LMFDB - Embedded newform 8001.2.a.p.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("8001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$63.8883066572$
Analytic rank:	$0$
Dimension:	$14$
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			1.1
Root			$-2.33388$ of defining polynomial
Character	$\chi$	$=$	8001.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.33388 q^{2} +3.44698 q^{4} +0.335557 q^{5} -1.00000 q^{7} -3.37706 q^{8} +O(q^{10})$	\(q-2.33388 q^{2} +3.44698 q^{4} +0.335557 q^{5} -1.00000 q^{7} -3.37706 q^{8} -0.783148 q^{10} +5.16521 q^{11} -5.99956 q^{13} +2.33388 q^{14} +0.987696 q^{16} -7.35302 q^{17} +5.24027 q^{19} +1.15666 q^{20} -12.0550 q^{22} -1.66716 q^{23} -4.88740 q^{25} +14.0022 q^{26} -3.44698 q^{28} -1.61282 q^{29} -6.38054 q^{31} +4.44897 q^{32} +17.1610 q^{34} -0.335557 q^{35} -11.1071 q^{37} -12.2302 q^{38} -1.13320 q^{40} +3.12952 q^{41} -8.78359 q^{43} +17.8044 q^{44} +3.89095 q^{46} +11.1208 q^{47} +1.00000 q^{49} +11.4066 q^{50} -20.6803 q^{52} +4.86128 q^{53} +1.73322 q^{55} +3.37706 q^{56} +3.76411 q^{58} -14.4216 q^{59} -9.80157 q^{61} +14.8914 q^{62} -12.3587 q^{64} -2.01319 q^{65} -4.47288 q^{67} -25.3457 q^{68} +0.783148 q^{70} +0.675275 q^{71} +13.2597 q^{73} +25.9226 q^{74} +18.0631 q^{76} -5.16521 q^{77} +10.6397 q^{79} +0.331428 q^{80} -7.30391 q^{82} -10.1444 q^{83} -2.46736 q^{85} +20.4998 q^{86} -17.4432 q^{88} +15.6055 q^{89} +5.99956 q^{91} -5.74667 q^{92} -25.9547 q^{94} +1.75841 q^{95} +13.0787 q^{97} -2.33388 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.33388	−1.65030	−0.825150	−	0.564914i	$-0.808909\pi$
$2$	−2.33388	−1.65030	−0.825150	+	0.564914i	$0.808909\pi$
$3$	0	0
$3$	0	0
$4$	3.44698	1.72349
$5$	0.335557	0.150066	0.0750328	−	0.997181i	$-0.476094\pi$
$5$	0.335557	0.150066	0.0750328	+	0.997181i	$0.476094\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−3.37706	−1.19397
$9$	0	0
$10$	−0.783148	−0.247653
$11$	5.16521	1.55737	0.778684	−	0.627416i	$-0.215887\pi$
$11$	5.16521	1.55737	0.778684	+	0.627416i	$0.215887\pi$
$12$	0	0
$13$	−5.99956	−1.66398	−0.831989	−	0.554793i	$-0.812798\pi$
$13$	−5.99956	−1.66398	−0.831989	+	0.554793i	$0.812798\pi$
$14$	2.33388	0.623755
$15$	0	0
$16$	0.987696	0.246924
$17$	−7.35302	−1.78337	−0.891685	−	0.452656i	$-0.850476\pi$
$17$	−7.35302	−1.78337	−0.891685	+	0.452656i	$0.850476\pi$
$18$	0	0
$19$	5.24027	1.20220	0.601101	−	0.799173i	$-0.294729\pi$
$19$	5.24027	1.20220	0.601101	+	0.799173i	$0.294729\pi$
$20$	1.15666	0.258636
$21$	0	0
$22$	−12.0550	−2.57013
$23$	−1.66716	−0.347627	−0.173814	−	0.984779i	$-0.555609\pi$
$23$	−1.66716	−0.347627	−0.173814	+	0.984779i	$0.555609\pi$
$24$	0	0
$25$	−4.88740	−0.977480
$26$	14.0022	2.74606
$27$	0	0
$28$	−3.44698	−0.651417
$29$	−1.61282	−0.299492	−0.149746	−	0.988724i	$-0.547846\pi$
$29$	−1.61282	−0.299492	−0.149746	+	0.988724i	$0.547846\pi$
$30$	0	0
$31$	−6.38054	−1.14598	−0.572989	−	0.819563i	$-0.694216\pi$
$31$	−6.38054	−1.14598	−0.572989	+	0.819563i	$0.694216\pi$
$32$	4.44897	0.786474
$33$	0	0
$34$	17.1610	2.94309
$35$	−0.335557	−0.0567195
$36$	0	0
$37$	−11.1071	−1.82600	−0.913000	−	0.407960i	$-0.866240\pi$
$37$	−11.1071	−1.82600	−0.913000	+	0.407960i	$0.866240\pi$
$38$	−12.2302	−1.98399
$39$	0	0
$40$	−1.13320	−0.179174
$41$	3.12952	0.488749	0.244374	−	0.969681i	$-0.421417\pi$
$41$	3.12952	0.488749	0.244374	+	0.969681i	$0.421417\pi$
$42$	0	0
$43$	−8.78359	−1.33948	−0.669742	−	0.742594i	$-0.733595\pi$
$43$	−8.78359	−1.33948	−0.669742	+	0.742594i	$0.733595\pi$
$44$	17.8044	2.68411
$45$	0	0
$46$	3.89095	0.573689
$47$	11.1208	1.62214	0.811071	−	0.584948i	$-0.198885\pi$
$47$	11.1208	1.62214	0.811071	+	0.584948i	$0.198885\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	11.4066	1.61314
$51$	0	0
$52$	−20.6803	−2.86785
$53$	4.86128	0.667748	0.333874	−	0.942618i	$-0.391644\pi$
$53$	4.86128	0.667748	0.333874	+	0.942618i	$0.391644\pi$
$54$	0	0
$55$	1.73322	0.233708
$56$	3.37706	0.451279
$57$	0	0
$58$	3.76411	0.494252
$59$	−14.4216	−1.87754	−0.938768	−	0.344551i	$-0.888031\pi$
$59$	−14.4216	−1.87754	−0.938768	+	0.344551i	$0.888031\pi$
$60$	0	0
$61$	−9.80157	−1.25496	−0.627481	−	0.778632i	$-0.715914\pi$
$61$	−9.80157	−1.25496	−0.627481	+	0.778632i	$0.715914\pi$
$62$	14.8914	1.89121
$63$	0	0
$64$	−12.3587	−1.54484
$65$	−2.01319	−0.249706
$66$	0	0
$67$	−4.47288	−0.546450	−0.273225	−	0.961950i	$-0.588090\pi$
$67$	−4.47288	−0.546450	−0.273225	+	0.961950i	$0.588090\pi$
$68$	−25.3457	−3.07362
$69$	0	0
$70$	0.783148	0.0936041
$71$	0.675275	0.0801404	0.0400702	−	0.999197i	$-0.487242\pi$
$71$	0.675275	0.0801404	0.0400702	+	0.999197i	$0.487242\pi$
$72$	0	0
$73$	13.2597	1.55193	0.775966	−	0.630775i	$-0.217263\pi$
$73$	13.2597	1.55193	0.775966	+	0.630775i	$0.217263\pi$
$74$	25.9226	3.01345
$75$	0	0
$76$	18.0631	2.07198
$77$	−5.16521	−0.588630
