LMFDB - Embedded newform 8001.2.a.u.1.2

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:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 6 x^{17} - 11 x^{16} + 123 x^{15} - 35 x^{14} - 982 x^{13} + 988 x^{12} + 3872 x^{11} + \cdots + 16 $ x^18 - 6x^17 - 11x^16 + 123x^15 - 35x^14 - 982x^13 + 988x^12 + 3872x^11 - 5421x^10 - 7882x^9 + 13376x^8 + 7948x^7 - 15795x^6 - 3858x^5 + 8199x^4 + 1453x^3 - 1610x^2 - 380*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2\cdot 5 $
Twist minimal:	no (minimal twist has level 2667)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.24824$ 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)

$f(q)$	$=$	$q-2.24824 q^{2} +3.05458 q^{4} +3.31434 q^{5} +1.00000 q^{7} -2.37095 q^{8} +O(q^{10})$	\(q-2.24824 q^{2} +3.05458 q^{4} +3.31434 q^{5} +1.00000 q^{7} -2.37095 q^{8} -7.45143 q^{10} -5.47711 q^{11} -4.39594 q^{13} -2.24824 q^{14} -0.778706 q^{16} +2.99559 q^{17} -0.499179 q^{19} +10.1239 q^{20} +12.3138 q^{22} +6.47149 q^{23} +5.98485 q^{25} +9.88313 q^{26} +3.05458 q^{28} -2.28220 q^{29} +4.20474 q^{31} +6.49261 q^{32} -6.73481 q^{34} +3.31434 q^{35} -7.84180 q^{37} +1.12227 q^{38} -7.85812 q^{40} +7.43947 q^{41} +6.74366 q^{43} -16.7303 q^{44} -14.5494 q^{46} +8.99909 q^{47} +1.00000 q^{49} -13.4554 q^{50} -13.4278 q^{52} +1.35946 q^{53} -18.1530 q^{55} -2.37095 q^{56} +5.13092 q^{58} +5.28556 q^{59} +0.533481 q^{61} -9.45326 q^{62} -13.0395 q^{64} -14.5696 q^{65} -14.0967 q^{67} +9.15028 q^{68} -7.45143 q^{70} +6.62684 q^{71} +6.36264 q^{73} +17.6302 q^{74} -1.52478 q^{76} -5.47711 q^{77} -15.8144 q^{79} -2.58089 q^{80} -16.7257 q^{82} -5.96805 q^{83} +9.92842 q^{85} -15.1614 q^{86} +12.9859 q^{88} -6.57010 q^{89} -4.39594 q^{91} +19.7677 q^{92} -20.2321 q^{94} -1.65445 q^{95} +12.5689 q^{97} -2.24824 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 6 q^{2} + 22 q^{4} + 10 q^{5} + 18 q^{7} + 21 q^{8}+O(q^{10}) $ 18 * q + 6 * q^2 + 22 * q^4 + 10 * q^5 + 18 * q^7 + 21 * q^8	$ 18 q + 6 q^{2} + 22 q^{4} + 10 q^{5} + 18 q^{7} + 21 q^{8} - 4 q^{10} + 9 q^{11} - 25 q^{13} + 6 q^{14} + 34 q^{16} + 17 q^{17} - 5 q^{19} + 21 q^{20} + 5 q^{22} + 14 q^{23} + 28 q^{25} + 8 q^{26} + 22 q^{28} + 17 q^{29} + 5 q^{31} + 53 q^{32} - 19 q^{34} + 10 q^{35} - 15 q^{37} + 22 q^{38} - q^{40} + 17 q^{41} + q^{43} + 33 q^{44} + 10 q^{46} + 31 q^{47} + 18 q^{49} + 35 q^{50} - 70 q^{52} + 35 q^{53} + 4 q^{55} + 21 q^{56} + 3 q^{58} + 46 q^{59} - 5 q^{61} + 10 q^{62} + 63 q^{64} + 12 q^{65} + 6 q^{67} + 56 q^{68} - 4 q^{70} + 22 q^{71} - 16 q^{73} - 18 q^{74} + 32 q^{76} + 9 q^{77} + 46 q^{79} + 30 q^{80} - 12 q^{82} + 46 q^{83} + 4 q^{85} - 18 q^{86} + 30 q^{88} + 42 q^{89} - 25 q^{91} + 48 q^{92} + 3 q^{94} + 2 q^{95} - 35 q^{97} + 6 q^{98}+O(q^{100}) $ 18 * q + 6 * q^2 + 22 * q^4 + 10 * q^5 + 18 * q^7 + 21 * q^8 - 4 * q^10 + 9 * q^11 - 25 * q^13 + 6 * q^14 + 34 * q^16 + 17 * q^17 - 5 * q^19 + 21 * q^20 + 5 * q^22 + 14 * q^23 + 28 * q^25 + 8 * q^26 + 22 * q^28 + 17 * q^29 + 5 * q^31 + 53 * q^32 - 19 * q^34 + 10 * q^35 - 15 * q^37 + 22 * q^38 - q^40 + 17 * q^41 + q^43 + 33 * q^44 + 10 * q^46 + 31 * q^47 + 18 * q^49 + 35 * q^50 - 70 * q^52 + 35 * q^53 + 4 * q^55 + 21 * q^56 + 3 * q^58 + 46 * q^59 - 5 * q^61 + 10 * q^62 + 63 * q^64 + 12 * q^65 + 6 * q^67 + 56 * q^68 - 4 * q^70 + 22 * q^71 - 16 * q^73 - 18 * q^74 + 32 * q^76 + 9 * q^77 + 46 * q^79 + 30 * q^80 - 12 * q^82 + 46 * q^83 + 4 * q^85 - 18 * q^86 + 30 * q^88 + 42 * q^89 - 25 * q^91 + 48 * q^92 + 3 * q^94 + 2 * q^95 - 35 * q^97 + 6 * 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 $

$2$	−2.24824	−1.58975	−0.794873	−	0.606776i	$-0.792462\pi$
$2$	−2.24824	−1.58975	−0.794873	+	0.606776i	$0.792462\pi$
$3$	0	0
$3$	0	0
$4$	3.05458	1.52729
$5$	3.31434	1.48222	0.741109	−	0.671385i	$-0.234300\pi$
$5$	3.31434	1.48222	0.741109	+	0.671385i	$0.234300\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−2.37095	−0.838256
$9$	0	0
$10$	−7.45143	−2.35635
$11$	−5.47711	−1.65141	−0.825705	−	0.564102i	$-0.809222\pi$
$11$	−5.47711	−1.65141	−0.825705	+	0.564102i	$0.809222\pi$
$12$	0	0
$13$	−4.39594	−1.21921	−0.609607	−	0.792703i	$-0.708673\pi$
$13$	−4.39594	−1.21921	−0.609607	+	0.792703i	$0.708673\pi$
$14$	−2.24824	−0.600867
$15$	0	0
$16$	−0.778706	−0.194676
$17$	2.99559	0.726538	0.363269	−	0.931684i	$-0.381661\pi$
$17$	2.99559	0.726538	0.363269	+	0.931684i	$0.381661\pi$
$18$	0	0
$19$	−0.499179	−0.114519	−0.0572597	−	0.998359i	$-0.518236\pi$
$19$	−0.499179	−0.114519	−0.0572597	+	0.998359i	$0.518236\pi$
$20$	10.1239	2.26378
$21$	0	0
$22$	12.3138	2.62532
$23$	6.47149	1.34940	0.674699	−	0.738093i	$-0.264273\pi$
$23$	6.47149	1.34940	0.674699	+	0.738093i	$0.264273\pi$
$24$	0	0
$25$	5.98485	1.19697
$26$	9.88313	1.93824
$27$	0	0
$28$	3.05458	0.577261
$29$	−2.28220	−0.423793	−0.211897	−	0.977292i	$-0.567964\pi$
$29$	−2.28220	−0.423793	−0.211897	+	0.977292i	$0.567964\pi$
$30$	0	0
$31$	4.20474	0.755193	0.377597	−	0.925970i	$-0.376751\pi$
$31$	4.20474	0.755193	0.377597	+	0.925970i	$0.376751\pi$
$32$	6.49261	1.14774
$33$	0	0
$34$	−6.73481	−1.15501
$35$	3.31434	0.560226
$36$	0	0
$37$	−7.84180	−1.28918	−0.644592	−	0.764526i	$-0.722973\pi$
$37$	−7.84180	−1.28918	−0.644592	+	0.764526i	$0.722973\pi$
$38$	1.12227	0.182057
$39$	0	0
