LMFDB - Embedded newform 8001.2.a.q.1.11

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:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 22 x^{13} + 186 x^{11} - 763 x^{9} - 7 x^{8} + 1588 x^{7} + 64 x^{6} - 1625 x^{5} - 185 x^{4} + \cdots - 13 $ x^15 - 22x^13 + 186x^11 - 763x^9 - 7x^8 + 1588x^7 + 64x^6 - 1625x^5 - 185x^4 + 726x^3 + 145x^2 - 83*x - 13
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 889)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$1.22206$ 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)

$f(q)$	$=$	$q+1.22206 q^{2} -0.506578 q^{4} -1.90746 q^{5} -1.00000 q^{7} -3.06318 q^{8} +O(q^{10})$	\(q+1.22206 q^{2} -0.506578 q^{4} -1.90746 q^{5} -1.00000 q^{7} -3.06318 q^{8} -2.33103 q^{10} +3.63447 q^{11} +3.18497 q^{13} -1.22206 q^{14} -2.73022 q^{16} -1.16987 q^{17} +4.72497 q^{19} +0.966279 q^{20} +4.44153 q^{22} -8.54271 q^{23} -1.36158 q^{25} +3.89221 q^{26} +0.506578 q^{28} -3.31603 q^{29} -3.21334 q^{31} +2.78987 q^{32} -1.42965 q^{34} +1.90746 q^{35} +5.38236 q^{37} +5.77418 q^{38} +5.84290 q^{40} +10.2439 q^{41} +7.65103 q^{43} -1.84114 q^{44} -10.4397 q^{46} -3.08075 q^{47} +1.00000 q^{49} -1.66393 q^{50} -1.61343 q^{52} -1.48934 q^{53} -6.93262 q^{55} +3.06318 q^{56} -4.05238 q^{58} -9.63337 q^{59} +3.71556 q^{61} -3.92689 q^{62} +8.86983 q^{64} -6.07521 q^{65} +14.4347 q^{67} +0.592632 q^{68} +2.33103 q^{70} -4.69680 q^{71} +8.70585 q^{73} +6.57754 q^{74} -2.39357 q^{76} -3.63447 q^{77} -9.45945 q^{79} +5.20780 q^{80} +12.5186 q^{82} -8.75335 q^{83} +2.23149 q^{85} +9.35000 q^{86} -11.1330 q^{88} -4.33011 q^{89} -3.18497 q^{91} +4.32755 q^{92} -3.76485 q^{94} -9.01271 q^{95} -3.61016 q^{97} +1.22206 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + 14 q^{4} - 7 q^{5} - 15 q^{7}+O(q^{10}) $ 15 * q + 14 * q^4 - 7 * q^5 - 15 * q^7	$ 15 q + 14 q^{4} - 7 q^{5} - 15 q^{7} + 10 q^{10} - 14 q^{11} + 6 q^{13} + 20 q^{16} - 10 q^{17} + 13 q^{19} - 8 q^{20} - 11 q^{22} - 15 q^{23} - 22 q^{26} - 14 q^{28} - 16 q^{29} + 22 q^{31} - 15 q^{34} + 7 q^{35} - 14 q^{37} + 6 q^{38} + 22 q^{40} - 19 q^{41} - q^{43} - 25 q^{44} - 28 q^{46} - 49 q^{47} + 15 q^{49} - 24 q^{50} - 17 q^{52} + 28 q^{53} + 39 q^{55} - 10 q^{58} - 43 q^{59} + 27 q^{61} - 14 q^{62} + 18 q^{64} + 8 q^{65} + 3 q^{67} - 13 q^{68} - 10 q^{70} - 55 q^{71} - 3 q^{73} + 12 q^{74} - 20 q^{76} + 14 q^{77} + 18 q^{79} - 29 q^{80} + 14 q^{82} - 17 q^{83} + 7 q^{85} - 4 q^{86} - 114 q^{88} - 36 q^{89} - 6 q^{91} - 45 q^{92} - 15 q^{94} - 59 q^{95} - 2 q^{97}+O(q^{100}) $ 15 * q + 14 * q^4 - 7 * q^5 - 15 * q^7 + 10 * q^10 - 14 * q^11 + 6 * q^13 + 20 * q^16 - 10 * q^17 + 13 * q^19 - 8 * q^20 - 11 * q^22 - 15 * q^23 - 22 * q^26 - 14 * q^28 - 16 * q^29 + 22 * q^31 - 15 * q^34 + 7 * q^35 - 14 * q^37 + 6 * q^38 + 22 * q^40 - 19 * q^41 - q^43 - 25 * q^44 - 28 * q^46 - 49 * q^47 + 15 * q^49 - 24 * q^50 - 17 * q^52 + 28 * q^53 + 39 * q^55 - 10 * q^58 - 43 * q^59 + 27 * q^61 - 14 * q^62 + 18 * q^64 + 8 * q^65 + 3 * q^67 - 13 * q^68 - 10 * q^70 - 55 * q^71 - 3 * q^73 + 12 * q^74 - 20 * q^76 + 14 * q^77 + 18 * q^79 - 29 * q^80 + 14 * q^82 - 17 * q^83 + 7 * q^85 - 4 * q^86 - 114 * q^88 - 36 * q^89 - 6 * q^91 - 45 * q^92 - 15 * q^94 - 59 * q^95 - 2 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.22206	0.864124	0.432062	−	0.901844i	$-0.357786\pi$
$2$	1.22206	0.864124	0.432062	+	0.901844i	$0.357786\pi$
$3$	0	0
$3$	0	0
$4$	−0.506578	−0.253289
$5$	−1.90746	−0.853043	−0.426522	−	0.904477i	$-0.640261\pi$
$5$	−1.90746	−0.853043	−0.426522	+	0.904477i	$0.640261\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−3.06318	−1.08300
$9$	0	0
$10$	−2.33103	−0.737136
$11$	3.63447	1.09583	0.547917	−	0.836533i	$-0.315421\pi$
$11$	3.63447	1.09583	0.547917	+	0.836533i	$0.315421\pi$
$12$	0	0
$13$	3.18497	0.883351	0.441675	−	0.897175i	$-0.354384\pi$
$13$	3.18497	0.883351	0.441675	+	0.897175i	$0.354384\pi$
$14$	−1.22206	−0.326608
$15$	0	0
$16$	−2.73022	−0.682556
$17$	−1.16987	−0.283736	−0.141868	−	0.989886i	$-0.545311\pi$
$17$	−1.16987	−0.283736	−0.141868	+	0.989886i	$0.545311\pi$
$18$	0	0
$19$	4.72497	1.08398	0.541991	−	0.840384i	$-0.317671\pi$
$19$	4.72497	1.08398	0.541991	+	0.840384i	$0.317671\pi$
$20$	0.966279	0.216067
$21$	0	0
$22$	4.44153	0.946937
$23$	−8.54271	−1.78128	−0.890639	−	0.454711i	$-0.849743\pi$
$23$	−8.54271	−1.78128	−0.890639	+	0.454711i	$0.849743\pi$
$24$	0	0
$25$	−1.36158	−0.272317
$26$	3.89221	0.763325
$27$	0	0
$28$	0.506578	0.0957343
$29$	−3.31603	−0.615772	−0.307886	−	0.951423i	$-0.599622\pi$
$29$	−3.31603	−0.615772	−0.307886	+	0.951423i	$0.599622\pi$
$30$	0	0
$31$	−3.21334	−0.577134	−0.288567	−	0.957460i	$-0.593179\pi$
$31$	−3.21334	−0.577134	−0.288567	+	0.957460i	$0.593179\pi$
$32$	2.78987	0.493185
$33$	0	0
$34$	−1.42965	−0.245183
$35$	1.90746	0.322420
$36$	0	0
$37$	5.38236	0.884854	0.442427	−	0.896804i	$-0.354118\pi$
$37$	5.38236	0.884854	0.442427	+	0.896804i	$0.354118\pi$
$38$	5.77418	0.936696
$39$	0	0
$40$	5.84290	0.923844
$41$	10.2439	1.59983	0.799913	−	0.600116i	$-0.204879\pi$
$41$	10.2439	1.59983	0.799913	+	0.600116i	$0.204879\pi$
$42$	0	0
$43$	7.65103	1.16677	0.583386	−	0.812195i	$-0.301728\pi$
$43$	7.65103	1.16677	0.583386	+	0.812195i	$0.301728\pi$
$44$	−1.84114	−0.277563
