LMFDB - Embedded newform 8001.2.a.w.1.16

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			1.16
Root			$-1.60447$ 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)

$f(q)$	$=$	$q+1.60447 q^{2} +0.574339 q^{4} -0.248535 q^{5} +1.00000 q^{7} -2.28744 q^{8} +O(q^{10})$	\(q+1.60447 q^{2} +0.574339 q^{4} -0.248535 q^{5} +1.00000 q^{7} -2.28744 q^{8} -0.398768 q^{10} -2.06439 q^{11} +4.61355 q^{13} +1.60447 q^{14} -4.81881 q^{16} +1.38495 q^{17} -2.86500 q^{19} -0.142743 q^{20} -3.31226 q^{22} -5.17882 q^{23} -4.93823 q^{25} +7.40233 q^{26} +0.574339 q^{28} +9.35535 q^{29} -7.06776 q^{31} -3.15679 q^{32} +2.22211 q^{34} -0.248535 q^{35} +9.60134 q^{37} -4.59682 q^{38} +0.568508 q^{40} -7.82024 q^{41} +10.4189 q^{43} -1.18566 q^{44} -8.30929 q^{46} +7.14182 q^{47} +1.00000 q^{49} -7.92327 q^{50} +2.64974 q^{52} -9.73442 q^{53} +0.513073 q^{55} -2.28744 q^{56} +15.0104 q^{58} +1.47644 q^{59} -11.9646 q^{61} -11.3400 q^{62} +4.57264 q^{64} -1.14663 q^{65} -14.5589 q^{67} +0.795429 q^{68} -0.398768 q^{70} -9.83898 q^{71} -4.36287 q^{73} +15.4051 q^{74} -1.64548 q^{76} -2.06439 q^{77} -2.06143 q^{79} +1.19764 q^{80} -12.5474 q^{82} -15.1003 q^{83} -0.344207 q^{85} +16.7169 q^{86} +4.72216 q^{88} +12.9515 q^{89} +4.61355 q^{91} -2.97440 q^{92} +11.4589 q^{94} +0.712053 q^{95} -5.88747 q^{97} +1.60447 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.60447	1.13453	0.567267	−	0.823534i	$-0.308001\pi$
$2$	1.60447	1.13453	0.567267	+	0.823534i	$0.308001\pi$
$3$	0	0
$3$	0	0
$4$	0.574339	0.287169
$5$	−0.248535	−0.111148	−0.0555740	−	0.998455i	$-0.517699\pi$
$5$	−0.248535	−0.111148	−0.0555740	+	0.998455i	$0.517699\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−2.28744	−0.808731
$9$	0	0
$10$	−0.398768	−0.126101
$11$	−2.06439	−0.622437	−0.311219	−	0.950338i	$-0.600737\pi$
$11$	−2.06439	−0.622437	−0.311219	+	0.950338i	$0.600737\pi$
$12$	0	0
$13$	4.61355	1.27957	0.639785	−	0.768554i	$-0.279023\pi$
$13$	4.61355	1.27957	0.639785	+	0.768554i	$0.279023\pi$
$14$	1.60447	0.428814
$15$	0	0
$16$	−4.81881	−1.20470
$17$	1.38495	0.335899	0.167950	−	0.985796i	$-0.446285\pi$
$17$	1.38495	0.335899	0.167950	+	0.985796i	$0.446285\pi$
$18$	0	0
$19$	−2.86500	−0.657277	−0.328638	−	0.944456i	$-0.606590\pi$
$19$	−2.86500	−0.657277	−0.328638	+	0.944456i	$0.606590\pi$
$20$	−0.142743	−0.0319183
$21$	0	0
$22$	−3.31226	−0.706177
$23$	−5.17882	−1.07986	−0.539930	−	0.841710i	$-0.681549\pi$
$23$	−5.17882	−1.07986	−0.539930	+	0.841710i	$0.681549\pi$
$24$	0	0
$25$	−4.93823	−0.987646
$26$	7.40233	1.45172
$27$	0	0
$28$	0.574339	0.108540
$29$	9.35535	1.73724	0.868622	−	0.495475i	$-0.165006\pi$
$29$	9.35535	1.73724	0.868622	+	0.495475i	$0.165006\pi$
$30$	0	0
$31$	−7.06776	−1.26941	−0.634703	−	0.772756i	$-0.718878\pi$
$31$	−7.06776	−1.26941	−0.634703	+	0.772756i	$0.718878\pi$
$32$	−3.15679	−0.558046
$33$	0	0
$34$	2.22211	0.381089
$35$	−0.248535	−0.0420100
$36$	0	0
$37$	9.60134	1.57845	0.789226	−	0.614103i	$-0.210482\pi$
$37$	9.60134	1.57845	0.789226	+	0.614103i	$0.210482\pi$
$38$	−4.59682	−0.745703
$39$	0	0
$40$	0.568508	0.0898889
$41$	−7.82024	−1.22132	−0.610658	−	0.791894i	$-0.709095\pi$
$41$	−7.82024	−1.22132	−0.610658	+	0.791894i	$0.709095\pi$
$42$	0	0
$43$	10.4189	1.58887	0.794437	−	0.607346i	$-0.207766\pi$
$43$	10.4189	1.58887	0.794437	+	0.607346i	$0.207766\pi$
$44$	−1.18566	−0.178745
$45$	0	0
$46$	−8.30929	−1.22514
$47$	7.14182	1.04174	0.520871	−	0.853636i	$-0.325607\pi$
$47$	7.14182	1.04174	0.520871	+	0.853636i	$0.325607\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−7.92327	−1.12052
$51$	0	0
$52$	2.64974	0.367453
$53$	−9.73442	−1.33713	−0.668563	−	0.743656i	$-0.733090\pi$
$53$	−9.73442	−1.33713	−0.668563	+	0.743656i	$0.733090\pi$
$54$	0	0
$55$	0.513073	0.0691827
$56$	−2.28744	−0.305672
$57$	0	0
$58$	15.0104	1.97096
$59$	1.47644	0.192217	0.0961083	−	0.995371i	$-0.469360\pi$
$59$	1.47644	0.192217	0.0961083	+	0.995371i	$0.469360\pi$
$60$	0	0
$61$	−11.9646	−1.53191	−0.765957	−	0.642892i	$-0.777734\pi$
$61$	−11.9646	−1.53191	−0.765957	+	0.642892i	$0.777734\pi$
$62$	−11.3400	−1.44019
$63$	0	0
$64$	4.57264	0.571580
$65$	−1.14663	−0.142222
$66$	0	0
$67$	−14.5589	−1.77865	−0.889325	−	0.457276i	$-0.848825\pi$
$67$	−14.5589	−1.77865	−0.889325	+	0.457276i	$0.848825\pi$
$68$	0.795429	0.0964599
$69$	0	0
$70$	−0.398768	−0.0476618
$71$	−9.83898	−1.16767	−0.583836	−	0.811872i	$-0.698449\pi$
$71$	−9.83898	−1.16767	−0.583836	+	0.811872i	$0.698449\pi$
$72$	0	0
$73$	−4.36287	−0.510636	−0.255318	−	0.966857i	$-0.582180\pi$
$73$	−4.36287	−0.510636	−0.255318	+	0.966857i	$0.582180\pi$
$74$	15.4051	1.79081
$75$	0	0
$76$	−1.64548	−0.188750
$77$	−2.06439	−0.235259
$78$	0	0
$79$	−2.06143	−0.231929	−0.115964	−	0.993253i	$-0.536996\pi$
$79$	−2.06143	−0.231929	−0.115964	+	0.993253i	$0.536996\pi$
$80$	1.19764	0.133900
$81$	0	0