$40$	−7.85812	−1.24248
$41$	7.43947	1.16185	0.580925	−	0.813957i	$-0.302691\pi$
$41$	7.43947	1.16185	0.580925	+	0.813957i	$0.302691\pi$
$42$	0	0
$43$	6.74366	1.02840	0.514200	−	0.857671i	$-0.328089\pi$
$43$	6.74366	1.02840	0.514200	+	0.857671i	$0.328089\pi$
$44$	−16.7303	−2.52218
$45$	0	0
$46$	−14.5494	−2.14520
$47$	8.99909	1.31265	0.656326	−	0.754477i	$-0.272110\pi$
$47$	8.99909	1.31265	0.656326	+	0.754477i	$0.272110\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−13.4554	−1.90288
$51$	0	0
$52$	−13.4278	−1.86209
$53$	1.35946	0.186736	0.0933679	−	0.995632i	$-0.470237\pi$
$53$	1.35946	0.186736	0.0933679	+	0.995632i	$0.470237\pi$
$54$	0	0
$55$	−18.1530	−2.44775
$56$	−2.37095	−0.316831
$57$	0	0
$58$	5.13092	0.673723
$59$	5.28556	0.688121	0.344061	−	0.938947i	$-0.388197\pi$
$59$	5.28556	0.688121	0.344061	+	0.938947i	$0.388197\pi$
$60$	0	0
$61$	0.533481	0.0683053	0.0341526	−	0.999417i	$-0.489127\pi$
$61$	0.533481	0.0683053	0.0341526	+	0.999417i	$0.489127\pi$
$62$	−9.45326	−1.20057
$63$	0	0
$64$	−13.0395	−1.62994
$65$	−14.5696	−1.80714
$66$	0	0
$67$	−14.0967	−1.72218	−0.861092	−	0.508449i	$-0.830219\pi$
$67$	−14.0967	−1.72218	−0.861092	+	0.508449i	$0.830219\pi$
$68$	9.15028	1.10963
$69$	0	0
$70$	−7.45143	−0.890616
$71$	6.62684	0.786461	0.393231	−	0.919440i	$-0.371357\pi$
$71$	6.62684	0.786461	0.393231	+	0.919440i	$0.371357\pi$
$72$	0	0
$73$	6.36264	0.744691	0.372345	−	0.928094i	$-0.378554\pi$
$73$	6.36264	0.744691	0.372345	+	0.928094i	$0.378554\pi$
$74$	17.6302	2.04947
$75$	0	0
$76$	−1.52478	−0.174904
$77$	−5.47711	−0.624174
$78$	0	0
$79$	−15.8144	−1.77926	−0.889628	−	0.456686i	$-0.849036\pi$
$79$	−15.8144	−1.77926	−0.889628	+	0.456686i	$0.849036\pi$
$80$	−2.58089	−0.288553
$81$	0	0
$82$	−16.7257	−1.84705
$83$	−5.96805	−0.655079	−0.327539	−	0.944838i	$-0.606219\pi$
$83$	−5.96805	−0.655079	−0.327539	+	0.944838i	$0.606219\pi$
$84$	0	0
$85$	9.92842	1.07689
$86$	−15.1614	−1.63489
$87$	0	0
$88$	12.9859	1.38430
$89$	−6.57010	−0.696429	−0.348215	−	0.937415i	$-0.613212\pi$
$89$	−6.57010	−0.696429	−0.348215	+	0.937415i	$0.613212\pi$
$90$	0	0
$91$	−4.39594	−0.460820
$92$	19.7677	2.06092
$93$	0	0
$94$	−20.2321	−2.08678
$95$	−1.65445	−0.169743
$96$	0	0
$97$	12.5689	1.27618	0.638092	−	0.769960i	$-0.279724\pi$
$97$	12.5689	1.27618	0.638092	+	0.769960i	$0.279724\pi$
$98$	−2.24824	−0.227106
$99$	0	0
$100$	18.2812	1.82812
$101$	4.31831	0.429688	0.214844	−	0.976648i	$-0.431076\pi$
$101$	4.31831	0.429688	0.214844	+	0.976648i	$0.431076\pi$
$102$	0	0
$103$	−7.67776	−0.756512	−0.378256	−	0.925701i	$-0.623476\pi$
$103$	−7.67776	−0.756512	−0.378256	+	0.925701i	$0.623476\pi$
$104$	10.4225	1.02201
$105$	0	0
$106$	−3.05638	−0.296862
$107$	6.15637	0.595159	0.297579	−	0.954697i	$-0.403821\pi$
$107$	6.15637	0.595159	0.297579	+	0.954697i	$0.403821\pi$
$108$	0	0
$109$	−15.1231	−1.44853	−0.724263	−	0.689524i	$-0.757820\pi$
$109$	−15.1231	−1.44853	−0.724263	+	0.689524i	$0.757820\pi$
$110$	40.8123	3.89130
$111$	0	0
$112$	−0.778706	−0.0735808
$113$	−1.57617	−0.148273	−0.0741366	−	0.997248i	$-0.523620\pi$
$113$	−1.57617	−0.148273	−0.0741366	+	0.997248i	$0.523620\pi$
$114$	0	0
$115$	21.4487	2.00010
$116$	−6.97115	−0.647255
$117$	0	0
$118$	−11.8832	−1.09394
$119$	2.99559	0.274606
$120$	0	0
$121$	18.9987	1.72715
$122$	−1.19939	−0.108588
$123$	0	0
$124$	12.8437	1.15340
$125$	3.26412	0.291951
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	16.3307	1.44345
$129$	0	0
$130$	32.7560	2.87289
$131$	−0.0249085	−0.00217626	−0.00108813	−	0.999999i	$-0.500346\pi$
$131$	−0.0249085	−0.00217626	−0.00108813	+	0.999999i	$0.500346\pi$
$132$	0	0
$133$	−0.499179	−0.0432843
$134$	31.6927	2.73783
$135$	0	0
$136$	−7.10239	−0.609025
$137$	−19.2352	−1.64337	−0.821685	−	0.569942i	$-0.806966\pi$
$137$	−19.2352	−1.64337	−0.821685	+	0.569942i	$0.806966\pi$
$138$	0	0
$139$	6.80780	0.577430	0.288715	−	0.957415i	$-0.406772\pi$
$139$	6.80780	0.577430	0.288715	+	0.957415i	$0.406772\pi$
$140$	10.1239	0.855627
$141$	0	0
$142$	−14.8987	−1.25027
$143$	24.0770	2.01342
$144$	0	0
$145$	−7.56397	−0.628154
$146$	−14.3047	−1.18387
$147$	0	0
$148$	−23.9534	−1.96896
$149$	11.2791	0.924021	0.462010	−	0.886874i	$-0.347128\pi$
$149$	11.2791	0.924021	0.462010	+	0.886874i	$0.347128\pi$
$150$	0	0
$151$	2.10031	0.170920	0.0854602	−	0.996342i	$-0.472764\pi$
$151$	2.10031	0.170920	0.0854602	+	0.996342i	$0.472764\pi$
$152$	1.18353	0.0959966
$153$	0	0
$154$	12.3138	0.992278
$155$	13.9359	1.11936
$156$	0	0
$157$	14.5687	1.16271	0.581353	−	0.813651i	$-0.302523\pi$
$157$	14.5687	1.16271	0.581353	+	0.813651i	$0.302523\pi$
$158$	35.5545	2.82856
$159$	0	0
$160$	21.5187	1.70120
$161$	6.47149	0.510025
$162$	0	0
$163$	−2.35015	−0.184078	−0.0920390	−	0.995755i	$-0.529338\pi$
$163$	−2.35015	−0.184078	−0.0920390	+	0.995755i	$0.529338\pi$
$164$	22.7245	1.77448
$165$	0	0
$166$	13.4176	1.04141
$167$	23.4699	1.81616	0.908079	−	0.418800i	$-0.137549\pi$
$167$	23.4699	1.81616	0.908079	+	0.418800i	$0.137549\pi$
$168$	0	0
$169$	6.32430	0.486485
$170$	−22.3215	−1.71198
$171$	0	0
$172$	20.5991	1.57066
$173$	−2.47947	−0.188511	−0.0942555	−	0.995548i	$-0.530047\pi$
$173$	−2.47947	−0.188511	−0.0942555	+	0.995548i	$0.530047\pi$
$174$	0	0
$175$	5.98485	0.452412
$176$	4.26505	0.321490
$177$	0	0
$178$	14.7712	1.10715
