LMFDB - Embedded newform 8001.2.a.p.1.10

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:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 5 x^{13} - 9 x^{12} + 76 x^{11} - 12 x^{10} - 414 x^{9} + 331 x^{8} + 959 x^{7} - 1067 x^{6} + \cdots + 44 $ x^14 - 5x^13 - 9x^12 + 76x^11 - 12x^10 - 414x^9 + 331x^8 + 959x^7 - 1067x^6 - 875x^5 + 1134x^4 + 301x^3 - 418x^2 - 42*x + 44
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 5 $
Twist minimal:	no (minimal twist has level 2667)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$1.75826$ 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)

$f(q)$	$=$	$q+1.75826 q^{2} +1.09149 q^{4} -1.44353 q^{5} -1.00000 q^{7} -1.59740 q^{8} +O(q^{10})$	\(q+1.75826 q^{2} +1.09149 q^{4} -1.44353 q^{5} -1.00000 q^{7} -1.59740 q^{8} -2.53810 q^{10} -4.64133 q^{11} -3.75930 q^{13} -1.75826 q^{14} -4.99163 q^{16} -2.76361 q^{17} +1.65863 q^{19} -1.57560 q^{20} -8.16068 q^{22} +6.34922 q^{23} -2.91623 q^{25} -6.60984 q^{26} -1.09149 q^{28} -0.687809 q^{29} -4.29992 q^{31} -5.58180 q^{32} -4.85916 q^{34} +1.44353 q^{35} -1.53230 q^{37} +2.91631 q^{38} +2.30589 q^{40} +9.88414 q^{41} +7.55579 q^{43} -5.06597 q^{44} +11.1636 q^{46} +0.153499 q^{47} +1.00000 q^{49} -5.12750 q^{50} -4.10324 q^{52} +4.56061 q^{53} +6.69988 q^{55} +1.59740 q^{56} -1.20935 q^{58} +0.908070 q^{59} +8.22355 q^{61} -7.56040 q^{62} +0.168978 q^{64} +5.42665 q^{65} -9.93107 q^{67} -3.01646 q^{68} +2.53810 q^{70} +8.33861 q^{71} +8.16747 q^{73} -2.69418 q^{74} +1.81038 q^{76} +4.64133 q^{77} -1.51078 q^{79} +7.20555 q^{80} +17.3789 q^{82} +8.23096 q^{83} +3.98935 q^{85} +13.2851 q^{86} +7.41405 q^{88} +1.44262 q^{89} +3.75930 q^{91} +6.93012 q^{92} +0.269891 q^{94} -2.39428 q^{95} -9.30424 q^{97} +1.75826 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + 5 q^{2} + 15 q^{4} + 4 q^{5} - 14 q^{7} + 12 q^{8}+O(q^{10}) $ 14 * q + 5 * q^2 + 15 * q^4 + 4 * q^5 - 14 * q^7 + 12 * q^8	$ 14 q + 5 q^{2} + 15 q^{4} + 4 q^{5} - 14 q^{7} + 12 q^{8} + 4 q^{10} + 3 q^{11} - 13 q^{13} - 5 q^{14} + 13 q^{16} + 5 q^{17} + 21 q^{19} + 3 q^{20} - 3 q^{22} + 10 q^{23} + 4 q^{25} + 6 q^{26} - 15 q^{28} + 15 q^{29} + 33 q^{31} + 29 q^{32} + 28 q^{34} - 4 q^{35} - 29 q^{37} + 15 q^{38} + 3 q^{40} + q^{41} - 25 q^{43} + 26 q^{44} - 4 q^{46} + 9 q^{47} + 14 q^{49} + 28 q^{50} - 13 q^{52} + 35 q^{53} + 14 q^{55} - 12 q^{56} - 23 q^{58} - 10 q^{59} + q^{61} + 43 q^{62} - 2 q^{64} + 24 q^{65} - 38 q^{67} + 2 q^{68} - 4 q^{70} + 10 q^{71} + 8 q^{73} + 25 q^{74} + 26 q^{76} - 3 q^{77} + 26 q^{79} + 48 q^{80} + 6 q^{82} + 30 q^{83} - 32 q^{85} + 50 q^{86} - 29 q^{88} - 4 q^{89} + 13 q^{91} + 32 q^{92} - 7 q^{94} + 32 q^{95} + 15 q^{97} + 5 q^{98}+O(q^{100}) $ 14 * q + 5 * q^2 + 15 * q^4 + 4 * q^5 - 14 * q^7 + 12 * q^8 + 4 * q^10 + 3 * q^11 - 13 * q^13 - 5 * q^14 + 13 * q^16 + 5 * q^17 + 21 * q^19 + 3 * q^20 - 3 * q^22 + 10 * q^23 + 4 * q^25 + 6 * q^26 - 15 * q^28 + 15 * q^29 + 33 * q^31 + 29 * q^32 + 28 * q^34 - 4 * q^35 - 29 * q^37 + 15 * q^38 + 3 * q^40 + q^41 - 25 * q^43 + 26 * q^44 - 4 * q^46 + 9 * q^47 + 14 * q^49 + 28 * q^50 - 13 * q^52 + 35 * q^53 + 14 * q^55 - 12 * q^56 - 23 * q^58 - 10 * q^59 + q^61 + 43 * q^62 - 2 * q^64 + 24 * q^65 - 38 * q^67 + 2 * q^68 - 4 * q^70 + 10 * q^71 + 8 * q^73 + 25 * q^74 + 26 * q^76 - 3 * q^77 + 26 * q^79 + 48 * q^80 + 6 * q^82 + 30 * q^83 - 32 * q^85 + 50 * q^86 - 29 * q^88 - 4 * q^89 + 13 * q^91 + 32 * q^92 - 7 * q^94 + 32 * q^95 + 15 * q^97 + 5 * 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 $

$2$	1.75826	1.24328	0.621640	−	0.783303i	$-0.286467\pi$
$2$	1.75826	1.24328	0.621640	+	0.783303i	$0.286467\pi$
$3$	0	0
$3$	0	0
$4$	1.09149	0.545745
$5$	−1.44353	−0.645565	−0.322782	−	0.946473i	$-0.604618\pi$
$5$	−1.44353	−0.645565	−0.322782	+	0.946473i	$0.604618\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.59740	−0.564766
$9$	0	0
$10$	−2.53810	−0.802618
$11$	−4.64133	−1.39941	−0.699706	−	0.714430i	$-0.746686\pi$
$11$	−4.64133	−1.39941	−0.699706	+	0.714430i	$0.746686\pi$
$12$	0	0
$13$	−3.75930	−1.04264	−0.521321	−	0.853361i	$-0.674560\pi$
$13$	−3.75930	−1.04264	−0.521321	+	0.853361i	$0.674560\pi$
$14$	−1.75826	−0.469916
$15$	0	0
$16$	−4.99163	−1.24791
$17$	−2.76361	−0.670275	−0.335137	−	0.942169i	$-0.608783\pi$
$17$	−2.76361	−0.670275	−0.335137	+	0.942169i	$0.608783\pi$
$18$	0	0
$19$	1.65863	0.380516	0.190258	−	0.981734i	$-0.439068\pi$
$19$	1.65863	0.380516	0.190258	+	0.981734i	$0.439068\pi$
$20$	−1.57560	−0.352314
$21$	0	0
$22$	−8.16068	−1.73986
$23$	6.34922	1.32390	0.661952	−	0.749546i	$-0.269728\pi$
$23$	6.34922	1.32390	0.661952	+	0.749546i	$0.269728\pi$
$24$	0	0
$25$	−2.91623	−0.583246
$26$	−6.60984	−1.29630
$27$	0	0
$28$	−1.09149	−0.206272
$29$	−0.687809	−0.127723	−0.0638614	−	0.997959i	$-0.520342\pi$
$29$	−0.687809	−0.127723	−0.0638614	+	0.997959i	$0.520342\pi$
$30$	0	0
$31$	−4.29992	−0.772289	−0.386144	−	0.922438i	$-0.626193\pi$
$31$	−4.29992	−0.772289	−0.386144	+	0.922438i	$0.626193\pi$
$32$	−5.58180	−0.986733
$33$	0	0
$34$	−4.85916	−0.833339
$35$	1.44353	0.244001
$36$	0	0
$37$	−1.53230	−0.251908	−0.125954	−	0.992036i	$-0.540199\pi$
$37$	−1.53230	−0.251908	−0.125954	+	0.992036i	$0.540199\pi$
$38$	2.91631	0.473088
$39$	0	0
$40$	2.30589	0.364593
$41$	9.88414	1.54364	0.771821	−	0.635839i	$-0.219346\pi$
