LMFDB - Embedded newform 8001.2.a.p.1.7

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			1.7
Root			$0.369092$ 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)

$f(q)$	$=$	$q+0.369092 q^{2} -1.86377 q^{4} +3.55869 q^{5} -1.00000 q^{7} -1.42609 q^{8} +O(q^{10})$	\(q+0.369092 q^{2} -1.86377 q^{4} +3.55869 q^{5} -1.00000 q^{7} -1.42609 q^{8} +1.31348 q^{10} +5.63711 q^{11} +0.337042 q^{13} -0.369092 q^{14} +3.20118 q^{16} -4.64942 q^{17} +1.06563 q^{19} -6.63257 q^{20} +2.08061 q^{22} -4.73319 q^{23} +7.66424 q^{25} +0.124399 q^{26} +1.86377 q^{28} +3.66220 q^{29} +3.89423 q^{31} +4.03371 q^{32} -1.71607 q^{34} -3.55869 q^{35} +5.81777 q^{37} +0.393317 q^{38} -5.07500 q^{40} +1.84704 q^{41} +0.154033 q^{43} -10.5063 q^{44} -1.74698 q^{46} -7.22906 q^{47} +1.00000 q^{49} +2.82881 q^{50} -0.628168 q^{52} +0.485681 q^{53} +20.0607 q^{55} +1.42609 q^{56} +1.35169 q^{58} +11.8050 q^{59} -11.0462 q^{61} +1.43733 q^{62} -4.91356 q^{64} +1.19942 q^{65} -2.43212 q^{67} +8.66546 q^{68} -1.31348 q^{70} +0.433452 q^{71} +9.50128 q^{73} +2.14729 q^{74} -1.98610 q^{76} -5.63711 q^{77} +10.9124 q^{79} +11.3920 q^{80} +0.681727 q^{82} +6.49033 q^{83} -16.5458 q^{85} +0.0568523 q^{86} -8.03902 q^{88} -0.313590 q^{89} -0.337042 q^{91} +8.82157 q^{92} -2.66819 q^{94} +3.79226 q^{95} +3.62785 q^{97} +0.369092 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.369092	0.260988	0.130494	−	0.991449i	$-0.458344\pi$
$2$	0.369092	0.260988	0.130494	+	0.991449i	$0.458344\pi$
$3$	0	0
$3$	0	0
$4$	−1.86377	−0.931885
$5$	3.55869	1.59149	0.795746	−	0.605630i	$-0.207079\pi$
$5$	3.55869	1.59149	0.795746	+	0.605630i	$0.207079\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.42609	−0.504198
$9$	0	0
$10$	1.31348	0.415360
$11$	5.63711	1.69965	0.849827	−	0.527062i	$-0.176707\pi$
$11$	5.63711	1.69965	0.849827	+	0.527062i	$0.176707\pi$
$12$	0	0
$13$	0.337042	0.0934785	0.0467393	−	0.998907i	$-0.485117\pi$
$13$	0.337042	0.0934785	0.0467393	+	0.998907i	$0.485117\pi$
$14$	−0.369092	−0.0986440
$15$	0	0
$16$	3.20118	0.800296
$17$	−4.64942	−1.12765	−0.563825	−	0.825894i	$-0.690671\pi$
$17$	−4.64942	−1.12765	−0.563825	+	0.825894i	$0.690671\pi$
$18$	0	0
$19$	1.06563	0.244473	0.122237	−	0.992501i	$-0.460993\pi$
$19$	1.06563	0.244473	0.122237	+	0.992501i	$0.460993\pi$
$20$	−6.63257	−1.48309
$21$	0	0
$22$	2.08061	0.443588
$23$	−4.73319	−0.986937	−0.493469	−	0.869764i	$-0.664271\pi$
$23$	−4.73319	−0.986937	−0.493469	+	0.869764i	$0.664271\pi$
$24$	0	0
$25$	7.66424	1.53285
$26$	0.124399	0.0243967
$27$	0	0
$28$	1.86377	0.352220
$29$	3.66220	0.680053	0.340027	−	0.940416i	$-0.389564\pi$
$29$	3.66220	0.680053	0.340027	+	0.940416i	$0.389564\pi$
$30$	0	0
$31$	3.89423	0.699424	0.349712	−	0.936857i	$-0.386279\pi$
$31$	3.89423	0.699424	0.349712	+	0.936857i	$0.386279\pi$
$32$	4.03371	0.713065
$33$	0	0
$34$	−1.71607	−0.294303
$35$	−3.55869	−0.601528
$36$	0	0
$37$	5.81777	0.956435	0.478217	−	0.878241i	$-0.341283\pi$
$37$	5.81777	0.956435	0.478217	+	0.878241i	$0.341283\pi$
$38$	0.393317	0.0638045
$39$	0	0
$40$	−5.07500	−0.802427
$41$	1.84704	0.288459	0.144229	−	0.989544i	$-0.453930\pi$
$41$	1.84704	0.288459	0.144229	+	0.989544i	$0.453930\pi$
$42$	0	0
$43$	0.154033	0.0234898	0.0117449	−	0.999931i	$-0.496261\pi$
$43$	0.154033	0.0234898	0.0117449	+	0.999931i	$0.496261\pi$
$44$	−10.5063	−1.58388
$45$	0	0
$46$	−1.74698	−0.257578
$47$	−7.22906	−1.05447	−0.527234	−	0.849720i	$-0.676771\pi$
$47$	−7.22906	−1.05447	−0.527234	+	0.849720i	$0.676771\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	2.82881	0.400054
$51$	0	0
$52$	−0.628168	−0.0871113
$53$	0.485681	0.0667134	0.0333567	−	0.999444i	$-0.489380\pi$
$53$	0.485681	0.0667134	0.0333567	+	0.999444i	$0.489380\pi$
$54$	0	0
$55$	20.0607	2.70499
$56$	1.42609	0.190569
$57$	0	0
$58$	1.35169	0.177485
$59$	11.8050	1.53688	0.768441	−	0.639920i	$-0.221033\pi$
$59$	11.8050	1.53688	0.768441	+	0.639920i	$0.221033\pi$
$60$	0	0
$61$	−11.0462	−1.41432	−0.707160	−	0.707054i	$-0.750024\pi$
$61$	−11.0462	−1.41432	−0.707160	+	0.707054i	$0.750024\pi$
$62$	1.43733	0.182541
$63$	0	0
$64$	−4.91356	−0.614195
$65$	1.19942	0.148770
$66$	0	0
$67$	−2.43212	−0.297130	−0.148565	−	0.988903i	$-0.547465\pi$
$67$	−2.43212	−0.297130	−0.148565	+	0.988903i	$0.547465\pi$
$68$	8.66546	1.05084
$69$	0	0
$70$	−1.31348	−0.156991
$71$	0.433452	0.0514413	0.0257206	−	0.999669i	$-0.491812\pi$
$71$	0.433452	0.0514413	0.0257206	+	0.999669i	$0.491812\pi$
$72$	0	0
$73$	9.50128	1.11204	0.556020	−	0.831169i	$-0.312328\pi$
$73$	9.50128	1.11204	0.556020	+	0.831169i	$0.312328\pi$
$74$	2.14729	0.249618
$75$	0	0
$76$	−1.98610	−0.227821
$77$	−5.63711	−0.642409
$78$	0	0
$79$	10.9124	1.22774	0.613871	−	0.789406i	$-0.289611\pi$
$79$	10.9124	1.22774	0.613871	+	0.789406i	$0.289611\pi$
$80$	11.3920	1.27367
$81$	0	0
$82$	0.681727	0.0752841
