LMFDB - Embedded newform 8001.2.a.w.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8001,2,Mod(1,8001)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8001, 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("8001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8001 = 3^{2} \cdot 7 \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8001.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:	$63.8883066572$
Analytic rank:	$1$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 8 x^{19} + 152 x^{17} - 274 x^{16} - 1061 x^{15} + 3125 x^{14} + 2977 x^{13} - 15474 x^{12} + \cdots + 31 $ x^20 - 8x^19 + 152x^17 - 274x^16 - 1061x^15 + 3125x^14 + 2977x^13 - 15474x^12 - 56x^11 + 39579x^10 - 17664x^9 - 52271x^8 + 35701x^7 + 32493x^6 - 25504x^5 - 8607x^4 + 6812x^3 + 609x^2 - 425x + 31
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 889)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$2.11976$ of defining polynomial
Character	$\chi$	$=$	8001.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.11976 q^{2} +2.49337 q^{4} -3.20666 q^{5} +1.00000 q^{7} -1.04583 q^{8} +O(q^{10})$	\(q-2.11976 q^{2} +2.49337 q^{4} -3.20666 q^{5} +1.00000 q^{7} -1.04583 q^{8} +6.79735 q^{10} -4.37963 q^{11} +3.90479 q^{13} -2.11976 q^{14} -2.76983 q^{16} +3.95821 q^{17} -4.13869 q^{19} -7.99541 q^{20} +9.28376 q^{22} -5.86310 q^{23} +5.28268 q^{25} -8.27720 q^{26} +2.49337 q^{28} -1.39389 q^{29} -0.976842 q^{31} +7.96305 q^{32} -8.39044 q^{34} -3.20666 q^{35} +2.06133 q^{37} +8.77303 q^{38} +3.35363 q^{40} +7.92706 q^{41} +7.54779 q^{43} -10.9201 q^{44} +12.4284 q^{46} -0.247316 q^{47} +1.00000 q^{49} -11.1980 q^{50} +9.73609 q^{52} -4.76242 q^{53} +14.0440 q^{55} -1.04583 q^{56} +2.95470 q^{58} +7.06605 q^{59} -4.24411 q^{61} +2.07067 q^{62} -11.3401 q^{64} -12.5213 q^{65} -3.83543 q^{67} +9.86929 q^{68} +6.79735 q^{70} +11.1273 q^{71} -14.6285 q^{73} -4.36952 q^{74} -10.3193 q^{76} -4.37963 q^{77} +8.46295 q^{79} +8.88192 q^{80} -16.8035 q^{82} -14.4831 q^{83} -12.6926 q^{85} -15.9995 q^{86} +4.58036 q^{88} -0.301233 q^{89} +3.90479 q^{91} -14.6189 q^{92} +0.524250 q^{94} +13.2714 q^{95} -14.8204 q^{97} -2.11976 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 8 q^{2} + 24 q^{4} - 3 q^{5} + 20 q^{7} - 24 q^{8}+O(q^{10}) $ 20 * q - 8 * q^2 + 24 * q^4 - 3 * q^5 + 20 * q^7 - 24 * q^8	$ 20 q - 8 q^{2} + 24 q^{4} - 3 q^{5} + 20 q^{7} - 24 q^{8} - 8 q^{10} - 26 q^{11} - 4 q^{13} - 8 q^{14} + 24 q^{16} - 4 q^{17} + q^{19} + 2 q^{20} + q^{22} - 31 q^{23} + 27 q^{25} - 4 q^{26} + 24 q^{28} - 16 q^{29} + 6 q^{31} - 41 q^{32} - 10 q^{34} - 3 q^{35} + 2 q^{37} - 3 q^{38} - 38 q^{40} - 25 q^{41} + 13 q^{43} - 66 q^{44} + 20 q^{46} - 19 q^{47} + 20 q^{49} + 4 q^{50} + 20 q^{52} - 24 q^{53} - 3 q^{55} - 24 q^{56} + 12 q^{58} - 23 q^{59} - 27 q^{61} - 7 q^{62} + 2 q^{64} - 26 q^{65} + 9 q^{67} + 25 q^{68} - 8 q^{70} - 63 q^{71} - 21 q^{73} - 21 q^{74} - 10 q^{76} - 26 q^{77} + 18 q^{79} + 23 q^{80} - 42 q^{82} + q^{83} - 41 q^{85} + 12 q^{86} + 57 q^{88} + 16 q^{89} - 4 q^{91} - 17 q^{92} + 7 q^{94} - 75 q^{95} - 32 q^{97} - 8 q^{98}+O(q^{100}) $ 20 * q - 8 * q^2 + 24 * q^4 - 3 * q^5 + 20 * q^7 - 24 * q^8 - 8 * q^10 - 26 * q^11 - 4 * q^13 - 8 * q^14 + 24 * q^16 - 4 * q^17 + q^19 + 2 * q^20 + q^22 - 31 * q^23 + 27 * q^25 - 4 * q^26 + 24 * q^28 - 16 * q^29 + 6 * q^31 - 41 * q^32 - 10 * q^34 - 3 * q^35 + 2 * q^37 - 3 * q^38 - 38 * q^40 - 25 * q^41 + 13 * q^43 - 66 * q^44 + 20 * q^46 - 19 * q^47 + 20 * q^49 + 4 * q^50 + 20 * q^52 - 24 * q^53 - 3 * q^55 - 24 * q^56 + 12 * q^58 - 23 * q^59 - 27 * q^61 - 7 * q^62 + 2 * q^64 - 26 * q^65 + 9 * q^67 + 25 * q^68 - 8 * q^70 - 63 * q^71 - 21 * q^73 - 21 * q^74 - 10 * q^76 - 26 * q^77 + 18 * q^79 + 23 * q^80 - 42 * q^82 + q^83 - 41 * q^85 + 12 * q^86 + 57 * q^88 + 16 * q^89 - 4 * q^91 - 17 * q^92 + 7 * q^94 - 75 * q^95 - 32 * q^97 - 8 * 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 $

$2$	−2.11976	−1.49890	−0.749448	−	0.662064i	$-0.769681\pi$
$2$	−2.11976	−1.49890	−0.749448	+	0.662064i	$0.769681\pi$
$3$	0	0
$3$	0	0
$4$	2.49337	1.24669
$5$	−3.20666	−1.43406	−0.717031	−	0.697041i	$-0.754500\pi$
$5$	−3.20666	−1.43406	−0.717031	+	0.697041i	$0.754500\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.04583	−0.369758
$9$	0	0
$10$	6.79735	2.14951
$11$	−4.37963	−1.32051	−0.660254	−	0.751042i	$-0.729551\pi$
$11$	−4.37963	−1.32051	−0.660254	+	0.751042i	$0.729551\pi$
$12$	0	0
$13$	3.90479	1.08299	0.541497	−	0.840703i	$-0.317858\pi$
$13$	3.90479	1.08299	0.541497	+	0.840703i	$0.317858\pi$
$14$	−2.11976	−0.566529
$15$	0	0
$16$	−2.76983	−0.692459
$17$	3.95821	0.960006	0.480003	−	0.877267i	$-0.340636\pi$
$17$	3.95821	0.960006	0.480003	+	0.877267i	$0.340636\pi$
$18$	0	0
$19$	−4.13869	−0.949481	−0.474741	−	0.880126i	$-0.657458\pi$
$19$	−4.13869	−0.949481	−0.474741	+	0.880126i	$0.657458\pi$
$20$	−7.99541	−1.78783
$21$	0	0
$22$	9.28376	1.97930
$23$	−5.86310	−1.22254	−0.611271	−	0.791422i	$-0.709341\pi$
$23$	−5.86310	−1.22254	−0.611271	+	0.791422i	$0.709341\pi$
$24$	0	0
$25$	5.28268	1.05654
$26$	−8.27720	−1.62329
$27$	0	0
$28$	2.49337	0.471203
$29$	−1.39389	−0.258838	−0.129419	−	0.991590i	$-0.541311\pi$
$29$	−1.39389	−0.258838	−0.129419	+	0.991590i	$0.541311\pi$
$30$	0	0
$31$	−0.976842	−0.175446	−0.0877230	−	0.996145i	$-0.527959\pi$
