LMFDB - Embedded newform 8001.2.a.u.1.1

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:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 6 x^{17} - 11 x^{16} + 123 x^{15} - 35 x^{14} - 982 x^{13} + 988 x^{12} + 3872 x^{11} + \cdots + 16 $ x^18 - 6x^17 - 11x^16 + 123x^15 - 35x^14 - 982x^13 + 988x^12 + 3872x^11 - 5421x^10 - 7882x^9 + 13376x^8 + 7948x^7 - 15795x^6 - 3858x^5 + 8199x^4 + 1453x^3 - 1610x^2 - 380*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2\cdot 5 $
Twist minimal:	no (minimal twist has level 2667)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.55461$ 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)

$f(q)$	$=$	$q-2.55461 q^{2} +4.52602 q^{4} +2.12871 q^{5} +1.00000 q^{7} -6.45298 q^{8} +O(q^{10})$	\(q-2.55461 q^{2} +4.52602 q^{4} +2.12871 q^{5} +1.00000 q^{7} -6.45298 q^{8} -5.43801 q^{10} +4.13995 q^{11} -4.75071 q^{13} -2.55461 q^{14} +7.43278 q^{16} +1.32611 q^{17} -5.03962 q^{19} +9.63456 q^{20} -10.5759 q^{22} -0.634865 q^{23} -0.468604 q^{25} +12.1362 q^{26} +4.52602 q^{28} +6.95445 q^{29} -3.31737 q^{31} -6.08189 q^{32} -3.38770 q^{34} +2.12871 q^{35} +7.32426 q^{37} +12.8742 q^{38} -13.7365 q^{40} -4.92704 q^{41} -0.118376 q^{43} +18.7375 q^{44} +1.62183 q^{46} -10.6971 q^{47} +1.00000 q^{49} +1.19710 q^{50} -21.5018 q^{52} -4.29353 q^{53} +8.81274 q^{55} -6.45298 q^{56} -17.7659 q^{58} +9.53007 q^{59} -9.70203 q^{61} +8.47457 q^{62} +0.671262 q^{64} -10.1129 q^{65} +2.95734 q^{67} +6.00202 q^{68} -5.43801 q^{70} -1.45245 q^{71} +7.08465 q^{73} -18.7106 q^{74} -22.8094 q^{76} +4.13995 q^{77} +5.36766 q^{79} +15.8222 q^{80} +12.5867 q^{82} +10.4852 q^{83} +2.82291 q^{85} +0.302403 q^{86} -26.7150 q^{88} +3.15112 q^{89} -4.75071 q^{91} -2.87341 q^{92} +27.3270 q^{94} -10.7279 q^{95} -16.7425 q^{97} -2.55461 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 6 q^{2} + 22 q^{4} + 10 q^{5} + 18 q^{7} + 21 q^{8}+O(q^{10}) $ 18 * q + 6 * q^2 + 22 * q^4 + 10 * q^5 + 18 * q^7 + 21 * q^8	$ 18 q + 6 q^{2} + 22 q^{4} + 10 q^{5} + 18 q^{7} + 21 q^{8} - 4 q^{10} + 9 q^{11} - 25 q^{13} + 6 q^{14} + 34 q^{16} + 17 q^{17} - 5 q^{19} + 21 q^{20} + 5 q^{22} + 14 q^{23} + 28 q^{25} + 8 q^{26} + 22 q^{28} + 17 q^{29} + 5 q^{31} + 53 q^{32} - 19 q^{34} + 10 q^{35} - 15 q^{37} + 22 q^{38} - q^{40} + 17 q^{41} + q^{43} + 33 q^{44} + 10 q^{46} + 31 q^{47} + 18 q^{49} + 35 q^{50} - 70 q^{52} + 35 q^{53} + 4 q^{55} + 21 q^{56} + 3 q^{58} + 46 q^{59} - 5 q^{61} + 10 q^{62} + 63 q^{64} + 12 q^{65} + 6 q^{67} + 56 q^{68} - 4 q^{70} + 22 q^{71} - 16 q^{73} - 18 q^{74} + 32 q^{76} + 9 q^{77} + 46 q^{79} + 30 q^{80} - 12 q^{82} + 46 q^{83} + 4 q^{85} - 18 q^{86} + 30 q^{88} + 42 q^{89} - 25 q^{91} + 48 q^{92} + 3 q^{94} + 2 q^{95} - 35 q^{97} + 6 q^{98}+O(q^{100}) $ 18 * q + 6 * q^2 + 22 * q^4 + 10 * q^5 + 18 * q^7 + 21 * q^8 - 4 * q^10 + 9 * q^11 - 25 * q^13 + 6 * q^14 + 34 * q^16 + 17 * q^17 - 5 * q^19 + 21 * q^20 + 5 * q^22 + 14 * q^23 + 28 * q^25 + 8 * q^26 + 22 * q^28 + 17 * q^29 + 5 * q^31 + 53 * q^32 - 19 * q^34 + 10 * q^35 - 15 * q^37 + 22 * q^38 - q^40 + 17 * q^41 + q^43 + 33 * q^44 + 10 * q^46 + 31 * q^47 + 18 * q^49 + 35 * q^50 - 70 * q^52 + 35 * q^53 + 4 * q^55 + 21 * q^56 + 3 * q^58 + 46 * q^59 - 5 * q^61 + 10 * q^62 + 63 * q^64 + 12 * q^65 + 6 * q^67 + 56 * q^68 - 4 * q^70 + 22 * q^71 - 16 * q^73 - 18 * q^74 + 32 * q^76 + 9 * q^77 + 46 * q^79 + 30 * q^80 - 12 * q^82 + 46 * q^83 + 4 * q^85 - 18 * q^86 + 30 * q^88 + 42 * q^89 - 25 * q^91 + 48 * q^92 + 3 * q^94 + 2 * q^95 - 35 * q^97 + 6 * 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 $

$2$	−2.55461	−1.80638	−0.903190	−	0.429241i	$-0.858781\pi$
$2$	−2.55461	−1.80638	−0.903190	+	0.429241i	$0.858781\pi$
$3$	0	0
$3$	0	0
$4$	4.52602	2.26301
$5$	2.12871	0.951987	0.475993	−	0.879449i	$-0.342089\pi$
$5$	2.12871	0.951987	0.475993	+	0.879449i	$0.342089\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−6.45298	−2.28147
$9$	0	0
$10$	−5.43801	−1.71965
$11$	4.13995	1.24824	0.624121	−	0.781328i	$-0.285457\pi$
$11$	4.13995	1.24824	0.624121	+	0.781328i	$0.285457\pi$
$12$	0	0
$13$	−4.75071	−1.31761	−0.658805	−	0.752313i	$-0.728938\pi$
$13$	−4.75071	−1.31761	−0.658805	+	0.752313i	$0.728938\pi$
$14$	−2.55461	−0.682747
$15$	0	0
$16$	7.43278	1.85820
$17$	1.32611	0.321630	0.160815	−	0.986985i	$-0.448588\pi$
$17$	1.32611	0.321630	0.160815	+	0.986985i	$0.448588\pi$
$18$	0	0
$19$	−5.03962	−1.15617	−0.578083	−	0.815978i	$-0.696199\pi$
$19$	−5.03962	−1.15617	−0.578083	+	0.815978i	$0.696199\pi$
$20$	9.63456	2.15435
$21$	0	0
$22$	−10.5759	−2.25480
$23$	−0.634865	−0.132378	−0.0661892	−	0.997807i	$-0.521084\pi$
$23$	−0.634865	−0.132378	−0.0661892	+	0.997807i	$0.521084\pi$
$24$	0	0
$25$	−0.468604	−0.0937208
$26$	12.1362	2.38011
$27$	0	0
$28$	4.52602	0.855336
$29$	6.95445	1.29141	0.645705	−	0.763587i	$-0.276564\pi$
$29$	6.95445	1.29141	0.645705	+	0.763587i	$0.276564\pi$
$30$	0	0
$31$	−3.31737	−0.595817	−0.297908	−	0.954594i	$-0.596289\pi$
$31$	−3.31737	−0.595817	−0.297908	+	0.954594i	$0.596289\pi$
$32$	−6.08189	−1.07514
$33$	0	0
$34$	−3.38770	−0.580986
$35$	2.12871	0.359817
$36$	0	0
$37$	7.32426	1.20410	0.602051	−	0.798458i	$-0.294350\pi$
$37$	7.32426	1.20410	0.602051	+	0.798458i	$0.294350\pi$
$38$	12.8742	2.08848
$39$	0	0