$41$	9.88414	1.54364	0.771821	+	0.635839i	$0.219346\pi$
$42$	0	0
$43$	7.55579	1.15225	0.576123	−	0.817363i	$-0.304565\pi$
$43$	7.55579	1.15225	0.576123	+	0.817363i	$0.304565\pi$
$44$	−5.06597	−0.763723
$45$	0	0
$46$	11.1636	1.64598
$47$	0.153499	0.0223901	0.0111951	−	0.999937i	$-0.496436\pi$
$47$	0.153499	0.0223901	0.0111951	+	0.999937i	$0.496436\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−5.12750	−0.725138
$51$	0	0
$52$	−4.10324	−0.569017
$53$	4.56061	0.626449	0.313224	−	0.949679i	$-0.398591\pi$
$53$	4.56061	0.626449	0.313224	+	0.949679i	$0.398591\pi$
$54$	0	0
$55$	6.69988	0.903412
$56$	1.59740	0.213461
$57$	0	0
$58$	−1.20935	−0.158795
$59$	0.908070	0.118221	0.0591103	−	0.998251i	$-0.481174\pi$
$59$	0.908070	0.118221	0.0591103	+	0.998251i	$0.481174\pi$
$60$	0	0
$61$	8.22355	1.05292	0.526459	−	0.850200i	$-0.323519\pi$
$61$	8.22355	1.05292	0.526459	+	0.850200i	$0.323519\pi$
$62$	−7.56040	−0.960171
$63$	0	0
$64$	0.168978	0.0211222
$65$	5.42665	0.673093
$66$	0	0
$67$	−9.93107	−1.21327	−0.606636	−	0.794979i	$-0.707482\pi$
$67$	−9.93107	−1.21327	−0.606636	+	0.794979i	$0.707482\pi$
$68$	−3.01646	−0.365799
$69$	0	0
$70$	2.53810	0.303361
$71$	8.33861	0.989611	0.494805	−	0.869004i	$-0.335239\pi$
$71$	8.33861	0.989611	0.494805	+	0.869004i	$0.335239\pi$
$72$	0	0
$73$	8.16747	0.955930	0.477965	−	0.878379i	$-0.341374\pi$
$73$	8.16747	0.955930	0.477965	+	0.878379i	$0.341374\pi$
$74$	−2.69418	−0.313193
$75$	0	0
$76$	1.81038	0.207665
$77$	4.64133	0.528928
$78$	0	0
$79$	−1.51078	−0.169976	−0.0849879	−	0.996382i	$-0.527085\pi$
$79$	−1.51078	−0.169976	−0.0849879	+	0.996382i	$0.527085\pi$
$80$	7.20555	0.805605
$81$	0	0
$82$	17.3789	1.91918
$83$	8.23096	0.903465	0.451733	−	0.892153i	$-0.350806\pi$
$83$	8.23096	0.903465	0.451733	+	0.892153i	$0.350806\pi$
$84$	0	0
$85$	3.98935	0.432706
$86$	13.2851	1.43257
$87$	0	0
$88$	7.41405	0.790340
$89$	1.44262	0.152917	0.0764587	−	0.997073i	$-0.475639\pi$
$89$	1.44262	0.152917	0.0764587	+	0.997073i	$0.475639\pi$
$90$	0	0
$91$	3.75930	0.394081
$92$	6.93012	0.722515
$93$	0	0
$94$	0.269891	0.0278372
$95$	−2.39428	−0.245648
$96$	0	0
$97$	−9.30424	−0.944702	−0.472351	−	0.881410i	$-0.656595\pi$
$97$	−9.30424	−0.944702	−0.472351	+	0.881410i	$0.656595\pi$
$98$	1.75826	0.177611
$99$	0	0
$100$	−3.18304	−0.318304
$101$	−16.3568	−1.62756	−0.813782	−	0.581171i	$-0.802595\pi$
$101$	−16.3568	−1.62756	−0.813782	+	0.581171i	$0.802595\pi$
$102$	0	0
$103$	9.68596	0.954386	0.477193	−	0.878798i	$-0.341654\pi$
$103$	9.68596	0.954386	0.477193	+	0.878798i	$0.341654\pi$
$104$	6.00510	0.588848
$105$	0	0
$106$	8.01876	0.778851
$107$	−9.34552	−0.903465	−0.451733	−	0.892153i	$-0.649194\pi$
$107$	−9.34552	−0.903465	−0.451733	+	0.892153i	$0.649194\pi$
$108$	0	0
$109$	2.34843	0.224939	0.112470	−	0.993655i	$-0.464124\pi$
$109$	2.34843	0.224939	0.112470	+	0.993655i	$0.464124\pi$
$110$	11.7802	1.12319
$111$	0	0
$112$	4.99163	0.471665
$113$	10.4237	0.980576	0.490288	−	0.871560i	$-0.336892\pi$
$113$	10.4237	0.980576	0.490288	+	0.871560i	$0.336892\pi$
$114$	0	0
$115$	−9.16527	−0.854666
$116$	−0.750737	−0.0697042
$117$	0	0
$118$	1.59663	0.146981
$119$	2.76361	0.253340
$120$	0	0
$121$	10.5419	0.958356
$122$	14.4592	1.30907
$123$	0	0
$124$	−4.69333	−0.421473
$125$	11.4273	1.02209
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	11.4607	1.01299
$129$	0	0
$130$	9.54148	0.836843
$131$	−12.5769	−1.09885	−0.549423	−	0.835545i	$-0.685152\pi$
$131$	−12.5769	−1.09885	−0.549423	+	0.835545i	$0.685152\pi$
$132$	0	0
$133$	−1.65863	−0.143822
$134$	−17.4614	−1.50844
$135$	0	0
$136$	4.41459	0.378548
$137$	9.20221	0.786198	0.393099	−	0.919496i	$-0.371403\pi$
$137$	9.20221	0.786198	0.393099	+	0.919496i	$0.371403\pi$
$138$	0	0
$139$	7.27644	0.617180	0.308590	−	0.951195i	$-0.400143\pi$
$139$	7.27644	0.617180	0.308590	+	0.951195i	$0.400143\pi$
$140$	1.57560	0.133162
$141$	0	0
$142$	14.6615	1.23036
$143$	17.4481	1.45909
$144$	0	0
$145$	0.992871	0.0824534
$146$	14.3606	1.18849
$147$	0	0
$148$	−1.67249	−0.137478
$149$	−12.6981	−1.04027	−0.520133	−	0.854085i	$-0.674118\pi$
$149$	−12.6981	−1.04027	−0.520133	+	0.854085i	$0.674118\pi$
$150$	0	0
$151$	−8.54298	−0.695218	−0.347609	−	0.937640i	$-0.613006\pi$
$151$	−8.54298	−0.695218	−0.347609	+	0.937640i	$0.613006\pi$
$152$	−2.64950	−0.214902
$153$	0	0
$154$	8.16068	0.657606
$155$	6.20705	0.498563
$156$	0	0
$157$	−17.7988	−1.42050	−0.710248	−	0.703951i	$-0.751417\pi$
$157$	−17.7988	−1.42050	−0.710248	+	0.703951i	$0.751417\pi$
$158$	−2.65635	−0.211328
$159$	0	0
$160$	8.05748	0.637000
$161$	−6.34922	−0.500389
$162$	0	0
$163$	−15.2011	−1.19064	−0.595320	−	0.803488i	$-0.702975\pi$
$163$	−15.2011	−1.19064	−0.595320	+	0.803488i	$0.702975\pi$
$164$	10.7884	0.842436
$165$	0	0
$166$	14.4722	1.12326
$167$	6.04443	0.467732	0.233866	−	0.972269i	$-0.424862\pi$
$167$	6.04443	0.467732	0.233866	+	0.972269i	$0.424862\pi$
$168$	0	0
$169$	1.13232	0.0871012
$170$	7.01433	0.537975
$171$	0	0
$172$	8.24707	0.628833
$173$	9.89841	0.752562	0.376281	−	0.926506i	$-0.377203\pi$
$173$	9.89841	0.752562	0.376281	+	0.926506i	$0.377203\pi$
$174$	0	0
$175$	2.91623	0.220446
$176$	23.1678	1.74634
$177$	0	0
$178$	2.53651	0.190119
$179$	11.9856	0.895847	0.447924	−	0.894072i	$-0.352164\pi$
