LMFDB - Embedded newform 8001.2.a.p.1.14

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			1.14
Root			$2.66839$ 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)

$f(q)$	$=$	$q+2.66839 q^{2} +5.12030 q^{4} +2.72934 q^{5} -1.00000 q^{7} +8.32617 q^{8} +O(q^{10})$	\(q+2.66839 q^{2} +5.12030 q^{4} +2.72934 q^{5} -1.00000 q^{7} +8.32617 q^{8} +7.28293 q^{10} -1.99935 q^{11} +3.01152 q^{13} -2.66839 q^{14} +11.9769 q^{16} +1.40099 q^{17} -3.11408 q^{19} +13.9750 q^{20} -5.33503 q^{22} +3.27197 q^{23} +2.44927 q^{25} +8.03590 q^{26} -5.12030 q^{28} +6.12906 q^{29} +6.08476 q^{31} +15.3066 q^{32} +3.73838 q^{34} -2.72934 q^{35} -5.42709 q^{37} -8.30959 q^{38} +22.7249 q^{40} -11.6265 q^{41} +9.33583 q^{43} -10.2373 q^{44} +8.73088 q^{46} +8.98796 q^{47} +1.00000 q^{49} +6.53561 q^{50} +15.4199 q^{52} -4.15642 q^{53} -5.45689 q^{55} -8.32617 q^{56} +16.3547 q^{58} -7.85361 q^{59} -3.02245 q^{61} +16.2365 q^{62} +16.8902 q^{64} +8.21944 q^{65} -7.97480 q^{67} +7.17347 q^{68} -7.28293 q^{70} -4.60611 q^{71} +6.75870 q^{73} -14.4816 q^{74} -15.9450 q^{76} +1.99935 q^{77} +0.469412 q^{79} +32.6889 q^{80} -31.0240 q^{82} +11.7285 q^{83} +3.82376 q^{85} +24.9116 q^{86} -16.6469 q^{88} -1.25580 q^{89} -3.01152 q^{91} +16.7534 q^{92} +23.9834 q^{94} -8.49938 q^{95} +5.52320 q^{97} +2.66839 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.66839	1.88684	0.943418	−	0.331606i	$-0.107590\pi$
$2$	2.66839	1.88684	0.943418	+	0.331606i	$0.107590\pi$
$3$	0	0
$3$	0	0
$4$	5.12030	2.56015
$5$	2.72934	1.22060	0.610298	−	0.792172i	$-0.291050\pi$
$5$	2.72934	1.22060	0.610298	+	0.792172i	$0.291050\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	8.32617	2.94375
$9$	0	0
$10$	7.28293	2.30306
$11$	−1.99935	−0.602826	−0.301413	−	0.953494i	$-0.597458\pi$
$11$	−1.99935	−0.602826	−0.301413	+	0.953494i	$0.597458\pi$
$12$	0	0
$13$	3.01152	0.835244	0.417622	−	0.908621i	$-0.362864\pi$
$13$	3.01152	0.835244	0.417622	+	0.908621i	$0.362864\pi$
$14$	−2.66839	−0.713157
$15$	0	0
$16$	11.9769	2.99422
$17$	1.40099	0.339789	0.169894	−	0.985462i	$-0.445657\pi$
$17$	1.40099	0.339789	0.169894	+	0.985462i	$0.445657\pi$
$18$	0	0
$19$	−3.11408	−0.714420	−0.357210	−	0.934024i	$-0.616272\pi$
$19$	−3.11408	−0.714420	−0.357210	+	0.934024i	$0.616272\pi$
$20$	13.9750	3.12491
$21$	0	0
$22$	−5.33503	−1.13743
$23$	3.27197	0.682252	0.341126	−	0.940018i	$-0.389192\pi$
$23$	3.27197	0.682252	0.341126	+	0.940018i	$0.389192\pi$
$24$	0	0
$25$	2.44927	0.489854
$26$	8.03590	1.57597
$27$	0	0
$28$	−5.12030	−0.967646
$29$	6.12906	1.13814	0.569069	−	0.822290i	$-0.307304\pi$
$29$	6.12906	1.13814	0.569069	+	0.822290i	$0.307304\pi$
$30$	0	0
$31$	6.08476	1.09286	0.546428	−	0.837506i	$-0.315987\pi$
$31$	6.08476	1.09286	0.546428	+	0.837506i	$0.315987\pi$
$32$	15.3066	2.70585
$33$	0	0
$34$	3.73838	0.641126
$35$	−2.72934	−0.461342
$36$	0	0
$37$	−5.42709	−0.892208	−0.446104	−	0.894981i	$-0.647189\pi$
$37$	−5.42709	−0.892208	−0.446104	+	0.894981i	$0.647189\pi$
$38$	−8.30959	−1.34799
$39$	0	0
$40$	22.7249	3.59313
$41$	−11.6265	−1.81575	−0.907875	−	0.419241i	$-0.862296\pi$
$41$	−11.6265	−1.81575	−0.907875	+	0.419241i	$0.862296\pi$
$42$	0	0
$43$	9.33583	1.42370	0.711850	−	0.702331i	$-0.247858\pi$
$43$	9.33583	1.42370	0.711850	+	0.702331i	$0.247858\pi$
$44$	−10.2373	−1.54332
$45$	0	0
$46$	8.73088	1.28730
$47$	8.98796	1.31103	0.655515	−	0.755182i	$-0.272452\pi$
$47$	8.98796	1.31103	0.655515	+	0.755182i	$0.272452\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	6.53561	0.924275
$51$	0	0
$52$	15.4199	2.13835
$53$	−4.15642	−0.570928	−0.285464	−	0.958389i	$-0.592148\pi$
$53$	−4.15642	−0.570928	−0.285464	+	0.958389i	$0.592148\pi$
$54$	0	0
$55$	−5.45689	−0.735806
$56$	−8.32617	−1.11263
$57$	0	0
$58$	16.3547	2.14748
$59$	−7.85361	−1.02245	−0.511227	−	0.859446i	$-0.670809\pi$
$59$	−7.85361	−1.02245	−0.511227	+	0.859446i	$0.670809\pi$
$60$	0	0
$61$	−3.02245	−0.386985	−0.193493	−	0.981102i	$-0.561982\pi$
$61$	−3.02245	−0.386985	−0.193493	+	0.981102i	$0.561982\pi$
$62$	16.2365	2.06204
$63$	0	0
$64$	16.8902	2.11128
$65$	8.21944	1.01950
$66$	0	0
$67$	−7.97480	−0.974276	−0.487138	−	0.873325i	$-0.661959\pi$
$67$	−7.97480	−0.974276	−0.487138	+	0.873325i	$0.661959\pi$
$68$	7.17347	0.869911
$69$	0	0
$70$	−7.28293	−0.870476
$71$	−4.60611	−0.546644	−0.273322	−	0.961923i	$-0.588123\pi$
$71$	−4.60611	−0.546644	−0.273322	+	0.961923i	$0.588123\pi$
$72$	0	0
$73$	6.75870	0.791046	0.395523	−	0.918456i	$-0.370563\pi$
$73$	6.75870	0.791046	0.395523	+	0.918456i	$0.370563\pi$
$74$	−14.4816	−1.68345
$75$	0	0
$76$	−15.9450	−1.82902
$77$	1.99935	0.227847
$78$	0	0
$79$	0.469412	0.0528130	0.0264065	−	0.999651i	$-0.491594\pi$
$79$	0.469412	0.0528130	0.0264065	+	0.999651i	$0.491594\pi$
$80$	32.6889	3.65473
$81$	0	0
$82$	−31.0240	−3.42602
$83$	11.7285	1.28737	0.643686	−	0.765290i	$-0.277404\pi$
$83$	11.7285	1.28737	0.643686	+	0.765290i	$0.277404\pi$
$84$	0	0
$85$	3.82376	0.414745
$86$	24.9116	2.68629
$87$	0	0