$31$	−0.976842	−0.175446	−0.0877230	+	0.996145i	$0.527959\pi$
$32$	7.96305	1.40768
$33$	0	0
$34$	−8.39044	−1.43895
$35$	−3.20666	−0.542025
$36$	0	0
$37$	2.06133	0.338881	0.169440	−	0.985540i	$-0.445804\pi$
$37$	2.06133	0.338881	0.169440	+	0.985540i	$0.445804\pi$
$38$	8.77303	1.42317
$39$	0	0
$40$	3.35363	0.530256
$41$	7.92706	1.23800	0.619000	−	0.785391i	$-0.287538\pi$
$41$	7.92706	1.23800	0.619000	+	0.785391i	$0.287538\pi$
$42$	0	0
$43$	7.54779	1.15103	0.575514	−	0.817792i	$-0.304802\pi$
$43$	7.54779	1.15103	0.575514	+	0.817792i	$0.304802\pi$
$44$	−10.9201	−1.64626
$45$	0	0
$46$	12.4284	1.83246
$47$	−0.247316	−0.0360748	−0.0180374	−	0.999837i	$-0.505742\pi$
$47$	−0.247316	−0.0360748	−0.0180374	+	0.999837i	$0.505742\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−11.1980	−1.58364
$51$	0	0
$52$	9.73609	1.35015
$53$	−4.76242	−0.654169	−0.327084	−	0.944995i	$-0.606066\pi$
$53$	−4.76242	−0.654169	−0.327084	+	0.944995i	$0.606066\pi$
$54$	0	0
$55$	14.0440	1.89369
$56$	−1.04583	−0.139755
$57$	0	0
$58$	2.95470	0.387972
$59$	7.06605	0.919922	0.459961	−	0.887939i	$-0.347863\pi$
$59$	7.06605	0.919922	0.459961	+	0.887939i	$0.347863\pi$
$60$	0	0
$61$	−4.24411	−0.543403	−0.271701	−	0.962382i	$-0.587586\pi$
$61$	−4.24411	−0.543403	−0.271701	+	0.962382i	$0.587586\pi$
$62$	2.07067	0.262975
$63$	0	0
$64$	−11.3401	−1.41751
$65$	−12.5213	−1.55308
$66$	0	0
$67$	−3.83543	−0.468572	−0.234286	−	0.972168i	$-0.575275\pi$
$67$	−3.83543	−0.468572	−0.234286	+	0.972168i	$0.575275\pi$
$68$	9.86929	1.19683
$69$	0	0
$70$	6.79735	0.812438
$71$	11.1273	1.32057	0.660285	−	0.751015i	$-0.270435\pi$
$71$	11.1273	1.32057	0.660285	+	0.751015i	$0.270435\pi$
$72$	0	0
$73$	−14.6285	−1.71214	−0.856071	−	0.516859i	$-0.827101\pi$
$73$	−14.6285	−1.71214	−0.856071	+	0.516859i	$0.827101\pi$
$74$	−4.36952	−0.507947
$75$	0	0
$76$	−10.3193	−1.18371
$77$	−4.37963	−0.499105
$78$	0	0
$79$	8.46295	0.952157	0.476078	−	0.879403i	$-0.342058\pi$
$79$	8.46295	0.952157	0.476078	+	0.879403i	$0.342058\pi$
$80$	8.88192	0.993029
$81$	0	0
$82$	−16.8035	−1.85563
$83$	−14.4831	−1.58972	−0.794862	−	0.606791i	$-0.792457\pi$
$83$	−14.4831	−1.58972	−0.794862	+	0.606791i	$0.792457\pi$
$84$	0	0
$85$	−12.6926	−1.37671
$86$	−15.9995	−1.72527
$87$	0	0
$88$	4.58036	0.488268
$89$	−0.301233	−0.0319307	−0.0159653	−	0.999873i	$-0.505082\pi$
$89$	−0.301233	−0.0319307	−0.0159653	+	0.999873i	$0.505082\pi$
$90$	0	0
$91$	3.90479	0.409333
$92$	−14.6189	−1.52413
$93$	0	0
$94$	0.524250	0.0540723
$95$	13.2714	1.36162
$96$	0	0
$97$	−14.8204	−1.50478	−0.752390	−	0.658718i	$-0.771099\pi$
$97$	−14.8204	−1.50478	−0.752390	+	0.658718i	$0.771099\pi$
$98$	−2.11976	−0.214128
$99$	0	0
$100$	13.1717	1.31717
$101$	15.8079	1.57295	0.786473	−	0.617624i	$-0.211905\pi$
$101$	15.8079	1.57295	0.786473	+	0.617624i	$0.211905\pi$
$102$	0	0
$103$	−6.09941	−0.600993	−0.300496	−	0.953783i	$-0.597152\pi$
$103$	−6.09941	−0.600993	−0.300496	+	0.953783i	$0.597152\pi$
$104$	−4.08376	−0.400445
$105$	0	0
$106$	10.0952	0.980531
$107$	2.31234	0.223542	0.111771	−	0.993734i	$-0.464348\pi$
$107$	2.31234	0.223542	0.111771	+	0.993734i	$0.464348\pi$
$108$	0	0
$109$	8.21987	0.787321	0.393661	−	0.919256i	$-0.371209\pi$
$109$	8.21987	0.787321	0.393661	+	0.919256i	$0.371209\pi$
$110$	−29.7699	−2.83844
$111$	0	0
$112$	−2.76983	−0.261725
$113$	9.92837	0.933982	0.466991	−	0.884262i	$-0.345338\pi$
$113$	9.92837	0.933982	0.466991	+	0.884262i	$0.345338\pi$
$114$	0	0
$115$	18.8010	1.75320
$116$	−3.47548	−0.322690
$117$	0	0
$118$	−14.9783	−1.37887
$119$	3.95821	0.362848
$120$	0	0
$121$	8.18116	0.743742
$122$	8.99648	0.814503
$123$	0	0
$124$	−2.43563	−0.218726
$125$	−0.906451	−0.0810754
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	8.11209	0.717014
$129$	0	0
$130$	26.5422	2.32790
$131$	−12.4577	−1.08844	−0.544219	−	0.838943i	$-0.683174\pi$
$131$	−12.4577	−1.08844	−0.544219	+	0.838943i	$0.683174\pi$
$132$	0	0
$133$	−4.13869	−0.358870
$134$	8.13018	0.702341
$135$	0	0
$136$	−4.13962	−0.354970
$137$	9.03759	0.772133	0.386067	−	0.922471i	$-0.373833\pi$
$137$	9.03759	0.772133	0.386067	+	0.922471i	$0.373833\pi$
$138$	0	0
$139$	−2.17778	−0.184717	−0.0923586	−	0.995726i	$-0.529441\pi$
$139$	−2.17778	−0.184717	−0.0923586	+	0.995726i	$0.529441\pi$
$140$	−7.99541	−0.675735
$141$	0	0
$142$	−23.5872	−1.97940
$143$	−17.1015	−1.43010
$144$	0	0
$145$	4.46972	0.371190
$146$	31.0090	2.56632
$147$	0	0
$148$	5.13967	0.422478
$149$	7.83362	0.641755	0.320878	−	0.947121i	$-0.396022\pi$
$149$	7.83362	0.641755	0.320878	+	0.947121i	$0.396022\pi$
$150$	0	0
$151$	−23.5469	−1.91622	−0.958111	−	0.286398i	$-0.907542\pi$
$151$	−23.5469	−1.91622	−0.958111	+	0.286398i	$0.907542\pi$
$152$	4.32838	0.351078
$153$	0	0
$154$	9.28376	0.748106
$155$	3.13240	0.251601
$156$	0	0
$157$	14.5119	1.15818	0.579090	−	0.815264i	$-0.303408\pi$
$157$	14.5119	1.15818	0.579090	+	0.815264i	$0.303408\pi$
$158$	−17.9394	−1.42718
$159$	0	0
$160$	−25.5348	−2.01870
$161$	−5.86310	−0.462077
$162$	0	0
$163$	10.2802	0.805211	0.402605	−	0.915374i	$-0.368105\pi$
$163$	10.2802	0.805211	0.402605	+	0.915374i	$0.368105\pi$
$164$	19.7651	1.54340
$165$	0	0
$166$	30.7006	2.38283
$167$	20.0932	1.55486	0.777428	−	0.628972i	$-0.216524\pi$
$167$	20.0932	1.55486	0.777428	+	0.628972i	$0.216524\pi$
$168$	0	0
$169$	2.24737	0.172874
$170$	26.9053	2.06354
$171$	0	0
$172$	18.8195	1.43497
