LMFDB - Embedded newform 8001.2.a.p.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8001,2,Mod(1,8001)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8001, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8001 = 3^{2} \cdot 7 \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8001.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$63.8883066572$
Analytic rank:	$0$
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			1.4
Root			$-1.00753$ 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)

$f(q)$	$=$	$q-1.00753 q^{2} -0.984892 q^{4} -2.84275 q^{5} -1.00000 q^{7} +3.00736 q^{8} +O(q^{10})$	\(q-1.00753 q^{2} -0.984892 q^{4} -2.84275 q^{5} -1.00000 q^{7} +3.00736 q^{8} +2.86415 q^{10} -3.37565 q^{11} -4.14428 q^{13} +1.00753 q^{14} -1.06020 q^{16} +4.31858 q^{17} +4.53137 q^{19} +2.79980 q^{20} +3.40106 q^{22} +7.48784 q^{23} +3.08124 q^{25} +4.17547 q^{26} +0.984892 q^{28} +1.64837 q^{29} -2.76165 q^{31} -4.94653 q^{32} -4.35108 q^{34} +2.84275 q^{35} -6.47989 q^{37} -4.56547 q^{38} -8.54916 q^{40} -8.87924 q^{41} -2.73621 q^{43} +3.32466 q^{44} -7.54419 q^{46} +2.45053 q^{47} +1.00000 q^{49} -3.10443 q^{50} +4.08167 q^{52} -7.67219 q^{53} +9.59615 q^{55} -3.00736 q^{56} -1.66077 q^{58} +9.16174 q^{59} -5.46532 q^{61} +2.78243 q^{62} +7.10416 q^{64} +11.7812 q^{65} -1.71562 q^{67} -4.25333 q^{68} -2.86415 q^{70} +3.84688 q^{71} -9.49128 q^{73} +6.52865 q^{74} -4.46291 q^{76} +3.37565 q^{77} -12.9615 q^{79} +3.01390 q^{80} +8.94607 q^{82} -16.6680 q^{83} -12.2766 q^{85} +2.75681 q^{86} -10.1518 q^{88} +14.4931 q^{89} +4.14428 q^{91} -7.37472 q^{92} -2.46897 q^{94} -12.8815 q^{95} -5.57380 q^{97} -1.00753 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + 5 q^{2} + 15 q^{4} + 4 q^{5} - 14 q^{7} + 12 q^{8}+O(q^{10}) $ 14 * q + 5 * q^2 + 15 * q^4 + 4 * q^5 - 14 * q^7 + 12 * q^8	$ 14 q + 5 q^{2} + 15 q^{4} + 4 q^{5} - 14 q^{7} + 12 q^{8} + 4 q^{10} + 3 q^{11} - 13 q^{13} - 5 q^{14} + 13 q^{16} + 5 q^{17} + 21 q^{19} + 3 q^{20} - 3 q^{22} + 10 q^{23} + 4 q^{25} + 6 q^{26} - 15 q^{28} + 15 q^{29} + 33 q^{31} + 29 q^{32} + 28 q^{34} - 4 q^{35} - 29 q^{37} + 15 q^{38} + 3 q^{40} + q^{41} - 25 q^{43} + 26 q^{44} - 4 q^{46} + 9 q^{47} + 14 q^{49} + 28 q^{50} - 13 q^{52} + 35 q^{53} + 14 q^{55} - 12 q^{56} - 23 q^{58} - 10 q^{59} + q^{61} + 43 q^{62} - 2 q^{64} + 24 q^{65} - 38 q^{67} + 2 q^{68} - 4 q^{70} + 10 q^{71} + 8 q^{73} + 25 q^{74} + 26 q^{76} - 3 q^{77} + 26 q^{79} + 48 q^{80} + 6 q^{82} + 30 q^{83} - 32 q^{85} + 50 q^{86} - 29 q^{88} - 4 q^{89} + 13 q^{91} + 32 q^{92} - 7 q^{94} + 32 q^{95} + 15 q^{97} + 5 q^{98}+O(q^{100}) $ 14 * q + 5 * q^2 + 15 * q^4 + 4 * q^5 - 14 * q^7 + 12 * q^8 + 4 * q^10 + 3 * q^11 - 13 * q^13 - 5 * q^14 + 13 * q^16 + 5 * q^17 + 21 * q^19 + 3 * q^20 - 3 * q^22 + 10 * q^23 + 4 * q^25 + 6 * q^26 - 15 * q^28 + 15 * q^29 + 33 * q^31 + 29 * q^32 + 28 * q^34 - 4 * q^35 - 29 * q^37 + 15 * q^38 + 3 * q^40 + q^41 - 25 * q^43 + 26 * q^44 - 4 * q^46 + 9 * q^47 + 14 * q^49 + 28 * q^50 - 13 * q^52 + 35 * q^53 + 14 * q^55 - 12 * q^56 - 23 * q^58 - 10 * q^59 + q^61 + 43 * q^62 - 2 * q^64 + 24 * q^65 - 38 * q^67 + 2 * q^68 - 4 * q^70 + 10 * q^71 + 8 * q^73 + 25 * q^74 + 26 * q^76 - 3 * q^77 + 26 * q^79 + 48 * q^80 + 6 * q^82 + 30 * q^83 - 32 * q^85 + 50 * q^86 - 29 * q^88 - 4 * q^89 + 13 * q^91 + 32 * q^92 - 7 * q^94 + 32 * q^95 + 15 * q^97 + 5 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00753	−0.712428	−0.356214	−	0.934404i	$-0.615933\pi$
$2$	−1.00753	−0.712428	−0.356214	+	0.934404i	$0.615933\pi$
$3$	0	0
$3$	0	0
$4$	−0.984892	−0.492446
$5$	−2.84275	−1.27132	−0.635659	−	0.771970i	$-0.719271\pi$
$5$	−2.84275	−1.27132	−0.635659	+	0.771970i	$0.719271\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	3.00736	1.06326
$9$	0	0
$10$	2.86415	0.905722
$11$	−3.37565	−1.01780	−0.508899	−	0.860826i	$-0.669947\pi$
$11$	−3.37565	−1.01780	−0.508899	+	0.860826i	$0.669947\pi$
$12$	0	0
$13$	−4.14428	−1.14942	−0.574708	−	0.818358i	$-0.694884\pi$
$13$	−4.14428	−1.14942	−0.574708	+	0.818358i	$0.694884\pi$
$14$	1.00753	0.269273
$15$	0	0
$16$	−1.06020	−0.265051
$17$	4.31858	1.04741	0.523704	−	0.851900i	$-0.324550\pi$
$17$	4.31858	1.04741	0.523704	+	0.851900i	$0.324550\pi$
$18$	0	0
$19$	4.53137	1.03957	0.519783	−	0.854298i	$-0.326013\pi$
$19$	4.53137	1.03957	0.519783	+	0.854298i	$0.326013\pi$
$20$	2.79980	0.626055
$21$	0	0
$22$	3.40106	0.725108
$23$	7.48784	1.56132	0.780662	−	0.624954i	$-0.214882\pi$
$23$	7.48784	1.56132	0.780662	+	0.624954i	$0.214882\pi$
$24$	0	0
$25$	3.08124	0.616247
$26$	4.17547	0.818877
$27$	0	0
$28$	0.984892	0.186127
$29$	1.64837	0.306094	0.153047	−	0.988219i	$-0.451091\pi$
$29$	1.64837	0.306094	0.153047	+	0.988219i	$0.451091\pi$
$30$	0	0
$31$	−2.76165	−0.496007	−0.248004	−	0.968759i	$-0.579774\pi$
$31$	−2.76165	−0.496007	−0.248004	+	0.968759i	$0.579774\pi$
$32$	−4.94653	−0.874431
$33$	0	0
$34$	−4.35108	−0.746204
$35$	2.84275	0.480513
$36$	0	0
$37$	−6.47989	−1.06529	−0.532644	−	0.846340i	$-0.678801\pi$
$37$	−6.47989	−1.06529	−0.532644	+	0.846340i	$0.678801\pi$
