LMFDB - Embedded newform 8001.2.a.z.1.27

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:	$32$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.27
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)

$f(q)$	$=$	$q+1.96743 q^{2} +1.87078 q^{4} -2.35101 q^{5} -1.00000 q^{7} -0.254227 q^{8} +O(q^{10})$	\(q+1.96743 q^{2} +1.87078 q^{4} -2.35101 q^{5} -1.00000 q^{7} -0.254227 q^{8} -4.62544 q^{10} +2.89015 q^{11} +6.85121 q^{13} -1.96743 q^{14} -4.24174 q^{16} -2.39968 q^{17} -4.06451 q^{19} -4.39822 q^{20} +5.68618 q^{22} +2.32861 q^{23} +0.527232 q^{25} +13.4793 q^{26} -1.87078 q^{28} -3.14236 q^{29} -6.35057 q^{31} -7.83687 q^{32} -4.72121 q^{34} +2.35101 q^{35} +5.72445 q^{37} -7.99665 q^{38} +0.597690 q^{40} -5.44826 q^{41} -4.92110 q^{43} +5.40685 q^{44} +4.58138 q^{46} +6.42833 q^{47} +1.00000 q^{49} +1.03729 q^{50} +12.8171 q^{52} +13.2746 q^{53} -6.79477 q^{55} +0.254227 q^{56} -6.18238 q^{58} +4.10912 q^{59} +10.0147 q^{61} -12.4943 q^{62} -6.93502 q^{64} -16.1072 q^{65} -9.78035 q^{67} -4.48929 q^{68} +4.62544 q^{70} +0.218239 q^{71} -12.3655 q^{73} +11.2625 q^{74} -7.60382 q^{76} -2.89015 q^{77} -2.97850 q^{79} +9.97236 q^{80} -10.7191 q^{82} -13.0162 q^{83} +5.64167 q^{85} -9.68193 q^{86} -0.734756 q^{88} -7.20192 q^{89} -6.85121 q^{91} +4.35632 q^{92} +12.6473 q^{94} +9.55570 q^{95} -18.5452 q^{97} +1.96743 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 30 q^{4} - 32 q^{7}+O(q^{10}) $ 32 * q + 30 * q^4 - 32 * q^7	$ 32 q + 30 q^{4} - 32 q^{7} - 16 q^{10} - 14 q^{13} + 18 q^{16} - 30 q^{19} - 10 q^{22} + 36 q^{25} - 30 q^{28} - 58 q^{31} - 34 q^{34} + 8 q^{37} - 34 q^{40} + 6 q^{43} - 36 q^{46} + 32 q^{49} - 56 q^{52} - 88 q^{55} - 22 q^{58} - 46 q^{61} + 20 q^{64} - 8 q^{67} + 16 q^{70} - 60 q^{73} - 128 q^{76} - 74 q^{79} - 52 q^{82} - 16 q^{85} - 64 q^{88} + 14 q^{91} - 58 q^{94} - 44 q^{97}+O(q^{100}) $ 32 * q + 30 * q^4 - 32 * q^7 - 16 * q^10 - 14 * q^13 + 18 * q^16 - 30 * q^19 - 10 * q^22 + 36 * q^25 - 30 * q^28 - 58 * q^31 - 34 * q^34 + 8 * q^37 - 34 * q^40 + 6 * q^43 - 36 * q^46 + 32 * q^49 - 56 * q^52 - 88 * q^55 - 22 * q^58 - 46 * q^61 + 20 * q^64 - 8 * q^67 + 16 * q^70 - 60 * q^73 - 128 * q^76 - 74 * q^79 - 52 * q^82 - 16 * q^85 - 64 * q^88 + 14 * q^91 - 58 * q^94 - 44 * 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 $

$2$	1.96743	1.39118	0.695592	−	0.718437i	$-0.255142\pi$
$2$	1.96743	1.39118	0.695592	+	0.718437i	$0.255142\pi$
$3$	0	0
$3$	0	0
$4$	1.87078	0.935391
$5$	−2.35101	−1.05140	−0.525701	−	0.850669i	$-0.676197\pi$
$5$	−2.35101	−1.05140	−0.525701	+	0.850669i	$0.676197\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−0.254227	−0.0898829
$9$	0	0
$10$	−4.62544	−1.46269
$11$	2.89015	0.871414	0.435707	−	0.900088i	$-0.356498\pi$
$11$	2.89015	0.871414	0.435707	+	0.900088i	$0.356498\pi$
$12$	0	0
$13$	6.85121	1.90018	0.950092	−	0.311970i	$-0.100989\pi$
$13$	6.85121	1.90018	0.950092	+	0.311970i	$0.100989\pi$
$14$	−1.96743	−0.525818
$15$	0	0
$16$	−4.24174	−1.06043
$17$	−2.39968	−0.582009	−0.291005	−	0.956722i	$-0.593989\pi$
$17$	−2.39968	−0.582009	−0.291005	+	0.956722i	$0.593989\pi$
$18$	0	0
$19$	−4.06451	−0.932464	−0.466232	−	0.884663i	$-0.654389\pi$
$19$	−4.06451	−0.932464	−0.466232	+	0.884663i	$0.654389\pi$
$20$	−4.39822	−0.983472
$21$	0	0
$22$	5.68618	1.21230
$23$	2.32861	0.485549	0.242775	−	0.970083i	$-0.421942\pi$
$23$	2.32861	0.485549	0.242775	+	0.970083i	$0.421942\pi$
$24$	0	0
$25$	0.527232	0.105446
$26$	13.4793	2.64350
$27$	0	0
$28$	−1.87078	−0.353545
$29$	−3.14236	−0.583522	−0.291761	−	0.956491i	$-0.594241\pi$
$29$	−3.14236	−0.583522	−0.291761	+	0.956491i	$0.594241\pi$
$30$	0	0
$31$	−6.35057	−1.14060	−0.570298	−	0.821438i	$-0.693172\pi$
$31$	−6.35057	−1.14060	−0.570298	+	0.821438i	$0.693172\pi$
$32$	−7.83687	−1.38538
$33$	0	0
$34$	−4.72121	−0.809681
$35$	2.35101	0.397393
$36$	0	0
$37$	5.72445	0.941095	0.470547	−	0.882375i	$-0.344057\pi$
$37$	5.72445	0.941095	0.470547	+	0.882375i	$0.344057\pi$
$38$	−7.99665	−1.29723
$39$	0	0
$40$	0.597690	0.0945031
$41$	−5.44826	−0.850875	−0.425438	−	0.904988i	$-0.639880\pi$
$41$	−5.44826	−0.850875	−0.425438	+	0.904988i	$0.639880\pi$
$42$	0	0
$43$	−4.92110	−0.750461	−0.375231	−	0.926931i	$-0.622436\pi$
$43$	−4.92110	−0.750461	−0.375231	+	0.926931i	$0.622436\pi$
$44$	5.40685	0.815113
$45$	0	0
$46$	4.58138	0.675488
$47$	6.42833	0.937668	0.468834	−	0.883286i	$-0.344674\pi$
$47$	6.42833	0.937668	0.468834	+	0.883286i	$0.344674\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	1.03729	0.146695
$51$	0	0
$52$	12.8171	1.77742
$53$	13.2746	1.82340	0.911702	−	0.410853i	$-0.134769\pi$
$53$	13.2746	1.82340	0.911702	+	0.410853i	$0.134769\pi$
$54$	0	0
$55$	−6.79477	−0.916207
$56$	0.254227	0.0339726
$57$	0	0
$58$	−6.18238	−0.811787
$59$	4.10912	0.534962	0.267481	−	0.963563i	$-0.413809\pi$
$59$	4.10912	0.534962	0.267481	+	0.963563i	$0.413809\pi$
$60$	0	0
$61$	10.0147	1.28225	0.641126	−	0.767436i	$-0.278468\pi$
$61$	10.0147	1.28225	0.641126	+	0.767436i	$0.278468\pi$
$62$	−12.4943	−1.58678