$40$	−13.7365	−2.17193
$41$	−4.92704	−0.769475	−0.384737	−	0.923026i	$-0.625708\pi$
$41$	−4.92704	−0.769475	−0.384737	+	0.923026i	$0.625708\pi$
$42$	0	0
$43$	−0.118376	−0.0180521	−0.00902606	−	0.999959i	$-0.502873\pi$
$43$	−0.118376	−0.0180521	−0.00902606	+	0.999959i	$0.502873\pi$
$44$	18.7375	2.82478
$45$	0	0
$46$	1.62183	0.239126
$47$	−10.6971	−1.56034	−0.780169	−	0.625568i	$-0.784867\pi$
$47$	−10.6971	−1.56034	−0.780169	+	0.625568i	$0.784867\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	1.19710	0.169295
$51$	0	0
$52$	−21.5018	−2.98176
$53$	−4.29353	−0.589762	−0.294881	−	0.955534i	$-0.595280\pi$
$53$	−4.29353	−0.589762	−0.294881	+	0.955534i	$0.595280\pi$
$54$	0	0
$55$	8.81274	1.18831
$56$	−6.45298	−0.862315
$57$	0	0
$58$	−17.7659	−2.33278
$59$	9.53007	1.24071	0.620355	−	0.784321i	$-0.286989\pi$
$59$	9.53007	1.24071	0.620355	+	0.784321i	$0.286989\pi$
$60$	0	0
$61$	−9.70203	−1.24222	−0.621109	−	0.783724i	$-0.713318\pi$
$61$	−9.70203	−1.24222	−0.621109	+	0.783724i	$0.713318\pi$
$62$	8.47457	1.07627
$63$	0	0
$64$	0.671262	0.0839078
$65$	−10.1129	−1.25435
$66$	0	0
$67$	2.95734	0.361297	0.180648	−	0.983548i	$-0.442180\pi$
$67$	2.95734	0.361297	0.180648	+	0.983548i	$0.442180\pi$
$68$	6.00202	0.727851
$69$	0	0
$70$	−5.43801	−0.649967
$71$	−1.45245	−0.172375	−0.0861873	−	0.996279i	$-0.527468\pi$
$71$	−1.45245	−0.172375	−0.0861873	+	0.996279i	$0.527468\pi$
$72$	0	0
$73$	7.08465	0.829195	0.414598	−	0.910005i	$-0.363922\pi$
$73$	7.08465	0.829195	0.414598	+	0.910005i	$0.363922\pi$
$74$	−18.7106	−2.17506
$75$	0	0
$76$	−22.8094	−2.61641
$77$	4.13995	0.471791
$78$	0	0
$79$	5.36766	0.603909	0.301954	−	0.953322i	$-0.402361\pi$
$79$	5.36766	0.603909	0.301954	+	0.953322i	$0.402361\pi$
$80$	15.8222	1.76898
$81$	0	0
$82$	12.5867	1.38996
$83$	10.4852	1.15090	0.575451	−	0.817836i	$-0.304826\pi$
$83$	10.4852	1.15090	0.575451	+	0.817836i	$0.304826\pi$
$84$	0	0
$85$	2.82291	0.306188
$86$	0.302403	0.0326090
$87$	0	0
$88$	−26.7150	−2.84783
$89$	3.15112	0.334018	0.167009	−	0.985955i	$-0.446589\pi$
$89$	3.15112	0.334018	0.167009	+	0.985955i	$0.446589\pi$
$90$	0	0
$91$	−4.75071	−0.498010
$92$	−2.87341	−0.299574
$93$	0	0
$94$	27.3270	2.81856
$95$	−10.7279	−1.10066
$96$	0	0
$97$	−16.7425	−1.69994	−0.849970	−	0.526831i	$-0.823380\pi$
$97$	−16.7425	−1.69994	−0.849970	+	0.526831i	$0.823380\pi$
$98$	−2.55461	−0.258054
$99$	0	0
$100$	−2.12091	−0.212091
$101$	18.6303	1.85378	0.926891	−	0.375331i	$-0.122471\pi$
$101$	18.6303	1.85378	0.926891	+	0.375331i	$0.122471\pi$
$102$	0	0
$103$	7.48405	0.737426	0.368713	−	0.929543i	$-0.379799\pi$
$103$	7.48405	0.737426	0.368713	+	0.929543i	$0.379799\pi$
$104$	30.6562	3.00609
$105$	0	0
$106$	10.9683	1.06533
$107$	11.2137	1.08407	0.542035	−	0.840356i	$-0.317654\pi$
$107$	11.2137	1.08407	0.542035	+	0.840356i	$0.317654\pi$
$108$	0	0
$109$	3.85001	0.368765	0.184382	−	0.982855i	$-0.440972\pi$
$109$	3.85001	0.368765	0.184382	+	0.982855i	$0.440972\pi$
$110$	−22.5131	−2.14654
$111$	0	0
$112$	7.43278	0.702332
$113$	−4.76521	−0.448273	−0.224137	−	0.974558i	$-0.571956\pi$
$113$	−4.76521	−0.448273	−0.224137	+	0.974558i	$0.571956\pi$
$114$	0	0
$115$	−1.35144	−0.126023
$116$	31.4759	2.92247
$117$	0	0
$118$	−24.3456	−2.24119
$119$	1.32611	0.121565
$120$	0	0
$121$	6.13918	0.558107
$122$	24.7849	2.24392
$123$	0	0
$124$	−15.0145	−1.34834
$125$	−11.6411	−1.04121
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	10.4490	0.923567
$129$	0	0
$130$	25.8344	2.26583
$131$	11.0346	0.964098	0.482049	−	0.876144i	$-0.339893\pi$
$131$	11.0346	0.964098	0.482049	+	0.876144i	$0.339893\pi$
$132$	0	0
$133$	−5.03962	−0.436990
$134$	−7.55485	−0.652639
$135$	0	0
$136$	−8.55738	−0.733790
$137$	10.6933	0.913589	0.456794	−	0.889572i	$-0.348997\pi$
$137$	10.6933	0.913589	0.456794	+	0.889572i	$0.348997\pi$
$138$	0	0
$139$	−3.73393	−0.316708	−0.158354	−	0.987382i	$-0.550619\pi$
$139$	−3.73393	−0.316708	−0.158354	+	0.987382i	$0.550619\pi$
$140$	9.63456	0.814269
$141$	0	0
$142$	3.71045	0.311374
$143$	−19.6677	−1.64470
$144$	0	0
$145$	14.8040	1.22940
$146$	−18.0985	−1.49784
$147$	0	0
$148$	33.1497	2.72489
$149$	11.4885	0.941174	0.470587	−	0.882354i	$-0.344042\pi$
$149$	11.4885	0.941174	0.470587	+	0.882354i	$0.344042\pi$
$150$	0	0
$151$	12.9894	1.05707	0.528533	−	0.848913i	$-0.322742\pi$
$151$	12.9894	1.05707	0.528533	+	0.848913i	$0.322742\pi$
$152$	32.5205	2.63776
$153$	0	0
$154$	−10.5759	−0.852234
$155$	−7.06171	−0.567210
$156$	0	0
$157$	12.6471	1.00934	0.504672	−	0.863311i	$-0.331613\pi$
$157$	12.6471	1.00934	0.504672	+	0.863311i	$0.331613\pi$
$158$	−13.7123	−1.09089
$159$	0	0
$160$	−12.9466	−1.02352
$161$	−0.634865	−0.0500344
$162$	0	0
$163$	6.11310	0.478815	0.239407	−	0.970919i	$-0.423047\pi$
$163$	6.11310	0.478815	0.239407	+	0.970919i	$0.423047\pi$
$164$	−22.2999	−1.74133
$165$	0	0
$166$	−26.7856	−2.07897
$167$	−18.3586	−1.42063	−0.710314	−	0.703885i	$-0.751447\pi$
$167$	−18.3586	−1.42063	−0.710314	+	0.703885i	$0.751447\pi$
$168$	0	0
$169$	9.56928	0.736098
$170$	−7.21143	−0.553091
$171$	0	0
$172$	−0.535770	−0.0408521
$173$	−18.7445	−1.42512	−0.712558	−	0.701613i	$-0.752464\pi$
$173$	−18.7445	−1.42512	−0.712558	+	0.701613i	$0.752464\pi$
$174$	0	0
$175$	−0.468604	−0.0354231
$176$	30.7713	2.31948
$177$	0	0
$178$	−8.04987	−0.603363
