LMFDB - Embedded newform 8037.2.a.w.1.5

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			1.5
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)

$f(q)$	$=$	$q-2.07430 q^{2} +2.30273 q^{4} -3.91523 q^{5} -3.22805 q^{7} -0.627946 q^{8} +O(q^{10})$	\(q-2.07430 q^{2} +2.30273 q^{4} -3.91523 q^{5} -3.22805 q^{7} -0.627946 q^{8} +8.12136 q^{10} +3.34770 q^{11} +1.97459 q^{13} +6.69595 q^{14} -3.30290 q^{16} +5.04374 q^{17} -1.00000 q^{19} -9.01570 q^{20} -6.94413 q^{22} +5.87098 q^{23} +10.3290 q^{25} -4.09590 q^{26} -7.43331 q^{28} +5.27669 q^{29} +4.59133 q^{31} +8.10711 q^{32} -10.4622 q^{34} +12.6385 q^{35} -4.20733 q^{37} +2.07430 q^{38} +2.45855 q^{40} +5.09816 q^{41} +2.78657 q^{43} +7.70883 q^{44} -12.1782 q^{46} +1.00000 q^{47} +3.42030 q^{49} -21.4255 q^{50} +4.54695 q^{52} +1.88830 q^{53} -13.1070 q^{55} +2.02704 q^{56} -10.9455 q^{58} +6.90091 q^{59} +5.06741 q^{61} -9.52381 q^{62} -10.2108 q^{64} -7.73099 q^{65} +5.63131 q^{67} +11.6144 q^{68} -26.2162 q^{70} +5.18280 q^{71} +5.54970 q^{73} +8.72728 q^{74} -2.30273 q^{76} -10.8065 q^{77} -2.05886 q^{79} +12.9316 q^{80} -10.5751 q^{82} +1.89779 q^{83} -19.7474 q^{85} -5.78019 q^{86} -2.10217 q^{88} +3.76522 q^{89} -6.37408 q^{91} +13.5193 q^{92} -2.07430 q^{94} +3.91523 q^{95} -2.23461 q^{97} -7.09472 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.07430	−1.46675	−0.733376	−	0.679823i	$-0.762057\pi$
$2$	−2.07430	−1.46675	−0.733376	+	0.679823i	$0.762057\pi$
$3$	0	0
$3$	0	0
$4$	2.30273	1.15136
$5$	−3.91523	−1.75094	−0.875472	−	0.483269i	$-0.839449\pi$
$5$	−3.91523	−1.75094	−0.875472	+	0.483269i	$0.839449\pi$
$6$	0	0
$7$	−3.22805	−1.22009	−0.610044	−	0.792368i	$-0.708848\pi$
$7$	−3.22805	−1.22009	−0.610044	+	0.792368i	$0.708848\pi$
$8$	−0.627946	−0.222012
$9$	0	0
$10$	8.12136	2.56820
$11$	3.34770	1.00937	0.504684	−	0.863304i	$-0.331609\pi$
$11$	3.34770	1.00937	0.504684	+	0.863304i	$0.331609\pi$
$12$	0	0
$13$	1.97459	0.547654	0.273827	−	0.961779i	$-0.411710\pi$
$13$	1.97459	0.547654	0.273827	+	0.961779i	$0.411710\pi$
$14$	6.69595	1.78957
$15$	0	0
$16$	−3.30290	−0.825726
$17$	5.04374	1.22329	0.611644	−	0.791133i	$-0.290509\pi$
$17$	5.04374	1.22329	0.611644	+	0.791133i	$0.290509\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−9.01570	−2.01597
$21$	0	0
$22$	−6.94413	−1.48049
$23$	5.87098	1.22418	0.612092	−	0.790787i	$-0.290328\pi$
$23$	5.87098	1.22418	0.612092	+	0.790787i	$0.290328\pi$
$24$	0	0
$25$	10.3290	2.06580
$26$	−4.09590	−0.803273
$27$	0	0
$28$	−7.43331	−1.40476
$29$	5.27669	0.979858	0.489929	−	0.871762i	$-0.337023\pi$
$29$	5.27669	0.979858	0.489929	+	0.871762i	$0.337023\pi$
$30$	0	0
$31$	4.59133	0.824628	0.412314	−	0.911042i	$-0.364721\pi$
$31$	4.59133	0.824628	0.412314	+	0.911042i	$0.364721\pi$
$32$	8.10711	1.43315
$33$	0	0
$34$	−10.4622	−1.79426
$35$	12.6385	2.13630
$36$	0	0
$37$	−4.20733	−0.691681	−0.345841	−	0.938293i	$-0.612406\pi$
$37$	−4.20733	−0.691681	−0.345841	+	0.938293i	$0.612406\pi$
$38$	2.07430	0.336496
$39$	0	0
$40$	2.45855	0.388731
$41$	5.09816	0.796199	0.398100	−	0.917342i	$-0.369670\pi$
$41$	5.09816	0.796199	0.398100	+	0.917342i	$0.369670\pi$
$42$	0	0
$43$	2.78657	0.424949	0.212474	−	0.977167i	$-0.431848\pi$
$43$	2.78657	0.424949	0.212474	+	0.977167i	$0.431848\pi$
$44$	7.70883	1.16215
$45$	0	0
$46$	−12.1782	−1.79557
$47$	1.00000	0.145865
$47$	1.00000	0.145865
$48$	0	0
$49$	3.42030	0.488614
$50$	−21.4255	−3.03002
$51$	0	0
$52$	4.54695	0.630548
$53$	1.88830	0.259378	0.129689	−	0.991555i	$-0.458602\pi$
$53$	1.88830	0.259378	0.129689	+	0.991555i	$0.458602\pi$
$54$	0	0
$55$	−13.1070	−1.76735
$56$	2.02704	0.270874
$57$	0	0
$58$	−10.9455	−1.43721
$59$	6.90091	0.898422	0.449211	−	0.893426i	$-0.351705\pi$
$59$	6.90091	0.898422	0.449211	+	0.893426i	$0.351705\pi$
$60$	0	0
$61$	5.06741	0.648815	0.324408	−	0.945917i	$-0.394835\pi$
$61$	5.06741	0.648815	0.324408	+	0.945917i	$0.394835\pi$
$62$	−9.52381	−1.20952
$63$	0	0
$64$	−10.2108	−1.27635
$65$	−7.73099	−0.958911
$66$	0	0
$67$	5.63131	0.687974	0.343987	−	0.938974i	$-0.388222\pi$
$67$	5.63131	0.687974	0.343987	+	0.938974i	$0.388222\pi$
$68$	11.6144	1.40845
$69$	0	0
$70$	−26.2162	−3.13343
$71$	5.18280	0.615086	0.307543	−	0.951534i	$-0.400493\pi$
$71$	5.18280	0.615086	0.307543	+	0.951534i	$0.400493\pi$
$72$	0	0
$73$	5.54970	0.649543	0.324771	−	0.945793i	$-0.394713\pi$
$73$	5.54970	0.649543	0.324771	+	0.945793i	$0.394713\pi$
$74$	8.72728	1.01453
$75$	0	0
$76$	−2.30273	−0.264141
$77$	−10.8065	−1.23152
$78$	0	0
$79$	−2.05886	−0.231639	−0.115820	−	0.993270i	$-0.536949\pi$
$79$	−2.05886	−0.231639	−0.115820	+	0.993270i	$0.536949\pi$
$80$	12.9316	1.44580
$81$	0	0
$82$	−10.5751	−1.16783
$83$	1.89779	0.208310	0.104155	−	0.994561i	$-0.466786\pi$
$83$	1.89779	0.208310	0.104155	+	0.994561i	$0.466786\pi$
$84$	0	0
$85$	−19.7474	−2.14191
