LMFDB - Embedded newform 8001.2.a.w.1.18

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			1.18
Root			$-1.86718$ of defining polynomial
Character	$\chi$	$=$	8001.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.86718 q^{2} +1.48637 q^{4} -2.52504 q^{5} +1.00000 q^{7} -0.959047 q^{8} +O(q^{10})$	\(q+1.86718 q^{2} +1.48637 q^{4} -2.52504 q^{5} +1.00000 q^{7} -0.959047 q^{8} -4.71472 q^{10} -2.14677 q^{11} +3.10489 q^{13} +1.86718 q^{14} -4.76345 q^{16} -5.91789 q^{17} +5.60200 q^{19} -3.75314 q^{20} -4.00841 q^{22} +7.11857 q^{23} +1.37585 q^{25} +5.79739 q^{26} +1.48637 q^{28} +7.43778 q^{29} +5.17447 q^{31} -6.97613 q^{32} -11.0498 q^{34} -2.52504 q^{35} -4.49744 q^{37} +10.4600 q^{38} +2.42164 q^{40} +6.07522 q^{41} -5.88817 q^{43} -3.19089 q^{44} +13.2917 q^{46} -3.24407 q^{47} +1.00000 q^{49} +2.56896 q^{50} +4.61500 q^{52} -10.8652 q^{53} +5.42069 q^{55} -0.959047 q^{56} +13.8877 q^{58} -11.7201 q^{59} -4.57418 q^{61} +9.66167 q^{62} -3.49880 q^{64} -7.83998 q^{65} +3.13289 q^{67} -8.79615 q^{68} -4.71472 q^{70} -15.6126 q^{71} -13.6309 q^{73} -8.39753 q^{74} +8.32663 q^{76} -2.14677 q^{77} +11.7382 q^{79} +12.0279 q^{80} +11.3435 q^{82} -11.0096 q^{83} +14.9429 q^{85} -10.9943 q^{86} +2.05885 q^{88} -5.48084 q^{89} +3.10489 q^{91} +10.5808 q^{92} -6.05728 q^{94} -14.1453 q^{95} -1.03456 q^{97} +1.86718 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.86718	1.32030	0.660148	−	0.751135i	$-0.270493\pi$
$2$	1.86718	1.32030	0.660148	+	0.751135i	$0.270493\pi$
$3$	0	0
$3$	0	0
$4$	1.48637	0.743183
$5$	−2.52504	−1.12923	−0.564617	−	0.825353i	$-0.690976\pi$
$5$	−2.52504	−1.12923	−0.564617	+	0.825353i	$0.690976\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−0.959047	−0.339074
$9$	0	0
$10$	−4.71472	−1.49092
$11$	−2.14677	−0.647275	−0.323638	−	0.946181i	$-0.604906\pi$
$11$	−2.14677	−0.647275	−0.323638	+	0.946181i	$0.604906\pi$
$12$	0	0
$13$	3.10489	0.861141	0.430571	−	0.902557i	$-0.358312\pi$
$13$	3.10489	0.861141	0.430571	+	0.902557i	$0.358312\pi$
$14$	1.86718	0.499025
$15$	0	0
$16$	−4.76345	−1.19086
$17$	−5.91789	−1.43530	−0.717649	−	0.696405i	$-0.754782\pi$
$17$	−5.91789	−1.43530	−0.717649	+	0.696405i	$0.754782\pi$
$18$	0	0
$19$	5.60200	1.28519	0.642594	−	0.766207i	$-0.277858\pi$
$19$	5.60200	1.28519	0.642594	+	0.766207i	$0.277858\pi$
$20$	−3.75314	−0.839228
$21$	0	0
$22$	−4.00841	−0.854595
$23$	7.11857	1.48432	0.742162	−	0.670220i	$-0.233800\pi$
$23$	7.11857	1.48432	0.742162	+	0.670220i	$0.233800\pi$
$24$	0	0
$25$	1.37585	0.275170
$26$	5.79739	1.13696
$27$	0	0
$28$	1.48637	0.280897
$29$	7.43778	1.38116	0.690581	−	0.723255i	$-0.257355\pi$
$29$	7.43778	1.38116	0.690581	+	0.723255i	$0.257355\pi$
$30$	0	0
$31$	5.17447	0.929361	0.464681	−	0.885478i	$-0.346169\pi$
$31$	5.17447	0.929361	0.464681	+	0.885478i	$0.346169\pi$
$32$	−6.97613	−1.23322
$33$	0	0
$34$	−11.0498	−1.89502
$35$	−2.52504	−0.426810
$36$	0	0
$37$	−4.49744	−0.739374	−0.369687	−	0.929156i	$-0.620535\pi$
$37$	−4.49744	−0.739374	−0.369687	+	0.929156i	$0.620535\pi$
$38$	10.4600	1.69683
$39$	0	0
$40$	2.42164	0.382894
$41$	6.07522	0.948791	0.474395	−	0.880312i	$-0.342667\pi$
$41$	6.07522	0.948791	0.474395	+	0.880312i	$0.342667\pi$
$42$	0	0
$43$	−5.88817	−0.897938	−0.448969	−	0.893547i	$-0.648209\pi$
$43$	−5.88817	−0.897938	−0.448969	+	0.893547i	$0.648209\pi$
$44$	−3.19089	−0.481044
$45$	0	0
$46$	13.2917	1.95975
$47$	−3.24407	−0.473197	−0.236598	−	0.971608i	$-0.576033\pi$
$47$	−3.24407	−0.473197	−0.236598	+	0.971608i	$0.576033\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	2.56896	0.363306
$51$	0	0
$52$	4.61500	0.639986
$53$	−10.8652	−1.49245	−0.746223	−	0.665697i	$-0.768135\pi$
$53$	−10.8652	−1.49245	−0.746223	+	0.665697i	$0.768135\pi$
$54$	0	0
$55$	5.42069	0.730925
$56$	−0.959047	−0.128158
$57$	0	0
$58$	13.8877	1.82354
$59$	−11.7201	−1.52583	−0.762915	−	0.646499i	$-0.776233\pi$
$59$	−11.7201	−1.52583	−0.762915	+	0.646499i	$0.776233\pi$
$60$	0	0
$61$	−4.57418	−0.585663	−0.292832	−	0.956164i	$-0.594598\pi$
$61$	−4.57418	−0.585663	−0.292832	+	0.956164i	$0.594598\pi$
$62$	9.66167	1.22703
$63$	0	0
$64$	−3.49880	−0.437350
$65$	−7.83998	−0.972430
$66$	0	0
$67$	3.13289	0.382743	0.191372	−	0.981518i	$-0.438706\pi$
$67$	3.13289	0.382743	0.191372	+	0.981518i	$0.438706\pi$
$68$	−8.79615	−1.06669
$69$	0	0
$70$	−4.71472	−0.563516
$71$	−15.6126	−1.85287	−0.926437	−	0.376449i	$-0.877145\pi$
$71$	−15.6126	−1.85287	−0.926437	+	0.376449i	$0.877145\pi$
$72$	0	0
$73$	−13.6309	−1.59538	−0.797690	−	0.603068i	$-0.793945\pi$
$73$	−13.6309	−1.59538	−0.797690	+	0.603068i	$0.793945\pi$
$74$	−8.39753	−0.976193
$75$	0	0
$76$	8.32663	0.955130
$77$	−2.14677	−0.244647
$78$	0	0
$79$	11.7382	1.32065	0.660326	−	0.750979i	$-0.270418\pi$
$79$	11.7382	1.32065	0.660326	+	0.750979i	$0.270418\pi$
$80$	12.0279	1.34476
$81$	0	0