$78$	0	0
$79$	10.6397	1.19707	0.598533	−	0.801098i	$-0.295751\pi$
$79$	10.6397	1.19707	0.598533	+	0.801098i	$0.295751\pi$
$80$	0.331428	0.0370548
$81$	0	0
$82$	−7.30391	−0.806582
$83$	−10.1444	−1.11349	−0.556746	−	0.830682i	$-0.687950\pi$
$83$	−10.1444	−1.11349	−0.556746	+	0.830682i	$0.687950\pi$
$84$	0	0
$85$	−2.46736	−0.267623
$86$	20.4998	2.21055
$87$	0	0
$88$	−17.4432	−1.85946
$89$	15.6055	1.65417	0.827087	−	0.562073i	$-0.189996\pi$
$89$	15.6055	1.65417	0.827087	+	0.562073i	$0.189996\pi$
$90$	0	0
$91$	5.99956	0.628924
$92$	−5.74667	−0.599132
$93$	0	0
$94$	−25.9547	−2.67702
$95$	1.75841	0.180409
$96$	0	0
$97$	13.0787	1.32794	0.663968	−	0.747761i	$-0.268871\pi$
$97$	13.0787	1.32794	0.663968	+	0.747761i	$0.268871\pi$
$98$	−2.33388	−0.235757
$99$	0	0
$100$	−16.8468	−1.68468
$101$	10.5362	1.04839	0.524195	−	0.851598i	$-0.324366\pi$
$101$	10.5362	1.04839	0.524195	+	0.851598i	$0.324366\pi$
$102$	0	0
$103$	5.13829	0.506291	0.253146	−	0.967428i	$-0.418535\pi$
$103$	5.13829	0.506291	0.253146	+	0.967428i	$0.418535\pi$
$104$	20.2609	1.98674
$105$	0	0
$106$	−11.3456	−1.10198
$107$	−5.60390	−0.541749	−0.270875	−	0.962615i	$-0.587313\pi$
$107$	−5.60390	−0.541749	−0.270875	+	0.962615i	$0.587313\pi$
$108$	0	0
$109$	−2.01660	−0.193155	−0.0965776	−	0.995325i	$-0.530790\pi$
$109$	−2.01660	−0.193155	−0.0965776	+	0.995325i	$0.530790\pi$
$110$	−4.04512	−0.385687
$111$	0	0
$112$	−0.987696	−0.0933285
$113$	7.50807	0.706300	0.353150	−	0.935567i	$-0.385111\pi$
$113$	7.50807	0.706300	0.353150	+	0.935567i	$0.385111\pi$
$114$	0	0
$115$	−0.559428	−0.0521669
$116$	−5.55934	−0.516172
$117$	0	0
$118$	33.6583	3.09850
$119$	7.35302	0.674051
$120$	0	0
$121$	15.6794	1.42540
$122$	22.8756	2.07106
$123$	0	0
$124$	−21.9936	−1.97508
$125$	−3.31779	−0.296752
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	19.9458	1.76298
$129$	0	0
$130$	4.69854	0.412089
$131$	−13.0995	−1.14451	−0.572256	−	0.820075i	$-0.693932\pi$
$131$	−13.0995	−1.14451	−0.572256	+	0.820075i	$0.693932\pi$
$132$	0	0
$133$	−5.24027	−0.454389
$134$	10.4392	0.901805
$135$	0	0
$136$	24.8316	2.12930
$137$	−0.780358	−0.0666705	−0.0333353	−	0.999444i	$-0.510613\pi$
$137$	−0.780358	−0.0666705	−0.0333353	+	0.999444i	$0.510613\pi$
$138$	0	0
$139$	13.0874	1.11006	0.555031	−	0.831830i	$-0.312707\pi$
$139$	13.0874	1.11006	0.555031	+	0.831830i	$0.312707\pi$
$140$	−1.15666	−0.0977554
$141$	0	0
$142$	−1.57601	−0.132256
$143$	−30.9890	−2.59143
$144$	0	0
$145$	−0.541192	−0.0449435
$146$	−30.9465	−2.56115
$147$	0	0
$148$	−38.2860	−3.14709
$149$	−3.99867	−0.327584	−0.163792	−	0.986495i	$-0.552373\pi$
$149$	−3.99867	−0.327584	−0.163792	+	0.986495i	$0.552373\pi$
$150$	0	0
$151$	9.14695	0.744368	0.372184	−	0.928159i	$-0.378609\pi$
$151$	9.14695	0.744368	0.372184	+	0.928159i	$0.378609\pi$
$152$	−17.6967	−1.43540
$153$	0	0
$154$	12.0550	0.971416
$155$	−2.14103	−0.171972
$156$	0	0
$157$	9.05863	0.722958	0.361479	−	0.932380i	$-0.382272\pi$
$157$	9.05863	0.722958	0.361479	+	0.932380i	$0.382272\pi$
$158$	−24.8319	−1.97552
$159$	0	0
$160$	1.49288	0.118023
$161$	1.66716	0.131391
$162$	0	0
$163$	−17.2229	−1.34900	−0.674502	−	0.738273i	$-0.735642\pi$
$163$	−17.2229	−1.34900	−0.674502	+	0.738273i	$0.735642\pi$
$164$	10.7874	0.842353
$165$	0	0
$166$	23.6758	1.83760
$167$	5.64920	0.437148	0.218574	−	0.975820i	$-0.429860\pi$
$167$	5.64920	0.437148	0.218574	+	0.975820i	$0.429860\pi$
$168$	0	0
$169$	22.9947	1.76882
$170$	5.75851	0.441657
$171$	0	0
$172$	−30.2768	−2.30859
$173$	16.2539	1.23576	0.617882	−	0.786271i	$-0.287991\pi$
$173$	16.2539	1.23576	0.617882	+	0.786271i	$0.287991\pi$
$174$	0	0
$175$	4.88740	0.369453
$176$	5.10165	0.384552
$177$	0	0
$178$	−36.4212	−2.72988
$179$	−1.52687	−0.114124	−0.0570618	−	0.998371i	$-0.518173\pi$
$179$	−1.52687	−0.114124	−0.0570618	+	0.998371i	$0.518173\pi$
$180$	0	0
$181$	5.39218	0.400798	0.200399	−	0.979714i	$-0.435776\pi$
$181$	5.39218	0.400798	0.200399	+	0.979714i	$0.435776\pi$
$182$	−14.0022	−1.03791
$183$	0	0
$184$	5.63011	0.415057
$185$	−3.72707	−0.274020
$186$	0	0
$187$	−37.9799	−2.77737
$188$	38.3333	2.79574
$189$	0	0
$190$	−4.10391	−0.297729
$191$	11.7059	0.847007	0.423503	−	0.905894i	$-0.360800\pi$
$191$	11.7059	0.847007	0.423503	+	0.905894i	$0.360800\pi$
$192$	0	0
$193$	7.31373	0.526454	0.263227	−	0.964734i	$-0.415213\pi$
$193$	7.31373	0.526454	0.263227	+	0.964734i	$0.415213\pi$
$194$	−30.5240	−2.19149
$195$	0	0
$196$	3.44698	0.246213
$197$	18.3378	1.30652	0.653259	−	0.757135i	$-0.273401\pi$
$197$	18.3378	1.30652	0.653259	+	0.757135i	$0.273401\pi$
$198$	0	0
$199$	11.5347	0.817672	0.408836	−	0.912608i	$-0.365935\pi$
$199$	11.5347	0.817672	0.408836	+	0.912608i	$0.365935\pi$
$200$	16.5051	1.16708
$201$	0	0
$202$	−24.5902	−1.73016
$203$	1.61282	0.113197
$204$	0	0
$205$	1.05013	0.0733444
$206$	−11.9921	−0.835532
$207$	0	0