$82$	−12.5474	−1.38563
$83$	−15.1003	−1.65747	−0.828735	−	0.559642i	$-0.810939\pi$
$83$	−15.1003	−1.65747	−0.828735	+	0.559642i	$0.810939\pi$
$84$	0	0
$85$	−0.344207	−0.0373345
$86$	16.7169	1.80263
$87$	0	0
$88$	4.72216	0.503384
$89$	12.9515	1.37286	0.686429	−	0.727197i	$-0.259177\pi$
$89$	12.9515	1.37286	0.686429	+	0.727197i	$0.259177\pi$
$90$	0	0
$91$	4.61355	0.483632
$92$	−2.97440	−0.310102
$93$	0	0
$94$	11.4589	1.18189
$95$	0.712053	0.0730551
$96$	0	0
$97$	−5.88747	−0.597782	−0.298891	−	0.954287i	$-0.596617\pi$
$97$	−5.88747	−0.597782	−0.298891	+	0.954287i	$0.596617\pi$
$98$	1.60447	0.162076
$99$	0	0
$100$	−2.83622	−0.283622
$101$	−12.2737	−1.22128	−0.610642	−	0.791907i	$-0.709088\pi$
$101$	−12.2737	−1.22128	−0.610642	+	0.791907i	$0.709088\pi$
$102$	0	0
$103$	10.0831	0.993514	0.496757	−	0.867890i	$-0.334524\pi$
$103$	10.0831	0.993514	0.496757	+	0.867890i	$0.334524\pi$
$104$	−10.5532	−1.03483
$105$	0	0
$106$	−15.6186	−1.51702
$107$	−11.0209	−1.06543	−0.532713	−	0.846296i	$-0.678828\pi$
$107$	−11.0209	−1.06543	−0.532713	+	0.846296i	$0.678828\pi$
$108$	0	0
$109$	−3.92874	−0.376305	−0.188152	−	0.982140i	$-0.560250\pi$
$109$	−3.92874	−0.376305	−0.188152	+	0.982140i	$0.560250\pi$
$110$	0.823212	0.0784902
$111$	0	0
$112$	−4.81881	−0.455335
$113$	−5.84341	−0.549702	−0.274851	−	0.961487i	$-0.588628\pi$
$113$	−5.84341	−0.549702	−0.274851	+	0.961487i	$0.588628\pi$
$114$	0	0
$115$	1.28712	0.120024
$116$	5.37314	0.498883
$117$	0	0
$118$	2.36892	0.218076
$119$	1.38495	0.126958
$120$	0	0
$121$	−6.73829	−0.612572
$122$	−19.1969	−1.73801
$123$	0	0
$124$	−4.05929	−0.364535
$125$	2.47000	0.220923
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	13.6503	1.20652
$129$	0	0
$130$	−1.83974	−0.161355
$131$	−5.23380	−0.457279	−0.228640	−	0.973511i	$-0.573428\pi$
$131$	−5.23380	−0.457279	−0.228640	+	0.973511i	$0.573428\pi$
$132$	0	0
$133$	−2.86500	−0.248427
$134$	−23.3593	−2.01794
$135$	0	0
$136$	−3.16798	−0.271652
$137$	2.25200	0.192402	0.0962008	−	0.995362i	$-0.469331\pi$
$137$	2.25200	0.192402	0.0962008	+	0.995362i	$0.469331\pi$
$138$	0	0
$139$	−21.4076	−1.81577	−0.907885	−	0.419220i	$-0.862304\pi$
$139$	−21.4076	−1.81577	−0.907885	+	0.419220i	$0.862304\pi$
$140$	−0.142743	−0.0120640
$141$	0	0
$142$	−15.7864	−1.32476
$143$	−9.52418	−0.796452
$144$	0	0
$145$	−2.32513	−0.193091
$146$	−7.00012	−0.579334
$147$	0	0
$148$	5.51442	0.453283
$149$	5.55086	0.454744	0.227372	−	0.973808i	$-0.426987\pi$
$149$	5.55086	0.454744	0.227372	+	0.973808i	$0.426987\pi$
$150$	0	0
$151$	−1.74871	−0.142308	−0.0711540	−	0.997465i	$-0.522668\pi$
$151$	−1.74871	−0.142308	−0.0711540	+	0.997465i	$0.522668\pi$
$152$	6.55351	0.531560
$153$	0	0
$154$	−3.31226	−0.266910
$155$	1.75658	0.141092
$156$	0	0
$157$	8.65739	0.690935	0.345467	−	0.938431i	$-0.387720\pi$
$157$	8.65739	0.690935	0.345467	+	0.938431i	$0.387720\pi$
$158$	−3.30751	−0.263132
$159$	0	0
$160$	0.784571	0.0620258
$161$	−5.17882	−0.408149
$162$	0	0
$163$	−8.20158	−0.642397	−0.321199	−	0.947012i	$-0.604086\pi$
$163$	−8.20158	−0.642397	−0.321199	+	0.947012i	$0.604086\pi$
$164$	−4.49146	−0.350724
$165$	0	0
$166$	−24.2280	−1.88046
$167$	5.68096	0.439606	0.219803	−	0.975544i	$-0.429459\pi$
$167$	5.68096	0.439606	0.219803	+	0.975544i	$0.429459\pi$
$168$	0	0
$169$	8.28488	0.637298
$170$	−0.552272	−0.0423573
$171$	0	0
$172$	5.98400	0.456276
$173$	10.8690	0.826356	0.413178	−	0.910650i	$-0.364419\pi$
$173$	10.8690	0.826356	0.413178	+	0.910650i	$0.364419\pi$
$174$	0	0
$175$	−4.93823	−0.373295
$176$	9.94791	0.749852
$177$	0	0
$178$	20.7804	1.55756
$179$	20.4990	1.53217	0.766083	−	0.642742i	$-0.222203\pi$
$179$	20.4990	1.53217	0.766083	+	0.642742i	$0.222203\pi$
$180$	0	0
$181$	8.86824	0.659171	0.329586	−	0.944126i	$-0.393091\pi$
$181$	8.86824	0.659171	0.329586	+	0.944126i	$0.393091\pi$
$182$	7.40233	0.548697
$183$	0	0
$184$	11.8462	0.873316
$185$	−2.38627	−0.175442
$186$	0	0
$187$	−2.85907	−0.209076
$188$	4.10182	0.299156
$189$	0	0
$190$	1.14247	0.0828835
$191$	18.2214	1.31846	0.659228	−	0.751943i	$-0.270883\pi$
$191$	18.2214	1.31846	0.659228	+	0.751943i	$0.270883\pi$
$192$	0	0
$193$	13.8127	0.994263	0.497132	−	0.867675i	$-0.334387\pi$
$193$	13.8127	0.994263	0.497132	+	0.867675i	$0.334387\pi$
$194$	−9.44629	−0.678204
$195$	0	0
$196$	0.574339	0.0410242
$197$	−8.13105	−0.579314	−0.289657	−	0.957131i	$-0.593541\pi$
$197$	−8.13105	−0.579314	−0.289657	+	0.957131i	$0.593541\pi$
$198$	0	0
$199$	−21.0663	−1.49335	−0.746676	−	0.665188i	$-0.768351\pi$
$199$	−21.0663	−1.49335	−0.746676	+	0.665188i	$0.768351\pi$
$200$	11.2959	0.798740
$201$	0	0
$202$	−19.6929	−1.38559
$203$	9.35535	0.656617
$204$	0	0
$205$	1.94360	0.135747
$206$	16.1780	1.12718
$207$	0	0
$208$	−22.2318	−1.54150
$209$	5.91449	0.409113
$210$	0	0
$211$	−22.9120	−1.57733	−0.788663	−	0.614826i	$-0.789226\pi$
$211$	−22.9120	−1.57733	−0.788663	+	0.614826i	$0.789226\pi$
