LMFDB - Embedded newform 8001.2.a.o.1.7

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:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 4 x^{12} - 10 x^{11} + 53 x^{10} + 19 x^{9} - 242 x^{8} + 61 x^{7} + 467 x^{6} - 211 x^{5} + \cdots - 8 $ x^13 - 4x^12 - 10x^11 + 53x^10 + 19x^9 - 242x^8 + 61x^7 + 467x^6 - 211x^5 - 372x^4 + 146x^3 + 116x^2 - 12x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 2667)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$0.307326$ 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)

$f(q)$	$=$	$q-0.307326 q^{2} -1.90555 q^{4} +0.988649 q^{5} +1.00000 q^{7} +1.20028 q^{8} +O(q^{10})$	\(q-0.307326 q^{2} -1.90555 q^{4} +0.988649 q^{5} +1.00000 q^{7} +1.20028 q^{8} -0.303838 q^{10} -2.00682 q^{11} -0.289737 q^{13} -0.307326 q^{14} +3.44222 q^{16} -0.186210 q^{17} +0.852422 q^{19} -1.88392 q^{20} +0.616750 q^{22} +3.33599 q^{23} -4.02257 q^{25} +0.0890436 q^{26} -1.90555 q^{28} +4.37763 q^{29} -5.73898 q^{31} -3.45844 q^{32} +0.0572272 q^{34} +0.988649 q^{35} -7.05580 q^{37} -0.261972 q^{38} +1.18665 q^{40} -3.81974 q^{41} +9.06384 q^{43} +3.82411 q^{44} -1.02524 q^{46} -1.34856 q^{47} +1.00000 q^{49} +1.23624 q^{50} +0.552108 q^{52} -3.75486 q^{53} -1.98405 q^{55} +1.20028 q^{56} -1.34536 q^{58} +4.26624 q^{59} +3.08411 q^{61} +1.76374 q^{62} -5.82158 q^{64} -0.286448 q^{65} -10.8349 q^{67} +0.354832 q^{68} -0.303838 q^{70} -13.0932 q^{71} +14.9066 q^{73} +2.16843 q^{74} -1.62433 q^{76} -2.00682 q^{77} +16.7593 q^{79} +3.40315 q^{80} +1.17391 q^{82} -15.1974 q^{83} -0.184096 q^{85} -2.78556 q^{86} -2.40875 q^{88} +0.526198 q^{89} -0.289737 q^{91} -6.35689 q^{92} +0.414448 q^{94} +0.842746 q^{95} +6.83150 q^{97} -0.307326 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 4 q^{2} + 10 q^{4} - 12 q^{5} + 13 q^{7} - 9 q^{8}+O(q^{10}) $ 13 * q - 4 * q^2 + 10 * q^4 - 12 * q^5 + 13 * q^7 - 9 * q^8	$ 13 q - 4 q^{2} + 10 q^{4} - 12 q^{5} + 13 q^{7} - 9 q^{8} + 6 q^{10} - 3 q^{11} + 21 q^{13} - 4 q^{14} + 8 q^{16} - 17 q^{17} + 5 q^{19} - 29 q^{20} + q^{22} - 4 q^{23} + q^{25} - 22 q^{26} + 10 q^{28} - 21 q^{29} - 7 q^{31} - 12 q^{32} + 2 q^{34} - 12 q^{35} + 7 q^{37} + 9 q^{38} + 29 q^{40} - 21 q^{41} - 9 q^{43} + 2 q^{44} - 28 q^{46} - 23 q^{47} + 13 q^{49} - 15 q^{50} + 15 q^{52} - 31 q^{53} - 8 q^{55} - 9 q^{56} - 25 q^{58} - 28 q^{59} + 29 q^{61} + 3 q^{62} + 9 q^{64} - 30 q^{65} - 18 q^{67} - 34 q^{68} + 6 q^{70} - 10 q^{71} + 24 q^{73} + 19 q^{74} - 3 q^{77} - 28 q^{79} - 26 q^{80} + 18 q^{82} - 26 q^{83} + 20 q^{85} + 2 q^{86} - 17 q^{88} - 44 q^{89} + 21 q^{91} - 6 q^{92} - 9 q^{94} + 2 q^{95} + 17 q^{97} - 4 q^{98}+O(q^{100}) $ 13 * q - 4 * q^2 + 10 * q^4 - 12 * q^5 + 13 * q^7 - 9 * q^8 + 6 * q^10 - 3 * q^11 + 21 * q^13 - 4 * q^14 + 8 * q^16 - 17 * q^17 + 5 * q^19 - 29 * q^20 + q^22 - 4 * q^23 + q^25 - 22 * q^26 + 10 * q^28 - 21 * q^29 - 7 * q^31 - 12 * q^32 + 2 * q^34 - 12 * q^35 + 7 * q^37 + 9 * q^38 + 29 * q^40 - 21 * q^41 - 9 * q^43 + 2 * q^44 - 28 * q^46 - 23 * q^47 + 13 * q^49 - 15 * q^50 + 15 * q^52 - 31 * q^53 - 8 * q^55 - 9 * q^56 - 25 * q^58 - 28 * q^59 + 29 * q^61 + 3 * q^62 + 9 * q^64 - 30 * q^65 - 18 * q^67 - 34 * q^68 + 6 * q^70 - 10 * q^71 + 24 * q^73 + 19 * q^74 - 3 * q^77 - 28 * q^79 - 26 * q^80 + 18 * q^82 - 26 * q^83 + 20 * q^85 + 2 * q^86 - 17 * q^88 - 44 * q^89 + 21 * q^91 - 6 * q^92 - 9 * q^94 + 2 * q^95 + 17 * q^97 - 4 * 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 $

$2$	−0.307326	−0.217312	−0.108656	−	0.994079i	$-0.534655\pi$
$2$	−0.307326	−0.217312	−0.108656	+	0.994079i	$0.534655\pi$
$3$	0	0
$3$	0	0
$4$	−1.90555	−0.952775
$5$	0.988649	0.442137	0.221069	−	0.975258i	$-0.429046\pi$
$5$	0.988649	0.442137	0.221069	+	0.975258i	$0.429046\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.20028	0.424362
$9$	0	0
$10$	−0.303838	−0.0960819
$11$	−2.00682	−0.605081	−0.302540	−	0.953137i	$-0.597835\pi$
$11$	−2.00682	−0.605081	−0.302540	+	0.953137i	$0.597835\pi$
$12$	0	0
$13$	−0.289737	−0.0803585	−0.0401792	−	0.999192i	$-0.512793\pi$
$13$	−0.289737	−0.0803585	−0.0401792	+	0.999192i	$0.512793\pi$
$14$	−0.307326	−0.0821364
$15$	0	0
$16$	3.44222	0.860556
$17$	−0.186210	−0.0451625	−0.0225813	−	0.999745i	$-0.507188\pi$
$17$	−0.186210	−0.0451625	−0.0225813	+	0.999745i	$0.507188\pi$
$18$	0	0
$19$	0.852422	0.195559	0.0977795	−	0.995208i	$-0.468826\pi$
$19$	0.852422	0.195559	0.0977795	+	0.995208i	$0.468826\pi$
$20$	−1.88392	−0.421257
$21$	0	0
$22$	0.616750	0.131491
$23$	3.33599	0.695601	0.347801	−	0.937569i	$-0.386929\pi$
$23$	3.33599	0.695601	0.347801	+	0.937569i	$0.386929\pi$
$24$	0	0
$25$	−4.02257	−0.804515
$26$	0.0890436	0.0174629
$27$	0	0
$28$	−1.90555	−0.360115
$29$	4.37763	0.812905	0.406452	−	0.913672i	$-0.366766\pi$
$29$	4.37763	0.812905	0.406452	+	0.913672i	$0.366766\pi$
$30$	0	0
$31$	−5.73898	−1.03075	−0.515376	−	0.856964i	$-0.672348\pi$
$31$	−5.73898	−1.03075	−0.515376	+	0.856964i	$0.672348\pi$
$32$	−3.45844	−0.611372
$33$	0	0
$34$	0.0572272	0.00981438
$35$	0.988649	0.167112
$36$	0	0
$37$	−7.05580	−1.15997	−0.579983	−	0.814629i	$-0.696941\pi$
$37$	−7.05580	−1.15997	−0.579983	+	0.814629i	$0.696941\pi$
$38$	−0.261972	−0.0424974
$39$	0	0
$40$	1.18665	0.187626
$41$	−3.81974	−0.596544	−0.298272	−	0.954481i	$-0.596410\pi$
$41$	−3.81974	−0.596544	−0.298272	+	0.954481i	$0.596410\pi$
