
$259$	0.250131	0.0155424
$260$	−37.1781	−2.30569
$261$	0	0
$262$	36.5813	2.26000
$263$	−0.227718	−0.0140417	−0.00702083	−	0.999975i	$-0.502235\pi$
$263$	−0.227718	−0.0140417	−0.00702083	+	0.999975i	$0.502235\pi$
$264$	0	0
$265$	40.4171	2.48280
$266$	−0.740078	−0.0453771
$267$	0	0
$268$	10.0074	0.611302
$269$	4.59627	0.280240	0.140120	−	0.990135i	$-0.455251\pi$
$269$	4.59627	0.280240	0.140120	+	0.990135i	$0.455251\pi$
$270$	0	0
$271$	15.6383	0.949959	0.474980	−	0.879997i	$-0.342455\pi$
$271$	15.6383	0.949959	0.474980	+	0.879997i	$0.342455\pi$
$272$	−8.87504	−0.538128
$273$	0	0
$274$	−7.29638	−0.440790
$275$	−14.6638	−0.884258
$276$	0	0
$277$	−11.6530	−0.700162	−0.350081	−	0.936719i	$-0.613846\pi$
$277$	−11.6530	−0.700162	−0.350081	+	0.936719i	$0.613846\pi$
$278$	51.4733	3.08716
$279$	0	0
$280$	5.54849	0.331586
$281$	1.03379	0.0616706	0.0308353	−	0.999524i	$-0.490183\pi$
$281$	1.03379	0.0616706	0.0308353	+	0.999524i	$0.490183\pi$
$282$	0	0
$283$	3.41346	0.202909	0.101455	−	0.994840i	$-0.467650\pi$
$283$	3.41346	0.202909	0.101455	+	0.994840i	$0.467650\pi$
$284$	19.2943	1.14491
$285$	0	0
$286$	−7.76857	−0.459365
$287$	−2.90322	−0.171372
$288$	0	0
$289$	−9.62623	−0.566249
$290$	38.7201	2.27372
$291$	0	0
$292$	−56.8431	−3.32649
$293$	−19.9243	−1.16399	−0.581994	−	0.813193i	$-0.697727\pi$
$293$	−19.9243	−1.16399	−0.581994	+	0.813193i	$0.697727\pi$
$294$	0	0
$295$	2.72049	0.158393
$296$	3.72429	0.216470
$297$	0	0
$298$	−25.7847	−1.49367
$299$	8.71799	0.504174
$300$	0	0
$301$	−1.47209	−0.0848501
$302$	−56.4301	−3.24719
$303$	0	0
$304$	−3.26833	−0.187451
$305$	−50.8552	−2.91196
$306$	0	0
$307$	14.7586	0.842315	0.421158	−	0.906987i	$-0.361624\pi$
$307$	14.7586	0.842315	0.421158	+	0.906987i	$0.361624\pi$
$308$	1.58025	0.0900430
$309$	0	0
$310$	−5.35223	−0.303986
$311$	−0.779877	−0.0442228	−0.0221114	−	0.999756i	$-0.507039\pi$
$311$	−0.779877	−0.0442228	−0.0221114	+	0.999756i	$0.507039\pi$
$312$	0	0
$313$	9.13476	0.516327	0.258164	−	0.966101i	$-0.416883\pi$
$313$	9.13476	0.516327	0.258164	+	0.966101i	$0.416883\pi$
$314$	−29.3311	−1.65525
$315$	0	0
$316$	−0.534996	−0.0300958
$317$	−29.2799	−1.64452	−0.822260	−	0.569112i	$-0.807287\pi$
$317$	−29.2799	−1.64452	−0.822260	+	0.569112i	$0.807287\pi$
$318$	0	0
$319$	5.33669	0.298797
$320$	−37.4654	−2.09438
$321$	0	0
$322$	−2.68856	−0.149827
$323$	2.71547	0.151093
$324$	0	0
$325$	−26.3497	−1.46162
$326$	−53.3848	−2.95671
$327$	0	0
$328$	−43.2270	−2.38681
$329$	−0.305320	−0.0168329
$330$	0	0
$331$	6.27903	0.345127	0.172563	−	0.984998i	$-0.444795\pi$
$331$	6.27903	0.345127	0.172563	+	0.984998i	$0.444795\pi$
$332$	29.5472	1.62161
$333$	0	0
$334$	2.78230	0.152240
$335$	10.3225	0.563981
$336$	0	0
$337$	3.96126	0.215784	0.107892	−	0.994163i	$-0.465590\pi$
$337$	3.96126	0.215784	0.107892	+	0.994163i	$0.465590\pi$
$338$	17.5517	0.954685
$339$	0	0
$340$	−42.0685	−2.28148
$341$	−0.737684	−0.0399479
$342$	0	0
$343$	4.24602	0.229264
$344$	−21.9185	−1.18177
$345$	0	0
$346$	7.34884	0.395076
$347$	11.9896	0.643636	0.321818	−	0.946802i	$-0.395706\pi$
$347$	11.9896	0.643636	0.321818	+	0.946802i	$0.395706\pi$
$348$	0	0
$349$	−25.7414	−1.37790	−0.688952	−	0.724807i	$-0.741929\pi$
$349$	−25.7414	−1.37790	−0.688952	+	0.724807i	$0.741929\pi$
$350$	8.12602	0.434354
$351$	0	0
$352$	−1.56230	−0.0832709
$353$	15.9106	0.846836	0.423418	−	0.905934i	$-0.360830\pi$
$353$	15.9106	0.846836	0.423418	+	0.905934i	$0.360830\pi$
$354$	0	0
$355$	19.9019	1.05628
$356$	62.5083	3.31293
$357$	0	0
$358$	−13.5629	−0.716823
$359$	−16.1001	−0.849733	−0.424867	−	0.905256i	$-0.639679\pi$