$179$	−9.59334	−0.717040	−0.358520	−	0.933522i	$-0.616718\pi$
$179$	−9.59334	−0.717040	−0.358520	+	0.933522i	$0.616718\pi$
$180$	0	0
$181$	1.01943	0.0757737	0.0378868	−	0.999282i	$-0.487937\pi$
$181$	1.01943	0.0757737	0.0378868	+	0.999282i	$0.487937\pi$
$182$	9.88313	0.732586
$183$	0	0
$184$	−15.3435	−1.13114
$185$	−25.9904	−1.91085
$186$	0	0
$187$	−16.4072	−1.19981
$188$	27.4884	2.00480
$189$	0	0
$190$	3.71959	0.269848
$191$	−9.93407	−0.718804	−0.359402	−	0.933183i	$-0.617019\pi$
$191$	−9.93407	−0.718804	−0.359402	+	0.933183i	$0.617019\pi$
$192$	0	0
$193$	15.2597	1.09842	0.549209	−	0.835685i	$-0.314929\pi$
$193$	15.2597	1.09842	0.549209	+	0.835685i	$0.314929\pi$
$194$	−28.2580	−2.02881
$195$	0	0
$196$	3.05458	0.218184
$197$	12.0045	0.855287	0.427644	−	0.903947i	$-0.359344\pi$
$197$	12.0045	0.855287	0.427644	+	0.903947i	$0.359344\pi$
$198$	0	0
$199$	22.0679	1.56435	0.782174	−	0.623060i	$-0.214111\pi$
$199$	22.0679	1.56435	0.782174	+	0.623060i	$0.214111\pi$
$200$	−14.1897	−1.00337
$201$	0	0
$202$	−9.70860	−0.683095
$203$	−2.28220	−0.160179
$204$	0	0
$205$	24.6569	1.72211
$206$	17.2614	1.20266
$207$	0	0
$208$	3.42314	0.237352
$209$	2.73405	0.189119
$210$	0	0
$211$	23.3444	1.60710	0.803548	−	0.595239i	$-0.202943\pi$
$211$	23.3444	1.60710	0.803548	+	0.595239i	$0.202943\pi$
$212$	4.15257	0.285200
$213$	0	0
$214$	−13.8410	−0.946151
$215$	22.3508	1.52431
$216$	0	0
$217$	4.20474	0.285436
$218$	34.0002	2.30279
$219$	0	0
$220$	−55.4497	−3.73842
$221$	−13.1685	−0.885807
$222$	0	0
$223$	7.33378	0.491106	0.245553	−	0.969383i	$-0.421030\pi$
$223$	7.33378	0.491106	0.245553	+	0.969383i	$0.421030\pi$
$224$	6.49261	0.433806
$225$	0	0
$226$	3.54360	0.235717
$227$	27.0309	1.79411	0.897053	−	0.441924i	$-0.145704\pi$
$227$	27.0309	1.79411	0.897053	+	0.441924i	$0.145704\pi$
$228$	0	0
$229$	−13.0556	−0.862737	−0.431368	−	0.902176i	$-0.641969\pi$
$229$	−13.0556	−0.862737	−0.431368	+	0.902176i	$0.641969\pi$
$230$	−48.2218	−3.17965
$231$	0	0
$232$	5.41096	0.355247
$233$	23.6719	1.55080	0.775398	−	0.631473i	$-0.217549\pi$
$233$	23.6719	1.55080	0.775398	+	0.631473i	$0.217549\pi$
$234$	0	0
$235$	29.8260	1.94564
$236$	16.1452	1.05096
$237$	0	0
$238$	−6.73481	−0.436553
$239$	−28.3744	−1.83539	−0.917693	−	0.397289i	$-0.869951\pi$
$239$	−28.3744	−1.83539	−0.917693	+	0.397289i	$0.869951\pi$
$240$	0	0
$241$	−6.89559	−0.444184	−0.222092	−	0.975026i	$-0.571288\pi$
$241$	−6.89559	−0.444184	−0.222092	+	0.975026i	$0.571288\pi$
$242$	−42.7136	−2.74573
$243$	0	0
$244$	1.62956	0.104322
$245$	3.31434	0.211745
$246$	0	0
$247$	2.19436	0.139624
$248$	−9.96921	−0.633045
$249$	0	0
$250$	−7.33851	−0.464128
$251$	−18.5391	−1.17018	−0.585090	−	0.810968i	$-0.698941\pi$
$251$	−18.5391	−1.17018	−0.585090	+	0.810968i	$0.698941\pi$
$252$	0	0
$253$	−35.4450	−2.22841
$254$	−2.24824	−0.141067
$255$	0	0
$256$	−10.6364	−0.664774
$257$	−14.7593	−0.920662	−0.460331	−	0.887747i	$-0.652269\pi$
$257$	−14.7593	−0.920662	−0.460331	+	0.887747i	$0.652269\pi$
$258$	0	0
$259$	−7.84180	−0.487266
$260$	−44.5041	−2.76003
$261$	0	0
$262$	0.0560002	0.00345971
$263$	−10.8798	−0.670877	−0.335439	−	0.942062i	$-0.608884\pi$
$263$	−10.8798	−0.670877	−0.335439	+	0.942062i	$0.608884\pi$
$264$	0	0
$265$	4.50570	0.276783
$266$	1.12227	0.0688110
$267$	0	0
$268$	−43.0594	−2.63027
$269$	−10.9718	−0.668960	−0.334480	−	0.942403i	$-0.608561\pi$
$269$	−10.9718	−0.668960	−0.334480	+	0.942403i	$0.608561\pi$
$270$	0	0
$271$	12.1714	0.739362	0.369681	−	0.929159i	$-0.379467\pi$
$271$	12.1714	0.739362	0.369681	+	0.929159i	$0.379467\pi$
$272$	−2.33269	−0.141440
$273$	0	0
$274$	43.2452	2.61254
$275$	−32.7796	−1.97669
$276$	0	0
$277$	18.1659	1.09149	0.545743	−	0.837953i	$-0.316247\pi$
$277$	18.1659	1.09149	0.545743	+	0.837953i	$0.316247\pi$
$278$	−15.3056	−0.917966
$279$	0	0
$280$	−7.85812	−0.469612
$281$	11.9874	0.715107	0.357554	−	0.933893i	$-0.383611\pi$
$281$	11.9874	0.715107	0.357554	+	0.933893i	$0.383611\pi$
$282$	0	0
$283$	−3.09385	−0.183910	−0.0919552	−	0.995763i	$-0.529312\pi$
$283$	−3.09385	−0.183910	−0.0919552	+	0.995763i	$0.529312\pi$
$284$	20.2422	1.20115
$285$	0	0
$286$	−54.1309	−3.20083
$287$	7.43947	0.439138
$288$	0	0
$289$	−8.02641	−0.472142
$290$	17.0056	0.998604
$291$	0	0
$292$	19.4352	1.13736
$293$	22.8848	1.33694	0.668471	−	0.743738i	$-0.266949\pi$
$293$	22.8848	1.33694	0.668471	+	0.743738i	$0.266949\pi$
$294$	0	0
$295$	17.5181	1.01995
$296$	18.5925	1.08067
$297$	0	0
$298$	−25.3581	−1.46896
$299$	−28.4483	−1.64521
$300$	0	0
$301$	6.74366	0.388698
$302$	−4.72199	−0.271720
$303$	0	0
$304$	0.388713	0.0222942
$305$	1.76814	0.101243
$306$	0	0
$307$	−27.6006	−1.57525	−0.787625	−	0.616155i	$-0.788689\pi$
$307$	−27.6006	−1.57525	−0.787625	+	0.616155i	$0.788689\pi$
$308$	−16.7303	−0.953295
$309$	0	0
$310$	−31.3313	−1.77950
$311$	−11.1839	−0.634182	−0.317091	−	0.948395i	$-0.602706\pi$
$311$	−11.1839	−0.634182	−0.317091	+	0.948395i	$0.602706\pi$
$312$	0	0
$313$	30.9402	1.74884	0.874422	−	0.485166i	$-0.161241\pi$
$313$	30.9402	1.74884	0.874422	+	0.485166i	$0.161241\pi$
$314$	−32.7539	−1.84841
$315$	0	0
$316$	−48.3062	−2.71744
$317$	−5.00488	−0.281102	−0.140551	−	0.990073i	$-0.544887\pi$
$317$	−5.00488	−0.281102	−0.140551	+	0.990073i	$0.544887\pi$
$318$	0	0
$319$	12.4998	0.699856
