LMFDB - Embedded newform 8001.2.a.r.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:	$1$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 5 x^{15} - 13 x^{14} + 98 x^{13} + 9 x^{12} - 712 x^{11} + 565 x^{10} + 2282 x^{9} - 3082 x^{8} + \cdots + 4 $ x^16 - 5x^15 - 13x^14 + 98x^13 + 9x^12 - 712x^11 + 565x^10 + 2282x^9 - 3082x^8 - 2747x^7 + 5821x^6 - 158x^5 - 3341x^4 + 1002x^3 + 416x^2 - 148*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 2667)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.84800$ 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)

$f(q)$	$=$	$q-1.84800 q^{2} +1.41512 q^{4} +3.49192 q^{5} -1.00000 q^{7} +1.08086 q^{8} +O(q^{10})$	\(q-1.84800 q^{2} +1.41512 q^{4} +3.49192 q^{5} -1.00000 q^{7} +1.08086 q^{8} -6.45308 q^{10} -1.44812 q^{11} +0.419006 q^{13} +1.84800 q^{14} -4.82768 q^{16} +3.49611 q^{17} -3.81133 q^{19} +4.94148 q^{20} +2.67612 q^{22} -7.19568 q^{23} +7.19347 q^{25} -0.774324 q^{26} -1.41512 q^{28} -1.43207 q^{29} +4.42140 q^{31} +6.75984 q^{32} -6.46082 q^{34} -3.49192 q^{35} +5.04489 q^{37} +7.04336 q^{38} +3.77427 q^{40} -9.79333 q^{41} -4.63261 q^{43} -2.04926 q^{44} +13.2976 q^{46} +6.75497 q^{47} +1.00000 q^{49} -13.2936 q^{50} +0.592943 q^{52} -10.6795 q^{53} -5.05670 q^{55} -1.08086 q^{56} +2.64647 q^{58} -7.12059 q^{59} -0.158644 q^{61} -8.17077 q^{62} -2.83687 q^{64} +1.46313 q^{65} +0.583961 q^{67} +4.94741 q^{68} +6.45308 q^{70} +9.97501 q^{71} +10.9043 q^{73} -9.32297 q^{74} -5.39349 q^{76} +1.44812 q^{77} +16.9790 q^{79} -16.8578 q^{80} +18.0981 q^{82} -13.3539 q^{83} +12.2081 q^{85} +8.56109 q^{86} -1.56521 q^{88} +13.2257 q^{89} -0.419006 q^{91} -10.1827 q^{92} -12.4832 q^{94} -13.3088 q^{95} -14.1419 q^{97} -1.84800 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 5 q^{2} + 19 q^{4} + q^{5} - 16 q^{7} - 6 q^{8}+O(q^{10}) $ 16 * q - 5 * q^2 + 19 * q^4 + q^5 - 16 * q^7 - 6 * q^8	$ 16 q - 5 q^{2} + 19 q^{4} + q^{5} - 16 q^{7} - 6 q^{8} - 12 q^{10} - 11 q^{11} + 18 q^{13} + 5 q^{14} + 25 q^{16} + 5 q^{17} - 11 q^{19} + q^{20} + q^{22} - 13 q^{23} + 33 q^{25} - 8 q^{26} - 19 q^{28} - 24 q^{29} - 42 q^{31} - 42 q^{32} + 9 q^{34} - q^{35} + 40 q^{37} - 38 q^{38} - 61 q^{40} - 9 q^{41} + 7 q^{43} - 3 q^{44} + 24 q^{46} - 31 q^{47} + 16 q^{49} - 6 q^{50} + 52 q^{52} - 66 q^{53} - 36 q^{55} + 6 q^{56} + 19 q^{58} + 7 q^{59} + 6 q^{61} - 52 q^{62} + 10 q^{64} - 51 q^{65} + 16 q^{67} - 14 q^{68} + 12 q^{70} - 46 q^{71} + 39 q^{73} - 72 q^{74} + 24 q^{76} + 11 q^{77} + 4 q^{79} + 2 q^{80} - 18 q^{82} - 15 q^{83} - 4 q^{85} - 14 q^{86} + 58 q^{88} + q^{89} - 18 q^{91} - 26 q^{92} + 5 q^{94} - 44 q^{95} + 41 q^{97} - 5 q^{98}+O(q^{100}) $ 16 * q - 5 * q^2 + 19 * q^4 + q^5 - 16 * q^7 - 6 * q^8 - 12 * q^10 - 11 * q^11 + 18 * q^13 + 5 * q^14 + 25 * q^16 + 5 * q^17 - 11 * q^19 + q^20 + q^22 - 13 * q^23 + 33 * q^25 - 8 * q^26 - 19 * q^28 - 24 * q^29 - 42 * q^31 - 42 * q^32 + 9 * q^34 - q^35 + 40 * q^37 - 38 * q^38 - 61 * q^40 - 9 * q^41 + 7 * q^43 - 3 * q^44 + 24 * q^46 - 31 * q^47 + 16 * q^49 - 6 * q^50 + 52 * q^52 - 66 * q^53 - 36 * q^55 + 6 * q^56 + 19 * q^58 + 7 * q^59 + 6 * q^61 - 52 * q^62 + 10 * q^64 - 51 * q^65 + 16 * q^67 - 14 * q^68 + 12 * q^70 - 46 * q^71 + 39 * q^73 - 72 * q^74 + 24 * q^76 + 11 * q^77 + 4 * q^79 + 2 * q^80 - 18 * q^82 - 15 * q^83 - 4 * q^85 - 14 * q^86 + 58 * q^88 + q^89 - 18 * q^91 - 26 * q^92 + 5 * q^94 - 44 * q^95 + 41 * 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 $

$2$	−1.84800	−1.30674	−0.653368	−	0.757040i	$-0.726645\pi$
$2$	−1.84800	−1.30674	−0.653368	+	0.757040i	$0.726645\pi$
$3$	0	0
$3$	0	0
$4$	1.41512	0.707560
$5$	3.49192	1.56163	0.780816	−	0.624761i	$-0.214804\pi$
$5$	3.49192	1.56163	0.780816	+	0.624761i	$0.214804\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.08086	0.382142
$9$	0	0
$10$	−6.45308	−2.04064
$11$	−1.44812	−0.436623	−0.218312	−	0.975879i	$-0.570055\pi$
$11$	−1.44812	−0.436623	−0.218312	+	0.975879i	$0.570055\pi$
$12$	0	0
$13$	0.419006	0.116211	0.0581056	−	0.998310i	$-0.481494\pi$
$13$	0.419006	0.116211	0.0581056	+	0.998310i	$0.481494\pi$
$14$	1.84800	0.493900
$15$	0	0
$16$	−4.82768	−1.20692
$17$	3.49611	0.847931	0.423965	−	0.905678i	$-0.360638\pi$
$17$	3.49611	0.847931	0.423965	+	0.905678i	$0.360638\pi$
$18$	0	0
$19$	−3.81133	−0.874379	−0.437190	−	0.899369i	$-0.644026\pi$
$19$	−3.81133	−0.874379	−0.437190	+	0.899369i	$0.644026\pi$
$20$	4.94148	1.10495
$21$	0	0
$22$	2.67612	0.570551
$23$	−7.19568	−1.50040	−0.750201	−	0.661210i	$-0.770043\pi$
$23$	−7.19568	−1.50040	−0.750201	+	0.661210i	$0.770043\pi$
$24$	0	0
$25$	7.19347	1.43869
$26$	−0.774324	−0.151858
$27$	0	0
$28$	−1.41512	−0.267433
$29$	−1.43207	−0.265928	−0.132964	−	0.991121i	$-0.542450\pi$
$29$	−1.43207	−0.265928	−0.132964	+	0.991121i	$0.542450\pi$
$30$	0	0
$31$	4.42140	0.794108	0.397054	−	0.917795i	$-0.370033\pi$
$31$	4.42140	0.794108	0.397054	+	0.917795i	$0.370033\pi$
$32$	6.75984	1.19498
$33$	0	0
$34$	−6.46082	−1.10802
$35$	−3.49192	−0.590241
$36$	0	0
$37$	5.04489	0.829374	0.414687	−	0.909964i	$-0.363891\pi$
$37$	5.04489	0.829374	0.414687	+	0.909964i	$0.363891\pi$
$38$	7.04336	1.14258
$39$	0	0
$40$	3.77427	0.596765
$41$	−9.79333	−1.52946	−0.764731	−	0.644350i	$-0.777128\pi$
$41$	−9.79333	−1.52946	−0.764731	+	0.644350i	$0.777128\pi$