$88$	−16.6469	−1.77457
$89$	−1.25580	−0.133115	−0.0665575	−	0.997783i	$-0.521202\pi$
$89$	−1.25580	−0.133115	−0.0665575	+	0.997783i	$0.521202\pi$
$90$	0	0
$91$	−3.01152	−0.315693
$92$	16.7534	1.74667
$93$	0	0
$94$	23.9834	2.47370
$95$	−8.49938	−0.872018
$96$	0	0
$97$	5.52320	0.560796	0.280398	−	0.959884i	$-0.409534\pi$
$97$	5.52320	0.560796	0.280398	+	0.959884i	$0.409534\pi$
$98$	2.66839	0.269548
$99$	0	0
$100$	12.5410	1.25410
$101$	−8.58267	−0.854007	−0.427004	−	0.904250i	$-0.640431\pi$
$101$	−8.58267	−0.854007	−0.427004	+	0.904250i	$0.640431\pi$
$102$	0	0
$103$	16.4007	1.61601	0.808003	−	0.589178i	$-0.200548\pi$
$103$	16.4007	1.61601	0.808003	+	0.589178i	$0.200548\pi$
$104$	25.0744	2.45875
$105$	0	0
$106$	−11.0910	−1.07725
$107$	−11.3225	−1.09458	−0.547292	−	0.836942i	$-0.684341\pi$
$107$	−11.3225	−1.09458	−0.547292	+	0.836942i	$0.684341\pi$
$108$	0	0
$109$	−3.94629	−0.377986	−0.188993	−	0.981978i	$-0.560522\pi$
$109$	−3.94629	−0.377986	−0.188993	+	0.981978i	$0.560522\pi$
$110$	−14.5611	−1.38835
$111$	0	0
$112$	−11.9769	−1.13171
$113$	5.99606	0.564062	0.282031	−	0.959405i	$-0.408992\pi$
$113$	5.99606	0.564062	0.282031	+	0.959405i	$0.408992\pi$
$114$	0	0
$115$	8.93029	0.832754
$116$	31.3826	2.91380
$117$	0	0
$118$	−20.9565	−1.92920
$119$	−1.40099	−0.128428
$120$	0	0
$121$	−7.00261	−0.636601
$122$	−8.06508	−0.730178
$123$	0	0
$124$	31.1558	2.79787
$125$	−6.96179	−0.622682
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	14.4565	1.27779
$129$	0	0
$130$	21.9327	1.92362
$131$	3.55675	0.310755	0.155377	−	0.987855i	$-0.450341\pi$
$131$	3.55675	0.310755	0.155377	+	0.987855i	$0.450341\pi$
$132$	0	0
$133$	3.11408	0.270025
$134$	−21.2799	−1.83830
$135$	0	0
$136$	11.6649	1.00025
$137$	8.28086	0.707482	0.353741	−	0.935343i	$-0.384909\pi$
$137$	8.28086	0.707482	0.353741	+	0.935343i	$0.384909\pi$
$138$	0	0
$139$	7.68160	0.651545	0.325772	−	0.945448i	$-0.394376\pi$
$139$	7.68160	0.651545	0.325772	+	0.945448i	$0.394376\pi$
$140$	−13.9750	−1.18110
$141$	0	0
$142$	−12.2909	−1.03143
$143$	−6.02106	−0.503507
$144$	0	0
$145$	16.7282	1.38921
$146$	18.0348	1.49257
$147$	0	0
$148$	−27.7883	−2.28419
$149$	−5.88761	−0.482332	−0.241166	−	0.970484i	$-0.577530\pi$
$149$	−5.88761	−0.482332	−0.241166	+	0.970484i	$0.577530\pi$
$150$	0	0
$151$	10.9793	0.893481	0.446741	−	0.894663i	$-0.352585\pi$
$151$	10.9793	0.893481	0.446741	+	0.894663i	$0.352585\pi$
$152$	−25.9284	−2.10307
$153$	0	0
$154$	5.33503	0.429909
$155$	16.6074	1.33393
$156$	0	0
$157$	−12.7199	−1.01516	−0.507578	−	0.861606i	$-0.669459\pi$
$157$	−12.7199	−1.01516	−0.507578	+	0.861606i	$0.669459\pi$
$158$	1.25257	0.0996494
$159$	0	0
$160$	41.7769	3.30275
$161$	−3.27197	−0.257867
$162$	0	0
$163$	−11.7452	−0.919954	−0.459977	−	0.887931i	$-0.652142\pi$
$163$	−11.7452	−0.919954	−0.459977	+	0.887931i	$0.652142\pi$
$164$	−59.5310	−4.64859
$165$	0	0
$166$	31.2962	2.42906
$167$	12.2804	0.950284	0.475142	−	0.879909i	$-0.342397\pi$
$167$	12.2804	0.950284	0.475142	+	0.879909i	$0.342397\pi$
$168$	0	0
$169$	−3.93077	−0.302367
$170$	10.2033	0.782556
$171$	0	0
$172$	47.8022	3.64489
$173$	−12.6768	−0.963797	−0.481898	−	0.876227i	$-0.660053\pi$
$173$	−12.6768	−0.963797	−0.481898	+	0.876227i	$0.660053\pi$
$174$	0	0
$175$	−2.44927	−0.185147
$176$	−23.9459	−1.80499
$177$	0	0
$178$	−3.35097	−0.251166
$179$	−4.76245	−0.355962	−0.177981	−	0.984034i	$-0.556957\pi$
$179$	−4.76245	−0.355962	−0.177981	+	0.984034i	$0.556957\pi$
$180$	0	0
$181$	−12.4933	−0.928621	−0.464311	−	0.885672i	$-0.653698\pi$
$181$	−12.4933	−0.928621	−0.464311	+	0.885672i	$0.653698\pi$
$182$	−8.03590	−0.595660
$183$	0	0
$184$	27.2430	2.00838
$185$	−14.8123	−1.08902
$186$	0	0
$187$	−2.80106	−0.204834
$188$	46.0211	3.35643
$189$	0	0
$190$	−22.6797	−1.64535
$191$	0.831219	0.0601449	0.0300725	−	0.999548i	$-0.490426\pi$
$191$	0.831219	0.0601449	0.0300725	+	0.999548i	$0.490426\pi$
$192$	0	0
$193$	1.79558	0.129248	0.0646242	−	0.997910i	$-0.479415\pi$
$193$	1.79558	0.129248	0.0646242	+	0.997910i	$0.479415\pi$
$194$	14.7380	1.05813
$195$	0	0
$196$	5.12030	0.365736
$197$	8.42435	0.600210	0.300105	−	0.953906i	$-0.402978\pi$
$197$	8.42435	0.600210	0.300105	+	0.953906i	$0.402978\pi$
$198$	0	0
$199$	−9.93932	−0.704579	−0.352290	−	0.935891i	$-0.614597\pi$
$199$	−9.93932	−0.704579	−0.352290	+	0.935891i	$0.614597\pi$
$200$	20.3931	1.44201
$201$	0	0
$202$	−22.9019	−1.61137
$203$	−6.12906	−0.430175
$204$	0	0
$205$	−31.7325	−2.21630
$206$	43.7634	3.04914
$207$	0	0
$208$	36.0685	2.50090
$209$	6.22613	0.430671
$210$	0	0
$211$	−10.1516	−0.698866	−0.349433	−	0.936961i	$-0.613626\pi$
$211$	−10.1516	−0.698866	−0.349433	+	0.936961i	$0.613626\pi$
$212$	−21.2821	−1.46166
$213$	0	0
$214$	−30.2128	−2.06530
$215$	25.4806	1.73776
$216$	0	0
$217$	−6.08476	−0.413061
$218$	−10.5302	−0.713197
$219$	0	0
$220$	−27.9409	−1.88377
$221$	4.21909	0.283807
$222$	0	0
$223$	−2.88646	−0.193292	−0.0966459	−	0.995319i	$-0.530811\pi$
$223$	−2.88646	−0.193292	−0.0966459	+	0.995319i	$0.530811\pi$