$45$	0	0
$46$	−10.4397	−1.53925
$47$	−3.08075	−0.449373	−0.224687	−	0.974431i	$-0.572136\pi$
$47$	−3.08075	−0.449373	−0.224687	+	0.974431i	$0.572136\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−1.66393	−0.235316
$51$	0	0
$52$	−1.61343	−0.223743
$53$	−1.48934	−0.204577	−0.102289	−	0.994755i	$-0.532616\pi$
$53$	−1.48934	−0.204577	−0.102289	+	0.994755i	$0.532616\pi$
$54$	0	0
$55$	−6.93262	−0.934794
$56$	3.06318	0.409335
$57$	0	0
$58$	−4.05238	−0.532104
$59$	−9.63337	−1.25416	−0.627079	−	0.778956i	$-0.715750\pi$
$59$	−9.63337	−1.25416	−0.627079	+	0.778956i	$0.715750\pi$
$60$	0	0
$61$	3.71556	0.475729	0.237864	−	0.971298i	$-0.423553\pi$
$61$	3.71556	0.475729	0.237864	+	0.971298i	$0.423553\pi$
$62$	−3.92689	−0.498715
$63$	0	0
$64$	8.86983	1.10873
$65$	−6.07521	−0.753537
$66$	0	0
$67$	14.4347	1.76348	0.881741	−	0.471734i	$-0.156372\pi$
$67$	14.4347	1.76348	0.881741	+	0.471734i	$0.156372\pi$
$68$	0.592632	0.0718672
$69$	0	0
$70$	2.33103	0.278611
$71$	−4.69680	−0.557408	−0.278704	−	0.960377i	$-0.589905\pi$
$71$	−4.69680	−0.557408	−0.278704	+	0.960377i	$0.589905\pi$
$72$	0	0
$73$	8.70585	1.01894	0.509471	−	0.860488i	$-0.329841\pi$
$73$	8.70585	1.01894	0.509471	+	0.860488i	$0.329841\pi$
$74$	6.57754	0.764624
$75$	0	0
$76$	−2.39357	−0.274561
$77$	−3.63447	−0.414186
$78$	0	0
$79$	−9.45945	−1.06427	−0.532136	−	0.846659i	$-0.678610\pi$
$79$	−9.45945	−1.06427	−0.532136	+	0.846659i	$0.678610\pi$
$80$	5.20780	0.582250
$81$	0	0
$82$	12.5186	1.38245
$83$	−8.75335	−0.960804	−0.480402	−	0.877048i	$-0.659509\pi$
$83$	−8.75335	−0.960804	−0.480402	+	0.877048i	$0.659509\pi$
$84$	0	0
$85$	2.23149	0.242039
$86$	9.35000	1.00824
$87$	0	0
$88$	−11.1330	−1.18679
$89$	−4.33011	−0.458991	−0.229496	−	0.973310i	$-0.573708\pi$
$89$	−4.33011	−0.458991	−0.229496	+	0.973310i	$0.573708\pi$
$90$	0	0
$91$	−3.18497	−0.333875
$92$	4.32755	0.451178
$93$	0	0
$94$	−3.76485	−0.388314
$95$	−9.01271	−0.924684
$96$	0	0
$97$	−3.61016	−0.366556	−0.183278	−	0.983061i	$-0.558671\pi$
$97$	−3.61016	−0.366556	−0.183278	+	0.983061i	$0.558671\pi$
$98$	1.22206	0.123446
$99$	0	0
$100$	0.689749	0.0689749
$101$	8.06646	0.802642	0.401321	−	0.915937i	$-0.368551\pi$
$101$	8.06646	0.802642	0.401321	+	0.915937i	$0.368551\pi$
$102$	0	0
$103$	−4.69757	−0.462866	−0.231433	−	0.972851i	$-0.574341\pi$
$103$	−4.69757	−0.462866	−0.231433	+	0.972851i	$0.574341\pi$
$104$	−9.75613	−0.956667
$105$	0	0
$106$	−1.82006	−0.176780
$107$	−20.2663	−1.95922	−0.979609	−	0.200916i	$-0.935608\pi$
$107$	−20.2663	−1.95922	−0.979609	+	0.200916i	$0.935608\pi$
$108$	0	0
$109$	−11.6473	−1.11561	−0.557806	−	0.829971i	$-0.688357\pi$
$109$	−11.6473	−1.11561	−0.557806	+	0.829971i	$0.688357\pi$
$110$	−8.47205	−0.807778
$111$	0	0
$112$	2.73022	0.257982
$113$	8.18123	0.769626	0.384813	−	0.922995i	$-0.374266\pi$
$113$	8.18123	0.769626	0.384813	+	0.922995i	$0.374266\pi$
$114$	0	0
$115$	16.2949	1.51951
$116$	1.67983	0.155968
$117$	0	0
$118$	−11.7725	−1.08375
$119$	1.16987	0.107242
$120$	0	0
$121$	2.20938	0.200852
$122$	4.54062	0.411089
$123$	0	0
$124$	1.62781	0.146182
$125$	12.1345	1.08534
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	5.25968	0.464895
$129$	0	0
$130$	−7.42424	−0.651149
$131$	−18.0189	−1.57432	−0.787161	−	0.616747i	$-0.788450\pi$
$131$	−18.0189	−1.57432	−0.787161	+	0.616747i	$0.788450\pi$
$132$	0	0
$133$	−4.72497	−0.409707
$134$	17.6400	1.52387
$135$	0	0
$136$	3.58353	0.307285
$137$	−12.9819	−1.10912	−0.554559	−	0.832145i	$-0.687113\pi$
$137$	−12.9819	−1.10912	−0.554559	+	0.832145i	$0.687113\pi$
$138$	0	0
$139$	−7.51765	−0.637639	−0.318820	−	0.947815i	$-0.603286\pi$
$139$	−7.51765	−0.637639	−0.318820	+	0.947815i	$0.603286\pi$
$140$	−0.966279	−0.0816655
$141$	0	0
$142$	−5.73976	−0.481670
$143$	11.5757	0.968006
$144$	0	0
$145$	6.32521	0.525280
$146$	10.6390	0.880493
$147$	0	0
$148$	−2.72658	−0.224124
$149$	−4.55948	−0.373527	−0.186763	−	0.982405i	$-0.559800\pi$
$149$	−4.55948	−0.373527	−0.186763	+	0.982405i	$0.559800\pi$
$150$	0	0
$151$	−10.7611	−0.875726	−0.437863	−	0.899042i	$-0.644264\pi$
$151$	−10.7611	−0.875726	−0.437863	+	0.899042i	$0.644264\pi$
$152$	−14.4734	−1.17395
$153$	0	0
$154$	−4.44153	−0.357909
$155$	6.12933	0.492320
$156$	0	0
$157$	−14.3426	−1.14466	−0.572332	−	0.820022i	$-0.693961\pi$
$157$	−14.3426	−1.14466	−0.572332	+	0.820022i	$0.693961\pi$
$158$	−11.5600	−0.919663
$159$	0	0
$160$	−5.32158	−0.420708
$161$	8.54271	0.673260
$162$	0	0
$163$	−7.93033	−0.621151	−0.310576	−	0.950549i	$-0.600522\pi$
$163$	−7.93033	−0.621151	−0.310576	+	0.950549i	$0.600522\pi$
$164$	−5.18933	−0.405218
$165$	0	0
$166$	−10.6971	−0.830255
$167$	−13.2631	−1.02633	−0.513164	−	0.858290i	$-0.671527\pi$
$167$	−13.2631	−1.02633	−0.513164	+	0.858290i	$0.671527\pi$
$168$	0	0
$169$	−2.85599	−0.219691
$170$	2.72701	0.209152
$171$	0	0
$172$	−3.87585	−0.295530
$173$	8.64516	0.657279	0.328640	−	0.944455i	$-0.393410\pi$
$173$	8.64516	0.657279	0.328640	+	0.944455i	$0.393410\pi$
$174$	0	0
$175$	1.36158	0.102926
$176$	−9.92291	−0.747968
$177$	0	0
$178$	−5.29164	−0.396626
$179$	7.52233	0.562245	0.281123	−	0.959672i	$-0.409293\pi$
$179$	7.52233	0.562245	0.281123	+	0.959672i	$0.409293\pi$
$180$	0	0
$181$	7.07959	0.526222	0.263111	−	0.964766i	$-0.415251\pi$