$173$	2.61965	0.199168	0.0995840	−	0.995029i	$-0.468249\pi$
$173$	2.61965	0.199168	0.0995840	+	0.995029i	$0.468249\pi$
$174$	0	0
$175$	5.28268	0.399333
$176$	12.1309	0.914397
$177$	0	0
$178$	0.638542	0.0478607
$179$	−6.41529	−0.479502	−0.239751	−	0.970834i	$-0.577066\pi$
$179$	−6.41529	−0.479502	−0.239751	+	0.970834i	$0.577066\pi$
$180$	0	0
$181$	14.0321	1.04300	0.521499	−	0.853252i	$-0.325373\pi$
$181$	14.0321	1.04300	0.521499	+	0.853252i	$0.325373\pi$
$182$	−8.27720	−0.613547
$183$	0	0
$184$	6.13183	0.452044
$185$	−6.60999	−0.485976
$186$	0	0
$187$	−17.3355	−1.26770
$188$	−0.616651	−0.0449739
$189$	0	0
$190$	−28.1321	−2.04092
$191$	−12.9836	−0.939462	−0.469731	−	0.882810i	$-0.655649\pi$
$191$	−12.9836	−0.939462	−0.469731	+	0.882810i	$0.655649\pi$
$192$	0	0
$193$	3.08904	0.222354	0.111177	−	0.993801i	$-0.464538\pi$
$193$	3.08904	0.222354	0.111177	+	0.993801i	$0.464538\pi$
$194$	31.4156	2.25551
$195$	0	0
$196$	2.49337	0.178098
$197$	3.17836	0.226449	0.113224	−	0.993569i	$-0.463882\pi$
$197$	3.17836	0.226449	0.113224	+	0.993569i	$0.463882\pi$
$198$	0	0
$199$	−0.614895	−0.0435888	−0.0217944	−	0.999762i	$-0.506938\pi$
$199$	−0.614895	−0.0435888	−0.0217944	+	0.999762i	$0.506938\pi$
$200$	−5.52480	−0.390662
$201$	0	0
$202$	−33.5090	−2.35768
$203$	−1.39389	−0.0978317
$204$	0	0
$205$	−25.4194	−1.77537
$206$	12.9293	0.900825
$207$	0	0
$208$	−10.8156	−0.749928
$209$	18.1259	1.25380
$210$	0	0
$211$	22.8164	1.57075	0.785373	−	0.619022i	$-0.212471\pi$
$211$	22.8164	1.57075	0.785373	+	0.619022i	$0.212471\pi$
$212$	−11.8745	−0.815544
$213$	0	0
$214$	−4.90160	−0.335067
$215$	−24.2032	−1.65065
$216$	0	0
$217$	−0.976842	−0.0663124
$218$	−17.4241	−1.18011
$219$	0	0
$220$	35.0169	2.36084
$221$	15.4560	1.03968
$222$	0	0
$223$	−13.9406	−0.933530	−0.466765	−	0.884381i	$-0.654581\pi$
$223$	−13.9406	−0.933530	−0.466765	+	0.884381i	$0.654581\pi$
$224$	7.96305	0.532053
$225$	0	0
$226$	−21.0457	−1.39994
$227$	15.9829	1.06083	0.530413	−	0.847740i	$-0.322037\pi$
$227$	15.9829	1.06083	0.530413	+	0.847740i	$0.322037\pi$
$228$	0	0
$229$	23.3450	1.54268	0.771339	−	0.636424i	$-0.219587\pi$
$229$	23.3450	1.54268	0.771339	+	0.636424i	$0.219587\pi$
$230$	−39.8535	−2.62786
$231$	0	0
$232$	1.45777	0.0957075
$233$	−1.20424	−0.0788926	−0.0394463	−	0.999222i	$-0.512559\pi$
$233$	−1.20424	−0.0788926	−0.0394463	+	0.999222i	$0.512559\pi$
$234$	0	0
$235$	0.793059	0.0517335
$236$	17.6183	1.14685
$237$	0	0
$238$	−8.39044	−0.543872
$239$	−22.0925	−1.42904	−0.714522	−	0.699613i	$-0.753356\pi$
$239$	−22.0925	−1.42904	−0.714522	+	0.699613i	$0.753356\pi$
$240$	0	0
$241$	−7.16340	−0.461435	−0.230718	−	0.973021i	$-0.574107\pi$
$241$	−7.16340	−0.461435	−0.230718	+	0.973021i	$0.574107\pi$
$242$	−17.3421	−1.11479
$243$	0	0
$244$	−10.5822	−0.677453
$245$	−3.20666	−0.204866
$246$	0	0
$247$	−16.1607	−1.02828
$248$	1.02161	0.0648725
$249$	0	0
$250$	1.92146	0.121524
$251$	−5.24585	−0.331115	−0.165557	−	0.986200i	$-0.552942\pi$
$251$	−5.24585	−0.331115	−0.165557	+	0.986200i	$0.552942\pi$
$252$	0	0
$253$	25.6782	1.61438
$254$	2.11976	0.133005
$255$	0	0
$256$	5.48445	0.342778
$257$	8.15232	0.508527	0.254264	−	0.967135i	$-0.418167\pi$
$257$	8.15232	0.508527	0.254264	+	0.967135i	$0.418167\pi$
$258$	0	0
$259$	2.06133	0.128085
$260$	−31.2204	−1.93620
$261$	0	0
$262$	26.4074	1.63146
$263$	17.1136	1.05527	0.527635	−	0.849471i	$-0.323079\pi$
$263$	17.1136	1.05527	0.527635	+	0.849471i	$0.323079\pi$
$264$	0	0
$265$	15.2715	0.938119
$266$	8.77303	0.537909
$267$	0	0
$268$	−9.56316	−0.584163
$269$	0.836884	0.0510257	0.0255129	−	0.999674i	$-0.491878\pi$
$269$	0.836884	0.0510257	0.0255129	+	0.999674i	$0.491878\pi$
$270$	0	0
$271$	25.2416	1.53332	0.766658	−	0.642056i	$-0.221918\pi$
$271$	25.2416	1.53332	0.766658	+	0.642056i	$0.221918\pi$
$272$	−10.9636	−0.664765
$273$	0	0
$274$	−19.1575	−1.15735
$275$	−23.1362	−1.39516
$276$	0	0
$277$	−23.9163	−1.43699	−0.718495	−	0.695532i	$-0.755169\pi$
$277$	−23.9163	−1.43699	−0.718495	+	0.695532i	$0.755169\pi$
$278$	4.61638	0.276872
$279$	0	0
$280$	3.35363	0.200418
$281$	3.32317	0.198244	0.0991218	−	0.995075i	$-0.468397\pi$
$281$	3.32317	0.198244	0.0991218	+	0.995075i	$0.468397\pi$
$282$	0	0
$283$	32.6108	1.93851	0.969254	−	0.246062i	$-0.0791366\pi$
$283$	32.6108	1.93851	0.969254	+	0.246062i	$0.0791366\pi$
$284$	27.7446	1.64634
$285$	0	0
$286$	36.2511	2.14357
$287$	7.92706	0.467920
$288$	0	0
$289$	−1.33260	−0.0783880
$290$	−9.47473	−0.556375
$291$	0	0
$292$	−36.4744	−2.13450
$293$	−20.4783	−1.19636	−0.598178	−	0.801363i	$-0.704108\pi$
$293$	−20.4783	−1.19636	−0.598178	+	0.801363i	$0.704108\pi$
$294$	0	0
$295$	−22.6584	−1.31923
$296$	−2.15581	−0.125304
$297$	0	0
$298$	−16.6054	−0.961924
$299$	−22.8942	−1.32400
$300$	0	0
$301$	7.54779	0.435048
$302$	49.9138	2.87222
$303$	0	0
$304$	11.4635	0.657477
$305$	13.6094	0.779273
$306$	0	0
$307$	16.5722	0.945823	0.472911	−	0.881110i	$-0.343203\pi$
$307$	16.5722	0.945823	0.472911	+	0.881110i	$0.343203\pi$
$308$	−10.9201	−0.622228
$309$	0	0
$310$	−6.63993	−0.377123
$311$	−18.2896	−1.03711	−0.518554	−	0.855045i	$-0.673530\pi$
$311$	−18.2896	−1.03711	−0.518554	+	0.855045i	$0.673530\pi$
$312$	0	0
$313$	−12.2337	−0.691487	−0.345744	−	0.938329i	$-0.612373\pi$
$313$	−12.2337	−0.691487	−0.345744	+	0.938329i	$0.612373\pi$
$314$	−30.7618	−1.73599
$315$	0	0
$316$	21.1013	1.18704
