LMFDB - Embedded newform 8001.2.a.y.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:	$28$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
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)

$f(q)$	$=$	$q-2.27499 q^{2} +3.17556 q^{4} +3.81133 q^{5} -1.00000 q^{7} -2.67439 q^{8} +O(q^{10})$	\(q-2.27499 q^{2} +3.17556 q^{4} +3.81133 q^{5} -1.00000 q^{7} -2.67439 q^{8} -8.67073 q^{10} +2.70441 q^{11} -3.11570 q^{13} +2.27499 q^{14} -0.266921 q^{16} -5.02843 q^{17} -0.979058 q^{19} +12.1031 q^{20} -6.15251 q^{22} -1.88287 q^{23} +9.52627 q^{25} +7.08817 q^{26} -3.17556 q^{28} +8.95898 q^{29} +6.70042 q^{31} +5.95603 q^{32} +11.4396 q^{34} -3.81133 q^{35} -5.22364 q^{37} +2.22734 q^{38} -10.1930 q^{40} -0.958363 q^{41} +0.988001 q^{43} +8.58804 q^{44} +4.28351 q^{46} -10.2364 q^{47} +1.00000 q^{49} -21.6721 q^{50} -9.89409 q^{52} -8.12501 q^{53} +10.3074 q^{55} +2.67439 q^{56} -20.3816 q^{58} -2.47516 q^{59} +9.26292 q^{61} -15.2434 q^{62} -13.0160 q^{64} -11.8750 q^{65} +4.94109 q^{67} -15.9681 q^{68} +8.67073 q^{70} +2.49095 q^{71} +5.66686 q^{73} +11.8837 q^{74} -3.10906 q^{76} -2.70441 q^{77} +10.2394 q^{79} -1.01733 q^{80} +2.18026 q^{82} -3.92342 q^{83} -19.1650 q^{85} -2.24769 q^{86} -7.23266 q^{88} +2.97581 q^{89} +3.11570 q^{91} -5.97919 q^{92} +23.2876 q^{94} -3.73152 q^{95} +10.6219 q^{97} -2.27499 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 30 q^{4} - 28 q^{7}+O(q^{10}) $ 28 * q + 30 * q^4 - 28 * q^7	$ 28 q + 30 q^{4} - 28 q^{7} + 4 q^{10} + 8 q^{13} + 42 q^{16} + 34 q^{19} - 10 q^{22} + 14 q^{25} - 30 q^{28} + 56 q^{31} - 6 q^{37} + 38 q^{40} + 18 q^{43} + 16 q^{46} + 28 q^{49} + 18 q^{52} + 48 q^{55} + 2 q^{58} + 36 q^{61} + 76 q^{64} - 4 q^{70} + 50 q^{73} + 132 q^{76} + 66 q^{79} - 36 q^{82} + 20 q^{85} + 6 q^{88} - 8 q^{91} + 54 q^{94} - 8 q^{97}+O(q^{100}) $ 28 * q + 30 * q^4 - 28 * q^7 + 4 * q^10 + 8 * q^13 + 42 * q^16 + 34 * q^19 - 10 * q^22 + 14 * q^25 - 30 * q^28 + 56 * q^31 - 6 * q^37 + 38 * q^40 + 18 * q^43 + 16 * q^46 + 28 * q^49 + 18 * q^52 + 48 * q^55 + 2 * q^58 + 36 * q^61 + 76 * q^64 - 4 * q^70 + 50 * q^73 + 132 * q^76 + 66 * q^79 - 36 * q^82 + 20 * q^85 + 6 * q^88 - 8 * q^91 + 54 * q^94 - 8 * q^97

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 $

$2$	−2.27499	−1.60866	−0.804329	−	0.594184i	$-0.797475\pi$
$2$	−2.27499	−1.60866	−0.804329	+	0.594184i	$0.797475\pi$
$3$	0	0
$3$	0	0
$4$	3.17556	1.58778
$5$	3.81133	1.70448	0.852240	−	0.523151i	$-0.175243\pi$
$5$	3.81133	1.70448	0.852240	+	0.523151i	$0.175243\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−2.67439	−0.945540
$9$	0	0
$10$	−8.67073	−2.74193
$11$	2.70441	0.815412	0.407706	−	0.913113i	$-0.366329\pi$
$11$	2.70441	0.815412	0.407706	+	0.913113i	$0.366329\pi$
$12$	0	0
$13$	−3.11570	−0.864139	−0.432069	−	0.901840i	$-0.642216\pi$
$13$	−3.11570	−0.864139	−0.432069	+	0.901840i	$0.642216\pi$
$14$	2.27499	0.608016
$15$	0	0
$16$	−0.266921	−0.0667304
$17$	−5.02843	−1.21957	−0.609787	−	0.792566i	$-0.708745\pi$
$17$	−5.02843	−1.21957	−0.609787	+	0.792566i	$0.708745\pi$
$18$	0	0
$19$	−0.979058	−0.224611	−0.112306	−	0.993674i	$-0.535824\pi$
$19$	−0.979058	−0.224611	−0.112306	+	0.993674i	$0.535824\pi$
$20$	12.1031	2.70634
$21$	0	0
$22$	−6.15251	−1.31172
$23$	−1.88287	−0.392606	−0.196303	−	0.980543i	$-0.562894\pi$
$23$	−1.88287	−0.392606	−0.196303	+	0.980543i	$0.562894\pi$
$24$	0	0
$25$	9.52627	1.90525
$26$	7.08817	1.39010
$27$	0	0
$28$	−3.17556	−0.600125
$29$	8.95898	1.66364	0.831821	−	0.555044i	$-0.187299\pi$
$29$	8.95898	1.66364	0.831821	+	0.555044i	$0.187299\pi$
$30$	0	0
$31$	6.70042	1.20343	0.601715	−	0.798711i	$-0.294484\pi$
$31$	6.70042	1.20343	0.601715	+	0.798711i	$0.294484\pi$
$32$	5.95603	1.05289
$33$	0	0
$34$	11.4396	1.96188
$35$	−3.81133	−0.644233
$36$	0	0
$37$	−5.22364	−0.858762	−0.429381	−	0.903123i	$-0.641268\pi$
$37$	−5.22364	−0.858762	−0.429381	+	0.903123i	$0.641268\pi$
$38$	2.22734	0.361323
$39$	0	0
$40$	−10.1930	−1.61166
$41$	−0.958363	−0.149671	−0.0748356	−	0.997196i	$-0.523843\pi$
$41$	−0.958363	−0.149671	−0.0748356	+	0.997196i	$0.523843\pi$
$42$	0	0
$43$	0.988001	0.150669	0.0753344	−	0.997158i	$-0.475998\pi$
$43$	0.988001	0.150669	0.0753344	+	0.997158i	$0.475998\pi$
$44$	8.58804	1.29470
$45$	0	0
$46$	4.28351	0.631569
$47$	−10.2364	−1.49313	−0.746565	−	0.665312i	$-0.768299\pi$
$47$	−10.2364	−1.49313	−0.746565	+	0.665312i	$0.768299\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−21.6721	−3.06490
$51$	0	0
$52$	−9.89409	−1.37206
$53$	−8.12501	−1.11606	−0.558028	−	0.829822i	$-0.688442\pi$
$53$	−8.12501	−1.11606	−0.558028	+	0.829822i	$0.688442\pi$
$54$	0	0
$55$	10.3074	1.38985
$56$	2.67439	0.357381
$57$	0	0
$58$	−20.3816	−2.67623
$59$	−2.47516	−0.322238	−0.161119	−	0.986935i	$-0.551510\pi$
$59$	−2.47516	−0.322238	−0.161119	+	0.986935i	$0.551510\pi$
$60$	0	0
$61$	9.26292	1.18600	0.592998	−	0.805204i	$-0.297944\pi$
$61$	9.26292	1.18600	0.592998	+	0.805204i	$0.297944\pi$
$62$	−15.2434	−1.93591
$63$	0	0
$64$	−13.0160	−1.62700
