LMFDB - Embedded newform 8037.2.a.j.1.6

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:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 2x^{6} - 7x^{5} + 16x^{4} + x^{3} - 13x^{2} + 2x + 1 $ x^7 - 2x^6 - 7x^5 + 16x^4 + x^3 - 13x^2 + 2*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2679)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$1.56422$ 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)

$f(q)$	$=$	$q+1.56422 q^{2} +0.446772 q^{4} -0.791846 q^{5} -0.0750818 q^{7} -2.42958 q^{8} +O(q^{10})$	\(q+1.56422 q^{2} +0.446772 q^{4} -0.791846 q^{5} -0.0750818 q^{7} -2.42958 q^{8} -1.23862 q^{10} -2.59597 q^{11} +2.80412 q^{13} -0.117444 q^{14} -4.69394 q^{16} +4.74232 q^{17} +1.00000 q^{19} -0.353774 q^{20} -4.06066 q^{22} +2.22663 q^{23} -4.37298 q^{25} +4.38626 q^{26} -0.0335444 q^{28} +7.42999 q^{29} -4.23480 q^{31} -2.48317 q^{32} +7.41801 q^{34} +0.0594532 q^{35} -2.97148 q^{37} +1.56422 q^{38} +1.92386 q^{40} -10.0802 q^{41} +7.63595 q^{43} -1.15981 q^{44} +3.48293 q^{46} -1.00000 q^{47} -6.99436 q^{49} -6.84029 q^{50} +1.25280 q^{52} -4.24470 q^{53} +2.05561 q^{55} +0.182417 q^{56} +11.6221 q^{58} +7.03939 q^{59} +7.22754 q^{61} -6.62415 q^{62} +5.50367 q^{64} -2.22043 q^{65} -3.35873 q^{67} +2.11873 q^{68} +0.0929976 q^{70} -2.75757 q^{71} -8.10896 q^{73} -4.64803 q^{74} +0.446772 q^{76} +0.194910 q^{77} -15.8710 q^{79} +3.71687 q^{80} -15.7676 q^{82} +2.95130 q^{83} -3.75518 q^{85} +11.9443 q^{86} +6.30713 q^{88} -3.70208 q^{89} -0.210539 q^{91} +0.994797 q^{92} -1.56422 q^{94} -0.791846 q^{95} +4.88638 q^{97} -10.9407 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 2 q^{2} + 4 q^{4} - 6 q^{5} - 3 q^{7} - 6 q^{8}+O(q^{10}) $ 7 * q + 2 * q^2 + 4 * q^4 - 6 * q^5 - 3 * q^7 - 6 * q^8	$ 7 q + 2 q^{2} + 4 q^{4} - 6 q^{5} - 3 q^{7} - 6 q^{8} - 10 q^{10} + 9 q^{11} - 8 q^{13} + 9 q^{14} + 14 q^{16} - 7 q^{17} + 7 q^{19} + 4 q^{20} - 7 q^{22} + 7 q^{23} - 9 q^{25} - q^{26} - 12 q^{28} + 12 q^{29} - 13 q^{31} - 15 q^{32} + 2 q^{34} - 4 q^{35} - 10 q^{37} + 2 q^{38} - 26 q^{40} + 8 q^{41} + 12 q^{43} + 14 q^{44} - 7 q^{46} - 7 q^{47} - 16 q^{49} + 12 q^{50} - 6 q^{52} + 17 q^{53} + 7 q^{55} + 38 q^{56} - 12 q^{58} - 12 q^{59} - 23 q^{61} - 23 q^{62} + 12 q^{64} + 13 q^{65} + q^{67} - 15 q^{68} + 8 q^{70} - 29 q^{73} - 5 q^{74} + 4 q^{76} - 4 q^{77} - 3 q^{79} - 15 q^{80} - 7 q^{82} - 19 q^{83} - 5 q^{85} + 23 q^{86} - 33 q^{88} + 14 q^{89} - 3 q^{91} + 5 q^{92} - 2 q^{94} - 6 q^{95} - 17 q^{97} - 38 q^{98}+O(q^{100}) $ 7 * q + 2 * q^2 + 4 * q^4 - 6 * q^5 - 3 * q^7 - 6 * q^8 - 10 * q^10 + 9 * q^11 - 8 * q^13 + 9 * q^14 + 14 * q^16 - 7 * q^17 + 7 * q^19 + 4 * q^20 - 7 * q^22 + 7 * q^23 - 9 * q^25 - q^26 - 12 * q^28 + 12 * q^29 - 13 * q^31 - 15 * q^32 + 2 * q^34 - 4 * q^35 - 10 * q^37 + 2 * q^38 - 26 * q^40 + 8 * q^41 + 12 * q^43 + 14 * q^44 - 7 * q^46 - 7 * q^47 - 16 * q^49 + 12 * q^50 - 6 * q^52 + 17 * q^53 + 7 * q^55 + 38 * q^56 - 12 * q^58 - 12 * q^59 - 23 * q^61 - 23 * q^62 + 12 * q^64 + 13 * q^65 + q^67 - 15 * q^68 + 8 * q^70 - 29 * q^73 - 5 * q^74 + 4 * q^76 - 4 * q^77 - 3 * q^79 - 15 * q^80 - 7 * q^82 - 19 * q^83 - 5 * q^85 + 23 * q^86 - 33 * q^88 + 14 * q^89 - 3 * q^91 + 5 * q^92 - 2 * q^94 - 6 * q^95 - 17 * q^97 - 38 * 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 $

$2$	1.56422	1.10607	0.553034	−	0.833159i	$-0.313470\pi$
$2$	1.56422	1.10607	0.553034	+	0.833159i	$0.313470\pi$
$3$	0	0
$3$	0	0
$4$	0.446772	0.223386
$5$	−0.791846	−0.354124	−0.177062	−	0.984200i	$-0.556659\pi$
$5$	−0.791846	−0.354124	−0.177062	+	0.984200i	$0.556659\pi$
$6$	0	0
$7$	−0.0750818	−0.0283782	−0.0141891	−	0.999899i	$-0.504517\pi$
$7$	−0.0750818	−0.0283782	−0.0141891	+	0.999899i	$0.504517\pi$
$8$	−2.42958	−0.858988
$9$	0	0
$10$	−1.23862	−0.391685
$11$	−2.59597	−0.782714	−0.391357	−	0.920239i	$-0.627994\pi$
$11$	−2.59597	−0.782714	−0.391357	+	0.920239i	$0.627994\pi$
$12$	0	0
$13$	2.80412	0.777724	0.388862	−	0.921296i	$-0.372868\pi$
$13$	2.80412	0.777724	0.388862	+	0.921296i	$0.372868\pi$
$14$	−0.117444	−0.0313883
$15$	0	0
$16$	−4.69394	−1.17348
$17$	4.74232	1.15018	0.575090	−	0.818090i	$-0.304967\pi$
$17$	4.74232	1.15018	0.575090	+	0.818090i	$0.304967\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−0.353774	−0.0791064
$21$	0	0
$22$	−4.06066	−0.865735
$23$	2.22663	0.464285	0.232143	−	0.972682i	$-0.425426\pi$
$23$	2.22663	0.464285	0.232143	+	0.972682i	$0.425426\pi$
$24$	0	0
$25$	−4.37298	−0.874596
$26$	4.38626	0.860216
$27$	0	0
$28$	−0.0335444	−0.00633930
$29$	7.42999	1.37972	0.689858	−	0.723945i	$-0.257673\pi$
$29$	7.42999	1.37972	0.689858	+	0.723945i	$0.257673\pi$
$30$	0	0
$31$	−4.23480	−0.760593	−0.380296	−	0.924865i	$-0.624178\pi$
$31$	−4.23480	−0.760593	−0.380296	+	0.924865i	$0.624178\pi$
$32$	−2.48317	−0.438966
$33$	0	0
$34$	7.41801	1.27218
$35$	0.0594532	0.0100494
$36$	0	0
$37$	−2.97148	−0.488508	−0.244254	−	0.969711i	$-0.578543\pi$
$37$	−2.97148	−0.488508	−0.244254	+	0.969711i	$0.578543\pi$
$38$	1.56422	0.253749
$39$	0	0
$40$	1.92386	0.304188
$41$	−10.0802	−1.57426	−0.787131	−	0.616786i	$-0.788434\pi$
$41$	−10.0802	−1.57426	−0.787131	+	0.616786i	$0.788434\pi$
$42$	0	0
$43$	7.63595	1.16447	0.582236	−	0.813020i	$-0.302178\pi$
$43$	7.63595	1.16447	0.582236	+	0.813020i	$0.302178\pi$