$82$	11.3435	1.25268
$83$	−11.0096	−1.20846	−0.604229	−	0.796811i	$-0.706519\pi$
$83$	−11.0096	−1.20846	−0.604229	+	0.796811i	$0.706519\pi$
$84$	0	0
$85$	14.9429	1.62079
$86$	−10.9943	−1.18554
$87$	0	0
$88$	2.05885	0.219474
$89$	−5.48084	−0.580968	−0.290484	−	0.956880i	$-0.593816\pi$
$89$	−5.48084	−0.580968	−0.290484	+	0.956880i	$0.593816\pi$
$90$	0	0
$91$	3.10489	0.325481
$92$	10.5808	1.10313
$93$	0	0
$94$	−6.05728	−0.624760
$95$	−14.1453	−1.45128
$96$	0	0
$97$	−1.03456	−0.105043	−0.0525217	−	0.998620i	$-0.516726\pi$
$97$	−1.03456	−0.105043	−0.0525217	+	0.998620i	$0.516726\pi$
$98$	1.86718	0.188614
$99$	0	0
$100$	2.04502	0.204502
$101$	−13.0504	−1.29856	−0.649280	−	0.760549i	$-0.724930\pi$
$101$	−13.0504	−1.29856	−0.649280	+	0.760549i	$0.724930\pi$
$102$	0	0
$103$	−9.26938	−0.913339	−0.456669	−	0.889636i	$-0.650958\pi$
$103$	−9.26938	−0.913339	−0.456669	+	0.889636i	$0.650958\pi$
$104$	−2.97773	−0.291991
$105$	0	0
$106$	−20.2872	−1.97047
$107$	8.65350	0.836565	0.418283	−	0.908317i	$-0.362632\pi$
$107$	8.65350	0.836565	0.418283	+	0.908317i	$0.362632\pi$
$108$	0	0
$109$	−1.20206	−0.115136	−0.0575682	−	0.998342i	$-0.518335\pi$
$109$	−1.20206	−0.115136	−0.0575682	+	0.998342i	$0.518335\pi$
$110$	10.1214	0.965038
$111$	0	0
$112$	−4.76345	−0.450103
$113$	1.24812	0.117413	0.0587067	−	0.998275i	$-0.481302\pi$
$113$	1.24812	0.117413	0.0587067	+	0.998275i	$0.481302\pi$
$114$	0	0
$115$	−17.9747	−1.67615
$116$	11.0553	1.02646
$117$	0	0
$118$	−21.8836	−2.01455
$119$	−5.91789	−0.542492
$120$	0	0
$121$	−6.39138	−0.581035
$122$	−8.54082	−0.773249
$123$	0	0
$124$	7.69115	0.690686
$125$	9.15114	0.818503
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	7.41936	0.655785
$129$	0	0
$130$	−14.6387	−1.28390
$131$	−10.6910	−0.934076	−0.467038	−	0.884237i	$-0.654679\pi$
$131$	−10.6910	−0.934076	−0.467038	+	0.884237i	$0.654679\pi$
$132$	0	0
$133$	5.60200	0.485755
$134$	5.84967	0.505335
$135$	0	0
$136$	5.67553	0.486673
$137$	−8.97622	−0.766891	−0.383445	−	0.923564i	$-0.625263\pi$
$137$	−8.97622	−0.766891	−0.383445	+	0.923564i	$0.625263\pi$
$138$	0	0
$139$	16.3441	1.38629	0.693144	−	0.720799i	$-0.256225\pi$
$139$	16.3441	1.38629	0.693144	+	0.720799i	$0.256225\pi$
$140$	−3.75314	−0.317198
$141$	0	0
$142$	−29.1516	−2.44634
$143$	−6.66548	−0.557395
$144$	0	0
$145$	−18.7807	−1.55965
$146$	−25.4514	−2.10637
$147$	0	0
$148$	−6.68484	−0.549490
$149$	−21.0349	−1.72325	−0.861623	−	0.507549i	$-0.830552\pi$
$149$	−21.0349	−1.72325	−0.861623	+	0.507549i	$0.830552\pi$
$150$	0	0
$151$	−19.6782	−1.60139	−0.800693	−	0.599075i	$-0.795535\pi$
$151$	−19.6782	−1.60139	−0.800693	+	0.599075i	$0.795535\pi$
$152$	−5.37258	−0.435774
$153$	0	0
$154$	−4.00841	−0.323007
$155$	−13.0658	−1.04947
$156$	0	0
$157$	8.58647	0.685275	0.342637	−	0.939468i	$-0.388680\pi$
$157$	8.58647	0.685275	0.342637	+	0.939468i	$0.388680\pi$
$158$	21.9174	1.74365
$159$	0	0
$160$	17.6150	1.39259
$161$	7.11857	0.561022
$162$	0	0
$163$	−7.72950	−0.605421	−0.302711	−	0.953082i	$-0.597892\pi$
$163$	−7.72950	−0.605421	−0.302711	+	0.953082i	$0.597892\pi$
$164$	9.03001	0.705125
$165$	0	0
$166$	−20.5569	−1.59552
$167$	1.90242	0.147214	0.0736070	−	0.997287i	$-0.476549\pi$
$167$	1.90242	0.147214	0.0736070	+	0.997287i	$0.476549\pi$
$168$	0	0
$169$	−3.35966	−0.258436
$170$	27.9012	2.13992
$171$	0	0
$172$	−8.75198	−0.667333
$173$	9.51257	0.723227	0.361614	−	0.932328i	$-0.382226\pi$
$173$	9.51257	0.723227	0.361614	+	0.932328i	$0.382226\pi$
$174$	0	0
$175$	1.37585	0.104005
$176$	10.2260	0.770815
$177$	0	0
$178$	−10.2337	−0.767050
$179$	−9.54411	−0.713360	−0.356680	−	0.934227i	$-0.616091\pi$
$179$	−9.54411	−0.713360	−0.356680	+	0.934227i	$0.616091\pi$
$180$	0	0
$181$	21.9419	1.63093	0.815465	−	0.578807i	$-0.196481\pi$
$181$	21.9419	1.63093	0.815465	+	0.578807i	$0.196481\pi$
$182$	5.79739	0.429731
$183$	0	0
$184$	−6.82704	−0.503296
$185$	11.3562	0.834926
$186$	0	0
$187$	12.7043	0.929033
$188$	−4.82188	−0.351672
$189$	0	0
$190$	−26.4119	−1.91612
$191$	20.8863	1.51128	0.755640	−	0.654987i	$-0.227326\pi$
$191$	20.8863	1.51128	0.755640	+	0.654987i	$0.227326\pi$
$192$	0	0
$193$	−14.8527	−1.06912	−0.534559	−	0.845131i	$-0.679522\pi$
$193$	−14.8527	−1.06912	−0.534559	+	0.845131i	$0.679522\pi$
$194$	−1.93171	−0.138689
$195$	0	0
$196$	1.48637	0.106169
$197$	−13.6942	−0.975669	−0.487834	−	0.872936i	$-0.662213\pi$
$197$	−13.6942	−0.975669	−0.487834	+	0.872936i	$0.662213\pi$
$198$	0	0
$199$	17.7755	1.26007	0.630035	−	0.776567i	$-0.283041\pi$
$199$	17.7755	1.26007	0.630035	+	0.776567i	$0.283041\pi$
$200$	−1.31951	−0.0933032
$201$	0	0
$202$	−24.3674	−1.71449
$203$	7.43778	0.522030
$204$	0	0
$205$	−15.3402	−1.07141
$206$	−17.3076	−1.20588
$207$	0	0
$208$	−14.7900	−1.02550
$209$	−12.0262	−0.831870
$210$	0	0
$211$	−7.22564	−0.497434	−0.248717	−	0.968576i	$-0.580009\pi$
