LMFDB - Embedded newform 8001.2.a.n.1.4

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:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 5 x^{11} - 3 x^{10} + 41 x^{9} - 11 x^{8} - 123 x^{7} + 44 x^{6} + 159 x^{5} - 39 x^{4} + \cdots - 1 $ x^12 - 5x^11 - 3x^10 + 41x^9 - 11x^8 - 123x^7 + 44x^6 + 159x^5 - 39x^4 - 71x^3 + 16x^2 + 7*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 889)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.84951$ 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)

$f(q)$	$=$	$q-0.849509 q^{2} -1.27834 q^{4} -1.65890 q^{5} +1.00000 q^{7} +2.78497 q^{8} +O(q^{10})$	\(q-0.849509 q^{2} -1.27834 q^{4} -1.65890 q^{5} +1.00000 q^{7} +2.78497 q^{8} +1.40925 q^{10} -2.00876 q^{11} -2.46735 q^{13} -0.849509 q^{14} +0.190811 q^{16} -4.15224 q^{17} -5.98425 q^{19} +2.12064 q^{20} +1.70646 q^{22} +6.20908 q^{23} -2.24804 q^{25} +2.09603 q^{26} -1.27834 q^{28} -4.07180 q^{29} -6.85536 q^{31} -5.73204 q^{32} +3.52737 q^{34} -1.65890 q^{35} -9.80028 q^{37} +5.08367 q^{38} -4.62001 q^{40} +6.54329 q^{41} -4.48183 q^{43} +2.56787 q^{44} -5.27467 q^{46} +4.80611 q^{47} +1.00000 q^{49} +1.90973 q^{50} +3.15410 q^{52} +6.94724 q^{53} +3.33234 q^{55} +2.78497 q^{56} +3.45903 q^{58} -3.25595 q^{59} -5.28396 q^{61} +5.82369 q^{62} +4.48780 q^{64} +4.09310 q^{65} +0.780600 q^{67} +5.30796 q^{68} +1.40925 q^{70} +6.78957 q^{71} +15.0145 q^{73} +8.32542 q^{74} +7.64987 q^{76} -2.00876 q^{77} -16.4442 q^{79} -0.316538 q^{80} -5.55858 q^{82} -0.329707 q^{83} +6.88817 q^{85} +3.80736 q^{86} -5.59435 q^{88} -17.5151 q^{89} -2.46735 q^{91} -7.93729 q^{92} -4.08283 q^{94} +9.92729 q^{95} +6.36680 q^{97} -0.849509 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 7 q^{2} + 9 q^{4} + 7 q^{5} + 12 q^{7} + 15 q^{8}+O(q^{10}) $ 12 * q + 7 * q^2 + 9 * q^4 + 7 * q^5 + 12 * q^7 + 15 * q^8	$ 12 q + 7 q^{2} + 9 q^{4} + 7 q^{5} + 12 q^{7} + 15 q^{8} - 2 q^{10} + 22 q^{11} + 7 q^{14} + 7 q^{16} + 6 q^{17} - 7 q^{19} + 8 q^{20} + 13 q^{22} + 29 q^{23} + 3 q^{25} + 9 q^{28} + 22 q^{29} - 16 q^{31} + 27 q^{32} - 5 q^{34} + 7 q^{35} - 4 q^{37} - 2 q^{38} + 16 q^{40} + 21 q^{41} + 11 q^{43} + 11 q^{44} + 31 q^{47} + 12 q^{49} + 21 q^{50} + 3 q^{52} + 38 q^{53} - 11 q^{55} + 15 q^{56} + 20 q^{58} + 15 q^{59} - 3 q^{61} + 4 q^{62} + 29 q^{64} + 32 q^{65} - q^{67} - 17 q^{68} - 2 q^{70} + 57 q^{71} - 7 q^{73} + 42 q^{74} - 44 q^{76} + 22 q^{77} - 18 q^{79} - q^{80} + 56 q^{82} + 21 q^{83} - 5 q^{85} + 32 q^{86} - 10 q^{88} - 6 q^{89} + 15 q^{92} + 35 q^{94} + 57 q^{95} + 4 q^{97} + 7 q^{98}+O(q^{100}) $ 12 * q + 7 * q^2 + 9 * q^4 + 7 * q^5 + 12 * q^7 + 15 * q^8 - 2 * q^10 + 22 * q^11 + 7 * q^14 + 7 * q^16 + 6 * q^17 - 7 * q^19 + 8 * q^20 + 13 * q^22 + 29 * q^23 + 3 * q^25 + 9 * q^28 + 22 * q^29 - 16 * q^31 + 27 * q^32 - 5 * q^34 + 7 * q^35 - 4 * q^37 - 2 * q^38 + 16 * q^40 + 21 * q^41 + 11 * q^43 + 11 * q^44 + 31 * q^47 + 12 * q^49 + 21 * q^50 + 3 * q^52 + 38 * q^53 - 11 * q^55 + 15 * q^56 + 20 * q^58 + 15 * q^59 - 3 * q^61 + 4 * q^62 + 29 * q^64 + 32 * q^65 - q^67 - 17 * q^68 - 2 * q^70 + 57 * q^71 - 7 * q^73 + 42 * q^74 - 44 * q^76 + 22 * q^77 - 18 * q^79 - q^80 + 56 * q^82 + 21 * q^83 - 5 * q^85 + 32 * q^86 - 10 * q^88 - 6 * q^89 + 15 * q^92 + 35 * q^94 + 57 * q^95 + 4 * q^97 + 7 * 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 $

$2$	−0.849509	−0.600693	−0.300347	−	0.953830i	$-0.597102\pi$
$2$	−0.849509	−0.600693	−0.300347	+	0.953830i	$0.597102\pi$
$3$	0	0
$3$	0	0
$4$	−1.27834	−0.639168
$5$	−1.65890	−0.741885	−0.370942	−	0.928656i	$-0.620965\pi$
$5$	−1.65890	−0.741885	−0.370942	+	0.928656i	$0.620965\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	2.78497	0.984637
$9$	0	0
$10$	1.40925	0.445645
$11$	−2.00876	−0.605664	−0.302832	−	0.953044i	$-0.597932\pi$
$11$	−2.00876	−0.605664	−0.302832	+	0.953044i	$0.597932\pi$
$12$	0	0
$13$	−2.46735	−0.684319	−0.342160	−	0.939642i	$-0.611158\pi$
$13$	−2.46735	−0.684319	−0.342160	+	0.939642i	$0.611158\pi$
$14$	−0.849509	−0.227041
$15$	0	0
$16$	0.190811	0.0477028
$17$	−4.15224	−1.00707	−0.503533	−	0.863976i	$-0.667967\pi$
$17$	−4.15224	−1.00707	−0.503533	+	0.863976i	$0.667967\pi$
$18$	0	0
$19$	−5.98425	−1.37288	−0.686440	−	0.727186i	$-0.740828\pi$
$19$	−5.98425	−1.37288	−0.686440	+	0.727186i	$0.740828\pi$
$20$	2.12064	0.474189
$21$	0	0
$22$	1.70646	0.363818
$23$	6.20908	1.29468	0.647342	−	0.762200i	$-0.275881\pi$
$23$	6.20908	1.29468	0.647342	+	0.762200i	$0.275881\pi$
$24$	0	0
$25$	−2.24804	−0.449607
$26$	2.09603	0.411066
$27$	0	0
$28$	−1.27834	−0.241583
$29$	−4.07180	−0.756114	−0.378057	−	0.925782i	$-0.623408\pi$
$29$	−4.07180	−0.756114	−0.378057	+	0.925782i	$0.623408\pi$
$30$	0	0
$31$	−6.85536	−1.23126	−0.615630	−	0.788036i	$-0.711098\pi$
$31$	−6.85536	−1.23126	−0.615630	+	0.788036i	$0.711098\pi$
$32$	−5.73204	−1.01329
$33$	0	0
$34$	3.52737	0.604938
$35$	−1.65890	−0.280406
$36$	0	0
$37$	−9.80028	−1.61116	−0.805578	−	0.592489i	$-0.798145\pi$
$37$	−9.80028	−1.61116	−0.805578	+	0.592489i	$0.798145\pi$
$38$	5.08367	0.824680
$39$	0	0
$40$	−4.62001	−0.730487
$41$	6.54329	1.02189	0.510945	−	0.859613i	$-0.329295\pi$
$41$	6.54329	1.02189	0.510945	+	0.859613i	$0.329295\pi$
$42$	0	0
$43$	−4.48183	−0.683473	−0.341737	−	0.939796i	$-0.611015\pi$