$86$	−5.78019	−0.623294
$87$	0	0
$88$	−2.10217	−0.224092
$89$	3.76522	0.399112	0.199556	−	0.979886i	$-0.436050\pi$
$89$	3.76522	0.399112	0.199556	+	0.979886i	$0.436050\pi$
$90$	0	0
$91$	−6.37408	−0.668186
$92$	13.5193	1.40948
$93$	0	0
$94$	−2.07430	−0.213948
$95$	3.91523	0.401694
$96$	0	0
$97$	−2.23461	−0.226890	−0.113445	−	0.993544i	$-0.536189\pi$
$97$	−2.23461	−0.226890	−0.113445	+	0.993544i	$0.536189\pi$
$98$	−7.09472	−0.716675
$99$	0	0
$100$	23.7849	2.37849
$101$	−10.2733	−1.02223	−0.511113	−	0.859513i	$-0.670767\pi$
$101$	−10.2733	−1.02223	−0.511113	+	0.859513i	$0.670767\pi$
$102$	0	0
$103$	−11.4571	−1.12891	−0.564453	−	0.825465i	$-0.690913\pi$
$103$	−11.4571	−1.12891	−0.564453	+	0.825465i	$0.690913\pi$
$104$	−1.23994	−0.121586
$105$	0	0
$106$	−3.91690	−0.380443
$107$	8.26111	0.798632	0.399316	−	0.916813i	$-0.369248\pi$
$107$	8.26111	0.798632	0.399316	+	0.916813i	$0.369248\pi$
$108$	0	0
$109$	5.26568	0.504360	0.252180	−	0.967680i	$-0.418852\pi$
$109$	5.26568	0.504360	0.252180	+	0.967680i	$0.418852\pi$
$110$	27.1878	2.59226
$111$	0	0
$112$	10.6619	1.00746
$113$	12.5771	1.18316	0.591579	−	0.806247i	$-0.298505\pi$
$113$	12.5771	1.18316	0.591579	+	0.806247i	$0.298505\pi$
$114$	0	0
$115$	−22.9862	−2.14348
$116$	12.1508	1.12817
$117$	0	0
$118$	−14.3146	−1.31776
$119$	−16.2814	−1.49252
$120$	0	0
$121$	0.207062	0.0188238
$122$	−10.5113	−0.951652
$123$	0	0
$124$	10.5726	0.949446
$125$	−20.8643	−1.86616
$126$	0	0
$127$	4.56182	0.404796	0.202398	−	0.979303i	$-0.435127\pi$
$127$	4.56182	0.404796	0.202398	+	0.979303i	$0.435127\pi$
$128$	4.96602	0.438938
$129$	0	0
$130$	16.0364	1.40648
$131$	0.883642	0.0772041	0.0386021	−	0.999255i	$-0.487710\pi$
$131$	0.883642	0.0772041	0.0386021	+	0.999255i	$0.487710\pi$
$132$	0	0
$133$	3.22805	0.279907
$134$	−11.6810	−1.00909
$135$	0	0
$136$	−3.16720	−0.271585
$137$	2.57899	0.220338	0.110169	−	0.993913i	$-0.464861\pi$
$137$	2.57899	0.220338	0.110169	+	0.993913i	$0.464861\pi$
$138$	0	0
$139$	7.59169	0.643919	0.321959	−	0.946753i	$-0.395659\pi$
$139$	7.59169	0.643919	0.321959	+	0.946753i	$0.395659\pi$
$140$	29.1031	2.45966
$141$	0	0
$142$	−10.7507	−0.902178
$143$	6.61034	0.552784
$144$	0	0
$145$	−20.6595	−1.71568
$146$	−11.5117	−0.952718
$147$	0	0
$148$	−9.68834	−0.796376
$149$	−12.8150	−1.04985	−0.524923	−	0.851149i	$-0.675906\pi$
$149$	−12.8150	−1.04985	−0.524923	+	0.851149i	$0.675906\pi$
$150$	0	0
$151$	−13.6404	−1.11004	−0.555019	−	0.831837i	$-0.687289\pi$
$151$	−13.6404	−1.11004	−0.555019	+	0.831837i	$0.687289\pi$
$152$	0.627946	0.0509331
$153$	0	0
$154$	22.4160	1.80633
$155$	−17.9761	−1.44388
$156$	0	0
$157$	21.3066	1.70045	0.850227	−	0.526416i	$-0.176464\pi$
$157$	21.3066	1.70045	0.850227	+	0.526416i	$0.176464\pi$
$158$	4.27069	0.339758
$159$	0	0
$160$	−31.7412	−2.50936
$161$	−18.9518	−1.49361
$162$	0	0
$163$	−20.6036	−1.61380	−0.806898	−	0.590691i	$-0.798855\pi$
$163$	−20.6036	−1.61380	−0.806898	+	0.590691i	$0.798855\pi$
$164$	11.7397	0.916714
$165$	0	0
$166$	−3.93660	−0.305539
$167$	−7.47869	−0.578719	−0.289359	−	0.957221i	$-0.593442\pi$
$167$	−7.47869	−0.578719	−0.289359	+	0.957221i	$0.593442\pi$
$168$	0	0
$169$	−9.10098	−0.700075
$170$	40.9621	3.14165
$171$	0	0
$172$	6.41672	0.489270
$173$	−17.3490	−1.31902	−0.659510	−	0.751696i	$-0.729236\pi$
$173$	−17.3490	−1.31902	−0.659510	+	0.751696i	$0.729236\pi$
$174$	0	0
$175$	−33.3426	−2.52046
$176$	−11.0571	−0.833461
$177$	0	0
$178$	−7.81019	−0.585399
$179$	−22.7509	−1.70048	−0.850241	−	0.526393i	$-0.823544\pi$
$179$	−22.7509	−1.70048	−0.850241	+	0.526393i	$0.823544\pi$
$180$	0	0
$181$	3.69679	0.274780	0.137390	−	0.990517i	$-0.456129\pi$
$181$	3.69679	0.274780	0.137390	+	0.990517i	$0.456129\pi$
$182$	13.2218	0.980063
$183$	0	0
$184$	−3.68666	−0.271784
$185$	16.4727	1.21109
$186$	0	0
$187$	16.8849	1.23475
$188$	2.30273	0.167944
$189$	0	0
$190$	−8.12136	−0.589186
$191$	−0.432710	−0.0313098	−0.0156549	−	0.999877i	$-0.504983\pi$
$191$	−0.432710	−0.0313098	−0.0156549	+	0.999877i	$0.504983\pi$
$192$	0	0
$193$	11.7967	0.849142	0.424571	−	0.905395i	$-0.360425\pi$
$193$	11.7967	0.849142	0.424571	+	0.905395i	$0.360425\pi$
$194$	4.63525	0.332792
$195$	0	0
$196$	7.87600	0.562572
$197$	19.6589	1.40064	0.700318	−	0.713831i	$-0.253042\pi$
$197$	19.6589	1.40064	0.700318	+	0.713831i	$0.253042\pi$
$198$	0	0
$199$	9.89024	0.701101	0.350550	−	0.936544i	$-0.385995\pi$
$199$	9.89024	0.701101	0.350550	+	0.936544i	$0.385995\pi$
$200$	−6.48606	−0.458634
$201$	0	0
$202$	21.3098	1.49935
$203$	−17.0334	−1.19551
$204$	0	0
$205$	−19.9605	−1.39410
$206$	23.7656	1.65583
$207$	0	0
$208$	−6.52189	−0.452212
$209$	−3.34770	−0.231565
$210$	0	0
$211$	−7.79645	−0.536730	−0.268365	−	0.963317i	$-0.586483\pi$
$211$	−7.79645	−0.536730	−0.268365	+	0.963317i	$0.586483\pi$
$212$	4.34823	0.298638
$213$	0	0