$42$	0	0
$43$	−4.63261	−0.706467	−0.353233	−	0.935535i	$-0.614918\pi$
$43$	−4.63261	−0.706467	−0.353233	+	0.935535i	$0.614918\pi$
$44$	−2.04926	−0.308937
$45$	0	0
$46$	13.2976	1.96063
$47$	6.75497	0.985314	0.492657	−	0.870224i	$-0.336026\pi$
$47$	6.75497	0.985314	0.492657	+	0.870224i	$0.336026\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−13.2936	−1.87999
$51$	0	0
$52$	0.592943	0.0822265
$53$	−10.6795	−1.46694	−0.733468	−	0.679724i	$-0.762100\pi$
$53$	−10.6795	−1.46694	−0.733468	+	0.679724i	$0.762100\pi$
$54$	0	0
$55$	−5.05670	−0.681845
$56$	−1.08086	−0.144436
$57$	0	0
$58$	2.64647	0.347498
$59$	−7.12059	−0.927021	−0.463511	−	0.886091i	$-0.653410\pi$
$59$	−7.12059	−0.927021	−0.463511	+	0.886091i	$0.653410\pi$
$60$	0	0
$61$	−0.158644	−0.0203123	−0.0101561	−	0.999948i	$-0.503233\pi$
$61$	−0.158644	−0.0203123	−0.0101561	+	0.999948i	$0.503233\pi$
$62$	−8.17077	−1.03769
$63$	0	0
$64$	−2.83687	−0.354609
$65$	1.46313	0.181479
$66$	0	0
$67$	0.583961	0.0713422	0.0356711	−	0.999364i	$-0.488643\pi$
$67$	0.583961	0.0713422	0.0356711	+	0.999364i	$0.488643\pi$
$68$	4.94741	0.599962
$69$	0	0
$70$	6.45308	0.771290
$71$	9.97501	1.18382	0.591908	−	0.806006i	$-0.298375\pi$
$71$	9.97501	1.18382	0.591908	+	0.806006i	$0.298375\pi$
$72$	0	0
$73$	10.9043	1.27626	0.638129	−	0.769930i	$-0.279709\pi$
$73$	10.9043	1.27626	0.638129	+	0.769930i	$0.279709\pi$
$74$	−9.32297	−1.08377
$75$	0	0
$76$	−5.39349	−0.618676
$77$	1.44812	0.165028
$78$	0	0
$79$	16.9790	1.91028	0.955142	−	0.296147i	$-0.0957017\pi$
$79$	16.9790	1.91028	0.955142	+	0.296147i	$0.0957017\pi$
$80$	−16.8578	−1.88476
$81$	0	0
$82$	18.0981	1.99860
$83$	−13.3539	−1.46578	−0.732892	−	0.680345i	$-0.761830\pi$
$83$	−13.3539	−1.46578	−0.732892	+	0.680345i	$0.761830\pi$
$84$	0	0
$85$	12.2081	1.32416
$86$	8.56109	0.923166
$87$	0	0
$88$	−1.56521	−0.166852
$89$	13.2257	1.40192	0.700958	−	0.713202i	$-0.252756\pi$
$89$	13.2257	1.40192	0.700958	+	0.713202i	$0.252756\pi$
$90$	0	0
$91$	−0.419006	−0.0439237
$92$	−10.1827	−1.06162
$93$	0	0
$94$	−12.4832	−1.28755
$95$	−13.3088	−1.36546
$96$	0	0
$97$	−14.1419	−1.43590	−0.717948	−	0.696097i	$-0.754919\pi$
$97$	−14.1419	−1.43590	−0.717948	+	0.696097i	$0.754919\pi$
$98$	−1.84800	−0.186677
$99$	0	0
$100$	10.1796	1.01796
$101$	1.82679	0.181772	0.0908862	−	0.995861i	$-0.471030\pi$
$101$	1.82679	0.181772	0.0908862	+	0.995861i	$0.471030\pi$
$102$	0	0
$103$	−11.7530	−1.15806	−0.579029	−	0.815307i	$-0.696568\pi$
$103$	−11.7530	−1.15806	−0.579029	+	0.815307i	$0.696568\pi$
$104$	0.452887	0.0444092
$105$	0	0
$106$	19.7357	1.91690
$107$	−8.74315	−0.845232	−0.422616	−	0.906309i	$-0.638888\pi$
$107$	−8.74315	−0.845232	−0.422616	+	0.906309i	$0.638888\pi$
$108$	0	0
$109$	−12.5844	−1.20536	−0.602682	−	0.797982i	$-0.705901\pi$
$109$	−12.5844	−1.20536	−0.602682	+	0.797982i	$0.705901\pi$
$110$	9.34480	0.890991
$111$	0	0
$112$	4.82768	0.456172
$113$	−0.626982	−0.0589815	−0.0294908	−	0.999565i	$-0.509389\pi$
$113$	−0.626982	−0.0589815	−0.0294908	+	0.999565i	$0.509389\pi$
$114$	0	0
$115$	−25.1267	−2.34308
$116$	−2.02655	−0.188160
$117$	0	0
$118$	13.1589	1.21137
$119$	−3.49611	−0.320488
$120$	0	0
$121$	−8.90296	−0.809360
$122$	0.293175	0.0265428
$123$	0	0
$124$	6.25682	0.561879
$125$	7.65942	0.685080
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	−8.27714	−0.731603
$129$	0	0
$130$	−2.70388	−0.237146
$131$	−17.2259	−1.50503	−0.752516	−	0.658574i	$-0.771160\pi$
$131$	−17.2259	−1.50503	−0.752516	+	0.658574i	$0.771160\pi$
$132$	0	0
$133$	3.81133	0.330484
$134$	−1.07916	−0.0932255
$135$	0	0
$136$	3.77881	0.324030
$137$	−16.4927	−1.40906	−0.704531	−	0.709673i	$-0.748843\pi$
$137$	−16.4927	−1.40906	−0.704531	+	0.709673i	$0.748843\pi$
$138$	0	0
$139$	7.22877	0.613137	0.306568	−	0.951849i	$-0.400819\pi$
$139$	7.22877	0.613137	0.306568	+	0.951849i	$0.400819\pi$
$140$	−4.94148	−0.417631
$141$	0	0
$142$	−18.4339	−1.54694
$143$	−0.606769	−0.0507405
$144$	0	0
$145$	−5.00066	−0.415282
$146$	−20.1513	−1.66773
$147$	0	0
$148$	7.13912	0.586832
$149$	−10.3005	−0.843853	−0.421926	−	0.906630i	$-0.638646\pi$
$149$	−10.3005	−0.843853	−0.421926	+	0.906630i	$0.638646\pi$
$150$	0	0
$151$	−11.3170	−0.920966	−0.460483	−	0.887668i	$-0.652324\pi$
$151$	−11.3170	−0.920966	−0.460483	+	0.887668i	$0.652324\pi$
$152$	−4.11952	−0.334137
$153$	0	0
$154$	−2.67612	−0.215648
$155$	15.4392	1.24010
$156$	0	0
$157$	17.5508	1.40071	0.700354	−	0.713795i	$-0.253025\pi$
$157$	17.5508	1.40071	0.700354	+	0.713795i	$0.253025\pi$
$158$	−31.3772	−2.49624
$159$	0	0
$160$	23.6048	1.86612
$161$	7.19568	0.567099
$162$	0	0
$163$	15.1587	1.18732	0.593662	−	0.804714i	$-0.297681\pi$
$163$	15.1587	1.18732	0.593662	+	0.804714i	$0.297681\pi$
$164$	−13.8587	−1.08219
$165$	0	0
$166$	24.6781	1.91539
$167$	−4.61494	−0.357114	−0.178557	−	0.983930i	$-0.557143\pi$
$167$	−4.61494	−0.357114	−0.178557	+	0.983930i	$0.557143\pi$
$168$	0	0
$169$	−12.8244	−0.986495
$170$	−22.5607	−1.73032
$171$	0	0
$172$	−6.55570	−0.499868
$173$	6.06659	0.461234	0.230617	−	0.973045i	$-0.425926\pi$
$173$	6.06659	0.461234	0.230617	+	0.973045i	$0.425926\pi$
$174$	0	0
$175$	−7.19347	−0.543776
$176$	6.99103	0.526969
$177$	0	0
$178$	−24.4411	−1.83194
$179$	−25.1683	−1.88117	−0.940583	−	0.339563i	$-0.889721\pi$
$179$	−25.1683	−1.88117	−0.940583	+	0.339563i	$0.889721\pi$