$63$	0	0
$64$	−6.93502	−0.866877
$65$	−16.1072	−1.99786
$66$	0	0
$67$	−9.78035	−1.19486	−0.597430	−	0.801921i	$-0.703811\pi$
$67$	−9.78035	−1.19486	−0.597430	+	0.801921i	$0.703811\pi$
$68$	−4.48929	−0.544406
$69$	0	0
$70$	4.62544	0.552846
$71$	0.218239	0.0259003	0.0129501	−	0.999916i	$-0.495878\pi$
$71$	0.218239	0.0259003	0.0129501	+	0.999916i	$0.495878\pi$
$72$	0	0
$73$	−12.3655	−1.44727	−0.723635	−	0.690183i	$-0.757530\pi$
$73$	−12.3655	−1.44727	−0.723635	+	0.690183i	$0.757530\pi$
$74$	11.2625	1.30924
$75$	0	0
$76$	−7.60382	−0.872218
$77$	−2.89015	−0.329364
$78$	0	0
$79$	−2.97850	−0.335107	−0.167553	−	0.985863i	$-0.553587\pi$
$79$	−2.97850	−0.335107	−0.167553	+	0.985863i	$0.553587\pi$
$80$	9.97236	1.11494
$81$	0	0
$82$	−10.7191	−1.18372
$83$	−13.0162	−1.42871	−0.714355	−	0.699783i	$-0.753280\pi$
$83$	−13.0162	−1.42871	−0.714355	+	0.699783i	$0.753280\pi$
$84$	0	0
$85$	5.64167	0.611926
$86$	−9.68193	−1.04403
$87$	0	0
$88$	−0.734756	−0.0783253
$89$	−7.20192	−0.763402	−0.381701	−	0.924286i	$-0.624662\pi$
$89$	−7.20192	−0.763402	−0.381701	+	0.924286i	$0.624662\pi$
$90$	0	0
$91$	−6.85121	−0.718202
$92$	4.35632	0.454178
$93$	0	0
$94$	12.6473	1.30447
$95$	9.55570	0.980394
$96$	0	0
$97$	−18.5452	−1.88298	−0.941491	−	0.337039i	$-0.890574\pi$
$97$	−18.5452	−1.88298	−0.941491	+	0.337039i	$0.890574\pi$
$98$	1.96743	0.198740
$99$	0	0
$100$	0.986336	0.0986336
$101$	4.67088	0.464770	0.232385	−	0.972624i	$-0.425347\pi$
$101$	4.67088	0.464770	0.232385	+	0.972624i	$0.425347\pi$
$102$	0	0
$103$	−5.40554	−0.532624	−0.266312	−	0.963887i	$-0.585805\pi$
$103$	−5.40554	−0.532624	−0.266312	+	0.963887i	$0.585805\pi$
$104$	−1.74176	−0.170794
$105$	0	0
$106$	26.1168	2.53669
$107$	−12.5665	−1.21485	−0.607425	−	0.794377i	$-0.707797\pi$
$107$	−12.5665	−1.21485	−0.607425	+	0.794377i	$0.707797\pi$
$108$	0	0
$109$	−16.6939	−1.59898	−0.799492	−	0.600676i	$-0.794898\pi$
$109$	−16.6939	−1.59898	−0.799492	+	0.600676i	$0.794898\pi$
$110$	−13.3682	−1.27461
$111$	0	0
$112$	4.24174	0.400807
$113$	−4.88624	−0.459658	−0.229829	−	0.973231i	$-0.573817\pi$
$113$	−4.88624	−0.459658	−0.229829	+	0.973231i	$0.573817\pi$
$114$	0	0
$115$	−5.47458	−0.510507
$116$	−5.87868	−0.545822
$117$	0	0
$118$	8.08440	0.744230
$119$	2.39968	0.219979
$120$	0	0
$121$	−2.64701	−0.240637
$122$	19.7032	1.78385
$123$	0	0
$124$	−11.8805	−1.06690
$125$	10.5155	0.940536
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	2.02957	0.179391
$129$	0	0
$130$	−31.6899	−2.77939
$131$	−21.4230	−1.87173	−0.935866	−	0.352356i	$-0.885381\pi$
$131$	−21.4230	−1.87173	−0.935866	+	0.352356i	$0.885381\pi$
$132$	0	0
$133$	4.06451	0.352438
$134$	−19.2422	−1.66227
$135$	0	0
$136$	0.610065	0.0523127
$137$	8.73744	0.746490	0.373245	−	0.927733i	$-0.378245\pi$
$137$	8.73744	0.746490	0.373245	+	0.927733i	$0.378245\pi$
$138$	0	0
$139$	−7.92793	−0.672438	−0.336219	−	0.941784i	$-0.609148\pi$
$139$	−7.92793	−0.672438	−0.336219	+	0.941784i	$0.609148\pi$
$140$	4.39822	0.371718
$141$	0	0
$142$	0.429371	0.0360320
$143$	19.8011	1.65585
$144$	0	0
$145$	7.38772	0.613517
$146$	−24.3282	−2.01342
$147$	0	0
$148$	10.7092	0.880291
$149$	−0.536374	−0.0439415	−0.0219707	−	0.999759i	$-0.506994\pi$
$149$	−0.536374	−0.0439415	−0.0219707	+	0.999759i	$0.506994\pi$
$150$	0	0
$151$	12.9566	1.05439	0.527196	−	0.849744i	$-0.323243\pi$
$151$	12.9566	1.05439	0.527196	+	0.849744i	$0.323243\pi$
$152$	1.03331	0.0838125
$153$	0	0
$154$	−5.68618	−0.458205
$155$	14.9302	1.19923
$156$	0	0
$157$	−1.78766	−0.142671	−0.0713353	−	0.997452i	$-0.522726\pi$
$157$	−1.78766	−0.142671	−0.0713353	+	0.997452i	$0.522726\pi$
$158$	−5.85998	−0.466195
$159$	0	0
$160$	18.4245	1.45659
$161$	−2.32861	−0.183520
$162$	0	0
$163$	−20.0026	−1.56672	−0.783362	−	0.621566i	$-0.786497\pi$
$163$	−20.0026	−1.56672	−0.783362	+	0.621566i	$0.786497\pi$
$164$	−10.1925	−0.795901
$165$	0	0
$166$	−25.6084	−1.98760
$167$	8.46145	0.654767	0.327383	−	0.944892i	$-0.393833\pi$
$167$	8.46145	0.654767	0.327383	+	0.944892i	$0.393833\pi$
$168$	0	0
$169$	33.9391	2.61070
$170$	11.0996	0.851301
$171$	0	0
$172$	−9.20631	−0.701975
$173$	−17.8933	−1.36040	−0.680202	−	0.733024i	$-0.738108\pi$
$173$	−17.8933	−1.36040	−0.680202	+	0.733024i	$0.738108\pi$
$174$	0	0
$175$	−0.527232	−0.0398550
$176$	−12.2593	−0.924078
$177$	0	0
$178$	−14.1693	−1.06203
$179$	2.60034	0.194359	0.0971793	−	0.995267i	$-0.469018\pi$
$179$	2.60034	0.194359	0.0971793	+	0.995267i	$0.469018\pi$
$180$	0	0
$181$	2.34080	0.173990	0.0869952	−	0.996209i	$-0.472274\pi$
$181$	2.34080	0.173990	0.0869952	+	0.996209i	$0.472274\pi$
$182$	−13.4793	−0.999151
$183$	0	0
$184$	−0.591997	−0.0436426
$185$	−13.4582	−0.989469
$186$	0	0
$187$	−6.93546	−0.507171
$188$	12.0260	0.877087
$189$	0	0
$190$	18.8002	1.36391
$191$	14.5839	1.05525	0.527626	−	0.849477i	$-0.323082\pi$
$191$	14.5839	1.05525	0.527626	+	0.849477i	$0.323082\pi$
$192$	0	0
$193$	4.35927	0.313787	0.156894	−	0.987615i	$-0.449852\pi$
$193$	4.35927	0.313787	0.156894	+	0.987615i	$0.449852\pi$
$194$	−36.4864	−2.61957