$214$	−17.1360	−1.17140
$215$	−10.9101	−0.744061
$216$	0	0
$217$	−14.8210	−1.00612
$218$	−10.9226	−0.739772
$219$	0	0
$220$	−30.1818	−2.03486
$221$	9.95934	0.669938
$222$	0	0
$223$	−21.8165	−1.46094	−0.730471	−	0.682943i	$-0.760700\pi$
$223$	−21.8165	−1.46094	−0.730471	+	0.682943i	$0.760700\pi$
$224$	−26.1701	−1.74857
$225$	0	0
$226$	−26.0888	−1.73540
$227$	7.34332	0.487393	0.243697	−	0.969852i	$-0.421640\pi$
$227$	7.34332	0.487393	0.243697	+	0.969852i	$0.421640\pi$
$228$	0	0
$229$	−9.79291	−0.647134	−0.323567	−	0.946205i	$-0.604882\pi$
$229$	−9.79291	−0.647134	−0.323567	+	0.946205i	$0.604882\pi$
$230$	47.6804	3.14395
$231$	0	0
$232$	−3.31348	−0.217540
$233$	−5.63277	−0.369015	−0.184507	−	0.982831i	$-0.559069\pi$
$233$	−5.63277	−0.369015	−0.184507	+	0.982831i	$0.559069\pi$
$234$	0	0
$235$	−3.91523	−0.255401
$236$	15.8909	1.03441
$237$	0	0
$238$	33.7726	2.18915
$239$	21.2173	1.37243	0.686216	−	0.727397i	$-0.259270\pi$
$239$	21.2173	1.37243	0.686216	+	0.727397i	$0.259270\pi$
$240$	0	0
$241$	6.47713	0.417228	0.208614	−	0.977998i	$-0.433105\pi$
$241$	6.47713	0.417228	0.208614	+	0.977998i	$0.433105\pi$
$242$	−0.429509	−0.0276099
$243$	0	0
$244$	11.6689	0.747022
$245$	−13.3912	−0.855535
$246$	0	0
$247$	−1.97459	−0.125640
$248$	−2.88311	−0.183077
$249$	0	0
$250$	43.2789	2.73719
$251$	3.90545	0.246510	0.123255	−	0.992375i	$-0.460667\pi$
$251$	3.90545	0.246510	0.123255	+	0.992375i	$0.460667\pi$
$252$	0	0
$253$	19.6542	1.23565
$254$	−9.46259	−0.593736
$255$	0	0
$256$	10.1205	0.632534
$257$	−8.61699	−0.537513	−0.268757	−	0.963208i	$-0.586613\pi$
$257$	−8.61699	−0.537513	−0.268757	+	0.963208i	$0.586613\pi$
$258$	0	0
$259$	13.5815	0.843912
$260$	−17.8023	−1.10405
$261$	0	0
$262$	−1.83294	−0.113239
$263$	23.3361	1.43896	0.719482	−	0.694512i	$-0.244380\pi$
$263$	23.3361	1.43896	0.719482	+	0.694512i	$0.244380\pi$
$264$	0	0
$265$	−7.39312	−0.454156
$266$	−6.69595	−0.410555
$267$	0	0
$268$	12.9674	0.792108
$269$	−2.82952	−0.172519	−0.0862593	−	0.996273i	$-0.527491\pi$
$269$	−2.82952	−0.172519	−0.0862593	+	0.996273i	$0.527491\pi$
$270$	0	0
$271$	−21.2266	−1.28942	−0.644711	−	0.764426i	$-0.723022\pi$
$271$	−21.2266	−1.28942	−0.644711	+	0.764426i	$0.723022\pi$
$272$	−16.6590	−1.01010
$273$	0	0
$274$	−5.34961	−0.323181
$275$	34.5784	2.08516
$276$	0	0
$277$	−9.08916	−0.546114	−0.273057	−	0.961998i	$-0.588035\pi$
$277$	−9.08916	−0.546114	−0.273057	+	0.961998i	$0.588035\pi$
$278$	−15.7475	−0.944469
$279$	0	0
$280$	−7.93632	−0.474286
$281$	−30.0278	−1.79131	−0.895654	−	0.444751i	$-0.853292\pi$
$281$	−30.0278	−1.79131	−0.895654	+	0.444751i	$0.853292\pi$
$282$	0	0
$283$	13.7371	0.816586	0.408293	−	0.912851i	$-0.366124\pi$
$283$	13.7371	0.816586	0.408293	+	0.912851i	$0.366124\pi$
$284$	11.9346	0.708187
$285$	0	0
$286$	−13.7118	−0.810798
$287$	−16.4571	−0.971433
$288$	0	0
$289$	8.43934	0.496432
$290$	42.8540	2.51647
$291$	0	0
$292$	12.7794	0.747860
$293$	16.5923	0.969332	0.484666	−	0.874699i	$-0.338941\pi$
$293$	16.5923	0.969332	0.484666	+	0.874699i	$0.338941\pi$
$294$	0	0
$295$	−27.0186	−1.57309
$296$	2.64198	0.153562
$297$	0	0
$298$	26.5822	1.53987
$299$	11.5928	0.670429
$300$	0	0
$301$	−8.99519	−0.518474
$302$	28.2943	1.62815
$303$	0	0
$304$	3.30290	0.189435
$305$	−19.8401	−1.13604
$306$	0	0
$307$	29.8454	1.70336	0.851682	−	0.524058i	$-0.175583\pi$
$307$	29.8454	1.70336	0.851682	+	0.524058i	$0.175583\pi$
$308$	−24.8845	−1.41792
$309$	0	0
$310$	37.2879	2.11781
$311$	24.6276	1.39650	0.698251	−	0.715853i	$-0.253962\pi$
$311$	24.6276	1.39650	0.698251	+	0.715853i	$0.253962\pi$
$312$	0	0
$313$	17.7495	1.00326	0.501630	−	0.865082i	$-0.332734\pi$
$313$	17.7495	1.00326	0.501630	+	0.865082i	$0.332734\pi$
$314$	−44.1964	−2.49415
$315$	0	0
$316$	−4.74098	−0.266701
$317$	−6.72627	−0.377785	−0.188893	−	0.981998i	$-0.560490\pi$
$317$	−6.72627	−0.377785	−0.188893	+	0.981998i	$0.560490\pi$
$318$	0	0
$319$	17.6648	0.989037
$320$	39.9775	2.23481
$321$	0	0
$322$	39.3118	2.19076
$323$	−5.04374	−0.280641
$324$	0	0
$325$	20.3956	1.13134
$326$	42.7380	2.36704
$327$	0	0
$328$	−3.20137	−0.176766
$329$	−3.22805	−0.177968
$330$	0	0
$331$	17.4875	0.961199	0.480599	−	0.876940i	$-0.340419\pi$
$331$	17.4875	0.961199	0.480599	+	0.876940i	$0.340419\pi$
$332$	4.37010	0.239840
$333$	0	0
$334$	15.5131	0.848837
$335$	−22.0479	−1.20460
$336$	0	0
$337$	10.1342	0.552045	0.276023	−	0.961151i	$-0.410984\pi$
$337$	10.1342	0.552045	0.276023	+	0.961151i	$0.410984\pi$
$338$	18.8782	1.02684
$339$	0	0
$340$	−45.4729	−2.46611
$341$	15.3704	0.832353
$342$	0	0
$343$	11.5555	0.623936
$344$	−1.74982	−0.0943438
$345$	0	0
$346$	35.9871	1.93468
$347$	32.0006	1.71788	0.858942	−	0.512072i	$-0.171122\pi$
$347$	32.0006	1.71788	0.858942	+	0.512072i	$0.171122\pi$
$348$	0	0
$349$	7.66656	0.410381	0.205191	−	0.978722i	$-0.434219\pi$