$320$	−43.2174	−2.41593
$321$	0	0
$322$	−14.5494	−0.810809
$323$	−1.49534	−0.0832028
$324$	0	0
$325$	−26.3090	−1.45936
$326$	5.28370	0.292637
$327$	0	0
$328$	−17.6386	−0.973928
$329$	8.99909	0.496136
$330$	0	0
$331$	10.7088	0.588609	0.294305	−	0.955712i	$-0.404912\pi$
$331$	10.7088	0.588609	0.294305	+	0.955712i	$0.404912\pi$
$332$	−18.2299	−1.00049
$333$	0	0
$334$	−52.7660	−2.88723
$335$	−46.7212	−2.55265
$336$	0	0
$337$	−8.32836	−0.453675	−0.226837	−	0.973933i	$-0.572839\pi$
$337$	−8.32836	−0.453675	−0.226837	+	0.973933i	$0.572839\pi$
$338$	−14.2185	−0.773387
$339$	0	0
$340$	30.3271	1.64472
$341$	−23.0298	−1.24713
$342$	0	0
$343$	1.00000	0.0539949
$344$	−15.9889	−0.862062
$345$	0	0
$346$	5.57445	0.299684
$347$	−4.09711	−0.219945	−0.109972	−	0.993935i	$-0.535076\pi$
$347$	−4.09711	−0.219945	−0.109972	+	0.993935i	$0.535076\pi$
$348$	0	0
$349$	24.8638	1.33093	0.665465	−	0.746429i	$-0.268233\pi$
$349$	24.8638	1.33093	0.665465	+	0.746429i	$0.268233\pi$
$350$	−13.4554	−0.719220
$351$	0	0
$352$	−35.5607	−1.89539
$353$	2.37283	0.126293	0.0631465	−	0.998004i	$-0.479886\pi$
$353$	2.37283	0.126293	0.0631465	+	0.998004i	$0.479886\pi$
$354$	0	0
$355$	21.9636	1.16571
$356$	−20.0689	−1.06365
$357$	0	0
$358$	21.5681	1.13991
$359$	8.59806	0.453788	0.226894	−	0.973919i	$-0.427143\pi$
$359$	8.59806	0.453788	0.226894	+	0.973919i	$0.427143\pi$
$360$	0	0
$361$	−18.7508	−0.986885
$362$	−2.29192	−0.120461
$363$	0	0
$364$	−13.4278	−0.703805
$365$	21.0880	1.10379
$366$	0	0
$367$	8.33703	0.435189	0.217595	−	0.976039i	$-0.430179\pi$
$367$	8.33703	0.435189	0.217595	+	0.976039i	$0.430179\pi$
$368$	−5.03938	−0.262696
$369$	0	0
$370$	58.4326	3.03777
$371$	1.35946	0.0705795
$372$	0	0
$373$	14.4028	0.745747	0.372873	−	0.927882i	$-0.378373\pi$
$373$	14.4028	0.745747	0.372873	+	0.927882i	$0.378373\pi$
$374$	36.8873	1.90740
$375$	0	0
$376$	−21.3364	−1.10034
$377$	10.0324	0.516695
$378$	0	0
$379$	13.5035	0.693626	0.346813	−	0.937934i	$-0.387264\pi$
$379$	13.5035	0.693626	0.346813	+	0.937934i	$0.387264\pi$
$380$	−5.05364	−0.259246
$381$	0	0
$382$	22.3342	1.14272
$383$	11.1740	0.570962	0.285481	−	0.958384i	$-0.407847\pi$
$383$	11.1740	0.570962	0.285481	+	0.958384i	$0.407847\pi$
$384$	0	0
$385$	−18.1530	−0.925162
$386$	−34.3075	−1.74620
$387$	0	0
$388$	38.3928	1.94910
$389$	29.1166	1.47627	0.738134	−	0.674654i	$-0.235707\pi$
$389$	29.1166	1.47627	0.738134	+	0.674654i	$0.235707\pi$
$390$	0	0
$391$	19.3860	0.980390
$392$	−2.37095	−0.119751
$393$	0	0
$394$	−26.9891	−1.35969
$395$	−52.4142	−2.63724
$396$	0	0
$397$	15.2799	0.766878	0.383439	−	0.923566i	$-0.374740\pi$
$397$	15.2799	0.766878	0.383439	+	0.923566i	$0.374740\pi$
$398$	−49.6138	−2.48692
$399$	0	0
$400$	−4.66043	−0.233022
$401$	−27.6205	−1.37930	−0.689652	−	0.724141i	$-0.742236\pi$
$401$	−27.6205	−1.37930	−0.689652	+	0.724141i	$0.742236\pi$
$402$	0	0
$403$	−18.4838	−0.920743
$404$	13.1906	0.656258
$405$	0	0
$406$	5.13092	0.254643
$407$	42.9504	2.12897
$408$	0	0
$409$	5.51095	0.272499	0.136249	−	0.990675i	$-0.456495\pi$
$409$	5.51095	0.272499	0.136249	+	0.990675i	$0.456495\pi$
$410$	−55.4347	−2.73772
$411$	0	0
$412$	−23.4523	−1.15541
$413$	5.28556	0.260085
$414$	0	0
$415$	−19.7801	−0.970969
$416$	−28.5411	−1.39934
$417$	0	0
$418$	−6.14681	−0.300650
$419$	26.9938	1.31873	0.659366	−	0.751822i	$-0.270825\pi$
$419$	26.9938	1.31873	0.659366	+	0.751822i	$0.270825\pi$
$420$	0	0
$421$	−22.9232	−1.11721	−0.558605	−	0.829434i	$-0.688663\pi$
$421$	−22.9232	−1.11721	−0.558605	+	0.829434i	$0.688663\pi$
$422$	−52.4839	−2.55487
$423$	0	0
$424$	−3.22320	−0.156532
$425$	17.9282	0.869644
$426$	0	0
$427$	0.533481	0.0258170
$428$	18.8051	0.908980
$429$	0	0
$430$	−50.2499	−2.42327
$431$	6.16714	0.297061	0.148530	−	0.988908i	$-0.452546\pi$
$431$	6.16714	0.297061	0.148530	+	0.988908i	$0.452546\pi$
$432$	0	0
$433$	−20.1082	−0.966340	−0.483170	−	0.875526i	$-0.660515\pi$
$433$	−20.1082	−0.966340	−0.483170	+	0.875526i	$0.660515\pi$
$434$	−9.45326	−0.453771
$435$	0	0
$436$	−46.1946	−2.21232
$437$	−3.23043	−0.154532
$438$	0	0
$439$	13.8552	0.661273	0.330636	−	0.943758i	$-0.392737\pi$
$439$	13.8552	0.661273	0.330636	+	0.943758i	$0.392737\pi$
$440$	43.0397	2.05184
$441$	0	0
$442$	29.6058	1.40821
$443$	14.4784	0.687890	0.343945	−	0.938990i	$-0.388237\pi$
$443$	14.4784	0.687890	0.343945	+	0.938990i	$0.388237\pi$
$444$	0	0
$445$	−21.7755	−1.03226
$446$	−16.4881	−0.780734
$447$	0	0
$448$	−13.0395	−0.616060
$449$	27.1832	1.28286	0.641428	−	0.767183i	$-0.278342\pi$
$449$	27.1832	1.28286	0.641428	+	0.767183i	$0.278342\pi$
$450$	0	0
$451$	−40.7468	−1.91869
$452$	−4.81452	−0.226456
$453$	0	0
$454$	−60.7720	−2.85217
$455$	−14.5696	−0.683035
$456$	0	0
$457$	5.56791	0.260456	0.130228	−	0.991484i	$-0.458429\pi$
$457$	5.56791	0.260456	0.130228	+	0.991484i	$0.458429\pi$
$458$	29.3521	1.37153
$459$	0	0
$460$	65.5168	3.05473
$461$	9.34013	0.435013	0.217507	−	0.976059i	$-0.430208\pi$
$461$	9.34013	0.435013	0.217507	+	0.976059i	$0.430208\pi$
$462$	0	0
$463$	−2.05307	−0.0954144	−0.0477072	−	0.998861i	$-0.515191\pi$
$463$	−2.05307	−0.0954144	−0.0477072	+	0.998861i	$0.515191\pi$
$464$	1.77716	0.0825025
$465$	0	0
$466$	−53.2200	−2.46537
$467$	32.0662	1.48384	0.741922	−	0.670486i	$-0.233914\pi$
$467$	32.0662	1.48384	0.741922	+	0.670486i	$0.233914\pi$
