LMFDB - Embedded newform 8001.2.a.w.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("8001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$63.8883066572$
Analytic rank:	$1$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 8 x^{19} + 152 x^{17} - 274 x^{16} - 1061 x^{15} + 3125 x^{14} + 2977 x^{13} - 15474 x^{12} + \cdots + 31 $ x^20 - 8x^19 + 152x^17 - 274x^16 - 1061x^15 + 3125x^14 + 2977x^13 - 15474x^12 - 56x^11 + 39579x^10 - 17664x^9 - 52271x^8 + 35701x^7 + 32493x^6 - 25504x^5 - 8607x^4 + 6812x^3 + 609x^2 - 425x + 31
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 889)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Root			$-0.343534$ of defining polynomial
Character	$\chi$	$=$	8001.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.343534 q^{2} -1.88198 q^{4} +4.21457 q^{5} +1.00000 q^{7} -1.33360 q^{8} +O(q^{10})$	\(q+0.343534 q^{2} -1.88198 q^{4} +4.21457 q^{5} +1.00000 q^{7} -1.33360 q^{8} +1.44785 q^{10} +0.252256 q^{11} -6.96098 q^{13} +0.343534 q^{14} +3.30583 q^{16} -2.67429 q^{17} -7.42951 q^{19} -7.93176 q^{20} +0.0866585 q^{22} -0.137677 q^{23} +12.7626 q^{25} -2.39134 q^{26} -1.88198 q^{28} +2.53359 q^{29} +2.20068 q^{31} +3.80286 q^{32} -0.918712 q^{34} +4.21457 q^{35} +7.40329 q^{37} -2.55229 q^{38} -5.62054 q^{40} +4.80680 q^{41} +2.71385 q^{43} -0.474741 q^{44} -0.0472967 q^{46} -5.23128 q^{47} +1.00000 q^{49} +4.38440 q^{50} +13.1005 q^{52} -7.52694 q^{53} +1.06315 q^{55} -1.33360 q^{56} +0.870377 q^{58} -5.54714 q^{59} +0.883716 q^{61} +0.756010 q^{62} -5.30525 q^{64} -29.3376 q^{65} -15.1517 q^{67} +5.03298 q^{68} +1.44785 q^{70} -6.41252 q^{71} +5.39437 q^{73} +2.54329 q^{74} +13.9822 q^{76} +0.252256 q^{77} -9.73258 q^{79} +13.9327 q^{80} +1.65130 q^{82} +4.76939 q^{83} -11.2710 q^{85} +0.932299 q^{86} -0.336407 q^{88} +7.64315 q^{89} -6.96098 q^{91} +0.259106 q^{92} -1.79712 q^{94} -31.3122 q^{95} -0.551708 q^{97} +0.343534 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 8 q^{2} + 24 q^{4} - 3 q^{5} + 20 q^{7} - 24 q^{8}+O(q^{10}) $ 20 * q - 8 * q^2 + 24 * q^4 - 3 * q^5 + 20 * q^7 - 24 * q^8	$ 20 q - 8 q^{2} + 24 q^{4} - 3 q^{5} + 20 q^{7} - 24 q^{8} - 8 q^{10} - 26 q^{11} - 4 q^{13} - 8 q^{14} + 24 q^{16} - 4 q^{17} + q^{19} + 2 q^{20} + q^{22} - 31 q^{23} + 27 q^{25} - 4 q^{26} + 24 q^{28} - 16 q^{29} + 6 q^{31} - 41 q^{32} - 10 q^{34} - 3 q^{35} + 2 q^{37} - 3 q^{38} - 38 q^{40} - 25 q^{41} + 13 q^{43} - 66 q^{44} + 20 q^{46} - 19 q^{47} + 20 q^{49} + 4 q^{50} + 20 q^{52} - 24 q^{53} - 3 q^{55} - 24 q^{56} + 12 q^{58} - 23 q^{59} - 27 q^{61} - 7 q^{62} + 2 q^{64} - 26 q^{65} + 9 q^{67} + 25 q^{68} - 8 q^{70} - 63 q^{71} - 21 q^{73} - 21 q^{74} - 10 q^{76} - 26 q^{77} + 18 q^{79} + 23 q^{80} - 42 q^{82} + q^{83} - 41 q^{85} + 12 q^{86} + 57 q^{88} + 16 q^{89} - 4 q^{91} - 17 q^{92} + 7 q^{94} - 75 q^{95} - 32 q^{97} - 8 q^{98}+O(q^{100}) $ 20 * q - 8 * q^2 + 24 * q^4 - 3 * q^5 + 20 * q^7 - 24 * q^8 - 8 * q^10 - 26 * q^11 - 4 * q^13 - 8 * q^14 + 24 * q^16 - 4 * q^17 + q^19 + 2 * q^20 + q^22 - 31 * q^23 + 27 * q^25 - 4 * q^26 + 24 * q^28 - 16 * q^29 + 6 * q^31 - 41 * q^32 - 10 * q^34 - 3 * q^35 + 2 * q^37 - 3 * q^38 - 38 * q^40 - 25 * q^41 + 13 * q^43 - 66 * q^44 + 20 * q^46 - 19 * q^47 + 20 * q^49 + 4 * q^50 + 20 * q^52 - 24 * q^53 - 3 * q^55 - 24 * q^56 + 12 * q^58 - 23 * q^59 - 27 * q^61 - 7 * q^62 + 2 * q^64 - 26 * q^65 + 9 * q^67 + 25 * q^68 - 8 * q^70 - 63 * q^71 - 21 * q^73 - 21 * q^74 - 10 * q^76 - 26 * q^77 + 18 * q^79 + 23 * q^80 - 42 * q^82 + q^83 - 41 * q^85 + 12 * q^86 + 57 * q^88 + 16 * q^89 - 4 * q^91 - 17 * q^92 + 7 * q^94 - 75 * q^95 - 32 * q^97 - 8 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0.343534	0.242916	0.121458	−	0.992597i	$-0.461243\pi$
$2$	0.343534	0.242916	0.121458	+	0.992597i	$0.461243\pi$
$3$	0	0
$3$	0	0
$4$	−1.88198	−0.940992
$5$	4.21457	1.88481	0.942407	−	0.334467i	$-0.108556\pi$
$5$	4.21457	1.88481	0.942407	+	0.334467i	$0.108556\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.33360	−0.471497
$9$	0	0
$10$	1.44785	0.457851
$11$	0.252256	0.0760579	0.0380290	−	0.999277i	$-0.487892\pi$
$11$	0.252256	0.0760579	0.0380290	+	0.999277i	$0.487892\pi$
$12$	0	0
$13$	−6.96098	−1.93063	−0.965314	−	0.261091i	$-0.915918\pi$
$13$	−6.96098	−1.93063	−0.965314	+	0.261091i	$0.915918\pi$
$14$	0.343534	0.0918134
$15$	0	0
$16$	3.30583	0.826458
$17$	−2.67429	−0.648611	−0.324306	−	0.945952i	$-0.605131\pi$
$17$	−2.67429	−0.648611	−0.324306	+	0.945952i	$0.605131\pi$
$18$	0	0
$19$	−7.42951	−1.70445	−0.852223	−	0.523179i	$-0.824746\pi$
$19$	−7.42951	−1.70445	−0.852223	+	0.523179i	$0.824746\pi$
$20$	−7.93176	−1.77360
$21$	0	0
$22$	0.0866585	0.0184757
$23$	−0.137677	−0.0287076	−0.0143538	−	0.999897i	$-0.504569\pi$
$23$	−0.137677	−0.0287076	−0.0143538	+	0.999897i	$0.504569\pi$
$24$	0	0
$25$	12.7626	2.55253
$26$	−2.39134	−0.468980
$27$	0	0
$28$	−1.88198	−0.355662
$29$	2.53359	0.470477	0.235238	−	0.971938i	$-0.424413\pi$
$29$	2.53359	0.470477	0.235238	+	0.971938i	$0.424413\pi$
$30$	0	0
$31$	2.20068	0.395254	0.197627	−	0.980277i	$-0.436677\pi$
$31$	2.20068	0.395254	0.197627	+	0.980277i	$0.436677\pi$
$32$	3.80286	0.672257
$33$	0	0
$34$	−0.918712	−0.157558
$35$	4.21457	0.712393
$36$	0	0
$37$	7.40329	1.21709	0.608547	−	0.793518i	$-0.291753\pi$
$37$	7.40329	1.21709	0.608547	+	0.793518i	$0.291753\pi$