$179$	−2.99701	−0.224007	−0.112004	−	0.993708i	$-0.535727\pi$
$179$	−2.99701	−0.224007	−0.112004	+	0.993708i	$0.535727\pi$
$180$	0	0
$181$	24.4052	1.81402	0.907012	−	0.421105i	$-0.138358\pi$
$181$	24.4052	1.81402	0.907012	+	0.421105i	$0.138358\pi$
$182$	12.1362	0.899595
$183$	0	0
$184$	4.09677	0.302018
$185$	15.5912	1.14629
$186$	0	0
$187$	5.49005	0.401472
$188$	−48.4154	−3.53106
$189$	0	0
$190$	27.4055	1.98820
$191$	15.5134	1.12251	0.561254	−	0.827643i	$-0.310319\pi$
$191$	15.5134	1.12251	0.561254	+	0.827643i	$0.310319\pi$
$192$	0	0
$193$	23.3101	1.67789	0.838947	−	0.544213i	$-0.183172\pi$
$193$	23.3101	1.67789	0.838947	+	0.544213i	$0.183172\pi$
$194$	42.7704	3.07074
$195$	0	0
$196$	4.52602	0.323287
$197$	22.0204	1.56889	0.784444	−	0.620200i	$-0.212949\pi$
$197$	22.0204	1.56889	0.784444	+	0.620200i	$0.212949\pi$
$198$	0	0
$199$	−1.90444	−0.135002	−0.0675011	−	0.997719i	$-0.521503\pi$
$199$	−1.90444	−0.135002	−0.0675011	+	0.997719i	$0.521503\pi$
$200$	3.02389	0.213821
$201$	0	0
$202$	−47.5930	−3.34863
$203$	6.95445	0.488107
$204$	0	0
$205$	−10.4882	−0.732530
$206$	−19.1188	−1.33207
$207$	0	0
$208$	−35.3110	−2.44838
$209$	−20.8637	−1.44318
$210$	0	0
$211$	14.5624	1.00252	0.501260	−	0.865297i	$-0.332870\pi$
$211$	14.5624	1.00252	0.501260	+	0.865297i	$0.332870\pi$
$212$	−19.4326	−1.33464
$213$	0	0
$214$	−28.6466	−1.95824
$215$	−0.251987	−0.0171854
$216$	0	0
$217$	−3.31737	−0.225198
$218$	−9.83527	−0.666129
$219$	0	0
$220$	39.8866	2.68915
$221$	−6.29999	−0.423783
$222$	0	0
$223$	2.83408	0.189784	0.0948919	−	0.995488i	$-0.469749\pi$
$223$	2.83408	0.189784	0.0948919	+	0.995488i	$0.469749\pi$
$224$	−6.08189	−0.406363
$225$	0	0
$226$	12.1732	0.809751
$227$	−1.45824	−0.0967865	−0.0483933	−	0.998828i	$-0.515410\pi$
$227$	−1.45824	−0.0967865	−0.0483933	+	0.998828i	$0.515410\pi$
$228$	0	0
$229$	21.1581	1.39817	0.699084	−	0.715040i	$-0.253591\pi$
$229$	21.1581	1.39817	0.699084	+	0.715040i	$0.253591\pi$
$230$	3.45240	0.227645
$231$	0	0
$232$	−44.8769	−2.94631
$233$	9.49469	0.622018	0.311009	−	0.950407i	$-0.399333\pi$
$233$	9.49469	0.622018	0.311009	+	0.950407i	$0.399333\pi$
$234$	0	0
$235$	−22.7711	−1.48542
$236$	43.1332	2.80774
$237$	0	0
$238$	−3.38770	−0.219592
$239$	25.7367	1.66477	0.832384	−	0.554199i	$-0.186975\pi$
$239$	25.7367	1.66477	0.832384	+	0.554199i	$0.186975\pi$
$240$	0	0
$241$	16.7397	1.07830	0.539149	−	0.842211i	$-0.318746\pi$
$241$	16.7397	1.07830	0.539149	+	0.842211i	$0.318746\pi$
$242$	−15.6832	−1.00815
$243$	0	0
$244$	−43.9115	−2.81115
$245$	2.12871	0.135998
$246$	0	0
$247$	23.9418	1.52338
$248$	21.4069	1.35934
$249$	0	0
$250$	29.7383	1.88082
$251$	5.36016	0.338330	0.169165	−	0.985588i	$-0.445893\pi$
$251$	5.36016	0.338330	0.169165	+	0.985588i	$0.445893\pi$
$252$	0	0
$253$	−2.62831	−0.165240
$254$	−2.55461	−0.160290
$255$	0	0
$256$	−28.0355	−1.75222
$257$	−13.1565	−0.820678	−0.410339	−	0.911933i	$-0.634590\pi$
$257$	−13.1565	−0.820678	−0.410339	+	0.911933i	$0.634590\pi$
$258$	0	0
$259$	7.32426	0.455108
$260$	−45.7710	−2.83860
$261$	0	0
$262$	−28.1891	−1.74153
$263$	15.6935	0.967702	0.483851	−	0.875150i	$-0.339238\pi$
$263$	15.6935	0.967702	0.483851	+	0.875150i	$0.339238\pi$
$264$	0	0
$265$	−9.13968	−0.561446
$266$	12.8742	0.789370
$267$	0	0
$268$	13.3850	0.817618
$269$	28.0155	1.70813	0.854067	−	0.520163i	$-0.174129\pi$
$269$	28.0155	1.70813	0.854067	+	0.520163i	$0.174129\pi$
$270$	0	0
$271$	−19.4489	−1.18143	−0.590717	−	0.806879i	$-0.701155\pi$
$271$	−19.4489	−1.18143	−0.590717	+	0.806879i	$0.701155\pi$
$272$	9.85672	0.597652
$273$	0	0
$274$	−27.3171	−1.65029
$275$	−1.94000	−0.116986
$276$	0	0
$277$	−15.0931	−0.906858	−0.453429	−	0.891292i	$-0.649799\pi$
$277$	−15.0931	−0.906858	−0.453429	+	0.891292i	$0.649799\pi$
$278$	9.53872	0.572094
$279$	0	0
$280$	−13.7365	−0.820913
$281$	15.7060	0.936943	0.468471	−	0.883479i	$-0.344805\pi$
$281$	15.7060	0.936943	0.468471	+	0.883479i	$0.344805\pi$
$282$	0	0
$283$	29.8370	1.77363	0.886813	−	0.462129i	$-0.152914\pi$
$283$	29.8370	1.77363	0.886813	+	0.462129i	$0.152914\pi$
$284$	−6.57383	−0.390085
$285$	0	0
$286$	50.2433	2.97095
$287$	−4.92704	−0.290834
$288$	0	0
$289$	−15.2414	−0.896554
$290$	−37.8184	−2.22077
$291$	0	0
$292$	32.0652	1.87648
$293$	−17.2017	−1.00493	−0.502467	−	0.864596i	$-0.667574\pi$
$293$	−17.2017	−1.00493	−0.502467	+	0.864596i	$0.667574\pi$
$294$	0	0
$295$	20.2867	1.18114
$296$	−47.2633	−2.74712
$297$	0	0
$298$	−29.3486	−1.70012
$299$	3.01606	0.174423
$300$	0	0
$301$	−0.118376	−0.00682306
$302$	−33.1829	−1.90946
$303$	0	0
$304$	−37.4584	−2.14838
$305$	−20.6528	−1.18258
$306$	0	0
$307$	−13.9522	−0.796294	−0.398147	−	0.917322i	$-0.630347\pi$
$307$	−13.9522	−0.796294	−0.398147	+	0.917322i	$0.630347\pi$
$308$	18.7375	1.06767
$309$	0	0
$310$	18.0399	1.02460
$311$	−4.04310	−0.229263	−0.114632	−	0.993408i	$-0.536569\pi$
$311$	−4.04310	−0.229263	−0.114632	+	0.993408i	$0.536569\pi$
$312$	0	0
$313$	3.51810	0.198855	0.0994274	−	0.995045i	$-0.468299\pi$
$313$	3.51810	0.198855	0.0994274	+	0.995045i	$0.468299\pi$
$314$	−32.3082	−1.82326
$315$	0	0
$316$	24.2941	1.36665
$317$	11.1408	0.625731	0.312865	−	0.949797i	$-0.398711\pi$
$317$	11.1408	0.625731	0.312865	+	0.949797i	$0.398711\pi$
$318$	0	0
$319$	28.7911	1.61199
$320$	1.42892	0.0798791
$321$	0	0