$208$	−5.92573	−0.410876
$209$	27.0671	1.87227
$210$	0	0
$211$	19.0652	1.31250	0.656250	−	0.754544i	$-0.272142\pi$
$211$	19.0652	1.31250	0.656250	+	0.754544i	$0.272142\pi$
$212$	16.7567	1.15086
$213$	0	0
$214$	13.0788	0.894049
$215$	−2.94739	−0.201011
$216$	0	0
$217$	6.38054	0.433139
$218$	4.70649	0.318764
$219$	0	0
$220$	5.97437	0.402792
$221$	44.1149	2.96749
$222$	0	0
$223$	−12.4055	−0.830733	−0.415367	−	0.909654i	$-0.636347\pi$
$223$	−12.4055	−0.830733	−0.415367	+	0.909654i	$0.636347\pi$
$224$	−4.44897	−0.297259
$225$	0	0
$226$	−17.5229	−1.16561
$227$	6.05595	0.401948	0.200974	−	0.979597i	$-0.435589\pi$
$227$	6.05595	0.401948	0.200974	+	0.979597i	$0.435589\pi$
$228$	0	0
$229$	−8.90167	−0.588239	−0.294120	−	0.955769i	$-0.595026\pi$
$229$	−8.90167	−0.588239	−0.294120	+	0.955769i	$0.595026\pi$
$230$	1.30564	0.0860910
$231$	0	0
$232$	5.44658	0.357586
$233$	−17.6760	−1.15799	−0.578997	−	0.815330i	$-0.696556\pi$
$233$	−17.6760	−1.15799	−0.578997	+	0.815330i	$0.696556\pi$
$234$	0	0
$235$	3.73168	0.243428
$236$	−49.7110	−3.23591
$237$	0	0
$238$	−17.1610	−1.11239
$239$	12.6176	0.816164	0.408082	−	0.912945i	$-0.366198\pi$
$239$	12.6176	0.816164	0.408082	+	0.912945i	$0.366198\pi$
$240$	0	0
$241$	−18.9487	−1.22059	−0.610297	−	0.792173i	$-0.708950\pi$
$241$	−18.9487	−1.22059	−0.610297	+	0.792173i	$0.708950\pi$
$242$	−36.5937	−2.35233
$243$	0	0
$244$	−33.7858	−2.16291
$245$	0.335557	0.0214379
$246$	0	0
$247$	−31.4393	−2.00044
$248$	21.5475	1.36827
$249$	0	0
$250$	7.74330	0.489729
$251$	−11.9290	−0.752954	−0.376477	−	0.926426i	$-0.622865\pi$
$251$	−11.9290	−0.752954	−0.376477	+	0.926426i	$0.622865\pi$
$252$	0	0
$253$	−8.61124	−0.541384
$254$	2.33388	0.146440
$255$	0	0
$256$	−21.8336	−1.36460
$257$	−4.45047	−0.277613	−0.138806	−	0.990320i	$-0.544327\pi$
$257$	−4.45047	−0.277613	−0.138806	+	0.990320i	$0.544327\pi$
$258$	0	0
$259$	11.1071	0.690163
$260$	−6.93943	−0.430365
$261$	0	0
$262$	30.5727	1.88879
$263$	8.75711	0.539987	0.269993	−	0.962862i	$-0.412979\pi$
$263$	8.75711	0.539987	0.269993	+	0.962862i	$0.412979\pi$
$264$	0	0
$265$	1.63124	0.100206
$266$	12.2302	0.749879
$267$	0	0
$268$	−15.4179	−0.941800
$269$	−4.58219	−0.279381	−0.139691	−	0.990195i	$-0.544611\pi$
$269$	−4.58219	−0.279381	−0.139691	+	0.990195i	$0.544611\pi$
$270$	0	0
$271$	17.3973	1.05681	0.528405	−	0.848992i	$-0.322790\pi$
$271$	17.3973	1.05681	0.528405	+	0.848992i	$0.322790\pi$
$272$	−7.26255	−0.440357
$273$	0	0
$274$	1.82126	0.110026
$275$	−25.2444	−1.52230
$276$	0	0
$277$	11.2062	0.673314	0.336657	−	0.941627i	$-0.390704\pi$
$277$	11.2062	0.673314	0.336657	+	0.941627i	$0.390704\pi$
$278$	−30.5444	−1.83193
$279$	0	0
$280$	1.13320	0.0677215
$281$	16.5245	0.985769	0.492885	−	0.870095i	$-0.335942\pi$
$281$	16.5245	0.985769	0.492885	+	0.870095i	$0.335942\pi$
$282$	0	0
$283$	15.7861	0.938385	0.469192	−	0.883096i	$-0.344545\pi$
$283$	15.7861	0.938385	0.469192	+	0.883096i	$0.344545\pi$
$284$	2.32766	0.138121
$285$	0	0
$286$	72.3244	4.27663
$287$	−3.12952	−0.184730
$288$	0	0
$289$	37.0670	2.18041
$290$	1.26307	0.0741703
$291$	0	0
$292$	45.7059	2.67474
$293$	4.45151	0.260060	0.130030	−	0.991510i	$-0.458493\pi$
$293$	4.45151	0.260060	0.130030	+	0.991510i	$0.458493\pi$
$294$	0	0
$295$	−4.83928	−0.281754
$296$	37.5095	2.18019
$297$	0	0
$298$	9.33241	0.540612
$299$	10.0022	0.578444
$300$	0	0
$301$	8.78359	0.506277
$302$	−21.3478	−1.22843
$303$	0	0
$304$	5.17580	0.296852
$305$	−3.28899	−0.188327
$306$	0	0
$307$	−13.3664	−0.762860	−0.381430	−	0.924398i	$-0.624568\pi$
$307$	−13.3664	−0.762860	−0.381430	+	0.924398i	$0.624568\pi$
$308$	−17.8044	−1.01450
$309$	0	0
$310$	4.99691	0.283805
$311$	31.7444	1.80006	0.900029	−	0.435831i	$-0.143545\pi$
$311$	31.7444	1.80006	0.900029	+	0.435831i	$0.143545\pi$
$312$	0	0
$313$	−1.31468	−0.0743100	−0.0371550	−	0.999310i	$-0.511830\pi$
$313$	−1.31468	−0.0743100	−0.0371550	+	0.999310i	$0.511830\pi$
$314$	−21.1417	−1.19310
$315$	0	0
$316$	36.6750	2.06313
$317$	−4.06070	−0.228071	−0.114036	−	0.993477i	$-0.536378\pi$
$317$	−4.06070	−0.228071	−0.114036	+	0.993477i	$0.536378\pi$
$318$	0	0
$319$	−8.33053	−0.466420
$320$	−4.14706	−0.231828
$321$	0	0
$322$	−3.89095	−0.216834
$323$	−38.5319	−2.14397
$324$	0	0
$325$	29.3222	1.62651
$326$	40.1962	2.22626
$327$	0	0
$328$	−10.5686	−0.583553
$329$	−11.1208	−0.613112
$330$	0	0
$331$	−33.6214	−1.84800	−0.923999	−	0.382396i	$-0.875099\pi$
$331$	−33.6214	−1.84800	−0.923999	+	0.382396i	$0.875099\pi$
$332$	−34.9675	−1.91909
$333$	0	0
$334$	−13.1845	−0.721425
$335$	−1.50091	−0.0820033
$336$	0	0
$337$	3.14343	0.171234	0.0856168	−	0.996328i	$-0.472714\pi$
$337$	3.14343	0.171234	0.0856168	+	0.996328i	$0.472714\pi$
$338$	−53.6667	−2.91908
$339$	0	0
$340$	−8.50493	−0.461244
$341$	−32.9568	−1.78471
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	29.6627	1.59931