$38$	−4.56547	−0.740617
$39$	0	0
$40$	−8.54916	−1.35174
$41$	−8.87924	−1.38670	−0.693352	−	0.720599i	$-0.743867\pi$
$41$	−8.87924	−1.38670	−0.693352	+	0.720599i	$0.743867\pi$
$42$	0	0
$43$	−2.73621	−0.417269	−0.208634	−	0.977994i	$-0.566902\pi$
$43$	−2.73621	−0.417269	−0.208634	+	0.977994i	$0.566902\pi$
$44$	3.32466	0.501211
$45$	0	0
$46$	−7.54419	−1.11233
$47$	2.45053	0.357446	0.178723	−	0.983899i	$-0.442803\pi$
$47$	2.45053	0.357446	0.178723	+	0.983899i	$0.442803\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−3.10443	−0.439032
$51$	0	0
$52$	4.08167	0.566025
$53$	−7.67219	−1.05386	−0.526928	−	0.849910i	$-0.676656\pi$
$53$	−7.67219	−1.05386	−0.526928	+	0.849910i	$0.676656\pi$
$54$	0	0
$55$	9.59615	1.29394
$56$	−3.00736	−0.401875
$57$	0	0
$58$	−1.66077	−0.218070
$59$	9.16174	1.19276	0.596379	−	0.802703i	$-0.296606\pi$
$59$	9.16174	1.19276	0.596379	+	0.802703i	$0.296606\pi$
$60$	0	0
$61$	−5.46532	−0.699762	−0.349881	−	0.936794i	$-0.613778\pi$
$61$	−5.46532	−0.699762	−0.349881	+	0.936794i	$0.613778\pi$
$62$	2.78243	0.353369
$63$	0	0
$64$	7.10416	0.888020
$65$	11.7812	1.46127
$66$	0	0
$67$	−1.71562	−0.209596	−0.104798	−	0.994494i	$-0.533420\pi$
$67$	−1.71562	−0.209596	−0.104798	+	0.994494i	$0.533420\pi$
$68$	−4.25333	−0.515792
$69$	0	0
$70$	−2.86415	−0.342331
$71$	3.84688	0.456541	0.228271	−	0.973598i	$-0.426693\pi$
$71$	3.84688	0.456541	0.228271	+	0.973598i	$0.426693\pi$
$72$	0	0
$73$	−9.49128	−1.11087	−0.555435	−	0.831560i	$-0.687448\pi$
$73$	−9.49128	−1.11087	−0.555435	+	0.831560i	$0.687448\pi$
$74$	6.52865	0.758941
$75$	0	0
$76$	−4.46291	−0.511930
$77$	3.37565	0.384692
$78$	0	0
$79$	−12.9615	−1.45828	−0.729141	−	0.684364i	$-0.760080\pi$
$79$	−12.9615	−1.45828	−0.729141	+	0.684364i	$0.760080\pi$
$80$	3.01390	0.336964
$81$	0	0
$82$	8.94607	0.987928
$83$	−16.6680	−1.82956	−0.914778	−	0.403958i	$-0.867634\pi$
$83$	−16.6680	−1.82956	−0.914778	+	0.403958i	$0.867634\pi$
$84$	0	0
$85$	−12.2766	−1.33159
$86$	2.75681	0.297274
$87$	0	0
$88$	−10.1518	−1.08218
$89$	14.4931	1.53627	0.768134	−	0.640289i	$-0.221185\pi$
$89$	14.4931	1.53627	0.768134	+	0.640289i	$0.221185\pi$
$90$	0	0
$91$	4.14428	0.434438
$92$	−7.37472	−0.768867
$93$	0	0
$94$	−2.46897	−0.254655
$95$	−12.8815	−1.32162
$96$	0	0
$97$	−5.57380	−0.565934	−0.282967	−	0.959130i	$-0.591319\pi$
$97$	−5.57380	−0.565934	−0.282967	+	0.959130i	$0.591319\pi$
$98$	−1.00753	−0.101775
$99$	0	0
$100$	−3.03469	−0.303469
$101$	2.13721	0.212660	0.106330	−	0.994331i	$-0.466090\pi$
$101$	2.13721	0.212660	0.106330	+	0.994331i	$0.466090\pi$
$102$	0	0
$103$	−16.7727	−1.65267	−0.826333	−	0.563182i	$-0.809577\pi$
$103$	−16.7727	−1.65267	−0.826333	+	0.563182i	$0.809577\pi$
$104$	−12.4633	−1.22213
$105$	0	0
$106$	7.72993	0.750797
$107$	−8.46145	−0.817999	−0.408999	−	0.912535i	$-0.634122\pi$
$107$	−8.46145	−0.817999	−0.408999	+	0.912535i	$0.634122\pi$
$108$	0	0
$109$	−4.17049	−0.399461	−0.199730	−	0.979851i	$-0.564007\pi$
$109$	−4.17049	−0.399461	−0.199730	+	0.979851i	$0.564007\pi$
$110$	−9.66837	−0.921843
$111$	0	0
$112$	1.06020	0.100180
$113$	−5.85865	−0.551135	−0.275568	−	0.961282i	$-0.588866\pi$
$113$	−5.85865	−0.551135	−0.275568	+	0.961282i	$0.588866\pi$
$114$	0	0
$115$	−21.2861	−1.98494
$116$	−1.62347	−0.150735
$117$	0	0
$118$	−9.23069	−0.849754
$119$	−4.31858	−0.395883
$120$	0	0
$121$	0.395045	0.0359132
$122$	5.50645	0.498530
$123$	0	0
$124$	2.71993	0.244257
$125$	5.45457	0.487871
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	2.73543	0.241780
$129$	0	0
$130$	−11.8698	−1.04105
$131$	−12.3282	−1.07712	−0.538558	−	0.842588i	$-0.681031\pi$
$131$	−12.3282	−1.07712	−0.538558	+	0.842588i	$0.681031\pi$
$132$	0	0
$133$	−4.53137	−0.392919
$134$	1.72853	0.149322
$135$	0	0
$136$	12.9875	1.11367
$137$	20.9497	1.78985	0.894927	−	0.446213i	$-0.147227\pi$
$137$	20.9497	1.78985	0.894927	+	0.446213i	$0.147227\pi$
$138$	0	0
$139$	−5.79454	−0.491486	−0.245743	−	0.969335i	$-0.579032\pi$
$139$	−5.79454	−0.491486	−0.245743	+	0.969335i	$0.579032\pi$
$140$	−2.79980	−0.236627
$141$	0	0
$142$	−3.87584	−0.325253
$143$	13.9897	1.16987
$144$	0	0
$145$	−4.68590	−0.389143
$146$	9.56270	0.791415
$147$	0	0
$148$	6.38199	0.524596
$149$	5.11902	0.419367	0.209683	−	0.977769i	$-0.432757\pi$
$149$	5.11902	0.419367	0.209683	+	0.977769i	$0.432757\pi$
$150$	0	0
$151$	4.90674	0.399305	0.199652	−	0.979867i	$-0.436019\pi$
$151$	4.90674	0.399305	0.199652	+	0.979867i	$0.436019\pi$
$152$	13.6274	1.10533
$153$	0	0
$154$	−3.40106	−0.274065
$155$	7.85069	0.630582
$156$	0	0
$157$	20.7566	1.65656	0.828280	−	0.560315i	$-0.189320\pi$
$157$	20.7566	1.65656	0.828280	+	0.560315i	$0.189320\pi$
$158$	13.0590	1.03892
$159$	0	0
$160$	14.0618	1.11168
$161$	−7.48784	−0.590125
$162$	0	0
$163$	−6.90596	−0.540916	−0.270458	−	0.962732i	$-0.587175\pi$
$163$	−6.90596	−0.540916	−0.270458	+	0.962732i	$0.587175\pi$
$164$	8.74510	0.682877
$165$	0	0
$166$	16.7935	1.30343
$167$	−0.429151	−0.0332087	−0.0166043	−	0.999862i	$-0.505286\pi$
$167$	−0.429151	−0.0332087	−0.0166043	+	0.999862i	$0.505286\pi$
$168$	0	0
$169$	4.17505	0.321158