$322$	1.62183	0.0903811
$323$	−6.68311	−0.371858
$324$	0	0
$325$	2.22620	0.123487
$326$	−15.6166	−0.864921
$327$	0	0
$328$	31.7941	1.75553
$329$	−10.6971	−0.589753
$330$	0	0
$331$	24.8759	1.36730	0.683650	−	0.729810i	$-0.260391\pi$
$331$	24.8759	1.36730	0.683650	+	0.729810i	$0.260391\pi$
$332$	47.4563	2.60450
$333$	0	0
$334$	46.8989	2.56619
$335$	6.29532	0.343950
$336$	0	0
$337$	−26.4887	−1.44293	−0.721465	−	0.692451i	$-0.756531\pi$
$337$	−26.4887	−1.44293	−0.721465	+	0.692451i	$0.756531\pi$
$338$	−24.4457	−1.32967
$339$	0	0
$340$	12.7765	0.692905
$341$	−13.7337	−0.743723
$342$	0	0
$343$	1.00000	0.0539949
$344$	0.763875	0.0411854
$345$	0	0
$346$	47.8848	2.57430
$347$	−22.3607	−1.20039	−0.600193	−	0.799855i	$-0.704910\pi$
$347$	−22.3607	−1.20039	−0.600193	+	0.799855i	$0.704910\pi$
$348$	0	0
$349$	6.65405	0.356183	0.178092	−	0.984014i	$-0.443008\pi$
$349$	6.65405	0.356183	0.178092	+	0.984014i	$0.443008\pi$
$350$	1.19710	0.0639876
$351$	0	0
$352$	−25.1787	−1.34203
$353$	−5.95514	−0.316960	−0.158480	−	0.987362i	$-0.550659\pi$
$353$	−5.95514	−0.316960	−0.158480	+	0.987362i	$0.550659\pi$
$354$	0	0
$355$	−3.09185	−0.164098
$356$	14.2620	0.755885
$357$	0	0
$358$	7.65618	0.404642
$359$	2.04929	0.108158	0.0540788	−	0.998537i	$-0.482778\pi$
$359$	2.04929	0.108158	0.0540788	+	0.998537i	$0.482778\pi$
$360$	0	0
$361$	6.39772	0.336722
$362$	−62.3457	−3.27682
$363$	0	0
$364$	−21.5018	−1.12700
$365$	15.0811	0.789383
$366$	0	0
$367$	−17.1828	−0.896936	−0.448468	−	0.893799i	$-0.648030\pi$
$367$	−17.1828	−0.896936	−0.448468	+	0.893799i	$0.648030\pi$
$368$	−4.71881	−0.245985
$369$	0	0
$370$	−39.8294	−2.07063
$371$	−4.29353	−0.222909
$372$	0	0
$373$	13.0837	0.677449	0.338725	−	0.940886i	$-0.390005\pi$
$373$	13.0837	0.677449	0.338725	+	0.940886i	$0.390005\pi$
$374$	−14.0249	−0.725211
$375$	0	0
$376$	69.0284	3.55987
$377$	−33.0386	−1.70157
$378$	0	0
$379$	−20.7955	−1.06819	−0.534096	−	0.845424i	$-0.679348\pi$
$379$	−20.7955	−1.06819	−0.534096	+	0.845424i	$0.679348\pi$
$380$	−48.5545	−2.49079
$381$	0	0
$382$	−39.6306	−2.02768
$383$	−6.61047	−0.337779	−0.168890	−	0.985635i	$-0.554018\pi$
$383$	−6.61047	−0.337779	−0.168890	+	0.985635i	$0.554018\pi$
$384$	0	0
$385$	8.81274	0.449139
$386$	−59.5480	−3.03091
$387$	0	0
$388$	−75.7767	−3.84698
$389$	−7.82979	−0.396986	−0.198493	−	0.980102i	$-0.563605\pi$
$389$	−7.82979	−0.396986	−0.198493	+	0.980102i	$0.563605\pi$
$390$	0	0
$391$	−0.841904	−0.0425769
$392$	−6.45298	−0.325924
$393$	0	0
$394$	−56.2534	−2.83401
$395$	11.4262	0.574913
$396$	0	0
$397$	−23.8727	−1.19814	−0.599068	−	0.800698i	$-0.704462\pi$
$397$	−23.8727	−1.19814	−0.599068	+	0.800698i	$0.704462\pi$
$398$	4.86509	0.243865
$399$	0	0
$400$	−3.48303	−0.174152
$401$	−19.4992	−0.973741	−0.486871	−	0.873474i	$-0.661862\pi$
$401$	−19.4992	−0.973741	−0.486871	+	0.873474i	$0.661862\pi$
$402$	0	0
$403$	15.7599	0.785055
$404$	84.3209	4.19512
$405$	0	0
$406$	−17.7659	−0.881706
$407$	30.3221	1.50301
$408$	0	0
$409$	−26.3931	−1.30505	−0.652526	−	0.757766i	$-0.726291\pi$
$409$	−26.3931	−1.30505	−0.652526	+	0.757766i	$0.726291\pi$
$410$	26.7933	1.32323
$411$	0	0
$412$	33.8729	1.66880
$413$	9.53007	0.468944
$414$	0	0
$415$	22.3200	1.09564
$416$	28.8933	1.41661
$417$	0	0
$418$	53.2987	2.60692
$419$	21.2813	1.03966	0.519831	−	0.854269i	$-0.325995\pi$
$419$	21.2813	1.03966	0.519831	+	0.854269i	$0.325995\pi$
$420$	0	0
$421$	7.10895	0.346469	0.173235	−	0.984881i	$-0.444578\pi$
$421$	7.10895	0.346469	0.173235	+	0.984881i	$0.444578\pi$
$422$	−37.2013	−1.81093
$423$	0	0
$424$	27.7061	1.34553
$425$	−0.621422	−0.0301434
$426$	0	0
$427$	−9.70203	−0.469514
$428$	50.7535	2.45326
$429$	0	0
$430$	0.643728	0.0310433
$431$	−22.4856	−1.08309	−0.541546	−	0.840671i	$-0.682161\pi$
$431$	−22.4856	−1.08309	−0.541546	+	0.840671i	$0.682161\pi$
$432$	0	0
$433$	−24.4030	−1.17273	−0.586367	−	0.810046i	$-0.699442\pi$
$433$	−24.4030	−1.17273	−0.586367	+	0.810046i	$0.699442\pi$
$434$	8.47457	0.406792
$435$	0	0
$436$	17.4252	0.834517
$437$	3.19947	0.153052
$438$	0	0
$439$	−8.99162	−0.429147	−0.214573	−	0.976708i	$-0.568836\pi$
$439$	−8.99162	−0.429147	−0.214573	+	0.976708i	$0.568836\pi$
$440$	−56.8684	−2.71109
$441$	0	0
$442$	16.0940	0.765513
$443$	−3.16638	−0.150439	−0.0752196	−	0.997167i	$-0.523966\pi$
$443$	−3.16638	−0.150439	−0.0752196	+	0.997167i	$0.523966\pi$
$444$	0	0
$445$	6.70781	0.317981
$446$	−7.23995	−0.342822
$447$	0	0
$448$	0.671262	0.0317142
$449$	3.58210	0.169050	0.0845248	−	0.996421i	$-0.473063\pi$
$449$	3.58210	0.169050	0.0845248	+	0.996421i	$0.473063\pi$
$450$	0	0
$451$	−20.3977	−0.960490
$452$	−21.5674	−1.01445
$453$	0	0
$454$	3.72522	0.174833
$455$	−10.1129	−0.474099
$456$	0	0
$457$	−12.8800	−0.602500	−0.301250	−	0.953545i	$-0.597404\pi$
$457$	−12.8800	−0.602500	−0.301250	+	0.953545i	$0.597404\pi$
$458$	−54.0507	−2.52562
$459$	0	0
$460$	−6.11665	−0.285190
$461$	17.0189	0.792648	0.396324	−	0.918111i	$-0.370286\pi$
$461$	17.0189	0.792648	0.396324	+	0.918111i	$0.370286\pi$
$462$	0	0
$463$	−26.4544	−1.22944	−0.614721	−	0.788745i	$-0.710731\pi$
$463$	−26.4544	−1.22944	−0.614721	+	0.788745i	$0.710731\pi$
$464$	51.6909	2.39969
$465$	0	0
$466$	−24.2552	−1.12360
$467$	−22.8012	−1.05511	−0.527557	−	0.849520i	$-0.676892\pi$
$467$	−22.8012	−1.05511	−0.527557	+	0.849520i	$0.676892\pi$