$349$	7.66656	0.410381	0.205191	+	0.978722i	$0.434219\pi$
$350$	69.1625	3.69689
$351$	0	0
$352$	27.1401	1.44657
$353$	−10.4310	−0.555187	−0.277593	−	0.960699i	$-0.589537\pi$
$353$	−10.4310	−0.555187	−0.277593	+	0.960699i	$0.589537\pi$
$354$	0	0
$355$	−20.2919	−1.07698
$356$	8.67026	0.459523
$357$	0	0
$358$	47.1922	2.49419
$359$	−21.3897	−1.12891	−0.564454	−	0.825465i	$-0.690913\pi$
$359$	−21.3897	−1.12891	−0.564454	+	0.825465i	$0.690913\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−7.66825	−0.403034
$363$	0	0
$364$	−14.6778	−0.769324
$365$	−21.7283	−1.13731
$366$	0	0
$367$	14.7115	0.767933	0.383966	−	0.923347i	$-0.374558\pi$
$367$	14.7115	0.767933	0.383966	+	0.923347i	$0.374558\pi$
$368$	−19.3913	−1.01084
$369$	0	0
$370$	−34.1693	−1.77638
$371$	−6.09552	−0.316463
$372$	0	0
$373$	−20.0767	−1.03953	−0.519766	−	0.854309i	$-0.673981\pi$
$373$	−20.0767	−1.03953	−0.519766	+	0.854309i	$0.673981\pi$
$374$	−35.0244	−1.81107
$375$	0	0
$376$	−0.627946	−0.0323838
$377$	10.4193	0.536623
$378$	0	0
$379$	22.6475	1.16332	0.581662	−	0.813431i	$-0.302403\pi$
$379$	22.6475	1.16332	0.581662	+	0.813431i	$0.302403\pi$
$380$	9.01570	0.462496
$381$	0	0
$382$	0.897572	0.0459238
$383$	−7.29079	−0.372542	−0.186271	−	0.982498i	$-0.559640\pi$
$383$	−7.29079	−0.372542	−0.186271	+	0.982498i	$0.559640\pi$
$384$	0	0
$385$	42.3100	2.15632
$386$	−24.4698	−1.24548
$387$	0	0
$388$	−5.14570	−0.261233
$389$	21.5387	1.09205	0.546027	−	0.837768i	$-0.316140\pi$
$389$	21.5387	1.09205	0.546027	+	0.837768i	$0.316140\pi$
$390$	0	0
$391$	29.6117	1.49753
$392$	−2.14776	−0.108478
$393$	0	0
$394$	−40.7784	−2.05439
$395$	8.06089	0.405588
$396$	0	0
$397$	36.9009	1.85200	0.926001	−	0.377520i	$-0.123223\pi$
$397$	36.9009	1.85200	0.926001	+	0.377520i	$0.123223\pi$
$398$	−20.5153	−1.02834
$399$	0	0
$400$	−34.1157	−1.70579
$401$	8.60472	0.429699	0.214850	−	0.976647i	$-0.431074\pi$
$401$	8.60472	0.429699	0.214850	+	0.976647i	$0.431074\pi$
$402$	0	0
$403$	9.06602	0.451611
$404$	−23.6565	−1.17695
$405$	0	0
$406$	35.3325	1.75352
$407$	−14.0849	−0.698161
$408$	0	0
$409$	−6.54121	−0.323442	−0.161721	−	0.986837i	$-0.551704\pi$
$409$	−6.54121	−0.323442	−0.161721	+	0.986837i	$0.551704\pi$
$410$	41.4040	2.04480
$411$	0	0
$412$	−26.3827	−1.29978
$413$	−22.2765	−1.09615
$414$	0	0
$415$	−7.43030	−0.364739
$416$	16.0083	0.784869
$417$	0	0
$418$	6.94413	0.339648
$419$	34.7216	1.69626	0.848130	−	0.529789i	$-0.177729\pi$
$419$	34.7216	1.69626	0.848130	+	0.529789i	$0.177729\pi$
$420$	0	0
$421$	−38.3905	−1.87104	−0.935518	−	0.353278i	$-0.885067\pi$
$421$	−38.3905	−1.87104	−0.935518	+	0.353278i	$0.885067\pi$
$422$	16.1722	0.787250
$423$	0	0
$424$	−1.18575	−0.0575850
$425$	52.0969	2.52707
$426$	0	0
$427$	−16.3578	−0.791611
$428$	19.0231	0.919515
$429$	0	0
$430$	22.6308	1.09135
$431$	16.9297	0.815475	0.407737	−	0.913099i	$-0.366318\pi$
$431$	16.9297	0.815475	0.407737	+	0.913099i	$0.366318\pi$
$432$	0	0
$433$	−6.92566	−0.332826	−0.166413	−	0.986056i	$-0.553219\pi$
$433$	−6.92566	−0.332826	−0.166413	+	0.986056i	$0.553219\pi$
$434$	30.7433	1.47573
$435$	0	0
$436$	12.1254	0.580702
$437$	−5.87098	−0.280847
$438$	0	0
$439$	36.7471	1.75384	0.876922	−	0.480633i	$-0.159593\pi$
$439$	36.7471	1.75384	0.876922	+	0.480633i	$0.159593\pi$
$440$	8.23048	0.392373
$441$	0	0
$442$	−20.6587	−0.982633
$443$	−24.8613	−1.18119	−0.590597	−	0.806967i	$-0.701108\pi$
$443$	−24.8613	−1.18119	−0.590597	+	0.806967i	$0.701108\pi$
$444$	0	0
$445$	−14.7417	−0.698823
$446$	45.2540	2.14284
$447$	0	0
$448$	32.9609	1.55726
$449$	−6.30800	−0.297693	−0.148846	−	0.988860i	$-0.547556\pi$
$449$	−6.30800	−0.297693	−0.148846	+	0.988860i	$0.547556\pi$
$450$	0	0
$451$	17.0671	0.803658
$452$	28.9617	1.36224
$453$	0	0
$454$	−15.2323	−0.714885
$455$	24.9560	1.16996
$456$	0	0
$457$	−30.1931	−1.41237	−0.706187	−	0.708025i	$-0.749586\pi$
$457$	−30.1931	−1.41237	−0.706187	+	0.708025i	$0.749586\pi$
$458$	20.3135	0.949185
$459$	0	0
$460$	−52.9310	−2.46792
$461$	−39.3684	−1.83357	−0.916784	−	0.399383i	$-0.869225\pi$
$461$	−39.3684	−1.83357	−0.916784	+	0.399383i	$0.869225\pi$
$462$	0	0
$463$	−23.1518	−1.07596	−0.537978	−	0.842959i	$-0.680812\pi$
$463$	−23.1518	−1.07596	−0.537978	+	0.842959i	$0.680812\pi$
$464$	−17.4284	−0.809094
$465$	0	0
$466$	11.6841	0.541253
$467$	11.2090	0.518692	0.259346	−	0.965784i	$-0.416493\pi$
$467$	11.2090	0.518692	0.259346	+	0.965784i	$0.416493\pi$
$468$	0	0
$469$	−18.1781	−0.839388
$470$	8.12136	0.374611
$471$	0	0
$472$	−4.33340	−0.199461
$473$	9.32860	0.428929
$474$	0	0
$475$	−10.3290	−0.473928
$476$	−37.4917	−1.71843
$477$	0	0
$478$	−44.0111	−2.01302
$479$	−22.3269	−1.02014	−0.510070	−	0.860133i	$-0.670380\pi$
$479$	−22.3269	−1.02014	−0.510070	+	0.860133i	$0.670380\pi$
$480$	0	0
$481$	−8.30777	−0.378802
$482$	−13.4355	−0.611971
$483$	0	0