$468$	0	0
$469$	−14.0967	−0.650925
$470$	−67.0561	−3.09307
$471$	0	0
$472$	−12.5318	−0.576822
$473$	−36.9358	−1.69831
$474$	0	0
$475$	−2.98751	−0.137076
$476$	9.15028	0.419402
$477$	0	0
$478$	63.7924	2.91780
$479$	25.0510	1.14461	0.572305	−	0.820041i	$-0.306049\pi$
$479$	25.0510	1.14461	0.572305	+	0.820041i	$0.306049\pi$
$480$	0	0
$481$	34.4721	1.57179
$482$	15.5029	0.706139
$483$	0	0
$484$	58.0330	2.63786
$485$	41.6578	1.89158
$486$	0	0
$487$	−24.1154	−1.09277	−0.546386	−	0.837534i	$-0.683997\pi$
$487$	−24.1154	−1.09277	−0.546386	+	0.837534i	$0.683997\pi$
$488$	−1.26486	−0.0572573
$489$	0	0
$490$	−7.45143	−0.336621
$491$	41.4602	1.87107	0.935535	−	0.353233i	$-0.114918\pi$
$491$	41.4602	1.87107	0.935535	+	0.353233i	$0.114918\pi$
$492$	0	0
$493$	−6.83653	−0.307902
$494$	−4.93345	−0.221966
$495$	0	0
$496$	−3.27425	−0.147018
$497$	6.62684	0.297254
$498$	0	0
$499$	6.32228	0.283024	0.141512	−	0.989937i	$-0.454804\pi$
$499$	6.32228	0.283024	0.141512	+	0.989937i	$0.454804\pi$
$500$	9.97050	0.445894
$501$	0	0
$502$	41.6804	1.86029
$503$	30.3857	1.35483	0.677416	−	0.735600i	$-0.263100\pi$
$503$	30.3857	1.35483	0.677416	+	0.735600i	$0.263100\pi$
$504$	0	0
$505$	14.3124	0.636891
$506$	79.6889	3.54260
$507$	0	0
$508$	3.05458	0.135525
$509$	43.5835	1.93181	0.965903	−	0.258906i	$-0.0833618\pi$
$509$	43.5835	1.93181	0.965903	+	0.258906i	$0.0833618\pi$
$510$	0	0
$511$	6.36264	0.281467
$512$	−8.74837	−0.386627
$513$	0	0
$514$	33.1825	1.46362
$515$	−25.4467	−1.12132
$516$	0	0
$517$	−49.2890	−2.16773
$518$	17.6302	0.774629
$519$	0	0
$520$	34.5438	1.51485
$521$	−3.41203	−0.149484	−0.0747419	−	0.997203i	$-0.523813\pi$
$521$	−3.41203	−0.149484	−0.0747419	+	0.997203i	$0.523813\pi$
$522$	0	0
$523$	−4.13861	−0.180969	−0.0904844	−	0.995898i	$-0.528842\pi$
$523$	−4.13861	−0.180969	−0.0904844	+	0.995898i	$0.528842\pi$
$524$	−0.0760849	−0.00332379
$525$	0	0
$526$	24.4604	1.06652
$527$	12.5957	0.548677
$528$	0	0
$529$	18.8801	0.820876
$530$	−10.1299	−0.440015
$531$	0	0
$532$	−1.52478	−0.0661076
$533$	−32.7035	−1.41655
$534$	0	0
$535$	20.4043	0.882155
$536$	33.4225	1.44363
$537$	0	0
$538$	24.6671	1.06348
$539$	−5.47711	−0.235916
$540$	0	0
$541$	30.5714	1.31437	0.657184	−	0.753730i	$-0.271747\pi$
$541$	30.5714	1.31437	0.657184	+	0.753730i	$0.271747\pi$
$542$	−27.3643	−1.17540
$543$	0	0
$544$	19.4492	0.833878
$545$	−50.1229	−2.14703
$546$	0	0
$547$	−34.9946	−1.49626	−0.748131	−	0.663551i	$-0.769049\pi$
$547$	−34.9946	−1.49626	−0.748131	+	0.663551i	$0.769049\pi$
$548$	−58.7553	−2.50990
$549$	0	0
$550$	73.6965	3.14243
$551$	1.13922	0.0485325
$552$	0	0
$553$	−15.8144	−0.672495
$554$	−40.8414	−1.73518
$555$	0	0
$556$	20.7950	0.881903
$557$	−2.88453	−0.122222	−0.0611108	−	0.998131i	$-0.519464\pi$
$557$	−2.88453	−0.122222	−0.0611108	+	0.998131i	$0.519464\pi$
$558$	0	0
$559$	−29.6448	−1.25384
$560$	−2.58089	−0.109063
$561$	0	0
$562$	−26.9505	−1.13684
$563$	12.0756	0.508924	0.254462	−	0.967083i	$-0.418102\pi$
$563$	12.0756	0.508924	0.254462	+	0.967083i	$0.418102\pi$
$564$	0	0
$565$	−5.22395	−0.219773
$566$	6.95572	0.292371
$567$	0	0
$568$	−15.7119	−0.659256
$569$	−4.27015	−0.179014	−0.0895070	−	0.995986i	$-0.528529\pi$
$569$	−4.27015	−0.179014	−0.0895070	+	0.995986i	$0.528529\pi$
$570$	0	0
$571$	30.6795	1.28390	0.641949	−	0.766747i	$-0.278126\pi$
$571$	30.6795	1.28390	0.641949	+	0.766747i	$0.278126\pi$
$572$	73.5452	3.07508
$573$	0	0
$574$	−16.7257	−0.698118
$575$	38.7309	1.61519
$576$	0	0
$577$	−1.83635	−0.0764482	−0.0382241	−	0.999269i	$-0.512170\pi$
$577$	−1.83635	−0.0764482	−0.0382241	+	0.999269i	$0.512170\pi$
$578$	18.0453	0.750585
$579$	0	0
$580$	−23.1047	−0.959372
$581$	−5.96805	−0.247596
$582$	0	0
$583$	−7.44589	−0.308377
$584$	−15.0855	−0.624241
$585$	0	0
$586$	−51.4504	−2.12540
$587$	38.6222	1.59411	0.797054	−	0.603908i	$-0.206390\pi$
$587$	38.6222	1.59411	0.797054	+	0.603908i	$0.206390\pi$
$588$	0	0
$589$	−2.09892	−0.0864843
$590$	−39.3850	−1.62145
$591$	0	0
$592$	6.10646	0.250974
$593$	19.1132	0.784886	0.392443	−	0.919776i	$-0.371630\pi$
$593$	19.1132	0.784886	0.392443	+	0.919776i	$0.371630\pi$
$594$	0	0
$595$	9.92842	0.407025
$596$	34.4529	1.41125
$597$	0	0
$598$	63.9585	2.61546
$599$	9.55259	0.390308	0.195154	−	0.980773i	$-0.437479\pi$
$599$	9.55259	0.390308	0.195154	+	0.980773i	$0.437479\pi$
$600$	0	0
$601$	39.7388	1.62098	0.810490	−	0.585752i	$-0.199201\pi$
$601$	39.7388	1.62098	0.810490	+	0.585752i	$0.199201\pi$
$602$	−15.1614	−0.617931
$603$	0	0
$604$	6.41555	0.261045
$605$	62.9681	2.56002
$606$	0	0
$607$	31.6818	1.28592	0.642962	−	0.765898i	$-0.277705\pi$
$607$	31.6818	1.28592	0.642962	+	0.765898i	$0.277705\pi$
$608$	−3.24097	−0.131439
$609$	0	0
$610$	−3.97520	−0.160951
$611$	−39.5595	−1.60041
$612$	0	0
$613$	−17.6030	−0.710980	−0.355490	−	0.934680i	$-0.615686\pi$
$613$	−17.6030	−0.710980	−0.355490	+	0.934680i	$0.615686\pi$
$614$	62.0528	2.50425
$615$	0	0
$616$	12.9859	0.523218
$617$	20.1237	0.810149	0.405075	−	0.914284i	$-0.367246\pi$
$617$	20.1237	0.810149	0.405075	+	0.914284i	$0.367246\pi$
$618$	0	0
$619$	−4.89268	−0.196653	−0.0983267	−	0.995154i	$-0.531349\pi$
$619$	−4.89268	−0.196653	−0.0983267	+	0.995154i	$0.531349\pi$
$620$	42.5684	1.70959
$621$	0	0
$622$	25.1441	1.00819
$623$	−6.57010	−0.263226