$468$	0	0
$469$	2.95734	0.136557
$470$	58.1712	2.68324
$471$	0	0
$472$	−61.4973	−2.83064
$473$	−0.490069	−0.0225334
$474$	0	0
$475$	2.36158	0.108357
$476$	6.00202	0.275102
$477$	0	0
$478$	−65.7471	−3.00720
$479$	10.8867	0.497424	0.248712	−	0.968578i	$-0.419993\pi$
$479$	10.8867	0.497424	0.248712	+	0.968578i	$0.419993\pi$
$480$	0	0
$481$	−34.7955	−1.58654
$482$	−42.7633	−1.94781
$483$	0	0
$484$	27.7860	1.26300
$485$	−35.6398	−1.61832
$486$	0	0
$487$	−17.0508	−0.772645	−0.386323	−	0.922364i	$-0.626255\pi$
$487$	−17.0508	−0.772645	−0.386323	+	0.922364i	$0.626255\pi$
$488$	62.6070	2.83408
$489$	0	0
$490$	−5.43801	−0.245664
$491$	27.2143	1.22817	0.614083	−	0.789241i	$-0.289526\pi$
$491$	27.2143	1.22817	0.614083	+	0.789241i	$0.289526\pi$
$492$	0	0
$493$	9.22240	0.415356
$494$	−61.1618	−2.75180
$495$	0	0
$496$	−24.6573	−1.10714
$497$	−1.45245	−0.0651515
$498$	0	0
$499$	3.74789	0.167779	0.0838893	−	0.996475i	$-0.473266\pi$
$499$	3.74789	0.167779	0.0838893	+	0.996475i	$0.473266\pi$
$500$	−52.6876	−2.35626
$501$	0	0
$502$	−13.6931	−0.611152
$503$	3.01178	0.134289	0.0671443	−	0.997743i	$-0.478611\pi$
$503$	3.01178	0.134289	0.0671443	+	0.997743i	$0.478611\pi$
$504$	0	0
$505$	39.6584	1.76478
$506$	6.71429	0.298487
$507$	0	0
$508$	4.52602	0.200809
$509$	−22.0551	−0.977576	−0.488788	−	0.872402i	$-0.662561\pi$
$509$	−22.0551	−0.977576	−0.488788	+	0.872402i	$0.662561\pi$
$510$	0	0
$511$	7.08465	0.313406
$512$	50.7218	2.24161
$513$	0	0
$514$	33.6096	1.48246
$515$	15.9314	0.702020
$516$	0	0
$517$	−44.2856	−1.94768
$518$	−18.7106	−0.822097
$519$	0	0
$520$	65.2582	2.86176
$521$	−3.30328	−0.144719	−0.0723596	−	0.997379i	$-0.523053\pi$
$521$	−3.30328	−0.144719	−0.0723596	+	0.997379i	$0.523053\pi$
$522$	0	0
$523$	29.4263	1.28672	0.643362	−	0.765562i	$-0.277539\pi$
$523$	29.4263	1.28672	0.643362	+	0.765562i	$0.277539\pi$
$524$	49.9428	2.18176
$525$	0	0
$526$	−40.0907	−1.74804
$527$	−4.39921	−0.191633
$528$	0	0
$529$	−22.5969	−0.982476
$530$	23.3483	1.01418
$531$	0	0
$532$	−22.8094	−0.988912
$533$	23.4070	1.01387
$534$	0	0
$535$	23.8707	1.03202
$536$	−19.0837	−0.824288
$537$	0	0
$538$	−71.5686	−3.08554
$539$	4.13995	0.178320
$540$	0	0
$541$	27.5971	1.18649	0.593246	−	0.805021i	$-0.297846\pi$
$541$	27.5971	1.18649	0.593246	+	0.805021i	$0.297846\pi$
$542$	49.6842	2.13412
$543$	0	0
$544$	−8.06528	−0.345796
$545$	8.19556	0.351059
$546$	0	0
$547$	−1.48013	−0.0632860	−0.0316430	−	0.999499i	$-0.510074\pi$
$547$	−1.48013	−0.0632860	−0.0316430	+	0.999499i	$0.510074\pi$
$548$	48.3980	2.06746
$549$	0	0
$550$	4.95593	0.211321
$551$	−35.0478	−1.49308
$552$	0	0
$553$	5.36766	0.228256
$554$	38.5570	1.63813
$555$	0	0
$556$	−16.8998	−0.716712
$557$	38.4105	1.62751	0.813753	−	0.581211i	$-0.197421\pi$
$557$	38.4105	1.62751	0.813753	+	0.581211i	$0.197421\pi$
$558$	0	0
$559$	0.562369	0.0237857
$560$	15.8222	0.668611
$561$	0	0
$562$	−40.1227	−1.69247
$563$	−43.0432	−1.81406	−0.907028	−	0.421070i	$-0.861655\pi$
$563$	−43.0432	−1.81406	−0.907028	+	0.421070i	$0.861655\pi$
$564$	0	0
$565$	−10.1437	−0.426750
$566$	−76.2218	−3.20384
$567$	0	0
$568$	9.37265	0.393268
$569$	1.66577	0.0698327	0.0349163	−	0.999390i	$-0.488884\pi$
$569$	1.66577	0.0698327	0.0349163	+	0.999390i	$0.488884\pi$
$570$	0	0
$571$	33.4302	1.39901	0.699505	−	0.714627i	$-0.253404\pi$
$571$	33.4302	1.39901	0.699505	+	0.714627i	$0.253404\pi$
$572$	−89.0164	−3.72196
$573$	0	0
$574$	12.5867	0.525357
$575$	0.297500	0.0124066
$576$	0	0
$577$	−11.3687	−0.473286	−0.236643	−	0.971597i	$-0.576047\pi$
$577$	−11.3687	−0.473286	−0.236643	+	0.971597i	$0.576047\pi$
$578$	38.9358	1.61952
$579$	0	0
$580$	67.0031	2.78215
$581$	10.4852	0.435000
$582$	0	0
$583$	−17.7750	−0.736165
$584$	−45.7171	−1.89179
$585$	0	0
$586$	43.9436	1.81529
$587$	29.5488	1.21961	0.609805	−	0.792551i	$-0.291248\pi$
$587$	29.5488	1.21961	0.609805	+	0.792551i	$0.291248\pi$
$588$	0	0
$589$	16.7183	0.688864
$590$	−51.8246	−2.13359
$591$	0	0
$592$	54.4397	2.23746
$593$	22.9545	0.942628	0.471314	−	0.881966i	$-0.343780\pi$
$593$	22.9545	0.942628	0.471314	+	0.881966i	$0.343780\pi$
$594$	0	0
$595$	2.82291	0.115728
$596$	51.9971	2.12988
$597$	0	0
$598$	−7.70485	−0.315075
$599$	−39.1632	−1.60016	−0.800082	−	0.599890i	$-0.795211\pi$
$599$	−39.1632	−1.60016	−0.800082	+	0.599890i	$0.795211\pi$
$600$	0	0
$601$	−16.5220	−0.673947	−0.336973	−	0.941514i	$-0.609403\pi$
$601$	−16.5220	−0.673947	−0.336973	+	0.941514i	$0.609403\pi$
$602$	0.302403	0.0123250
$603$	0	0
$604$	58.7904	2.39215
$605$	13.0685	0.531311
$606$	0	0
$607$	5.46104	0.221657	0.110828	−	0.993840i	$-0.464650\pi$
$607$	5.46104	0.221657	0.110828	+	0.993840i	$0.464650\pi$
$608$	30.6504	1.24304
$609$	0	0
$610$	52.7597	2.13618
$611$	50.8191	2.05592
$612$	0	0
$613$	−22.0546	−0.890779	−0.445389	−	0.895337i	$-0.646935\pi$
$613$	−22.0546	−0.890779	−0.445389	+	0.895337i	$0.646935\pi$
$614$	35.6424	1.43841
$615$	0	0
$616$	−26.7150	−1.07638
$617$	42.7536	1.72119	0.860597	−	0.509287i	$-0.170091\pi$
$617$	42.7536	1.72119	0.860597	+	0.509287i	$0.170091\pi$
$618$	0	0
$619$	43.8372	1.76197	0.880983	−	0.473147i	$-0.156882\pi$
$619$	43.8372	1.76197	0.880983	+	0.473147i	$0.156882\pi$
$620$	−31.9614	−1.28360
$621$	0	0
$622$	10.3285	0.414137
$623$	3.15112	0.126247
$624$	0	0