$170$	12.3690	0.948661
$171$	0	0
$172$	2.69488	0.205482
$173$	19.2117	1.46064	0.730318	−	0.683107i	$-0.239372\pi$
$173$	19.2117	1.46064	0.730318	+	0.683107i	$0.239372\pi$
$174$	0	0
$175$	−3.08124	−0.232920
$176$	3.57888	0.269768
$177$	0	0
$178$	−14.6022	−1.09448
$179$	−10.3741	−0.775396	−0.387698	−	0.921786i	$-0.626730\pi$
$179$	−10.3741	−0.775396	−0.387698	+	0.921786i	$0.626730\pi$
$180$	0	0
$181$	−0.393610	−0.0292568	−0.0146284	−	0.999893i	$-0.504657\pi$
$181$	−0.393610	−0.0292568	−0.0146284	+	0.999893i	$0.504657\pi$
$182$	−4.17547	−0.309506
$183$	0	0
$184$	22.5186	1.66009
$185$	18.4207	1.35432
$186$	0	0
$187$	−14.5780	−1.06605
$188$	−2.41350	−0.176023
$189$	0	0
$190$	12.9785	0.941559
$191$	−6.88341	−0.498066	−0.249033	−	0.968495i	$-0.580113\pi$
$191$	−6.88341	−0.498066	−0.249033	+	0.968495i	$0.580113\pi$
$192$	0	0
$193$	−12.9192	−0.929941	−0.464970	−	0.885326i	$-0.653935\pi$
$193$	−12.9192	−0.929941	−0.464970	+	0.885326i	$0.653935\pi$
$194$	5.61575	0.403187
$195$	0	0
$196$	−0.984892	−0.0703494
$197$	16.0626	1.14442	0.572208	−	0.820108i	$-0.306087\pi$
$197$	16.0626	1.14442	0.572208	+	0.820108i	$0.306087\pi$
$198$	0	0
$199$	19.5382	1.38503	0.692513	−	0.721406i	$-0.256504\pi$
$199$	19.5382	1.38503	0.692513	+	0.721406i	$0.256504\pi$
$200$	9.26637	0.655232
$201$	0	0
$202$	−2.15329	−0.151505
$203$	−1.64837	−0.115693
$204$	0	0
$205$	25.2415	1.76294
$206$	16.8990	1.17741
$207$	0	0
$208$	4.39378	0.304654
$209$	−15.2963	−1.05807
$210$	0	0
$211$	5.72786	0.394322	0.197161	−	0.980371i	$-0.436828\pi$
$211$	5.72786	0.394322	0.197161	+	0.980371i	$0.436828\pi$
$212$	7.55628	0.518967
$213$	0	0
$214$	8.52512	0.582766
$215$	7.77838	0.530481
$216$	0	0
$217$	2.76165	0.187473
$218$	4.20188	0.284587
$219$	0	0
$220$	−9.45117	−0.637198
$221$	−17.8974	−1.20391
$222$	0	0
$223$	−13.8279	−0.925984	−0.462992	−	0.886362i	$-0.653224\pi$
$223$	−13.8279	−0.925984	−0.462992	+	0.886362i	$0.653224\pi$
$224$	4.94653	0.330504
$225$	0	0
$226$	5.90274	0.392644
$227$	0.552953	0.0367008	0.0183504	−	0.999832i	$-0.494159\pi$
$227$	0.552953	0.0367008	0.0183504	+	0.999832i	$0.494159\pi$
$228$	0	0
$229$	−4.41743	−0.291912	−0.145956	−	0.989291i	$-0.546626\pi$
$229$	−4.41743	−0.291912	−0.145956	+	0.989291i	$0.546626\pi$
$230$	21.4463	1.41413
$231$	0	0
$232$	4.95723	0.325458
$233$	1.84832	0.121087	0.0605437	−	0.998166i	$-0.480717\pi$
$233$	1.84832	0.121087	0.0605437	+	0.998166i	$0.480717\pi$
$234$	0	0
$235$	−6.96624	−0.454427
$236$	−9.02333	−0.587368
$237$	0	0
$238$	4.35108	0.282038
$239$	−9.84050	−0.636529	−0.318264	−	0.948002i	$-0.603100\pi$
$239$	−9.84050	−0.636529	−0.318264	+	0.948002i	$0.603100\pi$
$240$	0	0
$241$	25.7558	1.65907	0.829537	−	0.558452i	$-0.188604\pi$
$241$	25.7558	1.65907	0.829537	+	0.558452i	$0.188604\pi$
$242$	−0.398018	−0.0255856
$243$	0	0
$244$	5.38275	0.344595
$245$	−2.84275	−0.181617
$246$	0	0
$247$	−18.7792	−1.19489
$248$	−8.30526	−0.527385
$249$	0	0
$250$	−5.49562	−0.347573
$251$	−9.54705	−0.602604	−0.301302	−	0.953529i	$-0.597421\pi$
$251$	−9.54705	−0.602604	−0.301302	+	0.953529i	$0.597421\pi$
$252$	0	0
$253$	−25.2764	−1.58911
$254$	1.00753	0.0632178
$255$	0	0
$256$	−16.9643	−1.06027
$257$	−24.2857	−1.51490	−0.757451	−	0.652892i	$-0.773555\pi$
$257$	−24.2857	−1.51490	−0.757451	+	0.652892i	$0.773555\pi$
$258$	0	0
$259$	6.47989	0.402641
$260$	−11.6032	−0.719598
$261$	0	0
$262$	12.4209	0.767368
$263$	27.0413	1.66744	0.833720	−	0.552188i	$-0.186207\pi$
$263$	27.0413	1.66744	0.833720	+	0.552188i	$0.186207\pi$
$264$	0	0
$265$	21.8101	1.33979
$266$	4.56547	0.279927
$267$	0	0
$268$	1.68970	0.103215
$269$	5.12254	0.312327	0.156163	−	0.987731i	$-0.450087\pi$
$269$	5.12254	0.312327	0.156163	+	0.987731i	$0.450087\pi$
$270$	0	0
$271$	9.83423	0.597387	0.298693	−	0.954349i	$-0.403449\pi$
$271$	9.83423	0.597387	0.298693	+	0.954349i	$0.403449\pi$
$272$	−4.57857	−0.277617
$273$	0	0
$274$	−21.1074	−1.27514
$275$	−10.4012	−0.627216
$276$	0	0
$277$	11.8515	0.712086	0.356043	−	0.934470i	$-0.384126\pi$
$277$	11.8515	0.712086	0.356043	+	0.934470i	$0.384126\pi$
$278$	5.83814	0.350149
$279$	0	0
$280$	8.54916	0.510910
$281$	7.95802	0.474736	0.237368	−	0.971420i	$-0.423715\pi$
$281$	7.95802	0.474736	0.237368	+	0.971420i	$0.423715\pi$
$282$	0	0
$283$	18.2111	1.08254	0.541268	−	0.840850i	$-0.317945\pi$
$283$	18.2111	1.08254	0.541268	+	0.840850i	$0.317945\pi$
$284$	−3.78877	−0.224822
$285$	0	0
$286$	−14.0949	−0.833451
$287$	8.87924	0.524125
$288$	0	0
$289$	1.65011	0.0970651
$290$	4.72117	0.277237
$291$	0	0
$292$	9.34788	0.547043
$293$	8.68964	0.507654	0.253827	−	0.967250i	$-0.418311\pi$
$293$	8.68964	0.507654	0.253827	+	0.967250i	$0.418311\pi$
$294$	0	0
$295$	−26.0446	−1.51637
$296$	−19.4873	−1.13268
$297$	0	0
$298$	−5.15755	−0.298769
$299$	−31.0317	−1.79461
$300$	0	0
$301$	2.73621	0.157713
$302$	−4.94366	−0.284476
$303$	0	0
$304$	−4.80417	−0.275538
$305$	15.5365	0.889620
$306$	0	0
$307$	15.8838	0.906537	0.453269	−	0.891374i	$-0.350258\pi$
$307$	15.8838	0.906537	0.453269	+	0.891374i	$0.350258\pi$
$308$	−3.32466	−0.189440
$309$	0	0
$310$	−7.90977	−0.449245