$345$	0	0
$346$	−37.9347	−2.03938
$347$	−14.4750	−0.777059	−0.388529	−	0.921436i	$-0.627017\pi$
$347$	−14.4750	−0.777059	−0.388529	+	0.921436i	$0.627017\pi$
$348$	0	0
$349$	4.01665	0.215006	0.107503	−	0.994205i	$-0.465714\pi$
$349$	4.01665	0.215006	0.107503	+	0.994205i	$0.465714\pi$
$350$	−11.4066	−0.609708
$351$	0	0
$352$	22.9799	1.22483
$353$	−6.95497	−0.370175	−0.185088	−	0.982722i	$-0.559257\pi$
$353$	−6.95497	−0.370175	−0.185088	+	0.982722i	$0.559257\pi$
$354$	0	0
$355$	0.226593	0.0120263
$356$	53.7916	2.85095
$357$	0	0
$358$	3.56352	0.188338
$359$	−27.2333	−1.43732	−0.718660	−	0.695362i	$-0.755244\pi$
$359$	−27.2333	−1.43732	−0.718660	+	0.695362i	$0.755244\pi$
$360$	0	0
$361$	8.46048	0.445288
$362$	−12.5847	−0.661436
$363$	0	0
$364$	20.6803	1.08394
$365$	4.44939	0.232892
$366$	0	0
$367$	10.7911	0.563292	0.281646	−	0.959518i	$-0.409120\pi$
$367$	10.7911	0.563292	0.281646	+	0.959518i	$0.409120\pi$
$368$	−1.64665	−0.0858375
$369$	0	0
$370$	8.69852	0.452215
$371$	−4.86128	−0.252385
$372$	0	0
$373$	−28.2991	−1.46527	−0.732636	−	0.680621i	$-0.761710\pi$
$373$	−28.2991	−1.46527	−0.732636	+	0.680621i	$0.761710\pi$
$374$	88.6404	4.58348
$375$	0	0
$376$	−37.5558	−1.93679
$377$	9.67618	0.498349
$378$	0	0
$379$	2.81598	0.144647	0.0723235	−	0.997381i	$-0.476959\pi$
$379$	2.81598	0.144647	0.0723235	+	0.997381i	$0.476959\pi$
$380$	6.06120	0.310933
$381$	0	0
$382$	−27.3200	−1.39782
$383$	4.44896	0.227331	0.113666	−	0.993519i	$-0.463741\pi$
$383$	4.44896	0.227331	0.113666	+	0.993519i	$0.463741\pi$
$384$	0	0
$385$	−1.73322	−0.0883331
$386$	−17.0693	−0.868807
$387$	0	0
$388$	45.0818	2.28868
$389$	18.8079	0.953601	0.476800	−	0.879012i	$-0.341796\pi$
$389$	18.8079	0.953601	0.476800	+	0.879012i	$0.341796\pi$
$390$	0	0
$391$	12.2587	0.619948
$392$	−3.37706	−0.170568
$393$	0	0
$394$	−42.7983	−2.15615
$395$	3.57024	0.179638
$396$	0	0
$397$	−37.1500	−1.86451	−0.932253	−	0.361807i	$-0.882160\pi$
$397$	−37.1500	−1.86451	−0.932253	+	0.361807i	$0.882160\pi$
$398$	−26.9205	−1.34940
$399$	0	0
$400$	−4.82727	−0.241363
$401$	13.0342	0.650898	0.325449	−	0.945560i	$-0.394485\pi$
$401$	13.0342	0.650898	0.325449	+	0.945560i	$0.394485\pi$
$402$	0	0
$403$	38.2804	1.90688
$404$	36.3180	1.80689
$405$	0	0
$406$	−3.76411	−0.186810
$407$	−57.3706	−2.84375
$408$	0	0
$409$	6.83929	0.338181	0.169091	−	0.985601i	$-0.445917\pi$
$409$	6.83929	0.338181	0.169091	+	0.985601i	$0.445917\pi$
$410$	−2.45088	−0.121040
$411$	0	0
$412$	17.7116	0.872587
$413$	14.4216	0.709642
$414$	0	0
$415$	−3.40402	−0.167097
$416$	−26.6918	−1.30868
$417$	0	0
$418$	−63.1713	−3.08981
$419$	−20.2068	−0.987165	−0.493582	−	0.869699i	$-0.664313\pi$
$419$	−20.2068	−0.987165	−0.493582	+	0.869699i	$0.664313\pi$
$420$	0	0
$421$	−19.9314	−0.971399	−0.485699	−	0.874126i	$-0.661435\pi$
$421$	−19.9314	−0.971399	−0.485699	+	0.874126i	$0.661435\pi$
$422$	−44.4957	−2.16602
$423$	0	0
$424$	−16.4168	−0.797272
$425$	35.9372	1.74321
$426$	0	0
$427$	9.80157	0.474331
$428$	−19.3165	−0.933699
$429$	0	0
$430$	6.87885	0.331728
$431$	−22.9047	−1.10328	−0.551641	−	0.834081i	$-0.685998\pi$
$431$	−22.9047	−1.10328	−0.551641	+	0.834081i	$0.685998\pi$
$432$	0	0
$433$	−2.26347	−0.108775	−0.0543876	−	0.998520i	$-0.517321\pi$
$433$	−2.26347	−0.108775	−0.0543876	+	0.998520i	$0.517321\pi$
$434$	−14.8914	−0.714810
$435$	0	0
$436$	−6.95117	−0.332901
$437$	−8.73639	−0.417918
$438$	0	0
$439$	33.4776	1.59780	0.798899	−	0.601465i	$-0.205416\pi$
$439$	33.4776	1.59780	0.798899	+	0.601465i	$0.205416\pi$
$440$	−5.85320	−0.279040
$441$	0	0
$442$	−102.959	−4.89724
$443$	10.7339	0.509984	0.254992	−	0.966943i	$-0.417927\pi$
$443$	10.7339	0.509984	0.254992	+	0.966943i	$0.417927\pi$
$444$	0	0
$445$	5.23652	0.248235
$446$	28.9529	1.37096
$447$	0	0
$448$	12.3587	0.583895
$449$	−27.3591	−1.29116	−0.645579	−	0.763694i	$-0.723384\pi$
$449$	−27.3591	−1.29116	−0.645579	+	0.763694i	$0.723384\pi$
$450$	0	0
$451$	16.1646	0.761162
$452$	25.8801	1.21730
$453$	0	0
$454$	−14.1338	−0.663334
$455$	2.01319	0.0943799
$456$	0	0
$457$	13.4551	0.629401	0.314701	−	0.949191i	$-0.398096\pi$
$457$	13.4551	0.629401	0.314701	+	0.949191i	$0.398096\pi$
$458$	20.7754	0.970771
$459$	0	0
$460$	−1.92833	−0.0899091
$461$	21.3424	0.994016	0.497008	−	0.867746i	$-0.334432\pi$
$461$	21.3424	0.994016	0.497008	+	0.867746i	$0.334432\pi$
$462$	0	0
$463$	34.6037	1.60817	0.804085	−	0.594514i	$-0.202656\pi$
$463$	34.6037	1.60817	0.804085	+	0.594514i	$0.202656\pi$
$464$	−1.59297	−0.0739518
$465$	0	0
$466$	41.2536	1.91104
$467$	12.1912	0.564142	0.282071	−	0.959394i	$-0.408979\pi$
$467$	12.1912	0.564142	0.282071	+	0.959394i	$0.408979\pi$
$468$	0	0
$469$	4.47288	0.206539
$470$	−8.70927	−0.401729
$471$	0	0
$472$	48.7028	2.24173
$473$	−45.3690	−2.08607
$474$	0	0
$475$	−25.6113	−1.17513
$476$	25.3457	1.16172
$477$	0	0
$478$	−29.4479	−1.34692