$625$	−22.4374	−0.897496
$626$	−8.98736	−0.359207
$627$	0	0
$628$	57.2408	2.28415
$629$	9.71281	0.387275
$630$	0	0
$631$	−12.7391	−0.507135	−0.253568	−	0.967318i	$-0.581604\pi$
$631$	−12.7391	−0.507135	−0.253568	+	0.967318i	$0.581604\pi$
$632$	−34.6374	−1.37780
$633$	0	0
$634$	−28.4604	−1.13031
$635$	2.12871	0.0844752
$636$	0	0
$637$	−4.75071	−0.188230
$638$	−73.5499	−2.91187
$639$	0	0
$640$	22.2428	0.879223
$641$	−6.94532	−0.274323	−0.137162	−	0.990549i	$-0.543798\pi$
$641$	−6.94532	−0.274323	−0.137162	+	0.990549i	$0.543798\pi$
$642$	0	0
$643$	−38.9483	−1.53597	−0.767986	−	0.640467i	$-0.778741\pi$
$643$	−38.9483	−1.53597	−0.767986	+	0.640467i	$0.778741\pi$
$644$	−2.87341	−0.113228
$645$	0	0
$646$	17.0727	0.671717
$647$	24.7855	0.974417	0.487209	−	0.873286i	$-0.338015\pi$
$647$	24.7855	0.974417	0.487209	+	0.873286i	$0.338015\pi$
$648$	0	0
$649$	39.4540	1.54871
$650$	−5.68707	−0.223065
$651$	0	0
$652$	27.6680	1.08356
$653$	23.6675	0.926182	0.463091	−	0.886311i	$-0.346740\pi$
$653$	23.6675	0.926182	0.463091	+	0.886311i	$0.346740\pi$
$654$	0	0
$655$	23.4895	0.917809
$656$	−36.6216	−1.42983
$657$	0	0
$658$	27.3270	1.06532
$659$	−13.2866	−0.517572	−0.258786	−	0.965935i	$-0.583322\pi$
$659$	−13.2866	−0.517572	−0.258786	+	0.965935i	$0.583322\pi$
$660$	0	0
$661$	−22.0055	−0.855915	−0.427958	−	0.903799i	$-0.640767\pi$
$661$	−22.0055	−0.855915	−0.427958	+	0.903799i	$0.640767\pi$
$662$	−63.5480	−2.46986
$663$	0	0
$664$	−67.6609	−2.62575
$665$	−10.7279	−0.416009
$666$	0	0
$667$	−4.41514	−0.170955
$668$	−83.0912	−3.21489
$669$	0	0
$670$	−16.0821	−0.621304
$671$	−40.1659	−1.55059
$672$	0	0
$673$	24.1513	0.930964	0.465482	−	0.885057i	$-0.345881\pi$
$673$	24.1513	0.930964	0.465482	+	0.885057i	$0.345881\pi$
$674$	67.6681	2.60648
$675$	0	0
$676$	43.3107	1.66580
$677$	19.8265	0.761993	0.380996	−	0.924577i	$-0.375581\pi$
$677$	19.8265	0.761993	0.380996	+	0.924577i	$0.375581\pi$
$678$	0	0
$679$	−16.7425	−0.642517
$680$	−18.2162	−0.698558
$681$	0	0
$682$	35.0843	1.34345
$683$	−11.5130	−0.440533	−0.220267	−	0.975440i	$-0.570693\pi$
$683$	−11.5130	−0.440533	−0.220267	+	0.975440i	$0.570693\pi$
$684$	0	0
$685$	22.7629	0.869725
$686$	−2.55461	−0.0975353
$687$	0	0
$688$	−0.879861	−0.0335444
$689$	20.3973	0.777077
$690$	0	0
$691$	8.79146	0.334443	0.167222	−	0.985919i	$-0.446520\pi$
$691$	8.79146	0.334443	0.167222	+	0.985919i	$0.446520\pi$
$692$	−84.8378	−3.22505
$693$	0	0
$694$	57.1229	2.16835
$695$	−7.94844	−0.301502
$696$	0	0
$697$	−6.53382	−0.247486
$698$	−16.9985	−0.643402
$699$	0	0
$700$	−2.12091	−0.0801628
$701$	7.46770	0.282051	0.141026	−	0.990006i	$-0.454960\pi$
$701$	7.46770	0.282051	0.141026	+	0.990006i	$0.454960\pi$
$702$	0	0
$703$	−36.9115	−1.39214
$704$	2.77899	0.104737
$705$	0	0
$706$	15.2130	0.572550
$707$	18.6303	0.700664
$708$	0	0
$709$	30.0859	1.12990	0.564950	−	0.825125i	$-0.308895\pi$
$709$	30.0859	1.12990	0.564950	+	0.825125i	$0.308895\pi$
$710$	7.89846	0.296424
$711$	0	0
$712$	−20.3341	−0.762052
$713$	2.10608	0.0788733
$714$	0	0
$715$	−41.8668	−1.56573
$716$	−13.5645	−0.506930
$717$	0	0
$718$	−5.23514	−0.195374
$719$	12.0466	0.449261	0.224631	−	0.974444i	$-0.427882\pi$
$719$	12.0466	0.449261	0.224631	+	0.974444i	$0.427882\pi$
$720$	0	0
$721$	7.48405	0.278721
$722$	−16.3437	−0.608248
$723$	0	0
$724$	110.458	4.10515
$725$	−3.25888	−0.121032
$726$	0	0
$727$	12.0016	0.445115	0.222557	−	0.974920i	$-0.428560\pi$
$727$	12.0016	0.445115	0.222557	+	0.974920i	$0.428560\pi$
$728$	30.6562	1.13620
$729$	0	0
$730$	−38.5264	−1.42593
$731$	−0.156980	−0.00580610
$732$	0	0
$733$	−2.59393	−0.0958090	−0.0479045	−	0.998852i	$-0.515254\pi$
$733$	−2.59393	−0.0958090	−0.0479045	+	0.998852i	$0.515254\pi$
$734$	43.8954	1.62021
$735$	0	0
$736$	3.86118	0.142325
$737$	12.2432	0.450986
$738$	0	0
$739$	−7.97887	−0.293508	−0.146754	−	0.989173i	$-0.546883\pi$
$739$	−7.97887	−0.293508	−0.146754	+	0.989173i	$0.546883\pi$
$740$	70.5661	2.59406
$741$	0	0
$742$	10.9683	0.402658
$743$	4.32447	0.158650	0.0793248	−	0.996849i	$-0.474724\pi$
$743$	4.32447	0.158650	0.0793248	+	0.996849i	$0.474724\pi$
$744$	0	0
$745$	24.4557	0.895986
$746$	−33.4238	−1.22373
$747$	0	0
$748$	24.8480	0.908534
$749$	11.2137	0.409740
$750$	0	0
$751$	2.34648	0.0856244	0.0428122	−	0.999083i	$-0.486368\pi$
$751$	2.34648	0.0856244	0.0428122	+	0.999083i	$0.486368\pi$
$752$	−79.5095	−2.89941
$753$	0	0
$754$	84.4006	3.07369
$755$	27.6507	1.00631
$756$	0	0
$757$	−26.4656	−0.961909	−0.480954	−	0.876746i	$-0.659710\pi$
$757$	−26.4656	−0.961909	−0.480954	+	0.876746i	$0.659710\pi$
$758$	53.1243	1.92956
$759$	0	0
$760$	69.2267	2.51111
$761$	40.6849	1.47483	0.737414	−	0.675441i	$-0.236047\pi$
$761$	40.6849	1.47483	0.737414	+	0.675441i	$0.236047\pi$
$762$	0	0
$763$	3.85001	0.139380
$764$	70.2138	2.54025
$765$	0	0
$766$	16.8871	0.610157
$767$	−45.2746	−1.63477
$768$	0	0
$769$	−9.94961	−0.358792	−0.179396	−	0.983777i	$-0.557414\pi$
$769$	−9.94961	−0.358792	−0.179396	+	0.983777i	$0.557414\pi$
$770$	−22.5131	−0.811315
$771$	0	0
$772$	105.502	3.79709
$773$	−31.8949	−1.14718	−0.573590	−	0.819142i	$-0.694450\pi$
$773$	−31.8949	−1.14718	−0.573590	+	0.819142i	$0.694450\pi$
$774$	0	0
$775$	1.55453	0.0558404
$776$	108.039	3.87837
$777$	0	0
$778$	20.0020	0.717108
$779$	24.8304	0.889641
$780$	0	0