$180$	0	0
$181$	14.3744	1.06844	0.534221	−	0.845345i	$-0.320605\pi$
$181$	14.3744	1.06844	0.534221	+	0.845345i	$0.320605\pi$
$182$	0.774324	0.0573968
$183$	0	0
$184$	−7.77752	−0.573367
$185$	17.6163	1.29518
$186$	0	0
$187$	−5.06277	−0.370226
$188$	9.55910	0.697169
$189$	0	0
$190$	24.5948	1.78429
$191$	8.81798	0.638047	0.319023	−	0.947747i	$-0.396645\pi$
$191$	8.81798	0.638047	0.319023	+	0.947747i	$0.396645\pi$
$192$	0	0
$193$	−7.57577	−0.545316	−0.272658	−	0.962111i	$-0.587903\pi$
$193$	−7.57577	−0.545316	−0.272658	+	0.962111i	$0.587903\pi$
$194$	26.1344	1.87634
$195$	0	0
$196$	1.41512	0.101080
$197$	15.6743	1.11675	0.558375	−	0.829589i	$-0.311425\pi$
$197$	15.6743	1.11675	0.558375	+	0.829589i	$0.311425\pi$
$198$	0	0
$199$	−11.7367	−0.831993	−0.415997	−	0.909366i	$-0.636567\pi$
$199$	−11.7367	−0.831993	−0.415997	+	0.909366i	$0.636567\pi$
$200$	7.77514	0.549786
$201$	0	0
$202$	−3.37591	−0.237529
$203$	1.43207	0.100512
$204$	0	0
$205$	−34.1975	−2.38846
$206$	21.7196	1.51328
$207$	0	0
$208$	−2.02282	−0.140258
$209$	5.51925	0.381774
$210$	0	0
$211$	−7.35010	−0.506002	−0.253001	−	0.967466i	$-0.581418\pi$
$211$	−7.35010	−0.506002	−0.253001	+	0.967466i	$0.581418\pi$
$212$	−15.1127	−1.03795
$213$	0	0
$214$	16.1574	1.10450
$215$	−16.1767	−1.10324
$216$	0	0
$217$	−4.42140	−0.300144
$218$	23.2560	1.57509
$219$	0	0
$220$	−7.15583	−0.482446
$221$	1.46489	0.0985391
$222$	0	0
$223$	4.71132	0.315493	0.157747	−	0.987480i	$-0.449577\pi$
$223$	4.71132	0.315493	0.157747	+	0.987480i	$0.449577\pi$
$224$	−6.75984	−0.451661
$225$	0	0
$226$	1.15867	0.0770733
$227$	−23.9529	−1.58981	−0.794905	−	0.606734i	$-0.792479\pi$
$227$	−23.9529	−1.58981	−0.794905	+	0.606734i	$0.792479\pi$
$228$	0	0
$229$	−2.82845	−0.186910	−0.0934548	−	0.995624i	$-0.529791\pi$
$229$	−2.82845	−0.186910	−0.0934548	+	0.995624i	$0.529791\pi$
$230$	46.4342	3.06178
$231$	0	0
$232$	−1.54787	−0.101622
$233$	−14.5002	−0.949941	−0.474970	−	0.880002i	$-0.657541\pi$
$233$	−14.5002	−0.949941	−0.474970	+	0.880002i	$0.657541\pi$
$234$	0	0
$235$	23.5878	1.53870
$236$	−10.0765	−0.655923
$237$	0	0
$238$	6.46082	0.418793
$239$	5.36336	0.346927	0.173463	−	0.984840i	$-0.444504\pi$
$239$	5.36336	0.346927	0.173463	+	0.984840i	$0.444504\pi$
$240$	0	0
$241$	−0.0791557	−0.00509887	−0.00254943	−	0.999997i	$-0.500812\pi$
$241$	−0.0791557	−0.00509887	−0.00254943	+	0.999997i	$0.500812\pi$
$242$	16.4527	1.05762
$243$	0	0
$244$	−0.224500	−0.0143722
$245$	3.49192	0.223090
$246$	0	0
$247$	−1.59697	−0.101613
$248$	4.77892	0.303462
$249$	0	0
$250$	−14.1547	−0.895219
$251$	22.5878	1.42573	0.712864	−	0.701302i	$-0.247398\pi$
$251$	22.5878	1.42573	0.712864	+	0.701302i	$0.247398\pi$
$252$	0	0
$253$	10.4202	0.655110
$254$	−1.84800	−0.115954
$255$	0	0
$256$	20.9699	1.31062
$257$	−0.699357	−0.0436247	−0.0218124	−	0.999762i	$-0.506944\pi$
$257$	−0.699357	−0.0436247	−0.0218124	+	0.999762i	$0.506944\pi$
$258$	0	0
$259$	−5.04489	−0.313474
$260$	2.07051	0.128407
$261$	0	0
$262$	31.8335	1.96668
$263$	−10.8149	−0.666874	−0.333437	−	0.942772i	$-0.608209\pi$
$263$	−10.8149	−0.666874	−0.333437	+	0.942772i	$0.608209\pi$
$264$	0	0
$265$	−37.2918	−2.29081
$266$	−7.04336	−0.431856
$267$	0	0
$268$	0.826375	0.0504789
$269$	8.14144	0.496392	0.248196	−	0.968710i	$-0.420162\pi$
$269$	8.14144	0.496392	0.248196	+	0.968710i	$0.420162\pi$
$270$	0	0
$271$	−10.6047	−0.644190	−0.322095	−	0.946707i	$-0.604387\pi$
$271$	−10.6047	−0.644190	−0.322095	+	0.946707i	$0.604387\pi$
$272$	−16.8781	−1.02338
$273$	0	0
$274$	30.4785	1.84127
$275$	−10.4170	−0.628168
$276$	0	0
$277$	0.284090	0.0170693	0.00853466	−	0.999964i	$-0.497283\pi$
$277$	0.284090	0.0170693	0.00853466	+	0.999964i	$0.497283\pi$
$278$	−13.3588	−0.801208
$279$	0	0
$280$	−3.77427	−0.225556
$281$	2.76057	0.164682	0.0823409	−	0.996604i	$-0.473760\pi$
$281$	2.76057	0.164682	0.0823409	+	0.996604i	$0.473760\pi$
$282$	0	0
$283$	−26.1803	−1.55626	−0.778128	−	0.628105i	$-0.783831\pi$
$283$	−26.1803	−1.55626	−0.778128	+	0.628105i	$0.783831\pi$
$284$	14.1158	0.837621
$285$	0	0
$286$	1.12131	0.0663045
$287$	9.79333	0.578082
$288$	0	0
$289$	−4.77723	−0.281013
$290$	9.24125	0.542665
$291$	0	0
$292$	15.4310	0.903029
$293$	17.8356	1.04197	0.520985	−	0.853566i	$-0.325565\pi$
$293$	17.8356	1.04197	0.520985	+	0.853566i	$0.325565\pi$
$294$	0	0
$295$	−24.8645	−1.44767
$296$	5.45282	0.316939
$297$	0	0
$298$	19.0354	1.10269
$299$	−3.01503	−0.174364
$300$	0	0
$301$	4.63261	0.267019
$302$	20.9139	1.20346
$303$	0	0
$304$	18.3999	1.05530
$305$	−0.553971	−0.0317203
$306$	0	0
$307$	−18.4266	−1.05166	−0.525832	−	0.850589i	$-0.676246\pi$
$307$	−18.4266	−1.05166	−0.525832	+	0.850589i	$0.676246\pi$
$308$	2.04926	0.116767
$309$	0	0
$310$	−28.5317	−1.62049
$311$	9.26368	0.525295	0.262648	−	0.964892i	$-0.415404\pi$
$311$	9.26368	0.525295	0.262648	+	0.964892i	$0.415404\pi$
$312$	0	0
$313$	−9.06608	−0.512445	−0.256223	−	0.966618i	$-0.582478\pi$
$313$	−9.06608	−0.512445	−0.256223	+	0.966618i	$0.582478\pi$
$314$	−32.4340	−1.83036
$315$	0	0
$316$	24.0273	1.35164
$317$	−4.01530	−0.225522	−0.112761	−	0.993622i	$-0.535969\pi$
$317$	−4.01530	−0.225522	−0.112761	+	0.993622i	$0.535969\pi$
$318$	0	0
$319$	2.07380	0.116111
$320$	−9.90611	−0.553768
$321$	0	0
$322$	−13.2976	−0.741049
$323$	−13.3248	−0.741413
$324$	0	0