$44$	−1.15981	−0.174847
$45$	0	0
$46$	3.48293	0.513531
$47$	−1.00000	−0.145865
$47$	−1.00000	−0.145865
$48$	0	0
$49$	−6.99436	−0.999195
$50$	−6.84029	−0.967363
$51$	0	0
$52$	1.25280	0.173733
$53$	−4.24470	−0.583054	−0.291527	−	0.956563i	$-0.594163\pi$
$53$	−4.24470	−0.583054	−0.291527	+	0.956563i	$0.594163\pi$
$54$	0	0
$55$	2.05561	0.277178
$56$	0.182417	0.0243766
$57$	0	0
$58$	11.6221	1.52606
$59$	7.03939	0.916451	0.458225	−	0.888836i	$-0.348485\pi$
$59$	7.03939	0.916451	0.458225	+	0.888836i	$0.348485\pi$
$60$	0	0
$61$	7.22754	0.925392	0.462696	−	0.886517i	$-0.346882\pi$
$61$	7.22754	0.925392	0.462696	+	0.886517i	$0.346882\pi$
$62$	−6.62415	−0.841267
$63$	0	0
$64$	5.50367	0.687959
$65$	−2.22043	−0.275411
$66$	0	0
$67$	−3.35873	−0.410334	−0.205167	−	0.978727i	$-0.565774\pi$
$67$	−3.35873	−0.410334	−0.205167	+	0.978727i	$0.565774\pi$
$68$	2.11873	0.256934
$69$	0	0
$70$	0.0929976	0.0111153
$71$	−2.75757	−0.327263	−0.163631	−	0.986522i	$-0.552321\pi$
$71$	−2.75757	−0.327263	−0.163631	+	0.986522i	$0.552321\pi$
$72$	0	0
$73$	−8.10896	−0.949082	−0.474541	−	0.880233i	$-0.657386\pi$
$73$	−8.10896	−0.949082	−0.474541	+	0.880233i	$0.657386\pi$
$74$	−4.64803	−0.540322
$75$	0	0
$76$	0.446772	0.0512483
$77$	0.194910	0.0222121
$78$	0	0
$79$	−15.8710	−1.78563	−0.892813	−	0.450428i	$-0.851271\pi$
$79$	−15.8710	−1.78563	−0.892813	+	0.450428i	$0.851271\pi$
$80$	3.71687	0.415559
$81$	0	0
$82$	−15.7676	−1.74124
$83$	2.95130	0.323947	0.161973	−	0.986795i	$-0.448214\pi$
$83$	2.95130	0.323947	0.161973	+	0.986795i	$0.448214\pi$
$84$	0	0
$85$	−3.75518	−0.407307
$86$	11.9443	1.28798
$87$	0	0
$88$	6.30713	0.672342
$89$	−3.70208	−0.392419	−0.196210	−	0.980562i	$-0.562863\pi$
$89$	−3.70208	−0.392419	−0.196210	+	0.980562i	$0.562863\pi$
$90$	0	0
$91$	−0.210539	−0.0220704
$92$	0.994797	0.103715
$93$	0	0
$94$	−1.56422	−0.161337
$95$	−0.791846	−0.0812416
$96$	0	0
$97$	4.88638	0.496137	0.248068	−	0.968743i	$-0.420204\pi$
$97$	4.88638	0.496137	0.248068	+	0.968743i	$0.420204\pi$
$98$	−10.9407	−1.10518
$99$	0	0
$100$	−1.95373	−0.195373
$101$	−13.8343	−1.37657	−0.688284	−	0.725442i	$-0.741636\pi$
$101$	−13.8343	−1.37657	−0.688284	+	0.725442i	$0.741636\pi$
$102$	0	0
$103$	−4.20257	−0.414092	−0.207046	−	0.978331i	$-0.566385\pi$
$103$	−4.20257	−0.414092	−0.207046	+	0.978331i	$0.566385\pi$
$104$	−6.81286	−0.668055
$105$	0	0
$106$	−6.63962	−0.644897
$107$	8.42853	0.814817	0.407408	−	0.913246i	$-0.366433\pi$
$107$	8.42853	0.814817	0.407408	+	0.913246i	$0.366433\pi$
$108$	0	0
$109$	−13.4505	−1.28833	−0.644163	−	0.764889i	$-0.722794\pi$
$109$	−13.4505	−1.28833	−0.644163	+	0.764889i	$0.722794\pi$
$110$	3.21541	0.306578
$111$	0	0
$112$	0.352429	0.0333014
$113$	−11.2844	−1.06154	−0.530771	−	0.847515i	$-0.678098\pi$
$113$	−11.2844	−1.06154	−0.530771	+	0.847515i	$0.678098\pi$
$114$	0	0
$115$	−1.76315	−0.164415
$116$	3.31951	0.308209
$117$	0	0
$118$	11.0111	1.01366
$119$	−0.356062	−0.0326401
$120$	0	0
$121$	−4.26094	−0.387358
$122$	11.3054	1.02355
$123$	0	0
$124$	−1.89199	−0.169906
$125$	7.42195	0.663840
$126$	0	0
$127$	2.62400	0.232842	0.116421	−	0.993200i	$-0.462858\pi$
$127$	2.62400	0.232842	0.116421	+	0.993200i	$0.462858\pi$
$128$	13.5753	1.19989
$129$	0	0
$130$	−3.47324	−0.304623
$131$	12.4897	1.09123	0.545613	−	0.838037i	$-0.316297\pi$
$131$	12.4897	1.09123	0.545613	+	0.838037i	$0.316297\pi$
$132$	0	0
$133$	−0.0750818	−0.00651042
$134$	−5.25378	−0.453857
$135$	0	0
$136$	−11.5219	−0.987991
$137$	−22.1067	−1.88870	−0.944352	−	0.328936i	$-0.893310\pi$
$137$	−22.1067	−1.88870	−0.944352	+	0.328936i	$0.893310\pi$
$138$	0	0
$139$	−10.3335	−0.876476	−0.438238	−	0.898859i	$-0.644397\pi$
$139$	−10.3335	−0.876476	−0.438238	+	0.898859i	$0.644397\pi$
$140$	0.0265620	0.00224490
$141$	0	0
$142$	−4.31343	−0.361975
$143$	−7.27942	−0.608736
$144$	0	0
$145$	−5.88341	−0.488590
$146$	−12.6842	−1.04975
$147$	0	0
$148$	−1.32757	−0.109126
$149$	11.9980	0.982913	0.491456	−	0.870902i	$-0.336465\pi$
$149$	11.9980	0.982913	0.491456	+	0.870902i	$0.336465\pi$
$150$	0	0
$151$	13.3367	1.08533	0.542664	−	0.839950i	$-0.317416\pi$
$151$	13.3367	1.08533	0.542664	+	0.839950i	$0.317416\pi$
$152$	−2.42958	−0.197065
$153$	0	0
$154$	0.304881	0.0245680
$155$	3.35331	0.269344
$156$	0	0
$157$	−9.34960	−0.746179	−0.373090	−	0.927795i	$-0.621702\pi$
$157$	−9.34960	−0.746179	−0.373090	+	0.927795i	$0.621702\pi$
$158$	−24.8257	−1.97502
$159$	0	0
$160$	1.96628	0.155448
$161$	−0.167180	−0.0131756
$162$	0	0
$163$	−7.60243	−0.595469	−0.297734	−	0.954649i	$-0.596231\pi$
$163$	−7.60243	−0.595469	−0.297734	+	0.954649i	$0.596231\pi$
$164$	−4.50355	−0.351668
$165$	0	0
$166$	4.61647	0.358307
$167$	−17.7868	−1.37639	−0.688193	−	0.725528i	$-0.741596\pi$
$167$	−17.7868	−1.37639	−0.688193	+	0.725528i	$0.741596\pi$
$168$	0	0
$169$	−5.13689	−0.395145
$170$	−5.87392	−0.450509
$171$	0	0
$172$	3.41153	0.260127
$173$	23.6447	1.79767	0.898836	−	0.438285i	$-0.144414\pi$
$173$	23.6447	1.79767	0.898836	+	0.438285i	$0.144414\pi$
$174$	0	0
$175$	0.328331	0.0248195
$176$	12.1853	0.918503
$177$	0	0
$178$	−5.79085	−0.434042
$179$	−6.79923	−0.508198	−0.254099	−	0.967178i	$-0.581779\pi$
$179$	−6.79923	−0.508198	−0.254099	+	0.967178i	$0.581779\pi$
$180$	0	0
$181$	−15.9378	−1.18464	−0.592322	−	0.805701i	$-0.701789\pi$