$211$	−7.22564	−0.497434	−0.248717	+	0.968576i	$0.580009\pi$
$212$	−16.1496	−1.10916
$213$	0	0
$214$	16.1577	1.10451
$215$	14.8679	1.01398
$216$	0	0
$217$	5.17447	0.351266
$218$	−2.24446	−0.152014
$219$	0	0
$220$	8.05713	0.543212
$221$	−18.3744	−1.23600
$222$	0	0
$223$	−1.99060	−0.133300	−0.0666501	−	0.997776i	$-0.521231\pi$
$223$	−1.99060	−0.133300	−0.0666501	+	0.997776i	$0.521231\pi$
$224$	−6.97613	−0.466112
$225$	0	0
$226$	2.33047	0.155021
$227$	−1.85235	−0.122945	−0.0614723	−	0.998109i	$-0.519580\pi$
$227$	−1.85235	−0.122945	−0.0614723	+	0.998109i	$0.519580\pi$
$228$	0	0
$229$	−4.03988	−0.266963	−0.133481	−	0.991051i	$-0.542616\pi$
$229$	−4.03988	−0.266963	−0.133481	+	0.991051i	$0.542616\pi$
$230$	−33.5621	−2.21302
$231$	0	0
$232$	−7.13318	−0.468316
$233$	−13.3853	−0.876902	−0.438451	−	0.898755i	$-0.644473\pi$
$233$	−13.3853	−0.876902	−0.438451	+	0.898755i	$0.644473\pi$
$234$	0	0
$235$	8.19143	0.534350
$236$	−17.4204	−1.13397
$237$	0	0
$238$	−11.0498	−0.716250
$239$	5.43938	0.351844	0.175922	−	0.984404i	$-0.443709\pi$
$239$	5.43938	0.351844	0.175922	+	0.984404i	$0.443709\pi$
$240$	0	0
$241$	−26.4016	−1.70067	−0.850337	−	0.526238i	$-0.823602\pi$
$241$	−26.4016	−1.70067	−0.850337	+	0.526238i	$0.823602\pi$
$242$	−11.9339	−0.767138
$243$	0	0
$244$	−6.79890	−0.435255
$245$	−2.52504	−0.161319
$246$	0	0
$247$	17.3936	1.10673
$248$	−4.96256	−0.315123
$249$	0	0
$250$	17.0868	1.08067
$251$	20.9117	1.31993	0.659967	−	0.751294i	$-0.270570\pi$
$251$	20.9117	1.31993	0.659967	+	0.751294i	$0.270570\pi$
$252$	0	0
$253$	−15.2819	−0.960767
$254$	−1.86718	−0.117157
$255$	0	0
$256$	20.8509	1.30318
$257$	24.3901	1.52141	0.760705	−	0.649097i	$-0.224853\pi$
$257$	24.3901	1.52141	0.760705	+	0.649097i	$0.224853\pi$
$258$	0	0
$259$	−4.49744	−0.279457
$260$	−11.6531	−0.722694
$261$	0	0
$262$	−19.9620	−1.23326
$263$	−8.75029	−0.539566	−0.269783	−	0.962921i	$-0.586952\pi$
$263$	−8.75029	−0.539566	−0.269783	+	0.962921i	$0.586952\pi$
$264$	0	0
$265$	27.4350	1.68532
$266$	10.4600	0.641341
$267$	0	0
$268$	4.65662	0.284448
$269$	−0.734176	−0.0447635	−0.0223818	−	0.999749i	$-0.507125\pi$
$269$	−0.734176	−0.0447635	−0.0223818	+	0.999749i	$0.507125\pi$
$270$	0	0
$271$	6.99503	0.424918	0.212459	−	0.977170i	$-0.431853\pi$
$271$	6.99503	0.424918	0.212459	+	0.977170i	$0.431853\pi$
$272$	28.1895	1.70924
$273$	0	0
$274$	−16.7602	−1.01252
$275$	−2.95363	−0.178111
$276$	0	0
$277$	3.60962	0.216881	0.108441	−	0.994103i	$-0.465414\pi$
$277$	3.60962	0.216881	0.108441	+	0.994103i	$0.465414\pi$
$278$	30.5174	1.83031
$279$	0	0
$280$	2.42164	0.144720
$281$	−24.7064	−1.47386	−0.736931	−	0.675968i	$-0.763726\pi$
$281$	−24.7064	−1.47386	−0.736931	+	0.675968i	$0.763726\pi$
$282$	0	0
$283$	23.5104	1.39755	0.698773	−	0.715343i	$-0.253730\pi$
$283$	23.5104	1.39755	0.698773	+	0.715343i	$0.253730\pi$
$284$	−23.2060	−1.37703
$285$	0	0
$286$	−12.4457	−0.735927
$287$	6.07522	0.358609
$288$	0	0
$289$	18.0214	1.06008
$290$	−35.0670	−2.05921
$291$	0	0
$292$	−20.2606	−1.18566
$293$	14.2321	0.831445	0.415723	−	0.909491i	$-0.363529\pi$
$293$	14.2321	0.831445	0.415723	+	0.909491i	$0.363529\pi$
$294$	0	0
$295$	29.5938	1.72302
$296$	4.31325	0.250703
$297$	0	0
$298$	−39.2760	−2.27520
$299$	22.1024	1.27821
$300$	0	0
$301$	−5.88817	−0.339389
$302$	−36.7427	−2.11430
$303$	0	0
$304$	−26.6848	−1.53048
$305$	11.5500	0.661351
$306$	0	0
$307$	−2.43954	−0.139232	−0.0696159	−	0.997574i	$-0.522177\pi$
$307$	−2.43954	−0.139232	−0.0696159	+	0.997574i	$0.522177\pi$
$308$	−3.19089	−0.181818
$309$	0	0
$310$	−24.3961	−1.38561
$311$	14.2486	0.807961	0.403981	−	0.914768i	$-0.367626\pi$
$311$	14.2486	0.807961	0.403981	+	0.914768i	$0.367626\pi$
$312$	0	0
$313$	1.01920	0.0576087	0.0288043	−	0.999585i	$-0.490830\pi$
$313$	1.01920	0.0576087	0.0288043	+	0.999585i	$0.490830\pi$
$314$	16.0325	0.904766
$315$	0	0
$316$	17.4473	0.981486
$317$	26.7112	1.50025	0.750124	−	0.661297i	$-0.229994\pi$
$317$	26.7112	1.50025	0.750124	+	0.661297i	$0.229994\pi$
$318$	0	0
$319$	−15.9672	−0.893991
$320$	8.83463	0.493871
$321$	0	0
$322$	13.2917	0.740716
$323$	−33.1520	−1.84463
$324$	0	0
$325$	4.27187	0.236960
$326$	−14.4324	−0.799336
$327$	0	0
$328$	−5.82642	−0.321710
$329$	−3.24407	−0.178852
$330$	0	0
$331$	3.21043	0.176461	0.0882305	−	0.996100i	$-0.471879\pi$
$331$	3.21043	0.176461	0.0882305	+	0.996100i	$0.471879\pi$
$332$	−16.3643	−0.898105
$333$	0	0
$334$	3.55217	0.194366
$335$	−7.91069	−0.432207
$336$	0	0
$337$	−15.9804	−0.870505	−0.435253	−	0.900308i	$-0.643341\pi$
$337$	−15.9804	−0.870505	−0.435253	+	0.900308i	$0.643341\pi$
$338$	−6.27310	−0.341212
$339$	0	0
$340$	22.2107	1.20454
$341$	−11.1084	−0.601553
$342$	0	0
$343$	1.00000	0.0539949
$344$	5.64703	0.304468
$345$	0	0
$346$	17.7617	0.954874
$347$	4.64687	0.249457	0.124729	−	0.992191i	$-0.460194\pi$