$179$	11.9856	0.895847	0.447924	+	0.894072i	$0.352164\pi$
$180$	0	0
$181$	−0.173431	−0.0128911	−0.00644553	−	0.999979i	$-0.502052\pi$
$181$	−0.173431	−0.0128911	−0.00644553	+	0.999979i	$0.502052\pi$
$182$	6.60984	0.489954
$183$	0	0
$184$	−10.1422	−0.747696
$185$	2.21191	0.162623
$186$	0	0
$187$	12.8268	0.937991
$188$	0.167543	0.0122193
$189$	0	0
$190$	−4.20977	−0.305409
$191$	−12.5727	−0.909728	−0.454864	−	0.890561i	$-0.650312\pi$
$191$	−12.5727	−0.909728	−0.454864	+	0.890561i	$0.650312\pi$
$192$	0	0
$193$	−21.0301	−1.51378	−0.756891	−	0.653541i	$-0.773283\pi$
$193$	−21.0301	−1.51378	−0.756891	+	0.653541i	$0.773283\pi$
$194$	−16.3593	−1.17453
$195$	0	0
$196$	1.09149	0.0779636
$197$	1.75543	0.125069	0.0625345	−	0.998043i	$-0.480082\pi$
$197$	1.75543	0.125069	0.0625345	+	0.998043i	$0.480082\pi$
$198$	0	0
$199$	6.90888	0.489758	0.244879	−	0.969554i	$-0.421252\pi$
$199$	6.90888	0.489758	0.244879	+	0.969554i	$0.421252\pi$
$200$	4.65838	0.329397
$201$	0	0
$202$	−28.7596	−2.02352
$203$	0.687809	0.0482747
$204$	0	0
$205$	−14.2680	−0.996522
$206$	17.0305	1.18657
$207$	0	0
$208$	18.7650	1.30112
$209$	−7.69825	−0.532499
$210$	0	0
$211$	−1.09079	−0.0750929	−0.0375464	−	0.999295i	$-0.511954\pi$
$211$	−1.09079	−0.0750929	−0.0375464	+	0.999295i	$0.511954\pi$
$212$	4.97787	0.341881
$213$	0	0
$214$	−16.4319	−1.12326
$215$	−10.9070	−0.743850
$216$	0	0
$217$	4.29992	0.291898
$218$	4.12917	0.279662
$219$	0	0
$220$	7.31286	0.493033
$221$	10.3892	0.698856
$222$	0	0
$223$	8.05233	0.539224	0.269612	−	0.962969i	$-0.413105\pi$
$223$	8.05233	0.539224	0.269612	+	0.962969i	$0.413105\pi$
$224$	5.58180	0.372950
$225$	0	0
$226$	18.3276	1.21913
$227$	−13.5976	−0.902505	−0.451252	−	0.892396i	$-0.649023\pi$
$227$	−13.5976	−0.902505	−0.451252	+	0.892396i	$0.649023\pi$
$228$	0	0
$229$	17.9502	1.18618	0.593090	−	0.805136i	$-0.297908\pi$
$229$	17.9502	1.18618	0.593090	+	0.805136i	$0.297908\pi$
$230$	−16.1150	−1.06259
$231$	0	0
$232$	1.09870	0.0721335
$233$	22.2276	1.45618	0.728089	−	0.685482i	$-0.240409\pi$
$233$	22.2276	1.45618	0.728089	+	0.685482i	$0.240409\pi$
$234$	0	0
$235$	−0.221580	−0.0144543
$236$	0.991150	0.0645184
$237$	0	0
$238$	4.85916	0.314973
$239$	24.2517	1.56871	0.784356	−	0.620311i	$-0.212994\pi$
$239$	24.2517	1.56871	0.784356	+	0.620311i	$0.212994\pi$
$240$	0	0
$241$	−6.72715	−0.433334	−0.216667	−	0.976246i	$-0.569519\pi$
$241$	−6.72715	−0.433334	−0.216667	+	0.976246i	$0.569519\pi$
$242$	18.5355	1.19151
$243$	0	0
$244$	8.97593	0.574625
$245$	−1.44353	−0.0922236
$246$	0	0
$247$	−6.23529	−0.396742
$248$	6.86869	0.436162
$249$	0	0
$250$	20.0922	1.27074
$251$	6.70087	0.422955	0.211478	−	0.977383i	$-0.432172\pi$
$251$	6.70087	0.422955	0.211478	+	0.977383i	$0.432172\pi$
$252$	0	0
$253$	−29.4688	−1.85269
$254$	−1.75826	−0.110323
$255$	0	0
$256$	19.8130	1.23831
$257$	30.6020	1.90890	0.954451	−	0.298368i	$-0.0964422\pi$
$257$	30.6020	1.90890	0.954451	+	0.298368i	$0.0964422\pi$
$258$	0	0
$259$	1.53230	0.0952124
$260$	5.92314	0.367337
$261$	0	0
$262$	−22.1134	−1.36617
$263$	−11.1210	−0.685749	−0.342875	−	0.939381i	$-0.611401\pi$
$263$	−11.1210	−0.685749	−0.342875	+	0.939381i	$0.611401\pi$
$264$	0	0
$265$	−6.58337	−0.404413
$266$	−2.91631	−0.178811
$267$	0	0
$268$	−10.8397	−0.662138
$269$	−7.32836	−0.446818	−0.223409	−	0.974725i	$-0.571719\pi$
$269$	−7.32836	−0.446818	−0.223409	+	0.974725i	$0.571719\pi$
$270$	0	0
$271$	−17.1720	−1.04312	−0.521562	−	0.853213i	$-0.674651\pi$
$271$	−17.1720	−1.04312	−0.521562	+	0.853213i	$0.674651\pi$
$272$	13.7949	0.836441
$273$	0	0
$274$	16.1799	0.977464
$275$	13.5352	0.816202
$276$	0	0
$277$	3.22679	0.193879	0.0969394	−	0.995290i	$-0.469095\pi$
$277$	3.22679	0.193879	0.0969394	+	0.995290i	$0.469095\pi$
$278$	12.7939	0.767327
$279$	0	0
$280$	−2.30589	−0.137803
$281$	31.1859	1.86039	0.930196	−	0.367063i	$-0.119637\pi$
$281$	31.1859	1.86039	0.930196	+	0.367063i	$0.119637\pi$
$282$	0	0
$283$	−13.3204	−0.791815	−0.395908	−	0.918290i	$-0.629570\pi$
$283$	−13.3204	−0.791815	−0.395908	+	0.918290i	$0.629570\pi$
$284$	9.10152	0.540076
$285$	0	0
$286$	30.6784	1.81405
$287$	−9.88414	−0.583442
$288$	0	0
$289$	−9.36244	−0.550732
$290$	1.74573	0.102513
$291$	0	0
$292$	8.91472	0.521695
$293$	33.5989	1.96287	0.981435	−	0.191795i	$-0.0614310\pi$
$293$	33.5989	1.96287	0.981435	+	0.191795i	$0.0614310\pi$
$294$	0	0
$295$	−1.31082	−0.0763191
$296$	2.44769	0.142269
$297$	0	0
$298$	−22.3266	−1.29334
$299$	−23.8686	−1.38036
$300$	0	0
$301$	−7.55579	−0.435508
$302$	−15.0208	−0.864351
$303$	0	0
$304$	−8.27927	−0.474849
$305$	−11.8709	−0.679727
$306$	0	0
$307$	15.6588	0.893694	0.446847	−	0.894610i	$-0.352547\pi$
$307$	15.6588	0.893694	0.446847	+	0.894610i	$0.352547\pi$
$308$	5.06597	0.288660
$309$	0	0
$310$	10.9136	0.619853
$311$	30.3734	1.72232	0.861160	−	0.508334i	$-0.169738\pi$
$311$	30.3734	1.72232	0.861160	+	0.508334i	$0.169738\pi$
$312$	0	0
$313$	30.6668	1.73339	0.866695	−	0.498839i	$-0.166240\pi$
$313$	30.6668	1.73339	0.866695	+	0.498839i	$0.166240\pi$
$314$	−31.2949	−1.76608
$315$	0	0
$316$	−1.64900	−0.0927636
$317$	9.14411	0.513585	0.256792	−	0.966467i	$-0.417334\pi$
$317$	9.14411	0.513585	0.256792	+	0.966467i	$0.417334\pi$
$318$	0	0
$319$	3.19235	0.178737
$320$	−0.243924	−0.0136358
$321$	0	0
$322$	−11.1636	−0.622123
$323$	−4.58382	−0.255050
$324$	0	0