$325$	3.01411	0.167193
$326$	−28.0134	−1.55152
$327$	0	0
$328$	−10.5852	−0.584471
$329$	−6.75497	−0.372414
$330$	0	0
$331$	−6.96401	−0.382777	−0.191388	−	0.981514i	$-0.561299\pi$
$331$	−6.96401	−0.382777	−0.191388	+	0.981514i	$0.561299\pi$
$332$	−18.8974	−1.03713
$333$	0	0
$334$	8.52842	0.466654
$335$	2.03914	0.111410
$336$	0	0
$337$	−22.6756	−1.23522	−0.617610	−	0.786485i	$-0.711899\pi$
$337$	−22.6756	−1.23522	−0.617610	+	0.786485i	$0.711899\pi$
$338$	23.6996	1.28909
$339$	0	0
$340$	17.2759	0.936920
$341$	−6.40270	−0.346726
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−5.00721	−0.269971
$345$	0	0
$346$	−11.2111	−0.602711
$347$	16.4715	0.884238	0.442119	−	0.896956i	$-0.354227\pi$
$347$	16.4715	0.884238	0.442119	+	0.896956i	$0.354227\pi$
$348$	0	0
$349$	31.8015	1.70230	0.851148	−	0.524926i	$-0.175907\pi$
$349$	31.8015	1.70230	0.851148	+	0.524926i	$0.175907\pi$
$350$	13.2936	0.710571
$351$	0	0
$352$	−9.78903	−0.521757
$353$	−15.7422	−0.837871	−0.418935	−	0.908016i	$-0.637597\pi$
$353$	−15.7422	−0.837871	−0.418935	+	0.908016i	$0.637597\pi$
$354$	0	0
$355$	34.8319	1.84868
$356$	18.7159	0.991940
$357$	0	0
$358$	46.5111	2.45819
$359$	−8.98942	−0.474443	−0.237222	−	0.971456i	$-0.576237\pi$
$359$	−8.98942	−0.474443	−0.237222	+	0.971456i	$0.576237\pi$
$360$	0	0
$361$	−4.47376	−0.235461
$362$	−26.5640	−1.39617
$363$	0	0
$364$	−0.592943	−0.0310787
$365$	38.0771	1.99304
$366$	0	0
$367$	−16.9185	−0.883140	−0.441570	−	0.897227i	$-0.645578\pi$
$367$	−16.9185	−0.883140	−0.441570	+	0.897227i	$0.645578\pi$
$368$	34.7384	1.81086
$369$	0	0
$370$	−32.5550	−1.69246
$371$	10.6795	0.554450
$372$	0	0
$373$	27.7577	1.43724	0.718621	−	0.695402i	$-0.244774\pi$
$373$	27.7577	1.43724	0.718621	+	0.695402i	$0.244774\pi$
$374$	9.35602	0.483788
$375$	0	0
$376$	7.30119	0.376530
$377$	−0.600045	−0.0309039
$378$	0	0
$379$	26.0214	1.33663	0.668314	−	0.743879i	$-0.267016\pi$
$379$	26.0214	1.33663	0.668314	+	0.743879i	$0.267016\pi$
$380$	−18.8336	−0.966144
$381$	0	0
$382$	−16.2957	−0.833759
$383$	−18.4705	−0.943799	−0.471900	−	0.881652i	$-0.656432\pi$
$383$	−18.4705	−0.943799	−0.471900	+	0.881652i	$0.656432\pi$
$384$	0	0
$385$	5.05670	0.257713
$386$	14.0000	0.712584
$387$	0	0
$388$	−20.0125	−1.01598
$389$	−23.4582	−1.18938	−0.594689	−	0.803956i	$-0.702725\pi$
$389$	−23.4582	−1.18938	−0.594689	+	0.803956i	$0.702725\pi$
$390$	0	0
$391$	−25.1569	−1.27224
$392$	1.08086	0.0545917
$393$	0	0
$394$	−28.9662	−1.45930
$395$	59.2892	2.98316
$396$	0	0
$397$	36.3769	1.82570	0.912851	−	0.408293i	$-0.133876\pi$
$397$	36.3769	1.82570	0.912851	+	0.408293i	$0.133876\pi$
$398$	21.6895	1.08720
$399$	0	0
$400$	−34.7278	−1.73639
$401$	13.6275	0.680524	0.340262	−	0.940331i	$-0.389484\pi$
$401$	13.6275	0.680524	0.340262	+	0.940331i	$0.389484\pi$
$402$	0	0
$403$	1.85259	0.0922843
$404$	2.58513	0.128615
$405$	0	0
$406$	−2.64647	−0.131342
$407$	−7.30558	−0.362124
$408$	0	0
$409$	31.4185	1.55355	0.776773	−	0.629780i	$-0.216855\pi$
$409$	31.4185	1.55355	0.776773	+	0.629780i	$0.216855\pi$
$410$	63.1971	3.12108
$411$	0	0
$412$	−16.6319	−0.819396
$413$	7.12059	0.350381
$414$	0	0
$415$	−46.6308	−2.28901
$416$	2.83241	0.138871
$417$	0	0
$418$	−10.1996	−0.498878
$419$	−8.75006	−0.427469	−0.213734	−	0.976892i	$-0.568563\pi$
$419$	−8.75006	−0.427469	−0.213734	+	0.976892i	$0.568563\pi$
$420$	0	0
$421$	28.5520	1.39154	0.695771	−	0.718264i	$-0.255063\pi$
$421$	28.5520	1.39154	0.695771	+	0.718264i	$0.255063\pi$
$422$	13.5830	0.661211
$423$	0	0
$424$	−11.5430	−0.560578
$425$	25.1492	1.21991
$426$	0	0
$427$	0.158644	0.00767732
$428$	−12.3726	−0.598052
$429$	0	0
$430$	29.8946	1.44165
$431$	1.57612	0.0759193	0.0379596	−	0.999279i	$-0.487914\pi$
$431$	1.57612	0.0759193	0.0379596	+	0.999279i	$0.487914\pi$
$432$	0	0
$433$	14.8219	0.712296	0.356148	−	0.934430i	$-0.384090\pi$
$433$	14.8219	0.712296	0.356148	+	0.934430i	$0.384090\pi$
$434$	8.17077	0.392210
$435$	0	0
$436$	−17.8084	−0.852867
$437$	27.4251	1.31192
$438$	0	0
$439$	1.75545	0.0837833	0.0418916	−	0.999122i	$-0.486662\pi$
$439$	1.75545	0.0837833	0.0418916	+	0.999122i	$0.486662\pi$
$440$	−5.46558	−0.260561
$441$	0	0
$442$	−2.70712	−0.128765
$443$	−27.5472	−1.30881	−0.654403	−	0.756146i	$-0.727080\pi$
$443$	−27.5472	−1.30881	−0.654403	+	0.756146i	$0.727080\pi$
$444$	0	0
$445$	46.1829	2.18928
$446$	−8.70654	−0.412267
$447$	0	0
$448$	2.83687	0.134030
$449$	−22.4661	−1.06024	−0.530120	−	0.847923i	$-0.677853\pi$
$449$	−22.4661	−1.06024	−0.530120	+	0.847923i	$0.677853\pi$
$450$	0	0
$451$	14.1819	0.667798
$452$	−0.887255	−0.0417330
$453$	0	0
$454$	44.2651	2.07746
$455$	−1.46313	−0.0685927
$456$	0	0
$457$	18.3869	0.860103	0.430052	−	0.902804i	$-0.358495\pi$
$457$	18.3869	0.860103	0.430052	+	0.902804i	$0.358495\pi$
$458$	5.22700	0.244241
$459$	0	0
$460$	−35.5573	−1.65787
$461$	4.12751	0.192237	0.0961187	−	0.995370i	$-0.469357\pi$
$461$	4.12751	0.192237	0.0961187	+	0.995370i	$0.469357\pi$
$462$	0	0
$463$	−1.59082	−0.0739317	−0.0369659	−	0.999317i	$-0.511769\pi$
$463$	−1.59082	−0.0739317	−0.0369659	+	0.999317i	$0.511769\pi$
$464$	6.91356	0.320954
$465$	0	0
$466$	26.7965	1.24132
$467$	16.8791	0.781072	0.390536	−	0.920588i	$-0.372290\pi$
$467$	16.8791	0.781072	0.390536	+	0.920588i	$0.372290\pi$
$468$	0	0
$469$	−0.583961	−0.0269648
$470$	−43.5904	−2.01067