$43$	−4.48183	−0.683473	−0.341737	+	0.939796i	$0.611015\pi$
$44$	2.56787	0.387121
$45$	0	0
$46$	−5.27467	−0.777707
$47$	4.80611	0.701043	0.350521	−	0.936555i	$-0.386004\pi$
$47$	4.80611	0.701043	0.350521	+	0.936555i	$0.386004\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	1.90973	0.270076
$51$	0	0
$52$	3.15410	0.437395
$53$	6.94724	0.954277	0.477139	−	0.878828i	$-0.341674\pi$
$53$	6.94724	0.954277	0.477139	+	0.878828i	$0.341674\pi$
$54$	0	0
$55$	3.33234	0.449333
$56$	2.78497	0.372158
$57$	0	0
$58$	3.45903	0.454193
$59$	−3.25595	−0.423889	−0.211945	−	0.977282i	$-0.567980\pi$
$59$	−3.25595	−0.423889	−0.211945	+	0.977282i	$0.567980\pi$
$60$	0	0
$61$	−5.28396	−0.676542	−0.338271	−	0.941049i	$-0.609842\pi$
$61$	−5.28396	−0.676542	−0.338271	+	0.941049i	$0.609842\pi$
$62$	5.82369	0.739609
$63$	0	0
$64$	4.48780	0.560975
$65$	4.09310	0.507686
$66$	0	0
$67$	0.780600	0.0953655	0.0476827	−	0.998863i	$-0.484816\pi$
$67$	0.780600	0.0953655	0.0476827	+	0.998863i	$0.484816\pi$
$68$	5.30796	0.643684
$69$	0	0
$70$	1.40925	0.168438
$71$	6.78957	0.805774	0.402887	−	0.915250i	$-0.368007\pi$
$71$	6.78957	0.805774	0.402887	+	0.915250i	$0.368007\pi$
$72$	0	0
$73$	15.0145	1.75732	0.878658	−	0.477451i	$-0.158439\pi$
$73$	15.0145	1.75732	0.878658	+	0.477451i	$0.158439\pi$
$74$	8.32542	0.967811
$75$	0	0
$76$	7.64987	0.877500
$77$	−2.00876	−0.228920
$78$	0	0
$79$	−16.4442	−1.85012	−0.925060	−	0.379821i	$-0.875986\pi$
$79$	−16.4442	−1.85012	−0.925060	+	0.379821i	$0.875986\pi$
$80$	−0.316538	−0.0353900
$81$	0	0
$82$	−5.55858	−0.613843
$83$	−0.329707	−0.0361900	−0.0180950	−	0.999836i	$-0.505760\pi$
$83$	−0.329707	−0.0361900	−0.0180950	+	0.999836i	$0.505760\pi$
$84$	0	0
$85$	6.88817	0.747127
$86$	3.80736	0.410558
$87$	0	0
$88$	−5.59435	−0.596359
$89$	−17.5151	−1.85659	−0.928297	−	0.371840i	$-0.878727\pi$
$89$	−17.5151	−1.85659	−0.928297	+	0.371840i	$0.878727\pi$
$90$	0	0
$91$	−2.46735	−0.258648
$92$	−7.93729	−0.827519
$93$	0	0
$94$	−4.08283	−0.421112
$95$	9.92729	1.01852
$96$	0	0
$97$	6.36680	0.646451	0.323226	−	0.946322i	$-0.395233\pi$
$97$	6.36680	0.646451	0.323226	+	0.946322i	$0.395233\pi$
$98$	−0.849509	−0.0858133
$99$	0	0
$100$	2.87374	0.287374
$101$	−6.70776	−0.667447	−0.333724	−	0.942671i	$-0.608305\pi$
$101$	−6.70776	−0.667447	−0.333724	+	0.942671i	$0.608305\pi$
$102$	0	0
$103$	1.56085	0.153795	0.0768977	−	0.997039i	$-0.475499\pi$
$103$	1.56085	0.153795	0.0768977	+	0.997039i	$0.475499\pi$
$104$	−6.87150	−0.673806
$105$	0	0
$106$	−5.90174	−0.573228
$107$	3.17782	0.307211	0.153606	−	0.988132i	$-0.450911\pi$
$107$	3.17782	0.307211	0.153606	+	0.988132i	$0.450911\pi$
$108$	0	0
$109$	−5.72634	−0.548484	−0.274242	−	0.961661i	$-0.588427\pi$
$109$	−5.72634	−0.548484	−0.274242	+	0.961661i	$0.588427\pi$
$110$	−2.83085	−0.269911
$111$	0	0
$112$	0.190811	0.0180300
$113$	−10.7097	−1.00748	−0.503741	−	0.863854i	$-0.668044\pi$
$113$	−10.7097	−1.00748	−0.503741	+	0.863854i	$0.668044\pi$
$114$	0	0
$115$	−10.3003	−0.960505
$116$	5.20512	0.483284
$117$	0	0
$118$	2.76596	0.254627
$119$	−4.15224	−0.380635
$120$	0	0
$121$	−6.96488	−0.633171
$122$	4.48877	0.406394
$123$	0	0
$124$	8.76345	0.786981
$125$	12.0238	1.07544
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	7.65166	0.676318
$129$	0	0
$130$	−3.47712	−0.304964
$131$	−13.1647	−1.15021	−0.575103	−	0.818081i	$-0.695038\pi$
$131$	−13.1647	−1.15021	−0.575103	+	0.818081i	$0.695038\pi$
$132$	0	0
$133$	−5.98425	−0.518900
$134$	−0.663126	−0.0572854
$135$	0	0
$136$	−11.5639	−0.991595
$137$	2.96915	0.253671	0.126836	−	0.991924i	$-0.459518\pi$
$137$	2.96915	0.253671	0.126836	+	0.991924i	$0.459518\pi$
$138$	0	0
$139$	−2.75112	−0.233347	−0.116674	−	0.993170i	$-0.537223\pi$
$139$	−2.75112	−0.233347	−0.116674	+	0.993170i	$0.537223\pi$
$140$	2.12064	0.179226
$141$	0	0
$142$	−5.76780	−0.484023
$143$	4.95632	0.414468
$144$	0	0
$145$	6.75473	0.560949
$146$	−12.7550	−1.05561
$147$	0	0
$148$	12.5280	1.02980
$149$	−13.8529	−1.13488	−0.567438	−	0.823416i	$-0.692066\pi$
$149$	−13.8529	−1.13488	−0.567438	+	0.823416i	$0.692066\pi$
$150$	0	0
$151$	9.20358	0.748976	0.374488	−	0.927232i	$-0.377818\pi$
$151$	9.20358	0.748976	0.374488	+	0.927232i	$0.377818\pi$
$152$	−16.6660	−1.35179
$153$	0	0
$154$	1.70646	0.137510
$155$	11.3724	0.913452
$156$	0	0
$157$	−6.10277	−0.487054	−0.243527	−	0.969894i	$-0.578304\pi$
$157$	−6.10277	−0.487054	−0.243527	+	0.969894i	$0.578304\pi$
$158$	13.9695	1.11135
$159$	0	0
$160$	9.50891	0.751745
$161$	6.20908	0.489344
$162$	0	0
$163$	−7.93770	−0.621729	−0.310864	−	0.950454i	$-0.600618\pi$
$163$	−7.93770	−0.621729	−0.310864	+	0.950454i	$0.600618\pi$
$164$	−8.36452	−0.653159
$165$	0	0
$166$	0.280089	0.0217391
$167$	−11.7083	−0.906016	−0.453008	−	0.891506i	$-0.649649\pi$
$167$	−11.7083	−0.906016	−0.453008	+	0.891506i	$0.649649\pi$
$168$	0	0
$169$	−6.91219	−0.531707
$170$	−5.85156	−0.448794
$171$	0	0
$172$	5.72929	0.436854
$173$	2.06843	0.157260	0.0786301	−	0.996904i	$-0.474945\pi$
$173$	2.06843	0.157260	0.0786301	+	0.996904i	$0.474945\pi$
$174$	0	0
$175$	−2.24804	−0.169936
$176$	−0.383294	−0.0288919
$177$	0	0
$178$	14.8792	1.11524
$179$	−8.03308	−0.600421	−0.300210	−	0.953873i	$-0.597057\pi$
$179$	−8.03308	−0.600421	−0.300210	+	0.953873i	$0.597057\pi$
$180$	0	0