$325$	10.9630	0.608116
$326$	−26.7275	−1.48030
$327$	0	0
$328$	−15.7889	−0.871797
$329$	−0.153499	−0.00846267
$330$	0	0
$331$	13.2057	0.725853	0.362926	−	0.931818i	$-0.381778\pi$
$331$	13.2057	0.725853	0.362926	+	0.931818i	$0.381778\pi$
$332$	8.98402	0.493062
$333$	0	0
$334$	10.6277	0.581522
$335$	14.3358	0.783246
$336$	0	0
$337$	−14.5053	−0.790157	−0.395078	−	0.918647i	$-0.629283\pi$
$337$	−14.5053	−0.790157	−0.395078	+	0.918647i	$0.629283\pi$
$338$	1.99091	0.108291
$339$	0	0
$340$	4.35434	0.236147
$341$	19.9573	1.08075
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−12.0696	−0.650749
$345$	0	0
$346$	17.4040	0.935645
$347$	−25.3965	−1.36336	−0.681678	−	0.731652i	$-0.738750\pi$
$347$	−25.3965	−1.36336	−0.681678	+	0.731652i	$0.738750\pi$
$348$	0	0
$349$	−5.78934	−0.309896	−0.154948	−	0.987923i	$-0.549521\pi$
$349$	−5.78934	−0.309896	−0.154948	+	0.987923i	$0.549521\pi$
$350$	5.12750	0.274076
$351$	0	0
$352$	25.9070	1.38085
$353$	−0.0262181	−0.00139545	−0.000697724	−	1.00000i	$-0.500222\pi$
$353$	−0.0262181	−0.00139545	−0.000697724		1.00000i	$0.500222\pi$
$354$	0	0
$355$	−12.0370	−0.638858
$356$	1.57461	0.0834540
$357$	0	0
$358$	21.0739	1.11379
$359$	−3.71767	−0.196211	−0.0981055	−	0.995176i	$-0.531278\pi$
$359$	−3.71767	−0.196211	−0.0981055	+	0.995176i	$0.531278\pi$
$360$	0	0
$361$	−16.2489	−0.855207
$362$	−0.304938	−0.0160272
$363$	0	0
$364$	4.10324	0.215068
$365$	−11.7900	−0.617115
$366$	0	0
$367$	−23.1949	−1.21076	−0.605382	−	0.795935i	$-0.706980\pi$
$367$	−23.1949	−1.21076	−0.605382	+	0.795935i	$0.706980\pi$
$368$	−31.6930	−1.65211
$369$	0	0
$370$	3.88913	0.202186
$371$	−4.56061	−0.236775
$372$	0	0
$373$	−10.6728	−0.552615	−0.276307	−	0.961069i	$-0.589111\pi$
$373$	−10.6728	−0.552615	−0.276307	+	0.961069i	$0.589111\pi$
$374$	22.5530	1.16619
$375$	0	0
$376$	−0.245199	−0.0126452
$377$	2.58568	0.133169
$378$	0	0
$379$	−20.3223	−1.04389	−0.521943	−	0.852981i	$-0.674792\pi$
$379$	−20.3223	−1.04389	−0.521943	+	0.852981i	$0.674792\pi$
$380$	−2.61333	−0.134061
$381$	0	0
$382$	−22.1061	−1.13105
$383$	−35.0560	−1.79128	−0.895639	−	0.444782i	$-0.853281\pi$
$383$	−35.0560	−1.79128	−0.895639	+	0.444782i	$0.853281\pi$
$384$	0	0
$385$	−6.69988	−0.341458
$386$	−36.9765	−1.88206
$387$	0	0
$388$	−10.1555	−0.515567
$389$	12.2210	0.619632	0.309816	−	0.950797i	$-0.399733\pi$
$389$	12.2210	0.619632	0.309816	+	0.950797i	$0.399733\pi$
$390$	0	0
$391$	−17.5468	−0.887380
$392$	−1.59740	−0.0806808
$393$	0	0
$394$	3.08650	0.155496
$395$	2.18085	0.109730
$396$	0	0
$397$	−24.4071	−1.22496	−0.612478	−	0.790487i	$-0.709827\pi$
$397$	−24.4071	−1.22496	−0.612478	+	0.790487i	$0.709827\pi$
$398$	12.1476	0.608906
$399$	0	0
$400$	14.5567	0.727837
$401$	−34.9516	−1.74540	−0.872700	−	0.488257i	$-0.837633\pi$
$401$	−34.9516	−1.74540	−0.872700	+	0.488257i	$0.837633\pi$
$402$	0	0
$403$	16.1647	0.805220
$404$	−17.8533	−0.888235
$405$	0	0
$406$	1.20935	0.0600190
$407$	7.11190	0.352524
$408$	0	0
$409$	22.5274	1.11391	0.556954	−	0.830543i	$-0.311970\pi$
$409$	22.5274	1.11391	0.556954	+	0.830543i	$0.311970\pi$
$410$	−25.0869	−1.23896
$411$	0	0
$412$	10.5721	0.520852
$413$	−0.908070	−0.0446832
$414$	0	0
$415$	−11.8816	−0.583245
$416$	20.9837	1.02881
$417$	0	0
$418$	−13.5356	−0.662046
$419$	−14.9294	−0.729349	−0.364675	−	0.931135i	$-0.618820\pi$
$419$	−14.9294	−0.729349	−0.364675	+	0.931135i	$0.618820\pi$
$420$	0	0
$421$	−3.08642	−0.150423	−0.0752115	−	0.997168i	$-0.523963\pi$
$421$	−3.08642	−0.150423	−0.0752115	+	0.997168i	$0.523963\pi$
$422$	−1.91789	−0.0933615
$423$	0	0
$424$	−7.28512	−0.353797
$425$	8.05933	0.390935
$426$	0	0
$427$	−8.22355	−0.397966
$428$	−10.2005	−0.493062
$429$	0	0
$430$	−19.1774	−0.924814
$431$	22.5358	1.08551	0.542755	−	0.839891i	$-0.317381\pi$
$431$	22.5358	1.08551	0.542755	+	0.839891i	$0.317381\pi$
$432$	0	0
$433$	−6.48812	−0.311799	−0.155900	−	0.987773i	$-0.549828\pi$
$433$	−6.48812	−0.311799	−0.155900	+	0.987773i	$0.549828\pi$
$434$	7.56040	0.362911
$435$	0	0
$436$	2.56329	0.122760
$437$	10.5310	0.503767
$438$	0	0
$439$	4.33971	0.207123	0.103561	−	0.994623i	$-0.466976\pi$
$439$	4.33971	0.207123	0.103561	+	0.994623i	$0.466976\pi$
$440$	−10.7024	−0.510216
$441$	0	0
$442$	18.2670	0.868874
$443$	−26.1111	−1.24058	−0.620288	−	0.784374i	$-0.712984\pi$
$443$	−26.1111	−1.24058	−0.620288	+	0.784374i	$0.712984\pi$
$444$	0	0
$445$	−2.08246	−0.0987182
$446$	14.1581	0.670406
$447$	0	0
$448$	−0.168978	−0.00798345
$449$	26.7556	1.26268	0.631338	−	0.775508i	$-0.282506\pi$
$449$	26.7556	1.26268	0.631338	+	0.775508i	$0.282506\pi$
$450$	0	0
$451$	−45.8755	−2.16019
$452$	11.3773	0.535145
$453$	0	0
$454$	−23.9082	−1.12207
$455$	−5.42665	−0.254405
$456$	0	0
$457$	5.82031	0.272263	0.136131	−	0.990691i	$-0.456533\pi$
$457$	5.82031	0.272263	0.136131	+	0.990691i	$0.456533\pi$
$458$	31.5611	1.47475
$459$	0	0
$460$	−10.0038	−0.466430
$461$	21.3340	0.993625	0.496813	−	0.867858i	$-0.334504\pi$
$461$	21.3340	0.993625	0.496813	+	0.867858i	$0.334504\pi$
$462$	0	0
$463$	19.1189	0.888530	0.444265	−	0.895896i	$-0.353465\pi$
$463$	19.1189	0.888530	0.444265	+	0.895896i	$0.353465\pi$
$464$	3.43329	0.159386
$465$	0	0
$466$	39.0820	1.81044
$467$	23.1736	1.07235	0.536173	−	0.844108i	$-0.319869\pi$
$467$	23.1736	1.07235	0.536173	+	0.844108i	$0.319869\pi$
$468$	0	0