$311$	−31.0378	−1.75999	−0.879996	−	0.474980i	$-0.842455\pi$
$311$	−31.0378	−1.75999	−0.879996	+	0.474980i	$0.842455\pi$
$312$	0	0
$313$	1.03300	0.0583886	0.0291943	−	0.999574i	$-0.490706\pi$
$313$	1.03300	0.0583886	0.0291943	+	0.999574i	$0.490706\pi$
$314$	−20.9128	−1.18018
$315$	0	0
$316$	12.7657	0.718125
$317$	−18.2990	−1.02778	−0.513888	−	0.857857i	$-0.671795\pi$
$317$	−18.2990	−1.02778	−0.513888	+	0.857857i	$0.671795\pi$
$318$	0	0
$319$	−5.56432	−0.311542
$320$	−20.1954	−1.12896
$321$	0	0
$322$	7.54419	0.420422
$323$	19.5691	1.08885
$324$	0	0
$325$	−12.7695	−0.708325
$326$	6.95793	0.385364
$327$	0	0
$328$	−26.7030	−1.47443
$329$	−2.45053	−0.135102
$330$	0	0
$331$	−10.9403	−0.601332	−0.300666	−	0.953729i	$-0.597209\pi$
$331$	−10.9403	−0.601332	−0.300666	+	0.953729i	$0.597209\pi$
$332$	16.4162	0.900957
$333$	0	0
$334$	0.432380	0.0236588
$335$	4.87707	0.266463
$336$	0	0
$337$	−8.39435	−0.457269	−0.228635	−	0.973512i	$-0.573426\pi$
$337$	−8.39435	−0.457269	−0.228635	+	0.973512i	$0.573426\pi$
$338$	−4.20647	−0.228802
$339$	0	0
$340$	12.0912	0.655736
$341$	9.32238	0.504835
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−8.22877	−0.443666
$345$	0	0
$346$	−19.3563	−1.04060
$347$	−9.75728	−0.523798	−0.261899	−	0.965095i	$-0.584349\pi$
$347$	−9.75728	−0.523798	−0.261899	+	0.965095i	$0.584349\pi$
$348$	0	0
$349$	−32.2382	−1.72567	−0.862836	−	0.505484i	$-0.831314\pi$
$349$	−32.2382	−1.72567	−0.862836	+	0.505484i	$0.831314\pi$
$350$	3.10443	0.165939
$351$	0	0
$352$	16.6978	0.889994
$353$	5.55740	0.295791	0.147895	−	0.989003i	$-0.452750\pi$
$353$	5.55740	0.295791	0.147895	+	0.989003i	$0.452750\pi$
$354$	0	0
$355$	−10.9357	−0.580409
$356$	−14.2742	−0.756529
$357$	0	0
$358$	10.4522	0.552414
$359$	20.6109	1.08780	0.543900	−	0.839150i	$-0.316947\pi$
$359$	20.6109	1.08780	0.543900	+	0.839150i	$0.316947\pi$
$360$	0	0
$361$	1.53328	0.0806989
$362$	0.396572	0.0208434
$363$	0	0
$364$	−4.08167	−0.213937
$365$	26.9813	1.41227
$366$	0	0
$367$	0.387988	0.0202528	0.0101264	−	0.999949i	$-0.496777\pi$
$367$	0.387988	0.0202528	0.0101264	+	0.999949i	$0.496777\pi$
$368$	−7.93864	−0.413830
$369$	0	0
$370$	−18.5593	−0.964854
$371$	7.67219	0.398320
$372$	0	0
$373$	7.54746	0.390793	0.195396	−	0.980724i	$-0.437401\pi$
$373$	7.54746	0.390793	0.195396	+	0.980724i	$0.437401\pi$
$374$	14.6877	0.759485
$375$	0	0
$376$	7.36960	0.380058
$377$	−6.83130	−0.351830
$378$	0	0
$379$	6.07471	0.312037	0.156018	−	0.987754i	$-0.450134\pi$
$379$	6.07471	0.312037	0.156018	+	0.987754i	$0.450134\pi$
$380$	12.6869	0.650826
$381$	0	0
$382$	6.93521	0.354836
$383$	12.8344	0.655810	0.327905	−	0.944711i	$-0.393657\pi$
$383$	12.8344	0.655810	0.327905	+	0.944711i	$0.393657\pi$
$384$	0	0
$385$	−9.59615	−0.489065
$386$	13.0164	0.662516
$387$	0	0
$388$	5.48959	0.278692
$389$	−9.85255	−0.499544	−0.249772	−	0.968305i	$-0.580356\pi$
$389$	−9.85255	−0.499544	−0.249772	+	0.968305i	$0.580356\pi$
$390$	0	0
$391$	32.3368	1.63534
$392$	3.00736	0.151894
$393$	0	0
$394$	−16.1835	−0.815315
$395$	36.8463	1.85394
$396$	0	0
$397$	22.1380	1.11107	0.555537	−	0.831492i	$-0.312513\pi$
$397$	22.1380	1.11107	0.555537	+	0.831492i	$0.312513\pi$
$398$	−19.6852	−0.986731
$399$	0	0
$400$	−3.26674	−0.163337
$401$	30.6080	1.52849	0.764246	−	0.644925i	$-0.223111\pi$
$401$	30.6080	1.52849	0.764246	+	0.644925i	$0.223111\pi$
$402$	0	0
$403$	11.4451	0.570119
$404$	−2.10492	−0.104724
$405$	0	0
$406$	1.66077	0.0824228
$407$	21.8739	1.08425
$408$	0	0
$409$	−10.7900	−0.533529	−0.266764	−	0.963762i	$-0.585955\pi$
$409$	−10.7900	−0.533529	−0.266764	+	0.963762i	$0.585955\pi$
$410$	−25.4314	−1.25597
$411$	0	0
$412$	16.5193	0.813849
$413$	−9.16174	−0.450820
$414$	0	0
$415$	47.3831	2.32595
$416$	20.4998	1.00509
$417$	0	0
$418$	15.4114	0.753798
$419$	−40.4740	−1.97728	−0.988641	−	0.150296i	$-0.951977\pi$
$419$	−40.4740	−1.97728	−0.988641	+	0.150296i	$0.951977\pi$
$420$	0	0
$421$	9.07418	0.442248	0.221124	−	0.975246i	$-0.429027\pi$
$421$	9.07418	0.442248	0.221124	+	0.975246i	$0.429027\pi$
$422$	−5.77097	−0.280926
$423$	0	0
$424$	−23.0730	−1.12052
$425$	13.3066	0.645463
$426$	0	0
$427$	5.46532	0.264485
$428$	8.33361	0.402820
$429$	0	0
$430$	−7.83692	−0.377930
$431$	30.1552	1.45252	0.726262	−	0.687418i	$-0.241256\pi$
$431$	30.1552	1.45252	0.726262	+	0.687418i	$0.241256\pi$
$432$	0	0
$433$	30.0801	1.44556	0.722778	−	0.691080i	$-0.242865\pi$
$433$	30.0801	1.44556	0.722778	+	0.691080i	$0.242865\pi$
$434$	−2.78243	−0.133561
$435$	0	0
$436$	4.10749	0.196713
$437$	33.9302	1.62310
$438$	0	0
$439$	15.8665	0.757268	0.378634	−	0.925546i	$-0.376394\pi$
$439$	15.8665	0.757268	0.378634	+	0.925546i	$0.376394\pi$
$440$	28.8590	1.37580
$441$	0	0
$442$	18.0321	0.857698
$443$	−23.9129	−1.13614	−0.568068	−	0.822981i	$-0.692309\pi$
$443$	−23.9129	−1.13614	−0.568068	+	0.822981i	$0.692309\pi$
$444$	0	0
$445$	−41.2003	−1.95308
$446$	13.9320	0.659697
$447$	0	0
$448$	−7.10416	−0.335640
$449$	−23.4161	−1.10507	−0.552537	−	0.833488i	$-0.686340\pi$
$449$	−23.4161	−1.10507	−0.552537	+	0.833488i	$0.686340\pi$
$450$	0	0