$224$	−15.3066	−1.02272
$225$	0	0
$226$	15.9998	1.06429
$227$	10.9582	0.727324	0.363662	−	0.931531i	$-0.381526\pi$
$227$	10.9582	0.727324	0.363662	+	0.931531i	$0.381526\pi$
$228$	0	0
$229$	10.3824	0.686085	0.343043	−	0.939320i	$-0.388542\pi$
$229$	10.3824	0.686085	0.343043	+	0.939320i	$0.388542\pi$
$230$	23.8295	1.57127
$231$	0	0
$232$	51.0316	3.35039
$233$	3.16488	0.207338	0.103669	−	0.994612i	$-0.466942\pi$
$233$	3.16488	0.207338	0.103669	+	0.994612i	$0.466942\pi$
$234$	0	0
$235$	24.5312	1.60024
$236$	−40.2129	−2.61763
$237$	0	0
$238$	−3.73838	−0.242323
$239$	22.4857	1.45448	0.727238	−	0.686385i	$-0.240804\pi$
$239$	22.4857	1.45448	0.727238	+	0.686385i	$0.240804\pi$
$240$	0	0
$241$	−9.57165	−0.616564	−0.308282	−	0.951295i	$-0.599754\pi$
$241$	−9.57165	−0.616564	−0.308282	+	0.951295i	$0.599754\pi$
$242$	−18.6857	−1.20116
$243$	0	0
$244$	−15.4759	−0.990741
$245$	2.72934	0.174371
$246$	0	0
$247$	−9.37811	−0.596715
$248$	50.6628	3.21709
$249$	0	0
$250$	−18.5768	−1.17490
$251$	−29.8157	−1.88195	−0.940975	−	0.338477i	$-0.890088\pi$
$251$	−29.8157	−1.88195	−0.940975	+	0.338477i	$0.890088\pi$
$252$	0	0
$253$	−6.54179	−0.411279
$254$	−2.66839	−0.167430
$255$	0	0
$256$	4.79510	0.299694
$257$	−16.4928	−1.02879	−0.514395	−	0.857553i	$-0.671984\pi$
$257$	−16.4928	−1.02879	−0.514395	+	0.857553i	$0.671984\pi$
$258$	0	0
$259$	5.42709	0.337223
$260$	42.0860	2.61006
$261$	0	0
$262$	9.49079	0.586343
$263$	−1.18225	−0.0729007	−0.0364504	−	0.999335i	$-0.511605\pi$
$263$	−1.18225	−0.0729007	−0.0364504	+	0.999335i	$0.511605\pi$
$264$	0	0
$265$	−11.3443	−0.696873
$266$	8.30959	0.509494
$267$	0	0
$268$	−40.8334	−2.49429
$269$	27.7889	1.69432	0.847159	−	0.531340i	$-0.178311\pi$
$269$	27.7889	1.69432	0.847159	+	0.531340i	$0.178311\pi$
$270$	0	0
$271$	−25.6122	−1.55583	−0.777914	−	0.628370i	$-0.783722\pi$
$271$	−25.6122	−1.55583	−0.777914	+	0.628370i	$0.783722\pi$
$272$	16.7794	1.01740
$273$	0	0
$274$	22.0966	1.33490
$275$	−4.89694	−0.295297
$276$	0	0
$277$	−27.1388	−1.63061	−0.815305	−	0.579032i	$-0.803431\pi$
$277$	−27.1388	−1.63061	−0.815305	+	0.579032i	$0.803431\pi$
$278$	20.4975	1.22936
$279$	0	0
$280$	−22.7249	−1.35807
$281$	−23.0038	−1.37229	−0.686145	−	0.727465i	$-0.740698\pi$
$281$	−23.0038	−1.37229	−0.686145	+	0.727465i	$0.740698\pi$
$282$	0	0
$283$	7.47796	0.444518	0.222259	−	0.974988i	$-0.428657\pi$
$283$	7.47796	0.444518	0.222259	+	0.974988i	$0.428657\pi$
$284$	−23.5847	−1.39949
$285$	0	0
$286$	−16.0665	−0.950035
$287$	11.6265	0.686289
$288$	0	0
$289$	−15.0372	−0.884543
$290$	44.6375	2.62120
$291$	0	0
$292$	34.6066	2.02520
$293$	27.3558	1.59814	0.799072	−	0.601235i	$-0.205324\pi$
$293$	27.3558	1.59814	0.799072	+	0.601235i	$0.205324\pi$
$294$	0	0
$295$	−21.4351	−1.24800
$296$	−45.1869	−2.62643
$297$	0	0
$298$	−15.7104	−0.910081
$299$	9.85358	0.569847
$300$	0	0
$301$	−9.33583	−0.538108
$302$	29.2970	1.68585
$303$	0	0
$304$	−37.2970	−2.13913
$305$	−8.24929	−0.472353
$306$	0	0
$307$	−32.6676	−1.86444	−0.932219	−	0.361894i	$-0.882130\pi$
$307$	−32.6676	−1.86444	−0.932219	+	0.361894i	$0.882130\pi$
$308$	10.2373	0.583322
$309$	0	0
$310$	44.3149	2.51692
$311$	−11.1239	−0.630779	−0.315390	−	0.948962i	$-0.602135\pi$
$311$	−11.1239	−0.630779	−0.315390	+	0.948962i	$0.602135\pi$
$312$	0	0
$313$	19.7840	1.11826	0.559130	−	0.829080i	$-0.311135\pi$
$313$	19.7840	1.11826	0.559130	+	0.829080i	$0.311135\pi$
$314$	−33.9416	−1.91543
$315$	0	0
$316$	2.40353	0.135209
$317$	22.6405	1.27162	0.635808	−	0.771847i	$-0.280667\pi$
$317$	22.6405	1.27162	0.635808	+	0.771847i	$0.280667\pi$
$318$	0	0
$319$	−12.2541	−0.686098
$320$	46.0991	2.57702
$321$	0	0
$322$	−8.73088	−0.486553
$323$	−4.36279	−0.242752
$324$	0	0
$325$	7.37602	0.409148
$326$	−31.3407	−1.73580
$327$	0	0
$328$	−96.8041	−5.34511
$329$	−8.98796	−0.495523
$330$	0	0
$331$	4.65062	0.255621	0.127811	−	0.991799i	$-0.459205\pi$
$331$	4.65062	0.255621	0.127811	+	0.991799i	$0.459205\pi$
$332$	60.0535	3.29587
$333$	0	0
$334$	32.7688	1.79303
$335$	−21.7659	−1.18920
$336$	0	0
$337$	−13.8999	−0.757175	−0.378587	−	0.925566i	$-0.623590\pi$
$337$	−13.8999	−0.757175	−0.378587	+	0.925566i	$0.623590\pi$
$338$	−10.4888	−0.570517
$339$	0	0
$340$	19.5788	1.06181
$341$	−12.1655	−0.658801
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	77.7317	4.19101
$345$	0	0
$346$	−33.8265	−1.81853
$347$	−6.36039	−0.341444	−0.170722	−	0.985319i	$-0.554610\pi$
$347$	−6.36039	−0.341444	−0.170722	+	0.985319i	$0.554610\pi$
$348$	0	0
$349$	−12.0881	−0.647063	−0.323531	−	0.946217i	$-0.604870\pi$
$349$	−12.0881	−0.647063	−0.323531	+	0.946217i	$0.604870\pi$
$350$	−6.53561	−0.349343
$351$	0	0
$352$	−30.6032	−1.63116
$353$	22.9178	1.21979	0.609897	−	0.792481i	$-0.291211\pi$
$353$	22.9178	1.21979	0.609897	+	0.792481i	$0.291211\pi$
$354$	0	0
$355$	−12.5716	−0.667232
$356$	−6.43010	−0.340794
$357$	0	0
$358$	−12.7081	−0.671642
$359$	16.3424	0.862519	0.431259	−	0.902228i	$-0.358069\pi$
$359$	16.3424	0.862519	0.431259	+	0.902228i	$0.358069\pi$