$359$	−16.1001	−0.849733	−0.424867	+	0.905256i	$0.639679\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−10.0555	−0.528504
$363$	0	0
$364$	2.83958	0.148835
$365$	−58.6329	−3.06898
$366$	0	0
$367$	14.1790	0.740140	0.370070	−	0.929004i	$-0.379334\pi$
$367$	14.1790	0.740140	0.370070	+	0.929004i	$0.379334\pi$
$368$	−11.8732	−0.618933
$369$	0	0
$370$	7.93818	0.412687
$371$	−3.08698	−0.160268
$372$	0	0
$373$	−5.94455	−0.307797	−0.153899	−	0.988087i	$-0.549183\pi$
$373$	−5.94455	−0.307797	−0.153899	+	0.988087i	$0.549183\pi$
$374$	−8.79045	−0.454543
$375$	0	0
$376$	−4.54602	−0.234443
$377$	9.58963	0.493891
$378$	0	0
$379$	7.34195	0.377131	0.188565	−	0.982061i	$-0.439616\pi$
$379$	7.34195	0.377131	0.188565	+	0.982061i	$0.439616\pi$
$380$	−15.4922	−0.794731
$381$	0	0
$382$	−55.1672	−2.82260
$383$	−6.29519	−0.321669	−0.160835	−	0.986981i	$-0.551419\pi$
$383$	−6.29519	−0.321669	−0.160835	+	0.986981i	$0.551419\pi$
$384$	0	0
$385$	1.63000	0.0830727
$386$	−28.6961	−1.46060
$387$	0	0
$388$	22.2438	1.12926
$389$	−15.7759	−0.799868	−0.399934	−	0.916544i	$-0.630967\pi$
$389$	−15.7759	−0.799868	−0.399934	+	0.916544i	$0.630967\pi$
$390$	0	0
$391$	9.86476	0.498882
$392$	31.3984	1.58586
$393$	0	0
$394$	−39.1281	−1.97125
$395$	−0.551840	−0.0277661
$396$	0	0
$397$	−10.9079	−0.547453	−0.273726	−	0.961808i	$-0.588256\pi$
$397$	−10.9079	−0.547453	−0.273726	+	0.961808i	$0.588256\pi$
$398$	−21.3626	−1.07081
$399$	0	0
$400$	35.8861	1.79430
$401$	5.25092	0.262219	0.131109	−	0.991368i	$-0.458146\pi$
$401$	5.25092	0.262219	0.131109	+	0.991368i	$0.458146\pi$
$402$	0	0
$403$	−1.32556	−0.0660310
$404$	55.1258	2.74261
$405$	0	0
$406$	−2.95736	−0.146771
$407$	1.09410	0.0542325
$408$	0	0
$409$	−4.82841	−0.238750	−0.119375	−	0.992849i	$-0.538089\pi$
$409$	−4.82841	−0.238750	−0.119375	+	0.992849i	$0.538089\pi$
$410$	−92.1367	−4.55031
$411$	0	0
$412$	2.54486	0.125376
$413$	−0.207785	−0.0102244
$414$	0	0
$415$	30.4776	1.49608
$416$	−2.80733	−0.137641
$417$	0	0
$418$	−3.23718	−0.158335
$419$	18.8825	0.922472	0.461236	−	0.887277i	$-0.347406\pi$
$419$	18.8825	0.922472	0.461236	+	0.887277i	$0.347406\pi$
$420$	0	0
$421$	−30.6995	−1.49620	−0.748101	−	0.663585i	$-0.769034\pi$
$421$	−30.6995	−1.49620	−0.748101	+	0.663585i	$0.769034\pi$
$422$	52.6444	2.56269
$423$	0	0
$424$	−45.9631	−2.23216
$425$	−29.8157	−1.44627
$426$	0	0
$427$	3.88421	0.187970
$428$	71.0874	3.43614
$429$	0	0
$430$	−46.7185	−2.25297
$431$	0.0992597	0.00478117	0.00239059	−	0.999997i	$-0.499239\pi$
$431$	0.0992597	0.00478117	0.00239059	+	0.999997i	$0.499239\pi$
$432$	0	0
$433$	−12.3590	−0.593937	−0.296968	−	0.954887i	$-0.595976\pi$
$433$	−12.3590	−0.593937	−0.296968	+	0.954887i	$0.595976\pi$
$434$	0.408792	0.0196227
$435$	0	0
$436$	−10.6280	−0.508989
$437$	3.63280	0.173781
$438$	0	0
$439$	−21.0303	−1.00372	−0.501860	−	0.864949i	$-0.667351\pi$
$439$	−21.0303	−1.00372	−0.501860	+	0.864949i	$0.667351\pi$
$440$	24.2697	1.15701
$441$	0	0
$442$	−15.7958	−0.751328
$443$	20.7240	0.984625	0.492313	−	0.870418i	$-0.336152\pi$
$443$	20.7240	0.984625	0.492313	+	0.870418i	$0.336152\pi$
$444$	0	0
$445$	64.4764	3.05648
$446$	−46.2039	−2.18782
$447$	0	0
$448$	2.86153	0.135195
$449$	28.6136	1.35036	0.675179	−	0.737654i	$-0.264066\pi$
$449$	28.6136	1.35036	0.675179	+	0.737654i	$0.264066\pi$
$450$	0	0
$451$	−12.6990	−0.597971
$452$	−61.2084	−2.87900
$453$	0	0
$454$	−50.5107	−2.37059
$455$	2.92899	0.137313
$456$	0	0
$457$	−20.3494	−0.951907	−0.475953	−	0.879470i	$-0.657897\pi$
$457$	−20.3494	−0.951907	−0.475953	+	0.879470i	$0.657897\pi$
$458$	−45.4601	−2.12421
$459$	0	0
$460$	−56.2800	−2.62407