$38$	−2.55229	−0.414036
$39$	0	0
$40$	−5.62054	−0.888685
$41$	4.80680	0.750696	0.375348	−	0.926884i	$-0.377523\pi$
$41$	4.80680	0.750696	0.375348	+	0.926884i	$0.377523\pi$
$42$	0	0
$43$	2.71385	0.413858	0.206929	−	0.978356i	$-0.433653\pi$
$43$	2.71385	0.413858	0.206929	+	0.978356i	$0.433653\pi$
$44$	−0.474741	−0.0715699
$45$	0	0
$46$	−0.0472967	−0.00697352
$47$	−5.23128	−0.763060	−0.381530	−	0.924356i	$-0.624603\pi$
$47$	−5.23128	−0.763060	−0.381530	+	0.924356i	$0.624603\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	4.38440	0.620048
$51$	0	0
$52$	13.1005	1.81671
$53$	−7.52694	−1.03390	−0.516952	−	0.856014i	$-0.672934\pi$
$53$	−7.52694	−1.03390	−0.516952	+	0.856014i	$0.672934\pi$
$54$	0	0
$55$	1.06315	0.143355
$56$	−1.33360	−0.178209
$57$	0	0
$58$	0.870377	0.114286
$59$	−5.54714	−0.722176	−0.361088	−	0.932532i	$-0.617595\pi$
$59$	−5.54714	−0.722176	−0.361088	+	0.932532i	$0.617595\pi$
$60$	0	0
$61$	0.883716	0.113148	0.0565741	−	0.998398i	$-0.481982\pi$
$61$	0.883716	0.113148	0.0565741	+	0.998398i	$0.481982\pi$
$62$	0.756010	0.0960134
$63$	0	0
$64$	−5.30525	−0.663157
$65$	−29.3376	−3.63888
$66$	0	0
$67$	−15.1517	−1.85107	−0.925537	−	0.378658i	$-0.876386\pi$
$67$	−15.1517	−1.85107	−0.925537	+	0.378658i	$0.876386\pi$
$68$	5.03298	0.610338
$69$	0	0
$70$	1.44785	0.173051
$71$	−6.41252	−0.761026	−0.380513	−	0.924776i	$-0.624253\pi$
$71$	−6.41252	−0.761026	−0.380513	+	0.924776i	$0.624253\pi$
$72$	0	0
$73$	5.39437	0.631364	0.315682	−	0.948865i	$-0.397767\pi$
$73$	5.39437	0.631364	0.315682	+	0.948865i	$0.397767\pi$
$74$	2.54329	0.295651
$75$	0	0
$76$	13.9822	1.60387
$77$	0.252256	0.0287472
$78$	0	0
$79$	−9.73258	−1.09500	−0.547500	−	0.836806i	$-0.684420\pi$
$79$	−9.73258	−1.09500	−0.547500	+	0.836806i	$0.684420\pi$
$80$	13.9327	1.55772
$81$	0	0
$82$	1.65130	0.182356
$83$	4.76939	0.523509	0.261754	−	0.965135i	$-0.415699\pi$
$83$	4.76939	0.523509	0.261754	+	0.965135i	$0.415699\pi$
$84$	0	0
$85$	−11.2710	−1.22251
$86$	0.932299	0.100532
$87$	0	0
$88$	−0.336407	−0.0358611
$89$	7.64315	0.810172	0.405086	−	0.914279i	$-0.367242\pi$
$89$	7.64315	0.810172	0.405086	+	0.914279i	$0.367242\pi$
$90$	0	0
$91$	−6.96098	−0.729709
$92$	0.259106	0.0270136
$93$	0	0
$94$	−1.79712	−0.185359
$95$	−31.3122	−3.21256
$96$	0	0
$97$	−0.551708	−0.0560174	−0.0280087	−	0.999608i	$-0.508917\pi$
$97$	−0.551708	−0.0560174	−0.0280087	+	0.999608i	$0.508917\pi$
$98$	0.343534	0.0347022
$99$	0	0
$100$	−24.0191	−2.40191
$101$	−7.52833	−0.749097	−0.374549	−	0.927207i	$-0.622202\pi$
$101$	−7.52833	−0.749097	−0.374549	+	0.927207i	$0.622202\pi$
$102$	0	0
$103$	−13.1782	−1.29848	−0.649241	−	0.760582i	$-0.724913\pi$
$103$	−13.1782	−1.29848	−0.649241	+	0.760582i	$0.724913\pi$
$104$	9.28313	0.910286
$105$	0	0
$106$	−2.58576	−0.251152
$107$	0.270585	0.0261584	0.0130792	−	0.999914i	$-0.495837\pi$
$107$	0.270585	0.0261584	0.0130792	+	0.999914i	$0.495837\pi$
$108$	0	0
$109$	−7.26005	−0.695386	−0.347693	−	0.937608i	$-0.613035\pi$
$109$	−7.26005	−0.695386	−0.347693	+	0.937608i	$0.613035\pi$
$110$	0.365229	0.0348232
$111$	0	0
$112$	3.30583	0.312372
$113$	8.77601	0.825577	0.412789	−	0.910827i	$-0.364555\pi$
$113$	8.77601	0.825577	0.412789	+	0.910827i	$0.364555\pi$
$114$	0	0
$115$	−0.580249	−0.0541085
$116$	−4.76819	−0.442715
$117$	0	0
$118$	−1.90563	−0.175428
$119$	−2.67429	−0.245152
$120$	0	0
$121$	−10.9364	−0.994215
$122$	0.303587	0.0274855
$123$	0	0
$124$	−4.14165	−0.371931
$125$	32.7162	2.92622
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	−9.42825	−0.833348
$129$	0	0
$130$	−10.0785	−0.883940
$131$	−3.72616	−0.325556	−0.162778	−	0.986663i	$-0.552045\pi$
$131$	−3.72616	−0.325556	−0.162778	+	0.986663i	$0.552045\pi$
$132$	0	0
$133$	−7.42951	−0.644220
$134$	−5.20513	−0.449654
$135$	0	0
$136$	3.56642	0.305818
$137$	−23.0959	−1.97322	−0.986608	−	0.163108i	$-0.947848\pi$
$137$	−23.0959	−1.97322	−0.986608	+	0.163108i	$0.947848\pi$
$138$	0	0
$139$	−12.2778	−1.04139	−0.520695	−	0.853743i	$-0.674327\pi$
$139$	−12.2778	−1.04139	−0.520695	+	0.853743i	$0.674327\pi$
$140$	−7.93176	−0.670356
$141$	0	0
$142$	−2.20292	−0.184865
$143$	−1.75595	−0.146840
$144$	0	0
$145$	10.6780	0.886761
$146$	1.85315	0.153368
$147$	0	0
$148$	−13.9329	−1.14528
$149$	−19.7330	−1.61659	−0.808297	−	0.588776i	$-0.799610\pi$
$149$	−19.7330	−1.61659	−0.808297	+	0.588776i	$0.799610\pi$
$150$	0	0
$151$	−0.694939	−0.0565533	−0.0282767	−	0.999600i	$-0.509002\pi$
$151$	−0.694939	−0.0565533	−0.0282767	+	0.999600i	$0.509002\pi$
$152$	9.90796	0.803641
$153$	0	0
$154$	0.0866585	0.00698314
$155$	9.27493	0.744981
$156$	0	0
$157$	−13.2219	−1.05522	−0.527609	−	0.849487i	$-0.676911\pi$
$157$	−13.2219	−1.05522	−0.527609	+	0.849487i	$0.676911\pi$
$158$	−3.34347	−0.265993
$159$	0	0
$160$	16.0274	1.26708
$161$	−0.137677	−0.0108505
$162$	0	0
$163$	7.04652	0.551926	0.275963	−	0.961168i	$-0.411003\pi$
$163$	7.04652	0.551926	0.275963	+	0.961168i	$0.411003\pi$
$164$	−9.04632	−0.706399
$165$	0	0
$166$	1.63845	0.127168
$167$	−20.0203	−1.54921	−0.774607	−	0.632442i	$-0.782053\pi$
$167$	−20.0203	−1.54921	−0.774607	+	0.632442i	$0.782053\pi$
$168$	0	0
$169$	35.4552	2.72733
$170$	−3.87198	−0.296967
$171$	0	0
$172$	−5.10741	−0.389437
$173$	−3.79139	−0.288254	−0.144127	−	0.989559i	$-0.546037\pi$
$173$	−3.79139	−0.288254	−0.144127	+	0.989559i	$0.546037\pi$
$174$	0	0
$175$	12.7626	0.964764