$360$	0	0
$361$	−9.30248	−0.489604
$362$	−33.3370	−1.75216
$363$	0	0
$364$	−15.4199	−0.808221
$365$	18.4468	0.965548
$366$	0	0
$367$	21.2723	1.11040	0.555202	−	0.831716i	$-0.312641\pi$
$367$	21.2723	1.11040	0.555202	+	0.831716i	$0.312641\pi$
$368$	39.1879	2.04281
$369$	0	0
$370$	−39.5251	−2.05481
$371$	4.15642	0.215791
$372$	0	0
$373$	−37.0144	−1.91653	−0.958266	−	0.285879i	$-0.907714\pi$
$373$	−37.0144	−1.91653	−0.958266	+	0.285879i	$0.907714\pi$
$374$	−7.47431	−0.386487
$375$	0	0
$376$	74.8354	3.85934
$377$	18.4578	0.950623
$378$	0	0
$379$	−14.0808	−0.723282	−0.361641	−	0.932317i	$-0.617783\pi$
$379$	−14.0808	−0.723282	−0.361641	+	0.932317i	$0.617783\pi$
$380$	−43.5194	−2.23250
$381$	0	0
$382$	2.21802	0.113484
$383$	10.2494	0.523719	0.261859	−	0.965106i	$-0.415664\pi$
$383$	10.2494	0.523719	0.261859	+	0.965106i	$0.415664\pi$
$384$	0	0
$385$	5.45689	0.278109
$386$	4.79130	0.243871
$387$	0	0
$388$	28.2804	1.43572
$389$	−28.1153	−1.42550	−0.712752	−	0.701416i	$-0.752551\pi$
$389$	−28.1153	−1.42550	−0.712752	+	0.701416i	$0.752551\pi$
$390$	0	0
$391$	4.58398	0.231822
$392$	8.32617	0.420535
$393$	0	0
$394$	22.4795	1.13250
$395$	1.28118	0.0644633
$396$	0	0
$397$	24.5374	1.23150	0.615750	−	0.787942i	$-0.288853\pi$
$397$	24.5374	1.23150	0.615750	+	0.787942i	$0.288853\pi$
$398$	−26.5220	−1.32943
$399$	0	0
$400$	29.3346	1.46673
$401$	−3.73523	−0.186528	−0.0932642	−	0.995641i	$-0.529730\pi$
$401$	−3.73523	−0.186528	−0.0932642	+	0.995641i	$0.529730\pi$
$402$	0	0
$403$	18.3244	0.912801
$404$	−43.9458	−2.18639
$405$	0	0
$406$	−16.3547	−0.811670
$407$	10.8506	0.537846
$408$	0	0
$409$	−6.79937	−0.336207	−0.168104	−	0.985769i	$-0.553764\pi$
$409$	−6.79937	−0.336207	−0.168104	+	0.985769i	$0.553764\pi$
$410$	−84.6748	−4.18179
$411$	0	0
$412$	83.9764	4.13722
$413$	7.85361	0.386451
$414$	0	0
$415$	32.0111	1.57136
$416$	46.0961	2.26005
$417$	0	0
$418$	16.6137	0.812605
$419$	−8.33477	−0.407180	−0.203590	−	0.979056i	$-0.565261\pi$
$419$	−8.33477	−0.407180	−0.203590	+	0.979056i	$0.565261\pi$
$420$	0	0
$421$	−36.9662	−1.80162	−0.900811	−	0.434211i	$-0.857027\pi$
$421$	−36.9662	−1.80162	−0.900811	+	0.434211i	$0.857027\pi$
$422$	−27.0885	−1.31865
$423$	0	0
$424$	−34.6071	−1.68067
$425$	3.43139	0.166447
$426$	0	0
$427$	3.02245	0.146267
$428$	−57.9744	−2.80230
$429$	0	0
$430$	67.9922	3.27887
$431$	28.8287	1.38863	0.694314	−	0.719672i	$-0.255708\pi$
$431$	28.8287	1.38863	0.694314	+	0.719672i	$0.255708\pi$
$432$	0	0
$433$	38.6827	1.85897	0.929487	−	0.368856i	$-0.120250\pi$
$433$	38.6827	1.85897	0.929487	+	0.368856i	$0.120250\pi$
$434$	−16.2365	−0.779377
$435$	0	0
$436$	−20.2062	−0.967701
$437$	−10.1892	−0.487414
$438$	0	0
$439$	8.17779	0.390305	0.195152	−	0.980773i	$-0.437480\pi$
$439$	8.17779	0.390305	0.195152	+	0.980773i	$0.437480\pi$
$440$	−45.4350	−2.16603
$441$	0	0
$442$	11.2582	0.535497
$443$	−6.23436	−0.296203	−0.148102	−	0.988972i	$-0.547316\pi$
$443$	−6.23436	−0.296203	−0.148102	+	0.988972i	$0.547316\pi$
$444$	0	0
$445$	−3.42751	−0.162480
$446$	−7.70220	−0.364710
$447$	0	0
$448$	−16.8902	−0.797988
$449$	28.8788	1.36287	0.681437	−	0.731877i	$-0.261355\pi$
$449$	28.8788	1.36287	0.681437	+	0.731877i	$0.261355\pi$
$450$	0	0
$451$	23.2454	1.09458
$452$	30.7016	1.44408
$453$	0	0
$454$	29.2409	1.37234
$455$	−8.21944	−0.385333
$456$	0	0
$457$	−23.1471	−1.08277	−0.541387	−	0.840774i	$-0.682101\pi$
$457$	−23.1471	−1.08277	−0.541387	+	0.840774i	$0.682101\pi$
$458$	27.7042	1.29453
$459$	0	0
$460$	45.7258	2.13198
$461$	−17.6455	−0.821835	−0.410918	−	0.911673i	$-0.634792\pi$
$461$	−17.6455	−0.821835	−0.410918	+	0.911673i	$0.634792\pi$
$462$	0	0
$463$	−9.24765	−0.429775	−0.214887	−	0.976639i	$-0.568938\pi$
$463$	−9.24765	−0.429775	−0.214887	+	0.976639i	$0.568938\pi$
$464$	73.4069	3.40783
$465$	0	0
$466$	8.44512	0.391213
$467$	−7.42242	−0.343468	−0.171734	−	0.985143i	$-0.554937\pi$
$467$	−7.42242	−0.343468	−0.171734	+	0.985143i	$0.554937\pi$
$468$	0	0
$469$	7.97480	0.368242
$470$	65.4587	3.01938
$471$	0	0
$472$	−65.3906	−3.00984
$473$	−18.6656	−0.858243
$474$	0	0
$475$	−7.62724	−0.349962
$476$	−7.17347	−0.328795
$477$	0	0
$478$	60.0005	2.74436
$479$	−27.9533	−1.27722	−0.638610	−	0.769531i	$-0.720490\pi$
$479$	−27.9533	−1.27722	−0.638610	+	0.769531i	$0.720490\pi$
$480$	0	0
$481$	−16.3438	−0.745211
$482$	−25.5409	−1.16336
$483$	0	0
$484$	−35.8555	−1.62979
$485$	15.0747	0.684505
$486$	0	0
$487$	−19.3574	−0.877168	−0.438584	−	0.898690i	$-0.644520\pi$
$487$	−19.3574	−0.877168	−0.438584	+	0.898690i	$0.644520\pi$
$488$	−25.1655	−1.13919
$489$	0	0
$490$	7.28293	0.329009
$491$	−19.1328	−0.863451	−0.431726	−	0.902005i	$-0.642095\pi$
$491$	−19.1328	−0.863451	−0.431726	+	0.902005i	$0.642095\pi$
$492$	0	0
$493$	8.58672	0.386726
$494$	−25.0245	−1.12590
$495$	0	0
$496$	72.8764	3.27225
$497$	4.60611	0.206612
$498$	0	0
$499$	−30.9594	−1.38593	−0.692966	−	0.720970i	$-0.743696\pi$
$499$	−30.9594	−1.38593	−0.692966	+	0.720970i	$0.743696\pi$