$181$	−15.9378	−1.18464	−0.592322	+	0.805701i	$0.701789\pi$
$182$	−0.329328	−0.0244114
$183$	0	0
$184$	−5.40979	−0.398815
$185$	2.35295	0.172992
$186$	0	0
$187$	−12.3109	−0.900263
$188$	−0.446772	−0.0325842
$189$	0	0
$190$	−1.23862	−0.0898588
$191$	−16.7865	−1.21463	−0.607313	−	0.794463i	$-0.707753\pi$
$191$	−16.7865	−1.21463	−0.607313	+	0.794463i	$0.707753\pi$
$192$	0	0
$193$	−4.09764	−0.294954	−0.147477	−	0.989065i	$-0.547115\pi$
$193$	−4.09764	−0.294954	−0.147477	+	0.989065i	$0.547115\pi$
$194$	7.64335	0.548761
$195$	0	0
$196$	−3.12489	−0.223206
$197$	−8.26711	−0.589007	−0.294504	−	0.955650i	$-0.595154\pi$
$197$	−8.26711	−0.589007	−0.294504	+	0.955650i	$0.595154\pi$
$198$	0	0
$199$	−15.5788	−1.10435	−0.552177	−	0.833727i	$-0.686203\pi$
$199$	−15.5788	−1.10435	−0.552177	+	0.833727i	$0.686203\pi$
$200$	10.6245	0.751267
$201$	0	0
$202$	−21.6399	−1.52258
$203$	−0.557857	−0.0391539
$204$	0	0
$205$	7.98196	0.557484
$206$	−6.57373	−0.458014
$207$	0	0
$208$	−13.1624	−0.912647
$209$	−2.59597	−0.179567
$210$	0	0
$211$	−7.08669	−0.487868	−0.243934	−	0.969792i	$-0.578438\pi$
$211$	−7.08669	−0.487868	−0.243934	+	0.969792i	$0.578438\pi$
$212$	−1.89641	−0.130246
$213$	0	0
$214$	13.1840	0.901243
$215$	−6.04649	−0.412367
$216$	0	0
$217$	0.317956	0.0215843
$218$	−21.0395	−1.42497
$219$	0	0
$220$	0.918388	0.0619177
$221$	13.2980	0.894523
$222$	0	0
$223$	−2.60966	−0.174756	−0.0873779	−	0.996175i	$-0.527849\pi$
$223$	−2.60966	−0.174756	−0.0873779	+	0.996175i	$0.527849\pi$
$224$	0.186440	0.0124571
$225$	0	0
$226$	−17.6512	−1.17414
$227$	8.49042	0.563529	0.281764	−	0.959484i	$-0.409080\pi$
$227$	8.49042	0.563529	0.281764	+	0.959484i	$0.409080\pi$
$228$	0	0
$229$	2.27526	0.150353	0.0751766	−	0.997170i	$-0.476048\pi$
$229$	2.27526	0.150353	0.0751766	+	0.997170i	$0.476048\pi$
$230$	−2.75795	−0.181854
$231$	0	0
$232$	−18.0518	−1.18516
$233$	18.9961	1.24448	0.622238	−	0.782828i	$-0.286224\pi$
$233$	18.9961	1.24448	0.622238	+	0.782828i	$0.286224\pi$
$234$	0	0
$235$	0.791846	0.0516543
$236$	3.14500	0.204722
$237$	0	0
$238$	−0.556957	−0.0361022
$239$	−16.8841	−1.09214	−0.546070	−	0.837740i	$-0.683877\pi$
$239$	−16.8841	−1.09214	−0.546070	+	0.837740i	$0.683877\pi$
$240$	0	0
$241$	−19.1812	−1.23557	−0.617785	−	0.786347i	$-0.711970\pi$
$241$	−19.1812	−1.23557	−0.617785	+	0.786347i	$0.711970\pi$
$242$	−6.66503	−0.428445
$243$	0	0
$244$	3.22906	0.206720
$245$	5.53845	0.353839
$246$	0	0
$247$	2.80412	0.178422
$248$	10.2888	0.653340
$249$	0	0
$250$	11.6095	0.734252
$251$	−8.47769	−0.535107	−0.267554	−	0.963543i	$-0.586215\pi$
$251$	−8.47769	−0.535107	−0.267554	+	0.963543i	$0.586215\pi$
$252$	0	0
$253$	−5.78027	−0.363403
$254$	4.10450	0.257539
$255$	0	0
$256$	10.2273	0.639206
$257$	4.50102	0.280766	0.140383	−	0.990097i	$-0.455167\pi$
$257$	4.50102	0.280766	0.140383	+	0.990097i	$0.455167\pi$
$258$	0	0
$259$	0.223104	0.0138630
$260$	−0.992027	−0.0615229
$261$	0	0
$262$	19.5365	1.20697
$263$	−16.1794	−0.997664	−0.498832	−	0.866699i	$-0.666238\pi$
$263$	−16.1794	−0.997664	−0.498832	+	0.866699i	$0.666238\pi$
$264$	0	0
$265$	3.36114	0.206473
$266$	−0.117444	−0.00720096
$267$	0	0
$268$	−1.50059	−0.0916629
$269$	31.4154	1.91543	0.957716	−	0.287717i	$-0.0928961\pi$
$269$	31.4154	1.91543	0.957716	+	0.287717i	$0.0928961\pi$
$270$	0	0
$271$	−4.09476	−0.248739	−0.124369	−	0.992236i	$-0.539691\pi$
$271$	−4.09476	−0.248739	−0.124369	+	0.992236i	$0.539691\pi$
$272$	−22.2601	−1.34972
$273$	0	0
$274$	−34.5797	−2.08904
$275$	11.3521	0.684559
$276$	0	0
$277$	−6.44779	−0.387410	−0.193705	−	0.981060i	$-0.562051\pi$
$277$	−6.44779	−0.387410	−0.193705	+	0.981060i	$0.562051\pi$
$278$	−16.1638	−0.969442
$279$	0	0
$280$	−0.144446	−0.00863233
$281$	6.72480	0.401168	0.200584	−	0.979677i	$-0.435716\pi$
$281$	6.72480	0.401168	0.200584	+	0.979677i	$0.435716\pi$
$282$	0	0
$283$	11.9474	0.710198	0.355099	−	0.934829i	$-0.384447\pi$
$283$	11.9474	0.710198	0.355099	+	0.934829i	$0.384447\pi$
$284$	−1.23200	−0.0731060
$285$	0	0
$286$	−11.3866	−0.673303
$287$	0.756839	0.0446748
$288$	0	0
$289$	5.48958	0.322916
$290$	−9.20292	−0.540414
$291$	0	0
$292$	−3.62285	−0.212012
$293$	18.7850	1.09743	0.548716	−	0.836009i	$-0.315117\pi$
$293$	18.7850	1.09743	0.548716	+	0.836009i	$0.315117\pi$
$294$	0	0
$295$	−5.57411	−0.324537
$296$	7.21945	0.419622
$297$	0	0
$298$	18.7674	1.08717
$299$	6.24375	0.361086
$300$	0	0
$301$	−0.573320	−0.0330456
$302$	20.8615	1.20045
$303$	0	0
$304$	−4.69394	−0.269216
$305$	−5.72309	−0.327703
$306$	0	0
$307$	−4.56173	−0.260351	−0.130176	−	0.991491i	$-0.541554\pi$
$307$	−4.56173	−0.260351	−0.130176	+	0.991491i	$0.541554\pi$
$308$	0.0870803	0.00496186
$309$	0	0
$310$	5.24530	0.297913
$311$	−22.7977	−1.29274	−0.646369	−	0.763025i	$-0.723713\pi$
$311$	−22.7977	−1.29274	−0.646369	+	0.763025i	$0.723713\pi$
$312$	0	0
$313$	1.22534	0.0692601	0.0346300	−	0.999400i	$-0.488975\pi$
$313$	1.22534	0.0692601	0.0346300	+	0.999400i	$0.488975\pi$
$314$	−14.6248	−0.825325
$315$	0	0
$316$	−7.09071	−0.398884
$317$	−1.88211	−0.105710	−0.0528549	−	0.998602i	$-0.516832\pi$
$317$	−1.88211	−0.105710	−0.0528549	+	0.998602i	$0.516832\pi$
$318$	0	0
$319$	−19.2880	−1.07992
$320$	−4.35806	−0.243623
$321$	0	0
$322$	−0.261505	−0.0145731
$323$	4.74232	0.263870
$324$	0	0
$325$	−12.2624	−0.680194
$326$	−11.8919	−0.658629