$451$	29.9733	1.41139
$452$	5.77014	0.271404
$453$	0	0
$454$	−0.557115	−0.0261467
$455$	−11.7812	−0.552309
$456$	0	0
$457$	−29.9749	−1.40217	−0.701085	−	0.713078i	$-0.747300\pi$
$457$	−29.9749	−1.40217	−0.701085	+	0.713078i	$0.747300\pi$
$458$	4.45067	0.207966
$459$	0	0
$460$	20.9645	0.977474
$461$	−23.1934	−1.08022	−0.540112	−	0.841593i	$-0.681618\pi$
$461$	−23.1934	−1.08022	−0.540112	+	0.841593i	$0.681618\pi$
$462$	0	0
$463$	−7.69863	−0.357786	−0.178893	−	0.983869i	$-0.557252\pi$
$463$	−7.69863	−0.357786	−0.178893	+	0.983869i	$0.557252\pi$
$464$	−1.74761	−0.0811306
$465$	0	0
$466$	−1.86223	−0.0862661
$467$	−28.2214	−1.30593	−0.652966	−	0.757387i	$-0.726476\pi$
$467$	−28.2214	−1.30593	−0.652966	+	0.757387i	$0.726476\pi$
$468$	0	0
$469$	1.71562	0.0792198
$470$	7.01866	0.323747
$471$	0	0
$472$	27.5526	1.26821
$473$	9.23652	0.424695
$474$	0	0
$475$	13.9622	0.640630
$476$	4.25333	0.194951
$477$	0	0
$478$	9.91455	0.453481
$479$	16.1900	0.739738	0.369869	−	0.929084i	$-0.379402\pi$
$479$	16.1900	0.739738	0.369869	+	0.929084i	$0.379402\pi$
$480$	0	0
$481$	26.8545	1.22446
$482$	−25.9496	−1.18197
$483$	0	0
$484$	−0.389077	−0.0176853
$485$	15.8449	0.719481
$486$	0	0
$487$	31.9374	1.44722	0.723611	−	0.690208i	$-0.242481\pi$
$487$	31.9374	1.44722	0.723611	+	0.690208i	$0.242481\pi$
$488$	−16.4362	−0.744030
$489$	0	0
$490$	2.86415	0.129389
$491$	−43.0899	−1.94462	−0.972311	−	0.233692i	$-0.924919\pi$
$491$	−43.0899	−1.94462	−0.972311	+	0.233692i	$0.924919\pi$
$492$	0	0
$493$	7.11861	0.320606
$494$	18.9206	0.851277
$495$	0	0
$496$	2.92791	0.131467
$497$	−3.84688	−0.172556
$498$	0	0
$499$	9.26093	0.414576	0.207288	−	0.978280i	$-0.433536\pi$
$499$	9.26093	0.414576	0.207288	+	0.978280i	$0.433536\pi$
$500$	−5.37216	−0.240250
$501$	0	0
$502$	9.61890	0.429312
$503$	−22.1552	−0.987851	−0.493926	−	0.869504i	$-0.664439\pi$
$503$	−22.1552	−0.987851	−0.493926	+	0.869504i	$0.664439\pi$
$504$	0	0
$505$	−6.07555	−0.270359
$506$	25.4666	1.13213
$507$	0	0
$508$	0.984892	0.0436975
$509$	4.62088	0.204817	0.102408	−	0.994742i	$-0.467345\pi$
$509$	4.62088	0.204817	0.102408	+	0.994742i	$0.467345\pi$
$510$	0	0
$511$	9.49128	0.419869
$512$	11.6211	0.513587
$513$	0	0
$514$	24.4685	1.07926
$515$	47.6807	2.10106
$516$	0	0
$517$	−8.27213	−0.363808
$518$	−6.52865	−0.286853
$519$	0	0
$520$	35.4301	1.55371
$521$	6.50694	0.285074	0.142537	−	0.989789i	$-0.454474\pi$
$521$	6.50694	0.285074	0.142537	+	0.989789i	$0.454474\pi$
$522$	0	0
$523$	39.6269	1.73276	0.866381	−	0.499384i	$-0.166440\pi$
$523$	39.6269	1.73276	0.866381	+	0.499384i	$0.166440\pi$
$524$	12.1419	0.530422
$525$	0	0
$526$	−27.2448	−1.18793
$527$	−11.9264	−0.519522
$528$	0	0
$529$	33.0678	1.43773
$530$	−21.9743	−0.954501
$531$	0	0
$532$	4.46291	0.193492
$533$	36.7981	1.59390
$534$	0	0
$535$	24.0538	1.03994
$536$	−5.15947	−0.222855
$537$	0	0
$538$	−5.16109	−0.222510
$539$	−3.37565	−0.145400
$540$	0	0
$541$	−8.00310	−0.344080	−0.172040	−	0.985090i	$-0.555036\pi$
$541$	−8.00310	−0.344080	−0.172040	+	0.985090i	$0.555036\pi$
$542$	−9.90824	−0.425595
$543$	0	0
$544$	−21.3620	−0.915887
$545$	11.8557	0.507841
$546$	0	0
$547$	−10.3663	−0.443230	−0.221615	−	0.975134i	$-0.571133\pi$
$547$	−10.3663	−0.443230	−0.221615	+	0.975134i	$0.571133\pi$
$548$	−20.6332	−0.881406
$549$	0	0
$550$	10.4795	0.446846
$551$	7.46936	0.318206
$552$	0	0
$553$	12.9615	0.551179
$554$	−11.9407	−0.507310
$555$	0	0
$556$	5.70699	0.242030
$557$	13.4886	0.571530	0.285765	−	0.958300i	$-0.407752\pi$
$557$	13.4886	0.571530	0.285765	+	0.958300i	$0.407752\pi$
$558$	0	0
$559$	11.3396	0.479616
$560$	−3.01390	−0.127360
$561$	0	0
$562$	−8.01791	−0.338215
$563$	−5.34446	−0.225242	−0.112621	−	0.993638i	$-0.535925\pi$
$563$	−5.34446	−0.225242	−0.112621	+	0.993638i	$0.535925\pi$
$564$	0	0
$565$	16.6547	0.700668
$566$	−18.3481	−0.771229
$567$	0	0
$568$	11.5689	0.485422
$569$	−42.9363	−1.79999	−0.899993	−	0.435905i	$-0.856428\pi$
$569$	−42.9363	−1.79999	−0.899993	+	0.435905i	$0.856428\pi$
$570$	0	0
$571$	8.85945	0.370757	0.185378	−	0.982667i	$-0.440649\pi$
$571$	8.85945	0.370757	0.185378	+	0.982667i	$0.440649\pi$
$572$	−13.7783	−0.576100
$573$	0	0
$574$	−8.94607	−0.373402
$575$	23.0718	0.962161
$576$	0	0
$577$	−30.4615	−1.26813	−0.634065	−	0.773280i	$-0.718615\pi$
$577$	−30.4615	−1.26813	−0.634065	+	0.773280i	$0.718615\pi$
$578$	−1.66252	−0.0691519
$579$	0	0
$580$	4.61511	0.191632
$581$	16.6680	0.691507
$582$	0	0
$583$	25.8987	1.07261
$584$	−28.5436	−1.18114
$585$	0	0
$586$	−8.75504	−0.361667
$587$	−20.8026	−0.858614	−0.429307	−	0.903159i	$-0.641242\pi$
$587$	−20.8026	−0.858614	−0.429307	+	0.903159i	$0.641242\pi$
$588$	0	0
$589$	−12.5141	−0.515632
$590$	26.2406	1.08031
$591$	0	0
$592$	6.87000	0.282355
$593$	26.1742	1.07485	0.537423	−	0.843313i	$-0.319398\pi$
$593$	26.1742	1.07485	0.537423	+	0.843313i	$0.319398\pi$
$594$	0	0
$595$	12.2766	0.503293
$596$	−5.04168	−0.206515
$597$	0	0
$598$	31.2652	1.27853
$599$	17.6679	0.721889	0.360945	−	0.932587i	$-0.382454\pi$
$599$	17.6679	0.721889	0.360945	+	0.932587i	$0.382454\pi$
$600$	0	0