$500$	−35.6465	−1.59416
$501$	0	0
$502$	−79.5599	−3.55093
$503$	37.7182	1.68177	0.840886	−	0.541213i	$-0.182035\pi$
$503$	37.7182	1.68177	0.840886	+	0.541213i	$0.182035\pi$
$504$	0	0
$505$	−23.4250	−1.04240
$506$	−17.4560	−0.776016
$507$	0	0
$508$	−5.12030	−0.227177
$509$	30.8055	1.36543	0.682715	−	0.730685i	$-0.260799\pi$
$509$	30.8055	1.36543	0.682715	+	0.730685i	$0.260799\pi$
$510$	0	0
$511$	−6.75870	−0.298987
$512$	−16.1178	−0.712313
$513$	0	0
$514$	−44.0091	−1.94116
$515$	44.7629	1.97249
$516$	0	0
$517$	−17.9701	−0.790322
$518$	14.4816	0.636284
$519$	0	0
$520$	68.4365	3.00114
$521$	−4.96460	−0.217503	−0.108751	−	0.994069i	$-0.534685\pi$
$521$	−4.96460	−0.217503	−0.108751	+	0.994069i	$0.534685\pi$
$522$	0	0
$523$	−10.8510	−0.474481	−0.237241	−	0.971451i	$-0.576243\pi$
$523$	−10.8510	−0.474481	−0.237241	+	0.971451i	$0.576243\pi$
$524$	18.2116	0.795578
$525$	0	0
$526$	−3.15471	−0.137552
$527$	8.52466	0.371340
$528$	0	0
$529$	−12.2942	−0.534532
$530$	−30.2709	−1.31488
$531$	0	0
$532$	15.9450	0.691305
$533$	−35.0133	−1.51660
$534$	0	0
$535$	−30.9028	−1.33605
$536$	−66.3995	−2.86802
$537$	0	0
$538$	74.1515	3.19690
$539$	−1.99935	−0.0861180
$540$	0	0
$541$	−5.98504	−0.257317	−0.128658	−	0.991689i	$-0.541067\pi$
$541$	−5.98504	−0.257317	−0.128658	+	0.991689i	$0.541067\pi$
$542$	−68.3432	−2.93559
$543$	0	0
$544$	21.4443	0.919418
$545$	−10.7707	−0.461368
$546$	0	0
$547$	25.6771	1.09787	0.548936	−	0.835865i	$-0.315033\pi$
$547$	25.6771	1.09787	0.548936	+	0.835865i	$0.315033\pi$
$548$	42.4005	1.81126
$549$	0	0
$550$	−13.0669	−0.557176
$551$	−19.0864	−0.813108
$552$	0	0
$553$	−0.469412	−0.0199614
$554$	−72.4168	−3.07669
$555$	0	0
$556$	39.3321	1.66805
$557$	−46.8158	−1.98365	−0.991825	−	0.127606i	$-0.959271\pi$
$557$	−46.8158	−1.98365	−0.991825	+	0.127606i	$0.959271\pi$
$558$	0	0
$559$	28.1150	1.18914
$560$	−32.6889	−1.38136
$561$	0	0
$562$	−61.3830	−2.58928
$563$	43.1085	1.81681	0.908403	−	0.418095i	$-0.137302\pi$
$563$	43.1085	1.81681	0.908403	+	0.418095i	$0.137302\pi$
$564$	0	0
$565$	16.3653	0.688492
$566$	19.9541	0.838733
$567$	0	0
$568$	−38.3513	−1.60918
$569$	−16.4177	−0.688267	−0.344134	−	0.938921i	$-0.611827\pi$
$569$	−16.4177	−0.688267	−0.344134	+	0.938921i	$0.611827\pi$
$570$	0	0
$571$	3.32790	0.139268	0.0696340	−	0.997573i	$-0.477817\pi$
$571$	3.32790	0.139268	0.0696340	+	0.997573i	$0.477817\pi$
$572$	−30.8297	−1.28905
$573$	0	0
$574$	31.0240	1.29491
$575$	8.01393	0.334204
$576$	0	0
$577$	14.0026	0.582937	0.291468	−	0.956580i	$-0.405856\pi$
$577$	14.0026	0.582937	0.291468	+	0.956580i	$0.405856\pi$
$578$	−40.1252	−1.66899
$579$	0	0
$580$	85.6537	3.55657
$581$	−11.7285	−0.486581
$582$	0	0
$583$	8.31013	0.344170
$584$	56.2741	2.32864
$585$	0	0
$586$	72.9960	3.01544
$587$	27.3082	1.12713	0.563565	−	0.826072i	$-0.309430\pi$
$587$	27.3082	1.12713	0.563565	+	0.826072i	$0.309430\pi$
$588$	0	0
$589$	−18.9485	−0.780758
$590$	−57.1973	−2.35478
$591$	0	0
$592$	−64.9995	−2.67146
$593$	48.2625	1.98190	0.990951	−	0.134222i	$-0.0428535\pi$
$593$	48.2625	1.98190	0.990951	+	0.134222i	$0.0428535\pi$
$594$	0	0
$595$	−3.82376	−0.156759
$596$	−30.1463	−1.23484
$597$	0	0
$598$	26.2932	1.07521
$599$	−0.325068	−0.0132819	−0.00664096	−	0.999978i	$-0.502114\pi$
$599$	−0.325068	−0.0132819	−0.00664096	+	0.999978i	$0.502114\pi$
$600$	0	0
$601$	29.3585	1.19756	0.598778	−	0.800915i	$-0.295653\pi$
$601$	29.3585	1.19756	0.598778	+	0.800915i	$0.295653\pi$
$602$	−24.9116	−1.01532
$603$	0	0
$604$	56.2172	2.28745
$605$	−19.1125	−0.777033
$606$	0	0
$607$	32.8877	1.33487	0.667434	−	0.744669i	$-0.267392\pi$
$607$	32.8877	1.33487	0.667434	+	0.744669i	$0.267392\pi$
$608$	−47.6661	−1.93311
$609$	0	0
$610$	−22.0123	−0.891252
$611$	27.0674	1.09503
$612$	0	0
$613$	−39.2994	−1.58729	−0.793643	−	0.608384i	$-0.791818\pi$
$613$	−39.2994	−1.58729	−0.793643	+	0.608384i	$0.791818\pi$
$614$	−87.1699	−3.51789
$615$	0	0
$616$	16.6469	0.670723
$617$	−29.4716	−1.18648	−0.593241	−	0.805025i	$-0.702152\pi$
$617$	−29.4716	−1.18648	−0.593241	+	0.805025i	$0.702152\pi$
$618$	0	0
$619$	24.7233	0.993712	0.496856	−	0.867833i	$-0.334488\pi$
$619$	24.7233	0.993712	0.496856	+	0.867833i	$0.334488\pi$
$620$	85.0346	3.41507
$621$	0	0
$622$	−29.6829	−1.19018
$623$	1.25580	0.0503127
$624$	0	0
$625$	−31.2474	−1.24990
$626$	52.7915	2.10997
$627$	0	0
$628$	−65.1296	−2.59895
$629$	−7.60327	−0.303162
$630$	0	0
$631$	46.9182	1.86778	0.933892	−	0.357555i	$-0.116389\pi$
$631$	46.9182	1.86778	0.933892	+	0.357555i	$0.116389\pi$
$632$	3.90841	0.155468
$633$	0	0
$634$	60.4136	2.39933
$635$	−2.72934	−0.108310
$636$	0	0
$637$	3.01152	0.119321
$638$	−32.6987	−1.29455
$639$	0	0
$640$	39.4566	1.55966
$641$	44.7058	1.76577	0.882887	−	0.469585i	$-0.155597\pi$
$641$	44.7058	1.76577	0.882887	+	0.469585i	$0.155597\pi$
$642$	0	0
$643$	7.91721	0.312224	0.156112	−	0.987739i	$-0.450104\pi$
$643$	7.91721	0.312224	0.156112	+	0.987739i	$0.450104\pi$
$644$	−16.7534	−0.660178
$645$	0	0
$646$	−11.6416	−0.458033