$461$	−16.2090	−0.754928	−0.377464	−	0.926024i	$-0.623204\pi$
$461$	−16.2090	−0.754928	−0.377464	+	0.926024i	$0.623204\pi$
$462$	0	0
$463$	30.2484	1.40576	0.702881	−	0.711307i	$-0.251897\pi$
$463$	30.2484	1.40576	0.702881	+	0.711307i	$0.251897\pi$
$464$	−13.0603	−0.606309
$465$	0	0
$466$	−5.76709	−0.267155
$467$	−1.14880	−0.0531601	−0.0265800	−	0.999647i	$-0.508462\pi$
$467$	−1.14880	−0.0531601	−0.0265800	+	0.999647i	$0.508462\pi$
$468$	0	0
$469$	−0.788415	−0.0364056
$470$	−9.68967	−0.446951
$471$	0	0
$472$	−3.09379	−0.142403
$473$	−6.43909	−0.296070
$474$	0	0
$475$	−10.9800	−0.503795
$476$	3.21311	0.147272
$477$	0	0
$478$	13.5661	0.620499
$479$	−33.0156	−1.50852	−0.754259	−	0.656576i	$-0.772004\pi$
$479$	−33.0156	−1.50852	−0.754259	+	0.656576i	$0.772004\pi$
$480$	0	0
$481$	1.96601	0.0896425
$482$	−43.5770	−1.98488
$483$	0	0
$484$	−35.7180	−1.62355
$485$	22.9442	1.04184
$486$	0	0
$487$	−19.9836	−0.905543	−0.452771	−	0.891627i	$-0.649565\pi$
$487$	−19.9836	−0.905543	−0.452771	+	0.891627i	$0.649565\pi$
$488$	57.8334	2.61800
$489$	0	0
$490$	66.9244	3.02334
$491$	−23.5537	−1.06296	−0.531482	−	0.847070i	$-0.678365\pi$
$491$	−23.5537	−1.06296	−0.531482	+	0.847070i	$0.678365\pi$
$492$	0	0
$493$	10.8511	0.488707
$494$	−5.81696	−0.261717
$495$	0	0
$496$	1.80531	0.0810607
$497$	−1.52006	−0.0681842
$498$	0	0
$499$	10.5390	0.471790	0.235895	−	0.971779i	$-0.424198\pi$
$499$	10.5390	0.471790	0.235895	+	0.971779i	$0.424198\pi$
$500$	92.6425	4.14310
$501$	0	0
$502$	−5.89819	−0.263249
$503$	4.96685	0.221461	0.110730	−	0.993850i	$-0.464681\pi$
$503$	4.96685	0.221461	0.110730	+	0.993850i	$0.464681\pi$
$504$	0	0
$505$	56.8615	2.53030
$506$	−11.7600	−0.522797
$507$	0	0
$508$	76.3587	3.38787
$509$	13.2728	0.588307	0.294154	−	0.955758i	$-0.404962\pi$
$509$	13.2728	0.588307	0.294154	+	0.955758i	$0.404962\pi$
$510$	0	0
$511$	4.47826	0.198107
$512$	33.5391	1.48223
$513$	0	0
$514$	0.164460	0.00725404
$515$	2.62499	0.115671
$516$	0	0
$517$	−1.33550	−0.0587354
$518$	−0.606302	−0.0266394
$519$	0	0
$520$	43.6108	1.91246
$521$	21.7297	0.951995	0.475998	−	0.879447i	$-0.342087\pi$
$521$	21.7297	0.951995	0.475998	+	0.879447i	$0.342087\pi$
$522$	0	0
$523$	−22.9393	−1.00307	−0.501533	−	0.865138i	$-0.667231\pi$
$523$	−22.9393	−1.00307	−0.501533	+	0.865138i	$0.667231\pi$
$524$	−58.4874	−2.55503
$525$	0	0
$526$	0.551973	0.0240672
$527$	−1.49993	−0.0653379
$528$	0	0
$529$	−9.80275	−0.426206
$530$	−97.9685	−4.25548
$531$	0	0
$532$	1.18326	0.0513008
$533$	−22.8191	−0.988404
$534$	0	0
$535$	73.3257	3.17015
$536$	−11.7390	−0.507047
$537$	0	0
$538$	−11.1411	−0.480326
$539$	9.22403	0.397307
$540$	0	0
$541$	25.8979	1.11344	0.556718	−	0.830702i	$-0.312060\pi$
$541$	25.8979	1.11344	0.556718	+	0.830702i	$0.312060\pi$
$542$	−37.9063	−1.62821
$543$	0	0
$544$	−3.17661	−0.136196
$545$	−10.9626	−0.469588
$546$	0	0
$547$	20.8541	0.891655	0.445827	−	0.895119i	$-0.352909\pi$
$547$	20.8541	0.891655	0.445827	+	0.895119i	$0.352909\pi$
$548$	11.6657	0.498334
$549$	0	0
$550$	35.5440	1.51560
$551$	3.99602	0.170236
$552$	0	0
$553$	0.0421485	0.00179233
$554$	28.2462	1.20007
$555$	0	0
$556$	−82.2972	−3.49018
$557$	43.1045	1.82640	0.913199	−	0.407514i	$-0.133604\pi$
$557$	43.1045	1.82640	0.913199	+	0.407514i	$0.133604\pi$
$558$	0	0
$559$	−11.5706	−0.489382
$560$	−3.98905	−0.168568
$561$	0	0
$562$	−2.50583	−0.105702
$563$	−25.5152	−1.07534	−0.537670	−	0.843156i	$-0.680695\pi$
$563$	−25.5152	−1.07534	−0.537670	+	0.843156i	$0.680695\pi$
$564$	0	0
$565$	−63.1357	−2.65614
$566$	−8.27401	−0.347783
$567$	0	0
$568$	−22.6327	−0.949649