$181$	7.07959	0.526222	0.263111	+	0.964766i	$0.415251\pi$
$182$	−3.89221	−0.288510
$183$	0	0
$184$	26.1679	1.92912
$185$	−10.2666	−0.754819
$186$	0	0
$187$	−4.25187	−0.310928
$188$	1.56064	0.113821
$189$	0	0
$190$	−11.0140	−0.799042
$191$	4.35725	0.315280	0.157640	−	0.987497i	$-0.449611\pi$
$191$	4.35725	0.315280	0.157640	+	0.987497i	$0.449611\pi$
$192$	0	0
$193$	5.45957	0.392989	0.196494	−	0.980505i	$-0.437044\pi$
$193$	5.45957	0.392989	0.196494	+	0.980505i	$0.437044\pi$
$194$	−4.41182	−0.316750
$195$	0	0
$196$	−0.506578	−0.0361841
$197$	−13.3278	−0.949569	−0.474785	−	0.880102i	$-0.657474\pi$
$197$	−13.3278	−0.949569	−0.474785	+	0.880102i	$0.657474\pi$
$198$	0	0
$199$	1.63961	0.116229	0.0581144	−	0.998310i	$-0.481491\pi$
$199$	1.63961	0.116229	0.0581144	+	0.998310i	$0.481491\pi$
$200$	4.17078	0.294919
$201$	0	0
$202$	9.85767	0.693583
$203$	3.31603	0.232740
$204$	0	0
$205$	−19.5398	−1.36472
$206$	−5.74070	−0.399974
$207$	0	0
$208$	−8.69567	−0.602936
$209$	17.1728	1.18786
$210$	0	0
$211$	13.9049	0.957250	0.478625	−	0.878019i	$-0.341135\pi$
$211$	13.9049	0.957250	0.478625	+	0.878019i	$0.341135\pi$
$212$	0.754469	0.0518171
$213$	0	0
$214$	−24.7666	−1.69301
$215$	−14.5941	−0.995307
$216$	0	0
$217$	3.21334	0.218136
$218$	−14.2337	−0.964028
$219$	0	0
$220$	3.51191	0.236773
$221$	−3.72601	−0.250638
$222$	0	0
$223$	−18.0316	−1.20749	−0.603744	−	0.797178i	$-0.706325\pi$
$223$	−18.0316	−1.20749	−0.603744	+	0.797178i	$0.706325\pi$
$224$	−2.78987	−0.186406
$225$	0	0
$226$	9.99793	0.665052
$227$	27.1480	1.80187	0.900937	−	0.433950i	$-0.142881\pi$
$227$	27.1480	1.80187	0.900937	+	0.433950i	$0.142881\pi$
$228$	0	0
$229$	−15.7946	−1.04373	−0.521867	−	0.853027i	$-0.674764\pi$
$229$	−15.7946	−1.04373	−0.521867	+	0.853027i	$0.674764\pi$
$230$	19.9133	1.31304
$231$	0	0
$232$	10.1576	0.666880
$233$	−1.45642	−0.0954131	−0.0477065	−	0.998861i	$-0.515191\pi$
$233$	−1.45642	−0.0954131	−0.0477065	+	0.998861i	$0.515191\pi$
$234$	0	0
$235$	5.87641	0.383335
$236$	4.88006	0.317665
$237$	0	0
$238$	1.42965	0.0926705
$239$	−1.95156	−0.126236	−0.0631181	−	0.998006i	$-0.520104\pi$
$239$	−1.95156	−0.126236	−0.0631181	+	0.998006i	$0.520104\pi$
$240$	0	0
$241$	−27.0572	−1.74291	−0.871453	−	0.490478i	$-0.836822\pi$
$241$	−27.0572	−1.74291	−0.871453	+	0.490478i	$0.836822\pi$
$242$	2.69998	0.173561
$243$	0	0
$244$	−1.88222	−0.120497
$245$	−1.90746	−0.121863
$246$	0	0
$247$	15.0489	0.957537
$248$	9.84305	0.625034
$249$	0	0
$250$	14.8290	0.937870
$251$	−19.7774	−1.24834	−0.624168	−	0.781290i	$-0.714562\pi$
$251$	−19.7774	−1.24834	−0.624168	+	0.781290i	$0.714562\pi$
$252$	0	0
$253$	−31.0482	−1.95199
$254$	1.22206	0.0766786
$255$	0	0
$256$	−11.3120	−0.707002
$257$	13.2607	0.827181	0.413590	−	0.910463i	$-0.364275\pi$
$257$	13.2607	0.827181	0.413590	+	0.910463i	$0.364275\pi$
$258$	0	0
$259$	−5.38236	−0.334443
$260$	3.07757	0.190863
$261$	0	0
$262$	−22.0202	−1.36041
$263$	27.8387	1.71661	0.858304	−	0.513142i	$-0.171518\pi$
$263$	27.8387	1.71661	0.858304	+	0.513142i	$0.171518\pi$
$264$	0	0
$265$	2.84087	0.174513
$266$	−5.77418	−0.354038
$267$	0	0
$268$	−7.31231	−0.446671
$269$	−29.1353	−1.77641	−0.888204	−	0.459449i	$-0.848047\pi$
$269$	−29.1353	−1.77641	−0.888204	+	0.459449i	$0.848047\pi$
$270$	0	0
$271$	14.8170	0.900070	0.450035	−	0.893011i	$-0.351412\pi$
$271$	14.8170	0.900070	0.450035	+	0.893011i	$0.351412\pi$
$272$	3.19401	0.193666
$273$	0	0
$274$	−15.8646	−0.958415
$275$	−4.94864	−0.298414
$276$	0	0
$277$	−15.8438	−0.951960	−0.475980	−	0.879456i	$-0.657907\pi$
$277$	−15.8438	−0.951960	−0.475980	+	0.879456i	$0.657907\pi$
$278$	−9.18700	−0.550999
$279$	0	0
$280$	−5.84290	−0.349180
$281$	21.1741	1.26314	0.631571	−	0.775318i	$-0.282411\pi$
$281$	21.1741	1.26314	0.631571	+	0.775318i	$0.282411\pi$
$282$	0	0
$283$	−31.3051	−1.86090	−0.930448	−	0.366423i	$-0.880582\pi$
$283$	−31.3051	−1.86090	−0.930448	+	0.366423i	$0.880582\pi$
$284$	2.37930	0.141185
$285$	0	0
$286$	14.1461	0.836478
$287$	−10.2439	−0.604677
$288$	0	0
$289$	−15.6314	−0.919494
$290$	7.72976	0.453907
$291$	0	0
$292$	−4.41019	−0.258087
$293$	−26.4183	−1.54338	−0.771688	−	0.636002i	$-0.780587\pi$
$293$	−26.4183	−1.54338	−0.771688	+	0.636002i	$0.780587\pi$
$294$	0	0
$295$	18.3753	1.06985
$296$	−16.4871	−0.958295
$297$	0	0
$298$	−5.57194	−0.322774
$299$	−27.2083	−1.57349
$300$	0	0
$301$	−7.65103	−0.440998
$302$	−13.1507	−0.756736
$303$	0	0
$304$	−12.9002	−0.739878
$305$	−7.08729	−0.405817
$306$	0	0
$307$	−16.9234	−0.965871	−0.482935	−	0.875656i	$-0.660429\pi$
$307$	−16.9234	−0.965871	−0.482935	+	0.875656i	$0.660429\pi$
$308$	1.84114	0.104909
$309$	0	0
$310$	7.49039	0.425426
$311$	−11.5785	−0.656559	−0.328279	−	0.944581i	$-0.606469\pi$
$311$	−11.5785	−0.656559	−0.328279	+	0.944581i	$0.606469\pi$
$312$	0	0
$313$	11.1549	0.630514	0.315257	−	0.949006i	$-0.397909\pi$
$313$	11.1549	0.630514	0.315257	+	0.949006i	$0.397909\pi$
$314$	−17.5275	−0.989133
$315$	0	0
$316$	4.79195	0.269568
$317$	−19.1556	−1.07589	−0.537944	−	0.842981i	$-0.680799\pi$
$317$	−19.1556	−1.07589	−0.537944	+	0.842981i	$0.680799\pi$
$318$	0	0
$319$	−12.0520	−0.674784
$320$	−16.9189	−0.945794
$321$	0	0
$322$	10.4397	0.581780
$323$	−5.52762	−0.307565
$324$	0	0
$325$	−4.33660	−0.240551
$326$	−9.69131	−0.536752
$327$	0	0