$83$	6.49033	0.712407	0.356203	−	0.934408i	$-0.384071\pi$
$83$	6.49033	0.712407	0.356203	+	0.934408i	$0.384071\pi$
$84$	0	0
$85$	−16.5458	−1.79465
$86$	0.0568523	0.00613054
$87$	0	0
$88$	−8.03902	−0.856962
$89$	−0.313590	−0.0332405	−0.0166202	−	0.999862i	$-0.505291\pi$
$89$	−0.313590	−0.0332405	−0.0166202	+	0.999862i	$0.505291\pi$
$90$	0	0
$91$	−0.337042	−0.0353316
$92$	8.82157	0.919713
$93$	0	0
$94$	−2.66819	−0.275203
$95$	3.79226	0.389077
$96$	0	0
$97$	3.62785	0.368352	0.184176	−	0.982893i	$-0.441038\pi$
$97$	3.62785	0.368352	0.184176	+	0.982893i	$0.441038\pi$
$98$	0.369092	0.0372839
$99$	0	0
$100$	−14.2844	−1.42844
$101$	5.49718	0.546989	0.273495	−	0.961874i	$-0.411820\pi$
$101$	5.49718	0.546989	0.273495	+	0.961874i	$0.411820\pi$
$102$	0	0
$103$	11.5802	1.14103	0.570516	−	0.821287i	$-0.306743\pi$
$103$	11.5802	1.14103	0.570516	+	0.821287i	$0.306743\pi$
$104$	−0.480651	−0.0471317
$105$	0	0
$106$	0.179261	0.0174114
$107$	−14.8335	−1.43401	−0.717005	−	0.697068i	$-0.754487\pi$
$107$	−14.8335	−1.43401	−0.717005	+	0.697068i	$0.754487\pi$
$108$	0	0
$109$	−13.1574	−1.26025	−0.630126	−	0.776493i	$-0.716997\pi$
$109$	−13.1574	−1.26025	−0.630126	+	0.776493i	$0.716997\pi$
$110$	7.40425	0.705967
$111$	0	0
$112$	−3.20118	−0.302483
$113$	−2.96469	−0.278895	−0.139448	−	0.990229i	$-0.544533\pi$
$113$	−2.96469	−0.278895	−0.139448	+	0.990229i	$0.544533\pi$
$114$	0	0
$115$	−16.8439	−1.57070
$116$	−6.82550	−0.633732
$117$	0	0
$118$	4.35714	0.401107
$119$	4.64942	0.426212
$120$	0	0
$121$	20.7770	1.88882
$122$	−4.07706	−0.369120
$123$	0	0
$124$	−7.25795	−0.651783
$125$	9.48119	0.848024
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	−9.88097	−0.873363
$129$	0	0
$130$	0.442698	0.0388272
$131$	−11.3901	−0.995158	−0.497579	−	0.867419i	$-0.665778\pi$
$131$	−11.3901	−0.995158	−0.497579	+	0.867419i	$0.665778\pi$
$132$	0	0
$133$	−1.06563	−0.0924022
$134$	−0.897675	−0.0775473
$135$	0	0
$136$	6.63048	0.568559
$137$	19.4593	1.66252	0.831261	−	0.555882i	$-0.187619\pi$
$137$	19.4593	1.66252	0.831261	+	0.555882i	$0.187619\pi$
$138$	0	0
$139$	8.52109	0.722750	0.361375	−	0.932421i	$-0.382307\pi$
$139$	8.52109	0.722750	0.361375	+	0.932421i	$0.382307\pi$
$140$	6.63257	0.560555
$141$	0	0
$142$	0.159984	0.0134255
$143$	1.89994	0.158881
$144$	0	0
$145$	13.0326	1.08230
$146$	3.50685	0.290229
$147$	0	0
$148$	−10.8430	−0.891288
$149$	14.5674	1.19341	0.596704	−	0.802461i	$-0.296476\pi$
$149$	14.5674	1.19341	0.596704	+	0.802461i	$0.296476\pi$
$150$	0	0
$151$	−12.2479	−0.996719	−0.498359	−	0.866971i	$-0.666064\pi$
$151$	−12.2479	−0.996719	−0.498359	+	0.866971i	$0.666064\pi$
$152$	−1.51969	−0.123263
$153$	0	0
$154$	−2.08061	−0.167661
$155$	13.8583	1.11313
$156$	0	0
$157$	−2.39792	−0.191375	−0.0956873	−	0.995411i	$-0.530505\pi$
$157$	−2.39792	−0.191375	−0.0956873	+	0.995411i	$0.530505\pi$
$158$	4.02769	0.320425
$159$	0	0
$160$	14.3547	1.13484
$161$	4.73319	0.373027
$162$	0	0
$163$	0.254058	0.0198994	0.00994969	−	0.999951i	$-0.496833\pi$
$163$	0.254058	0.0198994	0.00994969	+	0.999951i	$0.496833\pi$
$164$	−3.44245	−0.268811
$165$	0	0
$166$	2.39553	0.185929
$167$	8.29652	0.642004	0.321002	−	0.947079i	$-0.395980\pi$
$167$	8.29652	0.642004	0.321002	+	0.947079i	$0.395980\pi$
$168$	0	0
$169$	−12.8864	−0.991262
$170$	−6.10694	−0.468381
$171$	0	0
$172$	−0.287082	−0.0218898
$173$	6.69779	0.509224	0.254612	−	0.967043i	$-0.418052\pi$
$173$	6.69779	0.509224	0.254612	+	0.967043i	$0.418052\pi$
$174$	0	0
$175$	−7.66424	−0.579362
$176$	18.0454	1.36023
$177$	0	0
$178$	−0.115744	−0.00867535
$179$	−3.49182	−0.260991	−0.130495	−	0.991449i	$-0.541657\pi$
$179$	−3.49182	−0.260991	−0.130495	+	0.991449i	$0.541657\pi$
$180$	0	0
$181$	−10.5675	−0.785479	−0.392739	−	0.919650i	$-0.628473\pi$
$181$	−10.5675	−0.785479	−0.392739	+	0.919650i	$0.628473\pi$
$182$	−0.124399	−0.00922110
$183$	0	0
$184$	6.74994	0.497612
$185$	20.7036	1.52216
$186$	0	0
$187$	−26.2093	−1.91662
$188$	13.4733	0.982643
$189$	0	0
$190$	1.39969	0.101544
$191$	9.92021	0.717801	0.358901	−	0.933376i	$-0.383152\pi$
$191$	9.92021	0.717801	0.358901	+	0.933376i	$0.383152\pi$
$192$	0	0
$193$	−12.4266	−0.894490	−0.447245	−	0.894412i	$-0.647595\pi$
$193$	−12.4266	−0.894490	−0.447245	+	0.894412i	$0.647595\pi$
$194$	1.33901	0.0961353
$195$	0	0
$196$	−1.86377	−0.133126
$197$	−11.7779	−0.839138	−0.419569	−	0.907723i	$-0.637819\pi$
$197$	−11.7779	−0.839138	−0.419569	+	0.907723i	$0.637819\pi$
$198$	0	0
$199$	14.3199	1.01511	0.507557	−	0.861618i	$-0.330549\pi$
$199$	14.3199	1.01511	0.507557	+	0.861618i	$0.330549\pi$
$200$	−10.9299	−0.772859
$201$	0	0
$202$	2.02896	0.142757
$203$	−3.66220	−0.257036
$204$	0	0
$205$	6.57302	0.459080
$206$	4.27416	0.297795
$207$	0	0
$208$	1.07893	0.0748105
$209$	6.00710	0.415520
$210$	0	0
$211$	−1.57807	−0.108639	−0.0543193	−	0.998524i	$-0.517299\pi$