$484$	0.476807	0.0216730
$485$	8.74901	0.397272
$486$	0	0
$487$	−31.2665	−1.41682	−0.708410	−	0.705801i	$-0.750587\pi$
$487$	−31.2665	−1.41682	−0.708410	+	0.705801i	$0.750587\pi$
$488$	−3.18206	−0.144045
$489$	0	0
$490$	27.7775	1.25486
$491$	31.8973	1.43950	0.719752	−	0.694232i	$-0.244256\pi$
$491$	31.8973	1.43950	0.719752	+	0.694232i	$0.244256\pi$
$492$	0	0
$493$	26.6143	1.19865
$494$	4.09590	0.184283
$495$	0	0
$496$	−15.1647	−0.680917
$497$	−16.7303	−0.750458
$498$	0	0
$499$	18.0814	0.809434	0.404717	−	0.914442i	$-0.367370\pi$
$499$	18.0814	0.809434	0.404717	+	0.914442i	$0.367370\pi$
$500$	−48.0448	−2.14863
$501$	0	0
$502$	−8.10107	−0.361568
$503$	−42.2256	−1.88274	−0.941372	−	0.337370i	$-0.890463\pi$
$503$	−42.2256	−1.88274	−0.941372	+	0.337370i	$0.890463\pi$
$504$	0	0
$505$	40.2221	1.78986
$506$	−40.7688	−1.81240
$507$	0	0
$508$	10.5046	0.466067
$509$	12.8632	0.570153	0.285076	−	0.958505i	$-0.407981\pi$
$509$	12.8632	0.570153	0.285076	+	0.958505i	$0.407981\pi$
$510$	0	0
$511$	−17.9147	−0.792499
$512$	−30.9251	−1.36671
$513$	0	0
$514$	17.8742	0.788399
$515$	44.8573	1.97665
$516$	0	0
$517$	3.34770	0.147231
$518$	−28.1721	−1.23781
$519$	0	0
$520$	4.85464	0.212890
$521$	−44.1157	−1.93274	−0.966371	−	0.257152i	$-0.917216\pi$
$521$	−44.1157	−1.93274	−0.966371	+	0.257152i	$0.917216\pi$
$522$	0	0
$523$	−25.2475	−1.10400	−0.551998	−	0.833846i	$-0.686134\pi$
$523$	−25.2475	−1.10400	−0.551998	+	0.833846i	$0.686134\pi$
$524$	2.03478	0.0888900
$525$	0	0
$526$	−48.4060	−2.11060
$527$	23.1575	1.00876
$528$	0	0
$529$	11.4684	0.498626
$530$	15.3356	0.666134
$531$	0	0
$532$	7.43331	0.322275
$533$	10.0668	0.436042
$534$	0	0
$535$	−32.3441	−1.39836
$536$	−3.53615	−0.152739
$537$	0	0
$538$	5.86927	0.253042
$539$	11.4501	0.493191
$540$	0	0
$541$	−22.4854	−0.966721	−0.483360	−	0.875421i	$-0.660584\pi$
$541$	−22.4854	−0.966721	−0.483360	+	0.875421i	$0.660584\pi$
$542$	44.0303	1.89126
$543$	0	0
$544$	40.8902	1.75315
$545$	−20.6163	−0.883106
$546$	0	0
$547$	−10.5091	−0.449337	−0.224669	−	0.974435i	$-0.572130\pi$
$547$	−10.5091	−0.449337	−0.224669	+	0.974435i	$0.572130\pi$
$548$	5.93871	0.253689
$549$	0	0
$550$	−71.7260	−3.05841
$551$	−5.27669	−0.224795
$552$	0	0
$553$	6.64609	0.282620
$554$	18.8536	0.801015
$555$	0	0
$556$	17.4816	0.741384
$557$	−18.3153	−0.776044	−0.388022	−	0.921650i	$-0.626841\pi$
$557$	−18.3153	−0.776044	−0.388022	+	0.921650i	$0.626841\pi$
$558$	0	0
$559$	5.50235	0.232725
$560$	−41.7439	−1.76400
$561$	0	0
$562$	62.2867	2.62741
$563$	19.0107	0.801207	0.400603	−	0.916252i	$-0.368801\pi$
$563$	19.0107	0.801207	0.400603	+	0.916252i	$0.368801\pi$
$564$	0	0
$565$	−49.2424	−2.07164
$566$	−28.4949	−1.19773
$567$	0	0
$568$	−3.25452	−0.136557
$569$	34.6276	1.45166	0.725832	−	0.687872i	$-0.241455\pi$
$569$	34.6276	1.45166	0.725832	+	0.687872i	$0.241455\pi$
$570$	0	0
$571$	−1.20255	−0.0503252	−0.0251626	−	0.999683i	$-0.508010\pi$
$571$	−1.20255	−0.0503252	−0.0251626	+	0.999683i	$0.508010\pi$
$572$	15.2218	0.636455
$573$	0	0
$574$	34.1370	1.42485
$575$	60.6414	2.52892
$576$	0	0
$577$	3.00045	0.124910	0.0624552	−	0.998048i	$-0.480107\pi$
$577$	3.00045	0.124910	0.0624552	+	0.998048i	$0.480107\pi$
$578$	−17.5057	−0.728142
$579$	0	0
$580$	−47.5731	−1.97537
$581$	−6.12617	−0.254156
$582$	0	0
$583$	6.32145	0.261808
$584$	−3.48491	−0.144206
$585$	0	0
$586$	−34.4174	−1.42177
$587$	8.91815	0.368091	0.184046	−	0.982918i	$-0.441081\pi$
$587$	8.91815	0.368091	0.184046	+	0.982918i	$0.441081\pi$
$588$	0	0
$589$	−4.59133	−0.189183
$590$	56.0448	2.30733
$591$	0	0
$592$	13.8964	0.571139
$593$	45.3310	1.86152	0.930760	−	0.365630i	$-0.119146\pi$
$593$	45.3310	1.86152	0.930760	+	0.365630i	$0.119146\pi$
$594$	0	0
$595$	63.7456	2.61331
$596$	−29.5095	−1.20875
$597$	0	0
$598$	−24.0470	−0.983353
$599$	2.16775	0.0885718	0.0442859	−	0.999019i	$-0.485899\pi$
$599$	2.16775	0.0885718	0.0442859	+	0.999019i	$0.485899\pi$
$600$	0	0
$601$	41.4016	1.68881	0.844404	−	0.535707i	$-0.179955\pi$
$601$	41.4016	1.68881	0.844404	+	0.535707i	$0.179955\pi$
$602$	18.6587	0.760474
$603$	0	0
$604$	−31.4101	−1.27806
$605$	−0.810695	−0.0329594
$606$	0	0
$607$	10.7966	0.438220	0.219110	−	0.975700i	$-0.429685\pi$
$607$	10.7966	0.438220	0.219110	+	0.975700i	$0.429685\pi$
$608$	−8.10711	−0.328787
$609$	0	0
$610$	41.1543	1.66629
$611$	1.97459	0.0798835
$612$	0	0
$613$	17.1013	0.690716	0.345358	−	0.938471i	$-0.387758\pi$
$613$	17.1013	0.690716	0.345358	+	0.938471i	$0.387758\pi$
$614$	−61.9083	−2.49841
$615$	0	0
$616$	6.78591	0.273412
$617$	13.1209	0.528227	0.264113	−	0.964492i	$-0.414921\pi$
$617$	13.1209	0.528227	0.264113	+	0.964492i	$0.414921\pi$
$618$	0	0
$619$	−7.58198	−0.304746	−0.152373	−	0.988323i	$-0.548691\pi$
$619$	−7.58198	−0.304746	−0.152373	+	0.988323i	$0.548691\pi$
$620$	−41.3941	−1.66243
$621$	0	0
$622$	−51.0850	−2.04832