$781$	−6.01309	−0.215165
$782$	2.15073	0.0769100
$783$	0	0
$784$	7.43278	0.265457
$785$	26.9219	0.960883
$786$	0	0
$787$	−38.6972	−1.37941	−0.689703	−	0.724093i	$-0.742259\pi$
$787$	−38.6972	−1.37941	−0.689703	+	0.724093i	$0.742259\pi$
$788$	99.6646	3.55040
$789$	0	0
$790$	−29.1894	−1.03851
$791$	−4.76521	−0.169431
$792$	0	0
$793$	46.0916	1.63676
$794$	60.9854	2.16429
$795$	0	0
$796$	−8.61952	−0.305511
$797$	9.68211	0.342958	0.171479	−	0.985188i	$-0.445145\pi$
$797$	9.68211	0.342958	0.171479	+	0.985188i	$0.445145\pi$
$798$	0	0
$799$	−14.1856	−0.501852
$800$	2.85000	0.100763
$801$	0	0
$802$	49.8127	1.75895
$803$	29.3301	1.03504
$804$	0	0
$805$	−1.35144	−0.0476321
$806$	−40.2603	−1.41811
$807$	0	0
$808$	−120.221	−4.22935
$809$	25.8293	0.908111	0.454056	−	0.890973i	$-0.349977\pi$
$809$	25.8293	0.908111	0.454056	+	0.890973i	$0.349977\pi$
$810$	0	0
$811$	20.7060	0.727087	0.363544	−	0.931577i	$-0.381567\pi$
$811$	20.7060	0.727087	0.363544	+	0.931577i	$0.381567\pi$
$812$	31.4759	1.10459
$813$	0	0
$814$	−77.4610	−2.71501
$815$	13.0130	0.455825
$816$	0	0
$817$	0.596568	0.0208713
$818$	67.4239	2.35742
$819$	0	0
$820$	−47.4699	−1.65772
$821$	36.0955	1.25974	0.629871	−	0.776700i	$-0.283108\pi$
$821$	36.0955	1.25974	0.629871	+	0.776700i	$0.283108\pi$
$822$	0	0
$823$	45.5252	1.58691	0.793455	−	0.608629i	$-0.208280\pi$
$823$	45.5252	1.58691	0.793455	+	0.608629i	$0.208280\pi$
$824$	−48.2944	−1.68242
$825$	0	0
$826$	−24.3456	−0.847091
$827$	−19.6678	−0.683916	−0.341958	−	0.939715i	$-0.611090\pi$
$827$	−19.6678	−0.683916	−0.341958	+	0.939715i	$0.611090\pi$
$828$	0	0
$829$	−46.1390	−1.60247	−0.801237	−	0.598347i	$-0.795825\pi$
$829$	−46.1390	−1.60247	−0.801237	+	0.598347i	$0.795825\pi$
$830$	−57.0187	−1.97915
$831$	0	0
$832$	−3.18897	−0.110558
$833$	1.32611	0.0459471
$834$	0	0
$835$	−39.0800	−1.35242
$836$	−94.4296	−3.26592
$837$	0	0
$838$	−54.3654	−1.87802
$839$	45.6364	1.57554	0.787772	−	0.615966i	$-0.211234\pi$
$839$	45.6364	1.57554	0.787772	+	0.615966i	$0.211234\pi$
$840$	0	0
$841$	19.3644	0.667737
$842$	−18.1606	−0.625855
$843$	0	0
$844$	65.9098	2.26871
$845$	20.3702	0.700756
$846$	0	0
$847$	6.13918	0.210945
$848$	−31.9129	−1.09589
$849$	0	0
$850$	1.58749	0.0544504
$851$	−4.64992	−0.159397
$852$	0	0
$853$	14.5096	0.496799	0.248400	−	0.968658i	$-0.420095\pi$
$853$	14.5096	0.496799	0.248400	+	0.968658i	$0.420095\pi$
$854$	24.7849	0.848121
$855$	0	0
$856$	−72.3618	−2.47328
$857$	−3.03303	−0.103606	−0.0518031	−	0.998657i	$-0.516497\pi$
$857$	−3.03303	−0.103606	−0.0518031	+	0.998657i	$0.516497\pi$
$858$	0	0
$859$	54.2596	1.85131	0.925657	−	0.378365i	$-0.123513\pi$
$859$	54.2596	1.85131	0.925657	+	0.378365i	$0.123513\pi$
$860$	−1.14050	−0.0388906
$861$	0	0
$862$	57.4418	1.95648
$863$	48.8955	1.66442	0.832211	−	0.554460i	$-0.187075\pi$
$863$	48.8955	1.66442	0.832211	+	0.554460i	$0.187075\pi$
$864$	0	0
$865$	−39.9015	−1.35669
$866$	62.3401	2.11840
$867$	0	0
$868$	−15.0145	−0.509624
$869$	22.2218	0.753824
$870$	0	0
$871$	−14.0495	−0.476049
$872$	−24.8440	−0.841326
$873$	0	0
$874$	−8.17340	−0.276469
$875$	−11.6411	−0.393540
$876$	0	0
$877$	−25.9003	−0.874590	−0.437295	−	0.899318i	$-0.644063\pi$
$877$	−25.9003	−0.874590	−0.437295	+	0.899318i	$0.644063\pi$
$878$	22.9701	0.775202
$879$	0	0
$880$	65.5032	2.20811
$881$	1.06900	0.0360156	0.0180078	−	0.999838i	$-0.494268\pi$
$881$	1.06900	0.0360156	0.0180078	+	0.999838i	$0.494268\pi$
$882$	0	0
$883$	−29.3081	−0.986295	−0.493147	−	0.869946i	$-0.664154\pi$
$883$	−29.3081	−0.986295	−0.493147	+	0.869946i	$0.664154\pi$
$884$	−28.5139	−0.959025
$885$	0	0
$886$	8.08885	0.271750
$887$	40.0423	1.34449	0.672244	−	0.740330i	$-0.265331\pi$
$887$	40.0423	1.34449	0.672244	+	0.740330i	$0.265331\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	−17.1358	−0.574394
$891$	0	0
$892$	12.8271	0.429482
$893$	53.9095	1.80401
$894$	0	0
$895$	−6.37976	−0.213252
$896$	10.4490	0.349075
$897$	0	0
$898$	−9.15085	−0.305368
$899$	−23.0705	−0.769443
$900$	0	0
$901$	−5.69372	−0.189685
$902$	52.1081	1.73501
$903$	0	0
$904$	30.7498	1.02272
$905$	51.9515	1.72693
$906$	0	0
$907$	25.5478	0.848302	0.424151	−	0.905592i	$-0.360573\pi$
$907$	25.5478	0.848302	0.424151	+	0.905592i	$0.360573\pi$
$908$	−6.60000	−0.219029
$909$	0	0
$910$	25.8344	0.856403
$911$	−29.3845	−0.973553	−0.486777	−	0.873526i	$-0.661827\pi$
$911$	−29.3845	−0.973553	−0.486777	+	0.873526i	$0.661827\pi$
$912$	0	0
$913$	43.4083	1.43660
$914$	32.9033	1.08834
$915$	0	0
$916$	95.7619	3.16406
$917$	11.0346	0.364395
$918$	0	0
$919$	−10.8678	−0.358494	−0.179247	−	0.983804i	$-0.557366\pi$
$919$	−10.8678	−0.358494	−0.179247	+	0.983804i	$0.557366\pi$
$920$	8.72082	0.287517
$921$	0	0
$922$	−43.4766	−1.43182
$923$	6.90020	0.227123
$924$	0	0
$925$	−3.43218	−0.112849
$926$	67.5807	2.22084
$927$	0	0
$928$	−42.2962	−1.38844
$929$	−4.66688	−0.153115	−0.0765577	−	0.997065i	$-0.524393\pi$
$929$	−4.66688	−0.153115	−0.0765577	+	0.997065i	$0.524393\pi$
$930$	0	0
$931$	−5.03962	−0.165167
$932$	42.9731	1.40763
$933$	0	0
$934$	58.2481	1.90594
$935$	11.6867	0.382196
$936$	0	0
$937$	−14.0249	−0.458175	−0.229088	−	0.973406i	$-0.573574\pi$
$937$	−14.0249	−0.458175	−0.229088	+	0.973406i	$0.573574\pi$
$938$	−7.55485	−0.246674
$939$	0	0
$940$	−103.062	−3.36152