$42$	0	0
$43$	9.06384	1.38222	0.691112	−	0.722748i	$-0.257121\pi$
$43$	9.06384	1.38222	0.691112	+	0.722748i	$0.257121\pi$
$44$	3.82411	0.576506
$45$	0	0
$46$	−1.02524	−0.151163
$47$	−1.34856	−0.196708	−0.0983540	−	0.995151i	$-0.531358\pi$
$47$	−1.34856	−0.196708	−0.0983540	+	0.995151i	$0.531358\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	1.23624	0.174831
$51$	0	0
$52$	0.552108	0.0765636
$53$	−3.75486	−0.515770	−0.257885	−	0.966176i	$-0.583026\pi$
$53$	−3.75486	−0.515770	−0.257885	+	0.966176i	$0.583026\pi$
$54$	0	0
$55$	−1.98405	−0.267529
$56$	1.20028	0.160394
$57$	0	0
$58$	−1.34536	−0.176654
$59$	4.26624	0.555418	0.277709	−	0.960665i	$-0.410425\pi$
$59$	4.26624	0.555418	0.277709	+	0.960665i	$0.410425\pi$
$60$	0	0
$61$	3.08411	0.394880	0.197440	−	0.980315i	$-0.436737\pi$
$61$	3.08411	0.394880	0.197440	+	0.980315i	$0.436737\pi$
$62$	1.76374	0.223995
$63$	0	0
$64$	−5.82158	−0.727698
$65$	−0.286448	−0.0355295
$66$	0	0
$67$	−10.8349	−1.32370	−0.661850	−	0.749636i	$-0.730228\pi$
$67$	−10.8349	−1.32370	−0.661850	+	0.749636i	$0.730228\pi$
$68$	0.354832	0.0430297
$69$	0	0
$70$	−0.303838	−0.0363155
$71$	−13.0932	−1.55387	−0.776937	−	0.629579i	$-0.783227\pi$
$71$	−13.0932	−1.55387	−0.776937	+	0.629579i	$0.783227\pi$
$72$	0	0
$73$	14.9066	1.74469	0.872343	−	0.488894i	$-0.162600\pi$
$73$	14.9066	1.74469	0.872343	+	0.488894i	$0.162600\pi$
$74$	2.16843	0.252075
$75$	0	0
$76$	−1.62433	−0.186324
$77$	−2.00682	−0.228699
$78$	0	0
$79$	16.7593	1.88557	0.942785	−	0.333400i	$-0.108196\pi$
$79$	16.7593	1.88557	0.942785	+	0.333400i	$0.108196\pi$
$80$	3.40315	0.380484
$81$	0	0
$82$	1.17391	0.129636
$83$	−15.1974	−1.66813	−0.834067	−	0.551663i	$-0.813993\pi$
$83$	−15.1974	−1.66813	−0.834067	+	0.551663i	$0.813993\pi$
$84$	0	0
$85$	−0.184096	−0.0199680
$86$	−2.78556	−0.300374
$87$	0	0
$88$	−2.40875	−0.256773
$89$	0.526198	0.0557769	0.0278884	−	0.999611i	$-0.491122\pi$
$89$	0.526198	0.0557769	0.0278884	+	0.999611i	$0.491122\pi$
$90$	0	0
$91$	−0.289737	−0.0303726
$92$	−6.35689	−0.662752
$93$	0	0
$94$	0.414448	0.0427471
$95$	0.842746	0.0864639
$96$	0	0
$97$	6.83150	0.693634	0.346817	−	0.937933i	$-0.387262\pi$
$97$	6.83150	0.693634	0.346817	+	0.937933i	$0.387262\pi$
$98$	−0.307326	−0.0310446
$99$	0	0
$100$	7.66522	0.766522
$101$	−13.4337	−1.33670	−0.668350	−	0.743847i	$-0.732999\pi$
$101$	−13.4337	−1.33670	−0.668350	+	0.743847i	$0.732999\pi$
$102$	0	0
$103$	−2.29311	−0.225947	−0.112973	−	0.993598i	$-0.536037\pi$
$103$	−2.29311	−0.225947	−0.112973	+	0.993598i	$0.536037\pi$
$104$	−0.347764	−0.0341011
$105$	0	0
$106$	1.15397	0.112083
$107$	−4.90083	−0.473781	−0.236890	−	0.971536i	$-0.576128\pi$
$107$	−4.90083	−0.473781	−0.236890	+	0.971536i	$0.576128\pi$
$108$	0	0
$109$	2.58229	0.247339	0.123669	−	0.992323i	$-0.460534\pi$
$109$	2.58229	0.247339	0.123669	+	0.992323i	$0.460534\pi$
$110$	0.609749	0.0581373
$111$	0	0
$112$	3.44222	0.325260
$113$	1.77268	0.166760	0.0833799	−	0.996518i	$-0.473428\pi$
$113$	1.77268	0.166760	0.0833799	+	0.996518i	$0.473428\pi$
$114$	0	0
$115$	3.29812	0.307551
$116$	−8.34179	−0.774516
$117$	0	0
$118$	−1.31113	−0.120699
$119$	−0.186210	−0.0170698
$120$	0	0
$121$	−6.97265	−0.633878
$122$	−0.947829	−0.0858124
$123$	0	0
$124$	10.9359	0.982075
$125$	−8.92016	−0.797843
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	8.70601	0.769509
$129$	0	0
$130$	0.0880329	0.00772099
$131$	11.9789	1.04661	0.523303	−	0.852147i	$-0.324700\pi$
$131$	11.9789	1.04661	0.523303	+	0.852147i	$0.324700\pi$
$132$	0	0
$133$	0.852422	0.0739144
$134$	3.32986	0.287656
$135$	0	0
$136$	−0.223504	−0.0191653
$137$	13.4775	1.15146	0.575731	−	0.817639i	$-0.304718\pi$
$137$	13.4775	1.15146	0.575731	+	0.817639i	$0.304718\pi$
$138$	0	0
$139$	6.28920	0.533443	0.266721	−	0.963774i	$-0.414060\pi$
$139$	6.28920	0.533443	0.266721	+	0.963774i	$0.414060\pi$
$140$	−1.88392	−0.159220
$141$	0	0
$142$	4.02387	0.337676
$143$	0.581451	0.0486233
$144$	0	0
$145$	4.32794	0.359416
$146$	−4.58119	−0.379142
$147$	0	0
$148$	13.4452	1.10519
$149$	−2.21608	−0.181548	−0.0907740	−	0.995872i	$-0.528934\pi$
$149$	−2.21608	−0.181548	−0.0907740	+	0.995872i	$0.528934\pi$
$150$	0	0
$151$	−11.7145	−0.953316	−0.476658	−	0.879089i	$-0.658152\pi$
$151$	−11.7145	−0.953316	−0.476658	+	0.879089i	$0.658152\pi$
$152$	1.02314	0.0829879
$153$	0	0
$154$	0.616750	0.0496991
$155$	−5.67384	−0.455734
$156$	0	0
$157$	−3.95965	−0.316014	−0.158007	−	0.987438i	$-0.550507\pi$
$157$	−3.95965	−0.316014	−0.158007	+	0.987438i	$0.550507\pi$
$158$	−5.15058	−0.409758
$159$	0	0
$160$	−3.41918	−0.270310
$161$	3.33599	0.262913
$162$	0	0
$163$	22.5680	1.76766	0.883832	−	0.467805i	$-0.154955\pi$
$163$	22.5680	1.76766	0.883832	+	0.467805i	$0.154955\pi$
$164$	7.27872	0.568372
$165$	0	0
$166$	4.67056	0.362506
$167$	−15.6225	−1.20891	−0.604453	−	0.796641i	$-0.706608\pi$
$167$	−15.6225	−1.20891	−0.604453	+	0.796641i	$0.706608\pi$
$168$	0	0
$169$	−12.9161	−0.993543
$170$	0.0565776	0.00433930
$171$	0	0
$172$	−17.2716	−1.31695
$173$	−16.8019	−1.27743	−0.638713	−	0.769445i	$-0.720533\pi$
$173$	−16.8019	−1.27743	−0.638713	+	0.769445i	$0.720533\pi$
$174$	0	0
$175$	−4.02257	−0.304078
$176$	−6.90794	−0.520706
$177$	0	0
$178$	−0.161714	−0.0121210
$179$	−18.5086	−1.38340	−0.691699	−	0.722186i	$-0.743138\pi$
$179$	−18.5086	−1.38340	−0.691699	+	0.722186i	$0.743138\pi$