$328$	−31.3789	−1.73261
$329$	3.08075	0.169847
$330$	0	0
$331$	−0.563109	−0.0309513	−0.0154756	−	0.999880i	$-0.504926\pi$
$331$	−0.563109	−0.0309513	−0.0154756	+	0.999880i	$0.504926\pi$
$332$	4.43425	0.243361
$333$	0	0
$334$	−16.2082	−0.886876
$335$	−27.5337	−1.50433
$336$	0	0
$337$	−19.8824	−1.08306	−0.541532	−	0.840680i	$-0.682156\pi$
$337$	−19.8824	−1.08306	−0.541532	+	0.840680i	$0.682156\pi$
$338$	−3.49018	−0.189841
$339$	0	0
$340$	−1.13042	−0.0613059
$341$	−11.6788	−0.632443
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−23.4365	−1.26361
$345$	0	0
$346$	10.5649	0.567971
$347$	−15.5867	−0.836737	−0.418369	−	0.908277i	$-0.637398\pi$
$347$	−15.5867	−0.836737	−0.418369	+	0.908277i	$0.637398\pi$
$348$	0	0
$349$	15.6976	0.840274	0.420137	−	0.907461i	$-0.361982\pi$
$349$	15.6976	0.840274	0.420137	+	0.907461i	$0.361982\pi$
$350$	1.66393	0.0889410
$351$	0	0
$352$	10.1397	0.540449
$353$	4.76893	0.253824	0.126912	−	0.991914i	$-0.459493\pi$
$353$	4.76893	0.253824	0.126912	+	0.991914i	$0.459493\pi$
$354$	0	0
$355$	8.95898	0.475493
$356$	2.19354	0.116257
$357$	0	0
$358$	9.19271	0.485850
$359$	35.6678	1.88247	0.941237	−	0.337746i	$-0.109664\pi$
$359$	35.6678	1.88247	0.941237	+	0.337746i	$0.109664\pi$
$360$	0	0
$361$	3.32534	0.175018
$362$	8.65166	0.454721
$363$	0	0
$364$	1.61343	0.0845669
$365$	−16.6061	−0.869202
$366$	0	0
$367$	10.9438	0.571264	0.285632	−	0.958339i	$-0.407797\pi$
$367$	10.9438	0.571264	0.285632	+	0.958339i	$0.407797\pi$
$368$	23.3235	1.21582
$369$	0	0
$370$	−12.5464	−0.652257
$371$	1.48934	0.0773229
$372$	0	0
$373$	9.45090	0.489349	0.244674	−	0.969605i	$-0.421319\pi$
$373$	9.45090	0.489349	0.244674	+	0.969605i	$0.421319\pi$
$374$	−5.19603	−0.268680
$375$	0	0
$376$	9.43689	0.486670
$377$	−10.5615	−0.543943
$378$	0	0
$379$	4.41548	0.226808	0.113404	−	0.993549i	$-0.463825\pi$
$379$	4.41548	0.226808	0.113404	+	0.993549i	$0.463825\pi$
$380$	4.56564	0.234212
$381$	0	0
$382$	5.32481	0.272441
$383$	−21.9441	−1.12129	−0.560646	−	0.828056i	$-0.689447\pi$
$383$	−21.9441	−1.12129	−0.560646	+	0.828056i	$0.689447\pi$
$384$	0	0
$385$	6.93262	0.353319
$386$	6.67190	0.339591
$387$	0	0
$388$	1.82883	0.0928447
$389$	−15.0418	−0.762651	−0.381326	−	0.924441i	$-0.624532\pi$
$389$	−15.0418	−0.762651	−0.381326	+	0.924441i	$0.624532\pi$
$390$	0	0
$391$	9.99389	0.505413
$392$	−3.06318	−0.154714
$393$	0	0
$394$	−16.2874	−0.820546
$395$	18.0435	0.907869
$396$	0	0
$397$	−14.2802	−0.716702	−0.358351	−	0.933587i	$-0.616661\pi$
$397$	−14.2802	−0.716702	−0.358351	+	0.933587i	$0.616661\pi$
$398$	2.00370	0.100436
$399$	0	0
$400$	3.71743	0.185871
$401$	19.2682	0.962208	0.481104	−	0.876664i	$-0.340236\pi$
$401$	19.2682	0.962208	0.481104	+	0.876664i	$0.340236\pi$
$402$	0	0
$403$	−10.2344	−0.509811
$404$	−4.08629	−0.203301
$405$	0	0
$406$	4.05238	0.201116
$407$	19.5620	0.969653
$408$	0	0
$409$	20.6183	1.01951	0.509754	−	0.860320i	$-0.329736\pi$
$409$	20.6183	1.01951	0.509754	+	0.860320i	$0.329736\pi$
$410$	−23.8788	−1.17929
$411$	0	0
$412$	2.37969	0.117239
$413$	9.63337	0.474027
$414$	0	0
$415$	16.6967	0.819608
$416$	8.88566	0.435655
$417$	0	0
$418$	20.9861	1.02646
$419$	−16.7248	−0.817061	−0.408531	−	0.912745i	$-0.633959\pi$
$419$	−16.7248	−0.817061	−0.408531	+	0.912745i	$0.633959\pi$
$420$	0	0
$421$	10.8848	0.530491	0.265246	−	0.964181i	$-0.414547\pi$
$421$	10.8848	0.530491	0.265246	+	0.964181i	$0.414547\pi$
$422$	16.9925	0.827183
$423$	0	0
$424$	4.56213	0.221556
$425$	1.59288	0.0772661
$426$	0	0
$427$	−3.71556	−0.179808
$428$	10.2665	0.496248
$429$	0	0
$430$	−17.8348	−0.860069
$431$	14.0425	0.676402	0.338201	−	0.941074i	$-0.390182\pi$
$431$	14.0425	0.676402	0.338201	+	0.941074i	$0.390182\pi$
$432$	0	0
$433$	−6.47379	−0.311110	−0.155555	−	0.987827i	$-0.549717\pi$
$433$	−6.47379	−0.311110	−0.155555	+	0.987827i	$0.549717\pi$
$434$	3.92689	0.188497
$435$	0	0
$436$	5.90028	0.282572
$437$	−40.3641	−1.93087
$438$	0	0
$439$	−20.5353	−0.980094	−0.490047	−	0.871696i	$-0.663020\pi$
$439$	−20.5353	−0.980094	−0.490047	+	0.871696i	$0.663020\pi$
$440$	21.2359	1.01238
$441$	0	0
$442$	−4.55339	−0.216583
$443$	33.1338	1.57423	0.787117	−	0.616804i	$-0.211573\pi$
$443$	33.1338	1.57423	0.787117	+	0.616804i	$0.211573\pi$
$444$	0	0
$445$	8.25953	0.391539
$446$	−22.0357	−1.04342
$447$	0	0
$448$	−8.86983	−0.419060
$449$	10.6393	0.502100	0.251050	−	0.967974i	$-0.419224\pi$
$449$	10.6393	0.502100	0.251050	+	0.967974i	$0.419224\pi$
$450$	0	0
$451$	37.2311	1.75314
$452$	−4.14443	−0.194938
$453$	0	0
$454$	33.1763	1.55704
$455$	6.07521	0.284810
$456$	0	0
$457$	1.88587	0.0882171	0.0441085	−	0.999027i	$-0.485955\pi$
$457$	1.88587	0.0882171	0.0441085	+	0.999027i	$0.485955\pi$
$458$	−19.3018	−0.901915
$459$	0	0
$460$	−8.25464	−0.384875
$461$	7.86544	0.366330	0.183165	−	0.983082i	$-0.441366\pi$
$461$	7.86544	0.366330	0.183165	+	0.983082i	$0.441366\pi$
$462$	0	0
$463$	−7.85227	−0.364926	−0.182463	−	0.983213i	$-0.558407\pi$
$463$	−7.85227	−0.364926	−0.182463	+	0.983213i	$0.558407\pi$
$464$	9.05351	0.420299
$465$	0	0
$466$	−1.77982	−0.0824487
$467$	−18.0830	−0.836781	−0.418390	−	0.908267i	$-0.637406\pi$
$467$	−18.0830	−0.836781	−0.418390	+	0.908267i	$0.637406\pi$
$468$	0	0
$469$	−14.4347	−0.666533
$470$	7.18131	0.331249
$471$	0	0