$471$	0	0
$472$	−7.69636	−0.354254
$473$	6.70856	0.308460
$474$	0	0
$475$	−27.4167	−1.25796
$476$	−4.94741	−0.226764
$477$	0	0
$478$	−9.91151	−0.453342
$479$	−36.3592	−1.66130	−0.830648	−	0.556798i	$-0.812030\pi$
$479$	−36.3592	−1.66130	−0.830648	+	0.556798i	$0.812030\pi$
$480$	0	0
$481$	2.11384	0.0963827
$482$	0.146280	0.00666288
$483$	0	0
$484$	−12.5988	−0.572671
$485$	−49.3825	−2.24234
$486$	0	0
$487$	38.1782	1.73002	0.865010	−	0.501754i	$-0.167312\pi$
$487$	38.1782	1.73002	0.865010	+	0.501754i	$0.167312\pi$
$488$	−0.171472	−0.00776217
$489$	0	0
$490$	−6.45308	−0.291520
$491$	17.7847	0.802614	0.401307	−	0.915944i	$-0.368556\pi$
$491$	17.7847	0.802614	0.401307	+	0.915944i	$0.368556\pi$
$492$	0	0
$493$	−5.00667	−0.225489
$494$	2.95121	0.132781
$495$	0	0
$496$	−21.3451	−0.958423
$497$	−9.97501	−0.447440
$498$	0	0
$499$	−16.1862	−0.724592	−0.362296	−	0.932063i	$-0.618007\pi$
$499$	−16.1862	−0.724592	−0.362296	+	0.932063i	$0.618007\pi$
$500$	10.8390	0.484735
$501$	0	0
$502$	−41.7423	−1.86305
$503$	32.8060	1.46275	0.731373	−	0.681978i	$-0.238880\pi$
$503$	32.8060	1.46275	0.731373	+	0.681978i	$0.238880\pi$
$504$	0	0
$505$	6.37899	0.283862
$506$	−19.2565	−0.856057
$507$	0	0
$508$	1.41512	0.0627858
$509$	−10.9244	−0.484217	−0.242109	−	0.970249i	$-0.577839\pi$
$509$	−10.9244	−0.484217	−0.242109	+	0.970249i	$0.577839\pi$
$510$	0	0
$511$	−10.9043	−0.482380
$512$	−22.1982	−0.981033
$513$	0	0
$514$	1.29242	0.0570060
$515$	−41.0405	−1.80846
$516$	0	0
$517$	−9.78198	−0.430211
$518$	9.32297	0.409628
$519$	0	0
$520$	1.58144	0.0693508
$521$	−7.05981	−0.309296	−0.154648	−	0.987970i	$-0.549424\pi$
$521$	−7.05981	−0.309296	−0.154648	+	0.987970i	$0.549424\pi$
$522$	0	0
$523$	17.9412	0.784513	0.392256	−	0.919856i	$-0.371695\pi$
$523$	17.9412	0.784513	0.392256	+	0.919856i	$0.371695\pi$
$524$	−24.3767	−1.06490
$525$	0	0
$526$	19.9860	0.871429
$527$	15.4577	0.673348
$528$	0	0
$529$	28.7778	1.25121
$530$	68.9153	2.99349
$531$	0	0
$532$	5.39349	0.233837
$533$	−4.10346	−0.177741
$534$	0	0
$535$	−30.5303	−1.31994
$536$	0.631181	0.0272629
$537$	0	0
$538$	−15.0454	−0.648654
$539$	−1.44812	−0.0623747
$540$	0	0
$541$	−16.8290	−0.723536	−0.361768	−	0.932268i	$-0.617827\pi$
$541$	−16.8290	−0.723536	−0.361768	+	0.932268i	$0.617827\pi$
$542$	19.5975	0.841787
$543$	0	0
$544$	23.6331	1.01326
$545$	−43.9435	−1.88233
$546$	0	0
$547$	2.13355	0.0912240	0.0456120	−	0.998959i	$-0.485476\pi$
$547$	2.13355	0.0912240	0.0456120	+	0.998959i	$0.485476\pi$
$548$	−23.3391	−0.996996
$549$	0	0
$550$	19.2506	0.820849
$551$	5.45809	0.232522
$552$	0	0
$553$	−16.9790	−0.722020
$554$	−0.525000	−0.0223051
$555$	0	0
$556$	10.2296	0.433831
$557$	−29.0162	−1.22946	−0.614729	−	0.788738i	$-0.710735\pi$
$557$	−29.0162	−1.22946	−0.614729	+	0.788738i	$0.710735\pi$
$558$	0	0
$559$	−1.94109	−0.0820994
$560$	16.8578	0.712374
$561$	0	0
$562$	−5.10154	−0.215196
$563$	−45.2094	−1.90535	−0.952675	−	0.303991i	$-0.901681\pi$
$563$	−45.2094	−1.90535	−0.952675	+	0.303991i	$0.901681\pi$
$564$	0	0
$565$	−2.18937	−0.0921075
$566$	48.3813	2.03362
$567$	0	0
$568$	10.7816	0.452386
$569$	−11.2454	−0.471432	−0.235716	−	0.971822i	$-0.575744\pi$
$569$	−11.2454	−0.471432	−0.235716	+	0.971822i	$0.575744\pi$
$570$	0	0
$571$	−38.6605	−1.61789	−0.808946	−	0.587883i	$-0.799962\pi$
$571$	−38.6605	−1.61789	−0.808946	+	0.587883i	$0.799962\pi$
$572$	−0.858651	−0.0359020
$573$	0	0
$574$	−18.0981	−0.755401
$575$	−51.7619	−2.15862
$576$	0	0
$577$	40.7588	1.69681	0.848406	−	0.529347i	$-0.177563\pi$
$577$	40.7588	1.69681	0.848406	+	0.529347i	$0.177563\pi$
$578$	8.82833	0.367210
$579$	0	0
$580$	−7.07654	−0.293837
$581$	13.3539	0.554014
$582$	0	0
$583$	15.4651	0.640498
$584$	11.7861	0.487712
$585$	0	0
$586$	−32.9603	−1.36158
$587$	−41.4546	−1.71102	−0.855508	−	0.517790i	$-0.826755\pi$
$587$	−41.4546	−1.71102	−0.855508	+	0.517790i	$0.826755\pi$
$588$	0	0
$589$	−16.8514	−0.694351
$590$	45.9497	1.89172
$591$	0	0
$592$	−24.3551	−1.00099
$593$	8.51817	0.349799	0.174900	−	0.984586i	$-0.444040\pi$
$593$	8.51817	0.349799	0.174900	+	0.984586i	$0.444040\pi$
$594$	0	0
$595$	−12.2081	−0.500484
$596$	−14.5765	−0.597077
$597$	0	0
$598$	5.57179	0.227847
$599$	−35.6398	−1.45620	−0.728101	−	0.685470i	$-0.759597\pi$
$599$	−35.6398	−1.45620	−0.728101	+	0.685470i	$0.759597\pi$
$600$	0	0
$601$	21.3659	0.871534	0.435767	−	0.900059i	$-0.356477\pi$
$601$	21.3659	0.871534	0.435767	+	0.900059i	$0.356477\pi$
$602$	−8.56109	−0.348924
$603$	0	0
$604$	−16.0150	−0.651639
$605$	−31.0884	−1.26392
$606$	0	0
$607$	−7.55226	−0.306537	−0.153268	−	0.988185i	$-0.548980\pi$
$607$	−7.55226	−0.306537	−0.153268	+	0.988185i	$0.548980\pi$
$608$	−25.7640	−1.04487
$609$	0	0
$610$	1.02374	0.0414501
$611$	2.83037	0.114505
$612$	0	0
$613$	−2.34120	−0.0945601	−0.0472800	−	0.998882i	$-0.515055\pi$
$613$	−2.34120	−0.0945601	−0.0472800	+	0.998882i	$0.515055\pi$
$614$	34.0525	1.37425
$615$	0	0
$616$	1.56521	0.0630641
$617$	17.1585	0.690774	0.345387	−	0.938460i	$-0.387748\pi$
$617$	17.1585	0.690774	0.345387	+	0.938460i	$0.387748\pi$
$618$	0	0
$619$	−0.926565	−0.0372418	−0.0186209	−	0.999827i	$-0.505928\pi$
$619$	−0.926565	−0.0372418	−0.0186209	+	0.999827i	$0.505928\pi$
$620$	21.8483	0.877448
$621$	0	0
$622$	−17.1193	−0.686423
$623$	−13.2257	−0.529875
$624$	0	0