$180$	0	0
$181$	5.16665	0.384034	0.192017	−	0.981392i	$-0.438497\pi$
$181$	5.16665	0.384034	0.192017	+	0.981392i	$0.438497\pi$
$182$	0.0890436	0.00660035
$183$	0	0
$184$	4.00411	0.295187
$185$	−6.97571	−0.512864
$186$	0	0
$187$	0.373691	0.0273270
$188$	2.56975	0.187418
$189$	0	0
$190$	−0.258998	−0.0187897
$191$	1.27980	0.0926027	0.0463014	−	0.998928i	$-0.485257\pi$
$191$	1.27980	0.0926027	0.0463014	+	0.998928i	$0.485257\pi$
$192$	0	0
$193$	5.02979	0.362052	0.181026	−	0.983478i	$-0.442058\pi$
$193$	5.02979	0.362052	0.181026	+	0.983478i	$0.442058\pi$
$194$	−2.09950	−0.150735
$195$	0	0
$196$	−1.90555	−0.136111
$197$	5.69293	0.405604	0.202802	−	0.979220i	$-0.434995\pi$
$197$	5.69293	0.405604	0.202802	+	0.979220i	$0.434995\pi$
$198$	0	0
$199$	0.823780	0.0583962	0.0291981	−	0.999574i	$-0.490705\pi$
$199$	0.823780	0.0583962	0.0291981	+	0.999574i	$0.490705\pi$
$200$	−4.82820	−0.341406
$201$	0	0
$202$	4.12852	0.290482
$203$	4.37763	0.307249
$204$	0	0
$205$	−3.77639	−0.263754
$206$	0.704732	0.0491010
$207$	0	0
$208$	−0.997338	−0.0691530
$209$	−1.71066	−0.118329
$210$	0	0
$211$	−15.1264	−1.04135	−0.520674	−	0.853756i	$-0.674319\pi$
$211$	−15.1264	−1.04135	−0.520674	+	0.853756i	$0.674319\pi$
$212$	7.15508	0.491413
$213$	0	0
$214$	1.50615	0.102958
$215$	8.96096	0.611132
$216$	0	0
$217$	−5.73898	−0.389587
$218$	−0.793606	−0.0537498
$219$	0	0
$220$	3.78070	0.254895
$221$	0.0539518	0.00362919
$222$	0	0
$223$	4.14354	0.277472	0.138736	−	0.990329i	$-0.455696\pi$
$223$	4.14354	0.277472	0.138736	+	0.990329i	$0.455696\pi$
$224$	−3.45844	−0.231077
$225$	0	0
$226$	−0.544791	−0.0362390
$227$	17.7163	1.17587	0.587935	−	0.808908i	$-0.299941\pi$
$227$	17.7163	1.17587	0.587935	+	0.808908i	$0.299941\pi$
$228$	0	0
$229$	27.6285	1.82574	0.912871	−	0.408249i	$-0.133860\pi$
$229$	27.6285	1.82574	0.912871	+	0.408249i	$0.133860\pi$
$230$	−1.01360	−0.0668347
$231$	0	0
$232$	5.25437	0.344966
$233$	−21.6658	−1.41937	−0.709687	−	0.704517i	$-0.751164\pi$
$233$	−21.6658	−1.41937	−0.709687	+	0.704517i	$0.751164\pi$
$234$	0	0
$235$	−1.33325	−0.0869719
$236$	−8.12954	−0.529188
$237$	0	0
$238$	0.0572272	0.00370949
$239$	−3.33796	−0.215915	−0.107957	−	0.994156i	$-0.534431\pi$
$239$	−3.33796	−0.215915	−0.107957	+	0.994156i	$0.534431\pi$
$240$	0	0
$241$	−13.4703	−0.867701	−0.433851	−	0.900985i	$-0.642845\pi$
$241$	−13.4703	−0.867701	−0.433851	+	0.900985i	$0.642845\pi$
$242$	2.14288	0.137749
$243$	0	0
$244$	−5.87693	−0.376232
$245$	0.988649	0.0631625
$246$	0	0
$247$	−0.246978	−0.0157148
$248$	−6.88837	−0.437412
$249$	0	0
$250$	2.74140	0.173381
$251$	−3.19071	−0.201396	−0.100698	−	0.994917i	$-0.532108\pi$
$251$	−3.19071	−0.201396	−0.100698	+	0.994917i	$0.532108\pi$
$252$	0	0
$253$	−6.69474	−0.420895
$254$	0.307326	0.0192834
$255$	0	0
$256$	8.96758	0.560474
$257$	−9.04475	−0.564196	−0.282098	−	0.959386i	$-0.591030\pi$
$257$	−9.04475	−0.564196	−0.282098	+	0.959386i	$0.591030\pi$
$258$	0	0
$259$	−7.05580	−0.438426
$260$	0.545841	0.0338516
$261$	0	0
$262$	−3.68144	−0.227440
$263$	−28.1759	−1.73740	−0.868700	−	0.495339i	$-0.835044\pi$
$263$	−28.1759	−1.73740	−0.868700	+	0.495339i	$0.835044\pi$
$264$	0	0
$265$	−3.71224	−0.228041
$266$	−0.261972	−0.0160625
$267$	0	0
$268$	20.6465	1.26119
$269$	−28.9671	−1.76616	−0.883078	−	0.469226i	$-0.844533\pi$
$269$	−28.9671	−1.76616	−0.883078	+	0.469226i	$0.844533\pi$
$270$	0	0
$271$	−2.61077	−0.158593	−0.0792964	−	0.996851i	$-0.525267\pi$
$271$	−2.61077	−0.158593	−0.0792964	+	0.996851i	$0.525267\pi$
$272$	−0.640976	−0.0388649
$273$	0	0
$274$	−4.14199	−0.250227
$275$	8.07260	0.486796
$276$	0	0
$277$	−15.2284	−0.914988	−0.457494	−	0.889213i	$-0.651253\pi$
$277$	−15.2284	−0.914988	−0.457494	+	0.889213i	$0.651253\pi$
$278$	−1.93283	−0.115924
$279$	0	0
$280$	1.18665	0.0709161
$281$	16.7836	1.00122	0.500612	−	0.865672i	$-0.333108\pi$
$281$	16.7836	1.00122	0.500612	+	0.865672i	$0.333108\pi$
$282$	0	0
$283$	−5.84420	−0.347401	−0.173701	−	0.984798i	$-0.555573\pi$
$283$	−5.84420	−0.347401	−0.173701	+	0.984798i	$0.555573\pi$
$284$	24.9497	1.48049
$285$	0	0
$286$	−0.178695	−0.0105665
$287$	−3.81974	−0.225472
$288$	0	0
$289$	−16.9653	−0.997960
$290$	−1.33009	−0.0781054
$291$	0	0
$292$	−28.4053	−1.66229
$293$	−2.88058	−0.168285	−0.0841427	−	0.996454i	$-0.526815\pi$
$293$	−2.88058	−0.168285	−0.0841427	+	0.996454i	$0.526815\pi$
$294$	0	0
$295$	4.21782	0.245571
$296$	−8.46892	−0.492246
$297$	0	0
$298$	0.681058	0.0394526
$299$	−0.966557	−0.0558974
$300$	0	0
$301$	9.06384	0.522431
$302$	3.60018	0.207167
$303$	0	0
$304$	2.93423	0.168290
$305$	3.04911	0.174591
$306$	0	0
$307$	2.95818	0.168832	0.0844161	−	0.996431i	$-0.473098\pi$
$307$	2.95818	0.168832	0.0844161	+	0.996431i	$0.473098\pi$
$308$	3.82411	0.217899
$309$	0	0
$310$	1.74372	0.0990366
$311$	7.79466	0.441995	0.220997	−	0.975274i	$-0.429069\pi$
$311$	7.79466	0.441995	0.220997	+	0.975274i	$0.429069\pi$
$312$	0	0
$313$	7.56008	0.427321	0.213660	−	0.976908i	$-0.431461\pi$
$313$	7.56008	0.427321	0.213660	+	0.976908i	$0.431461\pi$
$314$	1.21690	0.0686738
$315$	0	0
$316$	−31.9357	−1.79653
$317$	−31.2502	−1.75519	−0.877593	−	0.479406i	$-0.840852\pi$
$317$	−31.2502	−1.75519	−0.877593	+	0.479406i	$0.840852\pi$
$318$	0	0
$319$	−8.78513	−0.491873
$320$	−5.75550	−0.321742
$321$	0	0
$322$	−1.02524	−0.0571341