$327$	0	0
$328$	24.4907	1.35227
$329$	0.0750818	0.00413939
$330$	0	0
$331$	−6.20759	−0.341200	−0.170600	−	0.985340i	$-0.554571\pi$
$331$	−6.20759	−0.341200	−0.170600	+	0.985340i	$0.554571\pi$
$332$	1.31856	0.0723652
$333$	0	0
$334$	−27.8224	−1.52238
$335$	2.65959	0.145309
$336$	0	0
$337$	−22.7383	−1.23864	−0.619318	−	0.785141i	$-0.712591\pi$
$337$	−22.7383	−1.23864	−0.619318	+	0.785141i	$0.712591\pi$
$338$	−8.03520	−0.437057
$339$	0	0
$340$	−1.67771	−0.0909866
$341$	10.9934	0.595327
$342$	0	0
$343$	1.05072	0.0567336
$344$	−18.5522	−1.00027
$345$	0	0
$346$	36.9854	1.98835
$347$	0.694353	0.0372748	0.0186374	−	0.999826i	$-0.494067\pi$
$347$	0.694353	0.0372748	0.0186374	+	0.999826i	$0.494067\pi$
$348$	0	0
$349$	−6.99928	−0.374663	−0.187331	−	0.982297i	$-0.559984\pi$
$349$	−6.99928	−0.374663	−0.187331	+	0.982297i	$0.559984\pi$
$350$	0.513581	0.0274521
$351$	0	0
$352$	6.44622	0.343585
$353$	28.1471	1.49812	0.749060	−	0.662502i	$-0.230505\pi$
$353$	28.1471	1.49812	0.749060	+	0.662502i	$0.230505\pi$
$354$	0	0
$355$	2.18357	0.115892
$356$	−1.65398	−0.0876610
$357$	0	0
$358$	−10.6355	−0.562101
$359$	−13.0891	−0.690818	−0.345409	−	0.938452i	$-0.612260\pi$
$359$	−13.0891	−0.690818	−0.345409	+	0.938452i	$0.612260\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−24.9301	−1.31030
$363$	0	0
$364$	−0.0940627	−0.00493023
$365$	6.42104	0.336093
$366$	0	0
$367$	3.29156	0.171818	0.0859089	−	0.996303i	$-0.472621\pi$
$367$	3.29156	0.171818	0.0859089	+	0.996303i	$0.472621\pi$
$368$	−10.4517	−0.544831
$369$	0	0
$370$	3.68052	0.191341
$371$	0.318699	0.0165460
$372$	0	0
$373$	−14.9095	−0.771983	−0.385991	−	0.922502i	$-0.626141\pi$
$373$	−14.9095	−0.771983	−0.385991	+	0.922502i	$0.626141\pi$
$374$	−19.2569	−0.995752
$375$	0	0
$376$	2.42958	0.125296
$377$	20.8346	1.07304
$378$	0	0
$379$	−27.1813	−1.39621	−0.698104	−	0.715996i	$-0.745973\pi$
$379$	−27.1813	−1.39621	−0.698104	+	0.715996i	$0.745973\pi$
$380$	−0.353774	−0.0181482
$381$	0	0
$382$	−26.2577	−1.34346
$383$	13.0270	0.665649	0.332824	−	0.942989i	$-0.391998\pi$
$383$	13.0270	0.665649	0.332824	+	0.942989i	$0.391998\pi$
$384$	0	0
$385$	−0.154339	−0.00786582
$386$	−6.40959	−0.326240
$387$	0	0
$388$	2.18310	0.110830
$389$	1.52129	0.0771323	0.0385662	−	0.999256i	$-0.487721\pi$
$389$	1.52129	0.0771323	0.0385662	+	0.999256i	$0.487721\pi$
$390$	0	0
$391$	10.5594	0.534012
$392$	16.9934	0.858296
$393$	0	0
$394$	−12.9315	−0.651482
$395$	12.5674	0.632333
$396$	0	0
$397$	16.0993	0.808001	0.404001	−	0.914759i	$-0.367619\pi$
$397$	16.0993	0.808001	0.404001	+	0.914759i	$0.367619\pi$
$398$	−24.3687	−1.22149
$399$	0	0
$400$	20.5265	1.02633
$401$	35.9203	1.79378	0.896888	−	0.442259i	$-0.145823\pi$
$401$	35.9203	1.79378	0.896888	+	0.442259i	$0.145823\pi$
$402$	0	0
$403$	−11.8749	−0.591531
$404$	−6.18079	−0.307506
$405$	0	0
$406$	−0.872609	−0.0433069
$407$	7.71386	0.382362
$408$	0	0
$409$	−11.3282	−0.560143	−0.280071	−	0.959979i	$-0.590358\pi$
$409$	−11.3282	−0.560143	−0.280071	+	0.959979i	$0.590358\pi$
$410$	12.4855	0.616615
$411$	0	0
$412$	−1.87759	−0.0925023
$413$	−0.528530	−0.0260073
$414$	0	0
$415$	−2.33697	−0.114717
$416$	−6.96311	−0.341394
$417$	0	0
$418$	−4.06066	−0.198613
$419$	−2.45815	−0.120088	−0.0600441	−	0.998196i	$-0.519124\pi$
$419$	−2.45815	−0.120088	−0.0600441	+	0.998196i	$0.519124\pi$
$420$	0	0
$421$	14.7828	0.720468	0.360234	−	0.932862i	$-0.382697\pi$
$421$	14.7828	0.720468	0.360234	+	0.932862i	$0.382697\pi$
$422$	−11.0851	−0.539615
$423$	0	0
$424$	10.3128	0.500836
$425$	−20.7381	−1.00594
$426$	0	0
$427$	−0.542656	−0.0262610
$428$	3.76563	0.182019
$429$	0	0
$430$	−9.45802	−0.456106
$431$	3.75243	0.180748	0.0903741	−	0.995908i	$-0.471194\pi$
$431$	3.75243	0.180748	0.0903741	+	0.995908i	$0.471194\pi$
$432$	0	0
$433$	15.1881	0.729896	0.364948	−	0.931028i	$-0.381087\pi$
$433$	15.1881	0.729896	0.364948	+	0.931028i	$0.381087\pi$
$434$	0.497353	0.0238737
$435$	0	0
$436$	−6.00931	−0.287794
$437$	2.22663	0.106514
$438$	0	0
$439$	−22.0222	−1.05106	−0.525531	−	0.850774i	$-0.676133\pi$
$439$	−22.0222	−1.05106	−0.525531	+	0.850774i	$0.676133\pi$
$440$	−4.99427	−0.238092
$441$	0	0
$442$	20.8010	0.989404
$443$	−33.0265	−1.56914	−0.784569	−	0.620042i	$-0.787116\pi$
$443$	−33.0265	−1.56914	−0.784569	+	0.620042i	$0.787116\pi$
$444$	0	0
$445$	2.93147	0.138965
$446$	−4.08207	−0.193292
$447$	0	0
$448$	−0.413225	−0.0195231
$449$	6.43102	0.303499	0.151749	−	0.988419i	$-0.451509\pi$
$449$	6.43102	0.303499	0.151749	+	0.988419i	$0.451509\pi$
$450$	0	0
$451$	26.1679	1.23220
$452$	−5.04153	−0.237134
$453$	0	0
$454$	13.2808	0.623301
$455$	0.166714	0.00781567
$456$	0	0
$457$	−1.61909	−0.0757378	−0.0378689	−	0.999283i	$-0.512057\pi$
$457$	−1.61909	−0.0757378	−0.0378689	+	0.999283i	$0.512057\pi$
$458$	3.55899	0.166301
$459$	0	0
$460$	−0.787726	−0.0367279
$461$	−5.18697	−0.241581	−0.120791	−	0.992678i	$-0.538543\pi$
$461$	−5.18697	−0.241581	−0.120791	+	0.992678i	$0.538543\pi$
$462$	0	0
$463$	34.7421	1.61460	0.807300	−	0.590141i	$-0.200928\pi$
$463$	34.7421	1.61460	0.807300	+	0.590141i	$0.200928\pi$
$464$	−34.8759	−1.61907
$465$	0	0
$466$	29.7140	1.37647
$467$	−41.7548	−1.93218	−0.966090	−	0.258205i	$-0.916869\pi$
$467$	−41.7548	−1.93218	−0.966090	+	0.258205i	$0.916869\pi$
$468$	0	0
$469$	0.252179	0.0116446