$176$	0.833915	0.0628587
$177$	0	0
$178$	2.62569	0.196803
$179$	3.60241	0.269257	0.134628	−	0.990896i	$-0.457016\pi$
$179$	3.60241	0.269257	0.134628	+	0.990896i	$0.457016\pi$
$180$	0	0
$181$	17.1084	1.27165	0.635827	−	0.771831i	$-0.280659\pi$
$181$	17.1084	1.27165	0.635827	+	0.771831i	$0.280659\pi$
$182$	−2.39134	−0.177258
$183$	0	0
$184$	0.183605	0.0135355
$185$	31.2017	2.29400
$186$	0	0
$187$	−0.674605	−0.0493320
$188$	9.84518	0.718033
$189$	0	0
$190$	−10.7568	−0.780382
$191$	8.57769	0.620660	0.310330	−	0.950629i	$-0.399560\pi$
$191$	8.57769	0.620660	0.310330	+	0.950629i	$0.399560\pi$
$192$	0	0
$193$	1.87983	0.135313	0.0676566	−	0.997709i	$-0.478448\pi$
$193$	1.87983	0.135313	0.0676566	+	0.997709i	$0.478448\pi$
$194$	−0.189531	−0.0136075
$195$	0	0
$196$	−1.88198	−0.134427
$197$	−2.92286	−0.208245	−0.104123	−	0.994564i	$-0.533203\pi$
$197$	−2.92286	−0.208245	−0.104123	+	0.994564i	$0.533203\pi$
$198$	0	0
$199$	−1.95596	−0.138654	−0.0693272	−	0.997594i	$-0.522085\pi$
$199$	−1.95596	−0.138654	−0.0693272	+	0.997594i	$0.522085\pi$
$200$	−17.0202	−1.20351
$201$	0	0
$202$	−2.58624	−0.181967
$203$	2.53359	0.177823
$204$	0	0
$205$	20.2586	1.41492
$206$	−4.52715	−0.315422
$207$	0	0
$208$	−23.0118	−1.59558
$209$	−1.87414	−0.129637
$210$	0	0
$211$	−15.1626	−1.04384	−0.521918	−	0.852995i	$-0.674783\pi$
$211$	−15.1626	−1.04384	−0.521918	+	0.852995i	$0.674783\pi$
$212$	14.1656	0.972896
$213$	0	0
$214$	0.0929552	0.00635429
$215$	11.4377	0.780045
$216$	0	0
$217$	2.20068	0.149392
$218$	−2.49408	−0.168920
$219$	0	0
$220$	−2.00083	−0.134896
$221$	18.6157	1.25223
$222$	0	0
$223$	27.8187	1.86288	0.931439	−	0.363898i	$-0.118554\pi$
$223$	27.8187	1.86288	0.931439	+	0.363898i	$0.118554\pi$
$224$	3.80286	0.254089
$225$	0	0
$226$	3.01486	0.200546
$227$	22.0539	1.46377	0.731885	−	0.681428i	$-0.238641\pi$
$227$	22.0539	1.46377	0.731885	+	0.681428i	$0.238641\pi$
$228$	0	0
$229$	−13.5945	−0.898349	−0.449174	−	0.893444i	$-0.648282\pi$
$229$	−13.5945	−0.898349	−0.449174	+	0.893444i	$0.648282\pi$
$230$	−0.199335	−0.0131438
$231$	0	0
$232$	−3.37879	−0.221828
$233$	−20.0163	−1.31131	−0.655657	−	0.755059i	$-0.727608\pi$
$233$	−20.0163	−1.31131	−0.655657	+	0.755059i	$0.727608\pi$
$234$	0	0
$235$	−22.0476	−1.43823
$236$	10.4396	0.679562
$237$	0	0
$238$	−0.918712	−0.0595512
$239$	−25.1646	−1.62776	−0.813880	−	0.581033i	$-0.802649\pi$
$239$	−25.1646	−1.62776	−0.813880	+	0.581033i	$0.802649\pi$
$240$	0	0
$241$	−9.69541	−0.624536	−0.312268	−	0.949994i	$-0.601089\pi$
$241$	−9.69541	−0.624536	−0.312268	+	0.949994i	$0.601089\pi$
$242$	−3.75702	−0.241510
$243$	0	0
$244$	−1.66314	−0.106472
$245$	4.21457	0.269259
$246$	0	0
$247$	51.7166	3.29065
$248$	−2.93482	−0.186361
$249$	0	0
$250$	11.2391	0.710825
$251$	10.9355	0.690244	0.345122	−	0.938558i	$-0.387838\pi$
$251$	10.9355	0.690244	0.345122	+	0.938558i	$0.387838\pi$
$252$	0	0
$253$	−0.0347297	−0.00218344
$254$	−0.343534	−0.0215553
$255$	0	0
$256$	7.37158	0.460723
$257$	0.906447	0.0565426	0.0282713	−	0.999600i	$-0.491000\pi$
$257$	0.906447	0.0565426	0.0282713	+	0.999600i	$0.491000\pi$
$258$	0	0
$259$	7.40329	0.460018
$260$	55.2128	3.42415
$261$	0	0
$262$	−1.28006	−0.0790827
$263$	−20.7216	−1.27775	−0.638873	−	0.769312i	$-0.720599\pi$
$263$	−20.7216	−1.27775	−0.638873	+	0.769312i	$0.720599\pi$
$264$	0	0
$265$	−31.7229	−1.94872
$266$	−2.55229	−0.156491
$267$	0	0
$268$	28.5152	1.74185
$269$	−16.7036	−1.01843	−0.509217	−	0.860638i	$-0.670065\pi$
$269$	−16.7036	−1.01843	−0.509217	+	0.860638i	$0.670065\pi$
$270$	0	0
$271$	8.63373	0.524462	0.262231	−	0.965005i	$-0.415542\pi$
$271$	8.63373	0.524462	0.262231	+	0.965005i	$0.415542\pi$
$272$	−8.84076	−0.536050
$273$	0	0
$274$	−7.93424	−0.479325
$275$	3.21945	0.194140
$276$	0	0
$277$	−24.7143	−1.48494	−0.742471	−	0.669879i	$-0.766346\pi$
$277$	−24.7143	−1.48494	−0.742471	+	0.669879i	$0.766346\pi$
$278$	−4.21785	−0.252970
$279$	0	0
$280$	−5.62054	−0.335891
$281$	24.0409	1.43416	0.717079	−	0.696992i	$-0.245479\pi$
$281$	24.0409	1.43416	0.717079	+	0.696992i	$0.245479\pi$
$282$	0	0
$283$	20.4819	1.21752	0.608762	−	0.793353i	$-0.291667\pi$
$283$	20.4819	1.21752	0.608762	+	0.793353i	$0.291667\pi$
$284$	12.0683	0.716119
$285$	0	0
$286$	−0.603228	−0.0356696
$287$	4.80680	0.283736
$288$	0	0
$289$	−9.84816	−0.579304
$290$	3.66827	0.215408
$291$	0	0
$292$	−10.1521	−0.594108
$293$	−5.21327	−0.304562	−0.152281	−	0.988337i	$-0.548662\pi$
$293$	−5.21327	−0.304562	−0.152281	+	0.988337i	$0.548662\pi$
$294$	0	0
$295$	−23.3788	−1.36117
$296$	−9.87299	−0.573856
$297$	0	0
$298$	−6.77898	−0.392696
$299$	0.958365	0.0554237
$300$	0	0
$301$	2.71385	0.156423
$302$	−0.238735	−0.0137377
$303$	0	0
$304$	−24.5607	−1.40865
$305$	3.72448	0.213263
$306$	0	0
$307$	17.4049	0.993351	0.496675	−	0.867936i	$-0.334554\pi$
$307$	17.4049	0.993351	0.496675	+	0.867936i	$0.334554\pi$
$308$	−0.474741	−0.0270509
$309$	0	0
$310$	3.18626	0.180967
$311$	8.51272	0.482712	0.241356	−	0.970437i	$-0.422408\pi$
$311$	8.51272	0.482712	0.241356	+	0.970437i	$0.422408\pi$
$312$	0	0
$313$	−19.1562	−1.08277	−0.541387	−	0.840774i	$-0.682100\pi$
$313$	−19.1562	−1.08277	−0.541387	+	0.840774i	$0.682100\pi$
$314$	−4.54216	−0.256329
$315$	0	0
$316$	18.3166	1.03039
$317$	30.1899	1.69564	0.847818	−	0.530288i	$-0.177916\pi$
$317$	30.1899	1.69564	0.847818	+	0.530288i	$0.177916\pi$