$181$	17.2384	1.28132	0.640661	−	0.767824i	$-0.278660\pi$
$181$	17.2384	1.28132	0.640661	+	0.767824i	$0.278660\pi$
$182$	2.09603	0.155368
$183$	0	0
$184$	17.2921	1.27479
$185$	16.2577	1.19529
$186$	0	0
$187$	8.34086	0.609944
$188$	−6.14382	−0.448084
$189$	0	0
$190$	−8.43332	−0.611817
$191$	18.4283	1.33342	0.666712	−	0.745315i	$-0.267701\pi$
$191$	18.4283	1.33342	0.666712	+	0.745315i	$0.267701\pi$
$192$	0	0
$193$	22.7061	1.63442	0.817209	−	0.576341i	$-0.195520\pi$
$193$	22.7061	1.63442	0.817209	+	0.576341i	$0.195520\pi$
$194$	−5.40865	−0.388319
$195$	0	0
$196$	−1.27834	−0.0913097
$197$	3.07641	0.219185	0.109592	−	0.993977i	$-0.465045\pi$
$197$	3.07641	0.219185	0.109592	+	0.993977i	$0.465045\pi$
$198$	0	0
$199$	−7.28695	−0.516558	−0.258279	−	0.966070i	$-0.583155\pi$
$199$	−7.28695	−0.516558	−0.258279	+	0.966070i	$0.583155\pi$
$200$	−6.26072	−0.442700
$201$	0	0
$202$	5.69830	0.400931
$203$	−4.07180	−0.285784
$204$	0	0
$205$	−10.8547	−0.758125
$206$	−1.32596	−0.0923838
$207$	0	0
$208$	−0.470798	−0.0326440
$209$	12.0209	0.831504
$210$	0	0
$211$	16.5261	1.13770	0.568851	−	0.822440i	$-0.307388\pi$
$211$	16.5261	1.13770	0.568851	+	0.822440i	$0.307388\pi$
$212$	−8.88091	−0.609943
$213$	0	0
$214$	−2.69958	−0.184540
$215$	7.43493	0.507058
$216$	0	0
$217$	−6.85536	−0.465372
$218$	4.86458	0.329471
$219$	0	0
$220$	−4.25985	−0.287199
$221$	10.2450	0.689155
$222$	0	0
$223$	−27.1661	−1.81918	−0.909589	−	0.415509i	$-0.863603\pi$
$223$	−27.1661	−1.81918	−0.909589	+	0.415509i	$0.863603\pi$
$224$	−5.73204	−0.382988
$225$	0	0
$226$	9.09797	0.605188
$227$	−25.6428	−1.70197	−0.850987	−	0.525187i	$-0.823995\pi$
$227$	−25.6428	−1.70197	−0.850987	+	0.525187i	$0.823995\pi$
$228$	0	0
$229$	5.98122	0.395250	0.197625	−	0.980278i	$-0.436677\pi$
$229$	5.98122	0.395250	0.197625	+	0.980278i	$0.436677\pi$
$230$	8.75017	0.576969
$231$	0	0
$232$	−11.3399	−0.744498
$233$	1.12759	0.0738710	0.0369355	−	0.999318i	$-0.488240\pi$
$233$	1.12759	0.0738710	0.0369355	+	0.999318i	$0.488240\pi$
$234$	0	0
$235$	−7.97287	−0.520093
$236$	4.16220	0.270936
$237$	0	0
$238$	3.52737	0.228645
$239$	22.2879	1.44169	0.720843	−	0.693098i	$-0.243755\pi$
$239$	22.2879	1.44169	0.720843	+	0.693098i	$0.243755\pi$
$240$	0	0
$241$	−7.06930	−0.455374	−0.227687	−	0.973734i	$-0.573116\pi$
$241$	−7.06930	−0.455374	−0.227687	+	0.973734i	$0.573116\pi$
$242$	5.91672	0.380341
$243$	0	0
$244$	6.75467	0.432424
$245$	−1.65890	−0.105984
$246$	0	0
$247$	14.7652	0.939489
$248$	−19.0920	−1.21234
$249$	0	0
$250$	−10.2143	−0.646010
$251$	26.9325	1.69997	0.849983	−	0.526810i	$-0.176612\pi$
$251$	26.9325	1.69997	0.849983	+	0.526810i	$0.176612\pi$
$252$	0	0
$253$	−12.4726	−0.784143
$254$	−0.849509	−0.0533029
$255$	0	0
$256$	−15.4757	−0.967234
$257$	11.0342	0.688293	0.344146	−	0.938916i	$-0.388168\pi$
$257$	11.0342	0.688293	0.344146	+	0.938916i	$0.388168\pi$
$258$	0	0
$259$	−9.80028	−0.608960
$260$	−5.23235	−0.324496
$261$	0	0
$262$	11.1835	0.690922
$263$	26.1243	1.61089	0.805447	−	0.592668i	$-0.201925\pi$
$263$	26.1243	1.61089	0.805447	+	0.592668i	$0.201925\pi$
$264$	0	0
$265$	−11.5248	−0.707964
$266$	5.08367	0.311700
$267$	0	0
$268$	−0.997868	−0.0609545
$269$	0.876881	0.0534644	0.0267322	−	0.999643i	$-0.491490\pi$
$269$	0.876881	0.0534644	0.0267322	+	0.999643i	$0.491490\pi$
$270$	0	0
$271$	−24.0185	−1.45902	−0.729510	−	0.683970i	$-0.760252\pi$
$271$	−24.0185	−1.45902	−0.729510	+	0.683970i	$0.760252\pi$
$272$	−0.792295	−0.0480399
$273$	0	0
$274$	−2.52232	−0.152379
$275$	4.51577	0.272311
$276$	0	0
$277$	3.56171	0.214002	0.107001	−	0.994259i	$-0.465875\pi$
$277$	3.56171	0.214002	0.107001	+	0.994259i	$0.465875\pi$
$278$	2.33710	0.140170
$279$	0	0
$280$	−4.62001	−0.276098
$281$	−5.96346	−0.355750	−0.177875	−	0.984053i	$-0.556922\pi$
$281$	−5.96346	−0.355750	−0.177875	+	0.984053i	$0.556922\pi$
$282$	0	0
$283$	−16.9015	−1.00469	−0.502344	−	0.864668i	$-0.667529\pi$
$283$	−16.9015	−1.00469	−0.502344	+	0.864668i	$0.667529\pi$
$284$	−8.67935	−0.515025
$285$	0	0
$286$	−4.21043	−0.248968
$287$	6.54329	0.386238
$288$	0	0
$289$	0.241120	0.0141835
$290$	−5.73820	−0.336959
$291$	0	0
$292$	−19.1936	−1.12322
$293$	−20.1551	−1.17748	−0.588738	−	0.808324i	$-0.700375\pi$
$293$	−20.1551	−1.17748	−0.588738	+	0.808324i	$0.700375\pi$
$294$	0	0
$295$	5.40132	0.314477
$296$	−27.2935	−1.58640
$297$	0	0
$298$	11.7682	0.681713
$299$	−15.3200	−0.885977
$300$	0	0
$301$	−4.48183	−0.258329
$302$	−7.81852	−0.449905
$303$	0	0
$304$	−1.14186	−0.0654902
$305$	8.76559	0.501916
$306$	0	0
$307$	31.2090	1.78119	0.890596	−	0.454796i	$-0.150288\pi$
$307$	31.2090	1.78119	0.890596	+	0.454796i	$0.150288\pi$
$308$	2.56787	0.146318
$309$	0	0
$310$	−9.66094	−0.548705
$311$	−19.6573	−1.11466	−0.557332	−	0.830290i	$-0.688175\pi$
$311$	−19.6573	−1.11466	−0.557332	+	0.830290i	$0.688175\pi$
$312$	0	0
$313$	21.7125	1.22727	0.613633	−	0.789592i	$-0.289708\pi$
$313$	21.7125	1.22727	0.613633	+	0.789592i	$0.289708\pi$
$314$	5.18435	0.292570
$315$	0	0
$316$	21.0212	1.18254
$317$	29.5123	1.65757	0.828787	−	0.559565i	$-0.189032\pi$
$317$	29.5123	1.65757	0.828787	+	0.559565i	$0.189032\pi$
$318$	0	0
$319$	8.17927	0.457951
$320$	−7.44483	−0.416178
$321$	0	0
$322$	−5.27467	−0.293946
$323$	24.8480	1.38258
$324$	0	0
$325$	5.54669	0.307675
$326$	6.74314	0.373468
$327$	0	0