$212$	−5.59085	−0.383981
$213$	0	0
$214$	−17.6827	−1.20876
$215$	−2.58947	−0.176600
$216$	0	0
$217$	−7.06776	−0.479791
$218$	−6.30356	−0.426931
$219$	0	0
$220$	0.294677	0.0198671
$221$	6.38953	0.429806
$222$	0	0
$223$	−0.0417173	−0.00279359	−0.00139680	−	0.999999i	$-0.500445\pi$
$223$	−0.0417173	−0.00279359	−0.00139680	+	0.999999i	$0.500445\pi$
$224$	−3.15679	−0.210922
$225$	0	0
$226$	−9.37560	−0.623656
$227$	−26.8446	−1.78174	−0.890870	−	0.454259i	$-0.849904\pi$
$227$	−26.8446	−1.78174	−0.890870	+	0.454259i	$0.849904\pi$
$228$	0	0
$229$	10.6928	0.706602	0.353301	−	0.935510i	$-0.385059\pi$
$229$	10.6928	0.706602	0.353301	+	0.935510i	$0.385059\pi$
$230$	2.06515	0.136172
$231$	0	0
$232$	−21.3998	−1.40496
$233$	−7.67199	−0.502609	−0.251304	−	0.967908i	$-0.580860\pi$
$233$	−7.67199	−0.502609	−0.251304	+	0.967908i	$0.580860\pi$
$234$	0	0
$235$	−1.77499	−0.115788
$236$	0.847979	0.0551987
$237$	0	0
$238$	2.22211	0.144038
$239$	−4.88761	−0.316153	−0.158077	−	0.987427i	$-0.550529\pi$
$239$	−4.88761	−0.316153	−0.158077	+	0.987427i	$0.550529\pi$
$240$	0	0
$241$	−21.2974	−1.37189	−0.685943	−	0.727656i	$-0.740610\pi$
$241$	−21.2974	−1.37189	−0.685943	+	0.727656i	$0.740610\pi$
$242$	−10.8114	−0.694984
$243$	0	0
$244$	−6.87175	−0.439918
$245$	−0.248535	−0.0158783
$246$	0	0
$247$	−13.2178	−0.841031
$248$	16.1671	1.02661
$249$	0	0
$250$	3.96304	0.250645
$251$	11.3807	0.718341	0.359170	−	0.933272i	$-0.383060\pi$
$251$	11.3807	0.718341	0.359170	+	0.933272i	$0.383060\pi$
$252$	0	0
$253$	10.6911	0.672145
$254$	−1.60447	−0.100674
$255$	0	0
$256$	12.7562	0.797263
$257$	−4.17847	−0.260646	−0.130323	−	0.991472i	$-0.541601\pi$
$257$	−4.17847	−0.260646	−0.130323	+	0.991472i	$0.541601\pi$
$258$	0	0
$259$	9.60134	0.596598
$260$	−0.658553	−0.0408417
$261$	0	0
$262$	−8.39749	−0.518799
$263$	19.8837	1.22608	0.613040	−	0.790052i	$-0.289947\pi$
$263$	19.8837	1.22608	0.613040	+	0.790052i	$0.289947\pi$
$264$	0	0
$265$	2.41934	0.148619
$266$	−4.59682	−0.281849
$267$	0	0
$268$	−8.36172	−0.510773
$269$	1.42998	0.0871876	0.0435938	−	0.999049i	$-0.486119\pi$
$269$	1.42998	0.0871876	0.0435938	+	0.999049i	$0.486119\pi$
$270$	0	0
$271$	28.2370	1.71527	0.857637	−	0.514255i	$-0.171932\pi$
$271$	28.2370	1.71527	0.857637	+	0.514255i	$0.171932\pi$
$272$	−6.67380	−0.404659
$273$	0	0
$274$	3.61328	0.218286
$275$	10.1944	0.614748
$276$	0	0
$277$	11.5810	0.695834	0.347917	−	0.937525i	$-0.386889\pi$
$277$	11.5810	0.695834	0.347917	+	0.937525i	$0.386889\pi$
$278$	−34.3480	−2.06005
$279$	0	0
$280$	0.568508	0.0339748
$281$	15.9817	0.953389	0.476694	−	0.879069i	$-0.341835\pi$
$281$	15.9817	0.953389	0.476694	+	0.879069i	$0.341835\pi$
$282$	0	0
$283$	−8.61636	−0.512190	−0.256095	−	0.966652i	$-0.582436\pi$
$283$	−8.61636	−0.512190	−0.256095	+	0.966652i	$0.582436\pi$
$284$	−5.65091	−0.335320
$285$	0	0
$286$	−15.2813	−0.903602
$287$	−7.82024	−0.461614
$288$	0	0
$289$	−15.0819	−0.887172
$290$	−3.73061	−0.219069
$291$	0	0
$292$	−2.50577	−0.146639
$293$	−5.27199	−0.307993	−0.153997	−	0.988071i	$-0.549214\pi$
$293$	−5.27199	−0.307993	−0.153997	+	0.988071i	$0.549214\pi$
$294$	0	0
$295$	−0.366948	−0.0213645
$296$	−21.9625	−1.27654
$297$	0	0
$298$	8.90621	0.515923
$299$	−23.8928	−1.38176
$300$	0	0
$301$	10.4189	0.600538
$302$	−2.80576	−0.161453
$303$	0	0
$304$	13.8059	0.791823
$305$	2.97362	0.170269
$306$	0	0
$307$	−4.86994	−0.277942	−0.138971	−	0.990296i	$-0.544379\pi$
$307$	−4.86994	−0.277942	−0.138971	+	0.990296i	$0.544379\pi$
$308$	−1.18566	−0.0675592
$309$	0	0
$310$	2.81839	0.160074
$311$	−8.28578	−0.469844	−0.234922	−	0.972014i	$-0.575483\pi$
$311$	−8.28578	−0.469844	−0.234922	+	0.972014i	$0.575483\pi$
$312$	0	0
$313$	−29.8516	−1.68731	−0.843657	−	0.536883i	$-0.819602\pi$
$313$	−29.8516	−1.68731	−0.843657	+	0.536883i	$0.819602\pi$
$314$	13.8906	0.783890
$315$	0	0
$316$	−1.18396	−0.0666029
$317$	−15.3754	−0.863570	−0.431785	−	0.901977i	$-0.642116\pi$
$317$	−15.3754	−0.863570	−0.431785	+	0.901977i	$0.642116\pi$
$318$	0	0
$319$	−19.3131	−1.08133
$320$	−1.13646	−0.0635300
$321$	0	0
$322$	−8.30929	−0.463059
$323$	−3.96788	−0.220779
$324$	0	0
$325$	−22.7828	−1.26376
$326$	−13.1592	−0.728822
$327$	0	0
$328$	17.8883	0.987716
$329$	7.14182	0.393741
$330$	0	0
$331$	−31.5246	−1.73275	−0.866374	−	0.499396i	$-0.833555\pi$
$331$	−31.5246	−1.73275	−0.866374	+	0.499396i	$0.833555\pi$
$332$	−8.67267	−0.475974
$333$	0	0
$334$	9.11496	0.498748
$335$	3.61839	0.197694
$336$	0	0
$337$	−9.40882	−0.512531	−0.256265	−	0.966606i	$-0.582492\pi$
$337$	−9.40882	−0.512531	−0.256265	+	0.966606i	$0.582492\pi$
$338$	13.2929	0.723037
$339$	0	0
$340$	−0.197692	−0.0107213
$341$	14.5906	0.790126
$342$	0	0
$343$	1.00000	0.0539949
$344$	−23.8327	−1.28497
$345$	0	0
$346$	17.4391	0.937530
$347$	−18.4781	−0.991955	−0.495978	−	0.868335i	$-0.665190\pi$