$317$	3.18961	0.179146	0.0895732	−	0.995980i	$-0.471450\pi$
$317$	3.18961	0.179146	0.0895732	+	0.995980i	$0.471450\pi$
$318$	0	0
$319$	6.10471	0.341798
$320$	36.3637	2.03279
$321$	0	0
$322$	12.4284	0.692605
$323$	−16.3818	−0.911508
$324$	0	0
$325$	20.6277	1.14422
$326$	−21.7916	−1.20693
$327$	0	0
$328$	−8.29038	−0.457760
$329$	−0.247316	−0.0136350
$330$	0	0
$331$	28.8166	1.58390	0.791950	−	0.610585i	$-0.209066\pi$
$331$	28.8166	1.58390	0.791950	+	0.610585i	$0.209066\pi$
$332$	−36.1117	−1.98189
$333$	0	0
$334$	−42.5927	−2.33057
$335$	12.2989	0.671962
$336$	0	0
$337$	31.9159	1.73857	0.869284	−	0.494312i	$-0.164580\pi$
$337$	31.9159	1.73857	0.869284	+	0.494312i	$0.164580\pi$
$338$	−4.76387	−0.259120
$339$	0	0
$340$	−31.6475	−1.71633
$341$	4.27821	0.231678
$342$	0	0
$343$	1.00000	0.0539949
$344$	−7.89373	−0.425602
$345$	0	0
$346$	−5.55301	−0.298532
$347$	−17.8708	−0.959356	−0.479678	−	0.877445i	$-0.659247\pi$
$347$	−17.8708	−0.959356	−0.479678	+	0.877445i	$0.659247\pi$
$348$	0	0
$349$	−4.32433	−0.231476	−0.115738	−	0.993280i	$-0.536923\pi$
$349$	−4.32433	−0.231476	−0.115738	+	0.993280i	$0.536923\pi$
$350$	−11.1980	−0.598558
$351$	0	0
$352$	−34.8752	−1.85885
$353$	−36.6788	−1.95221	−0.976107	−	0.217291i	$-0.930278\pi$
$353$	−36.6788	−1.95221	−0.976107	+	0.217291i	$0.930278\pi$
$354$	0	0
$355$	−35.6816	−1.89378
$356$	−0.751087	−0.0398075
$357$	0	0
$358$	13.5989	0.718723
$359$	−13.4119	−0.707852	−0.353926	−	0.935273i	$-0.615154\pi$
$359$	−13.4119	−0.707852	−0.353926	+	0.935273i	$0.615154\pi$
$360$	0	0
$361$	−1.87122	−0.0984851
$362$	−29.7446	−1.56334
$363$	0	0
$364$	9.73609	0.510310
$365$	46.9088	2.45532
$366$	0	0
$367$	−28.8995	−1.50854	−0.754270	−	0.656564i	$-0.772009\pi$
$367$	−28.8995	−1.50854	−0.754270	+	0.656564i	$0.772009\pi$
$368$	16.2398	0.846559
$369$	0	0
$370$	14.0116	0.728427
$371$	−4.76242	−0.247253
$372$	0	0
$373$	−3.07036	−0.158977	−0.0794886	−	0.996836i	$-0.525329\pi$
$373$	−3.07036	−0.158977	−0.0794886	+	0.996836i	$0.525329\pi$
$374$	36.7470	1.90014
$375$	0	0
$376$	0.258651	0.0133389
$377$	−5.44283	−0.280320
$378$	0	0
$379$	−30.6749	−1.57567	−0.787833	−	0.615889i	$-0.788797\pi$
$379$	−30.6749	−1.57567	−0.787833	+	0.615889i	$0.788797\pi$
$380$	33.0905	1.69751
$381$	0	0
$382$	27.5221	1.40815
$383$	−36.8833	−1.88465	−0.942323	−	0.334704i	$-0.891364\pi$
$383$	−36.8833	−1.88465	−0.942323	+	0.334704i	$0.891364\pi$
$384$	0	0
$385$	14.0440	0.715748
$386$	−6.54802	−0.333285
$387$	0	0
$388$	−36.9527	−1.87599
$389$	29.1092	1.47590	0.737949	−	0.674857i	$-0.235795\pi$
$389$	29.1092	1.47590	0.737949	+	0.674857i	$0.235795\pi$
$390$	0	0
$391$	−23.2074	−1.17365
$392$	−1.04583	−0.0528225
$393$	0	0
$394$	−6.73735	−0.339423
$395$	−27.1378	−1.36545
$396$	0	0
$397$	−34.6840	−1.74074	−0.870369	−	0.492399i	$-0.836120\pi$
$397$	−34.6840	−1.74074	−0.870369	+	0.492399i	$0.836120\pi$
$398$	1.30343	0.0653350
$399$	0	0
$400$	−14.6321	−0.731607
$401$	5.90891	0.295077	0.147539	−	0.989056i	$-0.452865\pi$
$401$	5.90891	0.295077	0.147539	+	0.989056i	$0.452865\pi$
$402$	0	0
$403$	−3.81436	−0.190007
$404$	39.4150	1.96097
$405$	0	0
$406$	2.95470	0.146639
$407$	−9.02786	−0.447495
$408$	0	0
$409$	−5.66900	−0.280314	−0.140157	−	0.990129i	$-0.544761\pi$
$409$	−5.66900	−0.280314	−0.140157	+	0.990129i	$0.544761\pi$
$410$	53.8830	2.66109
$411$	0	0
$412$	−15.2081	−0.749249
$413$	7.06605	0.347698
$414$	0	0
$415$	46.4423	2.27976
$416$	31.0940	1.52451
$417$	0	0
$418$	−38.4226	−1.87931
$419$	−0.127009	−0.00620481	−0.00310241	−	0.999995i	$-0.500988\pi$
$419$	−0.127009	−0.00620481	−0.00310241	+	0.999995i	$0.500988\pi$
$420$	0	0
$421$	28.1074	1.36987	0.684936	−	0.728603i	$-0.259830\pi$
$421$	28.1074	1.36987	0.684936	+	0.728603i	$0.259830\pi$
$422$	−48.3653	−2.35439
$423$	0	0
$424$	4.98070	0.241884
$425$	20.9099	1.01428
$426$	0	0
$427$	−4.24411	−0.205387
$428$	5.76553	0.278687
$429$	0	0
$430$	51.3050	2.47415
$431$	−33.4046	−1.60904	−0.804520	−	0.593925i	$-0.797578\pi$
$431$	−33.4046	−1.60904	−0.804520	+	0.593925i	$0.797578\pi$
$432$	0	0
$433$	9.16371	0.440380	0.220190	−	0.975457i	$-0.429332\pi$
$433$	9.16371	0.440380	0.220190	+	0.975457i	$0.429332\pi$
$434$	2.07067	0.0993953
$435$	0	0
$436$	20.4952	0.981543
$437$	24.2656	1.16078
$438$	0	0
$439$	2.26371	0.108041	0.0540204	−	0.998540i	$-0.482796\pi$
$439$	2.26371	0.108041	0.0540204	+	0.998540i	$0.482796\pi$
$440$	−14.6877	−0.700207
$441$	0	0
$442$	−32.7629	−1.55837
$443$	−39.4281	−1.87329	−0.936644	−	0.350284i	$-0.886085\pi$
$443$	−39.4281	−1.87329	−0.936644	+	0.350284i	$0.886085\pi$
$444$	0	0
$445$	0.965953	0.0457906
$446$	29.5507	1.39926
$447$	0	0
$448$	−11.3401	−0.535767
$449$	−25.8732	−1.22103	−0.610515	−	0.792004i	$-0.709038\pi$
$449$	−25.8732	−1.22103	−0.610515	+	0.792004i	$0.709038\pi$
$450$	0	0
$451$	−34.7176	−1.63479
$452$	24.7551	1.16438
$453$	0	0
$454$	−33.8800	−1.59007
$455$	−12.5213	−0.587009
$456$	0	0
$457$	26.8692	1.25689	0.628444	−	0.777855i	$-0.283692\pi$
$457$	26.8692	1.25689	0.628444	+	0.777855i	$0.283692\pi$
$458$	−49.4857	−2.31231
$459$	0	0
$460$	46.8779	2.18569
$461$	−7.20967	−0.335788	−0.167894	−	0.985805i	$-0.553697\pi$
$461$	−7.20967	−0.335788	−0.167894	+	0.985805i	$0.553697\pi$
$462$	0	0
$463$	−19.7372	−0.917266	−0.458633	−	0.888626i	$-0.651661\pi$
$463$	−19.7372	−0.917266	−0.458633	+	0.888626i	$0.651661\pi$
$464$	3.86084	0.179235
$465$	0	0