$328$	18.2229	1.00619
$329$	4.80611	0.264969
$330$	0	0
$331$	−7.99375	−0.439376	−0.219688	−	0.975570i	$-0.570504\pi$
$331$	−7.99375	−0.439376	−0.219688	+	0.975570i	$0.570504\pi$
$332$	0.421476	0.0231315
$333$	0	0
$334$	9.94631	0.544238
$335$	−1.29494	−0.0707502
$336$	0	0
$337$	30.8063	1.67813	0.839063	−	0.544035i	$-0.183104\pi$
$337$	30.8063	1.67813	0.839063	+	0.544035i	$0.183104\pi$
$338$	5.87196	0.319393
$339$	0	0
$340$	−8.80539	−0.477540
$341$	13.7708	0.745730
$342$	0	0
$343$	1.00000	0.0539949
$344$	−12.4818	−0.672973
$345$	0	0
$346$	−1.75715	−0.0944651
$347$	−21.4369	−1.15079	−0.575397	−	0.817874i	$-0.695152\pi$
$347$	−21.4369	−1.15079	−0.575397	+	0.817874i	$0.695152\pi$
$348$	0	0
$349$	−22.6352	−1.21164	−0.605818	−	0.795603i	$-0.707154\pi$
$349$	−22.6352	−1.21164	−0.605818	+	0.795603i	$0.707154\pi$
$350$	1.90973	0.102079
$351$	0	0
$352$	11.5143	0.613715
$353$	6.21806	0.330954	0.165477	−	0.986214i	$-0.447084\pi$
$353$	6.21806	0.330954	0.165477	+	0.986214i	$0.447084\pi$
$354$	0	0
$355$	−11.2633	−0.597791
$356$	22.3901	1.18667
$357$	0	0
$358$	6.82417	0.360669
$359$	26.8865	1.41901	0.709507	−	0.704698i	$-0.248917\pi$
$359$	26.8865	1.41901	0.709507	+	0.704698i	$0.248917\pi$
$360$	0	0
$361$	16.8112	0.884800
$362$	−14.6442	−0.769682
$363$	0	0
$364$	3.15410	0.165320
$365$	−24.9076	−1.30373
$366$	0	0
$367$	−19.1334	−0.998756	−0.499378	−	0.866384i	$-0.666438\pi$
$367$	−19.1334	−0.998756	−0.499378	+	0.866384i	$0.666438\pi$
$368$	1.18476	0.0617600
$369$	0	0
$370$	−13.8111	−0.718004
$371$	6.94724	0.360683
$372$	0	0
$373$	−9.16335	−0.474460	−0.237230	−	0.971453i	$-0.576240\pi$
$373$	−9.16335	−0.474460	−0.237230	+	0.971453i	$0.576240\pi$
$374$	−7.08564	−0.366390
$375$	0	0
$376$	13.3849	0.690273
$377$	10.0465	0.517424
$378$	0	0
$379$	−1.86855	−0.0959811	−0.0479906	−	0.998848i	$-0.515282\pi$
$379$	−1.86855	−0.0959811	−0.0479906	+	0.998848i	$0.515282\pi$
$380$	−12.6904	−0.651004
$381$	0	0
$382$	−15.6550	−0.800979
$383$	−1.55457	−0.0794349	−0.0397174	−	0.999211i	$-0.512646\pi$
$383$	−1.55457	−0.0794349	−0.0397174	+	0.999211i	$0.512646\pi$
$384$	0	0
$385$	3.33234	0.169832
$386$	−19.2890	−0.981784
$387$	0	0
$388$	−8.13891	−0.413191
$389$	−19.6326	−0.995412	−0.497706	−	0.867346i	$-0.665824\pi$
$389$	−19.6326	−0.995412	−0.497706	+	0.867346i	$0.665824\pi$
$390$	0	0
$391$	−25.7816	−1.30383
$392$	2.78497	0.140662
$393$	0	0
$394$	−2.61343	−0.131663
$395$	27.2794	1.37258
$396$	0	0
$397$	27.4003	1.37518	0.687592	−	0.726098i	$-0.258668\pi$
$397$	27.4003	1.37518	0.687592	+	0.726098i	$0.258668\pi$
$398$	6.19032	0.310293
$399$	0	0
$400$	−0.428951	−0.0214475
$401$	7.85600	0.392310	0.196155	−	0.980573i	$-0.437154\pi$
$401$	7.85600	0.392310	0.196155	+	0.980573i	$0.437154\pi$
$402$	0	0
$403$	16.9146	0.842575
$404$	8.57477	0.426611
$405$	0	0
$406$	3.45903	0.171669
$407$	19.6864	0.975820
$408$	0	0
$409$	−17.0487	−0.843006	−0.421503	−	0.906827i	$-0.638497\pi$
$409$	−17.0487	−0.843006	−0.421503	+	0.906827i	$0.638497\pi$
$410$	9.22116	0.455401
$411$	0	0
$412$	−1.99529	−0.0983010
$413$	−3.25595	−0.160215
$414$	0	0
$415$	0.546952	0.0268488
$416$	14.1430	0.693415
$417$	0	0
$418$	−10.2119	−0.499479
$419$	25.5894	1.25012	0.625062	−	0.780575i	$-0.285074\pi$
$419$	25.5894	1.25012	0.625062	+	0.780575i	$0.285074\pi$
$420$	0	0
$421$	21.6033	1.05288	0.526440	−	0.850212i	$-0.323526\pi$
$421$	21.6033	1.05288	0.526440	+	0.850212i	$0.323526\pi$
$422$	−14.0390	−0.683410
$423$	0	0
$424$	19.3479	0.939617
$425$	9.33439	0.452785
$426$	0	0
$427$	−5.28396	−0.255709
$428$	−4.06232	−0.196359
$429$	0	0
$430$	−6.31604	−0.304586
$431$	−8.73845	−0.420916	−0.210458	−	0.977603i	$-0.567496\pi$
$431$	−8.73845	−0.420916	−0.210458	+	0.977603i	$0.567496\pi$
$432$	0	0
$433$	32.3508	1.55468	0.777340	−	0.629081i	$-0.216569\pi$
$433$	32.3508	1.55468	0.777340	+	0.629081i	$0.216569\pi$
$434$	5.82369	0.279546
$435$	0	0
$436$	7.32019	0.350573
$437$	−37.1567	−1.77744
$438$	0	0
$439$	32.2137	1.53748	0.768738	−	0.639564i	$-0.220885\pi$
$439$	32.2137	1.53748	0.768738	+	0.639564i	$0.220885\pi$
$440$	9.28049	0.442430
$441$	0	0
$442$	−8.70324	−0.413971
$443$	−0.454044	−0.0215723	−0.0107861	−	0.999942i	$-0.503433\pi$
$443$	−0.454044	−0.0215723	−0.0107861	+	0.999942i	$0.503433\pi$
$444$	0	0
$445$	29.0558	1.37738
$446$	23.0779	1.09277
$447$	0	0
$448$	4.48780	0.212029
$449$	2.42560	0.114471	0.0572355	−	0.998361i	$-0.481771\pi$
$449$	2.42560	0.114471	0.0572355	+	0.998361i	$0.481771\pi$
$450$	0	0
$451$	−13.1439	−0.618923
$452$	13.6906	0.643950
$453$	0	0
$454$	21.7838	1.02236
$455$	4.09310	0.191887
$456$	0	0
$457$	16.4872	0.771240	0.385620	−	0.922658i	$-0.373988\pi$
$457$	16.4872	0.771240	0.385620	+	0.922658i	$0.373988\pi$
$458$	−5.08110	−0.237424
$459$	0	0
$460$	13.1672	0.613924
$461$	16.9374	0.788853	0.394426	−	0.918928i	$-0.370943\pi$
$461$	16.9374	0.788853	0.394426	+	0.918928i	$0.370943\pi$
$462$	0	0
$463$	−7.26450	−0.337610	−0.168805	−	0.985649i	$-0.553991\pi$
$463$	−7.26450	−0.337610	−0.168805	+	0.985649i	$0.553991\pi$
$464$	−0.776945	−0.0360688
$465$	0	0
$466$	−0.957899	−0.0443738
$467$	−2.62653	−0.121541	−0.0607706	−	0.998152i	$-0.519356\pi$
$467$	−2.62653	−0.121541	−0.0607706	+	0.998152i	$0.519356\pi$
$468$	0	0
$469$	0.780600	0.0360448
$470$	6.77302	0.312416
$471$	0	0