$469$	9.93107	0.458574
$470$	−0.389596	−0.0179707
$471$	0	0
$472$	−1.45055	−0.0667670
$473$	−35.0689	−1.61247
$474$	0	0
$475$	−4.83695	−0.221934
$476$	3.01646	0.138259
$477$	0	0
$478$	42.6409	1.95035
$479$	−5.24272	−0.239546	−0.119773	−	0.992801i	$-0.538217\pi$
$479$	−5.24272	−0.239546	−0.119773	+	0.992801i	$0.538217\pi$
$480$	0	0
$481$	5.76037	0.262650
$482$	−11.8281	−0.538755
$483$	0	0
$484$	11.5064	0.523019
$485$	13.4309	0.609867
$486$	0	0
$487$	27.3293	1.23841	0.619204	−	0.785230i	$-0.287455\pi$
$487$	27.3293	1.23841	0.619204	+	0.785230i	$0.287455\pi$
$488$	−13.1363	−0.594652
$489$	0	0
$490$	−2.53810	−0.114660
$491$	15.4480	0.697159	0.348579	−	0.937279i	$-0.386664\pi$
$491$	15.4480	0.697159	0.348579	+	0.937279i	$0.386664\pi$
$492$	0	0
$493$	1.90084	0.0856094
$494$	−10.9633	−0.493261
$495$	0	0
$496$	21.4636	0.963745
$497$	−8.33861	−0.374038
$498$	0	0
$499$	26.8940	1.20394	0.601970	−	0.798519i	$-0.294383\pi$
$499$	26.8940	1.20394	0.601970	+	0.798519i	$0.294383\pi$
$500$	12.4728	0.557800
$501$	0	0
$502$	11.7819	0.525852
$503$	19.1189	0.852469	0.426234	−	0.904613i	$-0.359840\pi$
$503$	19.1189	0.852469	0.426234	+	0.904613i	$0.359840\pi$
$504$	0	0
$505$	23.6115	1.05070
$506$	−51.8139	−2.30341
$507$	0	0
$508$	−1.09149	−0.0484271
$509$	−7.64323	−0.338780	−0.169390	−	0.985549i	$-0.554180\pi$
$509$	−7.64323	−0.338780	−0.169390	+	0.985549i	$0.554180\pi$
$510$	0	0
$511$	−8.16747	−0.361308
$512$	11.9150	0.526576
$513$	0	0
$514$	53.8064	2.37330
$515$	−13.9820	−0.616118
$516$	0	0
$517$	−0.712438	−0.0313330
$518$	2.69418	0.118376
$519$	0	0
$520$	−8.66852	−0.380140
$521$	10.4887	0.459520	0.229760	−	0.973247i	$-0.426206\pi$
$521$	10.4887	0.459520	0.229760	+	0.973247i	$0.426206\pi$
$522$	0	0
$523$	30.7504	1.34462	0.672311	−	0.740269i	$-0.265302\pi$
$523$	30.7504	1.34462	0.672311	+	0.740269i	$0.265302\pi$
$524$	−13.7275	−0.599690
$525$	0	0
$526$	−19.5536	−0.852579
$527$	11.8833	0.517646
$528$	0	0
$529$	17.3126	0.752723
$530$	−11.5753	−0.502799
$531$	0	0
$532$	−1.81038	−0.0784900
$533$	−37.1574	−1.60947
$534$	0	0
$535$	13.4905	0.583245
$536$	15.8639	0.685215
$537$	0	0
$538$	−12.8852	−0.555520
$539$	−4.64133	−0.199916
$540$	0	0
$541$	−33.7275	−1.45006	−0.725030	−	0.688717i	$-0.758174\pi$
$541$	−33.7275	−1.45006	−0.725030	+	0.688717i	$0.758174\pi$
$542$	−30.1929	−1.29690
$543$	0	0
$544$	15.4259	0.661382
$545$	−3.39003	−0.145213
$546$	0	0
$547$	−21.2477	−0.908488	−0.454244	−	0.890877i	$-0.650091\pi$
$547$	−21.2477	−0.908488	−0.454244	+	0.890877i	$0.650091\pi$
$548$	10.0441	0.429064
$549$	0	0
$550$	23.7984	1.01477
$551$	−1.14082	−0.0486006
$552$	0	0
$553$	1.51078	0.0642448
$554$	5.67354	0.241046
$555$	0	0
$556$	7.94217	0.336823
$557$	8.44884	0.357989	0.178994	−	0.983850i	$-0.442716\pi$
$557$	8.44884	0.357989	0.178994	+	0.983850i	$0.442716\pi$
$558$	0	0
$559$	−28.4045	−1.20138
$560$	−7.20555	−0.304490
$561$	0	0
$562$	54.8330	2.31299
$563$	−22.9926	−0.969022	−0.484511	−	0.874785i	$-0.661002\pi$
$563$	−22.9926	−0.969022	−0.484511	+	0.874785i	$0.661002\pi$
$564$	0	0
$565$	−15.0468	−0.633026
$566$	−23.4208	−0.984448
$567$	0	0
$568$	−13.3201	−0.558898
$569$	−37.8125	−1.58518	−0.792591	−	0.609753i	$-0.791269\pi$
$569$	−37.8125	−1.58518	−0.792591	+	0.609753i	$0.791269\pi$
$570$	0	0
$571$	18.8016	0.786822	0.393411	−	0.919363i	$-0.371295\pi$
$571$	18.8016	0.786822	0.393411	+	0.919363i	$0.371295\pi$
$572$	19.0445	0.796289
$573$	0	0
$574$	−17.3789	−0.725382
$575$	−18.5158	−0.772162
$576$	0	0
$577$	8.49450	0.353630	0.176815	−	0.984244i	$-0.443420\pi$
$577$	8.49450	0.353630	0.176815	+	0.984244i	$0.443420\pi$
$578$	−16.4616	−0.684714
$579$	0	0
$580$	1.08371	0.0449986
$581$	−8.23096	−0.341478
$582$	0	0
$583$	−21.1673	−0.876660
$584$	−13.0467	−0.539877
$585$	0	0
$586$	59.0758	2.44040
$587$	14.4771	0.597534	0.298767	−	0.954326i	$-0.403425\pi$
$587$	14.4771	0.597534	0.298767	+	0.954326i	$0.403425\pi$
$588$	0	0
$589$	−7.13199	−0.293868
$590$	−2.30477	−0.0948860
$591$	0	0
$592$	7.64867	0.314358
$593$	6.07736	0.249567	0.124784	−	0.992184i	$-0.460176\pi$
$593$	6.07736	0.249567	0.124784	+	0.992184i	$0.460176\pi$
$594$	0	0
$595$	−3.98935	−0.163547
$596$	−13.8598	−0.567721
$597$	0	0
$598$	−41.9673	−1.71617
$599$	9.16352	0.374411	0.187206	−	0.982321i	$-0.440057\pi$
$599$	9.16352	0.374411	0.187206	+	0.982321i	$0.440057\pi$
$600$	0	0
$601$	40.1533	1.63789	0.818945	−	0.573873i	$-0.194560\pi$
$601$	40.1533	1.63789	0.818945	+	0.573873i	$0.194560\pi$
$602$	−13.2851	−0.541459
$603$	0	0
$604$	−9.32459	−0.379412
$605$	−15.2175	−0.618681
$606$	0	0
$607$	−4.16187	−0.168925	−0.0844626	−	0.996427i	$-0.526917\pi$
$607$	−4.16187	−0.168925	−0.0844626	+	0.996427i	$0.526917\pi$
$608$	−9.25815	−0.375468
$609$	0	0
$610$	−20.8722	−0.845091
$611$	−0.577048	−0.0233449
$612$	0	0
$613$	32.3289	1.30575	0.652876	−	0.757465i	$-0.273562\pi$
$613$	32.3289	1.30575	0.652876	+	0.757465i	$0.273562\pi$
$614$	27.5323	1.11111
$615$	0	0
$616$	−7.41405	−0.298721
$617$	7.88446	0.317416	0.158708	−	0.987326i	$-0.449267\pi$
$617$	7.88446	0.317416	0.158708	+	0.987326i	$0.449267\pi$
$618$	0	0
$619$	25.4735	1.02386	0.511932	−	0.859026i	$-0.328930\pi$
$619$	25.4735	1.02386	0.511932	+	0.859026i	$0.328930\pi$
$620$	6.77494	0.272088
$621$	0	0
$622$	53.4045	2.14133
$623$	−1.44262	−0.0577974
$624$	0	0
$625$	−1.91446	−0.0765783