$318$	0	0
$319$	0.639114	0.0357835
$320$	−22.3594	−1.24993
$321$	0	0
$322$	−0.0472967	−0.00263574
$323$	19.8687	1.10552
$324$	0	0
$325$	−88.8404	−4.92798
$326$	2.42072	0.134071
$327$	0	0
$328$	−6.41032	−0.353951
$329$	−5.23128	−0.288410
$330$	0	0
$331$	21.9476	1.20635	0.603174	−	0.797610i	$-0.293902\pi$
$331$	21.9476	1.20635	0.603174	+	0.797610i	$0.293902\pi$
$332$	−8.97592	−0.492618
$333$	0	0
$334$	−6.87765	−0.376328
$335$	−63.8579	−3.48893
$336$	0	0
$337$	6.20544	0.338032	0.169016	−	0.985613i	$-0.445941\pi$
$337$	6.20544	0.338032	0.169016	+	0.985613i	$0.445941\pi$
$338$	12.1801	0.662510
$339$	0	0
$340$	21.2118	1.15037
$341$	0.555134	0.0300622
$342$	0	0
$343$	1.00000	0.0539949
$344$	−3.61917	−0.195133
$345$	0	0
$346$	−1.30247	−0.0700214
$347$	−30.2300	−1.62283	−0.811415	−	0.584471i	$-0.801302\pi$
$347$	−30.2300	−1.62283	−0.811415	+	0.584471i	$0.801302\pi$
$348$	0	0
$349$	27.2421	1.45824	0.729118	−	0.684387i	$-0.239930\pi$
$349$	27.2421	1.45824	0.729118	+	0.684387i	$0.239930\pi$
$350$	4.38440	0.234356
$351$	0	0
$352$	0.959292	0.0511304
$353$	13.1659	0.700751	0.350375	−	0.936609i	$-0.386054\pi$
$353$	13.1659	0.700751	0.350375	+	0.936609i	$0.386054\pi$
$354$	0	0
$355$	−27.0260	−1.43439
$356$	−14.3843	−0.762366
$357$	0	0
$358$	1.23755	0.0654067
$359$	−29.0391	−1.53262	−0.766312	−	0.642468i	$-0.777910\pi$
$359$	−29.0391	−1.53262	−0.766312	+	0.642468i	$0.777910\pi$
$360$	0	0
$361$	36.1976	1.90514
$362$	5.87731	0.308905
$363$	0	0
$364$	13.1005	0.686650
$365$	22.7350	1.19000
$366$	0	0
$367$	14.1830	0.740345	0.370172	−	0.928963i	$-0.379299\pi$
$367$	14.1830	0.740345	0.370172	+	0.928963i	$0.379299\pi$
$368$	−0.455136	−0.0237256
$369$	0	0
$370$	10.7189	0.557247
$371$	−7.52694	−0.390779
$372$	0	0
$373$	−16.5730	−0.858118	−0.429059	−	0.903276i	$-0.641155\pi$
$373$	−16.5730	−0.858118	−0.429059	+	0.903276i	$0.641155\pi$
$374$	−0.231750	−0.0119835
$375$	0	0
$376$	6.97640	0.359781
$377$	−17.6363	−0.908316
$378$	0	0
$379$	−1.16042	−0.0596066	−0.0298033	−	0.999556i	$-0.509488\pi$
$379$	−1.16042	−0.0596066	−0.0298033	+	0.999556i	$0.509488\pi$
$380$	58.9291	3.02300
$381$	0	0
$382$	2.94673	0.150768
$383$	−20.2297	−1.03369	−0.516844	−	0.856080i	$-0.672893\pi$
$383$	−20.2297	−1.03369	−0.516844	+	0.856080i	$0.672893\pi$
$384$	0	0
$385$	1.06315	0.0541831
$386$	0.645787	0.0328697
$387$	0	0
$388$	1.03830	0.0527119
$389$	−32.9176	−1.66899	−0.834494	−	0.551017i	$-0.814240\pi$
$389$	−32.9176	−1.66899	−0.834494	+	0.551017i	$0.814240\pi$
$390$	0	0
$391$	0.368188	0.0186201
$392$	−1.33360	−0.0673567
$393$	0	0
$394$	−1.00410	−0.0505860
$395$	−41.0187	−2.06387
$396$	0	0
$397$	20.3662	1.02215	0.511076	−	0.859536i	$-0.329247\pi$
$397$	20.3662	1.02215	0.511076	+	0.859536i	$0.329247\pi$
$398$	−0.671940	−0.0336813
$399$	0	0
$400$	42.1911	2.10956
$401$	18.2500	0.911363	0.455682	−	0.890143i	$-0.349396\pi$
$401$	18.2500	0.911363	0.455682	+	0.890143i	$0.349396\pi$
$402$	0	0
$403$	−15.3189	−0.763089
$404$	14.1682	0.704895
$405$	0	0
$406$	0.870377	0.0431961
$407$	1.86752	0.0925696
$408$	0	0
$409$	16.4498	0.813393	0.406696	−	0.913563i	$-0.366681\pi$
$409$	16.4498	0.813393	0.406696	+	0.913563i	$0.366681\pi$
$410$	6.95953	0.343707
$411$	0	0
$412$	24.8011	1.22186
$413$	−5.54714	−0.272957
$414$	0	0
$415$	20.1010	0.986717
$416$	−26.4716	−1.29788
$417$	0	0
$418$	−0.643830	−0.0314908
$419$	−27.0845	−1.32316	−0.661581	−	0.749874i	$-0.730114\pi$
$419$	−27.0845	−1.32316	−0.661581	+	0.749874i	$0.730114\pi$
$420$	0	0
$421$	6.35121	0.309539	0.154770	−	0.987951i	$-0.450536\pi$
$421$	6.35121	0.309539	0.154770	+	0.987951i	$0.450536\pi$
$422$	−5.20888	−0.253564
$423$	0	0
$424$	10.0379	0.487483
$425$	−34.1310	−1.65560
$426$	0	0
$427$	0.883716	0.0427660
$428$	−0.509237	−0.0246149
$429$	0	0
$430$	3.92924	0.189485
$431$	−14.7738	−0.711631	−0.355816	−	0.934556i	$-0.615797\pi$
$431$	−14.7738	−0.711631	−0.355816	+	0.934556i	$0.615797\pi$
$432$	0	0
$433$	−33.1359	−1.59241	−0.796204	−	0.605028i	$-0.793162\pi$
$433$	−33.1359	−1.59241	−0.796204	+	0.605028i	$0.793162\pi$
$434$	0.756010	0.0362896
$435$	0	0
$436$	13.6633	0.654353
$437$	1.02287	0.0489305
$438$	0	0
$439$	−34.8322	−1.66245	−0.831226	−	0.555935i	$-0.812360\pi$
$439$	−34.8322	−1.66245	−0.831226	+	0.555935i	$0.812360\pi$
$440$	−1.41781	−0.0675915
$441$	0	0
$442$	6.39513	0.304185
$443$	12.6381	0.600455	0.300227	−	0.953868i	$-0.402937\pi$
$443$	12.6381	0.600455	0.300227	+	0.953868i	$0.402937\pi$
$444$	0	0
$445$	32.2126	1.52702
$446$	9.55668	0.452522
$447$	0	0
$448$	−5.30525	−0.250650
$449$	−29.8026	−1.40647	−0.703236	−	0.710957i	$-0.748262\pi$
$449$	−29.8026	−1.40647	−0.703236	+	0.710957i	$0.748262\pi$
$450$	0	0
$451$	1.21254	0.0570964
$452$	−16.5163	−0.776862
$453$	0	0
$454$	7.57628	0.355572
$455$	−29.3376	−1.37537
$456$	0	0
$457$	−1.10190	−0.0515446	−0.0257723	−	0.999668i	$-0.508204\pi$
$457$	−1.10190	−0.0515446	−0.0257723	+	0.999668i	$0.508204\pi$
$458$	−4.67017	−0.218223
$459$	0	0
$460$	1.09202	0.0509157
$461$	−3.10488	−0.144609	−0.0723044	−	0.997383i	$-0.523035\pi$
$461$	−3.10488	−0.144609	−0.0723044	+	0.997383i	$0.523035\pi$
$462$	0	0
$463$	26.4830	1.23077	0.615384	−	0.788228i	$-0.289001\pi$
$463$	26.4830	1.23077	0.615384	+	0.788228i	$0.289001\pi$
$464$	8.37564	0.388829
$465$	0	0
$466$	−6.87630	−0.318539
$467$	−1.17656	−0.0544449	−0.0272225	−	0.999629i	$-0.508666\pi$