$195$	0	0
$196$	1.87078	0.133627
$197$	21.7166	1.54724	0.773620	−	0.633649i	$-0.218444\pi$
$197$	21.7166	1.54724	0.773620	+	0.633649i	$0.218444\pi$
$198$	0	0
$199$	−12.8349	−0.909842	−0.454921	−	0.890532i	$-0.650332\pi$
$199$	−12.8349	−0.909842	−0.454921	+	0.890532i	$0.650332\pi$
$200$	−0.134037	−0.00947783
$201$	0	0
$202$	9.18963	0.646580
$203$	3.14236	0.220551
$204$	0	0
$205$	12.8089	0.894612
$206$	−10.6350	−0.740977
$207$	0	0
$208$	−29.0610	−2.01502
$209$	−11.7471	−0.812562
$210$	0	0
$211$	−16.0684	−1.10619	−0.553097	−	0.833117i	$-0.686554\pi$
$211$	−16.0684	−1.10619	−0.553097	+	0.833117i	$0.686554\pi$
$212$	24.8338	1.70560
$213$	0	0
$214$	−24.7237	−1.69008
$215$	11.5695	0.789037
$216$	0	0
$217$	6.35057	0.431105
$218$	−32.8441	−2.22448
$219$	0	0
$220$	−12.7115	−0.857012
$221$	−16.4407	−1.10592
$222$	0	0
$223$	−21.4097	−1.43370	−0.716848	−	0.697229i	$-0.754416\pi$
$223$	−21.4097	−1.43370	−0.716848	+	0.697229i	$0.754416\pi$
$224$	7.83687	0.523623
$225$	0	0
$226$	−9.61333	−0.639469
$227$	−21.6474	−1.43679	−0.718395	−	0.695635i	$-0.755123\pi$
$227$	−21.6474	−1.43679	−0.718395	+	0.695635i	$0.755123\pi$
$228$	0	0
$229$	−24.4516	−1.61580	−0.807902	−	0.589316i	$-0.799397\pi$
$229$	−24.4516	−1.61580	−0.807902	+	0.589316i	$0.799397\pi$
$230$	−10.7709	−0.710209
$231$	0	0
$232$	0.798875	0.0524487
$233$	−5.42084	−0.355131	−0.177566	−	0.984109i	$-0.556822\pi$
$233$	−5.42084	−0.355131	−0.177566	+	0.984109i	$0.556822\pi$
$234$	0	0
$235$	−15.1130	−0.985867
$236$	7.68727	0.500398
$237$	0	0
$238$	4.72121	0.306031
$239$	13.9893	0.904895	0.452448	−	0.891791i	$-0.350551\pi$
$239$	13.9893	0.904895	0.452448	+	0.891791i	$0.350551\pi$
$240$	0	0
$241$	9.01626	0.580788	0.290394	−	0.956907i	$-0.406214\pi$
$241$	9.01626	0.580788	0.290394	+	0.956907i	$0.406214\pi$
$242$	−5.20780	−0.334770
$243$	0	0
$244$	18.7353	1.19941
$245$	−2.35101	−0.150200
$246$	0	0
$247$	−27.8468	−1.77185
$248$	1.61449	0.102520
$249$	0	0
$250$	20.6885	1.30846
$251$	18.3298	1.15697	0.578485	−	0.815693i	$-0.303644\pi$
$251$	18.3298	1.15697	0.578485	+	0.815693i	$0.303644\pi$
$252$	0	0
$253$	6.73005	0.423114
$254$	−1.96743	−0.123448
$255$	0	0
$256$	17.8631	1.11644
$257$	1.18621	0.0739939	0.0369970	−	0.999315i	$-0.488221\pi$
$257$	1.18621	0.0739939	0.0369970	+	0.999315i	$0.488221\pi$
$258$	0	0
$259$	−5.72445	−0.355700
$260$	−30.1331	−1.86878
$261$	0	0
$262$	−42.1482	−2.60392
$263$	−13.8991	−0.857054	−0.428527	−	0.903529i	$-0.640967\pi$
$263$	−13.8991	−0.857054	−0.428527	+	0.903529i	$0.640967\pi$
$264$	0	0
$265$	−31.2086	−1.91713
$266$	7.99665	0.490306
$267$	0	0
$268$	−18.2969	−1.11766
$269$	−24.4471	−1.49057	−0.745283	−	0.666748i	$-0.767686\pi$
$269$	−24.4471	−1.49057	−0.745283	+	0.666748i	$0.767686\pi$
$270$	0	0
$271$	5.36960	0.326180	0.163090	−	0.986611i	$-0.447854\pi$
$271$	5.36960	0.326180	0.163090	+	0.986611i	$0.447854\pi$
$272$	10.1788	0.617183
$273$	0	0
$274$	17.1903	1.03850
$275$	1.52378	0.0918875
$276$	0	0
$277$	−10.2489	−0.615799	−0.307899	−	0.951419i	$-0.599626\pi$
$277$	−10.2489	−0.615799	−0.307899	+	0.951419i	$0.599626\pi$
$278$	−15.5976	−0.935485
$279$	0	0
$280$	−0.597690	−0.0357188
$281$	−22.3072	−1.33074	−0.665369	−	0.746515i	$-0.731726\pi$
$281$	−22.3072	−1.33074	−0.665369	+	0.746515i	$0.731726\pi$
$282$	0	0
$283$	21.7199	1.29111	0.645557	−	0.763712i	$-0.276625\pi$
$283$	21.7199	1.29111	0.645557	+	0.763712i	$0.276625\pi$
$284$	0.408278	0.0242269
$285$	0	0
$286$	38.9572	2.30359
$287$	5.44826	0.321601
$288$	0	0
$289$	−11.2415	−0.661266
$290$	14.5348	0.853514
$291$	0	0
$292$	−23.1331	−1.35376
$293$	10.2743	0.600233	0.300117	−	0.953902i	$-0.402974\pi$
$293$	10.2743	0.600233	0.300117	+	0.953902i	$0.402974\pi$
$294$	0	0
$295$	−9.66057	−0.562460
$296$	−1.45531	−0.0845883
$297$	0	0
$298$	−1.05528	−0.0611306
$299$	15.9538	0.922633
$300$	0	0
$301$	4.92110	0.283648
$302$	25.4912	1.46685
$303$	0	0
$304$	17.2406	0.988817
$305$	−23.5446	−1.34816
$306$	0	0
$307$	−15.5926	−0.889915	−0.444957	−	0.895552i	$-0.646781\pi$
$307$	−15.5926	−0.889915	−0.444957	+	0.895552i	$0.646781\pi$
$308$	−5.40685	−0.308084
$309$	0	0
$310$	29.3742	1.66834
$311$	7.50153	0.425373	0.212686	−	0.977121i	$-0.431779\pi$
$311$	7.50153	0.425373	0.212686	+	0.977121i	$0.431779\pi$
$312$	0	0
$313$	28.8638	1.63148	0.815739	−	0.578420i	$-0.196330\pi$
$313$	28.8638	1.63148	0.815739	+	0.578420i	$0.196330\pi$
$314$	−3.51709	−0.198481
$315$	0	0
$316$	−5.57212	−0.313456
$317$	20.0007	1.12335	0.561675	−	0.827358i	$-0.310157\pi$
$317$	20.0007	1.12335	0.561675	+	0.827358i	$0.310157\pi$
$318$	0	0
$319$	−9.08192	−0.508490
$320$	16.3043	0.911437
$321$	0	0
$322$	−4.58138	−0.255310
$323$	9.75355	0.542702
$324$	0	0
$325$	3.61218	0.200368
$326$	−39.3537	−2.17960
$327$	0	0
$328$	1.38510	0.0764792
$329$	−6.42833	−0.354405
$330$	0	0
$331$	−11.3898	−0.626039	−0.313020	−	0.949747i	$-0.601341\pi$
$331$	−11.3898	−0.626039	−0.313020	+	0.949747i	$0.601341\pi$
$332$	−24.3504	−1.33640
$333$	0	0
$334$	16.6473	0.910901
$335$	22.9937	1.25628
$336$	0	0
$337$	22.2950	1.21448	0.607242	−	0.794517i	$-0.292276\pi$