$569$	0.138024	0.00578628	0.00289314	−	0.999996i	$-0.499079\pi$
$569$	0.138024	0.00578628	0.00289314	+	0.999996i	$0.499079\pi$
$570$	0	0
$571$	−25.2335	−1.05599	−0.527994	−	0.849248i	$-0.677056\pi$
$571$	−25.2335	−1.05599	−0.527994	+	0.849248i	$0.677056\pi$
$572$	12.4206	0.519333
$573$	0	0
$574$	7.03722	0.293728
$575$	−39.8880	−1.66344
$576$	0	0
$577$	−28.6288	−1.19183	−0.595916	−	0.803047i	$-0.703211\pi$
$577$	−28.6288	−1.19183	−0.595916	+	0.803047i	$0.703211\pi$
$578$	23.3334	0.970541
$579$	0	0
$580$	−61.9069	−2.57055
$581$	−2.32781	−0.0965740
$582$	0	0
$583$	−13.5028	−0.559227
$584$	66.6783	2.75917
$585$	0	0
$586$	48.2952	1.99506
$587$	17.8837	0.738139	0.369070	−	0.929402i	$-0.379676\pi$
$587$	17.8837	0.738139	0.369070	+	0.929402i	$0.379676\pi$
$588$	0	0
$589$	−0.552364	−0.0227598
$590$	−6.59429	−0.271483
$591$	0	0
$592$	−2.67755	−0.110047
$593$	40.5558	1.66543	0.832714	−	0.553704i	$-0.186786\pi$
$593$	40.5558	1.66543	0.832714	+	0.553704i	$0.186786\pi$
$594$	0	0
$595$	3.31427	0.135872
$596$	41.2254	1.68866
$597$	0	0
$598$	−21.1319	−0.864146
$599$	6.84085	0.279510	0.139755	−	0.990186i	$-0.455369\pi$
$599$	6.84085	0.279510	0.139755	+	0.990186i	$0.455369\pi$
$600$	0	0
$601$	−3.47193	−0.141623	−0.0708115	−	0.997490i	$-0.522559\pi$
$601$	−3.47193	−0.141623	−0.0708115	+	0.997490i	$0.522559\pi$
$602$	3.56826	0.145432
$603$	0	0
$604$	90.2223	3.67109
$605$	−36.8426	−1.49787
$606$	0	0
$607$	0.679654	0.0275863	0.0137932	−	0.999905i	$-0.495609\pi$
$607$	0.679654	0.0275863	0.0137932	+	0.999905i	$0.495609\pi$
$608$	−1.16982	−0.0474425
$609$	0	0
$610$	123.270	4.99104
$611$	−2.39980	−0.0970854
$612$	0	0
$613$	−8.78287	−0.354737	−0.177368	−	0.984145i	$-0.556758\pi$
$613$	−8.78287	−0.354737	−0.177368	+	0.984145i	$0.556758\pi$
$614$	−35.7738	−1.44371
$615$	0	0
$616$	−1.85367	−0.0746865
$617$	21.4616	0.864012	0.432006	−	0.901871i	$-0.357806\pi$
$617$	21.4616	0.864012	0.432006	+	0.901871i	$0.357806\pi$
$618$	0	0
$619$	42.8726	1.72320	0.861598	−	0.507592i	$-0.169464\pi$
$619$	42.8726	1.72320	0.861598	+	0.507592i	$0.169464\pi$
$620$	8.55732	0.343670
$621$	0	0
$622$	1.89037	0.0757970
$623$	−4.92458	−0.197299
$624$	0	0
$625$	40.6596	1.62639
$626$	−22.1421	−0.884976
$627$	0	0
$628$	46.8956	1.87134
$629$	2.22462	0.0887016
$630$	0	0
$631$	−28.6943	−1.14230	−0.571151	−	0.820845i	$-0.693503\pi$
$631$	−28.6943	−1.14230	−0.571151	+	0.820845i	$0.693503\pi$
$632$	0.627563	0.0249631
$633$	0	0
$634$	70.9725	2.81868
$635$	78.7630	3.12561
$636$	0	0
$637$	16.5749	0.656720
$638$	−12.9358	−0.512134
$639$	0	0
$640$	81.4611	3.22003
$641$	−0.713988	−0.0282008	−0.0141004	−	0.999901i	$-0.504488\pi$
$641$	−0.713988	−0.0282008	−0.0141004	+	0.999901i	$0.504488\pi$
$642$	0	0
$643$	29.1069	1.14786	0.573931	−	0.818904i	$-0.305418\pi$
$643$	29.1069	1.14786	0.573931	+	0.818904i	$0.305418\pi$
$644$	4.29855	0.169387
$645$	0	0
$646$	−6.58212	−0.258970
$647$	−36.9118	−1.45115	−0.725577	−	0.688141i	$-0.758427\pi$
$647$	−36.9118	−1.45115	−0.725577	+	0.688141i	$0.758427\pi$
$648$	0	0
$649$	−0.908875	−0.0356764
$650$	63.8699	2.50518
$651$	0	0
$652$	85.3534	3.34270
$653$	41.0210	1.60528	0.802638	−	0.596466i	$-0.203429\pi$
$653$	41.0210	1.60528	0.802638	+	0.596466i	$0.203429\pi$
$654$	0	0
$655$	−60.3289	−2.35725
$656$	31.0777	1.21338
$657$	0	0
$658$	0.740078	0.0288512
$659$	27.2262	1.06058	0.530291	−	0.847816i	$-0.322083\pi$
$659$	27.2262	1.06058	0.530291	+	0.847816i	$0.322083\pi$
$660$	0	0
$661$	−3.21448	−0.125029	−0.0625145	−	0.998044i	$-0.519912\pi$
$661$	−3.21448	−0.125029	−0.0625145	+	0.998044i	$0.519912\pi$
$662$	−15.2200	−0.591541
$663$	0	0