$347$	4.64687	0.249457	0.124729	+	0.992191i	$0.460194\pi$
$348$	0	0
$349$	−12.8959	−0.690300	−0.345150	−	0.938547i	$-0.612172\pi$
$349$	−12.8959	−0.690300	−0.345150	+	0.938547i	$0.612172\pi$
$350$	2.56896	0.137317
$351$	0	0
$352$	14.9761	0.798231
$353$	−6.04228	−0.321598	−0.160799	−	0.986987i	$-0.551407\pi$
$353$	−6.04228	−0.321598	−0.160799	+	0.986987i	$0.551407\pi$
$354$	0	0
$355$	39.4225	2.09233
$356$	−8.14654	−0.431766
$357$	0	0
$358$	−17.8206	−0.941847
$359$	−27.1794	−1.43447	−0.717237	−	0.696830i	$-0.754593\pi$
$359$	−27.1794	−1.43447	−0.717237	+	0.696830i	$0.754593\pi$
$360$	0	0
$361$	12.3824	0.651707
$362$	40.9695	2.15331
$363$	0	0
$364$	4.61500	0.241892
$365$	34.4187	1.80156
$366$	0	0
$367$	32.2253	1.68215	0.841073	−	0.540922i	$-0.181924\pi$
$367$	32.2253	1.68215	0.841073	+	0.540922i	$0.181924\pi$
$368$	−33.9089	−1.76763
$369$	0	0
$370$	21.2041	1.10235
$371$	−10.8652	−0.564091
$372$	0	0
$373$	−29.7787	−1.54188	−0.770941	−	0.636907i	$-0.780214\pi$
$373$	−29.7787	−1.54188	−0.770941	+	0.636907i	$0.780214\pi$
$374$	23.7213	1.22660
$375$	0	0
$376$	3.11122	0.160449
$377$	23.0935	1.18937
$378$	0	0
$379$	12.3300	0.633349	0.316675	−	0.948534i	$-0.397434\pi$
$379$	12.3300	0.633349	0.316675	+	0.948534i	$0.397434\pi$
$380$	−21.0251	−1.07857
$381$	0	0
$382$	38.9985	1.99534
$383$	36.4315	1.86156	0.930782	−	0.365574i	$-0.119127\pi$
$383$	36.4315	1.86156	0.930782	+	0.365574i	$0.119127\pi$
$384$	0	0
$385$	5.42069	0.276264
$386$	−27.7326	−1.41155
$387$	0	0
$388$	−1.53773	−0.0780666
$389$	35.0881	1.77904	0.889519	−	0.456898i	$-0.151040\pi$
$389$	35.0881	1.77904	0.889519	+	0.456898i	$0.151040\pi$
$390$	0	0
$391$	−42.1269	−2.13045
$392$	−0.959047	−0.0484392
$393$	0	0
$394$	−25.5695	−1.28817
$395$	−29.6395	−1.49133
$396$	0	0
$397$	21.7798	1.09310	0.546548	−	0.837428i	$-0.315942\pi$
$397$	21.7798	1.09310	0.546548	+	0.837428i	$0.315942\pi$
$398$	33.1900	1.66367
$399$	0	0
$400$	−6.55380	−0.327690
$401$	−18.4781	−0.922751	−0.461375	−	0.887205i	$-0.652644\pi$
$401$	−18.4781	−0.922751	−0.461375	+	0.887205i	$0.652644\pi$
$402$	0	0
$403$	16.0661	0.800312
$404$	−19.3976	−0.965068
$405$	0	0
$406$	13.8877	0.689234
$407$	9.65496	0.478578
$408$	0	0
$409$	−26.3563	−1.30324	−0.651618	−	0.758547i	$-0.725909\pi$
$409$	−26.3563	−1.30324	−0.651618	+	0.758547i	$0.725909\pi$
$410$	−28.6430	−1.41457
$411$	0	0
$412$	−13.7777	−0.678778
$413$	−11.7201	−0.576710
$414$	0	0
$415$	27.7997	1.36463
$416$	−21.6601	−1.06197
$417$	0	0
$418$	−22.4551	−1.09832
$419$	−7.75685	−0.378947	−0.189473	−	0.981886i	$-0.560678\pi$
$419$	−7.75685	−0.378947	−0.189473	+	0.981886i	$0.560678\pi$
$420$	0	0
$421$	−32.9882	−1.60775	−0.803873	−	0.594801i	$-0.797231\pi$
$421$	−32.9882	−1.60775	−0.803873	+	0.594801i	$0.797231\pi$
$422$	−13.4916	−0.656760
$423$	0	0
$424$	10.4202	0.506050
$425$	−8.14213	−0.394952
$426$	0	0
$427$	−4.57418	−0.221360
$428$	12.8623	0.621721
$429$	0	0
$430$	27.7611	1.33876
$431$	0.521602	0.0251247	0.0125623	−	0.999921i	$-0.496001\pi$
$431$	0.521602	0.0251247	0.0125623	+	0.999921i	$0.496001\pi$
$432$	0	0
$433$	14.6517	0.704117	0.352059	−	0.935978i	$-0.385482\pi$
$433$	14.6517	0.704117	0.352059	+	0.935978i	$0.385482\pi$
$434$	9.66167	0.463775
$435$	0	0
$436$	−1.78670	−0.0855675
$437$	39.8783	1.90764
$438$	0	0
$439$	11.7950	0.562946	0.281473	−	0.959569i	$-0.409177\pi$
$439$	11.7950	0.562946	0.281473	+	0.959569i	$0.409177\pi$
$440$	−5.19869	−0.247838
$441$	0	0
$442$	−34.3083	−1.63188
$443$	−4.46547	−0.212161	−0.106080	−	0.994358i	$-0.533830\pi$
$443$	−4.46547	−0.212161	−0.106080	+	0.994358i	$0.533830\pi$
$444$	0	0
$445$	13.8394	0.656049
$446$	−3.71680	−0.175996
$447$	0	0
$448$	−3.49880	−0.165303
$449$	−2.18869	−0.103291	−0.0516454	−	0.998665i	$-0.516447\pi$
$449$	−2.18869	−0.103291	−0.0516454	+	0.998665i	$0.516447\pi$
$450$	0	0
$451$	−13.0421	−0.614129
$452$	1.85517	0.0872597
$453$	0	0
$454$	−3.45867	−0.162323
$455$	−7.83998	−0.367544
$456$	0	0
$457$	−39.0672	−1.82749	−0.913743	−	0.406293i	$-0.866821\pi$
$457$	−39.0672	−1.82749	−0.913743	+	0.406293i	$0.866821\pi$
$458$	−7.54318	−0.352470
$459$	0	0
$460$	−26.7170	−1.24569
$461$	−23.5633	−1.09745	−0.548727	−	0.836002i	$-0.684887\pi$
$461$	−23.5633	−1.09745	−0.548727	+	0.836002i	$0.684887\pi$
$462$	0	0
$463$	11.8195	0.549298	0.274649	−	0.961544i	$-0.411438\pi$
$463$	11.8195	0.549298	0.274649	+	0.961544i	$0.411438\pi$
$464$	−35.4295	−1.64477
$465$	0	0
$466$	−24.9928	−1.15777
$467$	−16.5778	−0.767128	−0.383564	−	0.923514i	$-0.625303\pi$
$467$	−16.5778	−0.767128	−0.383564	+	0.923514i	$0.625303\pi$
$468$	0	0
$469$	3.13289	0.144663
$470$	15.2949	0.705501
$471$	0	0
$472$	11.2401	0.517370
$473$	12.6405	0.581213
$474$	0	0
$475$	7.70752	0.353645
$476$	−8.79615	−0.403171
$477$	0	0
$478$	10.1563	0.464538
$479$	−22.6906	−1.03676	−0.518380	−	0.855150i	$-0.673465\pi$
$479$	−22.6906	−1.03676	−0.518380	+	0.855150i	$0.673465\pi$