$337$	22.2950	1.21448	0.607242	+	0.794517i	$0.292276\pi$
$338$	66.7728	3.63196
$339$	0	0
$340$	10.5543	0.572390
$341$	−18.3541	−0.993932
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	1.25108	0.0674537
$345$	0	0
$346$	−35.2039	−1.89257
$347$	15.8395	0.850306	0.425153	−	0.905121i	$-0.360220\pi$
$347$	15.8395	0.850306	0.425153	+	0.905121i	$0.360220\pi$
$348$	0	0
$349$	17.2082	0.921133	0.460567	−	0.887625i	$-0.347646\pi$
$349$	17.2082	0.921133	0.460567	+	0.887625i	$0.347646\pi$
$350$	−1.03729	−0.0554456
$351$	0	0
$352$	−22.6498	−1.20724
$353$	−7.97989	−0.424727	−0.212363	−	0.977191i	$-0.568116\pi$
$353$	−7.97989	−0.424727	−0.212363	+	0.977191i	$0.568116\pi$
$354$	0	0
$355$	−0.513082	−0.0272316
$356$	−13.4732	−0.714079
$357$	0	0
$358$	5.11599	0.270388
$359$	−20.0243	−1.05684	−0.528420	−	0.848983i	$-0.677216\pi$
$359$	−20.0243	−1.05684	−0.528420	+	0.848983i	$0.677216\pi$
$360$	0	0
$361$	−2.47972	−0.130512
$362$	4.60536	0.242053
$363$	0	0
$364$	−12.8171	−0.671800
$365$	29.0713	1.52166
$366$	0	0
$367$	27.9152	1.45716	0.728581	−	0.684960i	$-0.240180\pi$
$367$	27.9152	1.45716	0.728581	+	0.684960i	$0.240180\pi$
$368$	−9.87736	−0.514893
$369$	0	0
$370$	−26.4781	−1.37653
$371$	−13.2746	−0.689182
$372$	0	0
$373$	8.01565	0.415035	0.207517	−	0.978231i	$-0.433462\pi$
$373$	8.01565	0.415035	0.207517	+	0.978231i	$0.433462\pi$
$374$	−13.6450	−0.705568
$375$	0	0
$376$	−1.63426	−0.0842804
$377$	−21.5290	−1.10880
$378$	0	0
$379$	−2.39197	−0.122867	−0.0614336	−	0.998111i	$-0.519567\pi$
$379$	−2.39197	−0.122867	−0.0614336	+	0.998111i	$0.519567\pi$
$380$	17.8766	0.917052
$381$	0	0
$382$	28.6927	1.46805
$383$	−19.5791	−1.00045	−0.500223	−	0.865896i	$-0.666749\pi$
$383$	−19.5791	−1.00045	−0.500223	+	0.865896i	$0.666749\pi$
$384$	0	0
$385$	6.79477	0.346294
$386$	8.57657	0.436536
$387$	0	0
$388$	−34.6941	−1.76132
$389$	3.39595	0.172181	0.0860907	−	0.996287i	$-0.472563\pi$
$389$	3.39595	0.172181	0.0860907	+	0.996287i	$0.472563\pi$
$390$	0	0
$391$	−5.58793	−0.282594
$392$	−0.254227	−0.0128404
$393$	0	0
$394$	42.7258	2.15250
$395$	7.00246	0.352332
$396$	0	0
$397$	−6.78359	−0.340459	−0.170229	−	0.985404i	$-0.554451\pi$
$397$	−6.78359	−0.340459	−0.170229	+	0.985404i	$0.554451\pi$
$398$	−25.2518	−1.26576
$399$	0	0
$400$	−2.23638	−0.111819
$401$	33.3109	1.66347	0.831734	−	0.555174i	$-0.187349\pi$
$401$	33.3109	1.66347	0.831734	+	0.555174i	$0.187349\pi$
$402$	0	0
$403$	−43.5091	−2.16734
$404$	8.73820	0.434742
$405$	0	0
$406$	6.18238	0.306827
$407$	16.5446	0.820083
$408$	0	0
$409$	20.2438	1.00099	0.500497	−	0.865738i	$-0.333151\pi$
$409$	20.2438	1.00099	0.500497	+	0.865738i	$0.333151\pi$
$410$	25.2006	1.24457
$411$	0	0
$412$	−10.1126	−0.498211
$413$	−4.10912	−0.202197
$414$	0	0
$415$	30.6011	1.50215
$416$	−53.6921	−2.63247
$417$	0	0
$418$	−23.1115	−1.13042
$419$	−28.0474	−1.37020	−0.685102	−	0.728447i	$-0.740242\pi$
$419$	−28.0474	−1.37020	−0.685102	+	0.728447i	$0.740242\pi$
$420$	0	0
$421$	20.8208	1.01474	0.507372	−	0.861727i	$-0.330617\pi$
$421$	20.8208	1.01474	0.507372	+	0.861727i	$0.330617\pi$
$422$	−31.6134	−1.53892
$423$	0	0
$424$	−3.37476	−0.163893
$425$	−1.26519	−0.0613708
$426$	0	0
$427$	−10.0147	−0.484646
$428$	−23.5092	−1.13636
$429$	0	0
$430$	22.7623	1.09769
$431$	8.67253	0.417741	0.208870	−	0.977943i	$-0.433021\pi$
$431$	8.67253	0.417741	0.208870	+	0.977943i	$0.433021\pi$
$432$	0	0
$433$	−3.89886	−0.187367	−0.0936837	−	0.995602i	$-0.529864\pi$
$433$	−3.89886	−0.187367	−0.0936837	+	0.995602i	$0.529864\pi$
$434$	12.4943	0.599746
$435$	0	0
$436$	−31.2306	−1.49568
$437$	−9.46467	−0.452757
$438$	0	0
$439$	8.02937	0.383221	0.191610	−	0.981471i	$-0.438629\pi$
$439$	8.02937	0.383221	0.191610	+	0.981471i	$0.438629\pi$
$440$	1.72742	0.0823514
$441$	0	0
$442$	−32.3460	−1.53854
$443$	−12.5240	−0.595036	−0.297518	−	0.954716i	$-0.596159\pi$
$443$	−12.5240	−0.595036	−0.297518	+	0.954716i	$0.596159\pi$
$444$	0	0
$445$	16.9318	0.802643
$446$	−42.1220	−1.99453
$447$	0	0
$448$	6.93502	0.327649
$449$	−8.48558	−0.400459	−0.200230	−	0.979749i	$-0.564169\pi$
$449$	−8.48558	−0.400459	−0.200230	+	0.979749i	$0.564169\pi$
$450$	0	0
$451$	−15.7463	−0.741465
$452$	−9.14108	−0.429960
$453$	0	0
$454$	−42.5898	−1.99884
$455$	16.1072	0.755119
$456$	0	0
$457$	−24.8695	−1.16335	−0.581673	−	0.813423i	$-0.697602\pi$
$457$	−24.8695	−1.16335	−0.581673	+	0.813423i	$0.697602\pi$
$458$	−48.1067	−2.24788
$459$	0	0
$460$	−10.2417	−0.477524
$461$	−16.4708	−0.767121	−0.383560	−	0.923516i	$-0.625302\pi$
$461$	−16.4708	−0.767121	−0.383560	+	0.923516i	$0.625302\pi$
$462$	0	0
$463$	40.8908	1.90035	0.950177	−	0.311710i	$-0.100902\pi$
$463$	40.8908	1.90035	0.950177	+	0.311710i	$0.100902\pi$
$464$	13.3291	0.618787
$465$	0	0
$466$	−10.6651	−0.494053
$467$	−24.6291	−1.13970	−0.569850	−	0.821749i	$-0.692999\pi$
$467$	−24.6291	−1.13970	−0.569850	+	0.821749i	$0.692999\pi$
$468$	0	0
$469$	9.78035	0.451615
$470$	−29.7339	−1.37152
$471$	0	0
$472$	−1.04465	−0.0480839
$473$	−14.2227	−0.653963
$474$	0	0