$347$	−18.4781	−0.991955	−0.495978	+	0.868335i	$0.665190\pi$
$348$	0	0
$349$	5.28963	0.283147	0.141574	−	0.989928i	$-0.454784\pi$
$349$	5.28963	0.283147	0.141574	+	0.989928i	$0.454784\pi$
$350$	−7.92327	−0.423516
$351$	0	0
$352$	6.51684	0.347349
$353$	14.4032	0.766605	0.383303	−	0.923623i	$-0.374787\pi$
$353$	14.4032	0.766605	0.383303	+	0.923623i	$0.374787\pi$
$354$	0	0
$355$	2.44533	0.129785
$356$	7.43855	0.394243
$357$	0	0
$358$	32.8901	1.73830
$359$	−6.83211	−0.360585	−0.180293	−	0.983613i	$-0.557704\pi$
$359$	−6.83211	−0.360585	−0.180293	+	0.983613i	$0.557704\pi$
$360$	0	0
$361$	−10.7918	−0.567987
$362$	14.2289	0.747853
$363$	0	0
$364$	2.64974	0.138884
$365$	1.08433	0.0567562
$366$	0	0
$367$	−36.1081	−1.88483	−0.942413	−	0.334451i	$-0.891449\pi$
$367$	−36.1081	−1.88483	−0.942413	+	0.334451i	$0.891449\pi$
$368$	24.9558	1.30091
$369$	0	0
$370$	−3.82870	−0.199045
$371$	−9.73442	−0.505386
$372$	0	0
$373$	−13.9607	−0.722858	−0.361429	−	0.932400i	$-0.617711\pi$
$373$	−13.9607	−0.722858	−0.361429	+	0.932400i	$0.617711\pi$
$374$	−4.58731	−0.237204
$375$	0	0
$376$	−16.3365	−0.842489
$377$	43.1614	2.22293
$378$	0	0
$379$	17.2040	0.883712	0.441856	−	0.897086i	$-0.354320\pi$
$379$	17.2040	0.883712	0.441856	+	0.897086i	$0.354320\pi$
$380$	0.408959	0.0209792
$381$	0	0
$382$	29.2358	1.49583
$383$	3.16174	0.161558	0.0807788	−	0.996732i	$-0.474259\pi$
$383$	3.16174	0.161558	0.0807788	+	0.996732i	$0.474259\pi$
$384$	0	0
$385$	0.513073	0.0261486
$386$	22.1622	1.12803
$387$	0	0
$388$	−3.38140	−0.171665
$389$	−3.60627	−0.182845	−0.0914226	−	0.995812i	$-0.529141\pi$
$389$	−3.60627	−0.182845	−0.0914226	+	0.995812i	$0.529141\pi$
$390$	0	0
$391$	−7.17240	−0.362724
$392$	−2.28744	−0.115533
$393$	0	0
$394$	−13.0461	−0.657251
$395$	0.512337	0.0257785
$396$	0	0
$397$	−1.35977	−0.0682447	−0.0341224	−	0.999418i	$-0.510864\pi$
$397$	−1.35977	−0.0682447	−0.0341224	+	0.999418i	$0.510864\pi$
$398$	−33.8004	−1.69426
$399$	0	0
$400$	23.7964	1.18982
$401$	−18.2667	−0.912195	−0.456097	−	0.889930i	$-0.650753\pi$
$401$	−18.2667	−0.912195	−0.456097	+	0.889930i	$0.650753\pi$
$402$	0	0
$403$	−32.6075	−1.62429
$404$	−7.04928	−0.350715
$405$	0	0
$406$	15.0104	0.744955
$407$	−19.8209	−0.982487
$408$	0	0
$409$	24.3580	1.20443	0.602213	−	0.798335i	$-0.294286\pi$
$409$	24.3580	1.20443	0.602213	+	0.798335i	$0.294286\pi$
$410$	3.11846	0.154010
$411$	0	0
$412$	5.79109	0.285307
$413$	1.47644	0.0726511
$414$	0	0
$415$	3.75294	0.184225
$416$	−14.5640	−0.714059
$417$	0	0
$418$	9.48964	0.464154
$419$	−4.53561	−0.221579	−0.110789	−	0.993844i	$-0.535338\pi$
$419$	−4.53561	−0.221579	−0.110789	+	0.993844i	$0.535338\pi$
$420$	0	0
$421$	−1.42660	−0.0695283	−0.0347642	−	0.999396i	$-0.511068\pi$
$421$	−1.42660	−0.0695283	−0.0347642	+	0.999396i	$0.511068\pi$
$422$	−36.7617	−1.78953
$423$	0	0
$424$	22.2669	1.08137
$425$	−6.83919	−0.331749
$426$	0	0
$427$	−11.9646	−0.579009
$428$	−6.32970	−0.305958
$429$	0	0
$430$	−4.15474	−0.200359
$431$	−26.6902	−1.28562	−0.642812	−	0.766024i	$-0.722232\pi$
$431$	−26.6902	−1.28562	−0.642812	+	0.766024i	$0.722232\pi$
$432$	0	0
$433$	−4.65866	−0.223881	−0.111940	−	0.993715i	$-0.535707\pi$
$433$	−4.65866	−0.223881	−0.111940	+	0.993715i	$0.535707\pi$
$434$	−11.3400	−0.544339
$435$	0	0
$436$	−2.25643	−0.108063
$437$	14.8373	0.709767
$438$	0	0
$439$	18.5623	0.885929	0.442964	−	0.896539i	$-0.353927\pi$
$439$	18.5623	0.885929	0.442964	+	0.896539i	$0.353927\pi$
$440$	−1.17362	−0.0559502
$441$	0	0
$442$	10.2518	0.487630
$443$	−26.2295	−1.24620	−0.623101	−	0.782142i	$-0.714127\pi$
$443$	−26.2295	−1.24620	−0.623101	+	0.782142i	$0.714127\pi$
$444$	0	0
$445$	−3.21890	−0.152591
$446$	−0.0669343	−0.00316943
$447$	0	0
$448$	4.57264	0.216037
$449$	11.5318	0.544220	0.272110	−	0.962266i	$-0.412279\pi$
$449$	11.5318	0.544220	0.272110	+	0.962266i	$0.412279\pi$
$450$	0	0
$451$	16.1440	0.760192
$452$	−3.35609	−0.157857
$453$	0	0
$454$	−43.0715	−2.02145
$455$	−1.14663	−0.0537548
$456$	0	0
$457$	25.5261	1.19406	0.597031	−	0.802218i	$-0.296347\pi$
$457$	25.5261	1.19406	0.597031	+	0.802218i	$0.296347\pi$
$458$	17.1564	0.801664
$459$	0	0
$460$	0.739241	0.0344673
$461$	7.68056	0.357719	0.178860	−	0.983875i	$-0.442759\pi$
$461$	7.68056	0.357719	0.178860	+	0.983875i	$0.442759\pi$
$462$	0	0
$463$	−0.801721	−0.0372591	−0.0186296	−	0.999826i	$-0.505930\pi$
$463$	−0.801721	−0.0372591	−0.0186296	+	0.999826i	$0.505930\pi$
$464$	−45.0817	−2.09286
$465$	0	0
$466$	−12.3095	−0.570227
$467$	31.8617	1.47439	0.737193	−	0.675682i	$-0.236151\pi$
$467$	31.8617	1.47439	0.737193	+	0.675682i	$0.236151\pi$
$468$	0	0
$469$	−14.5589	−0.672266
$470$	−2.84793	−0.131365
$471$	0	0
$472$	−3.37727	−0.155452
$473$	−21.5088	−0.988974
$474$	0	0
$475$	14.1480	0.649157
$476$	0.795429	0.0364584
$477$	0	0
$478$	−7.84205	−0.358687
$479$	−26.2469	−1.19925	−0.599627	−	0.800280i	$-0.704684\pi$