$480$	0	0
$481$	−13.9640	−0.636705
$482$	−49.2965	−2.24540
$483$	0	0
$484$	−9.49994	−0.431815
$485$	2.61231	0.118619
$486$	0	0
$487$	−21.8682	−0.990941	−0.495470	−	0.868625i	$-0.665004\pi$
$487$	−21.8682	−0.990941	−0.495470	+	0.868625i	$0.665004\pi$
$488$	4.38685	0.198583
$489$	0	0
$490$	−4.71472	−0.212989
$491$	−40.6492	−1.83447	−0.917235	−	0.398346i	$-0.869584\pi$
$491$	−40.6492	−1.83447	−0.917235	+	0.398346i	$0.869584\pi$
$492$	0	0
$493$	−44.0159	−1.98238
$494$	32.4770	1.46121
$495$	0	0
$496$	−24.6483	−1.10674
$497$	−15.6126	−0.700321
$498$	0	0
$499$	15.4222	0.690393	0.345197	−	0.938530i	$-0.387812\pi$
$499$	15.4222	0.690393	0.345197	+	0.938530i	$0.387812\pi$
$500$	13.6019	0.608297
$501$	0	0
$502$	39.0459	1.74270
$503$	−14.5992	−0.650945	−0.325473	−	0.945551i	$-0.605523\pi$
$503$	−14.5992	−0.650945	−0.325473	+	0.945551i	$0.605523\pi$
$504$	0	0
$505$	32.9528	1.46638
$506$	−28.5341	−1.26850
$507$	0	0
$508$	−1.48637	−0.0659469
$509$	−16.9749	−0.752397	−0.376199	−	0.926539i	$-0.622769\pi$
$509$	−16.9749	−0.752397	−0.376199	+	0.926539i	$0.622769\pi$
$510$	0	0
$511$	−13.6309	−0.602997
$512$	24.0937	1.06480
$513$	0	0
$514$	45.5407	2.00871
$515$	23.4056	1.03137
$516$	0	0
$517$	6.96428	0.306289
$518$	−8.39753	−0.368966
$519$	0	0
$520$	7.51891	0.329726
$521$	−16.7832	−0.735283	−0.367642	−	0.929968i	$-0.619835\pi$
$521$	−16.7832	−0.735283	−0.367642	+	0.929968i	$0.619835\pi$
$522$	0	0
$523$	−20.9119	−0.914414	−0.457207	−	0.889360i	$-0.651150\pi$
$523$	−20.9119	−0.914414	−0.457207	+	0.889360i	$0.651150\pi$
$524$	−15.8907	−0.694189
$525$	0	0
$526$	−16.3384	−0.712387
$527$	−30.6219	−1.33391
$528$	0	0
$529$	27.6741	1.20322
$530$	51.2262	2.22512
$531$	0	0
$532$	8.32663	0.361005
$533$	18.8629	0.817043
$534$	0	0
$535$	−21.8505	−0.944678
$536$	−3.00459	−0.129778
$537$	0	0
$538$	−1.37084	−0.0591011
$539$	−2.14677	−0.0924679
$540$	0	0
$541$	4.34159	0.186660	0.0933298	−	0.995635i	$-0.470249\pi$
$541$	4.34159	0.186660	0.0933298	+	0.995635i	$0.470249\pi$
$542$	13.0610	0.561018
$543$	0	0
$544$	41.2839	1.77003
$545$	3.03526	0.130016
$546$	0	0
$547$	−2.97082	−0.127023	−0.0635116	−	0.997981i	$-0.520230\pi$
$547$	−2.97082	−0.127023	−0.0635116	+	0.997981i	$0.520230\pi$
$548$	−13.3420	−0.569940
$549$	0	0
$550$	−5.51497	−0.235159
$551$	41.6665	1.77505
$552$	0	0
$553$	11.7382	0.499159
$554$	6.73982	0.286347
$555$	0	0
$556$	24.2933	1.03027
$557$	37.9755	1.60907	0.804536	−	0.593904i	$-0.202414\pi$
$557$	37.9755	1.60907	0.804536	+	0.593904i	$0.202414\pi$
$558$	0	0
$559$	−18.2821	−0.773252
$560$	12.0279	0.508272
$561$	0	0
$562$	−46.1314	−1.94594
$563$	25.8374	1.08892	0.544459	−	0.838787i	$-0.316735\pi$
$563$	25.8374	1.08892	0.544459	+	0.838787i	$0.316735\pi$
$564$	0	0
$565$	−3.15156	−0.132587
$566$	43.8981	1.84518
$567$	0	0
$568$	14.9732	0.628262
$569$	−7.36191	−0.308628	−0.154314	−	0.988022i	$-0.549317\pi$
$569$	−7.36191	−0.308628	−0.154314	+	0.988022i	$0.549317\pi$
$570$	0	0
$571$	−37.2421	−1.55854	−0.779268	−	0.626691i	$-0.784409\pi$
$571$	−37.2421	−1.55854	−0.779268	+	0.626691i	$0.784409\pi$
$572$	−9.90735	−0.414247
$573$	0	0
$574$	11.3435	0.473470
$575$	9.79410	0.408442
$576$	0	0
$577$	−8.48323	−0.353161	−0.176581	−	0.984286i	$-0.556504\pi$
$577$	−8.48323	−0.353161	−0.176581	+	0.984286i	$0.556504\pi$
$578$	33.6492	1.39962
$579$	0	0
$580$	−27.9150	−1.15911
$581$	−11.0096	−0.456754
$582$	0	0
$583$	23.3250	0.966023
$584$	13.0727	0.540952
$585$	0	0
$586$	26.5738	1.09775
$587$	7.57932	0.312832	0.156416	−	0.987691i	$-0.450006\pi$
$587$	7.57932	0.312832	0.156416	+	0.987691i	$0.450006\pi$
$588$	0	0
$589$	28.9874	1.19440
$590$	55.2571	2.27490
$591$	0	0
$592$	21.4233	0.880492
$593$	18.4304	0.756848	0.378424	−	0.925632i	$-0.376466\pi$
$593$	18.4304	0.756848	0.378424	+	0.925632i	$0.376466\pi$
$594$	0	0
$595$	14.9429	0.612600
$596$	−31.2656	−1.28069
$597$	0	0
$598$	41.2692	1.68762
$599$	6.29754	0.257311	0.128655	−	0.991689i	$-0.458934\pi$
$599$	6.29754	0.257311	0.128655	+	0.991689i	$0.458934\pi$
$600$	0	0
$601$	−0.979702	−0.0399629	−0.0199814	−	0.999800i	$-0.506361\pi$
$601$	−0.979702	−0.0399629	−0.0199814	+	0.999800i	$0.506361\pi$
$602$	−10.9943	−0.448094
$603$	0	0
$604$	−29.2490	−1.19012
$605$	16.1385	0.656125
$606$	0	0
$607$	30.7475	1.24800	0.624001	−	0.781424i	$-0.285506\pi$
$607$	30.7475	1.24800	0.624001	+	0.781424i	$0.285506\pi$
$608$	−39.0803	−1.58491
$609$	0	0
$610$	21.5660	0.873180
$611$	−10.0725	−0.407489
$612$	0	0
$613$	−5.30594	−0.214305	−0.107152	−	0.994243i	$-0.534173\pi$
$613$	−5.30594	−0.214305	−0.107152	+	0.994243i	$0.534173\pi$
$614$	−4.55506	−0.183827
$615$	0	0
$616$	2.05885	0.0829535
$617$	−31.1189	−1.25280	−0.626401	−	0.779501i	$-0.715473\pi$
$617$	−31.1189	−1.25280	−0.626401	+	0.779501i	$0.715473\pi$
$618$	0	0
$619$	2.21302	0.0889489	0.0444744	−	0.999011i	$-0.485839\pi$