$466$	2.55271	0.118252
$467$	−11.5910	−0.536368	−0.268184	−	0.963368i	$-0.586424\pi$
$467$	−11.5910	−0.536368	−0.268184	+	0.963368i	$0.586424\pi$
$468$	0	0
$469$	−3.83543	−0.177104
$470$	−1.68109	−0.0775430
$471$	0	0
$472$	−7.38991	−0.340148
$473$	−33.0565	−1.51994
$474$	0	0
$475$	−21.8634	−1.00316
$476$	9.86929	0.452358
$477$	0	0
$478$	46.8307	2.14199
$479$	−19.3197	−0.882741	−0.441371	−	0.897325i	$-0.645508\pi$
$479$	−19.3197	−0.882741	−0.441371	+	0.897325i	$0.645508\pi$
$480$	0	0
$481$	8.04906	0.367005
$482$	15.1847	0.691643
$483$	0	0
$484$	20.3987	0.927213
$485$	47.5239	2.15795
$486$	0	0
$487$	−41.5988	−1.88502	−0.942511	−	0.334176i	$-0.891542\pi$
$487$	−41.5988	−1.88502	−0.942511	+	0.334176i	$0.891542\pi$
$488$	4.43863	0.200927
$489$	0	0
$490$	6.79735	0.307073
$491$	26.3651	1.18984	0.594920	−	0.803785i	$-0.297184\pi$
$491$	26.3651	1.18984	0.594920	+	0.803785i	$0.297184\pi$
$492$	0	0
$493$	−5.51729	−0.248486
$494$	34.2568	1.54129
$495$	0	0
$496$	2.70569	0.121489
$497$	11.1273	0.499129
$498$	0	0
$499$	−12.7721	−0.571756	−0.285878	−	0.958266i	$-0.592285\pi$
$499$	−12.7721	−0.571756	−0.285878	+	0.958266i	$0.592285\pi$
$500$	−2.26012	−0.101076
$501$	0	0
$502$	11.1199	0.496307
$503$	20.4654	0.912508	0.456254	−	0.889850i	$-0.349191\pi$
$503$	20.4654	0.912508	0.456254	+	0.889850i	$0.349191\pi$
$504$	0	0
$505$	−50.6906	−2.25570
$506$	−54.4316	−2.41978
$507$	0	0
$508$	−2.49337	−0.110626
$509$	3.08629	0.136797	0.0683987	−	0.997658i	$-0.478211\pi$
$509$	3.08629	0.136797	0.0683987	+	0.997658i	$0.478211\pi$
$510$	0	0
$511$	−14.6285	−0.647129
$512$	−27.8499	−1.23080
$513$	0	0
$514$	−17.2809	−0.762229
$515$	19.5587	0.861861
$516$	0	0
$517$	1.08315	0.0476370
$518$	−4.36952	−0.191986
$519$	0	0
$520$	13.0952	0.574263
$521$	−21.7234	−0.951717	−0.475859	−	0.879522i	$-0.657863\pi$
$521$	−21.7234	−0.951717	−0.475859	+	0.879522i	$0.657863\pi$
$522$	0	0
$523$	14.7589	0.645361	0.322681	−	0.946508i	$-0.395416\pi$
$523$	14.7589	0.645361	0.322681	+	0.946508i	$0.395416\pi$
$524$	−31.0618	−1.35694
$525$	0	0
$526$	−36.2767	−1.58174
$527$	−3.86654	−0.168429
$528$	0	0
$529$	11.3760	0.494607
$530$	−32.3718	−1.40614
$531$	0	0
$532$	−10.3193	−0.447399
$533$	30.9535	1.34074
$534$	0	0
$535$	−7.41489	−0.320574
$536$	4.01122	0.173258
$537$	0	0
$538$	−1.77399	−0.0764822
$539$	−4.37963	−0.188644
$540$	0	0
$541$	36.5105	1.56971	0.784855	−	0.619680i	$-0.212738\pi$
$541$	36.5105	1.56971	0.784855	+	0.619680i	$0.212738\pi$
$542$	−53.5060	−2.29828
$543$	0	0
$544$	31.5194	1.35138
$545$	−26.3584	−1.12907
$546$	0	0
$547$	−13.3081	−0.569011	−0.284506	−	0.958674i	$-0.591829\pi$
$547$	−13.3081	−0.569011	−0.284506	+	0.958674i	$0.591829\pi$
$548$	22.5341	0.962609
$549$	0	0
$550$	49.0431	2.09120
$551$	5.76887	0.245762
$552$	0	0
$553$	8.46295	0.359881
$554$	50.6967	2.15390
$555$	0	0
$556$	−5.43003	−0.230285
$557$	10.0623	0.426353	0.213176	−	0.977014i	$-0.431619\pi$
$557$	10.0623	0.426353	0.213176	+	0.977014i	$0.431619\pi$
$558$	0	0
$559$	29.4725	1.24656
$560$	8.88192	0.375330
$561$	0	0
$562$	−7.04431	−0.297146
$563$	9.98158	0.420673	0.210337	−	0.977629i	$-0.432544\pi$
$563$	9.98158	0.420673	0.210337	+	0.977629i	$0.432544\pi$
$564$	0	0
$565$	−31.8369	−1.33939
$566$	−69.1269	−2.90562
$567$	0	0
$568$	−11.6373	−0.488291
$569$	25.7916	1.08124	0.540620	−	0.841267i	$-0.318190\pi$
$569$	25.7916	1.08124	0.540620	+	0.841267i	$0.318190\pi$
$570$	0	0
$571$	28.2727	1.18317	0.591587	−	0.806241i	$-0.298502\pi$
$571$	28.2727	1.18317	0.591587	+	0.806241i	$0.298502\pi$
$572$	−42.6405	−1.78289
$573$	0	0
$574$	−16.8035	−0.701363
$575$	−30.9729	−1.29166
$576$	0	0
$577$	−17.5871	−0.732161	−0.366081	−	0.930583i	$-0.619301\pi$
$577$	−17.5871	−0.732161	−0.366081	+	0.930583i	$0.619301\pi$
$578$	2.82478	0.117495
$579$	0	0
$580$	11.1447	0.462758
$581$	−14.4831	−0.600859
$582$	0	0
$583$	20.8576	0.863835
$584$	15.2990	0.633078
$585$	0	0
$586$	43.4091	1.79321
$587$	21.2557	0.877316	0.438658	−	0.898654i	$-0.355454\pi$
$587$	21.2557	0.877316	0.438658	+	0.898654i	$0.355454\pi$
$588$	0	0
$589$	4.04285	0.166583
$590$	48.0304	1.97738
$591$	0	0
$592$	−5.70955	−0.234661
$593$	15.7102	0.645140	0.322570	−	0.946546i	$-0.395453\pi$
$593$	15.7102	0.645140	0.322570	+	0.946546i	$0.395453\pi$
$594$	0	0
$595$	−12.6926	−0.520347
$596$	19.5321	0.800068
$597$	0	0
$598$	48.5301	1.98454
$599$	21.1995	0.866186	0.433093	−	0.901349i	$-0.357422\pi$
$599$	21.1995	0.866186	0.433093	+	0.901349i	$0.357422\pi$
$600$	0	0
$601$	−1.42150	−0.0579842	−0.0289921	−	0.999580i	$-0.509230\pi$
$601$	−1.42150	−0.0579842	−0.0289921	+	0.999580i	$0.509230\pi$
$602$	−15.9995	−0.652091
$603$	0	0
$604$	−58.7113	−2.38893
$605$	−26.2342	−1.06657
$606$	0	0
$607$	−20.4202	−0.828830	−0.414415	−	0.910088i	$-0.636014\pi$
$607$	−20.4202	−0.828830	−0.414415	+	0.910088i	$0.636014\pi$
$608$	−32.9566	−1.33657
$609$	0	0
$610$	−28.8487	−1.16805
$611$	−0.965717	−0.0390687
$612$	0	0
$613$	14.3253	0.578595	0.289297	−	0.957239i	$-0.406578\pi$
$613$	14.3253	0.578595	0.289297	+	0.957239i	$0.406578\pi$
$614$	−35.1290	−1.41769
$615$	0	0
$616$	4.58036	0.184548
$617$	−15.4453	−0.621804	−0.310902	−	0.950442i	$-0.600631\pi$
$617$	−15.4453	−0.621804	−0.310902	+	0.950442i	$0.600631\pi$
$618$	0	0
$619$	−5.61002	−0.225486	−0.112743	−	0.993624i	$-0.535964\pi$
$619$	−5.61002	−0.225486	−0.112743	+	0.993624i	$0.535964\pi$