$625$	−9.22130	−0.368852
$626$	16.7542	0.669631
$627$	0	0
$628$	24.8365	0.991085
$629$	17.6375	0.703252
$630$	0	0
$631$	−32.5006	−1.29383	−0.646915	−	0.762562i	$-0.723941\pi$
$631$	−32.5006	−1.29383	−0.646915	+	0.762562i	$0.723941\pi$
$632$	18.3519	0.730000
$633$	0	0
$634$	7.42030	0.294698
$635$	3.49192	0.138572
$636$	0	0
$637$	0.419006	0.0166016
$638$	−3.83239	−0.151726
$639$	0	0
$640$	−28.9031	−1.14249
$641$	−11.4244	−0.451235	−0.225617	−	0.974216i	$-0.572440\pi$
$641$	−11.4244	−0.451235	−0.225617	+	0.974216i	$0.572440\pi$
$642$	0	0
$643$	−1.81971	−0.0717623	−0.0358811	−	0.999356i	$-0.511424\pi$
$643$	−1.81971	−0.0717623	−0.0358811	+	0.999356i	$0.511424\pi$
$644$	10.1827	0.401256
$645$	0	0
$646$	24.6243	0.968832
$647$	−36.2813	−1.42636	−0.713182	−	0.700979i	$-0.752747\pi$
$647$	−36.2813	−1.42636	−0.713182	+	0.700979i	$0.752747\pi$
$648$	0	0
$649$	10.3114	0.404759
$650$	−5.57008	−0.218477
$651$	0	0
$652$	21.4514	0.840104
$653$	4.38103	0.171443	0.0857215	−	0.996319i	$-0.472680\pi$
$653$	4.38103	0.171443	0.0857215	+	0.996319i	$0.472680\pi$
$654$	0	0
$655$	−60.1513	−2.35031
$656$	47.2790	1.84594
$657$	0	0
$658$	12.4832	0.486647
$659$	5.14544	0.200438	0.100219	−	0.994965i	$-0.468046\pi$
$659$	5.14544	0.200438	0.100219	+	0.994965i	$0.468046\pi$
$660$	0	0
$661$	22.5345	0.876490	0.438245	−	0.898856i	$-0.355600\pi$
$661$	22.5345	0.876490	0.438245	+	0.898856i	$0.355600\pi$
$662$	12.8695	0.500188
$663$	0	0
$664$	−14.4337	−0.560137
$665$	13.3088	0.516095
$666$	0	0
$667$	10.3047	0.399000
$668$	−6.53069	−0.252680
$669$	0	0
$670$	−3.76835	−0.145584
$671$	0.229735	0.00886881
$672$	0	0
$673$	−25.8857	−0.997820	−0.498910	−	0.866654i	$-0.666266\pi$
$673$	−25.8857	−0.997820	−0.498910	+	0.866654i	$0.666266\pi$
$674$	41.9046	1.61411
$675$	0	0
$676$	−18.1481	−0.698004
$677$	17.1281	0.658287	0.329143	−	0.944280i	$-0.393240\pi$
$677$	17.1281	0.658287	0.329143	+	0.944280i	$0.393240\pi$
$678$	0	0
$679$	14.1419	0.542718
$680$	13.1953	0.506016
$681$	0	0
$682$	11.8322	0.453079
$683$	1.44948	0.0554627	0.0277314	−	0.999615i	$-0.491172\pi$
$683$	1.44948	0.0554627	0.0277314	+	0.999615i	$0.491172\pi$
$684$	0	0
$685$	−57.5909	−2.20044
$686$	1.84800	0.0705571
$687$	0	0
$688$	22.3647	0.852648
$689$	−4.47475	−0.170475
$690$	0	0
$691$	−46.8428	−1.78198	−0.890991	−	0.454020i	$-0.849989\pi$
$691$	−46.8428	−1.78198	−0.890991	+	0.454020i	$0.849989\pi$
$692$	8.58495	0.326351
$693$	0	0
$694$	−30.4395	−1.15547
$695$	25.2423	0.957494
$696$	0	0
$697$	−34.2386	−1.29688
$698$	−58.7693	−2.22445
$699$	0	0
$700$	−10.1796	−0.384754
$701$	6.33668	0.239333	0.119667	−	0.992814i	$-0.461817\pi$
$701$	6.33668	0.239333	0.119667	+	0.992814i	$0.461817\pi$
$702$	0	0
$703$	−19.2277	−0.725188
$704$	4.10811	0.154830
$705$	0	0
$706$	29.0916	1.09488
$707$	−1.82679	−0.0687035
$708$	0	0
$709$	−26.7976	−1.00641	−0.503203	−	0.864168i	$-0.667845\pi$
$709$	−26.7976	−1.00641	−0.503203	+	0.864168i	$0.667845\pi$
$710$	−64.3695	−2.41574
$711$	0	0
$712$	14.2951	0.535731
$713$	−31.8150	−1.19148
$714$	0	0
$715$	−2.11878	−0.0792381
$716$	−35.6162	−1.33104
$717$	0	0
$718$	16.6125	0.619972
$719$	12.0689	0.450093	0.225047	−	0.974348i	$-0.427747\pi$
$719$	12.0689	0.450093	0.225047	+	0.974348i	$0.427747\pi$
$720$	0	0
$721$	11.7530	0.437705
$722$	8.26752	0.307685
$723$	0	0
$724$	20.3415	0.755987
$725$	−10.3015	−0.382590
$726$	0	0
$727$	−29.2807	−1.08596	−0.542980	−	0.839746i	$-0.682704\pi$
$727$	−29.2807	−1.08596	−0.542980	+	0.839746i	$0.682704\pi$
$728$	−0.452887	−0.0167851
$729$	0	0
$730$	−70.3666	−2.60438
$731$	−16.1961	−0.599035
$732$	0	0
$733$	19.8440	0.732956	0.366478	−	0.930427i	$-0.380564\pi$
$733$	19.8440	0.732956	0.366478	+	0.930427i	$0.380564\pi$
$734$	31.2655	1.15403
$735$	0	0
$736$	−48.6416	−1.79295
$737$	−0.845643	−0.0311497
$738$	0	0
$739$	14.9392	0.549546	0.274773	−	0.961509i	$-0.411397\pi$
$739$	14.9392	0.549546	0.274773	+	0.961509i	$0.411397\pi$
$740$	24.9292	0.916416
$741$	0	0
$742$	−19.7357	−0.724520
$743$	18.3036	0.671495	0.335748	−	0.941952i	$-0.391011\pi$
$743$	18.3036	0.671495	0.335748	+	0.941952i	$0.391011\pi$
$744$	0	0
$745$	−35.9686	−1.31779
$746$	−51.2964	−1.87810
$747$	0	0
$748$	−7.16443	−0.261957
$749$	8.74315	0.319468
$750$	0	0
$751$	−45.1608	−1.64794	−0.823970	−	0.566634i	$-0.808245\pi$
$751$	−45.1608	−1.64794	−0.823970	+	0.566634i	$0.808245\pi$
$752$	−32.6108	−1.18919
$753$	0	0
$754$	1.10889	0.0403832
$755$	−39.5181	−1.43821
$756$	0	0
$757$	23.8554	0.867040	0.433520	−	0.901144i	$-0.357271\pi$
$757$	23.8554	0.867040	0.433520	+	0.901144i	$0.357271\pi$
$758$	−48.0876	−1.74662
$759$	0	0
$760$	−14.3850	−0.521799
$761$	−41.1513	−1.49173	−0.745866	−	0.666096i	$-0.767964\pi$
$761$	−41.1513	−1.49173	−0.745866	+	0.666096i	$0.767964\pi$
$762$	0	0
$763$	12.5844	0.455585
$764$	12.4785	0.451456
$765$	0	0
$766$	34.1336	1.23330
$767$	−2.98357	−0.107730
$768$	0	0
$769$	0.0467258	0.00168498	0.000842488	−	1.00000i	$-0.499732\pi$
$769$	0.0467258	0.00168498	0.000842488		1.00000i	$0.499732\pi$
$770$	−9.34480	−0.336763
$771$	0	0
$772$	−10.7206	−0.385844
$773$	−1.51524	−0.0544994	−0.0272497	−	0.999629i	$-0.508675\pi$
$773$	−1.51524	−0.0544994	−0.0272497	+	0.999629i	$0.508675\pi$
$774$	0	0
$775$	31.8053	1.14248
$776$	−15.2855	−0.548716
$777$	0	0
$778$	43.3508	1.55420
$779$	37.3256	1.33733
$780$	0	0