$619$	2.21302	0.0889489	0.0444744	+	0.999011i	$0.485839\pi$
$620$	−19.4205	−0.779946
$621$	0	0
$622$	26.6046	1.06675
$623$	−5.48084	−0.219585
$624$	0	0
$625$	−29.9863	−1.19945
$626$	1.90303	0.0760605
$627$	0	0
$628$	12.7626	0.509285
$629$	26.6153	1.06122
$630$	0	0
$631$	−30.3853	−1.20962	−0.604810	−	0.796370i	$-0.706751\pi$
$631$	−30.3853	−1.20962	−0.604810	+	0.796370i	$0.706751\pi$
$632$	−11.2575	−0.447799
$633$	0	0
$634$	49.8746	1.98077
$635$	2.52504	0.100203
$636$	0	0
$637$	3.10489	0.123020
$638$	−29.8137	−1.18033
$639$	0	0
$640$	−18.7342	−0.740535
$641$	29.0454	1.14722	0.573612	−	0.819127i	$-0.305542\pi$
$641$	29.0454	1.14722	0.573612	+	0.819127i	$0.305542\pi$
$642$	0	0
$643$	−46.5068	−1.83405	−0.917024	−	0.398833i	$-0.869415\pi$
$643$	−46.5068	−1.83405	−0.917024	+	0.398833i	$0.869415\pi$
$644$	10.5808	0.416942
$645$	0	0
$646$	−61.9008	−2.43546
$647$	40.6542	1.59828	0.799141	−	0.601144i	$-0.205288\pi$
$647$	40.6542	1.59828	0.799141	+	0.601144i	$0.205288\pi$
$648$	0	0
$649$	25.1604	0.987632
$650$	7.97635	0.312858
$651$	0	0
$652$	−11.4889	−0.449939
$653$	39.0488	1.52810	0.764049	−	0.645158i	$-0.223208\pi$
$653$	39.0488	1.52810	0.764049	+	0.645158i	$0.223208\pi$
$654$	0	0
$655$	26.9952	1.05479
$656$	−28.9390	−1.12988
$657$	0	0
$658$	−6.05728	−0.236137
$659$	43.6453	1.70018	0.850089	−	0.526639i	$-0.176548\pi$
$659$	43.6453	1.70018	0.850089	+	0.526639i	$0.176548\pi$
$660$	0	0
$661$	−8.95476	−0.348300	−0.174150	−	0.984719i	$-0.555718\pi$
$661$	−8.95476	−0.348300	−0.174150	+	0.984719i	$0.555718\pi$
$662$	5.99445	0.232981
$663$	0	0
$664$	10.5587	0.409757
$665$	−14.1453	−0.548531
$666$	0	0
$667$	52.9464	2.05009
$668$	2.82770	0.109407
$669$	0	0
$670$	−14.7707	−0.570641
$671$	9.81970	0.379085
$672$	0	0
$673$	24.6108	0.948678	0.474339	−	0.880342i	$-0.342687\pi$
$673$	24.6108	0.948678	0.474339	+	0.880342i	$0.342687\pi$
$674$	−29.8382	−1.14933
$675$	0	0
$676$	−4.99369	−0.192065
$677$	−28.2165	−1.08445	−0.542224	−	0.840234i	$-0.682418\pi$
$677$	−28.2165	−1.08445	−0.542224	+	0.840234i	$0.682418\pi$
$678$	0	0
$679$	−1.03456	−0.0397027
$680$	−14.3310	−0.549568
$681$	0	0
$682$	−20.7414	−0.794228
$683$	−3.13372	−0.119909	−0.0599543	−	0.998201i	$-0.519096\pi$
$683$	−3.13372	−0.119909	−0.0599543	+	0.998201i	$0.519096\pi$
$684$	0	0
$685$	22.6654	0.865999
$686$	1.86718	0.0712893
$687$	0	0
$688$	28.0480	1.06932
$689$	−33.7351	−1.28521
$690$	0	0
$691$	−15.3776	−0.584992	−0.292496	−	0.956267i	$-0.594486\pi$
$691$	−15.3776	−0.584992	−0.292496	+	0.956267i	$0.594486\pi$
$692$	14.1392	0.537490
$693$	0	0
$694$	8.67656	0.329358
$695$	−41.2696	−1.56544
$696$	0	0
$697$	−35.9525	−1.36180
$698$	−24.0789	−0.911401
$699$	0	0
$700$	2.04502	0.0772945
$701$	−0.753947	−0.0284762	−0.0142381	−	0.999899i	$-0.504532\pi$
$701$	−0.753947	−0.0284762	−0.0142381	+	0.999899i	$0.504532\pi$
$702$	0	0
$703$	−25.1946	−0.950234
$704$	7.51112	0.283086
$705$	0	0
$706$	−11.2820	−0.424605
$707$	−13.0504	−0.490810
$708$	0	0
$709$	−52.5784	−1.97462	−0.987312	−	0.158791i	$-0.949241\pi$
$709$	−52.5784	−1.97462	−0.987312	+	0.158791i	$0.949241\pi$
$710$	73.6090	2.76250
$711$	0	0
$712$	5.25639	0.196991
$713$	36.8348	1.37947
$714$	0	0
$715$	16.8306	0.629430
$716$	−14.1860	−0.530158
$717$	0	0
$718$	−50.7489	−1.89393
$719$	−42.2691	−1.57637	−0.788185	−	0.615439i	$-0.788979\pi$
$719$	−42.2691	−1.57637	−0.788185	+	0.615439i	$0.788979\pi$
$720$	0	0
$721$	−9.26938	−0.345210
$722$	23.1202	0.860446
$723$	0	0
$724$	32.6137	1.21208
$725$	10.2333	0.380054
$726$	0	0
$727$	6.60122	0.244826	0.122413	−	0.992479i	$-0.460937\pi$
$727$	6.60122	0.244826	0.122413	+	0.992479i	$0.460937\pi$
$728$	−2.97773	−0.110362
$729$	0	0
$730$	64.2660	2.37859
$731$	34.8456	1.28881
$732$	0	0
$733$	11.4503	0.422927	0.211464	−	0.977386i	$-0.432177\pi$
$733$	11.4503	0.422927	0.211464	+	0.977386i	$0.432177\pi$
$734$	60.1704	2.22093
$735$	0	0
$736$	−49.6601	−1.83049
$737$	−6.72559	−0.247740
$738$	0	0
$739$	−36.9250	−1.35831	−0.679154	−	0.733996i	$-0.737653\pi$
$739$	−36.9250	−1.35831	−0.679154	+	0.733996i	$0.737653\pi$
$740$	16.8795	0.620503
$741$	0	0
$742$	−20.2872	−0.744768
$743$	19.9823	0.733081	0.366541	−	0.930402i	$-0.380542\pi$
$743$	19.9823	0.733081	0.366541	+	0.930402i	$0.380542\pi$
$744$	0	0
$745$	53.1141	1.94595
$746$	−55.6022	−2.03574
$747$	0	0
$748$	18.8833	0.690442
$749$	8.65350	0.316192
$750$	0	0
$751$	8.90745	0.325037	0.162519	−	0.986705i	$-0.448038\pi$
$751$	8.90745	0.325037	0.162519	+	0.986705i	$0.448038\pi$
$752$	15.4530	0.563512
$753$	0	0
$754$	43.1197	1.57033
$755$	49.6882	1.80834
$756$	0	0
$757$	13.8060	0.501787	0.250893	−	0.968015i	$-0.419276\pi$
$757$	13.8060	0.501787	0.250893	+	0.968015i	$0.419276\pi$
$758$	23.0223	0.836209
$759$	0	0
$760$	13.5660	0.492091
$761$	−35.6626	−1.29277	−0.646383	−	0.763013i	$-0.723719\pi$