$624$	0	0
$625$	−19.1058	−0.764234
$626$	−69.5610	−2.78022
$627$	0	0
$628$	44.5012	1.77579
$629$	−23.4909	−0.936642
$630$	0	0
$631$	14.6120	0.581695	0.290848	−	0.956769i	$-0.406063\pi$
$631$	14.6120	0.581695	0.290848	+	0.956769i	$0.406063\pi$
$632$	37.4950	1.49147
$633$	0	0
$634$	11.2522	0.446880
$635$	3.31434	0.131526
$636$	0	0
$637$	−4.39594	−0.174174
$638$	−28.1026	−1.11259
$639$	0	0
$640$	54.1256	2.13950
$641$	−30.1442	−1.19062	−0.595312	−	0.803495i	$-0.702972\pi$
$641$	−30.1442	−1.19062	−0.595312	+	0.803495i	$0.702972\pi$
$642$	0	0
$643$	2.02706	0.0799396	0.0399698	−	0.999201i	$-0.487274\pi$
$643$	2.02706	0.0799396	0.0399698	+	0.999201i	$0.487274\pi$
$644$	19.7677	0.778955
$645$	0	0
$646$	3.36188	0.132271
$647$	−31.2618	−1.22903	−0.614514	−	0.788906i	$-0.710648\pi$
$647$	−31.2618	−1.22903	−0.614514	+	0.788906i	$0.710648\pi$
$648$	0	0
$649$	−28.9496	−1.13637
$650$	59.1490	2.32001
$651$	0	0
$652$	−7.17872	−0.281140
$653$	−40.2091	−1.57350	−0.786752	−	0.617270i	$-0.788239\pi$
$653$	−40.2091	−1.57350	−0.786752	+	0.617270i	$0.788239\pi$
$654$	0	0
$655$	−0.0825552	−0.00322570
$656$	−5.79316	−0.226185
$657$	0	0
$658$	−20.2321	−0.788730
$659$	39.3476	1.53276	0.766382	−	0.642385i	$-0.222055\pi$
$659$	39.3476	1.53276	0.766382	+	0.642385i	$0.222055\pi$
$660$	0	0
$661$	2.03809	0.0792724	0.0396362	−	0.999214i	$-0.487380\pi$
$661$	2.03809	0.0792724	0.0396362	+	0.999214i	$0.487380\pi$
$662$	−24.0760	−0.935739
$663$	0	0
$664$	14.1499	0.549123
$665$	−1.65445	−0.0641567
$666$	0	0
$667$	−14.7692	−0.571866
$668$	71.6907	2.77380
$669$	0	0
$670$	105.040	4.05807
$671$	−2.92193	−0.112800
$672$	0	0
$673$	−36.3745	−1.40213	−0.701067	−	0.713095i	$-0.747293\pi$
$673$	−36.3745	−1.40213	−0.701067	+	0.713095i	$0.747293\pi$
$674$	18.7241	0.721227
$675$	0	0
$676$	19.3181	0.743003
$677$	8.76754	0.336964	0.168482	−	0.985705i	$-0.446113\pi$
$677$	8.76754	0.336964	0.168482	+	0.985705i	$0.446113\pi$
$678$	0	0
$679$	12.5689	0.482352
$680$	−23.5397	−0.902708
$681$	0	0
$682$	51.7765	1.98262
$683$	−28.0592	−1.07366	−0.536828	−	0.843692i	$-0.680377\pi$
$683$	−28.0592	−1.07366	−0.536828	+	0.843692i	$0.680377\pi$
$684$	0	0
$685$	−63.7518	−2.43583
$686$	−2.24824	−0.0858382
$687$	0	0
$688$	−5.25133	−0.200205
$689$	−5.97609	−0.227671
$690$	0	0
$691$	49.3493	1.87734	0.938668	−	0.344821i	$-0.112060\pi$
$691$	49.3493	1.87734	0.938668	+	0.344821i	$0.112060\pi$
$692$	−7.57375	−0.287911
$693$	0	0
$694$	9.21129	0.349656
$695$	22.5634	0.855877
$696$	0	0
$697$	22.2856	0.844129
$698$	−55.8998	−2.11584
$699$	0	0
$700$	18.2812	0.690964
$701$	−43.6097	−1.64712	−0.823558	−	0.567232i	$-0.808014\pi$
$701$	−43.6097	−1.64712	−0.823558	+	0.567232i	$0.808014\pi$
$702$	0	0
$703$	3.91446	0.147637
$704$	71.4188	2.69170
$705$	0	0
$706$	−5.33469	−0.200774
$707$	4.31831	0.162407
$708$	0	0
$709$	−37.8878	−1.42291	−0.711454	−	0.702733i	$-0.751963\pi$
$709$	−37.8878	−1.42291	−0.711454	+	0.702733i	$0.751963\pi$
$710$	−49.3794	−1.85318
$711$	0	0
$712$	15.5774	0.583786
$713$	27.2109	1.01906
$714$	0	0
$715$	79.7995	2.98433
$716$	−29.3036	−1.09513
$717$	0	0
$718$	−19.3305	−0.721407
$719$	50.6110	1.88747	0.943736	−	0.330699i	$-0.107285\pi$
$719$	50.6110	1.88747	0.943736	+	0.330699i	$0.107285\pi$
$720$	0	0
$721$	−7.67776	−0.285935
$722$	42.1563	1.56890
$723$	0	0
$724$	3.11393	0.115728
$725$	−13.6586	−0.507267
$726$	0	0
$727$	19.5430	0.724809	0.362404	−	0.932021i	$-0.381956\pi$
$727$	19.5430	0.724809	0.362404	+	0.932021i	$0.381956\pi$
$728$	10.4225	0.386285
$729$	0	0
$730$	−47.4108	−1.75475
$731$	20.2013	0.747172
$732$	0	0
$733$	−17.8762	−0.660274	−0.330137	−	0.943933i	$-0.607095\pi$
$733$	−17.8762	−0.660274	−0.330137	+	0.943933i	$0.607095\pi$
$734$	−18.7436	−0.691840
$735$	0	0
$736$	42.0168	1.54876
$737$	77.2091	2.84403
$738$	0	0
$739$	23.3724	0.859768	0.429884	−	0.902884i	$-0.358554\pi$
$739$	23.3724	0.859768	0.429884	+	0.902884i	$0.358554\pi$
$740$	−79.3897	−2.91842
$741$	0	0
$742$	−3.05638	−0.112203
$743$	−21.1697	−0.776640	−0.388320	−	0.921525i	$-0.626944\pi$
$743$	−21.1697	−0.776640	−0.388320	+	0.921525i	$0.626944\pi$
$744$	0	0
$745$	37.3828	1.36960
$746$	−32.3809	−1.18555
$747$	0	0
$748$	−50.1171	−1.83246
$749$	6.15637	0.224949
$750$	0	0
$751$	49.7278	1.81459	0.907296	−	0.420492i	$-0.138142\pi$
$751$	49.7278	1.81459	0.907296	+	0.420492i	$0.138142\pi$
$752$	−7.00764	−0.255542
$753$	0	0
$754$	−22.5552	−0.821413
$755$	6.96113	0.253341
$756$	0	0
$757$	0.699544	0.0254254	0.0127127	−	0.999919i	$-0.495953\pi$
$757$	0.699544	0.0254254	0.0127127	+	0.999919i	$0.495953\pi$
$758$	−30.3590	−1.10269
$759$	0	0
$760$	3.92261	0.142288
$761$	−14.4262	−0.522949	−0.261474	−	0.965210i	$-0.584209\pi$
$761$	−14.4262	−0.522949	−0.261474	+	0.965210i	$0.584209\pi$
$762$	0	0
$763$	−15.1231	−0.547491
$764$	−30.3444	−1.09782
$765$	0	0
$766$	−25.1217	−0.907685
$767$	−23.2350	−0.838968
$768$	0	0
$769$	−23.0451	−0.831026	−0.415513	−	0.909587i	$-0.636398\pi$
$769$	−23.0451	−0.831026	−0.415513	+	0.909587i	$0.636398\pi$
$770$	40.8123	1.47077
$771$	0	0
$772$	46.6120	1.67760
$773$	1.43085	0.0514640	0.0257320	−	0.999669i	$-0.491808\pi$
$773$	1.43085	0.0514640	0.0257320	+	0.999669i	$0.491808\pi$
$774$	0	0
$775$	25.1647	0.903943
$776$	−29.8003	−1.06977
$777$	0	0
$778$	−65.4610	−2.34689
$779$	−3.71363	−0.133054