$781$	−14.4450	−0.516881
$782$	46.4900	1.66248
$783$	0	0
$784$	−4.82768	−0.172417
$785$	61.2860	2.18739
$786$	0	0
$787$	−10.0243	−0.357327	−0.178663	−	0.983910i	$-0.557177\pi$
$787$	−10.0243	−0.357327	−0.178663	+	0.983910i	$0.557177\pi$
$788$	22.1811	0.790168
$789$	0	0
$790$	−109.567	−3.89821
$791$	0.626982	0.0222929
$792$	0	0
$793$	−0.0664727	−0.00236052
$794$	−67.2246	−2.38571
$795$	0	0
$796$	−16.6089	−0.588685
$797$	34.0735	1.20694	0.603472	−	0.797384i	$-0.293783\pi$
$797$	34.0735	1.20694	0.603472	+	0.797384i	$0.293783\pi$
$798$	0	0
$799$	23.6161	0.835478
$800$	48.6268	1.71922
$801$	0	0
$802$	−25.1837	−0.889266
$803$	−15.7908	−0.557244
$804$	0	0
$805$	25.1267	0.885600
$806$	−3.42360	−0.120591
$807$	0	0
$808$	1.97450	0.0694628
$809$	−15.2746	−0.537026	−0.268513	−	0.963276i	$-0.586532\pi$
$809$	−15.2746	−0.537026	−0.268513	+	0.963276i	$0.586532\pi$
$810$	0	0
$811$	20.6806	0.726194	0.363097	−	0.931751i	$-0.381719\pi$
$811$	20.6806	0.726194	0.363097	+	0.931751i	$0.381719\pi$
$812$	2.02655	0.0711179
$813$	0	0
$814$	13.5007	0.473201
$815$	52.9331	1.85416
$816$	0	0
$817$	17.6564	0.617720
$818$	−58.0616	−2.03008
$819$	0	0
$820$	−48.3936	−1.68998
$821$	−46.0840	−1.60834	−0.804172	−	0.594397i	$-0.797391\pi$
$821$	−46.0840	−1.60834	−0.804172	+	0.594397i	$0.797391\pi$
$822$	0	0
$823$	26.8038	0.934323	0.467162	−	0.884172i	$-0.345277\pi$
$823$	26.8038	0.934323	0.467162	+	0.884172i	$0.345277\pi$
$824$	−12.7034	−0.442543
$825$	0	0
$826$	−13.1589	−0.457856
$827$	8.30934	0.288944	0.144472	−	0.989509i	$-0.453852\pi$
$827$	8.30934	0.288944	0.144472	+	0.989509i	$0.453852\pi$
$828$	0	0
$829$	32.2106	1.11872	0.559360	−	0.828925i	$-0.311047\pi$
$829$	32.2106	1.11872	0.559360	+	0.828925i	$0.311047\pi$
$830$	86.1739	2.99114
$831$	0	0
$832$	−1.18866	−0.0412095
$833$	3.49611	0.121133
$834$	0	0
$835$	−16.1150	−0.557681
$836$	7.81040	0.270128
$837$	0	0
$838$	16.1702	0.558589
$839$	28.8183	0.994918	0.497459	−	0.867487i	$-0.334266\pi$
$839$	28.8183	0.994918	0.497459	+	0.867487i	$0.334266\pi$
$840$	0	0
$841$	−26.9492	−0.929282
$842$	−52.7643	−1.81838
$843$	0	0
$844$	−10.4013	−0.358027
$845$	−44.7818	−1.54054
$846$	0	0
$847$	8.90296	0.305909
$848$	51.5570	1.77047
$849$	0	0
$850$	−46.4758	−1.59411
$851$	−36.3014	−1.24440
$852$	0	0
$853$	46.9004	1.60584	0.802919	−	0.596089i	$-0.203279\pi$
$853$	46.9004	1.60584	0.802919	+	0.596089i	$0.203279\pi$
$854$	−0.293175	−0.0100322
$855$	0	0
$856$	−9.45012	−0.322999
$857$	−39.1541	−1.33748	−0.668739	−	0.743497i	$-0.733166\pi$
$857$	−39.1541	−1.33748	−0.668739	+	0.743497i	$0.733166\pi$
$858$	0	0
$859$	14.0808	0.480432	0.240216	−	0.970720i	$-0.422782\pi$
$859$	14.0808	0.480432	0.240216	+	0.970720i	$0.422782\pi$
$860$	−22.8920	−0.780609
$861$	0	0
$862$	−2.91269	−0.0992065
$863$	−15.8939	−0.541036	−0.270518	−	0.962715i	$-0.587195\pi$
$863$	−15.8939	−0.541036	−0.270518	+	0.962715i	$0.587195\pi$
$864$	0	0
$865$	21.1840	0.720278
$866$	−27.3910	−0.930782
$867$	0	0
$868$	−6.25682	−0.212370
$869$	−24.5875	−0.834075
$870$	0	0
$871$	0.244683	0.00829077
$872$	−13.6019	−0.460620
$873$	0	0
$874$	−50.6817	−1.71433
$875$	−7.65942	−0.258936
$876$	0	0
$877$	−8.40763	−0.283906	−0.141953	−	0.989873i	$-0.545338\pi$
$877$	−8.40763	−0.283906	−0.141953	+	0.989873i	$0.545338\pi$
$878$	−3.24409	−0.109483
$879$	0	0
$880$	24.4121	0.822931
$881$	13.0385	0.439277	0.219639	−	0.975581i	$-0.429512\pi$
$881$	13.0385	0.439277	0.219639	+	0.975581i	$0.429512\pi$
$882$	0	0
$883$	−49.0892	−1.65198	−0.825991	−	0.563684i	$-0.809384\pi$
$883$	−49.0892	−1.65198	−0.825991	+	0.563684i	$0.809384\pi$
$884$	2.07299	0.0697224
$885$	0	0
$886$	50.9073	1.71026
$887$	−19.3140	−0.648500	−0.324250	−	0.945971i	$-0.605112\pi$
$887$	−19.3140	−0.648500	−0.324250	+	0.945971i	$0.605112\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	−85.3461	−2.86081
$891$	0	0
$892$	6.66708	0.223230
$893$	−25.7454	−0.861538
$894$	0	0
$895$	−87.8856	−2.93769
$896$	8.27714	0.276520
$897$	0	0
$898$	41.5174	1.38545
$899$	−6.33175	−0.211176
$900$	0	0
$901$	−37.3365	−1.24386
$902$	−26.2082	−0.872636
$903$	0	0
$904$	−0.677681	−0.0225393
$905$	50.1943	1.66851
$906$	0	0
$907$	−31.7689	−1.05487	−0.527434	−	0.849596i	$-0.676846\pi$
$907$	−31.7689	−1.05487	−0.527434	+	0.849596i	$0.676846\pi$
$908$	−33.8962	−1.12489
$909$	0	0
$910$	2.70388	0.0896326
$911$	−1.29050	−0.0427562	−0.0213781	−	0.999771i	$-0.506805\pi$
$911$	−1.29050	−0.0427562	−0.0213781	+	0.999771i	$0.506805\pi$
$912$	0	0
$913$	19.3380	0.639995
$914$	−33.9791	−1.12393
$915$	0	0
$916$	−4.00260	−0.132250
$917$	17.2259	0.568849
$918$	0	0
$919$	−48.2784	−1.59256	−0.796278	−	0.604931i	$-0.793201\pi$
$919$	−48.2784	−1.59256	−0.796278	+	0.604931i	$0.793201\pi$
$920$	−27.1585	−0.895388
$921$	0	0
$922$	−7.62766	−0.251204
$923$	4.17959	0.137573
$924$	0	0
$925$	36.2903	1.19322
$926$	2.93984	0.0966093
$927$	0	0
$928$	−9.68056	−0.317780
$929$	36.9922	1.21368	0.606838	−	0.794826i	$-0.292438\pi$
$929$	36.9922	1.21368	0.606838	+	0.794826i	$0.292438\pi$
$930$	0	0
$931$	−3.81133	−0.124911
$932$	−20.5195	−0.672140
$933$	0	0
$934$	−31.1927	−1.02066
$935$	−17.6788	−0.578157
$936$	0	0
$937$	−4.71575	−0.154057	−0.0770284	−	0.997029i	$-0.524543\pi$
$937$	−4.71575	−0.154057	−0.0770284	+	0.997029i	$0.524543\pi$
$938$	1.07916	0.0352359
$939$	0	0
$940$	33.3796	1.08872