$620$	7.81025	0.313667
$621$	0	0
$622$	38.7695	1.55452
$623$	−0.301233	−0.0120687
$624$	0	0
$625$	−23.5067	−0.940268
$626$	25.9324	1.03647
$627$	0	0
$628$	36.1837	1.44389
$629$	8.15917	0.325328
$630$	0	0
$631$	−37.5009	−1.49289	−0.746443	−	0.665449i	$-0.768240\pi$
$631$	−37.5009	−1.49289	−0.746443	+	0.665449i	$0.768240\pi$
$632$	−8.85084	−0.352067
$633$	0	0
$634$	−6.76120	−0.268522
$635$	3.20666	0.127252
$636$	0	0
$637$	3.90479	0.154713
$638$	−12.9405	−0.512320
$639$	0	0
$640$	−26.0127	−1.02824
$641$	−27.5551	−1.08836	−0.544180	−	0.838969i	$-0.683159\pi$
$641$	−27.5551	−1.08836	−0.544180	+	0.838969i	$0.683159\pi$
$642$	0	0
$643$	33.0013	1.30145	0.650723	−	0.759315i	$-0.274466\pi$
$643$	33.0013	1.30145	0.650723	+	0.759315i	$0.274466\pi$
$644$	−14.6189	−0.576066
$645$	0	0
$646$	34.7255	1.36626
$647$	31.6531	1.24441	0.622206	−	0.782854i	$-0.286237\pi$
$647$	31.6531	1.24441	0.622206	+	0.782854i	$0.286237\pi$
$648$	0	0
$649$	−30.9467	−1.21476
$650$	−43.7258	−1.71507
$651$	0	0
$652$	25.6325	1.00385
$653$	4.23558	0.165751	0.0828754	−	0.996560i	$-0.473590\pi$
$653$	4.23558	0.165751	0.0828754	+	0.996560i	$0.473590\pi$
$654$	0	0
$655$	39.9478	1.56089
$656$	−21.9567	−0.857263
$657$	0	0
$658$	0.524250	0.0204374
$659$	−27.7937	−1.08269	−0.541344	−	0.840801i	$-0.682084\pi$
$659$	−27.7937	−1.08269	−0.541344	+	0.840801i	$0.682084\pi$
$660$	0	0
$661$	−24.2927	−0.944878	−0.472439	−	0.881363i	$-0.656626\pi$
$661$	−24.2927	−0.944878	−0.472439	+	0.881363i	$0.656626\pi$
$662$	−61.0841	−2.37410
$663$	0	0
$664$	15.1469	0.587813
$665$	13.2714	0.514642
$666$	0	0
$667$	8.17250	0.316441
$668$	50.0998	1.93842
$669$	0	0
$670$	−26.0707	−1.00720
$671$	18.5876	0.717567
$672$	0	0
$673$	−29.0309	−1.11906	−0.559529	−	0.828811i	$-0.689018\pi$
$673$	−29.0309	−1.11906	−0.559529	+	0.828811i	$0.689018\pi$
$674$	−67.6539	−2.60593
$675$	0	0
$676$	5.60352	0.215520
$677$	−13.7360	−0.527919	−0.263959	−	0.964534i	$-0.585028\pi$
$677$	−13.7360	−0.527919	−0.263959	+	0.964534i	$0.585028\pi$
$678$	0	0
$679$	−14.8204	−0.568753
$680$	13.2744	0.509049
$681$	0	0
$682$	−9.06876	−0.347261
$683$	19.1701	0.733522	0.366761	−	0.930315i	$-0.380467\pi$
$683$	19.1701	0.733522	0.366761	+	0.930315i	$0.380467\pi$
$684$	0	0
$685$	−28.9805	−1.10729
$686$	−2.11976	−0.0809327
$687$	0	0
$688$	−20.9061	−0.797039
$689$	−18.5962	−0.708461
$690$	0	0
$691$	−46.8726	−1.78312	−0.891559	−	0.452905i	$-0.850388\pi$
$691$	−46.8726	−1.78312	−0.891559	+	0.452905i	$0.850388\pi$
$692$	6.53176	0.248300
$693$	0	0
$694$	37.8818	1.43797
$695$	6.98342	0.264896
$696$	0	0
$697$	31.3770	1.18849
$698$	9.16653	0.346958
$699$	0	0
$700$	13.1717	0.497843
$701$	27.4278	1.03593	0.517967	−	0.855400i	$-0.326689\pi$
$701$	27.4278	1.03593	0.517967	+	0.855400i	$0.326689\pi$
$702$	0	0
$703$	−8.53122	−0.321761
$704$	49.6653	1.87183
$705$	0	0
$706$	77.7501	2.92616
$707$	15.8079	0.594518
$708$	0	0
$709$	−37.4504	−1.40648	−0.703240	−	0.710952i	$-0.748264\pi$
$709$	−37.4504	−1.40648	−0.703240	+	0.710952i	$0.748264\pi$
$710$	75.6363	2.83858
$711$	0	0
$712$	0.315040	0.0118066
$713$	5.72732	0.214490
$714$	0	0
$715$	54.8388	2.05085
$716$	−15.9957	−0.597788
$717$	0	0
$718$	28.4299	1.06100
$719$	−42.3358	−1.57886	−0.789430	−	0.613841i	$-0.789624\pi$
$719$	−42.3358	−1.57886	−0.789430	+	0.613841i	$0.789624\pi$
$720$	0	0
$721$	−6.09941	−0.227154
$722$	3.96653	0.147619
$723$	0	0
$724$	34.9873	1.30029
$725$	−7.36346	−0.273472
$726$	0	0
$727$	−20.3897	−0.756213	−0.378107	−	0.925762i	$-0.623425\pi$
$727$	−20.3897	−0.756213	−0.378107	+	0.925762i	$0.623425\pi$
$728$	−4.08376	−0.151354
$729$	0	0
$730$	−99.4353	−3.68026
$731$	29.8757	1.10499
$732$	0	0
$733$	35.9902	1.32933	0.664664	−	0.747142i	$-0.268575\pi$
$733$	35.9902	1.32933	0.664664	+	0.747142i	$0.268575\pi$
$734$	61.2599	2.26114
$735$	0	0
$736$	−46.6881	−1.72095
$737$	16.7978	0.618753
$738$	0	0
$739$	12.4172	0.456775	0.228388	−	0.973570i	$-0.426655\pi$
$739$	12.4172	0.456775	0.228388	+	0.973570i	$0.426655\pi$
$740$	−16.4812	−0.605860
$741$	0	0
$742$	10.0952	0.370606
$743$	5.39300	0.197850	0.0989249	−	0.995095i	$-0.468460\pi$
$743$	5.39300	0.197850	0.0989249	+	0.995095i	$0.468460\pi$
$744$	0	0
$745$	−25.1198	−0.920317
$746$	6.50842	0.238290
$747$	0	0
$748$	−43.2238	−1.58042
$749$	2.31234	0.0844911
$750$	0	0
$751$	−24.0627	−0.878062	−0.439031	−	0.898472i	$-0.644678\pi$
$751$	−24.0627	−0.878062	−0.439031	+	0.898472i	$0.644678\pi$
$752$	0.685025	0.0249803
$753$	0	0
$754$	11.5375	0.420171
$755$	75.5070	2.74798
$756$	0	0
$757$	−39.3514	−1.43025	−0.715126	−	0.698996i	$-0.753631\pi$
$757$	−39.3514	−1.43025	−0.715126	+	0.698996i	$0.753631\pi$
$758$	65.0235	2.36176
$759$	0	0
$760$	−13.8797	−0.503468
$761$	−35.4137	−1.28375	−0.641873	−	0.766811i	$-0.721842\pi$
$761$	−35.4137	−1.28375	−0.641873	+	0.766811i	$0.721842\pi$
$762$	0	0
$763$	8.21987	0.297579
$764$	−32.3730	−1.17121
$765$	0	0
$766$	78.1836	2.82489
$767$	27.5914	0.996269
$768$	0	0
$769$	31.6281	1.14054	0.570269	−	0.821458i	$-0.306839\pi$
$769$	31.6281	1.14054	0.570269	+	0.821458i	$0.306839\pi$
$770$	−29.7699	−1.07283
$771$	0	0
$772$	7.70213	0.277206
$773$	−4.42026	−0.158986	−0.0794928	−	0.996835i	$-0.525330\pi$
$773$	−4.42026	−0.158986	−0.0794928	+	0.996835i	$0.525330\pi$
$774$	0	0
$775$	−5.16034	−0.185365
$776$	15.4996	0.556404
$777$	0	0
$778$	−61.7046	−2.21222
$779$	−32.8077	−1.17546