$323$	−0.158729	−0.00883194
$324$	0	0
$325$	1.16549	0.0646496
$326$	−6.93574	−0.384135
$327$	0	0
$328$	−4.58475	−0.253151
$329$	−1.34856	−0.0743486
$330$	0	0
$331$	12.3799	0.680461	0.340230	−	0.940342i	$-0.389495\pi$
$331$	12.3799	0.680461	0.340230	+	0.940342i	$0.389495\pi$
$332$	28.9595	1.58936
$333$	0	0
$334$	4.80120	0.262710
$335$	−10.7120	−0.585257
$336$	0	0
$337$	24.7975	1.35081	0.675403	−	0.737448i	$-0.263970\pi$
$337$	24.7975	1.35081	0.675403	+	0.737448i	$0.263970\pi$
$338$	3.96944	0.215909
$339$	0	0
$340$	0.350805	0.0190251
$341$	11.5171	0.623688
$342$	0	0
$343$	1.00000	0.0539949
$344$	10.8791	0.586563
$345$	0	0
$346$	5.16367	0.277601
$347$	36.3026	1.94883	0.974413	−	0.224763i	$-0.0721608\pi$
$347$	36.3026	1.94883	0.974413	+	0.224763i	$0.0721608\pi$
$348$	0	0
$349$	11.0963	0.593973	0.296986	−	0.954882i	$-0.404018\pi$
$349$	11.0963	0.593973	0.296986	+	0.954882i	$0.404018\pi$
$350$	1.23624	0.0660799
$351$	0	0
$352$	6.94049	0.369929
$353$	−10.7528	−0.572312	−0.286156	−	0.958183i	$-0.592378\pi$
$353$	−10.7528	−0.572312	−0.286156	+	0.958183i	$0.592378\pi$
$354$	0	0
$355$	−12.9445	−0.687025
$356$	−1.00270	−0.0531428
$357$	0	0
$358$	5.68818	0.300630
$359$	4.97551	0.262598	0.131299	−	0.991343i	$-0.458085\pi$
$359$	4.97551	0.262598	0.131299	+	0.991343i	$0.458085\pi$
$360$	0	0
$361$	−18.2734	−0.961757
$362$	−1.58785	−0.0834554
$363$	0	0
$364$	0.552108	0.0289383
$365$	14.7374	0.771391
$366$	0	0
$367$	−21.4750	−1.12099	−0.560493	−	0.828159i	$-0.689388\pi$
$367$	−21.4750	−1.12099	−0.560493	+	0.828159i	$0.689388\pi$
$368$	11.4832	0.598604
$369$	0	0
$370$	2.14382	0.111452
$371$	−3.75486	−0.194943
$372$	0	0
$373$	19.7237	1.02125	0.510627	−	0.859803i	$-0.329413\pi$
$373$	19.7237	1.02125	0.510627	+	0.859803i	$0.329413\pi$
$374$	−0.114845	−0.00593849
$375$	0	0
$376$	−1.61865	−0.0834754
$377$	−1.26836	−0.0653238
$378$	0	0
$379$	−0.520876	−0.0267556	−0.0133778	−	0.999911i	$-0.504258\pi$
$379$	−0.520876	−0.0267556	−0.0133778	+	0.999911i	$0.504258\pi$
$380$	−1.60590	−0.0823807
$381$	0	0
$382$	−0.393314	−0.0201237
$383$	−24.9559	−1.27519	−0.637594	−	0.770372i	$-0.720070\pi$
$383$	−24.9559	−1.27519	−0.637594	+	0.770372i	$0.720070\pi$
$384$	0	0
$385$	−1.98405	−0.101116
$386$	−1.54579	−0.0786785
$387$	0	0
$388$	−13.0178	−0.660877
$389$	−14.5891	−0.739696	−0.369848	−	0.929092i	$-0.620590\pi$
$389$	−14.5891	−0.739696	−0.369848	+	0.929092i	$0.620590\pi$
$390$	0	0
$391$	−0.621194	−0.0314151
$392$	1.20028	0.0606232
$393$	0	0
$394$	−1.74959	−0.0881429
$395$	16.5691	0.833681
$396$	0	0
$397$	−5.87002	−0.294608	−0.147304	−	0.989091i	$-0.547060\pi$
$397$	−5.87002	−0.294608	−0.147304	+	0.989091i	$0.547060\pi$
$398$	−0.253169	−0.0126902
$399$	0	0
$400$	−13.8466	−0.692330
$401$	−5.35527	−0.267429	−0.133715	−	0.991020i	$-0.542691\pi$
$401$	−5.35527	−0.267429	−0.133715	+	0.991020i	$0.542691\pi$
$402$	0	0
$403$	1.66279	0.0828296
$404$	25.5985	1.27358
$405$	0	0
$406$	−1.34536	−0.0667690
$407$	14.1598	0.701873
$408$	0	0
$409$	−35.4278	−1.75179	−0.875896	−	0.482500i	$-0.839729\pi$
$409$	−35.4278	−1.75179	−0.875896	+	0.482500i	$0.839729\pi$
$410$	1.16058	0.0573171
$411$	0	0
$412$	4.36963	0.215276
$413$	4.26624	0.209928
$414$	0	0
$415$	−15.0249	−0.737544
$416$	1.00204	0.0491289
$417$	0	0
$418$	0.525731	0.0257143
$419$	−15.6038	−0.762293	−0.381147	−	0.924515i	$-0.624471\pi$
$419$	−15.6038	−0.762293	−0.381147	+	0.924515i	$0.624471\pi$
$420$	0	0
$421$	25.4791	1.24177	0.620887	−	0.783900i	$-0.286773\pi$
$421$	25.4791	1.24177	0.620887	+	0.783900i	$0.286773\pi$
$422$	4.64875	0.226298
$423$	0	0
$424$	−4.50688	−0.218873
$425$	0.749043	0.0363339
$426$	0	0
$427$	3.08411	0.149251
$428$	9.33878	0.451407
$429$	0	0
$430$	−2.75394	−0.132807
$431$	−2.51062	−0.120932	−0.0604661	−	0.998170i	$-0.519259\pi$
$431$	−2.51062	−0.120932	−0.0604661	+	0.998170i	$0.519259\pi$
$432$	0	0
$433$	−26.5601	−1.27640	−0.638198	−	0.769872i	$-0.720320\pi$
$433$	−26.5601	−1.27640	−0.638198	+	0.769872i	$0.720320\pi$
$434$	1.76374	0.0846622
$435$	0	0
$436$	−4.92069	−0.235658
$437$	2.84367	0.136031
$438$	0	0
$439$	2.59419	0.123814	0.0619070	−	0.998082i	$-0.480282\pi$
$439$	2.59419	0.123814	0.0619070	+	0.998082i	$0.480282\pi$
$440$	−2.38141	−0.113529
$441$	0	0
$442$	−0.0165808	−0.000788668 0
$443$	−24.0954	−1.14481	−0.572404	−	0.819972i	$-0.693989\pi$
$443$	−24.0954	−1.14481	−0.572404	+	0.819972i	$0.693989\pi$
$444$	0	0
$445$	0.520225	0.0246610
$446$	−1.27342	−0.0602981
$447$	0	0
$448$	−5.82158	−0.275044
$449$	11.0719	0.522514	0.261257	−	0.965269i	$-0.415863\pi$
$449$	11.0719	0.522514	0.261257	+	0.965269i	$0.415863\pi$
$450$	0	0
$451$	7.66556	0.360957
$452$	−3.37793	−0.158885
$453$	0	0
$454$	−5.44467	−0.255531
$455$	−0.286448	−0.0134289
$456$	0	0
$457$	7.11118	0.332647	0.166324	−	0.986071i	$-0.446810\pi$
$457$	7.11118	0.332647	0.166324	+	0.986071i	$0.446810\pi$
$458$	−8.49095	−0.396756
$459$	0	0
$460$	−6.28473	−0.293027
$461$	−24.0587	−1.12052	−0.560262	−	0.828315i	$-0.689300\pi$
$461$	−24.0587	−1.12052	−0.560262	+	0.828315i	$0.689300\pi$
$462$	0	0
$463$	−18.9750	−0.881842	−0.440921	−	0.897546i	$-0.645348\pi$
$463$	−18.9750	−0.881842	−0.440921	+	0.897546i	$0.645348\pi$
$464$	15.0688	0.699550
$465$	0	0
$466$	6.65847	0.308448
$467$	−7.40649	−0.342732	−0.171366	−	0.985207i	$-0.554818\pi$
$467$	−7.40649	−0.342732	−0.171366	+	0.985207i	$0.554818\pi$