$472$	−9.06775	−0.417377
$473$	9.00293	0.413955
$474$	0	0
$475$	13.4528	0.617257
$476$	5.30796	0.243290
$477$	0	0
$478$	−18.9338	−0.866011
$479$	−10.2465	−0.468174	−0.234087	−	0.972216i	$-0.575210\pi$
$479$	−10.2465	−0.468174	−0.234087	+	0.972216i	$0.575210\pi$
$480$	0	0
$481$	24.1807	1.10255
$482$	6.00543	0.273540
$483$	0	0
$484$	8.90345	0.404702
$485$	−10.5619	−0.479592
$486$	0	0
$487$	14.9328	0.676668	0.338334	−	0.941026i	$-0.390137\pi$
$487$	14.9328	0.676668	0.338334	+	0.941026i	$0.390137\pi$
$488$	−14.7157	−0.666148
$489$	0	0
$490$	1.40925	0.0636636
$491$	28.9067	1.30454	0.652270	−	0.757987i	$-0.273817\pi$
$491$	28.9067	1.30454	0.652270	+	0.757987i	$0.273817\pi$
$492$	0	0
$493$	16.9071	0.761457
$494$	−12.5432	−0.564344
$495$	0	0
$496$	−1.30808	−0.0587345
$497$	6.78957	0.304554
$498$	0	0
$499$	−27.0830	−1.21240	−0.606200	−	0.795312i	$-0.707307\pi$
$499$	−27.0830	−1.21240	−0.606200	+	0.795312i	$0.707307\pi$
$500$	−15.3704	−0.687387
$501$	0	0
$502$	−22.8794	−1.02116
$503$	18.7640	0.836646	0.418323	−	0.908298i	$-0.362618\pi$
$503$	18.7640	0.836646	0.418323	+	0.908298i	$0.362618\pi$
$504$	0	0
$505$	11.1275	0.495169
$506$	10.5955	0.471030
$507$	0	0
$508$	−1.27834	−0.0567170
$509$	−15.2889	−0.677670	−0.338835	−	0.940846i	$-0.610033\pi$
$509$	−15.2889	−0.677670	−0.338835	+	0.940846i	$0.610033\pi$
$510$	0	0
$511$	15.0145	0.664203
$512$	−2.15655	−0.0953068
$513$	0	0
$514$	−9.37362	−0.413453
$515$	−2.58930	−0.114098
$516$	0	0
$517$	−9.65432	−0.424597
$518$	8.32542	0.365798
$519$	0	0
$520$	11.3992	0.499886
$521$	13.0293	0.570824	0.285412	−	0.958405i	$-0.407870\pi$
$521$	13.0293	0.570824	0.285412	+	0.958405i	$0.407870\pi$
$522$	0	0
$523$	37.0105	1.61836	0.809178	−	0.587564i	$-0.199913\pi$
$523$	37.0105	1.61836	0.809178	+	0.587564i	$0.199913\pi$
$524$	16.8289	0.735175
$525$	0	0
$526$	−22.1928	−0.967653
$527$	28.4651	1.23996
$528$	0	0
$529$	15.5527	0.676204
$530$	9.79043	0.425269
$531$	0	0
$532$	7.64987	0.331664
$533$	−16.1446	−0.699300
$534$	0	0
$535$	−5.27170	−0.227915
$536$	2.17395	0.0939003
$537$	0	0
$538$	−0.744918	−0.0321157
$539$	−2.00876	−0.0865235
$540$	0	0
$541$	38.9383	1.67409	0.837043	−	0.547137i	$-0.184282\pi$
$541$	38.9383	1.67409	0.837043	+	0.547137i	$0.184282\pi$
$542$	20.4039	0.876423
$543$	0	0
$544$	23.8008	1.02045
$545$	9.49946	0.406912
$546$	0	0
$547$	30.0610	1.28531	0.642657	−	0.766154i	$-0.277832\pi$
$547$	30.0610	1.28531	0.642657	+	0.766154i	$0.277832\pi$
$548$	−3.79556	−0.162138
$549$	0	0
$550$	−3.83618	−0.163575
$551$	24.3666	1.03805
$552$	0	0
$553$	−16.4442	−0.699280
$554$	−3.02570	−0.128550
$555$	0	0
$556$	3.51686	0.149148
$557$	26.9549	1.14212	0.571058	−	0.820910i	$-0.306533\pi$
$557$	26.9549	1.14212	0.571058	+	0.820910i	$0.306533\pi$
$558$	0	0
$559$	11.0582	0.467714
$560$	−0.316538	−0.0133762
$561$	0	0
$562$	5.06601	0.213697
$563$	−21.1532	−0.891503	−0.445751	−	0.895157i	$-0.647063\pi$
$563$	−21.1532	−0.891503	−0.445751	+	0.895157i	$0.647063\pi$
$564$	0	0
$565$	17.7663	0.747436
$566$	14.3579	0.603509
$567$	0	0
$568$	18.9088	0.793395
$569$	44.1177	1.84951	0.924754	−	0.380565i	$-0.124270\pi$
$569$	44.1177	1.84951	0.924754	+	0.380565i	$0.124270\pi$
$570$	0	0
$571$	−17.7127	−0.741254	−0.370627	−	0.928782i	$-0.620857\pi$
$571$	−17.7127	−0.741254	−0.370627	+	0.928782i	$0.620857\pi$
$572$	−6.33583	−0.264914
$573$	0	0
$574$	−5.55858	−0.232011
$575$	−13.9582	−0.582099
$576$	0	0
$577$	−45.1965	−1.88155	−0.940777	−	0.339025i	$-0.889903\pi$
$577$	−45.1965	−1.88155	−0.940777	+	0.339025i	$0.889903\pi$
$578$	−0.204834	−0.00851996
$579$	0	0
$580$	−8.63480	−0.358541
$581$	−0.329707	−0.0136785
$582$	0	0
$583$	−13.9554	−0.577972
$584$	41.8150	1.73032
$585$	0	0
$586$	17.1220	0.707302
$587$	−26.5654	−1.09647	−0.548236	−	0.836324i	$-0.684700\pi$
$587$	−26.5654	−1.09647	−0.548236	+	0.836324i	$0.684700\pi$
$588$	0	0
$589$	41.0242	1.69037
$590$	−4.58846	−0.188904
$591$	0	0
$592$	−1.87000	−0.0768567
$593$	45.0006	1.84795	0.923977	−	0.382448i	$-0.124919\pi$
$593$	45.0006	1.84795	0.923977	+	0.382448i	$0.124919\pi$
$594$	0	0
$595$	6.88817	0.282388
$596$	17.7087	0.725377
$597$	0	0
$598$	13.0144	0.532200
$599$	−19.7687	−0.807727	−0.403864	−	0.914819i	$-0.632333\pi$
$599$	−19.7687	−0.807727	−0.403864	+	0.914819i	$0.632333\pi$
$600$	0	0
$601$	−28.7745	−1.17374	−0.586868	−	0.809683i	$-0.699639\pi$
$601$	−28.7745	−1.17374	−0.586868	+	0.809683i	$0.699639\pi$
$602$	3.80736	0.155176
$603$	0	0
$604$	−11.7653	−0.478721
$605$	11.5541	0.469740
$606$	0	0
$607$	−23.9948	−0.973918	−0.486959	−	0.873425i	$-0.661894\pi$
$607$	−23.9948	−0.973918	−0.486959	+	0.873425i	$0.661894\pi$
$608$	34.3020	1.39113
$609$	0	0
$610$	−7.44644	−0.301498
$611$	−11.8583	−0.479737
$612$	0	0
$613$	4.53125	0.183016	0.0915078	−	0.995804i	$-0.470831\pi$
$613$	4.53125	0.183016	0.0915078	+	0.995804i	$0.470831\pi$
$614$	−26.5123	−1.06995
$615$	0	0
$616$	−5.59435	−0.225403
$617$	21.0239	0.846389	0.423194	−	0.906039i	$-0.360909\pi$
$617$	21.0239	0.846389	0.423194	+	0.906039i	$0.360909\pi$
$618$	0	0
$619$	6.66600	0.267929	0.133965	−	0.990986i	$-0.457229\pi$
$619$	6.66600	0.267929	0.133965	+	0.990986i	$0.457229\pi$
$620$	−14.5377	−0.583849
$621$	0	0
$622$	16.6990	0.669571
$623$	−17.5151	−0.701726
$624$	0	0
$625$	−8.70615	−0.348246
$626$	−18.4450	−0.737210