$601$	23.6438	0.964449	0.482225	−	0.876048i	$-0.339829\pi$
$601$	23.6438	0.964449	0.482225	+	0.876048i	$0.339829\pi$
$602$	−2.75681	−0.112359
$603$	0	0
$604$	−4.83261	−0.196636
$605$	−1.12302	−0.0456571
$606$	0	0
$607$	28.8121	1.16945	0.584724	−	0.811233i	$-0.301203\pi$
$607$	28.8121	1.16945	0.584724	+	0.811233i	$0.301203\pi$
$608$	−22.4145	−0.909029
$609$	0	0
$610$	−15.6535	−0.633790
$611$	−10.1557	−0.410854
$612$	0	0
$613$	9.68561	0.391198	0.195599	−	0.980684i	$-0.437335\pi$
$613$	9.68561	0.391198	0.195599	+	0.980684i	$0.437335\pi$
$614$	−16.0034	−0.645843
$615$	0	0
$616$	10.1518	0.409027
$617$	−11.9400	−0.480686	−0.240343	−	0.970688i	$-0.577260\pi$
$617$	−11.9400	−0.480686	−0.240343	+	0.970688i	$0.577260\pi$
$618$	0	0
$619$	48.9493	1.96744	0.983719	−	0.179713i	$-0.0575168\pi$
$619$	48.9493	1.96744	0.983719	+	0.179713i	$0.0575168\pi$
$620$	−7.73208	−0.310528
$621$	0	0
$622$	31.2714	1.25387
$623$	−14.4931	−0.580655
$624$	0	0
$625$	−30.9122	−1.23649
$626$	−1.04077	−0.0415977
$627$	0	0
$628$	−20.4430	−0.815766
$629$	−27.9839	−1.11579
$630$	0	0
$631$	−7.90610	−0.314737	−0.157368	−	0.987540i	$-0.550301\pi$
$631$	−7.90610	−0.314737	−0.157368	+	0.987540i	$0.550301\pi$
$632$	−38.9798	−1.55053
$633$	0	0
$634$	18.4367	0.732217
$635$	2.84275	0.112811
$636$	0	0
$637$	−4.14428	−0.164202
$638$	5.60620	0.221952
$639$	0	0
$640$	−7.77615	−0.307379
$641$	35.5066	1.40242	0.701212	−	0.712953i	$-0.252643\pi$
$641$	35.5066	1.40242	0.701212	+	0.712953i	$0.252643\pi$
$642$	0	0
$643$	−21.5269	−0.848938	−0.424469	−	0.905443i	$-0.639539\pi$
$643$	−21.5269	−0.848938	−0.424469	+	0.905443i	$0.639539\pi$
$644$	7.37472	0.290605
$645$	0	0
$646$	−19.7163	−0.775728
$647$	33.1235	1.30222	0.651110	−	0.758983i	$-0.274304\pi$
$647$	33.1235	1.30222	0.651110	+	0.758983i	$0.274304\pi$
$648$	0	0
$649$	−30.9269	−1.21399
$650$	12.8656	0.504631
$651$	0	0
$652$	6.80162	0.266372
$653$	14.1122	0.552253	0.276127	−	0.961121i	$-0.410949\pi$
$653$	14.1122	0.552253	0.276127	+	0.961121i	$0.410949\pi$
$654$	0	0
$655$	35.0459	1.36936
$656$	9.41381	0.367548
$657$	0	0
$658$	2.46897	0.0962504
$659$	35.1516	1.36931	0.684655	−	0.728867i	$-0.259953\pi$
$659$	35.1516	1.36931	0.684655	+	0.728867i	$0.259953\pi$
$660$	0	0
$661$	9.54497	0.371256	0.185628	−	0.982620i	$-0.440568\pi$
$661$	9.54497	0.371256	0.185628	+	0.982620i	$0.440568\pi$
$662$	11.0226	0.428406
$663$	0	0
$664$	−50.1267	−1.94529
$665$	12.8815	0.499525
$666$	0	0
$667$	12.3427	0.477912
$668$	0.422667	0.0163535
$669$	0	0
$670$	−4.91378	−0.189836
$671$	18.4490	0.712217
$672$	0	0
$673$	−6.47805	−0.249711	−0.124855	−	0.992175i	$-0.539847\pi$
$673$	−6.47805	−0.249711	−0.124855	+	0.992175i	$0.539847\pi$
$674$	8.45752	0.325771
$675$	0	0
$676$	−4.11197	−0.158153
$677$	14.2716	0.548503	0.274251	−	0.961658i	$-0.411570\pi$
$677$	14.2716	0.548503	0.274251	+	0.961658i	$0.411570\pi$
$678$	0	0
$679$	5.57380	0.213903
$680$	−36.9202	−1.41583
$681$	0	0
$682$	−9.39254	−0.359659
$683$	37.4549	1.43317	0.716586	−	0.697498i	$-0.245704\pi$
$683$	37.4549	1.43317	0.716586	+	0.697498i	$0.245704\pi$
$684$	0	0
$685$	−59.5548	−2.27547
$686$	1.00753	0.0384675
$687$	0	0
$688$	2.90095	0.110598
$689$	31.7957	1.21132
$690$	0	0
$691$	−21.7030	−0.825621	−0.412810	−	0.910817i	$-0.635453\pi$
$691$	−21.7030	−0.825621	−0.412810	+	0.910817i	$0.635453\pi$
$692$	−18.9214	−0.719285
$693$	0	0
$694$	9.83071	0.373169
$695$	16.4724	0.624835
$696$	0	0
$697$	−38.3457	−1.45245
$698$	32.4808	1.22942
$699$	0	0
$700$	3.03469	0.114700
$701$	−21.0817	−0.796245	−0.398122	−	0.917332i	$-0.630338\pi$
$701$	−21.0817	−0.796245	−0.398122	+	0.917332i	$0.630338\pi$
$702$	0	0
$703$	−29.3627	−1.10744
$704$	−23.9812	−0.903825
$705$	0	0
$706$	−5.59922	−0.210730
$707$	−2.13721	−0.0803780
$708$	0	0
$709$	24.3625	0.914954	0.457477	−	0.889222i	$-0.348753\pi$
$709$	24.3625	0.914954	0.457477	+	0.889222i	$0.348753\pi$
$710$	11.0180	0.413500
$711$	0	0
$712$	43.5860	1.63345
$713$	−20.6788	−0.774427
$714$	0	0
$715$	−39.7691	−1.48728
$716$	10.2174	0.381841
$717$	0	0
$718$	−20.7660	−0.774979
$719$	29.8109	1.11176	0.555880	−	0.831263i	$-0.312382\pi$
$719$	29.8109	1.11176	0.555880	+	0.831263i	$0.312382\pi$
$720$	0	0
$721$	16.7727	0.624649
$722$	−1.54482	−0.0574922
$723$	0	0
$724$	0.387663	0.0144074
$725$	5.07901	0.188630
$726$	0	0
$727$	8.46222	0.313846	0.156923	−	0.987611i	$-0.449842\pi$
$727$	8.46222	0.313846	0.156923	+	0.987611i	$0.449842\pi$
$728$	12.4633	0.461921
$729$	0	0
$730$	−27.1844	−1.00614
$731$	−11.8166	−0.437051
$732$	0	0
$733$	−5.76449	−0.212916	−0.106458	−	0.994317i	$-0.533951\pi$
$733$	−5.76449	−0.212916	−0.106458	+	0.994317i	$0.533951\pi$
$734$	−0.390907	−0.0144287
$735$	0	0
$736$	−37.0388	−1.36527
$737$	5.79133	0.213326
$738$	0	0
$739$	−33.2482	−1.22305	−0.611527	−	0.791223i	$-0.709445\pi$
$739$	−33.2482	−1.22305	−0.611527	+	0.791223i	$0.709445\pi$
$740$	−18.1424	−0.666928
$741$	0	0
$742$	−7.72993	−0.283775
$743$	49.9056	1.83086	0.915429	−	0.402480i	$-0.131852\pi$
$743$	49.9056	1.83086	0.915429	+	0.402480i	$0.131852\pi$
$744$	0	0
$745$	−14.5521	−0.533148
$746$	−7.60426	−0.278412
$747$	0	0