$211$	−1.57807	−0.108639	−0.0543193	+	0.998524i	$0.517299\pi$
$212$	−0.905198	−0.0621693
$213$	0	0
$214$	−5.47493	−0.374259
$215$	0.548154	0.0373838
$216$	0	0
$217$	−3.89423	−0.264357
$218$	−4.85630	−0.328910
$219$	0	0
$220$	−37.3886	−2.52074
$221$	−1.56705	−0.105411
$222$	0	0
$223$	21.6864	1.45223	0.726114	−	0.687574i	$-0.241324\pi$
$223$	21.6864	1.45223	0.726114	+	0.687574i	$0.241324\pi$
$224$	−4.03371	−0.269513
$225$	0	0
$226$	−1.09425	−0.0727881
$227$	3.34183	0.221805	0.110903	−	0.993831i	$-0.464626\pi$
$227$	3.34183	0.221805	0.110903	+	0.993831i	$0.464626\pi$
$228$	0	0
$229$	14.8808	0.983351	0.491676	−	0.870778i	$-0.336385\pi$
$229$	14.8808	0.983351	0.491676	+	0.870778i	$0.336385\pi$
$230$	−6.21696	−0.409934
$231$	0	0
$232$	−5.22262	−0.342882
$233$	21.3277	1.39723	0.698613	−	0.715499i	$-0.253801\pi$
$233$	21.3277	1.39723	0.698613	+	0.715499i	$0.253801\pi$
$234$	0	0
$235$	−25.7260	−1.67818
$236$	−22.0019	−1.43220
$237$	0	0
$238$	1.71607	0.111236
$239$	12.1410	0.785337	0.392669	−	0.919680i	$-0.371552\pi$
$239$	12.1410	0.785337	0.392669	+	0.919680i	$0.371552\pi$
$240$	0	0
$241$	−12.2122	−0.786656	−0.393328	−	0.919398i	$-0.628676\pi$
$241$	−12.2122	−0.786656	−0.393328	+	0.919398i	$0.628676\pi$
$242$	7.66864	0.492959
$243$	0	0
$244$	20.5876	1.31798
$245$	3.55869	0.227356
$246$	0	0
$247$	0.359163	0.0228530
$248$	−5.55351	−0.352648
$249$	0	0
$250$	3.49943	0.221324
$251$	−1.80820	−0.114133	−0.0570664	−	0.998370i	$-0.518175\pi$
$251$	−1.80820	−0.114133	−0.0570664	+	0.998370i	$0.518175\pi$
$252$	0	0
$253$	−26.6815	−1.67745
$254$	−0.369092	−0.0231589
$255$	0	0
$256$	6.18013	0.386258
$257$	−12.3720	−0.771746	−0.385873	−	0.922552i	$-0.626100\pi$
$257$	−12.3720	−0.771746	−0.385873	+	0.922552i	$0.626100\pi$
$258$	0	0
$259$	−5.81777	−0.361498
$260$	−2.23545	−0.138637
$261$	0	0
$262$	−4.20400	−0.259724
$263$	−7.65930	−0.472293	−0.236146	−	0.971718i	$-0.575884\pi$
$263$	−7.65930	−0.472293	−0.236146	+	0.971718i	$0.575884\pi$
$264$	0	0
$265$	1.72839	0.106174
$266$	−0.393317	−0.0241158
$267$	0	0
$268$	4.53291	0.276891
$269$	−14.6653	−0.894162	−0.447081	−	0.894493i	$-0.647536\pi$
$269$	−14.6653	−0.894162	−0.447081	+	0.894493i	$0.647536\pi$
$270$	0	0
$271$	−0.149863	−0.00910356	−0.00455178	−	0.999990i	$-0.501449\pi$
$271$	−0.149863	−0.00910356	−0.00455178	+	0.999990i	$0.501449\pi$
$272$	−14.8837	−0.902454
$273$	0	0
$274$	7.18229	0.433898
$275$	43.2042	2.60531
$276$	0	0
$277$	−3.18177	−0.191174	−0.0955869	−	0.995421i	$-0.530473\pi$
$277$	−3.18177	−0.191174	−0.0955869	+	0.995421i	$0.530473\pi$
$278$	3.14507	0.188629
$279$	0	0
$280$	5.07500	0.303289
$281$	14.3932	0.858629	0.429314	−	0.903155i	$-0.358755\pi$
$281$	14.3932	0.858629	0.429314	+	0.903155i	$0.358755\pi$
$282$	0	0
$283$	5.92097	0.351965	0.175983	−	0.984393i	$-0.443690\pi$
$283$	5.92097	0.351965	0.175983	+	0.984393i	$0.443690\pi$
$284$	−0.807855	−0.0479374
$285$	0	0
$286$	0.701253	0.0414660
$287$	−1.84704	−0.109027
$288$	0	0
$289$	4.61714	0.271596
$290$	4.81024	0.282467
$291$	0	0
$292$	−17.7082	−1.03629
$293$	15.4657	0.903515	0.451757	−	0.892141i	$-0.350797\pi$
$293$	15.4657	0.903515	0.451757	+	0.892141i	$0.350797\pi$
$294$	0	0
$295$	42.0103	2.44594
$296$	−8.29664	−0.482233
$297$	0	0
$298$	5.37672	0.311465
$299$	−1.59528	−0.0922574
$300$	0	0
$301$	−0.154033	−0.00887830
$302$	−4.52060	−0.260131
$303$	0	0
$304$	3.41129	0.195651
$305$	−39.3099	−2.25088
$306$	0	0
$307$	15.4572	0.882190	0.441095	−	0.897460i	$-0.354590\pi$
$307$	15.4572	0.882190	0.441095	+	0.897460i	$0.354590\pi$
$308$	10.5063	0.598651
$309$	0	0
$310$	5.11500	0.290512
$311$	−21.9042	−1.24207	−0.621037	−	0.783781i	$-0.713288\pi$
$311$	−21.9042	−1.24207	−0.621037	+	0.783781i	$0.713288\pi$
$312$	0	0
$313$	26.0319	1.47141	0.735706	−	0.677301i	$-0.236850\pi$
$313$	26.0319	1.47141	0.735706	+	0.677301i	$0.236850\pi$
$314$	−0.885052	−0.0499464
$315$	0	0
$316$	−20.3382	−1.14412
$317$	−5.86217	−0.329252	−0.164626	−	0.986356i	$-0.552642\pi$
$317$	−5.86217	−0.329252	−0.164626	+	0.986356i	$0.552642\pi$
$318$	0	0
$319$	20.6442	1.15585
$320$	−17.4858	−0.977486
$321$	0	0
$322$	1.74698	0.0973555
$323$	−4.95459	−0.275681
$324$	0	0
$325$	2.58317	0.143288
$326$	0.0937709	0.00519349
$327$	0	0
$328$	−2.63404	−0.145440
$329$	7.22906	0.398551
$330$	0	0
$331$	−16.2480	−0.893073	−0.446536	−	0.894765i	$-0.647343\pi$
$331$	−16.2480	−0.893073	−0.446536	+	0.894765i	$0.647343\pi$
$332$	−12.0965	−0.663881
$333$	0	0
$334$	3.06218	0.167555
$335$	−8.65513	−0.472881
$336$	0	0
$337$	14.7909	0.805713	0.402856	−	0.915263i	$-0.368017\pi$
$337$	14.7909	0.805713	0.402856	+	0.915263i	$0.368017\pi$
$338$	−4.75627	−0.258707
$339$	0	0
$340$	30.8376	1.67241
$341$	21.9522	1.18878
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−0.219664	−0.0118435
$345$	0	0
$346$	2.47210	0.132901
$347$	32.9538	1.76905	0.884527	−	0.466488i	$-0.154481\pi$