$475$	−2.14294	−0.0983249
$476$	4.48929	0.205766
$477$	0	0
$478$	27.5231	1.25888
$479$	−15.8536	−0.724369	−0.362184	−	0.932106i	$-0.617969\pi$
$479$	−15.8536	−0.724369	−0.362184	+	0.932106i	$0.617969\pi$
$480$	0	0
$481$	39.2194	1.78825
$482$	17.7389	0.807983
$483$	0	0
$484$	−4.95197	−0.225090
$485$	43.5999	1.97977
$486$	0	0
$487$	10.3699	0.469904	0.234952	−	0.972007i	$-0.424507\pi$
$487$	10.3699	0.469904	0.234952	+	0.972007i	$0.424507\pi$
$488$	−2.54601	−0.115253
$489$	0	0
$490$	−4.62544	−0.208956
$491$	38.5101	1.73794	0.868968	−	0.494869i	$-0.164784\pi$
$491$	38.5101	1.73794	0.868968	+	0.494869i	$0.164784\pi$
$492$	0	0
$493$	7.54068	0.339615
$494$	−54.7867	−2.46497
$495$	0	0
$496$	26.9375	1.20953
$497$	−0.218239	−0.00978937
$498$	0	0
$499$	−19.6920	−0.881536	−0.440768	−	0.897621i	$-0.645294\pi$
$499$	−19.6920	−0.881536	−0.440768	+	0.897621i	$0.645294\pi$
$500$	19.6722	0.879768
$501$	0	0
$502$	36.0627	1.60956
$503$	−10.8754	−0.484909	−0.242454	−	0.970163i	$-0.577953\pi$
$503$	−10.8754	−0.484909	−0.242454	+	0.970163i	$0.577953\pi$
$504$	0	0
$505$	−10.9813	−0.488660
$506$	13.2409	0.588630
$507$	0	0
$508$	−1.87078	−0.0830025
$509$	−36.5494	−1.62002	−0.810011	−	0.586415i	$-0.800539\pi$
$509$	−36.5494	−1.62002	−0.810011	+	0.586415i	$0.800539\pi$
$510$	0	0
$511$	12.3655	0.547016
$512$	31.0852	1.37379
$513$	0	0
$514$	2.33379	0.102939
$515$	12.7085	0.560002
$516$	0	0
$517$	18.5789	0.817098
$518$	−11.2625	−0.494844
$519$	0	0
$520$	4.09490	0.179573
$521$	30.1034	1.31885	0.659426	−	0.751769i	$-0.270799\pi$
$521$	30.1034	1.31885	0.659426	+	0.751769i	$0.270799\pi$
$522$	0	0
$523$	−33.0254	−1.44410	−0.722049	−	0.691842i	$-0.756800\pi$
$523$	−33.0254	−1.44410	−0.722049	+	0.691842i	$0.756800\pi$
$524$	−40.0777	−1.75080
$525$	0	0
$526$	−27.3455	−1.19232
$527$	15.2394	0.663837
$528$	0	0
$529$	−17.5776	−0.764242
$530$	−61.4008	−2.66708
$531$	0	0
$532$	7.60382	0.329667
$533$	−37.3272	−1.61682
$534$	0	0
$535$	29.5439	1.27730
$536$	2.48643	0.107397
$537$	0	0
$538$	−48.0980	−2.07365
$539$	2.89015	0.124488
$540$	0	0
$541$	−3.86545	−0.166189	−0.0830943	−	0.996542i	$-0.526480\pi$
$541$	−3.86545	−0.166189	−0.0830943	+	0.996542i	$0.526480\pi$
$542$	10.5643	0.453776
$543$	0	0
$544$	18.8060	0.806301
$545$	39.2474	1.68118
$546$	0	0
$547$	−4.24929	−0.181687	−0.0908433	−	0.995865i	$-0.528956\pi$
$547$	−4.24929	−0.181687	−0.0908433	+	0.995865i	$0.528956\pi$
$548$	16.3458	0.698260
$549$	0	0
$550$	2.99794	0.127832
$551$	12.7722	0.544113
$552$	0	0
$553$	2.97850	0.126659
$554$	−20.1641	−0.856689
$555$	0	0
$556$	−14.8314	−0.628993
$557$	42.3091	1.79269	0.896347	−	0.443352i	$-0.146211\pi$
$557$	42.3091	1.79269	0.896347	+	0.443352i	$0.146211\pi$
$558$	0	0
$559$	−33.7155	−1.42601
$560$	−9.97236	−0.421409
$561$	0	0
$562$	−43.8879	−1.85130
$563$	−32.4926	−1.36940	−0.684701	−	0.728825i	$-0.740067\pi$
$563$	−32.4926	−1.36940	−0.684701	+	0.728825i	$0.740067\pi$
$564$	0	0
$565$	11.4876	0.483286
$566$	42.7324	1.79618
$567$	0	0
$568$	−0.0554824	−0.00232799
$569$	13.2130	0.553917	0.276959	−	0.960882i	$-0.410673\pi$
$569$	13.2130	0.553917	0.276959	+	0.960882i	$0.410673\pi$
$570$	0	0
$571$	11.5633	0.483907	0.241954	−	0.970288i	$-0.422212\pi$
$571$	11.5633	0.483907	0.241954	+	0.970288i	$0.422212\pi$
$572$	37.0435	1.54887
$573$	0	0
$574$	10.7191	0.447405
$575$	1.22772	0.0511994
$576$	0	0
$577$	−31.1420	−1.29646	−0.648230	−	0.761445i	$-0.724490\pi$
$577$	−31.1420	−1.29646	−0.648230	+	0.761445i	$0.724490\pi$
$578$	−22.1169	−0.919942
$579$	0	0
$580$	13.8208	0.573878
$581$	13.0162	0.540002
$582$	0	0
$583$	38.3656	1.58894
$584$	3.14364	0.130085
$585$	0	0
$586$	20.2141	0.835035
$587$	35.5404	1.46691	0.733455	−	0.679738i	$-0.237907\pi$
$587$	35.5404	1.46691	0.733455	+	0.679738i	$0.237907\pi$
$588$	0	0
$589$	25.8120	1.06356
$590$	−19.0065	−0.782485
$591$	0	0
$592$	−24.2816	−0.997969
$593$	47.0247	1.93107	0.965536	−	0.260271i	$-0.0838119\pi$
$593$	47.0247	1.93107	0.965536	+	0.260271i	$0.0838119\pi$
$594$	0	0
$595$	−5.64167	−0.231286
$596$	−1.00344	−0.0411025
$597$	0	0
$598$	31.3880	1.28355
$599$	24.5946	1.00491	0.502453	−	0.864604i	$-0.332431\pi$
$599$	24.5946	1.00491	0.502453	+	0.864604i	$0.332431\pi$
$600$	0	0
$601$	−35.5297	−1.44929	−0.724643	−	0.689124i	$-0.757995\pi$
$601$	−35.5297	−1.44929	−0.724643	+	0.689124i	$0.757995\pi$
$602$	9.68193	0.394606
$603$	0	0
$604$	24.2390	0.986269
$605$	6.22313	0.253006
$606$	0	0
$607$	6.31589	0.256354	0.128177	−	0.991751i	$-0.459087\pi$
$607$	6.31589	0.256354	0.128177	+	0.991751i	$0.459087\pi$
$608$	31.8531	1.29181
$609$	0	0
$610$	−46.3224	−1.87554
$611$	44.0419	1.78174
$612$	0	0
$613$	−44.7009	−1.80545	−0.902727	−	0.430215i	$-0.858438\pi$
$613$	−44.7009	−1.80545	−0.902727	+	0.430215i	$0.858438\pi$
$614$	−30.6773	−1.23803
$615$	0	0
$616$	0.734756	0.0296042
$617$	2.00783	0.0808322	0.0404161	−	0.999183i	$-0.487132\pi$
$617$	2.00783	0.0808322	0.0404161	+	0.999183i	$0.487132\pi$
$618$	0	0
$619$	23.2979	0.936421	0.468211	−	0.883617i	$-0.344899\pi$
$619$	23.2979	0.936421	0.468211	+	0.883617i	$0.344899\pi$