$470$	1.23862	0.0571332
$471$	0	0
$472$	−17.1028	−0.787220
$473$	−19.8227	−0.911448
$474$	0	0
$475$	−4.37298	−0.200646
$476$	−0.159078	−0.00729134
$477$	0	0
$478$	−26.4103	−1.20798
$479$	32.1847	1.47056	0.735278	−	0.677766i	$-0.237052\pi$
$479$	32.1847	1.47056	0.735278	+	0.677766i	$0.237052\pi$
$480$	0	0
$481$	−8.33239	−0.379924
$482$	−30.0035	−1.36662
$483$	0	0
$484$	−1.90367	−0.0865304
$485$	−3.86926	−0.175694
$486$	0	0
$487$	13.6807	0.619930	0.309965	−	0.950748i	$-0.399683\pi$
$487$	13.6807	0.619930	0.309965	+	0.950748i	$0.399683\pi$
$488$	−17.5599	−0.794900
$489$	0	0
$490$	8.66334	0.391370
$491$	28.5961	1.29052	0.645261	−	0.763962i	$-0.276749\pi$
$491$	28.5961	1.29052	0.645261	+	0.763962i	$0.276749\pi$
$492$	0	0
$493$	35.2354	1.58692
$494$	4.38626	0.197347
$495$	0	0
$496$	19.8779	0.892544
$497$	0.207043	0.00928715
$498$	0	0
$499$	16.4688	0.737246	0.368623	−	0.929579i	$-0.379829\pi$
$499$	16.4688	0.737246	0.368623	+	0.929579i	$0.379829\pi$
$500$	3.31592	0.148292
$501$	0	0
$502$	−13.2609	−0.591865
$503$	8.44855	0.376702	0.188351	−	0.982102i	$-0.439686\pi$
$503$	8.44855	0.376702	0.188351	+	0.982102i	$0.439686\pi$
$504$	0	0
$505$	10.9547	0.487476
$506$	−9.04159	−0.401948
$507$	0	0
$508$	1.17233	0.0520137
$509$	9.77401	0.433225	0.216613	−	0.976258i	$-0.430499\pi$
$509$	9.77401	0.433225	0.216613	+	0.976258i	$0.430499\pi$
$510$	0	0
$511$	0.608835	0.0269333
$512$	−11.1528	−0.492889
$513$	0	0
$514$	7.04057	0.310546
$515$	3.32779	0.146640
$516$	0	0
$517$	2.59597	0.114171
$518$	0.348982	0.0153334
$519$	0	0
$520$	5.39473	0.236575
$521$	10.0261	0.439250	0.219625	−	0.975584i	$-0.429517\pi$
$521$	10.0261	0.439250	0.219625	+	0.975584i	$0.429517\pi$
$522$	0	0
$523$	44.6442	1.95215	0.976077	−	0.217425i	$-0.0697656\pi$
$523$	44.6442	1.95215	0.976077	+	0.217425i	$0.0697656\pi$
$524$	5.58003	0.243765
$525$	0	0
$526$	−25.3081	−1.10348
$527$	−20.0828	−0.874819
$528$	0	0
$529$	−18.0421	−0.784439
$530$	5.25756	0.228374
$531$	0	0
$532$	−0.0335444	−0.00145434
$533$	−28.2661	−1.22434
$534$	0	0
$535$	−6.67410	−0.288546
$536$	8.16031	0.352472
$537$	0	0
$538$	49.1405	2.11860
$539$	18.1572	0.782084
$540$	0	0
$541$	11.2249	0.482596	0.241298	−	0.970451i	$-0.422427\pi$
$541$	11.2249	0.482596	0.241298	+	0.970451i	$0.422427\pi$
$542$	−6.40509	−0.275122
$543$	0	0
$544$	−11.7760	−0.504890
$545$	10.6507	0.456227
$546$	0	0
$547$	−2.69402	−0.115188	−0.0575940	−	0.998340i	$-0.518343\pi$
$547$	−2.69402	−0.115188	−0.0575940	+	0.998340i	$0.518343\pi$
$548$	−9.87666	−0.421910
$549$	0	0
$550$	17.7572	0.757169
$551$	7.42999	0.316528
$552$	0	0
$553$	1.19162	0.0506729
$554$	−10.0857	−0.428502
$555$	0	0
$556$	−4.61672	−0.195792
$557$	45.6879	1.93586	0.967930	−	0.251219i	$-0.0808314\pi$
$557$	45.6879	1.93586	0.967930	+	0.251219i	$0.0808314\pi$
$558$	0	0
$559$	21.4121	0.905637
$560$	−0.279070	−0.0117928
$561$	0	0
$562$	10.5190	0.443719
$563$	6.80216	0.286677	0.143338	−	0.989674i	$-0.454216\pi$
$563$	6.80216	0.286677	0.143338	+	0.989674i	$0.454216\pi$
$564$	0	0
$565$	8.93547	0.375918
$566$	18.6883	0.785527
$567$	0	0
$568$	6.69974	0.281115
$569$	−1.47894	−0.0620002	−0.0310001	−	0.999519i	$-0.509869\pi$
$569$	−1.47894	−0.0620002	−0.0310001	+	0.999519i	$0.509869\pi$
$570$	0	0
$571$	9.50242	0.397664	0.198832	−	0.980034i	$-0.436285\pi$
$571$	9.50242	0.397664	0.198832	+	0.980034i	$0.436285\pi$
$572$	−3.25224	−0.135983
$573$	0	0
$574$	1.18386	0.0494133
$575$	−9.73702	−0.406062
$576$	0	0
$577$	24.1742	1.00639	0.503193	−	0.864174i	$-0.332158\pi$
$577$	24.1742	1.00639	0.503193	+	0.864174i	$0.332158\pi$
$578$	8.58688	0.357167
$579$	0	0
$580$	−2.62854	−0.109144
$581$	−0.221589	−0.00919304
$582$	0	0
$583$	11.0191	0.456365
$584$	19.7014	0.815250
$585$	0	0
$586$	29.3838	1.21384
$587$	−18.1377	−0.748621	−0.374311	−	0.927303i	$-0.622121\pi$
$587$	−18.1377	−0.748621	−0.374311	+	0.927303i	$0.622121\pi$
$588$	0	0
$589$	−4.23480	−0.174492
$590$	−8.71911	−0.358960
$591$	0	0
$592$	13.9479	0.573256
$593$	7.65993	0.314556	0.157278	−	0.987554i	$-0.449728\pi$
$593$	7.65993	0.314556	0.157278	+	0.987554i	$0.449728\pi$
$594$	0	0
$595$	0.281946	0.0115586
$596$	5.36036	0.219569
$597$	0	0
$598$	9.76658	0.399385
$599$	−24.2351	−0.990220	−0.495110	−	0.868830i	$-0.664872\pi$
$599$	−24.2351	−0.990220	−0.495110	+	0.868830i	$0.664872\pi$
$600$	0	0
$601$	−27.2692	−1.11233	−0.556167	−	0.831070i	$-0.687729\pi$
$601$	−27.2692	−1.11233	−0.556167	+	0.831070i	$0.687729\pi$
$602$	−0.896797	−0.0365507
$603$	0	0
$604$	5.95848	0.242447
$605$	3.37401	0.137173
$606$	0	0
$607$	−48.2935	−1.96017	−0.980086	−	0.198575i	$-0.936369\pi$
$607$	−48.2935	−1.96017	−0.980086	+	0.198575i	$0.936369\pi$
$608$	−2.48317	−0.100706
$609$	0	0
$610$	−8.95216	−0.362462
$611$	−2.80412	−0.113443
$612$	0	0
$613$	−14.3469	−0.579465	−0.289732	−	0.957108i	$-0.593566\pi$
$613$	−14.3469	−0.579465	−0.289732	+	0.957108i	$0.593566\pi$
$614$	−7.13552	−0.287966
$615$	0	0
$616$	−0.473550	−0.0190799
$617$	42.9975	1.73101	0.865507	−	0.500897i	$-0.166996\pi$
$617$	42.9975	1.73101	0.865507	+	0.500897i	$0.166996\pi$
$618$	0	0
$619$	12.2538	0.492522	0.246261	−	0.969204i	$-0.420798\pi$
$619$	12.2538	0.492522	0.246261	+	0.969204i	$0.420798\pi$
$620$	1.49816	0.0601677
$621$	0	0
$622$	−35.6605	−1.42986
$623$	0.277959	0.0111362
$624$	0	0
$625$	15.9879	0.639515