$627$	0	0
$628$	7.80138	0.311309
$629$	40.6932	1.62254
$630$	0	0
$631$	35.4589	1.41160	0.705798	−	0.708413i	$-0.250589\pi$
$631$	35.4589	1.41160	0.705798	+	0.708413i	$0.250589\pi$
$632$	−45.7967	−1.82170
$633$	0	0
$634$	−25.0709	−0.995693
$635$	−1.65890	−0.0658316
$636$	0	0
$637$	−2.46735	−0.0977599
$638$	−6.94836	−0.275088
$639$	0	0
$640$	−12.6934	−0.501750
$641$	−36.9481	−1.45936	−0.729681	−	0.683788i	$-0.760331\pi$
$641$	−36.9481	−1.45936	−0.729681	+	0.683788i	$0.760331\pi$
$642$	0	0
$643$	10.3236	0.407124	0.203562	−	0.979062i	$-0.434748\pi$
$643$	10.3236	0.407124	0.203562	+	0.979062i	$0.434748\pi$
$644$	−7.93729	−0.312773
$645$	0	0
$646$	−21.1086	−0.830508
$647$	8.02312	0.315421	0.157711	−	0.987485i	$-0.449589\pi$
$647$	8.02312	0.315421	0.157711	+	0.987485i	$0.449589\pi$
$648$	0	0
$649$	6.54043	0.256734
$650$	−4.71196	−0.184818
$651$	0	0
$652$	10.1470	0.397389
$653$	−19.8778	−0.777877	−0.388938	−	0.921264i	$-0.627158\pi$
$653$	−19.8778	−0.777877	−0.388938	+	0.921264i	$0.627158\pi$
$654$	0	0
$655$	21.8390	0.853321
$656$	1.24853	0.0487471
$657$	0	0
$658$	−4.08283	−0.159165
$659$	−30.9106	−1.20411	−0.602054	−	0.798456i	$-0.705651\pi$
$659$	−30.9106	−1.20411	−0.602054	+	0.798456i	$0.705651\pi$
$660$	0	0
$661$	−9.40738	−0.365905	−0.182952	−	0.983122i	$-0.558565\pi$
$661$	−9.40738	−0.365905	−0.182952	+	0.983122i	$0.558565\pi$
$662$	6.79076	0.263930
$663$	0	0
$664$	−0.918224	−0.0356340
$665$	9.92729	0.384964
$666$	0	0
$667$	−25.2821	−0.978928
$668$	14.9671	0.579096
$669$	0	0
$670$	1.10006	0.0424991
$671$	10.6142	0.409757
$672$	0	0
$673$	−28.2924	−1.09059	−0.545296	−	0.838243i	$-0.683583\pi$
$673$	−28.2924	−1.09059	−0.545296	+	0.838243i	$0.683583\pi$
$674$	−26.1702	−1.00804
$675$	0	0
$676$	8.83609	0.339850
$677$	10.7916	0.414756	0.207378	−	0.978261i	$-0.433507\pi$
$677$	10.7916	0.414756	0.207378	+	0.978261i	$0.433507\pi$
$678$	0	0
$679$	6.36680	0.244336
$680$	19.1834	0.735649
$681$	0	0
$682$	−11.6984	−0.447955
$683$	49.1220	1.87960	0.939801	−	0.341722i	$-0.111010\pi$
$683$	49.1220	1.87960	0.939801	+	0.341722i	$0.111010\pi$
$684$	0	0
$685$	−4.92553	−0.188195
$686$	−0.849509	−0.0324344
$687$	0	0
$688$	−0.855184	−0.0326036
$689$	−17.1413	−0.653031
$690$	0	0
$691$	−23.8404	−0.906932	−0.453466	−	0.891274i	$-0.649813\pi$
$691$	−23.8404	−0.906932	−0.453466	+	0.891274i	$0.649813\pi$
$692$	−2.64415	−0.100516
$693$	0	0
$694$	18.2108	0.691274
$695$	4.56385	0.173117
$696$	0	0
$697$	−27.1693	−1.02911
$698$	19.2288	0.727821
$699$	0	0
$700$	2.87374	0.108617
$701$	−24.8328	−0.937923	−0.468961	−	0.883219i	$-0.655372\pi$
$701$	−24.8328	−0.937923	−0.468961	+	0.883219i	$0.655372\pi$
$702$	0	0
$703$	58.6473	2.21192
$704$	−9.01491	−0.339762
$705$	0	0
$706$	−5.28229	−0.198802
$707$	−6.70776	−0.252271
$708$	0	0
$709$	−17.0114	−0.638878	−0.319439	−	0.947607i	$-0.603494\pi$
$709$	−17.0114	−0.638878	−0.319439	+	0.947607i	$0.603494\pi$
$710$	9.56823	0.359089
$711$	0	0
$712$	−48.7790	−1.82807
$713$	−42.5655	−1.59409
$714$	0	0
$715$	−8.22205	−0.307487
$716$	10.2690	0.383769
$717$	0	0
$718$	−22.8403	−0.852393
$719$	−5.29641	−0.197523	−0.0987613	−	0.995111i	$-0.531488\pi$
$719$	−5.29641	−0.197523	−0.0987613	+	0.995111i	$0.531488\pi$
$720$	0	0
$721$	1.56085	0.0581292
$722$	−14.2813	−0.531493
$723$	0	0
$724$	−22.0365	−0.818980
$725$	9.15355	0.339954
$726$	0	0
$727$	−9.10685	−0.337755	−0.168877	−	0.985637i	$-0.554014\pi$
$727$	−9.10685	−0.337755	−0.168877	+	0.985637i	$0.554014\pi$
$728$	−6.87150	−0.254675
$729$	0	0
$730$	21.1593	0.783139
$731$	18.6097	0.688303
$732$	0	0
$733$	0.328171	0.0121213	0.00606063	−	0.999982i	$-0.498071\pi$
$733$	0.328171	0.0121213	0.00606063	+	0.999982i	$0.498071\pi$
$734$	16.2540	0.599946
$735$	0	0
$736$	−35.5907	−1.31189
$737$	−1.56804	−0.0577595
$738$	0	0
$739$	−38.6597	−1.42212	−0.711060	−	0.703131i	$-0.751785\pi$
$739$	−38.6597	−1.42212	−0.711060	+	0.703131i	$0.751785\pi$
$740$	−20.7828	−0.763992
$741$	0	0
$742$	−5.90174	−0.216660
$743$	−36.2611	−1.33029	−0.665145	−	0.746714i	$-0.731630\pi$
$743$	−36.2611	−1.33029	−0.665145	+	0.746714i	$0.731630\pi$
$744$	0	0
$745$	22.9807	0.841948
$746$	7.78434	0.285005
$747$	0	0
$748$	−10.6624	−0.389857
$749$	3.17782	0.116115
$750$	0	0
$751$	10.1095	0.368901	0.184451	−	0.982842i	$-0.440949\pi$
$751$	10.1095	0.368901	0.184451	+	0.982842i	$0.440949\pi$
$752$	0.917060	0.0334417
$753$	0	0
$754$	−8.53463	−0.310813
$755$	−15.2679	−0.555654
$756$	0	0
$757$	22.9406	0.833790	0.416895	−	0.908955i	$-0.363118\pi$
$757$	22.9406	0.833790	0.416895	+	0.908955i	$0.363118\pi$
$758$	1.58735	0.0576552
$759$	0	0
$760$	27.6472	1.00287
$761$	−12.1774	−0.441430	−0.220715	−	0.975338i	$-0.570839\pi$
$761$	−12.1774	−0.441430	−0.220715	+	0.975338i	$0.570839\pi$
$762$	0	0
$763$	−5.72634	−0.207308
$764$	−23.5575	−0.852281
$765$	0	0
$766$	1.32062	0.0477160
$767$	8.03357	0.290076
$768$	0	0
$769$	20.2525	0.730324	0.365162	−	0.930944i	$-0.381014\pi$
$769$	20.2525	0.730324	0.365162	+	0.930944i	$0.381014\pi$
$770$	−2.83085	−0.102017
$771$	0	0
$772$	−29.0260	−1.04467
$773$	13.0856	0.470656	0.235328	−	0.971916i	$-0.424384\pi$
$773$	13.0856	0.470656	0.235328	+	0.971916i	$0.424384\pi$
$774$	0	0
$775$	15.4111	0.553583
$776$	17.7314	0.636520
$777$	0	0
$778$	16.6781	0.597937
$779$	−39.1567	−1.40293
$780$	0	0