$623$	−12.1543	−0.486952
$624$	0	0
$625$	30.0434	1.20174
$626$	−36.8178	−1.47154
$627$	0	0
$628$	49.0634	1.95784
$629$	−21.2207	−0.846125
$630$	0	0
$631$	−48.7771	−1.94179	−0.970893	−	0.239514i	$-0.923012\pi$
$631$	−48.7771	−1.94179	−0.970893	+	0.239514i	$0.923012\pi$
$632$	1.29285	0.0514268
$633$	0	0
$634$	13.9523	0.554117
$635$	−17.8606	−0.708775
$636$	0	0
$637$	6.75370	0.267591
$638$	−36.6420	−1.45067
$639$	0	0
$640$	−19.4431	−0.768556
$641$	−30.5026	−1.20478	−0.602391	−	0.798201i	$-0.705785\pi$
$641$	−30.5026	−1.20478	−0.602391	+	0.798201i	$0.705785\pi$
$642$	0	0
$643$	−16.2414	−0.640500	−0.320250	−	0.947333i	$-0.603767\pi$
$643$	−16.2414	−0.640500	−0.320250	+	0.947333i	$0.603767\pi$
$644$	−43.6408	−1.71969
$645$	0	0
$646$	10.4622	0.411631
$647$	−0.410840	−0.0161518	−0.00807589	−	0.999967i	$-0.502571\pi$
$647$	−0.410840	−0.0161518	−0.00807589	+	0.999967i	$0.502571\pi$
$648$	0	0
$649$	23.1021	0.906838
$650$	−42.3066	−1.65940
$651$	0	0
$652$	−47.4444	−1.85806
$653$	32.6819	1.27894	0.639471	−	0.768816i	$-0.279154\pi$
$653$	32.6819	1.27894	0.639471	+	0.768816i	$0.279154\pi$
$654$	0	0
$655$	−3.45966	−0.135180
$656$	−16.8387	−0.657442
$657$	0	0
$658$	6.69595	0.261035
$659$	4.13264	0.160985	0.0804924	−	0.996755i	$-0.474351\pi$
$659$	4.13264	0.160985	0.0804924	+	0.996755i	$0.474351\pi$
$660$	0	0
$661$	8.13273	0.316327	0.158163	−	0.987413i	$-0.449443\pi$
$661$	8.13273	0.316327	0.158163	+	0.987413i	$0.449443\pi$
$662$	−36.2743	−1.40984
$663$	0	0
$664$	−1.19171	−0.0462474
$665$	−12.6385	−0.490102
$666$	0	0
$667$	30.9794	1.19953
$668$	−17.2214	−0.666315
$669$	0	0
$670$	45.7339	1.76685
$671$	16.9641	0.654893
$672$	0	0
$673$	−35.5340	−1.36973	−0.684867	−	0.728668i	$-0.740140\pi$
$673$	−35.5340	−1.36973	−0.684867	+	0.728668i	$0.740140\pi$
$674$	−21.0214	−0.809713
$675$	0	0
$676$	−20.9571	−0.806041
$677$	−1.81280	−0.0696714	−0.0348357	−	0.999393i	$-0.511091\pi$
$677$	−1.81280	−0.0696714	−0.0348357	+	0.999393i	$0.511091\pi$
$678$	0	0
$679$	7.21343	0.276826
$680$	12.4003	0.475530
$681$	0	0
$682$	−31.8828	−1.22086
$683$	18.0644	0.691215	0.345608	−	0.938379i	$-0.387673\pi$
$683$	18.0644	0.691215	0.345608	+	0.938379i	$0.387673\pi$
$684$	0	0
$685$	−10.0973	−0.385800
$686$	−23.9695	−0.915160
$687$	0	0
$688$	−9.20379	−0.350891
$689$	3.72862	0.142049
$690$	0	0
$691$	−43.2508	−1.64534	−0.822669	−	0.568520i	$-0.807516\pi$
$691$	−43.2508	−1.64534	−0.822669	+	0.568520i	$0.807516\pi$
$692$	−39.9500	−1.51867
$693$	0	0
$694$	−66.3790	−2.51971
$695$	−29.7232	−1.12747
$696$	0	0
$697$	25.7138	0.973980
$698$	−15.9027	−0.601928
$699$	0	0
$700$	−76.7788	−2.90196
$701$	19.4892	0.736097	0.368048	−	0.929807i	$-0.380026\pi$
$701$	19.4892	0.736097	0.368048	+	0.929807i	$0.380026\pi$
$702$	0	0
$703$	4.20733	0.158683
$704$	−34.1826	−1.28830
$705$	0	0
$706$	21.6371	0.814322
$707$	33.1626	1.24721
$708$	0	0
$709$	48.6076	1.82550	0.912748	−	0.408523i	$-0.133956\pi$
$709$	48.6076	1.82550	0.912748	+	0.408523i	$0.133956\pi$
$710$	42.0914	1.57966
$711$	0	0
$712$	−2.36435	−0.0886078
$713$	26.9556	1.00950
$714$	0	0
$715$	−25.8810	−0.967894
$716$	−52.3891	−1.95787
$717$	0	0
$718$	44.3688	1.65583
$719$	−28.1831	−1.05105	−0.525526	−	0.850778i	$-0.676131\pi$
$719$	−28.1831	−1.05105	−0.525526	+	0.850778i	$0.676131\pi$
$720$	0	0
$721$	36.9842	1.37736
$722$	−2.07430	−0.0771975
$723$	0	0
$724$	8.51269	0.316371
$725$	54.5030	2.02419
$726$	0	0
$727$	−8.55450	−0.317269	−0.158634	−	0.987337i	$-0.550709\pi$
$727$	−8.55450	−0.317269	−0.158634	+	0.987337i	$0.550709\pi$
$728$	4.00258	0.148345
$729$	0	0
$730$	45.0711	1.66816
$731$	14.0548	0.519834
$732$	0	0
$733$	37.0482	1.36841	0.684203	−	0.729292i	$-0.260150\pi$
$733$	37.0482	1.36841	0.684203	+	0.729292i	$0.260150\pi$
$734$	−30.5160	−1.12637
$735$	0	0
$736$	47.5967	1.75444
$737$	18.8519	0.694419
$738$	0	0
$739$	−18.6690	−0.686751	−0.343376	−	0.939198i	$-0.611570\pi$
$739$	−18.6690	−0.686751	−0.343376	+	0.939198i	$0.611570\pi$
$740$	37.9320	1.39441
$741$	0	0
$742$	12.6439	0.464174
$743$	−29.7326	−1.09078	−0.545391	−	0.838182i	$-0.683619\pi$
$743$	−29.7326	−1.09078	−0.545391	+	0.838182i	$0.683619\pi$
$744$	0	0
$745$	50.1737	1.83822
$746$	41.6451	1.52474
$747$	0	0
$748$	38.8813	1.42164
$749$	−26.6673	−0.974401
$750$	0	0
$751$	−38.7415	−1.41370	−0.706848	−	0.707365i	$-0.749884\pi$
$751$	−38.7415	−1.41370	−0.706848	+	0.707365i	$0.749884\pi$
$752$	−3.30290	−0.120445
$753$	0	0
$754$	−21.6128	−0.787093
$755$	53.4052	1.94361
$756$	0	0
$757$	19.3140	0.701978	0.350989	−	0.936380i	$-0.385845\pi$
$757$	19.3140	0.701978	0.350989	+	0.936380i	$0.385845\pi$
$758$	−46.9777	−1.70631
$759$	0	0
$760$	−2.45855	−0.0891810
$761$	−7.52651	−0.272836	−0.136418	−	0.990651i	$-0.543559\pi$
$761$	−7.52651	−0.272836	−0.136418	+	0.990651i	$0.543559\pi$
$762$	0	0
$763$	−16.9979	−0.615364