$647$	24.8064	0.975239	0.487619	−	0.873056i	$-0.337865\pi$
$647$	24.8064	0.975239	0.487619	+	0.873056i	$0.337865\pi$
$648$	0	0
$649$	15.7021	0.616361
$650$	19.6821	0.771995
$651$	0	0
$652$	−60.1389	−2.35522
$653$	1.70618	0.0667680	0.0333840	−	0.999443i	$-0.489372\pi$
$653$	1.70618	0.0667680	0.0333840	+	0.999443i	$0.489372\pi$
$654$	0	0
$655$	9.70756	0.379306
$656$	−139.249	−5.43675
$657$	0	0
$658$	−23.9834	−0.934970
$659$	29.6522	1.15508	0.577542	−	0.816361i	$-0.304012\pi$
$659$	29.6522	1.15508	0.577542	+	0.816361i	$0.304012\pi$
$660$	0	0
$661$	−2.63972	−0.102673	−0.0513366	−	0.998681i	$-0.516348\pi$
$661$	−2.63972	−0.102673	−0.0513366	+	0.998681i	$0.516348\pi$
$662$	12.4097	0.482316
$663$	0	0
$664$	97.6537	3.78970
$665$	8.49938	0.329592
$666$	0	0
$667$	20.0541	0.776496
$668$	62.8792	2.43287
$669$	0	0
$670$	−58.0799	−2.24382
$671$	6.04293	0.233285
$672$	0	0
$673$	44.6496	1.72112	0.860559	−	0.509351i	$-0.170115\pi$
$673$	44.6496	1.72112	0.860559	+	0.509351i	$0.170115\pi$
$674$	−37.0903	−1.42866
$675$	0	0
$676$	−20.1267	−0.774105
$677$	−29.1270	−1.11944	−0.559721	−	0.828681i	$-0.689092\pi$
$677$	−29.1270	−1.11944	−0.559721	+	0.828681i	$0.689092\pi$
$678$	0	0
$679$	−5.52320	−0.211961
$680$	31.8373	1.22090
$681$	0	0
$682$	−32.4624	−1.24305
$683$	12.1721	0.465751	0.232876	−	0.972507i	$-0.425187\pi$
$683$	12.1721	0.465751	0.232876	+	0.972507i	$0.425187\pi$
$684$	0	0
$685$	22.6012	0.863549
$686$	−2.66839	−0.101880
$687$	0	0
$688$	111.814	4.26287
$689$	−12.5171	−0.476865
$690$	0	0
$691$	2.25104	0.0856337	0.0428168	−	0.999083i	$-0.486367\pi$
$691$	2.25104	0.0856337	0.0428168	+	0.999083i	$0.486367\pi$
$692$	−64.9088	−2.46746
$693$	0	0
$694$	−16.9720	−0.644248
$695$	20.9657	0.795273
$696$	0	0
$697$	−16.2885	−0.616972
$698$	−32.2558	−1.22090
$699$	0	0
$700$	−12.5410	−0.474005
$701$	−4.34622	−0.164154	−0.0820772	−	0.996626i	$-0.526155\pi$
$701$	−4.34622	−0.164154	−0.0820772	+	0.996626i	$0.526155\pi$
$702$	0	0
$703$	16.9004	0.637411
$704$	−33.7694	−1.27273
$705$	0	0
$706$	61.1537	2.30155
$707$	8.58267	0.322784
$708$	0	0
$709$	13.3197	0.500233	0.250116	−	0.968216i	$-0.419531\pi$
$709$	13.3197	0.500233	0.250116	+	0.968216i	$0.419531\pi$
$710$	−33.5460	−1.25896
$711$	0	0
$712$	−10.4560	−0.391857
$713$	19.9091	0.745603
$714$	0	0
$715$	−16.4335	−0.614578
$716$	−24.3852	−0.911316
$717$	0	0
$718$	43.6079	1.62743
$719$	−36.7196	−1.36941	−0.684704	−	0.728821i	$-0.740069\pi$
$719$	−36.7196	−1.36941	−0.684704	+	0.728821i	$0.740069\pi$
$720$	0	0
$721$	−16.4007	−0.610793
$722$	−24.8226	−0.923803
$723$	0	0
$724$	−63.9695	−2.37741
$725$	15.0117	0.557521
$726$	0	0
$727$	13.7988	0.511770	0.255885	−	0.966707i	$-0.417633\pi$
$727$	13.7988	0.511770	0.255885	+	0.966707i	$0.417633\pi$
$728$	−25.0744	−0.929319
$729$	0	0
$730$	49.2232	1.82183
$731$	13.0794	0.483758
$732$	0	0
$733$	−27.1217	−1.00176	−0.500881	−	0.865516i	$-0.666991\pi$
$733$	−27.1217	−1.00176	−0.500881	+	0.865516i	$0.666991\pi$
$734$	56.7627	2.09515
$735$	0	0
$736$	50.0827	1.84607
$737$	15.9444	0.587319
$738$	0	0
$739$	−37.7469	−1.38854	−0.694271	−	0.719713i	$-0.744273\pi$
$739$	−37.7469	−1.38854	−0.694271	+	0.719713i	$0.744273\pi$
$740$	−75.8436	−2.78807
$741$	0	0
$742$	11.0910	0.407162
$743$	−19.9166	−0.730668	−0.365334	−	0.930877i	$-0.619045\pi$
$743$	−19.9166	−0.730668	−0.365334	+	0.930877i	$0.619045\pi$
$744$	0	0
$745$	−16.0693	−0.588732
$746$	−98.7687	−3.61618
$747$	0	0
$748$	−14.3422	−0.524405
$749$	11.3225	0.413714
$750$	0	0
$751$	−38.5087	−1.40520	−0.702602	−	0.711583i	$-0.747978\pi$
$751$	−38.5087	−1.40520	−0.702602	+	0.711583i	$0.747978\pi$
$752$	107.648	3.92551
$753$	0	0
$754$	49.2525	1.79367
$755$	29.9661	1.09058
$756$	0	0
$757$	−27.4467	−0.997569	−0.498784	−	0.866726i	$-0.666220\pi$
$757$	−27.4467	−0.997569	−0.498784	+	0.866726i	$0.666220\pi$
$758$	−37.5731	−1.36472
$759$	0	0
$760$	−70.7673	−2.56700
$761$	−13.4306	−0.486857	−0.243429	−	0.969919i	$-0.578272\pi$
$761$	−13.4306	−0.486857	−0.243429	+	0.969919i	$0.578272\pi$
$762$	0	0
$763$	3.94629	0.142865
$764$	4.25609	0.153980
$765$	0	0
$766$	27.3493	0.988171
$767$	−23.6513	−0.853999
$768$	0	0
$769$	−19.4182	−0.700238	−0.350119	−	0.936705i	$-0.613859\pi$
$769$	−19.4182	−0.700238	−0.350119	+	0.936705i	$0.613859\pi$
$770$	14.5611	0.524746
$771$	0	0
$772$	9.19389	0.330895
$773$	−15.7160	−0.565264	−0.282632	−	0.959228i	$-0.591207\pi$
$773$	−15.7160	−0.565264	−0.282632	+	0.959228i	$0.591207\pi$
$774$	0	0
$775$	14.9032	0.535340
$776$	45.9871	1.65084
$777$	0	0
$778$	−75.0226	−2.68969
$779$	36.2058	1.29721
$780$	0	0
$781$	9.20921	0.329531
$782$	12.2318	0.437410
$783$	0	0
$784$	11.9769	0.427745
$785$	−34.7168	−1.23910
$786$	0	0
$787$	−3.83104	−0.136562	−0.0682809	−	0.997666i	$-0.521751\pi$
$787$	−3.83104	−0.136562	−0.0682809	+	0.997666i	$0.521751\pi$
$788$	43.1352	1.53663
$789$	0	0
$790$	3.41869	0.121632
$791$	−5.99606	−0.213196
$792$	0	0
$793$	−9.10217	−0.323227
$794$	65.4755	2.32364
$795$	0	0
$796$	−50.8923	−1.80383