$467$	−1.17656	−0.0544449	−0.0272225	+	0.999629i	$0.508666\pi$
$468$	0	0
$469$	−15.1517	−0.699640
$470$	−7.57411	−0.349368
$471$	0	0
$472$	7.39763	0.340504
$473$	0.684583	0.0314772
$474$	0	0
$475$	−94.8201	−4.35064
$476$	5.03298	0.230686
$477$	0	0
$478$	−8.64489	−0.395408
$479$	−34.3916	−1.57139	−0.785697	−	0.618612i	$-0.787695\pi$
$479$	−34.3916	−1.57139	−0.785697	+	0.618612i	$0.787695\pi$
$480$	0	0
$481$	−51.5342	−2.34976
$482$	−3.33071	−0.151709
$483$	0	0
$484$	20.5821	0.935549
$485$	−2.32521	−0.105582
$486$	0	0
$487$	−31.2720	−1.41707	−0.708536	−	0.705675i	$-0.750644\pi$
$487$	−31.2720	−1.41707	−0.708536	+	0.705675i	$0.750644\pi$
$488$	−1.17852	−0.0533490
$489$	0	0
$490$	1.44785	0.0654072
$491$	−38.1065	−1.71972	−0.859862	−	0.510527i	$-0.829450\pi$
$491$	−38.1065	−1.71972	−0.859862	+	0.510527i	$0.829450\pi$
$492$	0	0
$493$	−6.77557	−0.305156
$494$	17.7664	0.799350
$495$	0	0
$496$	7.27508	0.326661
$497$	−6.41252	−0.287641
$498$	0	0
$499$	−16.2596	−0.727879	−0.363939	−	0.931423i	$-0.618568\pi$
$499$	−16.2596	−0.727879	−0.363939	+	0.931423i	$0.618568\pi$
$500$	−61.5713	−2.75355
$501$	0	0
$502$	3.75673	0.167671
$503$	35.2949	1.57372	0.786860	−	0.617132i	$-0.211705\pi$
$503$	35.2949	1.57372	0.786860	+	0.617132i	$0.211705\pi$
$504$	0	0
$505$	−31.7287	−1.41191
$506$	−0.0119309	−0.000530392 0
$507$	0	0
$508$	1.88198	0.0834995
$509$	30.7019	1.36084	0.680420	−	0.732823i	$-0.261798\pi$
$509$	30.7019	1.36084	0.680420	+	0.732823i	$0.261798\pi$
$510$	0	0
$511$	5.39437	0.238633
$512$	21.3889	0.945264
$513$	0	0
$514$	0.311396	0.0137351
$515$	−55.5403	−2.44740
$516$	0	0
$517$	−1.31962	−0.0580368
$518$	2.54329	0.111746
$519$	0	0
$520$	39.1244	1.71572
$521$	25.7693	1.12897	0.564487	−	0.825442i	$-0.309074\pi$
$521$	25.7693	1.12897	0.564487	+	0.825442i	$0.309074\pi$
$522$	0	0
$523$	−11.3270	−0.495294	−0.247647	−	0.968850i	$-0.579657\pi$
$523$	−11.3270	−0.495294	−0.247647	+	0.968850i	$0.579657\pi$
$524$	7.01258	0.306346
$525$	0	0
$526$	−7.11857	−0.310384
$527$	−5.88527	−0.256366
$528$	0	0
$529$	−22.9810	−0.999176
$530$	−10.8979	−0.473374
$531$	0	0
$532$	13.9822	0.606206
$533$	−33.4600	−1.44931
$534$	0	0
$535$	1.14040	0.0493038
$536$	20.2062	0.872776
$537$	0	0
$538$	−5.73825	−0.247393
$539$	0.252256	0.0108654
$540$	0	0
$541$	30.3412	1.30447	0.652235	−	0.758017i	$-0.273832\pi$
$541$	30.3412	1.30447	0.652235	+	0.758017i	$0.273832\pi$
$542$	2.96598	0.127400
$543$	0	0
$544$	−10.1700	−0.436033
$545$	−30.5980	−1.31067
$546$	0	0
$547$	10.3866	0.444100	0.222050	−	0.975035i	$-0.428725\pi$
$547$	10.3866	0.444100	0.222050	+	0.975035i	$0.428725\pi$
$548$	43.4661	1.85678
$549$	0	0
$550$	1.10599	0.0471596
$551$	−18.8234	−0.801902
$552$	0	0
$553$	−9.73258	−0.413871
$554$	−8.49023	−0.360715
$555$	0	0
$556$	23.1066	0.979939
$557$	−13.1591	−0.557568	−0.278784	−	0.960354i	$-0.589931\pi$
$557$	−13.1591	−0.557568	−0.278784	+	0.960354i	$0.589931\pi$
$558$	0	0
$559$	−18.8910	−0.799005
$560$	13.9327	0.588763
$561$	0	0
$562$	8.25887	0.348379
$563$	24.2752	1.02308	0.511539	−	0.859260i	$-0.329076\pi$
$563$	24.2752	1.02308	0.511539	+	0.859260i	$0.329076\pi$
$564$	0	0
$565$	36.9871	1.55606
$566$	7.03624	0.295755
$567$	0	0
$568$	8.55170	0.358821
$569$	13.1589	0.551652	0.275826	−	0.961208i	$-0.411049\pi$
$569$	13.1589	0.551652	0.275826	+	0.961208i	$0.411049\pi$
$570$	0	0
$571$	28.5764	1.19589	0.597943	−	0.801539i	$-0.295985\pi$
$571$	28.5764	1.19589	0.597943	+	0.801539i	$0.295985\pi$
$572$	3.30466	0.138175
$573$	0	0
$574$	1.65130	0.0689240
$575$	−1.75712	−0.0732769
$576$	0	0
$577$	7.18377	0.299064	0.149532	−	0.988757i	$-0.452223\pi$
$577$	7.18377	0.299064	0.149532	+	0.988757i	$0.452223\pi$
$578$	−3.38318	−0.140722
$579$	0	0
$580$	−20.0959	−0.834435
$581$	4.76939	0.197868
$582$	0	0
$583$	−1.89871	−0.0786367
$584$	−7.19391	−0.297686
$585$	0	0
$586$	−1.79094	−0.0739829
$587$	30.8945	1.27515	0.637576	−	0.770387i	$-0.279937\pi$
$587$	30.8945	1.27515	0.637576	+	0.770387i	$0.279937\pi$
$588$	0	0
$589$	−16.3500	−0.673689
$590$	−8.03143	−0.330649
$591$	0	0
$592$	24.4740	1.00588
$593$	19.8985	0.817135	0.408568	−	0.912728i	$-0.366028\pi$
$593$	19.8985	0.817135	0.408568	+	0.912728i	$0.366028\pi$
$594$	0	0
$595$	−11.2710	−0.462066
$596$	37.1373	1.52120
$597$	0	0
$598$	0.329231	0.0134633
$599$	4.17501	0.170586	0.0852931	−	0.996356i	$-0.472817\pi$
$599$	4.17501	0.170586	0.0852931	+	0.996356i	$0.472817\pi$
$600$	0	0
$601$	26.8370	1.09471	0.547353	−	0.836902i	$-0.315636\pi$
$601$	26.8370	1.09471	0.547353	+	0.836902i	$0.315636\pi$
$602$	0.932299	0.0379977
$603$	0	0
$604$	1.30786	0.0532162
$605$	−46.0921	−1.87391
$606$	0	0
$607$	10.2559	0.416272	0.208136	−	0.978100i	$-0.433260\pi$
$607$	10.2559	0.416272	0.208136	+	0.978100i	$0.433260\pi$
$608$	−28.2534	−1.14583
$609$	0	0
$610$	1.27949	0.0518050
$611$	36.4148	1.47318
$612$	0	0
$613$	−6.73170	−0.271891	−0.135945	−	0.990716i	$-0.543407\pi$
$613$	−6.73170	−0.271891	−0.135945	+	0.990716i	$0.543407\pi$
$614$	5.97919	0.241300
$615$	0	0
$616$	−0.336407	−0.0135542
$617$	17.4699	0.703310	0.351655	−	0.936130i	$-0.385619\pi$
$617$	17.4699	0.703310	0.351655	+	0.936130i	$0.385619\pi$
$618$	0	0
$619$	40.6897	1.63546	0.817730	−	0.575603i	$-0.195232\pi$
$619$	40.6897	1.63546	0.817730	+	0.575603i	$0.195232\pi$
$620$	−17.4553	−0.701021
$621$	0	0
$622$	2.92441	0.117258
$623$	7.64315	0.306216
$624$	0	0