$620$	27.9312	1.12174
$621$	0	0
$622$	14.7587	0.591771
$623$	7.20192	0.288539
$624$	0	0
$625$	−27.3582	−1.09433
$626$	56.7875	2.26969
$627$	0	0
$628$	−3.34432	−0.133453
$629$	−13.7369	−0.547725
$630$	0	0
$631$	38.1985	1.52066	0.760329	−	0.649538i	$-0.225038\pi$
$631$	38.1985	1.52066	0.760329	+	0.649538i	$0.225038\pi$
$632$	0.757215	0.0301204
$633$	0	0
$634$	39.3499	1.56279
$635$	2.35101	0.0932969
$636$	0	0
$637$	6.85121	0.271455
$638$	−17.8680	−0.707403
$639$	0	0
$640$	−4.77154	−0.188612
$641$	48.1272	1.90091	0.950454	−	0.310864i	$-0.100618\pi$
$641$	48.1272	1.90091	0.950454	+	0.310864i	$0.100618\pi$
$642$	0	0
$643$	26.4929	1.04478	0.522390	−	0.852707i	$-0.325041\pi$
$643$	26.4929	1.04478	0.522390	+	0.852707i	$0.325041\pi$
$644$	−4.35632	−0.171663
$645$	0	0
$646$	19.1894	0.754998
$647$	22.9476	0.902164	0.451082	−	0.892483i	$-0.351038\pi$
$647$	22.9476	0.902164	0.451082	+	0.892483i	$0.351038\pi$
$648$	0	0
$649$	11.8760	0.466173
$650$	7.10671	0.278748
$651$	0	0
$652$	−37.4205	−1.46550
$653$	21.4210	0.838268	0.419134	−	0.907924i	$-0.362334\pi$
$653$	21.4210	0.838268	0.419134	+	0.907924i	$0.362334\pi$
$654$	0	0
$655$	50.3655	1.96794
$656$	23.1101	0.902298
$657$	0	0
$658$	−12.6473	−0.493043
$659$	48.0153	1.87041	0.935205	−	0.354106i	$-0.115215\pi$
$659$	48.0153	1.87041	0.935205	+	0.354106i	$0.115215\pi$
$660$	0	0
$661$	13.4557	0.523367	0.261683	−	0.965154i	$-0.415722\pi$
$661$	13.4557	0.523367	0.261683	+	0.965154i	$0.415722\pi$
$662$	−22.4086	−0.870935
$663$	0	0
$664$	3.30907	0.128417
$665$	−9.55570	−0.370554
$666$	0	0
$667$	−7.31735	−0.283329
$668$	15.8295	0.612463
$669$	0	0
$670$	45.2384	1.74771
$671$	28.9440	1.11737
$672$	0	0
$673$	−31.6378	−1.21955	−0.609773	−	0.792576i	$-0.708739\pi$
$673$	−31.6378	−1.21955	−0.609773	+	0.792576i	$0.708739\pi$
$674$	43.8638	1.68957
$675$	0	0
$676$	63.4927	2.44203
$677$	−9.63313	−0.370231	−0.185116	−	0.982717i	$-0.559266\pi$
$677$	−9.63313	−0.370231	−0.185116	+	0.982717i	$0.559266\pi$
$678$	0	0
$679$	18.5452	0.711700
$680$	−1.43427	−0.0550017
$681$	0	0
$682$	−36.1105	−1.38274
$683$	−11.6247	−0.444808	−0.222404	−	0.974955i	$-0.571390\pi$
$683$	−11.6247	−0.444808	−0.222404	+	0.974955i	$0.571390\pi$
$684$	0	0
$685$	−20.5418	−0.784861
$686$	−1.96743	−0.0751168
$687$	0	0
$688$	20.8740	0.795815
$689$	90.9469	3.46480
$690$	0	0
$691$	−12.6074	−0.479608	−0.239804	−	0.970821i	$-0.577083\pi$
$691$	−12.6074	−0.479608	−0.239804	+	0.970821i	$0.577083\pi$
$692$	−33.4745	−1.27251
$693$	0	0
$694$	31.1630	1.18293
$695$	18.6386	0.707003
$696$	0	0
$697$	13.0741	0.495217
$698$	33.8559	1.28147
$699$	0	0
$700$	−0.986336	−0.0372800
$701$	8.45875	0.319483	0.159741	−	0.987159i	$-0.448934\pi$
$701$	8.45875	0.319483	0.159741	+	0.987159i	$0.448934\pi$
$702$	0	0
$703$	−23.2671	−0.877536
$704$	−20.0433	−0.755409
$705$	0	0
$706$	−15.6999	−0.590873
$707$	−4.67088	−0.175667
$708$	0	0
$709$	−28.2276	−1.06011	−0.530056	−	0.847963i	$-0.677829\pi$
$709$	−28.2276	−1.06011	−0.530056	+	0.847963i	$0.677829\pi$
$710$	−1.00945	−0.0378841
$711$	0	0
$712$	1.83092	0.0686168
$713$	−14.7880	−0.553815
$714$	0	0
$715$	−46.5524	−1.74096
$716$	4.86467	0.181801
$717$	0	0
$718$	−39.3964	−1.47026
$719$	−20.8897	−0.779056	−0.389528	−	0.921015i	$-0.627362\pi$
$719$	−20.8897	−0.779056	−0.389528	+	0.921015i	$0.627362\pi$
$720$	0	0
$721$	5.40554	0.201313
$722$	−4.87868	−0.181566
$723$	0	0
$724$	4.37913	0.162749
$725$	−1.65676	−0.0615304
$726$	0	0
$727$	−21.4574	−0.795811	−0.397906	−	0.917426i	$-0.630263\pi$
$727$	−21.4574	−0.795811	−0.397906	+	0.917426i	$0.630263\pi$
$728$	1.74176	0.0645541
$729$	0	0
$730$	57.1958	2.11691
$731$	11.8091	0.436775
$732$	0	0
$733$	−48.1668	−1.77908	−0.889540	−	0.456857i	$-0.848975\pi$
$733$	−48.1668	−1.77908	−0.889540	+	0.456857i	$0.848975\pi$
$734$	54.9212	2.02718
$735$	0	0
$736$	−18.2490	−0.672668
$737$	−28.2667	−1.04122
$738$	0	0
$739$	33.3207	1.22572	0.612861	−	0.790191i	$-0.290019\pi$
$739$	33.3207	1.22572	0.612861	+	0.790191i	$0.290019\pi$
$740$	−25.1774	−0.925540
$741$	0	0
$742$	−26.1168	−0.958778
$743$	16.0300	0.588085	0.294043	−	0.955792i	$-0.404999\pi$
$743$	16.0300	0.588085	0.294043	+	0.955792i	$0.404999\pi$
$744$	0	0
$745$	1.26102	0.0462002
$746$	15.7702	0.577389
$747$	0	0
$748$	−12.9747	−0.474403
$749$	12.5665	0.459170
$750$	0	0
$751$	−31.2938	−1.14193	−0.570964	−	0.820975i	$-0.693430\pi$
$751$	−31.2938	−1.14193	−0.570964	+	0.820975i	$0.693430\pi$
$752$	−27.2673	−0.994336
$753$	0	0
$754$	−42.3568	−1.54254
$755$	−30.4610	−1.10859
$756$	0	0
$757$	6.16678	0.224136	0.112068	−	0.993701i	$-0.464253\pi$
$757$	6.16678	0.224136	0.112068	+	0.993701i	$0.464253\pi$
$758$	−4.70604	−0.170931
$759$	0	0
$760$	−2.42932	−0.0881207
$761$	−42.7410	−1.54936	−0.774680	−	0.632353i	$-0.782089\pi$
$761$	−42.7410	−1.54936	−0.774680	+	0.632353i	$0.782089\pi$
$762$	0	0
$763$	16.6939	0.604359
$764$	27.2832	0.987072
$765$	0	0
$766$	−38.5206	−1.39181
$767$	28.1524	1.01653
$768$	0	0
$769$	17.4128	0.627923	0.313961	−	0.949436i	$-0.398344\pi$