$479$	15.3651	0.702048	0.351024	−	0.936367i	$-0.385834\pi$
$479$	15.3651	0.702048	0.351024	+	0.936367i	$0.385834\pi$
$480$	0	0
$481$	66.6378	3.03842
$482$	44.2240	2.01435
$483$	0	0
$484$	54.0464	2.45666
$485$	4.38863	0.199278
$486$	0	0
$487$	−26.7956	−1.21422	−0.607112	−	0.794617i	$-0.707672\pi$
$487$	−26.7956	−1.21422	−0.607112	+	0.794617i	$0.707672\pi$
$488$	33.1005	1.49839
$489$	0	0
$490$	−0.783148	−0.0353790
$491$	−1.26699	−0.0571785	−0.0285893	−	0.999591i	$-0.509101\pi$
$491$	−1.26699	−0.0571785	−0.0285893	+	0.999591i	$0.509101\pi$
$492$	0	0
$493$	11.8591	0.534106
$494$	73.3755	3.30132
$495$	0	0
$496$	−6.30203	−0.282970
$497$	−0.675275	−0.0302902
$498$	0	0
$499$	−26.0496	−1.16614	−0.583070	−	0.812422i	$-0.698149\pi$
$499$	−26.0496	−1.16614	−0.583070	+	0.812422i	$0.698149\pi$
$500$	−11.4363	−0.511448
$501$	0	0
$502$	27.8409	1.24260
$503$	−14.0276	−0.625462	−0.312731	−	0.949842i	$-0.601244\pi$
$503$	−14.0276	−0.625462	−0.312731	+	0.949842i	$0.601244\pi$
$504$	0	0
$505$	3.53549	0.157327
$506$	20.0976	0.893446
$507$	0	0
$508$	−3.44698	−0.152935
$509$	−9.93465	−0.440346	−0.220173	−	0.975461i	$-0.570662\pi$
$509$	−9.93465	−0.440346	−0.220173	+	0.975461i	$0.570662\pi$
$510$	0	0
$511$	−13.2597	−0.586575
$512$	11.0653	0.489020
$513$	0	0
$514$	10.3868	0.458144
$515$	1.72419	0.0759769
$516$	0	0
$517$	57.4415	2.52627
$518$	−25.9226	−1.13898
$519$	0	0
$520$	6.79868	0.298142
$521$	32.4116	1.41998	0.709989	−	0.704212i	$-0.248700\pi$
$521$	32.4116	1.41998	0.709989	+	0.704212i	$0.248700\pi$
$522$	0	0
$523$	−10.7845	−0.471572	−0.235786	−	0.971805i	$-0.575766\pi$
$523$	−10.7845	−0.471572	−0.235786	+	0.971805i	$0.575766\pi$
$524$	−45.1538	−1.97255
$525$	0	0
$526$	−20.4380	−0.891140
$527$	46.9163	2.04370
$528$	0	0
$529$	−20.2206	−0.879155
$530$	−3.80710	−0.165370
$531$	0	0
$532$	−18.0631	−0.783135
$533$	−18.7757	−0.813267
$534$	0	0
$535$	−1.88043	−0.0812980
$536$	15.1052	0.652446
$537$	0	0
$538$	10.6943	0.461062
$539$	5.16521	0.222481
$540$	0	0
$541$	−24.1789	−1.03953	−0.519766	−	0.854309i	$-0.673981\pi$
$541$	−24.1789	−1.03953	−0.519766	+	0.854309i	$0.673981\pi$
$542$	−40.6031	−1.74405
$543$	0	0
$544$	−32.7134	−1.40257
$545$	−0.676684	−0.0289860
$546$	0	0
$547$	6.82760	0.291927	0.145963	−	0.989290i	$-0.453372\pi$
$547$	6.82760	0.291927	0.145963	+	0.989290i	$0.453372\pi$
$548$	−2.68988	−0.114906
$549$	0	0
$550$	58.9174	2.51225
$551$	−8.45160	−0.360050
$552$	0	0
$553$	−10.6397	−0.452448
$554$	−26.1538	−1.11117
$555$	0	0
$556$	45.1121	1.91318
$557$	18.3298	0.776658	0.388329	−	0.921521i	$-0.373052\pi$
$557$	18.3298	0.776658	0.388329	+	0.921521i	$0.373052\pi$
$558$	0	0
$559$	52.6976	2.22887
$560$	−0.331428	−0.0140054
$561$	0	0
$562$	−38.5661	−1.62681
$563$	36.5110	1.53875	0.769377	−	0.638795i	$-0.220567\pi$
$563$	36.5110	1.53875	0.769377	+	0.638795i	$0.220567\pi$
$564$	0	0
$565$	2.51939	0.105991
$566$	−36.8427	−1.54862
$567$	0	0
$568$	−2.28045	−0.0956855
$569$	15.6681	0.656842	0.328421	−	0.944531i	$-0.393484\pi$
$569$	15.6681	0.656842	0.328421	+	0.944531i	$0.393484\pi$
$570$	0	0
$571$	−32.3682	−1.35457	−0.677284	−	0.735722i	$-0.736843\pi$
$571$	−32.3682	−1.35457	−0.677284	+	0.735722i	$0.736843\pi$
$572$	−106.818	−4.46629
$573$	0	0
$574$	7.30391	0.304859
$575$	8.14809	0.339799
$576$	0	0
$577$	−9.05108	−0.376801	−0.188401	−	0.982092i	$-0.560330\pi$
$577$	−9.05108	−0.376801	−0.188401	+	0.982092i	$0.560330\pi$
$578$	−86.5097	−3.59833
$579$	0	0
$580$	−1.86548	−0.0774596
$581$	10.1444	0.420861
$582$	0	0
$583$	25.1095	1.03993
$584$	−44.7789	−1.85296
$585$	0	0
$586$	−10.3893	−0.429177
$587$	11.2035	0.462418	0.231209	−	0.972904i	$-0.425732\pi$
$587$	11.2035	0.462418	0.231209	+	0.972904i	$0.425732\pi$
$588$	0	0
$589$	−33.4358	−1.37770
$590$	11.2943	0.464978
$591$	0	0
$592$	−10.9705	−0.450883
$593$	−11.0604	−0.454195	−0.227097	−	0.973872i	$-0.572924\pi$
$593$	−11.0604	−0.454195	−0.227097	+	0.973872i	$0.572924\pi$
$594$	0	0
$595$	2.46736	0.101152
$596$	−13.7833	−0.564587
$597$	0	0
$598$	−23.3440	−0.954606
$599$	−19.8173	−0.809712	−0.404856	−	0.914380i	$-0.632678\pi$
$599$	−19.8173	−0.809712	−0.404856	+	0.914380i	$0.632678\pi$
$600$	0	0
$601$	−5.01781	−0.204681	−0.102340	−	0.994749i	$-0.532633\pi$
$601$	−5.01781	−0.204681	−0.102340	+	0.994749i	$0.532633\pi$
$602$	−20.4998	−0.835509
$603$	0	0
$604$	31.5293	1.28291
$605$	5.26132	0.213903
$606$	0	0
$607$	1.93279	0.0784497	0.0392248	−	0.999230i	$-0.487511\pi$
$607$	1.93279	0.0784497	0.0392248	+	0.999230i	$0.487511\pi$
$608$	23.3138	0.945500
$609$	0	0
$610$	7.67608	0.310796
$611$	−66.7201	−2.69921
$612$	0	0
$613$	−18.7587	−0.757655	−0.378828	−	0.925467i	$-0.623673\pi$
$613$	−18.7587	−0.757655	−0.378828	+	0.925467i	$0.623673\pi$
$614$	31.1955	1.25895
$615$	0	0
$616$	17.4432	0.702808
$617$	−16.2050	−0.652389	−0.326194	−	0.945303i	$-0.605766\pi$