$479$	−26.2469	−1.19925	−0.599627	+	0.800280i	$0.704684\pi$
$480$	0	0
$481$	44.2963	2.01974
$482$	−34.1711	−1.55645
$483$	0	0
$484$	−3.87006	−0.175912
$485$	1.46324	0.0664423
$486$	0	0
$487$	9.60895	0.435423	0.217712	−	0.976013i	$-0.430141\pi$
$487$	9.60895	0.435423	0.217712	+	0.976013i	$0.430141\pi$
$488$	27.3683	1.23891
$489$	0	0
$490$	−0.398768	−0.0180145
$491$	−5.73767	−0.258938	−0.129469	−	0.991584i	$-0.541327\pi$
$491$	−5.73767	−0.258938	−0.129469	+	0.991584i	$0.541327\pi$
$492$	0	0
$493$	12.9567	0.583539
$494$	−21.2077	−0.954179
$495$	0	0
$496$	34.0582	1.52926
$497$	−9.83898	−0.441339
$498$	0	0
$499$	−17.7188	−0.793201	−0.396601	−	0.917991i	$-0.629810\pi$
$499$	−17.7188	−0.793201	−0.396601	+	0.917991i	$0.629810\pi$
$500$	1.41861	0.0634423
$501$	0	0
$502$	18.2600	0.814983
$503$	−5.84637	−0.260677	−0.130338	−	0.991470i	$-0.541606\pi$
$503$	−5.84637	−0.260677	−0.130338	+	0.991470i	$0.541606\pi$
$504$	0	0
$505$	3.05045	0.135743
$506$	17.1536	0.762572
$507$	0	0
$508$	−0.574339	−0.0254822
$509$	42.4028	1.87947	0.939736	−	0.341902i	$-0.111071\pi$
$509$	42.4028	1.87947	0.939736	+	0.341902i	$0.111071\pi$
$510$	0	0
$511$	−4.36287	−0.193002
$512$	−6.83350	−0.302001
$513$	0	0
$514$	−6.70425	−0.295712
$515$	−2.50599	−0.110427
$516$	0	0
$517$	−14.7435	−0.648419
$518$	15.4051	0.676862
$519$	0	0
$520$	2.62284	0.115019
$521$	43.4989	1.90572	0.952861	−	0.303408i	$-0.0981246\pi$
$521$	43.4989	1.90572	0.952861	+	0.303408i	$0.0981246\pi$
$522$	0	0
$523$	14.8939	0.651264	0.325632	−	0.945497i	$-0.394423\pi$
$523$	14.8939	0.651264	0.325632	+	0.945497i	$0.394423\pi$
$524$	−3.00597	−0.131316
$525$	0	0
$526$	31.9028	1.39103
$527$	−9.78847	−0.426393
$528$	0	0
$529$	3.82022	0.166097
$530$	3.88177	0.168613
$531$	0	0
$532$	−1.64548	−0.0713407
$533$	−36.0791	−1.56276
$534$	0	0
$535$	2.73907	0.118420
$536$	33.3025	1.43845
$537$	0	0
$538$	2.29437	0.0989173
$539$	−2.06439	−0.0889196
$540$	0	0
$541$	−14.6373	−0.629305	−0.314652	−	0.949207i	$-0.601888\pi$
$541$	−14.6373	−0.629305	−0.314652	+	0.949207i	$0.601888\pi$
$542$	45.3055	1.94604
$543$	0	0
$544$	−4.37198	−0.187447
$545$	0.976428	0.0418256
$546$	0	0
$547$	10.2687	0.439059	0.219529	−	0.975606i	$-0.429548\pi$
$547$	10.2687	0.439059	0.219529	+	0.975606i	$0.429548\pi$
$548$	1.29341	0.0552518
$549$	0	0
$550$	16.3567	0.697453
$551$	−26.8031	−1.14185
$552$	0	0
$553$	−2.06143	−0.0876609
$554$	18.5814	0.789448
$555$	0	0
$556$	−12.2952	−0.521433
$557$	−16.1907	−0.686023	−0.343012	−	0.939331i	$-0.611447\pi$
$557$	−16.1907	−0.686023	−0.343012	+	0.939331i	$0.611447\pi$
$558$	0	0
$559$	48.0684	2.03308
$560$	1.19764	0.0506096
$561$	0	0
$562$	25.6422	1.08165
$563$	−16.6969	−0.703692	−0.351846	−	0.936058i	$-0.614446\pi$
$563$	−16.6969	−0.703692	−0.351846	+	0.936058i	$0.614446\pi$
$564$	0	0
$565$	1.45229	0.0610983
$566$	−13.8247	−0.581097
$567$	0	0
$568$	22.5061	0.944333
$569$	13.1417	0.550928	0.275464	−	0.961311i	$-0.411168\pi$
$569$	13.1417	0.550928	0.275464	+	0.961311i	$0.411168\pi$
$570$	0	0
$571$	5.86566	0.245470	0.122735	−	0.992439i	$-0.460833\pi$
$571$	5.86566	0.245470	0.122735	+	0.992439i	$0.460833\pi$
$572$	−5.47010	−0.228716
$573$	0	0
$574$	−12.5474	−0.523717
$575$	25.5742	1.06652
$576$	0	0
$577$	32.8604	1.36800	0.683998	−	0.729484i	$-0.260240\pi$
$577$	32.8604	1.36800	0.683998	+	0.729484i	$0.260240\pi$
$578$	−24.1986	−1.00653
$579$	0	0
$580$	−1.33541	−0.0554499
$581$	−15.1003	−0.626465
$582$	0	0
$583$	20.0956	0.832276
$584$	9.97980	0.412967
$585$	0	0
$586$	−8.45878	−0.349429
$587$	1.60407	0.0662069	0.0331035	−	0.999452i	$-0.489461\pi$
$587$	1.60407	0.0662069	0.0331035	+	0.999452i	$0.489461\pi$
$588$	0	0
$589$	20.2491	0.834352
$590$	−0.588758	−0.0242388
$591$	0	0
$592$	−46.2671	−1.90156
$593$	16.5682	0.680376	0.340188	−	0.940357i	$-0.389509\pi$
$593$	16.5682	0.680376	0.340188	+	0.940357i	$0.389509\pi$
$594$	0	0
$595$	−0.344207	−0.0141111
$596$	3.18807	0.130589
$597$	0	0
$598$	−38.3354	−1.56765
$599$	−21.7895	−0.890295	−0.445147	−	0.895457i	$-0.646849\pi$
$599$	−21.7895	−0.890295	−0.445147	+	0.895457i	$0.646849\pi$
$600$	0	0
$601$	19.1184	0.779856	0.389928	−	0.920845i	$-0.372500\pi$
$601$	19.1184	0.779856	0.389928	+	0.920845i	$0.372500\pi$
$602$	16.7169	0.681331
$603$	0	0
$604$	−1.00435	−0.0408665
$605$	1.67470	0.0680862
$606$	0	0
$607$	9.16116	0.371840	0.185920	−	0.982565i	$-0.440473\pi$
$607$	9.16116	0.371840	0.185920	+	0.982565i	$0.440473\pi$
$608$	9.04420	0.366791
$609$	0	0
$610$	4.77110	0.193176
$611$	32.9492	1.33298
$612$	0	0
$613$	−19.8971	−0.803636	−0.401818	−	0.915720i	$-0.631621\pi$
$613$	−19.8971	−0.803636	−0.401818	+	0.915720i	$0.631621\pi$
$614$	−7.81369	−0.315335
$615$	0	0
$616$	4.72216	0.190261
$617$	−10.7473	−0.432672	−0.216336	−	0.976319i	$-0.569411\pi$
$617$	−10.7473	−0.432672	−0.216336	+	0.976319i	$0.569411\pi$
$618$	0	0