$941$	−42.7164	−1.39252	−0.696258	−	0.717792i	$-0.745153\pi$
$941$	−42.7164	−1.39252	−0.696258	+	0.717792i	$0.745153\pi$
$942$	0	0
$943$	70.4697	2.29481
$944$	34.3759	1.11884
$945$	0	0
$946$	−12.3974	−0.403076
$947$	−39.7768	−1.29257	−0.646286	−	0.763096i	$-0.723679\pi$
$947$	−39.7768	−1.29257	−0.646286	+	0.763096i	$0.723679\pi$
$948$	0	0
$949$	4.56899	0.148316
$950$	50.6662	1.64383
$951$	0	0
$952$	−3.77881	−0.122472
$953$	−6.09718	−0.197507	−0.0987535	−	0.995112i	$-0.531486\pi$
$953$	−6.09718	−0.197507	−0.0987535	+	0.995112i	$0.531486\pi$
$954$	0	0
$955$	30.7916	0.996394
$956$	7.58980	0.245472
$957$	0	0
$958$	67.1920	2.17088
$959$	16.4927	0.532575
$960$	0	0
$961$	−11.4512	−0.369393
$962$	−3.90638	−0.125947
$963$	0	0
$964$	−0.112015	−0.00360776
$965$	−26.4539	−0.851582
$966$	0	0
$967$	29.0575	0.934427	0.467214	−	0.884144i	$-0.345258\pi$
$967$	29.0575	0.934427	0.467214	+	0.884144i	$0.345258\pi$
$968$	−9.62286	−0.309290
$969$	0	0
$970$	91.2590	2.93015
$971$	11.9169	0.382430	0.191215	−	0.981548i	$-0.438757\pi$
$971$	11.9169	0.382430	0.191215	+	0.981548i	$0.438757\pi$
$972$	0	0
$973$	−7.22877	−0.231744
$974$	−70.5535	−2.26068
$975$	0	0
$976$	0.765881	0.0245153
$977$	20.5871	0.658638	0.329319	−	0.944219i	$-0.393181\pi$
$977$	20.5871	0.658638	0.329319	+	0.944219i	$0.393181\pi$
$978$	0	0
$979$	−19.1523	−0.612109
$980$	4.94148	0.157850
$981$	0	0
$982$	−32.8663	−1.04880
$983$	−53.1167	−1.69416	−0.847080	−	0.531466i	$-0.821641\pi$
$983$	−53.1167	−1.69416	−0.847080	+	0.531466i	$0.821641\pi$
$984$	0	0
$985$	54.7335	1.74395
$986$	9.25234	0.294655
$987$	0	0
$988$	−2.25990	−0.0718971
$989$	33.3348	1.05998
$990$	0	0
$991$	−44.1874	−1.40366	−0.701829	−	0.712345i	$-0.747633\pi$
$991$	−44.1874	−1.40366	−0.701829	+	0.712345i	$0.747633\pi$
$992$	29.8880	0.948945
$993$	0	0
$994$	18.4339	0.584687
$995$	−40.9836	−1.29927
$996$	0	0
$997$	−26.6778	−0.844896	−0.422448	−	0.906387i	$-0.638829\pi$
$997$	−26.6778	−0.844896	−0.422448	+	0.906387i	$0.638829\pi$
$998$	29.9121	0.946851
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.r.1.4		16
3.2	odd	2		2667.2.a.o.1.13	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.o.1.13	✓	16	3.2	odd	2
8001.2.a.r.1.4		16	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)

\(f(q)\)	\(=\)	\(q-1.84800 q^{2} +1.41512 q^{4} +3.49192 q^{5} -1.00000 q^{7} +1.08086 q^{8} +O(q^{10})\)	\(q-1.84800 q^{2} +1.41512 q^{4} +3.49192 q^{5} -1.00000 q^{7} +1.08086 q^{8} -6.45308 q^{10} -1.44812 q^{11} +0.419006 q^{13} +1.84800 q^{14} -4.82768 q^{16} +3.49611 q^{17} -3.81133 q^{19} +4.94148 q^{20} +2.67612 q^{22} -7.19568 q^{23} +7.19347 q^{25} -0.774324 q^{26} -1.41512 q^{28} -1.43207 q^{29} +4.42140 q^{31} +6.75984 q^{32} -6.46082 q^{34} -3.49192 q^{35} +5.04489 q^{37} +7.04336 q^{38} +3.77427 q^{40} -9.79333 q^{41} -4.63261 q^{43} -2.04926 q^{44} +13.2976 q^{46} +6.75497 q^{47} +1.00000 q^{49} -13.2936 q^{50} +0.592943 q^{52} -10.6795 q^{53} -5.05670 q^{55} -1.08086 q^{56} +2.64647 q^{58} -7.12059 q^{59} -0.158644 q^{61} -8.17077 q^{62} -2.83687 q^{64} +1.46313 q^{65} +0.583961 q^{67} +4.94741 q^{68} +6.45308 q^{70} +9.97501 q^{71} +10.9043 q^{73} -9.32297 q^{74} -5.39349 q^{76} +1.44812 q^{77} +16.9790 q^{79} -16.8578 q^{80} +18.0981 q^{82} -13.3539 q^{83} +12.2081 q^{85} +8.56109 q^{86} -1.56521 q^{88} +13.2257 q^{89} -0.419006 q^{91} -10.1827 q^{92} -12.4832 q^{94} -13.3088 q^{95} -14.1419 q^{97} -1.84800 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 5 q^{2} + 19 q^{4} + q^{5} - 16 q^{7} - 6 q^{8}+O(q^{10}) \) 16 * q - 5 * q^2 + 19 * q^4 + q^5 - 16 * q^7 - 6 * q^8	\( 16 q - 5 q^{2} + 19 q^{4} + q^{5} - 16 q^{7} - 6 q^{8} - 12 q^{10} - 11 q^{11} + 18 q^{13} + 5 q^{14} + 25 q^{16} + 5 q^{17} - 11 q^{19} + q^{20} + q^{22} - 13 q^{23} + 33 q^{25} - 8 q^{26} - 19 q^{28} - 24 q^{29} - 42 q^{31} - 42 q^{32} + 9 q^{34} - q^{35} + 40 q^{37} - 38 q^{38} - 61 q^{40} - 9 q^{41} + 7 q^{43} - 3 q^{44} + 24 q^{46} - 31 q^{47} + 16 q^{49} - 6 q^{50} + 52 q^{52} - 66 q^{53} - 36 q^{55} + 6 q^{56} + 19 q^{58} + 7 q^{59} + 6 q^{61} - 52 q^{62} + 10 q^{64} - 51 q^{65} + 16 q^{67} - 14 q^{68} + 12 q^{70} - 46 q^{71} + 39 q^{73} - 72 q^{74} + 24 q^{76} + 11 q^{77} + 4 q^{79} + 2 q^{80} - 18 q^{82} - 15 q^{83} - 4 q^{85} - 14 q^{86} + 58 q^{88} + q^{89} - 18 q^{91} - 26 q^{92} + 5 q^{94} - 44 q^{95} + 41 q^{97} - 5 q^{98}+O(q^{100}) \) 16 * q - 5 * q^2 + 19 * q^4 + q^5 - 16 * q^7 - 6 * q^8 - 12 * q^10 - 11 * q^11 + 18 * q^13 + 5 * q^14 + 25 * q^16 + 5 * q^17 - 11 * q^19 + q^20 + q^22 - 13 * q^23 + 33 * q^25 - 8 * q^26 - 19 * q^28 - 24 * q^29 - 42 * q^31 - 42 * q^32 + 9 * q^34 - q^35 + 40 * q^37 - 38 * q^38 - 61 * q^40 - 9 * q^41 + 7 * q^43 - 3 * q^44 + 24 * q^46 - 31 * q^47 + 16 * q^49 - 6 * q^50 + 52 * q^52 - 66 * q^53 - 36 * q^55 + 6 * q^56 + 19 * q^58 + 7 * q^59 + 6 * q^61 - 52 * q^62 + 10 * q^64 - 51 * q^65 + 16 * q^67 - 14 * q^68 + 12 * q^70 - 46 * q^71 + 39 * q^73 - 72 * q^74 + 24 * q^76 + 11 * q^77 + 4 * q^79 + 2 * q^80 - 18 * q^82 - 15 * q^83 - 4 * q^85 - 14 * q^86 + 58 * q^88 + q^89 - 18 * q^91 - 26 * q^92 + 5 * q^94 - 44 * q^95 + 41 * q^97 - 5 * q^98

Introduction

L-functions

Modular forms

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Database

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