$65$	−11.8750	−1.47291
$66$	0	0
$67$	4.94109	0.603650	0.301825	−	0.953363i	$-0.402404\pi$
$67$	4.94109	0.603650	0.301825	+	0.953363i	$0.402404\pi$
$68$	−15.9681	−1.93642
$69$	0	0
$70$	8.67073	1.03635
$71$	2.49095	0.295621	0.147810	−	0.989016i	$-0.452777\pi$
$71$	2.49095	0.295621	0.147810	+	0.989016i	$0.452777\pi$
$72$	0	0
$73$	5.66686	0.663256	0.331628	−	0.943410i	$-0.392402\pi$
$73$	5.66686	0.663256	0.331628	+	0.943410i	$0.392402\pi$
$74$	11.8837	1.38145
$75$	0	0
$76$	−3.10906	−0.356634
$77$	−2.70441	−0.308197
$78$	0	0
$79$	10.2394	1.15202	0.576011	−	0.817442i	$-0.304608\pi$
$79$	10.2394	1.15202	0.576011	+	0.817442i	$0.304608\pi$
$80$	−1.01733	−0.113741
$81$	0	0
$82$	2.18026	0.240770
$83$	−3.92342	−0.430652	−0.215326	−	0.976542i	$-0.569081\pi$
$83$	−3.92342	−0.430652	−0.215326	+	0.976542i	$0.569081\pi$
$84$	0	0
$85$	−19.1650	−2.07874
$86$	−2.24769	−0.242375
$87$	0	0
$88$	−7.23266	−0.771005
$89$	2.97581	0.315435	0.157717	−	0.987484i	$-0.449586\pi$
$89$	2.97581	0.315435	0.157717	+	0.987484i	$0.449586\pi$
$90$	0	0
$91$	3.11570	0.326614
$92$	−5.97919	−0.623373
$93$	0	0
$94$	23.2876	2.40194
$95$	−3.73152	−0.382845
$96$	0	0
$97$	10.6219	1.07849	0.539247	−	0.842148i	$-0.318709\pi$
$97$	10.6219	1.07849	0.539247	+	0.842148i	$0.318709\pi$
$98$	−2.27499	−0.229808
$99$	0	0
$100$	30.2513	3.02513
$101$	−10.8599	−1.08060	−0.540298	−	0.841474i	$-0.681688\pi$
$101$	−10.8599	−1.08060	−0.540298	+	0.841474i	$0.681688\pi$
$102$	0	0
$103$	−13.6480	−1.34478	−0.672389	−	0.740198i	$-0.734732\pi$
$103$	−13.6480	−1.34478	−0.672389	+	0.740198i	$0.734732\pi$
$104$	8.33259	0.817078
$105$	0	0
$106$	18.4843	1.79535
$107$	15.6644	1.51433	0.757166	−	0.653223i	$-0.226584\pi$
$107$	15.6644	1.51433	0.757166	+	0.653223i	$0.226584\pi$
$108$	0	0
$109$	13.7185	1.31400	0.656999	−	0.753892i	$-0.271826\pi$
$109$	13.7185	1.31400	0.656999	+	0.753892i	$0.271826\pi$
$110$	−23.4493	−2.23580
$111$	0	0
$112$	0.266921	0.0252217
$113$	−4.82313	−0.453722	−0.226861	−	0.973927i	$-0.572846\pi$
$113$	−4.82313	−0.453722	−0.226861	+	0.973927i	$0.572846\pi$
$114$	0	0
$115$	−7.17626	−0.669190
$116$	28.4498	2.64150
$117$	0	0
$118$	5.63095	0.518371
$119$	5.02843	0.460955
$120$	0	0
$121$	−3.68614	−0.335104
$122$	−21.0730	−1.90786
$123$	0	0
$124$	21.2776	1.91078
$125$	17.2511	1.54299
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	17.6993	1.56441
$129$	0	0
$130$	27.0154	2.36941
$131$	10.8457	0.947592	0.473796	−	0.880635i	$-0.342883\pi$
$131$	10.8457	0.947592	0.473796	+	0.880635i	$0.342883\pi$
$132$	0	0
$133$	0.979058	0.0848951
$134$	−11.2409	−0.971067
$135$	0	0
$136$	13.4480	1.15316
$137$	8.53133	0.728881	0.364440	−	0.931227i	$-0.381260\pi$
$137$	8.53133	0.728881	0.364440	+	0.931227i	$0.381260\pi$
$138$	0	0
$139$	9.27627	0.786803	0.393401	−	0.919367i	$-0.371298\pi$
$139$	9.27627	0.786803	0.393401	+	0.919367i	$0.371298\pi$
$140$	−12.1031	−1.02290
$141$	0	0
$142$	−5.66687	−0.475553
$143$	−8.42613	−0.704629
$144$	0	0
$145$	34.1457	2.83564
$146$	−12.8920	−1.06695
$147$	0	0
$148$	−16.5880	−1.36353
$149$	13.2836	1.08824	0.544119	−	0.839008i	$-0.316864\pi$
$149$	13.2836	1.08824	0.544119	+	0.839008i	$0.316864\pi$
$150$	0	0
$151$	2.98128	0.242613	0.121306	−	0.992615i	$-0.461292\pi$
$151$	2.98128	0.242613	0.121306	+	0.992615i	$0.461292\pi$
$152$	2.61838	0.212379
$153$	0	0
$154$	6.15251	0.495783
$155$	25.5375	2.05122
$156$	0	0
$157$	−8.41439	−0.671542	−0.335771	−	0.941944i	$-0.608997\pi$
$157$	−8.41439	−0.671542	−0.335771	+	0.941944i	$0.608997\pi$
$158$	−23.2945	−1.85321
$159$	0	0
$160$	22.7004	1.79463
$161$	1.88287	0.148391
$162$	0	0
$163$	2.64731	0.207354	0.103677	−	0.994611i	$-0.466939\pi$
$163$	2.64731	0.207354	0.103677	+	0.994611i	$0.466939\pi$
$164$	−3.04334	−0.237645
$165$	0	0
$166$	8.92573	0.692771
$167$	13.6186	1.05384	0.526921	−	0.849914i	$-0.323346\pi$
$167$	13.6186	1.05384	0.526921	+	0.849914i	$0.323346\pi$
$168$	0	0
$169$	−3.29244	−0.253264
$170$	43.6002	3.34398
$171$	0	0
$172$	3.13746	0.239229
$173$	10.2063	0.775970	0.387985	−	0.921666i	$-0.373171\pi$
$173$	10.2063	0.775970	0.387985	+	0.921666i	$0.373171\pi$
$174$	0	0
$175$	−9.52627	−0.720118
$176$	−0.721866	−0.0544127
$177$	0	0
$178$	−6.76992	−0.507427
$179$	−5.93917	−0.443915	−0.221957	−	0.975056i	$-0.571245\pi$
$179$	−5.93917	−0.443915	−0.221957	+	0.975056i	$0.571245\pi$
$180$	0	0
$181$	21.7293	1.61513	0.807565	−	0.589779i	$-0.200785\pi$
$181$	21.7293	1.61513	0.807565	+	0.589779i	$0.200785\pi$
$182$	−7.08817	−0.525410
$183$	0	0
$184$	5.03554	0.371225
$185$	−19.9091	−1.46374
$186$	0	0
$187$	−13.5990	−0.994454
$188$	−32.5063	−2.37077
$189$	0	0
$190$	8.48915	0.615868
$191$	−0.308907	−0.0223517	−0.0111759	−	0.999938i	$-0.503557\pi$
$191$	−0.308907	−0.0223517	−0.0111759	+	0.999938i	$0.503557\pi$
$192$	0	0
$193$	7.75101	0.557930	0.278965	−	0.960301i	$-0.410009\pi$
$193$	7.75101	0.557930	0.278965	+	0.960301i	$0.410009\pi$
$194$	−24.1647	−1.73493
$195$	0	0
$196$	3.17556	0.226826
$197$	−11.3239	−0.806794	−0.403397	−	0.915025i	$-0.632171\pi$