$625$	74.0716	2.96286
$626$	−6.58082	−0.263022
$627$	0	0
$628$	24.8833	0.992952
$629$	−19.7986	−0.789421
$630$	0	0
$631$	20.5593	0.818454	0.409227	−	0.912433i	$-0.365798\pi$
$631$	20.5593	0.818454	0.409227	+	0.912433i	$0.365798\pi$
$632$	12.9793	0.516289
$633$	0	0
$634$	10.3713	0.411896
$635$	−4.21457	−0.167250
$636$	0	0
$637$	−6.96098	−0.275804
$638$	0.219558	0.00869237
$639$	0	0
$640$	−39.7361	−1.57071
$641$	−18.5559	−0.732914	−0.366457	−	0.930435i	$-0.619429\pi$
$641$	−18.5559	−0.732914	−0.366457	+	0.930435i	$0.619429\pi$
$642$	0	0
$643$	37.6418	1.48445	0.742224	−	0.670151i	$-0.233771\pi$
$643$	37.6418	1.48445	0.742224	+	0.670151i	$0.233771\pi$
$644$	0.259106	0.0102102
$645$	0	0
$646$	6.82557	0.268549
$647$	20.0686	0.788979	0.394490	−	0.918900i	$-0.370921\pi$
$647$	20.0686	0.788979	0.394490	+	0.918900i	$0.370921\pi$
$648$	0	0
$649$	−1.39930	−0.0549272
$650$	−30.5197	−1.19708
$651$	0	0
$652$	−13.2614	−0.519358
$653$	5.93099	0.232098	0.116049	−	0.993244i	$-0.462977\pi$
$653$	5.93099	0.232098	0.116049	+	0.993244i	$0.462977\pi$
$654$	0	0
$655$	−15.7042	−0.613613
$656$	15.8905	0.620419
$657$	0	0
$658$	−1.79712	−0.0700592
$659$	−20.5423	−0.800215	−0.400107	−	0.916468i	$-0.631027\pi$
$659$	−20.5423	−0.800215	−0.400107	+	0.916468i	$0.631027\pi$
$660$	0	0
$661$	4.10093	0.159508	0.0797539	−	0.996815i	$-0.474587\pi$
$661$	4.10093	0.159508	0.0797539	+	0.996815i	$0.474587\pi$
$662$	7.53975	0.293041
$663$	0	0
$664$	−6.36044	−0.246833
$665$	−31.3122	−1.21424
$666$	0	0
$667$	−0.348817	−0.0135063
$668$	37.6778	1.45780
$669$	0	0
$670$	−21.9374	−0.847515
$671$	0.222922	0.00860582
$672$	0	0
$673$	11.1787	0.430907	0.215454	−	0.976514i	$-0.430877\pi$
$673$	11.1787	0.430907	0.215454	+	0.976514i	$0.430877\pi$
$674$	2.13178	0.0821132
$675$	0	0
$676$	−66.7262	−2.56639
$677$	47.1724	1.81298	0.906491	−	0.422225i	$-0.138751\pi$
$677$	47.1724	1.81298	0.906491	+	0.422225i	$0.138751\pi$
$678$	0	0
$679$	−0.551708	−0.0211726
$680$	15.0310	0.576411
$681$	0	0
$682$	0.190708	0.00730258
$683$	34.5786	1.32311	0.661556	−	0.749896i	$-0.269896\pi$
$683$	34.5786	1.32311	0.661556	+	0.749896i	$0.269896\pi$
$684$	0	0
$685$	−97.3394	−3.71915
$686$	0.343534	0.0131162
$687$	0	0
$688$	8.97152	0.342036
$689$	52.3949	1.99609
$690$	0	0
$691$	15.1902	0.577862	0.288931	−	0.957350i	$-0.406700\pi$
$691$	15.1902	0.577862	0.288931	+	0.957350i	$0.406700\pi$
$692$	7.13533	0.271245
$693$	0	0
$694$	−10.3850	−0.394211
$695$	−51.7457	−1.96283
$696$	0	0
$697$	−12.8548	−0.486910
$698$	9.35860	0.354228
$699$	0	0
$700$	−24.0191	−0.907835
$701$	−42.5248	−1.60614	−0.803069	−	0.595886i	$-0.796801\pi$
$701$	−42.5248	−1.60614	−0.803069	+	0.595886i	$0.796801\pi$
$702$	0	0
$703$	−55.0028	−2.07447
$704$	−1.33828	−0.0504383
$705$	0	0
$706$	4.52294	0.170223
$707$	−7.52833	−0.283132
$708$	0	0
$709$	3.19096	0.119839	0.0599195	−	0.998203i	$-0.480916\pi$
$709$	3.19096	0.119839	0.0599195	+	0.998203i	$0.480916\pi$
$710$	−9.28437	−0.348436
$711$	0	0
$712$	−10.1929	−0.381994
$713$	−0.302983	−0.0113468
$714$	0	0
$715$	−7.40056	−0.276765
$716$	−6.77968	−0.253369
$717$	0	0
$718$	−9.97593	−0.372298
$719$	−4.87157	−0.181679	−0.0908394	−	0.995866i	$-0.528955\pi$
$719$	−4.87157	−0.181679	−0.0908394	+	0.995866i	$0.528955\pi$
$720$	0	0
$721$	−13.1782	−0.490780
$722$	12.4351	0.462787
$723$	0	0
$724$	−32.1977	−1.19662
$725$	32.3353	1.20090
$726$	0	0
$727$	−31.0150	−1.15028	−0.575141	−	0.818054i	$-0.695053\pi$
$727$	−31.0150	−1.15028	−0.575141	+	0.818054i	$0.695053\pi$
$728$	9.28313	0.344056
$729$	0	0
$730$	7.81025	0.289070
$731$	−7.25762	−0.268433
$732$	0	0
$733$	44.0090	1.62551	0.812754	−	0.582607i	$-0.197967\pi$
$733$	44.0090	1.62551	0.812754	+	0.582607i	$0.197967\pi$
$734$	4.87234	0.179841
$735$	0	0
$736$	−0.523565	−0.0192989
$737$	−3.82210	−0.140789
$738$	0	0
$739$	−7.91041	−0.290989	−0.145495	−	0.989359i	$-0.546477\pi$
$739$	−7.91041	−0.290989	−0.145495	+	0.989359i	$0.546477\pi$
$740$	−58.7211	−2.15863
$741$	0	0
$742$	−2.58576	−0.0949264
$743$	27.7485	1.01799	0.508997	−	0.860768i	$-0.330016\pi$
$743$	27.7485	1.01799	0.508997	+	0.860768i	$0.330016\pi$
$744$	0	0
$745$	−83.1663	−3.04698
$746$	−5.69340	−0.208450
$747$	0	0
$748$	1.26960	0.0464210
$749$	0.270585	0.00988696
$750$	0	0
$751$	−14.1357	−0.515819	−0.257910	−	0.966169i	$-0.583034\pi$
$751$	−14.1357	−0.515819	−0.257910	+	0.966169i	$0.583034\pi$
$752$	−17.2937	−0.630637
$753$	0	0
$754$	−6.05868	−0.220644
$755$	−2.92887	−0.106593
$756$	0	0
$757$	24.7885	0.900952	0.450476	−	0.892788i	$-0.351254\pi$
$757$	24.7885	0.900952	0.450476	+	0.892788i	$0.351254\pi$
$758$	−0.398643	−0.0144794
$759$	0	0
$760$	41.7578	1.51471
$761$	−10.8651	−0.393858	−0.196929	−	0.980418i	$-0.563097\pi$
$761$	−10.8651	−0.393858	−0.196929	+	0.980418i	$0.563097\pi$
$762$	0	0
$763$	−7.26005	−0.262831
$764$	−16.1431	−0.584036
$765$	0	0
$766$	−6.94959	−0.251099
$767$	38.6135	1.39425
$768$	0	0
$769$	29.6533	1.06932	0.534662	−	0.845066i	$-0.320439\pi$
$769$	29.6533	1.06932	0.534662	+	0.845066i	$0.320439\pi$
$770$	0.365229	0.0131619
$771$	0	0
$772$	−3.53781	−0.127329
$773$	−16.5327	−0.594640	−0.297320	−	0.954778i	$-0.596093\pi$
$773$	−16.5327	−0.594640	−0.297320	+	0.954778i	$0.596093\pi$
$774$	0	0
$775$	28.0865	1.00890
$776$	0.735755	0.0264120
$777$	0	0
$778$	−11.3083	−0.405423
$779$	−35.7121	−1.27952
$780$	0	0
$781$	−1.61759	−0.0578821