$664$	−34.6596	−1.34505
$665$	1.22052	0.0473296
$666$	0	0
$667$	14.5167	0.562090
$668$	−4.44843	−0.172115
$669$	0	0
$670$	−25.0212	−0.966653
$671$	16.9900	0.655890
$672$	0	0
$673$	−41.0478	−1.58227	−0.791137	−	0.611638i	$-0.790511\pi$
$673$	−41.0478	−1.58227	−0.791137	+	0.611638i	$0.790511\pi$
$674$	−9.60184	−0.369849
$675$	0	0
$676$	−28.0622	−1.07931
$677$	3.49959	0.134500	0.0672500	−	0.997736i	$-0.478577\pi$
$677$	3.49959	0.134500	0.0672500	+	0.997736i	$0.478577\pi$
$678$	0	0
$679$	−1.75243	−0.0672522
$680$	49.3474	1.89238
$681$	0	0
$682$	1.78810	0.0684699
$683$	−1.91261	−0.0731840	−0.0365920	−	0.999330i	$-0.511650\pi$
$683$	−1.91261	−0.0731840	−0.0365920	+	0.999330i	$0.511650\pi$
$684$	0	0
$685$	12.0330	0.459757
$686$	−10.2921	−0.392954
$687$	0	0
$688$	15.7582	0.600774
$689$	−24.2634	−0.924363
$690$	0	0
$691$	29.6083	1.12635	0.563176	−	0.826337i	$-0.309579\pi$
$691$	29.6083	1.12635	0.563176	+	0.826337i	$0.309579\pi$
$692$	−11.7496	−0.446652
$693$	0	0
$694$	−29.0621	−1.10318
$695$	−84.8884	−3.22000
$696$	0	0
$697$	−25.8207	−0.978030
$698$	62.3955	2.36170
$699$	0	0
$700$	−12.9921	−0.491057
$701$	−13.2309	−0.499724	−0.249862	−	0.968281i	$-0.580385\pi$
$701$	−13.2309	−0.499724	−0.249862	+	0.968281i	$0.580385\pi$
$702$	0	0
$703$	0.819242	0.0308983
$704$	12.5166	0.471739
$705$	0	0
$706$	−38.5663	−1.45146
$707$	−4.34297	−0.163334
$708$	0	0
$709$	1.54213	0.0579158	0.0289579	−	0.999581i	$-0.490781\pi$
$709$	1.54213	0.0579158	0.0289579	+	0.999581i	$0.490781\pi$
$710$	−48.2408	−1.81045
$711$	0	0
$712$	−73.3238	−2.74792
$713$	−2.00663	−0.0751489
$714$	0	0
$715$	12.8117	0.479131
$716$	21.6848	0.810401
$717$	0	0
$718$	39.0257	1.45643
$719$	31.3337	1.16855	0.584275	−	0.811556i	$-0.301379\pi$
$719$	31.3337	1.16855	0.584275	+	0.811556i	$0.301379\pi$
$720$	0	0
$721$	−0.200491	−0.00746668
$722$	−2.42394	−0.0902096
$723$	0	0
$724$	16.0770	0.597498
$725$	−43.8761	−1.62952
$726$	0	0
$727$	37.3293	1.38447	0.692233	−	0.721674i	$-0.256627\pi$
$727$	37.3293	1.38447	0.692233	+	0.721674i	$0.256627\pi$
$728$	−3.33090	−0.123451
$729$	0	0
$730$	142.122	5.26019
$731$	−13.0926	−0.484246
$732$	0	0
$733$	−6.63201	−0.244959	−0.122479	−	0.992471i	$-0.539085\pi$
$733$	−6.63201	−0.244959	−0.122479	+	0.992471i	$0.539085\pi$
$734$	−34.3691	−1.26859
$735$	0	0
$736$	−4.24973	−0.156647
$737$	−3.44861	−0.127031
$738$	0	0
$739$	39.0996	1.43830	0.719151	−	0.694853i	$-0.244531\pi$
$739$	39.0996	1.43830	0.719151	+	0.694853i	$0.244531\pi$
$740$	−12.6918	−0.466561
$741$	0	0
$742$	7.48264	0.274696
$743$	−26.9789	−0.989760	−0.494880	−	0.868961i	$-0.664788\pi$
$743$	−26.9789	−0.989760	−0.494880	+	0.868961i	$0.664788\pi$
$744$	0	0
$745$	42.5234	1.55794
$746$	14.4092	0.527559
$747$	0	0
$748$	14.0545	0.513882
$749$	−5.60047	−0.204637
$750$	0	0
$751$	−30.7822	−1.12326	−0.561630	−	0.827389i	$-0.689825\pi$
$751$	−30.7822	−1.12326	−0.561630	+	0.827389i	$0.689825\pi$
$752$	3.26833	0.119184
$753$	0	0
$754$	−23.2447	−0.846520
$755$	93.0630	3.38691
$756$	0	0
$757$	−33.3776	−1.21313	−0.606565	−	0.795034i	$-0.707453\pi$
$757$	−33.3776	−1.21313	−0.606565	+	0.795034i	$0.707453\pi$
$758$	−17.7964	−0.646395
$759$	0	0
$760$	18.1727	0.659193
$761$	−37.2028	−1.34860	−0.674300	−	0.738458i	$-0.735555\pi$
$761$	−37.2028	−1.34860	−0.674300	+	0.738458i	$0.735555\pi$
$762$	0	0
$763$	0.837304	0.0303124
$764$	88.2032	3.19108
$765$	0	0
$766$	15.2591	0.551335
$767$	−1.63318	−0.0589706
$768$	0	0
$769$	−8.36626	−0.301695	−0.150848	−	0.988557i	$-0.548200\pi$
$769$	−8.36626	−0.301695	−0.150848	+	0.988557i	$0.548200\pi$
$770$	−3.95103	−0.142385
$771$	0	0
$772$	45.8803	1.65127