$347$	32.9538	1.76905	0.884527	+	0.466488i	$0.154481\pi$
$348$	0	0
$349$	−14.6933	−0.786517	−0.393258	−	0.919428i	$-0.628652\pi$
$349$	−14.6933	−0.786517	−0.393258	+	0.919428i	$0.628652\pi$
$350$	−2.82881	−0.151206
$351$	0	0
$352$	22.7385	1.21196
$353$	−22.5672	−1.20113	−0.600566	−	0.799575i	$-0.705058\pi$
$353$	−22.5672	−1.20113	−0.600566	+	0.799575i	$0.705058\pi$
$354$	0	0
$355$	1.54252	0.0818684
$356$	0.584460	0.0309763
$357$	0	0
$358$	−1.28880	−0.0681153
$359$	35.0905	1.85200	0.926002	−	0.377519i	$-0.123223\pi$
$359$	35.0905	1.85200	0.926002	+	0.377519i	$0.123223\pi$
$360$	0	0
$361$	−17.8644	−0.940233
$362$	−3.90039	−0.205000
$363$	0	0
$364$	0.628168	0.0329250
$365$	33.8121	1.76980
$366$	0	0
$367$	−7.30134	−0.381127	−0.190563	−	0.981675i	$-0.561031\pi$
$367$	−7.30134	−0.381127	−0.190563	+	0.981675i	$0.561031\pi$
$368$	−15.1518	−0.789842
$369$	0	0
$370$	7.64153	0.397264
$371$	−0.485681	−0.0252153
$372$	0	0
$373$	10.6197	0.549865	0.274933	−	0.961464i	$-0.411344\pi$
$373$	10.6197	0.549865	0.274933	+	0.961464i	$0.411344\pi$
$374$	−9.67365	−0.500213
$375$	0	0
$376$	10.3093	0.531660
$377$	1.23431	0.0635704
$378$	0	0
$379$	31.5405	1.62013	0.810063	−	0.586343i	$-0.199433\pi$
$379$	31.5405	1.62013	0.810063	+	0.586343i	$0.199433\pi$
$380$	−7.06790	−0.362576
$381$	0	0
$382$	3.66147	0.187337
$383$	26.3521	1.34653	0.673265	−	0.739401i	$-0.264891\pi$
$383$	26.3521	1.34653	0.673265	+	0.739401i	$0.264891\pi$
$384$	0	0
$385$	−20.0607	−1.02239
$386$	−4.58658	−0.233451
$387$	0	0
$388$	−6.76147	−0.343262
$389$	−13.3973	−0.679268	−0.339634	−	0.940558i	$-0.610303\pi$
$389$	−13.3973	−0.679268	−0.339634	+	0.940558i	$0.610303\pi$
$390$	0	0
$391$	22.0066	1.11292
$392$	−1.42609	−0.0720283
$393$	0	0
$394$	−4.34712	−0.219005
$395$	38.8338	1.95394
$396$	0	0
$397$	27.1708	1.36366	0.681831	−	0.731509i	$-0.261184\pi$
$397$	27.1708	1.36366	0.681831	+	0.731509i	$0.261184\pi$
$398$	5.28538	0.264932
$399$	0	0
$400$	24.5346	1.22673
$401$	12.9744	0.647909	0.323955	−	0.946073i	$-0.394987\pi$
$401$	12.9744	0.647909	0.323955	+	0.946073i	$0.394987\pi$
$402$	0	0
$403$	1.31252	0.0653811
$404$	−10.2455	−0.509731
$405$	0	0
$406$	−1.35169	−0.0670832
$407$	32.7954	1.62561
$408$	0	0
$409$	23.6752	1.17066	0.585330	−	0.810795i	$-0.300965\pi$
$409$	23.6752	1.17066	0.585330	+	0.810795i	$0.300965\pi$
$410$	2.42605	0.119814
$411$	0	0
$412$	−21.5828	−1.06331
$413$	−11.8050	−0.580887
$414$	0	0
$415$	23.0970	1.13379
$416$	1.35953	0.0666563
$417$	0	0
$418$	2.21717	0.108446
$419$	−24.7078	−1.20706	−0.603528	−	0.797342i	$-0.706239\pi$
$419$	−24.7078	−1.20706	−0.603528	+	0.797342i	$0.706239\pi$
$420$	0	0
$421$	33.1633	1.61628	0.808141	−	0.588989i	$-0.200474\pi$
$421$	33.1633	1.61628	0.808141	+	0.588989i	$0.200474\pi$
$422$	−0.582452	−0.0283533
$423$	0	0
$424$	−0.692624	−0.0336368
$425$	−35.6343	−1.72852
$426$	0	0
$427$	11.0462	0.534563
$428$	27.6463	1.33633
$429$	0	0
$430$	0.202319	0.00975671
$431$	32.8229	1.58102	0.790512	−	0.612447i	$-0.209815\pi$
$431$	32.8229	1.58102	0.790512	+	0.612447i	$0.209815\pi$
$432$	0	0
$433$	−29.4329	−1.41445	−0.707227	−	0.706986i	$-0.750054\pi$
$433$	−29.4329	−1.41445	−0.707227	+	0.706986i	$0.750054\pi$
$434$	−1.43733	−0.0689940
$435$	0	0
$436$	24.5224	1.17441
$437$	−5.04385	−0.241280
$438$	0	0
$439$	8.19265	0.391014	0.195507	−	0.980702i	$-0.437365\pi$
$439$	8.19265	0.391014	0.195507	+	0.980702i	$0.437365\pi$
$440$	−28.6083	−1.36385
$441$	0	0
$442$	−0.578385	−0.0275110
$443$	4.41405	0.209718	0.104859	−	0.994487i	$-0.466561\pi$
$443$	4.41405	0.209718	0.104859	+	0.994487i	$0.466561\pi$
$444$	0	0
$445$	−1.11597	−0.0529020
$446$	8.00428	0.379014
$447$	0	0
$448$	4.91356	0.232144
$449$	28.4071	1.34062	0.670308	−	0.742083i	$-0.266162\pi$
$449$	28.4071	1.34062	0.670308	+	0.742083i	$0.266162\pi$
$450$	0	0
$451$	10.4120	0.490280
$452$	5.52551	0.259898
$453$	0	0
$454$	1.23344	0.0578884
$455$	−1.19942	−0.0562299
$456$	0	0
$457$	−17.4524	−0.816387	−0.408194	−	0.912895i	$-0.633841\pi$
$457$	−17.4524	−0.816387	−0.408194	+	0.912895i	$0.633841\pi$
$458$	5.49239	0.256642
$459$	0	0
$460$	31.3932	1.46372
$461$	13.0605	0.608287	0.304143	−	0.952626i	$-0.401630\pi$
$461$	13.0605	0.608287	0.304143	+	0.952626i	$0.401630\pi$
$462$	0	0
$463$	−35.1301	−1.63263	−0.816317	−	0.577604i	$-0.803988\pi$
$463$	−35.1301	−1.63263	−0.816317	+	0.577604i	$0.803988\pi$
$464$	11.7234	0.544244
$465$	0	0
$466$	7.87190	0.364659
$467$	−20.6369	−0.954963	−0.477482	−	0.878642i	$-0.658450\pi$
$467$	−20.6369	−0.954963	−0.477482	+	0.878642i	$0.658450\pi$
$468$	0	0
$469$	2.43212	0.112305
$470$	−9.49525	−0.437983
$471$	0	0
$472$	−16.8350	−0.774893
$473$	0.868300	0.0399245
$474$	0	0
$475$	8.16728	0.374740
$476$	−8.66546	−0.397181
$477$	0	0
$478$	4.48116	0.204963
$479$	27.1047	1.23844	0.619222	−	0.785216i	$-0.287448\pi$
$479$	27.1047	1.23844	0.619222	+	0.785216i	$0.287448\pi$