$748$	14.3578	0.524972
$749$	8.46145	0.309175
$750$	0	0
$751$	−26.1422	−0.953942	−0.476971	−	0.878919i	$-0.658265\pi$
$751$	−26.1422	−0.953942	−0.476971	+	0.878919i	$0.658265\pi$
$752$	−2.59806	−0.0947414
$753$	0	0
$754$	6.88271	0.250654
$755$	−13.9486	−0.507643
$756$	0	0
$757$	−39.4488	−1.43379	−0.716896	−	0.697181i	$-0.754438\pi$
$757$	−39.4488	−1.43379	−0.716896	+	0.697181i	$0.754438\pi$
$758$	−6.12043	−0.222304
$759$	0	0
$760$	−38.7394	−1.40523
$761$	48.1065	1.74386	0.871930	−	0.489631i	$-0.162868\pi$
$761$	48.1065	1.74386	0.871930	+	0.489631i	$0.162868\pi$
$762$	0	0
$763$	4.17049	0.150982
$764$	6.77941	0.245271
$765$	0	0
$766$	−12.9310	−0.467217
$767$	−37.9688	−1.37097
$768$	0	0
$769$	29.9286	1.07925	0.539627	−	0.841904i	$-0.318565\pi$
$769$	29.9286	1.07925	0.539627	+	0.841904i	$0.318565\pi$
$770$	9.66837	0.348424
$771$	0	0
$772$	12.7240	0.457946
$773$	1.58453	0.0569918	0.0284959	−	0.999594i	$-0.490928\pi$
$773$	1.58453	0.0569918	0.0284959	+	0.999594i	$0.490928\pi$
$774$	0	0
$775$	−8.50930	−0.305663
$776$	−16.7624	−0.601735
$777$	0	0
$778$	9.92669	0.355889
$779$	−40.2351	−1.44157
$780$	0	0
$781$	−12.9858	−0.464667
$782$	−32.5802	−1.16507
$783$	0	0
$784$	−1.06020	−0.0378644
$785$	−59.0059	−2.10601
$786$	0	0
$787$	33.5149	1.19468	0.597339	−	0.801989i	$-0.296225\pi$
$787$	33.5149	1.19468	0.597339	+	0.801989i	$0.296225\pi$
$788$	−15.8200	−0.563563
$789$	0	0
$790$	−37.1236	−1.32080
$791$	5.85865	0.208310
$792$	0	0
$793$	22.6498	0.804318
$794$	−22.3046	−0.791561
$795$	0	0
$796$	−19.2430	−0.682050
$797$	49.4146	1.75035	0.875177	−	0.483803i	$-0.160745\pi$
$797$	49.4146	1.75035	0.875177	+	0.483803i	$0.160745\pi$
$798$	0	0
$799$	10.5828	0.374392
$800$	−15.2414	−0.538866
$801$	0	0
$802$	−30.8384	−1.08894
$803$	32.0393	1.13064
$804$	0	0
$805$	21.2861	0.750236
$806$	−11.5312	−0.406169
$807$	0	0
$808$	6.42735	0.226113
$809$	7.97912	0.280531	0.140265	−	0.990114i	$-0.455204\pi$
$809$	7.97912	0.280531	0.140265	+	0.990114i	$0.455204\pi$
$810$	0	0
$811$	44.4634	1.56132	0.780661	−	0.624954i	$-0.214882\pi$
$811$	44.4634	1.56132	0.780661	+	0.624954i	$0.214882\pi$
$812$	1.62347	0.0569725
$813$	0	0
$814$	−22.0385	−0.772448
$815$	19.6319	0.687676
$816$	0	0
$817$	−12.3988	−0.433779
$818$	10.8712	0.380101
$819$	0	0
$820$	−24.8601	−0.868154
$821$	35.9378	1.25424	0.627118	−	0.778924i	$-0.284234\pi$
$821$	35.9378	1.25424	0.627118	+	0.778924i	$0.284234\pi$
$822$	0	0
$823$	43.8476	1.52843	0.764215	−	0.644961i	$-0.223126\pi$
$823$	43.8476	1.52843	0.764215	+	0.644961i	$0.223126\pi$
$824$	−50.4415	−1.75721
$825$	0	0
$826$	9.23069	0.321177
$827$	−26.4421	−0.919482	−0.459741	−	0.888053i	$-0.652058\pi$
$827$	−26.4421	−0.919482	−0.459741	+	0.888053i	$0.652058\pi$
$828$	0	0
$829$	9.14868	0.317747	0.158873	−	0.987299i	$-0.449214\pi$
$829$	9.14868	0.317747	0.158873	+	0.987299i	$0.449214\pi$
$830$	−47.7397	−1.65707
$831$	0	0
$832$	−29.4416	−1.02070
$833$	4.31858	0.149630
$834$	0	0
$835$	1.21997	0.0422187
$836$	15.0652	0.521042
$837$	0	0
$838$	40.7785	1.40867
$839$	35.5509	1.22735	0.613676	−	0.789558i	$-0.289690\pi$
$839$	35.5509	1.22735	0.613676	+	0.789558i	$0.289690\pi$
$840$	0	0
$841$	−26.2829	−0.906306
$842$	−9.14247	−0.315070
$843$	0	0
$844$	−5.64133	−0.194183
$845$	−11.8686	−0.408293
$846$	0	0
$847$	−0.395045	−0.0135739
$848$	8.13409	0.279326
$849$	0	0
$850$	−13.4067	−0.459846
$851$	−48.5204	−1.66326
$852$	0	0
$853$	6.98243	0.239074	0.119537	−	0.992830i	$-0.461859\pi$
$853$	6.98243	0.239074	0.119537	+	0.992830i	$0.461859\pi$
$854$	−5.50645	−0.188427
$855$	0	0
$856$	−25.4466	−0.869746
$857$	−7.96079	−0.271935	−0.135968	−	0.990713i	$-0.543414\pi$
$857$	−7.96079	−0.271935	−0.135968	+	0.990713i	$0.543414\pi$
$858$	0	0
$859$	26.7825	0.913806	0.456903	−	0.889517i	$-0.348959\pi$
$859$	26.7825	0.913806	0.456903	+	0.889517i	$0.348959\pi$
$860$	−7.66086	−0.261233
$861$	0	0
$862$	−30.3821	−1.03482
$863$	36.7172	1.24987	0.624933	−	0.780678i	$-0.285126\pi$
$863$	36.7172	1.24987	0.624933	+	0.780678i	$0.285126\pi$
$864$	0	0
$865$	−54.6140	−1.85693
$866$	−30.3065	−1.02986
$867$	0	0
$868$	−2.71993	−0.0923204
$869$	43.7535	1.48424
$870$	0	0
$871$	7.11000	0.240913
$872$	−12.5422	−0.424731
$873$	0	0
$874$	−34.1855	−1.15634
$875$	−5.45457	−0.184398
$876$	0	0
$877$	−35.5727	−1.20120	−0.600602	−	0.799548i	$-0.705072\pi$
$877$	−35.5727	−1.20120	−0.600602	+	0.799548i	$0.705072\pi$
$878$	−15.9859	−0.539499
$879$	0	0
$880$	−10.1739	−0.342961
$881$	27.9292	0.940958	0.470479	−	0.882411i	$-0.344081\pi$
$881$	27.9292	0.940958	0.470479	+	0.882411i	$0.344081\pi$
$882$	0	0
$883$	−44.5209	−1.49825	−0.749124	−	0.662429i	$-0.769525\pi$
$883$	−44.5209	−1.49825	−0.749124	+	0.662429i	$0.769525\pi$
$884$	17.6270	0.592860
$885$	0	0
$886$	24.0929	0.809416
$887$	5.99980	0.201454	0.100727	−	0.994914i	$-0.467883\pi$
$887$	5.99980	0.201454	0.100727	+	0.994914i	$0.467883\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	41.5104	1.39143
$891$	0	0
$892$	13.6190	0.455997
$893$	11.1042	0.371589
$894$	0	0
$895$	29.4910	0.985774
$896$	−2.73543	−0.0913843
$897$	0	0
$898$	23.5923	0.787286