$626$	1.91669	0.0766063
$627$	0	0
$628$	−4.17714	−0.166686
$629$	−14.0917	−0.561872
$630$	0	0
$631$	−36.9310	−1.47020	−0.735099	−	0.677960i	$-0.762864\pi$
$631$	−36.9310	−1.47020	−0.735099	+	0.677960i	$0.762864\pi$
$632$	38.5599	1.53383
$633$	0	0
$634$	−2.94403	−0.116922
$635$	−2.07780	−0.0824550
$636$	0	0
$637$	−19.6131	−0.777098
$638$	−30.1707	−1.19447
$639$	0	0
$640$	−10.7495	−0.424912
$641$	33.4992	1.32314	0.661569	−	0.749884i	$-0.269891\pi$
$641$	33.4992	1.32314	0.661569	+	0.749884i	$0.269891\pi$
$642$	0	0
$643$	−21.9291	−0.864801	−0.432400	−	0.901682i	$-0.642333\pi$
$643$	−21.9291	−0.864801	−0.432400	+	0.901682i	$0.642333\pi$
$644$	−0.0746911	−0.00294324
$645$	0	0
$646$	7.41801	0.291858
$647$	−14.8445	−0.583599	−0.291799	−	0.956480i	$-0.594254\pi$
$647$	−14.8445	−0.583599	−0.291799	+	0.956480i	$0.594254\pi$
$648$	0	0
$649$	−18.2740	−0.717319
$650$	−19.1810	−0.752341
$651$	0	0
$652$	−3.39655	−0.133019
$653$	−13.5290	−0.529432	−0.264716	−	0.964326i	$-0.585278\pi$
$653$	−13.5290	−0.529432	−0.264716	+	0.964326i	$0.585278\pi$
$654$	0	0
$655$	−9.88988	−0.386430
$656$	47.3158	1.84737
$657$	0	0
$658$	0.117444	0.00457845
$659$	−23.4875	−0.914945	−0.457472	−	0.889224i	$-0.651245\pi$
$659$	−23.4875	−0.914945	−0.457472	+	0.889224i	$0.651245\pi$
$660$	0	0
$661$	−33.6601	−1.30923	−0.654614	−	0.755964i	$-0.727169\pi$
$661$	−33.6601	−1.30923	−0.654614	+	0.755964i	$0.727169\pi$
$662$	−9.71000	−0.377390
$663$	0	0
$664$	−7.17042	−0.278266
$665$	0.0594532	0.00230549
$666$	0	0
$667$	16.5439	0.640581
$668$	−7.94666	−0.307465
$669$	0	0
$670$	4.16018	0.160722
$671$	−18.7625	−0.724317
$672$	0	0
$673$	47.2310	1.82062	0.910310	−	0.413928i	$-0.135843\pi$
$673$	47.2310	1.82062	0.910310	+	0.413928i	$0.135843\pi$
$674$	−35.5676	−1.37001
$675$	0	0
$676$	−2.29502	−0.0882699
$677$	11.3970	0.438023	0.219012	−	0.975722i	$-0.429717\pi$
$677$	11.3970	0.438023	0.219012	+	0.975722i	$0.429717\pi$
$678$	0	0
$679$	−0.366878	−0.0140795
$680$	9.12353	0.349872
$681$	0	0
$682$	17.1961	0.658472
$683$	−34.0807	−1.30406	−0.652030	−	0.758193i	$-0.726082\pi$
$683$	−34.0807	−1.30406	−0.652030	+	0.758193i	$0.726082\pi$
$684$	0	0
$685$	17.5051	0.668836
$686$	1.64356	0.0627512
$687$	0	0
$688$	−35.8427	−1.36649
$689$	−11.9027	−0.453455
$690$	0	0
$691$	14.8220	0.563857	0.281929	−	0.959435i	$-0.409026\pi$
$691$	14.8220	0.563857	0.281929	+	0.959435i	$0.409026\pi$
$692$	10.5638	0.401575
$693$	0	0
$694$	1.08612	0.0412285
$695$	8.18253	0.310381
$696$	0	0
$697$	−47.8035	−1.81069
$698$	−10.9484	−0.414402
$699$	0	0
$700$	0.146689	0.00554433
$701$	−33.9512	−1.28232	−0.641160	−	0.767407i	$-0.721547\pi$
$701$	−33.9512	−1.28232	−0.641160	+	0.767407i	$0.721547\pi$
$702$	0	0
$703$	−2.97148	−0.112071
$704$	−14.2874	−0.538475
$705$	0	0
$706$	44.0282	1.65702
$707$	1.03871	0.0390646
$708$	0	0
$709$	19.3006	0.724849	0.362424	−	0.932013i	$-0.381949\pi$
$709$	19.3006	0.724849	0.362424	+	0.932013i	$0.381949\pi$
$710$	3.41557	0.128184
$711$	0	0
$712$	8.99451	0.337083
$713$	−9.42935	−0.353132
$714$	0	0
$715$	5.76418	0.215568
$716$	−3.03770	−0.113524
$717$	0	0
$718$	−20.4742	−0.764092
$719$	6.35358	0.236949	0.118474	−	0.992957i	$-0.462200\pi$
$719$	6.35358	0.236949	0.118474	+	0.992957i	$0.462200\pi$
$720$	0	0
$721$	0.315537	0.0117512
$722$	1.56422	0.0582141
$723$	0	0
$724$	−7.12054	−0.264633
$725$	−32.4912	−1.20669
$726$	0	0
$727$	16.5900	0.615288	0.307644	−	0.951502i	$-0.400459\pi$
$727$	16.5900	0.615288	0.307644	+	0.951502i	$0.400459\pi$
$728$	0.511521	0.0189582
$729$	0	0
$730$	10.0439	0.371741
$731$	36.2121	1.33935
$732$	0	0
$733$	8.27651	0.305700	0.152850	−	0.988249i	$-0.451155\pi$
$733$	8.27651	0.305700	0.152850	+	0.988249i	$0.451155\pi$
$734$	5.14871	0.190042
$735$	0	0
$736$	−5.52910	−0.203805
$737$	8.71916	0.321174
$738$	0	0
$739$	−25.2125	−0.927457	−0.463729	−	0.885977i	$-0.653489\pi$
$739$	−25.2125	−0.927457	−0.463729	+	0.885977i	$0.653489\pi$
$740$	1.05123	0.0386441
$741$	0	0
$742$	0.498515	0.0183010
$743$	21.0749	0.773164	0.386582	−	0.922255i	$-0.373656\pi$
$743$	21.0749	0.773164	0.386582	+	0.922255i	$0.373656\pi$
$744$	0	0
$745$	−9.50054	−0.348073
$746$	−23.3216	−0.853865
$747$	0	0
$748$	−5.50017	−0.201106
$749$	−0.632829	−0.0231231
$750$	0	0
$751$	9.30849	0.339671	0.169836	−	0.985472i	$-0.445676\pi$
$751$	9.30849	0.339671	0.169836	+	0.985472i	$0.445676\pi$
$752$	4.69394	0.171170
$753$	0	0
$754$	32.5899	1.18685
$755$	−10.5606	−0.384341
$756$	0	0
$757$	20.6103	0.749096	0.374548	−	0.927208i	$-0.377798\pi$
$757$	20.6103	0.749096	0.374548	+	0.927208i	$0.377798\pi$
$758$	−42.5174	−1.54430
$759$	0	0
$760$	1.92386	0.0697856
$761$	15.0535	0.545690	0.272845	−	0.962058i	$-0.412035\pi$
$761$	15.0535	0.545690	0.272845	+	0.962058i	$0.412035\pi$
$762$	0	0
$763$	1.00989	0.0365604
$764$	−7.49972	−0.271330
$765$	0	0
$766$	20.3770	0.736253
$767$	19.7393	0.712746
$768$	0	0
$769$	−1.07627	−0.0388112	−0.0194056	−	0.999812i	$-0.506177\pi$
$769$	−1.07627	−0.0388112	−0.0194056	+	0.999812i	$0.506177\pi$
$770$	−0.241419	−0.00870013
$771$	0	0
$772$	−1.83071	−0.0658887
$773$	−15.2175	−0.547336	−0.273668	−	0.961824i	$-0.588237\pi$
$773$	−15.2175	−0.547336	−0.273668	+	0.961824i	$0.588237\pi$
$774$	0	0
$775$	18.5187	0.665212
$776$	−11.8719	−0.426175
$777$	0	0
$778$	2.37962	0.0853136
$779$	−10.0802	−0.361160