$773$	−38.4456	−1.38279	−0.691396	−	0.722476i	$-0.743004\pi$
$773$	−38.4456	−1.38279	−0.691396	+	0.722476i	$0.743004\pi$
$774$	0	0
$775$	6.06493	0.217859
$776$	−26.0926	−0.936669
$777$	0	0
$778$	38.2397	1.37096
$779$	−9.50876	−0.340687
$780$	0	0
$781$	−6.64891	−0.237917
$782$	−23.9115	−0.855075
$783$	0	0
$784$	−22.5736	−0.806201
$785$	48.3721	1.72648
$786$	0	0
$787$	−33.4570	−1.19261	−0.596306	−	0.802757i	$-0.703365\pi$
$787$	−33.4570	−1.19261	−0.596306	+	0.802757i	$0.703365\pi$
$788$	62.5593	2.22858
$789$	0	0
$790$	1.33763	0.0475906
$791$	4.82217	0.171457
$792$	0	0
$793$	30.5297	1.08414
$794$	26.4401	0.938325
$795$	0	0
$796$	34.1553	1.21060
$797$	21.6700	0.767592	0.383796	−	0.923418i	$-0.374617\pi$
$797$	21.6700	0.767592	0.383796	+	0.923418i	$0.374617\pi$
$798$	0	0
$799$	−2.71547	−0.0960663
$800$	12.8446	0.454125
$801$	0	0
$802$	−12.7279	−0.449438
$803$	19.5884	0.691259
$804$	0	0
$805$	4.43390	0.156274
$806$	3.21308	0.113176
$807$	0	0
$808$	−64.6639	−2.27487
$809$	−37.8952	−1.33232	−0.666161	−	0.745808i	$-0.732064\pi$
$809$	−37.8952	−1.33232	−0.666161	+	0.745808i	$0.732064\pi$
$810$	0	0
$811$	−14.2397	−0.500022	−0.250011	−	0.968243i	$-0.580434\pi$
$811$	−14.2397	−0.500022	−0.250011	+	0.968243i	$0.580434\pi$
$812$	4.72833	0.165932
$813$	0	0
$814$	−2.65203	−0.0929536
$815$	88.0409	3.08394
$816$	0	0
$817$	−4.82147	−0.168682
$818$	11.7038	0.409213
$819$	0	0
$820$	147.311	5.14433
$821$	38.6936	1.35042	0.675208	−	0.737627i	$-0.264054\pi$
$821$	38.6936	1.35042	0.675208	+	0.737627i	$0.264054\pi$
$822$	0	0
$823$	22.5776	0.787006	0.393503	−	0.919323i	$-0.371263\pi$
$823$	22.5776	0.787006	0.393503	+	0.919323i	$0.371263\pi$
$824$	−2.98518	−0.103994
$825$	0	0
$826$	0.503659	0.0175245
$827$	22.8877	0.795885	0.397942	−	0.917410i	$-0.369724\pi$
$827$	22.8877	0.795885	0.397942	+	0.917410i	$0.369724\pi$
$828$	0	0
$829$	44.4016	1.54213	0.771065	−	0.636756i	$-0.219724\pi$
$829$	44.4016	1.54213	0.771065	+	0.636756i	$0.219724\pi$
$830$	−73.8757	−2.56426
$831$	0	0
$832$	22.4914	0.779751
$833$	18.7551	0.649827
$834$	0	0
$835$	−4.58849	−0.158791
$836$	5.17570	0.179005
$837$	0	0
$838$	−45.7701	−1.58110
$839$	−15.8865	−0.548462	−0.274231	−	0.961664i	$-0.588423\pi$
$839$	−15.8865	−0.548462	−0.274231	+	0.961664i	$0.588423\pi$
$840$	0	0
$841$	−13.0319	−0.449374
$842$	74.4136	2.56446
$843$	0	0
$844$	−84.1696	−2.89724
$845$	−28.9457	−0.995764
$846$	0	0
$847$	2.81396	0.0966890
$848$	33.0448	1.13476
$849$	0	0
$850$	72.2714	2.47889
$851$	2.97614	0.102021
$852$	0	0
$853$	19.1400	0.655340	0.327670	−	0.944792i	$-0.393737\pi$
$853$	19.1400	0.655340	0.327670	+	0.944792i	$0.393737\pi$
$854$	−9.41509	−0.322178
$855$	0	0
$856$	−83.3873	−2.85012
$857$	26.4277	0.902754	0.451377	−	0.892333i	$-0.350933\pi$
$857$	26.4277	0.902754	0.451377	+	0.892333i	$0.350933\pi$
$858$	0	0
$859$	−25.0810	−0.855752	−0.427876	−	0.903837i	$-0.640738\pi$
$859$	−25.0810	−0.855752	−0.427876	+	0.903837i	$0.640738\pi$
$860$	74.6951	2.54708
$861$	0	0
$862$	−0.240599	−0.00819484
$863$	4.11395	0.140040	0.0700202	−	0.997546i	$-0.477694\pi$
$863$	4.11395	0.140040	0.0700202	+	0.997546i	$0.477694\pi$
$864$	0	0
$865$	−12.1195	−0.412076
$866$	29.9575	1.01800
$867$	0	0
$868$	−0.653591	−0.0221843
$869$	0.184362	0.00625404
$870$	0	0
$871$	−6.19688	−0.209973
$872$	12.4669	0.422183
$873$	0	0
$874$	−8.80568	−0.297857
$875$	−7.29863	−0.246739
$876$	0	0
$877$	19.4780	0.657725	0.328863	−	0.944378i	$-0.393335\pi$
$877$	19.4780	0.657725	0.328863	+	0.944378i	$0.393335\pi$
$878$	50.9760	1.72036
$879$	0	0
$880$	−17.4485	−0.588189
$881$	29.8775	1.00660	0.503299	−	0.864112i	$-0.332119\pi$