$764$	−0.996414	−0.0360490
$765$	0	0
$766$	15.1233	0.546427
$767$	13.6265	0.492024
$768$	0	0
$769$	−4.22000	−0.152177	−0.0760885	−	0.997101i	$-0.524243\pi$
$769$	−4.22000	−0.152177	−0.0760885	+	0.997101i	$0.524243\pi$
$770$	−87.7637	−3.16278
$771$	0	0
$772$	27.1645	0.977671
$773$	27.7154	0.996854	0.498427	−	0.866932i	$-0.333911\pi$
$773$	27.7154	0.996854	0.498427	+	0.866932i	$0.333911\pi$
$774$	0	0
$775$	47.4239	1.70352
$776$	1.40321	0.0503724
$777$	0	0
$778$	−44.6777	−1.60177
$779$	−5.09816	−0.182661
$780$	0	0
$781$	17.3504	0.620848
$782$	−61.4236	−2.19650
$783$	0	0
$784$	−11.2969	−0.403461
$785$	−83.4203	−2.97740
$786$	0	0
$787$	−30.2442	−1.07809	−0.539045	−	0.842277i	$-0.681215\pi$
$787$	−30.2442	−1.07809	−0.539045	+	0.842277i	$0.681215\pi$
$788$	45.2690	1.61264
$789$	0	0
$790$	−16.7207	−0.594897
$791$	−40.5996	−1.44356
$792$	0	0
$793$	10.0061	0.355326
$794$	−76.5436	−2.71643
$795$	0	0
$796$	22.7745	0.807222
$797$	−1.80381	−0.0638941	−0.0319470	−	0.999490i	$-0.510171\pi$
$797$	−1.80381	−0.0638941	−0.0319470	+	0.999490i	$0.510171\pi$
$798$	0	0
$799$	5.04374	0.178435
$800$	83.7384	2.96060
$801$	0	0
$802$	−17.8488	−0.630263
$803$	18.5787	0.655628
$804$	0	0
$805$	74.2006	2.61523
$806$	−18.8057	−0.662401
$807$	0	0
$808$	6.45104	0.226947
$809$	5.46426	0.192113	0.0960566	−	0.995376i	$-0.469377\pi$
$809$	5.46426	0.192113	0.0960566	+	0.995376i	$0.469377\pi$
$810$	0	0
$811$	44.8393	1.57452	0.787261	−	0.616621i	$-0.211499\pi$
$811$	44.8393	1.57452	0.787261	+	0.616621i	$0.211499\pi$
$812$	−39.2233	−1.37647
$813$	0	0
$814$	29.2163	1.02403
$815$	80.6677	2.82566
$816$	0	0
$817$	−2.78657	−0.0974899
$818$	13.5684	0.474409
$819$	0	0
$820$	−45.9635	−1.60512
$821$	−23.3653	−0.815456	−0.407728	−	0.913103i	$-0.633679\pi$
$821$	−23.3653	−0.815456	−0.407728	+	0.913103i	$0.633679\pi$
$822$	0	0
$823$	−0.281868	−0.00982529	−0.00491264	−	0.999988i	$-0.501564\pi$
$823$	−0.281868	−0.00982529	−0.00491264	+	0.999988i	$0.501564\pi$
$824$	7.19446	0.250631
$825$	0	0
$826$	46.2081	1.60779
$827$	28.5445	0.992591	0.496296	−	0.868154i	$-0.334693\pi$
$827$	28.5445	0.992591	0.496296	+	0.868154i	$0.334693\pi$
$828$	0	0
$829$	−13.9468	−0.484393	−0.242196	−	0.970227i	$-0.577868\pi$
$829$	−13.9468	−0.484393	−0.242196	+	0.970227i	$0.577868\pi$
$830$	15.4127	0.534982
$831$	0	0
$832$	−20.1621	−0.698997
$833$	17.2511	0.597715
$834$	0	0
$835$	29.2808	1.01330
$836$	−7.70883	−0.266615
$837$	0	0
$838$	−72.0230	−2.48799
$839$	14.7758	0.510119	0.255059	−	0.966925i	$-0.417905\pi$
$839$	14.7758	0.510119	0.255059	+	0.966925i	$0.417905\pi$
$840$	0	0
$841$	−1.15649	−0.0398791
$842$	79.6334	2.74435
$843$	0	0
$844$	−17.9531	−0.617971
$845$	35.6324	1.22579
$846$	0	0
$847$	−0.668406	−0.0229667
$848$	−6.23687	−0.214175
$849$	0	0
$850$	−108.065	−3.70659
$851$	−24.7012	−0.846745
$852$	0	0
$853$	−15.9220	−0.545158	−0.272579	−	0.962133i	$-0.587877\pi$
$853$	−15.9220	−0.545158	−0.272579	+	0.962133i	$0.587877\pi$
$854$	33.9311	1.16110
$855$	0	0
$856$	−5.18753	−0.177306
$857$	44.8108	1.53071	0.765353	−	0.643610i	$-0.222564\pi$
$857$	44.8108	1.53071	0.765353	+	0.643610i	$0.222564\pi$
$858$	0	0
$859$	−28.3758	−0.968170	−0.484085	−	0.875021i	$-0.660847\pi$
$859$	−28.3758	−0.968170	−0.484085	+	0.875021i	$0.660847\pi$
$860$	−25.1229	−0.856684
$861$	0	0
$862$	−35.1173	−1.19610
$863$	−2.60317	−0.0886128	−0.0443064	−	0.999018i	$-0.514108\pi$
$863$	−2.60317	−0.0886128	−0.0443064	+	0.999018i	$0.514108\pi$
$864$	0	0
$865$	67.9253	2.30953
$866$	14.3659	0.488174
$867$	0	0
$868$	−34.1288	−1.15841
$869$	−6.89242	−0.233809
$870$	0	0
$871$	11.1195	0.376771
$872$	−3.30656	−0.111974
$873$	0	0
$874$	12.1782	0.411933
$875$	67.3510	2.27688
$876$	0	0
$877$	−19.7386	−0.666527	−0.333263	−	0.942834i	$-0.608150\pi$
$877$	−19.7386	−0.666527	−0.333263	+	0.942834i	$0.608150\pi$
$878$	−76.2246	−2.57245
$879$	0	0
$880$	43.2911	1.45934
$881$	−3.54780	−0.119528	−0.0597642	−	0.998213i	$-0.519035\pi$
$881$	−3.54780	−0.119528	−0.0597642	+	0.998213i	$0.519035\pi$
$882$	0	0
$883$	23.7490	0.799217	0.399609	−	0.916686i	$-0.369146\pi$
$883$	23.7490	0.799217	0.399609	+	0.916686i	$0.369146\pi$
$884$	22.9336	0.771342
$885$	0	0
$886$	51.5697	1.73252
$887$	38.8059	1.30297	0.651487	−	0.758660i	$-0.274146\pi$
$887$	38.8059	1.30297	0.651487	+	0.758660i	$0.274146\pi$
$888$	0	0
$889$	−14.7258	−0.493887
$890$	30.5787	1.02500
$891$	0	0
$892$	−50.2375	−1.68208
$893$	−1.00000	−0.0334637
$894$	0	0
$895$	89.0750	2.97745
$896$	−16.0305	−0.535543
$897$	0	0
$898$	13.0847	0.436642
$899$	24.2271	0.808018
$900$	0	0
$901$	9.52409	0.317293
$902$	−35.4023	−1.17877
$903$	0	0
$904$	−7.89775	−0.262675
$905$	−14.4738	−0.481124
$906$	0	0
$907$	26.8852	0.892708	0.446354	−	0.894857i	$-0.352722\pi$
$907$	26.8852	0.892708	0.446354	+	0.894857i	$0.352722\pi$
$908$	16.9096	0.561166
$909$	0	0