$619$	30.7703	1.23676	0.618381	−	0.785878i	$-0.287789\pi$
$619$	30.7703	1.23676	0.618381	+	0.785878i	$0.287789\pi$
$620$	1.00887	0.0405173
$621$	0	0
$622$	−13.2943	−0.533054
$623$	12.9515	0.518891
$624$	0	0
$625$	24.0773	0.963091
$626$	−47.8962	−1.91432
$627$	0	0
$628$	4.97227	0.198415
$629$	13.2974	0.530200
$630$	0	0
$631$	30.5498	1.21617	0.608085	−	0.793872i	$-0.291938\pi$
$631$	30.5498	1.21617	0.608085	+	0.793872i	$0.291938\pi$
$632$	4.71539	0.187568
$633$	0	0
$634$	−24.6695	−0.979750
$635$	0.248535	0.00986280
$636$	0	0
$637$	4.61355	0.182796
$638$	−30.9874	−1.22680
$639$	0	0
$640$	−3.39256	−0.134103
$641$	14.6914	0.580276	0.290138	−	0.956985i	$-0.406299\pi$
$641$	14.6914	0.580276	0.290138	+	0.956985i	$0.406299\pi$
$642$	0	0
$643$	−10.9887	−0.433353	−0.216677	−	0.976243i	$-0.569522\pi$
$643$	−10.9887	−0.433353	−0.216677	+	0.976243i	$0.569522\pi$
$644$	−2.97440	−0.117208
$645$	0	0
$646$	−6.36636	−0.250481
$647$	−1.94049	−0.0762884	−0.0381442	−	0.999272i	$-0.512145\pi$
$647$	−1.94049	−0.0762884	−0.0381442	+	0.999272i	$0.512145\pi$
$648$	0	0
$649$	−3.04796	−0.119643
$650$	−36.5544	−1.43378
$651$	0	0
$652$	−4.71048	−0.184477
$653$	30.5882	1.19701	0.598504	−	0.801119i	$-0.295762\pi$
$653$	30.5882	1.19701	0.598504	+	0.801119i	$0.295762\pi$
$654$	0	0
$655$	1.30078	0.0508257
$656$	37.6843	1.47132
$657$	0	0
$658$	11.4589	0.446713
$659$	−6.09333	−0.237363	−0.118681	−	0.992932i	$-0.537867\pi$
$659$	−6.09333	−0.237363	−0.118681	+	0.992932i	$0.537867\pi$
$660$	0	0
$661$	31.6109	1.22952	0.614761	−	0.788714i	$-0.289253\pi$
$661$	31.6109	1.22952	0.614761	+	0.788714i	$0.289253\pi$
$662$	−50.5804	−1.96586
$663$	0	0
$664$	34.5409	1.34045
$665$	0.712053	0.0276122
$666$	0	0
$667$	−48.4497	−1.87598
$668$	3.26280	0.126241
$669$	0	0
$670$	5.80561	0.224290
$671$	24.6997	0.953520
$672$	0	0
$673$	−45.7773	−1.76458	−0.882292	−	0.470702i	$-0.844001\pi$
$673$	−45.7773	−1.76458	−0.882292	+	0.470702i	$0.844001\pi$
$674$	−15.0962	−0.581484
$675$	0	0
$676$	4.75833	0.183013
$677$	39.1821	1.50589	0.752944	−	0.658084i	$-0.228633\pi$
$677$	39.1821	1.50589	0.752944	+	0.658084i	$0.228633\pi$
$678$	0	0
$679$	−5.88747	−0.225940
$680$	0.787353	0.0301936
$681$	0	0
$682$	23.4103	0.896426
$683$	31.2759	1.19674	0.598369	−	0.801221i	$-0.295816\pi$
$683$	31.2759	1.19674	0.598369	+	0.801221i	$0.295816\pi$
$684$	0	0
$685$	−0.559701	−0.0213851
$686$	1.60447	0.0612591
$687$	0	0
$688$	−50.2069	−1.91412
$689$	−44.9103	−1.71094
$690$	0	0
$691$	−0.988316	−0.0375973	−0.0187986	−	0.999823i	$-0.505984\pi$
$691$	−0.988316	−0.0375973	−0.0187986	+	0.999823i	$0.505984\pi$
$692$	6.24250	0.237304
$693$	0	0
$694$	−29.6476	−1.12541
$695$	5.32053	0.201819
$696$	0	0
$697$	−10.8306	−0.410239
$698$	8.48707	0.321240
$699$	0	0
$700$	−2.83622	−0.107199
$701$	−10.9677	−0.414244	−0.207122	−	0.978315i	$-0.566410\pi$
$701$	−10.9677	−0.414244	−0.207122	+	0.978315i	$0.566410\pi$
$702$	0	0
$703$	−27.5079	−1.03748
$704$	−9.43972	−0.355773
$705$	0	0
$706$	23.1096	0.869740
$707$	−12.2737	−0.461602
$708$	0	0
$709$	46.4954	1.74617	0.873086	−	0.487567i	$-0.162116\pi$
$709$	46.4954	1.74617	0.873086	+	0.487567i	$0.162116\pi$
$710$	3.92347	0.147245
$711$	0	0
$712$	−29.6258	−1.11027
$713$	36.6027	1.37078
$714$	0	0
$715$	2.36709	0.0885241
$716$	11.7734	0.439991
$717$	0	0
$718$	−10.9620	−0.409096
$719$	23.9536	0.893319	0.446660	−	0.894704i	$-0.352614\pi$
$719$	23.9536	0.893319	0.446660	+	0.894704i	$0.352614\pi$
$720$	0	0
$721$	10.0831	0.375513
$722$	−17.3151	−0.644401
$723$	0	0
$724$	5.09337	0.189294
$725$	−46.1989	−1.71578
$726$	0	0
$727$	11.2434	0.416995	0.208498	−	0.978023i	$-0.433143\pi$
$727$	11.2434	0.416995	0.208498	+	0.978023i	$0.433143\pi$
$728$	−10.5532	−0.391128
$729$	0	0
$730$	1.73977	0.0643919
$731$	14.4297	0.533701
$732$	0	0
$733$	3.07023	0.113402	0.0567008	−	0.998391i	$-0.481942\pi$
$733$	3.07023	0.113402	0.0567008	+	0.998391i	$0.481942\pi$
$734$	−57.9345	−2.13840
$735$	0	0
$736$	16.3484	0.602612
$737$	30.0552	1.10710
$738$	0	0
$739$	0.267916	0.00985545	0.00492772	−	0.999988i	$-0.498431\pi$
$739$	0.267916	0.00985545	0.00492772	+	0.999988i	$0.498431\pi$
$740$	−1.37053	−0.0503815
$741$	0	0
$742$	−15.6186	−0.573378
$743$	−14.7553	−0.541320	−0.270660	−	0.962675i	$-0.587242\pi$
$743$	−14.7553	−0.541320	−0.270660	+	0.962675i	$0.587242\pi$
$744$	0	0
$745$	−1.37958	−0.0505439
$746$	−22.3996	−0.820107
$747$	0	0
$748$	−1.64208	−0.0600402
$749$	−11.0209	−0.402693
$750$	0	0
$751$	31.8578	1.16251	0.581253	−	0.813723i	$-0.302563\pi$
$751$	31.8578	1.16251	0.581253	+	0.813723i	$0.302563\pi$
$752$	−34.4151	−1.25499
$753$	0	0
$754$	69.2514	2.52199
$755$	0.434615	0.0158173
$756$	0	0
$757$	44.9482	1.63367	0.816835	−	0.576872i	$-0.195727\pi$
$757$	44.9482	1.63367	0.816835	+	0.576872i	$0.195727\pi$
$758$	27.6034	1.00260
$759$	0	0
$760$	−1.62878	−0.0590819
$761$	−2.11329	−0.0766067	−0.0383033	−	0.999266i	$-0.512195\pi$