$197$	−11.3239	−0.806794	−0.403397	+	0.915025i	$0.632171\pi$
$198$	0	0
$199$	18.9798	1.34544	0.672722	−	0.739895i	$-0.265125\pi$
$199$	18.9798	1.34544	0.672722	+	0.739895i	$0.265125\pi$
$200$	−25.4770	−1.80149
$201$	0	0
$202$	24.7060	1.73831
$203$	−8.95898	−0.628797
$204$	0	0
$205$	−3.65264	−0.255112
$206$	31.0490	2.16329
$207$	0	0
$208$	0.831646	0.0576643
$209$	−2.64778	−0.183151
$210$	0	0
$211$	−9.66373	−0.665279	−0.332639	−	0.943054i	$-0.607939\pi$
$211$	−9.66373	−0.665279	−0.332639	+	0.943054i	$0.607939\pi$
$212$	−25.8015	−1.77205
$213$	0	0
$214$	−35.6362	−2.43604
$215$	3.76560	0.256812
$216$	0	0
$217$	−6.70042	−0.454854
$218$	−31.2095	−2.11377
$219$	0	0
$220$	32.7319	2.20678
$221$	15.6671	1.05388
$222$	0	0
$223$	27.8419	1.86443	0.932217	−	0.361900i	$-0.117872\pi$
$223$	27.8419	1.86443	0.932217	+	0.361900i	$0.117872\pi$
$224$	−5.95603	−0.397954
$225$	0	0
$226$	10.9725	0.729883
$227$	−0.946014	−0.0627892	−0.0313946	−	0.999507i	$-0.509995\pi$
$227$	−0.946014	−0.0627892	−0.0313946	+	0.999507i	$0.509995\pi$
$228$	0	0
$229$	17.4517	1.15324	0.576621	−	0.817012i	$-0.304371\pi$
$229$	17.4517	1.15324	0.576621	+	0.817012i	$0.304371\pi$
$230$	16.3259	1.07650
$231$	0	0
$232$	−23.9598	−1.57304
$233$	−5.74198	−0.376170	−0.188085	−	0.982153i	$-0.560228\pi$
$233$	−5.74198	−0.376170	−0.188085	+	0.982153i	$0.560228\pi$
$234$	0	0
$235$	−39.0143	−2.54501
$236$	−7.86002	−0.511644
$237$	0	0
$238$	−11.4396	−0.741520
$239$	−17.4782	−1.13057	−0.565284	−	0.824896i	$-0.691233\pi$
$239$	−17.4782	−1.13057	−0.565284	+	0.824896i	$0.691233\pi$
$240$	0	0
$241$	−7.29629	−0.469995	−0.234998	−	0.971996i	$-0.575508\pi$
$241$	−7.29629	−0.469995	−0.234998	+	0.971996i	$0.575508\pi$
$242$	8.38593	0.539068
$243$	0	0
$244$	29.4150	1.88310
$245$	3.81133	0.243497
$246$	0	0
$247$	3.05045	0.194095
$248$	−17.9195	−1.13789
$249$	0	0
$250$	−39.2461	−2.48214
$251$	19.2194	1.21312	0.606559	−	0.795039i	$-0.292550\pi$
$251$	19.2194	1.21312	0.606559	+	0.795039i	$0.292550\pi$
$252$	0	0
$253$	−5.09207	−0.320136
$254$	−2.27499	−0.142745
$255$	0	0
$256$	−14.2335	−0.889594
$257$	−0.621578	−0.0387729	−0.0193865	−	0.999812i	$-0.506171\pi$
$257$	−0.621578	−0.0387729	−0.0193865	+	0.999812i	$0.506171\pi$
$258$	0	0
$259$	5.22364	0.324581
$260$	−37.7097	−2.33866
$261$	0	0
$262$	−24.6738	−1.52435
$263$	10.9716	0.676537	0.338268	−	0.941050i	$-0.390159\pi$
$263$	10.9716	0.676537	0.338268	+	0.941050i	$0.390159\pi$
$264$	0	0
$265$	−30.9671	−1.90230
$266$	−2.22734	−0.136567
$267$	0	0
$268$	15.6907	0.958465
$269$	−23.7880	−1.45038	−0.725191	−	0.688548i	$-0.758248\pi$
$269$	−23.7880	−1.45038	−0.725191	+	0.688548i	$0.758248\pi$
$270$	0	0
$271$	23.8085	1.44627	0.723133	−	0.690709i	$-0.242701\pi$
$271$	23.8085	1.44627	0.723133	+	0.690709i	$0.242701\pi$
$272$	1.34220	0.0813826
$273$	0	0
$274$	−19.4087	−1.17252
$275$	25.7630	1.55357
$276$	0	0
$277$	20.5806	1.23657	0.618285	−	0.785954i	$-0.287828\pi$
$277$	20.5806	1.23657	0.618285	+	0.785954i	$0.287828\pi$
$278$	−21.1034	−1.26570
$279$	0	0
$280$	10.1930	0.609148
$281$	−2.05854	−0.122802	−0.0614012	−	0.998113i	$-0.519557\pi$
$281$	−2.05854	−0.122802	−0.0614012	+	0.998113i	$0.519557\pi$
$282$	0	0
$283$	27.2291	1.61860	0.809302	−	0.587393i	$-0.199846\pi$
$283$	27.2291	1.61860	0.809302	+	0.587393i	$0.199846\pi$
$284$	7.91016	0.469382
$285$	0	0
$286$	19.1693	1.13351
$287$	0.958363	0.0565704
$288$	0	0
$289$	8.28511	0.487359
$290$	−77.6810	−4.56158
$291$	0	0
$292$	17.9955	1.05311
$293$	−0.251019	−0.0146647	−0.00733234	−	0.999973i	$-0.502334\pi$
$293$	−0.251019	−0.0146647	−0.00733234	+	0.999973i	$0.502334\pi$
$294$	0	0
$295$	−9.43365	−0.549248
$296$	13.9701	0.811994
$297$	0	0
$298$	−30.2201	−1.75060
$299$	5.86646	0.339266
$300$	0	0
$301$	−0.988001	−0.0569474
$302$	−6.78236	−0.390281
$303$	0	0
$304$	0.261331	0.0149884
$305$	35.3041	2.02151
$306$	0	0
$307$	−6.18795	−0.353165	−0.176583	−	0.984286i	$-0.556504\pi$
$307$	−6.18795	−0.353165	−0.176583	+	0.984286i	$0.556504\pi$
$308$	−8.58804	−0.489349
$309$	0	0
$310$	−58.0975	−3.29972
$311$	7.15969	0.405989	0.202994	−	0.979180i	$-0.434933\pi$
$311$	7.15969	0.405989	0.202994	+	0.979180i	$0.434933\pi$
$312$	0	0
$313$	22.2119	1.25549	0.627746	−	0.778419i	$-0.283978\pi$
$313$	22.2119	1.25549	0.627746	+	0.778419i	$0.283978\pi$
$314$	19.1426	1.08028
$315$	0	0
$316$	32.5159	1.82916
$317$	−7.05763	−0.396396	−0.198198	−	0.980162i	$-0.563509\pi$
$317$	−7.05763	−0.396396	−0.198198	+	0.980162i	$0.563509\pi$
$318$	0	0
$319$	24.2288	1.35655
$320$	−49.6085	−2.77320
$321$	0	0
$322$	−4.28351	−0.238711
$323$	4.92312	0.273930
$324$	0	0
$325$	−29.6810	−1.64640
$326$	−6.02260	−0.333561
$327$	0	0
$328$	2.56304	0.141520
$329$	10.2364	0.564350
$330$	0	0
$331$	−20.7009	−1.13782	−0.568911	−	0.822399i	$-0.692635\pi$
$331$	−20.7009	−1.13782	−0.568911	+	0.822399i	$0.692635\pi$
$332$	−12.4591	−0.683781
$333$	0	0
$334$	−30.9822	−1.69527
$335$	18.8321	1.02891
$336$	0	0
$337$	13.1535	0.716518	0.358259	−	0.933622i	$-0.383370\pi$
$337$	13.1535	0.716518	0.358259	+	0.933622i	$0.383370\pi$