$910$	−51.7663	−1.71603
$911$	37.9886	1.25862	0.629310	−	0.777155i	$-0.283338\pi$
$911$	37.9886	1.25862	0.629310	+	0.777155i	$0.283338\pi$
$912$	0	0
$913$	6.35323	0.210261
$914$	62.6296	2.07160
$915$	0	0
$916$	−22.5504	−0.745086
$917$	−2.85244	−0.0941958
$918$	0	0
$919$	−25.1923	−0.831016	−0.415508	−	0.909589i	$-0.636396\pi$
$919$	−25.1923	−0.831016	−0.415508	+	0.909589i	$0.636396\pi$
$920$	14.4341	0.475878
$921$	0	0
$922$	81.6619	2.68939
$923$	10.2339	0.336854
$924$	0	0
$925$	−43.4576	−1.42888
$926$	48.0238	1.57816
$927$	0	0
$928$	42.7787	1.40428
$929$	49.0836	1.61038	0.805190	−	0.593017i	$-0.202063\pi$
$929$	49.0836	1.61038	0.805190	+	0.593017i	$0.202063\pi$
$930$	0	0
$931$	−3.42030	−0.112096
$932$	−12.9707	−0.424870
$933$	0	0
$934$	−23.2509	−0.760793
$935$	−66.1083	−2.16197
$936$	0	0
$937$	49.2337	1.60839	0.804197	−	0.594363i	$-0.202596\pi$
$937$	49.2337	1.60839	0.804197	+	0.594363i	$0.202596\pi$
$938$	37.7069	1.23117
$939$	0	0
$940$	−9.01570	−0.294060
$941$	−31.6852	−1.03291	−0.516454	−	0.856315i	$-0.672748\pi$
$941$	−31.6852	−1.03291	−0.516454	+	0.856315i	$0.672748\pi$
$942$	0	0
$943$	29.9312	0.974694
$944$	−22.7930	−0.741850
$945$	0	0
$946$	−19.3503	−0.629133
$947$	−24.3822	−0.792317	−0.396158	−	0.918182i	$-0.629657\pi$
$947$	−24.3822	−0.792317	−0.396158	+	0.918182i	$0.629657\pi$
$948$	0	0
$949$	10.9584	0.355725
$950$	21.4255	0.695135
$951$	0	0
$952$	10.2239	0.331357
$953$	39.6041	1.28290	0.641452	−	0.767164i	$-0.278333\pi$
$953$	39.6041	1.28290	0.641452	+	0.767164i	$0.278333\pi$
$954$	0	0
$955$	1.69416	0.0548217
$956$	48.8576	1.58017
$957$	0	0
$958$	46.3126	1.49629
$959$	−8.32511	−0.268832
$960$	0	0
$961$	−9.91966	−0.319989
$962$	17.2328	0.555609
$963$	0	0
$964$	14.9150	0.480381
$965$	−46.1866	−1.48680
$966$	0	0
$967$	29.1134	0.936224	0.468112	−	0.883669i	$-0.344934\pi$
$967$	29.1134	0.936224	0.468112	+	0.883669i	$0.344934\pi$
$968$	−0.130024	−0.00417912
$969$	0	0
$970$	−18.1481	−0.582700
$971$	−9.77900	−0.313823	−0.156912	−	0.987613i	$-0.550154\pi$
$971$	−9.77900	−0.313823	−0.156912	+	0.987613i	$0.550154\pi$
$972$	0	0
$973$	−24.5063	−0.785637
$974$	64.8561	2.07812
$975$	0	0
$976$	−16.7372	−0.535744
$977$	−0.819395	−0.0262148	−0.0131074	−	0.999914i	$-0.504172\pi$
$977$	−0.819395	−0.0262148	−0.0131074	+	0.999914i	$0.504172\pi$
$978$	0	0
$979$	12.6048	0.402851
$980$	−30.8364	−0.985031
$981$	0	0
$982$	−66.1645	−2.11140
$983$	−62.0512	−1.97912	−0.989562	−	0.144105i	$-0.953970\pi$
$983$	−62.0512	−1.97912	−0.989562	+	0.144105i	$0.953970\pi$
$984$	0	0
$985$	−76.9689	−2.45243
$986$	−55.2061	−1.75812
$987$	0	0
$988$	−4.54695	−0.144658
$989$	16.3599	0.520215
$990$	0	0
$991$	−29.7352	−0.944571	−0.472286	−	0.881446i	$-0.656571\pi$
$991$	−29.7352	−0.944571	−0.472286	+	0.881446i	$0.656571\pi$
$992$	37.2224	1.18181
$993$	0	0
$994$	34.7038	1.10074
$995$	−38.7226	−1.22759
$996$	0	0
$997$	−13.7366	−0.435041	−0.217521	−	0.976056i	$-0.569797\pi$
$997$	−13.7366	−0.435041	−0.217521	+	0.976056i	$0.569797\pi$
$998$	−37.5062	−1.18724
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8037.2.a.w.1.5	yes	34
3.2	odd	2		8037.2.a.v.1.30	✓	34

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8037.2.a.v.1.30	✓	34	3.2	odd	2
8037.2.a.w.1.5	yes	34	1.1	even	1	trivial

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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)

\(f(q)\)	\(=\)	\(q-2.07430 q^{2} +2.30273 q^{4} -3.91523 q^{5} -3.22805 q^{7} -0.627946 q^{8} +O(q^{10})\)	\(q-2.07430 q^{2} +2.30273 q^{4} -3.91523 q^{5} -3.22805 q^{7} -0.627946 q^{8} +8.12136 q^{10} +3.34770 q^{11} +1.97459 q^{13} +6.69595 q^{14} -3.30290 q^{16} +5.04374 q^{17} -1.00000 q^{19} -9.01570 q^{20} -6.94413 q^{22} +5.87098 q^{23} +10.3290 q^{25} -4.09590 q^{26} -7.43331 q^{28} +5.27669 q^{29} +4.59133 q^{31} +8.10711 q^{32} -10.4622 q^{34} +12.6385 q^{35} -4.20733 q^{37} +2.07430 q^{38} +2.45855 q^{40} +5.09816 q^{41} +2.78657 q^{43} +7.70883 q^{44} -12.1782 q^{46} +1.00000 q^{47} +3.42030 q^{49} -21.4255 q^{50} +4.54695 q^{52} +1.88830 q^{53} -13.1070 q^{55} +2.02704 q^{56} -10.9455 q^{58} +6.90091 q^{59} +5.06741 q^{61} -9.52381 q^{62} -10.2108 q^{64} -7.73099 q^{65} +5.63131 q^{67} +11.6144 q^{68} -26.2162 q^{70} +5.18280 q^{71} +5.54970 q^{73} +8.72728 q^{74} -2.30273 q^{76} -10.8065 q^{77} -2.05886 q^{79} +12.9316 q^{80} -10.5751 q^{82} +1.89779 q^{83} -19.7474 q^{85} -5.78019 q^{86} -2.10217 q^{88} +3.76522 q^{89} -6.37408 q^{91} +13.5193 q^{92} -2.07430 q^{94} +3.91523 q^{95} -2.23461 q^{97} -7.09472 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 34 q + 5 q^{2} + 31 q^{4} + 6 q^{5} + 15 q^{8}+O(q^{10}) \) 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

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Database

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