$941$	−46.2542	−1.50784	−0.753921	−	0.656965i	$-0.771840\pi$
$941$	−46.2542	−1.50784	−0.753921	+	0.656965i	$0.771840\pi$
$942$	0	0
$943$	3.12801	0.101862
$944$	70.8350	2.30548
$945$	0	0
$946$	1.25193	0.0407039
$947$	−5.19190	−0.168714	−0.0843571	−	0.996436i	$-0.526884\pi$
$947$	−5.19190	−0.168714	−0.0843571	+	0.996436i	$0.526884\pi$
$948$	0	0
$949$	−33.6571	−1.09256
$950$	−6.03291	−0.195734
$951$	0	0
$952$	−8.55738	−0.277346
$953$	31.0054	1.00436	0.502181	−	0.864762i	$-0.332531\pi$
$953$	31.0054	1.00436	0.502181	+	0.864762i	$0.332531\pi$
$954$	0	0
$955$	33.0235	1.06861
$956$	116.485	3.76738
$957$	0	0
$958$	−27.8111	−0.898536
$959$	10.6933	0.345304
$960$	0	0
$961$	−19.9951	−0.645002
$962$	88.8887	2.86589
$963$	0	0
$964$	75.7640	2.44020
$965$	49.6203	1.59733
$966$	0	0
$967$	6.06932	0.195176	0.0975881	−	0.995227i	$-0.468887\pi$
$967$	6.06932	0.195176	0.0975881	+	0.995227i	$0.468887\pi$
$968$	−39.6159	−1.27330
$969$	0	0
$970$	91.0458	2.92330
$971$	8.98673	0.288398	0.144199	−	0.989549i	$-0.453939\pi$
$971$	8.98673	0.288398	0.144199	+	0.989549i	$0.453939\pi$
$972$	0	0
$973$	−3.73393	−0.119704
$974$	43.5581	1.39569
$975$	0	0
$976$	−72.1131	−2.30828
$977$	−55.0884	−1.76243	−0.881217	−	0.472711i	$-0.843275\pi$
$977$	−55.0884	−1.76243	−0.881217	+	0.472711i	$0.843275\pi$
$978$	0	0
$979$	13.0455	0.416935
$980$	9.63456	0.307765
$981$	0	0
$982$	−69.5219	−2.21853
$983$	−59.7557	−1.90591	−0.952956	−	0.303109i	$-0.901975\pi$
$983$	−59.7557	−1.90591	−0.952956	+	0.303109i	$0.901975\pi$
$984$	0	0
$985$	46.8750	1.49356
$986$	−23.5596	−0.750291
$987$	0	0
$988$	108.361	3.44742
$989$	0.0751526	0.00238971
$990$	0	0
$991$	−0.373535	−0.0118657	−0.00593286	−	0.999982i	$-0.501888\pi$
$991$	−0.373535	−0.0118657	−0.00593286	+	0.999982i	$0.501888\pi$
$992$	20.1759	0.640584
$993$	0	0
$994$	3.71045	0.117688
$995$	−4.05399	−0.128520
$996$	0	0
$997$	38.3145	1.21343	0.606716	−	0.794919i	$-0.292487\pi$
$997$	38.3145	1.21343	0.606716	+	0.794919i	$0.292487\pi$
$998$	−9.57439	−0.303072
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.u.1.1		18
3.2	odd	2		2667.2.a.p.1.18	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.p.1.18	✓	18	3.2	odd	2
8001.2.a.u.1.1		18	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)

\(f(q)\)	\(=\)	\(q-2.55461 q^{2} +4.52602 q^{4} +2.12871 q^{5} +1.00000 q^{7} -6.45298 q^{8} +O(q^{10})\)	\(q-2.55461 q^{2} +4.52602 q^{4} +2.12871 q^{5} +1.00000 q^{7} -6.45298 q^{8} -5.43801 q^{10} +4.13995 q^{11} -4.75071 q^{13} -2.55461 q^{14} +7.43278 q^{16} +1.32611 q^{17} -5.03962 q^{19} +9.63456 q^{20} -10.5759 q^{22} -0.634865 q^{23} -0.468604 q^{25} +12.1362 q^{26} +4.52602 q^{28} +6.95445 q^{29} -3.31737 q^{31} -6.08189 q^{32} -3.38770 q^{34} +2.12871 q^{35} +7.32426 q^{37} +12.8742 q^{38} -13.7365 q^{40} -4.92704 q^{41} -0.118376 q^{43} +18.7375 q^{44} +1.62183 q^{46} -10.6971 q^{47} +1.00000 q^{49} +1.19710 q^{50} -21.5018 q^{52} -4.29353 q^{53} +8.81274 q^{55} -6.45298 q^{56} -17.7659 q^{58} +9.53007 q^{59} -9.70203 q^{61} +8.47457 q^{62} +0.671262 q^{64} -10.1129 q^{65} +2.95734 q^{67} +6.00202 q^{68} -5.43801 q^{70} -1.45245 q^{71} +7.08465 q^{73} -18.7106 q^{74} -22.8094 q^{76} +4.13995 q^{77} +5.36766 q^{79} +15.8222 q^{80} +12.5867 q^{82} +10.4852 q^{83} +2.82291 q^{85} +0.302403 q^{86} -26.7150 q^{88} +3.15112 q^{89} -4.75071 q^{91} -2.87341 q^{92} +27.3270 q^{94} -10.7279 q^{95} -16.7425 q^{97} -2.55461 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 6 q^{2} + 22 q^{4} + 10 q^{5} + 18 q^{7} + 21 q^{8}+O(q^{10}) \) 18 * q + 6 * q^2 + 22 * q^4 + 10 * q^5 + 18 * q^7 + 21 * q^8	\( 18 q + 6 q^{2} + 22 q^{4} + 10 q^{5} + 18 q^{7} + 21 q^{8} - 4 q^{10} + 9 q^{11} - 25 q^{13} + 6 q^{14} + 34 q^{16} + 17 q^{17} - 5 q^{19} + 21 q^{20} + 5 q^{22} + 14 q^{23} + 28 q^{25} + 8 q^{26} + 22 q^{28} + 17 q^{29} + 5 q^{31} + 53 q^{32} - 19 q^{34} + 10 q^{35} - 15 q^{37} + 22 q^{38} - q^{40} + 17 q^{41} + q^{43} + 33 q^{44} + 10 q^{46} + 31 q^{47} + 18 q^{49} + 35 q^{50} - 70 q^{52} + 35 q^{53} + 4 q^{55} + 21 q^{56} + 3 q^{58} + 46 q^{59} - 5 q^{61} + 10 q^{62} + 63 q^{64} + 12 q^{65} + 6 q^{67} + 56 q^{68} - 4 q^{70} + 22 q^{71} - 16 q^{73} - 18 q^{74} + 32 q^{76} + 9 q^{77} + 46 q^{79} + 30 q^{80} - 12 q^{82} + 46 q^{83} + 4 q^{85} - 18 q^{86} + 30 q^{88} + 42 q^{89} - 25 q^{91} + 48 q^{92} + 3 q^{94} + 2 q^{95} - 35 q^{97} + 6 q^{98}+O(q^{100}) \) 18 * q + 6 * q^2 + 22 * q^4 + 10 * q^5 + 18 * q^7 + 21 * q^8 - 4 * q^10 + 9 * q^11 - 25 * q^13 + 6 * q^14 + 34 * q^16 + 17 * q^17 - 5 * q^19 + 21 * q^20 + 5 * q^22 + 14 * q^23 + 28 * q^25 + 8 * q^26 + 22 * q^28 + 17 * q^29 + 5 * q^31 + 53 * q^32 - 19 * q^34 + 10 * q^35 - 15 * q^37 + 22 * q^38 - q^40 + 17 * q^41 + q^43 + 33 * q^44 + 10 * q^46 + 31 * q^47 + 18 * q^49 + 35 * q^50 - 70 * q^52 + 35 * q^53 + 4 * q^55 + 21 * q^56 + 3 * q^58 + 46 * q^59 - 5 * q^61 + 10 * q^62 + 63 * q^64 + 12 * q^65 + 6 * q^67 + 56 * q^68 - 4 * q^70 + 22 * q^71 - 16 * q^73 - 18 * q^74 + 32 * q^76 + 9 * q^77 + 46 * q^79 + 30 * q^80 - 12 * q^82 + 46 * q^83 + 4 * q^85 - 18 * q^86 + 30 * q^88 + 42 * q^89 - 25 * q^91 + 48 * q^92 + 3 * q^94 + 2 * q^95 - 35 * q^97 + 6 * q^98

Introduction

L-functions

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Database

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