$338$	7.49025	0.407416
$339$	0	0
$340$	−60.8598	−3.30058
$341$	18.1207	0.981291
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−2.64230	−0.142463
$345$	0	0
$346$	−23.2192	−1.24827
$347$	−8.25856	−0.443343	−0.221671	−	0.975121i	$-0.571151\pi$
$347$	−8.25856	−0.443343	−0.221671	+	0.975121i	$0.571151\pi$
$348$	0	0
$349$	−2.76363	−0.147933	−0.0739667	−	0.997261i	$-0.523566\pi$
$349$	−2.76363	−0.147933	−0.0739667	+	0.997261i	$0.523566\pi$
$350$	21.6721	1.15842
$351$	0	0
$352$	16.1076	0.858536
$353$	36.4621	1.94068	0.970340	−	0.241744i	$-0.0777194\pi$
$353$	36.4621	1.94068	0.970340	+	0.241744i	$0.0777194\pi$
$354$	0	0
$355$	9.49383	0.503880
$356$	9.44987	0.500842
$357$	0	0
$358$	13.5115	0.714107
$359$	−9.33795	−0.492838	−0.246419	−	0.969163i	$-0.579254\pi$
$359$	−9.33795	−0.492838	−0.246419	+	0.969163i	$0.579254\pi$
$360$	0	0
$361$	−18.0414	−0.949550
$362$	−49.4340	−2.59819
$363$	0	0
$364$	9.89409	0.518591
$365$	21.5983	1.13051
$366$	0	0
$367$	15.6500	0.816926	0.408463	−	0.912775i	$-0.366065\pi$
$367$	15.6500	0.816926	0.408463	+	0.912775i	$0.366065\pi$
$368$	0.502579	0.0261988
$369$	0	0
$370$	45.2928	2.35466
$371$	8.12501	0.421830
$372$	0	0
$373$	−31.1590	−1.61335	−0.806675	−	0.590995i	$-0.798735\pi$
$373$	−31.1590	−1.61335	−0.806675	+	0.590995i	$0.798735\pi$
$374$	30.9374	1.59974
$375$	0	0
$376$	27.3761	1.41182
$377$	−27.9135	−1.43762
$378$	0	0
$379$	−7.47310	−0.383867	−0.191934	−	0.981408i	$-0.561476\pi$
$379$	−7.47310	−0.383867	−0.191934	+	0.981408i	$0.561476\pi$
$380$	−11.8497	−0.607875
$381$	0	0
$382$	0.702759	0.0359563
$383$	−8.60853	−0.439875	−0.219938	−	0.975514i	$-0.570585\pi$
$383$	−8.60853	−0.439875	−0.219938	+	0.975514i	$0.570585\pi$
$384$	0	0
$385$	−10.3074	−0.525315
$386$	−17.6334	−0.897518
$387$	0	0
$388$	33.7306	1.71241
$389$	−14.1861	−0.719265	−0.359633	−	0.933094i	$-0.617098\pi$
$389$	−14.1861	−0.719265	−0.359633	+	0.933094i	$0.617098\pi$
$390$	0	0
$391$	9.46790	0.478812
$392$	−2.67439	−0.135077
$393$	0	0
$394$	25.7617	1.29786
$395$	39.0258	1.96360
$396$	0	0
$397$	15.9819	0.802107	0.401054	−	0.916055i	$-0.368644\pi$
$397$	15.9819	0.802107	0.401054	+	0.916055i	$0.368644\pi$
$398$	−43.1789	−2.16436
$399$	0	0
$400$	−2.54277	−0.127138
$401$	−32.7461	−1.63526	−0.817631	−	0.575743i	$-0.804713\pi$
$401$	−32.7461	−1.63526	−0.817631	+	0.575743i	$0.804713\pi$
$402$	0	0
$403$	−20.8765	−1.03993
$404$	−34.4862	−1.71575
$405$	0	0
$406$	20.3816	1.01152
$407$	−14.1269	−0.700244
$408$	0	0
$409$	−24.3930	−1.20615	−0.603077	−	0.797683i	$-0.706059\pi$
$409$	−24.3930	−1.20615	−0.603077	+	0.797683i	$0.706059\pi$
$410$	8.30971	0.410387
$411$	0	0
$412$	−43.3401	−2.13521
$413$	2.47516	0.121794
$414$	0	0
$415$	−14.9535	−0.734037
$416$	−18.5572	−0.909840
$417$	0	0
$418$	6.02366	0.294627
$419$	25.7260	1.25680	0.628399	−	0.777891i	$-0.283711\pi$
$419$	25.7260	1.25680	0.628399	+	0.777891i	$0.283711\pi$
$420$	0	0
$421$	−20.9929	−1.02313	−0.511566	−	0.859244i	$-0.670934\pi$
$421$	−20.9929	−1.02313	−0.511566	+	0.859244i	$0.670934\pi$
$422$	21.9849	1.07021
$423$	0	0
$424$	21.7295	1.05528
$425$	−47.9022	−2.32360
$426$	0	0
$427$	−9.26292	−0.448264
$428$	49.7432	2.40443
$429$	0	0
$430$	−8.56670	−0.413123
$431$	2.73520	0.131750	0.0658751	−	0.997828i	$-0.479016\pi$
$431$	2.73520	0.131750	0.0658751	+	0.997828i	$0.479016\pi$
$432$	0	0
$433$	−24.0227	−1.15445	−0.577227	−	0.816583i	$-0.695865\pi$
$433$	−24.0227	−1.15445	−0.577227	+	0.816583i	$0.695865\pi$
$434$	15.2434	0.731705
$435$	0	0
$436$	43.5641	2.08634
$437$	1.84344	0.0881838
$438$	0	0
$439$	−38.5310	−1.83898	−0.919491	−	0.393110i	$-0.871399\pi$
$439$	−38.5310	−1.83898	−0.919491	+	0.393110i	$0.871399\pi$
$440$	−27.5661	−1.31416
$441$	0	0
$442$	−35.6424	−1.69533
$443$	−3.05768	−0.145275	−0.0726374	−	0.997358i	$-0.523142\pi$
$443$	−3.05768	−0.145275	−0.0726374	+	0.997358i	$0.523142\pi$
$444$	0	0
$445$	11.3418	0.537653
$446$	−63.3400	−2.99924
$447$	0	0
$448$	13.0160	0.614950
$449$	27.6945	1.30698	0.653492	−	0.756934i	$-0.273303\pi$
$449$	27.6945	1.30698	0.653492	+	0.756934i	$0.273303\pi$
$450$	0	0
$451$	−2.59181	−0.122044
$452$	−15.3161	−0.720411
$453$	0	0
$454$	2.15217	0.101006
$455$	11.8750	0.556707
$456$	0	0
$457$	−0.163115	−0.00763018	−0.00381509	−	0.999993i	$-0.501214\pi$
$457$	−0.163115	−0.00763018	−0.00381509	+	0.999993i	$0.501214\pi$
$458$	−39.7024	−1.85517
$459$	0	0
$460$	−22.7887	−1.06253
$461$	−21.3556	−0.994629	−0.497314	−	0.867570i	$-0.665680\pi$
$461$	−21.3556	−0.994629	−0.497314	+	0.867570i	$0.665680\pi$
$462$	0	0
$463$	−28.7735	−1.33722	−0.668610	−	0.743614i	$-0.733110\pi$
$463$	−28.7735	−1.33722	−0.668610	+	0.743614i	$0.733110\pi$
$464$	−2.39134	−0.111015
$465$	0	0
$466$	13.0629	0.605129
$467$	10.3578	0.479304	0.239652	−	0.970859i	$-0.422967\pi$
$467$	10.3578	0.479304	0.239652	+	0.970859i	$0.422967\pi$
$468$	0	0
$469$	−4.94109	−0.228158
$470$	88.7570	4.09406
$471$	0	0
$472$	6.61954	0.304689
$473$	2.67196	0.122857
$474$	0	0
$475$	−9.32677	−0.427941
$476$	15.9681	0.731897