$761$	−35.6626	−1.29277	−0.646383	+	0.763013i	$0.723719\pi$
$762$	0	0
$763$	−1.20206	−0.0435175
$764$	31.0447	1.12316
$765$	0	0
$766$	68.0243	2.45782
$767$	−36.3897	−1.31396
$768$	0	0
$769$	−3.21210	−0.115831	−0.0579156	−	0.998321i	$-0.518445\pi$
$769$	−3.21210	−0.115831	−0.0579156	+	0.998321i	$0.518445\pi$
$770$	10.1214	0.364750
$771$	0	0
$772$	−22.0765	−0.794550
$773$	35.7577	1.28611	0.643057	−	0.765818i	$-0.277666\pi$
$773$	35.7577	1.28611	0.643057	+	0.765818i	$0.277666\pi$
$774$	0	0
$775$	7.11930	0.255733
$776$	0.992190	0.0356175
$777$	0	0
$778$	65.5159	2.34886
$779$	34.0334	1.21937
$780$	0	0
$781$	33.5166	1.19932
$782$	−78.6586	−2.81283
$783$	0	0
$784$	−4.76345	−0.170123
$785$	−21.6812	−0.773836
$786$	0	0
$787$	−33.2160	−1.18402	−0.592012	−	0.805929i	$-0.701666\pi$
$787$	−33.2160	−1.18402	−0.592012	+	0.805929i	$0.701666\pi$
$788$	−20.3545	−0.725101
$789$	0	0
$790$	−55.3423	−1.96899
$791$	1.24812	0.0443781
$792$	0	0
$793$	−14.2023	−0.504339
$794$	40.6668	1.44321
$795$	0	0
$796$	26.4209	0.936463
$797$	8.21020	0.290820	0.145410	−	0.989371i	$-0.453550\pi$
$797$	8.21020	0.290820	0.145410	+	0.989371i	$0.453550\pi$
$798$	0	0
$799$	19.1981	0.679179
$800$	−9.59811	−0.339345
$801$	0	0
$802$	−34.5019	−1.21830
$803$	29.2625	1.03265
$804$	0	0
$805$	−17.9747	−0.633525
$806$	29.9984	1.05665
$807$	0	0
$808$	12.5159	0.440308
$809$	−49.0138	−1.72323	−0.861617	−	0.507559i	$-0.830548\pi$
$809$	−49.0138	−1.72323	−0.861617	+	0.507559i	$0.830548\pi$
$810$	0	0
$811$	−3.98700	−0.140002	−0.0700012	−	0.997547i	$-0.522300\pi$
$811$	−3.98700	−0.140002	−0.0700012	+	0.997547i	$0.522300\pi$
$812$	11.0553	0.387964
$813$	0	0
$814$	18.0276	0.631865
$815$	19.5173	0.683663
$816$	0	0
$817$	−32.9856	−1.15402
$818$	−49.2120	−1.72066
$819$	0	0
$820$	−22.8012	−0.796252
$821$	−24.3409	−0.849502	−0.424751	−	0.905310i	$-0.639638\pi$
$821$	−24.3409	−0.849502	−0.424751	+	0.905310i	$0.639638\pi$
$822$	0	0
$823$	−3.33724	−0.116329	−0.0581645	−	0.998307i	$-0.518525\pi$
$823$	−3.33724	−0.116329	−0.0581645	+	0.998307i	$0.518525\pi$
$824$	8.88977	0.309690
$825$	0	0
$826$	−21.8836	−0.761428
$827$	−8.40144	−0.292147	−0.146073	−	0.989274i	$-0.546664\pi$
$827$	−8.40144	−0.292147	−0.146073	+	0.989274i	$0.546664\pi$
$828$	0	0
$829$	−47.5492	−1.65145	−0.825726	−	0.564072i	$-0.809234\pi$
$829$	−47.5492	−1.65145	−0.825726	+	0.564072i	$0.809234\pi$
$830$	51.9070	1.80172
$831$	0	0
$832$	−10.8634	−0.376620
$833$	−5.91789	−0.205043
$834$	0	0
$835$	−4.80371	−0.166239
$836$	−17.8753	−0.618232
$837$	0	0
$838$	−14.4834	−0.500322
$839$	−32.1450	−1.10977	−0.554884	−	0.831927i	$-0.687238\pi$
$839$	−32.1450	−1.10977	−0.554884	+	0.831927i	$0.687238\pi$
$840$	0	0
$841$	26.3206	0.907606
$842$	−61.5949	−2.12270
$843$	0	0
$844$	−10.7400	−0.369684
$845$	8.48330	0.291834
$846$	0	0
$847$	−6.39138	−0.219611
$848$	51.7556	1.77730
$849$	0	0
$850$	−15.2028	−0.521453
$851$	−32.0153	−1.09747
$852$	0	0
$853$	−29.3210	−1.00393	−0.501966	−	0.864887i	$-0.667390\pi$
$853$	−29.3210	−1.00393	−0.501966	+	0.864887i	$0.667390\pi$
$854$	−8.54082	−0.292261
$855$	0	0
$856$	−8.29911	−0.283658
$857$	4.67171	0.159583	0.0797913	−	0.996812i	$-0.474575\pi$
$857$	4.67171	0.159583	0.0797913	+	0.996812i	$0.474575\pi$
$858$	0	0
$859$	19.8657	0.677810	0.338905	−	0.940821i	$-0.389944\pi$
$859$	19.8657	0.677810	0.338905	+	0.940821i	$0.389944\pi$
$860$	22.0992	0.753575
$861$	0	0
$862$	0.973925	0.0331720
$863$	51.5715	1.75552	0.877758	−	0.479105i	$-0.159039\pi$
$863$	51.5715	1.75552	0.877758	+	0.479105i	$0.159039\pi$
$864$	0	0
$865$	−24.0197	−0.816693
$866$	27.3574	0.929644
$867$	0	0
$868$	7.69115	0.261055
$869$	−25.1992	−0.854825
$870$	0	0
$871$	9.72727	0.329596
$872$	1.15283	0.0390398
$873$	0	0
$874$	74.4599	2.51865
$875$	9.15114	0.309365
$876$	0	0
$877$	−1.55320	−0.0524479	−0.0262240	−	0.999656i	$-0.508348\pi$
$877$	−1.55320	−0.0524479	−0.0262240	+	0.999656i	$0.508348\pi$
$878$	22.0235	0.743256
$879$	0	0
$880$	−25.8212	−0.870431
$881$	20.5004	0.690676	0.345338	−	0.938478i	$-0.387764\pi$
$881$	20.5004	0.690676	0.345338	+	0.938478i	$0.387764\pi$
$882$	0	0
$883$	0.206988	0.00696572	0.00348286	−	0.999994i	$-0.498891\pi$
$883$	0.206988	0.00696572	0.00348286	+	0.999994i	$0.498891\pi$
$884$	−27.3111	−0.918571
$885$	0	0
$886$	−8.33784	−0.280115
$887$	−7.01790	−0.235638	−0.117819	−	0.993035i	$-0.537590\pi$
$887$	−7.01790	−0.235638	−0.117819	+	0.993035i	$0.537590\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	25.8406	0.866180
$891$	0	0
$892$	−2.95876	−0.0990665
$893$	−18.1733	−0.608147
$894$	0	0
$895$	24.0993	0.805551
$896$	7.41936	0.247863
$897$	0	0
$898$	−4.08668	−0.136374
$899$	38.4865	1.28360
$900$	0	0
$901$	64.2988	2.14210
$902$	−24.3520	−0.810832
$903$	0	0
$904$	−1.19701	−0.0398119
$905$	−55.4043	−1.84170
$906$	0	0
$907$	48.5183	1.61102	0.805512	−	0.592579i	$-0.201890\pi$