$480$	0	0
$481$	1.96083	0.0894061
$482$	−4.50742	−0.205307
$483$	0	0
$484$	−38.7236	−1.76016
$485$	12.9104	0.586229
$486$	0	0
$487$	−23.2552	−1.05379	−0.526897	−	0.849929i	$-0.676645\pi$
$487$	−23.2552	−1.05379	−0.526897	+	0.849929i	$0.676645\pi$
$488$	15.7528	0.713097
$489$	0	0
$490$	1.31348	0.0593371
$491$	−11.4403	−0.516292	−0.258146	−	0.966106i	$-0.583112\pi$
$491$	−11.4403	−0.516292	−0.258146	+	0.966106i	$0.583112\pi$
$492$	0	0
$493$	−17.0271	−0.766863
$494$	0.132564	0.00596435
$495$	0	0
$496$	12.4661	0.559746
$497$	−0.433452	−0.0194430
$498$	0	0
$499$	−12.4731	−0.558375	−0.279187	−	0.960237i	$-0.590065\pi$
$499$	−12.4731	−0.558375	−0.279187	+	0.960237i	$0.590065\pi$
$500$	−17.6708	−0.790261
$501$	0	0
$502$	−0.667393	−0.0297872
$503$	11.0082	0.490829	0.245415	−	0.969418i	$-0.421076\pi$
$503$	11.0082	0.490829	0.245415	+	0.969418i	$0.421076\pi$
$504$	0	0
$505$	19.5627	0.870529
$506$	−9.84793	−0.437794
$507$	0	0
$508$	1.86377	0.0826915
$509$	−8.60414	−0.381372	−0.190686	−	0.981651i	$-0.561071\pi$
$509$	−8.60414	−0.381372	−0.190686	+	0.981651i	$0.561071\pi$
$510$	0	0
$511$	−9.50128	−0.420312
$512$	22.0430	0.974171
$513$	0	0
$514$	−4.56642	−0.201416
$515$	41.2103	1.81594
$516$	0	0
$517$	−40.7510	−1.79223
$518$	−2.14729	−0.0943466
$519$	0	0
$520$	−1.71048	−0.0750097
$521$	11.6367	0.509814	0.254907	−	0.966966i	$-0.417955\pi$
$521$	11.6367	0.509814	0.254907	+	0.966966i	$0.417955\pi$
$522$	0	0
$523$	18.4491	0.806723	0.403361	−	0.915041i	$-0.367842\pi$
$523$	18.4491	0.806723	0.403361	+	0.915041i	$0.367842\pi$
$524$	21.2285	0.927373
$525$	0	0
$526$	−2.82699	−0.123263
$527$	−18.1059	−0.788706
$528$	0	0
$529$	−0.596955	−0.0259545
$530$	0.637934	0.0277101
$531$	0	0
$532$	1.98610	0.0861083
$533$	0.622528	0.0269647
$534$	0	0
$535$	−52.7878	−2.28222
$536$	3.46841	0.149813
$537$	0	0
$538$	−5.41286	−0.233365
$539$	5.63711	0.242808
$540$	0	0
$541$	−8.37023	−0.359864	−0.179932	−	0.983679i	$-0.557588\pi$
$541$	−8.37023	−0.359864	−0.179932	+	0.983679i	$0.557588\pi$
$542$	−0.0553134	−0.00237592
$543$	0	0
$544$	−18.7544	−0.804089
$545$	−46.8231	−2.00568
$546$	0	0
$547$	13.4728	0.576056	0.288028	−	0.957622i	$-0.407000\pi$
$547$	13.4728	0.576056	0.288028	+	0.957622i	$0.407000\pi$
$548$	−36.2677	−1.54928
$549$	0	0
$550$	15.9463	0.679954
$551$	3.90257	0.166255
$552$	0	0
$553$	−10.9124	−0.464043
$554$	−1.17437	−0.0498940
$555$	0	0
$556$	−15.8814	−0.673520
$557$	−35.0898	−1.48680	−0.743401	−	0.668846i	$-0.766789\pi$
$557$	−35.0898	−1.48680	−0.743401	+	0.668846i	$0.766789\pi$
$558$	0	0
$559$	0.0519154	0.00219579
$560$	−11.3920	−0.481400
$561$	0	0
$562$	5.31243	0.224091
$563$	19.3082	0.813742	0.406871	−	0.913486i	$-0.366620\pi$
$563$	19.3082	0.813742	0.406871	+	0.913486i	$0.366620\pi$
$564$	0	0
$565$	−10.5504	−0.443859
$566$	2.18538	0.0918585
$567$	0	0
$568$	−0.618140	−0.0259366
$569$	32.9892	1.38298	0.691489	−	0.722387i	$-0.256955\pi$
$569$	32.9892	1.38298	0.691489	+	0.722387i	$0.256955\pi$
$570$	0	0
$571$	−10.2218	−0.427767	−0.213884	−	0.976859i	$-0.568611\pi$
$571$	−10.2218	−0.427767	−0.213884	+	0.976859i	$0.568611\pi$
$572$	−3.54106	−0.148059
$573$	0	0
$574$	−0.681727	−0.0284547
$575$	−36.2763	−1.51283
$576$	0	0
$577$	−27.6947	−1.15295	−0.576474	−	0.817116i	$-0.695572\pi$
$577$	−27.6947	−1.15295	−0.576474	+	0.817116i	$0.695572\pi$
$578$	1.70415	0.0708833
$579$	0	0
$580$	−24.2898	−1.00858
$581$	−6.49033	−0.269264
$582$	0	0
$583$	2.73784	0.113390
$584$	−13.5497	−0.560689
$585$	0	0
$586$	5.70826	0.235806
$587$	−36.0542	−1.48812	−0.744058	−	0.668115i	$-0.767101\pi$
$587$	−36.0542	−1.48812	−0.744058	+	0.668115i	$0.767101\pi$
$588$	0	0
$589$	4.14982	0.170990
$590$	15.5057	0.638359
$591$	0	0
$592$	18.6237	0.765431
$593$	7.17640	0.294699	0.147350	−	0.989084i	$-0.452926\pi$
$593$	7.17640	0.294699	0.147350	+	0.989084i	$0.452926\pi$
$594$	0	0
$595$	16.5458	0.678313
$596$	−27.1503	−1.11212
$597$	0	0
$598$	−0.588805	−0.0240780
$599$	37.8575	1.54682	0.773408	−	0.633908i	$-0.218550\pi$
$599$	37.8575	1.54682	0.773408	+	0.633908i	$0.218550\pi$
$600$	0	0
$601$	−24.0295	−0.980183	−0.490092	−	0.871671i	$-0.663037\pi$
$601$	−24.0295	−0.980183	−0.490092	+	0.871671i	$0.663037\pi$
$602$	−0.0568523	−0.00231713
$603$	0	0
$604$	22.8273	0.928828
$605$	73.9389	3.00604
$606$	0	0
$607$	−27.9293	−1.13362	−0.566808	−	0.823850i	$-0.691822\pi$
$607$	−27.9293	−1.13362	−0.566808	+	0.823850i	$0.691822\pi$
$608$	4.29846	0.174325
$609$	0	0
$610$	−14.5090	−0.587451
$611$	−2.43650	−0.0985701
$612$	0	0
$613$	22.0680	0.891320	0.445660	−	0.895202i	$-0.352969\pi$
$613$	22.0680	0.891320	0.445660	+	0.895202i	$0.352969\pi$
$614$	5.70514	0.230241
$615$	0	0
$616$	8.03902	0.323901
$617$	−45.7527	−1.84194	−0.920968	−	0.389639i	$-0.872600\pi$
$617$	−45.7527	−1.84194	−0.920968	+	0.389639i	$0.872600\pi$
$618$	0	0
$619$	30.9630	1.24451	0.622255	−	0.782815i	$-0.286217\pi$