$761$	−2.11329	−0.0766067	−0.0383033	+	0.999266i	$0.512195\pi$
$762$	0	0
$763$	−3.92874	−0.142230
$764$	10.4653	0.378620
$765$	0	0
$766$	5.07294	0.183293
$767$	6.81165	0.245955
$768$	0	0
$769$	−45.5871	−1.64391	−0.821956	−	0.569550i	$-0.807117\pi$
$769$	−45.5871	−1.64391	−0.821956	+	0.569550i	$0.807117\pi$
$770$	0.823212	0.0296665
$771$	0	0
$772$	7.93319	0.285522
$773$	−26.2249	−0.943244	−0.471622	−	0.881801i	$-0.656331\pi$
$773$	−26.2249	−0.943244	−0.471622	+	0.881801i	$0.656331\pi$
$774$	0	0
$775$	34.9022	1.25372
$776$	13.4672	0.483445
$777$	0	0
$778$	−5.78617	−0.207444
$779$	22.4050	0.802742
$780$	0	0
$781$	20.3115	0.726803
$782$	−11.5079	−0.411523
$783$	0	0
$784$	−4.81881	−0.172100
$785$	−2.15166	−0.0767961
$786$	0	0
$787$	26.9352	0.960137	0.480069	−	0.877231i	$-0.340612\pi$
$787$	26.9352	0.960137	0.480069	+	0.877231i	$0.340612\pi$
$788$	−4.66998	−0.166361
$789$	0	0
$790$	0.822031	0.0292466
$791$	−5.84341	−0.207768
$792$	0	0
$793$	−55.1994	−1.96019
$794$	−2.18171	−0.0774260
$795$	0	0
$796$	−12.0992	−0.428845
$797$	9.96337	0.352921	0.176460	−	0.984308i	$-0.443535\pi$
$797$	9.96337	0.352921	0.176460	+	0.984308i	$0.443535\pi$
$798$	0	0
$799$	9.89104	0.349920
$800$	15.5889	0.551152
$801$	0	0
$802$	−29.3084	−1.03492
$803$	9.00668	0.317839
$804$	0	0
$805$	1.28712	0.0453649
$806$	−52.3179	−1.84282
$807$	0	0
$808$	28.0754	0.987690
$809$	12.3868	0.435495	0.217747	−	0.976005i	$-0.430129\pi$
$809$	12.3868	0.435495	0.217747	+	0.976005i	$0.430129\pi$
$810$	0	0
$811$	45.3995	1.59419	0.797096	−	0.603853i	$-0.206368\pi$
$811$	45.3995	1.59419	0.797096	+	0.603853i	$0.206368\pi$
$812$	5.37314	0.188560
$813$	0	0
$814$	−31.8022	−1.11467
$815$	2.03838	0.0714013
$816$	0	0
$817$	−29.8503	−1.04433
$818$	39.0818	1.36646
$819$	0	0
$820$	1.11628	0.0389823
$821$	−36.9896	−1.29095	−0.645474	−	0.763782i	$-0.723340\pi$
$821$	−36.9896	−1.29095	−0.645474	+	0.763782i	$0.723340\pi$
$822$	0	0
$823$	−4.32363	−0.150712	−0.0753561	−	0.997157i	$-0.524009\pi$
$823$	−4.32363	−0.150712	−0.0753561	+	0.997157i	$0.524009\pi$
$824$	−23.0644	−0.803486
$825$	0	0
$826$	2.36892	0.0824252
$827$	−1.50024	−0.0521684	−0.0260842	−	0.999660i	$-0.508304\pi$
$827$	−1.50024	−0.0521684	−0.0260842	+	0.999660i	$0.508304\pi$
$828$	0	0
$829$	−39.3301	−1.36599	−0.682995	−	0.730423i	$-0.739323\pi$
$829$	−39.3301	−1.36599	−0.682995	+	0.730423i	$0.739323\pi$
$830$	6.02150	0.209009
$831$	0	0
$832$	21.0961	0.731377
$833$	1.38495	0.0479856
$834$	0	0
$835$	−1.41192	−0.0488614
$836$	3.39692	0.117485
$837$	0	0
$838$	−7.27727	−0.251389
$839$	3.56845	0.123197	0.0615983	−	0.998101i	$-0.480380\pi$
$839$	3.56845	0.123197	0.0615983	+	0.998101i	$0.480380\pi$
$840$	0	0
$841$	58.5225	2.01802
$842$	−2.28895	−0.0788823
$843$	0	0
$844$	−13.1592	−0.452959
$845$	−2.05908	−0.0708345
$846$	0	0
$847$	−6.73829	−0.231530
$848$	46.9083	1.61084
$849$	0	0
$850$	−10.9733	−0.376381
$851$	−49.7237	−1.70451
$852$	0	0
$853$	23.6630	0.810205	0.405102	−	0.914271i	$-0.367236\pi$
$853$	23.6630	0.810205	0.405102	+	0.914271i	$0.367236\pi$
$854$	−19.1969	−0.656906
$855$	0	0
$856$	25.2095	0.861644
$857$	31.5762	1.07862	0.539311	−	0.842107i	$-0.318685\pi$
$857$	31.5762	1.07862	0.539311	+	0.842107i	$0.318685\pi$
$858$	0	0
$859$	27.0910	0.924333	0.462167	−	0.886793i	$-0.347072\pi$
$859$	27.0910	0.924333	0.462167	+	0.886793i	$0.347072\pi$
$860$	−1.48723	−0.0507142
$861$	0	0
$862$	−42.8238	−1.45858
$863$	−25.8856	−0.881157	−0.440579	−	0.897714i	$-0.645227\pi$
$863$	−25.8856	−0.881157	−0.440579	+	0.897714i	$0.645227\pi$
$864$	0	0
$865$	−2.70133	−0.0918479
$866$	−7.47470	−0.254001
$867$	0	0
$868$	−4.05929	−0.137781
$869$	4.25560	0.144361
$870$	0	0
$871$	−67.1681	−2.27591
$872$	8.98675	0.304330
$873$	0	0
$874$	23.8061	0.805255
$875$	2.47000	0.0835011
$876$	0	0
$877$	32.1756	1.08649	0.543246	−	0.839574i	$-0.317195\pi$
$877$	32.1756	1.08649	0.543246	+	0.839574i	$0.317195\pi$
$878$	29.7827	1.00512
$879$	0	0
$880$	−2.47240	−0.0833446
$881$	14.9202	0.502673	0.251337	−	0.967900i	$-0.419130\pi$
$881$	14.9202	0.502673	0.251337	+	0.967900i	$0.419130\pi$
$882$	0	0
$883$	−37.8634	−1.27420	−0.637102	−	0.770780i	$-0.719867\pi$
$883$	−37.8634	−1.27420	−0.637102	+	0.770780i	$0.719867\pi$
$884$	3.66975	0.123427
$885$	0	0
$886$	−42.0846	−1.41386
$887$	3.91895	0.131585	0.0657927	−	0.997833i	$-0.479042\pi$
$887$	3.91895	0.131585	0.0657927	+	0.997833i	$0.479042\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	−5.16464	−0.173119
$891$	0	0
$892$	−0.0239598	−0.000802234 0
$893$	−20.4613	−0.684713
$894$	0	0
$895$	−5.09471	−0.170297
$896$	13.6503	0.456023
$897$	0	0
$898$	18.5025	0.617437
$899$	−66.1213	−2.20527
$900$	0	0
$901$	−13.4817	−0.449139
$902$	25.9027	0.862465
$903$	0	0
$904$	13.3664	0.444561
$905$	−2.20407	−0.0732656
$906$	0	0
$907$	−2.40898	−0.0799890	−0.0399945	−	0.999200i	$-0.512734\pi$