$782$	0.126485	0.00452310
$783$	0	0
$784$	3.30583	0.118065
$785$	−55.7245	−1.98889
$786$	0	0
$787$	13.3944	0.477460	0.238730	−	0.971086i	$-0.423269\pi$
$787$	13.3944	0.477460	0.238730	+	0.971086i	$0.423269\pi$
$788$	5.50078	0.195957
$789$	0	0
$790$	−14.0913	−0.501347
$791$	8.77601	0.312039
$792$	0	0
$793$	−6.15153	−0.218447
$794$	6.99650	0.248296
$795$	0	0
$796$	3.68109	0.130473
$797$	−46.0581	−1.63146	−0.815730	−	0.578433i	$-0.803665\pi$
$797$	−46.0581	−1.63146	−0.815730	+	0.578433i	$0.803665\pi$
$798$	0	0
$799$	13.9900	0.494929
$800$	48.5345	1.71595
$801$	0	0
$802$	6.26951	0.221384
$803$	1.36076	0.0480202
$804$	0	0
$805$	−0.580249	−0.0204511
$806$	−5.26257	−0.185366
$807$	0	0
$808$	10.0398	0.353197
$809$	−45.7555	−1.60868	−0.804339	−	0.594171i	$-0.797480\pi$
$809$	−45.7555	−1.60868	−0.804339	+	0.594171i	$0.797480\pi$
$810$	0	0
$811$	−17.1026	−0.600555	−0.300277	−	0.953852i	$-0.597079\pi$
$811$	−17.1026	−0.600555	−0.300277	+	0.953852i	$0.597079\pi$
$812$	−4.76819	−0.167330
$813$	0	0
$814$	0.641558	0.0224866
$815$	29.6981	1.04028
$816$	0	0
$817$	−20.1625	−0.705398
$818$	5.65109	0.197586
$819$	0	0
$820$	−38.1264	−1.33143
$821$	−8.41469	−0.293675	−0.146837	−	0.989161i	$-0.546909\pi$
$821$	−8.41469	−0.293675	−0.146837	+	0.989161i	$0.546909\pi$
$822$	0	0
$823$	−20.2122	−0.704552	−0.352276	−	0.935896i	$-0.614592\pi$
$823$	−20.2122	−0.704552	−0.352276	+	0.935896i	$0.614592\pi$
$824$	17.5743	0.612231
$825$	0	0
$826$	−1.90563	−0.0663054
$827$	−8.84183	−0.307460	−0.153730	−	0.988113i	$-0.549129\pi$
$827$	−8.84183	−0.307460	−0.153730	+	0.988113i	$0.549129\pi$
$828$	0	0
$829$	2.47179	0.0858487	0.0429244	−	0.999078i	$-0.486333\pi$
$829$	2.47179	0.0858487	0.0429244	+	0.999078i	$0.486333\pi$
$830$	6.90537	0.239689
$831$	0	0
$832$	36.9297	1.28031
$833$	−2.67429	−0.0926587
$834$	0	0
$835$	−84.3769	−2.91998
$836$	3.52709	0.121987
$837$	0	0
$838$	−9.30444	−0.321417
$839$	38.5081	1.32945	0.664723	−	0.747090i	$-0.268550\pi$
$839$	38.5081	1.32945	0.664723	+	0.747090i	$0.268550\pi$
$840$	0	0
$841$	−22.5809	−0.778652
$842$	2.18186	0.0751919
$843$	0	0
$844$	28.5358	0.982242
$845$	149.429	5.14050
$846$	0	0
$847$	−10.9364	−0.375778
$848$	−24.8828	−0.854479
$849$	0	0
$850$	−11.7252	−0.402170
$851$	−1.01926	−0.0349398
$852$	0	0
$853$	−27.6956	−0.948278	−0.474139	−	0.880450i	$-0.657241\pi$
$853$	−27.6956	−0.948278	−0.474139	+	0.880450i	$0.657241\pi$
$854$	0.303587	0.0103885
$855$	0	0
$856$	−0.360851	−0.0123336
$857$	30.9850	1.05843	0.529213	−	0.848489i	$-0.322487\pi$
$857$	30.9850	1.05843	0.529213	+	0.848489i	$0.322487\pi$
$858$	0	0
$859$	23.7710	0.811055	0.405528	−	0.914083i	$-0.367088\pi$
$859$	23.7710	0.811055	0.405528	+	0.914083i	$0.367088\pi$
$860$	−21.5256	−0.734016
$861$	0	0
$862$	−5.07533	−0.172866
$863$	36.3966	1.23895	0.619477	−	0.785015i	$-0.287345\pi$
$863$	36.3966	1.23895	0.619477	+	0.785015i	$0.287345\pi$
$864$	0	0
$865$	−15.9791	−0.543305
$866$	−11.3833	−0.386821
$867$	0	0
$868$	−4.14165	−0.140577
$869$	−2.45510	−0.0832835
$870$	0	0
$871$	105.471	3.57373
$872$	9.68196	0.327873
$873$	0	0
$874$	0.351391	0.0118860
$875$	32.7162	1.10601
$876$	0	0
$877$	3.30277	0.111527	0.0557633	−	0.998444i	$-0.482241\pi$
$877$	3.30277	0.111527	0.0557633	+	0.998444i	$0.482241\pi$
$878$	−11.9661	−0.403835
$879$	0	0
$880$	3.51460	0.118477
$881$	−0.422085	−0.0142204	−0.00711020	−	0.999975i	$-0.502263\pi$
$881$	−0.422085	−0.0142204	−0.00711020	+	0.999975i	$0.502263\pi$
$882$	0	0
$883$	−16.1977	−0.545098	−0.272549	−	0.962142i	$-0.587867\pi$
$883$	−16.1977	−0.545098	−0.272549	+	0.962142i	$0.587867\pi$
$884$	−35.0344	−1.17834
$885$	0	0
$886$	4.34163	0.145860
$887$	−32.0569	−1.07637	−0.538183	−	0.842828i	$-0.680889\pi$
$887$	−32.0569	−1.07637	−0.538183	+	0.842828i	$0.680889\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	11.0661	0.370938
$891$	0	0
$892$	−52.3543	−1.75295
$893$	38.8658	1.30059
$894$	0	0
$895$	15.1826	0.507500
$896$	−9.42825	−0.314976
$897$	0	0
$898$	−10.2382	−0.341654
$899$	5.57564	0.185958
$900$	0	0
$901$	20.1292	0.670602
$902$	0.416550	0.0138696
$903$	0	0
$904$	−11.7036	−0.389257
$905$	72.1045	2.39683
$906$	0	0
$907$	−51.3230	−1.70415	−0.852076	−	0.523417i	$-0.824657\pi$
$907$	−51.3230	−1.70415	−0.852076	+	0.523417i	$0.824657\pi$
$908$	−41.5051	−1.37740
$909$	0	0
$910$	−10.0785	−0.334098
$911$	−9.50265	−0.314837	−0.157418	−	0.987532i	$-0.550317\pi$
$911$	−9.50265	−0.314837	−0.157418	+	0.987532i	$0.550317\pi$
$912$	0	0
$913$	1.20311	0.0398170
$914$	−0.378540	−0.0125210
$915$	0	0
$916$	25.5846	0.845339
$917$	−3.72616	−0.123049
$918$	0	0
$919$	−33.8987	−1.11821	−0.559107	−	0.829096i	$-0.688856\pi$
$919$	−33.8987	−1.11821	−0.559107	+	0.829096i	$0.688856\pi$
$920$	0.773817	0.0255120
$921$	0	0
$922$	−1.06663	−0.0351277
$923$	44.6374	1.46926
$924$	0	0
$925$	94.4855	3.10666
$926$	9.09781	0.298972
$927$	0	0
$928$	9.63490	0.316281
$929$	0.983007	0.0322514	0.0161257	−	0.999870i	$-0.494867\pi$
$929$	0.983007	0.0322514	0.0161257	+	0.999870i	$0.494867\pi$
$930$	0	0
$931$	−7.42951	−0.243492
$932$	37.6704	1.23394
$933$	0	0
$934$	−0.404190	−0.0132255
$935$	−2.84317	−0.0929817
$936$	0	0
$937$	2.16120	0.0706032	0.0353016	−	0.999377i	$-0.488761\pi$
$937$	2.16120	0.0706032	0.0353016	+	0.999377i	$0.488761\pi$
$938$	−5.20513	−0.169953
$939$	0	0
$940$	41.4932	1.35336
$941$	2.19328	0.0714990	0.0357495	−	0.999361i	$-0.488618\pi$