$619$	30.9630	1.24451	0.622255	+	0.782815i	$0.286217\pi$
$620$	−25.8287	−1.03731
$621$	0	0
$622$	−8.08468	−0.324166
$623$	0.313590	0.0125637
$624$	0	0
$625$	−4.58062	−0.183225
$626$	9.60819	0.384020
$627$	0	0
$628$	4.46917	0.178339
$629$	−27.0493	−1.07852
$630$	0	0
$631$	17.4733	0.695599	0.347800	−	0.937569i	$-0.386929\pi$
$631$	17.4733	0.695599	0.347800	+	0.937569i	$0.386929\pi$
$632$	−15.5621	−0.619025
$633$	0	0
$634$	−2.16368	−0.0859308
$635$	−3.55869	−0.141222
$636$	0	0
$637$	0.337042	0.0133541
$638$	7.61962	0.301664
$639$	0	0
$640$	−35.1633	−1.38995
$641$	−41.0725	−1.62227	−0.811134	−	0.584861i	$-0.801149\pi$
$641$	−41.0725	−1.62227	−0.811134	+	0.584861i	$0.801149\pi$
$642$	0	0
$643$	35.6390	1.40547	0.702733	−	0.711454i	$-0.251963\pi$
$643$	35.6390	1.40547	0.702733	+	0.711454i	$0.251963\pi$
$644$	−8.82157	−0.347619
$645$	0	0
$646$	−1.82870	−0.0719492
$647$	−29.1831	−1.14731	−0.573653	−	0.819098i	$-0.694474\pi$
$647$	−29.1831	−1.14731	−0.573653	+	0.819098i	$0.694474\pi$
$648$	0	0
$649$	66.5462	2.61217
$650$	0.953427	0.0373965
$651$	0	0
$652$	−0.473506	−0.0185439
$653$	6.80472	0.266289	0.133145	−	0.991097i	$-0.457493\pi$
$653$	6.80472	0.266289	0.133145	+	0.991097i	$0.457493\pi$
$654$	0	0
$655$	−40.5338	−1.58379
$656$	5.91271	0.230852
$657$	0	0
$658$	2.66819	0.104017
$659$	44.4499	1.73152	0.865761	−	0.500458i	$-0.166835\pi$
$659$	44.4499	1.73152	0.865761	+	0.500458i	$0.166835\pi$
$660$	0	0
$661$	−14.4952	−0.563797	−0.281899	−	0.959444i	$-0.590964\pi$
$661$	−14.4952	−0.563797	−0.281899	+	0.959444i	$0.590964\pi$
$662$	−5.99702	−0.233081
$663$	0	0
$664$	−9.25578	−0.359194
$665$	−3.79226	−0.147057
$666$	0	0
$667$	−17.3339	−0.671170
$668$	−15.4628	−0.598274
$669$	0	0
$670$	−3.19454	−0.123416
$671$	−62.2686	−2.40385
$672$	0	0
$673$	−37.6427	−1.45102	−0.725509	−	0.688213i	$-0.758395\pi$
$673$	−37.6427	−1.45102	−0.725509	+	0.688213i	$0.758395\pi$
$674$	5.45921	0.210281
$675$	0	0
$676$	24.0173	0.923742
$677$	50.9327	1.95750	0.978752	−	0.205047i	$-0.0657348\pi$
$677$	50.9327	1.95750	0.978752	+	0.205047i	$0.0657348\pi$
$678$	0	0
$679$	−3.62785	−0.139224
$680$	23.5958	0.904858
$681$	0	0
$682$	8.10238	0.310256
$683$	−35.5266	−1.35939	−0.679694	−	0.733496i	$-0.737887\pi$
$683$	−35.5266	−1.35939	−0.679694	+	0.733496i	$0.737887\pi$
$684$	0	0
$685$	69.2496	2.64589
$686$	−0.369092	−0.0140920
$687$	0	0
$688$	0.493087	0.0187988
$689$	0.163695	0.00623627
$690$	0	0
$691$	−20.7423	−0.789073	−0.394537	−	0.918880i	$-0.629095\pi$
$691$	−20.7423	−0.789073	−0.394537	+	0.918880i	$0.629095\pi$
$692$	−12.4832	−0.474538
$693$	0	0
$694$	12.1630	0.461701
$695$	30.3239	1.15025
$696$	0	0
$697$	−8.58766	−0.325281
$698$	−5.42320	−0.205271
$699$	0	0
$700$	14.2844	0.539899
$701$	−18.7990	−0.710029	−0.355015	−	0.934861i	$-0.615524\pi$
$701$	−18.7990	−0.710029	−0.355015	+	0.934861i	$0.615524\pi$
$702$	0	0
$703$	6.19961	0.233823
$704$	−27.6983	−1.04392
$705$	0	0
$706$	−8.32939	−0.313481
$707$	−5.49718	−0.206743
$708$	0	0
$709$	48.5666	1.82396	0.911978	−	0.410239i	$-0.134555\pi$
$709$	48.5666	1.82396	0.911978	+	0.410239i	$0.134555\pi$
$710$	0.569332	0.0213666
$711$	0	0
$712$	0.447207	0.0167598
$713$	−18.4321	−0.690287
$714$	0	0
$715$	6.76129	0.252858
$716$	6.50795	0.243213
$717$	0	0
$718$	12.9516	0.483350
$719$	8.40818	0.313572	0.156786	−	0.987633i	$-0.449887\pi$
$719$	8.40818	0.313572	0.156786	+	0.987633i	$0.449887\pi$
$720$	0	0
$721$	−11.5802	−0.431269
$722$	−6.59362	−0.245389
$723$	0	0
$724$	19.6955	0.731976
$725$	28.0680	1.04242
$726$	0	0
$727$	−18.4608	−0.684675	−0.342337	−	0.939577i	$-0.611219\pi$
$727$	−18.4608	−0.684675	−0.342337	+	0.939577i	$0.611219\pi$
$728$	0.480651	0.0178141
$729$	0	0
$730$	12.4798	0.461897
$731$	−0.716164	−0.0264883
$732$	0	0
$733$	−44.5905	−1.64699	−0.823494	−	0.567325i	$-0.807978\pi$
$733$	−44.5905	−1.64699	−0.823494	+	0.567325i	$0.807978\pi$
$734$	−2.69487	−0.0994693
$735$	0	0
$736$	−19.0923	−0.703751
$737$	−13.7101	−0.505018
$738$	0	0
$739$	28.2628	1.03966	0.519832	−	0.854268i	$-0.325994\pi$
$739$	28.2628	1.03966	0.519832	+	0.854268i	$0.325994\pi$
$740$	−38.5868	−1.41848
$741$	0	0
$742$	−0.179261	−0.00658088
$743$	18.2135	0.668189	0.334094	−	0.942540i	$-0.391570\pi$
$743$	18.2135	0.668189	0.334094	+	0.942540i	$0.391570\pi$
$744$	0	0
$745$	51.8408	1.89930
$746$	3.91963	0.143508
$747$	0	0
$748$	48.8482	1.78607
$749$	14.8335	0.542005
$750$	0	0
$751$	7.97494	0.291010	0.145505	−	0.989358i	$-0.453519\pi$
$751$	7.97494	0.291010	0.145505	+	0.989358i	$0.453519\pi$
$752$	−23.1416	−0.843886
$753$	0	0
$754$	0.455575	0.0165911
$755$	−43.5864	−1.58627
$756$	0	0
$757$	−31.3158	−1.13819	−0.569097	−	0.822271i	$-0.692707\pi$
$757$	−31.3158	−1.13819	−0.569097	+	0.822271i	$0.692707\pi$
$758$	11.6413	0.422833
$759$	0	0
$760$	−5.40809	−0.196172
$761$	42.1779	1.52895	0.764474	−	0.644655i	$-0.222999\pi$