$468$	0	0
$469$	−10.8349	−0.500311
$470$	0.409744	0.0189001
$471$	0	0
$472$	5.12068	0.235698
$473$	−18.1895	−0.836356
$474$	0	0
$475$	−3.42893	−0.157330
$476$	0.354832	0.0162637
$477$	0	0
$478$	1.02584	0.0469209
$479$	4.92351	0.224961	0.112481	−	0.993654i	$-0.464120\pi$
$479$	4.92351	0.224961	0.112481	+	0.993654i	$0.464120\pi$
$480$	0	0
$481$	2.04432	0.0932131
$482$	4.13979	0.188562
$483$	0	0
$484$	13.2867	0.603943
$485$	6.75396	0.306681
$486$	0	0
$487$	9.85082	0.446383	0.223192	−	0.974775i	$-0.428352\pi$
$487$	9.85082	0.446383	0.223192	+	0.974775i	$0.428352\pi$
$488$	3.70179	0.167572
$489$	0	0
$490$	−0.303838	−0.0137260
$491$	31.4542	1.41951	0.709755	−	0.704449i	$-0.248806\pi$
$491$	31.4542	1.41951	0.709755	+	0.704449i	$0.248806\pi$
$492$	0	0
$493$	−0.815157	−0.0367128
$494$	0.0759027	0.00341503
$495$	0	0
$496$	−19.7549	−0.887020
$497$	−13.0932	−0.587309
$498$	0	0
$499$	−15.4537	−0.691801	−0.345900	−	0.938271i	$-0.612426\pi$
$499$	−15.4537	−0.691801	−0.345900	+	0.938271i	$0.612426\pi$
$500$	16.9978	0.760165
$501$	0	0
$502$	0.980590	0.0437659
$503$	−34.8449	−1.55366	−0.776829	−	0.629712i	$-0.783173\pi$
$503$	−34.8449	−1.55366	−0.776829	+	0.629712i	$0.783173\pi$
$504$	0	0
$505$	−13.2812	−0.591005
$506$	2.05747	0.0914656
$507$	0	0
$508$	1.90555	0.0845451
$509$	−22.9563	−1.01752	−0.508761	−	0.860908i	$-0.669896\pi$
$509$	−22.9563	−1.01752	−0.508761	+	0.860908i	$0.669896\pi$
$510$	0	0
$511$	14.9066	0.659429
$512$	−20.1680	−0.891307
$513$	0	0
$514$	2.77969	0.122607
$515$	−2.26708	−0.0998994
$516$	0	0
$517$	2.70633	0.119024
$518$	2.16843	0.0952754
$519$	0	0
$520$	−0.343817	−0.0150774
$521$	−23.8812	−1.04625	−0.523127	−	0.852255i	$-0.675235\pi$
$521$	−23.8812	−1.04625	−0.523127	+	0.852255i	$0.675235\pi$
$522$	0	0
$523$	−42.3363	−1.85124	−0.925620	−	0.378455i	$-0.876455\pi$
$523$	−42.3363	−1.85124	−0.925620	+	0.378455i	$0.876455\pi$
$524$	−22.8265	−0.997180
$525$	0	0
$526$	8.65919	0.377558
$527$	1.06866	0.0465514
$528$	0	0
$529$	−11.8712	−0.516139
$530$	1.14087	0.0495561
$531$	0	0
$532$	−1.62433	−0.0704238
$533$	1.10672	0.0479373
$534$	0	0
$535$	−4.84520	−0.209476
$536$	−13.0049	−0.561728
$537$	0	0
$538$	8.90235	0.383808
$539$	−2.00682	−0.0864401
$540$	0	0
$541$	−10.0291	−0.431186	−0.215593	−	0.976483i	$-0.569168\pi$
$541$	−10.0291	−0.431186	−0.215593	+	0.976483i	$0.569168\pi$
$542$	0.802357	0.0344642
$543$	0	0
$544$	0.643996	0.0276111
$545$	2.55298	0.109358
$546$	0	0
$547$	−38.5584	−1.64864	−0.824318	−	0.566127i	$-0.808441\pi$
$547$	−38.5584	−1.64864	−0.824318	+	0.566127i	$0.808441\pi$
$548$	−25.6821	−1.09708
$549$	0	0
$550$	−2.48092	−0.105787
$551$	3.73159	0.158971
$552$	0	0
$553$	16.7593	0.712679
$554$	4.68010	0.198838
$555$	0	0
$556$	−11.9844	−0.508251
$557$	27.3968	1.16084	0.580420	−	0.814317i	$-0.302888\pi$
$557$	27.3968	1.16084	0.580420	+	0.814317i	$0.302888\pi$
$558$	0	0
$559$	−2.62613	−0.111073
$560$	3.40315	0.143809
$561$	0	0
$562$	−5.15803	−0.217578
$563$	39.9956	1.68561	0.842807	−	0.538216i	$-0.180902\pi$
$563$	39.9956	1.68561	0.842807	+	0.538216i	$0.180902\pi$
$564$	0	0
$565$	1.75256	0.0737307
$566$	1.79607	0.0754946
$567$	0	0
$568$	−15.7154	−0.659405
$569$	39.2508	1.64548	0.822740	−	0.568418i	$-0.192444\pi$
$569$	39.2508	1.64548	0.822740	+	0.568418i	$0.192444\pi$
$570$	0	0
$571$	−14.5997	−0.610978	−0.305489	−	0.952196i	$-0.598820\pi$
$571$	−14.5997	−0.610978	−0.305489	+	0.952196i	$0.598820\pi$
$572$	−1.10798	−0.0463271
$573$	0	0
$574$	1.17391	0.0489979
$575$	−13.4192	−0.559621
$576$	0	0
$577$	22.3677	0.931181	0.465591	−	0.885000i	$-0.345842\pi$
$577$	22.3677	0.931181	0.465591	+	0.885000i	$0.345842\pi$
$578$	5.21389	0.216869
$579$	0	0
$580$	−8.24710	−0.342442
$581$	−15.1974	−0.630495
$582$	0	0
$583$	7.53535	0.312082
$584$	17.8921	0.740379
$585$	0	0
$586$	0.885278	0.0365705
$587$	−37.4645	−1.54633	−0.773164	−	0.634207i	$-0.781327\pi$
$587$	−37.4645	−1.54633	−0.773164	+	0.634207i	$0.781327\pi$
$588$	0	0
$589$	−4.89204	−0.201573
$590$	−1.29625	−0.0533656
$591$	0	0
$592$	−24.2876	−0.998216
$593$	−48.5027	−1.99177	−0.995883	−	0.0906456i	$-0.971107\pi$
$593$	−48.5027	−1.99177	−0.995883	+	0.0906456i	$0.971107\pi$
$594$	0	0
$595$	−0.184096	−0.00754721
$596$	4.22284	0.172974
$597$	0	0
$598$	0.297048	0.0121472
$599$	−26.4798	−1.08194	−0.540968	−	0.841043i	$-0.681942\pi$
$599$	−26.4798	−1.08194	−0.540968	+	0.841043i	$0.681942\pi$
$600$	0	0
$601$	44.0904	1.79849	0.899243	−	0.437449i	$-0.144118\pi$
$601$	44.0904	1.79849	0.899243	+	0.437449i	$0.144118\pi$
$602$	−2.78556	−0.113531
$603$	0	0
$604$	22.3227	0.908296
$605$	−6.89351	−0.280261
$606$	0	0
$607$	36.8393	1.49526	0.747630	−	0.664116i	$-0.231192\pi$
$607$	36.8393	1.49526	0.747630	+	0.664116i	$0.231192\pi$
$608$	−2.94805	−0.119559
$609$	0	0
$610$	−0.937070	−0.0379408
$611$	0.390728	0.0158071
$612$	0	0
$613$	−43.1987	−1.74478	−0.872389	−	0.488812i	$-0.837430\pi$
$613$	−43.1987	−1.74478	−0.872389	+	0.488812i	$0.837430\pi$
$614$	−0.909125	−0.0366893
$615$	0	0
$616$	−2.40875	−0.0970512
$617$	−8.35601	−0.336400	−0.168200	−	0.985753i	$-0.553795\pi$
$617$	−8.35601	−0.336400	−0.168200	+	0.985753i	$0.553795\pi$
$618$	0	0
$619$	−33.1859	−1.33385	−0.666927	−	0.745123i	$-0.732391\pi$
$619$	−33.1859	−1.33385	−0.666927	+	0.745123i	$0.732391\pi$
$620$	10.8118	0.434212
$621$	0	0
$622$	−2.39550	−0.0960509