$797$	22.1663	0.785170	0.392585	−	0.919716i	$-0.371581\pi$
$797$	22.1663	0.785170	0.392585	+	0.919716i	$0.371581\pi$
$798$	0	0
$799$	12.5920	0.445473
$800$	37.4900	1.32547
$801$	0	0
$802$	−9.96705	−0.351949
$803$	−13.5130	−0.476863
$804$	0	0
$805$	−8.93029	−0.314751
$806$	48.8965	1.72231
$807$	0	0
$808$	−71.4608	−2.51398
$809$	−15.5112	−0.545344	−0.272672	−	0.962107i	$-0.587907\pi$
$809$	−15.5112	−0.545344	−0.272672	+	0.962107i	$0.587907\pi$
$810$	0	0
$811$	44.6967	1.56951	0.784757	−	0.619804i	$-0.212788\pi$
$811$	44.6967	1.56951	0.784757	+	0.619804i	$0.212788\pi$
$812$	−31.3826	−1.10131
$813$	0	0
$814$	28.9537	1.01483
$815$	−32.0565	−1.12289
$816$	0	0
$817$	−29.0725	−1.01712
$818$	−18.1434	−0.634368
$819$	0	0
$820$	−162.480	−5.67405
$821$	16.7550	0.584754	0.292377	−	0.956303i	$-0.405554\pi$
$821$	16.7550	0.584754	0.292377	+	0.956303i	$0.405554\pi$
$822$	0	0
$823$	1.72459	0.0601155	0.0300577	−	0.999548i	$-0.490431\pi$
$823$	1.72459	0.0601155	0.0300577	+	0.999548i	$0.490431\pi$
$824$	136.555	4.75711
$825$	0	0
$826$	20.9565	0.729170
$827$	31.4958	1.09521	0.547607	−	0.836735i	$-0.315539\pi$
$827$	31.4958	1.09521	0.547607	+	0.836735i	$0.315539\pi$
$828$	0	0
$829$	−45.0565	−1.56488	−0.782438	−	0.622729i	$-0.786024\pi$
$829$	−45.0565	−1.56488	−0.782438	+	0.622729i	$0.786024\pi$
$830$	85.4179	2.96490
$831$	0	0
$832$	50.8652	1.76343
$833$	1.40099	0.0485413
$834$	0	0
$835$	33.5173	1.15991
$836$	31.8797	1.10258
$837$	0	0
$838$	−22.2404	−0.768282
$839$	−13.4601	−0.464693	−0.232347	−	0.972633i	$-0.574640\pi$
$839$	−13.4601	−0.464693	−0.232347	+	0.972633i	$0.574640\pi$
$840$	0	0
$841$	8.56533	0.295356
$842$	−98.6402	−3.39937
$843$	0	0
$844$	−51.9793	−1.78920
$845$	−10.7284	−0.369068
$846$	0	0
$847$	7.00261	0.240613
$848$	−49.7809	−1.70948
$849$	0	0
$850$	9.15629	0.314058
$851$	−17.7572	−0.608710
$852$	0	0
$853$	−16.0326	−0.548947	−0.274474	−	0.961595i	$-0.588504\pi$
$853$	−16.0326	−0.548947	−0.274474	+	0.961595i	$0.588504\pi$
$854$	8.06508	0.275981
$855$	0	0
$856$	−94.2729	−3.22218
$857$	−39.9267	−1.36387	−0.681934	−	0.731413i	$-0.738861\pi$
$857$	−39.9267	−1.36387	−0.681934	+	0.731413i	$0.738861\pi$
$858$	0	0
$859$	12.7996	0.436718	0.218359	−	0.975869i	$-0.429930\pi$
$859$	12.7996	0.436718	0.218359	+	0.975869i	$0.429930\pi$
$860$	130.468	4.44893
$861$	0	0
$862$	76.9261	2.62011
$863$	−28.2338	−0.961089	−0.480545	−	0.876970i	$-0.659561\pi$
$863$	−28.2338	−0.961089	−0.480545	+	0.876970i	$0.659561\pi$
$864$	0	0
$865$	−34.5991	−1.17641
$866$	103.221	3.50758
$867$	0	0
$868$	−31.1558	−1.05750
$869$	−0.938517	−0.0318370
$870$	0	0
$871$	−24.0162	−0.813759
$872$	−32.8575	−1.11269
$873$	0	0
$874$	−27.1887	−0.919671
$875$	6.96179	0.235352
$876$	0	0
$877$	−26.1834	−0.884151	−0.442076	−	0.896978i	$-0.645758\pi$
$877$	−26.1834	−0.884151	−0.442076	+	0.896978i	$0.645758\pi$
$878$	21.8215	0.736441
$879$	0	0
$880$	−65.3564	−2.20316
$881$	36.1780	1.21887	0.609434	−	0.792837i	$-0.291397\pi$
$881$	36.1780	1.21887	0.609434	+	0.792837i	$0.291397\pi$
$882$	0	0
$883$	43.6742	1.46975	0.734877	−	0.678200i	$-0.237240\pi$
$883$	43.6742	1.46975	0.734877	+	0.678200i	$0.237240\pi$
$884$	21.6030	0.726588
$885$	0	0
$886$	−16.6357	−0.558887
$887$	28.1057	0.943699	0.471849	−	0.881679i	$-0.343587\pi$
$887$	28.1057	0.943699	0.471849	+	0.881679i	$0.343587\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	−9.14593	−0.306572
$891$	0	0
$892$	−14.7795	−0.494856
$893$	−27.9893	−0.936625
$894$	0	0
$895$	−12.9983	−0.434486
$896$	−14.4565	−0.482958
$897$	0	0
$898$	77.0598	2.57152
$899$	37.2938	1.24382
$900$	0	0
$901$	−5.82309	−0.193995
$902$	62.0276	2.06529
$903$	0	0
$904$	49.9243	1.66046
$905$	−34.0985	−1.13347
$906$	0	0
$907$	17.5949	0.584228	0.292114	−	0.956383i	$-0.405641\pi$
$907$	17.5949	0.584228	0.292114	+	0.956383i	$0.405641\pi$
$908$	56.1095	1.86206
$909$	0	0
$910$	−21.9327	−0.727061
$911$	−34.8406	−1.15432	−0.577160	−	0.816631i	$-0.695839\pi$
$911$	−34.8406	−1.15432	−0.577160	+	0.816631i	$0.695839\pi$
$912$	0	0
$913$	−23.4494	−0.776061
$914$	−61.7654	−2.04302
$915$	0	0
$916$	53.1608	1.75648
$917$	−3.55675	−0.117454
$918$	0	0
$919$	−10.6961	−0.352832	−0.176416	−	0.984316i	$-0.556450\pi$
$919$	−10.6961	−0.352832	−0.176416	+	0.984316i	$0.556450\pi$
$920$	74.3552	2.45142
$921$	0	0
$922$	−47.0852	−1.55067
$923$	−13.8714	−0.456582
$924$	0	0
$925$	−13.2924	−0.437052
$926$	−24.6763	−0.810914
$927$	0	0
$928$	93.8150	3.07963
$929$	−52.4842	−1.72195	−0.860976	−	0.508645i	$-0.830146\pi$
$929$	−52.4842	−1.72195	−0.860976	+	0.508645i	$0.830146\pi$
$930$	0	0
$931$	−3.11408	−0.102060
$932$	16.2051	0.530816
$933$	0	0
$934$	−19.8059	−0.648069
$935$	−7.64502	−0.250019
$936$	0	0
$937$	23.2666	0.760088	0.380044	−	0.924968i	$-0.375909\pi$
$937$	23.2666	0.760088	0.380044	+	0.924968i	$0.375909\pi$
$938$	21.2799	0.694812
$939$	0	0
$940$	125.607	4.09685
$941$	−24.1022	−0.785709	−0.392855	−	0.919601i	$-0.628512\pi$
$941$	−24.1022	−0.785709	−0.392855	+	0.919601i	$0.628512\pi$
$942$	0	0
$943$	−38.0414	−1.23880