$477$	0	0
$478$	39.7626	1.81870
$479$	−12.6524	−0.578101	−0.289050	−	0.957314i	$-0.593340\pi$
$479$	−12.6524	−0.578101	−0.289050	+	0.957314i	$0.593340\pi$
$480$	0	0
$481$	16.2753	0.742089
$482$	16.5990	0.756062
$483$	0	0
$484$	−11.7056	−0.532072
$485$	40.4837	1.83827
$486$	0	0
$487$	40.7335	1.84581	0.922905	−	0.385028i	$-0.125808\pi$
$487$	40.7335	1.84581	0.922905	+	0.385028i	$0.125808\pi$
$488$	−24.7727	−1.12141
$489$	0	0
$490$	−8.67073	−0.391704
$491$	13.6796	0.617351	0.308676	−	0.951167i	$-0.400114\pi$
$491$	13.6796	0.617351	0.308676	+	0.951167i	$0.400114\pi$
$492$	0	0
$493$	−45.0496	−2.02893
$494$	−6.93972	−0.312233
$495$	0	0
$496$	−1.78848	−0.0803053
$497$	−2.49095	−0.111734
$498$	0	0
$499$	16.5033	0.738791	0.369395	−	0.929272i	$-0.379565\pi$
$499$	16.5033	0.738791	0.369395	+	0.929272i	$0.379565\pi$
$500$	54.7821	2.44993
$501$	0	0
$502$	−43.7238	−1.95149
$503$	9.47318	0.422388	0.211194	−	0.977444i	$-0.432265\pi$
$503$	9.47318	0.422388	0.211194	+	0.977444i	$0.432265\pi$
$504$	0	0
$505$	−41.3905	−1.84185
$506$	11.5844	0.514989
$507$	0	0
$508$	3.17556	0.140893
$509$	19.2130	0.851603	0.425801	−	0.904817i	$-0.359992\pi$
$509$	19.2130	0.851603	0.425801	+	0.904817i	$0.359992\pi$
$510$	0	0
$511$	−5.66686	−0.250687
$512$	−3.01750	−0.133356
$513$	0	0
$514$	1.41408	0.0623724
$515$	−52.0171	−2.29215
$516$	0	0
$517$	−27.6834	−1.21752
$518$	−11.8837	−0.522141
$519$	0	0
$520$	31.7583	1.39269
$521$	28.7115	1.25787	0.628937	−	0.777457i	$-0.283490\pi$
$521$	28.7115	1.25787	0.628937	+	0.777457i	$0.283490\pi$
$522$	0	0
$523$	9.29022	0.406233	0.203117	−	0.979155i	$-0.434893\pi$
$523$	9.29022	0.406233	0.203117	+	0.979155i	$0.434893\pi$
$524$	34.4412	1.50457
$525$	0	0
$526$	−24.9602	−1.08832
$527$	−33.6926	−1.46767
$528$	0	0
$529$	−19.4548	−0.845860
$530$	70.4498	3.06015
$531$	0	0
$532$	3.10906	0.134795
$533$	2.98597	0.129337
$534$	0	0
$535$	59.7021	2.58115
$536$	−13.2144	−0.570776
$537$	0	0
$538$	54.1174	2.33317
$539$	2.70441	0.116487
$540$	0	0
$541$	0.430598	0.0185128	0.00925642	−	0.999957i	$-0.497054\pi$
$541$	0.430598	0.0185128	0.00925642	+	0.999957i	$0.497054\pi$
$542$	−54.1641	−2.32655
$543$	0	0
$544$	−29.9495	−1.28407
$545$	52.2859	2.23968
$546$	0	0
$547$	−26.8884	−1.14966	−0.574832	−	0.818271i	$-0.694933\pi$
$547$	−26.8884	−1.14966	−0.574832	+	0.818271i	$0.694933\pi$
$548$	27.0918	1.15730
$549$	0	0
$550$	−58.6104	−2.49916
$551$	−8.77136	−0.373673
$552$	0	0
$553$	−10.2394	−0.435424
$554$	−46.8206	−1.98922
$555$	0	0
$556$	29.4574	1.24927
$557$	−29.9695	−1.26985	−0.634923	−	0.772575i	$-0.718968\pi$
$557$	−29.9695	−1.26985	−0.634923	+	0.772575i	$0.718968\pi$
$558$	0	0
$559$	−3.07831	−0.130199
$560$	1.01733	0.0429899
$561$	0	0
$562$	4.68316	0.197547
$563$	23.0329	0.970723	0.485361	−	0.874314i	$-0.338688\pi$
$563$	23.0329	0.970723	0.485361	+	0.874314i	$0.338688\pi$
$564$	0	0
$565$	−18.3825	−0.773360
$566$	−61.9459	−2.60378
$567$	0	0
$568$	−6.66177	−0.279522
$569$	−16.3318	−0.684665	−0.342332	−	0.939579i	$-0.611217\pi$
$569$	−16.3318	−0.684665	−0.342332	+	0.939579i	$0.611217\pi$
$570$	0	0
$571$	18.5774	0.777441	0.388721	−	0.921356i	$-0.372917\pi$
$571$	18.5774	0.777441	0.388721	+	0.921356i	$0.372917\pi$
$572$	−26.7577	−1.11880
$573$	0	0
$574$	−2.18026	−0.0910024
$575$	−17.9368	−0.748015
$576$	0	0
$577$	−4.61203	−0.192001	−0.0960007	−	0.995381i	$-0.530605\pi$
$577$	−4.61203	−0.192001	−0.0960007	+	0.995381i	$0.530605\pi$
$578$	−18.8485	−0.783995
$579$	0	0
$580$	108.432	4.50239
$581$	3.92342	0.162771
$582$	0	0
$583$	−21.9734	−0.910045
$584$	−15.1554	−0.627135
$585$	0	0
$586$	0.571064	0.0235905
$587$	0.400332	0.0165235	0.00826173	−	0.999966i	$-0.497370\pi$
$587$	0.400332	0.0165235	0.00826173	+	0.999966i	$0.497370\pi$
$588$	0	0
$589$	−6.56009	−0.270304
$590$	21.4614	0.883553
$591$	0	0
$592$	1.39430	0.0573055
$593$	−3.10794	−0.127628	−0.0638139	−	0.997962i	$-0.520326\pi$
$593$	−3.10794	−0.127628	−0.0638139	+	0.997962i	$0.520326\pi$
$594$	0	0
$595$	19.1650	0.785690
$596$	42.1830	1.72788
$597$	0	0
$598$	−13.3461	−0.545764
$599$	21.3788	0.873512	0.436756	−	0.899580i	$-0.356127\pi$
$599$	21.3788	0.873512	0.436756	+	0.899580i	$0.356127\pi$
$600$	0	0
$601$	28.0184	1.14290	0.571448	−	0.820638i	$-0.306382\pi$
$601$	28.0184	1.14290	0.571448	+	0.820638i	$0.306382\pi$
$602$	2.24769	0.0916090
$603$	0	0
$604$	9.46723	0.385216
$605$	−14.0491	−0.571178
$606$	0	0
$607$	47.5155	1.92860	0.964298	−	0.264821i	$-0.0853129\pi$
$607$	47.5155	1.92860	0.964298	+	0.264821i	$0.0853129\pi$
$608$	−5.83129	−0.236490
$609$	0	0
$610$	−80.3164	−3.25191
$611$	31.8935	1.29027
$612$	0	0
$613$	28.9219	1.16814	0.584072	−	0.811702i	$-0.301458\pi$
$613$	28.9219	1.16814	0.584072	+	0.811702i	$0.301458\pi$
$614$	14.0775	0.568122
$615$	0	0
$616$	7.23266	0.291412
$617$	6.14762	0.247494	0.123747	−	0.992314i	$-0.460509\pi$
$617$	6.14762	0.247494	0.123747	+	0.992314i	$0.460509\pi$
$618$	0	0
$619$	−22.6138	−0.908925	−0.454463	−	0.890766i	$-0.650169\pi$
$619$	−22.6138	−0.908925	−0.454463	+	0.890766i	$0.650169\pi$