$907$	48.5183	1.61102	0.805512	+	0.592579i	$0.201890\pi$
$908$	−2.75326	−0.0913703
$909$	0	0
$910$	−14.6387	−0.485267
$911$	37.7847	1.25186	0.625931	−	0.779878i	$-0.284719\pi$
$911$	37.7847	1.25186	0.625931	+	0.779878i	$0.284719\pi$
$912$	0	0
$913$	23.6350	0.782204
$914$	−72.9455	−2.41282
$915$	0	0
$916$	−6.00474	−0.198402
$917$	−10.6910	−0.353047
$918$	0	0
$919$	−30.8794	−1.01862	−0.509309	−	0.860584i	$-0.670099\pi$
$919$	−30.8794	−1.01862	−0.509309	+	0.860584i	$0.670099\pi$
$920$	17.2386	0.568340
$921$	0	0
$922$	−43.9970	−1.44896
$923$	−48.4754	−1.59559
$924$	0	0
$925$	−6.18780	−0.203454
$926$	22.0691	0.725237
$927$	0	0
$928$	−51.8869	−1.70327
$929$	38.4148	1.26035	0.630174	−	0.776454i	$-0.282984\pi$
$929$	38.4148	1.26035	0.630174	+	0.776454i	$0.282984\pi$
$930$	0	0
$931$	5.60200	0.183598
$932$	−19.8955	−0.651699
$933$	0	0
$934$	−30.9537	−1.01284
$935$	−32.0790	−1.04910
$936$	0	0
$937$	9.54026	0.311667	0.155833	−	0.987783i	$-0.450194\pi$
$937$	9.54026	0.311667	0.155833	+	0.987783i	$0.450194\pi$
$938$	5.84967	0.190999
$939$	0	0
$940$	12.1755	0.397120
$941$	45.4254	1.48082	0.740412	−	0.672153i	$-0.234630\pi$
$941$	45.4254	1.48082	0.740412	+	0.672153i	$0.234630\pi$
$942$	0	0
$943$	43.2469	1.40831
$944$	55.8282	1.81705
$945$	0	0
$946$	23.6022	0.767374
$947$	4.47693	0.145481	0.0727403	−	0.997351i	$-0.476826\pi$
$947$	4.47693	0.145481	0.0727403	+	0.997351i	$0.476826\pi$
$948$	0	0
$949$	−42.3225	−1.37385
$950$	14.3913	0.466917
$951$	0	0
$952$	5.67553	0.183945
$953$	29.6135	0.959274	0.479637	−	0.877467i	$-0.340768\pi$
$953$	29.6135	0.959274	0.479637	+	0.877467i	$0.340768\pi$
$954$	0	0
$955$	−52.7388	−1.70659
$956$	8.08491	0.261485
$957$	0	0
$958$	−42.3675	−1.36883
$959$	−8.97622	−0.289857
$960$	0	0
$961$	−4.22491	−0.136287
$962$	−26.0734	−0.840640
$963$	0	0
$964$	−39.2424	−1.26391
$965$	37.5036	1.20728
$966$	0	0
$967$	53.1670	1.70974	0.854868	−	0.518846i	$-0.173638\pi$
$967$	53.1670	1.70974	0.854868	+	0.518846i	$0.173638\pi$
$968$	6.12964	0.197014
$969$	0	0
$970$	4.87765	0.156612
$971$	−4.77380	−0.153199	−0.0765993	−	0.997062i	$-0.524406\pi$
$971$	−4.77380	−0.153199	−0.0765993	+	0.997062i	$0.524406\pi$
$972$	0	0
$973$	16.3441	0.523968
$974$	−40.8318	−1.30834
$975$	0	0
$976$	21.7889	0.697444
$977$	−20.3191	−0.650065	−0.325033	−	0.945703i	$-0.605375\pi$
$977$	−20.3191	−0.650065	−0.325033	+	0.945703i	$0.605375\pi$
$978$	0	0
$979$	11.7661	0.376046
$980$	−3.75314	−0.119890
$981$	0	0
$982$	−75.8993	−2.42205
$983$	2.20607	0.0703627	0.0351813	−	0.999381i	$-0.488799\pi$
$983$	2.20607	0.0703627	0.0351813	+	0.999381i	$0.488799\pi$
$984$	0	0
$985$	34.5784	1.10176
$986$	−82.1858	−2.61733
$987$	0	0
$988$	25.8533	0.822502
$989$	−41.9154	−1.33283
$990$	0	0
$991$	10.5849	0.336242	0.168121	−	0.985766i	$-0.446230\pi$
$991$	10.5849	0.336242	0.168121	+	0.985766i	$0.446230\pi$
$992$	−36.0977	−1.14610
$993$	0	0
$994$	−29.1516	−0.924631
$995$	−44.8839	−1.42291
$996$	0	0
$997$	29.5991	0.937413	0.468706	−	0.883354i	$-0.344720\pi$
$997$	29.5991	0.937413	0.468706	+	0.883354i	$0.344720\pi$
$998$	28.7961	0.911524
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.w.1.18		20
3.2	odd	2		889.2.a.d.1.3	✓	20
21.20	even	2		6223.2.a.l.1.3		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
889.2.a.d.1.3	✓	20	3.2	odd	2
6223.2.a.l.1.3		20	21.20	even	2
8001.2.a.w.1.18		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)

\(f(q)\)	\(=\)	\(q+1.86718 q^{2} +1.48637 q^{4} -2.52504 q^{5} +1.00000 q^{7} -0.959047 q^{8} +O(q^{10})\)	\(q+1.86718 q^{2} +1.48637 q^{4} -2.52504 q^{5} +1.00000 q^{7} -0.959047 q^{8} -4.71472 q^{10} -2.14677 q^{11} +3.10489 q^{13} +1.86718 q^{14} -4.76345 q^{16} -5.91789 q^{17} +5.60200 q^{19} -3.75314 q^{20} -4.00841 q^{22} +7.11857 q^{23} +1.37585 q^{25} +5.79739 q^{26} +1.48637 q^{28} +7.43778 q^{29} +5.17447 q^{31} -6.97613 q^{32} -11.0498 q^{34} -2.52504 q^{35} -4.49744 q^{37} +10.4600 q^{38} +2.42164 q^{40} +6.07522 q^{41} -5.88817 q^{43} -3.19089 q^{44} +13.2917 q^{46} -3.24407 q^{47} +1.00000 q^{49} +2.56896 q^{50} +4.61500 q^{52} -10.8652 q^{53} +5.42069 q^{55} -0.959047 q^{56} +13.8877 q^{58} -11.7201 q^{59} -4.57418 q^{61} +9.66167 q^{62} -3.49880 q^{64} -7.83998 q^{65} +3.13289 q^{67} -8.79615 q^{68} -4.71472 q^{70} -15.6126 q^{71} -13.6309 q^{73} -8.39753 q^{74} +8.32663 q^{76} -2.14677 q^{77} +11.7382 q^{79} +12.0279 q^{80} +11.3435 q^{82} -11.0096 q^{83} +14.9429 q^{85} -10.9943 q^{86} +2.05885 q^{88} -5.48084 q^{89} +3.10489 q^{91} +10.5808 q^{92} -6.05728 q^{94} -14.1453 q^{95} -1.03456 q^{97} +1.86718 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 8 q^{2} + 24 q^{4} - 3 q^{5} + 20 q^{7} - 24 q^{8}+O(q^{10}) \) 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

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Database

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