$472$	29.5088	1.35825
$473$	27.8075	1.27859
$474$	0	0
$475$	−6.43345	−0.295187
$476$	−0.592632	−0.0271633
$477$	0	0
$478$	−2.38492	−0.109084
$479$	−31.8492	−1.45523	−0.727614	−	0.685986i	$-0.759371\pi$
$479$	−31.8492	−1.45523	−0.727614	+	0.685986i	$0.759371\pi$
$480$	0	0
$481$	17.1426	0.781637
$482$	−33.0654	−1.50609
$483$	0	0
$484$	−1.11922	−0.0508737
$485$	6.88625	0.312688
$486$	0	0
$487$	28.9872	1.31353	0.656767	−	0.754093i	$-0.271923\pi$
$487$	28.9872	1.31353	0.656767	+	0.754093i	$0.271923\pi$
$488$	−11.3814	−0.515213
$489$	0	0
$490$	−2.33103	−0.105305
$491$	−38.1874	−1.72337	−0.861687	−	0.507440i	$-0.830592\pi$
$491$	−38.1874	−1.72337	−0.861687	+	0.507440i	$0.830592\pi$
$492$	0	0
$493$	3.87934	0.174717
$494$	18.3906	0.827431
$495$	0	0
$496$	8.77314	0.393926
$497$	4.69680	0.210680
$498$	0	0
$499$	9.78636	0.438098	0.219049	−	0.975714i	$-0.429705\pi$
$499$	9.78636	0.438098	0.219049	+	0.975714i	$0.429705\pi$
$500$	−6.14707	−0.274905
$501$	0	0
$502$	−24.1691	−1.07872
$503$	−7.12871	−0.317853	−0.158927	−	0.987290i	$-0.550803\pi$
$503$	−7.12871	−0.317853	−0.158927	+	0.987290i	$0.550803\pi$
$504$	0	0
$505$	−15.3865	−0.684689
$506$	−37.9427	−1.68676
$507$	0	0
$508$	−0.506578	−0.0224758
$509$	−0.732586	−0.0324713	−0.0162357	−	0.999868i	$-0.505168\pi$
$509$	−0.732586	−0.0324713	−0.0162357	+	0.999868i	$0.505168\pi$
$510$	0	0
$511$	−8.70585	−0.385124
$512$	−24.3433	−1.07583
$513$	0	0
$514$	16.2053	0.714787
$515$	8.96045	0.394845
$516$	0	0
$517$	−11.1969	−0.492439
$518$	−6.57754	−0.289001
$519$	0	0
$520$	18.6094	0.816078
$521$	−1.89073	−0.0828344	−0.0414172	−	0.999142i	$-0.513187\pi$
$521$	−1.89073	−0.0828344	−0.0414172	+	0.999142i	$0.513187\pi$
$522$	0	0
$523$	20.5748	0.899673	0.449837	−	0.893111i	$-0.351482\pi$
$523$	20.5748	0.899673	0.449837	+	0.893111i	$0.351482\pi$
$524$	9.12800	0.398759
$525$	0	0
$526$	34.0205	1.48336
$527$	3.75921	0.163754
$528$	0	0
$529$	49.9779	2.17295
$530$	3.47170	0.150801
$531$	0	0
$532$	2.39357	0.103774
$533$	32.6264	1.41321
$534$	0	0
$535$	38.6572	1.67130
$536$	−44.2161	−1.90985
$537$	0	0
$538$	−35.6049	−1.53504
$539$	3.63447	0.156548
$540$	0	0
$541$	3.70516	0.159297	0.0796487	−	0.996823i	$-0.474620\pi$
$541$	3.70516	0.159297	0.0796487	+	0.996823i	$0.474620\pi$
$542$	18.1072	0.777772
$543$	0	0
$544$	−3.26380	−0.139934
$545$	22.2169	0.951666
$546$	0	0
$547$	27.1022	1.15881	0.579403	−	0.815041i	$-0.303286\pi$
$547$	27.1022	1.15881	0.579403	+	0.815041i	$0.303286\pi$
$548$	6.57634	0.280927
$549$	0	0
$550$	−6.04752	−0.257867
$551$	−15.6682	−0.667486
$552$	0	0
$553$	9.45945	0.402257
$554$	−19.3620	−0.822612
$555$	0	0
$556$	3.80828	0.161507
$557$	5.75372	0.243793	0.121897	−	0.992543i	$-0.461102\pi$
$557$	5.75372	0.243793	0.121897	+	0.992543i	$0.461102\pi$
$558$	0	0
$559$	24.3683	1.03067
$560$	−5.20780	−0.220070
$561$	0	0
$562$	25.8759	1.09151
$563$	−32.0015	−1.34870	−0.674350	−	0.738411i	$-0.735576\pi$
$563$	−32.0015	−1.34870	−0.674350	+	0.738411i	$0.735576\pi$
$564$	0	0
$565$	−15.6054	−0.656524
$566$	−38.2566	−1.60805
$567$	0	0
$568$	14.3872	0.603672
$569$	−39.6835	−1.66362	−0.831810	−	0.555060i	$-0.812695\pi$
$569$	−39.6835	−1.66362	−0.831810	+	0.555060i	$0.812695\pi$
$570$	0	0
$571$	−17.3610	−0.726537	−0.363269	−	0.931684i	$-0.618339\pi$
$571$	−17.3610	−0.726537	−0.363269	+	0.931684i	$0.618339\pi$
$572$	−5.86398	−0.245185
$573$	0	0
$574$	−12.5186	−0.522517
$575$	11.6316	0.485072
$576$	0	0
$577$	−35.6745	−1.48515	−0.742575	−	0.669763i	$-0.766396\pi$
$577$	−35.6745	−1.48515	−0.742575	+	0.669763i	$0.766396\pi$
$578$	−19.1024	−0.794557
$579$	0	0
$580$	−3.20421	−0.133048
$581$	8.75335	0.363150
$582$	0	0
$583$	−5.41297	−0.224183
$584$	−26.6676	−1.10351
$585$	0	0
$586$	−32.2847	−1.33367
$587$	37.5366	1.54930	0.774650	−	0.632390i	$-0.217926\pi$
$587$	37.5366	1.54930	0.774650	+	0.632390i	$0.217926\pi$
$588$	0	0
$589$	−15.1830	−0.625603
$590$	22.4557	0.924485
$591$	0	0
$592$	−14.6950	−0.603962
$593$	34.0383	1.39779	0.698893	−	0.715226i	$-0.253676\pi$
$593$	34.0383	1.39779	0.698893	+	0.715226i	$0.253676\pi$
$594$	0	0
$595$	−2.23149	−0.0914822
$596$	2.30973	0.0946102
$597$	0	0
$598$	−33.2500	−1.35969
$599$	8.50168	0.347369	0.173685	−	0.984801i	$-0.444433\pi$
$599$	8.50168	0.347369	0.173685	+	0.984801i	$0.444433\pi$
$600$	0	0
$601$	9.66912	0.394412	0.197206	−	0.980362i	$-0.436813\pi$
$601$	9.66912	0.394412	0.197206	+	0.980362i	$0.436813\pi$
$602$	−9.35000	−0.381077
$603$	0	0
$604$	5.45134	0.221812
$605$	−4.21430	−0.171336
$606$	0	0
$607$	−33.6682	−1.36655	−0.683275	−	0.730161i	$-0.739445\pi$
$607$	−33.6682	−1.36655	−0.683275	+	0.730161i	$0.739445\pi$
$608$	13.1821	0.534604
$609$	0	0
$610$	−8.66107	−0.350676
$611$	−9.81208	−0.396954
$612$	0	0
$613$	23.9693	0.968109	0.484055	−	0.875038i	$-0.339164\pi$
$613$	23.9693	0.968109	0.484055	+	0.875038i	$0.339164\pi$
$614$	−20.6814	−0.834632
$615$	0	0
$616$	11.1330	0.448563
$617$	28.9856	1.16692	0.583458	−	0.812143i	$-0.301699\pi$
$617$	28.9856	1.16692	0.583458	+	0.812143i	$0.301699\pi$
$618$	0	0
$619$	−12.7777	−0.513579	−0.256789	−	0.966467i	$-0.582665\pi$
$619$	−12.7777	−0.513579	−0.256789	+	0.966467i	$0.582665\pi$
$620$	−3.10499	−0.124699
$621$	0	0
$622$	−14.1496	−0.567349
$623$	4.33011	0.173482
$624$	0	0
$625$	−16.3382	−0.653526