$623$	0.526198	0.0210817
$624$	0	0
$625$	11.2940	0.451758
$626$	−2.32341	−0.0928621
$627$	0	0
$628$	7.54531	0.301091
$629$	1.31386	0.0523870
$630$	0	0
$631$	−33.3707	−1.32847	−0.664234	−	0.747525i	$-0.731242\pi$
$631$	−33.3707	−1.32847	−0.664234	+	0.747525i	$0.731242\pi$
$632$	20.1158	0.800165
$633$	0	0
$634$	9.60400	0.381424
$635$	−0.988649	−0.0392333
$636$	0	0
$637$	−0.289737	−0.0114798
$638$	2.69990	0.106890
$639$	0	0
$640$	8.60718	0.340229
$641$	−11.9243	−0.470982	−0.235491	−	0.971877i	$-0.575670\pi$
$641$	−11.9243	−0.470982	−0.235491	+	0.971877i	$0.575670\pi$
$642$	0	0
$643$	−28.5011	−1.12397	−0.561987	−	0.827146i	$-0.689963\pi$
$643$	−28.5011	−1.12397	−0.561987	+	0.827146i	$0.689963\pi$
$644$	−6.35689	−0.250497
$645$	0	0
$646$	0.0487817	0.00191929
$647$	13.0053	0.511291	0.255645	−	0.966771i	$-0.417712\pi$
$647$	13.0053	0.511291	0.255645	+	0.966771i	$0.417712\pi$
$648$	0	0
$649$	−8.56161	−0.336072
$650$	−0.358184	−0.0140491
$651$	0	0
$652$	−43.0045	−1.68419
$653$	25.3698	0.992797	0.496398	−	0.868095i	$-0.334656\pi$
$653$	25.3698	0.992797	0.496398	+	0.868095i	$0.334656\pi$
$654$	0	0
$655$	11.8430	0.462743
$656$	−13.1484	−0.513359
$657$	0	0
$658$	0.414448	0.0161569
$659$	−38.0906	−1.48380	−0.741899	−	0.670511i	$-0.766075\pi$
$659$	−38.0906	−1.48380	−0.741899	+	0.670511i	$0.766075\pi$
$660$	0	0
$661$	29.6423	1.15295	0.576477	−	0.817114i	$-0.304427\pi$
$661$	29.6423	1.15295	0.576477	+	0.817114i	$0.304427\pi$
$662$	−3.80467	−0.147873
$663$	0	0
$664$	−18.2411	−0.707893
$665$	0.842746	0.0326803
$666$	0	0
$667$	14.6037	0.565458
$668$	29.7695	1.15182
$669$	0	0
$670$	3.29207	0.127184
$671$	−6.18928	−0.238934
$672$	0	0
$673$	13.5428	0.522038	0.261019	−	0.965334i	$-0.415941\pi$
$673$	13.5428	0.522038	0.261019	+	0.965334i	$0.415941\pi$
$674$	−7.62092	−0.293547
$675$	0	0
$676$	24.6122	0.946623
$677$	−15.8075	−0.607533	−0.303767	−	0.952746i	$-0.598244\pi$
$677$	−15.8075	−0.607533	−0.303767	+	0.952746i	$0.598244\pi$
$678$	0	0
$679$	6.83150	0.262169
$680$	−0.220967	−0.00847368
$681$	0	0
$682$	−3.53952	−0.135535
$683$	3.33700	0.127687	0.0638433	−	0.997960i	$-0.479664\pi$
$683$	3.33700	0.127687	0.0638433	+	0.997960i	$0.479664\pi$
$684$	0	0
$685$	13.3245	0.509104
$686$	−0.307326	−0.0117338
$687$	0	0
$688$	31.1998	1.18948
$689$	1.08792	0.0414465
$690$	0	0
$691$	18.9842	0.722191	0.361096	−	0.932529i	$-0.382403\pi$
$691$	18.9842	0.722191	0.361096	+	0.932529i	$0.382403\pi$
$692$	32.0169	1.21710
$693$	0	0
$694$	−11.1567	−0.423504
$695$	6.21781	0.235855
$696$	0	0
$697$	0.711274	0.0269414
$698$	−3.41019	−0.129078
$699$	0	0
$700$	7.66522	0.289718
$701$	−6.87293	−0.259587	−0.129793	−	0.991541i	$-0.541431\pi$
$701$	−6.87293	−0.259587	−0.129793	+	0.991541i	$0.541431\pi$
$702$	0	0
$703$	−6.01452	−0.226842
$704$	11.6829	0.440316
$705$	0	0
$706$	3.30461	0.124371
$707$	−13.4337	−0.505225
$708$	0	0
$709$	15.3156	0.575188	0.287594	−	0.957752i	$-0.407145\pi$
$709$	15.3156	0.575188	0.287594	+	0.957752i	$0.407145\pi$
$710$	3.97820	0.149299
$711$	0	0
$712$	0.631584	0.0236696
$713$	−19.1452	−0.716992
$714$	0	0
$715$	0.574851	0.0214982
$716$	35.2691	1.31807
$717$	0	0
$718$	−1.52911	−0.0570657
$719$	−28.3904	−1.05878	−0.529391	−	0.848378i	$-0.677579\pi$
$719$	−28.3904	−1.05878	−0.529391	+	0.848378i	$0.677579\pi$
$720$	0	0
$721$	−2.29311	−0.0853998
$722$	5.61589	0.209002
$723$	0	0
$724$	−9.84532	−0.365898
$725$	−17.6093	−0.653994
$726$	0	0
$727$	3.08701	0.114491	0.0572454	−	0.998360i	$-0.481768\pi$
$727$	3.08701	0.114491	0.0572454	+	0.998360i	$0.481768\pi$
$728$	−0.347764	−0.0128890
$729$	0	0
$730$	−4.52919	−0.167633
$731$	−1.68778	−0.0624247
$732$	0	0
$733$	−35.7181	−1.31928	−0.659638	−	0.751583i	$-0.729291\pi$
$733$	−35.7181	−1.31928	−0.659638	+	0.751583i	$0.729291\pi$
$734$	6.59983	0.243604
$735$	0	0
$736$	−11.5373	−0.425271
$737$	21.7438	0.800945
$738$	0	0
$739$	−19.6678	−0.723490	−0.361745	−	0.932277i	$-0.617819\pi$
$739$	−19.6678	−0.723490	−0.361745	+	0.932277i	$0.617819\pi$
$740$	13.2926	0.488644
$741$	0	0
$742$	1.15397	0.0423634
$743$	−38.7753	−1.42253	−0.711264	−	0.702925i	$-0.751877\pi$
$743$	−38.7753	−1.42253	−0.711264	+	0.702925i	$0.751877\pi$
$744$	0	0
$745$	−2.19092	−0.0802691
$746$	−6.06160	−0.221931
$747$	0	0
$748$	−0.712086	−0.0260365
$749$	−4.90083	−0.179072
$750$	0	0
$751$	−14.9648	−0.546074	−0.273037	−	0.962003i	$-0.588028\pi$
$751$	−14.9648	−0.546074	−0.273037	+	0.962003i	$0.588028\pi$
$752$	−4.64205	−0.169278
$753$	0	0
$754$	0.389800	0.0141957
$755$	−11.5816	−0.421497
$756$	0	0
$757$	−54.7349	−1.98937	−0.994687	−	0.102946i	$-0.967173\pi$
$757$	−54.7349	−1.98937	−0.994687	+	0.102946i	$0.967173\pi$
$758$	0.160079	0.00581432
$759$	0	0
$760$	1.01153	0.0366920
$761$	16.9679	0.615087	0.307543	−	0.951534i	$-0.400493\pi$
$761$	16.9679	0.615087	0.307543	+	0.951534i	$0.400493\pi$
$762$	0	0
$763$	2.58229	0.0934853
$764$	−2.43871	−0.0882296
$765$	0	0
$766$	7.66961	0.277114
$767$	−1.23609	−0.0446325
$768$	0	0
$769$	−12.4821	−0.450116	−0.225058	−	0.974345i	$-0.572257\pi$
$769$	−12.4821	−0.450116	−0.225058	+	0.974345i	$0.572257\pi$
$770$	0.609749	0.0219738
$771$	0	0
$772$	−9.58452	−0.344955
$773$	7.72207	0.277744	0.138872	−	0.990310i	$-0.455652\pi$
$773$	7.72207	0.277744	0.138872	+	0.990310i	$0.455652\pi$
$774$	0	0
$775$	23.0855	0.829255
$776$	8.19970	0.294352
$777$	0	0