$617$	−16.2050	−0.652389	−0.326194	+	0.945303i	$0.605766\pi$
$618$	0	0
$619$	−33.9811	−1.36582	−0.682908	−	0.730505i	$-0.739285\pi$
$619$	−33.9811	−1.36582	−0.682908	+	0.730505i	$0.739285\pi$
$620$	−7.38010	−0.296392
$621$	0	0
$622$	−74.0874	−2.97063
$623$	−15.6055	−0.625219
$624$	0	0
$625$	23.3237	0.932948
$626$	3.06829	0.122634
$627$	0	0
$628$	31.2249	1.24601
$629$	81.6709	3.25643
$630$	0	0
$631$	26.4905	1.05457	0.527285	−	0.849688i	$-0.323210\pi$
$631$	26.4905	1.05457	0.527285	+	0.849688i	$0.323210\pi$
$632$	−35.9311	−1.42926
$633$	0	0
$634$	9.47716	0.376386
$635$	−0.335557	−0.0133162
$636$	0	0
$637$	−5.99956	−0.237711
$638$	19.4424	0.769733
$639$	0	0
$640$	6.69296	0.264562
$641$	32.2324	1.27310	0.636552	−	0.771234i	$-0.280360\pi$
$641$	32.2324	1.27310	0.636552	+	0.771234i	$0.280360\pi$
$642$	0	0
$643$	−40.6658	−1.60370	−0.801851	−	0.597524i	$-0.796151\pi$
$643$	−40.6658	−1.60370	−0.801851	+	0.597524i	$0.796151\pi$
$644$	5.74667	0.226450
$645$	0	0
$646$	89.9286	3.53819
$647$	33.2527	1.30730	0.653649	−	0.756798i	$-0.273237\pi$
$647$	33.2527	1.30730	0.653649	+	0.756798i	$0.273237\pi$
$648$	0	0
$649$	−74.4907	−2.92402
$650$	−68.4345	−2.68422
$651$	0	0
$652$	−59.3670	−2.32499
$653$	−24.4363	−0.956265	−0.478132	−	0.878288i	$-0.658686\pi$
$653$	−24.4363	−0.956265	−0.478132	+	0.878288i	$0.658686\pi$
$654$	0	0
$655$	−4.39564	−0.171752
$656$	3.09101	0.120684
$657$	0	0
$658$	25.9547	1.01182
$659$	−37.1109	−1.44563	−0.722817	−	0.691039i	$-0.757153\pi$
$659$	−37.1109	−1.44563	−0.722817	+	0.691039i	$0.757153\pi$
$660$	0	0
$661$	−7.52161	−0.292557	−0.146278	−	0.989243i	$-0.546730\pi$
$661$	−7.52161	−0.292557	−0.146278	+	0.989243i	$0.546730\pi$
$662$	78.4681	3.04975
$663$	0	0
$664$	34.2583	1.32948
$665$	−1.75841	−0.0681882
$666$	0	0
$667$	2.68883	0.104112
$668$	19.4726	0.753419
$669$	0	0
$670$	3.50293	0.135330
$671$	−50.6272	−1.95444
$672$	0	0
$673$	−6.48410	−0.249944	−0.124972	−	0.992160i	$-0.539884\pi$
$673$	−6.48410	−0.249944	−0.124972	+	0.992160i	$0.539884\pi$
$674$	−7.33638	−0.282587
$675$	0	0
$676$	79.2621	3.04854
$677$	−3.08598	−0.118604	−0.0593018	−	0.998240i	$-0.518887\pi$
$677$	−3.08598	−0.118604	−0.0593018	+	0.998240i	$0.518887\pi$
$678$	0	0
$679$	−13.0787	−0.501913
$680$	8.33243	0.319534
$681$	0	0
$682$	76.9171	2.94531
$683$	12.1685	0.465614	0.232807	−	0.972523i	$-0.425209\pi$
$683$	12.1685	0.465614	0.232807	+	0.972523i	$0.425209\pi$
$684$	0	0
$685$	−0.261855	−0.0100050
$686$	2.33388	0.0891078
$687$	0	0
$688$	−8.67551	−0.330751
$689$	−29.1655	−1.11112
$690$	0	0
$691$	37.4639	1.42519	0.712596	−	0.701574i	$-0.247519\pi$
$691$	37.4639	1.42519	0.712596	+	0.701574i	$0.247519\pi$
$692$	56.0269	2.12982
$693$	0	0
$694$	33.7829	1.28238
$695$	4.39158	0.166582
$696$	0	0
$697$	−23.0114	−0.871620
$698$	−9.37437	−0.354825
$699$	0	0
$700$	16.8468	0.636748
$701$	30.5926	1.15547	0.577733	−	0.816226i	$-0.303938\pi$
$701$	30.5926	1.15547	0.577733	+	0.816226i	$0.303938\pi$
$702$	0	0
$703$	−58.2044	−2.19522
$704$	−63.8354	−2.40589
$705$	0	0
$706$	16.2320	0.610900
$707$	−10.5362	−0.396254
$708$	0	0
$709$	26.7057	1.00295	0.501476	−	0.865171i	$-0.332790\pi$
$709$	26.7057	1.00295	0.501476	+	0.865171i	$0.332790\pi$
$710$	−0.528841	−0.0198470
$711$	0	0
$712$	−52.7006	−1.97504
$713$	10.6374	0.398374
$714$	0	0
$715$	−10.3986	−0.388884
$716$	−5.26308	−0.196691
$717$	0	0
$718$	63.5592	2.37201
$719$	41.8742	1.56164	0.780822	−	0.624753i	$-0.214801\pi$
$719$	41.8742	1.56164	0.780822	+	0.624753i	$0.214801\pi$
$720$	0	0
$721$	−5.13829	−0.191360
$722$	−19.7457	−0.734859
$723$	0	0
$724$	18.5867	0.690770
$725$	7.88248	0.292748
$726$	0	0
$727$	32.4263	1.20263	0.601313	−	0.799013i	$-0.294644\pi$
$727$	32.4263	1.20263	0.601313	+	0.799013i	$0.294644\pi$
$728$	−20.2609	−0.750918
$729$	0	0
$730$	−10.3843	−0.384341
$731$	64.5859	2.38880
$732$	0	0
$733$	−17.8476	−0.659216	−0.329608	−	0.944118i	$-0.606917\pi$
$733$	−17.8476	−0.659216	−0.329608	+	0.944118i	$0.606917\pi$
$734$	−25.1851	−0.929601
$735$	0	0
$736$	−7.41715	−0.273400
$737$	−23.1034	−0.851024
$738$	0	0
$739$	−25.3278	−0.931697	−0.465848	−	0.884865i	$-0.654251\pi$
$739$	−25.3278	−0.931697	−0.465848	+	0.884865i	$0.654251\pi$
$740$	−12.8471	−0.472270
$741$	0	0
$742$	11.3456	0.416511
$743$	7.63434	0.280077	0.140038	−	0.990146i	$-0.455277\pi$
$743$	7.63434	0.280077	0.140038	+	0.990146i	$0.455277\pi$
$744$	0	0
$745$	−1.34178	−0.0491591
$746$	66.0466	2.41814
$747$	0	0
$748$	−130.916	−4.78676
$749$	5.60390	0.204762
$750$	0	0
$751$	−9.49290	−0.346401	−0.173200	−	0.984887i	$-0.555411\pi$
$751$	−9.49290	−0.346401	−0.173200	+	0.984887i	$0.555411\pi$
$752$	10.9840	0.400546
$753$	0	0
$754$	−22.5830	−0.822424
$755$	3.06932	0.111704
$756$	0	0
$757$	5.77991	0.210074	0.105037	−	0.994468i	$-0.466504\pi$
$757$	5.77991	0.210074	0.105037	+	0.994468i	$0.466504\pi$
$758$	−6.57214	−0.238711