$780$	0	0
$781$	−48.7335	−1.74382
$782$	49.1940	1.75917
$783$	0	0
$784$	−2.76983	−0.0989227
$785$	−46.5349	−1.66090
$786$	0	0
$787$	22.6944	0.808967	0.404483	−	0.914545i	$-0.367451\pi$
$787$	22.6944	0.808967	0.404483	+	0.914545i	$0.367451\pi$
$788$	7.92484	0.282311
$789$	0	0
$790$	57.5256	2.04667
$791$	9.92837	0.353012
$792$	0	0
$793$	−16.5723	−0.588501
$794$	73.5216	2.60919
$795$	0	0
$796$	−1.53316	−0.0543415
$797$	−18.2089	−0.644991	−0.322496	−	0.946571i	$-0.604522\pi$
$797$	−18.2089	−0.644991	−0.322496	+	0.946571i	$0.604522\pi$
$798$	0	0
$799$	−0.978928	−0.0346320
$800$	42.0662	1.48726
$801$	0	0
$802$	−12.5255	−0.442290
$803$	64.0676	2.26090
$804$	0	0
$805$	18.8010	0.662648
$806$	8.08552	0.284800
$807$	0	0
$808$	−16.5324	−0.581609
$809$	−31.5823	−1.11037	−0.555187	−	0.831726i	$-0.687353\pi$
$809$	−31.5823	−1.11037	−0.555187	+	0.831726i	$0.687353\pi$
$810$	0	0
$811$	−38.7953	−1.36229	−0.681144	−	0.732150i	$-0.738517\pi$
$811$	−38.7953	−1.36229	−0.681144	+	0.732150i	$0.738517\pi$
$812$	−3.47548	−0.121965
$813$	0	0
$814$	19.1369	0.670748
$815$	−32.9653	−1.15472
$816$	0	0
$817$	−31.2380	−1.09288
$818$	12.0169	0.420161
$819$	0	0
$820$	−63.3801	−2.21333
$821$	−35.7065	−1.24616	−0.623082	−	0.782156i	$-0.714120\pi$
$821$	−35.7065	−1.24616	−0.623082	+	0.782156i	$0.714120\pi$
$822$	0	0
$823$	−27.8047	−0.969210	−0.484605	−	0.874733i	$-0.661037\pi$
$823$	−27.8047	−0.969210	−0.484605	+	0.874733i	$0.661037\pi$
$824$	6.37896	0.222222
$825$	0	0
$826$	−14.9783	−0.521162
$827$	−49.4847	−1.72075	−0.860376	−	0.509661i	$-0.829771\pi$
$827$	−49.4847	−1.72075	−0.860376	+	0.509661i	$0.829771\pi$
$828$	0	0
$829$	49.0185	1.70248	0.851242	−	0.524773i	$-0.175850\pi$
$829$	49.0185	1.70248	0.851242	+	0.524773i	$0.175850\pi$
$830$	−98.4464	−3.41713
$831$	0	0
$832$	−44.2805	−1.53515
$833$	3.95821	0.137144
$834$	0	0
$835$	−64.4320	−2.22976
$836$	45.1948	1.56309
$837$	0	0
$838$	0.269229	0.00930036
$839$	−10.1163	−0.349253	−0.174626	−	0.984635i	$-0.555872\pi$
$839$	−10.1163	−0.349253	−0.174626	+	0.984635i	$0.555872\pi$
$840$	0	0
$841$	−27.0571	−0.933003
$842$	−59.5809	−2.05330
$843$	0	0
$844$	56.8899	1.95823
$845$	−7.20654	−0.247913
$846$	0	0
$847$	8.18116	0.281108
$848$	13.1911	0.452985
$849$	0	0
$850$	−44.3240	−1.52030
$851$	−12.0858	−0.414296
$852$	0	0
$853$	32.7118	1.12003	0.560015	−	0.828483i	$-0.310795\pi$
$853$	32.7118	1.12003	0.560015	+	0.828483i	$0.310795\pi$
$854$	8.99648	0.307853
$855$	0	0
$856$	−2.41832	−0.0826566
$857$	31.5800	1.07875	0.539376	−	0.842065i	$-0.318660\pi$
$857$	31.5800	1.07875	0.539376	+	0.842065i	$0.318660\pi$
$858$	0	0
$859$	−44.2094	−1.50840	−0.754202	−	0.656642i	$-0.771976\pi$
$859$	−44.2094	−1.50840	−0.754202	+	0.656642i	$0.771976\pi$
$860$	−60.3477	−2.05784
$861$	0	0
$862$	70.8096	2.41178
$863$	26.6665	0.907737	0.453869	−	0.891069i	$-0.350044\pi$
$863$	26.6665	0.907737	0.453869	+	0.891069i	$0.350044\pi$
$864$	0	0
$865$	−8.40032	−0.285619
$866$	−19.4248	−0.660083
$867$	0	0
$868$	−2.43563	−0.0826707
$869$	−37.0646	−1.25733
$870$	0	0
$871$	−14.9765	−0.507461
$872$	−8.59662	−0.291118
$873$	0	0
$874$	−51.4372	−1.73989
$875$	−0.906451	−0.0306436
$876$	0	0
$877$	−29.3537	−0.991205	−0.495602	−	0.868550i	$-0.665053\pi$
$877$	−29.3537	−0.991205	−0.495602	+	0.868550i	$0.665053\pi$
$878$	−4.79851	−0.161942
$879$	0	0
$880$	−38.8995	−1.31130
$881$	−9.62795	−0.324374	−0.162187	−	0.986760i	$-0.551855\pi$
$881$	−9.62795	−0.324374	−0.162187	+	0.986760i	$0.551855\pi$
$882$	0	0
$883$	−13.9119	−0.468172	−0.234086	−	0.972216i	$-0.575210\pi$
$883$	−13.9119	−0.468172	−0.234086	+	0.972216i	$0.575210\pi$
$884$	38.5375	1.29616
$885$	0	0
$886$	83.5781	2.80786
$887$	19.3299	0.649035	0.324518	−	0.945880i	$-0.394798\pi$
$887$	19.3299	0.649035	0.324518	+	0.945880i	$0.394798\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	−2.04759	−0.0686353
$891$	0	0
$892$	−34.7591	−1.16382
$893$	1.02357	0.0342523
$894$	0	0
$895$	20.5717	0.687635
$896$	8.11209	0.271006
$897$	0	0
$898$	54.8449	1.83020
$899$	1.36161	0.0454122
$900$	0	0
$901$	−18.8507	−0.628006
$902$	73.5929	2.45038
$903$	0	0
$904$	−10.3834	−0.345347
$905$	−44.9962	−1.49572
$906$	0	0
$907$	16.5382	0.549142	0.274571	−	0.961567i	$-0.411464\pi$
$907$	16.5382	0.549142	0.274571	+	0.961567i	$0.411464\pi$
$908$	39.8515	1.32252
$909$	0	0
$910$	26.5422	0.879865
$911$	−39.0056	−1.29231	−0.646157	−	0.763205i	$-0.723625\pi$
$911$	−39.0056	−1.29231	−0.646157	+	0.763205i	$0.723625\pi$
$912$	0	0
$913$	63.4305	2.09924
$914$	−56.9562	−1.88394
$915$	0	0
$916$	58.2077	1.92324
$917$	−12.4577	−0.411391
$918$	0	0
$919$	12.0334	0.396947	0.198473	−	0.980106i	$-0.436402\pi$
$919$	12.0334	0.396947	0.198473	+	0.980106i	$0.436402\pi$
$920$	−19.6627	−0.648260
$921$	0	0
$922$	15.2828	0.503311
$923$	43.4498	1.43017
$924$	0	0
$925$	10.8893	0.358039
$926$	41.8381	1.37489
$927$	0	0
$928$	−11.0996	−0.364362
$929$	−12.8613	−0.421965	−0.210982	−	0.977490i	$-0.567666\pi$
$929$	−12.8613	−0.421965	−0.210982	+	0.977490i	$0.567666\pi$
$930$	0	0
$931$	−4.13869	−0.135640
$932$	−3.00263	−0.0983544
$933$	0	0
$934$	24.5701	0.803960
$935$	55.5890	1.81796
$936$	0	0
$937$	26.8454	0.877000	0.438500	−	0.898731i	$-0.355510\pi$
$937$	26.8454	0.877000	0.438500	+	0.898731i	$0.355510\pi$
$938$	8.13018	0.265460
$939$	0	0
$940$	1.97739	0.0644954
$941$	−7.83560	−0.255433	−0.127717	−	0.991811i	$-0.540765\pi$