$778$	4.48361	0.160745
$779$	−3.25603	−0.116660
$780$	0	0
$781$	26.2757	0.940218
$782$	0.190909	0.00682689
$783$	0	0
$784$	3.44222	0.122937
$785$	−3.91470	−0.139722
$786$	0	0
$787$	4.74326	0.169079	0.0845396	−	0.996420i	$-0.473058\pi$
$787$	4.74326	0.169079	0.0845396	+	0.996420i	$0.473058\pi$
$788$	−10.8482	−0.386450
$789$	0	0
$790$	−5.09211	−0.181169
$791$	1.77268	0.0630293
$792$	0	0
$793$	−0.893580	−0.0317320
$794$	1.80401	0.0640220
$795$	0	0
$796$	−1.56975	−0.0556385
$797$	−46.3762	−1.64273	−0.821365	−	0.570403i	$-0.806787\pi$
$797$	−46.3762	−1.64273	−0.821365	+	0.570403i	$0.806787\pi$
$798$	0	0
$799$	0.251116	0.00888383
$800$	13.9118	0.491858
$801$	0	0
$802$	1.64581	0.0581157
$803$	−29.9149	−1.05568
$804$	0	0
$805$	3.29812	0.116243
$806$	−0.511020	−0.0179999
$807$	0	0
$808$	−16.1241	−0.567245
$809$	38.5949	1.35693	0.678463	−	0.734635i	$-0.262646\pi$
$809$	38.5949	1.35693	0.678463	+	0.734635i	$0.262646\pi$
$810$	0	0
$811$	6.69385	0.235053	0.117527	−	0.993070i	$-0.462503\pi$
$811$	6.69385	0.235053	0.117527	+	0.993070i	$0.462503\pi$
$812$	−8.34179	−0.292739
$813$	0	0
$814$	−4.35166	−0.152526
$815$	22.3119	0.781550
$816$	0	0
$817$	7.72622	0.270306
$818$	10.8879	0.380686
$819$	0	0
$820$	7.19610	0.251299
$821$	27.8784	0.972964	0.486482	−	0.873690i	$-0.338280\pi$
$821$	27.8784	0.972964	0.486482	+	0.873690i	$0.338280\pi$
$822$	0	0
$823$	−22.7329	−0.792419	−0.396209	−	0.918160i	$-0.629675\pi$
$823$	−22.7329	−0.792419	−0.396209	+	0.918160i	$0.629675\pi$
$824$	−2.75237	−0.0958832
$825$	0	0
$826$	−1.31113	−0.0456200
$827$	5.40384	0.187910	0.0939549	−	0.995576i	$-0.470049\pi$
$827$	5.40384	0.187910	0.0939549	+	0.995576i	$0.470049\pi$
$828$	0	0
$829$	38.4471	1.33532	0.667661	−	0.744466i	$-0.267296\pi$
$829$	38.4471	1.33532	0.667661	+	0.744466i	$0.267296\pi$
$830$	4.61755	0.160277
$831$	0	0
$832$	1.68672	0.0584767
$833$	−0.186210	−0.00645179
$834$	0	0
$835$	−15.4452	−0.534502
$836$	3.25975	0.112741
$837$	0	0
$838$	4.79544	0.165656
$839$	22.6601	0.782313	0.391157	−	0.920324i	$-0.372075\pi$
$839$	22.6601	0.782313	0.391157	+	0.920324i	$0.372075\pi$
$840$	0	0
$841$	−9.83638	−0.339186
$842$	−7.83038	−0.269853
$843$	0	0
$844$	28.8242	0.992170
$845$	−12.7694	−0.439282
$846$	0	0
$847$	−6.97265	−0.239583
$848$	−12.9251	−0.443849
$849$	0	0
$850$	−0.230200	−0.00789581
$851$	−23.5380	−0.806874
$852$	0	0
$853$	22.6319	0.774902	0.387451	−	0.921890i	$-0.373356\pi$
$853$	22.6319	0.774902	0.387451	+	0.921890i	$0.373356\pi$
$854$	−0.947829	−0.0324340
$855$	0	0
$856$	−5.88235	−0.201055
$857$	38.6289	1.31954	0.659770	−	0.751468i	$-0.270654\pi$
$857$	38.6289	1.31954	0.659770	+	0.751468i	$0.270654\pi$
$858$	0	0
$859$	−2.75056	−0.0938481	−0.0469240	−	0.998898i	$-0.514942\pi$
$859$	−2.75056	−0.0938481	−0.0469240	+	0.998898i	$0.514942\pi$
$860$	−17.0756	−0.582272
$861$	0	0
$862$	0.771579	0.0262801
$863$	18.8268	0.640871	0.320435	−	0.947270i	$-0.396171\pi$
$863$	18.8268	0.640871	0.320435	+	0.947270i	$0.396171\pi$
$864$	0	0
$865$	−16.6112	−0.564798
$866$	8.16261	0.277377
$867$	0	0
$868$	10.9359	0.371189
$869$	−33.6330	−1.14092
$870$	0	0
$871$	3.13928	0.106370
$872$	3.09947	0.104961
$873$	0	0
$874$	−0.873933	−0.0295612
$875$	−8.92016	−0.301556
$876$	0	0
$877$	5.91209	0.199637	0.0998186	−	0.995006i	$-0.468174\pi$
$877$	5.91209	0.199637	0.0998186	+	0.995006i	$0.468174\pi$
$878$	−0.797262	−0.0269063
$879$	0	0
$880$	−6.82953	−0.230223
$881$	−23.3170	−0.785569	−0.392785	−	0.919630i	$-0.628488\pi$
$881$	−23.3170	−0.785569	−0.392785	+	0.919630i	$0.628488\pi$
$882$	0	0
$883$	−20.2527	−0.681558	−0.340779	−	0.940143i	$-0.610691\pi$
$883$	−20.2527	−0.681558	−0.340779	+	0.940143i	$0.610691\pi$
$884$	−0.102808	−0.00345780
$885$	0	0
$886$	7.40515	0.248781
$887$	59.2402	1.98909	0.994545	−	0.104309i	$-0.0332631\pi$
$887$	59.2402	1.98909	0.994545	+	0.104309i	$0.0332631\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	−0.159879	−0.00535915
$891$	0	0
$892$	−7.89572	−0.264368
$893$	−1.14954	−0.0384680
$894$	0	0
$895$	−18.2985	−0.611652
$896$	8.70601	0.290847
$897$	0	0
$898$	−3.40268	−0.113549
$899$	−25.1231	−0.837903
$900$	0	0
$901$	0.699192	0.0232935
$902$	−2.35583	−0.0784404
$903$	0	0
$904$	2.12771	0.0707666
$905$	5.10801	0.169796
$906$	0	0
$907$	3.28126	0.108953	0.0544763	−	0.998515i	$-0.482651\pi$
$907$	3.28126	0.108953	0.0544763	+	0.998515i	$0.482651\pi$
$908$	−33.7592	−1.12034
$909$	0	0
$910$	0.0880329	0.00291826
$911$	53.0856	1.75880	0.879402	−	0.476080i	$-0.157943\pi$
$911$	53.0856	1.75880	0.879402	+	0.476080i	$0.157943\pi$
$912$	0	0
$913$	30.4986	1.00936
$914$	−2.18545	−0.0722883
$915$	0	0
$916$	−52.6475	−1.73952
$917$	11.9789	0.395580
$918$	0	0
$919$	−10.1491	−0.334787	−0.167394	−	0.985890i	$-0.553535\pi$
$919$	−10.1491	−0.334787	−0.167394	+	0.985890i	$0.553535\pi$
$920$	3.95866	0.130513
$921$	0	0
$922$	7.39386	0.243504
$923$	3.79357	0.124867
$924$	0	0
$925$	28.3825	0.933210
$926$	5.83151	0.191635
$927$	0	0
$928$	−15.1398	−0.496987
$929$	26.8433	0.880699	0.440350	−	0.897826i	$-0.354855\pi$
$929$	26.8433	0.880699	0.440350	+	0.897826i	$0.354855\pi$
$930$	0	0
$931$	0.852422	0.0279370
$932$	41.2853	1.35235
$933$	0	0
$934$	2.27621	0.0744798
$935$	0.369449	0.0120823
$936$	0	0
$937$	43.6904	1.42730	0.713651	−	0.700501i	$-0.247040\pi$
$937$	43.6904	1.42730	0.713651	+	0.700501i	$0.247040\pi$
$938$	3.32986	0.108724