$626$	13.6320	0.544843
$627$	0	0
$628$	7.26565	0.289931
$629$	−6.29668	−0.251065
$630$	0	0
$631$	−32.8309	−1.30698	−0.653489	−	0.756936i	$-0.726695\pi$
$631$	−32.8309	−1.30698	−0.653489	+	0.756936i	$0.726695\pi$
$632$	28.9760	1.15260
$633$	0	0
$634$	−23.4093	−0.929700
$635$	−1.90746	−0.0756954
$636$	0	0
$637$	3.18497	0.126193
$638$	−14.7283	−0.583097
$639$	0	0
$640$	−10.0327	−0.396575
$641$	37.2005	1.46933	0.734667	−	0.678428i	$-0.237339\pi$
$641$	37.2005	1.46933	0.734667	+	0.678428i	$0.237339\pi$
$642$	0	0
$643$	46.7929	1.84533	0.922666	−	0.385600i	$-0.126006\pi$
$643$	46.7929	1.84533	0.922666	+	0.385600i	$0.126006\pi$
$644$	−4.32755	−0.170529
$645$	0	0
$646$	−6.75506	−0.265774
$647$	41.8914	1.64692	0.823461	−	0.567373i	$-0.192040\pi$
$647$	41.8914	1.64692	0.823461	+	0.567373i	$0.192040\pi$
$648$	0	0
$649$	−35.0122	−1.37435
$650$	−5.29957	−0.207866
$651$	0	0
$652$	4.01733	0.157331
$653$	5.43552	0.212708	0.106354	−	0.994328i	$-0.466082\pi$
$653$	5.43552	0.212708	0.106354	+	0.994328i	$0.466082\pi$
$654$	0	0
$655$	34.3705	1.34297
$656$	−27.9681	−1.09197
$657$	0	0
$658$	3.76485	0.146769
$659$	2.51243	0.0978704	0.0489352	−	0.998802i	$-0.484417\pi$
$659$	2.51243	0.0978704	0.0489352	+	0.998802i	$0.484417\pi$
$660$	0	0
$661$	−9.20793	−0.358147	−0.179073	−	0.983836i	$-0.557310\pi$
$661$	−9.20793	−0.358147	−0.179073	+	0.983836i	$0.557310\pi$
$662$	−0.688151	−0.0267457
$663$	0	0
$664$	26.8131	1.04055
$665$	9.01271	0.349498
$666$	0	0
$667$	28.3279	1.09686
$668$	6.71879	0.259958
$669$	0	0
$670$	−33.6477	−1.29993
$671$	13.5041	0.521320
$672$	0	0
$673$	22.6226	0.872037	0.436019	−	0.899938i	$-0.356388\pi$
$673$	22.6226	0.872037	0.436019	+	0.899938i	$0.356388\pi$
$674$	−24.2974	−0.935902
$675$	0	0
$676$	1.44678	0.0556454
$677$	−11.9452	−0.459090	−0.229545	−	0.973298i	$-0.573724\pi$
$677$	−11.9452	−0.459090	−0.229545	+	0.973298i	$0.573724\pi$
$678$	0	0
$679$	3.61016	0.138545
$680$	−6.83546	−0.262128
$681$	0	0
$682$	−14.2722	−0.546509
$683$	3.57767	0.136896	0.0684478	−	0.997655i	$-0.478195\pi$
$683$	3.57767	0.136896	0.0684478	+	0.997655i	$0.478195\pi$
$684$	0	0
$685$	24.7625	0.946125
$686$	−1.22206	−0.0466583
$687$	0	0
$688$	−20.8890	−0.796387
$689$	−4.74351	−0.180713
$690$	0	0
$691$	−8.23171	−0.313149	−0.156574	−	0.987666i	$-0.550045\pi$
$691$	−8.23171	−0.313149	−0.156574	+	0.987666i	$0.550045\pi$
$692$	−4.37945	−0.166482
$693$	0	0
$694$	−19.0478	−0.723045
$695$	14.3396	0.543934
$696$	0	0
$697$	−11.9840	−0.453928
$698$	19.1834	0.726101
$699$	0	0
$700$	−0.689749	−0.0260701
$701$	21.0613	0.795475	0.397738	−	0.917499i	$-0.369795\pi$
$701$	21.0613	0.795475	0.397738	+	0.917499i	$0.369795\pi$
$702$	0	0
$703$	25.4315	0.959166
$704$	32.2371	1.21498
$705$	0	0
$706$	5.82790	0.219336
$707$	−8.06646	−0.303370
$708$	0	0
$709$	39.7533	1.49296	0.746482	−	0.665405i	$-0.231741\pi$
$709$	39.7533	1.49296	0.746482	+	0.665405i	$0.231741\pi$
$710$	10.9484	0.410885
$711$	0	0
$712$	13.2639	0.497086
$713$	27.4507	1.02804
$714$	0	0
$715$	−22.0802	−0.825751
$716$	−3.81065	−0.142411
$717$	0	0
$718$	43.5881	1.62669
$719$	3.44763	0.128575	0.0642874	−	0.997931i	$-0.479523\pi$
$719$	3.44763	0.128575	0.0642874	+	0.997931i	$0.479523\pi$
$720$	0	0
$721$	4.69757	0.174947
$722$	4.06376	0.151237
$723$	0	0
$724$	−3.58637	−0.133286
$725$	4.51506	0.167685
$726$	0	0
$727$	44.7014	1.65788	0.828941	−	0.559336i	$-0.188944\pi$
$727$	44.7014	1.65788	0.828941	+	0.559336i	$0.188944\pi$
$728$	9.75613	0.361586
$729$	0	0
$730$	−20.2936	−0.751099
$731$	−8.95074	−0.331055
$732$	0	0
$733$	−51.8174	−1.91392	−0.956960	−	0.290220i	$-0.906271\pi$
$733$	−51.8174	−1.91392	−0.956960	+	0.290220i	$0.906271\pi$
$734$	13.3740	0.493643
$735$	0	0
$736$	−23.8331	−0.878499
$737$	52.4626	1.93248
$738$	0	0
$739$	4.52813	0.166570	0.0832849	−	0.996526i	$-0.473459\pi$
$739$	4.52813	0.166570	0.0832849	+	0.996526i	$0.473459\pi$
$740$	5.20086	0.191187
$741$	0	0
$742$	1.82006	0.0668166
$743$	−38.6939	−1.41954	−0.709771	−	0.704432i	$-0.751202\pi$
$743$	−38.6939	−1.41954	−0.709771	+	0.704432i	$0.751202\pi$
$744$	0	0
$745$	8.69703	0.318635
$746$	11.5495	0.422858
$747$	0	0
$748$	2.15390	0.0787545
$749$	20.2663	0.740514
$750$	0	0
$751$	−40.5536	−1.47982	−0.739911	−	0.672705i	$-0.765132\pi$
$751$	−40.5536	−1.47982	−0.739911	+	0.672705i	$0.765132\pi$
$752$	8.41113	0.306722
$753$	0	0
$754$	−12.9067	−0.470034
$755$	20.5264	0.747032
$756$	0	0
$757$	−38.5957	−1.40278	−0.701392	−	0.712776i	$-0.747438\pi$
$757$	−38.5957	−1.40278	−0.701392	+	0.712776i	$0.747438\pi$
$758$	5.39597	0.195990
$759$	0	0
$760$	27.6075	1.00143
$761$	16.9389	0.614034	0.307017	−	0.951704i	$-0.400669\pi$
$761$	16.9389	0.614034	0.307017	+	0.951704i	$0.400669\pi$
$762$	0	0
$763$	11.6473	0.421662
$764$	−2.20729	−0.0798569
$765$	0	0
$766$	−26.8170	−0.968936
$767$	−30.6820	−1.10786
$768$	0	0
$769$	25.4302	0.917038	0.458519	−	0.888685i	$-0.348380\pi$
$769$	25.4302	0.917038	0.458519	+	0.888685i	$0.348380\pi$
$770$	8.47205	0.305312
$771$	0	0
$772$	−2.76570	−0.0995397
$773$	−8.65960	−0.311464	−0.155732	−	0.987799i	$-0.549774\pi$
$773$	−8.65960	−0.311464	−0.155732	+	0.987799i	$0.549774\pi$
$774$	0	0
$775$	4.37524	0.157163
$776$	11.0586	0.396980
$777$	0	0
$778$	−18.3820	−0.659025
$779$	48.4021	1.73418
$780$	0	0