$780$	0	0
$781$	−36.2959	−1.29877
$782$	−43.5843	−1.55857
$783$	0	0
$784$	−0.778706	−0.0278109
$785$	48.2855	1.72338
$786$	0	0
$787$	−50.3549	−1.79496	−0.897479	−	0.441057i	$-0.854604\pi$
$787$	−50.3549	−1.79496	−0.897479	+	0.441057i	$0.854604\pi$
$788$	36.6688	1.30627
$789$	0	0
$790$	117.840	4.19255
$791$	−1.57617	−0.0560420
$792$	0	0
$793$	−2.34515	−0.0832788
$794$	−34.3530	−1.21914
$795$	0	0
$796$	67.4080	2.38921
$797$	−17.2393	−0.610647	−0.305323	−	0.952249i	$-0.598765\pi$
$797$	−17.2393	−0.610647	−0.305323	+	0.952249i	$0.598765\pi$
$798$	0	0
$799$	26.9576	0.953692
$800$	38.8573	1.37381
$801$	0	0
$802$	62.0975	2.19274
$803$	−34.8489	−1.22979
$804$	0	0
$805$	21.4487	0.755967
$806$	41.5560	1.46375
$807$	0	0
$808$	−10.2385	−0.360189
$809$	−40.4761	−1.42306	−0.711532	−	0.702654i	$-0.751998\pi$
$809$	−40.4761	−1.42306	−0.711532	+	0.702654i	$0.751998\pi$
$810$	0	0
$811$	−9.44221	−0.331561	−0.165780	−	0.986163i	$-0.553014\pi$
$811$	−9.44221	−0.331561	−0.165780	+	0.986163i	$0.553014\pi$
$812$	−6.97115	−0.244639
$813$	0	0
$814$	−96.5628	−3.38452
$815$	−7.78920	−0.272844
$816$	0	0
$817$	−3.36629	−0.117772
$818$	−12.3899	−0.433204
$819$	0	0
$820$	75.3165	2.63017
$821$	26.8033	0.935440	0.467720	−	0.883877i	$-0.345076\pi$
$821$	26.8033	0.935440	0.467720	+	0.883877i	$0.345076\pi$
$822$	0	0
$823$	−11.0877	−0.386492	−0.193246	−	0.981150i	$-0.561901\pi$
$823$	−11.0877	−0.386492	−0.193246	+	0.981150i	$0.561901\pi$
$824$	18.2035	0.634150
$825$	0	0
$826$	−11.8832	−0.413469
$827$	−20.6815	−0.719167	−0.359584	−	0.933113i	$-0.617081\pi$
$827$	−20.6815	−0.719167	−0.359584	+	0.933113i	$0.617081\pi$
$828$	0	0
$829$	−5.98033	−0.207705	−0.103853	−	0.994593i	$-0.533117\pi$
$829$	−5.98033	−0.207705	−0.103853	+	0.994593i	$0.533117\pi$
$830$	44.4705	1.54359
$831$	0	0
$832$	57.3210	1.98725
$833$	2.99559	0.103791
$834$	0	0
$835$	77.7873	2.69194
$836$	8.35139	0.288839
$837$	0	0
$838$	−60.6885	−2.09645
$839$	28.0769	0.969321	0.484661	−	0.874702i	$-0.338943\pi$
$839$	28.0769	0.969321	0.484661	+	0.874702i	$0.338943\pi$
$840$	0	0
$841$	−23.7916	−0.820399
$842$	51.5369	1.77608
$843$	0	0
$844$	71.3074	2.45450
$845$	20.9609	0.721076
$846$	0	0
$847$	18.9987	0.652803
$848$	−1.05862	−0.0363530
$849$	0	0
$850$	−40.3068	−1.38251
$851$	−50.7481	−1.73962
$852$	0	0
$853$	−25.3187	−0.866894	−0.433447	−	0.901179i	$-0.642703\pi$
$853$	−25.3187	−0.866894	−0.433447	+	0.901179i	$0.642703\pi$
$854$	−1.19939	−0.0410424
$855$	0	0
$856$	−14.5964	−0.498895
$857$	4.04901	0.138312	0.0691558	−	0.997606i	$-0.477969\pi$
$857$	4.04901	0.138312	0.0691558	+	0.997606i	$0.477969\pi$
$858$	0	0
$859$	−0.0822329	−0.00280575	−0.00140287	−	0.999999i	$-0.500447\pi$
$859$	−0.0822329	−0.00280575	−0.00140287	+	0.999999i	$0.500447\pi$
$860$	68.2723	2.32806
$861$	0	0
$862$	−13.8652	−0.472251
$863$	27.8324	0.947425	0.473712	−	0.880680i	$-0.342914\pi$
$863$	27.8324	0.947425	0.473712	+	0.880680i	$0.342914\pi$
$864$	0	0
$865$	−8.21782	−0.279414
$866$	45.2081	1.53623
$867$	0	0
$868$	12.8437	0.435944
$869$	86.6170	2.93828
$870$	0	0
$871$	61.9682	2.09971
$872$	35.8559	1.21424
$873$	0	0
$874$	7.26278	0.245667
$875$	3.26412	0.110347
$876$	0	0
$877$	−45.7277	−1.54412	−0.772058	−	0.635553i	$-0.780772\pi$
$877$	−45.7277	−1.54412	−0.772058	+	0.635553i	$0.780772\pi$
$878$	−31.1498	−1.05126
$879$	0	0
$880$	14.1358	0.476519
$881$	−47.0656	−1.58568	−0.792840	−	0.609430i	$-0.791398\pi$
$881$	−47.0656	−1.58568	−0.792840	+	0.609430i	$0.791398\pi$
$882$	0	0
$883$	−31.6005	−1.06344	−0.531721	−	0.846920i	$-0.678454\pi$
$883$	−31.6005	−1.06344	−0.531721	+	0.846920i	$0.678454\pi$
$884$	−40.2241	−1.35288
$885$	0	0
$886$	−32.5509	−1.09357
$887$	30.9114	1.03790	0.518951	−	0.854804i	$-0.326323\pi$
$887$	30.9114	1.03790	0.518951	+	0.854804i	$0.326323\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	48.9566	1.64103
$891$	0	0
$892$	22.4016	0.750062
$893$	−4.49215	−0.150324
$894$	0	0
$895$	−31.7956	−1.06281
$896$	16.3307	0.545572
$897$	0	0
$898$	−61.1144	−2.03942
$899$	−9.59604	−0.320046
$900$	0	0
$901$	4.07238	0.135671
$902$	91.6085	3.05023
$903$	0	0
$904$	3.73700	0.124291
$905$	3.37874	0.112313
$906$	0	0
$907$	−16.1542	−0.536390	−0.268195	−	0.963365i	$-0.586427\pi$
$907$	−16.1542	−0.536390	−0.268195	+	0.963365i	$0.586427\pi$
$908$	82.5681	2.74012
$909$	0	0
$910$	32.7560	1.08585
$911$	30.7252	1.01797	0.508985	−	0.860775i	$-0.330021\pi$
$911$	30.7252	1.01797	0.508985	+	0.860775i	$0.330021\pi$
$912$	0	0
$913$	32.6876	1.08180
$914$	−12.5180	−0.414058
$915$	0	0
$916$	−39.8793	−1.31765
$917$	−0.0249085	−0.000822551 0
$918$	0	0
$919$	36.6178	1.20791	0.603955	−	0.797018i	$-0.293590\pi$
$919$	36.6178	1.20791	0.603955	+	0.797018i	$0.293590\pi$
$920$	−50.8537	−1.67660
$921$	0	0
$922$	−20.9988	−0.691560
$923$	−29.1312	−0.958865
$924$	0	0
$925$	−46.9320	−1.54311
$926$	4.61580	0.151685
$927$	0	0
$928$	−14.8174	−0.486405
$929$	3.45942	0.113500	0.0567500	−	0.998388i	$-0.481926\pi$
$929$	3.45942	0.113500	0.0567500	+	0.998388i	$0.481926\pi$
$930$	0	0
$931$	−0.499179	−0.0163599
$932$	72.3076	2.36851
$933$	0	0
$934$	−72.0924	−2.35893
$935$	−54.3790	−1.77838
$936$	0	0
$937$	27.2761	0.891073	0.445536	−	0.895264i	$-0.353013\pi$
$937$	27.2761	0.891073	0.445536	+	0.895264i	$0.353013\pi$
$938$	31.6927	1.03480
$939$	0	0