$769$	17.4128	0.627923	0.313961	+	0.949436i	$0.398344\pi$
$770$	13.3682	0.481758
$771$	0	0
$772$	8.15525	0.293514
$773$	29.8993	1.07540	0.537701	−	0.843135i	$-0.319293\pi$
$773$	29.8993	1.07540	0.537701	+	0.843135i	$0.319293\pi$
$774$	0	0
$775$	−3.34823	−0.120272
$776$	4.71470	0.169248
$777$	0	0
$778$	6.68129	0.239536
$779$	22.1445	0.793410
$780$	0	0
$781$	0.630746	0.0225699
$782$	−10.9939	−0.393140
$783$	0	0
$784$	−4.24174	−0.151491
$785$	4.20279	0.150004
$786$	0	0
$787$	−25.7598	−0.918238	−0.459119	−	0.888375i	$-0.651835\pi$
$787$	−25.7598	−0.918238	−0.459119	+	0.888375i	$0.651835\pi$
$788$	40.6269	1.44728
$789$	0	0
$790$	13.7769	0.490159
$791$	4.88624	0.173735
$792$	0	0
$793$	68.6129	2.43651
$794$	−13.3462	−0.473640
$795$	0	0
$796$	−24.0113	−0.851058
$797$	23.7025	0.839584	0.419792	−	0.907620i	$-0.362103\pi$
$797$	23.7025	0.839584	0.419792	+	0.907620i	$0.362103\pi$
$798$	0	0
$799$	−15.4260	−0.545731
$800$	−4.13185	−0.146083
$801$	0	0
$802$	65.5369	2.31419
$803$	−35.7381	−1.26117
$804$	0	0
$805$	5.47458	0.192954
$806$	−85.6011	−3.01517
$807$	0	0
$808$	−1.18747	−0.0417749
$809$	46.8159	1.64596	0.822981	−	0.568070i	$-0.192310\pi$
$809$	46.8159	1.64596	0.822981	+	0.568070i	$0.192310\pi$
$810$	0	0
$811$	−21.3447	−0.749515	−0.374758	−	0.927123i	$-0.622274\pi$
$811$	−21.3447	−0.749515	−0.374758	+	0.927123i	$0.622274\pi$
$812$	5.87868	0.206301
$813$	0	0
$814$	32.5503	1.14089
$815$	47.0262	1.64726
$816$	0	0
$817$	20.0019	0.699778
$818$	39.8283	1.39257
$819$	0	0
$820$	23.9627	0.836812
$821$	−25.9217	−0.904673	−0.452337	−	0.891847i	$-0.649409\pi$
$821$	−25.9217	−0.904673	−0.452337	+	0.891847i	$0.649409\pi$
$822$	0	0
$823$	18.6224	0.649137	0.324568	−	0.945862i	$-0.394781\pi$
$823$	18.6224	0.649137	0.324568	+	0.945862i	$0.394781\pi$
$824$	1.37424	0.0478738
$825$	0	0
$826$	−8.08440	−0.281292
$827$	3.32877	0.115753	0.0578763	−	0.998324i	$-0.481567\pi$
$827$	3.32877	0.115753	0.0578763	+	0.998324i	$0.481567\pi$
$828$	0	0
$829$	32.9044	1.14282	0.571409	−	0.820665i	$-0.306397\pi$
$829$	32.9044	1.14282	0.571409	+	0.820665i	$0.306397\pi$
$830$	60.2056	2.08977
$831$	0	0
$832$	−47.5133	−1.64723
$833$	−2.39968	−0.0831441
$834$	0	0
$835$	−19.8929	−0.688423
$836$	−21.9762	−0.760063
$837$	0	0
$838$	−55.1813	−1.90620
$839$	32.9296	1.13686	0.568428	−	0.822733i	$-0.307552\pi$
$839$	32.9296	1.13686	0.568428	+	0.822733i	$0.307552\pi$
$840$	0	0
$841$	−19.1255	−0.659502
$842$	40.9635	1.41170
$843$	0	0
$844$	−30.0605	−1.03472
$845$	−79.7910	−2.74490
$846$	0	0
$847$	2.64701	0.0909522
$848$	−56.3073	−1.93360
$849$	0	0
$850$	−2.48917	−0.0853780
$851$	13.3300	0.456948
$852$	0	0
$853$	19.4639	0.666431	0.333215	−	0.942851i	$-0.391866\pi$
$853$	19.4639	0.666431	0.333215	+	0.942851i	$0.391866\pi$
$854$	−19.7032	−0.674231
$855$	0	0
$856$	3.19475	0.109194
$857$	−27.8327	−0.950749	−0.475374	−	0.879784i	$-0.657687\pi$
$857$	−27.8327	−0.950749	−0.475374	+	0.879784i	$0.657687\pi$
$858$	0	0
$859$	50.4371	1.72089	0.860447	−	0.509541i	$-0.170185\pi$
$859$	50.4371	1.72089	0.860447	+	0.509541i	$0.170185\pi$
$860$	21.6441	0.738058
$861$	0	0
$862$	17.0626	0.581154
$863$	−33.6526	−1.14555	−0.572774	−	0.819714i	$-0.694133\pi$
$863$	−33.6526	−1.14555	−0.572774	+	0.819714i	$0.694133\pi$
$864$	0	0
$865$	42.0673	1.43033
$866$	−7.67074	−0.260662
$867$	0	0
$868$	11.8805	0.403252
$869$	−8.60831	−0.292017
$870$	0	0
$871$	−67.0072	−2.27045
$872$	4.24404	0.143721
$873$	0	0
$874$	−18.6211	−0.629868
$875$	−10.5155	−0.355489
$876$	0	0
$877$	−45.8712	−1.54896	−0.774480	−	0.632598i	$-0.781989\pi$
$877$	−45.8712	−1.54896	−0.774480	+	0.632598i	$0.781989\pi$
$878$	15.7972	0.533131
$879$	0	0
$880$	28.8216	0.971578
$881$	46.2258	1.55739	0.778694	−	0.627404i	$-0.215882\pi$
$881$	46.2258	1.55739	0.778694	+	0.627404i	$0.215882\pi$
$882$	0	0
$883$	25.5211	0.858854	0.429427	−	0.903102i	$-0.358716\pi$
$883$	25.5211	0.858854	0.429427	+	0.903102i	$0.358716\pi$
$884$	−30.7571	−1.03447
$885$	0	0
$886$	−24.6402	−0.827803
$887$	−54.5602	−1.83195	−0.915976	−	0.401232i	$-0.868582\pi$
$887$	−54.5602	−1.83195	−0.915976	+	0.401232i	$0.868582\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	33.3121	1.11662
$891$	0	0
$892$	−40.0528	−1.34107
$893$	−26.1280	−0.874342
$894$	0	0
$895$	−6.11342	−0.204349
$896$	−2.02957	−0.0678033
$897$	0	0
$898$	−16.6948	−0.557112
$899$	19.9558	0.665564
$900$	0	0
$901$	−31.8548	−1.06124
$902$	−30.9798	−1.03151
$903$	0	0
$904$	1.24221	0.0413154
$905$	−5.50324	−0.182934
$906$	0	0
$907$	13.5809	0.450947	0.225474	−	0.974249i	$-0.427607\pi$
$907$	13.5809	0.450947	0.225474	+	0.974249i	$0.427607\pi$
$908$	−40.4976	−1.34396
$909$	0	0
$910$	31.6899	1.05051
$911$	23.8052	0.788700	0.394350	−	0.918960i	$-0.370970\pi$
$911$	23.8052	0.788700	0.394350	+	0.918960i	$0.370970\pi$
$912$	0	0
$913$	−37.6188	−1.24500
$914$	−48.9290	−1.61843
$915$	0	0
$916$	−45.7435	−1.51141
$917$	21.4230	0.707448
$918$	0	0
$919$	−6.64193	−0.219097	−0.109549	−	0.993981i	$-0.534941\pi$
$919$	−6.64193	−0.219097	−0.109549	+	0.993981i	$0.534941\pi$
$920$	1.39179	0.0458859