$781$	−13.6386	−0.488029
$782$	21.9017	0.783203
$783$	0	0
$784$	0.190811	0.00681469
$785$	10.1239	0.361338
$786$	0	0
$787$	19.0664	0.679643	0.339822	−	0.940490i	$-0.389633\pi$
$787$	19.0664	0.679643	0.339822	+	0.940490i	$0.389633\pi$
$788$	−3.93268	−0.140096
$789$	0	0
$790$	−23.1741	−0.824497
$791$	−10.7097	−0.380793
$792$	0	0
$793$	13.0374	0.462971
$794$	−23.2768	−0.826063
$795$	0	0
$796$	9.31516	0.330167
$797$	−12.6227	−0.447119	−0.223559	−	0.974690i	$-0.571768\pi$
$797$	−12.6227	−0.447119	−0.223559	+	0.974690i	$0.571768\pi$
$798$	0	0
$799$	−19.9561	−0.705997
$800$	12.8858	0.455583
$801$	0	0
$802$	−6.67374	−0.235658
$803$	−30.1606	−1.06434
$804$	0	0
$805$	−10.3003	−0.363037
$806$	−14.3691	−0.506129
$807$	0	0
$808$	−18.6809	−0.657193
$809$	38.1883	1.34263	0.671315	−	0.741172i	$-0.265730\pi$
$809$	38.1883	1.34263	0.671315	+	0.741172i	$0.265730\pi$
$810$	0	0
$811$	22.3629	0.785268	0.392634	−	0.919695i	$-0.371564\pi$
$811$	22.3629	0.785268	0.392634	+	0.919695i	$0.371564\pi$
$812$	5.20512	0.182664
$813$	0	0
$814$	−16.7238	−0.586169
$815$	13.1679	0.461251
$816$	0	0
$817$	26.8204	0.938327
$818$	14.4831	0.506388
$819$	0	0
$820$	13.8759	0.484569
$821$	−27.3526	−0.954612	−0.477306	−	0.878737i	$-0.658387\pi$
$821$	−27.3526	−0.954612	−0.477306	+	0.878737i	$0.658387\pi$
$822$	0	0
$823$	48.5665	1.69292	0.846460	−	0.532452i	$-0.178729\pi$
$823$	48.5665	1.69292	0.846460	+	0.532452i	$0.178729\pi$
$824$	4.34693	0.151433
$825$	0	0
$826$	2.76596	0.0962401
$827$	−12.0606	−0.419389	−0.209695	−	0.977767i	$-0.567247\pi$
$827$	−12.0606	−0.419389	−0.209695	+	0.977767i	$0.567247\pi$
$828$	0	0
$829$	−30.5355	−1.06054	−0.530270	−	0.847829i	$-0.677910\pi$
$829$	−30.5355	−1.06054	−0.530270	+	0.847829i	$0.677910\pi$
$830$	−0.464640	−0.0161279
$831$	0	0
$832$	−11.0730	−0.383886
$833$	−4.15224	−0.143867
$834$	0	0
$835$	19.4230	0.672159
$836$	−15.3668	−0.531471
$837$	0	0
$838$	−21.7384	−0.750941
$839$	26.8390	0.926585	0.463293	−	0.886205i	$-0.346668\pi$
$839$	26.8390	0.926585	0.463293	+	0.886205i	$0.346668\pi$
$840$	0	0
$841$	−12.4205	−0.428291
$842$	−18.3522	−0.632458
$843$	0	0
$844$	−21.1259	−0.727183
$845$	11.4667	0.394465
$846$	0	0
$847$	−6.96488	−0.239316
$848$	1.32561	0.0455217
$849$	0	0
$850$	−7.92965	−0.271985
$851$	−60.8508	−2.08594
$852$	0	0
$853$	−2.90690	−0.0995304	−0.0497652	−	0.998761i	$-0.515847\pi$
$853$	−2.90690	−0.0995304	−0.0497652	+	0.998761i	$0.515847\pi$
$854$	4.48877	0.153603
$855$	0	0
$856$	8.85014	0.302492
$857$	31.3655	1.07143	0.535713	−	0.844400i	$-0.320043\pi$
$857$	31.3655	1.07143	0.535713	+	0.844400i	$0.320043\pi$
$858$	0	0
$859$	−24.6135	−0.839800	−0.419900	−	0.907570i	$-0.637935\pi$
$859$	−24.6135	−0.839800	−0.419900	+	0.907570i	$0.637935\pi$
$860$	−9.50434	−0.324095
$861$	0	0
$862$	7.42339	0.252842
$863$	−21.9379	−0.746774	−0.373387	−	0.927676i	$-0.621804\pi$
$863$	−21.9379	−0.746774	−0.373387	+	0.927676i	$0.621804\pi$
$864$	0	0
$865$	−3.43134	−0.116669
$866$	−27.4823	−0.933885
$867$	0	0
$868$	8.76345	0.297451
$869$	33.0325	1.12055
$870$	0	0
$871$	−1.92601	−0.0652604
$872$	−15.9477	−0.540058
$873$	0	0
$874$	31.5649	1.06770
$875$	12.0238	0.406479
$876$	0	0
$877$	46.7595	1.57896	0.789478	−	0.613778i	$-0.210351\pi$
$877$	46.7595	1.57896	0.789478	+	0.613778i	$0.210351\pi$
$878$	−27.3658	−0.923551
$879$	0	0
$880$	0.635849	0.0214344
$881$	23.2192	0.782276	0.391138	−	0.920332i	$-0.372082\pi$
$881$	23.2192	0.782276	0.391138	+	0.920332i	$0.372082\pi$
$882$	0	0
$883$	31.1884	1.04957	0.524787	−	0.851234i	$-0.324145\pi$
$883$	31.1884	1.04957	0.524787	+	0.851234i	$0.324145\pi$
$884$	−13.0966	−0.440486
$885$	0	0
$886$	0.385714	0.0129583
$887$	27.2182	0.913899	0.456950	−	0.889493i	$-0.348942\pi$
$887$	27.2182	0.913899	0.456950	+	0.889493i	$0.348942\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	−24.6832	−0.827382
$891$	0	0
$892$	34.7274	1.16276
$893$	−28.7609	−0.962448
$894$	0	0
$895$	13.3261	0.445443
$896$	7.65166	0.255624
$897$	0	0
$898$	−2.06056	−0.0687619
$899$	27.9137	0.930973
$900$	0	0
$901$	−28.8466	−0.961021
$902$	11.1659	0.371783
$903$	0	0
$904$	−29.8262	−0.992005
$905$	−28.5969	−0.950594
$906$	0	0
$907$	−42.3943	−1.40768	−0.703839	−	0.710359i	$-0.748533\pi$
$907$	−42.3943	−1.40768	−0.703839	+	0.710359i	$0.748533\pi$
$908$	32.7801	1.08785
$909$	0	0
$910$	−3.47712	−0.115265
$911$	34.7415	1.15104	0.575519	−	0.817788i	$-0.304800\pi$
$911$	34.7415	1.15104	0.575519	+	0.817788i	$0.304800\pi$
$912$	0	0
$913$	0.662302	0.0219190
$914$	−14.0060	−0.463279
$915$	0	0
$916$	−7.64601	−0.252631
$917$	−13.1647	−0.434737
$918$	0	0
$919$	0.253061	0.00834771	0.00417386	−	0.999991i	$-0.498671\pi$
$919$	0.253061	0.00834771	0.00417386	+	0.999991i	$0.498671\pi$
$920$	−28.6860	−0.945749
$921$	0	0
$922$	−14.3885	−0.473859
$923$	−16.7522	−0.551407
$924$	0	0
$925$	22.0314	0.724388
$926$	6.17125	0.202800
$927$	0	0
$928$	23.3397	0.766164
$929$	−30.0838	−0.987017	−0.493508	−	0.869741i	$-0.664286\pi$
$929$	−30.0838	−0.987017	−0.493508	+	0.869741i	$0.664286\pi$
$930$	0	0
$931$	−5.98425	−0.196126
$932$	−1.44144	−0.0472160
$933$	0	0
$934$	2.23126	0.0730089
$935$	−13.8367	−0.452508
$936$	0	0
$937$	27.8139	0.908640	0.454320	−	0.890839i	$-0.349882\pi$
$937$	27.8139	0.908640	0.454320	+	0.890839i	$0.349882\pi$
$938$	−0.663126	−0.0216518