$761$	42.1779	1.52895	0.764474	+	0.644655i	$0.222999\pi$
$762$	0	0
$763$	13.1574	0.476330
$764$	−18.4890	−0.668909
$765$	0	0
$766$	9.72636	0.351428
$767$	3.97878	0.143665
$768$	0	0
$769$	40.6114	1.46448	0.732241	−	0.681045i	$-0.238474\pi$
$769$	40.6114	1.46448	0.732241	+	0.681045i	$0.238474\pi$
$770$	−7.40425	−0.266831
$771$	0	0
$772$	23.1604	0.833562
$773$	39.1780	1.40914	0.704568	−	0.709636i	$-0.251141\pi$
$773$	39.1780	1.40914	0.704568	+	0.709636i	$0.251141\pi$
$774$	0	0
$775$	29.8463	1.07211
$776$	−5.17363	−0.185722
$777$	0	0
$778$	−4.94482	−0.177280
$779$	1.96827	0.0705205
$780$	0	0
$781$	2.44342	0.0874323
$782$	8.12246	0.290458
$783$	0	0
$784$	3.20118	0.114328
$785$	−8.53343	−0.304571
$786$	0	0
$787$	−25.9711	−0.925769	−0.462884	−	0.886419i	$-0.653185\pi$
$787$	−25.9711	−0.925769	−0.462884	+	0.886419i	$0.653185\pi$
$788$	21.9512	0.781980
$789$	0	0
$790$	14.3333	0.509955
$791$	2.96469	0.105412
$792$	0	0
$793$	−3.72303	−0.132209
$794$	10.0285	0.355899
$795$	0	0
$796$	−26.6891	−0.945969
$797$	−2.09192	−0.0740996	−0.0370498	−	0.999313i	$-0.511796\pi$
$797$	−2.09192	−0.0740996	−0.0370498	+	0.999313i	$0.511796\pi$
$798$	0	0
$799$	33.6110	1.18907
$800$	30.9153	1.09302
$801$	0	0
$802$	4.78874	0.169096
$803$	53.5598	1.89008
$804$	0	0
$805$	16.8439	0.593670
$806$	0.484439	0.0170637
$807$	0	0
$808$	−7.83945	−0.275791
$809$	−45.3379	−1.59400	−0.796999	−	0.603981i	$-0.793580\pi$
$809$	−45.3379	−1.59400	−0.796999	+	0.603981i	$0.793580\pi$
$810$	0	0
$811$	−14.1153	−0.495655	−0.247827	−	0.968804i	$-0.579717\pi$
$811$	−14.1153	−0.495655	−0.247827	+	0.968804i	$0.579717\pi$
$812$	6.82550	0.239528
$813$	0	0
$814$	12.1045	0.424263
$815$	0.904113	0.0316697
$816$	0	0
$817$	0.164143	0.00574262
$818$	8.73831	0.305528
$819$	0	0
$820$	−12.2506	−0.427810
$821$	17.8382	0.622559	0.311279	−	0.950318i	$-0.399243\pi$
$821$	17.8382	0.622559	0.311279	+	0.950318i	$0.399243\pi$
$822$	0	0
$823$	−35.0158	−1.22057	−0.610287	−	0.792180i	$-0.708946\pi$
$823$	−35.0158	−1.22057	−0.610287	+	0.792180i	$0.708946\pi$
$824$	−16.5144	−0.575306
$825$	0	0
$826$	−4.35714	−0.151604
$827$	−0.387394	−0.0134710	−0.00673550	−	0.999977i	$-0.502144\pi$
$827$	−0.387394	−0.0134710	−0.00673550	+	0.999977i	$0.502144\pi$
$828$	0	0
$829$	−40.9000	−1.42051	−0.710257	−	0.703942i	$-0.751421\pi$
$829$	−40.9000	−1.42051	−0.710257	+	0.703942i	$0.751421\pi$
$830$	8.52494	0.295905
$831$	0	0
$832$	−1.65607	−0.0574140
$833$	−4.64942	−0.161093
$834$	0	0
$835$	29.5247	1.02174
$836$	−11.1959	−0.387217
$837$	0	0
$838$	−9.11946	−0.315026
$839$	−40.9774	−1.41470	−0.707349	−	0.706864i	$-0.750109\pi$
$839$	−40.9774	−1.41470	−0.707349	+	0.706864i	$0.750109\pi$
$840$	0	0
$841$	−15.5883	−0.537527
$842$	12.2403	0.421829
$843$	0	0
$844$	2.94115	0.101239
$845$	−45.8587	−1.57759
$846$	0	0
$847$	−20.7770	−0.713907
$848$	1.55475	0.0533905
$849$	0	0
$850$	−13.1523	−0.451122
$851$	−27.5366	−0.943941
$852$	0	0
$853$	−47.2571	−1.61805	−0.809026	−	0.587773i	$-0.800005\pi$
$853$	−47.2571	−1.61805	−0.809026	+	0.587773i	$0.800005\pi$
$854$	4.07706	0.139514
$855$	0	0
$856$	21.1539	0.723025
$857$	57.5823	1.96697	0.983486	−	0.180983i	$-0.0579279\pi$
$857$	57.5823	1.96697	0.983486	+	0.180983i	$0.0579279\pi$
$858$	0	0
$859$	−39.1974	−1.33740	−0.668699	−	0.743534i	$-0.733148\pi$
$859$	−39.1974	−1.33740	−0.668699	+	0.743534i	$0.733148\pi$
$860$	−1.02163	−0.0348374
$861$	0	0
$862$	12.1147	0.412628
$863$	5.56776	0.189529	0.0947644	−	0.995500i	$-0.469790\pi$
$863$	5.56776	0.189529	0.0947644	+	0.995500i	$0.469790\pi$
$864$	0	0
$865$	23.8353	0.810426
$866$	−10.8634	−0.369155
$867$	0	0
$868$	7.25795	0.246351
$869$	61.5145	2.08674
$870$	0	0
$871$	−0.819724	−0.0277753
$872$	18.7636	0.635416
$873$	0	0
$874$	−1.86164	−0.0629710
$875$	−9.48119	−0.320523
$876$	0	0
$877$	−0.150328	−0.00507621	−0.00253811	−	0.999997i	$-0.500808\pi$
$877$	−0.150328	−0.00507621	−0.00253811	+	0.999997i	$0.500808\pi$
$878$	3.02384	0.102050
$879$	0	0
$880$	64.2180	2.16479
$881$	48.7777	1.64336	0.821680	−	0.569948i	$-0.193037\pi$
$881$	48.7777	1.64336	0.821680	+	0.569948i	$0.193037\pi$
$882$	0	0
$883$	−8.58743	−0.288990	−0.144495	−	0.989506i	$-0.546156\pi$
$883$	−8.58743	−0.288990	−0.144495	+	0.989506i	$0.546156\pi$
$884$	2.92062	0.0982311
$885$	0	0
$886$	1.62919	0.0547337
$887$	−57.0210	−1.91458	−0.957290	−	0.289131i	$-0.906634\pi$
$887$	−57.0210	−1.91458	−0.957290	+	0.289131i	$0.906634\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	−0.411895	−0.0138068
$891$	0	0
$892$	−40.4185	−1.35331
$893$	−7.70354	−0.257789
$894$	0	0
$895$	−12.4263	−0.415365
$896$	9.88097	0.330100
$897$	0	0
$898$	10.4849	0.349884
$899$	14.2614	0.475645
$900$	0	0
$901$	−2.25814	−0.0752295
$902$	3.84297	0.127957
$903$	0	0
$904$	4.22791	0.140618
$905$	−37.6065	−1.25008
$906$	0	0
$907$	−8.50463	−0.282392	−0.141196	−	0.989982i	$-0.545095\pi$