$939$	0	0
$940$	2.54058	0.0828647
$941$	9.59031	0.312635	0.156318	−	0.987707i	$-0.450038\pi$
$941$	9.59031	0.312635	0.156318	+	0.987707i	$0.450038\pi$
$942$	0	0
$943$	−12.7426	−0.414957
$944$	14.6854	0.477968
$945$	0	0
$946$	5.59012	0.181751
$947$	−3.89541	−0.126584	−0.0632918	−	0.997995i	$-0.520160\pi$
$947$	−3.89541	−0.126584	−0.0632918	+	0.997995i	$0.520160\pi$
$948$	0	0
$949$	−4.31899	−0.140200
$950$	1.05380	0.0341898
$951$	0	0
$952$	−0.223504	−0.00724379
$953$	0.685511	0.0222059	0.0111029	−	0.999938i	$-0.496466\pi$
$953$	0.685511	0.0222059	0.0111029	+	0.999938i	$0.496466\pi$
$954$	0	0
$955$	1.26527	0.0409431
$956$	6.36065	0.205718
$957$	0	0
$958$	−1.51312	−0.0488868
$959$	13.4775	0.435212
$960$	0	0
$961$	1.93592	0.0624489
$962$	−0.628274	−0.0202564
$963$	0	0
$964$	25.6684	0.826724
$965$	4.97270	0.160077
$966$	0	0
$967$	−38.2729	−1.23077	−0.615386	−	0.788226i	$-0.711000\pi$
$967$	−38.2729	−1.23077	−0.615386	+	0.788226i	$0.711000\pi$
$968$	−8.36912	−0.268994
$969$	0	0
$970$	−2.07567	−0.0666457
$971$	−2.76859	−0.0888482	−0.0444241	−	0.999013i	$-0.514145\pi$
$971$	−2.76859	−0.0888482	−0.0444241	+	0.999013i	$0.514145\pi$
$972$	0	0
$973$	6.28920	0.201622
$974$	−3.02741	−0.0970046
$975$	0	0
$976$	10.6162	0.339817
$977$	−25.2322	−0.807249	−0.403625	−	0.914925i	$-0.632250\pi$
$977$	−25.2322	−0.807249	−0.403625	+	0.914925i	$0.632250\pi$
$978$	0	0
$979$	−1.05599	−0.0337495
$980$	−1.88392	−0.0601796
$981$	0	0
$982$	−9.66671	−0.308477
$983$	−27.4799	−0.876472	−0.438236	−	0.898860i	$-0.644397\pi$
$983$	−27.4799	−0.876472	−0.438236	+	0.898860i	$0.644397\pi$
$984$	0	0
$985$	5.62831	0.179333
$986$	0.250519	0.00797816
$987$	0	0
$988$	0.470629	0.0149727
$989$	30.2369	0.961476
$990$	0	0
$991$	−26.6965	−0.848043	−0.424021	−	0.905652i	$-0.639382\pi$
$991$	−26.6965	−0.848043	−0.424021	+	0.905652i	$0.639382\pi$
$992$	19.8479	0.630172
$993$	0	0
$994$	4.02387	0.127629
$995$	0.814429	0.0258191
$996$	0	0
$997$	−4.92865	−0.156092	−0.0780459	−	0.996950i	$-0.524868\pi$
$997$	−4.92865	−0.156092	−0.0780459	+	0.996950i	$0.524868\pi$
$998$	4.74931	0.150337
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.o.1.7		13
3.2	odd	2		2667.2.a.l.1.7	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.l.1.7	✓	13	3.2	odd	2
8001.2.a.o.1.7		13	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)

\(f(q)\)	\(=\)	\(q-0.307326 q^{2} -1.90555 q^{4} +0.988649 q^{5} +1.00000 q^{7} +1.20028 q^{8} +O(q^{10})\)	\(q-0.307326 q^{2} -1.90555 q^{4} +0.988649 q^{5} +1.00000 q^{7} +1.20028 q^{8} -0.303838 q^{10} -2.00682 q^{11} -0.289737 q^{13} -0.307326 q^{14} +3.44222 q^{16} -0.186210 q^{17} +0.852422 q^{19} -1.88392 q^{20} +0.616750 q^{22} +3.33599 q^{23} -4.02257 q^{25} +0.0890436 q^{26} -1.90555 q^{28} +4.37763 q^{29} -5.73898 q^{31} -3.45844 q^{32} +0.0572272 q^{34} +0.988649 q^{35} -7.05580 q^{37} -0.261972 q^{38} +1.18665 q^{40} -3.81974 q^{41} +9.06384 q^{43} +3.82411 q^{44} -1.02524 q^{46} -1.34856 q^{47} +1.00000 q^{49} +1.23624 q^{50} +0.552108 q^{52} -3.75486 q^{53} -1.98405 q^{55} +1.20028 q^{56} -1.34536 q^{58} +4.26624 q^{59} +3.08411 q^{61} +1.76374 q^{62} -5.82158 q^{64} -0.286448 q^{65} -10.8349 q^{67} +0.354832 q^{68} -0.303838 q^{70} -13.0932 q^{71} +14.9066 q^{73} +2.16843 q^{74} -1.62433 q^{76} -2.00682 q^{77} +16.7593 q^{79} +3.40315 q^{80} +1.17391 q^{82} -15.1974 q^{83} -0.184096 q^{85} -2.78556 q^{86} -2.40875 q^{88} +0.526198 q^{89} -0.289737 q^{91} -6.35689 q^{92} +0.414448 q^{94} +0.842746 q^{95} +6.83150 q^{97} -0.307326 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 4 q^{2} + 10 q^{4} - 12 q^{5} + 13 q^{7} - 9 q^{8}+O(q^{10}) \) 13 * q - 4 * q^2 + 10 * q^4 - 12 * q^5 + 13 * q^7 - 9 * q^8	\( 13 q - 4 q^{2} + 10 q^{4} - 12 q^{5} + 13 q^{7} - 9 q^{8} + 6 q^{10} - 3 q^{11} + 21 q^{13} - 4 q^{14} + 8 q^{16} - 17 q^{17} + 5 q^{19} - 29 q^{20} + q^{22} - 4 q^{23} + q^{25} - 22 q^{26} + 10 q^{28} - 21 q^{29} - 7 q^{31} - 12 q^{32} + 2 q^{34} - 12 q^{35} + 7 q^{37} + 9 q^{38} + 29 q^{40} - 21 q^{41} - 9 q^{43} + 2 q^{44} - 28 q^{46} - 23 q^{47} + 13 q^{49} - 15 q^{50} + 15 q^{52} - 31 q^{53} - 8 q^{55} - 9 q^{56} - 25 q^{58} - 28 q^{59} + 29 q^{61} + 3 q^{62} + 9 q^{64} - 30 q^{65} - 18 q^{67} - 34 q^{68} + 6 q^{70} - 10 q^{71} + 24 q^{73} + 19 q^{74} - 3 q^{77} - 28 q^{79} - 26 q^{80} + 18 q^{82} - 26 q^{83} + 20 q^{85} + 2 q^{86} - 17 q^{88} - 44 q^{89} + 21 q^{91} - 6 q^{92} - 9 q^{94} + 2 q^{95} + 17 q^{97} - 4 q^{98}+O(q^{100}) \) 13 * q - 4 * q^2 + 10 * q^4 - 12 * q^5 + 13 * q^7 - 9 * q^8 + 6 * q^10 - 3 * q^11 + 21 * q^13 - 4 * q^14 + 8 * q^16 - 17 * q^17 + 5 * q^19 - 29 * q^20 + q^22 - 4 * q^23 + q^25 - 22 * q^26 + 10 * q^28 - 21 * q^29 - 7 * q^31 - 12 * q^32 + 2 * q^34 - 12 * q^35 + 7 * q^37 + 9 * q^38 + 29 * q^40 - 21 * q^41 - 9 * q^43 + 2 * q^44 - 28 * q^46 - 23 * q^47 + 13 * q^49 - 15 * q^50 + 15 * q^52 - 31 * q^53 - 8 * q^55 - 9 * q^56 - 25 * q^58 - 28 * q^59 + 29 * q^61 + 3 * q^62 + 9 * q^64 - 30 * q^65 - 18 * q^67 - 34 * q^68 + 6 * q^70 - 10 * q^71 + 24 * q^73 + 19 * q^74 - 3 * q^77 - 28 * q^79 - 26 * q^80 + 18 * q^82 - 26 * q^83 + 20 * q^85 + 2 * q^86 - 17 * q^88 - 44 * q^89 + 21 * q^91 - 6 * q^92 - 9 * q^94 + 2 * q^95 + 17 * q^97 - 4 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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