$921$	0	0
$922$	−32.4051	−1.06721
$923$	1.49520	0.0492152
$924$	0	0
$925$	3.01812	0.0992351
$926$	80.4497	2.64374
$927$	0	0
$928$	24.6263	0.808398
$929$	18.6804	0.612883	0.306441	−	0.951890i	$-0.400862\pi$
$929$	18.6804	0.612883	0.306441	+	0.951890i	$0.400862\pi$
$930$	0	0
$931$	−4.06451	−0.133209
$932$	−10.1412	−0.332187
$933$	0	0
$934$	−48.4561	−1.58553
$935$	16.3053	0.533241
$936$	0	0
$937$	36.7818	1.20161	0.600805	−	0.799396i	$-0.294847\pi$
$937$	36.7818	1.20161	0.600805	+	0.799396i	$0.294847\pi$
$938$	19.2422	0.628279
$939$	0	0
$940$	−28.2732	−0.922171
$941$	5.52071	0.179970	0.0899850	−	0.995943i	$-0.471318\pi$
$941$	5.52071	0.179970	0.0899850	+	0.995943i	$0.471318\pi$
$942$	0	0
$943$	−12.6869	−0.413142
$944$	−17.4298	−0.567292
$945$	0	0
$946$	−27.9823	−0.909782
$947$	−50.0953	−1.62788	−0.813940	−	0.580949i	$-0.802682\pi$
$947$	−50.0953	−1.62788	−0.813940	+	0.580949i	$0.802682\pi$
$948$	0	0
$949$	−84.7185	−2.75008
$950$	−4.21609	−0.136788
$951$	0	0
$952$	−0.610065	−0.0197723
$953$	19.8040	0.641514	0.320757	−	0.947162i	$-0.396063\pi$
$953$	19.8040	0.641514	0.320757	+	0.947162i	$0.396063\pi$
$954$	0	0
$955$	−34.2868	−1.10949
$956$	26.1710	0.846431
$957$	0	0
$958$	−31.1908	−1.00773
$959$	−8.73744	−0.282147
$960$	0	0
$961$	9.32975	0.300960
$962$	77.1615	2.48779
$963$	0	0
$964$	16.8675	0.543264
$965$	−10.2487	−0.329917
$966$	0	0
$967$	20.7804	0.668253	0.334127	−	0.942528i	$-0.391559\pi$
$967$	20.7804	0.668253	0.334127	+	0.942528i	$0.391559\pi$
$968$	0.672941	0.0216292
$969$	0	0
$970$	85.7798	2.75422
$971$	32.0142	1.02739	0.513693	−	0.857974i	$-0.328277\pi$
$971$	32.0142	1.02739	0.513693	+	0.857974i	$0.328277\pi$
$972$	0	0
$973$	7.92793	0.254158
$974$	20.4020	0.653723
$975$	0	0
$976$	−42.4798	−1.35974
$977$	30.7497	0.983772	0.491886	−	0.870660i	$-0.336308\pi$
$977$	30.7497	0.983772	0.491886	+	0.870660i	$0.336308\pi$
$978$	0	0
$979$	−20.8147	−0.665240
$980$	−4.39822	−0.140496
$981$	0	0
$982$	75.7659	2.41779
$983$	43.4959	1.38730	0.693651	−	0.720311i	$-0.256001\pi$
$983$	43.4959	1.38730	0.693651	+	0.720311i	$0.256001\pi$
$984$	0	0
$985$	−51.0558	−1.62677
$986$	14.8358	0.472467
$987$	0	0
$988$	−52.0954	−1.65737
$989$	−11.4593	−0.364386
$990$	0	0
$991$	−31.8498	−1.01174	−0.505871	−	0.862609i	$-0.668829\pi$
$991$	−31.8498	−1.01174	−0.505871	+	0.862609i	$0.668829\pi$
$992$	49.7686	1.58015
$993$	0	0
$994$	−0.429371	−0.0136188
$995$	30.1749	0.956610
$996$	0	0
$997$	20.8478	0.660257	0.330129	−	0.943936i	$-0.392908\pi$
$997$	20.8478	0.660257	0.330129	+	0.943936i	$0.392908\pi$
$998$	−38.7427	−1.22638
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.z.1.27	yes	32
3.2	odd	2	inner	8001.2.a.z.1.6	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8001.2.a.z.1.6	✓	32	3.2	odd	2	inner
8001.2.a.z.1.27	yes	32	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)

\(f(q)\)	\(=\)	\(q+1.96743 q^{2} +1.87078 q^{4} -2.35101 q^{5} -1.00000 q^{7} -0.254227 q^{8} +O(q^{10})\)	\(q+1.96743 q^{2} +1.87078 q^{4} -2.35101 q^{5} -1.00000 q^{7} -0.254227 q^{8} -4.62544 q^{10} +2.89015 q^{11} +6.85121 q^{13} -1.96743 q^{14} -4.24174 q^{16} -2.39968 q^{17} -4.06451 q^{19} -4.39822 q^{20} +5.68618 q^{22} +2.32861 q^{23} +0.527232 q^{25} +13.4793 q^{26} -1.87078 q^{28} -3.14236 q^{29} -6.35057 q^{31} -7.83687 q^{32} -4.72121 q^{34} +2.35101 q^{35} +5.72445 q^{37} -7.99665 q^{38} +0.597690 q^{40} -5.44826 q^{41} -4.92110 q^{43} +5.40685 q^{44} +4.58138 q^{46} +6.42833 q^{47} +1.00000 q^{49} +1.03729 q^{50} +12.8171 q^{52} +13.2746 q^{53} -6.79477 q^{55} +0.254227 q^{56} -6.18238 q^{58} +4.10912 q^{59} +10.0147 q^{61} -12.4943 q^{62} -6.93502 q^{64} -16.1072 q^{65} -9.78035 q^{67} -4.48929 q^{68} +4.62544 q^{70} +0.218239 q^{71} -12.3655 q^{73} +11.2625 q^{74} -7.60382 q^{76} -2.89015 q^{77} -2.97850 q^{79} +9.97236 q^{80} -10.7191 q^{82} -13.0162 q^{83} +5.64167 q^{85} -9.68193 q^{86} -0.734756 q^{88} -7.20192 q^{89} -6.85121 q^{91} +4.35632 q^{92} +12.6473 q^{94} +9.55570 q^{95} -18.5452 q^{97} +1.96743 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 30 q^{4} - 32 q^{7}+O(q^{10}) \) 32 * q + 30 * q^4 - 32 * q^7	\( 32 q + 30 q^{4} - 32 q^{7} - 16 q^{10} - 14 q^{13} + 18 q^{16} - 30 q^{19} - 10 q^{22} + 36 q^{25} - 30 q^{28} - 58 q^{31} - 34 q^{34} + 8 q^{37} - 34 q^{40} + 6 q^{43} - 36 q^{46} + 32 q^{49} - 56 q^{52} - 88 q^{55} - 22 q^{58} - 46 q^{61} + 20 q^{64} - 8 q^{67} + 16 q^{70} - 60 q^{73} - 128 q^{76} - 74 q^{79} - 52 q^{82} - 16 q^{85} - 64 q^{88} + 14 q^{91} - 58 q^{94} - 44 q^{97}+O(q^{100}) \) 32 * q + 30 * q^4 - 32 * q^7 - 16 * q^10 - 14 * q^13 + 18 * q^16 - 30 * q^19 - 10 * q^22 + 36 * q^25 - 30 * q^28 - 58 * q^31 - 34 * q^34 + 8 * q^37 - 34 * q^40 + 6 * q^43 - 36 * q^46 + 32 * q^49 - 56 * q^52 - 88 * q^55 - 22 * q^58 - 46 * q^61 + 20 * q^64 - 8 * q^67 + 16 * q^70 - 60 * q^73 - 128 * q^76 - 74 * q^79 - 52 * q^82 - 16 * q^85 - 64 * q^88 + 14 * q^91 - 58 * q^94 - 44 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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