$939$	0	0
$940$	10.1920	0.332427
$941$	−30.6670	−0.999715	−0.499857	−	0.866108i	$-0.666614\pi$
$941$	−30.6670	−0.999715	−0.499857	+	0.866108i	$0.666614\pi$
$942$	0	0
$943$	40.6278	1.32302
$944$	−0.621273	−0.0202207
$945$	0	0
$946$	−7.64807	−0.248660
$947$	11.7555	0.382002	0.191001	−	0.981590i	$-0.438827\pi$
$947$	11.7555	0.382002	0.191001	+	0.981590i	$0.438827\pi$
$948$	0	0
$949$	−37.0461	−1.20257
$950$	−11.4283	−0.370782
$951$	0	0
$952$	−11.5639	−0.374788
$953$	34.9073	1.13076	0.565379	−	0.824831i	$-0.308730\pi$
$953$	34.9073	1.13076	0.565379	+	0.824831i	$0.308730\pi$
$954$	0	0
$955$	−30.5708	−0.989247
$956$	−28.4914	−0.921479
$957$	0	0
$958$	8.70448	0.281229
$959$	2.96915	0.0958787
$960$	0	0
$961$	15.9960	0.516000
$962$	−20.5417	−0.662292
$963$	0	0
$964$	9.03694	0.291060
$965$	−37.6672	−1.21255
$966$	0	0
$967$	45.4810	1.46257	0.731285	−	0.682073i	$-0.238921\pi$
$967$	45.4810	1.46257	0.731285	+	0.682073i	$0.238921\pi$
$968$	−19.3970	−0.623443
$969$	0	0
$970$	8.97244	0.288088
$971$	24.4636	0.785073	0.392536	−	0.919736i	$-0.371598\pi$
$971$	24.4636	0.785073	0.392536	+	0.919736i	$0.371598\pi$
$972$	0	0
$973$	−2.75112	−0.0881969
$974$	−12.6855	−0.406470
$975$	0	0
$976$	−1.00824	−0.0322730
$977$	8.82161	0.282228	0.141114	−	0.989993i	$-0.454932\pi$
$977$	8.82161	0.282228	0.141114	+	0.989993i	$0.454932\pi$
$978$	0	0
$979$	35.1836	1.12447
$980$	2.12064	0.0677412
$981$	0	0
$982$	−24.5565	−0.783628
$983$	−1.08856	−0.0347197	−0.0173599	−	0.999849i	$-0.505526\pi$
$983$	−1.08856	−0.0347197	−0.0173599	+	0.999849i	$0.505526\pi$
$984$	0	0
$985$	−5.10346	−0.162610
$986$	−14.3627	−0.457402
$987$	0	0
$988$	−18.8749	−0.600491
$989$	−27.8281	−0.884881
$990$	0	0
$991$	−40.4080	−1.28360	−0.641801	−	0.766871i	$-0.721812\pi$
$991$	−40.4080	−1.28360	−0.641801	+	0.766871i	$0.721812\pi$
$992$	39.2952	1.24763
$993$	0	0
$994$	−5.76780	−0.182944
$995$	12.0883	0.383226
$996$	0	0
$997$	−3.01318	−0.0954285	−0.0477143	−	0.998861i	$-0.515194\pi$
$997$	−3.01318	−0.0954285	−0.0477143	+	0.998861i	$0.515194\pi$
$998$	23.0072	0.728281
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.n.1.4		12
3.2	odd	2		889.2.a.a.1.9	✓	12
21.20	even	2		6223.2.a.i.1.9		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
889.2.a.a.1.9	✓	12	3.2	odd	2
6223.2.a.i.1.9		12	21.20	even	2
8001.2.a.n.1.4		12	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)

\(f(q)\)	\(=\)	\(q-0.849509 q^{2} -1.27834 q^{4} -1.65890 q^{5} +1.00000 q^{7} +2.78497 q^{8} +O(q^{10})\)	\(q-0.849509 q^{2} -1.27834 q^{4} -1.65890 q^{5} +1.00000 q^{7} +2.78497 q^{8} +1.40925 q^{10} -2.00876 q^{11} -2.46735 q^{13} -0.849509 q^{14} +0.190811 q^{16} -4.15224 q^{17} -5.98425 q^{19} +2.12064 q^{20} +1.70646 q^{22} +6.20908 q^{23} -2.24804 q^{25} +2.09603 q^{26} -1.27834 q^{28} -4.07180 q^{29} -6.85536 q^{31} -5.73204 q^{32} +3.52737 q^{34} -1.65890 q^{35} -9.80028 q^{37} +5.08367 q^{38} -4.62001 q^{40} +6.54329 q^{41} -4.48183 q^{43} +2.56787 q^{44} -5.27467 q^{46} +4.80611 q^{47} +1.00000 q^{49} +1.90973 q^{50} +3.15410 q^{52} +6.94724 q^{53} +3.33234 q^{55} +2.78497 q^{56} +3.45903 q^{58} -3.25595 q^{59} -5.28396 q^{61} +5.82369 q^{62} +4.48780 q^{64} +4.09310 q^{65} +0.780600 q^{67} +5.30796 q^{68} +1.40925 q^{70} +6.78957 q^{71} +15.0145 q^{73} +8.32542 q^{74} +7.64987 q^{76} -2.00876 q^{77} -16.4442 q^{79} -0.316538 q^{80} -5.55858 q^{82} -0.329707 q^{83} +6.88817 q^{85} +3.80736 q^{86} -5.59435 q^{88} -17.5151 q^{89} -2.46735 q^{91} -7.93729 q^{92} -4.08283 q^{94} +9.92729 q^{95} +6.36680 q^{97} -0.849509 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 7 q^{2} + 9 q^{4} + 7 q^{5} + 12 q^{7} + 15 q^{8}+O(q^{10}) \) 12 * q + 7 * q^2 + 9 * q^4 + 7 * q^5 + 12 * q^7 + 15 * q^8	\( 12 q + 7 q^{2} + 9 q^{4} + 7 q^{5} + 12 q^{7} + 15 q^{8} - 2 q^{10} + 22 q^{11} + 7 q^{14} + 7 q^{16} + 6 q^{17} - 7 q^{19} + 8 q^{20} + 13 q^{22} + 29 q^{23} + 3 q^{25} + 9 q^{28} + 22 q^{29} - 16 q^{31} + 27 q^{32} - 5 q^{34} + 7 q^{35} - 4 q^{37} - 2 q^{38} + 16 q^{40} + 21 q^{41} + 11 q^{43} + 11 q^{44} + 31 q^{47} + 12 q^{49} + 21 q^{50} + 3 q^{52} + 38 q^{53} - 11 q^{55} + 15 q^{56} + 20 q^{58} + 15 q^{59} - 3 q^{61} + 4 q^{62} + 29 q^{64} + 32 q^{65} - q^{67} - 17 q^{68} - 2 q^{70} + 57 q^{71} - 7 q^{73} + 42 q^{74} - 44 q^{76} + 22 q^{77} - 18 q^{79} - q^{80} + 56 q^{82} + 21 q^{83} - 5 q^{85} + 32 q^{86} - 10 q^{88} - 6 q^{89} + 15 q^{92} + 35 q^{94} + 57 q^{95} + 4 q^{97} + 7 q^{98}+O(q^{100}) \) 12 * q + 7 * q^2 + 9 * q^4 + 7 * q^5 + 12 * q^7 + 15 * q^8 - 2 * q^10 + 22 * q^11 + 7 * q^14 + 7 * q^16 + 6 * q^17 - 7 * q^19 + 8 * q^20 + 13 * q^22 + 29 * q^23 + 3 * q^25 + 9 * q^28 + 22 * q^29 - 16 * q^31 + 27 * q^32 - 5 * q^34 + 7 * q^35 - 4 * q^37 - 2 * q^38 + 16 * q^40 + 21 * q^41 + 11 * q^43 + 11 * q^44 + 31 * q^47 + 12 * q^49 + 21 * q^50 + 3 * q^52 + 38 * q^53 - 11 * q^55 + 15 * q^56 + 20 * q^58 + 15 * q^59 - 3 * q^61 + 4 * q^62 + 29 * q^64 + 32 * q^65 - q^67 - 17 * q^68 - 2 * q^70 + 57 * q^71 - 7 * q^73 + 42 * q^74 - 44 * q^76 + 22 * q^77 - 18 * q^79 - q^80 + 56 * q^82 + 21 * q^83 - 5 * q^85 + 32 * q^86 - 10 * q^88 - 6 * q^89 + 15 * q^92 + 35 * q^94 + 57 * q^95 + 4 * q^97 + 7 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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