$626$	53.9203	2.15509
$627$	0	0
$628$	−19.4272	−0.775229
$629$	4.23468	0.168848
$630$	0	0
$631$	−14.9165	−0.593818	−0.296909	−	0.954906i	$-0.595956\pi$
$631$	−14.9165	−0.593818	−0.296909	+	0.954906i	$0.595956\pi$
$632$	2.41332	0.0959965
$633$	0	0
$634$	16.0778	0.638529
$635$	1.44353	0.0572846
$636$	0	0
$637$	−3.75930	−0.148949
$638$	5.61298	0.222220
$639$	0	0
$640$	−16.5439	−0.653953
$641$	−26.0895	−1.03047	−0.515237	−	0.857048i	$-0.672296\pi$
$641$	−26.0895	−1.03047	−0.515237	+	0.857048i	$0.672296\pi$
$642$	0	0
$643$	30.1981	1.19090	0.595449	−	0.803393i	$-0.296974\pi$
$643$	30.1981	1.19090	0.595449	+	0.803393i	$0.296974\pi$
$644$	−6.93012	−0.273085
$645$	0	0
$646$	−8.05956	−0.317099
$647$	38.0440	1.49566	0.747832	−	0.663888i	$-0.231095\pi$
$647$	38.0440	1.49566	0.747832	+	0.663888i	$0.231095\pi$
$648$	0	0
$649$	−4.21465	−0.165440
$650$	19.2758	0.756059
$651$	0	0
$652$	−16.5918	−0.649787
$653$	−11.9928	−0.469313	−0.234656	−	0.972078i	$-0.575396\pi$
$653$	−11.9928	−0.469313	−0.234656	+	0.972078i	$0.575396\pi$
$654$	0	0
$655$	18.1550	0.709376
$656$	−49.3380	−1.92632
$657$	0	0
$658$	−0.269891	−0.0105215
$659$	−9.87400	−0.384637	−0.192318	−	0.981333i	$-0.561601\pi$
$659$	−9.87400	−0.384637	−0.192318	+	0.981333i	$0.561601\pi$
$660$	0	0
$661$	3.73761	0.145376	0.0726880	−	0.997355i	$-0.476842\pi$
$661$	3.73761	0.145376	0.0726880	+	0.997355i	$0.476842\pi$
$662$	23.2192	0.902438
$663$	0	0
$664$	−13.1481	−0.510246
$665$	2.39428	0.0928462
$666$	0	0
$667$	−4.36705	−0.169093
$668$	6.59744	0.255263
$669$	0	0
$670$	25.2061	0.973795
$671$	−38.1682	−1.47347
$672$	0	0
$673$	−13.2086	−0.509153	−0.254577	−	0.967053i	$-0.581936\pi$
$673$	−13.2086	−0.509153	−0.254577	+	0.967053i	$0.581936\pi$
$674$	−25.5042	−0.982386
$675$	0	0
$676$	1.23591	0.0475351
$677$	26.0490	1.00114	0.500572	−	0.865695i	$-0.333123\pi$
$677$	26.0490	1.00114	0.500572	+	0.865695i	$0.333123\pi$
$678$	0	0
$679$	9.30424	0.357064
$680$	−6.37258	−0.244377
$681$	0	0
$682$	35.0903	1.34368
$683$	−12.5906	−0.481766	−0.240883	−	0.970554i	$-0.577437\pi$
$683$	−12.5906	−0.481766	−0.240883	+	0.970554i	$0.577437\pi$
$684$	0	0
$685$	−13.2836	−0.507542
$686$	−1.75826	−0.0671308
$687$	0	0
$688$	−37.7157	−1.43790
$689$	−17.1447	−0.653161
$690$	0	0
$691$	38.4906	1.46425	0.732126	−	0.681169i	$-0.238528\pi$
$691$	38.4906	1.46425	0.732126	+	0.681169i	$0.238528\pi$
$692$	10.8040	0.410707
$693$	0	0
$694$	−44.6538	−1.69503
$695$	−10.5037	−0.398430
$696$	0	0
$697$	−27.3159	−1.03466
$698$	−10.1792	−0.385288
$699$	0	0
$700$	3.18304	0.120308
$701$	46.3016	1.74879	0.874393	−	0.485218i	$-0.161260\pi$
$701$	46.3016	1.74879	0.874393	+	0.485218i	$0.161260\pi$
$702$	0	0
$703$	−2.54152	−0.0958552
$704$	−0.784281	−0.0295587
$705$	0	0
$706$	−0.0460983	−0.00173493
$707$	16.3568	0.615161
$708$	0	0
$709$	−21.1800	−0.795433	−0.397717	−	0.917508i	$-0.630197\pi$
$709$	−21.1800	−0.795433	−0.397717	+	0.917508i	$0.630197\pi$
$710$	−21.1642	−0.794280
$711$	0	0
$712$	−2.30444	−0.0863625
$713$	−27.3012	−1.02244
$714$	0	0
$715$	−25.1868	−0.941935
$716$	13.0822	0.488905
$717$	0	0
$718$	−6.53664	−0.243945
$719$	13.2651	0.494703	0.247352	−	0.968926i	$-0.420440\pi$
$719$	13.2651	0.494703	0.247352	+	0.968926i	$0.420440\pi$
$720$	0	0
$721$	−9.68596	−0.360724
$722$	−28.5699	−1.06326
$723$	0	0
$724$	−0.189299	−0.00703523
$725$	2.00581	0.0744938
$726$	0	0
$727$	12.2540	0.454474	0.227237	−	0.973839i	$-0.427031\pi$
$727$	12.2540	0.454474	0.227237	+	0.973839i	$0.427031\pi$
$728$	−6.00510	−0.222564
$729$	0	0
$730$	−20.7299	−0.767247
$731$	−20.8813	−0.772322
$732$	0	0
$733$	−9.16742	−0.338606	−0.169303	−	0.985564i	$-0.554152\pi$
$733$	−9.16742	−0.338606	−0.169303	+	0.985564i	$0.554152\pi$
$734$	−40.7827	−1.50532
$735$	0	0
$736$	−35.4401	−1.30634
$737$	46.0933	1.69787
$738$	0	0
$739$	50.7842	1.86813	0.934063	−	0.357108i	$-0.116237\pi$
$739$	50.7842	1.86813	0.934063	+	0.357108i	$0.116237\pi$
$740$	2.41428	0.0887509
$741$	0	0
$742$	−8.01876	−0.294378
$743$	50.0876	1.83753	0.918767	−	0.394800i	$-0.129186\pi$
$743$	50.0876	1.83753	0.918767	+	0.394800i	$0.129186\pi$
$744$	0	0
$745$	18.3300	0.671560
$746$	−18.7655	−0.687055
$747$	0	0
$748$	14.0004	0.511904
$749$	9.34552	0.341478
$750$	0	0
$751$	12.1849	0.444634	0.222317	−	0.974974i	$-0.428638\pi$
$751$	12.1849	0.444634	0.222317	+	0.974974i	$0.428638\pi$
$752$	−0.766209	−0.0279408
$753$	0	0
$754$	4.54630	0.165567
$755$	12.3320	0.448808
$756$	0	0
$757$	−0.977306	−0.0355208	−0.0177604	−	0.999842i	$-0.505654\pi$
$757$	−0.977306	−0.0355208	−0.0177604	+	0.999842i	$0.505654\pi$
$758$	−35.7319	−1.29784
$759$	0	0
$760$	3.82462	0.138733
$761$	48.0319	1.74115	0.870577	−	0.492032i	$-0.163746\pi$
$761$	48.0319	1.74115	0.870577	+	0.492032i	$0.163746\pi$
$762$	0	0
$763$	−2.34843	−0.0850190
$764$	−13.7230	−0.496480
$765$	0	0
$766$	−61.6377	−2.22706
$767$	−3.41371	−0.123262
$768$	0	0
$769$	10.4854	0.378111	0.189056	−	0.981966i	$-0.439457\pi$
$769$	10.4854	0.378111	0.189056	+	0.981966i	$0.439457\pi$
$770$	−11.7802	−0.424527
$771$	0	0
$772$	−22.9542	−0.826140
$773$	14.5077	0.521806	0.260903	−	0.965365i	$-0.415980\pi$
$773$	14.5077	0.521806	0.260903	+	0.965365i	$0.415980\pi$
$774$	0	0
$775$	12.5396	0.450434
$776$	14.8626	0.533535
$777$	0	0
$778$	21.4878	0.770376
$779$	16.3941	0.587381
$780$	0	0