$899$	−4.55222	−0.151825
$900$	0	0
$901$	−33.1330	−1.10382
$902$	−30.1988	−1.00551
$903$	0	0
$904$	−17.6190	−0.586001
$905$	1.11894	0.0371947
$906$	0	0
$907$	−15.6141	−0.518459	−0.259229	−	0.965816i	$-0.583469\pi$
$907$	−15.6141	−0.518459	−0.259229	+	0.965816i	$0.583469\pi$
$908$	−0.544599	−0.0180732
$909$	0	0
$910$	11.8698	0.393481
$911$	−15.3713	−0.509273	−0.254636	−	0.967037i	$-0.581956\pi$
$911$	−15.3713	−0.509273	−0.254636	+	0.967037i	$0.581956\pi$
$912$	0	0
$913$	56.2656	1.86212
$914$	30.2005	0.998945
$915$	0	0
$916$	4.35069	0.143751
$917$	12.3282	0.407112
$918$	0	0
$919$	23.6950	0.781625	0.390812	−	0.920470i	$-0.372194\pi$
$919$	23.6950	0.781625	0.390812	+	0.920470i	$0.372194\pi$
$920$	−64.0148	−2.11051
$921$	0	0
$922$	23.3680	0.769583
$923$	−15.9426	−0.524756
$924$	0	0
$925$	−19.9661	−0.656480
$926$	7.75657	0.254897
$927$	0	0
$928$	−8.15370	−0.267658
$929$	20.0879	0.659061	0.329531	−	0.944145i	$-0.393109\pi$
$929$	20.0879	0.659061	0.329531	+	0.944145i	$0.393109\pi$
$930$	0	0
$931$	4.53137	0.148510
$932$	−1.82040	−0.0596290
$933$	0	0
$934$	28.4338	0.930383
$935$	41.4417	1.35529
$936$	0	0
$937$	48.1128	1.57178	0.785889	−	0.618368i	$-0.212206\pi$
$937$	48.1128	1.57178	0.785889	+	0.618368i	$0.212206\pi$
$938$	−1.72853	−0.0564385
$939$	0	0
$940$	6.86099	0.223781
$941$	−35.2550	−1.14928	−0.574641	−	0.818406i	$-0.694858\pi$
$941$	−35.2550	−1.14928	−0.574641	+	0.818406i	$0.694858\pi$
$942$	0	0
$943$	−66.4864	−2.16509
$944$	−9.71331	−0.316141
$945$	0	0
$946$	−9.30603	−0.302565
$947$	35.5382	1.15484	0.577419	−	0.816448i	$-0.304060\pi$
$947$	35.5382	1.15484	0.577419	+	0.816448i	$0.304060\pi$
$948$	0	0
$949$	39.3345	1.27685
$950$	−14.0673	−0.456403
$951$	0	0
$952$	−12.9875	−0.420927
$953$	14.9360	0.483824	0.241912	−	0.970298i	$-0.422226\pi$
$953$	14.9360	0.483824	0.241912	+	0.970298i	$0.422226\pi$
$954$	0	0
$955$	19.5678	0.633200
$956$	9.69183	0.313456
$957$	0	0
$958$	−16.3118	−0.527011
$959$	−20.9497	−0.676501
$960$	0	0
$961$	−23.3733	−0.753977
$962$	−27.0566	−0.872339
$963$	0	0
$964$	−25.3666	−0.817004
$965$	36.7259	1.18225
$966$	0	0
$967$	18.2933	0.588274	0.294137	−	0.955763i	$-0.404968\pi$
$967$	18.2933	0.588274	0.294137	+	0.955763i	$0.404968\pi$
$968$	1.18804	0.0381851
$969$	0	0
$970$	−15.9642	−0.512579
$971$	−53.6940	−1.72312	−0.861562	−	0.507652i	$-0.830513\pi$
$971$	−53.6940	−1.72312	−0.861562	+	0.507652i	$0.830513\pi$
$972$	0	0
$973$	5.79454	0.185764
$974$	−32.1778	−1.03104
$975$	0	0
$976$	5.79435	0.185473
$977$	−25.7952	−0.825263	−0.412631	−	0.910898i	$-0.635390\pi$
$977$	−25.7952	−0.825263	−0.412631	+	0.910898i	$0.635390\pi$
$978$	0	0
$979$	−48.9238	−1.56361
$980$	2.79980	0.0894364
$981$	0	0
$982$	43.4142	1.38540
$983$	−27.5632	−0.879131	−0.439565	−	0.898211i	$-0.644867\pi$
$983$	−27.5632	−0.879131	−0.439565	+	0.898211i	$0.644867\pi$
$984$	0	0
$985$	−45.6621	−1.45492
$986$	−7.17218	−0.228409
$987$	0	0
$988$	18.4955	0.588421
$989$	−20.4883	−0.651491
$990$	0	0
$991$	−6.61637	−0.210176	−0.105088	−	0.994463i	$-0.533512\pi$
$991$	−6.61637	−0.210176	−0.105088	+	0.994463i	$0.533512\pi$
$992$	13.6606	0.433724
$993$	0	0
$994$	3.87584	0.122934
$995$	−55.5422	−1.76081
$996$	0	0
$997$	−17.8072	−0.563959	−0.281979	−	0.959420i	$-0.590991\pi$
$997$	−17.8072	−0.563959	−0.281979	+	0.959420i	$0.590991\pi$
$998$	−9.33062	−0.295356
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.p.1.4		14
3.2	odd	2		2667.2.a.m.1.11	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.m.1.11	✓	14	3.2	odd	2
8001.2.a.p.1.4		14	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8001.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00753 q^{2} -0.984892 q^{4} -2.84275 q^{5} -1.00000 q^{7} +3.00736 q^{8} +O(q^{10})\)	\(q-1.00753 q^{2} -0.984892 q^{4} -2.84275 q^{5} -1.00000 q^{7} +3.00736 q^{8} +2.86415 q^{10} -3.37565 q^{11} -4.14428 q^{13} +1.00753 q^{14} -1.06020 q^{16} +4.31858 q^{17} +4.53137 q^{19} +2.79980 q^{20} +3.40106 q^{22} +7.48784 q^{23} +3.08124 q^{25} +4.17547 q^{26} +0.984892 q^{28} +1.64837 q^{29} -2.76165 q^{31} -4.94653 q^{32} -4.35108 q^{34} +2.84275 q^{35} -6.47989 q^{37} -4.56547 q^{38} -8.54916 q^{40} -8.87924 q^{41} -2.73621 q^{43} +3.32466 q^{44} -7.54419 q^{46} +2.45053 q^{47} +1.00000 q^{49} -3.10443 q^{50} +4.08167 q^{52} -7.67219 q^{53} +9.59615 q^{55} -3.00736 q^{56} -1.66077 q^{58} +9.16174 q^{59} -5.46532 q^{61} +2.78243 q^{62} +7.10416 q^{64} +11.7812 q^{65} -1.71562 q^{67} -4.25333 q^{68} -2.86415 q^{70} +3.84688 q^{71} -9.49128 q^{73} +6.52865 q^{74} -4.46291 q^{76} +3.37565 q^{77} -12.9615 q^{79} +3.01390 q^{80} +8.94607 q^{82} -16.6680 q^{83} -12.2766 q^{85} +2.75681 q^{86} -10.1518 q^{88} +14.4931 q^{89} +4.14428 q^{91} -7.37472 q^{92} -2.46897 q^{94} -12.8815 q^{95} -5.57380 q^{97} -1.00753 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 5 q^{2} + 15 q^{4} + 4 q^{5} - 14 q^{7} + 12 q^{8}+O(q^{10}) \) 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

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Database

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