$781$	−17.0704	−0.610827
$782$	12.2131	0.436740
$783$	0	0
$784$	−2.73022	−0.0975079
$785$	27.3580	0.976449
$786$	0	0
$787$	32.4090	1.15526	0.577629	−	0.816300i	$-0.303978\pi$
$787$	32.4090	1.15526	0.577629	+	0.816300i	$0.303978\pi$
$788$	6.75159	0.240516
$789$	0	0
$790$	22.0502	0.784512
$791$	−8.18123	−0.290891
$792$	0	0
$793$	11.8339	0.420235
$794$	−17.4512	−0.619320
$795$	0	0
$796$	−0.830590	−0.0294395
$797$	−29.3410	−1.03931	−0.519655	−	0.854376i	$-0.673940\pi$
$797$	−29.3410	−1.03931	−0.519655	+	0.854376i	$0.673940\pi$
$798$	0	0
$799$	3.60409	0.127503
$800$	−3.79865	−0.134303
$801$	0	0
$802$	23.5468	0.831467
$803$	31.6411	1.11659
$804$	0	0
$805$	−16.2949	−0.574320
$806$	−12.5070	−0.440541
$807$	0	0
$808$	−24.7090	−0.869260
$809$	5.85818	0.205963	0.102981	−	0.994683i	$-0.467162\pi$
$809$	5.85818	0.205963	0.102981	+	0.994683i	$0.467162\pi$
$810$	0	0
$811$	20.4649	0.718621	0.359310	−	0.933218i	$-0.383012\pi$
$811$	20.4649	0.718621	0.359310	+	0.933218i	$0.383012\pi$
$812$	−1.67983	−0.0589505
$813$	0	0
$814$	23.9059	0.837901
$815$	15.1268	0.529869
$816$	0	0
$817$	36.1509	1.26476
$818$	25.1967	0.880982
$819$	0	0
$820$	9.89845	0.345669
$821$	−26.8212	−0.936065	−0.468033	−	0.883711i	$-0.655037\pi$
$821$	−26.8212	−0.936065	−0.468033	+	0.883711i	$0.655037\pi$
$822$	0	0
$823$	19.3603	0.674858	0.337429	−	0.941351i	$-0.390443\pi$
$823$	19.3603	0.674858	0.337429	+	0.941351i	$0.390443\pi$
$824$	14.3895	0.501283
$825$	0	0
$826$	11.7725	0.409619
$827$	−14.3641	−0.499488	−0.249744	−	0.968312i	$-0.580346\pi$
$827$	−14.3641	−0.499488	−0.249744	+	0.968312i	$0.580346\pi$
$828$	0	0
$829$	18.5710	0.644999	0.322500	−	0.946570i	$-0.395477\pi$
$829$	18.5710	0.644999	0.322500	+	0.946570i	$0.395477\pi$
$830$	20.4043	0.708243
$831$	0	0
$832$	28.2501	0.979396
$833$	−1.16987	−0.0405337
$834$	0	0
$835$	25.2989	0.875503
$836$	−8.69935	−0.300873
$837$	0	0
$838$	−20.4387	−0.706043
$839$	−17.3302	−0.598307	−0.299153	−	0.954205i	$-0.596704\pi$
$839$	−17.3302	−0.598307	−0.299153	+	0.954205i	$0.596704\pi$
$840$	0	0
$841$	−18.0039	−0.620825
$842$	13.3018	0.458410
$843$	0	0
$844$	−7.04390	−0.242461
$845$	5.44769	0.187406
$846$	0	0
$847$	−2.20938	−0.0759151
$848$	4.06624	0.139635
$849$	0	0
$850$	1.94659	0.0667675
$851$	−45.9799	−1.57617
$852$	0	0
$853$	3.71668	0.127257	0.0636284	−	0.997974i	$-0.479733\pi$
$853$	3.71668	0.127257	0.0636284	+	0.997974i	$0.479733\pi$
$854$	−4.54062	−0.155377
$855$	0	0
$856$	62.0793	2.12183
$857$	−13.5906	−0.464246	−0.232123	−	0.972686i	$-0.574567\pi$
$857$	−13.5906	−0.464246	−0.232123	+	0.972686i	$0.574567\pi$
$858$	0	0
$859$	−41.5268	−1.41687	−0.708437	−	0.705774i	$-0.750599\pi$
$859$	−41.5268	−1.41687	−0.708437	+	0.705774i	$0.750599\pi$
$860$	7.39303	0.252100
$861$	0	0
$862$	17.1607	0.584496
$863$	31.3333	1.06660	0.533299	−	0.845927i	$-0.320952\pi$
$863$	31.3333	1.06660	0.533299	+	0.845927i	$0.320952\pi$
$864$	0	0
$865$	−16.4903	−0.560688
$866$	−7.91134	−0.268838
$867$	0	0
$868$	−1.62781	−0.0552515
$869$	−34.3801	−1.16626
$870$	0	0
$871$	45.9741	1.55777
$872$	35.6779	1.20821
$873$	0	0
$874$	−49.3272	−1.66852
$875$	−12.1345	−0.410221
$876$	0	0
$877$	−33.8310	−1.14239	−0.571197	−	0.820813i	$-0.693521\pi$
$877$	−33.8310	−1.14239	−0.571197	+	0.820813i	$0.693521\pi$
$878$	−25.0952	−0.846923
$879$	0	0
$880$	18.9276	0.638049
$881$	−53.2516	−1.79409	−0.897046	−	0.441937i	$-0.854291\pi$
$881$	−53.2516	−1.79409	−0.897046	+	0.441937i	$0.854291\pi$
$882$	0	0
$883$	2.42041	0.0814532	0.0407266	−	0.999170i	$-0.487033\pi$
$883$	2.42041	0.0814532	0.0407266	+	0.999170i	$0.487033\pi$
$884$	1.88751	0.0634840
$885$	0	0
$886$	40.4914	1.36033
$887$	−14.1507	−0.475133	−0.237566	−	0.971371i	$-0.576350\pi$
$887$	−14.1507	−0.475133	−0.237566	+	0.971371i	$0.576350\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	10.0936	0.338339
$891$	0	0
$892$	9.13444	0.305844
$893$	−14.5564	−0.487113
$894$	0	0
$895$	−14.3486	−0.479620
$896$	−5.25968	−0.175714
$897$	0	0
$898$	13.0018	0.433877
$899$	10.6556	0.355383
$900$	0	0
$901$	1.74234	0.0580459
$902$	45.4985	1.51493
$903$	0	0
$904$	−25.0606	−0.833503
$905$	−13.5041	−0.448890
$906$	0	0
$907$	19.3125	0.641260	0.320630	−	0.947205i	$-0.396105\pi$
$907$	19.3125	0.641260	0.320630	+	0.947205i	$0.396105\pi$
$908$	−13.7526	−0.456395
$909$	0	0
$910$	7.42424	0.246111
$911$	24.1353	0.799639	0.399820	−	0.916594i	$-0.369073\pi$
$911$	24.1353	0.799639	0.399820	+	0.916594i	$0.369073\pi$
$912$	0	0
$913$	−31.8138	−1.05288
$914$	2.30463	0.0762305
$915$	0	0
$916$	8.00117	0.264366
$917$	18.0189	0.595038
$918$	0	0
$919$	−3.69050	−0.121738	−0.0608691	−	0.998146i	$-0.519387\pi$
$919$	−3.69050	−0.121738	−0.0608691	+	0.998146i	$0.519387\pi$
$920$	−49.9142	−1.64562
$921$	0	0
$922$	9.61201	0.316555
$923$	−14.9592	−0.492387
$924$	0	0
$925$	−7.32854	−0.240961
$926$	−9.59592	−0.315341
$927$	0	0
$928$	−9.25131	−0.303689
$929$	−38.0433	−1.24816	−0.624080	−	0.781361i	$-0.714526\pi$
$929$	−38.0433	−1.24816	−0.624080	+	0.781361i	$0.714526\pi$
$930$	0	0
$931$	4.72497	0.154855
$932$	0.737789	0.0241671
$933$	0	0
$934$	−22.0984	−0.723083
$935$	8.11029	0.265235
$936$	0	0
$937$	43.9200	1.43480	0.717402	−	0.696660i	$-0.245331\pi$
$937$	43.9200	1.43480	0.717402	+	0.696660i	$0.245331\pi$
$938$	−17.6400	−0.575968
$939$	0	0