$781$	−38.7022	−1.38487
$782$	−30.8519	−1.10326
$783$	0	0
$784$	−4.99163	−0.178272
$785$	25.6930	0.917023
$786$	0	0
$787$	41.3835	1.47516	0.737582	−	0.675258i	$-0.235968\pi$
$787$	41.3835	1.47516	0.737582	+	0.675258i	$0.235968\pi$
$788$	1.91603	0.0682559
$789$	0	0
$790$	3.83451	0.136426
$791$	−10.4237	−0.370623
$792$	0	0
$793$	−30.9148	−1.09782
$794$	−42.9141	−1.52296
$795$	0	0
$796$	7.54098	0.267283
$797$	39.0315	1.38257	0.691283	−	0.722584i	$-0.257046\pi$
$797$	39.0315	1.38257	0.691283	+	0.722584i	$0.257046\pi$
$798$	0	0
$799$	−0.424211	−0.0150075
$800$	16.2778	0.575508
$801$	0	0
$802$	−61.4541	−2.17002
$803$	−37.9079	−1.33774
$804$	0	0
$805$	9.16527	0.323033
$806$	28.4218	1.00111
$807$	0	0
$808$	26.1283	0.919192
$809$	−27.8671	−0.979755	−0.489878	−	0.871791i	$-0.662959\pi$
$809$	−27.8671	−0.979755	−0.489878	+	0.871791i	$0.662959\pi$
$810$	0	0
$811$	−25.7182	−0.903088	−0.451544	−	0.892249i	$-0.649127\pi$
$811$	−25.7182	−0.903088	−0.451544	+	0.892249i	$0.649127\pi$
$812$	0.750737	0.0263457
$813$	0	0
$814$	12.5046	0.438286
$815$	21.9432	0.768636
$816$	0	0
$817$	12.5323	0.438448
$818$	39.6091	1.38490
$819$	0	0
$820$	−15.5734	−0.543847
$821$	−37.0189	−1.29197	−0.645984	−	0.763351i	$-0.723553\pi$
$821$	−37.0189	−1.29197	−0.645984	+	0.763351i	$0.723553\pi$
$822$	0	0
$823$	−39.8987	−1.39078	−0.695390	−	0.718632i	$-0.744768\pi$
$823$	−39.8987	−1.39078	−0.695390	+	0.718632i	$0.744768\pi$
$824$	−15.4723	−0.539005
$825$	0	0
$826$	−1.59663	−0.0555537
$827$	−3.22603	−0.112180	−0.0560900	−	0.998426i	$-0.517863\pi$
$827$	−3.22603	−0.112180	−0.0560900	+	0.998426i	$0.517863\pi$
$828$	0	0
$829$	−32.6214	−1.13299	−0.566494	−	0.824066i	$-0.691701\pi$
$829$	−32.6214	−1.13299	−0.566494	+	0.824066i	$0.691701\pi$
$830$	−20.8910	−0.725137
$831$	0	0
$832$	−0.635238	−0.0220229
$833$	−2.76361	−0.0957535
$834$	0	0
$835$	−8.72530	−0.301952
$836$	−8.40257	−0.290609
$837$	0	0
$838$	−26.2498	−0.906785
$839$	−25.0716	−0.865568	−0.432784	−	0.901498i	$-0.642469\pi$
$839$	−25.0716	−0.865568	−0.432784	+	0.901498i	$0.642469\pi$
$840$	0	0
$841$	−28.5269	−0.983687
$842$	−5.42674	−0.187018
$843$	0	0
$844$	−1.19058	−0.0409816
$845$	−1.63453	−0.0562295
$846$	0	0
$847$	−10.5419	−0.362225
$848$	−22.7649	−0.781750
$849$	0	0
$850$	14.1704	0.486042
$851$	−9.72890	−0.333503
$852$	0	0
$853$	−31.6429	−1.08343	−0.541716	−	0.840562i	$-0.682225\pi$
$853$	−31.6429	−1.08343	−0.541716	+	0.840562i	$0.682225\pi$
$854$	−14.4592	−0.494783
$855$	0	0
$856$	14.9285	0.510246
$857$	42.5044	1.45192	0.725961	−	0.687735i	$-0.241395\pi$
$857$	42.5044	1.45192	0.725961	+	0.687735i	$0.241395\pi$
$858$	0	0
$859$	33.0019	1.12601	0.563005	−	0.826454i	$-0.309645\pi$
$859$	33.0019	1.12601	0.563005	+	0.826454i	$0.309645\pi$
$860$	−11.9049	−0.405953
$861$	0	0
$862$	39.6238	1.34959
$863$	−28.4511	−0.968485	−0.484243	−	0.874934i	$-0.660905\pi$
$863$	−28.4511	−0.968485	−0.484243	+	0.874934i	$0.660905\pi$
$864$	0	0
$865$	−14.2886	−0.485828
$866$	−11.4078	−0.387654
$867$	0	0
$868$	4.69333	0.159302
$869$	7.01202	0.237866
$870$	0	0
$871$	37.3338	1.26501
$872$	−3.75138	−0.127038
$873$	0	0
$874$	18.5163	0.626323
$875$	−11.4273	−0.386313
$876$	0	0
$877$	−14.4875	−0.489208	−0.244604	−	0.969623i	$-0.578658\pi$
$877$	−14.4875	−0.489208	−0.244604	+	0.969623i	$0.578658\pi$
$878$	7.63035	0.257512
$879$	0	0
$880$	−33.4433	−1.12737
$881$	−31.1121	−1.04819	−0.524096	−	0.851659i	$-0.675597\pi$
$881$	−31.1121	−1.04819	−0.524096	+	0.851659i	$0.675597\pi$
$882$	0	0
$883$	−0.797237	−0.0268291	−0.0134146	−	0.999910i	$-0.504270\pi$
$883$	−0.797237	−0.0268291	−0.0134146	+	0.999910i	$0.504270\pi$
$884$	11.3398	0.381398
$885$	0	0
$886$	−45.9102	−1.54238
$887$	32.7920	1.10105	0.550524	−	0.834819i	$-0.314428\pi$
$887$	32.7920	1.10105	0.550524	+	0.834819i	$0.314428\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	−3.66152	−0.122734
$891$	0	0
$892$	8.78904	0.294279
$893$	0.254598	0.00851980
$894$	0	0
$895$	−17.3016	−0.578328
$896$	−11.4607	−0.382876
$897$	0	0
$898$	47.0434	1.56986
$899$	2.95752	0.0986390
$900$	0	0
$901$	−12.6038	−0.419893
$902$	−80.6612	−2.68573
$903$	0	0
$904$	−16.6508	−0.553796
$905$	0.250353	0.00832201
$906$	0	0
$907$	14.1546	0.469997	0.234998	−	0.971996i	$-0.424492\pi$
$907$	14.1546	0.469997	0.234998	+	0.971996i	$0.424492\pi$
$908$	−14.8417	−0.492538
$909$	0	0
$910$	−9.54148	−0.316297
$911$	−1.50417	−0.0498352	−0.0249176	−	0.999690i	$-0.507932\pi$
$911$	−1.50417	−0.0498352	−0.0249176	+	0.999690i	$0.507932\pi$
$912$	0	0
$913$	−38.2026	−1.26432
$914$	10.2336	0.338499
$915$	0	0
$916$	19.5924	0.647352
$917$	12.5769	0.415325
$918$	0	0
$919$	−3.93208	−0.129707	−0.0648537	−	0.997895i	$-0.520658\pi$
$919$	−3.93208	−0.129707	−0.0648537	+	0.997895i	$0.520658\pi$
$920$	14.6406	0.482686
$921$	0	0
$922$	37.5109	1.23535
$923$	−31.3473	−1.03181
$924$	0	0
$925$	4.46853	0.146924
$926$	33.6160	1.10469
$927$	0	0
$928$	3.83921	0.126028
$929$	−2.31315	−0.0758919	−0.0379459	−	0.999280i	$-0.512081\pi$
$929$	−2.31315	−0.0758919	−0.0379459	+	0.999280i	$0.512081\pi$
$930$	0	0
$931$	1.65863	0.0543594
$932$	24.2612	0.794703
$933$	0	0
$934$	40.7453	1.33323
$935$	−18.5159	−0.605534
$936$	0	0
$937$	14.1234	0.461393	0.230696	−	0.973026i	$-0.425900\pi$
$937$	14.1234	0.461393	0.230696	+	0.973026i	$0.425900\pi$
$938$	17.4614	0.570136
$939$	0	0