$941$	2.19328	0.0714990	0.0357495	+	0.999361i	$0.488618\pi$
$942$	0	0
$943$	−0.661785	−0.0215507
$944$	−18.3379	−0.596848
$945$	0	0
$946$	0.235178	0.00764629
$947$	38.1039	1.23821	0.619104	−	0.785309i	$-0.287496\pi$
$947$	38.1039	1.23821	0.619104	+	0.785309i	$0.287496\pi$
$948$	0	0
$949$	−37.5501	−1.21893
$950$	−32.5740	−1.05684
$951$	0	0
$952$	3.56642	0.115588
$953$	23.2950	0.754600	0.377300	−	0.926091i	$-0.376852\pi$
$953$	23.2950	0.754600	0.377300	+	0.926091i	$0.376852\pi$
$954$	0	0
$955$	36.1513	1.16983
$956$	47.3593	1.53171
$957$	0	0
$958$	−11.8147	−0.381716
$959$	−23.0959	−0.745806
$960$	0	0
$961$	−26.1570	−0.843774
$962$	−17.7038	−0.570792
$963$	0	0
$964$	18.2466	0.587683
$965$	7.92269	0.255040
$966$	0	0
$967$	−33.1084	−1.06469	−0.532347	−	0.846526i	$-0.678690\pi$
$967$	−33.1084	−1.06469	−0.532347	+	0.846526i	$0.678690\pi$
$968$	14.5847	0.468770
$969$	0	0
$970$	−0.798790	−0.0256476
$971$	−49.1446	−1.57713	−0.788563	−	0.614954i	$-0.789175\pi$
$971$	−49.1446	−1.57713	−0.788563	+	0.614954i	$0.789175\pi$
$972$	0	0
$973$	−12.2778	−0.393608
$974$	−10.7430	−0.344229
$975$	0	0
$976$	2.92142	0.0935122
$977$	42.4242	1.35727	0.678636	−	0.734475i	$-0.262571\pi$
$977$	42.4242	1.35727	0.678636	+	0.734475i	$0.262571\pi$
$978$	0	0
$979$	1.92803	0.0616200
$980$	−7.93176	−0.253371
$981$	0	0
$982$	−13.0909	−0.417747
$983$	−62.2538	−1.98559	−0.992793	−	0.119839i	$-0.961762\pi$
$983$	−62.2538	−1.98559	−0.992793	+	0.119839i	$0.961762\pi$
$984$	0	0
$985$	−12.3186	−0.392504
$986$	−2.32764	−0.0741272
$987$	0	0
$988$	−97.3299	−3.09648
$989$	−0.373634	−0.0118809
$990$	0	0
$991$	−27.9157	−0.886772	−0.443386	−	0.896331i	$-0.646223\pi$
$991$	−27.9157	−0.886772	−0.443386	+	0.896331i	$0.646223\pi$
$992$	8.36888	0.265712
$993$	0	0
$994$	−2.20292	−0.0698724
$995$	−8.24354	−0.261338
$996$	0	0
$997$	10.7076	0.339115	0.169557	−	0.985520i	$-0.445766\pi$
$997$	10.7076	0.339115	0.169557	+	0.985520i	$0.445766\pi$
$998$	−5.58573	−0.176813
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.w.1.13		20
3.2	odd	2		889.2.a.d.1.8	✓	20
21.20	even	2		6223.2.a.l.1.8		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
889.2.a.d.1.8	✓	20	3.2	odd	2
6223.2.a.l.1.8		20	21.20	even	2
8001.2.a.w.1.13		20	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+0.343534 q^{2} -1.88198 q^{4} +4.21457 q^{5} +1.00000 q^{7} -1.33360 q^{8} +O(q^{10})\)	\(q+0.343534 q^{2} -1.88198 q^{4} +4.21457 q^{5} +1.00000 q^{7} -1.33360 q^{8} +1.44785 q^{10} +0.252256 q^{11} -6.96098 q^{13} +0.343534 q^{14} +3.30583 q^{16} -2.67429 q^{17} -7.42951 q^{19} -7.93176 q^{20} +0.0866585 q^{22} -0.137677 q^{23} +12.7626 q^{25} -2.39134 q^{26} -1.88198 q^{28} +2.53359 q^{29} +2.20068 q^{31} +3.80286 q^{32} -0.918712 q^{34} +4.21457 q^{35} +7.40329 q^{37} -2.55229 q^{38} -5.62054 q^{40} +4.80680 q^{41} +2.71385 q^{43} -0.474741 q^{44} -0.0472967 q^{46} -5.23128 q^{47} +1.00000 q^{49} +4.38440 q^{50} +13.1005 q^{52} -7.52694 q^{53} +1.06315 q^{55} -1.33360 q^{56} +0.870377 q^{58} -5.54714 q^{59} +0.883716 q^{61} +0.756010 q^{62} -5.30525 q^{64} -29.3376 q^{65} -15.1517 q^{67} +5.03298 q^{68} +1.44785 q^{70} -6.41252 q^{71} +5.39437 q^{73} +2.54329 q^{74} +13.9822 q^{76} +0.252256 q^{77} -9.73258 q^{79} +13.9327 q^{80} +1.65130 q^{82} +4.76939 q^{83} -11.2710 q^{85} +0.932299 q^{86} -0.336407 q^{88} +7.64315 q^{89} -6.96098 q^{91} +0.259106 q^{92} -1.79712 q^{94} -31.3122 q^{95} -0.551708 q^{97} +0.343534 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 8 q^{2} + 24 q^{4} - 3 q^{5} + 20 q^{7} - 24 q^{8}+O(q^{10}) \) 20 * q - 8 * q^2 + 24 * q^4 - 3 * q^5 + 20 * q^7 - 24 * q^8	\( 20 q - 8 q^{2} + 24 q^{4} - 3 q^{5} + 20 q^{7} - 24 q^{8} - 8 q^{10} - 26 q^{11} - 4 q^{13} - 8 q^{14} + 24 q^{16} - 4 q^{17} + q^{19} + 2 q^{20} + q^{22} - 31 q^{23} + 27 q^{25} - 4 q^{26} + 24 q^{28} - 16 q^{29} + 6 q^{31} - 41 q^{32} - 10 q^{34} - 3 q^{35} + 2 q^{37} - 3 q^{38} - 38 q^{40} - 25 q^{41} + 13 q^{43} - 66 q^{44} + 20 q^{46} - 19 q^{47} + 20 q^{49} + 4 q^{50} + 20 q^{52} - 24 q^{53} - 3 q^{55} - 24 q^{56} + 12 q^{58} - 23 q^{59} - 27 q^{61} - 7 q^{62} + 2 q^{64} - 26 q^{65} + 9 q^{67} + 25 q^{68} - 8 q^{70} - 63 q^{71} - 21 q^{73} - 21 q^{74} - 10 q^{76} - 26 q^{77} + 18 q^{79} + 23 q^{80} - 42 q^{82} + q^{83} - 41 q^{85} + 12 q^{86} + 57 q^{88} + 16 q^{89} - 4 q^{91} - 17 q^{92} + 7 q^{94} - 75 q^{95} - 32 q^{97} - 8 q^{98}+O(q^{100}) \) 20 * q - 8 * q^2 + 24 * q^4 - 3 * q^5 + 20 * q^7 - 24 * q^8 - 8 * q^10 - 26 * q^11 - 4 * q^13 - 8 * q^14 + 24 * q^16 - 4 * q^17 + q^19 + 2 * q^20 + q^22 - 31 * q^23 + 27 * q^25 - 4 * q^26 + 24 * q^28 - 16 * q^29 + 6 * q^31 - 41 * q^32 - 10 * q^34 - 3 * q^35 + 2 * q^37 - 3 * q^38 - 38 * q^40 - 25 * q^41 + 13 * q^43 - 66 * q^44 + 20 * q^46 - 19 * q^47 + 20 * q^49 + 4 * q^50 + 20 * q^52 - 24 * q^53 - 3 * q^55 - 24 * q^56 + 12 * q^58 - 23 * q^59 - 27 * q^61 - 7 * q^62 + 2 * q^64 - 26 * q^65 + 9 * q^67 + 25 * q^68 - 8 * q^70 - 63 * q^71 - 21 * q^73 - 21 * q^74 - 10 * q^76 - 26 * q^77 + 18 * q^79 + 23 * q^80 - 42 * q^82 + q^83 - 41 * q^85 + 12 * q^86 + 57 * q^88 + 16 * q^89 - 4 * q^91 - 17 * q^92 + 7 * q^94 - 75 * q^95 - 32 * q^97 - 8 * q^98

Introduction

L-functions

Modular forms

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Database

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