$759$	0	0
$760$	−5.93827	−0.215404
$761$	41.1682	1.49235	0.746173	−	0.665753i	$-0.231889\pi$
$761$	41.1682	1.49235	0.746173	+	0.665753i	$0.231889\pi$
$762$	0	0
$763$	2.01660	0.0730058
$764$	40.3499	1.45981
$765$	0	0
$766$	−10.3833	−0.375165
$767$	86.5233	3.12418
$768$	0	0
$769$	27.6708	0.997836	0.498918	−	0.866649i	$-0.333731\pi$
$769$	27.6708	0.997836	0.498918	+	0.866649i	$0.333731\pi$
$770$	4.04512	0.145776
$771$	0	0
$772$	25.2103	0.907337
$773$	47.5418	1.70996	0.854980	−	0.518660i	$-0.173569\pi$
$773$	47.5418	1.70996	0.854980	+	0.518660i	$0.173569\pi$
$774$	0	0
$775$	31.1843	1.12017
$776$	−44.1675	−1.58552
$777$	0	0
$778$	−43.8954	−1.57373
$779$	16.3995	0.587575
$780$	0	0
$781$	3.48794	0.124808
$782$	−28.6102	−1.02310
$783$	0	0
$784$	0.987696	0.0352748
$785$	3.03969	0.108491
$786$	0	0
$787$	39.6947	1.41496	0.707481	−	0.706732i	$-0.249831\pi$
$787$	39.6947	1.41496	0.707481	+	0.706732i	$0.249831\pi$
$788$	63.2101	2.25177
$789$	0	0
$790$	−8.33250	−0.296457
$791$	−7.50807	−0.266956
$792$	0	0
$793$	58.8051	2.08823
$794$	86.7036	3.07699
$795$	0	0
$796$	39.7598	1.40925
$797$	−5.19090	−0.183871	−0.0919356	−	0.995765i	$-0.529305\pi$
$797$	−5.19090	−0.183871	−0.0919356	+	0.995765i	$0.529305\pi$
$798$	0	0
$799$	−81.7718	−2.89288
$800$	−21.7439	−0.768763
$801$	0	0
$802$	−30.4202	−1.07418
$803$	68.4891	2.41693
$804$	0	0
$805$	0.559428	0.0197172
$806$	−89.3417	−3.14693
$807$	0	0
$808$	−35.5814	−1.25175
$809$	−30.8606	−1.08500	−0.542500	−	0.840056i	$-0.682522\pi$
$809$	−30.8606	−1.08500	−0.542500	+	0.840056i	$0.682522\pi$
$810$	0	0
$811$	50.3472	1.76793	0.883964	−	0.467555i	$-0.154865\pi$
$811$	50.3472	1.76793	0.883964	+	0.467555i	$0.154865\pi$
$812$	5.55934	0.195095
$813$	0	0
$814$	133.896	4.69305
$815$	−5.77927	−0.202439
$816$	0	0
$817$	−46.0284	−1.61033
$818$	−15.9621	−0.558100
$819$	0	0
$820$	3.61978	0.126408
$821$	26.2494	0.916110	0.458055	−	0.888924i	$-0.348546\pi$
$821$	26.2494	0.916110	0.458055	+	0.888924i	$0.348546\pi$
$822$	0	0
$823$	15.7568	0.549248	0.274624	−	0.961552i	$-0.411447\pi$
$823$	15.7568	0.549248	0.274624	+	0.961552i	$0.411447\pi$
$824$	−17.3523	−0.604498
$825$	0	0
$826$	−33.6583	−1.17112
$827$	−27.7286	−0.964218	−0.482109	−	0.876111i	$-0.660129\pi$
$827$	−27.7286	−0.964218	−0.482109	+	0.876111i	$0.660129\pi$
$828$	0	0
$829$	28.2734	0.981976	0.490988	−	0.871166i	$-0.336636\pi$
$829$	28.2734	0.981976	0.490988	+	0.871166i	$0.336636\pi$
$830$	7.94457	0.275760
$831$	0	0
$832$	74.1469	2.57058
$833$	−7.35302	−0.254767
$834$	0	0
$835$	1.89563	0.0656009
$836$	93.2997	3.22684
$837$	0	0
$838$	47.1601	1.62912
$839$	26.2104	0.904883	0.452442	−	0.891794i	$-0.350553\pi$
$839$	26.2104	0.904883	0.452442	+	0.891794i	$0.350553\pi$
$840$	0	0
$841$	−26.3988	−0.910304
$842$	46.5175	1.60310
$843$	0	0
$844$	65.7171	2.26208
$845$	7.71602	0.265439
$846$	0	0
$847$	−15.6794	−0.538750
$848$	4.80146	0.164883
$849$	0	0
$850$	−83.8729	−2.87682
$851$	18.5174	0.634767
$852$	0	0
$853$	46.9061	1.60603	0.803017	−	0.595956i	$-0.203227\pi$
$853$	46.9061	1.60603	0.803017	+	0.595956i	$0.203227\pi$
$854$	−22.8756	−0.782789
$855$	0	0
$856$	18.9247	0.646834
$857$	46.2259	1.57905	0.789524	−	0.613720i	$-0.210328\pi$
$857$	46.2259	1.57905	0.789524	+	0.613720i	$0.210328\pi$
$858$	0	0
$859$	−10.9914	−0.375022	−0.187511	−	0.982263i	$-0.560042\pi$
$859$	−10.9914	−0.375022	−0.187511	+	0.982263i	$0.560042\pi$
$860$	−10.1596	−0.346439
$861$	0	0
$862$	53.4568	1.82075
$863$	6.20970	0.211381	0.105690	−	0.994399i	$-0.466295\pi$
$863$	6.20970	0.211381	0.105690	+	0.994399i	$0.466295\pi$
$864$	0	0
$865$	5.45412	0.185446
$866$	5.28265	0.179512
$867$	0	0
$868$	21.9936	0.746511
$869$	54.9565	1.86427
$870$	0	0
$871$	26.8353	0.909280
$872$	6.81019	0.230622
$873$	0	0
$874$	20.3896	0.689690
$875$	3.31779	0.112162
$876$	0	0
$877$	3.02855	0.102267	0.0511334	−	0.998692i	$-0.483717\pi$
$877$	3.02855	0.102267	0.0511334	+	0.998692i	$0.483717\pi$
$878$	−78.1325	−2.63684
$879$	0	0
$880$	1.71190	0.0577080
$881$	−25.4949	−0.858945	−0.429473	−	0.903080i	$-0.641301\pi$
$881$	−25.4949	−0.858945	−0.429473	+	0.903080i	$0.641301\pi$
$882$	0	0
$883$	−1.99016	−0.0669742	−0.0334871	−	0.999439i	$-0.510661\pi$
$883$	−1.99016	−0.0669742	−0.0334871	+	0.999439i	$0.510661\pi$
$884$	152.063	5.11443
$885$	0	0
$886$	−25.0516	−0.841627
$887$	−14.8821	−0.499690	−0.249845	−	0.968286i	$-0.580380\pi$
$887$	−14.8821	−0.499690	−0.249845	+	0.968286i	$0.580380\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	−12.2214	−0.409662
$891$	0	0
$892$	−42.7615	−1.43176
$893$	58.2763	1.95014
$894$	0	0
$895$	−0.512352	−0.0171260
$896$	−19.9458	−0.666343
$897$	0	0
$898$	63.8528	2.13080
$899$	10.2906	0.343212
$900$	0	0
$901$	−35.7451	−1.19084
$902$	−37.7262	−1.25615
$903$	0	0
$904$	−25.3552	−0.843303
$905$	1.80938	0.0601460
$906$	0	0