$940$	−2.97686	−0.0970946
$941$	56.4020	1.83865	0.919326	−	0.393497i	$-0.128735\pi$
$941$	56.4020	1.83865	0.919326	+	0.393497i	$0.128735\pi$
$942$	0	0
$943$	−87.5106	−2.84974
$944$	26.3013	0.856033
$945$	0	0
$946$	33.9823	1.10486
$947$	−9.10507	−0.295875	−0.147937	−	0.988997i	$-0.547263\pi$
$947$	−9.10507	−0.295875	−0.147937	+	0.988997i	$0.547263\pi$
$948$	0	0
$949$	27.7278	0.900084
$950$	−7.86204	−0.255078
$951$	0	0
$952$	−3.58353	−0.116143
$953$	−15.7744	−0.510983	−0.255491	−	0.966811i	$-0.582237\pi$
$953$	−15.7744	−0.510983	−0.255491	+	0.966811i	$0.582237\pi$
$954$	0	0
$955$	−8.31130	−0.268947
$956$	0.988620	0.0319742
$957$	0	0
$958$	−38.9216	−1.25750
$959$	12.9819	0.419207
$960$	0	0
$961$	−20.6744	−0.666917
$962$	20.9493	0.675431
$963$	0	0
$964$	13.7066	0.441459
$965$	−10.4139	−0.335236
$966$	0	0
$967$	−20.5069	−0.659457	−0.329729	−	0.944076i	$-0.606957\pi$
$967$	−20.5069	−0.659457	−0.329729	+	0.944076i	$0.606957\pi$
$968$	−6.76772	−0.217523
$969$	0	0
$970$	8.41539	0.270202
$971$	18.0971	0.580765	0.290382	−	0.956911i	$-0.406218\pi$
$971$	18.0971	0.580765	0.290382	+	0.956911i	$0.406218\pi$
$972$	0	0
$973$	7.51765	0.241005
$974$	35.4240	1.13506
$975$	0	0
$976$	−10.1443	−0.324711
$977$	24.4031	0.780725	0.390363	−	0.920661i	$-0.372350\pi$
$977$	24.4031	0.780725	0.390363	+	0.920661i	$0.372350\pi$
$978$	0	0
$979$	−15.7377	−0.502978
$980$	0.966279	0.0308666
$981$	0	0
$982$	−46.6672	−1.48921
$983$	−21.7245	−0.692905	−0.346453	−	0.938067i	$-0.612614\pi$
$983$	−21.7245	−0.692905	−0.346453	+	0.938067i	$0.612614\pi$
$984$	0	0
$985$	25.4224	0.810024
$986$	4.74077	0.150977
$987$	0	0
$988$	−7.62343	−0.242534
$989$	−65.3606	−2.07835
$990$	0	0
$991$	−25.6152	−0.813692	−0.406846	−	0.913497i	$-0.633371\pi$
$991$	−25.6152	−0.813692	−0.406846	+	0.913497i	$0.633371\pi$
$992$	−8.96482	−0.284633
$993$	0	0
$994$	5.73976	0.182054
$995$	−3.12749	−0.0991482
$996$	0	0
$997$	12.2193	0.386989	0.193495	−	0.981101i	$-0.438018\pi$
$997$	12.2193	0.386989	0.193495	+	0.981101i	$0.438018\pi$
$998$	11.9595	0.378571
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.q.1.11		15
3.2	odd	2		889.2.a.b.1.5	✓	15
21.20	even	2		6223.2.a.j.1.5		15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
889.2.a.b.1.5	✓	15	3.2	odd	2
6223.2.a.j.1.5		15	21.20	even	2
8001.2.a.q.1.11		15	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)

\(f(q)\)	\(=\)	\(q+1.22206 q^{2} -0.506578 q^{4} -1.90746 q^{5} -1.00000 q^{7} -3.06318 q^{8} +O(q^{10})\)	\(q+1.22206 q^{2} -0.506578 q^{4} -1.90746 q^{5} -1.00000 q^{7} -3.06318 q^{8} -2.33103 q^{10} +3.63447 q^{11} +3.18497 q^{13} -1.22206 q^{14} -2.73022 q^{16} -1.16987 q^{17} +4.72497 q^{19} +0.966279 q^{20} +4.44153 q^{22} -8.54271 q^{23} -1.36158 q^{25} +3.89221 q^{26} +0.506578 q^{28} -3.31603 q^{29} -3.21334 q^{31} +2.78987 q^{32} -1.42965 q^{34} +1.90746 q^{35} +5.38236 q^{37} +5.77418 q^{38} +5.84290 q^{40} +10.2439 q^{41} +7.65103 q^{43} -1.84114 q^{44} -10.4397 q^{46} -3.08075 q^{47} +1.00000 q^{49} -1.66393 q^{50} -1.61343 q^{52} -1.48934 q^{53} -6.93262 q^{55} +3.06318 q^{56} -4.05238 q^{58} -9.63337 q^{59} +3.71556 q^{61} -3.92689 q^{62} +8.86983 q^{64} -6.07521 q^{65} +14.4347 q^{67} +0.592632 q^{68} +2.33103 q^{70} -4.69680 q^{71} +8.70585 q^{73} +6.57754 q^{74} -2.39357 q^{76} -3.63447 q^{77} -9.45945 q^{79} +5.20780 q^{80} +12.5186 q^{82} -8.75335 q^{83} +2.23149 q^{85} +9.35000 q^{86} -11.1330 q^{88} -4.33011 q^{89} -3.18497 q^{91} +4.32755 q^{92} -3.76485 q^{94} -9.01271 q^{95} -3.61016 q^{97} +1.22206 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 14 q^{4} - 7 q^{5} - 15 q^{7}+O(q^{10}) \) 15 * q + 14 * q^4 - 7 * q^5 - 15 * q^7	\( 15 q + 14 q^{4} - 7 q^{5} - 15 q^{7} + 10 q^{10} - 14 q^{11} + 6 q^{13} + 20 q^{16} - 10 q^{17} + 13 q^{19} - 8 q^{20} - 11 q^{22} - 15 q^{23} - 22 q^{26} - 14 q^{28} - 16 q^{29} + 22 q^{31} - 15 q^{34} + 7 q^{35} - 14 q^{37} + 6 q^{38} + 22 q^{40} - 19 q^{41} - q^{43} - 25 q^{44} - 28 q^{46} - 49 q^{47} + 15 q^{49} - 24 q^{50} - 17 q^{52} + 28 q^{53} + 39 q^{55} - 10 q^{58} - 43 q^{59} + 27 q^{61} - 14 q^{62} + 18 q^{64} + 8 q^{65} + 3 q^{67} - 13 q^{68} - 10 q^{70} - 55 q^{71} - 3 q^{73} + 12 q^{74} - 20 q^{76} + 14 q^{77} + 18 q^{79} - 29 q^{80} + 14 q^{82} - 17 q^{83} + 7 q^{85} - 4 q^{86} - 114 q^{88} - 36 q^{89} - 6 q^{91} - 45 q^{92} - 15 q^{94} - 59 q^{95} - 2 q^{97}+O(q^{100}) \) 15 * q + 14 * q^4 - 7 * q^5 - 15 * q^7 + 10 * q^10 - 14 * q^11 + 6 * q^13 + 20 * q^16 - 10 * q^17 + 13 * q^19 - 8 * q^20 - 11 * q^22 - 15 * q^23 - 22 * q^26 - 14 * q^28 - 16 * q^29 + 22 * q^31 - 15 * q^34 + 7 * q^35 - 14 * q^37 + 6 * q^38 + 22 * q^40 - 19 * q^41 - q^43 - 25 * q^44 - 28 * q^46 - 49 * q^47 + 15 * q^49 - 24 * q^50 - 17 * q^52 + 28 * q^53 + 39 * q^55 - 10 * q^58 - 43 * q^59 + 27 * q^61 - 14 * q^62 + 18 * q^64 + 8 * q^65 + 3 * q^67 - 13 * q^68 - 10 * q^70 - 55 * q^71 - 3 * q^73 + 12 * q^74 - 20 * q^76 + 14 * q^77 + 18 * q^79 - 29 * q^80 + 14 * q^82 - 17 * q^83 + 7 * q^85 - 4 * q^86 - 114 * q^88 - 36 * q^89 - 6 * q^91 - 45 * q^92 - 15 * q^94 - 59 * q^95 - 2 * q^97

Introduction

L-functions

Modular forms

Varieties

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Database

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