$941$	−7.83560	−0.255433	−0.127717	+	0.991811i	$0.540765\pi$
$942$	0	0
$943$	−46.4772	−1.51351
$944$	−19.5718	−0.637008
$945$	0	0
$946$	70.0719	2.27823
$947$	−26.6896	−0.867297	−0.433648	−	0.901082i	$-0.642774\pi$
$947$	−26.6896	−0.867297	−0.433648	+	0.901082i	$0.642774\pi$
$948$	0	0
$949$	−57.1213	−1.85424
$950$	46.3451	1.50363
$951$	0	0
$952$	−4.13962	−0.134166
$953$	−54.1049	−1.75263	−0.876315	−	0.481739i	$-0.840005\pi$
$953$	−54.1049	−1.75263	−0.876315	+	0.481739i	$0.840005\pi$
$954$	0	0
$955$	41.6341	1.34725
$956$	−55.0849	−1.78157
$957$	0	0
$958$	40.9532	1.32314
$959$	9.03759	0.291839
$960$	0	0
$961$	−30.0458	−0.969219
$962$	−17.0621	−0.550103
$963$	0	0
$964$	−17.8610	−0.575265
$965$	−9.90551	−0.318870
$966$	0	0
$967$	−21.5402	−0.692686	−0.346343	−	0.938108i	$-0.612577\pi$
$967$	−21.5402	−0.692686	−0.346343	+	0.938108i	$0.612577\pi$
$968$	−8.55612	−0.275004
$969$	0	0
$970$	−100.739	−3.23454
$971$	−0.261940	−0.00840606	−0.00420303	−	0.999991i	$-0.501338\pi$
$971$	−0.261940	−0.00840606	−0.00420303	+	0.999991i	$0.501338\pi$
$972$	0	0
$973$	−2.17778	−0.0698166
$974$	88.1794	2.82545
$975$	0	0
$976$	11.7555	0.376284
$977$	−61.3851	−1.96388	−0.981942	−	0.189181i	$-0.939417\pi$
$977$	−61.3851	−1.96388	−0.981942	+	0.189181i	$0.939417\pi$
$978$	0	0
$979$	1.31929	0.0421647
$980$	−7.99541	−0.255404
$981$	0	0
$982$	−55.8876	−1.78344
$983$	−55.0959	−1.75729	−0.878643	−	0.477478i	$-0.841551\pi$
$983$	−55.0959	−1.75729	−0.878643	+	0.477478i	$0.841551\pi$
$984$	0	0
$985$	−10.1919	−0.324742
$986$	11.6953	0.372455
$987$	0	0
$988$	−40.2947	−1.28195
$989$	−44.2535	−1.40718
$990$	0	0
$991$	28.0954	0.892481	0.446240	−	0.894913i	$-0.352763\pi$
$991$	28.0954	0.892481	0.446240	+	0.894913i	$0.352763\pi$
$992$	−7.77864	−0.246972
$993$	0	0
$994$	−23.5872	−0.748141
$995$	1.97176	0.0625090
$996$	0	0
$997$	−40.7067	−1.28920	−0.644598	−	0.764522i	$-0.722975\pi$
$997$	−40.7067	−1.28920	−0.644598	+	0.764522i	$0.722975\pi$
$998$	27.0737	0.857003
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.w.1.5		20
3.2	odd	2		889.2.a.d.1.16	✓	20
21.20	even	2		6223.2.a.l.1.16		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
889.2.a.d.1.16	✓	20	3.2	odd	2
6223.2.a.l.1.16		20	21.20	even	2
8001.2.a.w.1.5		20	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:	\( N \)	\(=\)	\( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8001.a (trivial)

\(f(q)\)	\(=\)	\(q-2.11976 q^{2} +2.49337 q^{4} -3.20666 q^{5} +1.00000 q^{7} -1.04583 q^{8} +O(q^{10})\)	\(q-2.11976 q^{2} +2.49337 q^{4} -3.20666 q^{5} +1.00000 q^{7} -1.04583 q^{8} +6.79735 q^{10} -4.37963 q^{11} +3.90479 q^{13} -2.11976 q^{14} -2.76983 q^{16} +3.95821 q^{17} -4.13869 q^{19} -7.99541 q^{20} +9.28376 q^{22} -5.86310 q^{23} +5.28268 q^{25} -8.27720 q^{26} +2.49337 q^{28} -1.39389 q^{29} -0.976842 q^{31} +7.96305 q^{32} -8.39044 q^{34} -3.20666 q^{35} +2.06133 q^{37} +8.77303 q^{38} +3.35363 q^{40} +7.92706 q^{41} +7.54779 q^{43} -10.9201 q^{44} +12.4284 q^{46} -0.247316 q^{47} +1.00000 q^{49} -11.1980 q^{50} +9.73609 q^{52} -4.76242 q^{53} +14.0440 q^{55} -1.04583 q^{56} +2.95470 q^{58} +7.06605 q^{59} -4.24411 q^{61} +2.07067 q^{62} -11.3401 q^{64} -12.5213 q^{65} -3.83543 q^{67} +9.86929 q^{68} +6.79735 q^{70} +11.1273 q^{71} -14.6285 q^{73} -4.36952 q^{74} -10.3193 q^{76} -4.37963 q^{77} +8.46295 q^{79} +8.88192 q^{80} -16.8035 q^{82} -14.4831 q^{83} -12.6926 q^{85} -15.9995 q^{86} +4.58036 q^{88} -0.301233 q^{89} +3.90479 q^{91} -14.6189 q^{92} +0.524250 q^{94} +13.2714 q^{95} -14.8204 q^{97} -2.11976 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 8 q^{2} + 24 q^{4} - 3 q^{5} + 20 q^{7} - 24 q^{8}+O(q^{10}) \) 20 * q - 8 * q^2 + 24 * q^4 - 3 * q^5 + 20 * q^7 - 24 * q^8	\( 20 q - 8 q^{2} + 24 q^{4} - 3 q^{5} + 20 q^{7} - 24 q^{8} - 8 q^{10} - 26 q^{11} - 4 q^{13} - 8 q^{14} + 24 q^{16} - 4 q^{17} + q^{19} + 2 q^{20} + q^{22} - 31 q^{23} + 27 q^{25} - 4 q^{26} + 24 q^{28} - 16 q^{29} + 6 q^{31} - 41 q^{32} - 10 q^{34} - 3 q^{35} + 2 q^{37} - 3 q^{38} - 38 q^{40} - 25 q^{41} + 13 q^{43} - 66 q^{44} + 20 q^{46} - 19 q^{47} + 20 q^{49} + 4 q^{50} + 20 q^{52} - 24 q^{53} - 3 q^{55} - 24 q^{56} + 12 q^{58} - 23 q^{59} - 27 q^{61} - 7 q^{62} + 2 q^{64} - 26 q^{65} + 9 q^{67} + 25 q^{68} - 8 q^{70} - 63 q^{71} - 21 q^{73} - 21 q^{74} - 10 q^{76} - 26 q^{77} + 18 q^{79} + 23 q^{80} - 42 q^{82} + q^{83} - 41 q^{85} + 12 q^{86} + 57 q^{88} + 16 q^{89} - 4 q^{91} - 17 q^{92} + 7 q^{94} - 75 q^{95} - 32 q^{97} - 8 q^{98}+O(q^{100}) \) 20 * q - 8 * q^2 + 24 * q^4 - 3 * q^5 + 20 * q^7 - 24 * q^8 - 8 * q^10 - 26 * q^11 - 4 * q^13 - 8 * q^14 + 24 * q^16 - 4 * q^17 + q^19 + 2 * q^20 + q^22 - 31 * q^23 + 27 * q^25 - 4 * q^26 + 24 * q^28 - 16 * q^29 + 6 * q^31 - 41 * q^32 - 10 * q^34 - 3 * q^35 + 2 * q^37 - 3 * q^38 - 38 * q^40 - 25 * q^41 + 13 * q^43 - 66 * q^44 + 20 * q^46 - 19 * q^47 + 20 * q^49 + 4 * q^50 + 20 * q^52 - 24 * q^53 - 3 * q^55 - 24 * q^56 + 12 * q^58 - 23 * q^59 - 27 * q^61 - 7 * q^62 + 2 * q^64 - 26 * q^65 + 9 * q^67 + 25 * q^68 - 8 * q^70 - 63 * q^71 - 21 * q^73 - 21 * q^74 - 10 * q^76 - 26 * q^77 + 18 * q^79 + 23 * q^80 - 42 * q^82 + q^83 - 41 * q^85 + 12 * q^86 + 57 * q^88 + 16 * q^89 - 4 * q^91 - 17 * q^92 + 7 * q^94 - 75 * q^95 - 32 * q^97 - 8 * q^98

Introduction

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