$940$	−0.241852	−0.00788835
$941$	−6.76692	−0.220595	−0.110298	−	0.993899i	$-0.535180\pi$
$941$	−6.76692	−0.220595	−0.110298	+	0.993899i	$0.535180\pi$
$942$	0	0
$943$	62.7566	2.04364
$944$	−4.53275	−0.147528
$945$	0	0
$946$	−61.6603	−2.00475
$947$	−25.7982	−0.838327	−0.419164	−	0.907911i	$-0.637677\pi$
$947$	−25.7982	−0.838327	−0.419164	+	0.907911i	$0.637677\pi$
$948$	0	0
$949$	−30.7040	−0.996693
$950$	−8.50463	−0.275927
$951$	0	0
$952$	−4.41459	−0.143078
$953$	−29.5671	−0.957773	−0.478886	−	0.877877i	$-0.658959\pi$
$953$	−29.5671	−0.957773	−0.478886	+	0.877877i	$0.658959\pi$
$954$	0	0
$955$	18.1490	0.587289
$956$	26.4705	0.856117
$957$	0	0
$958$	−9.21807	−0.297822
$959$	−9.20221	−0.297155
$960$	0	0
$961$	−12.5107	−0.403570
$962$	10.1282	0.326548
$963$	0	0
$964$	−7.34262	−0.236490
$965$	30.3576	0.977245
$966$	0	0
$967$	−29.8509	−0.959940	−0.479970	−	0.877285i	$-0.659352\pi$
$967$	−29.8509	−0.959940	−0.479970	+	0.877285i	$0.659352\pi$
$968$	−16.8396	−0.541247
$969$	0	0
$970$	23.6151	0.758235
$971$	8.42687	0.270431	0.135215	−	0.990816i	$-0.456827\pi$
$971$	8.42687	0.270431	0.135215	+	0.990816i	$0.456827\pi$
$972$	0	0
$973$	−7.27644	−0.233272
$974$	48.0521	1.53969
$975$	0	0
$976$	−41.0489	−1.31394
$977$	55.1342	1.76390	0.881950	−	0.471344i	$-0.156231\pi$
$977$	55.1342	1.76390	0.881950	+	0.471344i	$0.156231\pi$
$978$	0	0
$979$	−6.69567	−0.213995
$980$	−1.57560	−0.0503306
$981$	0	0
$982$	27.1617	0.866764
$983$	−7.72798	−0.246484	−0.123242	−	0.992377i	$-0.539329\pi$
$983$	−7.72798	−0.246484	−0.123242	+	0.992377i	$0.539329\pi$
$984$	0	0
$985$	−2.53401	−0.0807402
$986$	3.34217	0.106436
$987$	0	0
$988$	−6.80576	−0.216520
$989$	47.9734	1.52546
$990$	0	0
$991$	26.0766	0.828351	0.414175	−	0.910197i	$-0.364070\pi$
$991$	26.0766	0.828351	0.414175	+	0.910197i	$0.364070\pi$
$992$	24.0013	0.762043
$993$	0	0
$994$	−14.6615	−0.465034
$995$	−9.97316	−0.316170
$996$	0	0
$997$	−5.17142	−0.163781	−0.0818903	−	0.996641i	$-0.526096\pi$
$997$	−5.17142	−0.163781	−0.0818903	+	0.996641i	$0.526096\pi$
$998$	47.2867	1.49683
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.p.1.10		14
3.2	odd	2		2667.2.a.m.1.5	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.m.1.5	✓	14	3.2	odd	2
8001.2.a.p.1.10		14	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)

\(f(q)\)	\(=\)	\(q+1.75826 q^{2} +1.09149 q^{4} -1.44353 q^{5} -1.00000 q^{7} -1.59740 q^{8} +O(q^{10})\)	\(q+1.75826 q^{2} +1.09149 q^{4} -1.44353 q^{5} -1.00000 q^{7} -1.59740 q^{8} -2.53810 q^{10} -4.64133 q^{11} -3.75930 q^{13} -1.75826 q^{14} -4.99163 q^{16} -2.76361 q^{17} +1.65863 q^{19} -1.57560 q^{20} -8.16068 q^{22} +6.34922 q^{23} -2.91623 q^{25} -6.60984 q^{26} -1.09149 q^{28} -0.687809 q^{29} -4.29992 q^{31} -5.58180 q^{32} -4.85916 q^{34} +1.44353 q^{35} -1.53230 q^{37} +2.91631 q^{38} +2.30589 q^{40} +9.88414 q^{41} +7.55579 q^{43} -5.06597 q^{44} +11.1636 q^{46} +0.153499 q^{47} +1.00000 q^{49} -5.12750 q^{50} -4.10324 q^{52} +4.56061 q^{53} +6.69988 q^{55} +1.59740 q^{56} -1.20935 q^{58} +0.908070 q^{59} +8.22355 q^{61} -7.56040 q^{62} +0.168978 q^{64} +5.42665 q^{65} -9.93107 q^{67} -3.01646 q^{68} +2.53810 q^{70} +8.33861 q^{71} +8.16747 q^{73} -2.69418 q^{74} +1.81038 q^{76} +4.64133 q^{77} -1.51078 q^{79} +7.20555 q^{80} +17.3789 q^{82} +8.23096 q^{83} +3.98935 q^{85} +13.2851 q^{86} +7.41405 q^{88} +1.44262 q^{89} +3.75930 q^{91} +6.93012 q^{92} +0.269891 q^{94} -2.39428 q^{95} -9.30424 q^{97} +1.75826 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 5 q^{2} + 15 q^{4} + 4 q^{5} - 14 q^{7} + 12 q^{8}+O(q^{10}) \) 14 * q + 5 * q^2 + 15 * q^4 + 4 * q^5 - 14 * q^7 + 12 * q^8	\( 14 q + 5 q^{2} + 15 q^{4} + 4 q^{5} - 14 q^{7} + 12 q^{8} + 4 q^{10} + 3 q^{11} - 13 q^{13} - 5 q^{14} + 13 q^{16} + 5 q^{17} + 21 q^{19} + 3 q^{20} - 3 q^{22} + 10 q^{23} + 4 q^{25} + 6 q^{26} - 15 q^{28} + 15 q^{29} + 33 q^{31} + 29 q^{32} + 28 q^{34} - 4 q^{35} - 29 q^{37} + 15 q^{38} + 3 q^{40} + q^{41} - 25 q^{43} + 26 q^{44} - 4 q^{46} + 9 q^{47} + 14 q^{49} + 28 q^{50} - 13 q^{52} + 35 q^{53} + 14 q^{55} - 12 q^{56} - 23 q^{58} - 10 q^{59} + q^{61} + 43 q^{62} - 2 q^{64} + 24 q^{65} - 38 q^{67} + 2 q^{68} - 4 q^{70} + 10 q^{71} + 8 q^{73} + 25 q^{74} + 26 q^{76} - 3 q^{77} + 26 q^{79} + 48 q^{80} + 6 q^{82} + 30 q^{83} - 32 q^{85} + 50 q^{86} - 29 q^{88} - 4 q^{89} + 13 q^{91} + 32 q^{92} - 7 q^{94} + 32 q^{95} + 15 q^{97} + 5 q^{98}+O(q^{100}) \) 14 * q + 5 * q^2 + 15 * q^4 + 4 * q^5 - 14 * q^7 + 12 * q^8 + 4 * q^10 + 3 * q^11 - 13 * q^13 - 5 * q^14 + 13 * q^16 + 5 * q^17 + 21 * q^19 + 3 * q^20 - 3 * q^22 + 10 * q^23 + 4 * q^25 + 6 * q^26 - 15 * q^28 + 15 * q^29 + 33 * q^31 + 29 * q^32 + 28 * q^34 - 4 * q^35 - 29 * q^37 + 15 * q^38 + 3 * q^40 + q^41 - 25 * q^43 + 26 * q^44 - 4 * q^46 + 9 * q^47 + 14 * q^49 + 28 * q^50 - 13 * q^52 + 35 * q^53 + 14 * q^55 - 12 * q^56 - 23 * q^58 - 10 * q^59 + q^61 + 43 * q^62 - 2 * q^64 + 24 * q^65 - 38 * q^67 + 2 * q^68 - 4 * q^70 + 10 * q^71 + 8 * q^73 + 25 * q^74 + 26 * q^76 - 3 * q^77 + 26 * q^79 + 48 * q^80 + 6 * q^82 + 30 * q^83 - 32 * q^85 + 50 * q^86 - 29 * q^88 - 4 * q^89 + 13 * q^91 + 32 * q^92 - 7 * q^94 + 32 * q^95 + 15 * q^97 + 5 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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