$780$	0	0
$781$	7.15856	0.256153
$782$	16.5172	0.590653
$783$	0	0
$784$	32.8311	1.17254
$785$	7.40344	0.264240
$786$	0	0
$787$	29.4928	1.05130	0.525652	−	0.850699i	$-0.323821\pi$
$787$	29.4928	1.05130	0.525652	+	0.850699i	$0.323821\pi$
$788$	−3.69351	−0.131576
$789$	0	0
$790$	19.6581	0.699403
$791$	0.847249	0.0301247
$792$	0	0
$793$	20.2669	0.719699
$794$	25.1828	0.893704
$795$	0	0
$796$	−6.96019	−0.246697
$797$	50.6893	1.79551	0.897753	−	0.440498i	$-0.145198\pi$
$797$	50.6893	1.79551	0.897753	+	0.440498i	$0.145198\pi$
$798$	0	0
$799$	−4.74232	−0.167771
$800$	10.8588	0.383918
$801$	0	0
$802$	56.1871	1.98404
$803$	21.0506	0.742860
$804$	0	0
$805$	0.132380	0.00466579
$806$	−18.5749	−0.654274
$807$	0	0
$808$	33.6117	1.18245
$809$	−12.2497	−0.430676	−0.215338	−	0.976540i	$-0.569085\pi$
$809$	−12.2497	−0.430676	−0.215338	+	0.976540i	$0.569085\pi$
$810$	0	0
$811$	−36.1271	−1.26859	−0.634297	−	0.773090i	$-0.718710\pi$
$811$	−36.1271	−1.26859	−0.634297	+	0.773090i	$0.718710\pi$
$812$	−0.249235	−0.00874643
$813$	0	0
$814$	12.0661	0.422918
$815$	6.01995	0.210870
$816$	0	0
$817$	7.63595	0.267148
$818$	−17.7197	−0.619556
$819$	0	0
$820$	3.56611	0.124534
$821$	1.44772	0.0505257	0.0252629	−	0.999681i	$-0.491958\pi$
$821$	1.44772	0.0505257	0.0252629	+	0.999681i	$0.491958\pi$
$822$	0	0
$823$	−46.1434	−1.60846	−0.804229	−	0.594320i	$-0.797421\pi$
$823$	−46.1434	−1.60846	−0.804229	+	0.594320i	$0.797421\pi$
$824$	10.2105	0.355700
$825$	0	0
$826$	−0.826735	−0.0287658
$827$	−19.5417	−0.679531	−0.339765	−	0.940510i	$-0.610348\pi$
$827$	−19.5417	−0.679531	−0.339765	+	0.940510i	$0.610348\pi$
$828$	0	0
$829$	−11.9808	−0.416109	−0.208054	−	0.978117i	$-0.566713\pi$
$829$	−11.9808	−0.416109	−0.208054	+	0.978117i	$0.566713\pi$
$830$	−3.65553	−0.126885
$831$	0	0
$832$	15.4330	0.535042
$833$	−33.1695	−1.14925
$834$	0	0
$835$	14.0844	0.487411
$836$	−1.15981	−0.0401127
$837$	0	0
$838$	−3.84507	−0.132826
$839$	18.3871	0.634792	0.317396	−	0.948293i	$-0.397192\pi$
$839$	18.3871	0.634792	0.317396	+	0.948293i	$0.397192\pi$
$840$	0	0
$841$	26.2048	0.903614
$842$	23.1234	0.796886
$843$	0	0
$844$	−3.16613	−0.108983
$845$	4.06762	0.139930
$846$	0	0
$847$	0.319919	0.0109925
$848$	19.9243	0.684205
$849$	0	0
$850$	−32.4388	−1.11264
$851$	−6.61638	−0.226807
$852$	0	0
$853$	43.1717	1.47817	0.739086	−	0.673612i	$-0.235258\pi$
$853$	43.1717	1.47817	0.739086	+	0.673612i	$0.235258\pi$
$854$	−0.848832	−0.0290464
$855$	0	0
$856$	−20.4778	−0.699918
$857$	−26.1365	−0.892807	−0.446403	−	0.894832i	$-0.647295\pi$
$857$	−26.1365	−0.892807	−0.446403	+	0.894832i	$0.647295\pi$
$858$	0	0
$859$	0.457971	0.0156258	0.00781288	−	0.999969i	$-0.497513\pi$
$859$	0.457971	0.0156258	0.00781288	+	0.999969i	$0.497513\pi$
$860$	−2.70140	−0.0921171
$861$	0	0
$862$	5.86961	0.199920
$863$	23.1014	0.786380	0.393190	−	0.919457i	$-0.371371\pi$
$863$	23.1014	0.786380	0.393190	+	0.919457i	$0.371371\pi$
$864$	0	0
$865$	−18.7229	−0.636599
$866$	23.7575	0.807314
$867$	0	0
$868$	0.142054	0.00482163
$869$	41.2006	1.39763
$870$	0	0
$871$	−9.41829	−0.319127
$872$	32.6791	1.10666
$873$	0	0
$874$	3.48293	0.117812
$875$	−0.557253	−0.0188386
$876$	0	0
$877$	−24.3156	−0.821080	−0.410540	−	0.911843i	$-0.634660\pi$
$877$	−24.3156	−0.821080	−0.410540	+	0.911843i	$0.634660\pi$
$878$	−34.4475	−1.16255
$879$	0	0
$880$	−9.64889	−0.325264
$881$	−48.7676	−1.64302	−0.821511	−	0.570193i	$-0.806868\pi$
$881$	−48.7676	−1.64302	−0.821511	+	0.570193i	$0.806868\pi$
$882$	0	0
$883$	−56.1397	−1.88925	−0.944625	−	0.328152i	$-0.893574\pi$
$883$	−56.1397	−1.88925	−0.944625	+	0.328152i	$0.893574\pi$
$884$	5.94119	0.199824
$885$	0	0
$886$	−51.6606	−1.73557
$887$	−41.8563	−1.40540	−0.702699	−	0.711487i	$-0.748022\pi$
$887$	−41.8563	−1.40540	−0.702699	+	0.711487i	$0.748022\pi$
$888$	0	0
$889$	−0.197014	−0.00660765
$890$	4.58546	0.153705
$891$	0	0
$892$	−1.16592	−0.0390380
$893$	−1.00000	−0.0334637
$894$	0	0
$895$	5.38394	0.179965
$896$	−1.01925	−0.0340509
$897$	0	0
$898$	10.0595	0.335690
$899$	−31.4646	−1.04940
$900$	0	0
$901$	−20.1297	−0.670617
$902$	40.9322	1.36289
$903$	0	0
$904$	27.4163	0.911852
$905$	12.6202	0.419511
$906$	0	0
$907$	29.4776	0.978786	0.489393	−	0.872063i	$-0.337218\pi$
$907$	29.4776	0.978786	0.489393	+	0.872063i	$0.337218\pi$
$908$	3.79328	0.125884
$909$	0	0
$910$	0.260777	0.00864467
$911$	−35.1673	−1.16514	−0.582572	−	0.812779i	$-0.697953\pi$
$911$	−35.1673	−1.16514	−0.582572	+	0.812779i	$0.697953\pi$
$912$	0	0
$913$	−7.66148	−0.253558
$914$	−2.53260	−0.0837711
$915$	0	0
$916$	1.01652	0.0335868
$917$	−0.937746	−0.0309671
$918$	0	0
$919$	4.71270	0.155458	0.0777288	−	0.996975i	$-0.475233\pi$
$919$	4.71270	0.155458	0.0777288	+	0.996975i	$0.475233\pi$
$920$	4.28372	0.141230
$921$	0	0
$922$	−8.11354	−0.267205
$923$	−7.73256	−0.254520
$924$	0	0
$925$	12.9942	0.427247
$926$	54.3441	1.78586
$927$	0	0
$928$	−18.4499	−0.605648
$929$	52.9851	1.73839	0.869193	−	0.494473i	$-0.164639\pi$
$929$	52.9851	1.73839	0.869193	+	0.494473i	$0.164639\pi$
$930$	0	0
$931$	−6.99436	−0.229231
$932$	8.48692	0.277998
$933$	0	0
$934$	−65.3135	−2.13712
$935$	9.74834	0.318805
$936$	0	0
$937$	29.3048	0.957345	0.478672	−	0.877994i	$-0.341118\pi$
$937$	29.3048	0.957345	0.478672	+	0.877994i	$0.341118\pi$
$938$	0.394463	0.0128797
$939$	0	0