$620$	81.0960	3.25690
$621$	0	0
$622$	−16.2882	−0.653097
$623$	−2.97581	−0.119223
$624$	0	0
$625$	18.1185	0.724739
$626$	−50.5318	−2.01966
$627$	0	0
$628$	−26.7204	−1.06626
$629$	26.2667	1.04732
$630$	0	0
$631$	1.81941	0.0724294	0.0362147	−	0.999344i	$-0.488470\pi$
$631$	1.81941	0.0724294	0.0362147	+	0.999344i	$0.488470\pi$
$632$	−27.3842	−1.08928
$633$	0	0
$634$	16.0560	0.637666
$635$	3.81133	0.151248
$636$	0	0
$637$	−3.11570	−0.123448
$638$	−55.1202	−2.18223
$639$	0	0
$640$	67.4578	2.66650
$641$	−19.1895	−0.757939	−0.378969	−	0.925409i	$-0.623721\pi$
$641$	−19.1895	−0.757939	−0.378969	+	0.925409i	$0.623721\pi$
$642$	0	0
$643$	−32.9181	−1.29816	−0.649081	−	0.760719i	$-0.724847\pi$
$643$	−32.9181	−1.29816	−0.649081	+	0.760719i	$0.724847\pi$
$644$	5.97919	0.235613
$645$	0	0
$646$	−11.2000	−0.440660
$647$	3.85749	0.151654	0.0758269	−	0.997121i	$-0.475840\pi$
$647$	3.85749	0.151654	0.0758269	+	0.997121i	$0.475840\pi$
$648$	0	0
$649$	−6.69385	−0.262757
$650$	67.5238	2.64850
$651$	0	0
$652$	8.40671	0.329232
$653$	−19.1754	−0.750390	−0.375195	−	0.926946i	$-0.622424\pi$
$653$	−19.1754	−0.750390	−0.375195	+	0.926946i	$0.622424\pi$
$654$	0	0
$655$	41.3365	1.61515
$656$	0.255808	0.00998761
$657$	0	0
$658$	−23.2876	−0.907847
$659$	6.88263	0.268109	0.134055	−	0.990974i	$-0.457200\pi$
$659$	6.88263	0.268109	0.134055	+	0.990974i	$0.457200\pi$
$660$	0	0
$661$	18.7313	0.728564	0.364282	−	0.931289i	$-0.381314\pi$
$661$	18.7313	0.728564	0.364282	+	0.931289i	$0.381314\pi$
$662$	47.0942	1.83037
$663$	0	0
$664$	10.4928	0.407199
$665$	3.73152	0.144702
$666$	0	0
$667$	−16.8686	−0.653156
$668$	43.2469	1.67327
$669$	0	0
$670$	−42.8429	−1.65516
$671$	25.0508	0.967075
$672$	0	0
$673$	25.8570	0.996713	0.498356	−	0.866972i	$-0.333937\pi$
$673$	25.8570	0.996713	0.498356	+	0.866972i	$0.333937\pi$
$674$	−29.9241	−1.15263
$675$	0	0
$676$	−10.4553	−0.402128
$677$	−14.7489	−0.566846	−0.283423	−	0.958995i	$-0.591470\pi$
$677$	−14.7489	−0.566846	−0.283423	+	0.958995i	$0.591470\pi$
$678$	0	0
$679$	−10.6219	−0.407632
$680$	51.2548	1.96553
$681$	0	0
$682$	−41.2244	−1.57856
$683$	−48.9690	−1.87375	−0.936873	−	0.349670i	$-0.886294\pi$
$683$	−48.9690	−1.87375	−0.936873	+	0.349670i	$0.886294\pi$
$684$	0	0
$685$	32.5158	1.24236
$686$	2.27499	0.0868594
$687$	0	0
$688$	−0.263719	−0.0100542
$689$	25.3151	0.964428
$690$	0	0
$691$	32.1917	1.22463	0.612315	−	0.790614i	$-0.290238\pi$
$691$	32.1917	1.22463	0.612315	+	0.790614i	$0.290238\pi$
$692$	32.4107	1.23207
$693$	0	0
$694$	18.7881	0.713187
$695$	35.3550	1.34109
$696$	0	0
$697$	4.81906	0.182535
$698$	6.28721	0.237974
$699$	0	0
$700$	−30.2513	−1.14339
$701$	0.934961	0.0353130	0.0176565	−	0.999844i	$-0.494379\pi$
$701$	0.934961	0.0353130	0.0176565	+	0.999844i	$0.494379\pi$
$702$	0	0
$703$	5.11425	0.192888
$704$	−35.2008	−1.32668
$705$	0	0
$706$	−82.9507	−3.12189
$707$	10.8599	0.408427
$708$	0	0
$709$	33.2414	1.24841	0.624203	−	0.781262i	$-0.285424\pi$
$709$	33.2414	1.24841	0.624203	+	0.781262i	$0.285424\pi$
$710$	−21.5983	−0.810571
$711$	0	0
$712$	−7.95848	−0.298257
$713$	−12.6160	−0.472474
$714$	0	0
$715$	−32.1148	−1.20103
$716$	−18.8602	−0.704840
$717$	0	0
$718$	21.2437	0.792809
$719$	−40.1408	−1.49700	−0.748499	−	0.663136i	$-0.769225\pi$
$719$	−40.1408	−1.49700	−0.748499	+	0.663136i	$0.769225\pi$
$720$	0	0
$721$	13.6480	0.508278
$722$	41.0440	1.52750
$723$	0	0
$724$	69.0029	2.56447
$725$	85.3457	3.16966
$726$	0	0
$727$	3.40691	0.126355	0.0631776	−	0.998002i	$-0.479877\pi$
$727$	3.40691	0.126355	0.0631776	+	0.998002i	$0.479877\pi$
$728$	−8.33259	−0.308826
$729$	0	0
$730$	−49.1358	−1.81860
$731$	−4.96810	−0.183752
$732$	0	0
$733$	30.8370	1.13899	0.569495	−	0.821994i	$-0.307139\pi$
$733$	30.8370	1.13899	0.569495	+	0.821994i	$0.307139\pi$
$734$	−35.6036	−1.31415
$735$	0	0
$736$	−11.2144	−0.413370
$737$	13.3628	0.492223
$738$	0	0
$739$	−8.95236	−0.329318	−0.164659	−	0.986351i	$-0.552652\pi$
$739$	−8.95236	−0.329318	−0.164659	+	0.986351i	$0.552652\pi$
$740$	−63.2225	−2.32410
$741$	0	0
$742$	−18.4843	−0.678580
$743$	52.9670	1.94317	0.971585	−	0.236691i	$-0.0760630\pi$
$743$	52.9670	1.94317	0.971585	+	0.236691i	$0.0760630\pi$
$744$	0	0
$745$	50.6283	1.85488
$746$	70.8863	2.59533
$747$	0	0
$748$	−43.1844	−1.57898
$749$	−15.6644	−0.572363
$750$	0	0
$751$	8.80063	0.321139	0.160570	−	0.987025i	$-0.448667\pi$
$751$	8.80063	0.321139	0.160570	+	0.987025i	$0.448667\pi$
$752$	2.73231	0.0996371
$753$	0	0
$754$	63.5028	2.31264
$755$	11.3626	0.413529
$756$	0	0
$757$	17.6758	0.642437	0.321218	−	0.947005i	$-0.395908\pi$
$757$	17.6758	0.642437	0.321218	+	0.947005i	$0.395908\pi$
$758$	17.0012	0.617511
$759$	0	0
$760$	9.97954	0.361996
$761$	−47.1341	−1.70861	−0.854305	−	0.519772i	$-0.826017\pi$
$761$	−47.1341	−1.70861	−0.854305	+	0.519772i	$0.826017\pi$
$762$	0	0
$763$	−13.7185	−0.496644
$764$	−0.980954	−0.0354897
$765$	0	0
$766$	19.5843	0.707609
$767$	7.71184	0.278458
$768$	0	0
$769$	−15.9971	−0.576870	−0.288435	−	0.957499i	$-0.593135\pi$