$944$	−94.0617	−3.06145
$945$	0	0
$946$	−49.8070	−1.61936
$947$	−25.6098	−0.832207	−0.416104	−	0.909317i	$-0.636605\pi$
$947$	−25.6098	−0.832207	−0.416104	+	0.909317i	$0.636605\pi$
$948$	0	0
$949$	20.3539	0.660717
$950$	−20.3524	−0.660320
$951$	0	0
$952$	−11.6649	−0.378060
$953$	46.8742	1.51840	0.759202	−	0.650855i	$-0.225589\pi$
$953$	46.8742	1.51840	0.759202	+	0.650855i	$0.225589\pi$
$954$	0	0
$955$	2.26868	0.0734126
$956$	115.133	3.72368
$957$	0	0
$958$	−74.5903	−2.40990
$959$	−8.28086	−0.267403
$960$	0	0
$961$	6.02432	0.194333
$962$	−43.6115	−1.40609
$963$	0	0
$964$	−49.0097	−1.57850
$965$	4.90073	0.157760
$966$	0	0
$967$	8.35672	0.268734	0.134367	−	0.990932i	$-0.457100\pi$
$967$	8.35672	0.268734	0.134367	+	0.990932i	$0.457100\pi$
$968$	−58.3050	−1.87399
$969$	0	0
$970$	40.2251	1.29155
$971$	−10.5627	−0.338973	−0.169486	−	0.985533i	$-0.554211\pi$
$971$	−10.5627	−0.338973	−0.169486	+	0.985533i	$0.554211\pi$
$972$	0	0
$973$	−7.68160	−0.246261
$974$	−51.6531	−1.65507
$975$	0	0
$976$	−36.1995	−1.15872
$977$	−19.6629	−0.629072	−0.314536	−	0.949246i	$-0.601849\pi$
$977$	−19.6629	−0.629072	−0.314536	+	0.949246i	$0.601849\pi$
$978$	0	0
$979$	2.51079	0.0802451
$980$	13.9750	0.446415
$981$	0	0
$982$	−51.0538	−1.62919
$983$	−31.4186	−1.00210	−0.501049	−	0.865419i	$-0.667052\pi$
$983$	−31.4186	−1.00210	−0.501049	+	0.865419i	$0.667052\pi$
$984$	0	0
$985$	22.9929	0.732614
$986$	22.9127	0.729689
$987$	0	0
$988$	−48.0188	−1.52768
$989$	30.5465	0.971322
$990$	0	0
$991$	−30.8481	−0.979923	−0.489962	−	0.871744i	$-0.662989\pi$
$991$	−30.8481	−0.979923	−0.489962	+	0.871744i	$0.662989\pi$
$992$	93.1370	2.95710
$993$	0	0
$994$	12.2909	0.389843
$995$	−27.1277	−0.860007
$996$	0	0
$997$	−26.5734	−0.841588	−0.420794	−	0.907156i	$-0.638248\pi$
$997$	−26.5734	−0.841588	−0.420794	+	0.907156i	$0.638248\pi$
$998$	−82.6117	−2.61503
$999$	0	0

Twists

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

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

\(f(q)\)	\(=\)	\(q+2.66839 q^{2} +5.12030 q^{4} +2.72934 q^{5} -1.00000 q^{7} +8.32617 q^{8} +O(q^{10})\)	\(q+2.66839 q^{2} +5.12030 q^{4} +2.72934 q^{5} -1.00000 q^{7} +8.32617 q^{8} +7.28293 q^{10} -1.99935 q^{11} +3.01152 q^{13} -2.66839 q^{14} +11.9769 q^{16} +1.40099 q^{17} -3.11408 q^{19} +13.9750 q^{20} -5.33503 q^{22} +3.27197 q^{23} +2.44927 q^{25} +8.03590 q^{26} -5.12030 q^{28} +6.12906 q^{29} +6.08476 q^{31} +15.3066 q^{32} +3.73838 q^{34} -2.72934 q^{35} -5.42709 q^{37} -8.30959 q^{38} +22.7249 q^{40} -11.6265 q^{41} +9.33583 q^{43} -10.2373 q^{44} +8.73088 q^{46} +8.98796 q^{47} +1.00000 q^{49} +6.53561 q^{50} +15.4199 q^{52} -4.15642 q^{53} -5.45689 q^{55} -8.32617 q^{56} +16.3547 q^{58} -7.85361 q^{59} -3.02245 q^{61} +16.2365 q^{62} +16.8902 q^{64} +8.21944 q^{65} -7.97480 q^{67} +7.17347 q^{68} -7.28293 q^{70} -4.60611 q^{71} +6.75870 q^{73} -14.4816 q^{74} -15.9450 q^{76} +1.99935 q^{77} +0.469412 q^{79} +32.6889 q^{80} -31.0240 q^{82} +11.7285 q^{83} +3.82376 q^{85} +24.9116 q^{86} -16.6469 q^{88} -1.25580 q^{89} -3.01152 q^{91} +16.7534 q^{92} +23.9834 q^{94} -8.49938 q^{95} +5.52320 q^{97} +2.66839 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 5 q^{2} + 15 q^{4} + 4 q^{5} - 14 q^{7} + 12 q^{8}+O(q^{10}) \) 14 * q + 5 * q^2 + 15 * q^4 + 4 * q^5 - 14 * q^7 + 12 * q^8	\( 14 q + 5 q^{2} + 15 q^{4} + 4 q^{5} - 14 q^{7} + 12 q^{8} + 4 q^{10} + 3 q^{11} - 13 q^{13} - 5 q^{14} + 13 q^{16} + 5 q^{17} + 21 q^{19} + 3 q^{20} - 3 q^{22} + 10 q^{23} + 4 q^{25} + 6 q^{26} - 15 q^{28} + 15 q^{29} + 33 q^{31} + 29 q^{32} + 28 q^{34} - 4 q^{35} - 29 q^{37} + 15 q^{38} + 3 q^{40} + q^{41} - 25 q^{43} + 26 q^{44} - 4 q^{46} + 9 q^{47} + 14 q^{49} + 28 q^{50} - 13 q^{52} + 35 q^{53} + 14 q^{55} - 12 q^{56} - 23 q^{58} - 10 q^{59} + q^{61} + 43 q^{62} - 2 q^{64} + 24 q^{65} - 38 q^{67} + 2 q^{68} - 4 q^{70} + 10 q^{71} + 8 q^{73} + 25 q^{74} + 26 q^{76} - 3 q^{77} + 26 q^{79} + 48 q^{80} + 6 q^{82} + 30 q^{83} - 32 q^{85} + 50 q^{86} - 29 q^{88} - 4 q^{89} + 13 q^{91} + 32 q^{92} - 7 q^{94} + 32 q^{95} + 15 q^{97} + 5 q^{98}+O(q^{100}) \) 14 * q + 5 * q^2 + 15 * q^4 + 4 * q^5 - 14 * q^7 + 12 * q^8 + 4 * q^10 + 3 * q^11 - 13 * q^13 - 5 * q^14 + 13 * q^16 + 5 * q^17 + 21 * q^19 + 3 * q^20 - 3 * q^22 + 10 * q^23 + 4 * q^25 + 6 * q^26 - 15 * q^28 + 15 * q^29 + 33 * q^31 + 29 * q^32 + 28 * q^34 - 4 * q^35 - 29 * q^37 + 15 * q^38 + 3 * q^40 + q^41 - 25 * q^43 + 26 * q^44 - 4 * q^46 + 9 * q^47 + 14 * q^49 + 28 * q^50 - 13 * q^52 + 35 * q^53 + 14 * q^55 - 12 * q^56 - 23 * q^58 - 10 * q^59 + q^61 + 43 * q^62 - 2 * q^64 + 24 * q^65 - 38 * q^67 + 2 * q^68 - 4 * q^70 + 10 * q^71 + 8 * q^73 + 25 * q^74 + 26 * q^76 - 3 * q^77 + 26 * q^79 + 48 * q^80 + 6 * q^82 + 30 * q^83 - 32 * q^85 + 50 * q^86 - 29 * q^88 - 4 * q^89 + 13 * q^91 + 32 * q^92 - 7 * q^94 + 32 * q^95 + 15 * q^97 + 5 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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