$940$	0.353774	0.0115388
$941$	−44.8660	−1.46259	−0.731296	−	0.682061i	$-0.761084\pi$
$941$	−44.8660	−1.46259	−0.731296	+	0.682061i	$0.761084\pi$
$942$	0	0
$943$	−22.4449	−0.730906
$944$	−33.0425	−1.07544
$945$	0	0
$946$	−31.0070	−1.00812
$947$	20.5651	0.668277	0.334138	−	0.942524i	$-0.391555\pi$
$947$	20.5651	0.668277	0.334138	+	0.942524i	$0.391555\pi$
$948$	0	0
$949$	−22.7385	−0.738124
$950$	−6.84029	−0.221928
$951$	0	0
$952$	0.865082	0.0280375
$953$	−29.8726	−0.967669	−0.483834	−	0.875160i	$-0.660756\pi$
$953$	−29.8726	−0.967669	−0.483834	+	0.875160i	$0.660756\pi$
$954$	0	0
$955$	13.2923	0.430128
$956$	−7.54333	−0.243969
$957$	0	0
$958$	50.3438	1.62653
$959$	1.65981	0.0535981
$960$	0	0
$961$	−13.0665	−0.421498
$962$	−13.0337	−0.420222
$963$	0	0
$964$	−8.56962	−0.276009
$965$	3.24470	0.104450
$966$	0	0
$967$	−35.9826	−1.15712	−0.578561	−	0.815639i	$-0.696386\pi$
$967$	−35.9826	−1.15712	−0.578561	+	0.815639i	$0.696386\pi$
$968$	10.3523	0.332736
$969$	0	0
$970$	−6.05235	−0.194329
$971$	−35.9583	−1.15396	−0.576978	−	0.816760i	$-0.695768\pi$
$971$	−35.9583	−1.15396	−0.576978	+	0.816760i	$0.695768\pi$
$972$	0	0
$973$	0.775857	0.0248728
$974$	21.3995	0.685684
$975$	0	0
$976$	−33.9256	−1.08593
$977$	59.7656	1.91207	0.956035	−	0.293252i	$-0.0947375\pi$
$977$	59.7656	1.91207	0.956035	+	0.293252i	$0.0947375\pi$
$978$	0	0
$979$	9.61048	0.307152
$980$	2.47443	0.0790427
$981$	0	0
$982$	44.7304	1.42740
$983$	−13.2182	−0.421595	−0.210798	−	0.977530i	$-0.567606\pi$
$983$	−13.2182	−0.421595	−0.210798	+	0.977530i	$0.567606\pi$
$984$	0	0
$985$	6.54627	0.208582
$986$	55.1158	1.75524
$987$	0	0
$988$	1.25280	0.0398570
$989$	17.0025	0.540646
$990$	0	0
$991$	−11.0106	−0.349764	−0.174882	−	0.984589i	$-0.555954\pi$
$991$	−11.0106	−0.349764	−0.174882	+	0.984589i	$0.555954\pi$
$992$	10.5157	0.333874
$993$	0	0
$994$	0.323860	0.0102722
$995$	12.3360	0.391079
$996$	0	0
$997$	59.6654	1.88962	0.944811	−	0.327616i	$-0.106245\pi$
$997$	59.6654	1.88962	0.944811	+	0.327616i	$0.106245\pi$
$998$	25.7608	0.815444
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8037.2.a.j.1.6		7
3.2	odd	2		2679.2.a.k.1.2	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2679.2.a.k.1.2	✓	7	3.2	odd	2
8037.2.a.j.1.6		7	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)

\(f(q)\)	\(=\)	\(q+1.56422 q^{2} +0.446772 q^{4} -0.791846 q^{5} -0.0750818 q^{7} -2.42958 q^{8} +O(q^{10})\)	\(q+1.56422 q^{2} +0.446772 q^{4} -0.791846 q^{5} -0.0750818 q^{7} -2.42958 q^{8} -1.23862 q^{10} -2.59597 q^{11} +2.80412 q^{13} -0.117444 q^{14} -4.69394 q^{16} +4.74232 q^{17} +1.00000 q^{19} -0.353774 q^{20} -4.06066 q^{22} +2.22663 q^{23} -4.37298 q^{25} +4.38626 q^{26} -0.0335444 q^{28} +7.42999 q^{29} -4.23480 q^{31} -2.48317 q^{32} +7.41801 q^{34} +0.0594532 q^{35} -2.97148 q^{37} +1.56422 q^{38} +1.92386 q^{40} -10.0802 q^{41} +7.63595 q^{43} -1.15981 q^{44} +3.48293 q^{46} -1.00000 q^{47} -6.99436 q^{49} -6.84029 q^{50} +1.25280 q^{52} -4.24470 q^{53} +2.05561 q^{55} +0.182417 q^{56} +11.6221 q^{58} +7.03939 q^{59} +7.22754 q^{61} -6.62415 q^{62} +5.50367 q^{64} -2.22043 q^{65} -3.35873 q^{67} +2.11873 q^{68} +0.0929976 q^{70} -2.75757 q^{71} -8.10896 q^{73} -4.64803 q^{74} +0.446772 q^{76} +0.194910 q^{77} -15.8710 q^{79} +3.71687 q^{80} -15.7676 q^{82} +2.95130 q^{83} -3.75518 q^{85} +11.9443 q^{86} +6.30713 q^{88} -3.70208 q^{89} -0.210539 q^{91} +0.994797 q^{92} -1.56422 q^{94} -0.791846 q^{95} +4.88638 q^{97} -10.9407 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 2 q^{2} + 4 q^{4} - 6 q^{5} - 3 q^{7} - 6 q^{8}+O(q^{10}) \) 7 * q + 2 * q^2 + 4 * q^4 - 6 * q^5 - 3 * q^7 - 6 * q^8	\( 7 q + 2 q^{2} + 4 q^{4} - 6 q^{5} - 3 q^{7} - 6 q^{8} - 10 q^{10} + 9 q^{11} - 8 q^{13} + 9 q^{14} + 14 q^{16} - 7 q^{17} + 7 q^{19} + 4 q^{20} - 7 q^{22} + 7 q^{23} - 9 q^{25} - q^{26} - 12 q^{28} + 12 q^{29} - 13 q^{31} - 15 q^{32} + 2 q^{34} - 4 q^{35} - 10 q^{37} + 2 q^{38} - 26 q^{40} + 8 q^{41} + 12 q^{43} + 14 q^{44} - 7 q^{46} - 7 q^{47} - 16 q^{49} + 12 q^{50} - 6 q^{52} + 17 q^{53} + 7 q^{55} + 38 q^{56} - 12 q^{58} - 12 q^{59} - 23 q^{61} - 23 q^{62} + 12 q^{64} + 13 q^{65} + q^{67} - 15 q^{68} + 8 q^{70} - 29 q^{73} - 5 q^{74} + 4 q^{76} - 4 q^{77} - 3 q^{79} - 15 q^{80} - 7 q^{82} - 19 q^{83} - 5 q^{85} + 23 q^{86} - 33 q^{88} + 14 q^{89} - 3 q^{91} + 5 q^{92} - 2 q^{94} - 6 q^{95} - 17 q^{97} - 38 q^{98}+O(q^{100}) \) 7 * q + 2 * q^2 + 4 * q^4 - 6 * q^5 - 3 * q^7 - 6 * q^8 - 10 * q^10 + 9 * q^11 - 8 * q^13 + 9 * q^14 + 14 * q^16 - 7 * q^17 + 7 * q^19 + 4 * q^20 - 7 * q^22 + 7 * q^23 - 9 * q^25 - q^26 - 12 * q^28 + 12 * q^29 - 13 * q^31 - 15 * q^32 + 2 * q^34 - 4 * q^35 - 10 * q^37 + 2 * q^38 - 26 * q^40 + 8 * q^41 + 12 * q^43 + 14 * q^44 - 7 * q^46 - 7 * q^47 - 16 * q^49 + 12 * q^50 - 6 * q^52 + 17 * q^53 + 7 * q^55 + 38 * q^56 - 12 * q^58 - 12 * q^59 - 23 * q^61 - 23 * q^62 + 12 * q^64 + 13 * q^65 + q^67 - 15 * q^68 + 8 * q^70 - 29 * q^73 - 5 * q^74 + 4 * q^76 - 4 * q^77 - 3 * q^79 - 15 * q^80 - 7 * q^82 - 19 * q^83 - 5 * q^85 + 23 * q^86 - 33 * q^88 + 14 * q^89 - 3 * q^91 + 5 * q^92 - 2 * q^94 - 6 * q^95 - 17 * q^97 - 38 * q^98

Introduction

L-functions

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Database

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