$769$	−15.9971	−0.576870	−0.288435	+	0.957499i	$0.593135\pi$
$770$	23.4493	0.845053
$771$	0	0
$772$	24.6138	0.885870
$773$	54.6239	1.96469	0.982343	−	0.187091i	$-0.0599059\pi$
$773$	54.6239	1.96469	0.982343	+	0.187091i	$0.0599059\pi$
$774$	0	0
$775$	63.8300	2.29284
$776$	−28.4072	−1.01976
$777$	0	0
$778$	32.2732	1.15705
$779$	0.938293	0.0336178
$780$	0	0
$781$	6.73655	0.241053
$782$	−21.5393	−0.770245
$783$	0	0
$784$	−0.266921	−0.00953291
$785$	−32.0701	−1.14463
$786$	0	0
$787$	25.6663	0.914904	0.457452	−	0.889234i	$-0.348762\pi$
$787$	25.6663	0.914904	0.457452	+	0.889234i	$0.348762\pi$
$788$	−35.9598	−1.28101
$789$	0	0
$790$	−88.7831	−3.15876
$791$	4.82313	0.171491
$792$	0	0
$793$	−28.8605	−1.02486
$794$	−36.3585	−1.29032
$795$	0	0
$796$	60.2717	2.13627
$797$	−40.4005	−1.43106	−0.715529	−	0.698583i	$-0.753814\pi$
$797$	−40.4005	−1.43106	−0.715529	+	0.698583i	$0.753814\pi$
$798$	0	0
$799$	51.4730	1.82098
$800$	56.7387	2.00602
$801$	0	0
$802$	74.4969	2.63058
$803$	15.3255	0.540826
$804$	0	0
$805$	7.17626	0.252930
$806$	47.4937	1.67289
$807$	0	0
$808$	29.0435	1.02175
$809$	32.7719	1.15220	0.576100	−	0.817379i	$-0.304574\pi$
$809$	32.7719	1.15220	0.576100	+	0.817379i	$0.304574\pi$
$810$	0	0
$811$	31.9961	1.12354	0.561768	−	0.827295i	$-0.310121\pi$
$811$	31.9961	1.12354	0.561768	+	0.827295i	$0.310121\pi$
$812$	−28.4498	−0.998393
$813$	0	0
$814$	32.1385	1.12645
$815$	10.0898	0.353430
$816$	0	0
$817$	−0.967310	−0.0338419
$818$	55.4937	1.94029
$819$	0	0
$820$	−11.5992	−0.405062
$821$	32.5416	1.13571	0.567854	−	0.823129i	$-0.307774\pi$
$821$	32.5416	1.13571	0.567854	+	0.823129i	$0.307774\pi$
$822$	0	0
$823$	17.4893	0.609637	0.304819	−	0.952410i	$-0.401404\pi$
$823$	17.4893	0.609637	0.304819	+	0.952410i	$0.401404\pi$
$824$	36.5001	1.27154
$825$	0	0
$826$	−5.63095	−0.195926
$827$	26.4804	0.920814	0.460407	−	0.887708i	$-0.347704\pi$
$827$	26.4804	0.920814	0.460407	+	0.887708i	$0.347704\pi$
$828$	0	0
$829$	12.5075	0.434402	0.217201	−	0.976127i	$-0.430307\pi$
$829$	12.5075	0.434402	0.217201	+	0.976127i	$0.430307\pi$
$830$	34.0190	1.18082
$831$	0	0
$832$	40.5540	1.40596
$833$	−5.02843	−0.174225
$834$	0	0
$835$	51.9052	1.79625
$836$	−8.40819	−0.290803
$837$	0	0
$838$	−58.5263	−2.02176
$839$	−45.2917	−1.56364	−0.781821	−	0.623503i	$-0.785709\pi$
$839$	−45.2917	−1.56364	−0.781821	+	0.623503i	$0.785709\pi$
$840$	0	0
$841$	51.2634	1.76770
$842$	47.7586	1.64587
$843$	0	0
$844$	−30.6878	−1.05632
$845$	−12.5486	−0.431684
$846$	0	0
$847$	3.68614	0.126657
$848$	2.16874	0.0744748
$849$	0	0
$850$	108.977	3.73787
$851$	9.83546	0.337155
$852$	0	0
$853$	−33.9471	−1.16233	−0.581163	−	0.813787i	$-0.697402\pi$
$853$	−33.9471	−1.16233	−0.581163	+	0.813787i	$0.697402\pi$
$854$	21.0730	0.721104
$855$	0	0
$856$	−41.8927	−1.43186
$857$	30.5780	1.04452	0.522262	−	0.852785i	$-0.325088\pi$
$857$	30.5780	1.04452	0.522262	+	0.852785i	$0.325088\pi$
$858$	0	0
$859$	−14.6781	−0.500810	−0.250405	−	0.968141i	$-0.580564\pi$
$859$	−14.6781	−0.500810	−0.250405	+	0.968141i	$0.580564\pi$
$860$	11.9579	0.407762
$861$	0	0
$862$	−6.22255	−0.211941
$863$	−51.2590	−1.74488	−0.872438	−	0.488725i	$-0.837462\pi$
$863$	−51.2590	−1.74488	−0.872438	+	0.488725i	$0.837462\pi$
$864$	0	0
$865$	38.8996	1.32263
$866$	54.6512	1.85712
$867$	0	0
$868$	−21.2776	−0.722209
$869$	27.6916	0.939373
$870$	0	0
$871$	−15.3949	−0.521637
$872$	−36.6887	−1.24244
$873$	0	0
$874$	−4.19380	−0.141858
$875$	−17.2511	−0.583195
$876$	0	0
$877$	−17.8237	−0.601863	−0.300932	−	0.953646i	$-0.597298\pi$
$877$	−17.8237	−0.601863	−0.300932	+	0.953646i	$0.597298\pi$
$878$	87.6574	2.95830
$879$	0	0
$880$	−2.75127	−0.0927454
$881$	−12.1334	−0.408786	−0.204393	−	0.978889i	$-0.565522\pi$
$881$	−12.1334	−0.408786	−0.204393	+	0.978889i	$0.565522\pi$
$882$	0	0
$883$	−9.17746	−0.308846	−0.154423	−	0.988005i	$-0.549352\pi$
$883$	−9.17746	−0.308846	−0.154423	+	0.988005i	$0.549352\pi$
$884$	49.7518	1.67333
$885$	0	0
$886$	6.95618	0.233698
$887$	2.38679	0.0801406	0.0400703	−	0.999197i	$-0.487242\pi$
$887$	2.38679	0.0801406	0.0400703	+	0.999197i	$0.487242\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	−25.8024	−0.864900
$891$	0	0
$892$	88.4138	2.96031
$893$	10.0220	0.335374
$894$	0	0
$895$	−22.6362	−0.756644
$896$	−17.6993	−0.591291
$897$	0	0
$898$	−63.0046	−2.10249
$899$	60.0289	2.00208
$900$	0	0
$901$	40.8561	1.36111
$902$	5.89634	0.196326
$903$	0	0
$904$	12.8989	0.429012
$905$	82.8178	2.75296
$906$	0	0
$907$	18.0647	0.599827	0.299914	−	0.953966i	$-0.403042\pi$
$907$	18.0647	0.599827	0.299914	+	0.953966i	$0.403042\pi$
$908$	−3.00413	−0.0996955
$909$	0	0
$910$	−27.0154	−0.895551
$911$	−54.3279	−1.79996	−0.899982	−	0.435928i	$-0.856420\pi$
$911$	−54.3279	−1.79996	−0.899982	+	0.435928i	$0.856420\pi$
$912$	0	0
$913$	−10.6106	−0.351158
$914$	0.371083	0.0122743
$915$	0	0
$916$	55.4190	1.83110
$917$	−10.8457	−0.358156
$918$	0	0
$919$	−55.7356	−1.83855	−0.919274	−	0.393619i	$-0.871223\pi$
$919$	−55.7356	−1.83855	−0.919274	+	0.393619i	$0.871223\pi$
$920$	19.1921	0.632746