$940$	91.1060	2.97155
$941$	50.5133	1.64669	0.823343	−	0.567544i	$-0.192106\pi$
$941$	50.5133	1.64669	0.823343	+	0.567544i	$0.192106\pi$
$942$	0	0
$943$	48.1444	1.56780
$944$	−4.11589	−0.133961
$945$	0	0
$946$	83.0404	2.69988
$947$	−31.7991	−1.03333	−0.516666	−	0.856187i	$-0.672827\pi$
$947$	−31.7991	−1.03333	−0.516666	+	0.856187i	$0.672827\pi$
$948$	0	0
$949$	−27.9698	−0.907938
$950$	6.71663	0.217916
$951$	0	0
$952$	−7.10239	−0.230190
$953$	−32.8952	−1.06558	−0.532790	−	0.846247i	$-0.678856\pi$
$953$	−32.8952	−1.06558	−0.532790	+	0.846247i	$0.678856\pi$
$954$	0	0
$955$	−32.9249	−1.06542
$956$	−86.6718	−2.80317
$957$	0	0
$958$	−56.3206	−1.81964
$959$	−19.2352	−0.621136
$960$	0	0
$961$	−13.3202	−0.429683
$962$	−77.5016	−2.49875
$963$	0	0
$964$	−21.0631	−0.678397
$965$	50.5759	1.62809
$966$	0	0
$967$	60.1512	1.93433	0.967166	−	0.254147i	$-0.0817947\pi$
$967$	60.1512	1.93433	0.967166	+	0.254147i	$0.0817947\pi$
$968$	−45.0449	−1.44780
$969$	0	0
$970$	−93.6566	−3.00713
$971$	49.3664	1.58424	0.792122	−	0.610363i	$-0.208976\pi$
$971$	49.3664	1.58424	0.792122	+	0.610363i	$0.208976\pi$
$972$	0	0
$973$	6.80780	0.218248
$974$	54.2171	1.73723
$975$	0	0
$976$	−0.415425	−0.0132974
$977$	1.19493	0.0382293	0.0191147	−	0.999817i	$-0.493915\pi$
$977$	1.19493	0.0382293	0.0191147	+	0.999817i	$0.493915\pi$
$978$	0	0
$979$	35.9851	1.15009
$980$	10.1239	0.323396
$981$	0	0
$982$	−93.2124	−2.97453
$983$	−45.1609	−1.44041	−0.720204	−	0.693762i	$-0.755952\pi$
$983$	−45.1609	−1.44041	−0.720204	+	0.693762i	$0.755952\pi$
$984$	0	0
$985$	39.7871	1.26772
$986$	15.3702	0.489486
$987$	0	0
$988$	6.70285	0.213246
$989$	43.6415	1.38772
$990$	0	0
$991$	10.1543	0.322562	0.161281	−	0.986908i	$-0.448437\pi$
$991$	10.1543	0.322562	0.161281	+	0.986908i	$0.448437\pi$
$992$	27.2997	0.866767
$993$	0	0
$994$	−14.8987	−0.472559
$995$	73.1404	2.31871
$996$	0	0
$997$	−59.1360	−1.87286	−0.936428	−	0.350860i	$-0.885889\pi$
$997$	−59.1360	−1.87286	−0.936428	+	0.350860i	$0.885889\pi$
$998$	−14.2140	−0.449936
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.u.1.2		18
3.2	odd	2		2667.2.a.p.1.17	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.p.1.17	✓	18	3.2	odd	2
8001.2.a.u.1.2		18	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)

\(f(q)\)	\(=\)	\(q-2.24824 q^{2} +3.05458 q^{4} +3.31434 q^{5} +1.00000 q^{7} -2.37095 q^{8} +O(q^{10})\)	\(q-2.24824 q^{2} +3.05458 q^{4} +3.31434 q^{5} +1.00000 q^{7} -2.37095 q^{8} -7.45143 q^{10} -5.47711 q^{11} -4.39594 q^{13} -2.24824 q^{14} -0.778706 q^{16} +2.99559 q^{17} -0.499179 q^{19} +10.1239 q^{20} +12.3138 q^{22} +6.47149 q^{23} +5.98485 q^{25} +9.88313 q^{26} +3.05458 q^{28} -2.28220 q^{29} +4.20474 q^{31} +6.49261 q^{32} -6.73481 q^{34} +3.31434 q^{35} -7.84180 q^{37} +1.12227 q^{38} -7.85812 q^{40} +7.43947 q^{41} +6.74366 q^{43} -16.7303 q^{44} -14.5494 q^{46} +8.99909 q^{47} +1.00000 q^{49} -13.4554 q^{50} -13.4278 q^{52} +1.35946 q^{53} -18.1530 q^{55} -2.37095 q^{56} +5.13092 q^{58} +5.28556 q^{59} +0.533481 q^{61} -9.45326 q^{62} -13.0395 q^{64} -14.5696 q^{65} -14.0967 q^{67} +9.15028 q^{68} -7.45143 q^{70} +6.62684 q^{71} +6.36264 q^{73} +17.6302 q^{74} -1.52478 q^{76} -5.47711 q^{77} -15.8144 q^{79} -2.58089 q^{80} -16.7257 q^{82} -5.96805 q^{83} +9.92842 q^{85} -15.1614 q^{86} +12.9859 q^{88} -6.57010 q^{89} -4.39594 q^{91} +19.7677 q^{92} -20.2321 q^{94} -1.65445 q^{95} +12.5689 q^{97} -2.24824 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 6 q^{2} + 22 q^{4} + 10 q^{5} + 18 q^{7} + 21 q^{8}+O(q^{10}) \) 18 * q + 6 * q^2 + 22 * q^4 + 10 * q^5 + 18 * q^7 + 21 * q^8	\( 18 q + 6 q^{2} + 22 q^{4} + 10 q^{5} + 18 q^{7} + 21 q^{8} - 4 q^{10} + 9 q^{11} - 25 q^{13} + 6 q^{14} + 34 q^{16} + 17 q^{17} - 5 q^{19} + 21 q^{20} + 5 q^{22} + 14 q^{23} + 28 q^{25} + 8 q^{26} + 22 q^{28} + 17 q^{29} + 5 q^{31} + 53 q^{32} - 19 q^{34} + 10 q^{35} - 15 q^{37} + 22 q^{38} - q^{40} + 17 q^{41} + q^{43} + 33 q^{44} + 10 q^{46} + 31 q^{47} + 18 q^{49} + 35 q^{50} - 70 q^{52} + 35 q^{53} + 4 q^{55} + 21 q^{56} + 3 q^{58} + 46 q^{59} - 5 q^{61} + 10 q^{62} + 63 q^{64} + 12 q^{65} + 6 q^{67} + 56 q^{68} - 4 q^{70} + 22 q^{71} - 16 q^{73} - 18 q^{74} + 32 q^{76} + 9 q^{77} + 46 q^{79} + 30 q^{80} - 12 q^{82} + 46 q^{83} + 4 q^{85} - 18 q^{86} + 30 q^{88} + 42 q^{89} - 25 q^{91} + 48 q^{92} + 3 q^{94} + 2 q^{95} - 35 q^{97} + 6 q^{98}+O(q^{100}) \) 18 * q + 6 * q^2 + 22 * q^4 + 10 * q^5 + 18 * q^7 + 21 * q^8 - 4 * q^10 + 9 * q^11 - 25 * q^13 + 6 * q^14 + 34 * q^16 + 17 * q^17 - 5 * q^19 + 21 * q^20 + 5 * q^22 + 14 * q^23 + 28 * q^25 + 8 * q^26 + 22 * q^28 + 17 * q^29 + 5 * q^31 + 53 * q^32 - 19 * q^34 + 10 * q^35 - 15 * q^37 + 22 * q^38 - q^40 + 17 * q^41 + q^43 + 33 * q^44 + 10 * q^46 + 31 * q^47 + 18 * q^49 + 35 * q^50 - 70 * q^52 + 35 * q^53 + 4 * q^55 + 21 * q^56 + 3 * q^58 + 46 * q^59 - 5 * q^61 + 10 * q^62 + 63 * q^64 + 12 * q^65 + 6 * q^67 + 56 * q^68 - 4 * q^70 + 22 * q^71 - 16 * q^73 - 18 * q^74 + 32 * q^76 + 9 * q^77 + 46 * q^79 + 30 * q^80 - 12 * q^82 + 46 * q^83 + 4 * q^85 - 18 * q^86 + 30 * q^88 + 42 * q^89 - 25 * q^91 + 48 * q^92 + 3 * q^94 + 2 * q^95 - 35 * q^97 + 6 * q^98

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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