$907$	−2.40898	−0.0799890	−0.0399945	+	0.999200i	$0.512734\pi$
$908$	−15.4179	−0.511661
$909$	0	0
$910$	−1.83974	−0.0609866
$911$	2.93744	0.0973218	0.0486609	−	0.998815i	$-0.484505\pi$
$911$	2.93744	0.0973218	0.0486609	+	0.998815i	$0.484505\pi$
$912$	0	0
$913$	31.1728	1.03167
$914$	40.9560	1.35470
$915$	0	0
$916$	6.14130	0.202914
$917$	−5.23380	−0.172835
$918$	0	0
$919$	26.8776	0.886609	0.443304	−	0.896371i	$-0.353806\pi$
$919$	26.8776	0.886609	0.443304	+	0.896371i	$0.353806\pi$
$920$	−2.94420	−0.0970674
$921$	0	0
$922$	12.3233	0.405845
$923$	−45.3927	−1.49412
$924$	0	0
$925$	−47.4136	−1.55895
$926$	−1.28634	−0.0422718
$927$	0	0
$928$	−29.5328	−0.969463
$929$	51.6928	1.69599	0.847993	−	0.530007i	$-0.177811\pi$
$929$	51.6928	1.69599	0.847993	+	0.530007i	$0.177811\pi$
$930$	0	0
$931$	−2.86500	−0.0938967
$932$	−4.40632	−0.144334
$933$	0	0
$934$	51.1214	1.67274
$935$	0.710579	0.0232384
$936$	0	0
$937$	−45.0224	−1.47082	−0.735410	−	0.677623i	$-0.763010\pi$
$937$	−45.0224	−1.47082	−0.735410	+	0.677623i	$0.763010\pi$
$938$	−23.3593	−0.762710
$939$	0	0
$940$	−1.01945	−0.0332506
$941$	−7.32770	−0.238876	−0.119438	−	0.992842i	$-0.538109\pi$
$941$	−7.32770	−0.238876	−0.119438	+	0.992842i	$0.538109\pi$
$942$	0	0
$943$	40.4996	1.31885
$944$	−7.11471	−0.231564
$945$	0	0
$946$	−34.5103	−1.12203
$947$	−42.2672	−1.37350	−0.686749	−	0.726894i	$-0.740963\pi$
$947$	−42.2672	−1.37350	−0.686749	+	0.726894i	$0.740963\pi$
$948$	0	0
$949$	−20.1284	−0.653394
$950$	22.7002	0.736491
$951$	0	0
$952$	−3.16798	−0.102675
$953$	−52.4838	−1.70012	−0.850059	−	0.526688i	$-0.823434\pi$
$953$	−52.4838	−1.70012	−0.850059	+	0.526688i	$0.823434\pi$
$954$	0	0
$955$	−4.52865	−0.146544
$956$	−2.80714	−0.0907895
$957$	0	0
$958$	−42.1126	−1.36060
$959$	2.25200	0.0727210
$960$	0	0
$961$	18.9532	0.611394
$962$	71.0723	2.29146
$963$	0	0
$964$	−12.2319	−0.393963
$965$	−3.43295	−0.110510
$966$	0	0
$967$	27.7940	0.893796	0.446898	−	0.894585i	$-0.352529\pi$
$967$	27.7940	0.893796	0.446898	+	0.894585i	$0.352529\pi$
$968$	15.4134	0.495406
$969$	0	0
$970$	2.34773	0.0753811
$971$	−51.6768	−1.65839	−0.829194	−	0.558961i	$-0.811200\pi$
$971$	−51.6768	−1.65839	−0.829194	+	0.558961i	$0.811200\pi$
$972$	0	0
$973$	−21.4076	−0.686296
$974$	15.4173	0.494003
$975$	0	0
$976$	57.6553	1.84550
$977$	−51.0146	−1.63210	−0.816050	−	0.577981i	$-0.803841\pi$
$977$	−51.0146	−1.63210	−0.816050	+	0.577981i	$0.803841\pi$
$978$	0	0
$979$	−26.7370	−0.854518
$980$	−0.142743	−0.00455976
$981$	0	0
$982$	−9.20595	−0.293774
$983$	49.2089	1.56952	0.784761	−	0.619799i	$-0.212786\pi$
$983$	49.2089	1.56952	0.784761	+	0.619799i	$0.212786\pi$
$984$	0	0
$985$	2.02085	0.0643896
$986$	20.7886	0.662045
$987$	0	0
$988$	−7.59152	−0.241518
$989$	−53.9579	−1.71576
$990$	0	0
$991$	20.6408	0.655677	0.327839	−	0.944734i	$-0.393680\pi$
$991$	20.6408	0.655677	0.327839	+	0.944734i	$0.393680\pi$
$992$	22.3114	0.708388
$993$	0	0
$994$	−15.7864	−0.500714
$995$	5.23571	0.165983
$996$	0	0
$997$	48.8298	1.54645	0.773227	−	0.634129i	$-0.218641\pi$
$997$	48.8298	1.54645	0.773227	+	0.634129i	$0.218641\pi$
$998$	−28.4293	−0.899914
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.w.1.16		20
3.2	odd	2		889.2.a.d.1.5	✓	20
21.20	even	2		6223.2.a.l.1.5		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
889.2.a.d.1.5	✓	20	3.2	odd	2
6223.2.a.l.1.5		20	21.20	even	2
8001.2.a.w.1.16		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)

\(f(q)\)	\(=\)	\(q+1.60447 q^{2} +0.574339 q^{4} -0.248535 q^{5} +1.00000 q^{7} -2.28744 q^{8} +O(q^{10})\)	\(q+1.60447 q^{2} +0.574339 q^{4} -0.248535 q^{5} +1.00000 q^{7} -2.28744 q^{8} -0.398768 q^{10} -2.06439 q^{11} +4.61355 q^{13} +1.60447 q^{14} -4.81881 q^{16} +1.38495 q^{17} -2.86500 q^{19} -0.142743 q^{20} -3.31226 q^{22} -5.17882 q^{23} -4.93823 q^{25} +7.40233 q^{26} +0.574339 q^{28} +9.35535 q^{29} -7.06776 q^{31} -3.15679 q^{32} +2.22211 q^{34} -0.248535 q^{35} +9.60134 q^{37} -4.59682 q^{38} +0.568508 q^{40} -7.82024 q^{41} +10.4189 q^{43} -1.18566 q^{44} -8.30929 q^{46} +7.14182 q^{47} +1.00000 q^{49} -7.92327 q^{50} +2.64974 q^{52} -9.73442 q^{53} +0.513073 q^{55} -2.28744 q^{56} +15.0104 q^{58} +1.47644 q^{59} -11.9646 q^{61} -11.3400 q^{62} +4.57264 q^{64} -1.14663 q^{65} -14.5589 q^{67} +0.795429 q^{68} -0.398768 q^{70} -9.83898 q^{71} -4.36287 q^{73} +15.4051 q^{74} -1.64548 q^{76} -2.06439 q^{77} -2.06143 q^{79} +1.19764 q^{80} -12.5474 q^{82} -15.1003 q^{83} -0.344207 q^{85} +16.7169 q^{86} +4.72216 q^{88} +12.9515 q^{89} +4.61355 q^{91} -2.97440 q^{92} +11.4589 q^{94} +0.712053 q^{95} -5.88747 q^{97} +1.60447 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 8 q^{2} + 24 q^{4} - 3 q^{5} + 20 q^{7} - 24 q^{8}+O(q^{10}) \) 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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