$921$	0	0
$922$	48.5837	1.60002
$923$	−7.76103	−0.255458
$924$	0	0
$925$	−49.7618	−1.63616
$926$	65.4594	2.15113
$927$	0	0
$928$	53.3600	1.75163
$929$	−44.6854	−1.46608	−0.733041	−	0.680185i	$-0.761899\pi$
$929$	−44.6854	−1.46608	−0.733041	+	0.680185i	$0.761899\pi$
$930$	0	0
$931$	−0.979058	−0.0320873
$932$	−18.2340	−0.597275
$933$	0	0
$934$	−23.5640	−0.771036
$935$	−51.8302	−1.69503
$936$	0	0
$937$	21.5730	0.704758	0.352379	−	0.935857i	$-0.385373\pi$
$937$	21.5730	0.704758	0.352379	+	0.935857i	$0.385373\pi$
$938$	11.2409	0.367029
$939$	0	0
$940$	−123.892	−4.04092
$941$	33.0541	1.07753	0.538766	−	0.842455i	$-0.318891\pi$
$941$	33.0541	1.07753	0.538766	+	0.842455i	$0.318891\pi$
$942$	0	0
$943$	1.80448	0.0587619
$944$	0.660672	0.0215031
$945$	0	0
$946$	−6.07868	−0.197635
$947$	−39.1264	−1.27144	−0.635718	−	0.771921i	$-0.719296\pi$
$947$	−39.1264	−1.27144	−0.635718	+	0.771921i	$0.719296\pi$
$948$	0	0
$949$	−17.6562	−0.573145
$950$	21.2183	0.688412
$951$	0	0
$952$	−13.4480	−0.435852
$953$	−33.8471	−1.09642	−0.548208	−	0.836342i	$-0.684690\pi$
$953$	−33.8471	−1.09642	−0.548208	+	0.836342i	$0.684690\pi$
$954$	0	0
$955$	−1.17735	−0.0380981
$956$	−55.5030	−1.79510
$957$	0	0
$958$	28.7839	0.929967
$959$	−8.53133	−0.275491
$960$	0	0
$961$	13.8956	0.448244
$962$	−37.0261	−1.19377
$963$	0	0
$964$	−23.1698	−0.746250
$965$	29.5417	0.950980
$966$	0	0
$967$	42.9462	1.38106	0.690528	−	0.723305i	$-0.257378\pi$
$967$	42.9462	1.38106	0.690528	+	0.723305i	$0.257378\pi$
$968$	9.85819	0.316854
$969$	0	0
$970$	−92.0999	−2.95715
$971$	20.0092	0.642126	0.321063	−	0.947058i	$-0.395960\pi$
$971$	20.0092	0.642126	0.321063	+	0.947058i	$0.395960\pi$
$972$	0	0
$973$	−9.27627	−0.297384
$974$	−92.6681	−2.96928
$975$	0	0
$976$	−2.47247	−0.0791419
$977$	56.1118	1.79518	0.897588	−	0.440834i	$-0.145317\pi$
$977$	56.1118	1.79518	0.897588	+	0.440834i	$0.145317\pi$
$978$	0	0
$979$	8.04782	0.257209
$980$	12.1031	0.386621
$981$	0	0
$982$	−31.1209	−0.993107
$983$	16.3586	0.521757	0.260879	−	0.965372i	$-0.415988\pi$
$983$	16.3586	0.521757	0.260879	+	0.965372i	$0.415988\pi$
$984$	0	0
$985$	−43.1592	−1.37517
$986$	102.487	3.26386
$987$	0	0
$988$	9.68689	0.308181
$989$	−1.86028	−0.0591535
$990$	0	0
$991$	−18.2542	−0.579865	−0.289932	−	0.957047i	$-0.593633\pi$
$991$	−18.2542	−0.579865	−0.289932	+	0.957047i	$0.593633\pi$
$992$	39.9079	1.26708
$993$	0	0
$994$	5.66687	0.179742
$995$	72.3385	2.29328
$996$	0	0
$997$	−17.1564	−0.543347	−0.271674	−	0.962389i	$-0.587577\pi$
$997$	−17.1564	−0.543347	−0.271674	+	0.962389i	$0.587577\pi$
$998$	−37.5449	−1.18846
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.y.1.4	✓	28
3.2	odd	2	inner	8001.2.a.y.1.25	yes	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8001.2.a.y.1.4	✓	28	1.1	even	1	trivial
8001.2.a.y.1.25	yes	28	3.2	odd	2	inner

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)

\(f(q)\)	\(=\)	\(q-2.27499 q^{2} +3.17556 q^{4} +3.81133 q^{5} -1.00000 q^{7} -2.67439 q^{8} +O(q^{10})\)	\(q-2.27499 q^{2} +3.17556 q^{4} +3.81133 q^{5} -1.00000 q^{7} -2.67439 q^{8} -8.67073 q^{10} +2.70441 q^{11} -3.11570 q^{13} +2.27499 q^{14} -0.266921 q^{16} -5.02843 q^{17} -0.979058 q^{19} +12.1031 q^{20} -6.15251 q^{22} -1.88287 q^{23} +9.52627 q^{25} +7.08817 q^{26} -3.17556 q^{28} +8.95898 q^{29} +6.70042 q^{31} +5.95603 q^{32} +11.4396 q^{34} -3.81133 q^{35} -5.22364 q^{37} +2.22734 q^{38} -10.1930 q^{40} -0.958363 q^{41} +0.988001 q^{43} +8.58804 q^{44} +4.28351 q^{46} -10.2364 q^{47} +1.00000 q^{49} -21.6721 q^{50} -9.89409 q^{52} -8.12501 q^{53} +10.3074 q^{55} +2.67439 q^{56} -20.3816 q^{58} -2.47516 q^{59} +9.26292 q^{61} -15.2434 q^{62} -13.0160 q^{64} -11.8750 q^{65} +4.94109 q^{67} -15.9681 q^{68} +8.67073 q^{70} +2.49095 q^{71} +5.66686 q^{73} +11.8837 q^{74} -3.10906 q^{76} -2.70441 q^{77} +10.2394 q^{79} -1.01733 q^{80} +2.18026 q^{82} -3.92342 q^{83} -19.1650 q^{85} -2.24769 q^{86} -7.23266 q^{88} +2.97581 q^{89} +3.11570 q^{91} -5.97919 q^{92} +23.2876 q^{94} -3.73152 q^{95} +10.6219 q^{97} -2.27499 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 30 q^{4} - 28 q^{7}+O(q^{10}) \) 28 * q + 30 * q^4 - 28 * q^7	\( 28 q + 30 q^{4} - 28 q^{7} + 4 q^{10} + 8 q^{13} + 42 q^{16} + 34 q^{19} - 10 q^{22} + 14 q^{25} - 30 q^{28} + 56 q^{31} - 6 q^{37} + 38 q^{40} + 18 q^{43} + 16 q^{46} + 28 q^{49} + 18 q^{52} + 48 q^{55} + 2 q^{58} + 36 q^{61} + 76 q^{64} - 4 q^{70} + 50 q^{73} + 132 q^{76} + 66 q^{79} - 36 q^{82} + 20 q^{85} + 6 q^{88} - 8 q^{91} + 54 q^{94} - 8 q^{97}+O(q^{100}) \) 28 * q + 30 * q^4 - 28 * q^7 + 4 * q^10 + 8 * q^13 + 42 * q^16 + 34 * q^19 - 10 * q^22 + 14 * q^25 - 30 * q^28 + 56 * q^31 - 6 * q^37 + 38 * q^40 + 18 * q^43 + 16 * q^46 + 28 * q^49 + 18 * q^52 + 48 * q^55 + 2 * q^58 + 36 * q^61 + 76 * q^64 - 4 * q^70 + 50 * q^73 + 132 * q^76 + 66 * q^79 - 36 * q^82 + 20 * q^85 + 6 * q^88 - 8 * q^91 + 54 * q^94 - 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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