$881$	29.8775	1.00660	0.503299	+	0.864112i	$0.332119\pi$
$882$	0	0
$883$	36.5739	1.23081	0.615406	−	0.788211i	$-0.288992\pi$
$883$	36.5739	1.23081	0.615406	+	0.788211i	$0.288992\pi$
$884$	25.2548	0.849410
$885$	0	0
$886$	−50.2336	−1.68763
$887$	−15.0958	−0.506867	−0.253433	−	0.967353i	$-0.581560\pi$
$887$	−15.0958	−0.506867	−0.253433	+	0.967353i	$0.581560\pi$
$888$	0	0
$889$	−6.01576	−0.201762
$890$	−156.287	−5.23875
$891$	0	0
$892$	73.8724	2.47343
$893$	−1.00000	−0.0334637
$894$	0	0
$895$	22.3676	0.747667
$896$	−6.22184	−0.207857
$897$	0	0
$898$	−69.3575	−2.31449
$899$	−2.20726	−0.0736161
$900$	0	0
$901$	−27.4550	−0.914660
$902$	30.7815	1.02491
$903$	0	0
$904$	71.7990	2.38800
$905$	16.5832	0.551245
$906$	0	0
$907$	14.3763	0.477359	0.238679	−	0.971098i	$-0.423286\pi$
$907$	14.3763	0.477359	0.238679	+	0.971098i	$0.423286\pi$
$908$	80.7582	2.68006
$909$	0	0
$910$	−7.09969	−0.235353
$911$	−46.3502	−1.53565	−0.767825	−	0.640659i	$-0.778661\pi$
$911$	−46.3502	−1.53565	−0.767825	+	0.640659i	$0.778661\pi$
$912$	0	0
$913$	−10.1821	−0.336978
$914$	49.3258	1.63155
$915$	0	0
$916$	72.6831	2.40152
$917$	4.60780	0.152163
$918$	0	0
$919$	−34.2906	−1.13114	−0.565571	−	0.824700i	$-0.691344\pi$
$919$	−34.2906	−1.13114	−0.565571	+	0.824700i	$0.691344\pi$
$920$	66.0178	2.17654
$921$	0	0
$922$	39.2896	1.29393
$923$	−11.9476	−0.393260
$924$	0	0
$925$	−8.99524	−0.295762
$926$	−73.3202	−2.40945
$927$	0	0
$928$	−4.67462	−0.153452
$929$	1.29775	0.0425779	0.0212889	−	0.999773i	$-0.493223\pi$
$929$	1.29775	0.0425779	0.0212889	+	0.999773i	$0.493223\pi$
$930$	0	0
$931$	6.90678	0.226361
$932$	9.22061	0.302031
$933$	0	0
$934$	2.78461	0.0911154
$935$	14.4970	0.474102
$936$	0	0
$937$	21.5529	0.704102	0.352051	−	0.935981i	$-0.385484\pi$
$937$	21.5529	0.704102	0.352051	+	0.935981i	$0.385484\pi$
$938$	1.91107	0.0623986
$939$	0	0
$940$	15.4922	0.505299
$941$	−27.0311	−0.881189	−0.440595	−	0.897706i	$-0.645232\pi$
$941$	−27.0311	−0.881189	−0.440595	+	0.897706i	$0.645232\pi$
$942$	0	0
$943$	−34.5434	−1.12489
$944$	2.22426	0.0723933
$945$	0	0
$946$	15.6080	0.507459
$947$	7.56037	0.245679	0.122840	−	0.992427i	$-0.460800\pi$
$947$	7.56037	0.245679	0.122840	+	0.992427i	$0.460800\pi$
$948$	0	0
$949$	35.1988	1.14260
$950$	26.6147	0.863495
$951$	0	0
$952$	−3.76905	−0.122156
$953$	−20.4512	−0.662478	−0.331239	−	0.943547i	$-0.607467\pi$
$953$	−20.4512	−0.662478	−0.331239	+	0.943547i	$0.607467\pi$
$954$	0	0
$955$	90.9804	2.94406
$956$	−21.6899	−0.701503
$957$	0	0
$958$	80.0276	2.58557
$959$	−0.919056	−0.0296779
$960$	0	0
$961$	−30.6949	−0.990158
$962$	−4.76549	−0.153646
$963$	0	0
$964$	69.6724	2.24400
$965$	47.3249	1.52344
$966$	0	0
$967$	19.0265	0.611852	0.305926	−	0.952055i	$-0.401034\pi$
$967$	19.0265	0.611852	0.305926	+	0.952055i	$0.401034\pi$
$968$	41.8981	1.34666
$969$	0	0
$970$	−55.6153	−1.78570
$971$	−6.67903	−0.214340	−0.107170	−	0.994241i	$-0.534179\pi$
$971$	−6.67903	−0.214340	−0.107170	+	0.994241i	$0.534179\pi$
$972$	0	0
$973$	6.48360	0.207855
$974$	48.4389	1.55208
$975$	0	0
$976$	−41.5789	−1.33091
$977$	−1.84979	−0.0591801	−0.0295901	−	0.999562i	$-0.509420\pi$
$977$	−1.84979	−0.0591801	−0.0295901	+	0.999562i	$0.509420\pi$
$978$	0	0
$979$	−21.5406	−0.688441
$980$	−107.001	−3.41802
$981$	0	0
$982$	57.0927	1.82190
$983$	3.16449	0.100931	0.0504657	−	0.998726i	$-0.483929\pi$
$983$	3.16449	0.100931	0.0504657	+	0.998726i	$0.483929\pi$
$984$	0	0
$985$	64.5291	2.05607
$986$	−26.3023	−0.837635
$987$	0	0
$988$	9.30034	0.295883
$989$	−17.5155	−0.556959
$990$	0	0
$991$	17.5428	0.557267	0.278633	−	0.960398i	$-0.410119\pi$