$907$	−8.50463	−0.282392	−0.141196	+	0.989982i	$0.545095\pi$
$908$	−6.22841	−0.206697
$909$	0	0
$910$	−0.442698	−0.0146753
$911$	2.62136	0.0868497	0.0434248	−	0.999057i	$-0.486173\pi$
$911$	2.62136	0.0868497	0.0434248	+	0.999057i	$0.486173\pi$
$912$	0	0
$913$	36.5867	1.21084
$914$	−6.44153	−0.213067
$915$	0	0
$916$	−27.7344	−0.916371
$917$	11.3901	0.376134
$918$	0	0
$919$	22.4398	0.740222	0.370111	−	0.928988i	$-0.379320\pi$
$919$	22.4398	0.740222	0.370111	+	0.928988i	$0.379320\pi$
$920$	24.0209	0.791946
$921$	0	0
$922$	4.82052	0.158755
$923$	0.146091	0.00480866
$924$	0	0
$925$	44.5888	1.46607
$926$	−12.9662	−0.426097
$927$	0	0
$928$	14.7722	0.484923
$929$	−17.4614	−0.572890	−0.286445	−	0.958097i	$-0.592474\pi$
$929$	−17.4614	−0.572890	−0.286445	+	0.958097i	$0.592474\pi$
$930$	0	0
$931$	1.06563	0.0349248
$932$	−39.7500	−1.30206
$933$	0	0
$934$	−7.61693	−0.249234
$935$	−93.2707	−3.05028
$936$	0	0
$937$	5.65932	0.184882	0.0924409	−	0.995718i	$-0.470533\pi$
$937$	5.65932	0.184882	0.0924409	+	0.995718i	$0.470533\pi$
$938$	0.897675	0.0293101
$939$	0	0
$940$	47.9473	1.56387
$941$	49.6588	1.61883	0.809415	−	0.587237i	$-0.199784\pi$
$941$	49.6588	1.61883	0.809415	+	0.587237i	$0.199784\pi$
$942$	0	0
$943$	−8.74237	−0.284691
$944$	37.7900	1.22996
$945$	0	0
$946$	0.320483	0.0104198
$947$	−1.52880	−0.0496794	−0.0248397	−	0.999691i	$-0.507908\pi$
$947$	−1.52880	−0.0496794	−0.0248397	+	0.999691i	$0.507908\pi$
$948$	0	0
$949$	3.20233	0.103952
$950$	3.01448	0.0978026
$951$	0	0
$952$	−6.63048	−0.214895
$953$	−8.35049	−0.270499	−0.135249	−	0.990812i	$-0.543184\pi$
$953$	−8.35049	−0.270499	−0.135249	+	0.990812i	$0.543184\pi$
$954$	0	0
$955$	35.3029	1.14238
$956$	−22.6281	−0.731844
$957$	0	0
$958$	10.0041	0.323218
$959$	−19.4593	−0.628375
$960$	0	0
$961$	−15.8350	−0.510807
$962$	0.723727	0.0233339
$963$	0	0
$964$	22.7607	0.733073
$965$	−44.2225	−1.42357
$966$	0	0
$967$	8.83218	0.284024	0.142012	−	0.989865i	$-0.454643\pi$
$967$	8.83218	0.284024	0.142012	+	0.989865i	$0.454643\pi$
$968$	−29.6299	−0.952340
$969$	0	0
$970$	4.76511	0.152999
$971$	17.9338	0.575522	0.287761	−	0.957702i	$-0.407089\pi$
$971$	17.9338	0.575522	0.287761	+	0.957702i	$0.407089\pi$
$972$	0	0
$973$	−8.52109	−0.273174
$974$	−8.58332	−0.275027
$975$	0	0
$976$	−35.3609	−1.13187
$977$	−27.7655	−0.888296	−0.444148	−	0.895953i	$-0.646494\pi$
$977$	−27.7655	−0.888296	−0.444148	+	0.895953i	$0.646494\pi$
$978$	0	0
$979$	−1.76774	−0.0564973
$980$	−6.63257	−0.211870
$981$	0	0
$982$	−4.22252	−0.134746
$983$	45.3976	1.44796	0.723979	−	0.689822i	$-0.242311\pi$
$983$	45.3976	1.44796	0.723979	+	0.689822i	$0.242311\pi$
$984$	0	0
$985$	−41.9137	−1.33548
$986$	−6.28457	−0.200142
$987$	0	0
$988$	−0.669398	−0.0212964
$989$	−0.729066	−0.0231829
$990$	0	0
$991$	−40.4448	−1.28477	−0.642386	−	0.766381i	$-0.722055\pi$
$991$	−40.4448	−1.28477	−0.642386	+	0.766381i	$0.722055\pi$
$992$	15.7082	0.498735
$993$	0	0
$994$	−0.159984	−0.00507438
$995$	50.9601	1.61555
$996$	0	0
$997$	13.2931	0.420996	0.210498	−	0.977594i	$-0.432491\pi$
$997$	13.2931	0.420996	0.210498	+	0.977594i	$0.432491\pi$
$998$	−4.60374	−0.145729
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.p.1.7		14
3.2	odd	2		2667.2.a.m.1.8	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.m.1.8	✓	14	3.2	odd	2
8001.2.a.p.1.7		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)

\(f(q)\)	\(=\)	\(q+0.369092 q^{2} -1.86377 q^{4} +3.55869 q^{5} -1.00000 q^{7} -1.42609 q^{8} +O(q^{10})\)	\(q+0.369092 q^{2} -1.86377 q^{4} +3.55869 q^{5} -1.00000 q^{7} -1.42609 q^{8} +1.31348 q^{10} +5.63711 q^{11} +0.337042 q^{13} -0.369092 q^{14} +3.20118 q^{16} -4.64942 q^{17} +1.06563 q^{19} -6.63257 q^{20} +2.08061 q^{22} -4.73319 q^{23} +7.66424 q^{25} +0.124399 q^{26} +1.86377 q^{28} +3.66220 q^{29} +3.89423 q^{31} +4.03371 q^{32} -1.71607 q^{34} -3.55869 q^{35} +5.81777 q^{37} +0.393317 q^{38} -5.07500 q^{40} +1.84704 q^{41} +0.154033 q^{43} -10.5063 q^{44} -1.74698 q^{46} -7.22906 q^{47} +1.00000 q^{49} +2.82881 q^{50} -0.628168 q^{52} +0.485681 q^{53} +20.0607 q^{55} +1.42609 q^{56} +1.35169 q^{58} +11.8050 q^{59} -11.0462 q^{61} +1.43733 q^{62} -4.91356 q^{64} +1.19942 q^{65} -2.43212 q^{67} +8.66546 q^{68} -1.31348 q^{70} +0.433452 q^{71} +9.50128 q^{73} +2.14729 q^{74} -1.98610 q^{76} -5.63711 q^{77} +10.9124 q^{79} +11.3920 q^{80} +0.681727 q^{82} +6.49033 q^{83} -16.5458 q^{85} +0.0568523 q^{86} -8.03902 q^{88} -0.313590 q^{89} -0.337042 q^{91} +8.82157 q^{92} -2.66819 q^{94} +3.79226 q^{95} +3.62785 q^{97} +0.369092 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 5 q^{2} + 15 q^{4} + 4 q^{5} - 14 q^{7} + 12 q^{8}+O(q^{10}) \) 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

Fields

Representations

Groups

Database

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