$907$	−16.8522	−0.559567	−0.279784	−	0.960063i	$-0.590263\pi$
$907$	−16.8522	−0.559567	−0.279784	+	0.960063i	$0.590263\pi$
$908$	20.8747	0.692752
$909$	0	0
$910$	−4.69854	−0.155755
$911$	−3.07305	−0.101815	−0.0509074	−	0.998703i	$-0.516211\pi$
$911$	−3.07305	−0.101815	−0.0509074	+	0.998703i	$0.516211\pi$
$912$	0	0
$913$	−52.3980	−1.73412
$914$	−31.4024	−1.03870
$915$	0	0
$916$	−30.6839	−1.01382
$917$	13.0995	0.432585
$918$	0	0
$919$	−1.02033	−0.0336577	−0.0168289	−	0.999858i	$-0.505357\pi$
$919$	−1.02033	−0.0336577	−0.0168289	+	0.999858i	$0.505357\pi$
$920$	1.88922	0.0622859
$921$	0	0
$922$	−49.8106	−1.64042
$923$	−4.05135	−0.133352
$924$	0	0
$925$	54.2850	1.78488
$926$	−80.7607	−2.65396
$927$	0	0
$928$	−7.17537	−0.235543
$929$	10.7161	0.351582	0.175791	−	0.984427i	$-0.443752\pi$
$929$	10.7161	0.351582	0.175791	+	0.984427i	$0.443752\pi$
$930$	0	0
$931$	5.24027	0.171743
$932$	−60.9288	−1.99579
$933$	0	0
$934$	−28.4528	−0.931003
$935$	−12.7444	−0.416787
$936$	0	0
$937$	23.4875	0.767305	0.383652	−	0.923478i	$-0.374666\pi$
$937$	23.4875	0.767305	0.383652	+	0.923478i	$0.374666\pi$
$938$	−10.4392	−0.340850
$939$	0	0
$940$	12.8630	0.419545
$941$	−46.9801	−1.53151	−0.765753	−	0.643135i	$-0.777634\pi$
$941$	−46.9801	−1.53151	−0.765753	+	0.643135i	$0.777634\pi$
$942$	0	0
$943$	−5.21742	−0.169902
$944$	−14.2442	−0.463608
$945$	0	0
$946$	105.886	3.44264
$947$	18.1126	0.588579	0.294290	−	0.955716i	$-0.404917\pi$
$947$	18.1126	0.588579	0.294290	+	0.955716i	$0.404917\pi$
$948$	0	0
$949$	−79.5523	−2.58238
$950$	59.7737	1.93931
$951$	0	0
$952$	−24.8316	−0.804798
$953$	41.2912	1.33755	0.668777	−	0.743463i	$-0.266818\pi$
$953$	41.2912	1.33755	0.668777	+	0.743463i	$0.266818\pi$
$954$	0	0
$955$	3.92799	0.127107
$956$	43.4926	1.40665
$957$	0	0
$958$	−35.8602	−1.15859
$959$	0.780358	0.0251991
$960$	0	0
$961$	9.71129	0.313267
$962$	−155.524	−5.01431
$963$	0	0
$964$	−65.3158	−2.10368
$965$	2.45417	0.0790026
$966$	0	0
$967$	17.5957	0.565839	0.282919	−	0.959144i	$-0.408697\pi$
$967$	17.5957	0.565839	0.282919	+	0.959144i	$0.408697\pi$
$968$	−52.9503	−1.70189
$969$	0	0
$970$	−10.2425	−0.328868
$971$	4.59485	0.147456	0.0737278	−	0.997278i	$-0.476510\pi$
$971$	4.59485	0.147456	0.0737278	+	0.997278i	$0.476510\pi$
$972$	0	0
$973$	−13.0874	−0.419564
$974$	62.5375	2.00383
$975$	0	0
$976$	−9.68097	−0.309880
$977$	−15.6367	−0.500263	−0.250132	−	0.968212i	$-0.580474\pi$
$977$	−15.6367	−0.500263	−0.250132	+	0.968212i	$0.580474\pi$
$978$	0	0
$979$	80.6054	2.57616
$980$	1.15666	0.0369481
$981$	0	0
$982$	2.95700	0.0943617
$983$	55.4023	1.76706	0.883529	−	0.468376i	$-0.155161\pi$
$983$	55.4023	1.76706	0.883529	+	0.468376i	$0.155161\pi$
$984$	0	0
$985$	6.15339	0.196063
$986$	−27.6776	−0.881435
$987$	0	0
$988$	−108.371	−3.44773
$989$	14.6437	0.465641
$990$	0	0
$991$	36.1178	1.14732	0.573660	−	0.819094i	$-0.305523\pi$
$991$	36.1178	1.14732	0.573660	+	0.819094i	$0.305523\pi$
$992$	−28.3868	−0.901283
$993$	0	0
$994$	1.57601	0.0499880
$995$	3.87054	0.122704
$996$	0	0
$997$	37.4725	1.18677	0.593383	−	0.804921i	$-0.297792\pi$
$997$	37.4725	1.18677	0.593383	+	0.804921i	$0.297792\pi$
$998$	60.7965	1.92448
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.p.1.1		14
3.2	odd	2		2667.2.a.m.1.14	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.m.1.14	✓	14	3.2	odd	2
8001.2.a.p.1.1		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)

\(f(q)\)	\(=\)	\(q-2.33388 q^{2} +3.44698 q^{4} +0.335557 q^{5} -1.00000 q^{7} -3.37706 q^{8} +O(q^{10})\)	\(q-2.33388 q^{2} +3.44698 q^{4} +0.335557 q^{5} -1.00000 q^{7} -3.37706 q^{8} -0.783148 q^{10} +5.16521 q^{11} -5.99956 q^{13} +2.33388 q^{14} +0.987696 q^{16} -7.35302 q^{17} +5.24027 q^{19} +1.15666 q^{20} -12.0550 q^{22} -1.66716 q^{23} -4.88740 q^{25} +14.0022 q^{26} -3.44698 q^{28} -1.61282 q^{29} -6.38054 q^{31} +4.44897 q^{32} +17.1610 q^{34} -0.335557 q^{35} -11.1071 q^{37} -12.2302 q^{38} -1.13320 q^{40} +3.12952 q^{41} -8.78359 q^{43} +17.8044 q^{44} +3.89095 q^{46} +11.1208 q^{47} +1.00000 q^{49} +11.4066 q^{50} -20.6803 q^{52} +4.86128 q^{53} +1.73322 q^{55} +3.37706 q^{56} +3.76411 q^{58} -14.4216 q^{59} -9.80157 q^{61} +14.8914 q^{62} -12.3587 q^{64} -2.01319 q^{65} -4.47288 q^{67} -25.3457 q^{68} +0.783148 q^{70} +0.675275 q^{71} +13.2597 q^{73} +25.9226 q^{74} +18.0631 q^{76} -5.16521 q^{77} +10.6397 q^{79} +0.331428 q^{80} -7.30391 q^{82} -10.1444 q^{83} -2.46736 q^{85} +20.4998 q^{86} -17.4432 q^{88} +15.6055 q^{89} +5.99956 q^{91} -5.74667 q^{92} -25.9547 q^{94} +1.75841 q^{95} +13.0787 q^{97} -2.33388 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 5 q^{2} + 15 q^{4} + 4 q^{5} - 14 q^{7} + 12 q^{8}+O(q^{10}) \) 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

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Database

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