$991$	17.5428	0.557267	0.278633	+	0.960398i	$0.410119\pi$
$992$	0.646167	0.0205158
$993$	0	0
$994$	3.68454	0.116866
$995$	35.2307	1.11689
$996$	0	0
$997$	−9.70133	−0.307244	−0.153622	−	0.988130i	$-0.549094\pi$
$997$	−9.70133	−0.307244	−0.153622	+	0.988130i	$0.549094\pi$
$998$	−25.5459	−0.808640
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8037.2.a.w.1.3	yes	34
3.2	odd	2		8037.2.a.v.1.32	✓	34

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8037.2.a.v.1.32	✓	34	3.2	odd	2
8037.2.a.w.1.3	yes	34	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)

\(f(q)\)	\(=\)	\(q-2.42394 q^{2} +3.87547 q^{4} +3.99749 q^{5} -0.305320 q^{7} -4.54602 q^{8} +O(q^{10})\)	\(q-2.42394 q^{2} +3.87547 q^{4} +3.99749 q^{5} -0.305320 q^{7} -4.54602 q^{8} -9.68967 q^{10} -1.33550 q^{11} -2.39980 q^{13} +0.740078 q^{14} +3.26833 q^{16} -2.71547 q^{17} -1.00000 q^{19} +15.4922 q^{20} +3.23718 q^{22} -3.63280 q^{23} +10.9800 q^{25} +5.81696 q^{26} -1.18326 q^{28} -3.99602 q^{29} +0.552364 q^{31} +1.16982 q^{32} +6.58212 q^{34} -1.22052 q^{35} -0.819242 q^{37} +2.42394 q^{38} -18.1727 q^{40} +9.50876 q^{41} +4.82147 q^{43} -5.17570 q^{44} +8.80568 q^{46} +1.00000 q^{47} -6.90678 q^{49} -26.6147 q^{50} -9.30034 q^{52} +10.1106 q^{53} -5.33867 q^{55} +1.38799 q^{56} +9.68609 q^{58} +0.680548 q^{59} -12.7218 q^{61} -1.33890 q^{62} -9.37223 q^{64} -9.59317 q^{65} +2.58225 q^{67} -10.5237 q^{68} +2.95846 q^{70} +4.97858 q^{71} -14.6674 q^{73} +1.98579 q^{74} -3.87547 q^{76} +0.407757 q^{77} -0.138047 q^{79} +13.0651 q^{80} -23.0486 q^{82} +7.62417 q^{83} -10.8551 q^{85} -11.6869 q^{86} +6.07123 q^{88} +16.1292 q^{89} +0.732707 q^{91} -14.0788 q^{92} -2.42394 q^{94} -3.99749 q^{95} +5.73965 q^{97} +16.7416 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 34 q + 5 q^{2} + 31 q^{4} + 6 q^{5} + 15 q^{8}+O(q^{10}) \) 34 * q + 5 * q^2 + 31 * q^4 + 6 * q^5 + 15 * q^8	\( 34 q + 5 q^{2} + 31 q^{4} + 6 q^{5} + 15 q^{8} + 4 q^{10} + 6 q^{11} + 2 q^{13} + 12 q^{14} + 21 q^{16} + 4 q^{17} - 34 q^{19} + 20 q^{20} - 8 q^{22} + 26 q^{23} + 32 q^{25} + 29 q^{26} - 4 q^{28} + 14 q^{29} + 2 q^{31} + 35 q^{32} - 18 q^{34} + 50 q^{35} - 10 q^{37} - 5 q^{38} + 17 q^{40} + 18 q^{41} + 6 q^{43} + 6 q^{44} + 18 q^{46} + 34 q^{47} + 28 q^{49} + 41 q^{50} + 10 q^{52} + 40 q^{53} - 8 q^{55} + 76 q^{56} + 4 q^{58} + 62 q^{59} - 2 q^{61} + 50 q^{62} + 11 q^{64} + 32 q^{65} + 20 q^{67} + 28 q^{68} + 22 q^{70} + 52 q^{71} - 8 q^{73} + 10 q^{74} - 31 q^{76} + 36 q^{77} - 12 q^{79} + 92 q^{80} + 10 q^{82} + 82 q^{83} - 4 q^{85} + 40 q^{86} - 16 q^{88} + 58 q^{89} + 100 q^{92} + 5 q^{94} - 6 q^{95} - 6 q^{97} + 45 q^{98}+O(q^{100}) \) 34 * q + 5 * q^2 + 31 * q^4 + 6 * q^5 + 15 * q^8 + 4 * q^10 + 6 * q^11 + 2 * q^13 + 12 * q^14 + 21 * q^16 + 4 * q^17 - 34 * q^19 + 20 * q^20 - 8 * q^22 + 26 * q^23 + 32 * q^25 + 29 * q^26 - 4 * q^28 + 14 * q^29 + 2 * q^31 + 35 * q^32 - 18 * q^34 + 50 * q^35 - 10 * q^37 - 5 * q^38 + 17 * q^40 + 18 * q^41 + 6 * q^43 + 6 * q^44 + 18 * q^46 + 34 * q^47 + 28 * q^49 + 41 * q^50 + 10 * q^52 + 40 * q^53 - 8 * q^55 + 76 * q^56 + 4 * q^58 + 62 * q^59 - 2 * q^61 + 50 * q^62 + 11 * q^64 + 32 * q^65 + 20 * q^67 + 28 * q^68 + 22 * q^70 + 52 * q^71 - 8 * q^73 + 10 * q^74 - 31 * q^76 + 36 * q^77 - 12 * q^79 + 92 * q^80 + 10 * q^82 + 82 * q^83 - 4 * q^85 + 40 * q^86 - 16 * q^88 + 58 * q^89 + 100 * q^92 + 5 * q^94 - 6 * q^95 - 6 * q^97 + 45 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

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Database

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